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[ [ "MIMO Precoding in Underlay Cognitive Radio Systems with Completely\n Unknown Primary CSI" ], [ "Abstract This paper studies a novel underlay MIMO cognitive radio (CR) system, where the instantaneous or statistical channel state information (CSI) of the interfering channels to the primary receivers (PRs) is completely unknown to the CR.", "For the single underlay receiver scenario, we assume a minimum information rate must be guaranteed on the CR main channel whose CSI is known at the CR transmitter.", "We first show that low-rank CR interference is preferable for improving the throughput of the PRs compared with spreading less power over more transmit dimensions.", "Based on this observation, we then propose a rank minimization CR transmission strategy assuming a minimum information rate must be guaranteed on the CR main channel.", "We propose a simple solution referred to as frugal waterfilling (FWF) that uses the least amount of power required to achieve the rate constraint with a minimum-rank transmit covariance matrix.", "We also present two heuristic approaches that have been used in prior work to transform rank minimization problems into convex optimization problems.", "The proposed schemes are then generalized to an underlay MIMO CR downlink network with multiple receivers.", "Finally, a theoretical analysis of the interference temperature and leakage rate outage probabilities at the PR is presented for Rayleigh fading channels.We demonstrate that the direct FWF solution leads to higher PR throughput even though it has higher interference \"temperature (IT) compared with the heuristic methods and classic waterfilling, which calls into question the use of IT as a metric for CR interference." ], [ "INTRODUCTION", "Cognitive radios (CRs) have gained prominence as an efficient method of improving spectrum utilization by allowing coexistence with licensed networks, which is denoted as dynamic spectrum access (DSA).", "One popular variant of DSA is known as spectrum underlay, where underlay cognitive transmitters (UCTs) operate simultaneously with licensed (primary) users, but adapt their transmission parameters so as to confine the interference perceived at the primary receivers (PRs) to a pre-specified threshold [1].", "Therefore, a fundamental challenge for the CR is to balance between maximizing its own transmit rate and minimizing the interference it causes to the PRs.", "In CR networks with single-antenna nodes, this is usually achieved by exploiting some knowledge of the interfering cross-channels to the PRs at the UCT and performing some admission control algorithms with power control [2].", "If UCTs are equipped with multiple antennas, the available spatial degrees of freedom can be used to mitigate interference to the PRs during transmission to the underlay receivers.", "Multi-antenna CR networks have recently received extensive attention, assuming some knowledge of the interfering cross-channels to the PRs at the UCT, either perfect PR cross-channel state information (CSI) [3]-[5], perturbed PR CSI [6]-[8], or statistical PR CSI [9]-[12].", "However, the UCT may not have the luxury of knowing the CSI of the cross links to the PRs, as the primary system would not deliberately coordinate the collection of CSI for the CR system.", "In this work, we consider the novel scenario where both the realizations and distribution of the PR cross-channels are completely unknown at the CR, thereby precluding the overwhelming majority of existing spectrum underlay schemes in the literature [3], [4].", "Such a scenario of completely-unknown PR CSI is relevant in a number of instances, for example, when the PR transmits intermittently and therefore stymies attempts to learn the cross-channel, when channels are varying rapidly over time, when the PT and PR do not employ time-division duplexing as assumed in [10], [11] among others, or when there are a plurality of active PTs/UCTs and it is impossible to indirectly estimate specific channels.", "Specifically, we propose a rank minimization transmission strategy for the UCT while maintaining a minimum information rate on the CR link, and we present a simple solution referred to as frugal waterfilling (FWF) that uses the least amount of power required to achieve the rate constraint with a minimum-rank covariance matrix.", "In the context of MIMO interference channels (for which the CR underlay network is a special case), rank-minimization has been shown to be a reinterpretation of interference alignment [14], but this approach requires knowledge of interfering cross-channels and treats the overall system sum rate or degrees-of-freedom as the performance metric, assumptions which are both markedly different from the underlay CR scenario we consider.", "We also describe two heuristic approaches that have been used in prior work to transform rank minimization problems (RMP) into problems that can be solved via convex optimization.", "These approaches approximate the rank objective function with two relaxations, one based on the nuclear norm [22], and the other on a log-determinant function [23].", "We show theoretically and via numerical simulation that minimizing the rank of the UCT spatial covariance matrix leads to the highest PR throughput in general Rayleigh-fading channels, compared with spreading the transmit power over more dimensions.", "Furthermore, our simulations indicate that FWF provides a higher PR throughput than the nuclear-norm and log-det heuristic solutions, even though FWF has a higher interference “temperature” (IT).", "This suggests that the commonly used IT metric does not accurately capture the impact of the CR interference on PR performance.", "Instead, we propose a metric based on interference leakage (IL) rate that more accurately reflects the influence of the CR interference.", "This paper is organized as follows.", "The underlay system model is introduced in Section .", "PR CSI-unaware UCT transmit strategies for a single UCR are presented in Section .", "The generalization to the underlay downlink with multiple UCRs is shown in Section .", "A random matrix-theoretic analysis of the primary outage probability due to the proposed strategies is given in Section .", "The penultimate Section  presents numerical simulations for various underlay scenarios, and we conclude in Section .", "Notation: We will use $\\mathcal {CN}(\\mathbf {0},\\mathbf {Z})$ to denote a circularly symmetric complex Gaussian distribution with zero mean and covariance matrix $\\mathbf {Z}$ , $\\mathcal {E}\\lbrace \\cdot \\rbrace $ to denote expectation, $\\mathrm {vec}(\\cdot )$ the matrix column stacking operator, $(\\cdot )^T$ the transpose, $(\\cdot )^H$ the Hermitian transpose, $(\\cdot )^{-1}$ the matrix inverse, $\\mathrm {Tr}(\\cdot )$ the trace operator, $\\left| \\cdot \\right|$ or $\\det $ the matrix determinant, $\\mathrm {diag}(\\mathbf {a})$ a diagonal matrix with the elements of $\\mathbf {a}$ on the main diagonal, $|\\mathbf {A}|_{i,j}$ the $(i,j)$ element of $\\mathbf {A}$ , $\\Gamma (x)$ the gamma function, and $\\mathbf {I}$ is the identity matrix." ], [ "System Model", "A generic MIMO underlay CR network with $K$ multi-antenna underlay receivers is shown in Fig.", "1, where a primary system and an underlay CR system share the same spectral band.", "Since the UCT transmit strategies are independent of the cross-channels to primary users, the numbers of PRs and PTs and their array sizes can be made arbitrary; however to simplify notation we will consider a solitary multi-antenna PT-PR pair.", "To introduce the problem we first consider the scenario with a single UCR, and generalize to the case of $K>1$ in Sec. .", "We consider a multi-antenna UCT equipped with $N_a$ antennas, which transmits a signal vector $\\mathbf {x}_s \\in \\mathbb {C}^{N_a \\times 1}$ , to its $N_s$ -antenna underlay cognitive receiver (UCR).", "The PT is equipped with $N_p$ antennas and transmits signal $\\mathbf {x}_{p}\\in \\mathbb {C}^{N_p \\times 1}$ to the $N_r$ -antenna PR.", "Thus, the UCR observes ${{\\mathbf {y}}_s} = {{\\mathbf {G}}_1}{{\\mathbf {x}}_s} + {{\\mathbf {G}}_2}{{\\mathbf {x}}_p} + {{\\mathbf {n}}_s},$ where $\\mathbf {G}_1 \\in {\\mathbb {C}^{{N_s} \\times {N_a}}},\\mathbf {G}_2\\in {\\mathbb {C}^{{N_s} \\times {N_p}}}$ are the complex MIMO channels from the UCT and PT, and $\\mathbf {n}_{s}\\sim \\mathcal {CN}(\\mathbf {0},\\sigma _s^2\\mathbf {I})$ is complex additive white Gaussian noise.", "We assume Gaussian signaling with zero mean and second-order statistics $\\mathcal {E}\\lbrace \\mathbf {x}_s\\mathbf {x}_s^H\\rbrace = \\mathbf {Q}_s$ , and the average UCT transmit power is assumed to be bounded: $\\mathrm {Tr}(\\mathbf {Q}_s) \\le P_s.$ The signal at the PR is given by $\\mathbf {y}_{p}=\\mathbf {H}_1{\\mathbf {x}}_{p}+\\mathbf {H}_2\\mathbf {x}_s+\\mathbf {n}_{p},$ where $\\mathbf {H}_1\\in {\\mathbb {C}^{{N_p} \\times {N_r}}},\\mathbf {H}_2\\in {\\mathbb {C}^{{N_r} \\times {N_a}}}$ are the channels from the PU and CR transmitters (assumed to be full-rank), and $\\mathbf {n}_{p}\\sim \\mathcal {CN}(\\mathbf {0},\\sigma _p^2\\mathbf {I})$ is complex additive white Gaussian noise.", "The primary signal is also modeled as a zero-mean complex Gaussian signal with covariance matrix ${{{\\mathbf {Q}}_p}}$ and average power constraint $\\mathrm {Tr}(\\mathbf {Q}_p) \\le P_p$ .", "We will assume ${{{\\mathbf {Q}}_p}}$ is fixed and the channels are mutually independent and each composed of i.i.d.", "zero-mean circularly symmetric complex Gaussian entries, and focus our attention on the design of the UCT transmit signal.", "We assume there is no cooperation between the PT and UCT during transmission, and that both receivers treat interfering signals as noise.", "The network is essentially an asymmetric 2-user MIMO interference channel, where the UCT attempts to minimize the interference to the PR, but no such reciprocal gesture is made by the PT.", "The interference covariance matrix at the PR is $\\mathbf {K}_p = \\mathbf {H}_{2}\\mathbf {Q}_s\\mathbf {H}_{2}^H.$ Define the interference temperature at the PR as [3]-[12] ${T_p}\\left( {\\mathbf {Q}_s} \\right)=\\mathrm {Tr}\\left(\\mathbf {K}_p\\right).$ Without knowledge of $\\mathbf {H}_2$ or its distribution, the UCT cannot directly optimize the PR interference temperature or outage probability as in existing underlay proposals [3]-[12].", "To our best knowledge, precoding strategies and performance analyses for MIMO underlay systems with completely unknown primary CSI have not been presented in the literature thus far.", "In addition to [3]-[13] not being applicable, the blind interference alignment method for the 2-user MIMO interference channel [15] is also precluded since it requires knowledge of the cross-channel coherence intervals, which we assume is also unknown.", "In [13], a blind underlay precoding scheme is proposed where the MIMO CR iteratively updates its spatial covariance by observing the transmit power of a solitary PT.", "The UCT attempts to infer the least-harmful spatial orientation towards the PR, but requires that the PT employ a power control scheme monotonic in the interference caused by the CR, and that the cross-channel remains constant during the learning process.", "In contrast, we investigate simple non-iterative CR precoding strategies which do not impose any restrictions on the PT transmission strategy or number of PTs, or cross-channel coherence intervals.", "The PT achieves the following rate on its link: ${R_p}\\left( {{{\\mathbf {Q}}_s}} \\right) = {\\log _2}\\left| {{\\mathbf {I}} + {{\\mathbf {H}}_1}{{\\mathbf {Q}}_p}{\\mathbf {H}}_1^H{{{\\left( {{{\\mathbf {K}}_p} + \\sigma _p^2{\\mathbf {I}}} \\right)}^{ - 1}}}} \\right|.$ Similarly, the achievable rate on the CR link is ${R_s}\\left( {{{\\mathbf {Q}}_s}} \\right) = {\\log _2}\\left| {{\\mathbf {I}} + {{\\mathbf {G}}_1}{{\\mathbf {Q}}_s}{\\mathbf {G}}_1^H{{{\\left( {{{\\mathbf {K}}_s} + \\sigma _s^2{\\mathbf {I}}} \\right)}^{ - 1}}}} \\right|$ where ${{\\mathbf {K}}_s} = {{\\mathbf {G}}_2}{{\\mathbf {Q}}_p}{\\mathbf {G}}_2^H$ represents the interference from the PT." ], [ "A Rank Minimization strategy for CSI-Unaware Underlay Transmission", "We now expound on the fundamental motivation underlying the UCT transmission strategies proposed in this work.", "As we have seen, due to a lack of knowledge of $\\mathbf {H}_2$ or its distribution, the UCT cannot directly optimize the PU interference temperature.", "Hence, we propose an alternative transmission strategy where the UCT tries to minimize a measure of the interference caused to the PR in a “best-effort\" sense, while achieving a target data rate to the UCR.", "Assuming that $\\mathbf {Q}_p$ is fixed, we first show that in the clairvoyant case where the UCT has some knowledge of the channel to the PR ($\\mathbf {H}_2$ ), a rank-1 UCT covariance matrix $\\mathbf {Q}_s$ causes least interference to the primary link, which is described in the following proposition.", "Proposition 1 The optimal solution to the clairvoyant problem $\\max \\limits _{\\mathbf {Q}_s} &\\quad \\mathcal {E}\\left\\lbrace R_p(\\mathbf {Q}_s)\\right\\rbrace \\\\\\mathrm {s.t.}", "& \\quad {R_s}\\left( {{{\\mathbf {Q}}_s}} \\right) = {R_b} \\\\& \\quad \\mathrm {Tr}\\left( {{{\\mathbf {Q}}_s}} \\right) \\le {P_s}\\\\& \\quad \\mathbf {Q}_s \\succeq \\mathbf {0}.$ that maximizes the average PR rate is of rank one, i.e., $\\mathrm {rank}(\\mathbf {Q}_s^{\\star })=1$ .", "Since $\\mathbf {Q}_s \\succeq \\mathbf {0}$ , it can be expressed as $\\mathbf {Q}_s = \\mathbf {U}\\Lambda \\mathbf {U}^H$ , where $\\Lambda $ is the diagonal matrix of eigenvalues of $\\mathbf {Q}_s$ and $\\mathbf {U}$ is the unitary matrix with columns consisting of the eigenvectors of $\\mathbf {Q}_s$ .", "Defining $\\tilde{\\mathbf {H}}_2 = \\mathbf {H}_2\\mathbf {U}$ , it follow from Lemma 5 in [16] that the distribution of $\\tilde{\\mathbf {H}}_2$ is the same as that of $\\mathbf {H}_2$ .", "As a result, the average PU rate can be expressed as $\\begin{split}&\\quad R_p(\\mathbf {Q}_s) = \\Phi (\\Lambda )\\\\& = \\mathcal {E} \\bigg \\lbrace \\log _2 \\bigg [\\det \\left(\\mathbf {I} + \\mathbf {H}_1\\mathbf {Q}_p \\mathbf {H}_1^H {\\left( \\tilde{\\mathbf {H}}_2\\Lambda \\tilde{\\mathbf {H}}_2^H + \\sigma _p^2 \\mathbf {I}\\right)}^{ - 1} \\right)\\bigg ] \\bigg \\rbrace \\end{split}$ Thus, the problem we considered is essentially equivalent to constructing the diagonal matrix $\\Lambda $ with real nonnegative entries so as to maximize $\\Phi (\\Lambda )$ under the constraint $\\mathrm {Tr}(\\Lambda )= P_s$ .", "From [25],[26], we have that $\\Phi (\\Lambda )$ is a convex function of $\\Lambda $ .", "Note that given any permutation matrix $\\Pi $ , we see (using Lemma 5 in [16] again) that $\\Phi (\\Pi \\Lambda \\Pi ^{H} ) = \\Phi (\\Lambda ).$ From convexity, we have $\\Phi \\left(\\frac{1}{N_a!", "}\\sum _{\\Pi }\\Pi \\Lambda \\Pi ^{H}\\right) \\le \\frac{1}{N_a!", "}\\sum _{\\Pi } \\Phi (\\Pi \\Lambda \\Pi ^{H}) = \\Phi (\\Lambda )$ where we have used Jensen's inequality.", "From the transmit power constraint, we have $\\frac{1}{N_a!", "}\\sum _{\\Pi }\\Pi \\Lambda \\Pi ^{H} = (P_s/N_a)\\mathbf {I}_{N_a}$ .", "Thus, we have proved that the least PU rate is obtained by $\\Lambda =(P_s/N_a)\\mathbf {I}_{N_a}$ .", "Further, due to convexity, we can argue that the largest PU rate is obtain by a point farthest away from $\\Lambda =(P_s/N_a)\\mathbf {I}_{N_a}$ .", "Thus, we want $\\Lambda ^{\\star }=\\mathrm {diag}(\\lambda ^{\\star }_1,\\dots ,\\lambda ^{\\star }_{N_a})$ that satisfy [21] $\\max \\sum _{i=1}^{N_a} \\left(\\lambda _i-\\frac{P_s}{N_a} \\right)^2 ~~~~ \\mathrm {s.~t.~} \\sum _{i=1}^{N_a}\\lambda _i = P_s$ Now, $\\begin{split}\\sum _{i=1}^{N_a} \\left(\\lambda _i-\\frac{P_s}{N_a} \\right)^2 & = \\sum _{i=1}^{N_a} \\lambda _i^2 - 2\\frac{P_s}{N_a}\\sum _{i=1}^{N_a} \\lambda _i + \\frac{P_s^2}{N_a} \\\\& = P_s^2\\left( \\sum _{i=1}^{N_a} \\left(\\frac{\\lambda _i}{P_s}\\right)^2- \\frac{1}{N_a} \\right)\\\\& \\le P_s^2\\left( \\sum _{i=1}^{N_a} {\\frac{\\lambda _i}{P_s}} - \\frac{1}{N_a} \\right) \\\\& = P_s^2 \\left(1- \\frac{1}{N_a} \\right)\\end{split}$ where we used $\\sum _{i=1}^{N_a} \\left(\\frac{\\lambda _i}{P_s}\\right)^2 \\le \\sum _{i=1}^{N_a} {\\frac{\\lambda _i}{P_s}} =1$ , and the equality is satisfied by any $(\\lambda ^{\\star }_1,\\dots ,\\lambda ^{\\star }_{N_a})$ with all zeros except for one nonzero entry.", "Hence, we conclude that $\\mathrm {rank}(\\mathbf {Q}_s^{\\star })=\\mathrm {rank}(\\Lambda ^{\\star })=1$ .", "Therefore, in the clairvoyant case where the UCT has some knowledge of the primary CSI, a rank-1 $\\mathbf {Q}_s$ causes least interference to the primary link and full-rank $\\mathbf {Q}_s$ causes most interferenceThis notion has been echoed in prior art on MIMO interference channels [17], [18].", "Of course, the optimal $\\mathbf {Q}_s$ will depend on the PR CSI, which we have assumed is unavailable.", "Still, the result motivates the use of a low-rank transmit covariance at the UCT.", "It is evident that the UCT does not actually require knowledge of the PR CSI to minimize the rank of $\\mathbf {Q}_s$ needed to achieve a rate target $R_b$ on the CR link.", "To exploit this observation, we henceforth pose the UCT precoder design problem when the PR CSI is completely unknown as $(\\textrm {P0}):\\quad \\min &\\quad \\mathrm {rank}\\left( {{{\\mathbf {Q}}_s}} \\right) \\\\\\mathrm {s.t.}", "& \\quad {R_s}\\left( {{{\\mathbf {Q}}_s}} \\right) = {R_b} \\\\& \\quad \\mathrm {Tr}\\left( {{{\\mathbf {Q}}_s}} \\right) \\le {P_s}\\\\& \\quad \\mathbf {Q}_s \\succeq \\mathbf {0}.$ This is a rank-minimization problem (RMP), and in general is computationally hard to solve since the $\\mathrm {rank}$ function is quasi-concave and not convex.", "As explained below, however, in this case a simple waterfilling solution can be obtained.", "In [24], the mutual information of the CR link is maximized subject to an interference temperature constraint and arbitrary transmit covariance rank constraints, which implies knowledge of PR CSI and thus differs from this work." ], [ "Solutions for the Rank Minimization Design", "Notice that the design problem (P0) is ill-posed in the sense that there are potentially an infinite number of solutions.", "Suppose that we find one minimum-rank solution to (P0) such that $\\mathbf {Q}_s=\\mathbf {U}\\mathbf {\\Lambda }\\mathbf {U}^{H}$ for some unitary matrix $\\mathbf {U}$ and diagonal matrix $\\mathbf {\\Lambda }$ satisfies $R_s(\\mathbf {Q}_s)=R_b$ , $\\mathrm {Tr}(\\mathbf {\\Lambda })\\le P_s$ .", "If the required power $\\mathrm {Tr}(\\mathbf {\\Lambda })$ is strictly smaller than $P_s$ , then we could find an infinite number of solutions by making small perturbations to $\\mathbf {U}$ , which while leading to a higher power requirement, still would require less power than $P_s$ .", "Obviously, the solution with least power is more desirable for our underlay CR system in order to minimize the interference caused to the PR, and this solution can easily be found using the Frugal Waterfilling (FWF) approach described next." ], [ "FWF Approach", "The FWF solution seeks to find the least amount of power required to achieve the CR rate target of $R_b$ with the minimum rank transmit covariance $\\mathbf {Q}_s$ .", "The optimization problem can be solved using a combination of the classic waterfilling (CWF) algorithm and a simple bisection line search.", "The description for FWF is outlined as Algorithm IV-A.1 below.", "In brief, FWF cycles through the possible number of transmit dimensions in ascending order starting with a rank-one $\\mathbf {Q}_s$ , and at each step computes the transmit power required to meet the rate constraint $R_b$ based on CWF.", "This requires a simple line search over the transmit power for each step.", "Once a solution is found that satisfies the transmit power constraint, the algorithm terminates.", "If no feasible solution is found for all $N_a$ transmit dimensions, the CR link will be in outage.", "[htpb] Frugal Waterfilling for UCT Rank/Power Tradeoff [19] $P_s > 0, R_b > 0$ set $r=\\mathrm {rank}(\\mathbf {G}_1)$ $M=1$ to $r$ Solve: $\\begin{array}{c}p(M) = \\min \\mathrm {Tr}(\\mathbf {Q}_s) \\\\[10pt]\\mbox{\\rm s.~t.}", "\\; \\; {\\log _2}\\left| {{\\mathbf {I}} + {{\\mathbf {G}}_1}{{\\mathbf {Q}}_s}{\\mathbf {G}}_1^H{{{\\left( {{{\\mathbf {K}}_s} + \\sigma _s^2{\\mathbf {I}}} \\right)}^{ - 1}}}} \\right|=R_b \\; .\\end{array}$ $p(r) > P_s$ Declare outage CWF solution: ${\\displaystyle N = arg\\min _M p(M)};$ FWF solution: ${\\displaystyle N = arg\\min _M M};$ $\\mathbf {Q}_s$ determined by waterfilling $p(N)$ over $N$ largest singular values of ${{\\left( {{{\\mathbf {K}}_s} + \\sigma _s^2{\\mathbf {I}}} \\right)}^{ -1/2}}\\mathbf {G}_1$ The FWF algorithm was presented in brief without analysis by the authors in [19], and through simulation were shown to be an effective transmission strategy in conventional downlink, wiretap, and underlay networks.", "While FWF finds an efficient solution to (P0), in general rank minimization problems are difficult to solve and often require exponential-time complexity.", "Consequently, heuristic approximations to the matrix rank have been proposed as alternatives in order to yield simpler optimization problems.", "In particular, the nuclear norm [22] and log-determinant [23] heuristics have been proposed in order to convexify RMP problems like (P0) and provide approximate solutions with polynomial-time complexity.", "In the discussion below, we show how these approximations can be applied to the RMP we consider in this paper." ], [ "Nuclear Norm and Log-det Heuristic", "The nuclear norm heuristic is based on the fact the nuclear norm (sum of the singular values of a matrix) is the convex envelope of the rank function on the unit ball.", "When the matrix is positive semidefinite, the nuclear norm is the same as the trace function.", "As a result, the design problem (P0) can be formulated as follows: $\\begin{split}(\\textrm {P1}): \\quad \\min &\\quad \\mathrm {Tr}\\left( \\mathbf {Q}_s \\right) \\\\\\mathrm {s.t.}", "& \\quad {R_s}\\left( {{{\\mathbf {Q}}_s}} \\right) = {R_b} \\\\& \\quad \\mathrm {Tr}\\left( {{{\\mathbf {Q}}_s}} \\right) \\le {P_s}\\\\& \\quad \\mathbf {Q}_s \\succeq \\mathbf {0} \\; .\\end{split}$ The nuclear norm heuristic (P1) is a convex optimization problem and can be solved using the CWF algorithm together with a bisection line search (similar to FWF).", "It is well known that under the CWF algorithm, the lowest transmit power is achieved when $\\mathrm {rank}(\\mathbf {Q}_s)$ is chosen as large as possible (up to $\\mathrm {rank}(\\mathbf {G}_1)$ ).", "This is clearly contrary to the rank-minimization design formulation, which indicates that the nuclear norm approach for this problem is a poor approximation.", "Using the function $\\log \\det (\\mathbf {Q}_s+\\delta \\mathbf {I})$ as a smooth surrogate for $\\mathrm {rank}(\\mathbf {Q}_s)$ , the log-det heurisitic can be described as follows: $\\begin{split}(\\textrm {P2}):\\quad \\min &\\quad \\log \\det \\left( \\mathbf {Q}_s + \\delta \\mathbf {I} \\right) \\\\\\mathrm {s.t.}", "& \\quad {R_s}\\left( {{{\\mathbf {Q}}_s}} \\right) = {R_b} \\\\& \\quad \\mathrm {Tr}\\left( {{{\\mathbf {Q}}_s}} \\right) \\le {P_s}\\\\& \\quad \\mathbf {Q}_s \\succeq \\mathbf {0},\\end{split}$ where $\\delta \\ge 0$ can be interpreted as a small regularization constant (we choose $\\delta = 10^{-6}$ for numerical examples).", "Since the surrogate function $\\log \\det (\\mathbf {Q}_s+\\delta \\mathbf {I})$ is smooth on the positive definite cone, it can be minimized using a local minimization method.", "We use iterative linearization to find a local minimum to the optimization problem (P2) [23].", "Let $\\mathbf {Q}_s^{(k)}$ denote the $k$ th iteration of the optimization variable $\\mathbf {Q}_s$ .", "The first-order Taylor series expansion of $\\log \\det (\\mathbf {Q_s}+\\delta \\mathbf {I})$ about $\\mathbf {Q}_s^{(k)}$ is given by $\\begin{split}\\log \\det (\\mathbf {Q}_s+\\delta \\mathbf {I})~\\approx ~ & \\log \\det (\\mathbf {Q}_s^{(k)}+\\delta \\mathbf {I})+\\\\& \\mathrm {Tr}\\big [(\\mathbf {Q}_s^{(k)}+\\delta \\mathbf {I})^{-1}(\\mathbf {Q}_s-\\mathbf {Q}_s^{(k)})\\big ].\\end{split}$ Hence, we could attempt to minimize $\\log \\det (\\mathbf {Q}_s+\\delta \\mathbf {I})$ by iteratively minimizing the local linearization (REF ).", "This leads to $\\mathbf {Q}_s^{(k+1)} = \\mathrm {argmin} ~ \\mathrm {Tr}\\big [(\\mathbf {Q}_s^{(k)}+\\delta \\mathbf {I})^{-1}\\mathbf {Q}_s \\big ].$ If we choose $\\mathbf {Q}_s^{(0)}=\\mathbf {I}$ , the first iteration of (REF ) is equivalent to minimizing the trace of $\\mathbf {Q}_s$ .", "Therefore, this heuristic can be viewed as a refinement of the nuclear norm heuristic.", "As a result, we always pick $\\mathbf {Q}^{(0)}_s=\\mathbf {I}$ , so that $\\mathbf {Q}^{(1)}_s$ is the result of the trace heuristic, and the iterations that follow try to reduce the rank of $\\mathbf {Q}^{(1)}_s$ further.", "Note that at each iteration we will solve a weighted trace minimization problem, which is equivalent to the following optimization problem $\\begin{split}(\\textrm {P2-1}): \\quad \\min &\\quad \\mathrm {Tr}\\left( \\mathbf {F}^H \\mathbf {A}\\mathbf {F} \\right) \\\\\\mathrm {s.t.}", "& \\quad \\log _2\\det \\left(\\mathbf {I}+\\mathbf {F}^H \\mathbf {R}\\mathbf {F} \\right) = {R_b} \\\\& \\quad \\mathrm {Tr}\\left( \\mathbf {F}^H \\mathbf {F} \\right) \\le {P_s}.\\end{split}$ where $\\mathbf {A}=(\\mathbf {Q}_s^{(k)}+\\delta \\mathbf {I})^{-1}$ , $\\mathbf {R}=\\mathbf {G}_1^{H}(\\mathbf {G}_2\\mathbf {Q}_p\\mathbf {G}_2^{H}+\\sigma _s^2\\mathbf {I})^{-1}\\mathbf {G}_1$ .", "This is a Schur-concave optimization problem with multiple trace/log-det constraints.", "From Theorem 1 in [24], the optimal solution to problem (REF ) is $\\mathbf {F}^{\\star }=\\mathbf {A}^{-1/2}\\mathbf {U}\\mathbf {\\Sigma }$ , and $\\mathbf {Q}_s^{(k+1)}=\\mathbf {F}^{\\star }{\\mathbf {F}^{\\star }}^{H}$ is an optimal solution to (REF ), where $\\mathbf {A}^{-1/2}=\\mathbf {U}_{\\mathbf {A}}\\mathbf {\\Lambda }^{-1/2}_{\\mathbf {A}}\\mathbf {U}^{H}_{\\mathbf {A}}$ , $\\mathbf {U}_{\\mathbf {A}}$ and $\\mathbf {\\Lambda }_{\\mathbf {A}}$ are defined in the eigen-decomposition $\\mathbf {A}=\\mathbf {U}_{\\mathbf {A}}\\mathbf {\\Lambda }_{\\mathbf {A}}\\mathbf {U}^{H}_{\\mathbf {A}}$ , $\\mathbf {U}$ is a unitary matrix, and $\\mathbf {\\Sigma }=\\mathrm {diag}(\\sqrt{\\mathbf {}{p}) is a rectangular diagonal matrix.", "}Substituting the optimal solution structure $ F$ into (\\ref {pr:Log-detIter}), we have the following equivalent problem\\begin{equation}\\begin{split}(\\textrm {P2-2}): \\quad \\min &\\quad \\mathrm {Tr}\\left( \\mathbf {\\Sigma }\\mathbf {\\Sigma }^{H} \\right) \\\\\\mathrm {s.t.}", "& \\quad \\log _2\\det \\left(\\mathbf {I}+\\mathbf {\\Sigma }^H \\mathbf {U}^{H}\\tilde{\\mathbf {R}}\\mathbf {U}\\mathbf {\\Sigma } \\right) = {R_b} \\\\& \\quad \\mathrm {Tr}\\left( \\mathbf {U}^H \\mathbf {A}^{-1} \\mathbf {U} \\mathbf {\\Sigma }\\mathbf {\\Sigma }^{H} \\right) \\le {P_s}\\end{split}\\end{equation}where $ R=A-1/2RA-1/2$.", "It is found that the equivalent problem (\\ref {pr:Log-detEqu}) is essentially equivalent to the converse formulation\\begin{equation}\\begin{split}(\\textrm {P2-3}): \\quad \\max &\\quad \\log _2\\det \\left(\\mathbf {I}+\\mathbf {\\Sigma }^H \\mathbf {U}^{H}\\tilde{\\mathbf {R}}\\mathbf {U}\\mathbf {\\Sigma } \\right) \\\\\\mathrm {s.t.}", "& \\quad \\mathrm {Tr}\\left( \\mathbf {\\Sigma }\\mathbf {\\Sigma }^{H} \\right) = P_0 \\\\& \\quad \\mathrm {Tr}\\left( \\mathbf {U}^H \\mathbf {A}^{-1} \\mathbf {U} \\mathbf {\\Sigma }\\mathbf {\\Sigma }^{H} \\right) \\le {P_s}.\\end{split}\\end{equation}This is because both formulations (\\ref {pr:Log-detEqu}) and (\\ref {pr:Log-detEqu1}) describe the same tradeoff curve of performance versus power.", "Therefore, the quality-constrained problem (P2-2) can be numerically solved by iteratively solving the power-constrained problem (P2-3), combined with the bisection method.$ The problem formulation in (P2-3) is a Schur-convex optimization problem with two trace constraints.", "Using Theorem 1 in [24] again, if we let $\\tilde{\\mathbf {R}}=\\mathbf {U}_{\\tilde{\\mathbf {R}}}\\mathbf {\\Lambda }_{\\tilde{\\mathbf {R}}}\\mathbf {U}_{\\tilde{\\mathbf {R}}}^{H}$ denote the eigen-decomposition of $\\tilde{\\mathbf {R}}$ , then the optimal unitary matrix $\\mathbf {U}$ will be chosen as $\\mathbf {U}_{\\tilde{\\mathbf {R}}}$ .", "Denoting $\\mathbf {a}=\\mathrm {diag}(\\mathbf {U}^{H}\\mathbf {A}^{-1}\\mathbf {U})$ and letting $\\lambda _{\\tilde{\\mathbf {R}},1} \\ge \\lambda _{\\tilde{\\mathbf {R}},2} \\ge \\cdots \\ge \\lambda _{\\tilde{\\mathbf {R}},N_a}$ represent the diagonal elements of $\\mathbf {\\Lambda }_{\\tilde{\\mathbf {R}}}$ , the optimal power allocation can be shown to have the form of a multilevel waterfilling solution: $p_i = \\left(\\frac{1}{\\mu + a_i\\nu } - \\frac{1}{\\lambda _i} \\right)^{+}, ~~ i = 1,\\dots ,N_a$ where $a_i$ is the $i$ th element of $\\mathbf {a}$ , and $\\mu , \\nu $ can be shown to be the nonnegative Lagrange multipliers associated with the two power constraints.", "The algorithmic description for the log-det heuristic approach is outlined in Algorithm REF .", "[htpb] Iterative log-det heuristic Algorithm for rank-minimization problem $P_s > 0, R_b > 0$ , set $\\delta = 10^{-6}, \\Delta = 10^{-3}, k=0$ ,     $\\mathbf {Q}_s^{(0)}=\\mathbf {I}$ , $\\mathbf {R}=\\mathbf {G}_1^{H}(\\mathbf {K}_s+\\sigma _s^2\\mathbf {I})^{-1}\\mathbf {G}_1$ .", "$\\mathbf {A}=(\\mathbf {Q}_s^{(k)}+\\delta \\mathbf {I})^{-1}$ , $\\tilde{\\mathbf {R}}=\\mathbf {A}^{-1/2}\\mathbf {R}\\mathbf {A}^{-1/2}$ , $\\operatorname{eig}(\\tilde{\\mathbf {R}})=\\mathbf {U}_{\\tilde{\\mathbf {R}}}\\mathbf {\\Lambda }_{\\tilde{\\mathbf {R}}}\\mathbf {U}_{\\tilde{\\mathbf {R}}}^{H}$ .", "set $\\mathbf {U}=\\mathbf {U}_{\\tilde{\\mathbf {R}}}$ , $\\mathbf {a}=\\mathrm {diag}(\\mathbf {U}^{H}\\mathbf {A}^{-1}\\mathbf {U})$ .", "Solve: $\\begin{split}\\min ~~ & \\mathbf {1}^{T}\\mathbf {p} \\\\\\mbox{\\rm s.~t.}", "~~ & \\log _2 \\left(\\prod _{i}^{} (1+p_i \\lambda _{\\tilde{\\mathbf {R}},i})\\right)=R_b \\\\& \\mathbf {a}^T\\mathbf {p} \\le {P_s}.\\end{split}$ $\\mathbf {\\Sigma }=\\mathrm {diag}(\\mathbf {p})$ $\\mathbf {F}=\\mathbf {A}^{-1/2}\\mathbf {U}\\mathbf {\\Sigma }$ $\\mathbf {Q}_s^{(k+1)}=\\mathbf {F}\\mathbf {F}^{H}$ $\\log _2(\\det (\\mathbf {Q}_s^{(k)}+\\delta \\mathbf {I}))-\\log _2(\\det (\\mathbf {Q}_s^{(k+1)}+\\delta \\mathbf {I})) < \\Delta $ $\\mathbf {a}^T \\mathbf {p} > P_s$ Declare outage $\\rho = \\mathbf {a}^T \\mathbf {p} /P_s;$ $\\quad \\mathbf {Q}_s = \\mathbf {Q}_s^{(k+1)}$" ], [ "Underlay CR Downlink", "In this section we extend the blind underlay precoding paradigm to a MIMO underlay downlink network with $K$ UCRs.", "We consider a modified block-diagonalization precoding strategy [27] where multiple data streams are transmitted to each UCR.", "Let each UCR be equipped with $N_s$ antennas for simplicity, although the proposed precoding schemes hold for heterogeneous receiver array sizes as long as the total number of receive antennas does not exceed $N_a$ .", "The extension to the case where the UCT serves $N_a$ spatial streams regardless of the total number of receive antennas can be made using the coordinated beamforming approach [27], for example.", "The received signal at UCR $k$ is now ${{\\mathbf {y}}_k} = {{\\mathbf {G}}_{k,1}}{{\\mathbf {W}}_k}{{\\mathbf {s}}_{u,k}} + \\sum \\limits _{j \\ne k}^{{K_u}} {{{\\mathbf {G}}_{k,1}}{{\\mathbf {W}}_j}{{\\mathbf {s}}_{u,j}}} + {{{\\mathbf {G}}_{k,2}}{{\\mathbf {s}}_{p}}} + {{\\mathbf {n}}_k}$ where $\\mathbf {G}_{k,1} \\in \\mathbb {C}^{N_s \\times N_a}$ is the main channel, ${{\\mathbf {W}}_k}\\in \\mathbb {C}^{N_a \\times l_k}$ is the precoding matrix applied to signal $\\mathbf {s}_{u,k} \\in \\mathbb {C}^{l_k \\times 1}$ for user $k$ , $\\mathbf {s}_{p}$ is the PT signal received over interfering channel ${\\mathbf {G}}_{k,2}\\in \\mathbb {C}^{N_s \\times N_p}$ , and ${{\\mathbf {n}}_k}\\sim \\mathcal {CN}(0,\\sigma _k^2 \\mathbf {I})$ is additive Gaussian noise.", "The UCT transmit covariance per UCR is now $\\mathbf {Q}_{k,s}={\\mathbf {W}}_k{\\mathbf {W}}_k^H$ , and the overall UCT transmit covariance assuming independent messages is $\\mathbf {Q}_{s}=\\sum \\nolimits _{k=1}^K \\mathbf {Q}_{k,s}$ .", "We assume each UCR has a desired information rate of $R_k$ , and adopt the “BD for power control\" approach in [27].", "Letting ${{\\mathbf {W}}_k} = {{\\mathbf {T}}_k}{\\mathbf {\\Lambda }}_k^{{1 \\mathord {\\left\\bad.", "{\\vphantom{1 2}} \\right.\\hspace{0.0pt}} 2}}$ , it is possible to separately design the beamforming matrix ${\\mathbf {T}}_k$ and diagonal power allocation matrix ${\\mathbf {\\Lambda }}_k$ per user to achieve rate $R_k$ in a two-step process.", "Let ${{\\mathbf {G}}_{ - k}} = \\left[ {\\begin{array}{*{20}{l}}{{{\\mathbf {G}}_{1,1}}}& \\cdots &{{{\\mathbf {G}}_{k - 1,1}}}&{{{\\mathbf {G}}_{k + 1,1}}}& \\cdots &{{{\\mathbf {G}}_{{K,1}}}}\\end{array}} \\right]$ represent the the overall UCR downlink channel excluding the $k^{th}$ user.", "First, a closed-form solution for the unit-power beamforming matrix ${\\mathbf {T}}_k$ of user $k$ is obtained from the nullspace of ${{\\mathbf {G}}_{ - k}}$ .", "To achieve this, from the SVD ${{\\mathbf {G}}_{ - k}} = {{\\mathbf {U}}_{ - k}}{{\\mathbf {\\Sigma }}_{ - k}}{\\left[ {\\begin{array}{*{20}{c}}{{{\\mathbf {V}}_{ - k,1}}}&{{{\\mathbf {V}}_{ - k,0}}}\\end{array}} \\right]^H}$ , the last $(N_a-l_k)$ right singular vectors contained in ${\\mathbf {V}}_{ - k,0}$ can be used to construct $\\mathbf {T}_k$ [27].", "The BD strategy therefore completely eliminates intra-UCR interference on the underlay downlink, and the residual interference-plus-noise covariance matrix at UCR $k$ is ${{\\mathbf {Z}}_k} = {{\\mathbf {G}}_{k,2}}{{\\mathbf {Q}}_p}{\\mathbf {G}}_{k,2}^H + \\sigma _s^2{\\mathbf {I}}.$ Proceeding to the power allocation step, let $\\mathrm {rank}({{\\mathbf {G}}_{k,1}}{{\\mathbf {T}}_k})=r_k$ for user $k$ 's effective channel, and assume $l_k=r_k$ .", "Consider the SVD of user $k$ 's pre-whitened effective channel ${\\mathbf {Z}}_k^{ - {1 \\mathord {\\left\\bad.", "{\\vphantom{1 2}} \\right.\\hspace{0.0pt}} 2}}{{\\mathbf {G}}_{k,1}}{{\\mathbf {T}}_k} = {{\\mathbf {U}}_k}{{\\mathbf {\\Lambda }}_k}{\\mathbf {V}}_k^H$ where $\\mathbf {\\Lambda }_k=\\operatorname{diag}\\left(\\lambda _{k,1},\\ldots ,\\lambda _{k,r_k}\\right)$ is the power allocation matrix.", "While [27] computes $\\mathbf {\\Lambda }_k$ using the classic waterfilling algorithm in order to minimize the power required to achieve rate $R_k$ , we can instead apply any of the other schemes discussed in Sec.", "such as FWF.", "Due to the subadditivity of the $\\mathrm {rank}$ function, reducing the rank of the per-user transmit covariances via FWF effectively reduces the rank of the overall UCT transmit covariance $\\mathbf {Q}_{s}$ , which in turn mitigates the interference caused to the PR according to Proposition 1." ], [ "Primary Outage Probability", "In this section we characterize the impact of the classic and frugal waterfilling methods on the primary receiver performance assuming independent Rayleigh fading on all channels.", "Herein, the channels are mutually independent and are each composed of i.i.d.", "zero-mean circularly symmetric complex Gaussian entries, i.e., $\\mathrm {vec}\\left(\\mathbf {G}_1\\right) \\sim \\mathcal {CN}(\\mathbf {0},\\mathbf {I})$ , and the same distribution holds for $\\mathbf {H}_1$ ,$\\mathbf {H}_2$ , and $\\mathbf {G}_2$ .", "A first approach would be to directly analyze the PR rate outage probability $I_r=\\Pr \\left( {{R_p}\\left( {{{\\mathbf {Q}}_s}} \\right) \\le T} \\right)$ for a target rate $T$ , which is equivalent to $I_r=\\Pr \\left({\\log _2}\\left| {{\\mathbf {I}} + {{\\mathbf {H}}_1}{{\\mathbf {Q}}_p}{\\mathbf {H}}_1^H{{{\\left( {{{\\mathbf {K}}_p} + \\sigma _p^2{\\mathbf {I}}} \\right)}^{ - 1}}}} \\right|\\le T \\right)$ where ${\\mathbf {Q}}_s$ and ${\\mathbf {Q}}_p$ are obtained via one of the waterfilling methods on their noise-prewhitened channels and are therefore functions of random matrices $\\lbrace \\mathbf {G}_1,\\mathbf {G}_2\\rbrace $ and $\\lbrace \\mathbf {H}_1,\\mathbf {H}_2\\rbrace $ , respectively.", "Unfortunately, the computation of (REF ) is prohibitively complex since it is a non-linear function of the eigenvalues of four complex Gaussian random matrices (even if the PT applies uniform power allocation instead), and is an open problem to our best knowledge.", "Previous studies on the statistical distribution of MIMO capacity under interference usually circumvent this difficulty by assuming the MIMO transmitter and interferer adopt uniform or deterministic power allocation [28], [29], [37], which reduces the problem to one involving two complex Gaussian random matrices.", "As such, we are not aware of prior work on the statistics of MIMO capacity under interference where either one or both transmitters employ waterfilling as in our model.", "In light of the above, it is of interest to develop more tractable PR performance measures.", "One such candidate is the PR interference temperature outage probability (ITOP), which is the probability that ${T_p}\\left( {\\mathbf {Q}_s} \\right)$ [cf.", "(REF )] exceeds a threshold $\\eta $ : ${I_p}\\left( {{{\\mathbf {Q}}_s},\\eta } \\right) = \\Pr \\left(\\mathrm {Tr}\\left(\\mathbf {H}_{2}\\mathbf {Q}_s\\mathbf {H}_{2}^H\\right) \\ge \\eta \\right).$ The ITOP is appealing since the interference temperature metric is widely used in underlay systems, and can be considered to be the MIMO counterpart of efforts to characterize the statistical distribution of aggregate UCT interference in single-antenna networks as in [30].", "Paradoxically, however, it is seen in Sec.", "that FWF causes the highest ITOP, even though the average primary rate is the highest and the PR rate outage is the lowest when $\\mathbf {Q}_s$ is computed using FWF.", "Therefore, a more accurate surrogate for the PR rate outage $I_r$ is the interference leakage-rate outage probability (ILOP), defined as ${I_l}\\left( {{{\\mathbf {Q}}_s},\\eta } \\right) = \\Pr \\left(\\log _2\\left|\\sigma _p^2\\mathbf {I}+\\mathbf {H}_{2}\\mathbf {Q}_s\\mathbf {H}_{2}^H\\right| \\ge \\eta \\right),$ and it is verified in Sec.", "that UCT transmission schemes with the lowest ILOP also minimize $I_r$ .", "This is because the leakage rate has a direct impact on the PR rate: $R_p(\\mathbf {Q}_s)$ in (REF ) can be rewritten as ${R_p}\\left( {{{\\mathbf {Q}}_s}} \\right) &=& {\\log _2}\\left|\\sigma _p^2{\\mathbf {I}} + {{\\mathbf {H}}_1}{{\\mathbf {Q}}_p}{\\mathbf {H}}_1^H+\\mathbf {K}_p \\right| \\nonumber \\hfill \\\\&&\\:{-}{\\log _2}\\left|\\sigma _p^2{\\mathbf {I}} + \\mathbf {K}_p \\right|$ where the first term is the sum rate of the virtual PT/UCT multiple access channel (MAC) with optimal successive detection, and the second term is the leakage rate from the UCT.", "For the worst-case scenario where the PT is decoded first in the virtual MAC, decreasing the leakage rate improves the detection of the PT signal in the first term and simultaneously reduces the second term, thereby decreasing $I_r$ .", "On the other hand, the link between interference temperature and PR rate is more tenuous.", "Assume the UCT transmit covariance matrix $\\mathbf {Q}_s$ is of rank $k$ , $1 \\le k \\le \\min \\left( {{N_a},{N_s}} \\right)$ , where $k$ is determined by the choice of waterfilling scheme to achieve rate $R_b$ over the pre-whitened UCT channel $\\mathbf {\\tilde{G}}_1\\triangleq {{\\left( {{{\\mathbf {K}}_s} + \\sigma _s^2{\\mathbf {I}}} \\right)}^{ -1/2}}\\mathbf {G}_1$ .", "Assume $\\mathbf {\\tilde{G}}_1^H\\mathbf {\\tilde{G}}_1$ is of rank $d^{\\prime }$ , with non-zero ordered eigenvalues $\\left\\lbrace {{\\alpha _i}} \\right\\rbrace _{i = 1}^{d^{\\prime }}$ .", "Waterfilling yields a diagonal $\\mathbf {Q}_s$ with entries [16] ${\\left[ {{{\\mathbf {Q}}_s}} \\right]_{i,i}} = \\left[\\mu - \\frac{\\sigma _s^2}{{{\\alpha _i}}}\\right]^+,\\;i=1,\\ldots ,N_a,$ where the waterfilling level $\\mu $ is a function of $P_s$ , $\\sigma _s^2$ , and ${\\mathbf {\\alpha }} = \\left( {{\\alpha _1}, \\ldots ,{\\alpha _{d^{\\prime }}}} \\right)$ [31], [33].", "Similar arguments hold for the underlay downlink covariance $\\mathbf {Q}_s$ designed for sum-rate target $\\sum \\nolimits _k R_k$ and aggregate channel ${\\mathbf {\\tilde{G}}} = \\left[ {\\begin{array}{*{20}{c}}{\\left( {\\mathbf {Z}}_1^{ - {1 \\mathord {\\left\\bad.", "{\\vphantom{1 2}} \\right.\\hspace{0.0pt}} 2}}{{\\mathbf {G}}_{1,1}}{{\\mathbf {T}}_1}\\right)^T}&{\\ldots }&{\\left( {\\mathbf {Z}}_K^{ - {1 \\mathord {\\left\\bad.", "{\\vphantom{1 2}} \\right.\\hspace{0.0pt}} 2}}{{\\mathbf {G}}_{K,1}}{{\\mathbf {T}}_K}\\right)^T}\\end{array}} \\right]$ ." ], [ "Interference Temperature Outage Probability", "Noting that the ITOP and ILOP (REF )-(REF ) are still functions of four complex Gaussian matrices, we first develop bounds on the ITOP as follows.", "We assume $\\mathbf {H}_2$ is full rank such that $\\mathrm {rank}(\\mathbf {H}_2)=d=\\min (N_r,N_a)$ , which holds with probability 1 under i.i.d.", "Rayleigh fading.", "Define $r=\\min \\left( {{k},{d}} \\right)$ and let $\\lambda _i\\left( \\mathbf {A} \\right)$ denote the $i^{th}$ ordered eigenvalue of $\\mathbf {A}$ in descending order.", "Starting with the commutativity of the trace operator, ${I_p} &=& \\Pr \\left(\\mathrm {Tr}\\left( {{\\mathbf {H}}_2^H{{\\mathbf {H}}_2}{{\\mathbf {Q}}_s}} \\right) \\ge \\eta \\right)\\hfill \\\\&\\le & \\Pr \\left( {\\mathrm {Tr}\\left( {{\\mathbf {H}}_2^H{{\\mathbf {H}}_2}{{\\mathbf {Q}}_{s|{{\\mathbf {Q}}_p} = \\left( {{{{P_p}} \\mathord {\\left\\bad.", "{\\vphantom{{{P_p}} {{N_p}}}} \\right.\\hspace{0.0pt}} {{N_p}}}} \\right){\\mathbf {I}}}}} \\right) \\ge \\eta } \\right)\\\\&=&\\Pr \\left(\\sum \\limits _{i = 1}^r {{\\lambda _i}\\left( {{\\mathbf {H}}_2^H{{\\mathbf {H}}_2}{{\\mathbf {Q}}_s}} \\right)}\\ge \\eta \\right) \\hfill \\\\&\\approx & \\Pr \\left( {\\sum \\limits _{i = 1}^r {{\\lambda _i}\\left( {{\\mathbf {H}}_2^H{{\\mathbf {H}}_2}{{{\\mathbf {\\tilde{Q}}}}_s}} \\right)} \\ge \\eta } \\right)\\\\&\\le & \\Pr \\left(\\sum \\limits _{i = 1}^r {{\\lambda _i}\\left( {{\\mathbf {H}}_2^H{{\\mathbf {H}}_2}} \\right){\\lambda _i}\\left( {{{\\mathbf {\\tilde{Q}}}_s}} \\right)}\\ge \\eta \\right)\\\\&\\le & \\Pr \\left({\\lambda _1}\\left( {{{\\mathbf {\\tilde{Q}}}_s}}\\right)\\sum \\limits _{i = 1}^r {{\\lambda _i}\\left( {{\\mathbf {H}}_2^H{{\\mathbf {H}}_2}} \\right)}\\ge \\eta \\right)$ where in (REF ) we eliminate dependence on $\\mathbf {H}_1$ by assuming the PT adopts uniform power allocation $({{\\mathbf {Q}}_p} = \\left( {{{{P_p}} \\mathord {\\left\\bad.", "{\\vphantom{{{P_p}} {{N_p}}}} \\right.\\hspace{0.0pt}} {{N_p}}}} \\right){\\mathbf {I}})$ such that ${{\\mathbf {K}}_s} = \\left( {{{{P_p}} \\mathord {\\left\\bad.", "{\\vphantom{{{P_p}} {{N_p}}}} \\right.\\hspace{0.0pt}} {{N_p}}}} \\right){{\\mathbf {G}}_2}{\\mathbf {G}}_2^H$ , which is a worst-case interference scenario at the UCR according to Proposition 1 and potentially increases the power expended by the UCT; in () ${{{\\mathbf {\\tilde{Q}}}}_s}$ is the statistical waterfilling solution where $\\mu $ in (REF ) is a function of the statistics of $\\left\\lbrace {{\\alpha _i}} \\right\\rbrace _{i = 1}^d$ and offers nearly the same performance as instantaneous waterfilling [31]-[33]; and the inequality in () follows from the bound on the trace of a product of Hermitian matrices [34].", "We now define the ordered eigenvalue vectors ${\\mathbf {h}} = \\left( {{\\lambda _1}\\left( {{\\mathbf {H}}_2^H{{\\mathbf {H}}_2}} \\right), \\ldots ,{\\lambda _r}\\left( {{\\mathbf {H}}_2^H{{\\mathbf {H}}_2}} \\right)} \\right)$ and ${\\mathbf {q}} = \\left( {{\\lambda _1}\\left( {\\mathbf {\\tilde{Q}}}_s \\right), \\ldots ,{\\lambda _r}\\left( {\\mathbf {\\tilde{Q}}}_s \\right)} \\right)$ .", "Observe that the overall joint density of these random eigenvalues is given by the product of the individual joint densities: ${f_{{\\mathbf {h}},{\\mathbf {q}}}}\\left( {{\\mathbf {h}},{\\mathbf {q}}} \\right) = {f_{\\mathbf {h}}}\\left( {\\mathbf {h}} \\right){f_{\\mathbf {q}}}\\left( {\\mathbf {q}} \\right)$ due to the independence of the associated channel matrices.", "Thus, the bound in () can be rewritten as ${I_p} \\le {E_{\\mathbf {h}}}\\left\\lbrace {1 - {F_{{q_1}}}\\left( {{\\eta \\mathord {\\left\\bad.", "{\\vphantom{\\eta {\\sum \\nolimits _{i = 1}^r {{h_i}} }}} \\right.\\hspace{0.0pt}} {\\sum \\nolimits _{i = 1}^r {{h_i}} }}} \\right)} \\right\\rbrace $ where ${F_{{q_1}}}(x)$ is the cumulative distribution function (cdf) of the largest eigenvalue $q_1$ .", "From (REF ) we obtain ${F_{{q_1}}}\\left( x \\right) = {F_{{\\alpha _1}}}\\left( {{{\\sigma _s^2} \\mathord {\\left\\bad.", "{\\vphantom{{\\sigma _s^2} {\\left( {\\tilde{\\mu }- x} \\right)}}} \\right.\\hspace{0.0pt}} {\\left( {\\tilde{\\mu }- x} \\right)}}} \\right)$ where $\\bar{\\mu }$ is the statistical water level.", "The cdf of $\\alpha _1$ , the largest eigenvalue of $\\mathbf {\\tilde{G}}_1^H\\mathbf {\\tilde{G}}_1$ , is given next for the scenario $N_s\\ge N_a,N_s\\ge N_p$ .", "Lemma 1 [38] Given complex Gaussian matrices $\\mathbf {X}\\in \\mathbb {C}^{N_s \\times N_a}$ ,$\\mathbf {Y}\\in \\mathbb {C}^{N_s\\times N_p}$ , and $(N_p\\times N_p)$ diagonal matrix $\\mathbf {P}=\\mathrm {diag}(\\rho ,\\ldots ,\\rho )$ , the cdf of the largest eigenvalue $\\alpha _{max}$ of the quadratic form ${{\\mathbf {X}}^H}\\left( {{\\mathbf {YP}}{{\\mathbf {Y}}^H} + {\\sigma ^2}{\\mathbf {I}}} \\right){\\mathbf {X}}$ when $N_s\\ge N_a,N_s\\ge N_p$ is ${F_{{\\alpha _{max }}}}\\left( x \\right) = {K_1}\\left| {{{{\\mathbf {\\tilde{\\Delta }}}}_1}\\left( {x} \\right)} \\right|$ where $ {{{{\\mathbf {\\tilde{\\Delta }}}}_1}\\left( {x} \\right)} = {\\left[ {\\begin{array}{*{20}{c}}{{\\mathbf {\\tilde{Y}}}{{\\left( {x} \\right)}^T}}&{{{{\\mathbf {Z}}}^T}}\\end{array}} \\right]^T}$ , ${\\left[ {{{\\mathbf {\\tilde{Y}}}}\\left( {{x }} \\right)} \\right]_{i,j}} = \\left\\lbrace {\\begin{array}{*{20}{c}}{\\begin{array}{c}\\Gamma \\left( i \\right){I_{{N_a} - i}}\\left( \\rho \\right) - \\Gamma \\left( i \\right){e^{ - x}} \\hfill \\\\\\times \\sum \\limits _{k = 0}^{i - 1} {\\frac{{{x^k}}}{{k!", "}}{I_{{N_a} - i}}\\left( {\\frac{\\rho }{{1 + \\rho x}}} \\right),\\; i = 1, \\ldots ,{N_a},} \\hfill \\\\\\end{array}} \\\\{{{\\left( { - 1} \\right)}^{{N_s} - j}}{I_{{N_a} + {N_s} - i}}\\left( \\rho \\right),\\; i = {N_a} + 1, \\ldots ,{N_s},}\\end{array}} \\right.$ $I_a\\left( {b} \\right) = \\sum \\nolimits _{k = 0}^a {\\left( {\\begin{array}{*{20}{c}}a \\\\k\\end{array}} \\right){b^{j + k}}\\Gamma \\left( {j + k} \\right)}$ , and the normalization constant $K_1$ [38] and the entries ${\\left[ {{\\mathbf {Z}}} \\right]_{i,j}}$ [38] are functions of the array dimensions independent of $x$ .", "The cdf ${F_{{\\alpha _{1 }}}}\\left( x \\right)$ for other antenna array dimensions is of a similar form and can be found in [38].", "Now, in order to compute the expectation over $\\mathbf {h}$ in (REF ), we exploit the Gaussian distribution of $\\mathbf {H}_2$ based on the following lemma.", "Lemma 2 [16], [35] If $\\mathbf {X}$ is a $(N_r \\times N_a)$ matrix with i.i.d.", "zero-mean unit-variance complex Gaussian elements, then $\\mathbf {W}={\\mathbf {XX}}^H$ follows a central complex Wishart distribution if $N_r \\le N_a$ , otherwise $\\mathbf {W}={\\mathbf {X}^H\\mathbf {X}}$ is Wishart-distributed if $N_r > N_a$ .", "Given $c\\triangleq \\max (N_r,N_a)$ and $m\\triangleq \\min (N_r,N_a),$ the joint density of all $m$ ordered eigenvalues of ${{\\mathbf {W}}}$ is ${f_{\\mathbf {\\Lambda }}}\\left( {{\\lambda _{1}}, \\ldots ,{\\lambda _{{m}}}} \\right) = K_w{\\left| {{{\\mathbf {V}}_1}\\left( {\\mathbf {\\lambda }} \\right)} \\right|^2}\\prod \\limits _{i = 1}^{{m}} {\\frac{e^{ - {\\lambda _{i}}}}{\\left({\\lambda _{i}}\\right)^{{m}-{c}}}},$ where ${{{\\mathbf {V}}_1}\\left( {\\mathbf {\\lambda }} \\right)}$ is a Vandermonde matrix with entries ${\\left[ {{{\\mathbf {V}}_1}\\left( {\\mathbf {\\lambda }} \\right)} \\right]_{i,j}} = \\lambda _j^{i - 1}$ , and $K_w$ is a normalization constant independent of $\\mathbf {\\lambda }$ [35].", "For the case where $\\mathrm {rank}(\\mathbf {\\tilde{Q}}_s)=k$ is greater than $\\mathrm {rank}(\\mathbf {H}_2)=d$ , i.e., $r=d$ , the term ${\\sum \\nolimits _{i = 1}^r {{h_i}} }$ in (REF ) involves all $d$ ordered eigenvalues $\\left\\lbrace {{h_1}, \\ldots ,{h_d}} \\right\\rbrace $ and the associated joint density function $f_{\\mathbf {h}}\\left( {\\mathbf {h}} \\right)$ is given in Lemma REF .", "Thus (REF ) yields ${I_p} \\le 1 - K\\int { \\ldots \\int \\limits _\\mathfrak {D} {\\left| {{{{\\mathbf {\\tilde{\\Delta }}}}_1}\\left( {{{\\bar{h}}}} \\right)} \\right|} } \\left| {{{\\mathbf {V}}_1}\\left( {\\mathbf {h}} \\right)} \\right|\\prod \\limits _{i = 1}^d {\\xi \\left( {{h_i}} \\right)} d{\\mathbf {h}}$ where $K={K_1}{K_w}$ , ${{{\\bar{h}}}}={{{\\sigma _s^2} \\mathord {\\left\\bad.", "{\\vphantom{{\\sigma _s^2} {\\left( {\\tilde{\\mu }- {\\eta \\mathord {\\left\\bad.", "{\\vphantom{\\eta {\\sum \\nolimits _{i = 1}^d {{h_i}} }}} \\right.\\hspace{0.0pt}} {\\sum \\nolimits _{i = 1}^d {{h_i}} }}} \\right)}}} \\right.\\hspace{0.0pt}} {\\left( {\\tilde{\\mu }- {\\eta \\mathord {\\left\\bad.", "{\\vphantom{\\eta {\\sum \\nolimits _{i = 1}^d {{h_i}} }}} \\right.\\hspace{0.0pt}} {\\sum \\nolimits _{i = 1}^d {{h_i}} }}} \\right)}}}$ , $\\xi \\left( {{h_i}} \\right)={\\frac{e^{ - {h_{i}}}}{\\left({h_{i}}\\right)^{{m}-{c}}}}$ , and the integration region is $\\mathfrak {D} = \\left\\lbrace {\\infty \\ge {h_1} \\ge {h_2} \\ge \\ldots \\ge {h_d} \\ge 0} \\right\\rbrace $ .", "This multidimensional integral has a closed-form solution obtained from the following generalized Cauchy-Binet identity.", "Lemma 3 [40] For $\\mathbf {x}=\\left\\lbrace {{x_1}, \\ldots ,{x_M}} \\right\\rbrace $ , arbitrary integrable functions $c_i(\\cdot )$ , $u_i(\\cdot )$ , and ${\\varphi \\left( \\cdot \\right)}$ , $N \\times N$ matrix ${{\\mathbf {\\Phi }}\\left( {\\mathbf {x}} \\right)}$ and $M \\times M$ matrix ${{\\mathbf {\\Psi }}\\left( {\\mathbf {x}} \\right)}$ ($M\\le N$ ), where $\\Phi \\left( {\\mathbf {x}} \\right) = {\\left[ {\\begin{array}{*{20}{c}}{{{\\mathbf {C}}_1}{{\\left( \\mathbf {x} \\right)}^T}}&{{{\\mathbf {C}}_2}^T}\\end{array}} \\right]^T}$ , with entries ${{{\\left[ {{{\\mathbf {C}}_1}\\left( {\\mathbf {x}} \\right)} \\right]}_{i,j}} = {c_i}\\left( {{x_j}} \\right)}$ for $i=1,\\ldots ,N-M;j=1,\\ldots ,N$ , ${{{\\left[ {{{\\mathbf {C}}_2}} \\right]}_{i,j}} = {c_{2i,j}}}$ (constant scalars) for $i=N-M+1,\\ldots ,N;j=1,\\ldots ,N$ , and ${{{\\left[ {{{\\mathbf {\\Psi }}}\\left( {\\mathbf {x}} \\right)} \\right]}_{i,j}} = {u_i}\\left( {{x_j}} \\right)}$ , the following integral identity over domain $\\mathfrak {D} = \\left\\lbrace {b \\ge {x_1} \\ge {x_2} \\ge \\ldots \\ge {x_N} \\ge a} \\right\\rbrace $ holds: $\\int { \\cdots \\int \\limits _\\mathfrak {D} {\\left| {{\\mathbf {\\Phi }}\\left( {\\mathbf {x}} \\right)} \\right| \\cdot \\left| {{\\mathbf {\\Psi }}\\left( {\\mathbf {x}} \\right)} \\right|} } \\prod \\limits _{k = 1}^N {\\varphi \\left( {{x_k}} \\right)} d{\\mathbf {x}}= M!", "\\det \\mathbf {B}$ where ${\\left[ {\\mathbf {B}} \\right]_{i,j}} = \\left\\lbrace {\\begin{array}{*{20}{c}}{\\int _a^b {\\varphi \\left( x \\right){c_j}\\left( x \\right){u_i}\\left( x \\right)dx} },\\text{}{i=1,\\ldots ,N-M;\\forall j} \\\\{{c_{2i,j}}},\\quad {i=N-M+1,\\ldots ,N;\\forall j.", "}\\end{array}} \\right.$ A compact solution to (REF ) is then obtained by setting $\\mathbf {\\Phi }={{{\\mathbf {\\tilde{\\Delta }}}}_1}\\left( {{{\\bar{h}}}} \\right)$ with $\\rho =P_p/N_p$ , ${{\\mathbf {\\Psi }}\\left( {\\mathbf {x}} \\right)}={{{\\mathbf {V}}_1}\\left( {\\mathbf {h }} \\right)}$ , and $\\varphi \\left( {{x}} \\right)=\\xi \\left( {{h}} \\right)$ in Lemma REF : ${I_p} \\le 1 - d!K\\left| {\\mathbf {B}_1} \\right|;$ ${\\left[ {\\mathbf {B}_1} \\right]_{i,j}} = \\left\\lbrace {\\begin{array}{*{20}{c}}{\\int _0^\\infty {\\xi \\left( x \\right)x_j^{i-1} {\\left[ {{{\\mathbf {\\tilde{Y}}}}\\left( {{x }} \\right)} \\right]_{i,j}} dx} },\\text{}{i=1,\\ldots ,N-M;\\forall j} \\\\{\\left[ {{\\mathbf {Z}}} \\right]_{i,j}},\\quad {i=N-M+1,\\ldots ,N;\\forall j.", "}\\end{array}} \\right.$ and requires a computationally-inexpensive one-dimensional numerical integration of the product of elementary functions in (REF ).", "On the other hand, when $r=k<d$ , only the $k$ largest eigenvalues are included in the term ${\\sum \\nolimits _{i = 1}^r {{h_i}} }$ in (REF ), which necessitates invoking the corresponding joint density function given below in Lemma REF .", "Lemma 4 [39] The joint density function of the ordered subset of the $s$ largest eigenvalues of Wishart matrix $\\mathbf {W}$ having $m$ non-zero eigenvalues in total is ${\\begin{array}{c}{f_{{\\lambda _1}, \\ldots ,{\\mathbf {\\lambda } _s}}}\\left( {\\mathbf {\\lambda }} \\right) = K_w\\sum \\limits _{{\\mathbf {n}},{m},s} \\sum \\limits _{{\\mathbf {m}},{m},s} sgn\\left( {{\\mathbf {n}},{\\mathbf {m}}} \\right)\\left| {\\mathbf {D}\\left( {{\\lambda _s}} \\right)} \\right|\\\\\\quad \\qquad \\times \\prod \\limits _l^s e^{-1}\\lambda _l^{{N_r} - {N_a} + {n_l} - {m_l} + 2}\\end{array}}$ where $\\mathbf {n}=\\mathbf {m}=\\lbrace 1,\\ldots ,s\\rbrace $ , each summation is a $N_r$ -fold nested sum over permutations ${r_{i,{\\mathbf {n}}}},{r_{j,{\\mathbf {m}}}}$ of the index sets as defined in [39] with sign determined by $sgn\\left( {{\\mathbf {n}},{\\mathbf {m}}} \\right)\\in \\left\\lbrace { \\pm 1} \\right\\rbrace $ [39], ${\\left[ {\\mathbf {D}\\left( {{\\lambda _s}} \\right)} \\right]_{i,j}} = \\gamma \\left( {{N_a} - {N_r} + {r_{i,{\\mathbf {n}}}} + {r_{j,{\\mathbf {m}}}},{\\lambda _s}} \\right),$ and $\\gamma (a,b)$ is the incomplete gamma function [38].", "Substituting $\\varphi \\left( {{x_k}} \\right)=x_k^{{N_r} - {N_a} + {n_l} - {m_l} + 2}$ and associated expressions into (REF ) and invoking Lemma REF provides ${I_p} \\le 1 - \\frac{k!K}{e^k}\\sum \\limits _{{\\mathbf {n}},{d},k} \\sum \\limits _{{\\mathbf {m}},{d},k} sgn\\left( {{\\mathbf {n}},{\\mathbf {m}}} \\right)\\left| {\\mathbf {B}_2} \\right|$ where ${\\left[ {\\mathbf {B}_2} \\right]_{i,j}} = \\left\\lbrace {\\begin{array}{*{20}{c}}{\\int _0^\\infty {\\varphi \\left( {{x_k}} \\right){\\left[ {\\mathbf {D}\\left( {{x}} \\right)} \\right]_{i,j}} {\\left[ {{{\\mathbf {\\tilde{Y}}}}\\left( {{x }} \\right)} \\right]_{i,j}}dx} },\\text{}{i=1,\\ldots ,d-k,} \\\\{\\left[ {{\\mathbf {Z}}} \\right]_{i,j}},\\quad {i=d-k+1,\\ldots ,N;\\forall j.", "}\\end{array}} \\right.$ Since $\\gamma (a,b)$ is a standard function in MATLAB, the numerical integration of its product with two elementary functions in (REF ) is straightforward." ], [ "Interference Leakage Outage Probability", "Turning our attention to the ILOP, starting with the definition of $I_l$ we have ${I_l}\\left( {{{\\mathbf {Q}}_s},{\\mathbf {\\eta }}} \\right) &=& \\Pr \\left( {\\sum \\nolimits _{i = 1}^r {{{\\log }_2}\\left( {\\sigma _p^2 + {\\lambda _i}\\left( {{\\mathbf {Q}}_s}{\\mathbf {H}}_2^H{{{\\mathbf {H}}_2}} \\right)} \\right) \\ge \\eta } } \\right) \\nonumber \\hfill \\\\&\\approx & \\Pr \\left( {{{\\log }_2}\\prod \\nolimits _{i = 1}^r {{\\lambda _i}\\left( {{\\mathbf {Q}}_s}{\\mathbf {H}}_2^H{{{\\mathbf {H}}_2}} \\right) \\ge \\eta } } \\right) \\hfill \\\\&\\le & \\Pr \\left( {\\prod \\nolimits _{i = 1}^r {{\\lambda _i}\\left( {{{{\\mathbf {\\tilde{Q}}}}_s}} \\right){\\lambda _i}\\left( {{{\\mathbf {H}}_2}{\\mathbf {H}}_2^H} \\right) \\ge {2^\\eta }} } \\right) \\hfill \\\\&\\le & \\Pr \\left( {{{\\left( {{q_1}} \\right)}^r}\\prod \\nolimits _{i = 1}^r {{h_i} \\ge {2^\\eta }} } \\right)\\\\&=& {E_{\\mathbf {h}}}\\left\\lbrace {1 - {F_{{q_1}}}\\left( {{2^\\eta \\mathord {\\left\\bad.", "{\\vphantom{2^\\eta {\\prod \\nolimits _{i = 1}^r {{h_i}} }}} \\right.\\hspace{0.0pt}} {\\prod \\nolimits _{i = 1}^r {{h_i}} }}} \\right)^{1/r}} \\right\\rbrace $ where in (REF ) we consider the interference-limited scenario which is of interest, and () is due to [34].", "The computation of () closely parallels that of (REF ), by again separating the cases $r=d<k$ and $r=k<d$ , followed by invoking Lemmas REF –REF for the former and Lemmas REF ,REF and REF for the latter case, respectively.", "Therefore, the resulting closed-form bounds for the ILOP are of the form in (REF ) and (REF ), with $\\bar{h}=\\left( {{2^\\eta \\mathord {\\left\\bad.", "{\\vphantom{2^\\eta {\\prod \\nolimits _{i = 1}^r {{h_i}} }}} \\right.\\hspace{0.0pt}} {\\prod \\nolimits _{i = 1}^r {{h_i}} }}} \\right)^{1/r}$ and all other terms being unchanged." ], [ "Numerical Results", "In this section, we present some numerical examples to demonstrate the performance of the proposed rank-minimization UCT transmit covariance designs in MIMO cognitive radio networks.", "We consider MIMO cognitive radio networks with one primary user and one or more underlay receivers.", "In all simulations, the channel matrices and background noise samples were assumed to be composed of independent, zero-mean Gaussian random variables with unit variance.", "In situations where the desired rate for UCT cannot be achieved with the given $P_s$ , rather than indicate an outage, we simply assign all power to transmit signals.", "The performance is evaluated by averaging over 1000 independent channel realizations." ], [ "Single Underlay Receiver Scenario", "We first consider the single UCR scenario, where each node is equipped with 6 antennas, and $P_s=100$ , $P_p=10$ .", "Fig.", "REF illustrates the average fractional power $\\rho $ and the average number of subchannels $N$ required to achieve the UCR desired rates by CWF, FWF and log-det heuristic algorithms.", "It is shown that the trace and rank of the UCT transmit covariance matrix $\\mathbf {Q}_s$ are two competing objectives, and any scheme which requires more power occupies fewer spatial dimensions.", "Among the three methods, CWF demands the largest spatial footprint, while the FWF scheme offers the smallest feasible number of transmit dimension.", "We should point out that the log-det heuristic algorithm for matrix rank minimization does not always provide the smallest transmit dimension, compared to FWF.", "This is because the log-det algorithm is an approximate heuristic and can only give a local minimum.", "Figure: Achieved PU rate versus UCT desired rate.The achieved average primary user data rates for all the methods is depicted in Fig.", "REF .", "As expected, the lower-rank UCT transmit convariance will cause lesser degradation on average to the PU communication link, thus resulting a higher PR rate in accordance with Proposition 1.", "Compared to CWF, either of the proposed modified waterfilling algorithms or the log-det heuristic lead to more desirable PR rates, with the advantage of FWF being more pronounced as $R_b$ increases.", "Figure: Two metrics of PR Interference versus UCR desired rate.To obtain greater insight, Fig.", "REF compares two metrics of interference at PR using different algorithms, where one is the newly-defined UCT-PR leakage rate, the other one is the commonly-used interference temperature.", "We notice that an interesting phenomenon: the two metrics gives the opposite trend.", "It is worth to point out that the commonly-used interference temperature metric does not accurately capture the interference impact caused by the UCT on the primary mutual information, while the interference leakage rate remedies this defect.", "Figure: Empirical ccdf of interference temperature and leakage rate under CWF and FWF.For the statistical characterization of the proposed schemes, we exhibit the empirical complementary cdfs and select analytical upper bounds from Sec.", "of the interference temperature and leakage rate metrics in Fig.", "REF , for 6 antennas at all users and $P_s=200,R_b=8,P_p=40$ .", "An immediate observation is the conflicting trends of the leakage and temperature metrics: FWF causes a much greater interference temperature outage and much smaller leakage rate outage compared to CWF, and the superiority of one versus the other is not apparent.", "To resolve this dilemma, the corresponding empirical PR rate ccdfs are shown in Fig.", "REF , and it is clear that employing FWF leads to a very significant reduction in PR rate outage probability as compared to CWF.", "Furthermore, the interference temperature outage is again seen to be misleading regarding the true impact on the PR rate outage probability.", "Thus, FWF outperforms CWF in terms of both average PR rate and PR rate outage probability.", "Figure: Empirical ccdf of PR rate under CWF and FWF." ], [ "MIMO Underlay Downlink Scenario", "Next, we evaluate the performance of the proposed algorithms with the modified BD strategy of Sec.", "for a MIMO underlay downlink system, where there are $K=3$ UCRs, and $N_a=12$ , $N_p=N_r=N_s=4$ .", "Without loss of generality it is assumed that the desired rate targets for all UCRs are the same, i.e.", "$R_1=R_2=R_3$ .", "Fig.", "REF illustrates the achieved PU rate versus per UCT desired rate, when $P_s=100$ or $20dB$ and $P_p=10$ or $10dB$ .", "The benefit of minimizing the transmit covariance rank is seen to hold even for the multi-user downlink scenario.", "Figure: Achieved PR rate versus UCT transmit power in MIMO downlink system with identical target rates R 1 =R 2 =R 3 =5R_1=R_2=R_3=5bits/s/Hz.It is also of interest to see how the achieved PR rates under the various designs vary with the UCT transmit power, when the desired rate for each UCT is fixed.", "The simulation settings are the same as above, except that we fix $R_1=R_2=R_3=5$ and $P_s \\in [10dB,20dB]$ .", "The results are shown in Fig.", "REF , with the corresponding leakage rate and interference temperature metrics in Fig.", "REF .", "Once again, FWF offers the optimal average PR rate and PR rate outage probability in the downlink scenario.", "Figure: Two metrics of PR interference versus UCT transmit power in MIMO underlay downlink with identical target rates R 1 =R 2 =R 3 =5R_1=R_2=R_3=5." ], [ "Conclusion", "This paper has proposed a rank minimization precoding strategy for underlay MIMO CR systems with completely unknown primary CSI, assuming a minimum information rate must be guaranteed on the CR main channel.", "We presented a simple waterfilling approach can be used to find the minimum rank transmit covariance that achieves the desired CR rate with minimum power.", "We also presented two alternatives to FWF that are based on convex approximations to the minrank criterion, one that leads to conventional waterfilling for our CR problem, and another based on a log-det heuristic.", "The CWF approach turns out to be a poor approximation to the min-rank objective, while the log-det approach provides performance similar to FWF, although FWF consistently leads to the highest throughput for the primary link.", "We also observed that reducing the inteference temperature metric is surprisingly not consistent with improving the PR throughput; in particular, FWF has the highest interference temperature of the algorithms studied, but also leads to the highest PR rate.", "As an alternative, we proposed an interference leakage metric that is a better indicator of the impact of the CR on the primary link." ] ]

[ [ "Localization of water monomers inside ice-like clusters" ], [ "Abstract On the basis of the experimental data we suggest that water monomers could be trapped in channels running through ice-like clusters in water.", "Our argument relies on a simple model that describes the motion of a dipole particle inside a channel in the presence of an electric field with linear gradient.", "The model admits of both finite and infinite regimes of motion so that the finite one could correspond to the particle being confined to a channel." ], [ " Introduction: Water monomers trapped in channels of ice-like structures.", "It was W.C. Röntgen, [1] who suggested the presence of ice-like structural fragments in liquid water, and put forward the hypothesis of two fluid structure of water in equilibrium.", "Later, O.Ya.", "Samoilov, [2], and L.Pauling, [3],see also [4], extended the Röntgen model by introducing the concept of clathrate, low density and high density water.", "The idea has obtained experimental support that provides evidence for the existence of two kinds of water.", "Thus, using the technique of four-photon spectroscopy we have observed, for the first time to the best of our knowledge, the rotational resonances in water due to monomers $H_2O, \\; H_2O_2$ and $OH$ [5], [6].", "It is important that the mobility of monomers could explain the high permeation of water channels, in biological membranes, [7].", "As to the ice-like structural fragments, recently, Nielsson et al,[8], using x-ray spectroscopy, have found experimental evidence in favor of their existence.", "They visualized them as cluster structures, of several tens of Å and hexagonal ice symmetry $1h$ .", "Considering water as a mixture of ice-like clusters and monomers, makes for better understanding the physics of water.", "Nevertheless, there are still questions about the nature of coexistence of the monomers and the ice-like clusters.", "Specifically, the penetration of the $H_2 O$ monomers through channels of the ice-like clusters needs exploring.", "The key to the problem is provided by the study of beams of fast particles in crystalline media, effected many years ago for the needs of nuclear physics, [9].", "The main point is that the velocity of a particle is greatly enhanced if the beam is directed along a crystalline axis providing a kind of channel.", "Thus, one can claim the channeling effect, which appears to be of a significance reaching outside nuclear physics.", "In fact the problem is the old one.", "Years ago, Ya.I.", "Frenkel, [4], considered the motion of molecules and ions in channels inside crystalline structures, see also [10].", "The observation of rotational modes of water monomers suggests the existence of cavities that could accommodate the motion on a time scale larger than that due to the switching of hydrogen bonds in water, that is $1 - 2 \\; ps$ according to paper [11].", "The channels inside a cluster with hexagonal ice 1h structure, of diameter $ 5.7$ Å, could serve such cavities.", "It is worthwhile to notice that similar equipotential cavities can be provided through the freezing of water in a cryogenic matrix formed by argon, [12], and carbon nanotubes of diameter 14 Å, [13].", "It is reported,[13], that molecules of water preserve their mobility in carbon nanotubes down to temperatures $\\approx ~8 K$ .", "The main point about the dynamics of water monomers is that they could remain trapped in crystalline channels instead of forming complexes of hydrogen bonds and quitting the channels.", "In this paper we are considering a simple qualitative model that could accommodate the phenomenon.", "Our arguments essentially rely on the interplay of translational motion of a monomer and its rotational dynamics caused by the dipole moment of water molecule .", "Figure: Potential U=pE(x)cos(ϕ)U = p \\, E(x) \\, cos(\\varphi )" ], [ "A quasi-classical model for the motion of monomer ", "We shall draw a qualitative picture of the motion of a monomer as the dipole in the following potential $ U = p \\, E(x) \\, cos(\\varphi )$ .", "Potential surface is presented in Fig.REF .", "To that end we shall consider the latter as a two-dimensional rotator moving in direction of axis-$x$ in an external electric field, $E(x)$ , which mimics constraints imposed by the channel.", "Thus, its configuration is specified by angle $\\varphi $ describing its rotation and coordinate $x$ its position in axis-$x$ .", "Its dynamics is described by angular momentum $L$ and momentum $P$ .", "It is important to choose the right characteristic scales of the system.", "We shall take: (1) time scale $\\tau = 10^{-14} \\; sec$ corresponding to rotations of water molecules; (2) spatial scale $r = 3 \\cdot 10^{-8} \\; cm$ , close to the size of a water molecule; (3) mass scale $3.2 \\cdot 10^{-23}$ corresponding to a water molecule; (4) dipole moment $1 D = 10^{-18 } \\; CGS$ ; (5) electric field $10 \\; kV /cm$ .", "Hence we infer: (6) velocity and momentum scales $3 \\cdot 10^6 \\; cm / sec$ , $10^{-16} \\; gr \\, cm \\, sec^{-1}$ , respectfully; (7) moment of inertia $\\approx 3 \\cdot 10^{-38} gr \\, cm^2$ .", "We shall assume that the external electric field, $E(x)$ , is linear in $x$ $E(x) = A \\, x \\; + \\; W$ The scales are conducive to the use of quasi-classical approximation and numerical simulation.", "In fact, for the above characteristic scales we have the de Broglie wave length $\\lambda = \\frac{\\hbar }{P} \\approx 10^{-11} \\; cm .$ Assuming the size of a channel $ {\\cal L}\\approx 10$ Å, we get the parameter of quasi-classical expansion $\\mu = \\frac{\\lambda }{{\\cal L}} \\approx 10^{-4},$ that is sufficiently small.", "The usual constraint imposed on the interaction potential, [14], $\\left| \\frac{\\partial U(\\overline{x})}{\\partial \\overline{x}} \\right| \\gg \\frac{1}{2} \\left| \\frac{\\partial ^3 U(\\overline{x})}{\\partial \\overline{x}^3} \\right| \\; \\overline{\\Delta x^2},$ which means that the potential is smooth enough at the de Broglie length, is valid for the interaction between the dipole moment and the external field $U(x, \\varphi ) = p \\cdot \\ E(x) \\; cos(\\varphi )$ for $E(x)$ is linear in $x$ .", "The similar requirement for the rotational motion is also verified.", "We have the dimensional expansion parameter $\\mu _{\\varphi } = \\frac{\\hbar }{L} \\approx 3 \\cdot 10^{-4}, \\qquad L = I \\dot{\\varphi } \\approx 3 \\cdot 10^{-24} \\; erg \\cdot sec$ It should be noted that the requirements indicated above are not satisfied at return points where the size of de Broglie length rapidly changes.", "For example, this is the case even of the harmonic oscillator.", "Thus, the conventional quasi-classical approximation breaks down, [14].", "At these points one can change the x-representation for the p-representation, which, from the formal mathematical point of view, amounts to the Fourier transform.", "But the advanced theory, [15], [16], of the quasi-classical approximation claims that one can still use the classical equations of motion in conjunction with the Bohr-Sommerfeld quantization condition $\\int \\; p \\, dq = 2 \\, \\pi \\, n \\, \\hbar , \\qquad \\mbox{where } n \\mbox{ is an integer},$ provided the gradient of the Hamiltonian is not degenerate $\\left( \\frac{\\partial H}{\\partial x} \\right)^2 \\; + \\; \\left( \\frac{\\partial H}{\\partial p} \\right)^2 \\ne 0,$ where $x, \\; p$ are coordinates and momentum of the system.", "It should be noted that we are not constructing the wave function at turning points, but only find the quasi-classical trajectory by means of the classical Hamiltonian equations.", "It is the situation of the 'old' quantum theory by Niels Bohr.", "The arguments given above enable us to describe the dynamics of rotator within the framework of Lagrangian mechanics.", "The Lagrangian function reads ${\\cal L} = \\frac{I}{2} \\dot{\\varphi }^2 \\; + \\; \\frac{m}{2} v^2 \\; - \\; p \\cdot \\ E(x) \\; cos(\\varphi )$ Here $I = 3 \\cdot 10^{-38} \\; gr \\, cm^2$ is the moment of inertia of rotator equal by orders of magnitude to that of a molecule of water; $m = 3.2 \\cdot 10^{-23} \\; gr$ is the mass of rotator, close to the mass of a molecule of water; $p$ is the dipole moment of a water molecule.", "We take the electric field of the form $E = A \\, x \\; + \\; W$ It is worth noting that in our case both terms in the kinetic energy are $\\approx 10^{-10} \\; erg$ , whereas the potential energy is smaller by two orders of magnitude, that is $\\approx 10^{-12} \\; erg$ .", "Considering the shape of the potential energy, it is easy to come to the conclusion that Lagrangian function (REF ) admits both finite and infinite regimes of motion.", "The equations of motion corresponding to Lagrangian function (REF ) read $m \\, \\ddot{x} & = & p \\; \\frac{d E}{d x} \\; cos(\\varphi ) \\\\I \\, \\ddot{\\varphi }& = - & p \\; E(x) \\; sin(\\varphi ) \\nonumber $ Figure: Motion of a dipole particle represented in the window given by the variables:XX \\quad x-coordinate of the rotator;VV \\quad velocity of the rotator along x-axis;COSCOS \\quad cos(ϕ)cos(\\varphi ) where ϕ\\varphi is the phase of rotator.Dipole moment and rotator mass p=1.8Dp = 1.8 D, and 32·10 -24 gr32 \\cdot 10^{-24} \\; gr.Moment of inertia I=3·10 -38 grcm 2 I = 3 \\cdot 10^{-38} \\; gr \\, cm^2.Inserts A, B and C, D correspond to the confined and the infinite motions, respectfully.Equations (REF ) have only one integral of motion, the energy, and therefore one cannot solve it in a finite form, that is by writing down its solution by means of integrals.", "We have to employ numerical simulation for studying it.", "The key problem is the wise choice of visual representation of numerical results.", "In this respect, it should be noted that equations (REF ) are two equations of second order, and therefore their phase space is four dimensional comprising coordinates $x, \\; \\varphi $ and velocities $\\dot{x}, \\; \\dot{\\varphi }$ .", "We have used variables $x, \\; v = \\dot{x}$ and $cos( \\varphi )$ that provide a kind of three-dimensional window on the four-dimensional phase space of equations (REF ).", "It is important that due to the energy integral the values of angular velocity, $\\dot{\\varphi }$ , are bounded, and therefore we can infer the character of motion from the picture of a trajectory in the above window.", "The results of the simulation are illustrated in Fig.REF .", "We have employed the following values for coordinates and fields: A initial velocity, phase, and angular velocity: $3 \\cdot 10^6 \\; cm / sec, \\quad \\varphi = 0, \\quad 4 \\cdot 10^{14} \\; Hz$ , respectfully.", "Background field and field gradient $W = 5 \\; kV / cm$ and $A = 500 \\; kV / cm^2$ .", "Period of time considered $2 \\cdot 10^{-11} \\; sec$ .", "Amplitude of the particle's oscillation in $x$ several tens of Å.", "B initial velocity, phase, and angular velocity: $3 \\cdot 10^6 \\; cm / sec, \\quad \\varphi = 0, \\quad 4 \\cdot 10^{14} \\; Hz$ , respectfully.", "Background field and field gradient $W = 5.3 \\; kV / cm$ and $A = 500 \\; kV / cm^2$ .", "Period of time considered $ 10^{-10} \\; sec$ .", "Amplitude of the particle's oscillation in $x$ several tens of Å.", "C initial velocity, phase, and angular velocity: $ 3 \\cdot 10^6 \\; cm / sec, \\quad \\varphi = 1.7 \\; rad, \\quad 10^{13} \\; Hz$ , Background field and field gradient $W = 10 \\; kV / cm$ and $A = 10 \\; kV / cm^2$ .", "Infinite motion.", "D initial velocity, phase, and angular velocity: $ 3 \\cdot 10^6 \\; cm / sec, \\quad \\varphi = 1.7 \\; rad, \\quad 10^{12} \\; Hz$ , Background field and field gradient $W = 0.1\\; kV / cm$ and $A = 0.1 \\; kV / cm^2$ .", "Infinite motion.", "The motion being finite, that is the rotator confined to a finite region of x-axis, depends on values of initial coordinates and velocities, and values of the background field, $W$ , and the field gradient $A$ .", "The change of dynamical regimes has the threshold nature, so that small changes of fields and initial position may, generally, result in different regimes of motion." ], [ "Conclusions", "The numerical analysis of the dynamics of a rotator indicates that there exist regimes corresponding to the confinement of rotator to finite regions of phase space.", "The essential point is that the motion takes place in a field that increases linearly as regards the spatial coordinate, when a charged particle would move neglecting any boundaries.", "Thus, the finite motion is due to the interplay of translational and rotational degrees of freedom.", "In this respect it strongly resembles Maxwell's pendulum.", "The latter consists of a flywheel and two wires wound round the flywheel axis in the same direction and connected with a horizontal support.", "When released the flywheel goes down rotating round its axis due to the attached wires, until it arrives at the lowest point allowed by unwinding wires.", "Then it goes upwards rewinding in the opposite direction.", "The equations of motion are similar to those of the one dimensional rotator, so that one can consider Maxwell's pendulum as a kind of mechanical model for the confinement of a particle in channel.", "The model studied in this paper is the qualitative one.", "It provides the argument in favor of the claim that a monomer of water may be confined to a channel, or cavity, inside an ice-like cluster of liquid water hydration layers in the vicinity of \"membrane water channels\", [7].", "Specifically, we suggest that the $H$ and $H_30^{+}$ ions, which always exist in water, [2], [10], could produce a gradient electric field that may result in the channeling effect.", "The useful discussion with B.Yu.Sternin is gratefully acknowledged." ] ]

[ [ "Atmospheric dispersion effects in weak lensing measurements" ], [ "Abstract The wavelength dependence of atmospheric refraction causes elongation of finite-bandwidth images along the elevation vector, which produces spurious signals in weak gravitational lensing shear measurements unless this atmospheric dispersion is calibrated and removed to high precision.", "Because astrometric solutions and PSF characteristics are typically calibrated from stellar images, differences between the reference stars' spectra and the galaxies' spectra will leave residual errors in both the astrometric positions (dr) and in the second moment (width) of the wavelength-averaged PSF (dv) for galaxies.We estimate the level of dv that will induce spurious weak lensing signals in PSF-corrected galaxy shapes that exceed the statistical errors of the DES and the LSST cosmic-shear experiments.", "We also estimate the dr signals that will produce unacceptable spurious distortions after stacking of exposures taken at different airmasses and hour angles.", "We also calculate the errors in the griz bands, and find that dispersion systematics, uncorrected, are up to 6 and 2 times larger in g and r bands,respectively, than the requirements for the DES error budget, but can be safely ignored in i and z bands.", "For the LSST requirements, the factors are about 30, 10, and 3 in g, r, and i bands,respectively.", "We find that a simple correction linear in galaxy color is accurate enough to reduce dispersion shear systematics to insignificant levels in the r band for DES and i band for LSST,but still as much as 5 times than the requirements for LSST r-band observations.", "More complex corrections will likely be able to reduce the systematic cosmic-shear errors below statistical errors for LSST r band.", "But g-band effects remain large enough that it seems likely that induced systematics will dominate the statistical errors of both surveys, and cosmic-shear measurements should rely on the redder bands." ], [ "Introduction", "Weak gravitational lensing (WL) can be used for high-precision measurements of the expansion history of the Universe and the evolution of gravitational potentials within it [14], [17], [8].", "The WL effect by large-scale structure is best detected as a coherent elongation of background galaxy images, “cosmic shear,” which induces an RMS shift in axis ratios of only $\\approx 1\\%$ red or less for cosmologically distant source galaxies.", "Several ground-based surveys of $>1000$  deg$^2$ are commencing this year, including the 5000 deg$^2$ Dark Energy Survey (DES)http://www.darkenergysurvey.org/, with the goal of measuring the amplitude and redshift dependence of cosmic shear to high precision.", "A yet larger and deeper survey project for late in this decade, the Large Synoptic Survey Telescope (LSST)http://www.lsst.org/lsst/, is under intensive development.", "Ground-based visible observations are subject to atmospheric refraction, shifting each photon toward the zenith by an angle $R(\\lambda , z_a)$ that depends on wavelength and on apparent zenith angle $z_a$ .", "For finite-bandwidth observations, there are two measurable consequences: first, a centroid shift $\\overline{R} = \\frac{\\int d\\lambda R(\\lambda ,z_a) F(\\lambda ) S(\\lambda )}{\\int d\\lambda F(\\lambda ) S(\\lambda )}$ that depends on $z_a$ , the instrumental response function $F(\\lambda )$ , and the source spectrum $S(\\lambda )$ .", "Additionally the dispersion stretches the image along the zenith, which can be quantified by the photon-weighted second central moment of the dispersion kernel: $V \\equiv \\frac{\\int d\\lambda \\left[R(\\lambda ,z_a)-\\overline{R}\\right]^2 F(\\lambda ) S(\\lambda )}{\\int d\\lambda F(\\lambda ) S(\\lambda )}$ Like any other instrumental distortion or convolution of the galaxy images, this coherent vertical elongation of the galaxy images will be mistaken for WL shear if not removed.", "The standard strategy in WL observing is to use stellar images to estimate the instrumental point spread function (PSF), interpolating the stars' PSFs to the location of each galaxy, and then effect some limited form of deconvolution to extract the “pre-seeing” shape of the galaxy free of instrumental effects.", "In its simplest form, this instrumental correction amounts to subtracting the intensity-weighted second moments of the PSF from those observed for the galaxy to estimate the pre-seeing moments.", "If the atmospheric dispersion of the galaxies and stars are the same, then this stellar PSF correction will automatically correct the galaxy images for the dispersion $V$ .To be precise, the dispersion acts as a convolution, like the PSF, only if the galaxy's shape is constant across the band, i.e.", "the galaxy has homogeneous color.", "We will ignore in this work the complications of combining atmospheric dispersion with color gradients, as they produce shear errors at lower order.", "If, however, a galaxy's $V$ differs by an amount $\\mbox{$\\Delta V$}$ from the mean stellar $V$ used to estimate the PSF, then the inferred galaxy shape will be incorrect and propagate to cosmic shear errors.", "Our goal is to calculate the size of this effect, see if it will contribute a significant error to the cosmological results of DES or LSST, and evaluate some rudimentary techniques for eliminating the problem.", "Since cosmic-shear measurements are not affected by small shifts in the positions of galaxy images, we are less concerned with the centroid shift $\\overline{R}$ of the galaxy image than with the broadening $V$ .", "Suppose, however, that we intend to combine information from exposures at multiple hour angles and/or filters, for instance by stacking the images before measuring shapes.", "The registration of images will normally be done by forcing overlap of stellar images.", "Spectral differences between the galaxy and the mean calibrating star will induce a difference $\\Delta \\overline{R}$ in their mean refractions which depends on filter and hour angle.", "The stacked image will register the stars properly but not the galaxy, broadening the stacked galaxy image and hence producing a coherent shape error which again is confused with cosmic shear.", "Since both DES and LSST plan to combine many exposures to produce a single galaxy shape, we estimate $\\Delta \\overline{R}$ and determine whether it causes cosmic-shear errors comparable to the statistical errors of these surveys.", "This problem can be avoided by leaving the galaxy centroid free to differ among exposures in a shape analysis—but this typically requires that the galaxy have high signal-to-noise ratio in a single exposure, which precludes use of most of the galaxies detectable with the full combined survey data.", "The effect of atmospheric dispersion on weak lensing shear measurements was crudely estimated by [12], who determined that it should be unimportant to the measurement of galaxy cluster masses.", "In the intervening 15 years, however, WL observers have become far more ambitious, aiming to measure the much smaller cosmic-shear signal to high precision, so it is necessary to re-evaluate these limits, especially since neither the DES nor LSST optics incorporate atmospheric dispersion correctors.", "[10] investigate the impact of atmospheric dispersion on the LSST observations, but only with the criterion that the dispersion $V$ be a small contributor to the overall size of the PSF—they did not investigate the much stricter criterion that $V$ be correctable to a small fraction of the expected cosmic-shear signal.", "In the next section we derive requirements on $\\Delta \\overline{R}$ and $\\Delta V$ such that they will not bias cosmic-shear measurements at the level of the statistical errors of the DES and LSST surveys.", "Then § describes our methods for estimating these quantities, and § gives the resultant values in the $griz$ filters for a range of galaxy types and redshifts.", "Having demonstrated that atmospheric dispersion is indeed a significant issue in all but the $z$ band, we ask in § whether simple color-based corrections to $\\Delta \\overline{R}$ and $\\Delta V$ can recover the required accuracy.", "We conclude with an outlook in §.", "We use a simplified model of WL shear measurement to make rough estimates of the tolerable levels of $\\Delta \\overline{R}$ and $\\Delta V$ for cosmic-shear experiments.", "Galaxies are assigned shapes via their intensity-weighted second central moments $I_{\\mu \\nu } & \\equiv & \\frac{1}{f}\\int dx\\,dy\\, I(x,y) (\\mu - \\bar{\\mu })(\\nu - \\bar{\\nu }), \\\\\\bar{\\mu }& \\equiv & \\frac{1}{f}\\int dx\\,dy\\, I(x,y) \\mu , \\\\f & \\equiv & \\int dx\\,dy\\, I(x,y)$ where $I(x,y)$ is the surface brightness distribution of the source.", "In practice these integrals have divergent noise so must be bounded by some window function, which complicates the measurement but has little impact on our estimates of the effect of the (small) atmospheric dispersion on the measured shapes.", "The ellipticities are assigned as $e_1 & \\equiv & \\frac{I_{xx} - I_{yy}}{I_{xx}+I_{yy}} \\\\e_2 & \\equiv & \\frac{2I_{xy}}{I_{xx}+I_{yy}}.$ and we can also assign the galaxy a second-moment-based radius, usually quite similar to its half-light radius, $r^2 \\equiv \\left( I_{xx}+I_{yy} \\right).$ The two components $(\\gamma _1,\\gamma _2)$ of the applied gravitational lensing shear are estimated as $2\\gamma _i \\approx \\langle e_i \\rangle $ , where the average is taken over the ensemble of galaxies in a selected region of sky position and source redshift.", "These formulae require access to the galaxy's intensity $I^g(x,y)$ before any instrumental distortions, but we only have access to the observed $I^o$ after convolution with the PSF (including atmospheric dispersion components) $I^\\star $ .", "For present purposes it is a good approximation that: The second central moments of the observed image are the sum of those from the galaxy and the PSF, which means that the galaxy moments can be estimated as $I^g_{\\mu \\nu } = I^o_{\\mu \\nu } - I^\\star _{\\mu \\nu }.$ This holds exactly for shape measurements with unweighted second moments, which are not practical.", "However the additivity of second moments under convolution also holds for Gaussian galaxies and PSFs, hence we can expect this relation to be a good approximation for shape measurements based on Gaussian-weighted moments, such as [12].", "The effect of atmospheric dispersion is to boost slightly the second moment of the PSF in the zenith direction (call this the $x$ direction), $I^\\star _{xx}\\rightarrow I^\\star _{xx}+V$ .", "This relation is exact for unweighted second moments, but is also a very good approximation for Gaussian PSFs in the limit when the atmospheric dispersion is much smaller than the PSF itself—which is certainly true here.", "The same boost is given to $I^o_{xx}$ , and the subtraction will cancel $V$ in the estimate of intrinsic moments if our PSF is correctly calculated for the galaxy.", "Under these conditions we see that a small error $\\mbox{$\\Delta V$}\\ll r^2$ in the estimate of the $V$ for a galaxy's PSF will result in an error in the zenith second moment, $I^g_{xx}\\rightarrow I^g_{xx}+\\mbox{$\\Delta V$}$ , propagating to shape errors $e_1 & \\rightarrow & e_1\\left(1-\\frac{\\mbox{$\\Delta V$}}{r^2}\\right) + \\frac{\\mbox{$\\Delta V$}}{r^2} \\\\e_2 & \\rightarrow & e_2\\left(1-\\frac{\\mbox{$\\Delta V$}}{r^2}\\right).$ The shear estimate on a patch of sky is one-half the average galaxy ellipticity.", "In the shear-measurement literature it is typical to distinguish a multiplicative shear error $m$ and an additive shear error $c$ as $\\gamma _i^{\\rm est} = (1+m_i) \\gamma _i^{\\rm true} + c_i$ [7].", "Comparing to the above, we infer that the atmospheric dispersion errors propagate to a multiplicative shear error $m = \\left\\langle \\frac{\\mbox{$\\Delta V$}}{r^2} \\right\\rangle ,$ where the average is over the galaxies in a chosen bin of sky area and source redshift.", "The dispersion error also produces an additive shear error in the zenith direction of very similar amplitude, $c_1\\approx m/2$ ." ], [ "Shear measurement error budgets", "The multiplicative error $m$ in shear measurement propagates directly to a fractional error $2m$ in the cosmic-shear power spectrum.", "Particularly pernicious are multiplicative shear errors that vary as a function of source redshift, since most of the power of cosmic-shear experiments comes from tomographic analyses of detailed comparison between shear strength on different bins of source redshift.", "[9] (hereafter HTBJ) calculate the degradation of cosmological constraints for the DES and LSST cosmic-shear surveys due to multiplicative shear calibration errors with unknown redshift dependence.", "From their Figure 4 we can infer that a systematic error in $m$ of 0.008 (0.003) for DES (LSST) will raise the experiment's uncertainty in the dark-energy equation-of-state parameter $w$ by a factor of $\\sqrt{2}$ above the purely statistical errors.", "[It takes $\\approx 5\\times $ larger $m$ values to double the cosmological errors above the statistical uncertainties.]", "[1] carry out a similar analysis, concluding that $|m|<0.001$ is required for a space-based cosmic-shear survey somewhat more ambitious than LSST, consistent with HTBJ at factor-of-2 level.", "Since atmospheric dispersion is just one of several potential sources of shear calibration error, an experimental error budget would likely need to hold the contribution from atmospheric dispersion to $\\le \\frac{1}{2}$ of the total value in HTBJ.", "We can hence set a requirement that $\\left| \\left\\langle \\frac{\\mbox{$\\Delta V$}}{r^2} \\right\\rangle \\right| < \\left\\lbrace \\begin{array}{cc}0.004 & {\\rm (DES)} \\\\0.0015 & {\\rm (LSST)}\\end{array} \\right.$ Placing limits on additive shear errors is more complex because the limits on $c$ depend strongly on how the additive signals correlate across redshift bins.", "We can make a crude estimate, however, as follows: first, we note that the RMS value of cosmic shear is $\\approx 0.02$ , so the integrated shear power spectrum is $\\langle \\gamma ^2 \\rangle \\approx 4\\times 10^{-4}$ .", "The maximum tolerable LSST multiplicative shear error of $m<0.003$ produces an error in the integrated power spectrum of $2m \\langle \\gamma ^2 \\rangle \\approx 2.4\\times 10^{-6}$ .", "Spurious shear should not generate power larger than this value, $\\langle c^2 \\rangle < 2.4\\times 10^{-6}$ , suggesting that we need $c_{\\rm RMS} < 0.0015.$ If we again allocate half of this error budget to atmospheric dispersion, and keep in mind that the dispersion generates $c_1 = m/2$ , we find that the limits on $\\Delta V$ induced by additive shear are very similar to those induced by multiplicative shear (Equation (REF )).", "One issue that is more severe for additive shear than for multiplicative is that, for multiplicative errors, we are only sensitive to the mean $m$ over all galaxies in a given redshift slice [6], whereas for $c$ it is the variance across the sky, not the mean value, that is of particular concern.", "So we should be alert to aspects of the atmospheric dispersion systematic that will cause spurious shear fluctuating on the $\\sim 10$ scales that carry most cosmic-shear information.", "For instance, if $\\Delta V$ depends on source galaxy type within a redshift bin, we must be concerned that there will be variations in $c$ across the field as the mean background type fluctuates." ], [ "Requirements on atmospheric dispersion errors", "Our final step in setting requirements on $\\Delta V$ is to estimate the galaxy size to be used in Equation (REF ).", "The DES sets a goal of 10 lensing source galaxies per arcmin$^2$ .", "[11] produce a model of the background galaxy population sizes and colors using HST/COSMOS and other observations; from this catalog we note that there are 10 galaxies arcmin$^{-2}$ with half-light radii $r\\ge 04$ , hence the largest possible value of $\\langle 1/r^2\\rangle $ for DES would be to use this population.", "LSST is more ambitious, with a target source density of 30 arcmin$^{-2}$ , which will require reaching down to galaxies with $r\\ge 027$ .", "For this latter population, $\\langle 1/r^2\\rangle \\approx (035)^{-2}$ .", "It is also true, however, that many of the galaxies above these cutoff $r$ values will be unusable for weak lensing— too faint, crowded, or poor photo-z's—which will force the surveys to use galaxies smaller than these best-case cutoff sizes.", "As a rough guess, we therefore adopt estimates of $\\langle 1/r^2 \\rangle = (04)^{-2}$ for DES and $(027)^{-2}$ for LSST.", "The estimated requirements on $\\Delta V$ for the target galaxy population are therefore $\\left| \\left\\langle \\mbox{$\\Delta V$}\\right\\rangle \\right| < \\left\\lbrace \\begin{array}{cc}6\\times 10^{-4}\\,{\\rm arcsec}^2 & {\\rm (DES)} \\\\1\\times 10^{-4}\\,{\\rm arcsec}^2 & {\\rm (LSST)}\\end{array} \\right.$ Finally we consider requirements on $\\mbox{$\\Delta \\overline{R}$}$ , for galaxies where different exposures must be registered for shape measurement.", "In this case the mis-registration will cause a blurring of the galaxy that will be roughly equivalent to an added dispersion $\\mbox{$\\Delta V$}\\approx \\left\\langle (\\mbox{$\\Delta \\overline{R}$})^2\\right\\rangle $ .", "The details will depend upon the distribution of filters and zenith angles that one is trying to stack, but we can expect this variation to be of the same order as $\\Delta \\overline{R}$ itself.", "So we set rough requirements that $\\left\\langle (\\mbox{$\\Delta \\overline{R}$})^2\\right\\rangle $ be below the $\\Delta V$ requirements in equation (REF ), or: $\\mbox{$\\Delta \\overline{R}$}_{\\rm RMS} < \\left\\lbrace \\begin{array}{cc}25\\,{\\rm mas} & {\\rm (DES)} \\\\10\\,{\\rm mas} & {\\rm (LSST)}\\end{array} \\right.$ An additional point to make here is that two galaxies with opposite signs of $\\Delta \\overline{R}$ are both elongated in the same direction after stacking, hence our requirement is not on the population mean of $\\mbox{$\\Delta \\overline{R}$}$ ; it is on the RMS value of $\\Delta \\overline{R}$ of individual galaxies.", "Galaxies with negative $\\Delta \\overline{R}$ do not cancel the spurious shear of those with $\\mbox{$\\Delta \\overline{R}$}>0$ .", "To summarize, atmospheric refraction has the potential to significantly bias the results of cosmic-shear measurements whenever the stellar calibration results in errors for galaxies exceed any of the following limits: The dispersion error $\\langle \\mbox{$\\Delta V$}\\rangle $ of the galaxy population varies with redshift or other binning criteria by an amount exceeding $6(1)\\times 10^{-4}$  arcsec$^2$ for DES (LSST).", "This produces a multiplicative shear error that is too large.", "The galaxy-to-galaxy variation in $\\Delta V$ within a redshift bin substantially exceeds the above values, which could produce an additive shear fluctuating on arcminute scales that biases the cosmic shear power spectrum.", "The galaxy-by-galaxy error in refraction centroid $\\mbox{$\\Delta \\overline{R}$}_{\\rm RMS}$ acquires typical value exceeding 25 (10) mas in any redshift bin for DES (LSST).", "If this occurs, then stacks of images taken at different airmasses and/or filters will be misregistered and both multiplicative and additive shear errors will be produced.", "To quantify the effects of refraction on astronomical images, we first write the refraction angle as $R(\\lambda , z_a) = h(\\lambda ) g(z_a)$ where $h(\\lambda )$ and $g(z_a)$ encode its dependence on wavelength and altitude angle.", "As indicated by [4], $g(z_a)$ can be approximated as $g(z_a)=\\tan (z_a)$ .", "This functional form implies that $\\Delta \\overline{R}$ and $\\Delta V$ scale as $\\tan (z_a)$ and $\\tan ^2(z_a) $ , respectively.", "We will present results assuming $z_a = 45$ , since they are simple ($\\tan 45= 1$ ) and the $\\Delta V$ for a particular field of some survey should be scaled by $\\langle \\tan ^2 z_a \\rangle $ .", "The required scaling factor for DES is not, however, far from unity, so we will ignore it: we take a simulation of all the DES pointings produced by its scheduling software ObsTacProvided by Eric Neilsen [15] and bin them by declination.", "Figure REF shows the relevant quantity $\\langle \\tan ^2 z_a \\rangle $ vs declination expected for DES, showing that $\\langle \\tan ^2 z_a \\rangle $ is within about 20% of unity across the declination range of the survey.", "For LSST, Figure 3.3 of the LSST Science Bookhttp://http://www.lsst.org/lsst/scibook suggests that the LSST $i$ -band observations have $\\langle \\tan ^2 z_a \\rangle \\approx 0.6$ , so we can expect that the northern and southern extremes of the survey will have $\\langle \\tan ^2 z_a \\rangle $ at or above 0.8, slightly below our nominal value of unity.", "The calculation of shear artifacts induced by $\\Delta \\overline{R}$ in a stacking analysis is more complex, because $\\Delta \\overline{R}$ is actually a vector quantity and the direction to zenith will vary with hour angle of observations of a given field.", "The deleterious effects will scale with the two-dimensional variance of the $\\Delta \\overline{R}$ vectors over the course of the survey, so the relevant scaling factor will depend upon the hour-angle distribution as well as zenith-angle distribution, but will be $\\le \\langle \\tan ^2 z_a\\rangle $ .", "Our assumption of unity for this scaling factor is hence conservative, but we will see below that the shear systematics from $\\Delta V$ are larger than those from $\\Delta \\overline{R}$ even with this conservative upper bound on $\\Delta \\overline{R}$.", "We make use of values for $h (\\lambda )$ tabulated in [4], slightly corrected to take into account the average conditions of pressure and temperature (770 millibar and 11 C, respectively [18]) at Cerro Tololo, Chile (DES).", "As can be readily seen in Equations (REF ) and (REF ), $\\overline{R}$ and $V$ both depend on the spectrum of the source object, $S(\\lambda )$ .", "To make our calculations, we use the empirical galaxy Spectral Energy Distribution (SED) templates of [3] (CWW) and [13]These spectra are extended in the UV and IR regions by means of the synthetic [2] spectra..", "The CCW templates consist of 4 galaxies of type E, Im, Scb, Scd and the Kinney templates are representative of both quiescent and starbust galaxies.", "For the star spectra, we utilize the stellar libraries provided by [16].", "We calculate the values of $\\overline{R}$ and $V$ for each stellar spectral type, and for galaxy types at values of redshift $z \\in [0,1]$ with $\\Delta z = 0.02$ , according to Equations (REF ) and (REF ).", "Then, taking a G5V stellar template as our reference mean stellar spectrum, we calculate $\\Delta \\overline{R}$ and $\\Delta V$ for each object in each of the four $griz$ filters and compare our results to the requirements for cosmic shear derived in §." ], [ "Results for stellar and galactic spectra", "The top panels of Figures REF and REF show the calculated values of $\\Delta \\overline{R}$ and $\\Delta V$ in the $g$ filter as a function of redshift (at $z_a = 45$ ).", "The remaining panels show these same quantities after applying a correction linear in color that will be described in the next section.", "Similar calculations and plots were performed and obtained in the $r$ , $i$ , and $z$ filters.", "The results for stellar spectra at $z=0$ show how the effect of atmospheric refraction is greater for bluer sources, as expected.", "DES (LSST) requirements, as calculated in Equations (REF ) and (REF ), are also shown in the plots.", "To determine whether these requirements are satisfied, we first examine the galaxies expected to be detected by the DES according to the DES Data Challenge 6B.Simulations created by the DES Data Management Project http://cosmology.illinois.edu/DES/main/ To each of the $\\approx $ 250,000 galaxies detected in a 10 deg$^2$ subsample of the simulation we assign a $\\Delta V$ and $\\Delta \\overline{R}$ from the template that most closely matches its colors and redshift.", "Then we divide the galaxy population into four redshift bins, computing the mean value and the dispersion.", "of $\\Delta V$ and the RMS value of $\\mbox{$\\Delta \\overline{R}$}$ among the galaxy population.", "These are represented by the blue circular dots in Figures REF and REF (we assume that the LSST population has similar type distribution to the DES population).", "It can be seen that cosmic shear measurements performed in $g$ band will, in the absence of some correction scheme, have systematic errors due to atmospheric dispersion that dominate the statistical errors, with $\\Delta V$ up to $6\\times $ our threshold of importance for DES and $30\\times $ for LSST.", "The $\\mbox{$\\Delta \\overline{R}$}_{\\rm rms}$ values are also large enough to cause dominant systematic errors, but $\\Delta V$ induces worse problems, so we will focus on $\\Delta V$ henceforth.", "In $r$ band, the levels of $\\Delta V$ and $\\mbox{$\\Delta \\overline{R}$}_{\\rm RMS}$ remain large enough to dominate the LSST error budget but are $1.6\\times $ our nominal threshold for DES systematic errors, so may be acceptable for DES.", "For LSST, even in $i$ band, the dispersion effects are up to $2.6\\times $ larger than our derived requirements, so might be subdominant to statistical errors if our various approximations have been too conservative.", "But certainly the refraction effect requires attention in constructing an LSST error budget.", "Atmospheric dispersion is not a problem for either experiment in $z$ -band (and we have verified that this is also true, as expected, in $Y$ band).", "Figure: tan 2 z a \\tan ^2 z_a for DES exposures binned by field declination.", "The exposure lists are produced by simulations of the full DES survey using historical weather patterns and the ObsTac survey planning software .Figure: The differential centroid shift ΔR ¯\\Delta \\overline{R} as a function of redshift (Δz=0.02\\Delta z = 0.02) in the gg filter, for zenith angle 45.", "Standard templates of galaxies from CWW (E, Sb, Sc, Im galaxies ) and Kinney (quiescent—Sa, Sb—and starburst—SB, SB6—galaxies) were used.", "provides templates of stellar spectra (“ukxxyy\" labels).", "A G5V spectrum was used as the reference to set ΔR ¯=0\\mbox{$\\Delta \\overline{R}$}=0.", "The top panel shows the ΔR ¯\\Delta \\overline{R} vaues for each spectrum at different values of zz, whereas the lower panels show the same values after applying a linear correction in color, as described in the text.", "DES (LSST) requirements are that ΔR ¯ RMS \\mbox{$\\Delta \\overline{R}$}_{\\rm RMS} fall within the light grey (dark grey) shaded region (Equation ()).Figure: The differential PSF second moment ΔV\\Delta V as a function of redshift (Δz=0.02\\Delta z = 0.02) in the gg filter at zenith angle 45.", "Standard templates of galaxies from CWW (E, Sb, Sc, Im galaxies ) and Kinney (quiescent—Sa, Sb—and starburst—SB, SB6—galaxies) were used.", "provides templates of stellar spectra (“ukxxyy\" labels).", "A G5V spectrum was used as the reference to define ΔV=0\\mbox{$\\Delta V$}=0.", "The top panel shows the ΔV\\Delta V vaues for each spectrum at different values of zz, whereas the lower panels show the same values after applying a linear correction in color, as described in the text.", "DES (LSST) requirements are that ΔV\\Delta V fall within the light grey (dark grey) shaded region (Equation ()).Figure: ΔR ¯\\Delta \\overline{R} as a function of color for stellar and galaxy spectra at different redshift values (Δz=0.02\\Delta z= 0.02) for each of the grizgriz filters at z a =45z_a=45.", "The solid green (magenta) line represents a linear fit using only galaxy (stellar) spectra.", "Requirements for DES (LSST) on ΔR ¯ RMS \\mbox{$\\Delta \\overline{R}$}_{\\rm RMS} are shown by the light grey (dark grey) shaded regions.Figure: ΔV\\Delta V as a function of color for stellar and galaxy spectra at different redshift values (Δz=0.02\\Delta z= 0.02) for each of the grizgriz filters at z a =45z_a=45.", "The solid green (magenta) line represents a linear fit using only galaxy (stellar) spectra.", "Requirements for DES (LSST) on ΔV\\Delta V are shown by the light grey (dark grey) shaded regions." ], [ " Strategies for calibrating effects of atmospheric dispersion", "The previous section shows that the effects of atmospheric dispersion on shape measurements are non-negligible in the $g$ , $r$ and $i$ filters.", "As was pointed out above, $\\Delta \\overline{R}$ and $\\Delta V$ depend on the spectra of the observed objects, which cannot be measured in a photometric survey like DES.", "We would like to develop a calibration or correction that depends only on observable quantities, such as the color of the stars and galaxies.", "We calculate $\\Delta \\overline{R}$ and $\\Delta V$ of each star and galaxy at different redshift values and filters, and plot them as a function of color in Figures REF and REF .", "The plots suggest that a function of color can be defined that partially corrects the values of $\\Delta \\overline{R}$ and $\\Delta V$.", "We determine $\\Delta \\overline{R}$ and $\\Delta V$ functions that depend linearly on color and minimize the RMS residuals of the stellar templates—this is an operation that could be done empirically from the observations.", "The stars in Figures REF and REF give the $\\mbox{$\\Delta \\overline{R}$}_{\\rm RMS}$ , and mean and variance of $\\Delta V$, for the galaxy templates in 4 redshift bins after application of the star-based linear color correction to the galaxies.", "A linear function of color that minimizes the RMS residuals of $\\Delta \\overline{R}$ and $\\Delta V$ for the galaxy templates (crosses in Figures REF and REF ) produces a calibration with smaller residuals for galaxies, but note that such a calibration would have to rely on theoretical calculations since $\\Delta V$ cannot be measured from the data for the galaxies.", "The linear fits to both stars and galaxies are also shown in Figures REF and REF .", "The $\\mbox{$\\Delta \\overline{R}$}_{\\rm RMS}$ and $\\Delta V$ for the galaxies after applying the linear corrections at each filter are shown in Figures REF and REF .", "Either linear correction brings the atmospheric dispersion effects within requirements in the $r$ ($i$ ) band for DES (LSST).", "The LSST $r$ band systematic errors are still above our thresholds of concern after linear color corrections, by factors up to 5 depending on the redshift bin and type of correction—thus must still be considered an important contributor to the LSST error budget, although suggestive that more sophisticated correction schemes could bring atmopheric-dispersion systematics back below statistical errors in this case.", "For $g$ band, however, the linear corrections still leave errors well above our thresholds in both DES and LSST, and in fact the “corrected” data are worse in most of the redshift bins than uncorrected ones.", "Also we note that the galaxy-to-galaxy standard deviation of $\\Delta V$ within a redshift bin is typically much larger than the DES or LSST requirements we have derived, suggesting that spatial variations of the induced additive shear will be an issue.", "It appears that atmospheric dispersion is likely to create dominant systematic errors in $g$ band, and the surveys should plan on basing cosmic-shear measurements on the redder bands.", "We have verified that $u$ band is, as expected, even worse than $g$ .", "Figure: ΔR ¯ RMS \\mbox{$\\Delta \\overline{R}$}_{\\rm RMS} in four equally-spaced redshift bins for each of the four grizgriz filters when no correction (blue dots), a linear correction in color using only star spectra (magenta stars) and a linear correction using only galaxy spectra (green crosses) is applied.", "DES (LSST) requirements are that ΔR ¯ RMS \\mbox{$\\Delta \\overline{R}$}_{\\rm RMS} fall within the light grey (dark grey) shaded region (Equation ()).Figure: ΔV\\Delta V in four equally-spaced redshift bins for each of the four grizgriz filters when no correction (blue dots), a linear correction in color using only star spectra (magenta stars), and a linear correction using only galaxy spectra (green crosses) is applied.", "The vertical bars depict the standard deviation of ΔV\\Delta V between different galaxy templates and redshifts within each bin.", "They therefore represent the range of variation that the mean ΔV\\Delta V might accrue due to variations in the population mix of galaxies with redshift and with position on the sky.DES (LSST) requirements are that ΔV\\Delta V fall within the light grey (dark grey) shaded regions (Equation ())." ], [ "Conclusion", "Atmospheric dispersion can have a significant effect in PSF correction of shape measurements in broadband photometric surveys by affecting the first and second moments of the observed images.", "If uncorrected, it produces cosmic-shear spurious signals larger than the desired statistical uncertainties for the $g$ and $r$ bands of the DES, and for the $g$ , $r$ , and $i$ bands for LSST.", "This effect depends on the spectrum of the source and therefore is not easy to correct.", "A simple linear correction to the PSF position and size based on the observed color, calibrated empirically from stars or estimated theoretically for galaxies, should improve calibration of dispersion effects significantly.", "Even after applying this correction, however, weak lensing measurements in the $g$ band are significantly degraded by dispersion effects for both DES and LSST.", "Cosmic shear measurements can be performed with images taken in the $r$ band for the accuracy required by DES, but further work will be needed in the case of LSST.", "Shape and shear measurements in the $z$ and $Y$ band can safely be performed without attention to dispersion corrections.", "There are several possible tactics for further reduction of cosmic-shear systematic errors from atmospheric dispersion: Use an atmospheric dispersion corrector (ADC) during the imaging.", "While this option is available for KIDS cosmic-shear surveys being conducted on the VLT Survey Telescopehttp://www.eso.org/public/teles-instr/surveytelescopes/vst.html and for the HyperSuprime Camera to be installed on Subaruhttp://www.naoj.org/, an ADC is not part of the optical designs for DES and LSST.", "Observe at lower zenith angles—$\\Delta \\overline{R}$ and $\\Delta V$ scale as $\\tan (z_a)$ and $\\tan ^2(z_a)$ , respectively, so would be reduced by factors of $\\sqrt{3}$ and 3, respectively, if we were to take nominal zenith angles of 30 instead of the 45 assumed in our calculations.", "In practice it would not be easy to maintain such low airmasses throughout a large survey, especially if one wants to survey 15,000 deg$^2$ or more of low-Galactic-latitude sky from the latitudes of $\\approx -30$ of both Cerro Tololo (DES) and Cerro Pachón (LSST).", "Derive a more elaborate correction than the linear fit we have used, e.g., an approach that makes use of multi-color photometry to estimate the full spectrum of each galaxy—which of course is already done in the course of most template-fitting estimators for photometric redshifts.", "This technique is discussed by [5] in the context of calibrating wavelength-dependent PSF variation across the very broad band planned for the Euclid spacecraft.", "In the Euclid case it is telescope diffraction, not atmospheric refraction, that causes the wavelength dependence of the PSF, but the issue is similar.", "One could simulate the full effect in detail to estimate additive and multiplicative corrections that must be applied to the measured shear fields.", "This approach would have the usual limitations of precision calibration by simulation: it would be accurate only to the extent that the simulation captures the full multivariate distribution of the sizes, shapes, redshifts, and SEDs of galaxies in the true sky.", "The SED distribution will be particularly challenging to constrain for the faint end of the surveys' target galaxy populations.", "It seems likely, therefore, that both DES and LSST can make use of $r$ or redder bands for cosmic-shear surveys, but this will require explicit galaxy-by-galaxy correction for PSF and astrometric dependence on the source spectrum.", "For DES, we show that a simple linear color correction will suffice, but more work is needed to devise a correction that recovers the full cosmic-shear utility of the LSST $r$ band data.", "We note lateral color in telescope optics causes spurious spectrum-dependent cosmic-shear signals in a manner very similar to the atmospheric dispersion that we examine here.", "As with atmospheric dispersion, it is typical to specify an optical design that keeps this contribution well below the size of the PSF, but this is not a sufficient condition to guarantee that differences between galaxy and stellar spectra produce PSF calibration errors below the requirements of cosmic-shear surveys.", "This work was supported by NSF grant AST-090827 and DOE grant DE-FG02-95ER40893." ] ]

[ [ "Traditional formation scenarios fail to explain 4:3 mean motion\n resonances" ], [ "Abstract At least two multi-planetary systems in a 4:3 mean motion resonance have been found by radial velocity surveys.", "These planets are gas giants and the systems are only stable when protected by a resonance.", "Additionally the Kepler mission has detected at least 4 strong candidate planetary systems with a period ratio close to 4:3.", "This paper investigates traditional dynamical scenarios for the formation of these systems.", "We systematically study migration scenarios with both N-body and hydro-dynamic simulations.", "We investigate scenarios involving the in-situ formation of two planets in resonance.", "We look at the results from finely tuned planet-planet scattering simulations with gas disk damping.", "Finally, we investigate a formation scenario involving isolation-mass embryos.", "Although the combined planet-planet scattering and damping scenario seems promising, none of the above scenarios is successful in forming enough systems in 4:3 resonance with planetary masses similar to the observed ones.", "This is a negative result but it has important implications for planet formation.", "Previous studies were successful in forming 2:1 and 3:2 resonances.", "This is generally believed to be evidence of planet migration.", "We highlight the main differences between those studies and our failure in forming a 4:3 resonance.", "We also speculate on more exotic and complicated ideas.", "These results will guide future investigators toward exploring the above scenarios and alternative mechanisms in a more general framework." ], [ "Introduction", "To date, 777See e.g.", "http://exoplanetapp.com.", "extra-solar planets have been discovered via numerous detection techniques, including pulsar timing [71], radial velocity [36], Transits [11], and micro-lensing [7], while thousands of candidate systems from the Kepler transit mission await confirmation [2].", "A significant fraction ($\\sim 13\\%$ ) of the known planetary systems have been confirmed to possess systems of multiple planets.", "Multi-planet systems can provide valuable information on their history that single planet systems cannot.", "For example, multiple highly excited planets may indicate early dynamical instabilities [50].", "The existence of mean motion resonances (MMRs) on the other hand suggests that convergent migration occurred in the presence of dissipative forces.", "Numerous examples of resonant systems are known, both in extra-solar planetary systems and solar system satellites such as the 1:2:4 Laplace resonance in the Io-Europa-Ganymede system.", "The most studied planetary system in a MMR is Gliese 876 [33], [30], [31], [57], [45], [4], [62], [55].", "All these studies suggest that migration (albeit potentially driven by a variety of mechanisms) is an important factor in the sculpting of planetary systems.", "Convergent migration into closely spaced resonances (e.g.", "3:2) has been shown to be plausible, but difficult [56].", "The difficulties arise because planets that initially form far apart need to migrate through more widely spaced resonances (e.g.", "the 2:1 MMR) before subsequently being captured into the more closely spaced (e.g.", "3:2) MMR.", "To avoid the requirement for fine-tuning of initial locations, this requires relatively high migration rates [56].", "Even more closely spaced 4:3 resonances are suspected for some systems such as HD 200964 [26], and KOI 115 [8], suggesting that even more violent migration histories must have occurred to allow such systems to skip through the exterior 2:1 and 3:2 resonances before going on to be captured into the observed 4:3 resonance.", "We investigate the formation of 4:3 resonances in massive systems, i.e.", "planets with masses up to several times that of Jupiter.", "We examine mechanisms which include smooth migration, in-situ formation, scattering and damping.", "We demonstrate that the simple smooth migration mechanisms suspected to form the known 2:1 and 3:2 systems cannot plausibly form systems of massive planets in 4:3 resonances.", "We also study the formation of lower mass planets and find that these can readily form from a series of isolation-mass embryos in tightly-packed systems which then lead to 4:3 (and even closer) resonances.", "But these mechanisms underestimate the number of tightly packed systems in close resonances with multiple massive planets.", "The rate of detection of these multi-planetary systems (period ratios of 4:3 and closer) can therefore tell us valuable, not directly observable information about the formation history of extra-solar planetary systems and the solar system itself.", "In Section  we summarize observational results of closely packed planetary systems.", "Then we discuss the phase space and stability of massive systems in Section .", "The results of many different formation scenarios are presented in Section .", "This is the main part of our paper.", "We then extend the study to lower mass systems in 4:3 resonance and show that there is no difficulty in forming these in Section .", "Finally we summarize and discuss these results in Section ." ], [ "Observational evidence for 4:3 resonances", "As noted in Section , while many systems are thought to be in (or to be close to) a variety of mean-motion resonances, the population of systems suspected to occupy the closely spaced 4:3 MMR is much smaller.", "In this section we review the systems, both solar system satellites as well as exoplanets, which are suspected of populating 4:3 resonances." ], [ "4:3 resonances in the Solar System", "The best known pair of solar system satellites or planets which are locked in a 4:3 MMR with each other are Titan and Hyperion.", "With a mass of $5.6 \\cdot 10^{18}$  kg [60], Hyperion is 168 times less massive than Ceres and almost five orders of magnitude less massive than Titan, meaning that Hyperion effectively acts as a test particle and has a negligible effect on Titan's orbit.", "Hence, this resonance is dominated by a single term in the disturbing function (containing the longitude of pericenter of Hyperion and the mean longitudes of both satellites).", "The satellites' motion has been well-characterized by observations over several decades [59] and the stability afforded by the resonance has been considered in great detail [13], [6], [58].", "Although recently [5] suggested that Titan and Hyperion is an example of a system captured into resonance due to migration, [6] came to a different conclusion.", "The latter suggested that Hyperion was formed at its present location and cast doubt on a scenario where Titan and Hyperion achieved their current configuration through smooth differential tidal evolution across the chaotic zone.", "They further argued that a possible reason why Hyperion was accreted together in the 4:3 libration zone instead of the 3:2 libration zone is because in the former, more restricted region, the relative velocities of surviving planetesimals were small enough for coagulation to occur.", "Other cases of 4:3 resonances in the solar system include the asteroid Thule which is in the 4:3 resonance with Jupiter (for more objects see page 49 in [1]).", "Unfortunately, none of the above considerations translate easily into a 4:3 configuration with two massive planets.", "In this case, multiple terms in the disturbing function must be considered and stable libration zones are not as well characterized.", "However, as demonstrated later, capture into this resonance due to migration still proves to be difficult.", "There are currently two multi-planetary system discovered by the radial velocity method which are reported to be in or near a 4:3 mean motion resonance.", "HD 200964 consists of two planets with masses $m_1 = 1.8 M_{\\mathrm {Jup}}$ and $m_2=0.9 M_{\\mathrm {Jup}}$ [26].", "The period ratio is close to $1.33$ which suggests that the system is in a 4:3 mean motion resonance.", "The results of a Monte Carlo fitting routine that includes a penalty for unstable systems strongly favor systems in resonance [26].", "Another planetary system is suspected to also be in or near a 4:3 MMR (Giguere et al., in prep, private communication).", "Both systems consist of two massive (and most likely gaseous) planets on wide (a few AU) orbits.", "Both systems are subject to a short dynamical instability if the planets are not protected by a mean motion resonance.", "Their parameters are listed in Table REF .", "In the top panel of Figure REF we plot the cumulative distribution function of all the multi-planetary systems that have been discovered with the radial velocity method.", "In systems with more than two planets, each pair is treated independently.", "To compile this data set, we made use of the Open Exoplanet Cataloguehttps://github.com/hannorein/open_exoplanet_catalogue.", "One can clearly see the tendency for planets to pile up near integer ratios of the period ratio such as 2:1 and 3:1.", "This is usually attributed to resonant capture during the migration phase [31].", "The number of planets near the 4:3 resonance is far too small to make a statistical argument at this time.", "We also color code the total mass of the two planets.", "Note that mostly high mass planets get captured in the 2:1 and 3:2 resonances, whereas lower mass planets get preferentially captured into more closely spaced resonances.", "In this paper, we investigate the conditions under which systems on the far left side of this plot form.", "The reported best fit solutions of HD 200964 system puts it in a 4:3 resonance.", "However, there are other orbital solutions which are stable and cannot be ruled out with high confidence.", "For example, HD 200964 could also be in a 3:2 resonance [26].", "With the currently published RV data-points, this results in a higher $\\chi ^2$ value, therefore not being the best, but still a possible fit.", "This paper is concerned about the formation scenarios of the report systems.", "Fitting radial velocity data is a notoriously difficult job.", "We do not attempt to redo the analysis of [26].", "As we will show below, it is very difficult to get the system into the 4:3 resonance without fine-tuded initial conditions.", "One could therefore take our results and use it as a strong prior while fitting the RV light-curve, rejecting systems in 4:3 resonance.", "We do not want to go that far and think this is actually dangerous.", "If there is a new formation mechanism that we did not take into account, one can easily draw a wrong conclusion." ], [ "Kepler systems ", "In Table REF we list the four Kepler candidate systems which may occupy a 4:3 MMR [2].", "The first column lists the Kepler Object of Interest Number of the candidate system.", "The second and third columns list the planet radius in units of Earth-radii.", "The fourth and fifth columns list the orbital periods in days.", "The last column lists the ratio of the periods.", "We follow the procedure outlined in [65] and exclude any systems which are unlikely to be in resonance despite having period ratios close to 1.33.", "Note that all of the Kepler systems listed in Table REF are smaller and closer-in than the RV systems listed in Table REF : the largest of the KOIs has a radius less than that of Uranus, and all have orbital periods less than 8 days.", "It is thus not unreasonable to assume that their formation mechanism differs significantly from that of the massive planets at larger semi-major axes.", "We plot the cumulative distribution of all Kepler planet candidates [2] in the bottom panel of Figure REF .", "As in Section REF , for systems with more than two planets, every pair in that system is treated independently.", "Although there are clear features in this distribution near the 3:2 and 2:1 resonances, no statistically significant accumulation of planets can be seen near the 4:3 resonance.", "Furthermore, in contrast to the RV systems, there is no strong correlation between mass and the proximity to a resonance.", "It is worth pointing out the Kepler systems Kepler 36 and KOI 262.", "These systems are near the 6:7 MMR and 5:6 MMR, respectively [9], [19], i.e.", "even closer spaced than the 4:3 MMR that we study here.", "The masses of those planets are all below nine Earth masses." ], [ "Expansion of the disturbing function", "Resonant theory has been applied to dynamical problems in the solar system with great success [42], [40].", "However, properties of extra-solar systems typically do not adhere to the approximations which can be adopted for the solar system [4], [64].", "In particular, predicting the evolution of two massive bodies on non-circular orbits poses a rich dynamical challenge.", "Although much analytical progress has been made characterizing regimes of motion when these bodies are in the strong 2:1 MMR [3], [37], [38], few investigators have modeled in detail other first-order resonances, partly because of their close proximity to the chaotic resonant overlap region [68], [43].", "Further, detailed analyses of particular systems benefit from well-constrained observational data, showcasing another benefit of analyzing solar system bodies; the majority of confirmed extra-solar planet have unknown masses, bounded only from below.", "Orbital parameters for transiting systems are even less constrained, making detailed dynamical modeling almost impossible.", "One approach to tackling these issues is to consider when a planetary system cannot be in resonance, by confirming that any potential librating angle must circulate [65].", "This procedure can be carried out analytically by using a disturbing function with a sufficient number of terms to accurately sample the desired system.", "Here, we carry out the same procedure as in [65] by using both the disturbing functionThis is the same disturbing function which later appeared in [42].", "from [17] to fourth-order in eccentricities and the analytical formulas from [62].", "We assume coplanarity and use all resonant terms up to fourth-order (including the relevant 8:6, 12:9 and 16:12 terms) and secular terms up to fourth order.", "Additionally and separately we perform the same analysis for the 3:2 MMR, given its close proximity to the 4:3 MMR.", "The results are plotted in Figure REF .", "There are four different areas in these plots.", "In the dark-orange region to the bottom-right labeled 'NO 3:2', no 3:2 solutions are permitted (while 4:3 solutions may or may not exist).", "In the dark-red region to the far left, neither 3:2 nor 4:3 solutions are permitted.", "In the light-pink region labeled 'NO 4:3', no 4:3 solutions are permitted (while 3:2 solutions may or may not exist).", "In the central light-orange region no definitive statement can be made to exclude either the 3:2 or 4:3 MMRs (but this does not equate to a statement that both can definitely exist).", "Figure REF illustrates the excluded 4:3 and 3:2 MMR regions for the HD 200964 system, assuming $m_{\\star } = 1.44 M_{\\odot }$ , $m_1=1.8\\,M_{\\mathrm {Jup}}$ , $m_2=0.9\\,M_{\\mathrm {Jup}}$ and fixed outer planet values of $a_2=3.0$  AU and $e_2=0.01$ .", "This plot helps constrain the prospects for the system evolving in MMR given particular orbital parameters.", "Alternatively, if a MMR is assumed, then the plot helps constrain the planets' allowable orbital parameters.", "Figure REF over-plots a simple estimate of libration width from Eq.", "4.46 of [63] which assumes that just one disturbing function term is retained; this approach, often used in the solar system, poorly reproduces the allowed resonant motions for this extra-solar system.", "Figure REF demonstrates what the excluded regions look like in the limit of an inner planet mass of zero.", "In this case, both resonances become somewhat decoupled, and there is clear structure around each nominal commensurability (see also Section REF ).", "This plot may be compared to Figure 8.7 of [42]; differences arise because of the masses adopted here and the additional terms of the disturbing function used.", "This set of plots illustrates the difficulty in restricting resonance phase space analytically for two massive exoplanets.", "But because the system is only stable when it is protected by a resonance (see also next section) we can use such an analysis in the interpretation of orbital parameters which are weakly constrained from radial velocity data." ], [ "Direct $N$ -body simulations", "In the previous section we presented results of a resonant theory which was achieved by an expansion of the disturbing function.", "This allowed us to get an overview of the phase space structure of the HD 200964 system.", "We also run direct $N$ -body simulations of systems with two planets to investigate their stability.", "The freely available code REBOUND [54] is used for all integrations in this section.", "We choose the Wisdom-Holman type integrator [70] included in REBOUND.", "To verify the results, we implemented a 15th order adaptive Radau integrator [18].", "We find that the results do not depend on the integrator or any numerical parameters such as the timestep.", "For each run, we add two shadow particles to the simulation in addition to the two planets.", "This allows us to measure the long term stability of the system on short time-scales by calculating the maximum Lyapunov exponent.", "To do that, the position and velocity of each shadow particle is initially set to those of the planets.", "They are then perturbed by a small amount.", "At regular intervals the rate of divergence between the shadow particle and the planet is measured and rescaled to the initial displacement.", "By keeping track of the rescalings, we can estimate the maximum Lyapunov characteristic exponent.", "See [69] for more details on the numerical algorithm.", "If the numerically calculated Lyapunov timescale is smaller or comparable to the run time, we call the system system stable and unstable otherwise.", "The color scale in all stability plots was chosen to reflect this definition.", "White regions are unstable, dark regions are stable and have a Lyapunov timescale that satisfies the above criteria.", "The Lyapunov exponent gives us a good overview of the parameter space.", "In Figure REF we plot the results of $4\\cdot 10^5$ $N$ -body simulations, each running for approximately $2\\cdot 10^4$  dynamical times.", "The mass of the central object is that of the star HD 200964.", "We show the results for three different planetary masses.", "For Figure REF , we use the observed masses of the HD 200964 system (see Table REF ).", "We repeat the same calculation with planets that each have 3 Jupiter masses in Figure REF and for Earth mass planets in Figure REF .", "The initial semi-major axis and eccentricity of the inner orbit are $a_1=2, e_1=0.01$ .", "These values are the same for all simulations.", "The initial semi-major axis and eccentricity of the outer planet are varied and shown on the axis of the plot.", "All angles are chosen from a uniform distribution.", "The system is assumed to be coplanarWe discuss the formation of an inclined system in Appendix  but observe no qualitative difference to the coplanar case..", "In Figure REF , one can see bands and islands of stability for systems in mean motion resonances.", "Examples are located at $a_2=2.62$  (3:2), $a_2=3.17$  (2:1) and $a_2=3.68$  (5:2).", "The location of the 4:3 mean motion resonance is also visible at $a_2=2.42$ .", "This is a small stable island compared the other resonances mentioned.", "Also note that the eccentricity for these stable solutions is very high $e_2\\sim 0.64$ .", "However, this is also a function of the inner planet's eccentricity which has been kept fixed.", "We present a slice of the parameter space in the $e_1$ , $e_2$ plane in Appendix .", "The stable island that we attribute to the 4:3 MMR does not coincide with the best fit solution of [26].", "The eccentricities near the stable island are much higher than in the RV fit.", "Nevertheless, we verified that their solution is indeed stable.", "We conclude that their solution corresponds to fine tuned initial conditions.", "For systems with even higher mass planets than in the HD 200964 system, the stable regions get even smaller.", "This can be verified in Figure REF .", "Note that the reported masses for HD 200964 are the minimum masses and could be significantly larger if the system is inclined with respect to the line of sight.", "However, this result suggests that the masses can't be much larger in order for the system to be stable.", "Two additional plots showing a slice in the $a_1$ , $a_2$ plane of the the parameter space are presented in Appendix ." ], [ "Formation mechanisms", "In this section we investigate potential methods for the formation of a pair of massive planets in a 4:3 resonance.", "In Section REF we show that the long standing idea of convergent migration fails to produce closely packed resonances for massive planets.", "We then consider the in-situ formation of planets in Section REF , starting from small embryos in resonance which accrete mass from the protoplanetary disk.", "In Section REF scattering and simultaneous damping is considered as one possible alternative to the cold formation scenarios mentioned above.", "Finally, we discuss alternative formation scenarios in Section REF .", "We acknowledge that other plausible mechanisms may exist, some of which are mentioned in the discussion section." ], [ "Convergent migration in a disk", "Migration of planets through a disk depends on numerous parameters such as the planet and disk mass, disk viscosity, surface-density profile, disk scale height and the equation of state, to just mention a few.", "This allows for the possibility of convergent migration of planetary orbits, during which pairs of planets can pass through orbital period commensurabilities.", "If the convergent migration rate is sufficiently low, the planets can capture into resonance [22], [56].", "In Section REF we will simplify the migration process by assuming that we can describe it with only one semi-major axis and one eccentricity damping time-scale per planet.", "This allows us to understand the physical processes at work during resonance capture, study a wide parameter regime and not get lost in the complicated details of planetary migration.", "In Section REF we will then relax some of these simplifications and study the formation of a planetary system near a mean motion resonance using hydro-dynamical simulations.", "We use the $N$ -body code REBOUND [54] to model the orbital evolution of two massive planets.", "We apply non-conservative forces to the outer planets which damp its semi-major axis and eccentricity .", "We also refer the reader to the work by [56] on an investigation of the 3:2 resonance in the HD45364 system, which uses a similar methodology.", "We experimented with applying non-conservative forces to the inner planet simultaneously, but did not see any qualitative difference and will not investigate this further.", "Initially, the planets start far apart from each other on circular orbits.", "The assumption of circular orbits is reasonable in the currently favored core accretion model of planet formation.", "Furthermore, the eccentricity damping time-scale is much shorter than the migration time-scale in a standard disk unless the eccentricity is very large [44].", "The planets are also assumed to be coplanar.", "Even though convergent migration and resonant capture can theoretically excite inclination [61], the inclination damping time-scale is again much shorter than the migration time-scale as long as the planets are embedded in the disc, i.e.", "$i<10^\\circ $ [53].", "We also tried relaxing this condition and ran additional fully three dimensional simulations with finite relative initial inclination between the planets.", "The results are shown in Appendix .", "No significant changes can be observed.", "We therefore do not investigate this further.", "The outer planet migrates inwards on a time-scale $\\tau _a$ and might get captured into a resonance with the inner planet.", "In Figure REF we plot the final period ratio seen after the migration reached a quasi steady state and both planets migrate inwards self-similarly.", "The simulations are conducted with different semi-major axis damping rates and eccentricity damping rates which are the axes of the plot.", "Each pixel represents one N-body simulation.", "The color scheme has been chosen so that blue corresponds to a final period ratio close to 4:3, red is close to 3:2 and green is close to 2:1.", "The color black indicates that either planets are captured in another resonance or that at least one planet got ejected.", "The latter is more common for short migration timescales, $\\tau _a$ .", "We find that it is essentially impossible to form systems in a 4:3 resonance via a simple convergent migration scenario with the observed mass ratioNote that we use the minimum masses and ignore possibly even higher mass ratios if the system is inclined with respect to the line of sight..", "There are two fundamental problems.", "First, a very high migration rate is required to allow the planets to pass through the 2:1 and 3:2 resonances.", "Second, there is only a small stable island in parameter space around the 4:3 resonance and the migration scenarios tend not to put the planets into this specific location.", "To understand the evolutionary track, recall the stability plot in Figure REF .", "In the migration scenario the outer planet is initially on a circular orbit far away from the inner planet; that is the bottom right on the plot.", "Convergent migration brings the planets closer together.", "On the plot, this corresponds to moving to the left along a horizontal line.", "Depending on the migration speed, the planet feels the resonant interaction from the inner planet and starts gaining eccentricity, thus moving upwards in the plot.", "If the migration rate is slow enough, the planet will get captured into a resonance which will protect it from close encounters.", "However, if the migration rate is so fast that the planet slips through all resonances, it ends up in an unstable regime (red region in the bottom left of the plot).", "In fact, it has to go through an unstable region to get to the stable island that corresponds to the 4:3 resonance.", "This is why such a smooth migration scenario cannot form tight resonances for massive planets." ], [ "Hydro-dynamical simulations", "We now go beyond the simple $N$ -body model and use the two dimensional hydrodynamic code FARGO [34] to perform simulations of two gravitationally interacting planets that also undergo interactions with an accretion disk.", "This setup is used to further test the rapid migration hypothesis investigated with $N$ -body simulations and parameterized migration forces in Section REF .", "Again, the approach is conceptually similar to that of [56].", "We are in particular looking for more complicated effects such as varying migration rates that might be overlooked by running simplified $N$ -body simulations.", "Our simulations use a cylindrical grid with $N_\\phi =628$ and $N_r=512$ .", "The radial extent of the computational domain goes from $0.5\\,\\mathrm {AU}$ to $5\\,\\mathrm {AU}$ .", "We use non-reflecting boundary conditions to minimize the effects from the finite domain size.", "Both the disk and planet properties are varied in the simulations and are listed in Table REF .", "We run a total of over 50 different simulationsNot all are presented in the table.. All planets except those in simulations labeled inresonance (see Section REF for a discussion of those) start initially on circular orbits.", "Some of the simulations start with small mass planets and allow for accretion of mass from the proto-stellar disk.", "None of the simulation listed in Table REF results in a stable 4:3 resonance.", "The simulations either capture into a 2:1 or 3:2 resonance or scatter due to close encounters.", "The result is therefore in perfect agreement with the $N$ -body simulations of Section REF .", "In Figures REF we show the evolution of the semi-major axis of both planets as solid lines.", "The dashes line shows the nominal position of the 4:3 resonance of the inner planet and is thus the semi-major axis we are so desperately trying to reach.", "As can be seen easily, we do not achieve this.", "In Figures REF we also plot the surface density profile of the disk at the end of the simulation.", "Note that the final simulation time varies significantly between the simulations.", "Whenever a steady or adiabatic state was achieved, we decided to stop the simulation.", "There are several physical reasons why we cannot get the planets in a 4:3 resonance with these kind of migration scenarios.", "First, the planets are very massive and therefore the resonances are strong.", "This prefers more widely separated resonances as has already been shown by the $N$ -body simulations above.", "It can be verified in the simulations labeled vanilla, h0.07, slope0 and many others.", "Second, if we force the planets to have an extremely rapid (and most likely completely unphysical) migration rate, the planets do indeed get closer initially.", "This can be seen in the simulation sigma8 that has a very massive disk which leads to a fast migration rate.", "Once in resonance, the planets start migrating outwards again.", "The same happens in the Grand Tack Scenario [66].", "After a while the outer planets starts migrating out very quickly due to the heavy disk and a Type III migration regime.", "This breaks the resonance.", "Shortly after the resonance is lost, the planets start moving in again, recapture in resonance and the whole cycle repeats.", "We never observe the planets capturing in a 4:3 resonance in this process.", "Third, the inner planet is massive and will open a gap in any reasonable disk model.", "The location of the 4:3 resonance is a factor of 1.2 away from the inner planet in terms of its semi-major axis.", "Note that the gap cannot be smaller than a few scale heights or Hill radii [16].", "However, depending on the precise planet mass and disk model the outer planet might not open a gap on its own.", "In that case it might get trapped at the gap edge.", "Migration stops or at least slows down, preventing the outer planet getting closer.", "Fourth, as [48] point out, the interaction of the outer planet with the waves launched by the inner planet can cause additional torques if the outer planet does not open a gap on its own.", "This surfing effect tends to move the outer planet further away.", "This can be seen in simulations labeled sigma4_outer0.1_h0.03, sigma8_outer0.1, sigma4_ouer0.1 and a few others.", "Of course this cannot be a complete survey of all possible scenarios even though we ran over 50 hydro-dynamical simulations.", "Also, there are significant limitations and errors associated with such a numerical scheme.", "The resolution has been kept fixed and we did not check for convergence in every simulation.", "However, it seems very unlikely that this would lead to a completely different picture as not even one of our simulations comes even close to capturing the two planets in the 4:3 resonance for a significant amount of time.", "To conclude, the results of our hydro-dynamical simulations are in agreement with the $N$ -body simulations.", "They do not allow the capture of planets in the 4:3 resonance.", "In fact, due to gap opening and the surfing effect the problem of forming the resonance seems even harder to overcome than we would have estimated from $N$ -body simulations.", "While the presence of a MMR is generally taken as being the signature of convergent migration (see Section REF ), perhaps the most conceptually simple scenario to consider is that the two planets in the 4:3 resonance formed directly from embryos with 4:3 period ratios.", "At early times with much lower masses, their mutual interactions would have been much weaker and some period of relatively unperturbed growth could have taken place.", "There is then the possibility that the pair of 4:3 planets which formed in-situ could have migrated together through the disk to the currently-observed locations.", "We test this idea in a similar framework to that used in the last section.", "However, we do not start planets far apart from each other, but start them directly in the 4:3 MMR.", "To do this, we run an $N$ -body simulation using REBOUND with two ten Earth mass embryos.", "We choose a migration time-scale which results in a capture in the 4:3 MMR (see Section REF ).", "After the planets have reached an equilibrium state and migrate together adiabatically, we switch to a full hydrodynamic simulation as in Section REF .", "We let the embryos interact with the disk and accreted mass.", "The accretion not only changes the mass but adds an additional component to the torque that is felt by the planet.", "The interaction with the disk is turned on adiabatically over several orbits.", "We vary the accretion rate, the mass of the outer planet, and the slope of the initial surface density profile.", "The full list of initial conditions for these simulations is shown at the bottom of Table REF under seed_inresonance, seed_inresonance_slow, seed_inresonance_slope0, seed_inresonance_outer4 and seed_inresonance_slope0_veryslow.", "We plot the evolution of the semi-major axis and the mass of both planets in these simulations in Figure REF .", "One can see that the 4:3 MMR is lost a few hundred to a few thousand years after the planets are inserted into the disk.", "The path to loosing the resonance is as follows.", "While the planets are accreting mass, they are also migrating.", "They migrate initially in type-I, later in type-II when their mass is sufficiently large.", "As the planets grow in mass, their dynamical interaction becomes stronger.", "The eccentricities of both planet rise.", "Eventually the resonance is lost.", "None of the planets gets ejected, but the interactions push them several Hill radii apart.", "Note that slowing down the process, as in simulation seed_inresonance_slope0_versyslow does not prevent the resonance breaking.", "It merely delays it.", "There are clearly many more parameters that one could vary.", "However, it seems unlikely that we can keep the planets locked in resonance while they grow in mass by more than one order of magnitude.", "This is because the mass that the embryos accrete has to come from the proto-planetary disk.", "The planets always interact with the disk, exchange angular momentum and start migrating." ], [ "Scattering and simultaneous damping", "It has been shown in previous studies of planet-planet scattering [12], [52] that the formation of mean motion resonances in the aftermath of scattering between planets is possible, but that this is a relatively rare event (less than 1% level).", "The low probability of capture into resonance in these investigations is likely a simple consequence of the fact that these simulations are conducted in the absence of any disk-gas damping (or other dissipative forces) and hence there is no means for the system to damp into resonance.", "When planet-planet scattering simulations are conducted in the presence of a disk gas [35], [39], the dissipative gas component means that a much higher fraction of systems are observed to form MMRs of a wide variety of orders (see their Table 4).", "However, [35] did not explicitly observe the creation of 4:3 resonances.", "We therefore investigate the scattering scenario further to understand whether such systems are simply rare (and thus statistically unlikely to be seen in their simulations), or whether the formation of a 4:3 resonance in this manner is essentially impossible.", "We start from the assumption that an inner planet exists in an unperturbed, approximately circular orbit at a semi-major axis $a_1 \\sim 1.6\\,\\mathrm {AU}$ .", "External to this, we assume that there existed an outer ($a \\gg 2\\,\\mathrm {AU}$ ) population ($N \\ge 2$ ) of massive planets which became unstable and scattered a Jupiter-mass planet onto an orbit with a pericenter at $q_2\\sim 1.9\\,\\mathrm {AU}$ (the radial location of the 4:3 resonance with the inner planet at $1.6\\,\\mathrm {AU}$ ).", "We do not perform the lengthy, chaotic scattering simulations of this outer population, but rather assume that a scattering event has occurred with the outcome being that a $0.9\\,M_{\\mathrm {Jup}}$ planet has been placed onto an orbit with $q_2 \\sim 1.9\\,\\mathrm {AU}$ .", "We consider that the outer body interacts with a remnant gas disk of some form, causing its orbit to damp, reducing in eccentricity and semi-major axis.", "In some fraction of systems with suitable damping terms, while the outer body is simultaneously circularizing and migrating inwards, the two planets may become trapped into a 4:3 MMR.", "To investigate this scenario we perform $N$ -body integrations which include the effect of gas damping to investigate whether the system described above can be captured into a 4:3 resonance.", "The full details of the manner in which the simulations are specified are detailed in the following section." ], [ "Simulation methodology", "We initialize each simulation with conditions approximating some precursor of the HD 200964 system.", "The central star has a mass of $M_{\\star }\\sim 1.44M_{\\odot }$ .", "The inner planet has a of $m_1 = 1.8 M_{\\mathrm {Jup}}$ and is located at $a_1 = 1.6\\,\\mathrm {AU}$ on a circular orbit, $e_1 = 0$ .", "The outer planet has a mass of $m_2 = 0.9 M_{\\mathrm {Jup}}$ .", "The pericenter $q_2$ is randomly chosen from a flat distribution $1.6 < q_2 < 2.2\\,\\mathrm {AU}$ .", "Note that the pericenter lies close to the location of the 4:3 resonance with the inner planet.", "The outer planet's eccentricity is also randomly drawn from a flat distribution $e_{crit} < e_2 < 0.9$ , where $e_{crit} = 1 - q_2/2.54$ .", "Thus the outer planet is constrained to start with a semi-major axis outside the location of the 2:1 resonance with the inner planet (at $2.54\\,\\mathrm {AU}$ ).", "We parameterize the damping in the same manner as in Section REF , i.e.", "following the approach of [31] by calculating the Jacobi orbital elements.", "The semi-major axis $a$ and the eccentricity $e$ are then directly damped in the parameterized form $\\frac{\\dot{e}}{e}=K\\frac{\\dot{a}}{a}$ , where the damping rate $\\frac{\\dot{a}}{a}$ and the ratio of eccentricity damping to semi-major axis damping, $K$ , are varied as free parameters.", "We implement this damping in a modified version of the the Mercury integration package [10].", "Each simulation uses the hybrid-symplectic integrator in Mercury, and integrates the systems for a total time period $T_\\mathrm {final}=100 a/\\dot{a}$ .", "We perform 100 versions for each of the following 64 parameter combinations, thus performing a total of 6400 individual simulations.", "The damping rates $\\dot{a}/a$ that we choose are $10^{-6}~\\mathrm {yr}^{-1}$ , $10^{-5}~\\mathrm {yr}^{-1},$ $10^{-4}~\\mathrm {yr}^{-1}$ and $10^{-3}~\\mathrm {yr}^{-1}$ .", "We investigate four values of $K$ , $0.01,0.1,1$ and 10.", "We perform simulations in which the damping occurs on either only the outer planet or both planets.", "We also vary the location within which the gas damping operates: either over a wide range $r < 50\\,\\mathrm {AU}$ , or in a small disk $r < 2.5\\,\\mathrm {AU}$ .", "The latter implies that the outer planet will initially only be damped when very close to pericenter." ], [ "Simulation results", "We plot an example of a successful damping simulation in which an initially highly eccentric planet is subsequently made to damp and migrate into a 4:3 MMR with the inner planet in Figure REF .", "In the example shown, the outer planet initially had orbital parameters such that $a_2 = 3.59$ , $e_2 = 0.53$ and therefore $q_2 = 1.66$ .", "The damping model was such that $\\dot{a}/a = 10^{-5}\\;\\mathrm {yr}^{-1}$ and $K=1$ , with this damping operating on the outer body only, for all $r < 50\\,\\mathrm {AU}$ .", "Over the first $t\\sim 6\\times 10^5\\,\\mathrm {yr}$ of the simulation the outer planet can be seen to continuously suffer close encounters with the inner planet (kicking both semi-major axes and eccentricities to new values), while also experiencing rapid damping and inward migration due to the gas interactions.", "Finally at $t\\sim 6\\times 10^5\\,\\mathrm {yr}$ the damping component wins and the planet definitively captures into a 4:3 resonance with the inner planet.", "We plot the resonant angles $\\theta _3,\\,\\theta _4$ and $\\theta _{3-4}$ in bottom of Figure REF .", "They are defined as $\\theta _3 &=& 4(\\lambda _2 - \\varpi _2) - 3(\\lambda _1 - \\varpi _1) + 3(\\varpi _2-\\varpi _1) \\nonumber \\\\\\theta _4 &=& 4(\\lambda _2 - \\varpi _2) - 3(\\lambda _1 - \\varpi _1) + 4(\\varpi _2-\\varpi _1) \\\\\\theta _{3-4} &=& \\theta _3 - \\theta _4, \\nonumber $ where $\\lambda _i$ and $\\varpi _i$ are the mean longitude and longitude of periapse of the $i^{\\mathrm {th}}$ planet, respectively.", "We observe that the resonant angle $\\theta _3$ is librating.", "The results of all 6400 individual simulations performed are shown in Figure REF .", "The plots show the fraction of systems in a certain resonance as a function of the damping timescales.", "We marginalize over the other parameters.", "Approximately $1\\%$ of all systems are captured into the 4:3 resonance.", "Shorter damping time-scales in both semi-major axis and eccentricity tend to result in higher capture fractions into the 4:3 resonance.", "However, we note that if this model is at all realistic, then we also predict that the fraction of systems occupying a 1:1 resonance would be higher than the fraction of systems in a 4:3 resonance.", "One can see from Figure REF that the fraction of systems in a 1:1 resonance is more than 3 times as large as the fraction of systems in a 4:3 resonance.", "An example of a capture into a 1:1 resonance is illustrated in Appendix .", "Such resonances were examined by [29] and shown to be detectable by Kepler [21].", "This is consistent with simulations of multiple small mass planets embedded in a gas disk performed by [14].", "These authors find that 20% of their runs produce co-orbital planets which survive until the end of their integration.", "We reiterate the point that these simulations were deliberately initialized into a finely-tuned configuration that we hoped would lead to capture into a 4:3 resonance.", "Moreover, the model we use for our gas damping is a simple, parameterized one.", "We thus cannot claim to have fully explored whether such a scenario would be common (or even at all possible).", "Further study should address the likelihood of the initial scattering occurring.", "This is a notoriously hard task because the initial mass function of planets (and their position) is not well understood.", "Also, the realism of the damping scenario can only be checked when full hydro-dynamical simulations are employed.", "However, it is currently unfeasible to run thousands of hydrodynamic simulations for such a long time span ($t\\sim 10^6$ years).", "But what we can say is that the 4:3 resonance is clearly not a strong attractor in scattering simulations.", "Our results predict a large number of 1:1 resonances which are not observed.", "The overall likelihood to form a 4:3 resonance in our fine tuned simulations is only 1%.", "In a less fine tuned setup this rate would be much less then 1% and therefore much less than the observed occurrence rate." ], [ "Alternative means of forming planets in a 4:3 resonance", "One possible means of generating two planets in a 4:3 MMR is through the breaking of a resonant chain initially involving at least three planets.", "Such a resonant chain has been observed in planet-disk simulations of small mass planets by [15].", "The Kepler sample provides an interesting testing-ground for these systems [20].", "There are few investigations dedicated to measuring four-body resonances outside of the well-studied Laplace resonance involving Io, Europa and Ganymede.", "[46], [47] do so, but treat one of the bodies as small.", "[49] consider three equal mass bodies, but on near-circular orbits with semi-major axes in a geometrical ratio.", "As mentioned above, other studies consider the possible formation of resonant chains due to the presence of a disk.", "[51] find that a planetesimal disk can easily produce 4:2:1 four-body resonances.", "Like [35], [39] model planet-planet scattering during dissipation of a gas disk.", "The latter find that almost 45% of their 3-planet simulations which remain stable achieve a 9:6:4, 6:3:2, 3:2:1, or 4:2:1 four-body resonance.", "Although no pair of planets in these simulations reduces to a 4:3 MMR, this prevalence of resonant chains suggests that a 4:3 MMR may be formed in this manner.", "Another mechanism not directly considered in this paper is that of a chaotic migration scenario.", "Here the continued inward migration of planets in a 3:2 MMR causes the eccentricity of the planets to increase to such an extent that the planets pass out of the portion of parameter space in which stable libration is possible, and into a region of chaotic scattering.", "In such a scenario, it is generally envisaged that without the protection afforded by being in a mean motion resonance, the planets will scatter one another during subsequent close approaches and the stable resonant configuration will be destroyed.", "However, as the planets move into the region of chaotic scattering, an external perturbation (conceptually due to interactions with a disk or scattering from a small planetary embryo) might act to kick the planet from the chaotic overlap region down into the 4:3 MMR region.", "It is clear that in order for this scenario to realistically occur, the time-scale for a stochastic scattering event must be shorter than the time-scale over which chaotic scattering can start to disrupt the system.", "Detailed investigation would be required to understand whether such a scenario is at all possible, and more likely than the proposed scattering and damping scenario (see Section REF )." ], [ "Small mass planets", "In this section, we extend the previous calculations and consider systems with small mass planets.", "The motivation for this are the majority of systems in the Kepler sample which have a much smaller mass than those discovered by radial velocity (see Section REF ).", "We will show that such systems can easily form via a variety of mechanisms and the problems discussed in the last section do not arise.", "We first look at the stability map of small mass planetary systems in Section REF .", "Then we go on and study the traditional migration-capture scenario with $N$ -body simulations in Section REF .", "In Section REF we look at the possibility of growth of isolation mass embryos through direct collisions." ], [ "Stability map with direct $N$ -body simulations", "The stability plots in Section REF were dominated by large unstable regions and a small stable island associated with the 4:3 resonance.", "For two Earth mass planets, the stability plot looks very different, as shown in Figure REF .", "Here, the parameter space of stable regions is much larger.", "More structure is visible due to the presence of higher order resonances.", "In principle we can now capture into each of these resonance.", "In practice, the capture probability depends of course on the migration rate.", "We study this in the following section." ], [ "Convergent migration in a disk", "In Figure REF we plot the same quantity as in Figure REF but the mass of the outer planet has been reduced by a factor of ten.", "This leads to smaller critical migration rates required to pass through the 2:1 and 3:2 resonance.", "It also increases the stability of the system and thus enables us to form some stable systems which stay in the 4:3 resonance.", "It is again helpful to recall the stability plots in Figures REF and REF to understand the differences in the evolutionary track.", "The outer planet starts on the far right bottom of Figure REF .", "It migrates inwards (moving to the left) and encounters various commensurabilities.", "Depending on the migration rate it captures into any of these resonances and gains eccentricity in the process (moves up).", "In contrast to the high mass example in Figure REF , the resonances are stable (dark color) and the planet does not have to cross an unstable part of the parameter space to get there.", "Note that the migration rates required are still very high, $\\tau _a \\sim 200-800~\\mathrm {yr}$ .", "This corresponds to a type-III migration regime and is most likely not a realistic outcome of planet-disk interaction for these (reduced!)", "mass ratios.", "This is the opposite to the reasoning of [56], where the mass ratios are such that type-III migration is considered to be the most likely formation scenario." ], [ "Growth of isolation-mass embryos", "The end stage of the runaway [67], [27] and oligarchic growth [25], [28] phases of planetary embryo growth is envisaged to be a series of isolation-mass embryos.", "These are a series of embryos each having accreted all solid material within an annulus of width $c_\\mathrm {iso}\\,r_H$ of their location.", "Here, $c_\\mathrm {iso}$ is a dimensionless constant of order 10 and $r_H$ is the Hill radius of the embryo.", "For a power-law surface density profile $\\propto \\Sigma _0 a^{-\\alpha }$ , where $\\alpha $ denotes the slope of the power law, the mass of the isolation mass embryos can be written as [24] $m_\\mathrm {iso} &=& 5\\cdot 10^{-3} c_\\mathrm {iso}^{3/2} \\left(\\frac{\\Sigma _0}{10\\,\\mathrm {g cm}^{-2}}\\right)^{3/2}\\nonumber \\\\&&\\quad \\cdot \\left(\\frac{a}{1\\mathrm {AU}}\\right)^{2-\\alpha } \\left(\\frac{M_{\\star }}{M_{\\odot }}\\right)^{-1/2} M_{\\oplus }.$ We will use this as a starting point of our integrations." ], [ "Methods to simulate the growth of isolation-mass embryos", "We follow a method conceptually similar to that employed in [72] and [23].", "We consider that a series of isolation-mass embryos has formed as described above.", "We then simulate their subsequent evolution as they excite, scatter and collide over $\\sim 10^9\\,\\mathrm {yr}$ .", "The initial conditions used in Equation REF are set to be $\\Sigma _0 = 10\\,\\mathrm {g\\, cm}^{-2}$ , $c_\\mathrm {iso} = 7$ and $\\alpha = 3/2$ .", "We distribute the embryos over annuli with widths either $0.55\\,\\mathrm {AU} < a < 1.75\\,\\mathrm {AU}$ (Set A) or $0.1\\,\\mathrm {AU} < a < 2.2\\,\\mathrm {AU}$ (Set B).", "The embryos are damped for the first $10^6\\,\\mathrm {yr}$ of their evolution assuming an interaction with a gas disk of surface density of $\\Sigma _0 = 2400\\,\\mathrm {g\\,cm}^{-2}$ using the same eccentricity damping model as that employed in [32].", "The gas disk is then allowed to dissipate on a time-scale, $\\tau _d$ , with this simply being modeled as a reduction in the damping force by a factor $e^{-t/\\tau _d}$ .", "We use $\\tau _d = 10^5\\,\\mathrm {yr}$ , $10^6\\,\\mathrm {yr}$ and $10^7\\,\\mathrm {yr}$ for different subsets.", "Typically, this results in the number of embryos $N$ remaining approximately constant for the first $10^6\\,\\mathrm {yr}$ as the strong damping prevents many crossing-orbits developing.", "In the case of Set A, $N$ is approximately 40, in the case of Set B, $N \\sim 130$ .", "As the damping begins to decrease (the gas disk dissipates), the embryos begin to excite and collide with one another, reducing the number of bodies present in the simulation.", "The final number of planets varies across the simulations but is typically between 5 and 7." ], [ "Results from $N$ -body simulations of the growth of isolation-mass embryos", "The results of a simulation run from Set A with $\\tau = 10^6 \\mathrm {yr}$ are given in Figure REF .", "Figure REF shows the evolution of the periods as a function of time.", "At $\\sim 10^9\\,\\mathrm {yr}$ the 4th and 5th planets from the star (highlighted in the example) have period ratios $\\sim 1.33$ , i.e.", "close to the 4:3 MMR period ratio.", "In Figure REF we show the resonant angles $\\theta _3,\\,\\theta _4$ and the secular resonance angle $\\theta _{3-4}$ defined in Equation REF for the 4th and 5th bodies from the star.", "The resonant angles are observed to circulate over the full $360^{\\circ }$ range, while the secular angle librates for extended periods (although over very long time periods it too can be seen to circulate).", "To understand in detail the strengths of these resonances and the fraction of time that the planets spend in (or adjacent to) these MMRs as a function of (e.g.)", "the overlap of 3-body resonances requires much more detailed analysis in the manner of [49].", "Figure: Evolution of one damped embryo simulation.The damping time-scale is τ=10 6 yr \\tau = 10^6\\,\\mathrm {yr}.The orbital periods as a function of time is shown in black.As an illustration of the degree of eccentricity, the gray dashed lines show the the quantity a(1±e) 3/2 \\left(a\\,(1 \\pm e)\\right)^{3/2} for all of the bodies.The number of embryos reduces from ∼40\\sim 40 to 10, and a number of very close period ratios are present between various pairs of the planets.It is also of interest to note that in the simulations we performed, more closely spaced planetary configuration also arose.", "In Figure REF we illustrate the results of a simulation run from Set A with $\\tau = 10^7\\mathrm {yr}$ .", "In this system ten low mass bodies survive at $t\\sim 5\\times 10^8\\,\\mathrm {yr}$ .", "These have period ratios $1.34,1.24,1.23,1.32,1.37,1.17,1.33,1.32$ and $1.19$ .", "Hence multiple pairs are close to the 4:3 (1.33), 5:4 (1.25) and 7:6 (1.17) MMR period ratios.", "We have checked (but do not plot) the various possible two-body resonant angles for the respective pairs of planets close to the listed MMRs.", "We find that none of the bodies occupy exact 2-body resonances despite their period ratios being suggestively close.", "We thus find that the formation of rather closely spaced systems (i.e.", "with period ratios $1.15\\rightarrow 1.4$ ) is easily accomplished through the collisional evolution of isolation-mass embryos in an extended catastrophic collision phase of planet formation.", "But we emphasize that period commensurabilities do not equate to two-body resonances.", "The total embryo mass considered in these simulations was low.", "For Set A we have $\\sim 3.0 M_{\\oplus }$ and for Set B $\\sim 6.0 M_{\\oplus }$ .", "This corresponds to solid surface densities consistent with a minimum mass nebular model.", "The absolute masses of the planets formed at $t\\sim 10^9\\,\\mathrm {yr}$ are subsequently low ($0.5 M_{\\oplus }$ and $0.55 M_{\\oplus }$ for the example in Figure REF , $0.1 - 0.54M_{\\oplus }$ for the bodies in Figure REF ).", "While these masses are likely smaller than those listed in Table REF , they are still within the range of detectable masses for Kepler [41].", "All of the systems that we were able to form have multiple planets.", "We also note that [23] found nothing conceptually different in their simulations performed at much higher absolute surface density normalizations.", "These simulations resulted in mass-period distributions approximately resembling those of the observed Kepler systems.", "We emphasize that these simulations have run for $t \\sim 10^9\\,\\mathrm {yr}$ .", "For the majority of this time period the planets have been undamped.", "It is therefore realistic to consider these systems as being in an old, evolved state as might be observable by the (e.g.)", "Kepler mission.", "This mechanism may therefore explain the KOIs listed in Table REF .", "The growth of planets from isolation-mass embryos seems to provide a natural means by which low mass planets can be either captured into closely-spaced MMRs (i.e.", "4:3), or have period commensurabilities approximately similar to such closely-spaced MMRs while also remaining long-term stable.", "We emphasize that while such close spacing (period ratios less than $1.4$ ) does not happen in the majority of systems simulated, it was common enough for us to observe it in 2 out of 60 systems simulated." ], [ "Conclusion", "In this paper, we explored numerous methods of forming a 4:3 mean motion resonance in systems of both high-mass planets.", "We found that it is extremely difficult to form stable massive systems in a 4:3 resonance.", "The discovery of multiple such systems in radial velocity surveys leads us to conclude that traditional formation methods are failing to reproduce the observed fraction of systems that are in or close to this resonances.", "More precisely, we found that convergent migration due to torques imposed by a gas disk is not a viable mechanism for the formation of massive planets in a 4:3 resonances.", "In Section REF we showed that it can be ruled out to form the system HD 200964 in this way.", "There are four reason for this.", "First, the systems tend to lock into higher order resonances.", "Second, even with reduced masses the required migration timescale to pass through resonances such as 2:1 and 3:2 is unphysically short.", "Third, gap opening tends to stall migration at gap edges which prevents subsequent capture into close-in resonances.", "Fourth, the surfing effect pushes planets away from each other unless both planets are able to open a clean gap.", "We conducted a large survey of hydro-dynamical simulations.", "But we did not find a way to realistically form a 4:3 resonance in any of these hydro-dynamic models.", "It is important to note that these are planets detected by the radial velocity method and thus the masses are only minimum masses.", "However, we found that masses higher then the minimum masses make it even harder to form a stable resonance.", "In-situ formation where planet embryos start out in the 4:3 resonance is also not a viable formation mechanism.", "In Section REF we conducted hydro-dynamical simulations of such a scenario.", "There is always a phase when the planets migrate divergently during their mass accretion phase.", "As soon as the planets move apart from each other, outside of the 4:3 resonance, they cannot recapture into the resonance at a later stage because they preferentially capture in wider resonances.", "Gap opening and the tidal surfing effect are further preventing planets from staying or recapturing in a tight resonance while being embedded in a gas disk.", "A combined scattering and damping mechanism does seem to be a plausible means of forming giant planets in closely-spaced MMRs as we showed in Section REF .", "We found that for a wide range of damping parameters close-in resonance can form.", "However, the initial conditions are finely tuned to allow such a capture.", "We estimate that the fraction of planetary systems that might end up in such a configuration is much smaller than the currently observed fraction of approximately $4\\%$ .", "This scenario also predicts the formation a large number of planets in a 1:1 co-orbital resonance which is not consistent with current observations.", "We extended the study to include small mass planets in foresight of several Kepler planet candidates which are close to a 4:3 period ratio.", "We found that there is no problem in forming any of these smaller mass planets with a traditional migration scenario.", "We therefore expect that several of these systems will be confirmed by follow up observations.", "The main result in our paper is a negative one, i.e.", "we cannot explain the formation of the observed systems.", "One could use our results as evidence against the existence of massive planets in close in resonances and search for other explanations of the observed RV signals.", "But this requires a great amount of caution.", "Using the same argument, we could have ruled out the first discovered Hot Jupiter.", "Rather, we hope that this study will guide future investigations and spur interest in these systems among the community." ], [ "Acknowledgments", "We thank Alessandro Morbidelli for an extremely helpful and detailed referee report that greatly improved this paper.", "Hanno Rein would like to thank Scott Tremaine, Alice Quillen, Eiichiro Kokubo and Ilona Ruhl for helpful comments at various stages of the project.", "Eric Ford and Matthew Payne acknowledge enlightening discussions with Althea Moorhead and Richard Ruth, and Dimitri Veras thanks Alex Mustill for helpful comments.", "Hanno Rein was supported by the Institute for Advanced Study and the NSF grant AST-0807444.", "Matthew Payne was supported by NSF grant AST-0707203 and the NASA Origins of solar systems grant NNX09AB35G." ], [ "Additional stability plots", "In this appendix we show additional stability plots of the phase space of two massive planets.", "The procedure is described in Section REF .", "Figure REF shows two additional stability plots of the phase space in the $a_1$ -$a_2$ plane.", "The initial eccentricities are $0.1$ and $0.5$ for the top and bottom plot, respectively.", "All other angles are drawn from a uniform distribution.", "The system is assumed to be coplanar.", "One can see that there is clear evidence of resonances as the stability only depends on the period ratio (not the individual periods) and the eccentricities of the planets.", "Figure REF shows an additional stability plot of the phase space in the $e_1$ -$e_2$ plane.", "The planets' semi-major axes are initially at $a_1=2$ and $a_2=2.42$ , i.e.", "close to a 4:3 period ratio.", "We find stable islands for a wide variety of eccentricities as long as $e_1<0.55$ .", "One can see a clear anti-correlation of the eccentricity of the inner and outer planet." ], [ "Forced migration for inclined planets", "We show the final period ratio of two convergently migrating planets in Figure REF .", "The simulations are identical to those presented in Figure REF but are fully three dimensional and include a finite initial inclination of $i=5^\\circ $ between the planets.", "There is no inclination damping present in this simulation.", "However, in contrast to [61], we include explicit eccentricity damping which prohibits substantial eccentricity and inclination growth.", "One can see no qualitative or quantitative change in the results compared to the coplanar case.", "This allows us to restrict ourselves to the coplanar formation scenarios presented above which reduces the number of free parameters in the initial conditions." ], [ "Illustration of Capture into 1:1 Resonance", "In Figure REF we plot an example of the capture of two giant planets into a resonance 1:1 resonance.", "The simulation methodology is exactly as described in Section REF .", "The initial conditions for the inner planet are $m_1 = 1.8 M_{\\mathrm {Jup}}$ , $a_1 = 1.6 \\mathrm {AU}$ and $e_1 = 0$ , and for the outer planet $m_2 = 0.9 M_{\\mathrm {Jup}}$ , $a_2 = 3.6 \\mathrm {AU}$ and $e_2 = 0.46$ ($q_2 = 1.92$ ).", "The damping parameters are the same as those applied for Figure REF .", "The migration timescale is $\\tau = 10^5\\,\\mathrm {yr}$ and $K = 1$ .", "The disk removal timescale is $\\tau _d=10^7\\,\\mathrm {yr}$ .", "The damping is operating on the outer body only, for all $r > 1 \\mathrm {AU}$ .", "The outer planet migrates inwards over the first $10^6\\,\\mathrm {yr}$ of the simulation, while simultaneously suffering occasional strong perturbations from the inner planet.", "At $t \\sim 6\\times 10^5\\,\\mathrm {yr}$ , the pericenter of the outer planet drifts inside the apocenter of the inner planet and a period of strong interaction and rapid inward migration commences, leading to capture into the 1:1 resonance.", "The resonance remains stable until the end of the simulation." ] ]

[ [ "Relax and Localize: From Value to Algorithms" ], [ "Abstract We show a principled way of deriving online learning algorithms from a minimax analysis.", "Various upper bounds on the minimax value, previously thought to be non-constructive, are shown to yield algorithms.", "This allows us to seamlessly recover known methods and to derive new ones.", "Our framework also captures such \"unorthodox\" methods as Follow the Perturbed Leader and the R^2 forecaster.", "We emphasize that understanding the inherent complexity of the learning problem leads to the development of algorithms.", "We define local sequential Rademacher complexities and associated algorithms that allow us to obtain faster rates in online learning, similarly to statistical learning theory.", "Based on these localized complexities we build a general adaptive method that can take advantage of the suboptimality of the observed sequence.", "We present a number of new algorithms, including a family of randomized methods that use the idea of a \"random playout\".", "Several new versions of the Follow-the-Perturbed-Leader algorithms are presented, as well as methods based on the Littlestone's dimension, efficient methods for matrix completion with trace norm, and algorithms for the problems of transductive learning and prediction with static experts." ], [ "Introduction", "This paper studies the online learning framework, where the goal of the player is to incur small regret while observing a sequence of data on which we place no distributional assumptions.", "Within this framework, many algorithms have been developed over the past two decades, and we refer to the book of Cesa-Bianchi and Lugosi [7] for a comprehensive treatment of the subject.", "More recently, a non-algorithmic minimax approach has been developed to study the inherent complexities of sequential problems [2], [1], [14], [19].", "In particular, it was shown that a theory in parallel to Statistical Learning can be developed, with random averages, combinatorial parameters, covering numbers, and other measures of complexity.", "Just as the classical learning theory is concerned with the study of the supremum of empirical or Rademacher process, online learning is concerned with the study of the supremum of a martingale or a certain dyadic process.", "Even though complexity tools introduced in [14], [16], [15] provide ways of studying the minimax value, no algorithms have been exhibited to achieve these non-constructive bounds in general.", "In this paper, we show that algorithms can, in fact, be extracted from the minimax analysis.", "This observation leads to a unifying view of many of the methods known in the literature, and also gives a general recipe for developing new algorithms.", "We show that the potential method, which has been studied in various forms, naturally arises from the study of the minimax value as a certain relaxation.", "We further show that the sequential complexity tools introduced in [14] are, in fact, relaxations and can be used for constructing algorithms that enjoy the corresponding bounds.", "By choosing appropriate relaxations, we recover many known methods, improved variants of some known methods, and new algorithms.", "One can view our framework as one for converting a non-constructive proof of an upper bound on the value of the game into an algorithm.", "Surprisingly, this allows us to also study such “unorthodox” methods as Follow the Perturbed Leader [10], and the recent method of [8] under the same umbrella with others.", "We show that the idea of a random playout has a solid theoretical basis, and that Follow the Perturbed Leader algorithm is an example of such a method.", "It turns out that whenever the sequential Rademacher complexity is of the same order as its i.i.d.", "cousin, there is a family of randomized methods that avoid certain computational hurdles.", "Based on these developments, we exhibit an efficient method for the trace norm matrix completion problem, novel Follow the Perturbed Leader algorithms, and efficient methods for the problems of transductive learning and prediction with static experts.", "The framework of this paper gives a recipe for developing algorithms.", "Throughout the paper, we stress that the notion of a relaxation, introduced below, is not appearing out of thin air but rather as an upper bound on the sequential Rademacher complexity.", "The understanding of inherent complexity thus leads to the development of algorithms.", "One unsatisfying aspect of the minimax developments so far has been the lack of a localized analysis.", "Local Rademacher averages have been shown to play a key role in Statistical Learning for obtaining fast rates.", "It is also well-known that fast rates are possible in online learning, on the case-by-case basis, such as for online optimization of strongly convex functions.", "We show that, in fact, a localized analysis can be performed at an abstract level, and it goes hand-in-hand with the idea of relaxations.", "Using such localized analysis, we arrive at local sequential Rademacher and other local complexities.", "These complexities upper-bound the value of the online learning game and can lead to fast rates.", "What is equally important, we provide an associated generic algorithm to achieve the localized bounds.", "We further develop the ideas of localization, presenting a general adaptive (data-dependent) procedure that takes advantage of the actual moves of the adversary that might have been suboptimal.", "We illustrate the procedure on a few examples.", "Our study of localized complexities and adaptive methods follows from a general agenda of developing universal methods that can adapt to the actual sequence of data played by Nature, thus automatically interpolating between benign and minimax optimal sequences.", "This paper is organized as follows.", "In Section  we formulate the value of the online learning problem and present the (possibly computationally inefficient) minimax algorithm.", "In Section  we develop the idea of relaxations and the meta algorithm based on relaxations, and present a few examples.", "Section  is devoted to a new formalism of localized complexities, and we present a basic localized meta algorithm.", "We show, in particular, that for strongly convex objectives, the regret is easily bounded through localization.", "Next, in Section , we present a fully adaptive method that constantly checks whether the sequence being played by the adversary is in fact minimax optimal.", "We show that, in particular, we recover some of the known adaptive results.", "We also demonstrate how local data-dependent norms arise as a natural adaptive method.", "The remaining sections present a number of new algorithms, often with superior computational properties and regret guarantees than what is known in the literature." ], [ "Notation:", "A set $\\lbrace x_1,\\ldots ,x_t\\rbrace $ is often denoted by $x_{1:t}$ .", "A $t$ -fold product of $\\mathcal {X}$ is denoted by $\\mathcal {X}^t$ .", "Expectation with respect to a random variable $Z$ with distribution $p$ is denoted by $\\mathbb {E}_Z$ or $\\mathbb {E}_{Z\\sim p}$ .", "The set $\\lbrace 1,\\ldots ,T\\rbrace $ is denoted by $[T]$ , and the set of all distributions on some set ${\\mathcal {A}}$ by $\\Delta ({\\mathcal {A}})$ .", "The inner product between two vectors is written as $\\left\\langle a,b \\right\\rangle $ or as $a^{\\scriptscriptstyle \\mathsf {T}}b$ .", "The set of all functions from $\\mathcal {X}$ to $\\mathcal {Y}$ is denoted by $\\mathcal {Y}^\\mathcal {X}$ .", "Unless specified otherwise, $\\epsilon $ denotes a vector $(\\epsilon _1,\\ldots ,\\epsilon _T)$ of i.i.d.", "Rademacher random variables.", "An $\\mathcal {X}$ -valued tree $\\mathbf {x}$ of depth $d$ is defined as a sequence $(\\mathbf {x}_1,\\ldots ,\\mathbf {x}_d)$ of mappings $\\mathbf {x}_t:\\lbrace \\pm 1\\rbrace ^{t-1}\\mapsto \\mathcal {X}$ (see [14]).", "We often write $\\mathbf {x}_t(\\epsilon )$ instead of $\\mathbf {x}_t(\\epsilon _{1:t-1})$ ." ], [ "Value and The Minimax Algorithm", "Let $\\mathcal {F}$ be the set of learner's moves and $\\mathcal {X}$ the set of moves of Nature.", "The online protocol dictates that on every round $t=1,\\ldots ,T$ the learner and Nature simultaneously choose $f_t\\in \\mathcal {F}$ , $x_t\\in \\mathcal {X}$ , and observe each other's actions.", "The learner aims to minimize regret $\\mathbf {Reg}_T \\triangleq \\sum _{t=1}^T \\ell (f_t,x_t) - \\inf _{f \\in \\mathcal {F}} \\sum _{t=1}^T \\ell (f,x_t)$ where $\\ell :\\mathcal {F}\\times \\mathcal {X}\\rightarrow \\mathbb {R}$ is a known loss function.", "Our aim is to study this online learning problem at an abstract level without assuming convexity or other properties of the loss function and the sets $\\mathcal {F}$ and $\\mathcal {X}$ .", "We do assume, however, that $\\ell $ , $\\mathcal {F}$ , and $\\mathcal {X}$ are such that the minimax theorem in the space of distributions over $\\mathcal {F}$ and $\\mathcal {X}$ holds.", "By studying the abstract setting, we are able to develop general algorithmic and non-algorithmic ideas that are common across various application areas.", "The starting point of our development is the minimax value of the associated online learning game: $\\mathcal {V}_T(\\mathcal {F}) = \\inf _{q_1 \\in \\Delta (\\mathcal {F})} \\sup _{x_1 \\in \\mathcal {X}} \\underset{f_1 \\sim q_1}{\\mathbb {E}} \\ldots \\inf _{q_T \\in \\Delta (\\mathcal {F})} \\sup _{x_T \\in \\mathcal {X}} \\underset{f_T \\sim q_T}{\\mathbb {E}}\\left[\\sum _{t=1}^T \\ell (f_t,x_t) - \\inf _{f \\in \\mathcal {F}} \\sum _{t=1}^T \\ell (f,x_t) \\right]$ where $\\Delta (\\mathcal {F})$ is the set of distributions on $\\mathcal {F}$ .", "The minimax formulation immediately gives rise to the optimal algorithm that solves the minimax expression at every round $t$ .", "That is, after witnessing $x_1,\\ldots ,x_{t-1}$ and $f_1,\\ldots ,f_{t-1}$ , the algorithm returns $&\\underset{q \\in \\Delta (\\mathcal {F})}{\\mathrm {argmin}} \\ \\left\\lbrace \\sup _{x_t} \\underset{f_t \\sim q}{\\mathbb {E}} \\inf _{q_{t+1}} \\sup _{x_{t+1}} \\underset{f_{t+1}}{\\mathbb {E}}\\ldots \\inf _{q_T} \\sup _{x_T} \\underset{f_T}{\\mathbb {E}}\\left[\\sum _{i=t}^T \\ell (f_i,x_i) - \\inf _{f \\in \\mathcal {F}} \\sum _{i=1}^T \\ell (f,x_i) \\right] \\right\\rbrace \\\\&=\\underset{q \\in \\Delta (\\mathcal {F})}{\\mathrm {argmin}} \\ \\left\\lbrace \\sup _{x_t} \\underset{f_t \\sim q}{\\mathbb {E}} \\left[ \\ell (f_t,x_t) + \\inf _{q_{t+1}} \\sup _{x_{t+1}} \\underset{f_{t+1}}{\\mathbb {E}}\\ldots \\inf _{q_T} \\sup _{x_T} \\underset{f_T}{\\mathbb {E}}\\left[\\sum _{i=t+1}^T \\ell (f_i,x_i) - \\inf _{f \\in \\mathcal {F}} \\sum _{i=1}^T \\ell (f,x_i) \\right] \\right]\\right\\rbrace $ Henceforth, if the quantification in $\\inf $ and $\\sup $ is omitted, it will be understood that $x_t$ , $f_t$ , $p_t$ , $q_t$ range over $\\mathcal {X}$ , $\\mathcal {F}$ , $\\Delta (\\mathcal {X})$ , $\\Delta (\\mathcal {F})$ , respectively.", "Moreover, $\\mathbb {E}_{x_t}$ is with respect to $p_t$ while $\\mathbb {E}_{f_t}$ is with respect to $q_t$ .", "The first sum in (REF ) starts at $i=t$ since the partial loss $\\sum _{i=1}^{t-1} \\ell (f_i,x_i)$ has been fixed.", "We now notice a recursive form for defining the value of the game.", "Define for any $t \\in [T-1]$ and any given prefix $x_{1},\\ldots ,x_t \\in \\mathcal {X}$ the conditional value $\\mathcal {V}_T\\left(\\mathcal {F}| x_1,\\ldots ,x_t \\right) \\triangleq \\inf _{q \\in \\Delta (\\mathcal {F})} \\sup _{x \\in \\mathcal {X}} \\left\\lbrace \\underset{f \\sim q}{\\mathbb {E}}\\left[\\ell (f,x) \\right] + \\mathcal {V}_T(\\mathcal {F}| x_{1},\\ldots ,x_{t},x)\\right\\rbrace $ where $\\mathcal {V}_T\\left(\\mathcal {F}| x_1,\\ldots ,x_T \\right) \\triangleq - \\inf _{f \\in \\mathcal {F}} \\sum _{t=1}^T \\ell (f,x_t) ~~~~\\mbox{and}~~~~ \\mathcal {V}_T(\\mathcal {F}) = \\mathcal {V}_T(\\mathcal {F}| \\lbrace \\rbrace ).$ The minimax optimal algorithm specifying the mixed strategy of the player can be written succinctly $q_t = \\underset{q \\in \\Delta (\\mathcal {F})}{\\mathrm {argmin}} \\ \\sup _{x \\in \\mathcal {X}} \\Big \\lbrace \\mathbb {E}_{f \\sim q}\\left[ \\ell (f,x) \\right] + \\mathcal {V}_T(\\mathcal {F}| x_1,\\ldots ,x_{t-1},x)\\Big \\rbrace \\ .$ This recursive formulation has appeared in the literature, but now we have tools to study the conditional value of the game.", "We will show that various upper bounds on $\\mathcal {V}_T(\\mathcal {F}| x_1,\\ldots ,x_{t-1},x)$ yield an array of algorithms, some with better computational properties than others.", "In this way, the non-constructive approach of [14], [15], [16] to upper bound the value of the game directly translates into algorithms.", "The minimax algorithm in (REF ) can be interpreted as choosing the best decision that takes into account the present loss and the worst-case future.", "We then realize that the conditional value of the game serves as a “regularizer”, and thus well-known online learning algorithms such as Exponential Weights, Mirror Descent and Follow-the-Regularized-Leader arise as relaxations rather than a “method that just works”.", "The first step is to appeal to the minimax theorem and perform the same manipulation as in [1], [14], but only on the value from $t+1$ onwards: $\\mathcal {V}_T\\left(\\mathcal {F}| x_1,\\ldots ,x_t \\right) = \\sup _{p_{t+1}} \\underset{x_{t+1}}{\\mathbb {E}} \\ldots \\sup _{p_T} \\underset{x_T}{\\mathbb {E}}\\left[\\sum _{i=t+1}^T \\inf _{f_{i} \\in \\mathcal {F}} \\underset{x_i \\sim p_i}{\\mathbb {E}} \\ell (f_i,x_i) - \\inf _{f \\in \\mathcal {F}} \\sum _{i=1}^T \\ell (f,x_i) \\right]$ This expression is still unwieldy, and the idea is now to come up with more manageable, yet tight, upper bounds of the conditional value." ], [ "Relaxations and the Basic Meta-Algorithm", "A relaxation $\\mathbf {Rel}_{}\\left( \\right)$ is a sequence of functions $\\mathbf {Rel}_{T}\\left(\\mathcal {F} | x_{1},\\ldots ,x_{t} \\right)$ for each $t\\in [T]$ .", "We shall use the notation $\\mathbf {Rel}_{T}\\left(\\mathcal {F} \\right)$ for $\\mathbf {Rel}_{T}\\left(\\mathcal {F} | \\lbrace \\rbrace \\right)$ .", "A relaxation will be called admissible if for any $x_1,\\ldots ,x_T \\in \\mathcal {X}$ , $\\mathbf {Rel}_{T}\\left(\\mathcal {F} | x_{1},\\ldots ,x_{t} \\right) \\ge \\inf _{q \\in \\Delta (\\mathcal {F})} \\sup _{x \\in \\mathcal {X}} \\left\\lbrace \\underset{f \\sim q}{\\mathbb {E}}\\left[\\ell (f,x)\\right] + \\mathbf {Rel}_{T}\\left(\\mathcal {F} | x_{1},\\ldots ,x_{t},x \\right)\\right\\rbrace $ for all $t\\in [T-1]$ , and $ \\mathbf {Rel}_{T}\\left(\\mathcal {F} | x_{1},\\ldots ,x_{T} \\right) \\ge - \\inf _{f \\in \\mathcal {F}} \\sum _{t=1}^T \\ell (f,x_t) .$ A strategy $q$ that minimizes the expression in (REF ) defines an optimal algorithm for the relaxation $\\mathbf {Rel}_{}\\left( \\right)$ .", "This algorithm is given below under the name “Meta-Algorithm”.", "However, minimization need not be exact: any $q$ that satisfies the admissibility condition (REF ) is a valid method, and we will say that such an algorithm is admissible with respect to the relaxation $\\mathbf {Rel}_{}\\left( \\right)$.", "[h] Meta-Algorithm $\\mathbf {MetAlgo}$ Parameters: Admissible relaxation $\\mathbf {Rel}$ $t = 1$ to $T$ $q_t = \\arg \\min _{q \\in \\Delta (\\mathcal {F})} \\sup _{x \\in \\mathcal {X}} \\left\\lbrace \\mathbb {E}_{f \\sim q}\\left[ \\ell (f,x) \\right] + \\mathbf {Rel}_{T}\\left(\\mathcal {F} | x_1,\\ldots ,x_{t-1},x \\right)\\right\\rbrace $ Play $f_t \\sim q_t$ and receive $x_t$ from adversary Proposition 1 Let $\\mathbf {Rel}_{}\\left( \\right)$ be an admissible relaxation.", "For any admissible algorithm with respect to $\\mathbf {Rel}_{}\\left( \\right)$ , including the Meta-Algorithm, irrespective of the strategy of the adversary, $\\sum _{t=1}^T \\mathbb {E}_{f_t\\sim q_t}\\ell (f_t,x_t) - \\inf _{f\\in \\mathcal {F}} \\sum _{t=1}^T \\ell (f,x_t) \\le \\mathbf {Rel}_{T}\\left(\\mathcal {F} \\right) \\ ,$ and therefore, $\\mathbb {E}[\\mathbf {Reg}_T]\\le \\mathbf {Rel}_{T}\\left(\\mathcal {F} \\right) \\ .$ We also have that $\\mathcal {V}_T(\\mathcal {F}) \\le \\mathbf {Rel}_{T}\\left(\\mathcal {F} \\right) \\ .$ If $a\\le \\ell (f,x) \\le b$ for all $f\\in \\mathcal {F},x\\in \\mathcal {X}$ , the Hoeffding-Azuma inequality yields, with probability at least $1-\\delta $ , $\\mathbf {Reg}_T = \\sum _{t=1}^T \\ell (f_t,x_t) - \\inf _{f\\in \\mathcal {F}} \\sum _{t=1}^T \\ell (f,x_t) \\le \\mathbf {Rel}_{T}\\left(\\mathcal {F} \\right) + (b-a)\\sqrt{T/2\\cdot \\log (2/\\delta )} \\ .$ Further, if for all $t\\in [T]$ , the admissible strategies $q_t$ are deterministic, $\\mathbf {Reg}_T \\le \\mathbf {Rel}_{T}\\left(\\mathcal {F} \\right) \\ .$ The reader might recognize $\\mathbf {Rel}$ as a potential function.", "It is known that one can derive regret bounds by coming up with a potential such that the current loss of the player is related to the difference in the potentials at successive steps, and that the loss of the best decision in hindsight can be extracted from the final potential.", "The origin of “good” potential functions has always been a mystery (at least to the authors).", "One of the conceptual contributions of this paper is to show that they naturally arise as relaxations on the conditional value.", "The conditional value itself can be characterized as the tightest possible relaxation.", "In particular, for many problems a tight relaxation (sometimes within a factor of 2) is achieved through symmetrization.", "Define the conditional Sequential Rademacher complexity $\\mathfrak {R}_T (\\mathcal {F}| x_1,\\ldots ,x_t) = \\sup _{\\mathbf {x}} \\mathbb {E}_{\\epsilon _{t+1:T}} \\sup _{f\\in \\mathcal {F}} \\left[ 2\\sum _{s=t+1}^T \\epsilon _s\\ell (f,\\mathbf {x}_{s-t}(\\epsilon _{t+1:s-1})) - \\sum _{s=1}^t \\ell (f,x_s) \\right] \\ .$ Here the supremum is over all $\\mathcal {X}$ -valued binary trees of depth $T-t$ .", "One may view this complexity as a partially symmetrized version of the sequential Rademacher complexity $\\mathfrak {R}_T (\\mathcal {F}) \\triangleq \\mathfrak {R}_T (\\mathcal {F}~|~ \\lbrace \\rbrace ) = \\sup _{\\mathbf {x}} \\mathbb {E}_{\\epsilon _{1:T}} \\sup _{f\\in \\mathcal {F}} \\left[ 2\\sum _{s=1}^T \\epsilon _s\\ell (f,\\mathbf {x}_{s}(\\epsilon _{1:s-1})) \\right]$ defined in [14].", "We shall refer to the term involving the tree $\\mathbf {x}$ as the “future” and the term being subtracted off – as the “past”.", "This indeed corresponds to the fact that the quantity is conditioned on the already observed $x_1,\\ldots ,x_t$ , while for the future we have the worst possible binary tree.It is somewhat cumbersome to write out the indices on $\\mathbf {x}_{s-t}(\\epsilon _{t+1:s-1})$ in (REF ), so we will instead use $\\mathbf {x}_s(\\epsilon )$ for $s=1,\\ldots ,T-t$ , whenever this does not cause confusion.", "Proposition 2 The conditional Sequential Rademacher complexity is admissible.", "The proof of this proposition is given in the Appendix and it corresponds to one step of the sequential symmetrization proof in [14].", "We note that the factor 2 appearing in (REF ) is not necessary in certain cases (e.g.", "binary prediction with absolute loss).", "We now show that several well-known methods arise as further relaxations on the conditional sequential Rademacher complexity $\\mathfrak {R}_T$ ." ], [ "Exponential Weights", "Suppose $\\mathcal {F}$ is a finite class and $|\\ell (f,x)|\\le 1$ .", "In this case, a (tight) upper bound on sequential Rademacher complexity leads to the following relaxation: $\\mathbf {Rel}_{T}\\left(\\mathcal {F} | x_{1},\\ldots , x_t \\right) = \\inf _{\\lambda >0}\\left\\lbrace \\frac{1}{\\lambda }\\log \\left( \\sum _{f \\in \\mathcal {F}} \\exp \\left( - \\lambda \\sum _{i=1}^t \\ell (f,x_i) \\right) \\right) + 2 \\lambda (T-t) \\right\\rbrace $ Proposition 3 The relaxation (REF ) is admissible and $\\mathfrak {R}_T (\\mathcal {F}| x_1,\\ldots ,x_t)\\le \\mathbf {Rel}_{T}\\left(\\mathcal {F} | x_{1},\\ldots , x_t \\right).$ Furthermore, it leads to a parameter-free version of the Exponential Weights algorithm, defined on round $t+1$ by the mixed strategy $q_{t+1}(f) \\propto \\exp \\left(-\\lambda _t^* \\sum _{s=1}^{t}\\ell (f,x_s)\\right)$ with $\\lambda _t^*$ the optimal value in (REF ).", "The algorithm's regret is bounded by $\\mathbf {Rel}_{T}\\left(\\mathcal {F} \\right)\\le 2\\sqrt{2T\\log |\\mathcal {F}|} \\ .", "$ The Chernoff-Cramèr inequality tells us that (REF ) is the tightest possible relaxation.", "The proof of Proposition REF reveals that the only inequality is the softmax which is also present in the proof of the maximal inequality for a finite collection of random variables.", "In this way, exponential weights is an algorithmic realization of a maximal inequality for a finite collection of random variables.", "The connection between probabilistic (or concentration) inequalities and algorithms runs much deeper.", "We point out that the exponential-weights algorithm arising from the relaxation (REF ) is a parameter-free algorithm.", "The learning rate $\\lambda ^*$ can be optimized (via one-dimensional line search) at each iteration with almost no cost.", "This can lead to improved performance as compared to the classical methods that set a particular schedule for the learning rate." ], [ "Mirror Descent", "In the setting of online linear optimization, the loss is $\\ell (f,x)=\\left<f,x\\right>$ .", "Suppose $\\mathcal {F}$ is a unit ball in some Banach space and $\\mathcal {X}$ is the dual.", "Let $\\Vert \\cdot \\Vert $ be some $(2,C)$ -smooth norm on $\\mathcal {X}$ (in the Euclidean case, $C=2$ ).", "Using the notation $\\tilde{x}_{t-1}=\\sum _{s=1}^{t-1}x_s$ , a straightforward upper bound on sequential Rademacher complexity is the following relaxation: $\\mathbf {Rel}_{T}\\left(\\mathcal {F} | x_{1},\\ldots , x_t \\right)= \\sqrt{ \\left\\Vert \\tilde{x}_{t-1}\\right\\Vert ^2 + \\left<\\nabla \\left\\Vert \\tilde{x}_{t-1}\\right\\Vert ^2,x_t\\right> + C (T - t + 1) }$ Proposition 4 The relaxation (REF ) is admissible and $\\mathfrak {R}_T (\\mathcal {F}| x_1,\\ldots ,x_t)\\le \\mathbf {Rel}_{T}\\left(\\mathcal {F} | x_{1},\\ldots , x_t \\right) \\ .$ Furthermore, it leads to the Mirror Descent algorithm with regret at most $\\mathbf {Rel}_{T}\\left(\\mathcal {F} \\right)\\le \\sqrt{2C T}$ .", "An important feature of the algorithms we just proposed is the absence of any parameters, as the step size is tuned automatically.", "We had chosen Exponential Weights and Mirror Descent for illustration because these methods are well-known.", "Our aim at this point was to show that the associated relaxations arise naturally (typically with a few steps of algebra) from the sequential Rademacher complexity.", "More examples are included later in the paper.", "It should now be clear that upper bounds, such as the Dudley Entropy integral, can be turned into a relaxation, provided that admissibility is proved.", "Our ideas have semblance of those in Statistics, where an information-theoretic complexity can be used for defining penalization methods." ], [ "Localized Complexities and the Localized-Meta Algorithm", "The localized analysis plays an important role in Statistical Learning Theory.", "The basic idea is that better rates can be proved for empirical risk minimization when one considers the empirical process in the vicinity of the target hypothesis [11], [4].", "Through this, localization gives extra information by shrinking the size of the set which needs to be analyzed.", "What does it mean to localize in online learning?", "As we obtain more data, we can rule out parts of $\\mathcal {F}$ as those that are unlikely to become the leaders.", "This observation indeed gives rise to faster rates.", "Let us develop a general framework of localization and then illustrate it on examples.", "We emphasize that the localization ideas will be developed at an abstract level where no assumptions are placed on the loss function or the sets $\\mathcal {F}$ and $\\mathcal {X}$ .", "Given any $x_1,\\ldots ,x_{t} \\in \\mathcal {X}$ , for any $k \\ge 1$ define $\\mathcal {F}^k(x_1,\\ldots ,x_{t}) = \\left\\lbrace f \\in \\mathcal {F}: \\exists ~ x_{t+1},\\ldots ,x_{t+k} \\in \\mathcal {X}~\\textrm { s.t.", "}~ \\sum _{i=1}^{t+k} \\ell (f,x_i) = \\inf _{f \\in \\mathcal {F}} \\sum _{i=1}^{t+k} \\ell (f,x_i) \\right\\rbrace \\ .$ That is, given the instances $x_1,\\ldots ,x_t$ , the set $\\mathcal {F}^k(x_1,\\ldots ,x_{t})$ is the set of elements that could be the minimizers of cumulative loss on $t+k$ instances, the first $t$ of which are $x_1,\\ldots ,x_t$ and the remaining $k$ arbitrary.", "We shall refer to minimizers of cumulative loss as empirical risk minimizers (or, ERM).", "Importantly, $\\mathcal {V}_T(\\mathcal {F}|x_1,\\ldots ,x_t) = \\mathcal {V}_T\\left(\\mathcal {F}^{T-t}(x_1,\\ldots ,x_t)|x_1,\\ldots ,x_t\\right) \\ .$ Henceforth, we shall use the notation $\\tilde{k}_j \\triangleq \\sum _{i=1}^j k_i$ .", "We now consider subdividing $T$ into blocks of time $k_1,\\ldots ,k_m \\in [T]$ such that $\\tilde{k}_m = T$ .", "With this notation, $\\tilde{k}_i$ is the last time in the $i$ th block.", "We then have regret upper bounded as $\\sum _{t=1}^T \\ell (f_t,x_t) - \\inf _{f \\in \\mathcal {F}} \\sum _{t=1}^T \\ell (f,x_t) \\le \\sum _{t=1}^T \\ell (f_t,x_t) - \\sum _{i=1}^m \\inf _{f \\in \\mathcal {F}^{k_i}\\left(x_1,\\ldots ,x_{\\tilde{k}_{i-1}}\\right)} \\sum _{t=\\tilde{k}_{i-1}+1}^{\\tilde{k}_{i}} \\ell (f,x_t) \\ .$ The short inductive proof is given in Appendix, Lemma REF .", "We can now bound (REF ) by $&\\sum _{i=1}^m \\left( \\sum _{t=\\tilde{k}_{i-1}+1}^{\\tilde{k}_{i}} \\ell (f,x_t) - \\inf _{f \\in \\mathcal {F}^{k_i}\\left(x_1,\\ldots ,x_{\\tilde{k}_{i-1}}\\right)} ~\\sum _{t=\\tilde{k}_{i-1}+1}^{\\tilde{k}_{i}} \\ell (f,x_t) \\right)\\\\& ~~~~~~~\\le \\sum _{i=1}^m \\mathbf {Reg}_{k_i}(x_{\\tilde{k}_{i-1}},\\ldots ,x_{\\tilde{k}_{i}},f_{\\tilde{k}_{i-1}},\\ldots ,f_{\\tilde{k}_{i}},\\mathcal {F}^{k_i}(x_1,\\ldots ,x_{\\tilde{k}_{i-1}}))$ Hence, one can decompose the online learning game into blocks of $m$ successive games.", "The crucial point to notice is that at the $i^{th}$ block, we do not compete with the best hypothesis in all of $\\mathcal {F}$ but rather only $\\mathcal {F}^{k_i}(x_1,\\ldots ,x_{\\tilde{k}_{i-1}})$ .", "It is this localization based on history that could lead to possibly faster rates.", "While the “blocking” idea often appears in the literature (for instance, in the form of a doubling trick, as described below), the process is usually “restarted” from scratch by considering all of $\\mathcal {F}$ .", "Notice further that one need not choose all $k_1,\\ldots ,k_m$ in advance.", "The player can choose $k_i$ based on history $x_1,\\ldots ,x_{\\tilde{k}_{i-1}}$ and then use, for instance, the Meta-Algorithm introduced in previous section to play the game within the block $k_i$ using the localized class $\\mathcal {F}^{k_i}(x_1,\\ldots ,x_{\\tilde{k}_{i-1}})$ .", "Such adaptive procedures will be considered in Section , but presently we assume that the block sizes $k_1,\\ldots ,k_m$ are fixed.", "While the successive localizations using subsets $\\mathcal {F}^{k_i}(x_1,\\ldots ,x_{\\tilde{k}_{i-1}})$ can provide an algorithm with possibly better performance, specifying and analyzing the localized subset $\\mathcal {F}^{k_i}(x_1,\\ldots ,x_{\\tilde{k}_{i-1}})$ exactly might not be possible.", "In such a case, one can instead use $\\mathcal {F}_r(x_1,\\ldots ,x_{\\tilde{k}_{i-1}}) = \\left\\lbrace f \\in \\mathcal {F}: P\\left(f ~|~ x_1,\\ldots ,x_{\\tilde{k}_{i-1}}\\right) \\le r \\right\\rbrace $ where $P$ is some “property” of $f$ given data.", "This definition echoes the definition of the set of $r$ -minimizers of empirical or expected risk in Statistical Learning.", "Further, for a given $k$ define $r(k;x_1,\\ldots ,x_t) = \\inf \\lbrace r : \\mathcal {F}^{k}(x_1,\\ldots ,x_t) \\subset \\mathcal {F}_r(x_1,\\ldots ,x_{t})\\rbrace $ the smallest “radius” such that $\\mathcal {F}_r$ includes the set of potential minimizers over the next $k$ time steps.", "Of course, if the property $P$ does not enforce localization, the bounds are not going to exhibit any improvement, so $P$ needs to be chosen carefully for a particular problem of interest.", "We have the following algorithm: [h] Localized Meta-Algorithm Parameters : Relaxation $\\mathbf {Rel}$ Initialize $t=0$ and blocks $k_1,\\ldots ,k_m$ s.t.", "$\\sum _{i=1}^m k_i = T$ $i=1$ to $m$ Play $k_i$ rounds using $\\mathbf {MetAlgo}\\left(\\mathcal {F}_{r(k_i;x_1,\\ldots ,x_t)}\\right)$ and set $t = t + k_i$ Lemma 5 The regret of the Localized Meta-Algorithm is bounded as $\\mathbf {Reg}_T(x_1,\\ldots ,x_T) \\le \\sum _{i=1}^{m} \\mathbf {Rel}_{k_i}\\left(\\mathcal {F}_{r\\left(k_i;x_1,\\ldots ,x_{\\tilde{k}_{i-1}}\\right)} \\right)$ Note that the above lemma points to local sequential complexities for online learning problems that can lead to possibly fast rates.", "In particular, if sequential Rademacher complexity is used as the relaxation in the Localized Meta-Algorithm, we get a bound in terms of local sequential Rademacher complexities." ], [ "Local Sequential Complexities", "The following corollary is a direct consequence of Lemma REF .", "Corollary 6 (Local Sequential Rademacher Complexity) For any property $P$ and any $k_1,\\ldots ,k_m \\in \\mathbb {N}$ such that $\\sum _{i=1}^m k_i = T$ , we have that : $\\mathcal {V}_T(\\mathcal {F}) \\le \\sup _{x_1,\\ldots ,x_T} \\sum _{i=1}^m \\mathfrak {R}_{k_i}\\left(\\mathcal {F}_{r\\left(k_i;x_1,\\ldots ,x_{\\tilde{k}_{i-1}}\\right)}\\right)$ Clearly, the sequential Rademacher complexities in the above bound can be replaced with other sequential complexity measures of the localized classes that are upper bounds on the sequential Rademacher complexities.", "For instance, one can replace each Rademacher complexity $\\mathfrak {R}_{k_i}$ by covering number based bounds of the local classes, such as the analogues of the Dudley Entropy Integral bounds developed in the sequential setting in [14].", "Once can also use, for instance, fat-shattering dimension based complexity measures for these local classes." ], [ "Example : Doubling trick", "The doubling trick can be seen as a particular blocking strategy with $k_i = 2^{i-1}$ so that $\\mathbf {Reg}_T(x_1,\\ldots ,x_T) & \\le \\sum _{i=1}^{\\lceil \\log _2 T \\rceil + 1} \\mathbf {Rel}_{2^{i-1}}\\left(\\mathcal {F}_{r(2^{i-1};x_1,\\ldots ,x_{\\sum _{j=1}^{i-1} 2^{j-1}})} \\right)& \\le \\sum _{i=1}^{\\lceil \\log _2 T\\rceil + 1} \\mathbf {Rel}_{2^{i-1}}\\left(\\mathcal {F} \\right)$ for $\\mathcal {F}_r$ defined with respect to some property $P$ .", "The latter inequality is potentially loose, as the algorithm is “restarted” after the previous block is completed.", "Now if $\\mathbf {Rel}$ is such that for any $t$ , $\\mathbf {Rel}_{t}\\left(\\mathcal {F} \\right) \\le t^p$ for some $p$ then the regret is upper bounded by $\\frac{T^p - 2^{-p}}{1 - 2^{-p} }$ .", "The main advantage of the doubling trick is of course that we do not need to know $T$ in advance." ], [ "Example : Strongly Convex Loss", "To illustrate the idea of localization, consider online convex optimization with $\\lambda $ -strongly convex functions $x_t:\\mathcal {F}\\mapsto \\mathbb {R}$ (that is, $\\ell (f,x) = x(f)$ ).", "Define $\\mathbf {Rel}_{T}\\left(\\mathcal {F} | x_{1},\\ldots ,x_{t} \\right) = - \\inf _{f \\in \\mathcal {F}} \\sum _{i=1}^t x_i(f) + (T-t) \\inf _{f \\in \\mathcal {F}} \\sup _{f^{\\prime } \\in \\mathcal {F}} \\Vert f - f^{\\prime }\\Vert $ An easy Lemma REF in the Appendix shows that this relaxation is admissible.", "Notice that this relaxation grows linearly with block size and is by itself quite bad.", "However, with blocking and localization, the relaxation gives an optimal bound for strongly convex objectives.", "To see this note that for $k = 1$ , any minimizer of $\\sum _{i=1}^{t+1} x_i(f)$ has to be close to the minimizer $\\hat{f}_t$ of $\\sum _{i=1}^{t} x_i(f)$ , due to strong convexity of the functions.", "In other words, the property $P(f|x_1,\\ldots ,x_t) = \\Vert f-\\hat{f}_t\\Vert $ with $r=1/(\\lambda t)$ entails $\\mathcal {F}^1(x_1,\\ldots ,x_t) \\subseteq \\left\\lbrace f \\in \\mathcal {F}: \\Vert f - \\hat{f}_t\\Vert \\le 1/(\\lambda t)\\right\\rbrace = \\mathcal {F}_r(x_1,\\ldots ,x_t) .$ The relaxation for the block of size $k=1$ is $\\mathbf {Rel}_{1}\\left(\\mathcal {F}_r(x_1,\\ldots ,x_t) \\right) \\le \\inf _{f \\in \\mathcal {F}_r(x_1,\\ldots ,x_t)} \\sup _{f^{\\prime } \\in \\mathcal {F}_r(x_1,\\ldots ,x_t)} \\Vert f - f^{\\prime }\\Vert ,$ the radius of the smallest ball containing the localized set $\\mathcal {F}_r(x_1,\\ldots ,x_t)$ , and we immediately get $\\mathbf {Reg}_T(x_1,\\ldots ,x_T) \\le \\sum _{t=1}^T 1/(\\lambda t) \\le (1 + \\log (T))/\\lambda \\ .$ We remark that this proof is different in spirit from the usual proofs of fast rates for strongly convex functions, and it demonstrates the power of localization." ], [ "Adaptive Procedures", "There is a strong interest in developing methods that enjoy worst-case regret guarantees but also take advantage of the suboptimality of the sequence being played by Nature.", "An algorithm that is able to do so without knowing in advance that the sequence will have a certain property will be called adaptive.", "Imagine, for instance, running an experts algorithm, and one of the experts has gained such a lead that she is clearly the winner (that is, the empirical risk minimizer) at the end of the game.", "In this case, since we are to be compared with the leader at the end, we need not focus on anyone else, and regret for the remainder of the game is zero.", "There has been previous work on exploiting particular ways in which sequences can be suboptimal.", "Examples include the Adaptive Gradient Descent of [5] and Adaptive Hedge of [20].", "We now give a generic method which incorporates the idea of localization in order to adaptively (and constantly) check whether the sequence being played is of optimal or suboptimal nature.", "Notice that, as before, we present the algorithm at the abstract level of the online game with some decision sets $\\mathcal {F}$ , $\\mathcal {X}$ , and some loss function $\\ell $ .", "The adaptive procedure below uses a subroutine $\\mathbf {Block}(\\lbrace x_1,\\ldots ,x_t\\rbrace , \\tau )$ which, given the history $\\lbrace x_1,\\ldots ,x_t\\rbrace $ , returns a subdivision of the next $\\tau $ rounds into sub-blocks.", "The choice of the blocking strategy has to be made for the particular problem at hand, but, as we show in examples, one can often use very simple strategies.", "Let us describe the adaptive procedure.", "First, for simplicity of exposition, we start with the doubling-size blocks.", "Here is what happens within each of these blocks.", "During each round the learner decides whether to stay in the same sub-block or to start a new one, as given by the blocking procedure $\\mathbf {Block}$ .", "If started, the new sub-block uses the localized subset given history of adversary's moves up until last round.", "Choosing to start a new sub-block corresponds to the realization of the learner that the sequence being presented so far is in fact suboptimal.", "The learner then incorporates this suboptimality into the localized procedure.", "[h] Adaptive Localized Meta-Algorithm Parameters : Relaxation $\\mathbf {Rel}$ and block size calculator $\\mathbf {Block}$ .", "Initialize $t=1$ and ${\\tt nbl}= 1$ , and suppose $T=2^{c}-1$ for some $c\\ge 2$ .", "$i = 1$ to $c$ $G = \\mathbf {Rel}_{2^i}\\left(\\mathcal {F}_r(2^i;x_1,\\ldots ,x_{t-1}) \\right)$                             % guaranteed value of relaxation $m = 1, {\\tt curr}=1$ and $K_1 = 2^i$ ${\\tt curr}\\le 2^i$ and $t \\le T$ $(\\kappa _1,\\ldots ,\\kappa _{m^{\\prime }}) = \\mathbf {Block}\\left(\\lbrace x_1,\\ldots ,x_t\\rbrace ,2^i-{\\tt curr}\\right)$             % blocking for remainder of $2^i$ $G > \\sup _{x_{t+1},\\ldots ,x_{2^{i+1}-1}}\\sum _{j=1}^{m^{\\prime }} \\mathbf {Rel}_{\\kappa _j}\\left(\\mathcal {F}_{r(\\kappa _i;x_1,\\ldots ,x_{t+\\tilde{\\kappa }_{j-1}})} \\right)$ $k^*_{\\tt nbl}= \\kappa _1$ , $K = (\\kappa _2,\\ldots ,\\kappa _{m^{\\prime }})$ , $m = m^{\\prime }-1$       % if better value, accept new blocking $k^*_{\\tt nbl}= K_1$ , $K = (K_2,\\ldots ,K_m)$ , $m = m-1$       % else continue with current blocking Play $k^*_{\\tt nbl}$ rounds using $\\mathbf {MetAlgo}(\\mathcal {F}_{r(k^*_{\\tt nbl};x_1,\\ldots ,x_t)})$ ${\\tt nbl}= {\\tt nbl}+1$ , $t = t+k^*_{\\tt nbl}$ , ${\\tt curr}= {\\tt curr}+k^*_{\\tt nbl}$ Let $G = \\sup _{x_{t+1},\\ldots ,x_{2^{i+1}-1}}\\sum _{j=1}^{m} \\mathbf {Rel}_{K_j}\\left(\\mathcal {F}_{r(K_j;x_1,\\ldots ,x_{t+\\sum _{i=1}^{j-1} K_i})} \\right)$ Lemma 7 Given some admissible relaxation $\\mathbf {Rel}$ , the regret of the adaptive localized meta-algorithm (Algorithm ) is bounded as $\\mathbf {Reg}_T \\le \\sum _{i=1}^{{\\tt nbl}} \\mathbf {Rel}_{k^*_i}\\left(\\mathcal {F}_{r\\left(k^*_i;x_1,\\ldots ,x_{\\tilde{k}^*_{i-1}}\\right)} \\right)$ where ${\\tt nbl}$ is the number of blocks actually played and $k^*_i$ 's are adaptive block lengths defined within the algorithm.", "Further, irrespective of the blocking strategy $\\mathbf {Block} $ used, if the relaxation $\\mathbf {Rel}$ is such that for any $t$ , $\\mathbf {Rel}_{t}\\left(\\mathcal {F} \\right) \\le t^p$ for some $p \\in (0,1]$ , then the worst case regret is always bounded as $\\mathbf {Reg}_T \\le (T^p - 2^{-p})/(1 - 2^{-p}) \\ .$ We now demonstrate that the adaptive algorithm in fact takes advantage of sub-optimality in several situations that have been previously studied in the literature.", "On the conceptual level, adaptive localization allows us to view several fast rate results under the same umbrella." ], [ "Example: Adaptive Gradient Descent", "Consider the online convex optimization scenario.", "Following the setup of [5], suppose the learner encounters a sequence of convex functions $x_t$ with the strong convexity parameter $\\sigma _t$ , potentially zero, with respect to a $(2,C)$ -smooth norm $\\Vert \\cdot \\Vert $ .", "The goal is to adapt to the actual sequence of functions presented by the adversary.", "Let us invoke the Adaptive Localized Meta-Algorithm with a rather simple blocking strategy $\\mathbf {Block}\\left(\\lbrace x_1,\\ldots ,x_t\\rbrace ,k\\right) = \\left\\lbrace \\begin{array}{cc}(k) & \\textrm {if }\\sqrt{k} > \\sigma _{1:t}\\\\(1,1,\\ldots ,1) & \\textrm {otherwise }\\end{array}\\right.$ This blocking strategy either says “use all of the next $k$ rounds as one block”, or “make each of the next $k$ time step into separate blocks”.", "Let $\\hat{f}_t$ be the empirical minimizer at the start of the block (that is after $t$ rounds), and let $y_t = \\nabla x_t(f_t)$ .", "Then we can use the localization $\\mathcal {F}_{r(k;x_1,\\ldots ,x_t)} &= \\left\\lbrace f \\in \\mathcal {F}: \\Vert f - \\hat{f}_t\\Vert \\le 2 \\min \\left\\lbrace 1, k/\\sigma _{1:t}\\right\\rbrace \\right\\rbrace $ and relaxation $\\mathbf {Rel}_{k}\\left(\\mathcal {F}_{r(k;x_1,\\ldots ,x_t)} | y_1,\\ldots ,y_i \\right) &= - \\left<\\hat{f}_{t},\\tilde{y}_i\\right> + 2 \\min \\left\\lbrace 1, k/\\sigma _{1:t}\\right\\rbrace \\left( \\left\\Vert \\tilde{y}_{i-1}\\right\\Vert ^2 + \\left<\\nabla \\left\\Vert \\tilde{y}_{i-1}\\right\\Vert ^2,y_i\\right>+ C (k-i+1) \\right)^{1/2}$ where $\\tilde{y}_{i-1} = \\sum _{j=1}^{i-1} y_j$ .", "For the above relaxation we can show that the corresponding update at round $t+i$ is given by $f_{t+i} = \\hat{f}_t - \\max \\left\\lbrace 1 ,\\frac{k}{ \\sigma _{1:t} }\\right\\rbrace \\frac{- \\nabla \\left\\Vert \\bar{x}_{i-1}\\right\\Vert ^2}{\\sqrt{ \\left\\Vert \\bar{x}_{i-1}\\right\\Vert ^2 + C (k-i+1) }}$ where $k$ is the length of the current block.", "The next lemma shows that the proposed adaptive gradient descent recovers the results of [5].", "The method is a mixture of Follow the Leader -style algorithm and a Gradient Descent -style algorithm.", "Lemma 8 The relaxation specified above is admissible.", "Suppose the adversary plays 1-Lipchitz convex functions $x_1,\\ldots ,x_T$ such that for any $t \\in [T]$ , $\\sum _{i=1}^t x_i$ is $\\sigma _{1:t}$ -strongly convex, and further suppose that for some $B \\le 1$ , we have that $\\sigma _{1:t} = B t^{\\alpha }$ .", "Then, for the blocking strategy specified above, If $\\alpha \\le 1/2$ then $\\mathbf {Reg}_T \\le O\\left(\\sqrt{T}\\right)$ If $1 > \\alpha > 1/2$ then $\\mathbf {Reg}_T \\le O(\\frac{T^{1 - \\alpha }}{B})$ If $\\alpha = 1$ then $\\mathbf {Reg}_T \\le O\\left(\\frac{\\log T}{B}\\right)$" ], [ "Example: Adaptive Experts", "We now turn to the setting of Adaptive Hedge or Exponential Weights algorithm similar to the one studied in [20].", "Consider the following situation: for all time steps after some $\\tau $ , there is an element (or, expert) $f$ that is the best by a margin $k$ over the next-best choice in $\\mathcal {F}$ in terms of the (unnormalized) cumulative loss, and it remains to be the winner until the end.", "Let us use the localization $\\mathcal {F}_{r(k;x_1,\\ldots ,x_t)} = \\left\\lbrace f \\in \\mathcal {F}~:~ \\sum _{i=1}^t \\ell (f,x_i) - \\min _{f \\in \\mathcal {F}} \\sum _{i=1}^t \\ell (f,x_i) \\le k\\right\\rbrace \\ ,$ the set of functions closer than the margin to the ERM.", "Let $\\hat{\\mathcal {F}}_t = \\left\\lbrace f \\in \\mathcal {F}~:~ \\sum _{i=1}^t \\ell (f,x_i) = \\min _{f \\in \\mathcal {F}} \\sum _{i=1}^t \\ell (f , x_i)\\right\\rbrace $ be the set of empirical minimizers at time $t$ .", "We use the blocking strategy $\\mathbf {Block}(\\lbrace x_{1},\\ldots ,x_t\\rbrace ,k) = (j, k-j) ~~~\\text{where}~~~ j = \\left\\lfloor \\min _{ f \\notin \\hat{\\mathcal {F}}_t } \\sum _{i=1}^t \\ell (f,x_i) - \\min _{f \\in \\hat{\\mathcal {F}}_t}\\sum _{i=1}^t \\ell (f,x_i) \\right\\rfloor $ which says that the size of the next block is given by the gap between empirical minimizer(s) and non-minimizers.", "The idea behind the proof and the blocking strategy is simple.", "If it happens at the start a new block that there is a large gap between the current leader and the next expert, then for the number of rounds approximately equal to this gap we can play a new block and not suffer any extra regret.", "Consider the relaxation (REF ) used for the Exponential Weights algorithm.", "Lemma 9 Suppose that there exists a single best expert $\\hat{f}_T = \\arg \\min _{f \\in \\mathcal {F}} \\sum _{t=1}^T \\ell (f,x_t),$ and that for some $k\\ge 1$ there exists $\\tau \\in [T]$ such that for all $t > \\tau $ and all $f \\ne \\hat{f}_T$ the partial cumulative loss $\\sum _{i=1}^t \\ell (f,x_i) - \\sum _{i=1}^t \\ell (\\hat{f}_T,x_i) \\ge k \\ .$ Then the regret of Algorithm  with the Exponential Weights relaxation, the blocking strategy (REF ) and the localization mentioned above is bounded as $\\mathbf {Reg}_T \\le 4 \\min \\left\\lbrace \\tau , \\sqrt{\\tau \\log (|\\mathcal {F}|)}\\right\\rbrace $ While we demonstrated a very simple example, the algorithm is adaptive more generally.", "Lemma REF considers the assumption that a single expert becomes a clear winner after $\\tau $ rounds, with margin of $k$ .", "Even when there is no clear winner throughout the game, we can still achieve low regret.", "For instance, this happens if only a few elements of $\\mathcal {F}$ have low cumulative loss throughout the game and the rest of $\\mathcal {F}$ suffers heavy loss.", "Then the algorithm adapts to the suboptimality and gives regret bound with the dominating term depending logarithmically only on the cardinality of the “good” choices in the set $\\mathcal {F}$ .", "Similar ideas appear in [9], and will be investigated in more generality in the full version of the paper." ], [ "Example: Adapting to the Data Norm", "Recall that the set $\\mathcal {F}^k(x_1,\\ldots ,x_t)$ is the subset of functions in $\\mathcal {F}$ that are possible empirical risk minimizers when we consider $x_1,\\ldots ,x_{t+k}$ for some $x_{t+1},\\ldots ,x_{t+k}$ that can occur in the future.", "Now, given history $x_1,\\ldots ,x_{t}$ and a possible future sequence $x_{t+1},\\ldots ,x_{t+k}$ , if $\\hat{f}_{t+k}$ is an ERM for $x_{1} ,\\ldots ,x_{t+k}$ and $\\hat{f}_{t}$ is an ERM for $x_{1} ,\\ldots ,x_{t}$ then $\\sum _{i=1}^t \\ell (\\hat{f}_{t+k},x_i) -\\sum _{i=1}^t \\ell (\\hat{f}_{t},x_i) & = \\sum _{i=1}^{t+k} \\ell (\\hat{f}_{t+k},x_i) -\\sum _{i=1}^{t+k} \\ell (\\hat{f}_{t},x_i) + \\sum _{i=t+1}^{t+k} \\ell (\\hat{f}_{t},x_i) - \\sum _{i=t+1}^{t+k} \\ell (\\hat{f}_{t+k},x_i)\\\\& \\le 0 + \\sup _{x_{t+1} ,\\ldots ,x_{t+k}}\\left\\lbrace \\sum _{i=t+1}^{t+k} \\ell (\\hat{f}_{t},x_i) - \\sum _{i=t+1}^{t+k} \\ell (\\hat{f}_{t+k},x_i) \\right\\rbrace \\ .$ Hence, we see that it suffices to consider localizations $\\mathcal {F}_{r(k;x_1,\\ldots ,x_t)} = \\left\\lbrace f \\in \\mathcal {F}~:~ \\sum _{i=1}^t \\ell (f,x_i) -\\sum _{i=1}^t \\ell (\\hat{f}_{t},x_i) \\le \\sup _{x_{t+1} ,\\ldots ,x_{t+k}}\\left\\lbrace \\sum _{i=t+1}^{t+k} \\ell (\\hat{f}_{t},x_i) - \\sum _{i=t+1}^{t+k} \\ell (f,x_i) \\right\\rbrace \\right\\rbrace \\ .$ If we consider online convex Lipschitz learning problems where $\\mathcal {F}= \\lbrace f : \\left\\Vert f\\right\\Vert \\le 1\\rbrace $ and loss is convex in $f$ and is such that $\\left\\Vert \\nabla \\ell (f,x)\\right\\Vert _* \\le 1$ in the dual norm $\\Vert \\cdot \\Vert _*$ , using the above argument we can use localization $\\mathcal {F}_{r(k;x_1,\\ldots ,x_t)} = \\left\\lbrace f \\in \\mathcal {F}~:~ \\sum _{i=1}^t \\ell (f,x_i) -\\sum _{i=1}^t \\ell (\\hat{f}_{t},x_i) \\le k \\left\\Vert f - \\hat{f}_t\\right\\Vert \\right\\rbrace \\ .$ Further, using Taylor approximation we can pass to the localization $\\mathcal {F}_{r(k;x_1,\\ldots ,x_t)} = \\left\\lbrace f \\in \\mathcal {F}~:~ \\frac{1}{2}\\left\\Vert f - \\hat{f}_t\\right\\Vert ^2_{x_1,\\ldots ,x_T} \\le k \\left\\Vert f - \\hat{f}_t\\right\\Vert \\right\\rbrace $ where $\\left\\Vert f\\right\\Vert ^2_{x_1,\\ldots ,x_T} = f^\\top H_t f$ , and $H_t$ is the Hessian of the function $g(f) = \\sum _{i=1}^t \\ell (f,x_i)$ .", "Notice that the earlier example where we adapt to strong convexity of the loss is a special case of the above localization where we lower bound the data-dependent norm (Hessian-based norm) by the $\\ell _2$ norm times the smallest eigenvalue.", "If for instance we are faced with $\\eta $ -exp-concave losses, such as the squared loss, the data-dependent norm can be again lower bounded by $\\left\\Vert f\\right\\Vert ^2_{x_1,\\ldots ,x_T} \\ge \\eta f^\\top \\left(\\sum _{i=1}^t \\nabla _i \\right) \\left(\\sum _{i=1}^t \\nabla _i \\right)^\\top f$ and so we can use localization based on outer products of sum of gradients.", "We then do not “pay” for those directions in which the adversary has not played, thus adapting to the effective dimension of the sequence of plays.", "In general, for online convex optimization problems one can use localizations given in Equations (REF ) or (REF ).", "The localization in Equation (REF ) is applicable even in the linear setting, and if it so happens that the adversary mainly plays in a one dimensional sub-space, then the algorithm automatically adapts to the adversary and yields faster rates for regret.", "As already mentioned, the example of adaptive gradient descent is a special case of localization in Equation (REF ).", "Of course, one needs to provide also an appropriate blocking strategy.", "A possible general blocking strategy could be : $\\mathbf {Block}(\\lbrace x_1,\\ldots ,x_t\\rbrace , k) = (j,k-j), ~~~\\mbox{where}~~~ j = \\underset{j \\in \\lbrace 0,\\ldots ,k\\rbrace }{\\mathrm {argmin}} \\ \\left\\lbrace \\mathbf {Rel}_{j}\\left(\\mathcal {F}_{r(x_1,\\ldots ,x_t)} \\right) + \\sup _{x_{t+1},\\ldots ,x_{t+j}}\\mathbf {Rel}_{k-j}\\left(\\mathcal {F}_{r(x_1,\\ldots ,x_{t+k})} \\right) \\right\\rbrace \\ .$ In the remainder of the paper, we develop new algorithms to show the versatility of our approach.", "One could try to argue that the introduction of the notion of a relaxation has not alleviated the burden of algorithm development, as we simply pushed the work into magically coming up with a relaxation.", "We would like to stress that this is not so.", "A key observation is that a relaxation does not appear out of thin air, but rather as an upper bound on the sequential Rademacher complexity.", "Thus, a general recipe is to start with a problem at hand and develop a sequence of upper bounds until one obtains a computationally feasible one, or until other desired properties are satisfied.", "Exactly for this purpose, the proofs in the appendix derive the relaxations rather than just present them as something given.", "Since one would follow the same upper bounding steps to prove an upper bound on the value of the game, the derivation of the relaxation and the proof of the regret bound go hand-in-hand.", "For this reason, we sometimes omit the explicit mention of a regret bound for the sake of conciseness: the algorithms enjoy the same regret bound as that obtained by the corresponding non-constructive proof of the upper bound." ], [ "Classification", "We start by considering the problem of supervised learning, where $\\mathcal {X}$ is the space of instances and $\\mathcal {Y}$ the space of responses (labels).", "There are two closely related protocols for the online interaction between the learner and Nature, so let us outline them.", "The “proper” version of supervised learning follows the protocol presented in Section : at time $t$ , the learner selects $f_t\\in \\mathcal {F}$ , Nature simultaneously selects $(x_t,y_t)\\in \\mathcal {X}\\times \\mathcal {Y}$ , and the learner suffers the loss $\\ell (f(x_t),y_t)$ .", "The “improper” version is as follows: at time $t$ , Nature chooses $x_t\\in \\mathcal {X}$ and presents it to the learner as “side information”, the learner then picks $\\hat{y}_t\\in \\mathcal {Y}$ and Nature simultaneously chooses $y_t\\in \\mathcal {Y}$ .", "In the improper version, the loss of the learner is $\\ell (\\hat{y}_t,y_t)$ , and it is easy to see that we may equivalently state this protocol as the learner choosing any function $f_t\\in \\mathcal {Y}^\\mathcal {X}$ (not necessarily in $\\mathcal {F}$ ), and Nature simultaneously choosing $(x_t,y_t)$ .", "We mostly focus on the “improper” version of supervised learning, as the distinction does not make any difference in any of the bounds.", "For the improper version of supervised learning, we may write the value in (REF ) as $\\mathcal {V}_T(\\mathcal {F}) = \\sup _{x_1\\in \\mathcal {X}} \\inf _{q_1 \\in \\Delta (\\mathcal {Y})} \\sup _{y_1 \\in \\mathcal {X}} \\underset{\\hat{y}_1 \\sim q_1}{\\mathbb {E}} \\ldots \\sup _{x_T\\in \\mathcal {X}}\\inf _{q_T \\in \\Delta (\\mathcal {Y})} \\sup _{y_T \\in \\mathcal {X}} \\underset{\\hat{y}_T \\sim q_T}{\\mathbb {E}}\\left[\\sum _{t=1}^T \\ell (\\hat{y}_t,y_t) - \\inf _{f \\in \\mathcal {F}} \\sum _{t=1}^T \\ell (f(x_t),y_t) \\right]$ and a relaxation $\\mathbf {Rel}_{}\\left( \\right)$ is admissible if for any $(x_1,y_1)\\ldots ,(x_T,y_T) \\in \\mathcal {X}\\times \\mathcal {Y}$ , $\\sup _{x\\in \\mathcal {X}}\\inf _{q \\in \\Delta (\\mathcal {Y})} \\sup _{y \\in \\mathcal {Y}} \\left\\lbrace \\underset{\\hat{y} \\sim q}{\\mathbb {E}} \\ell (\\hat{y},y) + \\mathbf {Rel}_{T}\\left(\\mathcal {F} | \\lbrace (x_{i},y_i)\\rbrace _{i=1}^t,(x,y) \\right)\\right\\rbrace \\le \\mathbf {Rel}_{T}\\left(\\mathcal {F} | \\lbrace (x_{i},y_i)\\rbrace _{i=1}^t \\right)$ and $ \\mathbf {Rel}_{T}\\left(\\mathcal {F} | \\lbrace (x_{i},y_i)\\rbrace _{i=1}^T \\right) \\ge - \\inf _{f \\in \\mathcal {F}} \\sum _{t=1}^T \\ell (f(x_t),y_t) .$ Let us now focus on binary label prediction, that is $\\mathcal {Y}=\\lbrace \\pm 1\\rbrace $ .", "In this case, the supremum over $y$ in (REF ) becomes a maximum over two values.", "Let us now take the absolute loss $\\ell (\\hat{y},y)=|\\hat{y}-y| = 1-\\hat{y}y$ .", "We can see that the optimal randomized strategy, given the side information $x$ , is given by (REF ) as $\\underset{q \\in \\Delta (\\mathcal {Y})}{\\mathrm {argmin}} \\ \\max \\left\\lbrace 1-q + \\mathbf {Rel}_{T}\\left(\\mathcal {F} | \\lbrace (x_{i},y_i)\\rbrace _{i=1}^t,(x,1) \\right), 1+q+\\mathbf {Rel}_{T}\\left(\\mathcal {F} | \\lbrace (x_{i},y_i)\\rbrace _{i=1}^t,(x,-1) \\right) \\right\\rbrace $ which is achieved by setting the two expressions equal to each other: $q = \\frac{1}{2}\\left\\lbrace \\mathbf {Rel}_{T}\\left(\\mathcal {F} | \\lbrace (x_{i},y_i)\\rbrace _{i=1}^t,(x,1) \\right)-\\mathbf {Rel}_{T}\\left(\\mathcal {F} | \\lbrace (x_{i},y_i)\\rbrace _{i=1}^t,(x,-1) \\right) \\right\\rbrace $ This result will be specialized in the latter sections for particular relaxations $\\mathbf {Rel}_{}\\left( \\right)$ and extended beyond absolute loss.", "We remark that the extension to $k$ -class prediction is immediate and involves taking a maximum over $k$ terms in (REF )." ], [ "Algorithms Based on the Littlestone's Dimension", "Consider the problem of binary prediction, as described above.", "Further, assume that $\\mathcal {F}$ has a finite Littlestone's dimension $\\mathrm {Ldim}(\\mathcal {F})$ [12], [6].", "Suppose the loss function is $\\ell (\\hat{y},y) = |\\hat{y}-y|$ , and consider the “mixed” conditional Rademacher complexity $\\sup _{\\mathbf {x}} \\mathbb {E}_{\\epsilon } \\sup _{f \\in \\mathcal {F}} \\left\\lbrace 2 \\sum _{i=1}^{T-t} \\epsilon _i f(\\mathbf {x}_i(\\epsilon )) - \\sum _{i=1}^{t} |f(x_i)-y_i|\\right\\rbrace $ as a possible relaxation.", "Observe that the above complexity is defined with the loss function removed (in a contraction-style argument [14]) in the terms involving the “future”, in contrast with the definition (REF ).", "The latter is defined with loss functions on both the “future” and the “past” terms.", "In general, if we can pass from the sequential Rademacher complexity over the loss class $\\ell (\\mathcal {F})$ to the sequential Rademacher complexity of the base class $\\mathcal {F}$ , we may attempt to do so step-by-step by using the “mixed” type of sequential Rademacher complexity as in (REF ).", "This idea shall be used several times later in this paper.", "The admissibility condition (REF ) with the conditional sequential Rademacher (REF ) as a relaxation would require us to upper bound $\\sup _{x_t}\\inf _{q_t \\in [-1,1]} \\max _{y_t \\in \\lbrace \\pm 1\\rbrace } \\left\\lbrace \\underset{\\hat{y}_t \\sim q_t}{\\mathbb {E}} |\\hat{y}_t-y_t| + \\sup _{\\mathbf {x}} \\mathbb {E}_{\\epsilon } \\sup _{f \\in \\mathcal {F}} \\left\\lbrace 2 \\sum _{i=1}^{T-t} \\epsilon _i f(\\mathbf {x}_i(\\epsilon )) - \\sum _{i=1}^{t} |f(x_i)-y_i|\\right\\rbrace \\right\\rbrace $ We observe that the supremum over $\\mathbf {x}$ is preventing us from obtaining a concise algorithm.", "We need to further “relax” this supremum, and the idea is to pass to a finite cover of $\\mathcal {F}$ on the given tree $\\mathbf {x}$ and then proceed as in the Exponential Weights example for a finite collection of experts.", "This leads to an upper bound on (REF ) and gives rise to algorithms similar in spirit to those developed in [6], but with more attractive computational properties and defined more concisely.", "Define the function $g(d,t) = \\sum _{i=0}^d {t \\atopwithdelims ()i}$ , which is shown in [14] to be the maximum size of an exact (zero) cover for a function class with the Littlestone's dimension $\\mathrm {Ldim}=d$ .", "Given $\\lbrace (x_1,y_t),\\ldots ,(x_t,y_t)\\rbrace $ and $\\sigma =(\\sigma _1,\\ldots ,\\sigma _t)\\in \\lbrace \\pm 1\\rbrace ^t$ , let $\\mathcal {F}_t(\\sigma ) = \\lbrace f\\in \\mathcal {F}: f(x_i)=\\sigma _i ~~\\forall i\\le t\\rbrace ,$ the subset of functions that agree with the signs given by $\\sigma $ on the “past” data and let $\\mathcal {F}|_{x_1,\\ldots ,x_t}\\triangleq \\mathcal {F}|_{x^t}\\triangleq \\lbrace (f(x_1),\\ldots ,f(x_t)): f\\in \\mathcal {F}\\rbrace $ be the projection of $\\mathcal {F}$ onto $x_1,\\ldots ,x_t$ .", "Denote $L_t(f) = \\sum _{i=1}^{t} |f(x_i)-y_i|$ and $L_t(\\sigma ) = \\sum _{i=1}^{t} |\\sigma _i-y_i|$ for $\\sigma \\in \\lbrace \\pm 1\\rbrace ^{t}$ .", "The following proposition gives a relaxation and two algorithms, both of which achieve the $O(\\sqrt{\\mathrm {Ldim}(\\mathcal {F})T\\log T})$ regret bound proved in [6], yet both different from the algorithm in that paper.", "Proposition 10 The relaxation $\\mathbf {Rel}_{T}\\left(\\mathcal {F} | (x^t,y^t) \\right) = \\frac{1}{\\lambda }\\log \\left( \\sum _{\\sigma \\in \\mathcal {F}|_{x^t}} g(\\mathrm {Ldim}(\\mathcal {F}_t(\\sigma )),T-t) \\exp \\left\\lbrace - \\lambda L_{t}(\\sigma ) \\right\\rbrace \\right) + 2\\lambda (T-t) \\ .$ is admissible and leads to an admissible algorithm $q_t (+1)= \\frac{ \\sum _{(\\sigma ,+1) \\in \\mathcal {F}|_{x^t}} g(\\mathrm {Ldim}(\\mathcal {F}_t(\\sigma ,+1)),T-t) \\exp \\left\\lbrace - \\lambda L_{t-1}(\\sigma ) \\right\\rbrace }{\\sum _{(\\sigma ,\\sigma _t) \\in \\mathcal {F}|_{x^t}} g(\\mathrm {Ldim}(\\mathcal {F}_t(\\sigma ,\\sigma _t)),T-t) \\exp \\left\\lbrace - \\lambda L_{t-1}(\\sigma ) \\right\\rbrace },$ with $q_t(-1) = 1- q_t (+1)$ .", "An alternative method for the same relaxation and the same regret guarantee is to predict the label $y_t$ according to a distribution with mean $q_t = \\frac{1}{2\\lambda }\\log \\frac{ \\sum _{(\\sigma ,\\sigma _t) \\in \\mathcal {F}|_{x^t}} g(\\mathrm {Ldim}(\\mathcal {F}_t(\\sigma ,\\sigma _t)),T-t) \\exp \\left\\lbrace - \\lambda L_{t-1}(\\sigma ) \\right\\rbrace \\exp \\left\\lbrace - \\lambda (1-\\sigma _t) \\right\\rbrace }{ \\sum _{(\\sigma ,\\sigma _t) \\in \\mathcal {F}|_{x^t}} g(\\mathrm {Ldim}(\\mathcal {F}_t(\\sigma ,\\sigma _t)),T-t) \\exp \\left\\lbrace - \\lambda L_{t-1}(\\sigma ) \\right\\rbrace \\exp \\left\\lbrace - \\lambda (1+\\sigma _t) \\right\\rbrace }$ There is a very close correspondence between the proof of Proposition REF and the proof of the combinatorial lemma of [14], the analogue of the Vapnik-Chervonenkis-Sauer-Shelah result.", "The two algorithms presented above show two alternatives: one through employing the properties of exponential weights, and the other is through the solution in (REF ).", "The merits of the two approaches remain to be explored.", "In particular, it appears that the method based on (REF ) can lead to some non-trivial new algorithms, distinct from the more common exponential weighting technique." ], [ "Randomized Algorithms and Follow the Perturbed Leader", "We now develop a class of admissible randomized methods that arise through sampling.", "Consider the objective $\\inf _{q \\in \\Delta (\\mathcal {F})} \\sup _{x \\in \\mathcal {X}} \\left\\lbrace \\mathbb {E}_{f \\sim q}\\left[ \\ell (f,x) \\right] + \\mathbf {Rel}_{T}\\left(\\mathcal {F} | x_1,\\ldots ,x_{t-1},x \\right)\\right\\rbrace $ given by a relaxation $\\mathbf {Rel}_{}\\left( \\right)$ .", "If $\\mathbf {Rel}_{}\\left( \\right)$ is the sequential (or classical) Rademacher complexity, it involves an expectation over sequences of coin flips, and this computation (coupled with optimization for each sequence) can be prohibitively expensive.", "More generally, $\\mathbf {Rel}_{}\\left( \\right)$ might involve an expectation over possible ways in which the future might be realized.", "In such cases, we may consider a rather simple “random playout” strategy: draw the random sequence and solve only one optimization problem for that random sequence.", "The ideas of random playout have been discussed previously in the literature for estimating the utility of a move in a game (see also [3]).", "In this section we show that, in fact, the random playout strategy has a solid basis: for the examples we consider, it satisfies admissibility.", "Furthermore, we show that Follow the Perturbed Leader is an example of such a randomized strategy.", "Let us informally describe the general idea, as the key steps might be hard to trace in the proofs.", "Suppose our objective is of the form $ S(q) = \\sup _x \\left( \\mathbb {E}_{f\\sim q} \\Psi (f,x) + \\mathbb {E}_{w\\sim p} \\Phi (w,x)\\right)$ for some functions $\\Psi $ and $\\Phi $ , and $q$ a mixed strategy.", "We have in mind the situation where the first term is the instantaneous loss at the present round, and the second term is the expected cost for the future.", "Consider a randomized strategy $\\tilde{q}$ which is defined by first randomly drawing $w\\sim p$ and then computing $f(w) \\triangleq \\underset{f}{\\mathrm {argmin}} \\ \\sup _x(\\Psi (f,x) + \\Phi (w,x)) $ for the random draw $w$ .", "We then verify that $S(\\tilde{q}) &= \\sup _x\\left( \\mathbb {E}_{f\\sim \\tilde{q}} \\Psi (f,x) + \\mathbb {E}_{w\\sim p} \\Phi (w,x)\\right)= \\sup _x\\left( \\mathbb {E}_{w\\sim p} \\Psi (f(w),x) + \\mathbb {E}_{w\\sim p} \\Phi (w,x)\\right) \\\\&~~~~~\\le \\mathbb {E}_{w\\sim p} \\sup _x\\left( \\Psi (f(w),x) + \\Phi (w,x)\\right)= \\mathbb {E}_{w\\sim p} \\inf _{f} \\sup _x\\left( \\Psi (f,x) + \\Phi (w,x)\\right) \\ .$ What makes the proof of admissibility possible is that the infimum in the last expression is inside the expectation over $w$ rather than outside.", "We can then appeal to the minimax theorem to prove admissibility.", "In our examples, $\\Psi $ is the loss at round $t$ and $\\Phi $ is the relaxation term, such as the sequential Rademacher complexity.", "In Section REF we show that, if we can compute the “worst” tree $\\mathbf {x}$ , we can randomly draw a path and use it for our randomized strategy.", "Note that the worst-case trees are closely related to random walks of maximal variation, and our method thus points to an intriguing connection between regret minimization and random walks (see also [3], [13] for related ideas).", "Interestingly, in many learning problems it turns out that the sequential Rademacher complexity and the classical Rademacher complexity are within a constant factor of each other.", "In such cases, the function $\\Phi $ does not involve the supremum over a tree, and the randomized method only needs to draw a sequence of coin flips and compute a solution to an optimization problem slightly more complicated than ERM.", "In particular, the sequential and classical Rademacher complexities can be related for linear classes in finite-dimensional spaces.", "Online linear optimization is then a natural application of the randomized method we propose.", "Indeed, we show that Follow the Perturbed Leader (FPL) algorithm [10] arises in this way.", "We note that FPL has been previously considered as a rather unorthodox algorithm providing some kind of regularization via randomization.", "Our analysis shows that it arises through a natural relaxation based on the sequential (and thus the classical) Rademacher complexity, coupled with the random playout idea.", "As a new algorithmic contribution, we provide a version of the FPL algorithm for the case of the decision sets being $\\ell _2$ balls, with a regret bound that is independent of the dimension.", "We also provide an FPL-style method for the combination of $\\ell _1$ and $\\ell _\\infty $ balls.", "To the best of our knowledge, these results are novel.", "In the later sections, we provide a novel randomized method for the Trace Norm Completion problem, and a novel randomized method for the setting of static experts and transductive learning.", "In general, the techniques we develop might in future provide computationally feasible randomized algorithms where deterministic ones are too computationally demanding." ], [ "When Sequential and Classical Rademacher Complexities are Related", "The assumption below implies that the sequential Rademacher complexity and the classical Rademacher complexity are within constant factor $C$ of each other.", "We will later verify that this assumption holds in the examples we consider.", "Assumption 1 There exists a distribution $D \\in \\Delta (\\mathcal {X})$ and constant $C \\ge 2$ such that for any $t \\in [T]$ and given any $x_1,\\ldots ,x_{t-1}, x_{t+1},\\ldots ,x_T \\in \\mathcal {X}$ and any $\\epsilon _{t+1},\\ldots ,\\epsilon _{T} \\in \\lbrace \\pm 1\\rbrace $ , $\\sup _{p \\in \\Delta (\\mathcal {X})} & \\underset{x_t \\sim p}{\\mathbb {E}}\\sup _{f \\in \\mathcal {F}} \\left[ \\ C \\sum _{i=t+1}^{T} \\epsilon _i \\ell (f,x_i) - L_{t-1}(f) + \\mathbb {E}_{x \\sim p}\\left[ \\ell (f,x) \\right] - \\ell (f,x_t) \\right] \\\\& ~~~~~~~\\le \\underset{\\epsilon _{t}, x_{t} \\sim D }{\\mathbb {E}}\\sup _{f \\in \\mathcal {F}} \\left[ \\ C \\sum _{i=t}^{T} \\epsilon _i \\ell (f,x_i) - L_{t-1}(f) \\right]$ where $\\epsilon _t$ is an independent Rademacher random variable and $L_{t-1}(f)=\\sum _{i=1}^{t-1} \\ell (f,x_i)$ .", "Under the above assumption one can use the following relaxation $\\mathbf {Rel}_{T}\\left(\\mathcal {F} | x_1,\\ldots ,x_t \\right) = \\underset{x_{t+1},\\ldots x_T \\sim D}{\\mathbb {E}}\\mathbb {E}_{\\epsilon } \\sup _{f \\in \\mathcal {F}} \\left[C \\sum _{i=t+1}^T \\epsilon _i \\ell (f,x_i) - \\sum _{i=1}^t \\ell (f,x_i) \\right]$ which is a partially symmetrized version of the classical Rademacher averages.", "The proof of admissibility for the randomized methods based on this relaxation is quite curious – the forecaster can be seen as mimicking the sequential Rademacher complexity by sampling from the “equivalently bad” classical Rademacher complexity under the specific distribution $D$ given by the above assumption.", "Lemma 11 Under the Assumption REF , the relaxation in Eq.", "(REF ) is admissible and a randomized strategy that ensures admissibility is given by: at time $t$ , draw $x_{t+1},\\ldots ,x_{T} \\sim D$ and Rademacher random variables $\\epsilon =(\\epsilon _{t+1},\\ldots ,\\epsilon _T)$ and then : In the case the loss $\\ell $ is convex in its first argument and the set $\\mathcal {F}$ is convex and compact, define $f_t = \\underset{g \\in \\mathcal {F}}{\\mathrm {argmin}} \\ \\sup _{x \\in \\mathcal {X}} \\left\\lbrace \\ell (g,x) + \\sup _{f \\in \\mathcal {F}} \\left\\lbrace C \\sum _{i=t+1}^T \\epsilon _i \\ell (f,x_i) - \\sum _{i=1}^{t-1} \\ell (f,x_i) - \\ell (f,x) \\right\\rbrace \\right\\rbrace $ In the case of non-convex loss, sample $f_t$ from the distribution $\\hat{q}_t = \\underset{\\hat{q} \\in \\Delta (\\mathcal {F})}{\\mathrm {argmin}} \\ \\sup _{x \\in \\mathcal {X}} \\left\\lbrace \\mathbb {E}_{f \\sim \\hat{q}}\\left[ \\ell (f,x) \\right] + \\sup _{f \\in \\mathcal {F}} \\left\\lbrace C \\sum _{i=t+1}^T \\epsilon _i \\ell (f,x_i) - \\sum _{i=1}^{t-1} \\ell (f,x_i) - \\ell (f,x) \\right\\rbrace \\right\\rbrace $ The expected regret for the method is bounded by the classical Rademacher complexity: $\\mathbb {E}\\left[ \\mathbf {Reg}_T \\right] \\le C\\ \\mathbb {E}_{x_{1:T} \\sim D} \\mathbb {E}_{\\epsilon }\\left[ \\sup _{f \\in \\mathcal {F}} \\sum _{t=1}^T \\epsilon _t \\ell (f,x_t) \\right] ,$ Of particular interest are the settings of static experts and transductive learning, which we consider in Section .", "In the transductive case, the $x_t$ 's are pre-specified before the game, and in the static expert case – effectively absent.", "In these cases, as we show below, there is no explicit distribution $D$ and we only need to sample the random signs $\\epsilon $ 's.", "We easily see that in these cases, the expected regret bound is simply two times the transductive Rademacher complexity." ], [ "Linear Loss", "The idea of sampling from a fixed distribution is particularly appealing in the case of linear loss, $\\ell (f,x) = \\left\\langle f,x \\right\\rangle $ .", "Suppose $\\mathcal {X}$ is a unit ball in some norm $\\Vert \\cdot \\Vert $ in a vector space $B$ , and $\\mathcal {F}$ is a unit ball in the dual norm $\\Vert \\cdot \\Vert _*$ .", "Assumption REF then becomes Assumption 2 There exists a distribution $D \\in \\Delta (\\mathcal {X})$ and constant $C \\ge 2$ such that for any $t \\in [T]$ and given any $x_1,\\ldots ,x_{t-1}, x_{t+1},\\ldots ,x_T \\in \\mathcal {X}$ and any $\\epsilon _{t+1},\\ldots ,\\epsilon _{T} \\in \\lbrace \\pm 1\\rbrace $ , $\\sup _{p \\in \\Delta (\\mathcal {X})} \\underset{x_t \\sim p}{\\mathbb {E}} \\left\\Vert C \\sum _{i=t+1}^{T} \\epsilon _i x_i - \\sum _{i=1}^{t-1} x_i + \\underset{x \\sim p}{\\mathbb {E}} [x] - x_t \\right\\Vert \\le \\underset{\\epsilon _{t}, x_{t} \\sim D }{\\mathbb {E}}\\left\\Vert C \\sum _{i=t}^{T} \\epsilon _i x_i - \\sum _{i=1}^{t-1} x_i \\right\\Vert $ For (REF ) to hold it is enough to ensure that $\\sup _{p \\in \\Delta (\\mathcal {X})} \\underset{x_t \\sim p}{\\mathbb {E}} \\left\\Vert w + \\underset{x \\sim p}{\\mathbb {E}} [x] - x_t \\right\\Vert \\le \\underset{\\epsilon _{t}, x_{t} \\sim D }{\\mathbb {E}}\\left\\Vert w+ C\\epsilon _t x_t \\right\\Vert $ for any $w\\in B$ .", "At round $t$ , the generic algorithm specified by Lemma REF draws fresh Rademacher random variables $\\epsilon $ and $x_{t+1},\\ldots ,x_T \\sim D$ and picks $f_t = \\underset{f \\in \\mathcal {F}}{\\mathrm {argmin}} \\ \\sup _{x \\in \\mathcal {X}} \\left\\lbrace \\left<f,x\\right> + \\left\\Vert C \\sum _{i=t+1}^T \\epsilon _i x_i - \\sum _{i=1}^{t-1} x_i - x \\right\\Vert \\right\\rbrace $ We now look at specific examples of $\\ell _2/\\ell _2$ and $\\ell _1/\\ell _\\infty $ cases and provide closed form solution of the randomized algorithms.", "Example : $\\ell _1/\\ell _\\infty $ Follow the Perturbed Leader: Here, we consider the setting similar to that in [10].", "Let $\\mathcal {F}\\subset \\mathbb {R}^N$ be the $\\ell _1$ unit ball and $\\mathcal {X}$ the (dual) $\\ell _{\\infty }$ unit ball in $\\mathbb {R}^N$ .", "In [10], $\\mathcal {F}$ is the probability simplex and $\\mathcal {X}= [0,1]^N$ but these are subsumed by the $\\ell _1/\\ell _\\infty $ case.", "We claim that: Lemma 12 Assumption REF is satisfied with a distribution $D$ that is uniform on the vertices of the cube $\\lbrace \\pm 1\\rbrace ^N$ and $C=6$ .", "In fact, one can pick any symmetric distribution $D$ on the real line and use $D^N$ for the perturbation.", "Assumption REF is then satisfied, as we show in the following lemma.", "Lemma 13 If $D$ is any symmetric distribution over the real line, then Assumption REF is satisfied by using the product distribution $D^N$ .", "The constant $C$ required is any $ C \\ge 6/\\mathbb {E}_{x \\sim D} |x|$ .", "The above lemma is especially attractive when used with standard normal distribution because in that case as sum of normal random variables is again normal.", "Hence, instead of drawing $x_{t+1},\\ldots ,x_{T} \\sim N(0,1)$ on round $t$ , one can simply draw just one vector $X_t \\sim N(0,\\sqrt{T-t})$ and use it for perturbation.", "In this case constant $C$ is bounded by 8.", "While we have provided simple distributions to use for perturbation, the form of update in Equation (REF ) is not in a convenient form.", "The following lemma shows a simple Follow the Perturbed Leader type algorithm with the associated regret bound.", "Lemma 14 Suppose $\\mathcal {F}$ is the $\\ell ^N_1$ unit ball and $\\mathcal {X}$ is the dual $\\ell _\\infty ^N$ unit ball, and let $D$ be any symmetric distribution.", "Consider the randomized algorithm that at each round $t$ freshly draws Rademacher random variables $\\epsilon _{t+1} , \\ldots , \\epsilon _T$ and freshly draws $x_{t+1},\\ldots ,x_T \\sim D^N$ (each co-ordinate drawn independently from $D$ ) and picks $f_t = \\underset{f \\in \\mathcal {F}}{\\mathrm {argmin}} \\ \\left<f,\\sum _{i=1}^{t-1} x_i - C \\sum _{i=t+1}^T \\epsilon _i x_i \\right>$ where $C = 6/\\mathbb {E}_{x \\sim D}\\left[ |x| \\right]$ .", "The randomized algorithm enjoys a bound on the expected regret given by $\\mathbb {E}\\left[ \\mathbf {Reg}_T \\right] \\le C\\ \\underset{x_{1:T} \\sim D^N}{\\mathbb {E}}\\mathbb {E}_{\\epsilon } \\left\\Vert \\sum _{t=1}^T \\epsilon _t x_t\\right\\Vert _\\infty + 4 \\sum _{t=1}^T \\mathbf {P}_{y_{t+1:T} \\sim D}\\left( C \\left| \\sum _{i=t+1}^T y_i\\right| \\le 4 \\right)$ Notice that for $D$ being the $\\lbrace \\pm 1\\rbrace $ coin flips or standard normal distribution, the probability $\\mathbf {P}_{y_{t+1},\\ldots ,y_T \\sim D}\\left( C \\left| \\sum _{i=t+1}^T y_i\\right| \\le 4 \\right)$ is exponentially small in $T-t$ and so $\\sum _{t=1}^T \\mathbf {P}_{y_{t+1},\\ldots ,y_T \\sim D}\\left( C \\left| \\sum _{i=t+1}^T y_i\\right| \\le 4 \\right)$ is bounded by a constant.", "For these cases, we have $\\mathbb {E}\\left[ \\mathbf {Reg}_T \\right] \\le O\\left( \\underset{x_{1:T} \\sim D^N}{\\mathbb {E}}\\mathbb {E}_{\\epsilon } \\left\\Vert \\sum _{t=1}^T \\epsilon _t x_t\\right\\Vert _\\infty \\right) = O\\left(\\sqrt{T \\log N}\\right)$ This yields the logarithmic dependence on the dimension, matching that of the Exponential Weights algorithm.", "Example : $\\ell _2/\\ell _2$ Follow the Perturbed Leader: We now consider the case when $\\mathcal {F}$ and $\\mathcal {X}$ are both the unit $\\ell _2$ ball.", "We can use as perturbation the uniform distribution on the surface of unit sphere, as the following lemma shows.", "This result was already hinted at in [2], as the random draw from the unit sphere is likely to produce an orthogonal direction, yielding a strategy close to optimal.", "However, we do not require dimensionality to be high for the result to hold.", "Lemma 15 Let $\\mathcal {X}$ and $\\mathcal {F}$ be unit balls in Euclidean norm.", "Then Assumption REF is satisfied with a uniform distribution $D$ on the surface of the unit sphere with constant $C=4\\sqrt{2}$ .", "Again as in the previous example the form of update in Equation (REF ) is not in a convenient form and this is addressed in the following lemma.", "Lemma 16 Let $\\mathcal {X}$ and $\\mathcal {F}$ be unit balls in Euclidean norm, and $D$ be the uniform distribution on the surface of the unit sphere.", "Consider the randomized algorithm that at each round (say round $t$ ) freshly draws $x_{t+1},\\ldots ,x_T \\sim D$ and picks $f_t = \\frac{- \\sum _{i=1}^{t-1} x_i + C \\sum _{i=t+1}^T x_i }{\\sqrt{\\left\\Vert - \\sum _{i=1}^{t-1} x_i + C \\sum _{i=t+1}^T \\epsilon _i x_i \\right\\Vert _2^2 + 1}}$ where $C=4 \\sqrt{2}$ .", "The randomized algorithm enjoys a bound on the expected regret given by $\\mathbb {E}\\left[ \\mathbf {Reg}_T \\right] \\le C\\ \\mathbb {E}_{x_1,\\ldots ,x_T \\sim D} \\left\\Vert \\sum _{t=1}^T x_t\\right\\Vert _2 \\le 4 \\sqrt{2 T}$ Importantly, the bound does not depend on the dimensionality of the space.", "To the best of our knowledge, this is the first such result for Follow the Perturbed Leader style algorithms.", "Remark 1 The FPL methods developed in [10], [7] assume that the adversary is oblivious.", "With this simplification, the algorithms can reuse the same random perturbation drawn at the beginning of the game.", "It is then argued in [7] that the methods also work for non-oblivious opponents since the FPL strategy is fully determined by the outcomes played by the adversary [7].", "In contrast, our proofs directly deal with the adaptive adversary." ], [ "Supervised Learning", "For completeness, let us state a version of Assumption REF for the case of supervised learning.", "That is, the side information $x_t$ is presented to the learner, who then picks $\\hat{y}_t$ and observes the outcome $y_t$ .", "Assumption 3 There exists a distribution $D \\in \\Delta (\\mathcal {X}\\times \\mathcal {Y})$ and constant $C \\ge 2$ such that for any $t \\in [T]$ and given any $(x_1,y_1),\\ldots ,(x_{t-1},y_{t-1}), (x_{t+1},y_{t+1}),\\ldots ,(x_T,y_T) \\in \\mathcal {X}\\times \\mathcal {Y}$ and any $\\epsilon _{t+1},\\ldots ,\\epsilon _{T} \\in \\lbrace \\pm 1\\rbrace $ , $\\sup _{x_t \\in \\mathcal {X}}\\sup _{p_t \\in \\Delta (\\mathcal {Y})} & \\underset{y_t \\sim p_t}{\\mathbb {E}} \\sup _{f \\in \\mathcal {F}} \\left[ \\ C \\sum _{i=t+1}^{T} \\epsilon _i \\ell (f(x_i),y_i) -L_{t-1}(f) + \\mathbb {E}_{y \\sim p_t}\\left[ \\ell (f(x_t),y) \\right] - \\ell (f(x_t),y_t) \\right] \\\\& \\le \\underset{\\epsilon _{t}, (x_{t},y_t) \\sim D }{\\mathbb {E}}\\sup _{f \\in \\mathcal {F}} \\left[ \\ C \\sum _{i=t}^{T} \\epsilon _i \\ell (f(x_i),y_i) - L_{t-1}(f) \\right],$ where $\\epsilon _t$ is an independent Rademacher random variable and $L_{t-1}(f)=\\sum _{i=1}^{t-1} \\ell (f(x_i),y_i)$ .", "Under the Assumption REF , we can use the following relaxation: $\\mathbf {Rel}_{T}\\left(\\mathcal {F} | (x_1,y_1),\\ldots ,(x_t,y_t) \\right) = \\underset{\\underset{(x_{t+1},y_{t+1}),\\ldots (x_T,y_T) \\sim D}{\\epsilon _{t+1:T}}}{\\mathbb {E}} \\sup _{f \\in \\mathcal {F}} \\left[ C \\sum _{i=t+1}^T \\epsilon _i \\ell (f(x_i),y_i) - \\sum _{i=1}^t \\ell (f(x_i),y_i) \\right]$ Lemma 17 Under the Assumption REF , the relaxation in Eq.", "(REF ) is admissible and a randomized strategy that ensures admissibility is given by: at time $t$ , draw $(x_{t+1},y_{t+1}),\\ldots ,(x_{T},y_T) \\sim D$ and Rademacher random variables $\\epsilon _{t+1},\\ldots ,\\epsilon _T$ and then : In the case the loss $\\ell $ is convex in its first argument, define $\\hat{y}_t = \\underset{\\hat{y} \\in [-B,B]}{\\mathrm {argmin}} \\ \\sup _{y_t \\in \\mathcal {Y}} \\left\\lbrace \\ell (\\hat{y},y_t)+ \\sup _{f \\in \\mathcal {F}} \\left[ C \\sum _{i=t+1}^T \\epsilon _i \\ell (f(x_i),y_i) - \\sum _{i=1}^{t} \\ell (f(x_i),y_i) \\right] \\right\\rbrace $ and In the case of non-convex loss, pick $\\hat{y}_t$ from the distribution $\\hat{q}_t = \\underset{\\hat{q} \\in \\Delta ([-B,B])}{\\mathrm {argmin}} \\ \\sup _{y_t \\in \\mathcal {Y}} \\left\\lbrace \\mathbb {E}_{\\hat{y} \\sim \\hat{q}}\\left[ \\ell (\\hat{y},y_t) \\right]+ \\sup _{f \\in \\mathcal {F}} \\left[ C \\sum _{i=t+1}^T \\epsilon _i \\ell (f(x_i),y_i) - \\sum _{i=1}^{t} \\ell (f(x_i),y_i) \\right] \\right\\rbrace $ The expected regret bound of the method (in both cases) is $\\mathbb {E}\\left[ \\mathbf {Reg}_T \\right] \\le C\\ \\underset{(x_1,y_1),\\ldots ,(x_T,y_T) \\sim D}{\\mathbb {E}} \\mathbb {E}_{\\epsilon }\\left[ \\sup _{f \\in \\mathcal {F}} \\sum _{t=1}^T \\epsilon _t \\ell (f(x_t),y_t) \\right]$" ], [ "Random Walks with Trees", "We can also define randomized algorithms without the assumption that the classical and the sequential Rademacher complexities are close.", "Instead, we assume that we have a black-box access to a procedure that on round $t$ returns the “worst-case” tree $\\mathbf {x}^t$ of depth $T-t$ .", "Lemma 18 Given any $x_1,\\ldots ,x_{t-1}$ let $\\mathbf {x}^t \\triangleq \\underset{\\mathbf {x}}{\\mathrm {argmax}} \\ \\mathbb {E}_{\\epsilon } \\sup _{f \\in \\mathcal {F}} \\left[ 2 \\sum _{i=t+1}^T \\epsilon _i \\ell (f,\\mathbf {x}_i(\\epsilon )) - \\sum _{i=1}^t \\ell (f,x_i) \\right] \\ .$ Consider the randomized strategy where at round $t$ we first draw $\\epsilon _{t+1},\\ldots ,\\epsilon _T$ uniformly at random and then further draw our move $f_t$ according to the distribution $q_t(\\epsilon ) = \\underset{q \\in \\Delta (\\mathcal {F})}{\\mathrm {argmin}} \\ \\sup _{x_t}\\left\\lbrace \\mathbb {E}_{f_t \\sim q}\\left[ \\ell (f_t,x_t) \\right] + \\sup _{f \\in \\mathcal {F}} \\left[ 2 \\sum _{i=t+1}^T \\epsilon _i \\ell (f,\\mathbf {x}^t_i(\\epsilon )) - \\sum _{i=1}^t \\ell (f,x_i) \\right] \\right\\rbrace $ The expected regret of this randomized strategy is bounded by sequential Rademacher complexity: $\\mathbb {E}\\left[ \\mathbf {Reg}_T \\right] \\le \\mathfrak {R}_T(\\mathcal {F}) \\ .$ Thus, if for any given history $x_1,\\ldots ,x_{t-1}$ we can compute $\\mathbf {x}^t$ in (REF ), or even just draw directly a random path $\\mathbf {x}^t_1(\\epsilon ),\\ldots ,\\mathbf {x}^t_{T-t}(\\epsilon )$ on each round, then we obtain a randomized strategy that in expectation can guarantee a regret bound equal to sequential Rademacher complexity.", "Also notice that whenever the optimal strategy in (REF ) is deterministic (e.g.", "in the online convex optimization scenario), one does not need the double randomization.", "Instead, in such situations one can directly draw $\\epsilon _1,\\ldots ,\\epsilon _{T-t}$ and use $f_t(\\epsilon ) = \\underset{f_t \\in \\mathcal {F}}{\\mathrm {argmin}} \\ \\sup _{x_t}\\left\\lbrace \\ell (f_t,x_t) + \\sup _{f \\in \\mathcal {F}}\\left\\lbrace 2 \\sum _{i=t+1}^T \\epsilon _i \\ell (f,\\mathbf {x}^t_i(\\epsilon )) - \\sum _{i=1}^t \\ell (f,x_i) \\right\\rbrace \\right\\rbrace $" ], [ "Static Experts with Convex Losses and Transductive Online Learning", "We show how to recover a variant of the $R^2$ forecaster of [8], for static experts and transductive online learning.", "At each round, the learner makes a prediction $q_t\\in [-1,1]$ , observes the outcome $y_t\\in [-1,1]$ , and suffers convex $L$ -Lipschitz loss $\\ell (q_t,y_t)$ .", "Regret is defined as the difference between learner's cumulative loss and $\\inf _{f\\in F} \\sum _{t=1}^T \\ell (f[t],y_t)$ , where $F\\subset [-1,1]^T$ can be seen as a set of static experts.", "The transductive setting is equivalent to this: the sequence of $x_t$ 's is known before the game starts, and hence the effective function class is once again a subset of $[-1,1]^T$ .", "It turns out that in the static experts case, sequential Rademacher complexity boils down to the classical Rademacher complexity (see [16]), and thus the relaxation in (REF ) can be taken to be the classical, rather than sequential, Rademacher averages.", "This is also the reason that an efficient implementation by sampling is possible.", "Furthermore, for the absolute loss, the factor of 2 that appears in the sequential Rademacher complexity is not needed.", "For general convex loss, one possible relaxation is just a conditional version of the classical Rademacher averages: $\\mathbf {Rel}_{T}\\left(\\mathcal {F} | y_{1},\\ldots ,y_t \\right) = \\mathbb {E}_{\\epsilon _{t+1:T}} \\sup _{f\\in F} \\left[ 2L\\sum _{s=t+1}^T \\epsilon _s f[s] - L_t(f) \\right]$ where $L_t(f)=\\sum _{s=1}^t \\ell (f[s],y_s)$ .", "This relaxation can be shown to be admissible.", "First, consider the case of absolute loss $\\ell (q_t,y_t)=|q_t-y_t|$ and binary-valued outcomes $y_t\\in \\lbrace \\pm 1\\rbrace $ .", "In this case, the solution in (REF ) yields the algorithm $q_t &= \\frac{1}{2}\\mathbb {E}_{\\epsilon _{t+1:T}}\\left[ \\sup _{f\\in F} \\left( \\sum _{s=t+1}^T \\epsilon _s f[s] - L_{t-1}(f) + f[t] \\right) - \\sup _{f\\in F} \\left( \\sum _{s=t+1}^T \\epsilon _s f[s] - L_{t-1}(f) - f[t] \\right)\\right]$ which corresponds to the well-known minimax optimal forecaster for static experts with absolute loss [7].", "Plugging in this value of $q_t$ into Eq.", "(REF ) proves admissibility, and thus the regret guarantee of this method is equal to the classical Rademacher complexity.", "We now derive two variants of the $R^2$ forecaster for the more general case of $L$ -Lipschitz loss and $y_t\\in [-1,1]$ ." ], [ "First Alternative : ", "If (REF ) is used as a relaxation, the calculation of prediction $\\hat{y}_t$ involves a supremum over $f \\in F$ with (potentially nonlinear) loss functions of instances seen so far.", "In some cases this optimization might be hard and it might be preferable if the supremum only involves terms linear in $f$ .", "This is the idea behind he first method we present.", "To this end we start by noting that by convexity $\\sum _{t=1}^T \\ell (\\hat{y}_t,y_t) - \\inf _{f \\in \\mathcal {F}} \\sum _{t=1}^T \\ell (f(x_t),y_t) \\le \\sum _{t=1}^T \\partial \\ell (\\hat{y}_t,y_t) \\cdot \\hat{y}_t - \\inf _{f \\in \\mathcal {F}} \\sum _{t=1}^T \\partial \\ell (\\hat{y}_t,y_t) \\cdot f[t]$ Now given the above, one can consider an alternative online learning problem which, if we solve, also solves the original problem.", "That is, consider the online learning problem with the new loss $\\ell ^{\\prime }(\\hat{y},r) = r \\cdot \\hat{y}$ In this alternative game, we first pick prediction $\\hat{y}_t$ (deterministically), next the adversary picks $r_t$ (corresponding to $r_t = \\partial \\ell (\\hat{y}_t,y_t)$ for choice of $y_t$ picked by adversary).", "Now note that $\\ell ^{\\prime }$ is indeed convex in its first argument and is $L$ Lipschitz because $|\\partial \\ell (\\hat{y}_t,y_t)| \\le L$ .", "This is a one dimensional convex learning game where we pick $\\hat{y}_t$ and regret is given by $\\mathbf {Reg}_T = \\sum _{t=1}^T \\partial \\ell (\\hat{y}_t,y_t) \\cdot \\hat{y}_t - \\inf _{f \\in F} \\sum _{t=1}^T \\partial \\ell (\\hat{y}_t,y_t) \\cdot f[t]$ One can consider the relaxation $\\mathbf {Rel}_{T}\\left(\\mathcal {F} | \\partial \\ell (\\hat{y}_1,y_1) , \\ldots , \\partial \\ell (\\hat{y}_t,y_t) \\right) = \\mathbb {E}_{\\epsilon _{t+1:T}} \\sup _{f \\in F} \\left[ 2 L \\sum _{i=t+1}^T \\epsilon _i f[t] - \\sum _{i=1}^t \\partial \\ell (\\hat{y}_i,y_i) \\cdot f[i] \\right]$ as a linearized form of (REF ).", "At round $t$ , the prediction of the algorithm is then $\\hat{y}_t = \\mathbb {E}_{\\epsilon }\\left[ \\sup _{f \\in F} \\left\\lbrace \\sum _{i=t+1}^{T} \\epsilon _i f[i] - \\frac{1}{2L} \\sum _{i=1}^{t-1} \\partial \\ell (\\hat{y}_i,y_i) f[i] + \\frac{1}{2} f[t] \\right\\rbrace - \\sup _{f \\in F} \\left\\lbrace \\sum _{i=t+1}^{T} \\epsilon _i f[i] - \\frac{1}{2L} \\sum _{i=1}^{t-1} \\partial \\ell (\\hat{y}_i,y_i) f[i] - \\frac{1}{2} f[t] \\right\\rbrace \\right]$ Lemma 19 The relaxation in Equation (REF ) is admissible with respect to the prediction strategy specified in Equation (REF ).", "Further the regret of the strategy is bounded as $\\mathbf {Reg}_T \\le 2 L\\ \\mathbb {E}_{\\epsilon }\\left[ \\sup _{f \\in F} \\sum _{t=1}^T \\epsilon _t f[t] \\right]$ The presented algorithm is similar in principle to $R^2$ , with the main difference that $R^2$ computes the infima over a sum of absolute losses, while here we have a more manageable linearized objective.", "Note that while we need to evaluate the expectation over $\\epsilon $ 's on each round, we can estimate $\\hat{y}_t$ by sampling $\\epsilon $ 's and using McDiarmid's inequality to argue that, with enough draws, our estimate is close to $\\hat{y}_t$ with high probability.", "What is interesting, we can develop a randomized method that only draws one sequence of $\\epsilon $ 's per step, as shown next." ], [ "Second Alternative : ", "Consider the non-linearized relaxation $\\mathbf {Rel}_{T}\\left(\\mathcal {F} | y_1,\\ldots ,y_t \\right) = \\mathbb {E}_{\\epsilon }\\left[ \\sup _{f \\in F} \\ 2 L \\sum _{i=t+1}^{T} \\epsilon _i f[i] - \\sum _{i=1}^{t} \\ell (f[i],y_i) \\right]$ already given in (REF ).", "We now present a randomized method based on the ideas of Section : at round $t$ we first draw $\\epsilon _{t+1},\\ldots ,\\epsilon _T$ and predict $\\hat{y}_t(\\epsilon ) & = \\left(\\inf _{f \\in F} \\left\\lbrace - \\sum _{i=t+1}^{T} \\epsilon _i f[i] + \\frac{1}{2 L} \\sum _{i=1}^{t-1} \\ell (f[i],y_i) + \\frac{1}{2} f[t] \\right\\rbrace - \\inf _{f \\in F} \\left\\lbrace - \\sum _{i=t+1}^{T} \\epsilon _i f[i] + \\frac{1}{2 L}\\sum _{i=1}^{t-1} \\ell (f[i],y_i) - \\frac{1}{2} f[t] \\right\\rbrace \\right)$ We show that this predictor in expectation enjoys regret bound of the transductive Rademacher complexity.", "More specifically we have the following lemma.", "Lemma 20 The relaxation specified in Equation (REF ) is admissible w.r.t.", "the randomized prediction strategy specified in Equation (REF ).", "Further the expected regret of the randomized strategy is bounded as $\\mathbb {E}\\left[ \\mathbf {Reg}_T \\right] \\le 2 L\\ \\mathbb {E}_{\\epsilon }\\left[ \\sup _{f \\in F} \\sum _{t=1}^T \\epsilon _t f[t] \\right]$ In the next section, we employ both alternatives to develop novel algorithms for matrix completion." ], [ "Matrix Completion", "Consider the problem of predicting unknown entries in a matrix (as in collaborative filtering).", "We focus here on an online formulation, where at each round $t$ the adversary picks an entry in an $m\\times n$ matrix and a value $y_t$ for that entry (we shall assume without loss of generality that $n\\ge m$ ).", "The learner then chooses a predicted value $\\hat{y}_t$ , and suffers loss $\\ell (y_t,\\hat{y}_t)$ , which we shall assume to be $\\rho $ -Lipschitz.", "We define our regret with respect to the class $\\mathcal {F}$ which we will take to be the set of all matrices whose trace-norm is at most $B$ (namely, we can use any such matrix to predict just by returning its relevant entry at each round).", "Usually, one sets $B$ to be on the order of $\\sqrt{mn}$ .", "We consider here a transductive version, where the sequence of entry locations is known in advance, and only the entry values are unknown.", "We show how to develop an algorithm whose regret is bounded by the (transductive) Rademacher complexity of $\\mathcal {F}$ .", "We note that in Theorem 6 of [17], this complexity was shown to be at most order $B \\sqrt{n}$ independent of $T$ .", "Moreover, in [8], it was shown that for algorithms with such guarantees, and whose play each round does not depend on the order of future entries, under mild conditions on the loss function one can get the same regret even in the “fully” online case where the set of entry locations is unknown in advance.", "Algorithmically, all we need to do is pretend we are in a transductive game where the sequence of entries is all $m\\times n$ entries, in some arbitrary order.", "In this section we use the two alternatives provided for transductive learning problem in the previous subsection and provide two alternatives for the matrix completion problem.", "We note that both variants proposed here improve on the one provided by the $R^2$ forecaster in [8], since that algorithm competes against the smaller class $\\mathcal {F}^{\\prime }$ of matrices with bounded trace-norm and bounded individual entries.", "In contrast, our algorithm provides similar regret guarantees against the larger class of matrices only whose trace-norm is bounded.", "Moreover, the variants are also computationally more efficient." ], [ "First Alternative : ", "The algorithm we now present is obtained by using the first method for online tranductive learning proposed in the previous section.", "The relaxation in Equation (REF ) for the specific problem at hand is given by, $ \\mathbf {Rel}_{\\mathcal {F}}\\left(y_1,\\ldots ,y_t \\right) = B\\ \\mathbb {E}_{\\epsilon }\\left[ \\left\\Vert 2 \\rho \\sum _{i=t+1}^{T} \\epsilon _i x_i - \\sum _{i=1}^{t} \\partial \\ell (\\hat{y}_i,y_i) x_i \\right\\Vert _{\\mathrm {\\sigma }} \\right]$ In the above $\\left\\Vert \\cdot \\right\\Vert _{\\sigma }$ stands for the spectral norm and each $x_i$ is a matrix with a 1 at some specific position and 0 elsewhere.", "That is $x_i$ at round $i$ can be seen as the entry of the matrix which we are asked to fill in at round $i$ .", "The prediction at round $t$ returned by the algorithm is given by Equation (REF ) which for this problem is given by $\\hat{y}_t = B\\ \\mathbb {E}_{\\epsilon }\\left[ \\left( \\left\\Vert \\sum _{i=t+1}^{T} \\epsilon _i x_i - \\frac{1}{2 \\rho } \\sum _{i=1}^{t-1} \\partial \\ell (\\hat{y}_i,y_i) x_i + \\frac{1}{2} x_t\\right\\Vert _{\\sigma } -\\left\\Vert \\sum _{i=t+1}^{T} \\epsilon _i x_i - \\frac{1}{2\\rho } \\sum _{i=1}^{t-1} \\partial \\ell (\\hat{y}_i,y_i) x_i - \\frac{1}{2} x_t\\right\\Vert _{\\sigma } \\right) \\right]$ Notice that the algorithm only involves calculation of spectral norms on each round which can be done efficiently.", "Again as mentioned in previous subsection, one can evaluate the expectation over random signs by sampling $\\epsilon $ 's on each round." ], [ "Second Alternative : ", "The second algorithm is obtained from the second alternative for online transductive learning with convex losses in the previous section.", "The relaxation given in Equation (REF ) for the case of matrix completion problem with trace norm constraint is given by: $\\mathbf {Rel}_{T}\\left(\\mathcal {F} | y_1,\\ldots ,y_t \\right) = \\mathbb {E}_{\\epsilon }\\left[ \\sup _{f : \\left\\Vert f\\right\\Vert _{\\Sigma } \\le B} \\ 2 \\rho \\sum _{i=t+1}^{T} \\epsilon _i \\left<f,x_i\\right> - \\sum _{i=1}^{t} \\ell (\\left<f,x_i\\right>,y_i) \\right]$ where $\\left\\Vert \\cdot \\right\\Vert _{\\Sigma }$ stands for the race norm of the $m \\times n$ matrix $f$ and each $x_i$ is a matrix with a 1 at some specific position and 0 elsewhere.", "That is $x_i$ at round $i$ can be seen as the entry of the matrix which we are asked to fill in at round $i$ .", "We use $\\left<f,x\\right>$ to represent the generalized inner product of the two matrices.", "Since we only take inner products with respect to the matrices $x_i$ , each $\\left<f,x_i\\right>$ is simply the value of matrix $f$ at the position specified by $x_i$ 's.", "The prediction at a matrix entry corresponding to position $x_t$ is given by first drawing random $\\lbrace \\pm 1\\rbrace $ valued $\\epsilon $ 's and then applying Equation (REF ) to the problem at hand, yielding $\\hat{y}_t(\\epsilon ) & = \\inf _{\\left\\Vert f\\right\\Vert _{\\Sigma } \\le B} \\left\\lbrace - \\sum _{i=t+1}^{T} \\epsilon _i \\left<f,x_i\\right> + \\frac{1}{2 \\rho } \\sum _{i=1}^{t-1} \\ell (\\left<f,x_i\\right>,y_i) + \\frac{1}{2} \\left<f,x_t\\right> \\right\\rbrace - \\inf _{\\left\\Vert f\\right\\Vert _{\\Sigma } \\le B} \\left\\lbrace - \\sum _{i=t+1}^{T} \\epsilon _i \\left<f,x_i\\right> + \\frac{1}{2 \\rho }\\sum _{i=1}^{t-1} \\ell (\\left<f,x_i\\right>,y_i) - \\frac{1}{2} \\left<f,x_t\\right> \\right\\rbrace $ Notice that the above involves solving two trace norm constrained convex optimization problems per round.", "As a simple corollary of Lemma REF we get the following bound on expected regret of the algorithm.", "Corollary 21 For the randomized prediction strategy specified above, the expected regret is bounded as $\\mathbb {E}\\left[ \\mathbf {Reg}_{T} \\right] \\le 2 B\\ \\rho \\ \\mathbb {E}_{\\epsilon }\\left[ \\left\\Vert \\sum _{t=1}^T \\epsilon _t x_t\\right\\Vert _{\\sigma } \\right] \\le O\\left( B\\ \\rho \\ (\\sqrt{m} + \\sqrt{n}) \\right)$ The last inequality in the above corollary is using Theorem 6 in [17].", "Corollary 22 For the predictions $\\hat{y}_t$ specified above, the regret is bounded as $\\mathbf {Reg}_{T} \\le O\\left( B\\ \\rho \\ (\\sqrt{m} + \\sqrt{n}) \\right)$" ], [ "Constrained Adversaries", "We now show that algorithms can be also developed for situations when the adversary is constrained in the choices per step.", "Such constrained problems have been treated in a general non-algorithmic way in [16], and we picked the case of variation-constrained adversary for illustration.", "It is shown in [16] that the value of the game where the adversary is constrained to keep the next move $x_t$ within $\\sigma _t$ from the average of the past moves $\\frac{1}{t-1}\\sum _{s=1}^{t-1}x_s$ is upper bounded as $\\mathcal {V}_T & \\le 2 \\sup _{(\\mathbf {x},\\mathbf {x}^{\\prime }) \\in \\mathcal {T}} \\mathbb {E}_{\\epsilon }\\left[ \\sup _{f \\in \\mathcal {F}} \\sum _{t=1}^T \\epsilon _t\\left(\\left\\langle f,\\mathbf {x}_t(\\epsilon ) \\right\\rangle - \\frac{1}{t-1} \\sum _{\\tau =1}^{t-1} \\left\\langle f,\\chi _\\tau (\\epsilon _\\tau ) \\right\\rangle \\right) \\right]$ where the supremum is over $\\mathbf {x},\\mathbf {x}^{\\prime }$ trees satisfying the above mentioned constraint per step, and the selector $\\chi _t(\\epsilon _t)$ is defined as $\\mathbf {x}_t(\\epsilon )$ if $\\epsilon _t=-1$ and $\\mathbf {x}^{\\prime }_t(\\epsilon )$ otherwise.", "In our algorithmic framework, this leads to the following problem that needs to be solved at each step: $\\inf _{f_t}\\sup _{x_t} \\left\\lbrace \\left\\langle f_t,x_t \\right\\rangle +2 \\sup _{(\\mathbf {x},\\mathbf {x}^{\\prime }) \\in \\mathcal {T}} \\mathbb {E}_{\\epsilon }\\left[ \\sup _{f \\in \\mathcal {F}}\\left\\langle f, \\sum _{s=t+1}^T \\epsilon _s\\left(\\mathbf {x}_s(\\epsilon ) - \\frac{1}{s-t} \\sum _{\\tau =t+1}^{s-1} \\chi _\\tau (\\epsilon _\\tau ) \\right) - \\sum _{r=1}^t x_r \\right\\rangle \\right] \\right\\rbrace $ where the supremum is taken over $x_t$ such that the constraint $C(x_1,\\ldots ,x_t)$ is satisfied and $\\mathcal {T}$ is the set of trees that satisfy the constraints as continuation of the prefix $x_1,\\ldots ,x_t$ .", "While this expression gives rise to an algorithm, we are aiming for a more computationally feasible method.", "In fact, passing to an upper bound on the sequential Rademacher complexity yields the following result.", "Lemma 23 The following relaxation is admissible and upper bounds the constrained sequential complexity $\\mathbf {Rel}_{T}\\left(\\mathcal {F} | x_{1},\\ldots ,x_t \\right)= \\frac{2\\sqrt{2}R}{\\sqrt{\\lambda }} \\sqrt{ \\left\\Vert \\sum _{r=1}^t x_r \\right\\Vert ^2 + C\\sum _{s=t+1}^T\\sigma _s^2 }$ Furthermore, the an admissible algorithm for this relaxation is Mirror Descent with a step size given at time $t\\ge 2$ by $\\frac{\\left(1+\\frac{1}{t-1}\\right)^2}{2\\sqrt{\\Vert \\tilde{x}_{t-1}\\Vert ^2 + C\\sum _{s=t}^T\\sigma _s^2}}$" ], [ "Universal Mirror Descent", "In [18] it is shown that for the problem of general online convex optimization, the Mirror Descent algorithm is universal and near optimal (up to poly-log factors).", "Specifically, it is shown that there always exists an appropriate function $\\Psi $ such that the Mirror Descent algorithm using this function, along with an appropriate step size, gives the near optimal rate.", "Moreover, it is shown in [18] that one can use function $\\Psi $ whose convex conjugate is given by $\\Psi ^* (x) = \\sup _{\\mathbf {x}} \\mathbb {E}_{\\epsilon }\\left[ \\left\\Vert x + \\sum _{i=1}^{T-t} \\epsilon _i \\mathbf {x}_i(\\epsilon ) \\right\\Vert ^p - C \\sum _{i=1}^{T-t} \\mathbb {E}_{\\epsilon }\\left[ \\left\\Vert \\mathbf {x}_i(\\epsilon )\\right\\Vert ^p \\right] \\right],$ as the “universal regularizer\" for the Mirror Descent algorithm.", "We now show that this function arises rather naturally from the sequential Rademacher relaxation and, moreover, the Mirror Descent algorithm itself arises from this relaxation.", "Let us denote the convex cost functions chosen by the adversary as $\\ell _t$ , and let $x_t$ be the subgradients $x_t = \\nabla \\ell _t(f_t)$ of the convex functions.", "Lemma 24 The relaxation $\\mathbf {Rel}_{T}\\left(\\mathcal {F} | x_{1},\\ldots ,x_t \\right) = \\left( \\Psi ^*\\left(\\sum _{i=1}^{t-1} x_i \\right) +\\left<\\nabla \\Psi ^*\\left(\\sum _{i=1}^{t-1} x_i\\right),x_t\\right> + C (T-t+1) \\right)^{1/p}$ is an upper bound on the conditional sequential Rademacher complexity.", "Further, whenever for some $p^{\\prime } > p$ we have that $\\mathcal {V}_T(\\mathcal {F}) \\le (C T)^{1/p^{\\prime }}$ , then the relaxation is admissible and leads to a form of Mirror Descent algorithm with regret bounded as $\\mathbf {Reg}_T \\le (C T)^{1/p}$ It is remarkable that the universal regularizer and the Mirror Descent algorithm arise naturally, in a few steps of algebra, as upper bounds on the sequential Rademacher complexity." ], [ "PROOFS", "[Proof of Proposition  REF] By definition, $\\sum _{t=1}^T \\mathbb {E}_{f_t\\sim q_t}\\ell (f_t,x_t) - \\inf _{f\\in \\mathcal {F}} \\sum _{t=1}^T \\ell (f,x_t) \\le \\sum _{t=1}^T \\mathbb {E}_{f_t\\sim q_t}\\ell (f_t,x_t) + \\mathbf {Rel}_{T}\\left(\\mathcal {F} | x_{1},\\ldots ,x_{T} \\right) \\ .$ Peeling off the $T$ -th expected loss, we have $\\sum _{t=1}^T \\mathbb {E}_{f_t\\sim q_t}\\ell (f_t,x_t) + \\mathbf {Rel}_{T}\\left(\\mathcal {F} | x_{1},\\ldots ,x_{T} \\right) &\\le \\sum _{t=1}^{T-1} \\mathbb {E}_{f_t\\sim q_t}\\ell (f_t,x_t) + \\left\\lbrace \\mathbb {E}_{f_t\\sim q_t}\\ell (f_t,x_t) + \\mathbf {Rel}_{T}\\left(\\mathcal {F} | x_{1},\\ldots ,x_{T} \\right) \\right\\rbrace \\\\&\\le \\sum _{t=1}^{T-1} \\mathbb {E}_{f_t\\sim q_t}\\ell (f_t,x_t) + \\mathbf {Rel}_{T}\\left(\\mathcal {F} | x_{1},\\ldots ,x_{T-1} \\right)$ where we used the fact that $q_T$ is an admissible algorithm for this relaxation, and thus the last inequality holds for any choice $x_T$ of the opponent.", "Repeating the process, we obtain $\\sum _{t=1}^T \\mathbb {E}_{f_t\\sim q_t}\\ell (f_t,x_t) - \\inf _{f\\in \\mathcal {F}} \\sum _{t=1}^T \\ell (f,x_t) \\le \\mathbf {Rel}_{T}\\left(\\mathcal {F} \\right) \\ .$ We remark that the left-hand side of this inequality is random, while the right-hand side is not.", "Since the inequality holds for any realization of the process, it also holds in expectation.", "The inequality $\\mathcal {V}_T(\\mathcal {F}) \\le \\mathbf {Rel}_{T}\\left(\\mathcal {F} \\right)$ holds by unwinding the value recursively and using admissibility of the relaxation.", "The high-probability bound is an immediate consequences of (REF ) and the Hoeffding-Azuma inequality for bounded martingales.", "The last statement is immediate.", "[Proof of Proposition  REF] Denote $L_t(f) = \\sum _{s=1}^t \\ell (f,x_s)$ .", "The first step of the proof is an application of the minimax theorem (we assume the necessary conditions hold): $&\\inf _{q_t \\in \\Delta (\\mathcal {F})} \\sup _{x_t \\in \\mathcal {X}} \\left\\lbrace \\underset{f_t \\sim q_t}{\\mathbb {E}}\\left[\\ell (f_t,x_t)\\right] + \\sup _{\\mathbf {x}} \\mathbb {E}_{\\epsilon _{t+1:T}} \\sup _{f\\in \\mathcal {F}} \\left[ 2\\sum _{s=t+1}^T \\epsilon _s\\ell (f,\\mathbf {x}_{s-t}(\\epsilon _{t+1:s-1})) - L_t(f) \\right]\\right\\rbrace \\\\&=\\sup _{p_t \\in \\Delta (\\mathcal {X})} \\inf _{f_t \\in \\mathcal {F}} \\left\\lbrace \\underset{x_t \\sim p_t}{\\mathbb {E}}\\left[\\ell (f_t,x_t)\\right] + \\underset{x_t \\sim p_t}{\\mathbb {E}}\\sup _{\\mathbf {x}} \\mathbb {E}_{\\epsilon _{t+1:T}} \\sup _{f\\in \\mathcal {F}} \\left[ 2\\sum _{s=t+1}^T \\epsilon _s\\ell (f,\\mathbf {x}_{s-t}(\\epsilon _{t+1:s-1})) - L_t(f) \\right]\\right\\rbrace $ For any $p_t\\in \\Delta (\\mathcal {X})$ , the infimum over $f_t$ of the above expression is equal to $&\\underset{x_t \\sim p_t}{\\mathbb {E}}\\sup _{\\mathbf {x}} \\mathbb {E}_{\\epsilon _{t+1:T}} \\sup _{f\\in \\mathcal {F}} \\left[ 2\\sum _{s=t+1}^T \\epsilon _s\\ell (f,\\mathbf {x}_{s-t}(\\epsilon _{t+1:s-1})) - L_{t-1}(f) + \\inf _{f_t \\in \\mathcal {F}}\\underset{x_t \\sim p_t}{\\mathbb {E}}\\left[\\ell (f_t,x_t)\\right] - \\ell (f,x_t) \\right] \\\\&\\le \\underset{x_t \\sim p_t}{\\mathbb {E}}\\sup _{\\mathbf {x}} \\mathbb {E}_{\\epsilon _{t+1:T}} \\sup _{f\\in \\mathcal {F}} \\left[ 2\\sum _{s=t+1}^T \\epsilon _s\\ell (f,\\mathbf {x}_{s-t}(\\epsilon _{t+1:s-1})) - L_{t-1}(f) + \\underset{x_t \\sim p_t}{\\mathbb {E}}\\left[\\ell (f,x_t)\\right] - \\ell (f,x_t) \\right] \\\\&\\le \\underset{x_t,x^{\\prime }_t \\sim p_t}{\\mathbb {E}}\\sup _{\\mathbf {x}} \\mathbb {E}_{\\epsilon _{t+1:T}} \\sup _{f\\in \\mathcal {F}} \\left[ 2\\sum _{s=t+1}^T \\epsilon _s\\ell (f,\\mathbf {x}_{s-t}(\\epsilon _{t+1:s-1})) - L_{t-1}(f) + \\ell (f,x^{\\prime }_t) - \\ell (f,x_t) \\right]$ We now argue that the independent $x_t$ and $x^{\\prime }_t$ have the same distribution $p_t$ , and thus we can introduce a random sign $\\epsilon _t$ .", "The above expression then equals to $&\\underset{x_t,x^{\\prime }_t \\sim p_t}{\\mathbb {E}}\\underset{\\epsilon _t}{\\mathbb {E}} \\sup _{\\mathbf {x}} \\mathbb {E}_{\\epsilon _{t+1:T}} \\sup _{f\\in \\mathcal {F}} \\left[ 2\\sum _{s=t+1}^T \\epsilon _s\\ell (f,\\mathbf {x}_{s-t}(\\epsilon _{t+1:s-1})) - L_{t-1}(f) + \\epsilon _t(\\ell (f,x^{\\prime }_t) - \\ell (f,x_t)) \\right] \\\\&\\le \\sup _{x_t,x^{\\prime }_t \\in \\mathcal {X}}\\underset{\\epsilon _t}{\\mathbb {E}} \\sup _{\\mathbf {x}} \\mathbb {E}_{\\epsilon _{t+1:T}} \\sup _{f\\in \\mathcal {F}} \\left[ 2\\sum _{s=t+1}^T \\epsilon _s\\ell (f,\\mathbf {x}_{s-t}(\\epsilon _{t+1:s-1})) - L_{t-1}(f) + \\epsilon _t(\\ell (f,x^{\\prime }_t) - \\ell (f,x_t)) \\right]$ where we upper bounded the expectation by the supremum.", "Splitting the resulting expression into two parts, we arrive at the upper bound of $&2\\sup _{x_t\\in \\mathcal {X}}\\underset{\\epsilon _t}{\\mathbb {E}} \\sup _{\\mathbf {x}} \\mathbb {E}_{\\epsilon _{t+1:T}} \\sup _{f\\in \\mathcal {F}} \\left[ \\sum _{s=t+1}^T \\epsilon _s\\ell (f,\\mathbf {x}_{s-t}(\\epsilon _{t+1:s-1})) - \\frac{1}{2}L_{t-1}(f) + \\epsilon _t \\ell (f,x_t) \\right]= \\mathfrak {R}_T (\\mathcal {F}| x_1,\\ldots ,x_{t-1}) \\ .$ The last equality is easy to verify, as we are effectively adding a root $x_t$ to the two subtrees, for $\\epsilon _t=+1$ and $\\epsilon _t=-1$ , respectively.", "One can see that the proof of admissibility corresponds to one step minimax swap and symmetrization in the proof of [14].", "In contrast, in the latter paper, all $T$ minimax swaps are performed at once, followed by $T$ symmetrization steps.", "[Proof of Proposition  REF] Let us first prove that the relaxation is admissible with the Exponential Weights algorithm as an admissible algorithm.", "Let $L_t(f) = \\sum _{i=1}^t \\ell (f,x_i)$ .", "Let $\\lambda ^*$ be the optimal value in the definition of $\\mathbf {Rel}_{T}\\left(\\mathcal {F} | x_{1},\\ldots ,x_{t-1} \\right)$ .", "Then $&\\inf _{q_t \\in \\Delta (\\mathcal {F})} \\sup _{x_t \\in \\mathcal {X}} \\left\\lbrace \\underset{f \\sim q_t}{\\mathbb {E}}\\left[\\ell (f,x_t)\\right] + \\mathbf {Rel}_{T}\\left(\\mathcal {F} | x_{1},\\ldots ,x_t \\right)\\right\\rbrace \\\\&\\le \\inf _{q_t \\in \\Delta (\\mathcal {F})} \\sup _{x_t \\in \\mathcal {X}} \\left\\lbrace \\underset{f \\sim q_t}{\\mathbb {E}}\\left[\\ell (f,x_t)\\right] + \\frac{1}{\\lambda ^*}\\log \\left( \\sum _{f \\in \\mathcal {F}} \\exp \\left( - \\lambda ^* L_t(f) \\right) \\right) + 2\\lambda ^* (T-t) \\right\\rbrace $ Let us upper bound the infimum by a particular choice of $q$ which is the exponential weights distribution $q_t(f) = \\exp (-\\lambda ^* L_{t-1}(f))/Z_{t-1}$ where $Z_{t-1} = \\sum _{f \\in \\mathcal {F}} \\exp \\left( - \\lambda ^* L_{t-1}(f) \\right) $ .", "By [7], $\\frac{1}{\\lambda ^*}\\log \\left( \\sum _{f \\in \\mathcal {F}} \\exp \\left( - \\lambda ^* L_t(f) \\right) \\right) &= \\frac{1}{\\lambda ^*}\\log \\left( \\mathbb {E}_{f\\sim q_t} \\exp \\left( - \\lambda ^* \\ell (f,x_t) \\right) \\right) + \\frac{1}{\\lambda ^*}\\log Z_{t-1} \\\\&\\le -\\mathbb {E}_{f\\sim q_t} \\ell (f,x_t) + \\frac{\\lambda ^*}{2} + \\frac{1}{\\lambda ^*}\\log Z_{t-1}$ Hence, $\\inf _{q_t \\in \\Delta (\\mathcal {F})} \\sup _{x_t \\in \\mathcal {X}} \\left\\lbrace \\underset{f \\sim q_t}{\\mathbb {E}}\\left[\\ell (f,x_t)\\right] + \\mathbf {Rel}_{T}\\left(\\mathcal {F} | x_{1},\\ldots ,x_t \\right)\\right\\rbrace &\\le \\frac{1}{\\lambda ^*}\\log \\left( \\sum _{f \\in \\mathcal {F}} \\exp \\left( - \\lambda ^* L_{t-1}(f) \\right) \\right) + 2\\lambda ^* (T-t+1) \\\\&~~~ = \\mathbf {Rel}_{T}\\left(\\mathcal {F} | x_{1},\\ldots ,x_{t-1} \\right)$ by the optimality of $\\lambda ^*$ .", "The bound can be improved by a factor of 2 for some loss functions, since it will disappear from the definition of sequential Rademacher complexity.", "We conclude that the Exponential Weights algorithm is an admissible strategy for the relaxation (REF )." ], [ "Arriving at the relaxation", "We now show that the Exponential Weights relaxation arises naturally as an upper bound on sequential Rademacher complexity of a finite class.", "For any $\\lambda >0$ , $\\mathbb {E}_{\\epsilon }\\left[ \\sup _{f \\in \\mathcal {F}} \\left\\lbrace 2 \\sum _{i=1}^{T-t} \\epsilon _i \\ell (f,\\mathbf {x}_i(\\epsilon )) - L_t(f)\\right\\rbrace \\right]&\\le \\frac{1}{\\lambda }\\log \\left(\\mathbb {E}_{\\epsilon }\\left[ \\sup _{f \\in \\mathcal {F}} \\exp \\left( 2\\lambda \\sum _{i=1}^{T-t} \\epsilon _i \\ell (f,\\mathbf {x}_i(\\epsilon )) - \\lambda L_t(f) \\right) \\right] \\right)\\\\& \\le \\frac{1}{\\lambda }\\log \\left(\\mathbb {E}_{\\epsilon }\\left[ \\sum _{f \\in \\mathcal {F}} \\exp \\left( 2\\lambda \\sum _{i=1}^{T-t} \\epsilon _i \\ell (f,\\mathbf {x}_i(\\epsilon )) - \\lambda L_t(f) \\right) \\right] \\right)\\\\&= \\frac{1}{\\lambda }\\log \\left( \\sum _{f \\in \\mathcal {F}} \\exp \\left( - \\lambda L_t(f) \\right) \\mathbb {E}_{\\epsilon }\\left[ \\prod _{i=1}^{T-t} \\exp \\left( 2 \\lambda \\epsilon _i \\ell (f,\\mathbf {x}_i(\\epsilon )) \\right) \\right] \\right)$ We now upper bound the expectation over the “future” tree by the worst-case path, resulting in the upper bound $& \\frac{1}{\\lambda }\\log \\left( \\sum _{f \\in \\mathcal {F}} \\exp \\left( - \\lambda L_t(f) \\right) \\times \\exp \\left( 2 \\lambda ^2 \\max _{\\epsilon _1,\\ldots \\epsilon _{T-t} \\in \\lbrace \\pm 1\\rbrace }\\sum _{i=1}^{T-t} \\ell (f,\\mathbf {x}_i(\\epsilon ))^2 \\right) \\right)\\\\&\\le \\frac{1}{\\lambda }\\log \\left( \\sum _{f \\in \\mathcal {F}} \\exp \\left( - \\lambda L_t(f) + 2 \\lambda ^2 \\max _{\\epsilon _1,\\ldots \\epsilon _{T-t} \\in \\lbrace \\pm 1\\rbrace }\\sum _{i=1}^{T-t} \\ell (f,\\mathbf {x}_i(\\epsilon ))^2 \\right) \\right)\\\\& \\le \\frac{1}{\\lambda }\\log \\left( \\sum _{f \\in \\mathcal {F}} \\exp \\left( - \\lambda L_t(f) \\right) \\right) + 2 \\lambda \\sup _{\\mathbf {x}} \\sup _{f \\in \\mathcal {F}} \\max _{\\epsilon _1,\\ldots \\epsilon _{T-t} \\in \\lbrace \\pm 1\\rbrace }\\sum _{i=1}^{T-t} \\ell (f,\\mathbf {x}_i(\\epsilon ))^2$ The last term, representing the “worst future”, is upper bounded by $2\\lambda (T-t)$ .", "This removes the $\\mathbf {x}$ tree and leads to the relaxation (REF ) and a computationally tractable algorithm.", "[Proof of Proposition  REF] The argument can be seen as a generalization of the Euclidean proof in [2] to general smooth norms.", "The proof below not only shows that the Mirror Descent algorithm is admissible for the relaxation (REF ), but in fact shows that it coincides with the optimal algorithm for the relaxation, i.e.", "the one that attains the infimum over strategies.", "Let $\\tilde{x}_{t-1} = \\sum _{i=1}^{t-1} x_i$ .", "The optimal algorithm for the relaxation (REF ) is $f_t = \\underset{f \\in \\mathcal {F}}{\\mathrm {argmin}} \\ \\left\\lbrace \\sup _{x_t \\in \\mathcal {X}} \\left\\lbrace \\left<f,x_t\\right> + \\sqrt{ \\left\\Vert \\tilde{x}_{t-1}\\right\\Vert ^2 + \\left<\\nabla \\left\\Vert \\tilde{x}_{t-1}\\right\\Vert ^2,x_t\\right> + C (T - t + 1) } \\right\\rbrace \\right\\rbrace $ Now write any $f_t$ as $f_t = -\\alpha \\nabla \\Vert \\tilde{x}_{t-1}\\Vert ^2 + g$ for some $g \\in \\mathrm {Kernel}(\\nabla \\Vert \\tilde{x}_{t-1}\\Vert ^2) \\triangleq \\left\\lbrace h: \\left<\\nabla \\Vert \\tilde{x}_{t-1}\\Vert ^2,h\\right>=0\\right\\rbrace $ , and any $x_t$ as $x_t = \\beta \\tilde{x}_{t-1} + \\gamma y$ for some $y \\in \\mathrm {Kernel}(\\nabla \\Vert \\tilde{x}_{t-1}\\Vert ^2)$ .", "Hence we can write: $& \\left<f_t,x_t\\right> + \\left( \\Vert \\tilde{x}_{t-1} \\Vert ^2 +\\left<\\nabla \\Vert \\tilde{x}_{t-1}\\Vert ^2,x_t\\right> + C (T-t+1) \\right)^{1/2} \\\\& = -\\alpha \\beta \\Vert \\tilde{x}_{t-1}\\Vert ^2 + \\gamma \\left<g,y\\right> + \\left( \\Vert \\tilde{x}_{t-1}\\Vert ^2 + \\beta \\Vert \\tilde{x}_{t-1}\\Vert ^2 + C (T - t + 1) \\right)^{1/2}$ Given any $f_t = -\\alpha \\nabla \\left\\Vert \\tilde{x}_{t-1}\\right\\Vert ^2 + g$ , $x$ can be picked with $y \\in \\mathrm {Kernel}(\\nabla \\left\\Vert \\tilde{x}_{t-1}\\right\\Vert ^2)$ that satisfies $\\left<g,y\\right> \\ge 0$ .", "One can always do this because if for some $y^{\\prime }$ , $\\left<g,y^{\\prime }\\right> < 0$ by picking $y = -y^{\\prime }$ we can ensure that $\\left<g,y\\right> \\ge 0$ .", "Hence the minimizer $f_t$ must be once such that $f_t = -\\alpha \\nabla \\left\\Vert \\tilde{x}_{t-1}\\right\\Vert ^2$ and thus $\\left<g,y\\right>=0$ .", "Now, it must be that $\\alpha \\ge 0$ so that $x_t$ either increases the first term or second term but not both.", "Hence we have that $f_t = - \\alpha \\nabla \\left\\Vert \\tilde{x}_{t-1}\\right\\Vert ^2$ for some $\\alpha \\ge 0$ .", "Now given such an $f_t$ , the sup over $x_t$ can be written as supremum over $\\beta $ of a concave function, which gives rise to the derivative condition $-\\alpha \\left\\Vert \\tilde{x}_{t-1}\\right\\Vert ^2 + \\frac{\\left\\Vert \\tilde{x}_{t-1}\\right\\Vert ^2}{2 \\sqrt{ \\left\\Vert \\tilde{x}_{t-1}\\right\\Vert ^2 + \\beta \\left\\Vert \\tilde{x}_{t-1}\\right\\Vert ^2 + C (T - t + 1) }} = 0$ At this point it is clear that the value of $\\alpha = \\frac{1}{2 \\sqrt{\\left\\Vert \\tilde{x}_{t-1}\\right\\Vert ^2 + C (T - t + 1) }}$ forces $\\beta =0$ .", "Let us in fact show that this value is optimal.", "We have $\\frac{1}{4 \\alpha ^2 } = \\left\\Vert \\tilde{x}_{t-1}\\right\\Vert ^2 + \\beta \\left\\Vert \\tilde{x}_{t-1}\\right\\Vert ^2 + C (T - t + 1)$ Plugging this value of $\\beta $ back, we now aim to optimize $\\frac{1}{4 \\alpha } + \\alpha \\left\\Vert \\tilde{x}_{t-1}\\right\\Vert ^2 + \\alpha C (T - t + 1)$ over $\\alpha $ .", "We then obtain the value given in (REF ).", "With this value, we have the familiar update $f_t = -\\frac{\\nabla \\left\\Vert \\tilde{x}_{t-1}\\right\\Vert ^2}{2 \\sqrt{\\left\\Vert \\tilde{x}_{t-1}\\right\\Vert ^2 + C (T - t + 1)}} \\ .$ Plugging back the value of $\\alpha $ , we find that $\\beta = 0$ .", "With these values, $\\inf _{f \\in \\mathcal {F}}&\\left\\lbrace \\sup _{x \\in \\mathcal {X}} \\left\\lbrace \\left<f,x\\right> + \\left( \\Vert \\tilde{x}_{t-1} \\Vert ^2 +\\left<\\nabla \\Vert \\tilde{x}_{t-1}\\Vert ^2,x\\right> + C (T-t+1) \\right)^{1/2} \\right\\rbrace \\right\\rbrace = \\left( \\Vert \\tilde{x}_{t-1}\\Vert ^2 + C (T - t + 1) \\right)^{1/2} \\\\&~~~~~~~~~~~~~\\le \\left( \\Vert \\bar{x}_{t-2}\\Vert ^2 + \\left<\\nabla \\Vert \\bar{x}_{t-2}\\Vert ^2,x_{t-1}\\right> + C (T - t + 2) \\right)^{1/2} =\\mathbf {Rel}_{T}\\left(\\mathcal {F} | x_{1},\\ldots ,x_{t-1} \\right)$ We have shown that (REF ) is an optimal algorithm for the relaxation, and it is admissible." ], [ "Arriving at the Relaxation", "The derivation of the relaxation is immediate: $\\mathfrak {R}_T (\\mathcal {F}| x_1,\\ldots ,x_t) &= \\sup _{\\mathbf {x}} \\mathbb {E}_{\\epsilon _{t+1:T}} \\left\\Vert \\sum _{s=t+1}^T \\epsilon _s \\mathbf {x}_{s-t}(\\epsilon _{t+1:s-1}) - \\sum _{s=1}^t x_s \\right\\Vert \\\\&\\le \\sup _{\\mathbf {x}} \\sqrt{\\mathbb {E}_{\\epsilon _{t+1:T}} \\left\\Vert \\sum _{s=t+1}^T \\epsilon _s \\mathbf {x}_{s-t}(\\epsilon _{t+1:s-1}) - \\sum _{s=1}^t x_s \\right\\Vert ^2 }\\\\&\\le \\sup _{\\mathbf {x}} \\sqrt{ \\left\\Vert \\sum _{s=1}^t x_s \\right\\Vert ^2 + C \\mathbb {E}_{\\epsilon _{t+1:T}}\\sum _{s=t+1}^T\\left\\Vert \\epsilon _s \\mathbf {x}_{s-t}(\\epsilon _{t+1:s-1})\\right\\Vert ^2}$ where the last step is due to the smoothness of the norm and the fact that the first-order terms disappear under the expectation.", "The sum of norms is now upper bounded by $T-t$ , thus removing the dependence on the “future”, and we arrive at $\\sqrt{ \\left\\Vert \\sum _{s=1}^t x_s \\right\\Vert ^2 + C(T-t)} \\le \\sqrt{ \\left\\Vert \\sum _{s=1}^{t-1} x_s \\right\\Vert ^2 + \\left<\\nabla \\left\\Vert \\sum _{s=1}^{t-1} x_s\\right\\Vert ^2,x_t\\right> + C(T-t+1)}$ as a relaxation on the sequential Rademacher complexity.", "[Proof of Lemma  REF] We shall first establish the admissibility of the relaxation specified.", "To show admissibility, let us first check the initial condition: $\\mathbf {Rel}_{k}\\left(\\mathcal {F}_{r(k;x_1,\\ldots ,x_t)} | y_1,\\ldots ,y_k \\right) & = - \\left<\\hat{f}_{t},\\tilde{y}_k\\right> + 2\\min \\left\\lbrace 1, \\frac{k}{\\sigma _{1:t}}\\right\\rbrace \\sqrt{ \\left\\Vert \\sum _{j=1}^{k-1} y_j\\right\\Vert ^2 + \\left<\\nabla \\left\\Vert \\sum _{j=1}^{k-1}y_j\\right\\Vert ^2,y_k\\right>+ C }\\\\& \\ge - \\left<\\hat{f}_{t},\\tilde{y}_k\\right> + 2\\min \\left\\lbrace 1, \\frac{k}{\\sigma _{1:t}}\\right\\rbrace \\sqrt{ \\left\\Vert \\tilde{y}_k\\right\\Vert ^2}\\\\& \\ge - \\left<\\hat{f}_{t},\\tilde{y}_k\\right> + \\sup _{f : \\left\\Vert f - \\hat{f}_t\\right\\Vert \\le 2\\min \\lbrace 1 , \\frac{k}{\\sigma _{1:t}} \\rbrace } \\left<f - \\hat{f}_t,- \\tilde{y}_k\\right>\\\\& \\ge - \\inf _{f : \\left\\Vert f - \\hat{f}_t\\right\\Vert \\le 2\\min \\lbrace 1 , \\frac{k}{\\sigma _{1:t}} \\rbrace } \\sum _{j=1}^{k} \\left<f , y_j\\right>$ Now, for the recurrence, we have $& \\left<f_i,y_i\\right> + \\sup _{\\mathbf {y}} \\mathbb {E}_{\\epsilon }\\left[ \\sup _{f : \\Vert f -\\hat{f}_{t}\\Vert \\le 2\\min \\left\\lbrace 1,\\frac{k}{\\sigma _{1:t}}\\right\\rbrace } \\left<f,\\sum _{j=1}^{k-i} \\epsilon _j \\mathbf {y}_j(\\epsilon ) - \\sum _{j=1}^{i} y_j\\right> \\right] \\\\& = \\left<f_i,y_i\\right> - \\left<\\hat{f}_{t},\\sum _{j=1}^i y_j\\right>+ \\sup _{\\mathbf {y}} \\mathbb {E}_{\\epsilon }\\left[ \\sup _{f : \\Vert f -\\hat{f}_{t}\\Vert \\le 2\\min \\left\\lbrace 1, \\frac{k}{\\sigma _{1:t}}\\right\\rbrace } \\left<f - \\hat{f}_t,\\sum _{j=1}^{k-i} \\epsilon _j \\mathbf {y}_j(\\epsilon ) - \\sum _{j=1}^{i} y_j\\right> \\right] \\\\& \\le \\left<f_i,y_i\\right> - \\left<\\hat{f}_{t},\\sum _{j=1}^i y_j\\right> + 2\\min \\left\\lbrace 1, \\frac{k}{\\sigma _{1:t}}\\right\\rbrace \\sup _{\\mathbf {y}} \\mathbb {E}_{\\epsilon }\\left[ \\left\\Vert \\sum _{j=1}^{k-i} \\epsilon _j \\mathbf {y}_j(\\epsilon )- \\sum _{j=1}^{i} y_j\\right\\Vert \\right] \\\\\\multicolumn{2}{l}{\\text{}}\\\\& \\le \\left<f_i,y_i\\right> - \\left<\\hat{f}_{t},\\sum _{j=1}^{i} y_j\\right> + 2\\min \\left\\lbrace 1, \\frac{k}{\\sigma _{1:t}}\\right\\rbrace \\sqrt{ \\left\\Vert \\tilde{y}_i\\right\\Vert ^2 + C (k-i) }\\\\& \\le \\left<f_i,y_i\\right> - \\left<\\hat{f}_{t},\\tilde{y}_{i}\\right> + 2\\min \\left\\lbrace 1, \\frac{k}{\\sigma _{1:t}}\\right\\rbrace \\sqrt{ \\left\\Vert \\tilde{y}_i\\right\\Vert ^2 + \\left<\\nabla \\left\\Vert \\tilde{y}_{i-1}\\right\\Vert ^2,\\mathbf {y}_i\\right>+ C (k-i+1) }\\\\& = \\left<f_i - \\hat{f}_{t},y_i\\right> - \\left<\\hat{f}_{t},\\tilde{y}_{i-1}\\right> + 2\\min \\left\\lbrace 1, \\frac{k}{\\sigma _{1:t}}\\right\\rbrace \\sqrt{ \\left\\Vert \\tilde{y}_{i-1}\\right\\Vert ^2 + \\left<\\nabla \\left\\Vert \\tilde{y}_{i-1}\\right\\Vert ^2,\\mathbf {x}_i\\right>+ C (k-i+1) }$ and we start block at $\\hat{f}_t$ .", "For the first block, this value is 0 but later on it is the empirical risk minimizer.", "We therefore get a mixture of Follow the Leader (FTL) and Gradient Descent (GD) algorithms.", "If block size is 1, we get FTL only, and when the block size is $T$ we get GD only.", "In general, however, the resulting method is an interesting mixture of the two.", "Using the arguments of Proposition REF , the update in the block is given by $f_{t+i} = \\hat{f}_t - \\max \\left\\lbrace 1 ,\\frac{k}{ \\sigma _{1:t} }\\right\\rbrace \\frac{- \\nabla \\left\\Vert \\tilde{y}_{i-1}^2\\right\\Vert }{\\sqrt{ \\left\\Vert \\tilde{y}_{i-1}\\right\\Vert ^2 + C (k-i+1) }}$ Now that we have shown the admissibility of the relaxation and the form of update obtained by the relaxation we turn to the bounds on the regret specified in the lemma.", "We shall provide these bounds using Lemma REF .", "We will split the analysis to two cases, one when $\\alpha > 1/2$ and other when $\\alpha \\le 1/2$ .", "Case $\\alpha > \\frac{1}{2}$ : To start note that since we initialize the block lengths with the doubling trick, that is initialize block lengths as $1, 2, 4, \\ldots $ hence, after $t$ rounds the maximum length of current block say $k$ can be at most $2t$ and so $\\sqrt{k} \\le \\sqrt{2t}$ .", "Now let us first consider the case when $\\alpha > \\frac{1}{2}$ .", "In this case, since $\\sigma _{1:t} = B t^{\\alpha }$ , we can conclude that the condition $\\sigma _{1:t} \\ge \\sqrt{k}$ is satisfied as long as $t^{\\alpha - \\frac{1}{2}} \\ge \\frac{\\sqrt{2}}{B}$ .", "Since we are considering the case when $\\alpha > \\frac{1}{2}$ we can conclude that for all rounds larger than $\\sqrt{2}/B$ , the blocking strategy always picks block size of 1.", "Hence applying Lemma REF we conclude that in the case when $1 > \\alpha > 1/2$ (or when $\\alpha = 1/2$ and $B \\ge \\sqrt{2}$ ), $\\mathbf {Reg}_T \\le \\sum _{t=1}^T \\frac{1}{\\sigma _{1:t}} = \\sum _{t=1}^T \\frac{1}{B t^\\alpha } = O(T^{1-\\alpha }/B)$ Also note that for the case when $\\alpha = 1$ , the summation is bounded by $O(\\log T)$ and so $\\mathbf {Reg}_T \\le \\sum _{t=1}^T \\frac{1}{\\sigma _{1:t}} = \\sum _{t=1}^T \\frac{1}{B t^\\alpha } = O(\\log T / B)$ Case $\\alpha \\le \\frac{1}{2}$ : Now we consider the case when $\\alpha < 1/2$ .", "Say we are at start of some block $t = 2^m$ .", "The initial block length then is $2 t$ by the doubling trick initialization.", "Now within this block, the adaptive algorithm continues with this current block until the point when the square-root of the remaining number of rounds in the block say $k$ becomes smaller than $\\sigma _{1 : t + (2t - k)}$ .", "That is until $\\sqrt{k} \\le B (3t - k)^{\\alpha }$ The regret on this block can be bounded using Lemma REF (notice that here we use the lemma for the algorithm within a sub-block initialized by the doubling trick rather than on the entire $T$ rounds).", "The regret on this block is bounded as : $\\mathbf {Rel}_{2t - k}\\left(\\mathcal {F}_{r(x_1,\\ldots ,x_t)} \\right) + \\sum _{i=2t - k+1}^{2t} \\mathbf {Rel}_{1}\\left(\\mathcal {F}_{r(x_1,\\ldots ,x_{i})} \\right) & \\le \\sqrt{2t -k} + \\sum _{j=2t -k +1}^{2 t} \\frac{1}{B j^{\\alpha }} \\\\& \\le \\sqrt{2t} + \\sum _{j=2t -k +1}^{2 t} \\frac{1}{B j^{\\alpha }} \\\\& \\le \\sqrt{2t} + \\frac{1}{B}\\left( (2t+1)^{1 - \\alpha } - (2t - k +1)^{1 - \\alpha } \\right)\\\\& \\le \\sqrt{2t} + \\frac{k^{1 - \\alpha }}{B}\\\\& \\le \\sqrt{2t} + \\frac{B^{2(1 - \\alpha )} (3t)^{2\\alpha (1 - \\alpha )} }{B} & \\textrm {(using Eq.", "(\\ref {eq:int1}))}\\\\& \\le \\sqrt{2t} + B^{2(1 - \\alpha )- 1} \\sqrt{3t} \\\\& \\le \\sqrt{12\\ t}$ Hence overall regret is bounded as $\\mathbf {Reg}_T \\le \\sum _{i=1}^{\\lceil \\log _2 T \\rceil + 1} \\sqrt{12\\ \\times 2^{i-1}} \\le \\sqrt{12} \\sum _{i=1}^{\\lceil \\log _2 T \\rceil + 1} 2^{(i-1)/2} \\le O(\\sqrt{T})$ This concludes the proof.", "[Proof of Lemma REF . ]", "Notice that by doubling trick for at most first $2 \\tau $ rounds we simply play the experts algorithm, thus suffering a maximum regret that is minimum of $\\tau $ and $4 \\sqrt{\\tau \\log |\\mathcal {F}|}$ .", "After these initial number of rounds, consider any round $t$ at which we start a new block with the blocking strategy described above.", "The first sub-block given by the blocking strategy is of length at most $k$ , thanks to our assumption about the gap between the leader and the second-best action.", "Clearly the minimizer of cumulative loss up to $t$ rounds already played, $\\underset{f \\in \\mathcal {F}}{\\mathrm {argmin}} \\ \\sum _{i=1}^t \\ell (f,x_i)$ , is going to be the leader at least for the next $k$ rounds.", "Hence for this block we suffer no regret.", "Now when we use the same blocking strategy repeatedly, due to the same reasoning, we end up playing the same leader for the rest of the game only in chunks of size $k$ , and thus suffer no regret for the rest of the game.", "[Proof of Proposition  REF] We would like to show that, with the distribution $q^*_t$ defined in (REF ), $&\\max _{y_t \\in \\lbrace \\pm 1\\rbrace } \\left\\lbrace \\underset{\\hat{y}_t \\sim q^*_t}{\\mathbb {E}}|\\hat{y}_t-y_t| + \\mathbf {Rel}_{T}\\left(\\mathcal {F} | (x^t,y^t) \\right)\\right\\rbrace \\le \\mathbf {Rel}_{T}\\left(\\mathcal {F} | (x^{t-1},y^{t-1}) \\right)$ for any $x_t\\in \\mathcal {X}$ .", "Let $\\sigma \\in \\lbrace \\pm 1\\rbrace ^{t-1}$ and $\\sigma _t\\in \\lbrace \\pm 1\\rbrace $ .", "We have $&\\mathbf {Rel}_{T}\\left(\\mathcal {F} | (x^t,y^t) \\right) - 2\\lambda (T-t) \\\\&=\\frac{1}{\\lambda }\\log \\left( \\sum _{(\\sigma ,\\sigma _t) \\in \\mathcal {F}|_{x^t}} g(\\mathrm {Ldim}(\\mathcal {F}_t(\\sigma ,\\sigma _t)),T-t) \\exp \\left\\lbrace - \\lambda L_{t-1}(\\sigma ) \\right\\rbrace \\exp \\left\\lbrace - \\lambda |\\sigma _t-y_t| \\right\\rbrace \\right) \\\\&\\le \\frac{1}{\\lambda }\\log \\left( \\sum _{\\sigma _t\\in \\lbrace \\pm 1\\rbrace } \\exp \\left\\lbrace - \\lambda |\\sigma _t-y_t| \\right\\rbrace \\sum _{\\sigma : (\\sigma ,\\sigma _t) \\in \\mathcal {F}|_{x^t}} g(\\mathrm {Ldim}(\\mathcal {F}_t(\\sigma ,\\sigma _t)),T-t) \\exp \\left\\lbrace - \\lambda L_{t-1}(\\sigma ) \\right\\rbrace \\right)$ Just as in the proof of Proposition REF , we may think of the two choices $\\sigma _t$ as the two experts whose weighting $q_t^*$ is given by the sum involving the Littlestone's dimension of subsets of $\\mathcal {F}$ .", "Introducing the normalization term, we arrive at the upper bound $&\\frac{1}{\\lambda }\\log \\left( \\mathbb {E}_{\\sigma _t\\sim q^*_t} \\exp \\left\\lbrace - \\lambda |\\sigma _t-y_t| \\right\\rbrace \\right) + \\frac{1}{\\lambda }\\log \\left( \\sum _{\\sigma _t\\in \\lbrace \\pm 1\\rbrace }\\sum _{\\sigma : (\\sigma ,\\sigma _t) \\in \\mathcal {F}|_{x^t}} g(\\mathrm {Ldim}(\\mathcal {F}_t(\\sigma ,\\sigma _t)),T-t) \\exp \\left\\lbrace - \\lambda L_{t-1}(\\sigma ) \\right\\rbrace \\right) \\\\&\\le - \\mathbb {E}_{\\sigma _t\\sim q^*_t} |\\sigma _t-y_t| + 2\\lambda + \\frac{1}{\\lambda }\\log \\left( \\sum _{\\sigma _t\\in \\lbrace \\pm 1\\rbrace }\\sum _{\\sigma : (\\sigma ,\\sigma _t) \\in \\mathcal {F}|_{x^t}} g(\\mathrm {Ldim}(\\mathcal {F}_t(\\sigma ,\\sigma _t)),T-t) \\exp \\left\\lbrace - \\lambda L_{t-1}(\\sigma ) \\right\\rbrace \\right)$ The last step is due to Lemma A.1 in [7].", "It remains to show that the log normalization term is upper bounded by the relaxation at the previous step: $& \\frac{1}{\\lambda }\\log \\left( \\sum _{\\sigma _t\\in \\lbrace \\pm 1\\rbrace } \\sum _{\\sigma : (\\sigma ,\\sigma _t) \\in \\mathcal {F}|_{x^t}} g(\\mathrm {Ldim}(\\mathcal {F}_t(\\sigma ,\\sigma _t)),T-t) \\exp \\left\\lbrace - \\lambda L_{t-1}(\\sigma ) \\right\\rbrace \\right)\\\\&\\le \\frac{1}{\\lambda }\\log \\left( \\sum _{\\sigma \\in \\mathcal {F}|_{x^{t-1}}} \\exp \\left\\lbrace - \\lambda L_{t-1}(\\sigma ) \\right\\rbrace \\sum _{\\sigma _t\\in \\lbrace \\pm 1\\rbrace } g(\\mathrm {Ldim}(\\mathcal {F}_t(\\sigma ,\\sigma _t)),T-t) \\right) \\\\&\\le \\frac{1}{\\lambda }\\log \\left( \\sum _{\\sigma \\in \\mathcal {F}|_{x^{t-1}}} \\exp \\left\\lbrace - \\lambda L_{t-1}(\\sigma ) \\right\\rbrace g(\\mathrm {Ldim}(\\mathcal {F}_{t-1}(\\sigma )),T-t+1) \\right) \\\\&= \\mathbf {Rel}_{T}\\left(\\mathcal {F} | (x^{t-1},y^{t-1}) \\right)$ To justify the last inequality, note that $\\mathcal {F}_{t-1}(\\sigma )=\\mathcal {F}_{t}(\\sigma ,+1)\\cup \\mathcal {F}_t(\\sigma ,-1)$ and at most one of $\\mathcal {F}_t(\\sigma ,+1)$ or $\\mathcal {F}_t(\\sigma ,-1)$ can have Littlestone's dimension $\\mathrm {Ldim}(\\mathcal {F}_{t-1}(\\sigma ))$ .", "We now appeal to the recursion $g(d,T-t)+g(d-1,T-t) \\le g(d, T-t+1)$ where $g(d,T-t)$ is the size of the zero cover for a class with Littlestone's dimension $d$ on the worst-case tree of depth $T-t$ (see [14]).", "This completes the proof of admissibility." ], [ "Alternative Method", "Let us now derive the algorithm given in (REF ) and prove its admissibility.", "Once again, consider the optimization problem $&\\max _{y_t \\in \\lbrace \\pm 1\\rbrace } \\left\\lbrace \\underset{\\hat{y}_t \\sim q^*_t}{\\mathbb {E}}|\\hat{y}_t-y_t| + \\mathbf {Rel}_{T}\\left(\\mathcal {F} | (x^t,y^t) \\right)\\right\\rbrace $ with the relaxation $\\mathbf {Rel}_{T}\\left(\\mathcal {F} | (x^t,y^t) \\right) = \\frac{1}{\\lambda }\\log \\left( \\sum _{\\sigma \\in \\mathcal {F}|_{x^t}} g(\\mathrm {Ldim}(\\mathcal {F}_t(\\sigma )),T-t) \\exp \\left\\lbrace - \\lambda L_{t}(\\sigma ) \\right\\rbrace \\right) + \\frac{\\lambda }{2} (T-t)$ The maximum can be written explicitly, as in Section : $\\max &\\left\\lbrace 1-q^*_t + \\frac{1}{\\lambda }\\log \\left( \\sum _{(\\sigma ,\\sigma _t) \\in \\mathcal {F}|_{x^t}} g(\\mathrm {Ldim}(\\mathcal {F}_t(\\sigma ,\\sigma _t)),T-t) \\exp \\left\\lbrace - \\lambda L_{t-1}(\\sigma ) \\right\\rbrace \\exp \\left\\lbrace - \\lambda (1-\\sigma _t) \\right\\rbrace \\right) , \\right.\\\\&\\left.", "1+q^*_t + \\frac{1}{\\lambda }\\log \\left( \\sum _{(\\sigma ,\\sigma _t) \\in \\mathcal {F}|_{x^t}} g(\\mathrm {Ldim}(\\mathcal {F}_t(\\sigma ,\\sigma _t)),T-t) \\exp \\left\\lbrace - \\lambda L_{t-1}(\\sigma ) \\right\\rbrace \\exp \\left\\lbrace - \\lambda (1+\\sigma _t) \\right\\rbrace \\right) \\right\\rbrace $ where we have dropped the $\\frac{\\lambda }{2}(T-t)$ term from both sides.", "Equating the two values, we obtain $2q^*_t = \\frac{1}{\\lambda }\\log \\frac{ \\sum _{(\\sigma ,\\sigma _t) \\in \\mathcal {F}|_{x^t}} g(\\mathrm {Ldim}(\\mathcal {F}_t(\\sigma ,\\sigma _t)),T-t) \\exp \\left\\lbrace - \\lambda L_{t-1}(\\sigma ) \\right\\rbrace \\exp \\left\\lbrace - \\lambda (1-\\sigma _t) \\right\\rbrace }{ \\sum _{(\\sigma ,\\sigma _t) \\in \\mathcal {F}|_{x^t}} g(\\mathrm {Ldim}(\\mathcal {F}_t(\\sigma ,\\sigma _t)),T-t) \\exp \\left\\lbrace - \\lambda L_{t-1}(\\sigma ) \\right\\rbrace \\exp \\left\\lbrace - \\lambda (1+\\sigma _t) \\right\\rbrace }$ The resulting value becomes $&1+\\frac{\\lambda }{2}(T-t)+\\frac{1}{2\\lambda }\\log \\left\\lbrace \\sum _{(\\sigma ,\\sigma _t) \\in \\mathcal {F}|_{x^t}} g(\\mathrm {Ldim}(\\mathcal {F}_t(\\sigma ,\\sigma _t)),T-t) \\exp \\left\\lbrace - \\lambda L_{t-1}(\\sigma ) \\right\\rbrace \\exp \\left\\lbrace - \\lambda (1-\\sigma _t) \\right\\rbrace \\right\\rbrace \\\\&~~~~~~~~~~~~~~~~+ \\frac{1}{2\\lambda }\\log \\left\\lbrace \\sum _{(\\sigma ,\\sigma _t) \\in \\mathcal {F}|_{x^t}} g(\\mathrm {Ldim}(\\mathcal {F}_t(\\sigma ,\\sigma _t)),T-t) \\exp \\left\\lbrace - \\lambda L_{t-1}(\\sigma ) \\right\\rbrace \\exp \\left\\lbrace - \\lambda (1+\\sigma _t) \\right\\rbrace \\right\\rbrace \\\\&= 1+\\frac{\\lambda }{2}(T-t)+\\frac{1}{\\lambda }\\mathbb {E}_\\epsilon \\log \\left\\lbrace \\sum _{(\\sigma ,\\sigma _t) \\in \\mathcal {F}|_{x^t}} g(\\mathrm {Ldim}(\\mathcal {F}_t(\\sigma ,\\sigma _t)),T-t) \\exp \\left\\lbrace - \\lambda L_{t-1}(\\sigma ) \\right\\rbrace \\exp \\left\\lbrace - \\lambda (1-\\epsilon \\sigma _t) \\right\\rbrace \\right\\rbrace \\\\&\\le 1+\\frac{\\lambda }{2}(T-t)+ \\frac{1}{\\lambda }\\log \\left\\lbrace \\sum _{(\\sigma ,\\sigma _t) \\in \\mathcal {F}|_{x^t}} g(\\mathrm {Ldim}(\\mathcal {F}_t(\\sigma ,\\sigma _t)),T-t) \\exp \\left\\lbrace - \\lambda L_{t-1}(\\sigma ) \\right\\rbrace \\mathbb {E}_\\epsilon \\exp \\left\\lbrace - \\lambda (1-\\epsilon \\sigma _t) \\right\\rbrace \\right\\rbrace $ for a Rademacher random variable $\\epsilon \\in \\lbrace \\pm 1\\rbrace $ .", "Now, $\\mathbb {E}_\\epsilon \\exp \\left\\lbrace - \\lambda (1-\\epsilon \\sigma _t) \\right\\rbrace = e^{-\\lambda }\\mathbb {E}_\\epsilon e^{\\lambda \\epsilon \\sigma _t} \\le e^{-\\lambda } e^{\\lambda ^2/2}$ Substituting this into the above expression, we obtain an upper bound of $\\frac{\\lambda }{2}(T-t+1)+ \\frac{1}{\\lambda }\\log \\left\\lbrace \\sum _{(\\sigma ,\\sigma _t) \\in \\mathcal {F}|_{x^t}} g(\\mathrm {Ldim}(\\mathcal {F}_t(\\sigma ,\\sigma _t)),T-t) \\exp \\left\\lbrace - \\lambda L_{t-1}(\\sigma ) \\right\\rbrace \\right\\rbrace $ which completes the proof of admissibility using the same combinatorial argument as in the earlier part of the proof." ], [ "Arriving at the Relaxation", "Finally, we show that the relaxation we use arises naturally as an upper bound on the sequential Rademacher complexity.", "Fix a tree $\\mathbf {x}$ .", "Let $\\sigma \\in \\lbrace \\pm 1\\rbrace ^{t-1}$ be a sequence of signs.", "Observe that given history $x^t=(x_1,\\ldots ,x_t)$ , the signs $\\epsilon \\in \\lbrace \\pm 1\\rbrace ^{T-t}$ , and a tree $\\mathbf {x}$ , the function class $\\mathcal {F}$ takes on only a finite number of possible values $(\\sigma , \\sigma _t, \\omega )$ on $(x^t,\\mathbf {x}(\\epsilon ))$ .", "Here, $\\mathbf {x}(\\epsilon )$ denotes the sequences of values along the path $\\epsilon $ .", "We have, $\\sup _{\\mathbf {x}} \\mathbb {E}_{\\epsilon } \\sup _{f \\in \\mathcal {F}} \\left\\lbrace 2 \\sum _{i=1}^{T-t} \\epsilon _i f(\\mathbf {x}_i(\\epsilon )) - \\sum _{i=1}^{t} |f(x_i)-y_i|\\right\\rbrace &= \\sup _{\\mathbf {x}} \\mathbb {E}_{\\epsilon } \\max _{\\sigma _t\\in \\lbrace \\pm 1\\rbrace } \\max _{(\\sigma ,\\omega ): (\\sigma ,\\sigma _t,\\omega )\\in \\mathcal {F}|_{(x^t, \\mathbf {x}(\\epsilon ))}} \\left\\lbrace 2 \\sum _{i=1}^{T-t} \\epsilon _i \\omega _i - \\sum _{i=1}^{t} |\\sigma _i-y_i|\\right\\rbrace \\\\&\\le \\sup _{\\mathbf {x}} \\mathbb {E}_{\\epsilon } \\max _{\\sigma _t\\in \\lbrace \\pm 1\\rbrace } \\max _{\\sigma : (\\sigma ,\\sigma _t) \\in \\mathcal {F}|_{x^t}}\\max _{\\mathbf {v}\\in V(\\mathcal {F}(\\sigma ,\\sigma _t),\\mathbf {x})} \\left\\lbrace 2 \\sum _{i=1}^{T-t} \\epsilon _i \\mathbf {v}_i(\\epsilon ) - \\sum _{i=1}^{t} |\\sigma _i-y_i|\\right\\rbrace $ where $\\mathcal {F}|_{(x^t,\\mathbf {x}(\\epsilon ))}$ is the projection of $\\mathcal {F}$ onto $(x^t, \\mathbf {x}(\\epsilon ))$ , $\\mathcal {F}(\\sigma ,\\sigma _t)=\\lbrace f\\in \\mathcal {F}: f(x^t)=(\\sigma ,\\sigma _t)\\rbrace $ , and $V(\\mathcal {F}(\\sigma ,\\sigma _t),\\mathbf {x})$ is the zero-cover of the set $\\mathcal {F}(\\sigma ,\\sigma _t)$ on the tree $\\mathbf {x}$ .", "We then have the following relaxation: $\\frac{1}{\\lambda }\\log \\left( \\sup _{\\mathbf {x}} \\mathbb {E}_{\\epsilon } \\sum _{\\sigma _t\\in \\lbrace \\pm 1\\rbrace } \\sum _{\\sigma : (\\sigma ,\\sigma _t) \\in \\mathcal {F}|_{x^t}}\\sum _{\\mathbf {v}\\in V(\\mathcal {F}(\\sigma ,\\sigma _t),\\mathbf {x})} \\exp \\left\\lbrace 2\\lambda \\sum _{i=1}^{T-t} \\epsilon _i \\mathbf {v}_i(\\epsilon ) - \\lambda L_{t}(\\sigma ,\\sigma _t) \\right\\rbrace \\right)$ where $L_{t}(\\sigma ,\\sigma _t) = \\sum _{i=1}^{t} |\\sigma _i-y_i|$ .", "The latter quantity can be factorized: $&\\frac{1}{\\lambda }\\log \\left( \\sup _{\\mathbf {x}} \\sum _{\\sigma _t\\in \\lbrace \\pm 1\\rbrace } \\sum _{\\sigma : (\\sigma ,\\sigma _t) \\in \\mathcal {F}|_{x^t}} \\exp \\left\\lbrace - \\lambda L_{t}(\\sigma ,\\sigma _t) \\right\\rbrace \\mathbb {E}_{\\epsilon } \\sum _{\\mathbf {v}\\in V(\\mathcal {F}(\\sigma ,\\sigma _t),\\mathbf {x})} \\exp \\left\\lbrace 2\\lambda \\sum _{i=1}^{T-t} \\epsilon _i \\mathbf {v}_i(\\epsilon ) \\right\\rbrace \\right) \\\\&\\le \\frac{1}{\\lambda }\\log \\left( \\sup _{\\mathbf {x}} \\sum _{\\sigma _t\\in \\lbrace \\pm 1\\rbrace } \\sum _{\\sigma : (\\sigma ,\\sigma _t) \\in \\mathcal {F}|_{x^t}} \\exp \\left\\lbrace - \\lambda L_{t}(\\sigma ,\\sigma _t) \\right\\rbrace \\text{card}(V(\\mathcal {F}(\\sigma ,\\sigma _t),\\mathbf {x})) \\exp \\left\\lbrace 2\\lambda ^2 (T-t) \\right\\rbrace \\right) \\\\&\\le \\frac{1}{\\lambda }\\log \\left( \\sum _{\\sigma _t\\in \\lbrace \\pm 1\\rbrace } \\exp \\left\\lbrace - \\lambda |\\sigma _t-y_t| \\right\\rbrace \\sum _{\\sigma : (\\sigma ,\\sigma _t) \\in \\mathcal {F}|_{x^t}} g(\\mathrm {Ldim}(\\mathcal {F}(\\sigma ,\\sigma _t)),T-t) \\exp \\left\\lbrace - \\lambda L_{t-1}(\\sigma ) \\right\\rbrace \\right) + 2\\lambda (T-t) \\ .$ This concludes the derivation of the relaxation.", "[Proof of Lemma REF] We first exhibit the proof for the convex loss case.", "To show admissibility using the particular randomized strategy $q_t$ given in the lemma, we need to show that $\\sup _{x_t} \\left\\lbrace \\mathbb {E}_{f \\sim q_t}\\left[ \\ell (f,x_t) \\right] + \\mathbf {Rel}_{T}\\left(\\mathcal {F} | x_1,\\ldots ,x_t \\right)\\right\\rbrace \\le \\mathbf {Rel}_{T}\\left(\\mathcal {F} | x_1,\\ldots ,x_{t-1} \\right)$ The strategy $q_t$ proposed by the lemma is such that we first draw $x_{t+1},\\ldots ,x_T \\sim D$ and $\\epsilon _{t+1},\\ldots \\epsilon _T$ Rademacher random variables, and then based on this sample pick $f_t=f_t(x_{t+1:T},\\epsilon _{t+1:T})$ as in (REF ).", "Hence, $\\sup _{x_t} &\\left\\lbrace \\mathbb {E}_{f \\sim q_t}\\left[ \\ell (f,x_t) \\right] + \\mathbf {Rel}_{T}\\left(\\mathcal {F} | x_1,\\ldots ,x_t \\right)\\right\\rbrace \\\\& = \\sup _{x_t} \\left\\lbrace \\underset{\\underset{x_{t+1:T}}{\\epsilon _{t+1:T}}}{\\mathbb {E}} \\ell (f_t,x) + \\underset{\\underset{x_{t+1:T}}{\\epsilon _{t+1:T}}}{\\mathbb {E}} \\sup _{f \\in \\mathcal {F}} \\left[ C \\sum _{i=t+1}^T \\epsilon _i \\ell (f,x_i) - L_t(f) \\right] \\right\\rbrace \\\\& \\le \\underset{\\underset{x_{t+1:T}}{\\epsilon _{t+1:T}}}{\\mathbb {E}} \\sup _{x_t} \\left\\lbrace \\ell (f_t,x) + \\sup _{f \\in \\mathcal {F}} \\left[ C \\sum _{i=t+1}^T \\epsilon _i \\ell (f,x_i) - L_t(f) \\right]\\right\\rbrace $ where $L_t(f) = \\sum _{i=1}^t \\ell (f,x_i)$ .", "Observe that our strategy “matched the randomness” arising from the relaxation!", "Now, with $f_t$ defined as $f_t = \\underset{g \\in \\mathcal {F}}{\\mathrm {argmin}} \\ \\sup _{x_t\\in \\mathcal {X}} \\left\\lbrace \\ell (g,x_t) + \\sup _{f \\in \\mathcal {F}} \\left[ C \\sum _{i=t+1}^T \\epsilon _i \\ell (f,x_i) - L_t(f) \\right] \\right\\rbrace $ for any given $x_{t+1:T},\\epsilon _{t+1:T}$ , we have $\\sup _{x_t} & \\left\\lbrace \\ell (f_t,x_t) + \\sup _{f \\in \\mathcal {F}} \\left[ C \\sum _{i=t+1}^T \\epsilon _i \\ell (f,x_i) - L_t(f) \\right] \\right\\rbrace = \\inf _{g \\in \\mathcal {F}} \\sup _{x_t} \\left\\lbrace \\ell (g,x_t) + \\sup _{f \\in \\mathcal {F}} \\left[ C \\sum _{i=t+1}^T \\epsilon _i \\ell (f,x_i) - L_t(f) \\right] \\right\\rbrace $ We can conclude that for this choice of $q_t$ , $\\sup _{x_t} &\\left\\lbrace \\mathbb {E}_{f \\sim q_t}\\left[ \\ell (f,x_t) \\right] + \\mathbf {Rel}_{T}\\left(\\mathcal {F} | x_1,\\ldots ,x_t \\right)\\right\\rbrace \\le \\underset{\\underset{x_{t+1:T}}{\\epsilon _{t+1:T}}}{\\mathbb {E}} \\inf _{g \\in \\mathcal {F}} \\sup _{x_t} \\left\\lbrace \\ell (g,x_t) + \\sup _{f \\in \\mathcal {F}} \\left[ C \\sum _{i=t+1}^T \\epsilon _i \\ell (f,x_i) - L_t(f) \\right]\\right\\rbrace \\\\& = \\underset{\\underset{x_{t+1:T}}{\\epsilon _{t+1:T}}}{\\mathbb {E}} \\inf _{g \\in \\mathcal {F}} \\sup _{p_t \\in \\Delta (\\mathcal {X})} \\mathbb {E}_{x_t \\sim p_t}\\left[ \\ell (g,x_t) + \\sup _{f \\in \\mathcal {F}} \\left[ C \\sum _{i=t+1}^T \\epsilon _i \\ell (f,x_i) - L_t(f) \\right] \\right]\\\\& = \\underset{\\underset{x_{t+1:T}}{\\epsilon _{t+1:T}}}{\\mathbb {E}}\\sup _{p \\in \\Delta (\\mathcal {X}) } \\inf _{g \\in \\mathcal {F}} \\left\\lbrace \\mathbb {E}_{x_t \\sim p}\\left[ \\ell (g,x_t) \\right] + \\mathbb {E}_{x_t \\sim p}\\left[ \\sup _{f \\in \\mathcal {F}} C \\sum _{i=t+1}^T \\epsilon _i \\ell (f,x_i) - L_t(f) \\right] \\right\\rbrace $ In the last step we appealed to the minimax theorem which holds as loss is convex in $g$ and $\\mathcal {F}$ is a compact convex set and the term in the expectation is linear in $p_t$ , as it is an expectation.", "The last expression can be written as $&\\underset{\\underset{x_{t+1:T}}{\\epsilon _{t+1:T}}}{\\mathbb {E}} \\sup _{p \\in \\Delta (\\mathcal {X}) }\\mathbb {E}_{x_t \\sim p} \\sup _{f \\in \\mathcal {F}} \\left[ C \\sum _{i=t+1}^T \\epsilon _i \\ell (f,x_i) - L_{t-1}(f) + \\inf _{g \\in \\mathcal {F}} \\mathbb {E}_{x_t \\sim p}\\left[ \\ell (g,x_t) \\right] - \\ell (f,x_t) \\right] \\\\& \\le \\underset{\\underset{x_{t+1:T}}{\\epsilon _{t+1:T}}}{\\mathbb {E}} \\sup _{p \\in \\Delta (\\mathcal {X}) }\\mathbb {E}_{x_t \\sim p} \\sup _{f \\in \\mathcal {F}} \\left[ C \\sum _{i=t+1}^T \\epsilon _i \\ell (f,x_i) - L_{t-1}(f) + \\mathbb {E}_{x_t \\sim p}\\left[ \\ell (f,x_t) \\right] - \\ell (f,x_t) \\right] \\\\& \\le \\underset{\\underset{x_{t+1:T}}{\\epsilon _{t+1:T}}}{\\mathbb {E}} \\mathbb {E}_{x_t \\sim D}\\mathbb {E}_{\\epsilon _t} \\sup _{f \\in \\mathcal {F}} \\left[ C \\sum _{i=t+1}^T \\epsilon _i \\ell (f,x_i) - L_{t-1}(f) + C \\epsilon _t \\ell (f,x_t) \\right]\\\\& = \\mathbf {Rel}_{T}\\left(\\mathcal {F} | x_1,\\ldots ,x_{t-1} \\right)$ Last inequality is by Assumption REF , using which we can replace a draw from supremum over distributions by a draw from the “equivalently bad” fixed distribution $D$ by suffering an extra factor of $C$ multiplied to that random instance.", "The key step where we needed convexity was to use minimax theorem to swap infimum and supremum inside the expectation.", "In general the minimax theorem need not hold.", "In the non-convex scenario this is the reason we add the extra randomization through $\\hat{q}_t$ .", "The non-convex case has a similar proof except that we have expectation w.r.t.", "$\\hat{q}_t$ extra on each round which essentially convexifies our loss and thus allows us to appeal to the minimax theorem.", "[Proof of Lemma  REF] Let $w\\in \\mathbb {R}^N$ be arbitrary.", "Throughout this proof, let $\\epsilon \\in \\lbrace \\pm 1\\rbrace $ be a single Rademacher random variable, rather than a vector.", "To prove (REF ), observe that $\\sup _{p \\in \\Delta (\\mathcal {X})} \\underset{x_t \\sim p}{\\mathbb {E}} \\left\\Vert w + \\underset{x \\sim p}{\\mathbb {E}} [x] - x_t \\right\\Vert _\\infty &\\le \\sup _{p \\in \\Delta (\\mathcal {X})} \\underset{x,x^{\\prime } \\sim p}{\\mathbb {E}} \\left\\Vert w + x^{\\prime } - x \\right\\Vert _\\infty \\\\&= \\sup _{p \\in \\Delta (\\mathcal {X})} \\underset{x,x^{\\prime } \\sim p}{\\mathbb {E}}\\mathbb {E}_\\epsilon \\left\\Vert w + \\epsilon (x^{\\prime } - x) \\right\\Vert _\\infty \\\\&\\le \\sup _{x,x^{\\prime }\\in \\mathcal {X}} \\mathbb {E}_\\epsilon \\left\\Vert w + \\epsilon (x^{\\prime } - x) \\right\\Vert _\\infty \\\\&\\le \\sup _{x^{\\prime }\\in \\mathcal {X}} \\mathbb {E}_\\epsilon \\left\\Vert w/2 + \\epsilon x^{\\prime }\\right\\Vert _\\infty + \\sup _{x\\in \\mathcal {X}} \\mathbb {E}_\\epsilon \\left\\Vert w/2 - \\epsilon x\\right\\Vert _\\infty \\\\&= \\sup _{x\\in \\mathcal {X}} \\mathbb {E}_\\epsilon \\max _{i\\in [N]} \\left| w_i + 2\\epsilon x_i\\right|$ The supremum over $x\\in \\mathcal {X}$ is achieved at the vertices of $\\mathcal {X}$ since the expected maximum is a convex function.", "It remains to prove the identity $\\max _{x\\in \\lbrace \\pm 1\\rbrace ^N} \\mathbb {E}_\\epsilon \\max _{i\\in [N]} \\left| w_i + 2\\epsilon x_i\\right| \\le \\underset{x \\sim D }{\\mathbb {E}}\\underset{\\epsilon }{\\mathbb {E}}\\max _{i\\in [N]} \\left| w_i+ 6\\epsilon x_i \\right|$ Let $i^*=\\underset{i}{\\mathrm {argmax}} \\ |w_i|$ and $j^*=\\underset{i\\ne i^*}{\\mathrm {argmax}} \\ |w_i|$ be the coordinates with largest and second-largest magnitude.", "If $|w_{i^*}|-|w_{j^*}| \\ge 4$ , the statement follows since, for any $x\\in \\lbrace \\pm 1\\rbrace ^N$ and $\\epsilon \\in \\lbrace \\pm 1\\rbrace $ , $ \\max _{i\\ne i^*} \\left| w_i + 2\\epsilon x_i\\right|\\le \\max _{i\\ne i^*}\\left| w_{i} \\right| + 2 \\le \\left| w_{i^*} \\right| -2 \\le |w_{i^*}+2\\epsilon x_{i^*}|,$ and thus $\\max _{x\\in \\lbrace \\pm 1\\rbrace ^N} \\mathbb {E}_\\epsilon \\max _{i\\in [N]} \\left| w_i + 2\\epsilon x_i\\right| = \\max _{x\\in \\lbrace \\pm 1\\rbrace ^N} \\mathbb {E}_\\epsilon \\left| w_{i^*} + 2\\epsilon x_{i^*}\\right| = |w_{i^*}| = \\mathbb {E}_{x,\\epsilon } |w_{i^*}+6\\epsilon x_{i^*}| \\le \\mathbb {E}_{x,\\epsilon } \\max _{i}|w_{i}+6\\epsilon x_{i}|.$ It remains to consider the case when $|w_{i^*}|- |w_{j^*}| < 4$ .", "We have that $\\mathbb {E}_{x ,\\epsilon }\\max _{i\\in [N]} \\left| w_i+ 6\\epsilon x_i \\right| \\ge \\mathbb {E}_{x ,\\epsilon }\\max _{i\\in \\lbrace i^*,j^*\\rbrace } \\left| w_i+ 6\\epsilon x_i \\right| &\\ge \\frac{1}{2} (|w_{i^*}|+6)+ \\frac{1}{4} (|w_{i^*}|-6) + \\frac{1}{4} (|w_{j^*}|+6) \\ge |w_{i^*}|+2 \\\\&\\ge \\max _{x\\in \\lbrace \\pm 1\\rbrace ^N} \\mathbb {E}_\\epsilon \\max _{i\\in [N]} \\left| w_i + 2\\epsilon x_i\\right|,$ where $1/2$ is the probability that $\\epsilon x_{i^*} = sign(w_{i^*})$ , the second event of probability $1/4$ is the event that $\\epsilon x_{i^*} \\ne sign(w_{i^*})$ and $\\epsilon x_{j^*} \\ne sign(w_{j^*})$ , while the third event of probability $1/4$ is that $\\epsilon x_{i^*} \\ne sign(w_{i^*})$ and $\\epsilon x_{j^*} = sign(w_{j^*})$ .", "[Proof of Lemma REF] Let $w\\in \\mathbb {R}^N$ be arbitrary.", "Just as in the proof of Lemma REF , we need to show $\\max _{x\\in \\lbrace \\pm 1\\rbrace ^N} \\mathbb {E}_\\epsilon \\max _{i\\in [N]} \\left| w_i + 2\\epsilon x_i\\right| \\le \\underset{x \\sim D }{\\mathbb {E}}\\underset{\\epsilon }{\\mathbb {E}}\\max _{i\\in [N]} \\left| w_i+ C \\epsilon x_i \\right|$ Let $i^*=\\underset{i}{\\mathrm {argmax}} \\ |w_i|$ and $j^*=\\underset{i\\ne i^*}{\\mathrm {argmax}} \\ |w_i|$ be the coordinates with largest and second-largest magnitude.", "If $|w_{i^*}|-|w_{j^*}| \\ge 4$ , the statement follows exactly as in Lemma REF .", "It remains to consider the case when $|w_{i^*}|- |w_{j^*}| < 4$ .", "In this case first note that, $\\max _{x\\in \\lbrace \\pm 1\\rbrace ^N} \\mathbb {E}_\\epsilon \\max _{i\\in [N]} \\left| w_i + 2\\epsilon x_i\\right| \\le |w_{i^*}|+2$ On the other hand, since the distribution we consider is symmetric, with probability $1/2$ its sign is negative and with remaining probability positive.", "Define $\\sigma _{i^*} = \\mathrm {sign}(x_{i^*})$ , $\\sigma _{j^*} = \\mathrm {sign}(x_{j^*})$ , $\\tau _{i^*} = \\mathrm {sign}(w_{i^*})$ , and $\\tau _{j^*} = \\mathrm {sign}(w_{j^*})$ .", "Since each coordinate is drawn i.i.d., using conditional expectations we have, $&\\mathbb {E}_{x,\\epsilon } \\max _{i}|w_{i}+C \\epsilon x_{i}| = \\mathbb {E}_{x} \\max _{i}|w_{i}+C x_{i}| \\\\& \\ge \\frac{\\mathbb {E}_{x}\\left[ |w_{i^*} + C x_{i^*}|\\ |\\ \\sigma _{i^*} =\\tau _{i^*} \\right]}{2} + \\frac{\\mathbb {E}_{x}\\left[ |w_{j^*} + C x_{j^*}|\\ |\\ \\sigma _{i^*} \\ne \\tau _{i^*}, \\sigma _{j^*} = \\tau _{j^*} \\right]}{4} + \\frac{\\mathbb {E}\\left[ |w_{i^*} + C x_{i^*}|\\ |\\ \\sigma _{i^*} \\ne \\tau _{i^*}, \\sigma _{j^*} \\ne \\tau _{j^*} \\right]}{4}\\\\&\\ge \\frac{\\mathbb {E}_{x}\\left[ |w_{i^*}| + C |x_{i^*}|\\ |\\ \\sigma _{i^*} =\\tau _{i^*} \\right]}{2} + \\frac{\\mathbb {E}_{x}\\left[ |w_{j^*}| + C |x_{j^*}|\\ |\\ \\sigma _{i^*} \\ne \\tau _{i^*}, \\sigma _{j^*} = \\tau _{j^*} \\right]}{4} + \\frac{\\mathbb {E}\\left[ |w_{i^*}| - C |x_{i^*}|\\ |\\ \\sigma _{i^*} \\ne \\tau _{i^*}, \\sigma _{j^*} \\ne \\tau _{j^*} \\right]}{4}\\\\& = \\frac{\\mathbb {E}\\left[ |w_{i^*}| + C |x_{i^*}|\\ |\\ \\sigma _{i^*} =\\tau _{i^*} \\right]}{2} + \\frac{\\mathbb {E}\\left[ |w_{j^*}| + C |x_{j^*}| \\ | \\ \\sigma _{j^*} = \\tau _{j^*} \\right]}{4} + \\frac{\\mathbb {E}\\left[ |w_{i^*}| - C |x_{i^*}| \\ | \\ \\sigma _{i^*} \\ne \\tau _{i^*} \\right]}{4}\\\\& = \\frac{|w_{i^*}| + C \\mathbb {E}\\left[ |x_{i^*}| \\ |\\ \\sigma _{i^*} =\\tau _{i^*} \\right]}{2} + \\frac{|w_{j^*}| + C \\mathbb {E}\\left[ |x_{j^*}| \\ ~|~ \\ \\sigma _{j^*} = \\tau _{j^*} \\right]}{4} + \\frac{|w_{i^*}| - C \\mathbb {E}\\left[ |x_{i^*}| \\ ~|~\\ \\sigma _{i^*} \\ne \\tau _{i^*} \\right]}{4}\\\\& = \\frac{2 |w_{i^*}| + |w_{j^*}| + 3 C \\mathbb {E}\\left[ |x_{i^*}| \\ |\\ \\sigma _{i^*} =\\tau _{i^*} \\right]}{4} + \\frac{|w_{i^*}| - C \\mathbb {E}\\left[ |x_{i^*}| \\ |\\ \\sigma _{i^*} \\ne \\tau _{i^*} \\right]}{4}\\\\& = \\frac{3 |w_{i^*}| + |w_{j^*}| + 2 C \\mathbb {E}\\left[ |x_{i^*}| \\ |\\ \\sigma _{i^*} =\\tau _{i^*} \\right]}{4}$ Now since we are in the case when $|w_{i^*}|- |w_{j^*}| < 4$ we see that $\\mathbb {E}_{x,\\epsilon } \\max _{i}|w_{i}+C \\epsilon x_{i}| \\ge \\frac{3 |w_{i^*}| + |w_{j^*}| + 2 C \\mathbb {E}\\left[ |x_{i^*}| ~|~ \\sigma _{i^*} =\\tau _{i^*} \\right]}{4} \\ge \\frac{4 |w_{i^*}| + 2 C \\mathbb {E}\\left[ |x_{i^*}| ~|~ \\sigma _{i^*} =\\tau _{i^*} \\right] - 4}{4}$ On the other hand, as we already argued, $\\max _{x\\in \\lbrace \\pm 1\\rbrace ^N} \\mathbb {E}_\\epsilon \\max _{i\\in [N]} \\left| w_i + 2\\epsilon x_i\\right| \\le |w_{i^*}|+2$ Hence, as long as $\\frac{ C\\ \\mathbb {E}\\left[ |x_{i^*}|\\ |\\ \\sigma _{i^*} =\\tau _{i^*} \\right] - 2}{2} \\ge 2$ or, in other words, as long as $C \\ge 6 / \\mathbb {E}\\left[ |x_i|\\ |\\ \\mathrm {sign}(x_i) =\\mathrm {sign}(w_i) \\right] = 6/ \\mathbb {E}_{x}\\left[ |x| \\right]~,$ we have that $\\max _{x\\in \\lbrace \\pm 1\\rbrace ^N} \\mathbb {E}_\\epsilon \\max _{i\\in [N]} \\left| w_i + 2\\epsilon x_i\\right| \\le \\mathbb {E}_{x,\\epsilon } \\max _{i}|w_{i}+C \\epsilon x_{i}| \\ .$ This concludes the proof.", "Lemma 25 Consider the case when $\\mathcal {X}$ is the $\\ell _\\infty ^N$ ball and $\\mathcal {F}$ is the $\\ell _1^N$ unit ball.", "Let $f^* = \\underset{f \\in \\mathcal {F}}{\\mathrm {argmin}} \\ \\left<f,R\\right>$ , then for any random vector $R$ , $\\mathbb {E}_{R}\\left[ \\sup _{x \\in \\mathcal {X}} \\left\\lbrace \\left<f^*,x\\right> + \\left\\Vert R + x\\right\\Vert _\\infty \\right\\rbrace \\right] & \\le \\mathbb {E}_{R}\\left[ \\inf _{f \\in \\mathcal {F}} \\sup _{x} \\left\\lbrace \\left<f,x\\right> + \\left\\Vert R + x\\right\\Vert _\\infty \\right\\rbrace \\right] + 4\\ \\mathbf {P}\\left( \\left\\Vert R\\right\\Vert _{\\infty } \\le 4 \\right)$ Let $f^* = \\underset{f \\in \\mathcal {F}}{\\mathrm {argmin}} \\ \\left<f,R\\right>$ .", "We start by noting that for any $f^{\\prime } \\in \\mathcal {F}$ , $\\sup _{x \\in \\mathcal {X}} \\left\\lbrace \\left<f^{\\prime },x\\right> + \\left\\Vert R + x\\right\\Vert _\\infty \\right\\rbrace & = \\sup _{x \\in \\mathcal {X}} \\left\\lbrace \\left<f^{\\prime },x\\right> + \\sup _{f \\in \\mathcal {F}}\\left<f,R + x\\right>\\right\\rbrace \\\\& = \\sup _{f \\in \\mathcal {F}} \\sup _{x \\in \\mathcal {X}} \\left\\lbrace \\left<f^{\\prime },x\\right> + \\left<f,R + x\\right>\\right\\rbrace \\\\& = \\sup _{f \\in \\mathcal {F}} \\left\\lbrace \\sup _{x \\in \\mathcal {X}}\\left<f^{\\prime } + f,x\\right> + \\left<f,R \\right>\\right\\rbrace \\\\& = \\sup _{f \\in \\mathcal {F}} \\left\\lbrace \\left\\Vert f^{\\prime } + f\\right\\Vert _{1} + \\left<f,R \\right>\\right\\rbrace $ Hence note that $ \\inf _{f^{\\prime } \\in \\mathcal {F}}\\sup _{x \\in \\mathcal {X}} \\left\\lbrace \\left<f^{\\prime },x\\right> + \\left\\Vert R + x\\right\\Vert _\\infty \\right\\rbrace & = \\inf _{f^{\\prime } \\in \\mathcal {F}}\\sup _{f \\in \\mathcal {F}} \\left\\lbrace \\left\\Vert f^{\\prime } + f\\right\\Vert _1 + \\left<f,R \\right>\\right\\rbrace \\\\&\\ge \\inf _{f^{\\prime } \\in \\mathcal {F}}\\left\\lbrace \\left\\Vert f^{\\prime } - f^*\\right\\Vert _1 - \\left<f^*,R \\right>\\right\\rbrace \\ge \\inf _{f^{\\prime } \\in \\mathcal {F}} \\left\\lbrace \\left\\Vert f^{\\prime } - f^*\\right\\Vert _1 + \\left\\Vert R \\right\\Vert _{\\infty }\\right\\rbrace = \\left\\Vert R\\right\\Vert _\\infty $ On the other hand note that, $f^*$ is the vertex of the $\\ell _1$ ball (any one which given by $\\underset{i \\in [d]}{\\mathrm {argmin}} \\ |R[i]|$ with sign opposite as sign of $R[i]$ on that vertex).", "Since the $\\ell _1$ ball is the convex hull of the $2d$ vertices, any vector $f \\in \\mathcal {F}$ can be written as $f = \\alpha h-\\beta f^*$ some $h \\in \\mathcal {F}$ such that $\\left\\Vert h\\right\\Vert _1 = 1$ and $\\left<h,R\\right> = 0$ (which means that $h$ is 0 on the maximal co-ordinate of $R$ specified by $f^*$ ) and for some $\\beta \\in [-1,1]$ , $\\alpha \\in [0,1]$ s.t.", "$\\left\\Vert \\alpha h - \\beta f^*\\right\\Vert _1 \\le 1$ .", "Further note that the constraint on $\\alpha , \\beta $ imposed by requiring that $\\left\\Vert \\alpha h - \\beta f^*\\right\\Vert _1 \\le 1$ can be written as $\\alpha + |\\beta | \\le 1$ .", "Hence, $\\sup _{x \\in \\mathcal {X}} \\left\\lbrace \\left<f^*,x\\right> + \\left\\Vert R + x\\right\\Vert _\\infty \\right\\rbrace & = \\sup _{f \\in \\mathcal {F}} \\left\\lbrace \\left\\Vert f^* + f\\right\\Vert _1 + \\left<f,R\\right>\\right\\rbrace \\\\& = \\sup _{\\alpha \\in [0,1]}\\sup _{h \\perp f^*, \\left\\Vert h\\right\\Vert _1 = 1} \\sup _{\\beta \\in [-1,1] , \\left\\Vert \\alpha h - \\beta f^*\\right\\Vert _1 \\le 1} \\left\\lbrace \\left\\Vert (1 - \\beta )f^* + \\alpha h\\right\\Vert _1 + \\beta \\left<f^*,R \\right> + \\alpha \\left<h,R\\right> \\right\\rbrace \\\\& = \\sup _{\\alpha \\in [0,1]}\\sup _{h \\perp f^*, \\left\\Vert h\\right\\Vert _1 = 1} \\sup _{\\beta \\in [-1,1] , \\left\\Vert \\alpha h - \\beta f^*\\right\\Vert _1 \\le 1} \\left\\lbrace |1 - \\beta | \\left\\Vert f^*\\right\\Vert _1 + \\alpha \\left\\Vert h\\right\\Vert _1 + \\beta \\left\\Vert R\\right\\Vert _{\\infty } \\right\\rbrace \\\\& = \\sup _{\\alpha \\in [0,1]} \\sup _{\\beta \\in [-1,1] : |\\beta | + \\alpha \\le 1}\\left\\lbrace |1 - \\beta | + \\alpha + \\beta \\left\\Vert R\\right\\Vert _{\\infty } \\right\\rbrace \\\\& \\le \\sup _{\\beta \\in [-1,1] }\\left\\lbrace |1 - \\beta | + 1 - |\\beta | + \\beta \\left\\Vert R\\right\\Vert _{\\infty } \\right\\rbrace \\\\& \\le \\sup _{\\beta \\in [-1,1] }\\left\\lbrace 2|1 - \\beta | + \\beta \\left\\Vert R\\right\\Vert _{\\infty } \\right\\rbrace \\\\& = \\sup _{\\beta \\in \\lbrace -1,1\\rbrace }\\left\\lbrace 2|1 - \\beta | + \\beta \\left\\Vert R\\right\\Vert _{\\infty } \\right\\rbrace \\\\& = \\max \\left\\lbrace \\left\\Vert R\\right\\Vert _{\\infty } , 4 - \\left\\Vert R\\right\\Vert _{\\infty } \\right\\rbrace \\\\& \\le \\left\\Vert R\\right\\Vert _{\\infty } + 4\\ {\\bf 1}\\left\\lbrace \\left\\Vert R\\right\\Vert _{\\infty } \\le 4\\right\\rbrace $ Hence combining with equation REF we can conclude that $\\mathbb {E}_{R}\\left[ \\sup _{x} \\left\\lbrace \\left<f^*,x\\right> + \\left\\Vert R + x\\right\\Vert _\\infty \\right\\rbrace \\right] & \\le \\mathbb {E}_{R}\\left[ \\inf _{f \\in \\mathcal {F}} \\sup _{x} \\left\\lbrace \\left<f,x\\right> + \\left\\Vert R + x\\right\\Vert _\\infty \\right\\rbrace \\right] + 4\\ \\mathbb {E}_{R}\\left[ {\\bf 1}\\left\\lbrace \\left\\Vert R\\right\\Vert _{\\infty } \\le 4\\right\\rbrace \\right]\\\\& = \\mathbb {E}_{R}\\left[ \\inf _{f \\in \\mathcal {F}} \\sup _{x} \\left\\lbrace \\left<f,x\\right> + \\left\\Vert R + x\\right\\Vert _\\infty \\right\\rbrace \\right] + 4\\ \\mathbf {P}\\left( \\left\\Vert R\\right\\Vert _{\\infty } \\le 4 \\right)$ [Proof of Lemma REF] On any round $t$ , the algorithm draws $\\epsilon _{t+1} , \\ldots , \\epsilon _T$ and $x_{t+1},\\ldots ,x_T \\sim D^N$ and plays $f_t = \\underset{f \\in \\mathcal {F}}{\\mathrm {argmin}} \\ \\left<f,\\sum _{i=1}^{t-1} x_i - C \\sum _{i=t+1}^T x_i \\right>$ We shall show that this randomized algorithm is (almost) admissible w.r.t.", "the relaxation (with some small additional term at each step).", "We define the relaxation as $\\mathbf {Rel}_{T}\\left(\\mathcal {F} | x_1,\\ldots ,x_t \\right) = \\mathbb {E}_{x_{t+1},\\ldots x_T \\sim D}\\left[ \\left\\Vert \\sum _{i=1}^t x_i - C \\sum _{i=t+1}^T x_i\\right\\Vert _\\infty \\right]$ Proceeding just as in the proof of Lemma REF note that, for our randomized strategy, $\\sup _{x} &\\left\\lbrace \\mathbb {E}_{f \\sim q_t}\\left[ \\left<f,x\\right> \\right] + \\mathbf {Rel}_{T}\\left(\\mathcal {F} | x_1,\\ldots ,x_t \\right)\\right\\rbrace \\\\& = \\sup _{x} \\left\\lbrace \\mathbb {E}_{x_{t+1:T} \\sim D^N}\\left[ \\left<f_t,x\\right> \\right] + \\mathbb {E}_{x_{t+1:T} \\sim D^N}\\left[ \\left\\Vert \\sum _{i=1}^{t-1} x_i + x - C \\sum _{i=t+1}^T x_i\\right\\Vert _\\infty \\right] \\right\\rbrace \\\\& \\le \\mathbb {E}_{x_{t+1:T} \\sim D^N}\\left[ \\sup _{x} \\left\\lbrace \\left<f_t,x\\right> + \\left\\Vert \\sum _{i=1}^{t-1} x_i + x - C \\sum _{i=t+1}^T x_i \\right\\Vert _\\infty \\right\\rbrace \\right]$ In view of Lemma REF (with $R = \\sum _{i=1}^{t-1} x_i - C \\sum _{i=t+1}^T \\epsilon _i x_i$ ) we conclude that $& \\mathbb {E}_{x_{t+1},\\ldots ,x_{T}}\\left[ \\sup _{x \\in \\mathcal {X}} \\left\\lbrace \\left<f_t,x\\right> + \\left\\Vert \\sum _{i=1}^{t-1} x_i - C \\sum _{i=t+1}^T x_i + x\\right\\Vert _\\infty \\right\\rbrace \\right] \\\\&~~~~~~~~~~ \\le \\mathbb {E}_{x_{t+1},\\ldots ,x_{T}}\\left[ \\inf _{f \\in \\mathcal {F}} \\sup _{x} \\left\\lbrace \\left<f,x\\right> + \\left\\Vert \\sum _{i=1}^{t-1} x_i - C \\sum _{i=t+1}^T x_i + x\\right\\Vert _\\infty \\right\\rbrace \\right] + 4\\ \\mathbf {P}\\left( \\left\\Vert \\sum _{i=1}^{t-1} x_i - C \\sum _{i=t+1}^T x_i\\right\\Vert _{\\infty } \\le 4 \\right)\\\\&~~~~~~~~~~ = \\mathbb {E}_{x_{t+1},\\ldots ,x_{T}}\\left[ \\sup _{x} \\left\\lbrace \\left<f^*_t,x\\right> + \\left\\Vert \\sum _{i=1}^{t-1} x_i - C \\sum _{i=t+1}^T x_i + x\\right\\Vert _\\infty \\right\\rbrace \\right] + 4\\ \\mathbf {P}\\left( \\left\\Vert \\sum _{i=1}^{t-1} x_i - C \\sum _{i=t+1}^T x_i\\right\\Vert _{\\infty } \\le 4 \\right)$ where $f^*_t = \\underset{f \\in \\mathcal {F}}{\\mathrm {argmin}} \\ \\sup _{x} \\left\\lbrace \\left<f,x\\right> + \\left\\Vert \\sum _{i=1}^{t-1} x_i - C \\sum _{i=t+1}^T x_i + x\\right\\Vert _\\infty \\right\\rbrace $ Combining with Equation (REF ) we conclude that $\\sup _{x} &\\left\\lbrace \\mathbb {E}_{f \\sim q_t}\\left[ \\left<f,x\\right> \\right] + \\mathbf {Rel}_{T}\\left(\\mathcal {F} | x_1,\\ldots ,x_t \\right)\\right\\rbrace \\\\& \\le \\mathbb {E}_{x_{t+1},\\ldots ,x_{T}}\\left[ \\sup _{x} \\left\\lbrace \\left<f^*_t,x\\right> + \\left\\Vert \\sum _{i=1}^{t-1} x_i - C \\sum _{i=t+1}^T x_i + x\\right\\Vert _\\infty \\right\\rbrace \\right] + 4\\ \\mathbf {P}\\left( \\left\\Vert \\sum _{i=1}^{t-1} x_i - C \\sum _{i=t+1}^T x_i\\right\\Vert _{\\infty } \\le 4 \\right)$ Now, since $4\\ \\mathbf {P}\\left( \\left\\Vert \\sum _{i=1}^{t-1} x_i - C \\sum _{i=t+1}^T x_i\\right\\Vert _{\\infty } \\le 4 \\right) \\le 4\\ \\mathbf {P}\\left( C \\left\\Vert \\sum _{i=t+1}^T x_i\\right\\Vert _{\\infty } \\le 4 \\right)\\le 4\\ \\mathbf {P}_{y_{t+1},\\ldots ,y_T \\sim D}\\left( C \\left| \\sum _{i=t+1}^T y_i\\right| \\le 4 \\right)$ we have $\\sup _{x} &\\left\\lbrace \\mathbb {E}_{f \\sim q_t}\\left[ \\left<f,x\\right> \\right] + \\mathbf {Rel}_{T}\\left(\\mathcal {F} | x_1,\\ldots ,x_t \\right)\\right\\rbrace \\\\& \\le \\mathbb {E}_{x_{t+1},\\ldots ,x_{T}}\\left[ \\sup _{x} \\left\\lbrace \\left<f^*_t,x\\right> + \\left\\Vert \\sum _{i=1}^{t-1} x_i - C \\sum _{i=t+1}^T x_i + x\\right\\Vert _\\infty \\right\\rbrace \\right] + 4\\ \\mathbf {P}_{y_{t+1},\\ldots ,y_T \\sim D}\\left( C \\left| \\sum _{i=t+1}^T y_i\\right| \\le 4 \\right)$ In view of Lemma REF , Assumption REF is satisfied by $D^N$ with constant $C$ .", "Further in the proof of Lemma REF we already showed that whenever Assumption REF is satisfied, the randomized strategy specified by $f^*_t$ is admissible.", "More specifically we showed that $\\mathbb {E}_{x_{t+1},\\ldots ,x_{T}}\\left[ \\sup _{x} \\left\\lbrace \\left<f^*_t,x\\right> + \\left\\Vert \\sum _{i=1}^{t-1} x_i - C \\sum _{i=t+1}^T x_i + x\\right\\Vert _\\infty \\right\\rbrace \\right] \\le \\mathbf {Rel}_{T}\\left(F | x_1,\\ldots ,x_{t-1} \\right)$ and so using this in Equation (REF ) we conclude that for the randomized strategy in the statement of the lemma, $\\sup _{x} &\\left\\lbrace \\mathbb {E}_{f \\sim q_t}\\left[ \\left<f,x\\right> \\right] + \\mathbf {Rel}_{T}\\left(\\mathcal {F} | x_1,\\ldots ,x_t \\right)\\right\\rbrace \\\\& \\le \\mathbf {Rel}_{T}\\left(F | x_1,\\ldots ,x_{t-1} \\right) + 4\\ \\mathbf {P}_{y_{t+1},\\ldots ,y_T \\sim D}\\left( C \\left| \\sum _{i=t+1}^T y_i\\right| \\le 4 \\right)$ Or in other words the randomized strategy proposed is admissible with an additional additive factor of $4\\ \\mathbf {P}_{y_{t+1},\\ldots ,y_T \\sim D}\\left( C \\left| \\sum _{i=t+1}^T y_i\\right| \\le 4 \\right)$ at each time step $t$ .", "Hence by Proposition REF we have that for the randomized algorithm specified in the lemma, $\\mathbb {E}\\left[ \\mathbf {Reg}_T \\right] & \\le \\mathbf {Rel}_{T}\\left(F \\right) + 4 \\sum _{t=1}^T \\mathbf {P}_{y_{t+1},\\ldots ,y_T \\sim D}\\left( C \\left| \\sum _{i=t+1}^T y_i\\right| \\le 4 \\right) \\\\& = C\\ \\mathbb {E}_{x_1,\\ldots ,x_T \\sim D^N}\\left[ \\left\\Vert \\sum _{t=1}^T x_t\\right\\Vert _\\infty \\right] + 4 \\sum _{t=1}^T \\mathbf {P}_{y_{t+1},\\ldots ,y_T \\sim D}\\left( C \\left| \\sum _{i=t+1}^T y_i\\right| \\le 4 \\right)$ This concludes the proof.", "[Proof of Lemma REF] Instead of using $C = 4 \\sqrt{2}$ and drawing uniformly from surface of unit sphere we can equivalently think of the constant as being 1 and drawing uniformly from surface of sphere of radius $4 \\sqrt{2}$ .", "Let $\\left\\Vert \\cdot \\right\\Vert $ stand for the Euclidean norm.", "To prove (REF ), first observe that $\\sup _{p \\in \\Delta (\\mathcal {X})} \\underset{x_t \\sim p}{\\mathbb {E}} \\left\\Vert w + \\underset{x \\sim p}{\\mathbb {E}} [x] - x_t \\right\\Vert \\le \\sup _{x \\in \\mathcal {X}} \\underset{\\epsilon }{\\mathbb {E}} \\left\\Vert w + 2\\epsilon x \\right\\Vert $ for any $w\\in B$ .", "Further, using Jensen's inequality $\\sup _{x \\in \\mathcal {X}} \\underset{\\epsilon }{\\mathbb {E}} \\left\\Vert w + 2\\epsilon x \\right\\Vert \\le \\sup _{x \\in \\mathcal {X}} \\sqrt{\\underset{\\epsilon }{\\mathbb {E}} \\left\\Vert w + 2\\epsilon x \\right\\Vert ^2}\\le \\sup _{x \\in \\mathcal {X}} \\sqrt{ \\left\\Vert w\\right\\Vert ^2 + \\underset{\\epsilon }{\\mathbb {E}}\\left\\Vert 2\\epsilon x \\right\\Vert ^2}= \\sqrt{ \\left\\Vert w\\right\\Vert ^2 +4}$ To prove the lemma, it is then enough to show that for $r=4\\sqrt{2}$ $\\mathbb {E}_{x\\sim D}\\left\\Vert w+rx\\right\\Vert \\ge \\sqrt{ \\left\\Vert w\\right\\Vert ^2 +4}$ for any $w$ , where we omitted $\\epsilon $ since $D$ is symmetric.", "This fact can be proved with the following geometric argument.", "We define quadruplets $(w+z_1,w+z_2,w-z_1,w-z_2)$ of points on the sphere of radius $r$ .", "Each quadruplets will have the property that $\\frac{\\left\\Vert w+z_1\\right\\Vert +\\left\\Vert w+z_2\\right\\Vert +\\left\\Vert w-z_1\\right\\Vert +\\left\\Vert w-z_2\\right\\Vert }{4}\\ge \\sqrt{ \\left\\Vert w\\right\\Vert ^2 +4}$ for any $w$ .", "We then argue that the uniform distribution can be decomposed into these quadruplets such that each point on the sphere occurs in only one quadruplet (except for a measure zero set when $z_1$ is aligned with $-w$ ), thus concluding that (REF ) holds true.", "Figure: The two-dimensional construction for the proof of Lemma .Pick any direction $w^\\perp $ perpendicular to $w$ .", "A quadruplet is defined by perpendicular vectors $z_1$ and $z_2$ which have length $r$ and which lie in the plane spanned by $w,w^\\perp $ .", "Let $\\theta $ be the angle between $-w$ and $z_1$ .", "Since we are now dealing with a two dimensional plane spanned by $w$ and $w^\\perp $ , we may as well assume that $w$ is aligned with the positive $x$ -axis, as in Figure REF .", "We write $w$ for $\\Vert w\\Vert $ .", "The coordinates of the quadruplet are $(w-r\\cos (\\theta ), r\\sin (\\theta )),~~ (w+r\\cos (\\theta ), -r\\sin (\\theta )),~~ (w+r\\sin (\\theta ), r\\cos (\\theta )),~~ (w-r\\sin (\\theta ), -r\\cos (\\theta ))$ For brevity, let $s=\\sin (\\theta ), c=\\cos (\\theta )$ .", "The desired inequality (REF ) then reads $\\sqrt{w^2-8wc+r^2}+\\sqrt{w^2+8wc+r^2}+\\sqrt{w^2+8ws+r^2}+\\sqrt{w^2-8ws+r^2}\\ge 4\\sqrt{w^2+4}$ To prove that this inequality holds, we square both sides, keeping in mind that the terms are non-negative.", "The sum of four squares on the left hand side gives $4w^2+4r^2$ .", "For the six cross terms, we can pass to a lower bound by replacing $r^2$ in each square root by $r^2c^2$ or $r^2s^2$ , whichever completes the square.", "Then observe that $ |w+rs|\\cdot |w-rs|+|w+rc|\\cdot |w-rc| = 2w^2-r^2$ while the other four cross terms $(|w+rs|\\cdot |w-rc|+|w+rs|\\cdot |w+rc|)+(|w-rs|\\cdot |w+rc|+|w-rs|\\cdot |w-rc|) \\ge |w+rs|\\cdot 2w+|w-rs|\\cdot 2w \\ge 4w^2$ Doubling the cross terms gives a contribution of $2(6w^2-r^2)$ , while the sum of squares yielded $4w^2+4r^2$ .", "The desired inequality is satisfied as long as $16w^2+2r^2 \\ge 16(w^2+4)$ , or $r\\ge 4\\sqrt{2}$ .", "[Proof of Lemma REF] By Lemma REF , Assumption REF is satisfied by distribution $D$ with constant $C = 4 \\sqrt{2}$ .", "Hence by Lemma REF we can conclude that for the randomized algorithm which at round $t$ freshly draws $x_{t+1},\\ldots ,x_T \\sim D$ and picks $f^*_t = \\underset{f \\in \\mathcal {F}}{\\mathrm {argmin}} \\ \\sup _{x \\in \\mathcal {X}} \\left\\lbrace \\left<f,x\\right> + \\left\\Vert - \\sum _{i=1}^{t-1} x_i + 4 \\sqrt{2} \\sum _{i=t+1}^T x_i - x\\right\\Vert _2 \\right\\rbrace $ (we dropped the $\\epsilon $ 's as the distribution is symmetric to start with) the expected regret is bounded as $\\mathbb {E}\\left[ \\mathbf {Reg}_T \\right] \\le 4 \\sqrt{2}\\ \\mathbb {E}_{x_1,\\ldots ,x_T \\sim D}\\left[ \\left\\Vert \\sum _{t=1}^T x_t\\right\\Vert _2 \\right] \\le 4 \\sqrt{2 T}$ We claim that the strategy specified in the lemma that chooses $f_t = \\frac{- \\sum _{i=1}^{t-1} x_i + 4 \\sqrt{2} \\sum _{i=t+1}^T x_i }{\\sqrt{\\left\\Vert - \\sum _{i=1}^{t-1} x_i + 4 \\sqrt{2} \\sum _{i=t+1}^T \\epsilon _i x_i \\right\\Vert _2^2 + 1}}$ is the same as choosing $f^*_t$ .", "To see this let us start by defining $\\bar{x}_t = - \\sum _{i=1}^{t-1} x_i + 4 \\sqrt{2} \\sum _{i=t+1}^T x_i$ Now note that $f^*_t = \\underset{f \\in \\mathcal {F}}{\\mathrm {argmin}} \\ \\sup _{x \\in \\mathcal {X}} \\left\\lbrace \\left<f,x\\right> + \\left\\Vert - \\sum _{i=1}^{t-1} x_i + 4 \\sqrt{2} \\sum _{i=t+1}^T x_i - x\\right\\Vert _2 \\right\\rbrace & = \\underset{f \\in \\mathcal {F}}{\\mathrm {argmin}} \\ \\sup _{x \\in \\mathcal {X}} \\left\\lbrace \\left<f,x\\right> + \\left\\Vert \\bar{x}_t - x\\right\\Vert _2 \\right\\rbrace \\\\& = \\underset{f \\in \\mathcal {F}}{\\mathrm {argmin}} \\ \\sup _{x \\in \\mathcal {X}} \\left\\lbrace \\left<f,x\\right> + \\sqrt{\\left\\Vert \\bar{x}_t - x\\right\\Vert _2^2} \\right\\rbrace \\\\& = \\underset{f \\in \\mathcal {F}}{\\mathrm {argmin}} \\ \\sup _{x : \\left\\Vert x\\right\\Vert _2 \\le 1} \\left\\lbrace \\left<f,x\\right> + \\sqrt{\\left\\Vert \\bar{x}_t\\right\\Vert ^2 - 2 \\left<\\bar{x}_t,x\\right> + \\left\\Vert x\\right\\Vert _2^2} \\right\\rbrace \\\\& = \\underset{f \\in \\mathcal {F}}{\\mathrm {argmin}} \\ \\sup _{x : \\left\\Vert x\\right\\Vert _2 = 1} \\left\\lbrace \\left<f,x\\right> + \\sqrt{\\left\\Vert \\bar{x}_t\\right\\Vert ^2 - 2 \\left<\\bar{x}_t,x\\right> + 1} \\right\\rbrace $ However this argmin calculation is identical to the one in the proof of Proposition REF (with $C = 1$ and $T-t = 0$ ) and the solution is given by $f^*_t = f_t = \\frac{- \\sum _{i=1}^{t-1} x_i + 4 \\sqrt{2} \\sum _{i=t+1}^T x_i }{\\sqrt{\\left\\Vert - \\sum _{i=1}^{t-1} x_i + 4 \\sqrt{2} \\sum _{i=t+1}^T \\epsilon _i x_i \\right\\Vert _2^2 + 1}}$ Thus we conclude the proof.", "[Proof of Lemma  REF] We first prove the statement for the convex case.", "To show admissibility using the particular randomized strategy given in the lemma, we need to show that for the randomized strategy specified by $q_t$ , $\\sup _{y_t} \\left\\lbrace \\mathbb {E}_{\\hat{y}_t \\sim q_t}\\left[ \\ell (\\hat{y}_t,y_t) \\right] + \\mathbf {Rel}_{T}\\left(\\mathcal {F} | (x_1,y_1),\\ldots ,(x_t,y_t) \\right)\\right\\rbrace \\le \\mathbf {Rel}_{T}\\left(\\mathcal {F} | (x_1,y_1),\\ldots ,(x_{t-1},y_{t-1}) \\right)$ for any $x_t$ .", "The strategy $q_t$ proposed by the lemma is such that we first draw $(x_{t+1},y_{t+1}),\\ldots ,(x_T,y_T) \\sim D$ and $\\epsilon _{t+1},\\ldots \\epsilon _T$ Rademacher random variables, and then based on this sample pick $\\hat{y}_t=\\hat{y}_t(x_{t+1:T},y_{t+1:T},\\epsilon _{t+1:T})$ as in (REF ).", "Hence, $\\sup _{y_t} &\\left\\lbrace \\mathbb {E}_{\\hat{y}_t \\sim q_t}\\left[ \\ell (\\hat{y}_t,y_t) \\right] + \\mathbf {Rel}_{T}\\left(\\mathcal {F} | (x_1,y_1),\\ldots ,(x_t,y_t) \\right)\\right\\rbrace \\\\& = \\sup _{y_t} \\left\\lbrace \\underset{\\underset{(x_{t+1:T},y_{t+1:T})}{\\epsilon _{t+1:T}}}{\\mathbb {E}} \\ell (\\hat{y}_t,y_t) + \\underset{\\underset{(x_{t+1:T},y_{t+1:T})}{\\epsilon _{t+1:T}}}{\\mathbb {E}} \\sup _{f \\in \\mathcal {F}} \\left[ C \\sum _{i=t+1}^T \\epsilon _i \\ell (f(x_i),y_i) - L_t(f) \\right] \\right\\rbrace \\\\& \\le \\underset{\\underset{(x_{t+1:T},y_{t+1:T})}{\\epsilon _{t+1:T}}}{\\mathbb {E}} \\sup _{y_t} \\left\\lbrace \\ell (\\hat{y}_t,y_t) + \\sup _{f \\in \\mathcal {F}}\\left[ C \\sum _{i=t+1}^T \\epsilon _i \\ell (f(x_i),y_i) - L_t(f) \\right]\\right\\rbrace \\ .$ Now, with $\\hat{y}_t$ in (REF ), $\\sup _{y_t} & \\left\\lbrace \\ell (\\hat{y}_t,y_t) + \\sup _{f \\in \\mathcal {F}} \\left[ C \\sum _{i=t+1}^T \\epsilon _i \\ell (f(x_i),y_i) - L_t(f) \\right] \\right\\rbrace \\\\& = \\inf _{\\hat{y} \\in [-B,B]} \\sup _{y_t} \\left\\lbrace \\ell (\\hat{y},y_t) + \\sup _{f \\in \\mathcal {F}} \\left[ C \\sum _{i=t+1}^T \\epsilon _i \\ell (f(x_i),y_i) - L_t(f) \\right] \\right\\rbrace \\\\& = \\inf _{\\hat{y} \\in [-B,B]} \\sup _{p_t} \\mathbb {E}_{y_t \\sim p_t}\\left[ \\ell (\\hat{y},y_t) + \\sup _{f \\in \\mathcal {F}} \\left[ C \\sum _{i=t+1}^T \\epsilon _i \\ell (f(x_i),y_i) - L_t(f) \\right] \\right]$ Now we assume that the loss $\\ell (\\hat{y},y)$ is convex in the first argument (and bounded).", "Note that the term $\\mathbb {E}_{y_t \\sim p_t}\\left[ \\ell (\\hat{y},y_t) + \\sup _{f \\in \\mathcal {F}} \\left\\lbrace C \\sum _{i=t+1}^T \\epsilon _i \\ell (f(x_i),y_i) - L_{t}(f) \\right\\rbrace \\right]$ is linear in $p_t$ and, due to convexity of loss, is convex in $\\hat{y}_t$ .", "Hence by the minimax theorem, for this choice of $q_t$ , we conclude that $\\sup _{y_t} &\\left\\lbrace \\mathbb {E}_{\\hat{y}_t \\sim q_t}\\left[ \\ell (\\hat{y}_t,y_t) \\right] + \\mathbf {Rel}_{T}\\left(\\mathcal {F} | (x_1,y_1),\\ldots ,(x_t,y_t) \\right)\\right\\rbrace \\\\& \\le \\underset{\\underset{(x_{t+1},y_{t+1}),\\ldots , (x_T,y_T)}{\\epsilon _{t+1:T}}}{\\mathbb {E}} \\inf _{\\hat{y}_t \\in [-B,B]} \\sup _{p_t} \\mathbb {E}_{y_t \\sim p_t}\\left[ \\ell (\\hat{y}_t,y_t) + \\sup _{f \\in \\mathcal {F}} \\left\\lbrace C \\sum _{i=t+1}^T \\epsilon _i \\ell (f(x_i),y_i) - L_t(f) \\right\\rbrace \\right]\\\\& = \\underset{\\underset{(x_{t+1},y_{t+1}),\\ldots , (x_T,y_T)}{\\epsilon _{t+1:T}}}{\\mathbb {E}} \\sup _{p_t }\\inf _{\\hat{y}_t \\in [-B,B]} \\mathbb {E}_{y_t \\sim p_t}\\left[ \\ell (\\hat{y}_t,y_t) + \\sup _{f \\in \\mathcal {F}} \\left\\lbrace C \\sum _{i=t+1}^T \\epsilon _i \\ell (f(x_i),y_i) - L_t(f) \\right\\rbrace \\right]$ The last step above is due to the minimax theorem as the loss is convex in $\\hat{y}_t$ , the set $[-B,B]$ is compact, and the term is linear in $p_t$ .", "The above expression is equal to $& = \\mathbb {E}\\sup _{p_t }\\mathbb {E}_{y_t \\sim p_t} \\sup _{f \\in \\mathcal {F}} \\left\\lbrace C \\sum _{i=t+1}^T \\epsilon _i \\ell (f(x_i),y_i) - \\sum _{i=1}^{t-1} \\ell (f(x_i),y_i) + \\inf _{\\hat{y_t} \\in [-B,B]} \\mathbb {E}_{y_t \\sim p_t}\\left[ \\ell (\\hat{y}_t,y_t) \\right] - \\ell (f(x_t),y_t)\\right\\rbrace \\\\& \\le \\mathbb {E}\\sup _{p_t }\\mathbb {E}_{y_t \\sim p_t} \\sup _{f \\in \\mathcal {F}} \\left\\lbrace C \\sum _{i=t+1}^T \\epsilon _i \\ell (f(x_i),y_i) - L_{t-1}(f) + \\inf _{g \\in \\mathcal {F}} \\mathbb {E}_{y_t \\sim p_t}\\left[ \\ell (g(x_t),y_t) \\right] - \\ell (f(x_t),y_t)\\right\\rbrace \\\\& \\le \\mathbb {E}\\sup _{p_t }\\mathbb {E}_{y_t \\sim p_t} \\sup _{f \\in \\mathcal {F}} \\left\\lbrace C \\sum _{i=t+1}^T \\epsilon _i \\ell (f(x_i),y_i) - L_{t-1}(f) + \\mathbb {E}_{y_t \\sim p_t}\\left[ \\ell (f(x_t),y_t) \\right] - \\ell (f(x_t),y_t)\\right\\rbrace \\\\& \\le \\underset{\\underset{(x_{t+1},y_{t+1}),\\ldots , (x_T,y_T)}{\\epsilon _{t+1:T}}}{\\mathbb {E}} \\mathbb {E}_{(x_t,y_t) \\sim D}\\mathbb {E}_{\\epsilon _t} \\sup _{f \\in \\mathcal {F}} \\left\\lbrace C \\sum _{i=t+1}^T \\epsilon _i \\ell (f(x_i),y_i) - L_{t-1}(f) + C \\epsilon _t \\ell (f(x_t),y_t) \\right\\rbrace \\\\& = \\mathbf {Rel}_{T}\\left(\\mathcal {F} | (x_1,y_1),\\ldots ,(x_{t-1},y_{t-1}) \\right)$ The second part of the Lemma is proved analogously.", "[Proof of Lemma REF] Now let $q_t$ be the randomized strategy where we draw $\\epsilon _{t+1},\\ldots ,\\epsilon _T$ uniformly at random and pick $q_t(\\epsilon ) = \\underset{q \\in \\Delta (\\mathcal {F})}{\\mathrm {argmin}} \\ \\sup _{x_t}\\left\\lbrace \\mathbb {E}_{f_t \\in q}\\left[ \\ell (f_t,x_t) \\right] + \\sup _{f \\in \\mathcal {F}} \\left[ 2 \\sum _{i=t+1}^T \\epsilon _i \\ell (f,\\mathbf {x}^t_i(\\epsilon )) - \\sum _{i=1}^t \\ell (f,x_i) \\right] \\right\\rbrace $ With the definition of $\\mathbf {x}^t$ in (REF ), and with the notation $L_t(f)=\\sum _{i=1}^t \\ell (f,x_i) $ $\\sup _{x_t}& \\left\\lbrace \\mathbb {E}_{f_t \\sim q_t}\\left[ \\ell (f_t,x_t) \\right] + \\sup _{\\mathbf {x}} \\mathbb {E}_{\\epsilon } \\sup _{f \\in \\mathcal {F}} \\left[ 2 \\sum _{i=t+1}^T \\epsilon _i \\ell (f,\\mathbf {x}_i(\\epsilon )) - L_{t}(f)\\right] \\right\\rbrace \\\\& = \\sup _{x_t}\\left\\lbrace \\mathbb {E}_{\\epsilon }\\left[ \\mathbb {E}_{f_t \\sim q_t(\\epsilon )}\\left[ \\ell (f_t,x_t) \\right] \\right] + \\mathbb {E}_{\\epsilon }\\sup _{f \\in \\mathcal {F}} \\left[ 2 \\sum _{i=t+1}^T \\epsilon _i \\ell (f,\\mathbf {x}^t_i(\\epsilon )) - L_{t}(f)\\right] \\right\\rbrace \\\\& \\le \\mathbb {E}_{\\epsilon }\\left[ \\sup _{x_t}\\left\\lbrace \\mathbb {E}_{f_t \\sim q_t(\\epsilon )}\\left[ \\ell (f_t,x_t) \\right] + \\sup _{f \\in \\mathcal {F}} \\left[ 2 \\sum _{i=t+1}^T \\epsilon _i \\ell (f,\\mathbf {x}^t_i(\\epsilon )) - L_{t}(f)\\right] \\right\\rbrace \\right]\\\\& = \\mathbb {E}_{\\epsilon }\\left[ \\inf _{q_t \\in \\Delta (\\mathcal {F})} \\sup _{x_t}\\left\\lbrace \\mathbb {E}_{f_t \\sim q_t}\\left[ \\ell (f_t,x_t) \\right] + \\sup _{f \\in \\mathcal {F}} \\left[2 \\sum _{i=t+1}^T \\epsilon _i \\ell (f,\\mathbf {x}^t_i(\\epsilon )) - L_{t}(f)\\right] \\right\\rbrace \\right]$ where the last step is due to the way we pick our predictor $f_t(\\epsilon )$ given random draw of $\\epsilon $ 's in Equation (REF ).", "We now apply the minimax theorem, yielding the following upper bound on the term above: $&\\mathbb {E}_{\\epsilon }\\left[ \\sup _{p_t \\in \\Delta (\\mathcal {X})} \\inf _{f_t \\in \\mathcal {F}}\\left\\lbrace \\mathbb {E}_{x_t \\sim p_t}\\left[ \\ell (f_t,x_t) \\right] + \\mathbb {E}_{x_t \\sim p_t} \\sup _{f \\in \\mathcal {F}} \\left[ 2 \\sum _{i=t+1}^T \\epsilon _i \\ell (f,\\mathbf {x}^t_i(\\epsilon )) - L_{t}(f)\\right] \\right\\rbrace \\right]$ This expression can be re-written as $&\\mathbb {E}_{\\epsilon }\\left[ \\sup _{p_t \\in \\Delta (\\mathcal {X})} \\left\\lbrace \\mathbb {E}_{x_t \\sim p_t} \\sup _{f \\in \\mathcal {F}} \\left[ 2 \\sum _{i=t+1}^T \\epsilon _i \\ell (f,\\mathbf {x}^t_i(\\epsilon )) - L_{t-1}(f) + \\mathbb {E}_{x_t \\sim p_t}\\left[ \\ell (f ,x_t) \\right] - \\ell (f,x_t) \\right] \\right\\rbrace \\right]\\\\&\\le \\mathbb {E}_{\\epsilon }\\left[ \\sup _{p_t \\in \\Delta (\\mathcal {X})} \\left\\lbrace \\mathbb {E}_{x_t, x^{\\prime }_t \\sim p_t}\\mathbb {E}_{\\epsilon _t}\\sup _{f \\in \\mathcal {F}} \\left[ 2 \\sum _{i=t+1}^T \\epsilon _i \\ell (f,\\mathbf {x}^t_i(\\epsilon )) - L_{t-1}(f) + \\epsilon _t\\left(\\ell (f ,x_t) - \\ell (f,x_t) \\right) \\right] \\right\\rbrace \\right]$ By passing to the supremum over $x_t,x^{\\prime }_t$ , we get an upper bound $&\\mathbb {E}_{\\epsilon }\\left[ \\sup _{x_t , x^{\\prime }_t \\in \\mathcal {X}} \\left\\lbrace \\mathbb {E}_{\\epsilon _t} \\sup _{f \\in \\mathcal {F}} \\left[ 2 \\sum _{i=t+1}^T \\epsilon _i \\ell (f,\\mathbf {x}^t_i(\\epsilon )) - L_{t-1}(f) + \\epsilon _t\\left(\\ell (f ,x_t) - \\ell (f,x_t) \\right) \\right] \\right\\rbrace \\right]\\\\&\\le \\mathbb {E}_{\\epsilon }\\left[ \\sup _{x_t \\in \\mathcal {X}} \\left\\lbrace \\mathbb {E}_{\\epsilon _t} \\sup _{f \\in \\mathcal {F}} \\left[ 2 \\sum _{i=t+1}^T \\epsilon _i \\ell (f,\\mathbf {x}^t_i(\\epsilon )) - L_{t-1}(f) + 2 \\epsilon _t \\ell (f ,x_t) \\right] \\right\\rbrace \\right]\\\\&\\le \\sup _{\\tilde{\\mathbf {x}}} \\mathbb {E}_{\\epsilon }\\left[ \\sup _{x_t \\in \\mathcal {X}} \\left\\lbrace \\mathbb {E}_{\\epsilon _t}\\sup _{f \\in \\mathcal {F}} \\left[ 2 \\sum _{i=t+1}^T \\epsilon _i \\ell (f,\\tilde{\\mathbf {x}}_i(\\epsilon )) - L_{t-1}(f) + 2 \\epsilon _t \\ell (f ,x_t) \\right] \\right\\rbrace \\right] \\\\&\\le \\sup _{\\mathbf {x}}\\mathbb {E}_{\\epsilon } \\sup _{f \\in \\mathcal {F}}\\left[ 2 \\sum _{i=t}^T \\epsilon _i \\ell (f,\\mathbf {x}_i(\\epsilon )) - L_{t-1}(f) \\right]$ [Proof of Lemma REF] We shall start by showing that the relaxation is admissible for the game where we pick prediction $\\hat{y}_t$ and the adversary then directly picks the gradient $\\partial \\ell (\\hat{y}_t,y_t)$ .", "To this end note that $\\inf _{\\hat{y}_t} \\sup _{\\partial \\ell (\\hat{y}_t,y_t)}& \\left\\lbrace \\partial \\ell (\\hat{y}_t,y_t) \\cdot \\hat{y}_t + \\mathbf {Rel}_{T}\\left(\\mathcal {F} | \\partial \\ell (\\hat{y}_1,y_1) , \\ldots , \\partial \\ell (\\hat{y}_t,y_t) \\right) \\right\\rbrace \\\\& = \\inf _{\\hat{y}_t} \\sup _{\\partial \\ell (\\hat{y}_t,y_t)} \\left\\lbrace \\partial \\ell (\\hat{y}_t,y_t) \\cdot \\hat{y}_t + \\mathbb {E}_{\\epsilon }\\left[ \\sup _{f \\in \\mathcal {F}} 2 L \\sum _{i=t+1}^T \\epsilon _i f[t] - \\sum _{i=1}^t \\partial \\ell (\\hat{y}_i,y_i) \\cdot f[i] \\right] \\right\\rbrace \\\\& \\le \\inf _{\\hat{y}_t} \\sup _{r_t \\in [-L , L]} \\left\\lbrace r_t\\cdot \\hat{y}_t + \\mathbb {E}_{\\epsilon }\\left[ \\sup _{f \\in \\mathcal {F}} 2 L \\sum _{i=t+1}^T \\epsilon _i f[t] - L_{t-1}(f) - r_t \\cdot f[t] \\right] \\right\\rbrace $ Let us use the notation $L_{t-1}(f) = \\sum _{i=1}^{t-1} \\partial \\ell (\\hat{y}_i,y_i) \\cdot f[i]$ for the present proof.", "The supremum over $r_t\\in [-L,L]$ is achieved at the endpoints since the expression is convex in $r_t$ .", "Therefore, the last expression is equal to $&\\inf _{\\hat{y}_t} \\sup _{r_t \\in \\lbrace -L , L\\rbrace } \\left\\lbrace r_t\\cdot \\hat{y}_t + \\mathbb {E}_{\\epsilon } \\sup _{f \\in \\mathcal {F}}\\left[ 2 L \\sum _{i=t+1}^T \\epsilon _i f[t] - L_{t-1}(f) - r_t \\cdot f[t] \\right]\\right\\rbrace \\\\& = \\inf _{\\hat{y}_t} \\sup _{p_t \\in \\Delta (\\lbrace -L , L\\rbrace )} \\mathbb {E}_{r_t \\sim p_t}\\left[ r_t\\cdot \\hat{y}_t + \\mathbb {E}_{\\epsilon } \\sup _{f \\in \\mathcal {F}} \\left[ 2 L \\sum _{i=t+1}^T \\epsilon _i f[t] - L_{t-1}(f) - r_t \\cdot f[t]\\right] \\right] \\\\& = \\sup _{p_t \\in \\Delta (\\lbrace -L , L\\rbrace )} \\inf _{\\hat{y}_t} \\mathbb {E}_{r_t \\sim p_t}\\left[ r_t\\cdot \\hat{y}_t + \\mathbb {E}_{\\epsilon } \\sup _{f \\in \\mathcal {F}} \\left[ 2 L \\sum _{i=t+1}^T \\epsilon _i f[t] - L_{t-1}(f) - r_t \\cdot f[t] \\right] \\right]$ where the last step is due to the minimax theorem.", "The last quantity is equal to $& \\sup _{p_t \\in \\Delta (\\lbrace -L , L\\rbrace )} \\mathbb {E}_{\\epsilon }\\left[ \\mathbb {E}_{r_t \\sim p_t}\\left[ \\inf _{\\hat{y}_t} \\mathbb {E}_{r_t \\sim p_t}\\left[ r_t \\right] \\cdot \\hat{y}_t + \\sup _{f \\in \\mathcal {F}} \\left( 2 L \\sum _{i=t+1}^T \\epsilon _i f[t] - L_{t-1}(f) - r_t \\cdot f[t] \\right) \\right] \\right]\\\\& \\le \\sup _{p_t \\in \\Delta (\\lbrace -L , L\\rbrace )} \\mathbb {E}_{\\epsilon }\\left[ \\mathbb {E}_{r_t \\sim p_t}\\left[ \\sup _{f \\in \\mathcal {F}} \\left( 2 L \\sum _{i=t+1}^T \\epsilon _i f[t] - L_{t-1}(f) + (\\mathbb {E}_{r_t \\sim p_t}\\left[ r_t \\right] - r_t) \\cdot f[t] \\right) \\right] \\right]\\\\& \\le \\sup _{p_t \\in \\Delta (\\lbrace -L , L\\rbrace )} \\mathbb {E}_{r_t , r^{\\prime }_t \\sim p_t}\\left[ \\mathbb {E}_{\\epsilon } \\sup _{f \\in \\mathcal {F}} \\left[ 2 L \\sum _{i=t+1}^T \\epsilon _i f[t] - L_{t-1}(f) + (r^{\\prime }_t - r_t) \\cdot f[t] \\right] \\right]\\\\& = \\sup _{p_t \\in \\Delta (\\lbrace -L , L\\rbrace )} \\mathbb {E}_{r_t , r^{\\prime }_t \\sim p_t}\\left[ \\mathbb {E}_{\\epsilon } \\sup _{f \\in \\mathcal {F}} \\left[ 2 L \\sum _{i=t+1}^T \\epsilon _i f[t] - L_{t-1}(f) + \\epsilon _t (r^{\\prime }_t - r_t) \\cdot f[t] \\right] \\right]$ By passing to the worst-case choice of $r_t , r^{\\prime }_t$ (which is achieved at the endpoints because of convexity), we obtain a further upper bound $&\\sup _{r_t , r^{\\prime }_t \\in \\lbrace L, -L\\rbrace } \\mathbb {E}_{\\epsilon } \\sup _{f \\in \\mathcal {F}} \\left[ 2 L \\sum _{i=t+1}^T \\epsilon _i f[t] - L_{t-1}(f) + \\epsilon _t (r^{\\prime }_t - r_t) \\cdot f[t] \\right]\\\\& \\le \\sup _{r_t \\in \\lbrace L, -L\\rbrace } \\mathbb {E}_{\\epsilon } \\sup _{f \\in \\mathcal {F}} \\left[2 L \\sum _{i=t+1}^T \\epsilon _i f[t] - L_{t-1}(f) + 2 \\epsilon _t r_t \\cdot f[t] \\right]\\\\& = \\sup _{r_t \\in \\lbrace L, -L\\rbrace } \\mathbb {E}_{\\epsilon } \\sup _{f \\in \\mathcal {F}} \\left[ 2 L \\sum _{i=t}^T \\epsilon _i f[t] - L_{t-1}(f) \\right] \\\\& = \\mathbf {Rel}_{T}\\left(\\mathcal {F} | \\partial \\ell (\\hat{y}_1,y_1) , \\ldots , \\partial \\ell (\\hat{y}_{t-1},y_{t-1}) \\right)$ Thus we see that the relaxation is admissible.", "Now the corresponding prediction is given by $\\hat{y}_t & = \\underset{\\hat{y}}{\\mathrm {argmin}} \\ \\sup _{r_t \\in [-L,L]} \\left\\lbrace r_t \\hat{y} + \\mathbb {E}_{\\epsilon }\\left[ \\sup _{f \\in \\mathcal {F}} \\left\\lbrace 2 L \\sum _{i=t+1}^{T} \\epsilon _i f[i] - \\sum _{i=1}^{t-1} \\partial \\ell (\\hat{y}_i,y_i) f[i] - r_t f[t] \\right\\rbrace \\right] \\right\\rbrace \\\\& = \\underset{\\hat{y}}{\\mathrm {argmin}} \\ \\sup _{r_t \\in [-L,L]} \\left\\lbrace r_t \\hat{y} + \\mathbb {E}_{\\epsilon }\\left[ \\sup _{f \\in \\mathcal {F}} \\left\\lbrace 2 L \\sum _{i=t+1}^{T} \\epsilon _i f[i] - \\sum _{i=1}^{t-1} \\partial \\ell (\\hat{y}_i,y_i) f[i] - r_t f[t] \\right\\rbrace \\right] \\right\\rbrace \\\\& = \\underset{\\hat{y}}{\\mathrm {argmin}} \\ \\sup _{r_t \\in \\lbrace -L,L\\rbrace } \\left\\lbrace r_t \\hat{y} + \\mathbb {E}_{\\epsilon }\\left[ \\sup _{f \\in \\mathcal {F}} \\left\\lbrace 2 L \\sum _{i=t+1}^{T} \\epsilon _i f[i] - \\sum _{i=1}^{t-1} \\partial \\ell (\\hat{y}_i,y_i) f[i] - r_t f[t] \\right\\rbrace \\right] \\right\\rbrace $ The last step holds because of convexity of the term inside the supremum over $r_t$ is convex in $r_t$ and so the suprema is attained at the endpoints of the interval.", "The $\\hat{y}_t$ above is attained when both terms of the supremum are equalized, that is for $\\hat{y}_t$ is the prediction that satisfies : $\\hat{y}_t = \\mathbb {E}_{\\epsilon }\\left[ \\sup _{f \\in \\mathcal {F}} \\left\\lbrace \\sum _{i=t+1}^{T} \\epsilon _i f[i] - \\frac{1}{2L} \\sum _{i=1}^{t-1} \\partial \\ell (\\hat{y}_i,y_i) f[i] + \\frac{1}{2} f[t] \\right\\rbrace - \\sup _{f \\in \\mathcal {F}} \\left\\lbrace \\sum _{i=t+1}^{T} \\epsilon _i f[i] - \\frac{1}{2L} \\sum _{i=1}^{t-1} \\partial \\ell (\\hat{y}_i,y_i) f[i] - \\frac{1}{2} f[t] \\right\\rbrace \\right]$ Finally since the relaxation is admissible we can conclude that the regret of the algorithm is bounded as $\\mathbf {Reg}_T \\le \\mathbf {Rel}_{T}\\left(\\mathcal {F} \\right) = 2\\ L\\ \\mathbb {E}_{\\epsilon }\\left[ \\sup _{f \\in \\mathcal {F}} \\sum _{t=1}^T \\epsilon _t f[t] \\right]\\ .$ This concludes the proof.", "[Proof of Lemma  REF] The proof is similar to that of Lemma REF , with a few more twists.", "We want to establish admissibility of the relaxation given in (REF ) w.r.t.", "the randomized strategy $q_t$ we provided.", "To this end note that $& \\sup _{y_t} \\left\\lbrace \\mathbb {E}_{\\hat{y}_t \\sim q_t}\\left[ \\ell (\\hat{y}_t,y_t) \\right] + \\mathbb {E}_{\\epsilon }\\left[ \\sup _{f \\in \\mathcal {F}} \\left\\lbrace 2 L \\sum _{i=t+1}^{T} \\epsilon _i f[i] - L_{t}(f) \\right\\rbrace \\right] \\right\\rbrace \\\\& = \\sup _{y_t} \\left\\lbrace \\mathbb {E}_{\\epsilon }\\left[ \\ell (\\hat{y}_t(\\epsilon ),y_t) \\right] + \\mathbb {E}_{\\epsilon }\\left[ \\sup _{f \\in \\mathcal {F}} \\left\\lbrace 2 L \\sum _{i=t+1}^{T} \\epsilon _i f[i] - L_{t}(f) \\right\\rbrace \\right] \\right\\rbrace \\\\& \\le \\mathbb {E}_{\\epsilon }\\left[ \\sup _{y_t} \\left\\lbrace \\ell (\\hat{y}_t(\\epsilon ),y_t) + \\sup _{f \\in \\mathcal {F}} \\left\\lbrace 2 L \\sum _{i=t+1}^{T} \\epsilon _i f[i] - L_{t}(f) \\right\\rbrace \\right\\rbrace \\right]$ by Jensen's inequality, with the usual notation $L_{t}(f)= \\sum _{i=1}^{t} \\ell (f[i],y_i)$ .", "Further, by convexity of the loss, we may pass to the upper bound $& \\mathbb {E}_{\\epsilon }\\left[ \\sup _{y_t} \\left\\lbrace \\partial \\ell (\\hat{y}_t(\\epsilon ),y_t) \\hat{y}_t(\\epsilon ) + \\sup _{f \\in \\mathcal {F}} \\left\\lbrace 2 L \\sum _{i=t+1}^{T} \\epsilon _i f[i] - L_{t-1}(f) - \\partial \\ell (\\hat{y}_t(\\epsilon ),y_t) f[t] \\right\\rbrace \\right\\rbrace \\right] \\\\& \\le \\mathbb {E}_{\\epsilon }\\left[ \\sup _{y_t} \\left\\lbrace \\mathbb {E}_{r_t }\\left[ r_t \\cdot \\hat{y}_t(\\epsilon ) \\right] + \\sup _{f \\in \\mathcal {F}} \\left\\lbrace 2 L \\sum _{i=t+1}^{T} \\epsilon _i f[i] - L_{t-1}(f) - \\mathbb {E}_{r_t}\\left[ r_t \\cdot f[t] \\right] \\right\\rbrace \\right\\rbrace \\right] $ where $r_t$ is a $\\lbrace \\pm L\\rbrace $ -valued random variable with the mean $\\partial \\ell (\\hat{y}_t(\\epsilon ),y_t)$ .", "With the help of Jensen's inequality, and passing to the worst-case $r_t$ (observe that this is legal for any given $\\epsilon $ ), we have an upper bound $& \\mathbb {E}_{\\epsilon }\\left[ \\sup _{y_t} \\left\\lbrace \\mathbb {E}_{r_t \\sim \\partial \\ell (\\hat{y}_t(\\epsilon ),y_t)}\\left[ r_t \\cdot \\hat{y}_t(\\epsilon ) \\right] + \\mathbb {E}_{r_t \\sim \\partial \\ell (\\hat{y}_t(\\epsilon ),y_t)}\\left[ \\sup _{f \\in \\mathcal {F}} \\left\\lbrace 2 L \\sum _{i=t+1}^{T} \\epsilon _i f[i] - L_{t-1}(f) - r_t \\cdot f[t] \\right\\rbrace \\right] \\right\\rbrace \\right] \\\\& \\le \\mathbb {E}_{\\epsilon }\\left[ \\sup _{r_t \\in \\lbrace \\pm L\\rbrace } \\left\\lbrace r_t \\cdot \\hat{y}_t(\\epsilon ) + \\sup _{f \\in \\mathcal {F}} \\left\\lbrace 2 L \\sum _{i=t+1}^{T} \\epsilon _i f[i] - L_{t-1}(f) - r_t \\cdot f[t] \\right\\rbrace \\right\\rbrace \\right] $ Now the strategy we defined is $\\hat{y}_t(\\epsilon ) = \\underset{\\hat{y}_t}{\\mathrm {argmin}} \\ \\sup _{r_t \\in \\lbrace \\pm L\\rbrace } \\left\\lbrace r_t \\cdot \\hat{y}_t(\\epsilon ) + \\sup _{f \\in \\mathcal {F}} \\left\\lbrace 2 L \\sum _{i=t+1}^{T} \\epsilon _i f[i] - \\sum _{i=1}^{t-1} \\ell (f[i],y_i) - r_t \\cdot f[t] \\right\\rbrace \\right\\rbrace $ which can be re-written as $\\hat{y}_t(\\epsilon ) = \\left( \\sup _{f \\in \\mathcal {F}} \\left\\lbrace \\sum _{i=t+1}^{T} \\epsilon _i f[i] - \\frac{1}{2 L}L_{t-1}(f) + \\frac{1}{2} f[t] \\right\\rbrace - \\sup _{f \\in \\mathcal {F}} \\left\\lbrace \\sum _{i=t+1}^{T} \\epsilon _i f[i] - \\frac{1}{2 L} L_{t-1}(f) - \\frac{1}{2} f[t] \\right\\rbrace \\right)$ By this choice of $\\hat{y}_t(\\epsilon )$ , plugging back in Equation (REF ) we see that $\\sup _{y_t} & \\left\\lbrace \\mathbb {E}_{\\hat{y}_t \\sim q_t}\\left[ \\ell (\\hat{y}_t,y_t) \\right] + \\mathbb {E}_{\\epsilon }\\left[ \\sup _{f \\in \\mathcal {F}} \\left\\lbrace 2 L \\sum _{i=t+1}^{T} \\epsilon _i f[i] - L_{t}(f) \\right\\rbrace \\right] \\right\\rbrace \\\\& \\le \\mathbb {E}_{\\epsilon }\\left[ \\sup _{r_t \\in \\lbrace \\pm L\\rbrace } \\left\\lbrace r_t \\cdot \\hat{y}_t(\\epsilon ) + \\sup _{f \\in \\mathcal {F}} \\left\\lbrace 2 L \\sum _{i=t+1}^{T} \\epsilon _i f[i] - L_{t-1}(f) - r_t \\cdot f[t] \\right\\rbrace \\right\\rbrace \\right] \\\\& = \\mathbb {E}_{\\epsilon }\\left[ \\inf _{\\hat{y}_t} \\sup _{r_t \\in \\lbrace \\pm L\\rbrace } \\left\\lbrace r_t \\cdot \\hat{y}_t + \\sup _{f \\in \\mathcal {F}} \\left\\lbrace 2 L \\sum _{i=t+1}^{T} \\epsilon _i f[i] - L_{t-1}(f) - r_t \\cdot f[t] \\right\\rbrace \\right\\rbrace \\right] \\\\& = \\mathbb {E}_{\\epsilon }\\left[ \\inf _{\\hat{y}_t} \\sup _{p_t \\in \\Delta (\\lbrace \\pm L\\rbrace )} \\mathbb {E}_{r_t \\sim p_t} \\left\\lbrace r_t \\cdot \\hat{y}_t + \\sup _{f \\in \\mathcal {F}} \\left\\lbrace 2 L \\sum _{i=t+1}^{T} \\epsilon _i f[i] - L_{t-1}(f) - r_t \\cdot f[t] \\right\\rbrace \\right\\rbrace \\right]$ The expression inside the supremum is linear in $p_t$ , as it is an expectation.", "Also note that the term is convex in $\\hat{y}_t$ , and the domain $\\hat{y}_t \\in [ - \\sup _{f \\in \\mathcal {F}} |f[t]| , \\sup _{f \\in \\mathcal {F}} |f[t]|] $ is a bounded interval (hence, compact).", "We conclude that we can use the minimax theorem, yielding $& \\mathbb {E}_{\\epsilon }\\left[ \\sup _{p_t \\in \\Delta (\\lbrace \\pm L\\rbrace )} \\inf _{\\hat{y}_t} \\mathbb {E}_{r_t \\sim p_t}\\left[ r_t \\cdot \\hat{y}_t + \\sup _{f \\in \\mathcal {F}} \\left\\lbrace 2 L \\sum _{i=t+1}^{T} \\epsilon _i f[i] - L_{t-1}(f) - r_t \\cdot f[t] \\right\\rbrace \\right] \\right] \\\\& = \\mathbb {E}_{\\epsilon }\\left[ \\sup _{p_t \\in \\Delta (\\lbrace \\pm L\\rbrace )} \\left\\lbrace \\inf _{\\hat{y}_t} \\mathbb {E}_{r_t \\sim p_t}\\left[ r_t \\cdot \\hat{y}_t \\right] + \\mathbb {E}_{r_t \\sim p_t}\\left[ \\sup _{f \\in \\mathcal {F}} \\left\\lbrace 2 L \\sum _{i=t+1}^{T} \\epsilon _i f[i] - L_{t-1}(f) - r_t \\cdot f[t] \\right\\rbrace \\right] \\right\\rbrace \\right]\\\\& = \\mathbb {E}_{\\epsilon }\\left[ \\sup _{p_t \\in \\Delta (\\lbrace \\pm L\\rbrace )} \\left\\lbrace \\mathbb {E}_{r_t \\sim p_t}\\left[ \\sup _{f \\in \\mathcal {F}} \\left\\lbrace \\inf _{\\hat{y}_t} \\mathbb {E}_{r_t \\sim p_t}\\left[ r_t \\cdot \\hat{y}_t \\right] + 2 L \\sum _{i=t+1}^{T} \\epsilon _i f[i] - L_{t-1}(f) - r_t \\cdot f[t] \\right\\rbrace \\right] \\right\\rbrace \\right]\\\\& \\le \\mathbb {E}_{\\epsilon }\\left[ \\sup _{p_t \\in \\Delta (\\lbrace \\pm L\\rbrace )} \\left\\lbrace \\mathbb {E}_{r_t \\sim p_t}\\left[ \\sup _{f \\in \\mathcal {F}} \\left\\lbrace \\mathbb {E}_{r_t \\sim p_t}\\left[ r_t \\cdot f[t] \\right] + 2 L \\sum _{i=t+1}^{T} \\epsilon _i f[i] - L_{t-1}(f) - r_t \\cdot f[t] \\right\\rbrace \\right] \\right\\rbrace \\right]$ In the last step, we replaced the infimum over $\\hat{y}_t$ with $f[t]$ , only increasing the quantity.", "Introducing an i.i.d.", "copy $r^{\\prime }_t$ of $r_t$ , $& = \\mathbb {E}_{\\epsilon }\\left[ \\sup _{p_t \\in \\Delta (\\lbrace \\pm L\\rbrace )} \\left\\lbrace \\mathbb {E}_{r_t \\sim p_t}\\left[ \\sup _{f \\in \\mathcal {F}} \\left\\lbrace 2 L \\sum _{i=t+1}^{T} \\epsilon _i f[i] - L_{t-1}(f) + \\left(\\mathbb {E}_{r_t \\sim p_t}\\left[ r_t \\right]- r_t\\right) \\cdot f[t] \\right\\rbrace \\right] \\right\\rbrace \\right]\\\\& \\le \\mathbb {E}_{\\epsilon }\\left[ \\sup _{p_t \\in \\Delta (\\lbrace \\pm L\\rbrace )} \\left\\lbrace \\mathbb {E}_{r_t, r^{\\prime }_t \\sim p_t}\\left[ \\sup _{f \\in \\mathcal {F}} \\left\\lbrace 2 L \\sum _{i=t+1}^{T} \\epsilon _i f[i] - L_{t-1}(f) + \\left(r^{\\prime }_t- r_t\\right) \\cdot f[t] \\right\\rbrace \\right] \\right\\rbrace \\right]$ Introducing the random sign $\\epsilon _t$ and passing to the supremum over $r_t,r_t^{\\prime }$ , yields the upper bound $&\\mathbb {E}_{\\epsilon }\\left[ \\sup _{p_t \\in \\Delta (\\lbrace \\pm L\\rbrace )} \\left\\lbrace \\mathbb {E}_{r_t, r^{\\prime }_t \\sim p_t}\\mathbb {E}_{\\epsilon _t}\\left[ \\sup _{f \\in \\mathcal {F}} \\left\\lbrace 2 L \\sum _{i=t+1}^{T} \\epsilon _i f[i] - L_{t-1}(f) + \\left(r^{\\prime }_t- r_t\\right) \\cdot f[t] \\right\\rbrace \\right] \\right\\rbrace \\right]\\\\& \\le \\mathbb {E}_{\\epsilon }\\left[ \\sup _{r_t, r^{\\prime }_t \\in \\lbrace \\pm L\\rbrace } \\left\\lbrace \\mathbb {E}_{\\epsilon _t}\\left[ \\sup _{f \\in \\mathcal {F}} \\left\\lbrace 2 L \\sum _{i=t+1}^{T} \\epsilon _i f[i] - L_{t-1}(f) + \\epsilon _t\\left(r^{\\prime }_t- r_t\\right) \\cdot f[t] \\right\\rbrace \\right] \\right\\rbrace \\right] \\\\& \\le \\mathbb {E}_{\\epsilon }\\left[ \\sup _{r_t, r^{\\prime }_t \\in \\lbrace \\pm L\\rbrace } \\left\\lbrace \\mathbb {E}_{\\epsilon _t}\\left[ \\sup _{f \\in \\mathcal {F}} \\left\\lbrace L \\sum _{i=t+1}^{T} \\epsilon _i f[i] - \\frac{1}{2}L_{t-1}(f) + \\epsilon _t r^{\\prime }_t \\cdot f[t] \\right\\rbrace \\right] \\right\\rbrace \\right] \\\\& ~~~~~~~~~~~~~ + \\mathbb {E}_{\\epsilon }\\left[ \\sup _{r_t, r^{\\prime }_t \\in \\lbrace \\pm L\\rbrace } \\left\\lbrace \\mathbb {E}_{\\epsilon _t}\\left[ \\sup _{f \\in \\mathcal {F}} \\left\\lbrace L \\sum _{i=t+1}^{T} \\epsilon _i f[i] - \\frac{1}{2}L_{t-1}(f) - \\epsilon _t r_t \\cdot f[t] \\right\\rbrace \\right] \\right\\rbrace \\right]$ In the above we split the term in the supremum as the sum of two terms one involving $r_t$ and other $r^{\\prime }_t$ (other terms are equally split by dividing by 2), yielding $&\\mathbb {E}_{\\epsilon }\\left[ \\sup _{r_t \\in \\lbrace \\pm L\\rbrace } \\left\\lbrace \\mathbb {E}_{\\epsilon _t}\\left[ \\sup _{f \\in \\mathcal {F}} \\left\\lbrace 2 L \\sum _{i=t+1}^{T} \\epsilon _i f[i] - L_{t-1}(f) + 2 \\ \\epsilon _t\\ r_t \\cdot f[t] \\right\\rbrace \\right] \\right\\rbrace \\right]$ The above step used the fact that the first term only involved $r^{\\prime }_t$ and second only $r_t$ and further $\\epsilon _t$ and $- \\epsilon _t$ have the same distribution.", "Now finally noting that irrespective of whether $r_t$ in the above supremum is $L$ or $-L$ , since it is multiplied by $\\epsilon _t$ we obtain an upper bound $&\\mathbb {E}_{\\epsilon }\\left[ \\sup _{f \\in \\mathcal {F}} \\left\\lbrace 2 L \\sum _{i=t}^{T} \\epsilon _i f[i] - L_{t-1}(f) \\right\\rbrace \\right]$ We conclude that the relaxation $\\mathbf {Rel}_{T}\\left(\\mathcal {F} | y_1,\\ldots ,y_t \\right) = \\mathbb {E}_{\\epsilon }\\left[ \\sup _{f \\in \\mathcal {F}} \\left\\lbrace 2 L \\sum _{i=t+1}^{T} \\epsilon _i f[i] - L_{t}(f) \\right\\rbrace \\right]$ is admissible and further the randomized strategy where on each round we first draw $\\epsilon $ 's and then set $\\hat{y}_t(\\epsilon ) & = \\left( \\sup _{f \\in \\mathcal {F}} \\left\\lbrace \\sum _{i=t+1}^{T} \\epsilon _i f[i] - \\frac{1}{2 L}L_{t-1}(f) + \\frac{1}{2} f[t] \\right\\rbrace - \\sup _{f \\in \\mathcal {F}} \\left\\lbrace \\sum _{i=t+1}^{T} \\epsilon _i f[i] - \\frac{1}{2 L} L_{t-1}(f) - \\frac{1}{2} f[t] \\right\\rbrace \\right)\\\\& = \\left(\\inf _{f \\in \\mathcal {F}} \\left\\lbrace - \\sum _{i=t+1}^{T} \\epsilon _i f[i] + \\frac{1}{2 L} L_{t-1}(f) + \\frac{1}{2} f[t] \\right\\rbrace - \\inf _{f \\in \\mathcal {F}} \\left\\lbrace - \\sum _{i=t+1}^{T} \\epsilon _i f[i] + \\frac{1}{2 L}L_{t-1}(f) - \\frac{1}{2} f[t] \\right\\rbrace \\right)$ is an admissible strategy.", "Hence, the expected regret under the strategy is bounded as $\\mathbb {E}\\left[ \\mathbf {Reg}_T \\right] \\le \\mathbf {Rel}_{T}\\left(\\mathcal {F} \\right) = 2 L\\ \\mathbb {E}_{\\epsilon }\\left[ \\sup _{f \\in \\mathcal {F}} \\sum _{i=1}^{T} \\epsilon _i f[i] \\right]$ which concludes the proof.", "[Proof of Lemma  REF] The proof is almost identical to the proof of admissibility for the Mirror Descent relaxation, so let us only point out the differences.", "Let $\\tilde{x}_{t-1}=\\sum _{i=1}^t x_i$ and $\\mu _{t-1}=\\frac{1}{t-1}\\tilde{x}_{t-1}$ .", "Using the fact that $x_t$ is $\\sigma _t$ -close to $\\mu _{t-1}$ , we expand $\\left( \\left\\Vert \\tilde{x}_{t}\\right\\Vert ^2 + C\\sum _{s=t+1}^T\\sigma _s^2 \\right)^{1/2} &\\le \\left( \\left\\Vert \\tilde{x}_{t-1} \\left(\\frac{t}{t-1}\\right) \\right\\Vert ^2 + \\left<\\nabla \\left\\Vert \\left( \\frac{t}{t-1}\\right)\\tilde{x}_{t-1}\\right\\Vert ^2,x_t-\\mu _{t-1}\\right> + C\\sum _{s=t+1}^T\\sigma _s^2 \\right)^{1/2}\\\\$ As before, pick $x_t = \\beta \\tilde{x}_{t-1} + \\gamma y$ for some $y \\in \\mathrm {Kernel}(\\nabla \\Vert \\tilde{x}_{t-1}\\Vert ^2)$ .", "The above expression under the square root then becomes $\\left\\Vert \\tilde{x}_{t-1} \\right\\Vert ^2 +\\underbrace{\\left(\\frac{1}{(t-1)^2}+\\frac{2}{t-1} + \\left( \\frac{t}{t-1}\\right)^2\\left(\\beta -\\frac{1}{t-1}\\right)\\right)}_{\\beta ^{\\prime }}\\left\\Vert \\tilde{x}_{t-1} \\right\\Vert ^2 + C\\sum _{s=t+1}^T\\sigma _s^2 ,$ and the only difference from the expression in (REF ) is that we have a $\\beta ^{\\prime }$ instead of $\\beta $ under the square root.", "Taking the derivatives, we see that $ \\alpha = \\frac{\\left(1+\\frac{1}{t-1}\\right)^2}{2\\sqrt{\\Vert \\tilde{x}_{t-1}\\Vert ^2 + C\\sum _{s=t}^T\\sigma _s^2}}$ forces $\\beta ^{\\prime }=0$ and we conclude admissibility." ], [ "Arriving at the Relaxation", "We upper bound the sequential Rademacher complexity as $&\\frac{2}{\\alpha } \\sup _{(\\mathbf {x},\\mathbf {x}^{\\prime }) \\in \\mathcal {T}} \\mathbb {E}_{\\epsilon }\\left[ \\sup _{f \\in \\mathcal {F}}\\left\\langle f, \\alpha \\sum _{s=t+1}^T \\epsilon _s\\left(\\mathbf {x}_s(\\epsilon ) - \\frac{1}{s-t} \\sum _{\\tau =t+1}^{s-1} \\chi _\\tau (\\epsilon _\\tau ) \\right) - \\sum _{r=1}^t x_r \\right\\rangle \\right] \\\\& \\le \\frac{2 R^2}{\\alpha } + \\frac{\\alpha }{\\lambda } \\sup _{(\\mathbf {x},\\mathbf {x}^{\\prime })} \\mathbb {E}_\\epsilon \\left\\Vert \\sum _{s=t+1}^T\\epsilon _s \\left(\\mathbf {x}_s(\\epsilon ) - \\frac{1}{s-t} \\sum _{\\tau =t+1}^{s-1} \\chi _\\tau (\\epsilon _\\tau ) \\right) - \\sum _{r=1}^t x_r \\right\\Vert ^2 \\\\&\\le \\frac{2\\sqrt{2}R}{\\sqrt{\\lambda }} \\sqrt{\\sup _{(\\mathbf {x},\\mathbf {x}^{\\prime })} \\mathbb {E}_{\\epsilon }\\left[ \\left\\Vert \\sum _{s=t+1}^T \\epsilon _s\\left( \\mathbf {x}_s(\\epsilon ) - \\frac{1}{s-t} \\sum _{\\tau =t+1}^{s-1} \\chi _\\tau (\\epsilon _\\tau )\\right) - \\sum _{r=1}^t x_r \\right\\Vert ^2 \\right]}\\\\&\\le \\frac{2\\sqrt{2}R}{\\sqrt{\\lambda }} \\sqrt{ \\left\\Vert \\sum _{r=1}^t x_r \\right\\Vert ^2 + \\sup _{(\\mathbf {x},\\mathbf {x}^{\\prime })} C\\sum _{s=t+1}^T\\left\\Vert \\mathbf {x}_s(\\epsilon ) - \\frac{1}{s-t} \\sum _{\\tau =t+1}^{s-1} \\chi _\\tau (\\epsilon _\\tau ) \\right\\Vert ^2 }$ Since $(\\mathbf {x},\\mathbf {x}^{\\prime }) \\in \\mathcal {T}$ are pairs of tree such that for any $\\epsilon \\in \\lbrace \\pm 1\\rbrace ^T$ and any $t \\in [T]$ .", "$C(x_1,\\ldots ,x_t,\\chi _1(\\epsilon _1), \\ldots ,\\chi _{t-1}(\\epsilon _{t-1}), \\mathbf {x}_{t}(\\epsilon )) = 1$ we can conclude that for any $\\epsilon \\in \\lbrace \\pm 1\\rbrace ^T$ and any $t \\in [T]$ , $\\left\\Vert \\mathbf {x}_t(\\epsilon ) - \\frac{1}{t-1} \\sum _{\\tau =1}^{t-1} \\chi _{\\tau }(\\epsilon _\\tau ) \\right\\Vert \\le \\sigma _t$ [Proof of Lemma  REF] Then Sequential Rademacher complexity can be upper bounded as $\\sup _{\\mathbf {x}} \\mathbb {E}_{\\epsilon }\\left[ \\left\\Vert \\sum _{i=1}^t x_t + \\sum _{i=1}^{T-t} \\epsilon _i \\mathbf {x}_i(\\epsilon ) \\right\\Vert \\right] & \\le \\sup _{\\mathbf {x}} \\left( \\mathbb {E}_{\\epsilon }\\left[ \\left\\Vert \\sum _{i=1}^t x_t + \\sum _{i=1}^{T-t} \\epsilon _i \\mathbf {x}_i(\\epsilon ) \\right\\Vert ^p \\right] \\right)^{1/p}\\\\& \\le \\sup _{\\mathbf {x}} \\left( \\mathbb {E}_{\\epsilon }\\left[ \\left\\Vert \\sum _{i=1}^t x_t + \\sum _{i=1}^{T-t} \\epsilon _i \\mathbf {x}_i(\\epsilon ) \\right\\Vert ^p - C \\sum _{i=1}^{T-t} \\mathbb {E}_{\\epsilon }\\left[ \\left\\Vert \\mathbf {x}_i(\\epsilon )\\right\\Vert ^p \\right] \\right] + C (T - t) \\right)^{1/p} \\\\& = \\left( \\Psi ^*\\left(\\sum _{i=1}^t x_i \\right) + C (T-t) \\right)^{1/p} \\\\&\\le \\left( \\Psi ^*\\left(\\sum _{i=1}^{t-1} x_i \\right) +\\left<\\nabla \\Psi ^*\\left(\\sum _{i=1}^{t-1} x_i\\right),x_t\\right> + C (T-t+1) \\right)^{1/p}$ and admissibility is verified in a similar way to the 2-smooth case in the Section .", "Here we instead use $p$ -smoothness which follows from result in [18].", "The form of update specified by the relaxation in this case follows exactly the proof of Proposition REF , yielding $f_t = - \\frac{\\nabla \\Psi ^*(\\sum _{j=1}^{t-1} x_i)}{p \\left( \\Psi ^*(\\sum _{j=1}^{t-1} x_i) + C(T-t+1)\\right)^{1/p}}$ Lemma 26 The regret upper bound $\\sum _{t=1}^T \\ell (f_t,x_t) - \\inf _{f \\in \\mathcal {F}} \\sum _{t=1}^T \\ell (f,x_t) \\le \\sum _{t=1}^T \\ell (f_t,x_t) - \\sum _{i=1}^m \\inf _{f \\in \\mathcal {F}^{k_i}\\left(x_1,\\ldots ,x_{\\tilde{k}_{i-1}}\\right)} \\sum _{t=\\tilde{k}_{i-1}+1}^{\\tilde{k}_{i}} \\ell (f,x_t) \\ .$ is valid.", "[Proof of Lemma  REF] To prove this inequality, it is enough to show that it holds for subdividing $T$ into two blocks $k_1$ and $k_2$ .", "Observe, that the comparator term becomes only smaller if we pass to two instead of one infima, but we must check that no function $f$ that minimizes the loss over the first block is removed from being a potential minimizer over the second block.", "This is exactly the definition of $\\mathcal {F}^{k_2}(x_1,\\ldots ,x_{k_1})$ .", "Lemma 27 The relaxation $\\mathbf {Rel}_{T}\\left(\\mathcal {F} | x_{1},\\ldots ,x_{t} \\right) = - \\inf _{f \\in \\mathcal {F}} \\sum _{i=1}^t x_i(f) + (T-t) \\inf _{f \\in \\mathcal {F}} \\sup _{f^{\\prime } \\in \\mathcal {F}} \\Vert f - f^{\\prime }\\Vert $ is admissible.", "[Proof of Lemma  REF] First, $\\mathbf {Rel}_{T}\\left(\\mathcal {F} | x_{1},\\ldots ,x_T \\right) = - \\inf _{f \\in \\mathcal {F}} \\sum _{t=1}^T x_t(f).", "$ As for admissibility, $&\\inf _{f_t \\in \\mathcal {F}} \\sup _{x} \\left\\lbrace x(f_t) + \\mathbf {Rel}_{T}\\left(\\mathcal {F} | x_1,\\ldots ,x_{t-1},x \\right)\\right\\rbrace \\\\& = \\inf _{f_t \\in \\mathcal {F}} \\sup _{x} \\left\\lbrace x(f_t) - \\inf _{f \\in \\mathcal {F}} \\left\\lbrace \\sum _{i=1}^{t-1} x_i(f) + x(f) \\right\\rbrace \\right\\rbrace + (T-t) \\inf _{f \\in \\mathcal {F}} \\sup _{f^{\\prime } \\in \\mathcal {F}} \\left\\Vert f - f^{\\prime }\\right\\Vert \\\\& \\le \\inf _{f_t \\in \\mathcal {F}} \\sup _{x} \\left\\lbrace x(f_t) - \\inf _{f \\in \\mathcal {F}} \\sum _{i=1}^{t-1} x_i(f) - \\inf _{f \\in \\mathcal {F}} x(f)\\right\\rbrace + (T-t) \\inf _{f \\in \\mathcal {F}} \\sup _{f^{\\prime } \\in \\mathcal {F}} \\left\\Vert f - f^{\\prime }\\right\\Vert \\\\& \\le \\inf _{f_t \\in \\mathcal {F}} \\sup _{x} \\left\\lbrace \\sup _{f \\in \\mathcal {F}} \\left<\\nabla x, f_t - f\\right> - \\inf _{f \\in \\mathcal {F}} \\sum _{i=1}^{t-1} x_i(f) \\right\\rbrace + (T-t) \\inf _{f \\in \\mathcal {F}} \\sup _{f^{\\prime } \\in \\mathcal {F}} \\left\\Vert f - f^{\\prime }\\right\\Vert \\\\& \\le \\inf _{f_t \\in \\mathcal {F}} \\left\\lbrace \\sup _{f \\in \\mathcal {F}} \\left\\Vert f_t - f\\right\\Vert - \\inf _{f \\in \\mathcal {F}} \\sum _{i=1}^{t-1} x_i(f) \\right\\rbrace + (T-t) \\inf _{f \\in \\mathcal {F}} \\sup _{f^{\\prime } \\in \\mathcal {F}} \\left\\Vert f - f^{\\prime }\\right\\Vert \\\\& = \\mathbf {Rel}_{T}\\left(\\mathcal {F} | x_1,\\ldots ,x_{t-1} \\right)$" ], [ "Acknowledgements", "We gratefully acknowledge the support of NSF under grants CAREER DMS-0954737 and CCF-1116928." ] ]

[ [ "Adaptive Gaussian inverse regression with partially unknown operator" ], [ "Abstract This work deals with the ill-posed inverse problem of reconstructing a function $f$ given implicitly as the solution of $g = Af$, where $A$ is a compact linear operator with unknown singular values and known eigenfunctions.", "We observe the function $g$ and the singular values of the operator subject to Gaussian white noise with respective noise levels $\\varepsilon$ and $\\sigma$.", "We develop a minimax theory in terms of both noise levels and propose an orthogonal series estimator attaining the minimax rates.", "This estimator requires the optimal choice of a dimension parameter depending on certain characteristics of $f$ and $A$.", "This work addresses the fully data-driven choice of the dimension parameter combining model selection with Lepski's method.", "We show that the fully data-driven estimator preserves minimax optimality over a wide range of classes for $f$ and $A$ and noise levels $\\varepsilon$ and $\\sigma$.", "The results are illustrated considering Sobolev spaces and mildly and severely ill-posed inverse problems." ], [ "Introduction", "Let $(H,_H)$ and $(G,_G)$ be separable Hilbert spaces and $$ a compact linear operator from $H$ to $G$ with unknown singular values.", "This work deals with the reconstruction of a function $\\in H$ given noisy observations of the image $=$ on the one hand and of the unknown sequence of singular values $=(_j)_{j\\in }$ on the other hand.", "In other words, we consider a statistical inverse problem with partially unknown operator.", "There is a vast literature on statistical inverse problems.", "For the case where the operator is fully known, the reader may refer to [10], [11], [12], and [2] and the references therein.", "A typical illustration of such a situation is a deconvolution problem (cf.", "[6], [15], and [7] among many others).", "For a more detailed discussion and motivation of the case of a partially unknown operator which we consider in this work, we refer the reader to [3].", "[5] and [13] consider such a setting in the particular context of a density deconvolution problem.", "Let us describe in more detail the model we are going to consider.", "We suppose that $$ admits a singular value decomposition $(_j,_j,_j)_{j\\in }$ as follows.", "Denote by $^*$ the adjoint operator of $$ .", "Then, $^*$ is a compact operator on $H$ with eigenvalues $(_j^2)_{j\\in }$ whose associated orthonormal basis of eigenfunctions $\\lbrace _j\\rbrace $ we suppose to be known.", "Analogously, the operator $^*$ has eigenvalues $(_j^2)_{j\\in }$ and known orthonormal eigenfunctions $_j = {_j}_G^{-1}_j$ in $G$ .", "Projecting the inverse problem $=$ on the eigenfunctions, we obtain the system of equations ${}_j:={,_j}_G = _j {,_j}_H$ for $j\\in $ .", "As the operator $$ is compact, the sequence of singular values tends to zero and the inverse problem is called ill-posed.", "The solution $$ is characterized by its coefficients ${}_j:= {,_j}_H$ .", "Our objective is their estimation based on the following observations: $Y_j = {}_j+ \\sqrt{}\\,_j= _j{}_j+ \\sqrt{}_j \\qquad \\text{and}\\qquad X_j = _j + \\sqrt{}\\,_j\\qquad (j\\in ),$ where the $_j,_j$ are iid.", "standard normally distributed random variables and $,\\in (0,1)$ are noise levels.", "Thus we represent the problem at hand as a hierarchical Gaussian sequence space model.", "Of course $$ can only be reconstructed from such observations if all the $_j$ are non-zero which is the case if and only if the operator $$ is injective.", "We assume this from now on, which allows us to write $= \\sum _{j=1}^\\infty {}_j_j^{-1}\\,_j$ .", "Hence, an orthogonal series estimator of $$ is a natural approach: $\\qquad \\qquad _k := \\sum _{j=1}^k\\frac{Y_j}{X_j}_{[X_j^2\\ge \\,\\sigma ]}\\, _j.$ The threshold using the indicator function accounts for the uncertainty caused by estimating the $_j$ by $X_j$ .", "It corresponds to $X_j$ 's noise level as an estimator of $_j$ , which is a natural choice [13].", "Note that $_k$ depends on a dimension parameter $k$ whose choice essentially determines the estimation accuracy.", "Its optimal choice generally depends on both unknown sequences $({}_j)$ and $(_j)$ .", "Our purpose is to establish an adaptive estimation procedure for the function $$ which does not depend on these sequences.", "More precisely, assuming that the solution and the operator belong to given classes $\\in $ and $\\in $ , respectively, we shall measure the accuracy of an estimator $$ of $$ by the maximal weighted risk $_(,,) :=\\sup _{\\in }\\sup _{\\in }{-}_^2$ defined some weighted norm $_:=\\sum _{j\\in }_j|{\\cdot }_j|^2$ , where $:= (_j)_{j\\in }$ is a strictly positive weight sequences.", "This allows us to quantify the estimation accuracy in terms of the mean integrated square error (MISE) not only of $$ itself, but as well of its derivatives, for example.", "Given observations $Y=(Y_j)_{j\\in }$ and $X =(X_j)_{j\\in }$ with respective noise levels $$ and $$ according to (REF ), the minimax risk the classes $$ and $$ is then defined as $^*_(,,,) :=\\inf _{}_(,,)$ , where the infimum is taken over all possible estimators $$ of $$ .", "An estimator $$ is said to attain the minimax rate or to be minimax optimal $$ and $$ if there is a constant $C>0$ depending on the classes only such that $_(,,)\\le C\\,^*_(,,,)$ for all $,\\in (0,1)$ .", "An estimation procedure which is fully data-driven and minimax optimal for a wide range of classes $$ and $$ is called adaptive.", "In the next section, we show that for a wide range of classes $$ and $$ the orthogonal series estimator $_$ attains the minimax rate for an optimal choice $$ of the dimension parameter.", "We illustrate this result considering subsets of Sobolev spaces for $$ and distinguishing two types of operator classes $$ specifying the decay of the singular values: If $(_j)$ decays polynomially, the inverse problem is called mildly ill-posed and severely ill-posed if they decay exponentially.", "However, $$ is chosen subject to a classical variance-squared-bias trade-off and depends on properties of both classes $$ and $$ which are unknown in general.", "The last section is devoted to the development of a data-driven choice $$ of $k$ , following the general model selection scheme [1].", "This methodology requires the careful choice of a contrast function and a penalty term.", "In this work, we will use a contrast function inspired by the work of [8] who consider bandwidth selection for kernel estimators.", "Given a random sequence $(_k)_{k\\ge 1}$ of penalties, a random set $\\lbrace 1,\\dots ,_{,} \\rbrace $ of admissible dimension parameters and the random sequence of contrasts $_k := \\max _{k\\le j \\le _{,}} \\Big \\lbrace {_j -_k}^2_- _j \\Big \\rbrace \\qquad (k\\in ).$ The dimension parameter is selected as the minimizerFor a sequence $(b_k)_{k\\in }$ attaining a minimal value on $N\\subset $ , let $\\limits _{n\\in N}b_n := \\min \\lbrace n\\in N\\;|\\;b_n \\le b_k \\; \\forall k\\in N\\rbrace $ .", "of a penalized contrast $:= _{1\\le k \\le }\\Big \\lbrace _k + _k \\Big \\rbrace .$ We assess the accuracy of the fully data-driven estimator $_$ deriving an upper bound for $_(_,,)$ .", "Obviously this upper bound heavily depends the random sequence $(_k)$ and the random upper bound $$ .", "However, we construct these objects in such a way that the resulting fully data-driven estimator $_$ is minimax optimal over a wide range of classes and thus adaptive.", "The more technical proofs and some auxiliary results are deferred to the appendix.", "[9] also study adaptive estimation in linear inverse problems, but their method is limited to mildly ill-posed inverse problems with known degree of ill-posedness.", "Also, the theoretical framework is quite different: they focus on sparse representations and therefore consider estimators based on wavelet thresholding and show their rate-optimality and adaptivity properties over Besov spaces with respect to the corresponding norms.", "Adaptive estimation in a hierarchical Gaussian sequence space model has previously been considered by [3].", "Though, the authors restrict their investigation to the mildly ill-posed case and to noise levels satisfying $\\le $ .", "The new approach presented in this paper has the advantage of not requiring such restrictions.", "On the contrary, the influence of the two noise levels on the estimation accuracy is characterized.", "Moreover, the estimator presented in this paper can attain optimal convergence rates independently of whether the underlying inverse problem is mildly or severely ill-posed, for example, even when $\\ll $ .", "This is an important feature in applications where the reduction of the noise level $$ can be costly.", "In (satellite or medical) imaging, for example, the observation of the sequence $X$ may correspond to calibration measurements.", "In order to achieve an adequately high precision of these measures as to reduce the noise level $$ sufficiently, one might have to repeat expensive experiments.", "It is thus desirable to know how the estimator performs when $$ exceeds $$ ." ], [ "Minimax", "In this section we develop a minimax theory for Gaussian inverse regression with respect to the classes $\\begin{split}&:=\\bigg \\lbrace h \\in H\\;\\Big |\\;\\sum _{j\\in } _j|[h]_j|^2=:{h}_^2\\le \\bigg \\rbrace \\mbox{ and } \\\\&:=\\bigg \\lbrace T\\in C(H,G) \\;\\Big |\\; \\text{The eigenvalues$\\lbrace u_j\\rbrace $ of $T^*T$ satisfy}\\; 1/\\le \\frac{u_j^2}{_j}\\le \\quad \\forall \\, j\\in \\bigg \\rbrace ,\\end{split}$ where $C(H,G)$ denotes the set of all compact linear operators from $H$ to $G$ having $\\lbrace _j\\rbrace $ and $\\lbrace _j\\rbrace $ as eigenfunctions, respectively.", "The minimal regularity conditions on the solution, the operator and the weighted norm $_$ which we need in this section are summarized in the following assumption.", "Assumption 2.1 Let $:=(_j)_{j\\in }$ , $:=(_j)_{j\\in }$ and $:=(_j)_{j\\in }$ be strictly positive sequences of weights with $_1=_1=_1 =1$ such that $/$ and $$ are non-increasing, respectively.", "Illustration 2.2 As an illustration of the results below, we will consider weight sequences $_j = j^{2p}$ , for which $$ is a Sobolev space of $p$ -times differentiable functions if we consider the trigonometric basis in $H=L^2[0,1]$ .", "As for the operator, we will distinguish the cases $_j= j^{-2b}$ , referred to as mildly ill-posed ([m]) and $_j = \\exp (-j^{2b})$ , the severely ill-posed case ([s]).", "Concerning the weighted norm, we will consider sequences$b_\\rho \\sim c_\\rho $ means that $\\lim _{\\rho \\rightarrow 0} b_\\rho /c_\\rho $ exists in $(0,\\infty )$ .", "$_j\\sim j^{2s}$ , such that ${f}_= {f^{(s)}}_{L^2}$ for all $f\\in $ .", "We will assume that $b\\ge 0$ and $p\\ge s \\ge 0$ , such that Assumption REF is satisfied.", "The following result states lower risk bounds for the estimation of $$ and thus describes the complexity of the problem.", "Theorem 2.3 Suppose that we observe sequences $Y$ and $X$ according to the model (REF ).", "Consider sequences $$ , $$ , and $$ satisfying Assumption REF .", "For all $,\\in (0,1)$ , define $\\rho _{k,} := \\max \\Bigl (\\frac{_k}{_k},\\sum _{j=1}^k\\frac{_j}{_j}\\Bigr ),\\quad _:=\\min _{k\\in } \\rho _{k,}, \\quad :=\\limits _{k\\in } \\rho _{k,},\\quad _:=\\max _{k\\in } \\Bigr \\lbrace \\frac{_k}{_k}\\min \\Bigl (1,\\frac{}{_k}\\Bigr )\\Bigr \\rbrace .$ If $\\eta :=\\inf _{n\\in }\\lbrace _^{-1}\\min (_{}_{}^{-1},\\sum _{l=1}^{}{_l}{(_l)}^{-1}) \\rbrace >0$ , then $\\inf _{} _(,,)\\ge \\frac{1}{4d}\\min (\\eta ,r) \\min (r,1/(2d), (1-d^{-1/2})^2)\\;\\max (_, _),$ where the infimum is to be taken over all possible estimators $$ of $$ .", "It is noteworthy that apart from the unwieldy constant, the lower bound is given by two terms ($_$ and $_$ ), each of which depending only on one noise level.", "We show in the proof that $_$ is actually, up to a constant, a lower risk bound uniformly for any known operator $$ in the class $$ .", "Hence, in this case no supremum over the class $$ would be needed.", "The term $_$ only arises if the operator is unknown in $$ .", "The proof of this lower bound is based on a comparison of different inverse problems with different operators in $$ , whence the supremum over this class.", "The term $_$ quantifies to which extent the additional difficulty arising from the preliminary estimation of the eigenvalues $_j$ influences the possible estimation accuracy for $$ : As long as $_\\ge _$ , the same lower bound as in the case of known eigenvalues holds.", "Otherwise, the lower bound increases.", "Notice further that the term $\\rho _{k,}$ above corresponds to the MISE of the orthogonal series estimator $_k$ in the case of known eigenvalues $_j$ , and $$ is its minimizer $k$ .", "Under classical smoothness assumptions, the rates and $$ take the following forms.", "Illustration 2.4 In the special cases defined in Illustration REF above, the rates from (REF ) are [m]      $ _\\sim ^{2(p-s) / (2p+2b+1)},\\qquad \\sim ^{-1 / (2p+2b+1)}, \\qquad _\\sim ^{((p-s)\\wedge b)/b} $ [s]       $_\\sim |\\log |^{(p-s)/b},\\qquad \\sim |\\log |^{1/(2b)},\\qquad _\\sim |\\log |^{-(p-s)/b}$ .", "The following theorem shows that the orthogonal series estimator $_$ with optimal parameter $$ given in (REF ) actually attains the lower risk bound up to a constant and is thus minimax optimal.", "Theorem 2.5 Under the assumptions of Theorem REF , the estimator $_$ satisfies for all $,\\in (0,1)$ $\\sup _{\\in }\\sup _{\\in } \\left\\lbrace {_{}-}^2_\\right\\rbrace \\le 4(6+ )\\,\\max (_,_).$ To conclude this section, let us summarize the resulting optimal convergence rates under the classical smoothness assumptions introduced in Illustration REF .", "In order to characterize the influence of the second noise level $$ , we consider it as a function of the first noise level $$ .", "Illustration 2.6 Let $(_)_{\\in (0,1)}$ be a noise level in $X$ depending on the noise level $$ in $Y$ .", "[m] Let $p> 1/2$ , $b>1$ , and $0\\le s\\le p$ .", "If $q_1:=\\lim \\limits _{\\rightarrow 0} ^{-2((p-s)\\vee b)/(2p+2b+2)}_$ existsThe limit <<$\\infty $ >> meaning strict divergence is authorized., then $\\sup _{\\in }\\sup _{\\in }{^{(s)}_- ^{(s)}}^2_{L^2} ={\\left\\lbrace \\begin{array}{ll}O(^{2(p-s)/(2p+2b+1)}) & \\text{if $q_1<\\infty $} \\\\O(_^{((p-s)\\wedge b)/b}) & \\text{otherwise.}\\end{array}\\right.}", "$ [s] Let $p>1/2$ ,$b>0$ and $0\\le s \\le p$ .", "If $q_2:=\\lim \\limits _{\\rightarrow 0} |\\log | \\, |\\log _|^{-1}$ exists, then $\\sup _{\\in }\\sup _{\\in }{^{(s)}_- ^{(s)}}^2_{L^2} ={\\left\\lbrace \\begin{array}{ll}O(|\\log |^{(p-s)/b}) & \\text{if $q_2<\\infty $} \\\\O(|\\log _|^{(p-s)/b}) & \\text{otherwise.}\\end{array}\\right.", "}$ This illustration shows that often the same optimal rates as in the case of known eigenvalues hold even when $<$ ." ], [ "Adaptation", "In this section, we construct a fully data-driven estimator of $$ following the procedure sketched in (REF ) and (REF ).", "The following Lemma will be our key tool when controlling the risk of the adaptive estimator.", "Lemma 3.1 Let $$ be an arbitrary positive sequence and $K\\in $ .", "Consider the sequence $$ of contrasts $_k := \\max _{k\\le j \\le K} \\Big \\lbrace {_j -_k}^2_- _j \\Big \\rbrace $ and $:= _{1\\le j \\le K}\\lbrace _j+_j\\rbrace $ .", "Let further $(t)_+ := (t\\vee 0)$ .", "If $(_1,\\dots ,_K)$ is non-decreasing, then we have for all $1\\le k \\le K$ that ${_-}^2_\\le 7_k + 78^2_k +42\\max _{1\\le j\\le K}\\Big ({_j - _j}^2_-\\frac{1}{6}_j \\Big )_+,$ where we denote by $_j := \\sum _{k=1}^j {}_k\\,_k$ the projection of $$ on the first $j$ basis vectors in $H$ and by $_k:= \\sup _{j\\ge k}{- _j}_$ the bias due to the projection.", "In view of the definition of $$ , we have for all $1\\le k\\le K$ that $\\begin{split}{_- }_^2 &\\le 3 \\Big \\lbrace {_-_{k\\wedge }}_^2 + {_{k\\wedge } -_{k}}_^2+ {_k - }_^2 \\Big \\rbrace \\\\&\\le 3\\Big \\lbrace _k + _+ _+ _k+{_k -}_^2 \\Big \\rbrace \\\\&\\le 6\\Big \\lbrace _k + _k \\Big \\rbrace +3{_k -}_^2\\end{split}$ Since $(_1,\\dots ,_K)$ is non-decreasing and $4_k^2\\ge \\max _{k\\le j \\le K}{_k-_j}_^2$ , we have $_k\\le 6 \\max _{1\\le j \\le K}\\Big ({_j -_j}_^2 - \\frac{1}{6} _j \\Big )_+ + 12 _k^2.$ It easily verified that for all $1\\le k \\le K$ we have ${_k - }_^2 \\le \\frac{1}{3}_k + 2_k^2 + 2\\max _{1\\le j \\le K}\\Big ({_j -_j}_^2 - \\frac{1}{6} _j \\Big )_+.$ The result follows combining the last estimates with (REF ).", "$\\Box $ The Lemma being valid for any upper bound $K$ and any monotonic sequence of penalties $$ , we need to specify our choice.", "Let us first define some auxiliary quantities required in the construction of the random penalty sequence $$ and the upper bound $$ .", "Definition 3.2 For any sequence $\\alpha := (\\alpha _j)_{j\\in }$ , define $\\Delta _k^\\alpha := \\max _{1\\le j \\le k} _j\\,\\alpha _j^{-2} $       and       $\\delta _k^\\alpha := k\\Delta _k^\\alpha \\frac{\\log (\\Delta _k^\\alpha \\vee (k+2))}{\\log (k+2)}$ ; given $_k^+ := \\max _{1\\le j\\le k}_j$ , $N^\\circ _:= \\max \\lbrace 1\\le N \\le ^{-1} \\;|\\; _N^+ \\le ^{-1}\\rbrace $ , and $v_:= (8\\log (\\log (^{-1}+20)))^{-1}$ , let $N_^\\alpha := \\min \\Big \\lbrace 2\\le j \\le N^\\circ _\\;\\Big |\\;\\frac{\\alpha _j^2}{j_j^+} \\le |\\log | \\Big \\rbrace -1\\quad \\text{and} \\quad M_^\\alpha := \\min \\Big \\lbrace 2\\le j \\le ^{-1} \\;\\Big |\\;\\alpha _j^2 \\le ^{1-v_} \\Big \\rbrace -1,$ and $K_{,}^\\alpha := N_^\\alpha \\wedge M_^\\alpha $ .", "If the defining set is empty, set $N_^\\alpha = N^\\circ _$ or $M_^\\alpha = \\lfloor ^{-1}\\rfloor $ , respectively.", "Choosing appropriate sequences $\\alpha $ , these quantities allow us define the random penalty term needed for the data-driven choice of $k$ as well as its deterministic counterpart which will be used in the control of the risk.", "Using this definition and denoting by $X$ the sequence of random variables $(X_j)_{j\\in }$ , define $_{,}:=K^X_{,} \\quad \\text{and}\\quad _k:=600\\delta _k^X\\,.$ Substituting these definitions in (REF ) and (REF ) yields a choice of the dimension parameter $k$ depending exclusively on the observations and the noise levels, but not on any underlying smoothness classes.", "Consider the upper risk bound in Lemma REF .", "In order to control the risk of the data-driven estimator, we decompose it with respect to an event on which the randomized quantities $_k$ and $_{,}$ are close to some deterministic counterparts $^a_k$ , $K^-_{,}$ , and $K^+_{,}$ to be defined below in Propositions REF and REF .", "More precisely, consider the event $\\mho _{,} := \\lbrace ^a_k\\le _k \\le 30^a_k \\quad \\forall \\;1\\le k \\le K^+_{,}\\rbrace \\cap \\lbrace K^-_{,} \\le _{,} \\le K^+_{,}\\rbrace $ and the corresponding risk decomposition ${_-}^2_={_-}^2__{\\mho _{,}}+{_-}^2__{\\mho _{,}^c}.$ As the random sequence $_k$ is non-decreasing in $k$ by construction, we may apply Lemma REF and obtain for every $1\\le k \\le _{,}$ ${_-}^2_\\le 7\\,_k + 78^2_k +42\\max _{1\\le j\\le _{,}}\\Big ({_j - _j}^2_-\\frac{1}{6}_j \\Big )_+.$ On the event $\\mho _{,}$ , this implies that ${_-}^2__{\\mho _{,}}\\le 420\\min _{1\\le k \\le K^-_{,}}\\lbrace \\max ( ^a_k, ^2_k)\\rbrace + 42\\max _{1\\le j\\le K^+_{,}}\\Big ({_j - _j}^2_-\\frac{1}{6}^a_j \\Big )_+.$ The second term in the last inequality is controlled uniformly over $$ and $$ by the following Proposition.", "Proposition 3.3 Given $\\in $ with singular values $a:=(a_j)_{j\\in }$ , let $\\sqrt{4} := (\\sqrt{4_j})_{j\\in }$ and define $K^+_{,}:= K^{\\sqrt{4}}_{,}$ , $M^+_{,}:= M^{\\sqrt{4}}_{,}$ , and $^a_k:=60\\delta _k^\\,$ using Definition REF .", "There is a constant $C>0$ depending only on the class $$ such that $\\sup _{\\in }\\sup _{\\in }\\Big [\\max _{1\\le k \\le K^+_{,}}\\Big ({_k - _k}^2_- \\frac{1}{6}^a_k \\Big )_+\\Big ] \\le C\\,\\Big \\lbrace + _+ \\Big \\rbrace .$ Roughly speaking, the penalty term is an upper bound for the estimator's variation.", "Typically, it can be chosen as a multiple of the estimator's variance.", "Thus, inequality (REF ) actually features a bias variance decomposition of the risk with an additional third term which is controlled by the above proposition.", "Illustration 3.4 Note that for any operator $\\in $ with sequence $(_j)_{j\\ge 1}$ of singular values, the sequence $\\delta ^a$ appearing in the definition of the penalty term $^a$ satisfies $(\\zeta _)^{-1}\\le (\\delta ^_j/\\delta ^_j)\\le \\zeta _$ for all $j\\in $ , with $\\zeta _d =\\log (3)/\\log (3)$ .", "In the special cases defined in Illustration REF above, the sequence $\\delta ^$ takes the following form: [m]   $\\delta ^_k\\sim k^{2b+2s+1}$            [s]   $\\delta ^_k\\sim k^{2b+2s+1}\\exp (k^{2b})(\\log k)^{-1}$ .", "The next proposition ensures that the randomized upper bound and penalty sequence behave similarly to their deterministic counterparts with sufficiently high probability so as not to deteriorate the estimation risk.", "In view of Proposition REF , this justifies the choice of the penalty.", "Proposition 3.5 Let $K^-_{,}:=K^{\\sqrt{/(4)}}_{,}$ and $ M_^+ :=M_^{\\sqrt{4}}$ using Definition REF and suppose that there is a constant $L>0$ depending only on $$ and $$ such that $^{-7}_{^+_+1}^{-1/2}\\exp \\left( - {_{^+_+1}}/({72\\,})\\right) \\le L \\quad \\text{for all}\\quad \\in (0,1).$ Then, there is a constant $C>0$ depending only on the class $$ such that $\\sup _{\\in }\\sup _{\\in }[{_-}^2__{\\mho ^c_{,}} ] \\le C\\,(1+)\\,\\quad \\text{for all}\\quad ,\\in (0,1).$ Condition (REF ) is satisfied in particular under the classical smoothness assumptions considered in the illustrations.", "We are finally prepared to state the upper risk bound of the fully data-driven estimator $_$ of $$ , which is the main result of this article.", "Theorem 3.6 Under Assumption REF and supposing (REF ), there is a constant $C$ depending only on the class $$ such that for all $,\\in (0,1)$ the adaptive estimator $_$ satisfies $_(_,,)\\le C\\,(1+)\\,\\Big \\lbrace \\min _{1\\le k \\le K_{,}^-}\\lbrace \\max (_k/_{k}, \\delta ^_{k})\\rbrace +_+ + \\Big \\rbrace .$ Considering (REF ), note that for all $\\in $ , we have $^_k\\le 60\\zeta _\\delta ^_k$ with $\\zeta _d =\\log (3)/\\log (3)$ .", "On the other hand, it is easily seen that for all $\\in $ , one has $^2_k \\le \\,(_k/_k)$ .", "Thus, we can write $\\sup _{\\in }\\sup _{\\in }\\min _{1\\le k \\le K^-_{,}} \\lbrace \\max (^_k,^2_k)\\rbrace \\le C\\,(1+)\\,\\min _{1\\le k \\le K^-_{,}}\\lbrace \\max (_k/_k,\\delta ^_k) \\rbrace $ for some constant $C>0$ depending only on $$ .", "In view of (REF ), the rest of the proof is obvious using Propositions REF and REF .", "$\\Box $ A comparison with the lower bound from Theorem REF shows that this upper bound ensures minimax optimality of the adaptive estimator $_$ only if $^\\diamond _{,}:= \\min _{1\\le k \\le K^-_{,}}\\Big [\\max \\Big ( \\frac{_k}{_k} , \\delta ^_k\\Big )\\Big ]$ is at most of the same order as $\\max (_{},_)$ , whence the following corollary.", "Corollary 3.7 Under Assumption REF and if $\\sup _{,\\in (0,1)}\\lbrace ^\\diamond _{,} /\\max (_,_)\\rbrace <\\infty $ , we have $ _(_,,) \\le C\\,^*_(,) \\qquad \\forall \\; ,\\in (0,1).$ We conclude this article reconsidering the framework of the preceding Illustration REF .", "Notice that the adaptive estimator is minimax optimal over a wide range of cases, even when $<$ .", "Illustration 3.8 Let $(_)_{\\in (0,1)}$ be a noise level in $X$ depending on the noise level $$ in $Y$ and suppose that the limits $q_1$ and $q_2$ from Illustration REF exist in the respective cases.", "Some straightforward computations then show that the adaptive estimator attains the following rates of convergence.", "[m] If $p-s>b$ , the adaptive estimator $_^{(s)}$ attains the optimal rates (cf.", "Illustration REF ).", "In case $p-s\\le b$ , we have, supposing that $q^v_1:=\\lim \\limits _{\\rightarrow 0}^{-2b/(2p+2b+1)}_^{1-v_{_}}$ exists, $\\sup _{\\in }\\sup _{\\in }{^{(s)}_- ^{(s)}}^2_{L^2} ={\\left\\lbrace \\begin{array}{ll}O(^{2(p-s)/(2p+2b+1)}) & \\text{if $q_1<\\infty $ and $q^v_1<\\infty $, } \\\\O(_^{(p-s)/b}_^{-v_{_}}) & \\text{otherwise.}\\end{array}\\right.", "}$ [s] The adaptive estimator attains the optimal rates." ], [ "Lower risk bound", "Theorem REF The proof consists of two steps: (A) First, we show that $_$ yields a lower risk bound in the case where the eigenvalues $(_j)$ of the operator $$ are known.", "(B) Then, we show that another lower risk bound is given by $_$ .", "Step (A).", "Given $\\zeta :=\\eta \\min (,1/(2))$ and $\\alpha _:=_(\\sum _{j=1}^{}_j/_j)^{-1}$ we consider the function $:= (\\zeta \\alpha _)^{1/2}\\sum _{j=1}^{}_j^{-1/2}_j$ .", "We are going to show that for any $\\theta :=(\\theta _j)\\in \\lbrace -1,1\\rbrace ^{}$ , the function $_\\theta :=\\sum _{j=1}^{} \\theta _j{}_j _j$ belongs to $$ and is hence a possible candidate for the solution.", "For a fixed $\\theta $ and under the hypothesis that the solution is $_\\theta $ , the observation $Y_k$ is distributed according to $(_k{_\\theta }_k, )$ for any $k\\in $ .", "We denote by $_\\theta $ the distribution of the resulting sequence $\\lbrace Y_k\\rbrace $ and by $_\\theta $ the expectation this distribution.", "Furthermore, for $1\\le j\\le $ and each $\\theta $ , we introduce $\\theta ^{(j)}$ by $\\theta ^{(j)}_{l}=\\theta _{l}$ for $j\\ne l$ and $\\theta ^{(j)}_{j}=-\\theta _{j}$ .", "The key argument of this proof is the following reduction scheme.", "If $$ denotes an estimator of $$ then we conclude $\\begin{split}\\sup _{\\in } &{-}_^2 \\ge \\sup _{\\theta \\in \\lbrace -1,1\\rbrace ^{}} _\\theta {-_\\theta }_^2\\ge \\frac{1}{2^{{}}}\\sum _{\\theta \\in \\lbrace -1,1\\rbrace ^{2}}_\\theta {-_\\theta }_^2\\\\&\\ge \\frac{1}{2^{{}}}\\sum _{\\theta \\in \\lbrace -1,1\\rbrace ^{}}\\sum _{j=1}^{}_j_{{\\theta }}|[-_\\theta ]_j|^2\\\\&= \\frac{1}{2^{{}}}\\sum _{\\theta \\in \\lbrace -1,1\\rbrace ^{}}\\sum _{j=1}^{}\\frac{_j}{2}\\Bigl \\lbrace _{{\\theta }}|[-_\\theta ]_j|^2+_{{\\theta ^{(j)}}}|[-_{\\theta ^{(j)}}]_j|^2\\Bigr \\rbrace .\\end{split}$ Below we show furthermore that for all $\\in (0,1)$ we have $\\Bigl \\lbrace _{{\\theta }}|[-_\\theta ]_j|^2+_{{\\theta ^{(j)}}}|[-_{\\theta ^{(j)}}]_j|^2\\Bigr \\rbrace \\ge \\frac{\\,\\zeta \\alpha _}{2_j}.$ Combining the last lower bound and the reduction scheme gives $\\sup _{\\in } {-}^2_\\ge \\frac{1}{2^{{}}}\\sum _{\\theta \\in \\lbrace -1,1\\rbrace ^{}}\\sum _{j=1}^{}\\frac{_j}{2}\\frac{\\zeta \\alpha _}{2_j}= \\frac{\\zeta \\alpha _}{4} \\sum _{j=1}^{}\\frac{_j}{_j}= \\frac{\\zeta _}{4},$ which implies the lower bound given in the theorem by definition of $\\zeta $ .", "To complete the proof, it remains to check (REF ) and $_\\theta \\in $ for all $\\theta \\in \\lbrace -1,1\\rbrace ^{}$ .", "The latter is easily verified if $\\in $ , which can be seen recalling that $/$ is non-increasing and noticing that the definitions of $\\zeta $ , $\\alpha _$ and $\\eta $ imply ${}_^2 \\le \\zeta \\frac{_{}}{_{}}\\alpha _\\Bigl (\\sum _{j=1}^{}\\frac{_j}{_j}\\Bigr ) \\le { \\zeta /\\eta \\le }$ .", "It remains to show (REF ).", "Consider the Hellinger affinity $\\rho (_1,_{-1})= \\int \\sqrt{d_1\\;d_{-1}}$ , then we obtain for any estimator $$ of $$ that $\\rho ({_1,_{-1}})&\\leqslant \\int \\frac{|[-_{\\theta ^{(j)}}]_j|}{|[_\\theta -_{\\theta ^{(j)}}]_j|}\\sqrt{d_1\\;d_{-1}}+ \\int \\frac{|[-_{\\theta }]_j|}{|[_\\theta -_{\\theta ^{(j)}}]_j|}\\sqrt{d_1\\;d_{-1}}\\\\&\\leqslant \\Bigl ( \\int \\frac{|[-_{\\theta ^{(j)}}]_j|^2}{|[_\\theta -_{\\theta ^{(j)}}]_j|^2}{d_1}\\Bigr )^{1/2} + \\Bigl ( \\int \\frac{|[-_{\\theta }]_j|^2}{|[_\\theta -_{\\theta ^{(j)}}]_j|^2}d_{–1}\\Bigr )^{1/2}.$ Rewriting the last estimate we obtain $\\Bigl \\lbrace _{{\\theta }}|[-_\\theta ]_j|^2+_{{\\theta ^{(j)}}}|[-_{\\theta ^{(j)}}]_j|^2\\Bigr \\rbrace \\ge \\frac{1}{2} |[_\\theta -_{\\theta ^{(j)}}]_j|^2\\rho ^2(_1,_{-1}).", "$ Next, we bound the Hellinger affinity $\\rho ({_1,_{-1}})$ from below.", "Consider the Kullback-Leibler divergence of these two distributions first.", "The components of the two sequences corresponding to the distributions $_1$ and $_{-1}$ are pairwise equally distributed except for the $j$ -th component.", "Thus, we have $\\log ({d_\\theta }/{d_})=({2y_j_j\\theta _j{}_j}/{})$ , and taking the integral over $y_j$ $_\\theta $ , we find ${KL}({_1,_{-1}}) = \\frac{2}{} \\; _j^2{}_j^2 \\le \\frac{2}{}\\;{}_j^2_j = 2\\zeta \\alpha _\\le 1,$ Using the well-known relationship $\\rho (_1,_{-1}) \\ge 1 - (1/2)KL(_1,_{-1})$ between the Kullback-Leibler divergence and the Hellinger affinity, we obtain that $\\rho (_1,_{-1})\\ge 1/2$ .", "Using this estimate, (REF ) becomes $\\Bigl \\lbrace _{{\\theta }}|[-_\\theta ]_j|^2+_{{\\theta ^{(j)}}}|[-_{\\theta ^{(j)}}]_j|^2\\Bigr \\rbrace \\ge \\frac{1}{2} []_j^2$ , and combining this with (REF ) implies the result by construction of the solution $$ .", "Step (B).", "First, we construct two solutions $_\\theta \\in $ and operators $_\\theta \\in $ (with $\\theta \\in \\lbrace -1,1\\rbrace $ ) such that the resulting images $g_\\theta $ satisfy $g_{-1}=g_1$ .", "To this end, we define $:=_{j\\in } \\lbrace _j_j^{-1}\\min (1,_j^{-1})\\rbrace $ and $\\alpha _:=\\zeta \\min (1,^{1/2}_{}^{-1/2})$ with $\\zeta :=\\min (2^{-1},(1-^{-1/2}))$ .", "Observe that $1\\ge (1-\\alpha _)^2\\ge (1-(1-1/d^{1/2}))^2\\ge 1/$ and $1\\le (1+\\alpha _)^2\\le (1+(1-1/^{1/2}))^2=(2-1/^{1/2})^2 \\le $ , which implies $1/\\le (1+\\theta \\alpha _)^2\\le $ .", "These inequalities will be used below without further reference.", "We show below that for each $\\theta $ the function $_\\theta :=(1-\\theta \\alpha _) \\frac{}{}_{}^{-1/2}_{}$ belongs to $$ and that the operator $_\\theta $ with the singular values $^\\theta _k =[1+\\theta \\alpha _I{k=}] \\;\\sqrt{_k} $ is an element of $$ .", "We obviously have that $_1_f = {(1 - \\alpha _^2)(_{}/_)^{1/2}(/)_}=_{-1}_{-1}$ .", "For $\\theta \\in \\lbrace \\pm 1\\rbrace $ , denote by $_\\theta $ the joint distribution of the two sequences $(X_1,X_2,\\ldots )$ and $(Y_1,Y_2,\\ldots )$ , and let $_\\theta $ denote the expectation $_\\theta $ .", "Applying a reduction scheme as under Step (A) above, we deduce that for each estimator $$ of $$ $\\sup _{\\in }\\sup _{\\in }&{-}^2_\\ge \\max _{\\theta \\in \\lbrace -1,1\\rbrace }_{{\\theta }}{-_\\theta }^2_\\ge \\frac{1}{2}\\Bigl \\lbrace _{1}{-_1}^2_+_{-1}{-_{-1}}^2_\\Bigr \\rbrace .$ Below we show furthermore that $_{1}{-_1}^2_+_{-1}{-_{-1}}^2_\\ge {\\frac{1}{8}} {_{1}-_{-1}}^2_.$ Moreover, we have $ {_1-_{-1}}_^2 = {4\\alpha _^2(/)_{}_{}^{-1}=4 \\zeta ^2(/)_{}_{}^{-1}\\min \\Bigl (1,\\frac{}{_{}}\\Bigr ).", "}$ Combining the last lower bound with the reduction scheme and the definition of $$ implies the result of the theorem.", "To conclude the proof, it remains to check (REF ), $_\\theta \\in $ and $_\\theta \\in $ for both $\\theta $ .", "In order to show $_\\theta \\in $ , observe that ${_\\theta }^2_=_{}|[_\\theta ]_{}|^2\\le { _{}|(1-\\theta \\alpha _)(/) _{}^{-1/2}|^2 \\le }$ .", "To check that $_\\theta \\in $ , it remains to show that $1/\\le (^\\theta _j)^2/_j\\le $ for all $j\\ge 1$ .", "These inequalities are obviously satisfied for all $j\\ne $ , and as well for $j=$ by construction of the operator $$ .", "Finally consider (REF ).", "As in Step (A) above by employing the Hellinger affinity $\\rho (_1,_{-1})$ we obtain for any estimator $$ of $$ that $_{1}{-_{1}}_^2+_{-1}{-_{-1}}_^2\\ge \\frac{1}{2} {_{1}-_{-1}}^2_\\rho ^2(_1,_{-1}).$ Next, we bound the Hellinger affinity $\\rho (_1,_{-1})$ from below for all $\\in (0,1)$ , which proves (REF ).", "Notice that by construction of $_\\theta $ and $_\\theta $ , the distribution of $X_i$ and $Y_i$ does not depend on $\\theta $ , except for $X^\\theta _$ .", "It is thus easily seen that the Kullback-Leibler divergence can be controlled as follows, $KL(_1,_{-1}) ={\\frac{(_^1 - _^{-1})^2}{2} = \\frac{2\\alpha ^2_}{}\\,_\\le 1}$ Using $\\rho (_1,_{-1}) \\ge 1 - (1/2)KL(_1,_{-1})$ again, (REF ) is shown and so is the theorem.", "$\\Box $ The following proof uses Lemma REF from the auxiliary results section REF below.", "Theorem REF Define $:=\\sum _{j=1}^[]_j\\lbrace X_j^2\\ge \\rbrace _j$ and decompose the risk into two terms, ${-}_^2 ={-}_^2+{-}_^2=: A + B,$ which we bound separately.", "Consider first $A$ which we decompose further, ${- }_^2= \\sum _{j=1}^_j\\left[\\frac{(Y_j-Y_j)^2}{X_j^2} I{X_j^2\\geqslant }\\right] \\\\\\hfill +\\sum _{j=1}^_j |[]_j|^2\\left[\\frac{(X_j-X_j)^2}{X_j^2}\\lbrace X_j^2\\geqslant \\rbrace \\right]=: A_1 + A_2.$ As far as $A_1$ is considered, we use Lemma REF  (iii) from Section REF below and write $A_1 = \\sum _{j=1}^\\frac{_j}{[X_j]^2}\\; \\left[\\left(\\frac{[X_j]}{X_j} \\right)^2 I{X_j^2\\ge } \\right]\\le 4\\sum _{j=1}^\\frac{_j}{_j}\\le 4_.$ As for $A_2$ , we apply Lemma REF  (i) and obtain $A_2 \\le 8\\sum _{j=1}^_j |[]_j|^2\\min \\left(1,\\frac{}{_j}\\right)\\le 8d_$ Consider now $B$ which we decompose further into ${- }_^2=\\sum _{j\\in } _j|[]_j|^2 [(1-\\lbrace 1\\le j\\le \\rbrace \\lbrace X_j^2\\geqslant \\rbrace )^2] \\\\\\hfill = \\sum _{j > } _j|[]_j|^2+\\sum _{j = 1}^_j |[]_j|^2¶\\Bigl (X_j^2< \\Bigr )=: B_1 + B_2,$ where $B_1\\le {}^2__{}_{}^{-1}\\le _$ because $\\in $ .", "Moreover, $B_2\\le 4 _$ using Lemma REF  (ii).", "The result of the theorem follows now by combination of the decomposition (REF ) and the estimates of $A_1,A_2,B_1$ and $B_2$ .$\\Box $" ], [ "Adaptive estimation (Section ", "The proofs in this section use the Lemmas REF – REF from the auxiliary results section REF below.", "Proposition REF Using the model equation $Y_j ={}_j + \\sqrt{}\\,\\xi _j$ , we have for all $t\\in _k$ that ${_k - _k}_j= \\frac{\\sqrt{}\\,\\xi _j}{_j}+ \\left(\\frac{1}{X_j}_{[X_j^2\\ge ]} -\\frac{1}{_j}\\right)\\sqrt{}\\,\\xi _j+ \\left(\\frac{1}{X_j}_{[X_j^2\\ge ]} -\\frac{1}{_j}\\right){}_j.$ Thus, we may decompose the norm ${_k-_k}_^2$ in three terms according to ${_k - _k}_^2&\\le 3 \\sum _{j=1}^k\\frac{_j}{_j} {}\\,\\xi _j^2+ 3\\sum _{j=1}^k_j \\left(\\frac{1}{X_j}_{[X_j^2\\ge ]} -\\frac{1}{_j}\\right)^2 {}\\,\\xi _j^2+ 3\\sum _{j=1}^k_j \\left(\\frac{1}{X_j}_{[X_j^2\\ge ]}-\\frac{1}{_j}\\right)^2{}_j^2\\\\&=: 3\\,\\big \\lbrace T^{(1)}_k + T^{(2)}_k + T^{(3)}_k\\big \\rbrace .$ Define the event $\\Omega _:= \\bigg \\lbrace \\forall \\; 0<j\\le _^+ \\; \\bigg |\\quad \\Big |\\frac{1}{X_j}-\\frac{1}{_j}\\Big | \\le \\frac{1}{2\\,_j} \\quad \\text{and}\\quad X_j^2 \\ge \\bigg \\rbrace .$ Since $I{X_j^2\\ge }I{\\Omega _} = I{\\Omega _}$ , it follows that for all $1\\le j \\le K^+_{,}$ we have $\\bigg (\\frac{_j}{X_j}I{X_j^2\\ge } - 1\\bigg )^2\\,I{\\Omega _} = _j^2\\;I{\\Omega _}\\,\\bigg |\\frac{1}{X_j}-\\frac{1}{_j}\\bigg |^2\\le \\frac{1}{4}.$ Hence, $ T^{(2)}_k_{\\Omega _}\\le \\frac{1}{4} T^{(1)}_k $ for all $1\\le k\\le K_{,}^+$ , and thus $\\begin{split}\\max _{1\\le k \\le K_{,}^+}\\left({_k-_k}_^2 -\\frac{1}{6}^a_k\\right)_+ \\le 4&\\sum _{k=1}^{K_{,}^+}\\left(\\sum _{j=1}^k\\frac{_j}{_j}{}\\,\\xi _j^2 - 2\\delta _k\\right)_+ \\\\&+ 3\\max _{1\\le k \\le K_{,}^+} T^{(2)}_k _{\\Omega _^c}+ 3 \\max _{1\\le k \\le K_{,}^+} T^{(3)}_k.\\end{split}$ Keeping in mind that $¶[\\Omega _^c]\\le C(d)^2$ by virtue of Lemma REF , the result follows immediately using Lemmas REF , REF , and REF below.", "$\\Box $ Proposition REF Let $\\breve{}_{k}:=\\sum _{1\\le j\\le k} {}_j \\lbrace X_j^2\\ge \\rbrace _j$ .", "It is easy to see that ${_{k}-\\breve{}_{k}}^2 \\le {_{k^{\\prime }}-\\breve{}_{k^{\\prime }}}^2$ for all $ k^{\\prime }\\le k$ and ${\\breve{}_{k}-}^2\\le {}^2$ for all $k\\ge 1$ .", "Thus, using that $1\\le \\le ({_^\\circ }\\wedge ^{-1})$ , we can write ${_{}-}_^2I{\\mho _{,}^c} &\\le 2\\lbrace {_{}-\\breve{}_{}}_^2I{\\mho _{,}^c} +{\\breve{}_{}-}_^2I{\\mho _{,}^c}\\rbrace \\\\&\\le 2\\bigg \\lbrace {_{({_^\\circ }\\wedge \\lfloor ^{-1}\\rfloor )}-\\breve{}_{({_^\\circ }\\wedge \\lfloor ^{-1}\\rfloor )}}_\\omega ^2I{\\mho _{,}^c} +{}_\\omega ^2\\, ¶[\\mho _{,}^c] \\bigg \\rbrace .$ Moreover, using the Cauchy-Schwarz inequality, we conclude ${_{({_^\\circ }\\wedge \\lfloor ^{-1}\\rfloor )}&-\\breve{}_{({_^\\circ }\\wedge \\lfloor ^{-1}\\rfloor )}}_^2I{\\mho _{,}^c} \\\\&\\le 2 ^{-1}\\sum _{1\\le j\\le ({_^\\circ }\\wedge \\lfloor ^{-1}\\rfloor )}\\omega _j\\Bigl \\lbrace (Y_j -_j {}_j)^2I{\\mho _{,}^c} + (_j {}_j-X_j {}_j)^2I{\\mho _{,}^c}\\Bigr \\rbrace \\\\&\\le 2 ^{-1} \\Bigl \\lbrace \\sum _{1\\le j\\le ({_^\\circ }\\wedge \\lfloor ^{-1}\\rfloor )} _j \\Bigl [\\left(Y_j - {}_j\\right)^{4}\\Bigr ]^{1/2}¶[\\mho _{,}^c]^{1/2}\\\\&\\hspace{90.0pt} + \\sum _{1\\le <j\\le ({_^\\circ }\\wedge \\lfloor ^{-1}\\rfloor )} _j {_{j}}^2[(X_j-_j)^4]^{1/2}¶[\\mho _{,}^c]^{1/2} \\Bigr \\rbrace \\\\&\\le 2\\sqrt{3} ^{-1} \\Bigl \\lbrace ( ^{-1}{\\max _{1\\le j \\le {_^\\circ }}_j}) + {}_^2\\Bigr \\rbrace \\,¶[\\mho _{,}^c]^{1/2},$ which implies ${_{}-}_^2I{\\mho _{,}^c} \\le C\\bigg \\lbrace \\Bigl ( ^{-2}+ {}_^2\\Bigr ) \\,¶[\\mho _{,}^c]^{1/2} + {}_^2\\, ¶[\\mho _{,}^c] \\bigg \\rbrace .$ Lemma REF below yields, for some $C>0$ depending only on the class $$ , ${_{}-}_^2I{\\mho _{,}^c}\\le C\\,\\bigg \\lbrace + {}_\\omega ^2{^6}+ {}_\\omega ^2{^{12}} \\bigg \\rbrace $ which completes the proof due to $\\in $ .$\\Box $" ], [ "Auxiliary results", "Lemma 4.1 For every $j\\in $ , $R^{I}_j := \\bigg [\\;\\bigg (\\frac{_j}{X_j}-1\\bigg )^2I{X_j^2\\ge } \\;\\bigg ] \\le \\,\\min \\Big \\lbrace 1, \\frac{8}{_j^2}\\Big \\rbrace $ $R^{II}_j := ¶[X_j^2<] \\le \\,\\min \\Big \\lbrace 1, \\frac{4}{_j^2}\\Big \\rbrace $ $\\left[\\left(\\frac{[X_j]}{X_j}\\right)^2I{X_j^2\\ge }\\right] \\le 4 $ (i) It is easy to see that $R^I_j = \\bigg [ \\frac{|X_j - _j|^2}{X_j^2} \\;\\;I{X_j^2\\ge } \\bigg ]\\le ^{-1} (X_j) =1.$ On the other hand, using that $[(X_j - _j)^4] = 3^2$ , we obtain $R^I_j &\\le \\bigg [ \\frac{(X_j-_j)^2}{X_j^2} \\;\\;I{X_j^2\\ge } \\; 2\\bigg \\lbrace \\frac{(X_j-_j)^2}{_j^2} +\\frac{X_j^2}{_j^2} \\bigg \\rbrace \\bigg ]\\\\&\\le \\frac{2\\,[(X_j-_j)^4]}{_j^2}+\\frac{2\\; (X_j)}{_j^2}=\\frac{8}{^2}.$ Combining with (REF ) gives $R^I_j\\le \\,\\min \\Big \\lbrace 1,\\frac{8}{_j^2}\\Big \\rbrace $ , which completes the proof of (i).", "(ii) Trivially, $R_j^{II}\\le 1$ .", "If $1\\le 4/_j^2$ , then obviously $R^{II}_j\\le \\,\\min \\Big \\lbrace 1, \\frac{4}{_j^2}\\Big \\rbrace $ .", "Otherwise, we have $< _j^2/4$ and hence, using 's inequality, $R_j^{II}\\le ¶[|X_j-_j| > |_j|\\,/2\\,]\\le \\frac{4\\,(X_j)}{_j^2}\\le \\,\\min \\Big \\lbrace 1, \\frac{4}{_j^2}\\Big \\rbrace ,$ where we have used that $(X_j) = $ for all $j$ .", "(iii) $\\left[\\left(\\frac{[X_j]}{X_j}\\right)^2I{X_j^2\\ge }\\right] \\le 2\\left[ \\left(\\frac{X_j - [X_j]}{X_j}\\right)^2I{X_j^2\\ge } + I{X_j^2\\ge } \\right]\\le 4$ .", "$\\Box $ Lemma 4.2 Under Assumption REF , we have that $ \\delta _{^+_} \\le 32\\,^2 $ for all $\\in (0,1)$ , and there is a $_0\\in (0,1)$ such that for all $<_0$ , we have $\\min _{1\\le j \\le ^+_} _j^2\\ge 3.$ (i) For $_^+ = 0$ , we have $\\delta _{_^+} = 0$ and there is nothing to show.", "If $0<_^+\\le n$ , one can show that $_{_^+}^+/_{_^+}\\le 4/ (_^+|\\log |)$ , which we use in the following computation: $\\delta _{_^+} &= _^+ \\;\\frac{_{_^+}^+}{_{_^+}}\\;\\frac{\\log ((_{_^+}^+/_{_^+}) \\vee (_^+ + 2))}{\\log (_^+ + 2)}\\le \\frac{4}{|\\log |}\\; \\frac{\\log \\left( \\frac{4}{_^+|\\log |} \\vee (_^+ + 2)\\right)}{\\log (_^+ + 2)}\\\\[1em]&\\le ^{-1}\\,{\\left\\lbrace \\begin{array}{ll}4& (\\log (^{-1}+2)\\ge 4)\\\\4(4+ \\log (4))/(\\log (^{-1}+2)) & (\\text{otherwise}),\\end{array}\\right.", "}$ which implies $\\delta _{_^+} \\le 4(4+\\log (4))\\le 32^2$ for all $\\in (0, 1)$ .", "(ii) We have that $\\min _{1\\le j\\le _^+} _j^2\\ge \\min _{1\\le j\\le _^+} \\frac{_j}{}\\ge \\frac{^{1-v_}}{4^2}\\ge 3,$ where the last step holds for sufficiently small $$ as some algebra shows.$\\Box $ Lemma 4.3 We have that $\\sum _{ k=1}^{K_{,}^+}\\bigg (\\sum _{j=1}^k \\frac{_j}{_j}\\xi _j^2 - 2\\,\\delta ^_{k}\\bigg )_+\\le 6720\\;.$ Representing the expectation of the positive random variable by the integral over its tail probabilities and using $\\delta _k^\\ge \\sum _{j=1}^k(_j/_j^2)$ , we may write $\\sum _{ k=1}^{K_{,}^+}\\bigg (\\sum _{j=1}^k \\frac{_j}{_j}\\xi _j^2 - 2\\,\\delta ^_{k}\\bigg )_+\\le \\sum _{ k=1}^{K_{,}^+}\\int _0^\\infty ¶\\left[\\sum _{j=1}^{k}\\frac{_j}{_j^2}\\,(\\xi _j^2-1)\\ge x + 2\\delta ^_k - \\sum _{j=1}^{k}\\frac{_j}{_j^2}\\right] dx\\\\ \\le \\sum _{ k=1}^{K_{,}^+}\\int _0^\\infty ¶\\left[\\sum _{j=1}^{k}\\frac{_j}{_j^2}\\,(\\xi _j^2-1)\\ge x + \\delta ^_k \\right] dx$ Define $\\rho _k := (_k)/_k^2$ , $H_k := 4\\Delta ^_k$ , and $B_k := 2^2\\sum _{j=1}^{k}_j^2 /_j^4$ .", "It can be shown (see proof of Proposition A.1 in [4]) that for all $1\\le k^{\\prime }\\le k$ and $m\\ge 2$ , we have $\\Big |\\Big [\\Big (\\frac{_{k^{\\prime }}}{_{k^{\\prime }}^2}(\\xi _{k^{\\prime }}^2-1)\\Big )^m\\Big ]\\Big |\\le m!\\,\\rho _{k^{\\prime }}^2\\, H_k^{m-2}.$ Hence, the assumption of Theorem 2.8 from [14] is satisfied and splitting up the integral, we get the following bound: $\\sum _{k=1}^{K_{,}^+}\\bigg (\\sum _{j=1}^k \\frac{_j}{_j}\\xi _j^2 - 2\\,\\delta ^_{k}\\bigg )_+\\\\ \\le \\sum _{k=1}^{K_{,}^+}\\int _0^{B_{k}/H_{k} - \\delta ^_{k}} \\exp \\Big ( - \\frac{(x +\\delta ^_{k})^2}{4B_{k}}\\Big ) dx+ \\int _{B_{k}/H_{k} - \\delta ^_{k}}^\\infty \\exp \\Big (-\\frac{x +\\delta ^_{k}}{4H_{k}}\\Big ) dx$ The second integral is equal to $4H_{k}\\exp (-B_{k}/(4H_{k}^2))$ .", "Some computation shows that the first one is bounded from above by $4H_{k}\\big [\\exp \\big (-^2(\\delta ^_{k})^2/(4B_{k})\\big ) - \\exp \\big (-B_{k}/(4H_{k}^2)\\big )\\big ] $ .", "Thus, the two identical terms cancel, and we get $\\sum _{ k=1}^{K_{,}^+}\\bigg (\\sum _{j=1}^k \\frac{_j}{_j}\\xi _j^2 - 2\\,\\delta ^_{k}\\bigg )_+\\le 16\\;\\epsilon \\; \\sum _{ k=1}^{K_{,}^+} \\Delta ^_{k}\\exp \\left(-\\frac{(\\delta ^_{k})^2}{8k(\\Delta ^_{k})^2}\\right).$ To complete the proof, we bound the sum on the right hand side as follows, $\\sum _{ k=1}^{K_{,}^+} \\Delta ^_{k}&\\exp \\left(-\\frac{(\\delta ^_{k})^2}{8k(\\Delta ^_{k})^2}\\right)\\le \\sum _{k=1}^\\infty \\exp \\Big (-\\log (\\Delta ^_{k}\\vee (k+2))\\Big [\\frac{k}{8\\log (k+2)} -1 \\Big ] \\Big )\\\\&\\le e\\sum _{k=1}^\\infty \\exp \\Big (-\\frac{k}{8\\log (k+2)} \\Big )\\le e\\sum _{k=1}^\\infty \\exp \\Big (-\\frac{\\sqrt{k}}{8\\log (3)} \\Big )\\\\&\\le e\\int _{0}^\\infty \\exp \\Big (-\\frac{\\sqrt{x}}{8\\log (3)} \\Big )dx=128\\log ^2(3)\\,e,$ where we have used $\\log (k+2)\\le \\log (3)\\sqrt{k}$ for all $k\\ge 1$ .", "$\\Box $ Lemma 4.4 For every $k\\in $ and $\\in (0,1)$ , $\\bigg [\\sum _{j=1}^k_j{}_j^2\\left(\\frac{1}{X_j}_{[X_j\\ge ]}- \\frac{1}{_j}\\right)^2 \\bigg ] \\le 8\\;\\;\\;_(,,).$ Firstly, as $\\in $ , it is easily seen that $\\bigg [\\sum _{j=1}^k_j{}_j^2\\left(\\frac{1}{X_j}_{[X_j\\ge ]}- \\frac{1}{_j}\\right)^2 \\bigg ]\\le \\;\\max _{1\\le j\\le k}\\;\\frac{_j}{_j}\\,[|R_j|^2],$ where $R_j$ is defined as $R_j := \\left(\\frac{_j}{X_j}I{X_j^2\\ge ^2} -1\\right).$ In view of the definition of $_$ in Theorem REF , the result follows from $[|R_j|^2] \\le \\,\\min \\Big \\lbrace 1,\\frac{8}{_j}\\Big \\rbrace $ , which is a consequence of the decomposition $\\begin{split}|R_j|^2= \\bigg [\\;\\bigg (\\frac{_j}{X_j}-1\\bigg )^2I{X_j^2\\ge } \\;\\bigg ]+¶[X_j^2< ] \\end{split}$ and Lemma REF .", "$\\Box $ Lemma 4.5 We have that $\\bigg [\\sum _{j=1}^{K_{,}^+}_j\\left(\\frac{1}{X_j}_{[X_j\\ge ]}-\\frac{1}{_j}\\right)^2{}\\xi _j^2 _{\\Omega _^c}\\bigg ]{\\le 64\\,^3 (¶[\\Omega _^c])^{1/2}}.$ Given $R_j$ from (REF ), we begin our proof observing that $\\bigg [\\sum _{j=1}^{K_{,}^+}_j\\left(\\frac{1}{X_j}_{[X_j\\ge ]}-\\frac{1}{_j}\\right)^2\\sqrt{}\\xi _j^2 _{\\Omega _^c}\\bigg ]&\\le \\sum _{j=1}^{K_{,}^+}\\frac{_j}{_j^2}\\;[|R_j|^2_{\\Omega _^c}],\\multicolumn{2}{l}{\\text{{where we have used the independence of $X$ and $Y$ and$(Y_j) = $.", "Since $\\delta _k^\\ge \\sum _{j=1}^k \\frac{\\omega _j}{_j^2}$ for all $\\in $, the Cauchy-Schwarz inequality yields}}}\\\\\\bigg [\\sum _{j=1}^{K_{,}^+}_j\\left(\\frac{1}{X_j}_{[X_j\\ge ]}-\\frac{1}{_j}\\right)^2{}\\xi _j^2 _{\\Omega _^c}\\bigg ]&{\\le \\; d\\,(¶[\\Omega _^c])^{1/2}\\;{\\delta ^_{N^+_}}\\;\\max _{0<j\\le N^+_}([|R_j|^4])^{1/2}.", "}$ Proceeding analogously to (REF ) and (REF ), one can show that $[|R_j|^4]\\le 4$ .", "The result follows then using the definition of $^+_$ .$\\Box $ Lemma 4.6 For $k\\in $ , define the events $_k := \\bigg \\lbrace \\Big |\\frac{X_j}{_j} - 1\\Big | \\le \\frac{1}{3} \\quad \\forall \\, 1\\le j \\le k \\bigg \\rbrace $ and suppose that Assumption REF holds.", "For all $,\\in (0,1)$ , we have $ \\Omega _\\subseteq \\lbrace ^+_k\\le _k \\le 30^+_k \\quad \\forall \\;1\\le k \\le K^+_{,}\\rbrace $ , $_{_^++1}\\subseteq \\lbrace K^-_{,} \\le _{,} \\le K^+_{,}\\rbrace $ , $¶[_{_^+}^c]\\le C()\\,^2$ and $¶[\\Omega _^c]\\le C()\\,^2$ .", "If additionally condition (REF ) holds, then $¶[\\mho _{,}^c] \\le C(,) ^6$ .", "Consider (i).", "Notice first that $ \\delta ^_k \\le \\delta _k^\\, \\,\\zeta _$ for all $k\\ge 1$ with $\\zeta _:= (\\log (3d))/(\\log 3)$ .", "Observe that on $\\Omega _$ we have $ (1/2)\\Delta ^_k\\le \\Delta ^X_k\\le (3/2)\\Delta ^_k$ for all $1\\le k\\le _$ and hence $(1/2)[\\Delta ^_k\\vee (k+2)]\\le [ \\Delta ^X_k\\vee (k+2)]\\le (3/2)[\\Delta ^_k\\vee (k+2)]$ , which implies $\\begin{split}(1&/2) k \\Delta ^_k\\Bigl (\\frac{ \\log [ \\Delta ^_k\\vee (k+2)]}{\\log (k+2)}\\Bigr )\\Bigl (1-\\frac{\\log 2}{\\log (k+2)}\\frac{\\log (k+2)}{\\log (\\Delta ^_k \\vee [k+2])}\\Bigr )\\\\[1ex]&\\le \\delta ^X_k \\le (3/2) k \\Delta ^_k\\Bigl (\\frac{\\log (\\Delta ^_k\\vee [k+2])}{\\log ( k+2)}\\Bigr ) \\Bigl (1+\\frac{\\log 3/2}{\\log (k+2)} \\frac{\\log (k+2)}{\\log (\\Delta ^_k \\vee [k+2])}\\Bigr ).\\end{split}$ Using $ {\\log (\\Delta ^_k\\vee (k+2))}/{\\log ( k+2)}\\ge 1$ , we conclude from the last estimate that $\\begin{split}{ \\delta ^_k/10\\le }(\\log 3/2)/(2 \\log 3) \\delta ^_k&\\le (1/2) \\delta ^_k[1-(\\log 2)/\\log (k+2)] \\le {\\delta ^X_k} \\\\ &\\le (3/2) \\delta ^_k[1+(\\log 3/2)/\\log (k+2)]\\le { 3\\delta ^_k}.\\end{split}$ It follows that on $\\Omega _$ we have $^+_k\\le _k\\le 30^+_k$ for all $1\\le k\\le _^+$ as desired.", "Proof of (ii).", "Denoting by $X$ the random sequence $(X_j)_{j\\ge 1}$ , define sequences $N_^- :=N_^{\\sqrt{/(4)}}$ , $M_^- :=M_^{\\sqrt{/(4)}}$ and $_:= N_^{X}$ , $_:= M_^X$ .", "Note that by definition, $K_{,}^- = N_^- \\wedge M_^-$ and $_{,} = _\\wedge _$ .", "Define further the events $\\Omega _I:=\\lbrace K^-_{,}> _{,}\\rbrace $ and $\\Omega _{II}:=\\lbrace _{,}>K^+_{,}\\rbrace $ .", "Then we have $\\lbrace K^-_{,} \\le _{,} \\le K^+_{,}\\rbrace ^c=\\Omega _I\\cup \\Omega _{II}$ .", "Consider $\\Omega _I = \\lbrace _< K^-_{,}\\rbrace \\cup \\lbrace _< K^-_{,}\\rbrace $ first.", "By definition of $^-_$ , we have that $\\min _{1\\le j\\le ^-_}\\frac{_j^{2}}{j\\,_j^+}\\ge 4 \\,|\\log |$ , which implies, keeping in mind that $K^-_{,}\\le ^-_{,}$ , $\\lbrace _< K^-_{,}\\rbrace \\subset \\bigg \\lbrace \\exists \\, 1\\le j\\le K^-_{,}\\,\\bigg |\\,\\frac{X_j^2}{j\\,_j^+}<|\\log |\\bigg \\rbrace \\\\\\subset \\bigcup _{1\\le j\\le K^-_{,}}\\bigg \\lbrace \\frac{X_j}{_j}\\le \\frac{1}{2}\\bigg \\rbrace \\subset \\bigcup _{1\\le j\\le K^-_{,}}\\bigg \\lbrace \\left|\\frac{X_j}{_j}-1\\right|\\ge \\frac{1}{2}\\bigg \\rbrace .$ One can see that from $\\min _{1\\le j\\le ^-_}_j^{2}\\ge 4 ^{1- v_}$ it follows in the same way that $\\Big \\lbrace _< K^-_{,}\\Big \\rbrace \\subset \\bigcup _{1\\le j\\le K^-_{,}}\\bigg \\lbrace \\left|\\frac{X_j}{_j}-1\\right|\\ge \\frac{1}{2}\\bigg \\rbrace .\\hfill $ Therefore, $\\Omega _I \\subseteq \\bigcup _{1\\le j\\le ^+_}\\Bigl \\lbrace |X_j/_j-1|\\ge 1/2\\Bigr \\rbrace \\subseteq _{_^++1}^c$ , since $_^-\\le _^+$ .", "Consider $\\Omega _{II} = \\lbrace _> K^+_{,}\\rbrace \\cap \\lbrace _>K^+_{,}\\rbrace $ .", "In case $K^+_{,}= _^+$ , note that by definition of $_^+$ , we have $|\\log | / 4 \\ge \\frac{_{_^+ +1}^2}{(_^+ +1)\\,_{_^+ +1}^+}$ , such that $\\Omega _{II}\\subseteq \\lbrace _> _^+\\rbrace &\\subset \\Bigl \\lbrace \\forall 1\\le j\\le _^+ + 1\\quad \\bigg |\\quad \\frac{X_j^2}{j\\,_j^+}\\ge |\\log |\\Bigr \\rbrace \\\\[1ex] &\\subset \\Biggl \\lbrace \\frac{X_{_^++1}}{_{_^++1}}\\ge 2\\Biggr \\rbrace \\subset \\Biggl \\lbrace \\bigg |\\frac{X_{_^++1}}{_{_^++1}}-1\\bigg |\\ge 1\\Biggr \\rbrace .$ In case $K^+_{,}= _^+$ , it follows analogously from $^{1-v_} \\ge 4 \\max _{j\\ge _^++1} _j^2$ that $\\Omega _{II}\\subset \\lbrace _> _^+\\rbrace \\subset \\Bigl \\lbrace | X_{_^++1}/_{_^++1}-1|\\ge 1\\Bigr \\rbrace .\\hfill $ Therefore, we have $\\Omega _{II} \\subseteq \\Bigl \\lbrace |X_{K_{,}^++1}/_{K_{,}^++1}-1 | \\ge 1\\Bigr \\rbrace \\subseteq _{^+_{}+1}^c$ and (ii) is shown.", "Proof of (iii).", "For $Z\\sim (0,1)$ and $z\\ge 0$ , one has $¶[Z>z]\\le (2\\pi z^2)^{-1/2}\\exp (-z^2/2)$ .", "Hence, there is a constant $C()$ depending on $$ such that for every $ 1\\le j\\le ^+_$ , $¶[|X_j/_j -1|>1/3]\\le C()\\,\\left(\\frac{}{{_{_{}^+}}}\\right)^{1/2}\\exp \\bigg (-\\frac{_{_^+}}{18} \\bigg ).$ Consequently, as $_^+\\le ^{-1}$ and $_{_^+}>^{1-v_} / (4)$ , we have $¶[_{_^+}^c]\\le C() ^{2-v_}\\exp \\bigg (- \\frac{^{-v_}}{72^2} \\bigg ) $ which implies $¶[_{_^+}^c]\\le C()\\,^2$ using that $^{v_}\\,|\\log |\\rightarrow 0 $ as $\\rightarrow 0$ .", "As for the second assertion in (iii), we distinguish the cases $\\le _0$ and $>_0$ , where $_0$ is the constant from Lemma REF  (ii) depending only on $$ .", "The assertion is trivial for $>_0$ (keeping in mind that $¶[\\Omega _{}^c]\\le _0^{-2}^2$ ).", "Consider the case $\\le _0$ , where $_j^2\\ge 3$ for all $1\\le j \\le _^+$ due to Lemma REF  (ii).", "This yields for the complement of $\\Omega _$ $\\Omega _^c = \\bigg \\lbrace \\exists \\; 1\\le j\\le _^+ \\quad \\bigg |\\quad \\Big |\\frac{_j}{X_j}-1\\Big | > \\frac{1}{2} \\quad \\text{or}\\quad X_j^2 < \\bigg \\rbrace \\subseteq \\bigg \\lbrace \\exists \\; 1\\le j\\le _^+ \\;\\bigg |\\;\\bigg |\\frac{X_j}{_j} -1\\bigg |> \\frac{1}{3}\\bigg \\rbrace =^c_{_^+}.$ It follows with assertion (ii) that $\\mho _{,}^c \\subseteq _{_^+}^c$ for all $\\le _0$ , implying the second assertion of (iii).", "Proof of (iv).", "Following the proof of (iii) and using that $_^++1\\le ^{-1}$ , we obtain $¶[_{_^++1}^c]\\le C() (_{_{}^++1})^{-1/2}\\exp \\bigg (-\\frac{_{_^++1}}{18} \\bigg ).$ Note that $_{_^++1} \\subseteq \\Omega _$ , since trivially $_{_^++1} \\subseteq _{_^+}$ .", "Thus, (REF ) implies assertion (iv) by virtue of condition (REF ).$\\Box $" ] ]

[ [ "Geodesic restrictions of arithmetic eigenfunctions" ], [ "Abstract Let X be an arithmetic hyperbolic surface, \\psi a Hecke-Maass form, and l a geodesic segment on X.", "We obtain a power saving over the local bound of Burq-G\\'erard-Tzvetkov for the L^2 norm of \\psi restricted to l, by extending the technique of arithmetic amplification developed by Iwaniec and Sarnak.", "We also improve the local bounds for various Fourier coefficients of \\psi along l." ], [ "Introduction", "If $X$ is a compact Riemannian manifold and $\\psi $ is a Laplace eigenfunction on $X$ satisfying $\\Delta \\psi = \\lambda ^2 \\psi $ , it is an interesting problem to study the extent to which $\\psi $ can concentrate on small subsets of $X$ .", "Two well studied formulations of this problem are to normalise $\\psi $ by $\\Vert \\psi \\Vert _2 = 1$ , and either bound $\\Vert \\psi \\Vert _p$ for $2 \\le p \\le \\infty $ or bound the $L^p$ norms of $\\psi $ restricted to some submanifold.", "We shall be interested in both of these problems in the case where $X$ is two dimensional and the submanifold we restrict to is a geodesic segment $\\ell $ .", "The basic upper bound for $\\Vert \\psi \\Vert _p$ in this case was proven by Sogge [20] (see also Avakumović [1] and Levitan [14] when $p = \\infty $ ), and is $\\Vert \\psi \\Vert _p \\ll \\lambda ^{\\delta (p)}$ where $\\delta (p)$ is given by $\\delta (p) = \\bigg \\lbrace \\begin{array}{ll} 1/2 - 2/p & p \\ge 6\\\\ 1/4 - 1/2p & 2 \\le p \\le 6.", "\\end{array}$ The standard bound for $\\Vert \\psi |_\\ell \\Vert _p$ is due to Burq, Gérard and Tzvetkov [7], and is $\\Vert \\psi |_\\ell \\Vert _p \\ll \\lambda ^{\\delta ^{\\prime }(p)}$ where $\\delta ^{\\prime }(p)$ is given by $\\delta ^{\\prime }(p) = \\bigg \\lbrace \\begin{array}{ll} 1/2 - 1/p & p \\ge 4\\\\ 1/4 & 2 \\le p \\le 4.", "\\end{array}$ Both of these bounds are sharp when $X$ is the round 2-sphere, but can be strengthened under extra geometric assumptions on $X$ such as negative curvature, see for instance [21], [22], [23].", "It should be noted that all such improvements in the negatively curved case are by at most a power of $\\log \\lambda $ .", "We now let $X$ be a compact arithmetic hyperbolic surface and $\\psi $ a Hecke-Maass cusp form on $X$ , which we shall always assume to be $L^2$ -normalised.", "In this case, Iwaniec and Sarnak [13] have shown that the bound $\\Vert \\psi \\Vert _\\infty \\ll \\lambda ^{1/2}$ given by (REF ) may be strengthened by a power to $\\Vert \\psi \\Vert _\\infty \\ll _\\epsilon \\lambda ^{5/12 + \\epsilon }$ .", "Their approach, known as arithmetic amplification, is to construct a projection operator onto $\\psi $ using the Hecke operators as well as the wave group.", "It has been adapted by other authors to study the pointwise norms of arithmetic eigenfunctions in various aspects, see for instance [5], [12], [24] as well as the alternative approach taken in [3].", "In this paper we apply amplification to a new kind of semiclassical problem, namely improving the exponent in the bound (REF ) for $\\Vert \\psi |_\\ell \\Vert _2$ .", "Our main result is as follows.", "Theorem 1.1 Let $\\psi $ be a Hecke-Maass eigenfunction on $X$ with spectral parameter $t$ .", "For any geodesic segment $\\ell $ of unit length we have $\\Vert \\psi |_\\ell \\Vert _2 \\ll _\\epsilon t^{3/14 + \\epsilon },$ where the implied constant is independent of $\\ell $ .", "We may combine Theorem REF with a theorem of Bourgain [6] to give an improvement over the local bound $\\Vert \\psi \\Vert _4 \\ll t^{1/8}$ .", "Corollary 1.2 We have $\\Vert \\psi \\Vert _4 \\ll _\\epsilon t^{1/8 - 1/112 + \\epsilon }$ .", "Corollary REF is much weaker than the bound $\\Vert \\psi \\Vert _4 \\ll _\\epsilon t^\\epsilon $ announced by Sarnak and Watson ([18], Theorem 3), although their result may be conditional on the Ramanujan conjecture.", "See also [4] for results in the case of holomorphic eigenforms.", "Note that Bourgain's theorem actually gives an equivalence (up to factors of $t^\\epsilon $ ) between a sub-local bound for $\\Vert \\psi \\Vert _4$ and one for $\\Vert \\psi |_\\ell \\Vert _2$ that is uniform in $\\ell $ , and so the bound of Sarnak and Watson implies Theorem REF with an exponent of $1/8$ .", "However, we feel that our method is of interest as it does not rely on special value identities or summation formulas, and we hope to apply it to restriction problems on other groups by combining it with the techniques of [15].", "The methods we use to prove Theorem REF also allow us to prove bounds for periods of $\\psi $ along $\\ell $ .", "We let $\\ell : [0,1] \\rightarrow X$ be an arc length parametrisation of $\\ell $ , and let $b \\in C^\\infty _0(\\mathbb {R})$ be a function with $\\text{supp}(b) \\subset [0,1]$ .", "For $1/2 > \\delta > 0$ , let $I_\\delta = [-1+\\delta , -\\delta ] \\cup [\\delta , 1-\\delta ]$ .", "Theorem 1.3 For $\\lambda \\in \\mathbb {R}$ , denote the integral $\\int _{-\\infty }^\\infty e^{i\\lambda x} b(x) \\psi (\\ell (x)) dx$ by $\\langle \\psi , b e^{i\\lambda x} \\rangle $ .", "If $\\lambda = 0$ we have $\\langle \\psi , b e^{i\\lambda x} \\rangle \\ll _\\epsilon t^{-1/12 + \\epsilon }$ .", "If $1/2 > \\delta > 0$ and $\\lambda / t \\in I_\\delta $ , we have $\\langle \\psi , b e^{i\\lambda x} \\rangle \\ll _\\epsilon t^{-1/18 + \\epsilon }$ .", "Define $\\beta = \\min |\\lambda \\pm t|$ .", "If $\\beta \\le t^{2/3}$ , we have $\\langle \\psi , b e^{i\\lambda x} \\rangle \\ll _\\epsilon t^{5/24 + \\epsilon } (1 + \\beta )^{1/24}$ .", "All of these bounds are uniform in $\\lambda $ and $\\ell $ .", "Remark The bound $\\beta \\le t^{2/3}$ in Theorem REF could be replaced with $t^{1-\\delta }$ for any $\\delta > 0$ , however when $\\beta \\ge t^{1/7 + \\epsilon }$ the bound $\\langle \\psi , b e^{i\\lambda x} \\rangle \\ll _\\epsilon t^{5/24 + \\epsilon } (1 + \\beta )^{1/24}$ is weaker than the local bound of Proposition REF .", "When $\\ell $ is a closed geodesic instead of a segment, cases (a) and (b) of Theorem REF may be compared with the local bound $\\langle \\psi , b e^{i\\lambda x} \\rangle \\ll 1$ given in [17], [26], and the improvement $\\langle \\psi , b \\rangle = o(1)$ given in [8] in the case of negative curvature.", "These cases should correspond via a formula of Waldspurger [25] to a subconvex bound for certain $L$ -values of the form $L(1/2, \\psi \\otimes \\theta _\\chi )$ , where $\\chi $ is a Grossencharacter of a real quadratic field and $\\theta _\\chi $ is the associated theta series on $GL_2$ .", "As in [13], Theorems REF and REF can both be strengthened under the assumption that the Fourier coefficients of $\\psi $ are not small.", "In the case of Theorem REF and case (c) of Theorem REF , this assumption allows us to employ an amplifier of sufficient length that it becomes profitable to estimate the Hecke recurrence using spectral methods, rather than the standard diophantine ones.", "Let $\\lambda (n)$ be the automorphically normalised Hecke eigenvalues of $\\psi $ , and assume that they satisfy the bounds $\\sum _{N \\le p \\le 2N} | \\lambda (p)| \\gg _\\epsilon N^{1-\\epsilon }$ for all $N \\ge 2$ and $|\\lambda (p)| \\le 2 p^\\theta $ for some $\\theta < 1/2$ and $p$ prime.", "Note that (REF ) is known with $\\theta = 7/64$ , see [2].", "We then prove Theorem 1.4 If the normalised Hecke eigenvalues $\\lambda (n)$ satisfy (REF ) and (REF ) we have $\\Vert \\psi |_\\ell \\Vert _2 \\ll _\\epsilon t^{1/(8 - 8\\theta ) + \\epsilon },$ while if $\\beta = \\min |\\lambda \\pm t|$ and $\\beta \\le t^{2/3}$ we have $\\langle \\psi , b e^{i\\lambda x} \\rangle \\ll _\\epsilon t^{\\theta /2 + \\epsilon } (1 + \\beta )^{1/4 - \\theta /2},$ uniformly in $\\lambda $ and $\\ell $ .", "In particular, Theorem REF gives $\\langle \\psi , b e^{i\\lambda x} \\rangle \\ll _\\epsilon t^\\epsilon $ when $|\\lambda - t| \\ll t^\\epsilon $ under the assumption that $\\theta = 0$ .", "We note that (REF ) becomes weaker than the local bound of Proposition REF when $\\beta \\ge t^{1/2 + \\epsilon }$ .", "Acknowledgements.", "We would like to thank Xiaoqing Li, Peter Sarnak, Christopher Sogge, Nicolas Templier, Akshay Venkatesh, and Steve Zelditch for many helpful discussions." ], [ "Notation", "For simplicity, we shall restrict attention to $X$ that arise from a quaternion division algebra $A = ( \\frac{a,b}{\\mathbb {Q}} )$ over $\\mathbb {Q}$ .", "Here $a, b \\in \\mathbb {Z}$ are square free and we will assume that $a > 0$ .", "We choose a basis $1, \\omega , \\Omega , \\omega \\Omega $ for $A$ over $\\mathbb {Q}$ that satisfies $\\omega ^2 = a$ , $\\Omega ^2 = b$ and $\\omega \\Omega + \\Omega \\omega = 0$ .", "We denote the norm and trace by $N(\\alpha ) = \\alpha \\overline{\\alpha }$ and $\\text{tr}(\\alpha ) = \\alpha + \\overline{\\alpha }$ .", "We let $R$ be a maximal order in $A$ (or more generally an Eichler order, see [9]), and for $m \\ge 1$ let $R(m) = \\lbrace \\alpha \\in R | N(\\alpha ) = m \\rbrace .$ $R(1)$ is the group of elements of norm 1; it acts on $R(m)$ by multiplication on the left and $R(1) \\backslash R(m)$ is known to be finite [9].", "Fix an embedding $\\phi : A \\rightarrow M_2(F)$ , the $2 \\times 2$ matrices with entries in $F = \\mathbb {Q}( \\sqrt{a})$ by $\\phi (\\alpha ) = \\left( \\begin{array}{cc} \\overline{\\xi } & \\eta \\\\ b \\overline{\\eta } & \\xi \\end{array} \\right)$ where $\\alpha = x_0 + x_1 \\omega + (x_2 + x_3 \\omega )\\Omega = \\xi + \\eta \\Omega .$ We define the lattice $\\Gamma = \\phi (R(1)) \\subset SL(2,\\mathbb {R})$ , which is co-compact as we assumed $A$ to be a division algebra, and let $X = \\Gamma \\backslash \\mathbb {H}$ .", "We define the Hecke operators $T_n: L^2(X) \\rightarrow L^2(X)$ , $n \\ge 1$ , by $T_nf (z) = \\sum _{\\alpha \\in R(1) \\backslash R(n)} f( \\phi (\\alpha ) z).$ There is a positive integer $q$ (depending on $R$ ) such that for $(n,q) = 1$ , $T_n$ has the following properties (see [9]): $T_n = T_n^*, & \\quad \\text{that is $T_n$ is self-adjoint}, \\\\T_n T_m & = \\sum _{d | (n,m)} d T_{nm/d^2}.$ We let $\\lambda (n)$ be the normalised Hecke eigenvalues of $\\psi $ and $t$ be its spectral parameter, so that $T_n \\psi & = \\lambda (n) n^{1/2} \\psi , \\\\\\Delta \\psi & = (1/4 + t^2) \\psi .$ We let $K$ , $A$ , and $N$ be the standard subgroups of $PSL_2(\\mathbb {R})$ , with parametrisations $k(\\theta ) = \\left( \\begin{array}{cc} \\cos \\theta /2 & \\sin \\theta /2 \\\\ -\\sin \\theta /2 & \\cos \\theta /2 \\end{array} \\right), \\qquad a(y) = \\left( \\begin{array}{cc} e^y & 0 \\\\ 0 & 1 \\end{array} \\right), \\qquad n(x) = \\left( \\begin{array}{cc} 1 & x \\\\ 0 & 1 \\end{array} \\right).$ In particular, $k(\\theta )$ represents an anticlockwise rotation by $\\theta $ about the point $i$ .", "We denote the Lie algebra of $PSL_2(\\mathbb {R})$ by $\\mathfrak {g}$ , and equip $\\mathfrak {g}$ with the norm $\\Vert \\cdot \\Vert : \\left( \\begin{array}{cc} X_1 & X_2 \\\\ X_3 & -X_1 \\end{array} \\right) \\mapsto \\sqrt{X_1^2 + X_2^2 + X_3^2}.$ This norm defines a left-invariant metric on $PSL_2(\\mathbb {R})$ , which we denote by $d_G$ .", "We denote the Lie algebras of $K$ , $A$ , and $N$ by $\\mathfrak {k}$ , $\\mathfrak {a}$ , and $\\mathfrak {n}$ , and write the Iwasawa decomposition as $g = n(g) \\exp ( A(g)) k(g) = \\exp ( N(g)) \\exp ( A(g)) k(g).$ We define $H = \\left( \\begin{array}{cc} 1/2 & 0 \\\\ 0 & -1/2 \\end{array} \\right) \\in \\mathfrak {a}, \\quad X_\\mathfrak {n}= \\left( \\begin{array}{cc} 0 & 1 \\\\ 0 & 0 \\end{array} \\right) \\in \\mathfrak {n}, \\quad X_\\mathfrak {k}= \\left( \\begin{array}{cc} 0 & 1/2 \\\\ -1/2 & 0 \\end{array} \\right).$ We identify $\\mathfrak {a}\\simeq \\mathbb {R}$ under the map $H \\mapsto 1$ , and consider $A(g)$ as a function $A : PSL_2(\\mathbb {R}) \\rightarrow \\mathbb {R}$ under this identification, and likewise for $\\mathfrak {n}$ and $N(g)$ .", "We let $\\varphi _s$ denote the standard spherical function with spectral parameter $s$ on $\\mathbb {H}$ or $PSL_2(\\mathbb {R})$ , depending on the context.", "Throughout the paper, the notation $A \\ll B$ will mean that there is a positive constant $C$ such that $|A| \\le CB$ , and $A \\sim B$ will mean that there are positive constants $C_1$ and $C_2$ such that $C_1 B \\le A \\le C_2 B$ ." ], [ "Amplification of geodesic periods", "We now prove cases (a) and (b) of Theorem REF .", "As we may assume that $\\psi $ is real, we may also assume that $\\lambda \\ge 0$ .", "We shall fix $1/2 > \\delta > 0$ , and assume that either $\\lambda / t \\in [\\delta , 1-\\delta ]$ or $\\lambda = 0$ .", "Let $h \\in \\mathcal {S}(\\mathbb {R})$ be a real-valued function of Payley-Wiener type that is positive, even, and $\\ge 1$ in the interval $[-1, 1]$ .", "Define $h_t$ by $h_t(s) = h(s - t) + h( -s - t)$ , and let $k^0_t$ be the $K$ -biinvariant function on $\\mathbb {H}$ with Harish-Chandra transform $h_t$ (see [11] or [19] for definitions).", "The Payley-Wiener theorem of Gangolli [10] implies that $k^0_t$ is of compact support that may be chosen arbitrarily small.", "Let $K^0_t$ be the point-pair invariant on $\\mathbb {H}$ associated to $k^0_t$ , which is real-valued and satisfies $K_t^0(x,y) = K_t^0(y,x)$ .", "Let $A^0_t$ the operator on $X$ with integral kernel $A_t^0(x,y) = \\sum _{\\gamma \\in \\Gamma } K_t^0(x, \\gamma y).$ It follows that $A_t^0$ is a self-adjoint approximate spectral projector onto the eigenfunctions in $L^2(X)$ with spectral parameter near $t$ .", "Let $k_t$ be the $K$ -biinvariant function on $\\mathbb {H}$ with Harisch-Chandra transform $h_t^2$ , and let $K_t$ and $A_t$ be associated to $k_t$ in the same way.", "It follows that $A_t = (A_t^0)^2$ .", "Let $\\ell \\subset \\mathbb {H}$ be a unit length geodesic segment.", "By abuse of notation, we also let $\\ell : [0,1] \\rightarrow \\mathbb {H}$ be an arc length parametrisation of $\\ell $ .", "Let $b \\in C^\\infty _0(\\mathbb {R})$ be a function with $\\text{supp}(b) \\subset [0,1]$ , and let $\\lambda \\in \\mathbb {R}$ .", "Let $N \\ge 1$ be an integer, and let $\\alpha _n$ , $n \\le N$ , be a sequence of complex numbers.", "We define $\\mathcal {T}$ to be the Hecke operator $\\mathcal {T}= \\sum _{1 \\le n \\le N} \\frac{\\alpha _n}{\\sqrt{n}} T_n.$ We shall estimate $\\langle \\psi , b e^{i\\lambda x} \\rangle $ by estimating $\\langle \\mathcal {T}A_t^0 \\psi , b e^{i\\lambda x} \\rangle $ .", "We first take adjoints to obtain $\\langle \\mathcal {T}A_t^0 \\psi , b e^{i\\lambda x} \\rangle = \\langle \\psi , \\mathcal {T}^* A_t^0 b e^{i\\lambda x} \\rangle $ , where $A_t^0 b e^{i\\lambda x}$ is the function on $X$ given by $A_t^0 b e^{i\\lambda x}(y) = \\int _{-\\infty }^\\infty A_t^0(y,\\ell (x)) b(x) e^{i\\lambda x} dx.$ We then apply Cauchy-Schwarz to obtain $|\\langle \\psi , \\mathcal {T}^* A_t^0 b e^{i\\lambda x} \\rangle | & \\le \\langle \\mathcal {T}^* A_t^0 b e^{i\\lambda x}, \\mathcal {T}^* A_t^0 b e^{i\\lambda x} \\rangle ^{1/2} \\\\& = \\langle b e^{i\\lambda x}, \\mathcal {T}\\mathcal {T}^* A_t b e^{i\\lambda x} \\rangle .$ We have $\\mathcal {T}\\mathcal {T}^* = \\sum _{m,n \\le N} \\alpha _n \\overline{\\alpha }_m \\sum _{ d | (n,m) } \\frac{d}{\\sqrt{mn}} T_{nm/d^2},$ and so $\\mathcal {T}\\mathcal {T}^* A_t b e^{i\\lambda x}(y) = \\sum _{m,n \\le N} \\alpha _n \\overline{\\alpha }_m \\sum _{ d | (n,m) } \\frac{d}{\\sqrt{mn}} \\sum _{\\gamma \\in R(nm/d^2)} \\int _{-\\infty }^\\infty K_t(y, \\gamma \\ell (x)) b(x) e^{i\\lambda x} dx.$ If $\\ell _1$ and $\\ell _2$ are a pair of unit geodesic segments in $\\mathbb {H}$ with parametrisations $\\ell _i : [0,1] \\rightarrow \\mathbb {H}$ , we define $I(t, \\lambda , \\ell _1, \\ell _2) = \\iint _{-\\infty }^\\infty b(x_1) b(x_2) e^{i\\lambda (x_1-x_2)} K_t(\\ell _1(x_1), \\ell _2(x_2)) dx_1 dx_2.$ With this notation, we have $\\langle b e^{i\\lambda x}, \\mathcal {T}\\mathcal {T}^* A_t b e^{i\\lambda x} \\rangle = \\sum _{m,n \\le N} \\alpha _n \\overline{\\alpha }_m \\sum _{ d | (n,m) } \\frac{d}{\\sqrt{mn}} \\sum _{\\gamma \\in R(nm/d^2)} I(t, \\lambda , \\ell , \\gamma \\ell ).$ To estimate the integrals $I(t, \\lambda , \\ell , \\gamma \\ell )$ , we introduce two distance functions on pairs of unit geodesics.", "Let $\\ell _0$ be the upwards pointing unit geodesic based at $i$ , and let $\\ell _1 = g_1 \\ell _0$ and $\\ell _2 = g_2 \\ell _0$ .", "We define $d(\\ell _1, \\ell _2) = \\text{inf} \\lbrace d(p,q) | p \\in \\ell _1, q \\in \\ell _2 \\rbrace ,$ where $d(p,q)$ is the hyperbolic distance between points.", "We also define $n(\\ell _1, \\ell _2) = \\text{inf} \\lbrace d_G( g_1^{-1} g_2, a) | a \\in A \\rbrace .$ In particular, $n(\\ell _1, \\ell _2) = 0$ iff the infinite extensions of $\\ell _1$ and $\\ell _2$ coincide and have the same orientation.", "We assume that $k_t$ is supported in a ball of radius 1 about $i$ , so that $I(t, \\lambda , \\ell _1, \\ell _2) = 0$ unless $d(\\ell _1, \\ell _2) \\le 1$ .", "We shall prove the following bounds for $I(t, \\lambda , \\ell _1, \\ell _2)$ .", "Proposition 3.1 Suppose $d(\\ell _1, \\ell _2) \\le 1$ .", "If $\\lambda / t \\in [\\delta , 1-\\delta ]$ , we have $I(t, \\lambda , \\ell _1, \\ell _2) \\ll \\Bigg \\lbrace \\begin{array}{cc} (1 + t n(\\ell _1, \\ell _2) )^{-1/2} & n(\\ell _1, \\ell _2) \\le t^{-1/3} \\\\t^{-1/3} & n(\\ell _1, \\ell _2) \\ge t^{-1/3}.", "\\end{array}$ If $\\lambda = 0$ , we have $I(t, \\lambda , \\ell _1, \\ell _2) \\ll (1 + t n(\\ell _1, \\ell _2) )^{-1/2}.$ The second result we shall need is a bound for the counting function $M( \\ell , n, \\kappa ) = | \\lbrace \\gamma \\in R(n) | d( \\gamma \\ell , \\ell ) \\le 1, n(\\ell , \\gamma \\ell ) < \\kappa \\rbrace |.$ Lemma 3.2 We have the bound $M( \\ell , n, \\kappa ) \\ll _\\epsilon (\\kappa ^2 + \\kappa ^{1/2})n^{1+\\epsilon } + n^\\epsilon $ uniformly in $\\ell $ .", "This may be proven in exactly the same way as the corresponding Lemma 1.3 of [13].", "The only differences are that we must consider the quadratic form $[ \\alpha , \\beta , \\gamma ]$ associated to $\\ell $ with $\\beta ^2 - 4 \\alpha \\gamma = 1,$ and the subgroup $K_\\ell $ generated by translation along $\\ell $ which may be parametrized as $K_\\ell = \\left\\lbrace \\left[ \\begin{array}{cc} t - \\beta u & -2\\gamma u \\\\ 2 \\alpha u & t + \\beta u \\end{array} \\right] | t^2 - u^2 = 1 \\right\\rbrace .$ As $\\Gamma $ was cocompact, we may assume that $\\ell $ lies in a fixed compact set.", "If $d( \\ell , \\gamma \\ell ) \\le 1$ , we have $n( \\ell , \\gamma \\ell ) < \\kappa \\rightarrow \\gamma = z + O(\\kappa )\\quad \\text{with} \\quad z \\in K_\\ell .$ If we write $\\gamma $ as $\\gamma = \\frac{1}{\\sqrt{n}} \\left[ \\begin{array}{cc} x_0 - x_1 \\sqrt{a} & x_2 + x_3 \\sqrt{a} \\\\bx_2 - bx_3\\sqrt{a} & x_0 + x_1 \\sqrt{a} \\end{array} \\right]$ then $x_0$ and $x_1$ must satisfy the equations $\\big | x_0^2 - \\frac{a}{\\beta ^2} x_1^2 - n \\big | \\ll n \\kappa , \\quad |x_0| \\ll \\sqrt{n}, \\quad |x_1| \\ll \\sqrt{n},$ where the last two conditions come from the fact that the entries of $\\gamma $ must be bounded.", "The proof now proceeds exactly as in [13], with the difference that we must count ideals of a given norm in real quadratic fields rather than imaginary ones, and the presence of units intorduces an extra factor of $n^\\epsilon $ into our counting which we may ignore.", "With these results, we are ready to estimate the sum (REF ).", "We first consider the case in which $\\lambda / t \\in [\\delta , 1-\\delta ]$ .", "If we assume that $d(\\ell , \\gamma \\ell ) \\le 1$ then we have $n(\\ell , \\gamma \\ell ) \\in [0, 2]$ , and we cover $[0,2]$ with the intervals $I_0 = [0, t^{-1}]$ , $I_k = [e^{k-1} t^{-1}, e^k t^{-1} ]$ for $1 \\le k \\le \\tfrac{2}{3} \\log t$ , and $I_\\infty = [e^{-1} t^{-1/3}, 2 ]$ .", "When $n(\\ell , \\gamma \\ell ) \\in I_0$ we apply the bounds $|I(t, \\lambda , \\ell _1, \\ell _2)| & \\ll 1 \\\\M( \\ell , n, t^{-1}) & \\ll t^{-1/2} n^{1+\\epsilon } + n^\\epsilon $ from Proposition REF and Lemma REF to obtain $\\sum _{m,n \\le N} \\alpha _n \\overline{\\alpha }_m \\sum _{ d | (n,m) } \\frac{d}{\\sqrt{mn}} \\sum _{ \\begin{array}{c} \\gamma \\in R(nm/d^2),\\\\ n( \\ell , \\gamma \\ell ) \\in I_0 \\end{array} } I(t, \\lambda , \\ell , \\gamma \\ell ) & \\ll N^\\epsilon t^\\epsilon \\sum _{m,n \\le N} \\alpha _n \\overline{\\alpha }_m \\sum _{ d | (n,m) } \\frac{d}{\\sqrt{mn}} \\left( t^{-1/2} \\frac{nm}{d^2} + 1 \\right) \\\\& \\ll N^\\epsilon t^\\epsilon \\sum _{m,n \\le N} \\alpha _n \\overline{\\alpha }_m \\sum _{ d | (n,m) } \\frac{\\sqrt{mn}}{d} t^{-1/2} + \\frac{d}{\\sqrt{mn}}.$ When $n(\\ell , \\gamma \\ell ) \\in I_k$ we have $|I(t, \\lambda , \\ell _1, \\ell _2)| & \\ll e^{-k/2} \\\\M( \\ell , n, e^k t^{-1}) & \\ll t^{-1/2} e^{k/2} n^{1+\\epsilon } + n^\\epsilon ,$ which gives $\\sum _{m,n \\le N} \\alpha _n \\overline{\\alpha }_m \\sum _{ d | (n,m) } \\frac{d}{\\sqrt{mn}} \\sum _{ \\begin{array}{c} \\gamma \\in R(nm/d^2),\\\\ n( \\ell , \\gamma \\ell ) \\in I_k \\end{array} } I(t, \\lambda , \\ell , \\gamma \\ell ) & \\ll N^\\epsilon t^\\epsilon \\sum _{m,n \\le N} \\alpha _n \\overline{\\alpha }_m \\sum _{ d | (n,m) } \\frac{d}{\\sqrt{mn}} \\left( t^{-1/2} \\frac{nm}{d^2} + e^{-k/2} \\right) \\\\& \\ll N^\\epsilon t^\\epsilon \\sum _{m,n \\le N} \\alpha _n \\overline{\\alpha }_m \\sum _{ d | (n,m) } \\frac{\\sqrt{mn}}{d} t^{-1/2} + \\frac{d}{\\sqrt{mn}} e^{-k/2}.$ When $n(\\ell , \\gamma \\ell ) \\in I_\\infty $ we have $|I(t, \\lambda , \\ell _1, \\ell _2)| & \\ll t^{-1/3} \\\\M( \\ell , n, 10) & \\ll n^{1+\\epsilon },$ so that $\\sum _{m,n \\le N} \\alpha _n \\overline{\\alpha }_m \\sum _{ d | (n,m) } \\frac{d}{\\sqrt{mn}} \\sum _{ \\begin{array}{c} \\gamma \\in R(nm/d^2),\\\\ n( \\ell , \\gamma \\ell ) \\in I_\\infty \\end{array} } I(t, \\lambda , \\ell , \\gamma \\ell ) \\ll N^\\epsilon t^\\epsilon \\sum _{m,n \\le N} \\alpha _n \\overline{\\alpha }_m \\sum _{ d | (n,m) } \\frac{\\sqrt{mn}}{d} t^{-1/3}.$ Combining these, and noting that we are summing over $\\ll \\log t$ values of $k$ , we obtain $\\langle b e^{i\\lambda x}, \\mathcal {T}\\mathcal {T}^* A_t b e^{i\\lambda x} \\rangle \\ll N^\\epsilon t^\\epsilon \\sum _{m,n \\le N} \\alpha _n \\overline{\\alpha }_m \\sum _{ d | (n,m) } \\frac{\\sqrt{mn}}{d} t^{-1/3} + \\frac{d}{\\sqrt{mn}}.$ As in [13], p. 310, we have $\\sum _{m, n \\le N} \\sum _{d | (n,m) } \\frac{\\sqrt{nm}}{d} | \\alpha _n \\alpha _m| \\le N^{1+\\epsilon } \\left( \\sum _{n \\le N} |\\alpha _n| \\right)^2,$ and $\\sum _{m, n \\le N} \\sum _{d | (m,n)} \\frac{d}{ \\sqrt{mn}} |\\alpha _n \\alpha _m| \\ll N^\\epsilon \\sum _{n \\le N} |\\alpha _n|^2.$ Combining (REF ) with (REF ) and (REF ) gives $\\langle b e^{i\\lambda x}, \\mathcal {T}\\mathcal {T}^* A_t b e^{i\\lambda x} \\rangle \\ll N^\\epsilon t^\\epsilon \\left( \\sum _{n \\le N} |\\alpha _n|^2 + N t^{-1/3} \\left( \\sum _{n \\le N} |\\alpha _n| \\right)^2 \\right).$ If we choose $\\lbrace \\alpha _n \\rbrace $ to be the amplifier used in [13], it follows as on p. 311 there that $|\\langle \\psi , b e^{i\\lambda x} \\rangle |^2 \\ll N^\\epsilon t^\\epsilon ( N^{-1/2} + N t^{-1/3}),$ and choosing $N = t^{2/9}$ completes the proof.", "The proof in the case $\\lambda = 0$ is almost identical.", "We again perform a dyadic sum over $n(\\ell , \\gamma \\ell )$ and simplify to obtain $\\langle b, \\mathcal {T}\\mathcal {T}^* A_t b \\rangle \\ll N^\\epsilon t^\\epsilon \\left( \\sum _{n \\le N} |\\alpha _n|^2 + N t^{-1/2} \\left( \\sum _{n \\le N} |\\alpha _n| \\right)^2 \\right),$ and the result follows by using the same amplifier with $N = t^{1/3}$ ." ], [ "Bounds for $L^2$ norms", "To prove Theorem REF , it suffices to bound the $L^2$ norm of $b(x) \\psi (\\ell (x)) \\in L^2(\\mathbb {R})$ for $b \\in C^\\infty _0(\\mathbb {R})$ with $\\text{supp}(b) \\subseteq [0,1]$ , provided the bound is uniform in $\\ell $ .", "If $f \\in C^\\infty _0(\\mathbb {R})$ , define its Fourier transform $\\widehat{f}$ by $\\widehat{f}(\\xi ) = \\int _{-\\infty }^\\infty f(x) e^{-i \\xi x} dx,$ and extend this to an operator on $L^2(\\mathbb {R})$ .", "Let $\\beta $ be a parameter satisfying $1 \\le \\beta \\le t^{2/3}$ .", "Define $H_\\beta ^+$ , $H_\\beta ^- \\subset L^2(\\mathbb {R})$ to be the spaces of functions whose Fourier support lies in $[ \\pm t-\\beta , \\pm t+\\beta ]$ , and define $H_\\beta = H_\\beta ^+ + H_\\beta ^-$ .", "Let $\\Pi _\\beta $ be the orthogonal projection onto $H_\\beta $ , and likewise for $\\Pi _\\beta ^\\pm $ and $H_\\beta ^\\pm $ .", "We shall bound $\\Pi _\\beta b\\psi $ and $(1 - \\Pi _\\beta ) b\\psi $ separately, by applying amplification to the former and a local bound to the latter, and as $\\psi $ is real-valued it suffices to bound $\\Pi _\\beta ^+ b\\psi $ .", "The results we are obtain are the following.", "Proposition 4.1 We have $\\Vert \\Pi _\\beta ^+ b \\psi \\Vert _2 \\ll _\\epsilon t^{5/24 + \\epsilon } \\beta ^{1/24}$ , uniformly in $\\beta $ and $\\ell $ .", "Proposition 4.2 We have $\\Vert (1 - \\Pi _\\beta ) b \\psi \\Vert _2 \\ll _\\epsilon t^{1/4 + \\epsilon } \\beta ^{-1/4}$ , uniformly in $\\beta $ and $\\ell $ .", "Combining these two results with $\\beta = t^{1/7}$ gives Theorem REF .", "Note that we expect Proposition REF to be sharp on the round sphere." ], [ "Amplification of geodesic periods with $\\lambda \\sim t$", "We shall prove Proposition REF using the method of Section .", "As before, it suffices to estimate $\\langle \\psi , b \\phi \\rangle $ for $\\phi \\in H_\\beta ^+$ with $\\Vert \\phi \\Vert _2 = 1$ , and we have $|\\langle \\mathcal {T}A_t^0 \\psi , b \\phi \\rangle |^{1/2} \\le \\langle b \\phi , \\mathcal {T}\\mathcal {T}^* A_t b \\phi \\rangle .$ If $\\ell _1$ and $\\ell _2$ are a pair of unit geodesic segments in $\\mathbb {H}$ with parametrisations $\\ell _i : [0,1] \\rightarrow \\mathbb {H}$ , we define $I(t, \\phi , \\ell _1, \\ell _2) = \\iint _{-\\infty }^\\infty b(x_1) b(x_2) \\phi (x_1) \\overline{\\phi (x_2)} K_t(\\ell _1(x_1), \\ell _2(x_2)) dx_1 dx_2.$ With this notation, we again have $\\langle b \\phi , \\mathcal {T}\\mathcal {T}^* A_t b \\phi \\rangle = \\sum _{m,n \\le N} \\alpha _n \\overline{\\alpha }_m \\sum _{ d | (n,m) } \\frac{d}{\\sqrt{mn}} \\sum _{\\gamma \\in R(nm/d^2)} I(t, \\phi , \\ell , \\gamma \\ell ).$ We let the geodesic distance functions $d(\\ell _1, \\ell _2)$ and $n(\\ell _1, \\ell _2)$ be as in Section .", "The estimate for $I(t, \\phi , \\ell _1, \\ell _2)$ corresponding to Proposition REF in this case is as follows.", "Proposition 4.3 Suppse $d(\\ell _1, \\ell _2) \\le 1$ .", "We have $| I(t, \\phi , \\ell _1, \\ell _2) | \\ll t^{1/2}$ for all $\\ell _1$ and $\\ell _2$ , while if $n(\\ell _1, \\ell _2) \\ge t^{-1/2 + \\epsilon } \\beta ^{1/2}$ we have $| I(t, \\phi , \\ell _1, \\ell _2) | \\ll _{\\epsilon , A} t^{-A}.$ The implied constants in both bounds are independent of $\\phi $ and $\\beta $ .", "We shall prove Proposition REF in Section .", "Proposition REF implies that we only need to consider the terms in (REF ) with $d(\\ell , \\gamma \\ell ) \\le 1$ and $n(\\ell , \\gamma \\ell ) \\le t^{-1/2 + \\epsilon } \\beta ^{1/2}$ .", "Lemma REF gives $M(\\ell , n, t^{-1/2+\\epsilon } \\beta ^{1/2} ) \\ll _\\epsilon t^{-1/4 + \\epsilon } \\beta ^{1/4} n^{1+\\epsilon } + n^\\epsilon ,$ and so we have $\\sum _{m,n \\le N} \\alpha _n \\overline{\\alpha }_m \\sum _{ d | (n,m) } \\frac{d}{\\sqrt{mn}} \\sum _{ \\gamma \\in R(nm/d^2) } I(t, \\phi , \\ell , \\gamma \\ell ) & \\ll N^\\epsilon t^\\epsilon \\sum _{m,n \\le N} \\alpha _n \\overline{\\alpha }_m \\sum _{ d | (n,m) } \\frac{d}{\\sqrt{mn}} t^{1/2} M(\\ell , n, t^{-1/2+\\epsilon } \\beta ^{1/2} ) \\\\& \\ll N^\\epsilon t^\\epsilon \\sum _{m,n \\le N} \\alpha _n \\overline{\\alpha }_m \\sum _{ d | (n,m) } \\frac{\\sqrt{mn}}{d} t^{1/4} \\beta ^{1/4} + \\frac{d}{\\sqrt{mn}}t^{1/2}.$ Combining (REF ) with (REF ) and (REF ) gives $\\langle b \\phi , \\mathcal {T}\\mathcal {T}^* A_t b \\phi \\rangle \\ll N^\\epsilon t^\\epsilon \\left( t^{1/2} \\sum _{n \\le N} |\\alpha _n|^2 + N t^{1/4} \\beta ^{1/4} \\left( \\sum _{n \\le N} |\\alpha _n| \\right)^2 \\right),$ and Proposition REF now follows as in Section by choosing $N = t^{1/6} \\beta ^{-1/6}$ ." ], [ "Bounds Away from the Spectrum", "We now give the proof of Proposition REF .", "We are free to assume that $\\beta \\ge 2t^\\epsilon $ , as otherwise the result follows from the bound (REF ) of Burq-Gérard-Tzvetkov.", "As we will not be using Hecke operators, we are free to replace $\\Gamma $ by a finite index sublattice with $\\text{inj rad}(X) \\ge 10$ .", "It suffices to estimate $\\langle \\psi , b \\phi \\rangle $ for $\\phi \\in H_\\beta ^\\perp $ with $\\Vert \\phi \\Vert _2 = 1$ .", "Let $k_t$ , $K_t$ and $A_t$ be as in Section .", "It follows as before that $|\\langle A_t^0 \\psi , b \\phi \\rangle | \\le \\langle b \\phi , A_t b \\phi \\rangle ^{1/2},$ where $\\langle b \\phi , A_t b \\phi \\rangle = \\iint _{-\\infty }^\\infty b(x_1) b(x_2) \\phi (x_1) \\overline{\\phi (x_2)} \\sum _{\\gamma \\in \\Gamma } K_t(\\ell (x_1), \\gamma \\ell (x_2)) dx_1 dx_2.$ Our assumptions that $\\text{inj rad}(X) \\ge 10$ and $k_t$ is supported in a ball of radius 1 imply that only the term $\\gamma = e$ makes a contribution to the inner sum, so that $\\langle b \\phi , A_t b \\phi \\rangle = \\iint _{-\\infty }^\\infty b(x_1) b(x_2) \\phi (x_1) \\overline{\\phi (x_2)} K_t(\\ell (x_1), \\ell (x_2)) dx_1 dx_2.$ We have $K_t(\\ell (x_1), \\ell (x_2)) = k_t(a(x_1 - x_2))$ .", "Therefore, if we define $p_t(x) = k_t(a(x))$ and let $P_t$ be the operator on $\\mathbb {R}$ with integral kernel $P_t(x,y) = p_t(x-y)$ , we have $\\langle b \\phi , A_t b \\phi \\rangle = \\langle b \\phi , P_t b \\phi \\rangle $ .", "Define $I_\\beta = [ - t - \\beta /2, - t + \\beta /2] \\cup [ t - \\beta /2, t + \\beta /2],$ and write $b \\phi = \\phi _1 + \\phi _2$ , where the Fourier transform of $\\phi _2$ is supported on $I_\\beta $ and the transform of $\\phi _1$ is supported on $\\mathbb {R}\\setminus I_\\beta $ .", "Because $b$ was a fixed smooth function, we have $\\Vert \\phi _2 \\Vert _2 \\ll _A \\beta ^{-A} \\ll _{\\epsilon , A} t^{-A}$ .", "Because the kernel of $P_t$ is translation invariant, we have $\\langle b \\phi , P_t b \\phi \\rangle & = \\langle \\phi _1, P_t \\phi _1 \\rangle + \\langle \\phi _2, P_t \\phi _2 \\rangle \\\\& \\le \\underset{\\lambda \\notin I_\\beta }{\\sup } |\\widehat{p_t}(\\lambda )| + O_{\\epsilon ,A}(t^{-A}) \\Vert \\widehat{p_t} \\Vert _\\infty .$ By Lemma 2.6 of [15] (see also Lemma 4.1 of [7]) we have $p_t(x) \\ll t(1 + tx)^{-1/2},$ and this imples that $\\Vert \\widehat{p_t} \\Vert _\\infty \\ll t^{1/2}$ .", "It therefore suffices to prove the following estimate.", "Lemma 4.4 We have $| \\widehat{p_t}(\\lambda ) | \\ll _\\epsilon t^{1/2 + \\epsilon } \\beta ^{-1/2}$ for $\\lambda \\notin I_\\beta $ .", "Let $b_1\\in C^\\infty _0(\\mathbb {R})$ be a cutoff function that is equal to 1 on $[ -1, 1]$ and zero outside $[-2,2]$ .", "We wish to estimate the integral $\\int _{-\\infty }^\\infty p_t(x) e^{i \\lambda x} dx = \\int _{-\\infty }^\\infty b_1(x) k_t(a(x)) e^{i \\lambda x} dx$ for $\\lambda \\notin I_\\beta $ .", "Inverting the Harish-Chandra transform gives $\\int _{-\\infty }^\\infty b_1(x) k_t(a(x)) e^{i \\lambda x} dx = \\frac{1}{2\\pi } \\int _{-\\infty }^\\infty \\int _0^\\infty b_1(x) \\varphi _s(a(x)) e^{i \\lambda x} h_t^2(s) s \\tanh (\\pi s) ds dx,$ see for instance [19].", "If $s \\in [0,\\infty ) \\setminus [t-\\beta /4, t+\\beta /4]$ , our assumption that $\\beta \\ge 2t^\\epsilon $ implies that $(1+|s|) h_t(s) \\ll _{\\epsilon ,A} t^{-A}$ .", "As $s \\tanh (\\pi s) \\ll 1 + |s|$ , this gives $\\int _{-\\infty }^\\infty b_1(x) k_t(a(x)) e^{i \\lambda x} dx = \\frac{1}{2\\pi } \\int _{-\\infty }^\\infty \\int _{t-\\beta /4}^{t+\\beta /4} b_1(x) \\varphi _s(a(x)) e^{i \\lambda x} h_t^2(s) s \\tanh (\\pi s) ds dx + O(t^{-A}).$ It therefore suffices to prove the bound $\\int _{-\\infty }^\\infty b_1(x) \\varphi _s(a(x)) e^{i \\lambda x} dx \\ll _\\epsilon t^{-1/2 + \\epsilon } \\beta ^{-1/2}$ uniformly for $\\lambda \\notin I_\\beta $ and $s \\in [t-\\beta /4, t+\\beta /4]$ .", "We decompose the integral as $\\int _{-\\infty }^\\infty b_1( s^{-\\epsilon } \\beta x) \\varphi _s(a(x)) e^{i \\lambda x} dx + \\int _{-\\infty }^\\infty (b_1(x) - b_1( s^{-\\epsilon } \\beta x) ) \\varphi _s(a(x)) e^{i \\lambda x} dx.$ Our assumption that $\\beta \\ge 2 t^\\epsilon $ implies that $s^{-\\epsilon } \\beta \\ge 1$ for $t$ suficiently large.", "Theorem 1.3 of [15] gives the bound $\\varphi _s(a(x)) \\ll (1 + sx)^{-1/2}$ for $x \\in [-2,2]$ , and this implies that the first integral is $\\ll _\\epsilon t^{-1/2 + \\epsilon } \\beta ^{-1/2}$ .", "To bound the second integral, by combining Proposition 4.12 of [15] with either Lemma REF below or Proposition 4.13 of [15] and applying stationary phase, we may prove that $\\varphi _s(a(x)) = c_1(x) e^{isx} (sx)^{-1/2} + c_2(x) e^{-isx} (sx)^{-1/2} + O( (sx)^{-3/2}),$ where $c_i \\in C^\\infty (\\mathbb {R})$ and the error term is uniform for $x \\in [-2,2] \\setminus \\lbrace 0 \\rbrace $ .", "As we have $\\int _{s^{-1}}^1 (xs)^{-3/2} dx \\ll s^{-1} \\ll t^{-1/2 + \\epsilon } \\beta ^{-1/2},$ we may ignore the contribution to the second integral coming from the error term in (REF ).", "The two main terms in the asymptotic are identical, and so we shall treat the second one by estimating the integral $\\int _{-\\infty }^\\infty (b_1(x) - b_1( s^{-\\epsilon } \\beta x) ) e^{i (\\lambda -s) x} c_2(x) (sx)^{-1/2} dx.$ After changing variable from $x$ to $s^{-\\epsilon } \\beta x$ , this becomes $s^{-1/2+\\epsilon } \\beta ^{-1/2} \\int _{-\\infty }^\\infty (b_1(s^{-\\epsilon } \\beta x) - b_1(x) ) e^{i (\\lambda -s) s^\\epsilon \\beta ^{-1} x} c_2(s^\\epsilon \\beta ^{-1} x) x^{-1/2} dx.$ As $s^{-\\epsilon } \\beta \\ge 1$ , all derivatives of $b_1(s^{-\\epsilon } \\beta x) - b_1(x)$ and $c_2(s^\\epsilon \\beta ^{-1} x)$ are bounded.", "Moreover, all derivatives of $x^{-1/2}$ are bounded on the support of $b_1(s^{-\\epsilon } \\beta x) - b_1(x)$ .", "As $|\\lambda -s| s^\\epsilon \\beta ^{-1} \\gg t^\\epsilon $ , repeated integration by parts implies that this integral is $\\ll _{\\epsilon ,A} t^{-A}$ as required." ], [ "Spectral estimation of Hecke returns", "We now prove Theorem REF by improving the amplifier used in Proposition REF .", "Our new ingredient is a spectral method for estimating the number of times the Hecke operators map $\\ell $ close to itself, which allows us to prove the following result.", "Proposition 5.1 Assume that $\\psi $ satisfies (REF ) and (REF ).", "We have the bound $\\Vert \\Pi _\\beta b \\psi \\Vert _2 \\ll _\\epsilon t^{\\theta /2 + \\epsilon } \\beta ^{1/4 - \\theta /2}.$ Theorem REF follows by choosing $\\beta = t^{(1-2\\theta )/(2-2\\theta )}$ and combining this with Proposition REF .", "We maintain the notations of Section .", "Let $\\epsilon > 0$ be given, and let $N$ be an integer of size roughly $t^{1/2 + \\epsilon } \\beta ^{-1/2}$ .", "Define $\\mathcal {T}_1$ to be the operator $\\mathcal {T}_1 = \\sum _{N/2 < p < N} \\frac{\\lambda (p)}{\\sqrt{p}} T_p.$ It again suffices to bound the inner product $\\langle b \\phi , \\mathcal {T}_1 \\mathcal {T}_1^* A_t b \\phi \\rangle $ .", "After reducing $\\mathcal {T}_1 \\mathcal {T}_1^*$ using the Hecke relations, we have $\\langle b \\phi , \\mathcal {T}_1 \\mathcal {T}_1^* A_t b \\phi \\rangle = \\sum _{N/2 < p < N} I(t, \\ell , \\ell ) + \\sum _{N/2 < p_1,p_2 < N} \\lambda (p_1) \\overline{\\lambda (p_2)} \\frac{1}{\\sqrt{p_1p_2}} \\sum _{ \\gamma \\in R(p_1p_2)} I(t, \\phi , \\ell , \\gamma \\ell ).$ The key difference between the proof of Proposition REF and Proposition REF is that we shall now estimate the recurrences of $\\ell $ under a large collection of Hecke operators $T_n$ at once using spectral methods, rather than individually.", "This is carried out in the following proposition.", "Proposition 5.2 If $M$ and $1 > \\delta > 0$ satisfy $M \\ge \\delta ^{-2-\\epsilon }$ , we have $\\sum _{ \\begin{array}{c} M/2 < m < M \\\\ (m,q) = 1 \\end{array} } \\frac{1}{\\sqrt{m}} M(\\ell , m, \\delta ) \\ll _\\epsilon \\delta ^2 M^{3/2},$ where $q$ is the integer defined in Section .", "Let $b \\in C^\\infty _0(\\mathfrak {g})$ be a real non-negative function that is supported in the ball of radius 2 about the origin with respect to the norm $\\Vert \\cdot \\Vert $ defined in (REF ), and equal to 1 on the ball of radius 1.", "Let $C_1 > 0$ be a constant to be chosen later.", "Define $b_\\delta \\in C^\\infty _0(\\mathfrak {g})$ by $b_\\delta (X) = b(\\delta ^{-1} C_1 X)$ , and let $\\widetilde{b}_\\delta \\in C^\\infty _0(PSL_2(\\mathbb {R}))$ be the pushforward of $b_\\delta $ under $\\exp $ .", "Let $\\tilde{\\ell } \\subset PSL_2(\\mathbb {R})$ be the set obtained by extending $\\ell $ by three times its length in both directions and lifting to $PSL_2(\\mathbb {R})$ .", "Let $\\delta _{\\widetilde{\\ell }}$ be the length measure on $\\widetilde{\\ell }$ , and let $f = \\delta ^{-2} \\widetilde{b}_\\delta * \\delta _{\\widetilde{\\ell }}$ .", "If we choose $C_1$ to be small enough, the conditions $d(\\ell , g\\ell ) \\le 1$ and $n(\\ell , g\\ell ) \\le \\delta $ imply that $\\langle f, g f \\rangle \\gg 1$ , where the implied constant is independent of $\\delta $ and $\\ell $ .", "If we define $\\overline{f} \\in L^2(\\Gamma \\backslash PSL_2(\\mathbb {R}))$ by $\\overline{f}(g) = \\sum _{\\gamma \\in \\Gamma } f(\\gamma g),$ then $\\Vert \\overline{f} \\Vert _2 \\sim 1$ in $L^2(\\Gamma \\backslash PSL_2(\\mathbb {R}))$ .", "Choose $g \\in C^\\infty _0(0, \\infty )$ to be real, positive, and satisfy $g(x) = 1$ for $1/2 \\le x \\le 1$ .", "If we define $\\mathcal {S}= \\sum _{(m,q) = 1} \\frac{g(m/M)}{\\sqrt{m}} T_m,$ then we have $\\sum _{ \\begin{array}{c} M/2 < m < M \\\\ (m,q) = 1 \\end{array} } \\frac{1}{\\sqrt{m}} M(\\ell , m, \\delta ) \\ll \\sum _{(m,q) = 1} \\frac{g(m/M)}{\\sqrt{m}} \\sum _{\\gamma \\in R(m)} \\langle f, \\gamma f \\rangle = \\langle \\overline{f}, \\mathcal {S}\\overline{f} \\rangle $ and we may estimate the RHS spectrally.", "Expand $\\overline{f}$ with respect to a decomposition of $L^2(\\Gamma \\backslash PSL_2(\\mathbb {R}))$ into automorphic representations as $\\overline{f} = \\sum _i \\alpha _i \\psi _i,$ where $\\psi _i$ is an $L^2$ normalised vector in an automorphic representation with eigenvalue $\\mu _i$ under the Casimir operator $C$ .", "We have $\\Vert C^n \\overline{f} \\Vert _2 \\ll _n \\delta ^{- 2n}.$ Integration by parts then gives $\\langle \\overline{f}, \\psi _i \\rangle & = \\mu _i^{-n} \\langle \\overline{f}, C^n \\psi _i \\rangle \\\\& = \\mu _i^{-n} \\langle C^n \\overline{f}, \\psi _i \\rangle \\\\& \\ll _n |\\mu _i|^{-n} \\delta ^{-2n},$ which implies that $\\overline{f} = \\langle \\overline{f}, 1 \\rangle + \\mathop {\\mmlmultiscripts{\\sum {\\mmlnone }{\\prime }}}\\limits _{|\\mu _i| \\le \\delta ^{-2-\\epsilon /2} } \\alpha _i \\psi _i + O_{A, \\epsilon }(\\delta ^A).$ Note that we have normalised the volume of $\\Gamma \\backslash PSL_2(\\mathbb {R})$ to be 1, and $\\Sigma ^{\\prime }$ denotes the sum over the nontrivial representations.", "Substituting this into $\\langle \\overline{f}, \\mathcal {S}\\overline{f} \\rangle $ gives $\\langle \\overline{f}, \\mathcal {S}\\overline{f} \\rangle = \\langle \\overline{f}, 1 \\rangle ^2 \\sum _{(m,q) = 1} g(m/M) \\sqrt{m} + \\sum _{|\\mu _i| \\le \\delta ^{-2-\\epsilon } } |\\alpha _i|^2 \\sum _{(m,q) = 1} g(m/M) \\lambda _i(m) + O_{A, \\epsilon }(M^{3/2}\\delta ^A),$ where $\\lambda _i(m)$ are the Hecke eigenvalues of $\\psi _i$ .", "The result now follows from Lemma REF below, and the asymptotic $\\langle \\overline{f}, 1 \\rangle \\ll \\delta $ .", "(Note that our assumptions that $M \\ge \\delta ^{-2-\\epsilon }$ and $|\\mu _i| \\le \\delta ^{-2 - \\epsilon /2}$ guarantee that the hypothesis of the Lemma is satisfied.)", "Lemma 5.3 If $M \\ge |\\mu _i|^{1+\\epsilon }$ , we have $\\sum _{(m,q) = 1} g(m/M) \\lambda _i(m) \\ll _{A,\\epsilon } M^{-A},$ where the implied constant is uniform in $\\psi _i$ .", "We shall drop the subscript $i$ , and assume that $\\psi $ is a vector in a principal series representation as the discrete series case is similar.", "We first consider the case $q = 1$ .", "Let $r$ be the spectral parameter of $\\psi $ , so that $\\mu = 1/4 + r^2$ .", "By applying the functional equation and Stirling's formula, we see that the $L$ -function $L(s, \\psi )$ satisfies the estimate $L( -A + it, \\psi ) \\ll _{A, \\epsilon } ( t^2 + r^2 + 1)^{A + 1/2 + \\epsilon }$ for $A$ sufficiently large.", "If we let $\\widehat{g}(s)$ be the Mellin transform of $g$ , which is entire and decays rapidly in vertical strips, we obtain $\\sum _m g(m/M) \\lambda (m) = \\int _{(2)} L(s, \\psi ) \\widehat{g}(s) M^s ds.$ If we shift the line of integration to $\\sigma = -A$ , and apply (REF ) and the rapid decay of $\\widehat{g}$ , we have $\\sum _m g(m/M) \\lambda _i(m) & \\ll _{A,\\epsilon ^{\\prime }} M^{-A} ( 1 + r^2)^{A + 1/2 + \\epsilon ^{\\prime }} \\\\& \\ll _{A,\\epsilon ^{\\prime }} M^{-A} \\mu ^{A + 1/2 + \\epsilon ^{\\prime }} \\\\& \\ll _{A,\\epsilon ^{\\prime }} M^{-A} M^{(1 - \\epsilon )(A + 1/2 + \\epsilon ^{\\prime })} \\\\& \\ll _{B, \\epsilon } M^{-B}$ as required.", "In the case when $q > 1$ , we apply the same argument to the incomplete $L$ -function obtained by removing the local factors at primes dividing $q$ from $L(s, \\psi )$ .", "With these results, we are ready to estimate the RHS of (REF ).", "We begin by applying the trivial bound of Proposition REF to the first sum, and our assumption that $|\\lambda (p)| \\le 2p^\\theta $ to the second, which gives $\\langle b \\phi , \\mathcal {T}_1 \\mathcal {T}_1^* A_t b \\phi \\rangle \\ll Nt^{1/2} + N^{2\\theta } \\sum _{N/2 < p_1,p_2 < N} \\frac{1}{\\sqrt{p_1p_2}} \\sum _{ \\gamma \\in R(p_1p_2)} |I(t, \\ell , \\gamma \\ell )|.$ Enlarging the sum to one over all $N^2/4 < n < N^2$ with $(n,q) = 1$ gives $\\langle b \\phi , \\mathcal {T}_1 \\mathcal {T}_1^* A_t b \\phi \\rangle \\ll Nt^{1/2} + N^{2\\theta } \\sum _{ \\begin{array}{c} n \\sim N^2 \\\\ (n,q) = 1\\end{array} } \\frac{1}{\\sqrt{n}} \\sum _{ \\gamma \\in R(n)} |I(t, \\ell , \\gamma \\ell )|.$ By Proposition REF , we only need to consider the terms in the second sum with $d(\\ell , g\\ell ) \\le 1$ and $n(\\ell , g\\ell ) \\le t^{-1/2+\\epsilon } \\beta ^{1/2}$ , which gives $\\sum _{ \\begin{array}{c} n \\sim N^2 \\\\ (n,q) = 1\\end{array} } \\frac{1}{\\sqrt{n}} \\sum _{ \\gamma \\in R(n)} |I(t, \\ell , \\gamma \\ell )| \\ll \\sum _{ \\begin{array}{c} n \\sim N^2 \\\\ (n,q) = 1\\end{array} } \\frac{t^{1/2}}{\\sqrt{n}} M(\\ell , n, t^{-1/2+\\epsilon } \\beta ^{1/2}).$ The assumption that $N \\sim t^{1/2 + \\epsilon } \\beta ^{-1/2}$ implies that we may choose $\\delta = t^{-1/2+\\epsilon } \\beta ^{1/2}$ and $M = N^2$ in Proposition REF , so that $\\sum _{ \\begin{array}{c} n \\sim N^2 \\\\ (n,q) = 1\\end{array} } \\frac{t^{1/2}}{\\sqrt{n}} M(\\ell , n, t^{-1/2+\\epsilon } \\beta ^{1/2}) \\ll N^3 t^{-1 + \\epsilon } \\beta .$ Substituting this into (REF ) gives $| \\langle \\mathcal {T}_1 A_t^0 \\psi , b \\phi \\rangle |^2 \\le \\langle b \\phi , \\mathcal {T}_1 \\mathcal {T}_1^* A_t b \\phi \\rangle \\ll Nt^{1/2} + N^{3 + 2\\theta } t^{-1/2+\\epsilon } \\beta .$ If we estimate the action of $\\mathcal {T}_1$ on $\\psi $ using our assumption (REF ) and substitute $N \\sim t^{1/2 + \\epsilon } \\beta ^{-1/2}$ , we obtain $| \\langle \\psi , b \\phi \\rangle | \\ll _\\epsilon t^{\\theta /2 + \\epsilon } \\beta ^{1/4 - \\theta /2}$ as required.", "Remark The method we have used of estimating Hecke recurrences spectrally is unlikely to work in other situations.", "It requires us to choose an amplifier that makes the sums of eigenvalues in Proposition REF longer than the relevant analytic conductors, and in other cases (such as higher rank or when using the operators $T_{p^2}$ on $GL_2$ to give an unconditional theorem) this gives the amplifier so much mass that the `off-diagonal' term is worse than the trivial bound.", "The method also depends on the exponent of $\\kappa $ in Proposition REF being small, and fails to improve the $L^\\infty $ bound of [13] under the assumption (REF ) because the corresponding exponent in that case is larger." ], [ "Oscillatory Integrals When $\\lambda \\sim t$", "In this section, we prove Proposition REF by building up the integral $I(t, \\phi , \\ell _1, \\ell _2)$ in several steps.", "We begin with two calculations that we shall use repeatedly in this section and in Section .", "Lemma 6.1 Fix $g \\in PSL_2(\\mathbb {R})$ , and define $\\sigma : \\mathbb {R}/ 2\\pi \\mathbb {Z}\\rightarrow \\mathbb {R}/ 2\\pi \\mathbb {Z}$ by $k(\\theta ) g \\in NA k(\\sigma (\\theta ))$ .", "Then $\\sigma $ is a diffeomorphism.", "By using the Cartan decomposition, we may reduce to the case where $g = a(y)$ .", "Taking inverses gives $a(-y) k(-\\theta ) \\in k(-\\sigma (\\theta )) AN$ , and applying both sides to the point at infinity gives $e^{-y} \\cot (\\theta /2) = \\cot (\\sigma (\\theta )/2).$ This proves that $\\sigma $ is a bijection, and a diffeomorphism everywhere except at $\\theta = 0$ .", "Rewriting the equation as $e^y \\tan (\\theta /2) = \\tan ( \\sigma (\\theta )/2)$ proves it at $\\theta = 0$ also.", "Lemma 6.2 Let $g \\in PSL_2(\\mathbb {R})$ have Iwasawa decomposition $g = n a k(\\theta )$ .", "Then $\\frac{\\partial }{\\partial t} A( g a(t)) \\Big |_{t=0} = \\cos \\theta .$ If $H$ is as in (REF ), we have $A( g a(t)) & = A(a) + A( k(\\theta ) \\exp (tH) k(-\\theta ) ) \\\\& = A(a) + A( \\exp ( t \\text{Ad}(k(\\theta )) H) ),$ and therefore $\\frac{\\partial }{\\partial t} A( g a(t)) \\Big |_{t=0} & = H^*( \\text{Ad}(k(\\theta )) H ) \\\\& = \\cos \\theta .$" ], [ "Uniformisation results", "We shall need the following two uniformisation lemmas for the function $A$ .", "Lemma 6.3 Let $D > 0$ .", "There exists $\\delta > 0$ , $\\sigma > 0$ , and a real analytic function $\\xi : (-\\delta , \\delta ) \\times (-D,D)^2 \\rightarrow \\mathbb {R}$ such that $\\frac{\\partial }{\\partial y} A( k(\\theta ) n(x) a(y) ) = 1 - \\theta ^2 \\xi (\\theta , x, y)$ and $|\\xi (\\theta , x, y)| & \\ge \\sigma \\\\\\left| \\frac{\\partial ^n \\xi }{\\partial y^n} (\\theta , x, y) \\right| & \\ll _n 1$ for $(\\theta ,x,y) \\in (-\\delta , \\delta ) \\times (-D,D)^2$ .", "Define the function $\\alpha (\\theta , x, y) : \\mathbb {R}/ 2\\pi \\mathbb {Z}\\times (-2D, 2D)^2 \\rightarrow \\mathbb {R}/ 2\\pi \\mathbb {Z}$ by requiring that $k(\\theta ) n(x) a(y) \\in NA k(\\alpha (\\theta , x, y)).$ The analyticity of the Iwasawa decomposition implies that $\\alpha $ is analytic as a function of $(\\theta , x, y)$ .", "Lemma REF implies that $1 - \\frac{\\partial }{\\partial y} A( k(\\theta ) n(x) a(y) ) & = 1 - \\cos \\alpha \\\\& = 2 \\sin ^2(\\alpha /2).$ We choose $\\delta $ such that $\\sin (\\alpha /2)$ vanishes on $(-2\\delta , 2\\delta ) \\times (-2D,2D)^2$ iff $\\theta = 0$ .", "Lemma REF implies that $\\partial \\alpha / \\partial \\theta $ never vanishes on $\\lbrace 0 \\rbrace \\times (-2D, 2D)^2$ , and so because $\\alpha $ was analytic we see that there is a real analytic function $\\xi _0$ on $(-2\\delta , 2\\delta ) \\times (-2D,2D)^2$ such that $\\sin (\\alpha /2) = \\theta \\xi _0$ .", "Defining $\\xi = 2 \\xi _0^2$ and restricting the domain to $(-\\delta , \\delta ) \\times (-D, D)^2$ gives the result.", "Lemma 6.4 If $I \\subset \\mathbb {R}$ is a bounded open interval, there exists $\\delta > 0$ and a function $\\xi : I \\times (-\\delta , \\delta ) \\rightarrow \\mathbb {R}$ such that $\\xi (y,0) = 0$ for all $y \\in I$ , $A(k(\\theta ) a(y)) = y - y \\xi ^2(y,\\theta ),$ and the map $\\Xi : (y, \\theta ) \\mapsto (y, \\xi (y,\\theta ))$ gives a real-analytic diffeomorphism $\\Xi : I \\times (-\\delta , \\delta ) \\simeq U \\subset \\mathbb {R}^2$ .", "This follows in the same way as Lemma REF above, or Theorem 4.6 of [15]." ], [ "Constituent integrals of $I(t, \\phi , \\ell _1, \\ell _2)$", "We now estimate two one-dimensional integrals that appear in $I(t, \\phi , \\ell _1, \\ell _2)$ .", "Proposition 6.5 Let $C$ , $D$ and $\\epsilon $ be positive constants, and let $b \\in C^\\infty _0(\\mathbb {R})$ be a function supported in $[0,1]$ .", "If $x, y \\in [-D, D]$ and $|\\theta | \\ge C s^{-1/2 + \\epsilon } \\beta ^{1/2} \\quad \\text{and} \\quad |\\lambda - s| \\le \\beta $ for some $s$ and $\\beta $ satisfying $1 \\le \\beta \\ll s^{2/3}$ , then $\\int _{-\\infty }^\\infty b(z) \\exp ( i\\lambda z - is A(k(\\theta ) n(x) a(y + z))) dz \\ll _{A} s^{-A}$ uniformly in $\\lambda $ and $\\beta $ .", "By applying Lemma REF , we see that there is some $\\delta > 0$ and a nonvanishing real analytic function $\\xi $ on $(-\\delta , \\delta ) \\times (-D-2, D+2)^2$ such that $\\frac{\\partial }{\\partial z} A( k(\\theta ) n(x) a(y+z) ) = 1 - \\theta ^2 \\xi (\\theta , x, y+z)$ when $\\theta \\in (-\\delta , \\delta )$ , $x,y \\in [-D, D]$ and $z \\in [0,1]$ .", "If $Z(\\theta , x, y)$ is an antiderivative of $\\xi $ with respect to $y$ , we may integrate this to obtain $A( k(\\theta ) n(x) a(y) ) = y - \\theta ^2 Z(\\theta , x, y) + c(x,\\theta ).$ If $\\theta \\in (-\\delta , \\delta )$ , we may use this to rewrite the integral (REF ) as $\\int _{-\\infty }^\\infty b(z) \\exp ( i\\lambda z - is A( k(\\theta ) n(x) a(y+z) ) ) dy & = e^{i c(\\theta , x) - isy} \\int _{-\\infty }^\\infty b(z) \\exp ( i(\\lambda - s)z + is \\theta ^2 Z(\\theta , x, y+z)) dz \\\\& = e^{i c(\\theta , x) - isy} \\int _{-\\infty }^\\infty b(z) \\exp ( is \\theta ^2 \\Psi (z) ) dz,$ where we define $\\Psi (z) = Z(\\theta , x, y+z) + s^{-1} \\theta ^{-2}(s-\\lambda ) z$ .", "Our assumption (REF ) implies that $| s^{-1} \\theta ^{-2}(s-\\lambda ) | \\le s^{-1} \\theta ^{-2} \\beta \\ll s^{-2\\epsilon },$ so that $\\Psi = Z(\\theta , x, y+z) + O(s^{-2\\epsilon })z, \\quad \\text{and} \\quad \\frac{\\partial \\Psi }{\\partial z} = \\xi (\\theta , x, y+z) + O(s^{-2\\epsilon }).$ It follows from (REF ) and (REF ) that for $s$ sufficiently large, $|\\partial \\Psi / \\partial z| > \\sigma /2$ for all $\\theta \\in (-\\delta , \\delta )$ , $x,y \\in [-D, D]$ and $z \\in [0,1]$ .", "As (REF ) implies that $s \\theta ^2 \\gg s^{2\\epsilon } \\beta \\ge s^{2\\epsilon }$ , the bound (REF ) follows by integration by parts in (REF ).", "In the case where $\\theta \\notin (-\\delta , \\delta )$ , Lemma REF implies that $(\\partial / \\partial z) A( k(\\theta ) n(x) a(y+z) ) \\le 1 - c_1$ for some $c_1 > 0$ depending only on $\\delta $ , which gives $\\frac{\\partial }{\\partial z} ( i\\lambda s^{-1} z - A( k(\\theta ) n(x) a(y+z) ) \\gg 1.$ The result now follows by integration by parts.", "The second one-dimensional integral that we shall estimate is as follows.", "Proposition 6.6 Let $C$ , $D$ and $\\epsilon $ be positive constants, and let $b \\in C^\\infty _0(\\mathbb {R})$ be a function supported in $[0,1]$ .", "If $x, y \\in [-D, D]$ and $|x| \\ge C s^{-1/2 + \\epsilon } \\beta ^{1/2} \\quad \\text{and} \\quad |\\lambda - s| \\le \\beta $ for some $s$ and $\\beta $ satisfying $1 \\le \\beta \\ll s^{2/3}$ , then $\\int _{-\\infty }^\\infty b(z) e^{i\\lambda z} \\varphi _{-s}(n(x) a(y+z)) dz \\ll _{A} s^{-A}$ uniformly in $\\lambda $ and $\\beta $ .", "If we substitute the formula for $\\varphi _{-s}$ as an integral of plane waves into the LHS of (REF ), it becomes $\\int _{-\\infty }^\\infty \\int _0^{2\\pi } b(z) \\exp ( i\\lambda z + (1/2 - is) A( k(\\theta ) n(x) a(y+z) ) ) d\\theta dz.$ Let $f(x) \\in C^\\infty _0(\\mathbb {R})$ be a function with $\\text{supp}(f) \\subseteq [-2,2]$ and $f(x) = 1$ on $[-1,1]$ .", "Let $C_1$ be a positive constant to be chosen later.", "Define $b_1$ by $b_1(x) = f( C_1^{-1} s^{1/2 -\\epsilon } \\beta ^{-1/2} x)$ and set $b_2 = 1 - b_1$ , so that $1 = b_1(\\theta ) + b_2(\\theta )$ is a smooth partition of unity on $\\mathbb {R}/ 2\\pi \\mathbb {Z}$ with $\\text{supp}(b_1) & \\subseteq [-2C_1 s^{-1/2 +\\epsilon } \\beta ^{1/2}, 2C_1 s^{-1/2 +\\epsilon } \\beta ^{1/2}], \\\\\\text{supp}(b_2) & \\subseteq \\mathbb {R}/ 2\\pi \\mathbb {Z}\\setminus [-C_1 s^{-1/2 +\\epsilon } \\beta ^{1/2}, C_1 s^{-1/2 +\\epsilon } \\beta ^{1/2}].$ Proposition REF implies that $\\int _{-\\infty }^\\infty \\int _0^{2\\pi } b_2(\\theta ) b(z) \\exp ( i\\lambda z + (1/2 - is) A( k(\\theta ) n(x) a(y+z) ) ) d\\theta dz \\ll _A s^{-A},$ so that it suffices to estimate $\\int _{-\\infty }^\\infty \\int _0^{2\\pi } b_1(\\theta ) b(z) \\exp ( i\\lambda z + (1/2 - is) A( k(\\theta ) n(x) a(y+z) ) ) d\\theta dz.$ We shall do this by estimating the integrals $\\int _0^{2\\pi } b_1(\\theta ) \\exp ( - is A( k(\\theta ) n(x) a(y) ) ) d\\theta $ in $\\theta $ , where now $y \\in [-D, D+1]$ .", "If $X \\in \\mathfrak {g}$ , we let $X^*$ be the vector field on $\\mathbb {H}$ whose value at $p$ is $\\tfrac{\\partial }{\\partial t} \\exp (tX) p |_{t=0}$ .", "It may be shown that these vector fields satisfy $[X^*, Y^*] = -[X,Y]^*$ , where the first Lie bracket is on $\\mathbb {H}$ and the second is in $\\mathfrak {g}$ .", "We recall the vectors $X_\\mathfrak {n}$ and $X_\\mathfrak {k}$ defined in (REF ).", "It may be easily seen that the subset of $\\mathbb {H}$ where $X_\\mathfrak {k}^* A$ vanishes is exactly $A$ , and the following lemma implies that it vanishes to first order there.", "Lemma 6.7 We have $X_\\mathfrak {n}^* X_\\mathfrak {k}^* A(a(y)) = e^y$ for all $y$ .", "We have $X_\\mathfrak {n}^* X_\\mathfrak {k}^* A = X_\\mathfrak {k}^* X_\\mathfrak {n}^* A + [X_\\mathfrak {n}^*, X_\\mathfrak {k}^*] A.$ It may be seen that the first term vanishes, and we have $[X_\\mathfrak {n}^*, X_\\mathfrak {k}^*] = - [X_\\mathfrak {n}, X_\\mathfrak {k}]^* = H^*$ which implies the lemma.", "Lemma REF implies that there exist $\\sigma $ , $\\delta > 0$ such that if $|x| < \\sigma $ and $y \\in [-D-1, D+2]$ then we have $|X_\\mathfrak {k}A(n(x)a(y))| \\ge \\delta |x|$ .", "Define $B & = \\lbrace n(x) a(y) | \\; |x| \\le \\sigma /2, \\, y \\in [-D, D+1] \\rbrace , \\\\B^{\\prime } & = \\lbrace n(x) a(y) | \\; |x| < \\sigma , \\, y \\in [-D-1, D+2] \\rbrace .$ Let $p \\in B$ and assume that $|N(p)| \\ge C s^{-1/2+\\epsilon } \\beta ^{1/2}$ , where $N(p)$ is as in (REF ).", "If $s$ is sufficiently large and $C_1$ sufficiently small, and $|\\theta | \\le 2 C_1 s^{-1/2 +\\epsilon } \\beta ^{1/2}$ , we have $k(\\theta )p \\in B^{\\prime }$ and $|N(k(\\theta )p)| \\gg s^{-1/2+\\epsilon } \\beta ^{1/2}$ .", "It follows that $\\Big | \\frac{\\partial }{\\partial \\theta ^{\\prime }} A( k(\\theta ^{\\prime }) p) \\Big |_{\\theta ^{\\prime } = \\theta } \\Big | = | X_\\mathfrak {k}^* A( k(\\theta ) p) | \\ge \\delta |N( k(\\theta )p)| \\gg s^{-1/2+\\epsilon } \\beta ^{1/2}$ when $|\\theta | \\le 2 C_1 s^{-1/2 +\\epsilon } \\beta ^{1/2}$ .", "The proposition now follows by integration by parts.", "If $p = n(x) a(y)$ with $x \\in [-D, D]$ and $y \\in [-D, D+1]$ and $p \\notin B$ , then we have $|X_\\mathfrak {k}^* A(k(\\theta )p) | \\ge \\delta $ when $|\\theta | \\le 2 C_1 s^{-1/2 +\\epsilon } \\beta ^{1/2}$ and $s$ is sufficiently large.", "The proposition again follows by integration by parts." ], [ "Proof of Proposition ", "We now combine Propositions REF and REF to bound the integral (REF ) below, which will imply Proposition REF after integrating in the various spectral parameters.", "Proposition 6.8 Let $\\ell \\subset \\mathbb {H}$ be a unit geodesic segment with parametrisation $\\ell : [0,1] \\rightarrow \\mathbb {H}$ .", "Let $D \\subset PSL_2(\\mathbb {R})$ be a compact set, let $b_1, b_2 \\in C^\\infty _0(\\mathbb {R})$ be functions supported in $[0,1]$ , and let $\\epsilon > 0$ be given.", "If $g \\in D$ and $\\lambda _1$ , $\\lambda _2 \\in \\mathbb {R}$ satisfy $n(\\ell , g\\ell ) \\ge s^{-1/2 + \\epsilon } \\beta ^{1/2} \\quad \\text{and} \\quad \\lambda _i \\in [s-\\beta , s+\\beta ]$ for some $s$ and $1 \\le \\beta \\ll s^{2/3}$ , then $\\iint _{-\\infty }^\\infty b_1(x_1) b_2(x_2) e^{i(\\lambda _1 x_1 - \\lambda _2 x_2)} \\varphi _{-s}(\\ell (x_1), g\\ell (x_2)) dx_1 dx_2 \\ll _{A, \\epsilon } s^{-A}$ uniformly in $\\lambda _i$ and $\\beta $ .", "We begin by expressing $\\varphi _{-s}$ as an integral of plane waves.", "For $y, z \\in \\mathbb {H}$ we have $\\varphi _s(y,z) = \\int _0^{2\\pi } \\exp ( (1/2 - is)( A(k_z(\\sigma )y) - A( k_z(\\sigma )z)) d\\sigma ,$ where $K_z$ is the stabilizer of $z$ and $k_z : \\mathbb {R}/ 2\\pi \\mathbb {Z}\\rightarrow K_z$ is a parametrisation.", "Define the function $\\theta : \\mathbb {R}/ 2\\pi \\mathbb {Z}\\rightarrow \\mathbb {R}/ 2\\pi \\mathbb {Z}$ by $k_z(\\sigma ) \\in NA k(\\theta (\\sigma )).$ Lemma REF implies that $\\theta $ is a diffeomorphism.", "Because $A(k_z(\\sigma )y) - A( k_z(\\sigma )z) = A( k(\\theta (\\sigma )) y) - A( k(\\theta (\\sigma )) z)$ , we have $\\varphi _s(y,z) = \\int \\exp ( (1/2 - is)( A(k(\\theta )y) - A( k(\\theta )z)) \\frac{d\\sigma }{d\\theta } d\\theta .$ We may assume that $\\ell $ is the segment with one endpoint at $i$ and pointing upwards, so that $\\ell (x) = a(x) i$ .", "Substituting (REF ) into (REF ) gives $\\iint _{-\\infty }^\\infty b_1(x_1) b_2(x_2) e^{i(\\lambda _1 x_1 - \\lambda _2 x_2)} \\varphi _{-s}(\\ell (x_1), g\\ell (x_2)) dx_1 dx_2 = \\\\\\iint _{-\\infty }^\\infty \\int _0^{2\\pi } b_1(x_1) b_2(x_2) e^{i(\\lambda _1 x_1 - \\lambda _2 x_2)} \\exp ( (1/2 - is)( A(k(\\theta ) a(x_1)) - A( k(\\theta )g a(x_2))) \\frac{d\\sigma }{d\\theta } d\\theta dx_1 dx_2.$ Let $g = k(\\theta ^{\\prime })n(x^{\\prime }) a(y^{\\prime })$ , where $x^{\\prime }$ and $y^{\\prime }$ are bounded in terms of $D$ .", "We then have $k(\\theta )g a(x_2) = k(\\theta + \\theta ^{\\prime }) n(x^{\\prime }) a(y + y^{\\prime })$ .", "We integrate the RHS of (REF ) with respect to $x_1$ and $x_2$ with $\\theta $ fixed.", "Choose a constant $C > 0$ .", "If $\\theta \\notin [-C s^{-1/2+\\epsilon } \\beta ^{1/2}, C s^{-1/2+\\epsilon } \\beta ^{1/2}]$ , then Proposition REF implies that the integral is $\\ll s^{-A}$ , and likewise if $\\theta + \\theta ^{\\prime } \\notin [-C s^{-1/2+\\epsilon } \\beta ^{1/2}, C s^{-1/2+\\epsilon } \\beta ^{1/2}]$ .", "Combining these, we see that (REF ) will be $\\ll s^{-A}$ unless $|\\theta ^{\\prime }| \\le 2C s^{-1/2+\\epsilon } \\beta ^{1/2}$ .", "If $C$ is chosen sufficiently small, the condition $|\\theta ^{\\prime }| \\le 2C s^{-1/2+\\epsilon } \\beta ^{1/2}$ and our assumption that $n(\\ell , g\\ell ) \\ge s^{-1/2 + \\epsilon } \\beta ^{1/2}$ imply that $\\ell $ and $g \\ell $ are separated in the sense that there is a $C_1 > 0$ such that $d(p, g\\ell ) \\ge C_1 s^{-1/2 + \\epsilon } \\beta ^{1/2}$ for all $p \\in \\ell $ .", "The result now follows by applying Proposition REF to the integral of the LHS of (REF ) over $x_2$ for each fixed $x_1$ .", "Corollary 6.9 Let $\\ell \\subset \\mathbb {H}$ be a unit geodesic segment with parametrisation $\\ell : [0,1] \\rightarrow \\mathbb {H}$ .", "Let $D \\subset PSL_2(\\mathbb {R})$ be a compact set, and let $b_1, b_2 \\in C^\\infty _0(\\mathbb {R})$ be functions supported in $[0,1]$ .", "Let $\\epsilon > 0$ , $s > 0$ , and $1 \\le \\beta \\ll s^{2/3}$ be given.", "Let $\\phi \\in L^2(\\mathbb {R})$ be a function with $\\Vert \\phi \\Vert _2 = 1$ and such that $\\textup {supp}(\\widehat{\\phi }) \\subseteq [s-\\beta , s+\\beta ].$ If $g \\in D$ and $n(\\ell , g\\ell ) \\ge s^{-1/2 + \\epsilon } \\beta ^{1/2}$ , then $\\iint _{-\\infty }^\\infty b_1(x_1) b_2(x_2) \\phi (x_1) \\overline{\\phi (x_2)} \\varphi _{-s}(\\ell (x_1), g\\ell (x_2)) dx_1 dx_2 \\ll _{A, \\epsilon } s^{-A},$ where the implied constant is independent of $\\phi $ and $\\beta $ .", "This follows immediately from Proposition REF after inverting the Fourier transform of $\\phi $ and noting that $\\Vert \\widehat{\\phi } \\Vert _1 \\le \\Vert \\widehat{\\phi } \\Vert _2 (2\\beta )^{1/2} = (2\\pi )^{1/2} (2\\beta )^{1/2}$ .", "To prove the bound (REF ), observe that equation (REF ) implies that $\\int _{-\\infty }^\\infty |K_t(\\ell _1(x), p)|^2 dx \\ll t$ uniformly for $p \\in \\mathbb {H}$ .", "It follows that $K_t(\\ell _1(x_1), \\ell _2(x_2))$ has norm $\\ll t^{1/2}$ as an element of $L^2(\\mathbb {R}^2)$ , and the result follows by Cauchy-Schwarz.", "We now prove (REF ).", "Fix a unit geodesic segment $\\ell $ .", "We may assume without loss of generality that $\\ell _1 = \\ell $ , and we choose $g \\in PSL_2(\\mathbb {R})$ so that $g \\ell = \\ell _2$ .", "The assumption that $d(\\ell _1, \\ell _2) \\le 1$ implies that $g$ lies in the compact set $D := \\lbrace g \\in PSL_2(\\mathbb {R}) | d(\\ell , g \\ell ) \\le 1 \\rbrace $ .", "We have $I(t, \\phi , \\ell , g\\ell ) = \\iint _{-\\infty }^\\infty b(x_1) b(x_2) \\phi (x_1) \\overline{\\phi (x_2)} K_t(\\ell (x_1), g\\ell (x_2)) dx_1 dx_2.$ Inverting the Harish-Chandra transform of $k_t$ gives $I(t, \\phi , \\ell , g\\ell ) = \\frac{1}{2\\pi } \\iint _{-\\infty }^\\infty \\int _0^\\infty b(x_1) b(x_2) \\phi (x_1) \\overline{\\phi (x_2)} h_t^2(s) \\varphi _{-s}(\\ell (x_1), g \\ell (x_2)) s \\tanh (\\pi s) ds dx_1 dx_2.$ If we assume without loss of generality that $\\beta > t^\\epsilon $ , then we may restrict the domain of the Harish-Chandra transform to $[t-\\beta , t+\\beta ]$ as in Section REF to obtain $I(t, \\phi , \\ell , g\\ell ) = \\frac{1}{2\\pi } \\iint _{-\\infty }^\\infty \\int _{t-\\beta }^{t+\\beta } b(x_1) b(x_2) \\phi (x_1) \\overline{\\phi (x_2)} \\\\h_t^2(s) \\varphi _{-s}(\\ell (x_1), g \\ell (x_2)) s \\tanh (\\pi s) ds dx_1 dx_2 + O(t^{-A}).$ Applying Corollary REF with $2\\beta $ in place of $\\beta $ completes the proof." ], [ "Oscillatory Integrals When $\\lambda < t$", "We now prove Proposition REF .", "In this section, we assume that all geodesics we consider carry an orientation.", "When we refer to the unit tangent vector to a geodesic at a point, we shall always mean in the direction of its orientation.", "If $\\ell _1$ and $\\ell _2$ are two intersecting geodesics, we shall denote by $\\angle (\\ell _1, \\ell _2)$ the angle between their unit tangent vectors at the point of intersection measured in the counterclockwise direction from $\\ell _1$ to $\\ell _2$ .", "Let $\\ell $ be the vertical geodesic through $i$ .", "By slight abuse of notation, we take $a : \\mathbb {R}\\rightarrow \\ell $ to be a parametrisation of $\\ell $ , and define $\\ell _0 = a([0,1])$ which is a unit segment contained in $\\ell $ .", "We give the geodesic $\\ell $ the upwards-pointing orientation, which we transfer to $g \\ell $ for $g \\in PSL_2(\\mathbb {R})$ .", "As in the proof of Proposition REF , it suffices to bound the integral $I(s, \\lambda , g) = \\iint _{-\\infty }^\\infty e^{i \\lambda (x_1-x_2)} b_1(x_1)b_2(x_2) \\varphi _{-s}( \\ell (x_1), g\\ell (x_2)) dx_1 dx_2.$ After substituting the expression (REF ) for $\\varphi _{-s}( \\ell (x_1), g\\ell (x_2))$ , we obtain an oscillatory integral in the variables $\\theta $ , $x_1$ , and $x_2$ with phase function $\\phi (x_1, x_2, \\theta , g, \\rho ) = \\rho (x_1 - x_2) - A( k(\\theta ) \\ell (x_1)) + A( k(\\theta ) g \\ell (x_2)),$ where $\\rho = \\lambda / s \\ge 0$ .", "We first assume that $\\rho \\in [\\delta , 1-\\delta ]$ for some $1/2 > \\delta > 0$ .", "Define $\\alpha \\in [0,\\pi /2]$ to be the solution to $\\cos \\alpha = \\rho $ , which is bounded away from 0 and $\\pi /2$ .", "We shall study the critical points of $\\phi $ in Sections REF to REF , before deriving a bound for $I(s, \\lambda , g)$ from our results in Section REF .", "We shall write $\\phi (x_1, x_2, \\theta )$ when $g$ and $\\rho $ are not varying." ], [ "The critical points of $\\phi $", "Lemma 7.1 The phase function $\\phi $ has a critical point at $(x_1, x_2, \\theta , g, \\rho )$ exactly when $k(\\theta ) \\ell (x_1)$ and $k(\\theta ) g \\ell (x_2)$ lie on the same vertical geodesic $v$ , which we give the upwards-pointing orientation, and we have $\\angle (v, k(\\theta ) \\ell ), \\angle (v, k(\\theta ) g \\ell ) \\in \\lbrace \\pm \\alpha \\rbrace $ .", "Suppose that $(x_1^{\\prime }, x_2^{\\prime }, \\theta ^{\\prime })$ is a critical point of $\\phi $ .", "Define the functions $x(\\theta )$ , $y(\\theta )$ and $\\beta (\\theta )$ by $k(\\theta ) a(x_1^{\\prime }) = n(x(\\theta )) a( y(\\theta )) k(\\beta (\\theta )),$ and let $n^{\\prime } = n(x(\\theta ^{\\prime }))$ and $\\beta ^{\\prime } = \\beta (\\theta ^{\\prime })$ .", "It may be seen that $v := n(x^{\\prime }) \\ell $ is the upwards-pointing geodesic through $k(\\theta ^{\\prime }) \\ell (x_1^{\\prime })$ , and that $\\beta ^{\\prime } = \\angle (v, k(\\theta ^{\\prime }) \\ell )$ .", "Lemma REF then implies that $\\beta ^{\\prime } = \\pm \\alpha $ .", "The calculation in the case of $\\partial / \\partial x_2$ is identical.", "We have $A( k(\\theta ) a(x_1^{\\prime })) - A( k(\\theta ) g a(x_2^{\\prime })) & = A( k(\\theta ) a(x_1^{\\prime })) - A( k(\\theta ) a(x_1^{\\prime }) a(x_1^{\\prime })^{-1} g a(x_2^{\\prime })) \\\\& = y(\\theta ) - A( a(y(\\theta )) k(\\beta (\\theta )) a(x_1^{\\prime })^{-1} g a(x_2^{\\prime })) \\\\& = - A( k(\\beta (\\theta )) a(x_1^{\\prime })^{-1} g a(x_2^{\\prime }))$ and so $\\frac{\\partial \\phi }{\\partial \\theta } (x_1^{\\prime }, x_2^{\\prime }, \\theta ^{\\prime }) & = \\frac{\\partial }{\\partial \\theta } A( k(\\beta (\\theta )) a(x_1^{\\prime })^{-1} g a(x_2^{\\prime })) \\Big |_{\\theta = \\theta ^{\\prime }} \\\\& = \\frac{\\partial \\beta }{\\partial \\theta } (\\theta ^{\\prime }) \\frac{\\partial }{\\partial \\beta } A( k(\\beta ) a(x_1^{\\prime })^{-1} g a(x_2^{\\prime })) \\Big |_{\\beta = \\beta ^{\\prime }}.$ Because $\\partial \\beta /\\partial \\theta $ does not vanish by Lemma REF , and $\\frac{\\partial }{\\partial \\theta } A( k(\\theta ) g ) \\Big |_{\\theta =0} = 0$ iff $g \\in AK$ , we have $\\partial \\phi / \\partial \\theta = 0$ iff $k(\\beta ^{\\prime }) a(x_1^{\\prime })^{-1} g a(x_2^{\\prime }) \\in AK$ , i.e.", "$k(\\beta ^{\\prime }) a(x_1^{\\prime })^{-1} g a(x_2^{\\prime })i$ lies on the vertical geodesic through the origin.", "Because $k(\\beta ^{\\prime }) a(x_1)^{-1} = a^{\\prime -1} n^{\\prime -1} k(\\theta ^{\\prime })$ , this is equivalent to the condition that $k(\\theta ^{\\prime }) g a(x_2^{\\prime }) \\in n^{\\prime } AK$ , or that $k(\\theta ^{\\prime }) g \\ell (x_2^{\\prime })$ lies on the vertical geodesic $v$ passing through $k(\\theta ^{\\prime }) \\ell (x_1^{\\prime })$ .", "We finish with an observation that will be useful in calculating the Hessian of $\\phi $ .", "We have $k(\\beta ^{\\prime }) a(x_1^{\\prime })^{-1} g a(x_2^{\\prime }) \\in a(h)K$ for some $h \\in \\mathbb {R}$ , and it may be seen that $k(\\theta ^{\\prime }) \\ell (x_1^{\\prime }) \\in n^{\\prime } a^{\\prime } K$ and $k(\\theta ^{\\prime }) g \\ell (x_2^{\\prime }) \\in n^{\\prime } a^{\\prime } a(h) K$ , so that $h$ is the signed distance from $k(\\theta ^{\\prime }) \\ell (x_1^{\\prime })$ to $k(\\theta ^{\\prime }) g \\ell (x_2^{\\prime })$ along $v$ .", "Given a pair of geodesics $\\ell _1$ and $\\ell _2$ , we say that a geodesic $j$ is a critical geodesic for $(\\ell _1, \\ell _2)$ if $j$ meets $\\ell _1$ and $\\ell _2$ at angles of $\\pm \\alpha $ .", "We may therefore rephrase Lemma REF as saying that $(x_1, x_2, \\theta , g, \\rho )$ is a critical point of $\\phi $ exactly when $(\\ell , g\\ell )$ has a critical geodesic $j$ , $\\ell (x_1)$ and $g \\ell (x_2)$ both lie on $j$ , and $k(\\theta )j$ is vertical.", "As in Lemma REF , we define the aperture of a critical point to be the signed distance from $\\ell (x_1)$ to $g \\ell (x_2)$ on the geodesic $j$ .", "We shall now calculate the Hessian of $\\phi $ at its critical points.", "Let $(x_1^{\\prime }, x_2^{\\prime }, \\theta ^{\\prime })$ be a critical point of $\\phi $ , and define functions $\\beta _i(\\theta )$ by $k(\\theta ) a(x_1^{\\prime }) \\in NA k(\\beta _1(\\theta )), \\quad k(\\theta ) g a(x_2^{\\prime }) \\in NA k(\\beta _2(\\theta )).$ We let $\\beta _i^{\\prime } = \\beta _i(\\theta ^{\\prime })$ .", "It follows from Lemma REF that $\\beta _i^{\\prime } \\in \\lbrace \\pm \\alpha \\rbrace $ .", "Let $h$ be the aperture of the critical point, so that $k(\\beta _1^{\\prime }) a(x_1^{\\prime })^{-1} g a(x_2^{\\prime }) \\in \\left( \\begin{array}{cc} e^{h} & 0 \\\\ 0 & 1 \\end{array} \\right) K.$ We define $\\kappa = \\tfrac{\\partial \\beta _1}{\\partial \\theta } (\\theta ^{\\prime })$ , which is nonzero by Lemma REF .", "The Hessian of $\\phi $ at $(x_1^{\\prime }, x_2^{\\prime }, \\theta ^{\\prime })$ is given by the following proposition.", "Proposition 7.2 The Hessian of $\\phi $ at $(x_1^{\\prime }, x_2^{\\prime }, \\theta ^{\\prime })$ with respect to the co-ordinates $(x_1, x_2, \\theta )$ is $D = \\left( \\begin{array}{ccc} \\tfrac{1}{2} \\sin ^2 \\alpha & 0 & \\kappa \\sin \\beta _1^{\\prime } \\\\ 0 & -\\tfrac{1}{2} \\sin ^2 \\alpha & -\\kappa e^h \\sin \\beta _2^{\\prime } \\\\ \\kappa \\sin \\beta _1^{\\prime } & -\\kappa e^h \\sin \\beta _2^{\\prime } & \\kappa ^2 (1 - e^{2h})/2 \\end{array} \\right)$ The determinant of $D$ is $|D| = \\frac{3}{8} \\kappa ^2 \\sin ^4 \\alpha (1 - e^{2h}),$ which is nonzero unless $h = 0$ , i.e.", "the points $\\ell (x_1^{\\prime })$ and $g\\ell (x_2^{\\prime })$ coincide in $\\mathbb {H}$ .", "It is clear that $\\partial ^2 \\phi / \\partial x_1 \\partial x_2$ is identically 0.", "To calculate $\\partial ^2 \\phi / \\partial x_1^2$ , define $\\gamma : \\mathbb {R}\\rightarrow \\mathbb {R}/ 2\\pi \\mathbb {Z}$ by the condition that $k(\\theta ^{\\prime }) a(x_1^{\\prime } + t) \\in NA k (\\gamma (t))$ .", "Our assumption that we are at a critical point implies that $\\gamma (0) = \\beta _1^{\\prime } = \\pm \\alpha $ .", "Lemma REF gives $\\frac{\\partial }{\\partial t} \\phi (x_1^{\\prime } + t, x_2^{\\prime }, \\theta ^{\\prime }) = \\rho - \\cos \\gamma (t),$ and $\\frac{\\partial ^2 \\phi }{\\partial x_1^2} (x_1^{\\prime }, x_2^{\\prime }, \\theta ^{\\prime }) = \\sin \\beta _1^{\\prime } \\frac{\\partial \\gamma }{\\partial t} (0).$ We have $k(\\theta ^{\\prime }) a(x_1^{\\prime } + t) & \\in NA k (\\gamma (t)) \\\\NA k( \\beta _1^{\\prime }) a(t) & = NA k (\\gamma (t)) \\\\k( \\beta _1^{\\prime }) a(t) & \\in NA k (\\gamma (t)).$ Equation (REF ) then gives $\\tan (\\gamma (t)/2) = e^t \\tan ( \\beta _1^{\\prime }/2)$ , so that $\\frac{\\partial \\gamma }{\\partial t} \\sec ^2 (\\gamma (t)/2) & = e^t \\tan ( \\beta _1^{\\prime }/2) \\\\\\frac{\\partial \\gamma }{\\partial t} (0) & = \\cos ^2 (\\beta _1^{\\prime }/2) \\tan ( \\beta _1^{\\prime }/2) \\\\& = \\frac{1}{2} \\sin \\beta _1^{\\prime }.$ Substituting this into (REF ) gives $\\frac{\\partial ^2 \\phi }{\\partial x_1^2} (x_1^{\\prime }, x_2^{\\prime }, \\theta ^{\\prime }) = \\frac{1}{2} \\sin ^2 \\beta _1^{\\prime } = \\frac{1}{2} \\sin ^2 \\alpha .$ The calculation of $\\partial ^2 \\phi / \\partial x_2^2$ is identical, with the exception of a change in sign.", "To calculate $\\partial ^2 \\phi / \\partial \\theta \\partial x_1$ , we again have $\\frac{\\partial \\phi }{\\partial x_1} (x_1^{\\prime }, x_2^{\\prime }, \\theta ) = \\rho - \\cos \\beta _1(\\theta ),$ and $\\frac{\\partial ^2 \\phi }{\\partial \\theta \\partial x_1} (x_1^{\\prime }, x_2^{\\prime }, \\theta ^{\\prime }) = \\sin \\beta _1^{\\prime } \\frac{\\partial \\beta _1}{\\partial \\theta } (\\theta ^{\\prime }) = \\kappa \\sin \\beta _1^{\\prime }.$ We likewise have $\\frac{\\partial ^2 \\phi }{\\partial \\theta \\partial x_2} \\phi (x_1^{\\prime }, x_2^{\\prime }, \\theta ^{\\prime }) = -\\sin \\beta _2^{\\prime } \\frac{\\partial \\beta _2}{\\partial \\theta } (\\theta ^{\\prime }),$ and we shall express $\\tfrac{\\partial \\beta _2}{\\partial \\theta } (\\theta ^{\\prime })$ in terms of $\\kappa $ and $h$ .", "We recall that $k( \\beta _1^{\\prime }) a(x_1^{\\prime })^{-1} g a(x_2^{\\prime }) = a(h) k(\\theta _0)$ for some $\\theta _0$ , and so $k(\\theta ) a( x_1^{\\prime }) k( -\\beta _1^{\\prime }) a(h) k(\\theta _0) = k(\\theta ) g a(x_2^{\\prime }).$ Substituting both parts of (REF ) into this gives $NA k( \\beta _1(\\theta )) k( -\\beta _1^{\\prime }) a(h) k(\\theta _0) & = NA k( \\beta _2(\\theta )) \\\\k( \\beta _1(\\theta ) - \\beta _1^{\\prime })a(h) & \\in NA k( \\beta _2(\\theta ) -\\theta _0).$ By setting $\\theta = \\theta ^{\\prime }$ we see that $\\theta _0 = \\beta _1^{\\prime }$ .", "Equation (REF ) then gives $e^h \\tan ( (\\beta _1(\\theta ) - \\beta _1^{\\prime })/2) = \\tan ( (\\beta _2(\\theta ) - \\beta _2^{\\prime })/2),$ and differentiating both sides with respect to $\\theta $ and evaluating at $\\theta = \\theta ^{\\prime }$ gives $\\frac{\\partial \\beta _2}{\\partial \\theta } (\\theta ^{\\prime }) = \\kappa e^h.$ It follows that $\\frac{\\partial ^2 \\phi }{\\partial \\theta \\partial x_2} \\phi (x_1^{\\prime }, x_2^{\\prime }, \\theta ^{\\prime }) = - \\kappa e^h \\sin \\beta _2^{\\prime }.$ To calculate $\\partial ^2 \\phi / \\partial \\theta ^2$ , we have as in (REF ) that $\\frac{\\partial \\phi }{\\partial \\theta }(x_1^{\\prime }, x_2^{\\prime }, \\theta ) = \\frac{\\partial \\beta _1}{\\partial \\theta } \\frac{\\partial }{\\partial \\beta } A( k(\\beta ) a(x_1^{\\prime })^{-1} g a(x_2^{\\prime })) \\Big |_{\\beta = \\beta _1(\\theta )}.$ Because $\\frac{\\partial }{\\partial \\beta } A( k(\\beta ) a(x_1^{\\prime })^{-1} g a(x_2^{\\prime })) \\Big |_{\\beta = \\beta _1^{\\prime }} = 0,$ we have $\\frac{\\partial ^2 \\phi }{\\partial \\theta ^2} (x_1^{\\prime }, x_2^{\\prime }, \\theta ^{\\prime }) & = \\kappa ^2 \\frac{\\partial ^2}{\\partial \\beta ^2} A( k(\\beta ) a(x_1^{\\prime })^{-1} g a(x_2^{\\prime })) \\Big |_{\\beta = \\beta _1^{\\prime }} \\\\& = \\kappa ^2 \\frac{\\partial ^2}{\\partial \\beta ^2} A( k(\\beta - \\beta _1^{\\prime }) a(h) ) \\Big |_{\\beta = \\beta _1^{\\prime }}.$ It is a standard calculation (see for instance Proposition 4.4 of [15]) that $\\frac{\\partial ^2}{\\partial \\beta ^2} A( k(\\beta ) a(h) ) \\Big |_{\\beta = 0} = (1 - e^{2h})/2,$ and this completes the proof." ], [ "The function $\\psi $", "Define $\\mathcal {P}= \\mathbb {R}/ 2\\pi \\mathbb {Z}\\times PSL_2(\\mathbb {R}) \\times [\\delta , 1-\\delta ]$ , and define $\\mathcal {S}\\subset \\mathcal {P}$ to be the set where one of the geodesics $k(\\theta ) \\ell $ and $k(\\theta ) g \\ell $ is vertical.", "Note that $\\mathcal {S}$ is closed, and contains at most 4 values of $\\theta $ for each fixed $(g, \\rho )$ .", "We may define functions $\\xi _1, \\xi _2 : \\mathcal {P}\\setminus \\mathcal {S}\\rightarrow \\mathbb {R}$ by requiring that $k(\\theta ) a(\\xi _1(\\theta , g, \\rho ))$ is the unique point on $k(\\theta )\\ell $ at which the tangent vector to the geodesic makes an angle of $\\alpha $ with the upward pointing vector, and likewise for $\\xi _2(\\theta , g, \\rho )$ and $k(\\theta ) g \\ell $ .", "As $\\xi _1$ does not depend on $g$ , we will omit this argument of the function.", "We have $k(\\theta ) a(\\xi _1(\\theta , \\rho )) \\in NA k(\\epsilon _1 \\alpha ), \\quad k(\\theta ) g a(\\xi _2(\\theta , g, \\rho )) \\in NA k(\\epsilon _2 \\alpha )$ for $\\epsilon _i \\in \\lbrace \\pm 1 \\rbrace $ , and so equation (REF ) gives $e^{\\xi _1(\\theta , \\rho )} \\tan (\\theta /2) = \\tan ( \\epsilon _1 \\alpha /2), \\quad e^{\\xi _2(\\theta , g, \\rho )} \\tan (\\theta /2) = \\tan ( \\epsilon _2 \\alpha /2).$ Moreover, it may be seen that $\\epsilon _1 = 1$ iff the geodesic $k(\\theta ) \\ell $ runs from right to left in the upper half plane model of $\\mathbb {H}$ , which is equivalent to $\\theta \\in (0, \\pi )$ , and likewise for $\\epsilon _2$ .", "It follows from Lemma REF that $\\xi _1(\\theta , \\rho )$ and $\\xi _2(\\theta , g, \\rho )$ may also be characterised as the unique functions such that $\\frac{\\partial \\phi }{\\partial x_1}(\\xi _1(\\theta , \\rho ), x_2, \\theta ) = \\frac{\\partial \\phi }{\\partial x_2}(x_1, \\xi _2(\\theta , g, \\rho ), \\theta ) = 0.$ We define $\\psi : \\mathcal {P}\\setminus \\mathcal {S}& \\rightarrow \\mathbb {R}\\\\\\psi (\\theta , g, \\rho ) & = \\phi ( \\xi _1(\\theta , \\rho ), \\xi _2(\\theta , g, \\rho ), \\theta , g, \\rho ).$ Lemma 7.3 $(\\theta ^{\\prime }, g^{\\prime }, \\rho ^{\\prime })$ is a critical point of $\\psi $ exactly when $(\\xi _1(\\theta ^{\\prime }), \\xi _2(\\theta ^{\\prime }), \\theta ^{\\prime }, g^{\\prime }, \\rho ^{\\prime })$ is a critical point of $\\phi $ .", "If $(\\theta ^{\\prime }, g^{\\prime }, \\rho ^{\\prime })$ is a critical point of $\\psi $ , let $\\kappa $ and $h$ be the values associated to the corresponding critical point of $\\phi $ .", "We then have $\\frac{\\partial ^2 \\psi }{\\partial \\theta ^2} (\\theta ^{\\prime }, g^{\\prime }, \\rho ^{\\prime }) = -\\frac{3}{2} \\kappa ^2 (1 - e^{2h}).$ We shall fix $g$ and $\\rho $ , and omit them from the arguments of $\\phi $ and $\\psi $ .", "Let $D$ be the Hessian of $\\phi $ calculated in Proposition REF .", "If we apply the chain rule to $\\psi $ and substitute $\\theta = \\theta ^{\\prime }$ , we obtain $\\frac{\\partial ^2 \\psi }{\\partial \\theta ^2}(\\theta ^{\\prime }) = ( \\tfrac{ \\partial \\xi _1}{\\partial \\theta }(\\theta ^{\\prime }), \\tfrac{ \\partial \\xi _2}{\\partial \\theta }(\\theta ^{\\prime }), 1) D ( \\tfrac{ \\partial \\xi _1}{\\partial \\theta }(\\theta ^{\\prime }), \\tfrac{ \\partial \\xi _2}{\\partial \\theta }(\\theta ^{\\prime }), 1)^t.$ To calculate $\\tfrac{ \\partial \\xi _1}{\\partial \\theta }(\\theta ^{\\prime })$ and $\\tfrac{ \\partial \\xi _2}{\\partial \\theta }(\\theta ^{\\prime })$ , we differentiate (REF ) with respect to $\\theta $ and set $\\theta = \\theta ^{\\prime }$ to obtain $\\frac{\\partial ^2 \\phi }{\\partial \\theta \\partial x_1} (\\xi _1(\\theta ^{\\prime }), \\xi _2(\\theta ^{\\prime }), \\theta ^{\\prime }) + \\frac{ \\partial \\xi _1}{\\partial \\theta }(\\theta ^{\\prime }) \\frac{\\partial ^2 \\phi }{\\partial x_1^2} (\\xi _1(\\theta ^{\\prime }), \\xi _2(\\theta ^{\\prime }), \\theta ^{\\prime }) = 0.$ Substituting the second partial derivatives of $\\phi $ calculated in Proposition REF gives $\\frac{ \\partial \\xi _1}{\\partial \\theta }(\\theta ^{\\prime }) = \\frac{-2 \\kappa }{\\sin \\beta _1^{\\prime }},$ and likewise $\\frac{ \\partial \\xi _2}{\\partial \\theta }(\\theta ^{\\prime }) = \\frac{-2 \\kappa e^h}{\\sin \\beta _2^{\\prime }}.$ The lemma follows on substituting these into equation (REF ).", "It follows that the set of $(g,\\rho )$ for which the function $\\psi (\\theta , g, \\rho )$ has a degenerate critical point are exactly those for which either $\\ell = g \\ell $ or $\\angle (\\ell , g\\ell ) = \\pm 2\\alpha $ .", "Note that these two cases are distinct, as $\\alpha \\in (0, \\pi /2)$ .", "In the first case the function $\\psi (\\theta , g, \\rho )$ vanishes identically.", "In the second case, $\\psi (\\theta , g, \\rho )$ has only a single degenerate critical point, as no oriented geodesic can cross $\\ell $ and $g \\ell $ making an angle of $\\alpha $ with both except at their point of intersection.", "To determine this critical point, the condition that $\\angle (\\ell , g\\ell ) = \\pm 2\\alpha $ implies that $g \\in a(y) k(\\pm 2\\alpha ) A$ for some $y \\in \\mathbb {R}$ , so that $\\ell \\cap g \\ell = a(y)i$ .", "The angle bisector of the two geodesics at the point $a(y)i$ is $a(y) k(\\pm \\alpha ) \\ell $ , and the critical point of $\\psi (\\theta , g, \\rho )$ is the $\\theta $ such that the positive endpoint of $k(\\theta ) a(y) k(\\pm \\alpha ) \\ell $ is $i \\infty $ .", "This is equivalent to the condition $k(\\theta ) a(y) k(\\pm \\alpha ) \\in NA$ , and equation (REF ) then gives $\\cot \\theta /2 = \\mp e^y \\cot \\alpha /2$ .", "We define $\\mathcal {D}_1 & = \\lbrace (\\theta , g, \\rho ) \\in \\mathcal {P}\\setminus \\mathcal {S}| g \\in A \\rbrace \\\\\\mathcal {D}_2^\\pm & = \\lbrace (\\theta , g, \\rho ) \\in \\mathcal {P}\\setminus \\mathcal {S}| g \\in a(y) k(\\pm 2\\alpha ) A , \\cot \\theta /2 = \\mp e^y \\cot \\alpha /2 \\rbrace $ to be the three sets on which $\\psi $ has a degenerate critical point.", "We also define $\\overline{\\mathcal {P}} = PSL_2(\\mathbb {R}) \\times [\\delta , 1-\\delta ]$ , and define $\\overline{\\mathcal {D}}_1 & = A \\times [\\delta , 1-\\delta ] \\\\\\overline{\\mathcal {D}}_2^\\pm & = \\lbrace (g, \\rho ) \\in \\overline{\\mathcal {P}} | g \\in A k(\\pm 2\\alpha ) A \\rbrace $ to be the projections of $\\mathcal {D}_1$ and $\\mathcal {D}_2^\\pm $ to $\\overline{\\mathcal {P}}$ ." ], [ "The degenerate set $\\mathcal {D}_1$", "As $\\psi (\\theta , g, \\rho ) = \\psi (\\theta , ga, \\rho )$ for $a \\in A$ , we may study the degeneracy of $\\psi $ near $\\mathcal {D}_1$ by differentiating $\\psi (\\theta , \\exp (X), \\rho )$ at $X = 0$ as in the following proposition.", "Proposition 7.4 If $X = \\left( \\begin{array}{cc} 0 & X_1 \\\\ X_2 & 0 \\end{array} \\right) \\in \\mathfrak {g}$ , then $\\frac{\\partial }{\\partial t} \\psi (\\theta , \\exp (tX), \\rho ) \\Big |_{t=0} = \\epsilon \\sin \\alpha ( e^{-\\xi _2(\\theta , e, \\rho )} X_1 + e^{\\xi _2(\\theta , e, \\rho )} X_2 ),$ where $\\epsilon $ is 1 if $\\theta \\in (0,\\pi )$ and $-1$ otherwise.", "In particular, $\\partial \\psi / \\partial t (\\theta , \\exp (tX), \\rho ) |_{t=0}$ has no degenerate critical points unless $X = 0$ .", "Let $x_1^{\\prime } = \\xi _1(\\theta , \\rho )$ and $x_2^{\\prime } = x_2(\\theta , e, \\rho )$ .", "We have $\\frac{\\partial }{\\partial t} \\psi (\\theta , \\exp (tX), \\rho ) \\Big |_{t=0} & = \\frac{\\partial }{\\partial t} \\phi ( \\xi _1(\\theta , \\rho ), \\xi _2(\\theta , \\exp (tX), \\rho ), \\theta , \\exp (tX), \\rho ) \\Big |_{t=0} \\\\& = \\frac{\\partial \\phi }{\\partial x_2} ( \\xi _1(\\theta , \\rho ), \\xi _2(\\theta , e, \\rho ), \\theta , e, \\rho ) \\frac{\\partial }{\\partial t} \\xi _2(\\theta , \\exp (tX), \\rho ) \\Big |_{t=0} \\\\& \\qquad + \\frac{\\partial }{\\partial t} \\phi ( \\xi _1(\\theta , \\rho ), \\xi _2(\\theta , e, \\rho ), \\theta , \\exp (tX), \\rho ) \\Big |_{t=0}.$ The first term vanishes by (REF ), so we are left with $\\frac{\\partial }{\\partial t} \\psi (\\theta , \\exp (tX), \\rho ) \\Big |_{t=0} & = \\frac{\\partial }{\\partial t} \\phi ( \\xi _1(\\theta , \\rho ), \\xi _2(\\theta , e, \\rho ), \\theta , \\exp (tX), \\rho ) \\Big |_{t=0} \\\\& = \\frac{\\partial }{\\partial t} A( k(\\theta ) \\exp (tX) a(\\xi _2(\\theta , e, \\rho )) ) \\Big |_{t=0}.$ We shall abbreviate $\\xi _2(\\theta , e, \\rho )$ to $\\xi _2(\\theta )$ for the remainder of the proof.", "Write the first order approximation to the Iwasawa decomposition of $k(\\theta ) \\exp (tX) a(\\xi _2(\\theta ))$ as $k(\\theta ) \\exp (tX) a(\\xi _2(\\theta )) = n \\exp ( t X_N + O(t^2)) a \\exp ( t X_A + O(t^2)) k \\exp ( t X_K + O(t^2)),$ where $X_N \\in \\mathfrak {n}$ , $X_A \\in \\mathfrak {a}$ , and $X_K \\in \\mathfrak {k}$ .", "As in equation (REF ), we have $k = k(\\alpha )$ if $\\theta \\in (0,\\pi )$ and $k = k(-\\alpha )$ if $\\theta \\in (-\\pi , 0)$ .", "We first assume that $\\theta \\in (0, \\pi )$ .", "Rearranging and equating first order terms gives $X & = \\text{Ad}(a(\\xi _2(\\theta )) k(\\alpha )^{-1} a^{-1}) X_N + \\text{Ad}(a(\\xi _2(\\theta )) k(\\alpha )^{-1}) X_A + \\text{Ad}(a(\\xi _2(\\theta ))) X_K \\\\\\text{Ad}( k(\\alpha ) a(\\xi _2(\\theta ))^{-1}) X & = \\text{Ad}(a^{-1}) X_N + X_A + \\text{Ad}(k(\\alpha )) X_K$ As $\\text{Ad}(a^{-1}) X_N$ and $\\text{Ad}(k(\\alpha )) X_K$ lie in $\\mathfrak {a}^\\perp \\subset \\mathfrak {g}$ , we see that $X_A$ is the orthogonal projection of $\\text{Ad}( k(\\alpha ) a(\\xi _2(\\theta ))^{-1}) X$ to $\\mathfrak {a}$ .", "A calculation gives $X_A = \\sin \\alpha (e^{-\\xi _2(\\theta )} X_1 + e^{\\xi _2(\\theta )} X_2 ) \\left( \\begin{array}{cc} 1/2 & 0 \\\\ 0 & -1/2 \\end{array} \\right),$ so that $\\frac{\\partial }{\\partial t} A( k(\\theta ) \\exp (tX) a(\\xi _2(\\theta )) ) \\Big |_{t=0} = \\sin \\alpha (e^{-\\xi _2(\\theta )} X_1 + e^{\\xi _2(\\theta )} X_2 ).$ This proves (REF ) when $\\theta \\in (0,\\pi )$ , and the other case is identical.", "We now prove that $\\partial \\psi / \\partial t (\\theta , \\exp (tX), \\rho ) |_{t=0}$ has no degenerate critical points if $X \\ne 0$ and $\\theta \\in (0, \\pi )$ .", "We define $f(x) = \\sin \\alpha (X_1 e^{-x} + X_2 e^x)$ , so that $\\frac{\\partial }{\\partial t} A( k(\\theta ) \\exp (tX) a(\\xi _2(\\theta )) ) \\Big |_{t=0} = f( \\xi _2(\\theta )).$ Differentiating equation (REF ) gives $\\frac{ \\partial \\xi _2}{\\partial \\theta } = -\\frac{1}{2} e^{-\\xi _2(\\theta )} \\tan (\\alpha /2) \\csc ^2(\\theta /2),$ so that $\\partial \\xi _2 / \\partial \\theta $ is always nonzero.", "Suppose that $X \\ne 0$ , and that $\\theta $ is a degenerate critical point of $\\partial \\psi / \\partial t (\\theta , \\exp (tX), \\rho ) |_{t=0}$ .", "We then have $0 & = \\frac{\\partial ^2}{\\partial \\theta \\partial t} A( k(\\theta ) \\exp (tX) a(\\xi _2(\\theta )) ) \\Big |_{t=0} \\\\& = f^{\\prime }( \\xi _2(\\theta )) \\frac{ \\partial \\xi _2}{\\partial \\theta } \\\\& = f^{\\prime }( \\xi _2(\\theta )).$ Differentiating again with respect to $\\theta $ gives $0 & = \\frac{\\partial ^3}{\\partial ^2 \\theta \\partial t} A( k(\\theta ) \\exp (tX) a(\\xi _2(\\theta )) ) \\Big |_{t=0} \\\\& = f^{\\prime \\prime }( \\xi _2(\\theta )) \\left( \\frac{ \\partial \\xi _2}{\\partial \\theta } \\right)^2 \\\\& = f^{\\prime \\prime }( \\xi _2(\\theta )),$ but this is a contradiction as it may be easily checked that $f$ has no degenerate critical points unless $X = 0$ .", "The case of $\\theta \\in (-\\pi , 0)$ is identical.", "Define $P = \\mathbb {R}/ 2\\pi \\mathbb {Z}\\times \\mathfrak {a}^\\perp \\times [\\delta , 1-\\delta ]$ , and define $S = \\lbrace (\\theta , X, \\rho ) \\in P | (\\theta , \\exp (X), \\rho ) \\in \\mathcal {S}\\rbrace $ .", "$S$ is again closed, and contains at most 4 values of $\\theta $ for each fixed $(X, \\rho )$ .", "Lemma 7.5 There is an open neighbourhood $0 \\in U \\subset \\mathfrak {a}^\\perp $ such that for all $X \\in U$ and all $b \\in C^\\infty _0( P \\setminus S)$ we have $\\int b(\\theta , X, \\rho ) e^{is \\psi (\\theta , \\exp (X), \\rho )} d\\theta \\ll (1 + s \\Vert X\\Vert )^{-1/2},$ where $\\Vert X \\Vert $ is as in (REF ).", "Define the map $X : \\mathbb {R}\\times \\mathbb {R}/ 2\\pi \\mathbb {Z}\\rightarrow \\mathfrak {a}^\\perp $ by $X(r, \\gamma ) = \\left( \\begin{array}{cc} 0 & r \\sin \\gamma \\\\ r \\cos \\gamma & 0 \\end{array} \\right).$ We define $\\widetilde{P} = \\mathbb {R}/ 2\\pi \\mathbb {Z}\\times \\mathbb {R}\\times \\mathbb {R}/ 2\\pi \\mathbb {Z}\\times [\\delta , 1-\\delta ]$ and $\\widetilde{S} = \\lbrace (\\theta , r, \\gamma , \\rho ) \\subset \\widetilde{P} | (\\theta , X(r,\\gamma ), \\rho ) \\in S \\rbrace .$ We define $\\widetilde{b}(\\theta , r, \\gamma , \\rho ) \\in C^\\infty _0(\\widetilde{P} \\setminus \\widetilde{S})$ and $\\widetilde{\\psi }(\\theta , r, \\gamma , \\rho ) \\in C^\\infty (\\widetilde{P} \\setminus \\widetilde{S})$ to be the pullbacks of $b$ and $\\psi $ under $X$ .", "We know that $\\widetilde{\\psi }$ vanishes when $r = 0$ , and as $\\widetilde{\\psi }$ is smooth (in fact, real analytic) we have that $\\widetilde{\\psi }/r$ is again a smooth function.", "Proposition REF implies that $\\widetilde{\\psi }/r$ has no degenerate critical points when $r = 0$ , and so there is some $\\epsilon > 0$ such that it also has no degenerate critical points on the set $\\text{supp}(\\widetilde{b}) \\cap (\\mathbb {R}/ 2\\pi \\mathbb {Z}\\times [-\\epsilon , \\epsilon ] \\times \\mathbb {R}/ 2\\pi \\mathbb {Z}\\times [\\delta , 1-\\delta ])$ .", "If we define $U = X( (-\\epsilon , \\epsilon ) \\times \\mathbb {R}/ 2\\pi \\mathbb {Z})$ , the result now follows from stationary phase.", "Corollary 7.6 If $(a^{\\prime }, \\rho ^{\\prime }) \\in \\overline{\\mathcal {D}}_1$ , there is an open neighbourhood $(a^{\\prime }, \\rho ^{\\prime }) \\in U \\subset \\overline{\\mathcal {P}}$ such that for all $b \\in C^\\infty _0(\\mathcal {P}\\setminus \\mathcal {S})$ and all $(g, \\rho ) \\in U$ , we have $\\int _0^{2\\pi } b(\\theta , g, \\rho ) e^{is \\psi (\\theta , g, \\rho )} d\\theta \\ll (1 + s n(\\ell _0, g \\ell _0))^{-1/2}.$ Let $U_X \\subset \\mathfrak {a}^\\perp $ be as in Lemma REF .", "If $g = \\exp (X) a^{\\prime }$ for $X \\in U_X$ , we have $n(\\ell _0, g \\ell _0) \\sim \\Vert X \\Vert $ , where the implied constants depend on $a^{\\prime }$ .", "As $\\psi (\\theta , ga, \\rho ) = \\psi (\\theta , g, \\rho )$ for $a \\in A$ , the result follows from Lemma REF ." ], [ "The degenerate set $\\mathcal {D}_2^\\pm $", "The next proposition proves that $\\psi $ has a cubic degeneracy on $\\mathcal {D}_2^\\pm $ .", "Proposition 7.7 If $(\\theta ^{\\prime }, g^{\\prime }, \\rho ^{\\prime }) \\in \\mathcal {D}_2^\\pm $ , we have $\\partial ^3 \\psi / \\partial \\theta ^3(\\theta ^{\\prime }, g^{\\prime }, \\rho ^{\\prime }) \\ne 0$ .", "Figure: Two degenerating critical geodesics.Suppose $g^{\\prime } = a(y) k(2\\alpha ) a_2$ .", "Define $g = a(y) k(2\\alpha + \\epsilon ) a_2$ for some $\\epsilon > 0$ .", "If $\\epsilon $ is chosen sufficiently small, the pair $(\\ell , g \\ell )$ will have exactly two critical geodesics $\\ell _1$ and $\\ell _2$ as shown in Figure 1.", "The triangles $AB_1C_1$ and $AB_2C_2$ both have angular defect, and hence area, $\\epsilon $ .", "Our assumption that $\\alpha $ was bounded away from 0 and $\\pi /2$ then implies that $AB_1 = AB_2 \\sim \\epsilon ^{1/2}$ and $B_1C_1 = B_2C_2 \\sim \\epsilon ^{1/2}$ , where the implied constants depends only on $\\delta $ .", "The critical points $\\theta _i$ corresponding to $\\ell _i$ are the solutions to $\\cot \\theta _1/2 = -e^{y+AB_1} \\cot \\alpha /2, \\qquad \\cot \\theta _2/2 = -e^{y-AB_1} \\cot \\alpha /2.$ It follows that $0 > \\theta _1 > -\\alpha > \\theta _2 > -\\pi $ , and also that $\\theta _1 - \\theta _2 \\sim \\epsilon ^{1/2}$ .", "The apertures $h_i$ of the critical points $\\theta _i$ are given by $h_1 = - B_1C_1 \\sim -\\epsilon ^{1/2}$ and $h_2 = B_2C_2 \\sim \\epsilon ^{1/2}$ , so that Lemma REF gives $\\frac{\\partial ^2 \\psi }{\\partial \\theta ^2} (\\theta _1, g, \\rho ^{\\prime }) \\sim -\\epsilon ^{1/2}, \\quad \\frac{\\partial ^2 \\psi }{\\partial \\theta ^2} (\\theta _2, g, \\rho ^{\\prime }) \\sim \\epsilon ^{1/2}.$ It follows that there is $\\theta _0 \\in [\\theta _2, \\theta _1]$ at which $\\frac{\\partial ^3 \\psi }{\\partial \\theta ^3} (\\theta _0, g, \\rho ^{\\prime }) = \\frac{ \\frac{\\partial ^2 \\psi }{\\partial \\theta ^2} (\\theta _2, g, \\rho ^{\\prime }) - \\frac{\\partial ^2 \\psi }{\\partial \\theta ^2} (\\theta _1, g, \\rho ^{\\prime }) }{ \\theta _2 - \\theta _1} \\sim -1,$ and shrinking $\\epsilon $ to 0 gives the result.", "The case $g^{\\prime } \\in A k(-2\\alpha ) A$ is identical.", "Corollary 7.8 If $(g^{\\prime }, \\rho ^{\\prime }) \\in \\overline{\\mathcal {D}}_2^\\pm $ , there is an open neighbourhood $(g^{\\prime }, \\rho ^{\\prime }) \\in U \\subset \\overline{\\mathcal {P}}$ such that for all $b \\in C^\\infty _0(\\mathcal {P}\\setminus \\mathcal {S})$ and all $(g, \\rho ) \\in U$ , we have $\\int _0^{2\\pi } b(\\theta , g, \\rho ) e^{is \\psi (\\theta , g, \\rho )} d\\theta \\ll s^{-1/3}.$ By Proposition REF , there exists a neighbourhood $U_\\theta $ of $\\theta ^{\\prime }$ and $U$ of $(g^{\\prime }, \\rho ^{\\prime })$ such that $(U_\\theta \\times U) \\cap \\mathcal {S}= \\emptyset $ , and $|\\partial ^3 \\psi / \\partial \\theta ^3| \\ge \\sigma > 0$ on $U_\\theta \\times U$ .", "As $\\psi (\\theta , g^{\\prime }, \\rho ^{\\prime })$ only has a critical point at $\\theta ^{\\prime }$ , by shrinking $U$ we may also assume that $\\psi $ has no critical points on $(\\mathbb {R}/ 2\\pi \\mathbb {Z}\\setminus U_\\theta ) \\times U \\setminus \\mathcal {S}$ .", "The result then follows from Proposition 2, Section 1.2, Chapter VIII of [16]." ], [ "Bounds for $I(t, \\lambda , \\ell _1, \\ell _2)$", "We shall use the results of the previous sections to prove the follwing proposition, which implies Proposition REF in the case $\\lambda / t \\in [\\delta , 1-\\delta ]$ after inverting the Harish-Chandra transform.", "Proposition 7.9 Let $D \\subset PSL_2(\\mathbb {R})$ be a compact set, let $b_1, b_2 \\in C^\\infty _0(\\mathbb {R})$ be functions supported in $[0,1]$ , and let $1/2 > \\delta > 0$ .", "For $g \\in PSL_2(\\mathbb {R})$ and $\\lambda , s \\in \\mathbb {R}$ , define $I(s, \\lambda , g) = \\iint _{-\\infty }^\\infty e^{i \\lambda (x_1-x_2)} b_1(x_1)b_2(x_2) \\varphi _{-s}( \\ell (x_1), g\\ell (x_2)) dx_1 dx_2.$ If $g \\in D$ and $\\lambda / s \\in [\\delta , 1-\\delta ]$ , we have $I(s, \\lambda , g) \\ll \\Big \\lbrace \\begin{array}{ll} s^{-1}(1 + s n(\\ell _0, g\\ell _0))^{-1/2} & \\text{when} \\quad n(\\ell _0, g \\ell _0) \\le s^{-1/3} \\\\s^{-4/3} & \\text{when} \\quad n(\\ell _0, g \\ell _0) \\ge s^{-1/3}.\\end{array}$ If we substitute the expression (REF ) into (REF ), we obtain $\\int _0^{2\\pi } \\iint _{-\\infty }^\\infty b_1(x_1) b_2(x_2) e^{i\\lambda (x_1 - x_2)} \\exp ( (1/2 - is)( A(k(\\theta ) a(x_1)) - A( k(\\theta )g a(x_2))) \\frac{d\\sigma }{d\\theta } dx_1 dx_2 d\\theta .$ We let $b \\in C^\\infty _0(PSL_2(\\mathbb {R}))$ be a function that is equal to 1 on $D$ , and introduce a factor of $b(g)$ into the integral.", "When $g \\in D$ we then have $I(s, \\lambda , g) = \\int _0^{2\\pi } \\iint _{-\\infty }^\\infty e^{is \\phi (x_1, x_2, \\theta , g, \\rho )} c(x_1, x_2, \\theta , g, \\rho ) dx_1 dx_2 d\\theta ,$ where $c \\in C^\\infty _0(\\mathbb {R}^2 \\times \\mathcal {P})$ is the combination of all of the amplitude factors.", "The following lemma eliminates the variables $x_1$ and $x_2$ .", "Lemma 7.10 There is a function $c_1 \\in C^\\infty _0( \\mathcal {P}\\setminus \\mathcal {S})$ such that for all $(g, \\rho ) \\in D \\times [\\delta , 1-\\delta ]$ we have $I(s, \\lambda , g) = s^{-1} \\int _0^{2\\pi } e^{is \\psi (\\theta , g, \\rho )} c_1(\\theta , g, \\rho ) d\\theta + O(s^{-2}).$ We shall apply stationary phase in the $x_i$ variables.", "For fixed $(\\theta , g, \\rho )$ , the function $\\phi (x_1, x_2)$ has one critical point at $(\\xi _1(\\theta , \\rho ), \\xi _2(\\theta , g, \\rho ))$ if $(\\theta , g, \\rho ) \\notin \\mathcal {S}$ , and none otherwise.", "Moreover, it may be shown in the same way as the proof of Proposition REF that the Hessian at this critical point is $D = \\left( \\begin{array}{cc} \\tfrac{1}{2} \\sin ^2 \\alpha & 0 \\\\ 0 & -\\tfrac{1}{2} \\sin ^2 \\alpha \\end{array} \\right),$ so that the critical point is uniformly nondegenerate.", "Define $\\mathcal {P}_0 = \\lbrace (\\theta , g, \\rho ) \\in \\mathcal {P}\\setminus \\mathcal {S}| (\\xi _1(\\theta , \\rho ), \\xi _2(\\theta , g, \\rho ), \\theta , g, \\rho ) \\in \\text{supp}(c) \\rbrace ,$ so that $\\mathcal {P}_0$ is compact and $\\mathcal {P}_0 \\cap \\mathcal {S}= \\emptyset $ .", "If we define $c_1 \\in C^\\infty _0(\\mathcal {P}\\setminus \\mathcal {S})$ by $c_1(\\theta , g, \\rho ) = \\frac{2\\pi }{\\sin ^2 \\alpha } c( \\xi _1(\\theta , \\rho ), \\xi _2(\\theta , g, \\rho ), \\theta , g, \\rho ),$ then we have $\\text{supp}(c_1) \\subseteq \\mathcal {P}_0$ , and stationary phase gives $\\iint _{-\\infty }^\\infty e^{is \\phi (x_1, x_2, \\theta )} c(x_1, x_2, \\theta , g, \\rho ) dx_1 dx_2 = e^{is \\psi (\\theta , g, \\rho )} s^{-1} c_1(\\theta , g, \\rho ) + O(s^{-2})$ locally uniformly on $\\mathcal {P}\\setminus \\mathcal {S}$ .", "We also have $\\iint _{-\\infty }^\\infty e^{is \\phi (x_1, x_2, \\theta )} c(x_1, x_2, \\theta , g, \\rho ) dx_1 dx_2 \\ll _A s^{-A}$ locally uniformly on $\\mathcal {P}\\setminus \\mathcal {P}_0$ .", "Therefore, if we extend $c_1$ to a function in $C^\\infty (\\mathcal {P})$ by making it 0 on $\\mathcal {S}$ , then (REF ) holds locally uniformly on $\\mathcal {P}$ and the lemma follows.", "We now apply Corollaries REF and REF .", "Corollary REF implies that there is an open neighbourhood $U_1$ of $\\overline{\\mathcal {D}}_1 \\cap (D \\times [\\delta , 1-\\delta ])$ in $\\overline{\\mathcal {P}}$ such that $I(s, \\lambda , g) \\ll s^{-1}(1 + s n(\\ell _0, g\\ell _0))^{-1/2}$ when $(g, \\rho ) \\in U_1 \\cap (D \\times [\\delta , 1-\\delta ])$ , and Corollary REF implies that there is a neighbourhood $U_2$ of $\\overline{\\mathcal {D}}_2^\\pm \\cap (D \\times [\\delta , 1-\\delta ])$ such that $I(s, \\lambda , g) \\ll s^{-4/3}$ when $(g, \\rho ) \\in U_2 \\cap (D \\times [\\delta , 1-\\delta ])$ .", "As $\\psi $ has no degenerate critical points outside $\\overline{\\mathcal {D}}_1 \\cup \\overline{\\mathcal {D}}_2^\\pm $ , we also have $I(s, \\lambda , g) \\ll s^{-3/2}$ when $(g, \\rho ) \\in (D \\times [\\delta , 1-\\delta ]) \\setminus (U_1 \\cup U_2)$ .", "As the bound in Proposition REF is the maximum of these three bounds, this completes the proof.", "It remains to discuss the case when $\\lambda = 0$ , so that $\\alpha = \\pi /2$ .", "The proof proceeds as before, until the analysis of the degenerate critical points of $\\psi $ .", "These degeneracies now occur when $g \\in A \\cup \\left( \\begin{array}{cc} 0 & 1 \\\\ -1 & 0 \\end{array} \\right) A,$ and the function $\\psi $ vanishes identically at these points.", "These degeneracies may be treated in exactly the same way as $\\mathcal {D}_1$ in Section REF , which gives the bound $I(s, 0, g) \\ll s^{-1}(1 + s n(\\ell _0, g\\ell _0))^{-1/2}.$ Inverting the Harish-Chandra transform completes the proof." ], [ "Oscillatory Integrals When $\\lambda \\sim t$", "In this section, we prove Proposition REF by building up the integral $I(t, \\phi , \\ell _1, \\ell _2)$ in several steps.", "We begin with two calculations that we shall use repeatedly in this section and in Section .", "Lemma 6.1 Fix $g \\in PSL_2(\\mathbb {R})$ , and define $\\sigma : \\mathbb {R}/ 2\\pi \\mathbb {Z}\\rightarrow \\mathbb {R}/ 2\\pi \\mathbb {Z}$ by $k(\\theta ) g \\in NA k(\\sigma (\\theta ))$ .", "Then $\\sigma $ is a diffeomorphism.", "By using the Cartan decomposition, we may reduce to the case where $g = a(y)$ .", "Taking inverses gives $a(-y) k(-\\theta ) \\in k(-\\sigma (\\theta )) AN$ , and applying both sides to the point at infinity gives $e^{-y} \\cot (\\theta /2) = \\cot (\\sigma (\\theta )/2).$ This proves that $\\sigma $ is a bijection, and a diffeomorphism everywhere except at $\\theta = 0$ .", "Rewriting the equation as $e^y \\tan (\\theta /2) = \\tan ( \\sigma (\\theta )/2)$ proves it at $\\theta = 0$ also.", "Lemma 6.2 Let $g \\in PSL_2(\\mathbb {R})$ have Iwasawa decomposition $g = n a k(\\theta )$ .", "Then $\\frac{\\partial }{\\partial t} A( g a(t)) \\Big |_{t=0} = \\cos \\theta .$ If $H$ is as in (REF ), we have $A( g a(t)) & = A(a) + A( k(\\theta ) \\exp (tH) k(-\\theta ) ) \\\\& = A(a) + A( \\exp ( t \\text{Ad}(k(\\theta )) H) ),$ and therefore $\\frac{\\partial }{\\partial t} A( g a(t)) \\Big |_{t=0} & = H^*( \\text{Ad}(k(\\theta )) H ) \\\\& = \\cos \\theta .$" ], [ "Uniformisation results", "We shall need the following two uniformisation lemmas for the function $A$ .", "Lemma 6.3 Let $D > 0$ .", "There exists $\\delta > 0$ , $\\sigma > 0$ , and a real analytic function $\\xi : (-\\delta , \\delta ) \\times (-D,D)^2 \\rightarrow \\mathbb {R}$ such that $\\frac{\\partial }{\\partial y} A( k(\\theta ) n(x) a(y) ) = 1 - \\theta ^2 \\xi (\\theta , x, y)$ and $|\\xi (\\theta , x, y)| & \\ge \\sigma \\\\\\left| \\frac{\\partial ^n \\xi }{\\partial y^n} (\\theta , x, y) \\right| & \\ll _n 1$ for $(\\theta ,x,y) \\in (-\\delta , \\delta ) \\times (-D,D)^2$ .", "Define the function $\\alpha (\\theta , x, y) : \\mathbb {R}/ 2\\pi \\mathbb {Z}\\times (-2D, 2D)^2 \\rightarrow \\mathbb {R}/ 2\\pi \\mathbb {Z}$ by requiring that $k(\\theta ) n(x) a(y) \\in NA k(\\alpha (\\theta , x, y)).$ The analyticity of the Iwasawa decomposition implies that $\\alpha $ is analytic as a function of $(\\theta , x, y)$ .", "Lemma REF implies that $1 - \\frac{\\partial }{\\partial y} A( k(\\theta ) n(x) a(y) ) & = 1 - \\cos \\alpha \\\\& = 2 \\sin ^2(\\alpha /2).$ We choose $\\delta $ such that $\\sin (\\alpha /2)$ vanishes on $(-2\\delta , 2\\delta ) \\times (-2D,2D)^2$ iff $\\theta = 0$ .", "Lemma REF implies that $\\partial \\alpha / \\partial \\theta $ never vanishes on $\\lbrace 0 \\rbrace \\times (-2D, 2D)^2$ , and so because $\\alpha $ was analytic we see that there is a real analytic function $\\xi _0$ on $(-2\\delta , 2\\delta ) \\times (-2D,2D)^2$ such that $\\sin (\\alpha /2) = \\theta \\xi _0$ .", "Defining $\\xi = 2 \\xi _0^2$ and restricting the domain to $(-\\delta , \\delta ) \\times (-D, D)^2$ gives the result.", "Lemma 6.4 If $I \\subset \\mathbb {R}$ is a bounded open interval, there exists $\\delta > 0$ and a function $\\xi : I \\times (-\\delta , \\delta ) \\rightarrow \\mathbb {R}$ such that $\\xi (y,0) = 0$ for all $y \\in I$ , $A(k(\\theta ) a(y)) = y - y \\xi ^2(y,\\theta ),$ and the map $\\Xi : (y, \\theta ) \\mapsto (y, \\xi (y,\\theta ))$ gives a real-analytic diffeomorphism $\\Xi : I \\times (-\\delta , \\delta ) \\simeq U \\subset \\mathbb {R}^2$ .", "This follows in the same way as Lemma REF above, or Theorem 4.6 of [15]." ], [ "Constituent integrals of $I(t, \\phi , \\ell _1, \\ell _2)$", "We now estimate two one-dimensional integrals that appear in $I(t, \\phi , \\ell _1, \\ell _2)$ .", "Proposition 6.5 Let $C$ , $D$ and $\\epsilon $ be positive constants, and let $b \\in C^\\infty _0(\\mathbb {R})$ be a function supported in $[0,1]$ .", "If $x, y \\in [-D, D]$ and $|\\theta | \\ge C s^{-1/2 + \\epsilon } \\beta ^{1/2} \\quad \\text{and} \\quad |\\lambda - s| \\le \\beta $ for some $s$ and $\\beta $ satisfying $1 \\le \\beta \\ll s^{2/3}$ , then $\\int _{-\\infty }^\\infty b(z) \\exp ( i\\lambda z - is A(k(\\theta ) n(x) a(y + z))) dz \\ll _{A} s^{-A}$ uniformly in $\\lambda $ and $\\beta $ .", "By applying Lemma REF , we see that there is some $\\delta > 0$ and a nonvanishing real analytic function $\\xi $ on $(-\\delta , \\delta ) \\times (-D-2, D+2)^2$ such that $\\frac{\\partial }{\\partial z} A( k(\\theta ) n(x) a(y+z) ) = 1 - \\theta ^2 \\xi (\\theta , x, y+z)$ when $\\theta \\in (-\\delta , \\delta )$ , $x,y \\in [-D, D]$ and $z \\in [0,1]$ .", "If $Z(\\theta , x, y)$ is an antiderivative of $\\xi $ with respect to $y$ , we may integrate this to obtain $A( k(\\theta ) n(x) a(y) ) = y - \\theta ^2 Z(\\theta , x, y) + c(x,\\theta ).$ If $\\theta \\in (-\\delta , \\delta )$ , we may use this to rewrite the integral (REF ) as $\\int _{-\\infty }^\\infty b(z) \\exp ( i\\lambda z - is A( k(\\theta ) n(x) a(y+z) ) ) dy & = e^{i c(\\theta , x) - isy} \\int _{-\\infty }^\\infty b(z) \\exp ( i(\\lambda - s)z + is \\theta ^2 Z(\\theta , x, y+z)) dz \\\\& = e^{i c(\\theta , x) - isy} \\int _{-\\infty }^\\infty b(z) \\exp ( is \\theta ^2 \\Psi (z) ) dz,$ where we define $\\Psi (z) = Z(\\theta , x, y+z) + s^{-1} \\theta ^{-2}(s-\\lambda ) z$ .", "Our assumption (REF ) implies that $| s^{-1} \\theta ^{-2}(s-\\lambda ) | \\le s^{-1} \\theta ^{-2} \\beta \\ll s^{-2\\epsilon },$ so that $\\Psi = Z(\\theta , x, y+z) + O(s^{-2\\epsilon })z, \\quad \\text{and} \\quad \\frac{\\partial \\Psi }{\\partial z} = \\xi (\\theta , x, y+z) + O(s^{-2\\epsilon }).$ It follows from (REF ) and (REF ) that for $s$ sufficiently large, $|\\partial \\Psi / \\partial z| > \\sigma /2$ for all $\\theta \\in (-\\delta , \\delta )$ , $x,y \\in [-D, D]$ and $z \\in [0,1]$ .", "As (REF ) implies that $s \\theta ^2 \\gg s^{2\\epsilon } \\beta \\ge s^{2\\epsilon }$ , the bound (REF ) follows by integration by parts in (REF ).", "In the case where $\\theta \\notin (-\\delta , \\delta )$ , Lemma REF implies that $(\\partial / \\partial z) A( k(\\theta ) n(x) a(y+z) ) \\le 1 - c_1$ for some $c_1 > 0$ depending only on $\\delta $ , which gives $\\frac{\\partial }{\\partial z} ( i\\lambda s^{-1} z - A( k(\\theta ) n(x) a(y+z) ) \\gg 1.$ The result now follows by integration by parts.", "The second one-dimensional integral that we shall estimate is as follows.", "Proposition 6.6 Let $C$ , $D$ and $\\epsilon $ be positive constants, and let $b \\in C^\\infty _0(\\mathbb {R})$ be a function supported in $[0,1]$ .", "If $x, y \\in [-D, D]$ and $|x| \\ge C s^{-1/2 + \\epsilon } \\beta ^{1/2} \\quad \\text{and} \\quad |\\lambda - s| \\le \\beta $ for some $s$ and $\\beta $ satisfying $1 \\le \\beta \\ll s^{2/3}$ , then $\\int _{-\\infty }^\\infty b(z) e^{i\\lambda z} \\varphi _{-s}(n(x) a(y+z)) dz \\ll _{A} s^{-A}$ uniformly in $\\lambda $ and $\\beta $ .", "If we substitute the formula for $\\varphi _{-s}$ as an integral of plane waves into the LHS of (REF ), it becomes $\\int _{-\\infty }^\\infty \\int _0^{2\\pi } b(z) \\exp ( i\\lambda z + (1/2 - is) A( k(\\theta ) n(x) a(y+z) ) ) d\\theta dz.$ Let $f(x) \\in C^\\infty _0(\\mathbb {R})$ be a function with $\\text{supp}(f) \\subseteq [-2,2]$ and $f(x) = 1$ on $[-1,1]$ .", "Let $C_1$ be a positive constant to be chosen later.", "Define $b_1$ by $b_1(x) = f( C_1^{-1} s^{1/2 -\\epsilon } \\beta ^{-1/2} x)$ and set $b_2 = 1 - b_1$ , so that $1 = b_1(\\theta ) + b_2(\\theta )$ is a smooth partition of unity on $\\mathbb {R}/ 2\\pi \\mathbb {Z}$ with $\\text{supp}(b_1) & \\subseteq [-2C_1 s^{-1/2 +\\epsilon } \\beta ^{1/2}, 2C_1 s^{-1/2 +\\epsilon } \\beta ^{1/2}], \\\\\\text{supp}(b_2) & \\subseteq \\mathbb {R}/ 2\\pi \\mathbb {Z}\\setminus [-C_1 s^{-1/2 +\\epsilon } \\beta ^{1/2}, C_1 s^{-1/2 +\\epsilon } \\beta ^{1/2}].$ Proposition REF implies that $\\int _{-\\infty }^\\infty \\int _0^{2\\pi } b_2(\\theta ) b(z) \\exp ( i\\lambda z + (1/2 - is) A( k(\\theta ) n(x) a(y+z) ) ) d\\theta dz \\ll _A s^{-A},$ so that it suffices to estimate $\\int _{-\\infty }^\\infty \\int _0^{2\\pi } b_1(\\theta ) b(z) \\exp ( i\\lambda z + (1/2 - is) A( k(\\theta ) n(x) a(y+z) ) ) d\\theta dz.$ We shall do this by estimating the integrals $\\int _0^{2\\pi } b_1(\\theta ) \\exp ( - is A( k(\\theta ) n(x) a(y) ) ) d\\theta $ in $\\theta $ , where now $y \\in [-D, D+1]$ .", "If $X \\in \\mathfrak {g}$ , we let $X^*$ be the vector field on $\\mathbb {H}$ whose value at $p$ is $\\tfrac{\\partial }{\\partial t} \\exp (tX) p |_{t=0}$ .", "It may be shown that these vector fields satisfy $[X^*, Y^*] = -[X,Y]^*$ , where the first Lie bracket is on $\\mathbb {H}$ and the second is in $\\mathfrak {g}$ .", "We recall the vectors $X_\\mathfrak {n}$ and $X_\\mathfrak {k}$ defined in (REF ).", "It may be easily seen that the subset of $\\mathbb {H}$ where $X_\\mathfrak {k}^* A$ vanishes is exactly $A$ , and the following lemma implies that it vanishes to first order there.", "Lemma 6.7 We have $X_\\mathfrak {n}^* X_\\mathfrak {k}^* A(a(y)) = e^y$ for all $y$ .", "We have $X_\\mathfrak {n}^* X_\\mathfrak {k}^* A = X_\\mathfrak {k}^* X_\\mathfrak {n}^* A + [X_\\mathfrak {n}^*, X_\\mathfrak {k}^*] A.$ It may be seen that the first term vanishes, and we have $[X_\\mathfrak {n}^*, X_\\mathfrak {k}^*] = - [X_\\mathfrak {n}, X_\\mathfrak {k}]^* = H^*$ which implies the lemma.", "Lemma REF implies that there exist $\\sigma $ , $\\delta > 0$ such that if $|x| < \\sigma $ and $y \\in [-D-1, D+2]$ then we have $|X_\\mathfrak {k}A(n(x)a(y))| \\ge \\delta |x|$ .", "Define $B & = \\lbrace n(x) a(y) | \\; |x| \\le \\sigma /2, \\, y \\in [-D, D+1] \\rbrace , \\\\B^{\\prime } & = \\lbrace n(x) a(y) | \\; |x| < \\sigma , \\, y \\in [-D-1, D+2] \\rbrace .$ Let $p \\in B$ and assume that $|N(p)| \\ge C s^{-1/2+\\epsilon } \\beta ^{1/2}$ , where $N(p)$ is as in (REF ).", "If $s$ is sufficiently large and $C_1$ sufficiently small, and $|\\theta | \\le 2 C_1 s^{-1/2 +\\epsilon } \\beta ^{1/2}$ , we have $k(\\theta )p \\in B^{\\prime }$ and $|N(k(\\theta )p)| \\gg s^{-1/2+\\epsilon } \\beta ^{1/2}$ .", "It follows that $\\Big | \\frac{\\partial }{\\partial \\theta ^{\\prime }} A( k(\\theta ^{\\prime }) p) \\Big |_{\\theta ^{\\prime } = \\theta } \\Big | = | X_\\mathfrak {k}^* A( k(\\theta ) p) | \\ge \\delta |N( k(\\theta )p)| \\gg s^{-1/2+\\epsilon } \\beta ^{1/2}$ when $|\\theta | \\le 2 C_1 s^{-1/2 +\\epsilon } \\beta ^{1/2}$ .", "The proposition now follows by integration by parts.", "If $p = n(x) a(y)$ with $x \\in [-D, D]$ and $y \\in [-D, D+1]$ and $p \\notin B$ , then we have $|X_\\mathfrak {k}^* A(k(\\theta )p) | \\ge \\delta $ when $|\\theta | \\le 2 C_1 s^{-1/2 +\\epsilon } \\beta ^{1/2}$ and $s$ is sufficiently large.", "The proposition again follows by integration by parts." ], [ "Proof of Proposition ", "We now combine Propositions REF and REF to bound the integral (REF ) below, which will imply Proposition REF after integrating in the various spectral parameters.", "Proposition 6.8 Let $\\ell \\subset \\mathbb {H}$ be a unit geodesic segment with parametrisation $\\ell : [0,1] \\rightarrow \\mathbb {H}$ .", "Let $D \\subset PSL_2(\\mathbb {R})$ be a compact set, let $b_1, b_2 \\in C^\\infty _0(\\mathbb {R})$ be functions supported in $[0,1]$ , and let $\\epsilon > 0$ be given.", "If $g \\in D$ and $\\lambda _1$ , $\\lambda _2 \\in \\mathbb {R}$ satisfy $n(\\ell , g\\ell ) \\ge s^{-1/2 + \\epsilon } \\beta ^{1/2} \\quad \\text{and} \\quad \\lambda _i \\in [s-\\beta , s+\\beta ]$ for some $s$ and $1 \\le \\beta \\ll s^{2/3}$ , then $\\iint _{-\\infty }^\\infty b_1(x_1) b_2(x_2) e^{i(\\lambda _1 x_1 - \\lambda _2 x_2)} \\varphi _{-s}(\\ell (x_1), g\\ell (x_2)) dx_1 dx_2 \\ll _{A, \\epsilon } s^{-A}$ uniformly in $\\lambda _i$ and $\\beta $ .", "We begin by expressing $\\varphi _{-s}$ as an integral of plane waves.", "For $y, z \\in \\mathbb {H}$ we have $\\varphi _s(y,z) = \\int _0^{2\\pi } \\exp ( (1/2 - is)( A(k_z(\\sigma )y) - A( k_z(\\sigma )z)) d\\sigma ,$ where $K_z$ is the stabilizer of $z$ and $k_z : \\mathbb {R}/ 2\\pi \\mathbb {Z}\\rightarrow K_z$ is a parametrisation.", "Define the function $\\theta : \\mathbb {R}/ 2\\pi \\mathbb {Z}\\rightarrow \\mathbb {R}/ 2\\pi \\mathbb {Z}$ by $k_z(\\sigma ) \\in NA k(\\theta (\\sigma )).$ Lemma REF implies that $\\theta $ is a diffeomorphism.", "Because $A(k_z(\\sigma )y) - A( k_z(\\sigma )z) = A( k(\\theta (\\sigma )) y) - A( k(\\theta (\\sigma )) z)$ , we have $\\varphi _s(y,z) = \\int \\exp ( (1/2 - is)( A(k(\\theta )y) - A( k(\\theta )z)) \\frac{d\\sigma }{d\\theta } d\\theta .$ We may assume that $\\ell $ is the segment with one endpoint at $i$ and pointing upwards, so that $\\ell (x) = a(x) i$ .", "Substituting (REF ) into (REF ) gives $\\iint _{-\\infty }^\\infty b_1(x_1) b_2(x_2) e^{i(\\lambda _1 x_1 - \\lambda _2 x_2)} \\varphi _{-s}(\\ell (x_1), g\\ell (x_2)) dx_1 dx_2 = \\\\\\iint _{-\\infty }^\\infty \\int _0^{2\\pi } b_1(x_1) b_2(x_2) e^{i(\\lambda _1 x_1 - \\lambda _2 x_2)} \\exp ( (1/2 - is)( A(k(\\theta ) a(x_1)) - A( k(\\theta )g a(x_2))) \\frac{d\\sigma }{d\\theta } d\\theta dx_1 dx_2.$ Let $g = k(\\theta ^{\\prime })n(x^{\\prime }) a(y^{\\prime })$ , where $x^{\\prime }$ and $y^{\\prime }$ are bounded in terms of $D$ .", "We then have $k(\\theta )g a(x_2) = k(\\theta + \\theta ^{\\prime }) n(x^{\\prime }) a(y + y^{\\prime })$ .", "We integrate the RHS of (REF ) with respect to $x_1$ and $x_2$ with $\\theta $ fixed.", "Choose a constant $C > 0$ .", "If $\\theta \\notin [-C s^{-1/2+\\epsilon } \\beta ^{1/2}, C s^{-1/2+\\epsilon } \\beta ^{1/2}]$ , then Proposition REF implies that the integral is $\\ll s^{-A}$ , and likewise if $\\theta + \\theta ^{\\prime } \\notin [-C s^{-1/2+\\epsilon } \\beta ^{1/2}, C s^{-1/2+\\epsilon } \\beta ^{1/2}]$ .", "Combining these, we see that (REF ) will be $\\ll s^{-A}$ unless $|\\theta ^{\\prime }| \\le 2C s^{-1/2+\\epsilon } \\beta ^{1/2}$ .", "If $C$ is chosen sufficiently small, the condition $|\\theta ^{\\prime }| \\le 2C s^{-1/2+\\epsilon } \\beta ^{1/2}$ and our assumption that $n(\\ell , g\\ell ) \\ge s^{-1/2 + \\epsilon } \\beta ^{1/2}$ imply that $\\ell $ and $g \\ell $ are separated in the sense that there is a $C_1 > 0$ such that $d(p, g\\ell ) \\ge C_1 s^{-1/2 + \\epsilon } \\beta ^{1/2}$ for all $p \\in \\ell $ .", "The result now follows by applying Proposition REF to the integral of the LHS of (REF ) over $x_2$ for each fixed $x_1$ .", "Corollary 6.9 Let $\\ell \\subset \\mathbb {H}$ be a unit geodesic segment with parametrisation $\\ell : [0,1] \\rightarrow \\mathbb {H}$ .", "Let $D \\subset PSL_2(\\mathbb {R})$ be a compact set, and let $b_1, b_2 \\in C^\\infty _0(\\mathbb {R})$ be functions supported in $[0,1]$ .", "Let $\\epsilon > 0$ , $s > 0$ , and $1 \\le \\beta \\ll s^{2/3}$ be given.", "Let $\\phi \\in L^2(\\mathbb {R})$ be a function with $\\Vert \\phi \\Vert _2 = 1$ and such that $\\textup {supp}(\\widehat{\\phi }) \\subseteq [s-\\beta , s+\\beta ].$ If $g \\in D$ and $n(\\ell , g\\ell ) \\ge s^{-1/2 + \\epsilon } \\beta ^{1/2}$ , then $\\iint _{-\\infty }^\\infty b_1(x_1) b_2(x_2) \\phi (x_1) \\overline{\\phi (x_2)} \\varphi _{-s}(\\ell (x_1), g\\ell (x_2)) dx_1 dx_2 \\ll _{A, \\epsilon } s^{-A},$ where the implied constant is independent of $\\phi $ and $\\beta $ .", "This follows immediately from Proposition REF after inverting the Fourier transform of $\\phi $ and noting that $\\Vert \\widehat{\\phi } \\Vert _1 \\le \\Vert \\widehat{\\phi } \\Vert _2 (2\\beta )^{1/2} = (2\\pi )^{1/2} (2\\beta )^{1/2}$ .", "To prove the bound (REF ), observe that equation (REF ) implies that $\\int _{-\\infty }^\\infty |K_t(\\ell _1(x), p)|^2 dx \\ll t$ uniformly for $p \\in \\mathbb {H}$ .", "It follows that $K_t(\\ell _1(x_1), \\ell _2(x_2))$ has norm $\\ll t^{1/2}$ as an element of $L^2(\\mathbb {R}^2)$ , and the result follows by Cauchy-Schwarz.", "We now prove (REF ).", "Fix a unit geodesic segment $\\ell $ .", "We may assume without loss of generality that $\\ell _1 = \\ell $ , and we choose $g \\in PSL_2(\\mathbb {R})$ so that $g \\ell = \\ell _2$ .", "The assumption that $d(\\ell _1, \\ell _2) \\le 1$ implies that $g$ lies in the compact set $D := \\lbrace g \\in PSL_2(\\mathbb {R}) | d(\\ell , g \\ell ) \\le 1 \\rbrace $ .", "We have $I(t, \\phi , \\ell , g\\ell ) = \\iint _{-\\infty }^\\infty b(x_1) b(x_2) \\phi (x_1) \\overline{\\phi (x_2)} K_t(\\ell (x_1), g\\ell (x_2)) dx_1 dx_2.$ Inverting the Harish-Chandra transform of $k_t$ gives $I(t, \\phi , \\ell , g\\ell ) = \\frac{1}{2\\pi } \\iint _{-\\infty }^\\infty \\int _0^\\infty b(x_1) b(x_2) \\phi (x_1) \\overline{\\phi (x_2)} h_t^2(s) \\varphi _{-s}(\\ell (x_1), g \\ell (x_2)) s \\tanh (\\pi s) ds dx_1 dx_2.$ If we assume without loss of generality that $\\beta > t^\\epsilon $ , then we may restrict the domain of the Harish-Chandra transform to $[t-\\beta , t+\\beta ]$ as in Section REF to obtain $I(t, \\phi , \\ell , g\\ell ) = \\frac{1}{2\\pi } \\iint _{-\\infty }^\\infty \\int _{t-\\beta }^{t+\\beta } b(x_1) b(x_2) \\phi (x_1) \\overline{\\phi (x_2)} \\\\h_t^2(s) \\varphi _{-s}(\\ell (x_1), g \\ell (x_2)) s \\tanh (\\pi s) ds dx_1 dx_2 + O(t^{-A}).$ Applying Corollary REF with $2\\beta $ in place of $\\beta $ completes the proof." ], [ "Oscillatory Integrals When $\\lambda < t$", "We now prove Proposition REF .", "In this section, we assume that all geodesics we consider carry an orientation.", "When we refer to the unit tangent vector to a geodesic at a point, we shall always mean in the direction of its orientation.", "If $\\ell _1$ and $\\ell _2$ are two intersecting geodesics, we shall denote by $\\angle (\\ell _1, \\ell _2)$ the angle between their unit tangent vectors at the point of intersection measured in the counterclockwise direction from $\\ell _1$ to $\\ell _2$ .", "Let $\\ell $ be the vertical geodesic through $i$ .", "By slight abuse of notation, we take $a : \\mathbb {R}\\rightarrow \\ell $ to be a parametrisation of $\\ell $ , and define $\\ell _0 = a([0,1])$ which is a unit segment contained in $\\ell $ .", "We give the geodesic $\\ell $ the upwards-pointing orientation, which we transfer to $g \\ell $ for $g \\in PSL_2(\\mathbb {R})$ .", "As in the proof of Proposition REF , it suffices to bound the integral $I(s, \\lambda , g) = \\iint _{-\\infty }^\\infty e^{i \\lambda (x_1-x_2)} b_1(x_1)b_2(x_2) \\varphi _{-s}( \\ell (x_1), g\\ell (x_2)) dx_1 dx_2.$ After substituting the expression (REF ) for $\\varphi _{-s}( \\ell (x_1), g\\ell (x_2))$ , we obtain an oscillatory integral in the variables $\\theta $ , $x_1$ , and $x_2$ with phase function $\\phi (x_1, x_2, \\theta , g, \\rho ) = \\rho (x_1 - x_2) - A( k(\\theta ) \\ell (x_1)) + A( k(\\theta ) g \\ell (x_2)),$ where $\\rho = \\lambda / s \\ge 0$ .", "We first assume that $\\rho \\in [\\delta , 1-\\delta ]$ for some $1/2 > \\delta > 0$ .", "Define $\\alpha \\in [0,\\pi /2]$ to be the solution to $\\cos \\alpha = \\rho $ , which is bounded away from 0 and $\\pi /2$ .", "We shall study the critical points of $\\phi $ in Sections REF to REF , before deriving a bound for $I(s, \\lambda , g)$ from our results in Section REF .", "We shall write $\\phi (x_1, x_2, \\theta )$ when $g$ and $\\rho $ are not varying." ], [ "The critical points of $\\phi $", "Lemma 7.1 The phase function $\\phi $ has a critical point at $(x_1, x_2, \\theta , g, \\rho )$ exactly when $k(\\theta ) \\ell (x_1)$ and $k(\\theta ) g \\ell (x_2)$ lie on the same vertical geodesic $v$ , which we give the upwards-pointing orientation, and we have $\\angle (v, k(\\theta ) \\ell ), \\angle (v, k(\\theta ) g \\ell ) \\in \\lbrace \\pm \\alpha \\rbrace $ .", "Suppose that $(x_1^{\\prime }, x_2^{\\prime }, \\theta ^{\\prime })$ is a critical point of $\\phi $ .", "Define the functions $x(\\theta )$ , $y(\\theta )$ and $\\beta (\\theta )$ by $k(\\theta ) a(x_1^{\\prime }) = n(x(\\theta )) a( y(\\theta )) k(\\beta (\\theta )),$ and let $n^{\\prime } = n(x(\\theta ^{\\prime }))$ and $\\beta ^{\\prime } = \\beta (\\theta ^{\\prime })$ .", "It may be seen that $v := n(x^{\\prime }) \\ell $ is the upwards-pointing geodesic through $k(\\theta ^{\\prime }) \\ell (x_1^{\\prime })$ , and that $\\beta ^{\\prime } = \\angle (v, k(\\theta ^{\\prime }) \\ell )$ .", "Lemma REF then implies that $\\beta ^{\\prime } = \\pm \\alpha $ .", "The calculation in the case of $\\partial / \\partial x_2$ is identical.", "We have $A( k(\\theta ) a(x_1^{\\prime })) - A( k(\\theta ) g a(x_2^{\\prime })) & = A( k(\\theta ) a(x_1^{\\prime })) - A( k(\\theta ) a(x_1^{\\prime }) a(x_1^{\\prime })^{-1} g a(x_2^{\\prime })) \\\\& = y(\\theta ) - A( a(y(\\theta )) k(\\beta (\\theta )) a(x_1^{\\prime })^{-1} g a(x_2^{\\prime })) \\\\& = - A( k(\\beta (\\theta )) a(x_1^{\\prime })^{-1} g a(x_2^{\\prime }))$ and so $\\frac{\\partial \\phi }{\\partial \\theta } (x_1^{\\prime }, x_2^{\\prime }, \\theta ^{\\prime }) & = \\frac{\\partial }{\\partial \\theta } A( k(\\beta (\\theta )) a(x_1^{\\prime })^{-1} g a(x_2^{\\prime })) \\Big |_{\\theta = \\theta ^{\\prime }} \\\\& = \\frac{\\partial \\beta }{\\partial \\theta } (\\theta ^{\\prime }) \\frac{\\partial }{\\partial \\beta } A( k(\\beta ) a(x_1^{\\prime })^{-1} g a(x_2^{\\prime })) \\Big |_{\\beta = \\beta ^{\\prime }}.$ Because $\\partial \\beta /\\partial \\theta $ does not vanish by Lemma REF , and $\\frac{\\partial }{\\partial \\theta } A( k(\\theta ) g ) \\Big |_{\\theta =0} = 0$ iff $g \\in AK$ , we have $\\partial \\phi / \\partial \\theta = 0$ iff $k(\\beta ^{\\prime }) a(x_1^{\\prime })^{-1} g a(x_2^{\\prime }) \\in AK$ , i.e.", "$k(\\beta ^{\\prime }) a(x_1^{\\prime })^{-1} g a(x_2^{\\prime })i$ lies on the vertical geodesic through the origin.", "Because $k(\\beta ^{\\prime }) a(x_1)^{-1} = a^{\\prime -1} n^{\\prime -1} k(\\theta ^{\\prime })$ , this is equivalent to the condition that $k(\\theta ^{\\prime }) g a(x_2^{\\prime }) \\in n^{\\prime } AK$ , or that $k(\\theta ^{\\prime }) g \\ell (x_2^{\\prime })$ lies on the vertical geodesic $v$ passing through $k(\\theta ^{\\prime }) \\ell (x_1^{\\prime })$ .", "We finish with an observation that will be useful in calculating the Hessian of $\\phi $ .", "We have $k(\\beta ^{\\prime }) a(x_1^{\\prime })^{-1} g a(x_2^{\\prime }) \\in a(h)K$ for some $h \\in \\mathbb {R}$ , and it may be seen that $k(\\theta ^{\\prime }) \\ell (x_1^{\\prime }) \\in n^{\\prime } a^{\\prime } K$ and $k(\\theta ^{\\prime }) g \\ell (x_2^{\\prime }) \\in n^{\\prime } a^{\\prime } a(h) K$ , so that $h$ is the signed distance from $k(\\theta ^{\\prime }) \\ell (x_1^{\\prime })$ to $k(\\theta ^{\\prime }) g \\ell (x_2^{\\prime })$ along $v$ .", "Given a pair of geodesics $\\ell _1$ and $\\ell _2$ , we say that a geodesic $j$ is a critical geodesic for $(\\ell _1, \\ell _2)$ if $j$ meets $\\ell _1$ and $\\ell _2$ at angles of $\\pm \\alpha $ .", "We may therefore rephrase Lemma REF as saying that $(x_1, x_2, \\theta , g, \\rho )$ is a critical point of $\\phi $ exactly when $(\\ell , g\\ell )$ has a critical geodesic $j$ , $\\ell (x_1)$ and $g \\ell (x_2)$ both lie on $j$ , and $k(\\theta )j$ is vertical.", "As in Lemma REF , we define the aperture of a critical point to be the signed distance from $\\ell (x_1)$ to $g \\ell (x_2)$ on the geodesic $j$ .", "We shall now calculate the Hessian of $\\phi $ at its critical points.", "Let $(x_1^{\\prime }, x_2^{\\prime }, \\theta ^{\\prime })$ be a critical point of $\\phi $ , and define functions $\\beta _i(\\theta )$ by $k(\\theta ) a(x_1^{\\prime }) \\in NA k(\\beta _1(\\theta )), \\quad k(\\theta ) g a(x_2^{\\prime }) \\in NA k(\\beta _2(\\theta )).$ We let $\\beta _i^{\\prime } = \\beta _i(\\theta ^{\\prime })$ .", "It follows from Lemma REF that $\\beta _i^{\\prime } \\in \\lbrace \\pm \\alpha \\rbrace $ .", "Let $h$ be the aperture of the critical point, so that $k(\\beta _1^{\\prime }) a(x_1^{\\prime })^{-1} g a(x_2^{\\prime }) \\in \\left( \\begin{array}{cc} e^{h} & 0 \\\\ 0 & 1 \\end{array} \\right) K.$ We define $\\kappa = \\tfrac{\\partial \\beta _1}{\\partial \\theta } (\\theta ^{\\prime })$ , which is nonzero by Lemma REF .", "The Hessian of $\\phi $ at $(x_1^{\\prime }, x_2^{\\prime }, \\theta ^{\\prime })$ is given by the following proposition.", "Proposition 7.2 The Hessian of $\\phi $ at $(x_1^{\\prime }, x_2^{\\prime }, \\theta ^{\\prime })$ with respect to the co-ordinates $(x_1, x_2, \\theta )$ is $D = \\left( \\begin{array}{ccc} \\tfrac{1}{2} \\sin ^2 \\alpha & 0 & \\kappa \\sin \\beta _1^{\\prime } \\\\ 0 & -\\tfrac{1}{2} \\sin ^2 \\alpha & -\\kappa e^h \\sin \\beta _2^{\\prime } \\\\ \\kappa \\sin \\beta _1^{\\prime } & -\\kappa e^h \\sin \\beta _2^{\\prime } & \\kappa ^2 (1 - e^{2h})/2 \\end{array} \\right)$ The determinant of $D$ is $|D| = \\frac{3}{8} \\kappa ^2 \\sin ^4 \\alpha (1 - e^{2h}),$ which is nonzero unless $h = 0$ , i.e.", "the points $\\ell (x_1^{\\prime })$ and $g\\ell (x_2^{\\prime })$ coincide in $\\mathbb {H}$ .", "It is clear that $\\partial ^2 \\phi / \\partial x_1 \\partial x_2$ is identically 0.", "To calculate $\\partial ^2 \\phi / \\partial x_1^2$ , define $\\gamma : \\mathbb {R}\\rightarrow \\mathbb {R}/ 2\\pi \\mathbb {Z}$ by the condition that $k(\\theta ^{\\prime }) a(x_1^{\\prime } + t) \\in NA k (\\gamma (t))$ .", "Our assumption that we are at a critical point implies that $\\gamma (0) = \\beta _1^{\\prime } = \\pm \\alpha $ .", "Lemma REF gives $\\frac{\\partial }{\\partial t} \\phi (x_1^{\\prime } + t, x_2^{\\prime }, \\theta ^{\\prime }) = \\rho - \\cos \\gamma (t),$ and $\\frac{\\partial ^2 \\phi }{\\partial x_1^2} (x_1^{\\prime }, x_2^{\\prime }, \\theta ^{\\prime }) = \\sin \\beta _1^{\\prime } \\frac{\\partial \\gamma }{\\partial t} (0).$ We have $k(\\theta ^{\\prime }) a(x_1^{\\prime } + t) & \\in NA k (\\gamma (t)) \\\\NA k( \\beta _1^{\\prime }) a(t) & = NA k (\\gamma (t)) \\\\k( \\beta _1^{\\prime }) a(t) & \\in NA k (\\gamma (t)).$ Equation (REF ) then gives $\\tan (\\gamma (t)/2) = e^t \\tan ( \\beta _1^{\\prime }/2)$ , so that $\\frac{\\partial \\gamma }{\\partial t} \\sec ^2 (\\gamma (t)/2) & = e^t \\tan ( \\beta _1^{\\prime }/2) \\\\\\frac{\\partial \\gamma }{\\partial t} (0) & = \\cos ^2 (\\beta _1^{\\prime }/2) \\tan ( \\beta _1^{\\prime }/2) \\\\& = \\frac{1}{2} \\sin \\beta _1^{\\prime }.$ Substituting this into (REF ) gives $\\frac{\\partial ^2 \\phi }{\\partial x_1^2} (x_1^{\\prime }, x_2^{\\prime }, \\theta ^{\\prime }) = \\frac{1}{2} \\sin ^2 \\beta _1^{\\prime } = \\frac{1}{2} \\sin ^2 \\alpha .$ The calculation of $\\partial ^2 \\phi / \\partial x_2^2$ is identical, with the exception of a change in sign.", "To calculate $\\partial ^2 \\phi / \\partial \\theta \\partial x_1$ , we again have $\\frac{\\partial \\phi }{\\partial x_1} (x_1^{\\prime }, x_2^{\\prime }, \\theta ) = \\rho - \\cos \\beta _1(\\theta ),$ and $\\frac{\\partial ^2 \\phi }{\\partial \\theta \\partial x_1} (x_1^{\\prime }, x_2^{\\prime }, \\theta ^{\\prime }) = \\sin \\beta _1^{\\prime } \\frac{\\partial \\beta _1}{\\partial \\theta } (\\theta ^{\\prime }) = \\kappa \\sin \\beta _1^{\\prime }.$ We likewise have $\\frac{\\partial ^2 \\phi }{\\partial \\theta \\partial x_2} \\phi (x_1^{\\prime }, x_2^{\\prime }, \\theta ^{\\prime }) = -\\sin \\beta _2^{\\prime } \\frac{\\partial \\beta _2}{\\partial \\theta } (\\theta ^{\\prime }),$ and we shall express $\\tfrac{\\partial \\beta _2}{\\partial \\theta } (\\theta ^{\\prime })$ in terms of $\\kappa $ and $h$ .", "We recall that $k( \\beta _1^{\\prime }) a(x_1^{\\prime })^{-1} g a(x_2^{\\prime }) = a(h) k(\\theta _0)$ for some $\\theta _0$ , and so $k(\\theta ) a( x_1^{\\prime }) k( -\\beta _1^{\\prime }) a(h) k(\\theta _0) = k(\\theta ) g a(x_2^{\\prime }).$ Substituting both parts of (REF ) into this gives $NA k( \\beta _1(\\theta )) k( -\\beta _1^{\\prime }) a(h) k(\\theta _0) & = NA k( \\beta _2(\\theta )) \\\\k( \\beta _1(\\theta ) - \\beta _1^{\\prime })a(h) & \\in NA k( \\beta _2(\\theta ) -\\theta _0).$ By setting $\\theta = \\theta ^{\\prime }$ we see that $\\theta _0 = \\beta _1^{\\prime }$ .", "Equation (REF ) then gives $e^h \\tan ( (\\beta _1(\\theta ) - \\beta _1^{\\prime })/2) = \\tan ( (\\beta _2(\\theta ) - \\beta _2^{\\prime })/2),$ and differentiating both sides with respect to $\\theta $ and evaluating at $\\theta = \\theta ^{\\prime }$ gives $\\frac{\\partial \\beta _2}{\\partial \\theta } (\\theta ^{\\prime }) = \\kappa e^h.$ It follows that $\\frac{\\partial ^2 \\phi }{\\partial \\theta \\partial x_2} \\phi (x_1^{\\prime }, x_2^{\\prime }, \\theta ^{\\prime }) = - \\kappa e^h \\sin \\beta _2^{\\prime }.$ To calculate $\\partial ^2 \\phi / \\partial \\theta ^2$ , we have as in (REF ) that $\\frac{\\partial \\phi }{\\partial \\theta }(x_1^{\\prime }, x_2^{\\prime }, \\theta ) = \\frac{\\partial \\beta _1}{\\partial \\theta } \\frac{\\partial }{\\partial \\beta } A( k(\\beta ) a(x_1^{\\prime })^{-1} g a(x_2^{\\prime })) \\Big |_{\\beta = \\beta _1(\\theta )}.$ Because $\\frac{\\partial }{\\partial \\beta } A( k(\\beta ) a(x_1^{\\prime })^{-1} g a(x_2^{\\prime })) \\Big |_{\\beta = \\beta _1^{\\prime }} = 0,$ we have $\\frac{\\partial ^2 \\phi }{\\partial \\theta ^2} (x_1^{\\prime }, x_2^{\\prime }, \\theta ^{\\prime }) & = \\kappa ^2 \\frac{\\partial ^2}{\\partial \\beta ^2} A( k(\\beta ) a(x_1^{\\prime })^{-1} g a(x_2^{\\prime })) \\Big |_{\\beta = \\beta _1^{\\prime }} \\\\& = \\kappa ^2 \\frac{\\partial ^2}{\\partial \\beta ^2} A( k(\\beta - \\beta _1^{\\prime }) a(h) ) \\Big |_{\\beta = \\beta _1^{\\prime }}.$ It is a standard calculation (see for instance Proposition 4.4 of [15]) that $\\frac{\\partial ^2}{\\partial \\beta ^2} A( k(\\beta ) a(h) ) \\Big |_{\\beta = 0} = (1 - e^{2h})/2,$ and this completes the proof." ], [ "The function $\\psi $", "Define $\\mathcal {P}= \\mathbb {R}/ 2\\pi \\mathbb {Z}\\times PSL_2(\\mathbb {R}) \\times [\\delta , 1-\\delta ]$ , and define $\\mathcal {S}\\subset \\mathcal {P}$ to be the set where one of the geodesics $k(\\theta ) \\ell $ and $k(\\theta ) g \\ell $ is vertical.", "Note that $\\mathcal {S}$ is closed, and contains at most 4 values of $\\theta $ for each fixed $(g, \\rho )$ .", "We may define functions $\\xi _1, \\xi _2 : \\mathcal {P}\\setminus \\mathcal {S}\\rightarrow \\mathbb {R}$ by requiring that $k(\\theta ) a(\\xi _1(\\theta , g, \\rho ))$ is the unique point on $k(\\theta )\\ell $ at which the tangent vector to the geodesic makes an angle of $\\alpha $ with the upward pointing vector, and likewise for $\\xi _2(\\theta , g, \\rho )$ and $k(\\theta ) g \\ell $ .", "As $\\xi _1$ does not depend on $g$ , we will omit this argument of the function.", "We have $k(\\theta ) a(\\xi _1(\\theta , \\rho )) \\in NA k(\\epsilon _1 \\alpha ), \\quad k(\\theta ) g a(\\xi _2(\\theta , g, \\rho )) \\in NA k(\\epsilon _2 \\alpha )$ for $\\epsilon _i \\in \\lbrace \\pm 1 \\rbrace $ , and so equation (REF ) gives $e^{\\xi _1(\\theta , \\rho )} \\tan (\\theta /2) = \\tan ( \\epsilon _1 \\alpha /2), \\quad e^{\\xi _2(\\theta , g, \\rho )} \\tan (\\theta /2) = \\tan ( \\epsilon _2 \\alpha /2).$ Moreover, it may be seen that $\\epsilon _1 = 1$ iff the geodesic $k(\\theta ) \\ell $ runs from right to left in the upper half plane model of $\\mathbb {H}$ , which is equivalent to $\\theta \\in (0, \\pi )$ , and likewise for $\\epsilon _2$ .", "It follows from Lemma REF that $\\xi _1(\\theta , \\rho )$ and $\\xi _2(\\theta , g, \\rho )$ may also be characterised as the unique functions such that $\\frac{\\partial \\phi }{\\partial x_1}(\\xi _1(\\theta , \\rho ), x_2, \\theta ) = \\frac{\\partial \\phi }{\\partial x_2}(x_1, \\xi _2(\\theta , g, \\rho ), \\theta ) = 0.$ We define $\\psi : \\mathcal {P}\\setminus \\mathcal {S}& \\rightarrow \\mathbb {R}\\\\\\psi (\\theta , g, \\rho ) & = \\phi ( \\xi _1(\\theta , \\rho ), \\xi _2(\\theta , g, \\rho ), \\theta , g, \\rho ).$ Lemma 7.3 $(\\theta ^{\\prime }, g^{\\prime }, \\rho ^{\\prime })$ is a critical point of $\\psi $ exactly when $(\\xi _1(\\theta ^{\\prime }), \\xi _2(\\theta ^{\\prime }), \\theta ^{\\prime }, g^{\\prime }, \\rho ^{\\prime })$ is a critical point of $\\phi $ .", "If $(\\theta ^{\\prime }, g^{\\prime }, \\rho ^{\\prime })$ is a critical point of $\\psi $ , let $\\kappa $ and $h$ be the values associated to the corresponding critical point of $\\phi $ .", "We then have $\\frac{\\partial ^2 \\psi }{\\partial \\theta ^2} (\\theta ^{\\prime }, g^{\\prime }, \\rho ^{\\prime }) = -\\frac{3}{2} \\kappa ^2 (1 - e^{2h}).$ We shall fix $g$ and $\\rho $ , and omit them from the arguments of $\\phi $ and $\\psi $ .", "Let $D$ be the Hessian of $\\phi $ calculated in Proposition REF .", "If we apply the chain rule to $\\psi $ and substitute $\\theta = \\theta ^{\\prime }$ , we obtain $\\frac{\\partial ^2 \\psi }{\\partial \\theta ^2}(\\theta ^{\\prime }) = ( \\tfrac{ \\partial \\xi _1}{\\partial \\theta }(\\theta ^{\\prime }), \\tfrac{ \\partial \\xi _2}{\\partial \\theta }(\\theta ^{\\prime }), 1) D ( \\tfrac{ \\partial \\xi _1}{\\partial \\theta }(\\theta ^{\\prime }), \\tfrac{ \\partial \\xi _2}{\\partial \\theta }(\\theta ^{\\prime }), 1)^t.$ To calculate $\\tfrac{ \\partial \\xi _1}{\\partial \\theta }(\\theta ^{\\prime })$ and $\\tfrac{ \\partial \\xi _2}{\\partial \\theta }(\\theta ^{\\prime })$ , we differentiate (REF ) with respect to $\\theta $ and set $\\theta = \\theta ^{\\prime }$ to obtain $\\frac{\\partial ^2 \\phi }{\\partial \\theta \\partial x_1} (\\xi _1(\\theta ^{\\prime }), \\xi _2(\\theta ^{\\prime }), \\theta ^{\\prime }) + \\frac{ \\partial \\xi _1}{\\partial \\theta }(\\theta ^{\\prime }) \\frac{\\partial ^2 \\phi }{\\partial x_1^2} (\\xi _1(\\theta ^{\\prime }), \\xi _2(\\theta ^{\\prime }), \\theta ^{\\prime }) = 0.$ Substituting the second partial derivatives of $\\phi $ calculated in Proposition REF gives $\\frac{ \\partial \\xi _1}{\\partial \\theta }(\\theta ^{\\prime }) = \\frac{-2 \\kappa }{\\sin \\beta _1^{\\prime }},$ and likewise $\\frac{ \\partial \\xi _2}{\\partial \\theta }(\\theta ^{\\prime }) = \\frac{-2 \\kappa e^h}{\\sin \\beta _2^{\\prime }}.$ The lemma follows on substituting these into equation (REF ).", "It follows that the set of $(g,\\rho )$ for which the function $\\psi (\\theta , g, \\rho )$ has a degenerate critical point are exactly those for which either $\\ell = g \\ell $ or $\\angle (\\ell , g\\ell ) = \\pm 2\\alpha $ .", "Note that these two cases are distinct, as $\\alpha \\in (0, \\pi /2)$ .", "In the first case the function $\\psi (\\theta , g, \\rho )$ vanishes identically.", "In the second case, $\\psi (\\theta , g, \\rho )$ has only a single degenerate critical point, as no oriented geodesic can cross $\\ell $ and $g \\ell $ making an angle of $\\alpha $ with both except at their point of intersection.", "To determine this critical point, the condition that $\\angle (\\ell , g\\ell ) = \\pm 2\\alpha $ implies that $g \\in a(y) k(\\pm 2\\alpha ) A$ for some $y \\in \\mathbb {R}$ , so that $\\ell \\cap g \\ell = a(y)i$ .", "The angle bisector of the two geodesics at the point $a(y)i$ is $a(y) k(\\pm \\alpha ) \\ell $ , and the critical point of $\\psi (\\theta , g, \\rho )$ is the $\\theta $ such that the positive endpoint of $k(\\theta ) a(y) k(\\pm \\alpha ) \\ell $ is $i \\infty $ .", "This is equivalent to the condition $k(\\theta ) a(y) k(\\pm \\alpha ) \\in NA$ , and equation (REF ) then gives $\\cot \\theta /2 = \\mp e^y \\cot \\alpha /2$ .", "We define $\\mathcal {D}_1 & = \\lbrace (\\theta , g, \\rho ) \\in \\mathcal {P}\\setminus \\mathcal {S}| g \\in A \\rbrace \\\\\\mathcal {D}_2^\\pm & = \\lbrace (\\theta , g, \\rho ) \\in \\mathcal {P}\\setminus \\mathcal {S}| g \\in a(y) k(\\pm 2\\alpha ) A , \\cot \\theta /2 = \\mp e^y \\cot \\alpha /2 \\rbrace $ to be the three sets on which $\\psi $ has a degenerate critical point.", "We also define $\\overline{\\mathcal {P}} = PSL_2(\\mathbb {R}) \\times [\\delta , 1-\\delta ]$ , and define $\\overline{\\mathcal {D}}_1 & = A \\times [\\delta , 1-\\delta ] \\\\\\overline{\\mathcal {D}}_2^\\pm & = \\lbrace (g, \\rho ) \\in \\overline{\\mathcal {P}} | g \\in A k(\\pm 2\\alpha ) A \\rbrace $ to be the projections of $\\mathcal {D}_1$ and $\\mathcal {D}_2^\\pm $ to $\\overline{\\mathcal {P}}$ ." ], [ "The degenerate set $\\mathcal {D}_1$", "As $\\psi (\\theta , g, \\rho ) = \\psi (\\theta , ga, \\rho )$ for $a \\in A$ , we may study the degeneracy of $\\psi $ near $\\mathcal {D}_1$ by differentiating $\\psi (\\theta , \\exp (X), \\rho )$ at $X = 0$ as in the following proposition.", "Proposition 7.4 If $X = \\left( \\begin{array}{cc} 0 & X_1 \\\\ X_2 & 0 \\end{array} \\right) \\in \\mathfrak {g}$ , then $\\frac{\\partial }{\\partial t} \\psi (\\theta , \\exp (tX), \\rho ) \\Big |_{t=0} = \\epsilon \\sin \\alpha ( e^{-\\xi _2(\\theta , e, \\rho )} X_1 + e^{\\xi _2(\\theta , e, \\rho )} X_2 ),$ where $\\epsilon $ is 1 if $\\theta \\in (0,\\pi )$ and $-1$ otherwise.", "In particular, $\\partial \\psi / \\partial t (\\theta , \\exp (tX), \\rho ) |_{t=0}$ has no degenerate critical points unless $X = 0$ .", "Let $x_1^{\\prime } = \\xi _1(\\theta , \\rho )$ and $x_2^{\\prime } = x_2(\\theta , e, \\rho )$ .", "We have $\\frac{\\partial }{\\partial t} \\psi (\\theta , \\exp (tX), \\rho ) \\Big |_{t=0} & = \\frac{\\partial }{\\partial t} \\phi ( \\xi _1(\\theta , \\rho ), \\xi _2(\\theta , \\exp (tX), \\rho ), \\theta , \\exp (tX), \\rho ) \\Big |_{t=0} \\\\& = \\frac{\\partial \\phi }{\\partial x_2} ( \\xi _1(\\theta , \\rho ), \\xi _2(\\theta , e, \\rho ), \\theta , e, \\rho ) \\frac{\\partial }{\\partial t} \\xi _2(\\theta , \\exp (tX), \\rho ) \\Big |_{t=0} \\\\& \\qquad + \\frac{\\partial }{\\partial t} \\phi ( \\xi _1(\\theta , \\rho ), \\xi _2(\\theta , e, \\rho ), \\theta , \\exp (tX), \\rho ) \\Big |_{t=0}.$ The first term vanishes by (REF ), so we are left with $\\frac{\\partial }{\\partial t} \\psi (\\theta , \\exp (tX), \\rho ) \\Big |_{t=0} & = \\frac{\\partial }{\\partial t} \\phi ( \\xi _1(\\theta , \\rho ), \\xi _2(\\theta , e, \\rho ), \\theta , \\exp (tX), \\rho ) \\Big |_{t=0} \\\\& = \\frac{\\partial }{\\partial t} A( k(\\theta ) \\exp (tX) a(\\xi _2(\\theta , e, \\rho )) ) \\Big |_{t=0}.$ We shall abbreviate $\\xi _2(\\theta , e, \\rho )$ to $\\xi _2(\\theta )$ for the remainder of the proof.", "Write the first order approximation to the Iwasawa decomposition of $k(\\theta ) \\exp (tX) a(\\xi _2(\\theta ))$ as $k(\\theta ) \\exp (tX) a(\\xi _2(\\theta )) = n \\exp ( t X_N + O(t^2)) a \\exp ( t X_A + O(t^2)) k \\exp ( t X_K + O(t^2)),$ where $X_N \\in \\mathfrak {n}$ , $X_A \\in \\mathfrak {a}$ , and $X_K \\in \\mathfrak {k}$ .", "As in equation (REF ), we have $k = k(\\alpha )$ if $\\theta \\in (0,\\pi )$ and $k = k(-\\alpha )$ if $\\theta \\in (-\\pi , 0)$ .", "We first assume that $\\theta \\in (0, \\pi )$ .", "Rearranging and equating first order terms gives $X & = \\text{Ad}(a(\\xi _2(\\theta )) k(\\alpha )^{-1} a^{-1}) X_N + \\text{Ad}(a(\\xi _2(\\theta )) k(\\alpha )^{-1}) X_A + \\text{Ad}(a(\\xi _2(\\theta ))) X_K \\\\\\text{Ad}( k(\\alpha ) a(\\xi _2(\\theta ))^{-1}) X & = \\text{Ad}(a^{-1}) X_N + X_A + \\text{Ad}(k(\\alpha )) X_K$ As $\\text{Ad}(a^{-1}) X_N$ and $\\text{Ad}(k(\\alpha )) X_K$ lie in $\\mathfrak {a}^\\perp \\subset \\mathfrak {g}$ , we see that $X_A$ is the orthogonal projection of $\\text{Ad}( k(\\alpha ) a(\\xi _2(\\theta ))^{-1}) X$ to $\\mathfrak {a}$ .", "A calculation gives $X_A = \\sin \\alpha (e^{-\\xi _2(\\theta )} X_1 + e^{\\xi _2(\\theta )} X_2 ) \\left( \\begin{array}{cc} 1/2 & 0 \\\\ 0 & -1/2 \\end{array} \\right),$ so that $\\frac{\\partial }{\\partial t} A( k(\\theta ) \\exp (tX) a(\\xi _2(\\theta )) ) \\Big |_{t=0} = \\sin \\alpha (e^{-\\xi _2(\\theta )} X_1 + e^{\\xi _2(\\theta )} X_2 ).$ This proves (REF ) when $\\theta \\in (0,\\pi )$ , and the other case is identical.", "We now prove that $\\partial \\psi / \\partial t (\\theta , \\exp (tX), \\rho ) |_{t=0}$ has no degenerate critical points if $X \\ne 0$ and $\\theta \\in (0, \\pi )$ .", "We define $f(x) = \\sin \\alpha (X_1 e^{-x} + X_2 e^x)$ , so that $\\frac{\\partial }{\\partial t} A( k(\\theta ) \\exp (tX) a(\\xi _2(\\theta )) ) \\Big |_{t=0} = f( \\xi _2(\\theta )).$ Differentiating equation (REF ) gives $\\frac{ \\partial \\xi _2}{\\partial \\theta } = -\\frac{1}{2} e^{-\\xi _2(\\theta )} \\tan (\\alpha /2) \\csc ^2(\\theta /2),$ so that $\\partial \\xi _2 / \\partial \\theta $ is always nonzero.", "Suppose that $X \\ne 0$ , and that $\\theta $ is a degenerate critical point of $\\partial \\psi / \\partial t (\\theta , \\exp (tX), \\rho ) |_{t=0}$ .", "We then have $0 & = \\frac{\\partial ^2}{\\partial \\theta \\partial t} A( k(\\theta ) \\exp (tX) a(\\xi _2(\\theta )) ) \\Big |_{t=0} \\\\& = f^{\\prime }( \\xi _2(\\theta )) \\frac{ \\partial \\xi _2}{\\partial \\theta } \\\\& = f^{\\prime }( \\xi _2(\\theta )).$ Differentiating again with respect to $\\theta $ gives $0 & = \\frac{\\partial ^3}{\\partial ^2 \\theta \\partial t} A( k(\\theta ) \\exp (tX) a(\\xi _2(\\theta )) ) \\Big |_{t=0} \\\\& = f^{\\prime \\prime }( \\xi _2(\\theta )) \\left( \\frac{ \\partial \\xi _2}{\\partial \\theta } \\right)^2 \\\\& = f^{\\prime \\prime }( \\xi _2(\\theta )),$ but this is a contradiction as it may be easily checked that $f$ has no degenerate critical points unless $X = 0$ .", "The case of $\\theta \\in (-\\pi , 0)$ is identical.", "Define $P = \\mathbb {R}/ 2\\pi \\mathbb {Z}\\times \\mathfrak {a}^\\perp \\times [\\delta , 1-\\delta ]$ , and define $S = \\lbrace (\\theta , X, \\rho ) \\in P | (\\theta , \\exp (X), \\rho ) \\in \\mathcal {S}\\rbrace $ .", "$S$ is again closed, and contains at most 4 values of $\\theta $ for each fixed $(X, \\rho )$ .", "Lemma 7.5 There is an open neighbourhood $0 \\in U \\subset \\mathfrak {a}^\\perp $ such that for all $X \\in U$ and all $b \\in C^\\infty _0( P \\setminus S)$ we have $\\int b(\\theta , X, \\rho ) e^{is \\psi (\\theta , \\exp (X), \\rho )} d\\theta \\ll (1 + s \\Vert X\\Vert )^{-1/2},$ where $\\Vert X \\Vert $ is as in (REF ).", "Define the map $X : \\mathbb {R}\\times \\mathbb {R}/ 2\\pi \\mathbb {Z}\\rightarrow \\mathfrak {a}^\\perp $ by $X(r, \\gamma ) = \\left( \\begin{array}{cc} 0 & r \\sin \\gamma \\\\ r \\cos \\gamma & 0 \\end{array} \\right).$ We define $\\widetilde{P} = \\mathbb {R}/ 2\\pi \\mathbb {Z}\\times \\mathbb {R}\\times \\mathbb {R}/ 2\\pi \\mathbb {Z}\\times [\\delta , 1-\\delta ]$ and $\\widetilde{S} = \\lbrace (\\theta , r, \\gamma , \\rho ) \\subset \\widetilde{P} | (\\theta , X(r,\\gamma ), \\rho ) \\in S \\rbrace .$ We define $\\widetilde{b}(\\theta , r, \\gamma , \\rho ) \\in C^\\infty _0(\\widetilde{P} \\setminus \\widetilde{S})$ and $\\widetilde{\\psi }(\\theta , r, \\gamma , \\rho ) \\in C^\\infty (\\widetilde{P} \\setminus \\widetilde{S})$ to be the pullbacks of $b$ and $\\psi $ under $X$ .", "We know that $\\widetilde{\\psi }$ vanishes when $r = 0$ , and as $\\widetilde{\\psi }$ is smooth (in fact, real analytic) we have that $\\widetilde{\\psi }/r$ is again a smooth function.", "Proposition REF implies that $\\widetilde{\\psi }/r$ has no degenerate critical points when $r = 0$ , and so there is some $\\epsilon > 0$ such that it also has no degenerate critical points on the set $\\text{supp}(\\widetilde{b}) \\cap (\\mathbb {R}/ 2\\pi \\mathbb {Z}\\times [-\\epsilon , \\epsilon ] \\times \\mathbb {R}/ 2\\pi \\mathbb {Z}\\times [\\delta , 1-\\delta ])$ .", "If we define $U = X( (-\\epsilon , \\epsilon ) \\times \\mathbb {R}/ 2\\pi \\mathbb {Z})$ , the result now follows from stationary phase.", "Corollary 7.6 If $(a^{\\prime }, \\rho ^{\\prime }) \\in \\overline{\\mathcal {D}}_1$ , there is an open neighbourhood $(a^{\\prime }, \\rho ^{\\prime }) \\in U \\subset \\overline{\\mathcal {P}}$ such that for all $b \\in C^\\infty _0(\\mathcal {P}\\setminus \\mathcal {S})$ and all $(g, \\rho ) \\in U$ , we have $\\int _0^{2\\pi } b(\\theta , g, \\rho ) e^{is \\psi (\\theta , g, \\rho )} d\\theta \\ll (1 + s n(\\ell _0, g \\ell _0))^{-1/2}.$ Let $U_X \\subset \\mathfrak {a}^\\perp $ be as in Lemma REF .", "If $g = \\exp (X) a^{\\prime }$ for $X \\in U_X$ , we have $n(\\ell _0, g \\ell _0) \\sim \\Vert X \\Vert $ , where the implied constants depend on $a^{\\prime }$ .", "As $\\psi (\\theta , ga, \\rho ) = \\psi (\\theta , g, \\rho )$ for $a \\in A$ , the result follows from Lemma REF ." ], [ "The degenerate set $\\mathcal {D}_2^\\pm $", "The next proposition proves that $\\psi $ has a cubic degeneracy on $\\mathcal {D}_2^\\pm $ .", "Proposition 7.7 If $(\\theta ^{\\prime }, g^{\\prime }, \\rho ^{\\prime }) \\in \\mathcal {D}_2^\\pm $ , we have $\\partial ^3 \\psi / \\partial \\theta ^3(\\theta ^{\\prime }, g^{\\prime }, \\rho ^{\\prime }) \\ne 0$ .", "Figure: Two degenerating critical geodesics.Suppose $g^{\\prime } = a(y) k(2\\alpha ) a_2$ .", "Define $g = a(y) k(2\\alpha + \\epsilon ) a_2$ for some $\\epsilon > 0$ .", "If $\\epsilon $ is chosen sufficiently small, the pair $(\\ell , g \\ell )$ will have exactly two critical geodesics $\\ell _1$ and $\\ell _2$ as shown in Figure 1.", "The triangles $AB_1C_1$ and $AB_2C_2$ both have angular defect, and hence area, $\\epsilon $ .", "Our assumption that $\\alpha $ was bounded away from 0 and $\\pi /2$ then implies that $AB_1 = AB_2 \\sim \\epsilon ^{1/2}$ and $B_1C_1 = B_2C_2 \\sim \\epsilon ^{1/2}$ , where the implied constants depends only on $\\delta $ .", "The critical points $\\theta _i$ corresponding to $\\ell _i$ are the solutions to $\\cot \\theta _1/2 = -e^{y+AB_1} \\cot \\alpha /2, \\qquad \\cot \\theta _2/2 = -e^{y-AB_1} \\cot \\alpha /2.$ It follows that $0 > \\theta _1 > -\\alpha > \\theta _2 > -\\pi $ , and also that $\\theta _1 - \\theta _2 \\sim \\epsilon ^{1/2}$ .", "The apertures $h_i$ of the critical points $\\theta _i$ are given by $h_1 = - B_1C_1 \\sim -\\epsilon ^{1/2}$ and $h_2 = B_2C_2 \\sim \\epsilon ^{1/2}$ , so that Lemma REF gives $\\frac{\\partial ^2 \\psi }{\\partial \\theta ^2} (\\theta _1, g, \\rho ^{\\prime }) \\sim -\\epsilon ^{1/2}, \\quad \\frac{\\partial ^2 \\psi }{\\partial \\theta ^2} (\\theta _2, g, \\rho ^{\\prime }) \\sim \\epsilon ^{1/2}.$ It follows that there is $\\theta _0 \\in [\\theta _2, \\theta _1]$ at which $\\frac{\\partial ^3 \\psi }{\\partial \\theta ^3} (\\theta _0, g, \\rho ^{\\prime }) = \\frac{ \\frac{\\partial ^2 \\psi }{\\partial \\theta ^2} (\\theta _2, g, \\rho ^{\\prime }) - \\frac{\\partial ^2 \\psi }{\\partial \\theta ^2} (\\theta _1, g, \\rho ^{\\prime }) }{ \\theta _2 - \\theta _1} \\sim -1,$ and shrinking $\\epsilon $ to 0 gives the result.", "The case $g^{\\prime } \\in A k(-2\\alpha ) A$ is identical.", "Corollary 7.8 If $(g^{\\prime }, \\rho ^{\\prime }) \\in \\overline{\\mathcal {D}}_2^\\pm $ , there is an open neighbourhood $(g^{\\prime }, \\rho ^{\\prime }) \\in U \\subset \\overline{\\mathcal {P}}$ such that for all $b \\in C^\\infty _0(\\mathcal {P}\\setminus \\mathcal {S})$ and all $(g, \\rho ) \\in U$ , we have $\\int _0^{2\\pi } b(\\theta , g, \\rho ) e^{is \\psi (\\theta , g, \\rho )} d\\theta \\ll s^{-1/3}.$ By Proposition REF , there exists a neighbourhood $U_\\theta $ of $\\theta ^{\\prime }$ and $U$ of $(g^{\\prime }, \\rho ^{\\prime })$ such that $(U_\\theta \\times U) \\cap \\mathcal {S}= \\emptyset $ , and $|\\partial ^3 \\psi / \\partial \\theta ^3| \\ge \\sigma > 0$ on $U_\\theta \\times U$ .", "As $\\psi (\\theta , g^{\\prime }, \\rho ^{\\prime })$ only has a critical point at $\\theta ^{\\prime }$ , by shrinking $U$ we may also assume that $\\psi $ has no critical points on $(\\mathbb {R}/ 2\\pi \\mathbb {Z}\\setminus U_\\theta ) \\times U \\setminus \\mathcal {S}$ .", "The result then follows from Proposition 2, Section 1.2, Chapter VIII of [16]." ], [ "Bounds for $I(t, \\lambda , \\ell _1, \\ell _2)$", "We shall use the results of the previous sections to prove the follwing proposition, which implies Proposition REF in the case $\\lambda / t \\in [\\delta , 1-\\delta ]$ after inverting the Harish-Chandra transform.", "Proposition 7.9 Let $D \\subset PSL_2(\\mathbb {R})$ be a compact set, let $b_1, b_2 \\in C^\\infty _0(\\mathbb {R})$ be functions supported in $[0,1]$ , and let $1/2 > \\delta > 0$ .", "For $g \\in PSL_2(\\mathbb {R})$ and $\\lambda , s \\in \\mathbb {R}$ , define $I(s, \\lambda , g) = \\iint _{-\\infty }^\\infty e^{i \\lambda (x_1-x_2)} b_1(x_1)b_2(x_2) \\varphi _{-s}( \\ell (x_1), g\\ell (x_2)) dx_1 dx_2.$ If $g \\in D$ and $\\lambda / s \\in [\\delta , 1-\\delta ]$ , we have $I(s, \\lambda , g) \\ll \\Big \\lbrace \\begin{array}{ll} s^{-1}(1 + s n(\\ell _0, g\\ell _0))^{-1/2} & \\text{when} \\quad n(\\ell _0, g \\ell _0) \\le s^{-1/3} \\\\s^{-4/3} & \\text{when} \\quad n(\\ell _0, g \\ell _0) \\ge s^{-1/3}.\\end{array}$ If we substitute the expression (REF ) into (REF ), we obtain $\\int _0^{2\\pi } \\iint _{-\\infty }^\\infty b_1(x_1) b_2(x_2) e^{i\\lambda (x_1 - x_2)} \\exp ( (1/2 - is)( A(k(\\theta ) a(x_1)) - A( k(\\theta )g a(x_2))) \\frac{d\\sigma }{d\\theta } dx_1 dx_2 d\\theta .$ We let $b \\in C^\\infty _0(PSL_2(\\mathbb {R}))$ be a function that is equal to 1 on $D$ , and introduce a factor of $b(g)$ into the integral.", "When $g \\in D$ we then have $I(s, \\lambda , g) = \\int _0^{2\\pi } \\iint _{-\\infty }^\\infty e^{is \\phi (x_1, x_2, \\theta , g, \\rho )} c(x_1, x_2, \\theta , g, \\rho ) dx_1 dx_2 d\\theta ,$ where $c \\in C^\\infty _0(\\mathbb {R}^2 \\times \\mathcal {P})$ is the combination of all of the amplitude factors.", "The following lemma eliminates the variables $x_1$ and $x_2$ .", "Lemma 7.10 There is a function $c_1 \\in C^\\infty _0( \\mathcal {P}\\setminus \\mathcal {S})$ such that for all $(g, \\rho ) \\in D \\times [\\delta , 1-\\delta ]$ we have $I(s, \\lambda , g) = s^{-1} \\int _0^{2\\pi } e^{is \\psi (\\theta , g, \\rho )} c_1(\\theta , g, \\rho ) d\\theta + O(s^{-2}).$ We shall apply stationary phase in the $x_i$ variables.", "For fixed $(\\theta , g, \\rho )$ , the function $\\phi (x_1, x_2)$ has one critical point at $(\\xi _1(\\theta , \\rho ), \\xi _2(\\theta , g, \\rho ))$ if $(\\theta , g, \\rho ) \\notin \\mathcal {S}$ , and none otherwise.", "Moreover, it may be shown in the same way as the proof of Proposition REF that the Hessian at this critical point is $D = \\left( \\begin{array}{cc} \\tfrac{1}{2} \\sin ^2 \\alpha & 0 \\\\ 0 & -\\tfrac{1}{2} \\sin ^2 \\alpha \\end{array} \\right),$ so that the critical point is uniformly nondegenerate.", "Define $\\mathcal {P}_0 = \\lbrace (\\theta , g, \\rho ) \\in \\mathcal {P}\\setminus \\mathcal {S}| (\\xi _1(\\theta , \\rho ), \\xi _2(\\theta , g, \\rho ), \\theta , g, \\rho ) \\in \\text{supp}(c) \\rbrace ,$ so that $\\mathcal {P}_0$ is compact and $\\mathcal {P}_0 \\cap \\mathcal {S}= \\emptyset $ .", "If we define $c_1 \\in C^\\infty _0(\\mathcal {P}\\setminus \\mathcal {S})$ by $c_1(\\theta , g, \\rho ) = \\frac{2\\pi }{\\sin ^2 \\alpha } c( \\xi _1(\\theta , \\rho ), \\xi _2(\\theta , g, \\rho ), \\theta , g, \\rho ),$ then we have $\\text{supp}(c_1) \\subseteq \\mathcal {P}_0$ , and stationary phase gives $\\iint _{-\\infty }^\\infty e^{is \\phi (x_1, x_2, \\theta )} c(x_1, x_2, \\theta , g, \\rho ) dx_1 dx_2 = e^{is \\psi (\\theta , g, \\rho )} s^{-1} c_1(\\theta , g, \\rho ) + O(s^{-2})$ locally uniformly on $\\mathcal {P}\\setminus \\mathcal {S}$ .", "We also have $\\iint _{-\\infty }^\\infty e^{is \\phi (x_1, x_2, \\theta )} c(x_1, x_2, \\theta , g, \\rho ) dx_1 dx_2 \\ll _A s^{-A}$ locally uniformly on $\\mathcal {P}\\setminus \\mathcal {P}_0$ .", "Therefore, if we extend $c_1$ to a function in $C^\\infty (\\mathcal {P})$ by making it 0 on $\\mathcal {S}$ , then (REF ) holds locally uniformly on $\\mathcal {P}$ and the lemma follows.", "We now apply Corollaries REF and REF .", "Corollary REF implies that there is an open neighbourhood $U_1$ of $\\overline{\\mathcal {D}}_1 \\cap (D \\times [\\delta , 1-\\delta ])$ in $\\overline{\\mathcal {P}}$ such that $I(s, \\lambda , g) \\ll s^{-1}(1 + s n(\\ell _0, g\\ell _0))^{-1/2}$ when $(g, \\rho ) \\in U_1 \\cap (D \\times [\\delta , 1-\\delta ])$ , and Corollary REF implies that there is a neighbourhood $U_2$ of $\\overline{\\mathcal {D}}_2^\\pm \\cap (D \\times [\\delta , 1-\\delta ])$ such that $I(s, \\lambda , g) \\ll s^{-4/3}$ when $(g, \\rho ) \\in U_2 \\cap (D \\times [\\delta , 1-\\delta ])$ .", "As $\\psi $ has no degenerate critical points outside $\\overline{\\mathcal {D}}_1 \\cup \\overline{\\mathcal {D}}_2^\\pm $ , we also have $I(s, \\lambda , g) \\ll s^{-3/2}$ when $(g, \\rho ) \\in (D \\times [\\delta , 1-\\delta ]) \\setminus (U_1 \\cup U_2)$ .", "As the bound in Proposition REF is the maximum of these three bounds, this completes the proof.", "It remains to discuss the case when $\\lambda = 0$ , so that $\\alpha = \\pi /2$ .", "The proof proceeds as before, until the analysis of the degenerate critical points of $\\psi $ .", "These degeneracies now occur when $g \\in A \\cup \\left( \\begin{array}{cc} 0 & 1 \\\\ -1 & 0 \\end{array} \\right) A,$ and the function $\\psi $ vanishes identically at these points.", "These degeneracies may be treated in exactly the same way as $\\mathcal {D}_1$ in Section REF , which gives the bound $I(s, 0, g) \\ll s^{-1}(1 + s n(\\ell _0, g\\ell _0))^{-1/2}.$ Inverting the Harish-Chandra transform completes the proof." ] ]

[ [ "Circumbinary Planet Formation in the Kepler-16 system. I. N-body\n Simulations" ], [ "Abstract The recently discovered circumbinary planets (Kepler-16 b, Kepler-34 b, Kepler-35 b) represent the first direct evidence of the viability of planet formation in circumbinary orbits.", "We report on the results of N-body simulations investigating planetesimal accretion in the Kepler-16 b system, focusing on the range of impact velocities under the influence of both stars' gravitational perturbation and friction from a putative protoplanetary disk.", "Our results show that planet formation might be effectively inhibited for a large range in semi-major axis (1.75 < a_P < 4 AU), suggesting that the planetary core must have either migrated from outside 4 AU, or formed in situ very close to its current location." ], [ "Introduction", "The discovery of extrasolar planets around main-sequence stars is one of the major observational breakthroughs of the last decade.", "The size of the planetary census, propelled by radial velocity (RV) surveys and dedicated missions such as Kepler, has grown to include planetary systems where a variety of interesting dynamical interactions can be observed.", "Such systems include 61 exoplanets discovered around stellar binarieshttp://www.exoplanets.org, retrieved on February 14, 2012.", "(including both planets orbiting one of the stellar companions and circumbinary planets).", "While for the majority of these planets the binarity of the system represents only a weak perturbation on the gravitational pull of the central star, a few single-planet systems have been detected in binaries with $a_\\mathrm {bin} \\lesssim $ 30 AU (such as HD 41004, Gliese 86, HD196885 and $\\gamma $ Cephei), with each planet in a circumstellar (“S-type”) orbit.", "Only one multiple system with $a_\\mathrm {bin} \\lesssim 100$ AU has been found [14].", "The existence of these systems represents a major challenge to the current paradigm of planet formation.", "In fact, a number of simulations attempting to model the dynamics of the growth of planetary embryos from km-sized planetesimals in presence of a binary companion have hit significant difficulties [11], [31], [34], [32], [29], [19], [4].", "The most important parameter controlling planetesimal accretion is the mutual encounter velocity; indeed, runaway growth requires it to be less than the escape velocity for efficient accretion.", "The presence of the companion can stir up the relative velocity between planetesimals, interfering with runaway growth.", "Relative velocity is often excited beyond a fiducial threshold velocity at which all encounters are erosive, potentially slowing down planet formation or halting it altogether.", "Simulations taking into account a static background gas disk (representing an unperturbed protoplanetary disk at some point in time) initially posited that disk-planetesimals interaction induces a phasing of the orbits, making the environment more accretion-friendly [11].", "Nevertheless, if the protoplanets interact with the gas disk through aerodynamic drag alone, the phasing induced by the gas disk is clearly size-dependent, and protoplanets with different sizes will collide with large encounter speeds over the majority of the range in semi-major axis sampled [34].", "Finally, a misalignment between the orbital plane of the binary and the gas disk can significantly affect the dynamics of the planetesimals.", "Small inclinations ($i_\\mathrm {B} < 10^\\circ $ ) can favor planetesimal accretion somewhat [40].", "On the other hand, large inclinations ($30^\\circ < i_\\mathrm {B} < 50^\\circ $ ) can significantly perturb the planetesimal disk, causing planetesimals to “jump” inwards and pile up into a smaller inner disk, where encounter velocities are more favorable to accretion [39].", "Most of the works in the literature have focused on observed or plausible circumstellar configurations (e.g.", "a planet orbiting one of the two stellar components), in light of the lack of direct evidence of the existence of circumbinary planets orbiting main-sequence stars, outside the realm of science fiction.", "Therefore, only a handful of articles have considered planet formation in circumbinary orbits [17], [24], [26], [13], [21], [22], and they lacked a reference observed configuration.", "Kepler 16-b [3] is the first circumbinary planet that has been detected with Kepler.", "The presence of a third object was first hinted through deviations of the timing of the stellar eclipses from a linear ephemeris.", "The definitive characterization as a planetary object came from transits on both star A (tertiary eclipse) and star B (quaternary eclipse).", "The planet was determined to be a Saturn-mass planet ($\\mathcal {M}_\\mathrm {P} \\approx 0.33 \\mathcal {M}_\\mathrm {Jup}$ ) on a nearly circular 228-day orbit; long-term integrations have shown the planet to be stable, with an eccentricity oscillating between 0 and $\\approx 0.08$ .", "The binary stellar system is composed of two main-sequence stars in an eccentric 41-day orbit, with a mass of 0.69 and 0.2 $\\mathcal {M}_\\mathrm {\\odot }$ (mass ratio $\\mu \\approx 0.2$ ), respectively.", "The close coplanarity of the binary and planetary orbital planes suggests that the three bodies were formed in a common disk.", "This was bolstered by the measurement of the Rossiter-McLaughlin doppler shift by [37], which indicated that the spin of the primary is aligned as well.", "Recently, [36] reported the discovery of two additional circumbinary gas giants (Kepler-34 b and Kepler-35 b).", "The relative abundance of these systems among the more than 2,000 eclipsing binaries monitored by Kepler [27] implies a lower limit of $\\approx 1\\%$ in the frequency of circumbinary planets with comparable transit probabilities.", "Interestingly, all three planets lie just outside the stability boundary for test particles.", "Their pericenter distance is, respectively, only $\\approx $ 6% (Kepler-34 b), 9% (Kepler-16 b) and 20% (Kepler 35-b) larger than the critical semi-major axis, as estimated by the empirical fit in [7].", "This represents an important constraint for the formation of the planetary core.", "Indeed, a natural scenario would entail the planetary core migrating inwards until near the edge of the disk cavity (which will be comparable in extent to the stability boundary for test particles), where the steep gradient of the disk surface density can halt migration [20].", "[22] simulated the evolution of a 20 $\\mathcal {M}_\\mathrm {}$ core, initially placed at the edge of the cavity and free to accrete gas to become a Saturn-mass planet.", "They found that once the planet depletes the gas in the coorbital region, it will resume a slow inward migration, until its eccentricity is excited and a phase of runaway outward migration is experienced.", "This runaway migration appeared to stop once the planet crossed the 5:1 resonance with the binary, at which point slow migration is resumed.", "The ultimate fate of the planet in these simulations is uncertain, due to the long timescales involved.", "However, it is expected that disk dispersal will ultimately strand the planet on a circular orbit around the binary.", "Tantalizingly, Kepler-16b lies somewhat close (and outside of) the 5:1 period ratio with the binary.", "In this paper, we investigate the conditions for the formation of planetary cores in circumbinary orbits around the Kepler-16 binary system, using a simplified numerical model.", "We consider the evolution of a disk of $km$ -sized planetesimals and determine the impact velocities among planetesimals over $10^5$ years, the typical timescale for runaway and oligarchic accretion [9].", "These preliminary $N$ -body simulations will be used to assess the viability of core accretion as a function of the barycentric semi-major axis.", "The plan of the paper is as follows.", "In §, we briefly discuss our numerical model and limitations of our current approach.", "In § we discuss the results of our simulations in the context of planet formation, and conclude in §." ], [ "Numerical setup", "To conduct our simulations, we use a new hybrid code, Sphiga (described in Meschiari et al., 2012, in preparation).", "Sphiga is an $N$ -body code that evolves a system of non-interacting test particles (e.g.", "the planetesimals) subjected to the sum of gravitational forces of massive bodies (e.g.", "the binary system).", "In addition, it calculates the frictional force acting on the test particles caused by a putative protoplanetary disk.", "By default, this is accomplished by following the complete hydrodynamical evolution of the disk with the Smoothed Particle Hydrodynamics scheme [25], [23] in two and three dimensions.", "The same algorithm used to interpolate the hydrodynamical quantities can be used to interpolate the local gas density and flow and locate possible planetesimal impactors a single loop, leading to significant computational savings.", "Modelling the self-consistent perturbations from the binary on the disk can alter the planetesimal evolution and potentially increase impact velocities [13].", "Indeed, we expect that non-axisymmetric structure, such as global spiral patterns, will be imposed by the binary, adding a complex time-dependent term.", "The actual impact of the full hydrodynamical evolution is still uncertain, however.", "Even bulk quantities such as the disk eccentricity induced by a binary companion appear to depend sensitively on the computational scheme [19] and the amount of physics modeled [12], [10].", "Nevertheless, significant computational effort is still required to follow the evolution of the combined disk, binary and planetesimal system (with $N_\\mathrm {pl} + N_\\mathrm {gas} > 10^6$ particles) for at least $\\approx 10^5$ binary revolutions.", "Therefore, for the purpose of this paper, we will use an alternative code path that activates a fixed gas disk.", "The gas disk exerts a frictional acceleration at the location of the planetesimal given by $\\mathbf {f} = -K |\\delta \\mathbf {v}| \\delta \\mathbf {v}\\ ,$ In Equation (REF ), $\\delta \\mathbf {v} = \\mathbf {v}_\\mathrm {pl} - \\mathbf {v}_\\mathrm {gas}$ is the relative velocity of the planetesimal with respect to the Keplerian flow of the gas and $K$ is the drag parameter $K = \\frac{\\pi C \\rho _\\mathrm {g} \\mathcal {R}_\\mathrm {pl}^2}{2 \\mathcal {M}_\\mathrm {pl}}\\ .$ The drag parameter is defined in terms of the planetesimal radius $\\mathcal {R}_\\mathrm {pl}$ , the planetesimal mass $\\mathcal {M}_\\mathrm {pl}$ (calculated assuming $\\rho _\\mathrm {pl} = 3$ g/cm$^3$ ), and the dimensionless coefficient $C$ ($C \\approx 0.4$ for spherical bodies).", "We use the standard prescription of a minimum-mass solar nebula [5] for the disk parameters.", "In this configuration, our code and physical setup is functionally equivalent to that used by [26].", "To evaluate the collisional speeds among planetesimals, we follow the dynamical evolution of 30,000 test particles uniformly distributed with barycentric semi-major axes between 0.66 and 6 AU; this range includes the current location of the planet ($a_\\mathrm {P} \\approx 0.7$ AU).", "The inner boundary was determined by running a simulation with test particles in circular barycentric orbits covering semi-major axes in the range $(1.2 a_\\mathrm {b}; 5 a_\\mathrm {b})$ for $10^4$ years; we found very good agreement with the fit of [7].", "Particles that travel into the inner boundary or become unbound are removed from the simulation.", "The system is initially evolved to $10^5$ years.", "After this interval, planetesimal-planetesimal close encounters are recorded, with the most important parameter being $\\Delta v$ , the impact velocity.", "We follow [4] and [29] and adopt the prescription for classifying disruptive impacts for planetesimals presented in [28].", "The latter work offers a criterion for catastrophic disruption, the main parameters being the reduced kinetic energy, the masses of the impactors and material properties and constants derived from fits to numerical and laboratory data.", "Planetesimal collisions are tracked using the inflated radius prescription [2], [30], with $\\mathcal {R}_\\mathrm {infl} = 5\\times 10^{-5}$ AU.", "The code detects collisions by populating a tree structure at each timestep (as part of the SPH algorithm) and walking the tree to locate the nearest neighbors to each planetesimal with $d < 2\\mathcal {R}_\\mathrm {infl}$ [1], [6].", "In our simulation, we assign a planetesimal radius for each particle, randomly distributed between 1 and 10 km.", "We allow for a non-flat primordial distribution in planetesimal sizes by assigning a weight $f(\\mathcal {R}_\\mathrm {1}, \\mathcal {R}_\\mathrm {2})$ to each impact between planetesimals of radius $\\mathcal {R}_\\mathrm {1}$ and $\\mathcal {R}_\\mathrm {2}$ .", "Following [33], we use a Maxwellian weighting function centered around 5 km with $\\sigma = 1$ km.", "A priori, this choice should yield a more accretion-friendly environment, since it weighs collisions between same-sized planetesimals more than different-sized planetesimals." ], [ "Simulations", "As expected, the planetesimals are quickly perturbed from their initial low-eccentricity configuration by the gravitational stirring of the central binary.", "Their eccentricities initially oscillate around the forced eccentricity $e_\\mathrm {f} = \\frac{5}{4}(1 - 2\\mu ) \\frac{a_\\mathrm {B}}{a} e_\\mathrm {B}$ [17].", "The presence of gas drag tends to damp the eccentricity oscillations towards the forced eccentricity over longer timescales.", "Damping and periastron phasing will be more effective for smaller (since the gas drag coefficient is proportional to $\\mathcal {R}_\\mathrm {pl}^{-1}$ ) and close-in bodies (since $\\rho _{gas} \\propto a^{-2.75}$ ).", "However, the eccentricity spread remains somewhat large at small semi-major axes, where the gravitational perturbation of the central binary acts to pump eccentricities.", "At large semi-major axes, where the damping timescale is longer, the values of eccentricity tend to their counterparts in gas-free simulations.", "In the inner parts of the disk, planetesimals will spiral into the inner boundary due to radial drift.", "The radial drift timescale can be estimated by assuming the planetesimal loses angular momentum slowly due to the torque from the headwind of the gas [35].", "For the drag prescription of Equation REF , we find an estimate for the infall timescale (in units where $G\\mathcal {M}= 1$ ) is given by $\\tau _\\mathrm {rd} = \\frac{a_\\mathrm {pl}}{v_\\mathrm {rd}} \\approx \\frac{4}{3} C^{-1} \\mathcal {M}_* \\frac{\\rho _\\mathrm {pl}}{\\rho _\\mathrm {gas}} \\frac{\\mathcal {R}_\\mathrm {pl}}{a_\\mathrm {pl}^{1/2}} \\left(\\delta v\\right)^{-2}\\ ,$ where $\\rho _\\mathrm {pl}$ is the density of the planetesimal and $\\mathcal {M}_*$ is the total mass of the binary.", "In the case of planet formation around single stars, eccentricities are very low and $\\delta v \\sim h_\\mathrm {0}^2 v_\\mathrm {kep}$ is mainly determined by the local scale height $h_\\mathrm {0}$ , with a typical drift timescale at 1 AU of $10^6$ years for a 5-km planetesimal.", "In the circumbinary environment, on the other hand, the perturbation from the binary companion acts to raise eccentricities throughout the planetesimal disk, such that the dominant term contributing to $\\delta v$ is given by the time-varying speed of the planetesimal sampling different gas velocities at the apsides.", "Figure REF shows the distribution of planetesimals after $t = 10^5$ years, binned in semi-major axis.", "We find that inside $\\approx 1.5$ AU, the planetesimal disk is severely depleted.", "Indeed, in our setup, drift timescales are a strong function of semi-major axis ($\\propto a^{-5/2}$ ), such that radial drift from the outer parts of the disk cannot replenish the inner disk effectively.", "We compared the planetesimal distribution of our $N$ -body run with an analytic model based on Equation REF and REF .", "Assuming $\\delta v \\approx 0.5 e_\\mathrm {f} v_\\mathrm {kep}$ , we find good agreement between the two.", "Finally, the second panel of Figure REF shows that the distribution of planetesimal sizes is skewed towards larger planetesimals at small semi-major axes, since larger planetesimals are less affected by the gas drag.", "This can contribute to making the inner region more accretion-friendly for two reasons.", "Firstly, larger planetesimals can withstand larger impact velocities.", "Secondly, the spread in sizes will be reduced, which means that the spread in the phasing of the planetesimals will also be reduced.", "In the outer parts of the disk, where damping is less effective, planetesimals are initially weakly phased because the oscillations are coherent and spatially extended; therefore, impact velocities tend to be lower.", "However, the frequency of the oscillation around the forced eccentricity increases with time, ultimately leading to orbital crossing [32].", "The orbital crossing boundary $a_\\mathrm {cross}$ sweeps outwards in semi-major axis, increasing impact velocities.", "In our simulation, collisions are recorded for a small time window after $t = 10^5$ years.", "As evidenced in Figure REF , regions outside $\\approx $ 3.5 AU ($\\approx 13 a_\\mathrm {B}$ ) have not experienced orbital crossing.", "This is expected, since $a_\\mathrm {cross}$ is a weak function of time [32].", "Over longer timescales, the impact velocities will increase in the outer regions, as they are swept by the orbital crossing boundary.", "However, we expect the core of Kepler-16 b to be formed and accreting gas before significant gas dispersal has occurred [38]." ], [ "Implications for planet formation", "Figure REF shows the fraction of accreting encounters as a function of semi-major axis.", "We found that the following qualitative situation holds for different radial locations: in the region between the stability boundary and 1 AU (which includes the present-day location of the planet $a_P \\approx 0.7$ AU), eccentricities are pumped to high values by the central binary and planetesimal number density is low due to the fast radial drift.", "The majority of encounters are in the “uncertain” regime, with the potential of being accreting depending on the prescription for the outcome of disruptive collisions.", "for a small range in semi-major axis outside 1 AU, the spread in $e$ and $\\varpi $ is smaller and planetesimal distributions are skewed towards larger planetesimals.", "The majority of encounters are accreting.", "between 1.75 AU and 4 AUs, the magnitude of the eccentricity and the differential phasing raises the impact velocities, such that the majority of the encounters are erosive.", "outside 4 AUs, orbital crossing has not been realized yet and gas drag is weaker due to the steep radial dependence of the gas density; therefore, orbits are only weakly phased.", "The majority of encounters are accreting.", "We conclude that planet formation is likely inhibited for a large range in semi-major axis (location (c), between 1.75 and 4 AUs).", "This range in semi-major axis includes the nominal location of the ice line for an irradiated disk, estimated from the scaling $a_\\mathrm {ice} \\sim 2.7 \\mathrm {AU}\\ (\\mathcal {M}/\\mathcal {M}_\\mathrm {\\odot })^2 \\approx 2.3$ AU [8], assuming $\\mathcal {M}= \\mathcal {M}_\\mathrm {A} + \\mathcal {M}_\\mathrm {B}$ .", "What is the impact of this “forbidden region” for planet formation?", "It is instructive to refer to the predictions of the standard core accretion paradigm for single stars; in particular, the outcome of large-scale Monte-Carlo planet synthesis models [8], [16].", "[15] recently conducted a Monte-Carlo planet synthesis simulation for a variety of disk masses and metallicities, for the nominal case of a 1 $\\mathcal {M}_\\mathrm {\\odot }$ central star.", "In the core accretion paradigm, metallicity represents a threshold quantity for the formation of planetary cores.", "Accordingly, they found that the cores of giant planets ($\\mathcal {M}\\gtrsim \\mathcal {M}_\\mathrm {J}$ ) tend to preferentially form outside the ice line when the metallicity (which acts as a proxy for the solid content of the disk) is low.", "The actual location of the ice line scales with the disk mass, which contributes to the spread in semi-major axis.", "Figure: Initial location of embryos that grow to final masses 0.2<ℳ P <0.4ℳ J 0.2 < \\mathcal {M}_\\mathrm {P} < 0.4 \\mathcal {M}_\\mathrm {J} (black points) and ℳ final >1ℳ J \\mathcal {M}_\\mathrm {final} > 1 \\mathcal {M}_\\mathrm {J} (red dots) in the simulations of , for a range of metallicities and disk masses.", "The shaded region corresponds to the range in semi-major axis where embryo formation is disturbed in the Kepler-16 system.In Figure (REF ), we plot a different subset of the output of the simulations of [15]http://www.mpia-hd.mpg.de/homes/mordasini/Site7.html, focusing on the ensemble of embryos that acquire masses comparable to Kepler-16 b ($0.2 \\mathcal {M}_\\mathrm {J} < \\mathcal {M}< 0.4 \\mathcal {M}_\\mathrm {J}$ ).", "The initial location of the embryo is plotted as a function of metallicity.", "For disks of solar or super-solar metallicity, such planets are formed throughout the disk, with a substantial fraction formed inside 2 AU (about 40%).", "At subsolar metallicities comparable to Kepler-16 ([Fe/H] $\\approx -0.3 \\pm 0.2$ ), however, such cores are only found outside 2 AU, with a minority lying in location (c) (about 20%).", "While the synthetic population refers to the nominal 1 $\\mathcal {M}_\\mathrm {\\odot }$ single star case, with one embryo per disk, it suggests that in situ planet formation in location (a) might be hampered by the low surface density in solids at 1 AU.", "Our simulations place an additional dynamical constraint, indicating that less than 20% of encounters within 1 AU are accreting.", "This, compounded with the low planetesimal density in the region (Figure REF ), makes in situ formation of a substantial core difficult.", "Finally, it is also crucial to recognize that non-axisymmetric perturbations from the disk might play an important role in the dynamics of the inner disk.", "The eccentric central binary will likely excite spiral structures, which might act to pump the eccentricity of the inner planetesimals and alter the phasing of their orbits.", "Indeed, [13] conducted full 2D hydrodynamical simulations with a small number of tracer planetesimals embedded in the disk, and found significant oscillations in the eccentricity and longitude of pericenter around the equilibrium value." ], [ "Discussion", "Planet formation in presence of close binaries presents a number of challenges to the traditional core accretion paradigm.", "Historically, most of the theoretical effort has been expended to study pathways to planet formation in S-type orbits for planets that had been observed through RV surveys, or targets with observationally desirable properties (e.g., $\\alpha $ Centauri).", "With the launch of Kepler, however, we expect that the sample of planets in P-type orbits around eclipsing binaries will rapidly outnumber the handful of planets in circumstellar configurations detected with RV surveys.", "Indeed, a sample of 750 Kepler targets are eclipsing binaries for which eclipses of both stars are observed, and a subset of 18% exhibited deviations in the timing of the eclipses [36].", "Since the definitive determination of the planetary nature of a putative KOI relies on the detection of tertiary and quaternary eclipses, we expect that as the baseline of the observation increases, more KOIs will be confirmed as genuine circumbinary objects.", "In this paper, we have conducted a preliminary simulation of the feasibility of circumbinary planet formation in the Kepler-16 system.", "In accordance to an earlier study conducted by [26] for a different set of binary parameters, we have found that, for generous initial conditions that favor planetesimal accretion, planet formation appears to be feasible far enough from the central binary.", "However, we have identified a substantial radial span between 1.75 and 4 AU where planet formation is strongly inhibited.", "Within the planet accretion framework, the most likely sequence of event is the formation of a core outside the forbidden region, followed by inwards migration driven by tidal interaction with the protoplanetary disk [20].", "Although we measured impact velocities potentially favorable to accretion close to the present-day location of the planet, in situ formation of Kepler-16 b is less likely due to overall high encounter speeds, low planetesimal density, low metallicity of the star, and non-axisymmetric perturbations from the disk (not modeled in this simulation).", "We remark that the simulations presented in this paper only demonstrate that, choosing the most favorable conditions for planetesimal accretion and an assumed initial planetesimal size of 1-10 km, the formation of an embryo outside 4 AU is plausible, with traditional migration processes subsequently moving the planet to its current location.", "Our approach has several limitations introduced for the sake of simplicity and computational speed; chiefly, we disregarded the evolution of the protoplanetary disk and the collisional outcome of planetesimal impacts.", "For the former, we plan to follow approximately the hydrodynamical response of the disk with the SPH algorithm included in the Sphiga code in a follow-up paper.", "For the latter, a time-dependent distribution of planetesimal sizes would more accurately model the extent of the accretion-friendly regions, which depend sensitively on the planetesimal parameters.", "The numerical procedure of [18] represents a possible approach to following the collisional evolution of the planetesimal size distribution.", "We acknowledge support from the NASA Grant NNX11A145A.", "The author thanks Greg Laughlin and the referee, Philippe Thebault, for useful discussions." ] ]

[ [ "Discovery of High-energy and Very High Energy Gamma-ray Emission from\n the Blazar RBS 0413" ], [ "Abstract We report on the discovery of high-energy (HE; E > 0.1 GeV) and very high-energy (VHE; E > 100 GeV) gamma-ray emission from the high-frequency-peaked BL Lac object RBS 0413.", "VERITAS, a ground-based gamma-ray observatory, detected VHE gamma rays from RBS 0413 with a statistical significance of 5.5 standard deviations (sigma) and a gamma-ray flux of (1.5 \\pm 0.6stat \\pm 0.7syst) \\times 10^(-8) photons m^(-2) s^(-1) (\\sim 1% of the Crab Nebula flux) above 250 GeV.", "The observed spectrum can be described by a power law with a photon index of 3.18 \\pm 0.68stat \\pm 0.30syst.", "Contemporaneous observations with the Large Area Telescope (LAT) on the Fermi Gamma-ray Space Telescope detected HE gamma rays from RBS 0413 with a statistical significance of more than 9 sigma, a power-law photon index of 1.57 \\pm 0.12stat +0.11sys -0.12sys and a gamma-ray flux between 300 MeV and 300 GeV of (1.64 \\pm 0.43stat +0.31sys -0.22sys) \\times 10^(-5) photons m^(-2) s^(-1).", "We present the results from Fermi-LAT and VERITAS, including a spectral energy distribution modeling of the gamma-ray, quasi-simultaneous X-ray (Swift-XRT), ultraviolet (Swift-UVOT) and R-band optical (MDM) data.", "We find that, if conditions close to equipartition are required, both the combined synchrotron self-Compton/external-Compton and the lepto-hadronic models are preferred over a pure synchrotron self-Compton model." ], [ "Introduction", "Blazars are active galactic nuclei that have their jet axis oriented at a small angle with respect to the observer [37].", "They are observationally classified as either flat-spectrum radio quasars (FSRQ) or BL Lacertae (BL Lac) objects according to the broad line emission in their optical spectra.", "Recent studies interpreting the differing spectral high-energy (HE) $\\gamma $ -ray properties of the FSRQs and BL Lacs based on physical mechanisms can be found in e.g., [16].", "Blazars are known to emit non-thermal radiation characterized by a double-peaked spectral energy distribution (SED).", "The low-energy component, generally covering radio to UV/X-ray bands, is usually explained as due to synchrotron emission from relativistic electrons in the blazar jet.", "The origin of the HE component, occurring in the X-ray to $\\gamma $ -ray regime, is still not completely resolved and could be due to emission from a relativistic particle beam consisting of leptons and / or hadrons.", "In leptonic models, very high energy (VHE) photons are produced by inverse-Compton (IC) scattering of low-energy photons off the synchrotron-emitting electrons.", "The soft seed photons for the IC process can be the synchrotron photons (synchrotron self-Compton, SSC, e.g., [21]), or they may originate from ambient radiation (external-Compton, EC, e.g., [12]).", "Hadronic models include synchrotron emission from protons (e.g., [3]) and $\\pi ^\\mathrm {0}$ -decay from hadronic interactions with subsequent electromagnetic cascades (e.g., [25]).", "RBS 0413 was discovered in the X-ray band (1E 0317.0+1834) during the Einstein Medium Sensitivity Survey and was optically identified as a BL Lac [17].", "The object was also detected as a radio emitter with the Very Large Array of the National Radio Astronomy Observatory [35].", "It exhibits significant and variable optical polarization [34].", "Having a “featureless” optical spectrum [36] and an estimated synchrotron peak frequency log$(\\nu _\\mathrm {peak}/\\mathrm {Hz})=16.99$  [26], RBS 0413 is classified as a high-frequency-peaked BL Lac object (HBL, [29]).", "It is located at a redshift of 0.190 [17], [34].", "The MAGIC Collaboration observed RBS 0413 in 2004 December–2005 February for a livetime of 6.9 hr and reported a VHE flux upper limit of $4.2\\times 10^{-12}~\\textrm {erg}~\\textrm {cm}^{-2}~\\textrm {s}^{-1}$ , at 200 GeV, assuming a power-law spectrum with a photon index of 3.0 [5].", "VERITAS observed the source in the 2008–2009 season and obtained a marginal significance of $\\sim 3\\sigma $ .", "In 2009, Fermi-Large Area Telescope (LAT) detected HE emission from the direction of RBS 0413 [1], triggering new VERITAS observations.", "These new observations, combined with the previous data, resulted in the detection of RBS 0413 as a VHE $\\gamma $ -ray emitter in 2009 October [28]." ], [ "VERITAS Observations and Analysis Results", "VERITAS is a ground-based $\\gamma $ -ray observatory sensitive to $\\gamma $ rays with energy between 100 GeV and 30 TeV.", "Located at the Fred Lawrence Whipple Observatory (FLWO) near Amado, southern Arizona, USA (1.3 km above sea level, N $31^{\\circ }$ $40^{\\prime }$ , W $110^{\\circ }$ $57^{\\prime }$ ), the array consists of four imaging atmospheric Cherenkov telescopes, each having a diameter of 12 m and a field of view of $3^{\\circ }.5$  [19].", "During the 2009 annual shutdown (July - August) one of the telescopes was relocated to give a more symmetrical array layout.", "In addition, a new mirror-alignment system applied in 2009 Spring contributed to an improvement in the point-spread function (PSF), with a decrease of $25\\%-30\\%$ in the 80% containment radius [24].", "As a consequence, the sensitivity showed a significant improvement, and the observation time required for a $1\\%$ Crab Nebula detection dropped from $\\sim 48$ hr to less than 30 hr.", "For observations at $70^{\\circ }$ elevation, the energy resolution is $15\\%-20\\%$ , and the angular resolution, defined as the $68\\%$ containment radius, is less than $0.1^{\\circ }$  [30].", "VERITAS observed RBS 0413 for 48 hours in total, using wobble mode [4], with north, south, east and west wobble positions.", "After discarding observing runs compromised by bad weather, and a small number affected by hardware problems, 26 hours remained for analysis.", "One third of these data were obtained with the old array configuration (Sep 2008 - Feb 2009, MJD 54732–54883) and the rest with the new array (Sep 2009 - Jan 2010, MJD 55092–55485).", "Approximately 3 hours of data with the old array were taken under weak moonlight, which leads to a higher energy threshold for those observations.", "The source elevation in the data set ranges from $57^{\\circ }$ to $79^{\\circ }$ , with an average of $\\sim 70^{\\circ }$ .", "Data analysis steps consist of calibration, image parameterization [18], event reconstruction, background rejection and signal extraction as described in [11].", "For signal extraction, a $\\theta ^{2}$ cut [11] of 0.0169, optimized for a point source of 1% strength of the Crab Nebula, was used.", "RBS 0413 is a weak source in the VHE regime.", "Using a “reflected-region” background estimation [4], an excess of 180 events and a significance of $5.5\\sigma $ are obtained, for the source location at $\\textrm {RA}=03^{\\mathrm {h}}19^{\\mathrm {m}}47^{\\mathrm {s}}\\pm 4^{\\mathrm {s}}_\\mathrm {stat}\\pm 7^{\\mathrm {s}}_\\mathrm {syst}$ and $\\textrm {decl.", "}=18^{\\circ }45^{\\prime }.7\\pm 1^{\\prime }.0_\\mathrm {stat}\\pm 1^{\\prime }.8_\\mathrm {syst}$ (J2000 coordinates).", "The VERITAS signal is consistent with a point source, and we name the object VER J0319+187.", "The energy distribution of $\\gamma $ -ray events extends from $\\sim 250$  GeV to $\\sim 1.0$  TeV (see Table REF for a list of spectral data points) and is well described by a power-law function, $dN/dE=F_{0}E^{-\\Gamma }$ .", "The best fit is obtained with photon index $\\Gamma =3.18\\pm 0.68_\\mathrm {stat}\\pm 0.30_\\mathrm {syst}$ and flux normalization $F_0=(1.38\\pm 0.52_\\mathrm {stat}\\pm 0.60_\\mathrm {syst})\\times 10^{-7}$  $\\textrm {TeV}^{-1}$  $\\textrm {m}^{-2}$  $\\textrm {s}^{-1}$ at 0.3 TeV, with a value of $\\chi ^2$ per degree of freedom ($\\chi ^2$ /dof) of 0.14/2 (see Figure REF ).", "The integral flux above 250 GeV is $(1.5\\pm 0.6_\\mathrm {stat}\\pm 0.7_\\mathrm {syst})\\times 10^{-8}$  $\\textrm {m}^{-2}$  $\\textrm {s}^{-1}$ , corresponding to a flux level of approximately $1\\%$ the flux of the Crab Nebula.", "No significant flux variability is detected (see Figure REF top panel and caption for details of the light-curve analysis).", "An upper limit (99% confidence level) on the fractional variability amplitude ($F_\\mathrm {var}$ , [38]) yields $F_\\mathrm {var} < 3.2$ .", "Figure: VERITAS measured photon spectrum of RBS 0413.", "See the text for the parameters of the power-law fit shown.Table: Differential flux measurements of RBS 0413 above 250 GeV with VERITAS.", "The first column shows the mean energies, weighted by the spectral index.", "The errors are statistical only." ], [ "The LAT aboard the Fermi Gamma-ray Space Telescope is a pair-conversion $\\gamma $ -ray detector sensitive to photons in the energy range from below 20 MeV to more than 300 GeV [6].", "The present analysis includes the data taken between 2008 August 4 and 2011 January 4 (MJD 54682–55565), which covers the entire VERITAS observation interval.", "Events from the Pass 6 diffuse class with energy between 300 MeV and 300 GeV, with zenith angle $<100^{\\circ }$ , and from a square region of side $20^{\\circ }$ centered on RBS 0413, were selected for this analysis.", "The cut at 300 MeV was used to minimize larger systematic errors at lower energies.", "The time intervals when the source was close to the Sun (MJD 54954-54974 and 55320-55339) were excluded.", "The data were analyzed with the LAT Science Tools version v9r20p0http://fermi.gsfc.nasa.gov/ssc/data/analysis/scitools/overview.html and the post-launch instrument-response functions P6_V11_DIFFUSE.", "The binned maximum-likelihood tools were used for significance and flux calculation [10], [23].", "Sources from the 1FGL catalog [2] located within a square region of side $24^{\\circ }$ centered on RBS 0413 were included in the model of the region.", "The background model includes the standard Galactic and isotropic diffuse emission componentshttp://fermi.gsfc.nasa.gov/ssc/data/access/lat/BackgroundModels.html.", "A point source positionally consistent with RBS 0413 is detected with a significance of more than $9\\sigma $ (test statistic, TS=89; see [23]).", "The photon energy spectrum is best described by a power-law function.", "Replacing the power-law model with a log-parabola model does not significantly improve the likelihood fit.", "The time-averaged integral flux is $I(300\\,\\mathrm {MeV}<E<300\\,\\mathrm {GeV})=(1.64{\\pm 0.43_\\mathrm {stat}}^{+0.31}_{-0.22_\\mathrm {sys}})\\times 10^{-5}$  $\\textrm {m}^{-2}$  $\\textrm {s}^{-1}$ , and the spectral index is $1.57{\\pm 0.12_\\mathrm {stat}}^{+0.11}_{-0.12_\\mathrm {sys}}$ .", "The spectral points were calculated using the procedure presented in [1] (see Table REF ).", "In the energy range 100–300 GeV, no detection was obtained ($\\mathrm {TS}<9$ ) and an upper limit at the 95% confidence level was derived.", "Figure REF (bottom panel) shows the Fermi light curve with $\\sim 6$ month wide time bins.", "The upper limit point in the last time bin has 95% confidence level.", "Comparing the likelihood of a model in which the flux in each time bin is free to vary to one where it is assumed to be constant yields a null hypothesis probability of 11%; thus we find no evidence for variability.", "Details on the methodology can be found in [27].", "Using the same method as in Section  and including the flux value returned by the likelihood fit in the last bin, we estimate $F_\\mathrm {var} < 3.2$ with 99% confidence level.", "Table: Differential flux measurements of RBS 0413 with the Fermi-LAT.", "The energies correspond to the bin centers.", "The errors are statistical only.Figure: Top: 30-day light curve for the VERITAS data.", "A fit with a constant function gives a χ 2 \\chi ^2/dof value of 14/8, corresponding to a fit probability of 8%8\\%, consistent with the hypothesis of a constant flux.", "The negative flux point corresponding to the upper limit point in the light curve was included in the fit.Bottom: the light curve for the Fermi data using ∼6\\sim 6 month wide time bins.The shaded areas represent the time intervals that were excluded to avoid solar contamination.", "In both graphs, the dashed lines represent the constant fit function." ], [ "The VERITAS detection triggered a Swift [14] target-of-opportunity observation of RBS 0413 on 2009 November 11, with a total exposure of 2.4 ks.", "All Swift-XRT data were reduced using the standard Swift analysis pipeline described in [9] using the HEAsoft 6.8 package.", "Event files were calibrated and cleaned following the standard filtering criteria using the xrtpipeline task and applying calibration files current to 2010 March.", "All data were taken in photon-counting mode over the energy range 0.3–10 keV.", "Due to the moderate count rate of 0.3 counts s$^{-1}$ , the data are not affected by photon pile-up in the core of the PSF, and partial masking of the source is not necessary.", "Source events were extracted from a circular region with a radius of 30 pixels ($70.8$ ) centered on the source, and background events were extracted from a 40 pixel radius circle in a source-free region.", "Ancillary response files were generated using the xrtmkarf task, with corrections applied for the PSF losses and CCD defects.", "The latest response matrix from the XRT calibration files was applied.", "The extracted XRT energy spectrum was rebinned to contain a minimum of 20 counts in each bin.", "An absorbed power-law model, including the phabsThe phabs tool applies absorption using photoelectric cross-sections.", "model for photoelectric absorption, was fitted to the Swift-XRT photon spectrum.", "The cross-sections and abundances used the standard Xspec v12.5 values, as given in the Xspec Analysis Manualhttp://heasarc.nasa.gov/docs/software/lheasoft/xanadu/xspec/XspecManual.pdf.", "Using a fixed Galactic hydrogen column density, $N_{\\rm {H}} = 8.91 \\times 10^{20}$ cm$^{-2}$ [20], the best-fit model yields a $\\chi ^{2}$ /dof value of 25.9/26.", "Over the energy range 0.3–10 keV, the best-fit photon index is $\\Gamma = 2.22 \\pm 0.07$ , and the normalization at 1 keV is ($33.1 \\pm 2.2$ ) keV$^{-1}$ m$^{-2}$ s$^{-1}$ .", "The unabsorbed integral flux is $F$ (0.3-10 keV) = ($1.69\\pm 0.12)\\times 10^{-11}$  $\\mathrm {erg}$  $\\mathrm {cm}^{-2}$  $\\mathrm {s}^{-1}$ in the range 0.3–10 keV.", "The absorbed integral flux in the range 2–10 keV is $F$ (2-10 keV) = ($5.81\\pm 0.55)\\times 10^{-12}$  $\\mathrm {erg}$  $\\mathrm {cm}^{-2}$  $\\mathrm {s}^{-1}$ .", "No flux variability is evident over the 2.4 ks exposure.", "UVOT observations were taken in the photometric band UVM2 (2246 Å) [31].", "The uvotsource tool was used to extract counts, correct for coincidence losses, apply background subtraction, and calculate the source flux.", "The standard 5 radius source aperture was used, with a 20 background region.", "The source fluxes were dereddened using the procedure in [32].", "The measured flux is ($2.75 \\pm 0.11$ ) $\\times 10^{-12}$ $\\textrm {erg}$  $\\textrm {cm}^{-2}$  $\\textrm {s}^{-1}$ ." ], [ "MDM Observations", "The $R$ -band optical data were taken with the 1.3 m McGraw-Hill telescope at the MDM observatory on Kitt Peak, Arizona, between 2009 December 10 and 13 .", "All frames were bias corrected and flat fielded using standard routines in IRAFIRAF is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy (AURA) under cooperative agreement with the National Science Foundation.", "[7], and instrumental magnitudes of RBS 0413 and six comparison stars in the same field of view were extracted using DAOPHOT [22] within IRAF.", "Physical magnitudes were computed using the physical $R$ -band magnitudes of the six comparison stars from the NOMAD catalog [40], assuming that the magnitudes quoted in that catalog are exact, then were corrected for Galactic extinction using extinction coefficients calculated following [33], taken from NEDhttp://ned.ipac.caltech.edu/, and converted into $\\nu \\mathrm {F}_\\mathrm {\\nu }$ fluxes.", "The flux shows variations of up to $\\sim 30\\%$ from day to day, with an average of $(2.47\\pm 0.02)\\times 10^{-12}$  $\\mathrm {erg}$  $\\mathrm {cm}^{-2}$  $\\mathrm {s}^{-1}$ .", "In the case of RBS 0413, the host galaxy is expected to make a substantial contribution to the observed $R$ -band flux.", "We have taken this into account in our SED modeling by adding a phenomenological host galaxy SED to our model." ], [ "Modeling and Discussion", "The non-thermal continuum of RBS 0413 exhibits a double-peaked shape, as is typical for blazars.", "In this study, we applied three different time-independent models to the observed SED, using the contemporaneous X-ray, UV and optical ($R$ -band) data to complement the Fermi-LAT and VERITAS observations (see Figure REF ).", "It should be noted that these observations were not strictly simultaneous.", "For all of the models, the emission region was assumed to be a spherical blob of size $R_\\mathrm {b}$ , moving within the jet with a bulk Lorentz factor $\\Gamma $ .", "$R_\\mathrm {b}$ was constrained using the optical minimum variability timescale $\\mathrm {log}(\\Delta t_\\mathrm {min})=3.75$  [39], where $\\Delta t_\\mathrm {min}$ is in units of seconds.", "The angle between the line of sight of the observer and the jet axis, represented by $\\theta _\\mathrm {obs}$ , was chosen to be equal to 1/$\\Gamma $ .", "This is referred to as the critical or superluminal angle, for which the Doppler factor equals $\\Gamma $ .", "The synchrotron emission was assumed to originate from relativistic electrons with Lorentz factors distributed between $\\gamma _\\mathrm {min}$ and $\\gamma _\\mathrm {max}$ , following a power law with a spectral index $q_\\mathrm {e}$ , under the influence of a magnetic field $B$ .", "The particle-escape timescale is represented by $t_\\mathrm {esc}$ = $\\eta _\\mathrm {esc}R_\\mathrm {b}/c$ , where $\\eta _\\mathrm {esc}$ is the particle-escape parameter.", "For each model, the parameters were adjusted to describe the data and achieve an equilibrium between the acceleration of the injected particles, the radiative cooling and the particle escape.", "The best-fit parameters were used to calculate the relative partition between the magnetic field energy density and the kinetic luminosity of relativistic particles ($\\epsilon _\\mathrm {Be,p} \\equiv L_\\mathrm {B}/L_\\mathrm {e,p}$ ) for each model.", "All model spectra were corrected for extragalactic background light (EBL) absorption using the model of [13].", "For the optical band, a phenomenological SED reproducing the archival host galaxy spectral points was added to the model.", "Figure: RBS 0413 spectral energy distribution.", "Absorption in the VHE region due to the EBL is taken into account in the fits using the model of .", "The models are described in detail in the text.The first model we applied assumed a pure SSC scenario.", "The magnetic field energy density required in this model is only 6% of the value corresponding to equipartition with the relativistic electron distribution ($\\epsilon _\\mathrm {Be} = 0.06$ ).", "The model spectrum is too hard in the Fermi band (strongly curved, with $\\Gamma \\sim 1.5$ around $10^{23}$  Hz) and too soft in the VERITAS band ($\\Gamma =4.0$ ), albeit within the errors in both cases.", "On the other hand, while the X-ray measurements are well reproduced, the optical ($R$ -band) spectrum is not.", "Next, we tested a combined SSC+EC model.", "The external source of photons was assumed to be an isotropic thermal blackbody (BB) radiation field, which may be due to a torus of warm dust with temperature $T_\\mathrm {ext}=1.5\\times 10^{3}$ $\\textrm {K}$ .", "The assumed BB infrared (IR) radiation field corresponds to a $\\nu \\mathrm {F_{\\nu }}$ flux of $\\sim 5\\times 10^{8}\\times \\mathrm {R}^{2}\\,_{pc}$ JyHz, where $\\mathrm {R}_\\mathrm {pc}$ is the characteristic size of the IR emitter in units of parsecs.", "It should be noted that this quantity is far below the measured IR flux, thus consistent with our observations.", "The addition of an EC component improves the modeling for the optical and Fermi data compared with the pure SSC model and leads to values for the model parameters which are very close to equipartition ($\\epsilon _\\mathrm {Be} = 1.20$ ).", "However, the model tends to have too sharp a cutoff in the VHE band and therefore underpredicts the VERITAS flux measurements.", "This could be remedied by choosing a much weaker magnetic field and higher electron energies, but the resulting system would then be very far from equipartition, with $\\epsilon _\\mathrm {Be}$ reduced by at least two orders of magnitude.", "The last model we tested is a combined lepto-hadronic jet model as described in [8].", "In this case, the HE component of the non-thermal emission is dominated by a combination of synchrotron radiation from ultrarelativistic protons ($E_\\mathrm {max}\\gtrsim 10^{19}$  eV) and photons from decay of neutral pions.", "Secondary electrons that are produced in various electromagnetic cascades are the origin of the low-energy synchrotron emission.", "The kinetic energy of the relativistic proton population was assumed to have a single power-law distribution in the energy range $1.0\\times 10^{3}$  GeV $< E_\\mathrm {p} <$ $1.6\\times 10^{10}$  GeV, with a spectral index $q_\\mathrm {p}=2.4$ .", "The model is a good description of the overall SED, and the system is close to equipartition between the magnetic field and the total relativistic particle content dominated by protons ($\\epsilon _\\mathrm {Bp} = 0.95$ ).", "As is typical for lepto-hadronic models, the acceleration of protons to ultrarelativistic energies ($\\sim 10^{10}$ GeV) requires a high magnetic field, 30 G in this case.", "Although the lepto-hadronic model provides the best description for the data, it has two more free parameters than the SSC+EC model and is therefore less constraining.", "The best-fit parameters adopted for all three models are summarized in Table REF .", "Table: Best-fit parameters for the SED of RBS 0413 for the SSC, SSC+EC, and lepto-hadronic models.", "The parameters that were left free are marked with an asterisk.", "θ obs \\theta _\\mathrm {obs} is the superluminal angle (see Section ).Based on our calculations, all three models are good at describing the observed data.", "It appears that if the criterion of equipartition is taken as a reasonable measure of successful blazar emission models, SSC+EC is preferred over SSC for this HBL, which seems to be in contrast with some previous blazar studies.", "See [15] for arguments relating the presence of an EC component with the blazar sequence and [1] for a discussion of issues encountered in explaining blazar SEDs with a simple one-zone homogeneous SSC model.", "On the other hand, we cannot discriminate between leptonic and lepto-hadronic mechanisms, since the SSC+EC and lepto-hadronic models provide equally reasonable descriptions for the observed non-thermal continuum, and we did not detect variability in the HE and VHE regimes given the limited statistics.", "Since the synchrotron cooling timescales for electrons and protons are different, the detection of intraday variability would be harder to explain with a lepto-hadronic scenario and would accordingly favor a purely leptonic scenario.", "Therefore, any future observation of rapid variability would be helpful in distinguishing between the SSC+EC and lepto-hadronic models.", "The VERITAS research is supported by grants from the US Department of Energy Office of Science, the US National Science Foundation, the Smithsonian Institution, and the NASA Swift Guest Investigator Program, by NSERC in Canada, by Science Foundation Ireland (SFI 10/RFP/AST2748), and by STFC in the UK.", "We acknowledge the excellent work of the technical support staff at the FLWO and at the collaborating institutions in the construction and operation of the instrument.", "The Fermi-LAT Collaboration acknowledges generous ongoing support from a number of agencies and institutes that have supported both the development and the operation of the LAT as well as scientific data analysis.", "These include the National Aeronautics and Space Administration and the Department of Energy in the United States, the Commissariat à l'Energie Atomique and the Centre National de la Recherche Scientifique / Institut National de Physique Nucléaire et de Physique des Particules in France, the Agenzia Spaziale Italiana and the Istituto Nazionale di Fisica Nucleare in Italy, the Ministry of Education, Culture, Sports, Science and Technology (MEXT), High Energy Accelerator Research Organization (KEK) and Japan Aerospace Exploration Agency (JAXA) in Japan, and the K. A. Wallenberg Foundation, the Swedish Research Council and the Swedish National Space Board in Sweden.", "Additional support for science analysis during the operations phase is gratefully acknowledged from the Istituto Nazionale di Astrofisica in Italy and the Centre National d'Etudes Spatiales in France.", "Facilities: VERITAS, Fermi.", "2" ] ]

[ [ "Experimental signatures of the quantum-classical transition in a\n nanomechanical oscillator modeled as a damped driven double-well problem" ], [ "Abstract We demonstrate robust and reliable signatures for the transition from quantum to classical behavior in the position probability distribution of a damped double-well system using the Qunatum State Diffusion approach to open quantum systems.", "We argue that these signatures are within experimental reach, for example in a doubly-clamped nanomechanical beam." ], [ "Introduction", "The difference between quantum and classical behavior is of fundamental interest with many experimental consequences in contexts such as superconducting quantum interference devices (SQUIDs), cold atom systems, and significantly in nanoelectromechanical systems (NEMS).", "NEMS are small, oscillate at high frequencies and are typically maintained at low temperatures.", "Most of their parameters are adjustable, making them an ideal setting to explore the transition from quantum to classical behavior.", "Recent experimental results[1], [2], indicate that such systems can now be prepared and observed in their ground state, thus enabling the study of genuinely non-classical behavior in the dynamics.", "The exciting prospect of being able to observe the quantum-classical transition in detail has contributed to interest in quantum mechanical nanoelectromechanical systems (occasionally refered to as QEMS in the literature), whose study is an active and rapidly expanding field of research[3].", "A seminal theoretical study of the quantum–classical transition in NEMS was carried out by Katz et al[4], using a nonlinear resonator, which is a paradigmatic model for several common types of NEMS that yields quantum behavior distinct from the classical.", "They considered an isolated resonator as well as one modeled as an open system (coupled to the environment) in search of experimental signatures of such quantum behavior.", "Specifically, they examined the dynamics of the classical phase space distribution in such a system compared to that of the quantum-mechanical Wigner function, and looked at a zero temperature and a finite temperature environment.", "They were able to find differences in experimentally accessible signatures: In particular, the quantum mechanical version of the resonator has non-zero probability of being found in a position where the classical resonator has zero probability of being found.", "However, these differences are relatively small and blur away quickly when experimental noise or finite temperature effects through thermal noise are considered.", "In this paper, by considering a slightly different but equally accessible nonlinear oscillator, we are able to obtain more pronounced experimental signatures of the quantum-classical transition; these are arguably more visible at finite temperatures and in the presence of experimental noise.", "Further, we show that rather than comparing just quantum and classical behavior, there is insight to be gained from studying the continuous transition where the classical behavior emerges as the limiting case of the quantum dynamics.", "That is, although systems are fundamentally described by quantum mechanics, their behavior changes as we increase the size of the system (affecting the characteristic action scaled by $\\hbar $ ), the effect of the environment (decoherence) or other parameters of the system.", "For our system, we indeed recover the classical behavior as the limiting case.", "The details of the emergence of the classical limit are illuminating, and in particular, the signatures do not always change monotonically, thus establishing that comparing a single quantum solution with a classical solution leads to an incomplete picture.", "Katz et al considered the sytem with the Hamiltonian $H_{sys}=\\frac{1}{2}p^2+\\frac{1}{2}x^2+\\frac{1}{4}x^4-xF\\cos \\omega t.$ This quadratic nonlinear oscillator is within experimental reach in the near-quantum regime, and is an excellent system to study.", "However, even more interesting effects such as quantum tunneling obtain if we change the sign so that the $x^2$ term is negative[6].", "This yields the so-called double-well Duffing oscillator, which has the Hamiltonian $H_{sys}=\\frac{1}{2}p^2-\\frac{1}{2}x^2+\\frac{1}{4}x^4-xF\\cos \\omega t$ , and allows for clearer signatures of the classical to quantum transition as we demonstrate below.", "This paper is organized as follows: We start by motivating the modeling of a specific NEMS as a doubly-clamped beam to show the connection between experimental paramters and model parameters.", "We then sketch the quantum state diffusion (QSD) approach, where Lindblad operators act within stochastic Schrodinger equations to incorporate the effect of the environment, used to model the behavior of such an oscillator understood as an open quantum system.", "These two allow us to establish that changing the sign of the quadratic term is within current experimental capabilities and that such experiments are in or close to the realm where the quantum to classical transition can be explored.", "We then present results and conclude with our analysis." ], [ "NEMS modeled as a doubly-clamped beam", "Many nanoelectromechanical devices can be modeled as doubly-clamped beams driven near resonance and continuum mechanics continues to serve as an adequate model even at the submicron scale[7], [8], [9].", "Usually magnetomotive[10] or optical[7] actuation is used to study the resonance behaviors of NEMS such as dynamically induced bistability, hysteresis and effects of parameter attenuation on energy dissipation.", "The resonant behavior of the doubly-clamped NEMS structure has been experimentally identified with the fundamental bending mode[11] of the problem.", "We sketch here the theory allowing us to relate the resonance frequency to characteristic parameters of NEMS and to determine the scale of the system with respect to $\\hbar $ , establishing the regime of the transition to classical behavior in these systems.", "Starting with the assumption of an ideal beam of rectangular cross-sectional area, we can obtain the potential energy $V$ of a doubly-clamped beam under tension along the longitudinal direction.", "This consists of two parts[12]: elastic and bending potential energy, $V_{elastic}= T(\\int _{0}^{l_0}\\sqrt{1+(y^{\\prime })^2}dx-l_0)$ and $V_{bending}= \\frac{EI}{2}\\int _{0}^{l_0}(y^{^{\\prime \\prime }})^2 dx.$ $T$ is the external force on the beam in the form of the tension in the longitudinal direction along the beam, $E$ is the elastic modulus, $I$ is moment of inertia of the cross-section and $l_0$ is separation of two ends that clamp the beam.", "Note that $l_0$ need not be the length of the beam when it is neither stretched nor compressed because a static doubly-clamped beam may already be buckled due to inherent tension along the beam.", "$V_{elastic}$ is due to tension built up in the beam, which causes the elastic beam to deform and therefore gain energy.", "This is usually referred to as strain energy[13] while $V_{bending}$ is purely due to bending.", "The quantity $EI$ , termed “flexural rigidity”, denotes the force necessary to bend a beam by a unit of curvature.", "The tension $T$ along the beam has two parts[14]: firstly, inherent tension $T_0$ (either from compression at both ends of the beam or manufacturing process) and additional tension $\\Delta T$ (due to transverse motion that stretches the beam), whence $T=T_0+\\Delta T$ .", "When the beam is stretched, its length changes and therefore the additional tension $\\Delta T$ is given by fractional change in length multiplied by elastic modulus and cross section area $A$[14]:$\\Delta T = \\frac{l-l_0}{l_0}E A$ , where $l$ refers to the actual length of the buckled beam.", "We can rewrite this length as $l = l_0+\\Delta l = \\int _0^{l_0}dx \\sqrt{1+(y^{\\prime })^2}\\approx \\int _0^{l_0}dx(1+\\frac{1}{2}(y^{\\prime })^2) = l_0+\\frac{1}{2}\\int _0^{l_0}(y^{\\prime })^2 dx.$ This yields that the change in tension $\\Delta T=\\frac{E A}{2l_0}\\int _0^{l_0}(y^{\\prime })^2 dx$ whence the total tension $T = T_0+\\frac{E A}{2l_0}\\int _0^{l_0}(y^{\\prime })^2 dx.$ Substituting these expressions for $\\Delta l$ and $T$ in the expression for elastic potential energy above, we get $V_{elastic} =\\frac{T_0}{2}\\int _0^{l_0}(y^{\\prime })^2 dx+\\frac{E A}{4l_0}[\\int _0^{l_0}(y^{\\prime })^2 dx]^2.$ Thus, the total potential energy is $\\begin{array}{rcl}V &=& V_{elastic} + V_{bending}\\\\&=& \\frac{T_0}{2}\\int _{0}^{l_0}(y^{\\prime })^2 dx + \\frac{E A}{4 l_0}[\\int _0^{l_0}(y^{\\prime })^2 dx]^2 + \\frac{E I}{2}\\int _0^{l_0}(y^{\\prime \\prime })^2 dx.\\end{array}$ At low temperatures, as is often true for NEMS experiments, it is sufficient to consider only the fundamental mode.", "Mode amplitude solutions for this fundamental mode $y(x)$ are of the form [15] $y(x)=\\frac{1}{2}Y[1-\\cos (2\\pi \\frac{x}{l_0})]$ , where $Y$ is the transverse displacement of the center point of the beam.", "When we make this substitution, the fundamental mode amplitude $Y$ – that is, the location of the vibrating central point – is effectively the position variable of a single particle evolving in a potential $V$ , which as a function of $Y$ is given by $\\begin{array}{rcl}V(Y) &=& \\frac{E I \\pi ^4}{l_0^3}Y^2 + \\frac{\\pi ^2 T_0}{4 l_0}Y^2 + \\frac{E A \\pi ^4}{16 l_0^3} Y^4\\\\&=& Y^2(\\frac{E I \\pi ^4}{l_0^3}+\\frac{\\pi ^2 T_0}{4 l_0}) + Y^4 \\frac{E A \\pi ^4}{16 l_0^3}.\\end{array}$ Here $I$ , the moment of inertia of rectangular cross section is $\\frac{ab^3}{12}$[16], where $a$ is the thickness of the beam and $b$ is its width.", "$I$ varies with the geometry of the beam for specific experimental setups.", "Further, depending on whether inherent tension $T_0$ compresses or stretches the beam, the sign of $T_0$ can be negative or positive.", "Compressive inherent tension along the beam can arise from manufacturing processes, and can vary from 200 MPa to 1 GPa[15].", "In general, the inherent tension is not known a priori in an experimental setup.", "Instead, by considering the continuum mechanics model and fitting the experimentally measurable resonance frequency, the inherent tension can be calculated[11].", "For sufficiently large inherent compressive tension, $T_0<-\\frac{4\\pi ^2EI}{l_0^2}$ , we obtain a double-well potential and the beam is in a state of “Euler instability”[9], [17], meaning that it is buckled.", "Thus a single-well potential or a double-well potential can be experimentally achieved in doubly-clamped NEMS structures and these systems can then be studied for their dyamical behavior.", "The dynamics of a single-well NEMS actuated near the fundamental bending mode frequency, has been extensively studied[18], [19] but that of the double-well case has not been yet been fully explored.", "The results of Ref.", "[11] suggest that it within experimental reach and the above clarifies that in general tuning the characteristic parameters should enable switching from a single-well to a double-well potential." ], [ "Open Quantum Systems", "We now turn to the quantum version of the model for the problem and connect characteristic parameters with the degree of `quantumness' of the system.", "An open quantum system with weak system-environment coupling and a Markovian environment is modeled by a master equation $\\begin{split}\\dot{\\hat{\\rho }}(t)=& \\frac{-i}{\\hbar } \\left[\\hat{H},\\hat{\\rho }\\right]-\\frac{1}{2}\\sum _j \\left(\\hat{L_j}^\\dagger \\hat{L_j}\\hat{\\rho }+\\hat{\\rho }\\hat{L_j}^\\dagger \\hat{L_j}\\right)\\\\& + \\sum _j \\hat{L_j}\\hat{\\rho }\\hat{L_j}^\\dagger ,\\end{split}$ where $\\hat{L_j}$ is the Lindblad operator representing the system-environment interaction and $\\hat{\\rho }$ is the reduced density operator.", "The quantum state diffusion (QSD) approach corresponds to solving quantum trajectories which are unravellings of the master equation.", "This allows for numerical efficiencies, and physical insights not available via the master equation.", "Specifically, we can use the QSD numerical library[20] to solve a stochastic version of the Schrödinger equation $\\begin{split}d|\\psi \\rangle =&\\frac{i}{\\hbar }\\hat{H}|\\psi \\rangle dt\\\\& + \\sum _j \\left(\\langle \\hat{L_j}^{\\dagger }\\rangle \\hat{L_j} -\\frac{1}{2}\\hat{L_j}^{\\dagger }\\hat{L_j}-\\frac{1}{2}\\langle \\hat{L_j}^{\\dagger }\\rangle \\langle \\hat{L_j}\\rangle \\right)|\\psi \\rangle dt\\\\& +\\sum _j \\left( \\hat{L_j}-\\langle \\hat{L_j} \\rangle \\right) |\\psi \\rangle d\\xi _j.\\end{split}$ The solution $|\\psi (t)\\rangle $ to Eq.", "(REF ) is a “quantum trajectory”, and we obtain $\\hat{\\rho }=\\frac{1}{M}\\sum ^M_i|\\psi ^i\\rangle \\langle \\psi ^i|$ , as the mean over an ensemble of $M$ normalized pure state projection operators[20].", "This $\\hat{\\rho }$ satisfies the Lindblad master equation Eq.", "(REF ).", "That is, starting with an ensemble of identical pure states, we obtain the time evolution of multiple trajectories that evolve into an ensemble of different pure states due to interaction with the environment; this is the density matrix.", "For the open quantum double-well Duffing oscillator $\\hat{H}$ and $\\hat{L_j}$ in Eq.", "(REF ) are chosen as[21], [22], [23], [4], $&& \\hat{H}=\\hat{H_D}+\\hat{H_R}+\\hat{H_{ex}} \\\\&& \\hat{H_D}=\\frac{1}{2m}\\hat{p}^2 + \\frac{m\\omega _0^2}{4l^2}x^4-\\frac{m\\omega _0^2}{2}x^2, \\\\&& \\hat{H_R}=\\frac{\\gamma }{2}(\\hat{x}\\hat{p}+\\hat{p}\\hat{x}) \\\\&& \\hat{H_{ex}}=-gml\\omega _0^2\\hat{x}\\cos (\\omega t) \\\\&& \\hat{L_1}=\\sqrt{\\gamma \\left(1+\\bar{n}\\right)}\\left(\\sqrt{\\frac{m\\omega _0}{\\hbar }}\\hat{x}+i\\sqrt{\\frac{\\gamma }{m\\omega _0 \\hbar }}\\hat{p}\\right), \\\\&& \\hat{L_2}=\\sqrt{\\gamma \\bar{n}}\\left(\\sqrt{\\frac{m\\omega _0}{\\hbar }}\\hat{x}-i\\sqrt{\\frac{\\gamma }{m\\omega _0 \\hbar }}\\hat{p}\\right).", "$ where $\\gamma $ is the strength of coupling between the oscillator and thermal bath environment.", "We now define $\\hat{x}$ and $\\hat{p}$ in unitless forms $\\hat{Q}=\\sqrt{m\\omega _0/\\hbar }\\hat{x}$ and $\\hat{P}=\\sqrt{1/m\\omega _0\\hbar }\\hat{p}$ respectively[22], [23] yielding the dimensionless set of equations $&& \\hat{H^{^{\\prime }}_{D}}=\\frac{1}{2}\\hat{P}^2+\\frac{\\beta ^2}{4}\\hat{Q}^4-\\frac{1}{2}\\hat{Q}^2 \\\\&& \\hat{H^{^{\\prime }}_R}=\\frac{\\Gamma }{2}\\left(\\hat{Q}\\hat{P}+\\hat{P}\\hat{Q}\\right) \\\\&& \\hat{H^{^{\\prime }}_{ex}}=-\\frac{g}{\\beta }\\hat{Q}\\cos (\\Omega t) \\\\&&\\hat{L^{^{\\prime }}_1}=\\sqrt{\\Gamma (1+\\bar{n})}\\left(\\hat{Q}+i\\hat{P}\\right) \\\\&&\\hat{L^{^{\\prime }}_2}=\\sqrt{\\Gamma \\bar{n}}\\left(\\hat{Q}-i\\hat{P}\\right)$ where the time in measured in units of $\\omega _0^{-1}$ so that $\\Gamma =\\gamma /\\omega _0, \\Omega = \\omega /\\omega _0$ .", "In Eqs.", "(, ,,), $\\bar{n}=\\left(e^{\\hbar \\Omega /k_BT} - 1\\right)^{-1}$ is the Bose-Einstein distribution of thermal photons representing the effect of the environment at finite $T$ and is evaluated at the natural frequency of the oscillator[24].", "This dimensionless formulation isolates the scaling factor $\\beta =\\sqrt{\\frac{\\hbar }{S}}$ as the single length scale of the problem, where $S$ is the classical action, such that $\\beta ^2 = \\frac{\\hbar }{ml^2\\omega _0}$ .", "For the classical equation of motion for a damped Duffing oscillator, which is $\\frac{d^2x}{dt^2}+2\\Gamma \\frac{dx}{dt} +\\beta ^2x^3-x=\\frac{g}{\\beta }\\cos (\\Omega t)$ , solutions to the classical equation of motion are unchanged with respect to different values of $\\beta $ except for the length scale of the phase space.", "However, changing $\\beta $ changes the quantum solution considerably.", "Comparing the quartic potential Eq.", "(REF ) for the motion of the fundamental mode of the nanomechanical resonator with the potential term in the Duffing Hamiltonian Eq.", "(REF ), it is straightforward to see that for the quantum nanoelectromechanical resonator $\\beta ^2 \\equiv \\frac{\\hbar }{S}= \\frac{\\hbar l_0}{8\\pi ^2} \\sqrt{\\frac{A}{2E\\rho {I}^3[(-1)(1 + \\frac{T_0l_0^2}{4\\pi ^2EI})]^3}}$ where, as noted in section II, we restrict our consideration to resonators of the double well shape with $T_0<-\\frac{4\\pi ^2EI}{l_0^2}$ .", "The scaling parameter $\\beta ^2$ determines the way in which the action of the nanomechanical system scales relative to $\\hbar $ .", "As such, studying the system as $\\beta $ is changed allows us to explore the quantum to classical transition.", "Previous theoretical work[21], [22], [23] for the Duffing problem focused on the behavior of the quantum trajectories has confirmed that $\\beta \\rightarrow 0$ indeed recovers the classical limiting behavior, including the presence of 'strange attractors' of chaos in the phase space, while $\\beta =1$ shows behavior distinctive of the quantum regime.", "There is interesting physics in the intermediate regime[21], and in particular the details of the transition as $\\beta $ changes are informative (as below).", "As shown in Table (REF ), doubly-clamped nanoelectromechanical beams can be made of different materials including Si[15], [6], single-walled carbon nanotubes (SWNT) or multi-walled (MWNT) nanotubes[7] and even metals like gold[25].", "The analysis above for the Duffing oscillator also works for a doubly-clamped Pt nanowire[26], which has a high fundamental frequency (greater than 1 GHz).", "The characteristic parameters given in Table (REF ) for example show that the current experimental setups can be tuned within the range of $\\beta $ considered in this paper.", "Specifically, for the SWNT experiment of Witkamp et al[11], (see Table REF ), they used a device of radius $r=1.6 nm$ and length $l_0=1.15 \\mu m$ .", "This yields, using their other device parameters and an $I = \\frac{\\pi r^4}{4}$ that $\\beta ^2 = 5.34 X 10^{-6} X(\\frac{\\lambda }{4\\pi ^2}-1)^{-3/2}$ where $\\lambda = \\frac{T_0l_0^2}{EI}$ is greater than $4\\pi ^2$ to yield the double-well shape.", "If $\\lambda $ is in the range $(39.5 - 60)$ , we get $\\beta $ in the range $(0.65 - 0.04)$ , which as we see below, is precisely the range needed to map the transition.", "Witkamp et al report $\\lambda = 26$ , so this is clearly well within reach of current experiments.", "Table: Four common types of doubly-clamped structure of NEMS thathave been realized experimentally.", "The characteristic parameters can beused to find the fundamental mode frequency.There are other parameters of interest related to the system-environment interactions.", "In particular, the experimentally measurable quality factor $Q$ quantifies energy lost in comparison to total energy of the resonator during one complete driving period which can be related to the damping $\\Gamma $ .", "A comprehensive account for the mechanisms of quality factor, $Q$ , in NEMS has not been established[25].", "but for the case of doubly-clamped beam, $Q$ can arise from clamping, thermal elastic damping, as well as defects of crystalline structure of the material[8].", "Usually, $Q$ can be measured experimentally using magnetomotive techniques[10].", "For low temperatures (20 K and below) $Q$ can be as large as of the order of $10^3$[18].", "Lowering the temperature further to the milikelvin range further depresses energy dissipation and recent experiments[25], [27] have achieved quality factors on the order of $10^5$ .", "Although these experiments have pushed towards high $Q$ and correspondingly low $\\Gamma $ , it is clearly possible to make systems more dissipative, thus allowing the exploration of a large range in $\\Gamma $ ." ], [ "Results and Discussions", "The solutions to Eq.", "(REF ) yield “quantum trajectories” which have been previously discussed in detail in the context of the recovery of classical behavior from quantum mechanics[22], [21], [23].", "What we focus our attention on below is ensemble-and-time-averaged asymptotic behavior of the system, defined through the probability distributions $P_{avg}(x)=\\sum _{i=1}^{M}\\sum _{k=1}^{N}\\frac{1}{MN}\\langle \\psi ^i_k(t)|\\psi ^i_k(t)\\rangle .$ with $t_{\\rm transient} + 4\\pi > t> t_{\\rm transient}$ .", "In doing this we are averaging over multiple $M$ trajectories generated in QSD simulations to yield the behavior of density matrices accessible the laboratory.", "We also eliminate phase-dependent idiosyncracies of these distributions and correlate better with experimental capabilities regarding the measurement of the dynamics of the quadratures by time-averaging trajectories $|\\psi ^i_k(t)\\rangle $ with $N$ samples of each trajectory over two driving periods, taken after a suitably long transient time $t_{\\rm transient}$ .", "The time averaging is not necessary but is useful because (a) in exploring the quantum to classical transition, the properties that signal change most clearly as parameters are varied are asymptotic, global characteristics of the system rather than unrepresentative “snap shots” at specific times.", "Also (b), the time-averaging eliminates experimental difficulties in determining the phase of the driving relative to the observation, as well as allowing one to quickly build statistics.", "We have explored $P_{avg}(x)$ at both zero temperature (that is, $\\bar{n}=0$ ) and finite temperature for several $\\beta $ ; we report here $\\beta = 0.01, 0.3$ and $1.0$ .", "For each of these values, as we change the coupling strength $\\Gamma $ , we also looked for signatures of the quantum to classical transition through the changes in $P_{avg}(x)$ .", "Fig.", "(REF ) shows $P_{avg}(x)$ at zero temperature.", "With high dissipation ($\\Gamma =0.3$ ) we see what we term nonmonotonicity in the behavior of the probability distribution as a function of $\\beta $ .", "Specifically, the probability distribution changes as follows: (a) At $\\beta = 0.01$ it is an asymmetric object centered in a single well, followed by (b) at $\\beta = 0.3$ an essentially symmetric double-peak structure across both wells and then (c) at $\\beta = 0.1$ a symmetric single peak straddling the central potential barrier.", "We understand these results as follows: For $\\beta =0.01$ , the most classical case, the dissipation results in the particle being unable to overcome the central potential barrier even with driving and therefore being confined in a single well.", "The probability peaks at both ends of the probability distribution correspond to the classical turning points of the oscillator orbit in phase space.", "That is, the dynamics in this situation are identical to those that would be generated by a classical trajectory, and unsurprisingly this means that the probability distribution is also entirely explained by classical considerations.", "As we increase the quantum-ness of the system (decrease the scale of the system) to $\\beta =0.3$ , the particle acquires a nonzero probability of being in both wells.", "This is classically impossible given that it is a zero-temperature problem with high damping.", "The inter-well transitions could indeed occur because, as the system gets more quantum-mechanical with $\\beta $ increasing, the scale of the Lindblad term effectively also increases.", "It is therefore possible that the particle continues to act essentially classically but the increased effect of the noise leads to a classical noise-activated hopping over the barrier.", "However, our studies show that the particle is behaving entirely quantum-mechanically during the interwell transition, whence this transition is a quantum tunneling effect.", "Specifically, we have seen that the Wigner function for the particle shows classically forbidden negative regions throughout its dynamics.", "We have also seen that the energy expectation value remains negative while the expectation value of the position transits past the origin[21].", "Finally, we have found instances in time when a wave-packet has significant probability on either side of the barrier, which is incomensurate with it being a localized wavepacket behaving like a classical particle hopping over the barrier.", "Incidentally, the last two demonstrate one advantage of using the QSD wavefunction-based approach, since such information is unavailable in the master equation approach.", "The symmetric double-peak structure of $P_{avg}(x)$ at this value of $\\beta $ indicates that the ensemble average over the wave packet behavior leads to an asymptotically equal probability in either well.", "For $\\beta =1$ , the most quantum case, the probability is peaked at the origin.", "There are multiple ways of understanding this.", "The first is that the zero point energy is such that the lowest states of the potential are above the barrier.", "As such the probability is peaked in the middle of the quartic larger well, ignoring the quadratic bump at the bottom.", "Another way of understanding this situation is to realize that as $\\beta $ is increased above $0.3$ , the scale of the system changes such that the two peaks of the symmetric distribution from $\\beta =0.3$ come closer together until they merge.", "These three figures show not only that there is a clearer difference between the quantum and the classical probability behavior in the double-well Duffing oscillator compared to the single-well version, there are interesting things to be learned from studying the intermediate regime.", "As as alternative case, we also present results from slightly lower dissipation, with $\\Gamma = 0.125$ .", "In this case, even at the classical limit, the dissipation is not strong enough to confine the particle in one well.", "Specifically, the particle trajectory is that of a strange attractor across both wells[21], [22], [23] which yields the unusual shape for the probability distribution.", "The difference in the classical behavior does not persist, and the distributions at higher $\\beta $ for this $\\Gamma $ value follow the same pattern as those for $\\Gamma =0.3$ .", "As such, comparing the two sets of figures, the results at $\\Gamma =0.3$ show a more dramatic transition from the classical to the quantum regime.", "Figure: P avg (x)P_{avg}(x) plots for various parameters.", "The degree ofquantumness β\\beta increases from left to right as 0.01,0.3,1.00.01, 0.3, 1.0.The values for Γ=0.125,0.3\\Gamma = 0.125, 0.3.Figure: P avg (x)P_{avg}(x) plots for various parameters at finite temperature.For the first three rows, the number of thermal photonsn ¯=4\\bar{n}=4 and β\\beta increasesfrom left to right as 0.01,0.3,1.00.01, 0.3, 1.0.", "Γ\\Gamma increases down the gridas 0.03,0.125,0.30.03, 0.125, 0.3.", "The last row has Γ=0.3\\Gamma =0.3 and n ¯=0.5\\bar{n}=0.5and shows a smooth transition from the zero temperature case inFig.", "().At even lower values of $\\Gamma $ , the vanishing dissipation means that the wavefunctions do not localize in a basis as happens for greater dissipation[20].", "This leads to the standard semiclassical convergence problem for the calculations of $P_{avg}(x)$ .", "However we are able to compute $P_{avg}(x)$ at finite temperature for each set of $\\beta $ and $\\Gamma $ and these are presented in Fig.", "(REF ).", "We see quickly that the results are similar to those at zero temperature in Fig.", "(REF ) though we can see that temperatures corresponding to $\\simeq 4$ thermal photons begins to eliminate the structure of the distribution(s) at $\\beta = 0.3$ ; this is roughly where quantum effects are equal to thermal effects.", "As we can see, for large dissipation ($\\Gamma =0.3$ ) and the most classical case, the particle is still confined in one well; that is, thermal activation is not high enough to induce switching between wells.", "The probability distributions are of course broadened due to thermal effects, but the signature of the transition as $\\beta $ is increased are robust.", "The last row of Fig.", "(REF ) reassuringly shows the existence of a smooth transition away from zero temperature behavior.", "In conclusion, we have shown that in a open double-well quantum oscillator clear signatures of quantum behavior and in particular the transition away from classical behavior can be found in the changing shapes of an asymptotically obtained probability distribution $P_{avg}(x)$ .", "At the right parameters, this transition from classical through quantum mechanical behavior can be nonmonotonic and hence should be clearly visible.", "Further, these transitions are robust at finite temperatures.", "The experimentally tunable parameters in NEMS allow for flexibility in exploring the parameter landscape of the transition and are excellent candidates to study this transition.", "We believe that such experiments are viable in doubly-clamped NEMS,(see Table.REF ) Acknowledgements: The authors gratefully acknowledge funding from the Howard Hughes Medical Institute through Carleton College, and help with computing from Ryan Babbush." ] ]

[ [ "Radiation condition at infinity for the high-frequency Helmholtz\n equation: optimality of a non-refocusing criterion" ], [ "Abstract We consider the high frequency Helmholtz equation with a variable refraction index $n^2(x)$ ($x \\in \\R^d$), supplemented with a given high frequency source term supported near the origin $x=0$.", "A small absorption parameter $\\alpha_{\\varepsilon}>0$ is added, which somehow prescribes a radiation condition at infinity for the considered Helmholtz equation.", "The semi-classical parameter is $\\varepsilon>0$.", "We let $\\eps$ and $\\a_\\eps$ go to zero {\\em simultaneaously}.", "We study the question whether the indirectly prescribed radiation condition at infinity is satisfied {\\em uniformly} along the asymptotic process $\\eps \\to 0$, or, in other words, whether the conveniently rescaled solution to the considered equation goes to the {\\em outgoing} solution to the natural limiting Helmholtz equation.", "This question has been previously studied by the first autor.", "It is proved that the radiation condition is indeed satisfied uniformly in $\\eps$, provided the refraction index satisfies a specific {\\em non-refocusing condition}, a condition that is first pointed out in this reference.", "The non-refocusing condition requires, in essence, that the rays of geometric optics naturally associated with the high-frequency Helmholtz operator, and that are sent from the origin $x=0$ at time $t=0$, should not refocus at some later time $t>0$ near the origin again.", "In the present text we show the {\\em optimality} of the above mentionned non-refocusing condition, in the following sense.", "We exhibit a refraction index which {\\em does} refocus the rays of geometric optics sent from the origin near the origin again, and, on the other hand, we completely compute the asymptotic behaviour of the solution to the associated Helmholtz equation: we show that the limiting solution {\\em does not} satisfy the natural radiation condition at infinity.", "More precisely, we show that the limiting solution is a {\\em perturbation} of the outgoing solution to the natural limiting Helmholtz equation, and that the perturbing term explicitly involves the contribution of the rays radiated from the origin which go back to the origin.", "This term is also conveniently modulated by a phase factor, which turns out to be the action along the above rays of the hamiltonian associated with the semiclassical Helmholtz equation." ], [ "General introduction", "In this article, we study the convergence as $\\varepsilon $ approaches 0 of $w^{\\varepsilon }$ , solution to the following rescaled Helmholtz equation $i\\varepsilon \\alpha _{\\varepsilon } \\, w_\\varepsilon (x)+\\frac{\\Delta _{x}}{2}w_\\varepsilon (x)+n^2(\\varepsilon x)w_\\varepsilon (x)=S(x),\\quad x\\in {\\mathbb {R}}^d \\quad (d\\ge 3).$ Here $\\alpha _{\\varepsilon }$ is an absorption parameter, $n^2(x)$ is a space-dependent refraction indexHere and below we use the standard notation $n^2(x)$ , a squared term, assuming in doing so that the corresponding term is everywhere non-negative.", "This is a harmless abuse of notation, since the refraction index $n^2(x)$ that is eventually chosen in our analysis is negative for certain values of $x$ .", "The reader may safely skip this fact, since the Helmholtz equation also arises in the spectral analysis of Schrödinger operators, where the refraction index becomes $E-V(x)$ where $E$ is an energy and $V(x)$ is a space-dependent potential, and the term $E-V(x)$ may change sign in that context.", "and $S(x)$ is a given and smooth source term.", "In the sequel, we assume the following: The absorption parameter $\\alpha _{\\varepsilon }$ satisfiesThe limiting case ${\\alpha }_{\\varepsilon }=0^+$ can be considered along our analysis, see below.", "$\\alpha _{\\varepsilon } >0, \\qquad {\\alpha }_{\\varepsilon }\\mathop {\\longrightarrow }_{{\\varepsilon }\\rightarrow 0} 0.$ The smooth refraction index $n^2(x) \\in C^\\infty ({\\mathbb {R}}^d)$ is a possibly long-range perturbation of a positive constant $n_\\infty ^2>0$ at infinity, namely, for some $\\rho >0$ , we have $\\forall \\, \\alpha \\in {\\mathbb {N}}^d,\\quad \\exists C_{\\alpha }, \\quad \\forall \\, x\\in {\\mathbb {R}}^d, \\quad \\Big |\\partial _{x}^\\alpha \\left( n^2(x)-n^2_{\\infty } \\right)\\Big |\\le C_{\\alpha }\\langle x\\rangle ^{-\\rho -\\alpha },$ where we denote as usual $\\langle x \\rangle :=(1+|x|^2)^{1/2}$ .", "The source term $S(x)$ belongs to the Schwartz classThis assumption may be considerably relaxed at the price of some irrelevant technicalities.", "${\\mathcal {S}}({\\mathbb {R}}^d)$ .", "The question we raise is the following.", "Thanks to the absorption parameter ${\\alpha }_{\\varepsilon }>0$ in (REF ), the sequence of solutions $w_{\\varepsilon }$ is uniquely defined (see below for the limiting case ${\\alpha }_{\\varepsilon }=0^+$ ).", "On top of that, and as a consequence of specific homogeneous bounds obtained by Perthame and Vega in [14] (see [5] for extensions by Jecko and the first author, as well as [6]), it is clear that the sequence $w_{\\varepsilon }$ is bounded in some weighted $L^2$ space, uniformly in ${\\varepsilon }$ .", "Hence the sequence $w_{\\varepsilon }$ possesses a limit (up to subsequences), say in the distribution sense, and the limit $w=\\lim w_{\\varepsilon }$ satisifies in the distribution sense the Helmholtz equation $\\frac{\\Delta _{x}}{2} \\, w +n^2(0) \\, w=S,$ where the variable coefficients refraction index $n^2({\\varepsilon }x)$ in (REF ) has now coefficients frozen at the origin $x=0$ .", "Now, the difficulty is, the Helmholtz equation (REF ) does not have a uniquely defined solution.", "At least two distinct solutions exist, namely the outgoing solution, defined as $w_{out}(x):&=\\lim _{\\delta \\rightarrow 0^+}\\left(i\\delta +\\frac{\\Delta _{x}}{2}+n^2(0)\\right)^{-1}S(x),$ and the incoming solution, defined similarly as $\\displaystyle w_{in}=\\lim _{\\delta \\rightarrow 0^+}\\left(-i\\delta +\\frac{\\Delta _{x}}{2}+n^2(0)\\right)^{-1}S$ .", "Equivalently, the outgoing solution may be defined as the unique solution to the Helmholtz equation (REF ) which satisfies the so-called Sommerfeld radiation condition at infinity, namely $\\frac{x}{|x|}.\\nabla _{x}w_{out}(x)+i \\,\\sqrt{2} \\, n(0) \\, w_{out}(x)=O\\left(\\frac{1}{|x|^2}\\right),\\quad \\text{ as } |x|\\longrightarrow +\\infty .$ This formulation means that $w_{out}$ is required to oscillate like $w_{out} \\sim \\exp \\left(-i \\sqrt{2} \\, n(0) |x| \\right)/|x|$ as $|x| \\rightarrow \\infty $ .", "Similarly, the incoming solution satisfies the following radiation condition at infinity, namely $\\displaystyle ( x / |x|) \\, .\\nabla _{x}w_{in}- i \\, \\sqrt{2} \\, n(0) \\, w_{in}=O\\left(1/|x|^2\\right)$ , meaning that $w_{in} $ $\\sim $ $ \\exp \\left(+i \\sqrt{2} \\, n(0) |x| \\right)/|x|$ as $x \\rightarrow \\infty $ .", "In that perspective, and due to the positive absorption parameter ${\\alpha }_{\\varepsilon }>0$ in (REF ), it is natural to expect that the previously defined sequence $w_{\\varepsilon }$ goes to the outgoing solution $w_{out}$ to (REF ).", "This is the question we address here.", "It turns out that delicate analytical tools are needed to provide a clean understanding of the phenomena at hand, and to establish whether $w_{\\varepsilon }\\sim w_{out}$ as ${\\varepsilon }\\rightarrow 0$ .", "The basic difficulty is a conflict between a local and a global phenomenon.", "On the one hand, the obvious fact that $w_{\\varepsilon }$ goes to a solution to (REF ) is local: locally in $x$ , i.e.", "in the distribution sense, the variable refraction index $n^2({\\varepsilon }x)$ goes to the value $n^2(0)$ at the origin.", "On the other hand, the positive absorption parameter ${\\alpha }_{\\varepsilon }>0$ in (REF ) somehow asserts that $w_{\\varepsilon }$ is an outgoing solution to $\\Delta _x w_{\\varepsilon }/2+n^2({\\varepsilon }x) \\, w_{\\varepsilon }= S$ , hence introducing the value at infinity $\\displaystyle n_\\infty =\\lim _{x\\rightarrow \\infty } n({\\varepsilon }x)=\\lim _{x\\rightarrow \\infty } n(x)$ , the solution $w_{\\varepsilon }$ should roughly oscillate like $w_{\\varepsilon }\\sim \\exp \\left(-i \\, \\sqrt{2} \\, n_\\infty |x| \\right)/|x|$ at infinity.", "This is a global phenomenon.", "Now, all this is to be compared with the fact that $w_{out}$ oscillates like $w_{out}\\sim \\exp \\left(-i \\, \\sqrt{2} \\, n(0) |x| \\right)/|x|$ at infinity.", "Due to the fact that $n_\\infty \\ne n(0)$ , the radiation condition at infinity satisfied by $w_{\\varepsilon }$ for any positive value ${\\varepsilon }>0$ is a priori incompatible with the radiation condition at infinity satisfied by the expected limit $w_{out}$ : the radiation condition at infinity cannot be followed at once uniformly in ${\\varepsilon }$ , in any direct fashion (this is not in contradiction with the expected local convergence of $w_{\\varepsilon }$ towards $w_{out}$ .)", "Before going further, let us mention that the above question stems from a series of articles [1], [3] about the high-frequency Helmholtz equation (Equation (REF ) is a low-frequency equation) (see also [9] and [10] for similar considerations, in the case of a discontinuous refraction index, as well as [16] and [17] for the case of a variable absorption coefficient).", "These two papers investigate the high-frequency behaviour, in terms of semi-classical measures, of high-frequency Helmholtz equations of the form $i\\varepsilon \\alpha _{\\varepsilon }u_\\varepsilon (x)+\\frac{\\varepsilon ^2}{2} \\, \\Delta _{x} u_\\varepsilon (x)+n^2( x)u_\\varepsilon (x)=\\frac{1}{\\varepsilon ^{d/2}}S\\left(\\frac{x}{\\varepsilon }\\right)\\quad (x\\in {\\mathbb {R}}^d).$ The link between the low-frequency equation (REF ) that is the purpose of this article, and the high-frequency equation (REF ) is provided by the following basic observation: the function $w_{\\varepsilon }$ satisfies  (REF ) if and only if the rescaled function $u_{\\varepsilon }(x)=\\frac{1}{{\\varepsilon }^{d/2}} \\, w_{\\varepsilon }\\left(\\frac{x}{{\\varepsilon }}\\right)$ satisfies (REF ).", "In that picture, the main phenomenon to be described in (REF ) is the possibility of resonances between the high-frequency waves selected by the Helmholtz operator $\\varepsilon ^2 \\Delta _{x}/2 + n^2(x)$ , and the high-frequency waves carried by the rescaled source term $\\displaystyle {\\varepsilon }^{-d/2} \\, S(x/{\\varepsilon })$ , both having the same wavelength ${\\varepsilon }$ .", "Amongst others, it is established in [1], [3] that the semiclassical measure associated with $u_{\\varepsilon }$ can be completely computed provided $w_{\\varepsilon }$ indeed converges towards $w_{out}$ , this latter requirement being left as a conjecture in the cited papers.", "This is the motivation for the question we address here.", "In [4], the first positive convergence result $w_{\\varepsilon }\\rightarrow w_{out}$ is established.", "This results requires, amongst others, a specific and original non-refocusing condition on the refraction index $n^2(x)$ (called \"transversality condition\" in the original paper).", "This condition (see below for details) roughly asserts that the rays of geometric optics associated with the the semi-classical Helmholtz operator ${\\varepsilon }^2\\Delta _x/2+n^2(x)$ cannot focus at some positive time $t>0$ near the origin $x=0$ when issued from the origin at time $t=0$ .", "Later, X.-P. Wang and P. Zhang [19] proved a similar, positive result, using a so-called virial assumption which is stronger than the above non-refocusing condition.", "J.-F. Bony in [2] establishes along quite different lines a positive result that is similar in spirit, requiring a weaker non-refocusing condition.", "The goal of the present text is to prove in some sense the optimality of the non-refocusing condition pointed out in [4].", "We construct a refraction index $n^2(x)$ which violates the non-refocusing condition (rays of geometric optics issued from the origin do refocus close to the origin at some later time), and, by explicitly computing the asymptotic behaviour of $w_{\\varepsilon }$ thanks to an appropriate amplitude/phase representation developped in [4], we prove that $w_{\\varepsilon }\\mathop {\\sim }_{{\\varepsilon }\\rightarrow 0} w_{out} + \\underbrace{\\text{perturbation}}_{\\ne 0},$ where the perturbation is computed as well.", "It explicitly involves the contribution of the rays issued from the origin which go back to the origin at some positive time, modulated by a phase factor that is the action, along these rays, of the hamiltonian associated with the high-frequency Helmholtz operator." ], [ "The non-refocusing condition", "As already mentionned, the asymptotic behaviour of $w_{\\varepsilon }$ is dictated by that of the rescaled function $u_{\\varepsilon }(x)={\\varepsilon }^{-d/2} \\, w_{\\varepsilon }(x/{\\varepsilon })$ .", "The function $u_{\\varepsilon }$ is $w_{\\varepsilon }$ rescaled at the semi-classical scale, see (REF ) and (REF ).", "This is translated by the following identity, valid for any smooth test function $ \\phi \\in {\\mathcal {S}}({\\mathbb {R}}^d)$ , namely $\\forall \\, \\phi \\in {\\mathcal {S}}({\\mathbb {R}}^d), \\qquad \\langle w_{\\varepsilon },\\phi \\rangle =\\left\\langle u_{\\varepsilon },\\frac{1}{\\varepsilon ^{d/2}} \\, \\phi \\left( \\frac{x}{\\varepsilon } \\right) \\right\\rangle .\\,$ where we denote as usual $\\langle w_{\\varepsilon },\\phi \\rangle :=\\displaystyle \\int _{{\\mathbb {R}}^d} w_{\\varepsilon }(x) \\, \\phi ^\\ast (x) \\, dx$ , and $\\ast $ denotes complex conjugation.", "In other words, the weak limit $\\langle w_{\\varepsilon },\\phi \\rangle $ of $w_{\\varepsilon }$ can be computed as the weak limit at the semi-classical scale of $u^{\\varepsilon }$ , namely the limit of $\\langle u_{\\varepsilon },\\varepsilon ^{-d/2}\\phi (x/\\varepsilon )\\rangle $ .", "This first observation is the main reason why semi-classical tools play a key role in our analysis.", "Besides, the asymptotic study of $(\\ref {eqHelm})$ is done here by transforming the problem into a time-dependent problem.", "This approach, introduced in [4], has been used since by J.F.-Bony ($\\cite {MR2582438}$ ) to study the Wigner measure associated to $(\\ref {eqHelme})$ , or by J. Royer ([16]) when the absorption $\\alpha _{\\varepsilon }$ depends on $x$ .", "It consists in writing the solution $w_{\\varepsilon }$ as the integral over the whole time of the propagator associated with $i\\varepsilon \\alpha _{\\varepsilon }+\\Delta _{x}/2+n^2(\\varepsilon \\, x)$ , namely $w_{\\varepsilon }(x)=i\\int _{0}^{+\\infty }e^{-\\alpha _{\\varepsilon }t}e^{it\\left(\\frac{\\Delta _{x}}{2}+n^2(\\varepsilon \\,x)\\right)} \\, S(x) \\, dt.$ In the same way the outgoing solution can be written as $w_{out}(x):&=i\\int _{0}^{+\\infty }e^{it\\left(\\frac{\\Delta _{x}}{2}+n^2(0)\\right)} \\, S(x) \\, dt.$ In that picture, proving or disproving the convergence $w_{\\varepsilon }\\sim w_{out}$ reduces to passing to the limit in the above time integral.", "Combining the two above observations, the basic first step of our analysis consists in writing, for any given test function $\\phi $ , an in [4], $&\\nonumber \\langle w_{\\varepsilon }\\, , \\, \\phi \\rangle =\\big \\langle u^{\\varepsilon }\\, , \\, {\\varepsilon }^{-d/2} \\phi (x/{\\varepsilon }) \\big \\rangle \\\\&\\qquad \\quad \\,\\,=\\frac{i}{{\\varepsilon }} \\, \\int _0^{+\\infty }e^{-{\\alpha }_{\\varepsilon }\\, t} \\,\\Big \\langle U_{\\varepsilon }(t) \\, S_{\\varepsilon }\\, , \\, \\phi _{\\varepsilon }\\Big \\rangle \\, dt,$ where we use the notation $&S_{\\varepsilon }(x):= \\frac{1}{{\\varepsilon }^{d/2}} \\, S\\left(\\frac{x}{{\\varepsilon }} \\right), \\quad \\text{ and similarly } \\,\\phi _{\\varepsilon }(x):= \\frac{1}{{\\varepsilon }^{d/2}} \\, \\phi \\left(\\frac{x}{{\\varepsilon }} \\right),$ where the semi-classical propagator associated with the semi-classical Hamiltonian ${\\varepsilon }^2 \\Delta _x/2 + n^2(x)$ is $U_{\\varepsilon }(t)=\\exp \\left( i \\, \\frac{t}{{\\varepsilon }} \\, \\left( \\frac{{\\varepsilon }^2}{2} \\Delta _x + n^2(x) \\right) \\right)$ It is fairly clear on formula (REF ) that the asymptotics ${\\varepsilon }\\rightarrow 0$ in $\\langle w_{\\varepsilon }, \\phi \\rangle $ is dominated on the one hand by the concentration of the rescaled test function $\\phi _{\\varepsilon }$ close to the origin at the semi-classical scale ${\\varepsilon }$ , and on the other hand by the oscillations induced by the semi-classical propagator $U^{\\varepsilon }(t)$ at the semi-classical scale ${\\varepsilon }$ as well.", "The point is to measure the possible constructive intereference between both waves.", "As standard in semiclassical analysis we define the semiclassical symbol $h(x,\\xi )=\\frac{|\\xi |^2}{2}-n^2(x),$ associated with the semiclassical Schrödinger operator $-\\frac{\\varepsilon ^2}{2} \\, \\Delta _{x}-n^2(x)$ .", "The semi-classical propagator $U_{\\varepsilon }(t)$ is known to roughly propagate the information along the rays of geometric optics, defined as the solutions to the Hamiltonian ODE associated with $h$ , namely (see e.g.", "[8], [13], or [15]) ${\\left\\lbrace \\begin{array}{ll}\\begin{array}{ll}\\displaystyle \\vspace{5.69046pt}\\frac{\\partial }{\\partial t}X(t,x,\\xi )=\\Xi (t,x,\\xi ),& \\quad X(0,x,\\xi )=x,\\\\\\displaystyle \\frac{\\partial }{\\partial t}\\Xi (t,x,\\xi )=\\nabla _x n^2(X(t,x,\\xi )),&\\quad \\Xi (0,x,\\xi )=\\xi .\\end{array}\\end{array}\\right.", "}$ It is clear as well that the integral $\\int _0^{+\\infty } \\ldots $ in (REF ) carries most of its energy, semi-classically, over the zero energy level of $h$ , defined as $H_{0}:=\\left\\lbrace (x,\\xi )\\in {\\mathbb {R}}^{2d},\\ \\text{s.t.", "}\\ h(x,\\xi )=0\\right\\rbrace .$ In view of the integral (REF ) and of the above considerations, the following definitions are natural.", "The first definition is standard.", "Definition 1.1 [non-trapping condition] The refraction index $n^2$ is said non-trapping on the zero energy level whenever for each $(x,\\xi )\\in H_0$ , the associated trajectory $(X(t,x,\\xi ),\\Xi (t,x,\\xi ))$ satisfies $\\lim _{t\\rightarrow +\\infty }| X(t,x,\\xi )|=+ \\infty .$ When the refraction index is non-trapping, the rough idea is that any trajectory $X(t,x,\\xi )$ on the zero energy level leaves any given neighbourhood of the origin $x=0$ in finite time, making the above integral $\\int _0^{+\\infty } \\ldots $ in (REF ) converge with respect to the bound $t=+\\infty $ .", "The second definition comes from [4] (this assumption is called \"transversality condition\" in the original text).", "Definition 1.2 [non-refocusing condition] We say that $n^2$ satisfies the non-refocusing condition if the refocusing set, defined as $M:=\\left\\lbrace (t,\\xi ,\\eta )\\in ]0,+\\infty [\\times {\\mathbb {R}}^{2d}\\ \\ s.t.\\ \\frac{|\\eta |^2}{2}=n^2(0),\\ X(t,0,\\xi )=0,\\ \\Xi (t,0,\\xi )=\\eta \\right\\rbrace $ is such that $M$ is a submanifold of $]0,+\\infty [\\times {\\mathbb {R}}^{2d}$ and $M$ satisfies $\\text{dim}\\, M<d-1.$ When the non-refocusing condition is satisfied, the rough idea is that the trajectories $X(t,0,\\xi )$ on the zero energy level issued from the origin $x=0$ at time $t=0$ cannot accumulate in any given neighbourhood of the origin $x=0$ at later times $t>0$ (this is encoded in the requirement on ${\\rm dim} \\, M$ ).", "Technically speaking, an appropriate stationary phase argument in formula (REF ) allows to exploit in [4] the non-refocusing condition and to prove the weak convergence of $w_{\\varepsilon }$ towards $w_{out}$ under this assumption.", "The main result in [4] is the following: when the refraction index is both non-trapping and satisfies the above non-refocusing condition, then $w_{\\varepsilon }\\sim w_{out}$ as ${\\varepsilon }\\rightarrow 0$ weakly.", "Recently, J.F.", "Bony in [2] shows the convergence of the Wigner measure associated with $w_{\\varepsilon }$ .", "He requires a geometrical assumption on the index of refraction that is in the similar spirit, yet weaker, than the above non-refocusing condition, namely $\\text{meas}_{n-1}\\left\\lbrace \\xi \\in \\sqrt{2n^2(0)} \\, {\\mathbb {S}}^{d-1};\\quad \\exists \\ t>0\\quad X(t,0,\\xi )=0\\right\\rbrace =0,$ where $\\text{meas}_{n-1}$ is the Euclidian surface measure on $\\sqrt{2n^2(0)} \\, {\\mathbb {S}}^{d-1}$ and ${\\mathbb {S}}^{d-1}$ denotes the unit sphere in dimension $d$ .", "Besides, inspired by [4], he constructs a refraction index which is both non-trapping and does not satisfy condition (REF ), and in that case he proves the non-uniqueness of the limiting of the Wigner measure.", "The goal of this paper is to construct a refraction index that is both non-trapping and violates the non-refocusing condition, and to establish in that case that $w_{\\varepsilon }$ goes weakly to a function of the form \"$w_{out}+$ perturbation\", for some explicitly computed and non-zero perturbation.", "To be more accurate, we construct below a refraction index for which the above refocusing manifold $\\displaystyle M=\\big \\lbrace (t,\\xi ,\\eta )\\ \\text{s.t.}", "\\ \\frac{|\\eta |^2}{2}=n^2(0),\\ X(t,0,\\xi )=0,\\ \\Xi (t,0,\\xi )=\\eta \\big \\rbrace $ is smooth, yet has dimension $\\text{dim}\\, M=d-1$ , a critical case, and we prove $w_{\\varepsilon }\\sim $ \"$w_{out}+$ perturbation\" in that situation." ], [ "Construction of the refraction index and statement of our main result", "Let us first examine the case of dimension $d=2$ .", "Let $M_{s}$ be a circular mirror centered at the origin.", "Any standard ray issued from the origin $x=0$ hits the mirror and goes back to the origin at some later time: refocusing occurs in a strong fashion.", "However all rays are trapped inside the circular mirror, leading to a trapping situation, in the sense of definition REF .", "To recover a non-trapping and refocusing situation, it is necessary to consider an angular aperture of the circular mirror, with total aperture $<\\pi $ .", "This is shown in figure $\\ref {fig1}$ : the circular mirror with total aperture $<\\pi $ provides a (non-smooth) non-trapping and refocusing refraction index.", "Figure: Spherical mirror in dimension 2To transform the above paradigm into a smooth one, some regularizations need to be performed.", "The construction needs to be done in any dimension $d\\ge 2$ as well.", "Let us first introduce the hyperspherical coordinates $(r,\\theta _{1},\\ldots ,\\theta _{d-1})$ in dimension $d\\ge 2$ $x_{1}&=&r\\cos (\\theta _{1}),\\\\x_{2}&=&r\\sin (\\theta _{1})\\cos (\\theta _{2}),\\\\x_{3}&=&r\\sin (\\theta _{1})\\sin (\\theta _{2})\\cos (\\theta _{3}),\\\\\\vdots &&\\\\x_{d-1}&=&r\\sin (\\theta _{1})\\ldots \\sin (\\theta _{d-2})\\cos (\\theta _{d-1}),\\\\x_{d}&=&r\\sin (\\theta _{1})\\ldots \\sin (\\theta _{d-2})\\sin (\\theta _{d-1}),$ with $\\theta _{1}\\in [0,\\pi ], \\quad \\theta _{j}\\in [0,2\\pi ] \\, \\text{ whenever } \\, j\\ge 2 \\text{ when } d\\ge 3, \\qquad \\text{ and }\\theta _1\\in [-\\pi ,\\pi ] \\, \\text{ when } d=2.$ Next, we choose a fixed, smooth cut-off function $\\chi $ on ${\\mathbb {R}}$ such that $\\chi (t)= 1,\\quad \\forall \\ |t|\\le 1,\\qquad \\chi (t)= 0,\\quad \\forall \\ |t|\\ge 2,\\qquad \\chi (t)\\ge 0,\\quad \\forall \\, t\\in {\\mathbb {R}}.$ We choose a radius $R>0$ and define the radial function $f(x)\\equiv f(r):=\\chi \\left(2(r-R)\\right),\\quad \\forall \\ x=(r,\\theta _{1},\\ldots ,\\theta _{d-1}).$ We choose an angle (aperture) $\\theta _{0}\\in [0,\\pi /4[$ , and define the angular function $g(x)\\equiv g(\\theta _1):=\\chi \\left(\\frac{\\theta _{1}}{\\theta _{0}}\\right),\\quad \\forall \\ x=(r,\\theta _{1},\\ldots ,\\theta _{d-1}).$ a smooth version of the angular aperture $|\\theta _1|\\le \\theta _0$ .", "Finally, we choose two parameters $n^2_{\\infty }>0$ and $\\lambda >0$ such that $n^2_{\\infty }<\\lambda .$ We introduce the following Definition 1.3 [refraction index] We define the refraction index, retained in the whole subsequent analysis, as the following smooth version of the circular mirror with total aperture $\\theta _0<\\pi /4$ , namelyThe refraction index is negative in a bounded region of $x$ .", "As already mentioned, we still use the abuse of notation consisting in using the squared of $n$ .", "$n^2(x):=n^2_{\\infty }-\\lambda f(x)g(x)\\equiv n_\\infty ^2-\\lambda f(r) g(\\theta _1), \\qquad \\forall \\, x\\in {\\mathbb {R}}.$ Figure: The function n ∞ 2 -n 2 (x)=λf(x)g(x)n_\\infty ^2-n^2(x)=\\lambda \\, f(x) \\, g(x) in dimension d=2d=2We are now in position to state our main result.", "Let $(e_{1},\\ldots , e_{d})$ be the canonical basis of ${\\mathbb {R}}^d$ .", "Since the direction $e_1$ is a symmetry axis for our refraction index, we introduce for later purposes the space $M_{d}({\\mathbb {R}})$ of square matrices of dimension $d$ , we denote by ${\\mathbb {O}}_{d}({\\mathbb {R}})$ the space of orthogonal matrices, and we introduce the notation ${\\mathbb {O}}_{d,1}({\\mathbb {R}}):=\\left\\lbrace A\\in {\\mathbb {O}}_{d}({\\mathbb {R}}),\\ \\text{s.t.", "}\\ A e_{1}=e_{1}\\right\\rbrace .$ The refraction index $n^2(x)$ in (REF ) is invariant under the action of ${\\mathbb {O}}_{d,1}({\\mathbb {R}})$ .", "We last introduce a particular set of speeds, namely the set of initial speeds $\\xi $ such that the zero energy trajectory $X(t,0,\\xi )$ issued from the origin at time $t=0$ is reflected towards the origin at some later time $t>0$ .", "With the retained value of $n^2(x)$ , we arrive at the definition Definition 1.4 [reflected rays] The reflection set $I_{\\theta _0}$ is defined as $&\\nonumber I_{\\theta _{0}}=\\left\\lbrace \\xi :=(|\\xi |,\\theta _{1},\\ldots ,\\theta _{d-1}) \\in {\\mathbb {R}}^d \\, \\text{ s.t. }", "\\,\\theta _1\\in [-\\theta _0,+\\theta _0] \\, \\text{ and } \\, \\left|\\xi \\right|=\\sqrt{2 \\, n^2(0)}\\right\\rbrace .$ Note that the (intuitive) fact that a velocity $\\xi $ is such that $X(t,0,\\xi )$ hits the origin at some time $t>0$ if and only if $\\xi \\in I_{\\theta _0}$ , is proved later (see section REF ).", "Our main result in this text is the Theorem 1.5 [Main Result] Let $n^2$ be the refraction index defined in (REF ).", "Assume the aperture $\\theta _0<\\pi /4$ and the radius $R>0$ satisfy the smallness condition $1-\\cos (2\\theta _0) < \\frac{1}{2 R}.$ Assume $d\\ge 3$ .", "Then, the following holds: i) The index $n^2$ is non-trapping on the zero-energy level $H_{0}=\\lbrace (x,\\xi ) \\, \\text{ s.t. }", "\\, |\\xi |^2/2-n^2(x)=0\\rbrace $ .", "ii) The refocusing set $M=\\lbrace (t,\\xi ,\\eta ) \\, \\text{ s.t. }", "\\, |\\eta |^2=2 n^2(0), \\,X(t,0,\\xi )=0, \\,\\Xi (t,0,\\xi )=\\eta \\rbrace $ (see (REF )) is a smooth submanifold of $]0,+\\infty [\\times {\\mathbb {R}}^{2d}$ , with boundary, and its dimension has the critical value ${\\rm dim}(M)=d-1.$ iii) Assume the source term $S$ satisfies $S\\in {\\mathcal {S}}({\\mathbb {R}}^d)$ .", "Then, we have $\\forall \\,\\phi \\in {\\mathcal {S}}({\\mathbb {R}}^d),\\qquad \\big \\langle w_{\\varepsilon }-\\left( w_{out}+L_{\\varepsilon }\\right) \\, , \\,\\phi \\big \\rangle \\mathop {\\longrightarrow }_{{\\varepsilon }\\rightarrow 0} 0,$ where the distribution $L_{\\varepsilon }$ is defined for any $\\phi \\in {\\mathcal {S}}({\\mathbb {R}}^d)$ through $&\\nonumber \\langle L_{\\varepsilon }, \\phi \\rangle = \\\\&\\quad C_{n^2,d}\\int _{I_{\\theta _{0}}}\\exp \\left(\\frac{i}{\\varepsilon } \\int _{0}^{T_{R}} \\left(\\frac{\\left|\\Xi (s,0,\\xi )\\right|^2}{2}+n^2(X(s,0,\\xi ))\\right) \\, ds \\right) \\,\\widehat{S}(\\xi )\\widehat{\\phi ^*}(-\\xi ) \\, d\\sigma _{\\theta _0}(\\xi ).$ Here $d\\sigma _{\\theta _0}$ denote the natural Euclidean surface measure on $I_{\\theta _0}$ (see definition REF ), the return time $T_R>0$ is the unique timeThe fact that all these quantities exist and are well defined is part of the Theorem, and is proved in section REF .", "such that for any $\\xi \\in I_{\\theta _0}$ we have $X(T_R,0,\\xi )=0$ , and the constant $C_{n^2,d}\\ne 0$ can be explicitly computed and depends only on the index $n^2$ and on the dimension $d$ .", "Remark The condition (REF ) is technical, and requires the aperture $\\theta _0$ to be small: it ensures the trajectories cannot be trapped by the refraction index.", "Remark Note in passing that the constraint $d\\ge 3$ , which is also needed in reference [4], comes from a stationary phase argument.", "This constraint on the dimension is standard in the analysis of Schrödinger-like operators.", "It comes from the fact that the dispersion induced by the free Schrödinger operator acts like $t^{-d/2}$ , a factor that is integrable close to $t=+\\infty $ whenever $d\\ge 3$ .", "Remark Let $\\xi _0:=(\\sqrt{2} \\, n(0) , 0 ,\\ldots ,0)$ .", "The distribution $L_{\\varepsilon }$ can as well be written as $&\\nonumber \\langle L_{\\varepsilon } \\, , \\, \\phi \\rangle = \\\\&\\quad C_{n^2,d}\\,\\exp \\left(\\frac{i}{\\varepsilon } \\int _{0}^{T_{R}} \\left(\\frac{\\left|\\Xi (s,0,\\xi _0)\\right|^2}{2}+n^2(X(s,0,\\xi _0))\\right) \\, ds \\right) \\,\\left(\\int _{I_{\\theta _{0}}}\\widehat{S}(\\xi )\\widehat{\\phi ^*}(-\\xi ) \\, d\\sigma _{\\theta _0}(\\xi )\\right).$ This formulation illustrates in a clearer way the fact that if the source $S$ radiates towards the mirror, then $w_{\\varepsilon }$ converges towards a non-trivial perturbation of $w_{out}$ .", "Note in passing that in the present counter-example, as in the paper by J.-F. Bony [2] , only subsequences of $w_{\\varepsilon }$ converge, due to the above oscillatory factor $\\exp (i \\, {\\rm const.", "}/{\\varepsilon })$ .", "Remark In the chosen hyperspherical coordinates, the Euclidean measure $d\\sigma _{\\theta _0}(\\xi )$ coincides with $\\displaystyle d\\sigma _{\\theta _0}(\\xi )= n(0)^{d-1} \\, d\\sigma (\\theta _1, \\ldots , \\theta _{d-1})$ , where $d\\sigma (\\theta _1, \\ldots , \\theta _{d-1})$ denotes the standard euclidean surface measure on the unit sphere ${\\mathbb {S}}^{d-1}$ ." ], [ "Preliminary reduction of the proof", "Our main result contains three distinct statements.", "Items (i) and (ii) are of geometric nature, and merely concern the behavious of the classical trajectories associated with the retained refraction index.", "Their proof is performed in sections REF and REF , respectively.", "Item (iii) is the main item, and concerns the asymptotic analysis of $w_{\\varepsilon }$ .", "Since our analysis heavily relies on tools previously developped in [4], we briefly recall here some of these tools and indicate how the analysis of $w_{\\varepsilon }$ can be reduced to a simpler sub-problem.", "We postpone the analysis of the reduced subproblem, hence of item (iii) of our main result, to section below.", "As already indicated, given a smooth test function $\\phi $ , we start from the formulation $\\left\\langle w_{\\varepsilon },\\phi \\right\\rangle =\\frac{i}{\\varepsilon }\\int _{0}^{+\\infty } e^{-\\alpha _{\\varepsilon }t}\\big \\langle U_{\\varepsilon }(t)S_{\\varepsilon },\\phi _{\\varepsilon }\\big \\rangle \\,dt.$ (See above for the notation).", "The next step consists in splitting the above time integral into four time scales, namely very small, small, moderate, and large time scales.", "To do so, we take one small parameter $\\theta >0$ and two large parameters $T_0>0$ and $T_1>0$ , and split the above time integral into the four zones $0 \\le t \\le T_0 \\, {\\varepsilon }, \\quad T_0 \\, {\\varepsilon }\\le t \\le \\theta , \\quad \\theta \\le t \\le T_1, \\quad T_1\\le t \\le +\\infty \\quad (\\theta \\ll 1, \\quad T_0, T_1 \\gg 1).$ Technically, we use a smooth splitting, based on the already used cut-off function $\\chi $ (see (REF )).", "Besides, we also distinguish between the contribution of zero and non-zero energies, namely taking a small parameter $\\delta >0$ , we write, in the sense of functional caculus for self-adjoint operators, the identity $1=\\chi _\\delta (H_{\\varepsilon }) + (1-\\chi _\\delta )(H_{\\varepsilon }), \\, \\text{ where } \\, H_{\\varepsilon }:=\\frac{{\\varepsilon }^2}{2} \\Delta _x + n^2(x),\\, \\text{ and } \\, \\chi _\\delta (s):=\\chi \\left(\\frac{s}{\\delta } \\right) \\, (s \\in {\\mathbb {R}}, \\delta \\ll 1).$ The main intermediate result of the present subsection is the following Proposition 1.6 [Main intermediate result] Take a test function $\\phi \\in {\\mathcal {S}}({\\mathbb {R}}^d)$ .", "Define $\\widetilde{w_{\\varepsilon }}$ as $\\left\\langle \\widetilde{w_{\\varepsilon }},\\phi \\right\\rangle :=\\frac{i}{\\varepsilon }\\int _{\\theta }^{T_1}\\left( 1 - \\chi \\right)\\left( \\frac{t}{\\theta }\\right) \\,e^{-\\alpha _{\\varepsilon }t} \\,\\big \\langle U_{\\varepsilon }(t) \\,\\chi _\\delta (H_{\\varepsilon }) \\,S_{\\varepsilon },\\phi _{\\varepsilon }\\big \\rangle \\, dt.$ Then, there is a large $T_1>0$ such that for any small $\\delta >0$ , and any small $\\theta >0$ , there exists a constant $C_{\\theta ,\\delta }>0$ such that for any small ${\\varepsilon }>0$ , we have $\\Big |\\Big \\langle \\left( w_{\\varepsilon }- (w_{out} + \\widetilde{w_{\\varepsilon }}) \\right) \\, , \\, \\phi \\Big \\rangle \\Big |\\le C_{\\theta ,\\delta } \\, \\left(\\frac{1}{T_0^{d/2-1}}+\\frac{1}{T_0} + {\\alpha }_{\\varepsilon }^2+{\\varepsilon }\\right).$ This result roughly asserts that $w_{\\varepsilon }$ is asymptotic to $w_{out}+\\widetilde{w_{\\varepsilon }}$ as ${\\varepsilon }\\rightarrow 0$ , up to carefully choosing the various parameters $T_0$ , $T_1$ , etc.", "Hence the proof of item (iii) of our main result essentially reduces to proving that $\\widetilde{w_{\\varepsilon }}\\sim L_{\\varepsilon }$ as ${\\varepsilon }\\rightarrow 0$ .", "Proof of Proposition REF .", "The proof is obtained by gathering the statements of Proposition REF , Proposition REF , Proposition REF , Proposition REF below.", "$\\blacksquare $ The remainder part of this paragraph is devoted to a brief idea of proof of the above auxiliary Propositions that lead to Proposition REF ." ], [ "$\\bullet $ Contribution of very small times {{formula:aabbabd9-4b89-45c6-bd9c-1759f536f5c3}} .", "The contribution of very small times to $\\left\\langle w_{\\varepsilon },\\phi \\right\\rangle =i \\, {\\varepsilon }^{-1} \\displaystyle \\, \\int _{0}^{+\\infty } e^{-\\alpha _{\\varepsilon }t}\\left\\langle U_{\\varepsilon }(t)S_{\\varepsilon },\\phi _{\\varepsilon }\\right\\rangle \\,dt$ , is $\\frac{i}{\\varepsilon }\\int _{0}^{2T_{0}\\varepsilon }\\chi \\left(\\frac{t}{T_{0}\\varepsilon }\\right)e^{-\\alpha _{\\varepsilon }t}\\big \\langle U_{\\varepsilon }(t)S_{\\varepsilon },\\phi _{\\varepsilon }\\big \\rangle \\, dt.$ It is the main contribution to $w_{\\varepsilon }$ , provided $T_{0}$ is large enough.", "Indeed, we have the following fact, whose proof is based on a simple weak convergence argument.", "Proposition 1.7 (See [4]).", "Let $n^2(x)$ be any bounded and continuous refraction index.", "Then, if S and $\\phi $ belong to ${\\mathcal {S}}({\\mathbb {R}}^d)$ , we have (i) For all time $T_0>0$ , $\\frac{i}{\\varepsilon }\\int _0^{2T_0\\varepsilon } \\chi \\left(\\frac{t}{T_0\\varepsilon }\\right)e^{-\\alpha _\\varepsilon t}\\big \\langle U_\\varepsilon (t)S_\\varepsilon ,\\phi _\\varepsilon \\big \\rangle \\, dt\\underset{ \\varepsilon \\rightarrow 0}{\\longrightarrow }i\\int _0^{2T_0} \\chi \\left(\\frac{t}{T_0}\\right)\\left\\langle \\exp \\left(it\\left(\\frac{\\Delta _x}{2}+n^2(0)\\right)\\right)S,\\phi \\right\\rangle \\, dt.$ (ii) There exists $C_d>0$ which only depends on the dimension such that $\\left|\\left(\\frac{i}{\\varepsilon }\\int _0^{2T_0\\varepsilon } \\chi \\left(\\frac{t}{T_0}\\right)\\big \\langle \\exp (it(\\Delta _x/2+n^2(0)))S,\\phi \\big \\rangle \\, dt\\right)-\\left\\langle w_{out},\\phi \\right\\rangle \\right|\\\\\\le \\frac{C_d}{T_0^{d/2-1}}.$" ], [ "$\\bullet $ Contribution of small, up to large times, away from the zero-energy level.", "The contribution to $\\left\\langle w_{\\varepsilon },\\phi \\right\\rangle =i \\, {\\varepsilon }^{-1} \\displaystyle \\, \\int _{0}^{+\\infty } e^{-\\alpha _{\\varepsilon }t}\\big \\langle U_{\\varepsilon }(t)S_{\\varepsilon },\\phi _{\\varepsilon }\\big \\rangle \\,dt$ that is associated with small, up to large times, away from the zero-energy level, is $\\frac{i}{\\varepsilon }\\int _{T_{0}\\varepsilon }^{+\\infty } e^{-\\alpha _{\\varepsilon } t}(1-\\chi )\\left(\\frac{t}{T_{0}{\\varepsilon }}\\right)\\big \\langle (1-\\chi _{\\delta })(H_{\\varepsilon })U_{\\varepsilon }(t)S_{\\varepsilon },\\phi _{\\varepsilon }\\big \\rangle \\, dt.$ It is seen to be small, using a non-stationary phase argument in time, see [4] (this is the reason for the previous cut-off close to the initial time $t=0$ , where integrations by parts in time are forbidden).", "Indeed, we have the Proposition 1.8 (See [4]).", "Let $n^2$ be any long-range refraction index.", "Let $S$ and $\\phi $ belong to $L^2({\\mathbb {R}}^d)$ .", "Then there exists a constant $C_\\delta >0$ , which only depends on $\\delta >0$ , such that for any small $\\varepsilon >0$ and any $T_0>0$ , we have $\\left|\\frac{1}{\\varepsilon }\\int _{T_0\\varepsilon }^{+\\infty }(1-\\chi )\\left(\\frac{t}{T_0\\varepsilon }\\right)\\big \\langle \\left(1-\\chi _\\delta (H_\\varepsilon )\\right)U_\\varepsilon (t)S_\\varepsilon ,\\phi _\\varepsilon (t)\\big \\rangle \\, dt\\right|\\le C_\\delta \\left(\\frac{1}{T_0}+\\alpha _\\varepsilon ^2\\right).$" ], [ "$\\bullet $ Contribution of large times, near the zero-energy level.", "The contribution to $\\left\\langle w_{\\varepsilon },\\phi \\right\\rangle =i \\, {\\varepsilon }^{-1} \\displaystyle \\, \\int _{0}^{+\\infty } e^{-\\alpha _{\\varepsilon }t}\\big \\langle U_{\\varepsilon }(t)S_{\\varepsilon },\\phi _{\\varepsilon }\\big \\rangle \\,dt$ that is associated with large times, close to the zero-energy level, is $\\frac{i}{\\varepsilon }\\int _{T_{1}}^{+\\infty } e^{-\\alpha _{\\varepsilon }t}\\left\\langle \\chi _{\\delta }(H_{\\varepsilon })U_{\\varepsilon }(t)S_{\\varepsilon },\\phi _{\\varepsilon }\\right\\rangle dt.$ It is seen to be of order $O(\\varepsilon ^N)$ , for all $N\\in {\\mathbb {N}}$ , see [4].", "Indeed, the semiclassical support of $\\chi _{\\delta }(H_{\\varepsilon })U_{\\varepsilon }(t)S_{\\varepsilon }$ goes to infinity in the $x$ direction at speed of the order 1 (i.e.", "the semi-classical support lies in a region that is at distance of order $t$ from the origin – this uses an argument due to Wang, see [18]), while the semi-classical support of $\\phi _{\\varepsilon }$ remains close to the origin.", "This argument relies on the fact that for $T_{1}$ large enough, the semiclassical supports of the two functions are disconnected, which in turn uses the non-trapping behaviour of the refraction index.", "We arrive at Proposition 1.9 (See [4]).", "Let $n^2$ be any long-range refraction index that is non-trapping.", "Let S and $\\phi $ be in ${\\mathcal {S}}({\\mathbb {R}}^d)$ .", "Then there exist $\\delta _0>0$ and $T_1(\\delta _0)>0$ such that for all time $T_1\\ge T_1(\\delta _0)$ and any $0<\\delta <\\delta _0$ , there exists a constant $C_{\\delta }$ such that $\\left|\\frac{1}{\\varepsilon }\\int _{T_1}^{+\\infty }e^{-\\alpha _\\varepsilon t}\\big \\langle \\chi _\\delta (H_\\varepsilon )U_\\varepsilon (t)S_\\varepsilon ,\\phi _\\varepsilon \\big \\rangle \\, dt \\right|&\\le & C_{\\delta }\\varepsilon .$" ], [ "$\\bullet $ Contribution of small times near the zero-energy level", "The contribution to $\\left\\langle w_{\\varepsilon },\\phi \\right\\rangle =i \\, {\\varepsilon }^{-1} \\displaystyle \\, \\int _{0}^{+\\infty } e^{-\\alpha _{\\varepsilon }t}\\big \\langle U_{\\varepsilon }(t)S_{\\varepsilon },\\phi _{\\varepsilon }\\big \\rangle \\,dt$ that is associated with small times, close to the zero-energy level, is $\\frac{i}{\\varepsilon } \\int _{T_0 {\\varepsilon }}^{\\theta }e^{-\\alpha _{\\varepsilon }t} \\, (1-\\chi )\\left(\\frac{t}{T_0 {\\varepsilon }}\\right)\\chi \\left(\\frac{t}{\\theta }\\right) \\,\\big \\langle U_{\\varepsilon }(t)\\chi _{\\delta }(H_{\\varepsilon })S_{\\varepsilon },\\phi _{\\varepsilon }\\big \\rangle dt.$ Unlike in the previous case, the semiclassical supports of $U_{\\varepsilon }(t)\\chi _{\\delta }(H_{\\varepsilon })S_{\\varepsilon }$ and $\\phi _{\\varepsilon }$ may intersect for these values of time $t$ .", "The whole point in [4] lies, roughly speaking, in proving a dispersion estimate.", "The key is to prove that the variable coefficients Schrödinger propagator $U_{\\varepsilon }(t)$ has the same dispersive properties than the free Schrödinger propagator, corresponding to the case when $n^2\\equiv 0$ , at least for small values of $t$ such that $0 \\le t \\le \\theta $ (for later times, the semiclassical support of $U_{\\varepsilon }(t) \\, S_{\\varepsilon }$ is close to the classical trajectories $(X(t),\\Xi (t))$ , trajectories which in turn may come back close to the origin and contradict any dispersion effect).", "Indeed, for small times, the trajectory $(X(t),\\Xi (t))$ is close to its first order expansion in time, which is the key to obtaining dispersive effects similar to the one at hand in the free case.", "Technically speaking, the proof relies on establishing that the propagator $U_{\\varepsilon }(t)$ behaves like the free Schrödinger propagator for small times, a propagator whose symbol is $\\exp (i t |\\xi |^2/{\\varepsilon })$ , and which in turn has size $({\\varepsilon }/t)^{d/2}$ thanks to a stationary phase argument.", "To obtain the desired statement, a wave packet approach is actually introduced, which strongly uses the work by Combescure and Robert $(\\cite {MR1461126})$ .", "It allows to compute explicitly the propagator $U_{\\varepsilon }(t) \\, S_{\\varepsilon }$ , using the Hamiltonian flow and related, linearized, quantities, to obtain a representation of the form $\\nonumber &\\frac{i}{\\varepsilon } \\int _{T_0 {\\varepsilon }}^{\\theta }e^{-\\alpha _{\\varepsilon }t} \\, (1-\\chi )\\left(\\frac{t}{T_0 {\\varepsilon }}\\right)\\chi \\left(\\frac{t}{\\theta }\\right) \\,\\big \\langle U_{\\varepsilon }(t)\\chi _{\\delta }(H_{\\varepsilon })S_{\\varepsilon },\\phi _{\\varepsilon }\\big \\rangle \\, dt\\\\&\\qquad \\qquad =\\frac{1}{\\varepsilon ^{(5d+2)/2}} \\,\\int _{T_{0}\\varepsilon }^\\theta \\, \\int _{{\\mathbb {R}}^{6d}}e^{\\frac{i}{\\varepsilon } \\, \\psi (t,X)} \\, a_{N}(t,X)\\, dt \\, dX+ O_{\\theta ,\\delta }\\left({\\varepsilon }^N\\right),$ where $X=(q,p,x,y,\\xi ,\\eta ) \\in {\\mathbb {R}}^{6d}$ , where $N$ is a possibly large integer, and the remainder term $O_{\\theta ,\\delta }\\left({\\varepsilon }^N\\right)$ is upper bounded by $C_{\\theta ,\\delta } \\, {\\varepsilon }^N$ for some $C_{\\theta ,\\delta }>0$ independent of ${\\varepsilon }$ , which depends on the chosen $\\theta >0$ and $\\delta >0$ .", "Note that the amplitude $a_{N}$ is defined in (REF ) below, while the complex phase function $\\psi $ is defined in (REF ) below.", "We refer to section for details about the representation formula (REF ), which is a key ingredient in our proof of the main theorem.", "With this representation at hand, we arrive at the Proposition 1.10 (See [4]).", "Let $n^2$ be any long-range potential which is non-trapping.", "For $\\theta $ and $\\delta $ small enough, there exists $C_\\theta >0$ and $C_{\\theta ,\\delta }>0$ such that for all $\\varepsilon \\le 1$ we have $\\frac{1}{\\varepsilon }\\int _{T_0\\varepsilon }^{\\theta }\\chi \\left(\\frac{t}{\\theta }\\right)\\left(1-\\chi \\left(\\frac{t}{T_0\\varepsilon }\\right)\\right)e^{-\\alpha _\\varepsilon t}\\left\\langle U_\\varepsilon (t)\\chi _\\delta (H_\\varepsilon )S_\\varepsilon ,\\phi _\\varepsilon \\right\\rangle dt\\le \\frac{C_\\theta }{T_0^{d/2-1}}+C_{\\theta ,\\delta } \\, {\\varepsilon }.$" ], [ "Non-trapping behaviour", "The goal of this subsection is to prove item (i) of our main Theorem REF .", "We prove that the chosen refraction index $n^2(x)=n^2_{\\infty }-\\lambda f(r)g(\\theta _{1})$ in (REF ) is non-trapping on the zero-energy level $H_{0}=\\left\\lbrace (x,\\xi )\\in {\\mathbb {R}}^{2d},\\ \\text{s.t.", "}\\ \\xi ^2/2=n^2(x)\\right\\rbrace $ .", "We first observe that the zero energy level has the more explicit value $H_{0}=\\left\\lbrace (x,\\xi )\\in {\\mathbb {R}}^{2d},\\ \\text{s.t.", "}\\ x=(r,\\theta _{1},\\ldots ,\\theta _{d-1}),\\ \\frac{\\xi ^2}{2}=n^2_{\\infty }-\\lambda f(r)g(\\theta _{1})\\right\\rbrace .$ We readily define the following two regions.", "The first one is usually called the classically forbidden region: any trajectory living on the zero-energy level cannot reach the set $B_\\emptyset $ .", "The second one is sometimes called here the bump of the refraction index: it is the region where the refraction index actually varies with $x$ .", "Outside this region, the refraction index is constant and the Hamiltonian trajectories associated with $h(x,\\xi )=|\\xi |^2/2+n^2(x)$ are straight lines.", "Definition 2.1 (i) We denote by $B_{\\emptyset }$ the set (classically forbidden region) $B_{\\emptyset }:=\\left\\lbrace x\\in {\\mathbb {R}}^d, \\ \\text{s.t.", "}\\ n^2(x)<0\\right\\rbrace =\\left\\lbrace x=(r,\\theta _{1},\\ldots ,\\theta _{d-1}),\\ \\text{s.t.", "}\\ n^2_{\\infty }<\\lambda f(r)g(\\theta _{1}) \\right\\rbrace .$ (ii) We denote by $B_p$ the set (bump) $B_{p}:=\\left\\lbrace x=(r,\\theta _{1},,\\ldots ,\\theta _{d-1}),\\ \\text{s.t.", "}\\ R-1\\le r\\le R+1,\\,|\\theta _{1}| \\le 2\\theta _{0}\\right\\rbrace .$ Remark From the definition of $B_\\emptyset $ and the two functions $f(r)=\\chi (2(r-R))$ and $g(\\theta _1)=\\chi (\\theta _1/\\theta _0)$ it is clear that there exists $\\mu \\in ]1,2[$ such that $B_\\emptyset \\subset \\left\\lbrace R-\\frac{\\mu }{2}\\le r \\le R+\\frac{\\mu }{2}, \\, |\\theta _1|\\le \\mu \\theta _0\\right\\rbrace .$ It suffices to take $\\mu $ such that $0<\\chi (\\mu )<\\frac{n_\\infty }{\\sqrt{\\lambda }}\\, \\text{ (hence } \\mu \\in ]1,2[\\text{)}.$ Figure: Bump of refraction index, and classically forbiden regionOur main step lies in proving the following escape estimate Lemma 2.2 Select the refraction index $n^2(x)$ as in (REF ) and assume condition (REF ) is fulfilled, namely $1-\\cos (2 \\theta _0)<1/(2R)$ .", "Take a Hamiltonian trajectory $X(t,x,\\xi )\\equiv X(t)$ living on the zero-energy level and define $x_{0}:=(R,0,\\ldots ,0)$ in Cartesian coordinates.", "Then, there exists $\\alpha >0$ , as well as $\\beta \\in {\\mathbb {R}}$ and $\\gamma \\in {\\mathbb {R}}$ , such that $\\forall \\,t\\ge 0,\\quad \\left|X(t)-x_{0}\\right|^2\\ge \\alpha \\, t^2+\\beta \\, t+\\gamma .$ An immediate corollary of the above Lemma is Corollary 2.3 Assume condition (REF ) is fulfilled, namely $1-\\cos (2 \\theta _0)<1/(2R)$ .", "Then the refraction index $n^2(x)$ in (REF ) is non-trapping on the zero-energy level.", "Proof of Corollary REF .", "Apply the preceding lemma and let $t\\rightarrow +\\infty $ .", "$\\blacksquare $ Proof of Lemma REF ." ], [ "$\\bullet $ First step.", "We compute the second derivative of $\\left|X(t)-x_{0}\\right|^2$ and get $&\\nonumber \\frac{1}{2}\\frac{d^2}{dt^2}\\left|X(t)-x_{0}\\right|^2=\\left\\langle \\frac{d^2}{dt^2}X(t),X(t)-x_{0}\\right\\rangle +\\left|\\frac{d X}{dt}(t)\\right|^2,\\\\\\nonumber &\\qquad =\\left\\langle \\nabla n^2(X(t)),X(t)-x_{0}\\right\\rangle +\\left|\\frac{d X}{dt}(t)\\right|^2,\\\\&\\qquad =\\left\\langle \\nabla n^2(X(t)),X(t)-x_{0}\\right\\rangle +n^2(X(t)),$ where we have used the fact that the Hamiltonian trajectory $(X(t),\\Xi (t))$ belongs to $H_{0}$ .", "Letting $X(t)=r\\,\\vec{u}_{r}$ in hyperspherical coordinates and $x_{0}=(R,0,\\ldots ,0)$ in Cartesian coordinates, we obtain on the other hand $\\left\\langle \\nabla n^2(X(t)),X(t)-x_{0}\\right\\rangle &=\\left\\langle -\\lambda f^{\\prime }(r)g(\\theta _{1})\\vec{u}_{r}-\\lambda \\frac{f(r)}{r}g^{\\prime }(\\theta _{1})\\vec{u}_{\\theta _{1}},r\\,\\vec{u}_{r}-R\\,\\vec{e}_{1}\\right\\rangle ,\\\\&=F_{r}(r,\\theta _{1})+F_{\\theta }(r,\\theta _{1}),$ where $F_{r}(r,\\theta _{1})=-\\lambda f^{\\prime }(r)g(\\theta _{1})\\left(r-R\\cos (\\theta _{1})\\right),\\qquad F_{\\theta }(r,\\theta _{1})=-\\lambda \\frac{R}{r}f(r) g^{\\prime }(\\theta _{1}) \\sin (\\theta _{1}).$ Eventually we have $\\frac{1}{2}\\frac{d^2}{dt^2}\\left|X(t)-x_{0}\\right|^2=F_{r}(r,\\theta _{1})+F_{\\theta }(r,\\theta _{1})+n^2(X(t)).$ Therefore, the lemma is proved once we establish the existence of $\\alpha >0$ such that $F_{r}(x)+F_{\\theta }(x)+n^2(x)\\ge \\alpha >0$ whenever $x \\in \\Pi _x H_0={\\mathbb {R}}^d\\setminus B_\\emptyset $ (where $\\Pi _x$ denotes the projection $(x,\\xi )\\mapsto x$ from ${\\mathbb {R}}^{2d}$ to ${\\mathbb {R}}^d$ ).", "We readily notice that $n^2$ and $F_{\\theta }$ are clearly non-negative function on the whole of ${\\mathbb {R}}^d$ ." ], [ "$\\bullet $ Step two: non-negativity of {{formula:ad5a8543-a004-475c-aff7-789bfec9cbce}} .", "First, on ${\\mathbb {R}}^d\\setminus B_{p}$ , the function $F_{r}$ is zero, hence non-negative.", "In the same way on $B_p\\cap \\lbrace R-1/2\\le r\\le R+1/2\\rbrace $ , we have $f^{\\prime }\\equiv 0$ , hence $F_{r}\\equiv 0\\ge 0$ .", "There remains to study the non-negativity of $F_{r}$ on the two sets $\\left\\lbrace R-1\\le r\\le R-1/2, \\, |\\theta _{1}|\\le 2\\theta _{0}\\right\\rbrace $ and $\\left\\lbrace R+1/2\\le r\\le R+1, \\, |\\theta _{1}|\\le 2\\theta _{0}\\right\\rbrace $ .", "On $\\left\\lbrace R-1\\le r\\le R-1/2, \\, |\\theta _{1}|\\le 2\\theta _{0}\\right\\rbrace $ , we have $r-R\\cos (\\theta _{1})&\\le R-\\frac{1}{2}-R\\cos (2\\theta _{0})= R(1-\\cos (2\\theta _{0}))-\\frac{1}{2}< 0,$ thanks to our assumption (REF ).", "Since $f^{\\prime }\\ge 0$ on $\\left\\lbrace R-1\\le r\\le R-1/2\\right\\rbrace $ , we get $F_{r}\\ge 0$ on $\\left\\lbrace R-1\\le r\\le R-1/2, \\, |\\theta _{1}|\\le 2\\theta _{0}\\right\\rbrace $ .", "A similar computation proves that $F_{r}\\ge 0$ on the set $\\left\\lbrace R+1/2\\le r\\le R+1, \\, |\\theta _{1}|\\le 2\\theta _{0}\\right\\rbrace $ .", "We have obtained that $F_{r} \\ge 0$ on the whole of ${\\mathbb {R}}^d$ ." ], [ "$\\bullet $ Step three: decomposition of {{formula:c87d72ad-006b-406f-832c-2a3912733b5b}} .", "We have just proved that $F_r(x)+F_\\theta (x)+n^2(x) \\ge 0$ for all $x \\in {\\mathbb {R}}^d$ .", "We now wish to obtain a positive lower bound for $x\\notin B_\\emptyset $ .", "The argument relies on the fact that the refraction index $n^2$ is positive away from the boundary $\\partial B_{\\emptyset }$ , where $\\partial B_{\\emptyset }:=\\lbrace (r,\\theta _{1},\\ldots ,\\theta _{d-1}),\\quad f(r)g(\\theta _{1})=n^2_{\\infty }/\\lambda \\rbrace $ , while the term $F_r+F_\\theta $ stemming from the gradient of the refraction index in (REF ) is positive close to the boundary $\\partial B_\\emptyset $ .", "This is the reason for the decomposition we now introduce.", "We define the set (piece of ring) $C_{\\alpha ,\\beta }:=\\left\\lbrace R-\\alpha \\le r\\le R+\\alpha ,\\ -\\beta \\le \\theta _{1}\\le \\beta \\right\\rbrace .$ We know from the remark after Definition REF that there exist $\\mu \\in ]1,2[$ such that $B_{\\emptyset }\\subset C_{R+\\mu /2,\\mu \\theta _0}.$ We therefore decompose ${\\mathbb {R}}^d\\setminus B_\\emptyset = \\left({\\mathbb {R}}^d\\setminus C_{R+\\mu /2,\\mu \\theta _0}\\right) \\cup \\left(C_{R+\\mu /2,\\mu \\theta _0}\\setminus B_\\emptyset \\right).$ We readily observe that, by construction of $\\mu $ (namely $\\chi (\\mu )^2\\in ]0,n_\\infty ^2/\\lambda [$ – see (REF )), for any $x\\in {\\mathbb {R}}^d\\setminus C_{R+\\mu /2,\\mu \\theta _0}$ , we have the lower bound $n^2(x)=n_\\infty ^2-\\lambda f(r) g(\\theta _1)\\ge n_\\infty ^2-\\lambda \\chi (\\mu )^2=:c_{n^2}>0,$ There only remains to prove the existence of $c_\\nabla >0$ such that $F_r+F_\\theta \\ge c_\\nabla $ on $C_{R+\\mu /2,\\mu \\theta _0}\\setminus B_\\emptyset $ ." ], [ "$\\bullet $ Step four: positive lower bound for {{formula:8e1390cf-bb23-436f-aaa6-7ea66a8a1f9e}} on {{formula:150cda40-2e5f-4fee-83f5-7de215d968a2}} .", "Take $\\nu \\in ]1,2[$ such that $\\frac{n_\\infty }{\\sqrt{\\lambda }} < \\chi (\\nu ) < 1.$ where $\\chi $ is the truncation function defined in (REF ).", "With this choice of $\\nu $ , we clearly have, whenever $x \\in C_{R+\\nu /2,\\nu \\theta _0}$ , the relation $n^2(x)=n_\\infty ^2-\\lambda \\chi (2(r-R)) \\chi (\\theta _1/\\theta _0)\\le n_\\infty ^2-\\lambda \\chi (\\nu )^2<0$ , hence $C_{R+\\nu /2,\\nu \\theta _0}\\subset B_\\emptyset \\subset C_{R+\\mu /2,\\mu \\theta _0}.$ Therefore, it is enough to obtain a lower bound on $F_r+F_\\theta $ on the set $C_{R+\\mu /2,\\mu \\theta _0}\\setminus C_{R+\\nu /2,\\nu \\theta _0}$ .", "To this end, we decompose (see Figure REF ) $&C_{R+\\mu /2,\\mu \\theta _0}\\setminus C_{R+\\nu /2,\\nu \\theta _0}\\subset Z_{r}^1\\cup Z_{r}^2\\cup Z_{\\theta }^1\\cup Z_{\\theta }^2, \\, \\text{ with }\\\\&Z_{r}^1:=\\left\\lbrace R-\\mu /2\\le r \\le R-\\nu /2,\\ |\\theta _{1}| \\le \\nu \\theta _0 \\right\\rbrace ,\\\\&Z_{r}^2:=\\left\\lbrace R+\\nu /2\\le r \\le R+\\mu /2,\\ |\\theta _{1}|\\le \\nu \\theta _0\\right\\rbrace ,\\\\&Z_{\\theta }^1:=\\left\\lbrace R-\\mu /2 \\le r \\le R+\\mu /2,\\ -\\mu \\theta _0\\le \\theta _{1}\\le -\\nu \\theta _0\\right\\rbrace ,\\\\&Z_{\\theta }^2:=\\left\\lbrace R-\\mu /2\\le r \\le R+\\mu /2,\\ \\nu \\theta _{0}\\le \\theta _{1}\\le \\mu \\theta _0\\right\\rbrace .$ Figure: Zone of studyOn $Z_{r}^1.$ We use the structural hypothesis (REF ) to get $\\nonumber &F_{r}(x)=-\\lambda f^{\\prime }(r)g(\\theta _{1})(r-R\\cos (\\theta _{1}))\\ge -\\lambda f^{\\prime }(r)g(\\theta _{1})(R-\\frac{\\nu }{2}-R\\cos (2\\theta _{0}))\\\\&\\quad \\,\\ge \\lambda f^{\\prime }(r)g(\\theta _{1}) \\frac{\\nu -1}{2}\\ge \\lambda (\\nu -1) \\, \\left(\\min _{s \\in [-\\mu ,-\\nu ]} \\chi ^{\\prime }(s) \\right)\\left(\\min _{|s|\\le \\nu } \\chi (s) \\right)=:c_1>0.$ A similar proof establishes that, whenever $x \\in Z_r^2$ we have $F_{r}(x)\\ge \\lambda (\\nu -1) \\, \\left(\\min _{s \\in [\\nu ,\\mu ]} [-\\chi ^{\\prime }(s)] \\right)\\left(\\min _{|s|\\le \\nu } \\chi (s) \\right)=:c_2>0.$ On $Z_{\\theta }^1$ .", "The important term is now $F_\\theta $ .", "We have $&F_{\\theta }(x)=-\\lambda \\frac{R}{r}f(r)g^{\\prime }(\\theta _{1})\\sin (\\theta _{1})\\ge \\lambda \\frac{R}{r}f(r)g^{\\prime }(\\theta _{1})\\sin (\\nu \\theta _0)\\\\&\\quad \\,\\ge \\lambda \\frac{R}{\\theta _0 (R+\\mu /2)} \\left( \\min _{|s|\\le \\mu } \\chi (s) \\right)\\left( \\min _{s \\in [-\\mu ,-\\nu ]} \\chi ^{\\prime }(s) \\right)=:c_3>0.$ A similar argument establishes that, whenever $x\\in Z_\\theta ^2$ we have $F_{\\theta }(x)\\ge \\lambda \\frac{R}{\\theta _0 (R+\\mu /2)} \\left( \\min _{|s|\\le \\mu } \\chi (s) \\right)\\left( \\min _{s \\in [\\nu ,\\mu ]} [-\\chi ^{\\prime }(s)] \\right)=:c_4>0.$ Gathering all estimates, there exists a positive constant $c_{\\nabla }>0$ such that $\\forall \\, x\\,\\in C_{R+\\mu /2,\\mu \\theta _0}\\setminus C_{R+\\nu /2,\\nu \\theta _0},\\qquad F_r(x)+F_\\theta (x) \\ge c_{\\nabla }>0.$" ], [ "$\\bullet $ Step five: end of the proof.", "Putting all estimates together, we obtain $\\forall \\, x\\in \\Pi _{x}H_{0}={\\mathbb {R}}^d\\setminus B_\\emptyset ,\\quad F_r(x)+F_\\theta (x) +n^2(x)\\ge \\min (c_{n^2},c_{\\nabla })=:\\alpha >0.$ The lemma is proved.", "$\\blacksquare $" ], [ "Refocusing Set", "The goal of this subsection is to establish part (ii) of our main Theorem REF .", "Our main result is Proposition 2.4 Let $n^2$ be the potential defined in (REF ).", "Assume the structural hypothesis (REF ) is fulfilled, namely $1-\\cos (2\\theta _0)<1/(2R)$ .", "Then, the refocusing set defined in Definition REF as $M=\\left\\lbrace (t,\\xi ,\\eta )\\in ]0,+\\infty [\\times {\\mathbb {R}}^{2d} \\, \\text{ s.t. }", "\\,\\frac{|\\eta |^2}{2}=n^2(0), \\,X(t,0,\\xi )=0, \\,\\Xi (t,0,\\xi )=\\eta \\right\\rbrace $ satisfies $M=\\left\\lbrace (T_{R},\\xi ,\\eta ),\\ \\text{s.t.", "}\\ \\xi =-\\eta =(r,\\theta _{1},\\ldots ,\\theta _{d-1}),\\ r=\\sqrt{2n^2(0)},\\ |\\theta _{1}|\\le \\theta _{0}\\right\\rbrace ,$ where $T_{R}>0$ is the unique positive time such that $X(T_{R},0,(\\sqrt{2n^2(0)},0\\ldots ,0))=0$ .", "Proof of Proposition REF .", "Consider a trajectory $X(t,0,\\xi )\\equiv X(t)$ on the zero energy level, with $\\xi =(r,\\theta _{1},\\ldots ,\\theta _{d-1})$ in hyperspherical coordinates.", "If $|\\theta _1|\\ge 2\\theta _{0}$ , it is clear that $X(t)$ is a straight line which never enters $B_{p}$ , and the equation $X(t,0,\\xi )=0$ with $t>0$ has no solution.", "We need to understand the geometry when the trajectory reaches $B_{p}$ , i.e.", "when $|\\theta _1|< 2\\theta _0$ .", "We prove below that two cases occur.", "If $|\\theta _{1}|\\le \\theta _{0}$ , the trajectory remains along a line, and it is reflected by the refraction index towards the origin.", "If $\\theta _0<|\\theta _1|< 2 \\theta _0$ , the force acting on the trajectory has a non-vanishing component in the orthoradial direction, which prevents the trajectory to go back to the origin.", "The proposition follows.", "Let us come to a proof.", "$\\bullet $ First case: $|\\theta _1|\\le \\theta _0$ .", "Consider the trajectory $Y(t)$ defined in hyperspherical coordinates as $Y(t)=(r(t),\\theta _{1},\\ldots ,\\theta _{d-1})\\,,$ with $r(t)$ solution to the ordinary equation $r^{\\prime \\prime }=-\\lambda f^{\\prime }(r)$ with initial data $r(0)=0,\\qquad r^{\\prime }(0)=\\sqrt{2n^2(0)}.$ Then, $(Y(t),Y^{\\prime }(t))$ satisfies the Hamiltonian ODE (REF ) associated with $h(x,\\xi )=|\\xi |^2/2+n^2(x)$ .", "Since $Y(0)=X(0)=0$ , and $ Y^{\\prime }(0)= X^{\\prime }(0)=\\xi $ , uniqueness provides $X(t)=Y(t)$ for all $t$ .", "The trajectory $X(t)$ is radial.", "It is clear that the radial trajectory $t\\mapsto r(t)$ reaches the region $\\lbrace R-1\\le r\\le R+1\\rbrace $ at time $t_{e}=(R-1)/|\\xi |=(R-1)/\\sqrt{2 \\, n^2(0)}>0$ , where $t_{e}=\\inf \\left\\lbrace t>0, \\, X(t)\\in B_{p}\\right\\rbrace .$ Now, according to Corollary REF , the trajectory $r(t)$ necessarily leaves the region $\\lbrace R-1\\le r\\le R+1\\rbrace $ at some later time $t_s> t_e$ , where $t_{s}=\\inf \\left\\lbrace t>t_e, \\, X(t)\\notin B_{p}\\right\\rbrace .$ The trajectory can either leave the bump at $r=R-1$ or at $r=R+1$ .", "The case $r=R+1$ is forbidden, for in the contrary case, using continuity, there would exist a time $t_{c}$ such that $r(t_{c})=R$ , hence $X(t_{c})\\in B_{\\emptyset }$ , which is not allowed.", "Therefore, the trajectory leaves the bump $B_{p}$ at $X(t_s)$ where $|X(t_s)|=r(t_{s})=R-1$ .", "Energy conservation, together with the fact that the trajectory is radial, implies that $X^{\\prime }(t_s)=-\\xi $ .", "Therefore, the trajectory for later times $t\\ge t_s$ is a straight line with constant speed $-\\xi $ .", "We deduce that there exists a unique $T_R>t_s$ such that $X(T_R,0,\\xi )=0$ , and we have as desired $\\Xi (T_R,0,\\xi )=-\\xi $ .", "$\\bullet $ Second case: $\\theta _0<|\\theta _1|< 2\\theta _0$ .", "We first assume that $d=2$ , and next generalize the argument to $d\\ge 3$ using the symmetries of the system.", "To fix the ideas, we assume in the following that $\\theta _0<\\theta _1< 2\\theta _0$ , the proof being the same when $\\theta _1$ has the opposite sign.", "$*$ In dimension $d=2$ .", "Let $t_{e}=(R-1)/|\\xi |$ be the time when the trajectory enters $B_{p}$ , as in the preceding case.", "On the one hand, since the velocity $\\Xi (t_e)$ is radial and satisfies $\\Xi (t_e)=|\\xi | \\, \\vec{u}_{r}$ , there is an ${\\varepsilon }>0$ such that $R-1<|X(t)|<R+1$ whenever $t\\in ]t_e,t_e+{\\varepsilon }]$ .", "On the other hand, by assumption we have $\\theta _1(t_{e})=\\theta _1\\in ]\\theta _{0},2\\theta _0[$ , and continuity implies there is an ${\\varepsilon }>0$ such that $\\theta _0<\\theta _1(t)<2 \\theta _0$ whenever $t\\in [t_e,t_e+{\\varepsilon }]$ .", "Hence we may define $t_{s}:=\\sup \\lbrace t \\ge t_{e}, \\, \\, \\text{ s.t. }", "\\, \\forall t^{\\prime } \\in [t_e,t], \\quad \\theta _1(t^{\\prime })\\in ]\\theta _0,2\\theta _0[ \\text{ \\, { and } \\, }X(t^{\\prime })\\ne 0.\\rbrace .$ Now, Hamilton's equations of motion (REF ) can be written in polar coordinates as ${\\left\\lbrace \\begin{array}{ll}r^{\\prime \\prime }-r(\\theta _{1}^{\\prime })^2=-\\lambda f^{\\prime }(r)g(\\theta _{1}),\\\\2r^{\\prime }\\theta _{1}^{\\prime }+r\\theta _{1}^{\\prime \\prime }=-\\lambda \\frac{f(r)}{r}g^{\\prime }(\\theta _{1}).\\end{array}\\right.", "}$ Examining the second equation, we have $(r^2\\theta _{1}^{\\prime })^{\\prime }=2rr^{\\prime }\\theta _{1}^{\\prime }+r^2\\theta _{1}^{\\prime \\prime }=-\\lambda f(r) g(\\theta _{1})$ , and we get whenever $r(t)\\ne 0$ , $\\theta _{1}^{\\prime }(t)=-\\frac{\\lambda }{r^2(t)}\\int _{t_{e}}^tf(r(s))g^{\\prime }(\\theta _{1}(s))ds.$ Therefore, since $f(r)\\ge 0$ for any $r\\ge 0$ while $f(r)>0$ whenever $R-1<r<R+1$ , and since $g^{\\prime }(\\theta _1)\\le 0$ when $\\theta _0\\le \\theta _1 \\le 2\\theta _0$ , while $g^{\\prime }(\\theta _1)< 0$ when $\\theta _0< \\theta _1 < 2\\theta _0$ we get, with the above definitions and observations, $\\theta _1^{\\prime }(t)>0 \\quad \\forall t\\in ]t_e,t_s].$ With this observation at hand, two cases may occur.", "If $t_s=+\\infty $ , there is nothing to prove, for by definition of $t_s$ , we have $X(t)\\ne 0$ whenever $0<t\\le t_s=+\\infty $ .", "In the case $t_s<+\\infty $ , we already know $X(t)\\ne 0$ whenever $0<t\\le t_s$ .", "Besides, since $\\theta _1^{\\prime }(t)>0$ whenever $0<t\\le t_s$ , it is clear that the case $X(t_s)=0$ is impossible (for in that case the trajectory would be a straight line passing through the origin on some interval $[t_*,t_s]$ , in contradiction with $\\theta _1^{\\prime }(t)>0$ on $[t_*,t_s]$ ), hence $\\theta _1(t_s)=2\\theta _0$ and $\\theta _1^{\\prime }(t_s)>0$ .", "For that reason, the trajectory $X(t)$ for times $t> t_s$ is a straight line with constant velocity, which lies entirely in the set $2\\theta _0<\\theta _1<2\\theta _0+\\pi $ .", "In particular, since $\\theta _1^{\\prime }(t_s)>0$ , the trajectory cannot be radial and we have $X(t)\\ne 0$ whenever $t>t_s$ in that case.", "This concludes the proof.", "$*$ In dimension $d\\ge 3$.", "We use the invariance of $n^2$ under the action of ${\\mathbb {O}}_{d,1}({\\mathbb {R}})$ .", "Take $\\xi \\in {\\mathbb {R}}^d$ such that $|\\xi |=\\sqrt{2n^2(0)}$ .", "Write $\\xi =(\\sqrt{2n^2(0)},\\theta _{1},\\ldots ,\\theta _{d-1})$ in hyperspherical coordinates.", "There exists a matrix $A_\\xi \\in {\\mathbb {O}}_{d,1}({\\mathbb {R}})$ such that $A_\\xi \\, \\xi =(\\sqrt{2n^2(0)},\\theta _{1},0,\\ldots ,0)$ .", "On the other hand, denote by $(r(t),\\theta _{1}(t))$ the solution of Hamilton's equations of motion (REF ) with initial data $(\\sqrt{2n^2(0)},\\theta _{1})$ in dimension 2.", "We set $Y(t)=A_\\xi ^{-1}(r(t),\\theta _{1}(t),0\\ldots ,0)$ .", "Then $Y(t)$ satisfies Hamilton's equations of motion (REF ), with initial data $Y(0)=0,\\ Y^{\\prime }(0)=\\xi $ .", "Uniqueness provides $Y(t)=X(t)$ for any $t> 0$ .", "This, combined with the previous step, provides $X(t)\\ne 0$ for any $t> 0$ .", "$\\blacksquare $" ], [ "Convergence proof", "The goal of this section is to prove item (iii) of our main Theorem REF .", "The proof is performed in a number of steps.", "We begin by defining some necessary notation." ], [ "The linearized hamiltonian flow", "Let $\\varphi (t,x,\\xi )=\\left(X(t,x,\\xi ),\\Xi (t,x,\\xi )\\right)$ denote the flow associated with Hamilton's equations of motion (REF ).", "The linearized flow, written $F(t,x,\\xi )$ below, is $F(t,x,\\xi )= \\frac{D\\varphi (t,x,\\xi )}{D(x,\\xi )}:=\\begin{pmatrix}A(t,x,\\xi ) & B(t,x,\\xi )\\\\ C(t,x,\\xi )&D(t,x,\\xi )\\end{pmatrix},$ where $A(t)$ , $B(t)$ , $C(t)$ , $D(t)$ are by definition $&A(t,x,\\xi )=\\frac{DX(t,x,\\xi )}{Dx},\\ \\ B(t,x,\\xi )=\\frac{DX(t,x,\\xi )}{D\\xi },\\\\&C(t,x,\\xi )=\\frac{D\\Xi (t,x,\\xi )}{Dx},\\ \\ D(t,x,\\xi )=\\frac{D\\Xi (t,x,\\xi )}{D\\xi }.$ The linearisation of $(\\ref {eqHJ})$ leads to $\\left\\lbrace \\begin{array}{ll}\\displaystyle \\frac{\\partial }{\\partial t}A(t,x,\\xi )=C(t,x,\\xi ),& A(0,x,\\xi )=Id,\\\\\\displaystyle \\frac{\\partial }{\\partial t}C(t,x,\\xi )=\\frac{D^2n^2}{Dx^2}(X(t,x,\\xi ))A(t,x,\\xi ),&C(0,x,\\xi )=0,\\end{array}\\right.$ as well as $\\left\\lbrace \\begin{array}{ll}\\displaystyle \\frac{\\partial }{\\partial t}B(t,x,\\xi )=D(t,x,\\xi ),& B(0,x,\\xi )=Id,\\\\\\displaystyle \\frac{\\partial }{\\partial t}D(t,x,\\xi )=\\frac{D^2n^2}{Dx^2}(X(t,x,\\xi ))B(t,x,\\xi ),&D(0,x,\\xi )=0.\\end{array}\\right.$ Finally, we define for later purposes the matrix $\\Gamma (t,x,\\xi )$ as $\\Gamma (t,x,\\xi )=\\left( C(t,x,\\xi )+i \\, D(t,x,\\xi )\\right) \\, .", "\\, \\left( A(t,x,\\xi )+i B(t,x,\\xi )\\right)^{-1}.$" ], [ "A wave packet approach: preparing for a stationary phase argument", "The intermediate result in Proposition REF establishes roughly that $\\langle w_{\\varepsilon },\\phi \\rangle \\sim \\langle w_{out}+\\widetilde{w_{\\varepsilon }},\\phi \\rangle $ as ${\\varepsilon }\\rightarrow 0$ .", "Therefore, item (iii) of our main Theorem reduces to proving $\\langle \\widetilde{w_{\\varepsilon }},\\phi \\rangle \\sim \\langle L_{\\varepsilon },\\phi \\rangle $ as ${\\varepsilon }\\rightarrow 0$ .", "Therefore, this preliminary paragraph is devoted to express the quantity $\\langle \\widetilde{w_{\\varepsilon }},\\phi \\rangle =\\frac{1}{\\varepsilon }\\int _{\\theta }^{T_1}\\left(1-\\chi \\left(\\frac{t}{\\theta }\\right)\\right)e^{-\\alpha _\\varepsilon t}\\left\\langle U_\\varepsilon (t)\\chi _\\delta (H_\\varepsilon )S_\\varepsilon ,\\phi _\\varepsilon \\right\\rangle dt.$ as an appropriate oscillatory integral.", "Our approach uses the technique developped in [4], which in turn strongly uses a wave packet theorem due to M. Combescure and D. Robert (see [7]).", "We skip here the details of the proof, referring to [4].", "The main result in this paragraph is the following Proposition 3.1 (See [7])   Whenever $X=(q,p,x,\\xi ,y,\\eta )\\in {\\mathbb {R}}^{6d}$ and $t\\in {\\mathbb {R}}$ , define the complex phase $&\\nonumber \\psi (t,X):=\\int _{0}^t\\left(\\frac{p_{s}^2}{2}+n^2(q_{s})\\right)ds-p.(x-q)+p_{t}.", "(y-q_{t})\\\\&\\qquad \\qquad \\qquad +x.\\xi -y.\\eta +i\\frac{(x-q)^2}{2}+\\frac{1}{2}\\Gamma _{t}(y-q_{t}).", "(y-q_{t}),$ where $q_t:=X(t,q,p)$ , $p_t:=\\Xi (t,q,p)$ , and $\\Gamma _t:=\\Gamma (t,q,p)$ .", "Select an integer $N\\in {\\mathbb {N}}$ .", "Select two truncation functions $\\chi _0(q,p)$ and $\\chi _1(x,y)$ both lying in $C_0^\\infty ({\\mathbb {R}}^{2d})$ , and such that $&\\text{supp}\\ \\chi _{0}(q,p)\\subset \\left\\lbrace |q|\\le 2\\delta \\right\\rbrace \\cup \\left\\lbrace ||p|^2/2-n^2(q)|\\le 2\\delta \\right\\rbrace \\,,\\\\&\\chi _{0}(q,p)\\equiv 1\\ \\text{on}\\ \\left\\lbrace |q|\\le 3\\delta /2\\right\\rbrace \\cup \\left\\lbrace ||p|^2/2-n^2(q)|\\le 3\\delta /2\\right\\rbrace ,\\\\&\\chi _{1}(x,y)\\equiv 1 \\, \\text{ close to } \\, (0,0).$ Define the amplitude $a_{N}(t,X):=e^{-\\alpha _{\\varepsilon }t}(1-\\chi )\\left(\\frac{t}{\\theta }\\right)\\widehat{S}(\\xi )\\widehat{\\phi }^*(\\eta )\\chi _{0}(q,p)\\chi _{1}(x,y)P_{N}\\left(t,q,p,\\frac{y-q_{t}}{\\sqrt{\\varepsilon }}\\right),$ where $P_N(t,q,p,z)$ satisfies $P_{N}(t,q,p,x):=\\frac{1}{\\pi ^{d/4}}det(A(t,q,p)+iB(t,q,p))_c^{-1/2}\\mathcal {Q}_{N}(t,q,p,x),$ and the square root $det(A(t,q,p)+iB(t,q,p))_c^{-1/2}$ is defined by continuously following the argument of the relevant complex number, starting from the value $det(A(0,q,p)+iB(0,q,p)=1$ at time $t=0$ , while $\\mathcal {Q}_{N}(t,q,p,x)$ is a polynomial in the variable $x\\in {\\mathbb {R}}^d$ , whose coefficients vary smoothly with $(t,q,p)$ , and ${\\varepsilon }$ , and which satisfies $\\mathcal {Q}_{N}(t,q,p,x)=1+O(\\sqrt{{\\varepsilon }})$ in the relevant topology.", "More precisely, we have ${\\left\\lbrace \\begin{array}{ll}\\mathcal {Q}_N(t,q,p,x)=1+\\displaystyle \\sum _{(k,j)\\in I_N}\\varepsilon ^{\\frac{k}{2}-j}p_{k,j}(t,q,p,x),\\\\I_N=\\left\\lbrace 1\\le j\\le 2N-1,1\\le k-2j\\le 2N-1, \\, k\\ge 3j\\right\\rbrace ,\\end{array}\\right.", "}$ where each $p_{k,j}$ has at most degree $k$ in the variable $x$ .", "Then, the following holds $\\langle \\widetilde{w_{\\varepsilon }},\\phi \\rangle =\\frac{1}{\\varepsilon ^{(5d+2)/2}}\\int _{\\theta }^{T_{1}}\\int _{{\\mathbb {R}}^{6d}}e^{\\frac{i}{\\varepsilon }\\psi (t,X)}a_{N}(t,X)dtdX +O_{T_{1},\\delta }(\\varepsilon ^N).$ Sketch of proof of Proposition REF .", "Using the short-hand notation $\\widetilde{\\chi }_{\\delta }(t):=e^{-\\alpha _{\\varepsilon }t}(1-\\chi )\\left(t/\\theta \\right)$ , we have $\\langle \\widetilde{w_{\\varepsilon }},\\phi \\rangle =i/\\varepsilon \\int _{\\theta }^{T_{1}}\\widetilde{\\chi }_{\\delta }(t) \\,\\left\\langle \\chi _{\\delta }(H_{\\varepsilon })S_{\\varepsilon },U_{\\varepsilon }(-t)\\phi _{\\varepsilon }\\right\\rangle dt.$ To compute the term $U_{\\varepsilon }(-t)\\phi _{\\varepsilon }$ accurately, we use a projection over the overcomplete basis of $L^2({\\mathbb {R}}^d)$ obtained by using the so-called gaussian wave-packets, namely the family of functions indexed by $(q,p)\\in {\\mathbb {R}}^{2d}$ defined by $\\varphi _{q,p}^\\varepsilon (x,\\xi ):=\\frac{1}{(\\pi \\varepsilon )^{d/4}}\\exp \\left(\\frac{i}{\\varepsilon }p.\\left(x-\\frac{q}{2}\\right)\\right)\\exp \\left(-\\frac{(x-q)^2}{2\\varepsilon }\\right).$ The point indeed is that, as proved by Combescure and Robert in [7], we have $\\nonumber &U_{\\varepsilon }(-t)\\,\\varphi _{q,p}^\\varepsilon (x,\\xi )=O_{T_1,\\delta }({\\varepsilon }^N)+\\\\&\\nonumber \\qquad \\frac{1}{\\varepsilon ^{d/4}}\\exp \\left(\\frac{i}{\\varepsilon }p_t.\\left(x-\\frac{q_t}{2}\\right)\\right) \\,\\exp \\left( -\\frac{|x-q_t|^2}{2{\\varepsilon }}\\right) \\,\\\\&\\qquad \\quad \\exp \\left(\\frac{i}{{\\varepsilon }}\\left[\\int _0^t \\left(\\frac{p_s^2}{2}+n^2(q_s)\\right) ds-\\frac{q_t \\cdot p_t - q \\cdot p}{2}\\right]\\right)\\,P_N\\left(t,q,p,\\frac{x-q_t}{\\sqrt{{\\varepsilon }}}\\right) \\,$ in $L^\\infty ([0,T_1];L^2({\\mathbb {R}}^{d}))$ .", "In other words, we have a quite explicit complex-phase/amplitude representation of the Schrödinger propagator when acting on the gaussian wave packets.", "This observation leads to writing, successively, in (REF ) $\\ &\\left\\langle \\chi _{\\delta }(H_{\\varepsilon })S_{\\varepsilon },U_{\\varepsilon }(-t)\\phi _{\\varepsilon }\\right\\rangle =\\frac{1}{(2\\pi \\varepsilon )^d}\\int _{{\\mathbb {R}}^{2d}}\\left\\langle \\chi _{\\delta }(H_{\\varepsilon })S_{\\varepsilon },\\varphi _{q,p}^\\varepsilon \\right\\rangle \\left\\langle \\varphi _{q,p}^\\varepsilon ,U_{\\varepsilon }(-t)\\phi _{\\varepsilon }\\right\\rangle dq\\, dp\\,,\\\\&\\qquad \\qquad \\qquad \\qquad \\quad \\,\\,\\,\\,=\\frac{1}{(2\\pi \\varepsilon )^d}\\int _{{\\mathbb {R}}^{2d}}\\left\\langle \\chi _{\\delta }(H_{\\varepsilon })S_{\\varepsilon },\\varphi _{q,p}^\\varepsilon \\right\\rangle \\,\\left\\langle U_{\\varepsilon }(t)\\varphi _{q,p}^\\varepsilon ,\\phi _{\\varepsilon }\\right\\rangle dq\\, dp\\,.$ Now, the idea is to replace the factor $U_{\\varepsilon }(t)\\varphi _{q,p}^\\varepsilon $ by its approximation derived above.", "Yet a few preliminary steps are in order.", "The first one uses the truncation in energy $\\chi _{\\delta }(H_{\\varepsilon })$ , together with the functional calculus for pseudo-differential operators of Helffer and Robert (see [11] ), to replace this truncation by an explicit truncation near the set $p^2/2+n^2(q)=0$ , up to small error terms.", "The second step consists in using the Parseval formula to write (we want to exploit the source term $S_{\\varepsilon }$ on the Fourier side) $\\langle S_{\\varepsilon }\\, , \\phi _{q,p}^{\\varepsilon }\\rangle =\\frac{1}{(2\\pi {\\varepsilon })^{d/2}} \\int e^{i\\frac{x \\cdot \\xi }{{\\varepsilon }}} \\, \\widehat{S}(\\xi ) \\, \\phi _{q,p}^{\\varepsilon }(x) \\, dx \\, d\\xi =\\frac{1}{(2\\pi {\\varepsilon })^{d/2}} \\int \\widetilde{\\chi }(x) \\,e^{i\\frac{x \\cdot \\xi }{{\\varepsilon }}} \\, \\widehat{S}(\\xi ) \\, \\phi _{q,p}^{\\varepsilon }(x) \\, dx \\, d\\xi ,$ for some function $\\widetilde{\\chi }(x)$ that truncates close to $x=0$ , and similarly $\\left\\langle U_{\\varepsilon }(t)\\varphi _{q,p}^\\varepsilon ,\\phi _{\\varepsilon }\\right\\rangle =\\frac{1}{(2\\pi {\\varepsilon })^{d/2}} \\int \\widetilde{\\chi }(y) \\,e^{i\\frac{y \\cdot \\eta }{{\\varepsilon }}} \\, \\widehat{\\phi }(\\eta ) \\,\\left( U_{\\varepsilon }(t) \\varphi _{q,p}^{\\varepsilon }\\right) (y) \\, dy \\, d\\eta ,$ These two steps explain the truncation factors $\\chi _0$ and $\\chi _1$ in the Proposition, which act close to the zero energy-level in phase-space (this is where functional calculus is used) and close to the origin in physical space.", "The last step consists in exploiting formula (REF ) in the obtained representation.", "Eventually, one obtains the desired formula.", "$\\blacksquare \\\\$" ], [ "Preparing for a stationary phase argument", "This slightly technical paragraph is devoted to proving that the obtained phase $\\psi $ in Proposition REF satisfies the assumptions of the stationary phase Theorem.", "Our main result in this paragraph is the Proposition after the following Lemma.", "Lemma 3.2 Let $n^2$ be any smooth refraction index.", "Then, the following holds (i) The stationary set associated with the phase $\\psi $ in (REF ), defined as $M_X:=\\left\\lbrace (t,X)=(t,q,p,x,\\xi ,y,\\eta )\\in [\\theta ,T_1]\\times {\\mathbb {R}}^{6d} \\, \\text{ s.t. }", "\\,\\nabla _{t,X}\\psi (t,X)=0 \\, \\text{ and } \\, {\\rm Im} \\, \\psi (t,X)=0\\right\\rbrace $ satisfies $M_X=\\lbrace (t,q,p,x,\\xi ,y,\\eta ) \\, \\text{ s.t. }", "\\, x=y=q=0, \\, \\xi =p, \\, (t,p,\\eta ) \\in M\\rbrace ,$ where we recall that $M=\\lbrace (t,p,\\eta ), \\, X(t,0,p)=0, \\, \\Xi (t,0,p)=\\eta , \\, \\eta ^2/2=n^2(0)\\rbrace $ by definition.", "(ii) We have, whenever $m=(t,X)\\in M_X$ , the relation $&{\\rm Ker}(D^2\\psi _{|_{m}})=\\left\\lbrace (T,Q,P,X,\\Xi ,Y,H)\\in ]0,+\\infty [\\times {\\mathbb {R}}^{6d},\\ X=Y=Q=0,\\right.\\\\\\nonumber &\\hspace{85.35826pt} \\left.", "\\Xi =P,\\ \\eta ^TH=0,B_t(0,p) \\, P+T\\eta =0,\\ -H+D_t(0,p) \\, P+T\\nabla n^2(0)=0\\right\\rbrace .$ Note that this Lemma does not use the particular structure of our index.", "Proof of Lemma REF .", "A mere computation of $\\text{Im}\\, \\psi $ and $\\nabla \\,\\psi $ allows to write $(\\ref {stM})$ .", "Differentiating $\\nabla \\,\\psi $ once allows to write $(\\ref {ker})$ .", "For more details, the reader may check $\\cite {MR2139886}$ .", "$\\blacksquare $ With this Lemma at hand, our key result in this section is the following Proposition 3.3 Let $n^2$ be the refraction index defined in (REF ).", "We recall that the refocusing set $M$ is computed in Lemma REF and satisfies $M=\\left\\lbrace (T_R,\\xi ,\\eta ) \\, \\text{ s.t. }", "\\,\\xi =-\\eta =(r,\\theta _1,\\ldots ,\\theta _{d-1}), \\, r=\\sqrt{2 n^2(0)}, \\, |\\theta _1|\\le \\theta _0\\right\\rbrace .$ Now, take any $&m \\in \\overset{\\circ }{M}_{X}=\\Big \\lbrace (t,q,p,x,\\xi ,y,\\eta ) \\, \\text{ s.t. }", "\\, x=y=q=0, \\, \\xi =p, \\,\\\\&\\qquad \\qquad \\qquad (t,p,\\eta ) \\in M, \\, \\text{with} \\, p=(r,\\theta _1,\\ldots ,\\theta _{d-1}), \\, \\text{ and } \\, |\\theta _1|<\\theta _0\\Big \\rbrace $ Then, we have ${\\rm Ker} \\, D^2\\psi |_m=T_m M_X,$ where $T_m M_X$ denotes the space tangent to $M_x$ at point $m$ .", "The remainder part of this subsection is devoted to the proof of Proposition REF .", "We begin by proving the Proposition in the case $m=m_{0}:=(T_R,0,p_0,0,p_0,0,-p_{0}),\\quad \\text{where}\\quad p_{0}:=(\\sqrt{2n^2(0)},0,\\ldots ,0)\\,.$ We next generalize the result to other values of $m$ , using the symmetries of the problem." ], [ "Proof of Proposition ", "The computation of $T_{m_{0}}M_{X}$ on the one hand is rather easy Lemma 3.4 The space $T_{m_0} M_{X}$ is given by $T_{m_{0}}M_{X}=\\lbrace (T,Q,P,X,\\Xi ,Y,H)\\ \\text{s.t.", "}\\ X=Y=Q=T=0,\\ \\Xi =P=-H,\\ P.p_{0}=0\\rbrace .$ Proof of Lemma REF .", "This is a mere computation starting from the definition of the refocusing set $M$ , as $M=\\lbrace (t,p,\\eta ), \\, X(t,0,p)=0, \\, \\Xi (t,0,p)=\\eta , \\, \\eta ^2/2=n^2(0)\\rbrace $ .", "$\\blacksquare $ In order to determine ${\\rm Ker} \\, D^2\\psi _{|_{m_{0}}}$ the first step it to compute the matrices $B_t$ and $D_t$ involved in the linearized flow, see (REF ).", "Lemma 3.5 Let $n^2$ be the potential defined in (REF ).", "Then, we have $\\hspace{-8.2511pt}D(T_{R},0,p_{0}):=\\frac{\\partial \\Xi }{\\partial \\xi }(T_{R},0,p_{0})=-I_{d}, \\quad B(T_{R},0,p_{0}):=\\frac{\\partial X}{\\partial \\xi }(T_{R},0,p_{0})=\\begin{pmatrix}b_{11}&0&\\\\0&O_{d-1}\\end{pmatrix},$ where $I_{d}$ is the identity matrix, $b_{11}\\in {\\mathbb {R}}$ and $O_{d-1}$ is a square matrix of dimension $d-1$ equal to 0.", "Proof of Lemma REF .", "We consider $x_{0}(t,0,p)=(x_{0}^1(t,0,p),\\ldots ,x_{0}^d(t,0,p))$ the solution to (REF ) with initial data $x_{0}(0,0,p)=0$ and $x^{\\prime }_0(0,0,p)=p$ .", "We recall that the index $n^2$ is invariant under the action of ${\\mathbb {O}}_{d,1}({\\mathbb {R}}^d)$ .", "Thus we first compute the components of $D$ and $B$ that are invariant under ${\\mathbb {O}}_{d,1}({\\mathbb {R}}^d)$ , namely their first column.", "We next compute the other columns by using the symmetries again, in conjunction with a perturbation argument.", "$\\bullet $ Computation of $\\frac{\\partial \\Xi }{\\partial \\xi _{1}}(T_{R},0,p_{0})$ and $\\frac{\\partial X}{\\partial \\xi _{1}}(T_{R},0,p_{0})$ We start with $\\frac{\\partial \\Xi _{j}}{\\partial \\xi _{1}}(T_{R},0,p_{0})$ for $j\\ge 2$ .", "We have $\\frac{\\partial \\Xi _{j}}{\\partial \\xi _{1}}(T_{R},0,p_{0})&=\\lim _{\\varepsilon \\rightarrow 0}\\frac{\\Xi _{j}\\left(T_{R},0,(\\sqrt{2n^2(0)}+\\varepsilon ,0\\ldots ,0)\\right)-\\Xi _{j}\\left(T_{R},0,(\\sqrt{2n^2(0)},0\\ldots ,0)\\right)}{\\varepsilon }.$ Since the trajectory is radial we have $\\Xi _{j}\\left(T_{R},0,(\\sqrt{2n^2(0)}+\\varepsilon ,0\\ldots ,0)\\right)=\\Xi _{j}\\left(T_{R},0,(\\sqrt{2n^2(0)},0\\ldots ,0)\\right)=0,\\quad \\forall \\ j\\ge 2.$ Hence, $\\frac{\\partial \\Xi _{j}}{\\partial \\xi _{1}}(T_{R},0,p_{0})=0$ , $\\forall \\ j\\ge 2$ .", "A similar argument provides $\\frac{\\partial X_{j}}{\\partial \\xi _{1}}(T_{R},0,p_{0})=0$ , $\\forall \\ j\\ge 2$ .", "There remains to determine the first coefficient of $D$ , namely $\\frac{\\partial \\Xi _{1}}{\\partial \\xi _{1}}(T_{R},0,p_{0})$ .", "Since the trajectory is radial, and by conservation of the energy, we have for $\\varepsilon $ small enough $&\\Xi _{1}\\left(T_{R},0,(\\sqrt{2n^2(0)}+\\varepsilon ,0,\\ldots ,0)\\right)=-\\left(\\sqrt{2n^2(0)}+\\varepsilon \\right),\\\\&\\Xi _{1}\\left(T_{R},0,(\\sqrt{2n^2(0),0,\\ldots ,0)}\\right)=-\\sqrt{2n^2(0)}.$ Thus, $d_{11}:=\\lim _{\\varepsilon \\rightarrow 0^+}\\frac{\\Xi \\left(T_{R},0,(\\sqrt{2n^2(0)}+\\varepsilon ,0,\\ldots ,0)\\right)-\\Xi \\left(T_{R},0,(\\sqrt{2n^2(0)},0,\\ldots ,0)\\right)}{\\varepsilon }=-1.$ $\\bullet $ Computation of $\\frac{\\partial \\Xi (T_{R},0,p_{0})}{\\partial \\xi _{j}}$ and $\\frac{\\partial X(T_{R},0,p_{0})}{\\partial \\xi _{j}}$ ($j\\ge 2$ ) Considering the symmetries of the problem, it is enough to consider the case $j=2$ : the other components may be determined using the same argument.", "We perturb the initial speed along the direction $e_{2}$ , by a factor ${\\varepsilon }$ (see Figure $\\ref {Perturbation}$ ).", "Figure: Perturbation of the initial speedLet $X_{\\varepsilon }(t)$ be the solution of the perturbed problem ${\\left\\lbrace \\begin{array}{ll}X_{\\varepsilon }^{\\prime \\prime }(t)=\\nabla n^2(X_{\\varepsilon }(t)),\\qquad X_{\\varepsilon }(0)=0,\\qquad X_{\\varepsilon }^{\\prime }(0)=p_{0}+\\varepsilon \\,e_{2}.\\end{array}\\right.", "}$ We expand $X_{\\varepsilon }(t)$ with respect to $\\varepsilon $ and obtain $X_{\\varepsilon }(t)=X_{0}(t)+\\varepsilon X_{1}(t)+\\ldots $ .", "With this notation we have $X_{1}(t)=\\frac{\\partial X}{\\partial \\xi _{2}}(t)$ and $X_{1}^{\\prime }(t)=\\frac{\\partial \\Xi }{\\partial \\xi _{2}}(t)$ .", "To obtain the expansion in $\\varepsilon $ , we go back to the previous case ($j=1$ ) using a change of variables.", "Indeed, for $\\varepsilon $ small enough, the trajectory is radial along the direction $X_{\\varepsilon }^{\\prime }(0)$ .", "Let $(\\widetilde{e_{1}}, \\ldots , \\widetilde{e_{d}})$ be a new basis defined by $\\widetilde{e_{j}}:=O_{\\varepsilon }e_{j}$ , with $O_{\\varepsilon }:=\\begin{pmatrix}\\cos (\\theta _{\\varepsilon })& -\\sin (\\theta _{\\varepsilon })&0&\\ldots &0\\\\\\sin (\\theta _{\\varepsilon })&\\cos (\\theta _{\\varepsilon })&0&\\ldots &0\\\\0&0\\\\\\vdots &\\vdots &&I_{d-2}\\\\0&0\\end{pmatrix},\\qquad \\cos (\\theta _{\\varepsilon })=\\frac{p_{0}}{p_{0}^2+\\varepsilon ^2},\\qquad \\sin (\\theta _{\\varepsilon })=\\frac{\\varepsilon }{p_{0}^2+\\varepsilon ^2}.$ Let $\\widetilde{X_{\\varepsilon }}$ be the coordinates of $X_{\\varepsilon }$ in $(\\widetilde{e_{1}},\\ldots ,\\widetilde{e_{d}})$ .", "Since $O_{\\varepsilon }^{-1}\\nabla n^2(X_{\\varepsilon })=\\nabla n^2(\\widetilde{X_{\\varepsilon }})$ , we clearly have $\\widetilde{X_{\\varepsilon }}^{\\prime \\prime }(t)=\\nabla n^2(\\widetilde{X_{\\varepsilon }}(t)),\\quad \\widetilde{X_{\\varepsilon }}(0)=0,\\quad \\widetilde{X_{\\varepsilon }}^{\\prime }(0)=(\\sqrt{\\varepsilon ^2+p_{0}^2},0,\\ldots ,0)=p_{0}+O(\\varepsilon ^2).$ Hence it is clear that $\\widetilde{X_{\\varepsilon }}(t)=\\widetilde{X_{0}}(t)+O(\\varepsilon ^2)$ .", "Therefore, we recover $X_{0}(t)+\\varepsilon X_{1}(t)&=O_{\\varepsilon }\\left(\\widetilde{X_{0}}(t)+O(\\varepsilon ^2)\\right)=(I_{d}+\\varepsilon E+O(\\varepsilon ^2))(\\widetilde{X_{0}}(t)+O(\\varepsilon ^2)),$ with $E:=\\begin{pmatrix}0& -\\frac{1}{p_{0}}&0&\\ldots &0\\\\\\frac{1}{p_{0}}&0&0&\\ldots &0\\\\0&0\\\\\\vdots &\\vdots &&I_{d-2}\\\\0&0\\end{pmatrix}.$ In other words, we have $\\forall \\, t\\in {\\mathbb {R}},\\qquad X_{0}(t)=\\widetilde{X_{0}}(t)\\qquad \\text{and}\\qquad X_{1}(t)=E \\widetilde{X_{0}}(t).$ Since the Hamiltonian trajectory goes back to the origin at time $T_{R}$ , we deduce $\\frac{\\partial X}{\\partial \\xi _{2}}(T_{R},0,p_{0})=X_{1}(T_{R})=E\\widetilde{X_{0}}(T_{R},0,p_{0})=E\\times 0=0.$ In the same way, we have $\\frac{\\partial \\Xi }{\\partial \\xi _{2}}(T_{R},0,p_{0})&=X_{1}^{\\prime }(T_{R})=E\\widetilde{X_{0}}^{\\prime }(T_{R})={}^t\\left(-\\frac{x_{0,R}^{\\prime 2}(T_{R})}{p_{0}},\\frac{x_{0,R}^{\\prime 1}(T_{R})}{p_{0}},x_{0,R}^{\\prime 3}(T_{R}),\\ldots ,x_{0,R}^{\\prime d}(T_{R})\\right)\\,,\\\\&={}^t(0,1,0,\\ldots ,0).$ The columns of $B$ and $D$ (for $j\\ge 3$ ) are determined in the similar way.", "This leads to (REF ).", "$\\blacksquare $ At this stage, we deduce the Corollary 3.6 $\\text{Ker}\\, D^2\\psi _{|_{m_{0}}}=T_{m_{0}}M_{X}.$ Proof of Corollary REF .", "According to $(\\ref {ker})$ , we have $Ker(D^2\\psi _{|_{m}})=\\left\\lbrace (T,Q,P,X,\\Xi ,Y,H), \\, X=Y=Q=0,\\right.\\\\\\left.\\Xi =P,\\eta ^TH=0,B_{T_{R}}(0,p) \\, P+T\\eta =0,-H+D_{T_{R}}(0,p) \\, P+T\\nabla n^2(0)=0\\right\\rbrace .$ Since $\\eta =-p_{0}$ , we recover $H=(0,H_{2},\\ldots ,H_{d})$ (in Cartesian coordinates).", "Since $\\nabla n^2(0)=0$ , we deduce that $D_{T_{R}}(0,p) \\, P=H$ .", "According to Lemma $\\ref {lemlin}$ , we deduce that $H=-P$ .", "Finally, $B_{T_{R}(0,p) \\, }P=0$ hence $T=0$ .", "Thus, $\\text{Ker}\\, D^2\\psi _{|_{m_{0}}}=\\left\\lbrace (T,Q,P,X,\\Xi ,Y,H), \\, X=Y=Q=T=0,\\ P=\\Xi =-H,\\ P.p_{0}=0\\right\\rbrace .$ Using Lemma REF , the proof is complete.", "$\\blacksquare $" ], [ "Proof of Proposition ", "In this subsection, we prove the Lemma 3.7 $\\forall \\, m\\in \\overset{\\circ }{M}_{X}$ , we have $T_{m}M_{X}={\\rm Ker}\\, D^2\\psi _{|_{m}}$ .", "Proof of Lemma REF .", "The idea is to use a family of transformations which leave $\\overset{\\circ }{M}_{X}$ and $n^2$ invariant (in a sense we define later), next to transport the equality $\\text{Ker}\\, D^2\\psi _{|_{m_{0}}}= T_{m_{0}}M_{X}$ to any $m\\in \\overset{\\circ }{M}_{X}$ .", "Family of transformations.", "Let $m=(t,q,p,x,\\xi ,y,\\eta )\\in \\overset{\\circ }{M}_{X}$ .", "We write $m=(T_R,0,p,0,p,0,-p)$ for some $p\\in \\sqrt{2n^2(0)}\\,{\\mathbb {S}}^{d-1}$ Thus, there exists $R_{p}\\in {\\mathbb {O}}({\\mathbb {R}}^d)$ such that $R_{p}(p)=p_{0}$ .", "We define the map $\\widetilde{R}_{m}:{\\mathbb {R}}^{6d+1}\\longrightarrow {\\mathbb {R}}^{6d+1}$ by $\\widetilde{R}_{m}(t,q,p,x,\\xi ,y,\\eta )=\\left(t,R_{p}(q),R_{p}(y),R_p(x),R_{p}(\\xi ),R_p(y),R_{p}(\\eta )\\right).$ By construction we have $\\widetilde{R}_{m}\\,(m)=m_{0}$ .", "Action on the tangent place.", "We have identified that the set $M_{X}$ satisfies $&M_X=\\underbrace{\\lbrace (t,q,p,x,\\xi ,y,\\eta ), \\text{ s.t. }", "\\,t=T_R, \\, q=x=y=0, \\, p=\\xi =-\\eta , \\, p^2/2=n^2(0)\\rbrace }_{:=\\widetilde{M_X}}\\\\&\\qquad \\qquad \\cap \\lbrace p=(r,\\theta _1,\\ldots ,\\theta _{d-1})\\, \\text{ with } |\\theta _1|\\le \\theta _0\\rbrace .$ The set $\\widetilde{M_X}$ is clearly invariant under the action of $\\widetilde{R}_{m}$ .", "Therefore, by restricting the domain in the variable $\\theta _1$ , it is clear that whenever $m\\in \\overset{\\circ }{M}_{X}$ , there exists a neighbourhood $U$ of $m$ in $\\overset{\\circ }{M}_{X}$ such that $U_{0}:=\\widetilde{R}_{m}U\\subset \\overset{\\circ }{M}_{X}$ .", "Since the application $\\widetilde{R}_{m}$ is a linear map from $U$ to $U_{0}$ which satisfies $\\widetilde{R}_{m}\\, (m)=m_{0}$ , we deduce $\\widetilde{R}_{m}\\ (T_{m}M_{X})=T_{m_{0}}M_{X}.$       Action on the kernel.", "We now compute the set $\\widetilde{R}_{m}(\\text{Ker}(D^2\\psi _{|_{m}}))$ , as follows $\\widetilde{R}_{m}(\\text{Ker}(D^2\\psi _{|_{m}}))&=\\lbrace (T,R_{p}\\,Q,R_{p}\\,P,R_{p}\\,X, R_p \\, \\Xi ,R_{p}\\,Y,R_{p}\\,H),\\ \\text{s.t.", "}\\ X=Y=Q=0,\\\\&\\qquad p.H=0,\\ B_{T_{R}}(0,p) \\, P+Tp=0,\\ D_{T_{R}}(0,p) \\, P=H\\rbrace ,\\\\&=\\lbrace (T,Q,P,X,\\Xi ,Y,H),\\ \\text{s.t.", "}\\ X=Y=Q=0,\\\\&\\qquad p.R_{p}^{-1}H=0,\\ B_{T_{R}}(0,p) \\,R_{p}^{-1}P+Tp=0,\\ D_{T_{R}}(0,p) \\, R_{p}^{-1}P=R_{p}^{-1}H\\rbrace .\\\\&=\\lbrace (T,Q,P,X,\\Xi ,Y,H),\\ \\text{s.t.", "}\\ X=Y=Q=0,\\\\&\\qquad p_{0}.H=0,\\ R_{p}B_{T_{R}}(0,p) \\, R_{p}^{-1}P+Tp_{0}=0,\\ R_{p}D_{T_{R}}(0,p) \\, R_{p}^{-1}P=H\\rbrace .$ On the other hand, we claim that $R_{p}B_{T_{R}}(0,p)R_{p}^{-1}=B_{T_{R}}(0,p_{0}),\\qquad R_{p}D_{T_{R}}(0,p)R_{p}^{-1}=B_{T_{R}}(0,p_{0}).$ Assuming the above identity is proved, we immediately deduce $\\widetilde{R}_{m}\\, (\\text{Ker}(D^2\\psi _{|_{m}}))=\\text{Ker} \\, D^2\\psi _{|_{m_{0}}}.$ We conclude by writing $\\widetilde{R}_{m}\\, (\\text{Ker}(D^2\\psi _{|_{m}}))=\\text{Ker}D^2\\psi _{|_{m_{0}}}=T_{m_{0}}M_{X}=\\widetilde{R}_{m}\\ (T_{m}M_{X}).$ Thus, there only remains to prove (REF ).", "By construction of the potential we clearly have $R_p X(t,0,p)=X(t,0,p_0),\\text{ as well as }n^2\\left( R_p x) \\right) =n^2\\left( x\\right),$ whenever $x/|x|$ lies in the angular sector $|\\theta _1|\\le \\theta _0$ .", "This provides $\\frac{D^2n^2}{Dx^2}( X(t,0,p_{0}))=\\frac{D^2n^2}{Dx^2}(R_{p}\\, X(t,0,p))=R_{p}\\, \\frac{D^2n^2}{Dx^2}(X(t,0,p))\\, R_{p}^{-1}.$ Therefore, using the differential equation (REF ) relating the time evolution of $B_t$ and $D_t$ , we recover the following system $\\left\\lbrace \\begin{array}{ll}\\displaystyle \\vspace{5.69046pt}\\frac{\\partial }{\\partial t}R_p B(t,0,p) R_p^{-1}=R_p D(t,0,p) R_p^{-1},& R_p B(0,0,p) R_p^{-1}=Id,\\\\\\displaystyle \\frac{\\partial }{\\partial t} R_p D(t,0,p) R_p^{-1} =R_p \\frac{D^2n^2}{Dx^2}(X(t,0,p)) B(t,0,p) R_p^{-1},&\\\\\\displaystyle \\qquad \\qquad =\\frac{D^2n^2}{Dx^2}(R_p X(t,0,p)) R_p B(t,0,p) R_p^{-1}&\\\\\\displaystyle \\qquad \\qquad =\\frac{D^2n^2}{Dx^2}( X(t,0,p_0)) R_p B(t,0,p) R_p^{-1}&R_p D(0,0,p) R_p^{-1}=0.\\end{array}\\right.$ Uniqueness of solutions to a differential system then gives $\\forall t, \\quad R_{p}B_{t}(0,p)R_{p}^{-1}=B_t(0,p_0),\\quad R_{p}D_{t}(0,p)R_{p}^{-1}=D_t(0,p_0).$ Relation (REF ) is proved.", "$\\blacksquare $" ], [ "A useful byproduct of the proof of Proposition ", "Lemma 3.8 Let $n^2$ be the refraction index defined in (REF ).", "Take any $m \\in M_X$ , written as $m=(T_R,0,p,0,p,0,-p)$ with $p=\\left(\\sqrt{2 n^2(0)},\\theta _1,\\theta _2,\\ldots ,\\theta _{d-1}\\right)$ according to Lemma REF .", "Then, $\\psi (m) \\text{ is constant on the set } |\\theta _1|\\le \\theta _0.$ Proof of Lemma REF .", "Considering the actual value of $\\psi (m)$ , various terms need to be considered.", "The term $\\displaystyle \\int _0^t (p_s^2/2+n^2(q_s)) \\, ds$ is clearly constant whenever $|\\theta _1|\\le \\theta _0$ .", "The same statement holds for the factor $p_t \\cdot q_t$ .", "The only non-obvious factor is $\\Gamma _t q_t \\cdot q_t$ .", "As in the preceding proof we write $&\\Gamma _t(0,p) q_t(0,p) \\cdot q_t(0,p)=\\Gamma _t(0, p) q_t(0,R_p^{-1} p_0) \\cdot q_t(0,R_p^{-1} p_0)\\\\&\\qquad =R_p \\Gamma _t(0,p) R_p^{-1} q_t(0, p_0) \\cdot q_t(0, p_0).$ There remains to write $&R_p \\Gamma _t(0,p) R_p^{-1}=R_p \\left( C_t(0,p) + i D_t(0,p) \\right) \\cdot \\left( A_t(0,p) + i B_t(0, p) \\right)^{-1}R_p^{-1}\\\\&\\qquad =\\left( R_p C_t(0,p) R_p^{-1} + i R_p D_t(0,p) R_p^{-1} \\right) \\cdot \\left( R_p A_t(0,p) R_p^{-1} + i R_p B_t(0, p) R_p^{-1} \\right)^{-1}\\\\&\\qquad =\\Gamma _t(0,p_0)$ for we already know that $R_p B_t(0, p) R_p^{-1} =B_t(0,p_0)$ , $R_p D_t(0, p) R_p^{-1} =D_t(0,p_0)$ , and a similar proof establishes $R_p A_t(0, p) R_p^{-1} =B_t(0,p_0)$ , $R_p C_t(0, p) R_p^{-1} =D_t(0,p_0)$ .", "$\\blacksquare $" ], [ "The stationary phase argument: Proof of item (iii) of our main Theorem", "The main result of the present section is Proposition 3.9 Let $n^2$ be the potential constructed according to REF .", "Select a source $S\\in {\\mathcal {S}}({\\mathbb {R}}^d)$ .", "Then, the following holds.", "(i) If ${\\rm supp}\\left( \\widehat{S}(\\xi )\\right) \\cap \\partial I_{\\theta _0}=\\emptyset $ , we have $\\forall \\,\\phi \\in {\\mathcal {S}}({\\mathbb {R}}^d),\\quad \\langle \\widetilde{w_{\\varepsilon }}-L_{\\varepsilon },\\phi \\rangle =O_{T_1,\\delta }(\\sqrt{\\varepsilon }),$ where $\\langle L_{\\varepsilon }, \\phi \\rangle $ is defined in (REF ) above (see also the Remark after Theorem REF ), and $\\partial I_{\\theta _0}=\\lbrace \\xi =(|\\xi |,\\theta _1,\\ldots ,\\theta _{d-1}) \\, \\text{ such that } \\, \\theta _1=\\pm \\theta _0\\rbrace $ (see definition REF ).", "(ii) In the general case we have $\\forall \\,\\phi \\in {\\mathcal {S}}({\\mathbb {R}}^d),\\quad \\langle \\widetilde{w_{\\varepsilon }}-L_{\\varepsilon },\\phi \\rangle =o_{T_1,\\delta }({\\varepsilon }^0).$ Proof of Proposition REF .", "Due to the fact that the stationary set $M_X$ in the to-be-developped stationary phase argument has a boundary at $\\theta _1=\\pm \\theta _0$ , the argument is in two steps.", "This is the reason why the above Proposition distinguishes between two cases.", "$\\bullet \\bullet $ Proof of Proposition REF -part (i) Outside the stationary set $M_X$ associated with the complex phase $\\psi $ , the oscillatory integral (REF ) defining $\\langle \\widetilde{w_{\\varepsilon }},\\phi \\rangle $ is of order $O(\\varepsilon ^{\\infty })$ .", "On the stationary set $M_X$ and near the support of $a_{N}$ , the stationary set $M_X$ is a submanifold without boundary, having codimension $k=6d+1-(d-1)=5d+2$ .", "Indeed, thanks to the hypothesis on the support of $\\widehat{S}$ , we have $\\text{supp}\\, a_{N}\\cap \\partial M_{X}=\\emptyset $ .", "Let us now come to the explicit application of the stationary phase Theorem to the oscillatory integral (REF ).", "Writing $p=(r,\\theta _{1},\\ldots ,\\theta _{d-1})$ in hyperspherical coordinates, we define the application: $\\gamma :{\\mathbb {R}}^{6d+1}\\cap \\text{supp}\\ a_N&\\longrightarrow &{\\mathbb {R}}^{5d+2}\\times {\\mathbb {S}}^{d-1}\\\\(t,q,p,x,\\xi ,y,\\eta )&\\longmapsto & (\\underbrace{t-T_{R},q,x,y,\\xi -p,\\eta +p,r-\\sqrt{2n^2(0)}}_{=:\\alpha },\\underbrace{\\theta _{1},\\ldots ,\\theta _{d-1}}_{=:\\theta })$ The map $\\gamma $ is a $C^\\infty $ -diffeomorphism between $\\text{supp}\\ a_{N}$ and $\\gamma \\left(\\text{supp}\\ a_{N}\\right)$ .", "Furthermore, we have by construction $(t,X)\\in M_{X}\\cap \\text{supp}\\ a_{N}\\Longleftrightarrow \\alpha =0.$ The new coordinates $(\\alpha ,\\theta )$ are adapted to the stationary set $M_{X}$ associated with $\\psi $ .", "Making the change of variables $(t,X)=\\gamma ^{-1}(\\alpha ,\\theta )$ in the integral defining $\\langle \\widetilde{w_{\\varepsilon }},\\phi \\rangle $ we have $&\\langle \\widetilde{w_{\\varepsilon }},\\phi \\rangle =O_{\\delta ,T_{1}}(\\varepsilon ^N)+\\\\&\\nonumber \\frac{1}{\\varepsilon ^{(5d+2)/2}}\\int _{\\gamma (supp\\ a_{N})}e^{\\frac{i}{\\varepsilon }\\psi \\circ \\gamma ^{-1}(\\alpha ,\\theta )}\\left(\\widehat{S}(.", ")\\widehat{\\phi }^*(.)P_{N}\\left(.,.,.,\\frac{.", "}{\\sqrt{\\varepsilon }}\\right)\\right)\\circ \\gamma ^{-1}(\\alpha ,\\theta )\\chi _{3}(\\alpha ,\\theta )\\, r^{d-1} \\, d\\alpha \\, d\\sigma (\\theta ),$ where $d\\sigma (\\theta )$ denotes the standard euclidean surface measure on the unit sphere ${\\mathbb {S}}^{d-1}$ , and $\\chi _{3}$ is a truncation function on some compact set, a neighbourhood of $M_X$ , whose precise value is irrelevant.", "Here we have used the non-stationary phase Theorem to reduce the original integral to an integral on a given compact set.", "Since for all point $ m\\in M_{X}\\cap \\text{supp}\\, a_{N}$ we have $\\text{Ker}(D^2\\psi _{|_{m}})=T_{m}M_{X}$ (Lemma REF ), the function $D^2\\psi $ is non-degenerate in the normal direction to $M_{X}$ , which gives $\\text{det}\\left(\\frac{D^2\\psi \\circ \\gamma ^{-1}}{D\\alpha ^2}(0,\\theta )\\right)\\ne 0.$ Furthermore, the projection of $\\gamma (\\text{supp}\\ a_{N})$ onto the space variable $\\theta $ is the angular sector $\\Pi _\\theta I_{\\theta _0}:=\\left\\lbrace (\\theta _{1},\\ldots ,\\theta _{d-1}),\\quad \\theta _{1}\\in ]-\\theta _{0},\\theta _{0}[\\right\\rbrace ,$ where $\\Pi _\\theta $ denotes the projection $(r,\\theta _1,\\ldots ,\\theta _{d-1}) \\mapsto (\\theta _1,\\ldots ,\\theta _{d-1})$ .", "We can now apply the stationary phase Theorem in (REF ).", "Remembering that the codimension of the stationary set $M_{X}$ associated with $\\psi $ is $5d+2$ , we obtain that for any integer $L$ there exists a sequence $(Q_{2\\ell }(\\partial ))_{\\ell \\in \\lbrace 0,\\ldots ,L\\rbrace }$ of operators of order $2\\ell $ such that $&\\nonumber \\langle \\widetilde{w_{\\varepsilon }},\\phi \\rangle =\\\\&\\nonumber C_1 \\, \\int _{\\Pi _{\\theta }I_{\\theta _{0}}}\\,\\frac{ \\exp \\left( i \\frac{\\pi }{4} {\\rm sgn} \\frac{D^2 \\psi \\circ \\gamma ^{-1}}{D \\alpha ^2}(0,\\theta )\\right)}{\\left| {\\rm det} \\frac{D^2 \\psi \\circ \\gamma ^{-1}}{D \\alpha ^2}(0,\\theta )\\right|}\\,\\exp \\left(\\frac{i}{\\varepsilon }\\psi \\circ \\gamma ^{-1}(0,\\theta )\\right) \\,\\\\&\\nonumber \\qquad \\qquad \\qquad \\qquad \\qquad \\left(\\left(Q_{0}(.)\\widehat{S}(.", ")\\widehat{\\phi ^*}(.)P_{N}\\left(.,.,.,\\frac{.", "}{\\sqrt{\\varepsilon }}\\right)\\right)\\circ \\gamma ^{-1}\\chi _{3}\\right)(0,\\theta )\\, d\\sigma (\\theta )\\\\&\\nonumber +\\int _{\\Pi _{\\theta }I_{\\theta _{0}}}\\exp \\left(\\frac{i}{\\varepsilon }\\psi \\circ \\gamma ^{-1}(0,\\theta )\\right)\\sum _{\\ell =1}^L\\varepsilon ^\\ell \\, Q_{2\\ell }(\\partial )\\left(\\left(\\widehat{S}(.", ")\\widehat{\\phi ^*}(.)P_{N}\\left(.,.,.,\\frac{.", "}{\\sqrt{\\varepsilon }}\\right)\\right)\\circ \\gamma ^{-1}\\chi _{3}\\right)(0,\\theta ) \\,d\\sigma (\\theta )\\\\&+O\\left(\\varepsilon ^{L+1}\\sup _{K\\le 2L+d+3}\\left\\Vert \\partial ^K_{(\\alpha ,\\theta )}\\left(\\widehat{S}(.", ")\\widehat{\\phi ^*}(.)P_{N}\\left(.,.,.,\\frac{.", "}{\\sqrt{\\varepsilon }}\\right)\\circ \\gamma ^{-1}\\chi _{3}\\right)\\right\\Vert _{L^\\infty }\\right)+O_{\\delta ,T_{1}}(\\varepsilon ^N)\\\\&\\nonumber :=I_{\\varepsilon }+II_{\\varepsilon }+III_{\\varepsilon }+O_{\\delta ,T_{1}}(\\varepsilon ^N),$ with the value $C_1=(2\\pi )^{(5d+2)/2} \\, (2 n^2(0))^{(d-1)/2}.$ The last line in (REF ) serves as a definition of the three terms $I_{\\varepsilon }$ , $II_{\\varepsilon }$ and $III_{\\varepsilon }$ , and the $L^\\infty $ -norm in $III_{\\varepsilon }$ is evaluated on a compact set of values of $(\\alpha ,\\theta )$ , whose precise value is irrelevant.", "We compute these three contributions.", "Note that the retained value of the integer $L$ remains to be determined at this stage.", "$\\bullet $ Contribution of the remainder term $III_{\\varepsilon }$ in (REF ).", "This term is best studied by coming back to the original variables $(t,X)$ instead of $(\\alpha ,\\theta )$ .", "Expanding the $k$ -th order derivatives involved in this term, we clearly have $&III_{\\varepsilon }=O\\left(\\varepsilon ^{L+1}\\sup _{K\\le 2L+d+3}\\left\\Vert \\partial ^K_{(t,X)}\\left(\\widehat{S}(.", ")\\widehat{\\phi ^*}(.)P_{N}\\left(.,.,.,\\frac{.", "}{\\sqrt{\\varepsilon }}\\right)\\right)\\right\\Vert _{L^\\infty }\\right)\\\\&\\quad =O\\left(\\varepsilon ^{L+1}\\sup _{K\\le 2L+d+3}\\left\\Vert \\partial ^K_{(t,X)} P_{N}\\left(.,.,.,\\frac{.", "}{\\sqrt{\\varepsilon }}\\right)\\right\\Vert _{L^\\infty }\\right).$ Hence, since $P_{N}(t,q,p,x)=\\pi ^{-d/4}\\text{det}(A(t,q,p)+iB(t,q,p))_{c}^{-1/2}\\mathcal {Q}_{N}(t,q,p,x),$ we recover $&III_{\\varepsilon }=O\\left(\\varepsilon ^{L+1}\\sup _{K\\le 2L+d+3}\\left\\Vert \\partial ^K_{(t,q,p,y)} \\left(\\mathcal {Q}_N(t,q,p,(y-q_t)/\\sqrt{{\\varepsilon }})\\right)\\right\\Vert _{L^\\infty }\\right).$ Lastly, using (REF ) we have $\\mathcal {Q}_{N}(t,q,p,x):=1+\\sum _{(k,j)\\in I_{N}}\\varepsilon ^{\\frac{k}{2}-j}p_{k,j}(t,q,p,x),$ where $p_{k,j}$ has at most degree $k$ in $x$ .", "We deduce $&III_{\\varepsilon }=\\sum _{(k,j)\\in I_{N}}O\\left(\\varepsilon ^{\\frac{k}{2}-j+L+1}\\sup _{K\\le 2L+d+3}\\left\\Vert \\partial ^K_{(t,q,p,y)} \\left(p_{k,j}(t,q,p,(y-q_t)/\\sqrt{{\\varepsilon }})\\right)\\right\\Vert \\right)\\\\&\\quad =\\sum _{(k,j)\\in I_{N}}O\\left(\\varepsilon ^{\\frac{k}{2}-j+L+1-\\frac{k}{2}}\\right)=O\\left(\\varepsilon ^{L+1-(2N-1)}\\right),$ where we have used that $j\\le 2N-1$ whenever $(k,j)\\in I_N$ (see (REF )).", "There remains to chose $L=2N-1$ to recover $III_{\\varepsilon }=O({\\varepsilon }).$ $\\bullet $ Contribution of $II_{\\varepsilon }$ in (REF ).", "This estimate is more delicate.", "Firstly, we have $&II_{\\varepsilon }=\\sum _{\\ell =1}^L\\varepsilon ^\\ell O\\left(\\left\\Vert Q_{2\\ell }(\\partial )\\left(\\left(\\widehat{S}(.", ")\\widehat{\\phi ^*}(.)P_{N}\\left(.,.,.,\\frac{.", "}{\\sqrt{\\varepsilon }}\\right)\\right)\\circ \\gamma ^{-1}\\chi _{3}\\right)(0,\\theta )\\right\\Vert _{L^\\infty }\\right).$ Hence, going back to the $(t,X)$ variables again, and remembering that the relation $(\\alpha ,\\theta )=(0,\\theta )$ implies $y=q_t=0$ and $t=T_R$ , we recover the identity $&II_{\\varepsilon }=\\sum _{\\ell =1}^L\\varepsilon ^\\ell \\,O\\left(\\sup _{K\\le 2\\ell }\\left\\Vert \\partial ^K_{(t,q,p,y)}\\Big |_{y=q_t=0, t=T_R}\\left(P_{N}\\left(t,q,p,\\frac{y-q_t}{\\sqrt{\\varepsilon }}\\right)\\right)\\right\\Vert _{L^\\infty }\\right),$ where the $L^\\infty $ -norm is evaluated on some compact set of values of $p$ .", "Now, inserting the exact value of $P_N$ , we may write $&II_{\\varepsilon }=\\sum _{\\ell =1}^L\\varepsilon ^\\ell \\,O\\left(\\sum _{(k,j)\\in I_N}\\sup _{K\\le 2\\ell }\\left\\Vert \\partial ^K_{(t,q,p,y)}\\Big |_{y=q_t=0, t=T_R}\\left({\\varepsilon }^{\\frac{k}{2}-j} \\, p_{k,j}\\left(t,q,p,\\frac{y-q_t}{\\sqrt{\\varepsilon }}\\right)\\right)\\right\\Vert _{L^\\infty }\\right)\\\\&\\quad =\\sum _{\\ell =1}^L \\sum _{(k,j)\\in I_N}\\varepsilon ^\\ell \\,{\\varepsilon }^{\\frac{k}{2}-j} \\,O\\left(\\sup _{K\\le 2\\ell }\\left\\Vert \\partial ^K_{(t,q,p,y)}\\Big |_{y=q_t=0, t=T_R}\\left(p_{k,j}\\left(t,q,p,\\frac{y-q_t}{\\sqrt{\\varepsilon }}\\right)\\right)\\right\\Vert _{L^\\infty }\\right)$ Hence, using the fact that each $p_{k,j}$ is a polynomial in its last argument, so that the above derivatives evaluated at $y=q_t=0$ only leave the zero-th order term in the derived polynomial, we recover $&II_{\\varepsilon }=O\\left(\\sum _{\\ell =1}^L \\sum _{(k,j)\\in I_N}\\varepsilon ^\\ell \\,{\\varepsilon }^{\\frac{k}{2}-j} \\,\\sup _{K\\le 2\\ell }{\\varepsilon }^{-K/2}\\right)\\\\&\\quad =O\\left(\\sum _{\\ell =1}^L \\sum _{(k,j)\\in I_N}\\varepsilon ^\\ell \\,{\\varepsilon }^{\\frac{k}{2}-j} \\,{\\varepsilon }^{-\\ell }\\right)=O\\left(\\sum _{\\ell =1}^L \\sum _{(k,j)\\in I_N}{\\varepsilon }^{\\frac{k}{2}-j} \\, {\\varepsilon }^{-\\ell }\\right)=O\\left({\\varepsilon }^{1/2}\\right),$ where we have used that $k-2j\\ge 1$ whenever $(k,j)\\in I_N$ .", "$\\bullet $ Contribution of $I_{\\varepsilon }$ in (REF ).", "The integral defining $I_{\\varepsilon }$ has the following more explicit value, where $p=(\\sqrt{2 n^2(0)},\\theta _1,\\ldots ,\\theta _{d-1})$ , namely $I_{\\varepsilon }=C_1 \\, \\int _{\\Pi _\\theta I_{\\theta _0}}\\frac{\\displaystyle e^{i\\frac{\\pi }{4} \\text{sgn}\\left(\\frac{D^2\\psi \\circ \\gamma ^{-1}}{D\\alpha ^2}(0,\\theta )\\right)}}{\\text{det}\\left(\\frac{D^2\\psi \\circ \\gamma ^{-1}}{D\\alpha ^2}(0,\\theta )\\right) }\\,\\exp (\\left(\\frac{i}{\\varepsilon }\\psi (T_R,0,p,0,p,0,-p)\\right)\\\\\\text{det}(A(T_{R},0,p)+iB(T_{R},0,p))_{c}^{-1/2}\\, \\widehat{S}(p)\\, \\widehat{\\phi }^*(-p)\\, d\\theta _{1}\\ldots \\, d\\theta _{d-1},$ On top of that, we have $\\psi (T_R,0,p,0,p,0,-p)=\\int _{0}^{T_{R}}\\left(\\frac{|p_{s}(0,p)|^2}{2}+n^2(q_{s}(0,p))\\right) \\, ds,$ while the fact that $n^2$ is radial implies that $\\psi (T_R,0,p,0,p,0,-p)=\\psi (T_R,0,p_0,0,p_0,0,-p_0)$ whenever $p\\in I_{\\theta _{0}}$ .", "For the same reason, we also have whenever $\\theta \\in \\Pi _\\theta I_{\\theta _0}$ the relation $&\\frac{\\displaystyle e^{i\\frac{\\pi }{4} \\text{sgn}\\left(\\frac{D^2\\psi \\circ \\gamma ^{-1}}{D\\alpha ^2}(0,\\theta )\\right)}}{\\text{det}\\left(\\frac{D^2\\psi \\circ \\gamma ^{-1}}{D\\alpha ^2}(0,\\theta )\\right) }=\\frac{\\displaystyle e^{i\\frac{\\pi }{4} \\text{sgn}\\left(\\frac{D^2\\psi \\circ \\gamma ^{-1}}{D\\alpha ^2}(0,0)\\right)}}{\\text{det}\\left(\\frac{D^2\\psi \\circ \\gamma ^{-1}}{D\\alpha ^2}(0,0)\\right) }$ together with the identity, valid whenever $p \\in I_{\\theta _0}$ , $&\\text{det}(A(T_{R},0,p)+iB(T_{R},0,p))_{c}^{-1/2}=\\text{det}(A(T_{R},0,p_0)+iB(T_{R},0,p_0))_{c}^{-1/2}.$ Eventually, we have obtained $I_{\\varepsilon }=C_{n^2,d} \\,e^{\\left(\\displaystyle \\frac{i}{\\varepsilon }\\displaystyle \\int _{0}^{T_{R}}\\left(\\frac{|p_{s}(0,p_0)|^2}{2}+n^2(q_{s}(0,p_0))\\right)ds\\right)} \\,\\int _{I_{\\theta _{0}}}\\widehat{S}(p)\\widehat{\\phi ^*}(-p) \\, d\\sigma _{\\theta _0}(p),$ with $C_{T_{R},d}:=\\frac{(2\\pi )^{5d+2} e^{i\\frac{\\pi }{4} \\text{sgn}\\left(\\frac{D^2\\psi \\circ \\gamma ^{-1}}{D\\alpha ^2}(0,0)\\right)}}{\\text{det}\\left(\\frac{D^2\\psi \\circ \\gamma ^{-1}}{D\\alpha ^2}(0,0)\\right) }\\,\\text{det}(A(T_{R},0,p_{0})+iB(T_{R},0,p_{0}))_{c}^{-1/2}.$ This ends the proof of Proposition REF -part (i).", "$\\bullet \\bullet $ Proof of Proposition REF -part (ii) In that case, the argument is essentially the same (a stationary phase argument in the variable $\\alpha $ ), up to a convenient use of the dominated convergence Theorem (to deal with the variable $\\theta _1$ , and more specifically with the boundary $\\theta _1=\\pm \\theta _0$ ).", "Namely, we first write, as in the proof of part (i) of the Proposition, $&\\langle \\widetilde{w_{\\varepsilon }},\\phi \\rangle =O_{\\delta ,T_{1}}(\\varepsilon ^N)+\\\\&\\nonumber \\frac{1}{\\varepsilon ^{(5d+2)/2}}\\int _{\\gamma (supp\\ a_{N})}e^{\\frac{i}{\\varepsilon }\\psi \\circ \\gamma ^{-1}(\\alpha ,\\theta )}\\left(\\widehat{S}(.", ")\\widehat{\\phi }^*(.)P_{N}\\left(.,.,.,\\frac{.", "}{\\sqrt{\\varepsilon }}\\right)\\right)\\circ \\gamma ^{-1}(\\alpha ,\\theta )\\chi _{3}(\\alpha ,\\theta )\\, r^{d-1} \\, d\\alpha \\, d\\sigma (\\theta ),$ where $\\chi _{3}$ is a truncation function on some compact set, a neighbourhood of $M_X$ , whose precise value is irrelevant.", "Here we have used the non-stationary phase Theorem to reduce the original integral to an integral on a given compact set.", "The key point now lies in writing, $&\\langle \\widetilde{w_{\\varepsilon }},\\phi \\rangle =O_{\\delta ,T_{1}}(\\varepsilon ^N)+\\\\&\\nonumber \\int d\\sigma (\\theta ) \\,\\underbrace{\\left(\\frac{1}{\\varepsilon ^{(5d+2)/2}}\\int d\\alpha \\,e^{\\frac{i}{\\varepsilon }\\psi \\circ \\gamma ^{-1}(\\alpha ,\\theta )}\\left(\\widehat{S}(.", ")\\widehat{\\phi }^*(.)P_{N}\\left(.,.,.,\\frac{.", "}{\\sqrt{\\varepsilon }}\\right)\\right)\\circ \\gamma ^{-1}(\\alpha ,\\theta )\\chi _{3}(\\alpha ,\\theta )\\, r^{d-1}\\right)}_{=:J_{\\varepsilon }(\\theta )}.$ With this formulation in mind, our next objective is to prove that whenever $\\eta >0$ is a small parameter we have $&\\int _{|\\theta _1 \\pm \\theta _0|\\le \\eta } d\\sigma (\\theta ) \\,\\left| J_{\\varepsilon }(\\theta ) \\right| \\le C \\, \\eta ,$ for some $C>0$ independent of ${\\varepsilon }$ and $\\eta $ .", "It is clear indeed that the upper-bound (REF ), in conjunction with part (i) of the Proposition, provides a complete proof of Proposition REF -part (ii).", "Let us now concentraate on the case $|\\theta _1-\\theta _0|\\le \\eta $ (the proof in the case $|\\theta _1+\\theta _0|\\le \\eta $ is the same).", "In order to prove (REF ), we fix a value $(\\theta _2^0,\\ldots ,\\theta _{d-1}^0)$ and we prove that, given $(\\theta _2^0,\\ldots ,\\theta _{d-1}^0)$ , there is an $\\eta >0$ , and a $C>0$ independent of ${\\varepsilon }$ , such that $&\\forall \\theta \\, \\text{ such that } \\, |\\theta -(\\theta _0.\\theta _2^0,\\ldots ,\\theta _{d-1}^0)|\\le \\eta , \\quad \\text{ we have } \\,\\left| J_{\\varepsilon }(\\theta ) \\right| \\le C.$ Covering the whole set $\\lbrace \\theta \\in {\\mathbb {S}}^{d-1} ; |\\theta _1-\\theta _0|\\le \\eta \\rbrace $ by finitely many sets of the form $\\lbrace |\\theta -(\\theta _0.\\theta _2^0,\\ldots ,\\theta _{d-1}^0)|\\le \\eta \\rbrace $ clearly provides the desired relation (REF ) once (REF ) is proved.", "Now, relation (REF ) results from an application of the stationary phase Theorem, with complex phase and with parameter.", "Here $\\alpha $ is the variable used for the stationary phase itself, while $\\theta $ is the parameter, and $\\psi \\circ \\gamma ^{-1}$ is the complex phase.", "We introduce the short-hand notation $\\theta ^0=(\\theta _0,(\\theta ^{\\prime })^{0})=(\\theta _0,\\theta _2^0,\\ldots ,\\theta _{d-1}^0)$ for convenience.", "It has already been establishedStricto sensu, these relations have only be proved when $|\\theta _1|<\\theta _0$ , and we here extend the result to the case $\\theta _1=\\theta _0$ .", "This is allowed due to the invariance of the phase on the parameter $\\theta $ whenever $|\\theta _1|\\le \\theta _0$ – Lemma REF .", "that $&{\\rm Im}\\left(\\psi \\circ \\gamma ^{-1}\\right)(\\alpha ,\\theta ) \\ge 0, \\quad \\forall (\\alpha ,\\theta ),\\\\&{\\rm Im} \\left(\\psi \\circ \\gamma ^{-1}\\right)(\\alpha =0,\\theta =\\theta ^0)=0,\\\\&\\nabla _\\alpha \\left(\\psi \\circ \\gamma ^{-1}\\right)(\\alpha =0,\\theta =\\theta ^0)=0,\\\\&{\\rm det}\\left(\\frac{D^2 \\psi \\circ \\gamma ^{-1}}{D \\alpha ^2}\\right)(\\alpha =0,\\theta =\\theta ^0)\\ne 0$ Therefore, the stationary phase theorem with parameter ensures that close to $\\theta =\\theta ^0$ there is an expansion of the form $J_{\\varepsilon }(\\theta )=e^{i \\phi (\\theta )/{\\varepsilon }}\\,\\left(\\sum _{\\ell =0}^L{\\varepsilon }^\\ell \\,\\left(Q_{2\\ell }(\\partial _\\alpha )\\left(\\widehat{S}(.)", "\\,\\widehat{\\phi ^*}(.)", "\\,P_N\\left( ., ., \\frac{.", "}{\\sqrt{{\\varepsilon }}}\\right)\\circ \\gamma ^{-1} \\,\\chi _3(.", ")\\right)\\right)^0(\\theta )\\right)+R({\\varepsilon },L,\\theta ),$ for some smooth functions $\\phi $ and $R({\\varepsilon },L,\\theta )$ , where the $Q_{2\\ell }$ 's are differential operators of order $2\\ell $ in the variable $\\alpha $ , and, for any function $u(\\alpha ,\\theta )$ , the notation $u^0(\\theta )$ refers to any smooth function $u^0(\\theta )$ that belongs to the same residue class than the original function $u(\\alpha ,\\theta )$ modulo the ideal generated by $\\nabla _\\alpha \\psi \\circ \\gamma ^{-1}(\\alpha ,\\theta )$ (see Hörmander [12], sect.", "7.7, for the details).", "With this notation, we actually have $\\phi =\\left(\\psi \\circ \\gamma ^{-1}\\right)^0$ .", "Besides, the remainder term $R$ satisfies as the term $III_{\\varepsilon }$ in the previous step an estimate of the form $\\left|R({\\varepsilon },L,\\theta )\\right|\\le C_L \\, \\varepsilon ^{L+1} \\,\\left(\\sup _{K\\le 2 (L+1)}\\left\\Vert \\partial ^K_\\alpha \\left(\\left(\\widehat{S}(.)", "\\,\\widehat{\\phi ^*}(.)", "\\,P_{N}\\left(.,.,.,\\frac{.", "}{\\sqrt{\\varepsilon }}\\right) \\,\\right)\\circ \\gamma ^{-1} \\,\\chi _3(.", ")\\right)\\right\\Vert _{L^\\infty }\\right),$ for some constant $C_L>0$ independent of ${\\varepsilon }$ , and provided $\\theta $ is close to $\\theta ^0$ (independently of ${\\varepsilon }$ ).", "These two ingredients immediately provide, using the same estimates as we did for the terms $III_{\\varepsilon }$ and $II_{\\varepsilon }$ above, the upper-bound, valid for $\\theta $ close to $\\theta ^0$ , $\\left|J_{\\varepsilon }(\\theta )\\right|\\le C \\,\\left(\\sum _{\\ell =0}^L{\\varepsilon }^\\ell \\,\\left(Q_{2\\ell }(\\partial _\\alpha )\\left(\\widehat{S}(.)", "\\,\\widehat{\\phi ^*}(.)", "\\,P_N\\left( ., ., \\frac{.", "}{\\sqrt{{\\varepsilon }}}\\right)\\circ \\gamma ^{-1} \\,\\chi _3(.", ")\\right)\\right)^0(\\theta )\\right)+R({\\varepsilon },L,\\theta ),$ Gathering powers of ${\\varepsilon }$ as in the previous part of the proof, provides the upper bound $\\left|J_{\\varepsilon }(\\theta )\\right|\\le C,$ where $C$ does not depend on ${\\varepsilon }$ and $\\theta $ is close to $\\theta ^0$ , independently of ${\\varepsilon }$ .", "Point (REF ) is proved.", "We immediately deduce that (REF ) holds, and the proof of Proposition REF – part (ii) is complete.", "$\\blacksquare $" ], [ "Conclusion", "Gathering the intermediate result in Proposition REF , together with Proposition REF , gives item (iii) of Theorem REF , by conveniently choosing the parameters $\\delta $ , $\\theta $ , $T_{0}$ and $T_{1}$ ." ] ]

[ [ "Neutrino Mixing Anarchy: Alive and Kicking" ], [ "Abstract Neutrino mixing anarchy is the hypothesis that the leptonic mixing matrix can be described as the result of a random draw from an unbiased distribution of unitary three-by-three matrices.", "In light of the recent very strong evidence for a nonzero sin^2(theta_13), we show that the anarchy hypothesis is consistent with the choice made by the Nature -- the probability of a more unusual choice is 44%.", "We revisit anarchy's ability to make predictions, concentrating on correlations - or lack thereof - among the different neutrino mixing parameters, especially sin^2(theta_13) and sin^2(theta_23).", "We also comment on anarchical expectations regarding the magnitude of CP-violation in the lepton sector, and potential connections to underlying flavor models or the landscape." ], [ "Acknowledgments", "The work of AdG is sponsored in part by the DOE grant # DE-FG02-91ER40684.", "The work of HM was supported in part by the U.S. DOE under Contract DE-AC03-76SF00098, in part by the NSF under grant PHY-1002399, the Grant-in-Aid for scientific research (C) 23540289 from Japan Society for Promotion of Science (JSPS), and in part by World Premier International Research Center Initiative (WPI), MEXT, Japan." ] ]

[ [ "Quantum entanglement and phase transition in a two-dimensional\n photon-photon pair model" ], [ "Abstract We propose a two-dimensional model consisting of photons and photon pairs.", "In the model, the mixed gas of photons and photon pairs is formally equivalent to a two-dimensional system of massive bosons with non-vanishing chemical potential, which implies the existence of two possible condensate phases.", "Using the variational method, we discuss the quantum phase transition of the mixed gas and obtain the critical coupling line analytically.", "Moreover, we also find that the phase transition of the photon gas can be interpreted as second harmonic generation.", "We then discuss the entanglement between photons and photon pairs.", "Additionally, we also illustrate how the entanglement between photons and photon pairs can be associated with the phase transition of the system." ], [ "Introduction", "Bose-Einstein condensate (BEC) is the remarkable state of matter that spontaneously emerges when a system of bosons becomes cold enough that a significant fraction of them condenses into a single quantum state to minimize the system's free energy.", "Particles in that state then act collectively as a coherent wave.", "The phase transition for an atomic gas was first predicted by Einstein in 1924 and experimentally confirmed with the discovery of superfluid helium-4 in 1938.", "Obviously, atoms aren't the only option for a BEC.", "In recent years, with the development of techniques, the phenomenon of BEC was observed in several physical system [1-9], including exciton polaritions, solid-state quasiparticles and so on.", "We know that photons are the simplest of bosons, so that it would seem that they could in principle undergo this kind of condensation.", "The difficulty is that in the usual blackbody configuration, which consists of an empty three-dimensional (3D) cavity, the photon is massless and its chemical potential is zero, so that the BEC of photons under these circumstances would seem to be impossible.", "However, very recently, J. Klaers, etc.", "have overcome both obstacles using a simple approach [10,11]: By confining laser light within a two-dimensional (2D) cavity bounded by two concave mirrors, they create the conditions required for light to thermally equilibrate as a gas of conserved particles rather than as ordinary blackbody radiation.", "What is more, it is well known that there are many fascinated optical effects in the nonlinear medium, for instance, reduced fluctuation in one quadrature (squeezing) [12], sub-Poissonian statistics of the radiation field [13], or the collapse-revivals phenomenon [14].", "Especially, in the nonlinear medium, a photon from the laser beam can couple with other photons to form a photon-pair (PP) [15-18].", "The essence of PP has been investigated by many authors [19-21].", "However, inspired by the experimental discovery of BEC of photons, in this letter we construct another interesting 2D model consisting of photons and PPs.", "In this model, the mixed system of photons and PPs is formally equivalent to a 2D gas of massive bosons with non-vanishing chemical potential, which implies the existence of two possible condensate phases, the mixed photon-PP condensate phase and the pure PP condensate phase.", "By means of a variational method we investigate the quantum phase transition of the mixed photon gas.", "Especially, we find that the quantum phase transition of the photon gas can be interpreted as second harmonic generation.", "We then discuss the entanglement between photons and PPs.", "By investigating the entanglement in the ground state and the dynamics of entanglement, we also illustrate how the entanglement between photons and PPs can be associated with the phase transition of the system.", "The investigation of these questions is important both for its connection with quantum optics and for its practical applications to harmonic generation and quantum information.", "The remainder of this paper is organized as follows: In Sec.", "II, we theoretically investigate the phenomenon of BEC of photons and PPs in a 2D optical microcavity.", "The entanglement between photons and PPs is investigated in Sec.", "III.", "Finally, we make a simple conclusion.", "We start with the description of the PP.", "History speaking, the essence of the PP is presently still under discussion [19-21], and there exist many different ways to obtain it.", "However, here we will use the standard procedure [22] in the construction of harmonic generation to derive the PP.", "We know that the presence of an electromagnetic field in the nonlinear material causes a polarization of the medium and the polarization can be expanded in powers of the instantaneous electric field: ${\\bf {P}}({\\bf {r}},t) = \\chi ^{(1)} {\\bf {E}}({\\bf {r}},t) + \\chi ^{(2)} {\\bf {E}}^2 ({\\bf {r}},t) + ... .$ Here, the first term defines the usual linear susceptibility, and the second term defines the lowest order nonlinear susceptibility.", "Ignoring the high order parts (i.e.", "only expend the polarization to second order in electric field $E$ ), we find that the Hamiltonian describing the interaction of the radiation field with the dielectric medium is decomposed into two terms: $\\begin{array}{l}{\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} H_{{\\mathop {\\rm int}} } = H_{line} + H_{nonline} \\\\{\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} H_{line} = - \\int {\\chi ^{(1)} {\\bf {E}}^2 ({\\bf {r}},t)} d{\\bf {r}} \\\\H_{nonline} = - \\int {\\chi ^{(2)} {\\bf {E}}^3 ({\\bf {r}},t)} d{\\bf {r}} \\\\.\\end{array}$ where $H_{line}$ represents the energy of the linear interaction and $H_{nonline}$ the nonlinear interaction.", "It is well known that the electric field operator in a microcavity can be expanded in terms of normal modes [23] as ${\\bf {E}}(r,t) = i\\sum \\limits _{\\bf {k}} {{\\bf {e}}_{\\bf {k}} } \\left( {\\frac{{\\hbar \\omega _{\\bf {k}} }}{{2V\\varepsilon }}} \\right)^{1/2} \\left( {\\hat{a}_{\\bf {k}} e^{ - i\\omega _{\\bf {k}} t + i{\\bf {k}} \\cdot {\\bf {r}}} + \\hat{a}_{\\bf {k}}^ + e^{i\\omega _{\\bf {k}} t - i{\\bf {k}} \\cdot {\\bf {r}}} } \\right),$ where $\\hat{a}_{\\bf {k}}$ and $\\hat{a}_{\\bf {k}}^ +$ are the annihilation and creation operators of photons with frequency $\\omega _{\\bf {k}}$ , and they all obey the usual boson commutation rules.", "$V$ is the normalization volume, $\\varepsilon $ is the dielectric constant of the medium and ${\\bf {e}}_{\\bf {k}}$ is the unit polarization vector with the usual polarization indices omitted for simplicity.", "Substituting (3) into (2), for the linear interaction part, we find that it consists of two processes, dissipation and two-photon absorption(or emission).", "Here, dissipation is essentially also a two-photon process, in which one photon is absorbed by the medium, meanwhile another one is emitted.", "The linear interaction can be ignored, if the incident photon field frequency $\\omega _0$ is well below the electronic transition frequencies of the medium.", "In that case, we need only consider the nonlinear interaction, which has the simple form $H_{nonline} = \\frac{\\hbar }{{\\sqrt{V} }}\\sum \\limits _{{\\bf {k,k^{\\prime }}}} {\\chi _{{\\bf {k}},{\\bf {k^{\\prime }}}} \\left( {\\hat{b}_{{\\bf {k}} + {\\bf {k^{\\prime }}}}^ + \\hat{a}_{\\bf {k}} \\hat{a}_{{\\bf {k^{\\prime }}}} + H.c.} \\right)},$ under the requirements of phase matching.", "Above, the operator $\\hat{a}$ represents the normal photons, $\\hat{b}$ represents the coupling PP, and where $\\chi _{{\\bf {k}},{\\bf {k^{\\prime }}}}$ is the coupling matrix element.", "The interaction energy in (4) consists of two terms.", "The first term $b_{{\\bf {k}} + {\\bf {k^{\\prime }}}}^ + a_{\\bf {k}} a_{{\\bf {k^{\\prime }}}}$ describes the process in which two normal photon with wave-vector ${\\bf {k}}$ and ${\\bf {k^{\\prime }}}$ couple into a PP with wave-vector ${\\bf {K}} = {\\bf {k}} + {\\bf {k^{\\prime }}}$ , and the second term describe the opposite process.", "The energy is conserved in both the processes." ], [ "Free-photon dispersion relation inside the optical microcavity", "In this letter, we restrict out investigation inside a 2D optical microcavity.", "The microcavity, as shown in Fig.", "1, consists of two curved dielectric mirrors with high reflectivity (about 99.9), which ensure prefect reflection of the longitudinal component of the electromagnetic field within the cavity.", "In addition, the transverse size of the cavity is much larger than its longitudinal one.", "We know that for a free photon, its frequency as a function of transversal ($k_r$ ) and longitudinal ($k_z$ ) wave number is $\\omega = c\\left[ {k_z^2 + k_r^2 } \\right]^{1/2}$ .", "However, in the case of photons confined inside the microcavity, the vanishing of the electric field at the reflecting surfaces of the curved-mirrors imposes a quantization condition on the longitudinal mode number $k_z$ , $k_z = n\\pi /D(r)$ , where $n$ is an integer and where $D(r) = D_0 - 2(R - \\sqrt{R^2 - r^2 } )$ is the separation of two curved-mirrors at distance $r$ from the optical axis, with $D_0$ the mirror separation at distance $r = 0$ and $R$ the radius of curvature.", "In the present work, we consider to fix the longitudinal mode number of photons by inserting a circular filter into the cavity.", "The filter is filled with a dye solution, in which photons are repeatedly absorbed and re-emitted by the dye molecules.", "Thus, it also plays the role of photon reservoir.", "We know that the longitudinal size of the cavity (i.e., the distance between the mirrors) is very small.", "The small distance $D(r)$ between the mirrors causes a large frequency spacing between adjacent longitudinal modes, comparable with the spectral width of the dye.", "Modify spontaneous emission such that the emission of photons with a given longitudinal mode number, $n = q$ in our case, dominates over other emission processes.", "In this way, the longitudinal mode number is frozen out.", "For fixed longitudinal mode number $q$ and in paraxial approximation ($r \\ll R$ , $k_r \\ll k_z$ ), we also find that the dispersion relation of photons approximatively becomes $\\omega \\approx q\\pi c/D_0 + ck_r^2 D_0 /2q\\pi $ .", "The above frequency-wavevector relation, upon multiplication by $\\hbar $ , becomes the energy-momentum relation for the photon $E \\approx m_{{\\rm {ph}}} c^2 + \\frac{{(p_r )^2 }}{{2m_{{\\rm {ph}}} }},$ where $m_{{\\rm {ph}}} = \\hbar q\\pi /D_0 c = \\hbar \\omega _{eff} /c^2$ is the effective mass of the confined photons.", "At low temperatures, it is convenient to redefine the zero of energy, so that only the effective kinetic energy, $E \\approx \\frac{{(p_r )^2 }}{{2m_{{\\rm {ph}}} }},$ remains.", "The above analysis shows that for the photon confined inside the 2D microcavity, it is formally equivalent to a general boson having an effective mass $m_{{\\rm {ph}}} = \\hbar \\omega _{eff} /c^2$ , that is moving in the transverse resonator plane.", "Furthermore, we here consider the case that the microcavity (except the filter part) is filled with a Kerr nonlinear medium exhibiting significant third-order optical nonlinearity.", "Due to the nonlinear effect, photons can couple into PPs.", "If we connect the non-vanishing effective photon mass to the previous analysis of the PPs, in this case we then can rewrite the nonlinear interaction $H_{nonline}$ as $H_{nonline} = \\frac{\\hbar }{{\\sqrt{S} }}\\sum \\limits _{{\\bf {k}}_r {\\bf {,k^{\\prime }}}_r } {\\chi _{{\\bf {k}}_r ,{\\bf {k^{\\prime }}}_r } \\left( {b_{{\\bf {k}}_r + {\\bf {k^{\\prime }}}_r }^ + a_{{\\bf {k}}_r } a_{{\\bf {k^{\\prime }}}_r } + H.c.} \\right)}$ where $S$ is the surface area of the 2D cavity, and where $a_{{\\bf {k}}_r }$ and $a_{{\\bf {k^{\\prime }}}_r }$ are the annihilation operators of massive photons with transverse wavevectors ${\\bf {k}}_r$ and ${\\bf {k}}_{r^{\\prime }}$ , respectively, and $b_{{\\bf {k}}_r + {\\bf {k^{\\prime }}}_r }^ +$ are the creation operator of the massive PPs with transverse wave-vector $K_r = {\\bf {k}}_r + {\\bf {k^{\\prime }}}_r$ .", "Here, it should be remarked that the existence of effective photon mass makes the thermodynamics of this 2D mixed gas of photons and PPs different from the usual 3D photon gas.", "For the 2D system, thermalization is achieved in a photon-number-conserving way ($N = N_a + 2N_b $ ) with nonvanishing chemical potential $\\mu $ , by multiple scattering with the dye molecules, which acts as heat bath and equilibrates the transverse modal degrees of freedom of the photon gas to the temperature of dye molecules." ], [ "BEC of photons and photon pairs", "In virtue of the above analysis, we consider the following basic Hamiltonian, to give a simple model of PP formation (with $\\hbar = 1$ throughout this letter) $H_\\mu = H_a + H_b + H_{ab},$ with $\\begin{array}{l}H_a = \\sum \\limits _{\\bf {k}} {\\frac{{{\\bf {k}}^2 }}{{2m_{ph} }}a_{\\bf {k}}^ + a_{\\bf {k}}^{} } + \\frac{{u_{aa} }}{S}\\sum \\limits _{{\\bf {k}},{\\bf {k^{\\prime }}},{\\bf {k^{\\prime \\prime }}}} {a_{{\\bf {k}} + {\\bf {k^{\\prime }}} - {\\bf {k^{\\prime \\prime }}}}^ + a_{{\\bf {k^{\\prime \\prime }}}}^ + } a_{{\\bf {k^{\\prime }}}} a_{\\bf {k}} \\\\H_b = \\sum \\limits _{\\bf {k}} {\\left( {\\frac{{{\\bf {k}}^2 }}{{4m_{ph} }} - 2\\mu } \\right)b_{\\bf {k}}^ + b_{\\bf {k}}^{} } + \\frac{{u_{bb} }}{S}\\sum \\limits _{{\\bf {k}},{\\bf {k^{\\prime }}},{\\bf {k^{\\prime \\prime }}}} {b_{{\\bf {k}} + {\\bf {k^{\\prime }}} - {\\bf {k^{\\prime \\prime }}}}^ + b_{{\\bf {k^{\\prime \\prime }}}}^ + } b_{{\\bf {k^{\\prime }}}} b_{\\bf {k}} \\\\H_{ab} = \\frac{{u_{a,b} }}{S}\\sum \\limits _{{\\bf {k}},{\\bf {k^{\\prime }}}} {a_{\\bf {k}}^ + b_{{\\bf {k^{\\prime }}}}^ + } b_{{\\bf {k^{\\prime }}}} a_{\\bf {k}} - \\frac{1}{{\\sqrt{S} }}\\sum \\limits _{{\\bf {k}},{\\bf {k^{\\prime }}}} {\\chi _{{\\bf {k}},{\\bf {k^{\\prime }}}} (b_{{\\bf {k}} + {\\bf {k^{\\prime }}}}^ + } a_{{\\bf {k^{\\prime }}}} a_{\\bf {k}} + H.c.) \\\\\\end{array}$ Above, $H_a$ and $H_b$ denote the pure photon and PP contributions, and $H_{ab}$ refers to the interaction between them.", "In the dilute gas limit $u_{a,a}$ , $u_{b,b}$ , $u_{a,b}$ are proportional to the two-body s-wave photon-photon, photon-PP, and PP-PP scattering lengths [23], respectively, and $\\chi _{{\\bf {k}},{\\bf {k^{\\prime }}}}$ characterizes the coupling strength, encoding that PPs are composed of two massive photons.", "Note that in (9) we ignore the chemical potential term $\\mu N_a$ from the Hamiltonian [24].", "Additionally, in (9), we also drop the subscript of the transverse wave-vectors of photons and PPs, i.e.", "we take ${\\bf {k}}_r = {\\bf {k}}$ .", "It is well known that for a general massive Boson-Boson pairs gas, at absolute zero temperature, there exists a BEC consisting of two possible condensate phases [25-27]: (i) Both the single boson and the pair of bosons are condensed.", "(ii) The pair of bosons are condensed but the single boson is not.", "Now we know that in the case of photons (or PPs) confined inside the microcavity, the subsystem of photons (or PPs) is formally equivalent to a 2D gas of massive bosons with non-vanishing chemical potential.", "Thus, the feature should also survive for the 2D mixed gas of photons and PPs.", "In the condensate phase, a macroscopic number of particles occupy the zero-momentum state, and it is useful to separate out the condensate modes from the Hamiltonian.", "Follow the process, we find that at the BEC state, the grand canonical Hamiltonian of the mixed system has the form $H_\\mu = H_0 + \\delta H,$ where $\\begin{array}{l}H_0 = - \\mu _b b^ + b + \\frac{{u_{aa} }}{S}\\left( {a^ + a} \\right)^2 + \\frac{{u_{bb} }}{S}\\left( {b^ + b} \\right)^2 \\\\{\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} + \\frac{{u_{ab} }}{S}a^ + ab^ + b - \\frac{\\chi }{{\\sqrt{S} }}\\left( {a^ + a^ + b + b^ + aa} \\right) \\\\\\end{array}$ is the condensate part of the Hamiltonian with $\\mu _b = 2\\mu + u_{bb} /S$ the modified chemical potential of PPs, and where $\\delta H = H\\left( {a_{\\bf {k}} ,b_{\\bf {k}} } \\right)$ is the perturbation part and it is a complex function of the non-condensate modes.", "Here, we mention that at the condensate state we only need consider the case of single mode coupling, thus we can treat the coupling matrix element $\\chi $ as an adjustable constant.", "Note that in (11) we also drop the zero-momentum subscript of the creation and annihilation operators of photons and PPs.", "In the present work, we mainly aim to investigate the phenomenon of BEC of photons and PPs, thus, hereafter we will ignore the perturbation part and approximately write the Hamiltonian as the form $H_\\mu \\approx H_0$ .", "The Hamiltonian commutes with the total photon number $N = a^ + a + 2b^ + b$ and $n = N/S$ is the particle density.", "Up to now we have not made a careful distinction between the two possible condensate phases.", "However, for the mixed system, working out the ground-state phase diagram is very important.", "Here, we intend to employ the variational principle method for finding the ground-state configurations of the present system and examining their dependence from the microscopic parameters.", "In other word, we aim to work out the ground-state phase diagram of the mixed system starting from the study the semiclassical equation.", "Clearly, we know that the chemical potential $\\mu $ of the mixed system allows the total photon number (whether free or bound into PPs) to fluctuate around some constant average value $N$ , then the total number of photons need only be conserved on the average value.", "For convenience, hereafter we assume that $N$ is an even number and thus $M = N/2$ denotes the maximum number of the PPs.", "Furthermore, in this case we also introduce a new operation, namely the double photon creation operation with the relation $\\begin{array}{l}\\left( {c^ + } \\right)^m \\left| 0 \\right\\rangle \\equiv \\sqrt{\\frac{{m!", "}}{{\\left( {2m} \\right)!}}}", "\\left( {a^ + } \\right)^{2m} \\left| 0 \\right\\rangle \\\\{\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} c^ + c\\left| 0 \\right\\rangle \\equiv 2a^ + a\\left| 0 \\right\\rangle \\\\\\end{array},$ where $\\left| 0 \\right\\rangle $ is the vacuum state.", "Using the new operator, we construct the Gross-Pitaevskii (GP) states [28] $\\left| {\\psi _{{\\rm {GP}}} } \\right\\rangle = \\frac{1}{{\\sqrt{M!}", "}}\\left[ {\\alpha c^ + + \\beta b^ + } \\right]^M \\left| 0 \\right\\rangle $ as the trial macroscopic state.", "Here, $\\alpha = \\left| \\alpha \\right|e^{i\\theta _a }$ and $\\beta = \\left| \\beta \\right|e^{i\\theta _b }$ are complex amplitudes with $\\left| \\alpha \\right|^2 = N_a /N$ and $\\left| \\beta \\right|^2 = 2N_b /N$ the photon and PP densities, respectively.", "$\\theta _a$ and $\\theta _b$ (real valued) denoted the phases of each species.", "Obviously, the parameters $\\alpha $ and $\\beta $ satisfy the normalized condition $\\left| \\alpha \\right|^2 + \\left| \\beta \\right|^2 = 1$ .", "With the help of the GP state, the semiclassical model Hamiltonian $\\bar{H}(\\alpha ,\\beta )$ is given by $\\begin{array}{l}\\bar{H} = \\mathop {\\lim }\\limits _{N \\rightarrow \\infty } \\frac{{\\left\\langle {\\psi _{GP} } \\right|H_\\mu \\left| {\\psi _{GP} } \\right\\rangle }}{{M{\\chi }\\sqrt{2n} }} \\\\{\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} = \\left[ { - \\mu _b \\left| \\beta \\right|^2 + 2u_{aa} n\\left| \\alpha \\right|^4 + u_{bb} n\\left| \\beta \\right|^4 } \\right./2 \\\\{\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {{\\left.", "{ + u_{ab} n\\left| \\alpha \\right|^2 \\left| \\beta \\right|^2 } \\right]} \\mathord {\\left\\bad.{\\vphantom{{\\left.", "{ + u_{ab} n\\left| \\alpha \\right|^2 \\left| \\beta \\right|^2 } \\right]} {{\\chi }\\sqrt{2n} }}} \\right.\\hspace{0.0pt}} {{\\chi }\\sqrt{2n} }} - 2\\left| \\alpha \\right|^2 \\sqrt{\\left| \\beta \\right|^2 } \\cos \\theta \\\\\\end{array},$ where $\\theta = \\theta _b - 2\\theta _a$ is the phase difference.", "Considering the conserved condition $\\left| \\alpha \\right|^2 + \\left| \\beta \\right|^2 = 1$ , we next introduce a new variables $s = \\left| \\alpha \\right|^2 - \\left| \\beta \\right|^2$ .", "Using the new notation, we rewrite the model Hamiltonian as $\\bar{H} = - \\lambda s^2 - 2\\gamma s + \\xi - \\sqrt{2\\left( {1 - s} \\right)} (1 + s)\\cos \\theta ,$ with $\\begin{array}{l}\\lambda = \\frac{{\\sqrt{2n} }}{{\\chi }}\\left( {\\frac{{u_{ab} }}{4} - \\frac{{u_{aa} }}{2} - \\frac{{u_{bb} }}{8}} \\right) \\\\\\gamma = \\frac{{\\sqrt{2n} }}{{\\chi }}\\left( {\\frac{{u_{bb} }}{8} - \\frac{{u_{aa} }}{2} - \\frac{{\\mu _b }}{{4n}}} \\right) \\\\\\xi = \\frac{{\\sqrt{2n} }}{{\\chi }}\\left( {\\frac{{u_{aa} }}{2} + \\frac{{u_{ab} }}{4} + \\frac{{u_{bb} }}{8} - \\frac{{\\mu _b }}{n} } \\right) \\\\\\end{array}$ .", "According to the variational principle, we minimize the energy $\\bar{H}(\\alpha ,\\beta )$ with $s$ and $\\theta $ as variational parameters.", "We then obtain the optimum values [i.e.", "($\\bar{s}$ , $\\bar{\\theta }$ )] of parameters for the ground state as follows: $\\left( {\\bar{s},\\bar{\\theta }} \\right) = \\left\\lbrace \\begin{array}{l}( - 1,{\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} \\theta ),{\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} \\gamma - \\lambda + 1 < 0 \\\\(s,{\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} 0{\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} or{\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} \\pi ),{\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} \\gamma - \\lambda + 1 > 0 \\\\\\end{array} \\right.,$ where $- 1 < s < 1$ is the solution of the equation $\\lambda s + \\gamma = \\left( {3s - 1} \\right)/2\\sqrt{2(1 - s)}$ (the explicit value can be obtain by graphical solution method, it is generally too messy to be shown here).", "The result, together with the fact that the parameters $\\left| \\alpha \\right|^2$ and $\\left| \\beta \\right|^2$ denote the photon and PP densities, indicates that when $\\gamma - \\lambda + 1 < 0$ the system converts from the mixed photon-PP phase to the pure PP phase.", "We therefore can interpret this line $\\gamma - \\lambda + 1 = 0$ as the threshold coupling for the formation of a predominantly PP state.", "Here, it is to be mentioned that for the pure PP phase, $\\bar{s} = - 1$ thus the relative phase $\\theta $ cannot be defined." ], [ "Entanglement of the ground state", "At the BEC state, one may consider the mixed gas of photons and PPs as a bipartite system of two modes.", "For the present system, the entanglement of two modes is always closely associated with the phase transition of the system.", "Moreover, the two modes, be they spatially separated, and differing in some internal quantum number, are clearly distinguishable subsystems.", "Thus, the state of each mode can be characterized by its occupation number.", "By using the fact that the total number of photons $N$ is constant, a general state of the system (in the Heisenberg picture) can be written for even $N$ in terms of the Fock states by $\\left| \\psi \\right\\rangle = \\sum \\limits _{m = 0}^M {c_m } \\left| {2m,M - m} \\right\\rangle ,$ where $m$ is the half population of particles in photon mode $a$ , and $c_m$ is the coefficients of the state.", "In the Fock representation, the GP state also can be reexpressed as $\\left| {\\psi _{GP} } \\right\\rangle = \\sum \\limits _{m = 0}^M {g_m } \\left| {2m,M - m} \\right\\rangle $ , with coefficients $g_m = \\sqrt{\\frac{{M!", "}}{{m!\\left( {M - m} \\right)!}}}", "\\alpha ^m \\beta ^{M - m}$ .", "The standard measure of entanglement of the bipartite system is the entropy of entanglement $S(\\rho )$ $S(\\rho ) = - \\sum \\limits _{m = 0}^M {\\left| {c_m } \\right|^2 } \\log _2 \\left( {\\left| {c_m } \\right|^2 } \\right),$ which is the von Neumann entropy of the reduced density operator of either of the subsystems[29].", "In the present system, the maximal entanglement also can be obtain by optimizing the expression (18) with respect to $\\left| {c_m } \\right|^2$ .", "By imposing the normalization condition $\\sum \\limits _{m = 0}^M {\\left| {c_m } \\right|^2 } = 1$ , we finally get $S_{\\max } = \\log _2 \\left( {M + 1} \\right)$ , which is related to the dimension $M + 1$ of the Hilbert space of the individual modes.", "Using expression (18) and the coefficients $c_m$ obtained through exact diagonalization of the Hamiltonian (11) as done in the atom-molecule model [30], we plot in Fig.", "2 the entropy of entanglement of the ground state as a function of the parameters $\\lambda $ and $\\gamma $ .", "We note that in this letter we restrict our attentions to the repulsive case, i.e., we restrict $\\lambda \\le 0$ throughout the letter.", "From Fig.", "2, we observe that the entanglement entropy exhibits a sudden decrease close $\\gamma - \\lambda + 1 = 0$ .", "This is indicative of the fact that across the line $\\gamma - \\lambda + 1 = 0$ a quantum phase transition occurs.", "Figure: Variation in the entropy of entanglement of the ground state with respectto the parameters λ\\lambda and γ\\gamma .", "Here, we set N=200N=200 and u aa =u bb /4=0.25u_{aa} = u_{bb}/4 = 0.25.To gain more information associated with the quantum phase transition of the present system, we also depict in Fig.", "3 the entropy of entanglement (solid line) and the expectation value (dashed line) of the scaled PP number operator of the ground state as a function $\\gamma $ for fixed parameter value $\\lambda $ .", "From Fig.", "3, we see that the average value of the number of PPs increases as $\\gamma $ increases.", "Especially, when $\\gamma - \\lambda + 1 < 0$ , the average number of PPs is maximal.", "The result confirms that there indeed exists a phase transition for the present system in the ground state.", "Furthermore, we also find that the ground-state entanglement entropy is not maximal at the critical line, i.e.", "in the region $\\gamma - \\lambda + 1 > 0$ , the system is always strongly entangled.", "Ref.", "(28) gives the property responsible for the long-range correlation.", "Additionally, we also consider that the trait is associated with the symmetry-broking of the coupling term of the system.", "Due to the asymmetric form of the coupling term, the pure photon condensation will be forbidden.", "As a result, in the mixed condensate phase the imbalance $\\left( {2N_b - N_a } \\right)/N$ between the two modes is always very small, which is responsible for the strongly entanglement.", "Figure: The average photon pair occupation number and the entanglement entropy for the ground state as a function of the γ\\gamma for fixed parameter value λ=0\\lambda =0.", "Here, we set N=200N=200 and u aa =u bb /4=0.25u_{aa} = u_{bb}/4 = 0.25.In addition, if we connect the non-vanishing photon mass $m_{ph}$ to the longitudinal wave number $k_z$ by the relation $m_{{\\rm {ph}}} = \\hbar k_z /c$ with $k_z = q\\pi /D_0$ , then we find that the quantum phase transition of the photon system can be interpreted as second harmonic generation.", "When $\\gamma - \\lambda + 1 < 0$ , almost all photons with frequency $\\omega = ck_z$ couple into PPs with frequency $\\omega = 2ck_z$ .", "In this case, the entanglement between the photons and PPs is very small, and the entropy of entanglement is close to zero." ], [ "Dynamics of entanglement", "In the above analysis, we have investigated the entanglement of the ground state.", "We found that in the ground state, across the phase transition line the entanglement entropy exhibits a sudden change.", "To gain a better understanding of the influence of ground-state phase transition to entanglement, in this subsection we investigate the dynamics of entanglement.", "In studying the dynamics of the system, we first need express a general state in the form of temporal evolution (i.e., need change the expression of a general state from Heisenberg picture to Schrodinger picture).", "Following the standard procedure, we can obtain $\\begin{array}{l}\\left| {\\psi \\left( t \\right)} \\right\\rangle = U\\left( t \\right)\\left| {\\psi \\left( 0 \\right)} \\right\\rangle \\\\= \\sum \\limits _{m = 0}^M {c_m } \\left( t \\right)\\left| {2m,M - m} \\right\\rangle \\\\\\end{array},$ where, $U\\left( t \\right) = \\sum \\limits _{n = 0}^M {\\left| {\\psi _n } \\right\\rangle \\left\\langle {\\psi _n } \\right|} \\exp \\left( { - iE_n t} \\right)$ is the temporal operator with $\\left| {\\psi _n } \\right\\rangle $ the eigenstates of the system having energy $E_n$ , and $\\left| {\\psi \\left( 0 \\right)} \\right\\rangle $ is the initial state.", "Here, the time dependence of coefficients $c_m \\left( t \\right)$ are given by $c_m \\left( t \\right) = \\left\\langle {2m,M - m} \\right|U\\left( t \\right)\\left| {\\psi \\left( 0 \\right)} \\right\\rangle $ .", "Subsequently, the entanglement entropy given in (18) can be rewritten as $S(\\rho ) = - \\sum \\limits _{m = 0}^M {\\left| {c_m \\left( t \\right)} \\right|^2 } \\log _2 \\left( {\\left| {c_m \\left( t \\right)} \\right|^2 } \\right)$ .", "In this case, the entanglement entropy depends on both the choice of initial states and the value of microscopic parameters $\\left\\lbrace {\\lambda ,\\gamma } \\right\\rbrace $ .", "At the present work, we consider that the mixed system is in the BEC state, thus here choosing GP state as the initial state is suitable.", "By adjusting the GP coefficients $\\left\\lbrace {\\left| \\alpha \\right|^2 ,\\left| \\beta \\right|^2 } \\right\\rbrace $ and the microscopic parameters $\\left\\lbrace {\\lambda ,\\gamma } \\right\\rbrace $ , in this subsection we also want to know if the ground-state phase transition also characterizes different dynamics.", "The time evolution of the entanglement entropy for different initial state and interaction parameters is shown in Fig.", "4.", "From Fig.4 we observe the features of quantum dynamics, such as the collapse and revival of oscillations and non-periodic oscillations.", "Additionally, we also find that the amplitude of the entanglement entropy is smaller in the region $\\gamma - \\lambda + 1 < 0$ contrast with in the region $\\gamma - \\lambda + 1 > 0$ .", "Especially, we note that the greater the imbalance $\\left| \\beta \\right|^2 - \\left| \\alpha \\right|^2$ between the two modes in the initial state, the clearer the difference can be observed.", "Figure: Time evolution of the entanglement entropy for different initial states ψα 2 ,β 2 \\left| {\\psi \\left( {\\left| \\alpha \\right|^2 ,\\left| \\beta \\right|^2 } \\right)} \\right\\rangle and microscopic parameters λ,γ\\left\\lbrace {\\lambda ,\\gamma } \\right\\rbrace .", "Form top to bottom the GP coefficients used are α 2 ,β 2 =0.5,0.5\\left\\lbrace {\\left| \\alpha \\right|^2 ,\\left| \\beta \\right|^2 } \\right\\rbrace = \\left\\lbrace {0.5,0.5} \\right\\rbrace , α 2 ,β 2 =0.25,0.75\\left\\lbrace {\\left| \\alpha \\right|^2 ,\\left| \\beta \\right|^2 } \\right\\rbrace = \\left\\lbrace {0.25,0.75} \\right\\rbrace and α 2 ,β 2 =0,1\\left\\lbrace {\\left| \\alpha \\right|^2 ,\\left| \\beta \\right|^2 } \\right\\rbrace = \\left\\lbrace {0,1} \\right\\rbrace .", "From left to right the microscopic parameters used are γ,λ=-2,0\\left\\lbrace {\\gamma ,\\lambda } \\right\\rbrace = \\left\\lbrace { - 2,0} \\right\\rbrace and γ,λ=-0.5,0\\left\\lbrace {\\gamma ,\\lambda } \\right\\rbrace = \\left\\lbrace { - 0.5,0} \\right\\rbrace , respectively.", "Here, we set N=20N=20 and u aa =u bb /4=0.25u_{aa} = u_{bb}/4 = 0.25.To understand the physical reason for the above phenomenon, we also need rewrite the general state given in (19) in terms of the eigenstates of the system $\\left| {\\psi \\left( t \\right)} \\right\\rangle = \\sum \\limits _{n = 0}^M {c(n,t)} \\left| {\\psi _n } \\right\\rangle $ , where $\\left| {c(n,t)} \\right|^2 =\\left| {c(n,0)} \\right|^2 = \\left| {\\left\\langle {\\psi _n } \\right|\\exp ( - iE_n t)\\left| {\\psi \\left( 0 \\right)} \\right\\rangle } \\right|^2$ can be explained as the transition probability of the system from the initial state $\\left| {\\psi \\left( 0 \\right)} \\right\\rangle $ to the corresponding energy eigenstates $\\left| {\\psi _n } \\right\\rangle $ at any time $t$ .", "We have already known that for the ground state across the phase transition line $\\gamma - \\lambda + 1 = 0$ the entanglement entropy exhibits a sudden decrease.", "This is why in the region $\\gamma - \\lambda + 1 < 0$ the amplitude of the entanglement entropy becomes smaller.", "In addition, in this phase transition region we also investigative the dependence relation between the ground-state transition probability $\\left| {c\\left( {0,t} \\right)} \\right|^2$ and the imbalance $\\left| \\beta \\right|^2 - \\left| \\alpha \\right|^2$ of the initial state.", "The result is shown in Fig.", "5.", "From Fig.", "5, it is obvious that with the increasing of the initial-state imbalance the ground-state transition probability becomes greater.", "Thus the greater imbalance between the two modes in the initial state can lead the clearer difference of the amplitude for different region.", "Figure: Variation of the ground-state transition probability c0,t 2 \\left| {c\\left( {0,t} \\right)} \\right|^2 with respect to the initial-state imbalance β 2 -α 2 \\left| \\beta \\right|^2 - \\left| \\alpha \\right|^2.", "Here, we set N=20N=20 and u aa =u bb /4=0.25u_{aa} = u_{bb}/4 = 0.25." ], [ "Conclusion", "In this work, we have proposed a 2D model consisting of photons and PPs.", "In the model, the mixed gas of photons and PPs is formally equivalent to a 2D system of massive bosons with non-vanishing chemical potential, which implies the existence of two possible condensate phase.", "Based on the GP state and using the variational method, we have also discussed the quantum phase transition of the mixed gas and have obtained the critical coupling line analytically.", "Especially, we have found that the phase transition of the photon gas can be interpreted as second harmonic generation.", "Moreover, by investigating the entanglement entropy in the ground state and general state, we have illustrated how the entanglement between photons and PPs can be associated with the phase transition of the system." ] ]

[ [ "Effects of Radiation on Primordial Non-Gaussianity" ], [ "Abstract We study the non-Gaussian features in single-field slow-roll inflationary scenario where inflation is preceded by a radiation era.", "In such a scenario both bispectrum and trispectrum non-Gaussianities are enhanced.", "Interestingly, the trispectrum in this scenario does not depend up on the slow-roll parameters and thus $\\tau_{NL}$ is larger than $f_{NL}$ which can be a signature of such a pre-inflationary radiation era." ], [ "Introduction", "Study of non-Gaussian features in primordial perturbations generated during inflation has become a subject of great importance as the precise determination of these primordial non-Gaussianities can quantify the dynamics of the early universe [1].", "In generic single-field slow-roll inflationary scenario the preferred initial vacuum chosen for the inflaton perturbations is the Bunch-Davies vacuum.", "It is shown in [2], that if inflation is preceded by a radiation era then the inflaton fluctuations will have an initial thermal distribution where the initial vacuum will depart from the standard Bunch-Davies one.", "The presence of pre-inflationary radiation era enhances the power spectrum of scalar modes by an extra temperature depended factor $\\coth (k/2T)$ .", "The enhanced power spectrum is in accordance with the observations if the comoving temperature $T$ of the primordial perturbations is less than $10^{-3}$ Mpc$^{-1}$ [2].", "In this talk we will show that presence of pre-inflationary radiation era not only enhances the power spectrum but also generates large bispectrum and trispectrum and these non-Gaussianities will carry signatures of such pre-inflationary radiation era [3] which will be discussed in detail." ], [ "Bispectrum and Trispectrum in single field slow-roll inflation", "The derivations shown in this talk are done in spatially flat gauge.", "This gauge is preferred in the derivations as in this gauge the comoving curvature perturbation ${\\mathcal {R}}(t,{\\mathbf {x}})$ is proportional to the inflaton fluctuations $\\delta \\phi (t,{\\mathbf {x}})$ as ${\\mathcal {R}}(t,{\\mathbf {x}})=\\frac{H}{\\dot{\\phi }}\\delta \\phi (t,{\\mathbf {x}}),$ where $H$ is the Hubble parameter and the overdot represents derivative w.r.t.", "cosmic time $t$ .", "Thus, in this gauge, the comoving curvature power spectrum, i.e.", "the two-point correlation function of ${\\mathcal {R}}(t,{\\mathbf {k}})$ in Fourier space, is directly related to inflaton's power spectrum as ${\\cal P}_{\\cal R}(k)=\\frac{k^3}{2\\pi ^2}\\langle {\\mathcal {R}}(k){\\mathcal {R}}(k)\\rangle \\longleftrightarrow \\left(\\frac{H}{\\dot{\\phi }}\\right)^2\\langle \\delta \\phi (k)\\delta \\phi (k)\\rangle .$ The comoving curvature power spectrum is measured through observations of the $TT$ anisotropy spectrum of CMBR which is nearly scale-invariant.", "Bispectrum, the non-vanishing three-point correlation function of primordial fluctuations, is the lowest order departure from Gaussianity of those primordial perturbations.", "The non-Gaussianity arising from bispectrum is quantified by a non-linear parameter, $f_{NL}$ , which is constrained by several experiments as : (i) the WMAP 5yr data yields $-151 < f_{NL}^{eq}<253$ ($95 \\%$ CL) [4], (ii) the PLANCK mission is sensitive to probe bispectrum upto $f_{NL} \\sim 5$ [5] and (iii) in future experiments if the primordial non-Gaussianities imprinted in 21 cm background is measured then $f_{NL}< 0.1$ can be probed [6], [7].", "In a free theory, as the primordial perturbations are Gaussian in nature, the three-point correlation function vanishes yielding no non-Gaussianity.", "It is shown in [8] that self-interactions of inflaton field of the kind $V(\\phi )=\\lambda \\phi ^3$ generates non-vanishing bispectrum proportional to $\\lambda /H$ but the non-Gaussianity is too small $(\\sim \\mathcal {O}(10^{-7}))$ to be probed by any existing or future experiments.", "On the other hand, in a generic single-field slow-roll inflationary model non-linearities in the evolution of primordial perturbations ${\\mathcal {R}}(t,{\\mathbf {k}})$ can also generate primordial non-Gaussianities in CMBR.", "In the non-linear limit one can write ${\\mathcal {R}}_{NL}(t,{\\mathbf {x}})=\\frac{H}{\\dot{\\phi }}\\delta \\phi _L(t,{\\mathbf {x}})+\\frac{1}{2}\\frac{\\partial }{\\partial \\phi }\\left(\\frac{H}{\\dot{\\phi }}\\right)\\delta \\phi _L^2(t,{\\mathbf {x}})+{\\mathcal {O}}(\\delta \\phi _L^3),$ which yields a non-vanishing three-point correlation function of ${\\mathcal {R}}_{NL}$ in terms of four-point correlation function of inflaton perturbations $\\delta \\phi _L$ as $\\langle {\\mathcal {R}}_{NL}{\\mathcal {R}}_{NL}{\\mathcal {R}}_{NL}\\rangle \\simeq \\left(\\frac{H}{\\dot{\\phi }}\\right)^2\\frac{1}{2}\\frac{\\partial }{\\partial \\phi }\\left(\\frac{H}{\\dot{\\phi }}\\right)\\langle \\delta \\phi _L\\delta \\phi _L\\delta \\phi _L^2\\rangle ,$ even when the initial perturbations $\\delta \\phi _L$ are Gaussian in nature.", "The four-point correlation function on the R.H.S.", "can be written in terms of product of two two-point correlation functions and defining $f_{NL}$ as $\\left\\langle {\\mathcal {R}}({\\mathbf {k}}_1){\\mathcal {R}}({\\mathbf {k}}_2){\\mathcal {R}}({\\mathbf {k}}_3)\\right\\rangle =(2\\pi )^{-\\frac{3}{2}}\\delta ^3({\\mathbf {k}_1}+{\\mathbf {k}_2}+{\\mathbf {k}_3})\\frac{6}{5} f_{NL}\\left(\\frac{P_{\\cal R}(k_1)}{k_1^3}\\frac{P_{\\cal R}(k_2)}{k_2^3}+2\\,\\,{\\rm perms.", "}\\right),$ one can show that the non-linear parameter is of the order of slow-roll parameters $f_{NL}=\\frac{5}{6}(\\delta -\\epsilon )$ [9], which is also too small $\\left(\\mathcal {O}(10^{-2})\\right)$ to be detected by any present or forthcoming experiments.", "The delta-function in the above equation ensures that the three momenta form a triangle and $f_{NL}$ is determined in several such triangle configurations, some of them which we will consider in this talk are : (i) Squeezed configuration $(|{\\mathbf {k}}_1|\\approx |{\\mathbf {k}}_2|\\approx k\\gg |{\\mathbf {k}}_3|)$ , (ii) Equilateral configuration $(|{\\mathbf {k}}_1|=|{\\mathbf {k}}_2|=|{\\mathbf {k}}_3|=k)$ and (iii) Folded configuration $(|{\\mathbf {k}}_1|=|{\\mathbf {k}}_3|=\\frac{1}{2}|{\\mathbf {k}}_2|=k)$ .", "The connected part of four-point correlation function of primordial fluctuations is called the trispectrum $\\left\\langle {\\mathcal {R}}({\\mathbf {k}}_1){\\mathcal {R}}({\\mathbf {k}}_2){\\mathcal {R}}({\\mathbf {k}}_3){\\mathcal {R}}({\\mathbf {k}}_4)\\right\\rangle _c&\\equiv &\\left\\langle {\\mathcal {R}}({\\mathbf {k}}_1){\\mathcal {R}}({\\mathbf {k}}_2){\\mathcal {R}}({\\mathbf {k}}_3){\\mathcal {R}}({\\mathbf {k}}_4)\\right\\rangle -\\left(\\left\\langle {\\mathcal {R}}_{L}({\\mathbf {k}}_1){\\mathcal {R}}_{L}({\\mathbf {k}}_2)\\right\\rangle \\left\\langle {\\mathcal {R}}_{L}({\\mathbf {k}}_3){\\mathcal {R}}_{L}({\\mathbf {k}}_4)\\right\\rangle \\right.\\nonumber \\\\&&+2\\,\\,{\\rm perm}\\left.\\right).$ The non-linear parameter $\\tau _{NL}$ quantifies the non-Gaussianity arising from trispectrum and it is constrained by observations as (i) WMAP constraints trispectrum as $\\left|\\tau _{NL}\\right|<10^8$ [10], (ii) PLANCK is expected to reach the sensitivity upto $\\left|\\tau _{NL}\\right|\\sim 560$ [11] and (iii) future 21cm experiments can probe trispectrum up to the level $\\tau _{NL} \\sim 10$ [7].", "A free-scalar theory yields vanishing trispectrum like vanishing bispectrum.", "But non-linear evolution of $\\mathcal {R}$ as given in Eq.", "(REF ) yields a trispectrum where $\\tau _{NL}=\\left(\\frac{6}{5} f_{NL}\\right)^2$ [9], which being proportional to the square of slow-roll parameters is too small to be detected by any present or future experiments.", "It is to be noted that the generic single-field slow-roll inflation predicts a trispectrum which smaller than the bispectrum by orders of magnitude." ], [ "Inflation with prior radiation era and enhanced non-Gaussianity", "If inflation is preceded by a radiation era then the inflaton field will have an initial thermal distribution where the thermal vacuum $|\\Omega \\rangle $ will have finite occupation as $N_k|\\Omega \\rangle =n_k|\\Omega \\rangle $ (the number operator $N_k\\equiv a^\\dagger _{\\mathbf {k}}a_{\\mathbf {k}}$ ).", "Also there will be a probability of the system to be in an energy state $\\varepsilon _k\\equiv n_k k$ as $p(\\varepsilon _k)\\equiv \\frac{e^{-\\beta n_k k}}{\\sum _{n_k}e^{-\\beta n_kk}}=\\frac{e^{-\\beta n_k k}}{z}.$ Due to this probability distribution the correlation functions have to be thermal averaged.", "Taking into account the initial thermal distributions of the primordial fluctuations and the probability distribution due to pre-inflationary radiation era the thermal averaged inflaton's power spectrum will have an enhancement factor $1+2f_B(k)$ where $f_B(k)$ is the distribution function of primordial perturbations.", "For inflaton (scalar) perturbations the distribution function will be Bose-Einstein distribution function $\\left(f_B(k)\\equiv \\frac{1}{e^{\\beta k}-1}\\right)$ and thus the enhancement factor will be $1+2f_B(k)=\\coth (\\beta k/2)$ where $\\beta \\equiv \\frac{1}{T}$ [2], [3].", "This enhanced power spectrum is in accordance with the observations when $T<10^{-3}$ Mpc$^{-1}$ [2].", "As the two-point correlation function is thermal averaged due to the effects of pre-inflationary radiation era, the other higher-point correlation functions have also to be thermal averaged in a similar way.", "For three-point correlation function this will be $\\langle {\\mathcal {R}}_{NL}{\\mathcal {R}}_{NL}{\\mathcal {R}}_{NL}\\rangle _{\\beta }\\simeq \\left(\\frac{H}{\\dot{\\phi }}\\right)^2\\frac{1}{2}\\frac{\\partial }{\\partial \\phi }\\left(\\frac{H}{\\dot{\\phi }}\\right)\\langle \\delta \\phi \\delta \\phi \\delta \\phi ^2\\rangle _{\\beta },$ but now the probability of occupancy of a state with four energies $\\epsilon _r$ will be $p(k_1,k_2, k_3,k_4)\\equiv \\frac{\\prod _re^{-\\beta n_{k_r}k_r}}{\\prod _r\\sum _{n_k}e^{-\\beta n_{k_r}k_r}}.$ Due to thermal averaging $f_{NL}$ is enhanced.", "We compute the non-Gaussianity in this scenario arising from bispectrum in different triangle configurations : (i) Squeezed configuration : in this configuration the enhanced non-Gaussianity is $f_{NL}^{\\rm th}=f_{NL}\\times 2\\left(1+3.72\\coth \\left(\\frac{\\beta k}{2}\\right)\\right)$ where $f_{NL}$ is enhanced by a factor of 64.82, (ii) Equilateral configuration : in this configuration the enhanced non-Gaussianity is $f_{NL}^{\\rm th}=f_{NL}\\times \\left(3+\\frac{5}{4\\sinh ^2\\left(\\frac{\\beta k}{2}\\right)}\\right)$ where $f_{NL}$ is enhanced by a factor of 90.85, (iii) Folded configuration : in this configuration the enhanced non-Gaussianity is $f_{NL}^{\\rm th}=f_{NL}\\times \\left(3+\\frac{1}{\\sinh ^2\\left(\\frac{\\beta k}{2}\\right)}\\right)$ where $f_{NL}$ is enhanced by a factor of 73.28.", "It can be seen that $f_{NL}$ , which is enhanced due to effects of pre-inflationary radiation era, is within the sensitivity of future 21-cm experiments where primordial non-Gaussianities can be detected [6], [7].", "It is also to be noted that the maximum non-Gaussianity can arise in the Equilateral configuration when inflation is preceded by a radiation era.", "In a later work [12] similar analysis is done when perturbations are already present in the initial vacuum.", "It is found in [12] too that initial presence of quanta in the vacuum can significantly enhance non-Gaussianity arising from bispectrum which is in agreement with the results presented here and in [3].", "It is very interesting to note at this point that due to thermal averaging the four-point correlation function is not equal to the product of two two-point correlation functions which can yield a non-vanishing connected part as $\\left\\langle {\\mathcal {R}}({\\mathbf {k}}_1){\\mathcal {R}}({\\mathbf {k}}_2){\\mathcal {R}}({\\mathbf {k}}_3){\\mathcal {R}}({\\mathbf {k}}_4)\\right\\rangle _c&\\ne &\\left\\langle {\\mathcal {R}}({\\mathbf {k}}_1){\\mathcal {R}}({\\mathbf {k}}_2){\\mathcal {R}}({\\mathbf {k}}_3){\\mathcal {R}}({\\mathbf {k}}_4)\\right\\rangle _\\beta \\nonumber \\\\&-&\\left(\\left\\langle {\\mathcal {R}}_{L}({\\mathbf {k}}_1){\\mathcal {R}}_{L}({\\mathbf {k}}_2)\\right\\rangle _\\beta \\left\\langle {\\mathcal {R}}_{L}({\\mathbf {k}}_3){\\mathcal {R}}_{L}({\\mathbf {k}}_4)\\right\\rangle _\\beta +2{\\rm perm}\\right).$ Thus defining the trispectrum in such a situation as $\\left\\langle {\\mathcal {R}}({\\mathbf {k}}_1){\\mathcal {R}}({\\mathbf {k}}_2){\\mathcal {R}}({\\mathbf {k}}_3){\\mathcal {R}}({\\mathbf {k}}_4)\\right\\rangle _c=\\tau _{NL}\\left[\\frac{P_{\\mathcal {R}}(k_1)}{k_1^3}\\frac{P_{\\mathcal {R}}(k_2)}{k_2^3}\\delta ^3({\\mathbf {k}_1}+{\\mathbf {k}_3})\\delta ^3({\\mathbf {k}_2}+{\\mathbf {k}_4})+2\\,\\,{\\rm perm.", "}\\right],$ we see that, as the linear perturbations can generate non-vanishing connected part due to thermal averaging, the non-linear parameter $|\\tau _{NL}|$ will not depend up on slow-roll parameters and can be as large as 42.58 [3] which is within the detection range of future 21-cm background anisotropy experiments [7].", "Hence, we see that the presence of pre-inflationary radiation era yields larger trispectrum non-Gaussianity than bispectrum." ], [ "Conclusion", "The talk was focused on non-Gaussian features in a single-field slow-roll inflationary model where inflation is preceded by a radiation era.", "In a generic single-field slow-roll model of super-cool inflation non-linear evolution of primordial fluctuations generate bispectrum non-Gaussianity which is proportional to the slow-roll parameters [9] and thus too small to be detected by any present or future experiments.", "Non-linear evolution of primordial fluctuations also generates trispectrum non-Gaussianity where $\\tau _{NL}$ is proportional to the square of slow-roll parameters [9].", "Thus, this generic inflationary scenario predicts trispectrum non-Gaussianity which is much smaller than the bispectrum non-Gaussianity.", "We showed that if such a generic inflationary scenario is preceded by a radiation era it can yield large bispectrum and trispectrum non-Gaussianities [3] which are within the range of detection of future 21-cm background anisotropy experiments [6], [7].", "Due to presence of pre-inflationary radiation era the initial vacuum will contain thermal fluctuations and also the energy states will have a probability distribution.", "Accordingly, the thermal averaged three-point correlation function generates large non-Gaussianity where the enhancement of $f_{NL}$ over the generic scenario is largest in the equilateral configuration.", "An interesting situation arises in the case of trispectrum as the thermal averaged four-point correlation function is not equal to the product of two thermal-averaged two-point correlation functions.", "Thus the linear primordial perturbations can generate a non-vanishing connected part of four-point correlation function due to thermal averaging and $\\tau _{NL}$ in such a case will not depend up on the slow-roll parameters.", "We compute that in such a scenario $|\\tau _{NL}|$ can be as large as 43 [3].", "Thus a significant signature of such pre-inflationary radiation era is that it yields larger trispectrum than bispectrum.", "This signature can distinguish between an inflationary scenario preceded by a radiation era and the generic scenario of single-field slow-roll super-cool inflation.", "I thank my collaborator Subhendra Mohanty." ] ]

[ [ "Agglomerative Percolation on Bipartite Networks: A Novel Type of\n Spontaneous Symmetry Breaking" ], [ "Abstract Ordinary bond percolation (OP) can be viewed as a process where clusters grow by joining them pairwise, by adding links chosen randomly one by one from a set of predefined `virtual' links.", "In contrast, in agglomerative percolation (AP) clusters grow by choosing randomly a `target cluster' and joining it with all its neighbors, as defined by the same set of virtual links.", "Previous studies showed that AP is in different universality classes from OP for several types of (virtual) networks (linear chains, trees, Erdos-Renyi networks), but most surprising were the results for 2-d lattices: While AP on the triangular lattice was found to be in the OP universality class, it behaved completely differently on the square lattice.", "In the present paper we explain this striking violation of universality by invoking bipartivity.", "While the square lattice is a bipartite graph, the triangular lattice is not.", "In conformity with this we show that AP on the honeycomb and simple cubic (3-d) lattices -- both of which are bipartite -- are also not in the OP universality classes.", "More precisely, we claim that this violation of universality is basically due to a Z_2 symmetry that is spontaneously broken at the percolation threshold.", "We also discuss AP on bipartite random networks and suitable generalizations of AP on k-partite graphs." ], [ "Introduction", "Percolation was until recently considered a mature subject that held few surprises, but this has changed dramatically during the last few years.", "Recent discoveries that widened enormously the scope of different behaviors at the percolation threshold include infinite order transitions in growing networks [1], supposedly first order transitions in Achlioptas processes [2] (that are actually continuous [3], [4] but show very unusual finite size behavior [5]), and real first order transitions in interdependent networks [6], [7], [8], [9].", "Another class of “non-classical\" percolation models, inspired by attempts to formulate a renormalization group for networks [10], [11], was introduced in [12], [13], [14], [15], [16] and is called `agglomerative percolation' (AP).", "The prototype model in the ordinary percolation (OP) universality class is bond percolation [17].", "There one starts with a set of $N$ nodes and a set of `virtual' links between them, i.e.", "links that can be placed but that are not yet put down.", "One then performs a process where one repeatedly picks at random one of the virtual bonds and realizes it, i.e.", "actually links the two nodes.", "A giant cluster appears with probability one in the limit $N\\rightarrow \\infty $ , when the density $p=M/N$ of links ($M$ is here the number of realized links) exceeds a threshold $p_c$ whose value depends on the topology of the network.", "The behavior at $p\\approx p_c$ is governed by `universal' scaling laws, i.e.", "by scaling laws with exponents that depend only on few gross properties of the network.", "A typical example is that the universality class of OP on regular $d-$ dimensional lattices depends on $d$ but not on the lattice type.", "For example, OP on triangular and square lattices (both have $d=2$ ) are in the same universality class.", "AP differs from OP in that clusters do not grow by establishing links one by one.", "Rather, one picks a `target' cluster at random (irrespective of its mass; we are dealing here with model (a) in the classification of [13]) and joins it with all its neighbors, where neighborhoods are defined by the virtual links.", "The new combined cluster is then linked to all neighbors of its constituents.", "AP can be solved rigorously on 1-d linear chains [14], [15], where it is found to be in a different universality class from OP.", "Although a similarly complete mathematical analysis is not possible on random graphs, both numerics and non-rigorous analytical arguments show that the same is true for `critical' trees [12] and Erdös-Rényi graphs [16].", "In contrast to these cases that establish AP as a novel phenomenon but do not present big surprises, the behavior on 2-d regular lattices [13] is extremely surprising: While AP on the triangular lattice is clearly in the OP universality class (with only some minor caveats), it behaves completely different on the square lattice.", "There the average cluster size at criticality diverges as the system size $L$ increases (it stays finite for all realizations of OP on any regular lattice), the fractal dimension of the incipient giant cluster is $D_f=2$ ($D_f=91/48\\approx 1.90$ for OP), and the cluster mass distribution obeys a power law with power $\\tau =2$ ($\\tau =187/91\\approx 2.055$ for OP).", "This blatant violation of universality – one of the most cherished results of renormalization group theory – is, as far as we know, unprecedented.", "As we said above, gross topological features of the network (such as dimensionality in case of regular lattices, the correlations between links induced by growing networks [1], and finite ramification in hierarchical graphs [18]) are one set of properties that determine universality classes.", "The other features that determine universality classes in general are symmetries of the order parameter: The Ising and Heisenberg models are in different universality classes, e.g., because the order parameter is a scalar in the first and a 3-d vector in the second.", "Could it be that the non-universality of AP results from a similar symmetry?", "At first sight this seems unlikely, because the order parameter (the density of the giant cluster) is a scalar in any percolation model.", "Moreover, in order for a symmetry to affect the universality class it has to be broken spontaneously at the phase transition.", "In the following we show that it is indeed the latter scenario that leads to the non-universality of AP on square and triangular lattices, and the symmetry that is spontaneously broken at the AP threshold on the square lattice is a $Z_2$ symmetry resulting from bipartivity.", "A graph is bipartite, if the set ${\\cal N}$ of nodes can be split into two disjoint subsets, ${\\cal N} = {\\cal N}_1 \\sqcup {\\cal N}_2$ , such that all links are connecting a node in ${\\cal N}_1$ with a node in ${\\cal N}_2$ , and there are no links within ${\\cal N}_1$ or within ${\\cal N}_2$ .", "A square lattice is bipartite (as illustrated by the black/white colors of a checkerboard), but a triangular lattice is not.", "Following this example, we will in the following speak of the different colors of the sets ${\\cal N}_1$ and ${\\cal N}_2$ .", "The initial state of the AP process on a square lattice (where each site is a cluster) is color symmetric.", "If the AP cluster joining process is such that we can attribute a definite color to any cluster (even when it is not a single site), then the state remains color symmetric until we reach a state with a giant cluster.", "In this state the color symmetry is obviously broken.", "In Sec.", "2 we shortly review the evidence for non-universality given in [13], In Sec.", "3 we present new results which show that AP on square lattices behaves even more strange than found before.", "There we also present numerical results for the honeycomb and simple cubic lattices, both of which are bipartite and show similar anomalies as the square lattice.", "The detailed explanation why bipartivity leads to these results is given in Sec.", "4.", "Random bipartite networks are shortly treated in Sec. 5.", "Possible generalizations to $k$ -partite graphs with $k>2$ are discussed in Sec.", "6, while we finish with our conclusions in Sec.", "7." ], [ "Agglomerative Percolation: Definition, implementation, and review of previous results", "We start with a graph with $N$ nodes and $M$ links.", "Clusters are defined trivially in this initial state, i.e.", "each node is its own cluster.", "AP is then defined by repeating the following step until one single cluster is left: 1) Pick randomly one of the clusters with uniform probability; 2) Join this `target cluster' with all its neighboring clusters, where two clusters $C_1$ and $C_2$ are `neighbors', if there exist a pair of nodes $i\\in C_1$ and $j\\in C_2$ that are joined by a link; As described in [13], this is implemented most efficiently with the Newman-Ziff algorithm [19] that uses pointers to point to the “roots\" of clusters, augmented by a breadth-first search to find all neighbors of the target.", "In ordinary bond percolation one usually takes as control parameter $p$ the number of established (i.e.", "non-virtual) links, divided by the number of all possible links (including virtual ones).", "This is not practical for AP.", "Rather, we use as in [20], [13] the number $n$ of clusters per node.", "It was checked carefully in these papers that using $n$ instead of $p$ as a control parameter in ordinary bond percolation is perfectly legitimate, since one is a smooth monotonic (decreasing) function of the other.", "In [13], AP was studied on two different 2-d lattices.", "Helical boundary conditions were used for both, i.e.", "sites are labeled by a single index $i$ with $i \\equiv (i \\;\\;{\\rm mod}\\;\\; L^2$ , where $L$ is the lattice size).", "For the square lattice the four neighbors of site $i$ are $i\\pm 1$ and $i\\pm L$ , while there are two additional neighbors $i\\pm (L+1)$ for the triangular lattice.", "This seemingly minor difference has dramatic consequences.", "While AP on the triangular lattice is (within statistical errors, and with one minor caveat that was easily understood) in the universality class of OP, this is obviously not the case for the square lattice.", "Among other results, the following results were found: Figure: (Color online) Effective critical cluster densities for AP on finite 2-d lattices versusL -3/4 L^{-3/4}, where LL is the lattice size.", "For ordinary percolation, where n c (L)-n c ∼L -1/ν n_c(L) -n_c\\sim L^{-1/\\nu } with ν=4/3\\nu =4/3, this should give straight lines.", "For each lattice type (triangular:upper pair of curves; square: lower pair of curves) we show results obtained with two differentoperational definitions for the critical point: (i) Maximal range of the power lawP n (m)∼m -τ P_n(m) \\sim m^{-\\tau }, and (ii) the probability to have a cluster that wraps around a latticewith helical boundary conditions is equal to 1/2.", "The corresponding values of n c (L)n_c(L) are calledn c,τ (L)n_{c,\\tau }(L) and n c, wrap (L)n_{c,{\\rm wrap}}(L) (from Ref. ).", "The effective percolation threshold – measured either by the probability that a cluster wraps around the lattice, or via the best scaling law $P(m)\\sim m^{-\\tau } $ for the probability distribution of cluster masses $m$ – depends strongly on $L$ .", "For OP this dependence is governed by the correlation length exponent $\\nu $ via $p_c - p_c(L) \\propto n_c(L) - n_c \\propto L^{-1/\\nu }$ with $\\nu = 4/3$ and $n_c >0$ .", "The latter means, in particular, that the average cluster size is finite at criticality.", "For AP on the square lattice a parameterization like this would give $\\nu = 0$ .", "More precisely, $n_c(L)$ seems to decrease logarithmically to a value $n_c=0$ , i.e.", "the average cluster at criticality (and in the limit $L\\rightarrow \\infty $ ) is zero.", "This is summarized in Fig.", "REF .", "The exponent $\\tau $ in Eq.", "(REF ), which is $187/91 = 2.0549\\ldots $ for OP, seems to be $<2$ at first sight.", "But it slowly increases with $L$ , and it was argued that the exact value is $\\tau =2$ .", "Similarly, the fractal dimension of the largest cluster was measured as $\\approx 1.95$ , while it is $D_f= 91/48 = 1.9858 \\ldots $ for OP.", "It also increases slowly with $L$ , and it was conjectured that also $D_f = 2$ .", "Let $p_{\\rm wrap}(n,L)$ be the probability that there exists a cluster that wraps along the vertical direction on a lattice of size $L$ , when there are $nL^2$ clusters.", "The distribution $dp_{\\rm wrap}(n) /dn$ is universal for OP.", "For AP on triangular lattices it develops a weak tail for small $n$ (this is the easily explained caveat mentioned above), but for AP on the square lattice it has a very fat tail for small $n$ .", "Thus there is a high probability that even at very late stages in the agglomeration process, when only few clusters remain, none of them has yet wrapped." ], [ "Periodic boundary conditions", "Helical b.c.", "were used in [13] simply for convenience (they are more easy to code than periodic ones), and it was assumed that the small difference with strictly periodic b.c.", "should be without any consequences.", "This is not true.", "Not only is there a large difference between helical and strictly periodic b.c.", "'s (even for the largest values of $L$ that we could check), but for the latter there is an even stronger difference between even and odd $L$ .", "Figure: (Color online) Wrapping probability density dp wrap /dndp_{\\mathrm {wrap}}/dnfor 2D square lattices with different boundary condition, and with sizes differing by justone unit, compared to similar results for triangular lattices.", "All curves for square latticeshave peaks at smaller values than for triangular lattices and have more heavyleft hand tails, but this is most pronounced for strictly periodic b.c.", "with even LL.Figure: (Color online) Black triangles: Cluster densities nn at whichp wrap (n)=1/2p_{\\rm wrap}(n)=1/2, plotted against LL, for square lattices with periodic b.c.The upper curve is for odd sizes, and the lower curve is for even sizes.", "Red dots: Analogousresults for a slightly modified model, where each neighboring cluster agglomerates with thetarget cluster only with probability q<1q<1.", "In the present case q=0.999q=0.999.In Fig.", "REF we show $dp_{\\rm wrap}(n) /dn$ for four different cases: (i) Triangular lattices.", "The shape of this curve is practically indistinguishable from ordinary percolation, and serves as a reference for the latter.", "(ii) Square lattices of size $128\\times 128$ with helical b.c.", "Compared to the triangular lattice, there is a much fatter left hand tail of the distribution, i.e.", "there are many more realizations where no cluster has yet wrapped, although the number of clusters is very small.", "(iii) Square lattices of the same size, but with periodic b.c.", "Now the left hand tail is even more fat.", "Indeed, for this lattice size one finds realizations with $\\le 10$ clusters, none of which has yet wrapped.", "(iv) Square lattices of sizes $127\\times 127$ with periodic b.c.", "We see a huge difference, in spite of the small change in $L$ , making the results similar to those for helical b.c.", "Obviously, this indicates a distinction between even and odd $L$ .", "The last conclusion is confirmed by Fig.", "REF .", "There we show the values of $n$ where half of the configurations have wrapping clusters, $p_{\\rm wrap}(n)=1/2$ (black triangles; the red dots will be discussed in subsection REF ).", "These data confirm that the difference between even and odd $L$ persists even to our largest systems, where the difference in $L$ between the two is less than 0.025 per cent." ], [ "Finite agglomeration probability", "In AP, all neighbors of a chosen target are included in each agglomeration step.", "In contrast, bond percolation can be viewed as the limit $q\\rightarrow 0$ of a model where each neighbor is included with probability $q$ .", "One might then wonder where the cross-over from OP to AP happens.", "Is it for $q\\rightarrow 0$ (meaning that the model is in the AP universality class for any $q>0$ ), for $q\\rightarrow 1$ (in which case we have OP for any $q<1$ ), or for some $0<q<1$ ?", "The numerical answer is clear and surprising in its radicalness: For any $q<1$ we find OP, if we go to large enough $L$ , even if $q=0.999$ (see Fig.", "REF ).", "More precisely, we see that the difference between even and odd $L$ disappears rapidly when $L$ increases, and both curves seem to converge to a finite $n_{c,\\rm wrap}$ for $L\\rightarrow \\infty $ .", "It seems that even the slightest mistake in the agglomeration process completely destroys the phenomenon and places the model in the OP universality class.", "This is confirmed by looking at the order parameter $S = \\langle m_{\\rm max}\\rangle / N$ where $m_{\\rm max}$ is the size of the largest cluster.", "For infinite systems, $S=0$ for $n>n_c$ and $S>0$ for $n<n_c$ .", "For OP, one has $S \\sim (n_c-n)^\\beta $ for $n$ slightly below $n_c$ , and the usual finite size scaling (FSS) behavior $S \\sim L^{\\beta /\\nu } f[(n_c-n) L^{1/\\nu }] $ for finite systems.", "In Fig.", "REF we show $s$ versus $n$ for various cases, all with periodic b.c.", "In addition to two panels for other lattices discussed in later subsections (“honey\", “cubic\") we show results for the triangular lattice and for square lattices with $q=1$ (“square\") and with $q=0.999$ .", "We see that the results for $q<1$ are very similar to those for the triangular lattice, while they are completely different from those for $q=1$ .", "A data collapse for verifying the FSS ansatz would of course not be perfect, but it seems that the model with $q<1$ is in the OP universality class, for any $q<1$ .", "Figure: (Color online) The five panels of this figure correspond to(i) square lattices with q=1q=1, (ii) honeycomb lattices, (iii) triangular lattices, (iv)square lattices with q=0.999q=0.999, and (v) cubic lattices.", "In all cases, periodic b.c.", "wereused.", "In all cases except case (iv), q=1q=1.In addition to the square and triangular lattices we now study also the honeycomb lattice as a third lattice with $d=2$ , and the simple cubic lattice as an example of a 3-d lattice." ], [ "Honeycomb lattice", "Although we measured also other observables (such as the wrapping probabilities), we show here only the behavior of the order parameter $S$ .", "As seen in Fig.", "REF , the behavior here is very similar to that for the square lattice.", "In particular, we see no indication for the FSS ansatz with finite (non-zero) $n$ ." ], [ "3-d simple cubic lattice", "The behavior on the simple cubic lattice is more subtle.", "On the one hand, we clearly see in Fig.", "REF an indication for a non-zero value of $n_c$ , with $n_c \\approx 0.41$ .", "On the other hand, as for the square and honeycomb lattices we see that the slope $dS/dn$ is not monotonic.", "In all three cases the growth of the largest cluster slows down when $S \\approx 1/4$ , and accelerates again when $S>1/2$ .", "Alternatively, it seems as if two behaviors are superimposed: For large $n$ and $S < 1/3$ it seems as if the curves would extrapolate to $S=1/2$ for $n\\rightarrow 0$ , but then (as $n$ decreases further) $S$ rises again sharply, to reach $S=1$ .", "Although this scenario is too simplistic, we will see in the next section that it catches some of the relevant physics.", "Using the critical exponents for 3-dimensional OP, $\\beta = 0.4170(3), D_f = 2.5226(1),$ and $ \\nu = 0.8734(5)$ [21], one obtains an acceptable data collapse when plotting $m_{\\rm max}/L^{D_f}$ against $(n-n_c)L^{1/\\nu }$ , with $n_c = 0.411$ .", "But the behavior for large $L$ is not given by $S\\sim (n-n_c)^\\beta $ with the value of $\\beta $ given above, see Fig.", "REF .", "The latter plot is much improved, if we use instead $\\beta =0.435,\\;\\; D_f = 2.522,\\;\\; \\nu = 0.91, $ together with $n_c = 0.4110$ (see Fig.", "REF ).", "With these exponents we also obtain a good collapse of $m_{\\rm max}/L^{D_f}$ against $(n-n_c)L^{1/\\nu }$ , see Fig.", "REF .", "The main deviation from a perfect collapse in this plot is due to the smallest lattice, and is obviously a finite-size correction to the FSS ansatz.", "We do not quote error bars for the values in Eq.", "(REF ), as they are not yet our final estimates.", "Figure: (Color online) Plot of S/(n c -n) β S/(n_c-n)^\\beta against (n c -n)L 1/ν (n_c-n)L^{1/\\nu },using the exponents of ordinary 3-d percolation.", "According to the FSS ansatz, these curves shouldcollapse and should be horizontal in the limit where we first take L→∞L\\rightarrow \\infty and then n→n c n\\rightarrow n_c.This seems not to be the case.Figure: (Color online) Analogous to the previous plot, but using β=0.435,D f =2.5220\\beta =0.435,D_f = 2.5220, and ν=0.91\\nu = 0.91.Figure: (Color online) Plot of SL β/ν =m max /L D f S L^{\\beta /\\nu } = m_{\\rm max}/L^{D_f} versus(n-n c )L 1/ν (n-n_c)L^{1/\\nu } for AP on the simple cubic lattice, using n c =0.411n_c = 0.411 and the exponents given inEq.", "()." ], [ "Uniqueness of cluster colors", "Our first observation is that infinite square, honeycomb and cubic lattices are bipartite, while the triangular lattice is not.", "The next observation is that finite square lattices of size $L\\times L$ are still bipartite, if $L$ is even and periodic boundary conditions are used, but global bipartivity is lost when either $L$ is odd or helical b.c.", "are used.", "These observations strongly suggest that it is indeed bipartivity that is responsible for peculiarities of AP on these lattices.", "Figure: (Color online) Part of a square lattice with cluster boundariesindicated by black lines.", "Plaquettes correspond to nodes of the graph.", "Nine clusters are labeled withletters A-IA-I.", "Six of them (A,B,D-FA,B,D-F) are single nodes, one (CC) has five nodes, and two (H,IH,I) are very large.Three of them (D,H,ID,H,I) are blue, the other six are white.", "If cluster DD is chosen as target, itmerges with C,E,FC,E,F, and GG and the new cluster is white.", "If, however, FF is chosen as target, the new bluecluster would consist of D,HD, H, and II.In a bipartite graph, to each node can be assigned one of two colors.", "We now show that this is extended from single nodes to arbitrarily large clusters, if the rules of AP are strictly followed.", "Before we do this, we need two definitions: Definition: The surface of a cluster $C$ is the set of all nodes in $C$ that have at least one link to a node not contained in $C$ .", "Definition: If all surface nodes in $C$ have the same color, then we say that $C$ also has this color.", "Otherwise, the color of $C$ is not defined.", "We can now prove the following Theorem: (i) If clusters are grown according to the AP rules on a bipartite network, they always have a well defined color.", "(ii) All neighbors of a given cluster have the opposite color.", "(iii) If a target of color $c$ is chosen for agglomerating all its neighbors, the new cluster has the opposite color ${\\bar{c}}$ .", "For an illustration see Fig.", "REF .", "Proof: The proof follows by induction.", "First, the theorem is obviously true for the starting configuration, where all clusters are single nodes.", "Then, let us assume it is true for all agglomeration steps up to (and including) step $t$ .", "Let us call $c$ the color of the target cluster at step $t+1$ , and ${\\bar{c}}$ the opposite color.", "Then all neighbors of the target have color ${\\bar{c}}$ , so that after joining them the new cluster also has color ${\\bar{c}}$ , proving thereby (i) and (ii).", "On the other hand, all neighbors of the neighbors had color $c$ , and these form the neighbors of the new cluster, which proves (iii).", "$\\blacksquare $ Notice that it was crucial for the proof that all neighbors of the target were joined, so that none of the neighbors of the target is a neighbor of the new cluster.", "This shows why imperfect agglomeration as considered in Sec.", "REF leads to situations where the theorem does not hold." ], [ "Coexistence of large clusters", "A typical configuration on a $128\\times 128$ lattice with three large clusters, none of which has yet wrapped vertically, is shown in Fig.", "REF .", "Such a configuration would have an astronomically small probability in OP, since in OP the chance is very small to have more than one large cluster.", "If there were two large clusters at any time, they would immediately merge with very high probability.", "Obviously, in AP there exists a mechanism that prevents clusters of opposite color to merge fast, leading to the coexistence of large clusters of opposite colors.", "Figure: (Color online) A configurationwith three large clusters at n≈0.03n\\approx 0.03 and L=128L=128.", "For clusters of mass >1>1 the two colors arered and blue, with the larger clusters more bright and the smaller ones more dark.", "“Blue\" singletonsare colored white for better visibility and in order to distinguish them from “red\" singletons whichactually are indicated by black squares.", "Notice that no two clusters of the same color ever touch inthis figure.", "Wherever they seem to touch, there is indeed a small cluster of the opposite colorintervening.", "In spite of the size of the largest clusters, none of them has yet wrapped in the verticaldirection.This looks at first paradoxical.", "Take the two largest clusters in Fig.", "REF .", "If either of them were chosen as target, they would merge immediately.", "Why should this not happen?", "The crucial point is that each cluster is chosen as target with the same probability, and there are many more small clusters than large ones.", "The chances are thus overwhelming that neither of the large clusters is chosen as target, but a small cluster is picked instead.", "But in that case the two largest clusters cannot merge, because they have opposite color and all neighbors to be merged must have the same color.", "Thus its is most likely that a random agglomeration step merges one small cluster of color $c$ with several (small and large) clusters of color $\\bar{c}$ .", "Figure: (Color online) Cluster mass distributions for simple cubic latticeswith L=32L=32.", "The curve for n=0.4323n=0.4323 is subcritical, while the other curves are supercritical.", "Incontrast to the case of OP, where the mass distribution develops a single peak in the supercriticalphase, now (i.e.", "for AP) we see two peaks.", "They correspond to clusters of opposite colors.Figure: Average size of the second largest cluster on simple cubic latticeswith even size.", "In contrast to OP, where m max ,2 m_{\\rm max,2} peaks near the percolation transitionand decreases fast when one goes into the supercritical phase, here the second largest clustercontinues to grow far beyond the percolation transition n c ≈0.411n_{c}\\approx 0.411.In two dimensions this means also that the two large clusters of opposite color prevent each other from wrapping.", "In three dimensions this is not the case.", "Thus AP is in three dimensions more similar to OP, although it still should show several large clusters in the critical and supercritical regimes.", "To test this prediction we show two figures.", "Fig.", "REF shows that mass distributions in the supercritical phase have two peaks (in contrast to OP), corresponding to the fact that AP on bipartite graphs has two giant clusters of opposite colors.", "The same conclusion is drawn from Fig.", "REF , where we show the average normalized size $m_{\\rm max,2}$ of the second largest cluster as a function of $n$ .", "We see that $m_{\\rm max,2}$ starts to increase at $n_c$ and continues to grow as one goes deeper into the supercritical phase, while it would peak at $n_c$ in OP." ], [ "Surface color statistics", "While these two figures show that there is indeed more than one giant cluster in AP on bipartite lattices, they do not yet prove that these clusters have opposite colors.", "To verify also this prediction we denote the two colors as `+' and `$-$ ', and define $c_{ijk\\ldots }$ ($i,j,k\\ldots \\in \\lbrace +,-\\rbrace $ ) as the probabilities that the largest cluster has color $i$ , the second largest $j$ , etc.", "These probabilities are normalized such that $\\sum _{ijk\\ldots } c_{ijk\\ldots } = 1$ .", "Due to the symmetry under exchange of colors, $c_{ijk\\ldots }=c_{{\\bar{i}}{\\bar{j}}{\\bar{k}}\\ldots }$ .", "In Fig.", "REF we plot the four probabilities $c_{-jk}$ for square lattices with $L=512$ against $n$ .", "While they are all equal to $\\approx 1/8$ for large $n$ , this degeneracy is lifted as the agglomeration process proceeds.", "The most likely color pattern is $(-++)$ , followed by $(-+-)$ .", "Both have opposite colors for the two largest clusters.", "The least likely pattern has all colors the same.", "Figure: (Color online) Probabilities c s (n)c_{s}(n) of color patternss=(-++),(-+-),(--+),s = (-++),\\;(-+-),\\;(--+), and (---)(---) for the largest three clusters, plotted against nn.The first index (here always `-') gives the color of the largest cluster, while the other twoare for the second and third largest.", "The data are for square lattices of size L=512L=512.In Fig.", "REF we show how the $n$ -dependence of the probability $c_{--}$ that the two largest clusters have the same color changes with system size $L$ , both for 2 and 3 dimensions.", "There is a dramatic difference: While the data support our conclusion that there is no transition at any finite $n$ in case of the square lattice (the effective transition point moves to zero as $L$ increases), there is a clear indication for $n_c=0.411$ in case of the cubic lattice.", "Figure: (Color online) Probabilities that the two largest clusters have the same color.According to our theory, these probabilities should vanish in the supercritical phase, if L→∞L\\rightarrow \\infty .Panel (a) is for the square lattice, panel (b) for the cubic.", "The upper inset in panel (b) shows the regionclose to the critical point.", "The lower inset shows a data collapse plot, c -- (n)c_{--}(n) against(n-n c )L 1/ν (n-n_c) L^{1/\\nu } with n c =0.4109n_c = 0.4109 and ν=1.01\\nu = 1.01.More precisely, the lower inset in Fig.", "REF shows a nearly perfect data collapse when plotting $c_{--}(n)$ against $(n-n_c) L^{1/\\nu }$ , with $n_c = 0.4109$ and $\\nu = 1.01$ .", "The latter values are very close to the values obtained in Sec.", "REF from the data collapse for the ordinary order parameter, but sufficiently far from them to call for further, so far unnoticed, corrections to scaling.", "Combining both sets of parameters, accounting for such corrections by increasing the error estimates, and noticing that the system sizes in Fig.", "REF are much smaller than those in Figs.", "REF to REF , we get our final result $\\beta = 0.437(6),\\;\\; D_f = 2.523(3),\\;\\; \\nu = 0.918(13),$ and $n_c = 0.4110(1)$ .", "Since the differences between these exponents and those of OP are about three to four error bars, we conjecture that the two models are not in the same universality class.", "But more studies are needed to settle this question beyond reasonable doubts.", "In any case, Fig.", "REF should leave no doubt that $1/4 - c_{--}(n)$ is as good an order parameter for the symmetry breaking aspects of the transition, as $S$ is for the percolation aspects." ], [ "Lattices with local bipartite structure", "Let us finally discuss the case where we have locally a bipartite lattice, but where global bipartivity is broken by the boundary conditions.", "In that case the boundary conditions are irrelevant as long as the cluster does not wrap around the lattice.", "In particular, we expect that such a system is not in the OP universality class, if the globally bipartite system is not either.", "More precisely, we expect that clusters of size $<L$ are unaffected by the boundary conditions.", "Whether critical exponents like the order parameter exponent $\\beta $ are affected, which are defined through the behavior of the supercritical phase, is an open question." ], [ "Random bipartite graphs", "One minor problem in simulations of random bipartite networks is that we want connected graphs, but the most straightforward way of generating it leads to graphs that are not connected.", "We thus start with $N^*$ nodes, divide them into two equally large groups, and add $zN^*/2$ edges which have the two ends in different groups.", "Here $z$ is the average degree of the entire graph, which is chosen as $z = 2$ in the following.", "Figure: Fraction of nodes in the largest clusterm max /N\\left\\langle m_{\\mathrm {max}}\\right\\rangle /N for random bipartitenetworks.Figure: Scaling collapse for m max \\left\\langle m_{\\mathrm {max}}\\right\\rangle in the critical region for random bipartite networks.For this value of $z$ , the largest connected component of the network constructed this way has $\\approx 0.7968 N^*$ nodes.", "If we want to have a connected graph with $N$ nodes, we take $N^*=N/0.7968$ and discard all those graphs for which the size of the largest connected component is outside the range $N\\pm 0.01\\%$ , and for which any of the two components has size outside the range $N/2\\pm 0.01\\%$ .", "The $n-$ dependence of the size of the largest cluster is shown in Fig.", "REF .", "We assume again a FSS ansatz analogous to Eq.", "(REF ), with $L$ replaced by $N$ (now $D$ can of course not be interpreted as dimension, and $\\nu $ no longer is a correlation length exponent).", "The critical point and the critical exponents are found as (see Fig.", "REF ): $n_c=0.695,\\;\\; \\nu =4.88,\\;\\;D=0.567$ .", "Figure: Probabilities c -- (n)c_{--}(n) for the largest two clusters in arandom bipartite network to have the same color, for different system sizes.", "As also seen from theinset, these curves cross near the estimated critical point n c ≈0.696n_c\\approx 0.696.The probabilities $c_{--}(n)$ for both largest cluster to have the same color are shown in Fig.", "REF .", "As in the 3-d case (Fig.", "REF b) we see that the curves for different system sizes cross exactly at the critical point, suggesting again that $c_{--}(n)=0$ in the supercritical phase in the large system limit For very small and very large $N$ , $c_{--}(n)$ is ill defined, because in these limits there can be more than one largest (or second largest) cluster, but this affects only regions infinitesimally close to $n=0$ and $n=1$ in the limit $N\\rightarrow \\infty $ .. We also obtain a perfect data collapse if we plot $c_{--}(n)$ against $(n-n_c)N^{1/\\nu }$ , with slightly different parameters $n_c=0.696,\\; \\nu = 4.60$ (data not shown).", "Our best estimates for the critical parameters are the compromise $n_{c}=0.695(2),\\;\\; \\nu =4.7(2),\\;\\;D=0.567(6) \\;.$ The values for $\\nu $ and $D$ are very close to those for Erdös-Renyi networks ($\\nu =4.44,\\;D=0.60$ ), although they differ by more than one standard deviation.", "As in the 3-d case, more work would be needed to determine whether these differences are significant." ], [ "Generalizations to $k-$ partite graphs", "A graph is $k-$ partite for any $k\\ge 2$ , if the set of nodes can be divided into $k$ non-empty disjoint subsets ${\\cal N}_m,\\; m=1\\ldots k$ such that there are no links within any of the ${\\cal N}_m$ .", "As we saw in Sec.", ", the appearance of novel structures in AP on bipartite graphs depended on the fact that AP does not “mix\" colors: After each agglomeration step, one can still associate a unique color to the new cluster.", "This is no longer true on $k-$ partite graphs with $k>2$ .", "Assume a node $i$ has neighbors with two different colors, $c_1$ and $c_2$ say.", "Then, if $i$ is chosen as a target, the new cluster will display both colors on its surface.", "Figure: (Color online) Part of a a triangular lattice where sites(hexagons) are colored red, green and blue.", "The colors are arranged such that no two siteswith the same color are adjacent, i.e.", "if neighbors are connected by bonds the lattice istripartite.", "A modified AP process is defined such that a target with color R can join onlywith neighbors of color G, G can join only with B, and B can join only with R. This isindicated by the arrows and is denoted by R→G→B→RR\\rightarrow G\\rightarrow B\\rightarrow R. When node AA is chosen as target,it agglomerates with all G neighbors and becomes itself G, so that the new cluster is all Gon its surface.Figure: (Color online) Each circle represents a color, i.e.", "an element ofa partition of a k-k-partite network.", "An arrow from partition AA to partition BB means thata target with color AA joins with all nodes of color BB.", "Solid arrows indicate cycles thatare followed at each agglomeration step, while dotted arrows indicate random AP rules wheredifferent colors are chosen at different time steps.In order to arrive at non-trivial structures we have to generalize the AP rule.", "Assume we have a $k-$ partite graph with colors $c_1,\\ldots c_k$ .", "In Fig.", "REF we show the triangular lattice as an example of a tripartite graph with colors red (R), green (G) and blue (B).", "We define then a cycle $\\cal C$ in the set of colors as a closed non-intersecting directed path $c_{i_1}\\rightarrow c_{i_2}\\rightarrow \\ldots c_{i_k}\\rightarrow c_{i_1}$ .", "For tripartite graphs as in Fig.", "REF there are two possible cycles, $R\\rightarrow G\\rightarrow B\\rightarrow R$ and $R\\rightarrow B\\rightarrow G\\rightarrow R$ (up to circular shifts).", "For each cycle $\\cal C$ we define a modified AP rule AP$_{\\cal C}$ such that a target with color $c$ joins with all neighbors having the color that follows $c$ in $\\cal C$ , and only those.", "After that, the target is recolored to $c$ , so that the new cluster has a unique color.", "Alternatively, we can define a randomized rule AP$_{\\rm random}$ such that each target node $i$ chooses at random a color $c$ (different from its own) and joins with all neighbors of color $c$ .", "Obviously even more possibilities exist when $k>3$ .", "For instance we can choose the joined neighbors by following some subset of cycles.", "Different possibilities are illustrated in Fig.", "REF .", "We have not made any simulations, but we expect a rich variety of different behaviors resulting from different rules.", "It is not clear that in each case AP differs from OP in critical behavior.", "It is also not clear what happens, if one of the partitions of the network is finite.", "Naively one should expect that such finite components should not have any influence on critical behavior (which deals only with infinite clusters).", "But the example of finite $q$ in subsection REF might suggest otherwise: It could be that even a small number of nodes that do not follow the coloring and AP rules of the majority perturb the evolution sufficiently to change the universality class." ], [ "Discussion", "The main purpose of this paper was to explain in detail the reasons for the dramatic breakdown of universality in agglomerative percolation on 2-d lattices.", "In finding this reason – and demonstrating numerous other unexpected features of AP in these cases – we indeed uncovered a new class of models with non-trivial symmetries.", "In the present paper only the simplest of these, having a $Z_2$ symmetry due to bipartivity, is treated in detail, while more complex situations leading to higher symmetries are only sketched.", "Agglomerative percolation is a very natural extension of the standard percolation model, and we expect a number of applications (some of which were already mentioned in [13]).", "The main effect of bipartivity in AP is that the merging of large clusters is delayed, as compared to OP.", "It shares this feature with explosive percolation [2], but in contrast to the latter this delay is not imposed artificially, but is a natural consequence of the structure of the model.", "Also, the merging of large clusters is not delayed in all circumstances, but only subject to the symmetry structure imposed by bipartivity.", "The latter implies that clusters can have “colors\" (with $k$ colors in case of a $k-$ partite network), and only the merging of clusters with different colors is delayed.", "The effect of bipartivity is dramatic in case of 2-d lattices – shifting, in particular, the percolation threshold on infinite systems to the limit where the average cluster size diverges and the number of clusters per site is zero.", "It is much less dramatic for 3-d lattices (where we studied only the simple cubic lattice) and for random networks.", "In these cases we see a clear effect, and the simulations indicate that universality with OP is broken, but the percolation threshold is at finite values and the critical exponents are close to those of OP.", "Future work is needed to settle these questions of universality.", "In particular, it would be of interest to study high-dimensional ($3 < d \\le 6$ ) simple hypercubic lattices, in order to see how the lattice models cross over to the random graph model.", "Another interesting subject for future work is a modified AP model (discussed briefly in Sec. )", "on the triangular lattice, where the relevant group is $Z_3$ instead of $Z_2$ .", "Finally, there should exist a rich mathematical structure for modified AP models on $k-$ partite networks with $k>3$ , all of which is not yet understood." ], [ "Acknowledgements", "We thank Golnoosh Bizhani for numerous discussions, for helping with the simulations of AP on random bipartite graphs, and for carefully reading the manuscript." ] ]

[ [ "Shocking Tails in the Major Merger Abell 2744" ], [ "Abstract We identify four rare \"jellyfish\" galaxies in Hubble Space Telescope imagery of the major merger cluster Abell 2744.", "These galaxies harbor trails of star-forming knots and filaments which have formed in-situ in gas tails stripped from the parent galaxies, indicating they are in the process of being transformed by the environment.", "Further evidence for rapid transformation in these galaxies comes from their optical spectra, which reveal starburst, poststarburst and AGN features.", "Most intriguingly, three of the jellyfish galaxies lie near ICM features associated with a merging \"Bullet-like\" subcluster and its shock front detected in Chandra X-ray images.", "We suggest that the high pressure merger environment may be responsible for the star formation in the gaseous tails.", "This provides observational evidence for the rapid transformation of galaxies during the violent core passage phase of a major cluster merger." ], [ "Introduction", "The hierarchical nature of large-scale structure formation is spectacularly revealed by observations of approximately equal mass mergers between pairs of galaxy clusters.", "These major mergers can subject galaxies to an environment that can drive abnormal rates of galaxy evolution, particularly at times close to pericentric passage [38], [4].", "For example, it has been suggested that the core passage phase of a merger may be important in both triggering and truncating starbursts [10], [9], [37], [22], [28], although it remains unclear which merger-specific mechanisms are responsible.", "Simulations suggest that the high pressure environment a galaxy encounters during the core passage phase of a merger may result in the collapse of giant molecular clouds (GMCs) within a galaxy, leading to an initial burst of star formation [3], [27], [4] while the subsequent stripping of the interstellar medium terminates GMC formation, halting further star formation [18].", "Evidence for these processes should reveal itself in the vicinity of observed shock fronts which mark regions of severe merger activity.", "In this context, [11] find that the shock front in the “Bullet” cluster [30] has not had an appreciable effect on the star formation in the galaxies in its vicinity.", "Thus, while it appears that the intense core passage phase of a major merger may play a significant role in shaping the star-forming properties of the galaxies, more detailed observations are required to understand the dominant processes.", "In this vein, the merging cluster Abell 2744 [36] is an excellent candidate for testing the effects of major mergers on cluster galaxies.", "Recently, owers2011 combined Chandra X-ray and Anglo Australian Telescope AAOmega optical observations to constrain the dynamics of the merger in A2744.", "The new data allowed a refinement of the previous merger scenarios [25], [8] and indicate that a high velocity ($\\sim 4750$$\\,\\rm {km\\,s^{-1}}$ ), near head on, major merger along a roughly north-south axis at $\\sim 27^\\circ $ to our line of sight is shortly past core passage, along with an infalling group to the northwest [32].", "The Chandra data reveal evidence for a “Bullet-like” shock front driven by the remnant core of the less massive subcluster.", "Its higher velocity dispersion and strong lensing features around its brightest member (Figure 1) indicate that the northern-most subcluster is the remnant of the more massive subcluster.", "The properties of the major merger in A2744 make it a prime target for understanding how this extreme environment affects the resident galaxies.", "In this Letter we present a suggestive spatial coincidence between a merger-affected region revealed by the Chandra analysis of owers2011 and several peculiar galaxies discovered in archival Hubble Space Telescope (HST) imagery.", "These “jellyfish” galaxies exhibit one-sided trails of extremely blue knots and filaments reminiscent of those first noticed by [34] and [12].", "These knots and filaments are interpreted as the manifestation of hot, young stars formed in-situ within gas which has been stripped from the parent galaxy [39].", "We suggest that the major merger environment of A2744 is significantly influencing these `galaxies in turmoil'." ], [ "Observations", "HST/ACS observations of A2744 were taken during 2009 October using the F435W, F606W and F814W filters.", "The observations consist of north and south pointings each having exposure times of 6624 s for the F606W and F814W filters and 8081 s for the F435W filter.", "The data were retrieved from the archive and the pointings were combined using the multidrizzle task [26].", "The RGB color image is shown in Figure REF , where the F814W image is used for the red channel, F606W green and F435W blue.", "Overlaid are contours from our background-subtracted, exposure-corrected Chandra X-ray image owers2011.", "Plotted as a magenta arc is the position of the shock edge described in owers2011.", "We also make use of our AAOmega spectra.", "Figure: HST/ACS RGB image of A2744.", "The green contours show Chandra X-ray surface brightness.", "The red boxes highlight the “jellyfish” galaxies which are shown in more detail in Figure .", "From east to west, the four galaxies are F0083, F0237, the “central” jellyfish and F1228.", "The magenta dashed curve shows the shock edge reported in owers2011." ], [ "HST/ACS imaging", "Close inspection of the HST images revealed four galaxies (marked by red boxes in Figure REF ) with distinct trails of extremely blue knots and filaments, most conspicuous in the bluest F435W band and having magnitudes $24.7 < {\\rm F435W} < 28.5$ .", "This band corresponds to the Sloan $u$ -band in the cluster rest frame, which is known to be sensitive to light from young, hot OB stars, hence active regions of star formation [21].", "The top panels of Figure REF show images of these four galaxies, revealing their “jellyfish” morphology [nomenclature of][]smith2010, which is due to trails and filaments of bright star-forming regions.", "This interpretation is consistent with star-forming signatures in the optical spectra discussed below for three of the four galaxies.", "All 3 galaxies are spectroscopically confirmed cluster members owers2011.", "Galaxy F0083 is a luminous ($\\sim 3L^*_R$ ), nearly face-on late-type spiral with numerous blue star-forming regions in its disk and an unresolved bright nucleus.", "The disk structure is irregular with an asymmetry on the eastern side connecting to a faint tidal feature which connects to a nearby faint galaxy (F606W mag $\\sim 22.6$ ), indicating an interaction has taken place.", "There is a bright blue rim $\\sim 4$  kpc north of the galaxy center.", "A trail of blue knots and filaments extends $\\sim 35$  kpc from the galaxy center to the southwest – these knots and filaments are not associated with the disk.", "The spectrum (Figure REF ) reveals Balmer lines with narrow and broad components which are well fitted by two Gaussians having rest-frame velocity dispersions of $1078\\pm 13$$\\,\\rm {km\\,s^{-1}}$ ($1155\\pm 40$$\\,\\rm {km\\,s^{-1}}$ ) and $124\\pm 2$$\\,\\rm {km\\,s^{-1}}$ ($133\\pm 4$$\\,\\rm {km\\,s^{-1}}$ ), respectively, for $\\rm {H}{\\alpha }$   ($\\rm {H}{\\beta }$ ).", "The OIII doublet is also well fitted by broad and narrow components with dispersions $336\\pm 5$$\\,\\rm {km\\,s^{-1}}$ and $93\\pm 1$$\\,\\rm {km\\,s^{-1}}$ , respectively.", "The combination of broad and narrow lines indicate that F0083 hosts a Seyfert 1 nucleus.", "However, the ratios of the equivalent widths (EW) of the narrow components of log([NII]/$\\rm {H}{\\alpha }$ )=-0.51 and log([OIII]/$\\rm {H}{\\beta }$ )=0.67 lie close to the AGN/star-forming boundary, indicating that a fraction of the emission may be due to star formation [1].", "A bright X-ray point source is associated with this galaxy (Figure REF ).", "F0237 [14] is a major merger involving two $\\sim 0.7L^*_R$ late-type galaxies [13] the centers of which are separated by $\\sim 5$ kpc.", "The multi-color image (Figure REF ) reveals a faint tidal feature extending $\\sim 30$  kpc to the southeast, and a trail of $\\sim 6$ blue knots extending $\\sim 21$  kpc to the southwest.", "Unlike F0083, there are no blue knots within the galaxy disks.", "The AAOmega spectrum confirms [14]' poststarburst classification—there is strong Balmer absorption (EW ${\\rm H}\\delta \\sim 6$ ) with no measurable OII or $\\rm {H}{\\alpha }$ emission.", "The fiber aperture covers only a portion of F0237, leaving the extent of its poststarburst emission undetermined.", "However, the lack of blue star-forming regions within the system indicates a dearth of ongoing star formation.", "The faint ($\\sim 0.3L^*_R$ ), blue galaxy F1228 is the western-most of the highlighted galaxies in Figure REF .", "It hosts the least spectacular trail of blue knots, the most distant of which is $\\sim 11$  kpc from F1228's main component.", "The morphology is similar to the “tadpole” galaxies seen in the Hubble Ultra-Deep Field [41], [17] with a head that contains a number of bright knots of star formation, a blue ridge on its western side, and a tail pointing to the SE containing the blue knots.", "The spectrum (Figure REF ) reveals strong $\\rm {H}{\\alpha }$ emission with EW $\\sim -52Å$ .", "The emission line EW ratios of [NII]/$\\rm {H}{\\alpha }$ =-0.64 and [OIII]/$\\rm {H}{\\beta }$ =-0.04 are consistent with a star-forming origin [1].", "The jellyfish galaxy closest to the Bullet-like subcluster (right-most panel of Figure REF ) has a disk-like morphology, with no discernible bulge or spiral arms.", "The trail of extremely blue knots points to the NW and extends to a radius of $\\sim 26$  kpc.", "This galaxy is below the brightness limit used in owers2011 and thus does not have a spectrum or, therefore, a redshift.", "Figure: Top panel: Close up views of the jellyfish galaxies highlighted in Figure .", "The green contours show the surface brightness at ∼1σ\\sim 1 \\sigma above the background for an image generated by co-adding the F435W, F606W and F814W images using the SWarp tool .", "The white circle shows the AAOmega fiber aperture size.", "Bottom panel: AAOmega spectra (where available) for the jellyfish galaxies.", "We note that the broad feature at ∼8650\\sim 8650Å  seen in the F0237 and F1228 spectra is due to sky subtraction residuals." ], [ "Colors", "We estimate the ages of the knots and filaments by comparing their colors to those predicted by the solar metallicity models of [29].", "The models used correspond to the stellar populations produced by two extremal star formation histories – a single burst, and an exponentially decaying star formation with a decay timescale of 20 Gyrs.", "The colors of the knots and filaments and the color evolution tracks derived from the models are shown in Figure REF .", "There are two caveats.", "First, the contribution of emission lines is ignored, although this is exepected to be minimal within the broad bands of the filters [12].", "Second, we do not account for reddening.", "However, dust obscuration tends to make stellar populations appear older, thus our age estimates are upper limits.", "From Figure REF it can be seen that in the majority of cases the knots/filaments are bluer than predicted by the 100 Myr populations, thus we take this to be the upper limit on the times since the onset of star formation.", "Figure: Comparison of the colors of the knots and filaments in the jellyfish tails (black symbols with error bars) to those predicted by models of stellar populations produced by single burst (solid blue line) and exponentially declining (red dashed line) star formation histories." ], [ "Discussion and Interpretation", "We have presented HST observations which reveal four “jellyfish” galaxies in the merging cluster A2744.", "Low redshift jellyfish analogues are observed in nearby clusters [42], [49], [39], [20].", "However, at redshifts comparable to A2744, only three have been observed, all in separate clusters [34], [12].", "Indeed, [12] found only two examples in a survey of 13 intermediate redshift clusters.", "The rarity of jellyfish galaxies in intermediate redshift clusters emphasizes the significance of the observations presented here; we observe four such systems within A2744.", "While the statistics at intermediate redshifts are sparse, smith2010 compiled a sample of 13 Coma cluster galaxies which harbor ultraviolet asymmetries/tails.", "The orientation of the tails, which generally point away from the cluster center, suggested that these galaxies formed from an infalling population experiencing the cluster environment for the first time smith2010.", "However, there are two key distinctions to be made when comparing the low redshift jellyfish population with that of A2744.", "First, the knots in A2744 are bright ($-13 \\lesssim M_{I} \\lesssim -17$ ) when compared to prominent examples in Coma [48] and Virgo [20], [19] while only ESO 137-001 in the merging cluster A3627 has knots with comparable brightness [42], [46].", "Thus, the bright knots seen in A2744 are rare in low redshift clusters.", "Second, the orientation of the tails in A2744 are less well ordered than those seen in Coma.", "This does not preclude the existence of a faint, Coma-like infalling population of jellyfish in A2744, which may be revealed by deeper observations, but indicates that the picture outlined by smith2010 is unlikely in this case for the jellyfish observed in A2744.", "Thus, there appears to be a relatively larger number of jellyfish in A2744 compared to other intermediate redshift clusters and these jellyfish appear to be a different, brighter version of their low redshift counterparts.", "Is the major merger in A2744 driving the formation of an excess of these bright jellyfish galaxies?", "There is a strong spatial correlation between the jellyfish galaxies and features associated with the high speed Bullet-like subcluster [Figure REF and][]owers2011: the proximity of (i) F0083 and F0237 to the portion of the Bullet-driven shock front revealed as an edge in the Chandra observations, and (ii) the central jellyfish to the X-ray peak associated with the remnant gas core of the Bullet-like subcluster.", "While we cannot know the exact 3D locations of the jellyfish galaxies with respect to the ICM structures, the small projected distances suggest that the jellyfish galaxies may have recently been overrun by the shock front and/or the Bullet-like subcluster gas.", "This indicates that a mechanism related to an interaction with these ICM features may be responsible for either the stripping of the gas leading to the tails or the triggering of the star formation in the tails, or both.", "This assertion is supported by the young ages of the stellar populations in the knots and filaments, which suggest that the star formation was triggered $\\lesssim 100\\,{\\rm Myr}$ ago (Section REF ).", "On these timescales, a galaxy with velocity $\\sim 1000$$\\,\\rm {km\\,s^{-1}}$ travels $\\lesssim 100$  kpc, so we would expect there to still be a strong spatial coincidence between the jellyfish and the putative ICM features responsible for triggering the star formation.", "Furthermore, consideration of the peculiar velocity of the Bullet-like subcluster [$v_{pec} \\simeq 2500$$\\,\\rm {km\\,s^{-1}}$ ;][]owers2011 and of the three jellyfish galaxies with measured redshifts ($v_{pec} = -729, -2277\\, {\\rm and} -2528$$\\,\\rm {km\\,s^{-1}}$ ; lower panels, Figure REF ) indicates that the jellyfish galaxies are not members of the Bullet-like subcluster, and that if they have interacted with the shock or the Bullet-like subcluster's ICM, then the relative velocity of the interaction was high – of the order of the merger velocity $\\sim 4750$$\\,\\rm {km\\,s^{-1}}$ .", "Similarly, [34] suggested that the jellyfish-like galaxy C153 in A2125 may be a result of enhanced ram pressure stripping caused by a high velocity encounter with the ICM due to a cluster merger, while smith2010 find hints that some of their jellyfish galaxies are associated with merger-related enhancements in the ICM density.", "The proximity of the jellyfish to merger-related ICM features in this high speed merger suggests that the merger is responsible for the increased fraction and more extreme nature of the jellyfish in A2744.", "A high-speed, head-on merger creates much greater ram pressure and a powerful shock which can significantly enhance galaxy-ICM processes when compared with those felt by a galaxy falling into a relaxed cluster.", "Enhanced ram pressure may strip gas more efficiently leading to a higher incidence of gaseous tails, and therefore a higher likelihood of forming the observed jellyfish phenomena.", "This is consistent with the simulations of [15], who find the mass lost from a galaxy due to ram pressure during a major merger is substantially increased, and also with [45] who suggest the enhanced ram pressure stripping due to an interaction with the infalling M49 group explains the strong ram pressure stripping observed in NGC 4522.", "Furthermore, the increased ICM pressure due to the merger shock may promote star formation in the tails and may also drive higher star formation rates, hence higher surface brightness features, compared with jellyfish galaxies in lower pressure environments.", "The shock has a Mach number $M \\simeq 3$ owers2011 meaning the pressure jumps by a factor of $\\sim 11$ with respect to the pressure of the surrounding unshocked ICM, $P_{\\rm ICM}/{\\rm k_B} \\sim 10^5 {\\rm K/cm^3}$ .", "Thus, the static pressure due to the shock is $P_{\\rm Shock}/{\\rm k_B} \\sim 10^6 {\\rm K/cm^3}$ which is an order of magnitude higher than the threshold pressure required to trigger the collapse of GMCs leading to star formation [16], [4].", "Furthermore, there is a rapid increase in the ram-pressure the galaxy feels as it is overrun by the shock due to the high relative velocity and the factor of 3 increase in ICM density for a Mach 3 shock.", "If, like the Bullet cluster [31], the shock front is collisionless, it would be more abrupt than a collision-dominated shock (relaxation time $\\sim 10$  Myr), so the interaction is likely too fast ($\\sim 10$  Myr) for the shock's ram pressure to be responsible for stripping the gas leading to the observed tails.", "This is supported by the southwesterly orientation of the trails of material seen in F0083 and F0237.", "If ram pressure from the shock was responsible, the tails should point southeast toward the near part of the shock front.", "This high pressure shock-triggering scenario is consistent with the simulations of [43] and [24] who found that the star formation rates in ram-pressure stripped gaseous tails is largely driven by the ICM pressure.", "However, simulations specific to the scenario outlined above are required for confirmation.", "On the question of whether the increased ram pressure which occurs in mergers enhances or suppresses the star formation in cluster members [18], [3], [27], [4], the observations presented here show that both effects may be in play.", "Considering F0083, a number of blue knots are seen across the galaxy indicating that there is disk-wide star formation.", "There is evidence for an interaction with a smaller galaxy (Figure REF ), which may have driven gas towards the galaxy center, triggering the AGN activity.", "However, such a minor interaction is unlikely to trigger disk-wide star formation.", "Thus, we suggest that the high ICM pressure has compressed the GMCs and triggered disk-wide star formation [3].", "The asymmetric distribution in the star formation—particularly the arc-shaped star-forming region $\\sim 4$  kpc to the north of the galaxy center on the side opposing the tail of star-forming regions—is consistent with that expected due to the compression from ram pressure as a galaxy moves edge-on through the ICM [27].", "Conversely, the spectrum of F0237 (Figure REF ) shows strong Balmer absorption and an absence of emission lines indicating a starburst has recently ($<1$  Gyr ago) been abruptly truncated.", "Here, the major merger of two spiral galaxies is the most likely trigger for the initial starburst [33], [13], [2], [35].", "However, in the absence of external mechanisms the poststarburst phase is expected to occur at the later stages of the major merger after the galaxies have coalesced [5], [7], [40].", "The premature truncation of the starburst may be due to the enhanced ram pressure felt by a galaxy during a cluster merger, which strips gas more rapidly.", "Furthermore, the high pressure cluster merger environment may increase the star formation leading to a more rapid consumption of the gas.", "The difference in the star-forming properties of F0083 and F0237 may be attributed to the different mass and dynamical states of the two systems.", "Given its luminosity, F0083 is likely more massive than F0237, meaning it is less susceptible to complete stripping of gas due to ram pressure.", "Further, strong tidal forces due to the major merger in F0237 may unbind gas, making it more susceptible to complete ram pressure stripping [44], [23] thereby halting star formation in the disks." ], [ "Conclusions", "We have presented HST observations of four rare “jellyfish” galaxies in the merging cluster A2744.", "The knots and filaments are brighter than the majority of jellyfish analogues found in local clusters, indicating more vigorous star formation is occurring compared to that found in tails of low redshift jellyfish.", "Intriguingly, three of these jellyfish galaxies lie in close proximity to merger-related features in the ICM.", "This leads us to propose that the rise of the jellyfish in A2744 is due to the effects of the high pressure of the ICM during the ongoing cluster merger.", "In particular, we suggest that the star formation occurring in the tails of stripped gas has been triggered by the rapid, dramatic increase in pressure during an interaction with the shock.", "These observations support the hypothesis that galaxies undergo accelerated evolution in their star-forming properties during the violent core passage phase of a cluster merger.", "We thank the referee, Curtis Struck, for his helpful comments.", "M.S.O.", "and W.J.C acknowledge the financial support of the Australian Research Council.", "P.E.J.N.", "was partly supported by NASA grant NAS8-03060.", "Facilities: CXO (ACIS), AAT (AAOmega), HST (ACS)" ] ]

[ [ "Millimeter and sub-millimeter atmospheric performance at Dome C\n combining radiosoundings and ATM synthetic spectra" ], [ "Abstract The reliability of astronomical observations at millimeter and sub-millimeter wavelengths closely depends on a low vertical content of water vapor as well as on high atmospheric emission stability.", "Although Concordia station at Dome C (Antarctica) enjoys good observing conditions in this atmospheric spectral windows, as shown by preliminary site-testing campaigns at different bands and in, not always, time overlapped periods, a dedicated instrument able to continuously determine atmospheric performance for a wide spectral range is not yet planned.", "In the absence of such measurements, in this paper we suggest a semi-empirical approach to perform an analysis of atmospheric transmission and emission at Dome C to compare the performance for 7 photometric bands ranging from 100 GHz to 2 THz.", "Radiosoundings data provided by the Routine Meteorological Observations (RMO) Research Project at Concordia station are corrected by temperature and humidity errors and dry biases and then employed to feed ATM (Atmospheric Transmission at Microwaves) code to generate synthetic spectra in the wide spectral range from 100 GHz to 2 THz.", "To quantify the atmospheric contribution in millimeter and sub-millimeter observations we are considering several photometric bands in which atmospheric quantities are integrated.", "The observational capabilities of this site at all the selected spectral bands are analyzed considering monthly averaged transmissions joined to the corresponding fluctuations.", "Transmission and pwv statistics at Dome C derived by our semi-empirical approach are consistent with previous works.", "It is evident the decreasing of the performance at high frequencies.", "We propose to introduce a new parameter to compare the quality of a site at different spectral bands, in terms of high transmission and emission stability, the Site Photometric Quality Factor." ], [ "Introduction", "Astronomy and astrophysics in the 100-1000 GHz band allow the study of a large variety of processes, in the local and distant universe, which involve cool matter absorbing and re-radiating efficiently at these frequencies, in environments often unaccessed through observations at visible wavelengths.", "In fact, many key topics in modern astronomy and cosmology, such as galaxy formation and evolution, the amount and role of dark matter and dark energy in the universe, star formation, protoplanetary disks, or the properties of cold debris at the outskirts of the solar system, are related to radiative phenomena in this band.", "The field has undergone a huge development in the last two decades, thanks to the development of sensitive detectors, large cameras, polarization sensitive devices and spectroscopically capable instrumentation.", "Some key achievements range from the measurement of the intensity and polarization power spectra of the cosmic microwave background at millimeter (mm) wavelengths, to the discovery and the characterization of the optically elusive sub-millimeter (sub-mm) galaxy population (SCUBA, BLAST), and the recent galaxy cluster surveys through sub-arcminute resolution observations of the Sunyaev-Zel'dovich effect (ACT, SPT).", "Ground-based observations in the mm/sub-mm band are usually plagued by the transparency of the atmosphere (and its stability over time), mainly because of the presence of large, time-dependent pressure-broadened features in the emission (and absorption) spectrum of the water vapour.", "Of course, this issue is strongly mitigated when operating stratospheric balloon-borne or airborne detectors, and completely averted when moving detectors on spacecrafts.", "BOOMERANG, BLAST, SOFIA, $Planck$ and $Herschel$ have proven the effectiveness of mm and sub-mm observations from the stratosphere and from space, providing ground breaking advancements in their respective fields at the time of their operation.", "Anyway, the practical limitations on the telescope size, the weight and the accessibility of instrumentation still make substantially unfeasible the deployment of large (10m class) telescopes on balloons, aircrafts or satellites.", "As a matter of fact, the ground-based solution appears presently the only viable way to routinely perform high angular resolution observations of compact objects and/or small spatial and spectral features in cool diffuse media at sub-mm wavelengths.", "As a consequence, the last few years have witnessed increasing efforts in the design and construction of large telescopes in places of the planet which provide the potentially most attractive atmospheric features for mm and sub-mm astronomy.", "The community has realized the need to perform a thorough characterization of astronomical sites in terms of atmospheric opacity and stability across the whole mm/sub-mm spectral region, both for observation planning and for transparency monitoring during the observing sessions.", "At a time where bolometric detectors can easily approach the photon noise limit, and large cameras with hundreds or thousands of such detectors already allow to break this limitation, it is straightforward to realize that an improper characterization of the atmospheric properties may become the strongest restriction to the effective science return from ground-based instruments of the present (ACT, SPT) and next generation (CCAT).", "In order to continuously monitor the atmospheric transmission several approaches are possible: tippers or tau-meters, hygrometers, GPS, water vapour radiometers, radiosoundings and spectrometers.", "The first approaches allow a continuous data recording by simple instruments but with the drawback of single frequency observations, needing a synthetic atmospheric model to infer transmission at other frequencies.", "Dome C is considered one of the best sites in the world to perform observations in a wide range of the electromagnetic spectrum allowing also to explore Terahertz windows ([13] and [22]).", "Anyway a wide frequency coverage transmission measurements campaign at Dome C, employing the direct spectroscopic information derived by an interferometric experiment, was never carried out.", "The goal of this paper is to compensate the lack of those data by estimating the atmospheric performance with a semi-empirical approach.", "We test this method using the available dataset of radiosounding measurements recorded by the Routine Meteorological Observations (RMO) Research Project (www.climantartide.it) at Concordia station in the period from May 2005 to January 2007, carefully corrected for the main lag errors and dry biases.", "The profiles of air temperature, pressure and relative humidity allow to generate synthetic spectra, ranging from 100 GHz to 2 THz, with the ATM code ([14]).", "The paper is organized as follows.", "Atmospheric synthetic spectra, as derived by ATM code, are described in Sect.", ".", "In Sect.", "estimates of atmospheric transmission and emission corresponding to largely explored ground based telescope bands between 150 and 1500 GHz are analyzed.", "The effect of the filter bandwidths on the estimate of opacity is for the first time included in the relation showing a contribution up to a 30 per cent over-estimate on the opacity in the case of the highest frequency band.", "A parameter to rank the observational conditions for each of the selected spectral bands is introduced as the ratio between average transmission and the corresponding fluctuations.", "Finally a discussion on the analysis and the conclusions are summarized in Sect.", ".", "A detailed description of the correction procedure used to analyse the raw radiosounding data and determine the vertical profiles of the main thermodynamic parameters is reported in the Appendix.", "Figure: Daily values of precipitable water vapour (pwv)obtained through the present analysis from surface-levelto 8 km amsl, and plotted versus WW, the corresponding valuesof precipitable water derived by Tomasi et al.", "(2011a)over the altitude range from surface-levelto 12 km amsl.", "The data are best-fitted by a regressionline with intercept equal to 9.46610 -3 9.466 10^{-3} and slope coefficientequal to 0.9425, which was obtained with regression coefficientR = +0.993, and provided a standard error of estimate SEE = 0.04 mm.Figure: Atmospheric transmission spectra as modeled by ATMprogram for each radiosounding.", "Photometric bands inTable (gray) match the main transmission windows.Table: Characteristic spectral bands assumed in this work." ], [ "Synthetic spectra production", "At present, for the site of Dome C we can rely only on the atmospheric monitoring performed at a few individual frequencies, with no simultaneous measurements in different regions of the spectrum.", "In order to compensate for the lack of a continuous and spectrally wide atmospheric monitoring at Dome C, we predict the performance in the mm/sub-mm bands in the period from May 2005 to January 2007 by means of a semi-empirical approach.", "A set of raw radiosounding data was recorded for the present study, consisting of an overall number of 469 radiosounding measurements taken routinely at Dome C, at 12:00 UTC from May 2, 2005 to January 31, 2007 ranging from a minimum of 15 in May 2005 to a maximum of 30 in November 2006.", "In general, each radiosonde measurement consists of values of air pressure $P$ , air temperature $T$ and relative humidity $RH$ , taken at more than 800 standard and additional levels in the altitude range from surface to 10 km above mean sea level (amsl).", "Data provided by the radiosonde sensors are affected by lag and instrumental errors as well as by various dry biases.", "They were all corrected following the procedure described in the Appendix.", "The time-patterns of the daily pwv values are shown in Fig.", "REF .", "Two main features are evident in Fig.", "REF , showing that the majority of pwv values are lower than 0.3 mm during the austral autumn months, although presenting largely dispersed patterns (of 50 per cent or more), and, hence, low stability.", "As shown by [19], a limited contribution is given to the overall value of atmospheric pwv by the amount of water vapour present in the Upper Troposphere and Low Stratosphere (UTLS) region from 8 to 12 km amsl, while negligible fractions of pwv ranging mainly between 0.003 and 0.005 mm throughout the year are present in the stratosphere from 12 to 50 km above Dome C ([21]).", "To verify the reliability of the present estimates of pwv, a comparison is made in Fig.", "REF among the present daily values of pwv and those correspondingly determined by [19] (indicated as $W$ ) using a more advanced correction procedure from surface-level to 12 km amsl.", "The comparison showed that a close relationship exists between the present results and those of [19], defined by a regression line with nearly null intercept and slope coefficient of + 0.9425, having regression coefficient better than + 0.99, and providing a standard error of estimate equal to 0.04 mm.", "These findings clearly indicate that the present evaluations of pwv, as obtained over the altitude range from surface-level to 8 km amsl, are fully suitable for the purposes of our study, especially considering the intrinsic uncertainty of the simulation model.", "We have estimated synthetic spectra in emission and in opacity by means of the ATM code in the wide spectral range from 100 GHz to 2 THz.", "Each spectrum is derived considering the corrected radiosounding data.", "The transmission corresponding to each radiosounding dataset, estimated from optical depth spectra as $T = e^{-\\tau }$ , is shown in Fig.", "REF .", "In the period under consideration, the inferred pwv values show an average close to 0.3 mm with a mean dispersion of about 150 $\\mu $ m (see Fig.", "REF ).", "The same amount of pwv variation can contribute with a different weight to the total optical depth.", "As example in Fig.", "REF we represent the optical depth fluctuations derived by ATM, quantified as the maximum dispersion, due to fluctuations of pwv of the order of 150 $\\mu $ m around three different pwv values (0.15, 0.5 and 1.0 mm).", "It is worthy of note that for low pwv content, $\\Delta \\tau $ can be as high as 60 per cent in the high frequency windows.", "Figure: Optical depth fluctuations corresponding to a 150 μ\\mu m variation around three selected pwv values:1 mm (red line), 500 μ\\mu m (green line), 150 μ\\mu m (black line)." ], [ "Multi-band analysis", "A quantitative analysis is performed considering 7 photometric bands centered at the frequencies of several astrophysical and cosmological experiments: South Pole Telescope (SPT), Atacama Cosmology Telescope (ACT), Millimetre and Infrared Testagrigia Observatory (MITO) and BRAIN (B-mode RAdiation INterferometer) for Low Frequency (LF) atmospheric windows; Submillimetre Common-User Bolometer Array (SCUBA and SCUBA-2) and Two HUndred Micron PhotometER (THUMPER) for sub-mm bands (High Frequency, HF).", "The central frequency of each band, as well as the bandwidth, quantified with the FWHM (Full Width Half Maximum), are listed in Table REF (see also Fig.", "REF ).", "The band profiles are assumed to be top-hat assuming in this way the maximum rejection to off-band contributions.", "To assess the constraints on astronomical observations arising from the atmosphere emission above Dome C, we give an estimate of the $NEP$ (Noise Equivalent Power) and the $NEFD$ (Noise Equivalent Flux Density) for all the seven bands.", "In fact in such a wide spectral region both the quantities are normally employed: the power density, mainly for the low frequency bands, while the flux density, for the high frequency region.", "The quoted $NEP$ is the root of the sum of $NEP_{atm}^2$ , the term considering the atmospheric emission fluctuations, and $NEP_{tele}^2$ , i.e.", "the instrumental contribution to the photon noise.", "The atmospheric emissivity spectra are generated by ATM.", "The telescope is assumed a 10-m in diameter Al-mirror with a surface emissivity of the order 3 per cent at 150 GHz and depending on the frequency as $\\sqrt{\\nu }$ .", "The throughput of the telescope is assumed diffraction limited at each band.", "Focal plane optical efficiencies are taken as unitary for all the bands as well as telescope main beam efficiency.", "The dominant sky sources (CMB and dust) are not included, the instrument detector noise is assumed lower than the background noise and the spillover emission is neglected.", "In order to quantify the maximum variation of these quantities we plot in Fig.", "REF $NEP$ and $NEFD$ values for all the bands, for the extreme conditions occurred during the austral summers and winters at Dome C in the 2005-2007 period.", "Figure: Noise Equivalent Power (upper panel) for the seven bands in two extremeatmospheric conditions in the austral winter (red triangles) and summer (black diamonds).In the lower panel, the same for the Noise Equivalent Flux Density.Figure: Atmospheric transmission vs. cumulative time frequency for Dome C correspondingto the atmospheric windows listed in Table .Table: Transmission quartiles matching cumulative distributions in Fig.", ".Table: Pwv quartiles comparison." ], [ "Dome C statistics comparison", "To validate the proposed semi-empirical approach, we compare the derived atmospheric performance with the results available in literature.", "Fig.", "REF shows Dome C atmospheric transmission as a function of the cumulative time frequency derived by radiosounding data and ATM model for the bands listed in Table REF (the corresponding quartiles are reported in Table REF ).", "Transmission statistics at Dome C performed by [24], [13], [27], [22] and [1] are compared with our analysis.", "Low frequency atmospheric windows show high transparency during the whole period confirming that high quality mm observations can be performed from this site for most of the time.", "For instance the 150 GHz 50 per cent quartile transmission is about 97 per cent (see the green line in Fig.", "REF ).", "This is consistent with the 95 per cent value recently measured by [1] during the summer campaign in December 2009/January 2010, even considering their integrated in-band result.", "Median transparency for the 220 GHz atmospheric window is about 95 per cent (see the cyan line in Fig.", "REF ) as already derived by [24] by pwv measurements performed with a portable photometer in January 1997.", "Dome C 450 $\\mu m$ window remains above a transmission of 60 per cent for 50 per cent of the time.", "In [13] the atmospheric transmission at Dome C has been calculated through the 5-years pwv data from the South Pole available in [16] and extrapolating the corresponding atmospheric transmission at Dome C using the model in [10].", "They found that 450 $\\mu m$ median transmission at Dome C is about 70 per cent.", "Recently [27] measured a 450 $\\mu m$ median winter transmission at Dome C of about 60 per cent estimating pwv with the Microwave Humidity Sounder (MHS) sounding on the National Oceanic and Atmospheric Administration (NOAA) ozonesondeas in 2008.", "Figure: Monthly average values of transmission T\\left\\langle T\\right\\rangle and the relative monthly fluctuations σ T \\sigma _{T}, plotted as rms values, shown in different colors: austral winter months in the upper panel and summer months in the bottom panel.", "Monthly Site Photometric Quality Ratio (SPQR) values are shown in the right panel.Dome C median transmission for the 350 $\\mu m$ atmospheric window is about 50 per cent (see the bottom panel in Fig.", "REF ), as derived by [22] using the MOLIERE model and 200 $\\mu m$ optical depth measurements.", "They found also that the Dome C 200 $\\mu m$ window opens with a transmission of 10 per cent for less then 25 per cent of the time while [27] found that the transmission at 200 $\\mu m$ is about 13 per cent for 25 per cent of the time in 2008.", "The 200 $\\mu m$ transmission as a function of the cumulative frequency is the black solid line in the bottom panel of Fig.REF : the 25 per cent quartile transmission value is above 10 per cent.", "Pwv quartiles since May 2005 until January 2007 (see the right panel in Fig.", "REF ) have been compared with Dome C water vapour estimates performed in previous works in Table REF ." ], [ "High transmission and emission stability", "Following an observational approach, we report the statistics of integrated in-band quantities, like emission and transmission.", "Both monthly averages $\\left\\langle T\\right\\rangle $ and relative dispersions $\\sigma _{T}$ (rms values) of in-band transmissions, ranging from May 2005 until January 2007 and splitting between austral summer (from October to February) and winter months (from March to September) are shown in Fig.", "REF .", "Table: Seasonal averages of the SPQR.Monthly averaged transmission fluctuation is a good proxy of emission stability due to the fact that transmission and emission fluctuations are linearly correlated.", "In addition we assume that the estimated monthly averaged fluctuations, quantified in terms of the standard deviation, could be an underestimate of atmospheric stability because they derive from a daily data sampling, the time interval between two consecutive radiosoundings.", "We note that during the austral winter the atmospheric transmission in all the considered bands is generally higher, as expected.", "$\\left\\langle T\\right\\rangle $ shows values close to the unity in mm bands and decreases towards THz windows, while relative dispersions $\\sigma _{T}$ have the opposite trend.", "As an example, the best atmospheric conditions (in term of high transmission) occur when the atmospheric fluctuations $\\sigma _{T}$ are larger than others months (red dots in Fig.", "REF ).", "Referring to the atmospheric window centered at 200 $\\mu $ m, when the transmission has the maximum value, the large fluctuations at short time scales are likely to degrade the quality of a scientific observation.", "In addition it is not possible to identify the month with the best atmospheric performance as one can see from the gap between two consecutive years atmospheric transmission and fluctuations (red and blue dots in Fig.", "REF ).", "450$\\mu $ m and 350$\\mu $ m bands transmission show a reduction of few percent ranging from winter to summer months, while fluctuations are not sensitive to seasonal effects.", "All the considered bands are characterized by high stability in October (see the black or orange dots in the middle panel of Fig.", "REF ) with the exception of the 200 $\\mu $ m window, showing high stability especially during summer months like January or February, when atmospheric transparency is not suitable to perform astrophysical observations.", "To quantify the real capability of the observational site we need to study the atmospheric performance, mainly the stability, strongly affected by the weak reproducibility of weather conditions at long time scales.", "In order to highlight this issue we introduce a specific parameter, the Site Photometric Quality Ratio: $SPQR=\\frac{\\left\\langle T\\right\\rangle }{\\sigma _{T}}$ relating monthly averaged transmission to its fluctuations, sampled on a daily timescale, for all the considered atmospheric windows.", "SPQR amplitude provides information about atmospheric performance and it allows us to dermine if high transmission combined with high transmission (i.e.", "emission) stability conditions are both satisfied for each band.", "Even if we are not able to identify the desired SPQR threshold, this factor could represent a useful tool to compare several bands performance or sites.", "In the right panel of Fig.", "REF monthly values of the Site Photometric Quality Ratio are shown in different colors.", "The differences between the two years are more evident in SPQR, anyway a decrement of the Site Photometric Quality Ratio towards THz regime occurs in austral winter as well as in summer periods.", "Seasonal averaged values of the Site Photometric Quality Ratio in Table REF suggest the good quality of atmospheric conditions in the low frequency bands, notably during the austral winter.", "While SPQR appears useful for comparison among different bands at Dome C it could also be employed for comparison among different sites.", "It is worth reminding that it is difficult to quantify for SPQR a threshold value to discriminate the goodness of a site.", "Two outcomes can be gathered from this analysis.", "If we believe in the transmission values, as derived by this semi-empirical approach, a continuous atmospheric sampling is mandatory at least at high frequency to contrast the low transmission stability.", "Otherwise if we consider the data derived in this work not reliable enough for an accurate estimate of atmospheric properties, we need direct observational techniques.", "In either case continuous atmospheric transparency measurements in all the spectral range of interest are necessary.", "Figure: Best fit of the correlation between the narrow-band opacitydata and atmospheric water vapour for mm windows (top panel)and for the sub-mm bands (bottom panel).Table: Opacity-pwv relation best fit parameters evaluatedfor the bands of interest." ], [ "Effect of broadband filter on optical depth estimate", "The average of the optical depth over a band, $\\tau _{\\nu _0} (\\Delta \\nu )$ , is larger than its central value $\\tau _{\\nu _0}$ so the opacity is overestimated by broadband instruments like tippers, as remarked as example by [4].", "The determination of this effect is not unique because several pwv values could give the same in-band integrated opacity.", "Low-frequency instruments are less sensitive to this degeneracy even for large values of the bandwidth due to the flatness of the corresponding atmospheric windows.", "On the other hand a sub-mm broadband instrument overestimates the opacity (underestimates the transmission) and this difference depends on the filter shape as well as on the the atmospheric conditions.", "Little variations of atmospheric conditions give rise to a dispersion of this overestimate because of the relative shapes of the atmospheric window and the corresponding filter.", "For each band in Table REF we have included the effect in the relation between the integrated zenith opacity $\\tau _{\\nu _0} (\\Delta \\nu )$ and pwv values the effect of the instrumental bandwidth $\\Delta \\nu $ (see Fig.", "REF and Table REF ): $\\tau _{\\nu _0} (\\Delta \\nu )=(a_0 + a_1 \\Delta \\nu ) + (b_0 + b_1 \\Delta \\nu ) pwv$ $a_0$ and $b_0$ are the linear fit coefficients of the $\\tau _{\\nu 0}$ vs pwv relation referred to a narrow band filter matched to the central frequency and $a_1$ and $b_1$ take into account the dependency on the instrumental bandwidth $\\Delta \\nu $ , linearly approximated at least in the range within the maximum bandwidths as reported in Table REF .", "Realistic band profiles could highlight the effect instead of our approximation with top-hat profiles.", "The net result is that the optical depth can be overestimated at most of 30 per cent at 200 $\\mu m$ , assuming the pwv best quartile from Table REF , while low frequency windows are less sensitive to this effect, as expected (10 per cent at 150 GHz).", "The uncertainty related to the optical depth value due to the intrinsic scatter of the $\\tau _0$ vs pwv relation, can be approximated by a linear trend as a function of the instrumental bandwidth: $\\sigma _{\\tau _{\\nu _0}} (\\Delta \\nu )=c_0 + c_1 \\Delta \\nu $ The optical depth uncertainty turns out to be 0.002 at 150 GHz and rise up to 0.3 at 200 $\\mu m$ , assuming the dispersion independent on pwv value (see Fig.", "REF ).", "As a consequence the percentage uncertainty on optical depth estimate is about 15 per cent all over the considered atmospheric windows assuming the best pwv quartile and it remains above 10 per cent even assuming the 75 per cent quartile in Table REF .", "The six parameters corresponding to the seven bands are listed in Table REF .", "The Eq.", "REF is useful to infer the atmospheric opacity at the preferred frequency, with a specific bandwidth, when the pwv content is known, but it is important to remind that this relation is appropriate only in the environs of Dome C. In [22] the opacity is related to the atmospheric pwv by means of the MOLIERE model.", "The resulting linear regression of the pwv as a function of the 200 $\\mu m$ opacity and the corresponding best fit parameters in Table REF , neglecting $a_1$ and $b_1$ , gives less than 5 per cent difference in transmission for low pwv values.", "Such a gap could be easily included in the atmospheric performance variations observed at Dome C over the years.", "Also the difference in transmission evaluated for 220 GHz best fit parameters in Table REF and $\\tau _0(225GHz)$ -pwv linear fit in [24] is lower then 4 per cent." ], [ "Conclusions", "The quality of cosmological and astrophysical measurements performed from ground based observational sites in the mm and sub-mm wavelength regions are strongly dictated by the atmospheric performance.", "The simultaneous measurement of atmospheric transparency and transmission fluctuations, i.e.", "emission stability, is a necessary condition to determine the true capabilities of the site of interest.", "We try to monitor the atmosphere across a wide spectral range, mm and sub-mm, with a semi-empirical approach.", "The transmission at Dome C is inferred by generating ATM synthetic spectra as derived by radiosounding data in the period from May 2005 to January 2007.", "Excellent performance is evident in the low frequency bands while large emission fluctuations are present in the high frequency bands.", "In fact even if the median winter transmission is large in all the considered atmospheric windows, daily atmospheric emission fluctuations are not negligible and become remarkable in the sub-mm range.", "In addition, large timescales fluctuations of the atmospheric performance have been detected during two consecutive years.", "The ratio between monthly averaged transmission and the corresponding fluctuations, defined Site Photometric Quality Ratio, turns out to be an efficient estimator to rank the photometric performance of the atmosphere, in terms of stability, above Dome C, as well as any observational site.", "It allows to verify when high transmission as well as low skynoise requirements are satisfied for the atmospheric window of interest.", "The SPQR threshold for each band is not easily defined: it is depending on the detectors architecture and on the adopted observational strategy.", "We attempted to validate the proposed semi-empirical approach comparing pwv and transmission quartiles with other site-testing campaigns performed at Dome C during the last years also at different wavelengths.", "In the usual linearly dependent opacity-pwv relation, we include the effect due to the bandwidth of the monitor instrument.", "Anyway only direct and frequent measurements of atmospheric transmission in a wide spectral range can provide a perfect knowledge of atmospheric influence on astronomical observations.", "If the opacity measurements are done in a narrow (a few MHz) spectral coverage, it is impossible to distinguish between clear sky opacity, hydrometeors contributions, and systematic errors.", "A wide frequency coverage (several hundreds of GHz) is necessary to make sure we are in clear sky conditions and no instrumental offset is affecting our measurement and our analysis.", "In this way it is also possible to determine the dry and the wet continuum terms, see [15].", "A large spectral sampling can be achieved at the price of a bit complex instrument.", "The possibility to monitor the atmosphere towards different positions in the sky, also avoids bias due to a spatial model assuming the multi layers approximation.", "A dedicated spectrometer, like the one proposed for Dome C ([5]) and in operation at Testa Grigia station (3500 m a.s.l., Alps, Italy) in a spectrally limited version (100 $\\div $ 360 GHz), CASPER 2 ([7] and De Petris et al.", "in prep.", "), is a viable solution." ], [ "Acknowledgments", "We acknowledge Andrea Pellegrini and Paolo Grigioni for supplying us the radiosounding data and information obtained from IPEV/PNRA Project `Routine Meteorological Observation at Station Concordia - www.climantartide.it' and Daniela Galilei for contributing to the preliminary radiosoundings data analysis." ], [ "Correction of the radiosonde data and calculations of Precipitable Water Vapour.", "The meteorological data were obtained at Dome C using two Vaisala radiosonde models: (i) the RS92 model for 430 measurement days, i.e.", "for 94 per cent of the overall days, and (ii) the RS80-A model for 29 radiosonde launches only.", "Each triplet of signals giving the measurements of $P$ , $T$ and $RH$ at a certain level was sent by the transmitter onboard the radiosonde to the ground station every 2 s. Considering that the radiosonde ascent rate was in general 5 - 6 m s$^{-1}$ , the triplets of signals were recorded in altitude steps of 10 - 12 m. The main characteristics of the three sensors (Barocap, Thermocap, and Humicap) mounted on the two radiosonde models are available in Table 1 of [20], where their measurement range, resolution, accuracy, repeatability in calibration, and reproducibility in sounding are given.", "The measurements of $P$ , $T$ and $RH$ provided by the radiosonde sensors were all corrected following the procedure defined by [20], which consists of numerous steps adopted to minimize the errors due to: the not correct calibration of the Barocap sensors; the effects caused by solar and infrared radiation heating, heat conduction and ventilation on the Thermocap sensors; lag errors, ground-check errors, and dry biases of the Humicap sensors due to basic calibration model, chemical contamination, temperature dependence and sensor aging, corrected according to [25].", "This procedure substantially differs from that defined by [19] only in the parts regarding the correction of solar heating dry biases for both A- and H-Humicap sensors: (1) those of the A-Humicap sensor were corrected by [20] using the algorithm of [23], while [19] preferred to use the algorithm derived more recently by [3]; and (2) those of the H-Humicap sensor were corrected by [20] using the average correction factors proposed by [12] as a function of solar zenith angle, while [19] employed the pair of day-time and night-time correction algorithms of [11].", "A large part of the few percent discrepancies found in the comparison shown in Fig.", "REF between the present values of pwv and those determined by [19] arise from the use of these different algorithms in correcting the instrumental and solar heating dry biases affecting the field measurements of $RH$ .", "Using the correction procedures previously described, the daily vertical profiles of pressure $P(z)$ , temperature $T(z)$ and $RH(z)$ were determined at fixed levels above the surface-level, in regular steps of 25 m from 3.25 to 4 km, 50 m from 4 to 5 km, and 100 m from 5 to 8 km amsl.", "In order to calculate the values of absolute humidity $q(z)$ at the same fixed levels, the following procedure was adopted, consisting of the three steps: (i) calculation at each level of the saturation vapour pressure $E(T)$ in the pure phase over a plane surface of pure water, using the well-known [2] formula; (ii) calculation at each level of the water vapour partial pressure $e(z)$ as the product $E(T) \\times RH(z)$ ; (iii) calculation at each level of absolute humidity $q(z)$ (measured in $g\\ m^{-3}$ ) in terms of the well-known equation of state of water vapour, and, hence, as the ratio between $e(z)$ (in hPa) and the product $R_w \\times T(z)$ (in K), in which the water vapour gas constant $R_w = 0.4615\\ J\\ g^{-1}\\ K^{-1}$ is put in place of the constant $R_a = 0.287\\ J\\ g^{-1}\\ K^{-1}$ used in the equation of state for dry air.", "For all the 469 daily vertical profiles of $q(z)$ obtained using the above procedure, the values of pwv were then calculated by integrating each vertical profile of $q(z)$ from the surface-level to 8 km amsl (i.e.", "up to 4.767 km above the ground level)." ] ]

[ [ "Geometric phases in superconducting qubits beyond the\n two-level-approximation" ], [ "Abstract Geometric phases, which accompany the evolution of a quantum system and depend only on its trajectory in state space, are commonly studied in two-level systems.", "Here, however, we study the adiabatic geometric phase in a weakly anharmonic and strongly driven multi-level system, realised as a superconducting transmon-type circuit.", "We measure the contribution of the second excited state to the two-level geometric phase and find good agreement with theory treating higher energy levels perturbatively.", "By changing the evolution time, we confirm the independence of the geometric phase of time and explore the validity of the adiabatic approximation at the transition to the non-adiabatic regime." ], [ "Geometric phases in superconducting qubits beyond the two-level-approximation S. Berger [email protected] Department of Physics, ETH Zurich, CH-8093 Zurich, Switzerland M. Pechal Department of Physics, ETH Zurich, CH-8093 Zurich, Switzerland S. Pugnetti NEST, Scuola Normale Superiore and Istituto Nanoscienze – CNR, 56126 Pisa, Italy A. A. Abdumalikov Jr.", "Department of Physics, ETH Zurich, CH-8093 Zurich, Switzerland L. Steffen Department of Physics, ETH Zurich, CH-8093 Zurich, Switzerland A. Fedorov Department of Physics, ETH Zurich, CH-8093 Zurich, Switzerland A. Wallraff Department of Physics, ETH Zurich, CH-8093 Zurich, Switzerland S. Filipp Department of Physics, ETH Zurich, CH-8093 Zurich, Switzerland 03.65.Vf, 03.67.Lx, 42.50.Pq, 85.25.Cp Geometric phases, which accompany the evolution of a quantum system and depend only on its trajectory in state space, are commonly studied in two-level systems.", "Here, however, we study the adiabatic geometric phase in a weakly anharmonic and strongly driven multi-level system, realised as a superconducting transmon-type circuit.", "We measure the contribution of the second excited state to the two-level geometric phase and find good agreement with theory treating higher energy levels perturbatively.", "By changing the evolution time, we confirm the independence of the geometric phase of time and explore the validity of the adiabatic approximation at the transition to the non-adiabatic regime.", "When a quantum mechanical system evolves under a time-dependent Hamiltonian, its wavefunction acquires a geometric phase in addition to the dynamic phase.", "[1] If the evolution of the Hamiltonian is cyclic and adiabatic, the geometric phase is termed Berry's phase.", "[2] While the dynamic phase is proportional to the time integral of the system energy, Berry's phase simply depends on the geometry of the path the Hamiltonian traces out in its parameter space and is independent of time and energy.", "It has been observed in a variety of systems, ranging from photons in optical fibers [3] and nuclear magnetic resonance (NMR) [4] to superconducting circuits [5], [6], superconducting charge pumps[7] and electronic harmonic oscillators.", "[8] The purported resilience of the geometric phase against certain types of noise, studied both in theory [9], [10], [11] and experiment [5], [12], [13] has raised interest for implementing geometric gates in quantum information processing.", "Recently, the geometric aspects of multi-level systems have attracted increased attention.", "The geometric phase has been observed in NMR interferometry in a three-level system.", "[14] A superconducting phase qudit has been employed as an effective four-level system to show the symmetry of spinors under $2\\pi $ -rotations, which can be interpreted as a geometric phase.", "[6] It has also been proposed to use three-level systems to detect non-Abelian geometric phases [15] and schemes to perform non-adiabatic holonomic quantum computation have been studied theoretically.", "[16] Here, we present measurements of the effect of higher energy levels on the controlled accumulation of geometric phases in a superconducting qubit system.", "In contrast to previous measurements of the geometric phase in superconducting circuits, where a Cooper-pair-box (CPB) qubit was used,[5] we employ a transmon-type qubit.", "[17] It is characterised by an increased ratio $E_J/E_C$ of Josephson energy to charging energy of the qubit, a smaller charge dispersion [18] and a reduced anharmonicity, typically only a few hundred MHz.", "As a consequence, the $|1\\rangle \\leftrightarrow |2\\rangle $ transition of frequency $\\omega _{12}$ is in close vicinity to the $|0\\rangle \\leftrightarrow |1\\rangle $ transition of frequency $\\omega _{01}$ [Fig.", "REF (a)].", "Figure: (Color online)(a) Schematic energy level diagram of the transmon, with the computational subspace spanned by |0〉|0\\rangle and |1〉|1\\rangle , and the second excited state |2〉|2\\rangle with anharmonicity α 2 <0\\alpha _2<0.", "(b) Lumped element circuit diagram of the sample and the measurement setup (see text for details).", "(c) Optical microscope image of the sample with two transmons coupled to a coplanar waveguide resonator with individual capacitively coupled microwave gate lines.", "(d) Close-up of the transmon used in the experiments.Therefore, whereas the CPB qubit can safely be approximated as a two-level system, the transmon cannot.", "Higher transmon levels affect the qubit dynamics,[19], [20] but can also be employed as a resource for quantum gates,[21] to improve single-shot readout,[22] or to implement a single-photon router.", "[23] Quantum optical experiments involving three levels have been carried out [24], [25], [26] and their controlled preparation and tomography has been demonstrated.", "[27] Here, we find that the difference in the level structure between the transmon and an effective two-level system also significantly affects the geometric phase.", "Time-independent perturbation theory is used to successfully model the geometric phase of a transmon.", "Then, the dependence of the geometric phase on evolution time in both the adiabatic and non-adiabatic regime is analysed.", "Non-adiabatic corrections to the geometric phase are observed and explained by simulating unitary dynamics.", "For manipulation and readout, the transmon is coupled to a coplanar transmission line resonator of quality factor $Q=2155$ via a gate capacitance $C_{\\mathrm {g}}$ .", "The lumped-element circuit diagram of the measurement setup, as well as an optical microscope image of the sample, is shown in Fig.", "REF (b-d).", "The sample is mounted in a dilution refrigerator and operated at a temperature of $20\\, \\mathrm {mK}$ .", "The transition frequency of the transmon is tunable by externally applied magnetic flux $\\Phi $ , generated using superconducting coils mounted underneath the sample.", "The transmon state is manipulated using resonant and off-resonant microwave tones (of frequency $\\omega _{01}$ and $\\omega _{01}-\\Delta $ , respectively) created by AC modulation of an in-phase/quadrature-mixer.", "This provides individual control of both quadratures ($x$ and $y$ ) of the driving microwave signal, which couples capacitively to the transmon via a local microwave gate line.", "Figure: (Color online)(a) Sketch of the microwave pulse sequence used in a geometric phase measurement, consisting of a series of resonant (ω 01 \\omega _{01}) and off-resonant (ω 01 -Δ\\omega _{01}-\\Delta ) pulses applied either on the x-quadrature (blue solid line) or the y-quadrature (green dotted line).", "The geometric phase is generated by the adiabatic evolution of the qubit between the resonant pulses.", "(b) Simulated adiabatic evolution (blue line) of the ground state subjected to the off-resonant drive along the path C - C^-, visualised in the Hilbert space of a three-level system and on the Bloch sphere, the approximate two-level equivalent (see text for details).", "(c) Extracted phase γ\\gamma as a function of solid angle AA.", "Shown is the experimental data for C -+ C^{-+} (circles), C -- C^{--} (diamonds) and C +- C^{+-} (triangles), as well as the geometric phase obtained with second-order perturbation theory (solid lines) and the prediction for a two-level system (dashed lines).", "The off-resonant pulses were applied with detuning Δ/2π=-35 MHz \\Delta /2\\pi =-35\\, \\mathrm {MHz}.", "(d) Extracted phase γ\\gamma as a function of detuning Δ\\Delta .", "The experimental data for solid angles A≈π/4,3π/4,A\\approx \\pi /4,3\\pi /4, and 5π/45\\pi /4 [indicated by circles, triangles and diamonds, respectively, and also indicated by arrows in (c)] is shown alongside the geometric phase calculated using second-order perturbation theory (solid lines) and the prediction for a two-level system (dashed lines).From spectroscopic measurements, we have determined a maximum Josephson energy $E_J/h=13.96\\, \\mathrm {GHz}$ and a charging energy $E_C/h=0.36\\, \\mathrm {GHz}$ , which corresponds to a maximum transition frequency $\\omega _{01,\\mathrm {max}}/2\\pi =5.95\\, \\mathrm {GHz}$ , and a coupling strength $g/2\\pi =360\\, \\mathrm {MHz}$ to the fundamental mode of the resonator.", "To reduce the Purcell effect and optimise coherence properties, $\\omega _{01,\\mathrm {max}}$ was designed to lie below the fundamental mode of the resonator.", "[28] The experiment is carried out in the dispersive regime, where the transmon, biased at $\\omega _{01,\\mathrm {max}}$ , is non-resonantly coupled to the resonator (at frequency $\\omega _r/2\\pi =6.662\\, \\mathrm {GHz}$ with the qubit in the ground state) and can be read out via a state-dependent change in the microwave tone of frequency $\\omega _r$ transmitted through the resonator.", "[29] Since $E_J/E_C=39\\gg 1$ and the anharmonicity is $\\alpha _2/2\\pi =-423\\, \\mathrm {MHz}$ , our sample is operated well within the transmon regime.", "Charge dispersion is expected to amount to about $100\\, \\mathrm {kHz}$ for $\\omega _{01}$ and about $2.9\\, \\mathrm {MHz}$ for $\\omega _{12}$ .", "We have measured an energy relaxation time $T_1=0.84\\, \\mathrm {\\mu s}$ , a phase coherence time $T_2^*=1.03\\, \\mathrm {\\mu s}$ and a spin-echo phase coherence time $T_2^{\\mathrm {echo}}=1.11\\, \\mathrm {\\mu s}$ .", "We consider the Hamiltonian of a driven $n$ -level qubit in a frame corotating at the drive frequency $\\omega _{d}$ , $H(\\varphi )&=\\hbar \\sum _{j=0}^n (j\\Delta +\\alpha _j)|j\\rangle \\langle j| \\\\&\\quad +\\dfrac{1}{2}\\hbar \\Omega \\sum _{j=0}^{n-1}(\\sqrt{j+1}e^{-i\\varphi }|j+1\\rangle \\langle j| +\\mathrm {h.c.}), \\nonumber $ where $\\alpha _j$ is the anharmonicity defined through the energy of the $j$ th qubit energy level $\\omega _{0j}=j\\omega _{01}+\\alpha _j$ (with $\\alpha _0=\\alpha _1=0$ ), $\\Omega $ is the strength of the drive expressed in units of angular frequency and $\\varphi $ its phase.", "$\\Delta =\\omega _{01}-\\omega _d$ denotes the detuning between the frequency of the $|0\\rangle \\leftrightarrow |1\\rangle $ transition and the drive frequency.", "The increase of the coupling strength $\\propto \\sqrt{j+1}$ follows from the off-diagonal matrix elements of the charge operator.", "[17] In the two-level approximation, we restrict the Hamiltonian to the computational subspace of the qubit spanned by $|0\\rangle $ and $|1\\rangle $ : $H\\approx \\frac{\\hbar }{2}(\\Omega _x\\sigma _x+\\Omega _y\\sigma _y+\\Delta \\sigma _z)=\\frac{\\hbar }{2}\\sigma \\cdot \\mathbf {B},$ with $\\Omega _x=\\Omega \\cos \\varphi $ and $\\Omega _y=\\Omega \\sin \\varphi $ .", "This is precisely the Hamiltonian of a spin-half particle in an effective magnetic field $\\mathbf {B}=(\\Omega _x,\\Omega _y,\\Delta )$ .", "[5] By applying microwave frequency drives, we are able to experimentally control all three parameters of this effective field, which is our parameter space, to guide the qubit along a circular path $C$ with constant detuning $\\Delta $ .", "At first, the drive field is ramped up, tilting $\\mathbf {B}$ so that it forms an angle $\\vartheta =\\arctan (\\Omega /\\Delta )$ with respect to the $z$ -axis.", "Then, the phase of the drive is swept by $2\\pi $ , causing $\\mathbf {B}$ to rotate once around the $z$ -axis, either clockwise or anticlockwise.", "Finally, the drive is ramped back to zero.", "The time evolution of both quadratures of the drive is plotted in Fig.", "REF (a).", "During this sequence, the solid angle subtended by the path as seen from the origin is $A=2\\pi (1-\\cos \\vartheta )$ .", "We then repeat the measurement for different driving strengths $\\Omega $ , thereby changing the solid angle $A$ .", "To determine the geometric phase experimentally, we employ an interferometric measurement [Fig.", "REF (a)].", "The leading and trailing resonant $\\pi /2$ -pulses implement a Ramsey measurement, while the resonant spin-echo $\\pi $ -pulse in the centre serves to cancel the dynamic phase.", "[4], [5] After preparing an equal superposition of the $|0\\rangle $ and $|1\\rangle $ states, the qubit traverses the path $C^-$ and the relative phase $2(\\gamma _d-\\gamma _g)$ acquired between $|0\\rangle $ and $|1\\rangle $ comprises both a dynamic ($\\gamma _d$ ) and a geometric ($\\gamma _g$ ) contribution.", "The spin-echo $\\pi $ -pulse then effectively flips the sign of the phase.", "As it traverses the second loop in the opposite direction, $C^+$ , the qubit acquires the phase $2(\\gamma _d+\\gamma _g)$ since the dynamic phase, unlike the geometric phase, is independent of the direction of evolution.", "Thus, after following the contours $C^{-+}$ , dynamic phase contributions cancel out and the qubit state has acquired a phase $\\gamma =4\\gamma _g$ which is purely geometric.", "Tracing out the contours in opposite direction, $C^{+-}$ , simply inverts the sign of the phase, while following the contours twice in the same direction, $C^{++}$ or $C^{--}$ , results in zero phase and serves as a control experiment [Fig.", "REF (c)].", "During the off-resonant pulse sequences $C^{\\pm }$ , the drive $\\Omega $ is strong (corresponding to induced Rabi-frequencies $\\lesssim 110\\, \\mathrm {MHz}$ ) and therefore the higher levels of the qubit are populated as well.", "In order to visualise this population leakage, we consider the Hilbert space of a three-level system.", "Neglecting a global phase, any three-level state can be written as $|\\psi \\rangle =e^{{\\mathrm {i}}\\chi _1}\\sin {\\beta _1}\\cos {\\beta _2}|0\\rangle +e^{{\\mathrm {i}}\\chi _2}\\sin {\\beta _1}\\sin {\\beta _2}|1\\rangle +\\cos {\\beta _1}|2\\rangle $ , where $\\chi _{1,2}\\in [0,2\\pi ]$ are the phases of the ground and first excited state, respectively, relative to the phase of the second excited state, and $\\beta _{1,2}\\in [0,\\pi /2]$ parametrise the populations.", "Therefore, every state can be represented as a point in the product manifold of a torus and an octant of a unit sphere.", "[30] Observing that $\\beta _1\\ne \\pi /2$ while the ground state is subjected to the off-resonant drive $C^-$ [Fig.", "REF (b)], we conclude that the instantaneous ground state leaves the computational subspace.", "The population of second excited state reaches up to $\\approx 12\\%$ , showing the necessity to consider the higher levels in our experiment.", "It is important to note that all resonant pulses effectively act in the subspace spanned by $|0\\rangle $ and $|1\\rangle $ : in order to avoid exciting higher energy levels with the resonant pulses, we use an optimal control technique known as DRAG.", "[19] Furthermore, to ensure that the second excited state $|2\\rangle $ is depleted before the resonant pulses are applied, the off-resonant drive is adiabatically ramped down.", "After qubit manipulation, which takes approximately $700\\, \\mathrm {ns}$ , the population $P_z=(1-\\sigma _z)/2$ of the first excited state is extracted by a dispersive readout.", "[29] The phase $\\gamma $ the qubit has acquired during evolution is reconstructed with state tomography (ST).", "The second $\\pi /2$ -pulse of the Ramsey sequence rotates the qubit about either the $x$ or $y$ axis, and serves as tomography pulse.", "In the absence of decoherence, the phase $\\gamma $ is given by $\\arctan \\left(\\langle \\sigma _y\\rangle /\\langle \\sigma _x\\rangle \\right)$ with $\\langle \\sigma _x\\rangle =\\cos \\gamma $ , $\\langle \\sigma _y\\rangle =\\sin \\gamma $ and $\\langle \\sigma _z\\rangle =0$ .", "The same expression approximates $\\gamma $ well even in the presence of decoherence, which reduces the size of the Bloch vector $(\\sigma _x,\\sigma _y,\\sigma _z)$ to $0.47$ in our experiments, while keeping the ratio $\\langle \\sigma _y\\rangle /\\langle \\sigma _x\\rangle $ constant.", "Therefore, the geometric phase remains unaltered by the decoherence in our experimental setting.", "In keeping with Berry's predictions for a two-level system, $\\gamma =2A$ , we measure a phase $\\gamma $ which is approximately twice the solid angle $A$ subtended by the path [Fig.", "REF (c))].", "However, the data clearly shows that $\\gamma $ increasingly deviates from Berry's predictions as $A$ (and therefore also the drive $\\Omega $ ) increases: the measured geometric phase is observed to be up to $15\\%$ larger than expected.", "These deviations can be explained by the presence of higher transmon levels.", "Defining the operator $N=\\sum _{j=0}^n j|j\\rangle \\langle j|$ , the Hamiltonian in Eq.", "(REF ) can be rewritten as $H(\\varphi )=e^{-i\\varphi N}H(0)e^{i\\varphi N}$ .", "It follows that, given an eigenvector $|\\Phi (0)\\rangle $ of $H(0)$ , $|\\Phi (\\varphi )\\rangle =e^{-i\\varphi N}|\\Phi (0)\\rangle $ is an eigenvector of $H(\\varphi )$ .", "For the circular path $C$ described above, the geometric phase $\\gamma _{\\Phi (0)}$ acquired by the eigenvector $|\\Phi (0)\\rangle $ is then found to be [2] $\\gamma _{\\Phi (0)}=i\\int _0^{2\\pi }\\langle \\Phi (\\varphi )|\\dfrac{d}{d\\varphi }|\\Phi (\\varphi )\\rangle d\\varphi =2\\pi \\langle \\Phi (0)|N|\\Phi (0)\\rangle .$ From Eq.", "(REF ), one indeed recovers the expression $\\gamma _{\\pm }^{(0)}=\\pi (1\\pm \\cos \\vartheta )$ for the geometric phase of a two-level system, where the sign $\\pm $ corresponds to the positive and negative eigenvalue of $H(0)$ , respectively.", "To compute the geometric phase for a multi-level system, we divide the Hamiltonian $H(0)=H_0+V$ into a block-diagonal part $H_0$ coupling the lowest two transmon levels, and a perturbative part $V$ coupling the remaining levels: $H_0&=\\hbar \\sum _{j=0}^n(j\\Delta +\\alpha _j)|j\\rangle \\langle j|+\\dfrac{\\hbar \\Omega }{2} (|1\\rangle \\langle 0| +\\mathrm {h.c.}),\\nonumber \\\\V &=\\dfrac{\\hbar \\Omega }{2} \\sum _{j=1}^{n-1}(\\sqrt{j+1}|j+1\\rangle \\langle j|+\\mathrm {h.c.}).", "\\nonumber $ Substituting the expansion of the eigenvectors in V, $|\\Phi _j\\rangle =|\\Phi _j^{(0)}\\rangle +|\\Phi _j^{(1)}\\rangle +|\\Phi _j^{(2)}\\rangle +\\ldots $ , into Eq.", "(REF ), and retaining terms up to second order, the second excited state is found to contribute $\\Delta \\gamma _{\\pm }&=& \\gamma _{\\pm }-\\gamma _{\\pm }^{(0)} = \\pi k\\sin ^2\\vartheta \\\\ \\nonumber & & \\times \\frac{2k(1\\pm \\cos \\vartheta )+(2k\\mp (3k+2)\\cos \\vartheta )\\sin ^2\\vartheta }{(k\\mp (3k+2)\\cos \\vartheta )^2}.$ to the geometric phase $\\gamma _{\\pm }$ , where $k\\equiv \\Delta /\\alpha _2$ .", "In the experiment we measure the quantity $\\Delta \\gamma \\equiv 2(\\Delta \\gamma _{-}-\\Delta \\gamma _+)$ , see Fig.", "REF (c).", "We also note that $\\Delta \\gamma = 2\\pi k \\sin ^4\\vartheta /\\cos \\vartheta + \\mathcal {O}(k^2)$ vanishes for large $\\alpha _2$ as expected.", "Since in the experiment $k\\approx 1/8$ is small and $|\\Omega /\\alpha _2|\\ll 1$ , the expansion coefficients of $|\\Phi _j^{(1)}\\rangle $ and $|\\Phi _j^{(2)}\\rangle $ are small, and perturbative treatment is justified.", "To verify the validity of the perturbative treatment, we simulated the qubit evolution for the pulse sequence in Fig.", "REF (a), retaining four energy levels in a quantum master equation simulation, thereby taking into qubit population decay, decoherence and non-adiabatic effects arising from finite evolution time.", "[31] Also, the Hamiltonian in Eq.", "(REF ) with $n=4$ was numerically diagonalised to compute the geometric phase in the limit of perfect adiabaticity.", "We found that the perturbatively computed geometric phase $\\gamma $ using Eq.", "(REF ) differs from both simulations and numerical results by less than $2\\%$ .", "Figure: (Color online)(a) Measured phase γ\\gamma as function of phase sweep time τ\\tau for the solid angle A=π/4A=\\pi /4 at detuning Δ/2π=-45 MHz \\Delta /2\\pi =-45\\, \\mathrm {MHz}.", "The dashed line is the phase obtained by numerically calculating the unitary time evolution of a four level transmon using the Schrödinger equation.", "The adiabaticity parameters aa for a given τ\\tau are indicated on the upper horizontal axis.", "(b) Fidelity of the geometric phase gates shown in (a) as a function of τ\\tau .Furthermore, we have measured $\\gamma $ for a range of detunings $\\Delta $ and have found good agreement with the geometric phase computed using perturbation theory [Fig.", "REF (d)].", "We have verified that the geometric phase does not depend on the dispersive coupling of the transmon to the resonator by tuning $\\omega _{01}$ such that $\\delta =\\omega _r-\\omega _{01}=1.58\\, \\mathrm {GHz}$ and comparing the data to the case shown in Fig.", "REF (c), where $\\delta =0.71\\, \\mathrm {GHz}$ .", "Finally, the transition from the adiabatic regime to the non-adiabatic regime was examined by measuring the acquired phase $\\gamma $ at fixed solid angle $A=\\pi /4$ , changing only the phase sweep time $\\tau $ [Fig.", "REF (a)] but keeping the duration of the total sequence constant.", "The data in Fig.", "REF (a) shows that $\\gamma $ is constant, i.e., independent of evolution time for $\\tau $ larger than about $50\\, \\mathrm {ns}$ , or equivalently, for adiabaticity parameters $a=\\dot{\\varphi }\\sin (\\vartheta )/|\\mathbf {B}|\\lesssim 0.1$ .", "The measured $\\gamma $ also agrees well with the result obtained by numerically solving the Schrödinger equation.", "For values of $a>0.1$ , when entering the non-adiabatic regime, we observe that $\\gamma $ oscillates and varies by more than $50\\%$ .", "In this regime, $\\gamma $ is a combination of dynamic and geometric phase because the spin-echo technique fails to cancel the dynamic phase: for non-adiabatic evolution, the state after the spin-echo $\\pi $ -pulse does not necessarily correspond to the initial state with $|0\\rangle $ and $|1\\rangle $ interchanged.", "In the context of quantum information processing, the manipulation sequence could serve as a single qubit phase gate.", "Its performance can be assessed by computing the fidelity $F=\\mathrm {tr}\\sqrt{\\rho ^{1/2}\\sigma \\rho ^{1/2}}$ , where $\\rho $ is the experimental density matrix processed with maximum likelihood [32] and $\\sigma $ is the expected density matrix for perfectly adiabatic evolution.", "We find that the fidelity of the gate averages $F=90\\%$ in the adiabatic regime [Fig.", "REF (b)].", "There, about $8\\%$ of the loss in fidelity can be attributed to qubit decay, whereas inaccuracies in qubit preparation and qubit dephasing account for the remaining $2\\%$ .", "In the non-adiabatic regime a significant decrease in fidelity is observed, as expected.", "In conclusion, we have measured the geometric phase in a multi-level system with small anharmonicity and observed that the two-level approximation breaks down for strong drives, as evidenced in our experiment by deviations of the geometric phase from the expected linear dependance on solid angle.", "We have modelled the contributions from the second excited state to the adiabatic geometric phase using time-independent perturbation theory.", "By examining Berry's phase in the adiabatic limit and going to the non-adiabatic regime, we have shown that it is independent of evolution time for adiabaticity parameters $\\lesssim 0.1$ .", "The phase in the non-adiabatic limit could potentially inherit some of the adiabatic phase's noise resilience, suggesting further experimental tests on how it is affected by noise in the control parameters.", "This work was supported by the Swiss National Science Foundation (SNF) and the EU project GEOMDISS." ] ]

[ [ "Sealing the fate of a fourth generation of fermions" ], [ "Abstract The search for the effects of heavy fermions in the extension of the Standard Model with a fourth generation is part of the experimental program of the Tevatron and LHC experiments.", "Besides being directly produced, these states affect drastically the production and decay properties of the Higgs boson.", "In this note, we first reemphasize the known fact that in the case of a light and long-lived fourth neutrino, the present collider searches do not permit to exclude a Higgs boson with a mass below the WW threshold.", "In a second step, we show that the recent results from the ATLAS and CMS collaborations which observe an excess in the $\\gamma \\gamma$ and $4\\ell^\\pm$ search channels corresponding to a Higgs boson with a mass $M_H \\approx 125$ GeV, cannot rule out the fourth generation possibility if the $H \\to \\gamma \\gamma$ decay rate is evaluated when naively implementing the leading ${\\cal O}(G_F m_{f'}^2)$ electroweak corrections.", "Including the exact next-to-leading order electroweak corrections leads to a strong suppression of the $H \\to \\gamma \\gamma$ rate and makes this channel unobservable with present data.", "Finally, we point out that the observation by the Tevatron collaborations of a $\\gsim 2\\sigma$ excess in the mass range $M_H = 115$-135 GeV in the channel $q\\bar q \\to WH \\to Wb\\bar b$ can definitely not be accommodated by the fourth generation fermion scenario.", "All in all, if the excesses observed at the LHC and the Tevatron are indeed due to a Higgs boson, they unambiguously exclude the perturbative fermionic fourth generation case.", "In passing, we also point out that the Tevatron excess definitely rules out the fermiophobic Higgs scenario as well as scenarios in which the Higgs couplings to gauge bosons and bottom quarks are significantly reduced." ], [ " LPT–Orsay 12/31 CERN-PH-TH/2012–087 Sealing the fate of a fourth generation of fermions Abdelhak Djouadi$^{1,2}$ and Alexander Lenz$^{2}$ $^1$ Laboratoire de Physique Théorique, CNRS and Université Paris–Sud, Bât.", "210, F–91405 Orsay Cedex, France.", "$^2$ Theory Unit, Department of Physics, CERN, CH-1211 Geneva 23, Switzerland.", "The search for the effects of heavy fermions in the extension of the Standard Model with a fourth generation is part of the experimental program of the Tevatron and LHC experiments.", "Besides being directly produced, these states affect drastically the production and decay properties of the Higgs boson.", "In this note, we first reemphasize the known fact that in the case of a light and long–lived fourth neutrino, the present collider searches do not permit to exclude a Higgs boson with a mass below the $WW$ threshold.", "In a second step, we show that the recent results from the ATLAS and CMS collaborations which observe an excess in the $\\gamma \\gamma $ and $4\\ell ^\\pm $ search channels corresponding to a Higgs boson with a mass $M_H\\!", "\\approx \\!", "125$ GeV, cannot rule out the fourth generation possibility if the $H\\!\\rightarrow \\!\\gamma \\gamma $ decay rate is evaluated when naively implementing the leading ${\\cal O}(G_Fm_{f^{\\prime }}^2)$ electroweak corrections.", "Including the exact next-to-leading order electroweak corrections leads to a strong suppression of the $H\\!\\rightarrow \\!", "\\gamma \\gamma $ rate and makes this channel unobservable with present data.", "Finally, we point out that the observation by the Tevatron collaborations of a $\\raisebox {-0.13cm}{~{> \\\\[-0.07cm] \\sim }}~2\\sigma $ excess in the mass range $M_H\\!=\\!", "115$ –135 GeV in the channel $q\\bar{q}\\!", "\\rightarrow \\!", "WH\\!", "\\rightarrow \\!", "Wb\\bar{b}$ can definitely not be accommodated by the fourth generation fermion scenario.", "All in all, if the excesses observed at the LHC and the Tevatron are indeed due to a Higgs boson, they unambiguously exclude the perturbative fermionic fourth generation case.", "In passing, we also point out that the Tevatron excess definitely rules out the fermiophobic Higgs scenario as well as scenarios in which the Higgs couplings to gauge bosons and bottom quarks are significantly reduced.", "One of the most straightforward extensions of the Standard Model (SM) of particle physics is to assume a fourth generation of fermions: one simply adds to the known fermionic pattern with three generations, two quarks $t^{\\prime }$ and $b^{\\prime }$ with weak–isospin of respectively $\\frac{1}{2}$ and $-\\frac{1}{2}$ , a charged lepton $\\ell ^{\\prime }$ and a neutrino $\\nu ^{\\prime }$ .", "Such an extension that we will denote by SM4, besides of being rather simple, has been advocated as a possible solution of some problems of the SM; for recent reviews and motivations for a fourth fermion generation, see Refs.", "[1], [2], [3], [4].", "For instance, from a theoretical point of view, it provides new sources of CP–violation that could explain the baryon asymmetry in the universe [3] and, from the experimental side, it might soften some tensions in flavour physics [4].", "There are, however, severe constraints on this SM4 scenario.", "First, from the invisible width of the $Z$ boson, the LEP experiment has measured the number of light neutrinos to be $N_\\nu =3$ with a high precision [5] and, thus, the neutrino of SM4 should be rather heavy, $m_{\\nu ^{\\prime }} \\raisebox {-0.13cm}{~{> \\\\[-0.07cm] \\sim }}~\\frac{1}{2} M_Z$ , assuming that it has a very small mixing with the lighter SM leptons (not to be produced in association with its light partners which would lead to the stronger limit $m_{\\nu ^{\\prime }} \\raisebox {-0.13cm}{~{> \\\\[-0.07cm] \\sim }}~100$ GeV).", "A heavy charged lepton with a mass $m_{\\ell ^{\\prime }}\\!\\raisebox {-0.13cm}{~{< \\\\[-0.07cm] \\sim }}~\\!", "100$ GeV has also been excluded at LEP2 [5].", "In addition, the Tevatron and now the LHC experiments have excluded too light fourth generation quarks.", "In particular, direct searches performed by the ATLAS and CMS collaborations rule out heavy down-type and up–type quarks with masses $m_{b^{\\prime }} \\raisebox {-0.13cm}{~{< \\\\[-0.07cm] \\sim }}~600$ GeV and $m_{t^{\\prime }} \\raisebox {-0.13cm}{~{< \\\\[-0.07cm] \\sim }}~560$ GeV [6].", "On the other hand, high precision electroweak data severely constrain the mass splitting between the fourth generation quarks while data from B–meson physics constrain their mixing pattern [7].", "Finally, the requirement that SM4 remains unitary at very high energies suggests that fourth generation fermions should not be extremely heavy, $m_{q^{\\prime }} \\raisebox {-0.13cm}{~{< \\\\[-0.07cm] \\sim }}~500$ GeV [8].", "However, this bound should not be viewed as a strict limit but simply as an indication that strong dynamics takes place; a degenerate quark doublet with a mass $m_{q^{\\prime }} \\approx 700$ GeV has been considered in a simulation of a strong Yukawa coupling regime on the lattice [9].", "Thus, ATLAS and CMS direct searches for $t^{\\prime },b^{\\prime }$ SM4 quarks are closely approaching the masses required by the perturbative unitarity bound and we will assume here that $m_{t^{\\prime }}\\!", "\\approx \\!", "m_{b^{\\prime }}\\!", "\\pm \\!50\\;{\\rm GeV}\\!", "\\sim \\!", "650\\;$ GeV.", "Strong constraints on SM4 can be also obtained from Higgs searches at the Tevatron and the LHC.", "Indeed, it is known since a long time [10] that in the loop induced Higgs–gluon and Higgs-photon vertices, $Hgg$ and $H\\gamma \\gamma $ , any heavy particle coupling to the Higgs boson proportionally to its mass, as is the case in SM4, will not decouple from the amplitudes and would have a drastic impact.", "In particular, for the $gg\\rightarrow H$ process [11], [12] which is the leading mechanism for Higgs production at both the Tevatron and the LHC, the additional contribution of the two new SM4 quarks $t^{\\prime }$ and $b^{\\prime }$ will increase the rate by a factor of $K_{gg\\!\\rightarrow \\!H}^{\\rm SM4} \\approx 9$ .", "At leading order in the electroweak interaction, this factor is a very good approximation [13], as long as the heavy quarks are such that $m_{q^{\\prime }} \\raisebox {-0.13cm}{~{> \\\\[-0.07cm] \\sim }}~\\frac{1}{2} M_H$ which holds true for any $M_H$ value below the TeV scale, given the experimental bounds on the $q^{\\prime }$ masses.", "The Higgs searches at the Tevatron and the LHC, which are now becoming very sensitive, should therefore severely constrain the SM4 possibility [14].", "Indeed, the CDF and D0 experiments for instance exclude a Higgs boson in this scenario for masses $124\\;{\\rm GeV}\\!", "\\raisebox {-0.13cm}{~{< \\\\[-0.07cm] \\sim }}~\\!", "M_H \\raisebox {-0.13cm}{~{< \\\\[-0.07cm] \\sim }}~286\\;$ GeV by considering mainly the $gg\\!", "\\rightarrow \\!", "H \\!\\rightarrow \\!", "WW \\!", "\\rightarrow \\!", "2\\ell 2\\nu $ channel [15].", "The LHC experiments recently extended this exclusion limit up to $M_H\\!", "\\approx \\!600$ GeV (at 99% CL) by exploiting also the $gg \\!", "\\rightarrow \\!", "H \\!\\rightarrow ZZ \\!", "\\rightarrow \\!", "4\\ell , 2\\ell 2\\nu , 2\\ell 2j$ search channels [16].", "Nevertheless there are two caveats which might loosen these experimental limits.", "The first one is that the electroweak radiative corrections to the $gg\\rightarrow H$ process turn out to be significant [17], [18], [19].", "For a specific choice of fermion masses which approximately fulfills the electroweak precision constraints [2], $m_{b^{\\prime }}\\!", "= \\!", "m_{t^{\\prime }}+50\\;{\\rm GeV}= \\!", "m_{\\ell ^{\\prime }} \\!", "= \\!", "m_{\\nu ^{\\prime }}\\sim 600 $ GeV, they lead to an increase (decrease) of the cross section at low (high) Higgs masses, $M_H\\!\\approx \\!", "120\\;(600)$ GeV, by $\\approx 12\\%$ implying that the exclusion limits above need to be updated and changes in the excluded $M_H$ range up to 10 GeV are expected [19].", "The second caveat is that the Tevatron and LHC Higgs exclusion limits in SM4 are only valid for a heavy neutrino $\\nu ^{\\prime }$ .", "Indeed, if $m_{\\nu ^{\\prime }} \\raisebox {-0.13cm}{~{< \\\\[-0.07cm] \\sim }}~\\frac{1}{2}M_H$ , the Higgs boson will also decay into a neutrino pair [20] and the branching ratio BR($H\\rightarrow \\nu ^{\\prime } \\bar{\\nu }^{\\prime }$ ) can be sizable enough to suppress the rates for the visible channels such as $H\\rightarrow WW,ZZ$ by which the Higgs is searched for.", "This is particularly the case for a light Higgs, $M_H \\raisebox {-0.13cm}{~{< \\\\[-0.07cm] \\sim }}~160$ GeV, which mainly decays into $b$ –quark pairs and $W$ bosons (with one $W$ being virtual).", "The Higgs total width is small in this case, $\\Gamma _H \\raisebox {-0.13cm}{~{< \\\\[-0.07cm] \\sim }}~1$ GeV, making the invisible channel $H\\!\\rightarrow \\!", "\\nu ^{\\prime } \\bar{\\nu }^{\\prime }$ dominant.", "Using the program HDECAY [21] in which the Higgs decays in SM4, with all known QCD [13] as well as the leading ${\\cal O}(G_Fm_{f^{\\prime }}^2)$ electroweak and ${\\cal O}(G_F m_{q^{\\prime }}^2\\alpha _s)$ mixed corrections derived in Ref.", "[22] have been (naively) implementedFor the $H \\rightarrow gg, f\\bar{f}$ and $VV$ decays, the ${\\cal O}(G_F m_{f^{\\prime }}^2)$ terms when implemented by simply multiplying the couplings $g_{HXX}$ by the electroweak correction $1+\\delta _{EW}^X$ , should represent a good approximation [23].", "A fourth generation of fermions with degenerate $t^{\\prime },b^{\\prime },\\ell ^{\\prime }, \\nu ^{\\prime }$ masses $m_{f^{\\prime }} \\approx 300\\;(600)\\;$ GeV will suppress the $HVV$ coupling by $\\approx 10\\%\\;(40\\%)$ and, hence, the rate for the $H\\rightarrow VV$ decay (which grows like the square of the coupling) by $20\\%\\;(80\\%)$ [22].", "However, in the $H\\gamma \\gamma $ amplitude, this approximation leads to an unstable result and some reordering of the perturbative series is needed [19] as will be discussed later., we exemplify this feature in Fig.", "1 where the Higgs decay branching ratios into $VV$ states normalised to their SM values, BR$(H\\!", "\\rightarrow \\!", "VV)|_{\\rm SM4/SM}$ , are shown as a function of $m_{\\nu ^{\\prime }}$ for $M_H\\!=\\!125\\;$ GeV (with this normalisation, these ratios are the same for $V\\!=\\!W$ and $Z$ ).", "One first observes that for $m_{\\nu ^{\\prime }} \\raisebox {-0.13cm}{~{> \\\\[-0.07cm] \\sim }}~\\frac{1}{2} M_H$ , the ${\\cal O}(G_F m_{\\ell ^{\\prime }}^2)$ corrections suppress the rate for $H \\!\\rightarrow \\!", "VV$ decays while they increase the one for the $H\\rightarrow \\gamma \\gamma $ channel.", "In addition, one can see that for a heavy neutrino $\\nu ^{\\prime }$ , say $m_{\\nu ^{\\prime }}=300$ GeV, BR$(H\\rightarrow WW,ZZ)$ are suppressed by only a factor of $\\approx 5$ compared to their SM values, as a result of the additional $t^{\\prime },b^{\\prime }$ contributions.", "However, when the $H\\rightarrow \\nu ^{\\prime }\\bar{\\nu }^{\\prime }$ decay channel is kinematically allowed, i.e.", "for $\\frac{1}{2} M_Z\\raisebox {-0.13cm}{~{< \\\\[-0.07cm] \\sim }}~m_{\\nu ^{\\prime }} \\raisebox {-0.13cm}{~{< \\\\[-0.07cm] \\sim }}~\\frac{1}{2} M_H$ , BR$_{\\rm SM4/SM}$ is further suppressed and for a given neutrino mass, the suppression factor is comparable to or even larger than the factor $\\approx 9$ due to the increase of the $gg\\rightarrow H$ cross section by the $t^{\\prime },b^{\\prime }$ loop contributions.", "Thus, the rate for the processes $gg\\!\\rightarrow \\!", "H\\!", "\\rightarrow \\!", "WW,ZZ$ can be smaller in SM4 compared to the SM and, hence, the Tevatron and LHC exclusion limits of Refs.", "[15], [16], which are obtained using these processes, can be evadedNote that for larger Higgs mass values, $M_H \\raisebox {-0.13cm}{~{> \\\\[-0.07cm] \\sim }}~180$ GeV, the $H\\rightarrow WW,ZZ$ partial widths become large and these decays are by far dominant and thus not affected by the presence of the $H\\rightarrow \\nu ^{\\prime }\\bar{\\nu }^{\\prime }$ channel.", "The present Tevatron and LHC exclusion limits are valid in this case, modulo the impact of the electroweak corrections to the production and decay processes which need to be included..", "Figure: The decay branching ratios of a 125 GeV Higgs particleinto gg,bb ¯,γγgg,b\\bar{b}, \\gamma \\gamma and VVVV states (with V=W,ZV\\!=\\!W,Z) inSM4 normalised to their SM values as a function of the neutrinomass.", "The heavy quark masses are set to m b ' =m t ' +50 GeV =600m_{b^{\\prime }}\\!=\\!m_{t^{\\prime }}\\!+\\!50~{\\rm GeV}\\!=\\!600 GeV, while the charged lepton mass is m ℓ ' =m ν ' +50 GeV m_{\\ell ^{\\prime }}\\!=\\!m_{\\nu ^{\\prime }}\\!+\\!50~{\\rm GeV}.", "The electroweak corrections are included in a naive wayin H→γγH\\rightarrow \\gamma \\gamma .Let us now discuss, in the context of SM4, the excess of events recently observed by the ATLAS and CMS collaborations in the $H \\!", "\\rightarrow \\!", "ZZ \\!", "\\rightarrow \\!4\\ell ^\\pm $ and $H\\!\\rightarrow \\!", "\\gamma \\gamma $ channels corresponding to a SM–like Higgs boson with $M_H \\!", "\\approx \\!125$ GeV [24] .", "First of all, for the value $M_H= 125$ GeV, while BR$(H\\!", "\\rightarrow \\!", "ZZ)|_{\\rm SM4/SM}$ is different from unity as a result of the ${\\cal O}(G_F m_{f^{\\prime }}^2)$ corrections, the enhancement of the $H\\rightarrow gg$ rate by the $t^{\\prime },b^{\\prime }$ contributions and eventually the opening of the $H\\!\\rightarrow \\!", "\\nu ^{\\prime } \\bar{\\nu }^{\\prime }$ mode, the situation is more complicated in the case of BR$(H\\!", "\\rightarrow \\!", "\\gamma \\gamma )|_{\\rm SM4/SM}$ as there is another important effect.", "As a matter of fact, the $H\\!\\rightarrow \\!\\gamma \\gamma $ decay is mediated by $W$ boson and heavy fermion loops whose contributions interfere destructively.", "While this interference is mild in the SM, as the $W$ contribution is much larger than that of the top quark, it is very strong in SM4 because of the additional $t^{\\prime },b^{\\prime }$ and $\\ell ^{\\prime }$ contributions; the $W$ and all fermion contributions are then very close to each other but opposite in sign.", "This accidental cancellation makes BR$(H\\!\\rightarrow \\!\\gamma \\gamma )|_{\\rm SM4/SM}$ much smaller than BR$(H\\!", "\\rightarrow \\!", "VV)|_{\\rm SM4/SM}$ in general, with consequences summarized below.", "It is clear from Fig.", "1 that in the presence of a relatively light neutrino, $m_{\\nu ^{\\prime }} \\raisebox {-0.13cm}{~{< \\\\[-0.07cm] \\sim }}~\\frac{1}{2} M_H$ , the rates for the $H\\rightarrow ZZ$ and $H\\rightarrow \\gamma \\gamma $ decays are strongly suppressed by a factor that is larger than the one $K_{gg\\!\\rightarrow \\!", "H}^{\\rm SM4}\\!", "\\approx \\!", "9$ which enhances the $gg\\!\\rightarrow \\!", "H$ cross section.", "Thus, the $\\gamma \\gamma $ and $4\\ell ^\\pm $ excesses corresponding to a SM–like 125 GeV Higgs cannot occur in SM4 when the channel $H\\!\\rightarrow \\!", "\\nu ^{\\prime }\\nu ^{\\prime }$ is open.", "The possibility $m_{\\nu ^{\\prime }}\\!", "\\raisebox {-0.13cm}{~{< \\\\[-0.07cm] \\sim }}~\\!", "\\frac{1}{2} M_H$ is thus strongly disfavored.", "On the other hand, when this new decay channel is closed, the rates for $H\\!\\rightarrow \\!VV,\\gamma \\gamma $ decays increase significantly.", "If one assumes heavy leptons with $m_{\\nu ^{\\prime }}\\!\\approx \\!", "m_{\\ell ^{\\prime }}\\!\\approx \\!600$ GeV (and taking into account the fact that the electroweak corrections decrease BR$(H\\!\\rightarrow \\!", "VV)$ with increasing $m_{\\nu ^{\\prime }}$ ), one accidentally obtains a suppression rate BR$_{\\rm SM4/SM} \\approx 7.5$ that is the same in both cases.", "Recalling that in this case the rate for the main Higgs production process is enhanced by a factor $K_{gg\\!\\rightarrow \\!", "H}^{\\rm SM4}\\!", "\\approx \\!", "9.5$ , one obtains $gg\\!", "\\rightarrow \\!H\\!", "\\rightarrow \\!", "\\gamma \\gamma $ and $4\\ell $ rates for $M_H=125$ GeV that are a $\\approx 20\\%$ larger in SM4 than in the SM (see also Ref.[25]).", "It happens that the excesses observed by ATLAS and CMS in the $\\gamma \\gamma $ channel are stronger than what is expected in the SM, although within the errors bands.", "Therefore, not only a fourth generation with all heavy fermions having a mass close to $m_{f^{\\prime }}\\!=\\!600$ GeV could accommodate the excesses observed at the LHC, but it could also explain the substantial rate observed by ATLAS and CMS in the $H\\rightarrow \\gamma \\gamma $ signal that has the largest significance.", "There is, however, a serious flaw in the discussion above.", "As mentioned earlier, only the leading ${\\cal O}(G_Fm_{f^{\\prime }}^2)$ terms (and the ${\\cal O}(G_Fm_{q^{\\prime }}^2 \\alpha _s)$ ones) [22] are included in the electroweak corrections to BR($H\\rightarrow \\gamma \\gamma $ ) in Fig.", "1, by simply multiplying the $W,t$ and $f^{\\prime }$ amplitudes with the relevant correction $1+\\delta _{EW}^X$ .", "The exact next-to-leading order (NLO) electroweak corrections have been very recently calculated [19] and, because of the very strong interference between the $W$ and all fermion loop contributions, they have a drastic impact on the $H\\gamma \\gamma $ vertex.", "For $m_{f^{\\prime }}\\!", "\\approx \\!", "600$ GeV, these corrections suppress BR($H\\!\\rightarrow \\!\\gamma \\gamma ) |_{\\rm SM4}$ by almost an order of magnitude, compared to the case where the ${\\cal O}(G_Fm_{f^{\\prime }}^2)$ corrections are naively implemented in the amplitudes.", "Nevertheless, it has been shown [19] that by reordering the perturbative series and including subleading $M_H^2/4M_W^2$ terms in the $W$ amplitude, one can reproduce the relative NLO electroweak corrections of the exact result at the percent level.", "An adapted version of the program HDECAY implements this approximation of the full NLO electroweak corrections to the decay $H\\rightarrow \\gamma \\gamma $ in SM4 [26].", "Using this new version of HDECAY, we display in the left–hand side of Fig.", "2 the cross section times branching ratio $\\sigma (gg\\!", "\\rightarrow \\!", "H)\\!\\times \\!", "{\\rm BR}( H\\!", "\\rightarrow \\!", "\\gamma \\gamma )|_{\\rm SM4/SM}$ at $M_H\\!=\\!125$ GeV as a function of $m_{\\nu ^{\\prime }}\\!=\\!m_{\\ell ^{\\prime }}$ for the value $m_{b^{\\prime }}\\!=\\!m_{t^{\\prime }}\\!+\\!50\\!=\\!600\\;$ GeV (the change when varying $m_{b^{\\prime }}$ in the still allowed range 600–700 GeV should be mild).", "As can be seen, $\\sigma (gg\\!\\rightarrow \\!", "H)\\!\\times \\!", "{\\rm BR}( H\\!", "\\rightarrow \\!", "\\gamma \\gamma )$ in SM4 is a factor of 5 to 10 smaller than in the SM.", "The increase of $\\sigma (gg\\!\\rightarrow \\!", "H)$ by a factor of $\\approx 9.5$ in SM4 is thus not sufficient for the $\\gamma \\gamma $ signal to be observed by the ATLAS and CMS experiments, hence excluding the perturbative SM4 scenario if the $\\gamma \\gamma $ and $4\\ell ^\\pm $ excesses at the LHC are indeed due to a 125 GeV Higgs boson.", "Figure: Left: σ(gg→H)× BR (H→γγ)| SM 4/ SM \\sigma (gg\\!", "\\rightarrow \\!", "H)\\!\\times \\!", "{\\rm BR}( H\\!", "\\rightarrow \\!\\gamma \\gamma )|_{\\rm SM4/SM} for a 125 GeV Higgs boson as a function ofm ν ' =m ℓ ' m_{\\nu ^{\\prime }}\\!=\\!m_{\\ell ^{\\prime }} when the leading 𝒪(G F m f ' 2 ){\\cal O}(G_Fm_{f^{\\prime }}^2) electroweakcorrections are included in a naive way (“approx\" NLO) or in a way that mimicsthe exact NLO results (“exact\" NLO).", "Right: the HVVHVV coupling squared andσ(qq ¯→VH)× BR (H→bb ¯)\\sigma (q\\bar{q}\\!", "\\rightarrow \\!", "VH)\\!\\times \\!", "{\\rm BR}(H\\!\\rightarrow \\!", "b\\bar{b}) in SM4normalized to the SM values.A final argument against the existence of a fourth generation, and which is theoretically more robust than the argument above based on the LHC $\\gamma \\gamma $ signal that is subject to large cancellations in the $H\\rightarrow \\gamma \\gamma $ amplitude, is provided by the recently updated SM Higgs search by the CDF and D0 collaborations with up to 10 fb$^{-1}$ of data [27].", "In this search, a $\\approx 2.2 \\sigma $ excess of data has been observed in the Higgs mass range between 115 and 135 GeV and is mostly concentrated in the Higgs–strahlung channel $q\\bar{q}\\!", "\\rightarrow \\!", "VH \\rightarrow \\!", "V b\\bar{b}$ with $V\\!=\\!W,Z$ ; this excess thus strengthens the case for a $\\approx \\!", "125$ GeV Higgs boson at the LHC.", "In SM4, such an excess cannot occur for the following two reasons.", "First, compared to the SM, the $HVV$ coupling and hence the production cross sections $\\sigma (q\\bar{q} \\rightarrow VH) \\propto g_{HVV}^2$ are strongly suppressed by the leading ${\\cal O}(G_Fm_{f^{\\prime }}^2)$ corrections (which approximate well the full electroweak NLO corrections in this case [23]) as mentioned earlier.", "Second, the branching ratio BR$(H\\!", "\\rightarrow \\!", "b\\bar{b})$ in SM4 is significantly affected by the presence of the new $t^{\\prime },b^{\\prime }$ quarks and, as shown in Fig.", "1, is $\\approx 30\\%$ smaller than in the SM for $M_H\\!", "\\approx \\!125$ GeV.", "The ratio $\\sigma (q\\bar{q}\\!", "\\rightarrow \\!", "VH)\\!\\times \\!", "{\\rm BR}( H\\!", "\\rightarrow \\!", "b\\bar{b})|_{\\rm SM4/SM}$ is thus much smaller than unity as exemplified in Fig.", "2 (right) where it is displayed as a function of $m_{\\nu ^{\\prime }}\\!=\\!m_{\\ell ^{\\prime }}$ again for $m_{b^{\\prime }}\\!=\\!m_{t^{\\prime }}\\!+\\!50\\!=\\!600$ GeV.", "This reduction of the $Vb\\bar{b}$ signal rate by a factor 3 to 5 depending on the $m_{\\nu ^{\\prime }}$ value would make the Higgs signal unobservable at the Tevatron and, therefore, the $2.2$ excess seen by CDF and D0, if indeed due to a $\\approx 125$ GeV Higgs boson, unambiguously rules out the SM4 scenario with perturbative Yukawa couplings.", "Finally, one should note that the observation of the channel $q\\bar{q}\\rightarrow VH$ with $H \\rightarrow b \\bar{b}$ at the Tevatron would also definitely exclude the fermiophobic Higgs scenario.", "This possibility has been advocated to explain the excess of events at the LHC in the channel $\\gamma \\gamma $ (plus additional jets), although the fit probability is not larger than in the SM [28].", "The observation of the Tevatron excess in $\\ell \\nu b\\bar{b}$ events can occur only if the decay $H\\rightarrow b\\bar{b}$ is present.", "In fact, even cases in which the $Hb\\bar{b}$ coupling is non–zero but suppressed compared to its SM value are disfavored.", "Indeed, as the rate $\\sigma (WH)\\!", "\\times \\!BR(H\\!\\rightarrow \\!", "b \\bar{b})$ is, to a good approximation, $\\propto g_{HWW}^2 \\!", "\\times \\!\\Gamma (H \\!", "\\rightarrow \\!", "b\\bar{b}) / [\\Gamma (H\\!", "\\rightarrow \\!", "b\\bar{b})\\!", "+ \\!", "\\Gamma (H \\!\\rightarrow \\!", "WW^*)]$ with $\\Gamma (H\\!", "\\rightarrow \\!", "WW^*, b\\bar{b}) \\propto g_{HWW, Hbb}^2$ , a suppression by 10%, 50% and 90% of the couplings $g_{Hff}$ would lead to a suppression of $\\sigma (Wb\\bar{b})$ by, respectively, $\\approx 5\\%$ , 42% and 96%.", "This is exemplified in Fig.", "REF in which the cross section times branching ratio $\\sigma (q \\bar{q} \\rightarrow VH) \\times {\\rm BR}(H\\rightarrow b\\bar{b})$ , normalized to its SM value, is displayed when the fermionic Yukawa couplings $g_{Hff}|_{\\rm FP/SM}$ are collectively varied from zero (i.e.", "the pure fermiophobic case) to unity (the SM case).", "Figure: The rate σ(qq ¯→VH)× BR (H→bb ¯)\\sigma (q\\bar{q} \\!", "\\rightarrow \\!", "VH) \\!", "\\times \\!", "{\\rm BR}(H\\!", "\\rightarrow \\!", "b\\bar{b}) of a 125 GeV Higgs boson normalised to its SM value asa function of the ratio g Hff | FP / SM g_{Hff}|_{\\rm FP/SM} of couplings in a fermiophobicHiggs scenario.", "The program HDECAY , in which thefermiophobic Higgs scenario is implemented, has been used.This argument can be extended to many models in which either the $HVV$ coupling or the $H\\rightarrow b\\bar{b}$ branching fraction (or both) are significantly suppressed compared to their SM values as could be the case in, for instance, minimal composite Higgs models [29].", "In summary, we have pointed out that while the exclusion bounds on a light Higgs boson, $M_H\\raisebox {-0.13cm}{~{< \\\\[-0.07cm] \\sim }}~160$ GeV, in SM4 with a fourth generation can be evaded by assuming a light fourth neutrino, $m_{\\nu ^{\\prime }} \\raisebox {-0.13cm}{~{< \\\\[-0.07cm] \\sim }}~\\frac{1}{2} M_H$ , this possibility is excluded by the observation at the LHC of a Higgs signal at a mass $\\approx 125$ GeV in the $\\gamma \\gamma $ and $ZZ^*\\rightarrow 4\\ell ^\\pm $ final states.", "The ATLAS+CMS $4\\ell ^\\pm $ and $\\gamma \\gamma $ signals are compatible with the SM4 scenario (and larger signal rates than in the SM could even be accommodated) if the leading ${\\cal O}(G_F m_{f^{\\prime }}^2)$ electroweak corrections are naively included in the $H \\rightarrow \\gamma \\gamma $ rate.", "However, when including the full set of electroweak corrections at next-to-leading order [19], the $\\gamma \\gamma $ signal is suppressed by an order of magnitude compared to the previous approximation, hence strongly disfavoring a perturbative SM4.", "Finally, the observation by the CDF/D0 collaborations of a $\\raisebox {-0.13cm}{~{> \\\\[-0.07cm] \\sim }}~\\!", "2\\sigma $ excess corresponding to a Higgs boson with $M_H\\!=\\!", "115$ –135 GeV in the channel $q\\bar{q}\\!", "\\rightarrow \\!", "VH\\!", "\\rightarrow \\!", "Vb\\bar{b}$ can definitely not be accommodated in SM4.", "Hence, if the excesses observed at the LHC and the Tevatron are indeed the manifestations of a 125 GeV Higgs boson, the scenario with a perturbative fourth fermionic generation is unambiguously excluded.", "En passant, we also point out that the pure fermiophobic Higgs scenario cannot accommodate the Higgs signal in the $VH \\rightarrow Vb\\bar{b}$ channel observed at the Tevatron.", "In fact, many scenarios in which the $Hb\\bar{b}$ or $HWW$ couplings (or both) are suppressed compared to their SM values are disfavored if the Tevatron excess is indeed due to a Higgs particle.", "Note added: On July 4th, 2012 ATLAS and CMS announced new results for the Higgs boson search [30], which has severe implications on the fate of a fourth generation of fermions.", "Before this announcement a combined fit of electro-weak precision data and Higgs production and decay data yielded the result, that the SM4 is excluded by 3.1 standard deviations [31].", "The new data worsens the situation for the SM4 in several points: Both ATLAS and CMS see a $H \\rightarrow \\gamma \\gamma $ -signal with a statistical significance of more than 4 standard deviations.", "The total observed rate was higher than expected by the SM (a factor of $1.9 \\pm 0.5$ for ATLAS and a factor of $1.56 \\pm 0.43$ for CMS).", "In the SM4 one would expect instead a reduction of the rate by at least a factor of 5 compared to the standard model, see Fig.", "(REF ).", "Thus, both ATLAS and CMS individually see a $H \\rightarrow \\gamma \\gamma $ -signal, which is about 4 standard deviations away from the expectation of the SM4, which rules out the SM4.", "As discussed above, the theory prediction for $H \\rightarrow \\gamma \\gamma $ in the SM4 suffers from severe cancellations, so one might not want to rely on this decay channel alone.", "On July 2nd, 2012 also CDF and D0 updated their Higgs search [32] in the Higgs-strahlung channel, discussed above.", "There the statistical significance increased from 2.6 standard deviations in [27] to 2.9 standard deviations in [32].", "At $m_H = 125$ GeV Tevatron finds a signal strength of $1.97^{+0.74}_{-0.68}$ , so a little above the SM expectation, while the SM4 predicts values below 0.4, see Fig.", "(REF ).", "Again a stronger indication against the SM4, compared to the status of Moriond 2012.", "In the SM4 one would expect a sizeable enhancement of the $ H \\rightarrow \\tau \\tau $ channel, see e.g.", "[31], which is not observed [30] in the new data.", "A further argument against the SM4 at the 4 $\\sigma $ level.", "Both ATLAS and CMS see a $H \\rightarrow \\gamma \\gamma $ -signal with a statistical significance of more than 4 standard deviations.", "The total observed rate was higher than expected by the SM (a factor of $1.9 \\pm 0.5$ for ATLAS and a factor of $1.56 \\pm 0.43$ for CMS).", "In the SM4 one would expect instead a reduction of the rate by at least a factor of 5 compared to the standard model, see Fig.", "(REF ).", "Thus, both ATLAS and CMS individually see a $H \\rightarrow \\gamma \\gamma $ -signal, which is about 4 standard deviations away from the expectation of the SM4, which rules out the SM4.", "As discussed above, the theory prediction for $H \\rightarrow \\gamma \\gamma $ in the SM4 suffers from severe cancellations, so one might not want to rely on this decay channel alone.", "On July 2nd, 2012 also CDF and D0 updated their Higgs search [32] in the Higgs-strahlung channel, discussed above.", "There the statistical significance increased from 2.6 standard deviations in [27] to 2.9 standard deviations in [32].", "At $m_H = 125$ GeV Tevatron finds a signal strength of $1.97^{+0.74}_{-0.68}$ , so a little above the SM expectation, while the SM4 predicts values below 0.4, see Fig.", "(REF ).", "Again a stronger indication against the SM4, compared to the status of Moriond 2012.", "In the SM4 one would expect a sizeable enhancement of the $ H \\rightarrow \\tau \\tau $ channel, see e.g.", "[31], which is not observed [30] in the new data.", "A further argument against the SM4 at the 4 $\\sigma $ level.", "It is beyond the scope of this paper to make a precise statistical statement about the exclusion of the SM4.", "Nevertheless, we conclude that the standard model with a perturbative 4th generation and one Higgs doublet is ruled out by this new experimental developments.", "Acknowledgements: We thank R. Godbole, C. Grojean, H.-J.", "He and T. Volansky for discussions.", "Special thanks go to Michael Spira for his work in implementing SM4 in HDECAY and for valuable suggestions and comments.", "A.D. thanks the CERN TH Unit for hospitality and support and A.L.", "is supported by DFG through a Heisenberg fellowship." ] ]

[ [ "Searching for new OH megamasers out to redshifts z>1" ], [ "Abstract We have carried out a search for 18-cm OH megamaser (OHM) emission with the Green Bank Telescope.", "The targeted galaxies comprise a sample of 121 ULIRGs at 0.09<z<1.5, making this the first large, systematic search for OHMs at z>0.25.", "Nine new detections of OHMs are reported, all at redshifts z<0.25.", "For the remainder of the galaxies, observations constrain the upper limit on OH emission; this rules out OHMs of moderate brightness (L_OH > 10^3 L_sun) for 26% of the sample, and extremely bright OHM emission (L_OH > 10^4 L_sun) for 73% of the sample.", "Losses from RFI result in the OHM detection fraction being significantly lower than expected for galaxies with L_IR >10^12 L_sun.", "The new OHM detections are used to calculate an updated OH luminosity function, with \\Phi[L]\\simL_OH^{-0.66}; this slope is in agreement with previous results.", "Non-detections of OHMs in the COSMOS field constrain the predicted sky density of OHMs; the results are consistent with a galaxy merger rate evolving as (1+z)^m, where m<6." ], [ "Introduction", "OH megamasers (OHMs) trace of some of the most extreme physical conditions in the universe - in particular, the presence of an OHM signals specific stages in the merger process of gas-rich galaxies.", "OHMs can thus be used as probes of their environments, both directly and indirectly.", "Characteristics of the maser emission itself can be used to measure extragalactic magnetic fields (via Zeeman splitting) and gas kinematics, while the presence of an OHM is a signpost for phenomena associated with galaxy mergers, including extreme star formation and merging black holes.", "OHMs are a unique tool in this respect due to their extreme luminosities and ability to be seen at cosmic distances.", "The total number of OHMs detected to date is still low.", "As of 2012, there are $\\sim 113$ OHMs published in the literature, with roughly 50% discovered in the Arecibo survey of [3], [4], [5].", "No OHMs at a distance of greater than 1300 Mpc ($z=0.265$ ) have been detected, and no large, systematic searches for high-$z$ OHMs have been carried out.", "The association of OHMs with IR-bright merging galaxies, however, means that the density of OHMs is expected to be much higher at $z\\simeq 1-2$ , coinciding with an increase in both merging rate and cosmic star formation.", "We have conducted a search for high-redshift OHMs using the Green Bank Telescope (GBT)." ], [ "Sample selection and observations", "We constructed three samples of galaxies to search for high-redshift OHMs.", "None of the samples are fully complete or flux-limited, but draw on the catalogs of IR-luminous galaxies with well-defined redshifts that were available at the time.", "IRAS PSCz galaxies not visible from Arecibo: The first sample of galaxies consisted of IRAS sources included in the redshift catalog of the PSCz survey.", "Galaxies were selected according to similar criteria as in the flux-limited Arecibo sample, but included objects lying outside the declination limits of Arecibo ($-1^\\circ <\\delta <38^\\circ $ ).", "Galaxies selected for GBT observations had: a declination range of $-40^\\circ <\\delta <0^\\circ $ or $\\delta >37^\\circ $ , a redshift range of $0.10<z<0.25$ , and a lower-luminosity threshold of $L_{60\\mu m} > 10^{11.4} L_\\odot $ .", "153 galaxies in the PSCz met these criteria, of which 47 of the brightest candidates were observed according to the LST windows during the early commissioning phase of the GBT in 2002.", "Sub-mm and ULIRG galaxies from the field: The second sample of potential OHM hosts was assembled from flux-limited catalogs of ULIRGs at higher redshifts.", "We began with 35 galaxies in the FSC-FIRST catalog [10], which consists of targets detected in both the IRAS Faint Source Catalog and the 20-cm VLA FIRST survey.", "This was supplemented with 5 IR-bright galaxy pairs and 26 sub-millimetre galaxies.", "All galaxies have $L_{IR}>10^{11}L_\\odot $ , with more than half having $L_{IR}>10^{12}L_\\odot $ .", "The highest redshift in this sample is at $z=1.55$ .", "Starburst galaxies from COSMOS: The third group of 19 OHM candidates was the last observed in our program, and made explicit use of the results from the first two samples.", "Targets were selected from the COSMOS field, a 2-deg$^2$ survey with deep spectral coverage from X-ray through radio wavelengths.", "Recent infrared [11] studies show that while OHMs are found in infrared-bright galaxies, the OHM fraction is much higher for starburst-dominated galaxies vs. AGN.", "Selection of OHM candidates began with COSMOS galaxies detected by Spitzer at 70 $\\mu $ m, and then eliminating all targets except the LIRGs, ULIRGs, and HyLIRGs identified by [8].", "We removed all galaxies identified as AGN or with $L_{IR}<10^{12}L_\\odot $ .", "Finally, we culled the target list based on the expected RFI conditions near the observed frequency bands.", "By limiting the observed OH frequencies to cleaner regions, we have a broader margin for error on the galaxy redshift.", "The two windows used are at $\\nu _{obs}=825-830$  MHz and $960-1005$  MHz, equivalent to redshifted OH at $0.97<z<1.01$ and $0.67<z<0.73$ .", "Observations: We observed the OHM candidates in several sessions at the GBT from 2002–2010, totaling approximately 150 hours.", "The majority of observations used the maximum available bandwidth of 50 MHz and 8192 channels.", "Integration times were selected with the goal of achieving $\\sim 1$  mJy rms per channel for each galaxy.", "The data were reduced using standard routines in GBTIDL, including extensive flagging for RFI.", "After flagging, we fit the radio continuum around the expected line center with a polynomial function of order $n=5$ .", "This removed both intrinsic continuum structure from the target itself and any baseline structure not removed by the position-switching technique.", "After stacking, the spectra were smoothed to a rest-frame velocity resolution of 10 km s$^{-1}$ .", "Table: Properties of new OHM detections from the GBT survey" ], [ "Survey results", "OH megamasers: Out of 128 galaxies observed for OH, we detected new OH megamaser emission in nine objects (Figure REF ).", "Seven of the detections were PSCz galaxies from the first sample, while the other two were from the FSC-FIRST catalog in the second sample.", "All nine OHMs have redshifts near the lower end of the sample distribution, with the most distant lying at a redshift of $z=0.2427$ .", "The observed frequencies are in the range of $1300-1500$  MHz, which is covered by the L-band receiver and has relatively little RFI compared to the GBT prime focus bands.", "The OH emission has been confirmed with GBT follow-up observations for five of the galaxies.", "We also confirmed the detection of the previously-discovered OHM IRAS 09539+0857 [4].", "Table REF lists the 18-cm radio properties of the new OHM detections.", "We give the galaxy's optical redshift ($z_{hel}$ ), peak flux density of the OHM ($S_{1667}^{peak}$ ), the ratio of the integrated 1667 MHz emission to its peak flux density ($W_{1667}$ ), measured OH luminosity (log $L_{OH}$ ), and predicted OH luminosity ($L_{OH}^{pred}$ ) based on the $L_{OH}-L_{FIR}$ relationship in [5].", "OH non-detections: 112 galaxies showed no confirmed detections of OH.", "The $L_{OH}^{max}$  for each galaxy was (conservatively) derived assuming a boxcar line profile with a linewidth $\\Delta v=150$  km s$^{-1}$  and a 1.5$\\sigma $ detection.", "The rms was measured from baseline-subtracted continuum centered on the optical redshift of the galaxy and in a frequency range sufficient to cover the uncertainty in the optical redshift ($\\Delta \\nu _{obs}=\\Delta z\\times \\nu _{rest}/[1+z]$ )." ], [ "Updating the OH luminosity function", "One of the goals of performing a search for OHMs at higher redshifts was to improve the measurements of the OH megamaser luminosity function (LF).", "[6] used the results of the flux-limited Arecibo survey to construct a well-sampled LF between $10^{2.2}~L_\\odot <L_{OH}<10^{3.8}L_\\odot $ , which followed a power law in integrated line luminosity of $\\Phi [L]\\propto ~L_{OH}^{-0.64}$  Mpc$^{-3}$  dex$^{-1}$ .", "This measurement was limited to a narrow redshift range, spanning $0.1<z<0.23$ .", "We constructed a new OH LF by combining the GBT and Arecibo OHM detections, using the $1/V_a$ method and combining limits on both spectral line and continuum emission.", "We fit a power-law to all bins with more than one detection, yielding: ${\\rm log}~\\Phi [L]=(-0.66\\pm 0.14)~{\\rm log}~L_{OH} - (4.91\\pm 0.41),$ where $\\Phi $ is measured in Mpc$^{-3}$  dex$^{-1}$ and $L_{OH}$ in $L_\\odot $ .", "The new detections only change the slope measured by [6] by $-0.02$ and the offset by $+0.10$ .", "Both values are well within the uncertainties of the combined LF, as well as that of the original Arecibo LF.", "Assuming the Malmquist-corrected relationship of $L_{OH}\\propto L_{IR}^{1.2}$ from [5], this gives $\\Phi [L_{IR}]\\propto L_{IR}^{-0.83\\pm 0.18}$ .", "We compare this to the LF of ULIRGs in the local Universe using the AKARI measurements of [7].", "Folding in the OHM fraction derived from the combined Arecibo and GBT samples results in a slope of $(-0.6\\pm 0.2)$ , a much shallower value than that measured from the AKARI galaxies $(-2.6\\pm 0.1)$ .", "This inconsistency may suggest either that OHMs are highly saturated or that the maser strength is only weakly correlated with global properties such as $L_{IR}$ [5].", "The OHM LF of [2] assumed a quadratic OH-IR relation corresponding to unsaturated masing; a decrease in saturation could potentially steepen the OHM LF up to a slope of $-1.5$ ." ], [ "Constraining the evolution of the cosmic merger rate", "One of the ultimate goals of high-redshift OHM surveys is to use megamasers as tracers of the populations of merging galaxies as a function of redshift.", "Models of the merger rate as a function of redshift are typically parameterized with an evolutionary factor of $(1+z)^m$ .", "The value of $m$ , however, is not well-constrained [9], [1].", "Sufficiently deep surveys of OHMs can provide an independent measurement of the parametrization of the merging rate.", "We calculated the predicted sky density of OHMs as a function of redshift: $\\frac{dN}{d\\Omega d\\nu }[z] = \\frac{c D_L^2}{H_0\\nu _0\\sqrt{(1+z)^3\\Omega _M + \\Omega _\\Lambda }} \\left(\\frac{b}{a~{\\rm ln} 10}\\right) \\times \\nonumber \\\\\\left((L_{OH,max})^a - (L_{OH,min})^a\\right),$ where $a$ and $b$ are parameters of the OHM LF from Equation REF ($\\Phi [L_{OH}]=b L_{OH}^a$ ), $L_{OH,min}$ is the minimum OH luminosity that can be observed at a given sensitivity level, and $L_{OH,max}$ is the upper physical limit on OHM luminosity [6].", "Using the upper limit of zero OHMs detected in the COSMOS field, we place an upper limit on the merger rate of $m\\lesssim 6$ .", "While this is still within uncertainties for the highest estimated values of $m$ , the COSMOS limit is an important first step in using OHMs as an independent tracer.", "Further measurements of OH deep fields at higher redshifts will be crucial for a more accurate constraint." ], [ "Future OHM searches", "The final results for the GBT OHM survey yielded a much lower detection rate (9/121 = 7%) of megamasers than expected.", "Based on the high detection fraction for galaxies with $L_{IR}>10^{12}~L_\\odot $ from the Arecibo survey, in addition to our careful selection of starburst-dominated galaxies, we had predicted a success rate of 20–30%.", "We attribute one of the primary causes of the low OHM fraction to be the sensitivity of the observations.", "Figure REF shows the upper limits for OHM candidates from our survey, along with the necessary rms sensitivity to detect OH lines as a function of redshift.", "To detect median-luminosity OHMs at $z=1$ will require rms levels of 100 $\\mu $ Jy, and perhaps significantly more time devoted to individual targets.", "It must also be mentioned that RFI is a major culprit at $\\nu _{obs}<1$  GHz, significantly restricting the redshift path and increasing the noise in each receiver band.", "Future OHM searches may benefit from interferometric observations (for which celestial RFI is uncorrelated from dish to dish), or from observations in more radio-quiet environments.", "The GMRT (India) and ASKAP (Australia) are promising instruments for interferometry, while the upcoming 64-m Sardinia Radio Telescope (Italy) and 500-m FAST (China) will be options to continue the search for OHMs at high redshifts." ], [ "Acknowledgments", "This work was part of the Ph.D. thesis of KWW at the University of Colorado, and will be submitted in early 2012 as Willett, Darling, Kent, & Braatz (2012).", "Observations were funded in part by student support grants from NRAO.", "We are indebted to the staff of the Green Bank Telescope, whose expertise and assistance made these observations possible.", "The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc." ] ]

[ [ "Relativistic dissipative hydrodynamics with extended matching conditions\n for ultra-relativistic heavy-ion collisions" ], [ "Abstract Recently we proposed a novel approach to the formulation of relativistic dissipative hydrodynamics by extending the so-called matching conditions in the Eckart frame [Phys.", "Rev.", "{\\bf C 85}, (2012) 14906].", "We extend this formalism further to the arbitrary Lorentz frame.", "We discuss the stability and causality of solutions of fluid equations which are obtained by applying this formulation to the Landau frame, which is more relevant to treat the fluid produced in ultra-relativistic heavy-ion collisions.", "We derive equations of motion for a relativistic dissipative fluid with zero baryon chemical potential and show that linearized equations obtained from them are stable against small perturbations.", "It is found that conditions for a fluid to be stable against infinitesimal perturbations are equivalent to imposing restrictions that the sound wave, $c_s$, propagating in the fluid, must not exceed the speed of light $c$, i.e., $c_s < c$.", "This conclusion is equivalent to that obtained in the previous paper using the Eckart frame [Phys.", "Rev.", "{\\bf C 85}, (2012) 14906]." ], [ "Introduction", "It is known that one faces problems of instability and violation of causality in solutions obtained from the relativistic Naiver-Stokes (NS) equations (observed in the first-order theories [1], [2] which naively extend the non-relativistic NS equation).", "Israel and Stewart (IS) provide a phenomenological framework for a relativistic dissipative fluid [3], [4] accounting for these problems.", "In their model, a possible general form of the non-equilibrium entropy current is described by the dissipative part of the energy-momentum tensor and by the particle current up to second order in the deviation from the equilibrium state.", "After the stability and causality of the IS theory have been shown by Hiscock and Lindblom [5], [6], [7], the causal dissipative hydrodynamical model was adapted to study the dynamics of hot matter produced in ultra-relativistic heavy-ion collisions by Muronga [8], [9].", "In recent years, the IS model plays an important role in the analysis of the experimental data obtained by RHIC and LHC (see, for example, [10]).", "In parallel to the application of IS theory, investigations of the basis of relativistic dissipative hydrodynamic theory was continued [11], [12], [13], [14], [15], [16], [17], [18], [19].", "This is because the IS theory, in its present form, is too general and complex from the point of view of quantum chromodynamics (QCD), which is believed to be the fundamental theory for strongly interacting systems [11].", "Another reason is that the theory of relativistic dissipative hydrodynamics is not yet fully understood because, for example, the equation of motion of the fluid it uses depends on the choice of the Lorentz frame [16] (or on the definition of the hydrodynamical flow).", "Since the dissipative part of the energy-momentum tensor $\\delta T^{\\mu \\nu }$ and the particle current $\\delta N^{\\mu }$ cannot be determined uniquely by the second law of thermodynamics, one usually introduces some constraints to fix them.", "These constraints are known as matching conditions.", "The other reason to introduce these matching (or fitting) conditions [4] is the necessity of matching the energy density and baryonic charge density, $(\\varepsilon , n)$ , in a non-equilibrium state to the corresponding equilibrium densities: ($\\varepsilon _{\\rm eq}$ , $n_{\\rm eq}$ ), $\\varepsilon = \\varepsilon _{\\rm eq},~n=n_{\\rm eq}$ , or, equivalently, $u_{\\mu } u_{\\nu }\\delta T^{\\mu \\nu } =0, \\quad u_{\\mu } \\delta N^{\\mu } =0.$ Matching conditions allow to determine the thermodynamical pressure, $P_{\\rm eq}(\\varepsilon _{\\rm eq},n_{\\rm eq})$ , (defined as work done in isentropic expansion) via the equation of state for the equilibrium state.", "Here $P_{\\rm eq}$ should be distinguished from the bulk viscous contribution, $\\Pi \\equiv -\\frac{1}{3}\\Delta _{\\mu \\nu }\\delta T^{\\mu \\nu }$ , present in the energy-momentum tensor [4].", "Finally, matching conditions are also needed because they are necessary for the thermodynamical stability of the entropy current (see Appendix A in Ref.[20]).", "However, the matching conditions given by eq.", "(REF ) are not unique.", "So far, except of some recent works [16], [17], they were not investigated in detail.", "A state of a relativistic dissipative fluid is described by the energy-momentum tensor, $T^{\\mu \\nu }(x)$ , and by the baryon number current, $N^{\\mu }$ , which obey the conservation laws $&& T^{\\mu \\nu }_{;\\mu }=0, \\\\&& N^{\\mu }_{\\mu }=0,$ and the second law of the thermodynamics $S^{\\mu }_{;\\mu } \\ge 0.$ Because of the uncertainty in definition of the flow velocity $u^{\\mu }(x)$ for a non-equilibrium fluid, one needs, unlike in the case of perfect fluid, to fix the frame for the fluid considered.", "Two special frames can be defined: Landau and Eckart.", "The Landau frame [2] is defined by the vanishing of the energy flow, which consists of heat flow $q^{\\mu }$ and net baryon number flow $V^{\\mu }$ ; $W^{\\mu } &\\equiv &u_{\\nu } T^{\\nu \\lambda } \\Delta ^{\\mu }_{\\lambda } =0, \\quad q^{\\mu } =- \\frac{\\varepsilon _{\\rm eq}+P_{\\rm eq}}{n_{\\rm eq}} ~V^{\\mu },$ with $\\Delta ^{\\mu }_{\\nu }\\equiv g^{\\mu }_{\\nu } -u^{\\mu }u_{\\nu }$ being the projection operator orthogonal to the four vector $u^{\\mu }$ .", "In the Eckart frame, the hydrodynamic flow velocity $u^{\\mu }$ (with normalization $u^{\\mu }u_{\\mu }=1$ ) is defined by using baryon charge current $u^{\\mu } \\equiv N^{\\mu }/\\sqrt{N^{\\nu }N_{\\nu }}$ .", "In this frame one always has $V^{\\mu }\\equiv \\Delta ^{\\mu }_{\\lambda }N^{\\lambda }=0, \\quad \\quad \\quad q^{\\mu }= W^{\\mu }.", "\\quad \\quad \\quad \\quad $ Equations of motion for the fluid should not depend on the choice of Lorentz frame.", "Since the relativistic dissipative fluid dynamics with extended matching conditions has been already formulated in the Eckart frame [21], it is interesting to check it in another Lorentz frame, for example in the Landau frame.", "Also, the stability and causality conditions for a relativistic dissipative fluid should not depend on the Lorentz frame used.", "The purpose of this article is therefore to investigate how the stability and causality conditions should be imposed on a fluid depending on the Lorentz frame used.", "This paper is organized as follows: In Sec., we re-formulate relativistic dissipative hydrodynamics with extended matching condition (as originally introduced in ref.", "[21]) in an arbitrary Lorentz frame with $V^{\\mu }\\ne 0$ and $W^{\\mu }\\ne 0$ (neither the Eckart frame with $V^{\\mu }=0$ nor the Landau frame with $W^{\\mu }=0$ ).", "In such a general frame, the flow velocity field $u^{\\mu }(x)$ may be determined by using coexisting $W^{\\mu }$ and $V^{\\mu }$ .", "We then choose the Landau frame and consider a fluid with zero baryon chemical potential $\\mu _b=0$ , appearing in the central rapidity region of the ultra-relativistic heavy-ion collisions.", "In Sec., we check the stability and causality of the fluid obtained from our model in this frame.", "We close with Sec.", "containing a summary and some further discussion.", "The general off-equilibrium entropy current can be written in the following simple form using the vector $\\phi ^{\\mu }$ and 2-rank symmetric tensor $\\Phi ^{\\mu \\nu }$ , $S^{\\mu }(x) &\\equiv &-\\alpha \\phi ^{\\mu } + \\beta _{\\lambda }\\Phi ^{\\lambda \\mu },$ where $\\alpha \\equiv \\mu _b/T$ , $\\beta ^{\\mu }=\\beta u^{\\mu }$ with $\\beta \\equiv 1/T$ , $T$ and $\\mu _b$ are, respectively, the temperature and baryon chemical potential.", "In the local equilibrium case it is given by $S^{\\mu }_{\\rm eq} &\\equiv &-\\alpha \\phi ^{\\mu }_0 +\\beta _{\\lambda }\\Phi ^{\\lambda \\mu }_0,\\\\\\phi ^{\\mu }_0 &=& N^{\\mu }_{\\rm eq} ,\\quad \\Phi ^{\\lambda \\nu }_0 = T_{\\rm eq}^{\\lambda \\nu }- \\frac{g^{\\lambda \\nu }}{3}\\Delta _{\\alpha \\beta }T^{\\alpha \\beta }_{\\rm eq},$ where $T^{\\mu \\nu }_{\\rm eq}$ and $N^{\\mu }_{\\rm eq}$ are equilibrium energy-momentum tensor and baryon charge current: $T^{\\mu \\nu }_{\\rm eq} &=& \\varepsilon _{\\rm eq} u^{\\mu }u^{\\nu } -P_{\\rm eq}\\Delta ^{\\mu \\nu }, \\\\N^{\\mu }_{\\rm eq} &=& n_{\\rm eq} u^{\\mu }.$ In this case, the energy-momentum conservation, $T^{\\mu \\nu }_{{\\rm eq};\\mu }=0$ , and the baryon number conservation, $N^{\\mu }_{{\\rm eq};\\mu }=0$ , together with thermodynamic relations result in the locally conserved entropy current: $S^{\\mu }_{{\\rm eq};\\mu } &=& -\\alpha _{,\\mu }N^{\\mu }_{\\rm eq}+\\beta _{\\lambda ;\\mu } T^{\\lambda \\mu }_{\\rm eq} +[\\beta ^{\\mu }P_{\\rm eq}]_{;\\mu }=0.", "$ To this current we now introduce dissipative corrections by adding corresponding dissipative corrections $\\delta T^{\\mu \\nu }$ and $\\delta N^{\\mu }$ to the energy-momentum tensor and to the baryon number current appearing in $\\phi _0^{\\mu }$ and $\\Phi _0^{\\mu \\nu }$ .", "In this way one extends the expression of equilibrium entropy current towards the off-equilibrium entropy current: $&& S^{\\mu } \\equiv -\\alpha [\\phi ^{\\mu }_0+\\delta \\phi ^{\\mu }]+\\beta _{\\lambda }[\\Phi ^{\\lambda \\mu }_0+\\delta \\Phi ^{\\lambda \\mu }],\\\\&& \\delta \\phi ^{\\mu } =\\delta N^{\\mu }, \\quad \\delta \\Phi ^{\\lambda \\nu } = \\delta T^{\\lambda \\nu }+\\chi ~\\frac{g^{\\lambda \\nu } }{3}\\Delta _{\\alpha \\beta } \\delta T^{\\alpha \\beta }.", "\\quad $ Notice that term proportional to $\\Delta _{\\alpha \\beta } \\delta T^{\\alpha \\beta }$ , which appears due to the natural extension of eq.", "(), results in $\\lim _{\\Pi \\rightarrow 0} \\frac{d}{d\\Pi } (u_{\\mu }S^{\\mu }) ~\\ne 0,$ where $\\Pi $ is bulk pressure, cf.", "Eq.", "(REF ) below.", "It means than that entropy density is not maximal in spite of equilibrium state used, the entropy current eq.", "(REF ) is thermodynamically unstable [20].", "Therefore the term $\\Delta _{\\alpha \\beta }\\delta T^{\\alpha \\beta }$ is usually dropped (i.e., $\\chi $ should be put equal zero, $\\chi \\equiv 0$ ).", "However, the problem of thermodynamic instability can be avoided by simultaneously demanding the natural extension of the form of the entropy current (eq.", "(REF ) with ()) and the following general extension of matching conditions; $\\chi \\ne 0, \\quad u_{\\mu } u_{\\nu } \\delta T^{\\mu \\nu } \\ne 0 , \\quad u_{\\mu } \\delta N^{\\mu } \\ne 0.$" ], [ "Extended matching conditions", "To restore thermodynamical stability as discussed above, we propose to impose the following extended matching conditions on the dissipative correction of the energy momentum tensor and baryon charge current, $\\delta T^{\\mu \\nu }$ and $\\delta N^{\\mu }$ , respectively: $u_{\\mu }\\delta T^{\\mu \\nu }u_{\\nu } = \\Lambda , \\quad \\delta N^{\\mu } u_{\\mu } = \\delta n. $ With these conditions, off-equilibrium contributions for the energy momentum tensor and baryon charge vector in general Lorentz frame are $\\delta T^{\\mu \\nu } &=& \\Lambda u^{\\mu }u^{\\nu }-\\Pi \\Delta ^{\\mu \\nu } +W^{\\mu }u^{\\nu } + W^{\\nu }u^{\\mu } + \\pi ^{\\mu \\nu },\\quad \\\\\\delta N^{\\mu } &=&\\delta n u^{\\mu } + V^{\\lambda }\\Delta _{\\lambda }^{\\mu },$ where $\\Pi $ is bulk pressure, $\\pi ^{\\mu \\nu }$ is shear tensor and $V^{\\mu }$ is net flow of the baryonic charge.", "In this case, the off-equilibrium entropy current eq.", "(REF ) is given by $S^{\\mu } = S^{\\mu }_{\\rm eq}-\\alpha V^{\\lambda }\\Delta _{\\lambda }^{\\mu }+\\beta W^{\\mu }-\\beta [ \\mu _{\\rm b} \\delta n - \\Lambda +\\chi \\Pi ] u^{\\mu }.", "\\nonumber \\\\$ To ensure thermodynamic stability, one may impose a condition on the entropy current $S^{\\mu }$ , eq.", "(REF ), demanding that $\\frac{d}{d\\Pi } (u_{\\mu }S^{\\mu })=0.$ This requirement can be satisfied by the following unique condition, $\\chi \\Pi = -\\mu _{\\rm b} \\delta n + \\Lambda .$ Note that, when both $\\Lambda $ and $\\delta n$ are set equal to zero, $\\chi $ should also be zero, as so far considered widely in the literature.", "However, once one assumes that $\\Lambda \\ne 0$ and/or $\\delta n \\ne 0$ , there exists a term proportional to $\\chi $ .", "By the extended thermodynamical stability condition, eq.", "(REF ), the entropy current in the arbitrary Lorentz frame is $S^{\\mu } = S^{\\mu }_{\\rm eq}-\\alpha V^{\\lambda }\\Delta _{\\lambda }^{\\mu }+\\beta W^{\\mu }.$ One can obtain the entropy current corresponding to Eckart's or Landau's formulation in the limit $V^{\\mu }\\rightarrow 0$ and $W^{\\mu }\\rightarrow 0$ , respectively.", "The entropy current eq.", "(REF ) can be rewritten using eqs.", "(REF ), () and thermodynamical stability condition eq.", "(REF ), $S^{\\mu } &=& S^{\\mu }_{\\rm eq}-\\alpha \\delta N^{\\mu }+\\beta _{\\lambda } \\delta T^{\\lambda \\mu }-\\chi \\Pi u^{\\mu }.", "$ Thus the entropy production in this case is given by $S^{\\mu }_{;\\mu } = -\\alpha _{,\\mu } \\delta N^{\\mu }+\\beta _{\\lambda ;\\mu } \\delta T^{\\lambda \\mu }- [\\chi \\Pi \\beta ^{\\mu } ]_{;\\mu } .", "$ It should be noted here that eq.", "(REF ) is exactly the same as eq.", "(19) in Ref.", "[21] obtained in the Eckart frame when using extended matching condition.", "This is because $S^{\\mu }_{;\\mu }$ is a scaler and so it does not depend on Lorentz frame.", "This can be seen in the form of the entropy current, eq.", "(REF ).", "The expression for $S^{\\mu }$ has the same form both in Eckart frame ($V^{\\mu }=0$ but $\\delta N^{\\mu }\\ne 0$ ) and in Landau frame ($W^{\\mu }=0$ but $\\beta _{\\lambda }\\delta T^{\\lambda \\mu }\\ne 0$ ).", "The entropy production is explicitly given by $S^{\\mu }_{;\\mu } &=&-(\\nabla _{\\lambda }\\alpha )V^{\\lambda }+(\\nabla _{\\lambda }\\beta )W^{\\lambda }+\\beta \\frac{d u_{\\lambda }}{d\\tau } W^{\\lambda } \\nonumber \\\\&-&\\beta ~ \\Pi \\theta + \\beta ~\\nabla _{\\langle \\mu } u_{\\lambda \\rangle } \\pi ^{\\lambda \\mu }\\nonumber \\\\&+& [\\alpha \\frac{d\\delta n}{d\\tau }- \\beta \\frac{d\\Lambda }{d\\tau }]+[\\alpha \\delta n -\\beta \\Lambda ]\\theta ,$ where $\\theta $ is the divergence of the flow velocity field, $\\theta \\equiv u^{\\mu }_{;\\mu }$ .", "Using the definition that $\\alpha =\\mu _b \\beta $ one can also write the entropy production as $TS^{\\mu }_{;\\mu } &=&[\\frac{\\nabla _{\\mu }\\mu _b}{\\mu _b}-\\frac{\\nabla _{\\mu }T}{T}][-\\mu _b V^{\\mu }]\\nonumber \\\\&+&[~\\frac{du_{\\mu }}{d\\tau }~-\\frac{\\nabla _{\\mu }T}{T}] ~W^{\\mu } \\nonumber \\\\&-& \\Pi \\theta + \\nabla _{\\langle \\mu } u_{\\lambda \\rangle } \\pi ^{\\lambda \\mu }\\nonumber \\\\&+&[\\mu _b \\frac{d\\delta n}{d\\tau }-\\frac{d\\Lambda }{d\\tau }]+[\\mu _b \\delta n - \\Lambda ]\\theta .$ Note that the above equation can also be expressed in the following form: $TS^{\\mu }_{;\\mu } &=&[~\\frac{du_{\\mu }}{d\\tau }~-\\frac{\\nabla _{\\mu }T}{T}] ~\\tilde{W}^{\\mu }\\nonumber \\\\&+&[~\\frac{du_{\\mu }}{d\\tau }~- \\frac{\\nabla _{\\mu }\\mu _b}{\\mu _b}] ~\\tilde{V}^{\\mu }\\nonumber \\\\&-& \\Pi \\theta + \\nabla _{\\langle \\mu } u_{\\lambda \\rangle } \\pi ^{\\lambda \\mu }\\nonumber \\\\&+&[\\mu _b \\frac{d\\delta n}{d\\tau }-\\frac{d\\Lambda }{d\\tau }]+[\\mu _b \\delta n - \\Lambda ]\\theta ,$ where $\\tilde{W}^{\\mu }\\equiv W^{\\mu }-\\mu _b V^{\\mu }$ and $\\tilde{V}^{\\mu } \\equiv \\mu _b V^{\\mu }$ ." ], [ "Constitutive equations for a dissipative fluid in the Landau frame", "We shall now consider the Landau frame ($W^{\\mu }\\equiv 0$ ) which is more relevant in the context of ultra-relativistic heavy-ion collisions.", "In particular, we consider a fluid in the central rapidity region where it is expected that the baryon chemical potential $\\mu _b$ is small.", "In this paper, we assume that, for simplicity, $\\mu _b=0$ (this implies that $n_{\\rm eq}=0$ .)", "In this case, the entropy production takes the simple form; $TS^{\\mu }_{;\\mu }&=& -\\Pi \\theta + \\nabla _{\\langle \\mu } u_{\\lambda \\rangle } \\pi ^{\\lambda \\mu }-[\\frac{d\\Lambda }{d\\tau }+\\Lambda \\theta ] \\nonumber \\\\&=& -\\Pi \\theta + \\nabla _{\\langle \\mu } u_{\\lambda \\rangle } \\pi ^{\\lambda \\mu }-[\\frac{d(\\chi \\Pi )}{d\\tau }+(\\chi \\Pi )\\theta ] .$ In the second line of eq.", "(REF ) we have used eq.", "(REF ) with $\\mu _b=0$ .", "For the $\\chi $ in eq.", "(REF ) we then use $\\chi = \\kappa + \\xi \\Pi +\\xi ^{\\prime \\prime }\\frac{\\pi ^{\\mu \\nu }\\pi _{\\mu \\nu }}{\\Pi },$ i.e., the form of $\\chi $ used in eq.", "(21) in the ref.", "[21] with $W^{\\mu }\\equiv 0$ .", "The second law of thermodynamics is guaranteed (with $\\zeta $ and $\\eta $ being, respectively, the bulk pressure and shear viscosity, which are all positive constants) if the entropy production is given by $TS^{\\mu }_{;\\mu } = \\frac{\\Pi ^2}{\\zeta } + \\frac{\\pi ^{\\mu \\nu }\\pi _{\\mu \\nu }}{2\\eta }.$ This requirement determines the following constitutive equation for, respectively, bulk and shear pressure: $\\frac{\\Pi }{\\zeta } &=& -(1+\\chi ) \\theta -(\\frac{\\kappa }{\\Pi }+2\\xi )\\frac{d\\Pi }{d\\tau }, \\\\\\frac{\\pi _{\\mu \\nu }}{2\\eta } &=& \\nabla _{\\langle \\mu } u_{\\nu \\rangle }-\\xi ^{\\prime \\prime } \\frac{d\\pi _{\\mu \\nu }}{d\\tau }.$ Because eq.", "(REF ) includes term proportional to $1/\\Pi ~d\\Pi /d\\tau $ , one can introduce into the bulk pressure $\\Pi $ an arbitrary constant $z$ (with dimension [GeV]$^{4}$ ) and write $\\frac{\\Pi }{z} + 2\\zeta \\xi \\frac{d}{d\\tau }\\frac{\\Pi }{z}+\\frac{\\kappa \\zeta }{z}\\frac{d}{d\\tau } \\ln \\frac{\\Pi }{z}= -\\frac{\\zeta }{z} (1+\\chi )\\theta .$ We shall now consider small perturbations of $\\Pi $ fields.", "The bulk pressure $\\Pi $ can be written as $\\Pi = \\Pi _0 + \\delta \\Pi $ with the background reference field $\\Pi _0$ and its perturbation field $\\delta \\Pi $ .", "One can also regard $\\Pi _0$ as the value of $\\Pi $ at initial proper time $\\tau _0$ , $\\Pi _0 = \\Pi (\\tau _0)$ .", "Correspondingly, one can write $\\theta _0=u^{\\mu }_{0;\\mu }$ , obtained from the initial flow vector field $u^{\\mu }_0$ at $\\tau _0$ .", "In this case, the perturbation field $\\delta \\Pi $ can be interpreted as $\\delta \\Pi = \\Pi (\\tau )-\\Pi (\\tau _0)$ .", "In this sense, $\\Pi _0$ is a kind of parameter showing the degree of non-equilibrium at initial stage.", "Identifying the arbitrary constant $z$ with $\\Pi _0$ and noticing that $\\frac{d}{d\\tau }\\ln (1+\\frac{\\delta \\Pi }{\\Pi }) \\approx \\frac{1}{\\Pi _0}\\frac{d}{d\\tau }\\delta \\Pi $ , one can rewrite the above equation as: $\\Pi _0 &=& -(1+\\chi ) \\zeta \\theta _0, \\\\\\tau _{\\Pi } \\frac{d\\delta \\Pi }{d\\tau } +\\delta \\Pi &=& -\\zeta (1+\\chi )\\delta \\theta ,$ where the relaxation time $\\tau _{\\Pi }$ is given by $\\tau _{\\Pi } = \\zeta (2\\xi + \\kappa /\\Pi _0 )$ and $\\delta \\theta =\\theta -\\theta _0$ .", "It is interesting to note that $\\Pi _0$ contributes to the relaxation time $\\tau _{\\Pi }$ .", "This means that relaxation processes may depend on the initial condition.", "Note also that if $\\kappa =0$ then contribution $\\kappa /\\Pi _0$ in the relaxation time would disappear.", "A similar approach can also be applied to the $\\pi ^{\\mu \\nu }$ field, with perturbation of the shear viscosity around $\\pi ^{\\mu \\nu }= 0$ , leading to $\\tau _{\\pi } \\frac{d\\delta \\pi ^{\\mu \\nu }}{d\\tau } + \\delta \\pi ^{\\mu \\nu } =2\\eta \\nabla ^{\\langle \\mu } u^{\\nu \\rangle },$ where the relaxation time $\\tau _{\\pi }$ is given by $\\tau _{\\pi } = 2\\xi ^{\\prime \\prime } \\eta .$ The stability of a general class of dissipative relativistic fluid theories was investigated by Hiscock and Lindblom [5], [6], [7].", "Denoting by $\\delta V(x)$ the difference between the actual non-equilibrium value of a field $V(x)$ and the value in the background reference state, $V_0 (x)$ , we assume that variations $\\delta V$ are small enough so that their evolution is adequately described by the linearized equations of motion describing the background state.", "We shall now investigate the stability of the fluid obtained in our model following a prescription proposed in Ref.", "[5], [6], [7].", "In what follows: The background reference state is assumed to be homogeneous in space.", "Notice that, unlike in Ref.", "[5], [6], [7], in our case it is not an equilibrium state but rather a non-equilibrium one with $\\Pi =\\Pi _0$ and with $\\pi ^{\\mu \\nu }=0$ .", "Furthermore, the background space-time is assumed to be flat Minkowski space, so that all background field variables have vanishing gradients.", "We consider following plane wave form of perturbation propagating in $x$ direction $\\delta V = \\delta V_0 \\exp (ikx +\\Gamma \\tau ).", "$ Linearized equations for dissipative fluid dynamical model are given by $\\delta [T^{\\mu \\nu } ]_{;\\mu }&=&0,$ with the perturbed energy-momentum tensor: $\\delta [T^{\\mu \\nu }]&=&(\\varepsilon _{\\rm eq}^* + P_{\\rm eq}^*) (\\delta u^{\\mu }u^{\\nu } + u^{\\mu }\\delta u^{\\nu })+ (\\delta \\varepsilon _{\\rm eq}^* +\\kappa \\delta \\Pi ) u^{\\mu }u^{\\nu }\\nonumber \\\\&-&(\\delta P_{\\rm eq}^* +\\delta \\Pi ) \\Delta ^{\\mu \\nu } +\\delta \\pi ^{\\mu \\nu }.", "$ Here $\\varepsilon _{\\rm eq}^*$ and $P_{\\rm eq}^{*}$ are energy density and pressure in the background non-equilibrium state $\\varepsilon _{\\rm eq}^* \\equiv \\varepsilon _{\\rm eq} +\\kappa \\Pi _0 , \\quad P_{\\rm eq}^{*}\\equiv P_{\\rm eq}+\\Pi _0.", "\\quad $ However, since $\\delta [\\Pi _0]=0$ (it has vanishing gradient and is constant in $\\tau $ ), terms proportional to $\\Pi _0$ do not contribute to the linearized equations eq.", "(REF ) (we ignore terms like $\\Pi _0 \\delta u^{\\mu }$ in the linearized equation).", "Hence, one can replace in the eq.", "(REF ) $\\varepsilon _{\\rm eq}^*$ and $P^*_{\\rm eq}$ by the, respectively, $\\varepsilon _{\\rm eq}$ and $P_{\\rm eq}$ .", "For baryon charge current, since we deal with a fluid in the region where the baryon chemical potential can be considered $\\mu _b=0$ and the net baryon density $n_{\\rm eq}=0$ , one has $\\delta [N^{\\mu } ] \\equiv 0$ .", "The perturbed fluid dynamical fields must satisfy constraints $u_{\\mu }\\delta u^{\\mu }=0, \\quad \\delta \\pi ^{\\mu \\nu } u_{\\mu }=0.$ Hence, in the rest frame of fluid, $u^{\\mu }=(1,0,0,0)$ , the proper time $\\tau $ component of the flow velocity field vanishes, $\\delta u^{\\tau }\\equiv 0$ .", "We therefore obtain the following linearized equations for the energy-momentum tensor and baryon number current: $\\delta [T^{\\mu \\nu }]_{;\\mu }&=& (\\varepsilon _{\\rm eq} + P_{\\rm eq})((ik\\delta u^{x}) u^{\\nu } +\\Gamma \\delta u^{\\nu }) \\nonumber \\\\&+& (\\Gamma \\delta \\varepsilon _{\\rm eq}+\\kappa \\Gamma \\delta \\Pi ) u^{\\nu }- (\\nabla ^{\\nu } \\delta P_{\\rm eq}+\\nabla ^{\\nu }\\delta \\Pi ) \\nonumber \\\\&+& (ik)\\delta \\pi ^{x\\nu } =0.$ The linearized constitutive equations for $\\Pi $ and $\\pi ^{\\mu \\nu }$ have the form (with $\\tilde{\\kappa }\\equiv 1+\\kappa $ ) : $\\frac{(1+\\tau _{\\Pi }\\Gamma )}{\\zeta } \\delta \\Pi &=& -\\tilde{\\kappa }(ik)\\delta u^{x}, \\\\\\frac{(1+\\tau _{\\pi }\\Gamma )}{2\\eta } \\delta \\pi ^{\\mu \\nu }&=&-\\frac{1}{2}(ik)[ \\delta ^{\\mu }_{x}\\delta u^{\\nu }+\\delta ^{\\nu }_x\\delta u^{\\mu } -\\frac{2}{3} \\delta ^{\\mu \\nu }\\delta u^x].", "\\nonumber \\\\$ The parameters $\\xi $ and $\\xi ^{\\prime \\prime }$ introduced above have been absorbed in the expressions for relaxation time, $\\tau _{\\Pi }$ and $\\tau _{\\pi }$ , respectively.", "On the other hand, the parameter $\\kappa $ in the expression of the entropy production is kept and not absorbed in $\\tau _{\\Pi }$ .", "Its role will be discussed later.", "All perturbation equations can be expressed in concise matrix form: $M^A_B \\delta Y^B=0, $ where $\\delta Y^B$ represents the list of fields.", "The system matrix $M^A_B$ can be expressed in a block-diagonal form when one chooses the following set of perturbation variables [6] $\\delta Y^B &=&\\lbrace ~\\delta \\varepsilon _{\\rm eq}, \\delta u^x, \\delta \\Pi , \\delta \\pi ^{xx}, \\nonumber \\\\&& ~\\delta u^y,\\delta \\pi ^{xy}, ~\\delta u^z, \\delta \\pi ^{xz}, \\delta \\pi ^{yz}, \\delta \\pi ^{yy}-\\delta \\pi ^{zz} \\rbrace .\\quad $ In this case, ${\\bf M}= \\left( \\begin{array}{cccc}{\\bf Q} & & \\\\& {\\bf R} & & \\\\& & {\\bf R} & \\\\& & & {\\bf I} \\\\\\end{array} \\right) ,$ where the matrices ${\\bf Q}$ and ${\\bf R}$ are given by ${\\bf Q} &=&\\left(\\begin{array}{cccc}\\Gamma & ikh_{\\rm eq} & \\kappa \\Gamma & 0 \\\\ik\\frac{\\partial P_{\\rm eq}}{\\partial \\varepsilon _{\\rm eq}}&\\Gamma h_{\\rm eq}& ik & ik \\\\0 & ik \\tilde{\\kappa }& \\frac{1+\\tau _{\\Pi }\\Gamma }{\\zeta } & 0 \\\\0 & ik& 0 & \\frac{1+\\tau _{\\pi }\\Gamma }{4\\eta /3} \\\\\\end{array}\\right), \\nonumber \\\\{\\bf R}&=& \\left(\\begin{array}{cc}h_{\\rm eq} \\Gamma & ik \\\\ik & \\frac{1+\\tau _{\\pi }\\Gamma }{\\eta } \\\\\\end{array} \\right),$ respectively, and I is the $2\\times 2$ unit matrix.", "The $h_{\\rm eq}$ denotes the enthalpy density which is defined by $h_{\\rm eq} \\equiv \\varepsilon _{\\rm eq}+P_{\\rm eq}$ .", "For $\\Gamma $ and $k$ satisfying dispersion relation $[{\\rm det}{\\bf M}]=[{\\rm det}~{\\bf R}]^2[{\\rm det}~{\\bf Q}]=0,$ one has plane-wave solution such as eq.", "(REF ) for the linearized equations of the system eq.().", "In what follows we shall discuss in detail the stability of transverse and longitudinal modes separately." ], [ "Propagation of the transverse mode", "The dispersion relation obtained by setting $\\eta ~ {\\rm det} ({\\bf R}) =(\\tau _{\\pi } h_{\\rm eq}) \\Gamma ^2 + h_{\\rm eq} \\Gamma + \\eta k^2 =0$ corresponds to the solution of the perturbation equation which is referred to as the so-called transverse mode.", "The solution of the above equation (REF ) is given by $\\Gamma =\\frac{-h_{\\rm eq}\\pm \\sqrt{h_{\\rm eq}^2 -4\\eta (\\tau _{\\pi }h_{\\rm eq}) k^2}}{2(\\tau _{\\pi } h_{\\rm eq})}.$ Note that we have always ${\\rm Re} [\\Gamma ] <0$ independent of the value of $k$ , which means that any small perturbation propagating in the transverse direction (perpendicular to the $x$ axis, direction which the perturbation wave propagates) will be damped with time $\\tau $ .", "Since the general solution is given by a linear combination of those solutions of the transverse mode, one can say that the plane-wave solution of the mode is stable against small perturbation.", "Note also that when wave number $k \\ge k_c$ , where $k_c = \\sqrt{\\frac{h_{\\rm eq}}{4\\eta \\tau _{\\pi }}},$ the linear perturbation wave propagates towards the transverse direction, but waves with wave number $k<k_c$ are damped." ], [ "Propagation of the longitudinal mode", "Frequencies of the so-called longitudinal mode (propagating parallel to the $x$ direction) are given by the roots of the following dispersion relation: $\\left[\\frac{4\\eta }{3\\tau _{\\pi }}\\frac{\\zeta }{\\tau _{\\Pi }} \\right] {\\rm det}({\\bf Q})\\equiv \\sum _{n=0}^{n=4} q_n \\Gamma ^n =0,$ where the coefficients $q_n$ are given by $q_4 &=& h_{\\rm eq}, \\\\q_3 &=& h_{\\rm eq} \\left( \\frac{1}{\\tau _{\\Pi }}+\\frac{1}{\\tau _{\\pi }} \\right), \\\\q_2 &=& h_{\\rm eq} \\left( \\frac{1}{\\tau _{\\Pi }\\tau _{\\pi }}+\\frac{\\partial P_{\\rm eq}}{\\partial \\varepsilon _{\\rm eq}} k^2 \\right) \\nonumber \\\\&&\\quad + \\left( \\frac{4\\eta }{3\\tau _{\\pi }} +\\tilde{\\kappa }(1-\\kappa \\frac{\\partial P_{\\rm eq}}{\\partial \\varepsilon _{\\rm eq}})\\frac{\\zeta }{\\tau _{\\Pi }} \\right) k^2, \\\\q_1 &=& h_{\\rm eq} \\left( \\frac{1}{\\tau _{\\Pi }}+\\frac{1}{\\tau _{\\pi }} \\right)\\frac{\\partial P_{\\rm eq}}{\\partial \\varepsilon _{\\rm eq}}k^2 \\nonumber \\\\&&\\quad + \\left( \\frac{4\\eta }{3\\tau _{\\pi }}\\frac{1}{\\tau _{\\Pi }}+\\tilde{\\kappa }(1-\\kappa \\frac{\\partial P_{\\rm eq}}{\\partial \\varepsilon _{\\rm eq}})\\frac{\\zeta }{\\tau _{\\Pi }}\\frac{1}{\\tau _{\\pi }} \\right) k^2, \\\\q_0 &=& h_{\\rm eq} \\frac{\\partial P_{\\rm eq}}{\\partial \\varepsilon _{\\rm eq}}\\frac{1}{\\tau _{\\Pi }}\\frac{1}{\\tau _{\\pi }} k^2.$ When all coefficients $q_n$ of the fourth-order equation (REF ) have the same sign, the four (complex or real) solutions of the real part is definitely negative.", "In this case, the general solution which is a linear combination of those four solutions, is stable.", "Since $q_4$ , $q_3$ , and $q_0$ are positive defined, then the stability condition sought after is that $q_2$ and $q_1$ must be simultaneously positive.", "The condition is $1-\\kappa \\frac{\\partial P_{\\rm eq}}{\\partial \\varepsilon _{\\rm eq}}\\bigg |_{n_{\\rm eq}=0}>0.", "$ Using thermodyanmical relation $\\frac{\\partial P_{\\rm eq}}{\\partial \\varepsilon _{\\rm eq}}\\Big |_{n_{\\rm eq}}= c_s^2+\\frac{1}{T}\\frac{\\partial P_{\\rm eq}}{\\partial s_{\\rm eq}}\\Big |_{\\varepsilon _{\\rm eq}}$ , where $c_s^2\\equiv \\frac{\\partial P_{\\rm eq}}{\\partial \\varepsilon _{\\rm eq}}\\Big |_{s_{\\rm eq}} $ is the adiabatic velocity of sound, one can rewrite the stability condition eq.", "(REF ) with the following $c_s^2 + \\frac{1}{T}\\frac{\\partial P_{\\rm eq}}{\\partial s_{\\rm eq}}\\Big |_{\\varepsilon _{\\rm eq}} < \\frac{1}{\\kappa },$ which is exactly the same condition found in previous work [21] in the Eckart frame.", "Note that, when $\\kappa \\rightarrow 0$ , one finds that the speed of sound can exceed unity violating causality.", "On the other hand, when $\\kappa $ is a finite number restricted by $\\frac{1}{T}\\frac{\\partial P_{\\rm eq}}{\\partial s_{\\rm eq}}\\Big |_{\\varepsilon _{\\rm eq}}\\le \\frac{1}{\\kappa } \\le 1+\\frac{1}{T}\\frac{\\partial P_{\\rm eq}}{\\partial s_{\\rm eq}}\\Big |_{\\varepsilon _{\\rm eq}},$ then the velocity of sound satisfies $0\\le c_s \\le 1$ .", "Thus, when the condition for $\\kappa $ , eq.", "(REF ) is satisfied, the fluid is stable against small perturbations and evolves without violating causality." ], [ "Summary and concluding remarks", "We have proposed a novel formulation of the relativistic dissipative hydrodynamical model in an arbitrary frame by using extended matching conditions $u_{\\mu } u_{\\nu } \\delta T^{\\mu \\nu } =\\Lambda , \\quad u_{\\mu } \\delta N^{\\mu } =\\delta n.$ To apply the above extended matching conditions, we have also generalized the form of the entropy current for non-equilibrium state [cf.", "eq.", "(REF )] : $S^{\\mu } = S^{\\mu }_{\\rm eq}-\\alpha V^{\\lambda }\\Delta _{\\lambda }^{\\mu }+\\beta W^{\\mu }-\\beta [ \\mu _{\\rm b} \\delta n + \\Lambda -\\chi \\Pi ] u^{\\mu }.$ The phenomenological parameter $\\chi $ introduced in the generalization of the entropy current can be fixed by the extended thermodynamic stability condition, $\\mu _{\\rm b} \\delta n + \\Lambda -\\chi \\Pi =0.$ (Note that in the usual formulation $\\chi \\equiv 0$ , because of $\\Lambda =0$ and $\\delta n=0$ ).", "Taking the thermodynamical stability condition into account, the entropy current is given by $S^{\\mu } &=& S^{\\mu }_{\\rm eq}-\\alpha V^{\\lambda }\\Delta _{\\lambda }^{\\mu }+\\beta W^{\\mu } \\nonumber \\\\&=& S^{\\mu }_{\\rm eq}-\\alpha \\delta N^{\\mu }+\\beta _{\\lambda } \\delta T^{\\lambda \\mu }-\\chi \\Pi u^{\\mu }.$ As seen in the above equation, the last term $\\chi \\Pi u^{\\mu }$ is the new correction term.", "The corresponding entropy production evidently does not depend on the Lorentz frame considered.", "It is given by $S^{\\mu }_{;\\mu } = -\\alpha _{,\\mu } \\delta N^{\\mu }+ \\beta _{\\lambda ;\\mu } \\delta T^{\\lambda \\mu }- [\\chi \\Pi \\beta ]_{;\\mu } .$ In this paper, we chose the Landau frame and considered a dissipative fluid with zero chemical potential $\\mu _b\\equiv 0$ , for simplicity.", "For this case, the $\\chi $ was assumed as $\\chi = \\kappa + \\xi \\Pi + \\xi ^{\\prime \\prime } \\frac{\\pi ^{\\mu \\nu }\\pi _{\\mu \\nu }}{\\Pi }.$ The physical meaning of $\\kappa $ , $\\xi $ and $\\xi ^{\\prime \\prime }$ are revealed in the discussion on the stability and causality of the fluid in Sec..", "The $\\kappa $ is related to the bound for the speed of sound wave of the fluid and $\\xi $ and $\\xi ^{\\prime \\prime }$ are related to the relaxation of the off-equilibrium system.", "In the linearized field equations, the speed of sound $c_s$ is actually restricted so that $0 \\le c^2_s \\le 1$ when $\\kappa $ is chosen in the following range: $\\frac{1}{T}\\frac{\\partial P_{\\rm eq}}{\\partial s_{\\rm eq}}\\Big |_{\\varepsilon _{\\rm eq}}\\le \\frac{1}{\\kappa } \\le 1+\\frac{1}{T}\\frac{\\partial P_{\\rm eq}}{\\partial s_{\\rm eq}}\\Big |_{\\varepsilon _{\\rm eq}}.$ On the other hand, the relaxation time of the perturbed $\\Pi $ and $\\pi ^{\\mu \\nu }$ fields are respectively given by $\\tau _{\\Pi } = (2\\xi +\\kappa /\\Pi _0)\\zeta , \\quad \\tau _{\\pi } = 2\\xi ^{\\prime \\prime }\\eta .$ The parameter $\\kappa $ also contributes to the relaxation time and may bring in a contribution to the initial condition characterized by $\\Pi _0$ .", "We therefore conclude that, when the matching conditions $\\Lambda = \\chi \\Pi =\\kappa \\Pi + \\xi \\Pi ^2 + \\xi ^{\\prime \\prime } \\pi ^{\\mu \\nu }\\pi _{\\mu \\nu }$ and $\\delta n=0$ (also $\\mu _b=0$ ) are imposed, the relativistic dissipative fluid can be applied to an analysis of the phenomena observed in the ultra-relativistic heavy-ion collisions not only in the Eckart frame, as discussed in [21], but also in the Landau frame.", "The conditions that should be imposed seem to be independent of the Lorentz frame used, i.e., $\\frac{\\lambda T}{\\tau _w} \\le h_{\\rm eq} \\quad \\mbox{and}\\quad c_s^2 \\le \\frac{1}{\\kappa } -\\frac{1}{T}\\frac{\\partial P_{\\rm eq}}{\\partial s_{\\rm eq}}\\Big |_{\\varepsilon _{\\rm eq}},$ where $\\lambda $ and $\\tau _{w}$ are thermal energy conductivity and relaxation time of the thermal energy conduction, respectively [21].", "In the Landau frame, $\\frac{\\lambda T}{\\tau _w}$ should be regarded as 0 because of the definition of the frame, $W^{\\mu }=0.$ We gratefully acknowledge discussions with T. Koide.", "The author would like to warmly thank dr E. Infeld for reading this manuscript." ] ]

[ [ "Convergence and Equivalence results for the Jensen's inequality -\n Application to time-delay and sampled-data systems" ], [ "Abstract The Jensen's inequality plays a crucial role in the analysis of time-delay and sampled-data systems.", "Its conservatism is studied through the use of the Gr\\\"{u}ss Inequality.", "It has been reported in the literature that fragmentation (or partitioning) schemes allow to empirically improve the results.", "We prove here that the Jensen's gap can be made arbitrarily small provided that the order of uniform fragmentation is chosen sufficiently large.", "Non-uniform fragmentation schemes are also shown to speed up the convergence in certain cases.", "Finally, a family of bounds is characterized and a comparison with other bounds of the literature is provided.", "It is shown that the other bounds are equivalent to Jensen's and that they exhibit interesting well-posedness and linearity properties which can be exploited to obtain better numerical results." ], [ "Introduction", "The Jensen's Inequality [1] has had a tremendous impact on many different fields; e.g.", "convex analysis, probability theory, information theory, statistics, control and systems theory [2], [3], [4].", "It concerns the bounding of convex functions of integrals or sums: Lemma 1.1 Let $U$ be a given connected and compact set of $\\mathbb {R}$ , $f$ a function measurable over $U$ and $\\phi $ a convex function measurable over $f(U)$ .", "Then the inequality $\\phi \\left(\\int _Uf(s)d\\mu (s)\\right)\\le \\mu (U)\\int _U[\\phi \\circ f](s)d\\mu (s)$ holds where $\\mu $ is a given nonnegative measure, e.g.", "the Lebesgue measure, and $\\mu (U)=\\int _Ud\\mu (s)<+\\infty $ is the measure of the set $U$ .", "The discrete counterpart is given by: Lemma 1.2 Let $U$ be a given connected and compact set of $\\mathbb {Z}$ , $f$ a function measurable over $U$ and $\\phi $ a convex function measurable over $f(U)$ .", "Then the inequality $\\phi \\left(\\mu (U)^{-1}\\sum _{i\\in U}f_i\\mu _i\\right)\\le \\mu (U)^{-1}\\sum _{i\\in U}\\phi (f_i)\\mu _i$ holds where $\\lbrace \\mu _i\\rbrace _{i\\in U}$ is a given nonnegative measure, e.g.", "the counting measure, and $\\mu (U)=\\sum _{i\\in U}\\mu _i<+\\infty $ is the measure of the set $U$ .", "These inequalities have found applications in systems theory, for instance for the computation of an upper bound on the $\\mathcal {L}_2$ -gain of integral operators involved in time-delay systems analysis [2], [5], [6].", "Another application in time-delay systems [7], [3], [8], [4] concerns the bounding of integral quadratic terms of the form $-\\int _{t-h}^t\\dot{x}(s)^TR\\dot{x}(s)ds$ arising in some approaches based on Lyapunov-Krasovskii functionals (LKFs).", "Discrete counterparts, involving sums instead of integrals, also exist; see e.g.", "[9], [10].", "The objective of the paper is twofold.", "The first goal is to study the conservatism induced by the use of Jensen's inequality.", "Using the Grüss inequality, bounds on the gap will be obtained in the general case and refined by considering the eventual differentiability of the function $f$ .", "Recent results have reported an empirical conservatism reduction of LKFs when using 'delay-fragmentation' or 'delay-partitioning' procedures [3], [11], [12], [13], [4].", "This, however, remains to be proved theoretically and, as a first step towards this result, we will show that the fragmentation scheme reduces the gap of the Jensen's inequality.", "It will be also proved that the upper bound on the gap converges sublinearly to 0 as the fragmentation order increases.", "Finally, non-uniform fragmentation schemes are introduced and their accelerating effect on the convergence is illustrated.", "The second objective of the paper is to show the equivalence between several bounds provided in the literature.", "First, a complete family of bounds is characterized for which the equivalence with Jensen's is proved.", "This family involves additional variables, increasing then its computational complexity.", "Nevertheless, it contains bounds depending affinely on the measure of the interval of integration and remaining well-posed when the measure of interval of integration tends to 0.", "This is of great interest when numerical tools are sought, e.g.", "LMIs.", "This tightness and structural advantages prove their efficiency and motivate their use, e.g.", "for the analysis of time-delay and sampled-data systems.", "The paper is structured as follows, Section is devoted to the conservatism analysis of the Jensen's inequality through the use of the Grüss inequality.", "In Section , the fragmentation procedure is studied.", "Finally, Section concerns the derivation of a family of bounds equivalent to Jensen's and the comparison to existing bounds of the literature.", "Most of the notations are standard except maybe for $\\operatornamewithlimits{col}_{i=1}^N(u_i)$ defining the column vector $\\begin{bmatrix}u_1^T & \\ldots & u_N^T\\end{bmatrix}^T$ .", "The identity matrix is denoted by $I$ .", "For $f,g\\in \\mathbb {R}^n$ , we denote componentwise inequalities by $f\\le g$ .", "For Hermitian matrices $P$ and $Q$ , $P\\prec Q$ (resp.", "$P\\preceq Q$ ) stands for $P-Q$ negative definite (resp.", "negative semidefinite).", "In stability theory for linear systems, the associated convex function in (REF ) and (REF ) is very often $\\phi _Q(z)=z^TQz$ with $Q=Q^T\\succ 0$ .", "But letting $\\tilde{z}=Lz$ with $Q=L^TL$ , we get $\\phi _Q(z)=\\phi _I(\\tilde{z})$ .", "Hence, throughout the paper we will consider the function $\\phi \\equiv \\phi _I$ without loss of generality.", "Let us consider a general inner product space $\\mathcal {H}:=(\\mathcal {L}(U,\\mathbb {R}),\\langle \\cdot ,\\cdot \\rangle _U)$ over $\\mathbb {R}$ .", "A simple, but sufficient for our problem, version of the Grüss Inequality on inner product spaces [15] is defined as follows: Lemma 2.1 (Grüss Inequality) Assume there exist bounded $f^-,f^+,g^-,g^+\\in \\mathbb {R}$ such that $f^-\\le f^+$ , $g^-\\le g^+$ and functions $f,g\\in \\mathcal {L}(U,\\mathbb {R})$ satisfying $f^-\\le f\\le f^+$ and $g^-\\le g\\le g^+$ almost everywhere on $U$ .", "Then the following inequality $\\left|\\mu (U)^{-1}\\langle f,g\\rangle _U-\\mu (U)^{-2}\\langle f,\\mathbb {1}\\rangle _U\\langle g,\\mathbb {1}\\rangle _U\\right|\\le \\frac{1}{4}\\delta _f\\delta _g$ holds with $\\delta _f=f^+-f^-$ , $\\delta _g=g^+-g^-$ and $\\mathbb {1}(\\cdot )=1$ on $U$ , 0 otherwise.", "Moreover the constant term $1/4$ in the right-hand side is sharp and is obtained for the functions $f(s)=g(s)=\\operatornamewithlimits{sgn}\\left(s-(a+b)/2\\right)$ where $\\operatornamewithlimits{sgn}(\\cdot )$ is the signum function, $a=\\inf \\lbrace s\\in U\\rbrace $ and $b=\\sup \\lbrace s\\in U\\rbrace $ .", "More general versions of the Grüss inequality can be found in [16] and references therein, especially for complex functions, more general measure spaces or inner product spaces." ], [ "Conservatism of the Jensen's inequality", "In the continuous case, the function space is defined as: $\\mathcal {L}_c(U,\\mathbb {R}):=\\left\\lbrace f:U\\rightarrow \\mathbb {R},\\ f\\ \\mathrm {bounded, measurable\\ on\\ }U\\right\\rbrace $ where $U$ is a connected bounded set of $\\mathbb {R}$ .", "Let us also define the Hilbert space $\\mathcal {H}_c:=(\\mathcal {L}_c,\\langle \\cdot ,\\cdot \\rangle )$ with inner product $\\langle f,g\\rangle :=\\int _Uf(s)g(s)d\\mu (s)$ where $\\mu $ is a nonnegative measure.", "Using the Grüss inequality (Lemma REF ), the following result on the Jensen's inequality gap is obtained: Theorem 2.1 Given a function $f\\in \\mathcal {L}_c(U,\\mathbb {R}^n)$ and the convex function $\\phi (z)=z^Tz$ , then the Jensen's gap verifies $\\begin{array}{l}\\mu (U)\\int _U\\phi (f(s))d\\mu (s)-\\phi \\left(\\int _Uf(s)d\\mu (s)\\right)\\le \\frac{\\mu (U)^2}{4}\\phi (\\delta _f)\\end{array}$ where $f^-\\le f(\\cdot )\\le f^+$ almost everywhere on $U$ .", "Moreover the constant term $1/4$ in the right-hand side in sharp and is obtained for the functions $f_i(s)=\\operatornamewithlimits{sgn}\\left(s-(a+b)/2\\right)$ , $i=1,\\ldots ,n$ with $a=\\inf \\lbrace s\\in U\\rbrace $ and $b=\\sup \\lbrace s\\in U\\rbrace $ .", "Let us consider the Jensen's inequality (REF ) with $\\phi (z)=z^Tz$ and $f\\in \\mathcal {L}_c(U,\\mathbb {R}^n)$ .", "Simple calculations yield: $\\begin{array}{lcl}\\mathcal {I}_1&:=&\\int _Uf(s)^Tf(s)d\\mu (s)\\\\&=&\\sum _i\\langle f_i,f_i\\rangle \\\\\\mathcal {I}_2&:=&\\int _Uf(s)^Td\\mu (s)\\int _Uf(s)d\\mu (s)\\\\&=&\\sum _i\\langle f_i,\\mathbb {1}\\rangle \\langle f_i,\\mathbb {1}\\rangle .\\end{array}$ Note that the Jensen's inequality (REF ) writes $\\mathcal {I}_2\\le \\mu (U)\\mathcal {I}_1$ , hence we examine the difference $\\mu (U)\\mathcal {I}_1-\\mathcal {I}_2$ and the application of the Grüss inequality yields $0\\le \\mu (U)\\mathcal {I}_1-\\mathcal {I}_2\\le \\frac{\\mu (U)^2}{4}\\phi (f^+-f^-).$ The proof is complete.", "We have the following corollary when the continuous function $f$ is differentiable almost everywhere: Corollary 2.1 Given a continuous function $f\\in \\mathcal {L}_c(U,\\mathbb {R}^n)$ differentiable almost everywhere and the convex function $\\phi (z)=z^Tz$ ; then the Jensen's gap (REF ) is bounded from above by $\\frac{\\mu (U)^4}{4}\\phi \\left(\\sup _{U}|f^\\prime |\\right)$ whereThe derivative has to be understood here in a weak sense due to the possible presence of nonsmooth points.", "Hence the supremum at these points is taken over all possible values for the derivative.", "$\\sup _{U}|f^\\prime |=\\operatornamewithlimits{col}_i[\\sup _{U}\\lbrace |f_i^\\prime |\\rbrace ]$ .", "However, in such a case the coefficient 1/4 is not sharp anymore since the differentiability condition is not taken into account in the derivation of the Grüss inequality.", "If the function is differentiable almost everywhere then we have $f_i^+-f_i^-\\le \\mu (U)\\sup _U\\left|f^\\prime _i\\right|$ .", "The substitution inside (REF ) yields the result.", "Remark 2.1 A discrete-time counterpart of Theorem REF is easy to obtain.", "Both the upper bound and the worst-case function $f$ share a very similar expression with the continuous-time ones.", "It is however important to note that many works have been devoted to the obtention of tight upper bounds for the discrete Jensen's inequality, see e.g.", "[16], [17] and references therein." ], [ "Jensen's Bound Gap Reduction", "From the results of Theorem REF and Corollary REF , it is easily seen that the gap depends on the measure of the set $U$ and on the variability of the function $f$ .", "Since the constant term is sharp, at least for discontinuous functions, this means that the inequality is conservative when the integration domain is large.", "In [3], [13], [4], where time-delay systems are studied, a fragmentation of the integration domain is performed and this procedure refined the delay margin estimation.", "In the following, we will theoretically show, in the continuous-time, that the fragmentation procedure does asymptotically reduce the gap of the Jensen's inequality to 0.", "It will be also proved that the gap upper bound converges sublinearly.", "Finally, the convergence speed can be increased using non-uniform fragmentation schemes.", "Although, we will only consider the continuous-time case, the same conclusions can be drawn for the discrete-time case." ], [ "General results on gap reduction by fragmentation", "To partially explain this in the general continuous case, let us introduce the integrals: $\\begin{array}{lcl}\\mathcal {S}(U)&:=&-\\int _U[\\phi \\circ f](s)d\\mu (s)\\\\\\mathcal {S}_i(U)&:=&-\\int _{U_i}[\\phi \\circ f](s)d\\mu (s),\\ i=1,\\ldots ,N\\end{array}$ where the connected sets $U_i$ 's satisfy $\\bigcup _{k=1}^NU_k=U$ and $\\mu (U_j\\cap U_i)=0$ for all $i,j=1,\\ldots ,N$ , $i\\ne j$ .", "In such a case, we have $\\mathcal {S}(U)=\\sum _i\\mathcal {S}_i(U)$ .", "Then, rather than bounding $\\mathcal {S}(U)$ , each $\\mathcal {S}_i(U)$ will be bounded separately by $\\mathcal {J}_i$ and the respective bounds added up.", "This yields the following result: Theorem 3.1 Let us assume that the compact and connected set $U\\subset \\mathbb {R}$ is partitioned in $N$ disjoint parts as described above.", "Let us also consider the functions $\\phi (z)=z^Tz$ and $f\\in \\mathcal {L}_c$ .", "In such a case, the Jensen's gap is bounded from above by: $\\mathcal {S}(U)-\\sum _{i=1}^N\\mathcal {J}_i\\le e_1(N)=\\frac{1}{4}\\sum _{j=1}^N\\mu (U_j)^2\\phi (M^j-m^j)$ where $m^j=\\operatornamewithlimits{col}_i[m_i^j]$ , $m_i^j:=\\inf _{U_j}\\lbrace f_i\\rbrace $ , $M^j=\\operatornamewithlimits{col}_i[M_i^j]$ , $M_i^j:=\\sup _{U_j}\\lbrace f_i\\rbrace $ .", "Note that the term $1/4$ is also sharp in this case.", "The proof is similar as for Theorem REF .", "Corollary 3.1 Let us assume that the compact and connected set $U\\subset \\mathbb {R}$ is partitioned in $N$ disjoint parts as described above.", "Given a continuous function $f\\in \\mathcal {L}_c(U,\\mathbb {R}^n)$ which is differentiable almost everywhere and the convex function $\\phi (z)=z^Tz$ ; then the Jensen's gap is bounded from above by: $\\mathcal {S}(U)-\\sum _{i=1}^N\\mathcal {J}_i\\le \\ e_2(N)=\\frac{1}{4}\\sum _{j=1}^N\\mu (U_j)^4\\phi \\left(\\sup _{U_j}|f^\\prime |\\right)$ where $\\sup _{U_j}\\lbrace |f^\\prime |\\rbrace =\\operatornamewithlimits{col}_i[\\sup _{U_j}\\lbrace |f^\\prime _i|\\rbrace ]$ (in a weak sense).", "Remark 3.1 More general versions can also be defined using $\\phi _Q$ instead of $\\phi _I$ .", "Fragment-dependent $\\phi _{Q_j}$ can also be considered; see e.g.", "[3], [13], [4].", "The analysis follows the same lines and the results are qualitatively identical, this is thus omitted." ], [ "Equidistant fragmentation", "Let us consider the most simple case where the Lebesgue measure is considered together with a fragmentation of $U$ in $N$ parts of identical measure.", "Then the following corollary can be derived.", "Corollary 3.2 Assume that $f\\in \\mathcal {L}_c$ satisfies the assumptions of the previous results.", "Fragmenting the set $U$ in $N$ parts of identical Lebesgue measure yields the following bound for the Jensen's gap: $e_1(N)=\\frac{\\mu (U)^2}{4N}\\phi (\\theta )$ where $\\theta =\\operatornamewithlimits{col}_i(\\theta _i)$ and $\\theta _i=\\max _j\\lbrace M_i^j-m_i^j\\rbrace $ .", "Moreover, when a continuous function $f$ differentiable almost everywhere is considered, we get the bound: $e_2(N)=\\frac{\\mu (U)^4}{4N^3}\\phi (\\eta ).$ where $\\eta =\\max _j\\left\\lbrace \\sup _{U_j}|f^\\prime |\\right\\rbrace $ .", "The following proposition provides the result on the sublinear convergence of the gap upper bound when the fragmentation order increases: Proposition 3.1 When no continuity assumption is made on the function $f\\in \\mathcal {L}_c$ , the upper bound $e_1(N)$ on the gap satisfies $\\begin{array}{lcl}e_1(N)&=&\\left(1-N^{-1}\\right)e_1(N-1),\\ N>1\\\\e_1(1)&=&\\frac{\\mu (U)^2}{4}\\phi (\\theta ).\\end{array}$ When the continuous function $f\\in \\mathcal {L}_c$ is differentiable almost everywhere, the upper bound $e_2(N)$ on the gap obeys $\\begin{array}{lcl}e_2(N)&=&\\left(1+\\frac{-3N^2+3N-1}{N^3}\\right)e_2(N-1),\\ N>1\\\\&\\sim &\\left(1-3N^{-1}\\right)e_2(N-1)\\ \\mathrm {when\\ }N\\rightarrow +\\infty \\\\e_2(1)&=&\\frac{\\mu (U)^4}{4}\\phi (\\eta ).\\end{array}$ Hence, in both cases, the error tends asymptotically to 0 and the convergence is sublinear since $\\lim _{N\\rightarrow +\\infty }\\frac{e_i(N+1)}{e_i(N)}=1$ , $i\\in \\lbrace 1,2\\rbrace $ .", "Hence we can easily conclude that $\\sum _{i=1}^N\\mathcal {J}_i\\rightarrow \\mathcal {S}(U)\\ \\mathrm {when}\\ N\\rightarrow +\\infty $ in any case.", "Example 3.1 Let us consider the integral $J(\\alpha ):=-\\int _0^1e^{2\\alpha t}dt=\\frac{1}{2\\alpha }(1-e^{2\\alpha })$ for some $\\alpha \\in \\mathbb {R}$ (by continuity we have $J(0)=-1$ ).", "Now consider the following sum of Jensen's bounds taken over each $[ih,(i+1)h]$ with $h=1/N$ : $\\begin{array}{lcl}J_N(\\alpha )&:=&-\\frac{1}{h}\\sum _{i=0}^{N-1}\\left(\\int _{ih}^{(i+1)h}e^{\\alpha t}dt\\right)^2\\\\&=&-\\frac{N(1-e^{\\alpha /N})(1-e^{2\\alpha })}{\\alpha ^2(1+e^{\\alpha /N})}\\\\&\\rightarrow & \\frac{1}{2\\alpha }(1-e^{2\\alpha })\\ \\mathrm {as\\ }N\\rightarrow +\\infty .\\end{array}$ This shows the asymptotic convergence.", "To see the sublinear convergence, it is enough to remark that $\\lim _{N\\rightarrow +\\infty }\\frac{J_{N+1}(\\alpha )-J(\\alpha )}{J_{N}(\\alpha )-J(\\alpha )}=1.$ In Fig.", "REF , we can see the evolution of the normalized bound $J_N(\\alpha )/J(\\alpha )$ for different values for $\\alpha $ .", "For small positive value for $\\alpha <1$ , the convergence is very fast since the function is slowly varying.", "When $\\alpha $ increases the convergence becomes slower.", "This follows from Theorem REF and Corollary REF stating that the gap is depends quadratically on the variability of the function.", "In Fig.", "REF , the different upper bounds $e_1(N)$ (REF ) and $e_2(N)$ (REF ) are compared to the actual gap for the case $\\alpha =1$ .", "Figure: Evolution of the normalized bound J N (α)/J(α)J_N(\\alpha )/J(\\alpha ) (top) and the ratio J N+1 (α)/J N (α)J_{N+1}(\\alpha )/J_N(\\alpha ) (bottom) for different values for α\\alpha Figure: Evolution of the normalized bound J N (1)/J(1)J_N(1)/J(1) and comparison with upper bounds e 1 (N)e_1(N) () and e 2 (N)e_2(N) ()$\\diamond $" ], [ "Nonuniform fragmentation", "The second conclusion, difficult to consider when analyzing time-delay systems, concerns the fact that an adaptive fragmentation scheme could improve the efficiency of the method.", "Indeed, defining fragments whose measure is inversely proportional to the variability of the function should reduce the gap more efficiently than the naive equidistant fragmentation.", "This is an immediate consequence of the fact that Jensen's inequality is an equality for the set of constant functions $f$ (i.e.", "$f^-=f^+$ ).", "Example 3.2 Let us illustrate the above discussion by considering the critical function: $f(s)=\\operatornamewithlimits{sgn}\\left(s-(a+b)/2\\right),\\quad s\\in [a,b]$ and the Lebesgue measure $\\mu $ .", "Define also the intervals $U_1(\\epsilon )=\\left[a,(a+b-\\epsilon )/2\\right]$ , $U_2(\\epsilon )=\\left[(a+b-\\epsilon )/2,(a+b+\\epsilon )/2\\right]$ and $U_3(\\epsilon )=\\left[(a+b+\\epsilon )/2, b\\right]$ for some $\\epsilon \\in (0,a+b)$ .", "It is clear that $\\bigcup _{k=1}^3U_k(\\epsilon )=[a,b]$ and $\\mu (U_i(\\epsilon )\\cap U_j(\\epsilon ))=0$ , for all $i,j\\in \\lbrace 1,2,3\\rbrace $ , $i\\ne j$ and for any $\\epsilon \\in (0,a+b)$ .", "Since the function $f(s)$ is constant over $U_1(\\epsilon )$ and $U_2(\\epsilon )$ , for any $\\epsilon \\in (0,a+b)$ , then the Jensen's inequality is exact and does not introduce any conservatism.", "All the conservatism is concentrated on the interval $U_2(\\epsilon )$ where lies the discontinuity.", "Finally, using Theorem REF , the exact gap on this interval is $\\frac{\\mu (U_2(\\epsilon ))^2}{4}\\phi (f^+-f^-)=\\epsilon ^2.$ Thus the gap can be reduced to an arbitrary small value by choosing adequately the set $U_2(\\epsilon )$ .$\\diamond $ It is important to note that, when a uniform fragmentation scheme is used on the above discontinuous function, the gap does not converge monotonically.", "Indeed, by increasing $N$ , the measure of the interval where lies the discontinuity can be locally increasing.", "The non-monotonic gap is however bounded from above by the monotonic bounds $e_1$ and $e_2$ derived in Section REF .", "Example 3.3 Let us consider the function of Example REF .", "The idea is to use a nonuniform fragmentation to speed up the convergence.", "Since the slope of the function increases, then it seems natural to fragment the interval $[0,1]$ in such a way that the measure of the fragments decreases as we approach 1.", "We thus consider the following delimitating sequence of points $t_i:=(1-\\varepsilon ^{\\frac{i-1}{N}})(1-\\varepsilon )^{-1}$ where $\\varepsilon \\in (0,1)$ is a small positive scalar and $i=1,\\ldots ,N+1$ .", "Obviously we have $t_1=0$ , $t_{N+1}=1$ and the interval $[0,1]$ is nonuniformly partitioned in $N$ parts $U_i:=[t_i,t_{i+1}]$ , $i=1,\\ldots ,N$ .", "The Lebesgue measure of the interval $U_{i}$ satisfies $\\mu (U_i)=\\varepsilon ^{\\frac{i}{N}}\\kappa _0$ where $\\kappa _0=\\frac{\\varepsilon ^{-\\frac{1}{N}}-1}{1-\\varepsilon }$ .", "Choosing $\\varepsilon =10^{-4}$ and considering the exponential function of Example REF with $\\alpha =100$ , we obtain the result depicted on Fig.", "REF where we can see that the convergence speed has been increased quite spectacularly.", "Figure: Comparison of normalized bounds with uniform and nonuniform fragmentation for the exponential function with α=100\\alpha =100 and ε=10 -4 \\varepsilon =10^{-4}$\\diamond $ Unfortunately, despite of being very efficient for the exponential function, this is not of real interest for the analysis of time-delay and sampled-data systems since the trajectories of the system are not known a priori.", "To explain this, let us consider the time delay system $\\dot{x}(t)=-3/2\\cdot x(t-1)$ with functional initial condition $x(\\theta )=1,\\ \\theta \\in [-1,0]$ whose unique solution is oscillating and exponentially stable.", "Choose two different time instants $t_1<t_2$ and introduce the intervals $I_1:=[t_1-1,t_1]$ , $I_2:=[t_2-1, t_2]$ .", "Assuming that the exact solution of the system is known, then an adapted nonuniform partitioning of $I_1$ can be constructed.", "However, this partitioning fails almost surely to be a good one for the interval $I_2$ due to the oscillating behavior of the solution.", "This shows that even when the solution is known, it is, in general, not possible to find a good nonuniform fragmentation common to any interval of integration.", "Hence, it is natural to choose a uniform fragmentation which is the best tradeoff between all the non-uniform fragmentation schemes." ], [ "Equivalence between Jensen's bound and some bounds of the literature", "The goal of this section is to derive a complete family of bounds equivalent to Jensen's in terms of tightness but more complex from a computational point of view.", "Despite of their slight higher computational cost, this family has the nice properties of being affine in the measure of the interval of integration and leading to LMIs that remain well-posed when the measure of the interval of integration tends to 0.", "This is very convenient when this quantity is a (time-varying or uncertain [18], [19]) data of the problemThis is, for instance, the case when time-delay systems are studied.", "In such a case, the length of the interval of integration coincides with the delay itself..", "The latter feature is due to the convexity (affine) of the affine bound w.r.t.", "the measure of the interval integration.", "This is very interesting when LMI-based results are desired as it will be illustrated in Section REF .", "The upcoming results can be used to motivate the use of such affine bounds which are not worse than Jensen's in terms of tightness.", "It is finally shown that several bounds devised in the literature are elements of this general family.", "The results of this section rely on the following lemma: Lemma 4.1 ([4]) Given matrices $C=C^T\\succ 0$ , $A=A^T$ and $B$ , the following statements are equivalent: The matrix inequality $\\mathcal {M}_1:=A-B^TC^{-1}B\\prec 0$ holds.", "There exists a matrix $N$ such that the matrix inequality $\\mathcal {M}_2(N):=A+N^TB+B^TN+N^TCN\\prec 0$ holds.", "The statement $\\inf _N\\left\\lbrace N^TB+B^TN+N^TCN\\right\\rbrace =-B^TC^{-1}B.$ holds true in the partially ordered space of symmetric matrices with partial order '$\\prec $ '.", "Moreover the global minimizer $N^*$ is unique and is given by $N^*=-C^{-1}B$ .", "A proof is given in [4] and is quite involved.", "Here we provide an alternative one (some other proofs could rely on the elimination/projection lemma).", "To see the equivalence between 1) and 2) is enough to show that 3) holds.", "It is easy to see that (REF ) is convex in $N$ since $R\\succ 0$ .", "Hence completing the squares, we find that the minimum $-B^TC^{-1}B$ is attained for $N^*=-C^{-1}B$ .", "Thus, for any triplet $(A,B,C)$ satisfying the assumptions, we can always find $N^*$ such that $\\mathcal {M}_1=\\mathcal {M}_2(N^*)=\\inf _N\\left\\lbrace \\mathcal {M}_2(N)\\right\\rbrace .$ This concludes the proof.", "The interest of the above result is twofold: it can be used to transform complex nonlinear matrix inequalities [20], [8], [21] in a more convenient form [8]; or, what is of interest here, to prove equivalence between different results.", "This is stated in the following theorem: Theorem 4.1 Let us consider a vector function $z(\\cdot )$ integrable over $U$ , with Lebesgue measure $\\mu (U)$ , a real matrix $R=R^T\\succ 0$ , and a vector function $w(\\cdot )$ verifying $\\int _Uz(s)ds=Mw(\\cdot )$ , for some known matrix $M$ .", "Then the following statements are equivalent: The following inequality $-\\int _Uz(s)^TRz(s)ds\\le -\\mu (U)^{-1}w^TM^TRMw$ holds for all $w(\\cdot ),z(\\cdot )$ satisfying the above assumptions.", "There exists a matrix $N$ such that the inequality $-\\int _Uz(s)^TRz(s)ds\\le w^T\\mathcal {Q}(N)w$ holds for all $w(\\cdot ),z(\\cdot )$ satisfying the above assumptions and where $\\mathcal {Q}(N)=N^TM+M^TN+\\mu (U)N^TR^{-1}N.$ The proof is a consequence of Lemma REF .", "Remark 4.1 A discrete-time formulation can be obtained in the same way.", "This is omitted due to space limitations.", "In the sequel, we will apply Theorem REF and its discrete-counterpart in order to show the equivalence between different results of the literature." ], [ "A first Integral inequality", "Let us consider a differentiable function $x(t)$ verifying $\\int _{t_k}^t\\dot{x}(s)ds=Mw$ with $M=\\begin{bmatrix}I & -I\\end{bmatrix}$ and $w(\\cdot )=\\operatornamewithlimits{col}(x(t),x(t_k))$ .", "In [19], the following bound is used: $-\\int _{t_k}^tz(s)^TRz(s)ds\\le w^T\\mathcal {R}w,\\qquad R=R^T\\succ 0,\\ t>t_k$ where $\\mathcal {R}=N^TM+M^TN+(t-t_k)N^TR^{-1}N$ and $N$ is an additional matrix to be determined.", "Then according to Theorem REF , we can conclude on the equivalence with Jensen's.", "However, we will see in the next example that it is sometimes better suited to use the affine formulation." ], [ "A reason for using the affine formulation rather the rational one", "This discussion aims at illustrating the ill-posedness problem arising when the support of the integral varies in time and may vanish at some instants.", "The affine formulation does remain well-posed in such circumstances leading then to more appropriate numerical tools, like LMIs.", "In [19], aperiodic sampled-data systems are considered and an affine version of the Jensen's inequality is employed to provide an LMI condition [19].", "If the rational one was used, this would create a concave term in $(t-t_k)$ of the form $-(t-t_k)^{-1}M_2^TRM_2$ where $R=R^T\\succ 0$ , $M_2\\in \\mathbb {R}^{n\\times 2n}$ , $t\\in [t_k,t_{k+1}]$ , $t_{k+1}-t_k\\le \\tau _m$ , $t_k$ being the sampling instants (following the notation of [22]).", "This term is ill-posed when $t\\rightarrow t_k$ and a way to overcome this problem consists of bounding this term by $-\\tau _m^{-1}M_2^TRM_2$ .", "We compare now this 'result' to the Theorem 1 of [19] on the system [18], [19].", "Theorem 1 of [19], based on the affine formulation, yields a maximal $\\tau _m=1.6894$ while the 'result' based on the rational Jensen's inequality yields the lower value $\\tau _m=0.8691$ .", "Even though the bounds are initially equivalent, the desire of making the problem tractable (obtaining well-posed LMIs) introduces considerable conservatism.", "This illustrates the importance of the affine version of the Jensen's inequality since we have to favor tools in calculations that lead to better numerical solutions." ], [ "A second Integral inequality", "In [23], the following bound is considered: $-\\int _{t-\\tau }^t\\dot{x}(s)^TR\\dot{x}(s)ds\\le \\xi (t)^T({M}+\\tau N^TR^{-1}N)\\xi (t).$ Simple calculations on this upper bound shows that Theorem REF applies and that this bound is equivalent to the Jensen's inequality." ], [ "A sum inequality", "In [9], a bound of the form is introduced: $-\\sum _{i=k-h}^{k-1}y(i)^TRy(i)\\le \\xi (k)^T\\left[\\beth +hN^TR^{-1}N\\right]\\xi (k).$ In this case, the discrete-time version of Theorem REF can be applied, showing then equivalence with the discrete-time version of the Jensen's inequality." ], [ "Conclusion", "The conservatism of the Jensen's inequality has been analyzed using the Grüss inequality.", "Motivated by several results of the literature, a fragmentation scheme has been considered.", "It has been shown that the gap converges asymptotically to 0 as the order of fragmentation increases.", "Next, nonuniform fragmentation techniques have been introduced and their possible accelerating effect illustrated.", "Unfortunately, they can be applied in some very specific cases only.", "This showed that the best tradeoff lies in the use of uniform fragmentation schemes.", "The second part of the paper has been devoted to the characterization of a family of bounds, equivalent to Jensen's in terms of tightness but with a higher computational complexity.", "This family defines affine bounds in the measure of interval of integration (rational and nonconvex for Jensen's) for which the obtained matrix inequalities remain well-posed when the measure of the integral of integration tends to 0.", "This is of crucial interest when LMIs are sought.", "It has been shown that several bounds devised in the literature are elements of this family.", "As a final remark, this homogeneity suggests that Jensen's inequality and its companions could be the best bounds still preserving a tractable structure to the problem.", "This together with a (possibly adaptive) fragmentation scheme should lead to asymptotically exact well-posed approximants of integral terms, affine in the measure of the integration support." ], [ "Acknowledgments", "The author thanks the ACCESS team, the editor and the associate editor as well as the anonymous reviewers who helped to improve the quality of the paper." ] ]

[ [ "Column Reordering for Box-Constrained Integer Least Squares Problems" ], [ "Abstract The box-constrained integer least squares problem (BILS) arises in MIMO wireless communications applications.", "Typically a sphere decoding algorithm (a tree search algorithm) is used to solve the problem.", "In order to make the search algorithm more efficient, the columns of the channel matrix in the BILS problem have to be reordered.", "To our knowledge, there are currently two algorithms for column reordering that provide the best known results.", "Both use all available information, but they were derived respectively from geometric and algebraic points of view and look different.", "In this paper we modify one to make it more computationally efficient and easier to comprehend.", "Then we prove the modified one and the other actually give the same column reordering in theory.", "Finally we propose a new mathematically equivalent algorithm, which is more computationally efficient and is still easy to understand." ], [ "Introduction", "Given a real vector $y\\in \\mathbb {R}^m$ and a real matrix $H\\in \\mathbb {R}^{m \\times n}$ , integer vectors $l, u \\in \\mathbb {Z}^n$ with $l<u$ , the box-constrained integer least squares (BILS) problem is defined as: $\\min _{x\\in {\\cal B }} \\Vert y- Hx \\Vert _2,$ where ${\\cal B} = {\\cal B}_1 \\times \\cdots \\times {\\cal B}_n$ with ${\\cal B}_i = \\lbrace x_i \\in \\mathbb {Z} : l_i \\le x_i \\le u_i \\rbrace $ .", "This problem arises in wireless communications applications such as MIMO signal decoding.", "In this paper, we assume that $H$ has full column rank.", "The set $\\lbrace w=Hx : x\\in \\mathbb {Z}^n\\rbrace $ is referred to as the lattice generated by $H$ .", "Let $H$ have the QR factorization $H=[Q_1, Q_2] \\begin{bmatrix}R \\\\ 0 \\end{bmatrix},$ where $[\\underset{n}{Q_1}, \\underset{m-n}{Q_2}] \\in \\mathbb {R}^{m\\times m}$ is orthogonal and $R\\in \\mathbb {R}^{n\\times n}$ is upper triangular.", "Then, with $\\bar{y}=Q_1^Ty$ the BILS problem (REF ) is reduced to $\\min _{x \\in {\\cal B}} \\Vert {\\bar{y}}- Rx \\Vert _2.$ To solve this reduced problem sphere decoding search algorithms (see, e.g., [1], [2] and [3]) enumerate the elements in ${\\cal B}$ in some order to find the optimal solution.", "If we reorder the columns of $H$ , i.e., we apply a permutation matrix $P$ to $H$ from the right, then we will obtain a different R-factor, resulting in different search speed.", "A few algorithms have been proposed to find $P$ to minimize the complexity of the search algorithms.", "In [1], the well-known V-BLAST column reordering strategy originally given in [4] was proposed for this purpose.", "In [3], the SQRD column reordering strategy originally presented in [5] for the same purpose as V-BLAST, was proposed for this purpose.", "Both strategies use only the information of the matrix $H$ .", "In [6], Su and Wassell considered the geometry of the BILS problem for the case that $H$ is nonsingular and proposed a new column reordering algorithm (to be called the SW algorithm from here on for convenience) which uses all information of the BILS problem (REF ).", "Unfortunately, in our point of view, the geometric interpretation of this algorithm is hard to understand.", "Probably due to page limit, the description of the algorithm is very concise, making efficient implementation difficult for ordinary users.", "In this paper we will give some new insight of the SW algorithm from an algebraic point of view.", "We will make some modifications so that the algorithm becomes more efficient and easier to understand and furthermore it can handle a general full column rank $H$ .", "Independently Chang and Han in [3] proposed another column reordering algorithm (which will be referred to as CH).", "Their algorithm also uses all information of (REF ) and the derivation is based on an algebraic point of view.", "It is easy to see from the equations in the search process exactly what the CH column reordering is doing and why we should expect a reduced complexity in the search process.", "The detailed description of the CH column reordering is given in [3] and it is easy for others to implement the algorithm.", "But our numerical tests indicated CH has a higher complexity than SW, when SW is implemented efficiently.", "Our numerical tests also showed that CH and SW almost always produced the same permutation matrix $P$ .", "In this paper, we will show that the CH algorithm and the (modified) SW algorithm give the same column reordering in theory.", "This is interesting because both algorithms were derived through different motivations and we now have both a geometric justification and an algebraic justification for why the column reordering strategy should reduce the complexity of the search.", "Furthermore, using the knowledge that certain steps in each algorithm are equivalent, we can combine the best parts from each into a new algorithm.", "The new algorithm has a lower flop count than either of the originals.", "This is important to the successive interference cancellation decoder, which computes a suboptimal solution to (REF ).", "The new algorithm can be interpreted in the same way as CH, so it is easy to understand.", "In this paper, $e_i$ denotes the $i^{th}$ column of the identity matrix $I$ .", "For a set of integer numbers ${\\cal S}$ and real number $x$ , $\\lfloor x\\rceil _{\\cal S}$ denotes the nearest integer in ${\\cal S}$ to $x$ and if there is a tie it denotes the one which has smaller magnitude.", "For $z\\in {\\cal S}$ , ${\\cal S}\\backslash z$ denotes ${\\cal S}$ after $z$ is removed.", "We sometimes use MATLAB-like notation for matrices and vectors, e.g., $A_{1:m,1:n}$ denotes the matrix formed by the first $m$ rows and $n$ columns of the matrix $A$ and $A_{:,1:n}$ denote the matrix formed by the first $n$ columns of $A$ .", "The $j^{th}$ column of a matrix $A$ is demoted either by $a_j$ or $A_{:,j}$ ." ], [ "Search Process", "Both CH and SW column reordering algorithms use ideas that arise from the search process.", "Before the column reorderings are introduced, it is important to have an understanding of the sphere decoding search process.", "Consider the ILS problem (REF ).", "We would like to enumerate the elements in ${\\cal B}$ in an efficient manner in order to find the solution $x$ .", "One such enumeration strategy is described in [3].", "We will now describe it briefly.", "Suppose that the solution satisfies the following bound, $\\left\\Vert \\bar{y} - Rx \\right\\Vert _2^2 < \\beta .$ There are a few ways to choose a valid initial value for $\\beta $ , see, e.g., [3].", "The inequality (REF ) defines an ellipsoid in terms of $x$ or a hyper-sphere in terms of the lattice point $w=Rx$ with radius $\\beta $ .", "Define $c_k = (\\bar{y}_k - \\sum _{j=k+1}^nr_{kj}x_j)/r_{kk}, \\; k=n, n-1,\\ldots , 1,$ where when $k=n$ the sum in the right hand side does not exist.", "Then (REF ) can be rewritten as $\\sum _{k=1}^n r_{kk}^2(x_k-c_k)^2 < \\beta ,$ which implies the following set of inequalities: $\\text{level } k: \\ \\ r_{kk}^2(x_k-c_k)^2 < \\beta -\\sum _{i=k+1}^nr_{ii}^2(x_i-c_i)^2, $ for $k=n,n-1,\\ldots , 1$ .", "We begin the search process at level $n$ .", "Choose $x_n = \\lfloor c_n \\rceil _{{\\cal B}_n}$ , the nearest integer in ${\\cal B}_n$ to $c_n$ .", "If the inequality (REF ) with $k=n$ is not satisfied, it will not be satisfied for any integer, this means $\\beta $ was chosen to be too small, it must be enlarged.", "With $x_n$ fixed, we can move to level $n-1$ and choose $x_{n-1} = \\lfloor c_{n-1} \\rceil _{{\\cal B}_{n-1}}$ with $c_{n-1}$ calculated as in (REF ).", "At this point it is possible that the inequality (REF ) is no longer satisfied.", "If this is the case, we must move back to level $n$ and choose $x_n$ to be the second nearest integer to $c_n$ .", "We will continue this procedure until we reach level 1, moving back a level if ever the inequality for the current level is no longer satisfied.", "When we reach level 1, we will have found an integer point $\\hat{x}$ .", "We then update $\\beta = \\left\\Vert \\bar{y} - R\\hat{x} \\right\\Vert _2^2$ and try to find a better integer point which satisfies the box-constraint in the new ellipsoid.", "Finally in the search process, when we can no longer find any $x_n$ to satisfy (REF ) with $k=n$ , the search process is complete and the last integer point $\\hat{x}$ found is the solution.", "The above search process is actually a depth-first tree search, see Fig.", "REF , where the number in a node denote the step number at which the node is encountered.", "Figure: An example of the search process with solution x=[-1,3,1] T x = [-1,3,1]^T." ], [ "Column Reordering", "In this section we introduce the two orginal column reordering algorithms, CH and SW and explain their motivations.", "We give some new insight on SW and propose a modified version.", "We also give a complexity analysis for both algorithms." ], [ "Chang and Han's Algorithm", "The CH algorithm first computes the QR factorization $H$ , then tries to reorder the columns of $R$ .", "The motivation for this algorithm comes from observing equation (REF ).", "If the inequality is false we know that the current choice for the value of $x_k$ given $x_{k+1:n}$ are fixed is incorrect and we prune the search tree.", "We would like to choose the column permutations so that it is likely that the inequality will be false at higher levels in the search tree.", "The CH column reordering strategy does this by trying to maximize the left hand side of (REF ) with large values of $\\left| r_{kk}\\right|$ and minimize the right hand side by making $\\left| r_{kk}(x_k-c_k) \\right|$ large for values of $k = n,n-1, \\dots , 1$ .", "Here we describe step 1 of the CH algorithm, which determines the last column of the final $R$ (or equivalently the last column of the final $H$ ).", "Subsequent steps are the same but are applied to a subproblem that is one dimension smaller.", "In step 1, for $i = 1,\\dots ,n$ we interchange columns $i$ and $n$ of $R$ (thus entries of $i$ and $n$ in $x$ are also swapped), then return $R$ to upper-triangular by a series of Givens rotations applied to $R$ from the left, which are also applied to $\\bar{y}$ .", "To avoid confusion, we denote the new $R$ by $\\hat{R}$ and the new $\\bar{y}$ by $\\hat{y}$ .", "We then compute $c_n=\\hat{y}_n/\\hat{r}_{n,n}$ and $x_i^c=\\arg \\min _{x_i\\in {\\cal B}_i}\\left|\\hat{r}_{nn}(x_i- c_n) \\right| = \\left\\lfloor c_n \\right\\rceil _{{\\cal B}_i},$ where the superscript $c$ denotes the CH algorithm.", "Let $\\bar{x}_i^c$ be the second closest integer in ${{\\cal B}_i}$ to $c_n$ , i.e., $\\bar{x}_i^c= \\left\\lfloor c_n \\right\\rceil _{{\\cal B}_i\\backslash x_i^c}.$ Define $\\mathrm {dist}_i^c = |\\hat{r}_{nn}( \\bar{x}_i^c -c_n) |,$ which represents the partial residual given when $x_i$ is taken to be $\\bar{x}_i^c$ .", "Let $j = {\\arg \\max }_i \\mathrm {dist}_i^c$ .", "Then column $j$ of the original $R$ is chosen to be the $n^{th}$ column of the final $R$ .", "With the corresponding updated upper triangular $R$ and $\\bar{y}$ (here for convenience we have removed hats), the algorithm then updates $\\bar{y}_{1:n-1}$ again by setting $\\bar{y}_{1:n-1}: = \\bar{y}_{1:n-1} - r_{1:n-1,n}x_j$ where $x_j=x_j^c$ .", "Choosing $x_j$ to be $x_j^c$ here is exactly the same as what the search process does.", "We then continue to work on the subproblem $\\min _{\\tilde{x}\\in \\mathbb {Z}^{n-1}} \\left\\Vert \\bar{y}_{1:n-1}-R_{1:n-1,1:n-1}\\tilde{x} \\right\\Vert _2,$ where $\\tilde{x}=[x_1,\\ldots , x_{j-1}, x_n, x_{j+1}, \\ldots x_{n-1}]^T$ satisfies the corresponding box constraint.", "The pseudocode of the CH algorithm is given in Algorithm REF .", "To determine the last column, CH finds the permutation to maximize $\\left|r_{nn}(\\bar{x}_i^c-c_n) \\right|$ .", "Using $\\bar{x}_i^c$ instead of $x_i^c$ ensures that $\\left| \\bar{x}_i^c-c_n \\right|$ is never less than $0.5$ but also not very large.", "This means that usually if $\\left| r_{nn}(\\bar{x}_i^c-c_n)\\right|$ is large, $\\left| r_{nn} \\right|$ is large as well and the requirement to have large $|r_{nn}|$ is met.", "Using $x_i^c$ would not be a good choice because $\\left| x_i^c - c_n \\right|$ might be very small or even 0, then column $i$ would not be chosen to be column $n$ even if the corresponding $|r_{nn}|$ is large and on the contrary a column with small $|r_{nn}|$ but large $|x_i^c-c_n|$ may be chosen.", "Now we will consider the complexity of CH.", "The significant cost comes from line REF in Algorithm REF , which requires $6(k-i)^2$ flops.", "If we sum this cost over all loop iterations and add the cost of the QR factorization by Householder transformations, we get a total complexity of $0.5n^4+2mn^2$ flops.", "CH Algorithm - Returns $p$ , the column permutation vector [1] $p := 1:n$ $p^{\\prime } := 1:n$ Compute the QR factorization of $H$ : $\\left[{\\begin{matrix}Q_1^T \\\\ Q_2^T \\end{matrix}}\\right]H= \\left[{\\begin{matrix}R\\\\ 0 \\end{matrix}}\\right]$ and compute $\\bar{y} : = Q_1^Ty$ $k=n$ to 2 $maxDist := -1$ $i=1$ to $k$ $\\hat{y} := \\bar{y}_{1:k}$ $\\hat{R} := R_{1:k,1:k}$ swap columns $i$ and $k$ of $\\hat{R}$ , return it to upper triangular with Givens rotations, also apply the Givens rotations to $\\hat{y}$ $x_i^c := \\left\\lfloor \\hat{y}_k/\\hat{r}_{k,k}\\right\\rceil _{{\\cal B}_i}$ $\\bar{x}_i^c := \\left\\lfloor \\hat{y}_k/\\hat{r}_{k,k}\\right\\rceil _{{\\cal B}_i\\backslash x_i^c}$ $dist_i^c := \\left| \\hat{r}_{k,k}\\bar{x}_i^c - \\hat{y}_k\\right| $ $dist_i^c > maxDist$ $maxDist := dist_i^c$ $j := i$ $R^{\\prime } := \\hat{R}$ $y^{\\prime } := \\hat{y}$ $p_k := p^{\\prime }_j$ Interchange the intervals ${{\\cal B}_k}$ and ${{\\cal B}_{j}}$ Intechange entries $k$ and $j$ in $p^{\\prime }$ $R_{1:k,1:k} := R^{\\prime }$ $\\bar{y}_{1:k} := y^{\\prime } - R^{\\prime }_{1:k,k}x_j^c$ $p_1 := p^{\\prime }_1$" ], [ "Su and Wassell's Algorithm", "The motivation for the SW algorithm comes from examining the geometry of the search process.", "Figure: Geometry of the search with two different column ordering.Fig.", "REF shows a 2-D BILS problem; REF (a) represents the original column ordering and REF (b) is after the columns have been swapped.", "In the SW algorithm $H=[h_1,\\ldots , h_n]$ is assumed to be square and non-singular.", "Let $G =[g_1,\\ldots , g_n]= H^{-T}.$ For any integer $\\alpha $ , [6] defines the affine sets, $F_i(\\alpha ) = \\lbrace w \\ | \\ g_i^T(w-h_i\\alpha ) = 0\\rbrace $ .", "The lattice points generated by $H$ occur at the intersections of these affine sets.", "Let the orthogonal projection of a vector $s$ onto a vector $t$ be denoted as $\\mbox{proj}_t(s)$ , then the orthogonal projection of some vector $s$ onto $F_i(\\alpha )$ is $\\mbox{proj}_{F_i(\\alpha )}(s) = s -\\mbox{proj}_{g_i}(s-h_i\\alpha ).$ Therefore the orthogonal distance between $s$ and $F_i(\\alpha )$ is $\\mathrm {dist}(s,F_i(\\alpha )) = \\Vert s - \\mbox{proj}_{F_i(\\alpha )}(s) \\Vert _2$ .", "In [6], the points labeled $\\mbox{proj}_{F_2(1)}(y)$ and $\\mbox{proj}_{F_2(-1)}(y)$ in Fig.", "REF are called residual targets and “represent the components [of $y$ ] that remain after an orthogonal part has been projected away.” Note that $F_2(\\alpha )$ in Fig.", "REF is a sublattice of dimension 1.", "Algebraically it is the lattice generated by $H$ with column 2 removed.", "It can also be thought of as a subtree of the search tree where $x_2 = \\alpha $ has been fixed.", "In the first step of the search process for a general case, $x_n$ is chosen to be $x_n=\\arg \\min _{\\alpha \\in {\\cal B}_n}\\mathrm {dist}(y,F_n(\\alpha ))$ ; thus $F_n(x_n)$ is the nearest affine set to $y$ .", "Actually the value of $x_n$ is identical to $\\lfloor c_n \\rceil _{{\\cal B}_n}$ given in Section  , which will be proved later.", "Then $y$ is updated as $y := \\mbox{proj}_{F_n(x_n)}(y) - h_nx_n$ .", "If we look at Fig.", "REF , we see that the projection $\\mbox{proj}_{F_n(x_n)}(y)$ moves $y$ onto $F_n(x_n)$ , while the subtraction of $h_nx_n$ algebraically fixes the value of $x_n$ .", "This is necessary because in subsequent steps we will not consider the column $h_n$ .", "We now apply the same process to the new $n-1$ dimensional search space $F_n(x_n)$ .", "If at some level $i$ , $\\min _{\\alpha \\in {\\cal B}_i}\\mathrm {dist}(y,F_i(\\alpha ))$ exceeds the current search radius, we must move back to level $i+1$ .", "When the search process reaches level 1 and fixes $x_1$ , it updates the radius to $\\mathrm {dist}(y,F_1(x_1))$ and moves back up to level 2.", "Note that this search process is mathematically equivalent to the one described in section ; the difference is that it does projections because the generator matrix is not assumed to be upper-triangular.", "Computationally the former is more expensive than the latter.", "To see the motivation of the SW algorithm for choosing a particular column ordering, consider Fig.", "REF .", "Suppose the search algorithm has knowledge of the residual for the optimal solution (the radius of the circle in the diagram).", "With the column ordering chosen in (a), there are two possible choices for $x_2$ , leading to the two dashed lines $F_2(-1)$ and $F_2(1)$ which cross the circle.", "This means that we will need to find $x_1$ for both of these choices before we can determine which one leads to the optimum solution.", "In (b), there is only one possible choice for $x_1$ , leading to the only dashed line $F_1(-1)$ which crosses the circle, meaning we only need to find $x_2$ to find the optimum solution.", "Since the projection resulting from the correct choice of $x_2$ will always be within the sphere, it makes sense to choose the ordering which maximizes the distance to the second best choice for $x_2$ in hopes that the second nearest choice will result in a value for $\\min _{\\alpha \\in {\\cal B}_2}\\mathrm {dist}(y,F_2(\\alpha ))$ outside the sphere and the dimensionality can be reduced by one.", "For more detail on the geometry, see [6].", "The following will give an overview of the SW algorithm as given in [6] but described in a framework similar to what was used to describe CH.", "In the first step to determine the last column, for each $i = 1, \\dots , n$ , we compute $x_i^s \\!=\\!", "\\arg \\min _{\\alpha \\in {\\cal B}_i}\\mathrm {dist}(y,F_i(\\alpha ))\\!=\\!", "\\arg \\min _{\\alpha \\in {\\cal B}_i}| y^Tg_i - \\alpha | \\!=\\!", "\\left\\lfloor y^Tg_i \\right\\rceil _{{\\cal B}_i},$ where the superscript $s$ stands for the SW algorithm.", "Let $\\bar{x}_i^s$ be the second closest integer in ${{\\cal B}_i }$ to $y^Tg_i$ , i.e., $\\bar{x}_i^s = \\left\\lfloor y^Tg_i \\right\\rceil _{{\\cal B}_i\\backslash x_i^s}.$ Let $j = \\arg \\max _i \\mathrm {dist}(y,F_i(\\bar{x}_i^s))$ .", "Then SW chooses column $j$ as the last column of the final reordered $H$ , updates $y$ by setting $y:=\\mbox{proj}_{F_j(x_j^s)}(y) -h_jx_j^s$ and updates $G$ by setting $ g_i: = \\mbox{proj}_{F_j(0)}(g_i)$ for all $i\\ne j$ .", "After $G$ and $y$ have been updated, the algorithm continues to find column $n-1$ in the same way etc.", "The pseudo-code of the SW algorithm is given in Algorithm REF .", "SW Algorithm - Returns $p$ , the column permutation vector [1] $p := 1:n$ $p^{\\prime } := \\lbrace 1, 2, \\ldots , n\\rbrace $ $G := H^{-T}$ $k=n$ to 2 $maxDist := -1$ $i \\in p^{\\prime }$ $x_i^s := \\left\\lfloor y^Tg_i \\right\\rceil _{{\\cal B }_i}$ $\\bar{x}_i^s := \\left\\lfloor y^Tg_i \\right\\rceil _{{{\\cal B }_i}{\\backslash x_i^s}}$ $\\mathrm {dist}_i^s := \\mathrm {dist}(y,F_i(\\bar{x}_i^s))$ $dist_i^s > maxDist$ $maxDist := dist_i^s$ $j := i$ $p_k := j$ $p^{\\prime } := p^{\\prime } \\backslash j$ $y := \\mbox{proj}_{F_j(x_j^s)}(y) -h_jx_j^s$ $i \\in p^{\\prime }$ $g_i := \\mbox{proj}_{F_j(0)}(g_i) $ $p_1 := p^{\\prime }$ [6] did not say how to implement the algorithm and did not give a complexity analysis.", "The parts of the cost we must consider for implementation occur in lines REF and REF .", "Note that $ \\mathrm {dist}(y,F_i(\\bar{x}_i^s))=\\Vert \\mbox{proj}_{g_i}(y-h_i\\bar{x}_i^s)\\Vert _2$ and $\\mbox{proj}_{F_j(0)}(g_i)= g_i -\\mbox{proj}_{g_i} g_i$ , where $\\mbox{proj}_{g_i}=g_ig_i^\\dag = g_ig_i^T/\\Vert g_i\\Vert ^2$ .", "A naive implementation would first compute $\\mbox{proj}_{g_i}$ , requiring $n^2$ flops, then compute $\\Vert \\mbox{proj}_{g_i}(y-h_i\\bar{x}_i^s)\\Vert _2$ and $g_i -\\mbox{proj}_{g_i} g_i$ , each requiring $2n^2$ flops.", "Summing these costs over all loop iterations we get a total complexity of $2.5n^4$ flops.", "In the next subsection we will simplify some steps in Algorithm REF and show how to implement them efficiently." ], [ "Algebraic Interpretation and Modifications of SW", "In this section we give new algebraic interpretation of some steps in Algorithm 2, simplify some key steps to improve the efficiency, and extend the algorithm to handle a more general case.", "All line numbers refer to Algorithm 2.", "First we show how to efficiently compute $\\mathrm {dist}_i^s$ in line REF .", "Observing that $g_i^Th_i = 1$ , we have $\\mathrm {dist}_i^s = \\Vert g_ig_i^\\dag (y-h_i\\bar{x}_i^s) \\Vert _2= | y^Tg_i -\\bar{x}_i^s |/\\Vert g_i \\Vert _2.$ Note that $y^Tg_i$ and $\\bar{x}_i^s$ have been computed in lines REF and REF , respectively.", "So the main cost of computing $\\mathrm {dist}_i^s$ is the cost of computing $\\Vert g_i\\Vert _2$ , requiring only $2n$ flops.", "For $k=n$ in Algorithm 2, $y^Tg_i=y^TH^{-T}e_i=(H^{-1}y)^Te_i$ , i.e., $y^Tg_i$ is the $i^{th}$ entry of the real solution for $Hx=y$ .", "The interpretation can be generalized to a general $k$ .", "In line REF Algorithm 2, $g_i^{\\small \\mbox{new}} & \\equiv \\mbox{proj}_{F_j(0)}(g_i) \\nonumber \\\\& =(I- \\mbox{proj}_{g_j})g_i=g_i- g_j(g_j^Tg_i/\\Vert g_j\\Vert _2^2).", "$ Using the last expression for computation needs only $4n$ flops (note that $\\Vert g_j\\Vert _2$ has been computed before, see (REF )).", "We can actually show that the above is performing updating of $G$ , the Moore-Penrose generalized inverse of $H$ after we remove its $j^{th}$ column.", "For proof of this, see [8].", "In line REF of Algorithm 2, $y^{\\small \\mbox{new}} & \\!\\equiv \\!", "\\mbox{proj}_{F_j(x_j^s)}(y) - h_jx_j^s\\!=\\!", "(y - g_jg_j^\\dag (y-h_jx_j^s)) - h_jx_j^s \\nonumber \\\\&= (I-\\mbox{proj}_{g_j})(y-h_jx_j^s).", "$ This means that after $x_j$ is fixed to be $x_j^s$ , $h_jx_j^s$ is combined with $y$ (the same as CH does) and then the vector is projected to the orthogonal complement of the space spanned by $g_j$ .", "We can show that this guarantees that the updated $y$ is in the subspace spanned by the columns of $H$ which have not been chosen.", "This is consistent with the assumption that $H$ is nonsingular, which implies that the original $y$ is in the space spanned by the columns of $H$ .", "However, it is not necessary to apply the orthogonal projector $I- \\mbox{proj}_{g_j}$ to $y-h_jx_j^s$ in (REF ).", "The reason is as follows.", "In Algorithm 2, $y^{\\small \\mbox{new}}$ and $g_i^{\\small \\mbox{new}}$ will be used only for computing $(y^{\\small \\mbox{new}})^Tg_i^{\\small \\mbox{new}}$ (see line REF ).", "But from (REF ) and (REF ) $(y^{\\small \\mbox{new}})^Tg_i^{\\small \\mbox{new}}& =(y-h_jx_j^s)^T(I-\\mbox{proj}_{g_j})(I-\\mbox{proj}_{g_j})g_i \\\\&=(y-h_jx_j^s)^Tg_i^{\\small \\mbox{new}}.$ Therefore, line REF can be replaced by $y:=y-h_jx_j^s$ .", "This not only simplifies the computation but also is much easier to interpret—after $x_j$ is fixed to be $x_j^s$ , $h_jx_j^s$ is combined into $y$ as what the CH algorithm does.", "Let $H_{:,1:n-1}$ denote $H$ after its $j^{th}$ column is removed.", "We then continue to work on the subproblem $\\min _{\\check{x}\\in \\mathbb {Z}^{n-1}}\\Vert y-H_{:,1:n-1}\\check{x}\\Vert _2,$ where $\\check{x}=[x_1,\\ldots ,x_{j-1},x_{j+1},\\ldots ,x_n]^T$ satisfies the corresponding box constraint.", "Here $H_{:,1:n-1}$ is not square.", "But there is no problem to handle it, see the next paragraph.", "In [6], $H$ is assumed to be square and non-singular.", "In our opinion, this condition may cause confusion, since for each $k$ except $k=n$ in Algorithm 2, the remaining columns of $H$ which have not been chosen do not form a square matrix.", "Also the condition restricts the application of the algorithm to a general full column rank matrix $H$ , unless we transform $H$ to a nonsingular matrix $R$ by the QR factorization.", "To extend the algorithm to a general full column rank matrix $H$ , we need only replace line REF by $G:=(H^{\\dagger })^T$ .", "This extension has another benefit.", "We mentioned before that the updating of $G$ in line REF is actually the updating of the Moore-Pernrose generalized inverse of the matrix formed by the columns of $H$ which have not been chosen.", "So the extension makes all steps consistent.", "To reliably compute $G$ for a general full column rank $H$ , we can compute the QR factorization $H=Q_1R$ by the Householder transformations and then solve the triangular system $RG^T=Q_1^T$ to obtain $G$ .", "This requires $(5m-4n/3)n^2$ flops.", "Another less reliable but more efficient way to do this is to compute $G=H(H^TH)^{-1}$ .", "To do this efficiently we would compute the Cholesky factorization $H^TH = R^TR$ and solve $R^TRG^T = H^T$ for $G$ by using the triangular structure of $R$ .", "The total cost for computing $G$ by this method can be shown to be $3mn^2+\\frac{n^3}{3}$ .", "If $H$ is square and nonsingular, we would use the LU factorization with partial pivoting to compute $H^{-1}$ and the cost is $2n^3$ flops.", "For the rest part of the algorithm if we use the simplification and efficient implementations mentioned above, we can show that it needs $4mn^2$ flops.", "We see the modified SW algorithm is much more efficient than both the CH algorithm and the SW algorithm implemented in a naive way we mentioned in the previous subsection." ], [ "Equivalence of CH and SW", "In this section we prove that CH and the modified SW produce the same set of permutations for a general full column rank $H$ .", "To prove this it will suffice to prove that $x_i^s = x_i^c$ , $\\bar{x}_i^s =\\bar{x}_i^c$ , $\\mathrm {dist}_i^s = \\mathrm {dist}_i^c$ for $i=1, \\ldots , n$ in the first step which determines the last column of the final reordered $H$ and that the subproblems produced for the second step of each algorithm are equivalent.", "Proving $x_i^s = x_i^c$ is not difficult.", "The only effect the interchange of columns $i$ and $n$ of $R$ in CH has on the real LS solution is that elements $i$ and $n$ of the solution are swapped.", "Therefore $x_i^c$ is just the $i^{th}$ element of the real LS solution rounded to the nearest integer in ${{\\cal B}_i }$ .", "Thus, with (REF ) and (REF ), $x_i^c= \\lfloor (H^{\\dagger }y)_i \\rceil _{{\\cal B}_i }= \\lfloor e_i^T H^{\\dagger }y \\rceil _{{\\cal B}_i }= \\lfloor g_i^T y \\rceil _{{\\cal B}_i } =x_i^s.$ Therefore we also have $\\bar{x}_i^c=\\bar{x}_i^s$ .", "In CH, after applying a permutation $P$ to swap columns $i$ and $n$ of $R$ , we apply $V^T$ , a product of the Givens rotations, to bring $R$ back to a new upper triangular matrix, denoted by $\\hat{R}$ , and also apply $V$ to $\\bar{y}$ , leading to $\\hat{y} = V^T\\bar{y}$ .", "Thus $\\hat{R}=V^T RP$ and $\\hat{y} = V^T\\bar{y}=V^TQ_1^Ty$ .", "Then $H=Q_1R= Q_1V\\hat{R}P^T$ , $H^\\dag = P\\hat{R}^{-1}V^TQ_1^T$ , $g_i=(H^\\dag )^Te_i=Q_1V\\hat{R}^{-T}P^Te_i=Q_1V\\hat{R}^{-T}e_n$ , and $\\Vert g_i\\Vert _2=\\Vert \\hat{R}^{-T}e_n\\Vert _2=1/|\\hat{r}_{nn}|$ .", "Therefore, with (REF ) and (REF ) $\\mathrm {dist}_i^s&=\\frac{ | y^Tg_i - \\bar{x}_i^s |}{ \\Vert g_i \\Vert _2}=|\\hat{r}_{nn}||y^TQ_1V\\hat{R}^{-T}e_n- \\bar{x}_i^s | \\\\& = |\\hat{r}_{nn}|| \\hat{y}_n/\\hat{r}_{nn} - \\bar{x}_i^s |= |\\hat{r}_{nn}(c_n-\\bar{x}_i^c)| =\\mathrm {dist}_i^c.", "\\nonumber $ Now we consider the subproblem (REF ) in CH and the subproblem (REF ) in SW. We can easily show that $R_{1:n-1,1:n-1}$ in (REF ) is the $R$ -factor of the QR factorization of $H_{:,1:n-1}P$ , where $H_{:,1:n-1}$ is the matrix given in (REF ) and $P$ is a permutation matrix such that $\\check{x}=P\\tilde{x}$ , and that $\\bar{y}_{1:n-1}$ in (REF ) is the multiplication of the transpose of the $Q_1$ -factor of the QR factorization of $H_{:,1:n-1}P$ and $y$ in (REF ).", "Thus the two subproblems are equivalent." ], [ "New Algorithm", "Now that we know the two algorithms are equivalent, we can take the best parts from both and combine them to form a new algorithm.", "The main cost in CH is to interchange the columns of $R$ and return it to upper-triangular form using Givens rotations.", "When we determine the $k^{th}$ column, we must do this $k$ times.", "We can avoid all but one of these column interchanges by computing $x_i^c$ , $\\bar{x}_i^c$ and $\\mathrm {dist}_i^c$ directly.", "After the QR factorization of $H$ , we solve the reduced ILS problem (REF ).", "We need only consider how to determine the last column of the final $R$ .", "Other columns can be determined similarly.", "Here we use the ideas from SW. Let $G=R^{-T}$ , which is lower triangular.", "By (REF ), we compute for $i=1,\\ldots , n$ $& x_i = \\left\\lfloor \\bar{y}^TG_{:,i} \\right\\rceil _{{\\cal B}_i}=\\left\\lfloor \\bar{y}_{i:n}^T G_{i:n,i} \\right\\rceil _{{\\cal B}_i}, \\;\\bar{x}_i=\\left\\lfloor \\bar{y}_{i:n}^T G_{i:n,i} \\right\\rceil _{{\\cal B}_i\\backslash x_i}, \\\\& \\mathrm {dist}_i = |\\bar{y}_{i:n}^T G_{i:n,i}-\\bar{x}_i|/\\Vert G_{i:n,i}\\Vert _2.$ Let $j=\\arg \\max _{i} \\mathrm {dist}_i$ .", "We take a slightly different approach to permuting the columns than was used in CH.", "Once $j$ is determined, we set $\\bar{y}_{1:n-1} := \\bar{y}_{1:n-1} - r_{1:n-1,j}x_j$ .", "Then we simply remove the $j^{th}$ column from $R$ , and restore it to upper triangular using Givens rotations.", "We then apply the same Givens rotations to the new $\\bar{y}$ .", "In addition, we must also update the inverse matrix $G$ .", "This is very easy, we can just remove the $j^{th}$ column of $G$ and apply the same Givens rotations that were used to restore the upper triangular structure of $R$ .", "To see this is true notice that removing column $j$ of $R$ is mathematically equivalent to rotating $j$ to the last column and shifting columns $j, j+1, \\ldots , n$ to the left one position, since we will only consider columns $1, 2, \\ldots , n-1$ in subsequent steps.", "Suppose $P$ is the permutation matrix which will permute the columns as described, and $V^T$ is the product of Givens rotations to restore $R$ to upper-triangular.", "Let $\\hat{R} = V^TRP$ and set $\\hat{G} = \\hat{R}^{-T}$ .", "Then $\\hat{G} = (V^TRP)^{-T} = V^TR^{-T}P = V^TGP.$ This indicates that the same $V$ and $P$ , which are used to transform $R$ to $\\hat{R}$ , also transform $G$ to $\\hat{G}$ .", "Since $\\hat{G}$ is lower triangular, it is easy to verify that $\\hat{G}_{1:n-1,1:n-1} = \\hat{R}^{-T}_{1:n-1,1:n-1}$ .", "Both $\\hat{R}_{1:n-1,1:n-1}$ and $\\hat{G}_{1:n-1,1:n-1}$ will be used in the next step.", "After this, as in the CH algorithm, we continue to work on the subproblem of size $n-1$ .", "The advantages of using the ideas from CH are that we always have a lower triangular $G$ whose dimension is reduced by one at each step and the updating of $G$ is numerically stable as we use orthogonal transformations.", "We give the pseudocode of the new algorithm in Algorithm .", "New algorithm [1] Compute the QR factorization of $H$ by Householder transformations: $\\left[{\\begin{matrix}Q_1^T \\\\ Q_2^T \\end{matrix}}\\right]H= \\left[{\\begin{matrix}R\\\\ 0 \\end{matrix}}\\right]$ and compute $\\bar{y} : = Q_1^Ty$ ($2(m-n/3)n^2$ flops) $G := R^{-T}$ ($\\frac{n^3}{3}$ flops) $p := 1:n$ $p^{\\prime } := 1:n$ $k=n$ to 2 $maxDist := -1$ $i=1$ to $k$ $\\alpha =y_{i:k}^TG_{i:k,i}$ $x_i := \\left\\lfloor \\alpha \\right\\rceil _{{\\cal B}_i}$ ($2(k-i)$ flops) $\\bar{x}_i := \\left\\lfloor \\alpha \\right\\rceil _{{{\\cal B}_i}\\backslash x_i}$ $\\mathrm {dist}_i =|\\alpha -\\bar{x}_i|/ \\Vert G_{i:k,i} \\Vert _2$ ($2(k-i)$ flops) $dist_i > maxDist$ $maxDist : = dist_i$ $j:=i$ $p_k := p^{\\prime }_j$ Interchange the intervals ${{\\cal B}_k}$ and ${{\\cal B}_j}$ Interchange entries $k$ and $j$ in $p^{\\prime }$ Set $\\bar{y}:=\\bar{y}_{1:k-1} - R_{1:k-1,j}x_j$ Remove column $j$ of $R$ and $G$ , and return $R$ and $G$ to upper and lower triangular by Givens rotations, respectively, and then remove the last row of $R$ and $G$ .", "The same Givens rotations are applied to $\\bar{y}$ .", "($6k(k-j)$ flops) $p_1 = p^{\\prime }_1$ Here we consider the complexity analysis of our new algorithm.", "If we sum the costs in algorithm over all loop iterations, we get a total of $\\frac{7n^3}{3} + 2mn^2$ flops in the worst case.", "The worst case is very unlikely to occur, it arises when $j=1$ each iteration of the outer loop.", "In the average case however, $j$ is around $k/2$ and we get an average case complexity of $\\frac{4n^3}{3} + 2mn^2$ flops.", "In both cases, the complexity is less than the complexity of the modified SW algorithm." ], [ "Summary", "We showed that two algorithms for the column reordering of the box-constrained ILS problem are equivalent.", "To do that, we modified one algorithm and gave new insight.", "We proposed a new algorithm by combining the best ideas from both of the originals.", "Our new algorithm is more efficient than either of the originals and is easy to implement and understand.", "Since the three algorithms are theoretically equivalent and were derived through different motivations, we now have both geometrical and algebraic motivations for the algorithms." ] ]

[ [ "Fourier restriction Theorem and characterization of weak $L^2$\n eigenfunctions of the Laplace--Beltrami operator" ], [ "Abstract In this paper we prove the Fourier restriction theorem for $p=2$ on Riemannian symmetric spaces of noncompact type with real rank one which extends the earlier result proved in \\cite[Theorem 1.1]{KRS}.", "This result depends on the weak $L^2$ estimates of the Poisson transform of $L^2$ function.", "By using this estimate of the Poisson transform we also characterizes all weak $L^2$ eigenfunction of the Laplace--Beltrami operator of Riemannian symmetric spaces of noncompact type with real rank one and eigenvalue $-(\\lambda^2+\\rho^2)$ for $\\lambda\\in\\R\\setminus\\{0\\}$." ], [ "Introduction", "Let $X=G/K$ be a rank one symmetric space of non compact type.", "For $\\lambda \\in \\mathfrak {a^*_{} the Poisson transform\\mathcal {P}_\\lambda F of F\\in L^1(K/M), is given by\\mathcal {P}_\\lambda F(x)=\\int _{K/M} e^{(i\\lambda +\\rho )A(x,b)}F(b)db.For the meaning of symbols we refer the reader to section 2.It is well known that \\mathcal {P}_\\lambda F is an eigenfunction of the Laplace--Beltrami operator \\Delta with eigenvalue -(\\lambda ^2+\\rho ^2) \\cite [page 100]{He5}.", "A celebrated result of Helgason (and Kashiwara et al.for higher rank) says that if u is an eigenfunction of Laplace--Beltrami operator on X then u is thePoisson transform of an analytic functional T defined on K/M \\cite [Chapter V, Theorem 6.6]{He5}.", "}In this paper we are interested only in such eigenfunctions which are Poisson transforms of $ L1(K/M)$ functions.In \\cite {F} Frustenberg proved that $ u$ is a bounded harmonic function (i.e.", "$ u=0$) if and only if $ u$ is a Poisson integralof bounded function on $ K/M$ (the Poisson transform corresponding to $ =-i$ is known as the Poisson integral).The characterization of eigenfunctions which are Poisson transform of an $ Lp$ function was studied extensively from then onwards\\cite {LR,S,SOS,ACD,Str1,Io3,BS}.The following $ Lp$ analogue of Frustenberg^{\\prime }s result was proved in \\cite {LR,S} for other values of $ p$:$Let $1\\le p<2,$ $\\lambda =\\alpha +i\\gamma _{p^{\\prime }}\\rho ,$ $\\alpha \\in \\mathbb {R}$ and $\\Delta u=-(\\lambda ^2+\\rho ^2)u$ .", "Then $u\\in L^{p^{\\prime },\\infty }(X)$ if and only if $u=\\mathcal {P}_\\lambda F$ for some $F\\in L^{p^{\\prime }}(K/M)$ .", "Moreover, $\\Vert \\mathcal {P}_{\\alpha +i\\gamma _{p^{\\prime }}\\rho } F\\Vert _{p^{\\prime },\\infty }\\le \\Vert F\\Vert _{p^{\\prime }}.$ If $p=2$ then $u=\\mathcal {P}_{0} F$ with $F\\in L^2(K/M),$ if and only if $\\Delta u=-\\rho ^2u$ and the function $h(ka_t.o)=(1+t)^{-1}u(ka_t.o)$ is in $L^{2,\\infty }(X)$ .", "In the above $\\gamma _p=(\\frac{2}{p}-1),\\; 1\\le p\\le \\infty ,$ and hence $\\gamma _{p^{\\prime }}=-\\gamma _p$ where $\\frac{1}{p}+\\frac{1}{p^{\\prime }}=1$ .", "The estimate (REF ) played a fundamental role in the development of harmonic analysis on Riemannian symmetric spaces.", "By using the estimate (REF ) for $1<p<2$ , Lohoué and Rychener [21] obtained the following continuous inclusion $L^{p,1}(G)\\ast L^{p,1}(G/K)\\subseteq L^{p,1}(G).$ Cowling et al.", "in [5] proved an elegant generalization of this result.", "They proved a convolution relation of the following form $L^{p,r}(G)\\ast L^{p,s}(G)\\subseteq L^{p,t}(G),$ where $1<p<2$ , $\\frac{1}{r}+\\frac{1}{s}\\ge \\frac{1}{t}$ and $r,s,t \\in [1,\\infty ]$ .", "Finally in [15] Ionescu obtained the end point version of (REF ) $L^{2,1}(G)\\ast L^{2,1}(G)\\subseteq L^{2,\\infty }(G).$ Coming back to estimate REF , we note that this generalizes the well known behavior of the elementary spherical function $\\phi _\\lambda $ in terms of certain natural size estimates of the Poisson transform.", "For the reader's benefit we will explain this point in detail.", "For $\\lambda \\in the elementary spherical function $$ is given by$$\\phi _\\lambda (x)=\\mathcal {P}_\\lambda 1(x)=\\int _{K/M} e^{(i\\lambda +\\rho )A(x,b)}db.$$If $ 1p<2$ and $ R$ then the well known estimate $ |+ip'(at)|e-2t/p'(t0)$(\\cite [Lemma 3.2]{RS}) implies that $ +ip'$ belongs to $ Lp',(G)$ (the notation $ UW$ for twopositive functions mean that there exist positive constants $ c1,c2$such that $ c1U(x)W(x)c2 U(x)$ for appropriate values of $ x$).", "The estimate (\\ref {poissonp}) can bethought of as a generalization of this fact.", "However, the case $ p=2$ is little different.It is known from the work of Harish-Chandra and a subsequent refinement by Anker that$ 0(at)(1+t)-1e-t(t0)$(\\cite [page 656]{ADY}).", "Hence $ 0L2,(G)$.", "But it is not hard to show that $ 0Lq(G)$ for all $ q>2$.This was generalized in \\cite {Co1} in the form of the following estimate: for all $ q(2,]$ there exists a constant$ Cq>0$ such that for all $ FL2(K/M)$ one has theinequality$ P0FLq(X)CqFL2(K/M)$.We now focus on the case $ R{0}$.It follows from the Harish-Chandra series for elementaryspherical functions that for $ R{0}$ the function $$ satisfies the stronger estimate$ |(at)|Ce-t$ for all $ t0$ (\\cite [(3.11)]{Io2}).", "Hence $ L2,(G)$ for$ R{0}$.We will show that this also holds true for Poisson transforms $ PF$ for $ FL2(K/M)$ and $ R{0}$.As all radial eigenfunction of $$ with $ -(2+2)$ are constant multiple of$$ it follows that for a given $ R{0}$ all radial eigenfunctions of $$ with eigenvalue$ -(2+2)$ belongs to $ L2,(X)$.", "One of our aim in this paper is to characterize all weak $ L2$ eigenfunctions of$$ with eigenvalue $ -(2+2)$ for a given $ R{0}$.$ The main result we prove in this paper is the following: Theorem 1.1 If $\\lambda \\in \\mathbb {R}\\setminus {\\lbrace 0\\rbrace }$ then there exists a constant $C_\\lambda >0$ such that $\\Vert \\mathcal {P}_\\lambda F\\Vert _{2,\\infty }\\le C_\\lambda \\Vert F\\Vert _{L^2(K/M)},\\;\\;\\;\\ \\text{for all}\\;\\ F\\in L^2(K/M).$ If $\\Delta u=-(\\lambda ^2+\\rho ^2)u$ then $u\\in L^{2,\\infty }(X)$ if and only if $u=\\mathcal {P}_\\lambda F$ for some $F\\in L^2(K/M)$ .", "It is important to realize that Poisson transforms are really certain matrix coefficient of the class one principal series representation.", "For $\\lambda \\in \\mathfrak {a}^*_{ the class one principal series representation \\pi _\\lambda of a semisimple Liegroup G are realized on L^2(K/M) and are given by\\pi _\\lambda (g)f(kM)=e^{-(i\\lambda +\\rho )H(g^{-1}k)}F(k(g^{-1}k)).It is known that for \\lambda \\in \\mathfrak {a}^* the representation \\pi _\\lambda is unitary and irreducible.", "It is also known that\\pi _\\lambda and \\pi _{-\\lambda } are unitarily equivalent.", "Using the description of \\pi _\\lambda given abovethe Poisson transform of F\\in L^2(K/M) can also be written as\\mathcal {P}_\\lambda F(g\\cdot o)=\\left\\langle \\pi _\\lambda (g)1,\\bar{F} \\right\\rangle .Theorem \\ref {main} thus shows that |\\left\\langle \\pi _\\lambda (\\cdot )1, F \\right\\rangle | belongsto L^{2,\\infty }(G/K) for \\lambda \\in \\mathfrak {a}^*\\setminus \\lbrace 0\\rbrace and F\\in L^2(K/M).", "It is obvious from the estimates of \\phi _0 mentioned earlierthat such a result is not expected if \\lambda =0.", "}{\\bf Acknowledgement:} I would like to thank Swagato K. Ray and Rudra P. Sarkar for suggesting this problem to me.I would also like to thank them for several discussion on the subject.$" ], [ "Notation and Preliminaries", "In this section we summarize some standard results on noncompact semisimple Lie group and associated symmetric space which will be required.", "Most of our notation are standard and can be found in [10], [12].", "Let $G$ be a connected noncompact semisimple Lie group with finite center, and $\\mathfrak {g}$ be the Lie algebra of $G$ .", "Let $\\theta $ be a Cartan involution of $\\mathfrak {g}$ and $\\mathfrak {g}=\\mathfrak {k}\\oplus \\mathfrak {p}$ be the associated Cartan decomposition.", "Let $K=\\text{exp}\\;\\mathfrak {k}$ be a maximal compact subgroup of $G$ and let $X=G/K$ be the associated Riemannian symmetric space.", "If $o=eK$ denotes the identity coset then for $g\\in G$ the quantity $r(g)$ denotes the Riemannian distance of the coset $g.o$ from the identity coset.", "Let $\\mathfrak {a}$ be a maximal abelian subspace of $\\mathfrak {p}$ , $A=\\text{exp}\\;\\mathfrak {a}$ be the corresponding subgroup of $G,$ and $M$ the centralizer of $A$ in $K$ .", "Now onward we will assume that the group $G$ has real rank one that is dim $\\mathfrak {a}=1$ .", "In this case it is well known that the set of nonzero roots is either of the form $\\lbrace -\\alpha ,\\alpha \\rbrace $ or $\\lbrace -\\alpha ,-2\\alpha ,\\alpha ,2\\alpha \\rbrace $ .", "Let $\\mathfrak {g}_\\alpha $ and $\\mathfrak {g}_{2\\alpha }$ be the root spaces corresponding to the roots $\\alpha $ and $2\\alpha $ respectively.", "Let $\\mathfrak {n}=\\mathfrak {g}_\\alpha \\oplus \\mathfrak {g}_{2\\alpha }$ and $N=$ exp $\\mathfrak {n}$ .", "Let $H_0$ be the unique element of $\\mathfrak {a}$ such that $\\alpha (H_0)=1$ and $A=\\lbrace a_s : a_s=\\text{exp}\\;sH_0, s\\in \\mathbb {R}\\rbrace $ .", "We identify $\\mathfrak {a^*}$ (the dual of $\\mathfrak {a}$ ) and $\\mathfrak {a^*_{} (the complex dual of \\mathfrak {a}) by \\mathbb {R} and via the identification t\\mapsto t\\alpha and z\\mapsto z\\alpha , t\\in \\mathbb {R} and z\\in respectively.", "Letm_1=\\text{dim}\\;\\mathfrak {g}_\\alpha , m_2=\\text{dim}\\;\\mathfrak {g}_{2\\alpha } and \\rho =\\frac{1}{2}(m_1+2m_2)\\alpha be thehalf sum of positive roots.", "By abuse of notation we will denote \\rho (H_0)= \\frac{1}{2}(m_1+2m_2) by \\rho .Let G=KAN be the Iwasawa decomposition of G that is , any g\\in G can be uniquely written as\\begin{equation}g=k(g)\\;\\text{exp}H(g)\\;n(g)\\end{equation}where k(g)\\in K, H(g)\\in \\mathfrak {a} and n(g)\\in N.For F\\in L^1(K/M) and \\lambda \\in (=\\mathfrak {a}^*_{) the Poisson transform P_{\\lambda }F is a function on X defined bythe formula \\begin{equation}\\mathcal {P}_\\lambda F(x)=\\int _{K/M} e^{(i\\lambda +\\rho )A(x,b)}F(b)db,\\end{equation}where A(gK,kM)=-H(g^{-1}k).", "}Let \\bar{N}=\\text{exp}(\\mathfrak {g}_{-\\alpha }\\oplus \\mathfrak {g}_{-2\\alpha }) and G=\\bar{N}AK be the corresponding Iwasawadecomposition of G. Let dk,d\\bar{n} and dm be the normalized Haar measure on K,\\bar{N} and M such that\\begin{eqnarray}\\int _{K}1\\; dk=1;\\;\\;\\ \\int _M 1\\; dm=1; \\;\\;\\ \\int _{\\bar{N}} e^{-2\\rho (H(\\bar{n}))}d\\bar{n}=1\\nonumber &&\\\\ \\int _G f(g)\\;dg=c\\int _{\\bar{N}}\\int _{\\mathbb {R}}\\int _K f(\\bar{n}a_tk)e^{2\\rho t}\\;d\\bar{n}\\; dt\\; dk.\\end{eqnarray}We will also need the following change of variable formula relating integrals on K/M and integrals on \\bar{N} \\cite [Chapter I, Theorem 5.20]{H}\\begin{equation}\\int _{K/M}F(b)\\;db = \\int _{\\bar{N}} F(k(\\bar{n})M)e^{-2\\rho (H(\\bar{n}))}\\;d\\bar{n}.\\end{equation}We also have the cartan decomposition G=K\\overline{A^+}K where \\overline{A^+}=\\lbrace a_t\\in A : t\\ge 0\\rbrace .", "The functions defined onX can also be viewed as right K-invariant function on G. The K-biinvariant functions of G are called radial function.The Haar measure related to the Cartan decomposition is given by\\begin{equation}\\int _G f(g)\\; dg =C \\int _K\\int _0^\\infty \\int _K f(k_1a_tk_2)(\\sinh t)^{m_1}(\\sinh 2t)^{m_2}\\;dk\\;dt\\;dk.\\end{equation}The nilpotent subgroup \\bar{N} can be identified with \\mathbb {R}^{m_1}\\times \\mathbb {R}^{m_2} via anatural map \\bar{n}=\\exp (V+Z)\\rightarrow (V,Z),where m_1=\\text{dim}\\;\\mathfrak {g}_{-\\alpha }, m_2=\\text{dim}\\;\\mathfrak {g}_{-2\\alpha }, V\\in \\mathbb {R}^{m_1} and Z\\in \\mathbb {R}^{m_2}.For any t\\in \\mathbb {R} the dilation \\delta _t on subgroup \\bar{N} is an automorphism given by \\delta _t(\\bar{n})=a_{-t}\\bar{n}a_t.Writing \\bar{n}=(V,Z) it can be written more explicitly as: \\delta _t(V,Z)=(e^tV,e^{2t}Z).", "Wedefine the function |\\bar{n}| on \\bar{N} by\\begin{equation}|\\bar{n}|=|(V,Z)|=(c^2|V|^4+4c|Z|^2)^{1/4},\\end{equation}where c=\\frac{1}{4(m_1+m_2)}.", "This functionhas the property that |\\delta _t(\\bar{n})|=e^t|\\bar{n}|for any s\\in \\mathbb {R} and \\bar{n}\\in \\bar{N}.", "For \\lambda \\in \\mathbb {R}\\setminus \\lbrace 0\\rbrace we define the kernel K_\\lambda by the formula\\begin{equation}K_\\lambda (\\bar{n})=|\\bar{n}|^{-(Q+i2\\lambda )},\\;\\; \\text{for}\\;\\ |\\bar{n}| \\ne 0 \\end{equation}where Q=2\\rho is the homogeneous dimension of \\bar{N}.", "L^2 boundedness of the convolution operator defined by kernel K_\\lambda have been studied in \\cite [Section I]{KS} (see also \\cite [Theorem 6.19]{FS} ).", "We define the truncated kernels K_{\\lambda ,\\eta } byK_{\\lambda ,\\eta }(\\bar{n})= K_\\lambda (\\bar{n})\\chi _{\\lbrace \\bar{n}\\in \\bar{N}\\mid |\\bar{n}|\\ge \\eta \\rbrace }(\\bar{n}),\\;\\;\\;\\;\\bar{n}\\in \\bar{N},and the corresponding convolution operator T_\\eta byT_\\eta \\psi (\\bar{n})=\\int _{\\bar{N}}\\psi (\\bar{n}_1)K_{\\lambda ,\\eta }(\\bar{n}_1^{-1}\\bar{n})d\\bar{n}_1,\\;\\;\\;\\; \\psi \\in C_c^\\infty (N), \\bar{n}\\in \\bar{N}.We will need to consider the following maximal operator associated to the truncated kernel K_{\\lambda ,\\eta } defined byT_*\\psi (\\bar{n})=\\sup _{\\eta >0}|T_\\eta \\psi (\\bar{n})|=\\sup _{\\eta >0}\\left|\\int _{|\\bar{n}_1|\\ge \\eta }\\psi (\\bar{n}\\bar{n}_1)\\frac{1}{|\\bar{n}_1|^{Q+i2\\lambda }}\\;d\\bar{n}_1\\right|.By using the argument given in \\cite [page 33-page 36]{Stn} (see also \\cite [page 627]{Stn})we have the following result regarding the operator T_*\\begin{Theorem}If \\lambda \\in \\mathbb {R}\\setminus \\lbrace 0\\rbrace then there exists a constant C_{\\lambda }>0 such that\\begin{equation}\\Vert T_*\\psi \\Vert _{L^2(\\bar{N})}\\le C_\\lambda \\Vert \\psi \\Vert _{L^2(\\bar{N})},\\end{equation}for all \\psi \\in L^2(\\bar{N}).\\end{Theorem}In the following we collect some basic facts about Lorentz spaces which will be used in this paper (see \\cite {G, SW} for details).Let (M, m) be a \\sigma -finite measure space, f:M\\longrightarrow be a measurable function.The distribution function d_f: (0, \\infty )\\longrightarrow (0,\\infty ] and nonincreasing rearrangementf^*: (0, \\infty )\\longrightarrow (0,\\infty ] of f are defined by formulae d_f(s)= m(\\lbrace x \\in M): |f(x)|> s\\rbrace )\\;\\;\\;\\ \\text{and}\\;\\;\\ f^*(t)=\\inf \\lbrace s\\mid d_f(s)\\le t\\rbrace .", "}For $ p[1, )$, $ q[1, ]$ wedefine,\\begin{equation}\\Vert f\\Vert ^*_{p,q}={\\left\\lbrace \\begin{array}{ll}\\left(\\frac{q}{p}\\int _0^\\infty [f^*(t)t^{1/p}]^q\\frac{dt}{t}\\right)^{1/q}\\ \\textup { when } q<\\infty \\\\ \\\\ \\sup _{t>0}t^{1/p}f^*(t)=\\sup _{t>0}td_f(t)^{1/p}\\ \\ \\ \\ \\ \\ \\ \\ \\textup { when } q=\\infty .\\end{array}\\right.", "}\\end{equation}$ For $p\\in [1, \\infty )$ , $q\\in [1, \\infty ]$ we define the Lorentz space $L^{p,q}(M)$ as follows: $L^{p,q}(M) = \\lbrace f:M\\longrightarrow f \\quad \\text{measurable and}\\quad \\Vert f\\Vert ^*_{p,q}<\\infty \\rbrace .$ By $L^{\\infty , \\infty }(M)$ and $\\Vert \\cdot \\Vert _{\\infty , \\infty }$ we mean respectively the space $L^\\infty (M)$ and the norm $\\Vert \\cdot \\Vert _\\infty $ we also have $L^{p,p}(M)=L^p(M)$ .", "For $1<p<\\infty $ the space $L^{p,\\infty }$ is known as weak $L^p$ space and also $L^{p,q}\\subset L^{p,s}$ for all $1\\le q\\le s\\le \\infty $ .", "For $1<p<\\infty $ and $1\\le q<\\infty $ , the dual (the space of all continuous linear functional) of $L^{p,q}(M)$ is $L^{p^{\\prime }, q^{\\prime }}(M)$ .", "Everywhere in this paper any $p\\in [1, \\infty )$ is related to $p^{\\prime }$ by the relation $\\frac{1}{p}+\\frac{1}{p^{\\prime }}=1.$ We will follow the standard practice of using the letter C for constant, whose value may change from one line to another.", "Occasionally the constant C will be suffixed to show its dependency on related parameter.", "The letters $ and $ R$ will denote the set of complex and realnumbers respectively.$" ], [ "Proof of Theorem ", "The important part of the proof of Theorem REF is the following norm estimate.", "Our proof of the following Proposition uses an argument similar to one given in [24].", "Proposition 3.1 If $\\lambda \\in \\mathbb {R}\\setminus {\\lbrace 0\\rbrace }$ then there exists a constant $C_{\\lambda }>0$ such that for all $F\\in L^2(K/M)$ , $\\Vert \\mathcal {P}_\\lambda F\\Vert _{L^{2,\\infty }(X)}\\le C_\\lambda \\Vert F\\Vert _{L^2(K/M)}.$ In view of the Iwasawa decomposition $G=\\bar{N}AK$ the symmetric space $X$ can be identified with $\\bar{N}\\times \\mathbb {R}$ via the map $(\\bar{n},t)\\rightarrow \\bar{n}a_t\\cdot o$ .", "If $F\\in L^2(K/M)$ and $\\bar{n}a_t\\cdot o \\in X$ , then from the definition () of the Poisson transform and () we have, $\\mathcal {P}_\\lambda F(\\bar{n} a_t\\cdot o)=\\int _{\\bar{N}} e^{-(i\\lambda +\\rho )H(a_{-t}\\bar{n}^{-1}k(\\bar{m}))}F(k(\\bar{m})M)e^{-2\\rho (H(\\bar{m}))}d\\bar{m} .$ As $A$ normalizes $N$ it follows from the Iwasawa decomposition () that $-H(a_{-t}\\bar{n}^{-1}k(\\bar{m}))=-H(\\delta _t(\\bar{n}^{-1}\\bar{m}))+H(\\bar{m})+t$ (see [16]).", "By using the last two equality we get, $\\mathcal {P}_\\lambda F(\\bar{n} a_t\\cdot o)&=& e^{(i\\lambda +\\rho )t}\\int _{\\bar{N}}e^{-(i\\lambda +\\rho )H(\\delta _t(\\bar{n}^{-1}\\bar{m})}F(k(\\bar{m})M) e^{(i\\lambda -\\rho )(H(\\bar{m}))}d\\bar{m}\\nonumber \\\\&=& e^{(i\\lambda +\\rho )t}\\int _{\\bar{N}}e^{-(i\\lambda +\\rho )H(\\delta _t(\\bar{m}))}\\psi (\\bar{n} \\bar{m})\\; d\\bar{m},$ where $\\psi $ is a function defined on $\\bar{N}$ given by $\\psi (\\bar{m})=F(k(\\bar{m})M) e^{(i\\lambda -\\rho )(H(\\bar{m}))}$ .", "Using the integral formula () one has $\\Vert \\psi \\Vert _{L^2(\\bar{N})}=\\Vert F\\Vert _{L^2(K/M)}$ .", "If we write $\\bar{m}=(V,Z)$ then $e^{\\rho H(\\bar{m})}$ is given by the formula $ e^{\\rho (H(\\bar{m}))}=[(1+c|V|^2)^2+4c|Z|^2]^{\\frac{1}{4}(m_1+2m_2)},$ where $c^{-1}=4(m_1+4m_2)$ (see [13]).", "Thus from the above formula, definition of norm () and relation $\\delta _t(\\bar{m})=(e^tV,e^{2t}Z)$ we have, $e^{-(i\\lambda +\\rho )H(\\delta _t(\\bar{m}))}&=&\\frac{1}{[(1+c|e^tV|^2)^2+4c|e^{2t}Z|^2]^{(i\\lambda +\\rho )/2}}\\nonumber \\\\&=& \\frac{1}{e^{2(i\\lambda +\\rho )t}[(e^{-2t}+c|V|^2)^2+4c|Z|^2]^{(i\\lambda +\\rho )/2}}\\nonumber \\\\&=& \\frac{1}{e^{2(i\\lambda +\\rho )t}[(e^{-4t}+2ce^{-2t}|V|^2)+|\\bar{m}|^4]^{(i\\lambda +\\rho )/2}}.$ By using (REF ) in (REF ) we get the following expression of the Poisson transform $\\mathcal {P}_\\lambda F(\\bar{n} a_t\\cdot o)= e^{-(i\\lambda +\\rho )t}\\int _{\\bar{N}}\\frac{1}{[(e^{-4t}+2ce^{-2t}|V|^2)+|\\bar{m}|^4]^{(i\\lambda +\\rho )/2}} \\psi (\\bar{n}\\bar{m})\\;d\\bar{m}.", "$ We will now follow an argument of Sjögren [24] to dominate the above integral by two maximal functions whose $L^p$ behaviors are known.", "We first restrict the integral over the ball $ B(e^{-t})=\\lbrace \\bar{m}\\in \\bar{N} : |\\bar{m}|\\le e^{-t}\\rbrace $ in $\\bar{N}$ .", "Since the Haar measure of $\\bar{N}$ is the Lebesgue measure it follows that the measure of $B(e^{-t})$ is proportional to $e^{-Qt}$ .", "In this case we have, $\\left|\\int _{B(e^{-t})}\\frac{1}{[(e^{-4t}+2ce^{-2t}|V|^2)+|\\bar{m}|^4]^{(i\\lambda +\\rho )/2}} \\psi (\\bar{n}\\bar{m})\\;d\\bar{m}\\right|&\\le & \\frac{1}{e^{-Qt}} \\int _{B(e^{-t})} |\\psi (\\bar{n}\\bar{m})|\\;d\\bar{m}\\nonumber \\\\&\\le & C M_0\\psi (\\bar{n}),$ where $M_0$ is the standard Hardy–Littlewood maximal operator on $\\bar{N}$ .", "It is well known that the operator $M_0$ is bounded from $L^p(\\bar{N})$ to $L^p(\\bar{N})$ for $1<p\\le \\infty $ [26].", "For $|\\bar{m}|>e^{-t},$ we will compare the kernel with $|\\bar{m}|^{-(Q+i2\\lambda )}$ .", "We claim that, $\\left| \\frac{1}{[(e^{-4t}+2ce^{-2t}|V|^2)+|\\bar{m}|^4]^{(i\\lambda +\\rho )/2}} - \\frac{1}{|\\bar{m}|^{(Q+i2\\lambda )}}\\right|\\le C \\frac{e^{-t}}{|\\bar{m}|^{Q+1}},$ for all $\\bar{m} \\in B(e^{-t})$ .", "Consider the function $\\phi :(0,\\infty )\\longrightarrow defined by $ (r)=r-Q-2i$.", "If we take$ r=|m|$ and $ s=[(e-4t+2ce-2t|V|2)+|m|4]1/4$ then it follows from the mean value theorem that there exists$ r0 (r,s)$ such that$$|\\phi (s)-\\phi (r)|=|s-r||\\phi {^{\\prime }}(r_0)|=C|s-r|r_0^{-Q-1}.$$Since $ e-t<|m|$ we have $ |m|r03 |m|$ and $ |s-r|C e-t$.This proves our claim.", "We also have the following estimate\\begin{eqnarray}\\int _{|\\bar{m}|>e^{-t}} \\frac{e^{-t}}{|\\bar{m}|^{Q+1}}|\\psi (\\bar{n}\\bar{m})|\\; d\\bar{m}&=& \\sum _{j=0}^{\\infty }\\int _{2^j e^{-t}<|\\bar{m}|\\le 2^{j+1}e^{-t}}\\frac{e^{-t}}{|\\bar{m}|^{Q+1}}|\\psi (\\bar{n}\\bar{m})|\\; d\\bar{m}\\nonumber \\\\ &\\le & \\sum _{j=0}^{\\infty }\\frac{e^{-t}}{(2^j e^{-t})^{Q+1}}\\int _{B(2^{j+1}e^{-t})}|\\psi (\\bar{n}\\bar{m})|\\; d\\bar{m}\\nonumber \\\\&\\le & C_Q\\sum _{j=0}^{\\infty }\\frac{1}{2^j|B(2^{j+1}e^{-t})|}\\int _{B(2^{j+1}e^{-t})}|\\psi (\\bar{n}\\bar{m})|\\; d\\bar{m}\\nonumber \\\\&\\le & C_Q M_0\\psi (\\bar{n}), \\end{eqnarray}where $ |B(2j+1e-t)|$ denotes the Haar measure of $ B(2j+1e-t)$.If we combine the inequalities (\\ref {compare1}) and (\\ref {maximal}) we get\\begin{equation}\\left|\\int _{|\\bar{m}|>e^{-t}}\\frac{1}{[(e^{-4t}+2ce^{-2t}|V|^2)+|\\bar{m}|^4]^{(i\\lambda +\\rho )/2}} \\psi (\\bar{n}\\bar{m})\\;d\\bar{m}\\right|\\le C T\\psi (\\bar{n}),\\end{equation}where $ T(n)= M0(n)+T*(n)$.From (\\ref {proposition}), (\\ref {first}) and (\\ref {final}) we now have the following pointwise estimate of the Poisson transform$$|\\mathcal {P}_\\lambda F(\\bar{n} a_t\\cdot o)|\\le C e^{-\\rho t} T\\psi (\\bar{n})).$$We will use the above estimate to show that the Poisson transform belongs to weak $ L2$.In this regard, for any $ >0$ we have,$ $\\mu \\lbrace \\bar{n}a_t \\in X | |\\mathcal {P}_\\lambda F(\\bar{n} a_t\\cdot o)|>\\beta \\rbrace \\le \\int _{\\bar{N}}\\int _{-\\infty }^{t_0}e^{2\\rho t}dt\\; d\\bar{n}= \\frac{C}{\\beta ^2} \\int _{\\bar{N}} T\\psi (\\bar{n})^2\\;d\\bar{n},$ where $\\frac{C T\\psi (\\bar{n})}{\\beta ^2}=e^{2\\rho t_0}$ and $\\mu $ denotes the $G$ invariant measure on $X$ .", "We know from Theorem that the operator $T_*$ is bounded from $L^2(\\bar{N})$ to $L^2(\\bar{N})$ and hence so is $T$ .", "Therefore from () and (REF ) we get $\\Vert \\mathcal {P}_\\lambda F\\Vert _{L^{2,\\infty }(X)}\\le C_\\lambda \\Vert \\psi \\Vert _{L^2(\\bar{N})} = C_\\lambda \\Vert F\\Vert _{L^2(K/M)}.$ This complete the proof.", "To complete the proof of the Theorem REF we will now invoke a result of Ionescu [16].", "For a locally integrable function $u$ on $X$ define $M(u)=\\left(\\limsup _ {R\\rightarrow \\infty }\\frac{1}{R}\\int _{(B(o,R))}|u(x)|^2\\;dx\\right)^{1/2}.$ It was proved in [16] that if $\\Delta u=-(\\lambda ^2+\\rho ^2)u$ for some $\\lambda \\in \\mathbb {R}\\setminus \\lbrace 0\\rbrace $ and $M(u)<\\infty $ then $u=\\mathcal {P}_\\lambda F$ for some $F\\in L^2(K/M)$ (see also [3]).", "Theorem REF will follow immediately once we prove the following simple lemma (although it was proved in [20] but for the sake of completeness we provide the detail here).", "Lemma 3.2 If $\\Delta u=-(\\lambda ^2+\\rho ^2)u$ for $\\lambda \\in \\mathbb {R}\\setminus \\lbrace 0\\rbrace $ and $u\\in L^{2,\\infty }(X)$ then $M(u)\\le C \\Vert u\\Vert _{L^{2,\\infty }(X)}.$ It suffices to show that $\\int _{(B(o,R))}|u(x)|^2\\;dx\\le C R\\Vert u\\Vert _{L^{2,\\infty }(X)}^2 \\;\\;\\ \\text{for all}\\;\\ R>0 .$ Since the inequality (REF ) follows easily for $R\\le 1$ we will concentrate only in the case $R> 1$ .", "If $u^*$ denotes the decreasing rearrangement of $u$ then it follows from definition of weak $L^2$ spaces that $u^{*}(s)^2\\le \\frac{1}{s}\\Vert u\\Vert _{2,\\infty }^{2},\\quad \\mbox{for all $s>0$.", "}$ It follows from the similar argument given in [21] (see also [20]) that $|u(ka_t.o)|\\le C \\Vert u\\Vert _{L^{2,\\infty }(X)} e^{\\alpha t} \\;\\;\\;\\ \\text{for some $\\alpha >0$ and for all $t>0$}.$ Since $(\\chi _{B(o,R)})^{*}= \\chi _{(0, |B(o,R)|)}$ (see [9]) it follows from the above inequality that $(\\chi _{B(o,R)}|u|)^{*}(t)^2\\le C\\Vert u\\Vert _{2,\\infty }^2 e^{2\\alpha R}\\chi _{(0, |B(o,R)|)}(t).", "$ Using the fact that the $G$ -invariant measure $\\mu B(o,R)$ of the ball $\\mu B(o,R)$ is propositional to $e^{2\\rho R}$ for $R\\ge 1$ it follows from (REF ) and (REF ) that $\\int _{B(o,R)}|u(x)|^2dx&\\le & C \\int _0^{\\mu B(o,R)} u^*(s)^2 ds\\nonumber \\\\&\\le & C\\int _0^{\\mu B(o,R)}\\min \\left\\lbrace \\Vert u\\Vert _{2,\\infty }^{2}e^{2\\alpha R},\\Vert u\\Vert _{2,\\infty }^{2}\\frac{1}{t}\\right\\rbrace dt\\nonumber \\\\&\\le & C\\Vert u\\Vert _{2,\\infty }^{2}\\int _0^{e^{2\\rho R}}\\min \\left\\lbrace e^{2\\alpha R},\\frac{1}{t}\\right\\rbrace dt\\nonumber \\\\&=& C\\Vert u\\Vert _{2,\\infty }^{2}\\left(\\int _0^{e^{-2\\alpha R}}e^{2\\alpha R}dt+\\int _{e^{-2\\alpha R}}^{e^{2\\rho R}}\\frac{dt}{t}\\right)\\nonumber \\\\&=& C\\Vert u\\Vert _{2,\\infty }^{2}\\left(1+2\\rho R+2\\alpha R\\right)\\nonumber \\\\&\\le & C\\Vert u\\Vert _{2,\\infty }^{2}R, $ This completes the proof.", "An immediate consequence of the above results is [16]: Corollary 3.3 If $\\lambda \\in \\mathfrak {a}^*\\setminus \\lbrace 0\\rbrace $ and $F\\in L^2(K/M)$ then $\\left( \\sup _R \\frac{1}{R} \\int _{B(o,R)}|\\mathcal {P}_\\lambda F(x)|^2\\;dx\\right)^{1/2}\\le C_\\lambda \\Vert F\\Vert _{L^(K/M)}.$" ], [ "some consequences", "Theorem REF has certain consequences which are worth mentioning.", "One of them is related to the restriction (like) theorem for the Helgason Fourier transform on $X$ .", "The idea of Fourier restriction theorem on $\\mathbb {R}^n (n\\ge 2)$ originated in the work of Stein.", "The celebrated Tomas–Stein restriction Theorem says that the Fourier transform $\\hat{f}$ of a function $f\\in L^p(\\mathbb {R}^n)$ has a well defined restriction on the sphere $S^{n-1}$ via the inequality, $\\Vert \\widehat{f}|_{S^{n-1}}\\Vert _{L^2(S^{n-1})}\\le C_p\\Vert f\\Vert _{L^p(\\mathbb {R}^n)}\\;\\;\\;\\text{for all}\\;\\ 1\\le p\\le \\frac{2n+3}{n+3}$ (see [26]).", "We are interested in similar inequalities for the Helgason Fourier transform of suitable functions on $X$ .", "Given $f\\in C_c^\\infty (X)$ the Helgason Fourier transform $\\widetilde{f}$ of $f$ is defined by [13] $\\widetilde{f}(\\lambda ,b)=\\int _X f(x)e^{(i\\lambda +\\rho )A(x,b)}\\;dx,\\;\\;\\;\\ \\lambda \\in \\;\\;b\\in K/M.$ For fixed $\\lambda \\in the norm inequality of the form$$\\left(\\int _{K/M}|\\widetilde{f}(\\lambda ,b)|^qdb\\right)^{1/q}\\le C\\Vert f\\Vert _{L^p(X)}$$can be thought as an analogue of Fourier restriction theorem in the context of a symmetric space $ X$.", "For $ 1p<2$ the analogueof restriction theorem for $ Lp$ functions on $ X$ with rank~$ X=1$ was proved in \\cite {LR}.", "This result was extended for more generalspaces in \\cite {RS}, \\cite {KRS}.", "It was also shown in\\cite {KRS} that the best posible analogue of restriction theorem for $ R$ is the following: Given $ R{0}$ there exists a constant $ C,p>0$ suchthat the following inequality holds $$\\left(\\int _{K/M}|\\widetilde{f}(\\lambda ,b)|^2db\\right)^{1/2}\\le C_{\\lambda ,p}\\Vert f\\Vert _{L^p(X)},\\:\\:\\:\\:1\\le p<2 .$$It is the end point case of the above inequality which we are interested in.It turns out that this problem can be solved very easily by using estimates of the Poisson transform.We first observe that for a given $ and $F\\in C^\\infty (K/M)$ the Poisson transform $\\mathcal {P}_\\lambda F$ is related to the Helgason Fourier transform $\\widetilde{f}(\\lambda , \\cdot )$ as follows: $\\int _{K/M}\\widetilde{f}(\\lambda ,b)F(b)\\;db= \\int _X f(x)\\mathcal {P}_\\lambda F(x)\\;dx.$ By using the estimate (REF ) and the above equation we have the following version of the restriction Theorem : Theorem 4.1 If $\\lambda \\in \\mathbb {R}\\setminus \\lbrace 0\\rbrace $ then there exists a constant $C_\\lambda >0$ such that, $\\left(\\int _{K/M}|\\widetilde{f}(\\lambda , b)|^2 db\\right)^{1/2}\\le C_\\lambda \\Vert f\\Vert _{L^{2, 1}(X)},\\;\\;\\ \\text{for all}\\;\\;f\\in L^{2,1}(X).$ If $F\\in L^2(K/M)$ then using the fact that the dual of $L^{2,1}(X)$ is $L^{2,\\infty }(X)$ and the estimate (REF ) we get $\\left|\\int _{K/M}|\\widetilde{f}(\\lambda , b)F(b)db\\right|&=&\\left|\\int _X f(x)\\mathcal {P}_\\lambda F(x)\\;dx\\right|\\nonumber \\\\&\\le & \\Vert f\\Vert _{L^{2,1}(X)}\\Vert P_{\\lambda }F\\Vert _{L^{2,\\infty }(X)}\\nonumber \\\\&\\le & C_{\\lambda }\\Vert f\\Vert _{L^{2,1}(X)}\\Vert F\\Vert _{L^2(K/M)}.\\nonumber $ Theorem REF can be used to deduce an interesting analytic property of the spectral projection operator considered in [28].", "Authors of [6] used the Kunze Stein phenomenon to prove that the spectral projection operator $f\\mapsto f\\ast \\phi _{\\lambda }$ satisfies the estimate $\\Vert f\\ast \\phi _{\\lambda }\\Vert _{L^{p^{\\prime }}(X)}\\le C_p\\Vert f\\Vert _{L^p(X)}$ for $\\lambda \\in \\mathbb {R}$ and $1\\le p<2$ (the result is valid even if rank $X>1$ ).", "We now present the following end point estimate of the spectral projection operator which is closely related to the behavior of the noncentral Hardy Littlewood maximal operator on X (see [15]).", "Corollary 4.2 If $\\lambda \\in \\mathbb {R}\\setminus \\lbrace 0\\rbrace $ then the linear map $f\\longrightarrow f\\ast \\phi _\\lambda $ is restricted weak type $(2,2)$ .", "From [12] we have the relation $f\\ast \\phi _\\lambda (x)= \\mathcal {P}_{\\lambda }(\\widetilde{f}(-\\lambda , \\cdot ))(x)$ .", "Using (REF ), (REF ) and the above relation we have $\\Vert f\\ast \\phi _\\lambda \\Vert _{L^{2,\\infty }(X)}= \\Vert \\mathcal {P}_{\\lambda }(\\widetilde{f}(-\\lambda , \\cdot ))\\Vert _{L^{2,\\infty }(X)}\\le C_\\lambda \\Vert \\widetilde{f}(-\\lambda , \\cdot )\\Vert _{L^2(K/M)}\\le C_\\lambda \\Vert f\\Vert _{L^{2,1}(X)}.$ Remark 4.3 If $1\\le p<2$ and $\\lambda =\\alpha +i\\gamma _p\\rho $ ($\\alpha \\in \\mathbb {R}$ ) then it follows from [19] that $\\Vert f\\ast \\phi _\\lambda \\Vert _{L^{p^{\\prime },\\infty }(S)}\\le C_{\\lambda }\\Vert f\\Vert _{L^{p,1}(S)}$ .", "By using the standard estimate $|\\phi _{\\alpha +i\\gamma _{p^{\\prime }}\\rho }(x)|\\asymp \\kappa _p(x)$ where $\\kappa _p$ is the radial function defined by $\\kappa _p(x)=e^{\\frac{-2\\rho r(x)}{p^{\\prime }}}$ we get the estimate $\\Vert f\\ast \\kappa _p\\Vert _{L^{p^{\\prime },\\infty }(X)}\\le C_{\\lambda }\\Vert f\\Vert _{L^{p,1}(X)}\\;\\;\\; 1\\le p<2.$ For $p=2$ and $\\alpha \\in \\mathbb {R}\\setminus \\lbrace 0\\rbrace $ we have the estimate $|\\phi _{\\alpha }(x)|\\le C_{\\alpha }\\kappa _2(x)$ .", "Since the spectral projection operator, for $\\lambda \\in \\mathbb {R}\\setminus \\lbrace 0\\rbrace $ , is restricted weak type $(2,2)$ one may ask whether the same holds for the operator obtained by convolving with the larger kernel $\\kappa _2$ .", "By modifying an example given in [14] we will show that this is not the case.", "We first recall the following properties of the function $r(x)$ from [10] which will be needed $r(ka_t)=r(a_t)=t,\\;\\;\\; r(x)=r(x^{-1}), \\;\\;\\:r(xy)\\le r(x)+r(y)$ for all $k\\in K$ , $x,y\\in G$ and $t>0$ .", "We also use the functional equation for the elementary spherical functions [12] $\\int _K\\phi _0(xky)\\;dk=\\phi _0(x)\\phi _0(y),$ and the estimate [1] $\\phi _0(x)\\asymp (1+r(x))e^{-\\rho r(x)}.$ Consider the radial function $f:X\\longrightarrow (0,\\infty )$ given by $f(x)=e^{-\\rho r(x)}(1+r(x))^{-3/2}$ .", "It follows from the calculation [14] that $f\\in L^{2,1}(X)$ .", "We will show that $f\\ast \\kappa _2\\notin L^{2,\\infty }(X)$ .", "By using the Cartan decomposition () we have, $f\\ast \\kappa _2(a_s)=\\int _0^\\infty e^{-\\rho t}(1+t)^{-3/2}\\int _K e^{-\\rho r(a_{-t}ka_s)}dkJ(t)dt $ where $J(t)=(\\sinh t)^{m_1}(\\sinh 2t)^{m_2}$ .", "From (REF ), (REF ) and (REF ) and we get $\\int _K e^{-\\rho r(a_{-t}ka_s)}dk =\\int _K e^{-\\rho r(a_{-t}ka_s)}\\frac{(1+r( a_{-t}ka_s))}{(1+r( a_{-t}ka_s))}dk\\ge C\\frac{1}{1+t+s}\\phi _0(a_t)\\phi _0(a_s)$ If we use estimates above, (REF ) and $J(t)\\asymp e^{2\\rho t}$ (for $t\\ge 1$ ) then from (REF ) we have $f\\ast \\kappa _2(a_s)\\ge C \\frac{1+s}{1+2s}e^{-\\rho s} \\int _1^{s}e^{-2\\rho t}(1+t)^{-1/2}e^{2\\rho t}dt\\ge C s^{1/2} e^{-\\rho s}$ for large $s$ .", "It is now easy to see that $f\\ast \\kappa _2\\notin L^{2,\\infty }(X)$ .", "Comment regarding the Fourier transform on Damek–Ricci Spaces: The notion of Helgason Fourier transform is also meaningful for functions on Damek–Ricci space [2].", "It is well known that Damek–Ricci spaces include all Riemannian symmetric spaces of noncompact type with real rank one [1].", "The analogue of restriction theorem in this setup was proved in [22] and [19].", "By using arguments similar to symmetric spaces one can prove the following version of Theorem REF for Damek–Ricci spaces $S$ .", "Theorem 4.4 If $\\lambda \\in \\mathbb {R}\\setminus \\lbrace 0\\rbrace $ and $f\\in L^{2,1}(S)$ then, $\\left(\\int _N|\\widetilde{f}(\\lambda , n)|^2 dn\\right)^{1/2}\\le C_\\lambda \\Vert f\\Vert _{L^{2, 1}(S)}.$ The above theorem extends the following result proved in [19].", "Theorem 4.5 Let $f$ be a measurable function on $S$ and $\\alpha \\in \\mathbb {R}$ .", "For $f\\in L^{p,1}(S), 1\\le p<2$ and $p\\le q\\le p^{\\prime }$ , $\\left(\\int _N|\\widetilde{f}(\\alpha +i\\gamma _{q}\\rho ,n)|^q dn\\right)^{1/q}\\le C_{p,q}\\Vert f\\Vert _{p, 1}, \\ C_{1, q}=1.$ For $f\\in L^{p, \\infty }(S), 1<p<2$ , $p<q<p^{\\prime }$ , $\\left(\\int _N|\\widetilde{f}(\\alpha +i\\gamma _{q}\\rho , n)|^q dn\\right)^{1/q}\\le C_{p,q}\\Vert f\\Vert _{p, \\infty }.$ The constants $C_{p,q}>0$ are independent of $\\alpha $ and $f$ .", "Estimates i) and ii) are sharp." ] ]

[ [ "Boost-Invariant (2+1)-dimensional Anisotropic Hydrodynamics" ], [ "Abstract We present results of the application of the anisotropic hydrodynamics (aHydro) framework to (2+1)-dimensional boost invariant systems.", "The necessary aHydro dynamical equations are derived by taking moments of the Boltzmann equation using a momentum-space anisotropic one-particle distribution function.", "We present a derivation of the necessary equations and then proceed to numerical solutions of the resulting partial differential equations using both realistic smooth Glauber initial conditions and fluctuating Monte-Carlo Glauber initial conditions.", "For this purpose we have developed two numerical implementations: one which is based on straightforward integration of the resulting partial differential equations supplemented by a two-dimensional weighted Lax-Friedrichs smoothing in the case of fluctuating initial conditions; and another that is based on the application of the Kurganov-Tadmor central scheme.", "For our final results we compute the collective flow of the matter via the lab-frame energy-momentum tensor eccentricity as a function of the assumed shear viscosity to entropy ratio, proper time, and impact parameter." ], [ "Introduction", "The goal of ultrarelativistic heavy ion collision experiments at the Relativistic Heavy Ion Collider at Brookhaven National Laboratory (RHIC) and the Large Hadron Collider (LHC) at CERN is to create a tiny volume of matter ($\\sim $ 1000 fm$^3$ ) which has been heated to a temperature exceeding that necessary to create a quark-gluon plasma.", "Early on it was shown that ideal relativistic hydrodynamics is able to reproduce the soft collective flow of the matter and single particle spectra produced at RHIC [1], [2], [3], [4].", "Based on this there was a concerted effort to develop a more systematic framework for describing the soft collective motion.", "This effort resulted in a number of works dedicated to the development and application of relativistic viscous hydrodynamics to relativistic heavy ion collisions [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26].", "One of the weakness of the traditional viscous hydrodynamics approach is that it relies on an implicit assumption that the system is close to thermal equilibrium which implies that the system is also very close to being isotropic in momentum space.", "However, one finds during the application of these methods to relativistic heavy ion collisions that this assumption breaks down at the earliest times after the initial impact of the two nuclei due to large momentum-space anisotropies in the $p_T$ -$p_L$ plane which can persist for many fm/c [27].", "In addition, one finds that near the transverse and longitudinal edges of the system these momentum-space anisotropies are large at all times [27], [28], [29], [30], [31].", "Similar conclusions have been obtained in the context of strongly coupled systems where it has been shown using the AdS/CFT correspondence one achieves viscous hydrodynamical behavior at times when the system still possesses large momentum-space anisotropies and that these anisotropies remain large throughout the evolution [32], [33], [34], [35], [36], [37], [38].", "Based on these results one is motivated to obtain a dynamical framework that can accommodate potentially large momentum-space anisotropies.", "In this paper we follow up recent work which aims to extend the applicability of space-time evolution models for the bulk dynamics of a quark-gluon plasma to situations in which there can be large momentum-space anisotropies.", "Initial studies along this direction focused on boost-invariant expansion in systems which were transversally homogeneous [39], [40].", "The motivation and conceptual setup of Refs.", "[39], [40] were similar in the sense that they both relaxed the assumption of the system being nearly isotropic in momentum space; however, there was a key conceptual difference in the derivation of the resulting dynamical equations.", "In Ref.", "[39] an entropy source was postulated which satisfied the minimal constraints necessary in the limit of small momentum-space anisotropy and then the authors assumed a constant rate of isotropization regardless of the local typical momentum of the plasma constituents.", "In Ref.", "[40] the equations of motion were derived by taking moments of the Boltzmann equation and supplemented by a requirement that in the limit of small momentum-space anisotropy these equations reproduced those of 2nd-order Israel-Stewart viscous hydrodynamics [41], [42], [43].", "The result of this matching was that the relaxation rate of the system was necessarily proportional to the local hard momentum scale.In this context the hard momentum scale corresponds to the typical average momentum scale of the particles of the system.", "When one has local isotropic thermal equilibrium, the average momentum scale corresponds to the temperature of the system.", "This allowed the authors of Ref.", "[40] to smoothly match onto 2nd-order viscous hydrodynamics when the system was nearly isotropic in momentum space.", "The phenomenological consequence of these two different results for the relaxation rate is quite important.", "If the relaxation rate is proportional to the local hard momentum scale, then one expects a slower relaxation to isotropy when the local hard momentum scale is reduced.", "This occurs at late times in the one-dimensional case since the local hard momentum scale is dynamically lowered due to expansion.", "Even more importantly, having a relaxation rate which is proportional to the hard momentum scale has important consequences for the evolution of the matter near the longitudinal and transverse edges of the system where the local temperature is also initially lower.", "The first demonstration of this effect was in Ref.", "[28] which studied the one dimensional non-boost invariant evolution of a system which was transversally homogeneous.", "This work followed similar developments in Ref.", "[29] where a constant relaxation rate was assumed.", "A comparison of the results of these two papers shows that one sees much larger momentum-space anisotropies at large spatial rapidity being developed if one uses a relaxation rate which is proportional to the local hard momentum scale.", "Since these works were published, the anisotropic hydrodynamics methodology has been extended to include boost-invariant transverse dynamics [44], [45]; however, these papers once again assumed a fixed rate of relaxation to isotropy.", "In this paper we study the effect of using a more realistic relaxation rate which is proportional to the hard momentum scale [40], thereby allowing a smooth matching to 2nd-order viscous hydrodynamics.", "We present a derivation of the necessary equations and then proceed to numerical solutions of the resulting partial differential equations using both realistic smooth Glauber initial conditions and fluctuating Monte-Carlo Glauber initial conditions.", "For this purpose we have developed two numerical implementations: one which is based on straightforward integration of the resulting partial differential equations supplemented by a two-dimensional weighted Lax-Friedrichs smoothing in the case of fluctuating initial conditions; and another that is based on the application of the Kurganov-Tadmor central scheme.", "For our final results we compute the collective flow of the matter via the lab-frame energy-momentum tensor eccentricity as a function of the assumed shear viscosity to entropy ratio, proper time, and impact parameter.", "We also present results for the dependence of the momentum-space anisotropy in the full transverse plane and show that in regions where the temperature is low one can develop sizable momentum-space anisotropies.", "As a control test we compare with 2nd-order viscous hydrodynamics in the limit of small shear viscosities and demonstrate that the aHydro framework is able to reproduce the temperature and flow profiles obtained from 2nd-order viscous hydrodynamics in this limit.", "The structure of the paper is as follows: In Sec.", "we introduce the tensor basis we will use in the case that the system is anisotropic in momentum space and derive the partial differential equations necessary for the dynamical evolution by taking moments of the Boltzmann equation.", "In Sec.", "we present the types of smooth initial conditions we will use.", "In Sec.", "we introduce the three numerical algorithms (centered differences, weighted LAX, and hybrid Kurganov-Tadmor) we will we use to solve the resulting partial differential equations.", "In Sec.", "we compare with 2nd-order viscous hydrodynamics for non-central collisions and present our final results.", "In Sec.", "we present our conclusions and a future outlook.", "Finally, in three appendices we include a comparison of entropy production in 2nd-order viscous hydrodynamics and aHydro, some numerical checks of convergence etc., and a brief rederivation of the 0+1d Bjorken model using our tensor formalism." ], [ "Kinetic theory approach to anisotropic hydrodynamics", "In this section we describe our theoretical framework for describing relativistic plasmas which are anisotropic in momentum-space.", "Our setup is based on the kinetic theory approach to non-equilibrium systems [41].", "There are different methods for constructing approximate solutions of the relativistic Boltzmann equation [41].", "The most well-known approach is due to Israel and Stewart [46], [42].", "In this approach one expands the distribution function around a local thermal equilibrated distribution function, $f_{\\rm eq}(x,p)$ , in terms of a series of irreducible Lorentz tensors We point out that in the original approach by Israel and Stewart, the decomposition basis is not orthogonal and therefore, the exact form of the transport coefficients cannot be obtained once the expansion is truncated.", "Recently, Denicol et al.", "showed how to correct this and expand properly the distribution function in terms of a complete and orthogonal set of irreducible tensors of a particle with momentum $p^\\mu $ [26].", "of particle momentum $p^\\mu $ $f(x,p) &=& \\,f_{\\rm eq}(x,p)\\,(1+\\phi (x,t) )\\,, \\nonumber \\\\&=& f_{\\rm eq}(x,p)\\,(1+c(x,t) + c_\\mu p^{\\langle \\mu \\rangle } + c_{\\mu \\nu } p^{\\langle \\mu }p^{\\nu \\rangle }+ c_{\\mu \\nu \\lambda }p^{\\langle \\mu }p^\\nu p^{\\lambda \\rangle } + \\ldots \\; ) \\, ,$ where the angle brackets above stand for symmetrized tensors which are orthogonal to the fluid four-velocity $u^\\mu $ [41], [26].", "The thermal equilibrium distribution function has the functional form $f_{\\rm eq}= \\left[ \\exp \\!\\left(\\frac{p^\\mu u_\\mu (x)-\\mu (x)}{T(x)}\\right) + a\\right]^{-1} \\, ,$ where $a= \\pm 1$ gives Fermi-Dirac or Bose-Einstein statistics and $a=0$ gives Maxwell-Boltzmann statistics.", "The distribution function (REF ) is usually expanded until second order, i.e.", "just keeping the terms 1, $p^{\\langle \\mu \\rangle }$ , and $p^{\\langle \\mu }p^{\\nu \\rangle }$ .", "An important aspect in the construction of irreducible tensor basis is the decomposition of the four-momentum $p^\\mu $ of a particle in Minkowski space.", "One assumes the existence of a time-like normalized vector field $u^\\mu (x)$ (which is identified with the fluid velocity) and an operator $\\Delta _{\\mu \\nu }$ which is symmetric, traceless and orthogonal to $u^\\mu (x)$ such that $p^\\mu = E u^\\mu +\\Delta ^{\\mu \\nu }p_\\nu $ [41], [26].", "This decomposition allows one to have an irreducible $n$ th-rank tensor basis which is complete and orthogonal [41], [26].", "An alternative but equivalent treatment for expanding the distribution function in terms of an irreducible $n$ th-rank tensor basis was developed by Anderson [47].", "This method instead decomposes the four-momentum $p^\\mu $ of a particle as $p^\\mu = E u^\\mu + \\sum _{i=1}^{3} p_i x^\\mu _i \\, ,$ where $u^\\mu $ is the fluid velocity and $x^\\mu _i$ is a set of orthonormal vectors which are spacelike and orthogonal to $u^\\mu $ .", "With this decomposition one can also find a suitable irreducible tensor representation [47].", "We will follow this decomposition closely since it is the most convenient vector basis for a system which is anisotropic in momentum-space along some preferred direction(s).", "In the rest of this section, we use the vector basis decomposition (REF ) to construct 2nd-rank tensors.", "As a particular case, we construct the energy-momentum tensor for a (2+1)-dimensional boost invariant anisotropic plasma and derive the dynamical equations of motion by taking moments of the Boltzmann equation.", "Our discussion is restricted to the case of vanishing chemical potential." ], [ "Vector Basis", "In this paper we will concentrate on systems which possess a preferred direction associated with a single direction in momentum-space.", "It is possible to construct a tensor basis which allows for multiple anisotropy directions; however, we restrict our considerations to this simpler case since taking into account the momentum-space anisotropy along the beamline direction is of particular importance for heavy-ion phenomenology.", "To begin, we will specify a tensor basis which is completely general and not subject to any symmetry constraints and then add the necessary symmetry constraints when needed.", "A general tensor basis can be constructed by introducing four 4-vectors which in the local rest frame (LRF) are $&&X^\\mu _{0,{\\rm LRF}} \\equiv u^\\mu _{\\rm LRF} = (1,0,0,0) \\nonumber \\\\&&X^\\mu _{1,{\\rm LRF}} \\equiv x^\\mu _{\\rm LRF} = (0,1,0,0) \\nonumber \\\\&&X^\\mu _{2,{\\rm LRF}} \\equiv y^\\mu _{\\rm LRF} = (0,0,1,0) \\nonumber \\\\&&X^\\mu _{3,{\\rm LRF}} \\equiv z^\\mu _{\\rm LRF} = (0,0,0,1) \\, .$ These 4-vectors are orthonormal in all frames.", "The vector $X^\\mu _0$ is associated with the four-velocity of the local rest frame and is conventionally called $u^\\mu $ and one can also identify $X^\\mu _1 = x^\\mu $ , $X^\\mu _2 = y^\\mu $ , and $X^\\mu _3 = z^\\mu $ as indicated above.", "We will use the two different labels for these vectors interchangeably depending on convenience since the notation with numerical indices allows for more compact expressions in many cases.", "Note that, in the lab frame the three spacelike vectors $X^\\mu _i$ can be written entirely in terms of $X^\\mu _0=u^\\mu $ .", "This is because $X^\\mu _i$ can be obtained by a sequence of Lorentz transformations/rotations applied to the local rest frame expressions specified above.", "We will return to this issue and construct explicit lab-frame representations of these four-vectors later.", "Finally, we point out that one can express the metric tensor itself in terms of these 4-vectors as $g^{\\mu \\nu }= X^\\mu _0 X^\\nu _0 - \\sum _{i=1}^3 X^\\mu _i X^\\nu _i \\, .$ In addition, the standard transverse projection operator which is orthogonal to $X^\\mu _0$ can be rewritten in terms of the vector basis (REF ) as $\\Delta ^{\\mu \\nu } = g^{\\mu \\nu } - X^\\mu _0 X^\\nu _0 = - \\sum _{i=1}^3 X^\\mu _i X^\\nu _i \\, ,$ such that $u_\\mu \\Delta ^{\\mu \\nu } = u_\\nu \\Delta ^{\\mu \\nu } = 0$ .", "We note that the spacelike components of the tensor basis are eigenfunctions of this operator, i.e.", "$X_{i\\mu } \\Delta ^{\\mu \\nu } = X^\\nu _{i}$ ." ], [ "2nd-rank Tensors", "A general rank two tensor can be decomposed using the 4-vectors $X_\\alpha ^\\mu $ .", "In general there are sixteen possible terms $A^{\\mu \\nu }(t,{\\bf x}) &=& \\sum _{\\alpha ,\\beta =0}^3 c_{\\alpha \\beta } X^\\mu _\\alpha X^\\nu _\\beta \\, , \\nonumber \\\\&=& c_{00}X^\\mu _0X^\\nu _0 + \\sum _{i=1}^3 c_{ii} X^\\mu _i X^\\nu _i+ \\sum _{\\alpha ,\\beta =0 \\atop \\alpha \\ne \\beta }^3 c_{\\alpha \\beta } X^\\mu _\\alpha X^\\nu _\\beta \\, , \\nonumber \\\\&=& c_{00} g^{\\mu \\nu } + \\sum _{i=1}^3 \\underbrace{(c_{ii}+c_{00})}_{\\equiv \\, d_{ii}} X^\\mu _i X^\\nu _i+ \\sum _{\\alpha ,\\beta =0 \\atop \\alpha \\ne \\beta }^3 c_{\\alpha \\beta } X^\\mu _\\alpha X^\\nu _\\beta \\, ,$ where it is understood that the coefficients $c_{\\alpha \\beta }$ now contain all of the space-time dependence." ], [ "2nd-rank symmetric Tensors", "If a two tensor is symmetric under the interchange of $\\mu $ and $\\nu $ then $c_{\\alpha \\beta } = c_{\\beta \\alpha }$ and we can write $A^{\\mu \\nu }(t,{\\bf x}) = c_{00} g^{\\mu \\nu } + \\sum _{i=1}^3 d_{ii} X^\\mu _i X^\\nu _i+ \\sum _{\\alpha ,\\beta =0 \\atop \\alpha >\\beta }^3 c_{\\alpha \\beta } (X^\\mu _\\alpha X^\\nu _\\beta +X^\\mu _\\beta X^\\nu _\\alpha ) \\, .$ and there are only then ten independent terms." ], [ "Energy-Momentum Tensor for Ideal Hydrodynamics", "Since the energy-momentum tensor is a symmetric tensor of 2nd-rank, Eq.", "(REF ) can be used $T^{\\mu \\nu }(t,{\\bf x}) = t_{00} g^{\\mu \\nu } + \\sum _{i=1}^3 t_{ii} X^\\mu _i X^\\nu _i+ \\sum _{\\alpha ,\\beta =0 \\atop \\alpha >\\beta }^3 t_{\\alpha \\beta } (X^\\mu _\\alpha X^\\nu _\\beta +X^\\mu _\\beta X^\\nu _\\alpha ) \\, ,$ where we have relabeled the coefficients for this purpose.", "In the local rest frame we can identify the basis vectors via (REF ) and we have that $T^{00}_{\\rm LRF} = {\\cal E}$ and $T^{ii}_{\\rm LRF} = {\\cal P}_i$ where ${\\cal E}$ is the energy density and ${\\cal P}_i$ is the pressure in $i$ -direction and all other components vanish.", "If the system is locally isotropic as is the case for ideal hydrodynamics then ${\\cal P}_i \\equiv {\\cal P}$ .", "From (REF ) we have $T^{00}_{\\rm LRF} = {\\cal E} = t_{00}$ and $T^{ii}_{\\rm LRF} = {\\cal P} = -t_{00} + t_{ii}$ and since all off-diagonal components vanish we have $t_{\\alpha \\beta }$ = 0 for all $\\alpha \\ne \\beta $ .", "This allows us to write $T^{\\mu \\nu }(t,{\\bf x}) &=& {\\cal E} g^{\\mu \\nu } + ({\\cal P} + {\\cal E}) \\sum _{i=1}^3 X^\\mu _i X^\\nu _i \\, , \\nonumber \\\\&=& {\\cal E} g^{\\mu \\nu } + ({\\cal P} + {\\cal E}) (X^\\mu _0 X^\\nu _0 - g^{\\mu \\nu }) \\, \\nonumber \\\\&=& ({\\cal E} + {\\cal P}) X^\\mu _0 X^\\nu _0 - {\\cal P} g^{\\mu \\nu } \\, ,$ where in going from the first to second line we have used Eq.", "(REF ).", "Using the conventional notation that $X^\\mu _0 = u^\\mu $ we obtain $T^{\\mu \\nu } = ({\\cal E} +{\\cal P}) u^\\mu u^\\nu - {\\cal P} g^{\\mu \\nu } \\, ,$ in agreement with the expected result.", "For later use we also note that ${T^\\mu }_\\mu \\equiv {\\cal T} = {\\cal E} - 3 {\\cal P} \\, .$" ], [ "Energy-Momentum Tensor for Azimuthally-Symmetric Anisotropic Hydrodynamics", "In the bulk of this paper we will consider systems for which the momentum-space particle distribution is azimuthally symmetric while the rotational symmetry in the $p_\\perp $ -$p_L$ plane is broken.", "From here on we will refer to this as “azimuthally-symmetric” which only implies an assumed symmetry in momentum-space and not in configuration space.", "In the case of azimuthally-symmetric anisotropic hydrodynamics we have $T^{00}_{\\rm LRF} &=& {\\cal E} = t_{00} \\nonumber \\, , \\\\T^{xx}_{\\rm LRF} &=& {\\cal P}_\\perp = -t_{00} + t_{11}\\nonumber \\, , \\\\T^{yy}_{\\rm LRF} &=& {\\cal P}_\\perp = -t_{00} + t_{22}\\nonumber \\, , \\\\T^{zz}_{\\rm LRF} &=& {\\cal P}_L = -t_{00} + t_{33} \\, ,$ and due to the azimuthal symmetry in momentum-space we must have $ t_{11}= t_{22}$ which gives four equations for our four unknowns.", "Solving for the coefficients $t$ one obtains $T^{\\mu \\nu }(t,{\\bf x}) &=& {\\cal E} g^{\\mu \\nu } + ({\\cal P}_\\perp + {\\cal E}) \\sum _{i=1}^2 X^\\mu _i X^\\nu _i+ ({\\cal P}_L + {\\cal E}) X^\\mu _3 X^\\nu _3 \\, , \\nonumber \\\\&=& {\\cal E} g^{\\mu \\nu } + ({\\cal P}_\\perp + {\\cal E}) \\sum _{i=1}^3 X^\\mu _i X^\\nu _i+ ({\\cal P}_L - P_\\perp ) X^\\mu _3 X^\\nu _3 \\, , \\nonumber \\\\ &=& {\\cal E} g^{\\mu \\nu } + ({\\cal P}_\\perp + {\\cal E}) (X^\\mu _0 X^\\nu _0 - g^{\\mu \\nu })+ ({\\cal P}_L - P_\\perp ) X^\\mu _3 X^\\nu _3 \\, \\nonumber \\\\&=& ({\\cal E} + {\\cal P}_\\perp ) X^\\mu _0 X^\\nu _0 - {\\cal P}_\\perp g^{\\mu \\nu }+ ({\\cal P}_L - P_\\perp ) X^\\mu _3 X^\\nu _3\\, .$ Relabeling $X^\\mu _0 = u^\\mu $ and $X^\\mu _3 = z^\\mu $ to agree more closely with the notation of Ref.", "[45] we obtain $T^{\\mu \\nu } = ({\\cal E} +{\\cal P}_\\perp ) u^\\mu u^\\nu - {\\cal P}_\\perp g^{\\mu \\nu }+ ({\\cal P}_L - {\\cal P}_\\perp ) z^\\mu z^\\nu \\, ,$ which in the limit that ${\\cal P}_\\perp = {\\cal P}_L \\equiv {\\cal P}$ reduces to (REF ).", "We again note for later use that ${T^\\mu }_\\mu \\equiv {\\cal T} = {\\cal E} - 2 {\\cal P}_\\perp - {\\cal P}_L\\, .$" ], [ "Explicit Forms of the Basis Vectors", "In the lab frame the three spacelike vectors $X^\\mu _i$ can be written entirely in terms of $X^\\mu _0=u^\\mu $ .", "This is because $X^\\mu _i$ can be obtained by a sequence of Lorentz transformations/rotations applied to the local rest frame expressions specified above.", "To go from the lab frame to LRF we can apply a boost along the $z$ -axis followed by a rotation around the $z$ -axis and finally a boost along the $x$ -axis, i.e.", "$u_{\\rm LRF} = L_x(\\psi )R_z(\\theta ) L_z(\\vartheta ) u$ [48].", "This specific transformation is chosen in order to ensure that the four-vector $z^\\mu $ has no transverse components in all frames.", "To find the necessary vectors in the lab frame based on the LRF expressions (REF ) we apply the inverse operation $X_{\\alpha ,\\rm LAB}^\\mu = (L_x R_z L_z)^{-1} X_{\\alpha ,\\rm LRF}^\\mu = (L_z)^{-1} (R_z)^{-1} (L_x)^{-1} X_{\\alpha ,\\rm LRF}^\\mu $ which is explicitly given by $X_{\\alpha ,\\rm LAB}^\\mu \\!", "=\\underbrace{\\!\\!\\left(\\begin{array}{cccc}\\cosh \\vartheta & 0 & 0 & \\sinh \\vartheta \\\\0 & 1 & 0 & 0 \\\\0 & 0 & 1 & 0 \\\\\\sinh \\vartheta & 0 & 0 & \\cosh \\vartheta \\end{array}\\right)\\!\\!", "}_{(L_z)^{-1}}\\underbrace{\\!\\left(\\begin{array}{cccc}1 & 0 & 0 & 0 \\\\0 & \\cos \\phi & -\\sin \\phi & 0 \\\\0 & \\sin \\phi & \\cos \\phi & 0 \\\\0 & 0 & 0 & 1\\end{array}\\right)\\!\\!", "}_{(R_z)^{-1}}\\underbrace{\\!\\left(\\begin{array}{cccc}\\cosh \\psi & \\sinh \\psi & 0 & 0 \\\\\\sinh \\psi & \\cosh \\psi & 0 & 0 \\\\0 & 0 & 1 & 0 \\\\0 & 0 & 0 & 1\\end{array}\\right)\\!\\!", "}_{(L_x)^{-1}} \\, X_{\\alpha ,\\rm LRF}^\\mu \\, .$ which gives $\\begin{array}{ll}\\begin{aligned}u^0 &= \\cosh \\psi \\cosh \\vartheta \\, , \\\\u^1 &= \\sinh \\psi \\cos \\phi \\, , \\\\u^2 &= \\sinh \\psi \\sin \\phi \\, , \\\\u^3 &= \\cosh \\psi \\sinh \\vartheta \\, ,\\end{aligned}\\quad \\quad \\quad &\\begin{aligned}x^0 &= \\sinh \\psi \\cosh \\vartheta \\, , \\\\x^1 &= \\cosh \\psi \\cos \\phi \\, , \\\\x^2 &= \\cosh \\psi \\sin \\phi \\, , \\\\x^3 &= \\sinh \\psi \\sinh \\vartheta \\, ,\\end{aligned}\\\\&\\vspace{5.69054pt}\\\\\\begin{aligned}y^0 &= 0 \\, , \\\\y^1 &= -\\sin \\phi \\, , \\\\y^2 &= \\cos \\phi \\, , \\\\y^3 &= 0 \\, ,\\end{aligned}\\quad \\quad \\quad &\\begin{aligned}z^0 &= \\sinh \\vartheta \\, , \\\\z^1 &= 0 \\, , \\\\z^2 &= 0 \\, , \\\\z^3 &= \\cosh \\vartheta \\, .\\end{aligned}\\end{array}$ In the limit that the system is boost invariant one can identify $\\vartheta =\\varsigma $ , where $\\varsigma $ is the spatial rapidity defined through $t&=&\\tau \\cosh \\varsigma \\, , \\nonumber \\\\z&=&\\tau \\sinh \\varsigma \\, ,$ where $\\tau = \\sqrt{t^2 - z^2}$ is the proper time.", "In the remainder of the paper when we refer to a boost-invariant system we will use $\\tau $ and $\\varsigma $ as the longitudinal coordinates." ], [ "Dynamical Equations", "In this section we derive the dynamical equations of motion by taking moments of the Boltzmann equation [41] $p^\\mu \\partial _\\mu f(x,p) = - C[f] \\, .$ The moments are defined by multiplying the left and right hand sides of the Boltzmann equation by various powers of the four-momentum and then averaging in momentum space.", "This can be achieved via the $n^{\\rm th}$ moment integral operator ${\\hat{\\cal I}}_n \\, \\equiv \\int d\\chi \\; p^{\\mu _1} p^{\\mu _2} \\cdots p^{\\mu _n} \\, ,$ where $n\\ge 0$ is an integer and $\\int \\!", "d\\chi \\equiv \\int \\!", "\\!", "\\frac{d^4{\\bf p}}{(2\\pi )^3} \\, \\delta (p_\\mu p^\\mu - m^2) \\, 2 \\theta (p^0)= \\int \\!", "\\!", "\\frac{d^3{\\bf p}}{(2\\pi )^3} \\frac{1}{p^0} \\, .$" ], [ "Zeroth moment of the Boltzmann equation", "The zeroth moment of the Boltzmann equation results from applying ${\\hat{\\cal I}}_0$ to both sides of (REF ) $\\int d\\chi \\; p^\\mu \\partial _\\mu f&=& J_0 \\, ,\\nonumber \\\\\\partial _\\mu \\int \\!", "\\!", "\\frac{d^3{\\bf p}}{(2\\pi )^3} \\frac{p^\\mu }{p^0} \\; f&=& J_0 \\, ,\\nonumber \\\\\\partial _\\mu j^\\mu &=& J_0 \\, ,$ where $J_n \\equiv - {\\hat{\\cal I}}_n C[f]$ .", "Note that we can rewrite the left hand side of the last expression as $j^\\mu = n \\;\\!", "u^\\mu $ where $n$ is the particle number density in the local rest frame.", "Expanding we find $\\partial _\\mu j^\\mu = D n + n \\theta \\, ,$ where $D &\\equiv & u^\\mu \\partial _\\mu \\, , \\nonumber \\\\\\theta &\\equiv & \\partial _\\mu u^\\mu \\, ,$ allowing us to write a general expression for the zeroth moment of the Boltzmann equation $D n + n \\theta = J_0 \\, .$" ], [ "First moment of the Boltzmann equation", "The first moment of the Boltzmann equation is equivalent to the requirement of energy and momentum conservation [41] $\\partial _\\mu T^{\\mu \\nu } = 0 \\, ,$ where $T^{\\mu \\nu }$ is the energy momentum tensor.", "In the following we derive evolution equations under different assumptions about the degree of symmetry of $T^{\\mu \\nu }$ ." ], [ "Ideal hydrodynamics", "To begin we use the general form of the energy-momentum tensor for an isotropic system given in Eq.", "(REF ) to obtain $\\partial _\\mu T^{\\mu \\nu } =u^\\nu D ({\\cal E} +{\\cal P})+ u^\\nu ({\\cal E} +{\\cal P}) \\theta + ({\\cal E} +{\\cal P}) D u^\\nu - \\partial ^\\nu {\\cal P} \\, ,$ where $D$ and $\\theta $ are defined in Eq.", "(REF ).", "Canonically one takes projections of $\\partial _\\mu T^{\\mu \\nu } = 0$ parallel and perpendicular to $u^\\mu $ .", "The parallel projection is obtained via $u_\\nu \\partial _\\mu T^{\\mu \\nu }$ which gives $u_\\nu \\partial _\\mu T^{\\mu \\nu } &=& D ({\\cal E} +{\\cal P})+ ({\\cal E} +{\\cal P}) \\theta + ({\\cal E} +{\\cal P}) u_\\nu D u^\\nu - D {\\cal P} = 0\\nonumber \\\\&=& D{\\cal E} + ({\\cal E} +{\\cal P}) \\theta = 0$ where we have used $u_\\nu u^\\nu = 1$ and $u_\\nu D u^\\nu = \\frac{1}{2} D(u_\\nu u^\\nu ) = 0$ .", "This gives us our first equation for ideal hydrodynamics.", "For the transverse projection we use $\\Delta ^{\\mu \\nu }$ defined in Eq.", "(REF ) which satisfies $\\Delta _{\\alpha \\nu } u^\\nu = 0$ .", "This gives ${\\Delta ^\\alpha }_\\nu \\partial _\\mu T^{\\mu \\nu } = ({\\cal E} +{\\cal P}) {\\Delta ^\\alpha }_\\nu D u^\\nu - {\\Delta ^\\alpha }_\\nu \\partial ^\\nu {\\cal P} = 0\\, .$ Using the explicit form for ${\\Delta ^\\alpha }_\\nu = {g^\\alpha }_\\nu - u^\\alpha u_\\nu $ one obtains ${\\Delta ^\\alpha }_\\nu D u^\\nu = D u^\\alpha $ .", "We can additionally define $\\nabla ^\\alpha \\equiv {\\Delta ^\\alpha }_\\nu \\partial ^\\nu =- \\sum _{\\beta =1}^3 X_\\beta ^\\alpha X_{\\nu \\beta } \\partial ^\\nu \\, ,$ which is the gradient in the spacelike directions.", "Putting this together with Eq.", "(REF ) one obtains the following two equations $D{\\cal E} + ({\\cal E} +{\\cal P}) \\theta &=& 0 \\, , \\nonumber \\\\({\\cal E} +{\\cal P}) D u^\\alpha - \\nabla ^\\alpha {\\cal P} &=& 0 \\, .$ In the second case $\\alpha $ should be a spacelike index such that we have four equations in total which should be supplemented by the equation of state which can be expressed in the form of a constraint on the trace of the energy momentum tensor ${T^\\mu }_\\mu ={\\cal T}={\\cal E} - 3{\\cal P}$ ." ], [ "Ideal Boost Invariant Dynamics with Transverse Expansion", "In this section we briefly review what happens when the system is boost invariant and we allow for inhomogeneities and flow in the transverse direction.", "In this case we have from (REF ) $u^\\mu = (\\cosh \\psi \\cosh \\varsigma ,\\sinh \\psi \\cos \\phi ,\\sinh \\psi \\sin \\phi , \\cosh \\psi \\sinh \\varsigma ) \\, .$ It is convenient at this point to relabel the components of $u^\\mu $ as $u^\\mu = (u_0 \\cosh \\varsigma ,u_x,u_y, u_0 \\sinh \\varsigma ) \\, .$ where the constraint $u_0^2 = 1 + u_x^2 + u_y^2$ should be satisfied.", "Changing to proper time and spatial rapidity we obtain $u_\\tau = u_0$ , $u_\\varsigma = 0$ , and we have $D &=& u^\\mu \\partial _\\mu = u_0 \\partial _\\tau + {\\bf u}_\\perp \\!\\cdot \\nabla _\\perp \\, , \\nonumber \\\\\\theta &=& \\partial _\\mu u^\\mu = \\partial _\\tau u_0 + \\nabla _\\perp \\!\\cdot {\\bf u}_\\perp + \\frac{u_0}{\\tau } \\, .$ For the transverse gradient it is convenient to rewrite $\\nabla ^i = {\\Delta ^i}_\\nu \\partial ^\\nu = ({g^i}_\\nu - u^i u_\\nu ) \\partial ^\\nu = \\partial ^i - u^i D \\, ,$ such that the second equation in (REF ) can be expanded into three equations $({\\cal E} + {\\cal P}) D u_x + u_x D {\\cal P} + \\partial _x {\\cal P} &= 0 \\, , \\nonumber \\\\({\\cal E} + {\\cal P}) D u_y + u_y D {\\cal P} + \\partial _y {\\cal P} &= 0 \\, , \\nonumber \\\\({\\cal E} + {\\cal P}) D u_0 + u_0 D {\\cal P} - \\partial _\\tau {\\cal P} &= 0 \\, ,$ which together with $D{\\cal E} + ({\\cal E} + {\\cal P}) \\theta = 0 \\, ,$ would seem to give four equations for our four unknowns (${\\cal E}$ , ${\\cal P}$ , $u_x$ , and $u_y$ since $u_0^2 = 1 + u_x^2 + u_y^2$ ); however, upon inspection one finds that Eqs.", "(REF ) are not independent since $u_0$ times the third equation is equal to $u_x$ times the first plus $u_y$ times the second.", "We, therefore, have a choice of which equations to use and one can pick two of the three equations from (REF ), e.g.", "the first two.", "The final equation is then provided canonically by the equation of state which specifies, e.g., the energy density as a function of the pressure." ], [ "Azimuthally-Symmetric Anisotropic Hydrodynamics", "We now proceed to the derivation of the dynamical equation for azimuthally-symmetric anisotropic hydrodynamics.", "We remind the reader “azimuthally-symmetric” means that the momentum-space particle distribution is azimuthally symmetric while the rotational symmetry in the $p_\\perp $ -$p_L$ plane is broken.", "To begin we use the general form of the energy-momentum tensor for an azimuthally-symmetric anisotropic system given in Eq.", "(REF ) to obtain $\\partial _\\mu T^{\\mu \\nu } &=&u^\\nu D ({\\cal E} +{\\cal P}_\\perp )+ u^\\nu ({\\cal E} +{\\cal P}_\\perp ) \\theta + ({\\cal E} +{\\cal P}_\\perp ) D u^\\nu - \\partial ^\\nu {\\cal P}_\\perp \\nonumber \\\\&&\\hspace{14.22636pt}+ z^\\nu D_L ({\\cal P}_L - {\\cal P}_\\perp )+ z^\\nu ({\\cal P}_L - {\\cal P}_\\perp ) \\theta _L+ ({\\cal P}_L - {\\cal P}_\\perp ) D_L z^\\nu = 0\\, ,$ where $D_L &\\equiv & z^\\mu \\partial _\\mu \\, , \\nonumber \\\\\\theta _L &\\equiv & \\partial _\\mu z^\\mu \\, .$ As before we take projections of $\\partial _\\mu T^{\\mu \\nu } = 0$ parallel and perpendicular to $u^\\mu $ .", "The parallel projection is obtained via $u_\\nu \\partial _\\mu T^{\\mu \\nu }$ which gives $u_\\nu \\partial _\\mu T^{\\mu \\nu } = D {\\cal E}+ ({\\cal E} +{\\cal P}_\\perp ) \\theta + ({\\cal P}_L - {\\cal P}_\\perp ) u_\\nu D_L z^\\nu = 0 \\, ,$ where we have used $u_\\nu u^\\nu = 1$ , $u_\\nu D u^\\nu = \\frac{1}{2}D(u_\\nu u^\\nu ) = 0$ , and $u_\\nu z^\\nu = 0$ .", "This gives us our first equation for azimuthally-symmetric anisotropic hydrodynamics.", "For the transverse projection we use $\\Delta ^{\\mu \\nu }$ defined in Eq.", "(REF ) which satisfies $\\Delta _{\\alpha \\nu } u^\\nu = 0$ and $\\Delta _{\\alpha \\nu } z^\\nu = z^\\alpha $ .", "This gives ${\\Delta ^\\alpha }_\\nu \\partial _\\mu T^{\\mu \\nu }&=& ({\\cal E} +{\\cal P}_\\perp ) D u^\\alpha - {\\nabla }^\\alpha {\\cal P}_\\perp + z^\\alpha D_L ({\\cal P}_L - {\\cal P}_\\perp ) + z^\\alpha ({\\cal P}_L - {\\cal P}_\\perp ) \\theta _L\\nonumber \\\\&&\\hspace{14.22636pt}+ ({\\cal P}_L - {\\cal P}_\\perp ) D_L z^\\alpha - ({\\cal P}_L - {\\cal P}_\\perp ) u^\\alpha u_\\nu D_L z^\\nu = 0 \\, .$" ], [ "Boost Invariant Dynamics with Transverse Expansion", "In this case we have $z^\\tau = 0$ and $z^\\eta = 1/\\tau $ such that $D_L &=& z^\\mu \\partial _\\mu = \\frac{\\partial _\\varsigma }{\\tau } \\, , \\nonumber \\\\\\theta _L &=& \\partial _\\mu z^\\mu = 0 \\, .$ From the first line above we find $ u_\\nu D_L z^\\nu = u_0/\\tau $ .", "This allows us to simplify the parallel projection to $D {\\cal E} + ({\\cal E} +{\\cal P}_\\perp ) \\theta + ({\\cal P}_L - {\\cal P}_\\perp )\\frac{u_0}{\\tau } = 0 \\, .$ The transverse projections can also be simplified to $({\\cal E} +{\\cal P}_\\perp ) D u^\\alpha + u^\\alpha D {\\cal P}_\\perp + \\partial _\\alpha {\\cal P}_\\perp + ({\\cal P}_L - {\\cal P}_\\perp ) \\left( \\frac{\\partial _\\varsigma z^\\alpha }{\\tau }- \\frac{u_0}{\\tau } u^\\alpha \\right)= 0 \\, ,$ from which we can then obtain three equations $({\\cal E} +{\\cal P}_\\perp ) D u_x + u_x D {\\cal P}_\\perp + \\partial _x {\\cal P}_\\perp + ({\\cal P}_\\perp - {\\cal P}_L) \\frac{u_0 u_x}{\\tau } &=& 0 \\, , \\nonumber \\\\({\\cal E} +{\\cal P}_\\perp ) D u_y + u_y D {\\cal P}_\\perp + \\partial _y {\\cal P}_\\perp + ({\\cal P}_\\perp - {\\cal P}_L) \\frac{u_0 u_y}{\\tau } &=& 0 \\, , \\nonumber \\\\({\\cal E} +{\\cal P}_\\perp ) D u_0 + u_0 D {\\cal P}_\\perp - \\partial _\\tau {\\cal P}_\\perp + ({\\cal P}_\\perp - {\\cal P}_L) \\frac{u_\\perp ^2}{\\tau } &=& 0 \\, .$ As was the case with ideal hydrodynamics, we see that $u_0$ times the third equation is equal to $u_x$ times the first plus $u_y$ times the second so that it is redundant.", "This leaves us with the following three equations $D {\\cal E} + ({\\cal E} +{\\cal P}_\\perp ) \\theta + ({\\cal P}_L - {\\cal P}_\\perp )\\frac{u_0}{\\tau } &=& 0 \\, , \\nonumber \\\\({\\cal E} +{\\cal P}_\\perp ) D u_x + \\partial _x {\\cal P}_\\perp + u_x D {\\cal P}_\\perp + ({\\cal P}_\\perp - {\\cal P}_L) \\frac{u_0 u_x}{\\tau } &=& 0 \\, , \\nonumber \\\\({\\cal E} +{\\cal P}_\\perp ) D u_y + \\partial _y {\\cal P}_\\perp + u_y D {\\cal P}_\\perp + ({\\cal P}_\\perp - {\\cal P}_L) \\frac{u_0 u_y}{\\tau } &=& 0 \\, .$" ], [ "Distribution function for azimuthally-symmetric systems", "We next consider the one-particle distribution function $f$ in the local rest frame and show that in the case of a system that is locally azimuthally-symmetric in momentum space that it suffices to introduce one anisotropy parameter $\\xi $ and a single scale $\\Lambda $ [49].", "To begin we consider the general form $f(t,{\\bf x},{\\bf p}) = f_{\\rm iso}(\\sqrt{\\bar{p}_\\mu \\Xi ^{\\mu \\nu }(t,{\\bf x}) \\bar{p}_\\nu }) \\, .$ $\\Xi ^{\\mu \\nu }(t,{\\bf x}) $ is a symmetric tensor, $f_{\\rm iso}$ is an arbitrary isotropic distribution function, and ${\\bar{p}}^\\mu \\equiv p^\\mu /\\Lambda $ , where $\\Lambda (t,{\\bf x})$ is a momentum scale that can depend on space and time (the so-called hard momentum scale).", "In the case where the system is in thermal equilibrium, then $f_{\\rm iso}$ would be given by a Bose-Einstein or Fermi-Dirac distribution function.", "Note that the argument of the square root in $f_{\\rm iso}$ should remain greater than or equal to zero in order for $f$ to be a single-valued real function.", "If $\\Xi ^{\\mu \\nu }$ is a symmetric tensor and is diagonal in the local rest frame, we have $\\Xi ^{\\mu \\nu } = c_{00} u^\\mu u^\\nu + \\sum _{i=1}^3 c_{ii} X^\\mu _i X^\\nu _i \\, ,$ and if, additionally, the system is symmetric under $x \\leftrightarrow y$ then $c_{11} = c_{22} \\equiv c_{\\perp \\perp }$ and we have $\\Xi ^{\\mu \\nu } &=& c_{00} u^\\mu u^\\nu + c_{\\perp \\perp } \\sum _{i=1}^2 X^\\mu _i X^\\nu _i+ c_{33} X^\\mu _3 X^\\nu _3 \\, , \\nonumber \\\\&=& c_{00} u^\\mu u^\\nu - c_{\\perp \\perp } \\Delta ^{\\mu \\nu }+ (c_{33}-c_{\\perp \\perp }) X^\\mu _3 X^\\nu _3 \\, .$ Using our ability to redefine $\\Lambda \\rightarrow \\sqrt{c_{00}} \\Lambda $ in Eq.", "(REF ) we can rescale our coefficients.", "Defining $c_{\\perp \\perp }/c_{00} \\equiv \\Phi $ and $(c_{33}-c_{\\perp \\perp })/c_{00} \\equiv \\alpha $ we can write compactly $\\Xi ^{\\mu \\nu } = u^\\mu u^\\nu - \\Phi \\Delta ^{\\mu \\nu } + \\alpha z^\\mu z^\\nu \\, .$ Contracting with four-momenta on both sides we find $p_\\mu \\Xi ^{\\mu \\nu } p_\\nu &=& p_0^2 + \\Phi {\\bf p}^2 + \\alpha p_z^2 \\, ,\\nonumber \\\\&=& m^2 + (1+\\Phi ) {\\bf p}^2 + \\alpha p_z^2 \\, ,$ where we have used $p_0^2 = {\\bf p}^2 + m^2$ .", "If we have a system of massless particles then $p_\\mu \\Xi ^{\\mu \\nu } p_\\nu = (1+\\Phi )p_\\perp ^2 + (1 + \\Phi + \\alpha ) p_z^2 \\, ,$ and in this case we can once again use our ability to rescale $\\Lambda \\rightarrow \\sqrt{(1+\\Phi )} \\Lambda $ and defining $1+\\xi \\equiv (1 + \\Phi + \\alpha )/(1 + \\Phi )$ we obtain $p_\\mu \\Xi ^{\\mu \\nu } p_\\nu = p_\\perp ^2 + (1 + \\xi ) p_z^2 \\, ,$ which has the form of the argument of the original one-dimensional Romatschke-Strickland (RS) distribution function [49]." ], [ "Number density and Energy-Momentum Tensor with the RS distribution function", "Based on the results of the last section, the functional form of the RS distribution function for a locally azimuthally-symmetric expanding anisotropic plasma is $f({\\bf x},{\\bf p},\\tau )= f_{\\rm RS}({\\bf p},\\xi ,\\Lambda )= f_{\\rm iso}\\bigl (\\sqrt{[{\\bf p}_\\perp ^2+(1+\\xi )p^2_z]/\\Lambda ^2}\\bigr ) \\, ,$ where it is understood that on the right hand side $\\xi $ and $\\Lambda $ can depend on space and time.", "Using this distribution function the number density is given by [50], [51] $n(\\xi ,\\Lambda ) = \\int \\frac{d^3{\\bf p}}{(2 \\pi )^3} f _{\\rm RS} = \\frac{n_{\\rm iso}(\\Lambda )}{\\sqrt{1+\\xi }} \\, .$ where $n_{\\rm iso}(\\Lambda )$ is the number density one obtains in the isotropic limit.", "One can also evaluate the energy-momentum tensor in the LRF $T^{\\mu \\nu }= \\int \\frac{d^3{\\bf p}}{(2 \\pi )^3} \\frac{p^\\mu p^\\nu }{p_0} f(\\tau ,{\\bf x},{\\bf p}) \\, .$ By using the RS form (REF ) one gets the explicit components of the energy-momentum tensor [51] ${\\cal E}(\\Lambda ,\\xi ) &= T^{\\tau \\tau } = {\\cal R}(\\xi )\\,{\\cal E}_{\\rm iso}(\\Lambda )\\, ,\\\\{\\cal P}_\\perp (\\Lambda ,\\xi ) &= \\frac{1}{2}\\left( T^{xx} + T^{yy}\\right) = {\\cal R}_\\perp (\\xi ){\\cal P}_{\\rm iso}(\\Lambda )\\, , \\\\{\\cal P}_L(\\Lambda ,\\xi ) &= - T^{\\varsigma }_\\varsigma = {\\cal R}_{\\rm L}(\\xi ){\\cal P}_{\\rm iso}(\\Lambda )\\, ,$ where ${\\cal P}_{\\rm iso}(\\Lambda )$ and ${\\cal E}_{\\rm iso}(\\Lambda )$ are the isotropic pressure and energy density, respectively, and ${\\cal R}(\\xi ) &\\equiv \\frac{1}{2}\\left(\\frac{1}{1+\\xi } +\\frac{\\arctan \\sqrt{\\xi }}{\\sqrt{\\xi }} \\right) \\, , \\\\{\\cal R}_\\perp (\\xi ) &\\equiv \\frac{3}{2 \\xi } \\left( \\frac{1+(\\xi ^2-1){\\cal R}(\\xi )}{\\xi + 1}\\right) \\, , \\\\{\\cal R}_L(\\xi ) &\\equiv \\frac{3}{\\xi } \\left( \\frac{(\\xi +1){\\cal R}(\\xi )-1}{\\xi +1}\\right) \\, .$ The equation of state can be imposed as a relationship between ${\\cal E}_{\\rm iso}$ and ${\\cal P}_{\\rm iso}$ .", "In what follows we will assume an ideal equation of state which is appropriate for a conformal massless gas, i.e.", "${\\cal E}_{\\rm iso}= 3 {\\cal P}_{\\rm iso}$ ." ], [ "Relaxation time approximation", "As mentioned in previous sections the dynamical equations necessary can be obtained by taking moments of the Boltzmann equation $p^\\mu \\partial _\\mu f = - C[f]$ .", "Here we use the relaxation time approximation with relaxation rate $\\Gamma $ ${\\cal C}[f_{RS}] = p_\\mu u^\\mu \\, \\Gamma \\, \\left[ f_{\\rm RS}({\\bf p},\\xi ,\\Lambda ,\\varsigma ) - f_{\\rm eq}(|{\\bf p}|,T) \\right]\\,,$ where $\\varsigma $ is the spatial rapidity and we fix $\\Gamma $ such that the 2nd-order viscous hydrodynamical equations are reproduced in the one-dimensional transversally symmetric case [40].", "This requires that $\\Gamma &\\equiv & \\frac{2}{\\tau _\\pi }\\,, \\nonumber \\\\\\tau _\\pi &\\equiv & \\frac{5}{4}\\frac{\\eta }{\\cal P}\\, ,$ which for an ideal equation of state results in $\\Gamma = \\frac{2T(\\tau )}{5\\bar{\\eta }} = \\frac{2{\\cal R}^{1/4}(\\xi )\\Lambda }{5\\bar{\\eta }} \\, ,$ where $\\bar{\\eta }= \\eta /{\\cal S}$ with $\\eta $ being the shear viscosity and ${\\cal S}$ being the entropy density.", "We note that one could perform a matching to 2nd-order viscous hydrodynamics including transverse dynamics, but we have not attempted to do so.", "Instead we use the 1d matching above and in the results section we show that numerical results from viscous hydrodynamics codes which include transverse dynamics are reproduced for small $\\bar{\\eta }$ .", "That being said, we have no reason to expect that the linearized equations would not reproduce 2nd-order viscous hydrodynamics; however, this remains to be proven." ], [ "Dynamical Equations of Motion", "Based on the results of the previous sections, we can derive the explicit form of the dynamical equations of motion for a (2+1)-dimensional boost invariant system." ], [ "Zeroth moment of the Boltzmann Equation", "For the RS form the 0th moment of the Boltzmann equation (REF ) is written as $\\frac{1}{1+\\xi }D\\xi - 6D(\\log \\Lambda ) - 2 \\theta = 2 \\Gamma \\left(1 - {\\cal R}^{3/4}(\\xi ) \\sqrt{1+\\xi }\\right) \\, .$ where we used explicitly the functional form of particle density $n$  (REF ) and the scattering kernel for relaxation time approximation (REF )." ], [ "First moment of the Boltzmann Equation", "Using the RS form one finds the following three equations by requiring energy-momentum conservation ${\\cal R}^{\\prime }(\\xi ) D\\xi + 4 {\\cal R}(\\xi ) D(\\log \\Lambda ) &=&- \\left({\\cal R}(\\xi ) + \\frac{1}{3} {\\cal R}_\\perp (\\xi )\\right) \\Delta _\\perp - \\left({\\cal R}(\\xi ) + \\frac{1}{3} {\\cal R}_L(\\xi )\\right) \\frac{u_0}{\\tau } \\, ,\\nonumber \\\\\\left[3{\\cal R}(\\xi ) + {\\cal R}_\\perp (\\xi )\\right] D u_\\perp &=&-u_\\perp \\left[ {\\cal R}_\\perp ^{\\prime }(\\xi ) \\tilde{D} \\xi + 4 {\\cal R}_\\perp (\\xi ) \\tilde{D} (\\log \\Lambda ) + \\frac{u_0}{\\tau } ({\\cal R}_\\perp (\\xi )-{\\cal R}_L(\\xi )) \\right] ,\\nonumber \\\\u_y^2 \\left[3{\\cal R}(\\xi ) + {\\cal R}_\\perp (\\xi )\\right] D \\left( \\frac{u_x}{u_y} \\right) &=&{\\cal R}_\\perp ^{\\prime }(\\xi ) D_\\perp \\xi + 4 {\\cal R}_\\perp (\\xi ) D_\\perp (\\log \\Lambda ) \\, ,$ where $\\Delta _\\perp &\\equiv & \\partial _\\tau u_0 + \\nabla _\\perp \\cdot {\\bf u}_\\perp \\, , \\nonumber \\\\\\tilde{D} &\\equiv & u_0 \\partial _\\tau + \\frac{u_0^2}{u_\\perp ^2} {\\bf u}_\\perp \\cdot \\nabla _\\perp \\, , \\nonumber \\\\D_\\perp &\\equiv & \\hat{\\bf z} \\cdot ({\\bf u}_\\perp \\times \\nabla _T) = u_x \\partial _y - u_y \\partial _x \\, ,$ ${\\bf u}_\\perp \\equiv (u_x,u_y)$ , and $u_0^2 = 1 + u_\\perp ^2$ ." ], [ "Initial Conditions", "We consider collisions of symmetric nuclei, each containing $A$ nucleons.", "We will study both participant and binary collision type initial conditions [52] using a Woods-Saxon distribution for each nuclei's transverse profile [53].", "For an individual nucleus we take the density to be $n_A(r) = \\frac{n_0}{1 + e^{(r-R)/d}} \\, ,$ where $n_0 = 0.17\\;{\\rm fm}^{-3}$ is the central nucleon density, $R = (1.12 A^{1/3} - 0.86 A^{-1/3})\\;{\\rm fm}$ is the nuclear radius, and $d = 0.54\\;{\\rm fm}$ is the “skin depth”.", "The density is normalized such that $\\lim _{A\\rightarrow \\infty } \\int d^3r \\, n_A(r) = A$ , where $A$ is the total number of nucleons in the nucleus.", "The normalization condition fixes $n_0$ to the value specified above.", "From the nucleon density we first construct the thickness function in the standard way by integrating over the longitudinal direction, i.e.", "$T_A(x,y) = \\int _{-\\infty }^{\\infty } dz \\, n_A(\\sqrt{x^2+y^2+z^2}) \\, .$ With this in hand we can construct the overlap density between two nuclei whose centers are separated by an impact parameter vector $\\vec{b}$ which we choose to point along the $\\hat{x}$ direction, i.e.", "$\\vec{b} = b \\hat{x}$ .", "We choose to locate the origin of our coordinate system to lie halfway between the center of the two nuclei such that the overlap density can be written as $n_{AB}(x,y,b) = T_A(x+b/2,y) T_B(x-b/2,y) \\, .$ Another quantity of interest is the participant density which is given by $n_{\\rm part}(x,y,b) &=& T_A(x+b/2,y) \\left[ 1- \\left(1-\\frac{\\sigma _{NN}\\,T_B(x-b/2,y)}{B}\\right)^{\\!\\!B} \\right]\\nonumber \\\\&& \\hspace{56.9055pt} + \\; T_B(x-b/2,y) \\left[ 1- \\left(1-\\frac{\\sigma _{NN}\\,T_A(x+b/2,y)}{A}\\right)^{\\!\\!A} \\right] \\, .$ For LHC collisions at $\\sqrt{s_{NN}}=2.76$ TeV we use $\\sigma _{NN}$ = 62 mb and for RHIC collisions at $\\sqrt{s_{NN}}=200$ GeV we use $\\sigma _{NN}$ = 42 mb.", "From the participant density we construct our first possible initial condition for the transverse energy density profile at central rapidity ${\\cal E}_0^{\\rm part} = {\\cal E}_0 \\, \\frac{n_{\\rm part}(x,y,b)}{n_{\\rm part}(0,0,0)} \\, ,$ where ${\\cal E}_0$ is the central energy density obtained in a central collision between the two nuclei.", "As an alternative initial condition for energy density one could use the number of binary collisions which is defined as $n_{\\rm coll}(x,y,b) = \\sigma _{NN} \\, n_{AB}(x,y,b)\\, .$ from which we obtain the binary collision energy scaling ${\\cal E}_0^{\\rm coll} = {\\cal E}_0 \\, \\frac{n_{\\rm coll}(x,y,b)}{n_{\\rm coll}(0,0,0)}= {\\cal E}_0 \\, \\frac{n_{AB}(x,y,b)}{n_{AB}(0,0,0)} \\, .$" ], [ "Numerical Methods", "We consider both smooth and fluctuating initial conditions using three numerical algorithms.", "In the following two subsections we describe the implementation of each algorithm.", "In each case detailed below the code is implemented using the C programming language." ], [ "Centered Differences Algorithm", "In the first algorithm which we will refer to as the “centered-differences algorithm” we solve Eqs.", "(REF ) and Eqs.", "(REF ) by first analytically solving for the individual proper-time derivatives of the four dynamical variables: $\\xi $ , $\\Lambda $ , $u_x$ , and $u_y$ using Mathematica [54].", "We then had Mathematica output, in C format, the necessary right hand sides of the four update equations.", "We then discretize space on a regular square lattice with lattice spacing, $\\Delta x = a$ .", "For the spatial derivatives we use centered differences except on the edges of the lattice where we apply either a left- or right-handed first order derivative.", "For the temporal updates we use fourth-order Runge-Kutta (RK4) with a step size of $\\Delta t = \\epsilon $ .", "For smooth initial conditions the previous method suffices; however, for fluctuating initial conditions one finds that using centered differences introduces spurious oscillations in regions where there are large gradients.", "In order to damp these oscillations one could attempt to use a two-dimensional Lax-Friedrichs (LAX) update [55], [56].", "In practice this amounts to replacing the current value of a given dynamical variable by a local spatial average over neighboring sites and using this as a stand in for the current value of the variable, e.g.", "$\\xi _{\\rm LAX}(\\tau ,x,y) = \\left[ \\xi (\\tau ,x+a,y)+\\xi (\\tau ,x-a,y)+\\xi (\\tau ,x,y+a)+\\xi (\\tau ,y-a) \\right] /4,$ and now the $\\xi $ update for a temporal step of size $\\epsilon $ becomes schematically $\\xi (\\tau +\\epsilon ,x,y) = \\xi _{\\rm LAX}(\\tau ,x,y) + \\epsilon \\, {\\rm RHS}_\\xi (\\tau ,x,y) \\, ,$ where ${\\rm RHS}_\\xi $ stands for the (rather complicated) right hand size of the $\\xi $ update equation.", "However, such a scheme results in too much numerical dissipation.", "An alternative is to realize that the source of the spurious oscillations is the weak coupling between odd- and even-number lattice sites.", "The full LAX scheme above maximally couples these interleaving lattices; however, this need not be done.", "Instead one can weight the LAX-smoothed values with a weight $\\lambda $ and combine this with the current value of the variable in question, e.g.", "$\\xi _{\\rm wLAX}(\\tau ,x,y) = \\lambda \\xi _{\\rm LAX}(\\tau ,x,y) + (1-\\lambda ) \\xi (\\tau ,x,y) \\, .$ The smaller the value of $\\lambda $ , the less the numerical viscosity.", "In practice, we have found that for the aHydro equations one should take $\\lambda >0.02$ in order to achieve numerical stability.", "In the results section below we use $\\lambda = 0.05$ which represents a factor of twenty decrease in the dissipation induced by LAX-smoothing.", "Note that, when activated, wLAX smoothing is implemented for all dynamical variables ($\\Lambda $ , $\\xi $ , $u_x$ , and $u_y$ ) after each full time step of $\\epsilon $ and not within each RK4 substep.", "We will only need to use the wLAX method for fluctuating initial conditions; however, in App.", "we present numerical tests using it in the smooth initial conditions case in order to show that the amount of numerical viscosity in the wLAX case is not numerically significant.", "That being said, one would also like to have another method for handling the spurious oscillations caused by using higher-order centered differences.", "This has motivated us to also implement the Kurganov-Tadmor central scheme which we describe in the next subsection." ], [ "The MUSCL Algorithm", "As mentioned above, when there are large gradients present in a hyperbolic partial differential equation, the application of straightforward centered-differences scheme can lead to spurious oscillations.", "For smooth initial conditions and finite shear viscosity this is not an issue; however, for fluctuating initial conditions one needs a way to handle shocks and discontinuities.", "One way to proceed is to implement the LAX method as described previously; however, the LAX method introduces numerical viscosity into the algorithm which scales like the $(\\Delta x)^2/\\Delta t$ so that it is not possible to take the temporal step size to zero without having extremely small lattice spacing to reduce the numerical viscosity.", "As discussed above one can reduce the amount of numerical viscosity by instead using the weighted LAX (wLAX) prescription described above; however, it is desirable to have an alternative algorithm in order to be sure of the results.", "For this purpose we have also implemented a “Monotone Upstream-Centered Schemes for Conservation Laws” (MUSCL) scheme derived by Kurganov and Tadmor [57] which has been extended to include nonlinear sources [58].", "This method is particularly appealing because it can be shown that, although it does induce some numerical viscosity, the magnitude of the numerical viscosity induced scales like as a power of the lattice spacing with no power of the temporal step size in the denominator allowing one to take extremely small time steps without inducing large artificial numerical viscosity.", "Our implementation closely follows that introduced by Schenke et al.", "[59] to solve three-dimensional relativistic ideal hydrodynamics equations.", "They have also extended the method to 2nd-order three-dimensional relativistic viscous hydrodynamics [20], [21] with fluctuating initial conditions.", "To explain the algorithm let us consider the simpler case of a one dimensional system of hyperbolic partial differential equations which can be cast into “conservative” form, i.e.", "$\\partial _t u + F_x(u) = 0 \\, ,$ where $u$ is, in general, an n-dimensional vector, $F$ is a so-called flux variable or flux function, and $F_x(u) = \\partial _x F(u)$ .", "For example, if one were solving the advection equation, $\\partial _t u+ \\partial _x u= 0$ then we would have $F = u$ and if one were solving Burgers' equation $\\partial _t u + u \\partial _x u = 0$ this can be written in conservative form as $\\partial _t u + \\partial _x (u^2/2) = 0$ so that, in this case, $F = u^2/2$ .", "Given a partial differential equation of the form (REF ) Kurganov and Tadmor derived the following semi-discrete update equation $\\frac{d u_j}{d t} = - \\frac{H_{j+1/2}(t) - H_{j-1/2}(t)}{\\Delta x} \\, ,$ where the numerical flux function $H$ is given by $H_{j+1/2}(t) \\equiv \\frac{ F\\left(u^+_{j+1/2}(t)\\right) + F\\left(u^-_{j+1/2}(t)\\right) }{2}- \\frac{a^x_{j+1/2}(t)}{2} \\left[u^+_{j+1/2}(t) - u^-_{j+1/2}(t)\\right] \\, ,$ with $a^x_{j+1/2}(t)$ being the local propagation velocity in the $x$ -direction which is given by the maximum of the left and right half-site extrapolated spectral radius of $\\partial F/\\partial u$ which is defined as $\\rho $ $a^x_{j+1/2}(t) \\equiv {\\rm max} \\!", "\\left\\lbrace \\rho \\!\\left(\\frac{\\partial F}{\\partial u}\\left(u^+_{j+1/2}(t)\\right) \\right) , \\,\\rho \\!\\left(\\frac{\\partial F}{\\partial u}\\left(u^-_{j+1/2}(t)\\right) \\right)\\right\\rbrace \\, ,$ and finally, the half-site extrapolated intermediate values $u^\\pm _{j+1/2}$ are given by $u^+_{j+1/2} &\\equiv & u_{j+1}(t) - \\frac{\\Delta x}{2} (u_x)_{j+1}(t) \\, , \\nonumber \\\\u^-_{j+1/2} &\\equiv & u_j(t) + \\frac{\\Delta x}{2} (u_x)_j(t) \\, .$ For the derivatives, $u_x$ , appearing in (REF ) one should use a total variation diminishing “flux-limiter” so that spurious oscillators are avoided [60].", "We follow the original paper of Kurganov and Tadmor and use the three-argument minmod flux-limiter [61]: $(u_x)_j = {\\rm minmod}\\!\\left(\\theta \\frac{u_j - u_{j-1}}{\\Delta x} ,\\frac{u_{j+1}- u_{j-1}}{2 \\Delta x} ,\\frac{u_{j+1} - u_j}{\\Delta x} \\right), \\;\\;\\; 1 \\le \\theta \\le 2 \\, ,$ where $\\hbox{minmod}(x_1, x_2, \\cdots ) = \\left\\lbrace \\begin{array}{ll}\\hbox{min}_j\\lbrace x_j\\rbrace ,& \\hbox{if $x_j> 0$ $\\forall \\, j$}\\\\\\hbox{max}_j\\lbrace x_j\\rbrace ,& \\hbox{if $x_j< 0$ $\\forall \\, j$}\\\\0 & \\hbox{otherwise} \\;\\;\\;\\;\\;\\; .\\end{array}\\right.$ The value of $\\theta $ controls the dissipation of the flux limiter with $\\theta =1$ being the most dissipative and $\\theta =2$ being the least.", "In this paper we follow [59] and use $\\theta = 1.1$ .", "For details of the derivation of the Kurganov-Tadmor scheme we refer the reader to their original paper [57].", "As mentioned above one can extend the Kurganov-Tadmor scheme to accommodate nonlinear time-dependent sources.", "Including the possibility of a time-dependent source changes our one-dimensional example to $\\partial _t u + F_x(u) = J(t,u) \\, ,$ where $J$ is a source term.", "Naidoo and Baboolal [58] demonstrated that, in this case, only a simple modification of adding the source on the right hand side was necessary $\\frac{d u_j}{d t} = - \\frac{H_{j+1/2}(t) - H_{j-1/2}(t)}{\\Delta x} + J(t,u_j) \\, ,$ We note that to extend the method described thus far to multiple dimensions one introduces flux functions for each direction, e.g.", "$F_y$ and $F_z$ , and includes these in the update rule by defining new numerical flux functions (REF ) and propagation velocities (REF ) accordingly." ], [ "Applying MUSCL to ", "In the case of aHydro all of the evolution equations stem from conservative systems with sources, therefore we can apply the general method just described.", "For this purpose we need the first and second moments of the Boltzmann equation with the RS form for the one-particle distribution function.", "The zeroth moment can be written in a conservative form with sources in $\\tau $ -$\\varsigma $ coordinates as follows $\\partial _\\tau j^\\tau + \\nabla _\\perp \\cdot {\\bf j}^\\perp = -\\frac{j^\\tau }{\\tau } + J_0 \\, ,$ where $j^\\mu = n \\, u^\\mu $ is the particle four-current and $J_0 \\equiv \\Gamma \\, n_{\\rm iso}(\\Lambda ) \\left[ \\frac{1}{\\sqrt{1+\\xi }} - {\\cal R}^{3/4}(\\xi ) \\right] \\, ,$ is the zeroth-moment of the right-hand side of the Boltzmann equation in the relaxation time approximation used herein.", "The remaining three update equations necessary can be obtained from energy-momentum conservation, $\\partial _\\mu T^{\\mu \\nu } =0$ , giving $\\partial _\\tau T^{\\tau \\tau } + \\partial _x T^{\\tau x} + \\partial _y T^{\\tau y}&=&-\\frac{1}{\\tau } \\left[ T^{\\tau \\tau } + \\tau ^2 T^{\\varsigma \\varsigma }\\right] \\, , \\\\\\partial _\\tau T^{\\tau x} + \\partial _x T^{x x} + \\partial _y T^{x y}&=&-\\frac{T^{\\tau x}}{\\tau } \\, , \\\\\\partial _\\tau T^{\\tau y} + \\partial _x T^{x y} + \\partial _y T^{y y}&=&-\\frac{T^{\\tau y}}{\\tau } \\, .$ Once the dynamical variables $j^\\tau $ , $T^{\\tau \\tau }$ , $T^{\\tau x}$ , $T^{\\tau y}$ are updated via these equations, they can then can be used to construct the remaining components of $j^\\mu $ and $T^{\\mu \\nu }$ .", "In our case it is necessary to solve two simultaneous nonlinear equations for $\\xi $ and $\\Lambda $ which will then allow us to determine the rest of the information necessary to proceed with the solution.", "To see how this works in practice, we first use (REF ) to write the non-vanishing components of $T^{\\mu \\nu }$ and $j^\\mu = n \\, u^\\mu $ explicitly $T^{\\tau \\tau } &=& ({\\cal E}+{\\cal P}_\\perp ) u^0 u^0 - {\\cal P}_\\perp \\, , \\\\T^{\\tau i} &=& ({\\cal E}+{\\cal P}_\\perp ) u^0 u^i \\, , \\\\T^{ij} &=& ({\\cal E}+{\\cal P}_\\perp ) u^i u^j \\, , \\\\T^{ii} &=& ({\\cal E}+{\\cal P}_\\perp ) u^i u^i + {\\cal P}_\\perp \\, , \\\\T^{\\varsigma \\varsigma } &=& {\\cal P}_L/\\tau ^2 \\, , \\\\j^\\tau &=& n \\, u^0 \\, , \\\\j^i &=& n \\, u^i \\, ,$ where $i \\in \\lbrace x,y\\rbrace $ .", "Using these equations and the normalization condition $u_\\tau ^2 = 1 + u_x^2 + u_y^2$ one finds two nonlinear equations, similar to those obtained in Ref.", "[59], ${\\cal E}(\\Lambda ,\\xi ) = T^{\\tau \\tau } - \\frac{ (T^{\\tau x})^2 + (T^{\\tau y})^2 }{T^{\\tau \\tau }+ {\\cal P}_\\perp (\\Lambda ,\\xi )} \\, ,$ and $j^\\tau = n(\\Lambda ,\\xi ) \\left[ \\frac{T^{\\tau \\tau } + {\\cal P}_\\perp (\\Lambda ,\\xi )}{{\\cal E}(\\Lambda ,\\xi )+ {\\cal P}_\\perp (\\Lambda ,\\xi )} \\right] .$ From these two equations one can numerically solve for $\\Lambda $ and $\\xi $ .", "These values can then be used to determine $u^\\tau $ and $u^i$ via $u^\\tau &=& \\frac{j^\\tau }{n(\\Lambda ,\\xi )} \\, , \\\\u^i &=& \\frac{n(\\Lambda ,\\xi ) \\, T^{\\tau i} }{j^\\tau \\, [{\\cal E}(\\Lambda ,\\xi ) + {\\cal P}_\\perp (\\Lambda ,\\xi )]} \\, .$ Once determined, these components of the four-velocity together with the values of $\\Lambda $ and $\\xi $ can be used to determine all remaining variables in (REF ).", "The only remaining ingredient necessary for the Kurganov-Tadmor algorithm to be implemented fully is to determine the local propagation velocities $a^i_{j+1/2}(t)$ .", "These are obtained by evaluating the eigenvalues of the $4\\times 4$ Jacobian of $j^\\tau $ , $T^{\\tau \\tau }$ , $T^{\\tau x}$ , $T^{\\tau y}$ .", "As was the case in Ref.", "[59], with some work and a little bit of help from Mathematica, one finds that two of the four eigenvalues are degenerate and equal to $u^i/u^\\tau $ and the other two are given by $\\lambda ^\\pm _i = \\frac{A\\pm \\sqrt{B}}{D} \\, ,$ with $A &=& u^\\tau u^i ( 1 - v^2 ) \\, , \\\\B &=& \\left[u_\\tau ^2 - u_i^2 - (u_\\tau ^2-u_i^2-1) v^2 \\right] v^2 \\, , \\\\D &=& u_\\tau ^2 - (u_\\tau ^2-1) v^2 \\, ,$ and $v^2 = \\frac{\\partial {\\cal P}_\\perp }{\\partial {\\cal E}}+ \\frac{n}{{\\cal E}+{\\cal P}_\\perp } \\frac{\\partial {\\cal P}_\\perp }{\\partial n} \\, .$ Using an ideal equation of state for which ${\\cal E}_{\\rm iso} = 3 {\\cal P}_{\\rm iso}$ one obtains $v^2(\\xi ) = \\frac{1}{3} \\frac{2 {\\cal R}_\\perp (\\xi ) + 3 (1+\\xi ) {\\cal R}_\\perp ^\\prime (\\xi )}{2 {\\cal R}(\\xi ) + 3 (1+\\xi ) {\\cal R}^\\prime (\\xi )}+ \\frac{4(1+\\xi )}{3{\\cal R}(\\xi ) + {\\cal R}_\\perp (\\xi )}\\frac{{\\cal R}^\\prime (\\xi ) {\\cal R}_\\perp (\\xi ) - {\\cal R}(\\xi ) {\\cal R}_\\perp ^\\prime (\\xi )}{2 {\\cal R}(\\xi ) + 3 (1+\\xi ) {\\cal R}^\\prime (\\xi )} \\, .$ In this function both terms individually diverge in the limit that $\\xi \\rightarrow 0$ , however, these divergences cancel to give a finite result of $\\lim _{\\xi \\rightarrow 0} v^2 = 2/5$ .", "It has other limits of $\\lim _{\\xi \\rightarrow -1} v^2 = 0$ and $\\lim _{\\xi \\rightarrow \\infty } v^2 = 1/2$ .", "Using the now known eigenvalues one finds that the maximum value of the four eigenvalues is given by $\\rho = |{\\rm max}(\\lambda _i)| = \\frac{|A| + \\sqrt{B}}{D} \\, .$ Using the above scheme one can evolve the aHydro system with fluctuating initial conditions; however, there is a caveat, namely that the linearly interpolated intermediate values of $j^\\tau $ , $T^{\\tau \\tau }$ , $T^{\\tau x}$ , and $T^{\\tau y}$ determined via (REF ) may not have real-valued solutions for $\\Lambda $ and $\\xi $ using Eqs.", "(REF ) and (REF ).", "In practice, we find that it is necessary to use extremely fine lattices in order to ameliorate this problem.", "Alternatively, we have found that instead of extrapolating the four variables $j^\\tau $ , $T^{\\tau \\tau }$ , $T^{\\tau x}$ , and $T^{\\tau y}$ to the half-sites, one can instead extrapolate the current values of $\\Lambda $ and $\\xi $ to the half-sites for use in evaluating the flux functions.", "In addition, we have found that in practice it is necessary to use a “hybrid” algorithm in which the centered-differences scheme described in the previous subsection is used as the initial guess for the nonlinear root finder which solves Eqs.", "(REF ) and (REF ).", "This is necessary, in particular, in regions where $\\xi \\simeq 0$ since Eqs.", "(REF ) and (REF ) have two solutions which become very close together causing the root finder to oscillate between the two solutions.", "The predicted value from the centered-differences scheme predicts for the nonlinear root finder which solution to use in this case.", "We will refer to this method as “Hybrid Kurganov-Tadmor”." ], [ "Results", "In this section we present results for the time evolution of the matter generated in heavy ion collisions at LHC energies using the aHydro evolution equations (REF ) and (REF ).", "For the results presented here we assume a ideal gas of quarks and gluons with $N_f=2$ so that there are $N_{\\rm dof}= 37$ degrees of freedom.", "For our numerical tests and results we will concentrate on the spatial and momentum-space ellipticities, $\\epsilon _x$ $\\epsilon _x= \\frac{<\\!y^2 - x^2\\!>_{\\cal E}}{<\\!x^2 + y^2\\!>_{\\cal E}} \\, ,$ and $\\epsilon _p$ is defined in the lab frame via $\\epsilon _p = \\frac{<\\!T^{xx} - T^{yy}\\!>}{<\\!T^{xx} + T^{yy}\\!>} \\, ,$ where $<\\!\\!x^2\\!\\!>_{\\cal E}$ and $<\\!\\!y^2\\!\\!>_{\\cal E}$ are the proper-time dependent average values of $x^2$ and $y^2$ weighted by the energy density $<\\!x^2\\!>_{\\cal E} \\, \\equiv \\, {\\cal N} \\int _{x,y} x^2 {\\cal E}(\\tau ,x,y) \\, ,$ and the averages in the momentum-space ellipticity represent unweighted integrals over the transverse directions.", "Note that the normalization ${\\cal N}$ is arbitrary since it cancels in the ratio we are computing.", "These definitions are the conventional ones from the literature [62] which, unfortunately, are slightly inconsistent since $\\epsilon _x$ is defined in the local rest frame and $\\epsilon _p$ in the lab frame.", "It would be more consistent to weight the spatial average by $T^{\\tau \\tau }$ ; however, to be consistent with the existing literature we will use the definition weighted with the energy density in the local rest frame.", "Figure: (Color online) Comparison of aHydro isotropic temperature and flow profiles with 2nd-order viscous hydrodynamics code for4πη/𝒮=0.14 \\pi \\eta /{\\cal S}=0.1 and b=7b=7 fm.", "Lattice size used was 109×109109\\times 109 with a=0.394a=0.394 fm, ϵ=0.01\\epsilon =0.01 fm/c,τ 0 =0.25\\tau _0 = 0.25 fm/c, Λ 0 =\\Lambda _0 = 600 MeV, and ξ 0 =0\\xi _0 = 0.", "For the transverse profile Glauber binary collision scalingwas used.Figure: (Color online) Comparison of aHydro isotropic temperature and flow profiles with 2nd-order viscous hydrodynamics code for4πη/𝒮=104 \\pi \\eta /{\\cal S}=10 and b=7b=7 fm.", "Lattice size used was 109×109109\\times 109 with a=0.394a=0.394 fm, ϵ=0.01\\epsilon =0.01 fm/c,τ 0 =0.25\\tau _0 = 0.25 fm/c, Λ 0 =\\Lambda _0 = 600 MeV, and ξ 0 =0\\xi _0 = 0.", "For the transverse profile Glauber binary collision scalingwas used.Figure: (Color online) Spatial and momentum eccentricities as a function of proper time for (a) a Glauber wounded-nucleontransverse profile and (b) a Glauber binary-collision transverse profile with b=7b=7 fm, Λ 0 =T 0 =0.6\\Lambda _0=T_0=0.6 GeV,ξ 0 =0\\xi _0 = 0, and u ⊥,0 =0u_{\\perp ,0} = 0 at τ 0 =\\tau _0 = 0.25 fm/c.", "For the 4πη/𝒮=14\\pi \\eta /{\\cal S} = 1 runwe used Λ 0 =T 0 =0.6\\Lambda _0=T_0=0.6 GeV and for the 4πη/𝒮=104\\pi \\eta /{\\cal S} = 10 run we usedΛ 0 =T 0 =0.576\\Lambda _0=T_0=0.576 GeV for wounded-nucleon initial conditions and Λ 0 =T 0 =0.584\\Lambda _0=T_0=0.584 for binary-collisioninitial conditions.", "These adjustments were made in order to guarantee the same final particle number.In all cases we used the centered-differences algorithm with a lattice size of100 ×\\times 100, a lattice spacing of a=0.4a=0.4 fm, and a RK4 temporal step size ϵ=0.01\\epsilon =0.01 fm/c.Figure: (Color online) Visualization of the isotropic temperature and pressure anisotropy at three different times after thenuclear impact.", "For these plots we assumed a non-central collision with b=7b=7 fm, an isotropic Glauber wounded-nucleonprofile, and a b=0b=0 fm central temperature of 0.6 GeV at 0.25 fm/c.", "For this plot we used a value of 4πη/𝒮=14\\pi \\eta /{\\cal S} = 1and a lattice size of 200×200200\\times 200 with a lattice spacing of a=0.2a=0.2 fm and a RK4 temporal step size of ϵ=0.01\\epsilon =0.01fm/c.Figure: (Color online) Visualization of the isotropic temperature and pressure anisotropy at three different times after thenuclear impact.", "For these plots we assumed a non-central collision with b=7b=7 fm, an isotropic Glauber wounded-nucleon profile, and a b=0b=0 fm central temperature of 0.6 GeV at 0.25 fm/c.", "For this plot we used a value of 4πη/𝒮=104\\pi \\eta /{\\cal S} = 10 and a lattice size of 200×200200\\times 200 with a lattice spacing of a=0.2a=0.2 fm and a RK4 temporal step size of ϵ=0.01\\epsilon =0.01fm/c.We concentrate on the ellipticities since, as we will see, large momentum-space anisotropies are developed during the evolution of the system.", "Such large momentum-space anisotropies cast doubt on the naive application of Cooper-Frye [63] and linearly-corrected Cooper-Frye [64].", "We, therefore, postpone the implementation of freeze out until we can allow for large momentum-space anisotropies and, in the meantime, focus on quantities that are independent of the freeze-out prescription." ], [ "Smooth Initial Conditions", "We begin by presenting results using smooth initial conditions.", "For numerical tests of the various algorithms we refer the reader to App. .", "Therein we show scalings with lattice spacing, box size, and comparisons of the different algorithms employed for both smooth and fluctuating initial conditions.", "In order to demonstrate that aHydro reproduces known 2nd-order viscous hydrodynamics results, in Figs.", "REF and REF we compare the results of an aHydro run with results obtained using the latest version of the code of Romatschke and Luzum [12].", "In Fig.", "REF we assumed $4 \\pi \\eta /{\\cal S} = 0.1$ and in Fig.", "REF we assumed $4 \\pi \\eta /{\\cal S} = 10$ .", "In both cases we show the isotropic temperature profile, $T_{\\rm iso} ={\\cal R}^{1/4}(\\xi ) {\\cal E}_{\\rm iso}(\\Lambda )$ , in the left panel and the ratio of the y-component of the four velocity to the $\\tau $ -component in the right column.", "As can be seen from Fig.", "REF there are only small differences at large radii in the case that the shear viscosity to entropy ratio is small.", "This demonstrates that our code reproduces 2nd-order viscous hydrodynamics in the limit of small $\\eta /{\\cal S}$ .", "Fig.", "REF shows the case of large shear viscosity to entropy ratio.", "In this case we see only small deviations in the temperature profiles and substantial differences in the flow profiles.", "We therefore expect the aHydro and 2nd-order viscous hydrodynamics frameworks to give different flow observables for large $\\eta /{\\cal S}$ .", "We note that corrections near the edges are expected even for small values of $\\eta /{\\cal S}$ and that the relative magnitude of the aHydro flow and the viscous hydrodynamics flow is to be expected: since aHydro generates larger longitudinal pressure than viscous hydrodynamics one expects diminished radial flow.", "This pattern is also observed in simulations which use the lattice-boltzmann method [65].", "In Fig.", "REF a and REF b we compare the spatial and transverse momentum-space eccentricities as a function of proper time assuming two different values of the shear viscosity to entropy density ratio corresponding to typical strong-coupling ($4\\pi \\eta /{\\cal S} = 1$ ) and weak-coupling ($4\\pi \\eta /{\\cal S} = 10$ ) values.", "In Fig.", "REF a we used smooth Glauber wounded-nucleon initial conditions and in Fig.", "REF b we used smooth Glauber binary collision initial conditions.", "In both figures we assumed $b=7$ fm, $\\Lambda _0=T_0=0.6$ GeV, $\\xi _0 =0$ , and $u_{\\perp ,0} = 0$ at $\\tau _0 = $ 0.25 fm/c and used the centered-differences algorithm with a lattice size of 100 $\\times $ 100, a lattice spacing of $a=0.4$ fm, and a temporal step size of $\\epsilon =0.01$ fm/c.", "In both cases RK4 with a temporal step size of $\\epsilon =0.01$ fm/c was used for the updates.", "As can be seen from these figures increasing the shear viscosity to entropy ratio by a factor of ten only decreases the momentum-space eccentricity $\\epsilon _p$ at 5 fm/c by approximately 10% in both cases shown.", "We note, however, that the dynamical framework employed here, namely assuming that the local rest frame energy momentum tensor is azimuthally symmetric in momentum-space may underestimate the full effect of the shear viscosity.", "In Figs.", "REF and REF we present visualizations in the form of colormaps with contours of the proper-time dependence of the isotropic temperature and the pressure anisotropy defined by the ratio of the longitudinal and transverse pressures.", "Fig.", "REF shows the case of $4 \\pi \\eta /S =1$ and Fig.", "REF shows the case of $4 \\pi \\eta /S =10$ .", "In both cases we assumed a non-central collision with $b=7$ fm, a Glauber wounded-nucleon profile, and a $b=0$ fm central temperature of $\\Lambda _0 = T_0 = $ 0.6 GeV at $\\tau _0 = $ 0.25 fm/c.", "A lattice size of $200\\times 200$ with a lattice spacing of $a=0.2$ fm and a RK4 temporal step size of $\\epsilon =0.01$ fm/c was used in both cases.", "As we can see from this figure the magnitude of the momentum-space anisotropies can be large in the center of the fireball and grows towards the edges.", "In Fig.", "REF we see that assuming $4 \\pi \\eta /S =1$ at $\\tau $ = 1.5 fm/c the center still has a 25% momentum-space anisotropy and assuming $4 \\pi \\eta /S =10$ (Fig.", "REF ) one finds approximately 85% momentum-space anisotropy at $\\tau $ = 1.5 fm/c.", "In fact, in the case of $4 \\pi \\eta /S =10$ the system is highly anisotropic during the entire evolution.", "For such large shear viscosities the aHydro framework provides a dynamical framework which should be more reliable than the naive application of 2nd-order viscous hydrodynamics.", "Figure: (Color online) Momentum eccentricity at the freeze-out time as a function of impact for an isotropic Glauber wounded-nucleontransverse profile with ξ 0 =0\\xi _0 = 0, and u ⊥,0 =0u_{\\perp ,0} = 0 at τ 0 =\\tau _0 = 0.25 fm/c assumingT f =0.15T_f = 0.15 GeV.", "For the 4πη/𝒮=14\\pi \\eta /{\\cal S} = 1 runwe used Λ 0 =T 0 =0.6\\Lambda _0=T_0=0.6 GeV as the central temperature and for the 4πη/𝒮=104\\pi \\eta /{\\cal S} = 10 run we usedΛ 0 =T 0 =0.576\\Lambda _0=T_0=0.576 GeV in order to guarantee the same final particle number.We used the centered-differences algorithm with a lattice size of200 ×\\times 200, a lattice spacing of a=0.2a=0.2 fm, and aRK4 temporal step size ϵ=0.01\\epsilon =0.01 fm/c.In Fig.", "REF we plot the momentum space eccentricity, $\\epsilon _p$ , at the “freeze-out time” $\\tau _f$ as a function of the assumed impact parameter, $b$ .", "For this figure we used a Glauber wounded-nucleon transverse profile with $\\xi _0 = 0$ , and $u_{\\perp ,0} = 0$ at $\\tau _0 = $ 0.25 fm/c assuming $4\\pi \\eta /{\\cal S} = 1$ and $4\\pi \\eta /{\\cal S} = 10$ and a freeze-out temperature of $T_f = 0.15$ GeV.", "For the $4\\pi \\eta /{\\cal S} = 1$ run we used $\\Lambda _0=T_0=0.6$ GeV as the central temperature and for the $4\\pi \\eta /{\\cal S} = 10$ run we used $\\Lambda _0=T_0=0.576$ GeV in order to guarantee the same final particle number.", "We used the centered-differences algorithm with a lattice size of 200 $\\times $ 200, a lattice spacing of $a=0.2$ fm, and a RK4 temporal step size $\\epsilon =0.01$ fm/c.", "The freeze-out time $\\tau _f$ was determined by finding the time at which the maximum isotropic temperature $T_{\\rm iso}$ dropped below the freeze-out temperature of $T_f = 0.15$ GeV.", "This figure shows that changing the assumed value of the shear viscosity to entropy ratio from one to ten only makes a difference of   8% in the peak value of the momentum-space ellipticity.", "We should note, as a caveat which we will emphasize again in the conclusions, that because we assume that the energy momentum tensor is azimuthally symmetric in the local rest frame this places us somewhere between a full blown viscous hydrodynamical calculation and ideal hydrodynamics.", "Therefore, firm conclusions will have to wait until results with a completely general ellipsoidal energy-momentum tensor are available.", "Figure: (Color online) Visualization of the isotropic temperature and pressure anisotropy at three different times after thenuclear impact.", "For these plots we assumed a collision centrality of b=7b=7 fm with a sampled Monte-Carlo Glauber wounded-nucleon profile and an isotropic temperature of T=0.6T=0.6 GeV at 0.25 fm/c.", "For this plot we used a value of4πη/𝒮=14\\pi \\eta /{\\cal S} = 1.", "We used a lattice size of 200×200200\\times 200 with a lattice spacing of a=0.2a=0.2 fm and a RK4 temporal step size of ϵ=0.01\\epsilon =0.01 fm/c." ], [ "Fluctuating Initial Conditions", "For our fluctuating initial condition case we have implemented Monte-Carlo (MC) Glauber initial conditions [66].", "At a given impact parameter $b$ we statistically sample a Woods-Saxon distribution to determine the position of the nucleons in each colliding nuclei.", "We then compute the transverse distance between each pair of nucleons from nuclei A and B and assume that they collide if the transverse distance between the centers of the nucleons being compared is less than $d \\equiv \\sqrt{\\sigma _{NN}/\\pi }$ .", "If a collision is deemed to have occurred a two dimensional gaussian with width $\\sigma _0 = 0.46$ fm is added to the energy density.", "We then adjust the overall scale to match the smooth Glauber model results.", "In Fig.", "REF we present visualizations in the form of colormaps with contours of the proper-time dependence of the isotropic temperature and the pressure anisotropy defined by the ratio of the longitudinal and transverse pressures.", "In Fig.", "REF we assumed a central collision $b=7$ fm with a sampled Monte-Carlo Glauber wounded-nucleon profile, an isotropic temperature of $\\Lambda _0=T_0=0.6$ GeV at $\\tau _0=$ 0.25 fm/c, and $4\\pi \\eta /{\\cal S} = 1$ .", "We used a lattice size of $200\\times 200$ with a lattice spacing of $a=0.2$ fm and a RK4 temporal step size of $\\epsilon =0.01$ fm/c.", "As can be seen from this figure, fluctuations can induce large momentum-space anisotropies, particularly in regions where the initial temperature is lower and therefore the relaxation rate is smaller.", "In a 2nd-order viscous hydrodynamical approach one would have many “spots” with very large momentum-space anisotropies.", "Note that Fig.", "REF shows the case $4\\pi \\eta /{\\cal S} = 1$ and we do not include a similar figure for the case of $4\\pi \\eta /{\\cal S} = 10$ ; however, we note that similarly to the case of smooth initial conditions, for this large value of the shear viscosity to entropy ratio, one sees large persistent momentum-space anisotropies throughout the simulated region." ], [ "Conclusions", "In this paper we studied the application of anisotropic hydrodynamics to the evolution of the matter created in relativistic heavy ion collisions.", "We began by specifying a tensor basis for the energy-momentum tensor which was applicable when the system is azimuthally symmetric such that one has energy density, transverse pressure, and longitudinal pressure along the diagonal in the local rest frame.", "Microscopically we were able to demonstrate that if one assumes local momentum-space azimuthal symmetry, it suffices to introduce one scale $\\Lambda $ and an anisotropy parameter, $\\xi $ , which controls the transverse-longitudinal momentum-space anisotropy.", "We then used these results in the computation of moments of the Boltzmann equation.", "Using the zeroth and first moments of the Boltzmann equation we were able to determine dynamical equations for the plasma scale, $\\Lambda $ , anisotropy parameter, $\\xi $ , and the transverse flow components $u_x$ and $u_y$ .", "In order to solve the resulting partial differential equations we implemented three differencing schemes: centered differences, weighted LAX, and hybrid Kurganov-Tadmor.", "The first method is suitable for smooth initial conditions whereas the second two are required when one considers event-by-event simulations.", "Based on our analysis and benchmarks we find the weighted LAX scheme to be faster than the hybrid Kurganov-Tadmor scheme with both giving the same results within controllable numerical errors.", "We showed through explicit solution of the resulting partial differential equations that the pressure components remain positive definite and that plasma momentum-space anisotropies grow larger as one approaches the transverse edge.", "In addition, we studied fluctuating initial conditions and demonstrated that fluctuations can result in regions of high momentum-space anisotropy in the center of the simulated matter.", "As a cross check we demonstrated that in the limit of small $\\eta /{\\cal S}$ the solution of the aHydro dynamical equations reproduces results from publicly available 2nd-order viscous hydrodynamics codes.", "For smooth initial conditions we demonstrated that, subject to the assumption of momentum-space azimuthal symmetry in the local rest frame, one sees a relatively small variation of the final lab frame momentum-space eccentricity $\\epsilon _p$ as $\\eta /{\\cal S}$ is increased.", "Drawing quantitative conclusions from the results contained herein might be premature, however, since the impact of relaxing the assumption of azimuthal isotropy of the energy momentum tensor in the local rest frame is unknown.", "Removing this assumption will result in what we will term “ellipsoidal” anisotropic hydrodynamics.", "Work in this direction is currently underway.", "We note in closing that there have been a number of authors studying the behavior of anisotropic plasmas in strongly coupled gauge theories [67], [68], [69], [70], [71], [34], [36], [72], [73], [74].", "The aHydro framework agrees extremely well with existing 1st, 2nd, and 3rd order viscous hydrodynamical results which have been computed analytically for strongly-coupled ${\\cal N}=4$ supersymmetric Yang-Mills [75].", "It would be interesting to see if any of the results contained herein could be used in the context of strongly-coupled theories in order to develop useful phenomenological models.", "One open question first raised in Ref.", "[74] concerns whether or not the breaking of rotational symmetry in momentum-space requires the introduction of transverse and longitudinal transport coefficients.", "Mathematically this would seem to be the case in our formalism if one linearizes fluctuations around an anisotropic background.", "Such possibilities will be explored in the future.", "In the meantime, the progress made here opens up the possibility for phenomenological application to heavy ion observables such as collective flow, photon and dilepton production, quarkonium screening, jet energy loss, etc.", "in the presence of large momentum-space anisotropies." ], [ "Acknowledgements", "We thank Gabriel Denicol, Wojciech Florkowski, Sangjong Jeon, Harri Niemi, and Björn Schenke for useful conversations during the preparation of this work.", "M. M. and M. S. thank the H. Niewodniczański Institute of Nuclear Physics and the Frankfurt Institute of Advanced Studies where part of this work was done.", "M.S.", "also thanks the Institute for Nuclear Theory at University of Washington for allowing him to participate in the INT program “Gauge Field Dynamics In and Out of Equilibrium” where the final stages of this work were completed.", "M.S.", "was supported by NSF grant No.", "PHY-1068765 and the Helmholtz International Center for FAIR LOEWE program.", "M. M. was supported by Ministerio de Ciencia e Innovacion of Spain under project FPA2009-06867-E." ], [ "Particle production in the (0+1)-dimensional case", "In this appendix we discuss the issue of particle production in 2nd-order viscous hydrodynamics vs anisotropic hydrodynamics.", "To begin we note that there are two limits in which one expects particle production to go to zero: (a) the limit of ideal hydrodynamics and (b) the free-streaming limit.", "For small but non-vanishing shear viscosity we expect there to be additional particles associated with dissipation; however, as the shear viscosity to entropy ratio increases we should see a maximum in the particle production since it will eventually have to go to zero in the free-streaming limit.", "In contrast, second-order viscous hydrodynamics predicts that the excess in particle production is a monotonically increasing function of the assumed value of $\\eta /{\\cal S}$ .", "Figure: (Color online) Total particle number at τ=τ f \\tau = \\tau _f as a function of the assumed value of the shear viscosity toentropy ratio.", "For this figure we ignored transverse expansion making the system effectively (0+1)-dimensionaland we used initial values of Λ 0 =0.6\\Lambda _0=0.6 GeV and ξ 0 =0\\xi _0 = 0 at τ 0 =0.25\\tau _0=0.25 fm/c.In order to demonstrate the difference quantitatively, in Fig.", "REF we plot the quantity $\\tau /\\tau _0 \\, n/n_0 - 1$ at $\\tau =\\tau _f$ as a function of $4\\pi \\eta /{\\cal S}$ .", "We used a freeze out temperature of $T_f = 150$ MeV to determine $\\tau _f$ .", "This quantity should be zero if there are no particles produced during the evolution.", "As can be seen from these plots our expectations are confirmed, namely that one sees a maximum in entropy production at large values of $4\\pi \\eta /{\\cal S}$ with it returning to zero as $4\\pi \\eta /{\\cal S}$ increases above this point.", "Concentrating on the zoomed plot in Fig.", "REF one sees that for $4\\pi \\eta /{\\cal S}=10$ 2nd-order viscous hydrodynamics overestimates the entropy production by approximately 93%.", "We note that as the initial temperature is lowered, the excess particle production obtained from 2nd-order viscous hydrodynamics becomes larger.", "This will be important for phenomenology since one of the key constraints on $\\eta /{\\cal S}$ stems from having to reduce the assumed initial temperature in order to compensate for dissipative particle/entropy production." ], [ "Numerical Tests", "In Fig.", "REF we show the time evolution of the spatial and transverse momentum-space eccentricities as a function of proper time for a smooth Glauber wounded-nucleon transverse profile with $b=7$ fm, $\\Lambda _0=T_0=0.6$ GeV, $\\xi _0 = 0$ , and $u_{\\perp ,0} = 0$ at $\\tau _0 = $ 0.25 fm/c assuming $4\\pi \\eta /{\\cal S} = 1$ .", "In all three cases we used a RK4 temporal step size of $\\epsilon = 0.01$ fm/c.", "In this figure we have used the central-differences algorithm without wLAX smoothing and compare the effect of varying the lattice spacing and lattice volume.", "As can be seen from this figure, the systematics are well under control in this case.", "Knowing that the centered-differences algorithm systematics are under control we can now compare with the hybrid Kurganov-Tadmor algorithm.", "In Fig.", "REF we show such a comparison for the same conditions as shown in Fig.", "REF .", "As can be seen from this figure the naive centered-differences algorithm and the hybrid Kurganov-Tadmor algorithm give results that are indistinguishable by eye.", "In Fig.", "REF we present the spatial and momentum eccentricities as a function of proper time for a smooth Glauber wounded-nucleon transverse profile with $b=7$ fm, $\\Lambda _0=T_0=0.6$ GeV, $\\xi _0 = 0$ , and $u_{\\perp ,0} = 0$ at $\\tau _0 = $ 0.25 fm/c assuming $4\\pi \\eta /{\\cal S} = 1$ .", "In this plot we compare a run with the unsmeared centered-differences algorithm and the wLAX algorithm with two different lattice spacings.", "As can be seen from this figure the amount of numerical viscosity is small and can be reduced if one reduces the lattice spacing.", "To further illustrate the reliability of the wLAX algorithm in Fig.", "REF we compare a single MC Glauber wounded-nucleon run using both the wLAX and Hybrid Kurganov-Tadmor algorithms.", "Both codes were initialized with the same sampled MC initial condition (a visualization of the evolution of this configuration is shown in Fig.", "REF ).", "As can be seen from this figure, wLAX and Hybrid Kurganov-Tadmor give virtually indistinguishable results.", "We point out in this context that the wLAX algorithm take much less time to complete a run giving it a significant advantage when one wants to sample many different configurations.", "Based on our benchmarks the wLAX algorithm is approximately ten times faster than the Hybrid Kurganov-Tadmor algorithm." ], [ "Boost invariant 1d dynamics - The Bjorken solution", "In this section we briefly review what happens when the system is boost invariant, homogeneous in the transverse directions, and has conserved particle number, i.e.", "$J_0 =0$ .", "For this situation, it is convenient to switch to the comoving Milne coordinates defined as $t&=&\\tau \\cosh \\varsigma \\, , \\nonumber \\\\z&=&\\tau \\sinh \\varsigma \\, .$ In this coordinate system the metric $g_{\\mu \\nu }= {\\rm diag}\\,(1,-1,-1,-\\tau ^2)$ .", "In addition, the local rest frame four-velocity simplifies to $u^\\mu = (\\cosh \\varsigma ,0,0,\\sinh \\varsigma ) \\, ,$ such that $u_\\tau = 1$ , $u_\\varsigma = 0$ , and we have $D &=& u^\\mu \\partial _\\mu = \\partial _\\tau \\, , \\nonumber \\\\\\theta &=& \\partial _\\mu u^\\mu = \\frac{1}{\\tau } \\, .$ By applying the last two expressions to the zeroth moment of the Boltzmann equation (REF ) for an isotropic plasma we obtain $\\partial _\\tau n = - \\frac{n}{\\tau } \\, ,$ which has a solution of the form $n(\\tau ) = n_0\\,\\frac{\\tau _0}{\\tau } \\, .$ If now we apply again the expressions given in Eq.", "(REF ) to the first moment of the Boltzmann equation (Eq.REF ) one finds easily that $\\partial _\\tau {\\cal E} + \\frac{{\\cal E} +{\\cal P}}{\\tau } = 0 \\, .$ If the system has an ideal equation of state (EOS) then ${\\cal E} = 3{\\cal P}$ and one can further simplify this to $\\partial _\\tau {\\cal E} = - \\frac{4}{3} \\frac{{\\cal E}}{\\tau } \\, ,$ which has a solution ${\\cal E}_{\\rm ideal\\;gas} = {\\cal E}_0 \\left(\\frac{\\tau _0}{\\tau }\\right)^{4/3} \\, .$ If the system does not have an ideal EOS but instead has an equation of state corresponding to a constant speed of sound, i.e.", "$d{\\cal P}/d{\\cal E} = c_s^2$ , then it follows that ${\\cal P} = c_s^2 {\\cal E}$ where we have fixed the constant by demanding that the pressure goes to zero when the energy density goes to zero.", "In this case one finds instead ${\\cal E} = {\\cal E}_0 \\left(\\frac{\\tau _0}{\\tau }\\right)^{1+c_s^2} \\, ,$ which reduces to the ideal case when $c_s^2 = 1/3$ .", "If the EOS has varying speed of sound then one can express ${\\cal P}$ in terms of an integral of the speed of sound.", "Alternatively, one could calculate the pressure and energy density separately for e.g.", "an ideal massive Boltzmann gas [76] for which one finds ${\\cal E} &=& N_{\\rm dof} \\frac{e^{\\mu /T} m^2 T}{2 \\pi ^2}\\left[ 3 T K_2\\left(\\frac{m}{T}\\right) + m K_1\\left(\\frac{m}{T}\\right)\\right] ,\\nonumber \\\\{\\cal P} &=& N_{\\rm dof} \\frac{e^{\\mu /T} m^2 T^2}{2 \\pi ^2}K_2\\left(\\frac{m}{T}\\right) ,\\nonumber \\\\n &=& \\frac{\\cal P}{T} \\, ,$ and $c_s^2(T,\\mu =0) = \\left(3 + \\frac{m}{T} \\frac{K_2(m/T)}{K_3(m/T)} \\right)^{-1} \\, .$ Note that the thermodynamic relations above are consistent with Bjorken scaling for the number density, $n/n_0 = \\tau _0/\\tau $ , for all values of $m$ in the case of isotropic hydrodynamics." ] ]

[ [ "N=1-Supersymmetric Description of a Spin-1/2 charged Particle in a\n 5d-World" ], [ "Abstract We study the dynamics of a charged spin-1/2 particle in an external 5-dimensional electromagnetic field.", "We then consider that we are at the $TeV$\\ scale, so that we can access the fifth dimension and carry out our physical considerations in a 5-dimensional brane.", "In this brane, we focus our attention to the quantum-mechanical dynamics of a charged particle minimally coupled to the 5-dimensional electromagnetic field.", "We propose a way to identify the Abraham-Lorentz back reaction force as an effect of the extra (fifth) dimension.", "Also, a sort of dark matter behavior can be identified in a particular regime of the dynamics of the particle interacting with the bulk electric field." ], [ "N=1-Supersymmetric Description of a Spin-1/2 charged Particle in a 5d-World H. Belich $^{a,e}$ , D. Cocuroci$^{b}$ , G. S. Dias$^{c,d}$ , J.A.", "Helayël-Neto$^{b,e}$ , M.T.D.", "Orlando$^{a,e}$ $^{a}$Universidade Federal do Espírito Santo (UFES), Departamento de Física e Química, Av.", "Fernando Ferrari 514, Vitória, ES, CEP 29060-900, Brasil $^{b}$CBPF - Centro Brasileiro de Pesquisas Físicas, Rua Xavier Sigaud 150, Rio de Janeiro, RJ, CEP 22290-180, Brasil  $^{c}$Instituto Federal do Espírito Santo (IFES) - Campus Vitória,Av.", "Vitória 1729, Jucutuquara, Vitória - ES, 29040-780, CEP 29040-780, Brasil  $^{d}$Theoretical Physics Institute, Department of Physics, University of Alberta, Edmonton, Alberta, Canada T6G 2J1  eGrupo de Física Teórica José Leite Lopes, C.P.", "91933, CEP 25685-970, Petrópolis, RJ, Brasil [email protected], [email protected], [email protected], [email protected], [email protected] We study the dynamics of a charged spin-$\\frac{1}{2}$ particle in an external 5-dimensional electromagnetic field.", "We then consider that we are at the $TeV$ scale, so that we can access the fifth dimension and carry out our physical considerations in a $5-$ dimensional brane.", "In this brane, we focus our attention to the quantum-mechanical dynamics of a charged particle minimally coupled to the $5-$ dimensional electromagnetic field.", "We propose a way to identify the Abraham-Lorentz back reaction force as an effect of the extra ( fifth ) dimension.", "Also, a sort of dark matter behavior can be identified in a particular regime of the dynamics of the particle interacting with the bulk electric field.", "11.10.Kk, 11.30.Pb, 11.25.-w Supersymmetry (SUSY) has emerged as a viable response to obtain a unified picture of all interactions of Nature.", "The strengths of the three forces of the Standard Model evolve to similar values at high temperatures, but never converge to the same magnitude of the interactions.", "Including SUSY, the strengths of the three forces of the Standard Model (SM) converge to very close values at energies of the order of $10^{16}GeV$ .", "With the superpartner particles predicted by SUSY, the three forces approach the same strengths at very high temperatures, making grand-unification (GUT) possible.", "This extension is known as the Minimal Supersymmetric Standard Model (MSSM).", "Another aspect that SUSY contributes is a way-out to the gauge hierarchy problem, i.e., it justifies why the energy of electroweak symmetry breaking is so low as compared to the grand-unification scale.", "SUSY controls very well the divergences which emerge in the SM and take the breaking scale to the GUT region .", "Due to discoveries from Wilkinson Microwave Anisotropy Probe (WMAP), among others experiments, we have to include additional matter to the models of the universe - matter, whose nature we do not know and used to denominate as dark matter.", "Supersymmetry provides a natural explanation to dark matter if we take hand of SUSY breaking mechanism.", "We have a splitting of mass and this dark matter is not observed in current accessible energy scales.", "If Supersymmetry is, in fact, a symmetry of Nature, what we see today must be its low-energy remnant.", "Thus we can attribute the absence of supersymmetry through the breaking accomplished by means of mechanisms [1], [2], [3], such that, a field theory should be related to a supersymmetric Schrödinger equation in each particle number sector of the theory.", "To study these aspects of Supersymmetric Quantum Field Theory at low energies (non-relativistic limit) became a topic of special interest [4], [5], [6].", "In fact as we are interested in the regime in which particles are not generated or destroyed, therefore we deal with non relativistic limit, with Supersymmetric Quantum Mechanics description.", "On the other hand, the introduction of extra dimensions to explain the fundamental forces of Nature remains to the beginnings of General Relativity.", "The Kaluza-Klein Theory try to explain the Gravity from Electromagnetism in a space-time with more than four dimensions [7].", "Nowadays, multidimensional space-time approach have appeared connected with string models [8], and branes [9], [10], as a possible mechanism to overcome the hierarchy problem.", "This way of dealing with the problem is based on the idea that matter is restricted to a $3-D$ brane immersed in a manifold with more spatial dimensions.", "In this context, the extra dimensions are not compactified; this opens up the possibility that we can have experimental detections [11] observing directly the particle production at $TeV$ scale.", "To investigate the possibility that QED in $(1+3)-D$ , embedded in a submanifold of a $(1+4)-D$ brane, in this letter, we study Maxwells electrodynamics in the $5-D$ scenario.", "The analysis of this dynamics in $(1+4)-D$ dimensions gives us the possibility to describe a hidden sector of Electromagnetism which we can conjecture as dark energy [12], [13], [14], [15], and a particle coupled to this field is a candidate to dark matter.", "One question we address to in our study of the motion of a charged particle in a 5-dimensional supersymmetric scenario consists in checking how the electromagnetic interactions can control its motion in the 3-dimensional spatial sector corresponding to our world or may take it to the bulk [10].", "If it is driven to the bulk , $(x^{1}=x^{2}=x^{3}=0,x^{4}\\ne 0)$ , we can understand under what circumstances the external electromagnetic fields in $5-D$ determine that the particle behaves as dark matter.", "On the other hand, this discussion may guide us to relate the back radiation force in $(1+3)-D$ with the mechanical power of the magnetic force in $5-D$ .", "In this letter, we are going to pursue an investigation of a Maxwell electrodynamics in $(1+4)$ dimensions and, in this frame, we shall be discussing particular aspects of the dynamics of charged massive particles under the action of the $5-$ dimensional electric and magnetid fields.", "We pick up a particle of mass $m$ and spin-$1/2$ in an external field, [17], [18], [19],[20].", "We build up a superspace action for this model, read off the supersymmetry.transformations of the component coordinates.", "At the end, we study the dynamics of a particle in a $(1+4)-$ dimensional brane.", "The scenario we draw to pursue our investigation is as follows: we adopt the viewpoint of extra dimensions as large as $(TeV)^{-1}$ , according to the frame established by Dvali et al.", "[14].", "We then consider that we are at the $TeV$ scale, so that we can access the fifth dimension and carry out our physical considerations in a $5-$ dimensional brane.", "In this brane, we focus our attention to the quantum-mechanical dynamics of a charged particle minimally coupled to the $5-$ dimensional electromagnetic field.", "Clearly, this particle must be massive enough to an extent that it still makes sense to consider its dynamics described by Quantum Mechanics.", "Then, we suppose that the mass, $m$ , of our (charged) test particle is of the $TeV-$ order, but still larger than the energy scale we are considering.", "This means that there is not enough energy to penetrate the Compton wavelenght of the particle, so that a quantum-mechanical approach makes sense.", "So, we are considering a picture such that a charged massive particle is studied at an energy scale $\\varepsilon $ ( $\\varepsilon \\le 2m$ ) for which the fifth dimension shows up and, then, the dynamics of the particle is governed by a quantum-mechanical treatment and the particle feels the action of the $5-$ dimensional Maxwell field, which encompasses the electric $(\\vec{E})$ and magnetic $(\\vec{B})$ fields of the ordinary $4-$ dimensional Maxwell theory, and includes two extra electric- and magnetic-like fields genuinely connected to the fifth dimension.", "Supersymmetry (actually Supersymmetric Quantum Mechanics) is present in our approach for we know that the treatment of spin$-\\frac{1}{2}$ particles may be associated to a supersymmetric approach in which the spin variables are identified with the Grassmannian partner of the variables describing the particle position.", "The Supersymmetry we take about here is not a high-energy SUSY; it is simply the sort of dynamical symmetry which is underneath the description of the quantum-mechanics aspects of $s=\\frac{1}{2}$ -particles.", "We quote relevant references for such a discussion in [16].", "Once our physical framework has been settled down, we start our considerations on the dynamics of our test particle under the action of the 5-dimensional electromagnetic field.", "We intend to investigate the $5-D$ electrodynamics.", "The Maxwells electromagnetism in 4-dimensional space-time could be the derivative of a more fundamental theory in $5-D$ dimensions.", "This electrodynamics lives in a $5-D$ hypersurface of a brane.", "Vectors and tensors shall be decomposed in terms of $SO(3)$ indexes; we split the 4-th component which behave as a scalar under $SO(3)$ rotations.", "We consider the following action in $1+4$ dimensions, $L=\\overline{\\Psi }(i\\Gamma ^{\\mu }D_{\\mu }-m)\\Psi ; $ where we define: $\\mu &\\in &\\lbrace 0,1,2,3,4\\rbrace ,\\quad \\eta =(+,-,-,-,-),\\quad \\\\x^{\\mu } &=&(t,x=x^{1},y=x^{2},z=x^{3},x^{4}),\\quad F_{\\mu \\nu }=\\partial _{\\mu }A_{\\nu }-\\partial _{\\nu }A_{\\mu } \\\\D_{_{\\mu }} &=&\\partial _{\\mu }+ieA_{\\mu },\\quad A_{\\mu }=(A_{0},A_{i},A_{4}),\\quad \\left\\lbrace \\Gamma ^{\\mu },\\Gamma ^{\\nu }\\right\\rbrace =2\\eta ^{\\mu \\nu }, \\\\\\partial ^{i} &\\Longleftrightarrow &-\\overrightarrow{\\nabla },\\quad \\partial _{0}\\Longleftrightarrow \\frac{\\partial }{\\partial t},\\quad E^{i}\\Longleftrightarrow \\overrightarrow{E},\\quad B^{i}\\Longleftrightarrow \\overrightarrow{B},\\quad , \\\\F_{04} &=&-\\mathcal {E},\\quad F_{i4}=\\mathcal {B}_{i},\\quad F_{ij}=-\\epsilon _{ijk}B_{k},\\quad F_{0i}=E_{i},$ where $i,j,k\\in \\lbrace 1,2,3\\rbrace $ .", "Notice that the scalar, $\\mathcal {E}$ , and the vector, $\\mathcal {\\vec{B}}$ , accompany the vectors $\\overrightarrow{E}$ and $\\overrightarrow{B}$ , later on to be identified with the electric and magnetic fields, respectively.", "Our explicit representation for the gamma-matrices is given in the Appendix.", "We set: $\\Psi =\\left(\\begin{array}{c}\\psi _{1} \\\\\\psi _{2} \\\\\\psi _{3} \\\\\\psi _{4}\\end{array}\\right) ,\\quad \\overline{\\Psi }=\\Psi ^{\\dagger }\\Gamma ^{0}$ We now take the equation of motion for ${\\Psi }$ from the Euler-Lagrange equations (REF ).", "We suppose a stationary solution, $\\Psi (x,t)=\\exp (-i\\epsilon t)\\chi (x);$ here, we are considering that the external field does not depend on t, and $A_{0}=0$ .", "To observe the properties of the matter at low energy scale we have to obtain the non-relativistic limits.", "Then, considering the non-relativistic limit of the reduced theory, the equations of motions read: $(\\epsilon -m)\\chi _{1}+(i\\sigma ^{i}(\\partial _{i}+ieA_{i})+\\left( \\partial _{4}+ieA_{4}\\right) )\\chi _{2} &=&0, \\\\(i\\sigma ^{i}(\\partial _{i}+ieA_{i})-\\left( \\partial _{4}+ieA_{4}\\right))\\chi _{1}+(\\epsilon +m))\\chi _{2} &=&0.$ therefore, we see that $\\chi =\\left(\\begin{array}{c}\\chi _{1} \\\\\\chi _{2}\\end{array}\\right) $ looses two degrees of freedom, described by the \"weak\" spinor, $\\chi _{2}$ .", "So the Pauli-like Hamiltonian we get to reads as below: $H=-\\frac{1}{2m}\\left[ \\left( \\overrightarrow{\\nabla }-ie\\overrightarrow{A}\\right) ^{2}+\\left( \\partial _{4}-ieA^{4}\\right) ^{2}+e\\overrightarrow{\\sigma }\\left( \\overrightarrow{\\nabla }\\times \\overrightarrow{A}+2\\mathcal {\\vec{B}}\\right) +e\\phi \\right] $ where we used (REF ).", "If we define, $p_{i}=i\\overrightarrow{\\nabla }$ ($\\hbar =1$ ), then the Hamiltonian (REF ) will become $H &=&\\frac{1}{2m}\\left[ \\left( \\vec{p}-e\\vec{A}\\right) ^{2}+\\left(p^{4}-eA^{4}\\right) ^{2}\\right] -e\\left( \\vec{B}+2\\mathcal {\\vec{B}}\\right)\\vec{S}+e\\phi ; \\\\\\vec{S} &=&\\frac{e}{2m}\\vec{\\sigma },$ and we obtain the Pauli Hamiltonian revealing the extra-dimension contribution.", "Now it could be suitable to work out some specific aspect of Classical Electrodynamics in $(1+4)-D$ .", "We shall present the whole set of fields as written in terms of $SO(3)-$ tensor representations and to put in a manifest form Maxwell equations in five dimensions, but written in terms of the $SO(3)$ fields.", "Next, we written down the components of the energy-momentum tensor present its associated continuity equations and the meaning of the $\\Theta _{\\mu \\nu }-$ components.", "This tasks may be clarifying for the sake of our final discussion on the study of the classical dynamics of a $5-D$ charged particle in an electromagnetic field.", "We start by considering the equations of motion $\\epsilon ^{\\mu \\rho \\sigma \\beta \\gamma }\\partial _{\\sigma }F_{\\beta \\gamma }=0$ and $\\partial _{\\sigma }F^{\\sigma \\gamma }=\\rho ^{\\gamma }$ ; where $F_{\\beta \\gamma }$ is the field strength defined in (REF ) and $\\epsilon ^{\\mu \\rho \\sigma \\beta \\gamma }$ is the Levi-Civita tensor in 5 dimensions.", "Then, we have: $\\overrightarrow{\\nabla }\\cdot \\overrightarrow{E}-\\frac{\\partial \\mathcal {E}}{\\partial x^{4}} &=&0, \\\\\\overrightarrow{\\nabla }\\times \\overrightarrow{B}-\\frac{\\partial \\overrightarrow{E}}{\\partial t}-\\frac{\\partial \\mathcal {\\vec{B}}}{\\partial x^{4}} &=&0, \\\\\\frac{\\partial \\mathcal {E}}{\\partial t}+\\overrightarrow{\\nabla }\\cdot \\mathcal {\\vec{B}} &=&0, \\\\\\overrightarrow{\\nabla }\\times \\overrightarrow{E} &=&-\\frac{\\partial \\overrightarrow{B}}{\\partial t}, \\\\\\frac{\\partial \\mathcal {\\vec{B}}}{\\partial t}\\overrightarrow{+\\nabla }\\cdot \\mathcal {E}+\\frac{\\partial \\overrightarrow{E}}{\\partial x^{4}} &=&0, \\\\\\overrightarrow{\\nabla }\\times \\mathcal {\\vec{B}}-\\frac{\\partial \\overrightarrow{B}}{\\partial x^{4}} &=&0, \\\\\\overrightarrow{\\nabla }\\cdot \\overrightarrow{B} &=&0.$ We observe that, $\\overrightarrow{E} &\\Longleftrightarrow &F_{0i}\\equiv \\partial _{0}A_{i}-\\partial _{i}A_{0}=0; \\\\F_{04} &\\equiv &\\partial _{0}A_{4}-\\partial _{4}A_{0}=-\\mathcal {E}; \\\\F_{ij} &=&\\partial _{i}A_{j}-\\partial _{j}A_{i}\\equiv -\\epsilon _{ijk}B_{k};\\\\F_{i4} &=&\\partial _{i}A_{4}-\\partial _{4}A_{i}\\equiv \\mathcal {B}_{i}.$ with these equations we can better understand the influence of the fifth dimension in the particle motion.", "From the energy-momentum tensor, $\\Theta ^{\\mu }\\,_{\\kappa }=F^{\\mu \\alpha }F_{\\alpha \\kappa }+\\frac{1}{4}\\delta _{\\kappa }^{\\mu }F^{\\alpha \\beta }F_{\\alpha \\beta }$ , evaluating each set of separate components, we have:: $\\begin{array}{ccc}\\Theta ^{0}\\,_{0} & = & \\frac{1}{2}[(E^{2})+B^{2}+(\\mathcal {E}^{2})+\\mathcal {B}^{2}],\\end{array}$ which is interpreted as a new \"density\" of pressure toward our extra dimension $s$ .", "In our study the 5-D Electromagnetics this Poynting's theorem is summarized by the following expression: $\\frac{\\partial }{\\partial t}u+\\nabla \\cdot \\vec{S}+\\frac{\\partial }{\\partial s}\\xi =-\\vec{j}\\cdot \\vec{E}+j_{s}\\mathcal {E},$ where $S$ is the \"Poynting vector\" representing the energy flow, $\\overrightarrow{j}$ is the current density and $\\overrightarrow{E}$ is the electric field, and $u$ is the density of electromagnetic energy.", "$\\begin{array}{ccl}u & = & \\frac{1}{2}[(E^{2})+B^{2}+(\\mathcal {E}^{2})+\\mathcal {B}^{2}], \\\\\\vec{S} & = & (\\vec{E}\\times \\vec{B})+\\mathcal {E\\vec{B}}, \\\\\\xi & = & -\\overrightarrow{E}\\cdot \\mathcal {\\vec{B}}.\\end{array}$ The $5-D$ expression for the Lorentz force follows in connection with the following continuity equation ( in presence of external source ): $\\frac{1}{c}\\frac{\\partial \\overrightarrow{S}}{\\partial t}-\\nabla \\cdot \\sigma +\\frac{\\partial \\overrightarrow{\\chi }}{\\partial s}=-\\rho \\overrightarrow{E}-\\overrightarrow{j}\\times \\overrightarrow{B}+j_{s}\\mathcal {\\vec{B}}$ and $\\begin{array}{ccl}\\vec{S} & = & (\\vec{E}\\times \\vec{B})+\\mathcal {E\\vec{B}}, \\\\\\sigma & = & \\Theta ^{i}\\,_{j}, \\\\\\vec{\\chi } & = & (\\frac{1}{c^{2}}\\mathcal {E}\\vec{E}+\\mathcal {\\vec{B}}\\times \\vec{B}).\\end{array}$ In our scenario, 5-D, there is a third term which embraces the conservation of scalar moments, $-\\frac{1}{c^{2}}\\frac{\\partial \\xi }{\\partial t}-\\nabla \\cdot \\vec{\\chi }+\\frac{\\partial }{\\partial s}\\Omega =-\\rho b+\\vec{j}\\cdot \\mathcal {\\vec{B}}$ and $\\begin{array}{ccl}\\xi & = & -\\vec{E}\\cdot \\mathcal {\\vec{B}}, \\\\\\Omega & = & -\\frac{1}{2}[(E^{2}-B^{2}-(\\mathcal {E}^{2})+\\mathcal {B}^{2}].\\end{array}$ Now, we are back to the situation of a charged particle under the action of an external $5-D$ eletromagnetic field.", "We shall take this particular system in order to better understand how the $5-D$ fields may act upon charged particles dynamics may take place partly in $(1+3)-D$ .", "This will open up for us some interesting discussions on the back reaction of force and may even lead us to an interpretation of a possible sort of dark-matter-like charged particles.", "To render more systematic to our discussion, we think it is advisable to set up a superfield approach.", "We can define the N=1-supersymmetric model in analogy with the model presented above, eq.", "(REF ).", "It is not a trivial task, and to solve this question we start by defining the superfields as below: $\\Phi _{i}(t,\\theta )=x_{i}(t)+i\\theta \\psi _{i}(t),\\Phi _{4}(t,\\theta )=x_{4}(t)+i\\theta \\psi _{4}(t), $ The supercharge operators and the covariant derivatives are given by: $Q=\\partial _{\\theta }+i\\theta \\partial _{t}\\quad D=\\partial _{\\theta }-i\\theta \\partial _{t}\\quad \\text{and }H=i\\partial _{t}.$ We have to set up a Lagrangian in terms of the $\\Phi _{i}$ 's and $\\Phi _{4}$ so as to recover the Hamiltonian of eq.", "(REF ).", "The N=1-supersymmetric Lagrangian that generates the appropriate bosonic sector is given by: $\\mathcal {L}_{1}=\\frac{i}{2}m\\left( \\dot{\\Phi }_{i}D\\Phi _{i}+\\dot{\\Phi }_{4}D\\Phi _{4}\\right) +ie\\left[ (D\\Phi _{i})A_{i}(\\Phi _{j},\\Phi _{4})+\\left( D\\Phi _{4}\\right) A_{4}(\\Phi _{j},\\Phi _{4})\\right] .$ Integrating over the Grassman variable, we obtain: $L_{1}=\\frac{1}{2}m\\left( (\\dot{x}_{i})^{2}+(\\dot{x}_{4})^{2}\\right) -\\frac{i}{2}\\left( \\psi _{i}\\dot{\\psi }_{i}+\\psi _{4}\\dot{\\psi }_{4}\\right) +e\\left(A_{i}\\dot{x}_{i}+A_{4}\\dot{x}_{4}\\right) +e\\phi -\\frac{ie}{2}\\left( B_{i}+2\\mathcal {B}_{i}\\right) \\epsilon _{ijk}\\psi _{j}\\psi _{k},$ where the dot stands for a derivative with respect to time.", "We can also write $L_{1}=\\frac{1}{2}m\\left( (\\dot{x}_{i})^{2}+(\\dot{x}_{4})^{2}\\right) -\\frac{i}{2}\\left( \\psi _{i}\\dot{\\psi }_{i}+\\psi _{4}\\dot{\\psi }_{4}\\right) +e\\left(A_{i}\\dot{x}_{i}+A_{4}\\dot{x}_{4}\\right) +e\\phi +eB_{i}S_{i}+e2\\mathcal {B}_{i}S_{i}, $ where we define the spin by the product below: $S_{i}=-\\frac{i}{2}\\epsilon _{ijk}\\psi _{j}\\psi _{k}.$ $H_{1} &=&\\frac{1}{2}m\\left( (\\dot{x}_{i})^{2}+(\\dot{x}_{4})^{2}\\right)+i\\left( \\psi _{i}\\dot{\\psi }_{i}+\\psi _{4}\\dot{\\psi }_{4}\\right) +e\\phi +\\frac{ie}{2}\\left( B_{i}+2\\mathcal {B}_{i}\\right) \\epsilon _{ijk}\\psi _{j}\\psi _{k}, \\\\H_{1} &=&\\frac{1}{2m}\\left[ \\left( \\vec{p}-e\\vec{A}\\right) ^{2}+\\left(p^{4}-eA^{4}\\right) ^{2}\\right] +\\frac{ie}{2}\\left( B_{i}+2\\mathcal {B}_{i}field\\right) \\epsilon _{ijk}\\psi _{j}\\psi _{k}+e\\phi +i\\left( \\psi _{i}\\dot{\\psi }_{i}+\\psi _{4}\\dot{\\psi }_{4}\\right) ,$ where we observe a new Pauli coupling with the field $\\mathcal {B}_{i}$ and the contributions of the fermionic coordinates $\\psi _{j}$ and $\\psi _{4}$ .", "The eqs.", "of motion in the fermionic sector are: $\\dot{\\psi }_{4} &=&0,\\text{ } \\\\\\dot{\\psi }_{i} &=&e\\left( \\vec{B}+2\\mathcal {\\vec{B}}\\right) _{j}\\epsilon _{ijk}\\psi _{\\kappa },$ where it is manifest the Pauli-type coupling in the fermionic sector.", "For the coordinates $\\vec{x}$ and $x^{4}$ , the equations of motion assume the form: $m\\frac{\\partial ^{2}\\overrightarrow{x}}{\\partial t^{2}} &=&e\\overrightarrow{v}\\times \\overrightarrow{B}-e\\mathcal {\\vec{B}}\\dot{x}^{4}+e\\vec{E}, \\\\m\\frac{\\partial ^{2}x^{4}}{\\partial t^{2}} &=&-e\\mathcal {\\vec{B}}.\\overrightarrow{v}-e\\mathcal {E}, $ where the extended Lorentz force gets contribution from $\\mathcal {\\vec{B}}$ .", "To focus the possible novelties that this model may reveal, we have to pay attention to the extra-dimension contribution.", "To understand its exclusive effect, we set $\\vec{E}=0$ , $\\overrightarrow{B}=0$ , and $\\mathcal {E}=0;$ then, we get: $m\\frac{\\partial ^{2}\\overrightarrow{x}}{\\partial t^{2}} &=&-e\\mathcal {\\vec{B}}\\dot{x}_{4}, \\\\m\\frac{\\partial ^{2}x^{4}}{\\partial t^{2}} &=&-e\\mathcal {\\vec{B}}.\\overrightarrow{v}.", "$ Manipulating the equations above, we obtain: $\\frac{\\partial ^{3}x^{4}}{\\partial t^{3}} &=&\\alpha ^{2}.\\dot{x}_{4}, \\\\\\alpha &=&\\frac{e}{m}\\left|\\mathcal {\\vec{B}}\\right|.$ The solutions are: $(i)\\text{ \\ \\ }\\dot{x}^{4} &\\sim &\\exp \\left( +\\alpha t\\right) , \\\\(ii)\\text{ \\ }\\dot{x}^{4} &\\sim &\\exp \\left( -\\alpha t\\right) .$ Taking into account the run-away solution $(i),$ and replacing in (REF ), we obtain: $m\\frac{\\partial ^{2}\\overrightarrow{x}}{\\partial t^{2}} &\\sim &-\\mathcal {\\vec{B}}\\exp \\left( +\\alpha t\\right) , \\\\\\overrightarrow{x} &\\sim &-\\mathcal {\\vec{B}}\\exp \\left( +\\alpha t\\right) .$ In more explicit way, we have: $F_{+}=m\\frac{\\partial ^{2}\\overrightarrow{x_{+}}}{\\partial t^{2}}=-e\\mathcal {\\vec{B}}\\left|\\alpha \\right|\\exp \\left( +e\\frac{\\left|\\mathcal {\\vec{B}}\\right|}{m}t\\right) ,$ In this case, the particle moves in the opposite direction of $\\mathcal {\\vec{B}}$ and has an exponential growth force $F_{+}$ $F_{-}=m\\frac{\\partial ^{2}\\overrightarrow{x_{-}}}{\\partial t^{2}}=e\\mathcal {\\vec{B}}\\left|\\alpha \\right|\\exp \\left( -e\\frac{\\left|\\mathcal {\\vec{B}}\\right|}{m}t\\right) ,$ In this case, the particle moves in the same direction of $\\mathcal {\\vec{B}}$ and has a decreasing exponential strength $F_{-}$ .", "By (REF ), we can suppose that the extra term in the Lorentz force $(\\mathcal {\\vec{B}}\\dot{x}^{4})$ can be associated with,the back reaction or the Abraham-Lorentz force.", "Then, we could interpret the back reaction force as an effect of a fifth dimension of space-time where the $\\mathcal {\\vec{B}}$ -field is the 3-dimensional projection of the true magnetic field in (1+4)-D$.$ On the other hand taking into account $(ii),$ and replacing in (REF ), we obtain: $m\\frac{\\partial ^{2}\\overrightarrow{x}}{\\partial t^{2}} &\\sim &\\mathcal {\\vec{B}}\\exp \\left( -\\alpha t\\right) , \\\\\\overrightarrow{x} &\\sim &\\mathcal {\\vec{B}}\\exp \\left( -\\alpha t\\right) .$ But, if we study the case in which $\\mathcal {\\vec{B}}=0$ , and $\\mathcal {E}\\ne 0,$ from (), we get: $\\frac{\\partial ^{2}x^{4}}{\\partial t^{2}}=-\\frac{e\\mathcal {E}}{m},$ which shows that the particle indefinitely scapes to the extra dimension $x^{4}$ .", "Here, we can conclude that the particle may describe some component of dark matter, and the field $\\mathcal {E}$ , the piece remnant of the 5-D electric field drives the particle away from the Minkowski brane.", "So, to conclude, we would like to stress and comment on a few issues.", "We have studied an $N=1,$ $D=5$ -supersymmetric particle, in an external electromagnetic field by considering the non-relativistic regime, which, as already motivated in the introductory comments, is reasonable for we consider heavy particles with low kinetic energy.", "We admit that this model could be interesting to describe cold dark matter.", "We suppose that this particle is immersed in a (1+4)-dimensional brane.", "In this extended electromagnetic field, the ($\\mathcal {B}_{i}$ and $\\mathcal {E)}$ -fields appear in the Maxwell equations.", "The effect of the fifth coordinate $\\left( x^{4}\\right) $ is felt by means of an extension of the Lorentz force in 4 dimensions.", "And, by focusing on the evolution of the $x^{4}$ -coordinate, we have identified two possible relevant situations in connection with the time evolution of the $x^{4}$ -coordinate.", "In the situation, $x^{4}$ is of the run-away type and we associate its effect as the Abraham-Lorentz back reaction force in our $4-$ dimensional world.", "By adopting this point of view, we propose that the effect on an extra dimension may show up under the guise of the back reaction force in the dynamics of a charged particle subjected to an electromagnetic field.", "On the other hand, the presence of an electric field may drive the charged particle to the bulk and, by virtue of this mechanism, we propose that the charged particle, in this regime, has a similar behaviour to the dark matter particles.", "Of course, the particle is not electrically neutral; so, in this sense, it cannot be a genuine dark matter constituent.", "However, the extra electric field, also confined to the bulk, drives the test particle from the Minkowski brane and, once in the bulk, it interacts with the bulk field $\\mathcal {\\vec{B}}$ .", "Though charged, it escapes to the bulk and this is the reason we say it similar to the dark matter particles.", "In connection with this mechanism, we point out the results of the work of Ref.", "[21], where the authors discuss the stringent constraints on the existence of charged dark matters.", "The authors express their gratitude to CNPq for the financial support." ] ]

[ [ "On Spin Dependence of Relativistic Acoustic Geometry" ], [ "Abstract This work makes the first ever attempt to understand the influence of the black hole background space-time in determining the fundamental properties of the embedded relativistic acoustic geometry.", "To accomplish such task, the role of the spin angular momentum of the astrophysical black hole (the Kerr parameter $a$ -- a representative feature of the background black hole metric) in estimating the value of the acoustic surface gravity (the representative feature of the corresponding analogue space time) has been investigated for axially symmetric inflow of hydrodynamic fluid onto a rotating black hole.", "Since almost all astrophysical black holes are supposed to posses some degree of intrinsic rotation, the influence of the Kerr parameter on classical analogue models is very important to understand.", "For certain values of the initial boundary conditions describing the aforementioned flow, more than one acoustic horizons, namely two black hole type and one white hole type, may form, where the surface gravity may become formally infinite at the acoustic white hole.", "The connection between the corresponding analogue Hawking temperature with astrophysically relevant observables associated with the spectral signature has been discussed." ], [ "On Analogue Gravity Phenomena and Acoustic Surface Gravity", "Despite the remarkable resemblance in between a black hole and a usual thermodynamic system, black holes never radiate within the framework of the classical laws of physics.", "The introduction of the quantum mechanical effects radically changes the scenario – black holes radiate due to the Hawking effects [30], [31].", "Observational manifestation of the Hawking radiation for the astrophysical black holes is beyond the scope of the present day's experimental techniques.", "In addition, Hawking quanta may posses trans-Plankian frequencies, and physics beyond the Plank scale is yet to be realized within the framework of the contemporary knowledge of the theoretical physics.", "The aforementioned issues had been the prime motivation behind looking for an analogue version of the Hawking radiation in the laboratory set up and the formulation of the analogue gravity phenomena was thus introduced by establishing the profound similarities in between the propagating perturbation within an inhomogeneous dynamical continuum and certain kinematical features of the space time as conceived in the general theory of relativity [71], [72], [74], [55], [13], [5].", "Contemporary research in this direction has gained widespread popularity since it provides a possibility of verifying certain exotic features of the black hole physics (as manifested in the close proximity of the event horizon, including the Hawking radiation) directly some experimentally conceivable physical systems in the laboratory set up.", "Also since the smallest length scale encountered in the analogue gravity models are trans-Bohrian rather than trans-Plankian, quantum gravity effects usually significant beyond the plank scale may also show up on considerably larger length within the framework of the analogue gravity formalism.", "For the common-most example where the aforementioned perturbation is acoustic in nature, and the spacetime describing the fluid flow embedding such perturbation is Minkowskian, the acoustic surface gravity $\\kappa $ can be computed as [74]: $\\kappa =\\left[c_s\\frac{\\partial }{\\partial {\\eta }}\\left(c_s-u\\right)\\right]_{\\rm Evaluated~ at~ the~ acoustic~ horizon}$ where $c_s$ is the position dependent sound speed (the velocity of the propagation of the embedded perturbation in general), $u$ is the bulk velocity of the fluid normal to the acoustic horizon and $\\frac{\\partial }{\\partial {\\eta }}$ represents the derivative taken along the direction normal to the acoustic horizon.", "In this context it is to be pointed out that for the flat Minkowskian space time describing the fluid flow under the influence of the Newtonian gravity as well as for its relativistic generalization [10], [1], the acoustic perturbation propagates along the time like curves.", "Phonons construct the geodesics of acoustic propagation which are null with respect to the acoustic metric (the metric governing the propagation of the perturbation), and generate a null surface – the acoustic horizon.", "Such a horizon forms at the transonic point of the fluid flow.", "Hence the sonic surface is identical with the acoustic horizon, and the supersonic flow resembles the acoustic ergo region.", "It is evident from eq.", "(REF ) that the location of the acoustic/analogueHereafter the phrases `acoustic' and `analogue' will be used synonymously for the sake of brevity.", "horizon and the values of the speed of the propagation of the perturbation, the bulk velocity of the embedding continuum, and their space gradients – all evaluated on such a horizon – will provide the complete knowledge of the estimate of the analogue surface gravity.", "A more comprehensive expression for the acoustic surface gravity for a generalized analogue system (including the relativistic generalization) may be framed as [10], [1], : $\\kappa =\\left|\\frac{\\sqrt{{\\chi ^\\mu }{\\chi _\\mu }}}{\\left(1-{c_s}^2\\right)}\\frac{\\partial }{\\partial {\\eta }}\\left(u-{c_s}\\right)\\right|_{\\rm Evaluated~ at~ the~ acoustic~ horizon}$ where $\\chi ^\\mu $ is the Killing field which is null on the corresponding acoustic horizon and $\\frac{\\partial }{\\partial {\\eta }}=\\eta ^\\mu {\\partial _\\mu }$ .", "The algebraic expression corresponding to the $\\sqrt{\\chi ^\\mu {\\chi _\\mu }}$ can be evaluated once the space time metric describing the fluid flow as well as the propagation of the perturbation in a specified geometry is defined.", "In this context, eq.", "(REF ) will further be discussed in greater detail in the subsequent sections." ], [ "Accreting Astrophysical Black Holes as Analogue Systems", "Conventional works on the classical analogue gravity phenomena focus mainly on physical systems not directly subjected to the gravitational force.", "Gravity like effects rather appear to be an emergent phenomena in such configurations.", "In such systems, only the analogue Hawking effects may be observed and no source for the conventional Hawking radiation may therefore be realized.", "Quite recently, attempts have been made to study the analogue gravity phenomena in astrophysical systems where the strong gravity plays a significant role to influence the relevant features of the acoustic geometry for Newtonian, semi Newtonian and complete general relativistic flows, respectively [19], [35], [23], [1], [53], [21], [44], [43], [52] It is interesting to note that from the historical viewpoint, however, the first ever study on the analogue system was perhaps accomplished by Moncrief (1980)[48] which was essentially performed on the accreting astrophysical flows.. Special emphasis has been put to study such effects for accreting astrophysical black holes.", "Transonic accretion onto galactic and extra galactic black holes has recently been shown to be a very interesting example of the classical analogue systems found naturally in the Universe [19], [23], [1], [21].", "Such systems are unique in the sense that only for such configuration, both the gravitational as well as the acoustic horizons exist simultaneously.", "This makes the accreting black hole candidates the only analogue systems found in the Universe so far where the same flow encounters both kind of horizons.", "Such systems allow to perform a comprehensive study of the properties of space time in the close proximity of both the horizons.", "In this context, spherically symmetric as well as the axisymmetric accretion configuration has been studied.", "The non linear equations describing the stationary, inviscid, hydrodynamic accretion onto astrophysical black holes may be tailored to form a first order autonomous dynamical systems [62], [3], [60], [61], , , [9].", "The physical transonic solutions for such configuration may formally be realized as critical solutions on the radial Mach number versus radial distance phase portrait of the flow.", "For low angular momentum sub-Keplerian axisymmetric accretion of inviscid hydrodynamic fluid, multiple critical points (at most three) may appear and the the integral flow solution may contain stationary shock [37], [2], [49], [50], [51], [27], [40], [40], [28], [14], [33], [76], [41], [22].", "Any real physical global transonic solution joining the event horizon with the source of the accreting material will perforce have to pass through a saddle type critical point.", "Conventionally, a sonic point is defined at the radial distance where the radial Mach number becomes unity, i.e., $u$ becomes exactly equal to $c_s$ .", "Depending on the equation of state used to describe the flow and the flow geometry, a critical point may or may not coincide with a sonic point.", "For axisymmetric accretion flow governed by the adiabatic equation of state, critical points are not isomorphic with the sonic points when the flow configuration is assumed to be in vertical equilibrium [21], [22].", "A constant height flow or flow in conical equilibrium produces critical points which are identical with the sonic points when the adiabatic equation of state is used [1], [52].", "For isothermal flow, critical points and sonic points are found to be identical irrespective of the flow geometry used [52].", "Hence for adiabatic axisymmetric accretion in vertical equilibrium, the critical surface can not be considered as the acoustic horizon, rather the location of the acoustic horizon has to be computed by numerically integrating the differential equations describing the accretion solution starting from the critical point – detail procedure for which will further be discussed in greater detail in the subsequent sections.", "Usually for the low angular momentum accretion studied in this work, a complete multi-transonic solution may be referred to the configuration where two transonic solutions through two different saddle type critical/sonic points are connected through a discontinuous shock transition.", "Such a stationary shock usually formed in between the outer saddle type sonic point and the middle centre type sonic point.", "A comprehensive discussion about such flow configuration is available in [21] & [22].", "From the analogue gravity point of view, the aforementioned multi-transonicity may be realized through the presence of more than one acoustic horizon of different kind.", "As obvious that the acoustic horizon may be defined by the equation $u^2-c_s^2=0$ , for stationary axisymmetric flow configuration the acoustic horizon on the equatorial plane is a stationary circle of fixed radius.", "The measure of such radius depends on the initial boundary condition describing the accretion flow.", "Once the corresponding acoustic metric $G_{\\mu \\nu }$ is appropriately defined over the stationary background metric $g_{\\mu \\nu }$ , the discriminant of $G_{\\mu \\nu }$ as defined by ${\\cal D}=G^2_{t\\phi }-G_{tt}G_{\\phi \\phi }$ may be used as the marker for categorizing various sonic states of the flow, and can as well be used to determine whether the acoustic horizon is of black hole or of white hole type.", "Supersonic (subsonic) flow is characterized by the positivity (negativity) of ${\\cal D}$ [1], [21], and the change of sign of ${\\cal D}$ occurs at the acoustic horizon.", "Hence the sonic points are characterizes by ${\\cal D} <0 \\longrightarrow {\\cal D} > 0$ transition whereas ${\\cal D} >0 \\longrightarrow {\\cal D} < 0$ marks the presence of a stationary shock.", "Following the classification of Barceló, Liberati, Sonego & Visser (2004), it can be shown that for certain values of the initial boundary conditions describing the accretion flow, two acoustic black holes may form at the two sonic points and an acoustic white hole may form at the shock location flanked in between two such sonic points.", "The acoustic surface gravity becomes formally infinite at such white holes because both $c_s$ and $u$ changes discontinuously at the shock location." ], [ "Connection between the acoustic surface gravity and black hole\nspin", "In this work, we intend to study the acoustic geometry of an axially symmetric accretion flow in the Kerr metric for a configuration more complex and astrophycially relevant compared to the idealized disc model of uniform thickness as studied in [1].", "Our present work explores the low angular momentum relativistic accretion in vertical equilibrium (i.e., where the flow thickness is an analytically defined function of radial distance, as introduced in [22] using the dynamical systems approach).The essential motivation will be to investigate how the spin angular momentum of the astrophysical black holes influences the estimation of the acoustic surface gravity for multiple acoustic horizons.", "This is to understand how the properties of the curved space time governing the fluid flow (the Kerr parameter $a$ ) determines the basic features of the relevant acoustic metric (the corresponding acoustic surface gravity $\\kappa $ ).", "Such information will allow to probe a physical phenomena of profound significance – the role of the background black hole metric (as experienced by the embedding fluid flow) in constructing the perturbative curved manifold determining the properties of the space time at the close proximity of the acoustic horizon.", "Hence the nature of the dependence of the value of the acoustic surface gravity at the transonic surface of the accreting fluid on the Kerr parameter will manifest the interwining between the black hole metric and the acoustic metric.", "Along with the study of the black hole spin dependence of $\\kappa $ , we would also like to investigate whether a physical quantity as abstract as the analogue surface gravity may be associated with some observable phenomena in connection to the black hole astrophysics.", "As has already been explained, the low angular momentum accretion flow in the Kerr metric may generate two acoustic black holes at its two sonic points and an acoustic white hole at the shock location.", "At the shock, the accreting fluid will be heated up and will get considerably compressed.", "The amount of compression will depend on the strength of the shock, where the shock strength $R_M=\\frac{M_-}{M_+}$ is conventionally defined as the ratio of the pre- to the post-shock radial Mach number.", "Eventually, $\\frac{M_-}{M_+}$ depends on the initial boundary conditions, including the black hole spin.", "The shock strength $R_M$ is a significant marker of the spectral features of the accreting black hole systems [15], [68], [36], [11].", "Stronger is the shock, prominent is the spectral part indicating the radiation from the post shock flow.", "Hence $R_M$ is a good marker for the observational manifestation in the shock formation for sub-Keplarian black hole accretion.", "In this work we intend to study the co-relation between the value of the acoustic surface gravity with the shock strength (which will be different for different values of the initial boundary conditions governing the flow, including the Kerr parameter) to associate the concept of acoustic surface gravity with the spectral signature of the rotating black holes.", "The plan of the paper is as follows: In next section we will provide a brief account of the transonic black hole accretion system from the dynamical systems point of view and the details of such accretion in the Kerr metric will be provided.", "Section will be devoted to the calculation of the acoustic surface gravity $\\kappa $ .", "In section the dependence of $\\kappa $ on $a$ will clarified in detail, and the corresponding connection with the spectral signature will be described by providing the dependence of $\\kappa $ on various shock related quantities.", "Finally in section we will conclude." ], [ "Background Fluid Flow Configuration", "The acoustic geometry corresponding to the stationary solution of transonic, non self-gravitating, axisymmetric, inviscid hydrodynamic accretion of weakly rotating sub-Keplarian compressible inhomogeneous fluid onto a spinning black hole is considered on the background geometry in the Boyer-Lindquist [12] co-ordinate.", "The stationarity and the axial symmetry correspond to the two generators of the temporal and the axial isometries, respectively – leading to the fact that the total specific energy and angular momentum remains conserved along the streamline.", "The particular flow model for which the stationary axisymmetric solution of the energy momentum and the baryon number conservation equations is sought on the equatorial plane, is described in [22] in great detail.", "In this section such flow configuration will first be be introduced in brief for completeness, then the transonic structure for such configuration will be discussed at length.", "For further clarification regarding the transonic properties of general relativistic axially symmetric black hole accretion flow in general, the author may refer to, e.g., [56], [25], [4], ." ], [ "Metric Elements and Related Quantities", "In general any radial distance on the equatorial plane has been scaled in the units of $GM_{BH}/c^2$ and any velocity involved has been scaled by the velocity of light in vacuum, $c$ .", "$M_{BH}$ is the mass of the black hole considered, and $G=c=M_{BH}=1$ is used.", "$-+++$ signature is used along with a azimuthally Lorentz boosted orthonormal tetrad basis co-rotating with the accreting fluid.", "The energy momentum tensor $T^{\\mu \\nu }=\\left({\\epsilon }+p\\right)v^\\mu {v^\\nu }+pg^{\\mu \\nu }$ of a perfect fluid may be defined where $\\epsilon ,p$ and $v^\\mu $ are the total mass energy density, isotropic pressure in the rest frame, and the four velocity of the accreting fluid, respectively.", "If $\\rho $ is the corresponding rest mass density, the stationary solution of the energy momentum conservation equation $T^{\\mu \\nu }_{;\\mu }=0$ and the baryon number conservation equation $\\left({\\rho }v^\\mu \\right)_{;\\mu }=0$ will provide two first integrals of motion, e.g., the conserved specific energy ${\\cal E}$ (the relativistic Bernoulli's constant) and the mass accretion rate ${\\dot{M}}$ , respectively.", "The actual expression for ${\\cal E}$ and ${\\dot{M}}$ will depend on the specific form of the metric (and related co ordinate used), the geometric flow configuration, and the equation of state used to describe the accretion flow.", "As for the space time metric, Boyer-Lindquist line element on the equatorial plane can be expressed as [56] $ds^2=g_{{\\mu }{\\nu }}dx^{\\mu }dx^{\\nu }=-\\frac{r^2{\\Delta }}{A}dt^2+\\frac{A}{r^2}\\left(d\\phi -\\omega {dt}\\right)^2+\\frac{r^2}{\\Delta }dr^2+dz^2$ where $\\Delta =r^2-2r+a^2, A=r^4+r^2a^2+2ra^2,\\omega =2ar/A$ $a$ being the Kerr parameter related to the black holes spin angular momentum.", "The required metric elements are: $g_{rr}=\\frac{r^2}{A},~g_{tt}=\\left(\\frac{A\\omega ^2}{r^2}-\\frac{r^2{\\Delta }}{A}\\right),~g_{\\phi \\phi }=\\frac{A}{r^2},~ g_{t\\phi }=g_{\\phi {t}}=-\\frac{A\\omega }{r^2}$ The normal derivative ${\\partial }/{\\partial }{\\eta }{\\equiv }{\\eta ^\\mu }{\\partial _{\\mu }}=\\frac{1}{\\sqrt{g_{rr}}}d/dr$ .", "The specific angular momentum $\\lambda $ (angular momentum per unit mass) and the angular velocity $\\Omega $ can thus be expressed as $\\lambda =-\\frac{v_\\phi }{v_t}; \\;\\;\\;\\;\\;\\Omega =\\frac{v^\\phi }{v^t}=-\\frac{g_{t\\phi }+\\lambda {g}_{tt}}{{g_{\\phi {\\phi }}+\\lambda {g}_{t{\\phi }}}}\\, ,$ We also define $B=g_{\\phi \\phi }+2\\lambda {g_{t\\phi }}+\\lambda ^2{g_{tt}}$ which will be used in the subsequent section to calculate the value of the acoustic surface gravity.", "The flow is assumed to possess finite radial three velocity $u$ (will be designated as the `advective velocity') on the equatorial plane.", "Considering $v$ to be the magnitude of the three velocity, $u$ is the component of three velocity perpendicular to the set of timelike hypersurfaces $\\left\\lbrace \\Sigma _v\\right\\rbrace $ defined by $v^2={\\rm constant}$ .", "The normalization criteria $v^\\mu {v}_\\mu =-1$ leads to the expression for the temporal component $v_t$ of the four velocity $v_t=\\left[\\frac{Ar^2\\Delta }{\\left(1-u^2\\right)\\left\\lbrace A^2-4\\lambda arA+\\lambda ^2r^2\\left(4a^2-r^2\\Delta \\right)\\right\\rbrace }\\right]^{1/2} .$ Whereas the advective dynamics is assumed to be confined on the equatorial plane only, the thermodynamic profile of the flow is essentially obtained by vertically averaging the thermodynamic quantities over the radius dependent vertical thickness $H(r)=\\sqrt{\\frac{2}{\\gamma + 1}} r^{2} \\left[ \\frac{(\\gamma - 1)c^{2}_{s}}{\\lbrace \\gamma - (1+c^{2}_{s})\\rbrace \\lbrace \\lambda ^{2}v_t^2-a^{2}(v_{t}-1) \\rbrace }\\right] ^{\\frac{1}{2}} ,$ of the flow.", "The adiabatic equation of state of the form $p=K\\rho ^\\gamma $ has been used to describe the flow, where $\\gamma =c_p/c_v$ , the ratio of the specific heat at the constant pressure and at constant volume, respectively." ], [ "Conservation Equations and the Critical Point Conditions", "The two first integrals of motion, the conserved specific energy ${\\cal E}$ of the flow and the mass accretion rate ${\\dot{M}}$ can be computed as ${\\cal E} =\\left[ \\frac{(\\gamma -1)}{\\gamma -(1+c^{2}_{s})} \\right]\\sqrt{\\left(\\frac{1}{1-u^{2}}\\right)\\left[ \\frac{Ar^{2}\\Delta }{A^{2}-4\\lambda arA +\\lambda ^{2}r^{2}(4a^{2}-r^{2}\\Delta )} \\right] }$ ${\\dot{M}}=4{\\pi }{\\Delta }^{\\frac{1}{2}}H(r){\\rho }\\frac{u}{\\sqrt{1-u^2}} \\, ,$ by integrating the stationary part of the energy momentum conservation equation and the continuity equation, respectively.", "The corresponding entropy accretion rate ${\\dot{\\Xi }}$ that is conserved for the shock free polytropic accretion and increases discontinuously at the shock (if forms), can be expressed as ${\\dot{\\Xi }}= \\left( \\frac{1}{\\gamma } \\right)^{\\left( \\frac{1}{\\gamma -1} \\right)}4\\pi \\Delta ^{\\frac{1}{2}} c_{s}^{\\left( \\frac{2}{\\gamma - 1}\\right) } \\frac{u}{\\sqrt{1-u^2}}\\left[\\frac{(\\gamma -1)}{\\gamma -(1+c^{2}_{s})}\\right] ^{\\left( \\frac{1}{\\gamma -1} \\right) } H(r)$ The conservation equations for ${\\cal E}, {\\dot{M}}$ and ${\\dot{\\Xi }}$ may simultaneously be solved to obtain the complete accretion profile from the dynamical systems point of view (see, e.g., [29] for further detail).", "The relationship between the space gradient of the acoustic velocity and that of the dynamical velocity can now be computed by differentiating eq.", "(REF ) $\\frac{dc_s}{dr}=\\frac{c_s\\left(\\gamma -1-c_s^2\\right)}{1+\\gamma }\\left[\\frac{\\chi {\\psi _a}}{4} -\\frac{2}{r}-\\frac{1}{2u}\\left(\\frac{2+u{\\psi _a}}{1-u^2}\\right)\\frac{du}{dr} \\right]$ whereas $du/dr$ may be obtained by differentiating eq.", "(REF ) along with taking help of eq.", "(REF ) $\\frac{du}{dr}=\\frac{\\displaystyle \\frac{2c_{s}^2}{\\left(\\gamma +1\\right)}\\left[ \\frac{r-1}{\\Delta } + \\frac{2}{r} -\\frac{v_{t}\\sigma \\chi }{4\\psi }\\right] -\\frac{\\chi }{2}}{ \\displaystyle {\\frac{u}{\\left(1-u^2\\right)} -\\frac{2c_{s}^2}{ \\left(\\gamma +1\\right) \\left(1-u^2\\right) u }\\left[ 1-\\frac{u^2v_{t}\\sigma }{2\\psi } \\right] }}$ where $\\psi =\\lambda ^2{v_t^2}-a^2\\left(v_t-1\\right)~\\psi _a=\\left(1-\\frac{a^2}{\\psi }\\right)~\\sigma = 2\\lambda ^2v_{t}-a^2~& & \\nonumber \\\\\\chi =\\frac{1}{\\Delta } \\frac{d\\Delta }{dr} +\\frac{\\lambda }{\\left(1-\\Omega \\lambda \\right)} \\frac{d\\Omega }{dr} -\\frac{\\displaystyle {\\left( \\frac{dg_{\\phi \\phi }}{dr} + \\lambda \\frac{dg_{t\\phi }}{dr} \\right)}}{\\left( g_{\\phi \\phi } + \\lambda g_{t\\phi } \\right)}$ Eq.", "(REF ) as well as eq.", "(REF ) can readily be identified with a set of non-linear first order differential equations representing autonomous dynamical systems, and their integral solutions will provide phase trajectories on the radial Mach number M (where $M=u/c_s$ ) vs $r$ plane.", "The critical point condition for these integral solutions may be obtained by simultaneously making the numerator and the denominator of eq.", "(REF ) vanish, and the aforementioned critical point condition may thus be expressed as ${c_{s}}_{\\bf {{\\vert _{(r=r_c)}}}}={\\left[\\frac{u^2\\left(\\gamma +1\\right)\\psi }{2\\psi -u^2v_t\\sigma }\\right]^{1/2}_{\\bf {{\\vert _{(r=r_c)}}}} },~~u{\\bf {{\\vert _{(r=r_c)}}}}= {\\left[\\frac{\\chi \\Delta r}{2r\\left(r-1\\right)+ 4\\Delta } \\right]^{1/2}_{\\rm r=r_c} }$ It is to be noted that eq.", "(REF ) provides the critical point condition but not the location of the critical point(s).", "It is necessary to solve eq.", "(REF ) under the critical point condition for a set of initial boundary conditions as defined by $\\left[{\\cal E},\\lambda ,\\gamma ,a\\right]$ .", "The value of $c_s$ and $u$ , as obtained from eq.", "(REF ), may be substituted at eq.", "(REF ) to obtain a complicated non-polynomial algebraic expression for $r=r_c$ , $r_c$ being the location of the critical point.", "A particular set of values of $\\left[{\\cal E},\\lambda ,\\gamma ,a\\right]$ will then provide the numerical solution for expression to obtain the exact value of $r_c$ .", "It is thus important to know the astrophycially relevant domain of numerical values corresponding to ${\\cal E},\\lambda ,\\gamma $ and $a$ ." ], [ "Astrophysically Relevant Domain of the Initial Boundary Conditions", "${\\cal E}$ is scaled by the rest mass energy and includes the rest mass energy itself, hence ${\\cal E}=1$ corresponds to a flow with zero thermal energy at infinity, which is obviously not a realistic initial boundary condition to generate the acoustic perturbation.", "Similarly, ${\\cal E} <1$ is also not quite a good choice since such configuration with the negative energy accretion state requires a mechanism for dissipative extraction of energy to obtain a positive energy solutionA positive Bernoulii's constant flow is essential to study the accretion phenomena so that it can incorporate the accretion driven outflows ( [17] and references therein).. Any such dissipative mechanism is not preferred to study the acoustic geometry since dissipative terms in the energy momentum conservation equation may violate the Lorentzian invariance On the other hand, almost all ${\\cal E}>1$ solutions are theoretically allowed.", "However, large values of ${\\cal E}$ represents accretion with unrealistically hot flows in astrophysics.", "In particular, ${\\cal E}>2$ corresponds to with extremely large initial thermal energy which is not quite commonly observed in accreting black hole candidates.", "We thus set $1{\\; < \\over \\sim \\;}$ E$\\; < \\over \\sim \\;$ 2$.$ A somewhat intuitively obvious range for $\\lambda $ for our purpose is $0<\\lambda {\\le }2$ , since $\\lambda =0$ indicates spherically symmetric flow and for $\\lambda >2$ multi-critical behaviour does not show up in general.", "$\\gamma =1$ corresponds to isothermal accretion where the acoustic perturbation propagates with position independent speed.", "$\\gamma <1$ is not a realistic choice in accretion astrophysics.", "$\\gamma >2$ corresponds to the superdense matter with considerably large magnetic field and a direction dependent anisotropic pressure.", "The presence of a dynamically important magnetic field requires the solution of general relativistic magneto hydrodynamics equations which is beyond the scope of the present work.", "Hence a choice for $1{\\; < \\over \\sim \\;}$$\\; < \\over \\sim \\;$ 2$ seems tobe appropriate.", "However, preferred bound for realistic black holeaccretion is from $ =4/3$ (ultra-relativistic flow) to $ =5/3$(purely non relativistic flow), see, e.g., \\cite {fra02}for further detail.", "Thus we mainly concentrate on $ 4/35/3$.$ The domain for $a$ lies clearly in between the values of the Kerr parameters corresponding to the maximally rotating black hole for the prograde and the retrograde flow.", "Hence the obvious choice for $a$ is $-1{\\le }a{\\le }1$ .", "The allowed domains for the four parameter initial boundary conditions are thus $\\left[1{\\; < \\over \\sim \\;}\\right.", "{\\cal E}{\\; < \\over \\sim \\;}$ 2, 0<2,4/35/3,-1a1]$.The aforementioned four parameters may further be classifiedintro three different categories, according to the way they influence thecharacteristic properties of the accretion flow.$ [E,,]$ characterizesthe flow, and not the space–time since the accretion is assumed tobe non-self-gravitating.The Kerr parameter $ a$ exclusivelydetermines the nature of the space–time and hence can be thought ofas some sort of `inner boundary condition’ in qualitative sense\\footnote {The effect ofgravity is determined within the full general relativisticframework only up to several gravitational radii.", "Beyond a certainlength-scale it asymptotically follows the Newtonian regime.", "}.$ [E, ][E, ,]$determines the dynamical aspects of theflow, whereas $$ determines the thermodynamic properties.To follow a holistic approach, one needs to study the variation of thesalient features of the acoustic geometry on all of these four parameters.\\subsection {The Critical Velocity Gradients}Once the value of $ rc$ is computed for an astrophysically relevantset of $ [E,,,a]$, the nature of the critical point(s) can also be studiedto confirm whether it is a saddle type or a centre type criticalpoint (see, e.g., \\cite {gos07} for an analyticalscheme developed using the eigenvalue problemto accomplish such classification for axisymmetricaccretion in the Kerr metric).", "The space gradient for theadvective flow velocity at the critical pointcan also becomputed by solving the following quadratic equation\\begin{equation}\\alpha \\left(\\frac{du}{dr}\\right)_{\\bf {{\\vert _{(r=r_c)}}}}^2 +\\beta \\left(\\frac{du}{dr}\\right)_{\\bf {{\\vert _{(r=r_c)}}}} + \\zeta = 0,\\end{equation}where the respective co-efficients, all evaluated at the critical point $ rc$,are obtained as\\begin{eqnarray}\\alpha =\\frac{\\left(1+u^2\\right)}{\\left(1-u^2\\right)^2} - \\frac{2\\delta _1\\delta _5}{\\gamma +1},\\quad \\quad \\beta =\\frac{2\\delta _1\\delta _6}{\\gamma +1} + \\tau _6,\\quad \\quad \\zeta =-\\tau _5;& & \\nonumber \\\\\\delta _1=\\frac{c_s^2\\left(1-\\delta _2\\right)}{u\\left(1-u^2\\right)}, \\quad \\quad \\delta _2 = \\frac{u^2 v_t \\sigma }{2\\psi }, \\quad \\quad \\delta _3 = \\frac{1}{v_t} + \\frac{2\\lambda ^2}{\\sigma } - \\frac{\\sigma }{\\psi } ,\\quad \\quad \\delta _4 = \\delta _2\\left[\\frac{2}{u}+\\frac{u v_t \\delta _3}{1-u^2}\\right],& & \\nonumber \\\\~\\delta _5 = \\frac{3u^2-1}{u\\left(1-u^2\\right)} - \\frac{\\delta _4}{1-\\delta _2} -\\frac{u\\left(\\gamma -1-c_s^2\\right)}{a_s^2\\left(1-u^2\\right)},\\quad \\quad \\delta _6 = \\frac{\\left(\\gamma -1-c_s^2\\right)\\chi }{2c_s^2} +\\frac{\\delta _2\\delta _3 \\chi v_t}{2\\left(1-\\delta _2\\right)},& & \\nonumber \\\\\\tau _1=\\frac{r-1}{\\Delta } + \\frac{2}{r} - \\frac{\\sigma v_t\\chi }{4\\psi },\\quad \\quad \\tau _2=\\frac{\\left(4\\lambda ^2v_t-a^2\\right)\\psi - v_t\\sigma ^2}{\\sigma \\psi },& & \\nonumber \\\\\\tau _3=\\frac{\\sigma \\tau _2 \\chi }{4\\psi },\\quad \\quad \\tau _4 = \\frac{1}{\\Delta }- \\frac{2\\left(r-1\\right)^2}{\\Delta ^2}-\\frac{2}{r^2} - \\frac{v_t\\sigma }{4\\psi }\\frac{d\\chi }{dr},& & \\nonumber \\\\\\tau _5=\\frac{2}{\\gamma +1}\\left[c_s^2\\tau _4 -\\left\\lbrace \\left(\\gamma -1-c_s^2\\right)\\tau _1+v_tc_s^2\\tau _3\\right\\rbrace \\frac{\\chi }{2}\\right]- \\frac{1}{2}\\frac{d\\chi }{dr},& & \\nonumber \\\\\\tau _6=\\frac{2 v_t u}{\\left(\\gamma +1\\right)\\left(1-u^2\\right)}\\left[\\frac{\\tau _1}{v_t}\\left(\\gamma -1-c_s^2\\right) + c_s^2\\tau _3\\right].\\end{eqnarray}Note, however, that {\\it all} quantities defined in eq.", "(\\ref {eq19}) canfinally be reduced to an algebraic expression in $ rc$ withreal coefficients that are functions of $ [E,,,a]$.", "Hence$ (du/dr)r=rc$ is found to be an algebraic expressionin $ rc$ with constant co efficients those are non linear functionsof $ [E,,,a]$.", "Once $ rc$ is known for a set of values of $ [E,,,a]$,the critical slope, i.e., the space gradient for $ u$ at$ rc$ for the advective velocity can be computed as a purenumber, which may either be a real (for transonic accretion solution toexist) or an imaginary (no transonic solution may be found) number.The critical advective velocity gradientfor accretion solution may be computed as\\begin{equation}\\left(\\frac{du}{dr}\\right)_{\\rm r=r_c}=-\\frac{\\beta }{2\\alpha }{\\pm }\\sqrt{\\beta ^2-4\\alpha {\\zeta }}\\end{equation}by taking the positive sign.", "The negative sign corresponds tothe outflow/self-wind solution on which we would not like to concentratein this work.", "The critical acoustic velocity gradient$ (dcs/dr)r=rc$ can also be computed bysubstituting the value of $ (dudr)r=rc$in eq.", "(\\ref {eq14}) and byevaluating other quantities in eq.", "(\\ref {eq14}) at $ rc$.\\subsection {On Transonicity and Multi-transonic Solutions}As mentioned before, the critical point can be computed by putting thecritical point conditions in the expression on $ E$ as expressedin eq.", "(\\ref {eq11}).", "The number of critical points obtained is eitherone (saddle type), or three (one centre type flanked by two saddletype) depending on the particular value of $ [E,,,a]$used.Certain $ [E,,,a] $_{\\rm mc}{\\subset }$ $\\left[{\\cal E},\\lambda ,\\gamma ,a\\right]$ thus provides the multi criticality in accretion (as well as in outflow) solutions, where the subscript `mc' stands for multi critical.", "It is, however, to be noted that the radial Mach number at the critical point is a functional of $\\left[{\\cal E},\\lambda ,\\gamma ,a\\right]$ and is less than unity, which is obvious from eq.", "(REF ).", "Following our previous discussions, this is the consequence of the choice of the geometric configuration of the accretion flow and the equation of state to study such flow.", "As mentioned earlier, the acoustic horizon is defined as a time-like hypersurface defined by the equation $c_s^2-u^2=0$ The acoustic horizon are thus the collection of the `sonic' points where the radial Mach number becomes unity.", "Since critical points are not topologically isomorphic with the sonic points in general, acoustic horizon does not form at the critical point, and neither $\\left(du/dr\\right)_{\\rm r=r_c}$ nor $\\left(dc_s/dr\\right)_{\\rm r=r_c}$ can be used to evaluate the value of the acoustic surface gravity.", "Had it been the case that the critical points would be identical with the sonic points, the acoustic surface gravity could easily be evaluated by taking $r_c$ to be $r_h$ , the radius of the acoustic horizon, and by directly taking the value of $c_s, u, \\left(du/dr\\right)_{\\rm r=r_c}$ and $\\left(dc_s/dr\\right)_{\\rm r=r_c}$ as evaluated at $r_h$ .", "One thus understands that $r_h$ is actually a sonic point located on the combined integral solution of eq.", "(REF ) and eq.", "(REF ).", "For inviscid flow, a physically acceptable transonic solution can be realized to pass through a saddle type sonic point, resulting the hypothesis that every saddle type critical point is accompanied by its sonic point but no centre type critical point has its sonic counterpart.", "For axisymmetric flow in the Kerr metric, a multi-critical flow is thus a theoretical abstraction where three critical points are obtained as a mathematical solution of the energy conservation equation (through the critical point condition), whereas a multi-transonic flow is a practically realizable configuration where accretion solution passes through two different saddle type sonic points.", "One should, however, note that a smooth accretion solution can never encounter more than one regular sonic points, hence no continuous transonic solution exists which passes through two different acoustic horizons.", "The only way the multi transonicity could be realized as a combination of two different otherwise smooth solution passing through two different saddle type critical (and hence sonic) points and are connected to each other through a discontinuous shock transition.", "Such a shock has to be stationary and will be located in between two sonic points.", "For certain $\\left[{\\cal E},\\lambda ,\\gamma ,a\\right]$$_{\\rm nss}{\\subset }$$\\left[{\\cal E},\\lambda ,\\gamma ,a\\right]$$_{\\rm mc}$ , where `nss' stands for no shock solution, three critical points (two saddle embracing a centre one) are routinely obtained but no stationary shock forms.", "Hence no multi transonicity is observed even if the flow is multi-critical, and real physical accretion solution can have access to only one saddle type critical points (the outer one) out of the two.", "Thus multi critical accretion and multi transonic accretion are not topologically isomorphic in general.", "A true multi-transonic flow can only be realized for $\\left[{\\cal E},\\lambda ,\\gamma ,a\\right]$$_{\\rm ss}{\\subset }$$\\left[{\\cal E},\\lambda ,\\gamma ,a\\right]$$_{\\rm mc}$ , where `ss' stands for `shock solution', if the following criteria for forming a standing shock (the relativistic Rankine-Hugoniot condition, see, e.g., [1], [21] for further detail including the formulation) gets satisfied $\\left[\\left[{\\rho }u\\Gamma _{u}\\right]\\right]=0& & \\nonumber \\\\\\left[\\left[{\\large \\sf T}_{t\\mu }{\\eta }^{\\mu }\\right]\\right]=\\left[\\left[(p+\\epsilon )v_t u\\Gamma _{u} \\right]\\right]=0& & \\nonumber \\\\\\left[\\left[{\\large \\sf T}_{\\mu \\nu }{\\eta }^{\\mu }{\\eta }^{\\nu }\\right]\\right]=\\left[\\left[(p+\\epsilon )u^2\\Gamma _{u}^2+p \\right]\\right]=0$ where $\\Gamma _u=1/\\sqrt{1-u^2}$ is the Lorentz factor.", "The overall procedure to compute the value of the analogue surface gravity, is, however, a bit involved.", "For a set of $\\left[{\\cal E},\\lambda ,\\gamma ,a\\right]$ , $r_c,u{\\bf {{\\vert _{(r=r_c)}}}},{c_{s}}_{\\bf {{\\vert _{(r=r_c)}}}},\\left(du/dr\\right)_{\\rm r=r_c}, \\left(dc_s/dr\\right)_{\\rm r=r_c}$ can be calculated.", "These values are then used as the initial values to find the integral solution for eq.", "(REF - REF ).", "Along the integral solution, the radial Mach number $M$ is computed at every $r$ until one obtains the specific value of $r$ for which $M$ becomes exactly equal to unity.", "$r_{M=1}$ is thus equal to $r_h$ .", "We then calculate $u,c_s,du/dr$ and $dc_s/dr$ at $r_h$ and calculate the value of the acoustic surface gravity $\\kappa $ using such values.", "The exact expression of $\\kappa $ will be derived in the next section." ], [ "Acoustic Surface Gravity in terms of $\\left[{\\cal E},\\lambda ,\\gamma ,a\\right]$", "For non-dissipative, barotropic, relativistic irrotational flow, the wave equation describing the propagation of the acoustic perturbation can be described as [10] $\\frac{1}{\\sqrt{-\\left|G_{\\mu \\nu }\\right|}}\\partial _\\mu \\left({\\sqrt{-\\left|G_{\\mu \\nu }\\right|}}G_{\\mu \\nu }\\right)\\partial _\\nu {\\varphi }=0$ by linearizing the relativistic Euler equation and the continuity equation around some steady background.", "$\\varphi $ in eq.", "(REF ) represents the low amplitude linear perturbation of the velocity potential around the aforementioned steady background.", "The acoustic metric tensor and its inverse may be defined as ${G}_{\\mu \\nu } =\\frac{\\rho }{h c_s}\\left[g_{\\mu \\nu }+(1-c_s^2)v_{\\mu }v_{\\nu }\\right];\\;\\;\\;\\;G^{\\mu \\nu } =\\frac{h c_s}{\\rho }\\left[g^{\\mu \\nu }+(1-\\frac{1}{c_s^2})v^{\\mu }v^{\\mu }\\right] ,$ where $h$ is the relativistic enthalpy and $g_{\\mu \\nu }$ is the stationary background metric.", "For the steady axisymmetric fluid flow considered in this work, the acoustic metric is assumed to be stationary and any displacement along the projection of the flow velocity on ${\\Sigma _v}$ is assumed to be an isometry, where ${\\Sigma _v}$ is the hypersurface of constant $v$ .", "The acoustic ergo-region in such configuration may be defined as the region where the stationary Killing vector becomes spacelike, i.e., any supersonic region is an ergo region.", "The stationary limit surface ${\\Sigma _{c_s}}$ is realized as the boundary of the ergo region as defined by the equation $v^2-c_s^2=0$ In terms of the acoustic metric tensor, ${\\Sigma _{c_s}}$ is defined by the condition $G_{tt}=0$ .", "The acoustic horizon can now be defined as a timelike hypersurface satisfying the criteria $\\frac{\\left(\\eta ^\\mu v_\\mu \\right)^2}{\\left(\\eta ^\\mu {v_\\mu }\\right)^2 +\\eta ^\\mu \\eta _\\mu } - c_s^2 =0$ where $\\eta ^\\mu $ is the unit normal to the horizon.", "For axially symmetric accretion flow, the acoustic horizon does not co incide with the stationary limit surface in general, except where the three velocity component perpendicular to the horizon is considered to be the flow velocity (as measured by a co rotating observer) of interest.", "One thus identifies the advective velocity $u$ on the equatorial plane with $\\left(\\left(\\eta ^\\mu v_\\mu \\right)/\\sqrt{\\left(\\eta ^\\mu {v_\\mu }\\right)^2 +\\eta ^\\mu \\eta _\\mu }\\right)$ , and the acoustic horizon can essentially be defined by eq.", "(REF ) as explained earlier.", "This further confirms that the sonic horizon (the collection of all the points where $M=1$ ) is the acoustic horizon.", "One now constructs a Killing vector $\\chi ^\\mu = \\xi ^\\mu +\\Omega {\\phi ^\\mu }$ where the Killing vectors $\\xi ^\\mu $ and $\\phi ^\\mu $ are the two generators of the temporal (constant ${\\cal E}$ ) and axial (axisymmetric flow) isometry groups, respectively.", "When $\\Omega $ is computed at $r_h$ , $\\chi ^\\mu $ becomes null on the transonic surface.", "The norm of the Killing vector $\\chi _\\mu $ may be computed as $\\sqrt{\\chi ^\\mu {\\chi _\\mu }}=\\sqrt{\\left(g_{tt}+2\\Omega {g_{t\\phi }}+\\Omega ^2{g_{\\phi \\phi }}\\right)}=\\frac{\\sqrt{{\\Delta }B}}{g_{\\phi {\\phi }}+{\\lambda }g_{t{\\phi }}}$ With this, eq.", "(REF ) takes the form $\\kappa =\\left|\\frac{r\\sqrt{\\left(r^2-2r+a^2\\right)\\left(g_{\\phi \\phi }+2\\lambda {g_{t\\phi }}+\\lambda ^2{g_{tt}}\\right)}}{{\\sqrt{g_{rr}}}\\left(1-{c_s}^2\\right)\\left({r^3+a^2r+2a^2-2\\lambda {a}}\\right)}\\left(\\frac{du}{dr}-\\frac{dc_s}{dr}\\right)\\right|_{\\rm r=r_h}$ A knowledge of $u,c_s,du/dr$ and $dc_s/dr$ as evaluated at the sonic point is thus sufficient to calculate $\\kappa $ for a fixed set of values of $\\left[{\\cal E},\\lambda ,\\gamma ,a\\right]$ .", "Figure: Phase portrait corresponding to the mono-transonicaccretion ABCABC (and its associated self-wind DEFDEF) characterized byℰ=1.2,λ=2.0,γ=1.6,a=0.4\\left[{\\cal E}=1.2,\\lambda =2.0,\\gamma =1.6,a=0.4\\right].", "The inner typecritical point is formed at r=3.3276r=3.3276, while the sonic horizon islocated ar r=2.9503r=2.9503, both in units of GM BH /c 2 GM_{\\rm BH}/c^2.", "Seetext for further details.We now illustrate the procedure to calculate $\\kappa $ for two typical flow configuration – for a montransonic accretion solution passing through the inner type sonic point and a multi-transonic accretion solution with stationary shock.", "In figure 1, $ABC$ is the integral solution for transonic accretion obtained by simultaneously solving eq.", "(REF - REF ), and $DEF$ is the integral solution representing the `self wind'.", "The radial Mach number is plotted along the $Y$ axis and the logarithmic radial distance measured from the black hole event horizon (along the equatorial plane) has been plotted along the $X$ axis.", "The phase portrait is obtained for $\\left[{\\cal E}=1.2, \\lambda =2.0, \\gamma =1.6, a=0.4\\right]$ .", "The intersection of $ABC$ and $DEF$ is essentially the critical point $r_c$ located at $3.3276$ in units of $GM_{/rm BH}/c^2$ .", "The critical point $r_c$ is obtained by putting the critical point condition as obtained from eq.", "(REF ) in the energy conservation condition as stated in eq.", "(REF ), and by solving it numerically for $r_c$ .", "Such $r_c$ and its corresponding functions are then substituted back to eq.", "(REF ) to obtain the critical velocity and the critical sound speed.", "The critical space gradient for $u$ and $c_s$ are then obtained from eq.", "().", "$\\left[u,c_s,du/dr,dc_s/dr\\right]_{\\rm r=r_c}$ is then used as initial value to obtain the integral solution of eq.", "(REF - REF ).", "Mach number is calculated at each point along the integral accretion solution $ABC$ and the value of $r$ for which $M=1$ is identified as $r_h$ ($r_h<r_c$ for accretion for obvious reason).", "$\\left[u,c_s,du/dr,dc_s/dr\\right]_{\\rm r=r_h}$ is then computed, and their values are substituted at eq.", "(REF ) to obtain the surface gravity in terms of $\\left[{\\cal E},\\lambda ,\\gamma ,a\\right]$ .", "Similar exercises may be performed for the integral transonic solution for wind (branch $DEF$ ).", "However, in this work we are interested to study the acoustic geometry for the inward moving accretion flow only.", "$\\kappa $ can also be computed for the monotransonic integral accretion solution passing through the outer type sonic point.", "The acoustic surface gravity is way much higher for solutions passing through the inner type sonic point compared to that passing through the outer type sonic pointFor a full classification of the integral curves passing through various categories of sonic points, see, e.g., Das & Czerny 2012.. By varying any of the four parameters ${\\cal E},\\lambda ,\\gamma $ and $a$ , one can study the dependence of the acoustic surface gravity on the flow properties and on the property of the back ground space time itself.", "Figure: Phase portrait for the multi-transonic shocked accretioncharacterized by ℰ=1.00001,λ=2.6,γ=1.43,a=0.4\\left[{\\cal E}=1.00001,\\lambda =2.6,\\gamma =1.43,a=0.4\\right].The vertical line BGBG demonstrates the shock transition.", "The inner, middleand the outer critical points are located at r=5.2398r=5.2398, r=18.9471r=18.9471 and r=10829.8325r=10829.8325, respectively,in units of GM BH /c 2 GM_{\\rm BH}/c^2.", "The inner and the outer sonic horizonare located at r=4.6775r=4.6775 and r=8449.4640r=8449.4640, respectively.The shock is located at r=72.01r=72.01.", "The middle critical point is indicated by the symbol '+'.In figure 2, a multi-transonic flow with stationary shock has been depicted for $\\left[{\\cal E}=1.00001, \\lambda =2.6, \\gamma =1.43, a=0.4\\right]$ .", "$ABC$ is the integral accretion solution passing through the outer critical point $r_c^{\\rm out}$ and $DEGF$ is the lower half of the homoclinic orbit representing the transonic solution passing through the inner critical point $r_c^{\\rm in}$ .", "The vertical line $BG$ represents the shock transition.", "The shock location is obtained by solving the relativistic Rankine Hugoniot condition as stated in eq.", "(REF ).", "$CBGED$ is the combined multi-transonic integral solutions with stationary shock.", "Starting from $\\left[u,c_s,du/dr,dc_s/dr\\right]_{\\rm r=r_c^{out}}$ and $\\left[u,c_s,du/dr,dc_s/dr\\right]_{\\rm r=r_c^{in}}$ , one calculates $\\left[u,c_s,du/dr,dc_s/dr\\right]_{\\rm r=r_h^{out}}$ and $\\left[u,c_s,du/dr,dc_s/dr\\right]_{\\rm r=r_h^{in}}$ , respectively.", "The corresponding surface gravity $\\kappa _{out}$ and $\\kappa _{in}$ are then computed for the outer and the inner acoustic horizons $r_h^{\\rm out}$ and $r_h^{\\rm in}$ , respectively.", "The ratio of the surface gravities at the inner acoustic horizon to that of the outer acoustic horizon is then computed as $\\kappa _{\\rm io}=\\frac{\\kappa _{\\rm in}}{\\kappa _{\\rm out}}$ At the shock, the pre and the post shock Mach numbers $M_-$ and $M_+$ are the Mach numbers calculated at the shock location for the accretion solutions passing through the inner and the outer sonic points, respectively, and the shock strength $R_M=M_-/M_+$ can be computed for a fixed set of $\\left[{\\cal E},\\lambda ,\\gamma ,a\\right]$ .", "By varying any one of the four parameters ${\\cal E},\\lambda ,\\gamma $ or $a$ , the dependence of the ratio of two acoustic surface gravities on the shock strength can the studied by plotting $\\kappa _{io}$ as a function of $R_M$ .", "Similarly, the dependence of $\\kappa _{io}$ on ${\\cal E},\\lambda ,\\gamma $ and $a$ can be studied for the multi-transonic configuration." ], [ "Dependence of $\\kappa $ on {{formula:1a0dd95c-d3e0-4c12-95ae-18b1255de85a}} and on Various Shock Related Variables", "In this section we will illustrate how the properties of the background space time (the properties of the black hole metric manifested apparently through the Kerr parameter $a$ ) influences the properties of the perturbed manifold (the properties of the acoustic metric as manifested through the measure of the acoustic surface gravity $\\kappa $ , or the ratio $\\kappa _{\\rm io}$ of such $\\kappa $ 's at the inner and the outer acoustic horizon, respectively) for transonic accretion with and without shock.", "We will also investigate how the dynamical (manifested through the dependence of $\\kappa $ on $\\left[{\\cal E},\\lambda \\right]$ ) and the thermodynamic (manifested through the dependence of $\\kappa $ on $\\gamma $ )properties of the fluid flow influences the characteristic features of the acoustic geometry.", "We will consider both shocked and non shocked accretion of the prograde as well as the retrograde flow.", "As mentioned earlier, we would like to investigate the variation of $\\kappa $ with each of the parameters $\\left[{\\cal E},\\lambda ,\\gamma ,a\\right]$ .", "It is to be understood that while studying the dependence of $\\kappa $ on any one of the aforementioned four parameters, say ${\\cal E}$ , $\\lambda ,\\gamma $ and $a$ has to be kept fixed for the entire range of ${\\cal E}$ for which the value of $\\kappa $ has to be computed.", "Since for a limited non linear range of $\\left[{\\cal E},\\lambda ,\\gamma ,a\\right]$ one obtains a multi-transonic flow with shock, a continuous range of all parameters can not be explored for such flow.", "Whereas for the mono transonic flow, a wide range of parameters, sometimes even the entire range of the allowed values corresponding to such parameters, may be explored.", "We will illustrate this issue in the subsequent section in greater detail." ], [ "Black Hole Spin Dependence of $\\kappa $", "Astrophysical black holes can posses a wide range of spin angular momentum [47], [77], [69], [66], [46], [45].", "We would thus like to cover a sufficiently large domain of $a$ , from slowly rotating to the near extremally rotating Kerr black holes, while studying the spin dependence of the acoustic surface gravity for the prograde flow.", "However, multi-transonic flow characterized by a single set of $\\left[{\\cal E},\\lambda ,\\gamma \\right]$ can not cover the entire stretch of the aforementioned domain since the shock does not form (i.e., eq.", "(REF ) does not get satisfied) for all values of $a$ for a single set of $\\left[{\\cal E},\\lambda ,\\gamma \\right]$ .", "In this work we explore three different ranges of the Kerr parameters for the multi-transonic prograde flow.", "Flow with different ranges of $a$ as are characterized by three different $\\lambda $ (=2.6, 2.17 and 2.01 for the three different ranges of $a$ ), for the uppermost, middle and the lower panel, respectively) for fixed set of values of $\\left[{\\cal E}=1.00001,\\gamma =1.43\\right]$ .", "From recent theoretical and observational findings, the relevance of the counter rotating accretion in black hole astrophysics is being increasingly evident [24], [54], [70].", "It is thus instructive to study whether the characteristic features of the acoustic geometry remains invariant for a direct `spin flip'.", "In other words, whether the $\\kappa $ – $a$ profile changes significantly when the initial boundary condition is altered from $\\left[{\\cal E},\\lambda ,\\gamma ,a\\right]$ to $\\left[{\\cal E},\\lambda ,\\gamma ,-a\\right]$ .", "It is obvious that such investigation can not be performed for multi-transonic accretion since shock condition can never be satisfied for a certain set of $\\left[{\\cal E},\\lambda ,\\gamma ,a\\right]$ as well as for $\\left[{\\cal E},\\lambda ,\\gamma ,-a\\right]$ , i.e., for flows characterized by exactly the same value of $\\left[{\\cal E},\\lambda ,\\gamma ,\\right]$ and magnitude wise same but direction wise different values of $a$ .", "This is because the region of the four dimensional parameter space spanned by $\\left[{\\cal E},\\lambda ,\\gamma ,a\\right]$ responsible for the shock formation does not allow any parameter degeneracy.", "Monotransonic accretion solutions, however, is not constrained by this issue, and one can obtain such solutions when $\\left[{\\cal E},\\lambda ,\\gamma ,a\\right]$ gets directly altered to $\\left[{\\cal E},\\lambda ,\\gamma ,-a\\right]$ .", "However, dependence of $\\kappa $ on $a$ for multi-transonic shocked retrograde flow has also been performed for in this work, see subsequent sections for further details.", "We first calculate the acoustic surface gravity for mono-transonic accretion corresponding to the phase portrait as shown in figure 1.", "$\\left[{\\cal E}=1.2,\\lambda =2.0,\\gamma =1.6\\right]$ has been used to calculate $\\kappa $ for all values of the Kerr parameter ranges from $a=-1$ to $a=1$ to explore the entire range of the prograde as well as the retrograde flow.", "In figure 3, the location of the acoustic horizon $r_h$ is plotted as a function of the $a$ .", "Such variation is shown by the dashed curve.", "In the same figure the variation of $r_c$ with $a$ is shown by the solid curve.", "As a reference, location of $r_+=1+\\sqrt{1-a^2}$ as a function of $a$ has also been plotted in the same figure using the dotted curve.", "The location of the acoustic horizon anti-correlates with the black hole spin.", "$r_c$ anti-correlates with $a$ as well.", "We define ${\\Delta }r_{c_s}^{\\left[{\\cal E},\\lambda ,\\gamma \\right]}=\\left(r_c-r_h\\right)$ as the difference of the location of the critical point and the sonic point (acoustic horizon), respectively, for a fixed set of value of $\\left[{\\cal E},\\lambda ,\\gamma \\right]$ .", "${\\Delta }r_{c_s}^{\\left[{\\cal E},\\lambda ,\\gamma \\right]}$ non linearly anti correlates with $a$ .", "For very large value of the black hole spin for co rotating accretion, the critical point almost co incides with the sonic point.", "However, the critical point and the sonic point are never identical (${\\Delta }r_{c_s}^{\\left[{\\cal E},\\lambda ,\\gamma \\right]}{\\ne }0$ always) for any value of $a$ .", "Figure: The surface gravity κ\\kappa evaluated at theacoustic horizon as a function of black hole spin, aa, for mono-transonicaccretion characterized by a fixed set ofℰ=1.2,λ=2.0,γ=1.6\\left[{\\cal E}=1.2, \\lambda =2.0, \\gamma =1.6\\right].In figure 4, we plot the surface gravity as a function of the black hole spin.", "For retrograde flow, $\\kappa $ co relates with the black hole spin.", "For prograde flow, the surface gravity initially increases non linearly with the black hole spin and attains a maximum value for a moderately high value of $a$ , after which it falls off non linearly as $a$ is increased further.", "The location of the peak of the `$\\kappa - a$ ' graph on the abscissa, i.e., the value of $a$ for $\\kappa =\\kappa _{\\rm max}^{\\left[{\\cal E},\\lambda ,\\gamma \\right]}$ depends on the choice of $\\left[{\\cal E},\\lambda ,\\gamma \\right]$ .", "We find that $a_{\\rm max}^{\\left[{\\cal E},\\lambda ,\\gamma \\right]}$ nonlinear anti-correlates with $\\left[{\\cal E},\\lambda ,\\gamma \\right]$ whereas $\\kappa _{\\rm max}^{\\left[{\\cal E},\\lambda ,\\gamma \\right]}$ non-linearly correlates with $\\left[{\\cal E},\\lambda ,\\gamma \\right]$ .", "For the mono-transonic flow $\\kappa $ in general co-relates with $\\left[{\\cal E},\\lambda ,\\gamma \\right]$ .", "This is because $r_h$ anti-correlates with $\\left[{\\cal E},\\lambda ,\\gamma \\right]$ , and closer the acoustic horizon forms to the actual black hole event horizon, higher is the value of the associated surface gravity.", "It is interesting to note that similar non-monotonic behaviour of `$\\kappa - a$ ' dependence for the prograde flow is observed for multi-transonic shocked accretion as well.", "The surface gravity computed at the inner acoustic horizon for such flow exhibits a maximum for a certain high value of the black hole spin.", "Such results are presented in the subsequent sections." ], [ "Multi-transonic Shocked Accretion", "The typical phase diagram for a representative flow topology has already been shown in figure 2.", "For prograde flow, three different ranges of the black hole spin has been studied for same values of $\\left[{\\cal E}=1.00001,\\gamma =1.43\\right]$ but for three different values of $\\lambda $ - $\\lambda =2.6$ for $a$ ranging from $\\left[a=0.21067333~{\\rm to}~a=0.47828004\\right]$ , $\\lambda =2.17$ for $a$ ranging from $\\left[a=0.85614997~{\\rm to}~a=0.92420954\\right]$ , and $\\lambda =2.01$ for $a$ ranging from $\\left[a=0.96821904~{\\rm to}~a=0.98999715\\right]$ , respectively.", "For the retrograde flow, $\\left[{\\cal E}=1.00001,\\lambda =3.3,\\gamma =1.4\\right]$ has been used to study the range of $a$ from $\\left[a=-0.21~{\\rm to}~a=-.661200\\right]$ .", "Note that the values of $\\left[{\\cal E},\\lambda ,\\gamma \\right]$ used here are just some representative set of values for which the shock forms for a substantially large range of $a$ .", "Any other set of values of $\\left[{\\cal E},\\lambda ,\\gamma \\right]$ for which a reasonable range of Kerr parameter satisfies the relativistic Rankine Hugoniot condition can be used as well to study the dependence of the acoustic surface gravity on the black hole spin.", "Figure: The location of thethe outer acoustic horizon r h out r_{h}^{\\rm out} (upper panel),the stationary shock r sh r_{\\rm sh} (mid panel),and inner acoustic horizon r h in r_{h}^{\\rm in} (lowermost panel)as a function of black hole spin for multi-transonic shocked flowcharacterized byℰ=1.00001,λ=2.6,γ=1.43\\left[{\\cal E}=1.00001, \\lambda =2.6, \\gamma =1.43\\right].In figure 5, we plot the value of the location of the outer acoustic horizon (uppermost panel), the shock location (mid panel) and the inner acoustic horizon (lowermost panel) as a function of the Kerr parameter $a$ for prograde flow characterized by $\\left[{\\cal E}=1.00001,\\lambda =2.6,\\gamma =1.43\\right]$ .", "The location of the inner and the outer acoustic horizon non linearly anti-correlates with $a$ , whereas the shock location non linearly co relates with $a$ .", "The variation of the outer acoustic horizon is much insensitive on $a$ , which is probably expected since the outer horizon forms at a very large distance and $a$ being the inner boundary condition, does not play any significant role in influencing the variation of any accretion related quantity at that length scale.", "On the other hand, variation of the inner acoustic horizon as well as of the shock location are considerably sensitive to the black hole spin.", "Similar variation of the location of the inner acoustic horizon $r_h^{in}$ , the outer acoustic horizon $r_h^{\\rm out}$ and the shock location $r_{sh}$ on the black hole spin can be studied for other set of ranges of $a$ for both prograde as well as for the retrograde flow.", "The overall `$\\left[r_h^{\\rm in},r_{sh},r_h^{\\rm out}\\right] - a$ ' profile remains more or less the same for any range of $a$ used – although the numerical values of $\\left[r_h^{in},r_{sh},r_h^{out}\\right]$ differs for the different ranges of $a$ as characterized by by different set of $\\left[{\\cal E},\\lambda ,\\gamma \\right]$ .", "An acoustic black hole horizon is formed at the inner acoustic horizon as well as at the outer acoustic horizon, and an acoustic white hole is formed at the shock location.", "$\\kappa _{\\rm io}=\\kappa _{\\rm in}/\\kappa _{\\rm out}$ can be calculated by obtaining the value of $\\kappa _{\\rm in}$ at $r_h^{\\rm in}$ and $\\kappa _{out}$ at $r_h^{out}$ .", "However, it is observed that the value of $\\kappa _{\\rm out}$ is of the order of $10^5$ times less than that of $\\kappa _{\\rm in}$ .", "Also the variation of $\\kappa _{\\rm out}$ is quite insensitive on $a$ .", "This happens because the outer acoustic horizon forms at a very large distance from $r_+$ and the Kerr parameter is essentially an inner boundary condition.", "Hence the `$\\kappa _{\\rm io} - a$ ' profile is almost identical with the scaled version of the `$\\kappa _{\\rm in} - a$ ' profile.", "Figure: The surface gravity κ in \\kappa _{\\rm in} evaluated at the inneracoustic horizon r h in r_h^{\\rm in} has been plotted as a function of black holespin for three different ranges of the Kerr parameters.", "Multi-transonicshocked flow for a fixed set ofvalues of ℰ=1.00001,γ=1.43\\left[{\\cal E}=1.00001,\\gamma =1.43\\right]has been explored for three different values of the angular momentumλ=2.6\\lambda =2.6 (the uppermost panel), λ=2.17\\lambda =2.17 (mid panel)and λ=2.01\\lambda =2.01 (lowermost panel) respectively, to cover thedifferent ranges of the Kerr parameter.In figure 6, we plot $\\kappa _{\\rm in}$ as a function of $a$ for three different ranges of $a$ for flow characterized by same $\\left[{\\cal E},\\gamma \\right]$ but for three different values of $\\lambda =2.6$ (uppermost panel), $\\lambda =2.17$ (mid panel), and $\\lambda =2.01$ (lowermost panel), respectively.", "As observed for the mono transonic accretion, the surface gravity exhibits a maximum.", "For the multi-transonic flow, however, the $\\kappa _{\\rm max}^{\\left[{\\cal E},\\lambda ,\\gamma \\right]}$ at the inner acoustic horizon is characterized by a very large value of the black hole spin.", "Figure: The surface gravity κ in \\kappa _{\\rm in} evaluated at theinner acoustic horizon for multi-transonicshocked retrograde accretion characterized byℰ=1.00001,λ=3.3,γ=1.4\\left[{\\cal E}=1.00001,\\lambda =3.3,\\gamma =1.4\\right].In figure 7, we plot $\\kappa _{\\rm in}$ as a function of $a$ for retrograde flow characterized by $\\left[{\\cal E}=1.00001,\\lambda =3.3,\\gamma =1.4\\right]$ .", "$\\kappa _{\\rm in}$ non linearly anti correlates with $a$ for the retrograde flow.", "In our attempt to make a close connection between the salient features of the acoustic geometry and any astrophysically relevant observables, we study the relation between the analogue surface gravity with shock related accretion variables since such variables play a crucial role in determining the spectral signature of the astrophysical black hole candidates.", "Shock formation phenomena is a discontinuous event somewhat equivalent to the first order phase transition.", "Accretion variables will change discontinuously at the shock location.", "Such accretion variables like density, velocity and the flow temperature provides the characteristic profiles of the observed spectra [34].", "A shocked flow will provide distinctively different spectra in comparison to its shock free counterpart.", "Corresponding spectra for shocked flow exhibits additional rich features due to the presence of the shock.", "The ratio of the pre and the post shock flow variables will determine such complex features.", "The flux distribution for such spectra for axially symmetric accretion in the Kerr metric can be calculated in terms of such ratios (Das & Huang, in preparation).", "In subsequent sections, we will study the dependence of the values of the acoustic surface gravity on pre(post) to post(pre) shock ratio of various relevant quantities responsible to characterize the spectral feature – namely, the shock strength $R_M=M_-/M_+$ , which is the pre to post shock Mach number ratio, the ratio of the post to pre shock density $R_{\\rho }=\\rho _+/\\rho _-$ (the compression ratio) and the temperature $R_T=T_+/T_-$ , respectively, for both the prograde and the retrograde flow.", "In figure 8, we plot the ratio of the acoustic surface gravity $\\kappa _{\\rm io}=\\kappa _{\\rm in}/\\kappa _{\\rm out}$ (scaled by $10^{-5}$ ) with the shock strength $R_{\\rm M}=M_-/M_+$ for three different ranges of the Kerr parameter for the prograde flow (top left and right and the bottom left panel) and a particular range of the Kerr parameter for the retrograde flow (bottom right panel).", "The range of the Kerr parameters and the values of $\\left[{\\cal E},\\lambda ,\\gamma \\right]$ used are the same as has been used to produce the figure 6 and the figure 7.", "For low to moderately high values of $a$ for the prograde flow, $\\kappa _{\\rm io}$ anti-correlates with $R_M$ in general.", "However, for the substantially large value of $a$ , `$\\kappa _{\\rm io} - a$ ' profile is different.", "For such range of $a$ , $\\kappa _{\\rm io}$ initially correlates with $R_M$ and attains a maximum, then starts falling for larger $R_M$ non linearly as $a$ is increased further.", "For the retrograde flow, $\\kappa _{\\rm io}$ usually anti-correlates with $R_M$ .", "Figure: κ io \\kappa _{\\rm io}, the ratio of the surface gravitiesas evaluated at the inner acoustic horizon to that as evaluated at theouter acoustic horizon has been plotted (scaled by a factor of 10 -5 10^{-5})as a function of the shock strength R ρ R_{\\rho } for three different ranges of theblack hole spin for the prograde flow as characterized bya fixed set of calE=1.00001,γ=1.43\\left[{cal E}=1.00001,\\gamma =1.43\\right] andthree different values of λ=2.6\\lambda =2.6 (top left panel),λ=2.17\\lambda =2.17 (top right panel), λ=2.01\\lambda =2.01 (bottom left panel),and for retrograde flow as characterized byℰ=1.00001,λ=3.3,γ=1.4\\left[{\\cal E}=1.00001,\\lambda =3.3,\\gamma =1.4\\right](bottom right panel).Figure: κ io \\kappa _{io}, the ratio of the surface gravitiesas evaluated at the inner acoustic horizon to that as evaluated at theouter acoustic horizon has been plotted (scaled by a factor of 10 -5 10^{-5})as a function of the shock strength R T R_T for three different ranges of theblack hole spin for the prograde flow as characterized bya fixed set of calE=1.00001,γ=1.43\\left[{cal E}=1.00001,\\gamma =1.43\\right] andthree different values of λ=2.6\\lambda =2.6 (top left panel),λ=2.17\\lambda =2.17 (top right panel), λ=2.01\\lambda =2.01 (bottom left panel),and for retrograde flow as characterized byℰ=1.00001,λ=3.3,γ=1.4\\left[{\\cal E}=1.00001,\\lambda =3.3,\\gamma =1.4\\right](bottom right panel).Multi-transonic flow with shocks formed closer to the black hole are associated with higher values of the shock strength and larger compression ratio, and hence the higher value of $R_{\\rm T}$ as well.", "Hence the `$\\kappa _{\\rm io} - R_{\\rm M}$ ' profile should be similar with the `$\\kappa _{io} - R_{\\rho }$ ' and `$\\kappa _{\\rm io} - R_{\\rm T}$ ' profile.", "That this intuitively obvious conclusions follows in reality may further be formally demonstrated through figure 9 and figure 10 where we plot $\\kappa _{\\rm io}$ as a function of $R_{\\rho }$ (figure 9) and $R_{\\rm T}$ (figure 10) as well for the same set of values of the initial boundary conditions as has been used to produce figure 8.", "The overall conclusion is that except for certain span of $a$ corresponding to the near extremally rotating black holes, the ratio of the acoustic surface gravity at the inner and the outer acoustic horizon has relatively large value for weaker shocks formed in the prograde accretion.", "For retrograde accretion, however, the ratio $\\kappa _{\\rm io}$ assumes higher value for weak shocks in general.", "Since the black hole spin anti-correlates with $r_{sh}$ and correlates with $r_h^{in}$ , shocks formed at the larger distance leads to the formation of the inner acoustic horizon relatively closer to the black hole where the value of the acoustic surface gravity becomes substantially large.", "On the other hand, shocks formed closer to the black hole are rather strong shocks and thus produce greater compression ratio as well as the larger value of $R_T$ .", "Hence weaker shocks correspond to the inner acoustic horizons located relatively closer to the black hole where the acoustic surface gravity assumes a higher value.", "On the other hand, as explained earlier, change of the location of the outer acoustic horizon is less sensitive to the influence of the black hole spin, hence `$\\kappa _{io} - \\left[R_M,R_{\\rho },R_T\\right]$ ' variation is effectively equivalent with the scaled down version of `$\\kappa _i - \\left[R_M,R_{\\rho },R_T\\right]$ '.", "This explain why $\\kappa _{io}$ anti-correlates with $R_M,R_\\rho $ and $R_T$ in general.", "However, it is not quite clear at this stage why such `$\\kappa _{io} -- \\left[R_M,R_{\\rho },R_T\\right]$ ' profile exhibits opposite nature for some range of the black hole spin for prograde accretion onto nearly extremely rotating holes.", "The inherent complexity of the dependence of the value of $\\kappa $ on various quantities as evaluated on the horizon prohibits to make any conclusive remark in this case since the analytical expression for the quantities involved are not available within the framework of the formalism presented here." ], [ "Dependence of $\\kappa $ on {{formula:b3254016-68b9-4a7f-bb3f-f663b43a5eb9}}", "Both the inner and the outer acoustic horizon anti-correlates with $\\left[{\\cal E},\\lambda ,\\gamma \\right]$ .", "The variation of $r_h^{in}$ is quite insensitive to ${\\cal E}$ , moderately sensitive to $\\gamma $ and highly sensitive to $\\lambda $ .", "On the other hand, the variation of $r_h^{\\rm out}$ is hardly sensitive to $\\lambda $ , whereas it is moderately sensitive to $\\gamma $ and very much sensitive to ${\\cal E}$ .", "The shock location $r_{\\rm sh}$ correlates with $\\lambda $ but anti-correlates with $\\left[{\\cal E},\\gamma \\right]$ .", "As usual, stronger shocks forms closer to the black holes.", "In figure 11, the surface gravity evaluated at the inner acoustic horizon for multi-transonic shocked accretion is plotted as a function of ${\\cal E}$ (uppermost panel for flow characterized by $\\left[\\lambda =2.17,\\gamma =1.43,a=0.881049812\\right]$ ), $\\lambda $ (mid panel, for flow characterized by $\\left[{\\cal E}=1.000004,\\gamma =1.43,a=0.881049812\\right]$ ) and $\\gamma $ (lowermost panel, for flow characterized by $\\left[{\\cal E}=1.000004,\\lambda =2.28,a=0.881049812\\right]$ ).", "$\\kappa _{in}$ anti-correlates with $\\left[{\\cal E},\\lambda ,\\gamma \\right]$ .", "The variation of $\\kappa _{in}$ is insensitive to ${\\cal E}$ , moderately sensitive to $\\gamma $ and considerably sensitive to $\\lambda $ .", "$\\kappa _{out}$ also anti-correlates with $\\left[{\\cal E},\\lambda ,\\gamma \\right]$ in general (not shown in the figure).", "$\\kappa _{out}$ , however, is around 10$^{5}$ to 10$^6$ smaller in magnitude to $\\kappa _{in}$ .", "The variation of $\\kappa _{out}$ , however, is insensitive to $\\lambda $ , moderately sensitive to $\\gamma $ and quite sensitive to ${\\cal E}$ .", "In the context to the acoustic geometry, $\\lambda $ thus somewhat controls the properties of the flow as well as that of the acoustic geometry close to the actual black hole, whereas ${\\cal E}$ influences the related quantities mainly at the larger distance." ], [ "Discussion", "In this work we make attempt to understand the influence of the general relativistic background black hole spacetime metric on the embedded perturbative manifold, i.e., on the relativistic acoustic geometry.", "To accomplish that task, we consider a specific example of classical analogue model – the equatorial slice of the general relativistic hydrodynamic inviscid axisymmetric accretion onto a spinning astrophysical black hole.", "We found that such accretion configuration gives rise to a very interesting example of acoustic geometry where more than one acoustic horizons may form.", "We then calculate the acoustic surface gravity for such horizon(s) and studied how the value of such surface gravity, which is the characteristic feature of the perturbative manifold, depends on the spin angular momentum of the astrophysical black holes, the Kerr parameter, which is the main characteristic feature of the background space-time – as well as on various accretion parameters which are the characteristic features of the background fluid continuum.", "Visser and Weinfurtner [75] have demonstrated that the equatorial slice of the Kerr geometry is equivalent to certain analogue model based on a vortex geometry.", "Our work, in some sense, is complementary to such finding.", "We, however, for the first time in the literature provided a semi analytical formalism where the influence of background black hole space-time has explicitly been demonstrated on the embedded relativistic acoustic geometry.", "Since almost all astrophysical black holes are supposed to posses some degree of intrinsic rotation [47], [24], [77], [69], [54], [66], [46], [45], [70], the effect of the Kerr parameter on the classical analogue model is extremely important to understand.", "Our work precisely accomplished that task.", "One of the most interesting aspects of acoustic surface gravity is the existence of the associated analogue Hawking radiation.", "Acoustic horizon emits Hawking type radiation of thermal phonons.", "The corresponding analogue Hawking temperature $T_{AH}=\\kappa /{2{\\pi }}$ is the characteristic temperature of such radiation measured by an observer at infinity.", "Acoustic horizons explored in our work, and its associated analogue surface gravity are thus characterized by certain $T_{AH}$ .", "In this work, we have demonstrated how the measure of the acoustic surface gravity may be associated with certain observables through the spectral feature of the astrophysical black holes.", "This work, thus, makes the first ever attempt to obtain the observational signature of the analogue hawking temperature profile for a large scale relativistic classical fluid.", "One, however, should note that $T_{AH}$ for an astrophysical black hole being too low, might be masked by the background thermal noise, hence the accreting primordial black holes might serve as better candidates for the purpose of the possible measurement of the analogue Hawking temperature.", "The surface gravity can not be computed at the shock location since $u$ and $c_s$ changes discontinuously at the shock.", "As a result, $\\left(du/dr\\right)_{r_{sh}}$ and $\\left(dc_s/dr\\right)_{r_{sh}}$ diverges.", "The surface gravity and the associated analogue Hawking temperature thus becomes formally infinite at the shock.", "Had it been the case that viscosity and other dissipative effects would be included in the system, such discontinuities in $\\left(du/dr\\right)_{r_{sh}}$ and $\\left(dc_s/dr\\right)_{r_{sh}}$ would have been smeared out.", "Under that circumstances, $\\kappa $ as well as $T_{AH}$ would have a finite but extremely large value at the shock.", "This may be considered as a realistic manifestation of the general results obtained by [38].", "In our work, however, we consider only the inviscid flow.", "The effect of the viscous transport of the angular momentum, however, is quite a subtle issue in considering the analogue effects in black hole accretion.", "Thirty nine years after the discovery of the standard accretion disc theory [56], [67], realistic modelling of viscous transonic accretion flow with dissipation is still quite an arduous task.", "From the analogue point of view, viscosity is likely to destroy the Lorenz invariance, and the assumptions behind constructing an acoustic geometry may not be quite consistent for such case.", "Nevertheless, very large radial velocity close to the black hole implies that the infall time scale is substantially small compared to the viscous time scale.", "Large radial velocity even at larger distances are due to the fact that the rotational energy of the fluid is relatively low [6], [32], [59].", "Our assumption of inviscid flow is not unjustifed from the astrophysical point of view.", "One of the significant effects of inclusion of the viscosity would be the reduction of the angular momentum.", "As we demonstrate in this work, the location of the acoustic horizon anti-correlates with $\\lambda $ .", "Weakly rotating flow makes the dynamical velocity gradient steeper leading to the conclusion that for viscous flow the acoustic horizons will be pushed further out from the black hole and the flow would become supersonic at a larger distance for the same set of other initial boundary conditions.", "The value of the surface gravity (and the associated analogue Hawking temperature) anti-correlates with the location of the acoustic horizon.", "A viscous transonic accretion disc is thus expected to produce lower value of $\\kappa $ and $T_{AH}$ compared to its inviscid counterpart.", "An axially symmetric accretion flow in vertical equilibrium has been studied in our work where the disc height is a function of the radial distance on the equatorial plane.", "Axisymmetric accretion may also be studied for two other different flow configurations, namely, for disc with constant thickness and flow in conical equilibrium where the local flow thickness to the radial distance on the equatorial plane remains constant.", "For these two flow configurations, the critical points are isomorphic to the sonic points [1], [52].", "Hence the location of the acoustic horizon and the values of the dynamical velocity along with the sound speed and their space derivative as well can be computed analytically without taking recourse to the integral flow solution, hence by avoiding the process of numerical integration.", "The non monotonic behaviour of the `$\\kappa - a$ ' profile, especially the appearance of $\\kappa _{\\rm max}^{\\left[{\\cal E},\\lambda ,\\gamma \\right]}$ and the dependence of of the corresponding $a_{\\rm max}^{\\left[{\\cal E},\\lambda ,\\gamma \\right]}$ on $\\left[{\\cal E},\\lambda ,\\gamma \\right]$ will then be better understood since we can then directly differentiate the expression of $\\kappa $ with respect to a $a$ to find out for what value of $a$ (for a fixed set of $\\left[{\\cal E},\\lambda ,\\gamma \\right]$ ) the $\\kappa - a$ profile attains its maximum, and how such value of $a$ changes with the variation of $\\left[{\\cal E},\\lambda ,\\gamma \\right]$ .", "We plan to explore such analytical dependence in our future work for axisymmetric flow in conical equilibrium.", "Finally, we would like to emphasize that in the present work we did not aim to provide a formalism using which the phonon field generated in the system concerned could be quantized.", "To accomplish that task, one has to demonstrate that the effective action for the acoustic perturbation is equivalent to a field theoretical action in curved space, and the associated commutation and the dispersion relations should directly follow [72] & .", "Such considerations are rather involved and are beyond the scope of this paper.", "Our main motivation was rather to apply the analogy to describe the classical perturbation of the fluid flow in terms of a field satisfying the wave equation in an effective geometry and to study the relevant consequences.", "HP and IM would like to acknowledge the kind hospitality provided by HRI, Allahabad, India, under a visiting students research programme.", "TKD would like to acknowledge the professional support provided by S. N. Bose National Centre for Basic Science (by offering a long term sabbatical visiting professor position) where part of the work has been done.", "The research of HP and HC is partially supported by the Natioanl Science Council of the Republic of China under the grant NSC 99-2112-M-007-017-MY3.", "The research of TKD is partially supported by the astrophysics project under the XI th plan at HRI." ] ]

[ [ "Sparse Long Blocks and the Micro-Structure of the Longest Common\n Subsequences" ], [ "Abstract Consider two random strings having the same length and generated by an iid sequence taking its values uniformly in a fixed finite alphabet.", "Artificially place a long constant block into one of the strings, where a constant block is a contiguous substring consisting only of one type of symbol.", "The long block replaces a segment of equal size and its length is smaller than the length of the strings, but larger than its square-root.", "We show that for sufficiently long strings the optimal alignment corresponding to a Longest Common Subsequence (LCS) treats the inserted block very differently depending on the size of the alphabet.", "For two-letter alphabets, the long constant block gets mainly aligned with the same symbol from the other string, while for three or more letters the opposite is true and the block gets mainly aligned with gaps.", "We further provide simulation results on the proportion of gaps in blocks of various lengths.", "In our simulations, the blocks are \"regular blocks\" in an iid sequence, and are not artificially inserted.", "Nonetheless, we observe for these natural blocks a phenomenon similar to the one shown in case of artificially-inserted blocks: with two letters, the long blocks get aligned with a smaller proportion of gaps; for three or more letters, the opposite is true.", "It thus appears that the microscopic nature of two-letter optimal alignments and three-letter optimal alignments are entirely different from each other." ], [ "Introduction", "Let $x$ and $y$ be two finite strings.", "A common subsequence of $x$ and $y$ is a subsequence which is a subsequence of $x$ and at the same time a subsequence of $y$ , while a Longest Common Subsequence (LCS) of $x$ and $y$ is a common subsequence of maximal length.", "A LCS is often used as a measure of strings relatedness, and can be viewed as an alignment aligning same letter pairs.", "Every such alignment defines a common subsequence, and the length of the subsequence corresponding to an alignment, i.e., the number of aligned letter-pairs, is called the score of the alignment.", "The alignment representing a LCS is said to be optimal or called an optimal alignment.", "Longest Common Subsequences (LCS) and Optimal Alignments (OA) are important tools used in Computational Biology and Computational Linguistics for string matching [5], [10], [11], and, in particular, for the automatic recognition of related DNA pieces.", "The asymptotic behavior of the expectation and of the variance of the length of the LCSs of two independent random strings has been studied, among others, by probabilists, physicists, computer scientists and computational biologists.", "The LCS problem can be formulated as a last passage percolation problem with dependent weights; and finding the order of the fluctuations in such percolation problems has been open for quite a while.", "Throughout, $LC_n:=|LCS(X_1X_2\\ldots X_n;Y_1Y_2\\ldots Y_n)|$ is the length of the LCSs of two random strings where $\\lbrace X_n\\rbrace _{n\\ge 1}$ and $\\lbrace Y_n\\rbrace _{n\\ge 1}$ are two independent iid sequences uniformly distributed on an fixed alphabet of size $k$ .", "Clearly, $LC_n$ is super-additive and via the sub-additive ergodic theorem, Chvátal and Sankoff [6] showed (for stationary sequences) that $\\gamma ^*_k:=\\lim _{n\\rightarrow \\infty }\\frac{\\mathbb {E}\\, LC_n}{n}.$ However, even for the simplest distributions such as for binary equiprobable alphabet, the exact value of $\\gamma ^*_k$ is unknown.", "Nevertheless, extensive simulations have led to very good approximate values for these constants, e.g., in the iid case, $\\begin{array}{cc|c|c|c|c}k & &2 &3&4&\\cdots \\\\\\hline \\gamma ^*_k& &0.812 &0.717 &0.654 &\\cdots \\end{array}$ where the precision in the above table is around $\\pm 0.01$ (see [4]).", "Exact lower and upper bounds have also been obtained, an overview of those as well as new bounds are available in [9].", "Alexander [1] further established the speed of convergence of $\\mathbb {E}\\, LC_n/n$ to $\\gamma ^*_k$ , for iid sequences, showing that $\\gamma ^*_k n-C_L\\sqrt{n\\log n}\\le \\mathbb {E}\\, LC_n\\le \\gamma ^*_k n,$ where $C_L>0$ is a constant depending neither on $n$ nor on the distribution of the strings.", "Below, we also need to consider two sequences of different lengths but such that the two lengths are in a fixed proportion to each other.", "To do so, for $p\\in (-1,1)$ , let $\\gamma _k(n,p):=\\frac{\\mathbb {E}|LCS(X_1X_2\\ldots X_{n-np};Y_1Y_2\\ldots Y_{n+np})|}{n},$ where above, when real, the indices are understood to be roundings to the nearest positive integers, and let $\\gamma _k(p):=\\lim _{n\\rightarrow \\infty }\\gamma _k(n,p),$ which is again finite by standard super-additivity arguments.", "The function $\\gamma _k: p\\mapsto \\gamma _k(p)$ is called the mean LCS-function; it is clearly bounded, non-negative, symmetric around $p=0$ , and, as shown next, concave; therefore it has a maximum at $p=0$ .", "To prove the concavity property of $\\gamma _k$ , first by super-additivity, $ n\\gamma _k\\left(n, \\frac{p+q}{2}\\right) & =\\mathbb {E}|LCS(X_1\\ldots X_{n(1-(p+q)/2)};Y_1\\ldots Y_{n(1+(p+q)/2)})| \\\\&\\ge \\mathbb {E}|LCS(X_1\\ldots X_{n(1-p)/2};Y_1\\ldots Y_{n(1+p)/2})| \\\\&\\quad \\quad + \\mathbb {E}|LCS(X_1\\ldots X_{n(1-q)/2};Y_1\\ldots Y_{n(1+q)/2})|\\\\&= \\frac{n}{2}\\gamma _k\\left(\\frac{n}{2},p\\right)+ \\frac{n}{2}\\gamma _k\\left(\\frac{n}{2},q\\right).$ Therefore, $\\gamma _k\\left(\\frac{p+q}{2}\\right)\\ge \\frac{1}{2}\\gamma _k\\left(\\frac{n}{2},p\\right)+\\frac{1}{2}\\gamma _k\\left(\\frac{n}{2},q\\right),$ which by taking limits, as $n\\rightarrow \\infty $ , leads $\\gamma _k\\left(\\frac{p+q}{2}\\right)\\ge \\frac{\\gamma _k(p)+\\gamma _k(q)}{2}.$ The function $\\gamma _k$ corresponds to the wet-region-shape in first passage percolation.", "From our simulations it seems quite clear that $\\gamma _k$ is strictly concave in a neighborhood of $p=0$ , but this might be highly non trivial to prove.", "As a matter of fact, in first passage percolation, the corresponding problem, of showing the strict convexity of the asymptotic wet-region shape remains open.", "The main results of the present paper (Theorem REF and Theorem REF ) are concerned with sequences of length $n=2d$ .", "They describe the effect of replacing an iid piece, of length $d^\\beta $ , $1/2<\\beta <1$ , with a long constant block of equal length.", "It is shown that typically replacing an iid part by a long constant block leads to a decrease in the LCS.", "It is also shown that in the binary case, the long constant block gets mainly aligned with letters while with three or more letters the opposite is true.", "To illustrate our results, consider the sequences $01{\\bf 000 00}001$ and $00101 11010,$ where the bold faced letters are those of the replacing block.", "Theorem REF and Theorem REF respectively assert that the optimal alignments behave very differently depending upon the size of the alphabet: In the binary case the long constant block gets mainly aligned with bits, while with three or more equiprobable letters it gets mainly aligned with gaps.", "This phenomenon holds with high probability and assuming $d$ to be sufficiently large.", "In the above example, a (non unique) LCS is given by 00000 and it corresponds to the (non-unique) optimal alignment $\\begin{array}{c|c|c|c|c| c|c|c|c|c| c|c|c|c|c }0&1&{\\bf 0}&{\\bf 0}& &{\\bf 0}& & & &{\\bf 0}& &{\\bf 0}&0&0&1\\\\\\hline & &0 &0 &1&0 &1& 1&1&0 &1&0 & & &\\end{array}$ In the above example of optimal alignment all the zeros from the long constant block (in bold face) got aligned with zeros and not with gaps.", "Our results show that for binary sequences, the artificially inserted long constant block gets aligned with very few gaps; more precisely, the number of gaps has an order of magnitude smaller than the length of the long block.", "With three or more letters the opposite is true and the long constant block gets aligned almost exclusively with gaps.", "The situation for three or more letters is not surprising, but the binary one is rather counter-intuitive.", "Although our proof is for $d$ going to infinity, this phenomenon is observed in simulations for regular blocks which have not been artificially inserted: with binary sequences longer blocks tend to be aligned with a small proportion of gaps, while with more letter the opposite is true.", "(More examples of this type are given at the beginning of Section .)", "We thus seem to have uncovered an interesting phenomenon, in that the microstructure of the optimal alignment of iid sequences for binary sequences is fundamentally different from the case with more letters.", "It is another instance (see [7]) where the size of the alphabet in a subsequence problem plays an important role.", "Finally, let us described some differentiability conditions on $\\gamma _k$ which could be used to obtain our results.", "First, since it is concave, $\\gamma _k$ has non-increasing left and a right derivatives at any $p\\in (-1,1)$ , with $\\gamma _k^\\prime (p^-) \\ge \\gamma _k^\\prime (p^+)$ , while by symmetry, $\\gamma _k^\\prime (p^\\pm )=-\\gamma _k^\\prime ((-p)^\\mp )$ .", "Next, let $0\\le p_M < 1$ be the largest real for which $\\gamma _k$ is maximal.", "Hence, $[-p_M,p_M]$ is the largest interval on which $\\gamma _k$ is everywhere equal to its maximal value $\\gamma _k(0)$ , i.e., $[-p_M,p_M]=\\gamma ^{-1}(\\lbrace \\gamma _k(0)\\rbrace )$ .", "Our theorems will be verified under any one of the following four conditions: The mean LCS-function $\\gamma _k$ is strictly concave in a neighborhood of the origin and is differentiable at 0 (and so $p_M=0$ and $\\gamma _k^\\prime (0) = 0$ ).", "The function $\\gamma _k$ is differentiable at $p_M$ , i.e., $\\gamma _k^\\prime (p_M^+) =\\gamma _k^\\prime (p_M^-)$ and therefore (either by symmetry or since $\\gamma _k^\\prime (p_M^-) = 0$ if $p_M>0$ ) $\\gamma _k^\\prime (p_M) = 0$ .", "The absolute value of $\\gamma _k^\\prime (p_M^+)\\le 0$ is dominated by the absolute value of $\\gamma _k(0)-(2/k)$ : $\\left|\\frac{\\gamma _k^\\prime (p_M^+)}{2}\\right|<\\left|\\frac{\\gamma _k(0)}{2}-\\frac{1}{k}\\right|.$ The function $\\gamma _k$ is strictly concave in a neighborhood of the origin and its right derivative at the origin is such that: $\\left|\\frac{\\gamma _k^\\prime (0^+)}{2}\\right|<\\left|\\frac{\\gamma _k(0)}{2}-\\frac{1}{k}\\right|.$ Clearly, $1 \\Rightarrow 2\\Rightarrow 3, 1\\Rightarrow 4$ .", "In the present article, the main results are proved under the assumptions of Condition 2.", "But, in the summary of the proofs (Section 2) it is indicated how Condition 3 or 4 would also work.", "With Condition 3, the notations for the proofs would become even more cumbersome since an additional term would appear everywhere.", "From our simulations, we have no doubt that $p_M=0$ and that even Condition 1 holds true.", "Note also that Condition 3, unlike the others, can be verified up to a certain confidence level by Monte Carlo simulations, making it rather nice and important.", "As for the content of the paper, Section  presents some of the main ideas behind the proofs, while statements of the main results are given in Section .", "Section  presents many simulations and discusses the nature of two-letter and three-letter optimal alignments.", "The proofs of the main results are presented in Section  (Subsection REF for three or more letters and Subsection REF for binary alphabets.)", "In addition to its own interest, the present paper serves as background to showing that the variance of the LCS of two iid random strings with many added long blocks is linear in the length of the strings (see [3])." ], [ "Main Ideas", "This section outlines the main ideas behind the proofs of the results.", "Below, both strings $X$ and $Y$ have length $2d$ and approximately in its middle, the sequence $X$ contains a long constant block of approximate length $\\ell =d^\\beta $ (Actually, in many of the proofs we exactly take $\\ell =d^\\beta $ , $\\ell $ even, but it is clear that choosing a multiple of $d^\\beta $ or even multiplying $d^\\beta $ by a logarithmic factor of $d$ would work).", "Since we also believe that the phenomena we uncovered are cogent for naturally occurring long blocks we often interchange the symbols $\\ell $ and $d^\\beta $ .", "The two sequences are independent and except for the long constant block in $X$ , iid uniform.", "Besides combinatorial and concentration inequalities, the proofs results follow from the following two facts (the first of which also follows from Hoeffding's martingale inequality): First, $\\gamma _k(n,p)$ converges, uniformly in $p$ to $\\gamma _k(p)$ , at a rate of $\\sqrt{{\\ln n}/{n}}$ .", "More precisely, Alexander (see Example 1.4 and Theorem 4.2 in [2]), shows that there exists a constant $C_\\gamma >0$ independent of $n$ and $p\\in (-1,1)$ such that $|\\gamma _k(n,p)-\\gamma _k(p)|\\le C_\\gamma \\sqrt{\\frac{\\ln n}{n}},$ for all $n$ and all $p\\in (-1,1)$ .", "Second, when a string with only one symbol gets aligned with another iid string with equiprobable letters, a LCS is typically much shorter than for two iid strings with equiprobable letters.", "Let us illustrate this second point on an example.", "Let $v=000000$ , $w=100101$ , so $LCS(v,w)=000$ and $|LCS(v,w)|=3$ which is the number of zeros in the string $w$ .", "Now, if $w$ is an iid string with $k$ equiprobable letters and if $v$ consists only of zeros, both strings having the same length, then typically the LCS has length approximately equal to $|w|/k$ .", "This is typically much less than for two iid sequences with equiprobable letters, where the LCS length is approximated by $\\gamma ^*_k |w|$ .", "One then compares $1/k$ and $\\gamma ^*_k$ and see, that $1/k$ is smaller than $\\gamma ^*_k$ for $k\\ge 2$ .", "In the present article, we prove two fundamental properties of the optimal alignment with an inserted long constant block: First, replacing an iid part in one of the sequences by a long constant block causes an expected loss of the LCS.", "In fact, the expected effect of replacing an iid piece with a long constant block of equal length is linear in the length of block (as shown in Section ).", "The variance cannot make up for this loss since, by Hoeffding's inequality, the standard deviation is at most of order $\\sqrt{d}$ but $d^\\beta $ , $1/2 < \\beta < 1$ , has an order of magnitude greater that $\\sqrt{d}$ .", "Second, and still in the above setting, depending on whether ${\\gamma ^*_k}/{2}-{1}/{k}$ is positive or not, the long block gets mainly aligned with gaps or not.", "Parts I and II which follow, outline the proof estimating the number of gaps aligned with the long block.", "To start with, the string $X$ is made up of three concatenated strings $X^a$ , $B$ and $X^c$ , where $X^a$ and $X^c$ are iid strings and $B$ is a constant long block.", "This is written as: $X=X^a B X^c,$ where $X^a$ and $X^c$ have common length equal to: $d-d^\\beta /2=d+o(d)$ .", "Next, let $\\pi $ be an optimal alignment of $X$ with the iid string $Y=Y_1Y_2\\ldots Y_{2d}$ , and let $Y^a$ , $Y^b$ , and $Y^c$ denote pieces of $Y$ respectively aligned with $X^a$ , $B$ , and $X^c$ .", "Next, modify the alignment $\\pi $ to obtain a new alignment $\\bar{\\pi }$ .", "For this, align $X^a$ with $Y^aY^b$ instead of only with $Y^a$ .", "The block $B$ gets aligned exclusively with gaps under $\\bar{\\pi }$ , while the alignment of $X^c$ and $Y^c$ remains unchanged.", "Request also that $\\bar{\\pi }$ aligns $X^a$ and $Y^aY^b$ in an optimal way, so that the part of the alignment score of $\\bar{\\pi }$ coming from aligning $X^a$ with $Y^aY^b$ is equal to $|LCS(X^a,Y^aY^b)|$ .", "The alignments $\\pi $ and $\\bar{\\pi }$ are schematically represented via: $\\pi :\\;\\;\\;\\;\\begin{array}{c|c|c}X^a &B &X^c\\\\\\hline Y^a &Y^b & Y^c\\end{array}$ and $\\bar{\\pi }:\\;\\;\\;\\;\\begin{array}{c|c|c}X^a &B &X^c\\\\\\hline Y^aY^b & & Y^c\\end{array}$ As for the scores, they are given by: $&{\\tt score \\; of\\;}\\pi =|LCS(X^a,Y^a)|+|LCS(B,Y^b)|+|LCS(X^c,Y^c)|,\\\\&{\\tt score \\; of\\;}\\bar{\\pi }=|LCS(X^a,Y^aY^c)|+|LCS(X^c,Y^c)|.$ The difference between the two alignment scores has two sources: first the loss of those letters of the block $B$ which where aligned with letters, and not with gaps, under $\\pi $ (while under $\\bar{\\pi }$ , all the letters get aligned with gaps).", "If $h$ denotes the length of $Y^b$ , then this expected loss is typically $h/k$ .", "($B$ is only made up of letters of one type, while in the iid part, each letter has probability $1/k$ , and therefore a given letter is expected to appear $h/k$ times in the substring $Y^b$ ).", "The second source of change in score between $\\pi $ and $\\bar{\\pi }$ comes from “adding $Y^c$ to the alignment of $X^a$ and $Y^a$ ”.", "The amount gained is then $|LCS(X^a,Y^aY^b)|-|LCS(X^a,Y^a)|.$ Assuming, say, that Condition 2 holds and from the optimality of $\\pi $ , it is easy to see that $Y^a$ and $Y^c$ have length $d+o(d)$ , for $d$ large.", "Assume next that $Y^b$ has length $h=cd^\\beta $ , where $c>0$ is a constant not depending on $d$ .", "The increase given in (REF ) can be described as the increase in LCS, when adding $h=cd^\\beta $ iid letters to two iid strings of length $d+o(d)$ .", "Part I below analyzes the size of this increase and is then used in Part II to explain how to estimate the proportion of gaps the long block gets aligned with." ], [ "I) Effect of adding $h=cd^\\beta $ symbols to one sequence only. ", "Let $V$ and $W$ be two independent iid strings of length $d+o(d)$ with $k$ equiprobable letters.", "Let $d_1$ and $d_2$ be the respective length of $V$ and $W$ , and let $\\bar{d}&=\\frac{d_1+d_2}{2}\\\\\\underline{d}&=\\frac{d_2-d_1}{d_1+d_2}.$ Clearly, $\\lim _{d\\rightarrow \\infty } \\underline{d}=0.$ Now, increase the length of $W$ by appending $h=cd^\\beta $ iid equiprobable symbols (from the same alphabet as that of $V$ and $W$ ) to it.", "Let $\\Delta $ denote the size of the increase in the LCS score due to appending these $cd^\\beta $ letters, i.e., $\\Delta :=|LCS(V;W_1W_2\\ldots W_d\\ldots W_{d_2+cd^\\beta -1}W_{d_2+cd^\\beta })|-|LCS(V;W)|.$ First, for $d$ large, $\\mathbb {E}|LCS(V,W)|$ , is approximately equal to $\\gamma _k(0)d$ .", "Second, and as explain next, the expected gain $\\mathbb {E}\\Delta $ is approximately ${cd^\\beta }(\\gamma _k(0) + \\xi (d)\\gamma _k^\\prime (0^+))/2$ , where $\\xi (d)$ can take any value between $-1$ and 1.", "Indeed, by the very definition of $\\gamma _k(\\cdot ,\\cdot )$ , $\\mathbb {E}\\Delta =\\gamma _k\\left(\\bar{d}+\\frac{cd^\\beta }{2},\\frac{cd^\\beta /2}{\\bar{d}+(cd^\\beta /2)}+\\frac{d_2-d_1}{d_1+d_2+cd^\\beta }\\right)\\left(\\bar{d}+\\frac{cd^\\beta }{2}\\right) - \\gamma _k(\\bar{d},\\underline{d})\\bar{d},$ which, with the help of (REF ) and since, $\\frac{d_2-d_1}{d_1+d_2+cd^\\beta }-\\underline{d}=-\\frac{\\underline{d}cd^\\beta /2}{\\bar{d}+(cd^\\beta /2)},$ becomes $\\mathbb {E}\\Delta =\\gamma _k\\left(\\frac{cd^\\beta /2}{\\bar{d}+(cd^\\beta /2)}+\\frac{d_2-d_1}{d_1+d_2+cd^\\beta }\\right)\\left(\\bar{d}+\\frac{cd^\\beta }{2}\\right) - \\gamma _k(\\underline{d})\\bar{d}+O\\left(\\sqrt{d\\ln d}\\,\\right),$ i.e., $\\mathbb {E}\\Delta =\\left(\\gamma _k\\!\\left(\\frac{(1-\\underline{d})cd^\\beta /2}{\\bar{d}+(cd^\\beta /2)}+\\underline{d}\\right)-\\gamma _k(\\underline{d})\\right)\\!\\!\\left(\\bar{d}+\\frac{cd^\\beta }{2}\\right) + \\gamma _k(\\underline{d})\\!\\left(\\bar{d}+\\frac{cd^{\\beta }}{2}-\\bar{d}\\right)+O\\left(\\sqrt{d\\ln d}\\right),\\nonumber $ i.e., $\\mathbb {E}\\Delta =\\gamma _k(\\underline{d})\\frac{cd^{\\beta }}{2}+\\frac{\\gamma _k\\left(\\frac{(1-\\underline{d})cd^\\beta /2}{\\bar{d}+(cd^\\beta /2)}+\\underline{d}\\right)-\\gamma _k(\\underline{d})}{\\frac{(1-\\underline{d})cd^\\beta /2}{\\bar{d}+(cd^\\beta /2)}}\\,\\frac{(1-\\underline{d})cd^\\beta }{2}+O\\left(\\sqrt{d\\ln d}\\right),$ above the idea can be informally summarized as: $\\delta (d\\gamma _k)\\approx d\\delta (\\gamma _k) + \\gamma _k\\delta (d)$ .", "Now, $d^\\beta /(\\bar{d}+(cd^\\beta /2)) \\rightarrow 0$ , as $d\\rightarrow \\infty $ and $\\gamma _k$ is concave, therefore $\\gamma _k^\\prime (0^-)\\!\\ge \\limsup _{d\\rightarrow +\\infty }\\frac{\\gamma _k\\!\\left(\\frac{(1-\\underline{d})cd^\\beta /2}{\\bar{d}+(cd^\\beta /2)}+\\underline{d}\\right)-\\gamma _k(\\underline{d})}{\\frac{(1-\\underline{d})cd^\\beta /2}{\\bar{d}+(cd^\\beta /2)}}\\!\\ge \\!\\liminf _{d\\rightarrow +\\infty }\\frac{\\gamma _k\\!\\left(\\frac{(1-\\underline{d})cd^\\beta /2}{\\bar{d}+(cd^\\beta /2)}+\\underline{d}\\right)-\\gamma _k(\\underline{d})}{\\frac{(1-\\underline{d})cd^\\beta /2}{\\bar{d}+(cd^\\beta /2)}}\\!\\ge \\!\\gamma _k^\\prime (0^+).$ Hence, $\\gamma _k^\\prime (0^-)\\frac{cd^\\beta }{2} +o(d^\\beta )\\ge \\frac{\\gamma _k\\left(\\frac{(1-\\underline{d})cd^\\beta /2}{\\bar{d}+(cd^\\beta /2)}+\\underline{d}\\right)-\\gamma _k(\\underline{d})}{\\frac{(1-\\underline{d})cd^\\beta /2}{\\bar{d}+(cd^\\beta /2)}}\\, \\frac{cd^\\beta }{2}\\ge \\gamma _k^\\prime (0^+)\\frac{cd^\\beta }{2}+o(d^\\beta ).$ Now, using (REF ) with (REF ) yields the desired order of magnitude for the expected gain: $(\\gamma _k(0)+\\gamma _k^\\prime (0^-))\\frac{cd^\\beta }{2}+o(d^\\beta )\\ge \\mathbb {E}\\Delta \\ge (\\gamma _k(0)+\\gamma _k^\\prime (0^+))\\frac{cd^\\beta }{2}+o(d^\\beta ).$ In particular, when $\\gamma _k^\\prime (0^+)=\\gamma _k^\\prime (0^-)=0$ , the above inequality becomes $\\mathbb {E}\\Delta = \\gamma _k(0)\\frac{cd^\\beta }{2}+o(d^\\beta ).$ As shown next, the order of magnitude of $\\Delta $ is, with high probability, the same as the order of its expectation.", "Indeed, in our context, the random variable $\\Delta $ is a function of the iid entries $V_1V_2\\ldots V_d$ and $W_1W_2\\ldots W_{d+cd^\\beta }$ .", "Changing only one of its entries, changes $\\Delta $ by at most 2 and so by Hoeffding's martingale inequality and setting $u=2d+cd^\\beta $ , $\\mathbb {P}(|\\Delta -\\mathbb {E}\\Delta |\\ge t)\\le 2\\exp \\left( -\\frac{2t^2}{u}\\right),$ for all $t>0$ .", "Moreover, integrating out (REF ) gives $\\mathop {\\rm Var}\\Delta \\le u=2d+cd^\\beta .$" ], [ "II) On the proportion of gaps aligned with the long constant block. ", "The previous arguments can now be used to understand the gaps aligned with the long block by an optimal alignment.", "Unlike in Part I), we consider here string $X$ and $Y$ of length $2d$ , which are iid except for $X$ containing a long constant block in its middle.", "Again, $\\pi $ is an optimal alignment of $X=X^aBX^c$ and $Y=Y^aY^bY^c$ aligning $X^a$ with $Y^a$ , $B$ with $Y^b$ and $X^c$ with $Y^c$ .", "Then, $\\pi $ is modified to get $\\bar{\\pi }$ which aligns all of $B$ with gaps and $X^a$ with $Y^aY^b$ .", "Again, the length of $B$ is $d^\\beta $ , with $1/2<\\beta <1$ , while the length of $Y^b$ is $h=cd^\\beta $ .", "Now, a somewhat over-simplified summary of the details of Section  is presented.", "Assuming $\\gamma _k$ satisfies, say, Condition 2, it is shown in Section ) that with high probability $Y^a$ and $Y^c$ have approximately the same length as $X^a$ and $X^c$ .", "In other words, with high probability, the four strings $X^a$ , $X^c$ , $Y^a$ and $Y^c$ have length $d+o(d)$ .", "So, the result of Part I applies to $X^a$ and $Y^a$ , implying that, with high probability, $|LCS(X^a,Y^aY^b)|-|LCS(X^a,Y^a)|=cd^\\beta \\frac{\\gamma _k(0)}{2}+o(d^\\beta ).$ Now, in the new alignment $\\bar{\\pi }$ , the letters of the long block which were not aligned with gaps but with symbols from $Y^b$ are lost, and the loss is approximately ${h}/{k}={cd^\\beta }/{k}$ .", "Moreover as mentioned earlier, the difference of the scores between $\\pi $ and $\\bar{\\pi }$ is made up of the increase due to “adding $Y^b$ to the alignment of $X^a$ and $Y^a$ ” minus the loss in letters from the block $B$ .", "Assuming $\\gamma _k^\\prime (0)=0$ and using (REF ), this difference is equal to: $&{\\tt score\\;of\\;} \\bar{\\pi }-{\\tt score\\;of\\;}\\pi =\\\\&(|LCS(X^a,Y^aY^b)|-|LCS(X^a,Y^a)|)\\;\\;-\\;\\;|LCS(B,Y^b)|=c d^\\beta \\left(\\frac{\\gamma _k(0)}{2} -\\frac{1}{k}\\right)+o(d^\\beta ).$ Hence, whenever $\\gamma _k(0)/2> {1}/{k}$ , the change from $\\pi $ to $\\bar{\\pi }$ typically increases the number of aligned letters and therefore $\\pi $ cannot be an optimal alignment.", "In that case, the long constant block cannot be aligned with a piece $Y^b$ whose length-order is linear order $d^\\beta $ .", "In other words, the long constant block is, with high probability, mainly aligned with gaps.", "On the other hand, when ${\\gamma _k(0)}/{2}<1/k$ , then the score of $\\pi $ is larger than the score of $\\bar{\\pi }$ .", "So, an alignment like $\\bar{\\pi }$ cannot be optimal in that case.", "In other words, when ${\\gamma _k(0)}/{2}<1/k$ then, with high probability, any optimal alignment aligns most letters of the long block $B$ with letters and not with gaps.", "Here “most letters” indicates that at most $o(d^\\beta )$ letters from the long block could get aligned with gaps.", "These results are explained next, assuming $\\gamma _k$ strictly concave in a neighborhood of the origin and having a derivative (equal to zero) at the origin.", "The same arguments, with minor changes, work as well without the strict concavity, assuming only that $\\gamma _k^\\prime (p_M)$ exists (and is therefore equal to zero).", "Using, in our developments, (REF ) rather than (REF ), the weaker condition: $\\left|\\frac{\\gamma _k^\\prime (p_M^+)}{2}\\right|< \\left|\\frac{\\gamma _k(0)}{2}-\\frac{1}{k}\\right|$ will also do.", "The difference in score between the alignment $\\pi $ and $\\bar{\\pi }$ is then $&{\\tt score\\;of\\;} \\bar{\\pi }-{\\tt score\\;of\\;}\\pi =\\\\&|LCS(X^a\\!,Y^aY^b)|\\!-\\!|LCS(X^a\\!,Y^a)|\\!-\\!|LCS(B,Y^b)\\!|=\\!cd^\\beta \\!\\!\\left(\\!\\!\\frac{\\gamma _k(0)}{2}\\!", "+\\xi (d)\\frac{\\gamma _k^\\prime (p_M^+)}{2} -\\frac{1}{k}\\!\\right)\\!\\!+\\!o(d^\\beta ),$ where $\\xi (d)$ can take any value between $-1$ and 1." ], [ "III) For which $k$ do we have {{formula:c8db4a47-ebba-4b35-bd50-99538ff510e8}} ?", "Whether a long constant block gets mainly aligned with gaps or not depends on $\\gamma _k(0)$ being smaller or larger than $1/k$ .", "It turns out, that $\\gamma _k(0)$ is smaller than $2/k$ only for binary strings, that is when $k=2$ .", "For every $k\\ge 3$ , the opposite is true.", "Despite the exact values of $\\gamma _k(0)$ not being known, there are rigorous bounds available, precise enough to show our assertions.", "Anyhow, for large $k$ , Kiwi, Loebl and Matoušek [8] have shown that $\\gamma _k(0)$ is of linear in $1/\\sqrt{k}$ , making $\\gamma _k(0)/2$ strictly larger than $1/k$ when $k$ is large enough.", "The case $k=3$ is near critical as can also be seen in our simulations.", "Taking the value of $0.717$ in Table (REF ), then $ \\gamma _3(0)/2=0.3585$ , which is slightly larger than $1/3$ , specially since the order of magnitude of the precision by which the values $\\gamma _k(0)$ are known is around $0.01$ ." ], [ "Statements of Results", "In this section, we precisely state results indicating that the differentiability of the function $\\gamma _k$ at $p_M$ controls the proportion of symbols from the long constant block which get aligned with gaps.", "A kind of zero-one law holds true depending on the size of the alphabet.", "Below, both sequences $X$ and $Y$ have length $2d$ , while the long block has length approximately equal to $d^\\beta $ , with $1/2 <\\beta <1$ , with $d$ large enough.", "We start with an example explaining how the aligned gaps are counted.", "For this, let $x:=00011100$ and let $y:=00011001$ .", "The first block of $x$ consists of three zeros, its second block consists of three ones, the third block consists of two zeros and the LCS of $x$ and $y$ is $LCS(x;y)=0001100,$ which corresponds to the alignment $\\begin{array}{l|l}x&000111 00\\\\\\hline y&00011\\;\\;001\\\\\\hline LCS&00011\\;\\;00\\end{array}$ In this alignment, the first block of $x$ is only aligned with symbols, the second is aligned with one gap and so $1/3$ of its symbols gets aligned with gaps, while the last block of $x$ is only aligned with symbols and so the proportion of its symbols aligned with gaps is zero.", "Let us next present two more examples to illustrate how with two letters, long constant blocks tend to be aligned with a proportion of gaps close to zero, while with three and more letters the opposite is true: For this consider first the two binary strings: $x=100101111{\\bf 00000}101101101$ and $y=01111001011011011101001$ .", "The alignment corresponding to the LCS is $\\begin{array}{l|l}x& 100101111{\\bf 00\\;\\;0\\;\\;\\;0\\;\\;\\;\\;0}10110110\\;\\;1\\\\\\hline y& \\;\\;0\\;\\;1\\;\\;111\\;\\;00101101101\\;\\;1101\\;\\;001\\end{array}$ Above, every 0 from the long block is aligned with a 0.", "Let us next consider an example with six letters, and let $x=65 324 214 444 41 235 6631$ and $y=55 425 153 112 422 255 656$ .", "The strings $x$ and $y$ in the previous example are “generated\" in the following way: Roll a fair six-sided die independently to obtain the strings everywhere except in the location of the long block.", "For the long block, i.e., for the piece $x_{8}x_9x_{10}x_{11}x_{12}$ , decide in advance to artificially introduce a long constant block: $x_8=x_9=\\cdots =x_{12}$ .", "Outside that piece, roll the six-sided die independently, hence, for $x_1x_2x_3x_4x_5x_6x_7$ , the die is rolled independently seven times, eight times for $x_{13}x_{14}x_{15}x_{16}x_{17}x_{18}x_{19} x_{20}$ and similarly fifteen times for the whole string $y$ .", "The alignment corresponding to the LCS 542112566 is $\\begin{array}{l|r}x& \\;\\; 6 5\\;\\;32 42\\;\\; 1\\;\\;\\;\\;\\;\\;{\\bf 44444}123\\;\\;\\;\\;\\;\\;\\;\\;\\;56\\;\\;63 1\\\\ \\hline y&\\;\\;55\\;\\;\\;\\;425 1531{\\bf \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;}12\\;\\;422255656\\;\\;\\;\\;\\end{array}$ The long block of five fours in $x$ is solely aligned with gaps.", "At this stage, formally describe the model with one inserted long constant block and sequences of length $2d$ .", "Let $X=X_1X_2\\ldots X_{2d}$ and $Y=Y_1Y_2\\ldots Y_{2d}$ be two independent strings of length $2d$ .", "A long constant block of length $\\ell $ is artificially inserted in the middle of the string $X$ , replacing an iid part of equal length.", "Thus (assuming $\\ell $ even), $\\mathbb {P}\\left(X_{d-(\\ell /2)+1}=X_{d-(\\ell /2)+2}=\\ldots =X_{d+(\\ell /2)-1}=X_{d+(\\ell /2)}\\right)=1.$ while the rest the strings are iid with $k$ equiprobable letters.", "(Hence, $Y$ , $X_1X_2\\ldots X_{d-(\\ell /2)}$ and $X_{d+(\\ell /2)+1}X_{d+(\\ell /2)+2}\\ldots X_{2d-1}X_{2d}$ are three independent iid strings with $\\mathbb {P}(X_i=j)=\\mathbb {P}(Y_i=j)=(1/k)$ , $j=1,2,\\ldots ,k$ , $i=1,2,\\ldots ,2d$ .)", "Next, let $\\beta $ and $\\alpha $ be constants independent of $d$ and such that $\\frac{1}{2}<\\alpha <\\beta <1,$ and let the length of the long constant block be $\\ell =d^\\beta $ .", "To formulate our first main result we further need two definitions: Let $E^d$ be the event that the long constant block is mainly aligned with gaps.", "More precisely, $E^d$ is the event that at most $d^{\\alpha }-1$ symbols of the long block get aligned with letters in any LCS-alignment.", "In other words, the score does not decrease by more than $d^{\\alpha }-1$ , when cutting out the long block: $E^d:=\\left\\lbrace |LCS(X_1X_2\\ldots X_{d-(\\ell /2)-1}X_{d-(\\ell /2)}X_{d+(\\ell /2)+1}\\ldots X_{2d};Y)|+d^{\\alpha }> LCS(X;Y)\\right\\rbrace .$ Let $K^d$ be the event that replacing the long constant block with iid symbols approximately increases the LCS length by $\\gamma ^*_k/2$ times the length of the long constant block.", "Formally, let $\\gamma _k^a$ be any constant, independent of $d$ , and strictly smaller than $\\gamma ^*_k$ , then $K^d$ is the event that when replacing the long constant block with iid symbols the length of the LCS increases by at least $(\\gamma _k^a/2)d^\\beta -d^{\\alpha }$ : $K^d:=\\left\\lbrace |LCS(X^*;Y)|-|LCS(X;Y)|\\ge \\frac{\\gamma _k^a}{2} d^\\beta -d^{\\alpha }\\right\\rbrace ,$ where $X^*$ denotes the string obtained from $X$ by replacing the long constant block by iid symbols.", "In other words, for $i\\in [1,d-(\\ell /2)]\\cup [d+(\\ell /2)+1,2d]$ , $X_i^*:=X_i$ .", "Moreover, the whole string $X^*=X^*_1X_2^*\\ldots X^*_{2d}$ is iid.", "We are now ready to formulate our main result for three or more letters.", "Theorem 3.1 Let $k\\gamma ^*_k>{2}$ , and let also the mean LCS function $\\gamma _k:(-1,1)\\rightarrow \\mathbb {R}$ , be differentiable at $p_M$ .", "Let $1/2 < \\alpha <\\beta <1$ .", "Then, there exist constants $C_E>0, C_K>0$ , independent of $d$ , such that $\\mathbb {P}(E^d)\\ge 1-e^{-C_Ed^{2\\alpha -1}},$ and $\\mathbb {P}(K^d)\\ge 1-e^{-C_Kd^{2\\alpha -1}},$ for all $d\\ge 1$ .", "To give the result for the two-letter case, some more definitions are needed.", "Let $G^d$ be the event that the long constant block gets mainly aligned with symbols and not with gaps.", "More precisely, $G^d$ is the event that the long constant block has (in any optimal alignment) at most $d^{\\alpha }$ of its symbols aligned with gaps.", "Equivalently, leaving out $d^{\\alpha }$ symbols from the long constant block decreases the LCS by at least one unit.", "Hence, $G^d:=\\left\\lbrace |LCS(X;Y)|>|LCS(X_1X_2\\ldots X_{d-(\\ell /2)}X_{d-(\\ell /2)+d^{\\alpha }+1}X_{d-(\\ell /2)+d^{\\alpha }+2}\\ldots X_{2d};Y)|\\right\\rbrace .$ ($X_1X_2\\ldots X_{d-(\\ell /2)}X_{d-(\\ell /2)+d^{\\alpha }+1}X_{d-(\\ell /2)+d^{\\alpha }+2}\\ldots X_{2d}$ is simply the string $X_1X_2\\ldots X_{2d}$ from which the piece $X_{d-(\\ell /2)+1}X_{d-(\\ell /2)+2}\\ldots X_{d-(\\ell /2)+d^{\\alpha }}$ has been removed.)" ], [ "Let $H^d$ be the event that\nreplacing the long constant block with iid symbols increases the LCS\nby at least {{formula:0feee1bd-dd30-425b-a5a0-d3808d85a282}} .", "Here ${\\tilde{c}}_H>0$ is any constant independent of $d$ and such that ${\\tilde{c}}_H<\\frac{3\\gamma ^*_2}{2}-1,$ and so $H^d:=\\left\\lbrace |LCS(X^*;Y)|-|LCS(X;Y)|\\ge {\\tilde{c}}_Hd^\\beta \\right\\rbrace .$ Let us next formulate our second main result for the two-letter case.", "Theorem 3.2 Let $k\\gamma ^*_k < 2$ , and let the mean LCS function $\\gamma _2: (-1,1)\\rightarrow \\mathbb {R}$ , be differentiable at $p_M$ .", "Then, there exist constants $C_G>0,C_H>0$ , independent of $d$ , such that $\\mathbb {P}(G^d)\\ge 1-e^{-C_Gd^{2\\alpha -1}},$ and $\\mathbb {P}(H^d)\\ge 1-e^{-C_Hd^{2\\alpha -1}},$ for all $d\\ge 1$ .", "The situation encountered for two letters might seem counter-intuitive at first.", "Let us explain why: Consider for this two binary sequences of length $n$ where one string is made out only of ones while the other is made out of equiprobable zeros and ones.", "Then the length of the LCS is the number of ones in the sequence with both symbols.", "Since both symbols have probability $1/2$ , the length of the LCS is approximately $1/2$ times the length of the strings.", "However, for two binary iid sequences, the average length of the LCS is about $0.8$ times the length.", "Hence, the LCS is much greater for two iid sequences, than when one sequence is made up of only one letter (i.e., one sequence is just “a long constant block\").", "Thus, one would think that when within a sequence one gets an exceptionally long constant block, this should typically decrease the total LCS.", "Hence, since a long constant block “scores\" much less than a typical piece of string iid drawn, one would expect that the long constant block tends to be “left out\" and not used too much (and hence tends to be aligned with many gaps).", "But the opposite is true!", "Also, in optimal alignment, similar strings tend to be matched.", "Since a long constant block, is very different from an iid string, it thus would seem that a long block should be “left out\" and mainly matched with gaps.", "This typically happens with three or more letters, but with two letters, the opposite is true.", "Let us next further explain the binary situation on a illustrative example with two strings aligned in three different ways.", "Let $x=100101111 00000 101101101$ and $y=011110010 11011 011101001$ be two strings of total length 23 with $x$ containing a long constant block 00000 of length $\\ell =5$ .", "Consider now three alignments of $x$ and $y$ .", "First, an alignment aligning the long block only with digits and with no gap: $\\begin{array}{c|c|c|c|c| c|c|c|c|c| c|c|c|c|c| c|c|c|c|c| c|c|c|c}x& &1&0&0&1&0&1111& &00& &0& &0& & 0& &1&0&1101&1&0& &1\\\\\\hline y& & &0& &1& &111 & &00&1&0&11&0&11& 0& &1& &1101& &0&0&1\\end{array}$ Here the long block 00000 gets aligned with 0010110110 having a length of 10 which is twice the length of the long block.", "This is to be expected since the probability of 0 is $1/2$ , and so a string of length approximately $2\\ell $ is needed to get $\\ell $ zeros.", "The above alignment can be viewed as consisting of three parts: the part to the left of the long block in $x$ , the aligned long block, and the part to the right of the long block.", "The part to the left aligns 5 letter-pairs, the long block also gives 5 letter-pairs and the piece to its right gives 7 of them.", "The total number of aligned letter-pairs in this alignment is thus 17.", "Let us next try as second alignment, an alignment aligning the long block with a piece of string of similar size, e.g., of length 7.", "For example, the alignment: $\\begin{array}{c|c|c|c|c| c|c|c|c|c| c|c|c|c|c| c|c|c|c|c| c|c|c|c|c| c|c|c|c}y& &0&1& &111&0& &0&1& &0 &11&0&11& & & 0&1& &1&1&01& &0& & &0&1\\\\\\hline x& & &1&00& 1 &0&1 1& &1&1&000& &0& & & & 0& & &1& &01&1&0&1&1&0&1\\end{array}$ This second alignment gives $4+3+6=13$ aligned letter-pairs which, as predicted, is a fewer number than the previous one: Indeed, when aligning the long block entirely with letters and no gaps, the score tends to be higher.", "In the second alignment, the long block gets aligned with the piece of string 0110110 and with two of the zeros aligned with gaps.", "Let us show, next, how to slightly modify this second alignment to provide a third alignment with an increased the total score.", "For this, take the two zeros from the long block which are aligned with gaps and align them with zeros from the string $y$ .", "To do so, take a piece of $y$ to the left of $y_9$ containing two zeros, i.e., take $y_6y_7y_8=001$ .", "Now align the two “unused” zeros, $x_{10} and x_{11}$ from the long block, with $y_6$ and $y_7$ .", "Aligning the two “unused zeros” leads to a score-gain of two, but at the same time to a loss, since previously $y_6y_7y_8$ was aligned with $x_5x_6\\ldots x_8=0111$ .", "So, the previous alignment of $y_6y_7y_8$ with $x_5x_6x_7x_8$ has been destroyed creating a score-loss of two.", "However, $x_5x_6x_7x_8$ is now “free,” and so can be “included” into the alignment of $y_1y_2\\ldots y_5$ with $x_1\\ldots x_4$ , meaning that $x_7x_8$ is aligned with $y_4y_5$ .", "This addition gives a score-increase of two, and the total score-change is $2-2+2=2$ .", "Let us represent in “toy” form the three phases of the evolution between the second and third alignment (only the part of the alignment which is been modified is shown below): $\\begin{array}{c|c|c|c|c| c|c|c|c|c| c|c|c }y& &0&1& &111&0& &0&1& & &\\ldots \\\\\\hline x& & &1&00& 1 &0&1 1& &1&1&00&\\ldots \\end{array}$ The first phase consists in aligning the two unused bits $x_{10}x_{11}$ from the long block with $y_7y_8$ : $\\begin{array}{c|c|c|c|c| c|c|c|c|c| c|c|c|c }y& &0&1& &111& & & & & &00&1&\\ldots \\\\\\hline x& & &1&00& 1 &0&1 1& &1&1&00& &\\ldots \\end{array}$ This phase leads to a gain of two aligned letters since $x_{10}x_{11}$ gets aligned with $y_7y_8$ , but at the same time to a loss of two aligned letter-pairs, since previously $x_5x_6x_7x_8x_9$ had two aligned letters and has none now.", "Next, “bring the string $x_5x_6x_7x_8x_9=01111$ into the alignment\": $\\begin{array}{c|c|c|c|c| c }y& &0&1& &111\\\\\\hline x& & &1&00& 1\\end{array}$ Now, in the alignment (REF ), two ones on the right-end of $y$ are free (e.g., $y_4y_5$ ) providing, when aligned with two of the ones from $x_5x_6x_7x_8x_9$ , two additional aligned letter-pairs with end result: $\\begin{array}{c|c|c|c|c| c|c|c|c|c| c|c|c|c }y& &0&1& &1 & &11 & & & &00&1&\\ldots \\\\\\hline x& & &1&00& 1 &0&1 1& &1&1&00& &\\ldots \\end{array}$ The total score change is $2>0$ and therefore the alignment (REF ) cannot be optimal.", "In the second alignment, two zeros from the long constant block are not aligned with symbols.", "Assume that instead of just 2, we would have $j$ , where $j$ is not too small.", "Then, to align these $j$ zeros with zeros from the string $y$ would require a string of length approximately $2j$ in $y$ (in order to find $j$ zeros each having probability approximately $1/2$ of occurring).", "(Above, the piece of string from $y$ with which the free zeros from the long block were aligned was $y_6y_7y_8$ and had length 3.)", "Before changing the alignment, these $2j$ bits from $y$ were most likely aligned with about $2j$ bits from $x$ .", "(Above, these bits are: $x_5x_6x_7x_8x_9$ .)", "When aligning these additional bits which became free, an approximate score-gain of $2j\\gamma ^*_2/2=j\\gamma ^*_2 \\approx 0.8j$ is to be expected (with two sequences of length $j$ and so a total of $2j$ bits, the score is approximately equal to $j\\gamma ^*_2$ ).", "Hence, the ratio score/bits is $\\gamma ^*_2/2$ .", "(Using this average is a purely heuristic, since there is no proof that adding bits on one side of an alignment only produces an average increase of $\\gamma ^*_2/2$ per bit.)", "Summing up: a) The new alignment of the $j$ free bits from the long block leads to a score-increase of $j$ .", "b) Undoing the previous alignment of the piece of string of $y$ which now gets aligned with the free bits of the long block, leads to an approximate loss of $2j\\gamma ^*_2$ bits since that piece has approximate length $2j$ .", "c) Realigning the piece of $x$ , which was previously aligned with the piece of $y$ getting now aligned with the free bits of the long block, leads to a score-gain.", "Since this last piece has an approximate length of $2j$ , the score-gain is approximately $2j\\gamma ^*_2/2=j\\gamma ^*_2j$ .", "Therefore, the total score-change is $j-2j\\gamma ^*_2+j\\gamma ^*_2=j-j\\gamma ^*_2\\approx 0.2j>0.$ From the 2-letter-strings examples presented above, the tendency is to align a long constant block with barely any gap.", "How much is then gained by replacing the long constant block by an iid piece?", "Here is an heuristic answer: the long block gives $\\ell $ units, but uses a piece of length $2\\ell $ in the $y$ -string.", "This piece becoming free leads to a gain of $2\\ell $ bits plus the $\\ell $ -bits from the long block.", "Hence using $3\\ell $ bits to get $\\ell $ points, and believing in the “average point/bit number hypothesis” for $\\gamma ^*_2/2$ , lead to an approximate gain of $3\\ell \\gamma ^*_2/2$ aligned letter pairs.", "That is after replacing the long constant block by an iid piece, and realigning all the $3\\ell $ bits which were previously used with the long block.", "Hence, approximately $3\\ell \\gamma ^*_2/2-\\ell \\approx 0.215\\ell $ additional letter-pairs are available; for example, a long block of length 20 would lead to an approximate average gain of 4.", "Extensive simulations, listed in the next section, demonstrate something very close." ], [ "Simulations and the Nature of Alignments", "It is very unlikely in an iid sequence of length $2d$ to find a constant block of length $d^\\beta $ .", "Typically, the blocks reach a length whose order is linear in $\\ln d$ .", "Nonetheless, our results proved for artificially inserted long blocks can be observed in simulations for naturally occurring block-lengths.", "Our simulations are presented below.", "The first table gives estimates for the expected number of gaps in a block of length $\\ell $ placed in the middle of a string of length 1000 (except for $\\ell >100$ where the string has length 4000) as a function of $k$ , the number of letters.", "Since several optimal alignments might exist, we chose the one putting a maximum number of gaps into the long block.", "Inserting a constant block with naturally-occurring length is similar to finding a constant block of that length in an iid sequence.", "Indeed, assume that there is a constant block of length $\\ell $ , $\\ell $ not too small.", "Then, until such a block appears in an iid equiprobable binary sequence, it takes an expected $2^{\\ell }$ letters.", "But, the contribution to the optimal alignment score of such a block would be at most $\\ell $ , which is much smaller than the amount of symbols needed before encountering that block.", "So, heuristically, the constant block of length $\\ell $ has very little effect on the optimal alignment.", "Hence, the optimal alignment should more or less determine which parts get aligned with each other without regard to the long constant block.", "This could then indicate that in terms of the number of gaps it gets aligned with, this long constant block behaves as if it had been artificially inserted.", "$\\begin{array}{c|c|c|c|c|c|c|c|c|c|c|c}&\\ell =1 &\\ell =2 &\\ell =5 &\\ell =10&\\ell =20&\\ell =30&\\ell =50&\\ell =100&\\ell =200&\\ell =300&\\ell =400\\\\\\hline k=2&0.53&1.67&2.25&2.75&4.2 &6.17&8.16&14.68&12.26&14.2 &19.6\\\\\\hline k=3& & &2.85&4.6 &12.5&18 &32.3&70.64&152.6&226 &\\\\\\hline k=4&0.72&1.19&3.27&6.78&16.3&25.6&43.8&88.4 & & &\\\\\\hline k=5& &1.6 &3.36&7.76&16.3&27.1&49.7&96.2 & & &\\\\\\hline k=6& &1.43&3.67&8.32&17.2&28.2&47.7&97.1 & & & \\\\\\hline k=7& &1.53&3.82&8.6 &18.7&27.9&48.6&98.1 & & & \\\\\\hline k=9& & &4.23&8.7 &18.4&29.2&48.4& & & &\\end{array}$ For each entry 100 independent simulations are run.", "For each simulation, we find the number of gaps the block of length $\\ell $ gets aligned with and then compute the average of that number over the 100 simulations.", "This gives the entries of the above table.", "The next table provides estimates for the ratio of the expected number of gaps and the length of the block.", "Therefore, the next table is obtained from the previous one by dividing each entry by the value $\\ell $ corresponding to its column.", "The entries in the next table thus represent the “proportion of gaps” in the long blocks depending on the length of the long block: $\\begin{array}{c|c|c|c|c|c|c|c|c|c|c|c}&\\ell =1 &\\ell =2 &\\ell =5 &\\ell =10&\\ell =20&\\ell =30&\\ell =50&\\ell =100&\\ell =200&\\ell =300&\\ell =400\\\\\\hline k=2&0.53&0.83&0.45&0.27&0.21&0.20&0.16&0.14 &0.06 &0.04 &0.04\\\\\\hline k=3& & &0.19&0.15&0.62&0.6 &0.64&0.7 &0.76 &0.75 & \\\\\\hline k=4&0.72&0.59&0.65&0.67&0.81&0.85&0.87&0.88 & & &\\\\\\hline k=5& &0.8 &0.67&0.77&0.81&0.90&0.99&0.96 & & &\\\\\\hline k=6& &0.7 &0.67&0.83&0.86&0.94&0.95&0.97 & & & \\\\\\hline k=7& &0.75&0.76&0.86&0.93&0.93&0.97&0.98 & & & \\\\\\hline k=9& & &0.8 &0.87&0.92&0.97&0.96&\\end{array}$ As seen above, with two letters, the proportion of gaps decreases as the length of the block increases, while for $k\\ge 3$ the opposite is true ($k=3$ seems to be a close to the critical point, so this phenomenon kicks in only slowly).", "Even for small block-length such as $\\ell =5$ , this zero-one law seems to occur and, therefore, the micro-structure of the optimal alignment seems rather different for $k=2$ or $k\\ge 3$ .", "Which heuristic argument could explain that the result for artificially inserted long blocks result implies a similar one for iid sequences?", "The simulations show that for naturally occurring long constant blocks, the phenomenon proved for artificially inserted ones continue to hold.", "Now, for a block of length $\\ell _B$ much smaller than $d^\\beta $ take the neighborhood of size $\\ell _B^{1/\\beta }$ of that block.", "In the optimal alignment of $X$ and $Y$ , that neighborhood should also typically be aligned optimally.", "So for that part of the alignment our results should apply.", "Let us present an example: In the simulations when simulating the sequences $X$ and $Y$ of length 1000, replace in the sequence $X$ a piece of length 10 by a block of length 10 somewhere in the middle of $X$ , and then count the number of gaps it gets aligned with.", "An approach which would yield very similar results, would consist in finding the block of length 10 closest to the middle of $X$ and then counting the number of gaps that block is aligned with in an optimal alignment.", "In simulations, by repeating these two operations a great number of times, in order to estimate the expected number of gaps a block of length 10 gets aligned with, we find no significant difference between the two methods.", "Heuristically, this lead to an important consequence: Simulations seem to demonstrate that the results, on the proportion of aligned gaps in iid sequences with artificially inserted long blocks, for the naturally appearing long blocks appearing in an iid sequence continue to hold for the naturally appearing long blocks in an iid sequence.", "Let us next display results giving the different numbers of gaps obtained at each simulation run.", "This should provide the reader with a sense for the order of the variance of the number of gaps in long blocks, when the length of the long block is held fixed.", "Below $i$ is the result obtained with the $i$ -th simulation.", "Only blocks of length $\\ell =100$ are considered in the next table.", "$\\begin{array}{c|c|c|c|c| c|c|c|c|c|c}&i=1 &i=2 &i=3&i=4&i=5&i=6&i=7&i=8&i=9&i=10\\\\\\hline k=2&2 &1 &2 &31 &9 &0 &1 &3 &5 &7 \\\\\\hline k=3& 100&97 &30 &66 &76 &79 & 73&93 &74 &91 \\\\\\hline k=4&98 & 98 &99 &100&99 &93 &99&99 &100&60\\\\\\hline k=8&99 &100 &100&95 &100&99 &98&98 &100&99 \\\\\\hline \\end{array}$ $$ Let us further examine some of the entries in the table right above.", "For $k=4$ letters, two out of the ten simulations give 100 gaps, four out of the ten give 99 gaps, and once the much lower value of 60 gaps.", "This seems to indicate that the number of gaps has a strongly skewed distribution.", "Above the median estimate is $98.5$ which should be compared with the estimated expected number of gaps $88.4$ given in the first table.", "For the two-letter case, the respective estimates are $2.5$ and $14.68$ .", "It thus appears that to take into account this skewness, a median estimation might be more appropriate than an expectation estimate and the discrepancy between the two-letter situation and the situation with more letters becomes even more pronounced when looking at the median.", "The entries in the next table give the difference between the length of the LCSs when replacing the long block with iid entries in sequences of length $2d=1000$ .", "Again, $\\ell $ is the block length and $k$ the number of letters.", "For each entry 100 simulation runs are averaged.", "Here and below the results for the small values of $\\ell $ are displayed to show the progression the behavior as $\\ell $ increases.", "$\\begin{array}{c|c|c|c|c|c|c|c|c|c|c|c}&\\ell =1 &\\ell =2 &\\ell =5 &\\ell =10&\\ell =20&\\ell =30&\\ell =50&\\ell =100&\\ell =200&\\ell =300&\\ell =400\\\\\\hline k=2&-0.01&0.06&0.52&0.9 &2.88&4.7 &9.48&21.8 &44.3 &73.5 &88.5 \\\\\\hline k=3& & &0.45&1.36&4.55&7.62&14.5&32.8 &68.9 & & \\\\\\hline k=4&0.03&0.06&0.59&1.85&5.3 &8.86&14.32&31.4 & & &\\\\\\hline k=5& &0.2 &0.58&1.78&4.88&7.81&14.18&29.9 & & &\\\\\\hline k=6& &0.1 &0.53&1.86&4.42&7.7 &12.8 &27.9 & & & \\\\\\hline k=7& &0.13&0.7 &2.05&4.7&7.3 &13.1 &27.28& & &\\\\\\hline k=9& & &0.7 &1.85&4.33&7.26&11.6 & & & &\\end{array}$ The next tables display the values to expect, from our heuristic arguments, for the typical increase in LCS for long constant blocks and the values obtained through simulations.", "To start, let $k=2$ , in which case our predicted change in LCS due to the replacement of the long block of length $\\ell $ is $(\\gamma ^*_2/2)3\\ell -\\ell \\approx 0.215\\ell $ .", "$\\begin{array}{c|c|c|c|c|c|c|c|c|c|c|c}&\\ell =1 &\\ell =2 &\\ell =5 &\\ell =10&\\ell =20&\\ell =30&\\ell =50&\\ell =100&\\ell =200&\\ell =300&\\ell =400\\!\\!\\\\\\hline 0.215\\ell &0.215&0.43&1.07&2.15&4.3 &6.45&10.7&21.5 &43 &64.5 &86\\\\\\hline \\widehat{\\mathbb {E}}\\Delta &-0.01&0.06&0.52&0.9 &2.88&4.7 &9.48&21.8 &44.3 &73.5 &88.5 \\\\\\hline \\end{array}$ Let us next compare simulated values with predicted values for 4 letters alphabet where an increase of $(\\gamma ^*_4/2)\\ell \\approx 0.325\\ell $ is expected.", "$\\begin{array}{c|c|c|c|c|c|c|c|c|c|c|c}&\\ell =1 &\\ell =2 &\\ell =5 &\\ell =10&\\ell =20&\\ell =30&\\ell =50&\\ell =100&\\ell =200&\\ell =300&\\ell =400\\\\\\hline 0.325\\ell &0.325&0.65 &1.62&3.25&6.5&9.7&16.2&32.5 & & &\\\\\\hline \\widehat{\\mathbb {E}} \\Delta &0.03 &0.06&0.59&1.85&5.3 &8.86&14.32&31.4 & & &\\\\\\hline \\end{array}$ $$ Comparisons of simulated values with predicted values in case of 5 letters, where an increase of $(\\gamma ^*_5/2)\\ell \\approx 0.305\\ell $ is expected, are displayed next.", "$\\begin{array}{c|c|c|c|c|c|c|c|c|c|c|c}&\\ell =1 &\\ell =2 &\\ell =5 &\\ell =10&\\ell =20&\\ell =30&\\ell =50&\\ell =100&\\ell =200&\\ell =300&\\ell =400\\\\\\hline 0.305\\ell & &0.61&1.52&3.05&6.1 &9.15&15.2 &30.5& &&\\\\\\hline \\widehat{\\mathbb {E}}\\Delta & &0.2 &0.58&1.78&4.88 &7.81&14.18&29.9 & & &\\\\\\hline \\end{array}$ $$ Finally, for the 7-letter case the increase is expected to be $(\\gamma ^*_7/2)\\ell \\approx 0.27\\ell $ .", "$\\begin{array}{c|c|c|c|c|c|c|c|c|c|c|c}&\\ell =1 &\\ell =2 &\\ell =5 &\\ell =10&\\ell =20&\\ell =30&\\ell =50&\\ell =100&\\ell =200&\\ell =300&\\ell =400\\\\\\hline 0.27\\ell & &0.54 &1.35&2.70&5.4 &8.10&13.5&27.00& & &\\\\\\hline \\widehat{\\mathbb {E}}\\Delta & &0.13&0.7 &2.05&4.7&7.3 & 13.1 &27.28 & & &\\\\\\hline \\end{array}$ $$ As seen above, with more letters, the approximation is already quite good for blocks of lesser size." ], [ "The Proofs", "Throughout this section we are in the setting of Section : $X$ and $Y$ are two independent random sequences of length $2d$ , the string $Y$ is iid while the string $X$ has a long constant block of size $\\ell $ (even) in its middle: $\\mathbb {P}\\left(X_{d-(\\ell /2)+1}=X_{d-(\\ell /2)+2}=\\ldots =X_{d+(\\ell /2)-1}=X_{d+(\\ell /2)}\\right)=1,$ and is iid everywhere else.", "Moreover, the symbols are equally likely on an alphabet of size $k$ ." ], [ "Proofs For Three Letters or More", "In this subsection, assume that $k\\gamma ^*_k>2.$ To start with, some heuristic arguments are given to explain why under the condition (REF ), and in any optimal alignment, the artificially inserted long constant block is mainly aligned with gaps (see also Part I and Part II in Section ), the proofs then follow.", "As far as the heuristics is concerned, proceed by contradiction.", "Indeed, assume on the contrary that there is an optimal alignment $\\pi $ with $m$ symbols from the long constant block aligned with symbols.", "Then, by equiprobability, in order to get $m$ times the same letter in a contiguous substring of $Y$ , typically requires a piece of length approximately equal to $km$ .", "Therefore, if $m$ symbols from the long constant block get aligned with symbols, then typically a piece of $Y$ of length approximately equal to $km$ is required.", "Next, modify the alignment $\\pi $ .", "To do so, take the piece of $Y$ which was used for the $m$ symbols of the long constant block and align it otherwise.", "Let $\\bar{\\pi }$ be the new alignment obtained in this way.", "In this way, $m$ aligned letters from the long constant block are lost but realigning the $km$ symbols of $Y$ adds approximately $km(\\gamma ^*_k/2)$ aligned symbols elsewhere.", "So the change is approximately $\\frac{\\gamma ^*_k}{2}km-m=m\\left(\\frac{k\\gamma ^*_k}{2}-1\\right).$ But from (REF ), $k\\gamma ^*_k >2$ , and so the change due to realigning the $km$ symbols from $Y$ outside the long block, typically leads to an increase in the number of aligned symbols.", "Hence, $\\pi $ aligns fewer letter-pairs than $\\bar{\\pi }$ , and therefore $\\pi $ cannot be an optimal alignment.", "Let us now proceed to the formal arguments and to do so, recall that in Section  we defined: $E^d$ , the event that the long constant block is mainly aligned with gaps: $E^d=\\left\\lbrace \\!|LCS(X_1X_2\\ldots X_{d-(\\ell /2)-1}X_{d-(\\ell /2)}X_{d+(\\ell /2)+1}\\ldots X_{2d};Y)|+d^{\\alpha }\\!> LCS(X;Y)\\!\\right\\rbrace .$ $K^d$ , the event that replacing the long constant block with iid symbols leads to an approximate length-increase of $\\gamma ^*_k/2$ times the length of the long constant block: $K^d=\\left\\lbrace |LCS(X^*;Y)|-|LCS(X;Y)|\\ge \\frac{\\gamma _k^a}{2} d^\\beta -d^{\\alpha }\\right\\rbrace .$ We intend to prove that if $\\gamma ^*_k/2>1/k$ , then both events $E^d$ and $K^d$ occur with high probability, i.e., we intend to prove Theorem REF .", "For this proceed as follows: First define four events $B^d$ , $C^d$ , $D^d$ and $F^d$ and in Lemma REF prove that $B^d\\cap C^d\\cap D^d\\subset E^d,$ why Lemma REF , REF and REF respectively show that $B^d$ , $C^d$ and $D^d$ occur with high probability and thus so does $E^d$ .", "Next, Lemma REF show that $B^d\\cap C^d\\cap D^d\\cap F^d \\subset K^d,$ while Lemma REF shows that $F^d$ occurs with high probability and, thus, so does $K^d$ .", "Recall also that $\\alpha $ and $\\beta $ and reals independent of $d$ , such that $1/2 <\\alpha <\\beta <1$ ; that $d^{\\beta }$ is the length of the artificially inserted long constant block and that $d^{\\alpha }$ is the maximum number of symbols, from the long constant block, which can get aligned with symbols instead of gaps.", "Finally, for $p\\in (-1,1)$ , recall the definitions of $\\gamma _k(n,p)$ and $\\gamma _k(p)$ as respectively given in (REF ) and (REF ) and the definition of $p_M$ given towards the end of the introductory section.", "Let us next introduce some more notations.", "Let $\\kappa $ , $\\gamma _k^a$ , $\\gamma _k^b$ and $\\gamma _k^c$ be constants, independent of $d$ , such that $\\frac{2}{k}<\\frac{2}{\\kappa }<{\\gamma _k^a}<{\\gamma _k^b}<\\gamma _k^c<{\\gamma ^*_k}.$ One thinks of $\\kappa $ as being approximately equal to $k$ while $\\gamma _k^a$ , $\\gamma _k^b$ and $\\gamma _k^c$ are all very close to $\\gamma ^*_k$ .", "Let $q\\in (0,1)$ be such that $\\gamma _k(-q)=\\gamma _k(q)=\\gamma _k^c,$ with also $\\gamma _k^\\prime (q^+) = \\gamma _k^\\prime (q^-)$ .", "(The concavity of $\\gamma _k$ , clearly ensures that such a $q$ exists and is also such that $\\forall r\\in [-q,q],\\quad \\gamma _k(r)\\ge \\gamma _k^c.", ")$ Assume also, for $k\\ge 2$ , that for all $p_1,p_2\\in [-q,q]$ , $\\left| \\frac{\\gamma _k(p_2)-\\gamma _k(p_1)}{p_2-p_1} \\right|< \\frac{(\\gamma ^c_k-\\gamma ^b_k)}{16}\\,,$ Since the derivative at $p_M$ exists and is therefore zero, it is always possible to determine $\\gamma _k^b$ and $\\gamma _k^c$ so that (REF ) and (REF ) simultaneously hold.", "For this simply keep $\\gamma ^b_k$ fixed and let $\\gamma _k^c$ converge from below to $\\gamma ^*_k$ .", "When $\\gamma _k^c$ gets close enough to $\\gamma _k(0)=\\gamma ^*_k$ , then the conditions are fulfilled.", "In case $k=2$ , assume further that $\\gamma ^{II}_2$ is such that $\\gamma _2^\\prime (q)\\le \\frac{\\gamma ^{II}_2-\\gamma ^*_2}{16},$ and that $\\tilde{\\gamma _2}$ is such that $\\frac{|\\gamma _2(p_2) - \\gamma _2(p_1)|}{|p_2-p_1|}\\le \\frac{\\gamma _2^c-\\tilde{\\gamma _2}}{32},$ for all $p_1, p_2 \\in [-q,q]$ .", "Both the above conditions are satisfied from our assumptions on the derivative of $\\gamma _2$ (for example, take $\\tilde{\\gamma }_2$ close but smaller than $\\gamma _2^*$ .", "Then, let $q \\rightarrow 0$ and take $\\gamma _2^c$ closer and closer to $\\gamma _2^*$ till (REF ) is satisfied).", "Let $i_1$ and $i_2$ be the respective integer rounding of each right-hand side below: $i_1:=\\frac{1-q}{1+q}\\left(d-\\frac{\\ell }{2} \\right),$ and $i_2:=\\frac{1+q}{1-q}\\left(d-\\frac{\\ell }{2} \\right),$ where again $\\ell :=d^\\beta $ (is even) and $1/2<\\beta <1$ does not depend on $d$ .", "Clearly, both $i_1$ and $i_2$ both depend on $d$ .", "Moreover, whenever $i\\in [i_1,i_2]$ , then $\\frac{\\mathbb {E}|LCS(X_1X_2\\ldots X_{d-(\\ell /2)}; Y_1Y_2\\ldots Y_i)|}{d^*}= \\gamma _k(d^*,r),$ with $d^*=(d-(\\ell /2)+i)/2$ and $r=(i-d+\\ell /2)/(i+d-\\ell /2)$ .", "Since $r\\in [0,q]$ , if $i\\ge d -\\ell /2$ while $r\\in [-q,0]$ , if $i\\le d -\\ell /2$ , it also follows that $\\gamma _k(r) \\ge \\gamma _k^c$ ." ], [ "Let $B^d$ be the event that to find {{formula:a17937cb-20d8-44a9-ad3a-7dac17ccbaa5}} times the same symbol\nin {{formula:f4ebfa62-8f95-4cd6-ad4c-669f11d8037c}} , a piece of length at least {{formula:2ba539a2-5566-40cf-9caf-63bf38af93cf}} is needed.", "More precisely, let $B^d(i,h)$ be the event that, in the string $Y_iY_{i+1}\\ldots Y_{i+h}$ , every letter appears at most $h/\\kappa $ times.", "For this, let $r\\in \\lbrace 1,2,\\ldots ,k\\rbrace $ and let $W_j(r)$ be the Bernoulli random variable which is equal to one if $Y_j=r$ and zero otherwise.", "With these notations, let $B^d(i,h) := \\left\\lbrace \\forall r=1,2,\\ldots ,k: \\sum _{j=i}^{i+h}W_j(r)\\le \\frac{h}{\\kappa }\\right\\rbrace ,$ and let $B^d:=\\bigcap _{i\\in [1,2d]}B^{d}(i,h).$" ], [ "Let $C^d$ be the event that for every {{formula:09e9b4d6-5936-4781-9b42-a73490daecf6}} , the length of the optimal\nalignment between {{formula:2b56677b-743c-4909-a73f-632fa3a067c9}} and {{formula:4ee2518e-0840-4d19-9bc6-bc4ae520cdf5}} ,\n{{formula:12ce9239-0241-4ce4-b53c-036fc7e45b65}} , is larger, by at least {{formula:1b1a0f5e-ed1d-45c2-8ade-ae98774b9ce4}} , than the length of the optimal\nalignment between {{formula:f3547287-bd41-42b4-9327-5941ebd124f0}} and {{formula:6967fe2b-5cf5-4630-bde9-199cdab7b1c6}} .", "More precisely, let $C_R^d(i,h)$ be the event that by concatenating $h$ letters to the right of $Y_1Y_2\\ldots Y_i$ , the LCS with $X_1X_2\\ldots X_{d-(\\ell /2)}$ increases by at least $h\\gamma ^a_k/2$ .", "Hence the event $C_R^d(i,h)$ holds when $|LCS(X_1X_2\\ldots X_{d-(\\ell /2)};Y_1Y_2\\ldots Y_{i+h})|-|LCS(X_1X_2\\ldots X_{d-(\\ell /2)};Y_1Y_2\\ldots Y_i)|\\ge \\frac{\\gamma ^a_kh}{2},$ and $C_R^d:=\\bigcap _{i\\in [i_1,i_2]}C_R^d(i,h),$ where $i_1$ and $i_2$ are defined in (REF ).", "In a similar fashion, define $C_L^d(i,h)$ to be the event that by concatenating $h$ letters to the left of $Y_1Y_2\\ldots Y_i$ , the LCS with $X_1X_2\\ldots X_{d-(\\ell /2)}$ increases by at least $h\\gamma ^a_k/2$ , and then, as above, one defines $C_L^d$ .", "Finally, let $C^d = C_R^d \\cap C_L^d$ ." ], [ "Let $D^d$ be the event that any optimal alignment aligns\n{{formula:21bc8ab7-b684-45d9-afee-f0ba51260c0f}} into the interval {{formula:060f50cf-3168-46ce-a8ef-9b1545b04ed2}} .", "To define $D^d$ precisely, let us first set a convention: when an alignment $\\pi $ aligns $X_i$ with $Y_j$ , then $X_i$ is said to be aligned with $j$ under $\\pi $ .", "If $\\pi $ aligns $X_i$ with a gap, then let $i_I$ be the largest $m < i$ such that $X_m$ gets aligned with a symbol and not with a gap.", "If $X_{i_I}$ gets aligned with $Y_j$ , then $X_{i_I}$ is said to be aligned with $j$ under $\\pi $ .", "Next, let $D^d_I$ be the event that for any optimal alignment $\\pi $ of $X$ and $Y$ , if $i$ is the spot where $\\pi $ aligns $X_{d-(\\ell /2)}$ , then $i\\in [i_1,i_2]$ .", "Similarly, let $D^d_{II}$ be the event that for any optimal alignment $\\pi $ of $X$ and $Y$ , if $i$ designates the spot where $\\pi $ aligns $X_{d-(\\ell /2)+\\kappa d^{\\alpha }}$ , then $i\\in [i_1,i_2]$ .", "Finally, let $D^d:=D^d_I\\cap D^d_{II}.$" ], [ "Let $F^d$ be the event that for every {{formula:d96ab47f-a9b6-426d-ad38-c7236cc45838}} , the length of the optimal\nalignment between {{formula:5623022f-cd00-4af7-8740-b86407d7276d}} and {{formula:5d96c25c-e357-4c75-89c6-2bda2a3a7dcb}} \nis larger, by at least {{formula:28882f54-b781-411f-8bb1-56b4ee87da0d}} , than the length of the optimal\nalignment between {{formula:77e75588-aaae-448c-9d6f-b3bcaddf3d73}} and {{formula:74ee571f-90ae-4f56-b87c-ca007376957a}} .", "More precisely, $F^d := \\bigcap _{i\\in [i_1,i_2]}F^d_i,$ where $F^d_i:=\\lbrace |LCS(X_1^*X_2^*\\ldots X_{d-(\\ell /2)+d^\\beta }^*&;Y_1Y_2\\ldots Y_i)| - \\\\& |LCS(X_1^*X_2^*\\ldots X_{d-(\\ell /2)}^*;Y_1Y_2\\ldots Y_i)| \\ge \\frac{\\gamma ^a_kd^\\beta }{2} \\rbrace .$ We now prove the first combinatorial lemma of this subsection: Lemma 5.1 $B^d\\cap C^d\\cap D^d\\subset E^d.$ Proof.", "The proof is by contradiction and so assume that $E^d$ does not hold.", "Then, there is an optimal alignment $\\pi $ for which there are at least $d^{\\alpha }$ letters, from the long constant block, which are not aligned with gaps.", "Moreover, without loss of generality, assume that these letters are at the beginning of the block.", "However, when the event $B^d$ holds true, the first $d^{\\alpha }$ letters from the long block are aligned with a portion of $Y_1Y_2\\ldots Y_{2d}$ of length at least $\\kappa d^{\\alpha }$ .", "In other words, there exists $i\\in [1,2d]$ such that the optimal alignment $\\pi $ aligns the first $d^{\\alpha }$ letters from the long block with $Y_{i+1}\\ldots Y_{i+t}$ , where $t\\ge \\kappa d^{\\alpha }$ .", "Therefore, the optimal alignment $\\pi $ only aligns $Y_{i+1}\\ldots Y_{i+t}$ with those first $d^{\\alpha }$ letters from the long block.", "Next, since $D^d$ holds true, assume that $i\\in [i_1,i_2]$ .", "Now, modify the alignment $\\pi $ so as to no longer align these $d^{\\alpha }$ letters from the long block with $Y_{i+1}\\ldots Y_{i+t}$ .", "In doing so, $d^{\\alpha }$ aligned letters are lost.", "In turn, $Y_{i+1}\\ldots Y_{i+t}$ can be realigned with a part of $X$ outside the long block.", "In other words, align now $Y_1Y_2\\ldots Y_{i+t}$ entirely with $X_1X_2\\ldots X_{d-(\\ell /2)}$ .", "But, the event $C^d$ guarantees, since $t\\ge \\kappa d^{\\alpha }$ , a gain of at least $\\kappa d^{\\alpha }\\gamma _k^a/2$ .", "Summing up the losses and the gains, obtained in modifying $\\pi $ , lead to an increase of at least $\\frac{\\kappa d^{\\alpha }\\gamma _k^a}{2}-d^{\\alpha }=d^{\\alpha }\\left(\\frac{\\kappa \\gamma _k^a}{2}-1 \\right) > 0,$ by (REF ).", "Therefore, $\\pi $ is not optimal which is a contradiction.", "Let us now state and prove a second combinatorial lemma recalling that $X^*$ denotes the string obtained from $X$ by replacing the long constant block by iid symbols.", "Lemma 5.2 $B^d\\cap C^d\\cap D^d\\cap F^d\\subset K^d.$ Proof.", "By the previous lemma, when the events $B^d$ , $C^d$ and $D^d$ hold true, any optimal alignment aligns at most $d^{\\alpha }$ letters, from the long block, with letters.", "Let $\\pi $ be an optimal alignment of $X$ and $Y$ , then $\\pi $ aligns at least $d^\\beta -d^{\\alpha }$ symbols from the long block with gaps.", "Assume that $X_{d-(\\ell /2)}$ gets aligned with $Y_i$ by $\\pi $ .", "Now, transform $\\pi $ into a new alignment $\\bar{\\pi }$ aligning $X^*$ and $Y$ in the following manner: Instead of aligning $X_1^*X_2^*\\ldots X_{d-(\\ell /2)}^*$ with $Y_1Y_2\\ldots Y_i$ , concatenate $X^*_{d-(\\ell /2)+1}X^*_{d-(\\ell /2)+2}\\ldots X^*_{d+(\\ell /2)}$ to the $X$ -part, and align $Y_1Y_2\\ldots Y_i$ with $X_1^*X_2^*\\ldots X_{d+(\\ell /2)}^*$ in an optimal way, i.e., in such a way that any chosen alignment corresponds to a LCS of $Y_1Y_2\\ldots Y_i$ and $X_1^*X_2^*\\ldots X_{d+(\\ell /2)}^*$ .", "Next, for the remaining letters of the strings $X^*$ and $Y$ , use the alignment $\\pi $ .", "Hence, if $m\\in [i+1,2d]$ and $n\\in [d+(\\ell /2)+1,2d]$ and if $\\pi $ aligns $X_n$ with $Y_m$ , then $\\bar{\\pi }$ aligns $X_n^*$ with $Y_m$ .", "Since $D^d$ holds, $i\\in [i_1,i_2]$ and since $F^d$ holds, concatenating $X^*_{d-(\\ell /2)+1}X^*_{d-(\\ell /2)+2}\\ldots X^*_{d+(\\ell /2)}$ to the $X$ -part leads to a score-increase of at least $(\\gamma _k^a/2)d^\\beta $ .", "But the transformation of $\\pi $ into $\\bar{\\pi }$ could decrease the score.", "Indeed, up to $d^{\\alpha }$ letters, from the long block, could under $\\pi $ have not been aligned with gaps (and thus could have been aligned with letters).", "Therefore, replacing the long block by $X^*_{d-(\\ell /2)+1}X^*_{d-(\\ell /2)+2}\\ldots X^*_{d+(\\ell /2)}$ might lead to a loss not exceeding $d^{\\alpha }$ aligned letter-pairs.", "Summing up the losses and the gains, obtained in modifying $\\pi $ , lead to an increase of at least $\\frac{\\gamma _k^a}{2} \\; d^\\beta -d^{\\alpha }.$ This proves that the event $K^d$ holds and finishes this proof.", "Let us now show that $B^d$ , $C^d$ , $D^d$ and $F^d$ all occur with high probability.", "For $C^d$ , let us start with the following: Lemma 5.3 Let $\\ell :=d^\\beta $ .", "For all $d$ large enough and all $i\\in [i_1,i_2]$ , $\\mathbb {E}\\!\\left(|LCS(X_1^*X_2^*\\ldots X^*_{d-(\\ell /2)};Y_1Y_2\\ldots Y_{i+h})|-|LCS(X_1^*X_2^*\\ldots X^*_{d-(\\ell /2)};Y_1Y_2\\ldots Y_i)|\\right)\\!\\ge \\frac{h\\gamma ^b_k}{2},$ where $h=\\kappa d^{\\alpha }$ .", "Proof.", "Let $\\Delta := |LCS(X_1^*X_2^*\\ldots X^*_{d-(\\ell /2)};Y_1Y_2\\ldots Y_{i+h})|-|LCS(X_1^*X_2^*\\ldots X^*_{d-(\\ell /2)};Y_1Y_2\\ldots Y_i)|,$ and assume at first that $i\\in [i_1, d]$ (see (REF )).", "By definition, $\\mathbb {E}\\Delta =d_2\\gamma _k(d_2,p_2)-d_1\\gamma _k(d_1,p_1),$ where $d_2:=&\\frac{1}{2} \\left(i+d-\\frac{\\ell }{2}+h\\right)\\!, \\, \\quad p_2:=\\frac{i-d+(\\ell /2)+h}{i+d-(\\ell /2)+h},\\\\d_1:=&\\frac{1}{2} \\left(i+d-\\frac{\\ell }{2}\\right), \\quad \\quad \\quad p_1:=\\frac{i-d+(\\ell /2)}{i+d-(\\ell /2)}.$ From Alexander [2] (see also (REF )), there exists a constant $C_\\gamma >0$ (independent of $d$ and $p$ ) such that $|\\gamma _k(d,p)-\\gamma _k(p)|\\le C_\\gamma \\sqrt{\\frac{\\ln d}{d}}\\,,$ for all $p\\in (-1,1)$ and all $d\\ge 1$ .", "Using (REF ) and since $d_1,d_2\\le 2d$ , $\\mathbb {E}\\Delta \\ge d_2\\gamma _k(p_2)-d_1\\gamma _k(p_1)-2C_\\gamma \\sqrt{2d \\ln 2d}.$ Now, $d_2\\gamma _k(p_2)-d_1\\gamma _k(p_1)=\\frac{h\\gamma _k(p_1)}{2}+d_2{\\delta \\gamma _k},$ where $\\delta \\gamma _k=\\gamma _k(p_2)-\\gamma _k(p_1).$ By the concavity and the symmetry of $\\gamma _k$ , if $p_2\\le 0$ , and since $p_1 < p_2$ , then $\\delta \\gamma _k\\ge 0$ so that $d_2\\gamma _k(p_2)-d_1\\gamma _k(p_1)\\ge \\frac{h\\gamma _k(p_1)}{2}.$ If $p_2\\ge 0$ , then $\\delta p := p_2 -p_1 =\\frac{2h(d-(\\ell /2))}{(i+d-(\\ell /2))(i+d-(\\ell /2)+h)}\\le 2\\frac{h}{d},$ for $d$ large enough, e.g., $i\\ge \\ell /2$ , i.e., $(1-q)d \\ge \\ell $ .", "Since $i\\in [i_1,i_2]$ and $i+\\kappa d^{\\alpha }\\in [i_1,i_2]$ , then $p_1,p_2\\in [-q,q]$ and (REF ) lead to $\\frac{|\\delta \\gamma _k|}{\\delta p} <\\frac{(\\gamma ^c_k-\\gamma ^b_k)}{16}.$ Combining (REF ) with (REF ) and since $d_2\\le 2d$ , $\\frac{h\\gamma _k(p_1)}{2}+d_2\\,\\frac{|\\delta \\gamma _k|}{\\delta p}\\,\\delta p\\ge h\\left(\\frac{\\gamma _k(p_1)}{2}-\\frac{(\\gamma ^c_k-\\gamma ^b_k)}{4}\\right).$ Now, by the very definition of $i_1,i_2$ , $\\gamma _k(p_1)\\ge \\gamma _k^c$ , which yields $h\\left(\\frac{\\gamma _k(p_1)}{2}-\\frac{(\\gamma ^c_k-\\gamma ^b_k)}{4}\\right)\\ge h\\left(\\frac{\\gamma _k^c}{2}-\\frac{(\\gamma ^c_k-\\gamma ^b_k)}{4}\\right)= h\\left(\\frac{\\gamma ^b_k}{2}+\\frac{(\\gamma ^c_k-\\gamma ^b_k)}{4}\\right).$ Next, (REF ) together with (REF ), (REF ) and (REF ) lead to: $\\mathbb {E}\\Delta \\ge \\kappa d^{\\alpha }\\frac{\\gamma ^b_k}{2}+\\kappa d^\\alpha \\left(\\frac{\\gamma _k^c-\\gamma _k^b}{4}\\right)-2C_\\gamma \\sqrt{2d \\ln 2d}.$ Finally, since $\\alpha >1/2$ is independent of $d$ , and since $\\gamma _k^c - \\gamma _k^b>0$ , $2C_\\gamma \\sqrt{2d \\ln 2d}$ becomes “negligible\" when compared to $\\kappa d^{\\alpha }(\\gamma _k^c-\\gamma _k^b)/4$ .", "So, for large enough $d$ , (REF ) implies that $\\mathbb {E}\\Delta \\ge \\kappa d^{\\alpha }{\\gamma ^b_k}/{2}$ , which is what we intended to prove, at least for $i\\in [i_1, d]$ .", "As shown next, for $i\\in [d, i_2]$ and $d$ large enough, the inequality (REF ) remains valid.", "Indeed, at first, one only needs $i\\in [i_1, d]$ instead of $i\\in [i_1,i_2]$ , to obtain (REF ); more specifically one needs $i+\\kappa d^{\\alpha }\\in [i_1,i_2]$ .", "However, (REF ) is a strict inequality and so, for $d$ large enough, even if $i+\\kappa d^{\\alpha } \\notin [i_1,i_2]$ but as long as $i\\in [i_i,i_2]$ , by continuity ($p_2-p_1 \\le 2h/d$ ) and since $\\gamma _k^\\prime (q^+) = \\gamma _k^\\prime (q^-)$ , the inequality (REF ) still holds.", "This then implies (REF ).", "The next lemma shows that the event $C^d$ occurs with high probability: Lemma 5.4 For $d$ large enough, $\\mathbb {P}(C^d)\\ge 1- 2d\\exp \\left(-\\frac{d^{2\\alpha -1}\\kappa ^2(\\gamma ^a_k-\\gamma ^b_k)^2}{18}\\right).$ Proof.", "Let $i$ be a positive integer, let $h=\\kappa d^{\\alpha }$ and let $\\Delta :=|LCS(X_1^*X_2^*\\ldots X^*_{d-(\\ell /2)};Y_1Y_2\\ldots Y_{i+h})|-|LCS(X_1^*X_2^*\\ldots X^*_{d-(\\ell /2)};Y_1Y_2\\ldots Y_i)|.$ Recalling the very definition of the event $C_R^d(i,h)$ (see (REF )), note that when its complementary $(C_R^{d}(i,h))^c$ holds, then $\\Delta \\le \\frac{h\\gamma ^a_k}{2}.$ Now, $\\Delta $ is function of the iid entries $X^*_1,X^*_2,\\ldots ,X^*_{d-(\\ell /2)}$ and $Y_1,Y_2,\\ldots ,Y_{i+h}$ and changing one of them changes $\\Delta $ by at most 2.", "Assuming that $i\\in [i_1,d]$ , Lemma REF applies and $\\mathbb {E}\\Delta \\ge \\gamma _k^b h/2$ .", "Together with (REF ), this leads to $\\Delta -\\mathbb {E}\\Delta \\le \\frac{h\\gamma ^a_k}{2}-\\frac{h\\gamma ^b_k}{2}=\\frac{h(\\gamma ^a_k-\\gamma ^b_k)}{2}.$ In other words, the event $(C_R^{d}(i,h))^c$ implies that the inequality (REF ) holds true.", "Hence, $1-\\mathbb {P}(C_R^{d}(i,h))\\le \\mathbb {P}\\left(\\Delta -\\mathbb {E}\\Delta \\le d^*\\left(\\frac{h}{d^*}\\right)\\frac{(\\gamma ^a_k-\\gamma ^b_k)}{2}\\right)$ where $d^*:=d-(\\ell /2)+i+h$ .", "By assumption, $\\gamma ^a_k-\\gamma ^b_k<0$ and therefore by Hoeffding's inequality, the right-hand side of (REF ) is upper bounded by $\\exp \\left(-d^*\\left(\\frac{h}{d^*}\\right)^2\\frac{(\\gamma ^a_k-\\gamma ^b_k)^2}{2}\\right).$ Since $d^*\\le 3d$ and since, for $d$ large enough $d^*\\ge d$ , (REF ) becomes $1-\\mathbb {P}(C_R^{d}(i,h))\\le \\exp \\left(-d^{2\\alpha -1}\\kappa ^2\\frac{(\\gamma ^a_k-\\gamma ^b_k)^2}{18}\\right),$ with our choice of $h=\\kappa d^\\alpha $ , $1/2<\\alpha $ .", "Hence, since the interval $[i_1,d]$ contains at most $d$ elements, $1-\\mathbb {P}(C_R^{d})\\le d\\exp \\left(-d^{2\\alpha -1}\\kappa ^2\\frac{(\\gamma ^a_k-\\gamma ^b_k)^2}{18}\\right).$ A symmetric argument leads to the same bound for $C_L^d$ and, since $C^d = C^d_R\\cap C^d_L$ , $1-\\mathbb {P}(C^{d})\\le 2d \\exp \\left(-d^{2\\alpha -1}\\kappa ^2\\frac{(\\gamma ^a_k-\\gamma ^b_k)^2}{18}\\right).$ Next, the event $B^d$ is shown to hold with high probability.", "Lemma 5.5 For $d$ large enough, $\\mathbb {P}(B^d)\\ge 1-2dk\\exp \\left(-2h\\left(\\frac{1}{\\kappa }-\\frac{1}{k}\\right)^2\\right),$ where $h=\\kappa d^{\\alpha }$ .", "Proof.", "Let $B_r^d(i,h)$ be the event that $r\\in \\lbrace 1,\\ldots ,k\\rbrace $ appears at most $h/\\kappa $ times in the string $Y_iY_{i+1}\\ldots Y_{i+h}$ .", "Hence, $B^d(i,h)=\\bigcap _{r\\in \\lbrace 1,\\ldots ,k\\rbrace }B_r^d(i,h),$ and since all the symbols have equal probabilities: $\\mathbb {P}((B^d(i,h))^c)\\le k \\mathbb {P}((B_1^d(i,h))^c).$ Now if $B_1^{d}(i,h)$ does not hold true, then the letter 1 appears more than $h/\\kappa $ times in $Y_iY_{i+1}\\ldots Y_{i+h}$ .", "Let $W_j$ be the Bernoulli random variable which is equal to one if $Y_j=1$ and zero otherwise, and so if the event $(B_1^d(i,h))^c$ holds true then so does the event $\\sum _{j=i+1}^{i+h}W_j\\ge \\frac{h}{\\kappa },\\,$ where again by equiprobability, $\\mathbb {P}(W_j=1)=\\mathbb {E}W_j =1/k$ .", "Hence, $1-\\mathbb {P}(B_1^{d}(i,h))\\le \\mathbb {P}\\left(\\sum _{j=i+1}^{i+h}W_j-\\mathbb {E}\\left(\\sum _{j=i+1}^{i+h}W_j\\right)\\ge \\frac{h}{\\kappa } - \\frac{h}{k}\\right),$ and since $(1/\\kappa )-(1/k)>0$ , another use of Hoeffding's inequality leads to $1-\\mathbb {P}(B_1^{d}(i,h))\\le \\exp \\left(-2h\\left(\\frac{1}{\\kappa }-\\frac{1}{k}\\right)^2\\right).$ Since, $(B^d)^c=\\bigcup _{i\\in [1,2d]}\\bigcup _{r\\in \\lbrace 1,\\ldots ,k\\rbrace }(B_r^d(i,h))^c,$ then $\\mathbb {P}((B^d)^c)\\le \\sum _{i\\in [1,2d]}\\sum _{r\\in \\lbrace 1,\\ldots ,k\\rbrace }\\mathbb {P}((B_r^d(i,h))^c)\\le 2dk\\mathbb {P}((B_1^d(1,h))^c),$ which, with (REF ), lead to the announced result: $\\mathbb {P}((B^d)^c)\\le 2dk\\exp \\left(-2h\\left(\\frac{1}{\\kappa }-\\frac{1}{k}\\right)^2\\right).$ As shown now, the event $D^d$ occurs with high probability.", "Lemma 5.6 For $d$ large enough, $\\mathbb {P}(D^d)\\ge 1-4d\\exp \\left(-d\\frac{(\\gamma ^*_k-\\gamma _k^c)^2}{128}\\right).$ Proof.", "Recall that $D^d=D^d_I\\cap D^d_{II}$ , where $D^d_I(i)$ is the event that there exists an optimal alignment of $X$ and $Y$ aligning $X_{d-(\\ell /2)}$ to $i$ .", "Now, for $D^d_I$ to hold it is enough that none of the events $D^d_I(i)$ hold for all $i\\notin [i_1,i_2]$ .", "Hence, $\\bigcap _{i\\in [1,2d]\\backslash [i_1,i_2]}(D_I^d(i))^c\\subset D^d_I,$ so that $\\mathbb {P}((D^d_I)^c)\\le \\sum _{i\\in [1,2d]\\backslash [i_1,i_2]}\\mathbb {P}(D^{d}_I(i)).$ Let $L(i)$ be the maximal score obtained when leaving out the big block but giving as constraint that $X_{d-(\\ell /2)}$ gets aligned with $i$ , i.e., $L(i)&:=|LCS(X_1X_2\\cdots X_{d-(\\ell /2)};Y_1Y_2\\cdots Y_i)|\\\\&\\quad \\quad + \\quad |LCS(X_{d+(\\ell /2)+1}X_{d+(\\ell /2)+2}\\cdots X_{2d};Y_{i+1}Y_{i+2}\\cdots Y_{2d})|.$ As shown next, when $D^d_I(i)$ holds then, $L(i)+2d^\\beta \\ge LC_{2d}^*,$ where $LC_{2d}^*$ is the length of the LCS of $X_1^*X_2^*\\ldots X^*_{2d}$ and $Y_1Y_2\\ldots Y_{2d}$ .", "Indeed, less than $d^\\beta $ letters are changed between $X$ and $X^*$ , and so the length-difference between the LCS of $X$ and $Y$ and the LCS of $X^*$ and $Y$ is at most $d^\\beta $ .", "Also, if $D^d_I(i)$ holds then the difference between $L(i)$ and the length of the LCS of $X$ and $Y$ is at most $d^\\beta $ .", "Therefore, the difference between the lengths of $L(i)$ and $LC_{2d}^*$ is at most $2d^\\beta $ , when $D^d_I(i)$ holds.", "Hence, $\\mathbb {P}(D^d_I(i))\\le \\mathbb {P}(L(i)+2d^\\beta \\ge LC_{2d}^*),$ with $\\mathbb {P}(L(i)+2d^\\beta \\ge LC_{2d}^*)=\\mathbb {P}(L(i)-LC_{2d}^*-\\mathbb {E}L(i) +\\mathbb {E}LC_{2d}^* \\ge \\mathbb {E}LC_{2d}^* -\\mathbb {E}L(i) -2d^\\beta ).$ But as $d\\rightarrow \\infty $ , $\\mathbb {E}LC_{2d}^* /2d \\rightarrow \\gamma ^*_k$ , and via (REF ) $\\mathbb {E}LC_{2d}^*\\ge 2d\\gamma _k^*-C_L\\sqrt{2d\\ln 2d},$ for some constant $C_L>0$ .", "But, by definition, $\\mathbb {E}L(i)=d_1\\gamma _k(p_1,d_1)+d_2\\gamma _k(p_2,d_2),$ where $d_1:=&\\frac{1}{2}\\left(d-\\frac{\\ell }{2}+i\\right), \\quad p_1:=\\frac{i-d+(\\ell /2)}{d-(\\ell /2)+i}= \\frac{2i-2d+\\ell }{2i+2d-\\ell },\\\\d_2:=&\\frac{1}{2}\\left(3d-\\frac{\\ell }{2}-i\\right), \\quad p_2:=\\frac{i-d-(\\ell /2)}{3d-(\\ell /2)-i}= \\frac{2i-2d+\\ell }{6d-2i-\\ell }.$ Moreover, $\\gamma _k(p_2,d_2)\\le \\gamma ^*_k,$ and $d_1+d_2=2d-\\frac{\\ell }{2}\\le 2d.$ Now, if $i\\notin [i_1,i_2]$ , then by the very definition of $i_1$ and $i_2$ , $\\gamma _k(p_1)\\le \\gamma ^c_k.$ Next, by a sub-additivity argument, $\\gamma _k(p_1)=\\lim _{d\\rightarrow \\infty }\\gamma _k(p_1,d)\\ge \\gamma _k(p_1,d),$ for every $d\\ge 1$ , and therefore $\\gamma _k(p_1,d_1)\\le \\gamma _k(p_1).$ Applying (REF ), (REF ), (REF ) and (REF ) to (REF ), and assuming that $d$ is large enough so that $d_1\\ge d/4$ , lead to $\\mathbb {E}L(i) \\le 2d\\gamma ^*_k-\\frac{d(\\gamma ^*_k-\\gamma _k^c)}{4}\\,.$ Then, (REF ) and (REF ) give $\\mathbb {E}(LC_{2d}^*-L(i))-2d^\\beta \\ge \\frac{d(\\gamma ^*_k-\\gamma _k^c)}{4}-C_L\\sqrt{2d\\ln 2d}-2d^\\beta .$ By definition, $\\gamma ^*_k-\\gamma _k^c>0$ and $\\beta <1$ .", "So, for $d$ large enough, the right-hand side of (REF ) is at least $d(\\gamma ^*_k-\\gamma _k^c)/8$ , so that $\\mathbb {E}(LC_{2d}^*-L(i))-2d^\\beta \\ge \\frac{d(\\gamma ^*_k-\\gamma _k^c)}{8}\\,.$ Using (REF ) with (REF ) and (REF ), lead to: $\\mathbb {P}(D^d_I(i))\\le \\mathbb {P}\\left(L(i)-LC_{2d}^*-\\mathbb {E}L(i)+\\mathbb {E}L_{2d}^* \\ge d\\frac{(\\gamma ^*_k-\\gamma _k^c)}{8}\\right),$ for $d$ large enough.", "By Hoeffding's inequality, $\\mathbb {P}(D^d_I(i))\\le \\exp \\left(-d\\frac{(\\gamma ^*_k-\\gamma _k^c)^2}{128}\\right),$ for $i\\notin [i_1,i_2]$ .", "Combining (REF ) with (REF ) gives $\\mathbb {P}((D^d_I)^c)\\le \\sum _{i\\in [1,2d]\\backslash [i_1,i_2]}\\exp \\left(-d\\frac{(\\gamma ^*_k-\\gamma _k^c)^2}{128}\\right)\\le 2d\\exp \\left(-d\\frac{(\\gamma ^*_k-\\gamma _k^c)^2}{128}\\right).$ The same bound can be found for $\\mathbb {P}(D^{dc}_{II})$ and this finishes the proof.", "As the next lemma shows, the event $F^d$ also holds with high probability.", "Lemma 5.7 There exists a constant $C_F > 0$ , independent of $d$ , such that $\\mathbb {P}(F^d)\\ge 1-e^{-C_Fd^{2\\beta -1}},$ for all $d\\ge 1$ .", "Proof.", "The proof is very similar to the proof that $\\mathbb {P}(C^d)$ occurs with high probability (see Lemma REF and Lemma REF ).", "Nevertheless we present large parts of the proof, since many inequalities are reversed and the infinitesimal quantities are of different order.", "Let $\\Delta := |LCS(X_1^*X_2^*\\ldots X^*_{d-(\\ell /2)+d^\\beta };Y_1Y_2\\ldots Y_{i})|-|LCS(X_1^*X_2^*\\ldots X^*_{d-(\\ell /2)};Y_1Y_2\\ldots Y_i)|,$ and assume at first that $i\\in [d, i_2]$ (see (REF )).", "We first show that $\\mathbb {E}\\Delta \\!\\ge \\frac{\\gamma ^b_k}{2}d^\\beta .$ As in the proof of Lemma REF , with its notation, $\\mathbb {E}\\Delta =d_2\\gamma _k(d_2,p_2)-d_1\\gamma _k(d_1,p_1),$ where $d_2:=&\\frac{1}{2} \\left(i+d-\\frac{\\ell }{2}+d^\\beta \\right)\\!, \\,\\quad p_2:=\\frac{i-d+(\\ell /2)-d^\\beta }{i+d-(\\ell /2)+d^\\beta },\\\\d_1:=&\\frac{1}{2} \\left(i+d-\\frac{\\ell }{2}\\right), \\quad \\quad \\quad p_1:=\\frac{i-d+(\\ell /2)}{i+d-(\\ell /2)}.$ Again, for all $p\\in (-1,1)$ and all $d\\ge 1$ , and since $d_1,d_2\\le 2d$ , $\\mathbb {E}\\Delta \\ge d_2\\gamma _k(p_2)-d_1\\gamma _k(p_1)-2C_\\gamma \\sqrt{2d \\ln 2d}.$ Now, $d_2\\gamma _k(p_2)-d_1\\gamma _k(p_1)=\\frac{d^\\beta \\gamma _k(p_1)}{2}+d_2\\frac{\\delta \\gamma _k}{\\delta p}\\,\\delta p,$ where $\\delta \\gamma _k:=\\gamma _k(p_2)-\\gamma _k(p_1)$ and where $\\delta p:=p_2-p_1<0$ .", "Next, if $p_2\\ge 0$ , then $\\delta \\gamma _k\\ge 0$ and so $d_2\\gamma _k(p_2)-d_1\\gamma _k(p_1)\\ge \\frac{d^\\beta \\gamma _k(p_1)}{2}.$ If $p_2\\le 0$ , then $|\\delta p|=\\left|\\frac{-2id^\\beta }{(i+d-(\\ell /2))(i+d -(\\ell /2)+d^\\beta )}\\right|\\le 2\\frac{d^\\beta }{d},$ for $d$ large enough, e.g., $i\\ge \\ell /2$ , i.e., $(1-q)d \\ge \\ell $ .", "Since $i\\in [d,i_2]$ , $p_1\\in [0,q]$ .", "Moreover, $p_2<p_1$ and $|\\delta p|\\le {2d^\\beta }/{d}$ , for $d$ large enough, imply that $p_2\\in [-q,q]$ and therefore via (REF ), $\\frac{|\\delta \\gamma _k|}{|\\delta p|} <\\frac{(\\gamma ^c_k-\\gamma ^b_k)}{16}.$ Combining (REF ) with (REF ) and since $d_2\\le 2d$ , $\\frac{d^\\beta \\gamma _k(p_1)}{2}+d_2\\,\\frac{|\\delta \\gamma _k|}{\\delta p}\\,\\delta p\\ge d^\\beta \\left(\\frac{\\gamma _k(p_1)}{2}-\\frac{(\\gamma ^c_k-\\gamma ^b_k)}{4}\\right).$ Now, $i\\in [i_1,i_2]$ hence $p_1\\in [-q,q]$ and therefore, by the very definition of $i_1,i_2$ , $\\gamma _k(p_1)\\ge \\gamma _k^c$ , which yields $d^\\beta \\left(\\frac{\\gamma _k(p_1)}{2}-\\frac{(\\gamma ^c_k-\\gamma ^b_k)}{4}\\right)\\ge d^\\beta \\left(\\frac{\\gamma _k^b}{2}+\\frac{(\\gamma ^c_k-\\gamma ^b_k)}{4}\\right).$ Now, (REF ) together with (REF ), (REF ) and (REF ) imply that $\\mathbb {E}\\Delta \\ge d^{\\beta }\\frac{\\gamma ^c_k}{2}+d^\\beta \\frac{(\\gamma _k^c-\\gamma _k^b)}{4}-2C_\\gamma \\sqrt{2d \\ln 2d}.$ Finally, since $\\beta >1/2$ is independent of $d$ , and since $\\gamma _k^c-\\gamma _k^b>0$ , then $2C_\\gamma \\sqrt{2d \\ln 2d}$ becomes “negligible\" when compared to $d^{\\beta }(\\gamma _k^c-\\gamma _k^b)/4$ .", "So, for large enough $d$ , (REF ) implies that $\\mathbb {E}\\Delta \\ge d^{\\beta }{\\gamma ^b_k}/{2}$ , which is what we intended to prove for $i\\in [d, i_2]$ .", "Next, for $i\\in [i_1, d]$ , and $d$ large enough, the inequality (REF ) remains valid.", "Indeed, first, one only needs $i\\in [d,i_2]$ instead of $i\\in [i_1,i_2]$ , to obtain (REF ); more specifically one needs $p_2\\in [-q,q]$ .", "However, (REF ) is a strict inequality and so, for $d$ large enough, even if $p_2 \\notin [-q,q]$ but as long as $i\\in [i_i,i_2]$ , by continuity, recalling also that $|\\delta p| \\le 2d^\\beta /d$ , and since $\\gamma ^\\prime _k(q^+)=\\gamma ^\\prime _k(q^-)$ , the inequality (REF ) continues to hold.", "This will then imply (REF ) for all $i\\in [i_1,i_2]$ .", "We wish now to upper-bound the probability of the complement of $F^{d}_i$ when $i\\in [i_1,i_2]$ .", "From (REF ), $\\mathbb {P}((F^{d}_i)^c)=\\mathbb {P}\\left(\\Delta \\le d^\\beta \\frac{\\gamma ^a_k}{2}\\right)\\le \\mathbb {P}\\left(\\Delta -\\mathbb {E}\\Delta \\le d^\\beta \\frac{\\gamma ^a_k-\\gamma ^b_k}{2}\\right).$ Note that $\\Delta $ depends on $d^*:=d-(l/2)+d^\\beta +i$ , iid entries $X_1^*,X_2^*,\\ldots , X^*_{d-(\\ell /2)+d^\\beta }$ and $Y_1,Y_2,\\ldots , Y_{i}$ , and so by Hoeffding's inequality, $\\mathbb {P}\\left(\\Delta -\\mathbb {E}\\Delta \\le d^\\beta \\frac{\\gamma ^a_k-\\gamma ^b_k}{2}\\right)&\\le \\exp \\left(-\\frac{(d^\\beta (\\gamma ^a_k-\\gamma ^b_k))^2}{8d^*}\\right)\\\\&\\le \\exp \\left(-d^{2\\beta -1}\\frac{(\\gamma ^b_k-\\gamma ^a_k)^2}{32}\\right),$ since for $d$ large enough, $d^*\\le 4d$ .", "Hence, since $[i_1,i_2]$ contains less than $d$ elements (for $d$ large enough), $\\mathbb {P}((F^{d})^c)\\le \\sum _{i\\in [i_1,i_2]}\\mathbb {P}((F^{d}_i)^c)\\le d\\exp \\left(-d^{2\\beta -1}\\frac{(\\gamma ^b_k-\\gamma ^a_k)^2}{32}\\right).$ Finally, recall that $\\beta > 1/2$ ." ], [ "Proof of Theorem ", "This is the main theorem for three or more letters, i.e., for $\\gamma ^*_k/2>1/k$ .", "It states that the events $E^d$ and $K^d$ both hold high probability.", "Hence, for $d$ large enough, typically with three or more letters the long block gets mainly aligned with gaps.", "Moreover, this result asserts that replacing the long block with iid symbols typically leads to an increase in the LCS which is linear in the length of the long block.", "Let us first handle $E^d$ .", "By Lemma REF , $\\mathbb {P}((E^d)^c)\\le \\mathbb {P}((B^d)^c)+\\mathbb {P}((C^d)^c)+\\mathbb {P}((D^d)^c).$ By Lemma REF , $\\mathbb {P}((C^d)^c)$ is of exponential small order in $d^{2\\alpha -1}$ ; by Lemma REF , $\\mathbb {P}((B^d)^c)$ is of exponential small order in $d^{\\alpha }$ and by Lemma REF $\\mathbb {P}((D^d)^c)$ is exponentially small in $d$ .", "Therefore, for $\\alpha \\in \\,(1/2,1)$ , $\\mathbb {P}((E^d)^c)$ is also exponentially small in $d^{2\\alpha -1}$ .", "Hence, there exists a constant $C_E>0$ , independent of $d$ (but depending on $k$ ) such that $\\mathbb {P}((E^d)^c)\\le e^{-C_Ed^{2\\alpha -1}}.$ Let us, next, turn our attention to the event $K^d$ .", "From Lemma REF , $\\mathbb {P}((K^d)^c)\\le \\mathbb {P}((B^d)^c)+\\mathbb {P}((C^d)^c)+\\mathbb {P}((D^d)^c)+\\mathbb {P}((F^d)^c),$ and, as already seen, $\\mathbb {P}((B^d)^c)+\\mathbb {P}((C^d)^c)+\\mathbb {P}((D^d)^c)$ is of exponential small order in $d^{2\\alpha -1}$ .", "By Lemma REF , $\\mathbb {P}((F^d)^c)$ is of exponential small order in $d^{2\\beta -1}$ .", "Since $2\\alpha -1<2\\beta -1$ , the right side of (REF ) is thus exponentially small in $d^{2\\alpha -1}$ , and therefore there exists a constant $C_K>0$ , independent of $d$ , such that $\\mathbb {P}((K^d)^c)\\le e^{-C_Kd^{2\\alpha -1}}.$" ], [ "Proofs For Binary Strings", "The purpose of this subsection is to prove Theorem REF , and therefore throughout the rest of the article, $k\\gamma ^*_k<2$ , i.e., $k=2$ .", "Theorem REF states that typically, for $d$ large enough, the long block gets mainly aligned with symbols and not with gaps.", "The corresponding event $G^d$ was defined in Section .", "Theorem REF also asserts that replacing the long block with iid symbols typically increases the LCS linearly in the length of the long constant block.", "The corresponding event $H^d$ was also defined in Section .", "So, below, we intend to prove that both events hold with high probability and this is done in a way very similar to the 3-or more letter-case.", "Let $k_{II}$ and $\\gamma _2^{II}$ be two constants, independent of $d$ , such that $k_{II}>2$ and $\\gamma _2^{II}>\\gamma ^*_2$ , but also such that $k_{II}\\gamma _2^{II}<2$ (this last choice is certainly possible since $\\gamma ^*_2<1$ ).", "Actually, for the argument which follows, any values $k_{II}>2$ and $\\gamma _2^{II}>\\gamma ^*_2$ will do, provided the constants are close enough to their respective bounds and do not depend on $d$ .", "Let now $B^d_{II}$ be the event that in any piece of $Y$ of length $k_{II}d^{\\alpha }$ , there are at least $d^{\\alpha }$ ones and zeros.", "More precisely, let $B^d_{II}(i)$ be the event that $ \\sum _{j=i+1}^{i+h}Y_j\\ge d^{\\alpha },$ and that $\\sum _{j=i+1}^{i+h}|Y_j-1|\\ge d^{\\alpha },$ where $h:=k_{II}d^{\\alpha }$ .", "Finally, let $B^d_{II}=\\bigcap _{i=1}^{2d-h}B^d_{II}(i).$ Let $C^d_{II}$ be the event that an increase of the length of $Y_1\\ldots Y_i $ by $k_{II}d^{\\alpha }$ leads to an increase of the LCS of $Y_1\\ldots Y_i$ and $X_1\\ldots X_{d-(\\ell /2)}$ of no more than $k_{II}d^{\\alpha }\\gamma _2^{II}/2$ for all $i+k_{II}d^{\\alpha }\\in [i_1,i_2]$ .", "More precisely, for $h=k_{II}d^{\\alpha }$ , $C^d_{II}(i)\\!", ":=\\left\\lbrace |LCS(X_1\\ldots X_{d-(\\ell /2)};Y_1\\ldots Y_{i+h})|- |LCS(X_1\\ldots X_{d-(\\ell /2)};Y_1\\ldots Y_i)|\\le \\frac{h\\gamma _2^{II}}{2}\\right\\rbrace ,$ $C^d_{II}:=\\bigcap _{i+h\\in [i_1,i_2]} C^d_{II}(i).$ Recall finally that $G^d$ is the event that the long constant block gets mainly aligned with symbols and not with gaps; more precisely, $G^d=\\left\\lbrace |LCS(X;Y)|>|LCS(X_1X_2\\ldots X_{d-(\\ell /2)}X_{d-(\\ell /2)+d^{\\alpha }+1}X_{d-(\\ell /2)+d^{\\alpha }+2}\\ldots X_{2d};Y)|\\right\\rbrace .$ Lemma 5.8 $B_{II}^d\\cap D^d\\cap C^d_{II}\\subset G^d.$ Proof.", "The proof is by contradiction.", "Assume that $\\pi $ is an optimal alignment of $X$ and $Y$ aligning at least $d^{\\alpha }$ symbols from the long block with gaps, and assume that $\\pi $ aligns $X_{d-(\\ell /2)}$ with $j$ .", "Since $D^d$ holds, then $j\\in [i_1,i_2]$ .", "Now, let $i:=j-h$ (again, $h:=k_{II}d^{\\alpha }$ ), so that $i+h\\in [i_1,i_2]$ .", "Thus $C^d_{II}$ “applies” to $i$ , meaning that when “taking out” the piece $Y_{i+1}Y_{i+2}\\ldots Y_j$ from the alignment $\\pi $ , lose at most $h\\gamma _2^{II}/2$ .", "Now, because of $B^d_{II}$ , the string $Y_{i+1}Y_{i+2}\\ldots Y_{i+h}$ contains the symbols the long block is made of, at least $d^{\\alpha }$ times.", "Hence the $d^{\\alpha }$ symbols, from the long block, which are aligned by $\\pi $ with gaps, can be aligned with symbols contained in the piece of string $Y_iY_{i+1}\\ldots Y_{i+h}$ .", "Let $\\bar{\\pi }$ denote the new alignment obtained from modifying $\\pi $ in this way.", "Transforming $\\pi $ to $\\bar{\\pi }$ gained $d^{\\alpha }$ aligned symbols from the long block, which where aligned with gaps, and now are aligned with symbols.", "However, from the previously aligned symbols from $Y_i\\ldots Y_{i+h}$ we could lose as many as $h\\gamma _2^{II}/2$ aligned symbols pairs.", "Hence the change is at least $d^{\\alpha }-\\frac{d^{\\alpha }k_{II}\\gamma _2^{II}}{2}=d^{\\alpha }\\left( 1-\\frac{k_{II}\\gamma _2^{II}}{2}\\right) > 0,$ from the choices of $k_{II}$ and $\\gamma _2^{II}$ .", "Hence, (REF ) is strictly positive, and thus $\\bar{\\pi }$ aligns more letter-pairs than $\\pi $ .", "Therefore, $\\pi $ is not optimal, which is a contradiction, and it is not possible for $d^\\alpha $ symbols, of the long constant block, to get aligned with gaps when $B^d_{II}$ , $D^d$ and $C^d_{II}$ all hold.", "Hence, $B^d_{II}$ , $D^d$ and $C^d_{II}$ jointly imply $G^d$ ." ], [ "High probability of $G^d$ .", "From Lemma REF , $\\mathbb {P}((G^d)^c)\\le \\mathbb {P}((B_{II}^d)^c)+\\mathbb {P}((D^d)^c)+\\mathbb {P}((C^d_{II})^c).$ In Lemma REF , we already proved that $\\mathbb {P}((D^d)^c)$ is exponentially small in $d$ .", "Next, a simple application of Hoeffding's inequality shows that $\\mathbb {P}((B^{d}_{II})^{c})$ is exponentially small in $d^{\\alpha }$ and this is left to the reader.", "Let us now deal with $\\mathbb {P}(C^d_{II})$ and show that $\\mathbb {P}((C^d_{II})^c)$ is exponentially small in $d^{2\\alpha -1}$ .", "The proof is similar to the proof of Lemma REF .", "Using the notations there, but with $h=k_{II}d^{\\alpha }$ , let $\\Delta :=|LCS(X_1^*X_2^*\\ldots X^*_{d-(\\ell /2)};Y_1Y_2\\ldots Y_{i+h})|-|LCS(X_1^*X_2^*\\ldots X^*_{d-(\\ell /2)};Y_1Y_2\\ldots Y_{i})|.$ Again, $\\mathbb {E}\\Delta =d_2\\gamma _2(d_2,p_2)-d_1\\gamma _2(d_1,p_1)$ where $d_1,d_2,p_1,p_2$ are as in Lemma REF , and $\\left|\\mathbb {E}\\Delta -d_2\\gamma _2(p_2)+d_1\\gamma _2(p_1)\\right|\\le 2C_\\gamma \\sqrt{2d\\ln 2d}.$ Once more, $d_2\\gamma _2(p_2)-d_1\\gamma _2(p_1)=\\frac{h\\gamma _2(p_1)}{2}+d_2\\frac{\\delta \\gamma _2}{\\delta p}\\delta p,$ where $\\delta \\gamma _2:=\\gamma _2(p_2)-\\gamma _2(p_1),$ and where, for $d$ large enough, $0< \\delta p:= p_2 - p_1 \\le 2\\frac{h}{d}.$ Since $i\\in [i_1,i_2]$ then $p_1\\in [-q,q]$ .", "If $p_2$ would also be in $[-q,q]$ , then the inequality (REF ) would be enough to get our estimates.", "Now, $p_2$ might not be in $[-q,q]$ , but by continuity in $d$ and since $\\delta p\\rightarrow 0$ , as $d\\rightarrow \\infty $ , we have for large enough $d$ : $\\frac{|\\delta \\gamma _2|}{\\delta p}\\le \\frac{\\gamma ^{II}_2 - \\gamma _2^*}{16}$ Combining (REF ), (REF ), (REF ), (REF ) and (REF ) with the facts that $d_2\\le 2d$ and $\\gamma _2(p_1)\\le \\gamma _2^*$ lead to: $\\mathbb {E}\\Delta -\\frac{h\\gamma _2^*}{2} \\le \\frac{h}{4}(\\gamma _2^{II}-\\gamma _2^*) + 2C_\\gamma \\sqrt{2d\\ln 2d}.$ Applying Hoeffding's inequality to $\\Delta $ , which depends on $d-(\\ell /2)+i+h<4d$ iid entries and using (REF ) yield: $\\mathbb {P}((C^{d}_{II}(i))^c) = \\mathbb {P}\\left(\\!\\Delta - \\mathbb {E}\\Delta > \\frac{h\\gamma _2^{II}}{2} - \\mathbb {E}\\Delta \\!\\right)&\\le \\mathbb {P}\\left(\\!\\Delta - \\mathbb {E}\\Delta > \\frac{h(\\gamma _2^{II} - \\gamma _2^*)}{4} -2C_\\gamma \\!\\sqrt{2d\\ln 2d}\\right)\\\\&\\le \\exp \\left(-\\frac{k_{II}^2(\\gamma _2^{II}-\\gamma _2^*)^2}{512}d^{2\\alpha -1}\\right),$ since for $d$ large enough, the term $2C_\\gamma \\sqrt{2d\\ln 2d}$ becomes negligible when compared to $h=k_{II}d^{\\alpha }$ , $\\alpha > 1/2$ .", "Next, $[i_1,i_2]$ contains at most $2d$ integers and so $\\mathbb {P}((C^{d}_{II})^c)\\le 2d\\exp \\left(-k_{II}^2(\\gamma _2^{II}-\\gamma _2^*)^2d^{2\\alpha -1}/512\\right)$ .", "Finally, $0<2\\alpha -1<\\alpha <\\beta <1$ and, therefore, the orders of magnitude of $\\mathbb {P}((B^{d}_{II})^{c})$ , $\\mathbb {P}((C^{d}_{II})^{c})$ and $\\mathbb {P}((D^d)^c)$ together with (REF ) imply that $\\mathbb {P}((G^d)^c)\\le e^{-C_G d^{2\\alpha -1}},$ for all $d\\ge 1$ , where $C_G>0$ is a constant independent of $d$ .", "This finishes establishing that, with high probability, a small proportion of gaps is aligned with the long block.", "The rest of this subsection is devoted to analyzing the increase in the LCS when replacing the long constant block with iid symbols, thus showing that the event $H^d$ holds with high probability.", "Recall that $H^d$ states that the increase in the LCS is at least ${\\tilde{c}}_H>0$ times the length of the long block.", "(${\\tilde{c}}_H$ is any positive real, independent of $d$ , smaller than $3\\gamma ^*_2/2-1$ .)", "Again, $X$ contains a long block in $[d-\\ell /2 + 1,d+\\ell /2]$ , i.e., $\\mathbb {P}\\left(X_i=X_{i+1},\\quad \\forall i\\in \\left[d-\\left(\\frac{\\ell }{2}\\right)+1,d+\\left(\\frac{\\ell }{2}\\right)-1\\right]\\right)=1,$ while $X^*$ is obtained by replacing the long block in $X$ by iid symbols, i.e., $X^*_i=X_i$ for all $i\\notin [d-(\\ell /2)+1,d+(\\ell /2)]$ .", "Let now $\\tilde{k}_{II} < 2$ and $\\tilde{\\gamma }_2 < \\gamma ^*_2$ be two constants independent of $d$ such that $\\tilde{k}_{II}$ is extremely close to 2 while $\\tilde{\\gamma }_2$ extremely close to $\\gamma ^*_2$ , with moreover $\\left(\\frac{(1+\\tilde{k}_{II})\\tilde{\\gamma }_2}{2}-1\\right)>{\\tilde{c}}_H.$ (These choices are certainly possible since $3\\gamma ^*_2/2-1 \\approx 0.2$ and since (see (REF )) ${\\tilde{c}}_H < 3\\gamma ^*_2/2-1$ .)", "Next, let $\\tilde{B}^{d}_{II}$ be the event that in any piece of $Y$ of length $\\tilde{k}_{II}d^{\\beta }$ there are strictly less than $d^{\\beta }-d^{\\alpha }$ zeros and ones.", "More precisely, for $h:=\\tilde{k}_{II}d^\\beta $ , $\\tilde{B}^{d}_{II}(i):= \\left\\lbrace \\sum _{j=i+1}^{i+h}Y_j< d^{\\beta }-d^{\\alpha }, \\quad \\sum _{j=i+1}^{i+h}|Y_j-1|< d^{\\beta }-d^{\\alpha }\\right\\rbrace .$ Let also $\\tilde{B}^{d}_{II}=\\bigcap _{i=1}^{2d-h}\\tilde{B}^{d}_{II}(i).$ Let $\\tilde{C}^{d}_{II}$ be the event that an increase of the length of $Y_1\\ldots Y_i$ by $\\tilde{k}_{II}d^{\\beta }$ and an increase of the length of $X_1\\ldots X_{d-(\\ell /2)}$ by $d^\\beta $ leads to an increase of the LCS by at least $d^\\beta (1+\\tilde{k}_{II})\\tilde{\\gamma }_2/2$ , for all $i\\in [i_1,i_2]$ .", "More precisely, let $\\tilde{C}^{d}_{II}(i)$ be the event that $|LCS(X_1\\ldots X_{d-(\\ell /2)+d^\\beta };Y_1\\ldots Y_{i+h_y})|-|LCS(X_1\\ldots X_{d-(\\ell /2)};Y_1\\ldots Y_i)|\\ge d^\\beta \\!\\left(\\!\\!\\frac{(1+\\tilde{k}_{II})\\tilde{\\gamma }_2}{2}\\!\\!\\right)\\!\\!,$ where, again, $h:=\\tilde{k}_{II}d^\\beta $ , and let $\\tilde{C}^{d}_{II}:=\\bigcap _{i\\in [i_1,i_2]} \\tilde{C}^{d}_{II}(i).$ Once more, $H^d$ is the event that replacing the long constant block by iid symbols increases the LCS by at least ${\\tilde{c}}_H d^\\beta $ ; more precisely, $H^d=\\left\\lbrace |LCS(X^*;Y)|-|LCS(X;Y)|\\ge {\\tilde{c}}_H d^\\beta \\right\\rbrace .$ Lemma 5.9 $\\tilde{B}^{d}_{II}\\cap \\tilde{C}^{d}_{II}\\cap D^d\\subset H^d.$ Proof.", "Assume that $\\tilde{B}^{d}_{II}$ , $\\tilde{C}^{d}_{II}$ and $D^d$ all hold true and thus $G^d$ also holds.", "Let now $\\pi $ be an optimal alignment.", "Hence, $\\pi $ aligns at least $d^\\beta -d^{\\alpha }$ symbols from the long constant block with symbols.", "Let $[i,j]$ denote the interval on which the long block gets aligned to by $\\pi $ , meaning that $X_{d-(\\ell /2)+1}$ gets aligned to $i$ by $\\pi $ while $X_{d+(\\ell /2)}$ gets aligned to $j$ .", "Next, at least $d^\\beta -d^{\\alpha }$ symbols are aligned with symbols from the long block, it follows via the event $\\tilde{B}^{d}_{II}$ , that $j-1 \\ge \\tilde{k}_{II}d^{\\beta }$ (in order for $[i,j]$ to contain sufficiently many same symbols).", "Now modify the alignment $\\pi $ to obtain an alignment $\\bar{\\pi }$ aligning $X^*$ and $Y$ .", "The new alignment $\\bar{\\pi }$ is identical to $\\pi $ in the way it aligns $X_{d+(\\ell /2)+1}X_{d+(\\ell /2)+2}\\ldots X_{2d}$ with $Y_{j+1}Y_{j+2}\\ldots Y_{2d}$ , but instead of aligning the long block to $Y_iY_{i+1}\\ldots Y_j$ , it now aligns $X^*_1X_2^*\\ldots X^*_{d+(\\ell /2)}$ with $Y_1Y_2\\ldots Y_{j}$ .", "Since $D^d$ holds, then $i\\in [i_1,i_2]$ and therefore we can apply $\\tilde{C}^{d}_{II}$ to $i$ .", "This yields that the gain by aligning $X^*_1X_2^*\\ldots X^*_{d+(\\ell /2)}$ with $Y_1Y_2\\ldots Y_{j}$ , instead of just aligning $X^*_1X_2^*\\ldots X^*_{d-(\\ell /2)}$ with $Y_1Y_2\\ldots Y_{i}$ , is equal to $d^\\beta (1+\\tilde{k}_{II})\\tilde{\\gamma }_2/2$ .", "On the other hand, there is a loss of at most $d^\\beta $ symbols from the long constant block, so the overall gain is of at least: $d^\\beta \\left(\\frac{(1+\\tilde{k}_{II})\\tilde{\\gamma }_2}{2}-1\\right).$ Therefore the event $H^d$ holds true and this finishes this proof." ], [ "High probability of $H^d$ .", "From Lemma REF , $\\mathbb {P}((H^d)^c)\\le \\mathbb {P}((\\tilde{B}^d_{II})^{c})+ \\mathbb {P}((\\tilde{C}^{d}_{II})^{c})+\\mathbb {P}((D^d)^c),$ and clearly, with the help of Lemma REF , we only need to estimate $\\mathbb {P}(\\tilde{C}^d_{II})$ and $\\mathbb {P}(\\tilde{B}^d_{II})$ .", "A simple application of Hoeffding's inequality, left to the reader, shows that $\\mathbb {P}((\\tilde{B}^d_{II})^c)$ is exponentially small in $d^\\beta $ .", "For $\\mathbb {P}(\\tilde{C}^d_{II})$ , let $\\Delta :=|LCS(X_1^*X_2^*\\ldots X^*_{d-(\\ell /2)+d^\\beta };Y_1Y_2\\ldots Y_{i+h})|-|LCS(X_1^*X_2^*\\ldots X^*_{d-(\\ell /2)};Y_1Y_2\\ldots Y_{i})|,$ where $h=\\tilde{k}_{II}d^\\beta $ .", "Again, $\\mathbb {E}\\Delta =d_2\\gamma _2(d_2,p_2)-d_1\\gamma _2(d_1,p_1)$ , where $d_1,p_1$ are as in the proof of Lemma REF but where $d_2,p_2$ are different, i.e., $d_2:=&\\frac{1}{2} \\left(i+h+d-\\frac{\\ell }{2} +d^\\beta \\right)\\!, \\,\\quad p_2:=\\frac{i+h-d+ (\\ell /2)-d^\\beta }{i+h+d-(\\ell /2)+d^\\beta },\\\\d_1:=&\\frac{1}{2} \\left(i+d-\\frac{\\ell }{2}\\right), \\quad \\quad \\quad \\quad \\quad p_1:=\\frac{i-d+(\\ell /2)}{i+d-(\\ell /2)}.$ Once more, $\\left|\\mathbb {E}\\Delta -d_2\\gamma _2(p_2)+d_1\\gamma _2(p_1)\\right|\\le 2C_\\gamma \\sqrt{2d\\ln 2d},$ $d_2\\gamma _2(p_2)-d_1\\gamma _2(p_1)=\\frac{(d^\\beta +h)\\gamma _2(p_1)}{2}+d_2\\frac{\\delta \\gamma _2}{\\delta p}\\delta p,$ where, for $d$ large enough, $ 0< |\\delta p:= p_2 - p_1| \\le 2\\frac{h+d^\\beta }{d}=2(1+\\tilde{k}_{II})\\frac{d^\\beta }{d}.$ Since $i\\in [i_1,i_2]$ then $p_1\\in [-q,q]$ .", "If $p_2$ would also be in $[-q,q]$ , then the inequality (REF ) would be enough to get our estimates.", "Now, $p_2$ might not be in $[-q,q]$ , but by continuity in $d$ and since $\\delta p\\rightarrow 0$ , as $d\\rightarrow \\infty $ , we have for large enough $d$ : $\\frac{|\\delta \\gamma _2|}{\\delta p}\\le \\frac{(\\gamma _2^c-\\tilde{\\gamma _2})}{32}$ Combining (REF ), (REF ), (REF ), (REF ) and (REF ) with the facts that $d_2\\le 2d$ and $\\gamma _2(p_1)\\ge \\gamma _2^c$ lead to: $\\mathbb {E}\\Delta -\\frac{(h+d^\\beta )\\gamma _2^c}{2}\\ge \\mathbb {E}\\Delta -\\frac{(h+d^\\beta )\\gamma _2(p_1)}{2}\\ge -\\frac{h+d^\\beta }{8}(\\gamma _2^c-\\tilde{\\gamma }_2)-2C_\\gamma \\sqrt{2d\\ln 2d}.$ When $d$ is large enough the term $2C_\\gamma \\sqrt{2d\\ln 2d}$ becomes negligible when compared to $h=O(d^{\\beta })$ , $\\beta >1/2$ .", "Hence, for $d$ large enough, we find $\\mathbb {E}\\Delta -\\frac{(h+d^\\beta )\\tilde{\\gamma _2}}{2}\\ge \\frac{h+d^\\beta }{4}(\\gamma _2^c-\\tilde{\\gamma }_2).$ Applying Hoeffding's inequality to $\\Delta $ , which depends on $d-(\\ell /2)+d^\\beta +i+h<5d$ iid entries, yields: $\\mathbb {P}((\\tilde{C}^{d}_{II}(i))^c)\\le \\exp (-d^{2\\beta -1}(\\tilde{k}_{II}+1)^2(\\gamma _2^c-\\tilde{\\gamma }_2)^2/1024).$ Next, $[i_1,i_2]$ contains at most $2d$ integers, and so $\\mathbb {P}((\\tilde{C}^{d}_{II})^c)\\le 2d\\exp (-d^{2\\beta -1}(\\tilde{k}_{II}+1)^2(\\gamma _2^c-\\tilde{\\gamma }_2)^2/1024).$ Finally, $0<2\\alpha -1<\\alpha <\\beta <1$ and, therefore, the orders of magnitude of $\\mathbb {P}((\\tilde{B}^{d}_{II})^{c})$ , $\\mathbb {P}((\\tilde{C}^{d}_{II})^{c})$ and $\\mathbb {P}((D^d)^c)$ together with (REF ) imply that $\\mathbb {P}((H^d)^c)\\le e^{-C_H d^{2\\alpha -1}},$ for all $d\\ge 1$ , where $C_H>0$ is a constant independent of $d$ .", "This finishes establishing that, with high probability, replacing the long constant block by iid symbols increases the LCS.", "Acknowledgments.", "It is a pleasure to thank both referees for their numerous detailed and thoughtful comments on the manuscript leading to the current version.", "In particular, one of the referees suggested we replaced our differentiability condition at every maxima of the mean LCS function by a nicer non-tangential (cone) differentiability condition which can be verified up to a given degree of confidence using Monte Carlo simulations." ] ]

[ [ "The Herschel Exploitation of Local Galaxy Andromeda (HELGA) II: Dust and\n Gas in Andromeda" ], [ "Abstract We present an analysis of the dust and gas in Andromeda, using Herschel images sampling the entire far-infrared peak.", "We fit a modified-blackbody model to ~4000 quasi-independent pixels with spatial resolution of ~140pc and find that a variable dust-emissivity index (beta) is required to fit the data.", "We find no significant long-wavelength excess above this model suggesting there is no cold dust component.", "We show that the gas-to-dust ratio varies radially, increasing from ~20 in the center to ~70 in the star-forming ring at 10kpc, consistent with the metallicity gradient.", "In the 10kpc ring the average beta is ~1.9, in good agreement with values determined for the Milky Way (MW).", "However, in contrast to the MW, we find significant radial variations in beta, which increases from 1.9 at 10kpc to ~2.5 at a radius of 3.1kpc and then decreases to 1.7 in the center.", "The dust temperature is fairly constant in the 10kpc ring (ranging from 17-20K), but increases strongly in the bulge to ~30K.", "Within 3.1kpc we find the dust temperature is highly correlated with the 3.6 micron flux, suggesting the general stellar population in the bulge is the dominant source of dust heating there.", "At larger radii, there is a weak correlation between the star formation rate and dust temperature.", "We find no evidence for 'dark gas' in M31 in contrast to recent results for the MW.", "Finally, we obtained an estimate of the CO X-factor by minimising the dispersion in the gas-to-dust ratio, obtaining a value of (1.9+/-0.4)x10^20 cm^-2 [K kms^-1]^-1." ], [ "Introduction", "Astronomy at long infrared wavelengths (20–1000) is a relatively young field due to the need for space missions to avoid the absorption of the atmosphere in this waveband.", "This waveband, however, is vital for astronomical studies as this is where dust in the interstellar medium (ISM) radiates.", "This is important for studies of galaxy evolution as star formation regions tend to be dusty, and therefore the use of UV and optical measurements to trace the star formation rate can lead to it being underestimated [50], [10], [21], [63], [22].", "Calibrating the relationship between infrared emission and star formation rate has been difficult due to uncertainties from the contribution of the general stellar population to heating the dust, the fact that not all optical/UV emission is absorbed, and uncertainties in the properties of the dust.", "The dust emission is affected by the composition of the dust, and the proportion of non-equilibrium to equilibrium heating.", "Studies with previous space missions IRAS, ISO, Spitzer and AKARI tried to address these questions [91], [16], [22], [20].", "However, as they were limited to wavelengths less than 160, they were not sensitive to the cold dust ($\\le $  15 K) and missed up to 50% of the dust mass in galaxies.", "The continuum emission from the dust has been $^*$Herschel is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA.", "proposed as a potential method of measuring the total mass of the ISM [48], [46], [35], [34]; traditionally the amount of gas has been measured using Hi and CO surveys, but due to sensitivity and resolution issues this method is limited to low redshift and small numbers of galaxies.", "[81] found that for early-type galaxies (E/S0) the ISM was detected for 50% of objects through its dust emission but only 22% through its CO emission.", "In addition, the conversion of the CO tracer to molecular gas mass is highly uncertain and is believed to vary with metallicity [93], [15], [86], [49], [59].", "This is a topical area as recent studies using the Planck all-sky survey and IRAS maps have made an estimate of the amount of `dark gas' [67] in the Milky Way.", "The `dark gas' is thought to be molecular gas which is traced by dust, but not detected with the standard CO method.", "Previous works have also suggested the presence of `dark gas' by using $\\gamma $ -ray emission from cosmic-ray interactions with clouds of gas [43], [2] and by the kinematics of recycled dwarf galaxies [17].", "Herschel is one of the European Space Agency's flagship observatories and observes in the far-infrared (FIR) in the range 55–671 [66].", "Due to the large space mirror and cryogenic instruments, it has a high sensitivity and unprecedented angular resolution at these wavelengths.", "Herschel has the ability to target both large numbers of galaxies and map large areas of sky.", "It has two photometric instruments: PACS [71] which can observe in 3 broad bands around 70, 100 and 160 (70 and 100 cannot be used simultaneously) and SPIRE [44] which observes simultaneously in filter bands centered at 250, 350 and 500.", "SPIRE provides flux measurements on the longer wavelength side of the FIR dust peak ($\\sim $ 160), allowing us to obtain a full census of dust in nearby galaxies [32].", "Andromeda (M31) and the Milky Way (MW) are the only two large spirals in the Local Group.", "Studies of Andromeda therefore provide the best comparison to observations of our own Galaxy with the advantage that we get a `global' picture, whereas investigations of the Milky Way are limited by our location within the Galaxy.", "The total size of M31 and the scalelength of its disk are both approximately twice those of the MW [96].", "However, the star-formation rate of the MW is $\\sim $ 3–6 $M_{\\odot }\\rm \\,yr^{-1}$ [13] compared to only $\\sim $ 0.3–1.0 in M31 [5], [92], despite similar amounts of gas present [96].", "For this reason, M31 is often labelled as `quiescent'.", "The dust emission from M31 is dominated by a dusty star-forming ring at a radius of 10 kpc, and was first observed in the infrared by IRAS [47].", "Many projects to map the ISM in M31, have been undertaken using observations in the mid-infrared (MIR) with Spitzer [5], in the FIR with Spitzer [41], in the Hi atomic line [88], [19], [23] and in the CO($J$ =1-0) line [60].", "Studies of the gas kinematics and dust emission show M31 has a complicated structure [23], [41]; [12] attribute many of the features observed to density waves from a possible head-on collision with M32.", "[87] investigated the relation between gas, dust and star formation using FIR Spitzer data.", "[51] used Spitzer and gas observations to investigate the conversion between CO($J$ =1-0) line flux and molecular hydrogen column-density in a sample of nearby galaxies including M31.", "In their analysis they found M31 has the lowest dust temperatures in their sample and therefore would benefit most by including Herschel data.", "The Herschel Exploitation of Local Galaxy Andromeda (HELGA) is a survey covering a $\\sim 5.5^{\\circ }\\times 2.5^{\\circ }$ area centered on M31.", "We use PACS-SPIRE parallel mode, observing at 100, 160, 250, 350 and 500 simultaneously.", "Further details of the observations can be found in Section REF and in [38].", "HELGA observations have been used in other works to investigate structures in dust and Hi at large radii [38], the relationship between gas and star formation (G. P. Ford et al., in preparation), and the structure of M31 and the cloud-mass function (J. Kirk et al., in preparation).", "In this paper we use the Herschel data combined with the wealth of ancillary data to investigate the distribution, emission properties and the processes heating the dust in M31 on spatial scales of $\\sim $ 140 kpc.", "There have still been relatively few attempts to map the dust within a galaxy; recent studies with Herschel include [80], [14], [37], [8], and [56], but they are often limited to small numbers of independent pixels.", "The advantage of M31 over these other studies is the close proximity which allows us to investigate the dust at higher spatial scale and with many more independent pixels.", "We also apply the Planck method for finding `dark gas' [67] to M31 and use this method to determine a value of the X-factor — the relationship between the molecular hydrogen column density and the observed CO tracer.", "In Section we present the data used for this analysis and Section describes our method for fitting the spectral energy distribution of dust.", "Section presents our maps of the dust properties, including the distribution of dust surface density, temperature and spectral index.", "This section also describes a comparison of the distributions of gas and dust and of the search for an excess emission at long wavelengths.", "In Section we discuss the dust properties including the composition of the dust and the processes heating the dust.", "In this section we also search for `dark gas' and determine the value of the X-factor.", "The conclusions are presented in Section .", "Herschel observations of M31 were taken using both PACS and SPIRE in parallel mode covering an area of $\\sim 5.5^{\\circ } \\times 2.5^{\\circ }$ centered on M31.", "To observe an area this large, the observations were split into two halves with a cross-scan on each half, which produced data at 100, 160, 250, 350 and 500 simultaneously (observation IDs: 1342211294, 1342211309, 1342211319, 1342213207).", "Full details of the observing strategy and data reduction can be found in [38].", "The PACS data reduction was performed in two stages.", "The initial processing up to Level-1 (i.e., to the level where the pointed photometer timelines have been calibrated) is performed in HIPE v8.0 [62] using the standard pipeline.", "To remove low-frequency noise (or drifts) in the arrays, residual glitches and create the final maps we use SCANAMORPHOS [74].", "The final maps have a pixel scale of 2 and 3 and a spatial resolution of 12.5 and 13.3 full-width half maximum (FWHM) at 100 and 160 respectively.", "The SPIRE data were processed up to Level-1 with a custom pipeline script adapted from the official pipeline.", "We apply the latest flux correction factors from the SPIRE Instrument Control Centre to update the maps to the latest calibration product [82].", "For the baseline subtraction we use a custom method called BriGAdE (M. W. L. Smith et al., in preparation) which uses an alternative technique for correcting temperature drifts.", "The final maps were created using the naive mapper with pixel sizes of 6, 8and 12 with spatial resolution of 18.2, 24.5, 36.0 FWHM for the 250, 350 and 500 maps, respectively.", "All Herschel images are shown in Figure REF .", "In addition to the Herschel data, we make use of the 70 Spitzer MIPS [73] map published in [41].", "This observation covers a region of M31 approximately $3^{\\circ } \\times 1^{\\circ }$ in size and has an exposure time of $\\sim 40$ s pixel$^{-1}$ .", "The data was processed using the MIPS Data Analysis Tool, version 2.90 [40] and full details of the reduction can be found in [41]." ], [ "Gas Measurements", "To investigate the atomic hydrogen in Andromeda we use the Hi moment-zero map presented in [19].", "The observations were made with the Westerbork Synthesis Radio Telescope (WRST) covering a region of $\\sim 6^{\\circ } \\times 3.5^{\\circ }$ with a resolution of 18$\\times $ 15.", "In this work we present our results using a map which has not been corrected for opacity effects since this correction is uncertain.", "For the results which make use of the Hi map we also test how our results are affected by using a Hi map corrected for self-opacity using the prescription outlined in [19].", "The uncorrected Hi map has a sensitivity of $\\rm 4.2\\times 10^{18} cm^{-2}$ .", "For the molecular hydrogen gas content we use CO($J$ =1-0) observations presented in [60] made with the IRAM 30m telescope.", "This covers an area of $2^{\\circ } \\times 0.5^{\\circ }$ with a sensitivity of $\\sim $ 0.35 K km s$^{-1}$ .", "All maps other than the Herschel images used for this analysis are shown in Figure REF .", "Figure: Herschel images used in the analysis of this paper.", "The images have dimensions of approximately 4.5 ∘ ×1.3 ∘ \\rm 4.5^{\\circ }\\times 1.3^{\\circ },with a tick spacing of 30 ' 30^{\\prime } and centered on 10 h 43 m 02 s 10^{h} 43^{m} 02^{s}, +41 ∘ 17 ' 42 '' +41^{\\circ } 17^{\\prime } 42^{\\prime \\prime }.", "These imagesare all at their original resolution.Figure: Ancillary images for M31.", "The scale is the same as for the Herschel images presented in Figure .", "From top:Spitzer MIPS 70, the star formation rate (presented in G. P. Ford et al.", "in preparation), Spitzer IRAC 3.6 (presented in ),Hi and CO images as in Figure .", "The CO map (used as a tracer of H 2 \\rm H_{2}) only covers an area of2 ∘ ×0.5 ∘ \\rm 2^{\\circ } \\times 0.5^{\\circ }.", "These images are all at their original resolution." ], [ "The FIR-Submm Spectral Energy Distribution", "To investigate how the dust properties vary across M31 we undertake a pixel-by-pixel dust analysis, using the Spitzer 70, Herschel PACS and SPIRE images.", "We first convolve the data to the same resolution as the 500 map (our largest FWHM) by using a kernel which matches the point-spread function (PSF) in a particular band to the 500 band.", "This procedure is described in detail in [8].", "The images are then re-binned into 36 pixels, chosen to be about the same size as the 500 beam so that each pixel is approximately independent from each other.", "For each band we subtract a background value for the whole map, estimated from regions around the galaxy.", "The uncertainties on the flux in each pixel are found by measuring the standard deviation of the pixels in these background regions and adding this in quadrature with the calibration uncertainty.", "The flux errors in the majority of pixels are dominated by the calibration uncertainty which we take as 7% for Spitzer 70 [42], 10% for PACS [38] and 7% for SPIRE (see Section REF for more details)." ], [ "SED fitting", "For each pixel we fit the far-infrared–sub-millimetre SED with a one-temperature modified black-body model described by: $S_{\\nu } = {{\\kappa _{\\nu }M_dB({\\nu },T_d)}\\over {D^2}}$ where $M_d$ is the dust mass with dust temperature $T_d$ , $B(\\nu ,T_d)$ is the Planck function and $D$ is the distance to the galaxy.", "$\\kappa _{\\nu }$ is the dust absorption coefficient, described by a power law with dust emissivity index $\\beta $ such that $\\kappa _{\\nu } \\propto \\nu ^{\\beta }$ .", "We assume a typical value for the ISM of $\\kappa _{350\\mu \\rm m} = 0.192\\,\\rm m^2\\,kg^{-1}$ [30].", "While the absolute value of $\\kappa _{\\nu }$ is uncertain, its value will not affect trends with other parameters as it is a fixed scaling constant.", "The distance to Andromeda was taken in this work to be $D=0.785\\,\\rm Mpc$ [54].", "We initially used a fixed value of $\\beta $ across the whole galaxy, but we found that with a fixed value it was impossible to adequately fit the SEDs; we therefore let $\\beta $ be a free parameter.", "To ensure the simplex fitting routine did not get stuck in a local minimum, we ran the SED-fitter in two ways: first with all three parameters (M$_d$ , T$_d$ , $\\beta $ ) free to vary; second by fixing the value of $\\beta $ while allowing M$_d$ and T$_d$ to vary and repeating the process for all values of $\\beta $ in the range 0.20 to 5.90 in 0.01 intervals, selecting the result with the lowest $\\chi ^{2}$ .", "Both methods gave consistent results but we created the final maps of the dust properties by choosing the result with the lowest $\\chi ^{2}$ for each pixel.", "The SPIRE calibration has an overall systematic uncertainty of 5% due to the uncertainty in the prime calibrator Neptune, and a statistical uncertainty of 2% determined from instrumental reproducibility; the SPIRE Observer's Manual recommends linearly adding these to give an overall uncertainty of 7%.", "To implement this in practice in our fitting algorithm we increased the uncorrelated uncertainty to give the same overall uncertainty when the errors are added in quadrature.", "This is implemented in the SED fitting by using the full covariance matrix in the $\\chi ^{2}$ calculation.", "We apply our own color correction to the Herschel maps by removing the Herschel pipeline `K4' parameter and then convolving the SED with the filter transmission in the fitting process (for SPIRE the filters appropriate for extended sources are used).", "This method takes full account of all the wavelength-dependant effects associated with PACS and SPIRE.", "In previous work [80] we found that there is a significant contribution from a warmer component of dust at wavelengths $\\le $  70 and so the Spitzer flux at 70 is treated as an upper limit in the fitting (i.e, if the flux value is higher than the model it does not contribute to $\\chi ^{2}$ ).", "The omission of the warm component in the fitting process has a negligible effect on the derived dust mass as the cold component dominates the total dust mass.", "[8] suggest that a warmer component could influence the dust emission up to wavelengths of 160; to test this we repeated the SED-fitting by treating the Spitzer and PACS fluxes as upper limits and found it made a negligible difference to our results.", "To estimate the uncertainties on the results of our fits we use a monte-carlo technique.", "For each pixel we create a set of 1000 artificial SEDs, created by taking the original flux densities and adding a random value selected from a normal distribution with a mean of zero and a standard deviation equal to the uncertainty in the measured flux (the correlations between the calibration uncertainties for SPIRE are also included).", "We estimate the error in the derived parameters for each pixel from the 1000 fits.", "We find that for each pixel there is an uncertainty of 20% in our estimate of the surface density of the dust, of $\\pm $ 1.4 K in our estimate of the dust temperature, and $\\pm $ 0.31 in our estimate of $\\beta $ ." ], [ "Results of the Fits", "In producing the final dust mass, temperature and $\\beta $ images in this work, we only used pixels where the fluxes in all six bands (five Herschel & MIPS 70) have a signal-to-noise greater than 5$\\sigma $ .", "While this potentially causes us to miss the very coldest dust due to weak emission in the shortest wavelength bands, we choose it as a conservative approach to ensure we have accurate estimates of temperature.", "In practice it is the 100 map, which has the lowest sensitivity, which limits the number of pixels in our selection.", "Despite this very conservative selection, there are still $\\sim 4000$ pixels in our resultant maps.", "To see if our model is a statistically reasonable fit to the data, we created a histogram of the $\\chi ^{2}$ values for all pixels (Figure REF ).", "As the 70 flux is used as an upper limit (see Section REF ) and is usually higher than the best-fit model it does not usually contribute to $\\chi ^{2}$ and therefore we only have 1 degree of freedom ($5_{\\rm data\\ points} - 3_{\\rm parameters} - 1$ ).", "The 10% significance level for $\\chi ^{2}$ occurs at 2.71 which is shown by the red line in Figure REF .", "We find 9.8% of our fits have $\\chi ^{2}$ above this level showing our model is an adequate representation of the data.", "We have also checked to see if our radial gradients in temperature, $\\beta $ , and dust surface density are affected by lowering the criteria to include fluxes greater than 3$\\sigma $ and find no significant changes.", "Recent results from the Key Insight on Nearby Galaxies: A Far-Infrared Survey with Herschel (KINGFISH) presented in [26] have suggested that the one-temperature modified black-body model underestimates the dust mass compared to the [31] prescription.", "They attribute this difference to the contribution of warm dust emitting at shorter wavelengths.", "For our analysis this does not appear to be the case.", "First when we set the PACS fluxes ($\\le $  160) as upper limits there is no significant change in our results.", "Second, if multiple temperature components were present this would bias our $\\beta $ values to lower values [77], but we mostly find higher $\\beta $ values than expected (see Section ).", "Third the $\\chi ^{2}$ analysis suggests the model is consistent with the data.", "In addition, [56], using a similar analysis, have found in M51 that the dust mass distribution is similar when using the [31] or the single modified blackbody prescription.", "Figure: Examples of SED fits for pixels in different regions.", "The 70 point (blue) is used as an upper limitand the peak of distribution is shown by the dashed green line.We applied the same fitting technique as outlined in Section REF to the global flux densities, measured in an elliptical aperture with a semi-major axis of 2.0$^\\circ $ and semi-minor axis of 0.73$^\\circ $ .", "This produces a total dust mass of $10^{7.86\\pm 0.09} M_{\\odot }$ with a dust temperature of $16.1\\pm 1.1$  K and $\\beta = 1.9\\pm 0.3$ .", "The total dust mass from summing all the pixels in our pixel-by-pixel analysis gives a value $10^{7.46} M_{\\odot }$ , a factor of $\\sim $ 2.5 lower.", "This is expected as the fraction of 500 flux in the pixels which satisfy our signal to noise criterion is approximately half the global value.", "Combining our global dust mass with gas measurements, we find a global gas-to-dust mass ratio of 72.", "The global temperature is consistent with these for other spiral galaxies obtained using similar methods [27].", "The pixel-by-pixel analysis shows a large range of temperatures and $\\beta $ values as discussed in detail in Section .", "Examples of fits for individual pixels are shown in Figure REF ." ], [ "Simulations", "To help understand the significance of our results and any potential biases or degeneracies in the parameters, we ran a Monte-Carlo simulation.", "Assuming the dust emits as a single-component modified blackbody, we generated synthetic flux values for a range of temperatures and $\\beta $ values, with the same wavelengths as our real data.", "Noise was then added to the simulated fluxes with a value equal to the errors in the real fluxes (excluding the calibration errors) for 2000 repetitions per T, $\\beta $ combination.", "The calibration error was not included as it would systematically shift the fluxes for all pixels.", "We chose an input mass surface density of 0.5 $M_{\\odot }\\rm \\ pc^{-2}$ as this roughly corresponds to the values found in the 10 kpc ring.", "The quantity that is most important for our work is the dust mass, which from Equation REF is just a multiplicative term.", "Figure REF shows the mean and median mass returned for the 2000 repetitions as the input temperature is varied between 12 and 30 K for a $\\beta $ of 2; the error bars show the error on the mean.", "Between 15 and 30 K, the mean and median dust mass returned matches within the errors the input dust surface density of 0.50 $M_{\\odot }\\rm \\ pc^{-2}$ .", "At dust temperatures of 15 K and below, there are large errors on the returned mass, which is due to the fluxes at the PACS wavelengths not reaching the required sensitivity to be included in the fit.", "For the actual data, we only included pixels in which there is at least a 5$\\sigma $ detection in all bands, which will avoid the wildly incorrect estimates of the dust mass seen in the simulation.", "To fully estimate the dust mass and temperature of very cold dust (T $<$ 15 K), we need flux measurements at $>$ 500.", "Nevertheless, the lack of an excess at 500 (Section REF ) is circumstantial evidence that Andromeda does not contain very cold dust.", "We investigated if the results in Figure REF were different for a different choice of $\\beta $ but found no systematic differences.", "By plotting the difference in the resultant temperature in each pixel compared to the input model temperature, we find that between 15 K and 25 K, the temperature uncertainty is $\\sim $ 0.6 K. Above 25 K the uncertainty increases, although in the simulation we did not include a 70 point which would likely provide a constraint to our fits if the dust temperature was $>$ 25 K. The simulation suggests that for input $\\beta $ between 1.5 and 2.4, the uncertainty in the returned value of $\\beta $ for each pixel is $\\sim $ 0.1.", "As expected these uncertainties are lower than returned by the monte-carlo technique in Section REF as we have not included a calibration uncertainty.", "Figure: The range of mass surface densities returned from the SED fitting techniqueversus the input temperature for the simulated data in Section with β=2\\beta = 2 modified-blackbody.The mean of the returned masses is shown by the blue points and the medianis shown by the black points.", "The input mass surface density of 0.50 M ⊙ pc -2 M_{\\odot }\\rm \\ pc^{-2} is shown by the red dashed line.", "To avoid overlapping data points the bluehave been shifted by -0.1 K and the black by +0.1 K.A degeneracy is known to exist between temperature and $\\beta $ ; Figure REF shows that if there is an error in one parameter this is anti-correlated with the error in the other parameter.", "As the distribution is centered on the origin, there is no systematic offset in the returned value of T or $\\beta $ .", "As there is no systematic offset, the error in the mean values over many pixels will therefore be much smaller than the error for a single pixel.", "This simulation is based on the dust emission arising from a one-component modified blackbody.", "Fitting a one-component modified blackbody to pixels for which there are multiple temperature components would produce a bias towards smaller values of $\\beta $ [77], although we would hope to detect this by finding high $\\chi ^{2}$ values.", "In Section REF we show that different regions of M31 have different $\\beta - T$ relations and discuss why this is unlikely to be due to multiple temperature components.", "To summarise, while there is a $\\beta $ –T degeneracy from the fitting algorithm this does not create any systematic offsets in the value returned.", "If the dust temperature falls below 15 K we are unable to constrain the SED due to the lack of data points beyond 500.", "Figure: A density plot showing the correlated uncertainties between β\\beta and temperature.", "The uncertaintiesare taken from the simulated data with T=17T=17 K and β=1.8\\beta =1.8.", "A clear anti-correlation is observed with thedistribution centered on the correct values.To fully populate this graph we increased the simulation to 20,000 repetitions for this β\\beta , T combination.Figure: The distribution of dust surface density, temperature and β\\beta obtained from the SED fitting technique andthe distribution of the gas-to-dust ratio.", "The temperature and β\\beta images include a black ellipse showing a radius of 3.1 kpc.The ticks are plotted at 30 intervals." ], [ "Spatial Distribution of Dust Mass, Temperature and Emissivity Index", "By fitting SEDs to every pixel, we have created maps of dust surface density, temperature, $\\beta $ and gas-to-dust ratio which are shown in Figure REF (for details on how the gas surface density is calculated see Section REF ).", "The dust surface density distribution, un-surprisingly, is more similar to the maps of gas and star-formation rate than to the 3.6 image (See Figure REF & REF ), which traces the stellar mass distribution.", "The $\\beta $ and temperature maps show significant radial variations.", "Figure REF shows how the dust column density, temperature and $\\beta $ vary with radius (the physical radius is calculated assuming an inclination of 77$^{\\circ }$ and PA of 38$^{\\circ }$ , [38]).", "In the star-forming ring at 10 kpc, $\\beta $ has an average value of $\\sim $ 1.8 but increases with decreasing radius reaching a maximum of $\\sim $ 2.5 at a radius of $\\sim $ 3 kpc.", "This is higher than found in global studies of galaxies [67], [27], [32].", "However similarly high values have been reported recently in [37] and [18] for dust within galaxies.", "The value for the ring agrees well with early results from Planck [67] for dust in the galactic disk and the solar neighbourhood.", "The 10 kpc ring has an average dust surface density of $\\sim $ 0.6 $M_{\\odot } \\rm \\ pc^{-2}$ with a dust temperature of 18 K. Towards the very center of the galaxy the dust column density decreases to $\\sim $ 0.04 $M_{\\odot } \\rm \\ pc^{-2}$ , $\\beta $ values fall to $\\sim $ 1.9 and the dust temperature increases to $\\sim $ 30 K. Figure: Results from the SED fits for each pixel plotted versus radius.", "The dashed red lines represent the best-fit linear model (see Section ), thedashed green line represents the transition radius (3.1 kpc) found when fitting the β\\beta results.Figure: The distribution of atomic gas, molecular gas and gas-to-dust ratio versus radius.", "The twogas maps are plotted for all pixels >3σ>3\\sigma .", "The solid red linerepresents the best-fit exponential profile to the gas-to-dust ratio.In Figure REF we see a clear break in the radial variation for both temperature and $\\beta $ at a radius $\\sim $ 3 kpc.", "We fit a model with two straight lines and a transition radius (using a simplex routine) to the $\\beta $ results.", "The same method is used with the temperature values but we set the transition radius to the value obtained from the fit to the $\\beta $ values, which occurs at 3.1 kpc (shown by the dashed green line in Figure REF or black ellipse in Figure REF ).", "At radii smaller than the transition radius, the temperature decreases with radius from $\\sim $ 30 K in the center to $\\sim $ 17 K, with an associated increase in $\\beta $ from $\\sim $ 1.8 to $\\sim $ 2.5.", "At radii larger than the transition radius the dust temperature slowly increases with radius while $\\beta $ decreases with radius to $\\sim $ 1.7 at 13 kpc.", "The best-fit relationships between $T_d$ , $\\beta $ and $\\rm {\\it R}$ are shown in Figure REF and listed below: $\\ \\ \\, \\beta & = ~~0.15{\\rm \\it R} ({\\rm kpc}) + 1.98 ~~~~~~~~~~{\\rm \\it R}\\, < \\,3.1\\,{\\rm kpc} \\\\\\ \\ \\, & = -0.08 {\\rm \\it R} ({\\rm kpc}) + 2.70 ~~~~~~~~~~3.1\\le {\\rm \\it R}\\, < \\,20\\,{\\rm kpc}$ $T_d & = -3.24 {\\rm \\it R} ({\\rm kpc}) + 27.56 ~~~~~~~~{\\rm \\it R}\\, < \\,3.1\\,{\\rm kpc}\\\\& = ~~\\,0.12{\\rm \\it R} ({\\rm kpc}) +16.40~~~~~~~~3.1\\le {\\rm \\it R}\\, < \\,20\\,{\\rm kpc}$ We discuss the cause of the temperature and $\\beta $ variations in Section .", "For Sections REF & REF we consider the inner (${\\rm \\it R}$  $<$  3.1 $\\rm kpc$ ) and outer regions separately." ], [ "Radial Distribution of Gas and Dust", "In Figure REF the radial variations of the atomic gas, the molecular gas and the gas-to-dust ratio are shown.", "We assume an X-factor of $1.9\\times 10^{20}\\rm \\,cm^{-2}\\,[K\\, kms^{-1}]^{-1}$ [85] to convert a CO line flux to a H$_2$ column density, although this value is notoriously uncertain and has been found to vary with metallicity [86], [49].", "For our analysis of M31, the choice of X-factor makes very little difference as the total gas is dominated by the atomic phase.", "Out of the 3974 pixels plotted in the gas-to-dust figure, only 101 have a molecular fraction of $>$ 50% and globally the molecular gas only constitutes $\\sim $ 7% of the atmoic gas [60].", "To estimate the total gas surface density we include the contribution of the atomic or molecular gas in each pixel if the value is greater than 3$\\sigma $ in their respective maps.", "Only 86 of our 5$\\sigma $ dust pixels are not covered by the CO($J$ =1-0) observations; these pixels are in the outskirts of the galaxy and we assume the contribution of molecular gas is negligible.", "We find a tight relation between the gas-to-dust ratio and radius (Spearman Rank Coefficient of 0.80) as shown in Figure REF , which is described by an exponential profile, shown by the red line.", "The gas-to-dust ratio increases exponentially from low values of $\\sim $ 20 in the center of the galaxy to $\\sim $ 110 at 15 kpc typical of the Milky Way in the local environment [29].", "Note that the values are not corrected for helium in the ISM.", "Metallicity gradients have been estimated from oxygen line ratios [OIII]/H$\\beta $ , [OII]/[OIII] and R$_{23}$ by [39].", "They infer a radially decreasing metallicity gradient of $-0.06 \\pm 0.03 \\rm {\\ dex\\ kpc^{-1}}$ from the R$_{23}$ parameter.", "[89] calculate oxygen abundance gradients based on a set of 11 Hii regions from [11] and find gradients in the range of -0.027 to -0.013 $\\rm dex\\ kpc^{-1}$ depending on the calibration used.", "If a constant fraction of the metals in the ISM is incorporated into dust grains [36], one would expect the gas-to-dust gradient to be $-1\\times $ metallicity gradient.", "We find a gas-to-dust gradient of $0.0496 \\pm 0.0005\\rm {\\ dex\\ kpc^{-1}}$ consistent within the uncertainties to the gradient measured by [39] (if the Hi map corrected for self-opacity is used the gradient slightly increases to $0.0566 \\pm 0.0007\\rm {\\ dex\\ kpc^{-1}}$ ).", "This supports the claim that gas can be well traced by dust at a constant metallicity.", "To see if the uncertainty in the choice of X-factor could affect this result, we carried out the same procedure but limited the analysis to pixels where the molecular hydrogen contribution is less than 10% of the total gas mass.", "This produced only a small change in the gas-to-dust gradient to $0.0550 \\pm 0.0007\\rm {\\ dex\\ kpc^{-1}}$ which is still consistent with the metallicity gradient." ], [ "500 Excess", "Searches for a long-wavelength sub-mm excess (i.e.", "$>$ 500) has been made in both the Milky Way [65] and for nearby galaxies.", "A sub-mm excess is important as it could suggest very cold dust is present ($<$ 15 K) which would dominate the dust mass in galaxies.", "Other possible explanations of an excess include variations in the dust emissivity index with wavelength [95], [72], [65], different dust populations or contamination from a synchrotron radio source.", "Sub-mm excesses have been reported from observations of low-metallicity dwarf galaxies [61], [45], [26] and spiral galaxies [98], [6].", "Most detections have been made by combining FIR data with ground-based data at 850 or 1 mm data rather than from data only at $\\le $ 500.", "We searched for a sub-millimetre excess in M31 by comparing the 500 flux to our best-fit models.", "A 500 excess is defined to be any observed 500 flux that is greater than the expected flux at 500 from our modified blackbody fit to the data.", "Figure REF is a histogram of the ratio of the excess at 500 and the noise on the 500 map.", "The distribution of the excess is consistent with a Gaussian function with a mean of 0.54$\\sigma $ and standard deviation of 0.31$\\sigma $ .", "The fact the distribution of the histogram is a Gaussian suggests that what we are seeing is random noise with a fixed offset (not centered on zero).", "Two non-astronomical sources could explain a fixed offset, either an incorrect background subtraction or a calibration error.", "The background correction applied is quite small (0.2$\\sigma $ ), and thus an error in this is unlikely to be the entire explanation of the offset.", "The distribution of the excess is consistent with the 2% random SPIRE calibration uncertainty, so both factors together could explain the small offset.", "If the excess is generated from dust with a 10 K temperature and $\\beta $ of 2, the dust mass is $\\sim $ 10$^{6.58}$  $M_{\\odot }$ in our $>$ 5$\\sigma $ pixels, which corresponds to only 13% of the mass of the warmer dust.", "If we used a model containing dust at more then one temperature we would not get a useful constraint on the colder dust component without additional data at longer wavelengths.", "In particular observations at $\\sim $ 850 (e.g.", "with SCUBA2) would be useful to determine if any cold dust present." ], [ "Heating Mechanisms and Dust Distribution", "Recent studies by [7], [8] and [14] have used Herschel colors to confirm previous works [53], [91] that emission from dust in nearby galaxies at wavelengths longer than 160 is mostly from dust heated by the general stellar population.", "These authors conclude that at wavelengths shorter than 160, the emission tends to be dominated from warmer dust heated by newly formed stars.", "[58] in a study of M31 using SpitzerMIR/FIR, UV and optical data, also concluded that the dust is mostly heated by an old stellar population (a few Gyr old).", "To investigate the relation between our derived dust properties and the SFR and the general stellar population, we have used the 3.6 map presented in [5] to trace the general stellar population (rather than the luminous newly formed stars that dominate the UV) and a map of star-formation rate (SFR).", "The SFR map is created from far-UV and 24 images which have been corrected for the contribution of the general stellar population to the emission at these wavelengths (for details see G. P. Ford et al.", "in preparation).", "These maps were all convolved to the same resolution and binned to the same pixel size as all the other maps in our analysis.", "We have plotted the fluxes from these maps against the results of our SED-fits (see Figure REF ).", "In Section REF we showed there is a clear break in the dust properties at a radius of 3.1 kpc.", "Therefore in Figure REF the pixels at a radius less than 3.1 kpc are shown in blue and those at a radius above 3.1 kpc in green.", "For each graph (and both sets of pixels) the Spearman rank coefficient is computed and an estimate made of its significance (see Table REF ).", "With such a large number of pixels, all but one of our correlations are formally significant.", "In the discussion below we have concentrated on the strongest correlations as measured by the Spearman correlation coefficient.", "There are strong correlations in Figure REF between $\\beta $ and 3.6 emission.", "We believe these are most likely to be caused by radial variations in $\\beta $ seen in Figure REF and the decrease in 3.6 brightness with radius.", "We discuss the possible cause of the radial variation in $\\beta $ in Section REF .", "We find a strong correlation between dust surface-density and the SFR in the outer regions, but not with the surface density of total stars traced by the 3.6 emission.", "This correlation is expected as stars are formed in clouds of gas and dust.", "In the inner region there is an anti-correlation between the dust surface density and the 3.6 flux, which seem most likely to be explained by both quantities varying with radius: the 3.6 emission from the bulge increasing towards the center and the dust surface density decreasing towards the center.", "ccccc Spearman Correlation Coefficients for Properties of M31 0pt Property A Property B Region Spearman P-Value Coefficient 41.0cmDust Surface Density 2*3.6 flux Inner -0.73 $1.52\\times 10^{-28\\ }$ Outer -0.16 $1.53\\times 10^{-22\\ }$ 2*SFR Inner 0.31 $6.38\\times 10^{-5\\ \\ }$ Outer 0.74 0.00 (rl)1-5 4*Temperature 2*3.6 flux Inner 0.90 $6.17\\times 10^{-59\\ }$ Outer -0.09 $7.46\\times 10^{-9\\ \\ }$ 2*SFR Inner 0.04 $5.97\\times 10^{-1\\ \\ }$ Outer 0.14 $3.70\\times 10^{-19\\ }$ (rl)1-5 4*$\\beta $ 2*3.6 flux Inner -0.52 $8.78\\times 10^{-13\\ }$ Outer 0.56 $8.99\\times 10^{-311}$ 2*SFR Inner 0.16 $3.58\\times 10^{-2\\ \\ }$ Outer -0.24 $2.85\\times 10^{-52\\ }$ The Spearman rank-order correlation coefficient and P-value (for the the null hypothesis that the two data sets are uncorrelated) for the scatter plots shown in Figure REF (values were calculated using the scipy.stats package and checked with IDL r_correlate routine).", "The inner and outer region contain 164 and 3810 pixels, respectively.", "Figure: Scatter plots showing correlations between dust properties and 3.6 flux or SFR.", "The blue points are results at radii <<3.1 kpc andthe green data points are results at radii >>3.1 kpc.", "The Spearman rank-order coefficients for both sets of points are shown in the top-left cornerof each plot.Figure: Log(3.6) flux versus log(Temperature) for the inner 3.1 kpc.", "The best-fit linear model is shownby the red line.The strongest correlation is seen between the dust temperature and the 3.6 flux in the inner region.", "Figure REF shows a log-log graph of the two quantities and a linear fit to the points.", "The gradient represents the power n where $F_{3.6\\mu m} \\propto T^{n}_{dust}$ where $n=4.61\\pm 0.15$ .", "While the correlation suggests the dust in the bulge is heated by the general stellar population, for a modified blackbody with $\\beta = 2$ we would expect a gradient of 6.", "The difference is probably explained by the simplicity of our assumptions: that there is only a single stellar population in the bulge and that the bulge has a constant depth in the line of sight.", "If these assumptions are incorrect, the 3.6 surface brightness of M31 will only be an imperfect tracer of the intensity of the interstellar radiation field (ISRF).", "Figure: The color image shows the SFR image from G. P. Ford et al.", "(in preparation) which has been smoothed and re-gridded to match the maps presented inFigure .", "The contours are from the dust temperature map and drawn at 18.0, 19.5 and 21.0 K values.Looking at the temperature beyond 3.1 kpc, we find a weak, but still highly significant, correlation with SFR, suggesting that the ISRF has a significant contribution from star-forming regions.", "As most of the star formation in M31 occurs in the 10 kpc ring this is to be expected.", "This correlation can be seen in Figure REF where most but not all of the temperature peaks in the 10 kpc ring appear aligned with the peaks in the SFR map.", "In the same region there is a slight anti-correlation of temperature with the 3.6 flux which could be explained by the radial decrease in the 3.6 flux whilst the dust temperature increases slightly with radius.", "The fact that dust temperature increases slightly with radius, while the number-density of stars, traced by the 3.6, is falling with radius suggests that outside the bulge the dust is mainly heated by young stars.", "Nevertheless, the lack of a strong correlation between dust temperature and either SFR or 3.6 flux suggest the optical/UV light absorbed by a dust grain is from photons from a large range of distances (e.g., photons from the bulge heating dust in the disk).", "[8] have studied FIR colour ratios in M81, M83 and NGC4203.", "They find the 250/350 color ratio has the strongest positive correlation with 1.6 emission.", "An increase in the 250/350 ratio would indicate either an increase in dust temperature or a decrease of $\\beta $ .", "[8] conclude that the most likely explaination is the temperature effect, with the dust being heated by the general stellar population traced by the 1.6 emission.", "We find a similar correlation to [8], only we trace the stellar radiation field with the 3.6 band.", "However, our SED-fitting results suggest that this is caused by a combination of changes in temperature and $\\beta $ .", "Note that since [8] only use color ratios they are unable to discriminate between changes of temperature and $\\beta $ .", "At radii greater than 3.1 kpc our results suggest the variation in the 250/350 is mainly caused by a change in $\\beta $ .", "[8] found that the 70/160 color ratio has the greatest correlation with star-formation rate, which is evidence that there is dust at more than one temperature contributing to the 70–500 emission.", "One possible explanation of our failure to find a correlation between the 3.6 emission and dust temperature outside 3.1 kpc, instead of the negative correlation between 3.6 and $\\beta $ we find, might be if the Herschel emission at short wavelengths contains a contribution from a warmer dust component.", "This would mean our fits of a one-component modified blackbody would produce misleading results.", "However, as stated in Section REF when we attempted the same SED fitting process but using all flux densities $\\le $ 160 as upper limits, we see little difference in our results." ], [ "Dust Emissivity and Temperature Relation", "The dust emissivity index ($\\beta $ ) is related to the physical properties of the dust grains, including the grain composition, grain size, the nature of the absorption process and the equilibrium temperature of the dust.", "We would also expect to see a change in $\\beta $ due to environment from the processing of the grains via grain growth (e.g.", "coagulation, mantle accretion) or destruction through surface sputtering by ions/atoms or shattering by shocks.", "In M31, we detect an apparent inverse correlation between $\\rm T_{d}$ and $\\beta $ for the inner and outer regions of M31, as shown in Figure REF .", "We find the form of the relation is different for the two regions.", "Such an inverse relationship has been observed in the Milky Way with previous FIR-submm experiments and surveys including ARCHEOPS [28], which showed $\\beta $ ranging from 4 to 1 with the dust temperature varying between 7 and 27 K, and PRONAOS [33], which shows a variation of $\\beta $ from 2.4–0.8 for dust temperatures between 11 and 80 K. [90] used IR-mm data of Galactic high latitude clouds and found a similar trend, and more recently [64] with Herschel found a similar inverse relationship with $\\beta = 2.7-1.8$ for $\\rm T_d=\\rm 14-21\\,K$ for galactic longitude $59^{\\circ }$ (at longitude $30^{\\circ }$ $\\beta = 2.6-1.9$ for 18–23 K).", "Recently [18] used Herschel-ATLAS observations to investigate $\\beta $ variations in low-density, high-latitude galactic cirrus, measuring values of $\\beta $ ranging from 4.5–1.0 for $\\rm 10<T_d <28\\,K$ .", "These $T_d-\\beta $ relationships could be indicative of a problem with the temperature–$\\beta $ degeneracy arising from the SED fitting, the presence of dust with a range of temperatures along the line of sight [77], or real variations of the properties of the dust grains.", "On the assumption that the inverse correlations between $\\beta $ and $\\rm T_{d}$ in Figure REF are not simply caused by the two variables being separately correlated with radius, we looked for other possible causes of the relationships: The fitting can lead to a spurious inverse correlation between $\\beta $ and $\\rm T_{d}$ [76].", "The most striking feature in Figure REF is the clear separation in points between the inner 3.1 kpc and the outer regions.", "To test whether these different distributions might be produced by the fitting artifact, we used the Monte-Carlo simulations from Section REF to simulate the effect of fitting a modified blackbody for various combinations of $\\beta $ and $\\rm T_d$ .", "The grey lines in Figure REF show the best-fit relationships for different input $\\rm T_{d}$ and $\\beta $ combinations and clearly show that the two different relationships in the two regions cannot be obtained from a single population of dust grains.", "The green and blue data points represent the range of output $\\rm T_{d}$ and $\\beta $ for an input modified blackbody with $\\rm T = 17.0$  K (green) and $\\rm T = 25.0$  K (blue), with $\\rm \\beta = 2.0$ .", "A comparison of Figures REF and REF shows that in both regions of M31 there is a larger range of temperature and dust emissivity for the real data than that found in the Monte-Carlo simulation, indicating there are genuine variations of $\\rm T_{d}$ and $\\beta $ in both regions.", "Moreover, the fitting artifact cannot explain the observed relationships of $\\beta $ and temperature with radius (Figure REF ).", "Artificial inverse $\\rm T_{d} - \\beta $ relationships can also be produced if a one-component modified blackbody model is used to fit dust which contains a range of dust temperatures [77].", "Since we are averaging through the disk of a galaxy along the line of sight (LOS), it is obviously possible that the dust contains a range of dust temperatures.", "While we cannot fully address this issue, our $\\beta $ values are higher than expected, which is the opposite of what happens from a LOS averaging of temperatures.", "We also find no statistical evidence from our fits that there is more than one component of dust.", "Also [64] and [4] show inverse $T_d-\\beta $ relationships still exist in places where it is unlikely there is dust at more than one temperature.", "Variation of $\\beta $ with wavelength has been reported by some authors both from theoretical models and laboratory experiments and from observations [57], [24], with a transition around 500.", "In Section REF we show that there is no evidence for excess 500 emission, suggesting this is not an explanation of our results.", "Figure: The variation of the dust temperature with emissivity index across M31.", "Data points are colour-codedfor those within R<3.1 kpc R<3.1\\,\\rm kpc (blue) and those beyond this radius (green).", "Solid lines show thebest-fit relations for T d -βT_d-\\beta in M31.", "The T d -βT_d-\\beta relationships in the literature are indicatedby the dashed lines , , , .Figure: The variation of the dust temperature with emissivity index that arise from just the uncertainties in themeasurements.", "The data shown use the simulations of the SED-fitting method, describedin Section .", "The green and blue data points show the recovered values of β\\beta and T d \\rm T_{d}for an input model with T=17.0\\rm T = 17.0 K, β=2.0\\rm \\beta = 2.0 (green) and T=25.0\\rm T = 25.0 K, β=2.0\\rm \\beta = 2.0 (blue).We have carried out the same simulation for input values of T d \\rm T_{d} and β\\beta over the rangeT d \\rm T_{d} of 15–29 K in 2 K intervals and in β\\beta of 1.6–2.4 in 0.2 intervals.For each group of points we have fitted a line T d ∝β n \\rm T_{d} \\propto \\beta ^{n}, which are the grey dashed lines.In these cases we have not shown the recovered values of T d \\rm T_{d} and β\\beta , merely the lines that are thebest fit to the points.", "The red and black solid linesare the best fit models to the real data as shown in Figure .", "When compared with Figure it is clear that the uncertainties cannot account for the distribution in the real data.To describe the $T_d-\\beta $ relationships we use an empirical model of the form $\\beta = AT^{\\alpha }$ , commonly used in the literature [28], [64], [70] to fit the $T_d-\\beta $ anti-correlation.", "The observed anti-correlation between T and $\\beta $ may arise due to a change in the physical properties of grains including the grain optical constants changing with temperature for amorphous grains or changes in the dust emissivity with wavelength [97].", "Other possibilities include grain growth or quantum mechanical effects (though these latter grain properties only arise at lower dust temperatures than observed in M31).", "The best-fit relationship which describes $T_d-\\beta $ for R $< 3.1\\,\\rm kpc$ and for $\\rm 3.1 < R < 15\\,kpc$ are (shown in Figure REF ): $\\beta = \\left\\lbrace \\begin{array}{l l}2.30 (\\frac{T_d}{20})^{-0.61} & \\quad {\\rm R < 3.1\\,kpc}\\\\1.58 (\\frac{T_d}{20})^{-1.57} & \\quad {\\rm 3.1 \\le R < 15\\,kpc}\\\\\\end{array}\\right.$ where the steeper $\\rm T_d-\\beta $ relationship at R $>3.1\\,\\rm kpc$ agrees well with the relationship found in the plane of the Milky Way at longitudes of $59^{\\circ }$ [64] and in low-density, high-latitude cirrus [18].", "There is some evidence that the $\\rm T_d-\\beta $ relationship in M31 is slightly steeper, so that for the same temperature compared to the galactic plane, M31 has a lower $\\beta $ (but this is only at $\\sim 5-10\\%$ level).", "What could be a physical (or chemical) explanation of the different $\\rm \\beta - T_{d}$ relationships in the two regions?", "Typical values of $\\beta $ are in the range 1.5–2.0 for interstellar dust grains, and have been found in global extra-galactic studies [79], [81], [32] and average global values measured in the Milky Way , , [69].", "Low values of $\\beta $ for large grains would typically represent freshly-formed dust grains in circumstellar disks or stellar winds.", "Alternatively, $\\beta \\sim 1$ has been observed in regions where small grains dominate [75].", "High values of $\\beta $ ($>2$ ) might occur due to grain coagulation or to the growth of icy mantles on the surface of the grains in denser regions [1], [52], [84].", "Studies have also suggested high values of $\\beta $ are associated with very cold dust <12 K$ e.g.,\\endcsname {Desert2008}possibly caused by a change in the absorption properties due to quantum effects at low temperatures,increasing self-absorption in amorphous grains via tunnelling \\cite {Agladze1996, Mennella1998, Meny2007, Paradis2012}.$ The highest value of $\\beta $ is seen at the 3.1 kpc boundary between the two regions.", "This cannot be caused by changes in the quantum mechanical absorption, since this is only thought to be important for cold dust at temperatures $<12\\,\\rm K$ .", "The high $\\beta $ values could be due to efficient grain coagulation or mantle growth in dense molecular clouds, although this too seems unlikely as little CO($J=1-0$ ) is observed in this region.", "While there is no obvious explanation for the high $\\beta $ values at this radius, there are many indications that this 3 kpc “boundary” is an interesting regime, we discuss this further in Section REF .", "In the inner region, we suggest that the decrease in $\\beta $ with corresponding increase in $\\rm T_d$ might be caused by the increased intensity of the ISRF.", "Towards the center of M31, we would expect increased sputtering or sublimation of mantles from the increased ISRF, shown by the increased temperature of the dust and the increased X-ray emission observed in the center [78].", "The lack of gas in the central regions (Figure REF ) also suggests that dust is less likely to be shielded and thus more efficiently sputtered and leading to smaller grain sizes.", "As we mentioned above, a problem with this analysis is that it is difficult to determine which are the causal relationships.", "For example, we have argued that the radial variation in $\\rm T_d$ is due to the radial variation in the ISRF.", "The radial variation in $\\beta $ might then be due to a physical relationship between $\\rm T_d$ and $\\beta $ or it might be the case that there is no causal relationship between these parameters but the radial variation in $\\beta $ is caused by a different effect.", "For example, an interaction between M32 and M31 might have caused a wave of star formation which has moved out through the galaxy, which might have led (by a number of processes) to the radial variations in $\\beta $ .", "Therefore, we can rule out some hypotheses but we cannot conclusively determine which is the true explanation using this data set." ], [ "Why a Transition at 3.1 kpc?", "Interpreting the transition in dust properties seen at 3.1 kpc (Figure REF ) is difficult.", "One possible clue comes from previous gas kinematics studies.", "[23] found that the Hi rotation curve inside a 4 kpc radius is warped with respect to the rest of disk.", "[83] suggest the inner Hi data is consistent with a bar extended to 3.2 kpc, while a newer analysis by [9] explains the Hi distribution as the result of a triaxial rotating bulge.", "[12], using Spitzer IRAC observations, identified a new inner dust ring with dimensions of 1.5 $\\times $  1 kpc.", "By using the stellar and gas distributions and from the presence of the 10 kpc ring, they conclude that an almost head-on collision has occurred between M31 and M32 around 210 million years ago.", "This collision could explain the perturbation of the gas observed in the central 4 kpc.", "These other observations all show that the inner 3 kpc of M31 is an intriguing region, although it it not clear what are the causes of the difference in the dust properties.", "The perturbation of the gas may have lead to the processing of dust grains, or potentially material from M32 could have been deposited after the interaction.", "The total dust mass for the pixels in our selection within 3.1 kpc is $10^{4.2} M_{\\odot }$ which is a plausible amount to be deposited as recently dust masses of $\\sim $$10^{5} M_{\\odot }$ have been reported in Virgo dwarf ellipticals [45].", "Another possibility is the dust properties are affected by the differences between conditions in the bulge and disk.", "[25] decomposed the luminosity profile of Spitzer IRAC data into a bulge, disk and halo.", "From their Figure 16 we can see that our transition radius of $\\sim $ 3 kpc is approximately where the bulge emission begins to become a significant fraction of the optical disk emission.", "Whether the transition in the dust parameters is due to the changing contribution to the ISRF from the general stellar population and star formation or if there is another influence in the bulge is unknown." ], [ "Dark Gas and X factor", "The detection of `dark gas' in the Milky Way was a surprising early result from Planck [67], obtained by combining IRAS 100 data and the six-band Planck data from 350–3 mm.", "The Planck team compared the dust optical-depth with the total column density of hydrogen ($N_{H}^{Tot}$ ), where the optical depth at each wavelength is given by $\\tau _{\\nu } = \\frac{I_{\\nu }}{B(\\nu ,T_{dust})}$ where $I_{\\nu }$ is the flux density in that band and $B(\\nu ,T_{dust})$ is the blackbody function.", "They assumed that at low $N_{H}^{Tot}$ the atomic hydrogen dominates over the molecular component while at high column density the molecular hydrogen dominates the emission.", "For these two regimes they found a constant gas-to-dust ratio, but at intermediate column densities they found an excess of dust compared to the gas.", "This excess is attributed to gas traced by dust but not by the usual Hi and CO lines, and is found to be the equivalent 28% of the atomic gas or 118% of the molecular gas.", "The excess dust emission was typically found around molecular clouds, suggesting that the most likely cause is the presence of molecular gas not traced by the CO line.", "We attempted the same analysis as the [67] for M31 using our SED fitting results from Section .", "Instead of using Equation REF , we compare the column density of gas estimated from the Hi and CO with the column density of dust (for convenience we call this $\\Sigma _{dust}$ ).", "We use this parameter as it is calculated with data from all wavelengths, whereas if we used Equation REF , small errors in temperature would cause large uncertainties in $\\tau _{\\nu }$ for wavelengths close to the peak of emission.", "Figure: Radially corrected Σ dust \\Sigma _{dust} versus total column density of gas.", "The plot is shown using our best-fit value of the X-factor of(1.9±0.4)×10 20 cm -2 [K kms -1 ] -1 (1.9 \\pm 0.4)\\times 10^{20} \\rm \\ cm^{-2} [K\\ kms^{-1}]^{-1}.", "The red line represents the best fit model to the dataassuming that Σ dust ∝N H Tot \\Sigma _{dust} \\propto N_H^{Tot}.", "The plot shows that, unlikethe Planck data for the Milky Way , at intermediate gas column densities we do not find an excessin dust column density over the best-fit model which would indicate the presence of gas not traced by the Hi and CO (`dark gas').Figure: Radially corrected Σ dust \\Sigma _{dust} versus column density of gas for pixels where the molecular fractionis greater than 20%.", "The data points are colour-coded with the fraction of molecular gas compared to total gas.The Figure shows that the high column densities are not dominated by regions of molecular gas traced bythe CO.", "The red line is the fitted model from Figure .The Planck team found no radial variation in the gas-to-dust ratio in the Milky Way [68].", "In Andromeda we show that the gas-to-dust ratio does vary radially (Section REF and Figure REF ), as expected from the metallicity gradient.", "To determine if there is an excess at intermediate column density in dust compared to that expected from the gas we have to correct for the radial change in gas-to-dust ratio.", "To remove this dependence, we adjust dust column density by using the exponential fit (shown by the red line in Figure REF ) so the gas-to-dust ratio at all radii has the same value as the center of M31.", "To avoid biasing this correction by assuming a value for the X-factor which is highly uncertain, we estimate this relationship from pixels where the atomic hydrogen column density is $>95$ % of $N_{H}^{Tot}$ (1569 pixels out of the 3600).", "Figure: Map of Σ dust /N H Tot \\Sigma _{dust} / N_{H}^{Tot} ratio in M31.", "Higher values represent areas where there is less gas thanpredicted from dust measurements.", "The tick spacing represents 30.In Figure REF we show that the relationship between corrected dust column density ($\\Sigma _{dust}$ ) and gas column density is well represented by assuming the two quantities are directly proportional with no excess in dust column-density that could be attributed to `dark gas'.", "Although only a small proportion of the gas is molecular, we still need to use a value for the X-factor.", "We can estimate this quantity from the data itself by finding the values of the X-factor and the constant of proportionality between the gas column density and the corrected $\\Sigma _{dust}$ that gives the minimum $\\chi ^2$ value (the fitted line is shown in Figure REF .)", "We find a best value for the X-factor of $(1.9 \\pm 0.4)\\times 10^{20} \\rm \\ cm^{-2} [K\\ kms^{-1}]^{-1}$ (or expressed as $\\alpha _{CO} = 4.1 \\pm 0.9 M_{\\odot }\\rm \\,pc^{-2}\\,[K\\,kms^{-1}]^{-1}$ ), where the random error is estimated using a monte-carlo technique (similar to one used in Section REF ).", "For the dust column densities in each pixel we use the uncertainties provided by the SED fitter as explained in Section REF , which is on average $\\sim $ 22%.", "[60] quote a calibration error of 15% for the CO observations (which directly results in at least a 15% uncertainty in the X-factor) which we combine with the noise in each pixel of our processed moment-zero CO map.", "For the Hi observations we use an uncertainty map provided by Robert Braun (private communication) which has an average uncertainty of 12% on the raw 10 map.", "We also include a 5% systematic uncertainty (e.g., calibration uncertainties).", "If the opacity corrected Hi map is used our best fit X-factor is $(2.0 \\pm 0.4)\\times 10^{20} \\rm \\ cm^{-2} [K\\ kms^{-1}]^{-1}$ .", "[51] find values of the X-factor between $0.97$ to $4.6\\times 10^{20} \\rm \\ cm^{-2} [K\\ kms^{-1}]^{-1}$ when analysing the southern, northern and inner regions for M31 with Spitzer data.", "Our average value of the X-factor falls within their range of values.", "A full analysis of the spatial variations of the X-factor with our data will be undertaken in a future paper.", "There are two main problems with this method which could lead us to miss `dark gas'.", "First, unlike the Milky Way we have to average through the whole disk of M31 and, second, M31 has a significantly lower molecular gas fraction.", "The latter prevents us from fitting the model to pixels with just very high and low values of $N_{H}^{Tot}$ as the pixels with highest molecular gas fraction are not clustered to high $N_{H}$ values (this is illustrated in Figure REF ).", "This suggests Andromeda may not be the best galaxy for this analysis as the molecular contribution to the overall column density is quite low.", "However, we can also try an alternative method of looking for `dark gas' because we have one important advantage over the Planck team: we can see M31 from the outside.", "We can therefore make a map of the ratio of radially corrected dust column density ($\\Sigma _{dust}$ ) to gas column density to look for regions of enhancements in this ratio (Figure REF ).", "The image clearly shows spatial variations which could either suggest regions of `dark gas' or local variations in the metallicity or emissivity of dust.", "To distinguish between these scenarios an independent measurement of the `dark gas' is required.", "On the outskirts of molecular clouds, CO could be photodissociated and the carbon gas would therefore reside in C or C$^+$ [94].", "Planned observations of the C[II] 158 line in Andromeda with Herschel could then be a potential test for investigating whether `dark gas' exists in M31." ], [ "Conclusions", "In this paper we present the results of an analysis of dust and gas in Andromeda using new Herschel observations from the HELGA survey.", "We have $\\sim $ 4000 independent pixels with observations in the range of 70–500.", "We find the following results: We find that a variable dust emissivity index, $\\beta $ , is required to adequately fit all the pixels in Andromeda.", "When a variable $\\beta $ is used, the modified blackbody model with a single temperature is found to be a statistically reasonable fit to the data in the range 100–500.", "There is no significant evidence of an excess of dust emission at 500 above our model.", "There are two distinct regions with different dust properties, with a transition at $R=3.1$  kpc.", "In the center of Andromeda, the temperature peaks with a value of $\\sim $ 30 K and a $\\beta $ of $\\sim $ 1.9.", "The temperature then declines radially to a value of $\\sim $ 17 K at 3.1 kpc with a corresponding increase in $\\beta $ to $\\sim $ 2.5.", "At radii larger than 3.1 kpc $\\beta $ declines but only with a small associated increase in temperature.", "The drop in $\\beta $ towards the center of the galaxy may be caused by increased sputtering or sublimation of mantles from an increased ISRF.", "The origin of the high $\\beta $ values at 3.1 kpc from the center is less clear but may be indicative of either grain coagulation or an increase in the growth of icy mantles.", "The dust surface density for our pixels in which flux is detected at $>5\\sigma $ in all Herschel bands range is between $\\sim $ 0.1–2.0 $M_{\\odot }\\rm \\ pc^{-2}$ .", "We find the gas-to-dust ratio increases exponentially with radius.", "The gradient matches that predicted from the metallicity gradient assuming a constant fraction of metals are included into dust grains.", "The dust surface density is correlated with the star formation rate rather than with the stellar surface density.", "In the inner 3.1 kpc the dust temperature is correlated with the 3.6 flux.", "This suggests the heating of the dust in the bulge is dominated by the general stellar population.", "Beyond 3.1 kpc there is a weak correlation between dust temperature and the star formation rate.", "We find no evidence for `dark gas', using a similar technique as the Planck team.", "However, we find this technique may not be as effective for M31 due to poor angular resolution and line of sight effects.", "We have used an alternative technique by constructing a gas-to-dust map after correcting for the radial gradient.", "We do find regions with enhancements (i.e., higher values of $\\Sigma _{dust} / N_H^{Tot}$ ), which may show places where `dark gas' exists, or may be due to local variations in the gas-to-dust ratio.", "A detection of a potential component of CO-free molecular gas will be possible with future observations to measure the C[II] line planned with Herschel.", "By minimising the scatter between our radially corrected dust column-density and the column-density of gas inferred from the Hi and CO line we find a value for the X-factor of $(1.9 \\pm 0.4)\\times 10^{20} \\rm \\ cm^{-2} [K\\ kms^{-1}]^{-1}$ (or expressed as $\\alpha _{CO} = 4.1 \\pm 0.9 M_{\\odot }\\rm \\,pc^{-2}\\,[K\\,kms^{-1}]^{-1}$ ).", "Our results of Andromeda represent the largest resolved analysis of dust and gas in a single galaxy with Herschel.", "The results of this analysis on M31 is strikingly different from those obtained by the Planck team in the Milky Way, since we find no clear evidence for `dark gas', a radial gradient in the gas-to-dust ratio and evidence for radial variation in the dust emissivity index ($\\beta $ ).", "In future work it will be important to understand these differences between the two big spirals in the local group.", "We thank everyone involved with the Herschel Observatory.", "PACS has been developed by a consortium of institutes led by MPE (Germany) and including UVIE (Austria); KU Leuven, CSL, IMEC (Belgium); CEA, LAM (France); MPIA (Germany); INAF- IFSI/OAA/OAP/OAT, LENS, SISSA (Italy); IAC (Spain).", "This development has been supported by the funding agencies BMVIT (Austria), ESA-PRODEX (Belgium), CEA/CNES (France), DLR (Germany), ASI/INAF (Italy), and CICYT/MCYT (Spain).", "CVN, PR, DH, GR, YN and KE acknowledge support from the Belgian Federal Science Policy Office via the PRODEX Programme of ESA.", "SPIRE has been developed by a consortium of institutes led by Cardiff Univ.", "(UK) and including: Univ.", "Lethbridge (Canada); NAOC (China); CEA, LAM (France); IFSI, Univ.", "Padua (Italy); IAC (Spain); Stockholm Observatory (Sweden); Imperial College London, RAL, UCL-MSSL, UKATC, Univ.", "Sussex (UK); and Caltech, JPL, NHSC, Univ.", "Colorado (USA).", "This development has been supported by national funding agencies: CSA (Canada); NAOC (China); CEA, CNES, CNRS (France); ASI (Italy); MCINN (Spain); SNSB (Sweden); STFC, UKSA (UK); and NASA (USA).", "HIPE is a joint development by the Herschel Science Ground Segment Consortium, consisting of ESA, the NASA Herschel Science Center and the HIFI, PACS and SPIRE consortia.", "GG is a postdoctoral researcher of the FWO-Vlaanderen (Belgium) The research leading to these results has received funding from the European Community's Seventh Framework Programme (/FP7/2007-2013/) under grant agreement No 229517.", "Facilities: Herschel (PACS and SPIRE) , Spitzer (MIPS)." ] ]

[ [ "Classical and quantum massive cosmology for the open FRW universe" ], [ "Abstract In an open Friedmann-Robertson-Walker (FRW) space background, we study the classical and quantum cosmological models in the framework of the recently proposed nonlinear massive gravity theory.", "Although the constraints which are present in this theory prevent it from admitting the flat and closed FRW models as its cosmological solutions, for the open FRW universe, it is not the case.", "We have shown that, either in the absence of matter or in the presence of a perfect fluid, the classical field equations of such a theory adopt physical solutions for the open FRW model, in which the mass term shows itself as a cosmological constant.", "These classical solutions consist of two distinguishable branches: One is a contacting universe which tends to a future singularity with zero size, while another is an expanding universe having a past singularity from which it begins its evolution.", "A classically forbidden region separates these two branches from each other.", "We then employ the familiar canonical quantization procedure in the given cosmological setting to find the cosmological wave functions.", "We use the resulting wave function to investigate the possibility of the avoidance of classical singularities due to quantum effects.", "It is shown that the quantum expectation values of the scale factor, although they have either contracting or expanding phases like their classical counterparts, are not disconnected from each other.", "Indeed, the classically forbidden region may be replaced by a bouncing period in which the scale factor bounces from the contraction to its expansion eras.", "Using the Bohmian approach of quantum mechanics, we also compute the Bohmian trajectory and the quantum potential related to the system, which their analysis shows are the direct effects of the mass term on the dynamics of the universe." ], [ "Introduction", "General Relativity (GR) introduced by Einstein began a renaissance in scientific thought which changed our viewpoint on the concept of space-time geometry and gravity.", "The interpretation of gravitational force as a modification of geometrical structure of space-time made and makes this force distinguishable from other fundamental interactions, although there are arguments which support the idea that the other interactions may also have geometrical origin.", "Because of the unknown behavior of gravitational interaction at short distances, this distinction may have some roots in the heart of problems with quantum gravity.", "Therefore, any hope of dealing with such concepts would be in vain unless a reliable quantum theory of gravity can be constructed.", "In the absence of a full theory of quantum gravity, it would be then useful to describe its quantum aspects within the context of modified theories of gravity.", "From a field theory point of view, the gravitational force in GR can be represented as a field theory in which the space-time metric plays the role of the fields and the particle that is responsible to propagate gravity is named graviton.", "Then, naturally in comparison to the other field theories, one may ask about the different properties of such a particle.", "The answer to this question is deduced by the linearized form of GR and expansion of the space-time metric $g_{\\mu \\nu }$ , around a fixed background geometry $\\eta _{\\mu \\nu }$ , as $g_{\\mu \\nu }=\\eta _{\\mu \\nu }+h_{\\mu \\nu }$ , where $h_{\\mu \\nu }$ is the field representation of the graviton.", "Eventually, it is possible to show that the graviton is a massless spin-2 particle.", "Then, since our knowledge about the behavior of gravity at very long distances is also incomplete, a question arises: Is it possible to consider a small nonvanishing mass for the graviton, i.e., a massive spin-2 particle?", "In the first attempts to deal with this question, it seemed that adding a mass term to the action may be sufficient.", "This was done by Fierz and Pauli [1].", "However, it was shown that by considering the number of degrees of freedom, this model suffers from the existence of a ghost field, the so-called Boulware-Deser ghost [2], after studying the non-linear terms.", "This fact made massive gravity an abandoned theory for a while.", "Recently, de Rahm and Gabadadze proposed a new scenario in which they have shown that it is possible to have a ghost-free massive gravity even at the non-linear level [3].", "That was a positive signal in this area, and the early results in this subject have been followed by a number of works that address different aspects of massive gravity [4].", "As in the case of the other modified theories of gravity, it is important to seek cosmological solutions in the newly proposed massive theory of gravity.", "This is done by the authors of Ref.", "[5], who show that the existence of some constraints prevent the theory from having the nontrivial homogeneous and isotropic cosmological solutions.", "Indeed, what is shown in Ref.", "[5] is that, beginning with the flat FRW ansatz in the context of massive gravity, the corresponding field equations result in nothing but the Minkowski metric.", "However, by re-examination of the conditions, the authors of Refs.", "[6], [7] have shown that, for the open FRW model, this is not the case, and the nonlinear massive gravity admits the open FRW as a compatible solution for its field equations.", "Another progresses to find the massive cosmologies lie in the field of the bi-metric theories of gravity; see, for instance, Ref.", "[8] based on the works of Hassan and Rosen [9], in which they show that a bi-metric representation for massive gravity exists.", "Our purpose in the present paper is to continue the works of the authors of Ref.", "[6], [7] in greater detail, based on the Hamiltonian formalism of the open FRW cosmology in the framework of massive gravity.", "We obtain the solutions to the vacuum and perfect fluid classical field equations and investigate their different aspects, such as the roll of the graviton's mass as a cosmological constant, the appearance of singularities, and the late time expansion.", "We then consider the problem at hand in the context of canonical quantum cosmology to see how the classical picture will be modified.", "Our final results show that the singular behavior of the classical cosmology will be replaced by a bouncing one when quantum mechanical considerations are taken into account.", "This means that the quantization of the model suggests the existence of a minimal size for the corresponding universe.", "We shall also study the quantum model by the Bohmian approach of quantum mechanics to show how the mass term exhibits its direct effects on the evolution of the system.", "The structure of the paper is as follows.", "In section 2, we briefly present the basic elements of the issue of massive gravity and its canonical Hamiltonian for a given open FRW universe.", "In section 3, classical cosmological dynamics is introduced for the vacuum and perfect fluid.", "Quantization of the model is the subject of section 4, and in section 5, the Bohmian approach of quantum mechanics is applied to the model.", "Finally, the conclusions are summarized in section 6." ], [ "Preliminary set-up", "In this section, we start by briefly studying the nonlinear massive gravity action presented in Refs.", "[6], [7] for the open FRW model, where the metric is given by $ds^2=g_{\\mu \\nu }dx^{\\mu }dx^{\\nu }=-N^2(t)dt^2+a^2(t)\\left[dx^2+dy^2+dz^2-\\frac{|K|(xdx+ydy+zdz)^2}{1+|K|(x^2+y^2+z^2)}\\right],$ with $N(t)$ and $a(t)$ being the lapse function and the scale factor, respectively, and $K=-1$ denoting the curvature index.", "Here we work in units where $c=\\hbar =16\\pi G=1$ .", "In the massive gravity scenario one considers a metric perturbation as [5], [10] $g_{\\mu \\nu }=\\eta _{\\mu \\nu }+h_{\\mu \\nu }=\\eta _{ab}\\partial _{\\mu }\\phi ^a(x)\\partial _{\\nu }\\phi ^b(x)+H_{\\mu \\nu },$ where $\\eta _{ab}=\\mbox{diag}(-1,1,1,1)$ and $\\phi ^a(x)$ are four scalar fields known as Stückelberg scalars and are introduced to keep the principle of general covariance also in massive general relativity [11].", "It is clear that the first term in (REF ) is a representation of the Minkowski space-time in terms of the coordinate system $(\\phi ^0,\\phi ^i)$ and thus the tensor $H_{\\mu \\nu }$ is responsible for describing the propagation of gravity in this space.", "The action of the model consists of the gravitational part ${\\cal S}_g$ and the matter action ${\\cal S}_m$ as ${\\cal S}={\\cal S}_g+{\\cal S}_m.$ The matter part of the action is independent of the massive corrections to the gravity part.", "Also, the gravity part can be expressed in terms of the usual Einstein-Hilbert, with an additional correction term coming from the massive graviton; that is [5] ${\\cal S}_g=\\int \\sqrt{-g}\\left[R-\\frac{m^2}{4}{\\cal U}(g,H)\\right]d^4x,$ in which all of the modifications due to the mass and also the interactions between the tensor fields $H_{\\mu \\nu }$ and $g_{\\mu \\nu }$ are summarized in the potential ${\\cal U}(g,H)$ .", "By using ghost-free conditions for the theory in Ref.", "[11], we propose the following form for the potential term: [10] ${\\cal U}(g,H)=-4\\left({\\cal L}_2+\\alpha _3 {\\cal L}_3+\\alpha _4{\\cal L}_4\\right),$ where $\\left\\lbrace \\begin{array}{ll}{\\cal L}_2=\\frac{1}{2}\\left(<{\\cal K}>^2-<{\\cal K}^2>\\right),\\\\\\\\{\\cal L}_3=\\frac{1}{6}\\left(<{\\cal K}>^3-3<{\\cal K}><{\\cal K}^2>+2<{\\cal K}^3>\\right),\\\\\\\\{\\cal L}_4=\\frac{1}{24}\\left(<{\\cal K}>^4-6<{\\cal K}>^2<{\\cal K}^2>+3<{\\cal K}^2>^2+8<{\\cal K}><{\\cal K}^3>-6<{\\cal K}^4>\\right),\\end{array}\\right.$ in which the tensor ${\\cal K}_{\\mu \\nu }$ is defined as ${\\cal K}^{\\mu }_{\\nu }(g,H)=\\delta ^{\\mu }_{\\nu }-\\sqrt{\\eta _{ab}\\partial ^{\\mu }\\phi ^a\\partial _{\\nu }\\phi ^b},$ and the notations $<{\\cal K}>=g^{\\mu \\nu }{\\cal K}_{\\mu \\nu }$ , $<{\\cal K}^2>=g^{\\alpha \\beta }g^{\\mu \\nu }{\\cal K}_{\\alpha \\mu }{\\cal K}_{\\beta \\nu }$ ,... are used for the corresponding traces.", "Now, equations (REF )-(REF ) describe the gravitational part of the action for a massive gravity theory.", "Since its explicit form directly depends on the choice of scalar fields $\\phi ^a(x)$ , it is appropriate to concentrate on this point first.", "Interesting forms for such fields should involve terms which would describe a suitable coordinate transformation on the Minkowski space-time.", "In a flat FRW background, for instance, one may select $\\phi ^0=f(t)$ and $\\phi ^i=x^i$ , as is used in [5].", "Here, for the open FRW metric (REF ), we use the following ansatz proposed in [6] $\\phi ^0=f(t)\\sqrt{1+|K|x_ix^i},\\hspace{14.22636pt}\\phi ^i=\\sqrt{|K|}f(t)x^i.$ Upon substitution of these scalar fields and also the definition of the Ricci scalar into the relations (REF )-(REF ), we are led to a point-like form for the gravitational Lagrangian in the minisuperspace $\\lbrace N,a,f\\rbrace $ as ${\\cal L}_g=-\\frac{3 a \\dot{a}^2}{N}-3|K|Na+m^2\\left(L_2+\\alpha _3L_3+\\alpha _4 L_4\\right),$ where $\\left\\lbrace \\begin{array}{ll}L_2=3a\\left(a-\\sqrt{|K|}f\\right)\\left(2Na-a \\dot{f}-N\\sqrt{|K|}f\\right),\\\\\\\\L_3=\\left(a-\\sqrt{|K|}f\\right)^2\\left(4Na-3a\\dot{f}-N\\sqrt{|K|}f\\right),\\\\\\\\L_4=\\left(a-\\sqrt{|K|}f\\right)^3\\left(N-\\dot{f}\\right),\\end{array}\\right.$ in which an overdot represents differentiation with respect to the time parameter $t$ .", "It is seen that this Lagrangian does not involve $\\dot{N}$ , which means that the momentum conjugate to this variable vanishes.", "In the usual canonical formalism of general relativity, we know this issue as the primary constraint in the sense that the variable $N$ is not a dynamical variable but a Lagrange multiplier in the Hamiltonian formalism.", "On the other hand, Lagrangian (REF ) seems to show an additional constraint related to the Stückelberg scalars whose dynamics are encoded in the function $f(t)$ .", "We see that in spite of the common Lagrangians in which the first derivative of the configuration variables are of second order, $\\dot{f}$ appears linearly in the Lagrangian (REF ).", "Therefore, by computing the momentum conjugate to $f$ ; that is, $P_f=\\frac{\\partial {\\cal L}_g}{\\partial \\dot{f}}$ , we obtain $P_f=-m^2(a-\\sqrt{|K|}f)\\left[3a^2+3\\alpha _3a(a-\\sqrt{|K|}f)+\\alpha _4(a-\\sqrt{|K|}f)^2\\right].$ Now, it is clear that this relation is not invertible to obtain $\\dot{f}(f,P_f)$ .", "In such a case, the Lagrangian is said to be singular and the relations like (REF ), which hinder the inversion, are known as primary constraints.", "One may use the method of Lagrange multipliers to analyze the dynamics of the system by adding to the Lagrangian all of the primary constraints multiplied by arbitrary functions of time.", "However, to deal with our constrained system, we act differently and proceed as follows.", "We vary the Lagrangian (REF ) with respect to $f$ to obtain $\\left(\\dot{a}-\\sqrt{|K|}N\\right)\\left[|K|(\\alpha _3+\\alpha _4)f^2(t)-2\\sqrt{|K|}(1+2\\alpha _3+\\alpha _4)a(t)f(t)+(3+3\\alpha _3+\\alpha _4)a^2(t)\\right]=0.$ The solution $\\dot{a}=\\sqrt{|K|}N$ of this equation is nothing but what we obtain from the variation of the usual Einstein-Hilbert Lagrangian with respect to $N$ .", "Since its counterpart in massive gravity is $\\dot{a}=\\frac{\\sqrt{3|K|(\\alpha _3+\\alpha _4)^2+m^2a^2(t)\\left[2(1+\\alpha _3+\\alpha _3^2-\\alpha _4)^{3/2}-(1+\\alpha _3)(2+\\alpha _3+2\\alpha _3^2-3\\alpha _4)\\right]}}{\\sqrt{3}(\\alpha _3+\\alpha _4)}N,$ we cannot accept the relation $\\dot{a}=\\sqrt{|K|}N$ as a physical solution.", "Therefore, the constraint corresponding to the dynamic of $f(t)$ shows itself in the equation $\\left[|K|(\\alpha _3+\\alpha _4)f^2(t)-2\\sqrt{|K|}(1+2\\alpha _3+\\alpha _4)a(t)f(t)+(3+3\\alpha _3+\\alpha _4)a^2(t)\\right]=0,$ where using the same notation as in [6], its solutions can be written as $f(t)=\\frac{X_{\\pm }}{\\sqrt{|K|}}a(t)\\Rightarrow \\dot{f}=\\frac{X_{\\pm }}{\\sqrt{|K|}}\\dot{a},\\hspace{14.22636pt}X_{\\pm }\\equiv \\frac{1+2\\alpha _3+\\alpha _4 \\pm \\sqrt{1+\\alpha _3+\\alpha _3^2-\\alpha _4}}{\\alpha _3+\\alpha _4}.$ As is argued in [6], in the limit where $\\alpha _3$ and $\\alpha _4$ are of the order of a small quantity $\\epsilon $ , the expression of $X_{+}$ goes to infinity while $X_{-}\\rightarrow 3/2$ .", "Because of this limiting behavior, we use the subscript $-$ in the following for numerical values of constants with subscript $\\pm $ .", "Now we may insert the constraints (REF ) into the relations (REF ) to reduce the degrees of freedom of the system and obtain a minimal number of dynamical variables.", "If we do so, we obtain $\\left\\lbrace \\begin{array}{ll}L_2=3\\left(1-X_{\\pm }\\right)\\left[\\left(2-X_{\\pm }\\right)N-\\frac{X_{\\pm }}{\\sqrt{|K|}}\\dot{a}\\right]a^3,\\\\\\\\L_3=\\left(1-X_{\\pm }\\right)^2\\left[\\left(4-X_{\\pm }\\right)N-3\\frac{X_{\\pm }}{\\sqrt{|K|}}\\dot{a}\\right]a^3,\\\\\\\\L_4=\\left(1-X_{\\pm }\\right)^3\\left[N-\\frac{X_{\\pm }}{\\sqrt{|K|}}\\dot{a}\\right]a^3,\\end{array}\\right.$ in terms of which the Lagrangian (REF ) takes its reduced form with only one physical degree of freedom $a$ .", "The momentum conjugate to $a$ is $P_a=\\frac{\\partial {\\cal L}_g}{\\partial \\dot{a}}=-\\frac{6a\\dot{a}}{N}+m^2\\left(\\frac{\\partial L_2}{\\partial \\dot{a}}+\\alpha _3 \\frac{\\partial L_3}{\\partial \\dot{a}}+\\alpha _4 \\frac{\\partial L_4}{\\partial \\dot{a}}\\right).$ Noting that $\\frac{\\partial L_2}{\\partial \\dot{a}}=3\\frac{X_{\\pm }}{\\sqrt{|K|}}(X_{\\pm }-1)a^3,\\hspace{14.22636pt}\\frac{\\partial L_3}{\\partial \\dot{a}}=3\\frac{X_{\\pm }}{\\sqrt{|K|}}(X_{\\pm }-1)^2a^3,\\hspace{14.22636pt}\\frac{\\partial L_4}{\\partial \\dot{a}}=\\frac{X_{\\pm }}{\\sqrt{|K|}}(X_{\\pm }-1)^3a^3,$ one gets $P_a=-6\\frac{a\\dot{a}}{N}-\\frac{C_{\\pm }m^2}{\\sqrt{|K|}}a^3,\\hspace{14.22636pt}C_{\\pm }\\equiv X_{\\pm }(1-X_{\\pm })\\left[3+3\\alpha _3(1-X_{\\pm })+\\alpha _4(1-X_{\\pm })^2\\right].$ Now, the Hamiltonian of the model can be obtained from its standard definition $H=\\dot{a}P_a-{\\cal L}$ , with result $H_g=N{\\cal H}_g=N\\left[-\\frac{1}{12a}\\left(P_a+\\frac{C_{\\pm }m^2}{\\sqrt{|K|}}a^3\\right)^2+3|K|a+c_{\\pm }m^2a^3\\right],$ in which we have defined $c_{\\pm }=\\left(X_{\\pm }-1\\right)\\left[3(2-X_{\\pm })+\\alpha _3(1-X_{\\pm })(4-X_{\\pm })+\\alpha _4(1-X_{\\pm })^2\\right].$ We see that the lapse function enters in the Hamiltonian as a Lagrange multiplier as expected.", "Thus, when we vary the Hamiltonian with respect to $N$ , we get ${\\cal H}_g=0$ , which is called the Hamiltonian constraint.", "On a classical level this constraint is equivalent to the Friedmann equation, wherein our problem at hand can be easily checked by comparing it with the equation of motion (4.5) in [6].", "On a quantum level, on the other hand, the operator version of this constraint annihilates the wave function of the corresponding universe, leading to the so-called Wheeler-DeWitt equation.", "Now, let us deal with the matter field with which the action of the model is augmented.", "As we have mentioned, the matter part of the action is independent of modifications due to the mass terms.", "Therefore, the matter may come into play in a common way and the total Hamiltonian can be made by adding the matter Hamiltonian to the gravitational part of (REF ).", "To do this, we consider a perfect fluid whose pressure $p$ is linked to its energy density $\\rho $ by the equation of state $p=\\omega \\rho ,$ where $-1\\le \\omega \\le 1$ is the equation of the stated parameter.", "According to Schutz's representation for the perfect fluid [12], its Hamiltonian can be viewed as (see [13] for details) $H_m=N\\frac{P_T}{a^{3\\omega }},$ where $T$ is a dynamical variable related to the thermodynamical parameters of the perfect fluid and $P_T$ is its conjugate momentum.", "Finally, we are in a position in which can write the total Hamiltonian $H=H_g+H_m$ as $H=N{\\cal H}=N\\left[-\\frac{1}{12a}\\left(P_a+\\frac{C_{\\pm }m^2}{\\sqrt{|K|}}a^3\\right)^2+3|K|a+c_{\\pm }m^2a^3+\\frac{P_T}{a^{3\\omega }}\\right].$ The setup for constructing the phase space and writing the Lagrangian and Hamiltonian of the model is now complete.", "In the following section, we shall deal with classical and quantum cosmologies which can be extracted from a theory with the previously mentioned Hamiltonian." ], [ "Cosmological dynamics: classical point of view", "The classical dynamics are governed by the Hamiltonian equations.", "To achieve this purpose, we divide this section into two parts.", "We first consider the case in which the matter is absent, i.e., the vacuum, and then include the matter." ], [ "The vacuum classical cosmology", "In this case, we can construct the equations of motion by the Hamiltonian equations with use of the Hamiltonian (REF ).", "Equivalently, one may directly write the Friedmann equation from the Hamiltonian constraint $H=0$ which, as we mentioned previously, reflects the fact that the corresponding gravitational theory is a parameterized theory in the sense that its action is invariant under time reparameterization.", "Noting from (REF ) that $\\dot{a}=-\\frac{N}{6a}\\left(P_a+\\frac{C_{\\pm }m^2}{\\sqrt{|K|}}a^3\\right),$ equation (REF ) gives $3a\\dot{a}^2-3|K|a=c_{\\pm }m^2a^3,$ in which we have chosen the gauge $N=1$ , so that the time parameter $t$ becomes the cosmic time $\\tau $ .", "As is indicated in [6], this equation looks like the Friedmann equation for the open FRW universe with an effective cosmological constant $\\Lambda _{\\pm }=c_{\\pm }m^2$ and admits the following solutions $a_{\\pm }(\\tau )=\\sqrt{\\frac{3}{\\Lambda }}\\sinh \\left(\\pm \\sqrt{\\frac{\\Lambda }{3}}(\\tau -\\tau _{*})\\right),$ where $\\tau _{*}$ is an integration constant and we have taken $\\Lambda =c_{-}m^2$ .", "For a positive $\\tau _{*}$ , the condition $a(\\tau )\\ge 0$ implies that the expressions of $a_{+}(\\tau )$ and $a_{-}(\\tau )$ are valid for $\\tau \\ge \\tau _{*}$ and $\\tau \\le -\\tau _{*}$ respectively, such that $a_{\\pm }(\\tau _{*})=0$ .", "It is seen that the evolution of the corresponding universe with the scale factor $a_{+}(\\tau )$ begins with a big-bang-like singularity at $\\tau =\\tau _{*}$ and then follows an exponential law expansion at late time of cosmic evolution in which the mass term shows itself as a cosmological constant.", "For a universe with the scale factor $a_{-}(\\tau )$ , on the other hand, the behavior is opposite.", "The universe decreases its size from large values of scale factor at $\\tau =-\\infty $ and ends its evolution at $\\tau =-\\tau _{*}$ with a zero size.", "In figure REF we have plotted theses scale factors for typical values of the parameters.", "As this figure shows, although the behavior of $a_{+}(\\tau )$ ($a_{-}(\\tau )$ ) is like a de Sitter ($a(\\tau )\\sim e^{\\sqrt{\\Lambda /3}\\tau }$ ) universe at $\\tau \\rightarrow \\infty $ ($\\tau \\rightarrow -\\infty $ ), in spite of the de Sitter, it begins (ends) its evolution with a singularity.", "In summary, what we have shown previously is that in the framework of an open FRW background geometry, the vacuum solutions of the massive theory are equivalent to the solutions of the usual GR with a cosmological constant.", "Accordingly, the zero-size singularity of both theories has the same nature.", "In this sense we would like to emphasize that the metric (REF ) with the scale factor (REF ) is indeed a section of the de Sitter hyperboloid $-T^2+X^2+Y^2+Z^2+W^2=1,$ embedded in a 5-dimensional Minkowski space $ds^2=-dT^2+dX^2+dY^2+dZ^2+dW^2.$ To see this, one may parameterize the hyperboloid in terms of the spherical coordinates $(r,\\theta , \\phi )$ as [14] $\\left\\lbrace \\begin{array}{ll}T=\\sqrt{1+r^2}\\sinh \\tau ,\\\\\\\\X=\\cosh \\tau ,\\\\\\\\Y=r\\sinh \\tau \\cos \\phi \\cos \\theta ,\\\\\\\\Z=r\\sinh \\tau \\cos \\phi \\sin \\theta ,\\\\\\\\W=r\\sinh \\tau \\sin \\phi ,\\end{array}\\right.$ which, upon substitution into the metric (REF ), yields the open FRW metric with the scale factor $a(\\tau )=\\sinh \\tau $ .", "This means that the point $a=0$ can be viewed as a coordinate singularity.", "However, we have to note that in the presence of any kind of matter field the point $a(\\tau _{*})=0$ represents a true singularity.", "Thus, our following analysis to quantize the model is based on the minisuperspace coordinate system in terms of which the dynamical representation of the metric, i.e.", "(REF ), is written.", "In the next section, we shall see how the previous picture may be modified when one takes into account quantum mechanical considerations.", "Figure: The figures show the evolutionary behaviorof the universes based on ().", "We have used the numericalvalues Λ=1\\Lambda =1 and τ * =0\\tau _{*}=0." ], [ "Perfect fluid classical cosmology", "Now, we assume that a perfect fluid in its Schutz's representation is coupled with gravity.", "In this case the Hamiltonian (REF ) describes the dynamics of the system.", "The equations of motion for $T$ and $P_T$ read as $\\dot{T}=\\lbrace T,H\\rbrace =\\frac{N}{a^{3\\omega }},\\hspace{14.22636pt}\\dot{P_{T}}=\\lbrace P_T,H\\rbrace =0.$ A glance at the above equations shows that with choosing the gauge $N=a^{3\\omega }$ , we shall have $N=a^{3\\omega }\\Rightarrow T=t,$ which means that variable $T$ may play the role of time in the model.", "Therefore, the Friedmann equation $H=0$ can be written in the gauge $N=a^{3\\omega }$ as follows $3\\dot{a}^2=3|K|a^{6\\omega }+\\Lambda a^{6\\omega +2}+P_0a^{3\\omega -1},$ where we take $P_T=P_0=\\mbox{const.", "}$ from the second equation of (REF ).", "Since it is not possible to find the analytical solutions of the above differential equation for any arbitrary $\\omega $ , we present its solutions only in some special cases.", "$\\bullet $ $\\omega =-\\frac{1}{3}$ : cosmic string.", "In this case we obtain $a(t)=\\left[\\frac{\\Lambda }{3}(t-t_0)^2-\\frac{P_0+3|K|}{\\Lambda }\\right]^{1/2},$ where $t_0$ is an integration constant.", "We see that the evolution of the universe based on (REF ) has big-bang-like singularities at $t=t_0\\pm t_{*}$ where $t_{*}=\\frac{\\sqrt{3(P_0+3|K|)}}{\\Lambda }$ .", "Indeed, the condition $a^2(t)\\ge 0$ separates two sets of solutions $a^{(I)}(t)$ and $a^{(II)}(t)$ , each of which is valid for $t\\le t_0-t_{*}$ and $t\\ge t_0+t_{*}$ , respectively.", "For the former, we have a contracting universe which decreases its size according to a power law relation and ends its evolution in a singularity at $t=t_0-t_{*}$ , while for the latter, the evolution of the universe begins with a big-bang singularity at $t=t_0+t_{*}$ and then follows the power law expansion at late time of cosmic evolution.", "One may translate these results in terms of the cosmic time $\\tau $ .", "Using its relationship with the time parameter $t$ in this case, that is, $d\\tau =a^{-1}(t)dt$ , we are led to $a(\\tau )=\\left\\lbrace \\begin{array}{ll}a^{(I)}(\\tau )=\\frac{1}{\\sqrt{12\\Lambda }}\\left[e^{-\\sqrt{\\frac{\\Lambda }{3}}(\\tau -\\tau _0)}-3(P_0+3|K|)e^{\\sqrt{\\frac{\\Lambda }{3}}(\\tau -\\tau _0)}\\right],\\hspace{14.22636pt}\\tau \\le \\tau _0-\\tau _{*},\\\\\\\\\\\\a^{(II)}(\\tau )=\\frac{1}{\\sqrt{12\\Lambda }}\\left[e^{\\sqrt{\\frac{\\Lambda }{3}}(\\tau -\\tau _0)}-3(P_0+3|K|)e^{-\\sqrt{\\frac{\\Lambda }{3}}(\\tau -\\tau _0)}\\right],\\hspace{14.22636pt}\\tau \\ge \\tau _0+\\tau _{*},\\\\\\end{array}\\right.$ where $\\tau _{*}=\\frac{1}{2}\\sqrt{\\frac{3}{\\Lambda }}\\ln 3(P_0+3|K|)$ .", "Again, it is seen that there is a classically forbidden region $\\tau _0-\\tau _{*}<\\tau <\\tau _0+\\tau _{*}$ , for which we have no valid classical solutions.", "For $\\tau \\le \\tau _0-\\tau _{*}$ , the universe has a exponential decreasing behavior which ends its evolution in a singular point with zero size at $\\tau =\\tau _0-\\tau _{*}$ , while in the region $\\tau \\ge \\tau _0+\\tau _{*}$ it begins with the big-bang singularity at $\\tau =\\tau _0+\\tau _{*}$ and then grows exponentially forever.", "$\\bullet $ $\\omega =-1$ : cosmological constant.", "Performing the integration, we get the following implicit relation between $t$ and $a(t)$ : $\\frac{1}{\\sqrt{3}(P_0+\\Lambda )^2}\\left[-6|K|+(P_0+\\Lambda )a^2\\right]\\sqrt{3|K|+(P_0+\\Lambda )a^2}=t-t_0.$ In terms of the cosmic time $\\tau $ , it is easy to see that this solution returns to (REF ), in which the cosmological term is replaced by $\\Lambda \\rightarrow \\Lambda +\\mbox{cons}.$ This is expected because the solutions (REF ) were equivalent to an open FRW universe with a cosmological constant.", "Therefore, adding a new cosmological term (a perfect fluid with $\\omega =-1$ ) only makes a shift in the corresponding cosmological constant." ], [ "Cosmological dynamics: quantum point of view", "In this section we look for the quantization of the model presented above via the method of canonical quantization.", "As is well known, this procedure is based on the Wheeler-DeWitt equation $\\hat{{\\cal H}}\\Psi =0$ , where $\\hat{{\\cal H}}$ is the operator version of the Hamiltonian constraint and $\\Psi $ is the wave function of the universe, a function of the 3-geometries and the matter fields.", "As in the case of the classical cosmology, we consider the matter of free and perfect fluid quantum cosmology separately.", "Before going to the subject, a remark is in order related to the Hamiltonians (REF ) and (REF ).", "The term in the round bracket in these Hamiltonians is like the Hamiltonian of a charged particle moving in an electromagnetic field.", "From this analogy, one may define the transformation $P_a\\rightarrow \\Pi _a=P_a+\\frac{C_{\\pm }m^2}{\\sqrt{|K|}}a^3,\\hspace{14.22636pt}a\\rightarrow a,$ to simplify the form of the classical Hamiltonian.", "It is clear that this is a canonical transformation both classically and quantum mechanically [15].", "Since going back from a new set of variables to the old ones in a classical canonical transformation can be made without any ambiguity, applying this transformation may not be important for the classical dynamics presented in the previous section.", "In the context of quantum mechanics, on the other hand, the subject is of little difference.", "The transition to the quantum version of the theory is achieved by promoting observables to operators which are not necessarily commuting.", "Thus, by replacing the canonical variables $(a,P_a)$ by their operator counterparts $(\\hat{a},\\hat{P_a}=-id/da)$ , we obtain the quantum Hamiltonian $\\hat{{\\cal H}}=-\\frac{1}{12}\\hat{a}^{-1}\\hat{\\Pi _a}^2+...=-\\frac{1}{12}\\hat{a}^{-1}\\left(\\hat{P_a}+\\frac{C_{\\pm }m^2}{\\sqrt{|K|}}\\hat{a}^3\\right)^2+...,$ where $...$ denotes the terms out of the round bracket in expressions (REF ) or (REF ).", "When calculating the square, it should be noted that the operators $\\hat{a}$ and $\\hat{P_a}$ do not commute.", "Although the order of these operators does not matter in the classical analysis, quantum mechanically this issue is quite crucial.", "Indeed, this is the operator ordering problem and, unfortunately, there is no well defined principle which specifies the order of operators in the passage from classical to quantum theory.", "There are, however, some simple rules which one uses conventionally.", "If, for instance, we order the products of $\\hat{a}$ and $\\hat{P_a}$ in $\\hat{\\Pi _a}^2$ such that the momentum stands to the right of the scale factor, we obtain $\\hat{\\Pi _a}^2\\rightarrow \\hat{P_a}^2+\\frac{C_{\\pm }^2m^4}{|K|}a^6+2\\frac{C_{\\pm }m^2}{\\sqrt{|K|}}\\hat{a}^3\\hat{P_a}-3i\\frac{C_{\\pm }m^2}{\\sqrt{|K|}}\\hat{a}^2,$ in which we have used the commutation relation $[\\hat{a},\\hat{P_a}]=i$ .", "With this expression at hand, there is still another factor ordering ambiguity in the terms $\\hat{a}^{-1}\\hat{P_a}^2$ and $\\hat{a}^2\\hat{P_a}$ to construct the quantum Hamiltonian (REF ).", "As Hawking and Page have shown [16], the choice of different factor ordering will not affect semiclassical calculations in quantum cosmology, so for convenience one usually chooses a special place for it in the special models.", "However, in general, the behavior of the wave function depends on the chosen factor ordering [17].", "In what follows, as one usually does in the minisuperspace approximation to the cosmological models, we work in the framework of a special factor ordering in which, in addition to the expression (REF ) for $\\hat{\\Pi _a}^2$ , we also use the orderings $\\hat{a}^{-1}\\hat{P_a}^2=\\hat{P_a}\\hat{a}^{-1}\\hat{P_a}$ and $\\hat{a}^2\\hat{P_a}=\\hat{a}\\hat{P_a}\\hat{a}$ to make the Hamiltonian hermitianWith the canonical transformation (REF ) at hand, one may uses the transformed Hamiltonian ${\\cal H}=-\\frac{1}{12a}\\Pi _a^2+...,$ to quantize the system, where again ... denotes the terms out of the round bracket in expressions (REF ) or (REF ).", "Using this Hamiltonian in the hermitian form $a^{-1}\\Pi _a^2=\\Pi _a a^{-1}\\Pi _a$ and also representing $\\Pi _a$ by $-i\\partial _a$ , this is equivalent to our above treatment in which the last term in (REF ) is absent.", "Therefore, one may have some doubts on the validity of the main following results due to the effects of the chosen factor ordering.", "To overcome this problem, we have made some calculations based on the above mentioned transformed Hamiltonian and have verified that the general patterns of the resulting wave functions follow the behavior shown in following sections.." ], [ "The vacuum quantum cosmology", "In this case, with the help of the Hamiltonian (REF ) and use of the abovementioned choice of ordering, the Wheeler-DeWitt equation reads $\\left\\lbrace \\frac{d^2}{da^2}+\\left(-a^{-1}+2i\\frac{C_{\\pm }}{c_{\\pm }}\\Lambda a^3\\right)\\frac{d}{da}+\\left[\\left(36+2i\\frac{C_{\\pm }}{c_{\\pm }}\\Lambda \\right)a^2+12\\Lambda a^4-\\frac{C_{\\pm }^2}{c_{\\pm }^2}\\Lambda ^2a^6\\right]\\right\\rbrace \\Psi (a)=0.$ This equation does not seem to have analytical solutions.", "However, we can get some properties of its solutions in special regions where there is interest in classical and quantum regimes.", "First of all, let us rewrite this equation in the form $\\left\\lbrace \\frac{d^2}{da^2}-\\left(a^{-1}+6i\\Lambda a^3\\right)\\frac{d}{da}+\\left[\\left(36-6i\\Lambda \\right)a^2+12\\Lambda a^4-9\\Lambda ^2a^6\\right] \\right\\rbrace \\Psi (a)=0,$ in which we have used the numerical values $C_{-}=-9/4$ and $c_{-}=3/4$ [6].", "For large values of $a$ , the solution to this equation can easily be obtained in the Wentzel-Kramers-Brillouin (WKB) (semiclassical) approximation.", "In this regime we can neglect the term $a^{-1}$ in equation (REF ).", "Then, substituting $\\Psi (a)=\\Omega (a)e^{iS(a)}$ in this equation leads to the modified Hamilton-Jacobi equation $-\\left(\\frac{dS}{da}\\right)^2+6\\Lambda a^3\\frac{dS}{da}+\\left(36a^2+12\\Lambda a^4-9\\Lambda ^2a^6\\right)+{\\cal Q}=0,$ in which the quantum potential is defined as ${\\cal Q}=\\frac{1}{\\Omega }\\frac{d^2\\Omega }{da^2}$ .", "It is well-known that the quantum effects are important for small values of the scale factor and in the limit of the large scale factor can be neglected.", "Therefore, in the semiclassical approximation region we can omit the ${\\cal Q}$ term in (REF ) and obtain $\\frac{dS}{da}=3\\Lambda a^3\\pm a \\sqrt{36+12\\Lambda a^2}.$ In the WKB method, the correlation between classical and quantum solutions is given by the relation $P_a=\\frac{\\partial S}{\\partial a}$ .", "Thus, using the definition of $P_a$ in (REF ), the equation for the classical trajectories becomes $\\dot{a}=\\pm \\sqrt{1+\\frac{\\Lambda }{3}a^2},$ from which one finds $a(t)=\\sqrt{\\frac{3}{\\Lambda }}\\sinh \\left(\\pm \\sqrt{\\frac{\\Lambda }{3}}(t-\\delta )\\right),$ which shows that the late time behavior of the classical cosmology (REF ) is exactly recovered.", "The meaning of this result is that for large values of the scale factor, the effective action corresponding to the expanding and contracting universes is very large and the universe can be described classically.", "On the other hand, for small values of the scale factor we cannot neglect the quantum effects, and the classical description breaks down.", "Since the WKB approximation is no longer valid in this regime, one should go beyond the semiclassical approximation.", "In the quantum regime, if we neglect the term $\\Lambda ^2 a^6$ in (REF ), the two linearly independent solutions to this equation can be expressed in terms of the Hermite $H_{\\nu }(x)$ and hypergeometric $F_{1\\hspace{-14.22636pt}1}\\hspace{11.38092pt}(a,b;z)$ functions, leading to the following general solution: $\\Psi (a)=e^{-ia^2}\\left[c_1H_{-\\frac{1}{2}-\\frac{8}{3\\Lambda }i}\\left(\\frac{(1+i)(2+3\\Lambda a^2)}{2\\sqrt{3\\Lambda }}\\right)+c_2 \\,\\,\\,F_{1\\hspace{-14.22636pt}1}\\hspace{11.38092pt}\\left(\\frac{1}{4}+\\frac{4}{3\\Lambda }i,\\frac{1}{2};\\frac{i(2+3\\Lambda a^2)^2}{6\\Lambda }\\right)\\right].$ At this step we take a quick glance at the question of the boundary conditions on the solutions to the Wheeler-DeWitt equation.", "Note that the minisuperspace of the above model has only one degree of freedom denoted by the scale factor $a$ in the range $0<a<\\infty $ .", "According to [18], its nonsingular boundary is the line $a=0$ , while at the singular boundary this variable is infinite.", "Since the minisuperspace variable is restricted to the abovementioned domain, the minisuperspace quantization deals only with wave functions defined on this region.", "Therefore, to construct the quantum version of the model, one should take into account this issue.", "This is because in such cases, one usually has to impose boundary conditions on the allowed wave functions; otherwise the relevant operators, especially the Hamiltonian, will not be self-adjoint.", "The condition for the Hamiltonian operator $\\hat{{\\cal H}}$ associated with the classical Hamiltonian function (REF ) and (REF ) to be self-adjoint is $(\\psi _1,\\hat{{\\cal H}}\\psi _2)=(\\hat{{\\cal H}}\\psi _1,\\psi _2)$ or $\\int _0^\\infty \\psi _1^*(a)\\hat{{\\cal H}}\\psi _2(a)da= \\int _0^\\infty \\psi _2(a)\\hat{{\\cal H}}\\psi _1^*(a)da.$ Following the calculations in [19] and dealing only with square integrable wave functions, this condition yields a vanishing wave function at the nonsingular boundary of the minisuperspace.", "Hence, we impose the boundary condition on the solutions (REF ) such that at the nonsingular boundary (at $a=0$ ), the wave function vanishes.", "This makes the Hamiltonian hermitian and self-adjoint and can avoid the singularities of the classical theory, i.e.", "there is zero probability for observing a singularity corresponding to $a=0$ .Such a boundary condition is also suggested by DeWitt in the form $\\Psi [{\\cal G}^{(3)}]=0$ [20], where ${\\cal G}^{(3)}$ denotes all three-geometries which may play the roll of barriers, for instance singular three-geometries.", "As is argued in [20], with this boundary condition some kinds of classical singularities can be removed and a unique solution to the Wheeler-DeWitt equation may be obtained.", "Although in the presence of more fundamental proposals of the boundary condition in quantum cosmology (for example, Vilenkin's tunneling or Hawking's no boundary proposals), it is not clear that the above mentioned boundary condition is true, there are some evidences in quantum gravity models in which suitable wave packets obey such kind of boundary condition, see [21].", "Therefore, we require $\\Psi (a=0)=0\\Rightarrow \\frac{c_2}{c_1}=-\\frac{H_{-\\frac{1}{2}-\\frac{8}{3\\Lambda }i}\\left(\\frac{1+i}{\\sqrt{3\\Lambda }}\\right)}{F_{1\\hspace{-14.22636pt}1}\\hspace{11.38092pt}\\left(\\frac{1}{4}+\\frac{4i}{3\\Lambda },\\frac{1}{2};\\frac{2i}{3\\Lambda }\\right)}.$ Note that equation (REF ) is a Schrödinger-like equation for a fictitious particle with zero energy moving in the field of the superpotential with the real part $U(a)=-(36a^2+12\\Lambda a^4)$ .", "Usually, in the presence of such a potential the minisuperspace can be divided into two regions, $U>0$ and $U<0$ , which could be termed the classically forbidden and classically allowed regions, respectively.", "In the classically forbidden region the behavior of the wave function is exponential, while in the classically allowed region the wave function behaves oscillatorily.", "In the quantum tunneling approach [18], the wave function is so constructed as to create a universe emerging from nothing by a tunneling procedure through a potential barrier in the sense of usual quantum mechanics.", "Now, in our model, the superpotential is always negative, which means that there is no possibility of tunneling anymore, since a zero energy system is always above the superpotential.", "In such a case, tunneling is no longer required as classical evolution is possible.", "As a consequence the wave function always exhibits oscillatory behavior.", "In figure REF , we have plotted the square of the wave functions for typical values of the parameters.", "It is seen from this figure that the wave function has a well-defined behavior near $a=0$ and describes a universe emerging out of nothing without any tunneling.", "(See [22], in which such wave functions also appeared in the case study of the probability of quantum creation of compact, flat, and open de Sitter universes.)", "On the other hand, the emergence of several peaks in the wave function may be interpreted as a representation of different quantum states that may communicate with each other through tunneling.", "This means that there are different possible universes (states) from which the present universe could have evolved and tunneled in the past, from one universe (state) to another.", "Figure: The square of the wave function for thequantum universe.", "We take the numerical valueΛ=1.5\\Lambda =1.5." ], [ "Perfect fluid quantum cosmology", "In this case, the Wheeler-DeWitt equation can be constructed by means of the Hamiltonian (REF ).", "With the same approximations as we used in the previous subsection, we obtain $\\left\\lbrace \\frac{\\partial ^2}{\\partial a^2}-\\left(a^{-1}+6i\\Lambda a^3\\right)\\frac{\\partial }{\\partial a}+\\left[\\left(36-6i\\Lambda \\right)a^2+12\\Lambda a^4\\right]-ia^{1-3\\omega }\\frac{\\partial }{\\partial T}\\right\\rbrace \\Psi (a,T)=0.$ We separate the variables in this equation as $\\Psi (a,T)=e^{iET}\\psi (a),$ leading to $\\left\\lbrace \\frac{d^2}{d a^2}-\\left(a^{-1}+6i\\Lambda a^3\\right)\\frac{d}{d a}+\\left[\\left(36-6i\\Lambda \\right)a^2+12\\Lambda a^4+Ea^{1-3\\omega }\\right]\\right\\rbrace \\psi (a)=0.$ The solutions of the above differential equation may be written in the form $\\psi _E(a)=e^{-ia^2}\\left[c_1H_{-\\frac{1}{2}-\\frac{32+E}{12\\Lambda }i}\\left(\\frac{(1+i)(2+3\\Lambda a^2)}{2\\sqrt{3\\Lambda }}\\right)+c_2 \\,\\,\\,F_{1\\hspace{-14.22636pt}1}\\hspace{11.38092pt}\\left(\\frac{1}{4}+\\frac{32+E}{24\\Lambda }i,\\frac{1}{2};\\frac{i(2+3\\Lambda a^2)^2}{6\\Lambda }\\right)\\right],$ for $\\omega =-1/3$ and $\\psi _E(a)=e^{-i(1+\\frac{E}{12\\Lambda })a^2}\\left[c_1H_{-\\frac{1}{2}-\\frac{1152\\Lambda ^2-24E\\Lambda -E^2}{432\\Lambda ^3}i}\\left(\\frac{(1+i)\\left[E+6\\Lambda (2+3\\Lambda a^2)\\right]}{12\\Lambda \\sqrt{3\\Lambda }}\\right)+\\nonumber \\right.\\\\\\left.c_2\\,\\,\\,F_{1\\hspace{-14.22636pt}1}\\hspace{11.38092pt}\\left(\\frac{1}{4}+\\frac{1152\\Lambda ^2-24E\\Lambda -E^2}{864\\Lambda ^3}i,\\frac{1}{2};\\frac{i\\left[E+6\\Lambda (2+3\\Lambda a^2)\\right]^2}{216\\Lambda ^3}\\right)\\right],$ for $\\omega =-1$ .", "Now the eigenfunctions of the Wheeler-DeWitt equation can be written as $\\Psi _E(a,T)=e^{iET}\\psi _E(a).$ We may now write the general solution to the Wheeler-DeWitt equation as a superposition of its eigenfunctions; that is, $\\Psi (a,T)=\\int _0^\\infty A(E)\\Psi _E(a,T)dE,$ where $A(E)$ is a suitable weight function to construct the wave packets.", "The above relations seem to be too complicated to extract an analytical expression for the wave function.", "Therefore, in the following (for the case $\\omega =-1/3$ ), we present an approximate analytic method which is valid for very small values of scale factor, i.e., in the range that we expect the quantum effects to be important.", "In this regime if we keep only the $a^{-1}$ and $a^2$ terms in the second and third terms of (REF ), the solutions to this equation can be viewed as a superposition of the functions $\\sin \\left(\\frac{\\sqrt{36+E-6i\\Lambda }}{2}a^2\\right)$ and $\\cos \\left(\\frac{\\sqrt{36+E-6i\\Lambda }}{2}a^2\\right)$ .", "If we impose the boundary condition $\\psi (a=0)=0$ on these solutions, we are led to the following eigenfunctions: $\\Psi _E(a,T)=e^{iET}\\sin \\left(\\frac{\\sqrt{36+E-6i\\Lambda }}{2}a^2\\right).$ Now, by using the equality $\\int _0^\\infty e^{-\\gamma x}\\sin \\sqrt{mx}dx=\\frac{\\sqrt{\\pi m}}{2\\gamma ^{3/2}}e^{-(m/4\\gamma )},$ we can evaluate the integral over $E$ in (REF ), and the simple analytical expression for this integral is found if we choose the function $A(E)$ to be a quasi-Gaussian weight factor $A({\\cal E})=e^{-\\gamma {\\cal E}}$ ($\\gamma $ is an arbitrary positive constant and ${\\cal E}=36+E-6i\\Lambda $ ), which results in $\\Psi (a,T)=e^{-6(\\Lambda +6i)T}\\int _0^\\infty e^{-\\gamma {\\cal E}}e^{i{\\cal E}T}\\sin \\left(\\frac{\\sqrt{{\\cal E}}}{2}a^2\\right)d{\\cal E}.$ Using the relation (REF ) yields the following expression for the wave function $\\Psi (a,T)={\\cal N}e^{-6(\\Lambda +6i)T}\\frac{a^2}{(\\gamma -iT)^{3/2}}\\exp \\left(-\\frac{a^2}{8(\\gamma -iT)}\\right),$ where ${\\cal N}$ is a numerical factor.", "Now, having this expression for the wave function of the universe, we are going to obtain the predictions for the behavior of the dynamical variables in the corresponding cosmological model.", "To do this, one may calculate the time dependence of the expectation value of a dynamical variable $q$ as $<q>(T)=\\frac{<\\Psi |q|\\Psi >}{<\\Psi |\\Psi >}.$ Following this approach, we may write the expectation value for the scale factor as $<a>(T)=\\frac{\\int _0^\\infty \\Psi ^{*}(a,T)a\\Psi (a,T)da}{\\int _0^\\infty \\Psi ^{*}(a,T)\\Psi (a,T)da},$ which yields $<a>(T)=\\sqrt{\\frac{\\Lambda }{3}}\\left(\\gamma ^2+T^2\\right)^{1/2}.$ This relation may be interpreted as the quantum counterpart of the classical solutions (REF ).", "However, in spite of the classical solutions, for the wave function (REF ), the expectation value (REF ) of $a$ never vanishes, showing that these states are nonsingular.", "Indeed, in (REF ) $T$ varies from $-\\infty $ to $+\\infty $ , and any $T_0$ is just a specific moment without any particular physical meaning like big-bang singularity.", "The above result may be written in terms of the cosmic time $\\tau $ .", "By the definition $d\\tau =a^{-1}(T)dT$ , we obtain the quantum version of the relations (REF ) as $<a>(\\tau )=\\frac{1}{2}\\left(e^{\\sqrt{\\frac{\\Lambda }{3}}\\tau }+\\gamma ^2e^{-\\sqrt{\\frac{\\Lambda }{3}}\\tau }\\right).$ In figure REF , we have plotted the classical scale factors (REF ) and (REF ) and their quantum counterparts (REF ) and (REF ).", "As is clear from this figure, for a perfect fluid with $\\omega =-1/3$ , the corresponding classical cosmology admits two separate solutions which are disconnected from each other by a classically forbidden region.", "One of these solutions represents a contracting universe ending in a singularity while another describes an expanding universe which begins its evolution with a big-bang singularity.", "On the other hand, the evolution of the scale factor based on the quantum-mechanical considerations shows a bouncing behavior in which the universe bounces from a contraction epoch to a reexpansion era.", "Indeed, the classically forbidden region is where the quantum bounce has occurred.", "We see that in the late time of cosmic evolution in which the quantum effects are negligible, these two behaviors coincide with each other.", "This means that the quantum structure which we have constructed has a good correlation with its classical counterpart.", "Figure: Left: The figure shows qualitativebehavior of the classical scale factor () (solid lines: theleft branch for a (I) (t)a^{(I)}(t) and the right branch fora (II) (t)a^{(II)}(t)) and the expectation value of the scale factor() (dashed line).", "Right: The same figure in terms ofcosmic time.", "The left and right branches of the solid linesrepresent a (I) (τ)a^{(I)}(\\tau ) and a (II) (τ)a^{(II)}(\\tau ), respectively, in() while the dashed line represents the expectation value()." ], [ "Bohmian trajectories", "In the previous sections, we saw how the classical singular behavior of the universe was replaced with a bouncing one in a quantum picture.", "Now, a natural question may arise: Why will the bounce occur?", "Clearly, it is due to the quantum mechanical effects which show themselves when the size of the universe tends to very small values.", "However, we would like to know whether the massive correction to the underlying gravity theory has any contribution to this phenomenon.", "To deal with this question, let us return to the wave function (REF ) and write it in the polar form $\\Psi (a,T)=\\Omega (a,T)e^{iS(a,T)}$ , where $\\Omega (a,T)$ and $S(a,T)$ are real functions, which simple algebra gives as $\\Omega (a,T)=e^{-6\\Lambda T}\\frac{a^2}{(\\gamma ^2+T^2)^{3/4}}\\exp \\left[-\\frac{\\gamma a^2}{8(\\gamma ^2+T^2)}\\right],$ $S(a,T)=-36T+\\frac{3}{2}\\arctan \\frac{T}{\\gamma }-\\frac{Ta^2}{8(\\gamma ^2+T^2)}.$ According to the Bohm-de Broglie interpretation of quantum mechanics [23] and also its usage in quantum cosmology [24], upon using this form of the wave function in the corresponding wave equation, we arrive at the modified Hamilton-Jacobi equation as ${\\cal H}\\left(q_i,P_i=\\frac{\\partial S}{\\partial q_i}\\right)+{\\cal Q}=0,$ where $P_i$ are the momentum conjugate to the dynamical variables $q_i$ and ${\\cal Q}$ is the quantum potential.", "With beginning of the wave equation (REF ), for which we have used the same approximations as in the previous section, the above mentioned procedure gives the quantum potential as ${\\cal Q}=\\frac{1}{\\Omega }\\frac{\\partial ^2 \\Omega }{\\partial a^2}-\\frac{1}{a\\Omega }\\frac{\\partial \\Omega }{\\partial a}.$ On the other hand, the Bohmian equations of motion can be obtained by $P_a=\\frac{\\partial S}{\\partial a}$ , where by means of the relation (REF ) reads $-6a\\dot{a}+3\\Lambda a^2=-\\frac{T}{4(\\gamma ^2+T^2)}.$ The solution to this equation denotes the Bohmian representation of the scale factor; that is $a(t)=\\sqrt{ce^{\\Lambda T}+\\frac{1}{24}e^{\\Lambda T-i\\gamma \\Lambda }\\left[e^{2i\\gamma \\Lambda }\\mbox{ Ei}(1;-\\Lambda T-i\\gamma \\Lambda )+\\mbox{Ei}(1;-\\Lambda T+i\\gamma \\Lambda )\\right]},$ where $c$ is an integration constant and $\\mbox{Ei}(b;z)$ is the exponential integral function defined by $\\mbox{Ei}(b;z)=\\int _1^\\infty e^{-kz}k^{-b}dk.$ The bouncing behavior of the scale factor is again its main property near the classical singularities as we have shown in figure REF .", "To achieve an expression for the quantum potential in terms of the scale factor, we note that all of our above calculations are in the vicinity of $T\\sim 0$ , where the scale factor is small.", "In this regime, a numerical analysis shows that the Bohmian scale factor (REF ) behaves as $a(T)\\sim (\\gamma ^2+T^2)^{1/2}$ , in agreement with the expectation value (REF ).", "Thus, substituting in (REF ), we get the quantum potential from (REF ) as ${\\cal Q}(a)=\\frac{3}{4}\\left[\\gamma ^2 \\left(\\frac{1}{a^4 -\\gamma ^2 a^2}+\\frac{8\\Lambda }{(a^2 - \\gamma ^2)^{3/2}}\\right)+\\frac{-1+48 \\Lambda ^2 a^2}{a^2-\\gamma ^2}\\right].$ Figure: Left: The Bohmian trajectory of the scalefactor.", "Right: The horizontal line represents a typical energylevel.", "The solid curve is the quantum potential for Λ≠0\\Lambda \\ne 0, while the dashed curve denotes the quantum potential whenΛ=0\\Lambda =0.In figure REF we also have plotted the qualitative behavior of the quantum potential versus the scale factor.", "As this figure shows, this potential goes to zero for the large values of the scale factor.", "This behavior is expected, since in this regime the quantum effects can be neglected and the universe evolves classically.", "On the other hand, for the small values of the scale factor the potential takes a large magnitude and the quantum mechanical considerations come into the scenario.", "This is where the quantum potential can produce a huge repulsive force which may be interpreted as being responsible of the avoidance of singularity.", "In figure REF the horizontal line represents a constant energy level which in intersecting with the potential curves gives the turning points at which the bounce will occur.", "The solid curve in this figure is plotted in the case of $\\Lambda \\ne 0$ ; i.e., for the massive theory, while the dashed curve is for $\\Lambda =0$ ; i.e., for when the massive corrections are absent.", "It is seen that, although the mass term $\\Lambda $ is not the only reason for the bouncing behavior in the vicinity of the classical singularity, it may shift the bouncing point into the smaller values of the scale factor.", "This means that if we consider the bouncing point as the minimum size of the universe (which is suggested by quantum cosmology), then the massive version of the underlying gravity theory predicts a smaller value for this minimal size in comparison with the usual Einstein-Hilbert model.", "These facts and also other considerable possibilities such as quantum tunneling between different classically allowed regimes (as can be seen from figure REF ) through the potential barrier support the idea that the massive corrections to the classical cosmology are some signals from quantum gravity." ], [ "Conclusions", "In this paper we have applied the recently proposed nonlinear massive theory of gravity to an open FRW cosmological setting.", "Although the absence of homogeneous and isotropic solutions is one of the main challenges related to this kind of gravitational theory, we moved along the lines of [6], [7], in which the existence of open FRW cosmologies is investigated.", "By using the constraint corresponding to the Stückelberg scalars, we reduced the number of degrees of freedom, according to which the total Hamiltonian of the model is deduced.", "We then presented in detail, the classical cosmological solutions either for the empty universe or in the case where the universe is filled by a perfect fluid (in its Schutz representation) with the equation of state parameter $\\omega =-1/3,-1$ .", "We saw that in both of these cases, the solutions consist of a contraction universe which finalizes its evolution in a singular point and an expanding universe which begins its dynamic with a big-bang singularity.", "These two branches of solutions are disconnected from each other by a classically forbidden region.", "Also, the common feature of the vacuum and matter classical solutions is that the mass term plays a role which resembles the role of cosmological constant in the usual de Sitter universe.", "In this sense we may relate the massive corrections of GR to the problem of dark energy.", "In another part of the paper, we dealt with the quantization of the model described above via the method of canonical quantization.", "For an empty universe, we have shown that by applying the WKB approximation on the Wheeler-DeWitt equation, one can recover the late time behavior of the classical solutions.", "For the early universe, we obtained oscillatory quantum states free of classical singularities by which two branches of classical solutions may communicate with each other.", "In the presence of matter, we focused our attention on the approximate analytical solutions to the Wheeler-DeWitt equation in the domain of small scale factor, i.e.", "in the region which the quantum cosmology is expected to be dominant.", "Using Schutz’s representation for the perfect fluid, under a particular gauge choice, we led to the identification of a time parameter which allowed us to study the time evolution of the resulting wave function.", "Investigation of the expectation value of the scale factor shows a bouncing behavior near the classical singularity.", "In addition to singularity avoidance, the appearance of bounce in the quantum model is also interesting in its nature due to prediction of a minimal size for the corresponding universe.", "We know the idea of existence of a minimal length in nature is supported by almost all candidates of quantum gravity.", "Finally, we repeated the quantum calculations by means of the Bohmian approach to quantum mechanics.", "The analysis of the quantum potential shows the importance of the mass term in the action of the model.", "Indeed, we have shown that in the presence of the massive graviton, the quantum potential changes its behavior from an infinite barrier to a finite one, and hence the minimal size of the universe, from which the bounce occurs, will be shifted to the smaller values.", "Also, the massive theory of quantum cosmology exhibits some other possibilities; for example, tunneling between different classically allowed regions, for cosmic evolution in the early universe epoch." ] ]

[ [ "Self-improving Algorithms for Coordinate-wise Maxima" ], [ "Abstract Computing the coordinate-wise maxima of a planar point set is a classic and well-studied problem in computational geometry.", "We give an algorithm for this problem in the \\emph{self-improving setting}.", "We have $n$ (unknown) independent distributions $\\cD_1, \\cD_2, ..., \\cD_n$ of planar points.", "An input pointset $(p_1, p_2, ..., p_n)$ is generated by taking an independent sample $p_i$ from each $\\cD_i$, so the input distribution $\\cD$ is the product $\\prod_i \\cD_i$.", "A self-improving algorithm repeatedly gets input sets from the distribution $\\cD$ (which is \\emph{a priori} unknown) and tries to optimize its running time for $\\cD$.", "Our algorithm uses the first few inputs to learn salient features of the distribution, and then becomes an optimal algorithm for distribution $\\cD$.", "Let $\\OPT_\\cD$ denote the expected depth of an \\emph{optimal} linear comparison tree computing the maxima for distribution $\\cD$.", "Our algorithm eventually has an expected running time of $O(\\text{OPT}_\\cD + n)$, even though it did not know $\\cD$ to begin with.", "Our result requires new tools to understand linear comparison trees for computing maxima.", "We show how to convert general linear comparison trees to very restricted versions, which can then be related to the running time of our algorithm.", "An interesting feature of our algorithm is an interleaved search, where the algorithm tries to determine the likeliest point to be maximal with minimal computation.", "This allows the running time to be truly optimal for the distribution $\\cD$." ], [ "Introduction", "Given a set $P$ of $n$ points in the plane, the maxima problem is to find those points $p\\in P$ for which no other point in $P$ has a larger $x$ -coordinate and a larger $y$ -coordinate.", "More formally, for $p\\in \\mathbb {R}^2$ , let $x(p)$ and $y(p)$ denote the $x$ and $y$ coordinates of $p$ .", "Then $p^{\\prime }$ dominates $p$ if and only if $x(p^{\\prime })\\ge x(p)$ , $y(p^{\\prime })\\ge y(p)$ , and one of these inequalities is strict.", "The desired points are those in $P$ that are not dominated by any other points in $P$ .", "The set of maxima is also known as a skyline in the database literature [5] and as a Pareto frontier.", "This algorithmic problem has been studied since at least 1975 [17], when Kung et al.", "described an algorithm with an $O(n\\log n)$ worst-case time and gave an $\\Omega (n\\log n)$ lower bound.", "Results since then include average-case running times of $n+\\tilde{O}(n^{6/7})$ point-wise comparisons [14]; output-sensitive algorithms needing $O(n\\log h)$ time when there are $h$ maxima [19]; and algorithms operating in external-memory models [15].", "A major problem with worst-case analysis is that it may not reflect the behavior of real-world inputs.", "Worst-case algorithms are tailor-made for extreme inputs, none of which may occur (with reasonable frequency) in practice.", "Average-case analysis tries to address this problem by assuming some fixed distribution on inputs; for maxima, the property of coordinate-wise independence covers a broad range of inputs, and allows a clean analysis [7], but is unrealistic even so.", "The right distribution to analyze remains a point of investigation.", "Nonetheless, the assumption of randomly distributed inputs is very natural and one worthy of further research.", "The self-improving model.", "Ailon et al.", "introduced the self-improving model to address this issue [3].", "In this model, there is some fixed but unknown input distribution $\\mathcal {D}$ that generates independent inputs, that is, whole input sets $P$ .", "The algorithm initially undergoes a learning phase, where it processes inputs with a worst-case guarantee but tries to learn information about $\\mathcal {D}$ .", "The aim of the algorithm is to become optimal for the distribution $\\mathcal {D}$.", "After seeing some (hopefully small) number of inputs, the algorithm shifts into the limiting phase.", "Now, the algorithm is tuned for $\\mathcal {D}$ and the expected running time is (ideally) optimal for $\\mathcal {D}$ .", "A self-improving algorithm can be thought of as an algorithm that attains the optimal average-case running time for all, or at least a large class of, distributions $\\mathcal {D}$ .", "Following earlier self-improving algorithms, we assume the input has a product distribution.", "An input is a set of $n$ points $P = (p_1, p_2, \\ldots , p_n)$ in the plane.", "Each $p_i$ is generated independently from a distribution $\\mathcal {D}_i$ , so the probability distribution of $P$ is the product $\\prod _i \\mathcal {D}_i$ .", "The $\\mathcal {D}_i$ s themselves are arbitrary, and the only assumption made is their independence.", "There are lower bounds showing that some restriction on $\\mathcal {D}$ is necessary for a reasonable self-improving algorithm, as we explain later.", "The first self-improving algorithm was for sorting; this was extended to Delaunay triangulations, with these results eventually merged [11], [2].", "A self-improving algorithm for planar convex hulls was given by Clarkson et al.", "[10], however their analysis was recently discovered to be flawed.", "Our main result is a self-improving algorithm for planar coordinate-wise maxima over product distributions.", "We need some basic definitions before stating our main theorem.", "We explain what it means for a maxima algorithm to be optimal for a distribution $\\mathcal {D}$ .", "This in turn requires a notion of certificates for maxima, which allow the correctness of the output to be verified in $O(n)$ time.", "Any procedure for computing maxima must provide some “reason\" to deem an input point $p$ non-maximal.", "The simplest certificate would be to provide an input point dominating $p$ .", "Most current algorithms implicitly give exactly such certificates [17], [14], [19].", "Definition 1.1 A certificate $\\gamma $ has: (i) the sequence of the indices of the maximal points, sorted from left to right; (ii) for each non-maximal point, a per-point certificate of non-maximality, which is simply the index of an input point that dominates it.", "We say that a certificate $\\gamma $ is valid for an input $P$ if $\\gamma $ satisfies these conditions for $P$ .", "The model of computation that we use to define optimality is a linear computation tree that generates query lines using the input points.", "In particular, our model includes the usual CCW-test that forms the basis for many geometric algorithms.", "Let $\\ell $ be a directed line.", "We use $\\ell ^+$ to denote the open halfplane to the left of $\\ell $ and $\\ell ^-$ to denote the open halfplane to the right of $\\ell $ .", "Definition 1.2 A linear comparison tree $\\mathcal {T}$ is a binary tree such that each node $v$ of $\\mathcal {T}$ is labeled with a query of the form “$p \\in \\ell _v^+?$ ”.", "Here $p$ denotes an input point and $\\ell _v$ denotes a directed line.", "The line $\\ell _v$ can be obtained in three ways: (i) it can be a line independent of the input (but dependent on the node $v$ ); (ii) it can be a line with a slope independent of the input (but dependent on $v$ ) passing through a given input point; (iii) it can be a line through an input point and through a point $q$ independent of the input (but dependent on $v$ ); (iv) it can be the line defined by two distinct input points.", "A linear comparison tree is restricted if it only makes queries of type (i).", "A linear comparison tree $\\mathcal {T}$ computes the maxima for $P$ if each leaf corresponds to a certificate.", "This means that each leaf $v$ of $\\mathcal {T}$ is labeled with a certificate $\\gamma $ that is valid for every possible input $P$ that reaches $v$ .", "Let $\\mathcal {T}$ be a linear comparison tree and $v$ be a node of $\\mathcal {T}$ .", "Note that $v$ corresponds to a region $\\mathcal {R}_v \\subseteq \\mathbb {R}^{2n}$ such that an evaluation of $\\mathcal {T}$ on input $P$ reaches $v$ if and only if $P \\in \\mathcal {R}_v$ .", "If $\\mathcal {T}$ is restricted, then $\\mathcal {R}_v$ is the Cartesian product of a sequence $ (R_1, R_2, \\ldots , R_n)$ of polygonal regions.", "The depth of $v$ , denoted by $d_v$ , is the length of the path from the root of $\\mathcal {T}$ to $v$ .", "Given $\\mathcal {T}$ , there exists exactly one leaf $v(P)$ that is reached by the evaluation of $\\mathcal {T}$ on input $P$ .", "The expected depth of $\\mathcal {T}$ over $\\mathcal {D}$ , $d_\\mathcal {D}(\\mathcal {T})$ , is defined as $\\hbox{\\bf E}_{P \\sim \\mathcal {D}}[d_{v(P)}]$ .", "Consider some comparison based algorithm $A$ that is modeled by such a tree $\\mathcal {T}$ .", "The expected depth of $\\mathcal {T}$ is a lower bound on the number of comparisons performed by $A$ .", "Let $\\textbf {T}$ be the set of trees that compute the maxima of $n$ points.", "We define $\\text{OPT}_\\mathcal {D}= \\inf _{\\mathcal {T}\\in \\textbf {T}} d_\\mathcal {D}(\\mathcal {T})$ .", "This is a lower bound on the expected time taken by any linear comparison tree to compute the maxima of inputs distributed according to $\\mathcal {D}$ .", "We would like our algorithm to have a running time comparable to $\\text{OPT}_\\mathcal {D}$ .", "Theorem 1.3 Let $\\varepsilon > 0$ be a fixed constant and $\\mathcal {D}_1$ , $\\mathcal {D}_2$ , $\\ldots , \\mathcal {D}_n$ be independent planar point distributions.", "The input distribution is $\\mathcal {D}= \\prod _i \\mathcal {D}_i$ .", "There is a self-improving algorithm to compute the coordinate-wise maxima whose expected time in the limiting phase is $O(\\varepsilon ^{-1}(n + \\text{OPT}_\\mathcal {D}))$ .", "The learning phase lasts for $O(n^\\varepsilon )$ inputs and the space requirement is $O(n^{1+\\varepsilon })$ .", "There are lower bounds in [2] (for sorters) implying that a self-improving maxima algorithm that works for all distributions requires exponential storage, and that the time-space tradeoff (wrt $\\varepsilon $ ) in the above theorem is optimal.", "Figure: Examples of difficult distributionsChallenges.", "One might think that since self-improving sorters are known, an algorithm for maxima should follow directly.", "But this reduction is only valid for $O(n\\log n)$ algorithms.", "Consider Figure REF (i).", "The distributions $\\mathcal {D}_1$ , $\\mathcal {D}_2$ , $\\ldots , \\mathcal {D}_{n/2}$ generate the fixed points shown.", "The remaining distributions generate a random point from a line below $L$ .", "Observe that an algorithm that wishes to sort the $x$ -coordinates requires $\\Omega (n\\log n)$ time.", "On the other hand, there is a simple comparison tree that determines the maxima in $O(n)$ time.", "For all $p_j$ where $j > n/2$ , the tree simply checks if $p_{n/2}$ dominates $p_j$ .", "After that, it performs a linear scan and outputs a certificate.", "We stress that even though the points are independent, the collection of maxima exhibits strong dependencies.", "In Figure REF (ii), suppose a distribution $\\mathcal {D}_i$ generates either $p_h$ or $p_\\ell $ ; if $p_\\ell $ is chosen, we must consider the dominance relations among the remaining points, while if $p_h$ is chosen, no such evaluation is required.", "The optimal search tree for a distribution $\\mathcal {D}$ must exploit this complex dependency.", "Indeed, arguing about optimality is one of the key contributions of this work.", "Previous self-improving algorithms employed information-theoretic optimality arguments.", "These are extremely difficult to analyze for settings like maxima, where some points are more important to process that others, as in Figure REF .", "(The main error in the self-improving convex hull paper [10] was an incorrect consideration of dependencies.)", "We focus on a somewhat weaker notion of optimality—linear comparison trees—that nonetheless covers most (if not all) important algorithms for maxima.", "In Section , we describe how to convert linear comparison trees into restricted forms that use much more structured (and simpler) queries.", "Restricted trees are much more amenable to analysis.", "In some sense, a restricted tree decouples the individual input points and makes the maxima computation amenable to separate $\\mathcal {D}_i$ -optimal searches.", "A leaf of a restricted tree is associated with a sequence of polygons $(R_1,R_2,\\ldots , R_n)$ such that the leaf is visited if and only if every $p_i\\in R_i$ , and conditioned on that event, the $p_i$ remain independent.", "This independence is extremely important for the analysis.", "We design an algorithm whose behavior can be related to the restricted tree.", "Intuitively, if the algorithm spends many comparisons involving a single point, then we can argue that the optimal restricted tree must also do the same.", "We give more details about the algorithm in Section ." ], [ "Previous work", "Afshani et al.", "[1] introduced a model of instance-optimality applying to algorithmic problems including planar convex hulls and maxima.", "(However, their model is different from, and in a sense weaker than, the prior notion of instance-optimality introduced by Fagin et al. [13].)", "All previous (e.g., output sensitive and instance optimal) algorithms require expected $\\Omega (n\\log n)$ time for the distribution given in Figure REF , though an optimal self-improving algorithm only requires $O(n)$ expected time.", "(This was also discussed in [10] with a similar example.", "), We also mention the paradigm of preprocessing regions in order to compute certain geometric structures faster (see, e.g., [6], [12], [16], [20], [18]).", "Here, we are given a set $\\mathcal {R}$ of planar regions, and we would like to preprocess $\\mathcal {R}$ in order to quickly find the (Delaunay) triangulation (or convex hull) for any point set which contains exactly one point from each region in $\\mathcal {R}$ .", "This setting is adversarial, but if we only consider point sets where a point is randomly drawn from each region, it can be regarded as a special case of our setting.", "In this view, these results give us bounds on the running time a self-improving algorithm can achieve if $\\mathcal {D}$ draws its points from disjoint planar regions." ], [ "Preliminaries and notation", "Before we begin, let us define some basic concepts and agree on a few notational conventions.", "We use $c$ for a sufficiently large constant, and we write $\\log x$ to denote the logarithm of $x$ in base 2.", "All the probability distributions are assumed to be continuous.", "(It is not necessary to do this, but it makes many calculations a lot simpler.)", "Given a polygonal region $R \\subseteq \\mathbb {R}^2$ and a probability distribution $\\mathcal {D}$ on the plane, we call $\\ell $ a halving line for $R$ (with respect to $\\mathcal {D}$ ) if $\\Pr _{p \\sim \\mathcal {D}}[p \\in \\ell ^+ \\cap R]=\\Pr _{p \\sim \\mathcal {D}}[p \\in \\ell ^- \\cap R].$ Note that if $\\Pr _{p \\sim \\mathcal {D}}[p \\in R] = 0$ , every line is a halving line for $R$ .", "If not, a halving line exactly halves the conditional probability for $p$ being in each of the corresponding halfplanes, conditioned on $p$ lying inside $R$ .", "Define a vertical slab structure $\\textbf {S}$ as a sequence of vertical lines partitioning the plane into vertical regions, called leaf slabs.", "(We will consider the latter to be the open regions between the vertical lines.", "Since we assume that our distributions are continuous, we abuse notation and consider the leaf slabs to partition the plane.)", "More generally, a slab is the region between any two vertical lines of the $\\textbf {S}$ .", "The size of the slab structure is the number of leaf slabs it contains.", "We denote it by $|\\textbf {S}|$ .", "Furthermore, for any slab $S$ , the probability that $p_i \\sim \\mathcal {D}_i$ is in $S$ is denoted by $q(i,S)$ .", "A search tree $T$ over $\\textbf {S}$ is a comparison tree that locates a point within leaf slabs of $\\textbf {S}$ .", "Each internal node compares the $x$ -coordinate of the point with a vertical line of $\\textbf {S}$ , and moves left or right accordingly.", "We associate each internal node $v$ with a slab $S_v$ (any point in $S_v$ will encounter $v$ along its search)." ], [ "Tools from self-improving algorithms", "We introduce some tools that were developed in previous self-improving results.", "The ideas are by and large old, but our presentation in this form is new.", "We feel that the following statements (especially [lem:search-time]Lemma lem:search-time) are of independent interest.", "We define the notion of restricted searches, introduced in [10].", "This notion is central to our final optimality proof.", "(The lemma and formulation as given here are new.)", "Let $\\textbf {U}$ be an ordered set and $\\mathcal {F}$ be a distribution over $\\textbf {U}$ .", "For any element $j \\in \\textbf {U}$ , $q_j$ is the probability of $j$ according to $\\mathcal {F}$ .", "For any interval $S$ of $\\textbf {U}$ , the total probability of $S$ is $q_S$ .", "We let $T$ denote a search tree over $\\textbf {U}$ .", "It will be convenient to think of $T$ as (at most) ternary, where each node has at most 2 children that are internal nodes.", "In our application of the lemma, $\\textbf {U}$ will just be the set of leaf slabs of a slab structure $\\textbf {S}$ .", "We now introduce some definitions regarding restricted searches and search trees.", "Definition 1.4 Consider a distribution $\\mathcal {F}$ and an interval $S$ of $U$ .", "An $S$ -restricted distribution is given by the probabilities (for element $r \\in U$ ) $q^{\\prime }_r/\\sum _{j \\in U} q^{\\prime }_j$ , where the sequence $\\lbrace q^{\\prime }_j | j \\in U\\rbrace $ has the following property.", "For each $j \\in S$ , $0 \\le q^{\\prime }_j \\le q_j$ .", "For every other $j$ , $q^{\\prime }_j = 0$ .", "Suppose $j \\in S$ .", "An $S$ -restricted search is a search for $j$ in $T$ that terminates once $j$ is located in any interval contained in $S$ .", "For any sequence of numbers $\\lbrace q^{\\prime }_j | j \\in U\\rbrace $ and $S \\subseteq U$ , we use $q^{\\prime }_S$ to denote $\\sum _{j \\in S} q^{\\prime }_j$ .", "Definition 1.5 Let $\\mu \\in (0,1)$ be a parameter.", "A search tree $T$ over $\\textbf {U}$ is $\\mu $ -reducing if: for any internal node $S$ and for any non-leaf child $S^{\\prime }$ of $S$ , $q_{S^{\\prime }} \\le \\mu q_S$ .", "A search tree $T$ is $c$ -optimal for restricted searches over $\\mathcal {F}$ if: for all $S$ and $S$ -restricted distributions $\\mathcal {F}_S$ , the expected time of an $S$ -restricted search over $\\mathcal {F}_S$ is at most $c(-\\log q^{\\prime }_S + 1)$ .", "(The probabilities $q^{\\prime }$ are as given in [def:rest]Definition def:rest.)", "We give the main lemma about restricted searches.", "A tree that is optimal for searches over $\\mathcal {F}$ also works for restricted distributions.", "The proof is given in Appendix .", "Lemma 1.6 Suppose $T$ is a $\\mu $ -reducing search tree for $\\mathcal {F}$ .", "Then $T$ is $O(1/\\log (1/\\mu ))$ -optimal for restricted searches over $\\mathcal {F}$ .", "We list theorems about data structures that are built in the learning phase.", "Similar structures were first constructed in [2], and the following can be proved using their ideas.", "The data structures involve construction of slab structures and specialized search trees for each distribution $\\mathcal {D}_i$ .", "It is also important that these trees can be represented in small space, to satisfy the requirements of [thm:main]Theorem thm:main.", "The following lemmas give us the details of the data structures required.", "Because this is not a major contribution of this paper, we relegate the details to §.", "Lemma 1.7 We can construct a slab structure $\\textbf {S}$ with $O(n)$ leaf slabs such that, with probability $1-n^{-3}$ over the construction of $\\textbf {S}$ , the following holds.", "For a leaf slab $\\lambda $ of $\\textbf {S}$ , let $X_\\lambda $ denote the number of points in a random input $P$ that fall into $\\lambda $ .", "For every leaf slab $\\lambda $ of $\\textbf {S}$ , we have $\\hbox{\\bf E}[X^2_\\lambda ] = O(1)$ .", "The construction takes $O(\\log n)$ rounds and $O(n\\log ^2 n)$ time.", "Lemma 1.8 Let $\\varepsilon > 0$ be a fixed parameter.", "In $O(n^{\\varepsilon })$ rounds and $O(n^{1+\\varepsilon })$ time, we can construct search trees $T_1$ , $T_2$ , $\\ldots $ , $T_n$ over $\\textbf {S}$ such that the following holds.", "(i) the trees can be represented in $O(n^{1+\\varepsilon })$ total space; (ii) with probability $1-n^{-3}$ over the construction of the $T_i$ s, every $T_i$ is $O(1/\\varepsilon )$ -optimal for restricted searches over $\\mathcal {D}_i$ ." ], [ "Outline", "We start by providing a very informal overview of the algorithm.", "Then, we shall explain how the optimality is shown.", "If the points of $P$ are sorted by $x$ -coordinate, the maxima of $P$ can be found easily by a right-to-left sweep over $P$ : we maintain the largest $y$ -coordinate $Y$ of the points traversed so far; when a point $p$ is visited in the traversal, if $y(p)<Y$ , then $p$ is non-maximal, and the point $p_j$ with $Y=y(p_j)$ gives a per-point certificate for $p$ 's non-maximality.", "If $y(p)\\ge Y$ , then $p$ is maximal, and can be put at the beginning of the certificate list of maxima of $P$ .", "This suggests the following approach to a self-improving algorithm for maxima: sort $P$ with a self-improving sorter and then use the traversal.", "The self-improving sorter of [2] works by locating each point of $P$ within the slab structure $\\textbf {S}$ of [lem:slabstruct]Lemma lem:slabstruct using the trees $T_i$ of [lem:tree]Lemma lem:tree.", "While this approach does use $\\textbf {S}$ and the $T_i$ 's, it is not optimal for maxima, because the time spent finding the exact sorted order of non-maximal points may be wasted: in some sense, we are learning much more information about the input $P$ than necessary.", "To deduce the list of maxima, we do not need the sorted order of all points of $P$ : it suffices to know the sorted order of just the maxima!", "An optimal algorithm would probably locate the maximal points in $\\textbf {S}$ and would not bother locating “extremely non-maximal” points.", "This is, in some sense, the difficulty that output-sensitive algorithms face.", "As a thought experiment, let us suppose that the maximal points of $P$ are known to us, but not in sorted order.", "We search only for these in $\\textbf {S}$ and determine the sorted list of maximal points.", "We can argue that the optimal algorithm must also (in essence) perform such a search.", "We also need to find per-point certificates for the non-maximal points.", "We use the slab structure $\\textbf {S}$ and the search trees, but now we shall be very conservative in our searches.", "Consider the search for a point $p_i$ .", "At any intermediate stage of the search, $p_i$ is placed in a slab $S$ .", "This rough knowledge of $p_i$ 's location may already suffice to certify its non-maximality: let $m$ denote the leftmost maximal point to the right of $S$ (since the sorted list of maxima is known, this information can be easily deduced).", "We check if $m$ dominates $p_i$ .", "If so, we have a per-point certificate for $p_i$ and we promptly terminate the search for $p_i$ .", "Otherwise, we continue the search by a single step and repeat.", "We expect that many searches will not proceed too long, achieving a better position to compete with the optimal algorithm.", "Non-maximal points that are dominated by many maximal points will usually have a very short search.", "Points that are “nearly” maximal will require a much longer search.", "So this approach should derive just the “right\" amount of information to determine the maxima output.", "But wait!", "Didn't we assume that the maximal points were known?", "Wasn't this crucial in cutting down the search time?", "This is too much of an assumption, and because the maxima are highly dependent on each other, it is not clear how to determine which points are maximal before performing searches.", "The final algorithm overcomes this difficulty by interleaving the searches for sorting the points with confirmation of the maximality of some points, in a rough right-to-left order that is a more elaborate version of the traversal scheme given above for sorted points.", "The searches for all points $p_i$ (in their respective trees $T_i$ ) are performed “together\", and their order is carefully chosen.", "At any intermediate stage, each point $p_i$ is located in some slab $S_i$ , represented by some node of its search tree.", "We choose a specific point and advance its search by one step.", "This order is very important, and is the basis of our optimality.", "The algorithm is described in detail and analyzed in §.", "Arguing about optimality.", "A major challenge of self-improving algorithms is the strong requirement of optimality for the distribution $\\mathcal {D}$ .", "We focus on the model of linear comparison trees, and let $\\mathcal {T}$ be an optimal tree for distribution $\\mathcal {D}$ .", "(There may be distributions where such an exact $\\mathcal {T}$ does not exist, but we can always find one that is near optimal.)", "One of our key insights is that when $\\mathcal {D}$ is a product distribution, then we can convert $\\mathcal {T}$ to $\\mathcal {T}^{\\prime }$ , a restricted comparison tree whose expected depth is only a constant factor worse.", "In other words, there exists a near optimal restricted comparison tree that computes the maxima.", "In such a tree, a leaf is labeled with a sequence of regions $\\mathcal {R}= (R_1, R_2, \\ldots , R_n)$ .", "Any input $P = (p_1, p_2, \\ldots , p_n)$ such that $p_i \\in R_i$ for all $i$ , will lead to this leaf.", "Since the distributions are independent, we can argue that the probability that an input leads to this leaf is $\\prod _i \\Pr _{p_i \\sim \\mathcal {D}_i}[p_i \\in R_i]$ .", "Furthermore, the depth of this leaf can be shown to be $-\\sum _i \\log \\Pr [p_i \\in R_i]$ .", "This gives us a concrete bound that we can exploit.", "It now remains to show that if we start with a random input from $\\mathcal {R}$ , the expected running time is bounded by the sum given above.", "We will argue that for such an input, as soon as the search for $p_i$ locates it inside $R_i$ , the search will terminate.", "This leads to the optimal running time." ], [ "Reducing to restricted comparison trees", "We prove that when $P$ is generated probabilistically, it suffices to focus on restricted comparison trees.", "To show this, we provide a sequence of transformations, starting from the more general comparison tree, that results in a restricted linear comparison tree of comparable expected depth.", "The main lemma of this section is the following.", "Lemma 3.1 Let $\\mathcal {T}$ a finite linear comparison tree and $\\mathcal {D}$ be a product distribution over points.", "Then there exists a restricted comparison tree $\\mathcal {T}^{\\prime }$ with expected depth $d_{\\mathcal {D}}(\\mathcal {T}^{\\prime }) = O(d_\\mathcal {D}(\\mathcal {T}))$ , as $d_\\mathcal {D}(\\mathcal {T}) \\rightarrow \\infty $ .", "We will describe a transformation from $\\mathcal {T}$ into a restricted comparison tree with similar depth.", "The first step is to show how to represent a single comparison by a restricted linear comparison tree, provided that $P$ is drawn from a product distribution.", "The final transformation basically replaces each node of $\\mathcal {T}$ by the subtree given by the next claim.", "For convenience, we will drop the subscript of $\\mathcal {D}$ from $d_\\mathcal {D}$ , since we only focus on a fixed distribution.", "Claim 3.2 Consider a comparison $C$ as described in [def:opt]Definition def:opt, where the comparisons are listed in increasing order of simplicity.", "Let $\\mathcal {D}^{\\prime }$ be a product distribution for $P$ such that each $p_i$ is drawn from a polygonal region $R_i$ .", "Then either $C$ is the simplest, type (i) comparison, or there exists a restricted linear comparison tree $\\mathcal {T}^{\\prime }_C$ that resolves the comparison $C$ such that the expected depth of $\\mathcal {T}^{\\prime }_C$ (over the distribution $\\mathcal {D}^{\\prime }$ ) is $O(1)$ , and all comparisons used in $\\mathcal {T}^{\\prime }_C$ are simpler than $C$ .", "$v$ is of type (ii).", "This means that $v$ needs to determine whether an input point $p_i$ lies to the left of the directed line $\\ell $ through another input point $p_j$ with a fixed slope $a$ .", "We replace this comparison with a binary search.", "Let $R_j$ be the region in $\\mathcal {D}^{\\prime }$ corresponding to $p_j$ .", "Take a halving line $\\ell _1$ for $R_j$ with slope $a$ .", "Then perform two comparisons to determine on which side of $\\ell _1$ the inputs $p_i$ and $p_j$ lie.", "If $p_i$ and $p_j$ lie on different sides of $\\ell _1$ , we declare success and resolve the original comparison accordingly.", "Otherwise, we replace $R_j$ with the appropriate new region and repeat the process until we can declare success.", "Note that in each attempt the success probability is at least $1/4$ .", "The resulting restricted tree $\\mathcal {T}^{\\prime }_C$ can be infinite.", "Nonetheless, the probability that an evaluation of $\\mathcal {T}^{\\prime }_C$ leads to a node of depth $k$ is at most $2^{-\\Omega (k)}$ , so the expected depth is $O(1)$ .", "$v$ is of type (iii).", "Here the node $v$ needs to determine whether an input point $p_i$ lies to the left of the directed line $\\ell $ through another input point $p_j$ and a fixed point $q$ .", "We partition the plane by a constant-sized family of cones, each with apex $q$ , such that for each cone $V$ in the family, the probability that line $\\overline{q p_j}$ meets $V$ (other than at $q$ ) is at most $1/2$ .", "Such a family could be constructed by a sweeping a line around $q$ , or by taking a sufficiently large, but constant-sized, sample from the distribution of $p_j$ , and bounding the cones by all lines through $q$ and each point of the sample.", "Such a construction has a non-zero probability of success, and therefore the described family of cones exists.", "We build a restricted tree that locates a point in the corresponding cone.", "For each cone $V$ , we can recursively build such a family of cones (inside $V$ ), and build a tree for this structure as well.", "Repeating for each cone, this leads to an infinite restricted tree $\\mathcal {T}^{\\prime }_C$ .", "We search for both $p_i$ and $p_j$ in $\\mathcal {T}^{\\prime }_C$ .", "When we locate $p_i$ and $p_j$ in two different cones of the same family, then comparison between $p_i$ and $\\overline{q p_j}$ is resolved and the search terminates.", "The probability that they lie in the same cones of a given family is at most $1/2$ , so the probability that the evaluation leads to $k$ steps is at most $2^{-\\Omega (k)}$ .", "$v$ is of type (iv).", "Here the node $v$ needs to determine whether an input point $p_i$ lies to the left of the directed line $\\ell $ through input points $p_j$ and $p_k$ .", "We partition the plane by a constant-sized family of triangles and cones, such that for each region $V$ in the family, the probability that the line through $p_j$ and $p_k$ meets $V$ is at most $1/2$ .", "Such a family could be constructed by taking a sufficiently large random sample of pairs $p_j$ and $p_k$ and triangulating the arrangement of the lines through each pair.", "Such a construction has a non-zero probability of success, and therefore such a family exists.", "(Other than the source of the random lines used in the construction, this scheme goes back at least to [9]; a tighter version, called a cutting, could also be used [8].)", "When computing $C$ , suppose $p_i$ is in region $V$ of the family.", "If the line $\\overline{p_jp_k}$ does not meet $V$ , then the comparison outcome is known immediately.", "This occurs with probability at least $1/2$ .", "Moreover, determining the region containing $p_i$ can be done with a constant number of comparisons of type (i), and determining if $\\overline{p_jp_k}$ meets $V$ can be done with a constant number of comparisons of type (iii); for the latter, suppose $V$ is a triangle.", "If $p_j\\in V$ , then $\\overline{p_jp_k}$ meets $V$ .", "Otherwise, suppose $p_k$ is above all the lines through $p_j$ and each vertex of $V$ ; then $\\overline{p_jp_k}$ does not meet $V$ .", "Also, if $p_k$ is below all the lines through $p_j$ and each vertex, then $\\overline{p_j p_k}$ does not meet $V$ .", "Otherwise, $\\overline{p_j p_k}$ meets $V$ .", "So a constant number of type (i) and type (iii) queries suffice.", "By recursively building a tree for each region $V$ of the family, comparisons of type (iv) can be done via a tree whose nodes use comparisons of type (i) and (iii) only.", "Since the probability of resolving the comparison is at least $1/2$ with each family of regions that is visited, the expected number of nodes visited is constant.", "[of [lem:restrictedTree]Lemma lem:restrictedTree] We transform $\\mathcal {T}$ into a tree $\\mathcal {T}^{\\prime }$ that has no comparisons of type (iv), by using the construction of [clm:node]Claim clm:node where nodes of type (iv) are replaced by a tree.", "We then transform $\\mathcal {T}^{\\prime }$ into a tree $\\mathcal {T}^{\\prime \\prime }$ that has no comparisons of type (iii) or (iv), and finally transform $\\mathcal {T}^{\\prime \\prime \\prime }$ into a restricted tree.", "Each such transformation is done in the same general way, using one case of [clm:node]Claim clm:node, so we focus on the first one.", "We incrementally transform $\\mathcal {T}$ into the tree $\\mathcal {T}^{\\prime }$ .", "In each such step, we have a partial restricted comparison tree $\\mathcal {T}^{\\prime \\prime }$ that will eventually become $\\mathcal {T}^{\\prime }$ .", "Furthermore, during the process each node of $\\mathcal {T}$ is in one of three different states.", "It is either finished, fringe, or untouched.", "Finally, we have a function $S$ that assigns to each finished and to each fringe node of $\\mathcal {T}$ a subset $S(v)$ of nodes in $\\mathcal {T}^{\\prime \\prime }$ .", "The initial situation is as follows: all nodes of $\\mathcal {T}$ are untouched except for the root which is fringe.", "Furthermore, the partial tree $\\mathcal {T}^{\\prime \\prime }$ consists of a single root node $r$ and the function $S$ assigns the root of $\\mathcal {T}$ to the set $\\lbrace r\\rbrace $ .", "Now our transformation proceeds as follows.", "We pick a fringe node $v$ in $\\mathcal {T}$ , and mark $v$ as finished.", "For each child $v^{\\prime }$ of $v$ , if $v^{\\prime }$ is an internal node of $\\mathcal {T}$ , we mark it as fringe.", "Otherwise, we mark $v^{\\prime }$ as finished.", "Next, we apply [clm:node]Claim clm:node to each node $w \\in S(v)$ .", "Note that this is a valid application of the claim, since $w$ is a node of $\\mathcal {T}^{\\prime \\prime }$ , a restricted tree.", "Hence $\\mathcal {R}_w$ is a product set, and the distribution $\\mathcal {D}$ restricted to $\\mathcal {R}_w$ is a product distribution.", "Hence, replace each node $w \\in S(v)$ in $\\mathcal {T}^{\\prime \\prime }$ by the subtree given by [clm:node]Claim clm:node.", "Now $S(v)$ contains the roots of these subtrees.", "Each leaf of each such subtree corresponds to an outcome of the comparison in $v$ .", "(Potentially, the subtrees are countably infinite, but the expected number of steps to reach a leaf is constant.)", "For each child $v^{\\prime }$ of $v$ , we define $S(v^{\\prime })$ as the set of all such leaves that correspond to the same outcome of the comparison as $v^{\\prime }$ .", "We continue this process until there are no fringe nodes left.", "By construction, the resulting tree $\\mathcal {T}^{\\prime }$ is restricted.", "It remains to argue that $d_{\\mathcal {T}^{\\prime }} = O(d_{\\mathcal {T}})$ .", "Let $v$ be a node of $\\mathcal {T}$ .", "We define two random variables $X_v$ and $Y_v$ .", "The variable $X_v$ is the indicator random variable for the event that the node $v$ is traversed for a random input $P \\sim \\mathcal {D}$ .", "The variable $Y_v$ denotes the number of nodes traversed in $\\mathcal {T}^{\\prime }$ that correspond to $v$ (i.e., the number of nodes needed to simulate the comparison at $v$ , if it occurs).", "We have $d_{\\mathcal {T}} = \\sum _{v \\in \\mathcal {T}} \\hbox{\\bf E}[X_v]$ , because if the leaf corresponding to an input $P \\sim \\mathcal {D}$ has depth $d$ , exactly $d$ nodes are traversed to reach it.", "We also have $d_{\\mathcal {T}^{\\prime }} = \\sum _{v \\in \\mathcal {T}} \\hbox{\\bf E}[Y_v]$ , since each node in $\\mathcal {T}^{\\prime }$ corresponds to exactly one node $v$ in $\\mathcal {T}$ .", "[clm:YvBound]Claim clm:YvBound below shows that $\\hbox{\\bf E}[Y_v] = O(\\hbox{\\bf E}[X_v])$ , which completes the proof.", "Claim 3.3 $\\hbox{\\bf E}[Y_v] \\le c\\hbox{\\bf E}[X_v]$ Note that $\\hbox{\\bf E}[X_v] = \\Pr [X_v = 1] = \\Pr [P \\in \\mathcal {R}_v]$ .", "Since the sets $\\mathcal {R}_w$ , $w \\in S(v)$ , partition $\\mathcal {R}_v$ , we can write $\\hbox{\\bf E}[Y_v]$ as $\\hbox{\\bf E}[Y_v \\mid X_v = 0]\\Pr [X_v = 0] +\\\\\\sum _{w \\in S(v)} \\hbox{\\bf E}[Y_v \\mid P \\in \\mathcal {R}_w]\\Pr [P \\in \\mathcal {R}_w].$ Since $Y_v = 0$ if $P \\notin \\mathcal {R}_v$ , we have $\\hbox{\\bf E}[Y_v \\mid X_v = 0] = 0$ and also $\\Pr [P \\in \\mathcal {R}_v] = \\sum _{w \\in S(v)} \\Pr [P \\in \\mathcal {R}_w]$ .", "Furthermore, by [clm:node]Claim clm:node, we have $\\hbox{\\bf E}[Y_v \\mid P \\in \\mathcal {R}_w] \\le c$ .", "The claim follows." ], [ "Entropy-sensitive comparison trees", "Since every linear comparison tree can be made restricted, we can incorporate the entropy of $\\mathcal {D}$ into the lower bound.", "For this we define entropy-sensitive trees, which are useful because the depth of a node $v$ is related to the probability of the corresponding region $\\mathcal {R}_v$ .", "Definition 3.4 We call a restricted linear comparison tree entropy-sensitive if each comparison “$p_i \\in \\ell ^+?$ ” is such that $\\ell $ is a halving line for the current region $R_i$ .", "Lemma 3.5 Let $v$ be a node in an entropy-sensitive comparison tree, and let $\\mathcal {R}_v = R_1 \\times R_2 \\times \\cdots \\times R_n$ .", "Then $d_v = - \\sum _{i=1}^n \\log \\Pr [R_i]$ .", "We use induction on the depth of $v$ .", "For the root $r$ we have $d_r = 0$ .", "Now, let $v^{\\prime }$ be the parent of $v$ .", "Since $\\mathcal {T}$ is entropy-sensitive, we reach $v$ after performing a comparison with a halving line in $v^{\\prime }$ .", "This halves the measure of exactly one region in $\\mathcal {R}_v$ , so the sum increases by one.", "As in [lem:restrictedTree]Lemma lem:restrictedTree, we can make every restricted linear comparison tree entropy-sensitive without affecting its expected depth too much.", "Lemma 3.6 Let $\\mathcal {T}$ a restricted linear comparison tree.", "Then there exists an entropy-sensitive comparison tree $\\mathcal {T}^{\\prime }$ with expected depth $d_{\\mathcal {T}^{\\prime }} = O(d_\\mathcal {T})$ .", "The proof extends the proof of [lem:restrictedTree]Lemma lem:restrictedTree, via an extension to [clm:node]Claim clm:node.", "We can regard a comparison against a fixed halving line as simpler than an comparison against an arbitrary fixed line.", "Our extension of [clm:node]Claim clm:node is the claim that any type (i) node can be replaced by a tree with constant expected depth, as follows.", "A comparison $p_i \\in \\ell ^+$ can be replaced by a sequence of comparisons to halving lines.", "Similar to the reduction for type (ii) comparisons in [clm:node]Claim clm:node, this is done by binary search.", "That is, let $\\ell _1$ be a halving line for $R_i$ parallel to $\\ell $ .", "We compare $p_i$ with $\\ell $ .", "If this resolves the original comparison, we declare success.", "Otherwise, we repeat the process with the halving line for the new region $R_i^{\\prime }$ .", "In each step, the probability of success is at least $1/2$ .", "The resulting comparison tree has constant expected depth; we now apply the construction of [lem:restrictedTree]Lemma lem:restrictedTree to argue that for a restricted tree $\\mathcal {T}$ there is an entropy-sensitive version $\\mathcal {T}^{\\prime }$ whose expected depth is larger by at most a constant factor.", "Recall that $\\text{OPT}_\\mathcal {D}$ is the expected depth of an optimal linear comparison tree that computes the maxima for $P \\sim \\mathcal {D}$ .", "We now describe how to characterize $\\text{OPT}_\\mathcal {D}$ in terms of entropy-sensitive comparison trees.", "We first state a simple property that follows directly from the definition of certificates and the properties of restricted comparison trees.", "Proposition 3.7 Consider a leaf $v$ of a restricted linear comparison tree $\\mathcal {T}$ computing the maxima.", "Let $R_i$ be the region associated with non-maximal point $p_i \\in P$ in $\\mathcal {R}_v$ .", "There exists some region $R_j$ associated with an extremal point $p_j$ such that every point in $R_j$ dominates every point in $R_i$ .", "We now enhance the notion of a certificate ([def:cert]Definition def:cert) to make it more useful for our algorithm's analysis.", "For technical reasons, we want points to be “well-separated\" according to the slab structure $\\textbf {S}$ .", "By [prop:dom]Prop.", "prop:dom, every non-maximal point is associated with a dominating region.", "Definition 3.8 Let $\\textbf {S}$ be a slab structure.", "A certificate for an input $P$ is called $\\textbf {S}$ -labeled if the following holds.", "Every maximal point is labeled with the leaf slab of $\\textbf {S}$ containing it.", "Every non-maximal point is either placed in the containing leaf slab, or is separated from a dominating region by a slab boundary.", "We naturally extend this to trees that compute the $\\textbf {S}$ -labeled maxima.", "Definition 3.9 A linear comparison tree $\\mathcal {T}$ computes the $\\textbf {S}$ -labeled maxima of $P$ if each leaf $v$ of $\\mathcal {T}$ is labeled with a $\\textbf {S}$ -labeled certificate that is valid for every possible input $P \\in \\mathcal {R}_v$ .", "Lemma 3.10 There exists an entropy-sensitive comparison tree $\\mathcal {T}$ computing the $\\textbf {S}$ -labeled maxima whose expected depth over $\\mathcal {D}$ is $O(n + \\textup {\\text{OPT}}_\\mathcal {D})$ .", "Start with an optimal linear comparison tree $\\mathcal {T}^{\\prime }$ that computes the maxima.", "At every leaf, we have a list $M$ with the maximal points in sorted order.", "We merge $M$ with the list of slab boundaries of $\\textbf {S}$ to label each maximal point with the leaf slab of $\\textbf {S}$ containing it.", "We now deal with the non-maximal points.", "Let $R_i$ be the region associated with a non-maximal point $p_i$ , and $R_j$ be the dominating region.", "Let $\\lambda $ be the leaf slab containing $R_j$ .", "Note that the $x$ -projection of $R_i$ cannot extended to the right of $\\lambda $ .", "If there is no slab boundary separating $R_i$ from $R_j$ , then $R_i$ must intersect $\\lambda $ .", "With one more comparison, we can place $p_i$ inside $\\lambda $ or strictly to the left of it.", "All in all, with $O(n)$ more comparisons than $\\mathcal {T}^{\\prime }$ , we have a tree $\\mathcal {T}^{\\prime \\prime }$ that computes the $\\textbf {S}$ -labeled maxima.", "Hence, the expected depth is $\\text{OPT}_\\mathcal {D}+ O(n)$ .", "Now we apply Lemmas REF and REF to $\\mathcal {T}^{\\prime \\prime }$ to get an entropy-sensitive comparison tree $\\mathcal {T}$ computing the $\\textbf {S}$ -labeled maxima with expected depth $O(n + \\text{OPT}_\\mathcal {D})$ ." ], [ "The algorithm", "In the learning phase, the algorithm constructs a slab structure $\\textbf {S}$ and search trees $T_i$ , as given in Lemmas REF and REF .", "Henceforth, we assume that we have these data structures, and will describe the algorithm in the limiting (or stationary) phase.", "Our algorithm proceeds by searching progressively each point $p_i$ in its tree $T_i$ .", "However, we need to choose the order of the searches carefully.", "At any stage of the algorithm, each point $p_i$ is placed in some slab $S_i$ .", "The algorithm maintains a set $A$ of active points.", "An inactive point is either proven to be non-maximal, or it has been placed in a leaf slab.", "The active points are stored in a data structure $L(A)$ .", "This structure is similar to a heap and supports the operations delete, decrease-key, and find-max.", "The key associated with an active point $p_i$ is the right boundary of the slab $S_i$ (represented as an element of $[|\\textbf {S}|]$ ).", "We list the variables that the algorithm maintains.", "The algorithm is initialized with $A = P$ , and each $S_i$ is the largest slab in $\\textbf {S}$ .", "Hence, all points have key $|\\textbf {S}|$ , and we insert all these keys into $L(A)$ .", "$A, L(A)$ : the list $A$ of active points stored in data structure $L(A)$ .", "$\\widehat{\\lambda }, B$ : Let $m$ be the largest key among the active points.", "Then $\\widehat{\\lambda }$ is the leaf slab whose right boundary is $m$ and $B$ is a set of points located in $\\widehat{\\lambda }$ .", "Initially $B$ is empty and $m$ is $|S|$ , corresponding to the $+\\infty $ boundary of the rightmost, infinite, slab.", "$M, \\hat{p}$ : $M$ is a sorted (partial) list of currently discovered maximal points and $\\hat{p}$ is the leftmost among those.", "Initially $M$ is empty and $\\hat{p}$ is a “null” point that dominates no input point.", "The algorithm involves a main procedure Search, and an auxiliary procedure Update.", "The procedure Search chooses a point and proceeds its search by a single step in the appropriate tree.", "Occasionally, it will invoke Update to change the global variables.", "The algorithm repeatedly calls Search until $L(A)$ is empty.", "After that, we perform a final call to Update in order to process any points that might still remain in $B$ .", "Search.", "Let $p_i$ be obtained by performing a find-max in $L(A)$ .", "If the maximum key $m$ in $L(A)$ is less than the right boundary of $\\widehat{\\lambda }$ , we invoke Update.", "If $p_i$ is dominated by $\\hat{p}$ , we delete $p_i$ from $L(A)$ .", "If not, we advance the search of $p_i$ in $T_i$ by a single step, if possible.", "This updates the slab $S_i$ .", "If the right boundary of $S_i$ has decreased, we perform the appropriate decrease-key operation on $L(A)$ .", "(Otherwise, we do nothing.)", "Suppose the point $p_i$ reaches a leaf slab $\\lambda $ .", "If $\\lambda = \\widehat{\\lambda }$ , we remove $p_i$ from $L(A)$ and insert it in $B$ (in time $O(|B|)$ ).", "Otherwise, we leave $p_i$ in $L(A)$ .", "Update.", "We sort all the points in $B$ and update the list of current maxima.", "As [clm:order]Claim clm:order will show, we have the sorted list of maxima to the right of $\\widehat{\\lambda }$ .", "Hence, we can append to this list in $O(|B|)$ time.", "We reset $B = \\emptyset $ , set $\\widehat{\\lambda }$ to the leaf slab to the left of $m$ , and return.", "We prove some preliminary claims.", "We state an important invariant maintained by the algorithm, and then give a construction for the data structure $L(A)$ .", "Claim 4.1 At any time in the algorithm, the maxima of all points to the right of $\\widehat{\\lambda }$ have been determined in sorted order.", "The proof is by backward induction on $m$ , the right boundary of $\\widehat{\\lambda }$ .", "When $m = |S|$ , then this is trivially true.", "Let us assume it is true for a given value of $m$ , and trace the algorithm's behavior until the maximum key becomes smaller than $m$ (which is done in Update).", "When Search processes a point $p$ with a key of $m$ then either (i) the key value decreases; (ii) $p$ is dominated by $\\hat{p}$ ; or (iii) $p$ is eventually placed in $\\widehat{\\lambda }$ (whose right boundary is $m$ ).", "In all cases, when the maximum key decreases below $m$ , all points in $\\widehat{\\lambda }$ are either proven to be non-maximal or are in $B$ .", "By the induction hypothesis, we already have a sorted list of maxima to the right of $m$ .", "The procedure Update will sort the points in $B$ and all maximal points to the right of $m-1$ will be determined.", "Claim 4.2 Suppose there are $x$ find-max operations and $y$ decrease-key operations.", "We can implement the data structure $L(A)$ such that the total time for the operations is $O(n + x + y)$ .", "The storage requirement is $O(n)$ .", "We represent $L(A)$ as an array of lists.", "For every $k \\in [|\\textbf {S}|]$ , we keep a list of points whose key values are $k$ .", "We maintain $m$ , the current maximum key.", "The total storage is $O(n)$ .", "A find-max can trivially be done in $O(1)$ time, and an insert is done by adding the element to the appropriate list.", "A delete is done by deleting the element from the list (supposing appropriate pointers are available).", "We now have to update the maximum.", "If the list at $m$ is non-empty, no action is required.", "If it is empty, we check sequentially whether the list at $m-1, m-2, \\ldots $ is empty.", "This will eventually lead to the maximum.", "To do a decrease-key, we delete, insert, and then update the maximum.", "Note that since all key updates are decrease-keys, the maximum can only decrease.", "Hence, the total overhead for scanning for a new maximum is $O(n)$ .", "Running time analysis The aim of this section is to prove the following lemma.", "Lemma 4.3 The algorithm runs in $O(n+\\textup {\\text{OPT}}_\\mathcal {D})$ time.", "We can easily bound the running time of all calls to Update.", "Claim 4.4 The expected time for all calls to Update is $O(n)$ .", "The total time taken for all calls to Update is at most the time taken to sort points within leaf slabs.", "By [lem:slabstruct]Lemma lem:slabstruct, this takes expected time $\\hbox{\\bf E}\\Bigl [\\sum _{\\lambda \\in \\textbf {S}} X_\\lambda ^2\\Bigr ]= \\sum _{\\lambda \\in \\textbf {S}} \\hbox{\\bf E}\\bigl [X_\\lambda ^2\\bigr ]= \\sum _{\\lambda \\in \\textbf {S}} O(1)= O(n).$ The important claim is the following, since it allows us to relate the time spent by Search to the entropy-sensitive comparison trees.", "[lem:algo]Lemma lem:algo follows directly from this.", "Claim 4.5 Let $\\mathcal {T}$ be an entropy-sensitive comparison tree computing $\\textbf {S}$ -labeled maxima.", "Consider a leaf $v$ labeled with the regions $\\mathcal {R}_v = (R_1, R_2, \\ldots , R_n)$ , and let $d_v$ denote the depth of $v$ .", "Conditioned on $P \\in \\mathcal {R}_v$ , the expected running time of Search is $O(n + d_v)$ .", "For each $R_i$ , let $S_i$ be the smallest slab of $\\textbf {S}$ that completely contains $R_i$ .", "We will show that the algorithm performs at most an $S_i$ -restricted search for input $P \\in \\mathcal {R}_v$ .", "If $p_i$ is maximal, then $R_i$ is contained in a leaf slab (this is because the output is $\\textbf {S}$ -labeled).", "Hence $S_i$ is a leaf slab and an $S_i$ -restricted search for a maximal $p_i$ is just a complete search.", "Now consider a non-maximal $p_i$ .", "By the properties of $\\textbf {S}$ -labeled maxima, the associated region $R_i$ is either inside a leaf slab or is separated by a slab boundary from the dominating region $R_j$ .", "In the former case, an $S_i$ -restricted search is a complete search.", "In the latter case, we argue that an $S_i$ -restricted search suffices to process $p_i$ .", "This follows from [clm:order]Claim clm:order: by the time an $S_i$ -restricted search finishes, all maxima to the right of $S_i$ have been determined.", "In particular, we have found $p_j$ , and thus $\\hat{p}$ dominates $p_i$ .", "Hence, the search for $p_i$ will proceed no further.", "The expected search time taken conditioned on $P \\in \\mathcal {R}_v$ is the sum (over $i$ ) of the conditional expected $S_i$ -restricted search times.", "Let $\\mathcal {E}_i$ denote the event that $p_i \\in R_i$ , and $\\mathcal {E}$ be the event that $P \\in \\mathcal {R}_v$ .", "We have $\\mathcal {E}= \\bigwedge _i \\mathcal {E}_i$ .", "By the independence of the distributions and linearity of expectation $&\\hbox{\\bf E}_\\mathcal {E}[\\text{search time}]\\\\&=\\sum _{i=1}^n \\hbox{\\bf E}_\\mathcal {E}[\\text{$S_i$-restricted search time for $p_i$}] \\\\& = \\sum _{i=1}^n \\hbox{\\bf E}_{\\mathcal {E}_i} [\\text{$S_i$-restricted search time for $p_i$}].$ By [lem:search-time]Lemma lem:search-time, the time for an $S_i$ -restricted search conditioned on $p_i \\in R_i$ is $O(-\\log \\Pr [p_i \\in R_i] + 1)$ .", "By [lem:entropyDepth]Lemma lem:entropyDepth, $d_v = \\sum _i -\\log \\Pr [p_i \\in R_i]$ , completing the proof.", "We can now prove the main lemma.", "[of [lem:algo]Lemma lem:algo] By [lem:comp]Lemma lem:comp, there exists an entropy-sensitive comparison tree $\\mathcal {T}$ that computes the $\\textbf {S}$ -labeled maxima with expected depth $O(\\text{OPT}+n)$ .", "According to [clm:search]Claim clm:search, the expected running time of Search is $O(\\text{OPT}+n)$ .", "[clm:update]Claim clm:update tells us the expected time for Update is $O(n)$ , and we add these bounds to complete the proof.", "Data structures obtained during the learning phase Learning the vertical slab structure $\\textbf {S}$ is very similar to to learning the $V$ -list in Ailon et al. [2].", "We repeat the construction and proof for convenience: take the union of the first $k = \\log n$ inputs $P_1$ , $P_2$ , $\\ldots $ , $P_k$ , and sort those points by $x$ -coordinates.", "This gives a list $x_0,x_1,\\ldots ,x_{nk-1}$ .", "Take the $n$ values $x_0,x_k,x_{2k},\\ldots ,x_{(n-1)k}$ .", "They define the boundaries for $\\textbf {S}$ .", "We recall a useful and well-known fact [2].", "Claim 5.1 Let $Z = \\sum _i Z_i$ be a sum of nonnegative random variables such that $Z_i = O(1)$ for all $i$ , $\\hbox{\\bf E}[Z] = O(1)$ , and for all $i,j$ , $\\hbox{\\bf E}[Z_iZ_j] = \\hbox{\\bf E}[Z_i]\\hbox{\\bf E}[Z_j]$ .", "Then $\\hbox{\\bf E}[Z^2] = O(1)$ .", "Now let $\\lambda $ be a leaf slab in $\\textbf {S}$ .", "Recall that we denote by $X_\\lambda $ the number of points of a random input $P$ that end up in $\\lambda $ .", "Using [clm:indicator-square]Claim clm:indicator-square, we quickly obtain the following lemma.", "Lemma 5.2 With probability $1-n^{-3}$ over the construction of $\\textbf {S}$ , we have $\\hbox{\\bf E}[X_\\lambda ^2] = O(1)$ for all leaf slabs $\\lambda \\in \\textbf {S}$ .", "Consider two values $x_i$ , $x_j$ from the original list.", "Note that all the other $kn - 2$ values are independent of these two points.", "For every $r \\notin \\lbrace i,j\\rbrace $ , let $Y^{(r)}_t$ be the indicator random variable for $x_r \\in t :=[x_i, x_j)$ .", "Let $Y_t = \\sum _r Y^{(r)}_t$ .", "Since the $Y^{(r)}_t$ 's are independent, by Chernoff's bound [4], for any $\\beta \\in (0,1]$ , $\\Pr [Y_t \\le (1-\\beta )\\hbox{\\bf E}[Y_t]] \\le \\exp (-\\beta ^2\\hbox{\\bf E}[Y_t]/2).$ With probability at least $1 - n^{-5}$ , if $\\hbox{\\bf E}[Y_t] > 12\\log n$ , then $Y_t > \\log n$ .", "By applying the same argument for any pair $x_i, x_j$ and taking a union bound over all pairs, with probability at least $1 - n^{-3}$ the following holds: for any pair $t$ , if $Y_t \\le \\log n$ , then $\\hbox{\\bf E}[Y_t] \\le 12\\log n$ .", "For any leaf slab $\\lambda = [x_{ak},x_{(a+1)k}]$ , we have $Y_\\lambda \\le \\log n$ .", "Let $X^{(i)}_\\lambda $ be the indicator random variable for the event that $x_i \\sim \\mathcal {D}_i$ lies in $\\lambda $ , so that $X_\\lambda = \\sum _i X^{(i)}_\\lambda $ .", "Since $\\hbox{\\bf E}[Y_\\lambda ] \\ge (\\log n - 2) \\hbox{\\bf E}[X_\\lambda ]$ , we get $\\hbox{\\bf E}[X_\\lambda ] = O(1)$ .", "By independence of the ${\\mathcal {D}}_i$ 's, for all $i,j$ , $\\hbox{\\bf E}\\bigl [X^{(i)}_\\lambda X^{(j)}_\\lambda \\bigr ] =\\hbox{\\bf E}\\bigl [X^{(i)}_\\lambda \\bigr ] \\hbox{\\bf E}\\bigl [X^{(j)}_\\lambda \\bigr ]$ , so $\\hbox{\\bf E}[X^2_\\lambda ] = O(1)$ , by [clm:indicator-square]Claim clm:indicator-square.", "[lem:slabstruct]Lemma lem:slabstruct follows immediately from [lem:leaf-tail]Lemma lem:leaf-tail and the fact that sorting the $k$ inputs $P_1$ , $P_2$ , $\\ldots $ , $P_k$ takes $O(n \\log ^2 n)$ time.", "After the leaf slabs have been determined, the search trees $T_i$ can be found using essentially the same techniques as before [2].", "The main idea is to use $n^\\varepsilon \\log n$ rounds to find the first $\\varepsilon \\log n$ levels of $T_i$ , and to use a balanced search tree for searches that need to proceed to a deeper level.", "This only costs a factor of $\\varepsilon ^{-1}$ .", "We restate [lem:tree]Lemma lem:tree for convenience.", "Lemma 5.3 Let $\\varepsilon > 0$ be a fixed parameter.", "In $O(n^{\\varepsilon })$ rounds and $O(n^{1+\\varepsilon })$ time, we can construct search trees $T_1$ , $T_2$ , $\\ldots $ , $T_n$ over $\\textbf {S}$ such that the following holds.", "(i) the trees can be totally represented in $O(n^{1+\\varepsilon })$ space; (ii) probability $1-n^{-3}$ over the construction of the $T_i$ s: every $T_i$ is $O(1/\\varepsilon )$ -optimal for restricted searches over $\\mathcal {D}_i$ .", "Let $\\delta > 0$ be some sufficiently small constant and $c$ be sufficiently large .", "For $k = c\\delta ^{-2}n^{\\varepsilon }\\log n$ rounds and each $p_i$ , we record the leaf slab of $\\textbf {S}$ that contains it.", "We break the proof into smaller claims.", "Claim 5.4 Using $k$ inputs, we can compute estimates $\\hat{q}(i,S)$ for each index $i$ and slab $S$ .", "The following guarantee holds (for all $i$ and $S$ ) with probability $>1 - 1/n^3$ over the choice of the $k$ inputs.", "If at least $5\\log n$ instances of $p_i$ fell in $S$ , then $\\hat{q}(i,S) \\in [(1-\\delta )q(i,S),(1+\\delta )q(i,S)]$We remind the reader that this the probability that $p_i \\in S$ .. For a slab $S$ , let $N(S)$ be the number of times $p_i$ was in $S$ , and let $\\hat{q}(i,S) = N(S)/k$ be the empirical probability for this event ($\\hat{q}(i,S)$ is an estimate of $q(i,S)$ ).", "Fix a slab $S$ .", "If $q(i,S) \\le 1/2n^{\\varepsilon }$ , then by a Chernoff bound we get $\\Pr [N(S) \\ge 5\\log n \\ge 10kq(i,S)] \\le 2^{-5\\log n} = n^{-5}$ .", "Furthermore, if $q(i,S) \\ge 1/2n^{\\varepsilon }$ , then $q(i,S)k \\ge (c/2\\delta ^2)\\log n$ and $\\Pr [N(S) \\le (1-\\delta )q(i,S)k] \\le \\exp (-q(i,S)\\delta ^2k/4) \\le n^{-5}$ as well as $\\Pr [N(S) \\ge (1+\\delta )q(i,S)k] \\le \\exp (-\\delta ^2q(i,S)k/4) \\le n^{-5}$ .", "Thus, by taking a union bound, we get that with probability at least $1-n^{-3}$ for any slab $S$ , if $N(S) \\ge 5\\log n$ , then $q(i,S) \\ge n^{-\\varepsilon }/2$ and hence $\\hat{q}(i,S) \\in [(1-\\delta )q(i,S),(1+\\delta )q(i,S)]$ .", "We will henceforth assume that this claims holds for all $i$ and $S$ .", "Based on the values $\\hat{q}(i,S)$ , we construct the search trees.", "The tree $T_i$ is constructed recursively.", "We will first create a partial search tree, where some searches may end in non-leaf slabs (or, in other words, leaves of the tree may not be leaf slabs).", "The root is the just the largest slab.", "Given a slab $S$ , we describe how the create the sub-tree of $T_i$ rooted at $S$ .", "If $N(S) < 5\\log n$ , then we make $S$ a leaf.", "Otherwise, we pick a leaf slab $\\lambda $ such that for the slab $S_l$ consisting of all leaf slabs (strictly) to the left of $\\lambda $ and the slab $S_r$ consisting of all leaf slabs (strictly) to the right of $\\lambda $ we have $\\hat{q}(i,S_l) \\le (2/3)\\hat{q}(i,S)$ and $\\hat{q}(i,S_r) \\le (2/3)\\hat{q}(i,S)$ .", "We make $\\lambda $ a leaf child of $S$ .", "Then we recursively create trees for $S_l$ and $S_r$ and attach them as children to $S$ .", "For any internal node of the tree $S$ , we have $q(i,S) \\ge n^\\varepsilon /2$ , and hence the depth is at most $O(\\varepsilon \\log n)$ .", "Furthermore, this partial tree is $\\beta $ -reducing (for some constant $\\beta $ ).", "The partial tree $T_i$ is extended to a complete tree in a simple way.", "From each $T_i$ -leaf that is not a leaf slab, we perform a basic binary search for the leaf slab.", "This yields a tree $T_i$ of depth at most $(1+O(\\varepsilon ))\\log n$ .", "Note that we only need to store the partial $T_i$ tree, and hence the total space is $O(n^{1+\\varepsilon })$ .", "Let us construct, as a thought experiment, a related tree $T^{\\prime }_i$ .", "Start with the partial $T_i$ .", "For every leaf that is not a leaf slab, extend it downward using the true probabilities $q(i,S)$ .", "In other words, let us construct the subtree rooted at a new node $S$ in the following manner.", "We pick a leaf slab $\\lambda $ such that $q(i,S_l) \\le (2/3)q(i,S)$ and $q(i,S_r) \\le (2/3)q(i,S)$ (where $S_l$ and $S_r$ are as defined above).", "This ensures that $T^{\\prime }_i$ is $\\beta $ -reducing.", "By [lem:search-time]Lemma lem:search-time, $T^{\\prime }_i$ is $O(1)$ -optimal for restricted searches over $\\mathcal {D}_i$ (we absorb the $\\beta $ into $O(1)$ for convenience).", "Claim 5.5 The tree $T_i$ is $O(1/\\varepsilon )$ -optimal for restricted searches.", "Fix a slab $S$ and an $S$ -restricted distribution $\\mathcal {D}_S$ .", "Let $q^{\\prime }(i,\\lambda )$ (for each leaf slab $\\lambda $ ) be the series of values defining $\\mathcal {D}_S$ .", "Note that $q^{\\prime }(i,S) \\le q(i,S)$ .", "Suppose $q^{\\prime }(i,S) \\le n^{-\\varepsilon /2}$ .", "Then $-\\log q^{\\prime }(i,S) \\ge \\varepsilon (\\log n)/2$ .", "Since any search in $T_i$ takes at most $(1+O(\\varepsilon ))\\log n$ steps, the search time is at most $O(\\varepsilon ^{-1}(-\\log q^{\\prime }(i,S) + 1))$ .", "Suppose $q^{\\prime }(i,S) > n^{-\\varepsilon /2}$ .", "Consider a single search for some $p_i$ .", "We will classify this search based on the leaf of the partial tree that is encountered.", "By the construction of $T_i$ , any leaf $S^{\\prime }$ is either a leaf slab or has the property that $q(i,S^{\\prime }) \\le n^{-\\varepsilon }/2$ .", "The search is of Type 1 if the leaf of the partial tree actually represents a leaf slab (and hence the search terminates).", "The search is of Type 2 (resp.", "Type 3) if the leaf of the partial tree is a slab $S$ is an internal node of $T_i$ and the depth is at least (resp.", "less than) $\\varepsilon (\\log n)/3$ .", "When the search is of Type 1, it is identical in both $T_i$ and $T^{\\prime }_i$ .", "When the search is of Type 2, it takes at $\\varepsilon (\\log n)/3$ in $T^{\\prime }_i$ and at most (trivially) $(1+O(\\varepsilon ))(\\log n)$ in $T_i$ .", "The total number of leaves (that are not leaf slabs) of the partial tree at depth less than $\\varepsilon (\\log n)/3$ is at most $n^{\\varepsilon /3}$ .", "The total probability mass of $\\mathcal {D}_i$ inside such leaves is at most $n^{\\varepsilon /3}\\times n^{-\\varepsilon }/2 < n^{-2\\varepsilon /3}$ .", "Since $q^{\\prime }(i,S) > n^{-\\varepsilon /2}$ , in the restricted distribution $\\mathcal {D}_S$ , the probability of a Type 3 search is at most $n^{-\\varepsilon /6}$ .", "Choose a random $p \\sim \\mathcal {D}_S$ .", "Let $\\mathcal {E}$ denote the event that a Type 3 search occurs.", "Furthermore, let $X_p$ denote the depth of the search in $T_i$ and $X^{\\prime }_p$ denote the depth in $T^{\\prime }_i$ .", "When $\\mathcal {E}$ does not occur, we have argued that $X_p \\le O(X^{\\prime }_p/\\varepsilon )$ .", "Also, $\\Pr (\\mathcal {E}) \\le n^{-\\varepsilon /6}$ .", "The expected search time is just $\\hbox{\\bf E}[X_p]$ .", "By Bayes' rule, $\\hbox{\\bf E}[X_p] & = \\Pr (\\overline{\\mathcal {E}}) \\hbox{\\bf E}_{\\overline{\\mathcal {E}}}[X_p] + \\Pr (\\mathcal {E})\\hbox{\\bf E}_{\\mathcal {E}}[X_p] \\\\& \\le O(\\varepsilon ^{-1}\\hbox{\\bf E}_{\\overline{\\mathcal {E}}}[X^{\\prime }_p]) + n^{-\\varepsilon /6}(1+O(\\varepsilon ))\\log n \\\\\\hbox{\\bf E}[X^{\\prime }_p] & = \\Pr (\\overline{\\mathcal {E}}) \\hbox{\\bf E}_{\\overline{\\mathcal {E}}}[X^{\\prime }_p] + \\Pr (\\mathcal {E})\\hbox{\\bf E}_{\\mathcal {E}}[X_p] \\\\\\Longrightarrow \\quad &\\hbox{\\bf E}_{\\overline{\\mathcal {E}}}[X^{\\prime }_p] \\le \\hbox{\\bf E}[X^{\\prime }_p]/\\Pr (\\overline{\\mathcal {E}}) \\le 2\\hbox{\\bf E}[X^{\\prime }_p]$ Combining, the expected search time is $O(\\varepsilon ^{-1}(\\hbox{\\bf E}[X^{\\prime }_p] + 1))$ .", "Since $T^{\\prime }_i$ is $O(1)$ -optimal for restricted searches, $T_i$ is $O(\\varepsilon ^{-1})$ -optimal.", "Acknowledgments C. Seshadhri was funded by the Early-Career LDRD program at Sandia National Laboratories.", "Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.", "We would like to thank Eden Chlamtáč for suggesting a simple proof for [clm:YvBound]Claim clm:YvBound.", "Restricted searches Lemma 7.1 Given an interval $S$ in $\\textbf {U}$ , let $\\mathcal {F}_S$ be an $S$ -restricted distribution of $\\mathcal {F}$ .", "Let $T$ be a $\\mu $ -reducing search tree for $\\mathcal {F}$ .", "Conditioned on $j$ drawn from $\\mathcal {F}_S$ , the expected time of an $S$ -restricted search in $T$ for $j$ is at most $(b/\\log (1/\\mu ))(-\\log q^{\\prime }_S + 1)$ (for some absolute constant $b$ ).", "Now that we may assume that we are comparing against an entropy sensitive comparison tree, we need to think about how to make our searches entropy-sensitive.", "For this we proceed as follows.", "By [lem:slabstruct]Lemma lem:slabstruct, we have a vertical slab structure $\\textbf {S}$ such that each leaf slab contains only constantly many points in expectation.", "Now, for each distribution $\\mathcal {D}_i$ , we construct an optimal search tree $T_i$ for the leaf slabs of $\\textbf {S}$ .", "The recursion continues until $S_l$ or $S_r$ are empty.", "The search in $T_i$ proceeds in the obvious way.", "To find the leaf slab containing $p_i$ , we begin in at the root and check whether $p_i$ is contained in the corresponding leaf slab.", "If yes, the search stop.", "Otherwise, we branch to the appropriate child and continue.", "Each node in $T_i$ corresponds to a slab in $\\textbf {S}$ , and it is easily seen that if a node has depth $d$ , then $p_i$ is contained in the corresponding slab with probability at most $2^{-d}$ .", "From this, it quickly follows that $T_i$ is an asymptotically optimal search tree for $\\mathcal {D}_i$ .", "However, below we require a stronger result.", "Namely, we need a technical lemma showing how an optimal search tree for some distribution $\\mathcal {F}$ is also useful for some conditional distributions.", "Let $\\textbf {U}$ be an ordered set and $\\mathcal {F}$ be a distribution over $\\textbf {U}$ .", "For any element $j \\in \\textbf {U}$ , we let $p_j$ denote the probability of $j$ in $\\mathcal {F}$ .", "For any interval $S$ of $\\textbf {U}$ , the total probability of $S$ is $p_S$ .", "Let $T$ be a search tree over $\\textbf {U}$ with the following properties.", "For any internal node $S$ and a non-leaf child $S^{\\prime }$ , $p_{S^{\\prime }} \\le \\mu p_S$ .", "As a result, if $S$ has depth $k$ , then $p_S \\le \\mu ^k$ .", "Every internal node of $T$ has at most 2 internal children and at most 2 children that are leaves.", "Definition 7.2 Given an distribution $\\mathcal {F}$ and interval $S$ , an $S$ -restricted distribution $\\mathcal {F}_S$ is a conditional distribution of $\\mathcal {F}$ such that $i$ chosen from $\\mathcal {F}_S$ always falls in $S$ .", "For any $S$ -restriction $\\mathcal {F}_S$ of $\\mathcal {F}$ , there exist values $p^{\\prime }_j$ with the following properties.", "For each $j \\in S$ , $p^{\\prime }_j \\le p_j$ .", "For every other $j$ , $p^{\\prime }_j = 0$ .", "The probability of element $j$ in $\\mathcal {F}_S$ is $p^{\\prime }_j / \\sum _r p^{\\prime }_r$ .", "Henceforth, we will use the primed values to denote the probabilities in $\\mathcal {F}_S$ .", "For interval $R$ , we set $p^{\\prime }_R = \\sum _{r \\in R} p^{\\prime }_r$ .", "Suppose we perform a search for $j \\in S$ .", "This search is called $S$ -restricted if it terminates once we locate $j$ in any interval contained in $S$ .", "Lemma 7.3 Given an interval $S$ in $\\textbf {U}$ , let $\\mathcal {F}_S$ be an $S$ -restricted distribution.", "Conditioned on $j$ drawn from $\\mathcal {F}_S$ , the expected time of an $S$ -restricted search in $T$ for $j$ is $O(-\\log p^{\\prime }_S + 1)$ .", "We bound the number of visited nodes in an $S$ -restricted search.", "We will prove, by induction on the distance from a leaf, that for all visited nodes $V$ with $p_V \\le 1/2$ , the expected number of visited nodes below $V$ is $c_1 + c\\log (p_V/p^{\\prime }_V),$ for constants $c,c_1$ .", "This bound clearly holds for leaves.", "Moreover, since for $V$ at depth $k$ , $p_V \\le \\mu ^k$ , we have $p_V \\le 1/2$ for all but the root and at most $1/\\log (1/\\mu )$ nodes below it on the search path.", "We now examine all possible paths down $T$ that an $S$ -restricted search can lead to.", "It will be helpful to consider the possible ways that $S$ can intersect the nodes (intervals) that are visited in a search.", "Say that the intersection $S \\cap V$ of $S$ with interval $V$ is trivial if it is either empty, $S$ , or $V$ .", "Say that it is anchored if it shares at least one boundary line with $S$ .", "Suppose $S \\cap V = V$ .", "Then the search will terminate at $V$ , since we have certified that $j \\in S$ .", "Suppose $S \\cap V = S$ , so $S$ is contained in $V$ .", "There can be at most one child of $V$ that contains $S$ .", "If such a child exists, then the search will simply continue to this child.", "If not, then all possible children (to which the search can proceed to) are anchored.", "The search can possibly continue to any child, at most two of which are internal nodes.", "Suppose $V$ is anchored.", "Then at most one child of $V$ can be anchored with $S$ .", "Any other child that intersects $S$ must be contained in it.", "Refer to Figure REF .", "Figure: (α\\alpha ) The intersections S∩VS \\cap V in (i)-(iii) are trivial, theintersections in (iii) and (iv) are anchored; (β\\beta ) every node of T i T_ihas at most one non-trivial child, except for RR.Consider the set of all possible nodes that can be visited by an $S$ -restricted search (remove all nodes that are terminal, i.e., completely contained in $S$ ).", "These form a set of paths, that form some subtree of $S$ .", "In this subtree, there is only one possible node that has two children.", "This comes from some node $R$ that contains $S$ and has two anchored (non-leaf) children.", "Every other node of this subtree has a single child.", "Again, refer to Figure REF .", "From the above, it follows that for all visited nodes $V$ with $V\\ne R$ , there is at most one child $W$ whose intersection with $S$ is neither empty nor $W$ .", "Let $\\textrm {vis}(V)$ be the expected number of nodes visited below $V$ , conditioned on $V$ being visited.", "We have $\\textrm {vis}(V) \\le 1 + \\textrm {vis}(W)p^{\\prime }_W/p^{\\prime }_V$ , using the fact that when a search for $j$ shows that it is contained in a node contained in $S$ , the $S$ -restricted search is complete.", "Claim 7.4 For $V,W$ as above, with $p_V\\le 1/2$ , if $\\textrm {vis}(W)\\le c_1 + c\\log (p_W/p^{\\prime }_W)$ , then for $c\\ge c_1/\\log (1/\\mu )$ , with $\\mu \\in (0,1)$ , $ \\textrm {vis}(V) \\le 1 + c\\log (p_V/p^{\\prime }_V).$ By hypothesis, using $p_W\\le \\mu p_V$ , and letting $\\beta := p^{\\prime }_W/p^{\\prime }_V\\le 1$ , $\\textrm {vis}(V)$ is no more than $1 + (c_1 + c\\log (p_W/p^{\\prime }_W))p^{\\prime }_W/p^{\\prime }_V\\le 1 + (c_1 + c\\log (p_V/p^{\\prime }_W) + c\\log (\\mu ))\\beta \\\\= 1 + c_1\\beta + c\\log (p_V)\\beta + c\\log (1/p^{\\prime }_W)\\beta + c\\log (\\mu )\\beta .$ The function $x\\log (1/x)$ is increasing in the range $x \\in (0,1/2)$ .", "Hence, $p^{\\prime }_W\\log (1/p^{\\prime }_W)\\le p^{\\prime }_V\\log (1/p^{\\prime }_V)$ for $p^{\\prime }_V\\le p_V\\le 1/2$ .", "Since $\\beta \\le 1$ , we have $\\textrm {vis}(V) \\le 1 + c_1\\beta + c\\log (p_V)+ c\\log (1/p^{\\prime }_V) + c\\log (\\mu )\\beta \\\\= 1 + c\\log (p_V/p^{\\prime }_V) + \\beta (c_1 + c\\log (\\mu ))\\le 1 + c\\log (p_V/p^{\\prime }_V),$ for $c \\ge c_1/\\log (1/\\mu )$ .", "Only a slightly weaker statement can be made for the node $R$ having two nontrivial intersections at child nodes $R_1$ and $R_2$ .", "Claim 7.5 For $R,R_1,R_2$ as above, if $\\textrm {vis}(R_i)\\le c_1 + c\\log (p_{R_i}/p^{\\prime }_{R_i})$ , for $i=1,2$ , then for $c\\ge c_1/\\log (1/\\mu )$ , $\\textrm {vis}(R) \\le 1 + c\\log (p_R/p^{\\prime }_R) + c.$ We have $\\textrm {vis}(R) \\le 1 + \\textrm {vis}(R_1)p^{\\prime }_{R_1}/p^{\\prime }_R + \\textrm {vis}(R_2)p^{\\prime }_{R_2}/p^{\\prime }_R.$ Let $\\beta := (p^{\\prime }_{R_1} + p^{\\prime }_{R_2})/p^{\\prime }_R$ .", "With the given bounds for $\\textrm {vis}(R_i)$ , then using $p_{R_i}\\le \\mu p_R$ , $\\textrm {vis}(R)$ is bounded by $1 + \\sum _{i=1,2}[c_1 + c\\log (p_{R_i}/p^{\\prime }_{R_i})]p^{\\prime }_{R_i}/p^{\\prime }_R \\\\\\le 1 + c_1\\beta + c\\beta \\log (\\mu ) + c\\beta \\log (p_R)+ c\\sum _{i=1,2} (p^{\\prime }_{R_i}/p^{\\prime }_R)\\log (1/p^{\\prime }_{R_i}).$ The sum takes its maximum value when each $p^{\\prime }_{R_i} = p^{\\prime }_R/2$ , yielding $\\textrm {vis}(R)\\le 1 + c_1\\beta + c\\beta \\log (\\mu ) + c\\beta \\log (p_R) + c\\beta \\log (2/p^{\\prime }_R)\\\\\\le 1 + c\\log (p_R/p^{\\prime }_R) + \\beta (c_1 + c\\log (\\mu )) + c\\log (2)\\le 1 + c\\log (p_R/p^{\\prime }_R) + c\\log (2),$ for $c\\ge c_1/\\log (1/\\mu )$ , as in (REF ), except for the addition of $c\\log 2 = c$ .", "Now to complete the proof of [lem:search-time]Lemma lem:search-time.", "For the visited nodes below $R$ , we may inductively take $c_1 = 1$ and $c=1/\\log (1/\\mu )$ , using [clm:WA]Claim clm:WA.", "The hypothesis of [clm:WR]Claim clm:WR then holds for $R$ .", "For the visited node just above $R$ , we may apply [clm:WA]Claim clm:WA with $c_1 = 1 + 1/\\log (1/\\mu )$ and $c\\ge c_1/\\log (1/\\mu )$ .", "The result is that for the node $V$ just above $R$ , $\\textrm {vis}(V)\\le 1 + c\\log (p_1/p^{\\prime }_V)$ .", "This bound holds then inductively (with the given value of $c$ ) for nodes further up the tree, at least up until the $1+1/\\log (1/\\mu )$ top nodes.", "For the root $Q$ , note that $p^{\\prime }_Q = p^{\\prime }_S$ .", "Thus the expected number of visited nodes below $Q$ is at most $1/\\log (1/\\mu ) + 1 + c\\log (p_Q/p^{\\prime }_Q)= O(1 - \\log (p^{\\prime }_S)),$ as desired." ], [ "Data structures obtained during the learning phase", "Learning the vertical slab structure $\\textbf {S}$ is very similar to to learning the $V$ -list in Ailon et al. [2].", "We repeat the construction and proof for convenience: take the union of the first $k = \\log n$ inputs $P_1$ , $P_2$ , $\\ldots $ , $P_k$ , and sort those points by $x$ -coordinates.", "This gives a list $x_0,x_1,\\ldots ,x_{nk-1}$ .", "Take the $n$ values $x_0,x_k,x_{2k},\\ldots ,x_{(n-1)k}$ .", "They define the boundaries for $\\textbf {S}$ .", "We recall a useful and well-known fact [2].", "Claim 5.1 Let $Z = \\sum _i Z_i$ be a sum of nonnegative random variables such that $Z_i = O(1)$ for all $i$ , $\\hbox{\\bf E}[Z] = O(1)$ , and for all $i,j$ , $\\hbox{\\bf E}[Z_iZ_j] = \\hbox{\\bf E}[Z_i]\\hbox{\\bf E}[Z_j]$ .", "Then $\\hbox{\\bf E}[Z^2] = O(1)$ .", "Now let $\\lambda $ be a leaf slab in $\\textbf {S}$ .", "Recall that we denote by $X_\\lambda $ the number of points of a random input $P$ that end up in $\\lambda $ .", "Using [clm:indicator-square]Claim clm:indicator-square, we quickly obtain the following lemma.", "Lemma 5.2 With probability $1-n^{-3}$ over the construction of $\\textbf {S}$ , we have $\\hbox{\\bf E}[X_\\lambda ^2] = O(1)$ for all leaf slabs $\\lambda \\in \\textbf {S}$ .", "Consider two values $x_i$ , $x_j$ from the original list.", "Note that all the other $kn - 2$ values are independent of these two points.", "For every $r \\notin \\lbrace i,j\\rbrace $ , let $Y^{(r)}_t$ be the indicator random variable for $x_r \\in t :=[x_i, x_j)$ .", "Let $Y_t = \\sum _r Y^{(r)}_t$ .", "Since the $Y^{(r)}_t$ 's are independent, by Chernoff's bound [4], for any $\\beta \\in (0,1]$ , $\\Pr [Y_t \\le (1-\\beta )\\hbox{\\bf E}[Y_t]] \\le \\exp (-\\beta ^2\\hbox{\\bf E}[Y_t]/2).$ With probability at least $1 - n^{-5}$ , if $\\hbox{\\bf E}[Y_t] > 12\\log n$ , then $Y_t > \\log n$ .", "By applying the same argument for any pair $x_i, x_j$ and taking a union bound over all pairs, with probability at least $1 - n^{-3}$ the following holds: for any pair $t$ , if $Y_t \\le \\log n$ , then $\\hbox{\\bf E}[Y_t] \\le 12\\log n$ .", "For any leaf slab $\\lambda = [x_{ak},x_{(a+1)k}]$ , we have $Y_\\lambda \\le \\log n$ .", "Let $X^{(i)}_\\lambda $ be the indicator random variable for the event that $x_i \\sim \\mathcal {D}_i$ lies in $\\lambda $ , so that $X_\\lambda = \\sum _i X^{(i)}_\\lambda $ .", "Since $\\hbox{\\bf E}[Y_\\lambda ] \\ge (\\log n - 2) \\hbox{\\bf E}[X_\\lambda ]$ , we get $\\hbox{\\bf E}[X_\\lambda ] = O(1)$ .", "By independence of the ${\\mathcal {D}}_i$ 's, for all $i,j$ , $\\hbox{\\bf E}\\bigl [X^{(i)}_\\lambda X^{(j)}_\\lambda \\bigr ] =\\hbox{\\bf E}\\bigl [X^{(i)}_\\lambda \\bigr ] \\hbox{\\bf E}\\bigl [X^{(j)}_\\lambda \\bigr ]$ , so $\\hbox{\\bf E}[X^2_\\lambda ] = O(1)$ , by [clm:indicator-square]Claim clm:indicator-square.", "[lem:slabstruct]Lemma lem:slabstruct follows immediately from [lem:leaf-tail]Lemma lem:leaf-tail and the fact that sorting the $k$ inputs $P_1$ , $P_2$ , $\\ldots $ , $P_k$ takes $O(n \\log ^2 n)$ time.", "After the leaf slabs have been determined, the search trees $T_i$ can be found using essentially the same techniques as before [2].", "The main idea is to use $n^\\varepsilon \\log n$ rounds to find the first $\\varepsilon \\log n$ levels of $T_i$ , and to use a balanced search tree for searches that need to proceed to a deeper level.", "This only costs a factor of $\\varepsilon ^{-1}$ .", "We restate [lem:tree]Lemma lem:tree for convenience.", "Lemma 5.3 Let $\\varepsilon > 0$ be a fixed parameter.", "In $O(n^{\\varepsilon })$ rounds and $O(n^{1+\\varepsilon })$ time, we can construct search trees $T_1$ , $T_2$ , $\\ldots $ , $T_n$ over $\\textbf {S}$ such that the following holds.", "(i) the trees can be totally represented in $O(n^{1+\\varepsilon })$ space; (ii) probability $1-n^{-3}$ over the construction of the $T_i$ s: every $T_i$ is $O(1/\\varepsilon )$ -optimal for restricted searches over $\\mathcal {D}_i$ .", "Let $\\delta > 0$ be some sufficiently small constant and $c$ be sufficiently large .", "For $k = c\\delta ^{-2}n^{\\varepsilon }\\log n$ rounds and each $p_i$ , we record the leaf slab of $\\textbf {S}$ that contains it.", "We break the proof into smaller claims.", "Claim 5.4 Using $k$ inputs, we can compute estimates $\\hat{q}(i,S)$ for each index $i$ and slab $S$ .", "The following guarantee holds (for all $i$ and $S$ ) with probability $>1 - 1/n^3$ over the choice of the $k$ inputs.", "If at least $5\\log n$ instances of $p_i$ fell in $S$ , then $\\hat{q}(i,S) \\in [(1-\\delta )q(i,S),(1+\\delta )q(i,S)]$We remind the reader that this the probability that $p_i \\in S$ .. For a slab $S$ , let $N(S)$ be the number of times $p_i$ was in $S$ , and let $\\hat{q}(i,S) = N(S)/k$ be the empirical probability for this event ($\\hat{q}(i,S)$ is an estimate of $q(i,S)$ ).", "Fix a slab $S$ .", "If $q(i,S) \\le 1/2n^{\\varepsilon }$ , then by a Chernoff bound we get $\\Pr [N(S) \\ge 5\\log n \\ge 10kq(i,S)] \\le 2^{-5\\log n} = n^{-5}$ .", "Furthermore, if $q(i,S) \\ge 1/2n^{\\varepsilon }$ , then $q(i,S)k \\ge (c/2\\delta ^2)\\log n$ and $\\Pr [N(S) \\le (1-\\delta )q(i,S)k] \\le \\exp (-q(i,S)\\delta ^2k/4) \\le n^{-5}$ as well as $\\Pr [N(S) \\ge (1+\\delta )q(i,S)k] \\le \\exp (-\\delta ^2q(i,S)k/4) \\le n^{-5}$ .", "Thus, by taking a union bound, we get that with probability at least $1-n^{-3}$ for any slab $S$ , if $N(S) \\ge 5\\log n$ , then $q(i,S) \\ge n^{-\\varepsilon }/2$ and hence $\\hat{q}(i,S) \\in [(1-\\delta )q(i,S),(1+\\delta )q(i,S)]$ .", "We will henceforth assume that this claims holds for all $i$ and $S$ .", "Based on the values $\\hat{q}(i,S)$ , we construct the search trees.", "The tree $T_i$ is constructed recursively.", "We will first create a partial search tree, where some searches may end in non-leaf slabs (or, in other words, leaves of the tree may not be leaf slabs).", "The root is the just the largest slab.", "Given a slab $S$ , we describe how the create the sub-tree of $T_i$ rooted at $S$ .", "If $N(S) < 5\\log n$ , then we make $S$ a leaf.", "Otherwise, we pick a leaf slab $\\lambda $ such that for the slab $S_l$ consisting of all leaf slabs (strictly) to the left of $\\lambda $ and the slab $S_r$ consisting of all leaf slabs (strictly) to the right of $\\lambda $ we have $\\hat{q}(i,S_l) \\le (2/3)\\hat{q}(i,S)$ and $\\hat{q}(i,S_r) \\le (2/3)\\hat{q}(i,S)$ .", "We make $\\lambda $ a leaf child of $S$ .", "Then we recursively create trees for $S_l$ and $S_r$ and attach them as children to $S$ .", "For any internal node of the tree $S$ , we have $q(i,S) \\ge n^\\varepsilon /2$ , and hence the depth is at most $O(\\varepsilon \\log n)$ .", "Furthermore, this partial tree is $\\beta $ -reducing (for some constant $\\beta $ ).", "The partial tree $T_i$ is extended to a complete tree in a simple way.", "From each $T_i$ -leaf that is not a leaf slab, we perform a basic binary search for the leaf slab.", "This yields a tree $T_i$ of depth at most $(1+O(\\varepsilon ))\\log n$ .", "Note that we only need to store the partial $T_i$ tree, and hence the total space is $O(n^{1+\\varepsilon })$ .", "Let us construct, as a thought experiment, a related tree $T^{\\prime }_i$ .", "Start with the partial $T_i$ .", "For every leaf that is not a leaf slab, extend it downward using the true probabilities $q(i,S)$ .", "In other words, let us construct the subtree rooted at a new node $S$ in the following manner.", "We pick a leaf slab $\\lambda $ such that $q(i,S_l) \\le (2/3)q(i,S)$ and $q(i,S_r) \\le (2/3)q(i,S)$ (where $S_l$ and $S_r$ are as defined above).", "This ensures that $T^{\\prime }_i$ is $\\beta $ -reducing.", "By [lem:search-time]Lemma lem:search-time, $T^{\\prime }_i$ is $O(1)$ -optimal for restricted searches over $\\mathcal {D}_i$ (we absorb the $\\beta $ into $O(1)$ for convenience).", "Claim 5.5 The tree $T_i$ is $O(1/\\varepsilon )$ -optimal for restricted searches.", "Fix a slab $S$ and an $S$ -restricted distribution $\\mathcal {D}_S$ .", "Let $q^{\\prime }(i,\\lambda )$ (for each leaf slab $\\lambda $ ) be the series of values defining $\\mathcal {D}_S$ .", "Note that $q^{\\prime }(i,S) \\le q(i,S)$ .", "Suppose $q^{\\prime }(i,S) \\le n^{-\\varepsilon /2}$ .", "Then $-\\log q^{\\prime }(i,S) \\ge \\varepsilon (\\log n)/2$ .", "Since any search in $T_i$ takes at most $(1+O(\\varepsilon ))\\log n$ steps, the search time is at most $O(\\varepsilon ^{-1}(-\\log q^{\\prime }(i,S) + 1))$ .", "Suppose $q^{\\prime }(i,S) > n^{-\\varepsilon /2}$ .", "Consider a single search for some $p_i$ .", "We will classify this search based on the leaf of the partial tree that is encountered.", "By the construction of $T_i$ , any leaf $S^{\\prime }$ is either a leaf slab or has the property that $q(i,S^{\\prime }) \\le n^{-\\varepsilon }/2$ .", "The search is of Type 1 if the leaf of the partial tree actually represents a leaf slab (and hence the search terminates).", "The search is of Type 2 (resp.", "Type 3) if the leaf of the partial tree is a slab $S$ is an internal node of $T_i$ and the depth is at least (resp.", "less than) $\\varepsilon (\\log n)/3$ .", "When the search is of Type 1, it is identical in both $T_i$ and $T^{\\prime }_i$ .", "When the search is of Type 2, it takes at $\\varepsilon (\\log n)/3$ in $T^{\\prime }_i$ and at most (trivially) $(1+O(\\varepsilon ))(\\log n)$ in $T_i$ .", "The total number of leaves (that are not leaf slabs) of the partial tree at depth less than $\\varepsilon (\\log n)/3$ is at most $n^{\\varepsilon /3}$ .", "The total probability mass of $\\mathcal {D}_i$ inside such leaves is at most $n^{\\varepsilon /3}\\times n^{-\\varepsilon }/2 < n^{-2\\varepsilon /3}$ .", "Since $q^{\\prime }(i,S) > n^{-\\varepsilon /2}$ , in the restricted distribution $\\mathcal {D}_S$ , the probability of a Type 3 search is at most $n^{-\\varepsilon /6}$ .", "Choose a random $p \\sim \\mathcal {D}_S$ .", "Let $\\mathcal {E}$ denote the event that a Type 3 search occurs.", "Furthermore, let $X_p$ denote the depth of the search in $T_i$ and $X^{\\prime }_p$ denote the depth in $T^{\\prime }_i$ .", "When $\\mathcal {E}$ does not occur, we have argued that $X_p \\le O(X^{\\prime }_p/\\varepsilon )$ .", "Also, $\\Pr (\\mathcal {E}) \\le n^{-\\varepsilon /6}$ .", "The expected search time is just $\\hbox{\\bf E}[X_p]$ .", "By Bayes' rule, $\\hbox{\\bf E}[X_p] & = \\Pr (\\overline{\\mathcal {E}}) \\hbox{\\bf E}_{\\overline{\\mathcal {E}}}[X_p] + \\Pr (\\mathcal {E})\\hbox{\\bf E}_{\\mathcal {E}}[X_p] \\\\& \\le O(\\varepsilon ^{-1}\\hbox{\\bf E}_{\\overline{\\mathcal {E}}}[X^{\\prime }_p]) + n^{-\\varepsilon /6}(1+O(\\varepsilon ))\\log n \\\\\\hbox{\\bf E}[X^{\\prime }_p] & = \\Pr (\\overline{\\mathcal {E}}) \\hbox{\\bf E}_{\\overline{\\mathcal {E}}}[X^{\\prime }_p] + \\Pr (\\mathcal {E})\\hbox{\\bf E}_{\\mathcal {E}}[X_p] \\\\\\Longrightarrow \\quad &\\hbox{\\bf E}_{\\overline{\\mathcal {E}}}[X^{\\prime }_p] \\le \\hbox{\\bf E}[X^{\\prime }_p]/\\Pr (\\overline{\\mathcal {E}}) \\le 2\\hbox{\\bf E}[X^{\\prime }_p]$ Combining, the expected search time is $O(\\varepsilon ^{-1}(\\hbox{\\bf E}[X^{\\prime }_p] + 1))$ .", "Since $T^{\\prime }_i$ is $O(1)$ -optimal for restricted searches, $T_i$ is $O(\\varepsilon ^{-1})$ -optimal." ], [ "Acknowledgments", "C. Seshadhri was funded by the Early-Career LDRD program at Sandia National Laboratories.", "Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.", "We would like to thank Eden Chlamtáč for suggesting a simple proof for [clm:YvBound]Claim clm:YvBound." ], [ "Restricted searches", "Lemma 7.1 Given an interval $S$ in $\\textbf {U}$ , let $\\mathcal {F}_S$ be an $S$ -restricted distribution of $\\mathcal {F}$ .", "Let $T$ be a $\\mu $ -reducing search tree for $\\mathcal {F}$ .", "Conditioned on $j$ drawn from $\\mathcal {F}_S$ , the expected time of an $S$ -restricted search in $T$ for $j$ is at most $(b/\\log (1/\\mu ))(-\\log q^{\\prime }_S + 1)$ (for some absolute constant $b$ ).", "Now that we may assume that we are comparing against an entropy sensitive comparison tree, we need to think about how to make our searches entropy-sensitive.", "For this we proceed as follows.", "By [lem:slabstruct]Lemma lem:slabstruct, we have a vertical slab structure $\\textbf {S}$ such that each leaf slab contains only constantly many points in expectation.", "Now, for each distribution $\\mathcal {D}_i$ , we construct an optimal search tree $T_i$ for the leaf slabs of $\\textbf {S}$ .", "The recursion continues until $S_l$ or $S_r$ are empty.", "The search in $T_i$ proceeds in the obvious way.", "To find the leaf slab containing $p_i$ , we begin in at the root and check whether $p_i$ is contained in the corresponding leaf slab.", "If yes, the search stop.", "Otherwise, we branch to the appropriate child and continue.", "Each node in $T_i$ corresponds to a slab in $\\textbf {S}$ , and it is easily seen that if a node has depth $d$ , then $p_i$ is contained in the corresponding slab with probability at most $2^{-d}$ .", "From this, it quickly follows that $T_i$ is an asymptotically optimal search tree for $\\mathcal {D}_i$ .", "However, below we require a stronger result.", "Namely, we need a technical lemma showing how an optimal search tree for some distribution $\\mathcal {F}$ is also useful for some conditional distributions.", "Let $\\textbf {U}$ be an ordered set and $\\mathcal {F}$ be a distribution over $\\textbf {U}$ .", "For any element $j \\in \\textbf {U}$ , we let $p_j$ denote the probability of $j$ in $\\mathcal {F}$ .", "For any interval $S$ of $\\textbf {U}$ , the total probability of $S$ is $p_S$ .", "Let $T$ be a search tree over $\\textbf {U}$ with the following properties.", "For any internal node $S$ and a non-leaf child $S^{\\prime }$ , $p_{S^{\\prime }} \\le \\mu p_S$ .", "As a result, if $S$ has depth $k$ , then $p_S \\le \\mu ^k$ .", "Every internal node of $T$ has at most 2 internal children and at most 2 children that are leaves.", "Definition 7.2 Given an distribution $\\mathcal {F}$ and interval $S$ , an $S$ -restricted distribution $\\mathcal {F}_S$ is a conditional distribution of $\\mathcal {F}$ such that $i$ chosen from $\\mathcal {F}_S$ always falls in $S$ .", "For any $S$ -restriction $\\mathcal {F}_S$ of $\\mathcal {F}$ , there exist values $p^{\\prime }_j$ with the following properties.", "For each $j \\in S$ , $p^{\\prime }_j \\le p_j$ .", "For every other $j$ , $p^{\\prime }_j = 0$ .", "The probability of element $j$ in $\\mathcal {F}_S$ is $p^{\\prime }_j / \\sum _r p^{\\prime }_r$ .", "Henceforth, we will use the primed values to denote the probabilities in $\\mathcal {F}_S$ .", "For interval $R$ , we set $p^{\\prime }_R = \\sum _{r \\in R} p^{\\prime }_r$ .", "Suppose we perform a search for $j \\in S$ .", "This search is called $S$ -restricted if it terminates once we locate $j$ in any interval contained in $S$ .", "Lemma 7.3 Given an interval $S$ in $\\textbf {U}$ , let $\\mathcal {F}_S$ be an $S$ -restricted distribution.", "Conditioned on $j$ drawn from $\\mathcal {F}_S$ , the expected time of an $S$ -restricted search in $T$ for $j$ is $O(-\\log p^{\\prime }_S + 1)$ .", "We bound the number of visited nodes in an $S$ -restricted search.", "We will prove, by induction on the distance from a leaf, that for all visited nodes $V$ with $p_V \\le 1/2$ , the expected number of visited nodes below $V$ is $c_1 + c\\log (p_V/p^{\\prime }_V),$ for constants $c,c_1$ .", "This bound clearly holds for leaves.", "Moreover, since for $V$ at depth $k$ , $p_V \\le \\mu ^k$ , we have $p_V \\le 1/2$ for all but the root and at most $1/\\log (1/\\mu )$ nodes below it on the search path.", "We now examine all possible paths down $T$ that an $S$ -restricted search can lead to.", "It will be helpful to consider the possible ways that $S$ can intersect the nodes (intervals) that are visited in a search.", "Say that the intersection $S \\cap V$ of $S$ with interval $V$ is trivial if it is either empty, $S$ , or $V$ .", "Say that it is anchored if it shares at least one boundary line with $S$ .", "Suppose $S \\cap V = V$ .", "Then the search will terminate at $V$ , since we have certified that $j \\in S$ .", "Suppose $S \\cap V = S$ , so $S$ is contained in $V$ .", "There can be at most one child of $V$ that contains $S$ .", "If such a child exists, then the search will simply continue to this child.", "If not, then all possible children (to which the search can proceed to) are anchored.", "The search can possibly continue to any child, at most two of which are internal nodes.", "Suppose $V$ is anchored.", "Then at most one child of $V$ can be anchored with $S$ .", "Any other child that intersects $S$ must be contained in it.", "Refer to Figure REF .", "Figure: (α\\alpha ) The intersections S∩VS \\cap V in (i)-(iii) are trivial, theintersections in (iii) and (iv) are anchored; (β\\beta ) every node of T i T_ihas at most one non-trivial child, except for RR.Consider the set of all possible nodes that can be visited by an $S$ -restricted search (remove all nodes that are terminal, i.e., completely contained in $S$ ).", "These form a set of paths, that form some subtree of $S$ .", "In this subtree, there is only one possible node that has two children.", "This comes from some node $R$ that contains $S$ and has two anchored (non-leaf) children.", "Every other node of this subtree has a single child.", "Again, refer to Figure REF .", "From the above, it follows that for all visited nodes $V$ with $V\\ne R$ , there is at most one child $W$ whose intersection with $S$ is neither empty nor $W$ .", "Let $\\textrm {vis}(V)$ be the expected number of nodes visited below $V$ , conditioned on $V$ being visited.", "We have $\\textrm {vis}(V) \\le 1 + \\textrm {vis}(W)p^{\\prime }_W/p^{\\prime }_V$ , using the fact that when a search for $j$ shows that it is contained in a node contained in $S$ , the $S$ -restricted search is complete.", "Claim 7.4 For $V,W$ as above, with $p_V\\le 1/2$ , if $\\textrm {vis}(W)\\le c_1 + c\\log (p_W/p^{\\prime }_W)$ , then for $c\\ge c_1/\\log (1/\\mu )$ , with $\\mu \\in (0,1)$ , $ \\textrm {vis}(V) \\le 1 + c\\log (p_V/p^{\\prime }_V).$ By hypothesis, using $p_W\\le \\mu p_V$ , and letting $\\beta := p^{\\prime }_W/p^{\\prime }_V\\le 1$ , $\\textrm {vis}(V)$ is no more than $1 + (c_1 + c\\log (p_W/p^{\\prime }_W))p^{\\prime }_W/p^{\\prime }_V\\le 1 + (c_1 + c\\log (p_V/p^{\\prime }_W) + c\\log (\\mu ))\\beta \\\\= 1 + c_1\\beta + c\\log (p_V)\\beta + c\\log (1/p^{\\prime }_W)\\beta + c\\log (\\mu )\\beta .$ The function $x\\log (1/x)$ is increasing in the range $x \\in (0,1/2)$ .", "Hence, $p^{\\prime }_W\\log (1/p^{\\prime }_W)\\le p^{\\prime }_V\\log (1/p^{\\prime }_V)$ for $p^{\\prime }_V\\le p_V\\le 1/2$ .", "Since $\\beta \\le 1$ , we have $\\textrm {vis}(V) \\le 1 + c_1\\beta + c\\log (p_V)+ c\\log (1/p^{\\prime }_V) + c\\log (\\mu )\\beta \\\\= 1 + c\\log (p_V/p^{\\prime }_V) + \\beta (c_1 + c\\log (\\mu ))\\le 1 + c\\log (p_V/p^{\\prime }_V),$ for $c \\ge c_1/\\log (1/\\mu )$ .", "Only a slightly weaker statement can be made for the node $R$ having two nontrivial intersections at child nodes $R_1$ and $R_2$ .", "Claim 7.5 For $R,R_1,R_2$ as above, if $\\textrm {vis}(R_i)\\le c_1 + c\\log (p_{R_i}/p^{\\prime }_{R_i})$ , for $i=1,2$ , then for $c\\ge c_1/\\log (1/\\mu )$ , $\\textrm {vis}(R) \\le 1 + c\\log (p_R/p^{\\prime }_R) + c.$ We have $\\textrm {vis}(R) \\le 1 + \\textrm {vis}(R_1)p^{\\prime }_{R_1}/p^{\\prime }_R + \\textrm {vis}(R_2)p^{\\prime }_{R_2}/p^{\\prime }_R.$ Let $\\beta := (p^{\\prime }_{R_1} + p^{\\prime }_{R_2})/p^{\\prime }_R$ .", "With the given bounds for $\\textrm {vis}(R_i)$ , then using $p_{R_i}\\le \\mu p_R$ , $\\textrm {vis}(R)$ is bounded by $1 + \\sum _{i=1,2}[c_1 + c\\log (p_{R_i}/p^{\\prime }_{R_i})]p^{\\prime }_{R_i}/p^{\\prime }_R \\\\\\le 1 + c_1\\beta + c\\beta \\log (\\mu ) + c\\beta \\log (p_R)+ c\\sum _{i=1,2} (p^{\\prime }_{R_i}/p^{\\prime }_R)\\log (1/p^{\\prime }_{R_i}).$ The sum takes its maximum value when each $p^{\\prime }_{R_i} = p^{\\prime }_R/2$ , yielding $\\textrm {vis}(R)\\le 1 + c_1\\beta + c\\beta \\log (\\mu ) + c\\beta \\log (p_R) + c\\beta \\log (2/p^{\\prime }_R)\\\\\\le 1 + c\\log (p_R/p^{\\prime }_R) + \\beta (c_1 + c\\log (\\mu )) + c\\log (2)\\le 1 + c\\log (p_R/p^{\\prime }_R) + c\\log (2),$ for $c\\ge c_1/\\log (1/\\mu )$ , as in (REF ), except for the addition of $c\\log 2 = c$ .", "Now to complete the proof of [lem:search-time]Lemma lem:search-time.", "For the visited nodes below $R$ , we may inductively take $c_1 = 1$ and $c=1/\\log (1/\\mu )$ , using [clm:WA]Claim clm:WA.", "The hypothesis of [clm:WR]Claim clm:WR then holds for $R$ .", "For the visited node just above $R$ , we may apply [clm:WA]Claim clm:WA with $c_1 = 1 + 1/\\log (1/\\mu )$ and $c\\ge c_1/\\log (1/\\mu )$ .", "The result is that for the node $V$ just above $R$ , $\\textrm {vis}(V)\\le 1 + c\\log (p_1/p^{\\prime }_V)$ .", "This bound holds then inductively (with the given value of $c$ ) for nodes further up the tree, at least up until the $1+1/\\log (1/\\mu )$ top nodes.", "For the root $Q$ , note that $p^{\\prime }_Q = p^{\\prime }_S$ .", "Thus the expected number of visited nodes below $Q$ is at most $1/\\log (1/\\mu ) + 1 + c\\log (p_Q/p^{\\prime }_Q)= O(1 - \\log (p^{\\prime }_S)),$ as desired." ] ]

[ [ "Energy spectra of vortex distributions in two-dimensional quantum\n turbulence" ], [ "Abstract We theoretically explore key concepts of two-dimensional turbulence in a homogeneous compressible superfluid described by a dissipative two-dimensional Gross-Pitaeveskii equation.", "Such a fluid supports quantized vortices that have a size characterized by the healing length $\\xi$.", "We show that for the divergence-free portion of the superfluid velocity field, the kinetic energy spectrum over wavenumber $k$ may be decomposed into an ultraviolet regime ($k\\gg \\xi^{-1}$) having a universal $k^{-3}$ scaling arising from the vortex core structure, and an infrared regime ($k\\ll\\xi^{-1}$) with a spectrum that arises purely from the configuration of the vortices.", "The Novikov power-law distribution of intervortex distances with exponent -1/3 for vortices of the same sign of circulation leads to an infrared kinetic energy spectrum with a Kolmogorov $k^{-5/3}$ power law, consistent with the existence of an inertial range.", "The presence of these $k^{-3}$ and $k^{-5/3}$ power laws, together with the constraint of continuity at the smallest configurational scale $k\\approx\\xi^{-1}$, allows us to derive a new analytical expression for the Kolmogorov constant that we test against a numerical simulation of a forced homogeneous compressible two-dimensional superfluid.", "The numerical simulation corroborates our analysis of the spectral features of the kinetic energy distribution, once we introduce the concept of a {\\em clustered fraction} consisting of the fraction of vortices that have the same sign of circulation as their nearest neighboring vortices.", "Our analysis presents a new approach to understanding two-dimensional quantum turbulence and interpreting similarities and differences with classical two-dimensional turbulence, and suggests new methods to characterize vortex turbulence in two-dimensional quantum fluids via vortex position and circulation measurements." ], [ "Introduction", "Turbulence in three-dimensional (3D) classical fluids is associated with a cascade of energy from large length scales defined by the details of an energy-forcing mechanism, to small length scales where viscous damping removes kinetic energy from the fluid.", "This range of length scales, and the range of associated wavenumbers $k$ , define the inertial range of energy flux [1].", "As shown by Kolmogorov in 1941 [2], the energy cascade corresponds to a kinetic energy spectrum that is proportional to $k^{-5/3}$ in the inertial range.", "Turbulence in a 3D fluid is also often associated with the decay of large patches of vorticity into ever smaller regions of vorticity; this Richardson cascade of vorticity provides an important visual picture of the fluid dynamics involved in 3D turbulence [3].", "Remarkably, two-dimensional (2D) incompressible classical fluids exhibit very different turbulent flow characteristics due to the existence of an additional inviscid invariant: in the absence of forcing and dissipation, the total enstrophy [4] of a 2D fluid is conserved in addition to the fluid's kinetic energy [5], [6], [7], [8], [9].", "The fluid dynamics during forced 2D turbulence are highly distinctive when compared with 3D flows: rather than decaying into smaller patches, vorticity aggregates into larger coherent rotating structures [10] (see [11] for a more detailed picture in terms of turbulent stress imposed on small-scale vortices).", "Accompanying these 2D fluid dynamics is an inverse energy cascade in which energy moves from a small forcing scale to progressively larger length scales, defining an inertial range for energy transport with energy flux in a direction opposite that of 3D turbulence.", "Eventually, energy is transported into flows characterized by length scales that are on the order the system size [12], for which dissipation may occur.", "Additionally, there is an enstrophy cascade, in which enstrophy is transported from the forcing scale to progressively smaller scales [13].", "Thus in 2D turbulence, the kinetic energy distribution contains at least these two distinctly different spectral regimes.", "Quantum turbulence [14] involves chaotic flow in a superfluid [15], [16], [17], [18], [19] and is often associated with a random vortex tangle in 3D [15].", "In general, the quantization of circulation strongly constrains the velocity fields allowed in quantum turbulence, which must be irrotational everywhere within the fluid, yet inertial ranges with $k^{-5/3}$ spectral dependence are still found in 3D quantum turbulence [20].", "In an incompressible superfluid (such as HeII), the vortex core diameter can be neglected for all practical purposes, inspiring the study of point-vortex models of superfluid dynamics.", "Such a model was used by Onsager to first predict the aggregation of vortices within inviscid 2D fluids, and was the context for his prediction of the quantization of vortex circulation in a superfluid [21].", "Despite the historical importance of this approach in stimulating advances in 2D classical turbulence [22], characteristics of 2DQT remain little known, due in part to the difficulty of achieving the necessary 2D confinement for incompressible superfluids.", "The increasing relevance of 3D turbulence concepts to dilute-gas Bose-Einstein condensate (BEC) experiments [23], [24], [25] and recent theoretical work on 2DQT [26], [27], [28], [29], [30], [19], [31] have highlighted the need for a treatment of turbulence in 2D superfluid systems that incorporates the concept of compressibility from the outset.", "Motivated by recent experimental demonstrations of the confinement needed to study 2DQT in a dilute-gas BECs [32], [33] our aim in the present paper is to present a new approach to solving some of the open problems of 2DQT in the context of such a system.", "In a BEC, the vortex core size is non-negligible, and stems from the healing length $\\xi $ , a scale of fundamental importance in BEC dynamics that is typically about two orders of magnitude smaller than the system size [34].", "Compressibility also allows for a rich array of physical phenomena in these superfluids; in particular, a vortex dipole [32] (comprised of two vortices of opposite sign of circulation) can recombine, releasing vortex energy as a burst of acoustic waves.", "The opposite process of vortex dipole generation from sound may also efficiently occur.", "Recent theoretical studies of decaying quantum turbulence in 2D BECs have shown that when the vortex dipole annihilation process is dominant it sets up a direct cascade of energy over the scales associated with the dipole decay, suggesting that this annihilation mechanism could prohibit an inverse energy cascade from occurring in a compressible superfluid [29], [30].", "Moreover, enstrophy in a quantum fluid is associated with the number of vortex cores; if vortices annihilate, then enstrophy may not be conserved, bringing into question the existence of energy and enstrophy cascades in 2DQT, and the universal nature of 2D turbulence and its correspondence with 2DQT.", "The general characteristics of 2D quantum turbulence in compressible quantum fluids, including the capacity for these systems to show an inverse energy cascade, enstrophy conservation, and vortex aggregation have thus remained largely unknown.", "However, a recent study of the formation of vortex dipoles during the breakdown of superfluid flow around an obstacle in a highly oblate BEC experimentally and numerically observed aggregation of like-sign vortices into larger-scale coherent structures [32], and found time scales over which vortex number and hence enstrophy may remain constant.", "The vortex clustering effect inhibits the dipole-decay mechanism by keeping vortices distant from antivortices (vortices of opposite circulation), and suggests that an inverse cascade might be observed under the right conditions of forcing.", "System dynamics consistent with the existence of an inverse energy cascade were indeed found in a recent study of forced 2DQT in a BEC [33].", "In this article we address 2D quantum turbulence in a compressible quantum fluid from an analytical perspective.", "We determine the kinetic energy spectra of vortex distributions in a homogeneous compressible superfluid in a quasi-exact manner via an analytical treatment of the physics of the vortex core.", "We are thus able to study the properties of vortex configurations and their resulting spectra in BEC.", "We develop a technique to sample spatially localized vortex distributions with power-law behavior over a well defined scale range.", "We are thus able to identify the conditions for an inertial range in fully polarized and neutral systems.", "A polarized cluster is sampled using a specific exponent for the vortex locations relative to the cluster center, which is size and scale dependent.", "The specific radial exponent is shown to determine the velocity distribution in the classical limit and we thus identify an expanding inertial range with a steepening velocity distribution.", "Making use of the universality of the spectral region generated by the vortex core, we identify an analytical form of the Kolmogorov constant that we test against dynamical simulations of the damped GPE.", "The derivation of the Kolmogorov constant occurs for a highly idealized vortex distribution.", "Thus the complex flows generated by real forcing require that we introduce a new parameter called the clustered fraction, and modify our ansatz to account for imperfect clustering, based on the universality of the Kolmogorov constant.", "The modified ansatz agrees well with the numerical simulations of grid turbulence, supporting our analytical identification of the Kolmogorov constant." ], [ "Background", "The starting point for much of BEC theory is the Gross-Pitaevskii equation (GPE), which provides a capable description of trapped Bose-Einstein condensates at zero temperature [34].", "Our model, outlined below, consists of a damped GPE (dGPE) description of a finite-temperature BEC which can be derived from the stochastic GPE theory [35], [36].", "In this section we develop a link between the dGPE and the classical Navier-Stokes equation, identifying a quantum viscosity arising from the damping.", "The corresponding Reynolds number is defined in direct analogy with classical fluids.", "We then state some key properties of a single quantum vortex, and define the decomposition of kinetic energy into its compressible and incompressible components." ], [ "Damped Gross-Pitaevskii theory", "The damped Gross-Pitaevskii equation of motion for the quantum fluid wave function $\\psi (\\mathbf {r},t )$ has been obtained phenomenologically [37], within ZNG theory [38], and via a microscopic reservoir theory [39], [40], [35], and we will consider it within the context of the latter framework, for which the full equation of motion is the Stochastic Projected Gross-Pitaevskii equation (SPGPE).", "The SPGPE is derived by treating all atoms with energies above an appropriately chosen energy cutoff $\\epsilon _{\\rm {cut}}$ as thermalized, leading to a grand-canonical description of the atoms below $\\epsilon _{\\rm {cut}}$ .", "A dimensionless temperature-dependent rate $\\gamma $ describes Bose-enhanced collisions between thermal reservoir atoms and atoms in the BEC.", "Neglecting the noise, we obtain the equation of motion for the condensate wave function (in the frame rotating with the chemical potential $\\mu $ ) $i\\hbar \\frac{\\partial \\psi (\\mathbf {r},t )}{\\partial t}=(i\\gamma -1)(\\mu -{\\cal L})\\psi (\\mathbf {r},t ).$ For atoms of mass $m$ in an external potential $V(\\mathbf {r},t )$ , the operator ${\\cal L}$ gives the GPE evolution: ${\\cal L}\\psi (\\mathbf {r},t )\\equiv \\left(-\\frac{\\hbar ^2}{2m}\\nabla ^2+V(\\mathbf {r},t )+g|\\psi (\\mathbf {r},t )|^2\\right)\\psi (\\mathbf {r},t ),$ where the interaction parameter is $g=4\\pi \\hbar ^2 a/m$ , for $s$ -wave scattering length $a$ .", "This equation of motion has been used extensively in previous studies of vortex dynamics [38], [39], [41], [42] and provides a capable description of dynamical BEC phenomena.", "In general the damping parameter is small ($\\gamma \\ll 1$ ), and it is typically much smaller than any other rates characterizing the evolution.", "Defining the Gross-Pitaevskii Hamiltonian $H_{\\rm C}=\\int d ^3 \\mathbf {r} \\;\\,\\left\\lbrace \\frac{\\hbar ^2}{2m}|\\nabla \\psi (\\mathbf {r},t )|^2+V(\\mathbf {r},t )|\\psi (\\mathbf {r},t )|^2+\\frac{g}{2}|\\psi (\\mathbf {r},t )|^4\\right\\rbrace ,$ and condensate atom number $N_{\\rm C}=\\int d ^3 \\mathbf {r} \\;|\\psi (\\mathbf {r},t )|^2,$ the equation of motion (REF ) evolves the grand-canonical Hamiltonian $K_{\\rm C}=H_{\\rm C}-\\mu N_{\\rm C}$ according to $\\frac{dK_{\\rm C}}{dt}=-\\frac{2\\gamma }{\\hbar }\\int d ^3 \\mathbf {r} \\;|(\\mu -{\\cal L})\\psi (\\mathbf {r},t )|^2.$ The stationary solution minimizing $K_{\\rm C}$ is the ground state satisfying $\\mu \\psi _0(\\mathbf {r})\\equiv {\\cal L}\\psi _0(\\mathbf {r})$ .", "This is a consequence of the nonlinear form of the damping in (REF ).", "The damping term arises from collisions between high energy atoms that lead to a Bose-enhanced growth of the matter wave field, with instantaneous energy determined by ${\\cal L}$ .", "The equation of motion thus describes a system coupled to a thermal reservoir in the chosen frame of reference.", "The SPGPE provides a rigorous framework for the dGPE derivation, originating from a microscopic treatment of the reservoir interaction.", "In particular, $\\gamma $ can be calculated explicitly [36] for a system with well-defined reservoir parameters $\\mu , T,$ and $\\epsilon _{\\rm {cut}}$ , i.e.", "a system close to thermal equilibrium.", "In essence it is computed via a reduced Boltzmann collision integral that accounts for all irreversible $s$ -wave interactions that can change the condensate population by interacting with the thermal cloud.", "If the thermal cloud is 3D (i.e.", "$\\beta ^{-1} \\equiv k_BT$ is greater than the potential well mode spacing in each spatial dimension) the damping takes the explicit form $\\gamma =\\gamma _0\\sum _{j=1}^\\infty \\frac{e^{\\beta \\mu (j+1)}}{e^{2\\beta j\\epsilon _{\\rm cut} }}\\Phi \\left[\\frac{e^{\\beta \\mu }}{e^{\\beta j \\epsilon _{\\rm cut} }},1,j\\right]^2,$ where $\\Phi [z,s,\\alpha ]$ is the Lerch transcendent, and $\\gamma _0=8a^2/\\lambda _{dB}^2,$ with $\\lambda _{dB}\\equiv \\sqrt{2\\pi \\hbar ^2/m k_BT}$ the thermal deBroglie wavelength.", "The dimensionless rate $\\gamma _0$ provides a useful estimate of the full damping strength when the cutoff $\\epsilon _{\\rm {cut}}$ is unknown.", "Equation (REF ) is independent of position, and valid over the region $V(\\mathbf {r},t )\\le 2\\epsilon _{\\rm cut}/3$ , provided the potential can be treated semi-classically [36].", "The summation gives Bose-enhancement corrections due to the Bose-Einstein distributed reservoir atoms, and is typically of order 1-20 in SPGPE simulations with a consistently determined energy cutoff [43].", "Typicaly $\\gamma \\sim 5\\times 10^{-4}$ in $^{87}$ Rb experiments [42], [44]." ], [ "Heuristic derivation of a quantum Reynolds number", "In this section we consider the role of dissipation within the dGPE description, and show how to recover the celebrated Navier-Stokes equation (NSE).", "In doing so we find an explicit expression for the viscosity which has a microscopic quantum origin, stemming from s-wave scattering of incoherent reservoir particles with a coherent superfluid.", "While not offering a practical reformulation (the GPE and its generalizations are capable numerical workhorses), this indicates a connection between the dGPE and the NSE in the hydrodynamic regime, allowing the identification of a parameter analogous to the kinematic viscosity of classical fluids.", "The fluid dynamics interpretation of the Gross-Pitaevskii equation is based on the Madelung transformation, which we now apply to the damped GPE (REF ), writing $\\psi ( \\mathbf {r} ,t)=\\sqrt{\\rho ( \\mathbf {r} ,t)}\\exp {[i\\Theta ( \\mathbf {r} ,t)]}$ , where $\\rho ( \\mathbf {r} ,t)$ is the number density of the superfluid (number of atoms per unit volume), and $\\Theta ( \\mathbf {r} ,t)$ is the macroscopic phase of the quantum fluid.", "The velocity is then given by $\\mathbf {v( \\mathbf {r} },t)=\\hbar \\nabla \\Theta ( \\mathbf {r} ,t)/m$ .", "The resulting equations of motion (with implicit $t$ and $\\textbf {r}$ dependence) for density and velocity are then given by $\\frac{\\partial \\rho }{\\partial t}+\\nabla \\cdot (\\rho \\mathbf {v})&=&\\frac{2\\rho \\gamma }{\\hbar } (\\mu -U_{\\rm eff}),\\\\m\\frac{\\partial \\mathbf {v}}{\\partial t}&=&-\\nabla \\left(U_{\\rm eff}-\\frac{\\hbar \\gamma }{2\\rho }\\nabla \\cdot (\\rho \\mathbf {v})\\right),$ where an effective potential $U_{\\rm eff}$ is defined as $U_{\\rm eff}( \\mathbf {r} ,t)=\\frac{m \\mathbf {v}^2}{2}+V+g\\rho -\\frac{\\hbar ^2}{2m}\\frac{\\nabla ^2\\sqrt{\\rho }}{\\sqrt{\\rho }}.$ The last term is called the quantum pressure, which is very small except where $\\rho $ changes sharply, such as near vortex cores.", "By neglecting this term in the absence of dissipation we are considering the so-called hydrodynamic regime.", "We now consider the $\\gamma $ term in () $\\frac{\\hbar \\gamma }{2}\\nabla \\left(\\frac{1}{\\rho }\\nabla \\cdot (\\rho \\mathbf {v})\\right)=\\frac{\\hbar \\gamma }{2}\\left(\\nabla (\\nabla \\cdot \\mathbf {v})+\\nabla \\frac{\\mathbf {v}\\cdot \\nabla \\rho }{\\rho }\\right).$ Note that $\\mathbf {v}\\cdot \\nabla \\rho \\equiv 0$ for an isolated quantum vortex.", "In the absence of acoustic energy this will also be a good approximation for a system of vortices provided their cores are well separated, since the density gradient of each vortex is localized to a region where the velocity is dominated by the single-vortex velocity field.", "It should thus be a reasonable approximation to neglect the second term in (REF ).", "In a superfluid the curl term in the expansion $\\nabla (\\nabla \\cdot \\mathbf {v})=\\nabla \\times (\\nabla \\times \\mathbf {v})+\\nabla ^2\\mathbf {v}$ may also be consistently neglected away from vortex cores; similarly we neglect the curl term in $\\nabla (\\mathbf {v}\\cdot \\mathbf {v})=2\\,(\\mathbf {v}\\cdot \\nabla )\\mathbf {v}+2\\,\\mathbf {v}\\times (\\nabla \\times \\mathbf {v})$ when taking the gradient of (REF ).", "We then find that () reduces to a quantum Navier-Stokes equation for the velocity field: $\\frac{\\partial \\mathbf {v}}{\\partial t}+(\\mathbf {v}\\cdot \\nabla )\\mathbf {v}=-\\frac{1}{m}\\nabla \\left(V+g\\rho \\right)+\\nu _q\\nabla ^2\\mathbf {v}$ where the kinematic quantum viscosity is $\\nu _q\\equiv \\frac{\\hbar \\gamma }{2m},$ in analogy with classical fluids.", "In this regime, (REF ) is coupled to the continuity equation $\\frac{\\partial \\rho }{\\partial t}+\\nabla \\cdot (\\rho \\mathbf {v})&=&\\frac{2\\rho \\gamma }{\\hbar } (\\mu -U_H)$ with hydrodynamic potential $U_H( \\mathbf {r} )\\equiv \\frac{m \\mathbf {v}^2}{2}+V+g\\rho .$ The source term in (REF ) drives the system towards particle-number equilibrium with the reservoir.", "In the Thomas-Fermi regime $\\mu -U_H( \\mathbf {r} )\\approx 0$ , restoring approximate particle-number conservation.", "Making use of (REF ), we can give an order-of-magnitude estimate for the viscosity $\\nu _q^0\\equiv \\frac{\\hbar \\gamma _0}{2m}=\\frac{2a^2 k_BT}{\\pi \\hbar }.$ We can also estimate a quantum Reynolds number as $Re_q^0\\equiv \\frac{UL}{\\nu _q^0}=\\frac{\\pi \\hbar }{2a^2 }\\frac{UL}{k_BT}$ for BEC flow with characteristic speed $U$ and length scale $L$ .", "We can write the quantum Reynolds number as $Re_q^0=\\frac{\\lambda _{dB}^2}{a^2 }\\frac{mUL}{4\\hbar }.$ We note that temperature only enters the expression through the deBroglie wavelength of the matter wave field, in the ratio $\\lambda _{dB}/a $ , which is typically very large for a BEC.", "Note that strong scattering corresponds to strong damping, and hence low Reynolds number.", "A large deBroglie wavelength corresponds to a relatively cold system, which is hence expected to be weakly damped and have a high Reynolds number.", "We thus have a dimensionless ratio in the form $Re_q^0\\sim \\frac{[\\mbox{\\small deBroglie wavelength}]^2}{[\\mbox{\\small scattering length}]^2}\\cdot \\frac{[\\mbox{\\small flow momentum}]}{[\\mbox{\\small quantum momentum}]}\\;\\;\\;$ where $\\hbar /L$ is interpreted as the quantum momentum associated with the transverse length scale of the flow.", "A concrete example is provided by a recent experimental study of 2DQT generated by stirring a highly oblate, toroidally confined BEC [33].", "The initial system consists of $\\sim 2.6\\times 10^6$ atoms of $^{87}$ Rb at a temperature of $\\sim 100{\\rm nK}$ .", "Using these numbers in our analysis the dimensionless damping parameter $\\gamma _0\\sim 6\\times 10^{-4}$ gives a kinematic quantum viscosity $\\nu _q^0\\sim 6\\times 10^{-2}\\mu {\\rm m}^2\\,{\\rm s}^{-1}$ .", "The trapping potential confines the flow to an annular channel of width $L\\sim 30\\,\\mu {\\rm m}$ .", "The nominal flow speed can be estimated from numerical simulations of the dGPE [33], giving a value $U\\sim 5\\, \\mu {\\rm m\\, s}^{-1}$ as the peak value occurring in the bulk flow during the stirring sequence.", "These values give an estimate $Re_q^0\\sim 600$ .", "We can alternatively estimate a Reynolds number at the scale of the forcing in the experiment, which is of order of the size of the vortex dipoles nucleated, $d\\sim 10\\,\\xi $ , with healing length $\\xi \\sim 0.5\\,\\mu {\\rm m}$ .", "Such dipoles have a characteristic speed ${\\rm v}_d\\sim 146\\,\\mu {\\rm ms}^{-1}$ , for which we estimate the Reynolds number of the forcing scale as $\\sim {\\rm v}_dd/\\nu _q^0=6.2\\times 10^3$ .", "These large values suggest that turbulent flow in a finite-temperature BEC may exist across a wide range of length scales if the analogy is made with the classical Reynolds numbers that correspond to turbulent flow [1].", "This interpretation is broadly consistent with the experimental and numerical observations of chaotic vortex dynamics [33].", "We emphasize that the quantum Reynolds number estimate proposed here is applicable to a finite-temperature weakly interacting superfluid and may provide a general condition in analogy with classical fluids that is independent of dimension.", "However, taking the zero-temperature limit gives an infinite value for $Re_q^0$ , and in this regime the superfluid dissipation stems from vortex reconnections (or annihilation in two dimensions) and coupling to the sound field [45], [46], [47].", "The detailed description of criteria for superfluidity in the cross-over from the zero-temperature to high-temperature regimes is an open problem [48].", "Furthermore, due to the reversed direction of energy transfer in 2DQT, the scales of interest have to be reexamined; we do not pursue this here.", "Our aim is to establish a conceptual link between the dGPE and the NSE, given by Eq.", "(REF ).", "In doing so we have shown how to identify the equivalent viscosity in a finite temperature BEC." ], [ "Two-dimensional vortex wavefunction", "In the remainder of this work we limit our analysis to homogeneous compressible quantum fluids in two dimensions, and redefine our spatial and velocity coordinates accordingly: $\\mathbf {r}=(x,y)=r\\,(\\cos \\theta ,\\sin \\theta )$ and $\\mathbf {v}=(\\mathrm {v}_x\\,,\\mathrm {v}_y)$ .", "We thus confine our attention to the regime of an effective 2D GPE, with modified interaction parameter.", "While 2D BEC systems can be created through extremely tight confinement in one dimension, a regime of effective 2D vortex dynamics can also be accessed in less oblate systems, giving a 2D analysis wider applicability.", "For example, although the BECs of References [32] and [33] were three dimensional, the confinement along one dimension was strong enough to limit vortex motion to a plane and suppress vortex bending and tilting away from the tight-trapping direction.", "Aspects of BEC dimensionality in regards to vortices and Kelvin waves were analyzed in [44], further indicating that sufficiently oblate 3D BECs may be considered 2D as far as vortex dynamics and turbulence are concerned.", "At the same time, such systems can remain far enough away from the quasi-2D limits in which a Berezhkinski-Kosterlitz-Thouless (BKT) transition has been observed [49], and BKT physics may thus be neglected.", "For our analysis of kinetic energy spectra, we require certain properties of a quantized vortex, namely the asymptotic character of the wavefunction for large and small length scales.", "The Gross-Pitaevskii equation describing the homogeneous ($V=0$ ) 2D Bose gas is obtained from (REF ) by taking $\\gamma =0$ and using an interaction parameter $g_2 = g/l$ where $l$ is the characteristic thickness of the 3D system [30]: $i\\hbar \\frac{\\partial \\psi ({\\mathbf {r}},t)}{\\partial t}&=&\\left(-\\frac{\\hbar ^2\\nabla _\\perp ^2}{2m}+g_{2}|\\psi ({\\mathbf {r}},t)|^2\\right)\\psi ({\\mathbf {r}},t).$ For example, in a system with harmonic trapping in the $z$ -direction characterized by trapping frequency $\\omega _z$ , the length scale is $l=\\sqrt{2\\pi }l_z$ where $l_z=\\sqrt{\\hbar /m\\omega _z}$ is the $z$ -axis harmonic oscillator length, and the confinement is assumed sufficient to put the wavefunction into the $z$ -direction single-particle ground state.", "For solutions with chemical potential $\\mu $ containing a single vortex at the origin (with circulation normal to the plane of the quantum fluid) we can write [50] $\\psi _1({\\mathbf {r}},t)=\\sqrt{n_0}e^{-i\\mu t/\\hbar }\\chi \\!\\left(r/\\xi \\right)e^{\\pm i\\theta }$ where $\\xi =\\hbar /m c$ is the healing length for speed of sound $c=\\sqrt{\\mu /m}$ , and $n_0=\\mu /g_2$ is the 2D particle density for $r\\gg \\xi $ and is taken to be a constant.", "The vortex radial amplitude function $\\chi ({\\sigma })$ , where ${\\sigma }= r/\\xi $ is a scaled radial coordinate, is a solution of $\\left(-{\\sigma }^{-1}\\partial _{\\sigma }\\, {\\sigma }\\partial _{\\sigma }+{\\sigma }^{-2}\\right)\\chi =2(\\chi -\\chi ^3).$ The boundary conditions are $\\chi (0)=0$ , and the derivative $\\chi ^{\\prime } \\equiv d \\chi /d{\\sigma }$ evaluated at ${\\sigma }=0$ must be chosen such that it is consistent with $\\chi (\\infty )=1$ and $\\chi ^\\prime (\\infty )=0$ .", "The value $\\Lambda \\equiv \\chi ^\\prime (0)=\\lim _{r\\rightarrow 0}\\frac{\\xi }{\\sqrt{n_0}}\\left|\\frac{d\\psi _1}{dr}\\right|$ is determined numerically to be $\\Lambda =0.8249\\dots $ .", "The state (REF ) has the velocity field of a quantum vortex $\\mathbf {v}({\\mathbf {r}})=\\frac{\\hbar }{mr}(\\mp \\sin \\theta ,\\pm \\cos \\theta ).$" ], [ "Kinetic energy decomposition", "We make use of the decomposition of the kinetic energy into compressible and incompressible parts [51], [28].", "The 2D case of the Gross-Pitaevskii energy functional (REF ) can be decomposed as $E=E_{K}+E_{V}+E_{I}+E_Q$ , where $E_K&=&\\frac{m}{2}\\int d^2{\\mathbf {r}}\\; \\rho ({\\mathbf {r}},t)|\\mathbf {v}({\\mathbf {r}},t)|^2,\\\\E_V&=&\\int d^2{\\mathbf {r}}\\; \\rho ({\\mathbf {r}},t)V({\\mathbf {r}},t),\\\\E_I&=&\\frac{g_2}{2}\\int d^2{\\mathbf {r}}\\; \\rho ({\\mathbf {r}},t)^2,\\\\E_Q&=&\\frac{\\hbar ^2}{2m}\\int d^2{\\mathbf {r}}\\; |\\nabla \\!\\sqrt{\\rho ({\\mathbf {r}},t)}|^2.$ Respectively, these define the components of energy that can be attributed to kinetic energy, potential energy, interaction energy, and quantum pressure.", "We are interested in the kinetic energy, $E_K$ .", "We define a density-weighted velocity field $\\mathbf {u}({\\mathbf {r}},t)\\!\\equiv \\!\\!\\sqrt{\\rho ({\\mathbf {r}},t)}\\mathbf {v}({\\mathbf {r}},t)$ , then decompose this into $\\mathbf {u}({\\mathbf {r}},t)=\\mathbf {u}^i({\\mathbf {r}},t)+\\mathbf {u}^c({\\mathbf {r}},t)$ , where the incompressible field $\\mathbf {u}^i$ satisfies $\\nabla \\cdot \\mathbf {u}^i=0$ , and the compressible field $\\mathbf {u}^c$ satisfies $\\nabla \\times \\mathbf {u}^c=0$ .", "We can further decompose the kinetic energy as $E_K=E_{i}+E_{c}$ , where the portion of $E_K$ attributed to compressible or incompressible kinetic energy is defined as $E_{c,i}=\\frac{m}{2}\\int d^2{\\mathbf {r}}\\; |\\mathbf {u}^{c,i}({\\mathbf {r}},t)|^2.$ The compressible component is attributed to the kinetic energy contained in the sound field, while the incompressible part gives the contribution from quantum vortices.", "Our analysis below only involves $E_i$ .", "Because we focus on vortex configurations at instants in time, we drop the explicit time dependence from the remainder of our expressions.", "In $k-$ space, the total incompressible kinetic energy $E_i$ is given by $E_i=\\frac{m}{2}\\sum _{j=x,y}\\int d^2{\\mathbf {k}}\\;|{\\cal F}_j({\\mathbf {k}})|^2$ where ${\\cal F}_j({\\mathbf {k}})=\\frac{1}{2\\pi }\\int d^2{\\mathbf {r}}\\; e^{-i{\\mathbf {k}}\\cdot {\\mathbf {r}}}\\mathbf {u}^i_j({\\mathbf {r}}).$ The one-dimensional spectral density in $k$ -space is given in polar coordinates by integrating over the azimuthal angle to give $E_i(k)=\\frac{mk}{2}\\sum _{j=x,y} \\int _0^{2\\pi }d\\phi _k|{\\cal F}_j({\\mathbf {k}})|^2,$ which, when integrated over all $k$ , gives the total incompressible kinetic energy $E_i=\\int _0^\\infty dk\\; E_i(k)$ ." ], [ "Incompressible kinetic energy spectrum of a vortex", "We now consider the kinetic energy spectrum of a single quantum vortex in a 2D BEC.", "For an arbitrary wavefunction the decomposition into compressible and incompressible parts must be performed prior to carrying out the transformation to the spectral representation.", "However, for a quantum state containing a single vortex and no acoustic energy [i.e.", "the single vortex wavefunction $\\psi _1$ (REF )] we note that the wavefunction is automatically incompressible, i.e.", "the compressible part is identically zero: $\\nabla \\cdot (\\sqrt{\\rho ({\\mathbf {r}})}\\mathbf {v}({\\mathbf {r}}))=\\mathbf {v}\\cdot \\nabla \\sqrt{\\rho ({\\mathbf {r}})}+\\sqrt{\\rho ({\\mathbf {r}})}\\nabla \\cdot \\mathbf {v}\\equiv 0.$ The first term vanishes due to the orthogonality of the density gradient and velocity of a vortex, and the second due to the form of (REF ).", "Thus the incompressible spectrum is the entire spectrum for a single quantum vortex.", "For a single vortex we can thus ignore the incompressible decomposition and cast the kinetic energy spectrum in terms of the properties of the radial amplitude function $\\chi ({\\sigma }) = \\sqrt{\\rho ({\\sigma }\\xi )/n_0}$ obtained from (REF ).", "We have ${\\cal F}_x({\\mathbf {k}})&=&-\\frac{\\hbar }{2\\pi m}\\int d^2{\\mathbf {r}}\\; e^{-i{\\mathbf {k}}\\cdot {\\mathbf {r}}}\\frac{\\sqrt{\\rho ({\\mathbf {r}})}}{r}\\sin \\theta \\nonumber \\\\&=&\\frac{i\\hbar }{m}\\frac{d}{dk}\\int _0^\\infty dr\\frac{\\sqrt{\\rho (r)}}{r}J_0(kr)\\nonumber \\\\&=&\\frac{-i\\hbar \\sqrt{n_0}\\xi }{m}\\frac{1}{k\\xi }\\int _0^\\infty d{\\sigma }\\;\\chi ^\\prime ({\\sigma })J_0(k\\xi {\\sigma }),$ where $J_0$ is the zeroth-order Bessel function of the first kind.", "Similar analysis gives ${\\cal F}_y(k)=-{\\cal F}_x(k)$ .", "We can thus find the one-vortex spectrum [see (REF )] $E_i^1(k)=\\Omega \\xi ^3F(k\\xi ),$ where we define the dimensionless core spectral function $F(k\\xi )\\equiv \\frac{1}{k\\xi }\\left(\\int _0^\\infty d{\\sigma }\\;\\chi ^\\prime ({\\sigma })J_0(k\\xi {\\sigma })\\right)^2,$ and we have introduced the unit of enstrophy $\\Omega \\equiv \\frac{2\\pi \\hbar ^2 n_0}{m\\xi ^2},$ giving $\\Omega \\xi ^3$ as the natural unit for the kinetic energy density.", "The core spectral function has the small-$k\\xi $ asymptotic form $F(k\\xi )\\Big |_{k\\xi \\ll 1}=\\frac{1}{k\\xi }\\left(\\int _0^\\infty d{\\sigma }\\;\\chi ^\\prime ({\\sigma })\\right)^2&=&\\frac{1}{k\\xi }.$ For $k\\xi \\gg 1$ , $J_0(k\\xi {\\sigma })$ is highly oscillatory except at ${\\sigma }=0$ where it is unity, and the Taylor expansion of $\\chi ^\\prime ({\\sigma })$ can be truncated at zeroth order to give $F(k\\xi )\\Big |_{k\\xi \\gg 1}&=&\\frac{\\Lambda ^{2}}{k\\xi }\\left(\\int _0^\\infty d{\\sigma }\\;J_0(k\\xi {\\sigma })\\right)^2=\\frac{\\Lambda ^{2}}{(k\\xi )^3}.$ We thus have the asymptotic spectra for a single vortex $E_i^1(k)\\Big |_{k\\xi \\ll 1}&=& \\frac{\\Omega \\xi ^3}{ k\\xi }, \\\\E_i^1(k)\\Big |_{k\\xi \\gg 1}&=&\\Lambda ^{2}\\frac{\\Omega \\xi ^3}{(k\\xi )^3}.$ The $k\\xi \\ll 1$ regime arises purely from the irrotational velocity field of a quantum vortex, while the $k\\xi \\gg 1$ regime is a property of the core of a compressible quantum vortex.", "The $k\\xi \\gg 1$ regime explicitly depends on the slope of the wavefunction at the core of a vortex.", "The cross-over between these regions occurs in the vicinity of $k\\xi \\approx 1$ , hence we take $k\\xi =1$ as distinguishing the infrared ($k\\xi <1$ ) and ultraviolet ($k\\xi >1$ ) regimes in the remainder of our analysis.", "The scale $k\\xi =1$ thus serves to define an important length scale of the problem, namely $l_v\\equiv 2\\pi \\xi $ .", "In Fig.", "REF we see that at this distance from the vortex core the deviation of the amplitude from the background value is very small.", "This is the scale beyond which the details of the core structure are no longer important in characterizing the wavefunction, or equivalently that the fluid density has approximately reached its bulk value.", "The irrotational velocity field in Eq.", "(REF ) is the only remaining signature of a vortex at this range from its center and beyond.", "We note that our derivation of the $k^{-3}$ power-law stemming from the quantum vortex core structure is consistent with recent analysis of the Kelvin-wave cascade in 3D [52].", "Figure: Amplitude of the wavefunction for a single vortex solution of the Gross-Pitaevskii equation.", "The numerical solution of () (solid line) is compared with the ansatz () (dashed line).", "The inclined dashed line shows the slope Λ\\Lambda of the exact solution at the origin.", "The vertical line is the point σ=2π\\sigma =2\\pi .Now that we have identified the properties of a single vortex, it is natural to ask whether a unit of enstrophy can be attributed to a single quantum vortex, and to compare this with the quantity defined in (REF ).", "The point-vortex model suggests that this can be done, but gives a singular result, which is nevertheless known to be proportional to the number of vortices [30].", "The problem is also evident if we attempt to evaluate the enstrophy of a single vortex from the spectrum (REF ).", "Multiplying by $k^2$ to produce an enstrophy spectral density and integrating this over the ultraviolet regime $k\\xi >1$ , we are faced with the singular integral $\\int _1^\\infty dk/k$ .", "In this work we therefore define a new asymptotic quantity with units of enstrophy as $\\zeta \\equiv \\lim _{k\\rightarrow \\infty }k^3E_{i}(k).$ This quantity plays a fundamental role for a compressible superfluid because it completely specifies the large-$k$ region of the incompressible kinetic energy spectrum.", "Because the spectrum in this region of $k$ -space is determined by the core structure of quantized vortices, we call this unit the onstrophy to both recall Onsager's contribution to our understanding of quantized vorticity in a superfluid and emphasize the difference between enstrophy in classical and quantum fluids.", "For a single quantum vortex we find, using (REF ), the onstrophy $\\zeta _1=\\Lambda ^{2}\\Omega ,$ which differs from (REF ) by the factor $\\Lambda ^2=0.6805\\dots $ , a property of the vortex core in a compressible superfluid." ], [ "Vortex wavefunction ansatz and kinetic energy spectrum", "To study 2D kinetic energy spectra we will make extensive use of an algebraic ansatz for the wavefunction of a single vortex in a homogeneous superfluid.", "Numerical evaluation of the exact core function (REF ) is not straightforward due to the highly oscillatory integrand, and the need to determine the vortex amplitude $\\chi ({\\sigma })$ extremely accurately over a large range of length scales.", "In order to accurately represent the spectrum it will be crucial that our ansatz have the correct asymptotic properties for small and large length scales described immediately above (REF ).", "Making use of the slope at the origin computed for the exact solution in (REF ), we use the ansatz wavefunction: $\\phi _v({\\mathbf {r}})=\\sqrt{n_0}\\frac{re^{\\pm i\\theta }}{\\sqrt{r^2+(\\Lambda ^{-1}\\xi )^2}}.$ The general form of this ansatz has been previously used to describe the shape of a vortex core [50], but here we use a length scale $\\Lambda ^{-1}\\xi $ that enforces matching the slope of the ansatz density distribution to the exact value at the center of the core.", "The state (REF ) has the irrotational velocity field of a quantum vortex specified in (REF ) and reproduces the asymptotic slope of the exact solution near the origin, as shown in Figure REF .", "We now compute the kinetic energy spectrum for a single vortex by evaluating (REF ) using the form (REF ).", "Taking $\\Lambda ^{-1}\\xi =b$ for brevity, we have ${\\cal F}_x({\\mathbf {k}})&=&\\frac{i\\hbar \\sqrt{n_0}}{m}\\frac{d}{dk}\\int _0^\\infty \\frac{dr\\;J_0(kr)}{\\sqrt{r^2+b^2}}\\nonumber \\\\&=&i\\frac{\\hbar \\sqrt{n_0}b}{2m}\\left[I_1\\left(\\frac{kb}{2}\\right)K_0\\left(\\frac{kb}{2}\\right)-I_0\\left(\\frac{kb}{2}\\right)K_1\\left(\\frac{kb}{2}\\right)\\right]\\;\\;\\;\\;$ where $I_j$ and $K_j$ are modified Bessel functions of the first and second kind, respectively, of order $j$ .", "Since $|{\\cal F}_y|^2$ =$|{\\cal F}_x|^2$ , we find the incompressible energy spectrum of a single vortex $E_i^1(k)=\\Omega \\xi ^3F_\\Lambda (k\\xi ),$ where $F_\\Lambda (k\\xi )\\equiv \\Lambda ^{-1} f(k\\xi \\Lambda ^{-1}),$ and where we define $f(z)\\equiv (z/4)[I_1(z/2)K_0(z/2)-I_0(z/2)K_1(z/2)]^2.$ The function $f(z)$ has the following asymptotics: for $z\\ll 1$ $f(z)=\\frac{1}{z}+\\left(\\bar{\\gamma }+\\ln \\left(\\frac{z}{4}\\right)\\right)\\frac{z}{2}+\\dots ,$ where $\\bar{\\gamma }=0.57721...$ is the Euler-Masceroni constant; for $z\\gg $ 1 $f(z)=\\frac{1}{z^3}+ \\frac{3}{ z^5}+\\dots .$ The function $F_\\Lambda (k\\xi )$ thus has the asymtotics $F_\\Lambda (k\\xi )\\Big {|}_{k\\xi \\ll 1}&=&\\frac{1}{k\\xi }\\\\F_\\Lambda (k\\xi )\\Big {|}_{k\\xi \\gg 1}&=&\\frac{\\Lambda ^2}{(k\\xi )^3}$ which are identical to those of $F(k\\xi )$ .", "The two functions are very similar, with only small differences evident in the cross-over region $k\\xi \\sim 1$ , as seen in Figure REF .", "We use $F_\\Lambda (k\\xi )$ instead of $F(k\\xi )$ for describing the kinetic energy spectrum for a vortex core in the remainder of this work as it is numerically expedient and does not alter any of the physical consequences of our analysis.", "Towards the end of this paper we will compare the asymptotic results of our analysis with spectra determined numerically from forced dGPE dynamics.", "The spectrum of a single vortex is shown in Figure REF , and compared with the spectrum of a vortex-antivortex pair (a vortex dipole), and that for two vortices of the same sign (a vortex pair).", "These two-vortex spectra are analyzed in the following section.", "Figure: Comparison of the numerically computed kinetic energy spectrum [(), blue solid line] and that obtained from the core function F Λ (kξ)F_\\Lambda (k\\xi ) [(), red dashed line] in the cross-over regime kξ∼1k\\xi \\sim 1.", "The asymptotic expressions () and () are shown by the black and green lines respectively." ], [ "Two-vortex spectra", "Extension of the discussion in Section REF leads us to conclude that a wavefunction that only contains vortices (i.e., no sound field) separated by more than a few healing lengths will thus be approximately incompressible according to the decomposition.", "The approximation breaks down through the non-orthogonality of $\\mathbf {v}$ and $\\nabla \\sqrt{\\rho ({\\mathbf {r}})}$ near a vortex core due to the velocity field induced by the other vortices.", "However, in the close vicinity of a vortex core, where $\\nabla \\sqrt{\\rho ({\\mathbf {r}})}$ is significant, the velocity is dominated by the velocity field of that vortex core.", "An arrangement of vortices separated by more than a few healing lengths will thus be approximately incompressible.", "In the following analytical treatment we will neglect any compressible part that arises from an assembly of vortices described by the ansatz (REF ).", "Figure: Incompressible kinetic energy spectra for a single vortex (chain line), a vortex dipole (solid line), and a vortex pair (dashed line).", "The vortex dipole and pair are both shown for vortex separation d=20ξd=20\\xi , and the wavenumber k d ≡2π/dk_d\\equiv 2\\pi /d is shown as a vertical dashed line.", "The cross-over scale kξ=1k\\xi =1 is given by the solid vertical line.A two-vortex state in a homogeneous system with no boundaries has density-weighted velocity field $\\sqrt{\\rho ({\\mathbf {r}})}[\\mathbf {v}_1({\\mathbf {r}})+\\mathbf {v}_2({\\mathbf {r}})]$ , where $\\mathbf {v}_j$ is the velocity field around vortex $j = 1,2$ taken separately.", "If the vortex cores are separated by $d\\gg \\xi $ it will also be a very good approximation to write $\\rho ({\\mathbf {r}})=\\rho _1({\\mathbf {r}})\\rho _2({\\mathbf {r}})/n_0$ , with $\\rho _j({\\mathbf {r}})=\\frac{n_0\\;|{\\mathbf {r}}- {\\mathbf {r}}_j|^2}{|{\\mathbf {r}}- {\\mathbf {r}}_j|^2+(\\Lambda \\xi ^{-1})^2}.$ The density weighted two-vortex velocity field can then be written as $\\sqrt{\\frac{\\rho _1({\\mathbf {r}})\\rho _2({\\mathbf {r}})}{n_0}}[\\mathbf {v}_1({\\mathbf {r}})+\\mathbf {v}_2({\\mathbf {r}})]&=&\\sqrt{\\rho _1({\\mathbf {r}})}\\mathbf {v}_1({\\mathbf {r}})+\\sqrt{\\rho _2({\\mathbf {r}})}\\mathbf {v}_2({\\mathbf {r}})\\nonumber \\\\&&+K_{12}({\\mathbf {r}}).$ The final term is $K_{12}({\\mathbf {r}})&=&\\sqrt{\\frac{\\rho _1({\\mathbf {r}})\\rho _2({\\mathbf {r}})}{n_0}}\\Bigg (\\mathbf {v}_1({\\mathbf {r}})\\left[1-\\sqrt{\\frac{n_0}{\\rho _2({\\mathbf {r}})}}\\right]\\nonumber \\\\&&+\\mathbf {v}_2({\\mathbf {r}})\\left[1-\\sqrt{\\frac{n_0}{\\rho _1({\\mathbf {r}})}}\\right]\\Bigg )$ which is only significant when considering the velocity field of one vortex in the close vicinity of the other vortex core.", "$K_{12}$ is therefore a negligible correction to the spectrum for widely separated vortices.", "This approximation may be trivially generalized to arbitrary numbers of vortices provided the cores do not overlap appreciably.", "This approximation is central to our treatment, as it allows recovery of familiar point-like vortex physics in the infrared regime $k\\xi \\ll 1$ .", "Strict validity is limited to the regime where the intervortex spacing $d$ is bounded below by $\\sim l_v= 2\\pi \\xi $ , the scale at which the core structure becomes evident (see Fig.", "REF ).", "We require the transform ${\\cal F}_j^d({\\mathbf {k}})&=&\\frac{1}{2\\pi }\\int d^2{\\mathbf {r}}\\; e^{-i{\\mathbf {k}}\\cdot {\\mathbf {r}}}\\left[\\sqrt{\\rho _1({\\mathbf {r}})}\\mathrm {\\textbf {v}}_1({\\mathbf {r}})\\right]_j^i\\nonumber \\\\&&+\\frac{1}{2\\pi }\\int d^2{\\mathbf {r}}\\; e^{-i{\\mathbf {k}}\\cdot {\\mathbf {r}}}\\left[\\sqrt{\\rho _2({\\mathbf {r}})}\\mathrm {\\textbf {v}}_2({\\mathbf {r}})\\right]_j^i$ where the superscript $d$ denotes the case of a vortex dipole, and subscript $j = x,y$ indicates the $x$ and $y$ components of the density-weighted velocity fields of each vortex.", "As above, the subscripts 1 and 2 denote vortices 1 and 2, and the superscript $i$ denotes that it is the incompressible, or divergence-free portion of the density-weighted field that is of interest here.", "To account for the opposite signs of circulation for the two vortices, without loss of generality we choose vortex 1 as positively charged, located at ${\\mathbf {r}}_0=(d/2)\\hat{{\\mathbf {x}}}$ , so that $\\mathbf {v}_1({\\mathbf {r}})=\\mathbf {v}({\\mathbf {r}}-{\\mathbf {r}}_0)$ where $\\mathbf {v}({\\mathbf {r}})$ is the central vortex velocity field (REF ).", "Vortex 2 has velocity field $\\mathbf {v}_2({\\mathbf {r}})=-\\mathbf {v}({\\mathbf {r}}+{\\mathbf {r}}_0)$ .", "We then have ${\\cal F}_j^d({\\mathbf {k}})&=&\\frac{1}{2\\pi }\\int d^2{\\mathbf {r}}\\; e^{-i{\\mathbf {k}}\\cdot ({\\mathbf {r}}+{\\mathbf {r}}_0)}\\left[\\sqrt{\\rho ({\\mathbf {r}})}\\mathrm {\\textbf {v}}({\\mathbf {r}})\\right]_j^i\\nonumber \\\\&&-\\frac{1}{2\\pi }\\int d^2{\\mathbf {r}}\\; e^{-i{\\mathbf {k}}\\cdot ({\\mathbf {r}}-{\\mathbf {r}}_0)}\\left[\\sqrt{\\rho ({\\mathbf {r}})}\\mathrm {\\textbf {v}}({\\mathbf {r}})\\right]_j^i.$ For a vortex dipole we can then write ${\\cal F}_j^d({\\mathbf {k}})&=&{\\cal F}_j({\\mathbf {k}}) e^{-i{\\mathbf {k}}\\cdot {\\mathbf {r}}_0}-{\\cal F}_j({\\mathbf {k}}) e^{i{\\mathbf {k}}\\cdot {\\mathbf {r}}_0}$ where ${\\cal F}$ is the spectrum of a single vortex.", "Using (), and the fact that ${\\cal F}_y(k)=-{\\cal F}_x(k)$ , we find for a vortex dipole $E_i^d(k)=2\\Omega \\xi ^3F_\\Lambda (k\\xi )(1-J_0(k d)).$ The spectrum of a pair of vortices of the same circulation separated by $d$ is calculated similarly to be $E_i^p(k)=2\\Omega \\xi ^3 F_\\Lambda (k\\xi )(1+J_0(k d)).$ The spectra (REF ) and (REF ) are shown in Figure REF .", "It is clear that for scales less than the vortex separation distance $d$ there is interference in $k$ -space, leading to oscillations in the spectrum.", "The difference between the dipole and pair is that the interference fringes are offset, and the infrared asymptotics are different, a feature we discuss further below.", "The spectrum of the vortex pair is clearly similar to that of the single vortex in the far infrared region, but the additional kinetic energy of the vortex pair state is observed throughout the spectrum." ], [ "Kinetic energy spectrum of $N$ -vortex configurations", "Extending the above analysis, for a general system of $N$ singly quantized vortices with circulation signs $\\kappa _p=\\pm 1$ located at ${\\mathbf {r}}_p$ the kinetic energy spectrum is given by $E_{i}^N(k)=\\Omega \\xi ^3 F_\\Lambda (k\\xi )\\sum _{p=1,q=1}^N\\kappa _p\\kappa _q J_0(k|{\\mathbf {r}}_p-{\\mathbf {r}}_q |).$ We note the resemblance to point-vortex results which also have the Bessel function dependence [53], [29].", "The function $F_\\Lambda (k\\xi )$ gives the incompressible limit for small $k$ [$J_0(kr)$ approaches unity for small $k$ and finite $r$ ], and introduces the physics of compressible superfluids for $1\\lesssim k\\xi $ .", "We can write (REF ) as $E_{i}^N(k)=N\\Omega \\xi ^3F_\\Lambda (k\\xi )G_N(k)$ where $G_N(k)\\equiv 1+\\frac{2}{N}\\sum _{p=1}^{N-1}\\sum _{q=p+1}^N\\kappa _p\\kappa _q J_0(k|{\\mathbf {r}}_p-{\\mathbf {r}}_q|)$ is a purely configurational function involving summation of $M=N(N-1)/2$ distinct intervortex distances.", "This function has the limits $\\lim _{k\\rightarrow \\infty }G_N(k)&=&1,\\\\\\lim _{k\\rightarrow 0}G_N(k)&=&1+\\frac{2}{N}\\sum _{p=1}^{N-1}\\sum _{q=p+1}^N\\kappa _p\\kappa _q =\\frac{\\Gamma ^2}{N},$ where the total dimensionless circulation is defined by $\\Gamma \\equiv \\frac{m}{\\hbar }\\oint _C \\mathbf {v}\\cdot d\\mathbf {l}=\\sum _{p=1}^N\\kappa _p =N_+-N_-$ for any contour $C$ enclosing all $N_+$ positive and $N_-=N-N_+$ negative vortices.", "We then find that the onstrophy for the $N$ -vortex system is $\\zeta _N=\\lim _{k\\rightarrow \\infty }\\;k^3E_{i}^N(k)&=&N\\Lambda ^{2}\\Omega =N\\zeta _1,$ and consequently $E_{i}^N(k)\\Big {|}_{k\\xi \\gg 1}&=&\\frac{\\zeta _N}{k^3}=\\frac{N\\Lambda ^2\\Omega \\xi ^3}{(k\\xi )^3}.$ This is one of our central results: the ultraviolet regime $k\\xi \\gg \\,$ 1 has a universal asymptotic form that is independent of the vortex configuration, and that resembles the ultraviolet spectrum of classical 2D turbulence that is identified with a direct cascade of enstrophy.", "If we try to evaluate the classical definition of enstrophy, the result is singular, yet the onstrophy definition (REF ) gives a well-defined additive quantity that is singularity free and depends only on the total number of vortices in the system." ], [ "Infrared behavior", "When $\\Gamma \\ne 0$ the far infrared limit () gives $E_{i}^N(k)\\Big {|}_{k\\xi \\ll 1}&=&\\frac{\\Omega \\xi ^3\\Gamma ^2}{k\\xi }.$ This configuration-independent $k^{-1}$ power law arises from the far-field velocity distribution of a collection of point vortices, which becomes equivalent to that of a single vortex of charge $\\Gamma $ at sufficiently large scales.", "When $\\Gamma \\equiv 0$ we use the small-argument expansion $J_0(z)\\simeq 1-z^2/4$ , and the asymptotic form (REF ), to find the $k\\xi \\ll 1$ behavior determined by the configurational information contained in the intervortex distances $|{\\mathbf {r}}_p-{\\mathbf {r}}_q|$ : $E_{i}^N(k)\\Big {|}_{k\\xi \\ll 1}=-\\frac{\\Omega \\xi ^2k}{2}\\sum _{p=1}^{N-1}\\sum _{q=p+1}^N\\kappa _p\\kappa _q |{\\mathbf {r}}_p-{\\mathbf {r}}_q|^2.$ The simplest case involves a single vortex dipole and has only one length scale, namely the vortex separation, and the low-$k$ form $E_{i}^d(k)\\simeq \\Omega \\xi ^2d^2 k/2$ , as shown in Fig.", "REF .", "In general, when $\\Gamma =0$ the infrared region of the spectrum is sensitive to the vortex configuration, but approaches a power-law for low-$k$ that has a configuration-independent exponent.", "The linear decay of kinetic energy as $k\\rightarrow 0$ stems from the cancellation of the far-field velocity profiles for length scales greatly exceeding the largest intervortex separation in any neutral configuration of vortices." ], [ "Kolmogorov spectrum", "In the previous section we obtained an explicit expression (REF ) for the incompressible kinetic energy spectrum that incorporates the compressible nature of individual vortex cores through the function $F_\\Lambda (k\\xi )$ (derived via an ansatz for the vortex core profile), which captures the essential physics of the corresponding exact solution $F(k\\xi )$ defined in (REF ).", "For both functions, point-vortex physics is recovered at large length-scales ($k\\xi \\ll 1$ ).", "If the dynamical evolution is such that an inertial range associated with an inverse energy cascade develops, we should expect a Kolmogorov power law $E_i(k)\\propto k^{-5/3}$ over the inertial range.", "It is clear from the form of (REF ) that this law can only depend on the spatial configuration of the vortices.", "We now seek to understand the simplest situations that may show evidence for the existence of such an inertial range.", "We consider forcing occurring via vortex and energy injection at a forcing scale $k_F\\sim \\xi ^{-1}$ , and describe vortex configurations that do and do not lead to a Kolmogorov law for $k<\\xi ^{-1}$ .", "We now assume an idealized case in which the infrared spectrum is continuous with the universal $k^{-3}$ law of the ultraviolet spectrum at the scale $k\\xi \\approx 1$ .", "This constraint imposes a strong restriction on the infrared spectrum, completely determining its form in the case that it satisfies a power law.", "In this respect the universal ultraviolet region has significant physical consequences.", "The power-law approximation to the universal ultraviolet region based on (REF ) has the form $E_{i,{\\rm U}}^N(k)=\\Lambda ^{2}\\frac{N\\Omega \\xi ^3}{(k\\xi )^3}=\\zeta _Nk^{-3}.$ The number of vortices determines the $N$ -vortex onstrophy $\\zeta _N$ (REF ), from which the power-law approximation to the ultraviolet energy spectrum is completely determined.", "This power law is a very good approximation, as will be seen by sampling different vortex configurations below.", "The infra-red or configurational regime is then given by the $k\\xi \\ll 1$ regime of $F_\\Lambda (k\\xi )$ : $E_{i,\\rm {C}}^N(k)\\Big {|}_{k\\xi \\ll 1}=\\frac{N\\Omega \\xi ^3}{k\\xi }G_N(k).$ At this point we consider the consequences of assuming that a turbulent system will have a $k^{-5/3}$ law in the configurational regime, and that this power law is continuous with (REF ) at $k\\xi = 1$ .", "We suppose that $E_{i,{\\rm C}}^N(k)\\propto k^{-5/3}$ .", "Continuity at $k=1/\\xi $ then requires $E_{i,{\\rm C}}^N(1/\\xi )=E_{i,{\\rm U}}^N(1/\\xi )$ , and gives the infrared spectrum $E_{i,{\\rm C}}^N(k)=\\Lambda ^{2}\\frac{N\\Omega \\xi ^{3}}{(k\\xi )^{5/3}}=\\zeta _N \\,\\xi ^{4/3} \\, k^{-5/3}.$ Thus the constraint that the universal regime is continuous at the cross-over scale $k=\\xi ^{-1}$ completely constrains the form of the configurational spectrum.", "Physically this may correspond to an inertial range that extends upwards from the smallest scale of the infrared region given forcing at a wavenumber $k_F \\sim \\xi ^{-1}$ .", "Note that this expression (REF ) has no reference to the signs of the vortex circulations, the degree of circulation polarization, or vortex clustering.", "By assuming continuity at $k\\xi =1$ with a UV spectrum that has a universal $N$ -vortex form, we have implicitly assumed that all $N$ vortices are involved in determining the spectrum of the inertial range.", "We might expect that this will give a very good description for a completely polarized system exhibiting fully developed turbulence.", "When there is clustering in mixtures of different sign vortices, the spectrum may well still approach a Kolmogorov law, but there is no reason to expect that it will cross over so smoothly.", "We return to this problem when we compare our analysis with numerical simulations of forced turbulence in Sec.", "REF .", "It is useful at this point to give a simplified reiteration of Novikov's argument for the power law for the vortex distribution being $-1/3$ in Kolmogorov turbulence [53].", "To obtain power-law behavior we must consider the spectrum for a vortex distribution involving many length scales.", "For simplicity we assume all vortices have the same sign of circulation, $\\kappa _p\\equiv \\kappa $ .", "The configuration function (REF ) has $M=N(N-1)/2$ terms in the summation, and can be written as $G_N(k)=1+\\frac{2}{N}\\sum _{p=1}^M J_0(k s_p),$ in terms of an average over distinct vortex separations $s_i$ .", "We introduce the intervortex distance distribution $P(s)$ such that $P(s)ds$ is the fraction of intervortex distances in the range $[s,s+ds)$ .", "In the continuum limit $G_N(k)&\\propto & \\int P(s)J_0(ks)ds.$ We seek a distribution $P(s)$ that will generate a Kolmogorov law from the $N$ -vortex spectrum (REF ) for scales larger than the vortex core, $k\\ll \\xi ^{-1}$ , in the large-$N$ regime.", "We then find from (REF ) that $E_{i, \\rm {C}}^N(k)\\Big {|}_{k\\xi \\ll 1}&\\sim &\\frac{1}{k}\\int _{\\xi }^{\\infty } P(s)J_0(ks)ds\\\\&\\simeq &\\frac{1}{k}\\int _{0}^{\\infty } P(s)J_0(ks)ds.$ The scale invariance of turbulence naturally leads to the assumption that the intervortex separation distribution is a power-law $P(s)\\sim s^{-\\alpha }$ over the scale range of interest.", "The requirement of a power law in the kinetic energy then gives the scaling relation $E_{i,\\rm {C}}^N(k) &\\sim &\\frac{1}{k^{\\beta }}\\sim \\frac{1}{k}\\int _{0}^{\\infty } s^{-\\alpha }J_0(ks)ds\\nonumber \\\\&=&\\frac{1}{k^{2-\\alpha }}\\int _{0}^{\\infty } \\tau ^{-\\alpha }J_0(\\tau )d\\tau .$ The integral is convergent for $-1/2<\\alpha <1$ , allowing $1<\\beta <5/2.$ In particular, the universal Kolmogorov law $\\beta =5/3$ occurs for $P(s)\\sim s^{-1/3}$ as obtained by Novikov [53] for point-vortices.", "We will test this scaling argument for the exponent in numerical sampling of localized vortex configurations in the following sections.", "Testing if this vortex separation power-law holds in simulations and experiments may give a quantitative measure of fully developed 2D turbulence in a compressible superfluid, and a way to identify the inertial range as the scale range over which this power-law can be identified.", "In 2D classical turbulence a $k^{-3}$ region of the kinetic energy spectrum is often associated with a direct enstrophy cascade.", "We note that this exponent $\\beta =3$ is ruled out by (REF ).", "Hence, within this continuum analysis the $k^{-3}$ power-law spectrum cannot occur in the configurational region for 2D quantum turbulence, as long as the vortex distribution follows a simple power law.", "This result suggests that if a direct enstrophy cascade were to occur in the configurational region of the spectrum, a different type of vortex distribution would be necessary.", "This makes intuitive sense, since direct enstrophy cascades may be associated with the stretching of patches of vorticity in the 2D plane.", "Furthermore, direct enstrophy cascades and energy spectra proportional to $k^{-3}$ have been noted in simulations of superfluid helium thin films [54], [55].", "Nevertheless, since we have also shown that the $k\\xi >1$ range is determined entirely by the core structure, this gives a strong indication that a direct enstrophy cascade cannot occur in this region in compressible 2DQT.", "It is clear that configurations containing one or a few characteristic length scales, such as a vortex dipole or a vortex lattice, cannot lead to a power law spectrum for $E_{i,\\rm {C}}^N(k)$ .", "In the case of a vortex lattice the intervortex distance distribution has many discrete peaks [36].", "The vortex dipole and a vortex pair each have a single length scale and this leads to characteristic interference fringes in the energy spectrum seen in Fig.", "REF ." ], [ "Sampling spatial vortex distributions", "We now test our analysis of the spectrum by numerically sampling several vortex distributions $\\lbrace \\mathbf {r}_p,\\kappa _p\\rbrace _{p=1}^N$ and evaluating (REF ).", "A straightforward test of the statistical argument for the Kolmogorov power law to occur involves sampling the power law (REF ) and evaluating (REF ).", "Indeed, it is easily verified that this generates a $k^{-5/3}$ spectrum in the configurational region.", "However, the connection of such a sampling to particular spatial vortex distributions is not clear, and in fact the mapping is not unique.", "To make this connection concrete we require a way of sampling finite, localized, spatial vortex distributions for which the vortex separations are power-law distributed.", "In an ideal, infinitely extended vortex configuration exhibiting the power law (REF ), the system is translationally invariant and the coordinate origin can be placed at any particular vortex, yielding the same power law for the radial distribution of vortices from the origin.", "In a finite system the scale invariance can only persist up to scales of order the largest vortex separation.", "Furthermore, the vortices must be separated by a minimum distance to satisfy the assumptions used in deriving the energy spectrum from the point-vortex model.", "In practice it is necessary to use a self-consistent sampling scheme in order to generate the correct power-law distributions for localized finite configurations.", "Our sampling scheme for a configuration of $N$ vortices is: Sample the radial distance $r_p$ of each vortex from the coordinate origin according to a power law probability distribution $\\propto r_p^{-\\bar{\\alpha }}$ .", "The exponent $\\bar{\\alpha }$ is distinct from $\\alpha $ due to the finite system size and localization of the distribution.", "Assign each vortex a randomly chosen, uniformly distributed angle $\\theta _p\\in [0,2\\pi )$ .", "The cartesian coordinates for vortex $p$ are then $(x_p,y_p)&=&r_p(\\cos \\theta _p,\\sin \\theta _p).$ For a given $N$ , compute the kinetic energy spectrum, averaging over $n_s=100$ samples of vortex position data found via the foregoing routine.", "Iterate until the spectral power law of interest is found.", "We will sample a power-law distribution of vortex distances from the origin.", "In practice there is a lower ($r_{\\rm min}\\sim \\xi $ ) and upper ($r_{\\rm min}\\sim R=$ system size) cutoff for power-law scaling.", "We thus wish to sample the distribution $P_{\\bar{\\alpha }}(r)=\\frac{1-\\bar{\\alpha }}{r_{\\rm max}^{1-\\bar{\\alpha }}-r_{\\rm min}^{1-\\bar{\\alpha }}}r^{-\\bar{\\alpha }},$ that is normalized on the interval $r_{\\rm min}\\le r<r_{\\rm max}$ .", "We sample $r$ values from this distribution using uniform random variates $x\\in [0,1)$ via the transformation [56]: $r=\\left[r_{\\rm max}^{1-\\bar{\\alpha }}x+r_{\\rm min}^{1-\\bar{\\alpha }}(1-x)\\right]^{\\frac{1}{1-\\bar{\\alpha }}}.$ Note that when $\\bar{\\alpha }<1$ , as is always the case in this work, some kind of ultraviolet cutoff is required for the distribution to be normalizable.", "Here we have made a choice that gives a power-law distribution over a well-defined scale range (See e.g.", "Ref.", "[56] for other common choices).", "Figure: Locations of N=100N=100 vortices for particular samples (left), with velocity distributions [middle, with the analytical result () (dashed line)], and kinetic energy spectra (right, blue curves), averaged over n s =100n_s=100 samples distributed according to () and (), with r min =ξr_{\\rm min}=\\xi , and (a) r max =60πξr_{\\rm max}=60\\pi \\xi , α ¯=0.4\\bar{\\alpha }=0.4, (b) r max =400πξr_{\\rm max}= 400\\pi \\xi , α ¯=0.65\\bar{\\alpha }=0.65, and (c) r max =4000πξr_{\\rm max}=4000\\pi \\xi , α ¯=0.83\\bar{\\alpha }=0.83.", "The increasing scale range of power-law behavior gives an inertial range in the corresponding kinetic energy spectrum of ∼1\\sim 1, 1.5, and 2 decades respectively." ], [ "Classical velocity distribution of a large cluster", "A fully polarized configuration of vortices with a given radial power law forms a quantum analogue of the coherent vortices of forced 2D turbulence in classical fluids.", "As an arrangement of many vortices, the velocity distribution must approach a classical limit, according to Bohr's correspondence principle, in much the same way that a rotating Abrikosov lattice generates a velocity field that approaches that of a rotating rigid body [50].", "In what follows we find the classical velocity field, and identify the physical significance of the radial exponent $\\bar{\\alpha }$ .", "To determine the velocity field we first compute the fraction of vortices enclosed by a circular contour around the origin, with radius $r$ , for the distribution (REF ).", "Taking $r_{\\rm min}=\\xi , r_{\\rm max}=R$ , and considering scales $\\xi \\ll r\\ll R$ we have $f_r&=&\\int _{\\xi }^r P_{\\bar{\\alpha }}(u)du={\\cal N}\\int _{\\xi }^r \\frac{du}{u^{\\bar{\\alpha }}}=\\frac{{\\cal N} }{1-\\bar{\\alpha }}\\left(r^{1-\\bar{\\alpha }}-\\xi ^{1-\\bar{\\alpha }}\\right)\\nonumber \\\\&\\simeq &\\left(\\frac{r}{R}\\right)^{1-\\bar{\\alpha }},$ where we used the fact that ${\\cal N}^{-1}=\\int _{\\xi }^R s^{-\\bar{\\alpha }}ds=(1-\\bar{\\alpha })^{-1}(R^{1-\\bar{\\alpha }}-\\xi ^{1-\\bar{\\alpha }})\\simeq R^{1-\\bar{\\alpha }}/(1-\\bar{\\alpha })$ normalizes the distribution up to the largest scale $R$ .", "Considering the average azimuthal velocity component $ \\mathrm {v}_\\phi (r)$ , the circulation is $\\oint \\mathbf {v}\\cdot d\\mathbf {l}=\\frac{h}{m}n= \\mathrm {v}_\\phi (r)2\\pi r$ where $n=f_rN$ is the number of vortices enclosed by the contour of radius $r$ .", "Using (REF ), we obtain the velocity profile $\\mathrm {v}_\\phi (r)\\simeq \\frac{cN}{(R/\\xi )^{1-\\bar{\\alpha }}(r/\\xi )^{\\bar{\\alpha }}}.$ This inertial cluster has a power-law velocity profile determined by the specific radial exponent $\\bar{\\alpha }$ .", "In contrast, the velocity profile of an Abrikosov vortex lattice rotating at frequency $\\omega $ approaches that of a rigid body $ \\mathrm {v}_\\phi (r)=\\omega r$ , and a single quantum vortex has profile $ \\mathrm {v}_\\phi (r)=\\hbar /mr$ .", "The inertial cluster velocity profile (REF ) is compared with sampled distributions (see below) in Figure REF ." ], [ "Scale expansion of a large cluster", "We now illustrate the role of the radial power law exponent $\\bar{\\alpha }$ and the classical velocity distribution by sampling a large cluster.", "We vary our choice of scale range for the vortices ($r_{\\rm max}$ in Eq.", "REF ), and investigate how the range of $k^{-5/3}$ changes.", "A characteristic wave number measuring cluster size for a sample involving $n_c$ vortices in a given cluster is given by $\\bar{k}\\equiv 2\\pi /\\bar{r}$ Where $\\bar{r}=\\frac{1}{n_c}\\sum _{p=1}^{n_c}r_p$ .", "In the figure we plot the scale $\\bar{k}$ to give an indication of the range of $k^{-5/3}$ scaling.", "We also plot $k_{\\rm max}$ corresponding to the largest vortex separation scale in the system $k_{\\rm max}=2\\pi /{\\rm max}|{\\bf r}_p-{\\bf r}_q|.$ For $k<k_{\\rm max}$ the velocity field approaches that of a single vortex.", "In Figure REF vortex distributions are sampled for $N=100$ vortices of the same sign using the sampling scheme (REF ), (REF ), for $r_{\\rm min}=\\xi $ and different values of $r_{\\rm max}$ .", "Individual samples are shown to indicate the spread of vortices, and the velocity profiles and kinetic energy spectra are computed by averaging over $n_s=100$ samples.", "The mean azimuthal velocity compares well with Equation (REF ), showing that the specific radial exponent $\\bar{\\alpha }$ in (REF ) also determines the power law of the azimuthal velocity field, as seen in (REF ).", "For scales larger than $r_{\\rm max}$ , $v_\\phi (r)$ returns to the $r^{-1}$ scaling for a charge $N$ vortex [clearly seen in Fig.", "REF $(a)$ ].", "$N=100$ vortices distributed up to $r_{\\rm max}=60\\pi \\xi $ gives an inertial range in the kinetic energy spectrum of approximately one decade, for $\\bar{\\alpha }=0.4$ (as the number of vortices in a given scale range increases, $\\bar{\\alpha }\\rightarrow 1/3$ ).", "Increasing the upper scale cutoff to $r_{\\rm max}=400\\pi \\xi , 4000\\pi \\xi $ gives $\\bar{\\alpha }=0.65,0.83$ , with 1.5 and 2 decades of inertial range respectively.", "Thus expanding the scale range of power-law behavior expands the inertial range, but requires the azimuthal velocity profile to steepen.", "In the UV region of the spectrum the $k^{-3}$ law always holds [Eq.", "(REF )], while the inertial range holds for $ \\bar{k}\\lesssim k\\lesssim \\xi ^{-1}$ , and single vortex behavior is apparent for $k< k_{\\rm max}$ .", "Figure: Kinetic energy spectra for N + =100N_+=100, and N - =100N_-=100 vortices, averaged over n s =100n_s=100 samples (blue curves).", "The vortices are distributed (a) uniformly over the (600ξ) 2 (600\\xi )^2 square domain (inset), and clusters are sampled according to () and (), with (b) n c =5n_c=5 vortices in each cluster, r min =ξr_{\\rm min}=\\xi , r max =10πξr_{\\rm max}=10\\pi \\xi , and (c) n c =20n_c=20 vortices in each cluster and r max =100πξr_{\\rm max}= 100\\pi \\xi , with each cluster center uniformly distributed over the periodic domain as in (a).", "The power-law sampling in figures (b) and (c) requires a radial exponent of α ¯=0.8\\bar{\\alpha }= 0.8, and 0.750.75, and gives a Kolmogorov k -5/3 k^{-5/3} region in the corresponding kinetic energy spectrum of ∼1\\sim 1 and 2 decades respectively." ], [ "Clustering in a neutral distribution", "We now consider the role of increasing clustering in expanding the inertial range of the kinetic energy spectrum.", "First, in $n_s=100$ samples, we distribute $N_+=N_-=100$ vortices over a uniform periodic domain; one such sample is shown in the inset of Figure REF (a).", "The corresponding kinetic energy spectrum shows the correct UV-region spectrum given by Eq.", "(REF ), and also approaches the $E_i(k)\\propto k$ form for $k<k_{\\rm max}$ , given by Eq.", "(REF ).", "For $k_{\\rm max}\\lesssim k\\lesssim \\xi ^{-1}$ the spectrum is less steep than $k^{-5/3}$ and the system lacks an inertial range.", "In Figure REF (b) the vortices are sampled as 40 clusters of $n_c=5$ vortices of the same sign according to (REF ) and (REF ), with $\\bar{\\alpha }=0.8$ , and $r_{\\rm min}=\\xi $ , $r_{\\rm max}=10\\pi \\xi $ .", "The 10 +ve and 10 -ve cluster centers are uniformly distributed as in Fig.", "REF (a), as seen in the sample (inset).", "This distribution yields a $k^{-5/3}$ power-law kinetic energy spectrum over $\\sim 1$ decade of wave numbers.", "By further expanding the scale of clustering while reducing the number of clusters to preserve $N_+$ and $N_-$ , we find the inertial range can be extended.", "In Figure REF (c) samples consists of $n_c=20$ vortices in each cluster, distributed between $r_{\\rm min}=\\xi ^{-1}$ and $r_{\\rm max}=100\\pi \\xi $ , with $\\bar{\\alpha }=0.75$ , and giving $\\sim 2$ decades of inertial range.", "We note that, compared with Fig.", "REF , $\\bar{k}$ , shown in Fig.", "REF (b) and (c) does not correspond so well with the lower bound on the inertial range, presumably because of the significant space between clusters in the neutral system." ], [ "Scenario of forced turbulence in a compressible 2D superfluid", "The canonical model of 2D classical turbulence consists of a velocity field described by the 2D Navier-Stokes equation $\\frac{\\partial \\mathbf {v}}{\\partial t}+(\\mathbf {v}\\cdot \\nabla )\\mathbf {v}=-\\frac{1}{\\rho }\\nabla p+\\nu \\nabla ^2\\mathbf {v}-\\lambda \\mathbf {v}+\\mathbf {f}_\\mathbf {v}.$ Figure: Illustration of an inertial range (non-shaded region) for the incompressible portion of kinetic energy in forced compressible 2DQT.", "The E(k)∝k -3 E(k)\\propto k^{-3} region arises from the structure of the vortex core and thus is not a signature of vortex configurations and vortex turbulence.", "This ultraviolet region can thus not support energy cascades, nor does this region correspond to enstrophy cascades.", "Net energy injected at k F ∼ξ -1 k_F\\sim \\xi ^{-1} in the form of vortices can only move towards the infrared.", "The Kolmogorov law E(k)∝k -5/3 E(k)\\propto k^{-5/3} occurs in the inertial range of fully developed turbulence.", "The far-infrared region is given by E(k)∝kE(k)\\propto k for a system with no net vorticity, and is evident for k≪L -1 k\\ll L^{-1} where LL is the largest intervortex distance.", "For forcing at smaller wavenumbers, the spectrum may be more complex, possibly involving other forms of energy and enstrophy flux.The density $\\rho $ of the incompressible fluid is held constant by the pressure field $p$ , $\\nu $ is the kinematic viscosity, $\\mathbf {f}_\\mathbf {v}$ is a forcing term and $\\lambda $ represents linear frictional damping arising from irreducible 3D aspects of the system in which the 2D flow resides.", "If the fluid is subjected to suitable forcing it will develop an inverse energy cascade and a direct enstrophy cascade, with associated $k^{-5/3}$ and $k^{-3}$ power laws respectively [13].", "An inverse energy cascade induced by small-scale forcing can be steady because the $-\\lambda \\mathbf {v}$ term damps energy at large length scales [9].", "For a homogeneous compressible superfluid subject to forcing from an external potential, (REF ) can be written as $\\frac{\\partial \\mathbf {v}}{\\partial t}+(\\mathbf {v}\\cdot \\nabla )\\mathbf {v}=-\\frac{g}{m}\\nabla \\rho +\\nu _q\\nabla ^2\\mathbf {v}+\\mathbf {f}_\\mathbf {v},$ where the forcing $\\mathbf {f}_\\mathbf {v}\\equiv -\\nabla V( \\mathbf {r} ,t)/m$ is assumed to be spatially localized.", "The lack of a $-\\lambda \\mathbf {v}$ frictional damping term means, in the classical case, that if an inverse energy cascade develops as a result of steady forcing, it is not expected to be stationary.", "In the superfluid case, the compressibility of the fluid allows vortex-antivortex annihilation, which couples energy into the sound field.", "This interaction between the sound and vorticity fields renders the calculation of energy fluxes in compressible superfluid systems particularly difficult and somewhat ambiguous [30].", "As we have shown above, in contrast with the classical Kraichnan scenario utilizing a 2D Navier-Stokes analysis, a $k^{-3}$ spectrum for $k > \\xi ^{-1}$ for a 2D quantum fluid is not caused by a direct enstrophy cascade but is rather a consequence of the vortex core structure, and thus should not be interpreted in terms of vortex configuration dynamics (note that vortex core shape excitations can be neglected, since they constitute a component of the sound field).", "Given forcing at a wavenumber $k_F\\sim \\xi ^{-1}$ , and minimal vortex-antivortex annihilation, the incompressible kinetic energy can only move toward the infrared.", "This scenario is shown schematically in Figure REF .", "It has been shown that dipole recombination provides a route for a direct energy cascade to develop in 2D GPE dynamics [30].", "This mechanism provides a means for opposite-sign vortices to approach zero distance, coupling vortex energy to the sound field during vortex annihilation.", "However, if the forcing leads to significant clustering of like-sign vortices faster than recombination occurs, or prior to recombination occurring, dipole-decay will be strongly inhibited.", "This suggests that under the right conditions of forcing an inverse energy cascade can become the dominant mechanism of energy transport between distinct length scales." ], [ "Kolmogorov constant and clustered fraction", "By making use of the universal onstrophy and the condition of continuity at $k\\xi \\approx 1$ we have found that the $k^{-5/3}$ power-law given by (REF ) describes the spectrum of numerically sampled vortex configurations that exhibit a $s^{-1/3}$ power law for the vortex separation data.", "While individual spectra and configurations do not give information about dynamics, in particular, the direction in $k$ -space of any energy cascades, the power law suggests the existence of an inertial range comprised of vortices.", "In a cascade, such a configuration will transfer incompressible energy between scales while conserving energy.", "Assuming the infrared portion of our double-power-law analysis, namely (REF ), will also describe such a cascade, we can cast it as a statement about the Kolmogorov constant in terms of the one-vortex onstrophy and the slope of the radial wavefunction at the vortex core.", "To write (REF ) in standard form, we introduce the unique $N$ -vortex quantity with dimensions energy/mass/time that can be constructed from $\\Omega \\xi ^2$ , $m$ , and $\\hbar /\\Omega \\xi ^2$ : $\\epsilon _N\\equiv \\frac{(\\Omega \\xi ^2)^2}{m\\hbar }N^{3/2}.$ We then find $\\frac{E_{i,{\\rm C}}^N(k)}{m}=\\bar{C}_{2D}\\epsilon _N^{2/3}k^{-5/3}$ where the remaining quantities have been absorbed into the dimensionless Kolmogorov constant: $\\bar{C}_{2D}\\equiv \\Lambda ^{2} \\left(\\frac{\\mu }{\\Omega \\xi ^2}\\right)^{1/3},$ $\\mu =\\hbar ^2/m\\xi ^2$ in the homogeneous system, and the bar notation distinguishes the quantum system.", "In the dilute Bose gas the 2D interaction parameter is $\\mu /n_0=g_{2}=4\\pi \\hbar ^2a/ml$ , where $l$ is the characteristic thickness of the three-dimensional system [30], [57].", "In terms of this length we find $\\bar{C}_{2D}=\\Lambda ^{2} \\left(\\frac{mg_{2}}{2\\pi \\hbar ^2}\\right)^{1/3}=\\Lambda ^{2} \\left(\\frac{2a}{l}\\right)^{1/3}.$ Figure: Time evolution of grid turbulence in damped GPE.", "Left: particle density (rescaled to the peak density).", "Center: vortices colored by charge, with total (NN) and clustered (N c N_c) numbers of vortices.", "The field of view is (1024ξ) 2 (1024\\xi )^2.", "Right: Incompressible energy spectra (circles), with the Kolmogorov ansatz [red line, Eq.", "()], the ansatz for a polarized cluster of NN vortices [dashed line, Eq.", "()], and the universal k -3 k^{-3} region [blue line, Eq.", "()].Figure: Plot of C ¯ 2D * (k)\\bar{C}^*_{2D}(k) [see Eq.", "()] computed from the grid turbulence kinetic energy spectra of Figure .", "The horizontal dashed line gives C ¯ 2D \\bar{C}_{2D} from Eq.", "().We emphasize that the physical input needed to arrive at this form of the Kolmogorov constant is (i) accounting for the structure of a compressible quantum vortex in determining the ultraviolet spectrum, and (ii) imposing continuity of the ultraviolet spectrum at $k\\xi =1$ to a Kolmogorov power-law in the infrared.", "In classical turbulence, $C_{2D}\\simeq 7$  [58].", "To give an example of how $\\bar{C}_{2D}$ may be evaluated for a compressible superfluid exhibiting 2DQT, we consider a $^{87}$ Rb BEC that is homogeneous in the $x-y$ plane, and that is harmonically trapped in the $z$ -dimension with trap frequency $\\omega _z=2\\pi \\times 5000$ Hz.", "We use $m=1.44\\times 10^{-25} {\\rm kg}$ , $a=5.8{\\rm nm}$ , for which $l=\\sqrt{2\\pi \\hbar /m\\omega _z}=0.38$ $\\mu $ m, and $g_2=0.197 \\hbar ^2/m$ .", "For these values, $\\bar{C}_{2D} = 0.212$ .", "Note also that by defining the configurational rate constant (REF ), we have confined this discussion to a scale invariant distribution involving $N$ vortices.", "This expression suggests that such a configuration can support an inverse energy cascade at the rate $\\epsilon _N$ .", "The foregoing discussion involves an ideal distribution of $N$ vortices configured with the $\\alpha =1/3$ power law.", "It is clear from Fig.", "REF that the vortices do not have to all have the same sign of circulation, but they must be configured into clusters of vortices with the same sign.", "The universality of $C_{2D} $ in classical turbulence in incompressible fluids leads us to postulate that the condition that all $N$ vortices are power-law clustered can be further relaxed.", "For fully developed quantum turbulence involving $N$ vortices, we interpret $N$ as the participation number, representing the number of vortices in a scale-free turbulent configuration, which in the case of a fully polarized cluster is maximal.", "Imperfect clustering involves fewer vortices in power-law cluster configurations, and an effective participation number that is the number of clustered vortices $N_c<N$ , namely, the number with nearest-neighbors of the same sign.", "Making the replacement $N\\rightarrow N_c$ in (REF ), we propose the ansatz spectrum $E_{i,{\\rm C}}^N(k)=\\Lambda ^{2}\\frac{N_c\\Omega \\xi ^{3}}{(k\\xi )^{5/3}},$ as a more general definition for systems that have incomplete clustering in the inertial scale range.", "We test this hypothesis in the next section in dynamical simulations of the forced dGPE.", "It is important to note that the condition of continuity at $k\\xi =1$ is no longer exactly met, since only the value $N_c\\equiv N$ will produce an infrared spectrum with $k^{-5/3}$ that is continuous with the ultraviolet power law approximation at $k=1/\\xi $ .", "We also note that a more general measure of clustering, namely the polarization index ($P$ ) was introduced in Ref.", "[59] measuring the degree and type of spatial clustering of like-sign vortices in 3DCT.", "The Kolmogorov $k^{-5/3}$ spectrum was found to correspond to the partially polarized value $P=1/3$ , while other scaling laws yield differing polarizations.", "The clustered fraction used in the present work is a simpler (global) measure of polarization as it does not contain information about the spatial distribution of vortices, but it is only relevant for 2DQT.", "We can also write down an expression for $\\bar{C}_{2D}$ for a general energy spectrum that may be computed numerically from simulation data, by making use of the ansatz (REF ).", "This is equivalent to using (REF ) with $N\\rightarrow N_c$ , from which we can define the function $\\bar{C}_{2D}^*(k)=E_{i,{\\rm C}}^N(k)\\frac{(k\\xi )^{5/3}}{N_c\\Omega \\xi ^3}\\left(\\frac{\\mu }{\\Omega \\xi ^2}\\right)^{1/3}.$ In a region where the spectrum is approximately $k^{-5/3}$ , $\\bar{C}_{2D}^*(k)$ will be approximately constant, and it may be compared with the prediction (REF ).", "We test this numerically in the next section.", "We note that in 3DQT an energy bottleneck has been predicted for the direct energy cascade [59], [60], and also observed in GPE simulations [61].", "It occurs due a mismatch between the rates of energy transport at large length scales (hydrodynamic regime) and small length scales (Kelvin wave cascade).", "The mismatch causes energy to pile up at the length scale where the two cascades meet.", "This raises the possibility of a bottleneck in 2D, although Kelvin waves are disallowed in 2DQT, so this particular mechanism would not be relevant.", "However, for a given forcing mechanism, it is possible that the rate of transporting energy to large length scales may not be high enough to remove all of the vortex energy introduced at the forcing scale.", "Thus a bottleneck could still occur, and our assumption of continuity of the spectrum at $k\\xi \\simeq 1$ may not hold in general.", "We return to this question in the next subsection, where we find some indication of an energy bottleneck at the forcing scale in numerical simulations." ], [ "Damped Gross-Pitaevskii Dynamics", "We now consider a simulation of the forced dGPE that generates significant clustering of vortices of the same sign.", "The system consists of a homogeneous superfluid with periodic boundary conditions, stirred by dragging four Gaussian obstacle beams through it at a constant speed [32], thus modeling grid turbulence in a BEC.", "When an obstacle is dragged through a superfluid sufficiently rapidly, superfluidity cannot be maintained.", "For slowly moving obstacles, the superfluid will adapt to the forcing, and vortices are not formed.", "Above a critical velocity $\\mathrm {v}_c$  [62] vortex dipoles are periodically formed in the wake of the obstacle, injecting linear momentum into the superfluid.", "Sufficiently rapid motion ($\\mathrm {v}\\gg \\mathrm {v}_c$ ) causes many vortices to be nucleated behind the obstacles in a chaotic fashion [63], involving clustering of like sign vortices.", "Our choice of obstacle speed puts the system dynamics in the latter category.", "We work in units of $\\mu $ , $\\xi $ , and $\\xi /c$ for energy, length, and time, respectively.", "In these units the specific parameters we choose (see the previous subsection) are $g_2=0.197\\mu \\xi ^2$ , corresponding to homogeneous density $n_0=\\mu /g_2=5.26\\xi ^{-2}$ , and $N_{tot}=5.5\\times 10^6$ particles in a homogeneous 2D system of side length $L=1024\\xi $ .", "The Gaussian potentials each have fixed $1/e^2$ width of $w_0=\\sqrt{8}\\xi $ , and height $V_0=100\\mu $ , and are initially located at $x=-L/2+8\\xi $ , $y=\\pm L/8$ , $\\pm 3 L/8$ .", "Numerically, we proceed by first finding a ground state of (REF ), for a homogeneous system with periodic boundary conditions, subject to the localized obstacle beams.", "We then transform into a frame translating at ${\\rm v}_0=0.8 c$ , and maintain the obstacle locations relative to this frame, creating a dragging grid of stirring beams.", "A small amount of initial noise is added to the wave function to break the reflection symmetries of the system.", "We thus evolve the system according to (REF ) for the same potential, but with the Galilean transformed nonlinear operator ${\\cal L}\\rightarrow {\\cal L}+i\\hbar {\\rm v}_0\\partial _x$ .", "During evolution the dimensionless damping rate is set to $\\gamma =0.003$ .", "The time evolution of the system is shown in Figure REF at three times, at approximately $(1/3, 2/3,1)L/{\\rm v}_0$ , so as to avoid any periodic flow effects in the $x$ -direction.", "The four obstacle beams generate many vortices (up to $N\\sim 10^3$ ), and significant clustering ($N_c/N \\gtrsim 0.6$ ).", "The incompressible kinetic energy spectrum shows a wide region that is well described by the $k^{-5/3}$ form of Equation (REF ).", "For later times [Figure REF (b), (c)] the spectrum shows a significant pile up around the forcing scale $k_F\\sim \\xi ^{-1}$ , suggesting a mismatch between the rates of injection and transport of incompressible kinetic energy.", "In Figure REF we compare the function $\\bar{C}^*_{2D}(k)$ [Eq.", "(REF )], as numerically computed from our simulation data, with the analytical prediction of the Kolmogorov constant $\\bar{C}_{2D}$ [Eq.", "(REF )].", "The region of $k^{-5/3}$ appears as a broad flat region that is in close agreement with $\\bar{C}_{2D}=0.212$ pertaining to our simulation parameters." ], [ "Conclusions", "To summarize, we have investigated relationships between the concepts of 2D turbulence in classical fluids and the emerging topic of 2D quantum turbulence of vortices, specifically as it relates to Bose-Einstein condensates.", "We established a link between the hydrodynamic limit of the damped GPE and the Navier-Stokes equations, providing an estimate of a quantum Reynolds number for superfluid flows in BECs.", "We have given a theoretical treatment of the incompressible kinetic energy spectrum that explicitly incorporates the vortex core structure in a compressible superfluid.", "The incompressible kinetic energy spectrum for a compressible superfluid is deconstructed in terms of single-vortex contributions determining a unique ultraviolet power-law where the energy spectrum scales as $k^{-3}$ , and a contribution that depends on the configuration of vortices within the fluid that determines the infrared region of the spectrum.", "For the configurational regime we find: The spectrum only depends on the distribution of vortex separations and the sign of the circulation of each quantum vortex.", "If the distribution of vortex separation $s$ for a system of vortices of the same sign is a power-law $\\propto s^{-\\alpha }$ with exponent $\\alpha =1/3$ , the kinetic energy spectrum will take the universal Kolmogorov form $\\propto k^{-5/3}$ , as shown for point vortices [53].", "Localized clusters of $N$ vortices of the same circulation with this power law distribution can be constructed by sampling using a specific radial exponent $\\bar{\\alpha }$ that depends on the number of vortices and the scale range over which they are distributed.", "The azimuthal velocity field of a large cluster is determined by $\\bar{\\alpha }$ .", "By inflating the scale range of a cluster we find that $\\bar{\\alpha }$ increases, the velocity field is steepened and the inertial range expands to larger scales.", "In a neutral system the inertial range can be extended by increasing the size of clusters while decreasing their number.", "The universal form of the UV region of the kinetic energy spectrum imposes a strong constraint.", "If the Kolmogorov power law occurs in the infrared region, then the postulate of continuity between the infrared and ultraviolet regions completely determines the spectrum when the ultraviolet and infrared regions are approximated as power laws.", "Physically, this corresponds to the inertial range extending down to the smallest configurational scale of the system $\\sim \\xi $ .", "We note that the postulate of continuity may not be relevant for all systems or forcing mechanisms.", "We infer an analytical value for the Kolmogorov constant [Eq.", "(REF )] under the conditions of spectral continuity at the cross-over scale for a system of vortices of the same sign.", "To assess the validity of this inference for dynamical situations we compare our analytical results with spectra from a numerical simulation of the forced dGPE for the specific case of a dragging a grid of obstacles through an otherwise homogeneous BEC.", "We find reasonable agreement provided we introduce the concept of a clustered fraction $N_c/N\\le 1$ , which is the fraction of vortices that have same-sign nearest neighbors.", "This measure discounts all vortex dipoles from the configurational analysis.", "We then observe good agreement between our Kolmogorov ansatz [Eq.", "(REF )], and the spectrum calculated from the dGPE data.", "We also find that the predicted value of the Kolmogorov constant is in close agreement with the numerical simulations [Figure REF ].", "We note that while our analysis indicates that vortex positions and circulations are enough to determine an approximate incompressible kinetic energy spectrum, the reverse is not necessarily true: a Kolmogorov spectrum does not carry information about specific vortex distributions.", "Nevertheless, our analysis does indicate that the number of vortices in a quantum fluid can in principle be directly determined from the ultraviolet energy spectrum.", "Moreover, the concept of a cascade in turbulence implies system dynamics and energy transport, yet aside from our numerical simulation example, our analytical approach is an instantaneous measure.", "Importantly, one must determine means of characterizing vortex motion and relate such dynamics to the cascade concept.", "The field of 2D quantum vortex turbulence is relatively new, compared with the much longer histories of 3D superfluid turbulence, 2D classical turbulence, and even dilute-gas Bose-Einstein condensation.", "Point-vortex models have been extensively used in descriptions of superfluid dynamics as well as in 2D classical turbulence, although point-vortex distributions can only serve as approximate models of real 2D classical flows.", "Our approach merges concepts from each of the above subjects in order to develop a new understanding of 2D quantum turbulence.", "By considering the compressibility of a dilute-gas BEC, we find an analytical expression for the ultraviolet incompressible kinetic energy spectrum and an $N$ -vortex equivalent of enstrophy in a quantum fluid, which we term the onstrophy.", "For the infrared region, point vortex models are sufficient, and vortex configurations serve to identify spectra as summarized above.", "Taken together, the primary new outcome of our analysis is a link between vortex distributions, vortex core structure, and power-law spectra for 2D compressible quantum fluids.", "Future work on 2D quantum vortex turbulence will involve numerical simulations and comparisons with our analytical results, extension of this analysis to confined and inhomogeneous density distributions, characterization of vortex dynamics and the time-dependence of energy spectra particularly in relation to clustering [64], inverse-energy cascades, and the nonthermal fixed point [65], [31], and investigation of connections with weak-wave turbulence in BEC [66], [67], [16], [68], [69], [18], [19].", "We also believe that observing vortex distributions such as the $\\alpha =1/3$ power-law for localized clusters may provide a new means of quantitatively characterizing 2D quantum vortex turbulence through direct experimental observations of vortex locations in a forced 2D superfluid, and we are working towards realizing such experimental observations.", "We thank Matt Reeves for assistance with numerical simulations, and Aaron Clauset for providing code for sampling power-law distributions.", "We also thank Thomas Gasenzer, Gary Williams, Matt Reeves, Sam Rooney, Blair Blakie, Murray Holland, Giorgio Krstulovic, Michikazu Kobayashi, and Ewan Wright for useful discussions.", "We are supported by the Marsden Fund of New Zealand (contract UOO162) and The Royal Society of New Zealand (contract UOO004) (AB), and the US National Science Foundation grant PHY-0855467 (BA)." ] ]

[ [ "A photonic crystal cavity-optical fiber tip nanoparticle sensor for\n biomedical applications" ], [ "Abstract We present a sensor capable of detecting solution-based nanoparticles using an optical fiber tip functionalized with a photonic crystal cavity.", "When sensor tips are retracted from a nanoparticle solution after being submerged, we find that a combination of convective fluid forces and optically-induced trapping cause an aggregation of nanoparticles to form directly on cavity surfaces.", "A simple readout of quantum dot photoluminescence coupled to the optical fiber shows that nanoparticle presence and concentration can be detected through modified cavity properties.", "Our sensor can detect both gold and iron oxide nanoparticles and can be utilized for molecular sensing applications in biomedicine." ], [ "Main text", "Nanoparticles (NPs) have recently been the subject of much attention for their uses in nanomedicine [1], [2], molecular imaging [3], [4], and phototherapy [5], [6].", "Due to their small sizes (typically less than 100 nm), nanoparticles can infiltrate cancerous tissue through the enhanced permeability of the vasculature and can act as markers for imaging tumors [7].", "Active targeting of nanoparticles to specific tissue or cell surface proteins can be accomplished through proper ligand chemistry.", "Gold NPs, for example, can be detected through wide-field imaging of covalently linked Raman reporter molecules and have been used to image small tumors in mice models [7], [8], [9].", "Plasmonic gold nanoshells on the other hand have been used for photothermal ablation of tumors by locally heating tissue with a laser pump [5].", "Recently we developed a method to functionalize optical fiber tips with semiconductor photonic crystal (PC) cavities by using a simple epoxy transfer process [10].", "With this design, light can be coupled back and forth between the cavities and the optical fibers for efficient optical readout that avoids a bulky free-space setup.", "Furthermore, the optical fiber tip is well suited for remote sensing measurements in tough environments such as the body due to its compact form factor.", "In this letter, we show how our fiber photonic crystal, or fiberPC, can operate as a nanoparticle sensor for both gold and iron oxide and can even quantitatively determine NP concentration.", "In contrast to previous wide-field imaging techniques requiring high (20-200mW) pump powers and bulky optics [7], [8], [9], our method is fully embedded with the fiber which can be envisioned as an endoscopic tool that requires less than 1 mW.", "With this distinct modality, nanoparticles can be detected simply by their proximity to the cavity, allowing for a unique avenue of NP sensing (e.g., intraoperatively) in future biomedical studies.", "Devices were fabricated following our previous epoxy transfer methodology [10].", "Photonic crystal cavities were made out of a 220 nm thick membrane of GaAs with embedded high density InAs quantum dots (QDs) as internal light sources.", "Modified L3 defect cavities were patterned with a lattice constant a = 330 nm and hole radius r = 0.22a [11].", "In contrast to our former work, we generate just a single cavity at the center of our circular template rather than an array of many cavities.", "This is because we have improved our assembly process to have minimal alignment error (0-3 $\\mu $ m typical lateral offset) and because monitoring just a single cavity is preferable for sensing.", "Fig.", "1(a) shows a scanning electron microscope (SEM) of a completed fiberPC device.", "As in our previous work, the membrane matches and smoothly covers the optical fiber tip with epoxy regions located away from the PC cavity center.", "A schematic of our test setup is shown in Fig.", "1(b).", "The fiberPC pigtail is spliced to a patch cable and connected to a custom made wavelength division multiplexor (WDM) that is built to combine 1300 nm and 830 nm signals (Micro-Optics Inc.).", "We pump our devices with an 830 nm laser diode (LD) and collect the return photoluminescence (PL) with our spectrometer as we insert our sensors into various solutions.", "Laser pump powers ranged from 10 $\\mu $ W to 2.5 mW (measured prior to the fiberPC membrane) and integration times were typically several hundred ms.", "The brightness of our signals confirms the good quality of our device design as well as the high efficiency of collection.", "For our first experiments we use solutions of 15 nm gold nanoparticles that are colloid stabilized with carboxylic acid terminated polyethylene glycol.", "Fig.", "2(a) shows a PL spectrum of a fiberPC device in air before any testing when pumped with 250 $\\mu $ W of laser power.", "A fundamental cavity mode appears at 1278 nm along with several other peaks at longer wavelength which are likely PC band edge modes.", "The Q-factor of the fundamental mode is 800 prior to solution testing.", "We next insert our device into a 12.5 nM solution of gold nanoparticles as seen in Fig.", "2(b).", "Inside this solution the cavity mode redshifts by 12 nm, increases in quality factor to 1330, and decreases in emission intensity, all a result of the higher refractive index cladding provided by the water (versus the original air).", "Incidentally, the Q-factor goes up in this case due to the improved symmetry of the oxide and water claddings which reduces lossy TE-TM modal conversion [12].", "At this stage, metal nanoparticles are not detected since a control solution of water only has the exact same effect on our device.", "When we retract our device from the solution with the pump laser turned off during the retraction we find that the PL spectrum (observed 10 seconds after retraction with the pump then turned on) replicates the original air spectrum prior to solution testing (Fig.", "2(c)).", "An optical microscope image of the device shows that the cavity region is not modified and there is a circular coffee-ring like annulus where a small amount of NPs deposited as a droplet evaporated from the fiber tip [13] (see Supplementary material for a real-time video of a droplet evaporating on a fiber tip).", "A very interesting and different scenario results when we keep our laser pump on during the fiberPC sensor withdrawl from the solution (see Supplementary material for a real-time video of a droplet evaporating with the laser turned on).", "As shown in the PL spectrum from Fig.", "2(d), the cavity modes are now almost completely eliminated.", "The original fundamental mode is only barely visible now at 1298 nm, or 20 nm redshifted from air, and the Q-factor has dropped to 650.", "Examining the optical microscope picture it is clear that a large aggregation of metal NPs has formed directly over the cavity, suggesting that these nanoparticles are responsible for the change in cavity parameters.", "Likely, NPs cause some absorption of the optical field as well as perturb the uniform refractive index of the cavity cladding resulting in excess scattering [14].", "The higher refractive index of the particles result in a large cavity redshift as well.", "SEM close-up images of the metal aggregate are shown in Fig.", "3(a).", "An explanation of these results requires an accurate understanding of the numerous physical mechanisms at play here.", "Recent studies of nanoparticle aggregation in evaporating droplet systems both with and without optical illumination have shown that convection, hydrothermal effects, surface forces, and chemical interactions all take place [15], [13], [16].", "Our proposed model is visualized in Fig.", "3(b).", "When a fiberPC sensor is retracted from a NP solution, a nanoliter sized droplet is formed on the fiber tip.", "This droplet begins to quickly evaporate in air, which causes vigorous convection currents within the droplet [15], [13].", "Assisting in these evaporative convection currents is a hydrothermal contribution from the pump laser [15].", "Although the pump laser is meant to provide excitation for just the semiconductor cavity, we find that 55% of the light is transmitted beyond the membrane.", "This weakly focused light will be absorbed by both the water and nanoparticles resulting in a temperature rise in the center portion of the droplet of at most a few 10s of K (for our laser intensities) [15].", "Such a small temperature rise can still have a large impact on enhancing the Marangoni convective currents that circulate in a toroidal pattern, propelling NPs into the center of the droplet [15], [13].", "As for the QD photoluminescence, we don't believe it plays a significant role in NP aggregation since it is orders of magnitude weaker than our pump laser.", "The precise manner of NP aggregation is not quite as well understood in the literature, but several mechanisms are possible.", "Bahns et al.", "concluded that high temperature rises in excess of 100K were responsible for carbon to metal NP wetting interactions [15].", "However, our temperature rises are far too low to allow for gold-gold interactions.", "Pure optical trapping via the gradient optical force as observed by Yoshikawa et al.", "is ruled out here since our optical field is orders of magnitude too low for this [17].", "Instead we believe that light driven photochemical interactions are likely responsible for NP aggregation.", "Laser light could potentially remove repulsive capping ions from NP surfaces or could even cause ligands to dissociate from the NPs, allowing nanoparticle-nanoparticle Van der Waals attractive interactions [15], [18].", "More detailed studies need to be done to elucidate the precise attractive mechanisms.", "We also find that the aggregation process is reversible, and that a subsequent dip of the fiberPC tip in water or in the original NP solution with the pump laser turned off results in a clean washing of the NPs.", "Therefore the sensors are not limited to a one-time use.", "As further evidence that droplet evaporation and convection effects play an important role, we repeat the experiment with a bare fiber having no semiconductor membrane (see Supplementary material for a video of a droplet evaporating on a bare fiber tip).", "We observe a drastically weaker NP aggregation effect on these bare fiber tips.", "The contact angles measured on silica and GaAs surfaces were 10 and 50 degrees, respectively, suggesting that the much smaller droplet on silica surfaces like the bare fiber tip prevent effective convection of NPs into a central aggregate [19].", "Therefore not only does a PC cavity provide optical feedback for a sensing measurement, but the semiconductor surface itself is a necessary component for proper NP concentration.", "We next look at the power and concentration dependencies of gold NP aggregation on fiberPC sensors.", "Fig.", "4(a) shows a curve of the wavelength redshift for the same device as in Fig.", "2 but now in a very low concentration solution of 0.8 nM gold NPs.", "Unsurprisingly the curve is monotonically increasing in wavelength shift with pump power, suggesting that laser driven convection and aggregation are enhanced with higher power.", "Meanwhile, Fig.", "4(b) shows the wavelength shift of another fiberPC device when the pump power is held constant at our laser diode's maximum of 2.45 mW as the NP concentration is varied.", "We see a nearly linear dependence of wavelength shift with NP concentration, indicating that our sensor can be used to quantitatively measure NP concentration.", "Our detection limit for this device is at 100 pM, which corresponds to a 0.7 nm wavelength shift (equal to the cavity half width); however, we believe the detection limit could easily be improved by using higher pump powers or cavities with higher Q-factors.", "In Fig.", "4(c) we examine the other limit of a highly concentrated NP solution (25 nM) pumped at an extremely low power of only 12 $\\mu $ W. Even for this very small pump power, we observe a significant redshift of 15.2 nm and a Q-factor degradation from 1050 to 650.", "Judging from the optical microscope image in Fig.", "4(d) alone, one would conclude that no NPs had been detected; however, the fiberPC is sensitive to even miniscule aggregations of NPs that are not patently visible with the eye.", "Finally, we demonstrate the versatility of our sensor by switching detection to iron oxide NPs.", "We investigated the ultrasmall superparamagnetic iron oxide compound ferumoxytol (Feraheme, AMAG Pharmaceuticals Inc.).", "Ferumoxytol is an FDA-approved iron supplement that has been used in patients for intravenous treatment of iron deficiency anemia [20].", "Due to its superparamagnetic properties, ferumoxytol has also been used as a magnetic resonance (MR) contrast agent [21].", "Ferumoxytol NPs have a core diameter of 7 nm and are coated with carboxymethyl dextran for colloid stabilization.", "Fig.", "5(a) and (b) show PL spectra of a fiberPC sensor before and after submersion into a 400 $\\mu $ g/mL (or 533 nM) concentration solution of ferumoxytol when pumped at 1.75 mW.", "We chose this concentration because it is the value used when labeling cells for MR experiments [21].", "As seen in Fig.", "5, the cavity peaks once again redshift, this time by 21.8 nm and the Q-factor of the right-most mode decreases from 1000 to 770.", "The optical microscope picture in Fig.", "5(b) shows a very similar aggregation at the center, this time from ferumoxytol NPs rather than gold NPs.", "A close-up SEM in Fig.", "5(c) clearly shows the NP aggregation smothering the cavity.", "Contours of the aggregation highlight the fluidic deposition of the NPs much like wet sand dropped on a surface.", "We speculate that similar physical processes of convective concentration and photochemical binding are responsible for these NP aggregations.", "In summary, we have demonstrated a nanoparticle sensor using a semiconductor photonic crystal cavity-optical fiber tip device.", "The cavity-on-a-fiber platform provides robust optical feedback which can be used to sense changes in its external environment through wavelength, Q-factor, and intensity information.", "Nanoparticle concentrations can be quantitatively determined based on the cavity wavelength shift as well.", "In contrast to a sensor built on a large chip substrate, integration of a sensor on a fiber tip with fascile measurement optics could allow for remote testing in difficult environments such as in the human body.", "This modality of sensing could be applied to various areas in biomedicine and nanoscience and is likely to work for numerous other nanoparticles commonly found in research." ], [ "Acknowledgements", "Gary Shambat is supported by the Stanford Graduate Fellowship.", "Gary Shambat also acknowledges the NSF GRFP for support.", "The authors acknowledge the financial support from NCI ICMIC P50CA114747 (SSG), NCI CCNE-TR U54 CA119367 (SSG), and CCNE-T U54 U54CA151459 (SSG).", "Work was performed in part at the Stanford Nanofabrication Facility of NNIN supported by the National Science Foundation.", "We also thank Kelley Rivoire for helpful discussions." ], [ "List of captions", "FIG 1.", "(a) Tilted SEM images of a completed GaAs fiberPC device.", "The two small blurry circles are where epoxy has been applied.", "The circular semiconductor template consists of an outer release region surrounding a central photonic crystal of size 20 x 25 $\\mu $ m. At the center of the PC is an L3 defect cavity as seen in the second inset.", "(b) Schematic of the optical setup for measuring nanoparticles.", "Pump light from a laser diode is sent to the fiberPC tip through a custom 830/1300 nm WDM where it is absorbed above band by the GaAs semiconductor.", "Internal QDs embedded in the GaAs membrane emit PL both outward from the device and back into the fiberPC core where it can be subsequently detected by a spectrometer.", "FIG.", "2.", "(a) PL spectrum of a fiberPC device in air and prior to testing when pumped at 250 $\\mu $ W. Optical image on the right shows the fiber tip face before testing.", "Arrows indicate epoxy droplets.", "(b) PL spectrum of the same sensor now in a 12.5 nM gold NP solution.", "The cavity fundamental mode has redshifted, increased in quality factor, and dropped in intensity.", "(c) PL spectrum in air again after the device has been retracted with the pump laser turned off during retraction.", "The spectrum is almost identical to that in (a) and the fiber tip image shows only a slight circular deposition of NPs on the outer rim.", "(d) PL spectrum in air of the same device but after having retracted the fiber tip with the laser pump turned on.", "The cavity modes are almost completely removed from the spectrum and a large circular aggregation of nanoparticles (indicated by the arrow) is seen in the microscope image.", "FIG.", "3.", "(a) SEM images of a fiberPC sensor with a metal NP aggregation on the cavity.", "Inset shows a close-up image.", "(b) Schematic model of the nanoparticle aggregation effect.", "A solution droplet on the fiber tip begins to evaporate as indicated by the outward flowing arrows (the contact angle of the droplet here is exaggerated for clarity).", "Meanwhile, part of the optical pump transmits through the thin photonic crystal membrane.", "Water and nanoparticles in the weak focus absorb the pump laser light, raising the local temperature of the water and setting up hydrothermal gradients.", "Combined with evaporation, the hydrothermal gradients create Marangoni convective flow which circulates fluid in a toroidal pattern (shown by the circles).", "This circulation propels NPs into the center of the droplet where they begin to aggregate most likely due to photochemical processes.", "Eventually, all the water in solution evaporates and only a deposition of NPs on the cavity remains.", "During this whole process, the quantum dot PL back-coupled into the fiber is observed with a spectrometer.", "FIG.", "4.", "(a) Pump-power dependent wavelength shift of the fiberPC sensor from Fig.", "2 now in a 0.8 nM solution of metal NPs.", "(b) Concentration dependent wavelength shift of a different sensor when the pump power is held at 2.45 mW.", "(c) PL spectrum of a different device before testing and associated optical microscope image.", "(d) PL spectrum of the same device as in (c) after retraction with a 12 $\\mu $ W pump laser turned on in a 25 nM metal NP solution and associated image.", "FIG.", "5.", "(a) PL spectrum of a fiberPC device and the associated optical image of the fiber face prior to testing with ferumoxytol.", "(b) PL spectrum of the sensor after retraction from a 400 $\\mu $ g/mL solution of ferumoxytol with a 1.75 mW pump.", "The spectrum changes are similar to those caused by metal NPs with a large redshift, reduction in Q-factor, and reduction in peak intensity.", "The optical image shows a circular aggregate at the fiber center as seen previously.", "(c) Close-up SEM image of a ferumoxytol aggregate.", "The scale bar is 2 $\\mu $ m." ] ]

[ [ "Velocity of Light in Dark Matter with Charge" ], [ "Abstract We propose an interesting mechanism to reconcile the recent experiments of the Michelson-Morley type and slowdown of the velocity of light in dark matter with a fractional electric charge when the index of refraction of dark matter depends on the frequency of a photon.", "After deriving the formula for the velocity of light in a medium with the index of refraction $n(\\omega)$ in a relativistic regime, it is shown that the local anisotropy of the light speed is proportional to the second order in $n(\\omega) - 1$.", "This result implies that the experiments of the Michelson-Morley type do not give rise to a stringent constraint on the slowdown of the velocity of light in dark matter with electric charge." ], [ "Introduction", "It is nowadays well known that a large portion, about 23 percentages, of the total energy density in the present universe is occupied by unknown material, what we call \"dark matter\".", "The particles, of which dark matter is made, are usually thought to be weakly interacting massive ones, in other words, WIMPs, which are absent in the Standard Model (SM).", "These hypotheoretical particles are supposed to have mass of order ranging from GeV to TeV, and cross section of their annihilation process is comparable to that of the weak interaction.", "Thus they do not completely annihilate in the course of evolution of the universe and their mass density in the present universe could be of the order of the critical density $\\rho _c \\approx 0.52 \\times 10^{-5} GeV/cm^3$ (more precisely, $\\rho _{DM} \\approx 1.2 \\times 10^{-6} GeV/cm^3$ averaged over large distance scales) [1].", "The difficulty of terrestrial experiments for detecting dark matter by direct or indirect methods makes it impossible to clarify its physical properties and therefore answer various questions associated with dark matter such as the mechanism of the generation in the early universe.", "On the other hand, the density of dark matter in clusters of galaxies is determined by various methods of measurement of the gravitational potential.", "For instance, mass distribution in a cluster is obtained by the method of gravitational lensing.", "The result is that most of the mass is due to dark matter distributed smoothly over the cluster.", "Put differently, dark matter, like the usual matter, is more dense in galaxies and it appears as if luminous matter is embedded into the cloud of dark matter of larger size-galactic halo.", "It is easy to understand this concentration of mass of dark matter near clusters of galaxies from a physical viewpoint since the gravitational interaction attracts dark matter near the clusters compared to empty outer space.", "Given our current ignorance of the dark sector in the universe, it seems prudent not to restrict our interests only in WIMPs, though they must be most plausible, and open our mind to many possibilities.", "Indeed, though the particles consisting of WIMPs carry no electric charge, the notion of charged massive particles (CHAMPs) has also appeared periodically [2]-[6].", "Since CHAMPs, which normally carry a full unit of electric charge, receive stringent constraints from searches for heavy hydrogen to direct detection in the underground experiments, the possibility of CHAMPs carrying a fractional charge has been recently investigated [7], [8].", "In this article, we shall assume that the earth itself is embedded into a medium of dark matter with a fractional electric charge.", "Then, it is quite natural to think that the existence of such a dark matter manifests itself the index of refraction $n(\\omega )$ depending on the (angular) frequency of light since the dark matter can interact with the gauge field in a direct way [9].Precisely speaking, an electrically-neutral particle propagating through matter is also refracted if it couples to a charged particle even indirectly.", "For instance, a neutrino can couple to an electron via weak interaction and its refractive property has an important effect on neutrino propagation when the neutrino is massive [10].", "Moreover, we can ask ourselves if the recent experiments of the Michelson-Morley type [11], [12] shed some light on the possibility of detecting the dark matter on the earth through measuring the local anisotropy of velocity of light or not.", "One of motivations behind the present study comes from a recent claim [13] that in case of the OPERA superluminal neutrinos [14], such an assumption shows a clear tension and might be in conflict with the most recent experiments of the Michelson-Morley type about the local anisotropy of velocity of light [11], [12].", "In order to relax this tension, it is useful to recall that the index of refraction of a medium is in general dependent on the frequency of light.", "If we take account of this fact together with its property under the Lorentz transformation in a proper manner, a term depending on the frequency provides an additional contribution to the velocity of light and then the resulting constraint would become consistent with the experimental results of the Michelson-Morley type.", "This article is organized as follows: In the next section, we derive a formula of propagation of light in moving media based on special relativity.", "In Section 3, it is shown that using this formula the local anisotropy of the light speed is proportional to the second order in $n(\\omega ) - 1$ .", "Section 4 is devoted to discussion." ], [ "Velocity of propagation of light in moving media", "In this section, we review special relativity [15], in particular, on the basis of both the addition law of relativistic velocities and the invariance of phase of a plane wave, we shall derive the velocity formula for propagation of light in moving media.", "It is known that the Lorentz transformation for inertial coordinates can be used to derive the addition law of relativistic velocities.", "To show this fact explicitly, let us consider a physical setup where we have a moving point $P$ whose three-dimensional velocity vector $\\vec{u}^{\\prime }$ has spherical coordinates $(u^{\\prime }, \\theta ^{\\prime }, \\phi ^{\\prime })$ in the inertial frame $K^{\\prime }$ , and the frame $K^{\\prime }$ is moving with velocity $\\vec{v} = c_l \\vec{\\beta }$ in the direction of the $x_1$ axis with respect to the inertial reference frame $K$ .", "Here $c_l$ is a universal limiting speed, in other words, the maximum speed of all physical entities [16]-[21].", "Then, one wishes to calculate the velocity $\\vec{u}$ of the point $P$ as seen from the inertial frame $K$ .", "To do so, let us first note that the Lorentz transformation takes the form for the differential expressions $dx_0 &=& \\gamma (dx_0^{\\prime } + \\beta dx_1^{\\prime }), \\nonumber \\\\dx_1 &=& \\gamma (dx_1^{\\prime } + \\beta dx_0^{\\prime }), \\nonumber \\\\dx_2 &=& dx_2^{\\prime }, \\hspace{20.0pt} dx_3 = dx_3^{\\prime },$ where we have defined $\\gamma \\equiv \\frac{1}{\\sqrt{1 - (\\frac{v}{c_l})^2}}$ .", "Since the components of velocity are defined by $u_i^{\\prime } = c_l \\frac{dx_i^{\\prime }}{dx_0^{\\prime }}$ and $u_i = c_l \\frac{dx_i}{dx_0}$ , they transform as $u_1 = \\frac{u_1^{\\prime } + v}{1 + \\frac{v u_1^{\\prime }}{c_l^2}},\\hspace{20.0pt}u_2 = \\frac{u_2^{\\prime }}{\\gamma (1 + \\frac{v u_1^{\\prime }}{c_l^2})},\\hspace{20.0pt}u_3 = \\frac{u_3^{\\prime }}{\\gamma (1 + \\frac{v u_1^{\\prime }}{c_l^2})}.$ For later convenience, let us rewrite this transformation to a more general form $u_{||} = \\frac{u_{||}^{\\prime } + v}{1 + \\frac{\\vec{v} \\cdot \\vec{u}^{\\prime }}{c_l^2}},\\hspace{20.0pt}\\vec{u}_\\bot = \\frac{\\vec{u}_\\bot ^{\\prime }}{\\gamma (1 + \\frac{\\vec{v}\\cdot \\vec{u}^{\\prime }}{c_l^2})},$ where $u_{||}$ and $\\vec{u}_\\bot $ indicate the components of velocity parallel and perpendicular to $\\vec{v}$ , respectively.", "Moreover, it turns out that the magnitude $u$ of $\\vec{u}$ and its polar angles in the two inertial frames are related as follows: $\\phi &=& \\phi ^{\\prime }, \\nonumber \\\\\\tan \\theta &=& \\frac{u^{\\prime } \\sin \\theta ^{\\prime }}{\\gamma (u^{\\prime } \\cos \\theta ^{\\prime } + v)}, \\nonumber \\\\u &=& \\frac{\\sqrt{u^{\\prime 2} + v^2 + 2 u^{\\prime }v \\cos \\theta ^{\\prime } - (\\frac{u^{\\prime }v \\sin \\theta ^{\\prime }}{c_l})^2 }}{1 + \\frac{u^{\\prime }v}{c_l^2} \\cos \\theta ^{\\prime }}.$ The inverse transformation for $\\vec{u}^{\\prime }$ in terms of $\\vec{u}$ can be easily obtained by interchanging primed and unprimed quantities and simultaneously changing the sign of $v$ .", "In what follows, for simplicity, let us focus on the case of the parallel velocities, $\\theta ^{\\prime } = 0$ .", "In this case, the magnitude of $u$ takes the form $u = \\frac{u^{\\prime } + v}{1 + \\frac{u^{\\prime }v}{c_l^2}}.$ If one particularly sets $u^{\\prime } = c_l$ , this expression reads $u = c_l$ , which simply means the postulate of a universal limiting speed, which is an alternative fundamental principle for the principle of invariant speed of light in special relativity [16], [17], [18], [19].", "Next, let us choose $u_{||}^{\\prime } = \\frac{c_l}{n(\\omega ^{\\prime })}$ , thereby giving rise to $u_{||} = \\frac{\\frac{c_l}{n(\\omega ^{\\prime })} + v}{1 + \\frac{v}{n(\\omega ^{\\prime }) c_l}},$ where $\\omega ^{\\prime }$ is the (angular) frequency of light in the inertial frame $K^{\\prime }$ and $n(\\omega ^{\\prime })$ is the index of refraction of a medium.", "The important point is that the index of refraction of a medium in general depends on the magnitude of the frequency of the photon.", "In order to determine $n(\\omega ^{\\prime })$ , we make use of the fact that the phase of a plane wave is an invariant quantity under the Lorentz transformation since the phase $\\varphi $ can be identified with the mere counting of wave crests in a wave train, an operation that must be the same in every inertial frame $\\varphi \\equiv k_\\mu x^\\mu = \\omega t - \\vec{k} \\cdot \\vec{x}= \\omega ^{\\prime } t^{\\prime } - \\vec{k}^{\\prime } \\cdot \\vec{x}^{\\prime }.$ With the frequencies $\\omega = c_l k_0$ and $\\omega ^{\\prime } = c_l k_0^{\\prime }$ , the Lorentz transformation for the wave-number vector $k_\\mu $ reads $k_0^{\\prime } &=& \\gamma (k_0 - \\vec{\\beta } \\cdot \\vec{k}), \\nonumber \\\\k_{||}^{\\prime } &=& \\gamma (k_{||} - \\beta k_0), \\nonumber \\\\\\vec{k}_\\bot ^{\\prime } &=& \\vec{k}_\\bot .$ For light waves, $|\\vec{k}| = k_0$ and $|\\vec{k}^{\\prime }| = k_0^{\\prime }$ , so using $\\theta = 0$ coming from $\\theta ^{\\prime } = 0$ via Eq.", "(REF ) we have the relation $\\omega ^{\\prime } = \\gamma (1 - \\beta ) \\omega = \\sqrt{\\frac{1-\\beta }{1+\\beta }} \\omega \\approx (1 - \\beta ) \\omega ,$ where the last relation holds when $\\beta \\equiv \\frac{v}{c_l}$ is small.", "Thus, the index of refraction of a medium transforms to first order in $\\beta $ as $n(\\omega ^{\\prime }) = n(\\omega ) - \\beta \\omega \\frac{dn}{d\\omega }.$ Plugging this expression into Eq.", "(REF ) (and taking $\\pm v$ into consideration at the same time), it is easy to show that for medium flow at a speed $v$ parallel or antiparallel to the path of the light, the speed of the light, as observed in the laboratory ($K$ -frame), is given to first order in $v$ by $u_{||}^\\pm \\approx \\frac{c_l}{n(\\omega )} \\pm v \\left[ 1 - \\frac{1}{n(\\omega )^2}+ \\frac{\\omega }{n(\\omega )^2} \\frac{dn(\\omega )}{d \\omega } \\right],$ where superscripts $+$ and $-$ on $u_{||}$ denote the speed parallel and antiparallel to the light path, respectively.", "Formula (REF ) is the main result of this section and will be used in the next section." ], [ "The recent experiments of Michelson-Morley type", "We wish to apply the formula (REF ) to the problem that the recent experiments of the Michelson-Morley type could or could not give us useful information on detection of dark matter with electric charge.", "Before doing that, let us present some ideas behind our theory.", "Search for dark matter is underway, but no conclusive evidence has been obtained thus far.", "A direct way for the search is carried out in experiments trying to detect energy deposition in a detector caused by elastic scattering of dark matter off a nucleus.", "Besides the direct search, there are experiments to search for dark matter particles indirectly, which include the search for products of dark matter annihilation.", "One of the most promising indirect ways is to search for monoenergetic photons which are emitted in two-body annihilation processes $X + \\bar{X} \\rightarrow \\gamma + \\gamma , X + \\bar{X} \\rightarrow Z^0 + \\gamma $ where $X, \\bar{X}$ describe a dark matter particle and its antiparticle, respectively.", "By the crossing symmetry, these processes read $X + \\gamma \\rightarrow X + \\gamma , X + \\gamma \\rightarrow X + Z^0$ .", "If the photon energy is small compared to the weak energy scale (i.e., the energy of the photon beam is below the threshold for production of a single $Z^0$ ), only the former process is allowed to occur and it is nothing but the elastic Compton scattering.", "At this stage, let us recall that there is a well-known connection between the index of refraction and the Compton scattering amplitude where the standard formula takes the form [22] $n(\\omega ) = 1 + \\frac{2 \\pi c_l^2 N F(\\omega )}{\\omega ^2},$ where $N$ is the number of scattering centers per unit volume and $F(\\omega )$ is the forward Compton scattering amplitude for the photon scattering off the medium which is a function of the (angular) frequency $\\omega $ of a photon.", "Put differently, in this approach the dark matter with electric charge plays a role of a medium for the photons and provides a dispersive effect on light propagation.", "Since we assume in this article that dark matter carries a fractional electric charge associated with $U(1)_{EM}$ symmetry, it must be a stable Dirac particle with spin $\\frac{1}{2}$ like protons.", "Then, the amplitude for the forward scattering of a photon from a polarization state $\\vec{e}_1$ to the one $\\vec{e}_2$ by the dark matter must be of the general form [22] $F(\\omega ) = \\psi _f^* \\left[ F_1(\\omega ) \\vec{e}_2^* \\cdot \\vec{e}_1+ i F_2(\\omega ) \\vec{\\sigma } \\cdot (\\vec{e}_2^* \\times \\vec{e}_1) \\right] \\psi _i,$ where $\\psi _i (\\psi _f)$ is the wave function of the initial (final) dark matter and $\\vec{\\sigma }$ is the spin matrix of the dark matter.", "In general, both $F_1(\\omega )$ and $F_2(\\omega )$ are complex and have dispersive and absorptive parts.", "If one averages over dark matter spins in the amplitude one is left only with $F_1(\\omega )$ .", "The amplitudes $F_1(\\omega )$ and $F_2(\\omega )$ are separable if one can do experiments with polarized dark matters: $F_1(\\omega )$ corresponds to parallel and $F_2(\\omega )$ to perpendicular linear polarization vectors of the initial and final photons, respectively.", "Since we are interested in the spin-averaged forward amplitude, we will focus on only $F_1(\\omega )$ henceforth.", "(We therefore define $F = F_1$ .)", "With the assumption of causality and analyticity for the forward scattering amplitude $F(\\omega )$ , the once-subtracted dispersion relation is given by $Re F(\\omega ) = Re F(0) + \\frac{2 \\omega ^2}{\\pi } P \\int _{\\omega _0}^\\infty d \\omega ^{\\prime }\\frac{Im F(\\omega ^{\\prime })}{\\omega ^{\\prime } ( \\omega ^{\\prime 2} - \\omega ^2)},$ where $P$ denotes the principal value and $\\omega _0$ is the threshold for producing a single dark matter which is approximately taken to be $\\omega _0 \\approx M_X c_l^2 \\approx 100 GeV$ .", "Using the optical theorem, the imaginary part of the forward elastic scattering amplitude is related to the total cross section $\\sigma (\\omega )$ by $\\sigma (\\omega ) = \\frac{4 \\pi c_l}{\\omega } Im F(\\omega ).$ Furthermore, it is known that the forward elastic scattering amplitude at $\\omega =0$ is real and given by the Thomson formula $Re F(0) = - \\frac{(\\varepsilon e)^2}{M_X c_l^2},$ where $\\varepsilon e$ and $M_X$ are respectively the charge and mass of the dark matter particle.", "Thus, one can rewrite $Re F(\\omega )$ in Eq.", "(REF ) to the form $Re F(\\omega ) = - \\frac{(\\varepsilon e)^2}{M_X c_l^2} + \\frac{\\omega ^2}{2 \\pi ^2 c_l}P \\int _{\\omega _0}^\\infty d \\omega ^{\\prime } \\frac{\\sigma (\\omega ^{\\prime })}{\\omega ^{\\prime 2} - \\omega ^2}.$ Substituting this expression into the standard formula (REF ), we obtain $Re n(\\omega ) - 1 = - \\frac{2 \\pi N (\\varepsilon e)^2}{M_X} \\frac{1}{\\omega ^2}+ \\frac{c_l N}{\\pi } P \\int _{\\omega _0}^\\infty d \\omega ^{\\prime } \\frac{\\sigma (\\omega ^{\\prime })}{\\omega ^{\\prime 2} - \\omega ^2}.$ Since $Im n(\\omega )$ does not appear any more and $n(\\omega )$ in the previous section is in fact equivalent to $Re n(\\omega )$ in this section, for simplicity we rewrite $Re n(\\omega )$ as $n(\\omega )$ from now on.", "Now let us consider the case of the low photon energy of $\\omega \\ll \\omega _0$ .In the most recent Michelson-Morley experiments [11], [12], photons with the order $100 THz$ are utilized, which corresponds to the order $1 eV$ .", "It is then sufficient to limit ourselves to an expansion up to the constant order in $\\omega ^2$ Expanding in powers of $\\omega $ is known to be a useful tool for obtaining various interesting low-energy cross sections in the analysis of the Compton scattering [23].", "$\\delta (\\omega ) \\equiv n(\\omega ) - 1 \\approx \\frac{\\delta _0}{\\omega ^2} + \\delta _2,$ where $\\delta _0, \\delta _2$ are defined as $\\delta _0 = - \\frac{2 \\pi N (\\varepsilon e)^2}{M_X}, \\hspace{20.0pt}\\delta _2 = \\frac{c_l N}{\\pi } P \\int _{\\omega _0}^\\infty d \\omega ^{\\prime } \\frac{\\sigma (\\omega ^{\\prime })}{\\omega ^{\\prime 2}}.$ Note that observations of $\\gamma $ -ray burst (GRB) emission from radio energies to tens of GeV [24] require us that the energy dependence of the speed of the photon is so small that $\\delta _0$ must be very tiny.", "Moreover, the recent ICARUS result [25] implies that $\\delta _2$ are very small as well.Although $E = \\hbar \\omega $ corresponds to the energy of a photon, we will regard it as a typical energy scale controlling the system under consideration, in other words, the energy of the ICARUS neutrinos.", "However, this smallness is not a real problem for the present purpose.", "The key point is that if we regard dark matter as a kind of usual medium for light, the frequency dependence must be always given by (REF ) at least for the low energy photons.", "Then, an important relation, which will be utilized shortly, can be obtained $\\omega \\frac{d n(\\omega )}{d \\omega } = -2 (n - 1 - \\delta _2).$ Next, following the procedure in Ref.", "[13], let us calculate the local anisotropy of the speed of light whose experimental upper bound has been already obtained in the experiments of the Michelson-Morley type [11], [12] $\\left(\\frac{\\Delta c}{c}\\right)_{Exp} \\approx 1 \\times 10^{-17}.$ In these experiments, the parallel average velocity is measured $c_{||} = \\frac{2L}{\\frac{L}{u_{||}^+} + \\frac{L}{u_{||}^-}}.$ Using Eqs.", "(REF ) and (REF ), this quantity can be evaluated to be $c_{||} = \\frac{c_l}{n} - \\frac{v^2}{c_l} \\frac{\\left[(n-1)^2 + 2 \\delta _2 \\right]^2}{n^3}.$ Then, the theoretical local anisotropy of the speed of light is calculated to be $\\frac{\\Delta c}{c} = \\frac{\\frac{c_l}{n(\\omega )} - c_{||}}{\\frac{c_l}{n(\\omega )}}\\approx 4 \\left(\\frac{v}{c_l}\\right)^2 \\delta _2^2\\approx 4 \\left(\\frac{v}{c_l}\\right)^2 \\delta ^2,$ where we have used $\\delta \\approx 0, \\beta \\ll 1$ and $\\delta \\approx \\delta _2$ .", "It is worthwhile to notice that compared to the result in Ref.", "[13] which is proportional to the linear order in $\\delta $ , our result (REF ) becomes proportional to the second order in $\\delta $ since we have taken account of the frequency dependence and the Lorentz transformation of the index of refraction $n(\\omega )$ .", "In order to show concretely that Eq.", "(REF ) yields a very weak constraint on the local anisotropy of the speed of light, let us apply it to the OPERA result [14] together with the SN1987A result [26].", "Here note that the OPERA result has been recently refuted by the ICARUS group [25].", "The reason why we use the OPERA result is two-fold.", "First, Ref.", "[13] has discussed the OPERA result and shown that there is some tension between the recent Michelson-Morley experiments and the models based on CHAMPs owing to a large value of $\\delta $ in the OPERA result.", "But we wish to show that even in this large value of $\\delta $ there appears no tension between them since (REF ) becomes proportional to the second order in $\\delta $ in our calculation.", "And the second reason is that even if the OPERA main result, in particular the size of $\\delta $ might be wrong, the other result, i.e., $\\delta $ is independent of the neutrino energy, is consistent with that of GRB emission [24] and seems to be true.", "The result of SN1987A implies that the velocity of a neutrino is almost the same as a universal limiting speed $v_\\nu \\approx c_l.$ Then, we can obtain the following relation $\\frac{v_\\nu - \\frac{c_l}{n}}{\\frac{c_l}{n}} \\approx \\frac{c_l - \\frac{c_l}{n}}{\\frac{c_l}{n}}= n - 1 = \\delta \\approx 2.37 \\times 10^{-5},$ where we have used the OPERA results in the last step.", "The OPERA result suggests that $\\delta $ is independent of the neutrino energy, so this equation implies $\\delta _2 \\approx \\delta \\approx 2.37 \\times 10^{-5}.$ Substituting this value and the equatorial rotation speed of the earth at the observed place, $\\beta \\equiv \\frac{v}{c_l}= 1.55 \\times 10^{-6}$ [13] into Eq.", "(REF ), one arrives at the result $\\left(\\frac{\\Delta c}{c}\\right)_{OPERA} \\approx 4.6 \\times 10^{-21}.$ Note that this value is much smaller than the experimental upper bound (REF ), so it is not in conflict with the recent experimental results of the Michelson-Morley type.", "Of course, our result does not refute the result obtained in Ref.", "[13] directly since in our approach at hand the index of refraction depends on the frequency of a photon whereas in the approach considered in Ref.", "[13] the index of refraction is assumed to be independent of the photon frequency.", "Actually, when dark matter does not carry an electric charge like a neutrino, Eq.", "(REF ) means $\\delta _0 = 0$ , thereby making it impossible to get Eq.", "(REF ) owing to $\\omega \\frac{d n(\\omega )}{d \\omega } = 0$ .", "For completeness, let us move on to a general case of the arbitrary photon energy.", "In this case, we can proceed the argument in a perfectly similar way to the case of the low energy photons.", "The relation (REF ) is now replaced with $\\omega \\frac{d n(\\omega )}{d \\omega } = -2 (n - 1 - \\tilde{\\delta }),$ where $\\tilde{\\delta }$ is defined as $\\tilde{\\delta }= \\frac{c_l N}{\\pi } P \\int _{\\omega _0}^\\infty d \\omega ^{\\prime }\\frac{\\omega ^{\\prime 2} \\sigma (\\omega ^{\\prime })}{(\\omega ^{\\prime 2} - \\omega ^2)^2}.$ Then, it turns out that local anisotropy of the speed of light is exactly evaluated to be $\\frac{\\Delta c}{c} = \\left(\\frac{v}{c_l}\\right)^2\\frac{[ (n-1)^2 + 2 \\tilde{\\delta }]^2}{n^2}.$ With the reasonable assumptions as before, $\\delta \\equiv n - 1 \\approx 0, \\beta \\ll 1$ and $\\delta \\approx \\tilde{\\delta }$ , we can obtain a similar result to Eq.", "(REF ) $\\frac{\\Delta c}{c} \\approx 4 \\left(\\frac{v}{c_l}\\right)^2 \\tilde{\\delta }^2\\approx 4 \\left(\\frac{v}{c_l}\\right)^2 \\delta ^2.$ Compared to the low energy case, a slight modification here appears in the $\\tilde{\\delta }$ which is dependent on the photon frequency as can be seen in (REF ).", "However, except this fact, the essential feature remains unchanged so the conclusion obtained in the case of the low enery photons still holds even in this general case ." ], [ "Discussion", "Since the introduction of dispersion relations into elemetary particle physics, originally within the context of quantum field theory [22], a large number of literatures have grown up on their theoretical basis, on extensions and applications to new processes, and on their comparison with experiment with great success.", "The most advantageous point is that dispersion relations are very universal in the sense that they are formulated only on the basis of fundamental principles of quantum field theory such as causality, the Lorentz invariace and analyticity.", "As a simple application of the dispersion relations, in this article, we have studied the velocity of light in dark matter with a fractional electric charge and found that the recent experiments of the Michelson-Morley type [11], [12] do not provide a stringent condition on the slowdown of the velocity of light if the index of refraction of dark matter depends on the frequency of a photon.", "The experiments of the Michelson-Morley type have played a critical role in proving the non-existence of the $\\it {aether}$ , i.e., a medium that was once supposed to fill all space and to support the propagation of electromagnetic waves, and might reveal a breakdown of the Lorentz invariance at the Planck scale in future.", "But they do not seem to shed some light on the change of the velocity of light in dark matter with charge.", "Perhaps, if the ICARUS result is correct, this situation will not change even in the future experiments of the Michelson-Morley type.It is predicted in Ref.", "[11] that $\\left(\\frac{\\Delta c}{c}\\right)_{Exp}\\approx 1 \\times 10^{-20}$ regime is possible.", "Nevertheless, the experiments using the photons will provide important insight into the properties of CHAMPs in future.", "For instance, polarized Compton scattering will yield information on spin-structure of the dark matter.", "Moreover, if one increases the energy of the incoming photon beam, one can probe the internal structure of the dark matter [23].", "Finally, we wish to close this article by mentioning two future problems to be solved.", "An interesting application of the refractive effect in our model is calculation of the force acting on a macroscopic body when dark matter flux passes through.", "Simple calculation turns out to give a force proportional to $|n(\\omega ) -1|$ which could be measured in the Eotvos-Dicke experiment.", "Furthermore, the refractive effect in a high-temperature background such as in the early universe deserves study in future.", "It is of interest to notice that the presence of the electric charge might ensure the stability of the dark matter from another angle.", "Acknowledgements This work is supported in part by the Grant-in-Aid for Scientific Research (C) No.", "22540287 from the Japan Ministry of Education, Culture, Sports, Science and Technology." ] ]

[ [ "Approximability of the Vertex Cover Problem in Power Law Graphs" ], [ "Abstract In this paper we construct an approximation algorithm for the Minimum Vertex Cover Problem (Min-VC) with an expected approximation ratio of 2-f(beta) for random Power Law Graphs (PLG) in the (alpha,beta)-model of Aiello et.", "al., where f(beta) is a strictly positive function of the parameter beta.", "We obtain this result by combining the Nemhauser and Trotter approach for Min-VC with a new deterministic rounding procedure which achieves an approximation ratio of 3/2 on a subset of low degree vertices for which the expected contribution to the cost of the associated linear program is sufficiently large." ], [ "Introduction", "In recent years topological analyses have been applied to a variety of real world graphs such as the World-Wide Web, the Internet, Collaboration and Social Networks, Protein Interaction Networks and other large-scale graphs of biological systems.", "Typical statistical parameters such as the diameter, robustness, clustering coefficient and degree distribution have been measured and compared to the expected values of these parameters in uniform random graph models such as the classical $G(n,p)$ -Model due to Erdős and Rényi [10].", "It turned out that the real world graphs are significantly different from the random models with respect to these statistical and topological properties.", "In subsequent studies the aim was to describe the properties of real world networks mathematically and to propose new models in order to meet these conditions.", "As of 1999 Kumar et.", "al.", "[4], [22], Kleinberg et.", "al.", "[19], [20] and Faloutsos, Faloutsos and Faloutsos [11], [27] measured the degree sequence of the World-Wide Web and independently observed that it is well approximated by a power law distribution, i.e.", "the number of nodes $y_i$ of a given degree $i$ is proportional to $i^{-\\beta }$ where $\\beta > 0$ .", "This was later verified for a large number of existing real-world networks such as protein-protein interactions, gene regulatory networks, peer-to-peer networks, mobile call networks and social networks [16], [13], [28], [9].", "In order to analyze these graphs, some research has been directed towards finding suitable models for describing structural properties quantitatively and qualitatively.", "A number of Power Law Graph (PLG) models have been proposed, such as the Barabási-Albert model of Preferential Attachment [3], the Buckley-Osthus Model [5], the Cooper-Frieze Model [7] and the Copying Model due to Kumar et.", "al.", "[22].", "All these models describe a random growth process starting from a small seed graph and yielding – besides other features – a power law degree sequences in the limit.", "A different approach is to take a power law degree sequence as input and to generate a graph instance with this distribution in a random fashion.", "Among the most widely known models of this kind is the ACL-Model due to Aiello, Chung and Lu [2].", "Here, the number $y_i$ of vertices with degree $i$ is roughly given by $y_i \\approx e^{\\alpha } / i^{\\beta }$ , where $e^{\\alpha }$ is a normalization constant which determines the size of the graph.", "While this model is potentially less accurate than the detailed description of a growth process, it has the advantage of being robust and general, i.e., structural properties that are true in this model will be true for the majority of graphs with the given degree sequence.", "All of the above models are well motivated and there exists a large body of literature on mathematical foundations and applications [3], [1], [6], [9], [23].", "In this paper, we focus on the ACL-Model for random PLG which we will refer to as the $(\\alpha ,\\beta )$ -Model.", "Apart from having certain structural properties, such as high clustering coefficient, small-world characteristics and self similarity, there exists practical evidence that combinatorial optimization in PLG is easier than in general graphs [25], [14], [9], [18].", "Contrasting this Ferrante et.", "al.", "[12] and Shen et.", "al.", "[29] studied the approximation hardness of certain optimization problems in combinatorial Power Law Graphs and showed NP-hardness and APX-hardness of classical problems such as Minimum Vertex Cover (Min-VC), Maximum Independent Set (Max-IS) and Minimum Dominating Set (Min-DS).", "In this paper we study the approximability of the Minimum Vertex Cover problem in the random Power Law Graph model of Aiello et.", "al.", "[2].", "The Minimum Vertex Cover is one of the most well-studied problems in combinatorial optimization.", "A vertex cover of a graph $G=(V,E)$ is a set of vertices $C\\subseteq V$ such that each edge $e=\\lbrace u,v\\rbrace $ of $G$ has at least one endpoint in $C$ .", "The Minimum Vertex Cover problem (Min-VC) is the the problem of finding a cover of minimum cardinality in a graph.", "The problem is known to be NP-complete due to Karp's original proof [17] and APX-complete [26].", "Moreover, it cannot be approximated within a factor of $1.3606$ [8], unless $\\textsc {P}= \\textsc {NP}$ , and is inapproximable within $2-\\epsilon $ for any $\\epsilon > 0$ as long as the Unique Games Conjecture (UGC) holds true [21].", "Here, we show that the Min-VC problem can be approximated with an expected approximation ratio $<2$ in random Power Law Graphs: Theorem 1 There exists a polynomial time algorithm which approximates the Minimum Vertex Cover problem (Min-VC) in random Power Law Graphs in the $(\\alpha ,\\beta )$ -Model for $\\beta > 2$ (where graphs are given instance by instance) with an expected approximation ratio of $\\rho =2-\\frac{\\zeta (\\beta )-1-\\frac{1}{2^{\\beta }}}{2^{\\beta }\\zeta (\\beta -1)\\zeta (\\beta )}.$ We also give a refined analysis for the case $\\beta > 2.424$ and obtain the following improvement.", "Theorem 2 For $\\beta >2.424$ , the Minimum Vertex Cover problem (Min-VC) in the $(\\alpha ,\\beta )$ -Model can be approximated with expected asymptotic approximation ratio $\\rho ^{\\prime }=2-\\frac{\\left(\\zeta (\\beta )-1-\\frac{1}{2^{\\beta }}\\right) \\cdot \\zeta (\\beta -1)}{\\zeta (\\beta -1) \\cdot \\zeta (\\beta )}\\left[1-{\\left(\\frac{\\zeta (\\beta -1)-\\left(1+\\frac{1}{2^{\\beta -1}}\\right)}{\\zeta (\\beta -1)}\\right)}^{3}\\right].$ In fig:Comparison these two upper bounds $\\rho $ and $\\rho ^{\\prime }$ are shown as functions of the parameter $\\beta $ .", "Figure: Comparison of first ([baseline=-3pt](0,0) edge[dashed,bend left= 10] (.5,0);) and second ([baseline=-3pt](0,0) edge[bend left= 10] (.5,0);) analysis in terms of functions of the parameter β\\beta , for β>2\\beta > 2 and β>2.424\\beta > 2.424, respectively.The paper is organized as follows.", "In subsec:alphabeta we describe the $(\\alpha ,\\beta )$ -model for Power Law Graphs, describe the random generation process and give a formal description of the model parameters.", "In subsec:vertexcover we give some background on the Min-VC problem and briefly describe the half-integral solution method proposed by Nemhauser and Trotter.", "sec:vconplg presents our new approximation algorithm for Min-VC in Power Law Graphs.", "This algorithm basically consists of a deterministic rounding procedure on a half-integral solution for Min-VC.", "In sec:algorithm we show that this rounding procedure yields an approximation ratio of $\\frac{3}{2}$ in the subgraph induced by the low-degree vertices of the Power Law Graph and a 2-approximation in the residual graph.", "In subsec:bounds we construct upper and lower bounds on the expected size of the half-integral solution in the induced subgraph of low-degree vertices and finally prove our main theorems.", "We conclude the paper by giving a short summary and further research in sec:conclusion." ], [ "$(\\alpha ,\\beta )$ -Power Law Graphs", "In this section we describe the random PLG-model proposed by Aiello, Chung and Lu in [2], which we will denote as $M(\\alpha ,\\beta )$ .", "This model considers a random graph with the following degree distribution depending on two given values $\\alpha $ and $\\beta $ : For each $1\\leqslant i \\leqslant \\Delta = \\left\\lfloor {e^{\\frac{\\alpha }{\\beta }}} \\right\\rfloor $ there are $y_i$ vertices of degree $i$ with $y_i={\\left\\lbrace \\begin{array}{ll}\\left\\lfloor {\\frac{e^{\\alpha }}{i^{\\beta }}} \\right\\rfloor & \\text{if } i > 1 \\text{ or } \\sum ^{\\Delta }_{i=1} \\left\\lfloor {\\frac{e^{\\alpha }}{i^{\\beta }}} \\right\\rfloor \\text{ is even}\\\\\\left\\lfloor {e^{\\alpha }} \\right\\rfloor + 1 & \\text{otherwise.}\\end{array}\\right.", "}$ Here, $i$ and $y_i$ satisfy $\\log y_i = \\alpha - \\beta \\log i$ .", "The variable $\\alpha $ is the logarithm of the size of the graph and $\\beta $ is the $\\log $ -$\\log $ growth rate.", "Let $\\mathcal {G}_{(\\alpha ,\\beta )}$ be the set of all undirected graphs with multi-edges and self-loops on $n=\\sum _{i=1}^{\\Delta } y_i$ vertices which have $y_i$ vertices of degree $i$ ($1 \\leqslant i \\leqslant \\Delta $ ).", "Then $M(\\alpha ,\\beta )$ is the distribution on $\\mathcal {G}_{(\\alpha ,\\beta )}$ obtained in the following way [2]: Generate a set $L$ of $d(v)$ distinct copies of each vertex $v$ .", "Generate a random matching on the elements of $L$ .", "For each pair of vertices $u$ and $v$ , the number of edges joining $u$ and $v$ in $G$ is equal to the number of edges in the matching of $L$ which join copies of $u$ to copies of $v$ .", "decorations.pathreplacing,shapes.arrows,positioning [every label/.style=label distance=5pt, vertex/.style=draw,shading=ball,ball color=black,circle, inner sep=2pt, vertex2/.style=draw,shading=ball,ball color=black!25,circle, inner sep=2pt, every edge/.style=draw,densely dotted,in=-90,out=-90, every pin edge/.style=dotted,latex-,in=90,out=-45,shorten <=-4pt, every pin/.style=dashed,pin distance=1cm] [black!25,rounded corners=6pt] (-.25,.25) – (.25,0.25) – (.25,-1.25) – (-0.25,-1.25) – cycle; [black!25,rounded corners=6pt,xshift=1cm] (-0.25,0.25) – (.25,0.25) – (.25,-1.25) – (-0.25,-1.25) – cycle; [black!25,rounded corners=6pt,xshift=-1.5cm] (4.3,0.25) – (4.7,0.25) – (5,-1.25) – (4,-1.25) – cycle; [black!25,rounded corners=6pt] (4.3,0.25) – (4.7,0.25) – (5,-1.25) – (4,-1.25) – cycle; [black!25,rounded corners=6pt] (6.8,0.25) – (7.2,0.25) – (7.8,-1.25) – (6.2,-1.25) – cycle; [black!25,rounded corners=6pt,xshift=2cm] (6.8,0.25) – (7.2,0.25) – (7.8,-1.25) – (6.2,-1.25) – cycle; vertex,label=above:$v_{11}$ ] (v1) at (0,0) ; vertex,label=above:$v_{12}$ ] (v12) at (1,0) ; t (2,-0.5) $\\dots $ ; vertex,label=above:$v_{21}$ ] (v21) at (3,0) ; vertex,label=above:$v_{22}$ ] (v22) at (4.5,0) ; t (5.7,-0.5) $\\dots $ ; vertex,label=above:$v_{31}$ ] (v31) at (7,0) ; vertex,label=above:$v_{32}$ ] (v32) at (9,0) ; t (10.25,-0.5) $\\dots $ ; vertex2] (v11) at (0,-1) ; vertex2] (v12) at (1,-1) ; vertex2] (v211) at (2.75,-1) ; vertex2] (v212) at (3.25,-1) ; vertex2] (v221) at (4.25,-1) ; vertex2] (v222) at (4.75,-1) ; vertex2] (v311) at (6.5,-1) ; vertex2] (v312) at (7,-1) ; vertex2] (v313) at (7.5,-1) ; vertex2] (v321) at (8.5,-1) ; vertex2] (v322) at (9,-1) ; vertex2] (v323) at (9.5,-1) ; [rounded corners] (v11) – +(0,-.4) -| (v211); [rounded corners] (v12) – +(0,-.5) -| (v311); [rounded corners] (v221) – +(0,-.4) -| node (s) (v222); [rounded corners] (v313) – +(0,-.4) -| node (m1) (v322); [rounded corners] (v312) – +(0,-.5) -| node (m2) (v323); [rounded corners] (v212) – +(0,-.6) -| (v321); below left = .25 of m1] (lm) multi-edge; below left = .25 of s] (ls) self-loop; [-latex,dotted,rounded corners] (lm.east) ..controls +(1,0) and +(.5,-.5).. (m1.north); [-latex,dotted,rounded corners] (lm.east) ..controls +(.5,0) and +(.5,-.5).. (m2.north); [-latex,dotted,rounded corners] (ls.east) ..controls +(.5,0) and +(.5,-.5).. (s.north); As in [2], in the following we will work with the real numbers $\\frac{e^{\\alpha }}{i^{\\beta }}$ , $e^{\\frac{\\alpha }{\\beta }}$ instead of their integer counterparts.", "For $\\beta > 2$ the error is a lower order term (c.f.", "[2], remark on page 6).", "A graph $G \\in \\mathcal {G}_{(\\alpha ,\\beta )}$ has the following properties: The maximum degree of $G$ is $e^{\\frac{\\alpha }{\\beta }}$ , and for $\\beta > 2$ the number of vertices is $n=\\sum ^{e^{{\\alpha }{\\beta }}}_{i=1} \\frac{e^{\\alpha }}{i^{{\\beta }}} \\approx \\zeta (\\beta )e^{\\alpha }$ and the number of edges is $m=\\frac{1}{2}\\sum ^{e^{{\\alpha }{\\beta }}}_{i=1} i \\frac{e^{\\alpha }}{i^{{\\beta }}} \\approx \\frac{1}{2}\\zeta (\\beta -1)e^{\\alpha }$ where the error terms are $o(n)$ and $o(m)$ , respectively." ], [ "LP-Relaxation and Half-Integral Solution for Min-VC", "In this section we give a brief outline of the Nemhauser-Trotter Theorem stated in [24] and show how this is used to approximate Min-VC in a graph $G=(V,E)$ as described by Hochbaum et.", "al.", "in [15].", "Nemhauser and Trotter considered the following $\\textsf {LP}$ -relaxation, which applies to the more general weighted vertex cover problem: $\\text{minimize }\\quad & \\sum _{i=1}^n w_i x_i, &&&\\quad & \\\\\\text{subject to }\\quad & x_i + x_j &\\geqslant \\,& 1, && \\text{ for each edge } \\lbrace v_i,v_j\\rbrace \\in E, \\\\& x_i &\\geqslant \\,& 0, && \\text{ for each vertex }v_i\\in V,$ They show that there always exists an optimal solution $x$ for this LP which is half-integral, i.e.", "for all $i$ , $x_i \\in \\left\\lbrace 0,\\frac{1}{2},1 \\right\\rbrace $ .", "Then they partition the set of vertices into subsets $P, Q, R \\subseteq V$ , such that $v_i \\in P$ if $x_i=1$ , $v_i \\in Q$ if $x_i=\\frac{1}{2}$ and $v_i \\in R$ if $x_i=0$ in this solution.", "They show that at least one optimal vertex cover in $G$ contains the set $P$ , that each vertex in $R$ has all its neighbors in $P$ and – moreover – that each cover in $G$ has weight at least $\\mathrm {w}(P) + \\frac{1}{2}\\mathrm {w}(Q)$ .", "From this it follows that at least one optimal vertex cover in $G$ consists of the set $P$ and an optimal cover in the subgraph $H[Q]$ induced by $Q$ .", "Hochbaum et.", "al.", "[15] showed that an integer solution $y$ obtained by setting $y_i=1$ for all vertices $v_i \\in Q \\cup P$ and $y_i=0$ for all $v_i \\in R$ is a 2-approximate solution for the Min-VC problem in $G$ .", "Our approximation algorithm for Min-VC in random Power Law Graphs will make use of a half-integral solution $x$ of the LP-relaxation along with the properties described in the Nemhauser-Trotter Theorem in order to achieve an approximation ratio strictly less than 2." ], [ "Approximation of Min-VC in $(\\alpha ,\\beta )$ -PLG", "In this section we present our main result, namely an approximation algorithm with expected approximation ratio $2-\\frac{\\zeta (\\beta )-1-\\frac{1}{2^{\\beta }}}{2^{\\beta }\\zeta (\\beta -1)\\zeta (\\beta )}$ for the Min-VC problem in $(\\alpha ,\\beta )$ -PLG for $\\beta > 2$ .", "Furthermore a refined analysis yields an improved asymptotic approximation ratio for the case $\\beta > 2.424$ .", "Let us first give an outline of this algorithm.", "On instance $G\\in \\mathcal {G}_{(\\alpha ,\\beta )}$ the algorithm starts with a half-integral solution $x:V \\rightarrow \\left\\lbrace 0,\\frac{1}{2}, 1\\right\\rbrace $ of the associated LP and uses some deterministic rounding procedure to generate an integral solution $y: V \\rightarrow \\lbrace 0,1\\rbrace $ .", "We show that for the set $V^{*}= \\bigcup _{v:d(v)\\in \\lbrace 1,2\\rbrace } \\left(\\lbrace v\\rbrace \\cup \\mathcal {N}(v)\\right)$ of degree-1 and degree-2 nodes and their neighbors in $G$ , the rounding procedure satisfies $ y(V^{*}) \\leqslant \\frac{3}{2} \\cdot x(V^{*})$ and furthermore $x(V^{*})$ is sufficiently large (in expectation) with respect to $M(\\alpha ,\\beta )$ ." ], [ "Approximation Algorithm", "Now, we describe our deterministic rounding procedure ( alg:rounding) on $G=(V,E)$ for $G\\in \\mathcal {G}_{(\\alpha ,\\beta )}$ .", "First, the algorithm processes all nodes of the subset $V^{\\prime }=L \\cup \\mathcal {N}(L)$ where $L=\\left\\lbrace v \\in V \\vert \\left(d(v)=2, x(v)=\\frac{1}{2}\\right)\\vee \\left(d(v)=1\\right)\\right\\rbrace $ and provides a rounded integral solution $y$ with $y(V^{\\prime })\\leqslant \\frac{3}{2}\\cdot x(V^{\\prime })$ .", "Furthermore we show that $y(V^{*}\\setminus V^{\\prime }) \\leqslant \\frac{4}{3}\\cdot x(V^{*}\\setminus V^{\\prime })$ and $y(V\\setminus V^{*})\\leqslant 2\\cdot x(V\\setminus V^{*})$ .", "textbf() $G=(V,E) , x:V \\rightarrow \\left\\lbrace 0,\\frac{1}{2},1\\right\\rbrace $ .", "$y:V \\rightarrow \\lbrace 0,1\\rbrace $ .", "$v \\in V$ $y(v) := x(v)$ mark $v$ as unprocessed 0$\\texttt {compute}$ $G^{\\prime }=(V^{\\prime },E^{\\prime })$ induced by $V^{\\prime }=L\\cup \\mathcal {N}(L)$ where $L=\\left\\lbrace v \\in V \\vert \\left(d(v)=2, x(v)=\\frac{1}{2}\\right)\\vee \\left(d(v)=1\\right)\\right\\rbrace $ 1$v \\in V$ with $d(v)=1$ let $u$ be the neighbor of $v$ in $G$ set $y(v)=0$ ;   set $y(u)=1$ 2$P=uv_{1}v_{2}w \\subset G^{\\prime }$ unprocessed, $d(u)\\geqslant 3, d(v_{1})=d(v_{2})=2$ set $y(u)=y(w)=y(v_{1}) = 1$ set $y(v_{2})=0$ 3$v \\in V^{\\prime }$ unprocessed, $d(v)=2 \\wedge \\exists u\\in \\mathcal {N}(v), d(u)\\geqslant 3$ 3.1 $u$ unprocessed, $w$ processed set $y(v)=0$ ;   set $y(u)=1$ 3.2 both $u,w$ unprocessed set $y(v)=0$ ;   set $y(u)=y(w)=1$ *with $x(u)\\geqslant \\frac{1}{2}$ and $x(w)\\geqslant \\frac{1}{2}$ 3.3 both $u,w$ processed set $y(v)=0$ 3.4 $u$ processed, $w$ unprocessed set $y(v)=0$ ;   set $y(w)=1$ *$y(u)=1$ already set and $x(w)\\geqslant \\frac{1}{2}$ 4$v \\in V^{\\prime }$ unprocessed, $d(v)=2$ 4.1$u$ unprocessed, $w$ processed set $y(v)=0$ ;   set $y(u)=1$ 4.2both $u,w$ unprocessed set $y(v)=0$ ;   set $y(u)=y(w)=1$ *with $x(u)\\geqslant \\frac{1}{2}$ and $x(w)\\geqslant \\frac{1}{2}$ 4.3both $u,w$ processed set $y(v)=0$ 4.4$u$ processed, $w$ unprocessed set $y(v)=0$ ;   set $y(w)=1$ *$y(u)=1$ already set and $x(w)\\geqslant \\frac{1}{2}$ 5$v \\in V$ $x(v)=\\frac{1}{2}$ set $y(v)= 1$ *$y(v)= \\min \\lbrace 1,2\\cdot x(v)\\rbrace $ Deterministic Rounding An analysis of the algorithm is provided by the following lem:yInteger and lem:yThreeHalf.", "Lemma The assignment $y$ generated by alg:rounding is an integer solution and satisfies $y(u)=1$ for all $u \\in V^{\\prime }$ with $d(u) \\geqslant 3$ .", "Any high-degree neighbor of degree-1 vertices is set to 1 in step (1) of the algorithm.", "Since either step (3) or (4) is processing every single degree-2 vertex $v \\in V$ with $x(v)=\\frac{1}{2}$ , there are no leftover vertices $v\\in V^{\\prime }$ of degree 2 with fractional values.", "Assume that there is a vertex $u \\in V^{\\prime }, d(u)\\geqslant 3$ and $x(u)=y(u)=\\frac{1}{2}$ .", "Then $u$ has at least one degree 2 neighbor $v_1$ with $x(v_1)=\\frac{1}{2}$ .", "Because of step (3) and (4) of the algorithm, $v_1$ must have been processed by another degree 2 vertex $v_2$ , setting $y(v_1)=1$ .", "This again introduces another neighbor $w$ of $v_2$ with $y(w)=1$ and leads to the situation of a path $uv_{1}v_{2}w$ described in step (2).", "In this case, the algorithm sets $y(u)=1$ and thus we have a contradiction to the above assumption.", "Lemma The assignment $y$ generated by alg:rounding satisfies $y(V^{*})\\leqslant \\frac{3}{2}\\cdot x(V^{*})$ .", "The algorithm partitions the graph induced by $V^{*}$ into edge-disjoint subgraphs, namely stars whose leaves are degree-1 vertices and paths of length $\\leqslant 4$ whose internal nodes are degree-2 vertices.", "We show that for each such subgraph $P_i$ , $y(P_i)\\leqslant \\frac{3}{2} \\cdot x(P_i)$ and furthermore $y(v)=1$ for each $v\\in V^{*}$ which is contained in more than one such subgraph.", "In step (1) of the algorithm all degree-1 vertices and their neighbors are processed.", "In step (2) the subgraphs are unprocessed paths $P_i$ of length 3.", "Since $P_i= [baseline=-5pt] {\\node [vertex,label=below:\\tiny u] (u) {};\\node [vertex,label=below:\\tiny v_1,right= of u] (v1) {};\\node [vertex,label=below:\\tiny v_2,right= of v1] (v2) {};\\node [vertex,label=below:\\tiny w,right= of v2] (w) {};[draw] (u) -- (v1) -- (v2) -- (w);}$ contains two disjoint edges $\\lbrace u,v_1\\rbrace , \\lbrace v_2,w\\rbrace $ , $x(P_i)\\geqslant 2$ and particularly $x(v_2) + x(w) \\geqslant 1$ .", "Therefore $y(P_i)=3 \\leqslant \\frac{3}{2}\\cdot x(P_i)$ holds via mapping [baseline=0pt] vertex,label=above:$\\geqslant \\hspace{0.0pt}{1}{2}$ ] (u) ; vertex,label=above:12,right= of u] (v1) ; vertex,label=above:$\\geqslant \\hspace{0.0pt}{1}{2}$ ,right= of v1] (v2) ; vertex,label=above:$\\geqslant \\hspace{0.0pt}0$ ,right= of v2] (w) ; [draw] (u) – (v1) – (v2) – (w); right= of w, yshift=2pt] (mapsto) $\\mapsto $ ; vertex2,label=above:1,right= of mapsto, yshift=-2pt] (u) ; vertex2,label=above:1,right= of u] (v1) ; vertex2,label=above:0,right= of v1] (v2) ; vertex2,label=above:1,right= of v2] (w) ; [draw] (u) – (v1) – (v2) – (w); (where the gray color indicates a processed vertex) and $y$ restricted to $P_i$ (denoted as $y\\!\\upharpoonright \\!P_i$ ) is a vertex cover for $P_i$ .", "In step (3) all paths $P_i=[baseline=-5pt] {\\node [vertex,label=below:\\tiny u] (u) {};\\node [vertex,label=below:\\tiny v,right= of u] (v) {};\\node [vertex,label=below:\\tiny w,right= of v] (w) {};[draw] (u) -- (v) -- (w);}$ are processed, where at least one of $u,w$ is of degree $\\geqslant 3$ .", "In cases (3.1)-(3.4) the algorithm considers all possible combinations of some of these nodes being already processed.", "In case (3.1) $u$ is marked unprocessed, $w$ is already processed and $x(u)\\geqslant \\frac{1}{2}$ .", "The rounding algorithm sets $y(v)=0$ and $y(u)=1$ , mapping [baseline=0pt] vertex,label=above:$\\geqslant \\hspace{0.0pt}{1}{2}$ ] (u) ; vertex,label=above:12,right= of u] (v) ; vertex2,label=above:1,right= of v] (w) ; [draw] (u) – (v) – (w); right=5pt of w, yshift=2pt] (mapsto) $\\mapsto $ ; vertex2,label=above:1,right=5pt of mapsto, yshift=-2pt] (u) ; vertex2,label=above:0,right= of u] (v) ; vertex2,label=above:1,right= of v] (w) ; [draw] (u) – (v) – (w); , again yielding a vertex cover $y\\!\\upharpoonright \\!P_i$ for $P_i$ with $y(P_i) \\leqslant x(P_i)$ .", "In case (3.2) we have that both $u,w$ are marked as unprocessed and since $x(v)=\\frac{1}{2}$ we have that $x(u)\\geqslant \\frac{1}{2}$ and $x(w)\\geqslant \\frac{1}{2}$ .", "The rounding algorithm sets $y(v)=0, y(u)=y(w)=1$ , mapping [baseline=0pt] vertex,label=above:$\\geqslant \\hspace{0.0pt}{1}{2}$ ] (u) ; vertex,label=above:12,right= of u] (v) ; vertex,label=above:$\\geqslant \\hspace{0.0pt}{1}{2}$ ,right= of v] (w) ; [draw] (u) – (v) – (w); right=5pt of w, yshift=2pt] (mapsto) $\\mapsto $ ; vertex2,label=above:1,right=5pt of mapsto, yshift=-2pt] (u) ; vertex2,label=above:0,right= of u] (v) ; vertex2,label=above:1,right= of v] (w) ; [draw] (u) – (v) – (w); , and since $x(u)\\geqslant \\frac{1}{2}$ and $x(w)\\geqslant \\frac{1}{2}$ we have that $y(P_i) \\leqslant \\frac{4}{3}\\cdot x(P_i)$ .", "In case (3.3) both $u,w$ are marked as processed and therefore $y(u)=y(w)=1$ , since $u,w$ are adjacent to processed degree one or degree two vertices other than $v$ .", "The algorithm sets $y(v)=0$ , mapping [baseline=0pt] vertex2,label=above:1] (u) ; vertex,label=above:12,right= of u] (v) ; vertex2,label=above:1,right= of v] (w) ; [draw] (u) – (v) – (w); right=5pt of w, yshift=2pt] (mapsto) $\\mapsto $ ; vertex2,label=above:1,right=5pt of mapsto, yshift=-2pt] (u) ; vertex2,label=above:0,right= of u] (v) ; vertex2,label=above:1,right= of v] (w) ; [draw] (u) – (v) – (w); .", "Hence $y \\!\\upharpoonright \\!P_i$ is a vertex cover for $P_i$ with $y(P_i) \\leqslant x(P_i)$ .", "In case (3.4) $u$ is already processed and $w$ is still marked unprocessed.", "Since $x(v)=\\frac{1}{2}$ we have that $x(w)\\geqslant \\frac{1}{2}$ .", "The rounding algorithm sets $y(v)=0$ and $y(w)=1$ , mapping [baseline=0pt] vertex2,label=above:1] (u) ; vertex,label=above:12,right= of u] (v) ; vertex,label=above:$\\geqslant \\hspace{0.0pt}{1}{2}$ ,right= of v] (w) ; [draw] (u) – (v) – (w); right=5pt of w, yshift=2pt] (mapsto) $\\mapsto $ ; vertex2,label=above:1,right=5pt of mapsto, yshift=-2pt] (u) ; vertex2,label=above:0,right= of u] (v) ; vertex2,label=above:1,right= of v] (w) ; [draw] (u) – (v) – (w); , and since $x(w)\\geqslant \\frac{1}{2}$ it yields a vertex cover $y\\!\\upharpoonright \\!P_i$ for $P_i$ with $y(P_i) \\leqslant x(P_i)$ .", "Step (4) considers all remaining unprocessed vertices of degree 2.", "If $v$ is such a vertex with neighborhood $\\mathcal {N}(v)=\\lbrace u,w\\rbrace $ , the sub-cases (4.1)-(4.4) are treated analogously to cases (3.1)-(3.4) and the mapping $x\\mapsto y$ achieves $y(P_i) \\leqslant \\frac{4}{3}\\cdot x(P_i)$ on the considered paths $P_i$ .", "After steps (0)-(4) of the algorithm there may still be some remaining high-degree vertices $u \\in V^{*}, d(u)\\geqslant 3$ with $x(u)=y(u)=\\frac{1}{2}$ .", "These are treated separately (and rounded to $y(u)=1$ together with all other vertices in $V\\setminus (V^{\\prime }\\setminus V^{*})$ ) in step (5) of the algorithm.", "We have to argue that $y(V^{*}) \\leqslant \\frac{3}{2}\\cdot x(V^{*})$ still holds true.", "We consider first the case that $u\\in V^{\\prime }, d(u)\\geqslant 3$ and $x(u)=y(u)=\\frac{1}{2}$ .", "Then $u$ has a neighbor $v$ of degree $\\leqslant 2$ with $x(v)=\\frac{1}{2}$ and $y(v)=1$ , and since $y(u)=\\frac{1}{2}$ we have $d(v)=2$ .", "Let $v_2$ be the other neighbor of $v$ , then $d(v_2)=1$ (since otherwise the second neighbor $w$ of $v_2$ would give rise to a path of length 3, containing also $u$ and hence would have been processed in step (2)).", "But then locally on the set $\\lbrace u,v,v_2\\rbrace $ we have the mapping [baseline=0pt] vertex,label=above:12] (u) ; vertex,label=above:12,right= of u] (v) ; vertex,label=above:12,right= of v] (w) ; [draw] (u) – (v) – (w); right=5pt of w, yshift=2pt] (mapsto) $\\mapsto $ ; vertex,label=above:12,right=5pt of mapsto, yshift=-2pt] (u) ; vertex2,label=above:1,right= of u] (v) ; vertex2,label=above:0,right= of v] (w) ; [draw] (u) – (v) – (w); right=5pt of w, yshift=2pt] (mapsto) $\\mapsto $ ; vertex2,label=above:1,right=5pt of mapsto, yshift=-2pt] (u) ; vertex2,label=above:1,right= of u] (v) ; vertex2,label=above:0,right= of v] (w) ; [draw] (u) – (v) – (w); with a local ratio of $\\frac{4}{3}$ .", "Let us now assume $u \\in V^{*}\\setminus V^{\\prime }, d(u)\\geqslant 3$ and $x(u)=y(u)=\\frac{1}{2}$ .", "Then every degree-2 neighbor $v$ has $x(v)\\ne \\frac{1}{2}$ , hence $x(v)=1$ , and therefore $y(v)=1$ .", "We show that $v \\notin V^{\\prime }$ , i.e.", "that $v$ was not processed by the algorithm and can be treated as a part of a subgraph disjoint to $G^{\\prime }$ in $G$ .", "Let $w\\in \\mathcal {N}(v)$ be the second neighbor of $v$ besides $u$ .", "Then $x(w)=0$ since otherwise (in case $x(w)\\geqslant \\frac{1}{2}$ ) we could decrease $x(v)$ from 1 to $\\frac{1}{2}$ and still have a feasible half-integral solution, which would contradict the optimality of $x$ .", "Therefore $v,w\\notin V^{\\prime }$ , which means that $v,w$ are not processed by the algorithm.", "Rounding $y(u)=1$ , mapping [baseline=0pt] vertex,label=above:12] (u) ; vertex,label=above:1,right= of u] (v) ; [draw] (u) – (v); right=5pt of v, yshift=2pt] (mapsto) $\\mapsto $ ; vertex2,label=above:1,right=5pt of mapsto, yshift=-2pt] (u) ; vertex2,label=above:1,right= of u] (v) ; [draw] (u) – (v); , yields a vertex cover $y \\!\\upharpoonright \\!\\lbrace u,v\\rbrace $ with $y(\\lbrace u,v\\rbrace )\\leqslant \\frac{4}{3}\\cdot x(\\lbrace u,v\\rbrace )$ .", "We conclude that the assignment $y: V\\mapsto \\lbrace 0,1\\rbrace $ is a vertex cover of $G$ with $y(V^{*})\\leqslant \\frac{3}{2}\\cdot x(V^{*})$ and $y(V\\setminus V^{*}) \\leqslant 2 \\cdot x(V\\setminus V^{*})$ ." ], [ "Expected Approximation Ratio", "The following lemma shows how to retrieve an expected approximation ratio for our algorithm for Min-VC in $G$ .", "Lemma If the rounding scheme $x \\mapsto y$ satisfies $y(V^{*}) \\leqslant \\frac{3}{2} \\cdot x(V^{*})$ and $y(V \\setminus V^{*}) \\leqslant 2 \\cdot x(V\\setminus V^{*})$ then this gives an approximation ratio $\\frac{y(V)}{\\textsf {OPT}} \\leqslant \\frac{y(V)}{x(V)} \\leqslant \\frac{x(V^{*})}{x(V)} \\cdot \\frac{3}{2} + \\frac{x(V\\setminus V^{*})}{x(V)} \\cdot 2.$ In order to apply lem:ratiolemma and to derive an expected approximation ratio for the algorithm, in the following we will give a lower bound on $\\mathbb {E}[x(V^{*})]$ and an upper bound on $x(V)$ .", "The next lemma provides a lower bound on $x(V^{*})$ in terms of the number of high-degree vertices adjacent to degree-1 and degree-2 nodes.", "Lemma Let $G[V^{*}]$ be the subgraph of $G$ induced by $V^{*}$ .", "For every optimal half-integral solution $x$ for the Min-VC LP, the size of the half-integral solution restricted to $V^{*}$ is lower-bounded by the size of the high-degree neighborhood of degree-1 and degree-2 vertices: $x(V^{*})\\geqslant \\frac{1}{2} \\cdot \\big \\vert \\left\\lbrace u \\in V \\vert d(u)\\geqslant 3 \\wedge \\exists v \\in \\mathcal {N}(u), d(v)\\in \\lbrace 1,2\\rbrace \\right\\rbrace \\big \\vert $ Let $V^{*}= X \\cup Y, X=\\lbrace v \\in V \\vert d(v)\\in \\lbrace 1,2\\rbrace \\rbrace $ and $Y=\\lbrace u \\in V \\vert d(u)\\geqslant 3 \\wedge \\exists v \\in \\mathcal {N}(u), d(v)\\in \\lbrace 1,2\\rbrace \\rbrace $ .", "Choose some arbitrary function $f:Y \\rightarrow E(X,Y)$ such that for every $u \\in Y, f(u)=\\lbrace u,v\\rbrace $ for some $v\\in X$ adjacent to $u$ .", "$f(Y)$ consists of pairwise disjoint paths $Q_{1},\\dots ,Q_{m}$ of length $\\leqslant 2$ , such that each path contains one or two vertices from $Y$ .", "This implies $x(V^{*})\\geqslant m \\geqslant \\frac{\\vert Y \\vert }{2}$ ." ], [ "First Analysis", "We will now estimate the expected number of high-degree vertices adjacent to vertices of degree one or two, which – combined with the preceding lem:lemma5 – gives a lower bound on $\\mathbb {E}[x(V^{*})]$ .", "We prove the following theorem: Theorem 3 $\\mathbb {E}[x(V^{*})] &\\geqslant \\frac{1}{2}\\cdot \\mathbb {E}\\left[\\big \\vert \\left\\lbrace u \\in V \\vert d(u)\\geqslant 3 \\wedge \\exists v \\in \\mathcal {N}(u), d(v)\\in \\lbrace 1,2\\rbrace \\right\\rbrace \\big \\vert \\right]\\nonumber \\\\&= \\frac{1}{2}\\;\\cdot \\; \\sum _{\\mathchoice{\\hbox{t}o 0pt{\\hss \\displaystyle {u:d(u)\\geqslant 3}\\hss }}{\\hbox{t}o 0pt{\\hss \\textstyle {u:d(u)\\geqslant 3}\\hss }}{\\hbox{t}o 0pt{\\hss \\scriptstyle {u:d(u)\\geqslant 3}\\hss }}{\\hbox{t}o 0pt{\\hss \\scriptscriptstyle {u:d(u)\\geqslant 3}\\hss }}} \\eta (u)\\\\&\\geqslant \\frac{\\operatorname{e}^{\\alpha }}{2^{\\beta }}\\cdot \\frac{\\zeta (\\beta )-1-\\frac{1}{2^{\\beta }}}{\\zeta (\\beta -1)}$ where $\\eta (u)$ is the probability that $u\\in V$ has a neighbor in the set of vertices of degree one or two.", "In order to provide bounds on the probability $\\eta (u)$ for a vertex $u$ of degree $d$ of having a degree-1 or degree-2 neighbor, we consider how edges are generated in the random matching procedure of the distribution $M(\\alpha ,\\beta )$ : $d(u)$ copies of $u$ are randomly matched with the copies of the remaining vertices $v \\in V, v\\ne u$ .", "We use the following lower bound on $\\eta (u)$ .", "Lemma For every $u$ with $d(u)\\geqslant 3$ , $\\eta (u)\\geqslant \\frac{1}{2^{\\beta -1}\\cdot \\sum _{i=1}^{\\Delta } \\frac{1}{i^{\\beta -1}}}$ .", "$\\eta (u) &\\geqslant \\Pr (\\text{the first copy of $u$ is neighbor of a degree-2-node})\\\\&= \\frac{2\\cdot \\#\\text{deg-2-nodes}}{\\left(\\sum _{v\\in V}d(v)\\right)-1}\\\\&\\geqslant \\frac{2\\cdot \\frac{\\operatorname{e}^{\\alpha }}{2^{\\beta }}}{\\sum _{i=1}^{\\Delta }i\\cdot \\frac{\\operatorname{e}^{\\alpha }}{i^{\\beta }}} = \\frac{\\frac{1}{2^{\\beta -1}}}{\\sum _{i=1}^{\\Delta }\\frac{1}{i^{\\beta -1}}}, \\text{ where } \\Delta =\\operatorname{e}^{\\frac{\\alpha }{\\beta }} \\text{ is the maximum degree of }G.$ In Equation REF we substitute $\\eta (u)$ by the bound given in lem:etaLemma and obtain: $\\mathbb {E}[x(V^{*})] &\\geqslant \\frac{1}{2}\\;\\cdot \\;\\sum _{\\mathchoice{\\hbox{t}o 0pt{\\hss \\displaystyle {u:d(u)\\geqslant 3}\\hss }}{\\hbox{t}o 0pt{\\hss \\textstyle {u:d(u)\\geqslant 3}\\hss }}{\\hbox{t}o 0pt{\\hss \\scriptstyle {u:d(u)\\geqslant 3}\\hss }}{\\hbox{t}o 0pt{\\hss \\scriptscriptstyle {u:d(u)\\geqslant 3}\\hss }}}\\eta (u)\\nonumber \\\\&= \\frac{1}{2}\\cdot \\left(\\sum _{i=1}^{\\Delta }\\frac{\\operatorname{e}^{\\alpha }}{i^{\\beta }}-\\operatorname{e}^{\\alpha }-\\frac{\\operatorname{e}^{\\alpha }}{2^{\\beta }}\\right)\\cdot \\frac{1}{2^{\\beta -1}\\cdot \\sum _{i=1}^{\\Delta } \\frac{1}{i^{\\beta -1}}}\\nonumber \\\\&= \\frac{\\operatorname{e}^{\\alpha }}{2^{\\beta }}\\cdot \\frac{\\sum _{i=1}^{\\Delta }\\frac{1}{i^{\\beta }}-1-\\frac{1}{2^{\\beta }}}{\\sum _{i=1}^{\\Delta }\\frac{1}{i^{\\beta -1}}}$ We will now show that in Inequality REF we can replace the terns $\\sum _{i=1}^{\\Delta }\\frac{1}{i^{\\beta }}$ and $\\sum _{i=1}^{\\Delta }\\frac{1}{i^{\\beta -1}}$ by $\\zeta (\\beta )$ and $\\zeta (\\beta -1)$ , respectively.", "We make use of the following lemma.", "Lemma For $A,B,a,b > 0$ , $\\frac{A}{B}\\geqslant \\frac{A+a}{B+b} \\;\\Longleftrightarrow \\; \\frac{A}{B}\\geqslant \\frac{a}{b}$ .", "Therefore, in order to show $\\mathbb {E}[x(V^{*})] \\geqslant \\frac{e^{\\alpha }}{2^{\\beta }}\\cdot \\frac{\\zeta (\\beta )-1-\\frac{1}{2^{\\beta }}}{\\zeta (\\beta -1)},$ it is sufficient to show that there exists a $\\Delta _{0}$ such that for all $\\Delta \\geqslant \\Delta _{0}$ the following holds $\\frac{\\sum _{i=1}^{\\Delta }\\frac{1}{i^{\\beta }}-1-\\frac{1}{2^{\\beta }}}{\\sum _{i=1}^{\\Delta }\\frac{1}{i^{\\beta -1}}} \\geqslant \\frac{\\frac{1}{(\\Delta +1)^{\\beta }}}{\\frac{1}{(\\Delta +1)^{\\beta -1}}} = \\frac{1}{\\Delta +1}.$ This is provided by the following lemma.", "Lemma There exists a $\\Delta _{0}\\geqslant 8$ such that for all $\\Delta \\geqslant \\Delta _{0}$ , $\\frac{\\sum _{i=1}^{\\Delta }\\frac{1}{i^{\\beta }}-1-\\frac{1}{2^{\\beta }}}{\\sum _{i=1}^{\\Delta }\\frac{1}{i^{\\beta -1}}} \\geqslant \\frac{1}{\\Delta +1}$ .", "The above inequality is equivalent to $&\\phantom{\\Longleftrightarrow }& \\sum _{i=1}^{\\Delta }\\frac{1}{i^{\\beta }}-1-\\frac{1}{2^{\\beta }} &\\geqslant \\sum _{i=1}^{\\Delta }\\frac{1}{\\Delta +1}\\cdot \\frac{1}{i^{\\beta -1}}\\nonumber \\\\&\\Longleftrightarrow & \\sum _{i=1}^{\\Delta }\\left(\\frac{1}{i^{\\beta }}-\\frac{1}{\\Delta +1}\\cdot \\frac{1}{i^{\\beta -1}}\\right) &\\geqslant 1+ \\frac{1}{2^{\\beta }}\\nonumber \\\\&\\Longleftrightarrow & \\sum _{i=1}^{\\Delta }\\frac{\\Delta +1-i}{(\\Delta +1)i^{\\beta }} &\\geqslant 1+ \\frac{1}{2^{\\beta }}$ Suppose $\\Delta \\geqslant 8$ , then the sum on the left-hand side of the Inequality REF is bounded by the sum of the terms with indices $i=1,2,4,8$ : $\\sum _{i=1}^{\\Delta }\\frac{\\Delta +1-i}{(\\Delta +1)i^{\\beta }} \\geqslant \\frac{\\Delta }{\\Delta +1} + \\frac{\\Delta -1}{(\\Delta +1)2^{\\beta }} + \\frac{\\Delta -3}{(\\Delta +1)4^{\\beta }} + \\frac{\\Delta -7}{(\\Delta +1)8^{\\beta }}\\\\= \\frac{\\Delta 8^{\\beta } + (\\Delta -1)4^{\\beta } + (\\Delta -3)2^{\\beta } + \\Delta -7}{(\\Delta +1)8^{\\beta }}.$ Using Inequality REF and the fact that $1+\\frac{1}{2^{\\beta }} = \\frac{(\\Delta +1)8^{\\beta } + (\\Delta +1)4^{\\beta }}{(\\Delta +1)8^{\\beta }}$ , in order to prove Inequality REF it is sufficient to show the following: $&\\phantom{\\Longleftrightarrow }& \\frac{\\Delta 8^{\\beta } + (\\Delta -1)4^{\\beta } + (\\Delta -3)2^{\\beta } + \\Delta -7}{(\\Delta +1)8^{\\beta }} &\\stackrel{!", "}{\\geqslant } \\frac{(\\Delta +1)8^{\\beta } + (\\Delta +1)4^{\\beta }}{(\\Delta +1)8^{\\beta }}\\\\&\\Longleftrightarrow & (\\Delta -3)2^{\\beta } + \\Delta -7 &\\stackrel{!", "}{\\geqslant } 8^{\\beta } + 2\\cdot 4^{\\beta }.$ This is valid for $\\Delta \\geqslant \\frac{8^{\\beta }+2\\cdot 4^{\\beta }+6\\cdot 2^{\\beta }+7}{1+2^{\\beta }}$ .", "Hence we choose $\\Delta _{0}=\\left\\lceil {*} \\right\\rceil {\\frac{8^{\\beta }+2\\cdot 4^{\\beta }+6\\cdot 2^{\\beta }+7}{1+2^{\\beta }}}$ .", "This completes the proof of thm:ExVgood.", "The next lemma provides an upper bound for $x(V)$ : Lemma $x(V) \\leqslant \\frac{1}{2}\\zeta (\\beta )\\operatorname{e}^{\\alpha }$ In order to get an upper bound for $x(V)$ we construct a feasible half-integral solution for $G$ by setting $x(v)=\\frac{1}{2}$ for all $v\\in V$ where $\\frac{1}{2}\\sum _{v\\in V}\\leqslant \\frac{1}{2}\\zeta (\\beta )\\operatorname{e}^{\\alpha }$ .", "Now let us restate the main thm:main and finish the proof.", "Theorem For $\\beta > 2$ the Minimum Vertex Cover problem in $(\\alpha ,\\beta )$ -Power Law Graphs $G$ can be approximated with expected approximation ratio $\\rho \\leqslant 2-\\frac{\\zeta (\\beta )-1-\\frac{1}{2^{\\beta }}}{2^{\\beta }\\zeta (\\beta -1)\\zeta (\\beta )}$ .", "alg:rounding achieves an approximation ratio of $\\frac{3}{2}$ for Min-VC in the subgraph induced by $V^{*}$ in $G$ and a ratio of 2 in $G[V\\setminus V^{*}]$ , i.e.", "$\\rho \\leqslant \\mathbb {E}\\left[\\frac{3}{2}\\cdot \\frac{x(V^{*})}{x(V)} + 2\\cdot \\frac{x(V)-x(V^{*})}{x(V)} \\right] = \\mathbb {E}\\left[2-\\frac{1}{2}\\cdot \\frac{x(V^{*})}{x(V)}\\right].$ From thm:ExVgood and lem:UBV we have that $\\mathbb {E}[x(V^{*})] \\geqslant \\frac{1}{2} \\cdot \\frac{\\left( \\zeta (\\beta ) - 1 - \\frac{1}{2^{\\beta }}\\right) \\operatorname{e}^{\\alpha }}{2^{\\beta -1}\\zeta (\\beta -1)}$ and $x(V) \\leqslant \\frac{1}{2}\\cdot \\zeta (\\beta )\\operatorname{e}^{\\alpha }$ .", "This yields $\\mathbb {E}\\left[\\frac{x(V^{*})}{x(V)}\\right]\\geqslant \\frac{\\frac{1}{2} \\cdot \\frac{\\left( \\zeta (\\beta ) - 1 - \\frac{1}{2^{\\beta }}\\right) \\operatorname{e}^{\\alpha }}{2^{\\beta -1}\\zeta (\\beta -1)}}{\\frac{1}{2}\\cdot \\zeta (\\beta )\\operatorname{e}^{\\alpha }}= \\frac{\\zeta (\\beta )-1-\\frac{1}{2^{\\beta }}}{2^{\\beta -1}\\zeta (\\beta -1)\\zeta (\\beta )}$ and $\\rho \\leqslant 2-\\frac{1}{2}\\cdot \\frac{\\zeta (\\beta )-1-\\frac{1}{2^{\\beta }}}{2^{\\beta -1}\\zeta (\\beta -1)\\zeta (\\beta )}= 2-\\frac{\\zeta (\\beta )-1-\\frac{1}{2^{\\beta }}}{2^{\\beta }\\zeta (\\beta -1)\\zeta (\\beta )}$" ], [ "Refined Analysis for $\\beta > 2.424$", "We will now refine the analysis of alg:rounding by giving a better estimate on the probability $\\eta (u, U)$ of a high-degree node $u$ being adjacent to a vertex in the set $U$ , i.e.", "a vertex of degree one or two.", "However, this analysis will only apply to the more restricted range of $\\beta > 2.424$ .", "Again, we will first obtain a bound on the expected approximation ratio of the algorithm in terms of the partial sums $\\sum _{i=1}^{\\Delta }\\frac{1}{i^\\beta }$ and $ \\sum _{i=1}^{\\Delta }\\frac{1}{i^\\beta -1}$ and then show that these can be replaced by $\\zeta (\\beta )$ and $\\zeta (\\beta -1)$ , respectively.", "Lemma For every $u$ with $d(u)\\geqslant 3$ and $U\\subseteq V$ , $\\eta (u,U)\\geqslant \\frac{\\sum _{i = 1}^{\\Delta }\\frac{e^{\\alpha }}{i^{\\beta -1}}-e^{\\frac{\\alpha }{\\beta }}+1}{\\sum _{i = 1}^{\\Delta }\\frac{e^{\\alpha }}{i^{\\beta -1}}}\\left[1-\\left(\\frac{\\sum _{i = 1}^{\\Delta }\\frac{e^{\\alpha }}{i^{\\beta -1}}-d(U)-3+1}{\\sum _{i = 1}^{\\Delta }\\frac{e^{\\alpha }}{i^{\\beta -1}}-3+1}\\right)^{3}\\right].$ For a given set $U$ of vertices from $G$ we let $d(U)=\\sum _{v\\in U}d(v)$ .", "Furthermore let $\\eta (u,U)$ be the probability that $u$ is connected to at least one node in $U$ .", "We obtain $\\eta (u,U) &= \\Pr (u \\text{ matches to } U)\\\\&= \\sum _{j=1}^{d(u)} \\Pr (j\\text{-th copy is first one matching to } U)\\\\&= \\sum _{j=1}^{d(u)} \\frac{d(u)}{\\sum _{i = 1}\\frac{e^{\\alpha }}{i^{\\beta -1}}-(j-1)-1}\\prod _{k=1}^{j-1}\\left( 1- \\frac{d(U)}{\\sum _{i = 1}\\frac{e^{\\alpha }}{i^{\\beta -1}}-1-(k-1)} \\right)$ Now define $N=\\sum _{i = 1}\\frac{e^{\\alpha }}{i^{\\beta -1}}$ .", "We have: $\\hspace*{-21.25pt}\\eta (u,U) &= \\sum _{j=1}^{d(u)} \\frac{d(U)}{N-j}\\prod _{k=1}^{j-1}\\frac{N-d(U)-k}{N-k}\\hspace*{42.5pt}\\geqslant \\sum _{j=1}^{d(u)} \\frac{d(U)}{N-j}\\left(\\frac{N-d(U)-j+1}{N-j+1}\\right)^{j-1}\\\\&\\geqslant \\sum _{j=1}^{d(u)} \\frac{d(U)}{N-j}\\left(\\frac{N-d(U)-d(u)+1}{N-d(u)+1}\\right)^{j-1}\\geqslant \\sum _{j=1}^{d(u)} \\frac{d(U)}{N}\\left(\\frac{N-d(U)-d(u)+1}{N-d(u)+1}\\right)^{j-1}\\\\&= \\frac{d(U)}{N}\\left[\\frac{1-\\left(\\frac{N-d(U)-d(u)+1}{N-d(u)+1} \\right)^{d(u)}}{1-\\frac{N-d(U)-d(u)+1}{N-d(u)+1}}\\right]\\\\&= \\frac{d(U)}{N}\\left[1-\\left(\\frac{N-d(U)-d(u)+1}{N-d(u)+1} \\right)^{d(u)}\\right] \\cdot \\frac{N-d(u)+1}{d(U)}\\\\&= \\frac{N-d(u)+1}{N}\\left[1-\\left(\\frac{N-d(U)-d(u)+1}{N-d(u)+1}\\right)^{d(u)}\\right]$ Since the function $\\left(\\frac{N-d(U)-d(u)+1}{N-d(u)+1}\\right)^{d(u)}$ is monotone decreasing in $d(u)$ it follows that: $\\eta (u,U) &\\geqslant \\frac{N-\\Delta +1}{N}\\left[1-\\left(\\frac{N-d(U)-3+1}{N-3+1}\\right)^{3}\\right]\\\\&= \\frac{\\sum _{i = 1}\\frac{e^{\\alpha }}{i^{\\beta -1}}-e^{\\frac{\\alpha }{\\beta }}+1}{\\sum _{i = 1}\\frac{e^{\\alpha }}{i^{\\beta -1}}}\\left[1-\\left(\\frac{\\sum _{i = 1}\\frac{e^{\\alpha }}{i^{\\beta -1}}-d(U)-3+1}{\\sum _{i = 1}\\frac{e^{\\alpha }}{i^{\\beta -1}}-3+1}\\right)^{3}\\right]$ Because of Equation REF we have ${\\displaystyle \\mathbb {E}[x(V^{*})] \\geqslant \\frac{1}{2}\\;\\cdot \\;\\sum _{\\mathchoice{\\hbox{t}o 0pt{\\hss \\displaystyle {u:d(u)\\geqslant 3}\\hss }}{\\hbox{t}o 0pt{\\hss \\textstyle {u:d(u)\\geqslant 3}\\hss }}{\\hbox{t}o 0pt{\\hss \\scriptstyle {u:d(u)\\geqslant 3}\\hss }}{\\hbox{t}o 0pt{\\hss \\scriptscriptstyle {u:d(u)\\geqslant 3}\\hss }}}\\eta (u,U)}$ and we obtain the following approximation ratio: $\\hspace*{-4.25pt}\\rho &\\leqslant \\mathbb {E}\\left[2-\\frac{1}{2}\\cdot \\frac{x(V^{*})}{x(V)}\\right]\\nonumber \\\\&\\leqslant 2-\\frac{1}{2}\\cdot \\frac{\\left(\\sum _{i=1}^{\\Delta }\\frac{\\operatorname{e}^{\\alpha }}{i^{\\beta }}-\\operatorname{e}^{\\alpha }-\\frac{\\operatorname{e}^{\\alpha }}{2^{\\beta }}\\right)\\cdot \\frac{\\sum _{i = 1}^{\\Delta }\\frac{e^{\\alpha }}{i^{\\beta -1}}-e^{\\frac{\\alpha }{\\beta }}+1}{\\sum _{i = 1}^{\\Delta }\\frac{e^{\\alpha }}{i^{\\beta -1}}}\\left[1-\\left(\\frac{\\sum _{i = 1}^{\\Delta }\\frac{e^{\\alpha }}{i^{\\beta -1}}-d(U)-3+1}{\\sum _{i = 1}^{\\Delta }\\frac{e^{\\alpha }}{i^{\\beta -1}}-3+1}\\right)^{3}\\right]}{\\frac{1}{2}\\sum _{i=1}^{\\Delta }\\frac{\\operatorname{e}^{\\alpha }}{i^{\\beta }}}\\nonumber \\\\&= 2- \\underbrace{\\frac{\\left(\\sum _{i=1}^{\\Delta }\\frac{1}{i^{\\beta }}-1-\\frac{1}{2^{\\beta }}\\right) \\cdot \\left(\\sum _{i = 1}^{\\Delta } \\frac{1}{i^{\\beta -1}}-\\frac{\\Delta }{\\operatorname{e}^{\\alpha }} + \\frac{1}{\\operatorname{e}^{\\alpha }}\\right)}{\\left(\\sum _{i = 1}^{\\Delta }\\frac{1}{i^{\\beta -1}}\\right) \\cdot \\left(\\sum _{i=1}^{\\Delta }\\frac{1}{i^{\\beta }}\\right)}}_{F}\\left[1-{\\underbrace{\\left(\\frac{\\sum _{i = 1}^{\\Delta }\\frac{1}{i^{\\beta -1}}-\\frac{d(v)}{\\operatorname{e}^{\\alpha }}-\\frac{2}{\\operatorname{e}^{\\alpha }}}{\\sum _{i = 1}^{\\Delta }\\frac{1}{i^{\\beta -1}}-\\frac{2}{\\operatorname{e}^{\\alpha }}}\\right)}_{C}}^{3}\\right]$ Now we show that, in Inequality REF , we can replace the partial sums $\\sum _{i = 1}^{\\Delta }\\frac{1}{i^{\\beta }}$ and $\\sum _{i = 1}^{\\Delta }\\frac{1}{i^{\\beta -1}}$ by $\\zeta (\\beta )$ and $\\zeta (\\beta -1)$ respectively.", "First, we consider the term $C$ where $d(v)=\\operatorname{e}^{\\alpha }\\left(1+\\frac{1}{2^{\\beta -1}}\\right)$ , i.e.", "the number of copies of degree-1 and degree-2 vertices: $C=\\frac{\\sum _{i = 1}^{\\Delta }\\frac{1}{i^{\\beta -1}}-\\frac{\\operatorname{e}^{\\alpha }\\left(1+\\frac{1}{2^{\\beta -1}}\\right)}{\\operatorname{e}^{\\alpha }}-\\frac{2}{\\operatorname{e}^{\\alpha }}}{\\sum _{i = 1}^{\\Delta }\\frac{1}{i^{\\beta -1}}-\\frac{2}{\\operatorname{e}^{\\alpha }}}= \\frac{\\sum _{i = 1}^{\\Delta }\\frac{1}{i^{\\beta -1}}-\\left(1+\\frac{1}{2^{\\beta -1}}\\right)-\\frac{2}{\\Delta ^{\\beta }}}{\\sum _{i = 1}^{\\Delta }\\frac{1}{i^{\\beta -1}}-\\frac{2}{\\Delta ^{\\beta }}}$ We show that following inequality holds true: $&\\phantom{\\Longleftrightarrow }& \\frac{\\sum _{i = 1}^{\\Delta }\\frac{1}{i^{\\beta -1}}-\\left(1+\\frac{1}{2^{\\beta -1}}\\right)-\\frac{2}{\\Delta ^{\\beta }}}{\\sum _{i = 1}^{\\Delta }\\frac{1}{i^{\\beta -1}}-\\frac{2}{\\Delta ^{\\beta }}} &\\leqslant \\frac{\\frac{1}{(\\Delta +1)^{\\beta -1}} + \\frac{2}{\\Delta ^{\\beta }} - \\frac{2}{(\\Delta +1)^{\\beta }}}{\\frac{1}{(\\Delta +1)^{\\beta -1}} + \\frac{2}{\\Delta ^{\\beta }} - \\frac{2}{(\\Delta +1)^{\\beta }}}\\\\&\\Longleftrightarrow & \\frac{\\sum _{i = 1}^{\\Delta }\\frac{1}{i^{\\beta -1}}-\\left(1+\\frac{1}{2^{\\beta -1}}\\right)-\\frac{2}{\\Delta ^{\\beta }}}{\\sum _{i = 1}^{\\Delta }\\frac{1}{i^{\\beta -1}}-\\frac{2}{\\Delta ^{\\beta }}} &\\leqslant 1\\\\&\\Longleftrightarrow & \\sum _{i = 1}^{\\Delta }\\frac{1}{i^{\\beta -1}}-\\left(1+\\frac{1}{2^{\\beta -1}}\\right)-\\frac{2}{\\Delta ^{\\beta }} &\\leqslant \\sum _{i = 1}^{\\Delta }\\frac{1}{i^{\\beta -1}}-\\frac{2}{\\Delta ^{\\beta }}\\\\&\\Longleftrightarrow & \\sum _{i = 1}^{\\Delta }\\frac{1}{i^{\\beta -1}}-\\left(1+\\frac{1}{2^{\\beta -1}}\\right) &\\leqslant \\sum _{i = 1}^{\\Delta }\\frac{1}{i^{\\beta -1}} \\qquad \\square $ We have $F=\\frac{\\left(\\sum _{i=1}^{\\Delta }\\frac{1}{i^{\\beta }}-1-\\frac{1}{2^{\\beta }}\\right) \\cdot \\left(\\sum _{i = 1}^{\\Delta } \\frac{1}{i^{\\beta -1}}-\\frac{\\Delta }{\\operatorname{e}^{\\alpha }} + \\frac{1}{\\operatorname{e}^{\\alpha }}\\right)}{\\left(\\sum _{i = 1}^{\\Delta }\\frac{1}{i^{\\beta -1}}\\right) \\cdot \\left(\\sum _{i=1}^{\\Delta }\\frac{1}{i^{\\beta }}\\right)}.$ We let $S_{\\beta }=\\sum _{i=1}^{\\Delta }\\frac{1}{i^{\\beta }}$ and $S_{\\beta -1}=\\sum _{i=1}^{\\Delta }\\frac{1}{i^{\\beta -1}}$ and recall that $\\operatorname{e}^{\\alpha }=\\Delta ^{\\beta }$ .", "According to lem:ABgeqab it remains to show the following inequality: $\\frac{\\left({S_{\\beta }}-1-\\frac{1}{2^{\\beta }}\\right) \\cdot \\left(S_{\\beta -1}-\\frac{1}{\\Delta ^{\\beta -1}} + \\frac{1}{\\Delta ^{\\beta }}\\right)}{\\left(S_{\\beta -1}\\right) \\cdot \\left(S_{\\beta }\\right)}\\\\\\geqslant \\frac{1}{\\frac{1}{(\\Delta +1)^{\\beta }}\\left(S_{\\beta -1}\\right) + \\frac{1}{(\\Delta +1)^{\\beta -1}}\\left(S_{\\beta } + \\frac{1}{(\\Delta +1)^{\\beta }}\\right)} \\cdot \\left[\\frac{1}{(\\Delta +1)^{\\beta }}\\left(S_{\\beta -1}-\\frac{1}{\\Delta ^{\\beta -1}} + \\frac{1}{\\Delta ^{\\beta }}\\right)\\right.\\\\+ \\left.\\left(S_{\\beta }-1-\\frac{1}{2^{\\beta }}+\\frac{1}{(\\Delta +1)^{\\beta }}\\right) \\cdot \\left(-\\frac{1}{(\\Delta +1)^{\\beta }}+\\frac{1}{(\\Delta +1)^{\\beta -1}}+\\frac{1}{(\\Delta )^{\\beta -1}}-\\frac{1}{(\\Delta )^{\\beta }}\\right)\\right]$ which is equivalent to $\\left(S_{\\beta }-1-\\frac{1}{2^{\\beta }}\\right) \\left(S_{\\beta -1}-\\frac{\\Delta -1}{\\Delta ^{\\beta }}\\right) \\left[\\frac{1}{(\\Delta +1)^{\\beta }}S_{\\beta -1} + \\left(S_{\\beta }+\\frac{1}{(\\Delta +1)^{\\beta }}\\right)\\frac{1}{(\\Delta +1)^{\\beta -1}}\\right]\\\\\\geqslant \\left[\\frac{1}{(\\Delta +1)^{\\beta }}\\left(S_{\\beta -1}+\\frac{\\Delta -1}{\\Delta ^{\\beta }}\\right)\\cdot S_{\\beta }\\cdot S_{\\beta -1}\\right.\\\\+ \\left.\\left(S_{\\beta }-1-\\frac{1}{2^{\\beta }}+\\frac{1}{(\\Delta +1)^{\\beta }}\\right) \\cdot \\left(\\frac{\\Delta }{(\\Delta +1)^{\\beta }} + \\frac{\\Delta -1}{\\Delta ^{\\beta }}\\right)\\cdot S_{\\beta } \\cdot S_{\\beta -1}\\right].$ We rearrange terms and get $S_{\\beta -1}^{2}S_{\\beta }\\frac{1}{(\\Delta +1)^{\\beta }} + S_{\\beta }^{2}S_{\\beta -1}\\left(\\frac{\\Delta }{(\\Delta +1)^{\\beta }}+\\frac{\\Delta -1}{\\Delta ^{\\beta }}\\right)\\\\+ S_{\\beta }S_{\\beta -1}\\left[-\\frac{\\Delta -1}{\\Delta ^{\\beta }}\\cdot \\frac{1}{(\\Delta +1)^{\\beta }} + \\left(\\frac{\\Delta }{(\\Delta +1)^{\\beta }}+\\frac{\\Delta -1}{\\Delta ^{\\beta }}\\right) \\left(-1-\\frac{1}{2^{\\beta }}+\\frac{1}{(\\Delta +1)^{\\beta }}\\right)\\right] \\\\\\leqslant S_{\\beta -1}^{2}S_{\\beta }\\frac{1}{(\\Delta +1)^{\\beta }} + S_{\\beta -1}S_{\\beta }^{2}\\frac{1}{(\\Delta +1)^{\\beta -1}}\\\\+ S_{\\beta }S_{\\beta -1}\\left[\\frac{1}{(\\Delta +1)^{\\beta }}\\cdot \\frac{1}{(\\Delta +1)^{\\beta -1}} - \\frac{\\Delta -1}{\\Delta ^{\\beta }} - \\left(1+\\frac{1}{2^{\\beta }}\\right)\\frac{1}{(\\Delta +1)^{\\beta -1}}\\right]\\\\+ S_{\\beta }^{2}\\frac{1}{(\\Delta +1)^{\\beta -1}} + S_{\\beta -1}^{2}\\left(-1-\\frac{1}{2^{\\beta }}\\right)\\frac{1}{(\\Delta +1)^{\\beta }}\\\\+ S_{\\beta -1}\\left[-\\left(1+\\frac{1}{2^{\\beta }}\\right)\\frac{1}{(\\Delta +1)^{\\beta }}\\cdot \\frac{1}{(\\Delta +1)^{\\beta -1}}\\left(1+\\frac{1}{2^{\\beta }}\\right)\\frac{\\Delta -1}{\\Delta ^{\\beta }}\\cdot \\frac{1}{(\\Delta +1)^{\\beta }}\\right]\\\\+ S_{\\beta }\\left[\\left(-1-\\frac{1}{2^{\\beta }}\\right)\\frac{1-\\Delta }{\\Delta ^{\\beta }}\\cdot \\frac{1}{(\\Delta +1)^{\\beta -1}} - \\frac{\\Delta -1}{\\Delta ^{\\beta }}\\cdot \\frac{1}{(\\Delta +1)^{\\beta }}\\cdot \\frac{1}{(\\Delta +1)^{\\beta -1}}\\right]\\\\+ \\left(1+\\frac{1}{2^{\\beta }}\\right)\\frac{\\Delta -1}{\\Delta ^{\\beta }}\\cdot \\frac{1}{(\\Delta +1)^{\\beta }}\\cdot \\frac{1}{(\\Delta +1)^{\\beta -1}}$ The following lemma shows that in order to prove Inequality REF it is sufficient to show the respective inequality given by the terms of slowest convergence as $\\Delta $ goes to infinity.", "Lemma Let $f_{\\beta },g_{\\beta },F_{\\beta },G_{\\beta }$ be functions of $\\Delta $ depending on the parameter $\\beta $ with $|g_{\\beta }|,|G_{\\beta }|\\leqslant c$ for a constant $c$ depending only on $\\beta $ .", "Then $f_{\\beta }(\\Delta ) < F_{\\beta }(\\Delta )$ for almost all $\\Delta $ implies $f_{\\beta }(\\Delta )\\cdot \\frac{1}{\\Delta ^{\\beta -1}} + g_{\\beta }(\\Delta )\\cdot \\frac{1}{\\Delta ^{\\beta }} \\leqslant F_{\\beta }(\\Delta )\\cdot \\frac{1}{\\Delta ^{\\beta -1}} + G_{\\beta }(\\Delta )\\cdot \\frac{1}{\\Delta ^{\\beta }}$ for all but finitely many $\\Delta $ .", "Hence it remains to show that $S_{\\beta }^{2}S_{\\beta -1}\\left(\\frac{\\Delta }{(\\Delta +1)^{\\beta }}+\\frac{\\Delta -1}{\\Delta ^{\\beta }}\\right) - S_{\\beta }S_{\\beta -1}\\left(1+\\frac{1}{2^{\\beta }}\\right)\\left(\\frac{\\Delta }{(\\Delta +1)^{\\beta }} + \\frac{\\Delta -1}{\\Delta ^{\\beta }}\\right)\\\\\\leqslant S_{\\beta }S_{\\beta -1}(-1)\\cdot \\frac{\\Delta -1}{\\Delta ^{\\beta }} + S_{\\beta -1}S_{\\beta }^{2}\\frac{1}{(\\Delta +1)^{\\beta -1}}$ which holds true if and only if $S_{\\beta }^{2}S_{\\beta -1}\\left(\\frac{\\Delta }{(\\Delta +1)^{\\beta }}+\\frac{\\Delta -1}{\\Delta ^{\\beta }}\\right)\\\\\\leqslant S_{\\beta }S_{\\beta -1}\\left[\\left(1+\\frac{1}{2^{\\beta }}\\right)\\left(\\frac{\\Delta }{(\\Delta +1)^{\\beta }} + \\frac{\\Delta -1}{\\Delta ^{\\beta }}\\right)-\\frac{\\Delta -1}{\\Delta ^{\\beta }}\\right] + S_{\\beta -1}S_{\\beta }^{2}\\frac{1}{(\\Delta +1)^{\\beta -1}}$ which can be rewritten as $&\\phantom{\\Longleftrightarrow }& S_{\\beta }^{2}S_{\\beta -1}\\left(\\frac{1}{(\\Delta +1)^{\\beta }}+\\frac{\\Delta -1}{\\Delta ^{\\beta }}\\right) &\\leqslant S_{\\beta }S_{\\beta -1}\\left[\\left(1+\\frac{1}{2^{\\beta }}\\right)\\frac{\\Delta }{(\\Delta +1)^{\\beta }} + \\frac{1}{2^{\\beta }}\\cdot \\frac{\\Delta -1}{\\Delta ^{\\beta }}\\right]\\nonumber \\\\&\\Longleftrightarrow & S_{\\beta }\\left(\\frac{\\Delta -1}{\\Delta ^{\\beta }}\\right) &\\leqslant \\left(1+\\frac{1}{2^{\\beta }}\\right)\\frac{\\Delta }{(\\Delta +1)^{\\beta }} + \\frac{1}{2^{\\beta }}\\cdot \\frac{\\Delta -1}{\\Delta ^{\\beta }} \\nonumber \\\\&\\Longleftrightarrow & S_{\\beta }(\\Delta -1) &\\leqslant \\left(1+\\frac{1}{2^{\\beta }}\\right)\\frac{\\Delta ^{\\beta +1}}{(\\Delta +1)^{\\beta }} + \\frac{1}{2^{\\beta }}\\cdot (\\Delta -1)\\nonumber \\\\&\\Longleftrightarrow & \\left(S_{\\beta }-\\frac{1}{2^{\\beta }}\\right)(\\Delta -1) &\\leqslant \\left(1+\\frac{1}{2^{\\beta }}\\right)\\frac{\\Delta ^{\\beta +1}}{(\\Delta +1)^{\\beta }} $ Now Inequality REF follows from the observation that for all $\\beta >2.424$ , $S_{\\beta }-\\frac{1}{2^\\beta }< 1+\\frac{1}{2^\\beta }$ .", "Finally we have shown the following theorem.", "Theorem For all $\\beta >2.424$ the Minimum Vertex Cover problem on $(\\alpha ,\\beta )$ -Power Law Graphs $G$ can be approximated with expected approximation ratio $\\rho \\leqslant 2- \\frac{\\left(\\zeta (\\beta )-1-\\frac{1}{2^{\\beta }}\\right) \\cdot \\left(\\zeta (\\beta -1)-\\frac{\\Delta }{\\operatorname{e}^{\\alpha }} + \\frac{1}{\\operatorname{e}^{\\alpha }}\\right)}{\\zeta (\\beta -1) \\cdot \\zeta (\\beta )}\\left[1-{\\left(\\frac{\\zeta (\\beta -1)-\\left(1+\\frac{1}{2^{\\beta -1}}\\right)-\\frac{2}{\\operatorname{e}^{\\alpha }}}{\\zeta (\\beta -1)-\\frac{2}{\\operatorname{e}^{\\alpha }}}\\right)}^{3}\\right]$ This converges to $\\rho \\leqslant 2- \\frac{\\left(\\zeta (\\beta )-1-\\frac{1}{2^{\\beta }}\\right) \\cdot \\zeta (\\beta -1)}{\\zeta (\\beta -1) \\cdot \\zeta (\\beta )}\\left[1-{\\left(\\frac{\\zeta (\\beta -1)-\\left(1+\\frac{1}{2^{\\beta -1}}\\right)}{\\zeta (\\beta -1)}\\right)}^{3}\\right]$ as $\\alpha \\rightarrow \\infty $ ." ], [ "Conclusion", "In sec:vconplg we presented a new approximation algorithm for Min-VC in $(\\alpha ,\\beta )$ -PLG with expected approximation ratio of $\\rho \\leqslant 2-\\frac{\\zeta (\\beta )-1-\\frac{1}{2^{\\beta }}}{2^{\\beta }\\zeta (\\beta -1)\\zeta (\\beta )}$ in our first analysis of subsec:FirstAnalysis.", "Moreover, in our refined analysis we showed for $\\beta > 2.424$ an expected asymptotic approximation ratio of $\\rho ^{\\prime }\\leqslant 2-\\frac{\\left(\\zeta (\\beta )-1-\\frac{1}{2^{\\beta }}\\right) \\cdot \\zeta (\\beta -1)}{\\zeta (\\beta -1) \\cdot \\zeta (\\beta )}\\left[1-{\\left(\\frac{\\zeta (\\beta -1)-\\left(1+\\frac{1}{2^{\\beta -1}}\\right)}{\\zeta (\\beta -1)}\\right)}^{3}\\right]$ .", "The algorithm itself basically consists of a deterministic rounding procedure on a half-integral solution for Min-VC (c.f.", "alg:rounding).", "We showed that this rounding procedure yields an approximation ratio of $\\frac{3}{2}$ in the subgraph induced by the low-degree vertices of the $(\\alpha ,\\beta )$ -PLG and a 2-approximation in the residual graph.", "Further research will be directed towards extending the improved analysis also to the range $\\beta < 2.424$ and towards investigating the approximability of Min-VC in other PLG-Models, e.g.", "the Preferential Attachment Model in [3]." ], [ "Acknowledgements", "The first author is supported by the NRW State within the B-IT Research School.", "The authors would like to thank Marek Karpinski for helpful remarks and discussions." ] ]

[ [ "Supernova neutrino halo and the suppression of self-induced flavor\n conversion" ], [ "Abstract Neutrinos streaming from a supernova (SN) core occasionally scatter in the envelope, producing a small \"neutrino halo\" with a much broader angle distribution than the primary flux originating directly from the core.", "Cherry et al.", "(2012) have recently pointed out that, during the accretion phase, the halo actually dominates neutrino-neutrino refraction at distances exceeding some 100 km.", "However, the multiangle matter effect (which increases if the angle distribution is broader) still appears to suppress self-induced flavor conversion during the accretion phase." ], [ "Introduction", "Neutrino-neutrino refraction is responsible for the intriguing effect of self-induced flavor conversion [1], [2], [3], [4], [5] that can occur in the neutrino flux streaming from a supernova (SN) core [6].", "In this context, the angular neutrino distribution plays a crucial role.", "The current-current structure of low-energy weak interactions implies that the interaction energy between two relativistic particles involves a factor $(1-\\cos \\theta )$ where $\\theta $ is their relative direction of motion.", "If the neutrino-emitting region of a supernova core (“neutrino sphere”) has radius $R$ , then at distances $r\\gg R$ a typical neutrino-neutrino angle is $\\theta \\sim R/r$ and $\\langle 1-\\cos \\theta \\rangle \\propto (R/r)^2$ .", "The geometric flux dilution provides another factor $(R/r)^2$ , leading to an overall $(R/r)^4$ decrease of the neutrino-neutrino interaction energy [6].", "In a recent paper, Cherry et al.", "pointed out that this picture is not complete because neutrinos suffer residual collisions on their way out [7].", "Every layer of matter above the neutrino sphere is a secondary source, producing a wide-angle “halo” for the forward-peaked primary flux.", "While the halo flux is small, its broad angular distribution allows it to dominate the neutrino-neutrino interaction energy.", "We illustrate the halo with a numerical example, the 280 ms postbounce snapshot of a spherical $15\\,M_\\odot $ model.", "We recently used it as our benchmark case to study multiangle suppression of self-induced flavor conversion [8].", "In Fig.", "REF we show the angular dependence of the intensityWith “intenstiy” ${\\cal I}$ we mean the quantity “neutrinos per unit area per unit time per unit solid angle,” integrated over the energy spectrum.", "for the $\\bar{\\nu }_e$ radiation field, normalized to the forward direction, measured at the radial distances 300, 1000, 3000, and 10,000 km.", "(The angular distributions become noisy for $\\theta \\pi /2$ , where they are currently not well provided by our simulations.)", "The core and halo fluxes are two distinct components, the latter so small that it is not visible on a linear plot.", "If we use $\\theta _{\\rm c}\\sim 0.1$ as the edge of the core distribution for the 300 km case, we infer a radius of $R\\sim 30$  km for the region where neutrinos begin to stream almost freely.", "At larger distances, the angular scales are squeezed by a factor $R/r$ .", "Figure: Intensity for the ν ¯ e \\bar{\\nu }_e radiation field of our numerical model,normalized to theforward direction, measured at the radial distances 300, 1000,3000, and 10,000 km (right to left).The impact on neutrino-neutrino refraction is illustrated by the weak potential felt by a radially moving neutrino.", "In terms of the zenith angle $\\theta $ of the intensity ${\\cal I}(\\theta )$ we need the quantity $\\langle 1-\\cos \\theta \\rangle =\\frac{\\int _0^\\pi d\\theta \\,\\sin \\theta \\,(1-\\cos \\theta )\\,{\\cal I}(\\theta )}{\\int _0^\\pi d\\theta \\,\\sin \\theta \\,{\\cal I}(\\theta )}\\,.$ For small $\\theta $ , the integrand in the numerator expands as $(1-\\cos \\theta )\\sin \\theta =\\theta ^3/2$ , allowing the halo to contribute significantly at larger distances where the distributions are more squeezed.", "At 1000 km, for example, $\\langle 1-\\cos \\theta \\rangle $ is almost a factor of 10 larger than it would be without the halo, in agreement with Cherry et al. [7].", "The halo parts of the functions in Fig.", "REF decrease roughly as $\\theta ^{-3}$ and we find this implies that $\\langle 1-\\cos \\theta \\rangle $ decreases roughly as $r^{-1}$ instead of $r^{-2}$ .", "The neutrino-neutrino interaction energy then decreases roughly as $r^{-3}$ instead of $r^{-4}$ .", "The halo is important during the accretion phase when there is enough matter for the primary flux to scatter on [7].", "However, the same high matter density tends to suppress the self-induced flavor instability [9].", "During the early accretion phase, self-induced flavor conversions were found to be typically suppressed [8], [10].", "Does the halo change these conclusions?", "Its importance derives from its broad angle distribution, which also makes it susceptible to the multiangle matter suppression.", "To investigate this question, we study in Sec.", "the properties of the neutrino halo.", "In Sec.", "we perform a stability analysis along the lines of our recent study of an accretion-phase model [8], and we conclude in Sec. .", "Neutrinos stream almost freely and therefore, at larger distances, the angular distributions are simply squeezed to smaller angular scales (Fig.", "REF ).", "This behavior applies also to the halo flux which, once produced, streams almost freely.", "At larger distances, the halo distribution primarily gains at the edges: the newly available angular modes get populated by scattering.", "Figure: The 10 4 10^4 km numerical case(solid line) overlaid with the analytic fit (dashed line) ofEq. ().", "We also indicate the power-law behavior estimatedfrom an analytic argument (Sec.", ").The numerical angular distribution at the largest available radius essentially holds all the required information.", "The decreasing wings of ${\\cal I}(\\theta )$ look like power laws and we can fit the entire distribution by ${\\cal I}_{\\rm fit}(\\theta )&=&\\left[\\left(\\frac{0.9994}{[1+(\\theta /0.0029)^{4.5}]^2}\\right)^5\\right.\\nonumber \\\\&&\\hspace{20.0pt}{}+\\left.\\left(\\frac{0.0006}{[1+(\\theta /0.01)^{1.43}]^2}\\right)^5\\right]^{1/5}\\!\\!.$ We show this function overlaid with the $10^4$  km case in Fig.", "REF .", "For the other species, the situation is analogous with slightly different parameters." ], [ "Energy distribution", "The $\\bar{\\nu }_e$ flux spectrum emitted from the core roughly follows a thermal Maxwell-Boltzmann form $f_{\\rm c}(E)\\propto E^2\\,e^{-E/T}$ .", "In Fig.", "REF we show as a histogram (blue) the numerical spectrum of the core component (measured in the forward direction) together with a thermal fit with $T=4.8$  MeV (average energy 14.4 MeV).", "The halo component arises from scattering on nuclei with a cross section proportional to $E^2$ .", "Therefore, the halo spectrum should be $f_{\\rm h}(E)\\propto E^4\\,e^{-E/T}$ with the same $T$ .", "In Fig.", "REF we show as a red histogram the numerical halo spectrum (measured at a very large angle) together with such a fit.", "In the spirit of an overall consistency check, we find excellent agreement.", "Figure: Energy spectrum of core and halo components.Histogram: Numerical results.", "Smooth lines: Thermal spectra asdescribed in the text with the same TT." ], [ "Neutrino-neutrino refraction", "The impact on neutrino-neutrino refraction is quantified by $\\langle 1-\\cos \\theta \\rangle $ as defined in Eq.", "(REF ).", "Motivated by the numerical examples, we first consider truncated power-law intensity distributions of the form ${\\cal I}(\\theta )={\\left\\lbrace \\begin{array}{ll}1&\\hbox{for $\\theta \\le \\theta _{\\rm c}$}\\\\(\\theta _{\\rm c}/\\theta )^p&\\hbox{otherwise.}\\end{array}\\right.", "}$ We ask for the asymptotic behavior of $\\langle 1-\\cos \\theta \\rangle $ at large distance, corresponding to $\\theta _{\\rm c}\\ll 1$ .", "The integrand in Eq.", "(REF ) expands as $\\sin \\theta \\,(1-\\cos \\theta )\\rightarrow \\theta ^3/2$ .", "If $p>4$ , the integral is dominated by small angles and we find the result shown in Table REF .", "In other words, if ${\\cal I}(\\theta )$ falls off fast enough, we recover the classic $r^{-2}$ scaling, where we have used that $\\theta _{\\rm c}\\sim R/r$ .", "For $p\\le 4$ we can no longer extend the upper integration limit to $\\infty $ and can no longer expand the integrands.", "With Mathematica we find analytic results with coefficients for $\\theta _{\\rm c}\\ll 1$ that are given in Table REF .", "In particular, for $p=3$ the scaling is linear in $\\theta _{\\rm c}$ .", "Table: Average 〈1-cosθ〉\\langle 1-\\cos \\theta \\rangle for θ c ≪1\\theta _{\\rm c}\\ll 1 and different pp.Figure: Average 〈1-cosθ〉\\langle 1-\\cos \\theta \\rangle for our analyticangle distribution of Eq.", "(), representing the 280 ms numerical model.Figure: Same as Fig.", ", now showing theenhancement caused by the halo, i.e.", "the ratio total/corecontribution, where the core flux is attributed to the first term inEq.", "().Next we consider the analytic fit of Eq.", "(REF ) and show $\\langle 1-\\cos \\theta \\rangle $ in Fig.", "REF .", "We integrate up to $\\theta =\\pi $ for each radius, which means that we smoothly interpolate between small and large distances.", "At small radii (inside of the SN core) the distribution is isotropic and $\\langle 1-\\cos \\theta \\rangle =1$ .", "For $r200$  km we find approximately the naive $r^{-2}$ scaling.", "At larger distances, the halo becomes important and the scaling turns approximately to $r^{-1}$ .", "If the halo distribution would scale as $\\theta ^{-3}$ we would expect precisely $r^{-1}$ , but our fit corresponds to ${\\cal I}(\\theta )\\propto \\theta ^{-2.86}$ , explaining the less steep variation.", "In Fig.", "REF we show the enhancement of $\\langle 1-\\cos \\theta \\rangle $ caused by the halo, where we attribute the first part of the analytic fit of Eq.", "(REF ) to the core.", "The enhancement caused by the halo scales almost linearly at large distances.", "At $r\\sim 1000$  km the enhancement is about a factor of 8, roughly in agreement with Cherry et al.", "[7]." ], [ "Analytic halo estimate", "For an analytic estimate, we consider a total neutrino production rate $\\Phi =L/\\langle E\\rangle $ emitted from a pointlike source (neutrino luminosity $L$ ).", "It traverses a spherical matter distribution which we model as a decreasing power law of the form $n(r)=n_R(R/r)^m$ .", "We assume that multiple scatterings can be neglected.", "Every spherical shell is a neutrino source from scattering the primary flux.", "For the scattering cross section, we assume the form $d\\sigma /d\\Omega =\\sigma _{\\rm scatt}(1+a\\cos \\vartheta )/4\\pi $ , where $\\vartheta $ is the scattering angle.", "Elementary geometry (see the Appendix) reveals that at distance $r$ we expect a scattering flux (halo) of ${\\cal I}(r,\\theta )&=&\\frac{\\Phi \\,\\sigma _{\\rm scatt}n_R}{(4\\pi )^2R}\\,\\left(\\frac{R}{r\\,\\sin \\theta }\\right)^{m+1}\\nonumber \\\\&&{}\\times \\int _\\theta ^\\pi d\\vartheta \\,(1+a\\cos \\vartheta )\\,\\sin ^m \\vartheta \\,.$ For small angles, we find a power law ${\\cal I}(r,\\theta )\\propto \\theta ^{-(m+1)}$ , whereas the large-angle and backward flux depend on the detailed cross-section angular dependence.", "In our numerical example, the density outside the shock wave (at about 70 km) decreases roughly as $r^{-1.35}$ out to about 5000 km.", "With $m=1.35$ we expect the halo to vary as $\\theta ^{-2.35}$ as indicated in Fig.", "REF .", "While it would not provide a good global fit, it looks excellent for about the first decade of angles of the halo.", "Based on the analytic expression for ${\\cal I}(r,\\theta )$ we can estimate the inward-going flux.", "It is very small compared to the outward-going core flux, but its contribution to refraction need not be small.", "We consider a region at sufficiently large radius where neutrino-neutrino refraction is dominated by the halo and ask which fraction is caused by inward-going neutrinos, i.e.", "which fraction of the integral in Eq.", "(REF ) is contributed by $\\pi /2<\\theta \\le \\pi $ .", "For sufficiently small power-law index $m$ this fraction does not depend on the radius $r$ (to lowest order).", "Coherent neutrino-nucleus scattering is forward peaked and we use $a=1$ .", "For $m=1$ the contribution of the backward flux is 25%, for $m=2$ approximately 16%." ], [ "Stability Analysis", "All previous studies of collective flavor oscillations of SN neutrinos used the free-streaming approximation: it was assumed that collisions play no role in the relevant stellar region.", "On the other hand, the halo dominates neutrino-neutrino refraction at larger radii, so in some sense the free-streaming assumption gets worse with distance, not better.", "Therefore, in order to understand collective flavor conversion in this region, one may have to rethink the overall approach [7].", "On the other hand, the multiangle matter effect may still suppress the onset of collective flavor conversions, at least in some accretion-phase models.", "A linearized stability analysis would still be possible and self-consistent.", "If self-induced flavor conversions have not occurred up to some radius and if the matter effect is large in that region, then neutrinos continue to be in weak-interaction eigenstates up to small corrections caused by oscillations with the small in-matter effective mixing angle.", "Therefore, we can use the core+halo flux at that radius as an inner boundary condition for the subsequent evolution.", "In other words, collisions at smaller radii do not change the conceptual approach, they only provide a broader angular distribution than would have existed otherwise.", "A greater problem is how to deal with the inward-going flux which, for our conditions, contributes some 20% to neutrino-neutrino refraction.", "We will simply neglect it because it cannot be incorporated self-consistently in an approach based on outward-only streaming neutrinos.", "If the system is stable and stationary, then the picture remains self-consistent because neutrinos remain in flavor eigenstates, including the backward-moving ones.", "In this spirit we repeat our linearized flavor stability analysis for our 280 ms snapshot of a $15\\,M_\\odot $ accretion-phase model [8] in a simplified form.", "We model the angular distribution according to our analytic fit, Eq.", "(REF ), which we use at all distances in the form ${\\cal I}(r,\\theta )\\propto {\\cal I}_{\\rm fit}(\\theta \\,r/10^4~{\\rm km})\\,,$ because the curves shown in Fig.", "REF are almost identical up to an angle scaling.", "We cut the distribution at $\\theta =\\pi /2$ to avoid backward-going modes.", "We assume monoenergetic neutrinos with $\\omega _{\\rm c}=\\Delta m^2/2E_{\\rm c}=0.5~{\\rm km}^{-1}$ for the core flux, i.e.", "$E_{\\rm c}\\sim 12$  MeV.", "We have checked that this choice reproduces well the instability region in Fig.", "4 of our previous paper [8] where the full spectrum was used.", "The halo energies are larger as described in Sec.", "REF , implying $\\langle E^{-1}_{\\rm c}\\rangle =2\\langle E^{-1}_{\\rm h}\\rangle $ , thus we use monoenergetic halo neutrinos with $\\omega _{\\rm h}=0.25~{\\rm km}^{-1}$ .", "The system is unstable in flavor space if the linearized equation of motion discussed in Ref.", "[8] has eigenvalues with imaginary part $\\kappa $ , the radial growth rate (in units of km$^{-1}$ ) of the unstable mode.", "Whether this growth rate is large or small depends on the available distance.", "In other words, at a given distance $r$ , the instability is important if $\\kappa r\\gg 1$ and unimportant if $\\kappa r\\ll 1$ .", "Therefore, in Fig.", "REF we show contours of constant $\\kappa r$ in a two-dimensional parameter space consisting of radius $r$ of our model and an assumed electron density $n_e$ which causes the multiangle matter effect.", "Figure: Contours for the indicated κr\\kappa r values in the parameterspace of radius rr and assumed electron density n e n_e.", "We use asimplified energy spectrum as described in the text.", "The core fluxalone is responsible for κr≫1\\kappa r\\gg 1 for r700r700 km (blueshading), the halo adds the red-shaded region at larger rr.", "We alsoshow the density profile of our 280 ms numericalmodel.If we use the core flux alone, this figure contains the same information as Fig.", "4 in our previous paper [8].", "In the absence of matter, the instability would set in at about 150 km, whereas the presence of matter suppresses the instability entirely: the SN density profile never overlaps with the instability region.", "The most dangerous region of closest approach is at around 600 km.", "In Fig.", "REF one clearly sees how the halo flux extends the instability region to larger radii (red shading), but only far away from the SN density profile.", "In other words, the multiangle matter effect strongly suppresses the would-be instability caused by the halo.", "This finding does not contradict the importance of the halo flux for neutrino-neutrino refraction.", "It is true that the neutrino-neutrino interaction energy caused by the halo was found to decrease roughly as $r^{-3}$ , instead of $r^{-4}$ by the core flux alone.", "However, the multiangle matter effect is enhanced in a similar way because it depends on the same neutrino angular distribution.", "A more appropriate comparison is between the neutrino and electron densities.", "The former decreases as $r^{-2}$ whereas in our model the latter decreases roughly as $r^{-1.35}$ and thus becomes relatively more important with distance.", "Therefore, it is no surprise that the multiangle matter effect would be more important with increasing distance." ], [ "Conclusions", "The halo distribution of neutrinos causes neutrino-neutrino refraction to decrease more slowly with distance than had been thought, but still faster than the matter density during the accretion phase.", "Repeating our linearized stability analysis including the halo, the system remains stable for the chosen example.", "As anticipated, the multiangle matter effect is a crucial ingredient for collective flavor conversions especially in those regions where the halo flux is important.", "Thus it remains possible that self-induced flavor conversions are generically suppressed during the early accretion phase.", "The early signal rise would then encode a measurable imprint of the neutrino mass hierarchy [11].", "However, more systematic studies are required to understand if such conclusions are generic.", "For example, we have assumed that the relatively small backward flux can be neglected for the stability analysis, but its possible role needs to be better understood.", "Likewise, the role of multi-D effects in SN modeling remains to be explored.", "The topic of collective flavor oscillations of SN neutrinos remains a fiendishly complicated problem." ], [ "Acknowledgements", "This work was partly supported by the Deutsche Forschungsgemeinschaft under Grants No.", "TR-7 and No.", "EXC-153, and by the European Union FP7 ITN INVISIBLES (Marie Curie Actions, PITN-GA-2011-289442).", "I.T.", "thanks the Alexander von Humboldt Foundation for support.", "Figure: Geometric definitions for the analytic haloestimate.", "*" ], [ "Analytic Halo Estimate", "In order to derive the analytic halo estimate of Eq.", "(REF ) we consider a radial distribution of scattering targets, $n(\\ell )$ , and a differential scattering cross section $d\\sigma /d\\Omega $ .", "The scattering rate per volume and per solid angle for neutrinos with number density $n_{\\nu }(\\ell )$ at radius $\\ell $ is $\\frac{d q_\\mathrm {scatt}}{d \\Omega } =n_{\\nu }(\\ell )\\,n(\\ell )\\,c\\,\\frac{d \\sigma }{d \\Omega } \\,,$ where we have reinstated the speed of light.", "The “number intensity” ${\\cal I}(r,\\theta )$ at radius $r$ of neutrinos scattered to angle direction $\\theta $ relative to the radius vector at $r$ can be obtained by adding up all scattering contributions along the path $s$ as shown in Fig.", "REF .", "It depends on the total neutrino emission rate $\\Phi = L/\\left\\langle E\\right\\rangle $ of the pointlike source assumed to be located at $r = \\ell = 0$ , which determines the neutrino number density $n_{\\nu }(\\ell ) = \\Phi /(4\\pi \\ell ^2 c)$ .", "With this result and Eq.", "(REF ) one finds ${\\cal I}(r,\\theta ) &=& \\int _0^\\infty ds\\,\\frac{d q_\\mathrm {scatt}}{d \\Omega }(\\alpha +\\theta ) \\nonumber \\\\&=& \\int _0^\\infty ds\\,\\frac{\\Phi }{4\\pi \\ell ^2}\\,n(\\ell )\\,\\frac{d\\sigma }{d\\Omega }(\\alpha +\\theta )\\,,$ where $\\vartheta =\\alpha +\\theta $ is the scattering angle and $\\ell = r\\sin \\theta /\\sin \\vartheta $ and $s = r\\sin \\alpha /\\sin \\vartheta $ , see Fig.", "REF .", "Using a decreasing power-law distribution of the target density, $n(\\ell ) = n_R(R/\\ell )^m$ with $m \\ge 0$ , an assumed dependence of the scattering cross section on the scattering angle $\\vartheta $ of the form $d\\sigma (\\vartheta )/d\\Omega = \\sigma _\\mathrm {scatt}(1 +a\\cos \\vartheta )/4\\pi $ , one can transform the integral of Eq.", "(REF ) into the form of Eq.", "(4)." ] ]

[ [ "Leo IV & V - A possible dwarf galaxy pair ?" ], [ "Abstract The last few years have seen the discovery of many faint and ultra-faint dwarf spheroidal galaxies around the Milky Way.", "Among these is a pair of satellites called Leo IV and Leo V. This pair is found at large distances from the Milky Way (154 and 175 kpc respectively).", "The rather small difference in radial distance, and the fact that they also show a close projected distance on the sky, has led to the idea that we might be seeing a new pair of bound galaxies - like the Magellanic Clouds.", "In this paper we investigate this speculation by means of a simple integration code (confirming the results with full N-body simulations).", "As the luminous mass of both faint dwarfs is far too low to allow them to be bound, we simulate the pair assuming extended dark matter haloes.", "Our results show that the minimum dark matter mass required for the pair to be bound is rather high - ranging from 1.6 x 10^10 Msun to 5.4 x 10^10 Msun (within the virial radii).", "Computing the mass of dark matter within a commonly adopted radius of 300 pc shows that our models are well within the predicted range of dark matter content for satellites so faint.", "We therefore conclude that it could be possible that the two galaxies constitute a bound pair." ], [ "Introduction", "The last decade has seen the discovery of many new faint dwarf spheroidal (dSph) galaxies of the Milky Way (MW) [37], [1], [39], [2], [36].", "Many of these dwarfs are less luminous than a globular cluster (or even an open cluster) and exhibit high velocity dispersions (given their luminous mass) [31], [16], [10].", "Should these objects be in virial equilibrium, they are the most dark matter (DM) dominated objects known in the universe.", "They would exhibit mass-to-light (M/L) ratios of more than a thousand [31], [9], [10].", "$\\Lambda $ CDM simulations [4], [18] predict that a galaxy like our MW should be surrounded by hundreds, if not thousands of small DM haloes which could host a dwarf galaxy.", "The discrepancy between the known number of MW satellites and these predicted values is known as the `missing satellite problem' [15], [24].", "The discovery of new faint dwarfs in the SDSS catalogue doubled the number of known satellites.", "Extrapolating to areas of the sky and distances not covered by the survey [17], [20], [21] may suggest that the missing satellite problem is now solved.", "Amongst these ultra-faint galaxies are dwarfs which are extremely metal-poor [14], some show complex star formation histories [12] and many of them show unusual morphologies [5], [29].", "Others seem to show signs of tidal disruption [40], [8], [26].", "Signs of tidal disruption and the fact that all known dwarfs seem to be aligned in a disc-like structure around the MW [22], [23] has given rise to alternative explanations for the existence of those galaxies.", "These theories imply that most, if not all, of the MW dwarfs are in fact no more than disrupting star clusters or tidal dwarf galaxies hosting no DM at all [28].", "Table: Observational properties of Leo IV and Leo V. The data ismainly taken from .In this paper we focus on two of these new ultra-faint dwarfs, namely Leo IV [2] and Leo V [3].", "Their properties have been studied by many authors: for Leo IV: [25], [32], [30] and for Leo V: [34], [35].", "We summarize a selection of the observational data in Tab.", "REF .", "The pair of galaxies is found at a rather large distance from the MW (154 and 175 kpc respectively).", "The two galaxies are not only very close to each other in radial distance (22 kpc) [25], [13] but also in projected distance on the sky.", "Their radial velocity differs only by about 50 km s$^{-1}$ .", "In the discovery paper of Leo V the authors speculate that the two dwarfs could be a bound pair similar to the Magellanic Clouds.", "In particular, the smaller dwarf, Leo V, exhibits deformed elongated contours [34], which could be signs of tidal interaction.", "Furthermore, there is a tentative stellar bridge between the two satellites [13], which now is more likely a foreground stream of the Virgo overdensity [11].", "[13] argued that, to form a bound object, the twin system would need a lot of DM - much more than is seen in similar faint satellites.", "Nevertheless, the authors claimed that it is highly unlikely that the two satellites are a simple by-chance alignment.", "They also rule out the possibility that the two faint dwarfs are not galaxies at all but simple density enhancements of a stellar stream by orbital arguments.", "They conclude that they might be a `tumbling pair' of galaxies that have fallen into the MW together.", "With our paper we want to investigate the hypothesis of a bound pair further.", "Using a simple two-body integration method we investigate a large part of the possible parameter space, searching for the minimal total mass the system needs, in order to form a tightly bound pair.", "We assume that both galaxies have their own DM halo, orbiting each other.", "We describe the setup of our simulations in the next section.", "We then report and verify (using full N-body simulations) our results in Sect.", "and finally discuss our findings in Sect. .", "We use the positions and distances, as reported in the discovery papers of [2], [3].", "This enables us to transform the positions into a Cartesian coordinate system.", "The coordinates for Leo IV and Leo V are shown in the first three lines in Tab.", "REF .", "Current observations only provide radial velocities for the pair.", "Nothing is known so far about their proper motions.", "We therefore adopt two velocity cases in our simulations.", "First, we investigate the case that the radial velocity is the only velocity component the galaxies have (line 4-6 in Tab.", "REF ) and second, we adopt an outward tangential velocity of the same magnitude as the radial velocity (line 7-9 in Tab.", "REF ).", "Here `outward tangential' means that the velocity vector is perpendicular to the radial velocity and points away from the other dwarf.", "The latter choice could be regarded as the worst case scenario.", "With these assumptions we restrict the possible parameter space of initial conditions significantly, but we are able to deduce how strong the influence of a tentative tangential velocity is on our results.", "To restrict our parameter space further we adopt just two cases for the mass-ratio between the two satellites.", "The observed absolute magnitudes give a luminosity ratio of $1.8$ .", "Adopting a mass-follows-light scenario for one of our cases, we use the same ratio for the two DM haloes: $\\frac{L_{\\rm LeoIV}}{L_{\\rm LeoV}} = \\frac{1.8 \\cdot 10^4}{1 \\cdot 10^4} = 1.8 = \\frac{M^*_{\\rm LeoIV}}{M^*_{\\rm LeoV}} = \\frac{M^{\\rm DM}_{\\rm LeoIV}}{M^{\\rm DM}_{\\rm LeoV}}.$ [35] suggest that the faint and ultra-faint dwarfs reside in similar DM haloes of a certain minimum mass.", "Therefore, we investigate equal mass DM haloes as the other case.", "For the four cases described above we search for solutions adopting halo concentrations of $c = 5$ , 10 and 20, as those are values typically adopted for dwarf galaxies [19].", "This gives a total of 12 different solutions to the problem.", "The haloes are described as NFW-profiles with $c = r_{\\rm vir} /r_{\\rm scale}$ , $r_{\\rm vir} = r_{200}$ being the virial radius in which the density is 200 times the critical density of the universe, using a standard value of the Hubble constant of $H_{0} =70$  km s$^{-1}$  Mpc$^{-1}$ .", "Table: This table shows the minimum bound mass for each of ourcases.", "The first column gives the number of the case, the secondis the adopted concentration of the haloes.", "Then we give the massof the DM halo and its virial radius for Leo IV and Leo V. Thenext column gives the total mass in DM of the whole system, thenext column shows the `ratio' by which the maximum distancediffers between the full N-body simulation and the two-body code.The last column is a short explanation for the cases: rad.", "vel.", "=only radial velocity, rad.", "& tan.", "vel.", "= radial and tangentialvelocity adopted; mass ratio 1.8 = the two haloes have a fixedmass ratio of 1.8; equal mass = the two haloes have the same mass" ], [ "Method", "[13] investigated the minimum mass for the two DM haloes of the satellites, by assuming they were point masses.", "We use a simple two-body integration programme modelling the system in the following way: Both satellites are represented by analytical, rigid [27] potentials.", "The force on the centre of mass of one halo is computed using the exact force according to its position with respect to the potential of the other halo.", "To be a tightly bound pair we adopt a rigid distance criterion which requires that neither centre of the two haloes leaves the halo of the other dwarf, i.e.", "their separation is always smaller than the virial radius of the (smaller) halo.", "For each case we choose the total mass of the system and set up the two haloes according to their mass-ratio and concentration.", "Then we run the two-body code to determine if our distance criterion is fulfilled.", "If the maximum distance is larger or smaller, we alter the the total mass respectively and use the code again.", "We iterate this process until we find a maximum separation equal to our distance criterion.", "The reason why we choose a distance criterion instead of computing the escape velocity (i.e.", "the velocity the two dwarfs need to separate from each other to infinity) is that if we were to adopt such a criterion, we would also include bound cases in which the maximum separation between the two dwarfs easily exceeds their distance to the MW.", "As this would not make sense, we exclude these solutions by imposing a very rigid distance criterion.", "Furthermore, an estimation of the tidal radii of Leo IV and V gives values that are similar to our distance criterion.", "Using $r_{\\rm tidal} & \\approx & \\left( \\frac{m_{\\rm dwarf}}{3 M_{\\rm MW}(D)} \\right)^{{1/3}} \\cdot D$ with $m_{\\rm dwarf} \\approx 4 \\times 10^{10}$  M$_{\\odot }$ , $M_{\\rm MW}(D)\\approx 10^{12}$  M$_{\\odot }$ and $D \\approx 165$  kpc, we get 40 kpc as tidal radius.", "This is slightly lower than the distance criterion used, but is also a rather rough estimate.", "However, it shows that the distance criterion is a sensible way to restrict the solutions.", "Hence, the choice of our distance criterion allows us to treat the galaxy pair as isolated, i.e.", "to neglect the potential of the MW.", "The simulations are always computed forward in time, starting from our current view of the dwarfs.", "We thus ascertain the next maximum separation to assess if the two dwarfs are bound to each other now.", "As we do not know where they came from, nor the details of their orbit around the MW, we cannot predict their future fate using these models." ], [ "Full N-body simulations", "To ensure that the results are reasonable, we perform full N-body simulations as a check of each of the 12 solutions obtained with our simple code.", "We use the particle-mesh code Superbox [6].", "It is fast and enables simulations of galaxies on normal desktop computers.", "It has two levels of higher resolution grids, which stay focused on the simulated objects, providing high resolution only in the areas where it is needed.", "Each object (halo) is modeled using 1,000,000 particles.", "We use NFW distributions for the haloes according to the results obtained with the two-body code.", "The haloes extend all the way to their virial radius.", "The resolution of the grids is such that we try to keep about 15 cell-lengths per scale length, $r_{\\rm sc}$ , of the haloes.", "A particle-mesh code has no softening-length like a Tree-code but previous studies [33] showed that the length of one grid-cell is approximately the equivalent of the softening-length.", "Furthermore the particles in a particle-mesh code are not stars or in our case DM-particles, they rather represent tracer-particles of the phase-space of the simulated object.", "Densities are derived on a grid, and then a smoothed potential is calculated.", "The number of particles is chosen according to the adopted grid-resolution to ensure smooth density distributions.", "A detailed discussion about the particle-mesh code Superbox can be found in [6].", "To clarify we state once more, that the N-body simulations are only used to verify the results of the simple method.", "This means we check the next maximum distance of the two galaxies.", "The simulations do not represent full-scale simulations of the past, present and future of the two Leo galaxies.", "Such simulations are more demanding and are beyond the simple scope of this paper." ], [ "Point Mass Case", "We first recalculate the minimal bound mass assuming both galaxies are point masses, following the methodology by [13].", "We assume that the relative velocity between the two dwarfs is equal to their escape velocity (i.e.", "the velocity required for the haloes to separate to infinity).", "Because point masses do not have any characteristic radius we cannot apply any meaningful distance criterion here.", "Using only the observed radial velocities (case 0a) we confirm the total DM mass of the system obtained by [13].", "Our result is a factor of two lower, however, those authors used an approximation obtained from energy arguments, while we perform the full escape velocity calculation.", "Given the wide range in possible results, as shown later in the text, and taking the observational uncertainties into account, a difference of a factor of two is still a very good match.", "Furthermore, we calculate the minimum bound mass assuming the two satellites also have tangential velocities (according to Tab.", "REF ).", "In this case (case 0b) the total mass required to keep the galaxies bound is $1.47\\times 10^{10}$  M$_{\\odot }$ .", "That is, three times more DM is required to keep them bound.", "While this may seem a large mass, we refer the reader to the following sections to put this result into context.", "The results are shown in Tab.", "REF in the first two lines and in Fig.", "REF at $1/c = 0$ ." ], [ "Two-body Integrator", "We now have a look at our results, obtained by the two-body integrator we use.", "As explained in Sect.", "the two haloes are rigid, analytical potentials acting on the centre of mass of the other galaxy.", "Additionally, we now introduce a very strict distance criterion of the form that neither halo centre should leave the other halo (i.e.", "separations larger than the (smaller) virial radius).", "This way we make sure that we are really dealing with a tightly bound pair.", "The results are two-fold: Of course we see an immediate large increase in the required minimum mass (compared to the point-mass cases) just by introducing the rigid distance criterion.", "We plot the total mass in DM against $1/c$ in Fig.", "REF , to include the point mass cases ($c=\\infty $ ).", "We see that the bound mass is larger for lower values of the concentration.", "This can be easily understood as with higher concentrations we have more of the total mass of the halo concentrated towards the centre and therefore the gravitational pull on the other dwarf is larger.", "Secondly, we also find that including the additional tangential velocity roughly triples the required mass.", "While the cases with radial velocities (cases 1–3, 1a–3a) require masses in the range of about $1.6$ –$2.1 \\times 10^{10}$  M$_{\\odot }$ , the additional tangential velocity increases the necessary masses up to much larger values of $4.2$ –$5.4 \\times 10^{10}$  M$_{\\odot }$ (cases 4–6, 4a–6a).", "As a little side-remark, we see that we need slightly less massive haloes in the equal halo mass cases than if we adopt a mass-ratio of $1.8$ between the two haloes.", "As these differences are small compared with the differences of the unknown concentration, and even more so with the unknown tangential velocity, we can easily neglect them and assume that distributing the mass differently between the two haloes has no strong effect on our results.", "The possible range of DM masses for the two galaxies spans about half an order of magnitude.", "However, given the large observational uncertainties it is the best we can do.", "The masses themselves are rather large and taken at face value would imply that the two dwarfs are among the most DM dominated objects in the observed universe." ], [ "Comparison with observationally obtained data", "We now put our results into context and compare them with observations.", "[31] measured the velocity dispersion of Leo IV and derived a total dynamical mass, within their optical radius (97 pc), of $1.4 \\pm 1.5 \\times 10^{6}$  M$_{\\odot }$ .", "This implies a M/L-ratio of 151.", "This value is quite similar to most of the other known dSph galaxies of the MW.", "We, therefore, use our results and compute the mass of our Leo IV haloes within the same 97 pc.", "The resulting masses and derived M/L-ratios are shown in Tab.", "REF .", "Our M/L-ratios are in the range of 35–328 and encompass the results of [31].", "Furthermore, should the results of [31], which are based on very few stars, prove to be correct, our results mean that we can rule out DM haloes with high concentrations (i.e.", "$c=20$ ).", "[34] report a central velocity dispersion of $\\sigma =2.4^{+2.4}_{-1.4}$  km s$^{-1}$ based on five stars for Leo V. Using this value they calculate a dynamical mass, within an adopted $r_{\\rm h} = 67.4$  pc, of $3.3^{+9.1}_{-2.5} \\times 10^{5}$  M$_{\\odot }$ .", "Calculating the mass within this radius in our models gives a range from $2.7 \\times 10^{5}$  M$_{\\odot }$ to $2.5 \\times 10^{6}$  M$_{\\odot }$ , again encompassing the results derived from observations by [34].", "Another way to compare our results with observations is by computing the total mass within a `standard' radius of 300 pc, as adopted by [35].", "We can then infer the M/L-ratio inside this radius and compare our results with the observationally derived results reported by [38].", "Given the low luminous masses of the two dwarfs, our results point to M/L-ratios in the order of $\\log _{10}(M/L) = 2.5$ –$3.7$ (see also Tab.", "REF ).", "Despite being quite high, plotting these values together with observationally derived values of other dwarfs (Fig.", "REF ) we see that they follow the general trend of higher M/L-ratios with lower luminous masses.", "In fact, if we fit a line through the observational results, our values would intersect that linear fit.", "This means, not only do our results follow and confirm the observed trend of the known MW dSphs, we can further conclude that the two satellites do not need unreasonable amounts of DM to form a bound pair." ], [ "Comparison to N-body simulations", "Since we now have the results of all our cases, we have to make sure that they still hold if we resimulate them with a full N-body code.", "Of course the full N-body simulations will differ significantly from the ones above, we simply want to know if our conclusions remain valid.", "In the full N-body simulations, the two live haloes are interacting with each other.", "They experience dynamical friction which shrinks their orbits around each other until they finally merge.", "This cannot be reproduced by the simple code but we can determine if the next maximum separation of the orbit is smaller than the extent of the other halo, as our distance criterion in the simple simulations requires.", "What we find is that the results differ by a few per cent (max.", "6.5%) in the simulations using a mass ratio of $1.8$ between the haloes.", "The haloes get slowed down and, therefore, turn around at a smaller separation.", "If the haloes have the same mass and we only adopt radial velocities, the orbit overshoots the maximum distance by approximately 20 per cent.", "If we add the additional tangential velocities, our restricted results differ by about 28–45 per cent, in the sense that the maximum separation is larger in the `full' simulation than in the restricted one.", "This may seem odd given that dynamical friction should act in the opposite direction, however, there are other mechanisms at work.", "We see an expansion of the haloes as orbital energy is transformed into internal energy, furthermore, we see that the two haloes get deformed – particles from one halo get dragged along by the gravitational force of the other.", "We give the ratio of the maximum separations between the full and restricted simulations in the second to last column of Tab.", "REF (labeled `ratio').", "We plot, in Fig.", "REF , the results of case 4a, the case with the largest discrepancy in separation between the two haloes compared with the restricted prediction.", "In the left panel we see the contours of the Leo V halo at the time of maximum separation, with the cross marking the position of the centre of Leo IV.", "We see that the contours are slightly elongated towards the other halo and that they show a clear deformation.", "This deformation is caused by the gravitational pull of the other halo, which has dragged particles of the dwarf towards it.", "In the middle panel we see the large discrepancy between the distance criterion (horizontal line) and the actual first maximum separation of the orbit.", "But as the total mass of a NFW profile only increases with the logarithm of the radius, even a large discrepancy in radius as in our case 4a amounts only to a few per cent error in the mass of the halo.", "Given the fact that our results span almost an order of magnitude, and the large uncertainties from the observations (luminous mass, distance, etc.", "), we claim that the results of the restricted code are verified.", "Finally, the right panel of Fig.", "REF shows the Lagrangian radii of Leo IV (solid lines) and Leo V (dashed lines).", "We see that the interaction of the two haloes causes the Lagrangian radii to expand.", "At the time of the maximum separation (marked with a cross) the halo of Leo IV is just outside the 90 % mass-radius and, as shown in the left panel, is still within the expanded and deformed halo of Leo V. In some sense, this matches the original distance criterion, which said that neither halo-centre should leave the other halo.", "We have presented possible scenarios for a twin system consisting of the faint dwarf spheroidal galaxies Leo IV and Leo V. The simulations were performed using a simple two-body code to rapidly find the solutions in the vast parameter space and the results were verified using a full N-body code.", "From this we find the minimum DM masses required for the two galaxies to form a tightly bound pair.", "The parameter space is restricted by assuming two independent DM haloes orbiting each other.", "Two perfectly shaped haloes would only be seen before the first close passage.", "This is a strong simplification of the real geometry of the problem.", "But as our results show (i.e.", "the comparison with the real N-body simulations) the resulting error of this simplification is in the order of 5-20% and therefore much smaller than the mass-range of our results, stemming from e.g.", "the unknown tangential velocity.", "A further restriction is the maximum distance criterion we adopt.", "We find this criterion sensible given the satellites' large distances from the MW.", "Smaller maximum separations would lead to higher required masses for the system to be bound.", "Larger separations would lead to lower masses, but since the expected tidal radius of the system (with respect to the gravitational force of the MW) is of the order of our distance criterion, we feel confident with our choice.", "As our distance criterion is of the order of the tidal radius, we are able to simplify even further and treat the system of the two dwarfs as isolated (i.e.", "we do not simulate the potential of the MW).", "As our aim is to determine whether the two galaxies are bound now (and make no predictions about their future or past), we do not need to take their orbit around the MW into account.", "Regarding the relative velocity we adopt two cases.", "In one the restriction is that the measured difference in radial velocity is the only relative velocity the dwarfs have.", "In the other case the two satellites are given an additional tangential velocity of the same magnitude as the radial velocity.", "We also adopt two mass ratios.", "First, mass-follows-light, i.e.", "the two haloes have a mass ratio of $1.8$ like the luminous components.", "Second, mimicking the fact (claimed by [35]) that almost all dSph galaxies reside in DM haloes of the same minimum mass (i.e a minimum halo mass to carry a luminous component), a mass ratio of $1.0$ .", "Moreover, to span the full range of proposed concentrations for dwarf galaxy dark haloes we take three values for the concentration into account $c = 5, 10, 20$ .", "If we assume that the bound system consists of two DM haloes orbiting each other, we infer masses of about $1.7 - 2.1 \\times 10^{10}$  M$_{\\odot }$ for the whole system.", "If we add an additional tangential velocity, which cannot be observationally verified, we obtain $\\approx 4.2 - 5.4 \\times 10^{10}$  M$_{\\odot }$ .", "These are indeed very high masses for the two faint satellites and would put them amongst the most DM dominated objects known.", "Still, these results do not infer that the scenario is impossible.", "Another point to take away is that if we add the same amount of relative velocity tangentially, i.e.", "increasing the total relative velocity by a factor $\\sqrt{(}2)$ , the required mass more than doubles.", "This shows quite a strong dependence on the relative velocity.", "Still, if we double the tangential velocity the mass would vary within an order of magnitude, an uncertainty we find in our results anyway.", "We compute the DM mass within the adopted optical radius of [31] for Leo IV (i.e.", "97 pc) and find that the M/L-ratios we obtain span the observationally (measured velocity dispersion) derived results.", "Our results are also in agreement with the measured velocity dispersion of Leo V [34] and the inferred dynamical mass.", "Taking the observations at face value, our results could, therefore, restrict the possible concentrations of the real DM haloes, once we know their relative tangential velocity.", "Furthermore, we checked our results against the trend for dSph galaxies published by [38].", "They give the M/L-ratios within a radius of 300 pc [35].", "Our simulations predict M/L-ratios, using the same radius, in the range $\\log _{10}{M/L} = 2.5$ –$3.7$ .", "These values are high but encompass the predictions for faint dSph galaxies, if we extrapolate the known values to the magnitudes of the Leos.", "Comparing the results of our two-body code with full N-body simulations we find differences in the maximum separations of only a few per cent in most of the simulations.", "Only the simulations with equal mass haloes and additional tangential velocity have rather large discrepancies.", "Because the mass of an NFW halo increases with radius, proportional to $\\ln (r)$ , the uncertainty in the masses is much lower.", "The simple integration programme used cannot predict any deformations of the haloes due to their mutual interactions.", "In the full simulations we see those deformations and, even though the initial distance criterion is not fulfilled anymore, the haloes still stay within the deformations of the other.", "In that sense the distance criterion is still obeyed.", "Our final remark is that we wanted to search for the necessary total dark matter mass of the pair of satellites to ensure that they are bound to each other.", "Even though the comparison between the simple code and the full N-body results deviate somewhat from our distance criterion, they do not change the conclusions of the simulations.", "A bound pair in the restricted case is still a bound pair in the full simulations.", "Just by looking at our different cases (i.e.", "radial velocity only or radial plus tangential velocity) our results differ by about half an order of magnitude in total mass.", "In that respect a mass uncertainty of even 20–30 per cent does not change the conclusions of this paper, nor would it alter the inferred M/L-ratios significantly.", "Summing up, assuming that the two Leos do, in fact, consist of a tightly bound pair, we find their inferred dark matter masses to be high but still within reasonable values.", "Therefore, it is possible that the two galaxies form a bound pair, making them an ultra-faint counterpart of the Magellanic Clouds.", "MF acknowledges funding through FONDECYT grant 1095092 and BASAL.", "RS is funded through a Comite Mixto grant." ] ]

[ [ "Adaptive Wavelet Collocation Method for Simulation of Time Dependent\n Maxwell's Equations" ], [ "Abstract This paper investigates an adaptive wavelet collocation time domain method for the numerical solution of Maxwell's equations.", "In this method a computational grid is dynamically adapted at each time step by using the wavelet decomposition of the field at that time instant.", "In the regions where the fields are highly localized, the method assigns more grid points; and in the regions where the fields are sparse, there will be less grid points.", "On the adapted grid, update schemes with high spatial order and explicit time stepping are formulated.", "The method has high compression rate, which substantially reduces the computational cost allowing efficient use of computational resources.", "This adaptive wavelet collocation method is especially suitable for simulation of guided-wave optical devices." ], [ "Introduction", "The numerical solution of Maxwell's equations is an active area of computational research.", "Typically, Maxwell's equations are solved either in the frequency domain or in the time domain, where each of these approaches has its own relative merits.", "We are specifically interested in efficient algorithms for light propagation problems in guided wave photonic applications [1], and work in the time domain.", "The most popular class of methods in this area is the finite difference time domain (FDTD) method [2].", "Due to the structured grid requirement of these methods, they become cumbersome while dealing with optical devices having curved interfaces and different length scales.", "To overcome these difficulties, a discontinuous Galerkin time domain (DGTD) method has been investigated [3].", "For a time dependent wave propagation problem, all these methods use a fixed grid/mesh for discretization.", "In general, such a grid can under-sample the temporal dynamics, or over-sample the field propagation causing high computational costs.", "If the spatial grid adapts itself according to the temporal evolution of the field, then the computational resources will be used much more efficiently.", "We propose an adaptive-grid method which represents propagating fields at each time step by a compressed wavelet decomposition, and which automatically adapts the computational mesh to the changing shape of the signal.", "In the initial studies of the wavelet formulation, the interpolating scaling functions were used for frequency domain waveguide analysis [4].", "To the best of our knowledge, the suitability of the wavelet decompositions for time dependent Maxwell problems has not yet been investigated.", "Vasilyev and his co-authors developed the adaptive wavelet collocation time domain (AWC-TD) method as a general scheme to solve evolution equations, and they successfully verified the scheme's effectiveness in the area of computational fluid dynamics [5], [6].", "Based on these studies, we present in this work a proof-of-concept for an AWC-TD for the time dependent Maxwell's equations.", "The paper is organized as follows.", "In Sec.", ", we provide a brief account on Maxwell's equations and some of the related concepts for their numerical solutions.", "We start Sec.", "with an introduction to (interpolating) wavelets, and how they can be used to discretize partial differential equations.", "Also in this section we explain the structure of AWC-TD method in the context of Maxwell's equations.", "Sec.", "gives algorithmic details of the method.", "Numerical results of the AWC-TD method are given in Sec.", "which contains our numerical experiments of propagating a 2D Gaussian peak in homogeneous environment.", "Finally we close the paper with concluding remarks in Sec.", "." ], [ "Time domain Maxwell's equations", "Propagation of optical waves in a linear, non-magnetic dielectric medium with no charges and currents is governed by the following time dependent Maxwell's equations $- \\dfrac{\\partial }{\\partial t}\\vec{\\mathcal {B}}(\\vec{r}, t) = \\nabla \\times \\vec{\\mathcal {E}}(\\vec{r}, t),\\quad \\dfrac{\\partial }{\\partial t}\\vec{\\mathcal {D}}(\\vec{r}, t) = \\nabla \\times \\vec{\\mathcal {H}}(\\vec{r},t),\\quad \\nabla \\cdot \\vec{\\mathcal {D}}(\\vec{r}, t) = 0,\\ \\text{ and }\\ \\nabla \\cdot \\vec{\\mathcal {B}}(\\vec{r}, t) = 0,$ where the electric field $\\vec{\\mathcal {E}}$ and the electric flux density $\\vec{\\mathcal {D}}$ , as well as the magnetic field $\\vec{\\mathcal {H}}$ and the magnetic flux density $\\vec{\\mathcal {B}}$ , are related by the constitutive relations $\\vec{\\mathcal {D}}(\\vec{r}, t)= \\varepsilon _0 \\varepsilon _r(\\vec{r}) \\vec{\\mathcal {E}}(\\vec{r}, t)\\ \\text{ and }\\ \\vec{\\mathcal {B}}(\\vec{r}, t) = \\mu _0 \\vec{\\mathcal {H}}(\\vec{r}, t).$ Here $\\varepsilon _0$ is the free space permittivity, $\\varepsilon _r$ is the relative permittivity and $\\mu _0$ is free space permeability.", "For illustration purpose, we restrict ourselves to a 2D setting where the fields and the material properties are assumed to be invariant in the $y$ -direction, i. e. $\\vec{r} = (x, z)$ and the partial derivatives of all fields with respect to $y$ vanish identically.", "We suppress the explicit function dependence on $\\vec{r}$ and $t$ .", "Then Maxwell's equations (REF ) decouple into a pair of independent sets of equations, $\\frac{\\partial \\mathcal {E}_{x}}{\\partial t} = -\\frac{1}{\\varepsilon _0 \\varepsilon _r}\\frac{\\partial \\mathcal {H}_{y}}{\\partial z},~\\hfill \\frac{\\partial \\mathcal {E}_{z}}{\\partial t} = \\frac{1}{\\varepsilon _0 \\varepsilon _r} \\frac{\\partial \\mathcal {H}_{y}}{\\partial x},~\\hfill \\frac{\\partial \\mathcal {H}_{y}}{\\partial t} = \\frac{1}{\\mu _{0}}\\left(\\frac{\\partial \\mathcal {E}_{x}}{\\partial z}-\\frac{\\partial \\mathcal {E}_{z}}{\\partial x} \\right),$ identified as transverse electric (TE)$_y$ setting, and $\\frac{\\partial \\mathcal {H}_x}{\\partial t} = \\frac{1}{\\mu _{0}} \\frac{\\partial \\mathcal {E}_y}{\\partial z}, ~\\hfill \\quad \\frac{\\partial \\mathcal {H}_z}{\\partial t} = -\\frac{1}{\\mu _{0}} \\frac{\\partial \\mathcal {E}_y}{\\partial x},~\\hfill \\frac{\\partial \\mathcal {E}_y}{\\partial t} = \\frac{1}{\\varepsilon _0 \\varepsilon _r} \\left(\\frac{\\partial \\mathcal {H}_x}{\\partial z}-\\frac{\\partial \\mathcal {H}_z}{\\partial x} \\right),$ identified as transverse magnetic (TM)$_y$ setting.", "Here $\\mathcal {E}_{x}$ , $\\mathcal {E}_{z}$ , $\\cdots $ etc.", "denote the respective field components.", "Originally, Maxwell's equations are formulated for a whole space.", "For numerical computations we need to restrict them to a bounded computational domain $\\Omega $ as shown in Fig.", "REF .", "This is done with a transparent boundary condition, which is realized in our case with perfectly matched layer (PML) [7], [8].", "The principle of PML is that (outgoing) waves scattered from the scatterer $\\Omega _{\\mathrm {s}}$ pass through the interface between $\\Omega $ and PML without reflections, and attenuate significantly inside the PML.", "The waves virtually vanish before reaching the outermost boundary of the PML, where the perfectly electric boundary (PEB) condition is employed.", "Implementation details about the PML technique specific for the method discussed in this paper can be found in Ref. [9].", "For the sake of clarity, we work with the general formulation given by Eq.", "(REF )-(REF ).", "Figure: Typical simulation setting with acomputational domain Ω\\Omega surrounded by the perfectly matchedlayer.", "Here just for the sake of illustration, weshow the scatterer Ω s \\Omega _{\\rm {s}} completely enclosed inside Ω\\Omega .", "Otherconfigurations like incoming-outgoing waveguides are alsopossible .As in the case of the standard FDTD method [2], in our approach we use the central difference scheme for the time derivatives in Eq.", "(REF )-(REF ), but we will construct a different discretization scheme of the spatial derivatives.", "This is done with interpolating scaling functions and lifted interpolating wavelets (explained in Sec. ).", "The induced multiresolution approximation [10], [11] enables us to decompose fields into various resolution levels, and thus allows to discard unimportant features.", "As a result, we will obtain a variant of the FDTD method, which is constructed with respect to a locally refined grid.", "In the next section we describe this numerical scheme in detail." ], [ "Adaptive wavelet collocation method", "The adaptive wavelet collocation (AWC) method was proposed by Vasilyev and co-authors in a series of papers [12], [13], [5], [6] as a general scheme to solve evolution equations.", "In the present section, we tailor the AWC method to tackle Eq.", "(REF )-(REF ).", "In contrast to the originally formulated AWC method, we do not need to utilize second generation wavelets, which have been mainly invented to implement boundary constraints, and to find wavelet decompositions on irregular domains.", "Since we use the PML method, we can identify field values outside the PML region with zero, and therefore we are not forced to adapt our wavelets to the boundary restrictions.", "Hence, we consider only the first generation wavelets, which are generated by the shifts and the dilations of a single function.", "Now we outline the essential steps for computing spatial derivatives of functions in wavelet representations." ], [ "Preliminaries", "A starting point of the AWC method is a wavelet decomposition of a function $f\\in L^2(\\mathbb {R})$ : $f = \\sum _{k \\in \\mathbb {Z}} \\alpha _{j_0,k} \\phi _{j_0,k} + \\sum _{j=j_0}^{+\\infty }\\sum _{m \\in \\mathbb {Z}} \\beta _{j,m} \\psi _{j,m}$ where $j_{0} \\in \\mathbb {Z}$ , $\\phi $ is the scaling function and $\\psi $ is the wavelet function [14], [15].", "For all $j,n \\in \\mathbb {Z}$ , by $\\phi _{j,n}$ and $\\psi _{j,n}$ we abbreviate the dilated and translated versions of $\\phi $ and $\\psi $ , i.e.", "$\\phi _{j,n} (\\cdot ) = 2^{j/2} \\phi (2^j\\cdot -n)$ , $\\psi _{j,n} (\\cdot ) = 2^{j/2} \\psi (2^j\\cdot -n)$ .", "The first (single) sum in (REF ) represents rough or low frequency information of $f$ , while the second (double) sum contains the detail information at various resolution levels starting from the level $j_0$ to $+\\infty $ .", "The absolute magnitude of the coefficients $\\alpha _{j_0,k}$ and $\\beta _{j,m}$ measure the contributions of $\\phi _{j_0,k}$ and $\\psi _{j,m}$ to $f$ .", "By discarding terms in the double sum for which the wavelet coefficients $\\beta _{j,m}$ are absolutely less than a given threshold, one can efficiently compress the representation of $f$ .", "This wavelet decomposition compression principle is exploited in the AWC method to enhance the computational efficiency.", "There are various families of the scaling functions $\\phi $ and wavelet functions $\\psi $ allowing representations like (REF ).", "As in [5], [6], we work with the interpolating scaling functions [16] and the corresponding lifted interpolating wavelets [17], [18].", "Due to their interpolation property, we have $\\phi (k)=\\delta _{0,k}={\\left\\lbrace \\begin{array}{ll}1\\ :&\\!", "k=0,\\\\[1mm]0\\ : &\\!", "k\\in \\mathbb {Z}\\setminus \\lbrace 0\\rbrace ,\\end{array}\\right.", "}$ and as a result, there exits a unique grid associated with the family $\\lbrace \\phi _{j,k}\\rbrace $ .", "The resulting numerical scheme can be seen as a variant of the well known finite difference method.", "We exploit this interpolating property in Sec.", "REF and Sec.", "REF .", "In particular, we use the interpolating scaling function (ISF) family developed by Deslauriers and Dubuc [19], [16].", "They constructed the interpolating functions by the iterative interpolation method, which does not require the concept of wavelets.", "Later Sweldens [17], [18] constructed the corresponding wavelet by lifting the Donoho wavelet [20].", "We use $\\displaystyle D \\hspace{-1.42262pt} D_N$ to denote ISF of order $N$ , and $\\displaystyle D \\hspace{-0.56905pt} l_{\\widetilde{N}}$ to denote the lifted interpolating wavelet of order $\\widetilde{N}$ .", "Here the order $N$ means that any polynomial $p$ of degree $k \\le 2N-1$ can be expressed as $p(\\cdot ) = \\sum _m c_m \\displaystyle D \\hspace{-1.42262pt} D_N(\\cdot - m)$ with suitable coefficients $\\lbrace c_m\\rbrace $ .", "The order $\\widetilde{N}$ is half the number of the vanishing moments of the lifted interpolating wavelet, i.e., $\\int x^k \\displaystyle D \\hspace{-0.56905pt} l_{\\widetilde{N}} (x) \\mathrm {d}x =0, \\quad k=0, 1, \\dots , 2\\widetilde{N}-1.$ Further details can be found in [17], [18], [9].", "We normally choose same orders for the ISF and the lifted interpolating wavelet, i.e., $N={\\widetilde{N}}$ .", "It is easy to see that $\\displaystyle D \\hspace{-1.42262pt} D_N$ and $\\displaystyle D \\hspace{-0.56905pt} l_{\\widetilde{N}}$ have compact supports, which increase with the order $N$ .", "For the TM$_y$ setting in Eq.", "(REF ), the electric and magnetic fields depend on the spatial variables $(x,z)$ .", "As usual, see, e.g., [11], [15], we represent 2D fields by expansions of 2D scaling functions and wavelets which are defined by ${\\displaystyle \\phi }_N (x,z) & := \\displaystyle D \\hspace{-1.42262pt} D_N (x) \\displaystyle D \\hspace{-1.42262pt} D_N (z), \\\\[2mm]{\\displaystyle \\psi }^{\\nu }_N (x,z) & := \\left\\lbrace \\!", "\\begin{array}{ll}\\displaystyle D \\hspace{-0.56905pt} l_N(x) \\displaystyle D \\hspace{-1.42262pt} D_N(z) &: \\quad \\nu = 1,\\\\[1mm]\\displaystyle D \\hspace{-1.42262pt} D_N(x) \\displaystyle D \\hspace{-0.56905pt} l_N(z) &: \\quad \\nu = 2,\\\\[1mm]\\displaystyle D \\hspace{-0.56905pt} l_N(x) \\displaystyle D \\hspace{-0.56905pt} l_N(z) &: \\quad \\nu = 3, \\end{array} \\right.$ and use the following abbreviations $({\\displaystyle \\phi }_N)_{j,m,n} (x,z) & := {(\\displaystyle D \\hspace{-1.42262pt} D_N)}_{j,m} (x) {(\\displaystyle D \\hspace{-1.42262pt} D_N)}_{j,n} (z), \\\\[2mm]({\\displaystyle \\psi }_N^{\\nu })_{j,m,n} (x,z) & := \\left\\lbrace \\!", "\\begin{array}{ll}{(\\displaystyle D \\hspace{-0.56905pt} l_N)}_{j,m} (x) {(\\displaystyle D \\hspace{-1.42262pt} D_N)}_{j+1,2n} (z) &: \\quad \\nu = 1, \\\\[1mm]{(\\displaystyle D \\hspace{-1.42262pt} D_N)}_{j+1,2m} (x) {(\\displaystyle D \\hspace{-0.56905pt} l_N)}_{j,n} (z) &: \\quad \\nu = 2, \\\\[1mm]{(\\displaystyle D \\hspace{-0.56905pt} l_N)}_{j,m} (x) {(\\displaystyle D \\hspace{-0.56905pt} l_N)}_{j,n} (z) &: \\quad \\nu = 3.", "\\end{array} \\right.$ Let $j_{\\min }$ and $j_{\\max }$ (with $j_{\\min } < j_{\\max }$ ) be the coarsest and the finest spatial resolution levels.", "Let us consider $f\\in L^2(\\mathbb {R}^2)$ with exact resolution level $j_{\\max }$ , that is, $f = \\sum _{m,n} \\alpha _{j_{\\max },m,n} ({\\displaystyle \\phi }_N)_{j_{\\max },m,n}.$ Then the wavelet representation of $f$ with coarsest resolution level $j_{\\min }$ is given by $f = \\sum _{m,n} \\alpha _{j_{\\min },m,n} ({\\displaystyle \\phi }_N)_{j_{\\min },m,n} +\\sum _{\\nu = 1}^{3} \\sum _{j = j_{\\min }}^{j_{\\max } - 1} \\sum _{m,n}\\beta ^{\\nu }_{j,m,n} ({\\displaystyle \\psi }_N^{\\nu })_{j,m,n}$ where the scaling coefficients $\\lbrace \\alpha _{j_{\\min },m,n}\\rbrace $ and the wavelet coefficients $\\lbrace \\beta ^{\\nu }_{j,m,n}\\rbrace $ can be calculated from the level $j_{\\max }$ scaling coefficients $\\lbrace \\alpha _{j_{\\max },m,n}\\rbrace $ by the normalized 2D forward wavelet transform (FWT): $d^1_{j,m,n} &= \\frac{1}{2} \\Big ( c_{j+1,2m+1,2n} - \\sum _l 2 \\tilde{s}_{-l} c_{j+1,2m+2l,2n}\\Big ),\\\\d^2_{j,m,n} &= \\frac{1}{2} \\Big ( c_{j+1,2m,2n+1} - \\sum _l 2 \\tilde{s}_{-l} c_{j+1,2m,2n+2l}\\Big ),\\\\d^3_{j,m,n} &= \\frac{1}{4} \\Big ( c_{j+1,2m+1,2n+1} - \\sum _l 2 \\tilde{s}_{-l} c_{j+1,2m+2l,2n+1}- \\sum _{l^{\\prime }} 2 \\tilde{s}_{-l^{\\prime }} c_{j+1,2m+1,2n+2l^{\\prime }} \\nonumber \\\\& + \\sum _l \\sum _{l^{\\prime }} (2 \\tilde{s}_{-l})(2 \\tilde{s}_{-l^{\\prime }}) c_{j+1,2m+2l,2n+2l^{\\prime }} \\Big ),\\\\c_{j,m,n} &= c_{j+1,2m,2n} + \\sum _l s_{-l} d^1_{j,m+l,n} + \\sum _{l^{\\prime }} s_{-l^{\\prime }} d^2_{j,m,n+l^{\\prime }} + \\sum _l\\sum _{l^{\\prime }} s_{-l} s_{-l^{\\prime }} d^3_{j,m+l,n+l^{\\prime }},$ with the following normalization conventions $c_{j,m,n} = 2^j \\alpha _{j,m,n}, \\quad d^1_{j,m,n} = 2^{j+1/2} \\beta ^1_{j,m,n}, \\quad d^2_{j,m,n} = 2^{j+1/2} \\beta ^2_{j,m,n} \\text{ and } d^3_{j,m,n} = 2^j \\beta ^3_{j,m,n}.$ The coefficients $2\\tilde{s}_{l}$ and $s_{l}$ are Lagrangian interpolation weights.", "For example, when $N=2$ , these weights are $s_{-2}=-1/16,\\quad s_{-1}=9/16,\\quad s_0=9/16,\\quad s_1=-1/16, \\quad 2\\tilde{s}_{-1}=-1/16,$ and $2\\tilde{s}_0=9/16,\\quad 2\\tilde{s}_1=9/16, \\quad 2\\tilde{s}_2=-1/16.$ Readers may consult [16], [21] and [17] for an explanation of how and why Lagrangian weights enter the iterative interpolation process.", "We also can compute back from the wavelet representation (REF ) to the scaling function representation (REF ) by the inverse wavelet transform (IWT): $c_{j+1,2m,2n} &= c_{j,m,n} - \\sum _l s_{-l} d^1_{j,m+l,n} + \\sum _{l^{\\prime }} s_{-l^{\\prime }} d^2_{j,m,n+l^{\\prime }} + \\sum _l \\sum _{l^{\\prime }} s_{-l} s_{-l^{\\prime }} d^3_{j,m+l,n+l^{\\prime }},\\\\c_{j+1,2m+1,2n}& = 2d^1_{j,m,n} + \\sum _l 2 \\tilde{s}_{-l} c_{j+1,2m+2l,2n}, \\\\c_{j+1,2m,2n+1} &= 2d^2_{j,m,n} + \\sum _l 2 \\tilde{s}_{-l} c_{j+1,2m,2n+2l}, \\\\c_{j+1,2m+1,2n+1}& = 4d^3_{j,m,n} + \\sum _l 2 \\tilde{s}_{-l}c_{j+1,2m+2l,2n+1} + \\sum _{l^{\\prime }} 2 \\tilde{s}_{-l^{\\prime }} c_{j+1,2m+1,2n+2l^{\\prime }}\\nonumber \\\\& - \\sum _l \\sum _{l^{\\prime }} (2 \\tilde{s}_{-l})(2 \\tilde{s}_{-l^{\\prime }}) c_{j+1,2m+2l,2n+2l^{\\prime }}.", "$" ], [ "Adaptive grid refinement wavelet compression", "We thin out the triple sum in (REF ) by discarding small wavelet coefficients, which corresponds to small scale details.", "For a given threshold $\\zeta >0$ , let $f_{\\zeta } := \\sum _{m,n} \\alpha _{j_{\\min },m,n} ({\\displaystyle \\phi }_N)_{j_{\\min },m,n} + \\sum _{\\nu = 1}^{3} \\sum _{j =j_{\\min }}^{j_{\\max } - 1} \\sum _{m,n}T^{\\nu }_{\\zeta }(\\beta ^{\\nu }_{j,m,n}) ({\\displaystyle \\psi }_N^{\\nu })_{j,m,n},$ where the threshold function $T^{\\nu }_{\\zeta }\\colon \\mathbb {R}\\rightarrow \\mathbb {R}$ is defined by $T^{\\nu }_{\\zeta }(x) = \\left\\lbrace \\!\\begin{array}{rl}x &: \\text{ for } \\nu \\in \\lbrace 1,2\\rbrace \\mbox{ and } \\ |x| \\ge 2^{-j - 1/2}\\zeta ,\\\\x &: \\text{ for } \\nu =3 \\mbox{ and } \\ |x| \\ge 2^{-j} \\zeta , \\\\0 &:\\ \\text{otherwise.}", "\\end{array} \\right.$ Note that we have defined the uniform threshold $\\zeta $ in terms of the normalized wavelet coefficients $d^{\\nu }_{j,m,n}$ defined in Eq.", "(REF ) i.e., if $|d^{\\nu }_{j,m,n}| < \\zeta $ then $d^{\\nu }_{j,m,n}=0$ in $f_\\zeta $ in Eq.", "(REF ).", "Then the compression error is proportional to $\\zeta $ [5]: $\\Vert f-f_{\\zeta }\\Vert _{\\infty } \\le C \\zeta .$ Our basis functions in (REF ), which are translates and dilates of ${\\displaystyle \\phi }_N$ and ${\\displaystyle \\psi }_N^\\nu $ , are interpolating at the corresponding grid points.", "Let $x_{j,m}:=\\frac{m}{2^j}\\ \\text{ and }\\ z_{j,n}:=\\frac{n}{2^j} \\text{for } m,n \\in \\mathbb {Z},$ then we have the following one-to-one correspondence between the basis functions and the grid points: $&({\\displaystyle \\phi }_N)_{j,m,n}\\longleftrightarrow (x_{j,m},\\,z_{j,n}), &\\quad & ({\\displaystyle \\psi }^1_N)_{j,m,n}\\longleftrightarrow (x_{j+1,2m+1}, \\,z_{j+1,2n}),\\\\&({\\displaystyle \\psi }^2_N)_{j,m,n}\\longleftrightarrow (x_{j+1,2m}, \\,z_{j+1,2n+1}), & & ({\\displaystyle \\psi }^3_N)_{j,m,n}\\longleftrightarrow (x_{j+1,2m+1},\\, z_{j+1,2n+1}).$ Here this correspondence means the validity of the interpolation property.", "For instance, we have that $({\\displaystyle \\phi }_N)_{j,m,n}(x_{j,m^{\\prime }},\\,z_{j,n^{\\prime }}) =\\delta _{m,m^{\\prime }} \\delta _{n,n^{\\prime }}.$ With this explanations, we justified the synonymous usage of compression of the wavelet representation and compression/adaption of the grid points." ], [ "Adjacent zone", "With the above described wavelet compression, the grid gets suitably sampled only for the current state of the fields.", "For a meaningful (i.e.", "physical) field evolution in the next time-step, the grid need to be supplemented by additional grid points, on which the fields may become significant in the next time step.", "This allows the grid to capture correctly the propagation of a wave.", "To this end Vasilyev [5], [6] has introduced a concept of an adjacent zone.", "To each point $P=(x_{j,m}, z_{j,n})$ in the current grid, we attach an adjacent zone which is defined as the set of points $(x_{j^{\\prime },m^{\\prime }}, z_{j^{\\prime },n^{\\prime }})$ which satisfy $|j^{\\prime }-j| \\le L, \\quad |2^{j^{\\prime }-j}m-m^{\\prime }| \\le M, \\quad |2^{j^{\\prime }-j}n-n^{\\prime }| \\le M,$ where $L$ is the width of the adjacent levels and $M$ is the width of the physical space.", "As in [5], we verified that $L=M=1$ is a computationally sufficient choice.", "Then the adjacent zone for a point $P$ can be depicted as in Fig.", "REF .", "Figure: Description of the adjacent zone of a grid pointPP.Note that the concept of adjacent zone is reasonable only for continuously propagating waves, as in case of our guided-wave applications, where in each time step the propagating waves do not travel far from the current position due to their finite propagation speed." ], [ "Reconstruction check", "In this work we use the wavelet decompositions of the fields only to determine the adaptive grid.", "We do not propagate fields in their wavelet representations (cf.", "the statement in the first paragraph of Sec. ).", "Thus at each time step, after adapting the grid using the FWT, and adding the adjacent zone, we need to restore the fields in the physical space by performing the inverse wavelet transformation (IWT).", "To this end, we may need to augment the adaptive grid with additional neighboring points (e.g.", "see Fig REF ).", "This process of adding neighboring points needed to calculate the wavelet coefficients in the next time step is called reconstruction check.", "Fig REF shows various possible scenarios, and the corresponding minimal set of the grid points required for calculation of the wavelet coefficients.", "The values of the wavelet coefficients at these newly added points are set to zero.", "Figure: ×\\times : The point corresponding to d j,m,n 3 d^3_{j,m,n}; •\\bullet , ∘\\circ : The neighboring points needed to calculate d j,m,n 3 d^3_{j,m,n}.The efficiency of the wavelet transform depends on the number of the finest grid points only at the beginning; however, after the first compression, it depends solely on the cardinality (= number of grid points) of the adaptive grid." ], [ "Calculation of the spatial derivatives on the adaptive grid", "After the adjacent zone correction and the reconstruction check, we are in a position to calculate the derivative of $f_{\\zeta }$ at a grid point in the adaptive grid.", "For this we need to know the density level of this point, which is defined as the maximum of the $x$ -level and the $z$ -level of that point.", "We illustrate this concept explicitly only for the $x$ -level, the $z$ -level can be determined analogously.", "For a point $Q = (x_0,z_0)$ in the adaptive grid $\\mathcal {G}$ , let $Q^{\\prime } = (x_1, z_0)\\in \\mathcal {G}$ be the nearest point to $Q$ .", "Then the $x$ -level $Levelx$ of $Q$ relative to $\\mathcal {G}$ is $Levelx := j_{\\max } - \\log _2 (\\mathrm {dist}(Q,Q^{\\prime })/\\Delta x)$ where $\\Delta x$ is the smallest computational mesh size along the $x$ axis, and $\\mathrm {dist}(Q,Q^{\\prime }) = |x_1 - x_0|$ .", "For $\\mathrm {dist}(Q,Q^{\\prime }) =\\Delta x$ , the level $Levelx$ of $Q$ attains its maximum $j_{\\max }$ .", "For $\\mathrm {dist}(Q,Q^{\\prime }) = 2 \\Delta x$ , we have $Levelx=j_{\\max }-1$ , etc.", "See Fig.", "REF for an example of describing the density level of a grid point.", "Figure: Description of the density level of a point QQ in an adaptivegrid: the xx-level of QQ is j max -1j_{\\max }-1 and the zz-level of QQis j max j_{\\max }, thus, the density level of QQ isj max j_{\\max }.Now we continue to discuss the derivative calculations.", "Suppose $j_0$ to be the density level of $Q$ in $\\mathcal {G}$ .", "Then, we can represent $f_{\\zeta }$ by a finite sum ${\\bf P}_{j_0} f$ locally in some neighborhood $\\Omega _0$ of $Q$ .", "$ {\\bf P}_{j_0} f (x,z) = \\sum _{m,n} \\alpha _{j_0,m,n} ({\\displaystyle \\phi }_N)_{j_0,m,n} (x,z), \\quad (x,z) \\in \\Omega _0$ We differentiate ${\\bf P}_{j_0} f$ with respect to $x$ to approximate the $x$ -derivative of $f$ at $Q$ .", "If any points in the sum (REF ) are not present in $\\mathcal {G}$ , then we interpolate the values at these points by the IWT using the values of the coarser levels.", "From the interpolation property of $({\\displaystyle \\phi }_N)_{j_0,m,n}$ we know that $\\alpha _{j_0,m,n} = 2^{-j_0} ({\\bf P}_{j_0} f) \\Big ( \\frac{m}{2^{j_0}},\\frac{n}{2^{j_0}} \\Big ), \\quad \\text{for } m,n \\in \\mathbb {Z}.$ Thus, we have $ ({\\bf P}_{j_0} f) (x,z) = \\sum _{m,n} ({\\bf P}_{j_0} f) \\Big (\\frac{m}{2^{j_0}}, \\frac{n}{2^{j_0}} \\Big ) \\displaystyle D \\hspace{-1.42262pt} D_N \\big ( 2^{j_0}x - m\\big ) \\displaystyle D \\hspace{-1.42262pt} D_N \\big ( 2^{j_0}z - n \\big ), \\quad (x,z) \\in \\Omega _0.$ Differentiate both sides of (REF ) with respect to $x$ gives $ \\frac{\\partial ({\\bf P}_{j_0} f)}{\\partial x} (x,z) = \\sum _{m,n} ({\\bf P}_{j_0} f) \\Big ( \\frac{m}{2^{j_0}}, \\frac{n}{2^{j_0}} \\Big )\\frac{\\mathrm {d} \\displaystyle D \\hspace{-1.42262pt} D_N \\big ( 2^{j_0}x - m \\big )}{\\mathrm {d}x} \\displaystyle D \\hspace{-1.42262pt} D_N\\big ( 2^{j_0}z - n \\big ), \\quad (x,z) \\in \\Omega _0.$ The derivatives of $\\displaystyle D \\hspace{-1.42262pt} D_N^{\\prime }$ can be calculated exactly at the integers using the difference filters shown in Table REF (see Ref.", "[16] for details of the derivation).", "Table: Difference filters {DD N ' (i)} i∈ℤ \\lbrace \\displaystyle D \\hspace{-1.42262pt} D_N^{\\prime }(i)\\rbrace _{i\\in \\mathbb {Z}} with consistency order 2N2N.", "Note that DD N ' (i)=-DD N ' (-i)\\displaystyle D \\hspace{-1.42262pt} D_N^{\\prime }(i)=-\\displaystyle D \\hspace{-1.42262pt} D_N^{\\prime }(-i).Since the density level of $Q$ is $j_0$ , there exist $m^{\\prime },n^{\\prime }\\in \\mathbb {Z}$ such that $Q=\\big (\\frac{m^{\\prime }}{2^{j_0}}, \\frac{n^{\\prime }}{2^{j_0}}\\big )$ and it is easy to see that $\\frac{\\partial ({\\bf P}_{j_0} f)}{\\partial x}\\Big (\\frac{m^{\\prime }}{2^{j_0}},\\frac{n^{\\prime }}{2^{j_0}} \\Big ) &= \\sum _{m,n} ({\\bf P}_{j_0} f) \\Big ( \\frac{m}{2^{j_0}}, \\frac{n}{2^{j_0}} \\Big )\\frac{\\mathrm {d} \\displaystyle D \\hspace{-1.42262pt} D_N \\big ( m^{\\prime } - m \\big )}{\\mathrm {d}x} \\displaystyle D \\hspace{-1.42262pt} D_N \\big ( n^{\\prime }- n \\big ) \\nonumber \\\\[1mm]&= 2^{j_0} \\sum _{m} ({\\bf P}_{j_0} f) \\Big ( \\frac{m}{2^{j_0}},\\frac{n^{\\prime }}{2^{j_0}} \\Big ) \\displaystyle D \\hspace{-1.42262pt} D_N^{\\prime } \\big ( m^{\\prime }- m \\big ).$ Similarly, $\\frac{\\partial ({\\bf P}_{j_0} f)}{\\partial z}\\Big (\\frac{m^{\\prime }}{2^{j_0}},\\frac{n^{\\prime }}{2^{j_0}} \\Big ) = 2^{j_0}\\sum _{n} ({\\bf P}_{j_0} f) \\Big ( \\frac{m^{\\prime }}{2^{j_0}},\\frac{n}{2^{j_0}} \\Big ) \\displaystyle D \\hspace{-1.42262pt} D_N^{\\prime } \\big ( n^{\\prime } - n \\big ).$ This finishes the general discussion about the adaptive wavelet collocation method; in the next section, we apply it to Maxwell's equations." ], [ "AWC-TD method for Maxwell's equations", "In this section we formulate the update scheme for Maxwell's equations, and then elaborate on algorithmic issues related with the AWC-TD method.", "In the present formulation we represent the electric and magnetic fields in the physical space, and not in the wavelet space.", "To unleash the full power of adaptivity, however, the field representation and the update in wavelet space are advantageous.", "We illustrate the method for the transverse magnetic (TM)$_y$ setting given by (REF ).", "Similar procedure can also be formulated for TE$_y$ setting in (REF ).", "Unlike the standard FDTD method, here the electric field and the magnetic field components are evaluated on same spatial grid, and their spatial derivatives are approximated at the same grid point.", "But the electric field components are sampled at integer time-steps, whereas the magnetic field components are sampled at half-integer time-steps." ], [ "Update scheme for the spatial derivative", "For a point $Q$ in the adapted grid $\\mathcal {G}$ , let $\\mathcal {H}_x|^{k+1/2}_{Q}$ , $\\mathcal {H}_z|^{k+1/2}_{Q}$ and $\\mathcal {E}_y|^{k}_{Q}$ denote the discretized value of $\\mathcal {H}_x$ , $\\mathcal {H}_z$ and $\\mathcal {E}_y$ at the point $Q$ , and at a time $(k+1/2) \\Delta t$ for the magnetic field components and at a time $k \\Delta t$ for the electric field component where $\\Delta t>0$ is the time step size (Note that, the electric field components are sampled at integer time-steps, whereas the magnetic field components are sampled at half-integer time-steps.).", "Assume $j(Q)$ to be the density level of $Q$ relative to $\\mathcal {G}$ .", "Then we can represent the point $Q$ as $(x_{j(Q),m^{\\prime }},z_{j(Q),n^{\\prime }})$ for some $m^{\\prime },n^{\\prime } \\in \\mathbb {Z}$ .", "Let $L$ be the length of the computational domain $\\Omega $ .", "We rescale the wavelet decomposition (REF ) with the factor $L$ .", "Then using the central difference scheme for the time derivatives and using (REF )-(REF ) for the spatial derivatives, we get the following difference equations $\\mathcal {H}_x|^{k+\\frac{1}{2}}_{Q} &= \\mathcal {H}_x|^{k-\\frac{1}{2}}_{Q} + \\frac{\\Delta t}{\\mu _{0}}\\frac{2^{j(Q)}}{L} \\sum _{n} \\mathcal {E}_y|^k_{(x_{j(Q),m^{\\prime }}, z_{j(Q),n})}\\displaystyle D \\hspace{-1.42262pt} D_N^{\\prime }(n^{\\prime }-n),\\\\\\mathcal {H}_z|^{k+\\frac{1}{2}}_{Q} &= \\mathcal {H}_z|^{k-\\frac{1}{2}}_{Q} + \\frac{\\Delta t}{\\mu _{0}}\\frac{2^{j(Q)}}{L} \\sum _{m} \\mathcal {E}_y|^k_{(x_{j(Q),m}, z_{j(Q),n^{\\prime }})}\\displaystyle D \\hspace{-1.42262pt} D_N^{\\prime }(m^{\\prime }-m),\\\\\\mathcal {E}_y|^{k+1}_{Q} &= \\mathcal {E}_y|^{k-1}_{Q} + \\frac{\\Delta t}{\\varepsilon _{0}} \\frac{1}{\\varepsilon _r|_{Q}}\\frac{2^{j(Q)}}{L} \\Big (\\sum _{n} \\mathcal {H}_x|^{k+\\frac{1}{2}}_{(x_{j(Q),m^{\\prime }}, z_{j(Q),n})}\\displaystyle D \\hspace{-1.42262pt} D_N^{\\prime }(n^{\\prime }-n) \\nonumber \\\\& - \\sum _{m} \\mathcal {H}_z|^{k+\\frac{1}{2}}_{(x_{j(Q),m},z_{j(Q),n^{\\prime }})} \\displaystyle D \\hspace{-1.42262pt} D_N^{\\prime }(m^{\\prime }-m) \\Big ),$ The first time step ($k=0$ ) is an explicit Euler step with step size $\\Delta t/2$ using initial conditions for the fields at the time $t=0$ .", "If not explicitly mentioned, otherwise the fields are set zero at the beginning for all our numerical experiments in Sec.  .", "The update equations for the PML assisted Maxwell's equations can be found in Ref. [9].", "From the form of these update equations, it is clear that the AWC-TD method can be thought as an variant of high order FDTD method.", "The AWC-TD method is defined with respect to a locally adapted mesh, and unlike the FDTD method, it does not require a static (fixed), structured mesh.", "This will lead to efficient use of the computational resources.", "In the next section, we elaborate on algorithmic aspects of the method." ], [ "Update scheme for the time derivative", "Several choices are available for time stepping.", "As in case of the standard FDTD method, we use in (REF ) the central difference scheme for the discretization of the time derivatives.", "For this explicit scheme, the smallest spatial step-size restricts the maximal time-step according to the Courant–Friedrichs–Lewy (CFL) stability condition.", "Using a uniform spatial mesh in the update equations (REF ) with a mesh size $\\Delta $ in both coordinate directions the CFL condition reads $\\Delta t \\le \\frac{\\Delta }{\\sqrt{2} \\mathrm {c} \\sum _{l=0}^{l_0-1}|\\displaystyle D \\hspace{-1.42262pt} D_N^{\\prime }(l)|},$ see [9] and [22], where $\\mathrm {c}$ is the speed of light in vacuum and $\\lbrace \\displaystyle D \\hspace{-1.42262pt} D_N^{\\prime }(l)\\rbrace $ is the known derivative filter of the ISF as in Table REF .", "Due to the local adaptive grid strategy of the AWC-TD method, we cannot define a global stability criteria as above.", "But choosing $\\Delta $ to be the smallest step size in the adaptive grid, we get a conservative bound for $\\Delta t$ via (REF ) for the AWC-TD method.", "In the simulation tests (in this paper, and in [9]) we did not experience any stability related issues with this modus operandi." ], [ "Grid management", "In AWC method the computational grid is changed with the state (spatial localization) of the propagating field.", "Thus the grid management is one of the important steps in the implementation of this method.", "This is done as following: We store the information of the adaptive grid into a 2D Boolean array called a grid mask or simply a mask, whose size is square the number of the finest grid points along one direction.", "We use 2D arrays of real numbers with the size of the grid mask to store the fields such as $\\mathcal {E}_y$ , $\\mathcal {H}_x$ , and $\\mathcal {H}_z$ etc.", "Note that the computational effort for updating the fields at each time step is proportional to the cardinality (i.e.", "the number of entries in the mask with value 1) of the adaptive grid.", "If the value of an entry of a mask is $true$ or 1, then the corresponding grid point is included in the adaptive grid; otherwise, it is not included in the grid.", "Thus by forcing the value of an entry of a mask to 1, we can include the corresponding point to the grid, or by forcing the entry to 0, we can exclude the corresponding point from the grid." ], [ "Algorithmic procedures", "Algorithm REF outlines the main function awcm_main() of AWC-TD method for TM$_y$ setting.", "It mainly consists of two blocks of operations: The first block is initialization, and the second block is time stepping.", "In the time stepping block, at each time step the routines awcm_adaptive() and awcm_update() are called.", "The former routine optimally adapts the computational grid for the field updates at the next time step, whereas the latter routine calculates the spatial derivatives on the non-equidistant, adaptive grid, and updates the field values.", "[!h] awcm_main() for TM$_y$ settings Initialization awcm_initialize() ———————————————————————————————————– Time stepping of $\\mathcal {E}_y$ , $\\mathcal {H}_x$ and $\\mathcal {H}_z$ $t \\le T$ Adapt the grid for $t + \\Delta t$ according to $\\mathcal {E}_{y}^{t}$ , see Algorithm REF .", "awcm_adaptive() —————————————————————————————————— Update $\\mathcal {H}_{x}^{t + \\Delta t/2}$ , $\\mathcal {H}_{z}^{t + \\Delta t/2}$ and $\\mathcal {E}_{y}^{t + \\Delta t}$ , see Algorithm REF .", "awcm_update() —————————————————————————————————— Go to the next time step.", "$t = t + \\Delta t$ The initialization subroutine awcm_initialize() ensures that various required inputs for the AWC method are systematically prepared.", "It consists of checking the given initial data (i.e.", "for time step $k=0$ ) ${\\mathcal {H}_x}^{-\\frac{1}{2}}$ , ${\\mathcal {H}_z}^{-\\frac{1}{2}}$ and ${\\mathcal {E}_y}^0$ at the finest resolution level $j_{max}$ , the threshold $\\zeta $ , the maximum and the minimum spatial resolution levels $j_{max}$ and $j_{min}$ respectively, and the number of time steps $k_{max}$ .", "The time step $\\Delta t$ is chosen such that it satisfies the CFL condition given by (REF ).", "The adaptivity procedure in Algorithm REF handled by a subroutine awcm_adaptive() is outlined in Algorithm REF .", "It is done by means of a 2D array $\\mathcal {E}_y$ with a mask $Mask0$ .", "For later use, we store a copy of $Mask0$ in $pMask0$ , since $Mask0$ will be modified by the subsequent subroutines.", "The duplicate $pMask0$ serves as a reference for finding those points which need to be interpolated before we can update the fields.", "We perform the fast wavelet transform of $\\mathcal {E}_y$ on $Mask0$ .", "Note that $Mask0$ is either fully 1 (as at the beginning) or a reconstruction check has been performed in the previous time step.", "In any case, FWTs on $Mask0$ are always possible.", "By the FWT applied to $\\mathcal {E}_y$ we obtain the scaling coefficients on the coarsest level $j_{\\min }$ , and the wavelet coefficients on levels from $j_{\\min }$ to $j_{\\max }-1$ .", "For each wavelet coefficient, we compare its absolute value with the given tolerance $\\zeta $ .", "If it is less than $\\zeta $ , we remove the corresponding point from $Mask0$ .", "Next, we determine the adjacent zone for each point in $Mask0$ , and then modify $Mask0$ to include all points in these adjacent zones.", "Finally, a reconstruction check is applied to $Mask0$ so that the FWT in the next time step is well defined.", "The latter two processes are done in the subroutine Maskext($Mask0$ ) as shown in Algorithm REF .", "[!h] awcm_adaptive() for TM$_y$ settings Store $Mask0$ into $pMask0$ .", "$pMask0$ : The adaptive grid for $\\mathcal {E}_y$ at current time step.", "$pMask0 = Mask0$ ———————————————————————————————————— Fast wavelet transform of $\\mathcal {E}_y$ on $Mask0$ with $\\zeta $ .", "$\\mathcal {E}_y$ is converted into coefficients of wavelet domain, $Mask0$ is thinned.", "FWT($\\mathcal {E}_y$ , $Mask0$ , $\\zeta $ ) ———————————————————————————————————— Add adjacent zone and perform a reconstruction check to $Mask0$ .", "Maskext($Mask0$ ) ———————————————————————————————————— Add points needed to calculate $\\frac{\\partial \\mathcal {E}_y}{\\partial x}$ and $\\frac{\\partial \\mathcal {E}_y}{\\partial z}$ on $Mask0$ .", "1.", "Determine the density level of each point in $Mask0$ .", "$Level0 =$ Level($Mask0$ ) 2.", "Initialize $Mask1$ with $Mask0$ .", "$Mask1 = Mask0$ 3.", "Update $Mask1$ .", "gMaskext($Mask1$ , $Level0$ ) ———————————————————————————————————— Add points needed to calculate $\\frac{\\partial \\mathcal {H}_x}{\\partial z}$ and $\\frac{\\partial \\mathcal {H}_z}{\\partial x}$ on $Mask1$ .", "1.", "Determine the density level of each point in $Mask1$ .", "$Level1 =$ Level($Mask1$ ) 2.", "Initialize $Mask2$ with $Mask1$ .", "$Mask2 = Mask1$ 3.", "Update $Mask2$ .", "gMaskext($Mask2$ , $Level1$ ) ———————————————————————————————————— Inverse wavelet transform of the values $\\mathcal {E}_y$ in the wavelet domain on $Mask2$ .", "$\\mathcal {E}_y$ is reconstructed from the values in the wavelet domain on $Mask2$ .", "IWT($\\mathcal {E}_y$ , $Mask2$ ) After the above adaptation of the grid is done, we still need to make further reconstructions on this grid, so that it will allow computation of the field derivatives required for the field update.", "For updating $\\mathcal {H}_x$ and $\\mathcal {H}_z$ , we need $\\frac{\\partial \\mathcal {E}_y}{\\partial z}$ and $\\frac{\\partial \\mathcal {E}_y}{\\partial x}$ (see (REF ) or (REF )).", "To calculate these spatial derivatives of the electric field, we interpolate values of $\\mathcal {E}_y$ at those neighbors of points in $Mask0$ which are not already in $Mask0$ .", "We store the information of $Mask0$ into $Mask1$ .", "Further, we add all points to $Mask1$ needed in the calculations of spatial derivatives according to the density levels of the points in $Mask1$ .", "These density levels are computed in subroutine Level($Mask1$ ) and stored in the 2D array $Level0$ .", "Again a reconstruction check of $Mask1$ is required to enable IWTs.", "This is done by the subroutine gMaskext($Mask1$ , $Level0$ ).", "Then we need to follow the same procedure as above for updating $\\mathcal {E}_y$ using the spatial derivatives $\\frac{\\partial \\mathcal {H}_x}{\\partial z}$ and $\\frac{\\partial \\mathcal {H}_z}{\\partial x}$ .", "Again we add the neighboring points needed for calculations of the spatial derivatives of the magnetic field.", "We copy $Mask1$ to $Mask2$ , and calculate the density level array $Level1$ of $Mask2$ .", "The necessary reconstruction check is then done by calling gMaskext($Mask2$ ,$Level1$ ).", "The call of IWT($\\mathcal {E}_y$ , $Mask2$ ) to reconstruct $\\mathcal {E}_y$ in the physical domain finishes the routine awcm_adaptive() in Algorithm REF .", "Next, we update the field values on the adaptive grid, which is described by Algorithms REF .", "Since the adaptive grid may change with time, we need to interpolate the field values at points in the adaptive grid of the current time step, which are not included in the adaptive grid of the previous time step.", "For example, consider the update of $\\mathcal {H}_x$ about a grid point $Q$ at a time $(k+1/2)\\Delta t$ in (REF ).", "Since $Q$ is not necessarily in the adaptive grid of previous time $(k-1/2)\\Delta t$ , the value $\\mathcal {H}_x|_{Q}^{k-1/2}$ in (REF ) must be interpolated.", "Once this is done, Algorithm REF calculates the spatial derivatives of each field components on the adaptive grid, and then the fields are updated.", "[!t] awcm_update() for TM$_y$ settings To update $\\mathcal {H}_x$ : Interpolate $\\mathcal {H}_x$ on points in $Mask1$ which are not in $pMask0$ using inverse wavelet transform.", "interpolate($\\mathcal {H}_x$ , $pMask0$ , $Mask1$ ) Calculate $\\frac{\\partial \\mathcal {E}_y}{\\partial z}$ on $Mask1$ using Algorithm REF .", "$dA_z =$ diffz($\\mathcal {E}_y$ , $Mask1$ , $Level1$ , dfilter, $dz$ ) linkcolor=Black Update $\\mathcal {H}_x$ on $Mask1$ using $dA_z$ as per formulation in (REF ).", "———————————————————————————————————— To update $\\mathcal {H}_z$ : Interpolate $\\mathcal {H}_z$ on points in $Mask1$ which are not in $pMask0$ using inverse wavelet transform.", "interpolate($\\mathcal {H}_z$ , $pMask0$ , $Mask1$ ) Calculate $\\frac{\\partial \\mathcal {E}_y}{\\partial x}$ on $Mask1$ using Algorithm REF .", "$dA_x =$ diffx($\\mathcal {E}_y$ , $Mask1$ , $Level1$ , dfilter, $dx$ ) Update $\\mathcal {H}_z$ on $Mask1$ using $dA_x$ as per formulation in ().", "———————————————————————————————————— To update $\\mathcal {E}_y$ : Interpolate $\\mathcal {E}_y$ on points in $Mask0$ which are not in $pMask0$ using inverse wavelet transform.", "interpolate($\\mathcal {E}_y$ , $pMask0$ , $Mask0$ ) Calculate $\\frac{\\partial \\mathcal {H}_x}{\\partial z}$ and $\\frac{\\partial \\mathcal {H}_z}{\\partial x}$ on $Mask0$ , see the Algorithm REF .", "diffx() is defined in Algorithm REF .", "diffz() is similarly defined.", "$dA_z =$ diffz($\\mathcal {H}_x$ , $Mask0$ , $Level0$ , dfilter, $dz$ ) $dA_x =$ diffx($\\mathcal {H}_z$ , $Mask0$ , $Level0$ , dfilter, $dx$ ) Update $\\mathcal {E}_y$ on $Mask0$ using $dA_z$ and $dA_x$ as per formulation in ().", "[!h] diffx($A$ , $Mask$ , $Level$ , dfilter, $dx$ ) InputInputOutputReturn a 2D array of $\\frac{\\partial A}{\\partial x}$ on $Mask$ Initialize a 2D array $dA$ for the storage of $\\frac{\\partial A}{\\partial x}$ .", "$dA = 0$ $\\mathcal {K}_{j}=\\lbrace (x_{j,m}, y_{j,n}) \\,|\\, m,n=0,1, \\dots ,2^{j}\\rbrace $ , where $x_{j,m}=\\dfrac{mL}{2^j},y_{j,n}=\\dfrac{nL}{2^j}$ , for $j_{\\min } \\le j \\le j_{\\max }$ .", "$Q=(x_{j_{\\max },m},y_{j_{\\max },n}) \\in \\mathcal {K}_{j_{\\max }}$ $Q \\in Mask$ Read the density level of $Q$ from $Level$ .", "$j(Q)=Level[n][m]$ linkcolor=Black Calculate $dA$ at point $Q$ using dfilter and values of $A$ at neighbor points in the level $j(Q)$ as described in (REF )." ], [ "Numerical results: Gaussian pulse propagation", "In this section we demonstrate the applicability of the AWC-TD method.", "The method has been implemented in C++, and the computations have been performed on 32 GB RAM, Linux system with AMD Opteron processors.", "As an example, we consider propagation of a spatial Gaussian pulse in free space ($\\varepsilon _r=1$ ).", "We solve a system of TM$_y$ equations within a square domain $\\Omega =[-L/2, L/2] \\times [-L/2, L/2]$ in the $XZ$ plane.", "We set the domain length $L = 6.0$  $\\mu $ m, the PML width $d=L/4$ , and the initial spatial Gaussian excitation $\\mathcal {E}_y(x,z,0) = \\exp (-(x^2+z^2)/(2\\sigma ^{2}))$ with the Gaussian pulse width $\\sigma = 1/(4\\,\\sqrt{2})$ $\\mu $ m. Implementation details about the PML can be found in Ref. [9].", "Our minimum and maximum resolution levels are $j_{\\min } = 3$ and $j_{\\max }=9$ inducing the smallest mesh size $\\Delta = \\Delta x =\\Delta z= L/2^{j_{max}} = 11.71875$ nm.", "The temporal error of the AWC-TD method is controlled by $O(\\Delta t^2)$ if we do not consider the compression, which is the consistency order of the central difference discretization of the time derivatives.", "Accordingly, a reasonable choice for the threshold $\\zeta $ is a value slightly larger than the discretization error.", "As the orders of the underlying interpolating scaling function/wavelet pair is $N=\\tilde{N}=4$ , we set $\\Delta t=\\Delta /\\mathrm {c}/1.6$ , which is just below the maximal step size from the CFL condition (REF ).", "For this setting, a choice of wavelet threshold $\\zeta =5.0 \\times 10^{-4}$ experimentally turned out to be sufficient concerning both adaptivity and accuracy.", "Figure: Evolution of the initial excitationℰ y (x,z,0)=exp(-(x 2 +z 2 )/(2σ 2 ))\\mathcal {E}_y(x,z,0) = \\exp (-(x^2+z^2)/(2\\sigma ^{2})) in the XZXZplane with σ=1/(42)\\sigma = 1/(4\\,\\sqrt{2}) μ\\mu m, N=N ˜=4N=\\tilde{N}=4 and ζ=5.0×10 -4 \\zeta =5.0 \\times 10^{-4}.", "On top of each time frame, time andgrid compression rate cp are given (cp is the ratio of the cardinalityof the adaptive grid and the cardinality of the full grid with a uniformstep size Δ\\Delta (= the smallest mesh size) in the both coordinatedirections).", "The adaptive grid systematically follows and resolves thewavefront.", "In regions where the field is small or not present only grid points of the coarsest levelare assigned.For an animation movie, see the YouTube channel:www.youtube.com/user/HaojunLi#p/u/1/2Yzpjf7Xnp4.The Gaussian pulse, launched in the center of the computational domain, spreads away from the center as time evolves.", "Fig.", "REF illustrates how the adaptive grid systematically follows and resolves the wavefront.", "Since the electromagnetic field energy is spreading in all directions, the field's amplitude is decreasing (unlike as in 1D, where during the propagation the amplitude stays at half of the initial value, see [9]).", "The AWC method generates a detailed mesh only in the regions where the field is localized, the mesh gets coarse in other parts of the computational domain.", "As seen in the snapshots for $t = 200 \\Delta t$ or $t = 920 \\Delta t$ , it is evident that depending on the extend of the field localization, the density of the grid points varies accordingly.", "A figure of merit for the performance of the AWC-TD method is the compression rate cp, which is defined as a ratio of the cardinality of the adaptive grid and the cardinality of the full grid with a uniform step size $\\Delta $ (= the smallest mesh size) in the both coordinate directions.", "The percentage cp on the top of each time frame in Fig.", "REF shows the grid compression rate.", "Since the extent of a spatial localization of a pulse depends on its frequency contents, the compression rate cp for the test case in Fig.", "REF varies (also seen in Fig.", "REF ).", "Nevertheless, for all time steps the number of grid points in the adapted grid is substantially less than that of in the full grid; but still the AWC method resolves the pulse very well with an optimal (with respect to the given threshold $\\zeta $ ) allocation of the grid points.", "Figure: Relative Error in ℰ y \\mathcal {E}_y between the adaptivewavelet collocation method and the full grid waveletmethod.The relative maximal error of $\\mathcal {E}_y$ field values over $\\Omega $ between the adaptive and the full grid methods as the time evolves is shown in Fig.", "REF .", "Despite of grid compression (which can be quite significant at some time instants, as seen in Fig.", "REF ), the solution by the AWC method is quite close to that of by the full grid method.", "As mentioned earlier, as the pulse spreads in all the direction, the field becomes weak, and the real performance gain by the adaptivity effectively reduces.", "It is reflected in the apparent increase in the relative maximal error (with respect to the full grid method) in Fig.", "REF .", "Note that when the field has completely left the computational domain $\\Omega $ roughly after 800 time steps, the error over $\\Omega $ is not defined meaningfully any more.", "Fig.", "REF demonstrates that (the major part of) the computational effort of the AWC-TD method per time step is indeed proportional to the cardinality of the adapted grid at that time instant.", "To this end, we recorded the CPU time for every ten time steps (Fig.", "REF top).", "For comparison, we also plotted the grid compression rate as a function of the time step (Fig.", "REF bottom).", "Both functions progress in parallel, thus validating the above assertion about the numerical effort of the AWC-TD method.", "Figure: CPU time (top) and gridcompression rate cp (bottom) as functions of the time step.", "Bothfunctions progress in parallel which illustrates the fact that the numerical effort of the AWC-TD method for each time step is proportional to the number of points in the actual grid." ], [ "Conclusions", "In this paper we investigated an adaptive wavelet collocation time domain method for the numerical solution of Maxwell's equations.", "In this method a computational grid is dynamically adapted at each time step by using the wavelet decomposition of the field at that time instant.", "With additional amendments (e.g.", "adjacent zone corrections, reconstruction check, etc.)", "to the adapted grid, we formulated explicit time stepping update scheme for the field evolution, which is a variant of high order FDTD method, and is defined with respect to the locally adapted mesh.", "We illustrated that the AWC-TD method has high compression rate.", "Since (the major part of) the computational cost of the method per time step is proportional to the cardinality of the adapted grid at that time instant, it allows efficient use of computational resources.", "This method is especially suitable for simulation of guided-wave phenomena as in the case of integrated optics devices.", "Initial studies for simulation of integrated optics microring resonators can be found in [9].", "In the present feasibility study we represented the electric and magnetic fields in the physical space, and not in the wavelet space.", "To unleash the full power of adaptivity, however, the field representation and the update in wavelet space are mandatory." ], [ "Acknowledgments", "This work is funded by the Deutsche Forschungsgemeinschaft (German Research Foundation) through the Research Training Group 1294 `Analysis, Simulation and Design of Nanotechnological Processes' at the Karlsruhe Institute of Technology." ] ]

[ [ "Disentangling correlated scatter in cluster mass measurements" ], [ "Abstract The challenge of obtaining galaxy cluster masses is increasingly being addressed by multiwavelength measurements.", "As scatters in measured cluster masses are often sourced by properties of or around the clusters themselves, correlations between mass scatters are frequent and can be significant, with consequences for errors on mass estimates obtained both directly and via stacking.", "Using a high resolution 250 Mpc/h side N-body simulation, combined with proxies for observational cluster mass measurements, we obtain mass scatter correlations and covariances for 243 individual clusters along ~96 lines of sight each, both separately and together.", "Many of these scatters are quite large and highly correlated.", "We use principal component analysis (PCA) to characterize scatter trends and variations between clusters.", "PCA identifies combinations of scatters, or variations more generally, which are uncorrelated or non-covariant.", "The PCA combination of mass measurement techniques which dominates the mass scatter is similar for many clusters, and this combination is often present in a large amount when viewing the cluster along its long axis.", "We also correlate cluster mass scatter, environmental and intrinsic properties, and use PCA to find shared trends between these.", "For example, if the average measured richness, velocity dispersion and Compton decrement mass for a cluster along many lines of sight are high relative to its true mass, in our simulation the cluster's mass measurement scatters around this average are also high, its sphericity is high, and its triaxiality is low.", "Our analysis is based upon estimated mass distributions for fixed true mass.", "Extensions to observational data would require further calibration from numerical simulations, tuned to specific observational survey selection functions and systematics." ], [ "Introduction", "Although there is no question that galaxy clusters are the most massive virialized objects in the Universe, identifying the mass of any particular cluster remains a challenge.", "A cluster's mass is however one of its core properties, important for using cluster samples statistically to constrain cosmological parameters, for understanding clusters as hosts for galaxy evolution, and for studying the growth and other properties of the clusters themselves.", "Recent reviews include [120], [19], [76], [2].", "In simulations, the cluster mass is the sum of the masses of the simulation particles which are cluster members, for whatever cluster member (and thus mass) definition is used (comparisons of some mass definitions are found in, e.g., [125], [127], [59], [35], [68]).", "From a given cosmological set of parameters, simulations predict well defined and directly measurable masses, accurate to the extent that the simulation captures the required physics and has the requisite resolution.", "These theoretical mass definitions cannot be directly applied to observational data, as observations instead measure properties of galaxies both within the cluster and near the cluster, or gas within and sometimes near the cluster, or the bending of space by mass in and around the cluster.", "These observational mass proxies are converted, via physical modeling and assumptions, to mass measurements for comparison with theory.", "Improvements in the noisy mappings between observationally accessible cluster properties and theoretically calculable cluster properties is much sought after by both theorists and observers.", "Complications for observations include projection (and more generally the lack of three dimensional informationThere is a long history, e.g.", "for optical cluster richness starting with [1] and continuing with, for example, [36], [69], [116], [124], [24], [25], [92], [14], for cluster weak lensing (e.g., [90], [74], [55], [87], [73], [12], [56]), for cluster Sunyaev-Zel'dovich [113], [114] (SZ) flux measurements, (e.g., [126], [57], [54], [100], [3]) and for cluster velocity dispersions (e.g., [21], [115], [64], [15], [128], [95]).)", "and reliance of the mapping between mass and observational proxy upon simplifying assumptions such as hydrostatic equilibrium.", "The simulation based theoretical approaches, for their part, find it challenging to capture the directly observable baryonic physics, including galaxy properties.", "In order to alleviate systematics and reduce errors in observationally obtained cluster masses, it is becoming common to combine measurements from different (often multiwavelength) observational techniques.", "The advantages of complementary information and crosschecks are unfortunately mitigated by the fact that scatters from different observational methods are often correlated.", "Essentially, as physical properties of the clusters themselves and their environments are often the causes of mass measurement scatters, more than one measurement technique can be affected.", "It is important include these correlations in order to properly estimate the errors in mass measurements of any individual cluster and to avoid a bias when stacking clusters on one property and measuring another.", "(For discussion see [93], [25], [108], [128], additional simulated examples of correlated scatters using different observational methods include [73], [89]; analyses are beginning to include these, e.g.", "[91], [70], [13].).", "A recent application to an observational cluster sample, resolving some questions raised by earlier analyses is found in [3].)", "Here we consider multiwavelength mass measurements for clusters “observed” in a cosmological dark matter simulation.", "Our primary focus is on mass scatters for individual clusters viewed along several different lines of sight.", "We measure and characterize the multiwavelength correlations and covariances, and study their relation to other cluster properties using both correlations and PCA, principal component analysis.", "This extends recent work using PCA to compare relationships between cluster [103], [61] properties such as concentration, mass and ellipticity in simulations, and some supercluster counterparts, [42], [43], in observations.Comparing cluster mass scatters to physical cluster properties has a long history, recent studies include [131], [12], [11], [7], as well as papers mentioned above.", "The mass observables we simulate are red galaxy richness, phase space richness, velocity dispersions, Sunyaev-Zel'dovich decrement and weak lensing $\\zeta $ statistic [45], [63], techniques in use or planned for large volume current and upcoming cluster surveys such as Atacama Cosmology Telescope (ACT www.physics.princeton.edu/act/), South Pole Telescope (SPTpole.chicago.edu), Blanco Cosmology Survey (BCScosmology.uiuc.edu/BCS/), Dark Energy Survey (DESwww.darkenergysurvey.org) and Large Synoptic Survey Telescope (LSSTwww.lsst.org/).", "The mock simulation measurements, their scatters, and general information about PCA are in §2.", "(Much of §2 summarizes work on the same simulation detailed in [128] (hereafter WCS), and further studied in [78] and [23].)", "In §3, mass scatters for each cluster, along $\\sim 96$ lines of sight, are correlated, and their covariances are analyzed via PCA.", "Distributions of the scatter properties are considered, and, cluster by cluster, the PCA direction of largest scatters is compared to special physical cluster directions.", "In §4 cluster properties, including the individual cluster mass scatter distributions, and environmental and intrinsic properties, are intercompared using correlations and PCA.", "In §5 PCA is instead applied to mass scatter for the whole sample of clusters at once, to analyze scatter including both line of sight and cluster-to-cluster variation, with some discussion of possible extensions to observations.", "§6 discusses outliers and §7 summarizes.", "While we were preparing this work for publication, [3] appeared.", "They consider correlated mass scatter in multiwavelength measurements, in a 4.1 Gpc side simulation which also includes X-ray.", "Our simulation data are the outputs of an N-body simulation of M. White, described in detail in WCS.", "His TreePM [127] code was run with $2048^3$ particles in a periodic box with side length 250 $h^{-1} Mpc$ .", "The 45 outputs are equally spaced in $\\ln (a)$ from $z=10$ to $z=0$ .", "Cosmological parameters were taken to be $(h,n,\\Omega _m,\\sigma _8)=(0.7,0.95,0.274,0.8)$ , consistent with a large number of cosmological observations.", "We focus here on mock observations at $z=0.1$ , where our methods have been most closely tuned to and tested with observational data, as reported in WCS.", "Halos are found via Friends of Friends (FoF) [37], with linking length $b=0.168$ times the mean interparticle spacing (connecting regions with density at least roughly 100 times the mean background density).", "Clusters are halos with FoF masses $M\\ge 10^{14} h^{-1}M_\\odot $ ($M$ hereon will mean this $b=0.168$ FoF mass, we will also write this as $M_{\\rm true}$ when comparing to estimates).", "There are 243 clusters in the box.", "Note that because we have a periodic box we do not need to worry about clusters located near the edge, similarly, because we are using FoF as a halo finder, every particle is uniquely assigned to a single halo.", "Galaxies are taken to be resolved subhalos, which are found via Fof6d [39], with the implementation as described in the appendix of WCS.", "Subhalos are tracked (see [122], [123] for particular details) from their infall into their host halos in order to assign luminosities via subhalo abundance matching [33].", "The resulting galaxy catalogue minimum luminosity at $z=0.1$ is $0.2L_*$ (again see WCS for more discussion and validation tests of the catalogue galaxy properties with observations)." ], [ "Cluster mass measurements and scatters", "We consider five cluster mass measurement methods with this simulation (see WCS for specifics): $N_{\\rm red}$ : Richness using the [65] MaxBCG algorithm based upon colors.", "Galaxy colors are assigned using the algorithm of [102] with evolution of [30], [31], [32].", "Galaxies are taken to be “red” if they have $g-r$ within 0.05 of the peak of the red galaxy $g-r$ distribution specified by [102] for their observed $M_r$ , again see WCS for more detail.", "$N_{\\rm ph}$ : Richness based upon spectroscopy, with cluster membership assigned via the criteria of [132].", "SZ: SZ flux (Compton decrement) is assigned to every particle by giving it a temperature based upon the mass of its halo.", "For every cluster, its measured SZ flux is then the flux within an annulus of radius $r_{180b}$ (the radius within which the average mass is greater than or equal to 180 times background density), through the length of the box, apodized at the edges.", "This was shown in e.g.", "[126] to well approximate hydrodynamic simulation results for SZ at the scales appropriate for two cluster surveys mentioned earlier, SPT and ACT.", "Vel: Velocity dispersions calculated via the method detailed in WCS, and based on [38], [15], [129].", "Phase space information is used to reject outliers and the mass estimate includes the harmonic radius (calculated as part of the outlier rejection, more details and definitions in WCS).", "WL: Weak lensing using a singular isothermal sphere (SIS) or NFW model to assume a cluster lens profile and then fitting the projected mass, using the $\\zeta $ statistic [45], [63], in a cylinder with radius $r_{180b}$ and (apodized) length of the box (again WCS describes fitting models, etc.).", "The red galaxy richness, phase space richness and velocity dispersions measured in our simulations are expected to include the majority of systematics that are present in real observations.", "The weak lensing and Compton decrement (SZ) observations however do not include all known systematics, such as miscentering, shape measurement and source redshift errors for lensing, and foreground and point source removal for Compton decrement.", "The relatively small box size (250 $h^{-1} Mpc$ on a side) also means that line of sight scatter is underestimated (e.g., [126], [54], [57], [25], [3] for SZ and [90], [74], [55], [87], [73], [12], [7], [56] for lensing).", "We extend the cluster sample used in WCS to a lower mass range , $M\\ge 10^{14} h^{-1} M_\\odot $ , as in [78], [23].", "The five observables listed above are found along 96 lines of sight for each cluster, each time placing the cluster at the center of the periodic box.", "Just as in WCS, lines of sight for clusters are removed for all measurement methods when a more massive cluster has its center within $r_{180b}$ along the line of sight (this removes $\\sim 400$ of the original $\\sim $ 23000 lines of sight).", "In addition, to allow fair intercomparisons, only lines of sight which have reliable mass measurements for all methods are included; the $\\sim 90$ lines of sight with fewer than 8 galaxies making the cut for a velocity dispersion estimate, or either richness $<1.1$ are also removed.", "These cuts will have some effect on the scatters we consider but would be expected to be identifiable observationally.", "We take the logarithm of these observables and that of the true mass $M$ to find the mean relations for all clusters with $M \\ge 10^{14}h^{-1} M_\\odot $ .", "We find relations for bins of $M$ vs. observables, because of the large scatter at low mass.", "The fits are done by throwing out 3 $\\sigma $ outliers for three iterations.", "This gives us our map between the observables and mass estimates $M_{N_{\\rm red}},M_{N_{\\rm ph}}, M_{\\rm SZ},M_{\\rm Vel},M_{\\rm WL}$ .", "The distribution of the fractional mass scatters, $(M_{\\rm est}-M_{\\rm true})/M_{\\rm true}$ , for the five mass measurement methods along $\\sim 96$ lines of sight for each of the 243 clusters, is shown in Fig.", "REF .", "Figure: The fractional mass scatter, M est ,i /M true -1M_{\\rm est,i}/M_{\\rm true}-1 forall 243 clusters along ∼\\sim 96 lines of sight each, for the five massmeasurement methods we consider.", "Solid lines are the massesestimated via N red N_{\\rm red},N ph N_{\\rm ph},SZ (Compton decrement), velocity dispersion and WL (weak lensing) asdescribedin section .", "An approximately 0.2-0.5 standard deviation isfound.", "The (for the most part difficult todistinguish) dashed line is a least squares Gaussian fit, givingσ fit \\sigma _{\\rm fit}, which is much smaller than the standard deviationfor SZ.The WL and SZ mass scatter is expectedto be strongly underestimated compared to true observations,as the 250 Mpc/h box size is too small toinclude all contributions to mass scatter along the line of sight tothe observer, and in these two cases several known systematics are notincluded as well.", "See text for more information.The scatters range from $\\sim $ 0.2-0.5, with the smallest mass scatters associated with Compton decrementRecall the caveats for both SZ and weak lensing measurements mentioned earlier, however.", "and $N_{\\rm ph}$ .", "Note that this work follows a theoretical approach where all the mass measurements are related to the known true mass, a quantity inaccessible in observations.", "In particular, the sample is cut on the unobservable value of $M_{\\rm true}$ and the distributions we consider are $M_{\\rm est}(M_{\\rm true})$ , not vice versa.We thank E. Rozo for extremely helpful discussions on this.", "For an observational sample based upon some measurement $M_{\\rm est}$ , such as richness mass or weak lensing mass, often the quantity of interest is the scatter in $P(M_{\\rm true}|M_{\\rm est})$ (see also extended comments in §).For example, the richness mass scatter in Fig.", "REF , looks essentially Gaussian, unlike the double peaked distribution found and studied in e.g.", "[24], [44].", "If looking at $P(M_{\\rm true}|M_{\\rm est})$ , one will find clusters which are “blends”, i.e.", "several halos which contribute to one apparent halo, often with a higher $M_{\\rm est}$ than any of the contributing halos.", "These blends (see also e.g.", "[51] for discussion) are the source of the bimodal mass distribution reported and used in these other papers." ], [ "Filaments and Galaxy Subgroups", "In section §, cluster filament properties calculated in [78] and cluster galaxy subgroup properties calculated in [23] for this simulation are used.", "Detailed background can be found in those two papers, but we briefly summarize some key aspects here.", "Filaments are found in [78] using a modification of the dark matter halo based filament finder of [134].Although the cosmic web was noted years ago [133], [97], [41], [18], no unique filament finder exists.", "A variety of finders are in use, based on a wide range of dark matter, halo and/or galaxy properties, including for example [9], [72], [94], [96], [29], [101], [27], [84], [85], [109], [79], [4], [28], [53], [119], [105], [110], [16], [49], [52], [86], [106], [111], [130], [134], [5], [6], [17], [75], [107], [121], [99], [98], [50], [104], [60].", "This filament finder searches for bridges 10 $h^{-1} Mpc$ or smaller between halos above $3 \\times 10^{10} h^{-1} M_\\odot $ , starting with the most massive halos as potential bridge endpoints.", "Some clusters (16/243) end up within filaments because of the finder, such as less massive clusters located between two close ($< 10 h^{-1}Mpc$ ) massive clusters and clusters closer than $3 h^{-1} Mpc$ to a larger cluster.", "The rest of the clusters each lie at the center of a filament map extending out to 10 $h^{-1} Mpc$ .", "We found that filaments, halo mass (halos with mass $\\ge 3 \\times 10^{10} h^{-1} M_\\odot $ ) and galaxy richness all tended to lie in a planar region around each cluster.", "We characterized these regions by taking a fiducial 3 $h^{-1} Mpc$ high disk centered on the cluster which extends out to the edge of the radius 10 $h^{-1} Mpc$ sphere.", "We use the planes related to halo and filament mass below.", "For halo mass, we randomly sampled 10,000 orientations to maximize the halo mass fraction in the plane (relative to the halo mass in the 10 $h^{-1} Mpc$ sphere).", "For filaments, we considered planes spanned by the cluster and pairs of filament endpoints, and then took the plane which enclosed the most filament mass.", "This plane was not found for clusters lying within filaments.", "See [78] for more details.", "We consider four quantities from this analysis below: $f_{M_{hplane}}$ (halo mass fraction in plane relative to that in 10 $h^{-1}M_\\odot $ sphere), $f_{M_{fplane}}$ (cluster filament mass fraction in plane relative to sphere), and the respective plane normal directions $\\hat{n}_{\\rm mass},\\hat{n}_{\\rm fil}$ .", "Galaxy subgroups were characterized in [23] for this simulation.", "These are groups of galaxies that fell into a cluster as part of a shared halo at an earlier time.", "Within the clusters, they share some coherence in space and time which can remain for several Gyr.", "We will use for each cluster its largest (richest) galaxy subgroup, in particular its fractional richness relative to that of the cluster, $f_{R_{\\rm sub}}$ , its displacement relative to the cluster center, divided by the cluster long axis $f_{D_{\\rm Sub}}$ , and the directions of its position and average velocity relative to the cluster center, $\\hat{r}_{\\rm sub}, \\hat{v}_{\\rm sub}$ ." ], [ "Principal Component Analysis", "For context and background, we summarize PCA and our notation here (see, e.g.", "[62] for extensive discussion).", "PCA can be used when there are several correlated or covariant quantities.", "It is essentially a rotation of axes to find linearly independent bases (i.e.", "quantities which are not covariant or correlated), and is based on a model where some underlying average linear relation is present.", "We will apply PCA in a few different contexts.", "Our starting application will be for individual clusters.", "For each individual cluster and line of sight, we have several different methods to estimate the true cluster mass.", "Each line of sight can thus be associated with five numbers, where each number is the mass measured in one method.", "These numbers can then be thought of as coordinates in some five dimensional abstract space, with each axis in this space corresponding to a different measurement method.", "All of the different lines of sight considered together then give a cloud of points in this space of measurement methods.", "PCA gives the properties of the “shape” of the mass scatters in this space, around their average values for the combined observations, for each cluster.", "We will consider these shapes and how they relate to other cluster properties.", "In addition, correspondences can be found between large mass scatters and physical properties or directions of the cluster.", "The PCA direction with smallest mass scatter is useful as well.", "Taking the ensemble of clusters, groups of properties which change together can be inferred by using PCA on the full set of correlations (this latter approach was pioneered by [61], [103], [42]).", "There are other uses of PCA, and caveats as well.", "PCA is often used to find the minimum set of variables needed to describe a system to some accuracy, for instance the dominant contributing basis vectors composing a galaxy spectrum.", "As far as caveats go, one concern is that if correlated variables are not scattered around a linear relation, a simple rotation of basis using PCA will not usefully separate them.", "For this reason, sometimes other functions are used besides the variables themselves, e.g.", "logarithms, when it is suspected that variables might be related by power laws.", "For illustration, we take a set hypothetical measurements for two methods, as shown in Fig.", "REF .", "Each pair of measurements, by the two methods, is a position, i.e.", "a dot, in this plane labeled by two coordinates.", "We take one coordinate to be the shifted mass using red galaxy richness, $M_{N_{\\rm red}}^\\alpha =M^\\alpha _{N_{red},est}- \\langle M_{N_{red},est}\\rangle $ , and the other to be the shifted weak lensing mass, $M_{\\rm WL}^\\alpha = M^\\alpha _{\\rm WL, est} -\\langle M_{\\rm WL, est}\\rangle $ .", "Here, $\\alpha $ denotes which particular point is being measured and the average is over all the points shown, all $\\alpha $ values, i.e.", "$\\langle M_{N_{red},est}\\rangle =\\frac{1}{N_\\alpha }\\sum _\\alpha M^\\alpha _{N_{red},est}$ .", "The vector $\\vec{M}^\\alpha _{\\rm obs}$ denotes $(M_{N_{\\rm red}}^\\alpha ,M_{\\rm WL}^\\alpha )$ and $N_\\alpha $ is the number of measurements (points) indexed by $\\alpha $ .", "For our first application below, all different values of $\\alpha $ pertain to the same cluster, but label different lines of sight.", "The shift by the average over all the points (all $\\alpha $ ) guarantees that $\\langle M_{\\rm WL}\\rangle =\\langle M_{N_{\\rm red}}\\rangle = 0$ .", "Diagonalizing the covariance matrix for $(M_{N_{\\rm red}},M_{\\rm WL})$ , i.e.", "found by summing over all $\\alpha $ , produces orthonormal eigenvectors $\\hat{PC}_i$ (principal components) with eigenvalues $\\lambda _i$ .", "The eigenvectors are illustrated in Fig.", "REF and are the axes of a new coordinate system in the space of measurement methods in which the measurements have zero covariance.", "Figure: Principal component analysis takes a set of correlatedmeasurements (using two methods shown as basis elements e ^ red ,e ^ WL \\hat{e}_{\\rm red}, \\hat{e}_{WL}) androtates their coordinates toa new basis, shown as dotted lines, where the measurements areuncorrelated.For example, the point marked as a star (i.e.", "α=☆\\alpha = \\star ) isM → ☆ =10 14 e ^ red +10 14 e ^ WL =2×10 14 PC ^ 0 +0PC ^ 1 \\vec{M}^\\star = 10^{14} \\hat{e}_{\\rm red} + 10^{14} \\hat{e}_{\\rm WL}= \\sqrt{2}\\times 10^{14} \\hat{PC}_0 + 0\\; \\hat{PC}_1.The generalization to more measurement methods and thus a higherdimensional space is immediate.", "Amongst the orthonormalPC ^ i \\hat{PC}_i, we choosePC ^ 0 \\hat{PC}_0 to be along the direction of largest scatter(i.e.", "to correspond to the largest variance, λ 0 \\lambda _0), PC ^ 1 \\hat{PC}_1 to be along the direction of second largestvariance, etc., as described in the text.The $\\hat{PC}_i$ can be expressed in terms of the original basis directions, $\\hat{PC}_i = \\beta _{\\rm red, \\; i} \\hat{e}_{\\rm red} +\\beta _{\\rm WL, \\; i}\\hat{e}_{\\rm WL} \\; ,$ which identifies the mass scatters contributing most to each $\\hat{PC}_i$ .", "For example, if $\\beta _{\\rm red, \\; 0}$ is large, then most of the scatter in the direction of $\\hat{PC}_0$ also lies in the direction of $\\hat{e}_{\\rm red }$ .", "One also sees how different scatters are related.", "In this simple example the biggest scatters come from increases or decreases in both ${M}_{\\rm red},{M}_{\\rm WL}$ simultaneously.", "This implies, in this case, that $\\hat{PC}_0$ might be related to an average overall mass shift, amongst methods, as well.", "Note that the overall signs for the $\\hat{PC}_i$ are arbitrary.", "A point (or vector from origin, labeled by $\\alpha $ ) in the original space can then be rewritten in the basis spanned by $\\hat{PC}_0,\\hat{PC}_1$ : $\\vec{M}^\\alpha = M^\\alpha _{\\rm red} \\hat{e}_{\\rm red} +M^\\alpha _{\\rm WL} \\hat{e}_{\\rm WL}= a^\\alpha _0 \\hat{PC}_0 + a^\\alpha _1 \\hat{PC}_1 \\;.$ For example, the point marked by a star in Fig.", "REF has coordinates $\\vec{M}^\\star = 10^{14} \\hat{e}_{\\rm red} + 10^{14} \\hat{e}_{\\rm WL}= \\sqrt{2}\\times 10^{14} \\hat{PC}_0 + 0\\; \\hat{PC}_1$ .", "That is, $(M^\\star _{\\rm red},M^\\star _{\\rm WL})=(1,1)\\times 10^{14}$ (assuming measurements are in units of $h^{-1} M_\\odot $ ) and $(a^\\star _0,a^\\star _1)=(\\sqrt{2},0)\\times 10^{14}$ .", "The variances in each of the new directions, associated with the coefficients $a^\\alpha _i$ , are the eigenvalues of the principal components.", "Thus, $\\lambda _0$ is the eigenvalue associated with $\\hat{PC}_0$ , etc.", "For all PCA eigensystems we consider here, we will order $i<j$ if $\\lambda _i > \\lambda _j$ and define $\\sum {\\lambda } =\\sum _i \\lambda _{i}$ and $\\prod \\lambda = \\prod _{i} \\lambda _{i}$ .", "Generally, if there are $N_{\\rm method}$ measurement methods, there are $N_{\\rm method}$ $\\hat{PC}_i$ , spanning an $N_{\\rm method}$ dimensional space.", "In Fig.", "REF , $N_{\\rm method}$ =2; when we consider the five different mass measurement methods below, for example, $N_{\\rm method}=5$ .", "For PCA applied to different cluster properties, in §REF below, we have $N_{\\rm method}=24$ .", "We will apply PCA to covariance and correlation matrices, using the Pearson covariance ${\\rm Cov} (x y) =\\frac{1}{N_\\alpha -1}\\sum _{\\alpha =1}^{N_\\alpha } (x_\\alpha - \\bar{x})(y_\\alpha - \\bar{y}) $ where $\\bar{x}$ is the average of the $N_\\alpha $ points $x_\\alpha $ , etc., and ${\\rm Cov}(x,y)/\\sqrt{{\\rm Cov}(x,x) {\\rm Cov}(y,y)}$ its associated correlation, for $N_{\\alpha }$ measurements.For the cluster mass scatters, as this covariance can be affected by outliers, we experimented with several outlier rejection schemes.", "As the resulting values were somewhat similar, and our analysis is in part just to provide an example, we use the untrimmmed Pearson covariances and correlations hereon.", "We also considered for some properties (as did some of the earlier cluster PCA work) the Spearman correlation.", "The Spearman correlation coefficient uses the ranking of the measurements rather than the raw measurements themselves.", "Trends were similar to the Pearson covariances and correlations.", "The sum of the $\\lambda _i$ , $\\sum \\lambda $ , is the sum of the variances of the measurement methods, their product, $\\prod \\lambda $ , is related to the “volume” in this space of scatters, i.e.", "how the measurements are spread out in the space of measurement methods (specifically, for the example in Fig.", "REF , $\\sqrt{\\lambda _0 \\lambda _1}$ is proportional to the area of the ellipse).", "A small eigenvalue $\\lambda _i$ means that the scatter in the corresponding $\\hat{PC}_i$ direction is small, i.e.", "that the volume of scatters is roughly confined to a lower dimension.", "In particular, if most of the scatter is due to $\\lambda _0$ , then there is a close to linear relation present.", "Such lower dimensionality was used in [42] to come up with scaling relations for superclusters.", "For PCA of correlations, a large $\\lambda _0$ occurs if the initial measurements via different methods have strong correlations; for covariances, a large $\\lambda _0$ can also occur if an individual measurement method has large scatter.", "Measurement methods $M^\\alpha _{{\\rm obs},j}$ (in the example $M_{N_{\\rm red}}, M_{\\rm WL}$ ) with the largest correlation or covariance with $a^\\alpha _i$ (their projection on $\\hat{PC}_i$ ) can be thought of as those dominating the scatter in the direction of $\\hat{PC}_i$ .", "This covariance or correlation is not unrelated to the measurement method's contribution to $\\hat{PC}_i$ , ($\\beta _{{\\rm obs}, j, i}$ in Eq.", "REF ).", "The covariance or correlation is largest when $\\beta _{{\\rm obs}, j, i}$ is large, and when the eigenvalue $\\lambda _i$ is large relative to the other $\\beta _{{\\rm obs},j}$ 's and $\\lambda _{i}$ .", "(For PCA on correlations in particular, $\\langle M_{{\\rm obs},j} a_i\\rangle / \\sqrt{\\langle M_{{\\rm obs},j}M_{{\\rm obs},j}\\rangle \\langle a_i a_i\\rangle } \\sim \\beta _{{\\rm obs},j, i}\\lambda _i/\\sqrt{\\lambda _i \\sum _k \\lambda _k\\beta _{{\\rm obs},j,k}^2}$ ; if $\\lambda _0, \\beta _{{\\rm obs}j,0}$ are large and $a_0$ is considered then this approaches 1.)", "To summarize, PCA is a method taking a set of measurements via different methods which are correlated and separating them into uncorrelated combinations.", "In particular, if one denotes each set of measurements as a position in some space, along axes corresponding to each type of measurement method, then PCA is a rotation of coordinates in this space.", "The volume and shape traced out by the points representing the measurements are related to the PC eigenvectors and eigenvalues, and the direction in the space corresponding to the largest PC eigenvector is the combination of measurement methods with the largest scatter in its distribution." ], [ "Variations for a single cluster due to line of sight effects", "As mentioned in the introduction, many of the mass measurement method scatters in Fig.", "REF are comparable in size because they are due to similar properties of the cluster or its environment.", "Correlations between the scatters are thus expected, and as noted above, these correlations and their consequences become increasingly important as multiwavelength studies become more common.", "We first consider each cluster and its line of sight mass scatters separately.", "In this way, the “true” object and its true mass remained fixed; all variations in scatters are due to changes in line of sight.", "Several examples of correlations for these scatters were already noted and illustrated in WCSAn example of correlations between velocity dispersion and weak lensing measurements are shown in WCS Fig.", "14, and correlations with various physical properties (discussed below) were further studied in [78], [23].", "Here we statistically describe these correlations and covariances.", "We characterize cluster to cluster trends and variations in line of sight mass scatters and predicted masses, and then apply PCA to these scatters." ], [ "Correlated mass scatters for different cluster observables", "To give an idea of the correlations and covariances for our five different mass measurement methods, we start with an example: the 10 pairs of mass measurements for a single cluster $(M=4.8\\times 10^{14} h^{-1}M_\\odot )$ shown in Fig.", "REF .", "Each panel shows a different pair of mass estimates along all lines of sight (i.e.", "$N_\\alpha = 95$ ), and correlations and covariances are listed at the top of each.", "We use $M_{\\rm est}/M_{\\rm true}-1$ to focus on fractional mass scatter.", "As can be seen, many of the correlations are large.", "Figure: Cluster mass scatters (M est /M true -1M_{\\rm est}/M_{\\rm true} -1) forone cluster of mass4.8×10 14 h -1 M ⊙ 4.8 \\times 10^{14} h^{-1} M_\\odot , along 95 lines ofsight.", "The correlation and covariance for each mass measurementmethod pair is shown at the top of each panel (the y-axis is for alargerscale to allow room for these numbers).", "Large correlations arepresent for many pairs of mass measurement methods.For all 243 clusters, the correlations for the same pairs of mass measurement methods are compiled in Fig.", "REF , with medians and averages given in Table REF .", "Figure: Correlations of M est /M true -1M_{\\rm est}/M_{\\rm true}-1 for pairs of mass measurement methods(M N red M_{N_{\\rm red}}, M N ph M_{N_{\\rm ph}}, M SZ M_{\\rm SZ}, M Vel M_{\\rm Vel}, M WL M_{\\rm WL}).The solid line is the distribution of cluster mass scattercorrelations, for all clusters individually, for the same pairs as in Fig.", ".The dotted line corresponds tothe 70 clusters with mass ≥2×10 14 h -1 M ⊙ \\ge 2\\times 10^{14} h^{-1}M_\\odot .The vertical dashed lines are at the median values which are listed inTable , along with the average values.", "Note that the x-axis,the range of correlations, has a scale which varies widely betweendifferent types of measurement method pairs.Strong correlations are frequent.", "For each cluster, at least one pair of mass measurement methods has correlation $>$ 0.4, and the largest pair correlation is often larger, $\\sim 0.7$ .", "Within our cluster sample, the mass scatters for ($M_{N_{\\rm red}}$ , $M_{N_{\\rm ph}}$ ) are most often the highest correlated pair.", "The other measurement method pairs which frequently have the highest correlation (but not as often as ($M_{N_{\\rm red}}$ , $M_{N_{\\rm ph}}$ )) are ($M_{N_{\\rm red}}$ , $M_{\\rm SZ}$ ), ($M_{N_{\\rm red}}$ , $M_{\\rm WL}$ ), ($M_{N_{\\rm ph}}$ , $M_{\\rm SZ}$ ) and ($M_{N_{\\rm ph}}$ , $M_{\\rm WL}$ ).", "The pair with the minimum correlation is most frequently ($M_{\\rm Vel}$ , $M_{\\rm WL}$ ) (closely followed by ($M_{\\rm Vel}$ , $M_{\\rm SZ}$ ), and ($M_{\\rm Vel}$ , $M_{N_{\\rm ph}}$ )).", "Figure: Covariances of M est /M true -1M_{\\rm est}/M_{\\rm true}-1 for pairs of mass measurement methods(M N red ,M N ph ,M SZ ,M Vel ,M WL M_{N_{\\rm red}},M_{N_{\\rm ph}},M_{\\rm SZ},M_{\\rm Vel},M_{\\rm WL}).The solid line shows the distribution of cluster mass scattercovariances, for all clusters individually, and the same massobservation pairs as in Figs. ,.", "The dotted line restricts tothe 70 clusters with mass ≥2×10 14 h -1 M ⊙ \\ge 2\\times 10^{14} h^{-1}M_\\odot .As the characteristic mass scatter for mostclusters is about 0.3, the range for covariances shown is±0.05\\pm 0.05.", "The vertical dashed lines are at the median values, whichare also listed in Table .Note that the x-axis,the range of covariances, has a scale which varies with types ofmass measurement method pairs.Table: Average and median values for the distribution(for 243 clusters) of pairs of massmeasurement method correlations shown inFig.", "and covariances shown Fig.", ".For our PCA analysis below, we will use covariances instead, shown in Fig.", "REF , with medians and averages given in Table REF .", "These are more relevant for understanding the actual mass scatters and how they change together, rather than, for example, how much a relatively large $M_{\\rm SZ}$ mass scatter corresponds to a relatively large $M_{N_{\\rm red}}$ mass scatter.", "As the fractional mass fluctuations (the $\\sigma $ values in Fig.", "REF ) tend to be about 0.3, the covariance sizes should be compared to $\\sim $ 0.09.", "The largest covariances are between ($M_{N_{\\rm red}}$ , $M_{\\rm WL}$ and $M_{\\rm Vel}$ ).", "Each cluster has at least one covariance $\\ge $ 0.01.", "In contrast, the minimum covariance between measurement methods tends to be between the pair ($M_{N_{\\rm ph}}$ , $M_{\\rm SZ}$ ).If the abovementioned neglected weak lensing and SZ scatter is uncorrelated with local properties of the cluster, as expected, when this scatter is included the majority of the covariances will not change (except possibly the one between weak lensing and SZ, as they can possibly have correlated changes in their scatter due to structure outside the box).", "However, the correlations will change as they are divided by the correlations of SZ with itself or weak lensing with itself.", "Some of these correlation and covariance trends are understandable (larger covariances tend to go with quantities with larger scatter more generally and vice versa), but others rely upon the interplay between different measurement methods and the causes of the scatter.", "The relative importance of different contributions to these were not a priori obvious to us, although reasons could be found for trends.", "For instance, the two richnesses often are the most correlated mass measurement pair.", "This is perhaps because both use the same objects (galaxy counts), so that an enhancement or decrement of $M_{N_{\\rm ph}}$ (galaxies making the spectroscopic cut) might be more likely accompanied by a similar change in $M_{N_{\\rm red}}$ (galaxies making the red sequence cut), than by changes in the mass measurements by other methods.", "Other mass measurement methods are correlated with richness less directly (for $M_{\\rm SZ}$ and $M_{\\rm WL}$ the measurements and thus presumably the scatter are more directly tied to the dark matter distribution rather than the biased galaxies; $M_{\\rm Vel}$ just seems to be weakly correlated with most things).", "In section §REF we compare the PCA results (scatters which occur together) with physical cluster directions to get some idea of which properties might be driving covariant mass measurement method scatters." ], [ "PCA for individual clusters", "We now apply PCA to the covariances for $\\vec{M}_{\\rm obs}/M_{\\rm true}$ for each individual cluster, to get a new basis, $\\frac{\\vec{M}_{\\rm obs}}{M_{\\rm true}} =\\frac{\\vec{M}_{\\rm est}^\\alpha }{M_{\\rm true}}-\\frac{\\vec{M}_{\\rm ave}}{M_{\\rm true}} = \\sum _ia^\\alpha _i \\hat{PC}_{i,M} \\; .$ The subtracted offset $\\vec{M}_{\\rm ave} \\equiv \\langle \\vec{M}_{\\rm est} \\rangle $ , where the average is over all the lines of sight for each cluster of interest, i.e., there is a different $\\vec{M}_{\\rm ave}$ for each cluster.", "The median (and rms around zero) values for the ensemble of clusters, for $|\\vec{M}_{\\rm ave}/M_{\\rm true}- 1|$ are [0.21 (0.32), 0.11 (0.17), 0.08 (0.15), 0.11 (0.18), 0.18 (0.27)] for $M_{N_{\\rm red}}, M_{N_{\\rm ph}},M_{\\rm SZ},M_{\\rm Vel},M_{\\rm WL}$ respectively.", "The relative sizes of the line of sight scatters around the average mass, $\\vec{M}_{\\rm obs}/M_{\\rm true}- \\vec{M}_{\\rm ave}/M_{\\rm true}$ compared to the line of sight averaged mass around the true mass $\\vec{M}_{\\rm ave}/M_{\\rm true}- 1$ varies widely cluster to cluster.", "Except for velocity dispersions, the rms scatters of $(\\vec{M}_{\\rm obs}/M_{\\rm true}- \\vec{M}_{\\rm ave}/M_{\\rm true})$ is $ \\ge |\\vec{M}_{\\rm ave}/M_{\\rm true}- 1|$ for 50-60 percent of the clusters and the median value of $|\\vec{M}_{\\rm obs}/M_{\\rm true}- \\vec{M}_{\\rm ave}/M_{\\rm true}|$ is $\\ge |\\vec{M}_{\\rm ave}/M_{\\rm true}- 1|$ for about 30-40 percent of the clusters, for velocity dispersions the numbers are closer to 90 and 70 percent respectively.", "As in section §REF , and by the definition in Eq.", "REF , $M_{\\rm obs}$ refers to mass measurements which have zero average when summed over the sample of interest.", "Any vector in this space can of course be written in terms of the orthonormal basis $\\hat{PC}_{i,M}$ ; what is special for $\\vec{M}_{\\rm obs}$ is that the variances of the $a^\\alpha _i$ are equal to $\\lambda _i$ for gaussian scatter.", "Because there are five mass scatters, there are five PC vectors $\\hat{PC}_{i,M}$ per cluster, with eigenvalues $\\lambda _{i,M}$ , again ordered $\\lambda _{0,M}>\\lambda _{1,M}>\\lambda _{2,M}$ and so on.", "We use the subscript $M$ to distinguish these PC vectors from others which will be considered in section , and take $M_{{\\rm obs},i}$ , with $i=0,1,2,3,4$ , to correspond to $(M_{N_{\\rm red}},M_{N_{\\rm ph}},M_{\\rm SZ},M_{\\rm Vel},M_{\\rm WL})$ and similarly for $\\hat{e}_{{\\rm obs},i}$ .", "We also have, as in Eq.", "REF , $\\hat{PC}_{i,M} =\\sum _{{\\rm obs},j = 0}^4 \\beta _{{\\rm obs},j,i} \\hat{e}_{{\\rm obs},j} \\; .$ We first consider the PC eigenvalues, the $\\lambda _{i,M}$ .", "There are some trends: the fractional scatter in the largest direction ($\\frac{\\lambda _{0,M}}{\\sum {\\lambda }}$ ) $ \\sim 0.7$ , but can vary from 0.4 to $\\sim 1$ , as shown in Fig.", "REF .", "The relatively large contribution from $\\lambda _{0,M}$ means that the variance is strongly dominated by the single combination of mass scatters in the direction of $\\hat{PC}_{0,M}$ .", "As seen in Fig.", "REF , bottom, $\\lambda _{0,M},\\lambda _{1,M},\\lambda _{2,M}$ together comprise almost all the variance for most clusters.", "The presence of some mass measurement methods with small scatter suggests that there are some directions of the combined measurement methods which would also have small scatter and thus small $\\lambda _{i,M}$ , and this is seen in the much smaller values of $\\lambda _{4,M}$ and sometimes $\\lambda _{3,M}$ .", "The distribution of covariance matrices shown in Fig.", "REF determine the $\\lambda _{i,M}$ when combined with the relation of the covariances to each other, cluster by cluster.", "The combination of measurement methods in $\\hat{PC}_{4,M}$ which has the smallest variance, $\\lambda _{4,M}$ in our case, is also interesting, we return to this in §.", "Figure: (Top) Peaks from right to left:fraction of covariance (i.e.", "scatter) λ 0,M ∑λ\\frac{\\lambda _{0,M}}{\\sum {\\lambda }},λ 1,M ∑λ\\frac{\\lambda _{1,M}}{\\sum {\\lambda }}, λ 2,M ∑λ\\frac{\\lambda _{2,M}}{\\sum {\\lambda }}, etc.On average λ 0,M ∑λ∼0.7\\frac{\\lambda _{0,M}}{\\sum {\\lambda }} \\sim 0.7.", "(Bottom) Peaks from left to right: fraction of covariance from scatterindirection of PC ^ 0,M ,PC ^ 0,M \\hat{PC}_{0,M}, \\hat{PC}_{0,M}or PC ^ 1,M \\hat{PC}_{1,M}, etc.", "Most ofthe scatter is in the directions spanned byPC ^ 0,M ,PC ^ 1,M ,PC ^ 2,M \\hat{PC}_{0,M},\\hat{PC}_{1,M},\\hat{PC}_{2,M}, with a substantial fractionin the direction of the largest scatter.The sum and product of the $\\lambda _{i,M}$ for all clusters are shown on a logarithmic plot in Fig.", "REF .", "The sum of scatters can vary by a factor of $\\sim 30$ from cluster to cluster, and tends to be dominated by $\\lambda _{0,M}$ , while the product of the $\\lambda _{i,M}$ can be made very small (its size varies by over $10^7$ ) if some directions, especially $\\hat{PC}_{4,M}$ , have very little scatter.", "Figure: Sum and product of covariances for all clusters: the sum isdominatedby the eigenvalue in the direction of largest scatter, λ 0,M \\lambda _{0,M},while the product, related to the volume in the space of scatters, can be made very small by small values ofe.g.", "λ 4,M \\lambda _{4,M}.The sum of eigenvalues peaks at ∼0.3\\sim 0.3, the product peaks around10 -8 10^{-8}.In § below, these scalar properties of the cluster mass scatter will be compared to physical cluster properties such as triaxiality and mass.", "Turning to the PC vectors, many clusters have similar $\\hat{PC}_{0,M}$ (i.e.", "the combination of mass scatters that dominates is similar for many of the clusters).", "To quantify this more generally, we took $\\hat{PC}_{i,M,minsq}$ as the direction which minimizes $(\\hat{PC}_{i,M} \\cdot \\hat{PC}_{i,M,minsq})^2$ for the full ensemble of 243 clusters ($\\hat{PC}_{0,M,minsq}$ is shown in the first line of Table REF ).", "Fewer than 20 percent of the clusters have their $\\hat{PC}_{0,M}$ pointing more than $45^\\circ $ away from $\\hat{PC}_{0,M,minsq}$ ; about 25 percent have their $\\hat{PC}_{1,M}$ pointing more than $45^\\circ $ away from $\\hat{PC}_{1,M,minsq}$ .", "We found that for almost all the clusters the projection upon $\\hat{PC}_{1,M,minsq}$ is close in size to the projection on $\\hat{PC}_{0,M,minsq}$ , (i.e.", "0.62 correlation).", "Continuing to $\\hat{PC}_{4,M}$ , the direction of least scatter, $\\sim $ 70 percent of the clusters are within $45^\\circ $ of $\\hat{PC}_{4,M,minsq} $ given in Table REF .", "This minimum scatter direction is (not surprisingly) dominated by the $\\hat{e}_{\\rm N_{ph}},\\hat{e}_{\\rm SZ}$ directions, since these are the mass measurement methods with the smallest scatter in our sample.Again recall these particular coefficients do not carry over directly to observations, most importantly because they are based on scatter in $P(M_{\\rm est}|M_{\\rm true})$ and not vice versa.", "The similar forms of the $\\hat{PC}_i$ suggest that the mass scatter combinations they correspond to might have similar physical origins.", "For 75 percent of the clusters, the coefficients of $\\hat{PC}_{0,M}$ are also all the same sign, that is that the dominant combination of scatter also has all the scatters increasing together relative to their average values, but 16/243 have large ($<-0.1$ ) opposite sign coefficients for some mass measurement methods.", "As a large fraction of the variance is captured by $\\hat{PC}_{0,M}$ , as seen in the large values of $\\lambda _{0,M}/\\sum \\lambda $ , the coefficient of $\\hat{PC}_{0,M}$ for the any given line of sight is indicative of the size of scatter from the average along that line of sight.", "Table: A comparison betweendifferent characteristic directions in the space of mass scatters,(e ^ N red \\hat{e}_{\\rm N_{\\rm red}}, e ^ N ph \\hat{e}_{\\rm N_{\\rm ph}},e ^ SZ \\hat{e}_{\\rm SZ}, e ^ vel \\hat{e}_{\\rm vel}, e ^ WL \\hat{e}_{\\rm WL}).The direction which minimizes thedot product squared with PC ^ i,M \\hat{PC}_{i,M} of all 243 clusters isPC ^ i,M,minsq \\hat{PC}_{i,M,minsq}.PC ^ i,M,total \\hat{PC}_{i,M,total} are the vectors found by PCA analysis of allobserved cluster masses (normalized by M true M_{\\rm true})and all lines of sight, jointly, discussed in §;PC ^ i,M,total,massive \\hat{PC}_{i,M,total,massive} restricts to the 70 clusterswithM≥2×10 14 h -1 M ⊙ M \\ge 2 \\times 10^{14} h^{-1} M_\\odot .These vectors are derived from our estimates of M est (M true )M_{\\rm est}(M_{\\rm true}),whichas mentioned earlier, neglects some systematics for WL and SZ and relies upon oursimulation calibrated massdefinitions in terms of observables.The correlations of the five mass measurement methods $M_{{\\rm obs},i}$ with the $\\hat{PC}_j$ coefficients $a^\\alpha _j$ , for all clusters, are shown in Fig.", "REF .", "Figure: The correlations between the observed mass scatters and their projectedvalues on PC ^ i,M \\hat{PC}_{i,M}.Velocity dispersions are very strongly correlated with the directionof the largest scatter, PC ^ 0,M \\hat{PC}_{0,M} in part because the component ofPC ^ 0,M \\hat{PC}_{0,M} in the velocity dispersion direction, β vel,0 \\beta _{vel, \\; 0},tends to be large.", "The dotted line marks the median value of thecorrelations.Note that the y-axis scale varies with mass measurement method andPC ^ i \\hat{PC}_i,the x-axis does not.For the ensemble of cluster measurements, the largest correlation with $a^\\alpha _0$ , i.e.", "with the coefficient of $\\hat{PC}_{0,M}$ , is for velocity dispersions (in part because most of the scatter is due to velocity dispersions).", "Correlations of $a^\\alpha _0$ with the other mass measurement methods are relatively smaller, and of similar size to each other.", "Taking instead the fraction of the mass scatter vector due to $\\hat{PC}_0$ , i.e., $a^\\alpha _0/\\sqrt{\\sum _i(a^\\alpha _i)^2}$ , the correlations are weaker.", "In addition, weaker correlations arise with the coefficients of $\\hat{PC}_{1,M}$ , and for the direction with the smallest direction of scatter, $\\hat{PC}_{4,M}$ , only $M_{N_{\\rm ph}}$ has a noticeable correlation on average.", "The direction of $\\hat{PC}_{0,M}$ can also be compared to that of $\\vec{M}_{\\rm ave}/M_{\\rm true}$ : the average inner product $\\sim 0.7$ , but the peak is closer to 0.8, and there is a broad range of values.", "(The rest of the vector $\\frac{\\vec{M}_{\\rm ave}}{M_{\\rm true}}$ seems to lie in the $\\hat{PC}_{1,M},\\hat{PC}_{2,M}$ plane).", "That is, the direction of largest mass measured is close to that of largest scatter; presumably this is because this direction has large scatters generally.", "We now compare the above quantities to line of sight dependent cluster properties.", "We will follow this in section § with properties depending on the entire cluster rather than a given line of sight." ], [ "Relation to cluster line of sight properties", "To get more understanding of the PCA decomposition, we compare values of PC quantities along lines of sight to cluster properties along those lines of sight.", "The PCA line of sight dependent properties we consider are the coefficients $a^\\alpha _i$ of the $\\hat{PC}_{i,M}$ , $a^{\\alpha }_i/\\sqrt{\\sum _j (a^{\\alpha }_j)^2}$ (the fraction of scatter in different $\\hat{PC}_i$ directions, $i$ and $j$ each run from 0 to 4), and the total scatter for a given line of sight ($\\sum _i(a^{\\alpha }_i)^2/\\lambda _{i,M}$ ).", "We correlate these with the angle $\\theta _{\\rm obs}$ between the line of sight and six specific physical cluster directions, listed below.", "Previously, for this simulated data set, increased mass scatter was found by observing along certain physical cluster directions by WCS; [78], [23].", "Our extension using PCA includes six physical directions: the long axis $\\hat{\\ell }$ of the cluster, calculated using the dark matter particles in the simulation with a FoF finder as mentioned above, the plane normal containing the most halo mass $\\hat{n}_{\\rm mass}$ , or connected filament mass $\\hat{n}_{\\rm fil}$ centered on the cluster (see §REF and [78] for more details), the direction of the largest subgroup of galaxies $\\hat{r}_{\\rm sub}$ which originated from the same infall host halo (see §REF ), and the velocity direction $\\hat{v}_{\\rm sub}$ of this largest subgroup.We also measured correlations with another direction dependent quantity, the amount of substructure found via the Dressler-Schectman [40] test, as described in [128], [23]; however, the correlations with $\\hat{PC}_{0,M}$ were much weaker with this directional dependent quantity than with the ones reported here.", "Many of these special cluster directions are similar to each other, e.g.", "[64], [128], [23], as expected.", "The correlation of $a^{\\alpha }_i/\\sqrt{\\sum _j(a^{\\alpha }_j)^2}$ (the fraction of the mass scatter in the $\\hat{PC}_{i,M}$ direction, for the line of sight indexed by $\\alpha $ ) with $|\\cos \\theta _{obs}|$ for each observation is shown in Fig.", "REF .", "The medians and the averages of these correlation coefficient distributions are shown in Table REF .", "The largest average correlations of angles with the physical cluster axes are with $\\hat{PC}_{0,M}$ , the direction of the combination of mass scatters that dominates the scatter.", "The average correlations with $\\hat{PC}_{1,M}$ are slightly smaller, for the rest of the $\\hat{PC}_{i,M}$ the average correlations tend to zero (as can be seen, the individual clusters can have larger correlations).", "For $\\hat{PC}_{0,M}$ , the largest correlation is with the direction of the cluster long axis $\\hat{\\ell }$ .", "The next largest signals are with the direction of the mass plane normal $\\hat{n}_{\\rm mass}$ , filament plane normal $\\hat{n}_{\\rm fil}$ , and the direction of the largest substructure $\\hat{r}_{\\rm sub}$ .", "The velocity of the largest substructure $\\hat{v}_{\\rm sub}$ and the direction of the most massive filament $\\hat{r}_{\\rm fil}$ are more weakly correlated.", "That is, scatter dominated by the mass scatter combination in $\\hat{PC}_{0,M}$ tends to occur more often when the direction of observation is more aligned with the long axis of the cluster.", "(As the other cluster directions are not linearly independent, strong correlations with them are possible and seen as well.)", "As most of the scatter occurs along $\\hat{PC}_{0,M}$ and $\\hat{PC}_{1,M}$ , both of which are most correlated with looking along the long axis, it suggests that most of the scatter is due to looking along the long axis $\\hat{\\ell }$ .", "Figure: Correlations betweenthe fraction of the line of sightmass scatter in thePC ^ 0,M \\hat{PC}_{0,M} direction and |cosθ obs ||\\cos \\theta _{obs}|.", "The clusterphysical directions are ℓ ^\\hat{\\ell }, the long axis of the cluster,n ^ mass \\hat{n}_{\\rm mass}, the direction perpendicular to the mass diskwith radius 10 h -1 Mpch^{-1} Mpc and width 3 h -1 Mpch^{-1} Mpc centered on thecluster, containing the majority of the mass in halos, n ^ fil \\hat{n}_{\\rm fil}, similar to n ^ mass \\hat{n}_{\\rm mass} in shape and volume, butoriented to contain the majority of thefilamentary mass ending on the cluster, r ^ fil \\hat{r}_{\\rm fil}, the direction to themost massive halo filament surrounding the cluster, and v ^ sub ,r ^ sub \\hat{v}_{\\rm sub},\\hat{r}_{\\rm sub}, the relative velocity anddirection of the largest galaxy subgroup in the cluster.See text and § for moredetailed definitions.Only the 227 clusters which are filament endpoints are included in thecomparisons concerning filament planes.Vertical dotted lines denote medians, and the averages and medians areshown in Table .The largestcontributions to PC ^ 0 \\hat{PC}_0, for most clusters, seem to come whenobserving along the long axis of the cluster, but there is a widescatter.", "The correlations with the mass and filament planes aremaximized for observations along the planes, i.e.", "perpendicular to thenormal vectors n ^ mass ,n ^ fil \\hat{n}_{\\rm mass}, \\hat{n}_{\\rm fil}, as expected.Table: The median of correlation coefficients betweenthe fraction of the line of sight mass scatter covariance in each PC ^ i,M \\hat{PC}_{i, M} direction and|cosθ obs ||\\cos \\theta _{obs}|.", "The average is shown in the parenthesis.", "The full distribution is shown inFig.", ".The correlation with $a^\\alpha _0$ , the full contribution from $\\hat{PC}_{0,M}$ , rather than the fractional contribution above, was much weaker.", "One other line of sight quantity, $\\sum _i (a^{\\alpha }_i)^2/\\lambda _{i,M}$ (the weighted scatter of $M^\\alpha _{\\rm obs}$ ), also tends to have correlations with special cluster directions as seen in Fig.", "REF .", "The largest fraction of clusters with correlations $> 0.20$ occurs with the long axis.Correlations with the cluster long axis and external filaments or nearby clusters, both possible causes of mass scatters, have been seen in many other works too, e.g.", "[117], [112], [26], [22], [80], [46], [118], [58], [8], [47], [64], [10], [66], [67], [81], [83], [88], [34], [20], [82], correlations with the long axis and mass scatters specifically have been discussed recently in, e.g., [12], [71], [11], [7], [48].", "Figure: Correlations of∑ i (a i α ) 2 /λ i,M \\sum _i (a^{\\alpha }_i)^2/\\lambda _{i,M}(the weighted scatter ofM obs α M^\\alpha _{\\rm obs}) with angle betweenline of sight and various cluster axes, as described in the text andin Fig.", ".Vertical lines are at average value.To summarize this section, each cluster's line of sight mass scatter variations and correlations were analyzed and characterized separately using PCA.", "Both similarities (similar large scatter directions) and differences (raw size of scatter, amount of variation with line of sight) were found.", "The scatter was usually dominated (large $\\lambda _{0,M}/\\sum {\\lambda }$ ) by one combination of mass scatters, which also tended to become an increasing component of the total mass scatter as the angle of observation became more aligned along the cluster's long axis." ], [ "Cluster to cluster variations", "Now we turn to line-of-sight independent properties, PCA related or otherwise.", "The scatters of each cluster are also characterized by several numbers, i.e.", "scalars, which are not dependent upon the line of sight of observation (e.g.", "$\\lambda _0$ ).", "Here we take some characteristic mass scatter scalars for each cluster and compare them with other cluster properties, intrinsic to the cluster or due to its environment, combining to give 24 quantities in total.", "We make use of several quantities obtained previously (WCS; [78], [23]) for this simulation and described in §REF .", "We use correlations rather than covariances, to take out the dimensional dependence.", "After considering properties of the pairwise correlations in §REF , we consider the full ensemble of correlations using PCA in §REF ." ], [ "Cluster quantities", "For each cluster we consider the following (note that mass scatters are for $\\vec{M}_{\\rm obs} = \\vec{M}_{\\rm est} - \\langle \\vec{M}_{\\rm est} \\rangle =\\vec{M}_{\\rm est} -\\vec{M}_{\\rm ave}$ ): $\\Delta M_i \\equiv ({M}_{ave,i}-M_{\\rm true})/M_{\\rm true}$ , observed average mass offsets for each of the five observables considered earlier: red galaxy richness, phase richness, SZ, velocity dispersions and weak lensing respectively.", "Note that these correlations are identical to those of $M_{ave,i}/M_{\\rm true}$ .", "$|\\cos \\theta _{0,minsq}|$ , i.e.", "$|\\hat{PC}_{0,M} \\cdot \\hat{PC}_{0,M,minsq}|$ , the angle between the largest mass scatter direction for each cluster, $\\hat{PC}_{0,M}$ , and the direction $\\hat{PC}_{0,M,minsq}$ , given in Table REF , which minimizes the dot product with $\\hat{PC}_{0,M}$ for all the clusters.", "$\\sum {\\lambda }$ , the sum of the cluster's mass scatters in §REF .", "$\\prod \\lambda $ , the product of the cluster's scatters in §REF .", "$\\frac{ \\lambda _{0,M}}{\\sum {\\lambda }}$ fraction of total (mass scatter) variance along direction of largest variance in §REF .", "$\\lambda _{1,M}$ , variance along the direction with second largest mass scatter variance in §REF .", "$\\lambda _{4,M}$ , variance along the direction with smallest mass scatter variance in §REF .", "$T = \\frac{l_{1}^{2} - l_{2}^{2}}{l_{1}^{2} - l_{3}^{2}}$ , triaxiality, where the $l_1,l_2,l_3$ are the axes of cluster, calculated using the dark matter particles in the FoF halo.", "$S = \\frac{l_{3}}{l_{1}} \\ \\ \\ (l_{1} > l_{2} > l_{3})$ , sphericity.", "$f_{M_{\\rm fplane}}$ , the fractional connected filamentary mass in the local plane around clusters defined in §REF .", "$f_{M_{\\rm hplane}}$ , the fractional halo mass in the local plane around clusters, see §REF .", "$M_{\\rm sphere}$ , the mass in halos above $5 \\times 10^{13}h^{-1}M_\\odot $ within a 10 $h^{-1} Mpc$ radius sphere of the cluster.", "We used the sum of large halo masses in a 10 $h^{-1} Mpc$ radius sphere around the central cluster rather than the total mass because the former is already known to be correlated with the mass of the central halo (e.g., [78]).", "$M_{\\rm true}$ , the cluster (FoF $b=0.168$ ) mass (also called $M$ ).", "$f_{R_{\\rm sub}}$ , the fractional richness of the largest galaxy subgroup (see §REF ).", "$f_{D_{\\rm sub}}$ , the ratio of the distance to the largest galaxy subgroup in the cluster to the length of the longest cluster axis (see §REF ).", "$c_{\\rm vir}$ , the concentration (not scatter from mean concentration of a given mass), from fitting all of the Friends of Friends halo particles to an NFW [77] profile.", "This was found to be very important in the previous studies [61], [103], [42] inspiring this work.", "$t_{1:3}$ , the time of most recent $\\ge $ 1:3 merger (often taken as the threshold for a merger-driven starbust) $t_{1:10}$ , the time of most recent $\\ge $ 1:10 merger (often taken as a threshold for merger-driven AGN feeding) $x_{\\rm off}$ , the distance between the central galaxy position and the average of the galaxy positions.", "This is similar to the “relaxedness” considered by [103], but in that case they use the offset between the most bound particle and the halo center of mass.", "We also considered scaling by $M^{1/3}$ , the results did not change significantly.", "$\\cos \\theta _{\\Delta ,0}$ , the cosine of the angle between $\\hat{PC}_{0}$ and $\\vec{M}_{\\rm ave}$ , i.e., the largest mass scatter direction vs. the average mass offset direction.", "Figure: Correlations between cluster properties:0.5<corr0.5<corr (filled hexagons), 0.35<corr≤0.50.35<corr\\le 0.5 (filled squares), 0.2<corr≤0.350.2< corr \\le 0.35(filled triangles), opposite signs are open versionsof the same symbols,i.e.", "corr<-0.5corr<-0.5 (open hexagon), etc.", "Auto-correlations are notshown.Cluster properties aredescribed insection .A blank space means that a correlation hasabsolute value ≤\\le 0.2.", "The horizontal andvertical lines distinguish the types of cluster properties.", "ΔM\\Delta M labels the five components of ΔM →\\Delta \\vec{M}.", "λ M \\lambda _M refers tomeasurements for that cluster's mass scatter correlations.", "“phys1”refers to cluster properties which are environmental or shaperelated.", "“phys2” refers to cluster properties more related tosubstructure ormerging (concentration, distance to largest substructure, etc.", "),λ 0 \\lambda _0refers to dot product or correlation with PC ^ 0,M \\hat{PC}_{0,M}.Only half of the correlations are shown, as they are symmetricacross thediagonal axis.We thus have a space of dimension $N_{\\rm method}=24$ for the correlation and PCA analysis below.", "Several of these quantities might be expected to be related.", "For instance, sphericity and triaxiality both characterize departures from perfect spheres, but triaxiality measures prolateness and oblateness while sphericity is sensitive to flatness.", "Similarly the sum and product of the eigenvalues probe the size of the largest eigenvalue and in principle how the largest and smallest eigenvalue change together, respectively.", "A priori, it isn't clear which of our large set of quantities have the strongest or most illuminating relations to each other, so we start with a large set." ], [ "Correlations", "Fig.", "REF summarizes the pairwise correlations.", "For simplicity, only correlations which have absolute value $>0.2$ , in six ranges, $\\pm 0.2, \\pm 0.35, \\pm 0.5$ , are shown.", "The correlations of measurements with themselves are omitted.", "Filled dark (open red) symbols are positive (negative) correlations.", "Properties in the list are grouped by type: offsets of cluster average mass measurements from their true mass (“$\\Delta M$ ”, defined in §REF ), PCA related scalars for each cluster from §REF (“$\\lambda _M$ ”), cluster environment or shape (“phys1”), cluster history (“phys2”), and $|\\cos \\theta _{\\Delta ,0}|$ .", "Relations between measurements of the same sort (e.g.", "concentration and time of last merger, etc.)", "can be seen in the diagonal boxes.", "The off-diagonal boxes thus correlate different sorts of cluster properties.", "We concentrate on these.", "The first thing to notice is that many large correlations are visible.", "We start with correlations with $\\Delta \\vec{M}$ , each cluster's five average fractional mass measurement offsets, corresponding to the five measurement methods.", "The different components of $\\Delta \\vec{M}$ often have similar correlations with other quantities.", "In particular, they are all strongly correlated with sphericity, most are anticorrelated with triaxiality and all are anticorrelated with the offset of the average galaxy position.", "All but the weak lensing mass offset are also correlated with the smallest mass variance $\\lambda _{4,M}$ , that is, when the average mass offsets $\\Delta M_i$ are large, the variance in the direction of minimum scatter around the average measured value tends to be large as well: the line of sight averaged scatter often is increasing for clusters with larger mass scatters around these averages.", "There is also a trend of these average mass offsets being anticorrelated with signatures of relaxedness (fractional richness of the largest subgroup, offset of the average galaxy position relative to the galaxy center, etc.).", "Turning to other correlations with the mass scatter variances, an increase in the mass in halos around the cluster tends to be accompanied by an increase many of the measures of mass scatter, i.e.", "properties associated with the $\\lambda _{i,M}$ .", "This might indicate the presence of a supercluster.", "The size of the smallest mass scatter variance $\\lambda _{4,M}$ also seems to increase with increasing fractional halo mass in the plane of the cluster, $f_{M_{hplane}}$ .", "Lastly, triaxiality is correlated with indicators of substructure, (the offset of the galaxy average position from the central galaxy and the fractional richness of the largest galaxy subgroup), while in contrast sphericity is anticorrelated with these, as well with the fractional distance of the largest galaxy subgroup from the cluster center.", "Physically, one might expect a rich subgroup in the cluster to lie along the long axis of the cluster (e.g.", "coming from a filament feeding matter into the cluster) and be correlated with a lengthened (more triaxial) cluster as a result." ], [ "PCA", "By using PCA on these pairwise correlations, more general groupings can be investigated.", "This application of PCA to our sample is an extension of PCA of several cluster properties studied by [61], [103] and by [42] for superclusters, which inspired this study.", "Properties correlated by [61], [103] included virial mass, concentration, age, relaxedness, sphericity, triaxiality, substructure, spin, and environment.", "[61] used 1867 halos from several boxes with masses ranging from $ \\sim 10^{11} h^{-1} M_\\odot $ up and found that the dominant PC vector was most correlated with halo concentration.", "[103] used several boxes with halos with masses $ \\ge 10^{10} h^{-1}M_\\odot $ , and also found halo concentration to be important for clusters, as well as halo mass and degree of relaxedness.For superclusters in SDSS DR7, [42] considered weighted luminosity, volume, diameter, density of the highest density peak of galaxies and the number of galaxies, as well as shape parameters, and used the two largest $\\hat{PC}_i$ to find scaling relations.", "They divided up the sample into two sets of superclusters based upon where they lay in the planes spanned by pairs of $\\hat{PC}_{0},\\hat{PC}_{1}, \\hat{PC}_2$ .", "Our sample is closest to the high mass tail of these samples.", "Figure: Correlation coefficients between the cluster propertieslisted in § and their projection on the first four principal components.Horizontal dotted lines at ±0.4\\pm 0.4 are to guide the eye to largerpositive or negative correlations.The top box is for PC ^ 0 \\hat{PC}_0, the second is for PC ^ 1 \\hat{PC}_1 and soon; the expansion of the PC ^ i \\hat{PC}_i vectors is given by thesecorrelations divided by the λ i \\lambda _i eigenvalues, 5.23, 3.0, 2.07, 1.94respectively for PC ^ 0 ,PC ^ 1 ,PC ^ 2 ,PC ^ 3 \\hat{PC}_0,\\hat{PC}_1,\\hat{PC}_2,\\hat{PC}_3.The 24 PC eigenvaluesSome of the previous analyses consider logarithms of the scalar quantities, we redid the analysis taking logarithms of $\\sum {\\lambda }$ , $\\prod \\lambda $ , $\\lambda _{1,M}$ , $\\lambda _{4,M}$ , $f_{M_{\\rm fplane}}$ , $f_{M_{hplane}}$ , $f_{R{\\rm sub}}$ , $f_{{D_{\\rm sub}}}$ , $c$ , $M_{\\rm sphere}$ , $ M$ and found very similar results for correlations with the first 2 $\\hat{PC}_i$ as below, for $\\hat{PC}_2$ , correlations roughly increased for the merger related quantities, and decreased for the $M_{\\rm sphere}$ and plane environment quantities.", "are $\\lambda _i/\\sum {\\lambda }$ =(0.22, 0.12, 0.09, 0.08, 0.07, 0.05 (3 times), 0.04, 0.03 (4 times), 0.02 (3 times), 0.01 (6 times), $<0.005$ (2 times)).", "Unlike the case of mass scatter, where $\\lambda _{0,M}$ is relatively large, (on average $\\lambda _{1,M}/\\lambda _{0,M}<0.4$ ), here $\\lambda _1/\\lambda _0 \\sim 0.6$ , and $(\\lambda _1,\\lambda _2,\\lambda _3,\\lambda _4) = (5.23,3.00,2.07,1.95)$ , fairly close to each other.", "This makes interpretation less straightforward.", "However, subsets of properties with strong correlations with individual $\\hat{PC}_i$ may indicate these properties change together.", "To better identify relations, one can take these subsets and do PCA on the correlations within this subset alone.", "An example is given below.", "The expansion coefficients of the $\\hat{PC}_i$ for the different measured quantities, the $\\beta _i$ in Eq.", "REF , are given by the correlations divided by the eigenvalues.", "(For example, the expansion coefficients of the $\\hat{PC}_{0}$ in the order listed in §REF are (0.19, 0.26, 0.34, 0.35, 0.30, 0.03, 0.23, 0.16, 0.06, 0.19, 0.22, -0.20, 0.35, 0.09, 0.10, 0.11, -0.11, -0.22, -0.14, 0.05, -0.06, -0.03, -0.33, -0.12).)", "The strongest correlation is between the velocity dispersion mass offset $\\Delta M_{\\rm Vel}$ and $\\hat{PC}_0$ ($\\sim $ 0.8), 9 other properties have correlations with absolute value above or equal to 0.7 with at least one of the $\\hat{PC}_i$ ($i \\le 6$ ).", "Considering smaller correlations, 21/24 of the properties have at least one correlation of absolute value $\\ge 0.5$ with one of the $\\hat{PC}_i$ , $i \\le 11$ , and all properties have at least one correlation $\\ge 0.4$ with at least one $\\hat{PC}_i$ .", "The correlations of the first four principal component vectors with the 24 properties are shown in Fig.", "REF .", "Lines are drawn at $\\pm 0.4$ to guide the eye to the larger correlations.", "The sign of the correlations with the eigenvectors depends upon the overall sign of the eigenvectors, which is arbitrary, but the relative signs of the correlations of the different properties with each eigenvector are not arbitrary.", "We begin with $\\hat{PC}_0$ , the direction of largest variation.", "The $\\Delta M_i$ 's, the sum of the mass scatter variances $\\sum {\\lambda }$ , the individual mass scatter variances $\\lambda _{1, M},\\lambda _{4,M}$ and sphericity $S$ all have large correlations ($\\ge 0.4$ ) with their projections onto $\\hat{PC}_0$ .", "These correlations are in the opposite sense of those of $T$ , $f_{R_{sub}}$ and $x_{off}$ (these latter three might indicate less relaxed halos).", "A relation between large triaxiality and an increased offset in the average galaxy position or the presence a rich subgroup is somewhat intuitive, but the sign of the relation to the average mass measured for the cluster is surprising.", "One interpretation of the relation between $\\Delta M_i$ and $T$ seen both here and in the correlations is that clusters which are more triaxial have fewer lines of sight along the elongated and presumably large measured mass direction, so that fewer lines of sight result in a large $\\vec{M}_{\\rm est}/M_{\\rm true}-1$ .", "See also related discussion in [89].We thank E. Rasia for discussions about this.", "To go further, we considered the subset of properties which correlate strongly with $\\hat{PC}_0$ ($\\Delta M_i$ ,$S$ ,$T$ , $f_{R_{sub}}$ ,$x_{off}$ ,$\\sum \\lambda $ , $\\lambda _{1,M}$ ,$\\lambda _{4,M}$ ) and applied PCA to their intercorrelations.", "The trends with $\\hat{PC}_0$ for these quantities appear with the direction $\\hat{PC}_{0,{\\rm subset}}$ as well.", "In this case, however, $\\lambda _{0,{\\rm subset}}$ is relatively larger, $\\lambda _{0,{\\rm subset}}/\\sum \\lambda = 0.39$ .", "To help in understanding, we took three groups of properties (1) $\\Delta M_i$ (except $\\Delta M_{\\rm WL}$ ); (2) the mass scatter measures $\\sum \\lambda ,\\lambda _{1,M},\\lambda _{4,M}$ ; and (3) $T$ ,$f_{R_{Sub}}$ ,$x_{off}$ , $S$ , $\\Delta M_{\\rm WL}$ .", "The properties in (1) and (2) have large same sign correlations with $\\hat{PC}_0$ .", "For (3), sphericity $S$ and average weak lensing mass offset $\\Delta M_{\\rm WL}$ have this sign correlation as well, while $T$ , $f_{R_{Sub}}$ ,$x_{off}$ have the opposite sign.", "These three groups (roughly) can be seen in correlations with $\\hat{PC}_{1,{\\rm subset}}$ (0.17 of the sum of eigenvalues), but in this case the relative behavior between properties in (2) and (3) reverses (i.e.", "increased mass scatter now has the opposite correlation with $\\hat{PC}_1$ as increased sphericity does), and changes in (1) all have correlations $<0.07$ with $\\hat{PC}_{1,{\\rm subset}}$ .", "(Note that the general orthogonality of the $\\hat{PC}_i$ means relations between properties, i.e.", "relative sizes and signs of correlations, do have to change from from one $\\hat{PC}_i$ vector to another.)", "The correlation strengths change as well, going from $\\hat{PC}_0$ to $\\hat{PC}_1$ , increasing for (2) and decreasing (below 0.4) for (3).", "Considering the trends in broad brush, for a given cluster, large measured average (over lines of sight) mass offsets from true mass, aside from weak lensing, seem to tend to come with high sphericity and large mass measurement scatters, and low triaxiality, $f_{R_{sub}},x_{off}$ .", "That is, when a cluster's mass offsets are very large, there is a tendency for the vector of all its properties to be lying far along $\\hat{PC}_{0,{\\rm subset}}$ , implying trends for the other properties.", "When average mass offsets (besides $\\Delta M_{\\rm WL}$ ) are small, it is not clear whether a given cluster's variations are lying along $\\hat{PC}_{0,{\\rm subset}}$ or $\\hat{PC}_{1,{\\rm subset}}$ or some other direction.", "As the relation between large sphericity (and small triaxiality, $f_{R_{sub}},x_{off}$ ), i.e., (3), and the size of mass measurement scatters, i.e., (2), reverses between these two PCA vectors, small mass offsets makes it difficult to estimate the relation between them (i.e., between (2) and (3)).", "Continuing with the full set of 24 properties, and considering correlations with $\\hat{PC}_1$ , the large sphericity/small triaxiality, (relatively) small richness in biggest subgroup, (relatively) small distance of largest subgroup from center and small offset of average galaxy positions with central galaxy position relation trends are also seen, but more weakly.", "Fluctuations in the direction of $\\hat{PC}_1$ have an increase in $|\\hat{PC}_{0,M} \\cdot \\hat{PC}_{0,M,minsq}|$ tied to a decrease in overall mass scatter ($\\sum \\lambda ,\\prod \\lambda , \\lambda _{1,M},\\lambda _{4,M}$ ) and a decrease in mass in nearby large halos $M_{\\rm sphere}$ .", "This last perhaps indicates that the alignment of $\\hat{PC}_{0,M}$ is better, and the mass scatter smaller, when the cluster is not in a supercluster.", "Low concentration and relative distance of the largest subgroup from the cluster center are accompanied by earlier major mergers for variations in the direction of $\\hat{PC}_2$ .", "The fraction of mass in the filament and mass planes seem to change together for changes along $\\hat{PC}_3$ , along with the amount of mass in nearby large halos.", "Again, these last 3 properties can be studied alone via PCA.", "In this case the associated $\\lambda _{0,{\\rm subset}}/\\sum \\lambda =0.65$ .", "Note this is a different subset of PC vectors than considered earlier, and corresponds to all three quantities, normalized by covariance, changing by about the same amount with the same sign.", "This may imply that clusters which are in a richer environment, that is, with more nearby halos, may have the mass around them distributed in a more planar shape.", "We experimented with a larger set of properties than shown here, including for example the quantities shown in the first column of Fig.", "REF for each cluster, i.e.", "the correlations with projections on $\\hat{PC}_{0,M}$ and physical directions.", "In this case the correlations with $\\hat{PC}_0$ for the other quantities remained essentially as shown in the top line of Fig.", "REF but the $\\hat{PC}_{0,M}$ -long axis projections and the $\\hat{PC}_{0,M}$ -direction of largest subgroup projections had large correlations with $\\hat{PC}_0$ in the same sense as $T$ , cluster triaxiality." ], [ "An ensemble of clusters", "The previous sections considered cluster directional mass scatters and variations in properties, cluster to cluster.", "One can also consider the joint ensemble of mass measurements of all the simulated clusters and all their lines of sight, for all five methods.", "A comparison is interesting between the PC vectors of the individual clusters, for many lines of sight, and those for all the clusters together." ], [ "Trends for all clusters considered together", "We take the union of the estimated masses of all clusters and all lines of sight (aside from those discarded as discussed in §2) and repeat the analysis of §3.", "Now $\\vec{M}_{\\rm obs}/M_{\\rm true}= \\vec{M}_{\\rm est}/M_{\\rm true}- \\langle \\vec{M}_{\\rm est}/M_{\\rm true}\\rangle $ .", "The average $\\langle \\vec{M}_{\\rm est}/M_{\\rm true}\\rangle $ is over all clusters and lines of sight, and for our case is within a few percent of 1.", "The resulting covariance matrix for these mass scatters is shown in Table REF .", "The covariances for the combined sample are larger than the averages for individual clusters, the latter are shown in Table REF in parentheses for comparison, and were plotted individually in Fig.", "REF .", "The increased scatter is not unexpected as it is a combination of scatters from objects which are all themselves scattered around different central values.", "Table: Covariance matrix for full set of measurements M → obs /M true \\vec{M}_{\\rm obs}/M_{\\rm true}, for all clusters, and, in parentheses, the average of the individualcluster covariances inFig. .", "The median values for the theindividual cluster covariances tendto be smaller than the average values.", "The covariances for the full sample tend to be larger than theaverages cluster to cluster.The corresponding PC eigenvalues are $\\lambda _{i,M,total}$ =(0.31, 0.18, 0.094, 0.061, 0.022).", "The direction of largest scatter has $\\lambda _{0,M,total}/\\sum {\\lambda }\\sim $ 0.47, compared to the average of 0.66 for individual clusters shown in Fig.", "REF .", "The total mass scatter variance $\\sum {\\lambda } = 0.67$ , which is relatively high compared to the average for individual clusters; the product $\\prod \\lambda $ is also relatively larger (7$\\times 10^{-6}$ ), as to be expected from the increased covariances in Table REF .", "For the 70 clusters with mass $\\ge 2\\times 10^{14} h^{-1} M_\\odot $ , the total scatter $\\sum {\\lambda }$ goes down (to 0.44 from 0.67 for the sample with $M \\ge 10^{14} h^{-1} M_\\odot $ ), and the direction of the $\\hat{PC}_{0,M,total}$ slightly rotates as seen in Table REF .", "The fraction of variance in $\\lambda _{0,M}$ increases, i.e.", "the direction of the largest scatter has more of the scatter.", "The directions of largest scatters are similar to those for the clusters considered separately (Table REF ), although $\\hat{PC}_{4,M,total}$ has a much smaller SZ component than $\\hat{PC}_{4,M,minsq}$ .", "The correlations with $\\hat{PC}_{0,M,total}$ of the projections of the different mass observables ($M_{N_{\\rm red}}$ , $M_{N_{\\rm ph}}$ , $M_{\\rm SZ}$ , $M_{\\rm Vel}$ , $M_{\\rm WL}$ ) are (0.69, 0.57, 0.57, 0.79, 0.57), similar to the median of their individual cluster counterparts, (0.69, 0.51, 0.48, 0.87, 0.47), shown in Fig.", "REF .", "The correlations are fairly large for the other $\\hat{PC}_i$ components, but these by definition contribute less to the total mass scatter.", "$\\hat{PC}_{0,M,total}$ and $\\Delta \\vec{M}$ are $23^\\circ $ apart, similar to their counterparts for the individual clusters.", "Again, just as for individual clusters, the closer one is to the long axis of the cluster, the larger the fraction of scatter due to $\\hat{PC}_{0,M,total}$ ($\\sim $ 0.4 correlated, see, e.g.", "Fig.", "REF for the individual cluster distribution).", "The next leading correlations of $\\hat{PC}_{0,M,total}$ are with the position of the largest subgroup and the direction perpendicular to the cluster's dominant filamentary plane or mass plane.", "These trends were very close to those found for clusters individually.", "These relations are still however based on the estimated mass as a function of true mass, and thus not immediately applicable to observational samples, as we now discuss." ], [ "Future extensions to observational samples", "One can also consider using some variant of our PCA analysis for an observational sample, which would have some range of true cluster masses and some observations, as we do in our box.", "However, our implementation of PCA doesn't directly carry over, and our measurement sample in hand is not appropriate.", "We discuss these limitations and possible ways forward here.", "Observations are concerned with $M_{\\rm true}$ as a function of estimated mass, $M_{\\rm true}(M_{\\rm est})$ .", "An observer does not have $M_{\\rm true}$ .", "We have instead been calculating $M_{\\rm est}(M_{\\rm true})$ and in addition dividing by $M_{\\rm true}$ .", "Applying PCA in a way useful to observations (so that one can take 5 methods to measure mass of one cluster and compare to the PC vectors calculated in simulation) requires PC vectors calculated, from simulation, for a representative range of $M_{\\rm est}$ and $M_{\\rm true}$ , and a proxy for the unobservable $M_{\\rm true}$ .Using no proxy for $M_{\\rm true}$ , i.e.", "doing PCA on $M_{\\rm est}$ alone, differently weights the scatter of high and low mass clusters.", "In our sample, caveats below, most of the scatter then corresponds to all mass estimates increasing or decreasing together in equal amounts.", "Projections on this combination of mass scatters are weakly correlated with observations along the long axis of the cluster, and less correlated with other directions.", "As a proxy for $M_{\\rm true}$ , one possibility is the likelihood mass.", "This would come from simulations, which are already required to calibrate covariances and offsets.", "Another possible mass proxy uses the principal component vectors directly, in principle encoding similar information.", "That is, one has from the definition of PC vectors, $\\frac{\\vec{M}_{\\rm obs}^\\alpha }{M_{\\rm true}} =\\frac{\\vec{M}_{\\rm est}^\\alpha }{M_{\\rm true}}-\\left\\langle \\frac{\\vec{M}_{\\rm est}}{M_{\\rm true}}\\right\\rangle = \\sum _ia^\\alpha _i \\hat{PC}_{i,M} \\; .$ The quantities aside from $M_{\\rm true}$ are either measured , e.g., $\\vec{M}_{\\rm est}^\\alpha $ , or calculated from simulations, e.g., $\\langle \\vec{M}_{\\rm est}/M_{\\rm true}\\rangle , \\hat{PC}_{i,M}$ .", "In particular, we have $\\hat{PC}_{4,M,total} \\cdot \\left(\\frac{\\vec{M}_{\\rm est}^\\alpha }{M_{\\rm true}}-\\left\\langle \\frac{\\vec{M}_{\\rm est}}{M_{\\rm true}}\\right\\rangle \\right) =a_4^\\alpha $ but the variance of $a_4$ is given by $\\lambda _{4,M,total}$ .", "If $\\lambda _{4,M,total}$ is very small then we can try the approximation $a_4 \\sim 0$ , so that projecting on $\\hat{PC}_{4,M,total}$ gives the approximationAn extreme and unphysical limit of this case would be if one measurement method had tiny scatter and no correlation with the other measurement methods.", "In this case $\\hat{PC}_{4,M,total}$ would be proportional to a measurement via this one method, and the last equation would basically give that the estimated mass in this method is the true mass; up to any overall biases that might exist, an unbiased mass estimator would make the denominator 1.", "$M_4 \\sim \\frac{ \\hat{PC}_{4,M,total} \\cdot \\vec{M}^\\alpha _{\\rm est}}{\\hat{PC}_{4,M,total} \\cdot \\langle \\frac{\\vec{M}_{\\rm est}}{M_{\\rm true}}\\rangle }\\; .$ The general use of this approach depends upon the size of $\\lambda _{4,M,total}$ for the system; the smaller $\\lambda _{4,M,total}$ is, the better it appears this approximation should work.", "As there tend to be some minus signs in $\\hat{PC}_4$ , some catastrophic failures will occur where $M_4 \\ll M_{\\rm est,i}$ , for all $i$ .", "However, the relevance of testing these possible $M_{\\rm true}$ proxies on our sample seems limited.", "In particular, the $P(M_{\\rm est}|M_{\\rm true})$ for our sample (all $M \\ge 10^{14} h^{-1}M_\\odot $ halos observed along $\\sim 96$ lines of sight each) is not representative of any expected observed sample, nor is our $P(M_{\\rm true})$ , which is required for likelihoods.", "Observing our few high mass clusters along many lines of sight is not a good approximation to observing many clusters along one line of sight: the few high mass clusters observed along many lines of sight in particular do not well sample the realistic population of high mass clusters.", "There are no clusters appearing fewer than $\\sim 96$ times; in particular, rare high mass clusters which would be expected from the number of lower mass clusters “present” are missing.", "Halos with the same $M_{\\rm est}$ (the only observable) but a lower $M_{\\rm true}$ will also occur in an observational sample, and might contribute differently to the scatter as well.", "It would also be important to include the neglected, in our simulation, line of sight larger scale scatter for SZ and weak lensing and the systematics mentioned above.", "These would be very interesting directions to pursue in future work.", "In summary, in this section we considered all the clusters in the box, along all lines of sight, to see how cluster-to-cluster variation altered mass scatter relations found in earlier sections for individual clusters; the trends remained but changed in strength.", "In particular, the dominant mass scatter combination, similar in form to that for many of the individual clusters, still seems to be more prevalent when looking down the long axis of the cluster or perpendicular to the mass or filament plane of the cluster.", "In the second subsection, we mentioned two possible methods for extending our analysis which do not require prior knowledge of $M_{\\rm true}$ , replacing it with the likelihood mass or a mass derived from $\\hat{PC}_4$ .", "These would be interesting to apply to an observational sample, but would need a very closely matched simulation.", "It would be interesting to see if the resulting PC vectors and projections on them by estimated mass measurement methods have correlations with cluster orientation or perhaps $M_{\\rm proxy}/M_{\\rm true}$ ." ], [ "Outliers", "We have focussed on general trends above, but not all clusters (or all lines of sight) fell on the general trends.", "We searched for properties of outliers or tails in the distributions of the various quantities related to $\\lambda _{i,M}$ , outliers in average cluster measured mass vs. true cluster mass, clusters with different maximum or minimum covariance mass measurement method pairs than the majority of clusters, and clusters where $\\hat{PC}_{0,M}$ had at least one opposite sign coefficient with absolute value $\\ge 0.1$ (i.e.", "largest direction of scatter not corresponding to all mass scatters increasing together, mentioned earlier).", "The various outliers did not seem to follow any pattern.", "Some outliers were common to a cluster, e.g.", "sometimes a large $\\lambda _{4,M}$ outlier occurred with a large $\\prod \\lambda $ , as might be expected, as $\\lambda _{4,M}$ is the smallest of the $\\lambda _{i,M}$ .", "Some of the clusters which had $\\hat{PC}_{0,M}$ not aligned with the most likely $\\hat{PC}_{0,M,minsq}$ had much larger contributions to mass scatter from Compton decrement than the average cluster, often due to close massive halos, and $\\lambda _{0,M}/\\sum {\\lambda }$ tending to be smaller than for the usual cluster.", "The clusters having different correlations of lines-of-sight properties with mass measurement methods don't seem to have a clear relation to the outliers in mass scatter properties." ], [ "Summary and discussion", "The scatters between estimated and true cluster masses, for different observational methods, are often correlated.", "Understanding these correlations is becoming more and more important as reliance on multiwavelength measurements increases.", "For instance, correlations and covariances in scatters affect both error estimates in multiple measurements of individual clusters and produce a bias in measurements of stacked objects (see [93], [108], [128], [3] for detailed discussion).", "(Using the full covariance matrices has been done in some cluster analyses, e.g.", "by [91], [70], [13].)", "We characterized the scatter and correlations of clusters in two ways.", "We started by considering clusters individually to identify mass scatter properties due to line of sight effects for 243 clusters in an N-body simulation, each along $\\sim 96$ lines of sight.", "We used five observational mass proxies: red galaxy richness, phase space galaxy richness, Sunyaev-Zel'dovich flux, velocity dispersions and weak lensing.", "It would also be very interesting to include X-ray observation as well, but our attempt at a proxy, based on assigning fractional X-ray flux to cluster galaxy subgroups, was not illuminating.", "These are employed to find cluster masses in current and upcoming large scale cluster surveys.", "For each cluster, we characterized the “shape” and “volume” of the mass scatters of $M_{\\rm est}$ calculated as a function of $M_{\\rm true}$ , using PCA, or Principal Component Analysis, to obtain a set of non-covariant measurements.", "Most clusters had one combination of observational mass scatters contributing the majority of the mass scatter, and this combination was similar for many of the clusters, i.e.", "they had a similar largest principal component $\\hat{PC}_{0,M}$ .", "This scatter combination was a larger fraction of the total line of sight mass scatter when the cluster was observed along the long axis of the cluster.", "Weaker relations with observations along other cluster intrinsic and environmental axes were seen.", "Identifying the long axis of course requires the clusters to have non-spherical shapes.", "In our case the cluster member dark matter particles were determined using the FoF finder with linking length $b=0.168$ .", "Individual cluster mass scatter properties due to line of sight effects were then compared to several intrinsic and environmental cluster properties, including triaxiality, planarity of filament or halo mass in the immediate neighborhood of the cluster and relative richness of largest galaxy subgroup within the cluster.", "For example, pairwise correlations, and their combined effects using PCA, showed that clusters with average mass measurements (over lines of sight) which are large tend to also have large mass scatter around their average, relatively high sphericity and small triaxiality, richness in the largest subgroup, and offset of the galaxy position average from the cluster center.", "Relations were also seen for other quantities such as concentration, recent major merger time and fraction of halo mass near the cluster within a 3 $h^{-1} Mpc$ plane.", "Finally, instead of considering each cluster individually, we considered the sample of all clusters and all lines of sight together, and found that most of the trends for the analysis of $\\vec{M}_{\\rm est}/M_{\\rm true}$ remained, albeit at different strengths.", "The projection on the direction of largest scatter was more weakly correlated with the observation angle relative to the cluster long axis.", "It is interesting to think about applying these methods directly to observation.", "One would need more information, in particular estimates of $M_{\\rm true}(M_{\\rm est})$ , rather than the opposite which we have here.", "This requires calibrations from simulations which better sample an expected observational sample at the high mass end, and which also include estimated masses from lower mass halos (as well as faithfully reproducing the observational systematics and selection function).", "Such a simulation would provide correlations and covariances and predicted likelihood masses $M_{\\rm like}$ (and a variant, $M_4$ considered above, based on a narrow direction of scatter in our sample).", "PCA could then be applied to $M_{\\rm est}/M_{\\rm like}$ or $M_{\\rm est}/M_4$ , rather than $M_{\\rm est}/M_{\\rm true}$ as we did here.", "It would be interesting to see if these PC vectors also had relations to cluster orientation such as we found, or perhaps other quantities such as $M_{\\rm like}/M_{\\rm true}$ .", "They might also give some idea of which follow up mass measurement methods would together provide the most constraining power.", "One could for example identify the measurement method with potentially the least covariance with measurements already in hand.", "More generally one could design a combination of measurement methods with smaller covariant scatter (and hopefully smaller scatter as well) using calculated PC vectors as a guide, if the simulations are faithful enough.", "It is a major challenge to accurately capture the systematics and selection function of observational surveys with numerical simulations.", "Another technical issue is to improve estimates of the correlations and covariances, so that sets of inconsistent measurements can be more easily recognized.", "It would be very interesting to do such analyses on a larger box, and/or with other measurement methods." ], [ "Acknowledgements", "YN thanks E. Rasia for discussions and the Essential Cosmology for the Next Generation school for the opportunity to present results of this work.", "She was supported in part by NSF.", "JDC thanks A. Ross for suggestions and M. White for numerous helpful discussions and suggestions.", "We both thank Z. Lukic, E. Rozo and the anonymous referee for extremely helpful criticisms and comments on the draft." ] ]

[ [ "Measurements of the top quark mass at the tevatron" ], [ "Abstract The mass of the top quark (\\mtop) is a fundamental parameter of the standard model (SM).", "Currently, its most precise measurements are performed by the CDF and D0 collaborations at the Fermilab Tevatron $p\\bar p$ collider at a centre-of-mass energy of $\\sqrt s=1.96 \\TeV$.", "We review the most recent of those measurements, performed on data samples of up to 8.7 \\fb\\ of integrated luminosity.", "The Tevatron combination using up to 5.8 fb$^{-1}$ of data results in a preliminary world average top quark mass of $m_{\\rm top} = 173.2 \\pm 0.9$ GeV.", "This corresponds to a relative precision of about 0.54%.", "We conclude with an outlook of anticipated precision the final measurement of \\mtop at the Tevatron." ], [ "Introduction", "The pair-production of the top quark was discovered in 1995 by the CDF and D0 experiments [1] at the Fermilab Tevatron proton-antiproton collider.", "Observation of the electroweak production of single top quarks was presented only two years ago [2].", "The large top quark mass and the resulting Yukawa coupling of almost unity indicates that the top quark could play a crucial role in electroweak symmetry breaking.", "Precise measurements of the properties of the top quark provide a crucial test of the consistency of the SM and could hint at physics beyond the SM.", "In the following, we review measurements of the top quark mass at the Tevatron, which is a fundamental parameter of the SM.", "Its precise knowledge, together with the mass of the $W$  boson ($m_W$ ), provides an important constraint on the mass of the postulated SM Higgs boson.", "This is illustrated in the $m_{\\rm top}$ ,$m_W$ plane in Fig.", "REF , which includes the recent, most precise measurements of $m_W$  [3].", "A detailed review of measurements of the top quark mass is provided in Ref. [4].", "Recent measurements of properties of the top quark other than $m_{\\rm top}$ at the Tevatron are reviewed in Ref. [5].", "The full listing of top quark measurements at the Tevatron is available at public web pages [6], [7].", "At the Tevatron, top quarks are mostly produced in pairs via the strong interaction.", "By the end of Tevatron operation, about 10 fb$^{-1}$ of integrated luminosity per experiment were recorded by CDF and DØ, which corresponds to about 80k produced $t\\bar{t}$ pairs.", "In the framework of the SM, the top quark decays to a $W$  boson and a $b$  quark nearly 100% of the time, resulting in a $W^+W^-b\\bar{b}$ final state from top quark pair production.", "Thus, $t\\bar{t}$ events are classified according to the $W$ boson decay channels as “dileptonic”, “all–jets”, or “lepton+jets”.", "More details on the channels and their experimental challenges can be found in Ref.", "[8], while the electroweak production of single top quarks is reviewed in Ref. [9].", "Figure: (a) The constraint on mass of the SM Higgs boson from direct m top m_{\\rm top} and m W m_W measurements in the m top m_{\\rm top},m W m_W plane  9 ^9.", "The red ellipsis indicates the 68% CL contour.", "(b) The anticipated precision on m top m_{\\rm top} measurements at D0 and the Tevatron combination versus integrated luminosity." ], [ "Direct measurements of the top quark mass in $\\ell +{\\rm jets}$ final states", "D0's most precise measurement of $m_{\\rm top}$ is performed in $\\ell +4{\\,\\rm jets}$ final state using the so-called matrix element (ME) method in 3.6 ${\\rm fb}^{-1}$ of data [11].", "This technique was pioneered by DØ in Run I of the Tevatron [12], and it calculates the probability that a given event, characterised by a set of measured observables $x$ , comes from the $t\\bar{t}$ production given an $m_{\\rm top}$ hypothesis, or from a background process: $\\mathcal {P}_{\\rm evt}(x)\\propto f\\mathcal {P}_{\\rm sig}(x,m_{\\rm top})+(1-f)\\mathcal {P}_{\\rm bgr}$ .", "The dependence on $m_{\\rm top}$ is explicitly introduced by calculating $\\mathcal {P}_{\\rm sig}$ using the differential cross section ${\\rm d}\\sigma (y,m_{\\rm top})\\propto |\\mathcal {M}_{t\\bar{t}}|^2(m_{\\rm top})$ , where $\\mathcal {M}_{t\\bar{t}}$ is the leading order (LO) matrix element for $t\\bar{t}$ production: $\\mathcal {P}_{\\rm sig}(x,m_{\\rm top},k_{\\rm JES}) = \\frac{1}{\\sigma _{t\\bar{t}}^{\\rm observed}}\\cdot \\int W(x,y,k_{\\rm JES})~{\\rm d}\\sigma (y,m_{\\rm top})\\,.$ Since ${\\rm d}\\sigma (y,m_{\\rm top})$ is defined for a set of parton-level observables $y$ , the transfer function $W(x,y,k_{\\rm JES})$ is used to map them to the reconstruction-level set $x$ .", "This accounts for detector resolutions and acceptance cuts, and introduces explicitly the dependence on the jet energy scale (JES) via an overall scaling factor $k_{\\rm JES}$ .", "The uncertainty on the JES, which is almost fully correlated with $m_{\\rm top}$ , is around 2% or larger.", "Therefore, an in situ calibration is performed by requiring that the mass of the dijet system assigned to the parton pair from the hadronically decaying $W$ boson be $m_{jj}=80.4~\\textnormal {GeV}$ .", "Thus, $m_{\\rm top}$ and $k_{\\rm JES}$ are extracted simultaneously.", "This reduces the uncertainty from the JES to about 0.5%, decreasing with the number of selected $t\\bar{t}$ events.", "The measurement is performed in events with four jets, resulting in 24 possible jet-parton assignments.", "All 24 assignments are summed over, weighted according to the consistency of a given assignment with the $b$ -tagging information.", "$\\mathcal {P}_{\\rm bgr}$ is calculated using the VECBOS matrix element for $W+4$  jets production.", "Generally, the ME technique offers a superior statistical sensitivity as it uses the full topological and kinematic information in the event in form of 4-vectors.", "The drawback of this method is the high computational demand.", "D0 measures $m_{\\rm top}=174.9 \\pm 0.8~({\\rm stat}) \\pm 0.8~({\\rm JES}) \\pm 1.0~({\\rm syst})~\\textnormal {GeV}$ , corresponding to a relative uncertainty of 0.9%.", "The dominant systematic uncertainties are from modeling of underlying event activity and hadronisation, as well as the colour reconnection effects.", "On the detector modeling side, diffential uncertainties on the JES which are compatible with the overall $k_{\\rm JES}$ value from in situ calibration, and the difference between the JES for light and b-quark jets are dominant.", "This picture is representative for all $m_{\\rm top}$ measurements in $\\ell +{\\rm jets}$ final states shown here.", "CDF employs the ME technique similar to that used at D0 to measure $m_{\\rm top}$ on a dataset corresponding to 5.6 ${\\rm fb}^{-1}$ and finds $m_{\\rm top}=173.0 \\pm 0.7~({\\rm stat}) \\pm 0.6~({\\rm JES}) \\pm 0.9~({\\rm syst})~\\textnormal {GeV}$  [13].", "Most notable differences from the D0 measurement are: (i) background events present in the data sample are accounted for on average rather than on an event-by-event basis using a likelihood based on a neural network output, (ii) the contribution of “mismeasured” signal events, where one of the jets cannot be matched to a parton, is reduced with a cut on the aforementioned likelihood.", "Currently, the world's best single measurement of $m_{\\rm top}$ is performed by CDF in $\\ell +{\\rm jets}$ final states using the so-called template method to analyse the full dataset of 8.7 ${\\rm fb}^{-1}$  [14].", "The basic idea of the template method is to construct “templates”, i.e.", "distributions in a set of variables $x$ , which are sensitive to $m_{\\rm top}$ , for different mass hypotheses, and extract $m_{\\rm top}$ by matching them to the distribution found in data, e.g.", "via a maximum likelihood fit.", "CDF minimises a $\\chi ^2$ -like function to kinematically reconstruct the event for jet-parton assignments consistent with the $b$ -tagging information.", "To extract $m_{\\rm top}$ and calibrate the JES in-situ, three-dimensional templates are defined in the fitted $m_{\\rm top}$ of the best jet-parton assignment, the fitted $m_{\\rm top}$ of the second-best assignment, and the fitted invariant mass of the dijet system from the hadronically decaying $W$ boson.", "CDF finds $m_{\\rm top}=172.9~ \\pm 0.7~({\\rm stat}) \\pm 0.8~({\\rm syst})~\\textnormal {GeV}$ ." ], [ "Direct measurement of the top quark mass in all-hadronic final states", "The third most statistically significant contribution to the current Tevatron average of $m_{\\rm top}$ comes from a measurement in $6\\le N_{\\rm jets}\\le 8$ final states by CDF using 5.8 ${\\rm fb}^{-1}$ of data [15].", "The main challenge is the high level of the background contribution from QCD multijet production: the $S:B$ ratio is about $1:1200$ after a multijet trigger requirement.", "Therefore, a discrimination variable $\\mathcal {D}_{\\rm NN}$ is constructed with a multilayered neural network (NN).", "Beyond typical kinematic and topological variables, also jet shape variables which provide discrimination between quark and gluon jets, are used as inputs.", "To enhance the purity of the sample and to reduce the number of combinatoric possibilities, $b$ tagging is applied.", "For each jet–parton assignment, a $\\chi ^2$ is constructed which accounts for: the consistency of the two dijet pairs with the reconstructed $m_W$ , the consistency of the $jjb$ combinations with the reconstructed $m_{\\rm top}$ , and the consistency of the individual fitted jet momenta with the measured ones, within experimental resolutions.", "The final sample for top mass extraction is defined by $\\mathcal {D}_{\\rm NN}>0.97~(0.84)$ for events with 1 ($\\ge 2$ ) $b$  tags, yielding a signal to background ratio of $1:3~(1:1)$ .", "The measured value is $m_{\\rm top}=172.5~ \\pm 1.4~({\\rm stat}) \\pm 1.4~({\\rm syst})~\\textnormal {GeV}$ .", "Beyond systematic uncertainties relevant in $\\ell +{\\rm jets}$ final states, potential biases from the data-driven background model pose a notable contribution to the total uncertainty." ], [ "Direct measurement of the top quark mass in dilepton final states", "The world's most precise measurement of $m_{\\rm top}$ in dilepton final states is performed by D0 using 5.4 ${\\rm fb}^{-1}$ of data [16].", "Leaving $m_{\\rm top}$ as a free parameter, dilepton final states are kinematically underconstrained by one degree of freedom, and the so-called neutrino weighting algorithm is applied for kinematic reconstruction.", "It postulates distributions in rapidities of the neutrino and the antineutrino, and calculates a weight, which depends on the consistency of the reconstructed $\\vec{p_{\\rm T}}^{\\,\\nu \\bar{\\nu }}\\equiv \\vec{p_{\\rm T}}^{\\,\\nu }+\\vec{p_{\\rm T}}^{\\,\\bar{\\nu }}$ with the measured missing transverse momentum $/\\!\\!\\!\\!", "{p}_{\\rm T}$ vector, versus $m_{\\rm top}$ .", "D0 uses the first and second moment of this weight distribution to define templates and extract $m_{\\rm top}$ .", "To reduce the systematic uncertainty, the in situ JES calibration in $\\ell +{\\rm jets}$ final states [11] is applied, accounting for differences in jet multiplicity, luminosity, and detector ageing.", "After calibration and all corrections, $m_{\\rm top}=174.0~ \\pm 2.4~({\\rm stat}) \\pm 1.4~({\\rm syst})~\\textnormal {GeV}$ is found." ], [ "Measurement of $m_{\\rm top}$ from the {{formula:3e228c93-50d6-4bdc-bca6-51ad03b26b6b}} production cross-section", "The $t\\bar{t}$ production cross section ($\\sigma _{t\\bar{t}}$ ) is correlated to $m_{\\rm top}$ .", "This can be used to extract $m_{\\rm top}$ by comparing the measured $\\sigma _{t\\bar{t}}$ to the most complete to–date, fully inclusive theoretical predictions, assuming the validity of the SM.", "Such calcualtions offer the advantage of using mass definitions in well-defined renormalisation schemes like $m_{\\rm top}^{\\overline{\\rm MS}}$ or $m_{\\rm top}^{\\rm pole}$ .", "In contrast, the main methods using kinematic fits utilise the mass definition in MC generators $m_{\\rm top}^{\\rm MC}$ , which cannot be translated into $m_{\\rm top}^{\\overline{\\rm MS}}$ or $m_{\\rm top}^{\\rm pole}$ in a straightforward way.", "D0 uses 5.3 ${\\rm fb}^{-1}$ of data to measure $\\sigma _{t\\bar{t}}$ and extracts $m_{\\rm top}$  [17] using theoretical calculations for $\\sigma _{t\\bar{t}}$ like the next-to-leading order (NLO) calculation with next-to-leading logarithmic (NLL) terms resummed to all orders [18], an approximate NNLO calculation [19], and others.", "For this, a correction is derived to account for the weak dependence of measured $\\sigma _{t\\bar{t}}$ on $m_{\\rm top}^{\\rm MC}$ .", "The results for $m_{\\rm top}^{\\rm pole}$ are presented in Fig.", "REF , and can be summarised as follows: $m_{\\rm top}^{\\rm pole}=163.0^{+5.1}_{-4.6}~\\textnormal {GeV}$ and $m_{\\rm top}^{\\rm pole}=167.5^{+5.2}_{-4.7}~\\textnormal {GeV}$ for Ref.", "[18] and [19], respectively." ], [ "Measurements of the mass difference between the $t$ and {{formula:c856b47e-285b-4f1a-b88e-5db26b3de75e}} quarks", "The invariance under $\\mathcal {CPT}$ transformations is a fundamental property of the SM.", "$m_{\\rm particle}\\ne m_{\\rm antiparticle}$ would constitute a violation of $\\mathcal {CPT}$ , and has been tested extensively in the charged lepton sector.", "Given its short decay time, the top quark offers a possiblity to test $m_t=m_{\\bar{t}}$ at the percent level, which is unique in the quark sector.", "D0 applies the ME technique to measure $m_t$ and $m_{\\bar{t}}$ directly and independently using 3.6 ${\\rm fb}^{-1}$ of data, and finds $\\Delta m\\equiv m_t-m_{\\bar{t}}=0.8\\pm 1.8~\\textnormal {GeV}$  [20], in agreement with the SM prediction.", "The results are illustrated in Fig.", "REF .", "With 0.5 $\\textnormal {GeV}$ , the systematic uncertainty on $\\Delta m$ is much smaller than that on $m_{\\rm top}$ due to cancellations in the difference, and is dominated by the uncertainty on the difference in calorimeter response to $b$ and $\\bar{b}$ quark jets.", "CDF uses a template-based method and a kinematic reconstruction similar to that in Ref.", "[14] to measure $\\Delta m$ directly given the constraint $\\frac{m_t+m_{\\bar{t}}}{2}\\equiv 172.5~\\textnormal {GeV}$ from 8.7 ${\\rm fb}^{-1}$ of data, and finds $\\Delta m=-2.0\\pm 1.3~\\textnormal {GeV}$  [21].", "Figure: (a) σ tt ¯ \\sigma _{t\\bar{t}} measured by D0 using 5.3  fb -1 {\\rm fb}^{-1} (black line) and theoretical NLO+NNLL  17 ^{17} (green solid line) and approximate NNLO  18 ^{18} (red solid line) predictions as a function of m top pole m_{\\rm top}^{\\rm pole}.", "The experimental acceptance correction assumes m top MC =m top pole m_{\\rm top}^{\\rm MC}=m_{\\rm top}^{\\rm pole}.", "The gray band corresponds to the total uncertainty on measured σ tt ¯ \\sigma _{t\\bar{t}}.", "The dashed lines indicate theoretical uncertainties from the choice of scales and parton distribution functions.", "(b) m t m_t and m t ¯ m_{\\bar{t}} measured by D0 directly and independently using 3.6  fb -1 {\\rm fb}^{-1}in e+ jets e+{\\rm jets} final states.", "The solid, dashed, and dash-dotted lines represent the 1, 2, and 3 SD contours.", "(c) same as (b) but for μ+ jets \\mu +{\\rm jets}." ], [ "Tevatron combination and outlook", "Currently, the world's most precise measurements of $m_{\\rm top}$ are performed by CDF and D0 collaborations in $\\ell +{\\rm jets}$ final states.", "The preliminary Tevatron combination using up to 5.8 fb$^{-1}$ of data results in $m_{\\rm top} = 173.2 \\pm 0.9$  GeV [22], corresponding to a relative uncertainty of 0.54%.", "With about 10.5 ${\\rm fb}^{-1}$ recorded, the precision on $m_{\\rm top}$ is expected to further improve, especially at D0, where only 3.6 ${\\rm fb}^{-1}$ are used in the flagship measurement in $\\ell +{\\rm jets}$ final states.", "This applies not only to the statistical uncertainty, but also to several systematic uncertainties due to the limited size of calibration samples, like e.g.", "some components of the JES.", "Moreover, efforts are underway to better understand systematic uncertainties from the modeling of $t\\bar{t}$ signal, in particular the dominating uncertainty from different hadronisation and underlying event models.", "We look forward to exciting updates of $m_{\\rm top}$ measurements presented here.", "With uncertainties approaching $\\mathcal {O}(\\textnormal {GeV})$ at the LHC [23], we strongly advocate to start preparations towards the first world-wide combination of the measurements of the top quark mass including ATLAS and CMS results." ], [ "Acknowledgments", "I would like to thank my collaborators from the CDF and D0 experiments for their help in preparing this article.", "I also thank the staffs at Fermilab and collaborating institutions, as well as the CDF and D0 funding agencies." ] ]

[ [ "Ageing effects in single particle trajectory averages" ], [ "Abstract We study time averages of single particle trajectories in scale free anomalous diffusion processes, in which the measurement starts at some time t_a>0 after initiation of the process at the time origin, t=0.", "Using ageing renewal theory we show that for such non-stationary processes a large class of observables are affected by a unique ageing function, which is independent of boundary conditions or the external forces.", "We quantify the weakly non-ergodic nature of this process in terms of the distribution of time averages and the ergodicity breaking parameter which both explicitly depend on the ageing time t_a.", "Consequences for the interpretation of single particle tracking data are discussed." ], [ "Ageing effects in single particle trajectory averages Johannes Schulz Physics Department T30g, Technical University of Munich, 85747 Garching, Germany Eli Barkai Department of Physics, Bar Ilan University, Ramat-Gan 52900, Israel Ralf Metzler Institute for Physics & Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany Physics Department, Tampere University of Technology, FI-33101 Tampere, Finland We study time averages of single particle trajectories in scale free anomalous diffusion processes, in which the measurement starts at some time $t_a>0$ after initiation of the process at the time origin, $t=0$ .", "Using ageing renewal theory we show that for such non-stationary processes a large class of observables are affected by a unique ageing function, which is independent of boundary conditions or the external forces.", "We quantify the weakly non-ergodic nature of this process in terms of the distribution of time averages and the ergodicity breaking parameter which both explicitly depend on the ageing time $t_a$ .", "Consequences for the interpretation of single particle tracking data are discussed.", "87.10.Mn,02.50.-r,05.40.-a,05.10.Gg Ergodicity in the Boltzmann sense, the equivalence of sufficiently long time and ensemble averages of a physical observable, and time translational invariance are hallmarks of many classical systems.", "In contrast, ergodicity violation and ageing effects are found in the dynamics of various complex systems including glasses [1], blinking quantum dots [2], weakly chaotic maps [3], and single particle tracking experiments in living biological cells [4].", "As originally pointed out by Bouchaud [1], when the lifetimes $t$ of states of a physical system are power-law distributed, $\\psi (t)\\simeq \\tau ^{\\alpha }/t^{1+\\alpha }$ ($0<\\alpha <1$ ), with an infinite mean sojourn time $\\langle t\\rangle $ , both weak ergodicity breaking and ageing in the following sense occur: time averages of physical observables remain random even in the limit of long measurement times and differ from the corresponding ensemble averages [5], [7], [6], [8], [10], [9], [11], [12], [13], [14].", "Furthermore ensemble averaged correlation functions of observables, taken at two time instants $t_2$ and $t_1$ , are not solely functions of the time difference $|t_2-t_1|$ .", "With this breakdown of stationarity, the statistical properties of such systems are no longer time translation invariant.", "Consider, for example, a particle undergoing a random walk in a random environment, such that the particle hops a finite distance to the left or right with equal probability, but sojourn times between jumps are distributed like $\\psi (t)$ .", "Such a model, called the continuous time random walk model (CTRW), was introduced by Scher and Montroll in the context of charge carriers in disordered systems [15].", "Its intriguing statistical properties become apparent when studying time averages, like the time averaged mean squared displacement (TAMSD), which is commonly used to analyze the diffusive properties measured in single particle tracking assays.", "Assume that the process starts at time $t=0$ .", "From a single trajectory $x(t)$ , observed in the time interval $(t_a,t_a+T)$ , the TAMSD is typically determined through the definition $\\overline{\\delta ^2(\\Delta ;t_a,T)}=\\int _{t_a}^{t_a+T-\\Delta }\\frac{\\left[x(t+\\Delta )-x(t)\\right]^2}{T-\\Delta }dt,$ with the lag time $\\Delta $ and the measurement time $T$ .", "For Brownian motion, in the limit of large $T$ , we obtain the expected behavior, namely, $\\overline{\\delta ^2}\\rightarrow 2K_1\\Delta $ .", "In this case, the process does not age: the result does not depend on the choice of $t_a$ .", "Moreover Brownian motion is ergodic: the result for the TAMSD is not random, and the same as found for an ensemble of Brownian particles, $\\langle x^2(\\Delta )\\rangle =2K_1\\Delta $ [16].", "Thus the diffusion constant $K_1$ can be determined either from an ensemble of trajectories as originally done by Perrin [17] or, as conceived by Nordlund [18], from a single particle trajectory.", "For anomalous diffusion, defined in terms of $\\langle x^2(\\Delta )\\rangle =2K_\\alpha \\Delta ^\\alpha /\\Gamma (1+\\alpha )$ with the generalized diffusion constant $K_{\\alpha }$ , the equivalence between time and ensemble averages as well as the time translation invariance generally breaks down.", "One of our central results is that for CTRW processes with long tailed $\\psi (t)$ the aged TAMSD (REF ) for $\\Delta \\ll T$ follows $\\left<\\overline{\\delta ^2(\\Delta ;t_a,T)}\\right>=\\frac{\\Lambda _{\\alpha }(t_a/T)}{\\Gamma (1+\\alpha )}\\frac{g(\\Delta )}{T^{1-\\alpha }},$ where $\\Lambda _\\alpha (z)=(1+z)^\\alpha -z^\\alpha $ , and $g$ is a function of $\\Delta $ only.", "Another important finding is that $\\overline{\\delta ^2}$ remains a random variable, whose statistical properties explicitly depend on both process age $t_a$ and measurement time $T$ .", "Previous work revealed deviations from standard ergodic statistical mechanics by studying time averages in the interval $(t_a=0,T)$ [7], [6], [8], [10], [9], [11], [12].", "Namely, in these works the start of the measurement coincides with the start of the process [19].", "However, in experiment, the observed particle may be immersed in the medium long before we start our observation.", "In fact in some cases we may not even know when the process was initiated.", "In this Letter, we derive the ageing renewal theory for CTRW processes and study the dependence of time averages of physical observables on the starting time $t_a$ of a measurement after system preparation at $t=0$ .", "According to Eq.", "(REF ), the time intervals $(0,T)$ and $(t_a,t_a+T)$ are not equivalent: In complete contrast to Brownian motion, the statistical properties of CTRW trajectories will depend on the observation window, the process exhibits ageing.", "The remarkable property of Eq.", "(REF ) is that corrections due to ageing enter in terms of a unique prefactor depending on $t_a$ and the measurement time $T$ .", "We call this $\\Lambda _{\\alpha }$ the ageing depression function, which is independent of $\\Delta $ .", "This function contains all the information on ageing, and is universal since the formula applies for any external force or boundary condition, and, as shown below, also holds for a large class of physical observables.", "The function $g(\\Delta )$ contains the complete $\\Delta $ -dependence.", "For instance, we find $g(\\Delta )\\simeq \\Delta $ for free motion [7], [6] and $g(\\Delta )\\simeq \\Delta ^{1-\\alpha }$ at long times under confinement [10], [9].", "Ageing renewal theory.", "In a CTRW, the position coordinate $x$ of a random walker is an accumulation of random jump lengths, $x(n)=\\sum _{i=0}^n\\delta x_i$ .", "In the simplest, unbiased version of the model, the $\\delta x_i$ are independent, identically distributed (IID) random variables with zero mean, and we assume also finite variance $\\sigma ^2$ .", "Jumps are separated by random IID waiting times, drawn from the common distribution $\\psi (t)\\simeq \\tau ^\\alpha /t^{1+\\alpha }$ , with $0<\\alpha <1$ .", "This implicitly defines a counting process $n(t)$ , the number of steps up to time $t$ .", "The statistics of the overall diffusion process $x(t)=x(n(t))$ are to be derived from the properties of both of its constituents, a method commonly called subordination [20], [22], [21].", "For a large variety of physical applications of CTRW and non-aged renewal theory see [23], [15], [24].", "We first discuss the counting process $n(t)$ , emphasizing the role of ageing.", "Since waiting times are independent, the counting process $n(t)$ is a renewal process, which we assume to start at $t=0$ .", "To study the ageing properties of the system, we consider $n_a(t_a,t)=n(t+t_a)-n(t_a)$ , the number of renewals in the interval $(t_a,t_a+t)$ .", "The corresponding probability density in double Laplace space, $(t_a,t)\\rightarrow (s_a,s)$ , in the scaling limit of large times becomes [25], [26] $p(n_a;s_a,s)=\\delta (n_a)\\left(\\frac{1}{s_as}-\\frac{h(s_a,s)}{s}\\right)+\\frac{h(s_a,s)}{s^{1-\\alpha }}\\tau ^\\alpha e^{-n_a(s\\tau )^\\alpha }.$ Here the probability of the waiting time for the first jump to occur after start of the measurement at $t_a$ is [26], [27], [28] $h(s_a,s)=\\frac{s_a^\\alpha -s^\\alpha }{s_a^\\alpha (s_a-s)}\\,\\,\\,\\Leftrightarrow \\,\\,\\, h(t_a,t)=\\frac{t_a^\\alpha }{t^\\alpha (t_a+t)}.$ In the Brownian limit $\\alpha \\rightarrow 1$ , the number of jumps $n$ and real time $t$ are equivalent, $p(n_a;t_a,t)=\\delta (n_a-t/\\tau )$ .", "This formalism allows for a direct calculation of the average of any function of $n_{a}(t_a,t)$ .", "For instance, the $q$ th order moment becomes [29] $\\nonumber \\langle n_a^q(s_a,s)\\rangle &=& \\int _0^\\infty n_a^q\\, p(n_a;s_a,s) \\,dn_a\\\\&=&\\Gamma (q+1)\\frac{s_a^\\alpha -s^\\alpha }{s_a^\\alpha (s_a-s)}\\frac{\\tau ^\\alpha }{s^{1+\\alpha q}}.$ After double Laplace inversion, we find $\\nonumber \\langle n_a^q(t_a,t)\\rangle &=&\\Gamma (q+1)/\\left[\\Gamma (\\alpha )\\Gamma (1+\\alpha q-\\alpha )\\right] \\hspace{56.9055pt}\\\\&&\\hspace*{-22.76228pt}\\times \\left(\\frac{t+t_a}{\\tau }\\right)^{\\alpha q}B\\left(\\frac{t}{t+t_a};1+\\alpha q-\\alpha ,\\alpha \\right),$ where $B(z;a,b)$ is the incomplete beta function [30].", "Thus, the number of steps taken during a time interval of length $t$ is not stationary in distribution: the moments for the period $[0,t]$ are clearly different from those for $[t_a,t_a+t]$ .", "Indeed we see from Eqs.", "(REF ) and (REF ) that the process gets slower and eventually stalls as $t_a\\rightarrow \\infty $ .", "Concurrently the $t$ -dependence changes with $t_a$ : $\\langle n_{a}^q(0,t)\\rangle \\simeq t^{\\alpha q}$ at $t_a=0$ , but $\\langle n_{a}^q(t_a,t)\\rangle \\simeq t_a^{\\alpha -1}t^{1-\\alpha +\\alpha q}$ for large but finite $t_a/t$ .", "For the probability density (REF ) we obtain $\\nonumber p(n_a;t_a,t)&\\sim &\\left[1-m_\\alpha (t/t_a)\\right]\\delta (n_a)+m_\\alpha (t/t_a)\\Gamma (2-\\alpha )\\\\&&\\hspace*{-28.45274pt} \\times \\frac{1}{(t/\\tau )^{\\alpha }}H^{1,0}_{1,1}\\left[\\frac{n_a}{(t/\\tau )^{\\alpha }}\\left|\\begin{array}{l}(2-2\\alpha ,\\alpha )\\\\(0,1)\\end{array}\\right.\\right]$ for $t_a\\gg t$ , in terms of an $H$ -function [31], [32] and the probability to make at least one step during $[t_a,t_a+t]$ , $m_{\\alpha }(t/t_a)=\\frac{B\\left([1+t_a/t]^{-1},1-\\alpha ,\\alpha \\right)}{\\Gamma (1-\\alpha )\\Gamma (\\alpha )}.$ Again, we emphasize the explicit dependence on $t_a$ .", "When compared to the form for $t_a=0$ [6], [27], the most striking difference in Eq.", "(REF ) is the occurrence of the $\\delta (n_a)$ -term: For a non-aged process ($t_a=0$ ), we have $m_{\\alpha }=1$ , while an aged process always has a nonzero probability of not performing any steps at all within the chosen time window, its amplitude approaching 1 algebraically, $m_{\\alpha }\\sim (t/t_a)^{1-\\alpha }/[\\Gamma (\\alpha )\\Gamma (2-\\alpha )]$ , as $t_a\\gg t$ .", "Ageing CTRW.", "We proceed by adding the position coordinate to our description.", "For IID jump distances $\\delta x_i$ , the jump process $x(n)$ converges in distribution to free Brownian motion in the scaling limit of many jumps.", "The anomalous diffusion process $x(t)=x(n(t))$ inherits the ageing properties of the counting process discussed above.", "To see this, consider the $q$ th order TA moments $\\left<\\overline{\\delta ^q(\\Delta ;t_a,T)}\\right>=\\int _{t_a}^{t_a+T-\\Delta }\\frac{\\langle \\left|x(t+\\Delta )-x(t)\\right|^q\\rangle }{T-\\Delta }dt,$ which are useful to characterize experimental data [33].", "For free Brownian motion $x(n)$ we know that $\\langle \\left|x(n_2)-x(n_1)\\right|^q\\rangle =\\frac{2\\Gamma (q)}{\\Gamma (q/2)}\\frac{\\sigma ^q}{2^{q/2}}|n_2-n_1|^{q/2},$ Since $x(n)$ and $n(t)$ are independent processes, we use conditional averaging to evaluate the integrand of Eq.", "(REF ).", "From above result (REF ), it follows that $\\nonumber \\left<\\left|x(t+\\Delta )-x(t)\\right|^q\\right>&=&\\frac{2\\Gamma (q)}{\\Gamma (q/2)}\\frac{\\sigma ^q}{2^{q/2}}\\langle n_a^{q/2}(t,\\Delta )\\rangle \\\\\\nonumber &&\\hspace*{-102.43008pt}=\\Gamma (q+1)/\\left[\\Gamma (\\alpha )\\Gamma (1+\\alpha q/2-\\alpha )\\right]\\\\&&\\hspace*{-102.43008pt}\\times \\left[K_\\alpha \\left(t+\\Delta \\right)^\\alpha \\right]^{q/2}B\\left(\\frac{\\Delta }{t+\\Delta },1-\\alpha +\\alpha q/2,\\alpha \\right),$ where we identify $K_\\alpha =\\sigma ^2/(2\\tau ^\\alpha )$ [23].", "The $q$ th order TA moment (REF ) in the limit $\\Delta \\ll T$ then becomes $\\left<\\overline{\\delta ^q(\\Delta ;t_a,T)}\\right>=\\frac{\\Lambda _{\\alpha }(t_a/T)\\left[K_\\alpha \\Delta ^\\alpha \\right]^{q/2}\\Gamma (q+1)}{\\Gamma (\\alpha +1)\\Gamma (2-\\alpha +\\alpha q/2)}\\left(\\frac{\\Delta }{T}\\right)^{1-\\alpha },$ which is of the general shape (REF ) with $g(\\Delta )\\sim \\Delta ^{1-\\alpha +\\alpha q/2}$ .", "Interestingly, we see the special role of the TAMSD ($q=2$ ), for which the $\\Delta $ -scaling is independent of $\\alpha $ .", "Fig.", "REF shows simulations results for the TAMSD.", "If evaluated during the initial time period, $t_a=0$ , the TAMSD for individual trajectories scatter around the ensemble average $\\langle \\overline{\\delta ^2}\\rangle $ , Eq.", "(REF ).", "In contrast, for the aged process ($t_a\\gg T$ ) the ensemble average $\\langle \\overline{\\delta ^2}\\rangle $ appears much lower than the shown individual trajectories.", "This is due to the fact that a significant fraction $1-m_{\\alpha }$ of particles do not move during the entire measurement time.", "Such trajectories are naturally not visible in a logarithmic plot, while being included in the calculation of $\\langle \\overline{\\delta ^2}\\rangle $ .", "Figure: Time averaged mean squared displacement δ 2 (Δ;t a ,T) ¯\\overline{\\delta ^2(\\Delta ;t_a,T)} for individual free CTRW trajectories (full symbols) and the averagesaccording to Eq.", "() (bold black lines).", "Left: Measurementstarts at t a =0t_a=0, so that m α =1m_\\alpha =1.", "Right: Aged process, t a =10 11 t_a=10^{11}(a.u.", "), so that a large fraction 1-m α ≈94%1-m_\\alpha \\approx 94\\% of trajectories issuppressed in the log-log plot.", "The parameters are α=1/2\\alpha =1/2,τ=1\\tau =1, σ 2 =1\\sigma ^2=1 and T=10 9 T=10^9.In a biological cell, the diffusive motion of a tracer particle is spatially confined, or a charge carrier in a disordered semiconductor experiences a drift force.", "To address such systems we now determine the TAMSD in the presence of an external potential.", "We start with a Hookean force $-\\lambda x$ and address the more general case below.", "To that end the continuum approximation of the process is represented by the Langevin equation [34] $dx/dn=-\\lambda x(n)+\\xi (n),$ where $\\xi (n)$ is white Gaussian noise with $\\langle \\xi (n_1)\\xi (n_2)\\rangle =\\sigma ^2\\delta (n_2-n_1)$ .", "In other words, $x(n)$ is a stationary Ornstein-Uhlenbeck process.", "Its increments are Gaussian variables, characterized by the variance $\\langle [x(n_2)-x(n_1)]^2\\rangle =\\sigma ^2[1-\\exp (-\\lambda |n_2-n_1|)]/\\lambda $ .", "To calculate the TAMSD, we follow the approach outlined for the free particle, a general approach to compute this and similar quantities will be provided below.", "The result reads $\\left<\\overline{\\delta ^2(\\Delta ;t_a,T)}\\right>=\\frac{\\Lambda _\\alpha (t_a/T)}{\\Gamma (1+\\alpha )}\\frac{2K_\\alpha \\Delta }{T^{1-\\alpha }}E_{\\alpha ,2}\\left(-\\lambda _{\\alpha } \\Delta ^\\alpha \\right)$ in terms of the generalized Mittag-Leffler function [35], where $\\lambda _{\\alpha }=\\lambda /\\tau ^\\alpha $ .", "We deduce the limiting behavior $\\left<\\overline{\\delta ^2}\\right>\\sim \\frac{2\\Lambda _{\\alpha }(t_a/T)K_{\\alpha }}{\\Gamma (1+\\alpha )T^{1-\\alpha }}\\left\\lbrace \\begin{array}{ll}\\Delta , & \\Delta \\ll \\lambda _{\\alpha }^{-1/\\alpha }\\\\\\frac{\\Delta ^{1-\\alpha }/\\lambda _{\\alpha }}{\\Gamma (2-\\alpha )}, &\\Delta \\gg \\lambda _{\\alpha }^{-1/\\alpha }\\end{array}\\right..$ Eq.", "(REF ) is another special case of Eq.", "(REF ).", "Interestingly the entire dynamics are multiplied by the unique factor $\\Lambda _{\\alpha }$ .", "Let us now test the generality of this feature.", "Consider the time average of some observable $F(x_2,x_1)$ along the trajectory, $\\left<\\overline{F(\\Delta ;t_a,T)}\\right>=\\int _{t_a}^{t_a+T-\\Delta }\\frac{\\langle F(x(t+\\Delta ),x(t))\\rangle }{T-\\Delta }dt.$ $F$ may represent moments ($\\overline{F}=\\overline{\\delta ^q}, F(x_2,x_1)=|x_2-x_1|^q$ ) or the TA of a correlation function.", "We only require that the jump process $x(n)$ and the function $F$ fulfill $\\langle F(x(n_2),x(n_1))\\rangle =f(n_2-n_1).$ For instance, we have $f(n)=\\sigma ^2 n$ for the second moment of unbounded motion [cf.", "(REF )], or $f(n)=\\sigma ^2[1-\\exp (-\\lambda n)]/\\lambda $ for the TAMSD in an harmonic potential.", "Condition (REF ) is fulfilled whenever $x(n)$ is a stationary process (e.g., equilibrated Brownian motion).", "Alternatively, one may consider a process with stationary increments (e.g., unbounded Brownian motion), when $F(x_2,x_1)=F(x_2-x_1)$ .", "In these cases we find $\\left<\\overline{F(\\Delta ;t_a,T)}\\right>=\\int _{t_a}^{t_a+T-\\Delta }\\int _0^\\infty \\frac{f(n_a)p(n_a;t,\\Delta )}{T-\\Delta }dn_adt,$ where $p(n_a;t_a,t)$ is defined in Eq.", "(REF ).", "We obtain $\\left<\\overline{F(\\Delta ;t_a,T)}\\right>=C+\\frac{\\Lambda _{\\alpha }(t_a/T)}{\\Gamma (1+\\alpha )}\\frac{g(\\Delta /\\tau )}{(T/\\tau )^{1-\\alpha }},$ at short lag times $\\Delta \\ll T$ , with the constant $C=f(0)$ .", "The function $g$ in Laplace space is defined as [25] $g(s)=s^{2\\alpha -2}{L}\\left\\lbrace f(n)-f(0);n\\rightarrow s^\\alpha \\right\\rbrace .$ Comparing with our specific results (REF ), (REF ), and (REF ), we identify all relevant terms in TAs: (i) In the limit $\\Delta \\rightarrow 0$ , the TA reduces to the constant $C$ , which equals the expectation value of the observable when measured at identical positions.", "For example, if we study correlations in an equilibrated process, $F(x_2,x_1)=x_2x_1$ , then $C=\\langle x^2\\rangle $ is the thermal value of $x^2$ .", "Conversely, $C$ naturally vanishes if we are interested in TA moments of displacements, $F(x_2,x_1)=|x_2-x_1|^q$ , so it did not appear previously.", "(ii) The lag time dependence enters through the function $g(\\Delta )$ .", "For example, if $f(n)\\sim n^q$ , then $C=0$ , and in Laplace space $g(s)\\sim s^{\\alpha -2-\\alpha q}$ , which implies $g(\\Delta )\\sim \\Delta ^{1-\\alpha +\\alpha q}$ as in Eq.", "(REF ).", "(iii) The ageing depression function $\\Lambda _{\\alpha }$ only depends on the ratio $t_a/T$ and the parameter $\\alpha $ , and due to a factor $T^{\\alpha -1}$ any TA converges to the constant $C$ as $T\\rightarrow \\infty $ .", "Note that this dependence on ageing and measurement time $t_a$ and $T$ is universal in the sense that it is indifferent to the specific choice of observable $F$ or model of the jump process $x(n)$ , but is directly deduced from the nature of the ageing counting process $n(t)$ .", "Also note that in the Brownian limit $\\alpha =1$ , Eq.", "(REF ) reduces to $\\langle \\overline{F(\\Delta ;t_a,T)}\\rangle =f(\\Delta /\\tau )$ , restoring the ergodic equivalence of ensemble and time averages, and the stationarity of the process.", "Figure: Numerical validation of Eq.", "(), for variousα\\alpha , boundary conditions, and t a t_a (see key).", "Each point in thegraph represents an individual trajectory.", "Parameters are τ=1\\tau =1 (a.u.", "),σ 2 =1\\sigma ^2=1, Δ=100\\Delta =100, T=2×10 6 T=2\\times 10^6.", "t a t_a is either 0(non-aged), or for specific α\\alpha chosen such that m α =0.054m_\\alpha =0.054(aged).Distribution of TAMSD.", "Due to the scale-free nature of the distribution $\\psi (t)$ of waiting times all TAs of physical observables, e.g.", "$\\overline{\\delta ^2}$ , remain random quantities, however, with a limiting distribution $\\phi (\\xi )$ for the dimensionless ratio $\\xi =\\overline{\\delta ^2}/\\langle \\overline{\\delta ^2}\\rangle $ [6], [10], [36].", "As contributions to time averages of the form (REF ) occur at time instants when the particle performs a jump, we expect that in the sense of distributions both $\\overline{\\delta ^2}$ and $n_a$ should be equivalent, $\\overline{\\delta ^2}\\overset{d}{=}cn_a$ , for some non-random, positive $c$ .", "In other words, $\\xi =\\frac{\\overline{\\delta ^2(\\Delta ;t_a,T)}}{\\langle \\overline{\\delta ^2(\\Delta ;t_a,T)}\\rangle }\\overset{d}{=}\\frac{n_a(t_a,T)}{\\langle n_a(t_a,T)\\rangle },$ for $\\Delta \\ll T$ .", "We may thus deduce the statistics directly from the underlying counting process.", "Fig.", "REF provides numerical evidence for this argument in the case of a free particle and a particle in a box for several values of $\\alpha $ .", "The distribution $\\phi (\\xi )$ for $t_a=0$ is related to a one-sided Lévy stable law [6].", "In the opposite case $t_a\\gg T$ , combination of Eqs.", "(REF ) and (REF ), yields $\\nonumber \\phi (\\xi )&\\sim &\\left[1-m_{\\alpha }(T/t_a)\\right]\\delta (\\xi )+m_{\\alpha }(T/t_a)\\Gamma (2-\\alpha )\\\\&&\\hspace*{-28.45274pt}\\times \\frac{\\left(T/t_a\\right)^{1-\\alpha }}{\\Gamma (\\alpha )}H^{1,0}_{1,1}\\left[\\xi \\frac{(T/t_a)^{1-\\alpha }}{\\Gamma (\\alpha )}\\left|\\begin{array}{ll}(2-2\\alpha ,\\alpha )\\\\(0,1)\\end{array}\\right.\\right],$ for $\\Delta \\ll T\\ll t_a$ .", "In $\\phi (\\xi )$ , $m_{\\alpha }(T/t_a)$ is the weight of the continuous part.", "The probability for not moving during the whole measurement period, ($\\xi =0$ ), approaches unity as $\\simeq (T/t_a)^{1-\\alpha }$ .", "If conditioned to measurements with $\\xi >0$ , we also find the scaling $\\xi \\sim (t_a/T)^{1-\\alpha }$ .", "In Fig.", "REF we demonstrate excellent agreement of Eq.", "(REF ) with numerical simulations.", "Figure: Scatter density φ(ξ)\\phi (\\xi ) for different α\\alpha and m α m_\\alpha ,see text.", "Lines: Eq.", "(7) from (Left) and Eq. ()(Right).", "Symbols: Simulations of free CTRW.", "Note that the area under the curvesfor the aged process in the right panel is not unity, since the fraction 1-m α 1-m_{\\alpha } of immobile events is not shown.Same parameters as in Fig.", ".Deviations from ergodic behavior are quantified by the ergodicity breaking parameter [6], for which we obtain $\\mathrm {EB}=\\frac{\\left<\\overline{\\delta ^2}^2\\right>}{\\left<\\overline{\\delta ^2}\\right>^2}-1=2\\alpha \\frac{B\\left([1+t_a/T]^{-1},1+\\alpha ,\\alpha \\right)}{\\left[1-(1+T/t_a)^{-\\alpha }\\right]^2}-1,$ depending only on the ratio $t_a/T$ .", "At $t_a=0$ EB reduces to the result of Ref.", "[6], while for $t_a\\gg T$ , we find $\\mathrm {EB}\\sim 2(t_a/T)^{1-\\alpha }/[\\alpha (1+\\alpha )]$ .", "In the non-aged case EB is bounded, $0\\le \\mathrm {EB}\\le 1$ , depending on the value of $\\alpha $ only.", "In contrast we find that EB may diverge in the limit $t_a/T\\rightarrow \\infty $ .", "This implies that the non ergodic fluctuations are much larger in the aged regime under investigation.", "We show the behavior of EB in Fig.", "REF .", "Figure: Ergodicity breaking parameter () as function of α\\alpha (Left) and t a /Tt_a/T (Right).", "Notice that the non-ergodicfluctuations become larger with increasing t a t_a.Conclusions.", "We investigated the effects of ageing on TAs of physical observables.", "Previous calculations of TAs tacitly neglect the fact that often the preparation of the system and start of the measurement do not coincide.", "While this does not cause any problems for ergodic systems with rapid memory loss of the initial conditions, in general this cannot be taken for granted in processes of anomalous diffusion.", "Here we showed for the case of CTRW with diverging characteristic waiting time that TAs of arbitrary physical observables carry the common factor $\\Lambda _{\\alpha }$ .", "This ageing depression function is universal in the sense that it only depends on the process age $t_a$ and the measurement time $T$ .", "All details such as confinement effects enter through a single function, $g(\\Delta )$ .", "The structure of this result was shown to hold for a large class of physical observables.", "We also see that the ageing of the process has a pronounced statistical effect, splitting the population into two: the mobile fraction $m_{\\alpha }$ and the immobile one whose amplitude $1-m_{\\alpha }$ grows with $t_a/T$ .", "Knowledge of this effect is significant for the quantitative physical interpretation of experimental data.", "Finally, since renewal theory is applicable to many systems, our results with minor changes should be relevant more generally, for instance, to the Aaronson-Darling-Kac theorem in infinite ergodic theory or for counting the number of renewals in blinking quantum dots.", "We acknowledge funding from the CompInt graduate school, the Academy of Finland (FiDiPro scheme), and the Israel Science Foundation." ] ]

[ [ "Formes modulaires modulo 2 : l'ordre de nilpotence des op\\'erateurs de\n Hecke" ], [ "Abstract The nilpotence order of the mod 2 Hecke operators.", "Let $\\Delta=\\sum_{m=0}^\\infty q^{(2m+1)^2} \\in F_2[[q]]$ be the reduction mod 2 of the $\\Delta$ series.", "A modular form f modulo 2 of level 1 is a polynomial in $\\Delta$.", "If p is an odd prime, then the Hecke operator Tp transforms f in a modular form Tp(f) which is a polynomial in $\\Delta$ whose degree is smaller than the degree of f, so that Tp is nilpotent.", "The order of nilpotence of f is defined as the smallest integer g = g(f) such that, for every family of g odd primes p1, p2, ..., pg, the relation Tp1Tp2... Tpg (f) = 0 holds.", "We show how one can compute explicitly g(f); if f is a polynomial of degree d in $\\Delta$, one finds that g(f) << d^(1/2)." ], [ "Introduction", "Soit $\\displaystyle \\Delta (q)=q\\prod _{n=1}^\\infty (1-q^n)^{24}=\\sum _{n=1}^\\infty \\tau (n) q^n$ où $\\tau $ est la fonction de Ramanujan.", "Soit $k$ un entier $\\geqslant 0$ .", "On écrit $\\displaystyle \\Delta ^k(q)=\\sum _{n=k}^\\infty \\tau _k(n)q^n.$ Les congruences connues sur $\\tau (n) \\pmod {2}$ (cf.", "[5]), montrent que $\\displaystyle \\Delta (q)\\equiv \\sum _{m=0}^\\infty q^{(2m+1)^2} \\pmod {2}$ , ce qui entraîne $n\\lnot \\equiv k\\pmod {8}\\quad \\Longrightarrow \\quad \\tau _k(n)\\equiv 0 \\pmod {2}.$ Une forme modulaire modulo 2 de niveau 1 est un polynôme $f(\\Delta )$ à coefficients dans $\\hbox{\\bf F}_2$ (cf.", "par exemple [2], [4]) ; nous l'identifierons à une série formelle en la variable $q$ , à coefficients dans $\\hbox{\\bf F}_2$ .", "Nous ne nous intéresserons qu'aux formes paraboliques (celles dont le terme constant est 0).", "À partir de maintenant (sauf mention expresse du contraire), toutes les séries considérées sont à coefficients mod 2, et nous nous permettrons d'écrire $\\Delta =\\Delta (q)=\\sum _{m=0}^\\infty q^{(2m+1)^2} \\ \\in \\hbox{\\bf F}_2[[q]].$ Soit $\\mathcal {F}$ le sous-espace de $ \\hbox{\\bf F}_2[\\Delta ]$ engendré par $\\Delta ,\\Delta ^3,\\Delta ^5,\\ldots $ .", "Compte tenu de (REF ), on a $\\mathcal {F}=\\mathcal {F}_1\\oplus \\mathcal {F}_3\\oplus \\mathcal {F}_5\\oplus \\mathcal {F}_7$ où, pour $i\\in \\lbrace 1,3,5,7\\rbrace $ , $\\mathcal {F}_i$ a pour base $\\lbrace \\Delta ^i,\\Delta ^{i+8},\\Delta ^{i+16},\\ldots \\rbrace .$ Puisque $\\Delta ^{2k}(q)=\\Delta ^k(q^2)$ , toute forme parabolique $f$ modulo 2 peut s'écrire comme une somme finie $f=\\sum _{s\\geqslant 0} f_s^{2^s} \\quad {\\rm avec} \\quad f_s\\in \\mathcal {F}.$" ], [ "Opérateurs de Hecke", "Soit $f(q)=\\sum _{n\\geqslant 0} c_n q^n$ une forme modulaire modulo 2 et soit $p$ un nombre premier $> 2$ .", "L'opérateur de Hecke $T_p$ transforme $f$ en la forme $T_p|f=\\sum _{n\\geqslant 0} \\gamma _n q^n \\;\\; {\\rm avec } \\;\\;\\gamma (n)=\\left\\lbrace \\begin{array}{ll}c(pn)& {\\rm si} \\;p \\; {\\rm ne \\; divise \\; pas\\; } n\\\\c(pn)+c(n/p)\\;\\;& {\\rm si} \\;p \\;{\\rm divise}\\; n.\\end{array}\\right.$ [Nous écrirons parfois $T_p(f)$ à la place de $T_p|f$ . ]", "Si $f$ est de degré $\\leqslant k$ (comme polynôme en $\\Delta $ ), alors il en est de même de $T_p|f$  ; on peut écrire $T_p|\\Delta ^k$ sous la forme $T_p|\\Delta ^k=\\sum _{j=0}^k \\mu _j \\Delta ^j,\\quad {\\rm avec } \\; \\mu _j\\in \\hbox{\\bf F}_2.$ Supposons maintenant $k$ impair.", "Les formules (REF ) et (REF ) entraînent que $j\\lnot \\equiv pk \\pmod {8}\\quad \\Longrightarrow \\quad \\mu _j=0.$ En particulier, on a $T_p(\\mathcal {F}_i) \\subset \\mathcal {F}_j$ si $j \\equiv pi \\ $ (mod $8)$ .", "L'opérateur de Hecke $T_p$ commute avec les opérations $f \\mapsto f^{2^s}$ de sorte que, si l'on connaît l'action de $T_p$ sur $\\mathcal {F}$ , par (REF ), on la connaît sur toutes les formes paraboliques." ], [ "Nilpotence des opérateurs de Hecke modulo 2", "L'une des propriétés essentielles des opérateurs de Hecke modulo 2 est qu'ils sont nilpotents (cf.", "par exemple [1], [3], [4]).", "Cela implique que, dans (REF ), le coefficient $\\mu _k$ est nul.", "Par (REF ) et (REF ), on a donc pour tout $p$ premier $\\geqslant 3$ , et tout $k$ impair positif, $T_p|\\Delta ^k=\\sum _{\\begin{array}{c}j\\equiv pk \\hspace{-5.69054pt}\\pmod {8}\\\\1\\leqslant j \\leqslant k-2\\end{array}}\\mu _j \\Delta ^j ,\\quad {\\rm avec }\\;\\; \\mu _j\\in \\hbox{\\bf F}_2.$ Exemples : (i) $T_p|\\Delta =0$ pour tout $p$ premier $ > 2$ .", "(ii) Si $p\\equiv 3 \\pmod {8}$ , on a $T_p|\\Delta ^3=\\Delta ;$ sinon, $T_p|\\Delta ^3=0$ .", "(iii) Si $p\\equiv 5 \\pmod {8}$ , on a $T_p|\\Delta ^5=\\Delta ;$ sinon, $T_p|\\Delta ^5=0$ .", "(iv) On a : $\\;\\; T_p|\\Delta ^7=\\left\\lbrace \\begin{array}{llll}0 \\quad {\\rm si} \\quad p \\equiv 1 \\pmod {8} & \\!", "{\\rm ou \\ si } \\quad p \\equiv -1 \\pmod {16}\\\\\\Delta ^5 \\ {\\rm si} \\quad p \\equiv 3 \\pmod {8}\\\\\\Delta ^3 \\ {\\rm si} \\quad p \\equiv 5 \\pmod {8}\\\\\\Delta \\ \\ {\\rm si} \\quad p \\equiv 7 \\pmod {16}.\\end{array}\\right.", "$" ], [ "L'ordre de nilpotence", "Par définition, l'ordre de nilpotence d'une forme modulaire $f\\in \\hbox{\\bf F}_2[\\Delta ]$ est le plus petit entier $g=g(f)$ tel que, pour toute suite de $g$ nombres premiers impairs $p_1,p_2,\\ldots ,p_g$ , on ait $T_{p_1}T_{p_2}\\ldots T_{p_g}|f=0$ .", "[Comme les $T_p$ commutent entre eux, l'ordre dans lequel on écrit les $T_{p_i}$ n'a pas d'importance.", "Noter aussi que l'on ne suppose pas que les $p_i$ soient distincts.]", "Lorsque $f$ = 0, on convient que $g(f)=-\\infty $ .", "Nous désignerons par $g(k)=g(\\Delta ^k)$ l'ordre de nilpotence de $\\Delta ^k$ .", "Comme chaque $T_p$ abaisse le degré en $\\Delta $ d'au moins 2 unités, on a $g(k) \\leqslant \\frac{k+1}{2}\\cdot $ Soit $p$ un nombre premier impair ; il résulte de la définition de l'ordre de nilpotence d'une forme modulaire $f\\in \\mathcal {F}$ que l'on a $g(f) \\geqslant g(T_p|f) +1.$ Exemples : $g(0)=-\\infty ,\\ \\ g(\\Delta )=1,\\ \\ g(\\Delta ^3)= g(\\Delta ^3+\\Delta )=2,\\; $ $g(\\Delta ^5)=g(\\Delta ^5+\\Delta )=g(\\Delta ^5+\\Delta ^3)=g(\\Delta ^5+\\Delta ^3+\\Delta )=2.", "$" ], [ "Calcul des $T_p|\\Delta ^k :$ une récurrence linéaire", "Soit $p$ un nombre premier $> 2$ .", "Théorème 3.1 Il existe un unique polynôme symétrique $F_p(X,Y) \\in \\hbox{\\bf F}_2[X,Y]$ , $F_p(X,Y)=Y^{p+1}+s_1(X) Y^p +\\ldots +s_p(X) Y +s_{p+1}(X)$ de degré $p+1$ tel que $T_p(\\Delta ^k) = \\sum _{r=1}^{p+1} s_r (\\Delta ) \\ T_p (\\Delta ^{k-r})$ pour tout $k \\geqslant p+1$ .", "De plus, pour $1 \\leqslant r \\leqslant p+1$ , $s_r(X)$ est une somme de monômes en $X$ dont les degrés sont congrus à $pr$ modulo 8 et sont $\\leqslant r$ .", "Esquisse de démonstration.", "On définit les $s_r(\\Delta ), \\ 1 \\leqslant i \\leqslant p+1$ , comme les fonctions symétriques élémentaires des $p+1$ séries $f_0 = \\Delta (q^p), \\quad f_i = \\Delta (z^iq^{1/p}), \\quad i = 1,...,p,$ où $z$ est une racine primitive $p$ -ième de l'unité dans une extension finie de $\\hbox{\\bf F}_2$ .", "On déduit (REF ) de la formule : $ T_p|\\Delta ^k = \\sum _{i=0}^p \\ (f_i)^k , \\ \\ k= 0,1,...$ Exemples Une table des polynômes $F_p$ pour $p \\leqslant 257$ , calculée avec SAGE par Marc Deléglise, se trouve sur le site http ://math.univ-lyon1.fr/$\\sim $ nicolas/polHecke.html Pour $p=3$ on a $F_3(X,Y)=(X+Y)^4 + XY =X^4+XY+Y^4.$ Vu (REF ), cela donne un procédé de calcul des $T_3|\\Delta ^k$  ; si $t$ est une indéterminée, on a : $\\sum _{k=1}^{\\infty } T_3(\\Delta ^k)t^k = \\frac{ \\Delta t^3}{1+\\Delta ^3t+\\Delta ^4t^4}\\cdot $ De même, pour $p=5$ , on a : $F_5(X,Y)=(X+Y)^6 + XY = X^6 + X^4 Y^2 + X^2 Y^4 +X Y+Y^{6}$ et $\\sum _{k=1}^{\\infty } T_5(\\Delta ^k)t^k = \\frac{ \\Delta t^5}{1+\\Delta ^2t^2+\\Delta ^4t^4+\\Delta ^5t^5+\\Delta ^6t^6}\\cdot $" ], [ "Les nombres $n_3(k), n_5(k)$ \net {{formula:f2f0841a-9b2e-47d3-8c31-364982953e9d}}", "Soit $k$ un nombre entier $\\geqslant 0$ .", "Ecrivons-le sous forme dyadique : $\\displaystyle k=\\sum _{i=0}^\\infty \\beta _i 2^i$ avec $\\beta _i=0$ ou 1.", "Posons : $ n_3(k)=\\sum _{i=0}^\\infty \\beta _{2i+1} 2^i=\\sum _{\\begin{array}{c}i=1\\\\ i\\;\\rm { impair}\\;\\end{array}}^\\infty \\beta _i 2^{\\frac{i-1}{2}},\\quad n_5(k)=\\sum _{i=0}^\\infty \\beta _{2i+2} 2^i=\\sum _{\\begin{array}{c}i=1\\\\ i\\;\\rm { pair}\\;\\end{array}}^\\infty \\beta _i 2^{\\frac{i-2}{2}},\\quad h(k)=n_3(k)+n_5(k).$ L'entier $h(k) $ est du même ordre de grandeur que $k^{1/2}$  : si $k$ est impair $> 0$ on a $ \\frac{1}{2} k^{1/2} \\ < \\ h(k)+1 \\ < \\ \\frac{3}{2} k^{1/2}.$ Notons que l'on a pour $\\ell \\geqslant 0$ $n_3(2\\ell +1)=n_3(2\\ell ),\\quad n_5(2\\ell +1)=n_5(2\\ell ),\\quad h(2\\ell +1)=h(2\\ell ).$ Nous appellerons $[n_3(k),n_5(k)]$ le code du nombre $k$ .", "L'application $k \\mapsto [n_3(k),n_5(k)]$ est une bijection de l'ensemble des nombres impairs (resp.", "pairs) $\\geqslant 0$ sur $\\hbox{\\bf N}^2$ ." ], [ "Relation de domination", "Nous utiliserons la relation d'ordre suivante sur l'ensemble des nombres entiers naturels pairs (ou impairs) : Définition 4.1 Si $k$ et $\\ell $ ont même parité, on dit que $\\ell $ domine $k$ et on écrit $k\\prec \\ell $ ou $\\ell \\succ k$ si l'on a $h(k) < h(\\ell )$ ou bien $h(k)=h(\\ell )$ et $n_5(k)< n_5(\\ell )$ .", "La relation $k \\preccurlyeq \\ell $ définie par $k\\prec \\ell $ ou $k=\\ell $ , est une relation d'ordre total sur l'ensemble des entiers pairs $($ resp.", "impairs$) \\geqslant 0$ .", "À partir de maintenant, nous écrirons une forme modulaire $f\\in \\mathcal {F}$ , $f\\ne 0$ sous la forme $f=\\Delta ^{m_1}+\\Delta ^{m_2} \\ldots +\\Delta ^{m_r} \\;\\; {\\rm avec }\\;\\;m_1 \\succ m_2 \\succ \\ldots \\succ m_r.$" ], [ "La fonction $h$ pour les formes modulaires mod 2", "Définition 4.2 Soit $f\\in \\mathcal {F}$ .", "Si $f\\ne 0$ , on écrit $f$ sous la forme (REF ).", "On dit que $m_1$ est l'exposant dominant de $f$ et l'on définit $h(f)$ par $h(f)=h(m_1)=\\max _{1 \\leqslant i \\leqslant r} h(m_i).", "$ Si $f=0$ , on pose $h(f)=-\\infty $ ." ], [ "Le cas de $T_3|f$", "Proposition 4.3 Soit $f\\in \\mathcal {F}$ , $f\\ne 0$ et soit $m_1$ son exposant dominant.", "(i) On a $h(T_3|f) \\leqslant h(f)-1=h(m_1)-1$ .", "(ii) Lorsque $n_3(m_1)\\geqslant 1$ , on a $h(T_3|f)=h(m_1)-1$ et l'exposant dominant de $T_3|f$ a pour code $[n_3(m_1)-1, n_5(m_1)]$ .", "Démonstration : On considère d'abord le cas où $f=\\Delta ^k$ .", "On raisonne alors par récurrence sur $k$ en utilisant les relations (REF ), (REF ) et (REF ).", "La démonstration est assez longue et technique." ], [ "Le cas de $T_5|f$", "Proposition 4.4 Soit $f\\in \\mathcal {F}$ , $f\\ne 0$ et soit $m_1$ son exposant dominant.", "(i) On a $h(T_5|f) \\leqslant h(f)-1=h(m_1)-1$ .", "(ii) Lorsque $n_5(m_1)\\geqslant 1$ , on a $h(T_5|f)=h(m_1)-1$ et l'exposant dominant de $T_5|f$ a pour code $[n_3(m_1), n_5(m_1)-1]$ .", "Démonstration : Même méthode que pour la proposition REF  ; on utilise (REF ) au lieu de (REF ).", "Théorème 5.1 Soit $f\\in \\mathcal {F}$ , $f\\ne 0$ , que l'on écrit comme en (REF ).", "(i) On a $T_3^{n_3(m_1)} T_5^{n_5(m_1)} |f=\\Delta .$ (ii) La valeur de l'ordre de nilpotence $g(f)$ (cf.", "§ REF ) est donnée par $g(f) = h(f)+1.$ Démonstration : (i) Soit $m$ l'exposant dominant de $\\varphi =T_3^{n_3(m_1)} T_5^{n_5(m_1)}|f$ .", "En appliquant $n_3(m_1)$ fois la proposition REF (ii) et $n_5(m_1)$ fois la proposition REF  (ii), on voit que $m$ a pour code $[0,0]$  ; comme $m$ est impair, on a $m=1$ , d'où $\\varphi =\\Delta $ , ce qui démontre (REF ).", "Notons que (REF ) implique $g(f) \\geqslant n_3(m_1)+n_5(m_1)+1=h(m_1)+1=h(f)+1.$ (ii) Soit $d=\\max (m_1,m_2,\\ldots ,m_r)$ le degré de $f$  ; on va démontrer (REF ) par récurrence sur le nombre impair $d$ .", "Si $d=1,3$ ou 5, (REF ) résulte de (REF ) et (REF ).", "Soit $d \\geqslant 7$ et supposons (REF ) vraie pour toute forme de degré $\\leqslant d-2$ .", "Pour $d \\geqslant 7$ , on a $h(d) \\geqslant 2$ et la définition de l'exposant dominant entraîne $h(f)=h(m_1) \\geqslant h(d) \\geqslant 2$ .", "Par (REF ), on a $g(f)\\geqslant h(f)+1 \\geqslant 3$  ; donc il existe des nombres premiers impairs $p_1,p_2,\\ldots , p_s$ avec $s=g(f)-1 \\geqslant 2$ et $T_{p_1} T_{p_2}\\ldots T_{p_s}|f \\ne 0.$ Posons $\\varphi =T_{p_s}|f$ , et calculons $g(\\varphi )$ .", "De (REF ), on déduit $T_{p_1} T_{p_2}\\ldots T_{p_{s-1}}|\\varphi =T_{p_1} T_{p_2}\\ldots T_{p_s}|f \\ne 0$ , ce qui implique $g(\\varphi ) \\geqslant s$ .", "Mais (REF ) entraîne $g(\\varphi )=g(T_p|f)\\leqslant g(f)-1=s$ .", "On en déduit $g(\\varphi )= s =g(f)-1\\geqslant 2.$ Observons que (REF ) et $s\\geqslant 2$ entraînent $\\varphi \\ne 0$ .", "Par (REF ), le degré de $\\varphi $ est $\\leqslant d-2$  ; on peut donc appliquer à $\\varphi $ l'hypothèse de récurrence, ce qui donne $g(\\varphi )=h(\\varphi )+1$ .", "En désignant par $j$ l'exposant dominant de $\\varphi $ , avec (REF ), il vient $g(\\varphi )=h(\\varphi )+1=h(j)+1=s\\geqslant 2.$ Soit $[u,v]$ le code de $j$ , avec $u\\geqslant 0$ , $v\\geqslant 0$ et $u+v=s-1$ .", "En appliquant (i) à $\\varphi $ et en posant $q_1=q_2=\\ldots =q_u=3$ et $q_{u+1}=q_{u+2}=\\ldots = q_{u+v}=5$ , il vient $T_{q_1} T_{q_2}\\ldots T_{q_{s-1}}|\\varphi =T_{q_1} T_{q_2}\\ldots T_{q_{s-1}}T_{p_s}|f = \\Delta .$ Posons $\\psi =T_{q_{s-1}}|f$  ; on a $T_{q_1} T_{q_2}\\ldots T_{q_{s-2}}T_{p_s}|\\psi =T_{q_1} T_{q_2}\\ldots T_{q_{s-1}}T_{p_s}|f = \\Delta .$ Cette formule montre que $g(\\psi )\\geqslant s$ .", "Mais (REF ) entraîne $g(\\psi )=g(T_{q_{s-1}}|f)\\leqslant g(f)-1=s$    et    $g(\\psi )=s$ .", "Par (REF ), le degré de $\\psi $ est $\\leqslant d-2$ et l'hypothèse de récurrence donne $g(\\psi )=h(\\psi )+1$ .", "On a ainsi $g(\\psi )= s =g(f)-1=h(\\psi )+1.$ Par la proposition REF (i) lorsque $q_{s-1}=3$ , et par la proposition REF (i) lorsque $q_{s-1}=5$ , on a $h(T_{q_{s-1}}|f) \\leqslant h(f)-1$ , d'où, par (REF ), $s-1=g(f)-2 = h(\\psi )=h(T_{q_{s-1}}|f) \\leqslant h(f)-1$ ce qui implique $g(f) \\leqslant h(f)+1$  ; vu (REF ), cela entraîne (REF ).", "Corollaire 5.2 Soit $f\\in \\mathcal {F}$ , $f\\ne 0$ , et soit $p$ un nombre premier tel que $p\\equiv \\pm 1 \\pmod {8}$ .", "Alors, on a $g(T_p|f)\\leqslant g(f)-2.$ Démonstration : On observe que, pour $p\\equiv \\pm 1 \\pmod {8}$ , on a $h(T_p|f) \\equiv h(f)\\pmod {2}$ , ce qui, par le théorème REF , entraîne $g(T_p|f)\\equiv g(f) \\pmod {2}$ .", "Corollaire 5.3 Soit $f\\in \\mathcal {F}$ , $f\\ne 0$ .", "Si $T_3|f = T_5|f = 0,$ alors $f=\\Delta $ .", "Démonstration : En effet, d'après (i), on a $n_3(m_1)=n_5(m_1)=0$ , d'où $m_1=1$ et $f=\\Delta $ ." ] ]

[ [ "Constructing regular ultrafilters from a model-theoretic point of view" ], [ "Abstract This paper contributes to the set-theoretic side of understanding Keisler's order.", "We consider properties of ultrafilters which affect saturation of unstable theories: the lower cofinality $\\lcf(\\aleph_0, \\de)$ of $\\aleph_0$ modulo $\\de$, saturation of the minimum unstable theory (the random graph), flexibility, goodness, goodness for equality, and realization of symmetric cuts.", "We work in ZFC except when noted, as several constructions appeal to complete ultrafilters thus assume a measurable cardinal.", "The main results are as follows.", "First, we investigate the strength of flexibility, detected by non-low theories.", "Assuming $\\kappa > \\aleph_0$ is measurable, we construct a regular ultrafilter on $\\lambda \\geq 2^\\kappa$ which is flexible (thus: ok) but not good, and which moreover has large $\\lcf(\\aleph_0)$ but does not even saturate models of the random graph.", "We prove that there is a loss of saturation in regular ultrapowers of unstable theories, and give a new proof that there is a loss of saturation in ultrapowers of non-simple theories.", "Finally, we investigate realization and omission of symmetric cuts, significant both because of the maximality of the strict order property in Keisler's order, and by recent work of the authors on $SOP_2$.", "We prove that for any $n < \\omega$, assuming the existence of $n$ measurable cardinals below $\\lambda$, there is a regular ultrafilter $D$ on $\\lambda$ such that any $D$-ultrapower of a model of linear order will have $n$ alternations of cuts, as defined below.", "Moreover, $D$ will $\\lambda^+$-saturate all stable theories but will not $(2^{\\kappa})^+$-saturate any unstable theory, where $\\kappa$ is the smallest measurable cardinal used in the construction." ], [ "Introduction", "The motivation for our work is a longstanding, and far-reaching, problem in model theory: namely, determining the structure of Keisler's order on countable first-order theories.", "Introduced by Keisler in 1967, this order suggests a way of comparing the complexity of first-order theories in terms of the difficulty of producing saturated regular ultrapowers.", "Much of the power of this order comes from the interplay of model-theoretic structure and set-theoretic constraints.", "However, this interplay also contributes to its difficulty: progress requires advances in model-theoretic analysis on the one hand, and advances in ultrapower construction on the other.", "Our work in this paper is of the second kind and is primarily combinatorial set theory, though the model-theoretic point of view is fundamental.", "As might be expected from a problem of this scope, surprising early results were followed by many years of little progress.", "Results of Shelah in [17], Chapter VI (1978) had settled Keisler's order for stable theories, as described in §REF below.", "Apart from this work, and the result on maximality of $SOP_3$ in [20], the problem of understanding Keisler's order on unstable theories was dormant for many years and seemed difficult.", "Very recently, work of Malliaris and Shelah has led to considerable advances in the understanding of how ultrafilters and theories interact (Malliaris [11]-[13], Malliaris and Shelah [15]-[16]).", "In particular, we now have much more information about properties of ultrafilters which have model-theoretic significance.", "However, the model-theoretic analysis gave little information about the relative strength of the ultrafilter properties described.", "In the current paper, we substantially clarify the picture.", "We establish various implications and non-implications between model-theoretic properties of ultrafilters, and we develop a series of tools and constraints which help the general problem of constructing ultrafilters with a precise degree of saturation.", "Though we have framed this as a model-theoretically motivated project, it naturally relates to questions in combinatorial set theory, and our results answer some questions there.", "Moreover, an interesting and unexpected phenomenon in this paper is the relevance of measurable cardinals in the construction of regular ultrafilters, see REF below.", "This paper begins with several introductory sections which frame our investigations and collect the implications of the current work.", "We give two extended examples in §, the first historical, the second involving results from the current paper.", "Following this, we give definitions and fix notation in §.", "§ gives an overview of our main results in this paper.", "§ includes context for, and implications of, our constructions.", "Sections §-§ contain the main proofs.", "In this paper, we focus on product constructions and cardinality constraints.", "In a related paper in preparation [14] we will focus on constructions via families of independent functions." ], [ "Background and examples", ".", "In this section we give two extended examples.", "The first is historical; we motivate the problem of Keisler's order, i.e.", "of classifying first-order theories in terms of saturation of ultrapowers, by explaining the classification for the stable case.", "The second involves a proof from the current paper: we motivate the idea that model-theoretic properties can give a useful way of calibrating the “strength” of ultrafilters by applying saturation arguments to prove that consistently flexible (=OK) does not imply good.", "Some definitions will be given informally; formal versions can be found in § below." ], [ "Infinite and pseudofinite sets: Theories through the lens of ultrafilters.", "This first example is meant to communicate some intuition for the kinds of model-theoretic “complexity” to which saturation of ultrapowers is sensitive.", "First, recall that questions of saturation and expressive power already arise in the two fundamental theorems of ultrapowers.", "Theorem A (Łos' theorem for first-order logic) Let $\\mathcal {D}$ be an ultrafilter on $\\lambda \\ge \\aleph _0$ , $M$ an $\\mathcal {L}$ -structure, $\\varphi (\\overline{x})$ an $\\mathcal {L}$ -formula, and $\\overline{a} \\subseteq N = M^\\lambda /\\mathcal {D}$ , $\\ell ({\\overline{a}}) = \\ell (\\overline{x})$ .", "Fixing a canonical representative of each $\\mathcal {D}$ -equivalence class, write $\\overline{a}[i]$ for the value of $\\overline{a}$ at index $i$ .", "Then $ N \\models \\varphi (\\overline{a}) \\iff \\lbrace i \\in \\lambda : M \\models \\varphi (\\overline{a}[i]) \\rbrace \\in \\mathcal {D}$ Theorem B (Ultrapowers commute with reducts) Let $M$ be an $\\mathcal {L}^\\prime $ -structure, $\\mathcal {L}\\subseteq \\mathcal {L}^\\prime $ , $\\mathcal {D}$ an ultrafilter on $\\lambda \\ge \\aleph _0$ , $N = M^\\lambda /\\mathcal {D}$ .", "Then $ \\left( M^\\lambda /\\mathcal {D}\\right)|_{\\mathcal {L}} = \\left( M|_{\\mathcal {L}} \\right)^\\lambda /\\mathcal {D}$ That is: By itself, Theorem REF may appear only to guarantee that $M \\equiv M^\\lambda /\\mathcal {D}$ .", "Yet combined with Theorem REF , it has consequences for saturation of ultrapowers, as we now explain.", "Consider the following three countable models in the language $\\mathcal {L}= \\lbrace E, = \\rbrace $ , for $E$ a binary relation symbol, interpreted as an equivalence relation.", "In $M_1$ , $E$ is an equivalence relation with two countable classes.", "In $M_2$ , $E$ is an equivalence relation with countably many countable classes.", "In $M_3$ , $E$ is an equivalence relation with exactly one class of size $n$ for each $n \\in \\mathbb {N}$ .", "What variations are possible in ultrapowers of these models?", "That is, for $N_i = M_i^\\lambda /\\mathcal {D}$ , what can we say about: (a) the number of $E^{N_i}$ -classes, (b) the possible sizes of $E^{N_i}$ -classes, (b)$^\\prime $ if two $E^{N_i}$ -classes can have unequal sizes?", "Observation 1.1 For any index set $I$ and ultrafilter $\\mathcal {D}$ on $I$ , $N_1 = (M_1)^I/\\mathcal {D}$ will have two $E$ -classes each of size $|N_1|$ $N_2 = (M_2)^I/\\mathcal {D}$ will have $|N_2|$ $E$ -classes each of size $|N_2|$ (1) Two classes follows by Łos' theorem, so we prove the fact about size.", "By Theorem REF $(M_{11}^\\lambda /\\mathcal {D})|_\\mathcal {L}= M_1^\\lambda /\\mathcal {D}$ , where $M_{11}$ is the expansion of $M_1$ to $\\mathcal {L}^\\prime = \\mathcal {L}\\cup \\lbrace f \\rbrace $ and $f$ is interpreted as a bijection between the equivalence classes.", "By Łos' theorem, $f$ will remain a bijection in $N_1$ , but Theorem REF means that whether we forget the existence of $f$ before or after taking the ultrapower, the result is the same.", "(2) Similarly, $M_2$ admits an expansion to a language with a bijection $f_1$ between $M_2$ and a set of representatives of $E$ -classes; a bijection $f_2$ between $M_2$ and a fixed $E$ -class; and a parametrized family $f_3(x,y,z)$ where for each $a,b$ , $f_3(x,a,b)$ is a bijection between the equivalence class of $b$ and that of $b$ .", "So once more, by Theorems REF and REF , the ultrapowers of $M_2$ are in a sense one-dimensional: if $N_2 = M_2^\\lambda /\\mathcal {D}$ is an ultrapower, it will be an equivalence relation with $|N_2|$ classes each of which has size $|N_2|$ .", "Now for $M_3$ , the situation is a priori less clear.", "Any nonprincipal ultrapower will contain infinite (pseudofinite) sets by Łos' theorem, but it is a priori not obvious whether induced bijections between these sets exist.", "It is easy to choose infinitely many distinct pseudofinite sets (let the $n$ th set project a.e.", "to a class whose size is a power of the $n$ th prime) which do not clearly admit bijections to each other in the index model $M$ , nor to $M$ itself.", "We have reached the frontier of what Theorem REF can control, and a property of ultrafilters comes to the surface: Definition 1.2 ([18] Definition III.3.5) Let $\\mathcal {D}$ be an ultrafilter on $\\lambda $ .", "$ \\mu (\\mathcal {D}) := \\operatorname{min} \\left\\lbrace \\rule {0pt}{15pt}\\prod _{t<\\lambda }~ n_t /\\mathcal {D}~: ~ n_t < \\aleph _0, ~\\prod _{t<\\lambda }~n_t/\\mathcal {D}\\ge \\aleph _0 \\right\\rbrace $ be the minimum value of the product of an unbounded sequence of cardinals modulo $\\mathcal {D}$ .", "Observation 1.3 Let $\\mathcal {D}$ be an ultrafilter on $\\lambda $ , let $M_3$ be the model defined above, and $N_3 = (M_3)^\\lambda /\\mathcal {D}$ .", "Then: $N_3$ will have $|N_3|$ $E$ -classes.", "$E^{N_3}$ will contain only classes of size $\\ge \\mu (\\mathcal {D})$ , and will contain at least one class of size $\\mu (\\mathcal {D})$ .", "(1) As for the number of classes, Theorem REF still applies.", "(2) Choose a sequence of cardinals $n_t$ witnessing $\\mu (\\mathcal {D})$ , and consider the class whose projection to the $t$ th index model has cardinality $n_t$ .", "Definition REF isolates a well-defined set-theoretic property of ultrafilters, and indeed, an early theorem of the second author proved that one could vary the size of $\\mu (\\mathcal {D})$ : Theorem C (Shelah, [18].VI.3.12) Let $\\mu (\\mathcal {D})$ be as in Definition REF .", "Then for any infinite $\\lambda $ and $\\nu = \\nu ^{\\aleph _0} \\le 2^{\\lambda }$ there exists a regular ultrafilter $\\mathcal {D}$ on $\\lambda $ with $\\mu (\\mathcal {D}) = \\nu $ .", "Whereas the saturation of $(M_1)^\\lambda /\\mathcal {D}$ and of $(M_2)^\\lambda /\\mathcal {D}$ will not depend on $\\mu (\\mathcal {D})$ , $N_3 = (M_3)^\\lambda /\\mathcal {D}$ will omit a type of size $\\le \\kappa $ of the form $\\lbrace E(x,a) \\rbrace \\cup \\lbrace \\lnot x=a^\\prime : N_3 \\models E(a^\\prime , a) \\rbrace $ if and only if $\\mu (\\mathcal {D}) \\le \\kappa $ .", "Restricting to regular ultrafilters, so that saturation of the ultrapower does not depend on saturation of the index model but only on its theory, the same holds if we replace each $M_i$ by some elementarily equivalent model, and is thus a statement about their respective theories.", "This separation of theories by means of their sensitivity to $\\mu (\\mathcal {D})$ is, in fact, characteristic within stability.", "Recall that a formula $\\varphi (x;y)$ has the finite cover property with respect to a theory $T$ if for all $n < \\omega $ , there are $a_0, \\dots a_n$ in some model $M \\models T$ such that the set $\\Sigma _n = \\lbrace \\varphi (x;a_0), \\dots \\varphi (x;a_n) \\rbrace $ is inconsistent but every $n$ -element subset of $\\Sigma _n$ is consistent.", "Theorem D (Shelah [18] VI.5) Let $T$ be a countable stable theory, $M \\models T$ , and $\\mathcal {D}$ a regular ultrafilter on $\\lambda \\ge \\aleph _0$ .", "Then: If $T$ does not have the finite cover property, then $M^\\lambda /\\mathcal {D}$ is always $\\lambda ^+$ -saturated.", "If $T$ has the finite cover property, then $M^\\lambda /\\mathcal {D}$ is $\\lambda ^+$ -saturated if and only if $\\mu (\\mathcal {D}) \\ge \\lambda ^+$ .", "Thus Keisler's order on stable theories has exactly two classes, linearly ordered.", "(Sketch) This relies on a characterization of saturated models of stable theories: $N$ is $\\lambda ^+$ -saturated if and only if it is $\\kappa (T)$ -saturated and every maximal indiscernible set has size $\\ge \\lambda ^+$ .", "[This relies heavily on uniqueness of nonforking extensions: given a type $p$ one hopes to realize over some $A$ , $|A| \\le \\lambda $ , restrict $p$ to a small set over which it does not fork, and use $\\kappa (T)$ -saturation to find a countable indiscernible sequence of realizations of the restricted type.", "By hypothesis, we may assume this indiscernible sequence extends to one of size $\\lambda ^+$ , and by uniqueness of nonforking extensions, any element of this sequence which does not fork with $A$ will realize the type.]", "Returning to ultrapowers: for countable theories, $\\kappa (T) \\le \\aleph _1$ and any nonprincipal ultrapower is $\\aleph _1$ -saturated.", "So it suffices to show that any maximal indiscernible set is large, and the theorem proves, by a coding argument, that this is true whenever the size of every pseudofinite set is large.", "Discussion.", "As mentioned above, in this paper we construct ultrafilters with “model-theoretically significant properties.” The intent of this example was to motivate our work by showing what “model-theoretically significant” might mean.", "However, the example also illustrates what kinds of properties may fit the bill.", "We make two general remarks.", "“Only formulas matter”: The fact that $\\mu (\\mathcal {D})$ was detected by a property of a single formula, the finite cover property, is not an accident.", "For $\\mathcal {D}$ a regular ultrafilter and $M \\models T$ any countable theory, $M^\\lambda /\\mathcal {D}$ is $\\lambda ^+$ -saturated if and only if it is $\\lambda ^+$ -saturated for $\\varphi $ -types, for all formulas $\\varphi $ , Malliaris [10] Theorem 12.", "Thus, from the point of view of Keisler's order, it suffices to understand properties of regular ultrafilters which are detected by formulas.", "The role of pseudofinite structure is fundamental, reflecting the nature of the objects involved (regular ultrapowers, first-order theories).", "On one hand, pseudofinite phenomena can often be captured by a first-order theory.", "On the other, saturation of regular ultrapowers depends on finitely many conditions in each index model, since by definition regular ultrafilters $\\mathcal {D}$ on $I$ , $|I| = \\lambda $ contain regularizing families, i.e.", "$\\lbrace X_i : i < \\lambda \\rbrace $ such that for each $t \\in I$ , $|\\lbrace i < \\lambda : t \\in X_i \\rbrace | < \\aleph _0$ ." ], [ "Flexibility without goodness: Ultrafilters through the lens of theories.", "Our second example takes the complementary point of view.", "The following is a rich and important class of ultrafilters introduced by Keisler: Definition 1.4 (Good ultrafilters, Keisler [4]) The filter $\\mathcal {D}$ on $I$ is said to be $\\mu ^+$ -good if every $f: {\\mathcal {P}}_{\\aleph _0}(\\mu ) \\rightarrow \\mathcal {D}$ has a multiplicative refinement, where this means that for some $f^\\prime : {\\mathcal {P}}_{\\aleph _0}(\\mu ) \\rightarrow \\mathcal {D}$ , $u \\in {\\mathcal {P}}_{\\aleph _0}(\\mu ) \\Rightarrow f^\\prime (u) \\subseteq f(u)$ , and $u,v \\in {\\mathcal {P}}_{\\aleph _0}(\\mu ) \\Rightarrow f^\\prime (u) \\cap f^\\prime (v) = f^\\prime (u \\cup v)$ .", "Note that we may assume the functions $f$ are monotonic.", "$\\mathcal {D}$ is said to be good if it is $|I|^+$ -good.", "It is natural to ask for meaningful weakenings of this notion, e.g.", "by requiring only that certain classes of functions have multiplicative refinements.", "An important example is the notion of $OK$ , which appeared without a name in Keisler [4], was named and studied by Kunen [8] and investigated generally by Dow [3] and by Baker and Kunen [1].", "We follow the definition from [3] 1.1.", "Definition 1.5 (OK ultrafilters) The filter $\\mathcal {D}$ on $I$ is said to be $\\lambda $ -OK if each monotone function $g: {\\mathcal {P}}_{\\aleph _0}(\\lambda ) \\rightarrow \\mathcal {D}$ with $g(u) = g(v)$ whenever $|u| = |v|$ has a multiplicative refinement $f: {\\mathcal {P}}_{\\aleph _0}(\\lambda ) \\rightarrow \\mathcal {D}$ .", "It is immediate that $\\lambda ^+$ -good implies $\\lambda $ -OK.", "Though OK is an a priori weaker notion, the relative strength of OK and good was not clear.", "For instance, in [3] 3.10 and 4.7, Dow raises the problem of constructing ultrafilters which are $\\lambda ^+$ -OK but not $\\lambda ^+$ -good; to our knowledge, even the question of constructing $\\lambda $ -OK not $\\lambda ^+$ -good ultrafilters on $\\lambda $ was open.", "Before discussing how a model-theoretic perspective can help with such questions, we define the main objects of interest in this paper: Definition 1.6 (Regular filters) A filter $\\mathcal {D}$ on an index set $I$ of cardinality $\\lambda $ is said to be $\\lambda $ -regular, or simply regular, if there exists a $\\lambda $ -regularizing family $\\langle X_i : i<\\lambda \\rangle $ , which means that: for each $i<\\lambda $ , $X_i \\in \\mathcal {D}$ , and for any infinite $\\sigma \\subset \\lambda $ , we have $\\bigcap _{i \\in \\sigma } X_i = \\emptyset $ Equivalently, for any element $t \\in I$ , $t$ belongs to only finitely many of the sets $X_i$ .", "Now we make a translation.", "As Keisler observed, good regular ultrafilters can be characterized as those regular ultrafilters able to saturate any countable theory.", "(By “$\\mathcal {D}$ saturates $T$ ” we will always mean: $\\mathcal {D}$ is a regular ultrafilter on the infinite index set $I$ , $T$ is a countable complete first-order theory and for any $M \\models T$ , we have that $M^I/\\mathcal {D}$ is $\\lambda ^+$ -saturated, where $\\lambda = |I|$ .)", "We state this as a definition and an observation, which together say simply that the distance between consistency of a type (i.e.", "finite consistency, reflected by Łos' theorem) and realization of a type in a regular ultrapower can be explained by whether or not certain monotonic functions have multiplicative refinements.", "Definition 1.7 Let $T$ be a countable complete first-order theory, $M \\models T$ , $\\mathcal {D}$ a regular ultrafilter on $I$ , $|I| = \\lambda $ , $N = M^\\lambda /\\mathcal {D}$ .", "Let $p(x) = \\lbrace \\varphi _i(x;a_i) : i < \\lambda \\rbrace $ be a consistent partial type in the ultrapower $N$ .", "Then a distribution of $p$ is a map $d : {\\mathcal {P}}_{\\aleph _0}(\\lambda ) \\rightarrow \\mathcal {D}$ which satisfies: For each $\\sigma \\in [\\lambda ]^{<\\aleph _0}$ , $d(\\sigma ) \\subseteq \\lbrace t \\in I : M \\models \\exists x \\bigwedge \\lbrace \\varphi _i(x;a_i[t]) : i \\in \\sigma \\rbrace \\rbrace $ .", "Informally speaking, $d$ refines the Łos map.", "$d$ is monotonic, meaning that $\\sigma , \\tau \\in [\\lambda ]^{<\\aleph _0}$ , $\\sigma \\subseteq \\tau $ implies $d(\\sigma ) \\supseteq d(\\tau )$ The set $\\lbrace d(\\sigma ) : \\sigma \\in [\\lambda ]^{<\\aleph _0} \\rbrace $ is a regularizing family, i.e.", "each $t \\in I$ belongs to only finitely many elements of this set.", "Observation 1.8 Let $T$ be a countable complete first-order theory, $M \\models T$ , $\\mathcal {D}$ a regular ultrafilter on $\\lambda $ , $N = M^\\lambda /\\mathcal {D}$ .", "Then the following are equivalent: For every consistent partial type $p$ in $N$ of size $\\le \\lambda $ , some distribution $d$ of $p$ has a multiplicative refinement.", "$N$ is $\\lambda ^+$ -saturated.", "The obstacle to realizing the type is simply that, while Łos' theorem guarantees each finite subset of $p$ is almost-everywhere consistent, there is no a priori reason why, at an index $t \\in I$ at which $M \\models \\exists x \\bigwedge \\lbrace \\varphi _i(x;a_i[t]) : i \\in \\sigma \\rbrace $ , $M \\models \\exists x \\bigwedge \\lbrace \\varphi _i(x;a_j[t]) : j \\in \\tau \\rbrace $ , these two sets should have a common witness.", "The statement that $d$ has a multiplicative refinement is precisely the statement that there is, in fact, a common witness almost everywhere, in other words $t \\in d(\\sigma ) \\cap d(\\tau ) \\Rightarrow t \\in d(\\sigma \\cup \\tau )$ .", "When this happens, we may choose at each index $t$ an element $c_t$ such that $\\sigma \\in [\\lambda ]^{<\\aleph _0} \\wedge t \\in d(\\sigma ) \\Rightarrow M \\models \\bigwedge \\lbrace \\varphi _i(c_t : a_i[t]) : i \\in \\sigma \\rbrace $ , by REF (1).", "Then by Łos' theorem and REF (1), $\\prod _{t < \\lambda } c_t$ will realize $p$ in $N$ .", "The other direction is clear (choose a realization $a$ and use Łos' theorem to send each finite subset of the type to the set on which it is realized by $a$ ).", "Keisler's characterization of good ultrafilters then follows from showing that there are first order theories which can “code” enough possible patterns to detect whether any $f: {\\mathcal {P}}_{\\aleph _0}(\\lambda ) \\rightarrow \\mathcal {D}$ fails to have a multiplicative refinement.", "Note that first-order theories correspond naturally to monotonic functions of a certain kind (depending, very informally speaking, on some notion of the pattern-complexity inherent in the theory) and thus, were one to succeed in building ultrafilters which were able to saturate certain theories and not saturate others, this would likewise show a meaningful weakening of goodness.", "In this context we mention a property which arose in the study of certain unstable simple theories, called non-low.", "The original definition is due to Buechler.", "Definition 1.9 The formula $\\varphi (x;y)$ is called non-low with respect to the theory $T$ if in some sufficiently saturated model $M \\models T$ , for arbitrarily large $k < \\omega $ , there exists an infinite indiscernible sequence $\\lbrace a_i : i < \\omega \\rbrace $ , with $i < \\omega \\Rightarrow \\ell (a_i) = \\ell (y)$ , such that every $k$ -element subset of $ \\lbrace \\varphi (x;a_i) : i < \\omega \\rbrace $ is consistent, but every $k+1$ -element subset is inconsistent.", "Here we make a second translation.", "Recall from Definition REF above that the characteristic objects of regular filters $\\mathcal {D}$ on $\\lambda $ are $(\\lambda )$ -regularizing families, i.e.", "sets of the form $\\lbrace X_i : i < \\lambda \\rbrace $ with $t \\in I \\Rightarrow | \\lbrace i < \\lambda : t \\in X_i \\rbrace | = n_t < \\aleph _0$ .", "Malliaris had noticed in [11] that non-low formulas could detect the size (i.e.", "the nonstandard integer whose $t$ th coordinate is $n_t$ ) of the regularizing families in $\\mathcal {D}$ , and thus had defined and studied the “flexibility” of a filter, Definition REF .", "Definition 1.10 (Flexible ultrafilters, Malliaris [11], [12]) We say that the filter $\\mathcal {D}$ is $\\lambda $ -flexible if for any $f \\in {^I \\mathbb {N}}$ with $n \\in \\mathbb {N} \\Rightarrow n <_{\\mathcal {D}} f$ , we can find $X_\\alpha \\in \\mathcal {D}$ for $\\alpha < \\lambda $ such that for all $t \\in I$ $ f(t) \\ge | \\lbrace \\alpha : t \\in X_\\alpha \\rbrace | $ Informally, given any nonstandard integer, we can find a $\\lambda $ -regularizing family below it.", "Specifically, Malliaris had shown that if $\\mathcal {D}$ is not $\\lambda $ -flexible then it fails to $\\lambda ^+$ -saturate any theory containing a non-low formula.", "(Note that by Keisler's observation about good ultrafilters, any property of ultrafilters which can be shown to be detected by formulas must necessarily hold of good ultrafilters.)", "Moreover, we mention a useful convergence.", "Kunen had brought the definition of “OK” filters to Malliaris' attention in 2010; the notions of “$\\lambda $ -flexible” and “$\\lambda $ -OK” are easily seen to be equivalent, Observation REF below.", "We now sketch the proof from § below that consistently flexible need not imply good.", "(This paper and its sequel [14] contain at least three distinct proofs of that fact, of independent interest.)", "The numbering of results follows that in §.", "To begin, we use a diagonalization argument to show that saturation decays in ultrapowers of the random graph, i.e.", "the Rado graph, Definition REF below.", "(“The random graph” means, from the set-theoretic point of view, that the function which fails to have a multiplicative refinement will code the fact that there are two sets $A,B$ in the final ultrapower N, $|A| = |B| = \\lambda $ , which are disjoint in $N$ but whose projections to the index models cannot be taken to be a.e.", "disjoint.)", "Claim REF .", "Assume $\\lambda \\ge \\kappa \\ge \\aleph _0$ , $T=T_{rg}$ , $M$ a $\\lambda ^+$ -saturated model of $T$ , $E$ a uniform ultrafilter on $\\kappa $ such that $|\\kappa ^\\kappa /E| = 2^\\kappa $ Then $M^\\kappa /E$ is not $(2^{\\kappa })^+$ -saturated.", "Note that the hypothesis of the claim will be satisfied when $E$ is regular, and also when $E$ is complete.", "Our strategy will be to take a product of ultrafilters $D \\times E$ , where $D$ is a regular ultrafilter on $\\lambda $ and $E$ is an ultrafilter on $\\kappa $ .", "Then $D \\times E$ will be regular, and if $\\lambda \\ge 2^\\kappa $ , it will fail to saturate the random graph, thus fail to be good.", "What remains is to ensure flexibility, and for this we will need $E$ to be $\\aleph _1$ -complete.", "In the following Corollary, $\\operatorname{lcf}(\\aleph _0, \\mathcal {D})$ is the coinitiality of $\\mathbb {N}$ in $(\\mathbb {N}, <)^I/\\mathcal {D}$ , i.e.", "the cofinality of the set of $\\mathcal {D}$ -nonstandard integers.", "Corollary REF Let $\\lambda , \\kappa \\ge \\aleph _0$ and let $\\mathcal {D}_1$ , $E$ be ultrafilters on $\\lambda , \\kappa $ respectively where $\\kappa > \\aleph _0$ is measurable.", "Let $\\mathcal {D}= \\mathcal {D}_1 \\times E$ be the product ultrafilter on $\\lambda \\times \\kappa $ .", "Then: If $\\mathcal {D}_1$ is $\\lambda $ -flexible and $E$ is $\\aleph _1$ -complete, then $\\mathcal {D}$ is $\\lambda $ -flexible.", "If $\\lambda \\ge \\kappa $ and $\\operatorname{lcf}(\\aleph _0, \\mathcal {D}_1) \\ge \\lambda ^+$ , then $\\operatorname{lcf}(\\aleph _0, \\mathcal {D}) \\ge \\lambda ^+$ , so in particular, $\\mathcal {D}= D \\times E$ will $\\lambda ^+$ -saturate any countable stable theory.", "(Sketch) (1) We first show that the following are equivalent: (i) any $\\mathcal {D}$ -nonstandard integer projects $E$ -a.e.", "to a $\\mathcal {D}_1$ -nonstandard integer, (ii) $E$ is $\\aleph _1$ -complete.", "Then, since we have assumed (ii) holds, let some $\\mathcal {D}$ -nonstandard integer $n_*$ be given.", "By (ii), for $E$ -almost all $t \\in \\kappa $ , $n_*[t]$ is $\\mathcal {D}_1$ -nonstandard and by the flexibility of $\\mathcal {D}_1$ there is a regularizing family $\\lbrace X^t_i : i < \\lambda \\rbrace \\subseteq \\mathcal {D}_1$ below any such $n_*[t]$ .", "Let $X_i = \\lbrace (s,t) : s \\in X^t_i \\rbrace \\subseteq \\mathcal {D}$ .", "It follows that $\\lbrace X_i : i <\\lambda \\rbrace $ is a regularizing family in $\\mathcal {D}$ below $m_*$ and thus below $n_*$ .", "(2) From the first sentence of (1), we show that if $E$ is $\\aleph _1$ -complete and in addition $\\operatorname{lcf}(\\aleph _0, \\mathcal {D}) \\ge \\lambda ^+$ , then the $\\mathcal {D}_1$ -nonstandard integers (under the diagonal embedding) are cofinal in the $\\mathcal {D}$ -nonstandard integers.", "This suffices.", "For the second clause, see Theorem REF , § below.", "Thus we obtain: Theorem REF .", "Assume $\\aleph _0 < \\kappa < \\lambda = \\lambda ^\\kappa $ , $2^\\kappa \\le \\lambda $ , $\\kappa $ measurable.", "Then there exists a regular uniform ultrafilter $\\mathcal {D}$ on $\\lambda $ such that $\\mathcal {D}$ is $\\lambda $ -flexible, yet for any model $M$ of the theory of the random graph, $M^\\lambda /\\mathcal {D}$ is not $(2^\\kappa )^+$ -saturated.", "Thus $\\mathcal {D}$ is not good, and will fail to $(2^\\kappa )^+$ -saturate any unstable theory.", "However, $\\mathcal {D}$ will $\\lambda ^+$ -saturate any countable stable theory.", "Note that the model-theoretic failure of saturation is quite strong, more so than simply “not good.” The random graph is known to be minimum among unstable theories in Keisler's order (meaning that any regular $\\mathcal {D}$ which fails to saturate the random graph will fail to saturate any other unstable theory).", "This is the strongest failure of saturation one could hope for given that $\\operatorname{lcf}(\\aleph _0, \\mathcal {D})$ is large, see Section for details.", "Theorem REF has the following immediate corollary in the language of OK and good: Corollary 1.11 Assume $\\aleph _0 < \\kappa < 2^\\kappa \\le \\mu _1 \\le \\mu _2 < \\lambda = \\lambda ^\\kappa $ and $\\kappa $ is measurable.", "Then there exists a regular uniform ultrafilter $\\mathcal {D}$ on $\\lambda $ such that $\\mathcal {D}$ is $\\lambda $ -flexible, thus $\\lambda $ -OK, but not $(2^\\kappa )^+$ -good.", "In particular, $\\mathcal {D}$ is $(\\mu _2)^+$ -OK but not $(\\mu _1)^+$ -good.", "In particular, this addresses the problem raised by Dow in [3] 3.10 and 4.7, namely, the problem of constructing ultrafilters which are $\\alpha ^+$ -OK and not $\\alpha ^+$ -good.", "Discussion.", "The intent of this example was to show that model theory can contribute to calibrating ultrafilters.", "Note that in terms of determining the strength of a priori weakenings of goodness, the model-theoretic perspective has given both positive and negative results: On one hand, Theorem REF applies model-theoretic arguments to show that multiplicative refinements for size-uniform functions $f: {\\mathcal {P}}_{\\aleph _0}(\\lambda ) \\rightarrow \\mathcal {D}$ are not enough to guarantee multiplicative refinements for all such functions.", "On the other, the second author's proof of the maximality of strict order (see Theorem REF (6) below) does isolate an a priori weaker class of functions which have such a guarantee – namely, those corresponding to distributions of types in linear order.", "The set-theoretic question of why these functions suffice appears to be deep.", "The model-theoretic formulation of “determine a minimum such set of functions” is: determine a necessary condition for maximality in Keisler's order.", "This concludes our two examples.", "We now fix definitions and notation, before giving a summary of our results in §." ], [ "Definitions and conventions", "This section contains background, most definitions, and conventions.", "Note that the definition of $\\mu (\\mathcal {D})$ was given in Definition REF , and the definitions of good, regular and flexible filters were Definitions REF , REF and REF above.", "(Recall that a filter is said to be $\\lambda $ -regular if it contains a family of $\\lambda $ sets any countable number of which have empty intersection, REF above.)", "Let $I = \\lambda \\ge \\aleph _0$ and fix $f: {\\mathcal {P}}_{\\aleph _0}(\\lambda )\\rightarrow I$ .", "Then $\\lbrace \\lbrace s\\in I : \\eta \\in f^{-1}(s) \\rbrace : \\eta < \\lambda \\rbrace $ can be extended to a regular filter on $I$ , so regular ultrafilters on $\\lambda \\ge \\aleph _0$ always exist, see [2].", "Keisler proposed in 1967 [5] that saturation properties of regular ultrapowers might be used to classify countable first-order theories.", "His preorder $\\trianglelefteq $ on theories is often thought of as a partial order on the $\\trianglelefteq $ -equivalence classes, and so known as “Keisler's order.” Definition 2.1 (Keisler [5]) Given countable theories $T_1, T_2$ , say that: $T_1 \\trianglelefteq _\\lambda T_2$ if for any $M_1 \\models T_1, M_2 \\models T_2$ , and $\\mathcal {D}$ a regular ultrafilter on $\\lambda $ , if $M^{\\lambda }_2/\\mathcal {D}$ is $\\lambda ^+$ -saturated then $M^{\\lambda }_1/\\mathcal {D}$ must be $\\lambda ^+$ -saturated.", "(Keisler's order) $T_1 \\trianglelefteq T_2$ if for all infinite $\\lambda $ , $T_1 \\trianglelefteq _\\lambda T_2$ .", "Question 2.2 Determine the structure of Keisler's order.", "The hypothesis regular justifies the quantification over all models: when $T$ is countable and $\\mathcal {D}$ is regular, saturation of the ultrapower does not depend on the choice of index model.", "Theorem E (Keisler [5] Corollary 2.1 p. 30; see also Shelah [18].VI.1) Suppose that $M_0 \\equiv M_1$ , the ambient language is countable, and $\\mathcal {D}$ is a regular ultrafilter on $\\lambda $ .", "Then ${M_0}^\\lambda /\\mathcal {D}$ is $\\lambda ^+$ -saturated iff ${M_1}^\\lambda /\\mathcal {D}$ is $\\lambda ^+$ -saturated.", "More information on Keisler's order, including many examples and a summary of all known results through early 2010, may be found in the introduction to the first author's paper [12].", "More recent results are incorporated (in a somewhat denser form) into the various summary theorems of § below.", "On the opposite end of the spectrum to “regular,” we have: Definition 2.3 (Complete ultrafilters) The ultrafilter $\\mathcal {E}$ on $\\kappa $ is said to be $\\mu $ -complete if for any $\\lbrace X_i : i < \\mu ^\\prime < \\mu \\rbrace \\subseteq \\mathcal {E}$ , $\\bigcap \\lbrace X_i : i < \\mu ^\\prime \\rbrace \\in \\mathcal {E}$ .", "Working with complete ultrafilters, we are obliged to make large cardinal hypotheses.", "We will use measurable, normal and to a lesser extent, weakly compact cardinals.", "Their utility for our arguments will be clear from the choice of definitions: Definition 2.4 (Measurable, weakly compact) The uncountable cardinal $\\kappa $ is said to be measurable if there is a $\\kappa $ -complete nonprincipal ultrafilter on $\\kappa $ .", "The uncountable cardinal $\\kappa $ is said to be weakly compact if $\\kappa \\rightarrow (\\kappa )^2_2$ .", "Fact 2.5 If $\\kappa > \\aleph _0$ is weakly compact, $n < \\aleph _0$ and $\\rho < \\kappa $ , then for any $\\alpha : [\\kappa ]^n \\rightarrow \\rho $ there exists $\\mathcal {U}\\subseteq \\kappa $ , $|\\mathcal {U}| = \\kappa $ such that $\\langle \\alpha (\\epsilon _1, \\dots \\epsilon _n) : \\epsilon _1 < \\dots < \\epsilon _n ~\\mbox{from $\\mathcal {U}$} \\rangle $ is constant.", "Definition 2.6 (Normal ultrafilters) A filter $D$ on $\\kappa $ is normal when, for any sequence $\\langle A_i : i < \\kappa \\rangle $ with $i < \\kappa \\Rightarrow A_i \\in D$ , $ \\lbrace \\alpha < \\kappa : (\\forall j < 1+\\alpha ) (\\alpha \\in A_j) \\rbrace \\in D $ Fact 2.7 Let $\\kappa $ be a measurable cardinal.", "Then there exists a normal, $\\kappa $ -complete, uniform ultrafilter $D$ on $\\kappa $ .", "for any $f: \\kappa \\rightarrow \\kappa $ which is regressive on $X \\in D$ , there is a set $Y \\in D$ , $Y \\subseteq X$ on which $f$ is constant.", "Discussion 2.8 An interesting and unexpected phenomenon visible in this work is the relevance of measurable cardinals, and in particular $\\kappa $ -complete nonprincipal ultrafilters, in the construction of regular ultrafilters.", "In the 1960s, model theorists pointed out regularity as a central property of ultrafilters, and generally concentrated on this case.", "Regularity ensures that saturation [of ultrapowers of models of complete countable theories] does not depend on the saturation of the index model, and that the cardinality of ultrapowers can be settled ($M^I/\\mathcal {D}= M^{|I|}$ ).", "Meanwhile, the construction of various non-regular ultrafilters was investigated by set theorists.", "However, many questions about regular ultrafilters remained opaque from the model-theoretic point of view.", "For example, from the point of view of Theorem REF , p. REF above, the regular ultrafilters with large $\\operatorname{lcf}(\\aleph _0)$ – a condition which implies that these ultrafilters saturate ultrapowers of stable theories – appeared to look alike.", "Moreover, it was not clear whether various a priori weakenings of goodness (e.g.", "flexible/ok) were indeed weaker.", "Here, in several different constructions, we combine both lines of work, using $\\kappa $ -complete ultrafilters to construct regular ultrafilters on $\\lambda > \\kappa $ with model-theoretically meaningful properties, i.e.", "presence or absence of some specific kind of saturation.", "We mention several other relevant properties.", "Definition 2.9 (Good for equality, Malliaris [13]) Let $\\mathcal {D}$ be a regular ultrafilter.", "Say that $\\mathcal {D}$ is good for equality if for any set $X \\subseteq N = M^I/\\mathcal {D}$ , $|X| \\le |I|$ , there is a distribution $d: X \\rightarrow \\mathcal {D}$ such that $t \\in \\lambda , t \\in d(a) \\cap d(b)$ implies that $(M \\models a[t] = b[t]) \\iff (N \\models a = b)$ .", "Definition 2.10 (Lower cofinality, $\\operatorname{lcf}(\\kappa , D)$ ) Let $D$ be an ultrafilter on $I$ and $\\kappa $ a cardinal.", "Let $N = (\\kappa , <)^I/\\mathcal {D}$ .", "Let $X \\subset N$ be the set of elements above the diagonal embedding of $\\kappa $ .", "We define $\\operatorname{lcf}(\\kappa , D)$ to be the cofinality of $X$ considered with the reverse order.", "Note: this is sometimes called the coinitiality of $\\kappa $ .", "Definition 2.11 (Product ultrafilters) Let $I_1, I_2$ be infinite sets and let $D_1, D_2$ be ultrafilters on $I_1, I_2$ respectively.", "Then the product ultrafilter $D = D_1 \\times D_2$ on $I_1 \\times I_2$ is defined by: $ X \\in D \\iff \\lbrace t \\in I_2 ~:~ \\lbrace s \\in I_1 ~:~ (s,t) \\in X \\rbrace \\in D_1 \\rbrace \\in D_2 $ for any $X \\subseteq I_1 \\times I_2$ .", "Finally, it will be useful to have a name for functions, or relations, to which Łos' theorem applies since they are visible in an expanded language: Definition 2.12 (Induced structure) Let $N = M^\\lambda /\\mathcal {D}$ be an ultrapower and $X \\subseteq N^{m}$ .", "Say that $X$ is an induced function, or relation, if there exists a new function, or relation, symbol $P$ of the correct arity, and an expansion $M^\\prime _t$ of each index model $M_t$ to $\\mathcal {L}\\cup \\lbrace P \\rbrace $ , so that $P^N \\equiv X \\mathcal {}\\mod {\\mathcal {D}}$ .", "Equivalently, $X$ is the ultraproduct modulo $\\mathcal {D}$ of its projections to the index models.", "Definition 2.13 (Cuts in regular ultrapowers of linear orders) For a model $M$ expanding the theory of linear order, a $(\\kappa _1, \\kappa _2)$ -cut in $M$ is given by sequences $\\langle a_i : i < \\kappa _1 \\rangle $ , $\\langle b_j : j < \\kappa _2 \\rangle $ of elements of $M$ such that $i_1 < i_2 < \\kappa _1 \\Rightarrow a_{i_1} < a_{i_2}$ $j_1 < j_2 < \\kappa _2 \\Rightarrow b_{j_2} < b_{j_1}$ $i < \\kappa _1$ , $j < \\kappa _2$ implies $a_i < b_j$ and the type $\\lbrace a_i < x < b_j : i < \\kappa _1, j<\\kappa _2 \\rbrace $ is omitted in $M$ .", "For $\\mathcal {D}$ a (regular) ultrafilter on $I$ we define: $ \\mathcal {C}(\\mathcal {D}) = \\left\\lbrace (\\kappa _1, \\kappa _2) \\in (\\operatorname{Reg} \\cap {|I|}^+) \\times (\\operatorname{Reg} \\cap {|I|}^+) :~\\mbox{$(\\mathbb {N}, <)^I/\\mathcal {D}$ has a $(\\kappa _1, \\kappa _2)$-cut} \\right\\rbrace $ Here we list the main model-theoretic properties of formulas used in this paper.", "For $TP_1$ /$SOP_2$ and $TP_2$ , see §.", "The finite cover property is from Keisler [5] and the order property, independence property and strict order property are from Shelah [18].II.4.", "Definition 2.14 (Properties of formulas) Let $\\varphi =\\varphi (x;y)$ be a formula of $T$ and $M \\models T$ be any sufficiently saturated model.", "Note that $\\ell (x), \\ell (y)$ are not necessarily 1.", "Say that the formula $\\varphi (x;y)$ has: not the finite cover property, written nfcp, if there exists $k<\\omega $ such that: for any $A \\subseteq M$ and any set $X = \\lbrace \\varphi (x;a) : a \\in A \\rbrace $ of instances of $\\varphi $ , $k$ -consistency implies consistency.", "(This does not depend on the model chosen.)", "the finite cover property, written fcp, if it does not have nfcp: that is, for arbitrarily large $k < \\omega $ there exist $a_0, \\dots a_k \\in M$ such that $\\lbrace \\varphi (x;a_0), \\dots \\varphi (x;a_k) \\rbrace $ is inconsistent, but every $k$ -element subset is consistent.", "the order property if there exist elements $a_i$ $(i<\\omega )$ such that for each $n<\\omega $ , the following partial type is consistent: $ \\lbrace \\lnot \\varphi (x;a_i) : i < n \\rbrace \\cup \\lbrace \\varphi (x;a_j) : j \\ge n \\rbrace $ Formulas with the order property are called $\\emph {unstable}$ .", "the independence property if there exist elements $a_i$ $(i<\\omega )$ such that for each $\\sigma , \\tau \\in [\\omega ]^{<\\aleph _0}$ with $\\sigma \\cap \\tau = \\emptyset $ , the following partial type is consistent: $ \\lbrace \\lnot \\varphi (x;a_i) : i \\in \\sigma \\rbrace \\cup \\lbrace \\varphi (x;a_j) : j \\in \\tau \\rbrace $ Note that the independence property implies the order property.", "the strict order property if there exist elements $a_i$ $(i<\\omega )$ such that for all $j \\ne i < \\omega $ , $ \\left( \\exists x ( \\lnot \\varphi (x;a_j) \\wedge \\varphi (x;a_i)) \\iff j<i \\right) $ Note that $(4)$ , $(5)$ each imply $(3)$ .", "A theory $T$ is said to have the finite cover property, the order property, the independence property or the strict order property iff one of its formulas does, and to have $nfcp$ if all of its formulas do.", "The “random graph” is known to be minimum in Keisler's order among the unstable theories, and so will feature in our proofs with some regularity.", "Definition 2.15 The random graph, i.e.", "the Rado graph, is (the unique countable model of) the complete theory in the language with equality and a binary relation $R$ axiomatized by saying that there are infinitely many elements, and that for each $n$ , and any two disjoint subsets of size $n$ , there is an element which $R$ -connects to all elements in the first set and to none in the second set.", "We conclude this section with some conventions which hold throughout the paper.", "Convention 2.16 (Conventions) The letters $D, E, \\mathcal {D}, \\mathcal {E}$ are used for filters.", "Generally, we reserve $\\mathcal {D}$ for a regular filter or ultrafilter, and $\\mathcal {E}$ for a $\\kappa $ -complete ultrafilter where $\\kappa \\ge \\aleph _0$ , though this is always stated in the relevant proof.", "Throughout, tuples of variables may be written without overlines, that is: when we write $\\varphi = \\varphi (x;y)$ , neither $x$ nor $y$ are necessarily assumed to have length 1.", "For transparency, all languages are assumed to be countable.", "As mentioned in §REF , by “$\\mathcal {D}$ saturates $T$ ” we will always mean: $\\mathcal {D}$ is a regular ultrafilter on the infinite index set $I$ , $T$ is a countable complete first-order theory and for any $M \\models T$ , we have that $M^I/\\mathcal {D}$ is $\\lambda ^+$ -saturated, where $\\lambda = |I|$ .", "We will also say that the ultrafilter $D$ is “good” (or: “not good”)  for the theory $T$ to mean that $D$ saturates (or: does not saturate)  the theory $T$ .", "We reserve the word cut in models of linear order for omitted types.", "A partial type in a model $M$ given by some pair of sequences $( \\langle a_\\alpha : \\alpha < \\kappa _1 \\rangle , \\langle b_\\beta : \\beta < \\kappa _2 \\rangle )$ with $\\alpha < \\alpha ^\\prime < \\kappa _1, \\beta < \\beta ^\\prime < \\kappa _2 \\Rightarrow M \\models a_\\alpha < a_{\\alpha ^\\prime } < b_{\\beta ^\\prime } < b_\\beta $ , which may or may not have a realization in $M$ , is called a pre-cut.", "See also Definition REF ." ], [ "Description of results", "In this section, we describe the main results of the paper.", "Some notes: For relevant definitions and conventions (“$\\mathcal {D}$ saturates $T$ ,” “good for,” “pre-cut”) see § above, in particular REF .", "Lists of the properties mentioned from the point of view of Keisler's order can be found in Theorems REF -REF , §.", "The reader unused to phrases of the form “not good for the random graph therefore not good” is referred to §REF , in particular the half-page following Definition REF .", "The first main result, Theorem REF , was discussed in §REF above; we will give a few more details here.", "We begin by showing that for $E$ an ultrafilter on $\\kappa $ , if $E$ is $\\kappa $ -regular or $E$ is $\\kappa $ -complete then $M^\\kappa /E$ will not be $(2^\\kappa )^+$ -saturated for any $\\kappa ^+$ -saturated model of $T_{rg}$ , the theory of the random graph.", "This is essentially a diagonalization argument, and the saturation of $M$ is not important if $E$ is regular.", "Note that with this result in hand, we have a useful way of producing regular ultrafilters which are not good: for any $\\lambda $ -regular $\\mathcal {D}_1$ on $\\lambda $ , the product ultrafilter $\\mathcal {D}= \\mathcal {D}_1 \\times E$ will remain regular but will not be good for $T_{rg}$ when $2^\\kappa \\le \\lambda $ .", "We then show, as sketched above, that when $E$ is $\\aleph _1$ -complete and $\\operatorname{lcf}(\\aleph _0, \\mathcal {D}_1)$ is large the nonstandard integers of $\\mathcal {D}_1$ will be cofinal in those of $\\mathcal {D}$ (under the diagonal embedding) and thus that $\\mathcal {D}$ will inherit both the large $\\operatorname{lcf}(\\aleph _0)$ and the flexibility of $\\mathcal {D}_1$ .", "This yields Theorem REF : for any $\\lambda \\ge 2^\\kappa $ , $\\kappa $ measurable there is a regular ultrafilter on $\\lambda $ which is flexible but not good, which has large $\\operatorname{lcf}(\\aleph _0)$ and thus saturates all stable theories, but does not saturate any unstable theory.", "In at least one sense, this is a surprising reversal.", "From the model-theoretic point of view, flexible ultrafilters had appeared “close” in power to ultrafilters capable of saturating any first-order theory.", "By Theorem REF , consistently flexibility cannot guarantee the saturation of any unstable theory, since the random graph is minimum among the unstable theories in Keisler's order.", "Thus the space between flexible and good is potentially quite large.", "Discussion.", "In fact we obtain several different flexible-not-good theorems, including Conclusion REF and a related theorem in [14].", "These results have other advantages, and less dramatic failures of saturation.", "In fact, if the stated failures of saturation can be shown to be sharp, this could be quite useful for obtaining further dividing lines within the unstable theories.", "Second, we give a new proof that there is a loss of saturation in ultrapowers of non-simple theories, Conclusion REF .", "Specifically, we show that if $M$ is a model of a non simple theory and $\\mathcal {D}$ a regular ultrafilter on $\\lambda = \\lambda ^{<\\lambda }$ , then for some formula $\\varphi $ $M^\\lambda /\\mathcal {D}$ is not $\\lambda ^{++}$ -saturated for $\\varphi $ -types.", "(The second author's book [18].VI contains a first proof of this result.)", "Unlike the unstable case, we do not need diagonalization per se, but instead show that both forms of the tree property allow for large consistent types to be built from minimally compatible constellations of formulas in each (quite saturated) index model.", "Since a degree of inconsistency just above what is visible to Łos' theorem has been hard-coded in to the choice of type, and the size of the type is larger than the size of the index set, we can use Fodor's lemma to push any possible realization into one of the index models to obtain a contradiction.", "The remainder of the paper focuses on realization and omission of symmetric cuts, and here complete ultrafilters are very useful; see Discussion REF above.", "This work is best understood in the context of two facts which highlight the importance of linear order for calibrating saturation.", "First, a surprising early result on Keisler's order was the second author's proof that any theory with the strict order property is maximal.", "Second, some key properties of ultrafilters which reflect much lower levels of saturation such as $\\mu (\\mathcal {D})$ , $\\operatorname{lcf}(\\aleph _0, \\mathcal {D})$ (see the comprehensive theorems of Section ) are detected by ultrapowers of linear order.", "Thus, information about a given ultrafilter's ability to realize or omit types in linear order is, in principle, very informative.", "The focus on symmetric cuts is underlined by a recent result of the authors in [16] that any regular ultrafilter which saturates some theory with $SOP_2$ (Definition REF below) will realize all symmetric pre-cuts.", "Thus omission of symmetric cuts means the ultrafilter is not good for a theory which is not known to be maximal.", "Returning to the synopsis of results, we prove in Claim REF that if the ultrafilter $\\mathcal {E}$ is $\\kappa $ -complete not $\\kappa ^+$ -complete, any $\\mathcal {E}$ -ultrapower of a sufficiently saturated model of linear order will have no $(\\kappa , \\kappa )$ -cuts.", "This is a fairly direct proof, and we also show there that if we further assume that $\\mathcal {E}$ is normal then it follows that $\\mathcal {E}$ is good (though not regular: see the Appendix e.g.", "REF ).", "We then prove, in Claim REF , that if $\\mathcal {E}$ is $\\kappa $ -complete and normal on $\\kappa $ , then any ultrapower of a sufficiently saturated model of linear order will contain a $(\\kappa ^+, \\kappa ^+)$ -cut.", "Here the proof is to build the cut in question.", "We work by induction on $\\alpha < \\kappa ^+$ , and the intent is on one hand, to define a pair of elements $(f_\\alpha , g_\\alpha )$ which, in the ultrapower, continue the current pre-cut; and on the other, to ensure globally that the sequence of such intervals is a type with no realization.", "To accomplish the second, informally speaking, we make the inductive choice of $(f_\\alpha (\\epsilon ), g_\\alpha (\\epsilon ))$ at index $\\epsilon $ to be compatible with a relatively short, linearly ordered segment of the cut so far, but no more.", "We then use normality to show that over the length of the index set, an interval chosen in this way will in fact continue to define a pre-cut.", "However, at the end of the construction, this enforced bound on compatibility will prevent realization of the type, since by Fodor's lemma any potential realization of the cut will reflect the existence of a long linearly ordered set in some index model.", "Finally, we leverage these proofs for a last existence result, Theorem REF .", "There, assuming $n$ measurable cardinals below $\\lambda $ , we show how to take products of ultrafilters to produce any number of finite alternations of cuts in an ultrafilter on $\\lambda $ with clearly described saturation properties.", "The proof is by downward induction.", "Note that if the first ultrafilter in our $(n+1)$ -fold product is regular, the final ultrafilter will be regular.", "By a result mentioned at the beginning of this section, if the first ultrafilter is flexible with large $\\operatorname{lcf}(\\aleph _0)$ and all remaining ultrafilters are at least $\\aleph _1$ -complete, the final ultrafilter will inherit flexibility and large $\\operatorname{lcf}(\\aleph _0)$ , thus be able to saturate any stable theory.", "By the result about loss of saturation for the random graph, it will fail to $(2^\\kappa )^+$ -saturate any unstable theory, where $\\kappa $ is the smallest measurable cardinal used in the construction.", "This completes the summary of our main results.", "In the appendix, we collect some easy observations and extensions of previous results.", "We discuss goodness for non-regular ultrafilters, give the equivalence of flexible and OK, and extend the second author's proof from [18].VI.4.8, that an $\\aleph _1$ -incomplete ultrafilter with small $\\operatorname{lcf}(\\aleph _0, D)$ fails to saturate unstable theories, to the case of $\\kappa $ -complete, $\\kappa ^+$ -incomplete ultrafilters with small $\\operatorname{lcf}(\\kappa , D)$ ." ], [ "Equivalences and implications", "This section justifies the phrase “properties of ultrafilters with model-theoretic significance.” We state and prove several comprehensive theorems which give the picture of Keisler's order, Definition REF , in light of our current work and of the recent progress mentioned in the introduction.", "Recall our conventions in REF , especially with respect to “saturates” and “good”.", "Minimum, maximum, etc.", "refer to Keisler's order.", "The first theorem collects the currently known correspondences between properties of regular ultrafilters and properties of first-order theories.", "Theorem F In the following table, for each of the rows (1),(3),(5),(6) the regular ultrafilter $\\mathcal {D}$ on $\\lambda $ fails to have the property in the left column if and only if it omits a type in every formula with the property in the right column.", "For rows (2) and (4), left to right holds: if $\\mathcal {D}$ fails to have the property on the left then it omits a type in every formula with the property on the right.", "Table: NO_CAPTION(Discussion - Sketch) (1) $\\leftrightarrow $ (A) Shelah [18].VI.5, see §REF above.", "(2) $\\leftarrow $ (B) Shelah [18].VI.4.8, see also Theorem REF below which generalizes that result.", "(3) $\\leftrightarrow $ (C) Straightforward by quantifier elimination, see [12].", "More generally, Malliaris [13] shows that the random graph, as the minimum non-simple theory, and $T_{feq}$ , as the minimum $TP_2$ theory, are in a natural sense characteristic of “independence properties” seen by ultrafilters.", "(4) $\\leftarrow $ (D) Malliaris [11], see §REF above, or [14].", "(5) $\\leftrightarrow $ (E) Malliaris [12] §6, which proves the existence of a minimum $TP_2$ -theory, the theory $T^*_{feq}$ of a parametrized family of independent (crosscutting) equivalence relations.", "(6) $\\leftrightarrow $ (F) Keisler observed that good ultrafilters can saturate any countable theory, and proved that goodness is equivalent to the saturation of certain (“versatile”) formulas [5], thus establishing the existence of a maximum class in Keisler's order; see §REF above.", "The result (6) $\\leftrightarrow $ (F) follows from Shelah's proof in [18].VI.3 that any theory with the strict order property is maximum in Keisler's order.", "Thus any ultrafilter able to saturate $SOP$ -types must be good, and by Keisler's observation the reverse holds.", "In fact, $SOP_3$ is known to be sufficient for maximality by [20]-[21], but this formulation is more suggestive here given our focus on order-types and cuts.", "A model-theoretic characterization of the maximum class is not known.", "Remark 4.1 Moreover, by work of the authors in [16] if $\\mathcal {D}$ on $\\lambda $ has “treetops,” i.e.", "it realizes a certain set of $SOP_2$ -types then it must realize all symmetric pre-cuts, that is, there can be no $(\\kappa , \\kappa )$ -cuts in ultrapowers of linear order for $\\kappa \\le \\lambda $ .", "So we will also be interested in the property of realizing symmetric cuts.", "As rows (2) and (4) of Theorem REF suggest, there are subtleties to the correspondence.", "If $T$ is not Keisler-maximal then any formula $\\varphi $ of $T$ with the order property has the independence property, as does any non-low formula.", "Yet consistently neither (4) nor (2) imply (3), as the rest of this section explains.", "So while we have model-theoretic sensitivity to properties (2) and (4), this is not enough for a characterization: in fact it follows from the theorems below that there is consistently no theory (and no formula $\\varphi $ ) which is saturated by $\\mathcal {D}$ if and only if (2), or (4).", "Theorem G Using the enumeration of properties of ultrafilters from Theorem REF , we have that: is necessary and sufficient for saturating stable theories, is necessary for saturating unstable theories, is necessary and sufficient for saturating the minimum unstable theory, is necessary for saturating non-low theories, is necessary and sufficient for saturating the minimum $TP_2$ theory, is necessary and sufficient for saturating any Keisler-maximum theory, e.g.", "$Th(\\mathbb {Q}, <)$ ; note that the identity of the maximum class is not known.", "The sources follow those of Theorem REF , but we make some additional remarks.", "(1) Note that Shelah's proof of (1) in [18].VI.5, quoted and sketched as Theorem REF , §REF above, gives the only two known equivalence classes in Keisler's order.", "(2) By Shelah [18].VI.4.8 (or Theorem REF below) if $\\mathcal {D}$ is regular and $\\operatorname{lcf}(\\mathcal {D}, \\aleph _0) \\le \\lambda ^+$ then any $\\mathcal {D}$ -ultrapower will omit a $\\lambda $ -type in some unstable formula, i.e., a formula with the order property.", "From the set-theoretic point of view, (2) $\\lnot \\rightarrow $ (3) of Theorem REF shows that $\\operatorname{lcf}(\\aleph _0, \\mathcal {D}) \\ge \\lambda ^+$ is weaker than ensuring $\\lambda ^+$ -saturation for the random graph (or equivalently, for some formula with the independence property).", "From the model-theoretic point of view, since any unstable theory has either the strict order property or the independence property, this gap is not visible.", "(3), (5) In fact what the characterization in Malliaris [12] shows is that a necessary and sufficient condition for an ultrafilter $\\mathcal {D}$ on $\\lambda $ to saturate the minimum $TP_2$ theory is that it be “good for equality,” meaning that for any set $X \\subseteq N = M^\\lambda /\\mathcal {D}$ , $|X| = \\lambda $ , there is a distribution $d: X \\rightarrow \\mathcal {D}$ such that $t \\in \\lambda , t \\in d(a) \\cap d(b)$ implies that $(M \\models a[t] = b[t]) \\iff (N \\models a = b)$ .", "By contrast, saturation of the minimum unstable theory asks only that for any two disjoint sets $X, Y \\subseteq M^\\lambda /\\mathcal {D}$ , $|X| = |Y| = \\lambda $ , there is a distribution $d: X \\cup Y \\rightarrow \\mathcal {D}$ such that for any $t \\in \\lambda $ , $a \\in X, b \\in Y$ , $t \\in d(a) \\cap d(b)$ implies $M \\models a[t] \\ne b[t]$ .", "(4) This was discussed in §REF above.", "Note that by work of Shelah [19] non-simple theories have an inherent structure/randomness “dichotomy” of $TP_1$ versus $TP_2$ , analogous to the structure/randomness dichotomy for unstable theories of $SOP$ versus $IP$ ; see § below.", "We know from [12] that any ultrafilter which saturates the minimum $TP_2$ -theory must be flexible, however we do not know whether an ultrafilter which saturates some $SOP_2$ theory must be flexible.", "(6) See the proof of Theorem REF (6).", "With the progress in this paper, and in other recent work of the authors, we may summarize the current picture of implications as follows: Theorem 4.2 Let properties $(1)-(6)$ be as in Theorem REF .", "Assume that $\\mathcal {D}$ is a regular ultrafilter on $\\lambda $ (note that not all of these properties imply regularity).", "Then: $(1) \\leftarrow (2) \\leftarrow (3) \\leftarrow (5) \\leftarrow (6)$ , with $(1) \\lnot \\rightarrow (2)$ , consistently $(2) \\lnot \\rightarrow (3)$ , $(3) \\lnot \\rightarrow (5)$ , and whether $(5)$ implies $(6)$ is open.", "Moreover $(1) \\leftarrow (4) \\leftarrow (5) \\leftarrow (6)$ , where $(3) \\lnot \\rightarrow (4)$ thus $(2) \\lnot \\rightarrow (4)$ , $(4) \\lnot \\rightarrow (3)$ , consistently $(4) \\lnot \\rightarrow (5)$ , consistently $(4) \\lnot \\rightarrow (6)$ ; and $(4)$ implies $(2)$ is open.", "$(6) \\rightarrow (x)$ Since good ultrafilters saturate any countable theory and properties (1)-(5) are all detected by formulas via Theorem REF , property (6) implies all the others.", "$(5) \\rightarrow (3)$ By Theorem REF and the fact that the random graph is minimum among unstable theories.", "$(3) \\rightarrow (2)$ By Theorem REF lines (2)-(3), i.e.", "Shelah [18].VI.4.8.", "$(2) \\rightarrow (1)$ Clearly the failure of (1) implies the failure of (2).", "$(1) \\lnot \\rightarrow (2)$ Shelah [18].VI.5, see Theorem REF , §REF above.", "$(2) \\lnot \\rightarrow (3)$ Consistently (assuming an $\\aleph _1$ -complete ultrafilter) by Theorem REF below.", "$(5) \\rightarrow (4)$ Malliaris [12] §6.", "$(4) \\rightarrow (1)$ Proved in [14].", "$(3) \\lnot \\rightarrow (4)$ Proved in a paper of the authors on excellent ultrafilters [15].", "$(4) \\lnot \\rightarrow (3)$ Consistently (assuming an $\\aleph _1$ -complete ultrafilter) by Theorem REF below.", "$(4) \\lnot \\rightarrow (5), (6)$ We give several proofs of independent interest (each assuming the existence of an $\\aleph _1$ -complete ultrafilter): Theorem REF proves $(4) \\lnot \\rightarrow (3)$ thus a fortiori $(4) \\lnot \\rightarrow (5)$ , and in [14], we give a different proof that $(4) \\lnot \\rightarrow (6)$ .", "See also Conclusion REF .", "At this point, the constructions begin.", "We remind the reader of the table of contents on p. REF , the definitions in §, and the overview of results in §." ], [ "$M^\\lambda /\\mathcal {D}$ is not {{formula:e05e5f3b-0754-41fd-a2a0-d060c2c3b022}} -saturated for {{formula:998f9645-9bc5-44d4-b418-70e936644806}} unstable", "In this section and the next we prove that flexibility does not imply saturation of the random graph, and thus a fortiori that flexibility does not imply goodness for equality.", "This gives a proof (assuming the existence of an $\\aleph _1$ -compact ultrafilter) that flexible need not mean good.", "Fact 5.1 ([18] Conclusion 1.13 p. 332) If $\\kappa $ is an infinite cardinal and $\\mathcal {D}$ is a regular ultrafilter on $I$ then $\\kappa ^I/\\mathcal {D}= \\kappa ^{|I|}$ .", "Claim 5.2 Assume $\\lambda \\ge \\kappa \\ge \\aleph _0$ , $T=T_{rg}$ , $M$ a $\\lambda ^+$ -saturated model of $T$ , $E$ a uniform ultrafilter on $\\kappa $ such that $|\\kappa ^\\kappa /E| = 2^\\kappa $ (i.e.", "we can find a sequence $\\langle g_\\alpha : \\alpha < 2^\\kappa \\rangle $ of members of ${^\\kappa \\kappa }$ such that $\\alpha < \\beta \\Rightarrow g_\\alpha \\ne g_\\beta \\mod {E}$).", "Then $M^\\kappa /E$ is not $(2^{\\kappa })^+$ -saturated.", "Let $\\mathcal {F}= \\lbrace f ~: ~ f: \\kappa \\times \\kappa \\rightarrow \\lbrace 0, 1\\rbrace \\rbrace $ , so $|\\mathcal {F}| = 2^\\kappa $ , and let $\\langle f_\\alpha : \\alpha < 2^\\kappa \\rangle $ list $\\mathcal {F}$ .", "Let $\\langle g_\\alpha : \\alpha < 2^\\kappa \\rangle $ be the distinct sequence given by hypothesis.", "First, for each $\\alpha < 2^\\kappa $ , we define $\\textbf {t}_\\alpha \\in \\lbrace 0,1\\rbrace $ by: $ \\textbf {t}_\\alpha = 1 \\iff \\lbrace i : f_\\alpha (i, g_\\alpha (i)) = 1 \\rbrace \\notin E $ Second, since $|M| \\ge \\kappa $ , we may fix some distinguished sequence $\\langle a_i : i <\\kappa \\rangle $ of elements of $M$ .", "Let $\\hat{g}_\\alpha \\in {^\\kappa M}$ be give by $\\hat{g}_\\alpha (i) = a_{g_\\alpha (i)}$ .", "Together these give a set $ p(x) = \\lbrace (xR{\\hat{g}_\\alpha /E})^{\\operatorname{if} \\textbf {t}_\\alpha } ~: ~\\alpha < 2^{\\kappa } \\rbrace $ We check that $p(x)$ is a consistent partial type in $M^\\kappa /E$ .", "Since each $\\hat{g}_\\alpha /E \\in M^\\kappa /E$ , $p$ is a set of formulas in $M^\\kappa /E$ .", "Since $\\alpha < \\beta \\Rightarrow g_\\alpha /E \\ne g_\\beta /E \\Rightarrow \\hat{g}_\\alpha /E \\ne \\hat{g}_\\beta /E$ , the parameters are distinct and so the type is consistent (note that for the given sequence of parameters, any choice of exponent sequence $\\langle \\textbf {t}_\\alpha : \\alpha < 2^\\kappa \\rangle $ would produce a consistent partial type).", "Moreover, $|p| = 2^\\kappa $ again by the choice of $\\langle g_\\alpha : \\alpha < 2^\\kappa \\rangle $ .", "We now show that $p(x)$ is omitted in $M^\\kappa /E$ .", "Towards a contradiction, suppose that $h \\in {^\\kappa M}$ were such that $h/E$ realized $p$ .", "Let $f: \\kappa \\times \\kappa \\rightarrow \\lbrace 0, 1 \\rbrace $ be defined by $f(i,j) = 1 \\iff M \\models h(i) R a_j$ .", "Then $f \\in \\mathcal {F}$ , hence for some $\\alpha _* < 2^\\kappa $ we have that $f_{\\alpha _*} = f$ .", "Thus: $ \\begin{array}{llll}\\textbf {t}_{\\alpha _*} = 1 &\\operatorname{iff}& M^\\kappa /E \\models (h/E) R (\\hat{g}_{\\alpha _*}/E) & \\mbox{(by choice of $p$, since $h/E \\models p$)} \\\\& \\operatorname{iff}& \\lbrace i < \\kappa : M\\models h(i) R {a_{g_{\\alpha _*}(i)}} \\rbrace \\in E & \\mbox{(by Łos' theorem)} \\\\& \\operatorname{iff}& \\lbrace i : f(i, g_{\\alpha _*}(i)) = 1 \\rbrace \\in E & \\mbox{(by the choice of $f$)} \\\\& \\operatorname{iff}& \\lbrace i : f_{\\alpha _*}(i, g_{\\alpha _*}(i)) = 1 \\rbrace \\in E & \\mbox{(as $f_{\\alpha _*} = f$)} \\\\\\end{array}$ But by definition of the truth values $\\textbf {t}$ , $ \\textbf {t}_{\\alpha _*} = 1 \\iff \\lbrace i : f_{\\alpha _*}(i, g_\\alpha (i)) = 1 \\rbrace \\notin E $ This contradiction completes the proof.", "Corollary 5.3 If $\\mathcal {D}$ is a $\\lambda $ -regular ultrafilter on $\\lambda $ , then $M^\\lambda /\\mathcal {D}$ is not $(2^\\lambda )^+$ -saturated for $M$ a model of any unstable theory.", "Without loss of generality, $M$ is $\\lambda ^+$ -saturated so of cardinality $\\ge \\kappa $ .", "The result follows by Claim REF and the fact that the random graph is minimum among the unstable theories in Keisler's order.", "(That is: by the Claim and Theorem REF , $\\mathcal {D}$ is not $(2^\\kappa )^+$ -good for formulas with the independence property, thus not $(2^\\kappa )^+$ -good, thus also not $(2^\\kappa )^+$ -good for formulas with the strict order property.)", "Observation 5.4 The hypothesis of Claim REF is satisfied when $E$ is a regular ultrafilter on $\\kappa $ and when $E$ is a $\\kappa $ -complete ultrafilter on $\\kappa $ .", "That is, we want to show that we can find a sequence $\\langle g_\\alpha : \\alpha < 2^\\kappa \\rangle $ of members of ${^\\kappa \\kappa }$ so that $\\alpha < \\beta \\Rightarrow g_\\alpha \\ne g_\\beta \\mod {E}$ .", "When $E$ is regular, this follows from Fact REF .", "We give two proofs for the complete case.", "Suppose then that $\\kappa $ is measurable, thus inaccessible.", "For each $\\alpha < \\kappa $ , let $\\Gamma _\\alpha = \\langle \\gamma _\\eta : \\eta \\in {^\\alpha 2} \\rangle $ be a sequence of pairwise distinct ordinals $< \\kappa $ .", "For each $\\eta \\in {^\\kappa 2}$ let $g_\\eta : \\kappa \\rightarrow \\kappa $ be given by $g_\\eta (\\alpha ) = \\gamma _{\\eta |_\\alpha }$ .", "So $\\lbrace g_\\eta : \\eta \\in {^\\kappa 2} \\rbrace \\subseteq {^\\kappa \\kappa }$ .", "By construction, all we need is one point of difference to know the functions diverge: $\\eta \\ne \\nu \\in {^\\kappa 2}, \\eta (\\beta ) \\ne \\nu (\\beta ) \\Rightarrow \\lbrace \\alpha < \\kappa : g_\\eta (\\alpha ) = g_\\nu (\\alpha ) \\rbrace \\subseteq \\lbrace \\alpha : \\alpha < \\beta \\rbrace = \\emptyset \\mod {E}$ as $E$ is uniform.", "Suppose then that $\\kappa $ is measurable, thus inaccessible.", "So we may choose $M$ , $|M| = \\kappa $ to be a $\\kappa $ -saturated model of the theory of the random graph.", "To show that $|M^\\kappa /E| \\ge 2^\\kappa $ , it will suffice to show that $2^\\kappa $ -many distinct types over the diagonal embedding of $M$ in the ultrapower $N$ are realized.", "Let $p(x) = \\lbrace xRf^0_\\alpha \\wedge \\lnot xRf^1_\\alpha : \\alpha < \\kappa \\rbrace $ be such a type, with each $f^i_\\alpha = {^\\kappa \\lbrace m\\rbrace }$ for some $m \\in M$ and of course $\\alpha , \\beta < \\kappa \\Rightarrow f^0_\\alpha \\ne f^1_\\beta $ .", "For each $t \\in \\kappa $ , let $p_t(x) = \\lbrace xRf^0_\\alpha (t) \\wedge \\lnot xRf^1_\\alpha (t) : \\alpha < t \\rbrace $ .", "Note that since the elements $f^i_\\alpha $ are constant, for each $t\\in \\kappa $ we have that $p_t(x)$ is a consistent partial type in $M$ .", "Choose a new element $h \\in {^\\kappa M}$ so that $t \\in \\kappa $ implies $h(t)$ satisfies $p_t(x)$ in $M$ .", "By the saturation of $M$ , some such $h$ exists.", "By uniformity of $E$ $h$ realizes the type $p(x)$ , that is, for $\\alpha < \\kappa $ , $ | \\kappa \\setminus \\lbrace t \\in \\kappa : M \\models h(t) R f^0_\\alpha (t) \\wedge \\lnot h(t) R f^1_\\alpha (t) \\rbrace | \\le \\alpha < \\kappa $ As no such $h$ can realize two distinct types over $M$ in $N$ , we finish." ], [ "For $\\kappa $ measurable, {{formula:5567349d-79cc-4375-96ea-483163bdd595}} there is {{formula:f19b2783-3074-4193-af44-02231548f073}} on {{formula:f15c6fc2-3678-4962-a7cf-a10b3a79dff0}} flexible but not good for {{formula:f5e7a279-cc4e-440f-a88e-ddce95d1827b}}", "We begin by characterizing when flexibility is preserved under products of ultrafilters, Definition REF .", "The first observation says that $\\lambda $ -flexibility of the first ultrafilter ensures there are $\\lambda $ -regularizing families in $\\mathcal {D}$ below certain nonstandard integers, namely those which are a.e.", "$\\mathcal {D}_1$ -nonstandard.", "Observation 6.1 Let $\\lambda , \\kappa \\ge \\aleph _0$ and let $\\mathcal {D}_1$ , $E$ be ultrafilters on $\\lambda , \\kappa $ respectively.", "Let $\\mathcal {D}= \\mathcal {D}_1 \\times E$ be the product ultrafilter on $\\lambda \\times \\kappa $ .", "Suppose that we are given $n_* \\in {^{\\lambda \\times \\kappa }\\mathbb {N}}$ such that: $n \\in \\mathbb {N} \\Rightarrow \\lbrace (s,t) \\in \\lambda \\times \\kappa ~:~ n_*(s,t) > n \\rbrace \\in \\mathcal {D}$ , i.e.", "$n_*$ is $\\mathcal {D}$ -nonstandard $N := \\lbrace t \\in \\kappa : n \\in \\mathbb {N} \\Rightarrow \\lbrace s \\in \\lambda : n_*(s,t) > n \\rbrace \\in \\mathcal {D}_1 \\rbrace \\in E$ i.e.", "$E$ -almost all of its projections are $\\mathcal {D}_1$ -nonstandard Then $(a) \\Rightarrow (b)$ , where: $\\mathcal {D}_1$ is $\\lambda $ -flexible there is a regularizing set $\\langle X_i : i < \\lambda \\rangle \\subseteq \\mathcal {D}$ below $n_*$ , i.e.", "such that for all $(s,t) \\in \\lambda \\times \\kappa $ , $|\\lbrace i < \\lambda : (s,t) \\in X_i \\rbrace | \\le n_*(s,t)$ For each $t \\in N$ , let $\\langle X^t_i : i < \\lambda \\rangle \\subseteq \\mathcal {D}_1$ be a regularizing family below $n_*(-,t)$ , that is, such that for each $s \\in \\lambda $ , $| \\lbrace i < \\lambda : s \\in X^t_i \\rbrace | \\le n_*(s,t)$ .", "Such a family is guaranteed by the $\\lambda $ -flexibility of $\\mathcal {D}_1$ along with the definition of $N$ , since the latter ensures that $n_*(-,t) \\in {^\\lambda \\lambda }$ is $\\mathcal {D}_1$ -nonstandard.", "Now define $\\langle X_i : i < \\lambda \\rangle $ by $X_i = \\lbrace (s,t) : s \\in X^t_i \\rbrace $ .", "We verify that: $\\langle X_i : i <\\lambda \\rangle \\subseteq \\mathcal {D}$ , as $\\lbrace t \\in \\kappa : \\lbrace s \\in \\lambda : (s,t) \\in X_i \\rbrace \\in \\mathcal {D}_1 \\rbrace \\supseteq N$ and $N \\in E$ by hypothesis.", "$\\langle X_i : i < \\lambda \\rangle $ is below $n_*$ , since for each $(s,t) \\in \\lambda \\times \\kappa $ , $| \\lbrace i : (s,t) \\in X_i \\rbrace | = | \\lbrace i : (s,t) \\in X^t_i \\rbrace | \\le n_*(s,t)$ by construction.", "This completes the proof.", "The next claim shows that $\\mathcal {D}$ -nonstandard integers project $E$ -a.e.", "to $\\mathcal {D}_1$ -nonstandard integers precisely when the second ultrafilter $E$ is at least $\\aleph _1$ -complete.", "Claim 6.2 Let $\\lambda , \\kappa \\ge \\aleph _0$ and let $\\mathcal {D}_1$ , $E$ be uniform ultrafilters on $\\lambda , \\kappa $ respectively.", "Let $\\mathcal {D}= \\mathcal {D}_1 \\times E$ be the product ultrafilter on $\\lambda \\times \\kappa $ .", "Then the following are equivalent.", "If $n_* \\in {^{\\lambda \\times \\kappa }\\mathbb {N}}$ is such that $n \\in \\mathbb {N} \\Rightarrow \\lbrace (s,t) \\in \\lambda \\times \\kappa ~:~ n_*(s,t) > n \\rbrace \\in \\mathcal {D}$ , then $N \\in E$ where $N := \\lbrace t \\in \\kappa : n \\in \\mathbb {N} \\Rightarrow \\lbrace s \\in \\lambda : n_*(s,t) > n \\rbrace \\in \\mathcal {D}_1 \\rbrace $ .", "$E$ is $\\aleph _1$ -complete.", "(1) $\\rightarrow $ (2) Suppose $E$ is not $\\aleph _1$ -complete, so it is countably incomplete and we can find $\\langle X_n : n < \\omega \\rangle \\subseteq E$ such that $\\bigcap \\lbrace X_n : n \\in \\omega \\rangle \\rbrace = \\emptyset \\mathcal {}\\mod {\\mathcal {D}}$ .", "Without loss of generality, $n < \\omega \\rightarrow X_{n+1} \\subsetneq X_n$ .", "Let $n_* \\in {^{\\lambda \\times \\kappa } \\mathbb {N}}$ be given by: $ t \\in \\kappa \\wedge t \\in X_n \\setminus X_{n+1} \\Rightarrow n_*(-,t) = {^\\lambda \\lbrace n\\rbrace } $ Then $n_*$ is $\\mathcal {D}$ -nonstandard but its associated set $N$ is empty (as a subset of $\\kappa $ , so a fortiori empty modulo $\\mathcal {D}$ ).", "(2) $\\rightarrow $ (1) Suppose on the other hand that $E$ is $\\aleph _1$ -complete, and let some $\\mathcal {D}$ -nonstandard $n_*$ be given.", "For each $n \\in \\mathbb {N}$ , define $X_n = \\lbrace t \\in \\kappa : \\lbrace s \\in \\lambda : n_*(s,t) > n \\rbrace \\in \\mathcal {D}_1 \\rbrace $ .", "Then by completeness, $N \\supseteq \\bigcap \\lbrace X_n : n \\in \\mathbb {N} \\rbrace \\in E$ .", "Corollary 6.3 Let $\\lambda , \\kappa \\ge \\aleph _0$ and let $\\mathcal {D}_1$ , $E$ be ultrafilters on $\\lambda , \\kappa $ respectively where $\\kappa > \\aleph _0$ is measurable.", "Let $\\mathcal {D}= \\mathcal {D}_1 \\times E$ be the product ultrafilter on $\\lambda \\times \\kappa $ .", "Then: If $\\mathcal {D}_1$ is $\\lambda $ -flexible and $E$ is $\\aleph _1$ -complete, then $\\mathcal {D}$ is $\\lambda $ -flexible.", "If $\\lambda \\ge \\kappa $ and $\\operatorname{lcf}(\\aleph _0, \\mathcal {D}_1) \\ge \\lambda ^+$ , then $\\operatorname{lcf}(\\aleph _0, \\mathcal {D}) \\ge \\lambda ^+$ , so in particular, $\\mathcal {D}= D \\times E$ will $\\lambda ^+$ -saturate any countable stable theory.", "(1) By Claim REF and Observation REF .", "(2) Let us show that the $\\mathcal {D}_1$ -nonstandard integers are cofinal in the $\\mathcal {D}$ -nonstandard integers.", "Let $M = (\\mathbb {N}, <)^\\lambda /\\mathcal {D}_1$ , $N = M^\\kappa /E$ .", "Let $n_* \\in N$ be $\\mathcal {D}$ -nonstandard.", "By Claim REF , the set $X = \\lbrace t \\in \\kappa : n_*(t)~\\mbox{is a $\\mathcal {D}_1$-nonstandard element of $M$} \\rbrace \\in E$ .", "Since $\\operatorname{lcf}(\\aleph _0, \\mathcal {D}_1) \\ge \\lambda ^+ > \\kappa $ , there is $m_* \\in M$ which is $\\mathcal {D}_1$ -nonstandard and such that $t \\in X \\Rightarrow M \\models n_*(t) > m_*$ .", "Then the diagonal embedding of $m_*$ in $N$ will be $\\mathcal {D}$ -nonstandard but below $n_*$ , as desired.", "The statement about stable theories follows by §, Theorem REF and Theorem REF (2)$\\rightarrow $ (1).", "Theorem 6.4 Assume $\\aleph _0 < \\kappa < \\lambda = \\lambda ^\\kappa $ , $2^\\kappa \\le \\lambda $ , $\\kappa $ measurable.", "Then there exists a regular uniform ultrafilter $\\mathcal {D}$ on $\\lambda $ such that $\\mathcal {D}$ is $\\lambda $ -flexible, yet for any model $M$ of the theory of the random graph, $M^\\lambda /\\mathcal {D}$ is not $(2^\\kappa )^+$ -saturated.", "However, $\\mathcal {D}$ will $\\lambda ^+$ -saturate any countable stable theory.", "A fortiori, $\\mathcal {D}$ is neither good nor good for equality, and it will fail to $(2^\\kappa )^+$ -saturate any unstable theory.", "Let $\\mathcal {E}$ be a uniform $\\aleph _1$ -complete ultrafilter on $\\kappa $ .", "Let $\\mathcal {D}_1$ be any $\\lambda $ -flexible (thus, $\\lambda $ -regular) ultrafilter on $\\lambda $ , e.g.", "a regular $\\lambda ^+$ -good ultrafilter on $\\lambda $ .", "Let $\\mathcal {D}= \\mathcal {D}_1 \\times \\mathcal {E}$ be the product ultrafilter on $I = \\lambda \\times \\kappa $ .", "Then $|I| = \\lambda $ , and we have that $\\mathcal {D}$ is $\\lambda $ -flexible by Corollary REF (1), it saturates countable stable theories by REF (2), and it fails to $(2^\\kappa )^+$ -saturate the random graph by Claim REF .", "The last line of the Theorem follows from the fact that the random graph is minimum among unstable theories in Keisler's order, along with the fact that both goodness and goodness for equality are necessary conditions on regular ultrafilters for saturating some unstable theory (the theory of dense linear orders and the theory $T^*_{feq}$ of infinitely many independent equivalence relations, respectively).", "Corollary 6.5 In the construction just given, by Claim REF , $\\operatorname{lcf}(\\aleph _0, \\mathcal {D}) = \\operatorname{lcf}(\\aleph _0, \\mathcal {D}_1) \\ge \\lambda ^+$ since $\\mathcal {D}_1$ is $\\lambda ^+$ -good and the nonstandard integers of $\\mathcal {D}_1$ are cofinal in the nonstandard integers of $\\mathcal {D}$ .", "Thus consistently, a regular ultrafilter on $\\lambda > \\kappa $ may have large lower cofinality of $\\aleph _0$ while failing to $(2^\\kappa )^+$ -saturate the random graph.", "By [18].VI.4, a necessary condition for a regular ultrafilter $\\mathcal {D}$ on $\\lambda $ to saturate some unstable theory is that $\\operatorname{lcf}(\\aleph _0, \\mathcal {D}) \\ge \\lambda ^+$ ; Corollary REF shows it is not sufficient." ], [ "$M^\\lambda /\\mathcal {D}$ is not {{formula:e7c8438e-071e-4f53-a5e7-7d6481ab7183}} -saturated for {{formula:464c980e-e287-4c4e-b5de-0af5175c96fa}} regular and {{formula:346ebe53-023b-411c-bfb3-2ef2511149ea}} non-simple", "In this section we prove that there is a loss of saturation in ultrapowers of non-simple theories.", "As mentioned above, this is a new proof of a result from [18].VI.4.7, which reflects an interest (visible elsewhere in this paper e.g.", "REF ) in controlling the distribution of sets of indices.", "Definition 7.1 A first-order theory has $TP_2$ if there is a formula $\\varphi (x;\\overline{y})$ which does, where this means that in any $\\aleph _1$ -saturated model $M \\models T$ there exists an array $A = \\lbrace \\overline{a}^i_j : i <\\omega , j<\\omega \\rbrace $ of tuples, $\\ell (\\overline{a}^i_j) = \\ell (\\overline{y})$ such that: for any finite $X \\subseteq \\omega \\times \\omega $ , the partial type $ \\lbrace \\varphi (x;\\overline{a}^i_j): (i,j) \\in X \\rbrace $ is consistent if and only if $ (i,j) \\in X \\wedge (i^\\prime , j^\\prime ) \\in X \\wedge i=i^\\prime \\Rightarrow j=j^\\prime $ Informally speaking, by choosing no more than one tuple of parameters from each column we form a consistent partial type.", "Definition 7.2 A first-order theory has $TP_1$ , or equivalently $SOP_2$ , if there is a formula $\\varphi (x;\\overline{y})$ which does, where this means that in any $\\aleph _1$ -saturated model $M \\models T$ there exist $\\langle \\overline{a}_\\eta : \\eta \\in {^{\\aleph _0 > } 2 } \\rangle $ such that: for $\\eta , \\rho \\in {^{\\aleph _0 > } 2}$ incomparable, i.e.", "$\\lnot (\\eta \\trianglelefteq \\rho ) \\wedge \\lnot (\\rho \\trianglelefteq \\eta )$ , we have that $\\lbrace \\varphi (x;\\overline{a}_\\eta ), \\varphi (x;\\overline{a}_\\rho ) \\rbrace $ is inconsistent.", "for $\\eta \\in {^{\\aleph _0 } 2}$ , $\\lbrace \\varphi (x;\\overline{a}_{\\eta |_i}) : i < \\aleph _0 \\rbrace $ is a consistent partial $\\varphi $ -type.", "Fact 7.3 (See [18].III.7, [19]) If $T$ is not simple then $T$ contains either a formula with $TP_1$ (equivalently $SOP_2$ ) or a formula with $TP_2$ .", "Fact 7.4 Suppose that $\\mathcal {D}$ is a regular uniform ultrafilter on $\\lambda $ , $\\lambda = \\lambda ^{<\\lambda }$ or just $(\\aleph _1, \\aleph _0) \\rightarrow (\\lambda ^+, \\lambda )$ (see ).", "Let $\\kappa = \\aleph _0$ .", "Then for each $\\epsilon < \\lambda $ we may choose a sequence of sets $\\overline{u}_\\epsilon = \\langle u_{\\epsilon , \\alpha } : \\alpha < \\lambda ^+ \\rangle $ such that: $u_{\\epsilon , \\alpha } \\subseteq \\alpha $ $ | u_{\\epsilon , \\alpha } | < \\lambda $ $\\alpha \\in u_{\\epsilon , \\beta } \\Rightarrow u_{\\epsilon , \\alpha } = u_{\\epsilon , \\beta } \\cap \\alpha $ if $ u \\subseteq \\lambda ^+$ , $|u| < \\kappa $ then $ \\lbrace \\epsilon < \\lambda : \\exists \\alpha (u \\subseteq u_{\\epsilon , \\alpha }) \\rbrace \\in \\mathcal {D}$ By Kennedy-Shelah-Vaananen [6] p. 3 this is true when $\\lambda $ satisfies the stated hypothesis and $\\mathcal {D}$ is regular.", "Note that as briefly mentioned there, in the case of singular $\\lambda $ , the claim may be true; but it is also consistent that it may fail.", "We will use $\\trianglelefteq $ to indicate comparability in the $TP_1$ tree, i.e.", "$\\eta \\trianglelefteq \\rho $ means $\\eta $ is before $\\rho $ in the partial tree order.", "Claim 7.5 Given $\\lambda \\ge \\aleph _0$ regular, let $\\mathcal {D}$ be a regular uniform ultrafilter on $\\lambda $ .", "Suppose $T$ has $TP_1$ , as witnessed by $\\varphi $ , and let $M \\models T$ be $\\lambda ^{++}$ -saturated.", "Then $M^\\lambda /\\mathcal {D}$ is not $\\lambda ^{++}$ -saturated, and in particular is not $\\lambda ^{++}$ -saturated for $\\varphi $ -types.", "Fix $\\varphi =\\varphi (x;\\overline{y})$ a formula with $TP_1$ .", "By the hypothesis of $TP_1$ and saturation of $M$ , we may choose parameters $\\overline{a}_\\eta \\in M$ ($\\ell (a_\\eta ) = \\ell (y)$ ) for $\\eta \\in ^{^{\\lambda ^+ > } \\lambda }$ such that: $\\lbrace \\varphi (x; a_{\\eta |_i}) : i < \\lambda ^+ \\rbrace $ is a consistent partial type for each $\\eta \\in ^{^{\\lambda ^+ > } \\lambda }$ if $\\eta , \\nu \\in ^{^{\\lambda ^+ > } \\lambda }$ are $\\trianglelefteq $ -incomparable then the set $\\lbrace \\varphi (x; a_\\eta ), \\varphi (x;a_\\nu ) \\rbrace $ is contradictory.", "Let $\\langle \\overline{u_\\epsilon } : \\epsilon < \\lambda \\rangle $ be as given by Fact REF .", "For each $\\epsilon < \\lambda $ , we choose by induction on $\\alpha < \\lambda ^+$ indices $\\langle \\eta _{\\epsilon , \\alpha } : \\alpha < \\lambda ^+ \\rangle $ such that: $\\eta _{\\epsilon , \\alpha } \\in {^\\alpha \\lambda }$ $\\eta _{\\epsilon , \\alpha } \\triangleleft \\eta _{\\epsilon , \\beta }$ iff $\\alpha \\in u_{\\epsilon , \\beta }$ Informally, we choose indices for nodes of the tree so that consistency of the associated formulas reflects the structure of the sets $\\langle \\overline{u_\\epsilon } : \\epsilon < \\lambda \\rangle $ , which we can do by the assumption on saturation of $M$ and the downward coherence condition, Fact REF (3).", "For each $\\alpha < \\lambda ^+$ we thus have an element $f_\\alpha \\in M^\\lambda $ given by $f_\\alpha (\\epsilon ) = a_{\\eta _{\\epsilon , \\alpha }} \\in M$ .", "The sequence $\\langle f_\\alpha /\\mathcal {D}: \\alpha < \\lambda ^+ \\rangle $ is a sequence of $\\lambda ^+$ members of $M^\\lambda /\\mathcal {D}$ .", "Moreover, by the downward coherence condition, if $n<\\omega $ , $\\alpha _0 < \\dots < \\alpha _n < \\lambda ^+$ and for some $\\alpha $ , $\\alpha _n < \\alpha < \\lambda $ we have that $\\lbrace \\alpha _0, \\dots \\alpha _n \\rbrace \\subseteq \\alpha $ , then in fact $ \\ell < n \\Rightarrow \\alpha _\\ell \\in u_{\\alpha _{\\ell + 1 }}$ ; note here that the identity of $\\alpha $ is not important, only its existence.", "Thus for any $n < \\omega $ and any $\\alpha _0 < \\dots < \\alpha _n < \\lambda ^+$ , Fact REF (4) implies that $ \\lbrace \\epsilon < \\lambda : \\eta _{\\epsilon , \\alpha _0} \\triangleleft \\eta _{\\epsilon , \\alpha _1}\\triangleleft \\dots \\triangleleft \\eta _{\\epsilon , \\alpha _n} \\rbrace \\in \\mathcal {D}$ and therefore $ \\lbrace \\epsilon < \\lambda : M \\models \\exists x \\bigwedge _{\\ell } \\varphi (x; a_{\\epsilon , \\alpha _\\ell } ) \\rbrace \\in \\mathcal {D}$ Since $n$ , $\\alpha _0, \\dots \\alpha _n$ were arbitrary, we have verified that $p = \\lbrace \\varphi (x; f_\\alpha /\\mathcal {D}) : \\alpha < \\lambda ^+ \\rbrace $ is a consistent partial type.", "Assume towards a contradiction that $p$ is realized, say by $f \\in {^\\lambda M}$ .", "For each $\\alpha < \\lambda ^+$ , define $J_\\alpha = \\lbrace \\epsilon < \\lambda : M \\models \\varphi (f(\\epsilon ), f_\\alpha (\\epsilon )) \\rbrace $ and note that $J_\\alpha \\in \\mathcal {D}$ since we assumed $f$ realizes the type.", "By definition of $f_\\alpha $ , if $\\epsilon \\in J_\\alpha $ then $M \\models \\varphi (f(\\epsilon ), a_{\\eta _{\\epsilon , \\alpha }})$ .", "For each $\\alpha < \\lambda ^+$ , as $J_\\alpha \\ne \\emptyset $ we may choose some $\\epsilon _\\alpha \\in J_\\alpha $ .", "Since $\\lambda ^+ > \\lambda $ is regular, there is some $\\epsilon _* < \\lambda $ such that $S = \\lbrace \\alpha < \\lambda ^+ : \\epsilon _\\alpha = \\epsilon _* \\rbrace \\subseteq \\lambda ^+$ is unbounded in $\\lambda ^+$ .", "By definition of $J_\\alpha $ , this means that $p_* = \\lbrace \\varphi (x,a_{\\eta _{\\epsilon _*, \\alpha }}) : \\alpha \\in S \\rbrace $ is a consistent partial type in $M$ .", "By definition of $TP_1$ , if $p_*$ is a consistent partial type it must be that $X = \\lbrace \\eta _{\\epsilon _*, \\alpha } : \\alpha \\in S \\rbrace $ has no two $\\trianglelefteq $ -incomparable elements.", "Since $|S| = \\lambda ^+$ , we may choose some $\\alpha \\in S$ such that $|S \\cap \\alpha | = \\lambda $ .", "But this contradicts the choice of indices $\\eta _{\\epsilon , \\alpha }$ in (i)-(ii) above, in light of Fact REF (2).", "We now consider $TP_2$ .", "Observation 7.6 Let $|I| = \\lambda \\ge \\aleph _0$ and let $\\mathcal {D}$ be a regular ultrafilter on $\\lambda $ .", "There are functions $\\langle \\nu _\\alpha : \\lambda \\le \\alpha < \\lambda ^+ \\rangle $ such that: $\\nu _\\alpha : I \\rightarrow \\alpha $ for any $\\alpha < \\beta < \\lambda ^+$ , $ \\lbrace i \\in I : \\nu _\\alpha (i) = \\nu _\\beta (i) \\rbrace \\notin \\mathcal {D}$ This follows e.g.", "from the statement that for any regular ultrafilter $\\mathcal {D}$ on $\\lambda $ , $|\\mathbb {N}^\\lambda /\\mathcal {D}| = 2^\\lambda $ .", "Claim 7.7 Given $\\lambda \\ge \\aleph _0$ , let $\\mathcal {D}$ be a regular uniform ultrafilter on $I$ , $|I| = \\lambda $ .", "Suppose $T$ has $TP_2$ , as witnessed by $\\varphi $ , and let $M \\models T$ be $\\lambda ^{++}$ -saturated.", "Then $M^\\lambda /\\mathcal {D}$ is not $\\lambda ^{++}$ -saturated, and in particular is not $\\lambda ^{++}$ -saturated for $\\varphi $ -types.", "Let $\\langle \\nu _\\alpha : \\alpha < \\lambda ^+ \\rangle $ be as given by Observation REF .", "By the definition of $TP_2$ , for some formula $\\varphi = \\varphi (x;\\overline{y})$ we have an array $\\lbrace {a}_{\\alpha , \\beta } : \\alpha , \\beta < \\lambda ^+ \\rbrace \\subseteq M$ , with $\\ell ({a}) = \\ell (\\overline{y})$ , such that: for $\\beta < \\gamma < \\lambda ^+$ , $\\lbrace \\varphi (x;a_{\\alpha , \\beta }), \\varphi (x; a_{\\alpha , \\gamma }) \\rbrace $ is contradictory for any $n < \\omega $ and $\\alpha _0 < \\dots < \\alpha _n < \\lambda ^+$ , and any $\\lbrace \\beta _0, \\dots \\beta _n \\rbrace \\subseteq \\lambda ^+$ , the set $\\lbrace \\varphi (x;a_{\\alpha _i, \\beta _i}) : i \\le n \\rbrace $ is a consistent partial type Informally, the columns consist of parameters for pairwise contradictory instances of $\\varphi $ , whereas choosing any sequence of parameters with no more than one in any given column produces a consistent partial type.", "Now we define for each $\\alpha $ , $\\lambda \\le \\alpha < \\lambda ^+$ functions $f_\\alpha \\in M^I$ by: $f_\\alpha (i) = a_{\\nu _\\alpha (i), \\alpha }$ .", "Let us verify that $p(x) = \\lbrace \\varphi (x; f_\\alpha /\\mathcal {D}) : \\alpha < \\lambda ^+ \\rbrace $ is a consistent partial type.", "For any $n < \\omega $ and $\\alpha _0 < \\dots < \\alpha _n < \\lambda ^+$ , we have by the second item in Observation REF that $\\lbrace i \\in I : \\bigwedge \\lbrace v_{\\alpha _j}(i) \\ne v_{\\alpha _k}(i) : j < k \\le n \\rbrace \\in \\mathcal {D}$ , and thus by definition of the $TP_2$ array, $\\lbrace \\varphi (x;f_{\\alpha _j}/\\mathcal {D}: j \\le n \\rbrace $ is consistent.", "Suppose for a contradiction that $p$ were realized in $M^\\lambda /\\mathcal {D}$ , say by $f \\in {^IM}$ .", "We proceed analogously to Claim REF .", "For each $\\alpha < \\lambda ^+$ , define $J_\\alpha = \\lbrace \\epsilon < \\lambda : M \\models \\varphi (f(\\epsilon ), f_\\alpha (\\epsilon )) \\rbrace $ , which again will be in $\\mathcal {D}$ (and in particular, nonempty) since we assumed $f$ realizes the type.", "For each $\\alpha < \\lambda ^+$ , choose some $\\epsilon _\\alpha \\in J_\\alpha $ , and let $S \\subseteq \\lambda ^+$ be a stationary set on which this choice is constant, call it $\\epsilon _*$ .", "Now we look at $p_* = \\lbrace \\varphi (x,a_{\\nu _\\alpha (\\epsilon _*), \\alpha }) : \\alpha \\in S \\rbrace $ .", "By definition of $S$ , $f(\\epsilon _*)$ realizes this type, so in particular it is consistent.", "But since $S$ is stationary and $\\nu _\\alpha (\\epsilon ) < \\alpha $ , by Fodor's lemma there are $\\alpha \\ne \\beta $ from $S$ such that $\\nu _\\alpha (\\epsilon _*) = \\nu _\\beta (\\epsilon _*) =: \\gamma $ .", "But then $\\lbrace \\varphi (x, a_{\\gamma , \\alpha }), \\varphi (x, a_{\\gamma , \\beta }) \\rbrace \\subseteq p_*$ is contradictory, and this completes the proof.", "Conclusion 7.8 Let $\\lambda = \\lambda ^{<\\lambda }$ and let $\\mathcal {D}$ be a regular ultrafilter on $\\lambda $ .", "If $T$ is not simple and $M \\models T$ , then there is $\\varphi $ such that $M^\\lambda /\\mathcal {D}$ is not $\\lambda ^{++}$ -saturated for $\\varphi $ -types.", "By Claim REF , Claim REF and Fact REF .", "Remark 7.9 Let $D_1, D_2$ be ultrafilters on $\\lambda , \\kappa $ respectively and suppose that $\\kappa = \\kappa ^{<\\kappa }$ .", "If $\\lambda \\ge \\kappa ^+$ and $D_2$ is regular, then by Theorem REF (5) and Conclusion REF , $D_1 \\times D_2$ cannot be good for equality." ], [ "$\\kappa $ -complete not {{formula:76e1bca4-72a1-4950-9c7f-883f501c11f5}} -complete implies no {{formula:2a94f2da-1c16-48e2-a509-f3bd897f5d13}} -cuts", "Claim 8.1 Suppose that $\\mathcal {E}$ is a $\\kappa $ -complete but not $\\kappa ^+$ -complete ultrafilter on $I$ and $M_1$ is a $\\kappa ^+$ -saturated model in which a linear order $L$ and tree $T$ are interpreted.", "Then in $M_2 = M_1^I/\\mathcal {E}$ : the linear order $L^{M_2}$ has no $(\\kappa , \\kappa )$ -cut, and moreover no $(\\theta , \\sigma )$ -cut for $\\theta , \\sigma < \\kappa $ both regular.", "the tree $T^{M_2}$ has no branch (i.e.", "maximal linearly ordered set) of cofinality $\\le \\kappa $ .", "Remark 8.2 In the statement of Claim REF : in (a), the $\\kappa $ -saturation of $M_1$ is necessary in the following sense: if there is a sequence $\\overline{\\theta } = \\langle \\theta _t : t \\in I \\rangle $ , which certainly need not be distinct, such that $M$ has a $(\\theta _t, \\theta _t)$ -cut for each $t \\in I$ and $(\\prod _{t \\in I} \\theta _t, <_\\mathcal {E})$ has cofinality $\\kappa $ , then the conclusion of Claim REF (a) is false.", "By this Claim, we may add to the conclusion of Theorem REF that $(\\kappa , \\kappa ) \\notin \\mathcal {C}(\\mathcal {E})$ , Definition REF , since in that Theorem the ultrafilter $E$ is a $\\kappa $ -complete uniform ultrafilter on $\\kappa $ and thus not $\\kappa ^+$ -complete.", "(of Claim REF ) (a) The “moreover” clause in (a) follows from the fact that $M_1$ and $M_2$ are $L_{\\kappa , \\kappa }$ -equivalent, by the completeness of $\\mathcal {E}$ , and the hypothesis on saturation of $M_1$ .", "So we consider a potential $(\\kappa , \\kappa )$ -cut in $M_2$ , i.e.", "a $(\\kappa , \\kappa )$ -pre-cut given by $\\langle f_\\alpha : \\alpha < \\kappa \\rangle $ , $\\langle g_\\alpha : \\alpha < \\kappa \\rangle $ where if $\\alpha < \\beta < \\kappa $ then $ M_2 \\models (f_\\alpha /\\mathcal {E}) <_L (f_\\beta /\\mathcal {E}) <_L < (g_\\beta /\\mathcal {E}) <_L (g_\\alpha /\\mathcal {E}) $ For $0 < \\gamma < \\kappa $ let $ A_\\gamma = \\lbrace t : ~\\mbox{if}~ \\alpha < \\beta < \\gamma ~\\mbox{then} ~M_1 \\models f_\\alpha (t) <_L f_\\beta (t) <_L g_\\beta (t) <_L g_\\alpha (t) \\rbrace $ Let $A_0 = A_1 = I$ .", "Then $\\overline{A} = \\langle A_\\gamma : \\gamma < \\kappa \\rangle $ is a continuously decreasing sequence of elements of $\\mathcal {E}$ , i.e.", ": $\\gamma _1 < \\gamma _2 \\Rightarrow A_{\\gamma _1} \\supseteq A_{\\gamma _2}$ for limit $\\delta < \\kappa $ , $A_\\delta = \\bigcap \\lbrace A_\\gamma : \\gamma < \\delta \\rbrace $ each $A_\\gamma \\in \\mathcal {E}$ , by choice of the functions and $\\kappa $ -completeness As we assumed $\\mathcal {E}$ is $\\kappa $ -complete but not $\\kappa ^+$ -complete, there is a sequence $\\overline{B} = \\langle B_\\gamma : \\gamma < \\kappa \\rangle $ of elements of $\\mathcal {E}$ such that $\\bigcap \\lbrace B_\\gamma : \\gamma < \\kappa \\rbrace = \\emptyset $ .", "We may furthermore assume that $\\overline{B}$ is a continuously decreasing sequence (if necessary, inductively replace $B_\\delta $ by $\\bigcap \\lbrace B_\\gamma : \\gamma < \\delta \\rbrace $ using $\\kappa $ -completeness).", "Thus given $\\overline{A}, \\overline{B}$ , for each $t \\in I$ we may define $ \\gamma (t) = \\operatorname{min} \\lbrace \\alpha : t \\notin A_{\\alpha +1} \\cap B_{\\alpha +1} \\rbrace $ By choice of $\\overline{B}$ , $t \\mapsto \\gamma (t)$ is a well-defined function from $I$ to $\\kappa $ , and $t \\in A_{\\gamma (t)} \\cap B_{\\gamma (t)}$ .", "Recall that we want to show that our given $(\\kappa , \\kappa )$ -sequence is not a cut.", "Choose $f_\\kappa , g_\\kappa \\in {^IM}$ so that first, for each $t \\in I$ , $f_\\kappa (t), g_\\kappa (t) \\in L^{M_1}$ , and second, for each $t \\in I$ and all $\\alpha < \\gamma (t)$ , $ M_1 \\models f_\\alpha (t) \\le _{L} f_\\kappa (t) <_L g_\\kappa (t) \\le _L g_\\alpha (t) $ This we can do by the choice of $\\overline{A}$ as a continuously decreasing sequence (so the function values $f_\\alpha , g_\\beta $ below $\\gamma (t)$ in each index model are correctly ordered) and the saturation hypothesis on $M_1$ .", "Thus for each $\\alpha < \\kappa $ , we have that $ \\lbrace t : f_\\alpha (t) \\le _L f_\\kappa (t) <_L g_\\kappa (t) \\le _L g_\\alpha (t) \\rbrace \\supseteq A_{\\alpha +1} \\cap B_{\\alpha +1} \\in \\mathcal {E}$ which completes the proof.", "(b) Similar proof, but we only need to use one sequence $\\langle f_\\alpha : \\alpha < \\kappa \\rangle $ which we choose to potentially witness that the cofinality of the branch is at most $\\kappa $ .", "We prove a related fact for normal filters, Definition REF .", "Claim 8.3 Assume $\\mathcal {E}$ is a normal filter on $\\lambda $ and $M$ is a $\\lambda ^+$ -saturated dense linear order.", "Then $M^I/\\mathcal {E}$ is $\\lambda ^+$ -saturated.", "Suppose that $\\langle f_\\alpha /\\mathcal {E}: \\alpha < \\kappa _1 \\rangle $ is increasing in $M^I/\\mathcal {E}$ , and $\\langle g_\\beta /\\mathcal {E}: \\beta < \\kappa _2 \\rangle $ is decreasing in $M^I/\\mathcal {E}$ , with $\\kappa _1, \\kappa _2 \\le \\lambda $ and $f_\\alpha /\\mathcal {E}< g_\\beta /\\mathcal {E}$ for $\\alpha <\\kappa _1, \\beta <\\kappa _2$ .", "Let $ X_{\\alpha , \\beta } = \\lbrace t \\in \\lambda : f_\\alpha (t) < g_\\beta (t) \\rbrace \\in \\mathcal {E}$ for $\\alpha < \\beta < \\lambda $ .", "Without loss of generality, suppose $\\kappa _1 \\le \\kappa _2$ .", "For each $\\beta < \\kappa _2$ , let $ Y_\\beta = \\lbrace \\alpha \\in \\lambda : (\\forall j < (1+\\alpha ) \\cap \\beta ) (j \\in X_{\\alpha , \\beta }) \\rbrace \\in \\mathcal {E}$ by normality.", "For $\\beta \\ge \\kappa _2$ , let $Y_\\beta = I$ .", "Finally, define $ Z = \\lbrace \\beta \\in \\lambda : (\\forall k < (1+\\beta ) \\cap \\kappa _2 ) ( j \\in Y_\\beta ) \\rbrace \\in \\mathcal {E}$ Now if $t \\in Z$ (so $t$ plays the role of $\\beta $ ) we have that $ p_t = \\lbrace f_\\alpha (t) < x < g_t(t) : \\alpha < t \\rbrace $ is a consistent partial type, realized in $M$ by the saturation hypothesis.", "Choose $h \\in {^\\lambda M}$ such that for each $t \\in Z$ , $h \\models p_t$ .", "Then $h$ realizes the type.", "Note that Claim REF implies by Fact REF of the Appendix that the relevant $\\mathcal {E}$ is good.", "The use of the additional hypothesis “normal” in this section comes in Step 3 of Claim REF , and consequently in later results which rely on it.", "Recall Definition REF and Fact REF .", "Claim 8.4 Assume $\\kappa $ measurable, $\\mathcal {E}$ a normal $\\kappa $ -complete ultrafilter on $\\kappa $ , $\\lambda \\ge \\kappa $ , $M_1$ a $\\lambda $ -saturated model with ${(L_M, <_M)}$ a dense linear order.", "Let $M_2 = M_1^\\kappa /\\mathcal {E}$ .", "Then $L_{M_2}$ has a $(\\kappa ^+, \\kappa ^+)$ -cut.", "The proof has several steps.", "Step 1: Fixing sequences of indices.", "For each $\\alpha < \\kappa ^+$ choose $\\overline{\\mathcal {U}}_\\alpha = \\langle u_{\\alpha , \\epsilon } : \\epsilon < \\kappa \\rangle $ so that: this sequence is $\\subseteq $ -increasing and continuous, and for each $\\alpha < \\kappa ^+$ , $u_{\\alpha , 0} = \\emptyset $ for each $\\epsilon < \\kappa $ , $|u_{\\alpha , \\epsilon }| < \\kappa $ $\\bigcup \\lbrace u_{\\alpha , \\epsilon } : \\epsilon < \\kappa \\rbrace = \\alpha $ (for coherence) for $\\beta < \\alpha < \\kappa ^+$ , $ \\beta \\in u_{\\alpha , \\epsilon } \\Rightarrow u_{\\beta , \\epsilon } \\subseteq u_{\\alpha , \\epsilon } $ Such a sequence will always exist as $|\\alpha | \\le \\kappa $ .", "[Details: Clearly such a sequence exists for $\\alpha \\le \\kappa $ : let $u_{\\kappa , \\epsilon } = \\epsilon \\cap \\alpha $ , so for arbitrary $\\kappa \\le \\alpha < \\kappa ^+$ , fixing a bijection to $\\kappa $ let $\\overline{V}_\\alpha = \\langle v_{\\alpha , \\epsilon } : \\epsilon < \\kappa \\rangle $ be the preimage of the sequence for $\\kappa $ .", "Having thus fixed, for each $\\alpha < \\kappa ^+$ , a sequence satisfying (a)-(c) we may then inductively pad these sequences to ensure coherence.", "For $\\beta = 0$ and each $\\epsilon < \\kappa $ , let $u_{\\beta , \\epsilon } = v_{\\beta , \\epsilon }$ .", "For $0 < \\beta < \\kappa ^+$ , for each $\\epsilon < \\kappa $ let $u_{\\beta , \\epsilon } = \\bigcup \\lbrace u_{\\alpha , \\epsilon } : \\alpha \\in v_{\\beta , \\epsilon } \\rbrace $ , and note that this will preserve (b), (a), (c) and ensure (d).]", "Step 2: The inductive construction of the (pre-)cut.", "We now construct a cut.", "We will first describe the construction, and then show that it is in fact a cut (i.e.", "we will show that we have indeed constructed a pre-cut, and that this pre-cut is not realized).", "By induction on $\\alpha < \\kappa ^+$ we will choose $f_\\alpha , g_\\alpha \\in {^\\kappa (L_{M_1})}$ .", "The intention is that each $u_{\\alpha , \\epsilon }$ represents a small set of prior functions which we take into account when choosing the values for $f_\\alpha , g_\\alpha $ at the index $\\epsilon \\in \\kappa $ .", "At stage $\\alpha $ , for each index $\\epsilon < \\kappa $ define $w_{\\alpha , \\epsilon } = \\lbrace \\beta \\in u_{\\alpha , \\epsilon } :&\\langle f_\\gamma (\\epsilon ) : \\gamma \\in u_{\\alpha , \\epsilon } \\cap (\\beta + 1) \\rangle ~\\mbox{is $<_{L(M_1)}$-increasing}, \\\\& \\langle g_\\gamma (\\epsilon ) : \\gamma \\in u_{\\alpha , \\epsilon } \\cap (\\beta + 1) \\rangle ~\\mbox{is $<_{L(M_1)}$-decreasing}, \\\\& \\mbox{and}~ f_\\beta (\\epsilon ) <_{L(M_1)} g_\\beta (\\epsilon ) ~~\\rbrace \\\\$ Our aims in defining $f_\\alpha , g_\\alpha $ are, on the one hand, to continue describing a pre-cut, and on the other, to stay as close to the linearly ordered $w_{\\alpha , \\epsilon }$ as possible, as we now describe.", "That is, for fixed $\\alpha $ for each $\\epsilon $ , we will choose $f_\\alpha , g_\\alpha $ so that: For all $\\beta \\in w_{\\alpha , \\epsilon }$ , $M_1 \\models f_{\\beta }(\\epsilon ) < f_\\alpha (\\epsilon ) < g_\\alpha (\\epsilon ) < g_{\\beta }(\\epsilon )$ i.e.", "locally we continue the pre-cut described by $w_{\\alpha , \\epsilon }$ .", "For all $\\beta < \\alpha $ , neither $M_1 \\models f_\\beta (\\epsilon ) < f_\\alpha (\\epsilon ) < g_\\beta (\\epsilon ) < g_\\alpha (\\epsilon )$ nor $M_1 \\models f_\\alpha (\\epsilon ) < f_\\beta (\\epsilon ) < g_\\alpha (\\epsilon ) < g_\\beta (\\epsilon )$ i.e.", "the intervals are either nested or disjoint.", "If $\\gamma \\in \\alpha $ and $f_\\gamma (\\epsilon ), g_\\gamma (\\epsilon )$ satisfy: $ \\beta \\in w_{\\alpha , \\epsilon } \\Rightarrow f_\\beta (\\epsilon ) < f_\\gamma (\\epsilon ) < g_\\gamma (\\epsilon ) < g_\\beta (\\epsilon ) $ then the intervals $[f_\\alpha (\\epsilon ), g_\\alpha (\\epsilon )]_{L(M_1)}$ , $[f_\\beta (\\epsilon ), g_\\beta (\\epsilon )]_{L(M_1)}$ are disjoint i.e.", "inside the pre-cut given by $w_{\\alpha , \\epsilon }$ we avoid any further refinements: we realize exactly the intitial segment given by $w_{\\alpha , \\epsilon }$ .", "We will show in step 3 that by the hypothesis on $\\kappa $ , (a)-(g) imply the further condition that for each fixed $\\alpha < \\kappa ^+$ , For all $\\beta < \\alpha $ , $\\lbrace \\epsilon : f_{\\beta }(\\epsilon ) < f_\\alpha (\\epsilon ) < g_\\alpha (\\epsilon ) < g_{\\beta }(\\epsilon ) \\rbrace \\in \\mathcal {E}$ , i.e.", "the functions chosen will ultimately describe a pre-cut.", "At each index $\\epsilon < \\kappa $ , we may choose $f_\\alpha (\\epsilon ), g_\\alpha (\\epsilon )$ to satisfy (e),(f),(g) simply because $L_{M_1}$ is dense and $\\lambda ^+$ -saturated; the definition of $w_{\\alpha , \\epsilon }$ ensures (e) describes a pre-cut; and since (f) is inductively satisfied, (g) is possible.", "Step 3: For $\\beta < \\alpha $ , $f_\\beta < f_\\alpha < g_\\alpha < g_\\beta $ .", "In this step we verify that for the objects constructed in the previous step, for each $\\alpha < \\kappa ^+$ and all $\\beta < \\alpha $ , $ X_{\\alpha , \\beta } = \\lbrace \\epsilon < \\kappa :\\beta \\in u_{\\alpha , \\epsilon }, f_{\\beta }(\\epsilon ) < f_\\alpha (\\epsilon ) < g_\\alpha (\\epsilon ) < g_{\\beta }(\\epsilon ) \\rbrace \\in \\mathcal {E}$ (By conditions (a)-(d) requiring $\\beta \\in u_{\\alpha , \\epsilon }$ does not affect membership in $\\mathcal {E}$ .)", "Suppose this is not the case, so let $\\alpha < \\kappa ^+$ be minimal for which there is $\\beta < \\alpha $ with $X_{\\alpha , \\beta } \\notin \\mathcal {E}$ , and having fixed $\\alpha $ , let $\\beta < \\alpha $ be minimal such that $X_{\\alpha , \\beta } \\notin \\mathcal {E}$ .", "For the remainder of this step we fix this choice of $\\alpha , \\beta $ .", "Since $\\beta < \\alpha $ , by construction (that is, by (e),(f),(g) of Step 2) $ X_{\\alpha , \\beta } \\subseteq \\lbrace \\epsilon < \\kappa : \\beta \\in u_{\\alpha , \\epsilon } \\setminus w_{\\alpha , \\epsilon } \\rbrace $ Define a function $\\mathbf {x}: \\kappa \\rightarrow \\kappa $ by $t \\mapsto \\max \\lbrace \\epsilon \\le t :& \\langle f_\\gamma (t) : \\gamma \\in u_{\\alpha , \\epsilon } \\rangle ~\\mbox{is $<_{L(M_1)}$-increasing}, \\\\& \\langle g_\\gamma (t) : \\gamma \\in u_{\\alpha , \\epsilon }\\rangle ~\\mbox{is $<_{L(M_1)}$-decreasing}, \\\\& \\mbox{and $\\gamma \\in u_{\\alpha , \\epsilon } \\Rightarrow $}~ f_\\gamma (t) <_{L(M_1)} g_\\gamma (t) ~~\\rbrace \\\\$ This is well defined by Step 1, condition (a): 0 belongs to the set on the righthand side, and by continuity, there are no new conditions at limits.", "For each $\\epsilon < \\kappa $ , the set $\\lbrace t < \\kappa : \\mathbf {x}(t) > \\epsilon \\rbrace \\in \\mathcal {E}$ .", "This is because: by (c) $|u_{\\alpha , \\epsilon }| < \\kappa $ by minimality of $\\alpha $ , for any $\\gamma < \\gamma ^\\prime < \\alpha $ (e.g.", "any two elements of $u_{\\alpha , \\epsilon }$ ) we have that $X_{\\gamma , \\gamma ^\\prime } \\in \\mathcal {E}$ $\\mathcal {E}$ is $\\kappa $ -complete by (a) $\\epsilon ^\\prime < \\epsilon \\Rightarrow u_{\\alpha , \\epsilon ^\\prime } \\subseteq u_{\\alpha , \\epsilon }$ Notice that for any $t < \\kappa $ , $\\mathbf {x}(t) = t$ implies $u_{\\alpha , t} = w_{\\alpha , t}$ .", "So if $\\mathbf {x}(t) = t$ on an $\\mathcal {E}$ -large set, $X_{\\alpha , \\beta } \\in \\mathcal {E}$ , which would finish the proof.", "Suppose, then, that $Y = \\lbrace t < \\kappa : \\mathbf {x}(t) < t \\rbrace \\in \\mathcal {E}$ .", "By normality (Fact REF ), there is $Z \\subseteq Y$ , $Z \\in \\mathcal {E}$ on which $\\mathbf {x}(t) = \\epsilon _*$ for some fixed $\\epsilon _* < \\kappa $ .", "But this contradicts the first sentence of the previous paragraph.", "These contradictions prove that for no $\\alpha , \\beta $ can it happen that $X_{\\alpha , \\beta } \\notin \\mathcal {E}$ , which finishes the proof of Step 3.", "Step 4: The pre-cut is not realized, i.e.", "it is indeed a cut.", "In this step we assume, for a contradiction, that there is $h \\in {^\\kappa {M_1}}$ such that for each $\\alpha < \\kappa ^+$ $ f_\\alpha /\\mathcal {E}<_L h/\\mathcal {E}<_L g_\\alpha /\\mathcal {E}$ i.e.", "$h$ realizes the type.", "Fixing such an $h$ , let $ A_\\alpha = \\lbrace \\epsilon < \\kappa : f_\\alpha (\\epsilon ) <_L h(\\epsilon ) <_L g_\\alpha (\\epsilon ) \\rbrace \\in \\mathcal {E}$ Since to each $\\alpha $ we may associate a choice of index in $A_\\alpha $ , by Fodor's lemma for some $\\epsilon _* < \\kappa $ , $ S_1 = \\lbrace \\delta : \\delta < \\kappa ^+, ~ \\operatorname{cf}(\\delta ) = \\kappa , ~ \\epsilon _* \\in A_\\delta \\rbrace $ is stationary.", "Furthermore, since $|u_{\\epsilon _*, \\alpha }| < \\kappa $ , there is some $w_* \\subseteq u_{\\epsilon , \\alpha }$ for which $ S_2 = \\lbrace \\delta \\in S_1 : w_{\\epsilon _*, \\delta } = w_* \\rbrace \\subseteq \\kappa ^+ $ is stationary.", "Let $\\delta _* \\in S_2$ be such that $|\\delta _* \\cap S_2| = \\kappa $ .", "As $|w_{\\epsilon _*, \\delta _*}| \\le |u_{\\epsilon _*, \\delta _*}| < \\kappa $ , we may choose $\\gamma _* \\in S_2 \\cap \\lbrace \\delta _* \\setminus w_{\\epsilon _*, \\delta _*} \\rbrace $ .", "Now $w_{\\epsilon _*, \\delta _*} = w_{\\epsilon _*, \\gamma _*} = w_*$ since $\\delta _*, \\gamma _* \\in S_2$ , and note $\\gamma _* < \\delta _*$ .", "The definition of the sets $w$ (here, $w_*$ ) and Step 3, condition (e) means that when choosing $f_{\\delta _*}(\\epsilon _*), g_{\\delta _*}(\\epsilon _*)$ we would have ensured that $ \\beta \\in w_* \\Rightarrow f_{\\beta }(\\epsilon _*) < f_{\\delta _*}(\\epsilon _*) < g_{\\delta _*}(\\epsilon _*)< g_{\\beta }(\\epsilon _*) $ and likewise that $ \\beta \\in w_* \\Rightarrow f_{\\beta }(\\epsilon _*) < f_{\\gamma _*}(\\epsilon _*) < g_{\\gamma _*}(\\epsilon _*)< g_{\\beta }(\\epsilon _*) $ On the other hand, $\\gamma _* < \\delta _*$ , and $\\gamma _* \\notin w_*$ .", "So when choosing $f_{\\gamma _*}(\\epsilon ),g_{\\gamma _*}(\\epsilon )$ , Step 3, condition (g) would have meant we chose the intervals $[f_{\\delta _*}(\\epsilon _*), g_{\\delta _*}(\\epsilon _*)]_{L(M_1)}$ , $[f_{\\gamma _*}(\\epsilon _*), g_{\\gamma _*}(\\epsilon _*)]_{L(M_1)}$ to be disjoint.", "But we also know that $\\gamma _*, \\delta _* \\in S_1$ , so $h(\\epsilon _*)$ must belong to both intervals.", "This contradiction completes Step 4 and the proof.", "Remark 8.5 We know that if $D$ is any ultrafilter on $\\kappa $ and $M$ is a model whose theory is not simple, then $M^\\kappa /\\mathcal {D}$ is not $\\kappa ^{++}$ -saturated.", "Still, Claim REF gives more precise information about the size of the cut: we are guaranteed a cut of type $(\\kappa ^+, \\kappa ^+)$ as opposed to e.g.", "$(\\kappa ^+, \\kappa )$ .", "On the importance of symmetric cuts, see [16].", "Claim 8.6 Assume $\\kappa $ measurable, $\\mathcal {E}$ a $\\kappa $ -complete filter on $\\kappa $ , $\\lambda \\ge \\kappa $ , $M_1$ a $\\lambda $ -saturated model with ${(L_M, <_M)}$ a dense linear order.", "Let $M_2 = M_1^\\kappa /\\mathcal {E}$ .", "Then Suppose for a contradiction that there were such a cut given by $\\langle f_\\alpha : \\alpha < \\theta \\rangle $ , $\\langle g_\\beta : \\beta < \\sigma \\rangle $ with $\\alpha _1, \\alpha _2 < \\theta , \\beta _1, \\beta _2 < \\sigma \\Rightarrow \\lbrace t : M_1 \\models f_{\\alpha _1}(t) <_L f_{\\alpha _2}(t) <_L g_{\\alpha _2}(t) <_L g_{\\alpha _1}(t) \\rbrace \\in \\mathcal {E}$ .", "Expand the language to add constants $\\lbrace c_\\alpha : \\alpha < \\theta \\rbrace $ where in the $t$ th copy of the index model $M_1$ , denoted $M_1[t]$ , $c_\\alpha $ is interpreted as $f_\\alpha (t)$ .", "Then in the ultrapower (which in the expanded language is an ultraproduct), $\\langle c_\\alpha : \\alpha < \\theta \\rangle $ forms the lower half of the supposed cut.", "For each $\\beta < \\sigma $ , the set $ A_\\beta := \\bigcap \\lbrace \\lbrace t ~:~ M_1[t] \\models c_\\alpha <_L g_{\\beta }[t] \\rbrace : \\alpha < \\theta \\rbrace \\in \\mathcal {E}$ by $\\kappa $ -completeness.", "But recall that $M_1$ is a $\\lambda $ -saturated model, and $\\sigma < \\lambda $ .", "Since for each $t$ , we have $ | \\lbrace \\beta < \\sigma : t \\in A_\\beta \\rbrace | \\le \\sigma < \\lambda $ we may choose a new element $h \\in {^IM_1}$ so that for each $t$ , $h(t)$ satisfies $\\alpha < \\theta \\Rightarrow M_1[t] \\models c_\\alpha <_L h(t)$ and $t \\in A_\\beta \\Rightarrow M_1[t] \\models h(t) <_L g_\\beta (t)$ .", "By Łos' theorem $h$ realizes our cut, which is the desired contradiction.", "In a forthcoming paper the authors have shown that: Theorem H (Malliaris and Shelah [16]) If $\\mathcal {D}$ is a regular ultrafilter on $\\lambda $ which saturates some theory with $SOP_2$ , and $M$ is a model of linear order, then among other things: for all $\\mu \\le \\lambda $ $M^\\lambda /\\mathcal {D}$ has no $(\\mu , \\mu )$ -cut, for all $\\mu \\le \\lambda $ there is at most one $\\rho \\le \\lambda $ such that $M^\\lambda /\\mathcal {D}$ has a $(\\mu , \\rho )$ -cut Conclusion 8.7 Let $\\kappa < \\lambda $ and suppose $\\kappa $ is measurable.", "Then there exists a regular ultrafilter $\\mathcal {D}$ on $I$ , $|I| = \\lambda $ which is flexible but not good, specifically not good for any theory with $SOP_2$ .", "Let $D$ be a $\\lambda ^+$ -good, $\\lambda $ -regular ultrafilter on $\\lambda $ .", "Let $\\mathcal {E}$ be a normal $\\kappa $ -complete, not $\\kappa ^+$ -complete ultrafilter on $\\kappa $ .", "Let $\\mathcal {D}= D \\times \\mathcal {E}$ be the product ultrafilter.", "Then $\\mathcal {D}$ is flexible by Claim REF .", "On the other hand, by Claim REF , any $\\mathcal {D}$ -ultrapower of linear order will omit a $(\\kappa ^+, \\kappa ^+)$ -cut.", "By Theorem REF , $\\mathcal {D}$ cannot saturate any theory with $SOP_2$ .", "Remark 8.8 On one hand, the advantage of Conclusion REF over Theorem REF is in the greater range of cardinals: we ask only that $\\kappa < \\lambda $ , not $2^\\kappa \\le \\lambda $ .", "On the other hand, Theorem REF gives an a priori stronger failure of goodness, since the random graph is minimum among unstable theories in Keisler's order." ], [ "Finite alternations of symmetric cuts", "In this section we iterate the results of §, § to produce regular ultrafilters $\\mathcal {D}$ whose library of cuts, $\\mathcal {C}(\\mathcal {D})$ , contains any fixed finite number of alternations (or gaps).", "The following definition is stated for regular ultrafilters only so that the choice of index model will not matter.", "Definition 9.1 Let $\\kappa $ be a cardinal.", "Say that the regular ultrafilter $\\mathcal {D}$ on $\\lambda \\ge \\aleph _0$ has $\\kappa $ alternations of cuts if there exist cardinals $\\langle \\mu _\\ell : \\ell < \\kappa \\rangle $ , $\\langle \\rho _\\ell : \\ell < \\kappa \\rangle $ such that: $\\ell _1 < \\ell _2 < \\kappa \\Rightarrow \\aleph _0 < \\rho _{\\ell _1} < \\mu _{\\ell _1} < \\rho _{\\ell _2} < \\mu _{\\ell _2} < \\lambda $ for each $0 \\le \\ell < \\kappa $ , $(\\rho _\\ell , \\rho _\\ell ) \\in \\mathcal {C}(\\mathcal {D})$ , i.e.", "$(\\mathbb {N}, <)^\\lambda /\\mathcal {D}$ has some $(\\rho _\\ell , \\rho _\\ell )$ -cut for each $0 \\le \\ell < \\kappa $ , $(\\mu _\\ell , \\mu _\\ell ) \\notin \\mathcal {C}(\\mathcal {D})$ , i.e.", "$(\\mathbb {N}, <)^\\lambda /\\mathcal {D}$ has no $(\\mu _\\ell , \\mu _\\ell )$ -cut We will start by proving a theorem for products of complete ultrafilters, Theorem REF , and then extend it to regular ones in Theorem REF by adding one more iteration of the ultrapower.", "First we observe that taking ultrapowers will not fill symmetric cuts whose cofinality is larger than the size of the index set.", "Observation 9.2 Suppose $M$ is a $\\lambda $ -saturated model of linear order, $\\kappa < \\lambda $ , $D$ an ultrafilter on $\\kappa $ .", "If $M$ contains a $(\\kappa _*, \\kappa _*)$ -cut, where $\\kappa _* = \\operatorname{cf}(\\kappa _*) > \\kappa $ , then $M^\\kappa /D$ will also contain a $(\\kappa _*, \\kappa _*)$ -cut.", "More precisely, the image of the cut from $M$ under the diagonal embedding will remain unrealized in $M^\\kappa /D$ .", "Let the cut in $M$ be given by $(\\langle f_\\alpha : \\alpha < \\kappa _* \\rangle , \\langle g_\\beta : \\beta < \\kappa _* \\rangle )$ , and we consider the pre-cut given by $(\\langle f_\\alpha /D : \\alpha < \\kappa _* \\rangle , \\langle g_\\beta /D : \\beta < \\kappa _* \\rangle )$ in the ultrapower $M^\\kappa /D$ .", "Suppose for a contradiction that there were a realization $h \\in {^\\kappa M}$ .", "Let $\\mathbf {x}: \\kappa _* \\rightarrow \\kappa $ be a function which to each $\\alpha < \\kappa _*$ associates some index $\\epsilon < \\kappa $ for which $M \\models f_\\alpha (\\epsilon ) < h(\\epsilon ) < g_\\alpha (\\epsilon )$ .", "By Fodor's lemma, there is a stationary subset $X \\subseteq \\kappa _*$ on which $\\mathbf {x}$ is constant and equal to, say, $\\epsilon _*$ .", "Then in $M$ , $(\\langle f_\\alpha (\\epsilon _*) : \\alpha \\in X \\rangle , \\langle g_\\beta (\\epsilon _*): \\beta \\in X \\rangle )$ will be cofinal in the original cut, but by choice of $X$ it will be realized by $h(\\epsilon _*)$ , contradiction.", "Since the proof of Theorem REF involves an inductive construction, it will be convenient to index the cardinals $\\kappa _\\ell $ in reverse order of size.", "Theorem 9.3 Suppose that we are given: $n<\\omega $ and $\\kappa _{n} < \\dots < \\kappa _0 < \\kappa _{-1} = \\lambda $ $\\mathcal {E}_\\ell $ a normal $\\kappa _\\ell $ -complete ultrafilter on $\\kappa _\\ell $ , for $\\ell \\le n$ .", "$M_0$ a $\\lambda $ -saturated model which is, or contains, a dense linear order $<$ $M_{\\ell + 1} = (M_\\ell )^{\\kappa _\\ell }/\\mathcal {E}_\\ell $ for $\\ell \\le n$ Then: for $\\ell \\le n$ , $M_{\\ell + 1}$ is $\\kappa _\\ell $ -saturated if $\\ell < k \\le n+1$ then $M_k$ has a $({\\kappa _\\ell }^+, {\\kappa _\\ell }^+)$ -cut for $i, \\ell \\le n+1$ , $M_\\ell $ has no $(\\kappa _i, \\kappa _i)$ -cut for $\\ell \\le n+1$ , $M_\\ell $ has no $(\\theta , \\theta )$ -cut for $\\theta < \\lambda $ weakly compact Thus, for each $\\ell \\le n$ , $({\\kappa _\\ell }^+, {\\kappa _\\ell }^+) \\in \\mathcal {C}(M_{n+1})$ , and for any weakly compact $\\theta < \\lambda $ , in particular $\\theta = \\kappa _\\ell $ , $(\\theta , \\theta ) \\notin \\mathcal {C}(M_{n+1})$ .", "The “thus” clause summarizes ($\\alpha $ )-($\\delta $ ).", "Recall that for transparency all languages are countable.", "Note that condition (b) implies the cardinals $\\kappa _\\ell $ are measurable cardinals, thus limit cardinals, so condition ($\\gamma $ ) can never contradict condition ($\\beta $ ).", "($\\alpha $ ) By induction on $-1 \\le \\ell \\le n$ we verify that $M_{\\ell + 1}$ is $\\kappa _\\ell $ -saturated.", "For $\\ell = -1$ , $M_0$ is $\\lambda $ -saturated and $\\lambda > \\kappa _0$ .", "For $\\ell > -1$ , use the fact that $M_{\\ell +1} = ({M_\\ell })^{\\kappa _\\ell }/\\mathcal {E}_\\ell $ thus $M_{\\ell +1} \\equiv _{L_{\\infty , \\kappa _\\ell }} M_\\ell $ by Łos' theorem for $L_{\\kappa _i, \\kappa _i}$ .", "($\\beta $ ) By Claim REF and Observation REF .", "($\\gamma $ ) Follows from ($\\delta $ ) as measurable implies weakly compact.", "($\\delta $ ) We prove this by induction on $\\ell \\le n$ .", "For $\\ell = -1$ , $M_0$ is $\\lambda $ -saturated.", "For $\\ell > -1$ , let $\\theta < \\lambda $ be given and suppose we have a pre-cut in $M_{\\ell +1}$ given by $(\\langle f_\\alpha : \\alpha < \\theta \\rangle , \\langle g_\\beta : \\beta < \\theta \\rangle )$ .", "There are three cases.", "If $\\theta < \\kappa _\\ell $ , then use ($\\alpha $ ).", "If $\\theta = \\kappa _\\ell $ , use Claim REF .", "So we may assume $\\kappa _\\ell < \\theta $ .", "Since $\\theta $ is weakly compact, therefore inaccessible, ${2^{\\kappa _\\ell }} < \\theta $ .", "For $\\alpha < \\beta < \\theta $ let $ A_{\\alpha , \\beta } = \\lbrace \\epsilon < \\kappa _\\ell : f_\\alpha (\\epsilon ) < f_\\beta (\\epsilon ) < g_\\beta (\\epsilon ) < g_\\alpha (\\epsilon ) \\rbrace \\in \\mathcal {E}_\\ell $ As $\\theta $ is weakly compact, by Fact REF the function $\\mathbf {x}: \\theta \\times \\theta \\rightarrow {2^{\\kappa _\\ell }} < \\theta $ is constant on some $\\mathcal {U}\\in [\\theta ]^{\\theta }$ .", "Call this constant value $A_*$ .", "Now for $\\epsilon \\in A_*$ , the sequence $(\\langle f_\\alpha (\\epsilon ) : \\alpha \\in \\mathcal {U}\\rangle , \\langle g_\\beta (\\epsilon ) : \\beta \\in \\mathcal {U}\\rangle )$ is a pre-cut in $M_\\ell $ , meaning that $\\alpha < \\beta \\in \\mathcal {U}\\Rightarrow f_\\alpha (\\epsilon ) < f_\\beta (\\epsilon ) < g_\\beta (\\epsilon ) < g_\\alpha (\\epsilon )$ .", "Let $B_* = \\lbrace \\epsilon \\in A_* : ~\\mbox{in $M_\\ell $ there is $c$ such that $\\alpha \\in \\mathcal {U}\\Rightarrow f_\\alpha (\\epsilon ) <_{M_\\ell } c <_{M_\\ell } g_\\alpha (\\epsilon )$} \\rbrace $ .", "Now if $A_* \\setminus B_* \\ne \\emptyset $ , for any $\\epsilon \\in A_* \\setminus B_*$ we have that $(\\langle f_\\alpha (\\epsilon ) : \\alpha \\in \\mathcal {U}\\rangle , \\langle g_\\beta (\\epsilon ) : \\beta \\in \\mathcal {U}\\rangle )$ is not just a pre-cut but also a cut in $M_\\ell $ , contradicting the inductive hypothesis.", "Thus for every $\\epsilon \\in A_*$ we may choose a realization $c(\\epsilon )$ of the relevant pre-cut.", "For $\\epsilon \\in \\kappa \\setminus A_*$ , let $c(\\epsilon )$ be arbitrary.", "Then $\\langle c(\\epsilon ) : \\epsilon < \\kappa _\\ell \\rangle /\\mathcal {E}_\\ell \\in M_{\\ell +1}$ realizes $(\\langle f_\\alpha : \\alpha < \\theta \\rangle , \\langle g_\\beta : \\beta < \\theta \\rangle )$ , as desired.", "By appending the construction of Theorem REF to a suitable regular ultrafilter, we may produce regular ultrapowers with $n$ alternations of cuts for any finite $n$ .", "Theorem 9.4 Let $\\lambda $ be an infinite cardinal, $n<\\omega $ and suppose that there exist measurable cardinals $\\kappa _n < \\dots < \\kappa _0 <\\lambda $ .", "Then there is a regular ultrafilter $\\mathcal {D}$ on $I$ , $|I| = \\lambda $ such that: $({\\kappa _\\ell }^+, {\\kappa _\\ell }^+) \\in \\mathcal {C}(\\mathcal {D})$ for $\\ell \\le n$ $(\\theta , \\theta ) \\notin \\mathcal {C}(\\mathcal {D})$ for $\\theta < \\lambda $ weakly compact, in particular $\\ell \\le n$ , $\\theta = \\kappa _\\ell $ $\\mathcal {D}$ is ${\\kappa _n}^+$ -good $\\mathcal {D}$ is $\\lambda $ -flexible $\\mathcal {D}$ is $\\lambda ^+$ -good for countable stable theories $\\mathcal {D}$ is not $(2^{\\kappa _n})^+$ -good for unstable theories Let $\\mathcal {D}_1$ be a $\\lambda $ -regular, $\\lambda ^+$ -good ultrafilter on $\\lambda $ .", "Let $\\mathcal {E}$ be the ultrafilter on $\\kappa _n$ given by $\\mathcal {E}_0 \\times \\dots \\times \\mathcal {E}_n$ , where the $\\mathcal {E}_\\ell $ are as in the statement of Theorem REF .", "Let $\\mathcal {D}= \\mathcal {D}_1 \\times \\mathcal {E}$ .", "We will show that $\\mathcal {D}$ has the desired properties.", "This follows from having chosen $M_0$ in Theorem REF to be a $\\mathcal {D}_1$ -ultrapower, thus $\\lambda ^+$ -saturated by the $\\lambda ^+$ -goodness (and regularity) of $\\mathcal {D}_1$ .", "By Observation REF .", "By induction on $\\ell \\le n$ , using Claim REF (1) and the completeness of $\\mathcal {E}_\\ell $ , $\\ell \\le n$ .", "By induction on $\\ell \\le n$ , using Claim REF (2).", "Since $\\mathcal {D}_1$ is regular and $\\lambda ^+$ -good, $\\operatorname{lcf}(\\aleph _0, \\mathcal {D}) \\ge \\lambda ^+$ .", "By Claim REF .", "Question 9.5 Can Theorem REF be generalized to any number of alternations, not necessarily finite?" ], [ "Appendix", "In this appendix, we collect several easy proofs or extensions of proofs relevant to the material in the paper.", "Observation 9.6 If $\\mathcal {E}$ is a $\\kappa $ -complete ultrafilter on $\\kappa $ then $\\mathcal {E}$ is $\\kappa ^+$ -good.", "We adapt the proof of [18] Claim 3.1 p. 334 that any ultrafilter is $\\aleph _1$ -good.", "Suppose we are given a monotonic function $f : {\\mathcal {P}}_{\\aleph _0}(\\kappa ) \\rightarrow \\mathcal {E}$ .", "Define $g(u) = \\bigcap \\lbrace f(w) : \\max {w} \\le \\max {u} \\rbrace $ .", "For each $u \\in {\\mathcal {P}}_{\\aleph _0}(\\kappa )$ , $\\max (u) < \\kappa $ and moreover the number of possible finite $w \\subseteq \\lbrace \\gamma : \\gamma < \\max (u) \\rbrace $ is $< \\kappa $ .", "Thus by $\\kappa $ -completeness, $g(u) \\in \\mathcal {E}$ .", "Clearly $g$ refines $f$ , and since $\\max (u \\cup v) = \\max \\lbrace \\max (u), \\max (v) \\rbrace $ , $g$ is multiplicative as desired.", "Note, however, that while a good regular ultrafilter produces saturated ultrapowers, this need not be the case when the ultrafilter is complete, unless the index models are also saturated.", "That is, the goodness of the ultrafilter $D$ on $I$ is equivalent to saturation of the ultrapower $M^I/D$ when (a) we have regularity of the ultrafilter, or (b) we assume the model $M$ is saturated.", "Fact 9.7 Let $D$ be an ultrafilter on $I$ and $\\lambda $ a cardinal.", "Then the following are equivalent: $D$ is $\\lambda $ -good for any model $M$ in a countable signature which is $\\lambda $ -saturated, $M^I/D$ is $\\lambda $ -saturated and if $D$ is regular, (1) is equivalent to: for any model $M$ in a countable signature, $M^I/D$ is $\\lambda $ -saturated See [18] VI.2 in particular Theorems 2.2-2.3, Claim 2.4 and Lemma 2.11.", "By re-presenting the proof of [18] Theorem VI.4.8 p. 379 to emphasize the role of incompleteness, we obtain a more general result.", "Theorem 9.8 Let $M$ be a $\\lambda ^+$ -saturated model of an unstable theory $T$ , $\\varphi $ an unstable formula and $E$ a $\\kappa $ -complete, $\\kappa ^+$ -incomplete ultrafilter on $\\lambda $ .", "Let $\\delta = \\operatorname{lcf}(\\kappa , \\mathcal {E})$ .", "Then $M^I/\\mathcal {E}$ is not $(\\kappa + \\operatorname{lcf}(\\kappa , \\mathcal {E}))^+$ -saturated for $\\varphi $ -types.", "First consider the countably incomplete case.", "We build a correspondence between a $\\varphi $ -type and a $<$ -type in an expanded language.", "Let $\\varphi = \\varphi (x;y)$ , where without loss of generality $\\ell (x)=1$ but $\\ell (y)$ need not be 1.", "Choose, for each $m < n <\\omega $ , sequences $\\overline{a}^n_m$ where $\\ell (\\overline{a}^n_m) = \\ell (y)$ , so that $\\varphi $ has the order property over each $\\langle \\overline{a}^n_m : m < n \\rangle $ : i.e., $k < n < \\omega $ implies $\\lbrace \\varphi (x;\\overline{a}^n_m)^{if (m > k)} : m < n \\rbrace $ is consistent.", "Let $\\langle b_n : n < \\omega \\rangle $ be a sequence of distinct elements.", "Let $P$ be a new unary relation symbol, $<$ a new binary relation symbol, and for each $\\ell < \\ell (y)$ let $F_\\ell $ be a new binary function symbol.", "Let $M_1$ denote the expansion of $M$ by these new symbols, as follows.", "$P^{M_1} = \\lbrace b_n : n < \\omega \\rbrace $ and $<^{M_1} = \\lbrace \\langle b_k, b_n \\rangle : k < n < \\omega \\rbrace $ .", "Finally, interpret the functions $F_\\ell $ so that for each $k<n<\\omega $ , $\\overline{a}^n_k = \\langle F_0(b_k, b_n), \\dots F_{\\ell (y)-1}(b_k, b_n) \\rangle $ .", "Now we take the ultrapower $N = M^\\lambda /E$ and let $N_1$ denote the corresponding ultrapower in the expanded language.", "In $N_1$ , $P$ is a linear order and so by the hypothesis on $E$ we have a $\\delta $ -cut over the diagonal embedding of the sequence $\\langle b_n : n < \\omega \\rangle $ .", "Choose a sequence $\\langle c_i : i < \\delta \\rangle $ witnessing this, so (1) each $c_i \\in P^{N_1}$ , (2) $n <\\omega $ and $i<j<\\delta $ implies $N_1 \\models b_n < c_i < c_j$ , (3) for no $c \\in P^{N_1}$ is it the case that for each $n<\\omega , i <\\delta $ , $N_1 \\models b_n < c < c_i$ .", "We now translate back to a $\\varphi $ -type.", "Consider: $ q(x) = \\lbrace \\lnot \\varphi (x,F_0(b_n, c_0), \\dots F_{\\ell (y)-1}(b_n, c_0)) : n < \\omega \\rbrace \\cup \\lbrace \\varphi (x,F_0(c_i, c_0), \\dots F_{\\ell (y)-1}(c_i, c_0)) : 0 < i < \\delta \\rbrace $ This is a consistent partial $\\varphi $ -type by $Łos^{\\prime } $ theorem, since for $n < \\omega , i < \\delta $ we have that $b_n < c_i \\mod {E}$ .", "We will show that $q$ is omitted.", "Suppose it were realized, say by $a$ .", "Let $\\langle X_n : n < \\omega \\rangle $ be a sequence of elements of $E$ witnessing that $E$ is $\\aleph _1$ -incomplete.", "Let $\\langle Y_n : n < \\omega \\rangle $ be a sequence of elements of $E$ given by $ Y_n = \\lbrace t \\in \\lambda : M \\models \\bigwedge \\lbrace \\lnot \\varphi (a[t],F_0(b_k, c_0[t]), \\dots F_{\\ell (y)-1}(b_k, c_0[t])) : k \\le n \\rbrace $ which exists by Łos' theorem.", "(We write $b_k$ rather than $b_k[t]$ since these are essentially constant elements.", "Note that in each index model, $c_0[t]$ is simply one of the $b_n$ s.) For each $t \\in I$ define $\\rho (t) = \\min \\lbrace n : t \\notin X_{n+1} \\cap Y_{n+1} \\rbrace $ .", "By the assumption of $\\aleph _1$ -incompleteness, $\\rho $ is well defined.", "Define a new element $b \\in {^\\lambda M}$ by: $b[t] = b_{\\rho (t)}$ .", "By Łos' theorem $P^{N_1}(b)$ , so let us determine its place in the linear order $<^{N_1}$ .", "First, for each $n<\\omega $ , we have $b > b_n \\mod {E}, as $ E is $\\omega $ -complete (i.e.", "a filter) thus: $ \\lbrace t : c[t] > b_n[t] \\rbrace \\supseteq \\bigcap _{j \\le n} ( X_{j+1} \\cap Y_{j+1} ) \\in \\mathcal {D}$ On the other hand, suppose that for some $i < \\delta $ we had $c_i \\le b \\mod {E}$ .", "Then by Łos' theorem and definition of $\\rho $ , it would have to be the case that $ N_1 \\models \\lnot \\varphi (a[t],F_0(c_i[t], c_0[t]), \\dots F_{\\ell (y)-1}(c_i[t], c_0[t])) $ contradicting the definition of $q$ .", "So for each $i < \\delta $ , we have that $b < c_i \\mod {E}$ .", "Thus from a realization $a \\models q$ we could construct a realization $b$ of the $(\\aleph _0, \\delta )$ -cut in $P^N$ .", "Since the latter is omitted, $q$ must be as well, which completes the proof for countably incomplete filters.", "Now for the general case: If $\\lambda = \\kappa $ , modify the argument of Theorem REF by replacing $\\omega $ with $\\kappa $ everywhere in the proof just given, and Łos' theorem by Łos' theorem for $L_{\\kappa , \\kappa }$ .", "If $\\lambda > \\kappa $ , begin by choosing a surjective map $\\mathbf {h}: \\lambda \\rightarrow \\kappa $ so that $\\mathcal {E}= h(E)$ is a nonprincipal ultrafilter on $\\kappa $ , thus $\\kappa $ -complete not $\\kappa ^+$ -complete.", "For completeness, we verify that $\\lambda $ -flexible corresponds to $\\lambda $ -OK, Definition REF above.", "Observation 9.9 Suppose that $\\mathcal {D}$ is an $\\aleph _1$ -incomplete ultrafilter on $I$ .", "Then the following are equivalent.", "$\\mathcal {D}$ is $\\lambda $ -O.K.", "$\\mathcal {D}$ is $\\lambda $ -flexible.", "(1) $\\rightarrow $ (2) Let $M$ be given with $(\\mathbb {N}, <) \\preceq M$ and let $h_0 \\in {^I M}$ be any $\\mathcal {D}$ -nonstandard integer.", "Let $\\lbrace Z_n : n < \\omega \\rbrace \\subseteq \\mathcal {D}$ witness that $\\mathcal {D}$ is $\\aleph _1$ -incomplete.", "Without loss of generality, $n < n^\\prime \\Rightarrow Z_n \\supseteq Z_{n^\\prime }$ .", "Let $h_1 \\in { ^I \\mathbb {N} }$ be given by $h_1(t) = \\max \\lbrace n : t \\in Z_n \\rbrace $ .", "Define $h \\in {^I \\mathbb {N}}$ by $ h(t) = \\min \\lbrace h_0(t), h_1(t) \\rbrace $ Then for each $n\\in \\mathbb {N}$ , $X_n := \\lbrace t : h(t) \\ge n \\rbrace \\in \\mathcal {D}$ and $X_n \\subseteq Z_n$ , thus $\\bigcap \\lbrace X_n : n \\in \\mathbb {N} \\rbrace = \\emptyset $ .", "Define a function $f: {\\mathcal {P}}_{\\aleph _0}(\\lambda ) \\rightarrow \\mathcal {D}$ by $f(u) = X_{|u|}$ .", "As $\\mathcal {D}$ is $\\lambda $ -OK, we may choose $g$ to be a multiplicative refinement of $f$ , and consider $\\mathbf {Y} = \\lbrace Y_i : i < \\lambda \\rbrace $ given by $Y_i = g(\\lbrace i \\rbrace )$ .", "First, we verify that $\\mathbf {Y}$ is a regularizing family, by showing that each $t \\in I$ can only belong to finitely many elements of $\\mathbf {Y}$ .", "Given $t \\in I$ , let $m = h(t) + 1 < \\omega $ , so $t \\notin X_m$ .", "Suppose there were $i_1 < \\dots < i_m < \\lambda $ such that $t \\in g(\\lbrace i_1 \\rbrace ) \\cap \\dots \\cap g(\\lbrace i_m \\rbrace )$ .", "As $g$ is multiplicative and refines $f$ , this would imply $t \\in g(\\lbrace i_1, \\dots i_m \\rbrace ) \\subseteq f(\\lbrace i_1, \\dots i_m \\rbrace ) = X_m$ , a contradiction.", "Thus $\\mathbf {Y}$ is a regularizing family.", "Moreover, as $t$ was arbitrary, we have shown that $ \\left| \\lbrace i < \\lambda : t \\in g(\\lbrace i \\rbrace ) \\rbrace \\right| \\le h(t) \\le h_0(t) $ and thus that $\\mathbf {Y}$ is a regularizing family below $h_0$ .", "As $h_0$ was an arbitrary nonstandard integer, this completes the proof.", "(2) $\\rightarrow $ (1) Let $f: {\\mathcal {P}}_{\\aleph _0}(\\lambda ) \\rightarrow \\mathcal {D}$ be such that $|u| = |v| \\Rightarrow f(u) = f(v)$ , and we will construct a multiplicative refinement for $f$ .", "Let $\\langle Z_n : n < \\omega \\rangle $ witness the $\\aleph _1$ -incompleteness of $\\mathcal {D}$ , and as before, we may assume $n < n^\\prime \\Rightarrow Z_n \\supseteq Z_{n^\\prime }$ .", "For each $t \\in I$ , let $\\rho \\in {^I\\mathbb {N}}$ be given by $\\rho (t) = \\max \\lbrace n \\in \\mathbb {N} : t \\in f(n) \\cap Z_n \\rbrace $ , which is well defined by the choice of the $Z_n$ .", "Now for each $m \\in \\mathbb {N}$ $ \\lbrace t \\in I : \\rho (t) > m \\rbrace \\supseteq \\bigcap \\lbrace f(n) \\cap Z_n : n \\le m \\rbrace \\in \\mathcal {D}$ so $\\rho $ is $\\mathcal {D}$ -nonstandard.", "Applying the hypothesis of flexibility, let $\\lbrace Y_i : i < \\lambda \\rbrace $ be a $\\lambda $ -regularizing family below $\\rho $ .", "Let $g: {\\mathcal {P}}_{\\aleph _0}(\\lambda ) \\rightarrow \\mathcal {D}$ be given by $g(\\lbrace i \\rbrace ) = f(\\lbrace i\\rbrace ) \\cap Y_i$ and for $|u| > 1$ , $g(u) = \\bigcap \\lbrace g(\\lbrace i\\rbrace ) : i \\in u \\rbrace $ .", "Thus $g$ is multiplicative, by construction.", "Let us show that it refines $f$ .", "Given any $n<\\omega $ and $i_1 < \\dots < i_n <\\lambda $ , observe that by definition of “below $\\rho $ ” we have $t \\in Y_{i_1} \\cap \\dots \\cap Y_{i_n} \\Rightarrow \\rho (t) \\ge n$ .", "Applying this fact and the definitions of $g$ and $f$ , $ g( \\lbrace i_1, \\dots i_n \\rbrace ) \\subseteq \\bigcap \\lbrace Y_{i_j} : 1 \\le j \\le n \\rbrace \\subseteq \\lbrace t \\in I : \\rho (t) \\ge n \\rbrace \\subseteq f(n) =f(\\lbrace i_1, \\dots i_n \\rbrace ) $ thus $g$ refines $f$ , which completes the proof." ] ]

[ [ "A Note on the Balanced ST-Connectivity" ], [ "Abstract We prove that every YES instance of Balanced ST-Connectivity has a balanced path of polynomial length." ], [ "Introduction", "Kintali [2] introduced new kind of connectivity problems called graph realizability problems, motivated by the study of AuxPDAs [1].", "In this paper, we study one such graph realizability problem called Balanced ST-Connectivity and prove that every YES instance of Balanced ST-Connectivity has a balanced path of polynomial length.", "Let $\\mathcal {G}(V,E)$ be a directed graph and let $n = |V|$ .", "Let $\\mathcal {G^{\\prime }}(V,E^{\\prime })$ be the underlying undirected graph of $\\mathcal {G}$ .", "Let $P$ be a path in $\\mathcal {G^{\\prime }}$ .", "Let $e = (u,v)$ be an edge along the path $P$ .", "Edge $e$ is called neutral edge if both $(u,v)$ and $(v,u)$ are in $E$ .", "Edge $e$ is called forward edge if $(u,v) \\in E$ and $(v,u) \\notin E$ .", "Edge $e$ is called backward edge if $(u,v) \\notin E$ and $(v,u) \\in E$ .", "A path (say $P$ ) from $s \\in V$ to $t \\in V$ in $\\mathcal {G^{\\prime }}(V,E^{\\prime })$ is called balanced if the number of forward edges along $P$ is equal to the number of backward edges along $P$ .", "A balanced path might have any number of neutral edges.", "By definition, if there is a balanced path from $s$ to $t$ then there is a balanced path from $t$ to $s$ .", "Balanced ST-Connectivity : Given a directed graph $\\mathcal {G}(V,E)$ and two distinguished nodes $s$ and $t$ , decide if there is balanced path between $s$ and $t$ .", "A balanced path may not be a simple path.", "The example in Figure REF shows an instance of Balanced ST-Connectivity where the only balanced path between $s$ and $t$ is of length $\\Theta (n^2)$ .", "The directed simple path from $s$ to $t$ is of length $n/2$ .", "There is a cycle of length $n/2$ at the vertex $v$ .", "All the edges (except $(v,u)$ ) in this cycle are undirected.", "The balanced path from $s$ to $t$ is obtained by traversing from $s$ to $v$ , traversing the cycle clockwise for $n/2$ times and then traversing from $v$ to $t$ ." ], [ "Length of Balanced Paths", "We now prove that every YES instance of Balanced ST-Connectivity has a balanced path of polynomial length.", "We need the following lemma.", "Lemma 2.1 Let $c_1 < c_2 < \\dots < c_r \\in [n]$ and $k \\in [n]$ .", "If $m_1, m_2, \\dots , m_r$ are integers such that $m_1c_1 + m_2c_2 + \\dots + m_rc_r = k$ , then there exist integers $m^{\\prime }_1, m^{\\prime }_2, \\dots ,m^{\\prime }_r$ satisfying $m^{\\prime }_1c_1 + m^{\\prime }_2c_2 + \\dots + m^{\\prime }_rc_r = k$ such that $|m^{\\prime }_1|+|m^{\\prime }_2|+\\dots +|m^{\\prime }_r| \\le O(nr)$ .", "Let $a_{i} = \\lfloor {\\frac{m_{i}}{c_{r}}}\\rfloor $ and $m_i = a_ic_r+b_i$ for $1 \\le i \\le r-1$ .", "We have, $(a_1c_r+b_1)c_1 + (a_2c_r+b_2)c_2 + \\dots + (a_{r-1}c_r+b_{r-1})c_{r-1}+ m_rc_r = k$ Rearranging we get, $b_1c_1 + b_2c_2 + \\dots + b_{r-1}c_{r-1} + (m_r+a_1c_1+a_2c_2+\\dots +a_{r-1}c_{r-1})c_r = k$ .", "Note that $|b_i| < c_r < n$ for $1 \\le i \\le r-1$ .", "Hence, $b_1c_1 + b_2c_2 + \\dots + b_{r-1}c_{r-1} - k = O(n\\sum _{i=1}^{r-1}{c_{i}})$ .", "Hence, $m_r+a_1c_1+a_2c_2+\\dots +a_{r-1}c_{r-1} = O(n\\sum _{i=1}^{r-1}{c_{i}})/c_{r} = O(nr)$ .", "Setting $m^{\\prime }_i = b_i$ for $1 \\le i \\le r-1$ and $m^{\\prime }_{r} = m_r+a_1c_1+a_2c_2+\\dots +a_{r-1}c_{r-1}$ we get the desired result.", "Theorem 2.2 Let $G(V,E)$ be a directed graph with two distinguished vertices $s,t \\in V$ and let $P$ be a balanced path from $s$ to $t$ .", "Then there exists a balanced path $Q$ from $s$ to $t$ such that the length of $Q$ is $O(n^3)$ .", "We decompose $P$ into a simple path (say $P^{\\prime }$ ) from $s$ to $t$ and a set of cycles $\\mathcal {C} = \\lbrace C_1,C_2,\\dots ,C_l\\rbrace $ .", "Let $c_1,\\dots ,c_r$ be the distinct lengths of the cycles in $\\mathcal {C}$ .", "Let $-k$ denote the number of forward edges minus the number of backward edges along $P^{\\prime }$ from $s$ to $t$ .", "Since there is a balanced path from $s$ to $t$ using the path $P^{\\prime }$ and the cycles from $\\mathcal {C}$ , there exist integers $m_1,\\dots ,m_r$ satisfying $m_1c_1+\\dots +m_rc_r=k$ .", "Applying Lemma REF there exist integers $m^{\\prime }_1,\\dots ,m^{\\prime }_r$ satisfying $m^{\\prime }_1c_1+\\dots +m^{\\prime }_rc_r=k$ such that $|m^{\\prime }_1|+\\dots +|m^{\\prime }_r| \\le O(nr)$ .", "We now construct a balanced path $Q$ from $s$ to $t$ as follows : For every $m^{\\prime }_i$ we walk $m^{\\prime }_i$ times around the cycle of length $c_i$ (if there are several cycles of this length, we choose one of them arbitrarily).", "Note that these cycles may not be connected to each other.", "We now choose an arbitrary vertex from each cycle and connect it to $t$ by simple paths (say $P_1, P_2, \\dots , P_r$ ).", "The new balanced path $Q$ starts from $s$ and follows the simple path $P^{\\prime }$ from $s$ to $t$ and uses $P_i$ to reach the cycle of length $c_i$ and walks around it $m^{\\prime }_i$ times and comes back to $t$ using $P_{i}$ .", "This is repeated for $1 \\le i \\le r$ .", "Since each $P_i$ is used once while going away from $t$ and once while coming back to $t$ , the paths $P_1, P_2, \\dots , P_r$ do not modify the balancedness of the path $Q$ .", "The combined length of paths $P_1, P_2, \\dots , P_r$ is $O(nr)$ .", "Since $|m^{\\prime }_1|+\\dots +|m^{\\prime }_r| \\le O(nr)$ the overall length of the balanced path $Q$ is $O(nr) = O(n^{2})$ ." ] ]

[ [ "On the origin of brane cosmological constant in two-brane warped\n geometry model" ], [ "Abstract In the backdrop of generalised Randall-Sundrum braneworld scenario, we look for the possible origin of an effective four dimensional cosmological constant ($\\Omega_{vis}$) on the visible 3-brane due to the effects of bulk curvature and the modulus field that can either be a constant or dependent on extra dimensional co-ordinate $y$ or a time dependent quantity.", "In case of constant or $y$ dependent modulus field, the induced $\\Omega_{vis}$ leads to an exponentially expanding Universe.", "For such modulus fields the presence of vacuum energy densities on either of the 3-branes as well as a non-vanishing bulk curvature $l$ ($l \\sim {\\Lambda_5}^{-1}$) are essential to generate an effective $\\Omega_{vis}$.", "In particular for constant modulus field the Hubble constant turns out to be equal to the visible brane cosmological constant which agrees with the present result.", "In an alternative scenario, a time dependent modulus field is found to be capable of accelerating the Universe.", "The Hubble parameter in this case is determined for a slowly time-varying modulus field." ], [ "Introduction", "The introduction of extra dimensions to four dimensional spacetime dates back to early twentieth century when Kaluza and Klein tried to unify gravity and electromagnetic theory in the presence of an extra spatial dimension.", "In recent times with the advent of String theory this concept once again witnessed revival.", "Moreover, our inability to explain many problems in Particle physics and Cosmology has stimulated widespread interests for exploring higher dimensional theories in recent times.", "To start with, the Standard model of Particle physics suffers from a discrepancy known as Higgs mass hierarchy problem or alternatively gauge hierarchy problem [1].", "The Higgs boson receives quantum corrections from higher order self energy corrections typically of the order of Planck energy scale, but to obtain a theoretical predicted value of Higgs mass (of the order of 100 GeV), an extreme fine tuning is necessary, often called the naturalness problem.", "One therefore needs to look beyond the Standard model to seek a possible explanation to this fine tuning condition.", "There also exists another fine-tuning known as cosmological constant problem which relates with the mismatch between observed and theoretically estimated values of cosmological constant and needs to be resolved.", "Extra dimensional theories offer possible solutions to these problems and address many more which include various phenomenological and cosmological issues including the existence of dark energy, inflation, CMB anisotropies etc.", "Typically in a braneworld model, the branes are hypersurfaces embedded in a higher dimensional bulk.", "All standard model particles are confined to the branes while gravity is allowed to propagate everywhere including in higher dimensions.", "Among various extra dimensional models, the warped geometry model assumes that the extra dimensions are compactified on a closed path and their sizes are required to be extremely small (Planck length $\\sim 10^{-33}$ cm) such that no intermediate scale enters in the theory.", "On the contrary the large extra dimensional models like ADD model [2] tries to solve the naturalness problem by lowering the higher dimensional Planck scale to TeV scale with the help of two flat and compact extra dimensions whose sizes are quite large compared to the Planck length.", "In this work we focus our attention to the well known warped geometry model proposed by Randall and Sundrum (RS) [3] and it's subsequent generalization to include non-flat 3-branes [4].", "The Randall Sundrum model consisting of two $Z_2$ symmetric 3-branes embedded in a five dimensional AdS bulk offers a possible solution to the gauge hierarchy problem by exponentially suppressing mass of Higgs boson and it's vev on TeV brane.", "The naturalness problem can be avoided in this scenario without invoking any intermediate scale in the theory.", "In this model, the effect of bulk cosmological constant is exactly counterbalanced by the brane tensions leading to zero brane cosmological constant and therefore the 3-branes are flat and static [3].", "So far various applications of RS model in the cosmology/phenomenological issues of Particle physics and String theory have been explored [5], [6].", "The flat brane RS model described above was generalised [4] to include 3-branes which possess a net induced non-zero cosmological constant.", "We try to investigate the possible origin of this 3-brane cosmological constant by analysing the effective gravity models derived by embedding 3-branes at orbifold fixed points in a bulk spacetime [7] in presence of bulk cosmological constant and a modulus field which can be constant, time dependent or extra dimensional co-ordinate dependent.", "In section and in section , we briefly discuss the works by Shiromizu et.al [7] and the generalised Randall Sundrum model [4].", "In section , the induced cosmological constant in the presence of a constant radion field, leading to an exponentially expanding Universe, has been shown to emerge due to the presence of both brane matter as well as bulk curvature.", "The Hubble parameter on the visible brane is determined in this case.", "It is shown to be proportional to the induced visible brane cosmological constant which agrees with the present results without any fine tuning.", "In section , we incorporate extra-dimensional coordinate ($y$ ) dependence in the modulus field called radion and once again try to generate an effective cosmological constant on the Universe located on the visible brane ($y = r \\pi $ ).", "An exponential solution of the scale factor is also found in this case and we have further examined the origin of visible brane cosmological constant.", "It is to be noted that many results of these two sections are similar in nature.", "Finally in section , we find that a time dependent radion field can be responsible for an accelerated expanding phase of our Universe even in absence of any brane matter.", "For a slowly time varying radion field, we determine the exact solution of Hubble's expansion of the Universe.", "Such a scenario indicates that a slowly time evolving inter-brane separation (modulus) may result into an effective accelerating scale factor of our 3-brane Universe." ], [ "Low energy effective Einstein's equation in the presence of two branes", "In this section we review the analysis done in [7].", "Let us consider a system of two 3-branes placed at the orbifold fixed points and embedded in the bulk which is a five dimensional AdS spacetime containing the bulk cosmological constant $\\Lambda _5$ only [7].", "Each of the 3-branes is $Z_2$ symmetric.", "The 3-brane located at $y = 0 $ hypersurface has positive tension ${\\cal V}_{pl} $ while the other brane situated at $y = r \\pi $ (where $r$ is the stable value of the modulus) is characterised by a negative brane tension $ {\\cal V}_{vis}$ commonly called the visible brane where our Universe is located.", "The most general metric is taken by incorporating a radion field which is a function of both spacetime co-ordinates $x^{\\mu }$ and extra dimensional co-ordinate $y$ [7].", "Metric Ansatz : $ds^2\\,=\\,e^{2 \\phi (y,x)}dy^2\\,+\\,q_{\\mu \\nu }(y,x)dx^{\\mu }dx^{\\nu } $ The proper distance between the two branes within the fixed interval $y\\,=\\,0 $ to $y\\,=\\,r\\pi $ is given by: $d_0(x)\\,=\\,\\displaystyle \\int _{0}^{r\\pi }dy \\,e^{\\phi {(y,x)}} $ The effective Einstein's equations on a 3-brane is given by Gauss-Codacci equations as follows : $^{(4)}G^{\\mu }_{\\nu }\\,=\\,\\displaystyle \\frac{3}{l^2}\\,\\delta ^{\\mu }_{\\nu }\\,+\\,K K^{\\mu }_{\\nu }\\,-\\,K^{\\mu }_{\\alpha }K^{\\alpha }_{\\nu }\\,+\\,\\displaystyle \\frac{1}{2}\\,\\delta ^{\\mu }_{\\nu }\\left(K^2\\,-\\,K^{\\alpha }_{\\beta }K^{\\beta }_{\\alpha }\\right)\\,-\\,E^{\\mu }_{\\nu }$ and $D_{\\nu }K^{\\nu }_{\\mu }\\,-\\,D_{\\mu }K\\,=\\,0$ where $D_{\\mu }$ is the covariant derivative with respect to the induced metric $q_{\\mu \\nu }$ on a brane and $l$ is the bulk curvature radius which is related to five dimensional bulk cosmological constant $\\Lambda _5$ as $l = \\sqrt{\\displaystyle \\frac{-3}{\\kappa ^2 \\Lambda _5}}$ .", "$K_{\\mu \\nu }$ is the extrinsic curvature on $y=$ constant hypersurface and is given by, $K_{\\mu \\nu }\\,=\\,\\nabla _{\\mu }\\,n_{\\nu }+\\,n_{\\mu }D_{\\nu }\\phi $ where $n = e^{-\\phi }\\partial _{y}$ and $E^{\\mu }_{\\nu }$ is the projected part of the five dimensional Weyl tensor.", "In a single brane system, $ E^{\\mu }_{\\nu } $ gets contribution from Kaluza-Klein modes whose contribution in the low energy effective theory can be neglected [8].", "However, in the presence of two branes $ E^{\\mu }_{\\nu } $ does not vanish in the low energy limit due to the existence of radion fields .", "Therefore the evolution equations of five dimensional projected Weyl tensor and extrinsic curvature have to be solved in the bulk for determining the effective Einstein's equations.", "Now the junction conditions on the 3-branes are as follows : $\\left[K^{\\mu }_{\\nu }\\,-\\,\\delta ^{\\mu }_{\\nu } K\\right]_{y=0}\\,=\\,-\\displaystyle \\frac{\\kappa ^2}{2}\\left(-{\\cal V}_{pl}\\, \\delta ^{\\mu }_{\\nu }\\,+\\,T_1^{\\mu }\\,_{\\nu }\\right)$ $\\left[K^{\\mu }_{\\nu }\\,-\\,\\delta ^{\\mu }_{\\nu } K\\right]_{y=r \\pi }\\,=\\,\\displaystyle \\frac{\\kappa ^2}{2}\\left(-{\\cal V}_{vis}\\, \\delta ^{\\mu }_{\\nu }\\,+\\,T_2^{\\mu }\\,_{\\nu }\\right)$ where $T_1^{\\mu }\\,_{\\nu }$ and $T_2^{\\mu }\\,_{\\nu }$ are respective energy momentum tensors on positive and negative tension branes.", "In order to derive the low energy effective theory, a perturbative scheme is adopted in which the dimensionless perturbation parameter is $\\epsilon = (\\frac{l}{L})^2$ such that $L>>l$ , where $L$ is the brane curvature scale.", "In order to determine $K^{\\mu }_{\\nu } $ and $ E^{\\mu }_{\\nu } $ in the bulk, these are expanded as, $K^{\\mu }_{\\nu }\\,=\\,^{(\\,0\\,)}K^{\\mu }_{\\nu }\\,+\\,^{(\\,1\\,)}K^{\\mu }_{\\nu }\\,+....$ and $E^{\\mu }_{\\nu }\\,=\\,^{(1)}E^{\\mu }_{\\nu }\\,+....$ At the zero-th order we have $ ^{(0)}E^{\\mu }_{\\nu } = 0$ [7], [8], [9].", "There is only one evolution equation corresponding to $^{(0)}K^{\\mu }_{\\nu } $ whose solution, satisfying the Coddacci equation, is given by, $^{(0)}K^{\\mu }_{\\nu }\\,=\\,-\\displaystyle \\frac{1}{l}\\delta ^{\\mu }_{\\nu }$ The junction conditions at the zero-th order are given by : $\\left[^{(0)} K^{\\mu }_{\\nu }\\,-\\,\\delta ^{\\mu }_{\\nu }\\, ^{(0)} K\\right]_{y=0}\\,=\\,\\displaystyle \\frac{\\kappa ^2}{2}\\,{\\cal V}_{pl}\\,\\delta ^{\\mu }_{\\nu }$ and $\\left[^{(0)} K^{\\mu }_{\\nu }\\,-\\,\\delta ^{\\mu }_{\\nu }\\, ^{(0)} K\\right]_{y=r \\pi }\\,=\\,-\\displaystyle \\frac{\\kappa ^2}{2}\\,{\\cal V}_{vis}\\,\\delta ^{\\mu }_{\\nu }$ implying that the relation between bulk curvature radius $l$ and brane tensions is similar to the fine-tuning condition of the RS model [3] and is given by : $\\displaystyle \\frac{1}{l}\\,=\\,\\displaystyle \\frac{1}{6}\\kappa ^2{\\cal V}_{pl}\\,=\\,-\\displaystyle \\frac{1}{6}\\kappa ^2{\\cal V}_{vis} $ In this perturbative approach, at the zero-th order, the effect of bulk curvature and the brane tensions on the 3-branes exactly counterbalance one another and as a result we have : $^{(4)}G^{\\mu }_{\\nu }\\,=\\,0$ .", "This is the static RS model [3] which considers both static and flat 3-branes.", "So the curvature of four dimensional spacetime can emerge only from higher order corrections to $K^{\\mu }_{\\nu }$ and $E^{\\mu }_{\\nu }$ .", "Now at the first order we have two evolution equations, one for $^{(1)}E^{\\mu }_{\\nu }$ and the other for $^{(1)} K^{\\mu }_{\\nu }$ which is solved using the solution of evolution equation of $^{(1)}E^{\\mu }_{\\nu }$ .", "The corresponding junction conditions at the first order now become : $\\left[^{(1)} K^{\\mu }_{\\nu }\\,-\\,\\delta ^{\\mu }_{\\nu }\\, ^{(1)} K\\right]_{y=0}\\,=\\,-\\displaystyle \\frac{\\kappa ^2}{2}\\,T_1^{\\mu }\\,_{\\nu } $ and $\\left[^{(1)} K^{\\mu }_{\\nu }\\,-\\,\\delta ^{\\mu }_{\\nu }\\, ^{(1)} K\\right]_{y=r \\pi }\\,=\\,\\displaystyle \\frac{\\kappa ^2}{2}\\,T_2^{\\mu }\\,_{\\nu } $ while the Gauss equation at the first order on the visible brane is given by : $^{(4)}G^{\\mu }_{\\nu }\\,=\\,-\\displaystyle \\frac{2}{l} \\left(^{(1)} K^{\\mu }_{\\nu }(y_0,x)\\,-\\,\\delta ^{\\mu }_{\\nu }\\, ^{(1)} K(y_0,x)\\right)\\,-\\,^{(1)}E^{\\mu }_{\\nu }(y_0,x) \\,=\\,-\\displaystyle \\frac{\\kappa ^2}{l}\\,T_2^{\\mu }\\,_{\\nu }\\,-\\,^{(1)}E^{\\mu }_{\\nu }(y_0,x) $ where $y_0= r\\pi $ .", "Now using the bulk solution of $^{(1)}K^{\\mu }_{\\nu }$ which contains $^{(1)}E^{\\mu }_{\\nu }$ and eqn.", "(REF ), we can rewrite eqn.", "(REF ) from which $^{(1)}E^{\\mu }_{\\nu }$ can be explicitly determined on the visible brane.", "Substituting $^{(1)}E^{\\mu }_{\\nu }$ in eqn.", "(REF ), the Einstein's equation on visible brane is found to be : $\\begin{array}{rcl}^{(4)}G^{\\mu }_{\\nu } & = & \\displaystyle \\frac{\\kappa ^2}{l} \\displaystyle \\frac{1}{\\Phi }\\,T_2^{\\mu }\\,_{\\nu }\\,+\\,\\displaystyle \\frac{\\kappa ^2}{l}\\displaystyle \\frac{(1\\,+\\,\\Phi )^2}{\\Phi }\\,T_1^{\\mu }\\,_{\\nu }\\\\ [4mm]& + & \\displaystyle \\frac{1}{\\Phi }\\,(D^{\\mu }D_{\\nu }\\Phi \\,-\\delta ^{\\mu }_{\\nu } D^2 \\Phi \\,)\\\\ [4mm]& + & \\displaystyle \\frac{\\omega (\\Phi )}{\\Phi ^2}\\left(D^{\\mu } \\Phi D_{\\nu }\\Phi \\,-\\,\\frac{1}{2}\\,\\delta ^{\\mu }_{\\nu }(D \\Phi )^2\\right) \\end{array}$ where, $\\Phi \\,=\\,e^{2d_0(x)/l}\\,-\\,1, \\qquad \\omega (\\Phi )\\,=\\,-\\displaystyle \\frac{3}{2} \\displaystyle \\frac{\\Phi }{1 \\,+\\,\\Phi }$ So $\\Phi $ is a function of the brane co-ordinates $x$ .", "From RHS of eqn.", "(REF ) we note that an effective brane matter on a 3-brane can originate from : Explicit matter distribution on the brane through $T_i^{\\mu }\\,_{\\nu }$ , where $i=1,2$ .", "Even a matter distribution on the hidden brane i.e.", "$T_1^{\\mu }\\,_{\\nu }$ can induce a non-zero matter distribution on the visible brane via the bulk curvature.", "and/or A time dependent modulus field $e^{2 \\phi (t)}$ can serve as an induced non-vanishing energy momentum tensor on the visible brane even in the absence of any explicit brane matter on either of the 3-branes.", "Whether the above two scenarios may result in generating an effective cosmological constant on the visible 3-brane will be explored in the following sections." ], [ "The generalized Randall Sundrum braneworld scenario", "We now briefly review a previous work by one of the authors which deals with non-flat brane scenario with constant radion field [4].", "We shall subsequently follow this analysis as well as the analysis given in [7] to explore the possible origin of brane cosmological constant in an expanding Universe.", "We now consider a situation in which the inter-brane distance is constant.", "We know that in the Randall Sundrum model (RS) [3] the induced cosmological constant on both the 3-branes identically vanishes to zero and therefore to any four dimensional observer these branes appear flat as well as static.", "We can however construct a more generalized scenario [4] with a general warp factor that brings in curvature on both the branes and at the same time resolves the gauge hierarchy problem without introducing any unnatural fine tuning.", "Since the radion field is constant, therefore metric in eqn.", "(REF ) suggests $\\phi (y,x) = 0$ and in this case the metric is taken to be, $ds^2\\,=\\,e^{-2A(y)}g_{\\mu \\nu } dx^{\\mu }dx^{\\nu }\\,+\\,dy^2 $ The induced metric $q_{\\mu \\nu }(x,y)$ in the previous section is now taken as : $e^{-2A(y)}g_{\\mu \\nu }(x)$ .", "The action is : $S\\,=\\,\\displaystyle \\int d^5x \\sqrt{-G}(M^3\\,{\\cal R}\\,-\\,\\Lambda _5)\\,+\\,\\int d^4x \\sqrt{-g_i}\\, {\\cal V}_{i}$ where $\\Lambda _5$ is AdS five dimensional bulk cosmological constant, ${\\cal R}$ is the five dimensional Ricci scalar and $g_{\\mu \\nu }$ is the effective four dimensional metric.", "The two 3-branes only contain the brane tensions ${\\cal V}_{i}$ (where $i = vis, pl$ ) and therefore the energy-momentum tensors : $T_1^{\\mu }\\,_{\\nu } = T_2^{\\mu }\\,_{\\nu } = 0$ .", "Now by varying the action the corresponding bulk Einstein's equations are as follows : $^{(4)}\\,G_{\\mu \\nu }\\,-\\,g_{\\mu \\nu }e^{-2A}\\,\\left(-6A^{\\prime 2}\\,+\\,3A^{\\prime \\prime }\\right)\\,=\\,-\\displaystyle \\frac{\\Lambda _5}{2 M^3}g_{\\mu \\nu }e^{-2A} $ and $-\\displaystyle \\frac{1}{2}e^{2A} \\,^{(4)}\\, R\\,+\\,6A^{\\prime 2}\\,=\\,-\\displaystyle \\frac{\\Lambda _5}{2 M^3} $ with the boundary conditions : $A^{\\prime }(y)\\,=\\,\\displaystyle \\frac{\\epsilon _i}{12 M^3} {\\cal V}_i \\qquad \\epsilon _{pl}\\,=\\,-\\epsilon _{vis}\\,=\\,1 $ where $^{\\prime }$ represents derivative with respect to $y$ .", "Dividing both sides of eqn.", "(REF ) by $g_{\\mu \\nu }$ and after rearranging terms we get, $^{(4)}G_{\\mu \\nu }\\,=\\,-\\Omega g_{\\mu \\nu } $ So eqn.", "(REF ) gives the effective four dimensional Einstein's equation on the Planck brane which is a $y=0$ hypersurface embedded in five dimensional AdS bulk.", "This equation being a function of $x^{\\mu }$ alone, is defined on the Planck brane with metric $g_{\\mu \\nu }(x)$ .", "The arbitrary proportionality constant $\\Omega $ therefore behaves as a induced cosmological constant on the Planck brane.", "Symbolically, we can write, $\\frac{^{(4)}G^{(pl)}_{\\mu \\nu }}{g^{(pl)}_{\\mu \\nu }}\\,=\\,-\\Omega _{pl} $ where $-\\Omega _{pl}\\,=\\,e^{-2A}\\left[-6A^{\\prime 2}\\,+\\,3A^{\\prime \\prime }\\,-\\,\\displaystyle \\frac{\\Lambda _5}{2M^3}\\right]$ calculated at $y = 0$ .", "The induced metric on the visible brane is : $g^{(vis)}_{\\mu \\nu } = e^{-2A(\\pi )}\\,g^{(pl)}_{\\mu \\nu }$ .", "However the Einstein's tensor is invariant under the multiplicative factor $e^{-2 A(\\pi )}$ therefore, $^{(4)}G^{(pl)}_{\\mu \\nu }\\,=\\, ^{(4)}G^{(vis)}_{\\mu \\nu } $ Now using eqn.", "(REF ) and eqn.", "(REF ), the Einstein's equation $^{(4)}G^{(vis)}_{\\mu \\nu }$ of visible brane located at $y=r\\pi $ orbifold fixed point is related to visible brane cosmological constant as : $\\frac{^{(4)}G^{(vis)}_{\\mu \\nu }}{e^{-2 A(\\pi )}g^{(pl)}_{\\mu \\nu }}\\,=\\,-\\Omega _{pl}\\,e^{2 A(\\pi )}\\,=\\,-\\Omega _{vis}$ which in turn implies that the induced cosmological constants of our Universe and that of the Planck brane are interrelated in the following way : $\\Omega _{vis}\\,=\\,e^{2 A(\\pi )}\\,\\Omega _{pl} $ Till this part of our analysis, the origin of induced cosmological constant on both the 3-branes is unknown.", "We shall however explore the possible origin of induced brane cosmological constant in the upcoming sections.", "Now using Einstein's equations one can write : $6A^{\\prime 2}\\,=\\,-\\displaystyle \\frac{\\Lambda _5}{2 M^3}\\,+\\,2\\Omega _{pl} e^{2A} $ $3A^{\\prime \\prime }\\,=\\,\\Omega _{pl} e^{2A}$ Solution of Eqn.", "(REF ) yields a warp factor which is different from that of the original RS model.", "The sign of cosmological constant decides whether the visible brane will be de-Sitter or Anti de-Sitter.", "Case-1 : $\\omega ^2$ is negative We define a dimensionless quantity on the Planck brane $ \\omega _{pl}^2 = -\\Omega _{pl}/3\\tilde{k}^2 \\ge 0 $ in the background of AdS bulk ($\\Lambda _5 < 0$ ).", "On solving eqn.", "(REF ) the warp factor becomes, $e^{-A}\\,=\\,\\omega _{pl} \\cosh \\left(\\ln \\displaystyle \\frac{\\omega _{pl}}{c_1}\\,+\\,\\tilde{k}|y|\\right) $ where $\\tilde{k} = \\displaystyle \\sqrt{\\frac{-\\Lambda _5}{12 M^3}}$ .", "The constant of integration $c_1\\,=\\,1\\,+\\,\\sqrt{1\\,-\\,\\omega _{pl}^2}$ is fixed by normalising the warp factor to unity at the orbifold point $ y = 0 $ .", "The RS result of $A = \\tilde{k}|y|$ is obtained in the limit $\\omega _{pl} \\longrightarrow 0$ .", "It is to be noted that $\\Omega _{pl} \\longrightarrow 0$ implies $\\Omega _{vis} \\longrightarrow 0$ from eqn.", "(REF ).", "Using the boundary conditions the brane tensions on both the branes can be determined [4].", "Now all mass scales on the visible brane get exponentially warped.", "If we define $ e^{-A(\\tilde{k} r \\pi )} = m/m_0 = 10^{-16} $ and $ \\omega _{vis}^2 = 10^{-124} $ (say), in anti de-Sitter spacetime, the gauge hierarchy problem is resolved for two different values of modulus namely $\\tilde{k} r_1\\,\\simeq \\,36.84\\,+\\,10^{-93}$ and $\\tilde{k} r_2\\,= 79.60$ instead of one as in the RS case.", "Case-2 : $\\omega ^2$ is positive Present cosmological observations suggest that our Universe is endowed with a positive cosmological constant i.e.", "has a de-Sitter character.", "We therefore focus into Case-2 more critically.", "By solving eqn.", "(REF ), the warp factor is, $e^{-A}\\,=\\,\\omega _{pl} \\sinh \\left(\\ln \\displaystyle \\frac{c_2}{\\omega _{pl}}\\,-\\,\\tilde{k}|y|\\right) $ where $ \\omega _{pl}^2 = \\Omega _{pl}/3\\tilde{k}^2 $ with $ c_2 = 1\\,+\\,\\sqrt{1 + \\omega _{pl}^2} $ .", "However the induced cosmological constant on the Planck brane ($\\omega _{pl}$ ) can be taken to be much smaller compared to 1, so that $c_2 \\approx 2$ .", "For $ m/m_0 = 10^{-n}$ , eqn.", "(REF ) now becomes, $e^{-\\tilde{k} r \\pi }\\,=\\,\\displaystyle \\frac{10^{-n}}{2}\\left[1\\,+\\,\\sqrt{1\\,+\\,\\omega _{pl}^2 10^{2n}}\\right] $ A positive induced cosmological constant on the visible brane ($\\omega _{vis}^2 = \\omega _{pl}^2 10^{2n}$ ) does not have any upper bound and therefore can be of arbitrary magnitude.", "However, using the present estimated value of $\\omega _{vis}^2 \\sim 10^{-124}$ and $n=16$ we find $\\tilde{k}r\\pi = 36.84$ which is quite close to RS value.", "Therefore the hierarchy problem is also solved for small positive brane cosmological constant.", "Using eqn.", "(REF ) the brane tensions on both the branes are given by : ${\\cal V}_{vis}\\,=\\,-12 M^3 \\tilde{k} \\displaystyle \\left[\\frac{c_2^2+\\omega _{vis}^2}{c_2^2-\\omega _{vis}^2}\\right],\\qquad {\\cal V}_{pl}\\,=\\,12 M^3 \\tilde{k} \\displaystyle \\left[\\frac{c_2^2\\,+\\,{\\omega _{pl}^2}}{c_2^2\\,-\\,{\\omega _{pl}^2}}\\right] $ where on the visible brane : $\\omega ^2_{vis} = \\displaystyle \\frac{\\Omega _{vis}}{3\\tilde{k}^2} = e^{2\\tilde{k}r\\pi } \\omega ^2_{pl}$ .", "But we have seen from the previous section that the contribution to non-zero value of induced cosmological constant $\\omega _{vis}^2$ on the visible brane comes from the first order correction to the extrinsic curvature and projected Weyl tensor.", "Therefore in the zero-th order i,e.", "in the absence of any matter on the visible brane the RS fine-tuning condition is exactly valid.", "As a result in this order $\\omega _{vis}^2 = 0$ and from eqn.", "(REF ) one finds that ${\\cal V}_{vis}\\,=\\,-12 M^3 \\tilde{k}$ and ${\\cal V}_{pl}\\,=\\,12 M^3 \\tilde{k}$ .", "These lead to the original flat brane RS model.", "However in order to explore the origin of cosmological constant on our Universe (i,e.", "$\\omega ^2_{vis} \\ne 0$ ) which is located at the $y= r \\pi $ 3-brane, we assume that the Universe is described by FRW metric embedded in five dimensional AdS bulk spacetime containing the bulk cosmological constant $\\Lambda _5$ .", "The inter-brane separation, known as the radion field may be constant, time dependent or $y$ -dependent.", "We examine each case separately to explore the possibility of obtaining a non-vanishing $\\omega ^2$ on the visible 3-brane." ], [ "Warped cosmological metric in constant radion field ", "We first consider a constant modulus scenario where the 3-branes are endowed with matter densities $\\rho _{vis}$ , $\\rho _{pl}$ and brane pressures $p_{vis}, p_{pl}$ .", "We look for a FRW solution on a visible brane with appropriate warp factor.", "Metric ansatz : $ds^2\\,=\\,e^{-2A(y)}\\left[-dt^2\\,+\\,v^2(t)\\delta _{ij}\\,dx^{i}dx^{j}\\right]\\,+\\,dy^2 $ It is to be noted that for $\\phi =0$ (i.e.", "for a constant radion field) and $q_{\\mu \\nu }(x,y) = e^{-2A(y)}g_{\\mu \\nu }(x)$ where $g_{\\mu \\nu }(x)$ is a FRW metric with a flat spatial curvature.", "The metric given in eqn.", "(REF ) reduces to that in eqn.", "(REF ).", "The location of the two 3-branes are $y = 0, r\\pi $ .", "Since the radion field is constant therefore the proper distance along the $y$ direction between the interval $y = 0$ to $y = r \\pi $ is given by: $d_0\\,=\\,\\int _{0}^{r \\pi }dy\\,=\\,r\\pi $ The five dimensional bulk-brane action is : $S\\,=\\,\\displaystyle \\int d^5x \\sqrt{-G}(M^3\\,{\\cal R}\\,-\\,\\Lambda _5)\\,+\\,\\displaystyle \\int d^5x \\left[\\sqrt{-g_{pl}}\\left({\\cal L}_1\\,-\\,{\\cal V}_{pl}\\right)\\delta (y)\\,+\\, \\sqrt{-g_{vis}}\\left({\\cal L}_2\\,-\\,{\\cal V}_{vis}\\right)\\delta (y-\\pi )\\right] $ where ${\\cal L}_1$ , ${\\cal L}_2$ and ${\\cal V}_{pl}$ , ${\\cal V}_{vis}$ are Lagrangian densities and the constant brane tensions on the Planck and the visible brane respectively.", "Notations: Non-compact brane co-ordinates : $x^{\\mu }$ where $\\mu \\,=\\,0,1,2,3$ .", "Bulk co-ordinates: $M,N$ run over $M,N\\,=\\,0,1,2,3,y$ .", "By varying the action, the five dimensional Einstein's equations can be written as : $\\sqrt{-G}\\,G_{M N}\\,=\\,-\\displaystyle \\frac{\\Lambda _5}{2 M^3}\\,\\sqrt{-G}\\,g_{M N}\\,+\\,\\frac{1}{2 M^3}\\left[\\tilde{T_1}^{\\gamma }\\,_{\\mu }\\,\\delta (y)\\sqrt{-g_{pl}}\\,+\\,\\tilde{T_2}^{\\gamma }\\,_{\\mu }\\,\\delta (y-\\pi )\\sqrt{-g_{vis}}\\right] \\, g_{\\gamma \\nu }\\,\\delta ^{\\mu }_M\\,\\delta ^{\\nu }_{N} $ where $ g_{\\gamma \\nu }$ is the four-dimensional effective metric on the Planck brane.", "The energy momentum tensors derived from the Lagrangian densities and brane tensions on the two 3-branes placed at the edges of the bulk are given by : On the Planck brane ($y=0$ ) : $\\tilde{T_1}^{\\gamma }\\,_{\\mu }\\,=\\,diag(-\\rho _{pl} - {\\cal V}_{pl}\\,,p_{pl} - {\\cal V}_{pl}\\,,p_{pl} - {\\cal V}_{pl}\\,,p_{pl} - {\\cal V}_{pl},0)$ On the visible brane ($y=\\pi $ ) : $\\tilde{T_2}^{\\gamma }\\,_{\\mu }\\,=\\,diag(-\\rho _{vis}- {\\cal V}_{vis}\\,,p_{vis}- {\\cal V}_{vis}\\,,p_{vis}- {\\cal V}_{vis}\\,,p_{vis}- {\\cal V}_{vis},0)$ On substituting eqn.", "(REF ) in eqn.", "(REF ) we obtain the Einstein's equations for the above metric.", "Now considering the bulk part only, we have : tt component : $\\begin{array}{rcl}3\\displaystyle \\frac{\\dot{v}^2}{v^2}\\,+\\, e^{-2 A(y)}\\,\\left(\\,3A^{\\prime \\prime }\\,-\\,6 A^{\\prime 2}\\right) \\,=\\,-\\displaystyle \\frac{\\Lambda _5}{2 M^3}\\,(-1)\\,e^{-2 A(y)} \\end{array}$ ii component : $\\begin{array}{rcl}\\displaystyle \\left(-2\\frac{\\ddot{v}}{v}\\,-\\,\\frac{\\dot{v}^2}{v^2}\\right)\\,+\\, e^{-2 A(y)}\\left[-3A^{\\prime \\prime }\\,+\\,6 A^{\\prime 2}\\right] \\,= \\,-\\displaystyle \\frac{\\Lambda _5}{2 M^3}\\, e^{-2 A(y)} \\end{array}$ yy component : $6\\,A^{\\prime 2}\\,-\\,3 \\,e^{2 A} \\,\\frac{\\dot{v}^2\\,+\\,\\ddot{v}}{v^2} \\,=\\,-\\displaystyle \\frac{\\Lambda _5}{2 M^3} $ where dot represents derivative with respect to time.", "With a little rearrangement of terms in $tt$ and $ii$ components we can write, $\\displaystyle \\frac{^{(4)}G_{\\mu \\nu }}{g_{\\mu \\nu }}\\,=\\, e^{-2 A(y)}\\,\\left[-\\displaystyle \\frac{\\Lambda _5}{2 M^3}\\,+\\,3A^{\\prime \\prime }\\,-\\,6 A^{\\prime 2}\\right]\\,=\\,-\\Omega $ The left hand side is the function of time while right hand side is function of $y$ alone for every component of $g_{\\mu \\nu }$ .", "This separation of variables enables us to write the effective Einstein equation on a 3-brane as : $\\displaystyle \\frac{^{(4)}G_{\\mu \\nu }}{g_{\\mu \\nu }}\\,=\\,-\\Omega \\,$ Once again, this is the effective Einstein's equation on the Planck brane whose induced metric is $g^{(pl)}_{\\mu \\nu }$ .", "Therefore the above equation may be written as, $\\frac{^{(4)}G^{(pl)}_{\\mu \\nu }}{g^{(pl)}_{\\mu \\nu }}\\,=\\,-\\Omega _{pl} $ such that $-\\Omega _{pl}\\,=\\,e^{-2 A(y)}\\,\\left[-\\displaystyle \\frac{\\Lambda _5}{2 M^3}\\,+\\,3A^{\\prime \\prime }\\,-\\,6 A^{\\prime 2}\\right]$ calculated at $y=0$ .", "Here $\\Omega _{pl}$ is a separation constant and is interpreted as induced cosmological constant on the Planck brane on which the warp factor is equal to unity.", "Since all length scales are exponentially warped on the visible brane, therefore the effective metric on the visible brane located at $y=r\\pi $ hypersurface is : $g^{(vis)}_{\\mu \\nu } = e^{-2A(\\pi )}\\,g^{(pl)}_{\\mu \\nu }$ .", "From the fact that Einstein's tensor remains invariant under a constant multiplicative factor, on our Universe, using Eqn.", "(REF ), we have, $\\frac{^{(4)}G^{(vis)}_{\\mu \\nu }}{e^{-2 A(\\pi )}g^{(pl)}_{\\mu \\nu }}\\,=\\,-\\Omega _{pl}\\,e^{2 A(\\pi )}\\,=\\,-\\Omega _{vis} $ which implies $\\Omega _{vis}\\,=\\,e^{2 A(\\pi )}\\,\\Omega _{pl} $ Now in order to determine $\\Omega _{vis}$ we must find the warp factor on the visible brane which will be determined very shortly.", "But before that let us first find the solution of the scale factor.", "From $tt$ and $ii$ components, we have, $-\\displaystyle \\frac{\\dot{v}^2}{v^2}\\,+\\,\\displaystyle \\frac{\\ddot{v}}{v}\\,=\\,0$ On solving we get, $v(t)\\,=\\,e^{H_0 t} $ where $H_0$ is an integration constant.", "It is to be noted that $H_0$ is not the Hubble parameter of the effective theory which will be determined later.", "Now using eqn.", "(REF ), eqn.", "(REF ), eqn.", "(REF ) and the scale factor solution given by eqn.", "(REF ) we get, $\\Omega _{pl}\\,=\\,3 H_0^2 $ which indicates a de-Sitter spacetime.", "When $H_0 \\longrightarrow 0$ , eqn.", "(REF ) shows that the induced cosmological constants on both the branes vanish leading to a static and flat Universe.", "Now substituting eqn.", "(REF ) in eqn.", "(REF ), the $yy$ component becomes, $A^{\\prime 2}\\,=\\,\\tilde{k}^2\\,+\\,\\omega _{pl}^2\\,\\tilde{k}^2\\,e^{2 A} $ where $\\tilde{k}\\,=\\,\\sqrt{\\displaystyle \\frac{-\\Lambda _5}{12 M^3}}$ and $\\omega _{pl}^2\\,=\\,\\displaystyle \\frac{\\Omega _{pl}}{3\\tilde{k}^2}$ is a dimensionless quantity.", "Similarly one can also define $\\omega _{vis}$ on the visible brane such that : $\\omega _{vis}^2\\,=\\,\\displaystyle \\frac{\\Omega _{vis}}{3\\tilde{k}^2}= \\displaystyle \\frac{e^{2 A(\\pi )}\\Omega _{pl}}{3\\tilde{k}^2}$ .", "Now on solving eqn.", "(REF ), the solution of the warp factor consistent with $Z_2$ symmetry is given by : $e^{-A(y)}\\,=\\,\\omega _{pl}\\,\\sinh \\left[-\\tilde{k}\\,|y|\\,+\\,\\ln \\displaystyle \\frac{c_2}{\\omega _{pl}}\\right] $ where $c_2$ is the integration constant.", "We normalize the warp factor on the Planck brane, such that $e^{-A(y = 0)}\\,=\\, 1$ so that eqn.", "(REF ) can be written as, $\\omega _{pl}\\,\\sinh \\left[\\ln \\displaystyle \\frac{c_2}{\\omega _{pl}}\\right]\\,=\\,1$ After a little rearrangement of terms we finally get : $c_2\\,=\\,1\\,+\\,\\sqrt{1\\,+\\,\\omega ^2_{pl}} $ Taking the value of $\\omega ^2_{pl}$ much less compared to 1, $c_2 \\approx 2$ .", "Now substituting eqn.", "(REF ) and eqn.", "(REF ) with $c_2\\,=\\,2$ in eqn.", "(REF ), the resulting 5D metric becomes : $ds^2\\,=\\,\\omega _{pl}^2\\,\\sinh ^2 \\left[-\\tilde{k} \\,|y|\\,+\\,\\ln \\displaystyle \\frac{2}{\\omega _{pl}}\\right]\\left(-dt^2\\,+\\,e^{2 H_0 t}\\delta _{ij}dx^{i}\\,dx^{j}\\right)\\,+\\,dy^2 $ The above metric describes exponential expansion in the spatial three dimensions.", "In the static limit, $\\omega _{pl} \\longrightarrow 0$ and $\\omega _{vis} \\longrightarrow 0$ that imply $H_0$ also becomes zero.", "Eqn.", "(REF ) now corresponds to the static RS metric [3].", "Since the radion is independent of spacetime as well as extra-dimensional co-ordinates, all terms involving covariant derivatives in eqn.", "(REF ) vanish.", "Therefore the effective Einstein's equations on the visible brane are only related to energy momentum tensors of the two 3-branes and bulk curvature radius $l$ : $^{(4)}G^{\\mu }_{\\nu } \\,= \\, \\displaystyle \\frac{\\kappa ^2}{l} \\displaystyle \\frac{1}{\\Phi }\\,T_2^{\\mu }\\,_{\\nu }\\,+\\,\\displaystyle \\frac{\\kappa ^2}{l}\\displaystyle \\frac{(1\\,+\\,\\Phi )^2}{\\Phi }\\,T_1^{\\mu }\\,_{\\nu } $ where $\\Phi \\,=\\,(e^{2 r \\pi /l}\\,-\\,1) $ , $\\kappa ^2$ is related to the five dimensional gravitational constant.", "Now from eqn.", "(REF ) and eqn.", "(REF ) the visible brane cosmological constant is : $-\\Omega _{vis}\\,=\\,\\displaystyle \\frac{\\kappa ^2}{4 l (e^{2 r\\pi /l}-1)}\\,\\left[e^{4 r \\pi /l}\\,T_1^{\\mu }\\,_{\\nu }\\,\\delta ^{\\nu }_{\\mu }\\,+\\,T_2^{\\mu }\\,_{\\nu }\\,\\delta ^{\\nu }_{\\mu }\\right] $ So the existence of induced cosmological constant on the visible brane is directly related to the presence of matter density and pressure of both the 3-branes.", "The effects of extra dimension on the induced brane cosmological constant however shows up through the multiplicative factor $\\displaystyle \\frac{\\kappa ^2}{4 l (e^{2 r \\pi /l}-1)}$ .", "Furthermore it is interesting to find that even if the visible brane matter $T_2^{\\mu }\\,_{\\nu } = 0$ , there can be a net cosmological constant in the Universe solely due to the matter content of the hidden brane.", "We now determine the effective Hubble parameter of our Universe." ], [ "Effective Hubble parameter on the visible brane :", "Now, we determine the Hubble parameter on the 4D world following the method adopted in [10].", "The induced metric of the four dimensional spacetime is : $ds_{(4)}^2 = -d\\tilde{t}^2\\,+\\,e^{2 H(y)\\tilde{t}}\\delta _{ij}\\,d\\tilde{x}^i\\,d\\tilde{x}^j $ Here, the effective Hubble parameter is determined on orbifold fixed points i.e.", "at $y = 0, r\\pi $ .", "In order to get the induced metric from 5D metric, we define following co-ordinate transformations : $d\\tilde{t}\\,=\\,\\omega _{pl}\\sinh \\left[-\\tilde{k}\\,|y|\\,+\\,\\ln \\displaystyle \\frac{2}{\\omega _{pl}}\\right]\\,dt $ and $d\\tilde{x}^i\\,=\\,\\omega _{pl}\\sinh \\left[-\\tilde{k}\\,|y|\\,+\\,\\ln \\displaystyle \\frac{2}{\\omega _{pl}}\\right]\\,dx^i$ Now for a fixed $y$ (on a 3-brane) integrating eqn.", "(REF ) results into : $\\tilde{t}\\,=\\,\\omega _{pl}\\sinh \\left[-\\tilde{k}\\,|y|\\,+\\,\\ln \\displaystyle \\frac{2}{\\omega _{pl}}\\right]\\,t$ On comparing the 5D metric and 4D effective metric given by eqn.", "(REF ) and eqn.", "(REF ) we obtain, $\\omega _{pl}^2 \\sinh ^2\\left[-\\tilde{k}\\,|y|\\,+\\,\\ln \\displaystyle \\frac{2}{\\omega _{pl}}\\right]\\,e^{2H_0 t}\\,\\delta _{ij}\\,dx^i\\,dx^j\\,=\\,e^{2\\,H(y)\\tilde{t}}\\delta _{ij}\\,d\\tilde{x}^i\\,d\\tilde{x}^j$ Once again, if we determine the Hubble parameter on a 3-brane by comparing the terms in the exponential, $H$ should be a function of $y$ because $t$ depends on $\\tilde{t}$ through $y$ .", "Therefore the effective 4D Hubble parameter on a 3-brane for a given value of $y$ is given by : $H(y)\\,=\\,\\displaystyle \\frac{H_0}{\\omega _{pl}}\\,cosech\\left[-\\tilde{k}\\,|y|\\,+\\,\\ln \\displaystyle \\frac{2}{\\omega _{pl}}\\right]$ On the visible brane ($y\\,=\\,r \\pi $ ) : $H(\\pi )\\equiv H_{vis}\\,=\\,\\displaystyle \\frac{H_0}{\\omega _{pl}\\,\\sinh \\left[-\\tilde{k} r \\pi \\,+\\,\\ln \\displaystyle \\frac{2}{\\omega _{pl}}\\right]}$ Since $\\omega _{pl}=\\displaystyle \\frac{H_0}{\\tilde{k}}$ and $\\omega _{vis} = \\omega _{pl} \\,e^{A(\\pi )}$ , therefore the effective visible brane Hubble parameter becomes, $H_{vis}\\,=\\,\\tilde{k}\\,cosech\\left[-\\tilde{k} r \\pi \\,+\\,\\ln \\displaystyle \\frac{2}{\\omega _{pl}}\\right]\\,=\\,\\omega _{vis}\\tilde{k} $ The above equation suggests that $H_{vis} \\simeq \\omega _{vis}$ (in Planckian unit).", "This result is consistent with the present result relating the Hubble parameter and cosmological constant of the Universe.", "Thus when the brane separation is constant, present evolution of the Universe is indeed governed by constant vacuum energy densities on the 3-branes.", "As expected, a vanishing $H_{vis}$ implies a static as well as flat Universe, devoid of any matter.", "Such a Universe does not undergo exponential expansion but is described by static RS model where both the branes are flat possessing brane tensions only [3].", "Energy-momentum tensor on 3-branes : Visible brane: $T_2^{\\mu }\\,_{\\nu }\\,=\\,diag(-\\rho _{vis},p_{vis},p_{vis},p_{vis})$ Hidden brane: $T_1^{\\mu }\\,_{\\nu }\\,=\\,diag(-\\rho _{pl},p_{pl},p_{pl},p_{pl})$ Substituting the components of $T_i\\,^{\\mu }\\,_{\\nu }$ (where $i = 1, 2$ ) in eqn.", "(REF ), the visible brane induced cosmological constant is : $-\\Omega _{vis}\\,=\\,\\displaystyle \\frac{\\kappa ^2}{4\\,l (e^{2 r \\pi /l}-1)}\\,\\left[-\\left(e^{4 r \\pi /l}\\,\\rho _{pl}\\,+\\,\\rho _{vis}\\right)\\,+\\,3\\left(e^{4 r \\pi /l}\\,p_{pl}\\,+\\,p_{vis}\\right)\\right] $ In case of vacuum energy dominated Universe, the energy density and pressure are related as : $\\rho _{vis}\\,=\\,-p_{vis}$ Similarly, if the same equation of state is valid on Planck brane, then from eqn.", "(REF ) the induced cosmological constant on the visible brane becomes : $\\Omega _{vis}\\,=\\,\\displaystyle \\frac{\\kappa ^2}{l (e^{2 r\\pi /l}-1)}\\,\\left[e^{4 r \\pi /l}\\,\\rho _{pl}\\,+\\,\\rho _{vis}\\right] $ From the above equation we find that the induced $\\Omega _{vis}$ is constant only when the visible brane contains matter in the form of constant vacuum energy density.", "Most importantly the non-zero value of brane matter as well as the bulk cosmological constant ($1/l$ ) are essential to inject an effective cosmological constant on the visible brane.", "Thus in a braneworld consisting of two 3-branes one cannot generate an effective $\\Omega _{vis}$ in the 4D spacetime only out of the intrinsic brane tensions and bulk cosmological constant without any brane matter when modulus field is independent of spacetime co-ordinates.", "If we further focus on eqn.", "(REF ), we discover that an absence of matter in the visible brane i,e.", "$\\rho _{vis} = 0$ does not necessarily imply a vanishing 4D cosmological constant as long as $\\rho _{pl}\\ne 0$ .", "This can be seen from eqn.", "(REF ) by putting $\\rho _{vis} = 0$ , $\\Omega _{vis}\\,=\\,\\displaystyle \\frac{\\kappa ^2}{l (e^{2 r \\pi /l}-1)}\\,{e^{4 r \\pi /l}\\,\\rho _{pl}} $ So from eqn.", "(REF ), we can infer that the Planck brane matter $\\rho _{pl}$ mediates an effective cosmological constant on the visible brane due to the curvature in bulk spacetime and therefore the vacuum energy density of the Planck brane and five dimensional bulk cosmological constant $\\Lambda _5$ may be the possible origins of an effective cosmological constant on our Universe (i,e.", "the visible brane) which can result into an exponentially expanding Universe." ], [ "Extra dimension dependent radion field", "Let us now assume that the inter-brane distance depends only on the extra dimension $y$ i.e.", "$\\phi $ in eqn.", "(REF ) is a function of $y$ only.", "This modifies the proper distance between the two branes as defined in eqn.", "(REF ).", "we explore the possibility of having an effective cosmological constant on our Universe ( visible 3-brane ) placed at $y=r \\pi $ .", "In such a scenario, we consider the metric, Metric ansatz : $ds^2\\,=\\,e^{-2A(y)}\\left[-dt^2\\,+\\,v^2(t)\\delta _{ij}\\,dx^{i}dx^{j}\\right]\\,+\\,e^{2 \\phi (y)}dy^2 $ The proper distance along the $y$ co-ordinate between the orbifold fixed points $y = 0$ to $y = r \\pi $ is now given by: $d_0\\,=\\,\\int _{0}^{r \\pi }dy\\, e^{\\phi (y)} $ Substituting eqn.", "(REF ) in eqn.", "(REF ), Einstein's equations in the bulk are : tt component : $\\begin{array}{rcl}3\\displaystyle \\frac{\\dot{v}^2}{v^2} \\,+\\, e^{-2 A}\\,e^{-2 \\phi }\\left(3A^{\\prime \\prime }\\,-\\,6 A^{\\prime 2}\\,-\\,3 A^{\\prime }\\phi ^{\\prime }\\right)\\,=\\,-\\displaystyle \\frac{\\Lambda _5}{2 M^3}\\,(-1)\\,e^{-2 A} \\end{array}$ ii component : $\\begin{array}{rcl}-\\displaystyle \\left(2\\ddot{v}v\\,+\\,\\dot{v}^2\\right)\\,+ \\, e^{-2 A}\\,e^{-2 \\phi }v^2\\left(-3A^{\\prime \\prime }\\,+\\,6 A^{\\prime 2}\\,+\\,3\\,A^{\\prime } \\phi ^{^{\\prime }}\\right)\\,=\\, -\\displaystyle \\frac{\\Lambda _5}{2 M^3}\\,e^{-2 A}v^2\\end{array}$ yy component : $6\\,A^{\\prime 2}\\,-\\,3 \\,e^{2 (A\\,+\\,\\phi )} \\,\\left[\\displaystyle \\frac{\\dot{v}^2}{v^2}\\,+\\,\\displaystyle \\frac{\\ddot{v}}{v}\\right] \\,=\\,-\\displaystyle \\frac{\\Lambda _5}{2 M^3}\\,e^{2 \\phi } $ From $tt$ and $ii$ components, the solution of the scale factor is found to be : $v(t)\\,=\\,e^{H_0 t} $ where $H_0$ is a constant.", "Dividing eqn.", "(REF ) by $g_{tt}$ , eqn.", "(REF ) by $g_{ii}$ and using the solution of the scale factor, we obtain the effective Einstein's equation on the Planck brane as, $\\displaystyle \\frac{^{(4)}G_{\\mu \\nu }^{(pl)}}{g_{\\mu \\nu }^{(pl)}}\\,= \\,\\left[\\displaystyle \\frac{-\\Lambda _5}{2 M^3}\\,e^{-2 A}\\,+\\,\\left(-6 A^{\\prime 2}\\,-\\,3 A^{\\prime }\\phi ^{\\prime }\\,+\\,3 A^{\\prime \\prime }\\right)\\,e^{-2 (A + \\phi )}\\right]\\,=\\,-\\Omega _{pl}\\,=\\,-3 H_0^2 $ However, the induced cosmological constant on the visible brane is related to the Planck brane as $\\Omega _{vis} = \\Omega _{pl}\\,e^{2 A(\\pi )}$ .", "Since the covariant derivatives are evaluated on $y=r\\pi $ hypersurface therefore using eqn.", "(REF ) the effective cosmological constant on the visible brane is : $-\\Omega _{vis}\\,=\\,\\displaystyle \\frac{\\kappa ^2}{4 l (e^{2 d_0/l}-1)}\\,\\left[e^{4 d_0/l}\\,T_1^{\\mu }\\,_{\\nu }\\,\\delta ^{\\nu }_{\\mu }\\,+\\,T_2^{\\mu }\\,_{\\nu }\\,\\delta ^{\\nu }_{\\mu }\\right] $ where $d_0$ is constant.", "Therefore (REF ) and (REF ) once again suggest that even a y-dependent radion field cannot produce an exponential expansion along spatial three dimensions unless a constant energy density is given explicitly on 3-branes.", "The bulk effect however appears through the bulk curvature radius $l$ .", "It may be further noted that even when visible brane matter $T_2^{\\mu }\\,_{\\nu } = 0$ , the Planck brane matter and $\\Lambda _5$ together can induce visible brane cosmological constant $\\Omega _{vis}$ that triggers an exponentially expanding Universe.", "However $\\Omega _{vis}$ differs from (REF ) in terms of proper length as defined in eqn.", "(REF )." ], [ "Radion driven acceleration without brane matter - time dependent radion field", "So far we were studying an exponentially expanding model in the presence of brane (visible and/or hidden) matter leading to an effective $\\Omega _{vis}$ on the visible brane modulated by bulk effects.", "In this section we show that if the radion field is time dependent then there can be an effective acceleration of the Universe even in the absence of any explicit brane matter.", "In this case the dynamics of the radion field produces the effective acceleration of 3-brane scale factor.", "For this we recall the original RS model with a time dependent radion field and look for a possible cosmological solution in the effective 4D theory.", "Following [11] we assume that in presence of a time-varying field the metric has a form: $ds^2\\,=\\,e^{-2 k b(t)|y|}\\left[-dt^2\\,+\\,v^2(t)\\delta _{ij}\\,dx^{i}dx^{j}\\right]\\,+\\,b^2(t)\\,dy^2 $ where $b(t) = e^{2 \\phi (t)}$ .", "In this case the proper distance between the interval $y=0$ to $y=r \\pi $ is given by : $d_0(t)\\,=\\,\\int _{0}^{r\\pi }\\,e^{2 \\phi (t)}dy\\,=\\,r \\pi e^{2 \\phi (t)} $ Such a metric is considered in [11] where the warp factor is : $e^{-k b(t)|y|}$ , $b(t)$ is the radion field.In the static limit, when the inter-brane separation is a constant, this warp factor reduces to that of the warp factor of RS metric [3] where both the 3-branes are flat and static.", "However it is to be noted that in general this warp factor does not lead to Einstein's equations separable solely in terms of $t$ and $y$ variables.", "In order to study the time evolution of our Universe located on the visible brane, we therefore consider the effective 4D metric in the an appropriate choice of co-ordinates as : $ds^2_{(4)}\\,=\\, -dt^2\\,+\\,v^2(t)\\,\\delta _{ij}\\, dx^{i} dx^{j}$ , where $v(t) = v(t,\\pi )$ .", "With the above effective metric, the components of effective Einstein's equations on the visible brane together with eqn.", "(REF ) become : tt component : $^{(4)}G^{t}_{t}\\,=\\,-\\frac{3\\dot{v}^2}{v^2}\\,=\\,\\frac{\\kappa ^2}{l (e^{2 d_0/l}-1)}\\,\\left[T_2\\,^{t}\\,_{t}\\,+\\,e^{4 d_0/l}\\,T_1\\,^{t}\\,_{t}\\right]\\,+\\,\\frac{3\\, e^{2 d_0/l} }{l^2 (e^{2 d_0/l}-1)} \\,\\dot{d_0}^2 $ ii component : $^{(4)}G^{i}_{i}= -\\displaystyle \\left(2\\frac{\\ddot{v}}{v} + \\frac{\\dot{v}^2}{v^2}\\right)= \\frac{\\kappa ^2}{l (e^{2 d_0/l}-1)} \\left[T_2\\,^{i}\\,_{i} + e^{4 d_0/l}\\,T_1\\,^{i}\\,_{i}\\right]\\,+\\,\\frac{2\\, e^{2 d_0/l} }{l (e^{2 d_0/l}-1)} \\displaystyle \\left(\\frac{3\\,\\dot{v}}{v}\\dot{d_0} + \\ddot{d_0}\\right) +\\, \\frac{e^{2 d_0/l} }{l^2 (e^{2 d_0/l}-1)} \\,\\dot{d_0}^2 $ where $d_0(t)$ is given by eqn.", "(REF ) and the spatial component of the covariant derivative $D_{i}(e^{2 d_0(t)/l}-1) = 0$ on the visible brane.", "Now subtracting eqn.", "(REF ) from eqn.", "(REF ), we get, $\\begin{array}{rcl}\\displaystyle \\frac{\\ddot{v}}{v}\\, -\\, \\displaystyle \\frac{\\dot{v}^2}{v^2}\\,=\\,\\frac{d}{dt}H(t) & = &\\displaystyle \\frac{\\kappa ^2}{2 \\,l (e^{2 d_0/l}-1)}\\,\\left[(T_2\\,^{t}\\,_{t} - T_2\\,^{i}\\,_{i}) \\,+\\,e^{4 d_0/l}\\,(T_1\\,^{t}\\,_{t} - T_1\\,^{i}\\,_{i}) \\right] \\\\[4mm]& -& \\displaystyle \\frac{\\, e^{2 d_0/l} }{l (e^{2 d_0/l}-1)}\\,\\displaystyle \\left(3\\,\\frac{\\dot{v}}{v}\\,\\dot{d_0}\\,+\\,\\ddot{d_0}\\right)\\, + \\, \\displaystyle \\frac{e^{2 d_0/l} }{2\\,l^2\\, (e^{2 d_0/l}-1)} \\,\\dot{d_0}^2 \\end{array}$ where $H(t) = \\displaystyle \\frac{\\dot{v}(t)}{v(t)}$ is the Hubble parameter of the Universe located on the visible brane.", "Eqn.", "(REF ) reveals that a time dependent modulus field can itself produce dynamical evolution of the Universe even in the absence of matter on the 3-branes i,e.", "when $T_1\\,^{\\mu }\\,_{\\nu } = T_2\\,^{\\mu }\\,_{\\nu } = 0$ .", "The signature of bulk in the evolution is evident from the presence of bulk curvature radius $l$ .", "Now in case of a constant radion field when all terms involving $\\dot{d_0},\\ddot{d_0}$ drop out, from eqn.", "(REF ) we retrieve the exponential solution of the scale factor $v(t)$ on the visible brane as obtained earlier when both the 3-brane energy-momentum tensors are described by vacuum energy densities for which the equation of state is $\\rho _{i}= -p_{i}$ where $i = 1,2$ .", "Let us now assume a slowly time varying radion field.", "Then in absence of any brane matter i.e.", "$T_1\\,^{\\mu }\\,_{\\nu } = T_2\\,^{\\mu }\\,_{\\nu } = 0$ , eqn.", "(REF ) in the leading order of $\\dot{d_0}$ becomes : $\\displaystyle \\frac{\\ddot{v}}{v}\\, -\\, \\displaystyle \\frac{\\dot{v}^2}{v^2}\\,=\\,\\frac{d}{dt}H(t)\\,=\\,\\frac{3\\,e^{2 d_0/l}\\,\\dot{d_0}}{l (e^{2 d_0/l}-1)}\\,H(t)$ which implies $\\frac{d}{dt} \\ln H(t)\\,=\\,\\frac{d}{dt} \\left[ \\frac{3}{2}\\,\\ln (e^{2 d_0/l}-1)\\right]$ After integration we get, $H(t)\\,=\\,(e^{2 d_0(t)/l}-1)^{3/2} $ Since $\\dot{H(t)} > 0$ , therefore an accelerating phase of our Universe is obtained from a slowly time-varying modulus field.", "Thus a time-varying radion field indeed leads to an accelerating phase of visible 3-brane even when the energy momentum tensors vanish on both the branes.", "In the presence of appropriate modulus potential, such a scenario can be envisioned as the tunnelling of the modulus from a metastable vacuum state to a stable vacuum state leading to an accelerating phase of the Universe located on the 3-brane." ], [ "Conclusion", "In the background of the work by Shiromizu et.al [7] and generalised Randall Sundram braneworld scenario [4], our focus in the present work is to study the role of brane energy momentum tensors $T_i\\,^{\\mu }\\,_{\\nu }$ , the bulk curvature and the modulus field in inducing an effective four dimensional cosmological constant on the visible 3-brane embedded in a five dimensional AdS ($\\Lambda _5 < 0$ ) bulk.", "We have considered three different cases to study the dynamical evolution of our Universe.", "Our results can be summarised as follows : Constant modulus field In this case an effective cosmological constant $\\Omega $ appears on the visible 3-brane due to the combined effects of the bulk curvature and brane matters in either of the two 3-branes.", "On solving Einstein's equations, we obtain an exponential solution of the scale factor $v(t) = e^{H_0 t}$ and $\\Omega _{vis} = 3 H_0^2 \\,e^{2 A(\\pi )} $ which indicates four dimensional de-Sitter spacetime.", "In the static limit $H_0\\longrightarrow 0$ and therefore $\\Omega _{vis} \\longrightarrow 0$ .", "This leads to static RS model with flat and static visible 3-brane [3].", "The same argument applies for hidden brane also.", "The nature of the warp factor is determined.", "It is found that the sine hyperbolic solution of the warp factor depends on $\\Omega _{pl}$ , modulus length and bulk cosmological constant.", "The induced $\\Omega _{vis}$ on the visible brane is further related to energy momentum tensors (see eqn.", "(REF )) of both the 3-branes.", "Therefore, absence of matter on both the 3-branes corresponds to zero value of effective 4D cosmological constant and hence reproduces RS warp factor [3] which signifies a static and flat visible 3-brane.", "On the visible brane, from eqn.", "(REF ) it is seen that the effective brane cosmological constant $\\Omega _{vis}$ depends on energy momentum tensors of both the 3-branes through constant $\\rho _{vis}$ and $\\rho _{pl}$ .", "The effects of bulk curvature and the proper length given by $d_0 = r\\pi $ on $\\Omega _{vis}$ appear through the factor $\\displaystyle \\frac{\\kappa ^2}{4 l (e^{2r\\pi /l}-1)}$ .", "Therefore in order to generate a non-zero $\\Omega _{vis}$ , the contribution of non-zero $T_i\\,^{\\mu }\\,_{\\nu }$ (where $i = 1,2$ ) as well as bulk cosmological constant $\\Lambda _5$ are essential which can be seen from eqn.", "(REF ).", "Furthermore eqn.", "(REF ) suggests that one can have a scenario where $\\Omega _{vis} \\ne 0$ , even if our Universe is devoid of any matter i,e.", "when $\\rho _{vis} = 0$ but $\\rho _{pl} \\ne 0$ .", "Thus the exponential expansion of our Universe may be due to the presence of bulk cosmological constant and Planck brane energy density $\\rho _{pl}$ .", "We have determined the Hubble parameter of the Universe for a constant radion field.", "From eqn.", "(REF ), $H_{vis}$ is found to depend on bulk cosmological constant $\\Lambda _5$ , $\\Omega _{vis}$ and constant brane separation distance $r$ .", "This implies that even if there is no matter on the visible brane, one can still obtain a non-zero Hubble parameter on the Universe with $\\rho _{pl} \\ne 0$ as $\\Omega _{vis} \\ne 0$ .", "The value of Hubble parameter of our Universe from eqn.", "(REF ) is found to be proportional to $\\omega _{vis}$ of the visible brane which matches very well with the present cosmological results.", "Extra dimensional co-ordinate dependent modulus field In case the modulus field is $y$ dependent, the expression of proper length changes as shown in (REF ).", "In this situation, we can generate a four dimensional effective cosmological constant on the visible brane and hence in our Universe.", "This results into the scale factor which varies exponentially with time as given by eqn.", "(REF ).", "Such a scale factor indicates exponential expansion of the Universe in the presence of extra dimensional co-ordinate dependent modulus field which guarantees four dimensional de Sitter spacetime.", "However, such a radion field cannot produce a dynamical evolution by itself unless there are non-zero contributions from energy momentum tensors on the 3-branes and bulk cosmological constant $\\Lambda _5$ .", "It may further be noted that, just as in the previous case, an effective 4D cosmological constant appears on the visible brane due to a non-vanishing Planck brane matter even if the visible brane matter $T_2\\,^{\\mu }\\,_{\\nu } = 0$ (see (REF )).", "Time-varying modulus field An effective energy-momentum density is induced on our Universe in a situation when the inter-brane separation varies with time.", "The dynamical evolution of the Universe is now possible in the absence of any matter on the 3-branes i,e.", "when $T_1\\,^{\\mu }\\,_{\\nu } = T_2\\,^{\\mu }\\,_{\\nu } = 0$ .", "Actually this time dependent modulus field and bulk cosmological constant together trigger the time evolution of the Universe leading to a non-zero Hubble parameter.", "In case of a slowly time varying radion field, the Hubble parameter on our Universe has been determined in (REF ) purely in terms of time varying proper length $d_0(t)$ (defined in (REF )) modulated by bulk curvature $l$ .", "The fact that $\\dot{H(t)} > 0$ further suggests an accelerating nature of the Universe driven solely by time dependent radion field.", "Furthermore our result reveals that a solely cosmological constant driven expansion is possible only when either of the brane energy-momentum tensors is non-zero and the radion field is time independent i.e.", "a time dependent radion field can not give rise to a constant 3-brane vacuum energy.", "This can be easily seen from eqn.", "(REF )." ] ]

[ [ "Maximally inhomogeneous G\\\"{o}del-Farnsworth-Kerr generalizations" ], [ "Abstract It is pointed out that physically meaningful aligned Petrov type D perfect fluid space-times with constant zero-order Riemann invariants are either the homogeneous solutions found by G\\\"{o}del (isotropic case) and Farnsworth and Kerr (anisotropic case), or new inhomogeneous generalizations of these with non-constant rotation.", "The construction of the line element and the local geometric properties for the latter are presented." ], [ "Introduction", "The results by Milson and Pelavas [1], [2] reopened the question whether the so far theoretically determined Karlhede upper bounds for given Weyl-Petrov and/or Ricci-Segre type are sharp as well.", "In any case, a necessary condition is that the space-time is curvature homogeneous of order zero (further denoted by CH$_0$ ), i.e., its zero order Cartan-Riemann invariants are all constant.", "It is well known that ample families of pure radiation metrics satisfy this property I thank Michael Bradley for reminding me of this after the talk.", "Here we revise the situation for electrovac fields and their Einstein space limits, and present a theorem classifying all CH$_0$ genuine (i.e.", "non-Einstein space) perfect fluids, in the case where the Weyl tensor is of aligned Petrov type $D$ .", "We will focus on the physically relevant models, which turn out to be exhausted by the celebrated homogeneous Gödel universe, its anisotropic generalizations found by Farnsworth and Kerr, and a new family constituted by generalizations of these known solutions, the members of which have non-constant rotation.", "We mention the construction of good coordinates, summarize the local properties and end with concrete and more general conclusions of this investigation." ], [ "Inhomogeneous CH$_0$ perfect fluid solutions of Petrov type {{formula:cae24f9d-07d4-4878-9127-3f576883ff43}}", "In general, CH$_0$ Petrov type $D$ space-times are characterized by the existence of a Weyl principal complex null frame $(k^a,l^a,m^a,\\overline{m}^a)$ relative to which $\\Psi _0=\\Psi _1=\\Psi _3=\\Psi _4=0,\\quad \\Psi _2=const\\ne 0.$ $k^a$ and $l^a$ spanning the Weyl principal null directions (PND's).", "All double aligned Petrov type $D$ , non-null Einstein-Maxwell `electrovac' fields have been classified and integrated by Debever, Kamran and McLenaghan [3], [4] and independently by Garcia [5].", "From their results, or from the invariant approaches in [6], [7] (see also [8] for a more recent account in the vacuum case, making use of the GHP formalism) one readily infers that the only CH$_0$ solutions in this class are given by $&&\\textrm {d}s^2=\\frac{\\textrm {d}x^2}{P(x)}+P(x)\\textrm {d}\\phi ^2+\\frac{\\textrm {d}y^2}{Q(y)}-Q(y)\\textrm {d}\\psi ^2,\\\\&&P(x)=1-(\\Lambda +\\Phi _0)x^2,\\quad Q(y)=1-(\\Lambda -\\Phi _0)y^2.$ They are gravito-electric ($\\Psi _2=-\\Lambda /3$ ), homogeneous, have a complete group $G_6$ of isometries and are attributed to Levi-Civita [9], Robinson [10] and Bertotti [11], the last author giving the more general case with $\\Lambda \\ne 0$ .", "Putting the electromagnetic field parameter $\\Phi _0$ equal to zero one gets all CH$_0$ Einstein spaces (notice that $\\Lambda =0$ gives the Petrov type $O$ Minkowski space-time), having the same isometry group.", "A Petrov type $D$ space-time represents an aligned perfect fluid if its Einstein tensor has the structure $G_{ab}=(w+p)u_au_b+p g_{ab},\\quad w+p\\ne 0\\quad u^a=\\frac{qk^a+l^a}{\\sqrt{2q}}$ where $q>0$ , $w=w^{\\prime }+\\Lambda $ and $p=p^{\\prime }+\\Lambda $ are the (geometric or effective) energy density, resp.", "pressure of the fluid (in which the cosmological constant $\\Lambda $ has been absorbed) and $u^a$ is the fluid 4-velocity, lying in the PND-plane.", "Space-times of this nature are CH$_0$ if and only if the conditions (REF ) hold and $w$ and $p$ are constant.", "Then, by energy-momentum conservation and $w+p\\ne 0$ , $u^a$ is non-accelerating and non-expanding.", "The only explicitly known CH$_0$ examples are given by $A^2\\textrm {d}s^2=&&\\textrm {d}u^2-2(\\textrm {d}t+e^x\\textrm {d}y)^2+\\frac{e^v}{\\cosh v}(\\cos \\,t\\,\\textrm {d}x+\\sin \\, t\\, e^x\\textrm {d}y)^2\\nonumber \\\\&&+\\frac{e^{-v}}{\\cosh v}(-\\sin \\, t\\,\\textrm {d}x +\\cos \\, t \\,e^x\\textrm {d}y)^2$ where $A$ and $v$ are constant.", "For $v=0$ one obtains the $G_5$ shearfree homogeneous Gödel universe [12], and for $v\\ne 0$ the anisotropic and shearing, yet still homogeneous generalizations discovered by Farnsworth and Kerr [13].", "Let us list the local geometric properties of these space-times: (A) the vector field ${\\mathbf {v}}\\equiv \\frac{q{\\mathbf {k}}- {\\mathbf {l}}}{\\sqrt{2q}}=\\frac{\\partial }{\\partial {u}}$ is covariantly constant; (B) the vorticity and shear of $u^a$ are given by $&&\\omega ^a=\\omega \\,\\, v^a,\\quad \\omega =a\\cosh v\\,(=\\textrm {i}\\sqrt{2q}\\,\\rho =\\textrm {i}\\sqrt{{2}/{q}}\\,\\mu ),\\\\&&\\sigma _{ab}=V\\overline{m}_{(a}\\overline{m}_{b)}+\\overline{V}m_{(a}m_{b)},\\quad V=a\\sinh v\\,e^{\\textrm {i}\\varphi _V}\\,(=-\\sqrt{2q}\\,\\sigma =\\sqrt{{2}/{q}}\\,\\overline{\\lambda }),$ where the relation with the NP spin coefficients $\\rho $ , $\\mu $ , $\\sigma $ and $\\lambda $ has been added, and where $e^{\\textrm {i}\\varphi _V}$ is a spin gauge field; (C) the space-time represents gravito-electric `stiff dust', satisfying $A^2\\equiv p=w=-3\\Psi _2=\\omega _a\\omega ^a-\\sigma _{ab}\\sigma ^{ab}=\\omega ^2-V\\overline{V}.$ Notice that perfect fluids with $p=w$ satisfy the dominant energy condition only when $\\Lambda \\le 0$ and, when $w$ is moreover constant, may be interpreted as e.g.", "dust space-times ($\\Lambda =-w$ ) or stiff fluids ($\\Lambda =0$ ).", "The question arises whether the Gödel-Farnsworth-Kerr space-times are the only CH$_0$ aligned Petrov type $D$ perfect fluids.", "In contrast to the Einstein space case mentioned above, however, homogeneity of the curvature invariants does not imply homogeneity of the space-time here.", "One can deduce the following classification result [14]: Theorem.", "Any CH$_0$ aligned Petrov type $D$ perfect fluid satisfies property (C).", "It is either an unphysical member $w=p<0$ of Ellis' LRS II non-rotating dust family [15] or the Stephani [16]-Barnes [17] rotating dust family, or it satisfies properties (A) and (B) as well.", "Let us focus on the physically relevant class of space-times satisfying (A)-(C).", "When $\\omega $ is constant one recovers the homogeneous Gödel-Farnsworth-Kerr solutions; when $\\omega $ is non-constant, however, new inhomogeneous solutions arise as follows (see [14] for more details).", "At each point the variables $\\omega $ , $V$ and $\\overline{V}$ are constrained by the hyperbolic equation $\\omega ^2-V\\overline{V}=A^2$ .", "Thus they can be parametrized as in (REF )-(), but this does not give suitable coordinates.", "A better choice is the parametrization $\\omega =A\\frac{\\cosh x}{\\cos y},\\quad V=A\\frac{\\sin y+\\textrm {i}\\sinh x}{\\cos y}.$ Also, the Jacobi identities allow to set the NP spin coefficients $\\alpha $ , $\\beta $ , $\\gamma $ and $\\epsilon $ to zero.", "On rectifying the vector field ${\\mathbf {u}}=a\\,\\partial _t$ , the integration of the Cartan equations is quite straightforward.", "On using $x$ , $y$ , $t$ and $u$ as coordinates, where ${\\mathbf {v}}=\\frac{\\partial }{\\partial {u}}$ , one obtains $ A^2\\textrm {d}s^2=&&\\textrm {d}u^2-\\left(\\textrm {d}t+\\frac{1}{2}\\frac{\\partial F(x,y)}{\\partial x}\\textrm {d}y-\\frac{1}{2}\\frac{\\partial F(x,y)}{\\partial y}\\textrm {d}x\\right)^2\\nonumber \\\\&&+e^{F(x,y)}e^x\\left(\\cos \\left(t-\\frac{y}{2}\\right)\\,\\textrm {d}x+\\sin \\left(t-\\frac{y}{2}\\right)\\,\\textrm {d}y\\right)^2\\nonumber \\\\&&+e^{F(x,y)}e^{-x}\\left(-\\sin \\left(t+\\frac{y}{2}\\right)\\,\\textrm {d}x+\\cos \\left(t+\\frac{y}{2}\\right)\\,\\textrm {d}y\\right)^2,$ $F(x,y)$ being a solution of $\\frac{\\partial ^2 F(x,y)}{\\partial x{}^2}+\\frac{\\partial ^2F(x,y)}{\\partial y{}^2}=4\\cosh (x)e^{F(x,y)}.$ We emphasize that this is the general line element for inhomogeneous aligned Petrov type $D$ perfect fluids with positive effective energy density $w$ .", "Finally, denote $t_k$ for the number of functionally independent components of the Riemann tensor and its first $k$ covariant derivatives, relative to the canonically fixed frame at step $k$ in the Karlhede space-time classification algorithm [18], and let $q$ be the Karlhede bound.", "As there is at least one Killing vector field $\\frac{\\partial }{\\partial {u}}$ one has $t_q\\le 3$ .", "The CH$_0$ assumption means precisely $t_0=0$ , and by direct calculation one finds that $t_1=1$ and $2\\le t_2$ , whence $2\\le t_2\\le t_3\\le 3$ .", "Further investigation then shows that $t_2=t_3$ , such that $q=3$ for any member of the new family.", "The generic situation is $t_2=3$ , in which case $\\frac{\\partial }{\\partial {u}}$ is the only Killing vector field; only a very specific subfamily satisfies $t_2=2$ , in which case the complete isometry group is abelian $G_2$ ." ], [ "Conclusions", "For physically relevant CH$_0$ aligned Petrov type D perfect fluids the constant energy density $w$ equals the pressure $p$ on the one hand, and the difference between the vorticity and shear amplitudes on the other.", "In contrast to the homogeneous Gödel-Farnsworth-Kerr models, these amplitudes can be non-constant, resulting in the non-constancy of higher-order curvature invariants and, correspondingly, a dramatic drop of the isometry group dimension of the relevant 3D part of the metric.", "A special choice of (non-invariantly defined) coordinates leads to the metric (REF )-(REF ).", "Put in a much broader context, the results exemplify a generation technique, valid for any metric-based gravitation theory in any space-time dimension: investigate whether a set of (possibly higher-order) invariant relations, valid for a well-known family of space-times, singles out this family.", "If not, new interesting families may arise." ] ]

[ [ "Complete asymptotic expansion of the integrated density of states of\n multidimensional almost-periodic pseudo-differential operators" ], [ "Abstract We obtain a complete asymptotic expansion of the integrated density of states of operators of the form H =(-\\Delta)^w +B in R^d.", "Here w >0, and B belongs to a wide class of almost-periodic self-adjoint pseudo-differential operators of order less than 2w.", "In particular, we obtain such an expansion for magnetic Schr\\\"odinger operators with either smooth periodic or generic almost-periodic coefficients." ], [ "Introduction", "In [5], two of the authors of this paper have obtained the complete power asymptotic expansion of the integrated density of states of Schrödinger operators $H= -\\Delta +V$ acting in $ \\mathbb {R}^d$ assuming that the real-valued potential $V$ is either smooth periodic, or generic quasi-periodic, or belongs to a reasonably wide class of almost-periodic functions (see [5] for a complete set of conditions on $V$ as well as the previous history of the subject).", "The main aim of the current paper is to extend the results of [5] to a more general class of operators.", "We give a detailed description of this new class in the next section; here, we list the main properties of the operators belonging to it.", "(i) We consider perturbations of the Laplacian, or any positive power of the Laplacian.", "More precisely, we work with operators of the form $H= (-\\Delta )^w +B,$ where $B$ is a differential or pseudo-differential operator of order $\\kappa <2w$ .", "Here $H$ is self-adjoint and belongs to the standard algebra of almost-periodic pseudo-differential operators, see e.g.", "[7] and [8].", "(ii) If $B$ is a differential operator, we assume that its coefficients satisfy the same conditions the potential $V$ had to satisfy in [5] (for example, the coefficients can be smooth periodic, or generic quasi-periodic functions).", "In particular, periodic magnetic Schrödinger operators are covered by our results.", "(iii) If $B$ is pseudo-differential, we assume that it is a classical pseudo-differential operator, or, more generally, the operator of classical type.", "By the latter we mean that the symbol of $B$ admits an asymptotic decomposition in powers of $|\\xi |$ when $|\\xi |\\rightarrow \\infty $ ; however, these powers do not have to be integer.", "Note that operators with the relativistic kinetic energy $\\sqrt{(-i\\nabla + \\mathbf {A})^2+ m^2}$ are admissible for (almost-)periodic smooth $\\mathbf {A}$ and $m\\geqslant 0$ .", "Under these assumptions we prove that the integrated density of states $N(\\lambda )$ has the complete asymptotic expansion (REF ).", "This expansion contains powers of $\\lambda $ and powers of $\\ln \\lambda $ ; the values of the exponents in the powers of $\\lambda $ depend on the form of $B$ , whereas logarithms are raised to integer powers smaller than $d$ .", "Sometimes (as in the case of the magnetic Schrödinger operator) we can guarantee that the logarithmic terms are absent (i.e., the corresponding coefficients are zero).", "Remark 1.1 The main reason why we need assumption (iii) is to match asymptotic expansions in different intervals $I_n$ in Section .", "If we did not have assumption (iii), we would have obtained the asymptotic expansions containing the general `phase volumes' (like in [9]), and it is not clear how to relate the expansions obtained in different intervals $I_n$ .", "One immediate and slightly unexpected corollary of (REF ) is as follows: Corollary 1.2 Suppose, $H =(-\\Delta )^w +B$ with $B$ being periodic and either differential, or pseudo-differential operator of classical type.", "Then for sufficiently large $\\lambda $ the spectrum of $H$ is purely absolutely continuous.", "Since $H$ is periodic, the general Floquet-Bloch theory implies that the spectrum of $H$ is absolutely continuous with the possible exception of eigenvalues of finite multiplicity.", "If $\\lambda $ is such an eigenvalue, the integrated density of states has a jump at least $|\\Gamma ^{\\dagger }|$ at $\\lambda $ , where $\\Gamma ^{\\dagger }$ is the lattice dual to the lattice of periods of $H$ .", "Due to (REF ), this cannot happen for large $\\lambda $ .", "The approach of our paper is similar to the one of [5].", "In particular, we use the method of gauge transform developed in [9], [10], and [6].", "Nevertheless, there are plenty of new (mostly technical, but sometimes ideological) difficulties arising because the operator $B$ is no longer bounded and no longer local.", "One example of the new methods employed in this paper is the proof of Lemma REF : not only this proof works for unbounded $B$ , it also makes Condition D from [5] redundant.", "The biggest increase in technical difficulties comes in Section where we express the contribution to the density of states from various regions in the momentum space as certain complicated integrals and then try to compute these integrals.", "As a result, our paper is technically more complicated than [5] (which already was quite difficult to read).", "Thus, we have reluctantly abandoned the idea of making our paper completely self-contained; we will skip all parts of the argument which are identical (or close) to corresponding parts of [5] and refer the reader to that paper.", "Nevertheless, we will present all the definitions and properties of the important objects.", "Remark 1.3 Throughout the article we employ the convention that, if some statement is given without a proof, then an analogous statement can be found in [5], and the proof is the same up to obvious modifications.", "It comes without saying that the reader is strongly encouraged to read the article [5] first, before attempting to read this paper.", "Aknowlegements.", "SM and LP were partially supported by the EPSRC grant EP/ F029721/1.", "SM was also supported by the Lundbeck Foundation and the European Research Council under the European Community's Seventh Framework Program (FP7/2007–2013)/ERC grant agreement 202859.", "RS was partially supported by the NSF grant DMS-0901015.", "The authors would like to thank Gerassimos Barbatis for participation in preliminary discussions which led to this paper.", "SM would like to express his thanks for hospitality to the University of Athens and ESI Vienna, where part of this work was made." ], [ "Preliminaries", "For $w> 0$ we consider the operator $H =(-\\Delta )^w+ B$ acting in $\\textup {{\\textsf {L}}}_{2}( \\mathbb {R}^d)$ .", "The action of the pseudo-differential operator $B$ on functions from the Schwarz class $\\textup {{\\textsf {S}}}( \\mathbb {R}^d)$ is defined by the formula $(Bf)(\\mathbf {x}):= (2\\pi )^{-d/2} \\int b(\\mathbf {x}, \\xi )e^{i\\xi \\mathbf {x}} (\\mathcal {F}f)(\\xi ) d\\xi .$ Here $\\mathcal {F}$ is the Fourier transform $(\\mathcal {F} f)(\\xi ):= (2\\pi )^{-d/2} \\int e^{-i\\xi \\mathbf {x}}f(\\mathbf {x}) d\\mathbf {x},\\qquad \\xi \\in \\mathbb {R}^d,$ the integration is over $ \\mathbb {R}^d$ , and $b$ is the symbol of $B$ .", "We assume that $b(\\mathbf {x}, \\xi )$ , $\\mathbf {x}, \\xi \\in \\mathbb {R}^d$ , is a smooth almost-periodic in $\\mathbf {x}$ complex-valued function and, moreover, that for some countable set $\\Theta $ of frequencies we have $b(\\mathbf {x}, \\xi ) = \\sum \\limits _{\\theta \\in \\Theta }\\hat{b}(\\theta , \\xi )\\mathbf {e}_{\\theta }(\\mathbf {x})$ where $\\mathbf {e}_{\\theta }(\\mathbf {x}):=e^{i\\theta \\mathbf {x}},$ and $\\hat{b}(\\theta , \\xi ):=\\mathbf {M}_\\mathbf {x}\\big (b(\\mathbf {x},\\xi )\\mathbf {e}_{-\\theta }(\\mathbf {x})\\big )$ are the Fourier coefficients of $b$ (here $\\mathbf {M}_\\mathbf {x}$ is the mean of an almost-periodic function of $\\mathbf {x}$ ).", "We assume that the series (REF ) converges absolutely, and that $b$ satisfies the symmetry condition $\\hat{b}(\\theta , \\xi ) = \\overline{\\hat{b}(-\\theta , \\xi +\\theta )},$ so that the operator $B$ is formally self-adjoint.", "For $R>0$ let $\\operatorname{\\mathbb {1}}_{\\mathcal {B}_R}$ be the indicator function of the ball $\\mathcal {B}_R:= \\big \\lbrace \\xi : |\\xi |< R\\big \\rbrace $ .", "We assume that there exists a constant $C_0$ such that $\\Vert b\\operatorname{\\mathbb {1}}_{\\mathcal {B}_{C_0}}\\Vert _{L_\\infty ( \\mathbb {R}^d\\times \\mathbb {R}^d)}< \\infty ,$ and that $\\big (1- \\operatorname{\\mathbb {1}}_{\\mathcal {B}_{C_0}}(\\xi )\\big )b(\\mathbf {x}, \\xi )= \\sum _{\\iota \\in J}|\\xi |^\\iota b_\\iota \\big (\\mathbf {x}, \\xi /|\\xi |\\big ),$ where $J$ is a discrete subset of $(-\\infty , \\varkappa ]$ with $0\\leqslant \\varkappa <2w$ (the first inequality here is assumed for convenience without loss of generality), and $b_\\iota (\\mathbf {x}, \\eta )$ are smooth functions on $ \\mathbb {R}^d\\times \\mathbb {S}^{d- 1}$ almost-periodic with respect to $\\mathbf {x}$ .", "Let $\\tilde{w} :=(w+ \\varkappa )/2.$ We introduce $\\chi \\in \\textup {{\\textsf {C}}}^{\\infty }( \\mathbb {R}_+)$ so that $\\chi (r)= {\\left\\lbrace \\begin{array}{ll}r, & r\\geqslant C_0,\\\\ 0, &r\\leqslant C_0/2.\\end{array}\\right.", "}$ Remark 2.1 Increasing $C_0$ if necessary, we can guarantee that for any $\\widetilde{J}\\subset J$ and any $\\widetilde{\\Theta }\\subset \\Theta $ the operator $\\widetilde{B}$ with the symbol $\\tilde{b}$ given by $\\tilde{b}(\\mathbf {x}, \\xi ):= \\sum _{\\iota \\in \\widetilde{J}}\\Big (\\chi \\big (|\\xi |\\big )\\Big )^\\iota \\sum _{\\theta \\in \\widetilde{\\Theta }}\\hat{b}_\\iota \\big (\\theta , \\xi /|\\xi |\\big )\\mathbf {e}_{\\theta }(\\mathbf {x})$ satisfies $(-\\Delta )^{\\tilde{w}}- |\\widetilde{B}|\\geqslant 0.$ We also assume that the coefficients in the expansion $b_\\iota (\\mathbf {x}, \\eta )= \\sum _{\\theta \\in \\Theta }\\hat{b}_\\iota (\\theta , \\eta )\\mathbf {e}_{\\theta }(\\mathbf {x}), \\qquad \\mathbf {x}\\in \\mathbb {R}^d, \\quad \\eta \\in \\mathbb {S}^{d- 1}, \\quad \\iota \\in J$ can be represented by a series $\\hat{b}_\\iota (\\theta , \\eta _1, \\dots , \\eta _d)= \\sum _{\\tau \\in \\mathbb {N}_0^d}\\hat{b}_\\iota ^{(\\tau )}(\\theta )\\eta _1^{\\tau _1}\\cdots \\eta _d^{\\tau _d}$ which converges absolutely in a ball of radius greater than one of $ \\mathbb {R}^d$ .", "Under the above assumptions $H$ is a selfadjoint operator on the Sobolev space $\\textup {{\\textsf {H}}}^{2w}( \\mathbb {R}^d)$ .", "We are interested in the asymptotic behaviour of its integrated density of states $N(\\lambda )$ as the spectral parameter $\\lambda $ tends to infinity.", "Definition 2.2 Let $e(\\lambda ;\\mathbf {x},\\mathbf {y})$ be the kernel of the spectral projection of $H$ .", "We define the integrated density of states as $N(\\lambda ) :=\\mathbf {M}_{\\mathbf {x}}\\big (e(\\lambda ;\\mathbf {x},\\mathbf {x})\\big ).$ It was proved in Theorem 4.1 of [8] that for differential operators this definition agrees with the traditional one (at least at its continuity points).", "The following lemma is proved at the end of Section 4 of [5].", "Lemma 2.3 If $A\\geqslant B$ , then $N(\\lambda ; A)\\leqslant N(\\lambda ; B)$ .", "Suppose, $A= a(\\mathbf {x}, D)$ and $U= u(\\mathbf {x}, D)$ are two pseudo-differential operators with almost-periodic coefficients.", "Let operator $A$ be elliptic self-adjoint and operator $U$ be unitary.", "Then $N(\\lambda ; A)= N(\\lambda ; U^{-1}AU)$ .", "Without loss of generality we assume that $\\Theta $ (recall (REF )) spans $ \\mathbb {R}^d$ , contains $\\mathbf {0}$ and is symmetric about $\\mathbf {0}$ ; we also put $\\Theta _k :=\\Theta +\\Theta +\\dots +\\Theta $ (algebraic sum taken $k$ times) and $\\Theta _{\\infty }:=\\cup _k\\Theta _k=Z(\\Theta )$ , where for a set $S\\subset \\mathbb {R}^d$ by $Z(S)$ we denote the set of all finite linear combinations of elements in $S$ with integer coefficients.", "The set $\\Theta _\\infty $ is countable and non-discrete (unless $B$ is periodic).", "We will need" ], [ "Suppose that $\\theta _1,\\dots ,\\theta _d\\in \\Theta _\\infty $ .", "Then $Z(\\theta _1,\\dots ,\\theta _d)$ is discrete.", "It is easy to see that this condition can be reformulated like this: suppose, $\\theta _1,\\dots ,\\theta _d\\in \\Theta _\\infty $ .", "Then either $\\lbrace \\theta _j\\rbrace $ are linearly independent, or $\\sum _{j=1}^d n_j\\theta _j=0$ , where $n_j\\in \\mathbb {Z}$ and not all $n_j$ are zeros.", "This reformulation shows that Condition A is generic: indeed, if we are choosing frequencies of $b$ one after the other, then on each step we have to avoid choosing a new frequency from a countable set of hyperplanes, and this is obviously a generic restriction.", "Condition A is obviously satisfied for periodic $B$ , but it becomes meaningful if $B$ is quasi-periodic (i.e., if it is a linear combination of finitely many exponentials).", "If $\\Theta $ and $J$ are finite, Condition A is all we need.", "If, however, any (or both) of these sets is infinite, we need other conditions which describe, how well $B$ can be approximated by operators with quasi-periodic symbols.", "In the proof we are going to work with quasi-periodic approximations of $B$ , and we need these conditions to make sure that all estimates in the proof are uniform with respect to these approximations.", "We introduce $\\textsf {b}_\\iota (\\theta ):= \\sup _{|\\eta |= 1}\\big |\\hat{b}_\\iota (\\theta , \\eta )\\big |, \\quad \\theta \\in \\Theta .$" ], [ "Let $k$ be a positive integer.", "Then there exists $R_0\\geqslant C_0$ such that for each $\\rho > R_0$ there exist a finite symmetric set $\\widetilde{\\Theta }\\subset \\big (\\Theta \\cap \\mathcal {B}(\\rho ^{1/k})\\big )$ (where $\\mathcal {B}(r)$ is the ball of radius $r$ centered at 0) and a finite subset $\\widetilde{J}\\subset J$ with $\\operatorname{{card}}\\widetilde{J}\\leqslant \\rho ^{1/k}$ such that $\\sum _{(\\theta , \\iota )\\in (\\Theta \\times J)\\setminus (\\widetilde{\\Theta }\\times \\widetilde{J})}\\big (1+ |\\theta |^2\\big )^{\\varkappa /4}|R_0|^{\\iota - \\varkappa }\\textup {\\textsf {b}}_\\iota (\\theta )\\leqslant \\rho ^{- k}.$ The last condition we need is a version of the Diophantine condition on the frequencies of $b$ .", "First, we need some definitions.", "We fix a natural number $\\tilde{k}$ (the choice of $\\tilde{k}$ will be determined later by the order of the remainder in the asymptotic expansion) and denote $\\widetilde{\\Theta }^{\\prime }_{\\tilde{k}}:= \\widetilde{\\Theta }_{\\tilde{k}}\\setminus \\lbrace 0\\rbrace $ (see (REF ) for the notation).", "We say that $\\mathfrak {V}$ is a quasi-lattice subspace of dimension $m$ , if $\\mathfrak {V}$ is a linear span of $m$ linearly independent vectors $\\theta _1,\\dots ,\\theta _m$ from $\\widetilde{\\Theta }_{\\tilde{k}}$ .", "Obviously, the zero space (which we will denote by $\\mathfrak {X}$ ) is a quasi-lattice subspace of dimension 0, and $ \\mathbb {R}^d$ is a quasi-lattice subspace of dimension $d$ .", "We denote by $\\mathcal {V}_m$ the collection of all quasi-lattice subspaces of dimension $m$ and put $\\mathcal {V}:=\\cup _m\\mathcal {V}_m$ .", "If $\\xi \\in \\mathbb {R}^d$ and $\\mathfrak {V}$ is a linear subspace of $ \\mathbb {R}^d$ , we denote by $\\xi _{\\mathfrak {V}}$ the orthogonal projection of $\\xi $ onto $\\mathfrak {V}$ , and put $\\mathfrak {V}^\\perp $ to be an orthogonal complement of $\\mathfrak {V}$ , so that $\\xi = \\xi _{\\mathfrak {V}}+ \\xi _{\\mathfrak {V}^\\perp }$ .", "Let $\\mathfrak {V},\\mathfrak {U}\\in \\mathcal {V}$ .", "We say that these subspaces are strongly distinct, if neither of them is a subspace of the other one.", "This condition is equivalent to stating that if we put $\\mathfrak {W}:=\\mathfrak {V}\\cap \\mathfrak {U}$ , then $\\dim \\mathfrak {W}$ is strictly less than dimensions of $\\mathfrak {V}$ and $\\mathfrak {U}$ .", "We put $\\phi = \\phi (\\mathfrak {V}, \\mathfrak {U})\\in [0, \\pi /2]$ to be the angle between them, i.e.", "the angle between $\\mathfrak {V}\\ominus \\mathfrak {W}$ and $\\mathfrak {U}\\ominus \\mathfrak {W}$ , where $\\mathfrak {V}\\ominus \\mathfrak {W}$ is the orthogonal complement of $\\mathfrak {W}$ in $\\mathfrak {V}$ .", "This angle is non-zero iff $\\mathfrak {V}$ and $\\mathfrak {W}$ are strongly distinct.", "We put $s= s(\\rho )= s(\\widetilde{\\Theta }_{\\tilde{k}}):= \\inf \\sin \\big (\\phi (\\mathfrak {V},\\mathfrak {U})\\big )$ , where the infimum is over all strongly distinct pairs of subspaces from $\\mathcal {V}$ , $R= R(\\rho ):= \\sup _{\\theta \\in \\widetilde{\\Theta }_{\\tilde{k}}}|\\theta |$ , and $r= r(\\rho ):= \\inf _{\\theta \\in \\widetilde{\\Theta }^{\\prime }_{\\tilde{k}}}|\\theta |$ .", "Obviously, $R(\\rho )= O(\\rho ^{1/k}),$ where the implied constant can depend on $k$ and $\\tilde{k}$ ." ], [ "For each fixed $k$ and $\\tilde{k}$ the sets $\\widetilde{\\Theta }_{\\tilde{k}}$ can be chosen in such a way that for sufficiently large $\\rho $ the number of elements in $\\widetilde{\\Theta }_{\\tilde{k}}$ satisfies $\\operatorname{{card}}\\widetilde{\\Theta }_{\\tilde{k}}\\leqslant \\rho ^{1/k}$ and we have $s(\\rho )\\geqslant \\rho ^{-1/k}$ and $r(\\rho )\\geqslant \\rho ^{-1/k},$ where the implied constant (i.e.", "how large should $\\rho $ be) can depend on $k$ and $\\tilde{k}$ .", "Remark 2.4 Note that Condition C is automatically satisfied for quasi-periodic and smooth periodic $B$ ; see [5] for further discussion of this condition.", "Condition A implies the following statement, which will be used crucially in our constructions.", "Corollary 2.5 Suppose, $\\theta _1, \\dots , \\theta _{l}\\in \\widetilde{\\Theta }_{\\tilde{k}}$ , $l\\leqslant d- 1$ .", "Let $\\mathfrak {V}$ be the span of $\\theta _1, \\dots , \\theta _{l}$ .", "Then each element of the set $\\widetilde{\\Theta }_{\\tilde{k}}\\cap \\mathfrak {V}$ is a linear combination of $\\theta _1, \\dots , \\theta _{l}$ with rational coefficients.", "Since the set $\\widetilde{\\Theta }_{\\tilde{k}}\\cap \\mathfrak {V}$ is finite, this implies that the set $Z(\\widetilde{\\Theta }_{\\tilde{k}}\\cap \\mathfrak {V})$ is discrete and is, therefore, a lattice in $\\mathfrak {V}$ .", "From now on, we always assume that $B$ satisfies all the conditions from this section; we will also denote $\\rho := \\lambda ^{1/2w}.$ Now we can formulate our main theorem.", "Theorem 2.6 Let $H$ be an operator (REF ) satisfying Conditions A, B and C. Then for each $K\\in \\mathbb {R}$ there exists a finite positive integer $L$ and a finite subset $J_0\\subset J$ such that $\\begin{split}&N(\\rho ^{2w})\\\\ &= \\sum _{q= 0}^{d- 1}\\sum _{h= 0}^{L}\\sum _{\\iota _1, \\dots , \\iota _h\\in J_0}\\sum _{j= 0}^{[K+ d+ (2- 2w)h+ \\iota _1+ \\cdots + \\iota _h]}C_{q\\, h\\, j}^{\\iota _1\\cdots \\iota _h}\\rho ^{d+ (2- 2w)h+ \\iota _1+ \\cdots + \\iota _h- j}\\ln ^q\\rho + O(\\rho ^{-K}).\\end{split}$ as $\\rho \\rightarrow \\infty $ .", "Remark 2.7 The powers of $\\rho $ present in (REF ) are equal to $d+ (2- 2w)h+ \\iota _1+ \\cdots + \\iota _h- j$ , and the first impression is that there are far too many of them (indeed, a priori the set of all such powers can be dense in $ \\mathbb {R}$ , for instance).", "However, many of these powers are, in fact, spurious (i.e.", "the corresponding coefficients $C_{q\\, h\\, j}^{\\iota _1\\cdots \\iota _h}$ are zero).", "This happens, for example, when $d+ (2- 2w)h+ \\iota _1+ \\cdots + \\iota _h- j>d$ (for obvious reasons).", "Equally obviously, these powers do not `multiply' when we increase $K$ .", "This means that if $K_1<K_2$ , then expansion (REF ) with $K=K_2$ does not contain extra terms with $d+ (2- 2w)h+ \\iota _1+ \\cdots + \\iota _h- j>-K_1$ , compared to this expansion for $K=K_1$ .", "In the case of magnetic Schrödinger operators, Theorem REF and calculations similar to those of [1] and [5] imply that most of the terms in (REF ) will indeed disappear: Corollary 2.8 For each $K\\in \\mathbb {N}$ we have: $N(\\lambda )=\\lambda ^{d/2}\\bigg (C_d+\\sum \\limits _{j=1}^{K}e_j\\lambda ^{-j}+o(\\lambda ^{-K})\\bigg )$ as $\\lambda \\rightarrow \\infty $ .", "Remark 2.9 By taking the Laplace transform of (REF ), one can obtain an asymptotic expansion of the (regularised) heat trace as $t\\rightarrow 0$ .", "However, it seems that using the approach of [1] and [2], it is possible to obtain even stronger results (the pointwise asymptotic expansion of the heat kernel).", "Remark 2.10 Of course, formula (REF ) cannot be differentiated; moreover, we do not even know if in the almost periodic case $N(\\lambda )$ is strictly increasing.", "However, in the periodic Schrödinger case there are some results on the high-energy behaviour of the (non-integrated) density of states, see e. g. [11].", "Given Conditions B and C, we want to introduce the following definition.", "We say that a non-negative function $f= f(\\rho )= f(\\rho ; k, \\tilde{k})$ satisfies the estimate $f(\\rho )\\leqslant \\rho ^{0+}$ (resp.", "$f(\\rho )\\geqslant \\rho ^{0-}$ ), if for each positive $\\varepsilon $ and for each $\\tilde{k}$ we can achieve $f(\\rho )\\leqslant \\rho ^{\\varepsilon }$ (resp.", "$f(\\rho )\\geqslant \\rho ^{-\\varepsilon }$ ) for sufficiently large $\\rho $ by choosing parameter $k$ from Conditions B and C sufficiently large.", "For example, we have $R(\\rho )\\leqslant \\rho ^{0+},$ $\\operatorname{{card}}\\widetilde{\\Theta }\\leqslant \\rho ^{0+}$ , $s(\\rho )\\geqslant \\rho ^{0-}$ , and $r(\\rho )\\geqslant \\rho ^{0-}$ .", "Throughout the paper, we always assume that the value of $k$ is chosen sufficiently large so that all inequalities of the form $\\rho ^{0+}\\leqslant \\rho ^{\\varepsilon }$ or $\\rho ^{0-}\\geqslant \\rho ^{-\\varepsilon }$ we encounter in the proof are satisfied.", "The next statement proved in [5] is an example of how this new notation is used.", "Lemma 2.11 Suppose, $\\theta , \\mu _1,\\dots ,\\mu _d\\in \\widetilde{\\Theta }^{\\prime }_{\\tilde{k}}$ , the set $\\lbrace \\mu _j\\rbrace $ is linearly independent, and $\\theta =\\sum _{j=1}^db_j\\mu _j$ .", "Then each non-zero coefficient $b_j$ satisfies $\\rho ^{0-}\\leqslant |b_j| \\leqslant \\rho ^{0+}.$ In this paper, by $C$ or $c$ we denote positive constants, the exact value of which can be different each time they occur in the text, possibly even in the same formula.", "On the other hand, the constants which are labeled (like $C_1$ , $c_3$ , etc) have their values being fixed throughout the text.", "Given two positive functions $f$ and $g$ , we say that $f\\gtrsim g$ , or $g\\lesssim f$ , or $g=O(f)$ if the ratio $g/f$ is bounded.", "We say $f\\asymp g$ if $f\\gtrsim g$ and $f\\lesssim g$ .", "We will also need a number of auxiliary constants.", "Let us choose numbers $\\lbrace \\alpha _j\\rbrace _{j= 1}^d$ , $\\beta $ , $\\vartheta $ , and $\\varsigma $ satisfying $\\max \\lbrace 1- w + \\varkappa /2, 1/2\\rbrace < \\beta < \\alpha _1< \\alpha _2< \\cdots < \\alpha _d< \\vartheta < \\varsigma < 1$ (recall (REF )), and set $ \\alpha := \\varkappa /\\beta .$" ], [ "Reduction to a finite interval of spectral parameter", "To begin with, we choose sufficiently large $\\rho _0> C_0$ (to be fixed later on) and for $n\\in \\mathbb {N}$ put $\\rho _n:= 2\\rho _{n- 1}= 2^n\\rho _0$ .", "We also define the intervals $I_n:= [\\rho _n, 4\\rho _n]$ .", "The proof of Theorem REF will be based on the following lemma: Lemma 3.1 For each $M\\in \\mathbb {R}$ there exist $L> 0$ and a finite subset $J_0\\subset J$ such that for every $n\\in \\mathbb {N}$ and $\\rho \\in I_n$ $N(\\rho ^{2w})= \\sum _{q= 0}^{d- 1}\\sum _{h= 0}^{L}\\sum _{\\iota _1, \\dots , \\iota _h\\in J_0}\\sum _{j= 0}^{[\\frac{d+ M}{1- \\varsigma }]}C_{q\\, h\\, j}^{\\iota _1\\cdots \\iota _h}(n, M)\\rho ^{d+ (2- 2w)h+ \\iota _1+ \\cdots + \\iota _h- j}\\ln ^q\\rho + O(\\rho _n^{-M}).$ Here, $C_{q\\, h\\, j}^{\\iota _1\\cdots \\iota _h}(n, M)$ are some real numbers satisfying $C_{q\\, h\\, j}^{\\iota _1\\cdots \\iota _h}(n, M)= O(\\rho _n^{-2\\beta h+ \\varsigma j}).$ The constants in the $O$ -terms do not depend on $n$ (but they may depend on $M$ ).", "Remark 3.2 Note that (REF ) is not a `proper' asymptotic formula, since the coefficients are allowed to grow with $n$ (and, therefore, with $\\rho $ ).", "Some of the powers of $\\rho $ on the right hand side of (REF ) may coincide.", "In order to avoid the ambiguity let us redefine coefficients $C_{q\\, h\\, j}^{\\iota _1\\cdots \\iota _h}(n, M)$ in such a way that, for any given values of $q$ and $d+ (2- 2w)h+ \\iota _1+ \\cdots + \\iota _h- j$ , only the coefficient with the minimal possible value of $h$ and maximal possible values of $j$ , $\\iota _1, \\dots , \\iota _h$ (in this order) is nonzero.", "Note that these new coefficients still satisfy (REF ).", "Let us prove Theorem REF assuming that we have proved Lemma REF .", "Let $M$ be fixed.", "Denote the sum on the right hand side of (REF ) by $N_n(\\rho ^{2w})$ .", "Then, for $n\\geqslant 1$ , whenever $\\rho \\in I_{n-1}\\cap I_n=[\\rho _n,2\\rho _n]$ , we have: $\\begin{split}&N_n(\\rho ^{2w})- N_{n- 1}(\\rho ^{2w})\\\\ &= \\sum _{q= 0}^{d- 1}\\sum _{h= 0}^{L}\\sum _{\\iota _1, \\dots , \\iota _h\\in J_0}\\sum _{j= 0}^{[\\frac{d+ M}{1- \\varsigma }]} t_{q\\, h\\, j}^{\\iota _1\\cdots \\iota _h}(n, M)\\rho ^{d+ (2- 2w)h+ \\iota _1+ \\cdots + \\iota _h- j}\\ln ^q\\rho + O(\\rho _n^{-M}),\\end{split}$ where $t_{q\\, h\\, j}^{\\iota _1\\cdots \\iota _h}(n, M):= C_{q\\, h\\, j}^{\\iota _1\\cdots \\iota _h}(n, M)- C_{q\\, h\\, j}^{\\iota _1\\cdots \\iota _h}(n- 1, M).$ On the other hand, since for $\\rho \\in I_{n- 1}\\cap I_n$ we have both $N(\\rho ^{2w})= N_n(\\rho ^{2w})+ O(\\rho _n^{-M})$ and $N(\\rho ^{2w})= N_{n- 1}(\\rho ^{2w})+ O(\\rho _n^{-M})$ , this implies that $\\sum _{q= 0}^{d- 1}\\sum _{h= 0}^{L}\\sum _{\\iota _1, \\dots , \\iota _h\\in J_0}\\sum _{j= 0}^{[\\frac{d+ M}{1- \\varsigma }]} t_{q\\, h\\, j}^{\\iota _1\\cdots \\iota _h}(n, M)\\rho ^{d+ (2- 2w)h+ \\iota _1+ \\cdots + \\iota _h- j}\\ln ^q\\rho = O(\\rho _n^{-M}).$ Claim 3.3 For each combination of indices present on the right hand side of (REF ) we have: $t_{q\\, h\\, j}^{\\iota _1\\cdots \\iota _h}(n, M)= O(\\rho _n^{j- M- d+ (2w- 2)h- \\iota _1- \\cdots - \\iota _h}\\ln ^{d- 1- q}\\rho _n).$ Put $y:= \\rho _n/\\rho $ and let $\\tau _{p\\, h\\, j}^{\\iota _1\\cdots \\iota _h}(n, M):= \\rho _n^{M+ d+ (2- 2w)h+ \\iota _1+ \\cdots + \\iota _h- j}\\sum _{q= p}^{d- 1}\\binom{q}{p}(-1)^p t_{q\\, h\\, j}^{\\iota _1\\cdots \\iota _h}(n, M)\\ln ^{q- p}\\rho _n.$ Then by (REF ) for $y\\in [1/2, 1]$ $P(y):= \\sum _{p= 0}^{d- 1}\\sum _{h= 0}^{L}\\sum _{\\iota _1, \\dots , \\iota _h\\in J_0}\\sum _{j= 0}^{[\\frac{d+ M}{1- \\varsigma }]}\\tau _{p\\, h\\, j}^{\\iota _1\\cdots \\iota _h}(n, M)y^{j- d+ (2w- 2)h- \\iota _1- \\cdots - \\iota _h}\\ln ^p y= O(1).$ Let us denote by $h_1, \\dots , h_T$ the functions $y^{j- d+ (2w- 2)h- \\iota _1- \\cdots - \\iota _h}\\ln ^p y$ entering the sum in (REF ) with non-zero coefficients; these functions are linearly independent on the interval $[1/2, 1]$ .", "Therefore, there exist points $y_1,...,y_{T}\\in [1/2, 1]$ such that the determinant of the matrix $\\big (h_j(y_l)\\big )_{j, l= 1}^{T}$ is non-zero.", "Now (REF ) and the Cramer's Rule imply that the values of $\\tau _{p\\, h\\, j}^{\\iota _1\\cdots \\iota _h}(n, M)$ are fractions with a bounded expression in the numerator and a fixed non-zero number in the denominator.", "Therefore, $\\tau _{p\\, h\\, j}^{\\iota _1\\cdots \\iota _h}(n, M)= O(1).$ Thus, choosing $p= d- 1$ in (REF ), we obtain $t_{d- 1\\, h\\, j}^{\\iota _1\\cdots \\iota _h}(n, M)= O(\\rho _n^{j- M- d+ (2w- 2)h- \\iota _1- \\cdots - \\iota _h}).$ Now we can put $p= d- 2$ into (REF ) and obtain $t_{d- 1\\, h\\, j}^{\\iota _1\\cdots \\iota _h}(n, M)= O(\\rho _n^{j- M- d+ (2w- 2)h- \\iota _1- \\cdots - \\iota _h}\\ln \\rho _n).$ Continuing this process until $p= 0$ , we obtain (REF ).", "Thus, for $j< M+ d+ (2- 2w)h+ \\iota _1+ \\cdots + \\iota _h$ , the series $\\sum _{m=0}^\\infty t_{q\\, h\\, j}^{\\iota _1\\cdots \\iota _h}(m, M)$ is absolutely convergent; moreover, for such $j$ we have: $\\begin{split}& C_{q\\, h\\, j}^{\\iota _1\\cdots \\iota _h}(n, M)= C_{q\\, h\\, j}^{\\iota _1\\cdots \\iota _h}(0, M)+ \\sum _{m= 1}^nt_{q\\, h\\, j}^{\\iota _1\\cdots \\iota _h}(m, M)\\\\ &= C_{q\\, h\\, j}^{\\iota _1\\cdots \\iota _h}(0, M)+ \\sum _{m= 1}^\\infty t_{q\\, h\\, j}^{\\iota _1\\cdots \\iota _h}(m, M)+ O(\\rho _n^{j- M- d+ (2w- 2)h- \\iota _1- \\cdots - \\iota _h}\\ln ^{d- 1- q}\\rho _n)\\\\ & =: C_{q\\, h\\, j}^{\\iota _1\\cdots \\iota _h}(M)+ O(\\rho _n^{j- M- d+ (2w- 2)h- \\iota _1- \\cdots - \\iota _h}\\ln ^{d- 1- q}\\rho _n),\\end{split}$ where we have denoted $C_{q\\, h\\, j}^{\\iota _1\\cdots \\iota _h}(M):= C_{q\\, h\\, j}^{\\iota _1\\cdots \\iota _h}(0, M)+ \\sum _{m =1}^\\infty t_{q\\, h\\, j}^{\\iota _1\\cdots \\iota _h}(m, M)$ .", "For bigger values of $j$ we use (REF ) and (REF ) to obtain $\\begin{split} &\\sum _{q= 0}^{d- 1}\\sum _{h= 0}^{L}\\sum _{\\iota _1, \\dots , \\iota _h\\in J_0}\\sum _{\\begin{array}{c}j\\geqslant M+ d+ (2- 2w)h+ \\iota _1+ \\cdots +\\iota _h\\end{array}}^{[\\frac{d+ M}{1- \\varsigma }]} \\big |C_{q\\, h\\, j}^{\\iota _1\\cdots \\iota _h}(n, M)\\big |\\rho ^{d+ (2- 2w)h+ \\iota _1+ \\cdots + \\iota _h- j}\\ln ^q\\rho \\\\ &\\lesssim \\sum _{q= 0}^{d- 1} \\sum _{h= 0}^{L}\\sum _{\\iota _1, \\dots , \\iota _h\\in J_0}\\rho _n^{\\varsigma d+ (2\\varsigma - 2\\beta - 2\\varsigma w+ \\varsigma \\varkappa )h- (1- \\varsigma )M}\\ln ^q\\rho _n\\lesssim \\rho _n^{\\varsigma d- (1- \\varsigma )M}\\ln ^{d- 1}\\rho _n.\\end{split}$ Thus, when $\\rho \\in I_n$ , we have: $\\begin{split}N(\\rho ^{2w})&= \\sum _{q= 0}^{d- 1}\\sum _{h= 0}^{L}\\sum _{\\iota _1,\\dots , \\iota _h\\in J_0}\\sum _{j= 0}^{[M+ d+ (2- 2w)h+ \\iota _1+ \\cdots +\\iota _h]} C_{q\\, h\\, j}^{\\iota _1\\cdots \\iota _h}(M)\\rho ^{d+ (2- 2w)h+\\iota _1+ \\cdots + \\iota _h- j}\\ln ^q\\rho \\\\ &+ O(\\rho ^{-M}\\ln ^{d-1}\\rho )+ O(\\rho ^{\\varsigma d- (1- \\varsigma )M}\\ln ^{d- 1}\\rho ).\\end{split}$ Since the constants in $O$ terms do not depend on $n$ , it is sufficient to choose $M:= \\big [(\\varsigma d+ K)/(1- \\varsigma )\\big ]+ 1$ to get (REF ) for all $\\rho \\geqslant \\rho _0$ .", "The rest of the paper is devoted to proving Lemma REF .", "The first step of the proof is fixing $n$ and fixing large $\\tilde{k}$ and $k$ .", "The precise value of $\\tilde{k}$ will be chosen later; the only restriction on it will be to satisfy inequality (REF ) (it says that the more asymptotic terms we want to have in (REF ), the bigger $\\tilde{k}$ we need to choose; note that the choice of $\\tilde{k}$ does not depend on $k$ ).", "We will have several requirements on how large $k$ should be (most of them will be of the form $\\rho _n^{0+}< \\rho _n^{\\varepsilon }$ or $\\rho _n^{0-}>\\rho _n^{-\\varepsilon }$ ); each time we have such an inequality, we assume that $k$ is chosen sufficiently large to satisfy it.", "Remark 3.4 Our choice of $k$ will only depend on $M$ , $w$ , $\\varkappa $ , and the constants introduced in (REF ).", "The set $J_0$ in Lemma REF can be chosen to be $J_0:= J\\cap [\\varkappa -d -M -1, \\varkappa ].$ The first requirement on $k$ we have is that $k> d+ M+ \\varkappa (d+ M)/(w- \\varkappa ) -2w.$ After fixing $\\tilde{k}$ and $k$ we get $R_0$ from Condition B.", "Then, taking $\\rho _0\\geqslant R_0$ and fixing $n$ , we choose $\\widetilde{\\Theta }$ and $\\widetilde{J}$ so that Conditions B and C are satisfied for $\\rho := 4\\rho _n$ .", "Without loss of generality we may assume that $\\widetilde{J} \\supset J_0$ .", "Then we introduce an auxiliary pseudo–differential operator $\\widetilde{B}$ with the symbol $\\tilde{b}$ given by (REF ).", "From now on we prove Lemma REF for $B= \\widetilde{B}$ and with $J_0$ replaced by $\\widetilde{J}$ .", "However, in view of (REF ) and (REF ), the results with $\\widetilde{J}$ and $J_0$ are equivalent.", "Afterwards, in Section  we will prove that the asymptotics (REF ) for the original $B$ follows from Condition B and (REF ) for $\\widetilde{B}$ ." ], [ "Pseudo-differential operators", "Most of the material in this and several subsequent sections is very similar to the corresponding sections of [5] and [6], as are the proofs of most of the statements.", "Therefore, we will often omit the proofs, instead referring the reader to [5], [9], and [6]." ], [ "Classes of PDO's", "Before we define the pseudo-differential operators (PDO's), we introduce the relevant classes of symbols.", "Let $b = b(\\mathbf {x}, \\xi )$ , $\\mathbf {x}, \\xi \\in \\mathbb {R}^d$ , be an almost-periodic (in $\\mathbf {x}$ ) complex-valued function and, moreover, for some countable set $\\hat{\\Theta }$ of frequencies (we always assume that $\\hat{\\Theta }$ is symmetric and contains 0; starting from the middle of this section, $\\hat{\\Theta }$ will be assumed to be finite) $b(\\mathbf {x}, \\xi ) = \\sum \\limits _{\\theta \\in \\hat{\\Theta }}\\hat{b}(\\theta , \\xi )\\mathbf {e}_{\\theta }(\\mathbf {x}),$ where $\\hat{b}(\\theta , \\xi ):=\\mathbf {M}_\\mathbf {x}\\big (b(\\mathbf {x},\\xi )\\mathbf {e}_{-\\theta }(\\mathbf {x})\\big )$ are the Fourier coefficients of $b(\\cdot , \\xi )$ (recall that $\\mathbf {M}$ is the mean of an almost-periodic function).", "We always assume that (REF ) converges absolutely.", "Let us now define the classes of symbols we will consider and operators associated with them.", "For $\\xi \\in \\mathbb {R}^d$ let $\\langle \\xi \\rangle := \\sqrt{1+|\\xi |^2}$ .", "We notice that $\\langle \\xi + \\eta \\rangle \\leqslant 2\\langle \\xi \\rangle \\langle \\eta \\rangle , \\ \\forall \\xi , \\eta \\in \\mathbb {R}^d.$ We say that a symbol $b$ belongs to the class $\\mathbf {S}_{ \\alpha }= \\mathbf {S}_{ \\alpha }(\\beta )= \\mathbf {S}_{ \\alpha }(\\beta , \\hat{\\Theta })$ , if for any $l\\geqslant 0$ and any non-negative $s\\in \\mathbb {Z}$ the conditions ${\\,\\vrule depth3pt height9pt width1pt}\\, b {\\,\\vrule depth3pt height9pt width1pt}\\,^{( \\alpha )}_{l, s}:= \\max _{|\\mathbf {s}| \\leqslant s}\\sum \\limits _{\\theta \\in \\hat{\\Theta }}\\langle \\theta \\rangle ^{l}\\sup _{\\xi }\\langle \\xi \\rangle ^{(- \\alpha + |\\mathbf {s}|)\\beta }\\big |\\mathbf {D}_{\\xi }^{\\mathbf {s}}\\hat{b}(\\theta , \\xi )\\big |< \\infty , \\quad |\\mathbf {s}|= s_1+ s_2+ \\dots + s_d,$ are fulfilled.", "The quantities (REF ) define norms on the class $\\mathbf {S}_ \\alpha $ .", "Note that $\\mathbf {S}_ \\alpha $ is an increasing function of $ \\alpha $ , i.e.", "$\\mathbf {S}_{ \\alpha }\\subset \\mathbf {S}_{\\gamma }$ for $ \\alpha < \\gamma $ .", "Given $\\theta \\in \\mathbb {R}^d$ , let us introduce a linear map $\\nabla _{\\theta }$ on symbols which acts according to the rule $\\widehat{(\\nabla _{\\theta } a)}(\\phi , \\xi ):= \\hat{a}(\\phi , \\xi + \\theta )- \\hat{a}(\\phi , \\xi ).$ If the Fourier transform of the symbol is factorized, i.e.", "$\\hat{a}(\\phi , \\xi )= \\prod _{q= 1}^Q\\hat{a}_q(\\phi , \\xi ),$ then the action of $\\nabla _{\\theta }$ can be written as a sum of actions on each factor separately: $\\widehat{(\\nabla _{\\theta } a)}(\\phi , \\xi )= \\sum _{q= 1}^Q\\prod _{l= 1}^{q- 1}\\hat{a}_l(\\phi , \\xi + \\theta )\\widehat{(\\nabla _{\\theta } a_q)}(\\phi , \\xi )\\prod _{s= q- 1}^Q\\hat{a}_s(\\phi , \\xi ).$ For later reference we mention here the following convenient bound that follows from definition (REF ) and property (REF ): $\\sum \\limits _{\\theta \\in \\hat{\\Theta }}\\langle \\theta \\rangle ^{l}\\sup _{\\xi }\\, \\langle \\xi \\rangle ^{(- \\alpha + s+ 1)\\beta }\\Big (\\big |\\mathbf {D}^{\\mathbf {s}}_{\\xi }\\widehat{(\\nabla _{\\eta } b)}(\\theta , \\xi )\\big |\\Big )\\leqslant C{\\,\\vrule depth3pt height9pt width1pt}\\, b{\\,\\vrule depth3pt height9pt width1pt}\\,^{( \\alpha )}_{l, s+ 1} \\langle \\eta \\rangle ^{| \\alpha - s- 1|\\beta }|\\eta |, \\ s= |\\mathbf {s}|, $ with a constant $C$ depending only on $ \\alpha , s$ , and $\\beta $ .", "The estimate (REF ) implies that for all $\\eta $ with $|\\eta |\\leqslant C$ we have a uniform bound ${\\,\\vrule depth3pt height9pt width1pt}\\, \\nabla _{\\eta } b{\\,\\vrule depth3pt height9pt width1pt}\\,^{( \\alpha -1)}_{l, s}\\leqslant C {\\,\\vrule depth3pt height9pt width1pt}\\, b{\\,\\vrule depth3pt height9pt width1pt}\\,^{( \\alpha )}_{l, s+1}|\\eta |.$ Now we define the PDO $\\operatorname{{Op}}(b)$ in the usual way: $\\operatorname{{Op}}(b)u(\\mathbf {x}) = (2\\pi )^{-d/2} \\int b(\\mathbf {x}, \\xi )e^{i\\xi \\mathbf {x}} (\\mathcal {F}u)(\\xi ) d\\xi ,$ the integral being over $ \\mathbb {R}^d$ .", "Under the condition $b\\in \\mathbf {S}_ \\alpha $ the integral on the r.h.s.", "is clearly finite for any $u$ from the Schwarz class $\\textup {{\\textsf {S}}}( \\mathbb {R}^d)$ .", "Moreover, the property $b\\in \\mathbf {S}_0$ guarantees the boundedness of $\\operatorname{{Op}}(b)$ in $\\textup {{\\textsf {L}}}_{2}( \\mathbb {R}^d)$ , see Proposition REF .", "Unless otherwise stated, from now on $\\textup {{\\textsf {S}}}( \\mathbb {R}^d)$ is taken as a natural domain for all PDO's when they act in $\\textup {{\\textsf {L}}}_{2}( \\mathbb {R}^d)$ .", "Applying the standard regularization procedures to definition (REF ) (see, e.g., [7]), we can also consider the action of $\\operatorname{{Op}}(b)$ on the exponentials $\\mathbf {e}_{\\nu }$ , $\\nu \\in \\mathbb {R}^d$ .", "Namely, we have $\\operatorname{{Op}}(b)\\mathbf {e}_{\\nu }=\\sum _{\\theta \\in \\hat{\\Theta }}\\hat{b}(\\theta , \\nu )\\mathbf {e}_{\\nu +\\theta }.$ This action can be extended by linearity to all quasi-periodic functions (i.e.", "finite linear combinations of $\\mathbf {e}_{\\nu }$ with different $\\nu $ ).", "By taking the closure, we can extend this action of $\\operatorname{{Op}}(b)$ to the Besicovitch space $\\textup {{\\textsf {B}}}_2( \\mathbb {R}^d)$ .", "This is the space of all formal sums $\\sum _{j=1}^\\infty a_{j}\\mathbf {e}_{\\theta _j}(\\mathbf {x}), \\quad \\textrm {with}\\quad \\sum _{j=1}^\\infty |a_{j}|^2<+\\infty .$ It is known (see [7]) that the spectra of $\\operatorname{{Op}}(b)$ acting in $\\textup {{\\textsf {L}}}_{2}( \\mathbb {R}^d)$ and $\\textup {{\\textsf {B}}}_2( \\mathbb {R}^d)$ are the same, although the types of the spectra can be entirely different.", "It is very convenient, when working with the gauge transform constructions, to assume that all the operators involved act in $\\textup {{\\textsf {B}}}_2( \\mathbb {R}^d)$ , although in the end we will return to operators acting in $\\textup {{\\textsf {L}}}_{2}( \\mathbb {R}^d)$ .", "This trick (working with operators acting in $\\textup {{\\textsf {B}}}_2( \\mathbb {R}^d)$ ) is similar to working with fibre operators in the periodic case in the sense that we can freely consider the action of an operator on one, or finitely many, exponentials (REF ), despite the fact that these exponentials do not belong to our original function space.", "Moreover, if the order $\\alpha =0$ then by continuity this action can be extended to all of $\\textup {{\\textsf {B}}}_2( \\mathbb {R}^d)$ , and the extension has the same norm as $\\operatorname{{Op}}(b)$ acting in $\\textup {{\\textsf {L}}}_{2}$ (see [7]).", "Thus, in what follows, when we speak about a pseudo-differential operator with almost-periodic symbol acting in $\\textup {{\\textsf {B}}}_2$ , we mean that its domain is either whole $\\textup {{\\textsf {B}}}_2$ (when the order is non-positive), or the space of all quasi-periodic functions (for operators with positive order).", "And, when we make a statement about the norm of a pseudo-differential operator with almost-periodic symbol, we will not specify whether the operator acts in $\\textup {{\\textsf {L}}}_{2}( \\mathbb {R}^d)$ or $\\textup {{\\textsf {B}}}_2( \\mathbb {R}^d)$ , since these norms are the same." ], [ "Some basic results on the calculus of almost-periodic PDO's", "We begin by listing some elementary results for almost-periodic PDO's.", "The proofs are very similar (with obvious changes) to the proof of analogous statements in [9].", "Proposition 4.1 Suppose that $b$ satisfies (REF ) and that ${\\,\\vrule depth3pt height9pt width1pt}\\, b{\\,\\vrule depth3pt height9pt width1pt}\\,^{(0)}_{0, 0}<\\infty $ .", "Then $\\operatorname{{Op}}(b)$ is bounded in both $\\textup {{\\textsf {L}}}_{2}( \\mathbb {R}^d)$ and $\\textup {{\\textsf {B}}}_2( \\mathbb {R}^d)$ and $\\big \\Vert \\operatorname{{Op}}(b)\\big \\Vert \\leqslant {\\,\\vrule depth3pt height9pt width1pt}\\, b {\\,\\vrule depth3pt height9pt width1pt}\\,^{(0)}_{0, 0}$ .", "In what follows, if we need to calculate a product of two (or more) operators with some symbols $b_j\\in \\mathbf {S}_{ \\alpha _j}(\\hat{\\Theta }_j)$ we will always consider that $b_j\\in \\mathbf {S}_{ \\alpha _j}(\\sum _j\\hat{\\Theta }_j)$ where, of course, all extra terms are assumed to have zero coefficients in front of them.", "Since $\\operatorname{{Op}}(b) u\\in \\textup {{\\textsf {S}}}( \\mathbb {R}^d)$ for any $b\\in \\mathbf {S}_{ \\alpha }$ and $u\\in \\textup {{\\textsf {S}}}( \\mathbb {R}^d)$ , the product $\\operatorname{{Op}}(b) \\operatorname{{Op}}(g)$ , $b\\in \\mathbf {S}_{ \\alpha }(\\hat{\\Theta }_1), g\\in \\mathbf {S}_{\\gamma }(\\hat{\\Theta }_2)$ , is well defined on $\\textup {{\\textsf {S}}}( \\mathbb {R}^d)$ .", "A straightforward calculation leads to the following formula for the symbol $b\\circ g$ of the product $\\operatorname{{Op}}(b)\\operatorname{{Op}}(g)$ : $(b\\circ g)(\\mathbf {x}, \\xi ) = \\sum _{\\theta \\in \\hat{\\Theta }_1,\\, \\phi \\in \\hat{\\Theta }_2}\\hat{b}(\\theta , \\xi +\\phi ) \\hat{g}(\\phi , \\xi ) e^{i(\\theta +\\phi )\\mathbf {x}},$ and hence $\\widehat{(b\\circ g)}(\\chi , \\xi ) =\\sum _{\\theta +\\phi = \\chi } \\hat{b} (\\theta , \\xi +\\phi ) \\hat{g}(\\phi ,\\xi ),\\ \\chi \\in \\hat{\\Theta }_1+\\hat{\\Theta }_2,\\ \\xi \\in \\mathbb {R}^d.$ We have Proposition 4.2 Let $b\\in \\mathbf {S}_{ \\alpha }(\\hat{\\Theta }_1)$ , $g\\in \\mathbf {S}_{\\gamma }(\\hat{\\Theta }_2)$ .", "Then $b\\circ g\\in \\mathbf {S}_{ \\alpha +\\gamma }(\\hat{\\Theta }_1+\\hat{\\Theta }_2)$ and ${\\,\\vrule depth3pt height9pt width1pt}\\, b\\circ g{\\,\\vrule depth3pt height9pt width1pt}\\,^{( \\alpha +\\gamma )}_{l,s} \\leqslant C {\\,\\vrule depth3pt height9pt width1pt}\\, b{\\,\\vrule depth3pt height9pt width1pt}\\,^{( \\alpha )}_{l,s} {\\,\\vrule depth3pt height9pt width1pt}\\, g{\\,\\vrule depth3pt height9pt width1pt}\\,^{(\\gamma )}_{l+(| \\alpha |+s)\\beta ,s},$ with the constant $C$ depending only on $l$ , $\\alpha $ , and $s$ .", "We are also interested in the estimates for symbols of commutators.", "For PDO's $A, \\Psi _l, \\ l = 1, 2, \\dots ,N$ , denote $\\operatorname{{ad}}(A; \\Psi _1, \\Psi _2, \\dots , \\Psi _N):= i\\bigl [\\operatorname{{ad}}(A; \\Psi _1, \\Psi _2, \\dots , \\Psi _{N-1}), \\Psi _N\\bigr ],\\\\\\operatorname{{ad}}(A; \\Psi ):= i[A, \\Psi ],\\quad \\operatorname{{ad}}^N(A; \\Psi ):= \\operatorname{{ad}}(A; \\Psi , \\Psi , \\dots , \\Psi ),\\quad \\operatorname{{ad}}^0(A; \\Psi ):= A.$ For the sake of convenience, we use the notation $\\operatorname{{ad}}(a; \\psi _1, \\psi _2, \\dots , \\psi _N)$ and $\\operatorname{{ad}}^N(a, \\psi )$ for the symbols of multiple commutators.", "Let $\\operatorname{{supp}}\\hat{b}:= \\big \\lbrace \\theta \\in \\mathbb {R}^d: \\hat{b}(\\theta , \\cdot )\\lnot \\equiv 0\\big \\rbrace .$ It follows from (REF ) that the Fourier coefficients of the symbol $\\operatorname{{ad}}(b,g)$ are given by $\\widehat{\\operatorname{{ad}}(b, g)}(\\chi , \\xi )= i\\!\\!\\!\\sum _{\\theta \\in (\\operatorname{{supp}}\\hat{b})\\cup (\\chi - \\operatorname{{supp}}\\hat{g})}\\!\\!\\!\\bigl [\\widehat{(\\nabla _{\\chi - \\theta } b)}(\\theta , \\xi )\\hat{g}(\\chi - \\theta , \\xi )- \\hat{b}(\\theta , \\xi )\\widehat{(\\nabla _{\\theta }g)}(\\chi - \\theta , \\xi )\\bigr ].$ Proposition 4.3 Let $b\\in \\mathbf {S}_{ \\alpha }(\\hat{\\Theta })$ and $g_j\\in \\mathbf {S}_{\\gamma _j}(\\hat{\\Theta }_j)$ , $j = 1, 2, \\dots , N$ .", "Then $\\operatorname{{ad}}(b; g_1, \\dots , g_N) \\in \\mathbf {S}_{\\gamma }(\\hat{\\Theta }+\\sum _j\\hat{\\Theta }_j)$ with $\\gamma = \\alpha +\\sum _{j=1}^N(\\gamma _j-1),$ and ${\\,\\vrule depth3pt height9pt width1pt}\\, \\operatorname{{ad}}(b; g_1, \\dots , g_N){\\,\\vrule depth3pt height9pt width1pt}\\,^{(\\gamma )}_{l,s} \\leqslant C {\\,\\vrule depth3pt height9pt width1pt}\\, b{\\,\\vrule depth3pt height9pt width1pt}\\,^{( \\alpha )}_{p,s +N}\\prod _{j =1}^N {\\,\\vrule depth3pt height9pt width1pt}\\, g_j{\\,\\vrule depth3pt height9pt width1pt}\\,^{(\\gamma _j)}_{p, s +N -j +1},$ where $C$ and $p$ depend on $l, s, N, \\alpha $ and $\\gamma _j$ ." ], [ "Resonant regions", "We now define resonant regions and mention some of their properties.", "This material is essentially identical to Section 5 of [5], where the reader can find the proofs of all the statements of this section.", "Recall the definition of the set $\\Theta = \\widetilde{\\Theta }$ as well as of the quasi-lattice subspaces from Section .", "As before, by $\\Theta _{\\tilde{k}}$ we denote the algebraic sum of $\\tilde{k}$ copies of $\\Theta $ ; remember that we consider $\\tilde{k}$ fixed.", "We also put $\\Theta ^{\\prime }_{\\tilde{k}}:=\\Theta _{\\tilde{k}}\\setminus \\lbrace 0\\rbrace $ .", "For each $\\mathfrak {V}\\in \\mathcal {V}$ we put $S_{\\mathfrak {V}}:= \\big \\lbrace \\xi \\in \\mathfrak {V},\\ |\\xi |=1\\big \\rbrace $ .", "For each non-zero $\\theta \\in \\mathbb {R}^d$ we put $\\mathbf {n}(\\theta ):=\\theta |\\theta |^{-1}$ .", "Let $\\mathfrak {V}\\in \\mathcal {V}_m$ .", "We say that $\\mathfrak {F}$ is a flag generated by $\\mathfrak {V}$ , if $\\mathfrak {F}$ is a sequence $\\mathfrak {V}_j\\in \\mathcal {V}_j$ ($j= 0, 1, \\dots , m$ ) such that $\\mathfrak {V}_{j- 1}\\subset \\mathfrak {V}_j$ and $\\mathfrak {V}_m= \\mathfrak {V}$ .", "We say that $\\lbrace \\nu _j\\rbrace _{j= 1}^m$ is a sequence generated by $\\mathfrak {F}$ if $\\nu _j\\in \\mathfrak {V}_j\\ominus \\mathfrak {V}_{j- 1}$ and $\\Vert \\nu _j\\Vert = 1$ (obviously, this condition determines each $\\nu _j$ up to multiplication by $-1$ ).", "We denote by $\\mathcal {F}(\\mathfrak {V})$ the collection of all flags generated by $\\mathfrak {V}$ .", "We put $L_j:= \\rho _n^{\\alpha _j},$ recall (REF ).", "Let $\\theta \\in \\Theta ^{\\prime }_{\\tilde{k}}$ .", "The resonant region generated by $\\theta $ is defined as $\\Lambda (\\theta ):= \\Big \\lbrace \\xi \\in \\mathbb {R}^d,\\ \\big |\\langle \\xi , \\mathbf {n}(\\theta )\\rangle \\big |\\leqslant L_1\\Big \\rbrace .$ Suppose, $\\mathfrak {F}\\in \\mathcal {F}(\\mathfrak {V})$ is a flag and $\\lbrace \\nu _j\\rbrace _{j= 1}^m$ is a sequence generated by $\\mathfrak {F}$ .", "We define $\\Lambda (\\mathfrak {F}):= \\Big \\lbrace \\xi \\in \\mathbb {R}^d,\\ \\big |\\langle \\xi ,\\nu _j\\rangle \\big |\\leqslant L_j\\Big \\rbrace .$ If $\\dim \\mathfrak {V}= 1$ , definition (REF ) is reduced to (REF ).", "Obviously, if $\\mathfrak {F}_1\\subset \\mathfrak {F}_2$ , then $\\Lambda (\\mathfrak {F}_2)\\subset \\Lambda (\\mathfrak {F}_1)$ .", "Suppose, $\\mathfrak {V}\\in \\mathcal {V}_j$ .", "We denote $\\Xi _1(\\mathfrak {V}) :=\\cup _{\\mathfrak {F}\\in \\mathcal {F}(\\mathfrak {V})}\\Lambda (\\mathfrak {F}).$ Note that $\\Xi _1(\\mathfrak {X})= \\mathbb {R}^d$ and $\\Xi _1(\\mathfrak {V})= \\Lambda (\\theta )$ if $\\mathfrak {V}\\in \\mathcal {V}_1$ is spanned by $\\theta $ .", "Finally, we put $\\Xi (\\mathfrak {V}):= \\Xi _1(\\mathfrak {V})\\setminus \\big (\\cup _{\\mathfrak {U}\\supsetneq \\mathfrak {V}}\\Xi _1(\\mathfrak {U})\\big )= \\Xi _1(\\mathfrak {V})\\setminus \\big (\\cup _{\\mathfrak {U}\\supsetneq \\mathfrak {V}}\\cup _{\\mathfrak {F}\\in \\mathcal {F}(\\mathfrak {U})}\\Lambda (\\mathfrak {F})\\big ).$ We call $\\Xi (\\mathfrak {V})$ the resonance region generated by $\\mathfrak {V}$ .", "Very often, the region $\\Xi (\\mathfrak {X})$ is called the non-resonance region.", "We, however, will omit using this terminology since we will treat all regions $\\Xi (\\mathfrak {V})$ in the same way.", "The first set of properties follows immediately from the definitions.", "Lemma 5.1 (i) We have $\\cup _{\\mathfrak {V}\\in \\mathcal {V}}\\Xi (\\mathfrak {V}) = \\mathbb {R}^d.$ (ii) $\\xi \\in \\Xi _1(\\mathfrak {V})$ iff $\\xi _{\\mathfrak {V}}\\in \\Omega (\\mathfrak {V})$ , where $\\Omega (\\mathfrak {V})\\subset \\mathfrak {V}$ is a certain bounded set (more precisely, $\\Omega (\\mathfrak {V}) =\\Xi _1(\\mathfrak {V})\\cap \\mathfrak {V}\\subset \\mathcal {B}(m L_m)$ if $\\dim \\mathfrak {V}=m$ ).", "(iii) $\\Xi _1( \\mathbb {R}^d) =\\Xi ( \\mathbb {R}^d)$ is a bounded set, $\\Xi ( \\mathbb {R}^d)\\subset \\mathcal {B}(d L_d)$ ; all other sets $\\Xi _1(\\mathfrak {V})$ are unbounded.", "Now we move to slightly less obvious properties.", "From now on we always assume that $\\rho _0$ (and thus $\\rho _n$ ) is sufficiently large.", "We also assume, as we always do, that the value of $k$ is sufficiently large so that, for example, $L_j\\rho _n^{0+}< L_{j+ 1}$ .", "Lemma 5.2 Let $\\mathfrak {V}, \\mathfrak {U}\\in \\mathcal {V}$ .", "Then $\\big (\\Xi _1(\\mathfrak {V})\\cap \\Xi _1(\\mathfrak {U})\\big )\\subset \\Xi _1(\\mathfrak {W})$ , where $\\mathfrak {W}:= \\mathfrak {V}+ \\mathfrak {U}$ (algebraic sum).", "Corollary 5.3 (i) We can re-write definition (REF ) like this: $\\Xi (\\mathfrak {V}) :=\\Xi _1(\\mathfrak {V})\\setminus \\big (\\cup _{\\mathfrak {U}\\lnot \\subset \\mathfrak {V}}\\Xi _1(\\mathfrak {U})\\big ).$ (ii) If $\\mathfrak {V}\\ne \\mathfrak {U}$ , then $\\Xi (\\mathfrak {V})\\cap \\Xi (\\mathfrak {U}) =\\emptyset $ .", "(iii) We have $ \\mathbb {R}^d =\\sqcup _{\\mathfrak {V}\\in \\mathcal {V}}\\Xi (\\mathfrak {V})$ (the disjoint union).", "Lemma 5.4 Let $\\mathfrak {V}\\in \\mathcal {V}_m$ and $\\mathfrak {V}\\subset \\mathfrak {W}\\in \\mathcal {V}_{m +1}$ .", "Let $\\mu $ be (any) unit vector from $\\mathfrak {W}\\ominus \\mathfrak {V}$ .", "Then, for $\\xi \\in \\Xi _1(\\mathfrak {V})$ , we have $\\xi \\in \\Xi _1(\\mathfrak {W})$ if and only if the estimate $\\big |\\langle \\xi ,\\mu \\rangle \\big |= \\big |\\langle \\xi _{\\mathfrak {V}^{\\perp }},\\mu \\rangle \\big |\\leqslant L_{m +1}$ holds.", "Lemma 5.5 We have $\\Xi _1(\\mathfrak {V})\\cap \\cup _{\\mathfrak {U}\\supsetneq \\mathfrak {V}}\\Xi _1(\\mathfrak {U})= \\Xi _1(\\mathfrak {V})\\cap \\cup _{\\mathfrak {W}\\supsetneq \\mathfrak {V}, \\ \\dim \\mathfrak {W}= 1+ \\dim \\mathfrak {V}}\\Xi _1(\\mathfrak {W}).$ Corollary 5.6 We can re-write (REF ) as $\\Xi (\\mathfrak {V}):=\\Xi _1(\\mathfrak {V})\\setminus \\big (\\cup _{\\mathfrak {W}\\supsetneq \\mathfrak {V}, \\dim \\mathfrak {W}=1 +\\dim \\mathfrak {V}}\\Xi _1(\\mathfrak {W})\\big ).$ Lemma 5.7 Let $\\mathfrak {V}\\in \\mathcal {V}$ and $\\theta \\in \\Theta _{\\tilde{k}}$ .", "Suppose that $\\xi \\in \\Xi (\\mathfrak {V})$ and both points $\\xi $ and $\\xi +\\theta $ are inside $\\Lambda (\\theta )$ .", "Then $\\theta \\in \\mathfrak {V}$ and $\\xi +\\theta \\in \\Xi (\\mathfrak {V})$ .", "Definition 5.8 Let $\\theta , \\theta _1, \\theta _2, \\dots , \\theta _l$ be some vectors from $\\Theta ^{\\prime }_{\\tilde{k}}$ , which are not necessarily distinct.", "We say that two vectors $\\xi , \\eta \\in \\mathbb {R}^d$ are $\\theta $ -resonant congruent if both $\\xi $ and $\\eta $ are inside $\\Lambda (\\theta )$ and $(\\xi - \\eta ) =l\\theta $ with $l\\in \\mathbb {Z}$ .", "In this case we write $\\xi \\leftrightarrow \\eta \\mod {\\theta }$ .", "For each $\\xi \\in \\mathbb {R}^d$ we denote by $\\Upsilon _{\\theta }(\\xi )$ the set of all points which are $\\theta $ -resonant congruent to $\\xi $ .", "For $\\theta \\ne \\mathbf {0}$ we say that $\\Upsilon _{\\theta }(\\xi ) = \\varnothing $ if $\\xi \\notin \\Lambda (\\theta )$ .", "We say that $\\xi $ and $\\eta $ are $\\theta _1, \\theta _2, \\dots , \\theta _l$ -resonant congruent, if there exists a sequence $\\xi _j\\in \\mathbb {R}^d, j=0, 1, \\dots , l$ such that $\\xi _0 = \\xi $ , $\\xi _l = \\eta $ , and $\\xi _j \\in \\Upsilon _{\\theta _j}(\\xi _{j-1})$ for $j =1, 2, \\dots , l$ .", "We say that $\\eta \\in \\mathbb {R}^d$ and $\\xi \\in \\mathbb {R}^d$ are resonant congruent, if either $\\xi =\\eta $ or $\\xi $ and $\\eta $ are $\\theta _1, \\theta _2, \\dots , \\theta _l$ -resonant congruent with some $\\theta _1, \\theta _2, \\dots , \\theta _l \\in \\Theta _{\\tilde{k}}^{\\prime }$ .", "The set of all points, resonant congruent to $\\xi $ , is denoted by $\\Upsilon (\\xi )$ .", "For points $\\eta \\in \\Upsilon (\\xi )$ (note that this condition is equivalent to $\\xi \\in \\Upsilon (\\eta )$ ) we write $\\eta \\leftrightarrow \\xi $ .", "Note that $\\Upsilon (\\xi )= \\lbrace \\xi \\rbrace $ for any $\\xi \\in \\Xi (\\mathfrak {X})$ .", "Now Lemma REF immediately implies Corollary 5.9 For each $\\xi \\in \\Xi (\\mathfrak {V})$ we have $\\Upsilon (\\xi )\\subset \\Xi (\\mathfrak {V})$ and thus $\\Xi (\\mathfrak {V})= \\sqcup _{\\xi \\in \\Xi (\\mathfrak {V})}\\Upsilon (\\xi ).$ Lemma 5.10 The diameter of $\\Upsilon (\\xi )$ is bounded above by $mL_m$ , if $\\xi \\in \\Xi (\\mathfrak {V})$ , $\\mathfrak {V}\\in \\mathcal {V}_m$ .", "Lemma 5.11 For each $\\xi \\in \\Xi (\\mathfrak {V}),\\ \\mathfrak {V}\\ne \\mathbb {R}^d$ , the set $\\Upsilon (\\xi )$ is finite, and $\\operatorname{{card}}\\Upsilon (\\xi )$ is bounded uniformly in $\\xi \\in \\mathbb {R}^d\\setminus \\Xi ( \\mathbb {R}^d)$ ." ], [ "Description of the approach", "We first prove (REF ) assuming that the symbol $b$ of $B$ is replaced by $\\tilde{b}$ which satisfies (REF ).", "In particular, it belongs to the class $\\mathbf {S}_\\alpha $ .", "At the end, in Section , we will use (REF ) to show that Theorem REF holds as stated.", "For any set $\\mathcal {C}\\subset \\mathbb {R}^d$ by $\\mathcal {P}(\\mathcal {C})$ we denote the orthogonal projection onto $\\mathrm {span}\\lbrace \\mathbf {e}_{\\xi }\\rbrace _{\\xi \\in \\mathcal {C}}$ in $\\textup {{\\textsf {B}}}_2( \\mathbb {R}^d)$ and by $\\mathcal {P}^{L}(\\mathcal {C})$ the same projection considered in $\\textup {{\\textsf {L}}}_{2}( \\mathbb {R}^d)$ , i.e.", "$\\mathcal {P}^{L}(\\mathcal {C})=\\mathcal {F}^*\\operatorname{\\mathbb {1}}_{\\mathcal {C}}\\mathcal {F},$ where $\\mathcal {F}$ is the Fourier transform and $\\operatorname{\\mathbb {1}}_{\\mathcal {C}}$ is the operator of multiplication by the indicator function of $\\mathcal {C}$ .", "Obviously, $\\mathcal {P}^{L}(\\mathcal {C})$ is a well-defined (respectively, non-zero) projection iff $\\mathcal {C}$ is measurable (respectively, has non-zero measure).", "Let us fix sufficiently large $n$ , and denote (recall that $\\lambda _n =\\rho _n^{2w}$ ) $\\mathcal {X}_n:= \\Big \\lbrace \\xi \\in \\mathbb {R}^d,\\,|\\xi |^{2w}\\in \\big [(5/6)^{2w}\\lambda _n, 5^{2w}\\lambda _n\\big ]\\Big \\rbrace .$ We also put $\\mathcal {A}= \\mathcal {A}_n:= \\cup _{\\xi \\in \\mathcal {X}_n}\\Upsilon (\\xi ).$ Lemma REF implies that, if $\\rho _0$ is big enough, $\\textrm {for each \\xi \\in \\mathcal {A} we have |\\xi |^{2w}\\in \\big [(2/3)^{2w}\\lambda _n, 6^{2w}\\lambda _n\\big ].", "}$ In particular, we have $\\mathcal {A}\\cap \\Xi ( \\mathbb {R}^d)= \\varnothing .$ Let us define $\\hat{\\mathcal {A}}:=\\big \\lbrace \\xi \\notin \\mathcal {A},\\ |\\xi |^{2w} <\\lambda _n\\big \\rbrace $ and $\\check{\\mathcal {A}}:=\\big \\lbrace \\xi \\notin \\mathcal {A},\\ |\\xi |^{2w}>\\lambda _n\\big \\rbrace .$ We now plan to apply the gauge transform as in Sections 8 and 9 of [5] to the operator $H$ .", "The details of this procedure will be explained in Sections  and ; here, we just mention that we are going to introduce two operators: $H_1$ and $H_2$ .", "The operator $H_1$ is unitary equivalent to $H$ : $H_1= U^{-1}HU$ , where $U=e^{i\\Psi }$ with a bounded pseudo-differential operator $\\Psi $ with almost-periodic coefficients (then Lemma REF implies that the densities of states of $H$ and $H_1$ are the same).", "Moreover, $H_1= H_2+ R_{\\tilde{k}}$ , where $\\Vert R_{\\tilde{k}}\\Vert \\lesssim \\rho _n^{-M+ 2w- d}$ and $H_2= (-\\Delta )^{w}+ W_{\\tilde{k}}$ is a self-adjoint pseudo-differential operator with symbol $|\\xi |^{2w}+ w_{\\tilde{k}}(\\mathbf {x}, \\xi )$ which satisfies the following property: $ \\hat{w}_{\\tilde{k}}(\\theta , \\xi )= 0, \\ \\mathrm {if} \\ \\big (\\xi \\notin \\Lambda (\\theta )\\ \\&\\ \\xi \\in \\mathcal {A}\\big ), \\ \\mathrm {or} \\ \\big (\\xi +\\theta \\notin \\Lambda (\\theta )\\ \\&\\ \\xi \\in \\mathcal {A}\\big ), \\ \\mathrm {or}\\ (\\theta \\notin \\Theta _{\\tilde{k}}).", "$ We can now use a simple statement which follows from Lemma REF and Remark REF : Lemma 6.1 Suppose, $H_1$ and $H_2$ are two elliptic self-adjoint pseudo-differential operators with almost-periodic coefficients such that $\\Vert H_1- H_2\\Vert \\lesssim \\rho _n^{-M+2w- d}$ .", "Suppose that $N(H_2; \\rho ^{2w})$ satisfies asymptotic expansion (REF ).", "Then $N(H_1; \\rho ^{2w})$ also satisfies (REF ) with the same coefficients.", "This means that it is enough to establish the asymptotic expansion (REF ) for the operator $H_2$ instead of $H$ .", "Condition (REF ) implies that for each $\\xi \\in \\mathcal {A}$ the subspace $\\mathcal {P}\\big (\\Upsilon (\\xi )\\big )\\textup {{\\textsf {B}}}_2( \\mathbb {R}^d)$ is an invariant subspace of $H_2$ ; its dimension is finite by Lemma REF .", "We put $H_2(\\xi ):= H_2|_{\\mathcal {P}(\\Upsilon (\\xi ))\\textup {{\\textsf {B}}}_2( \\mathbb {R}^d)}.$ Note that the subspaces $\\mathcal {P}(\\hat{\\mathcal {A}})\\textup {{\\textsf {B}}}_2( \\mathbb {R}^d)$ and $\\mathcal {P}(\\check{\\mathcal {A}})\\textup {{\\textsf {B}}}_2( \\mathbb {R}^d)$ are invariant as well; by $H_2(\\hat{\\mathcal {A}})$ and $H_2(\\check{\\mathcal {A}})$ we denote the restrictions of $H_2$ to these subspaces; we also denote by $H_2(\\mathcal {A})$ the restriction of $H_2$ to $\\mathcal {P}(\\mathcal {A})\\textup {{\\textsf {B}}}_2( \\mathbb {R}^d)$ .", "If we consider the operator $H_2$ acting in $\\textup {{\\textsf {L}}}_{2}( \\mathbb {R}^d)$ , then $\\mathcal {P}^L(\\hat{\\mathcal {A}})\\textup {{\\textsf {L}}}_{2}( \\mathbb {R}^d)$ , $\\mathcal {P}^L(\\check{\\mathcal {A}})\\textup {{\\textsf {L}}}_{2}( \\mathbb {R}^d)$ , and $\\mathcal {P}^L(\\mathcal {A})\\textup {{\\textsf {L}}}_{2}( \\mathbb {R}^d)$ are still invariant subspaces.", "It follows from (REF ) – (REF ) that $UH_2(\\hat{\\mathcal {A}})U^*< (5/6)^{2w}\\lambda _n I$ and $UH_2(\\check{\\mathcal {A}})U^*> 5^{2w}\\lambda _n I$ .", "For each $\\xi \\in \\mathcal {A}$ the operator $H_2(\\xi )$ is a finite-dimensional self-adjoint operator, so its spectrum is purely discrete; we denote its eigenvalues (counting multiplicities) by $\\lambda _1(\\xi )\\leqslant \\lambda _2(\\xi )\\leqslant \\dots \\leqslant \\lambda _{\\operatorname{{card}}\\Upsilon (\\xi )}(\\xi )$ .", "Next, we list all points $\\eta \\in \\Upsilon (\\xi )$ in increasing order of their absolute values; thus, we have put into correspondence to each point $\\eta \\in \\Upsilon (\\xi )$ a natural number $t= t(\\eta )$ so that $t(\\eta )< t(\\eta ^{\\prime })$ if $|\\eta |< |\\eta ^{\\prime }|$ .", "If two points $\\eta = (\\eta _1,\\dots ,\\eta _d)$ and $\\eta ^{\\prime }= (\\eta ^{\\prime }_1, \\dots , \\eta ^{\\prime }_d)$ have the same absolute values, we put them in the lexicographic order of their coordinates, i.e.", "we say that $t(\\eta )< t(\\eta ^{\\prime })$ if $\\eta _1< \\eta ^{\\prime }_1$ , or $\\eta _1= \\eta ^{\\prime }_1$ and $\\eta _2< \\eta ^{\\prime }_2$ , etc.", "Now we define the map $g: \\mathcal {A}\\rightarrow \\mathbb {R}$ which to each point $\\eta \\in \\mathcal {A}$ brings into correspondence the number $\\lambda _{t(\\eta )}\\big (\\Upsilon (\\eta )\\big )$ .", "This map is an injection from $\\mathcal {A}$ onto the set of eigenvalues of $H_2$ , counting multiplicities (recall that we consider the operator $H_2$ acting in $\\textup {{\\textsf {B}}}_2( \\mathbb {R}^d)$ , so there is nothing miraculous about its spectrum consisting of eigenvalues and their limit points).", "Moreover, all eigenvalues of $H_2$ inside the interval $\\big [(7/8)^{2w}\\lambda _n, (9/2)^{2w}\\lambda _n\\big ]$ have a pre-image under $g$ .", "We define $g(\\xi ):= |\\xi |^{2w}, \\quad \\textrm {for} \\quad \\xi \\in \\mathbb {R}^d\\setminus \\mathcal {A}.$ Arguments similar to the ones used in [6] show that $g$ is a measurable function.", "We introduce $\\mathcal {G}_{\\lambda }:= \\big \\lbrace \\xi \\in \\mathbb {R}^d,\\,g(\\xi )\\leqslant \\lambda \\big \\rbrace .$ Lemma 6.2 For $\\lambda \\in [\\lambda _n, 4^{2w}\\lambda _n]$ being a continuity point of $N(\\lambda ; H_2)$ we have: $N(\\lambda ; H_2)= (2\\pi )^{-d}\\operatorname{{vol}}\\mathcal {G}_{\\lambda }.$ Since points of continuity of $N(\\lambda )$ are dense, the asymptotic expansion proven for such $\\lambda $ can be extended to all $\\lambda \\in [\\lambda _n, 4^{2w}\\lambda _n]$ by taking the limit.", "Thus, our next task is to compute $\\operatorname{{vol}}\\mathcal {G}_{\\lambda }$ .", "Let us put $\\mathcal {A}^+(\\rho ):= \\big \\lbrace \\xi \\in \\mathbb {R}^d,\\,g(\\xi )<\\rho ^{2w}<|\\xi |^{2w}\\big \\rbrace $ and $\\mathcal {A}^-(\\rho ):= \\big \\lbrace \\xi \\in \\mathbb {R}^d,\\,|\\xi |^{2w}<\\rho ^{2w}<g(\\xi )\\big \\rbrace .$ Lemma 6.3 $\\operatorname{{vol}}(\\mathcal {G}_{\\lambda })= \\omega _d\\rho ^d+ \\operatorname{{vol}}\\mathcal {A}^+(\\rho )- \\operatorname{{vol}}\\mathcal {A}^-(\\rho ),$ where $\\omega _d$ is the volume of the unit ball in $ \\mathbb {R}^d$ .", "We obviously have $\\mathcal {G}_{\\lambda }= \\mathcal {B}(\\rho )\\cup \\mathcal {A}^+(\\rho )\\setminus \\mathcal {A}^-(\\rho )$ .", "Since $\\mathcal {A}^-(\\rho )\\subset \\mathcal {B}(\\rho )$ and $\\mathcal {A}^+(\\rho )\\cap \\mathcal {B}(\\rho )=\\emptyset $ , this implies (REF ).", "Remark 6.4 Properties of the mapping $g$ imply that $\\mathcal {A}^+(\\rho )\\cup \\mathcal {A}^-(\\rho )\\subset \\mathcal {A}$ .", "Thus, in order to compute $N(\\lambda )$ , we need to analyze the behavior of $g$ only inside $\\mathcal {A}$ .", "We will compute volumes of $\\mathcal {A}^{\\pm }(\\rho )$ by means of integrating their characteristic functions in a specially chosen set of coordinates.", "The next section is devoted to introducing these coordinates." ], [ "Coordinates", "In this section, we do some preparatory work before computing $\\operatorname{{vol}}\\mathcal {A}^{\\pm }(\\rho )$ .", "Namely, we are going to introduce a convenient set of coordinates in $\\Xi (\\mathfrak {V})$ .", "Let $\\mathfrak {V}\\in \\mathcal {V}_m$ be fixed; since $\\mathcal {A}^{\\pm }(\\rho )\\cap \\Xi ( \\mathbb {R}^d)= \\emptyset $ , we will assume that $m<d$ .", "Then, as we have seen, $\\xi \\in \\Xi _1(\\mathfrak {V})$ if and only if $\\xi _{\\mathfrak {V}}\\in \\Omega (\\mathfrak {V})$ .", "Let $\\lbrace \\mathfrak {U}_j\\rbrace $ be a collection of all subspaces $\\mathfrak {U}_j\\in \\mathcal {V}_{m+ 1}$ such that each $\\mathfrak {U}_j$ contains $\\mathfrak {V}$ .", "Let $\\mu _j= \\mu _j(\\mathfrak {V})$ be (any) unit vector from $\\mathfrak {U}_j\\ominus \\mathfrak {V}$ .", "Then it follows from Lemma REF that for $\\xi \\in \\Xi _1(\\mathfrak {V})$ , we have $\\xi \\in \\Xi _1(\\mathfrak {U}_j)$ if and only if the estimate $\\big |\\langle \\xi , \\mu _j\\rangle \\big |= \\big |\\langle \\xi _{\\mathfrak {V}^{\\perp }}, \\mu _j\\rangle \\big |\\leqslant L_{m+ 1}$ holds.", "Thus, formula (REF ) implies that $\\Xi (\\mathfrak {V})= \\Big \\lbrace \\xi \\in \\mathbb {R}^d,\\ \\xi _{\\mathfrak {V}}\\in \\Omega (\\mathfrak {V})\\ \\&\\ \\forall j \\ \\big |\\langle \\xi _{\\mathfrak {V}^{\\perp }},\\mu _j(\\mathfrak {V})\\rangle \\big | > L_{m+ 1}\\Big \\rbrace .$ The collection $\\big \\lbrace \\mu _j(\\mathfrak {V})\\big \\rbrace $ obviously coincides with $\\big \\lbrace \\mathbf {n}(\\theta _{\\mathfrak {V}^{\\perp }}),\\ \\theta \\in \\Theta _{\\tilde{k}}\\setminus \\mathfrak {V}\\big \\rbrace .$ The set $\\Xi (\\mathfrak {V})$ is, in general, disconnected; it consists of several connected components which we will denote by $\\big \\lbrace \\Xi (\\mathfrak {V})_p\\big \\rbrace _{p=1}^P$ .", "Let us fix a connected component $\\Xi (\\mathfrak {V})_p$ .", "Then for some vectors $\\big \\lbrace \\tilde{\\mu }_j(p)\\big \\rbrace _{j=1}^{J_p}\\subset \\lbrace \\pm \\mu _j\\rbrace $ we have $\\Xi (\\mathfrak {V})_p= \\big \\lbrace \\xi \\in \\mathbb {R}^d,\\ \\xi _{\\mathfrak {V}}\\in \\Omega (\\mathfrak {V})\\ \\&\\ \\forall j \\ \\langle \\xi _{\\mathfrak {V}^{\\perp }},\\tilde{\\mu }_j(p)\\rangle > L_{m +1}\\big \\rbrace ;$ we assume that $\\big \\lbrace \\tilde{\\mu }_j(p)\\big \\rbrace _{j =1}^{J_p}$ is the minimal set with this property, so that each hyperplane $\\big \\lbrace \\xi \\in \\mathbb {R}^d,\\ \\xi _{\\mathfrak {V}}\\in \\Omega (\\mathfrak {V})\\ \\ \\&\\ \\ \\langle \\xi _{\\mathfrak {V}^{\\perp }}, \\tilde{\\mu }_j(p)\\rangle = L_{m+ 1}\\big \\rbrace ,\\ j= 1, \\dots , J_p$ has a non-empty intersection with the boundary of $\\Xi (\\mathfrak {V})_p$ .", "It is not hard to see that $J_p\\geqslant d- m$ .", "Indeed, otherwise $\\Xi (\\mathfrak {V})_p$ would have non-empty intersection with $\\Xi _1(\\mathfrak {V}^{\\prime })$ for some $\\mathfrak {V}^{\\prime }$ , $\\mathfrak {V}\\subsetneq \\mathfrak {V}^{\\prime }$ .", "We also introduce $\\tilde{\\Xi }(\\mathfrak {V})_p:= \\big \\lbrace \\xi \\in \\mathfrak {V}^{\\perp },\\ \\forall j \\ \\langle \\xi ,\\tilde{\\mu }_j(p)\\rangle > 0\\big \\rbrace .$ Note that our assumption that $\\Xi (\\mathfrak {V})_p$ is a connected component of $\\Xi (\\mathfrak {V})$ implies that for any $\\xi \\in \\tilde{\\Xi }(\\mathfrak {V})_p$ and any $\\theta \\in \\Theta _{\\tilde{k}}\\setminus \\mathfrak {V}$ we have $\\langle \\xi ,\\theta \\rangle = \\langle \\xi ,\\theta _{\\mathfrak {V}^{\\perp }}\\rangle \\ne 0.$ We also put $K:= d- m- 1.$ Without loss of generality we may (and will) assume that the number $J_p$ of `defining planes' is the minimal possible, i.e.", "$J_p= K+ 1$ .", "Indeed, the argument presented in Section 11 of [5] explains how to derive the result for arbitrary $\\Xi (\\mathfrak {V})_p$ , assuming we have proved it in the case $J_p= K+ 1$ .", "If $J_p= K+ 1$ , then the set $\\big \\lbrace \\tilde{\\mu }_j(p)\\big \\rbrace _{j= 1}^{K+ 1}$ is linearly independent.", "Let $\\mathbf {a}= \\mathbf {a}(p)$ be a unique point from $\\mathfrak {V}^\\perp $ satisfying the following conditions: $\\langle \\mathbf {a},\\tilde{\\mu }_j(p)\\rangle = L_{m+ 1}$ , $j= 1, \\dots , K+ 1$ .", "Then, since the determinant of the Gram matrix of vectors $\\tilde{\\mu }_j(p)$ is $\\gtrsim \\rho _n^{0-}$ by (REF ), we have $|\\mathbf {a}|\\lesssim L_{m+ 1}\\rho _n^{0+}= \\rho _n^{\\alpha _{m+ 1}+ 0+}.$ We introduce shifted cylindrical coordinates in $\\Xi (\\mathfrak {V})_p$ .", "These coordinates will be denoted by $\\xi = (r; \\mathbf {\\Phi }; \\mathbf {X})$ .", "Here, $\\mathbf {X}= (X_1, \\dots , X_m)$ is an arbitrary set of cartesian coordinates in $\\Omega (\\mathfrak {V})$ .", "These coordinates do not depend on the choice of the connected component $\\Xi (\\mathfrak {V})_p$ .", "The rest of the coordinates $(r, \\mathbf {\\Phi })$ are shifted spherical coordinates in $\\mathfrak {V}^{\\perp }$ , centered at $\\mathbf {a}$ .", "This means that $r(\\xi )= |\\xi _{\\mathfrak {V}^{\\perp }}- \\mathbf {a}|$ and $\\mathbf {\\Phi }= \\mathbf {n}(\\xi _{\\mathfrak {V}^{\\perp }}- \\mathbf {a})\\in S_{\\mathfrak {V}^{\\perp }}.$ More precisely, $\\mathbf {\\Phi }\\in \\mathcal {M}_p$ , where $\\mathcal {M}_p:= \\big \\lbrace \\mathbf {n}(\\xi _{\\mathfrak {V}^{\\perp }}- \\mathbf {a}),\\ \\xi \\in \\Xi (\\mathfrak {V})_p\\big \\rbrace \\subset S_{\\mathfrak {V}^{\\perp }}$ is a $K$ -dimensional spherical simplex with $K+ 1$ sides.", "Note that $\\begin{split}\\mathcal {M}_p&= \\big \\lbrace \\mathbf {n}(\\xi _{\\mathfrak {V}^{\\perp }}-\\mathbf {a}),\\ \\xi \\in \\Xi (\\mathfrak {V})_p\\big \\rbrace = \\big \\lbrace \\mathbf {n}(\\xi _{\\mathfrak {V}^{\\perp }}-\\mathbf {a}),\\ \\forall j \\ \\langle \\xi _{\\mathfrak {V}^{\\perp }},\\tilde{\\mu }_j(p)\\rangle > L_{m+1}\\big \\rbrace \\\\&= \\big \\lbrace \\mathbf {n}(\\eta ),\\ \\eta := \\xi _{\\mathfrak {V}^{\\perp }}-\\mathbf {a}\\in \\mathfrak {V}^\\perp ,\\ \\forall j \\ \\langle \\eta ,\\tilde{\\mu }_j(p)\\rangle > 0\\big \\rbrace = S_{\\mathfrak {V}^{\\perp }}\\cap \\tilde{\\Xi }(\\mathfrak {V})_p.\\end{split}$ We will denote by $d\\mathbf {\\Phi }$ the spherical Lebesgue measure on $\\mathcal {M}_p$ .", "For each non-zero vector $\\mu \\in \\mathfrak {V}^{\\perp }$ , we denote $\\mathcal {W}(\\mu ):= \\big \\lbrace \\eta \\in \\mathfrak {V}^\\perp ,\\ \\langle \\eta ,\\mu \\rangle = 0\\big \\rbrace .$ Thus, the sides of the simplex $\\mathcal {M}_p$ are intersections of $\\mathcal {W}\\big (\\tilde{\\mu }_j(p)\\big )$ with the sphere $S_{\\mathfrak {V}^{\\perp }}$ .", "Each vertex $\\mathbf {v}= \\mathbf {v}_t$ , $t= 1, \\dots , K+ 1$ of $\\mathcal {M}_p$ is an intersection of $S_{\\mathfrak {V}^{\\perp }}$ with $K$ hyperplanes $\\mathcal {W}\\big (\\tilde{\\mu }_j(p)\\big )$ , $j= 1, \\dots , K+ 1$ , $j\\ne t$ .", "This means that $\\mathbf {v}_t$ is a unit vector from $\\mathfrak {V}^{\\perp }$ which is orthogonal to $\\big \\lbrace \\tilde{\\mu }_j(p)\\big \\rbrace $ , $j= 1, \\dots , K+ 1$ , $j\\ne t$ ; this defines $\\mathbf {v}$ up to a multiplication by $-1$ .", "Lemma 7.1 Let $\\mathfrak {U}_1$ and $\\mathfrak {U}_2$ be two strongly distinct subspaces each of which is a linear combination of some of the vectors from $\\big \\lbrace \\tilde{\\mu }_j(p)\\big \\rbrace $ .", "Then the angle between them is not smaller than $s(\\rho _n)$ .", "In particular, all non-zero angles between two sides of any dimensions of $\\mathcal {M}_p$ as well as all the distances between two vertexes $\\mathbf {v}_t$ and $\\mathbf {v}_{\\tau }$ , $t\\ne \\tau $ , are bounded below by $s(\\rho _n)$ .", "Lemma 7.2 Let $p$ be fixed.", "Suppose, $\\theta \\in \\Theta _{\\tilde{k}}\\setminus \\mathfrak {V}$ and $\\theta _{\\mathfrak {V}^{\\perp }}=\\sum _{j=1}^{K+1} b_j\\tilde{\\mu }_j(p)$ .", "Then either all coefficients $b_j$ are non-positive, or all of them are non-negative.", "By taking sufficiently large $\\tilde{k}$ we can assure that the diameter of $\\mathcal {M}_p$ does not exceed $(100d^2)^{-1}$ .", "We put $\\Phi _q:= \\frac{\\pi }{2}-\\phi \\big (\\xi _{\\mathfrak {V}^\\perp }- \\mathbf {a}, \\tilde{\\mu }_q(p)\\big )$ , $q= 1, \\dots , K+ 1$ .", "The geometrical meaning of these coordinates is simple: $\\Phi _q$ is the spherical distance between $\\mathbf {\\Phi }= \\mathbf {n}(\\xi _{\\mathfrak {V}^{\\perp }}-\\mathbf {a})$ and $\\mathcal {W}\\big (\\tilde{\\mu }_q(p)\\big )$ .", "The reason why we have introduced $\\Phi _q$ is that in these coordinates some important objects will be especially simple (see e.g.", "Lemma REF below) which is very convenient for integration.", "At the same time, the set of coordinates $\\big (r, \\lbrace \\Phi _q\\rbrace \\big )$ contains $K+ 2$ variables, whereas we only need $K+ 1$ coordinates in $\\mathfrak {V}^{\\perp }$ .", "Thus, we have one constraint for variables $\\Phi _j$ .", "Namely, let $\\lbrace \\mathbf {e}_j\\rbrace $ , $j= 1, \\dots , K+ 1$ be a fixed orthonormal basis in $\\mathfrak {V}^{\\perp }$ chosen in such a way that the $K+1$ -st axis is directed along $\\mathbf {a}$ , and thus passes through $\\mathcal {M}_p$ .", "Then we have $\\mathbf {e}_j= \\sum _{l= 1}^{K+ 1}a_{jl}\\tilde{\\mu }_l$ with some matrix $\\lbrace a_{jl}\\rbrace $ , $j,l= 1, \\dots , K+ 1$ , and $\\tilde{\\mu }_l= \\tilde{\\mu }_l(p)$ .", "Therefore (recall that we denote $\\eta := \\xi _{\\mathfrak {V}^{\\perp }}- \\mathbf {a}$ ), $\\eta _j= \\langle \\eta ,\\mathbf {e}_j\\rangle = r\\sum _{q= 1}^{K+ 1}a_{jq}\\sin \\Phi _q$ and, since $r^2(\\xi )=|\\eta |^2=\\sum _{j=1}^{K+1}\\eta _j^2$ , this implies that $\\sum _{j= 1}^{K+ 1}\\Big (\\sum _{q= 1}^{K+ 1} a_{jq}\\sin \\Phi _q\\Big )^2= 1,$ which is our constraint.", "Let us also put $\\eta _j^{\\prime }:= \\frac{\\eta _j}{|\\eta |}= \\sum _{q= 1}^{K+ 1}a_{jq}\\sin \\Phi _q.$ Then we can write the surface element $d\\mathbf {\\Phi }$ in the coordinates $\\lbrace \\eta _j^{\\prime }\\rbrace $ as $d\\mathbf {\\Phi }= \\frac{d\\eta _1^{\\prime }\\dots d\\eta _K^{\\prime }}{\\eta _{K+ 1}}= \\frac{d\\eta _1^{\\prime }\\dots d\\eta _K^{\\prime }}{\\big (1- \\sum _{j= 1}^K(\\eta _{j}^{\\prime })^2\\big )^{1/2}},$ where the denominator is bounded below by $1/2$ by our choice of the basis $\\lbrace \\mathbf {e}_j\\rbrace $ .", "It follows from our choice of the coordinates and (REF ) that $\\langle \\mathbf {a}, \\mathbf {\\Phi }\\rangle = \\langle \\mathbf {a}, \\mathbf {n}(\\eta )\\rangle = |\\mathbf {a}|\\eta _{K+ 1}^{\\prime }= |\\mathbf {a}|\\sum _{q= 1}^{K+ 1}a_{K+ 1\\, q}\\sin \\Phi _q.$ Lemma 7.3 For each $p,l$ we have $|a_{pl}|\\leqslant s(\\rho _n)^{-1}$ .", "Lemma 7.4 We have $\\max _j\\sin \\Phi _j(\\eta )\\geqslant s(\\rho _n) d^{-3/2}$ .", "The next lemma describes the dependence on $r$ of all possible inner products $\\langle \\xi ,\\theta \\rangle $ , $\\theta \\in \\Theta _{\\tilde{k}}$ , $\\xi \\in \\Xi (\\mathfrak {V})_p$ .", "Lemma 7.5 Let $\\xi \\in \\Xi (\\mathfrak {V})_p$ , $\\mathfrak {V}\\in \\mathcal {V}_m$ , and $\\theta \\in \\Theta _{\\tilde{k}}$ .", "(i) If $\\theta \\in \\mathfrak {V}$ , then $\\langle \\xi ,\\theta \\rangle $ does not depend on $r$ .", "(ii) If $\\theta \\notin \\mathfrak {V}$ and $\\theta _{\\mathfrak {V}^{\\perp }}=\\sum _{q}b_q\\tilde{\\mu }_q(p)$ , then $\\langle \\xi ,\\theta \\rangle =\\langle \\mathbf {X},\\theta _{\\mathfrak {V}}\\rangle +L_{m+1}\\sum _{q}b_q+r(\\xi )\\sum _{q}b_q\\sin \\Phi _q.$ In the case (ii) all the coefficients $b_q$ are either non-positive or non-negative and each non-zero coefficient $b_q$ satisfies $\\rho _n^{0-}\\lesssim |b_q| \\lesssim \\rho _n^{0+}.$" ], [ "Partition of the perturbation", "The symbols we are going to construct in this section will depend on $\\rho _n$ ; this dependence will usually be omitted from the notation.", "Let $\\varpi \\in \\textup {{\\textsf {C}}}^{\\infty }( \\mathbb {R})$ be such that $0\\leqslant \\varpi \\leqslant 1,\\ \\ \\varpi (z)={\\left\\lbrace \\begin{array}{ll}& 1,\\ z \\leqslant 1;\\\\& 0,\\ z \\geqslant 21/20.\\end{array}\\right.", "}$ For $\\theta \\in \\Theta ^{\\prime }$ we define several $\\textup {{\\textsf {C}}}^{\\infty }$ -cut-off functions: ${\\left\\lbrace \\begin{array}{ll}e_{\\theta }(\\xi )&:= \\varpi \\Big (\\big |2|2\\xi + \\theta |/\\rho _n- 15\\big |/13\\Big ),\\\\\\ell ^{>}_{\\theta }(\\xi )&:= 1- \\varpi \\Big (\\big (2|2\\xi + \\theta |/\\rho _n- 15\\big )/13\\Big ),\\\\\\ell ^{<}_{\\theta }(\\xi )&:= 1- \\varpi \\Big (\\big (15- 2|2\\xi + \\theta |/\\rho _n\\big )/13\\Big ),\\end{array}\\right.", "}$ and ${\\left\\lbrace \\begin{array}{ll}\\zeta _{\\theta }(\\xi )&:= \\varpi \\biggl (\\dfrac{\\big |\\langle \\theta , \\xi + \\theta /2\\rangle \\big |}{\\rho _n^\\beta |\\theta |}\\biggr ),\\\\\\varphi _{\\theta }(\\xi )&:= 1- \\zeta _{\\theta }(\\xi ).\\end{array}\\right.", "}$ Remark 8.1 Note that $e_{\\theta }+ \\ell ^{>}_{\\theta }+ \\ell ^{<}_{\\theta }= 1$ .", "The function $\\ell ^{>}_{\\theta }$ is supported on the set $|\\xi + \\theta /2|\\geqslant 7\\rho _n$ , and $\\ell ^{<}_{\\theta }$ is supported on the set $|\\xi + \\theta /2|\\leqslant \\rho _n/2$ .", "The function $e_{\\theta }$ is supported in the shell $\\rho _n/3\\leqslant |\\xi + \\theta /2|\\leqslant 8\\rho _n$ .", "Using the notation $\\ell _{\\theta }$ for any of the functions $\\ell ^{>}_{\\theta }$ or $\\ell ^{<}_{\\theta }$ , we point out that ${\\left\\lbrace \\begin{array}{ll}e_{\\theta }(\\xi )= e_{-\\theta }(\\xi + \\theta ), &\\ell _{\\theta }(\\xi )= \\ell _{-\\theta }(\\xi + \\theta ),\\\\\\varphi _{\\theta }(\\xi )= \\varphi _{-\\theta }(\\xi + \\theta ), &\\zeta _{\\theta }(\\xi )= \\zeta _{-\\theta }(\\xi + \\theta ).\\end{array}\\right.", "}$ Note that the above functions satisfy the estimates ${\\left\\lbrace \\begin{array}{ll}\\big |\\mathbf {D}^{\\mathbf {s}}_{\\xi }e_{\\theta }(\\xi )\\big |+ \\big |\\mathbf {D}^{\\mathbf {s}}_{\\xi }\\ell _{\\theta }(\\xi )\\big |\\lesssim \\rho _n^{-|\\mathbf {s}|},\\\\\\big |\\mathbf {D}^{\\mathbf {s}}_{\\xi }\\varphi _{\\theta }(\\xi )\\big |+ \\big |\\mathbf {D}^{\\mathbf {s}}_{\\xi }\\zeta _{\\theta }(\\xi )\\big |\\lesssim \\rho _n^{-\\beta |\\mathbf {s}|}.\\end{array}\\right.", "}$ Now for any symbol $b\\in \\mathbf {S}_{ \\alpha }(\\beta )$ we introduce five new symbols: $\\begin{split}b^{{\\mathcal {L E}}}(\\mathbf {x}, \\xi ; \\rho _n)&:= \\sum _{\\theta \\in \\Theta ^{\\prime }}\\hat{b}(\\theta , \\xi )\\ell ^{>}_{\\theta }(\\xi )e^{i\\theta \\mathbf {x}}, \\\\b^{{\\mathcal {NR}}}(\\mathbf {x}, \\xi ; \\rho _n)&:= \\sum _{\\theta \\in \\Theta ^{\\prime }}\\hat{b}(\\theta , \\xi )\\varphi _{\\theta }(\\xi )e_{\\theta }(\\xi ) e^{i\\theta \\mathbf {x}}, \\\\b^{{\\mathcal {R}}}(\\mathbf {x}, \\xi ; \\rho _n)&:= \\sum _{\\theta \\in \\Theta ^{\\prime }}\\hat{b}(\\theta , \\xi )\\zeta _{\\theta }(\\xi )e_{\\theta }(\\xi )e^{i\\theta \\mathbf {x}}, \\\\b^{{\\mathcal {S E}}}(\\mathbf {x}, \\xi ; \\rho _n)&:= \\sum _{\\theta \\in \\Theta ^{\\prime }}\\hat{b}(\\theta , \\xi )\\ell ^{<}_{\\theta }(\\xi )e^{i\\theta \\mathbf {x}}, \\\\b^o(\\mathbf {x}, \\xi ; \\rho _n)&= b^o(\\xi ; \\rho _n):= \\hat{b}(0, \\xi ).\\end{split}$ The superscripts here are chosen to mean, respectively: `large energy', `non-resonant', `resonant', `small energy' and 0-th Fourier coefficient.", "The corresponding operators are denoted by $B^{{\\mathcal {L E}}}$ , $B^{{\\mathcal {NR}}}$ , $B^{{\\mathcal {R}}}$ , $B^{{\\mathcal {S E}}}$ , and $B^o$ .", "By definitions (REF ), (REF ) and (REF ) $b= b^o+ b^{{\\mathcal {S E}}}+ b^{{\\mathcal {R}}}+ b^{{\\mathcal {NR}}}+ b^{{\\mathcal {L E}}}.$ The role of each of these operators is easy to explain.", "Note that on the support of the functions $\\hat{b}^{{\\mathcal {NR}}}(\\theta , \\cdot ; \\rho _n)$ and $\\hat{b}^{{\\mathcal {R}}}(\\theta , \\cdot ; \\rho _n)$ we have (using (REF )) $\\rho _n/3 - O(\\rho _n^{0+})\\leqslant |\\xi |\\leqslant 8\\rho _n + O(\\rho _n^{0+}).$ On the support of $b^{{\\mathcal {S E}}}(\\theta , \\cdot ; \\rho _n)$ we have $|\\xi |\\leqslant \\rho _n/2 + O(\\rho _n^{0+}).$ On the support of $b^{{\\mathcal {L E}}}(\\theta , \\ \\cdot \\ ; \\rho _n)$ we have $|\\xi |\\geqslant 7\\rho _n- O(\\rho _n^{0+}).$ The introduced symbols play a central role in the proof of Lemma REF .", "As we have seen in Section , due to (REF ) and (REF ) the symbols $b^{\\mathcal {S E}}$ and $b^{\\mathcal {L E}}$ make only a negligible contribution to the spectrum of the operator $H$ near $\\lambda = \\rho ^{2w}$ for $\\rho \\in I_n$ .", "The only significant components of $b$ are the symbols $b^{\\mathcal {NR}}, b^{{\\mathcal {R}}}$ and $b^o$ .", "The symbol $b^o$ will remain as it is, and the symbol $b^{\\mathcal {NR}}$ will be transformed in the next section to another symbol, independent of $\\mathbf {x}$ .", "Under the condition $b\\in \\mathbf {S}_{ \\alpha }(\\beta )$ the above symbols belong to the same class $\\mathbf {S}_{ \\alpha }(\\beta )$ and the following bounds hold: ${\\,\\vrule depth3pt height9pt width1pt}\\, b^{{\\mathcal {R}}}{\\,\\vrule depth3pt height9pt width1pt}\\,^{( \\alpha )}_{l, s}+ {\\,\\vrule depth3pt height9pt width1pt}\\, b^{{\\mathcal {NR}}}{\\,\\vrule depth3pt height9pt width1pt}\\,^{( \\alpha )}_{l, s}+ {\\,\\vrule depth3pt height9pt width1pt}\\, b^{{\\mathcal {L E}}}{\\,\\vrule depth3pt height9pt width1pt}\\,^{( \\alpha )}_{l, s}+ {\\,\\vrule depth3pt height9pt width1pt}\\, b^{o}{\\,\\vrule depth3pt height9pt width1pt}\\,^{( \\alpha )}_{l, s}+ {\\,\\vrule depth3pt height9pt width1pt}\\, b^{{\\mathcal {S E}}}{\\,\\vrule depth3pt height9pt width1pt}\\,^{( \\alpha )}_{l, s}\\lesssim {\\,\\vrule depth3pt height9pt width1pt}\\, b{\\,\\vrule depth3pt height9pt width1pt}\\,^{( \\alpha )}_{l, s}.$ If $b$ symmetric, then so are the symbols on the right hand side of (REF ).", "Let us mention some other elementary properties of the introduced operators.", "In the lemma below we use the projection $\\mathcal {P}(\\mathcal {C})$ , $\\mathcal {C}\\subset \\mathbb {R}^d$ which was defined in Section .", "Lemma 8.2 Let $b\\in \\mathbf {S}_{ \\alpha }(\\beta )$ with some $ \\alpha \\in \\mathbb {R}$ .", "Then: (i) The operator $B^{{\\mathcal {S E}}}$ is bounded and $\\Vert B^{{\\mathcal {S E}}}\\Vert \\lesssim {\\,\\vrule depth3pt height9pt width1pt}\\, b {\\,\\vrule depth3pt height9pt width1pt}\\,^{( \\alpha )}_{0, 0} \\rho _n^{\\beta \\max ( \\alpha ,0)}.$ Moreover, $\\Big (I- \\mathcal {P}\\big (\\mathcal {B}(2\\rho _n/3)\\big )\\Big ) B^{{\\mathcal {S E}}}= B^{{\\mathcal {S E}}}\\Big (I- \\mathcal {P}\\big (\\mathcal {B}(2\\rho _n/3)\\big )\\Big )= 0.$ (ii) The operator $B^{\\mathcal {R}}$ satisfies the relations $\\mathcal {P}\\big (\\mathcal {B}(\\rho _n/6)\\big )B^{{\\mathcal {R}}}= B^{{\\mathcal {R}}}\\mathcal {P}\\big (\\mathcal {B}(\\rho _n/6)\\big )= \\Big (I- \\mathcal {P}\\big (\\mathcal {B}(9\\rho _n)\\big )\\Big )B^{{\\mathcal {R}}}= B^{{\\mathcal {R}}}\\Big (I- \\mathcal {P}\\big (9\\rho _n)\\big )\\Big )= 0,$ and similar relations hold for the operator $B^{{\\mathcal {NR}}}$ as well.", "Moreover, $b^{{\\mathcal {NR}}}, b^{{\\mathcal {R}}}\\in \\mathbf {S}_{\\gamma }$ for any $\\gamma \\in \\mathbb {R}$ , and for all $l$ and $s$ ${\\,\\vrule depth3pt height9pt width1pt}\\, b^{{\\mathcal {NR}}}{\\,\\vrule depth3pt height9pt width1pt}\\,^{(\\gamma )}_{l, s} + {\\,\\vrule depth3pt height9pt width1pt}\\, b^{{\\mathcal {R}}}{\\,\\vrule depth3pt height9pt width1pt}\\,^{(\\gamma )}_{l, s} \\lesssim \\rho _n^{\\beta ( \\alpha - \\gamma )}{\\,\\vrule depth3pt height9pt width1pt}\\, b{\\,\\vrule depth3pt height9pt width1pt}\\,^{( \\alpha )}_{l, s},$ with the implied constant independent of $b$ and $n\\geqslant 1$ .", "In particular, the operators $B^{{\\mathcal {NR}}}, B^{{\\mathcal {R}}}$ are bounded and $\\Vert B^{{\\mathcal {NR}}}\\Vert + \\Vert B^{{\\mathcal {R}}}\\Vert \\lesssim \\rho _n^{\\beta \\alpha }{\\,\\vrule depth3pt height9pt width1pt}\\, b{\\,\\vrule depth3pt height9pt width1pt}\\,^{( \\alpha )}_{0, 0}.$ (iii) $\\mathcal {P}\\bigl (\\mathcal {B}(6\\rho _n)\\bigr )B^{{\\mathcal {L E}}}= B^{{\\mathcal {L E}}}\\mathcal {P}\\bigl (\\mathcal {B}(6\\rho _n)\\bigr ) = 0.$" ], [ "Preparation", "As mentioned at the end of Section , we assume that the symbol $b$ of $B$ satisfies (REF ), and thus belongs to the class $\\mathbf {S}_\\alpha (\\beta )$ with $\\alpha $ defined in (REF ).", "Our strategy is to find a unitary operator which reduces $H= H_0+ B$ , $H_0:= (-\\Delta )^w$ , to another PDO, whose symbol, essentially, depends only on $\\xi $ .", "More precisely, we want to find operators $H_1$ and $H_2$ with the properties discussed in Section .", "Repeating the calculations of Subsection 9.1 of [5] we find that $H$ is unitarily equivalent to $H_1 =H_0 +Y^{(o)}_{\\tilde{k}} +Y_{\\tilde{k}}^{{\\mathcal {R}}} +Y_{{\\tilde{k}}}^{{\\mathcal {S E}}, {\\mathcal {L E}}} +R_{\\tilde{k}},$ where $Y_{\\tilde{k}} &:=\\sum _{l =1}^{{\\tilde{k}}}B_l +\\sum _{l =2}^{{\\tilde{k}}}T_l, \\\\B_1 &:=\\operatorname{{Op}}(b),\\\\B_l &:=\\sum _{j =1}^{l -1}\\frac{1}{j!", "}\\sum _{k_1+ k_2+ \\dots + k_j= l- 1} \\operatorname{{ad}}\\big (\\operatorname{{Op}}(b); \\Psi _{k_1}, \\Psi _{k_2}, \\dots , \\Psi _{k_j}\\big ),\\ l\\geqslant 2, \\\\T_l &:=\\sum _{j =2}^l \\frac{1}{j!}", "\\sum _{k_1+ k_2 +\\dots +k_j =l}\\operatorname{{ad}}(H_0; \\Psi _{k_1}, \\Psi _{k_2}, \\dots , \\Psi _{k_j}),\\ l\\geqslant 2,\\\\R_{\\tilde{k}} &:=\\int _0^1dt_1\\int _0^{t_1}dt_2\\cdots \\int _0^{t_{\\tilde{k}}}\\exp (-it\\Psi )\\operatorname{{ad}}^{{\\tilde{k}}+ 1}(H; \\Psi )\\exp (it\\Psi )dt\\\\ &+\\sum _{j =1}^{\\tilde{k}} \\frac{1}{j!", "}\\sum _{\\begin{array}{c}k_1+ k_2 +\\dots +k_j\\geqslant {\\tilde{k}} +1,\\\\ k_q\\leqslant \\tilde{k}, \\ q =1, \\dots , j\\end{array}} \\operatorname{{ad}}(H; \\Psi _{k_1}, \\Psi _{k_2}, \\dots , \\Psi _{k_j}),\\\\\\Psi &:= \\sum _{p =1}^{\\tilde{k}}\\Psi _p.$ The symbols $\\psi _j$ of PDO $\\Psi _j$ are found from the following system of commutator equations: $\\operatorname{{ad}}(H_0; \\Psi _1) +B_1^{{\\mathcal {NR}}} =0,\\\\\\operatorname{{ad}}(H_0; \\Psi _l) +B_l^{{\\mathcal {NR}}} +T_l^{{\\mathcal {NR}}} =0,\\ l\\geqslant 2.$ By Lemma REF (ii), the operators $B_l^{{\\mathcal {NR}}}$ , $T_l^{{\\mathcal {NR}}}$ are bounded.", "This, in view of (REF ) and (), implies boudedness of the commutators $\\operatorname{{ad}}(H_0; \\Psi _l)$ , $l\\geqslant 1$ .", "Below we denote by $y_{\\tilde{k}}$ the symbol of the PDO $Y_{\\tilde{k}}$ ." ], [ "Commutator equations", "Put $\\tilde{\\chi }_{\\theta }(\\xi ) :=e_{\\theta }(\\xi )\\varphi _{\\theta }(\\xi )\\big (|\\xi +\\theta |^{2w} -|\\xi |^{2w}\\big )^{-1}$ when $\\theta \\ne {\\bf 0}$ , and $\\tilde{\\chi }_{\\bf 0}(\\xi ) :=0$ .", "We have Lemma 9.1 Let $A =\\operatorname{{Op}}(a)$ be a symmetric PDO with $a\\in \\mathbf {S}_{\\omega }$ .", "Then the PDO $\\Psi $ with the Fourier coefficients of the symbol $\\psi (\\mathbf {x}, \\xi )$ given by $\\hat{\\psi }(\\theta , \\xi ) :=i\\,{\\hat{a}}(\\theta , \\xi )\\tilde{\\chi }_{\\theta }(\\xi )$ solves the equation $\\operatorname{{ad}}(H_0; \\Psi ) +\\operatorname{{Op}}(a^{{\\mathcal {NR}}})= 0.$ Moreover, the operator $\\Psi $ is bounded and self-adjoint, its symbol $\\psi $ belongs to $\\mathbf {S}_{ \\gamma }$ with any $\\gamma \\in \\mathbb {R}$ and the following bound holds: ${\\,\\vrule depth3pt height9pt width1pt}\\,\\psi {\\,\\vrule depth3pt height9pt width1pt}\\,^{(\\gamma )}_{l, s} \\lesssim \\rho _n^{\\beta (\\omega -\\gamma -1)- 2w+ 2}r(\\rho _n)^{-1}{\\,\\vrule depth3pt height9pt width1pt}\\, a{\\,\\vrule depth3pt height9pt width1pt}\\,^{(\\omega )}_{l -1, s} \\lesssim \\rho _n^{\\beta (\\omega -\\gamma -1)- 2w+ 2+ 0+}{\\,\\vrule depth3pt height9pt width1pt}\\, a{\\,\\vrule depth3pt height9pt width1pt}\\,^{(\\omega )}_{l -1, s}.$ The proof of this lemma is analogous to that of Lemma 4.1 of [6] and is based on the estimate $|\\xi +\\theta |^{2w} -|\\xi |^{2w}= |\\xi |^{2w}\\Big (\\big (1+ |\\xi |^{-2}(2\\xi + \\theta )\\cdot \\theta \\big )^w- 1\\Big )\\asymp \\rho ^{2w- 2}\\big |\\theta \\cdot (\\xi + \\theta /2)\\big |$ which holds for $\\xi $ in the support of $e_{\\theta }\\varphi _{\\theta }$ .", "Using Propositions REF , REF , REF , Lemma REF , and repeating arguments from the proof of Lemma 4.2 from [6] (with $\\sigma _j:=j\\big ( \\alpha - 2- (2w- 2)\\beta ^{-1}\\big )+ 1$ ), we obtain the following Lemma 9.2 Let $b\\in \\mathbf {S}_{\\alpha }(\\beta )$ be a symmetric symbol.", "Suppose that $k$ is large enough so that $r(\\rho _n)^{-1}\\lesssim \\rho _n^{0+}\\lesssim \\rho _n^{w+ \\beta - \\frac{\\alpha \\beta }{2} -1}$ and $\\tilde{k}$ satisfies $ {\\tilde{k}}> 2(M+ \\alpha \\beta +d- 2w)/(2w+ 2\\beta - \\alpha \\beta -2).", "$ Then $\\psi _j,\\,b_j,\\,t_j\\in \\mathbf {S}_\\gamma (\\beta )$ for any $\\gamma \\in \\mathbb {R}$ and there exists sufficiently large $\\rho _0$ , such that $\\Vert R_{\\tilde{k}}\\Vert \\lesssim \\rho _n^{-M+ 2w- d}.$ Remark 9.3 Note that the expression in the denominator of (REF ) is positive by (REF ) and (REF ).", "Now Lemmas REF and REF imply that the contribution of $R_{\\tilde{k}}$ to the integrated density of states can be neglected.", "More precisely, let $W_{\\tilde{k}}$ be the operator with symbol $w_{\\tilde{k}}(\\mathbf {x}, \\xi ):= y_{\\tilde{k}}(\\mathbf {x}, \\xi )- y_{\\tilde{k}}^{{\\mathcal {NR}}}(\\mathbf {x}, \\xi ),\\ \\ \\hbox{i.e.", "}\\ \\ \\hat{w}_{\\tilde{k}}(\\theta , \\xi )= \\hat{y}_{\\tilde{k}}(\\theta , \\xi )\\big (1- e_{\\theta }(\\xi )\\varphi _{\\theta }(\\xi )\\big ).$ We introduce $H_2:= (-\\Delta )^w+ W_{\\tilde{k}}$ .", "Then, by (REF ) and (REF ), $\\Vert H_1- H_2\\Vert \\lesssim \\rho _n^{-M+ 2w- d}$ and, moreover, the symbol $w_{\\tilde{k}}$ satisfies (REF ).", "This means that all the constructions of Section  are valid, and all we need to do is to compute $\\operatorname{{vol}}\\mathcal {G}_{\\lambda }$ .", "Until this point, the material in our paper was quite similar to the corresponding parts of [5].", "From now on, the differences will be substantial." ], [ "Computing the symbol of the operator after gauge transform", "The following lemma provides us with a more explicit form of the symbol $\\hat{y}_{\\tilde{k}}$ .", "Lemma 9.4 We have $\\hat{y}_{\\tilde{k}}(\\theta ,\\xi )=0$ for $\\theta \\notin \\Theta _{\\tilde{k}}$ .", "Otherwise, $\\begin{split}&\\hat{y}_{\\tilde{k}}(\\theta , \\xi )= \\hat{b}(\\theta , \\xi )\\\\ &+ \\sum \\limits _{s= 1}^{\\tilde{k}- 1}\\sum _{\\begin{array}{c}\\theta _j, \\theta _{s+ 1}\\in \\Theta \\\\ \\phi _j, \\phi _{s+ 1}, \\phi _j^{\\prime }\\in \\Theta _{s+ 1}\\\\ \\theta _j^{\\prime }\\in \\Theta _{s+ 1}^{\\prime }\\\\ 1\\leqslant j\\leqslant s\\end{array}}\\sum _{p= 1}^s\\sum _{\\begin{array}{c}\\theta _q^{\\prime \\prime }, \\phi _q^{\\prime \\prime }\\in \\Theta ^{\\prime }_{s+ 1}\\\\ 1\\leqslant q\\leqslant p- 1\\end{array}} \\sum _{\\begin{array}{c}\\nu _1, \\dots , \\nu _{2s+ p}\\geqslant 0\\\\ \\sum \\nu _i= s\\end{array}}\\prod _{q= 1}^{p- 1}\\widehat{(\\nabla ^{\\nu _q}e_{\\theta _q^{\\prime \\prime }}\\varphi _{\\theta _q^{\\prime \\prime }})}(\\xi + \\phi _q^{\\prime \\prime })\\\\ &\\times \\widehat{(\\nabla ^{\\nu _{p}}b)}(\\theta _{s+ 1}, \\xi + \\phi _{s+ 1})\\prod \\limits _{j= 1}^s \\widehat{(\\nabla ^{\\nu _{p+ j}}b)}(\\theta _j, \\xi + \\phi _j)\\widehat{(\\nabla ^{\\nu _{p+ s+ j}}\\tilde{\\chi }_{\\theta _j^{\\prime }})}(\\xi + \\phi _j^{\\prime }).\\end{split}$ Here for $\\nu \\in \\mathbb {N}$ $\\nabla ^\\nu := \\sum _{\\eta _1, \\dots , \\eta _\\nu \\in \\Theta }C^{(s, p)}_{\\eta _1, \\dots , \\eta _\\nu }\\big (\\lbrace \\theta , \\phi \\rbrace \\big )\\nabla _{\\eta _1}\\cdots \\nabla _{\\eta _\\nu }; \\quad \\nabla ^0:= C^{(s, p)}\\big (\\lbrace \\theta , \\phi \\rbrace \\big ),$ and, for $\\theta \\in \\mathbb {R}^d$ , the action of $\\nabla _{\\theta }$ on symbols of PDO is defined in (REF ), whereas for any function $f$ on $ \\mathbb {R}^d$ $(\\nabla _{\\theta }f)(\\xi ):= f(\\xi + \\theta )- f(\\xi ).$ The coefficients $C^{(s, p)}\\big (\\lbrace \\theta , \\phi \\rbrace \\big )$ and $C^{(s, p)}_{\\eta _1, \\dots , \\eta _\\nu }\\big (\\lbrace \\theta , \\phi \\rbrace \\big )$ depend on $s,\\ p$ and all vectors $\\theta $ , $\\theta _j$ , $\\theta _{s+ 1}$ , $\\phi _j$ , $\\phi _{s+ 1}$ , $\\theta _j^{\\prime }$ , $\\phi _j^{\\prime }$ , $\\theta _q^{\\prime \\prime }$ , $\\phi _q^{\\prime \\prime }$ (and on $\\eta _1, \\dots , \\eta _\\nu $ if these subscripts are present).", "Moreover, these coefficients can differ for each particular $\\nabla ^\\nu $ , $\\nu \\in \\mathbb {N}_0$ .", "At the same time, they are uniformly bounded by a constant which depends on $\\tilde{k}$ only.", "We apply the convention that $\\prod _{q= 1}^{0}\\widehat{(\\nabla ^{\\nu _q}e_{\\theta _q^{\\prime \\prime }}\\varphi _{\\theta _q^{\\prime \\prime }})}(\\xi + \\phi _q^{\\prime \\prime })= 1$ .", "We will prove the lemma by induction.", "Namely, let $\\ell \\geqslant 2$ .", "We claim that: 1) For any $m= 1, \\dots , \\ell - 1$ , $\\hat{\\psi }_m(\\theta , \\xi )= 0$ for $\\theta \\notin \\Theta _m$ .", "Otherwise, $\\begin{split}\\hat{\\psi }_m(\\theta , \\xi )&= \\sum _{\\begin{array}{c}\\theta _j\\in \\Theta \\\\ \\phi _j, \\phi _j^{\\prime }\\in \\Theta _{m}\\\\ \\theta _j^{\\prime }\\in \\Theta _{m}^{\\prime }\\\\ 1\\leqslant j\\leqslant m\\end{array}}\\sum _{p= 1}^{m}\\sum _{\\begin{array}{c}\\theta _q^{\\prime \\prime }, \\phi _q^{\\prime \\prime }\\in \\Theta ^{\\prime }_{m}\\\\ 1\\leqslant q\\leqslant p- 1\\end{array}} \\sum _{\\begin{array}{c}\\nu _1, \\dots , \\nu _{2m+ p- 1}\\geqslant 0\\\\ \\sum \\nu _i= m- 1\\end{array}}\\prod _{q= 1}^{p- 1}\\widehat{(\\nabla ^{\\nu _q}e_{\\theta _q^{\\prime \\prime }}\\varphi _{\\theta _q^{\\prime \\prime }})}(\\xi + \\phi _q^{\\prime \\prime })\\\\ &\\times \\prod \\limits _{j= 1}^m \\widehat{(\\nabla ^{\\nu _{p- 1+ j}}b)}(\\theta _j, \\xi + \\phi _j)\\widehat{(\\nabla ^{\\nu _{p- 1+ m+ j}}\\tilde{\\chi }_{\\theta _j^{\\prime }})}(\\xi + \\phi _j^{\\prime }).\\end{split}$ 2) For any $s= 1, \\dots , \\ell - 1$ and any $k_1, \\dots , k_p$ $(p\\geqslant 1)$ such that $k_1+ \\dots + k_p= s$ , $\\widehat{\\operatorname{{ad}}\\big (\\operatorname{{Op}}(b); \\Psi _{k_1}, \\dots , \\Psi _{k_p}\\big )}(\\theta , \\xi )= 0$ for $\\theta \\notin \\Theta _{s+ 1}$ .", "Otherwise, $\\begin{split}&\\widehat{\\operatorname{{ad}}\\big (\\operatorname{{Op}}(b); \\Psi _{k_1}, \\dots , \\Psi _{k_p}\\big )}(\\theta , \\xi )\\\\ &= \\sum _{\\begin{array}{c}\\theta _j, \\theta _{s+ 1}\\in \\Theta \\\\ \\phi _j, \\phi _{s+ 1}, \\phi _j^{\\prime }\\in \\Theta _{s+ 1}\\\\ \\theta _j^{\\prime }\\in \\Theta _{s+ 1}^{\\prime }\\\\ 1\\leqslant j\\leqslant s\\end{array}}\\sum _{p= 1}^s\\sum _{\\begin{array}{c}\\theta _q^{\\prime \\prime }, \\phi _q^{\\prime \\prime }\\in \\Theta ^{\\prime }_{s+ 1}\\\\ 1\\leqslant q\\leqslant p- 1\\end{array}} \\sum _{\\begin{array}{c}\\nu _1, \\dots , \\nu _{2s+ p}\\geqslant 0\\\\ \\sum \\nu _i= s\\end{array}}\\prod _{q= 1}^{p- 1}\\widehat{(\\nabla ^{\\nu _q}e_{\\theta _q^{\\prime \\prime }}\\varphi _{\\theta _q^{\\prime \\prime }})}(\\xi + \\phi _q^{\\prime \\prime })\\\\ &\\times \\widehat{(\\nabla ^{\\nu _{p}}b)}(\\theta _{s+ 1}, \\xi + \\phi _{s+ 1})\\prod \\limits _{j= 1}^s \\widehat{(\\nabla ^{\\nu _{p+ j}}b)}(\\theta _j, \\xi + \\phi _j)\\widehat{(\\nabla ^{\\nu _{p+ s+ j}}\\tilde{\\chi }_{\\theta _j^{\\prime }})}(\\xi + \\phi _j^{\\prime }).\\end{split}$ 3) For any $s= 2, \\dots , \\ell $ and any $k_1, \\dots , k_p$ $(p\\geqslant 2)$ such that $k_1+ \\dots + k_p= s$ , $\\widehat{\\operatorname{{ad}}(H_0; \\Psi _{k_1}, \\dots , \\Psi _{k_p})}(\\theta , \\xi )= 0$ for $\\theta \\notin \\Theta _{s}$ .", "Otherwise, $\\begin{split}&\\widehat{\\operatorname{{ad}}(H_0; \\Psi _{k_1}, \\dots , \\Psi _{k_p})}(\\theta , \\xi )\\\\ &= \\sum _{\\begin{array}{c}\\theta _j, \\theta _s\\in \\Theta \\\\ \\phi _j, \\phi _s, \\phi _j^{\\prime }\\in \\Theta _{s}\\\\ \\theta _j^{\\prime }\\in \\Theta _{s}^{\\prime }\\\\ 1\\leqslant j\\leqslant s- 1\\end{array}}\\sum _{p= 1}^s\\sum _{\\begin{array}{c}\\theta _q^{\\prime \\prime }, \\phi _q^{\\prime \\prime }\\in \\Theta _{s}\\\\ 1\\leqslant q\\leqslant p- 1\\end{array}} \\sum _{\\begin{array}{c}\\nu _1, \\dots , \\nu _{2s+ p- 2}\\geqslant 0\\\\ \\sum \\nu _i= s- 1\\end{array}}\\prod _{q= 1}^{p- 1}\\widehat{(\\nabla ^{\\nu _q}e_{\\theta _q^{\\prime \\prime }}\\varphi _{\\theta _q^{\\prime \\prime }})}(\\xi + \\phi _q^{\\prime \\prime })\\\\ &\\times \\widehat{(\\nabla ^{\\nu _{p}}b)}(\\theta _{s}, \\xi + \\phi _{s})\\prod \\limits _{j= 1}^{s- 1} \\widehat{(\\nabla ^{\\nu _{p+ j}}b)}(\\theta _j, \\xi + \\phi _j)\\widehat{(\\nabla ^{\\nu _{p+ s- 1+ j}}\\tilde{\\chi }_{\\theta _j^{\\prime }})}(\\xi + \\phi _j^{\\prime }).\\end{split}$ For $\\ell =2$ assumptions 1)–3) can be easily checked.", "Indeed, by (REF ), (REF ) and (REF ), $\\hat{\\psi }_1(\\theta ,\\xi ) =i\\hat{b}(\\theta ,\\xi )\\tilde{\\chi }_{\\theta }(\\xi ),$ $\\begin{split}&\\widehat{\\operatorname{{ad}}\\big (\\operatorname{{Op}}(b); \\Psi _1\\big )}(\\theta , \\xi )= \\sum _{\\chi \\in \\Theta \\cup (\\theta - \\Theta )}\\big (\\hat{b}(\\theta , \\xi )\\hat{b}(\\theta - \\chi , \\xi + \\chi )\\widehat{(\\nabla _{\\chi }\\tilde{\\chi }_{\\theta - \\chi })}(\\xi )\\\\ &+\\hat{b}(\\theta , \\xi )\\widehat{(\\nabla _{\\chi }b)}(\\theta - \\chi , \\xi )\\tilde{\\chi }_{\\theta - \\chi }(\\xi )- \\widehat{(\\nabla _{\\theta - \\chi }b)}(\\chi , \\xi )\\hat{b}(\\theta - \\chi , \\xi )\\tilde{\\chi }_{\\theta - \\chi }(\\xi )\\big ),\\end{split}$ $\\begin{split}\\widehat{\\operatorname{{ad}}(H_0; \\Psi _1, \\Psi _1)}(\\theta , \\xi )&= \\sum _{\\chi \\in \\Theta \\cup (\\theta - \\Theta )}\\big (\\hat{b}(\\chi , \\xi + \\theta - \\chi )\\widehat{(\\nabla _{\\theta - \\chi }\\varphi _{\\chi } e_{\\chi })}(\\xi )\\hat{b}(\\theta - \\chi , \\xi )\\tilde{\\chi }_{\\theta - \\chi }(\\xi )\\\\ &+ \\widehat{(\\nabla _{\\theta - \\chi }b)}(\\chi , \\xi )\\varphi _{\\chi }(\\xi )e_{\\chi }(\\xi )\\hat{b}(\\theta - \\chi , \\xi )\\tilde{\\chi }_{\\theta - \\chi }(\\xi )\\\\ &- \\varphi _{\\theta }(\\xi )e_{\\theta }(\\xi )\\hat{b}(\\theta , \\xi )\\hat{b}(\\theta - \\chi , \\xi + \\chi )\\widehat{(\\nabla _{\\chi }\\tilde{\\chi }_{\\theta - \\chi })}(\\xi )\\\\ &- \\varphi _{\\theta }(\\xi )e_{\\theta }(\\xi )\\hat{b}(\\theta , \\xi )\\widehat{(\\nabla _{\\chi }b)}(\\theta - \\chi , \\xi )\\tilde{\\chi }_{\\theta - \\chi }(\\xi )\\big ).\\end{split}$ Now, we complete the induction in several steps.", "Step 1.", "First of all, notice that due to (), (), for any $m =2, \\dots , \\ell $ the symbol of $B_m$ admits a representation of the form (REF ) with $s= m- 1$ , and symbol of $T_m$ admits a representation of the form (REF ) with $s= m$ .", "Then it follows from Lemma REF and () that $\\Psi _\\ell $ admits a representation of the form (REF ).", "Step 2.", "Proof of (REF ) with $s =\\ell $ .", "Let $k_1 +\\dots +k_p =\\ell $ .", "If $p \\geqslant 2$ .", "Then $\\operatorname{{ad}}\\big (\\operatorname{{Op}}(b); \\Psi _{k_1}, \\dots , \\Psi _{k_p}\\big )= \\operatorname{{ad}}\\Big (\\operatorname{{ad}}\\big (\\operatorname{{Op}}(b); \\Psi _{k_1}, \\dots , \\Psi _{k_{p- 1}}\\big ); \\Psi _{k_p}\\Big ).$ Since $k_1+ \\dots + k_{p- 1}\\leqslant \\ell - 1$ and $k_p\\leqslant \\ell - 1$ we can apply (REF ) and (REF ).", "Combined with (REF ) it gives a representation of the form (REF ).", "If $p= 1$ then $\\operatorname{{ad}}\\big (\\operatorname{{Op}}(b); \\Psi _\\ell \\big )$ satisfies (REF ) because of (REF ) and step 1.", "Step 3.", "Proof of (REF ) with $s= \\ell + 1$ .", "Let $k_1+ \\dots + k_p= \\ell + 1$ , $p\\geqslant 2$ .", "If $p\\geqslant 3$ , then (cf.", "step 2) $\\operatorname{{ad}}(H_0; \\Psi _{k_1}, \\dots , \\Psi _{k_p})= \\operatorname{{ad}}\\big (\\operatorname{{ad}}(H_0; \\Psi _{k_1}, \\dots , \\Psi _{k_{p- 1}}); \\Psi _{k_p}\\big ).$ Since $k_1+ \\dots + k_{p- 1}\\leqslant \\ell $ , $p- 1\\geqslant 2$ and $k_p\\leqslant \\ell - 1$ we can apply (REF ) and (REF ).", "Together with (REF ) it gives a representation of the form (REF ).", "If $p= 2$ then (see ()) $\\operatorname{{ad}}(H_0; \\Psi _{k_1}, \\Psi _{k_2}) =\\operatorname{{ad}}\\big (\\operatorname{{ad}}(H_0; \\Psi _{k_1}); \\Psi _{k_2}\\big ) = -\\operatorname{{ad}}(B_{k_1}^{{\\mathcal {NR}}} +T_{k_1}^{{\\mathcal {NR}}}; \\Psi _{k_2}).$ Since $k_1 \\leqslant \\ell $ and $k_2 \\leqslant \\ell $ , the representation of the form (REF ) follows from (REF ) and step 1.", "(Formally exceptional case $k_1 =1$ , $k_2 =\\ell $ can be treated separately in the same way using (REF ) instead of ().)", "Induction is complete.", "Now, (REF ), (REF ) and (REF ), (), () prove the lemma." ], [ "Contribution from various resonant regions", "Let us fix a subspace $\\mathfrak {V}\\in \\mathcal {V}_m$ , $m <d$ , and a component $\\Xi _p$ of the resonant region $\\Xi (\\mathfrak {V})$ .", "Our aim is to compute the contribution to the density of states from each component $\\Xi _p$ .", "Therefore, we define $\\mathcal {A}^+_p(\\rho ):= \\mathcal {A}^+(\\rho )\\cap \\Xi _p\\quad \\textrm {and} \\quad \\mathcal {A}^-_p(\\rho ):= \\mathcal {A}^-(\\rho )\\cap \\Xi _p$ and try to compute $\\operatorname{{vol}}\\mathcal {A}^+_p(\\rho )- \\operatorname{{vol}}\\mathcal {A}^-_p(\\rho ).$ Since formulas (REF ) and (REF ) obviously imply that $\\operatorname{{vol}}(\\mathcal {G}_{\\lambda })= \\omega _d\\rho ^d+ \\sum _{m= 0}^{d -1}\\sum _{\\mathfrak {V}\\in \\mathcal {V}_m}\\sum _{p}\\bigl (\\operatorname{{vol}}\\mathcal {A}^+_p(\\rho )- \\operatorname{{vol}}\\mathcal {A}^-_p(\\rho )\\bigr ),$ Lemma REF would be proved if we manage to compute (REF ) (or at least prove that this expression admits a complete asymptotic expansion in $\\rho $ ).", "Note that if $\\xi \\in \\Xi _p$ , then we also have that $\\Upsilon (\\xi )\\subset \\Xi _p$ .", "We denote $H_2(\\xi ) :=H_2|_{\\textup {{\\textsf {H}}}^{}_{\\xi }}, \\quad \\textup {{\\textsf {H}}}^{}_{\\xi } :=\\mathcal {P}\\big (\\Upsilon (\\xi )\\big )\\textup {{\\textsf {B}}}_2( \\mathbb {R}^d)$ (recall that $\\textup {{\\textsf {H}}}^{}_{\\xi }$ is an invariant subspace of $H_2$ acting in $\\textup {{\\textsf {B}}}_2( \\mathbb {R}^d)$ ).", "Suppose now that two points $\\xi $ and $\\eta $ have the same coordinates $\\mathbf {X}$ and $\\mathbf {\\Phi }$ and different coordinates $r$ .", "Then $\\xi \\in \\Xi _p$ implies $\\eta \\in \\Xi _p$ and $\\Upsilon (\\eta )= \\Upsilon (\\xi )+ (\\eta - \\xi )$ .", "This shows that two spaces $\\textup {{\\textsf {H}}}^{}_{\\xi }$ and $\\textup {{\\textsf {H}}}^{}_{\\eta }$ have the same dimension and, moreover, there is a natural isometry $F_{\\xi ,\\eta }: \\textup {{\\textsf {H}}}^{}_{\\xi }\\rightarrow \\textup {{\\textsf {H}}}^{}_{\\eta }$ given by $F: \\mathbf {e}_{\\nu }\\mapsto \\mathbf {e}_{\\nu + (\\eta - \\xi )}$ , $\\nu \\in \\Upsilon (\\xi )$ .", "This isometry allows us to `compare' operators acting in $\\textup {{\\textsf {H}}}^{}_{\\xi }$ and $\\textup {{\\textsf {H}}}^{}_{\\eta }$ .", "Thus, abusing slightly our notation, we can assume that $H_2(\\xi )$ and $H_2(\\eta )$ act in the same (finite dimensional) Hilbert space $\\textup {{\\textsf {H}}}^{}(\\mathbf {X}, \\mathbf {\\Phi })$ .", "We will fix the values $(\\mathbf {X}, \\mathbf {\\Phi })$ and study how these operators depend on $r$ .", "Thus, we denote by $H_2(r) =H_2(r; \\mathbf {X}, \\mathbf {\\Phi })$ the operator $H_2(\\xi )$ with $\\xi =(\\mathbf {X}, r, \\mathbf {\\Phi })$ , acting in $\\textup {{\\textsf {H}}}^{}(\\mathbf {X}, \\mathbf {\\Phi })$ .", "Let $W_{\\tilde{k}}(r)$ be the operator in $\\textup {{\\textsf {H}}}^{}(\\mathbf {X}, \\mathbf {\\Phi })$ with the symbol $w_{\\tilde{k}}\\big (\\mathbf {x},\\xi (\\mathbf {X}, r, \\mathbf {\\Phi })\\big )$ .", "According to formula (REF ), for any $s\\leqslant \\tilde{k}- 1$ and $\\theta \\in \\Theta _{s+ 1}$ $|\\xi + \\phi |^2= r^2+ 2r|\\mathbf {a}|\\sum _{q= 1}^{K+ 1}a_{K+ 1\\, q}\\sin \\Phi _q+ 2\\langle \\xi , \\phi \\rangle + |\\mathbf {X}|^2+ |\\mathbf {a}|^2+ |\\phi |^2.$ This, together with (REF ), (REF ) and (REF ), implies that for $|\\xi + \\phi |> C_0$ the coefficients $\\hat{b}(\\theta ,\\xi + \\phi )$ can be represented as the absolutely convergent series $\\begin{split}\\hat{b}(\\theta ,\\xi + \\phi )= \\sum _{\\iota \\in \\widetilde{J}}\\sum _{l= 0}^\\infty \\sum _{\\begin{array}{c}n_1, \\dots , n_{K+ 1}\\geqslant 0\\\\ n_1+ \\cdots + n_{K+ 1}\\leqslant l\\\\ j_1, \\dots , j_d\\geqslant 0\\\\ j_1+ \\cdots + j_d\\leqslant l\\end{array}}C^{\\iota \\, j_1\\cdots j_d}_{l\\, n_1\\cdots n_{K+ 1}}(\\mathbf {X}; \\theta )r^{\\iota - l}\\phi _1^{j_1}\\cdots \\phi _d^{j_d}\\prod _{a= 1}^{K+ 1}(\\sin \\Phi _a)^{n_a},\\end{split}$ where the coefficients satisfy $\\big |C^{\\iota \\, j_1\\cdots j_d}_{l\\, n_1\\cdots n_{K+ 1}}(\\mathbf {X}; \\theta )\\big |\\lesssim \\rho _n^{(l- j_1- \\cdots - j_d)(\\alpha _{m+ 1}+ 0+)}$ In the next lemma, to facilitate the expansion of the RHS of (REF ) in a suitable form, we transform the denominator of $\\tilde{\\chi }_{\\theta ^{\\prime }}$ (recall (REF )).", "In the subsequent calculations we will use the generalized binomial coefficeints: $\\binom{p}{j}:= {\\left\\lbrace \\begin{array}{ll}1, & j= 0;\\\\ \\displaystyle \\frac{1}{j!", "}\\prod _{k= 0}^{j- 1}(p- k), & j\\in \\mathbb {N}.\\end{array}\\right.", "}$ Lemma 10.1 For $s\\leqslant \\tilde{k}- 1$ , $\\phi ^{\\prime }\\in \\Theta _{s+ 1}$ , $\\theta ^{\\prime }\\in \\Theta _{s+ 1}^{\\prime }$ , and $\\xi $ in the support of $e_{\\theta ^{\\prime }}\\varphi _{\\theta ^{\\prime }}$ let $\\begin{split}D&:= \\frac{1}{w}\\sum _{j= 2}^\\infty \\binom{w}{j}r^{2- 2j}\\sum _{k= 0}^{j- 1}\\binom{j}{k}\\Big (2r|\\mathbf {a}|\\sum _{q= 1}^{K+ 1}a_{K+ 1\\, q}\\sin \\Phi _q+ 2\\langle \\xi , \\phi ^{\\prime }\\rangle + |\\mathbf {X}|^2+ |\\mathbf {a}|^2+ |\\phi ^{\\prime }|^2\\Big )^{k}\\\\ &\\times \\big (2\\langle \\xi , \\theta ^{\\prime }\\rangle + 2\\langle \\phi ^{\\prime }, \\theta ^{\\prime }\\rangle + |\\theta ^{\\prime }|^2\\big )^{j-k- 1}.\\end{split}$ Then $|D|\\lesssim \\rho _n^{-1+ \\alpha _{m+ 1}+ 0+}$ and $\\big (|\\xi + \\phi ^{\\prime }+ \\theta ^{\\prime }|^{2w}-|\\xi + \\phi ^{\\prime }|^{2w}\\big )^{-1}= w^{-1}r^{2- 2w}\\big (2\\langle \\xi , \\theta ^{\\prime }\\rangle + 2\\langle \\phi ^{\\prime }, \\theta ^{\\prime }\\rangle + |\\theta ^{\\prime }|^2\\big )^{-1}\\sum _{a= 0}^\\infty (-D)^a.$ We introduce a shorthand $N:= 2r|\\mathbf {a}|\\sum _{q= 1}^{K+ 1}a_{K+ 1\\, q}\\sin \\Phi _q+ 2\\langle \\xi , \\phi ^{\\prime }\\rangle + |\\mathbf {X}|^2+ |\\mathbf {a}|^2+ |\\phi ^{\\prime }|^2.$ Then by (generalized) binomial formula and (REF ) we obtain $\\begin{split}&|\\xi + \\phi ^{\\prime }+ \\theta ^{\\prime }|^{2w}-|\\xi + \\phi ^{\\prime }|^{2w}\\\\ &= \\big (|\\xi |^2+ 2\\langle \\xi , \\phi ^{\\prime }+ \\theta ^{\\prime }\\rangle + |\\phi ^{\\prime }+ \\theta ^{\\prime }|^2\\big )^w- \\big (|\\xi |^2+ 2\\langle \\xi , \\phi ^{\\prime }\\rangle + |\\phi ^{\\prime }|^2\\big )^w\\\\ &= \\big (r^2+ N+ 2\\langle \\xi , \\theta ^{\\prime }\\rangle + 2\\langle \\phi ^{\\prime }, \\theta ^{\\prime }\\rangle + |\\theta ^{\\prime }|^2\\big )^w- (r^2+ N)^w\\\\ &= r^{2w}\\sum _{j= 1}^\\infty \\binom{w}{j}r^{-2j}\\Big (\\big (N+ 2\\langle \\xi , \\theta ^{\\prime }\\rangle + 2\\langle \\phi ^{\\prime }, \\theta ^{\\prime }\\rangle + |\\theta ^{\\prime }|^2\\big )^j- N^j\\Big )\\\\ &= wr^{2w- 2}\\big (2\\langle \\xi , \\theta ^{\\prime }\\rangle + 2\\langle \\phi ^{\\prime }, \\theta ^{\\prime }\\rangle + |\\theta ^{\\prime }|^2\\big )(1+ D).\\end{split}$ The estimate on $|D|$ follows from estimates (REF ) and (REF ), and Lemmas REF and REF .", "Now (REF ) follows from (REF ).", "As we have seen from the previous sections, the symbol of the operator $H_2$ satisfies $h_2(\\mathbf {x},\\xi )= |\\xi |^{2w}+ {w}_{\\tilde{k}}(\\mathbf {x},\\xi )= \\big (r^2+ 2r\\langle \\mathbf {a}, \\mathbf {\\Phi }\\rangle + |\\mathbf {a}|^2+ |\\mathbf {X}|^2\\big )^w+ {w}_{\\tilde{k}}(\\mathbf {x},\\xi ),$ where $w_{\\tilde{k}}$ are given by (REF ) and (REF ).", "Remark 10.2 In this section we assume that $\\xi \\in \\mathcal {A}$ , so by (REF ) $2\\rho _n/3\\leqslant |\\xi |\\leqslant 6\\rho _n$ , and by Remark REF all functions $e_{\\theta }(\\xi + \\cdot )$ from (REF ) and (REF ) are equal to 1.", "Note that if $\\theta \\in \\Theta _{\\tilde{k}}$ , $\\phi \\in \\Theta _{\\tilde{k}}$ , and $\\theta \\notin \\mathfrak {V}$ , then (see Lemma REF and (REF )) $\\varphi _{\\theta }(\\xi + \\phi )= 1$ .", "This means that all cut-off functions from (REF ) and (REF ) are equal to 1 unless $\\theta \\in \\mathfrak {V}$ .", "If, on the other hand, $\\theta \\in \\mathfrak {V}$ , then $\\varphi _{\\theta }(\\xi + \\phi )$ depends only on the projection $\\xi _{\\mathfrak {V}}$ and thus is a function only of the coordinates $\\mathbf {X}$ .", "By Proposition REF , (REF ), Lemma REF , formulas (REF ) and (REF ), Lemma REF , and Remark REF , for $r\\asymp \\rho _n$ $\\Big \\Vert \\frac{d^l}{dr^l}W_{\\tilde{k}}(r)\\Big \\Vert \\lesssim \\rho _n^{\\varkappa - l+ 0+}, \\qquad l\\geqslant 0.$ This, together with (REF ), implies Lemma 10.3 The operator $H_2(r)$ is monotonically increasing in $r$ ; in particular, all its eigenvalues $\\lambda _j\\big (H_2(r)\\big )$ are increasing in $r$ .", "Thus the function $g\\big (\\xi (\\mathbf {X}, r, \\mathbf {\\Phi })\\big )$ (defined in Section ) is an increasing function of $r$ if we fix the other coordinates of $\\xi $ , so the equation $g(\\xi )= \\rho ^{2w}$ has a unique solution for fixed values of $\\mathbf {X}$ and $\\mathbf {\\Phi }$ ; we denote the $r$ -coordinate of this solution by $\\tau = \\tau (\\rho )= \\tau (\\rho ; \\mathbf {X}, \\mathbf {\\Phi })$ , so that $g\\big (\\xi (\\mathbf {X}, \\tau , \\mathbf {\\Phi })\\big )= \\rho ^{2w}.$ By $\\tau _0= \\tau _0(\\rho )= \\tau _0(\\rho ; \\mathbf {X}, \\mathbf {\\Phi })$ we denote the value of $\\tau $ for $(-\\Delta )^w$ , i.e.", "$\\tau _0$ is a unique solution of the equation $\\big |\\xi (\\mathbf {X}, \\tau _0, \\mathbf {\\Phi })\\big |= \\rho .$ Obviously, we can write down a precise analytic expression for $\\tau _0$ (and we have done this in [4] in the two-dimensional case) and show that it allows an expansion in powers of $\\rho $ and $\\ln \\rho $ , but we will not need it.", "The definition (REF ) of the sets $\\mathcal {A}^{\\pm }_p(\\rho )$ implies that the intersection $\\mathcal {A}^+_p(\\rho )\\cap \\big \\lbrace \\xi (\\mathbf {X}, r, \\mathbf {\\Phi }),\\ r\\in \\mathbb {R}_+\\big \\rbrace $ consists of points with $r$ -coordinate belonging to the interval $\\big [\\tau _0(\\rho ), \\tau (\\rho )\\big ]$ (where we assume the interval to be empty if $\\tau _0> \\tau $ ).", "Similarly, the intersection $\\mathcal {A}^-_p(\\rho )\\cap \\big \\lbrace \\xi (\\mathbf {X}, r, \\mathbf {\\Phi }),\\ r\\in \\mathbb {R}_+\\big \\rbrace $ consists of points with $r$ -coordinate belonging to the interval $\\big [\\tau (\\rho ), \\tau _0(\\rho )\\big ]$ .", "Therefore, $\\mathcal {A}^+_p(\\rho )= \\Big \\lbrace \\xi = \\xi (\\mathbf {X}, r, \\mathbf {\\Phi }), \\mathbf {X}\\in \\Omega (\\mathfrak {V}), \\mathbf {\\Phi }\\in \\mathcal {M}_p, r\\in \\big [\\tau _0(\\rho ; \\mathbf {X}, \\mathbf {\\Phi }), \\tau (\\rho ; \\mathbf {X}, \\mathbf {\\Phi })\\big ]\\Big \\rbrace $ and $\\mathcal {A}^-_p(\\rho )= \\Big \\lbrace \\xi = \\xi (\\mathbf {X}, r, \\mathbf {\\Phi }), \\mathbf {X}\\in \\Omega (\\mathfrak {V}), \\mathbf {\\Phi }\\in \\mathcal {M}_p, r\\in \\big [\\tau (\\rho ; \\mathbf {X}, \\mathbf {\\Phi }),\\tau _0(\\rho ; \\mathbf {X}, \\mathbf {\\Phi })\\big ]\\Big \\rbrace .$ This implies that (recall that $K= d- m- 1$ ) $\\begin{split}&\\operatorname{{vol}}\\mathcal {A}^+_p(\\rho )- \\operatorname{{vol}}\\mathcal {A}^-_p(\\rho ) =\\int _{\\Omega (\\mathfrak {V})}d\\mathbf {X}\\int _{\\mathcal {M}_p}d\\mathbf {\\Phi }\\int _{\\tau _0(\\rho ; \\mathbf {X}, \\mathbf {\\Phi })}^{\\tau (\\rho ; \\mathbf {X}, \\mathbf {\\Phi })}r^{K}dr\\\\&= (K+ 1)^{-1}\\int _{\\mathcal {M}_p}d\\mathbf {\\Phi }\\int _{\\Omega (\\mathfrak {V})}d\\mathbf {X}\\big (\\tau (\\rho ; \\mathbf {X}, \\mathbf {\\Phi })^{K+ 1}- \\tau _0(\\rho ; \\mathbf {X}, \\mathbf {\\Phi })^{K+ 1}\\big ).\\end{split}$ Remark 10.4 Note that in the case $K= 0$ the simplex $\\mathcal {M}_p$ is degenerate and there is no integration in $d\\mathbf {\\Phi }$ .", "Obviously, it is enough to compute the part of (REF ) containing $\\tau $ , since the second part (containing $\\tau _0$ ) can be computed analogously.", "We start by considering $\\int _{\\Omega (\\mathfrak {V})}\\tau (\\rho ; \\mathbf {X}, \\mathbf {\\Phi })^{K+ 1}d\\mathbf {X}.$ First of all, we notice that if $\\xi ,\\eta \\in \\Xi (\\mathfrak {V})$ are resonant congruent points then, according to Lemma REF , all vectors $\\theta _j$ from Definition REF of equivalence belong to $\\mathfrak {V}$ .", "This naturally leads to the definition of equivalence for projections $\\xi _{\\mathfrak {V}}$ and $\\eta _{\\mathfrak {V}}$ .", "Namely, we say that two points $\\nu $ and $\\mu $ from $\\Omega (\\mathfrak {V})$ are $\\mathfrak {V}$ -equivalent (and write $\\nu \\leftrightarrow _{\\mathfrak {V}}\\mu $ ) if $\\nu $ and $\\mu $ are equivalent in the sense of Definition REF with an additional requirement that all $\\theta _j\\in \\mathfrak {V}$ .", "Then $\\xi \\leftrightarrow \\eta $ implies $\\xi _{\\mathfrak {V}} \\leftrightarrow _{\\mathfrak {V}}\\eta _{\\mathfrak {V}}$ .", "For $\\nu \\in \\Omega (\\mathfrak {V})$ we denote by $\\Upsilon _{\\mathfrak {V}}(\\nu )$ the class of equivalence of $\\nu $ generated by $\\leftrightarrow _{\\mathfrak {V}}$ .", "Then $\\Upsilon _{\\mathfrak {V}}(\\xi _{\\mathfrak {V}})$ is a projection of $\\Upsilon (\\xi )$ to $\\mathfrak {V}$ and is, therefore, finite.", "Since $\\Upsilon _{\\mathfrak {V}}(\\nu )$ is a finite set for each $\\nu \\in \\Omega (\\mathfrak {V})$ , we can re-write (REF ) as $\\int _{\\Omega (\\mathfrak {V})}\\tau (\\rho ; \\mathbf {X}, \\mathbf {\\Phi })^{K+ 1}d\\mathbf {X}= \\int _{\\Omega (\\mathfrak {V})}\\big (\\operatorname{{card}}\\Upsilon _{\\mathfrak {V}}(\\nu )\\big )^{-1}\\sum _{\\mathbf {X}\\in \\Upsilon _{\\mathfrak {V}}(\\nu )}\\tau (\\rho ; \\mathbf {X}, \\mathbf {\\Phi })^{K+ 1}d\\nu $ and try to compute $\\sum _{\\mathbf {X}\\in \\Upsilon _{\\mathfrak {V}}(\\nu )}\\tau (\\rho ; \\mathbf {X}, \\mathbf {\\Phi })^{K+ 1}.$ Remark REF , together with equations (REF ), (REF ), and (REF ), shows that $H_2(r)$ depends on $r$ analytically, so we can and will consider the family $H_2(z)$ with complex values of the parameter $z$ with $\\Re e~z\\asymp \\rho $ .", "Likewise, we analytically continue the function $\\xi (\\mathbf {X}, r, \\mathbf {\\Phi })$ to $\\xi (\\mathbf {X}, z, \\mathbf {\\Phi }):= \\mathbf {X}+ \\mathbf {a}+ z\\mathbf {\\Phi }.$ We also introduce the analytic continuation $|\\cdot |_{ of the modulus of vectors, so that\\begin{equation}|\\xi |_{^{2}:= z^2+ 2z\\langle \\mathbf {a}, \\mathbf {\\Phi }\\rangle + |\\mathbf {a}|^2+ |\\mathbf {X}|^2.", "}Formulas (\\ref {eq:newy}) and (\\ref {symbol equation}) give matrix elements of H_2(z) in an orthonormal basis even for complex z.\\end{equation}We choose a contour\\begin{equation}\\gamma := \\bigg \\lbrace z\\in \\mathbb {C}:\\, |z- \\rho |= t\\rho _n:= \\Big (8\\max \\big \\lbrace (2w- 2)/3, 1\\big \\rbrace \\Big )^{-1}\\rho _n\\bigg \\rbrace \\end{equation}to be a circle in the complex plane going in the positive direction.", "}Estimates (\\ref {early derivative estimate}) remain valid after the analytic continuation: for all $ z$ inside and on $$\\begin{equation}\\Big \\Vert \\frac{d^l}{dz^l}W_{\\tilde{k}}(z)\\Big \\Vert \\lesssim \\rho _n^{\\varkappa - l+ 0+}, \\qquad l\\geqslant 0.\\end{equation}$ Lemma 10.5 For $\\rho \\in I_n= [\\rho _n, 4\\rho _n]$ all $\\tau (\\rho ; \\mathbf {X}, \\mathbf {\\Phi })$ lie inside $\\gamma $ .", "These are the only zeros of the function $\\det \\big (H_2(z)- \\rho ^{2w}I\\big )$ inside the contour.", "Let $r:= \\Re e~z$ , $y:= \\Im m~z$ .", "For $y= 0$ the operator $H_2(r)$ is self-adjoint.", "Thus it has $\\operatorname{{card}}\\Upsilon _{\\mathfrak {V}}(\\nu )$ real eigenvalues.", "Now for $r\\geqslant \\rho + t\\rho _n\\geqslant (1+ t/4)\\rho $ relations (REF ), (REF ), Lemma REF (ii), and () imply $H_2(r)\\geqslant \\Big (\\big ((1+ t/4)\\rho \\big )^{2w}\\big (1- O(\\rho ^{\\alpha _{m+ 1}- 1+ 0+})\\big )- O(\\rho ^{\\varkappa + 0+})\\Big )I.$ Thus by (REF ) and (REF ) for big $\\rho $ no eigenvalue of $H_2(r)$ can coincide with $\\rho ^{2w}$ .", "Likewise for $r\\leqslant \\rho - t\\rho _n\\leqslant (1- t/4)\\rho $ for big $\\rho $ we have $H_2(r)\\leqslant \\Big (\\big ((1- t/4)\\rho \\big )^{2w}\\big (1+ O(\\rho ^{\\alpha _{m+ 1}- 1+ 0+})\\big )+ O(\\rho ^{\\varkappa + 0+})\\Big )I,$ and no eigenvalue of $H_2(r)$ can coincide with $\\rho ^{2w}$ .", "This implies that all the eigenvalues of $H_2(r)$ lie in the real interval $(\\rho - t\\rho _n, \\rho + t\\rho _n)$ .", "By (REF ) and Lemma REF these eigenvalues coincide with $\\big \\lbrace \\tau (\\rho ; \\mathbf {X}, \\mathbf {\\Phi }): \\mathbf {X}\\in \\Upsilon _{\\mathfrak {V}}(\\nu )\\big \\rbrace $ .", "It remains to show that $H_2(r+ iy)$ is invertible for any nonzero $y$ such that $r+ iy$ is inside or on $\\gamma $ .", "Relation (REF ), Lemma REF (ii), definition (REF ), and bound (REF ) imply that inside and on the contour $\\xi = (r+ iy)\\big (1+ O(\\rho _n^{-1+ \\alpha _{m+ 1}+ 0+})\\big )$ and $\\mathrm {arg\\,}|\\xi |_{\\leqslant \\big (1+ o(1)\\big )\\arcsin (t\\rho _n/\\rho )\\leqslant \\big (1+ o(1)\\big )\\arcsin t\\leqslant t\\big (1+ o(1)\\big ).", "}Hence\\begin{equation*}\\big ||\\xi |_{^{2w}\\big |= \\big ||\\xi |_{^2\\big |^w\\asymp \\rho ^{2w}\\quad \\textrm {and}\\quad \\mathrm {arg\\,}|\\xi |_{^{2w}= w\\arcsin \\frac{2y\\big (r+ \\langle \\mathbf {a}, \\mathbf {\\Phi }\\rangle \\big )}{\\big ||\\xi |_{^2\\big |}\\asymp y\\rho ^{-1},}which implies that\\begin{equation}\\Big |\\mathrm {Im}\\big (|\\xi |_{^{2w}\\big )\\Big |\\gtrsim |y|\\rho ^{2w- 1}.", "}Now for any \\Psi \\in \\textup {{\\textsf {H}}}^{}(\\mathbf {X}, \\mathbf {\\Phi }) with \\Vert \\Psi \\Vert = 1 we have by (\\ref {Im of main symbol}) and (\\ref {derivative estimate})\\begin{equation*}\\begin{split}\\Big \\Vert \\big (H_2(z)- \\rho ^{2w}I\\big )\\Psi \\Big \\Vert &\\geqslant \\Big |\\textrm {Im}\\langle \\big (H_2(z)- \\rho ^{2w}I\\big )\\Psi , \\Psi \\rangle \\Big |\\\\ &\\geqslant \\Big |\\textrm {Im}\\big (|\\xi |_{^{2w}\\big )\\Big |- |y|\\underset{t\\in [0, y]}{\\textrm {sup}}\\big \\Vert W^{\\prime }(r+ it)\\big \\Vert \\gtrsim |y|\\rho ^{2w- 1},}\\end{split}where we have used that for y= 0 the quadratic form of W(z) is real-valued.", "So the kernel of H_2(r+ iy)- \\rho ^{2w} is trivial for y\\ne 0.\\end{equation*}\\end{equation}\\begin{lem}For z\\in \\gamma and l\\in \\mathbb {N}\\begin{equation}(z^{2w}- \\rho ^{2w})^{-l}= \\rho ^{-2wl}\\sum _{j= 0}^\\infty A_{l\\, j}\\Big (\\frac{z- \\rho }{\\rho }\\Big )^{j- l},\\end{equation}where\\begin{equation*}A_{l\\, j}={\\left\\lbrace \\begin{array}{ll}(2w)^{-l}, & j= 0;\\\\\\displaystyle \\frac{1}{(2w)^l}\\sum _{p= 1}^j\\frac{1}{(2w)^{p}}\\binom{-l}{p}\\sum _{\\begin{array}{c}q_1, \\dots , q_p\\geqslant 1\\\\ q_1+ \\cdots +q_p= j\\end{array}}\\binom{2w}{q_1+ 1}\\binom{2w}{q_2+ 1}\\cdots \\binom{2w}{q_p+ 1}, & j> 0.\\end{array}\\right.", "}\\end{equation*}The series in (\\ref {denominator representation}) converges absolutely.\\end{lem}}{\\begin{xmlelement*}{proof}A striaghtforward calculation gives\\begin{equation}\\begin{split}(z^{2w}- \\rho ^{2w})^{-l}&= \\frac{1}{\\rho ^{2wl}}\\bigg (\\Big (1+ \\frac{z- \\rho }{\\rho }\\Big )^{2w}- 1\\bigg )^{-l}= \\frac{1}{\\rho ^{2wl}}\\bigg (\\sum _{q= 1}^\\infty \\binom{2w}{q}\\Big (\\frac{z- \\rho }{\\rho }\\Big )^q\\bigg )^{-l}\\\\ &= \\frac{\\rho ^{-2wl}}{(2w)^l}\\Big (\\frac{z- \\rho }{\\rho }\\Big )^{-l}\\bigg (1+ \\frac{1}{2w}\\sum _{q= 1}^\\infty \\binom{2w}{q+ 1}\\Big (\\frac{z- \\rho }{\\rho }\\Big )^q\\bigg )^{-l}.\\end{split}\\end{equation}If 2w\\in \\mathbb {N}, then the series on the right hand side is finite.", "Otherwise, by (\\ref {contour}) and (\\ref {binomial coefficient}), for z\\in \\gamma the ratio of absolute values of any two sequential terms of the series satisfies\\begin{equation*}\\bigg |\\frac{z-\\rho }{\\rho }\\binom{2w}{q+ 2}\\binom{2w}{q+ 1}^{-1}\\bigg |= \\Big |\\frac{z-\\rho }{\\rho }\\Big |\\frac{|2w- q- 1|}{q+ 2}\\leqslant \\frac{1}{8}, \\qquad q\\geqslant 1.\\end{equation*}So, again by (\\ref {contour}) and (\\ref {binomial coefficient}), we have\\begin{equation*}\\bigg |\\sum _{q= 1}^\\infty \\binom{2w}{q+ 1}\\Big (\\frac{z- \\rho }{\\rho }\\Big )^q\\bigg |< \\bigg |\\binom{2w}{2}\\bigg |\\frac{|z- \\rho |}{\\rho }\\sum _{q= 0}^\\infty \\frac{1}{8^q}\\leqslant \\frac{4w}{7}.\\end{equation*}Thus we can decompose the expression on the right hand side of (\\ref {before progression}) into an absolutely converging series obtaining\\begin{equation*}\\begin{split}&(z^{2w}- \\rho ^{2w})^{-l}= \\frac{\\rho ^{-2wl}}{(2w)^l}\\Big (\\frac{z- \\rho }{\\rho }\\Big )^{-l}\\sum _{p= 0}^\\infty \\binom{-l}{p}\\frac{1}{(2w)^p}\\bigg (\\sum _{q= 1}^\\infty \\binom{2w}{q+ 1}\\Big (\\frac{z- \\rho }{\\rho }\\Big )^q\\bigg )^p\\\\ &= \\frac{\\rho ^{-2wl}}{(2w)^l}\\Big (\\frac{z- \\rho }{\\rho }\\Big )^{-l}\\bigg (1+ \\sum _{j= 1}^\\infty \\Big (\\frac{z- \\rho }{\\rho }\\Big )^{j}\\sum _{p= 1}^j\\frac{1}{(2w)^p}\\binom{-l}{p}\\sum _{\\begin{array}{c}q_1, \\dots , q_p\\geqslant 1\\\\ q_1+ \\cdots +q_p= j\\end{array}}\\binom{2w}{q_1+ 1}\\cdots \\binom{2w}{q_p+ 1}\\bigg ),\\end{split}\\end{equation*}which finishes the proof.\\end{xmlelement*}}}Let S(z):= H_2(z)- z^{2w}I in \\textup {{\\textsf {H}}}^{}(\\mathbf {X}, \\mathbf {\\Phi }).", "Then by (\\ref {eq:nn1}) on \\gamma the symbol of S(z) admits the representaion\\begin{equation}s(z)= \\sum _{v= 1}^\\infty \\binom{w}{v}z^{2w- v}\\Big (2\\langle \\mathbf {a}, \\mathbf {\\Phi }\\rangle + z^{-1}\\big (|\\mathbf {a}|^2+|\\mathbf {X}|^2\\big )\\Big )^v+ w_{\\tilde{k}}(z).\\end{equation}}Relations (\\ref {symbol for S}), (\\ref {derivative estimate}), (\\ref {bound on a}), Lemma~\\ref {lem:propBUps}(ii), and (\\ref {beta and alphas}) imply that everywhere inside and on \\gamma \\begin{equation}\\Big \\Vert \\frac{d^l}{dz^l}S(z)\\Big \\Vert \\lesssim \\rho _n^{2w- 1+ \\alpha _{m+ 1}- l+ 0+},\\ \\ l\\geqslant 0.\\end{equation}\\end{equation*}A version of the Jacobi^{\\prime }s formula states that for any differentiable invertible matrix-valued function F(z) we have\\operatorname{{tr}}\\big [F^{\\prime }(z)F^{-1}(z)\\big ]=\\Big (\\det \\big [F(z)\\big ]\\Big )^{\\prime }\\Big (\\det \\big [F(z)\\big ]\\Big )^{-1}(it can be proved, for example, using the expansion of the determinant along rows and the induction in the size of F).$ Then by Lemma  and the residue theorem $\\begin{split}&\\sum _{\\mathbf {X}\\in \\Upsilon _{\\mathfrak {V}}(\\nu )}\\tau (\\rho ;\\mathbf {X},\\mathbf {\\Phi })^{K+ 1}\\\\&= \\frac{1}{2\\pi i}\\oint _\\gamma z^{K+ 1}\\Big (\\det \\big [H_2(z)- \\rho ^{2w}I\\big ]\\Big )^{\\prime }\\Big (\\det \\big [H_2(z)- \\rho ^{2w}I\\big ]\\Big )^{-1}dz\\\\&= \\frac{1}{2\\pi i}\\oint _\\gamma \\operatorname{{tr}}\\Big [z^{K+ 1} H_2^{\\prime }(z)\\big (H_2(z)- \\rho ^{2w}I\\big )^{-1}\\Big ]dz\\\\&= \\frac{1}{2\\pi i}\\oint _\\gamma \\operatorname{{tr}}\\Big [\\big (2wz^{2w+ K}I+ z^{K+ 1}S^{\\prime }(z)\\big )\\sum _{l=0}^\\infty (-1)^lS^l(z)(z^{2w}- \\rho ^{2w})^{-1- l}\\Big ]dz\\\\&= \\frac{1}{2\\pi i}\\oint _\\gamma \\operatorname{{tr}}\\Big [\\big (2wz^{2w+ K}I+ z^{K+ 1}S^{\\prime }(z)\\big )\\\\ &\\qquad \\times \\sum _{l= -\\infty }^\\infty (z- \\rho )^{-1- l}\\sum _{j= 0}^\\infty (-1)^{l+ j}A_{1+ l+ j\\, j}\\rho ^{1+ l- 2w(1+ l+ j)}S^{l+ j}(z)\\Big ]dz\\\\&= \\sum _{l= 0}^\\infty \\frac{1}{l!", "}\\operatorname{{tr}}\\frac{d^l}{dr^l}\\Big [\\big (2wr^{2w+ K}I+ r^{K+ 1}S^{\\prime }(r)\\big )\\sum _{j= 0}^\\infty (-1)^{l+ j}A_{1+ l+ j\\, j}\\rho ^{1+ l- 2w(1+ l+ j)}S^{l+ j}(r)\\Big ]\\Big |_{r= \\rho }.\\end{split}$ We can restrict the summation on the RHS of (REF ) to $l+ j\\leqslant l_0:= \\big (M+ K+ d+ 1+ (d- 1)\\alpha _{d- 1}- 2w\\big )/(1- \\alpha _{m+ 1}).$ Indeed, using the trivial fact that for any linear operator $A$ in the finite dimensional Hilbert space spanned by $\\mathbf {e}_{\\theta }$ with $\\theta \\in \\Upsilon _{\\mathfrak {V}}(\\nu )$ $|\\operatorname{{tr}}A|\\leqslant \\Vert A\\Vert \\operatorname{{card}}\\Upsilon _{\\mathfrak {V}}(\\nu ),$ estimate (), and relation (REF ) we can see that the sum of the terms in (REF ) with $l+ j> l_0$ contributes only to the order $O(\\rho _n^{-M+ 2w- d})$ in (REF ), and thus after integration in $\\mathbf {\\Phi }$ the corresponding term can be included into the remainder $R_{\\tilde{k}}$ of Section .", "Formula (REF ) shows that in order to compute the contribution to the density of states from $\\Xi (\\mathfrak {V})_p$ , we need to integrate the RHS of (REF ) against $d\\nu $ and $d\\mathbf {\\Phi }$ .", "We are going to integrate against $d\\mathbf {\\Phi }$ first: $\\begin{split}&\\int _{\\mathcal {M}_p}d\\mathbf {\\Phi }\\int _{\\Omega (\\mathfrak {V})}\\big (\\operatorname{{card}}\\Upsilon _{\\mathfrak {V}}(\\nu )\\big )^{-1}\\sum _{\\mathbf {X}\\in \\Upsilon _{\\mathfrak {V}}(\\nu )}\\tau (\\rho ; \\mathbf {X}, \\mathbf {\\Phi })^{K+ 1}\\\\ &=\\int _{\\Omega (\\mathfrak {V})}d\\nu \\big (\\operatorname{{card}}\\Upsilon _{\\mathfrak {V}}(\\nu )\\big )^{-1}\\sum _{l=0}^{l_0}\\sum _{j= 0}^{l_0- l}\\frac{(-1)^{l+ j}}{l!", "}A_{1+ l+ j\\,j}\\rho ^{1+ l- 2w(1+ l+ j)}\\\\ &\\times \\operatorname{{tr}}\\frac{d^l}{dr^l}\\Big [\\int _{\\mathcal {M}_p}d\\mathbf {\\Phi }\\big (2wr^{2w+ K}I+r^{K+ 1}S^{\\prime }(r)\\big )S^{l+ j}(r)\\Big ]\\Big |_{r= \\rho }+ O(\\rho _n^{-M+ 2w-d})\\\\ &= O(\\rho _n^{-M+ 2w- d})+\\int _{\\Omega (\\mathfrak {V})}\\frac{d\\nu }{\\operatorname{{card}}\\Upsilon _{\\mathfrak {V}}(\\nu )}\\sum _{l=0}^{l_0}\\sum _{j= 0}^{l_0- l}\\frac{(-1)^{l+ j}}{l!", "}A_{1+ l+ j\\, j}\\rho ^{1+ l- 2w(1+ l+ j)}\\\\&\\times \\operatorname{{tr}}\\bigg [\\frac{d^l}{dr^l}\\Big (2wr^{2w+ K}\\int _{\\mathcal {M}_p} S^{l+j}(r)d\\mathbf {\\Phi }- \\frac{(K+1)r^{K}}{l+ j+ 1}\\int _{\\mathcal {M}_p}S^{l+ j+1}(r)d\\mathbf {\\Phi }\\Big )\\\\ &+ \\frac{d^{l+ 1}}{dr^{l+1}}\\Big (\\frac{r^{K+ 1}}{l+ j+ 1}\\int _{\\mathcal {M}_p}S^{l+ j+1}(r)d\\mathbf {\\Phi }\\Big )\\bigg ]\\bigg |_{r= \\rho }.\\end{split}$ We will prove that the integrand of the exterior integral in (REF ) is a convergent series of products of powers of $\\rho $ and $\\ln \\rho $ .", "The coefficients in front of all terms will be bounded functions of $\\mathbf {X}$ , so afterwards we will just integrate these coefficients to obtain the desired asymptotic expansion.", "Let us discuss, how $S(r)$ depends on $\\rho $ , $\\mathbf {X}$ and $\\mathbf {\\Phi }$ .", "In order to do this, we first look again at (REF ).", "As follows from Remark REF , the product $e_{\\theta _q^{\\prime \\prime }}\\varphi _{\\theta _q^{\\prime \\prime }}$ does not depend on $r$ and $\\mathbf {\\Phi }$ , and by (REF ) $\\Vert \\widehat{\\nabla ^{\\nu }e_{\\theta _q^{\\prime \\prime }}\\varphi _{\\theta _q^{\\prime \\prime }}}\\Vert _{\\textup {{\\textsf {L}}}_{\\infty }( \\mathbb {R}^d)}\\lesssim \\rho _n^{-\\nu \\beta }.$ For any $\\eta \\in \\Theta _{s+ 1}$ the application of the finite difference operator $\\nabla _{\\eta }$ to a polynomial decreases its degree by 1.", "Hence formula (REF ) ensures that $\\begin{split}\\widehat{(\\nabla ^{\\nu }b)}(\\theta , \\xi + \\phi )= \\sum _{\\iota \\in \\widetilde{J}}\\sum _{i= \\nu }^\\infty \\sum _{\\begin{array}{c}n_1, \\dots , n_{K+ 1}\\geqslant 0\\\\ n_1+ \\cdots + n_{K+ 1}\\leqslant i\\\\ j_1, \\dots , j_d\\geqslant 0\\\\ j_1+ \\cdots + j_d\\leqslant i- \\nu \\end{array}}\\widetilde{C}^{\\iota \\, j_1\\cdots j_d}_{i\\, n_1\\cdots n_{K+ 1}}(\\mathbf {X}; \\theta )r^{\\iota - i}\\phi _1^{j_1}\\cdots \\phi _d^{j_d}\\prod _{a= 1}^{K+ 1}(\\sin \\Phi _a)^{n_a}.\\end{split}$ Here $\\widetilde{C}^{\\iota \\, j_1\\cdots j_d}_{i\\, n_1\\cdots n_{K+ 1}}(\\mathbf {X}; \\theta )$ depend on the coefficients of (REF ) and satisfy a uniform estimate $\\big |\\widetilde{C}^{\\iota \\, j_1\\cdots j_d}_{i\\, n_1\\cdots n_{K+ 1}}(\\mathbf {X}; \\theta )\\big |\\lesssim \\rho _n^{(i- \\nu - j_1- \\cdots - j_d)(\\alpha _{m+ 1}+ 0+)}.$ Now $\\widehat{(\\nabla ^{\\nu }\\tilde{\\chi }_{\\theta })}(\\xi )= \\sum _{\\tilde{\\nu }= 0}^{\\nu }\\widehat{(\\nabla ^{\\tilde{\\nu }}e_{\\theta }\\varphi _{\\theta })}\\Big (\\xi + \\sum _{p= 1}^{\\tilde{\\nu }}\\eta _k\\Big )\\widehat{\\Big (\\nabla ^{\\nu - \\tilde{\\nu }}\\big (|\\cdot + \\theta |_{^{2w}- |\\cdot |_{^{2w}\\big )^{-1}\\Big )}(\\xi ).", "}The factors \\widehat{(\\nabla ^{\\tilde{\\nu }}e_{\\theta }\\varphi _{\\theta })} satisfy the estimate (\\ref {Delta cutoffs}).", "For \\eta \\in \\Theta _{s+ 1} we have\\begin{equation*}\\begin{split}&\\widehat{\\Big (\\nabla _{\\eta }\\big (|\\cdot + \\theta |_{^{2w}- |\\cdot |_{^{2w}\\big )^{-1}\\Big )}(\\xi )\\\\ &= \\big (|\\xi + \\eta + \\theta |_{^{2w}- |\\xi + \\eta |_{^{2w}\\big )^{-1}\\big (|\\xi + \\theta |_{^{2w}- |\\xi |_{^{2w}\\big )^{-1}G(\\xi ; \\theta , \\eta ),}}where\\begin{equation*}\\begin{split}&G(\\xi ; \\theta , \\eta ):= |\\xi + \\theta |_{^{2w}- |\\xi |_{^{2w}- |\\xi + \\eta + \\theta |_{^{2w}+ |\\xi + \\eta |_{^{2w}\\\\ &= -2w\\langle \\eta , \\theta \\rangle |\\xi |_{^{2w- 2}\\\\ &+ \\sum _{j= 2}^\\infty \\binom{w}{j}|\\xi |_{^{2w- 2j}\\Big (\\big (2\\langle \\xi , \\theta \\rangle + |\\theta |^2\\big )^j- \\big (2\\langle \\xi , \\eta + \\theta \\rangle + |\\eta + \\theta |^2\\big )^j+ \\big (2\\langle \\xi , \\eta \\rangle + |\\eta |^2\\big )^j\\Big ).", "}}In analogy to (\\ref {Delta b}) we have\\begin{equation*}\\begin{split}\\widehat{\\big (\\nabla ^{\\nu }G(\\cdot ; \\theta , \\eta )\\big )}(\\xi )= \\sum _{i= \\nu }^\\infty \\sum _{\\begin{array}{c}n_1, \\dots , n_{K+ 1}\\geqslant 0\\\\ n_1+ \\cdots + n_{K+ 1}\\leqslant i+ 2\\end{array}}\\widetilde{C}^i_{n_1\\cdots n_{K+ 1}}(\\mathbf {X}; \\theta , \\eta )r^{2w- 2- i}\\prod _{a= 1}^{K+ 1}(\\sin \\Phi _a)^{n_a},\\end{split}\\end{equation*}with\\begin{equation}\\big |\\widetilde{C}^i_{n_1\\cdots n_{K+ 1}}(\\mathbf {X}; \\theta , \\eta )\\big |\\lesssim \\rho _n^{(i- \\nu )(\\alpha _{m+ 1}+ 0+)+ 0+}.\\end{equation}Altogether, applying relations (\\ref {Delta cutoffs}) -- (\\ref {Delta G coefficients}) to (\\ref {eq:newy}) and (\\ref {symbol equation}) we obtain\\begin{equation}\\begin{split}&w_{\\tilde{k}}(\\theta , \\xi )\\\\ &= \\sum _{s= 0}^{\\tilde{k}- 1}\\sum _{\\iota _0, \\dots , \\iota _s\\in \\widetilde{J}}\\sum _{\\mu = 0}^s\\sum _{\\begin{array}{c}\\eta _1, \\dots , \\eta _{s+ \\mu }\\in \\Theta _{s+ 1}\\\\ \\theta _1, \\dots , \\theta _{s+ \\mu }\\in \\Theta ^{\\prime }_{s+ 1}\\end{array}}\\sum _{p= 0}^{s- \\mu }\\sum _{i= 0}^\\infty \\sum _{\\begin{array}{c}n_1, \\dots , n_{K+ 1}\\geqslant 0\\\\ n_1+ \\cdots + n_{K+ 1}\\leqslant 2\\mu + p+ i\\end{array}} C_{s\\, \\mu \\, p\\, i\\, \\iota _0\\cdots \\iota _s\\, n_1\\cdots n_{K+ 1}}^{\\eta _1\\cdots \\eta _{s+ \\mu }\\, \\theta _1\\cdots \\theta _{s+ \\mu }}(\\mathbf {X}; \\theta )\\\\ &\\times r^{(2w- 2)\\mu + \\iota _0+ \\cdots + \\iota _s- p- i}\\prod _{a= 1}^{K+ 1}(\\sin \\Phi _a)^{n_a}\\prod _{v= 1}^{s+ \\mu }\\big (|\\xi + \\eta _v+ \\theta _v|_{^{2w}- |\\xi + \\eta _v|_{^{2w}\\big )^{-1},}}where\\begin{equation*}\\big |C_{s\\, \\mu \\, p\\, i\\, \\iota _0\\cdots \\iota _s\\, n_1\\cdots n_{K+ 1}}^{\\eta _1\\cdots \\eta _{s+ \\mu }\\, \\theta _1\\cdots \\theta _{s+ \\mu }}(\\mathbf {X}; \\theta )\\big |\\lesssim \\rho _n^{i(\\alpha _{m+ 1}+ 0+)- (s- \\mu - p)\\beta + 0+}.\\end{equation*}According to Lemma~\\ref {denominator lemma},\\begin{equation}\\begin{split}&\\big (|\\xi + \\eta _v+ \\theta _v|_{^{2w}- |\\xi + \\eta _v|_{^{2w}\\big )^{-1}\\\\ &= r^{2- 2w}\\big (2\\langle \\xi , \\theta _v\\rangle + 2\\langle \\eta _v, \\theta _v\\rangle + |\\theta _v|^2\\big )^{-1}\\\\ &\\times \\sum _{i= 0}^\\infty \\sum _{\\begin{array}{c}n_1, \\dots , n_{K+ 1}\\geqslant 0\\\\ n_1+ \\cdots + n_{K+ 1}\\leqslant i\\end{array}}C_{n_1\\cdots n_{K+ 1}}^i(\\mathbf {X}; \\eta _v, \\theta _v)r^{-i}\\prod _{a= 1}^{K+ 1}(\\sin \\Phi _a)^{n_a},}}and here\\begin{equation*}\\big |C_{n_1\\cdots n_{K+ 1}}^i(\\mathbf {X}; \\eta _v, \\theta _v)\\big |\\lesssim \\rho _n^{i(\\alpha _{m+ 1}+ 0+)}.\\end{equation*}If now subsitute (\\ref {old to new denominator}) to (\\ref {W series}), we obtain\\begin{equation}\\begin{split}&w_{\\tilde{k}}(\\theta , \\xi )\\\\ &= \\sum _{s= 0}^{\\tilde{k}- 1}\\sum _{\\iota _0, \\dots , \\iota _s\\in \\widetilde{J}}\\sum _{\\mu = 0}^s\\sum _{\\begin{array}{c}\\eta _1, \\dots , \\eta _{s+ \\mu }\\in \\Theta _{s+ 1}\\\\ \\theta _1, \\dots , \\theta _{s+ \\mu }\\in \\Theta ^{\\prime }_{s+ 1}\\end{array}}\\sum _{p= 0}^{s- \\mu }\\sum _{i= 0}^\\infty \\sum _{\\begin{array}{c}n_1, \\dots , n_{K+ 1}\\geqslant 0\\\\ n_1+ \\cdots + n_{K+ 1}\\\\ \\leqslant 2\\mu + p+ i\\end{array}} C_{s\\, \\mu \\, p\\, i\\, \\iota _0\\cdots \\iota _s\\, n_1\\cdots n_{K+ 1}}^{\\eta _1\\cdots \\eta _{s+ \\mu }\\, \\theta _1\\cdots \\theta _{s+ \\mu }}(\\mathbf {X}; \\theta , \\eta _v, \\theta _v)\\\\ &\\times r^{(2- 2w)s+ \\iota _0+ \\cdots + \\iota _s- p- i}\\prod _{a= 1}^{K+ 1}(\\sin \\Phi _a)^{n_a}\\prod _{v= 1}^{s+ \\mu }\\big (2\\langle \\xi , \\theta _v\\rangle + 2\\langle \\eta _v, \\theta _v\\rangle + |\\theta _v|^2\\big )^{-1},\\end{split}\\end{equation}with\\begin{equation*}\\big |C_{s\\, \\mu \\, p\\, i\\, \\iota _0\\cdots \\iota _s\\, n_1\\cdots n_{K+ 1}}^{\\eta _1\\cdots \\eta _{s+ \\mu }\\, \\theta _1\\cdots \\theta _{s+ \\mu }}(\\mathbf {X}; \\theta , \\eta _v, \\theta _v)\\big |\\lesssim \\rho _n^{i(\\alpha _{m+ 1}+ 0+)- (s- \\mu - p)\\beta + 0+}.\\end{equation*}\\end{split}The first sum in (\\ref {symbol for S}) can be written in the form\\begin{equation}\\begin{split}&\\sum _{v= 1}^\\infty \\binom{w}{v}z^{2w- v}\\Big (2\\langle \\mathbf {a}, \\mathbf {\\Phi }\\rangle + z^{-1}\\big (|\\mathbf {a}|^2+ |\\mathbf {X}|^2\\big )\\Big )^v\\\\ &= \\sum _{i= 0}^\\infty \\sum _{\\begin{array}{c}n_1, \\dots , n_{K+ 1}\\geqslant 0\\\\ n_1+ \\cdots + n_{K+ 1}\\leqslant i+ 1\\end{array}}C^i_{n_1\\cdots n_{K+ 1}}(\\mathbf {X})z^{2w- 1- i}\\prod _{a= 1}^{K+ 1}(\\sin \\Phi _a)^{n_a},\\end{split}\\end{equation}where\\begin{equation*}\\big |C^i_{n_1\\cdots n_{K+ 1}}(\\mathbf {X})\\big |\\lesssim \\rho _n^{(i+ 1)(\\alpha _{m+ 1}+ 0+)}.\\end{equation*}\\end{equation}Substituting (\\ref {new W series}) and (\\ref {first sum for S}) into (\\ref {symbol for S}) we can calculate the series for the symbol of the operator S^f for f\\in \\mathbb {N}:\\begin{equation}\\begin{split}&\\widehat{s^f}(\\theta , \\xi )= \\sum _{\\begin{array}{c}\\theta _1, \\dots , \\theta _f\\in \\Theta _{\\tilde{k}}\\\\ \\phi _1, \\dots , \\phi _{f}\\in \\Theta _{\\tilde{k}}\\end{array}}C_{\\phi _1\\cdots \\phi _{f}}^{\\theta _1\\cdots \\theta _{f}}(\\theta )\\prod _{g= 1}^f\\hat{s}(\\theta _g, \\xi + \\phi _g)\\\\ &= \\sum _{\\nu = 0}^{f}\\sum _{h= \\nu }^{\\nu \\tilde{k}}\\sum _{\\iota _1, \\dots , \\iota _h\\in \\widetilde{J}}\\sum _{\\mu = 0}^{h- \\nu }\\sum _{\\begin{array}{c}\\phi _1, \\dots , \\phi _{h- \\nu + \\mu }\\in \\Theta _{\\tilde{k}}\\\\ \\theta _1, \\dots , \\theta _{h- \\nu + \\mu }\\in \\Theta ^{\\prime }_{\\tilde{k}}\\end{array}}\\sum _{p= 0}^{h- \\nu - \\mu }\\sum _{i= 0}^\\infty \\sum _{\\begin{array}{c}n_1, \\dots , n_{K+ 1}\\geqslant 0\\\\ n_1+ \\cdots + n_{K+ 1}\\leqslant 2\\mu + p+ i+ f- \\nu \\end{array}}C(\\mathbf {X}; \\theta , \\dots )\\\\ &\\times r^{(2- 2w)h+ (2w- 1)f- \\nu + \\iota _1+ \\cdots + \\iota _h- p- i}\\prod _{a= 1}^{K+ 1}(\\sin \\Phi _a)^{n_a}\\prod _{v= 1}^{h- \\nu + \\mu }\\big (2\\langle \\xi , \\theta _v\\rangle + 2\\langle \\phi _v, \\theta _v\\rangle + |\\theta _v|^2\\big )^{-1},\\end{split}\\end{equation}with\\begin{equation*}\\big |C(\\mathbf {X}; \\theta , \\dots )\\big |\\lesssim \\rho _n^{(f- \\nu + i)(\\alpha _{m+ 1}+ 0+)- (h- \\nu - \\mu - p)\\beta + 0+}.\\end{equation*}\\end{split}Note that the last product on the right hand side of (\\ref {S^f}) is of the form\\begin{equation*}\\prod _{t =1}^T\\Big (l_t +\\rho \\sum _{q} b_q^t\\sin \\Phi _q\\Big )^{-k_t}.\\end{equation*}Here we have expanded the inner products \\langle \\xi ,\\theta _v\\rangle using Lemma \\ref {lem:products}(ii).The coefficients \\lbrace b_q^t\\rbrace in the decomposition (\\theta _v)_{\\mathfrak {V}^{\\perp }}= \\sum _q b_q^t\\tilde{\\mu }_q are all of the same sign and satisfy (\\ref {eq:n10}).", "Without loss of generality we may assume that all b_q^t are non-negative.The numbersl_t= l(b_1^t, \\dots , b_{K+ 1}^t):= 2L_{m+ 1}\\sum _{q}b_q^t+ 2\\langle \\mathbf {X}, (\\theta _v)_{\\mathfrak {V}}\\rangle + 2\\langle \\phi _v, \\theta _v\\rangle + |\\theta _v|^2satisfy\\rho _n^{\\alpha _{m+1}}\\rho _n^{0-}\\lesssim l_t\\lesssim \\rho _n^{\\alpha _{m+1}}\\rho _n^{0+}, since\\big |2\\langle \\mathbf {X},(\\theta _v)_{\\mathfrak {V}}\\rangle + 2\\langle \\phi _v, \\theta _v\\rangle + |\\theta _v|^2\\big |\\lesssim \\rho _n^{\\alpha _m+ 0+}.This numbers depend on \\mathbf {X}, but not on \\mathbf {\\Phi } or \\rho .", "The numbers k_t= k(b_1^t, \\dots , b_{K+ 1}^t) are positive, integer, and independent of \\xi .\\end{equation}The following lemma is identical to Lemma 10.4 of \\cite {ParSht2}, where for our purposes we have replaced the explicit constants 1/2 and 2/3 by \\vartheta and \\varsigma , respectively.", "}\\begin{lem}For 1\\leqslant K\\leqslant d-1; n_1, \\dots , n_{K+ 1}\\in \\mathbb {N}_0; k_1, \\dots , k_T\\in \\mathbb {N} let Q:= \\sum _{t= 1}^Tk_t,\\begin{equation*}\\hat{J}_K:= \\int \\limits _{\\mathcal {M}_p}\\frac{(\\sin \\Phi _1)^{n_1}\\dots (\\sin \\Phi _K)^{n_K}(\\sin \\Phi _{K+ 1})^{n_{K+ 1}}\\,d\\mathbf {\\Phi }}{\\prod _{t= 1}^T \\big (l_t+ \\rho \\sum _{j= 1}^{K+ 1} b_j^t \\sin \\Phi _j\\big )^{k_t}}.\\end{equation*}Then there exist positive numbers \\delta _0, p_K, and q_K depending only on the constants (\\ref {beta and alphas}) and K such that\\begin{equation*}\\hat{J}_K= \\sum _{q= 0}^K(\\ln \\rho )^q\\sum _{p= 0}^\\infty e(p,q){\\rho }^{-p},\\end{equation*}where\\begin{equation*}\\big |e(p,q)\\big |\\lesssim \\rho _n^{(\\varsigma - p_K)p}\\rho _n^{-Q\\beta }.\\end{equation*}These estimates are uniform in the following regions of variables:\\begin{equation*}\\rho _n^{\\beta }\\lesssim l_t\\lesssim \\rho _n^{\\vartheta },\\ \\ \\rho _n^{-\\delta _0}\\lesssim b_j^t\\lesssim \\rho _n^{\\delta _0},\\ \\ \\rho _n^{\\varsigma - q_K}<{\\rho }.\\end{equation*}\\end{lem}}Now using Lemma~\\ref {lem:integral1} we can compute the integrals of (\\ref {S^f}) over the domain \\lbrace \\mathbf {\\Phi }\\in \\mathcal {M}_p\\rbrace (recall that this integration is not needed for K= 0 by Remark~\\ref {K= 0 remark}).", "Substituting the result into (\\ref {integration in Phi}), integrating in d\\nu over \\Omega (\\mathfrak {V}), and taking into account (\\ref {eq:n4}) and (\\ref {eq:n6}) we obtain in the region 2\\rho _n/3< \\rho < 6\\rho _n\\begin{equation*}\\begin{split}&\\operatorname{{vol}}\\mathcal {A}^+_p(\\rho )- \\operatorname{{vol}}\\mathcal {A}^-_p(\\rho )\\\\ &= \\sum _{q= 0}^K\\sum _{h= 0}^{(l_0+ 1)\\tilde{k}}\\sum _{\\iota _1, \\dots , \\iota _h\\in \\widetilde{J}}\\sum _{j= 0}^\\infty C_{q\\, h\\, j}^{\\iota _1\\cdots \\iota _h}\\rho ^{K+ 1+ (2- 2w)h+ \\iota _1+ \\cdots + \\iota _h- j}(\\ln \\rho )^q+ O(\\rho _n^{-M+ 2w- d}),\\end{split}\\end{equation*}with the coefficients satisfying\\begin{equation*}|C_{q\\, h\\, j}^{\\iota _1\\cdots \\iota _h}|\\lesssim \\rho _n^{-2\\beta h+ \\varsigma j}.\\end{equation*}This, together with equations (\\ref {eq:n2}), (\\ref {eq:densityh3}), Lemma~\\ref {H1H2 lemma}, relation (\\ref {beta and alphas}), Section~11 of \\cite {ParSht2}, and the observation that the number of different quasi-lattice subspaces \\mathfrak {V} is \\lesssim \\rho _n^{0+}, completes the proof of Lemma \\ref {main_lem} and, thus, of our main theorem in the case of B= \\widetilde{B} with the symbol satisfying (\\ref {tilde b}).", "As explained at the end of Section~\\ref {reduction section}, the summation over \\widetilde{J} may be replaced by summation over J_0.", "}It remains to relax the assumptions on B.", "This will be done in the subsequent section.", "}\\end{split}\\section {Approximation}\\end{equation*}In this section we prove Lemma~\\ref {main_lem} and thus Theorem~\\ref {main_thm} for general B using the fact that the proof is complete for \\widetilde{B} whose symbol fulfills the extra assumption (\\ref {tilde b}).", "}}\\subsection *{1.", "}Given B satisfying the hypothesis of Theorem~\\ref {main_thm} and the number M, we fix the values of k and \\tilde{k} in such a way that Lemma~\\ref {main_lem} holds true for H= (-\\Delta )^w+ \\widetilde{B}, where the symbol \\tilde{b} of \\widetilde{B} satisfying (\\ref {tilde b}) is constructed at the end of Section~\\ref {reduction section}.For R> 0 let us define (recall (\\ref {CP}))\\begin{equation*}\\mathcal {P}_R:= \\mathcal {P}^L(\\mathcal {B}_{R}), \\quad \\mathcal {P}_R^c:= \\mathcal {P}^L( \\mathbb {R}^d\\setminus \\mathcal {B}_{R})\\end{equation*}}We start by estimating the quadratic form of B- \\widetilde{B}.", "For any \\psi \\in \\textup {{\\textsf {H}}}^{2w}( \\mathbb {R}^d)\\begin{equation}\\begin{split}\\big |\\langle \\psi , (B- \\widetilde{B})\\psi \\rangle \\big |&\\leqslant \\big |\\langle \\psi , \\mathcal {P}_{R_0}(B- \\widetilde{B})\\mathcal {P}_{R_0}\\psi \\rangle \\big |+ \\big |\\langle \\psi , \\mathcal {P}_{R_0}(B- \\widetilde{B})\\mathcal {P}_{R_0}^c\\psi \\rangle \\big |\\\\ &+ \\big |\\langle \\psi , \\mathcal {P}_{R_0}^c(B- \\widetilde{B})\\mathcal {P}_{R_0}\\psi \\rangle \\big |+ \\big |\\langle \\psi , \\mathcal {P}_{R_0}^c(B- \\widetilde{B})\\mathcal {P}_{R_0}^c\\psi \\rangle \\big |.\\end{split}\\end{equation}By Condition (\\ref {eq:condB2}), the symbol of (B- \\widetilde{B})\\mathcal {P}_{R_0}^c satisfies\\begin{equation*}{\\,\\vrule depth4pt height11pt width1pt}\\,(b- \\tilde{b})\\operatorname{\\mathbb {1}}_{ \\mathbb {R}^d\\setminus \\mathcal {B}_{R_0}}{\\vrule depth4pt height11pt width1pt\\,}_{\\varkappa /2,\\, 0}^{(\\varkappa /\\beta )}< \\rho _n^{-k}.\\end{equation*}Now Propositions~\\ref {bound:prop} and \\ref {product:prop} imply that\\begin{equation}\\big \\Vert (-\\Delta + 1)^{-\\varkappa /4}(B- \\widetilde{B})(-\\Delta +1)^{-\\varkappa /4}\\mathcal {P}_{R_0}^c\\big \\Vert \\leqslant C\\rho _n^{-k}.\\end{equation}Hence\\begin{equation}\\begin{split}&\\big |\\langle \\psi , \\mathcal {P}_{R_0}(B- \\widetilde{B})\\mathcal {P}_{R_0}^c\\psi \\rangle \\big |\\\\ &= \\big |\\langle (-\\Delta + 1)^{\\varkappa /4}\\psi , \\mathcal {P}_{R_0}(-\\Delta + 1)^{-\\varkappa /4}(B- \\widetilde{B})\\mathcal {P}_{R_0}^c(-\\Delta + 1)^{-\\varkappa /4}(-\\Delta + 1)^{\\varkappa /4}\\psi \\rangle \\big |\\\\ &\\leqslant C\\rho _n^{-k}\\langle \\psi , (-\\Delta + 1)^{\\varkappa /2}\\psi \\rangle ,\\end{split}\\end{equation}and the analogous estimates hold for the last two terms in (\\ref {B correction estimate}).", "Thus (\\ref {B correction estimate}) implies\\begin{equation}|B- \\widetilde{B}|\\leqslant B^{(k)},\\end{equation}where B^{(k)} is the operator of multiplication by the function\\begin{equation}b^{(k)}(\\xi ):= {\\left\\lbrace \\begin{array}{ll}\\Vert b\\Vert _{\\textup {{\\textsf {L}}}_{\\infty }( \\mathbb {R}^d\\times \\mathcal {B}_{R_0})}+ \\Vert \\tilde{b}\\Vert _{\\textup {{\\textsf {L}}}_{\\infty }( \\mathbb {R}^d\\times \\mathcal {B}_{R_0})}, &|\\xi |\\leqslant R_0,\\\\ C\\rho _n^{-k}\\big (1+ |\\xi |^2\\big )^{\\varkappa /2}, &|\\xi |> R_0\\end{array}\\right.", "}\\end{equation}in the momentum space.", "}In view of Lemma~\\ref {norms lemma}(a), we conclude that\\begin{equation}N\\big ((-\\Delta )^w+ B, \\lambda \\big )\\gtrless N\\big ((-\\Delta )^w+ \\widetilde{B} \\pm B^{(k)}, \\lambda \\big ).\\end{equation}So to prove (\\ref {eq:main_lem1}) it will be sufficient to show that for \\rho \\in I_n (which we assume everywhere below) the right hand side of (\\ref {up and down}) does not differ from N\\big ((-\\Delta )^w+ \\widetilde{B}, \\rho ^{2w}\\big ) by more than O(\\rho _n^{-M}).By (\\ref {eq:main_lem1}) and Remark~\\ref {spurious remark}, it is enough to prove that\\begin{equation}N\\big ((-\\Delta )^w +\\widetilde{B} \\pm B^{(k)}, \\lambda \\big ) =N\\big ((-\\Delta )^w +\\widetilde{B}, \\lambda +O(\\rho _n^{2w -d -M})\\big ).\\end{equation}\\end{split}\\subsection *{2.", "}We note that for\\begin{equation}R_*:= (4\\rho _n^{d+ M})^{1/(w- \\varkappa )}\\end{equation}we have\\begin{equation}N\\big ((-\\Delta )^w+ \\widetilde{B} \\pm B^{(k)}, \\lambda \\big )= N\\big ((-\\Delta )^w+ \\widetilde{B} \\pm \\mathcal {P}_{R_0}B^{(k)} \\pm \\mathcal {P}_{R_*}^cB^{(k)}, \\lambda + O(\\rho _n^{2w -d -M})\\big ).\\end{equation}Indeed,\\begin{equation*}\\big \\Vert (\\mathcal {P}_{R_*}- \\mathcal {P}_{R_0})B^{(k)}\\big \\Vert = C\\rho _n^{-k}(1+ R_*^2)^{\\varkappa /2}= O(\\rho _n^{2w -d -M})\\end{equation*}in view of (\\ref {k fist condition}).\\end{equation*}}\\subsection *{3.", "}Now we are going to prove that\\begin{equation}N\\big ((-\\Delta )^w+ \\widetilde{B} \\pm \\mathcal {P}_{R_0}B^{(k)} \\pm \\mathcal {P}_{R_*}^cB^{(k)}, \\lambda \\big )= N\\big ((-\\Delta )^w+ \\widetilde{B} \\pm \\mathcal {P}_{R_*}^cB^{(k)}, \\lambda + O(\\rho _n^{2w -d -M})\\big ).\\end{equation}This will be done with the help of the following lemma, which is a development of Lemma~3.1 from \\cite {Par}.$ Lemma 10.6 Let $H_0$ , $V$ , $A$ be pseudo–differential operators with almost–periodic coefficients.", "Suppose that $H:= H_0+ V$ is elliptic, selfadjoint and bounded below, and there exists a collection of orthogonal projections $\\lbrace P_l\\rbrace _{l= 0}^L$ commuting with $H_0$ such that $\\sum _{l =0}^LP_l =I \\quad \\textrm {and} \\quad V_{n\\, l} :=P_nVP_l =0 \\quad \\textrm {for} \\quad |l -n|> 1.$ Suppose that $A= P_0A$ and that $a:= \\Vert A\\Vert < \\infty .$ At last, suppose that for $\\lambda \\in \\mathbb {R}$ $D_l:= \\operatorname{{dist}}\\big (\\lambda , \\sigma (P_lHP_l)\\big ) -(4+ 2^{5 -L})a >0, \\quad l =0, \\dots , L -1$ and $\\max _{0 \\leqslant l \\leqslant L -1}\\big (a+ \\Vert V_{l\\, l -1}\\Vert + \\Vert V_{l\\, l+ 1}\\Vert \\big )/D_l\\leqslant 1/4.$ Then for $\\varepsilon := 2^{4 -L}a$ we have $N(H, \\lambda -\\varepsilon ) \\leqslant N(H+ A, \\lambda ) \\leqslant N(H, \\lambda +\\varepsilon ).$ We will prove the first inequality; the second follows by interchanging the roles of $H_0$ and $H_0 +A$ .", "Let $E_\\lambda $ be the spectral projection of $(-\\infty , \\lambda ]$ for $H$ .", "By Lemma 4.1 of [5] it is enough to prove that $\\langle \\phi , (H+ A)\\phi \\rangle \\leqslant \\lambda \\Vert \\phi \\Vert ^2 \\quad \\textrm {for every}\\quad \\phi \\in E_{\\lambda - \\varepsilon }\\textup {{\\textsf {L}}}_{2}( \\mathbb {R}^d).$ Let $\\delta :=\\min \\lbrace a, 2^{-3 -L}\\min _{0 \\leqslant l \\leqslant L -1}D_l\\rbrace , \\quad K :=[2a/\\delta ] +2,$ so that $2a \\leqslant (K -1)\\delta \\leqslant 3a$ and by (REF ) $K -1\\leqslant 3a/\\delta \\leqslant 3\\max \\lbrace 1, 2^{L +3}a/\\min _{0 \\leqslant l \\leqslant L -1}D_l\\rbrace \\leqslant 2^{L +3}.$ For $\\phi \\in E_{\\lambda - \\varepsilon }$ introduce $\\begin{split}\\phi ^k &:=(E_{\\lambda -\\varepsilon -(k -1)\\delta } -E_{\\lambda -\\varepsilon -k\\delta })\\phi , \\quad k =1, \\dots , K -1, \\\\ \\phi ^K&:= E_{\\lambda -\\varepsilon -(K- 1)\\delta }\\phi , \\quad \\phi ^{\\prime } :=\\phi -\\phi ^K =\\sum _{k =1}^{K -1}\\phi ^k.\\end{split}$ Then $\\phi =\\sum _{k =1}^K\\oplus \\phi ^k$ and, letting $\\eta ^k :=H\\phi ^k -\\big (\\lambda -\\varepsilon -(k -1)\\delta \\big )\\phi ^k, \\quad k =1, \\dots , K -1,$ we have $\\Vert \\eta ^k\\Vert \\leqslant \\delta \\Vert \\phi ^k\\Vert .$ Let $P_{-1} := P_{L +1} := 0$ .", "Projecting (REF ) with $P_l$ we obtain $\\eta _l^k= V_{l\\, l -1}\\phi _{l -1}^k+ \\Big (P_lHP_l- \\big (\\lambda -\\varepsilon -(k -1)\\delta \\big )\\Big )\\phi ^k_l+ V_{l\\, l +1}\\phi ^k_{l +1}, \\quad l =0, \\dots , L,$ and thus by (REF ), (REF ) and (REF ) $\\begin{split}\\Vert \\phi ^k_l\\Vert &\\leqslant \\big (\\Vert \\eta ^k_l\\Vert +\\Vert V_{l\\, l -1}\\Vert \\Vert \\phi _{l -1}^k\\Vert +\\Vert V_{l\\, l +1}\\Vert \\Vert \\phi _{l +1}^k\\Vert \\big )/D_l \\\\ &\\leqslant 2^{-3 -L}\\Vert \\phi ^k\\Vert +\\Vert \\phi ^k_{l -1}\\Vert /4 +\\Vert \\phi ^k_{l +1}\\Vert /4, \\quad l =0, \\dots , L -1.\\end{split}$ By induction, starting from $l =0$ we obtain $\\Vert \\phi ^k_l\\Vert \\leqslant 2^{-2 -L}\\Vert \\phi ^k\\Vert +3\\Vert \\phi ^k_{l +1}\\Vert /8, \\quad l =0, \\dots , L -1.$ Again by induction, using that $\\Vert \\phi ^k_L\\Vert \\leqslant \\Vert \\phi ^k\\Vert $ , we get $\\Vert \\phi ^k_l\\Vert \\leqslant 2^{l -L}\\Vert \\phi ^k\\Vert $ , $l =1, \\dots , L$ and thus $\\Vert \\phi ^k_0\\Vert \\leqslant 2^{-L}\\Vert \\phi ^k\\Vert $ .", "Therefore, for $k =1, \\dots , K -1$ , $\\Vert A\\phi ^k\\Vert =\\Vert A\\phi ^k_0\\Vert \\leqslant 2^{-L}a\\Vert \\phi ^k\\Vert ,$ and thus $\\Vert A\\phi ^{\\prime }\\Vert \\leqslant \\sum _{k =1}^{K -1}\\Vert A\\phi ^k\\Vert \\leqslant 2^{-L}\\sqrt{K -1}a\\Vert \\phi ^{\\prime }\\Vert $ and $\\big |\\langle \\phi ^{\\prime }, A\\phi ^{\\prime }\\rangle \\big | =\\Big |\\sum _{k, m =1}^{K -1}\\langle \\phi ^k_0, A\\phi ^m_0\\rangle \\Big | \\leqslant 2^{-2L}(K -1)a\\Vert \\phi ^{\\prime }\\Vert ^2.$ Hence $\\begin{split}\\langle \\phi , (H +A)\\phi \\rangle &=\\langle \\phi ^{\\prime }, H\\phi ^{\\prime }\\rangle +\\langle \\phi ^{\\prime }, A\\phi ^{\\prime }\\rangle +2\\Re \\langle \\phi ^K, A\\phi ^{\\prime }\\rangle +\\langle \\phi ^K, H\\phi ^K\\rangle +\\langle \\phi ^K, A\\phi ^K\\rangle \\\\ &\\leqslant (\\lambda - \\varepsilon )\\Vert \\phi ^{\\prime }\\Vert ^2 +2^{-2L}(K -1)a\\Vert \\phi ^{\\prime }\\Vert ^2+ 2^{1 -L}\\sqrt{K -1}a\\Vert \\phi ^{\\prime }\\Vert \\Vert \\phi ^K\\Vert \\\\ &+ \\big (\\lambda -\\varepsilon -(K -1)\\delta \\big )\\Vert \\phi ^K\\Vert ^2+ a\\Vert \\phi ^K\\Vert ^2\\\\ &\\leqslant \\big (\\lambda -\\varepsilon +2^{1 -2L}(K -1)a\\big )\\Vert \\phi ^{\\prime }\\Vert ^2 +\\big (\\lambda -\\varepsilon -(K -1)\\delta +2a\\big )\\Vert \\phi ^K\\Vert ^2\\\\ &\\leqslant \\lambda \\Vert \\phi \\Vert ^2,\\end{split}$ where the last inequality follows from (REF ) and (REF ).", "We now want to apply Lemma REF to $H_0^\\pm :=(-\\Delta )^w \\pm \\mathcal {P}_{R_*}^cB^{(k)}, \\quad V :=\\widetilde{B}, \\quad A^\\pm :=\\pm \\mathcal {P}_{R_0}B^{(k)}.$ Note that $a :=\\Vert b\\Vert _{\\textup {{\\textsf {L}}}_{\\infty }( \\mathbb {R}^d\\times \\mathcal {B}_{R_0})}+ \\Vert \\tilde{b}\\Vert _{\\textup {{\\textsf {L}}}_{\\infty }( \\mathbb {R}^d\\times \\mathcal {B}_{R_0})}$ does not depend on $\\rho _n$ .", "For $L :=\\big [4 +\\log _2a +(M +d -2w)\\log _2\\rho _n\\big ] +1$ we let $R_l :=R_0 +l\\rho _n^{2/k}, \\quad l =0, \\dots , L -1,$ and introduce a family of projections $P_0 :=\\mathcal {P}_{R_0}, \\quad P_l :=\\mathcal {P}_{R_l} -\\mathcal {P}_{R_{l -1}}, \\quad l =1, \\dots , L -1, \\quad P_L :=\\mathcal {P}_{R_{L -1}}^c.$ Let us check that the hypothesis of Lemma REF is satisfied.", "Relation (REF ) follows from (REF ) and (REF ).", "It follows from (REF ) that for $l\\leqslant L -1$ $\\Vert P_lHP_l\\Vert \\leqslant \\Big \\Vert P_{L -1}\\big ((-\\Delta )^w+ \\widetilde{B}\\big )P_{L -1}\\Big \\Vert \\leqslant 2\\big \\Vert P_{L -1}(-\\Delta )^wP_{L -1}\\big \\Vert \\leqslant 2(R_{L -1})^{2w}.$ Also, for $l\\leqslant L -1$ $\\Vert V_{l\\, l -1}\\Vert + \\Vert V_{l\\, l +1}\\Vert \\leqslant 2(R_{L -1} +\\rho _n^{2/k})^{2\\tilde{w}}.$ Since by (REF ) and (REF ) we have $R_{L -1} =R_0 +\\big [4 +\\log _2a +(M +d -2w)\\log _2\\rho _n\\big ]\\rho _n^{2/k}\\lesssim \\rho _n^{2/k}\\log \\rho _n,$ relations (REF ) and (REF ) follow from (REF ) and (REF ) if $\\rho _n$ is big enough.", "Applying Lemma REF , we get (REF ) with $\\varepsilon =2^{4- L}a \\leqslant \\rho _n^{-M +2w -d},$ which implies ().", "4.", "It remains to prove that $N\\big ((-\\Delta )^w+ \\widetilde{B} \\pm \\mathcal {P}_{R_*}^cB^{(k)}, \\lambda \\big )= N\\big ((-\\Delta )^w+ \\widetilde{B}, \\lambda + O(\\rho _n^{2w -d -M})\\big ).$ Choose $\\varepsilon :=\\rho _n^{-d -M}.$ In view of (REF ), we have $(-\\Delta )^{\\tilde{w}} +\\widetilde{B} \\lessgtr \\mathcal {P}_{R_*}(1 \\pm \\varepsilon )\\big ((-\\Delta )^{\\tilde{w}} +\\widetilde{B}\\big )\\mathcal {P}_{R_*} \\oplus \\mathcal {P}_{R_*}^c(1 \\pm 1/\\varepsilon )\\big ((-\\Delta )^{\\tilde{w}} +\\widetilde{B}\\big )\\mathcal {P}_{R_*}^c.$ Therefore, $\\begin{split}(-\\Delta )^w+ \\widetilde{B} \\pm \\mathcal {P}_{R_*}^cB^{(k)} &\\lessgtr \\mathcal {P}_{R_*}\\big ((-\\Delta )^w \\pm \\varepsilon (-\\Delta )^{\\tilde{w}} +(1 \\pm \\varepsilon )\\widetilde{B}\\big )\\mathcal {P}_{R_*} \\\\ &\\oplus \\mathcal {P}_{R_*}^c\\big ((-\\Delta )^w \\pm (-\\Delta )^{\\tilde{w}}/\\varepsilon +(1 \\pm 1/\\varepsilon )\\widetilde{B} \\pm B^{(k)}\\big )\\mathcal {P}_{R_*}^c.\\end{split}$ Using (REF ) again and recalling the definitions (REF ), () and (), we can estimate the last term on the right hand side of (REF ) from below: $\\begin{split}&\\mathcal {P}_{R_*}^c\\big ((-\\Delta )^w \\pm (-\\Delta )^{\\tilde{w}}/\\varepsilon +(1 \\pm 1/\\varepsilon )\\widetilde{B} \\pm B^{(k)}\\big )\\mathcal {P}_{R_*}^c \\\\ &>\\big ((-\\Delta )^w -2(-\\Delta )^{\\tilde{w}}/\\varepsilon \\big )\\mathcal {P}_{R_*}^c \\geqslant (R_*^{2w} -2R_*^{2\\tilde{w}}/\\varepsilon )\\mathcal {P}_{R_*}^c \\geqslant (5\\rho _n)^{2w}\\mathcal {P}_{R_*}^c,\\end{split}$ so it does not contribute to the density of states for $\\rho \\in I_n$ .", "For the first term we have $\\mathcal {P}_{R_*}\\big ((-\\Delta )^w \\pm \\varepsilon (-\\Delta )^{\\tilde{w}} +(1 \\pm \\varepsilon )\\widetilde{B}\\big )\\mathcal {P}_{R_*} \\lessgtr \\mathcal {P}_{R_*}(1 \\pm \\varepsilon )\\big ((-\\Delta )^w +\\widetilde{B}\\big )\\mathcal {P}_{R_*},$ so $\\begin{split}&N\\Big ((-\\Delta )^w+ \\widetilde{B} \\pm \\mathcal {P}_{R_*}^cB^{(k)}, \\lambda \\Big )\\\\&\\gtrless N\\Big (\\mathcal {P}_{R_*}\\big ((-\\Delta )^w \\pm \\varepsilon (-\\Delta )^{\\tilde{w}} +(1 \\pm \\varepsilon )\\widetilde{B}\\big )\\big |_{\\mathcal {P}_{R_*}\\textup {{\\textsf {L}}}_{2}( \\mathbb {R}^d)}, \\lambda \\Big )\\\\ &\\gtrless N\\Big (\\mathcal {P}_{R_*}\\big ((-\\Delta )^w +\\widetilde{B}\\big )\\big |_{\\mathcal {P}_{R_*}\\textup {{\\textsf {L}}}_{2}( \\mathbb {R}^d)}, \\lambda /(1 \\pm \\varepsilon )\\Big ),\\end{split}$ and the same estimates hold true for $B^{(k)}$ replaced by 0.", "Combining these two versions of (REF ), we obtain $\\begin{split}&N\\big ((-\\Delta )^w+ \\widetilde{B}, \\lambda \\big )\\lessgtr N\\big ((-\\Delta )^w+ \\widetilde{B} \\mp \\mathcal {P}_{R_*}^cB^{(k)}, \\lambda \\big )\\\\ &\\lessgtr N\\Big (\\mathcal {P}_{R_*}\\big ((-\\Delta )^w +\\widetilde{B}\\big )\\big |_{\\mathcal {P}_{R_*}\\textup {{\\textsf {L}}}_{2}( \\mathbb {R}^d)}, \\lambda /(1 \\mp \\varepsilon )\\Big ) \\lessgtr N\\big ((-\\Delta )^w+ \\widetilde{B}, (1 \\pm \\varepsilon )\\lambda /(1 \\mp \\varepsilon )\\big ).\\end{split}$ Recalling that $\\lambda =\\rho ^{2w} \\leqslant (4\\rho _n)^{2w}$ and (REF ), we arrive at (REF ).", "Combining (), () and (REF ), we get ()." ] ]

[ [ "Classical Duals, Legendre Transforms and the Vainshtein Mechanism" ], [ "Abstract We show how to generalize the classical duals found by Gabadadze {\\it et al} to a very large class of self-interacting theories.", "This enables one to adopt a perturbative description beyond the scale at which classical perturbation theory breaks down in the original theory.", "This is particularly relevant if we want to test modified gravity scenarios that exhibit Vainshtein screening on solar system scales.", "We recognise the duals as being related to the Legendre transform of the original Lagrangian, and present a practical method for finding the dual in general; our methods can also be applied to self-interacting theories with a hierarchy of strong coupling scales, and with multiple fields.", "We find the classical dual of the full quintic galileon theory as an example." ], [ "Introduction", "For any sensible theory of gravity, the dynamics is described by a complicated coupled system of non-linear partial differential equations.", "This is true of General Relativity (GR), and must therefore be true of any theory that hopes to mimic GR at some appropriate scale.", "The complexity of the governing equations renders it difficult to find exact solutions, and typically one can only make progress by imposing certain symmetries in order to reduce the phase space, or else to consider perturbations about a known background solution.", "Perturbation theory works best far away from the relevant excitation of the background, and will inevitably break down as we move closer and closer to the source.", "In classical GR this breakdown occurs at the Schwarzschild radius of the source.", "This is good enough for the most part, since we cannot experimentally probe the dynamics within the Schwarzschild radius since it would lie behind an event horizon.", "In modified theories of gravity the situation can be more subtle.", "Large distance modifications of GR are often considered in order to address an outstanding problem in cosmology such as the dark matter problem [1], the dark energy problem [2], and/or the cosmological constant problem [3] (see [4] for a general review and a more complete list of references).", "This requires an ${\\cal O}(1)$ deviation from GR on astrophysical or cosmological scales, and for the theory to be phenomenologically viable, this deviation must reduce to $\\lesssim {\\cal O}(10^{-5})$ on solar system scales.", "Such dramatic suppression can sometimes be achieved through non-linearities, either from non-linear matter couplings, as in the chameleon mechanism [5], or via non-linear self-interactions, as in the Vainshtein mechanism [6], [7]; here we are primarily interested in the latter.", "Such self couplings have two important effects.", "At the quantum level, the interactions become strongly coupled above some particular scale, $\\Lambda $ , and one can no longer trust the classical background solution on distances $\\lesssim \\Lambda ^{-1}$ .", "At the classical level, the interactions lead to the breakdown of the standard linearized theory around a heavy source.", "In a generic modified gravity theory, with a non-relativistic source of mass $M$ , the linearized perturbations break down at the Vainshtein scale, $r_V$ , which typically takes the form [8] $r_V \\sim \\left(\\frac{M}{M_{pl}}\\right)^\\frac{1}{1+4(1-\\alpha )} \\Lambda ^{-1}, \\qquad 0 <\\alpha <1$ For the Sun, this scale must lie beyond the edges of the solar system, making it much larger than the Schwarzschild radius.", "This is both a blessing and a curse.", "If the Vainshtein mechanism is effective the non-linearities help to suppress the modifications of GR making the theory compatible with observation.", "However, in the absence of a perturbative description below the Vainshtein scale, it is difficult to test any corrections to the leading order effect.", "Recently, Gabadadze et al [9] considered two examples of a classical theory with derivative self-interactions, each admitting a standard perturbative description above the relevant Vainshtein scale.", "They then presented a “classical dual\" of each theory, describing exactly the same physics, but admitting a perturbative description below the Vainshtein scale.", "This is not a duality in the usual sense of strong versus weak coupling.", "Rather it is a classical analogue in the sense that the classical expansion parameter, $r/r_V$ , is inverted, with the two descriptions giving us the flexibility to do perturbation theory over a much larger range of scales, provided the classical effective theory is valid.", "This is reminiscent of Vainshtein's original approach to massive gravity [6] in which he made use of an expansion in $r_V/r$ above the scale $r_V \\sim \\left(\\frac{r_s}{m^4}\\right)^{\\frac{1}{5}}$ , and an expansion in $r/r_V$ below the scale $r_V$ , where $m$ is the graviton mass and $r_s$ is the Schwarzschild radius of the source.", "Indeed, this inversion of the expansion parameter was a big factor in motivating Gabadadze et al's recent work.", "In this paper we give a general prescription for finding the classical duals of a much larger class of theories with self-interactions.", "Our methods work well for those theories that remain weakly coupled at low energies, and whose classical high energy dynamics is dominated by $N$ -point interactions with finite $N$ only.", "We also require that the interactions become subdominant as the fields tend to zero.", "We begin by observing that the classical duals presented in [9] actually make use of Legendre transforms of the interaction terms.", "Running with this idea, we are able to generalize their method.", "Indeed, it is no coincidence that some mathematicians refer to the Legendre transform as the Legendre dual[10].", "The duality is only useful if it admits a new perturbative description, and this is always true for the broad class of theories under consideration.", "Whilst the Legendre transform picture is certainly instructive, it is not always straightforward to arrive at a workable dual theory.", "This is down to technical difficulties in inverting the expression for the transformed variables.", "To alleviate this problem we will also present a more practical method for finding the dual, using Lagrange multipliers, which are then integrated out.", "This should really be considered as the working method for finding the classical dual of your favourite theory.", "It also helps us to identify exactly when a useful dual can be found.", "Both methods can be applied to theories with multiple scales, and multiple fields, in contrast to the examples given in [9].", "The rest of this paper is organised as follows: We start with a recap of the cubic galileon case covered in [9], showing that the dual theory is just the Legendre transform of the original, and we then extend this idea in section to the general case, as well as proposing a practical implementation in section REF .", "We see how models with multiple scales are dealt with in section REF and multiple fields are examined in section REF .", "As an example of the method we work out the dual of the full galileon model in section , allowing for multiple scales, and then finish the paper with a discussion." ], [ "The cubic galileon and its classical dual", "Let us begin by reviewing one of the examples considered by Gabadadze et al [9], namely, the cubic galileon theory [11], ${\\cal L}&=&-\\frac{1}{2}(\\partial \\phi )^2-\\frac{1}{\\Lambda ^3}\\Box \\phi (\\partial \\phi )^2,$ where $\\Lambda $ corresponds to the scale at which the interaction term becomes strongly coupled.", "This theory arises in the decoupling limit of the DGP model [14], [12], [13], as well as in recent models of ghost-free massive gravity [15].", "Let us assume that matter couples to the scalar via an interaction of the form, $\\phi \\frac{T}{M_{pl}} ,$ where $T$ is the trace of the energy momentum tensor.", "It follows that, classically, the linearised theory around a non-relativistic source of mass $M$ breaks down at the scale, $r_V \\sim \\left(\\frac{M}{M_{pl}}\\right)^{1/3} \\frac{1}{\\Lambda }$ .", "To see this we note that the field equations can be schematically written as $p^2 \\phi -\\frac{1}{\\Lambda ^3}(p^2 \\phi )^2&\\sim &\\frac{T}{M_{pl}},$ where $p$ corresponds to the momentum operator.", "For a non-relativistic source of mass, $M$ , we get the right schematic behaviour by takingStrictly speaking, for an approximately point-like source, we would take $T =- M \\delta ^{(3)}(x)$ and integrate up the differential equations.", "However, our schematic trick yields exactly the same behaviour, as of course it had to on dimensional grounds.", "$T \\sim M p^3$ .", "In the linearised theory, $p^2 \\phi \\sim -\\frac{T}{M_{pl}}$ and we obtain $\\phi \\sim \\frac{M}{M_{pl}} p \\sim \\frac{M}{M_{pl}} \\frac{1}{r}$ .", "It is clear that the linearised theory breaks down when $p^2 \\phi \\sim \\Lambda ^3$ , and is only valid for $r \\gg r_V$ .", "Above the Vainshtein scale we can explore corrections to the leading order behaviour by expansion in $r/r_V$ .", "Gabadadze et al [9] proposed the following classical dual to the cubic galileon theory ${\\cal L}^{\\prime }&=&-\\frac{1}{2}(\\partial \\phi )^2-b^\\mu \\partial _\\mu \\phi -\\lambda \\Box \\phi +\\Lambda ^{3/2} \\sqrt{\\lambda b^2}.$ A significant feature of the dual formulation is that the strong coupling scale, $\\Lambda $ , enters with positive powers.", "Thus the dual Lagrangian is well defined even in the limit $\\Lambda \\rightarrow 0$ , in contrast to the original Lagrangian.", "We also note that the two interaction terms are related by a Legendre transformation, $-\\frac{1}{\\Lambda ^3}\\Box \\phi (\\partial \\phi )^2 \\rightarrow -\\Lambda ^{3/2} \\sqrt{\\lambda b^2},$ with $\\partial _\\mu \\phi $ conjugate to $-b^\\mu $ and $\\Box \\phi $ conjugate to $-\\lambda $ .", "With this observation, it now seems natural to include the canonical kinetic term in the transformation.", "To this end we define, $b_{(1)}^\\mu &=&\\frac{\\partial {\\cal L}}{\\partial \\;\\partial _\\mu \\phi }=-\\partial ^\\mu \\phi \\left(1+\\frac{2}{\\Lambda ^3} \\Box \\phi \\right) ,\\\\b_{(2)}&=&\\frac{\\partial {\\cal L}}{\\partial \\;\\Box \\phi }=-\\frac{1}{\\Lambda ^3} (\\partial \\phi )^2 .$ The Legendre transform of the Lagrangian is given by $f(b_{(1)}, b_{(2)}) &=&b_{(1)}^\\mu \\partial _\\mu \\phi +b_{(2)}\\Box \\phi -{\\cal L}=\\pm \\Lambda ^{3/2} \\sqrt{-b_{(2)} b_{(1)}^2}-\\frac{\\Lambda ^3}{2} b_{(2)}.$ The ambiguity in the sign reflects the ambiguity in inverting the relations (REF ) and ().", "This stems from the fact that the cubic galileon theory admits two distinct branches.", "The classical dual is now given by ${\\cal L}^{\\prime }=b_{(1)}^\\mu \\partial _\\mu \\phi +b_{(2)}\\Box \\phi -f(b_{(1)}, b_{(2)}),$ and describes the same physics.", "In the presence of the source (REF ), we obtain the following field equations $\\partial _\\mu b_{(1)}^\\mu -\\Box b_{(2)} &=& \\frac{T}{M_{pl}} , \\\\\\partial _\\mu \\phi \\pm \\Lambda ^{3/2}\\frac{b_{(2)} b_{(1)}{}_\\mu }{\\sqrt{-b_{(2)}b_{(1)}^2}} &=& 0, \\\\\\Box \\phi \\pm \\Lambda ^{3/2}\\frac{ b_{(1)}^2}{\\sqrt{-b_{(2)}b_{(1)}^2}}+\\frac{\\Lambda ^3}{2} &=&0.$ These can be solved order by order in an expansion in $\\Lambda ^{3/2}$ .", "For a non-relativistic source of mass, $M$ , schematically we have $p b_{(1)}+p^2 b_{(2)} &=& \\frac{M}{M_{pl}}p^3, \\\\p \\phi +\\Lambda ^{3/2} \\sqrt{ |b_{(2)}| } \\frac{b_{(1)}}{|b_{(1)}|} &=&0, \\\\p^2 \\phi +\\Lambda ^{3/2} \\frac{b_{(1)}}{\\sqrt{ |b_{(2)}| } }+\\Lambda ^3 &=&0.$ It follows that ${ b_{(1)} } & \\sim & \\frac{M}{M_{pl}}p^2+ \\Lambda ^{3/2}\\sqrt{\\frac{M}{M_{pl}}}p^{1/2}+\\ldots , \\\\b_{(2)} &\\sim & \\frac{M}{M_{pl}}p + \\Lambda ^{3/2}\\sqrt{\\frac{M}{M_{pl}}}p^{-1/2}+\\ldots , \\\\\\phi &\\sim & \\Lambda ^{3/2}\\sqrt{\\frac{M}{M_{pl}}}p^{-1/2}+\\ldots ,$ corresponding to a perturbative expansion in $\\left(\\frac{\\Lambda }{p}\\right)^{3/2}\\left(\\frac{M}{M_{pl}}\\right)^{-1/2} \\sim \\left(\\frac{r}{r_V} \\right)^{3/2}$ ." ], [ "A general theory with self-interactions and its classical dual", "We will now consider general theories involving self-interactions, extending the ideas initiated in [9].", "In the interests of clarity we present our analysis for a general theory involving a single field, $\\phi $ , of any type (we suppress tensor indices) with interactions all becoming strong in the UV, at the same scale $\\Lambda $ .", "Our generalization can be applied to theories with multiple scales, and multiple fields, and we will sketch how this should be done in sections REF and REF .", "We start with the Lagrangian density for the field $\\phi $ propagating on Minkowski spacetime ${\\cal L} (\\partial ^{(k)} \\phi )\\equiv {\\cal L}(\\phi ,\\partial _\\mu \\phi ,\\partial _\\mu \\partial _\\nu \\phi , \\ldots ),$ emphasizing once again that we are suppressing tensor indices on the field – $\\phi $ does not have to correspond to a scalar.", "We will assume that the field is “canonically normalised\" in some appropriate way, and that the propagator scales like $1/p^2$ in the UV.", "The various interactions are characterised by the number of fields involved in the interaction, $N$ , and the number of derivatives, $D$ , and will schematically have the form $\\frac{ \\partial ^D \\phi ^N}{\\Lambda ^{D+N-4}},$ where $D+N > 4$ .", "This follows from the fact that we only consider theories that are always weakly coupled at low energies, so our interactions should disappear in the limit $\\Lambda \\rightarrow \\infty $ .", "We also require that the interactions become subdominant as $\\phi \\rightarrow 0$ so we have $N>2$ .", "If we further assume that the field couples to a source $J$ via an interaction $\\phi J,$ then the field equations are schematically given by $p^2 \\phi +\\sum _i \\frac{1}{\\Lambda ^{D_i+N_i-4}}p^{D_i} \\phi ^{N_i-1} \\sim J.$ Now we can probe the low energy physics by taking $\\Lambda \\rightarrow \\infty $ , in which case we have a good linearised theory with $p^2 \\phi \\sim J$ .", "To probe the high energy physics, we must take the opposite limit $\\Lambda \\rightarrow 0$ .", "Which term dominates the dynamics?", "Naively one might expect the dynamics to be dominated by the interaction containing the largest power of $1/\\Lambda $ .", "However, things are a little more subtle than that.", "It turns out that the dynamics is dominated by the term (or terms) with largest $t$ , where $t=\\frac{D+N-4}{N-1}$ To see this, suppose that the $j$ th interaction dominates the dynamics at high energies.", "It follows that as $\\Lambda \\rightarrow 0$ , $\\phi \\propto \\Lambda ^{t_j}$ , since $J$ is independent of $\\Lambda $ .", "At the level of the equations of motion, the $k$ th interaction now scales as $p^{D_k} \\left(\\frac{\\Lambda ^{t_j}}{\\Lambda ^{t_k}}\\right)^{N_k-1}.$ Since $t_j \\ge t_k$ for all $k$ , this will not diverge as $\\Lambda \\rightarrow 0$ .", "Now, if the largest value of $t$ occurs at finite values of $N$ , then it is possible to identify the dominant UV behaviour and to expand around it.", "This controls whether or not we can find a classical dual that admits a useful perturbative description.", "Therefore, assuming that $\\textrm {t=t_{max} for finite values of N only} $ we proceed to dualize the theory." ], [ "The Legendre dual", "To dualize our general theory, we simply compute the Legendre transform for the Lagrangian.", "To this end we define, $a_{(0)}&=&\\frac{\\partial {\\cal L}}{\\partial \\phi },\\\\a_{(1)}^\\mu &=&\\frac{\\partial {\\cal L}}{\\partial \\;\\partial _\\mu \\phi },\\\\a_{(2)}^{\\mu \\nu }&=&\\frac{\\partial {\\cal L}}{\\partial \\;\\partial _\\mu \\partial _\\nu \\phi },\\\\&\\vdots & \\nonumber $ The Legendre transform of the Lagrangian is given by $f(a_{(k)})=a_{(0)}\\phi +a_{(1)}^\\mu \\partial _\\mu \\phi +a_{(2)}^{\\mu \\nu }\\partial _\\mu \\partial _\\nu \\phi +\\ldots -{\\cal L}.$ The precise form of this depends on the inversion of the relation $a_{(k)}=\\frac{\\partial {\\cal L}}{\\partial \\; \\partial ^k \\phi }$ .", "This may be multivalued, as it was for the cubic galileon.", "In any event, choosing some particular branch for the inverse, the dual theory is given by ${\\cal L}^{\\prime } &=&a_{(0)}\\phi +a_{(1)}^\\mu \\partial _\\mu \\phi +a_{(2)}^{\\mu \\nu }\\partial _\\mu \\partial _\\nu \\phi +...-f(a_{(k)}),$ with the following field equations $a_{(0)}-\\partial _\\mu a_{(1)}^\\mu +\\partial _\\mu \\partial _\\nu a_{(2)}^{\\mu \\nu } +\\ldots &=& -J, \\\\\\phi &=&\\frac{\\partial f}{\\partial a_{(0)}},\\\\\\partial _\\mu \\phi &=&\\frac{\\partial f}{\\partial a_{(1)}^\\mu },\\\\\\partial _\\mu \\partial _\\nu \\phi &=&\\frac{\\partial f}{\\partial a_{(2)}^{\\mu \\nu }}, \\\\&\\vdots & \\nonumber $ Now the original Lagrangian ${\\cal L}\\left(\\partial ^{(k)}\\phi \\right)$ is such that all interactions remain weakly coupled at low energies, in the limit $\\Lambda \\rightarrow \\infty $ .", "It turns out the dual Lagrangian is well behaved in the opposite limit, $\\Lambda \\rightarrow 0$ .", "To see why, we note that schematically $a_{(k)} \\sim \\sum _i \\frac{1}{\\Lambda ^{D_i+N_i-4}}p^{D_i-k} \\phi ^{N_i-1} \\sim \\sum _i p^{D_i-k} \\left(\\frac{\\phi }{\\Lambda ^{t_i}}\\right)^{N_i-1},$ This is inverted to find that the dominant $\\Lambda $ scaling is $\\phi \\sim \\Lambda ^{t_{max}}$ , from which it follows that $\\partial ^{(k)}\\phi \\sim \\Lambda ^{t_{max}} O^{(k)} (a)$ where the operators $O^{(k)}$ remain well behaved in the limit $\\Lambda \\rightarrow 0$ .", "The Legendre transform is therefore given by $f(a) \\sim \\Lambda ^{t_{max}} \\left[\\sum _k a_{(k)} O^{(k)} (a)- \\frac{{\\cal L}( \\Lambda ^{t_{max}} O^{(k)} (a))}{\\Lambda ^{t_{max}}}\\right].$ Now ${{\\cal L}( \\partial ^{(k)}\\phi })\\sim \\phi \\sum _i p^{D_i} \\left(\\frac{\\phi }{\\Lambda ^{t_i}}\\right)^{N_i-1}$ and so $ \\frac{{\\cal L}( \\Lambda ^{t_{max}} O^{(k)} (a))}{\\Lambda ^{t_{max}}} \\sim \\sum _i \\left(\\frac{ \\Lambda ^{t_{max}} }{\\Lambda ^{t_i}}\\right)^{N_i-1}{\\cal F}_i (O^{(k)} (a)).$ This is well behaved as $\\Lambda \\rightarrow 0$ .", "It follows that the Legendre transform is well behaved as $\\Lambda \\rightarrow 0$ , provided $t_{max} \\ge 0$ .", "This is indeed the case since we know that the quadratic term has $t=0$ .", "We therefore confirm our assertion that the dual theory is well behaved in the limit $\\Lambda \\rightarrow 0$ .", "We see that the Legendre transform causes the expansion parameter to be inverted, the original theory working best at large $\\Lambda $ , with the dual working best at small $\\Lambda $ .", "There is a characteristic scale depending on $J$ and $\\Lambda $ that acts as a pivot about which the duality is performed.", "This is precisely what you mean by the Vainshtein scale in certain modified gravity scenarios.", "On one side of the pivot we have the standard linearised theory with corrections that go like negative powers of $\\Lambda $ .", "On the other side we have the leading order short distance dynamics with corrections going like positive powers of $\\Lambda $ .", "The dual theory gives us the means to study the latter using ordinary perturbative methods.", "The dual theory describes exactly the same physics as the original theory.", "This is manifestly true when one is considering excitations due to a source.", "When one is interested in freely propagating modes care must be taken to perform a non-covariant decomposition of the conjugate variablesWe thank Gregory Gabadadze for pointing this out., as emphasized in [9]." ], [ "A practical method for finding the dual", "Now, in general, one cannot explicitly invert the relation $a_{(k)}=\\frac{\\partial {\\cal L}}{\\partial \\; \\partial ^{(k)} \\phi }$ .", "For example, this is already true for the full galileon theory, including quartic and/or quintic interactions.", "[11].", "This makes it difficult to find the dual theory using the Legendre transform method.", "Fortunately, however, we may use Lagrange multipliers to arrive at an equivalent dual theory with the same useful properties; we will now describe that method.", "Introducing some Lagrange multipliers, $\\zeta _{(k)}$ , and auxiliary fields, $A_{(k)}$ , we begin with a new Lagrangian, ${\\cal L}^{\\prime \\prime }&=&\\zeta _{(0)}(\\phi -A_{(0)})+\\zeta _{(1)}^\\mu (\\partial _\\mu \\phi -A_{(1)\\mu })+\\zeta _{(2)}^{\\mu \\nu }(\\partial _\\mu \\partial _\\nu \\phi -A_{(2)\\mu \\nu })+\\ldots +{\\cal L}(A_{(k)}).$ This is obviously entirely equivalent to our starting Lagrangian, $\\cal L$ , given by (REF ).", "However, the equations of motion for $A_{(k)}$ now correspond to constraints that we can use to integrate out the Lagrange multipliers.", "In particular, we find $\\zeta _{(0)}&=&\\frac{\\partial {\\cal L}}{\\partial A_{(0)}},\\\\\\zeta _{(1)}^\\mu &=&\\frac{\\partial {\\cal L}}{\\partial A_{(1)\\mu }},\\\\\\zeta _{(2)}^{\\mu \\nu }&=&\\frac{\\partial {\\cal L}}{\\partial A_{(2)\\mu \\nu }}, \\\\&\\vdots & \\nonumber $ Plugging this back into the action we obtain, ${\\cal L}^{\\prime \\prime }&=&\\frac{\\partial {\\cal L}}{\\partial A_{(0)}}\\phi +\\frac{\\partial {\\cal L}}{\\partial A_{(1)\\mu }}\\partial _\\mu \\phi +\\frac{\\partial {\\cal L}}{\\partial A_{(2)\\mu \\nu }}\\partial _\\mu \\partial _\\nu \\phi +\\ldots \\\\\\nonumber &~&+{\\cal L}(A_{(k)}) -A_{(0)}\\frac{\\partial {\\cal L}}{\\partial A_{(0)}}-A_{(1)\\mu }\\frac{\\partial {\\cal L}}{\\partial A_{(1)\\mu }}-A_{(2)\\mu \\nu }\\frac{\\partial {\\cal L}}{\\partial A_{(2)\\mu \\nu }} -\\ldots $ This is now of the same form as (REF ), but by using the variables $A_{(k)}$ instead of $a_{(k)}$ we are able to get an explicit expression for the dual Lagrangian.", "Note that $a_{(k)}&=&\\frac{\\partial {\\cal L}}{\\partial A_{(k)}},$ which is difficult to invert in general.", "As it stands, there is no guarantee that the Lagrangian ${\\cal L}^{\\prime \\prime }$ is well behaved as $\\Lambda \\rightarrow 0$ .", "We can fix this by rescaling the auxiliary fields $A_{(k)} = \\Lambda ^{t_{max}} \\hat{A}_{(k)}$ .", "To see why this helps consider the generic interaction $\\frac{ \\partial ^D \\phi ^N}{\\Lambda ^{D+N-4}} =\\Lambda ^4 \\prod _i\\left(\\frac{\\partial ^{(i)}\\phi }{\\Lambda ^{i+1}}\\right)^{n_i} \\subset {\\cal L}(\\partial ^{(k)}\\phi ),$ where $D=\\sum _i in_i$ and $N=\\sum _i n_i$ .", "It follows that $\\frac{1}{\\Lambda ^{(N-1)t}} \\prod _i A_{(i)} ^{n_i} \\subset {\\cal L}(A_{(k)}), \\qquad \\frac{n_j}{\\Lambda ^{(N-1)t} A_{(j)} } \\prod _i A_{(i)} ^{n_i} \\subset \\frac{\\partial {\\cal L}}{\\partial A_{(j)}},$ where we recall that $t=\\frac{D+N-4}{N-1}$ .", "Rescaling our variables, we obtain $\\Lambda ^{t_{max}}\\left(\\frac{\\Lambda ^{t_{max}}}{\\Lambda ^{t}} \\right)^{N-1} \\prod _i \\hat{A}_{(i)} ^{n_i} \\subset {\\cal L}(A_{(k)}), \\qquad n_j\\left(\\frac{\\Lambda ^{ t_{max}}}{\\Lambda ^{t}} \\right)^{N-1} \\frac{1}{\\hat{A}_{(j)} } \\prod _i \\hat{A}_{(i)} ^{n_i} \\subset \\frac{\\partial {\\cal L}}{\\partial A_{(j)}}.$ Since $t_{max} \\ge t$ and $t_{\\max } \\ge 0$ , it is clear that as long as we replace $A_{(k)}$ with $\\hat{A}_{(k)}$ , then both ${\\cal L}(A_{(k)})$ and $\\frac{\\partial {\\cal L}}{\\partial A_{(j)}}$ are well behaved as $\\Lambda \\rightarrow 0$ .", "As a result, ${\\cal L}^{\\prime \\prime }$ is also well behaved in this limit.", "Let us see how this method works when applied to the cubic galileon.", "Note that the canonical kinetic term has $t=0$ , while the interaction has $t=3/2$ , so we have $t_{max}=\\frac{3}{2}$ .", "Before rescaling we have ${\\cal L}^{\\prime \\prime }&=&\\left(-\\frac{2}{\\Lambda ^3}A_{(2)}A_{(1)}^\\mu -A_{(1)}^\\mu \\right)\\partial _\\mu \\phi -\\frac{1}{\\Lambda ^3}A^2_{(1)}\\Box \\phi +\\frac{1}{2}A^2_{(1)}+\\frac{2}{\\Lambda ^3}A_{(2)}A_{(1)}^2.$ If we set $A=\\Lambda ^{3/2}\\hat{A}$ we obtain the dual Lagrangian ${\\cal L}^{\\prime \\prime }&=&\\left(-2\\hat{A}_{(2)}\\hat{A}_{(1)}^\\mu -\\Lambda ^{3/2}\\hat{A}_{(1)}^\\mu \\right)\\partial _\\mu \\phi -\\hat{A}^2_{(1)}\\Box \\phi +\\frac{1}{2}\\Lambda ^3\\hat{A}^2_{(1)}+2\\Lambda ^{3/2}\\hat{A}_{(2)}\\hat{A}_{(1)}^2,$ which is clearly well behaved as $\\Lambda \\rightarrow 0$ , as desired." ], [ "Multiple scales", "Our method for dualising can be easily adapted to deal with theories with more than one strong coupling scale.", "To illustrate how, we consider a theory that depends on two scales $\\Lambda \\ll \\bar{\\Lambda }$ , which we may schematically write as, ${\\cal L} \\sim -\\frac{1}{2} (\\partial \\phi )^2 +\\sum _i \\frac{1}{\\Lambda ^{D_i+N_i-4}}\\partial ^{D_i} \\phi ^{N_i} +\\sum _j \\frac{1}{\\bar{\\Lambda }^{\\bar{D}_j+\\bar{N}_j-4}}\\partial ^{\\bar{D}_j} \\phi ^{\\bar{N}_j}.$ We see that we have explicitly separated the interactions into two families – those that become strong at $\\Lambda $ , and those that become strong at $\\bar{\\Lambda }$ .", "Again, in order to guarantee that all interactions remain weakly coupled at low energies we assume $D+N>4$ , and $\\bar{D}+\\bar{N}>4$ .", "We also assume $N, ~\\bar{N} >2$ in order to ensure that the interactions become subdominant as $\\phi \\rightarrow 0$ .", "For a source interaction of the form $\\phi J$ , the equations of motion are schematically given by $p^2 \\phi +\\sum _i \\frac{1}{\\Lambda ^{D_i+N_i-4}}p^{D_i} \\phi ^{N_i-1} ++\\sum _j \\frac{1}{\\bar{\\Lambda }^{\\bar{D}_j+\\bar{N}_j-4}}p^{\\bar{D}_j} \\phi ^{\\bar{N}_j-1} \\sim J.$ To probe the low energy physics we simply take $\\Lambda , ~\\bar{\\Lambda }\\rightarrow \\infty $ , and truncate to the linearised theory, $p^2 \\phi \\sim J$ .", "We can perturb about the leading order solution using inverse powers of $\\Lambda $ and $\\bar{\\Lambda }$ .", "At higher energies we now have two distinct regimes.", "We first encounter an intermediate regime by taking $\\Lambda \\rightarrow 0$ , and $\\bar{\\Lambda }\\rightarrow \\infty $ .", "In contrast, the high energy regime is obtained by taking both $\\Lambda , ~\\bar{\\Lambda }\\rightarrow 0$ .", "The importance of multiple scales and multiple classical regimes was emphasized in general galileon theories in [16] Let us first consider the intermediate regime.", "We can obtain corrections to the leading order behaviour in terms of positive powers of $\\Lambda $ and negative powers of $\\bar{\\Lambda }$ .", "Introducing both $t=\\frac{D+N-4}{N-1}$ and $\\bar{t}=\\frac{\\bar{D}-\\bar{N}-4}{\\bar{N}-1}$ , it is clear from our previous discussion that the intermediate scale dynamics is dominated by the term (or terms) with largest $t$ (with the $\\bar{t}$ playing no role).", "To see this explicitly note that the leading order behaviour has $\\phi \\sim \\Lambda ^{t_{max}}$ .", "At the level of the field equations, the other interactions go like $p^D \\left(\\frac{\\Lambda ^{t_{max}}}{\\Lambda ^t}\\right)^{N-1}, \\qquad p^{\\bar{D} }\\left(\\frac{\\Lambda ^{t_{max}}}{\\bar{\\Lambda }^{\\bar{t}}}\\right)^{\\bar{N}-1}.$ Now since $t, ~\\bar{t} \\ge 0$ , and $t_{max} \\ge t$ , it is clear that none of these terms diverge as $\\Lambda \\rightarrow 0, ~\\bar{\\Lambda }\\rightarrow \\infty $ .", "It is hopefully now obvious how to obtain a suitable dual in the intermediate regime.", "The key point is that the interactions $\\frac{1}{\\bar{\\Lambda }^{\\bar{D}+\\bar{N}-4}}\\partial ^{\\bar{D}} \\phi ^{\\bar{N}}$ are already in the correct form to admit an expansion in terms of inverse powers of $\\bar{\\Lambda }$ .", "Thus we leave these interactions alone, and focus on dualizing the truncated Lagrangian ${\\cal L}_{truncated} \\sim -\\frac{1}{2} (\\partial \\phi )^2 +\\sum _i \\frac{1}{\\Lambda ^{D_i+N_i-4}}\\partial ^{D_i} \\phi ^{N_i}.$ This can be obtained using either Legendre transforms, or our“practical\" method, with $t_{max}$ playing the same critical role as outlined in previous sections.", "We denote the resulting “truncated' dual as ${\\cal L}_{truncated}^{\\prime }$ and note that it is well behaved in the limit $\\Lambda \\rightarrow 0$ .", "Combining it with the other interactions, we obtain the following dual for the full theory ${\\cal L}^{\\prime } &=& {\\cal L}_{truncated}^{\\prime }+{\\cal L}-{\\cal L}_{truncated},\\\\&\\sim & {\\cal L}_{truncated}^{\\prime }+\\sum _j \\frac{1}{\\bar{\\Lambda }^{\\bar{D}_j+\\bar{N}_j-4}}\\partial ^{\\bar{D}_j} \\phi ^{\\bar{N}_j}.$ This theory is well behaved as $\\Lambda \\rightarrow 0$ and $\\bar{\\Lambda }\\rightarrow \\infty $ , and beyond this limit we can expand our classical solution in positive powers of $\\Lambda $ and negative powers of $\\bar{\\Lambda }$ .", "In a modified gravity model with two Vainshtein radii[16], $r_V$ and $\\bar{r}_V< r_V$ , this expansion will ultimately be equivalent to an expansion in $r/r_V$ , and $\\bar{r}_V/r$ .", "The expansion works well in the intermediate regime $\\bar{r}_V<r< r_V$ .", "We now turn to the high energy regime.", "Corrections to the leading order behaviour are now expressed in terms of positive powers of both $\\Lambda $ and $\\bar{\\Lambda }$ .", "Which term (or terms) dominates the dynamics?", "To answer this we must introduce $\\bar{t}_{max}$ , the largest of the $\\bar{t}$ 's, in addition to $t_{max}$ , the largest of the $t$ 's.", "The term that dominates the high energy regime depends on the ratio of $\\Lambda ^{t_{max}}$ and $\\bar{\\Lambda }^{\\bar{t}_{max}}$ in the limit $\\Lambda , ~\\bar{\\Lambda }\\rightarrow 0$ .", "If $\\Lambda ^{t_{max}} / \\bar{\\Lambda }^{\\bar{t}_{max}} \\nrightarrow \\infty $ then the dominant terms stem from $\\Lambda $ -type interactions, $\\frac{ \\partial ^D \\phi ^N}{\\Lambda ^{D+N-4}} $ , with $t=t_{max}$ .", "In contrast, if $ \\bar{\\Lambda }^{\\bar{t}_{max}}/\\Lambda ^{t_{max}} \\nrightarrow \\infty $ then the dominant terms stem from $\\bar{\\Lambda }$ -type interactions, $\\frac{ \\partial ^{\\bar{D}} \\phi ^{\\bar{N}}}{\\bar{\\Lambda }^{{\\bar{D}}+{\\bar{N}}-4}}$ , with $\\bar{t}=\\bar{t}_{max}$ .", "To see why such terms dominate when they do, we consider the two possibilities separately.", "If $\\Lambda ^{t_{max}}/\\Lambda ^{\\bar{t}_{max}}$ is not divergent, then the $\\Lambda $ -type interactions with $t=t_{max}$ dominate and we have $\\phi \\sim \\Lambda ^{t_{max}}$ .", "To verify that this is correct, we need to show that none of the other interactions lead to divergences in the field equations in the desired limit.", "Indeed, at the level of the field equations, the interactions go like $p^D \\left(\\frac{\\Lambda ^{t_{max}}}{\\Lambda ^t}\\right)^{N-1}, \\qquad p^{\\bar{D} }\\left(\\frac{\\Lambda ^{t_{max}}}{\\bar{\\Lambda }^{\\bar{t}}}\\right)^{\\bar{N}-1}.$ The first of these will certainly not diverge as $\\Lambda , ~\\bar{\\Lambda }\\rightarrow 0$ since $t_{max}\\ge t$ .", "The second term is more subtle.", "To see how it behaves we rewrite it suggestively as $p^{\\bar{D} }\\left(\\frac{\\Lambda ^{t_{max}}}{\\bar{\\Lambda }^{\\bar{t}_{max}}}\\right)^{\\bar{N}-1}\\left(\\frac{\\bar{\\Lambda }^{\\bar{t}_{max}}}{ \\bar{\\Lambda }^{\\bar{t}}}\\right)^{\\bar{N}-1}.$ Since $\\Lambda ^{t_{max}} / \\bar{\\Lambda }^{\\bar{t}_{max}}\\nrightarrow \\infty $ and $\\bar{t}_{max}\\ge \\bar{t}$ , it is clear that this will not diverge as $\\Lambda , ~\\bar{\\Lambda }\\rightarrow 0$ .", "Now consider the alternative scenario, in which, $ \\bar{\\Lambda }^{\\bar{t}_{max}}/\\Lambda ^{t_{max}}$ is not divergent.", "Using an entirely analogous argument one can easily prove that the $\\bar{\\Lambda }$ -type interactions with $\\bar{t}=\\bar{t}_{max}$ dominate.", "It should now be clear how to take arrive at the dual theory in the high energy regime.", "One can simply take the Legendre transform of the full Lagrangian.", "The dominant terms in the expansion of the transform then depend on the ratios of $\\Lambda ^{t_{max}}$ and $\\bar{\\Lambda }^{\\bar{t}_{max}}$ in the limit.", "In any event, the resulting theory is well behaved as $\\Lambda , ~\\bar{\\Lambda }\\rightarrow 0$ , and admits a perturbative expansion in positive powers of $\\Lambda $ and $\\bar{\\Lambda }$ .", "In applying our “practical\" method, we need to be sure to rescale the $A_{(k)}$ appropriately, depending on the dominant interaction.", "In particular, if $\\Lambda ^{t_{max}} / \\bar{\\Lambda }^{\\bar{t}_{max}}$ is not divergent, we introduce $A_{(k)} = \\Lambda ^{t_{max}} \\hat{A}_{(k)}$ , whilst if $\\bar{\\Lambda }^{\\bar{t}_{max}}/\\Lambda ^{t_{max}} $ is not divergent, we introduce $A_{(k)} = \\bar{\\Lambda }^{\\bar{t}_{max}} \\hat{A}_{(k)}$ .", "In both cases, the resulting dual theory is guaranteed to be well behaved as $\\Lambda , ~\\bar{\\Lambda }\\rightarrow 0$ , and to admit a perturbative expansion in positive powers of $\\Lambda $ and $\\bar{\\Lambda }$ .", "In a modified gravity model with two Vainshtein radii, $r_V$ and $\\bar{r}_V <r_V$ , the dual theory would lend itself to an expansion in both $r/r_V$ and $r/\\bar{r}_V$ .", "The generalization of the ideas presented in this section to theories with even more scales should now be obvious." ], [ "Multiple fields", "We shall now explain how our method should be generalized to deal with more than one field.", "We will assume a single strong coupling scale for brevity, so that the theory may be schematically written as ${\\cal L}\\sim \\sum _\\alpha -\\textstyle {1\\over 2}(\\partial \\phi _\\alpha )^2+\\sum _i\\frac{\\prod _\\alpha \\partial ^{D_{i, \\alpha }} (\\phi _\\alpha )^{N_{i, \\alpha }}}{\\Lambda ^{D_i+N_i-4}},$ where the index $\\alpha $ labels the field and $D_i=\\sum _\\alpha D_{i, \\alpha },~N_i=\\sum _\\alpha N_{i, \\alpha }$ .", "We have that $D_i+N_i>4$ , as well as $N_{i, \\alpha }\\ge 0$ and $N_i>2$ .", "The first condition is required for the theory to remains weakly coupled at low energies, while the two latter conditions are required in order to guarantee that the interactions become subdominant as $\\phi _\\alpha \\rightarrow 0$ .", "We assume interactions of the form $\\phi _\\alpha J_\\alpha $ , so that the equations of motion can be schematically written as $p^2 \\phi _\\beta + \\sum _i \\frac{N_{i, \\beta }}{\\phi _\\beta } \\frac{\\prod _\\alpha p^{D_{i, \\alpha } } \\phi ^{N_{i, \\alpha } }}{\\Lambda ^{D_i+N_i-4} } \\sim J_\\beta .$ Now, generically, if a particular interaction (or interactions) dominates the dynamics as $\\Lambda \\rightarrow 0$ , then we expect $\\phi _\\alpha $ to scale the same way for each value of $\\alpha $This is easily seen by taking the ratio of the $\\beta =\\beta _1$ and $\\beta =\\beta _2$ equations of motion.", "Assuming the term with $i=j$ is dominant, we have $\\frac{N_{j, \\beta _1} }{N_{j, \\beta _2} } \\frac{\\phi _{\\beta _2} }{\\phi _{\\beta _1}}\\sim \\frac{J_{\\beta _1}}{J_{\\beta _2}}$ .", "The right hand side is independent of $\\Lambda $ , and so neglecting the special case where some of the $N_{j, \\beta } $ vanish, we conclude that $\\phi _{\\beta _1} \\sim \\phi _{\\beta _2}$.", "We then claim that generically the term with largest $t_i=\\frac{D_i+N_i-4}{N_i-1}$ dominates the dynamics.", "To prove this we must show that none of the other interactions will give a divergent contribution to the equations of motion as $\\Lambda \\rightarrow 0$ .", "To this end, we note that if our claim is true, each field scales as $\\phi _\\alpha \\sim \\Lambda ^{t_{max}}$ .", "At the level of the equations of motion, the interactions will now go like $\\frac{1}{\\Lambda ^{t_{max}} } \\frac{\\prod _\\alpha p^{D_{i, \\alpha } } \\Lambda ^{t_{max} N_{i, \\alpha } }}{\\Lambda ^{D_i+N_i-4} } =p^D \\left(\\frac{\\Lambda ^{t_{max}}}{\\Lambda ^{t_i}}\\right)^{N_i-1},$ where we have used the fact that $D_i=\\sum _\\alpha D_{i, \\alpha },~N_i=\\sum _\\alpha N_{i, \\alpha }$ .", "Since $t_{max} \\ge t_i$ and $N_i>1$ it is clear that this does not diverge as $\\Lambda \\rightarrow 0$ .", "It is now clear that generically we can take the dual of this theory in complete analogy with the single field case.", "Again, $t_{max}$ plays a critical role, particularly when applying the “practical\" method." ], [ "The full galileon theory and its classical dual", "As an example of our method, we now consider the full galileon theory ${\\cal L}=\\sum _{n=1}^{n=4}\\frac{\\alpha _n}{\\Lambda _{(n)}^{3n-3}} \\phi \\delta ^{[\\nu _1}_{\\mu _1}...\\delta ^{\\nu _n]}_{\\mu _n}{\\partial ^{\\mu _1}\\partial _{\\nu _1}}\\phi \\ldots {\\partial ^{\\mu _n}\\partial _{\\nu _n}}\\phi ,$ where each $\\alpha _n ={\\cal O}(1)$ , and in principle we have a hierarchy of as many as three different scales, $\\Lambda _{(n)}$ , $n=2, 3, 4$ .", "The interaction terms have $t_n=3\\frac{n-1}{n}$ which will be important in establishing how to rescale our conjugate variables.", "Our aim is to find the dual theory that is well behaved as $\\Lambda _{(n)} \\rightarrow 0$ .", "Adopting the practical method, we first arrive at an equivalent Lagrangian, ${\\cal L}^{\\prime }&=&\\sum _n\\frac{\\alpha _n}{\\Lambda _{(n)}^{3n-3}} {\\phi } \\delta ^{[\\nu _1}_{\\mu _1}... \\delta ^{\\nu _n]}_{\\mu _n}{A^{\\;\\;\\mu _1}_{(2)\\nu _1}}\\ldots {A^{\\;\\;\\mu _n}_{(2)\\nu _n}}\\\\\\nonumber &~&+\\sum _n\\frac{n\\alpha _n }{\\Lambda _{(n)}^{3n-3}} A_{(0)}\\delta ^{[\\nu _1}_{\\mu _1}...\\delta ^{\\nu _n]}_{\\mu _n}{(\\partial ^{\\mu _1}\\partial _{\\nu _1}\\phi )}A^{\\;\\;\\mu _2}_{(2)\\nu _2}\\ldots A^{\\;\\;\\mu _n}_{(2)\\nu _n} \\\\\\nonumber &~&-\\sum _n\\frac{n\\alpha _n}{\\Lambda _{(n)}^{3n-3} } {A_{(0)}}\\delta ^{[\\nu _1}_{\\mu _1}...\\delta ^{\\nu _n]}_{\\mu _n}{A^{\\;\\;\\mu _1}_{(2)\\nu _1}}\\ldots {A^{\\;\\;\\mu _n}_{(2)\\nu _n}}.$ We need to rescale the variables according to the rules outlined in section REF .", "How we do this depends on the ratios of the following in the limit where $\\Lambda _{(n)} \\rightarrow 0$ , $\\Lambda _{(2)}^{t_2}, \\qquad \\Lambda _{(3)}^{t_3}, \\qquad \\Lambda _{(4)}^{t_4},$ where we recall that $t_n=3\\frac{n-1}{n}$ .", "Let us assume, that we are interested in the case where $\\Lambda _{(n_*)}^{t_{n_*}}/\\Lambda _{(n)}^{t_n}$ is not divergent in this limit for some particular choice of $n_*$ and for any $n$ .", "Then we perform the rescaling $A=\\Lambda _{(n_*)}^{t_{n_*}}\\hat{A}$ ${\\cal L}^{\\prime }&=&\\sum _n\\alpha _n\\left( \\frac{\\Lambda _{(n_*)}^{t_{n_*}}}{\\Lambda _{(n)}^{t_n}}\\right)^n {\\phi } \\delta ^{[\\nu _1}_{\\mu _1}... \\delta ^{\\nu _n]}_{\\mu _n}{\\hat{A}^{\\;\\;\\mu _1}_{(2)\\nu _1}}\\ldots {\\hat{A}^{\\;\\;\\mu _n}_{(2)\\nu _n}}\\\\\\nonumber &~&+\\sum _n n\\alpha _n\\left( \\frac{\\Lambda _{(n_*)}^{t_{n_*}}}{\\Lambda _{(n)}^{t_n}}\\right)^n \\hat{A}_{(0)}\\delta ^{[\\nu _1}_{\\mu _1}...\\delta ^{\\nu _n]}_{\\mu _n}{(\\partial ^{\\mu _1}\\partial _{\\nu _1}\\phi )}\\hat{A}^{\\;\\;\\mu _2}_{(2)\\nu _2}\\ldots \\hat{A}^{\\;\\;\\mu _n}_{(2)\\nu _n} \\\\\\nonumber &~&-\\sum _n n \\alpha _n \\Lambda _{(n_*)}^{t_{n_*}} \\left( \\frac{\\Lambda _{(n_*)}^{t_{n_*}}}{\\Lambda _{(n)}^{t_n}}\\right)^n {\\hat{A}_{(0)}}\\delta ^{[\\nu _1}_{\\mu _1}...\\delta ^{\\nu _n]}_{\\mu _n}{\\hat{A}^{\\;\\;\\mu _1}_{(2)\\nu _1}}\\ldots {\\hat{A}^{\\;\\;\\mu _n}_{(2)\\nu _n}}.$ Because we are assuming that $\\Lambda _{(n_*)}^{t_{n_*}}/\\Lambda _{(n)}^{t_n}$ does not diverge as we take the $\\Lambda _{(n)} \\rightarrow 0$ , it follows that this dual action is well behaved in the limit.", "Assuming that matter couples as in the cubic galileon case, we have the following equations of motion in the dual theory, $\\nonumber &~&\\sum _{n=1}^{n=4}\\alpha _n(\\lambda _{(n)})^n\\delta ^{[\\nu _1}_{\\mu _1}... \\delta ^{\\nu _n]}_{\\mu _n}\\hat{A}^{\\;\\;\\mu _1}_{(2)\\nu _1}\\ldots \\hat{A}^{\\;\\;\\mu _n}_{(2)\\nu _n}\\\\&&+\\sum _{n=1}^{n=4}n\\alpha _n(\\lambda _{(n)})^n\\delta ^{[\\nu _1}_{\\mu _1}... \\delta ^{\\nu _n]}_{\\mu _n}\\partial ^{\\mu _1}\\partial _{\\nu _1}\\left(\\hat{A}_{(0)}\\hat{A}^{\\;\\;\\mu _2}_{(2)\\nu _2}\\ldots \\hat{A}^{\\;\\;\\mu _n}_{(2)\\nu _n}\\right)+J=0,\\\\&~&\\sum _{n=1}^{n=4}\\alpha _n(\\lambda _{(n)})^n\\delta ^{[\\nu _1}_{\\mu _1}... \\delta ^{\\nu _n]}_{\\mu _n}\\left(\\partial ^{\\mu _1}\\partial _{\\nu _1}\\phi -\\Lambda ^{t_{n_\\star }}_{(n_\\star )}\\hat{A}^{\\;\\;\\mu _1}_{(2)\\nu _1}\\right)\\hat{A}^{\\;\\;\\mu _2}_{(2)\\nu _2}\\ldots \\hat{A}^{\\;\\;\\mu _n}_{(2)\\nu _n}=0,\\\\\\nonumber &~&\\sum _{n=1}^{n=4}n\\alpha _n(\\lambda _{(n)})^n\\phi \\delta ^{[\\rho \\nu _2}_{\\;\\sigma \\mu _2}... \\delta ^{\\nu _{n}]}_{\\mu _{n}}\\hat{A}^{\\;\\;\\mu _2}_{(2)\\nu _2}\\ldots \\hat{A}^{\\;\\;\\mu _{n}}_{(2)\\nu _{n}}\\\\\\nonumber &&+\\sum _{n=1}^{n=4}n(n-1)\\alpha _n(\\lambda _{(n)})^n\\hat{A}_{(0)}\\delta ^{[\\rho \\nu _2}_{\\;\\sigma \\mu _2}... \\delta ^{\\nu _{n}]}_{\\mu _{n}}\\partial ^{\\mu _2}\\partial _{\\nu _2}\\phi \\hat{A}^{\\;\\;\\mu _3}_{(2)\\nu _3}\\ldots \\hat{A}^{\\;\\;\\mu _{n}}_{(2)\\nu _{n}}\\\\&&-\\sum _{n=1}^{n=4}n^2\\alpha _n\\Lambda _{(n_\\star )}^{t_{n_\\star }}(\\lambda _n)^n\\hat{A}_{(0)}\\delta ^{[\\rho \\nu _2}_{\\;\\sigma \\mu _2}... \\delta ^{\\nu _{n}]}_{\\mu _{n}}\\hat{A}^{\\;\\;\\mu _2}_{(2)\\nu _2}\\ldots \\hat{A}^{\\;\\;\\mu _{n}}_{(2)\\nu _{n}}=0,$ where $\\lambda _{(n)}=\\Lambda ^{t_{n_\\star }}_{(n_\\star )}/\\Lambda ^{t_{n}}_{(n)}$ .", "We now find the background solution in the following high energy limit: $\\Lambda _{(n)} \\rightarrow 0$ with $\\lambda _{(n)}\\rightarrow {\\left\\lbrace \\begin{array}{ll}1 & n=n_* \\\\ 0 & n \\ne n_* \\end{array}\\right.", "}$ .", "Taking the source to be $J\\sim \\frac{M}{M_{pl}}p^3$ , as before, the background becomes $A_{(2)}\\sim \\left(\\frac{M}{M_{pl}}\\right)^{1/n_\\star }p^{\\frac{3}{n_\\star }},\\\\A_{(0)}\\sim A_{(2)}p^{-2}\\sim \\left(\\frac{M}{M_{pl}}\\right)^{1/n_\\star }p^{\\frac{3-2n_\\star }{n_\\star }},\\\\\\phi \\sim \\Lambda _{(n_\\star )}^{t_{n_\\star }}A_{(2)}p^{-2}\\sim \\Lambda _{(n_\\star )}^{t_{n_\\star }}\\left(\\frac{M}{M_{pl}}\\right)^{1/n_\\star }p^{\\frac{3-2n_\\star }{n_\\star }},$ with the higher order terms being by expansions in $\\Lambda ^{t_{n_\\star }}_{(n_\\star )}$ and in $\\lambda _{(n)}^{n}$ for $ n \\ne n_*$ .", "These correspond to expansions in $\\left(\\frac{\\Lambda _{(n_\\star )}}{p}\\right)^{t_{n_\\star }} \\left(\\frac{M}{M_{pl}} \\right)^{-t_{n_*}/3} \\sim \\left(\\frac{r}{r_{n_*}}\\right)^{t_{n_\\star }}$ and $\\lambda _{(n)}^{n} p^{\\frac{3(n-n_*)}{n_*}} \\left(\\frac{M}{M_{pl}}\\right)^{\\frac{n-n_*}{n_*}} \\sim \\left( \\frac{r}{r_n}\\right)^{\\frac{3(n_*-n)}{n_*}}, \\qquad n \\ne n_*$ respectively.", "The theory admits up to three critical radii given by $r_n \\sim {\\left\\lbrace \\begin{array}{ll} \\left(\\frac{M}{M_{pl}}\\right)^{1/3}\\Lambda _{(n_\\star )}^{-1} & n=n_* \\\\\\left(\\frac{M}{M_{pl}}\\right)^{1/3} \\lambda _{(n)}^\\frac{nn_*}{3(n-n_*)} &n \\ne n_*\\end{array}\\right.", "}$ thereby generalising the Vainshtein radius for the multiscale theory, as expected [16]." ], [ "Discussion", "Whenever classical perturbation theory breaks down at some particular scale, to continue making predictions beyond that scale one would like to have a dual theory.", "This should describe the same physics, but admit a perturbative description that works best in the opposite regime to that in the original theory.", "This is exactly what Gabadadze et al [9] achieved by identifying the duals to two particular classical theories that exhibit Vainhstein screening.", "In each case, perturbation theory works best above the Vainshtein radius in the original theory, and below the Vainshtein radius in the dual.", "In this paper, we have recognised these examples as being nothing more than the Legendre transform of the original Lagrangian.", "This has enabled us to generalize the idea, and outline how one can find the classical dual for a much larger class of self-interacting theories.", "We have also presented a more “user-friendly\" method for finding the dual for the case where it is difficult to compute the Legendre transform explicitly.", "Of course, a dual is only of any use if it admits a complementary perturbative description, as in the examples given in [9].", "If the original theory admits a good perturbative description as some dimensionful scale $\\Lambda \\rightarrow \\infty $ , this amounts to the dual theory admitting a good perturbative description in the opposite limit $\\Lambda \\rightarrow 0$ .", "That this should happen places certain restrictions on the form of the theory one can successfully dualize.", "The key ingredient is that there must exist a finite interaction (or interactions) that dominates the classical dynamics as $\\Lambda \\rightarrow 0$ , and about which we turn the theory on its head.", "This logic applies even when there are multiple scales or multiple fields.", "We have presented the classical dual of the full galileon theory as an example.", "Of course, there are many more examples one could consider, and one can find their duals using the methods we have discussed.", "We have also shown explicitly that a useful dual can only be found if certain specific criteria are met: namely that there exists a finite $N$ -point interaction (or interactions) with largest $t$ , where $t=\\frac{D+N-4}{N-1}$ and $D$ is the number of derivatives.", "Let us end with an example of a Lagrangian for which we cannot find a useful classical dual.", "Consider the Einstein-Hilbert action, written in terms of the metric expanded around Minkowski space, $g_{\\mu \\nu }=\\eta _{\\mu \\nu }+\\frac{1}{M_{pl}} h_{\\mu \\nu }$ .", "Schematically we have $S_{EH}=\\int d^4 x ~h \\partial ^2 h +\\sum _{i=3}^\\infty \\frac{\\partial ^2 h^i}{M_{pl}^i}.$ As is well known, this theory is well behaved as $M_{pl} \\rightarrow \\infty $ , as the interactions vanish.", "Each interaction has $D_i=2$ derivatives, $N_i=i$ fields and so $t_i=\\frac{D_i-N_i-4}{N_i-1}=-\\frac{i+2}{i-1}$ .", "Recall that the dynamics as $M_{pl} \\rightarrow 0$ is dominated by the term (or terms) with largest $t_i$ .", "However, this does not occur at a finite value of $i$ , and so we cannot find a classical dual that is well behaved as $M_{pl}\\rightarrow 0$ .", "This does not mean that one cannot find a classical dual to GR.", "Such a dual can be found but only if we describe the dynamics using something other than $h_{\\mu \\nu }$ .", "This is currently a work in progress [17]." ], [ "Acknowledgements", "We thank Gregory Gabadadze for a number of insightful comments.", "AP is supported by a Royal Society University Research Fellowship, and PMS by STFC." ] ]

[ [ "The Viewing Angles of Broad Absorption Line Versus Unabsorbed Quasars" ], [ "Abstract It was recently shown that there is a significant difference in the radio spectral index distributions of broad absorption line (BAL) quasars and unabsorbed quasars, with an overabundance of BAL quasars with steeper radio spectra.", "This result suggests that source orientation does play into the presence or absence of BAL features.", "In this paper we provide more quantitative analysis of this result based on Monte-Carlo simulations.", "While the relationship between viewing angle and spectral index does indeed contain a lot of scatter, the spectral index distributions are different enough to overcome that intrinsic variation.", "Utilizing two different models of the relationship between spectral index and viewing angle, the simulations indicate that the difference in spectral index distributions can be explained by allowing BAL quasar viewing angles to extend about 10 degrees farther from the radio jet axis than non-BAL sources, though both can be seen at small angles.", "These results show that orientation cannot be the only factor determining whether BAL features are present, but it does play a role." ], [ "INTRODUCTION", "A long-time popular explanation for the presence of broad absorption lines (BALs) seen in approximately 20% of quasar spectra (Knigge et al.", "2008) has been a simple orientation model, in which BAL quasars are seen only from a more “edge-on\" perspective, or at larger viewing angles (Elvis 2000).", "Understanding the geometry of the outflows producing these lines is an important part of modeling the role they play in the evolution of the quasar itself, as well as the effects they may have on the surrounding environment and host galaxy via feedback effects.", "For example, it has been shown that it is possible for AGN feedback to affect star formation rates in the host galaxy, both theoretically (Hopkins & Elvis 2010) and now it seems observationally (Cano-Díaz et al.", "2012).", "The similarity of the emission lines in BAL and non-BAL quasars (Weymann et al.", "1991), as well as their optical polarization properties (Ogle et al.", "1999) have been used to support simple orientation models.", "However, this scheme fails to explain various other observations, particularly at radio frequencies.", "Short timescale radio variability has been identified in around 20 BAL quasars, which is argued to indicate a viewing angle near the radio jet axis to explain the derived brightness temperatures (Ghosh & Punsly 2007, Zhou et al.", "2006).", "In general, BAL quasars are more compact than non-BALs in radio maps (Becker et al.", "2000), at least at low to intermediate resolution, and in small samples they do not show a significant difference in radio spectral index distribution compared to non-BAL sources (Becker et al.", "2000, Montenegro-Montes et al.", "2008, Fine et al.", "2011).", "Radio spectral index ($\\alpha $ ; $S_{\\nu } \\propto \\nu ^{\\alpha }$ , where $S_{\\nu }$ is the radio flux and $\\nu $ is the frequency) is generally considered an orientation indicator, with steeper spectrum ($\\alpha < -0.5$ ) sources seen more edge-on because they are dominated by optically thin lobe emission, which has a steep spectrum due to little synchrotron self-absorption.", "More pole-on sources are dominated by core emission because of relativistic beaming effects, and have a flatter spectrum because they are optically thick and thus significantly self-absorbed.", "Due in large part to these observations, other explanations based on pure evolution (e.g.", "Gregg et al.", "2006), and not orientation, have also come into favor.", "DiPompeo et al.", "(2011) expanded greatly the number of BAL quasars with multi-frequency radio data and presented a sample of 74 BAL quasars, along with a sample of 74 individually matched unabsorbed quasars, with flux measurements at 4.9 and 8.4 GHz (observed frame) from the Very Large Array (VLA)/Expanded Very Large Array (EVLA).", "These data provided quasi-simultaneous flux measurements to remove the effects of radio variability in the spectral index measurements.", "The distributions of $\\alpha _{8.4}^{4.9}$ were significantly different, with BAL quasars showing an overabundance of steep spectrum sources, but both samples show a wide range of spectral indices.", "Analysis of other measurements of $\\alpha $ (including $\\alpha _{fit}$ , a simple linear fit to available literature fluxes at various frequencies, in addition to the new measurements), a variety of statistical tests, and a restriction to only unresolved sources in both samples all also show that the difference is present, although the significance does vary.", "Some of this variation could be due to the fact that the other measures of $\\alpha $ included non-simultaneous flux measurements, or due to variation in the number of sources included in the tests.", "Because the two samples are one-to-one matched in redshift, among other properties, use of rest-frame spectral indices will not effect the results.", "These results indicate that while BAL quasars likely span a range of orientations, viewing angle does plays a role in their presence.", "The next step, presented here, is to quantify this difference and provide the most likely viewing angles to these sources, at least in a general sense.", "Our aim is to test if the difference seen in spectral index distributions can be explained by differences in viewing angle, and we recognize that it may be possible to develop more sophisticated models and simulations in the future." ], [ "THE $\\alpha $ -{{formula:0d66111d-509a-4d0b-ba2b-f07f1fb755f4}} RELATIONSHIP", "We have based our modeling off of two relationships between $\\alpha $ and viewing angle ($\\theta $ , defined as 0 along the radio jet axis); one purely observational, and one from semi-empirical simulations.", "We began developing the observational relationship with the sample of Wills & Brotherton (1995), which included the 29 quasars of Ghisellini et al.", "(1993) with viewing angles calculated from superluminal motion seen in VLBI maps, as well as 4 additional sources from Vermeulen & Cohen (1994) with superluminal motion measurements and X-ray data at 1 keV available.", "We then searched NEDThe NASA/IPAC Extragalactic Database (NED) is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration for multi-frequency radio fluxes in order to measure $\\alpha $ for these sources.", "We initially attempted to use fluxes at 4.85 and 8.4 GHz to better match the observations in DiPompeo et al.", "(2011), as well as a linear fit to all available radio fluxes, but given the small number of sources and the scatter in the $\\alpha $ -$\\theta $ relationship we were unable to get a reasonable fit using these data.", "One complication is that in general the radio flux measurements are not simultaneous, and variability almost certainly exaggerates the scatter since the majority of these sources are seen from small viewing angles.", "In the end we settled on using the spectral index between 15 and 8.4 GHz, which was available for 27 of the 33 sources (26 from Ghisellini et al.", "(1993) and 1 from Vermeulen & Cohen 1994), in order to build a useable model.", "We note that a necessary assumption in this analysis is that on average the radio spectra are reasonably approximated by a power law over a large frequency range, which may be an oversimplification in some cases.", "However, if you inspect the spectra presented in DiPompeo et al.", "(2011) for both BAL and non-BAL samples, when data is available at more frequencies the spectra are often well behaved across a large frequency range.", "We adopt the values of the viewing angles from Ghisellini et al.", "(1993) and follow their method to calculate the viewing angle for the additional Vermeulen & Cohen (1994) source.", "Two sources that were large outliers (1830+285 and 1845+797, from Ghisellini et al.", "(1993), which both have highly inverted radio spectra) were excluded in order to allow us to make a fit to the data that could possibly reproduce our observed range of spectral indices.", "The final list of sources, their viewing angles and their spectral indices are given in Table .", "While the relationship is likely more complex in reality, we made a simple linear fit to the data.", "Above a viewing angle of 30 we assume a flat relationship where all the scatter is due to intrinsic variation in radio sources for several reasons.", "First, there is no observational data above 30 to constrain the shape of the relationship.", "Second, if we simply extend the linear relationship, we would see large numbers of sources with extremely steep spectra ($\\alpha $ extending to $-3$ or $-4$ ), which is simply not observed in most radio sources.", "Finally, although it is dependent on the Lorentz factor of the the emitting material, the increase in observed flux due to Doppler boosting is expected to be small at larger viewing angles.", "Because variability almost certainly exaggerates the scatter in the relationship, we use the standard deviation of the spectral index distribution of the quasars in the 3CR catalog (as presented in Smith & Spinrad 1980) to model the scatter.", "Because this catalog was built using low frequency data, it is dominated by lobe emission and therefore consists of mostly steep spectrum, edge-on sources.", "Shown in the top of Figure REF is the data used from Ghisellini et al (1993) and Vermeulen & Cohen (1994) and the fit used to model the $\\alpha $ -$\\theta $ relationship; the bottom of Figure REF shows the result of our Monte-Carlo simulation of 10,000 randomly distributed jet viewing angles and their spectral indices based on the above model.", "The semi-empirical model used is from the simulations of Wilman et.", "al (2008), which is the extragalactic portion of the SKA Simulated Skies (S$^3$ ) project.", "We pulled from their simulations the fluxes at all available frequencies (151 MHz, 610 MHz, 1.4 GHz, 4.86 GHz, and 18 GHz, all observed frame) for all FR I and FR II type sources between redshifts of 1.5 and 3.5 and with values of $S_{1.4} \\ge 10$ mJy.", "Ideally we would use the spectral index between 4.86 and 18 GHz from these simulations; however, there are issues with their high frequency results.", "The source populations are drawn from a 151 MHz luminosity function and extrapolated to high frequencies, which almost certainly does not accurately predict the high frequency source population (see for example, Mahony et al.", "2011).", "Additionally, there may be problems with the functional form assumed for the SEDs of radio cores, causing them to fall off too steeply at high frequency (Wilman, private communication).", "This causes the maximum spectral index found to drop well below 0 as you go to higher frequencies, which has no bearing on physical reality.", "The simulations also apply a lower limit on the lobe spectral indices at $\\alpha =-0.75$ , which is problematic because we clearly see a high fraction of our sources with steeper spectra than this (the steep spectrum overabundance in BAL sources is most prominent between $-2 \\le \\alpha \\le -1$ ).", "However, if we again simply assume that in general radio spectra of these objects obey a simple power law, we can use the lower frequency spectral index distribution from these simulations and assume that a similar relationship holds between $\\theta $ and $\\alpha $ regardless of which part of the radio spectrum is considered.", "Given the scatter in the relationship it is unlikely that this assumption will significantly effect our results.", "The relationship between $\\theta $ and $\\alpha _{610}^{151}$ from these simulations is shown in the upper panel of Figure REF .", "In order to use this in our simulations, we made two fits to different ranges of $\\theta $ .", "First, we did a simple linear fit from $10 \\le \\theta \\le 20$ , where the data can reasonably be approximated as linear.", "This allows us to have sources with steeper spectra than $\\alpha =-0.75$ , which is needed to reproduce what is actually observed despite the limit applied in the simulations.", "The distribution in $\\alpha $ is then calculated by normalizing the data by the fit so all the variation is in $\\alpha $ , assuming the distribution is Gaussian and calculating the standard deviation.", "The data is clearly non-linear at small $\\theta $ , and so between $5 \\le \\theta \\le 20$ we also make a separate linear fit in $\\log (\\alpha + 0.75)$ -$\\theta $ space.", "Our final model is then based on a combination of these fits; below viewing angles of 10 the fit in logarithmic space is used, and above 10 the linear fit is used.", "Both lines are shown in the upper panel of Figure REF .", "The lower panel of Figure REF shows the result of our Monte-Carlo simulation of 10,000 randomly distributed jet viewing angles and their spectral indices based on this semi-empirical model.", "The overall general shape of the two models used is similar; however, the amount of scatter in the observational model is significantly more than that in the empirical model.", "We also do not consider the possible effects of redshift in either model, though it is possible that the wide range of redshifts in the observed sample could have an effect on the spectral index distributions.", "Again however, since the BAL and non-BAL samples are well matched in redshift, we feel that any effects of redshift will not change the general trends found." ], [ "SIMULATIONS", "The simulations were performed using IDL.", "We first generate a random vector in 3-D space, utilizing IDL's uniformly distributed random number generator to create x, y, and z coordinates between -1 and 1, which are combined to create vectors within a “unit cube”.", "This will produce a distribution of vector directions biased toward the corners of the cube, and so we then reject any vector that does not fall within the unit sphere (has a magnitude greater than 1), resulting in vectors uniformly distributed in random directions in 3-D space.", "This vector is taken to represent one side of a bipolar radio jet.", "We next choose a viewing direction along the z-axis, and compute the viewing angle to the randomly generated vector.", "If the vector is pointing in the negative z direction, we reflect it about the origin by multiplying all of its components by $-1$ , to represent the opposite side of the bipolar jet, so that viewing angles range from 0 to 90.", "Once the viewing angle is known, we can assign a value of $\\alpha $ based on one of the models in the previous section.", "This is done utilizing IDL's normally distributed random number generator, so that the assigned values of $\\alpha $ are normally distributed about the model fit with the assumed standard deviation.", "The bottom panels of Figures REF and REF were generated by repeating this process $10^4$ times.", "In order to compare the Monte-Carlo simulations to the real data and quantify the viewing angles to BAL and normal quasars, we used the simulation to virtually repeat our observational program while restricting what source viewing angles were allowed.", "We systematically stepped through all possible ranges of allowed viewing angles between some $\\theta _{min}$ and $\\theta _{max}$ , assigning values of $\\alpha $ for each random viewing angle between the allowed $\\theta _{min}$ and $\\theta _{max}$ until there were 74 virtual “observations” (to compare to the 74 objects really observed in each sample).", "We then compared the distribution of simulated spectral indices to the observed spectral index distributions for the BAL and non-BAL samples, using both K-S and R-S tests, to check whether the real data were well reproduced.", "Our criterion was a p-value of less than 0.05 to indicate that the distributions are from a different parent population, and are not well matched.", "We then repeat this experiment $10^5$ times, in order to get a statistical sense of how often the real observations are well reproduced for a given range of viewing angles.", "Once this is complete, the allowed viewing angle range is changed and the process is repeated until all possible $\\theta $ ranges are tested.", "We apply an upper limit to $\\theta _{max}$ of 45  as above this value it is likely that most quasars are obscured by dust (Barthel 1989).", "As the results will show in the next section, this upper limit has no effect on our findings." ], [ "RESULTS", "The probabilities of each set of simulations producing a match (based on the K-S tests) to the observed data as a function of $\\theta _{min}$ and $\\theta _{max}$ are shown in Figures REF -REF .", "The z-axis represents the percentage (out of the $10^5$ runs) that the resulting $\\alpha $ distribution matched the observed distribution, the x-axis is $\\theta _{min}$ and the y-axis is $\\theta _{max}$ .", "In each figure, the left panel shows the probabilities for the non-BAL sample, and the right panel shows the probabilities for the BAL sample.", "Figures showing the probability distributions using an R-S test for comparison are not included to save space, since the results are not significantly different than using K-S tests.", "The results are all summarized in Tables  (K-S test comparison) and  (R-S test comparison); Column (1) is the sample that the simulations were compared to, Column (2) is the model used in the simulations (“Obs” is the model based on the sample in Wills & Brotherton (1995), “Emp” is the model based on the semi-empirical simulations of Wilman et al.", "(2008)), Column (3) is the maximum probability reached, Column (4) is the value of $\\theta _{min}$ at $P_{max}$ , and Column (5) is the value of $\\theta _{max}$ at $P_{max}$ .", "The final two columns, (6) and (7) to the right of the vertical line, indicate the values of $\\theta _{min}$ and $\\theta _{max}$ with the largest separation that also have a probability of reproducing the observed results of greater than 90%.", "In general, the simulations are able to accurately reproduce our observations, with more than half of them having probability distributions peaking at or above 99%.", "The only exception is trying to reproduce the $\\alpha _{fit}$ distribution for non-BALs using the semi-empirical model and comparing using a K-S test, where the probability peaks at only 71%.", "However, the shape of the probability distribution and the location of the peak is still basically the same as other simulations.", "The probability distributions are also generally quite flat in the $\\theta _{min}$ direction, except for the cases where the semi-empirical model and a K-S test are used; in those situations the distributions are well-peaked.", "This flatness is the reason we include the final two columns in Tables  and , as well as as the fact that the viewing angle ranges suggested by considering only the peaks can often be unrealistically narrow.", "While there is some variation in the results depending on which model or statistical test is used, they are usually quite consistent.", "In all cases, BAL quasars can have small viewing angles, but always extend farther from the radio jet axis when compared to non-BAL quasars.", "If we take an average of all simulations, we find an average viewing angle range of 0-22 for non-BALs and 1-32 for BAL quasars." ], [ "DISCUSSION", "With maybe the exception of the simulations using the observational model of the $\\alpha $ -$\\theta $ relationship and comparing to $\\alpha _{fit}$ with a K-S test, the general trend seen is that BAL quasars cover the same range of viewing angles as normal quasars, but extend about 10 farther from the jet axis.", "These results suggest that objects such as “polar” BALs (Ghosh & Punsly 2007, Zhou et al.", "2006) are indeed real and that for small to intermediate viewing angles it is possible to observe a quasar either with or without BALs, but at the largest viewing angles one will only see BAL features.", "While constraining the $\\alpha $ -$\\theta $ relationship can be difficult, it is interesting and encouraging that the results are quite similar using either of the two models or statistical comparisons here.", "We suggest that more effort be placed on constraining this relationship (both observationally and theoretically) in the future, as sample sizes grow to significant enough numbers to use radio spectral index as a statistical orientation indicator when no other method is available.", "One problem that is readily apparent in these results is that none of the samples extend to viewing angles as large as one might expect from, for example, the results of Barthel (1989).", "This could indicate a problem with the models used.", "However, the fact that the general result seems mostly independent of which model is used, we believe that while a change in the model may affect the absolute numbers it will not change the main conclusion.", "Also, as mentioned in DiPompeo et al.", "(2011) there may be a slight biasing of our original sample toward more face-on sources (due to the requirement that all sources have 1.4 GHz FIRST fluxes greater than 10 mJy), but this bias should effect both BAL and non-BAL quasars equally.", "So while it is possible that $\\theta _{max}$ may be higher for both samples, the end result should remain the same.", "Another consideration in this analysis is whether we can interpret the shape of the radio spectrum in a similar way in unresolved and extended radio sources.", "To build models of the $\\alpha $ -$\\theta $ relationship, we are required to utilize large scale, extended radio sources because these are the only sources in which $\\theta $ can be measured directly, at least in sufficient numbers to build a useable model.", "In contrast, the samples of DiPompeo et al.", "(2011) consist of high luminosity, generally compact radio sources (86% and 78% of BAL and non-BAL quasars, respectively, at 5 resolutions), although there is generally not data at high enough resolution or enough frequencies to classify them as belonging to the special classes of compact steep-spectrum (CSS) or especially gigahertz peaked-spectrum (GPS) sources.", "It is possible that the shape of the spectrum in the samples is affected by evolution, as is the case for CSS sources, and not just geometry.", "However, because CSS sources are selected to have steep spectra by definition, a comparison of spectral index distributions to test this theory would not be useful.", "Instead, more radio data are needed, in particular at lower frequency, to look for turnovers in the radio spectra before comparisons with this class of object can be reasonably made.", "There are also indications that compact sources may have lower bulk velocities and thus lower Doppler factors (e.g., Polatidis & Conway 2003).", "If this is the case in these quasars, it is possible then that the core component is not enhanced at the same level as other sources and therefore the lobe component remains dominant to smaller viewing angles.", "However, the fact that DiPompeo et al.", "(2011) still see a significant difference in spectral index distributions even when restricting the BAL and non-BAL samples to only unresolved sources suggests that there are still significant orientation effects.", "Additionally, VLBI studies of BAL quasars (such as Doi et al.", "2009, Kunert-Bajraszewska et al.", "2010) do in fact show that compact BAL quasars have highly collimated radio jets, which means that viewing angle should effect the steepness of the radio spectrum in these sources in a similar way as more extended sources, as supported for example by the results of Jiang & Wang (2003).", "Regardless of the complications mentioned above, these simulations should lay to rest any claims that BAL quasars are only seen edge-on, and move the argument away from simple dichotomies toward a more complex explanation that includes both orientation and other factors.", "Of course with the data used here, we can only make these claims for radio-loud BAL quasars, though the similarities in other geometrically dependent properties between radio-loud and radio-quiet BAL quasars (for example, optical polarization; DiPompeo et al.", "2010) suggest they may extend to radio-quiet BALs as well.", "It is clear that to explain the spectral index distributions seen, there needs to be a large overlap in viewing angles to BAL and non-BAL quasars.", "One simple explanation could be that BAL winds are launched at a variety of angles in different sources, or that they have a wide range of opening angles.", "It is also possible that BALs can be explained by a combination of orientation and evolution.", "The evolutionary schemes put forth for example by Gregg et al.", "(2002, 2006) suggest that BAL quasars begin enshrouded in a cocoon of gas and dust, which is blown out as radio jets develop and luminosity increases.", "We may be able to draw on this description with these results, by making the clearing out of the gas and dust orientation dependent.", "It is possible that the polar regions around the developing radio jets are cleared out first, which would explain the low numbers of seemingly “polar” BALs found as well as the spectral index distribution we observe.", "The most equatorial regions then are never completely cleared of absorbing material (or are replenished by a disk wind), causing only BAL quasars to be seen beyond a particular viewing angle.", "Whether a BAL is seen at small or intermediate angles is determined by the evolutionary status of the quasar, but in the latest stages of the quasar lifetime the presence of BALs is only orientation dependent.", "We will attempt to develop this picture further in future papers, considering more factors than just the spectral index distributions modeled here." ], [ "SUMMARY", "We have performed Monte-Carlo simulations of randomly oriented bi-polar radio jets and their corresponding radio spectral indices, in order to quantify the results of DiPompeo et al.", "(2011) that show a significant difference in the spectral index distributions of BAL versus non-BAL quasars.", "The simulations utilize two different models of the relationship between spectral index and viewing angle; one based on observations and one based on semi-empirical simulations.", "By limiting the allowed source viewing angles and comparing the resulting spectral index distributions with our observations, we can constrain the viewing angles to BAL and unabsorbed sources.", "The results are mostly independent of which spectral index-viewing angle model is used, which spectral index measurement they are compared to, and which statistical test is used to compare the simulations to reality, at least in a general sense.", "We find that there is a large overlap between viewing angles to the two samples, with both probably extending all the way down to 0  or along the jet axis, which supports the claim that there are BAL quasars with polar outflows.", "However, viewing angles to BAL sources generally extend farther from the jet axis, with about a 10 span in which BALs will always be seen.", "It is clear that a simple orientation-only model for BAL quasars cannot explain all of their observed properties, though orientation does play a role.", "We need to move away from simple dichotomies in order to fully understand them, and it is likely that a combination of previous models will prevail in explaining this important subclass of quasar.", "This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.", "M. DiPompeo would like to thank the European Southern Observatory for funding visits to collaborate with C. De Breuck, as well as the Wyoming NASA Space Grant Consortium for providing funding to perform all of the radio observations.", "M. DiPompeo would also like to thank Dr. Ruben Gamboa of the University of Wyoming Computer Science Department for assisting with the simulations, as well as Dr. Adam Myers for providing computing facilities to run the simulations.", "Finally, we thank NRAO for their support in performing the radio observations that are the basis of these simulations; The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. ccc|ccc 0pt Sources Used in the Observational Model Source $\\theta $ ($$ ) $\\alpha _{15}^{8.4}$ Source $\\theta $ ($$ ) $\\alpha _{15}^{8.4}$ 0016$+$ 731 5.0 0.91 1040$+$ 123 17.0 -0.80 0106$+$ 013 3.0 0.47 1150$+$ 812 13.2 0.46 0153$+$ 744 26.1 -1.53 1156$+$ 295 2.2 1.12 0212$+$ 735 6.2 0.28 1226$+$ 023 6.3 -1.04 0333$+$ 321 3.0 1.02 1253$-$ 055 3.2 -0.06 0430$+$ 052 9.9 0.37 1641$+$ 399 5.6 0.56 0552$+$ 398 20.2 -1.11 1828$+$ 487a 6.8 0.74 0615$+$ 820 21.7 -0.56 1928$+$ 738 7.4 0.11 0836$+$ 710 4.7 0.21 2134$+$ 004 0.1 -0.53 0850$+$ 581 12.5 -0.93 2223$-$ 052 1.6 -0.31 0906$+$ 430 0.9 0.69 2230$+$ 114 4.0 0.52 0923$+$ 392 4.4 0.42 2251$+$ 158 5.8 0.82 1039$+$ 811 24.6 -0.32 aThis source is from the sample of Vermeulen & Cohen et al.", "(1994).", "The viewing angle was calculated using the method of Ghisellini et al.", "(1993), using x-ray data from Wilkes et al.", "(1994) and additional radio data from Polatidis et al (1993).", "All sources and viewing angles are taken from Ghisellini et al.", "(1993) unless noted otherwise.", "The spectral index $\\alpha _{15}^{8.4}$ is measured from data gathered via NED.", "cccccc|cc Simulation Results Using K-S Tests Sample Model $\\alpha $ $P_{max}$ $\\theta _{min, P_{max}}$ $\\theta _{max, P_{max}}$ $\\theta _{min, P>0.9}$ $\\theta _{max, P>0.9}$ non-BAL Obs $\\alpha _{8.4}^{4.9}$ 0.99 12 19 0 24 BAL Obs $\\alpha _{8.4}^{4.9}$ 0.99 13 27 0 34 non-BAL Emp $\\alpha _{8.4}^{4.9}$ 0.93 0 24 0 25 BAL Emp $\\alpha _{8.4}^{4.9}$ 0.96 1 37 0 39 non-BAL Obs $\\alpha _{fit}$ 0.99 9 17 0 22 BAL Obs $\\alpha _{fit}$ 0.99 17 18 5 25 non-BAL Emp $\\alpha _{fit}$ 0.71 0 18 0a 19a BAL Emp $\\alpha _{fit}$ 0.99 4 27 0 30 aIn this simulation the probability never reaches above 0.9, and so these values give the maximum range in $\\theta $ where the probability is over 70% See section 4 for a detailed explanation of the entries in this table.", "cccccc|cc Simulation Results using R-S Tests Sample Model $\\alpha $ $P_{max}$ $\\theta _{min, P_{max}}$ $\\theta _{max, P_{max}}$ $\\theta _{min, P>0.9}$ $\\theta _{max, P>0.9}$ non-BAL Obs $\\alpha _{8.4}^{4.9}$ 0.99 15 17 0 24 BAL Obs $\\alpha _{8.4}^{4.9}$ 0.99 18 24 0 34 non-BAL Emp $\\alpha _{8.4}^{4.9}$ 0.99 14 17 0 25 BAL Emp $\\alpha _{8.4}^{4.9}$ 1.00 21 27 0 40 non-BAL Obs $\\alpha _{fit}$ 0.98 13 14 0 21 BAL Obs $\\alpha _{fit}$ 0.97 15 20 0 27 non-BAL Emp $\\alpha _{fit}$ 1.00 7 14 0 21 BAL Emp $\\alpha _{fit}$ 0.99 17 28 0 29 See section 4 for a detailed explanation of the entries in this table.", "Figure: (Top) Observational data to constrain the α\\alpha -θ\\theta relationship, with the derived linear fit.", "Here α\\alpha is measured between 15 and 8.4 GHz due to constraints on available data, and we assume that this can be extended to lower frequencies.", "Due to a lack of data and the fact that Doppler boosting does not likely effect the relationship above 30, we use a flat line there with some intrinsic scatter.", "(Bottom) Our monte-carlo simulation of 10,000 randomly distributed radio jets and their corresponding spectral index, determined by the observational data at the top.Figure: (Top) Simulation data from Wilman et al.", "(2008) showing the α\\alpha -θ\\theta relationship for all FRI and FRII sources between 1.5≤z≤3.51.5 \\le z \\le 3.5 with S 1.4 S_{1.4} greater than 10 mJy.", "Here alpha is measured between 151 and 610 MHz, due to issues with the Wilman et al.", "(2008) simulations at higher frequency.", "The linear fit is to the points between 10 and 20 before the data bottom out at α=-0.75\\alpha =-0.75 (a lower limit used in the models) and flatten out at low θ\\theta .", "The curved line is a linear fit to log(α+0.75)\\log (\\alpha +0.75) and θ\\theta between 5 and 20.", "Below 10 the curved fit is used, above the linear fit is used.", "(Bottom) Our Monte-Carlo simulation of 10,000 randomly distributed radio jets and their corresponding spectral indicies, determined by the simulation data at the top.Figure: Comparison (via K-S tests) of simulations to observations of non-BALs (left) and BALs (right) for the observed α 8.4 4.9 \\alpha _{8.4}^{4.9}, using the observationally constrained model.", "The z-axis is the probability of the simulations matching the observations, as a function of θ min \\theta _{min} (x-axis) and θ max \\theta _{max} (y-axis).Figure: Comparison (via K-S tests) of simulations to observations of non-BALs (left) and BALs (right) for α 8.4 4.9 \\alpha _{8.4}^{4.9}, using the semi-empirical model.", "The axes are the same as in Figure .Figure: Comparison (via K-S tests) of simulations to observations of non-BALs (left) and BALs (right) for α fit \\alpha _{fit}, using the observational model.", "The axes are the same as in Figure .Figure: Comparison (via K-S tests) of simulations to observations of non-BALs (left) and BALs (right) for α fit \\alpha _{fit}, using the semi-empirical model.", "The axes are the same as in Figure ." ] ]

[ [ "Image Processing Variations with Analytic Kernels" ], [ "Abstract Let $f\\in L^1(\\R^d)$ be real.", "The Rudin-Osher-Fatemi model is to minimize $\\|u\\|_{\\dot{BV}}+\\lambda\\|f-u\\|_{L^2}^2$, in which one thinks of $f$ as a given image, $\\lambda > 0$ as a \"tuning parameter\", $u$ as an optimal \"cartoon\" approximation to $f$, and $f-u$ as \"noise\" or \"texture\".", "Here we study variations of the R-O-F model having the form $\\inf_u\\{\\|u\\|_{\\dot{BV}}+\\lambda \\|K*(f-u)\\|_{L^p}^q\\}$ where $K$ is a real analytic kernel such as a Gaussian.", "For these functionals we characterize the minimizers $u$ and establish several of their properties, including especially their smoothness properties.", "In particular we prove that on any open set on which $u \\in W^{1,1}$ and $\\nabla u \\neq 0$ almost every level set $\\{u =c\\}$ is a real analytic surface.", "We also prove that if $f$ and $K$ are radial functions then every minimizer $u$ is a radial step function." ], [ "Introduction", "Several $BV$ variational models have been proposed as image decomposition models (see Section 2 for the definition of $BV$ ).", "First, Rudin-Osher-Fatemi [26] proposed the minimization $\\inf _{u\\in BV} \\left\\lbrace \\Vert u\\Vert _{\\dot{BV}} + \\lambda \\Vert f-u\\Vert ^2_{L^2}\\right\\rbrace .$ In (REF ), $f\\in L^1({\\mathbb {R}}^d)$ is a real function and one thinks of $u$ as the “cartoon” component of $f$ and $f-u$ as the “noise+texture” component of $f$ .", "By the strict convexity of the functional $\\Vert f-u\\Vert ^2_{L^2}$ , problem (REF ) has a unique minimizer $u$ .", "However, one limitation of model (REF ) is illustrated by the following example from [22] and [13]: if $d =2$ and $f=\\alpha \\chi _D$ where $D$ a disk centered at the origin and of radius $R$ , then $u=(\\alpha -(\\lambda R)^{-1})\\chi _D$ and $v=f-u=(\\lambda R)^{-1}\\chi _D$ if $\\lambda R\\ge 1/{\\alpha }$ , but $u =0$ if $\\lambda R\\le 1/{\\alpha }$ .", "Thus $u\\ne f$ can occur even though $f\\in BV$ is already a cartoon without texture or noise (note that $f$ and $u$ still have the same set of discontinuity).", "To overcome this limitation and also to attempt to separate noise from texture, many authors have introduced alternate forms of (REF ) by replacing $\\Vert f-u\\Vert ^2_{L_2}$ by other expressions.", "We mention the book [22] and the papers [30], [31], [29], [12], [1], [2], [3], [33], [25], [21], [20], [6], [7], [17], [9]), [18], [8].", "Among these, the papers of Chan and Esedoglu [12] and Allard [1], [2], [3] are closest to the present work.", "Chan and Esedoglu [12] considered the minimization $\\inf _{u\\in BV}\\Big \\lbrace \\Vert u\\Vert _{\\dot{BV}}+\\lambda \\int |f-u|dx\\Big \\rbrace $ (see also Alliney [4] for the one-dimensional discrete case).", "For this problem minimizers always exist but they may not be unique.", "For the example $d=2$ and $f=\\chi _{B(0,R)},$ [12] gives $u=f$ if $R> { {2}\\over {\\lambda } } $ and $u=0$ if $R< { {2}\\over {\\lambda } } $ .", "W. Allard [1], [2], [3] analyzed extremals for the problem $\\inf _{u\\in BV}\\Big \\lbrace \\Vert u\\Vert _{\\dot{BV}}+\\lambda \\int \\gamma (u-f)dx\\Big \\rbrace $ where $\\gamma (0)=0$ , $\\gamma \\ge 0$ , and $\\gamma $ is locally Lipschitz.", "Then minimizers $u$ exist although they may not be unique.", "Moreover, the minimizers $u$ satisfy the smoothness condition $\\partial ^*(\\lbrace u>t\\rbrace )\\in C^{1+\\alpha },\\ \\ \\ \\alpha \\in (0,1)$ where $\\partial ^*$ denotes “measure theoretic boundary\".", "Allard also gave mean curvature estimates on $\\partial ^*(\\lbrace u>t\\rbrace ).$ In this paper we study a cartoon+texture decomposition model defined with a positive, real analytic convolution kernel $K$ : $\\inf _{u\\in BV} \\Big \\lbrace \\Vert u\\Vert _{\\dot{BV}} + \\lambda \\Vert K*(f-u)\\Vert ^q_{L^p}\\Big \\rbrace $ where $1 \\le p, q < \\infty .$ We choose the kernel $K$ in (REF ) so that the Fourier transform $\\widehat{K}(\\xi )$ decays rapidly as $|\\xi |\\rightarrow \\infty $ .", "The motivation is that we expect $v=f-u$ to be oscillatory, so that $\\widehat{v}(\\xi )$ is large when $|\\xi |$ is large.", "Thus, $\\widehat{K}\\cdot \\widehat{v}= \\widehat{(K * v)}$ dampens high frequencies of $v$ , which suggests that $\\Vert K*v\\Vert _{L^p}^q$ is small for oscillatory $v$ .", "We also want the cartoon component $u$ to be very simple, for example, to be piecewise constant or to have real analytic level sets, and for that reason we choose $K$ to be real analytic.", "Examples of such $K$ are the Gaussian kernel where $\\widehat{K}(\\xi ) = e^{-\\pi t|\\xi |^2}$ or the Poisson kernel where $\\widehat{K}(\\xi ) = e^{-\\pi t|\\xi |}$ , for some $t>0$ .", "By comparison [12] takes $p = q =1$ and $K = {\\rm identity}$ and our choices of $K$ yield more precise results about the minimizers for (REF ).", "In comparison with Allard's paper [1] we note that for many choices of the kernel $K$ our functional $||K*(f-u)||^q_{L^p}$ is admissible in the sense of [1] so that the regularity results from section 1.5 of that paper hold for the minimizers $u$ of (REF ).", "However, because of the analyticity of $K$ our minimizers have greater smoothness than those from [1].", "Moreover the functional in (REF ) is not local in the sense of [1], so that the conclusions of section 1.6 of [1] need not hold for the minimizers of (REF )." ], [ "The Variational Problems", "To begin we recall the definition of $BV = BV({\\mathbb {R}}^d).$ Definition 1 Let $u \\in L^1_{\\rm {loc}}({\\mathbb {R}}^d)$ be real.", "We say $u \\in BV$ if $\\sup \\Bigl \\lbrace \\int u {\\rm {div}}\\varphi dx: \\varphi \\in C^1_0({\\mathbb {R}}^d), \\sup |\\varphi (x)| \\le 1\\Bigr \\rbrace =\\Vert u\\Vert _{\\dot{BV}} < \\infty .$ If $u \\in BV$ there is an ${\\mathbb {R}}^d$ -valued measure $\\vec{\\mu }$ such that ${{\\partial u} \\over {\\partial x_j}} = (\\vec{\\mu })_j$ as distributions and we write $Du = \\vec{\\mu }.$ The vector measure $\\mu $ has a polar decomposition $\\vec{\\mu }= \\vec{\\rho }\\mu $ where $\\mu $ is a finite positive Borel measure and $\\vec{\\rho }:{\\mathbb {R}}^d\\rightarrow S^{d-1}$ is a Borel function, and $\\Vert u\\Vert _{\\dot{BV}} = \\int d\\mu .", "$ (see for example Evans-Gariepy [16]).", "We assume $K$ is a positive, even, bounded and real analytic kernel on ${\\mathbb {R}}^d$ such that $\\int Kdx =1$ and such that $K*u$ determines $u$ (i.e.", "the map $L^p \\ni u \\rightarrow K*u$ is injective).", "For example we may take $K$ to be a Gaussian or a Poisson kernel.", "We fix $\\lambda > 0$ , $1 \\le p < \\infty $ and $1 \\le q < \\infty .$ For real $f(x) \\in L^1$ we consider the extremal problem: $m_{p,q,\\lambda } = \\inf \\lbrace \\Vert u\\Vert _{\\dot{BV}} + {\\cal F}_{p,q,\\lambda }(f-u): u \\in BV\\rbrace $ where ${\\cal F}_{p,q,\\lambda }(h) = \\lambda \\Vert K * h\\Vert ^q_{L^p}.$ Since $BV \\subset L^{{d} \\over {d-1}}$ and $K \\in L^{\\infty }$ , a weak-star compactness argument shows that (REF ) has at least one minimizer $u$ .", "Our objective is to describe, given $f$ , the set ${\\cal M}_{p,q,\\lambda }(f)$ of minimizers $u$ of (REF )." ], [ "Convexity", "Since the functional in (REF ) is convex, the set of minimizers ${\\cal M}_{p,q,\\lambda }(f)$ is a convex subset of $BV$ .", "If $p > 1$ or if $q > 1$ , then the functional (REF ) is strictly convex and the problem (REF ) has a unique minimizer because $K*u$ determines $u$ .", "When $p = q =1$ minimizers may not be unique, but they satisfy the relations given in (REF ) and (REF ) below.", "Lemma 1 Let $p = q =1$ and assume $u_1 \\in {\\cal M}_{p,q,\\lambda }(f)$ and $u_2 \\in {\\cal M}_{p,q,\\lambda }(f)$ .", "For $j =1,2$ write $Du_j = \\vec{\\mu }_j = \\vec{\\rho }_j \\mu _j$ with $|\\vec{\\rho }_j| =1$ and $\\mu _j \\ge 0$ and write ${{d\\vec{\\mu }_j} \\over {d\\mu _k}}$ for the Radon-Nikodym derivative of (the absolutely continuous part of) $\\vec{\\mu }_j$ with respect to $\\mu _k.$ Then ${{K*(f - u_1)} \\over {|K*(f-u_1)|}} ={{K*(f-u_2)} \\over {|K * (f-u_2)|}}~~~almost ~ everywhere$ on $\\lbrace |K * (f -u_j)| > 0\\rbrace , j = 1, 2;$ and $\\vec{\\rho }_k \\cdot {{d\\vec{\\mu }_j} \\over {d\\mu _k}} = \\Big |{{d\\vec{\\mu }_j} \\over {d\\mu _k}}\\Big |, ~j \\ne k.$ Proof: Since ${\\cal M}_{p,q,\\lambda }(f)$ is a convex subset of $BV$ , ${{u_1 + u_2} \\over {2}}$ is also a minimizer.", "This implies $\\begin{split}\\left\\Vert \\frac{u_1+u_2}{2}\\right\\Vert _{\\dot{BV}} +\\lambda \\left\\Vert K*\\left(f-\\frac{u_1+u_2}{2}\\right)\\right\\Vert _1 &= \\frac{1}{2}\\bigl (\\Vert u_1\\Vert _{\\dot{BV}}+ \\Vert u_2\\Vert _{\\dot{BV}}\\bigr ) \\\\&+ \\frac{\\lambda }{2}\\bigl (\\Vert K * (f -u_1)\\Vert _1 + \\Vert K * (f -u_2)\\Vert _1\\bigr ).\\end{split}$ On the other hand, using the convexity of $\\Vert \\cdot \\Vert _{\\dot{BV}}$ and $\\Vert \\cdot \\Vert _{L^1}$ we have $\\left\\Vert \\frac{u_1+u_2}{2}\\right\\Vert _{\\dot{BV}}\\le \\frac{1}{2}\\bigl (\\Vert u_1\\Vert _{\\dot{BV}}+ \\Vert u_2\\Vert _{\\dot{BV}}\\bigr )$ and $\\left\\Vert K*\\left(f-\\frac{u_1+u_2}{2}\\right)\\right\\Vert _1\\le \\frac{1}{2}\\bigl (\\Vert K * (f -u_1)\\Vert _1 + \\Vert K * (f -u_2)\\Vert _1\\bigr )$ Combining (REF ), (REF ), and (REF ) we obtain the equality $\\Big |\\Big |K*(f - {{u_1 + u_2} \\over {2}})\\Big |\\Big |_1 = {{1} \\over {2}}\\bigl (\\Vert K * (f -u_1)\\Vert _1 + \\Vert K*(f -u_2)\\Vert _1\\bigr ),$ which implies (REF ).", "We also obtain $\\left\\Vert u_1+u_2\\right\\Vert _{\\dot{BV}}= \\Vert u_1\\Vert _{\\dot{BV}}+\\Vert u_2\\Vert _{\\dot{BV}}$ and for $k\\ne j$ equation (REF ) implies $\\int \\Big |\\vec{\\rho }_k + {{d\\vec{\\mu }_j} \\over {d\\mu _k}}\\Big | d\\mu _k = \\int d\\mu _k + \\int \\Big |{{d \\vec{\\mu }_j}\\over {d\\mu _k}}\\Big | d\\mu _k,$ which yields (REF ).", "$\\square $" ], [ "Properties of $u \\in {\\cal M}_{p,q,\\lambda }(f)$", "Lemma 2 Let $f\\in L^1$ , let $u \\in BV$ be a minimizer of (REF ) with $\\Vert u-f\\Vert _1 \\ne 0$ and write $Du = \\vec{\\mu } = \\vec{\\rho }\\cdot \\mu .$ Then whenever $h \\in BV$ is real, $Dh = \\vec{\\nu }$ and $ \\vec{\\nu }= {{d\\vec{\\nu }} \\over {d\\mu }} \\mu + \\vec{\\nu }_s$ is the Lebesgue decomposition of $\\vec{\\nu }$ with respect to $\\mu $ (so that $\\vec{\\nu }_s$ is singular to $\\mu $ ), we have $\\left| \\int \\vec{\\rho } \\cdot {{d \\vec{\\nu }} \\over {d\\mu }} d\\mu - \\lambda \\int h (K * J_{p,q}) dx\\right| \\le \\Vert \\vec{\\nu }_s\\Vert ,$ where $J_{p,q} = q{{F |F|^{p-2}} \\over {\\Vert F\\Vert _p^{p-q}}},~~ F = K *(f-u)$ and $\\Vert \\vec{\\nu }_s\\Vert $ denotes the norm of the vector measure $\\vec{\\nu }_s$ .", "Conversely, if $u \\in BV$ , $\\Vert u -f\\Vert _1 \\ne 0$ and if (REF ) and (REF ) hold for all $h$ , then $u \\in {\\cal M}_{p,q,\\lambda }(f).$ Note that because $\\Vert u - f\\Vert _1 \\ne 0$ and $K *(f-u)$ is real analytic and bounded, $J_{p,q}$ is defined almost everywhere, and that by Lemma REF , $J_{p,q}$ is independent of $u \\in {\\cal M}_{p,q,\\lambda }$ in the case $p = q =1$ .", "Proof: Let $|\\epsilon |$ be sufficiently small.", "Since $u$ is extremal, we have $\\Vert u + \\epsilon h\\Vert _{\\dot{BV}} - \\Vert u\\Vert _{\\dot{BV}} + {\\cal F}_{p,q,\\lambda }(f - u - \\epsilon h)-{\\cal F}_{p,q,\\lambda }(f -u) \\ge 0.$ On the other hand we have $\\left|\\vec{\\rho } + \\epsilon {{d\\vec{\\nu }} \\over {d\\mu }}\\right| = \\left(1 + 2\\epsilon \\vec{\\rho }\\cdot \\frac{d\\vec{\\nu }}{d\\mu } + \\epsilon ^2\\left\\Vert \\frac{d\\vec{\\nu }}{d\\mu } \\right\\Vert ^2\\right)^{1/2} = 1 + \\epsilon \\vec{\\rho }\\cdot \\frac{d\\vec{\\nu }}{d\\mu } + o(|\\epsilon |),$ where in the last equality, we use the estimate $(1+\\alpha )^{1/2} = 1 + \\frac{\\alpha }{2} + o(|\\alpha |)$ .", "This implies, $\\begin{split}\\Vert u + \\epsilon h\\Vert _{\\dot{BV}} - \\Vert u\\Vert _{\\dot{BV}} =|\\epsilon | \\Vert \\vec{\\nu }_s\\Vert + \\int \\left(\\left|\\vec{\\rho } + \\epsilon {{d\\vec{\\nu }} \\over {d\\mu }}\\right| - 1\\right) d\\mu =|\\epsilon | \\Vert \\vec{\\nu }_s\\Vert + \\epsilon \\int \\vec{\\rho } \\cdot {{d\\vec{\\nu }} \\over {d\\mu }} d\\mu + o(|\\epsilon |).\\end{split}$ Moreover $K * (f -u)$ is bounded and non-zero almost everywhere, since $K$ is real analytic.", "Hence we also have $\\begin{split}{\\cal F}_{p,q,\\lambda }(f -u -\\epsilon h) - {\\cal F}_{p,q,\\lambda }(f-u) &=-\\lambda \\epsilon \\int (K * h) J_{p,q} dx + o(|\\epsilon |) \\\\&= - \\lambda \\epsilon \\int h (K*J_{p,q}) dx + o(|\\epsilon |)\\end{split}$ since $K(-x) = K(x)$ .", "Thus by (REF ), we have $-\\epsilon \\left[ \\int \\vec{\\rho } \\cdot {{d\\vec{\\nu }} \\over {d\\mu }} d\\mu - \\lambda \\int h (K*J_{p,q}) dx\\right] \\le |\\epsilon |\\Vert \\vec{\\nu }_s\\Vert + o(|\\epsilon |).$ Taking $\\pm \\epsilon $ and noting that the right side of the above inequality does not depend on the sign of $\\epsilon $ , we see that (REF ) holds.", "The converse statement holds because the functional (REF ) is convex.", "$\\square $ Lemma REF does not hold for the Chan-Esedoglu [12] functional because in that case one can have $f-u =0$ on a set of positive measure, and this yields the additional term $\\int _{\\lbrace |f -u| = 0\\rbrace } |h| dx$ on the right side of (REF ).", "Later we will need the following alternate characterization of minimizers, due to Meyer [22] in the case of the Rudin-Osher-Fatemi model.", "Define $\\Vert v\\Vert _* = \\inf \\left\\lbrace \\left\\Vert |u|\\right\\Vert _{\\infty }: v = \\sum _{j=1}^d{{\\partial u_j} \\over {\\partial x_j}}, |u|^2 = \\sum _{i=1}^d |u_j|^2\\right\\rbrace $ so that $\\Vert v\\Vert _*$ is (isometrically) the norm of the dual of $W^{1,1}\\subset BV$ when $W^{1,1}$ is given the norm of $BV$ .", "By the weak-star density of $W^{1,1}$ in $BV$ , $\\left|\\int hv dx\\right| \\le \\Vert h\\Vert _{\\dot{BV}}\\Vert v\\Vert _*$ whenever $v \\in L^2$ .", "The lemma characterizes minimizers in terms of $\\Vert \\cdot \\Vert _*$ .", "Lemma 3 Let $u \\in BV$ such that $u \\ne f$ , and let $J_{p,q}$ be defined as in Lemma 2.", "Then $u$ is a minimizer for the problem (REF ) if and only if $\\Vert K * J_{p,q}\\Vert _* = {{1} \\over {\\lambda }}$ and $\\int u (K * J_{p,q}) dx = {{1} \\over {\\lambda }} \\Vert u\\Vert _{\\dot{BV}}.$ Proof: The short proof is the same as in [22], but we include it for the reader's convenience.", "Let $u$ is a minimizer for (REF ).", "Then for any $h \\in W^{1,1}$ , (REF ) yields $\\left|\\int h (K * J_{p,q}) dx\\right| \\le {{\\Vert h\\Vert _{\\dot{BV}}} \\over {\\lambda }}$ by the definition of $\\vec{\\nu }_s$ .", "Hence by the definition of $\\Vert \\cdot \\Vert _*$ , $\\Vert K *J_{p,q}\\Vert _* \\le {{1} \\over {\\lambda }}.$ But setting $h =u$ in (REF ) gives (REF ), so that (REF ) follows.", "Conversely, assume $u \\in BV$ satisfies (REF ) and (REF ) and note that $u$ determines $J_{p,q}$ .", "Still following Meyer [22], we let $h \\in BV$ be real.", "Then for small $\\epsilon > 0$ , (REF ), (REF ) and (REF ) give $\\Vert u + \\epsilon h\\Vert _{\\dot{BV}}+ \\lambda \\Vert K * (f - u -\\epsilon h)\\Vert _1&\\ge & \\lambda \\int (u + \\epsilon h) (K * J_{p,q})dx+ \\lambda \\Vert K * (f -u)\\Vert _1 \\\\&-&\\epsilon \\lambda \\int h (K * J_{p,q}) dx + o(\\epsilon )\\\\&=& \\Vert u\\Vert _{\\dot{BV}} + \\epsilon \\lambda \\int h (K * J_{p,q}) dx \\\\& -&\\epsilon \\lambda \\int h (K *J_{p,q})dx + o(\\epsilon ) \\\\&\\ge & 0.$ Therefore $u$ is a local minimizer for the functional (REF ), and by convexity that means $u$ is a global minimizer.", "$\\square $ Lemma 4 Assume $f \\in L^1, ~ u \\in {\\cal M}_{p,q,\\lambda }(f)$ , and $\\Vert u -f\\Vert _1 \\ne 0$ .", "Let $U$ be an open set on which $Du = \\vec{\\mu }$ is absolutely continuous to Lebesgue measure and has Radon-Nikodym derivative ${{d \\vec{\\mu }} \\over {dx}} \\ne 0$ almost everywhere.", "Then as distributions on $U$ ${\\rm {div}}\\left( {{\\nabla u} \\over {|\\nabla u|}} \\right)= -\\lambda K *J_{p,q},$ and $u \\in W^{1,1}(U).$ In particular, if $u \\in C^2(U)$ then the level set $\\lbrace u =c\\rbrace $ is locally a $C^2$ surface having mean curvature $-\\lambda K * J_{p,q}(x)$ at $x \\in U.$ Proof: Since $Du$ is absolutely continuous on $U$ we have $u \\in W^{1,1}(U)$ and $\\vec{\\mu }= \\nabla u dx$ there.", "Let $h \\in C^{\\infty }$ have compact support contained in $U$ .", "Then by the hypotheses, $\\vec{\\nu }= Dh = \\nabla h dx$ is absolutely continuous to $Du$ so that by (REF ) $\\int _U \\nabla h \\cdot {{\\nabla u} \\over {|\\nabla u|}} dx = \\lambda \\int _U h (K * J_{p,q}) dx.$ This implies (REF ).", "Also, if $u \\in C^2(U)$ then (REF ) holds pointwise and gives the mean curvature of $\\lbrace u =c\\rbrace $ inside $U$ .", "$\\square $ Known results on mean curvature equations can now be used to show that almost every level set $U \\cap \\lbrace u =c\\rbrace $ is a real analytic surface, even without the assumption $u \\in C^2(U)$ .", "Below we write $\\Lambda _{d-1}$ for $d-1$ dimensional Hausdorff measure.", "Theorem 1 Assume $f \\in L^1, ~ u \\in {\\cal M}_{p,q,\\lambda }(f)$ , and $\\Vert u -f\\Vert _1 \\ne 0$ .", "Let $U$ be an open set on which $Du = \\vec{\\mu }$ is absolutely continuous to Lebesgue measure and on which the Radon-Nikodym derivative ${{d \\vec{\\mu }} \\over {dx}} \\ne 0$ almost everywhere.", "Then for almost all $c \\in {\\mathbb {R}}$ and for $\\Lambda _{d-1}$ almost every $x_0 \\in U \\cap \\lbrace u =c\\rbrace $ there exists a $C^1$ -hypersurface $S$ with continuous unit normal $\\vec{n}(x) = {{\\nabla u} \\over {|\\nabla u|}}$ and a neighborhood $V$ of $x$ such that $\\Lambda _{d-1}((V \\cap \\lbrace u =c\\rbrace ) \\Delta S) = 0.$ After a rotation $S = \\lbrace x_d = \\varphi (y): ~ y = (x_1, \\dots , x_{d-1}) \\in V_0\\rbrace $ , where $V_0 \\subset {{\\mathbb {R}}}^{d-1}$ is open, $\\varphi \\in C^1(V_0)$ , and $\\vec{n}(y, \\varphi (y)) = (1 + |\\nabla \\varphi |^2)^{-1/2} (\\nabla \\varphi , -1).$ Moreover, as a distribution on $V_0$ ${\\rm {div}} \\left( {{\\nabla \\varphi } \\over {(1 + |\\nabla \\varphi |^2)^{1/2}}}\\right)= -\\lambda K *J_{p,q}(y,\\varphi (y))dy,$ and the function $\\varphi $ and the surface $S$ are real analytic.", "Proof: That $S$ and $\\varphi $ exist almost everywhere follows from standard properties of BV functions and the hypothesis that $|\\nabla u| > 0$ a. e. on $U$ .", "See the proof of Theorem 4 below and Chapter 5 of [16].", "To prove (REF ) we may assume $c =0$ .", "Let $h \\in C^{\\infty }_0(V_0)$ , let $$$$ (t) = 1 $\\chi $ (t )$ where $$\\chi $ (t) = $\\chi $ (-t) 0$is $ C(-1,1)$ and $$\\chi $ dt =1,$ and define$$H_{\\epsilon }(x) = $$$ (h(x1, ..., xd-1) - xd) h(x1, ..., xd-1).", "$Then by (\\ref {eq(Lap)}),\\begin{eqnarray*}& &\\int \\Biggl ( \\sum _{j=1}^{d-1}\\bigl (\\chi \\end{eqnarray*}_{\\epsilon }(h(x_1, \\dots , x_{d-1}) - x_d)+ $ $$ '(h(x1, ..., xd-1) - xd)h(x1, ..., xd-1)) h xj 1 |u| u xj) - ( $\\chi $ '(h(x1, ..., xd-1) - xd)h(x1, ..., xd-1) 1 |u| u xd) dx = V0 H(x) K*Jp,q(x) dx.", "Now for almost every $c$ the right side of this equation tends to $\\lambda \\int _{V_0} h (K * J_{p,q})(y) dy$ and, by the fine properties of BV functions in Chapter 5 of [16] or Chapter 3 of [5], the left side tends to $\\int _{V_0} \\nabla h \\cdot {{\\nabla \\varphi } \\over {(1 + |\\nabla \\varphi |^2)^{1/2}}}dy.$ That proves (REF ).", "To prove the real analyticity of $\\varphi $ , and hence of $S$ , we invoke three theorems.", "First, since $\\varphi \\in C^1$ , the results on mean curvature equations in Section 7.7 of [5] show that $\\varphi \\in W^{2,2} \\cap C^{1 + \\alpha }$ whenever $0 < \\alpha < 1.$ Next, since $\\varphi \\in W^{2,2}$ we can rewrite (REF ) as $\\sum _{j,k} {{\\delta _{j,k} - \\varphi _j \\varphi _k} \\over {(1 + |\\nabla \\varphi |^2)^{3/2}}} \\varphi _{j,k} = \\lambda K*J_{p,q}(y,\\varphi (y)).$ Indeed, (REF ) is clear if $\\varphi \\in C^2$ , and if we set $\\varphi ^{\\epsilon }= $$$ *C2$ then in the norms of $ C1 + $ and$ W2,2$, $ $ as $ 0.$ Hence for each $ j$$$\\int _{V_0} h_j \\sum _k {{\\delta _{j,k} - \\varphi ^{\\epsilon }_j \\varphi ^{\\epsilon }_k} \\over {(1 + |\\nabla \\varphi ^{\\epsilon }|^2)^{3/2}}} \\varphi ^{\\epsilon }_{j,k} dy \\rightarrow \\int _{V_0} h_j \\sum _k {{\\delta _{j,k} - \\varphi _j \\varphi _k} \\over {(1 + |\\nabla \\varphi |^2)^{3/2}}} \\varphi _{j,k} dy$$as $ 0,$ and consequently (\\ref {eq(Nab1)}) also holds with $ W2,2.$ We may assume $ || 1/2$ because $$ locally parametrizes a $ C1$ surface,and then (\\ref {eq(Nab1)}) becomes an elliptic equation with $ C$ coefficients(which depend on $$).", "It then follows by Schauder^{\\prime }s theorem (see \\cite {Caff})that $ C2 + (V0)$ for some $ > 0.$Finally, by the analyticity of the right side of (\\ref {eq(Nab1)}),the function $$, and hence thesurface $ S$, is real analytic by a theorem ofHopf \\cite {Hopf} (see also \\cite {Morrey}).$$\\square $ See Theorem 5 below for a related result for the case $q =1.$" ], [ "Radial Functions", "Assume $K$ is radial, $K(x)= K(|x|)$ and assume $f$ is radial and $f \\notin {\\cal M}_{p,q,\\lambda }(f).$ Then averaging over rotations shows that every $u \\in {\\cal M}_{p,q,\\lambda }(f)$ is radial and $Du = \\rho (|x|) {x \\over |x|} \\mu $ where $\\mu $ is invariant under rotations and where $\\rho (|x|) = \\pm 1$ a.e.", "$d\\mu $ .", "Let $H \\in L^1(\\mu )$ be radial and satisfy $\\int H d\\mu =0$ and $H = 0$ on $|x| < \\epsilon $ , and define $h(x) = \\int _{B(0,|x|)} H(|y|) {{1}\\over {|y|^{d-1}}} d\\mu .$ Then $h \\in BV$ is radial and $Dh = \\vec{\\nu }=H(|x|) {x \\over |x|} \\mu .$ Consequently $\\vec{\\nu }_s =0$ and (REF ) gives $\\int \\rho H d\\mu &=& \\lambda \\int K* J_{p,q}(x) \\int _{B(0,|x|)} {{H(y)}\\over {|y|^{d-1}}} d\\mu (y) dx \\\\&=& \\lambda \\int \\Bigl (\\int _{|x| > |y|} K* J_{p,q}(x) dx\\Bigr ){{H(|y|)} \\over {|y|^{d-1}}} d\\mu (y),$ so that a.e.", "$d\\mu $ , $\\rho (|y|) = {{\\lambda } \\over {|y|^{d-1}}} \\int _{|x| > |y|}K *J_{p,q}(x)dx.$ But the right side of (REF ) is real analytic in $|y|$ , with a possible pole at $|y| =0,$ and $\\rho (|y|) = \\pm 1$ almost everywhere $\\mu .$ Therefore there is a finite set $\\lbrace r_1 < r_2 < \\dots < r_n\\rbrace $ of radii such that $Du = {{x} \\over {|x|}} \\sum _{j=1}^n c_j \\Lambda _{d-1}|\\lbrace |x| = r_j\\rbrace |$ for real constants $c_1, \\dots , c_n.$ By Lemma REF , $J_{p,q}$ is uniquely determined by $f$ , and hence the set (REF ) is also unique.", "Moreover, it follows from Lemma REF that for each $j$ , either $c_j \\ge 0$ for all $u \\in {\\cal M}_{p,1,\\lambda }(f)$ or $c_j \\le 0$ for all $u \\in {\\cal M}_{p,1,\\lambda }(f)$ .", "We have proved: Theorem 2 Suppose $K$ and $f$ are both radial.", "If $f \\notin {\\cal M}_{p,q,\\lambda }(f)$ , then there is a finite set (REF ) such that all $u \\in {\\cal M}_{p,q,\\lambda }(f)$ have the form $\\sum _{j=1}^n c_j \\chi $ B(0,rj).", "Moreover, there is $X^+ \\subset \\lbrace 1, 2, \\dots , n\\rbrace $ such that $c_j \\ge 0$ if $j \\in X^+$ while $c_j \\le 0$ if $j \\notin X^+.$ Note that by convexity ${\\cal M}_{p,q,\\lambda }(f)$ consists of a single function unless $p = q = 1.$ In Section REF we will say more about the solutions of the form (REF )." ], [ "Example", "Unfortunately, Theorem REF does not hold more generally.", "The reason is that when $u$ is not radial it is difficult to produce $BV$ functions satisfying $Dh = \\vec{\\nu }<< \\mu .$ For simplicity we take $d =2$ and $p = q =1$ and define $J(x,y)=\\left\\lbrace \\begin{array}{rcr}1& \\mbox{ if }& 0 < x\\le 1\\\\-1 & \\mbox{ if }& -1 < x\\le 0\\end{array}\\right.$ and $J(x+2,y) = J(x,y).$ Choose $\\lambda >0$ so that $U = \\lambda K *J$ satisfies $\\Vert U\\Vert _* =1,$ and note that ${{U} \\over {|U|}} =J.$ Also notice that $u \\in C^2$ solves the curvature equation ${\\rm {div}} \\Bigl ({{\\nabla u} \\over {|\\nabla u|}}\\Bigr ) = U$ if and only if the level sets $\\lbrace u =a\\rbrace $ are curves $y = y(x)$ that satisfy the simple ODE $y^{\\prime \\prime } = U(x,0) (1 + (y^{\\prime })^2)^{3/2}$ on the line.", "Consequently (REF ) has infinitely many solutions $u$ and both $u$ and $J$ satisfy (REF ) and (REF ).", "Hence by Lemma REF , $u$ is a minimizer for $f$ provided that $J = {{K *(f-u)} \\over {|K *(f-u)|}},$ and there are many $f$ that satisfy (REF ).", "For example, one can choose $u$ and $f$ so that $f-u=J$ .", "Note that in this example $u$ can be real analytic except on $U^{-1}(0)$ and not piecewise constant.", "Similar examples can be made when $(p,q) \\ne (1,1).$" ], [ "Further Properties of Minimizers when q =1", "When $q =1$ the minimizers $u \\in {\\cal M}_{p,1,\\lambda }(f)$ have several additional properties.", "The results of the next two sections do not depend on the real analyticity of the kernel $K$ .", "They also hold when $K =I$ , i.e.", "when ${\\cal F}_{p,q,\\lambda }(h) = \\lambda \\Vert h\\Vert _p$ , and in the case $K =I$ somewhat stronger results have already been proved by Allard in [1].", "However, since the arguments in [1] do not apply to the case $K \\ne I$ we include complete but brief proofs." ], [ "Layer Cake Decomposition", "Here we have been inspired by the paper of Strang [28].", "Lemma 5 If $q =1$ and $u \\in {\\cal M}_{p,1,\\lambda }(f)$ , then $u \\in {\\cal M}_{p,1,\\lambda }(u).$ Proof: If $\\Vert h\\Vert _{\\dot{BV}} + \\lambda \\Vert K*(u-h)\\Vert _p < \\Vert u\\Vert _{\\dot{BV}},$ then by the triangle inequality $\\Vert h\\Vert _{\\dot{BV}} + \\lambda \\Vert K*(f -h)\\Vert _p < \\Vert u\\Vert _{\\dot{BV}} + \\lambda \\Vert K*(f-u)\\Vert _p$ so that $u$ is not a minimizer for $f$ .", "$\\square $ We write ${\\cal M}= {\\cal M}_{p,1,\\lambda } = \\bigcup _f {\\cal M}_{p,1,\\lambda }(f).$ Lemma 6 Let $u \\in BV$ .", "Then $u \\in {\\cal M}$ if and only if $\\left| \\int \\rho \\cdot {{d \\vec{\\nu }} \\over {d \\mu }} d\\mu \\right|\\le \\Vert \\vec{\\nu }_s\\Vert + \\lambda \\Vert K * h\\Vert _p$ for all $h \\in BV$ , where $Dh = \\vec{\\nu }$ and $\\vec{\\nu }_s$ is the part of $\\vec{\\nu }$ singular to $\\mu .$ Proof: By Lemma REF we may take $f =u.$ Then for $|\\epsilon |$ small we have $0 &\\le & \\Vert u + \\epsilon h\\Vert _{\\dot{BV}} - \\Vert u\\Vert _{\\dot{BV}} + \\lambda \\Vert \\epsilon K * h\\Vert _p \\\\&=& |\\epsilon | \\Vert \\vec{\\nu }_s\\Vert + \\epsilon \\int \\rho \\cdot {{d \\vec{\\nu }} \\over {d \\mu }} d\\mu + |\\epsilon | \\lambda \\Vert K * h\\Vert _p + o(|\\epsilon |) \\\\$ and the Lemma follows from the proof of Lemma REF .", "$\\square $ Let $a < b$ be such that $\\mu (\\lbrace u = a\\rbrace \\cup \\lbrace u =b\\rbrace ) =0.$ Then $u_{a,b} = {\\rm {Min}}\\lbrace (u-a)^+, (b -a)\\rbrace \\in BV$ and $D(u_{a,b}) = $$$ a < u < b .$$ Lemma 7 Assume $q =1$ .", "(a) If $u \\in {\\cal M}$ , then $u_{a,b} \\in {\\cal M}.$ (b) More generally, if $u\\in {\\cal M}$ and if $v\\in BV$ satisfies $\\mu _v<<\\mu _u$ and $\\rho _v=\\rho _u$ a.e.", "$d\\mu _v$ , then $v\\in {\\cal M}$ .", "Proof: To prove (a) we verify (REF ).", "Write $\\mu _{a,b} = $$$ (a,b) $ so that$ D(ua,b) = a,b.$ Let $ h BV$ and write $ Dh = .$ Thenby (\\ref {eq(6.2)})$$\\vec{\\nu } = $$$ a < u < b d d + (()s + $\\chi $ u(x) [a,b] d d ) $is the Lebesgue decomposition of $ $ with respect to $ a,b,$ and$$\\int \\vec{\\rho } \\cdot {{d \\vec{\\nu }} \\over {d\\mu _{a,b}}} d\\mu _{a,b} =\\int \\vec{\\rho } \\cdot {{d \\vec{\\nu }} \\over {d\\mu }} d\\mu - \\int _{g(x) \\notin [a,b]}\\vec{\\rho } \\cdot {{d \\vec{\\nu }} \\over {d\\mu }} d\\mu .$$Then (\\ref {eq(6.1)}) for $$ and $ a,b$ follows from (\\ref {eq(6.1)}) for $$ and $$.", "The proof of (b) is similar.\\hfill $$\\bigskip $ For simplicity we assume $u \\ge 0.$ Write $E_t =\\lbrace x: u(x) > t\\rbrace .$ Then by Evans-Gariepy [16], $E_t$ has finite perimeter for almost every $t$ , $\\Vert u\\Vert _{\\dot{BV}} = \\int _0^{\\infty } \\Vert \\chi $ EtBV dt, and $u(x) = \\int _0^{\\infty } \\chi $ Et(x) dt.", "Moreover, almost every set $E_t$ has a measure theoretic boundary $\\partial _* E_t$ such that $\\Lambda _{d-1}(\\partial _* E_t) = \\Vert \\chi $ EtBV and a measure theoretic outer normal $\\vec{n}_t: \\partial _* E_t \\rightarrow S^{d-1}$ so that $D(\\chi $ Et) = ntd-1 | * Et.", "Theorem 3 Assume $q =1$ .", "(a) If $u \\in {\\cal M}$ , then for almost every $t$ , $$$$ Et M. $$ (b) If $u \\in {\\cal M}$ and $u \\ge 0$ , then for all nonnegative $c_1,...,c_n$ and for almost all $t_1<...<t_n$ , $\\sum c_j\\chi _{E_{t_j}}\\in {\\cal M}$ .", "Proof: Suppose (a) is false.", "Then there is $\\beta < 1$ , and a compact set $A \\subset (0,\\infty )$ with $|A| > 0$ such that for all $t \\in A$ (REF ) and (REF ) hold and there exists $h_t \\in BV$ such that $\\Vert \\chi $ Et - htBV + K *htp $\\chi $ EtBV.", "Choose an interval $I = (a,b)$ such that (REF ) holds and $|I \\cap A| \\ge {{|I|} \\over {2}}.$ Define $h_t =0$ for $t \\in I \\setminus A,$ and take finite sums such that $\\sum _{j=1}^{N_n} \\chi $ Et(n)j tj(n) ua,b    (n ), $\\sum _{j=1}^{N_n} \\Vert \\chi $ Et(n)jBV tj(n) ua,b    (n ), and $t_j^{(n)} \\in A$ whenever possible.", "Write $h^{(n)} = \\sum _{j=1}^{N_n} h_{t_j^{(n)}} \\Delta t_j^{(n)}$ .", "Then by (REF ) and (REF ) $\\lbrace h^{(n)}\\rbrace $ has a weak-star limit $h \\in BV$ , and by (REF ), (REF ) and (REF ), $\\Vert u_{a,b} -h\\Vert _{\\dot{BV}} + \\lambda \\Vert K*h\\Vert _p \\le {{1 + \\beta } \\over {2}} \\Vert u_{a,b}\\Vert _{\\dot{BV}},$ contradicting Lemma REF .", "The proof of (b) is similar.", "$\\square $ We believe that the converse of Theorem REF is false, but we have no counterexample.", "In the case $K =I$ and $p =1$ the converse of this Theorem is true.", "See [1], Theorem 5.3." ], [ "Characteristic Functions", "Still assuming $q =1$ we let $E$ be such that $$$$ E M.$ Thenby Evans-Gariepy \\cite {EvansGariepy} $ * E = N Kj$, where $ D($\\chi $ E)(N) = n-1(N) =0,$$ Kj$ is compact and $ Kj Sj$, where $ Sj$ is a $ C1-$hypersurface with continuousunit normal $ nj(x), x Sj,$ and $ nj$ is the measure theoretic outer normalof $ E$.", "After a coordinate change write $ Sj = {xd = j(y)}, y = (x1, ..., xd-1)$with $ j$ continuous and $ nj(y,j(y)) = (1 + |j|2)-1/2 (j, -1).$Assume $ y =0$ is a point of Lebesgue density of $ (y,j)-1(Kj)$, let$ V Rd-1$ be a neighborhood of $ y =0$, let $ g C0(V)$with $ g 0,$ and consider the variation$ u = $\\chi $ E$where $ > 0$ and$$E_{\\epsilon } = E \\cup \\lbrace 0 \\le x_d \\le \\epsilon g(y), y \\in V\\rbrace .$$Then $ E E$, and writing $ u0 = $\\chi $ E$, we have\\begin{equation}\\Vert u_{\\epsilon }\\Vert _{\\dot{BV}} - \\Vert u_0\\Vert _{\\dot{BV}} = \\int _V \\sqrt{(1 + |\\nabla (\\varphi _j + \\epsilon g)|^2)}- \\sqrt{(1 + |\\nabla f\\varphi _j|^2)} dy + o(\\epsilon )\\end{equation}because by \\cite {EvansGariepy} page 203$$\\Lambda _{d-1}((\\partial _* E) \\cup (E_\\epsilon \\setminus E)) = o(\\epsilon )$$$ d-1$ a.e.", "on $ Kj.$Hence\\begin{equation}\\Vert u_{\\epsilon }\\Vert _{\\dot{BV}} - \\Vert u_0\\Vert _{\\dot{BV}} =\\epsilon \\int _V \\nabla g \\cdot {{\\nabla \\varphi _j} \\over {\\sqrt{1 + |\\nabla \\varphi _j|^2}}} dy + o(\\epsilon ).\\end{equation}Also, a careful calculation gives\\begin{equation}\\lambda \\Vert K * (u_{\\epsilon } - u_0)\\Vert _p = \\lambda |\\epsilon |\\left\\Vert \\int _VK(x - (y,\\varphi _j(y)) g(y) dy\\right\\Vert _{L^p(dx)} + o(\\epsilon ).\\end{equation}Together (\\ref {eq(7.1)}) and (\\ref {eq(7.2)}) show\\begin{equation} - \\int _V \\nabla g \\cdot \\Bigl ({{\\nabla \\varphi _j} \\over {\\sqrt{1 +|\\nabla \\varphi _j|^2}}}\\Bigr ) dy\\le \\lambda \\Vert K\\Vert _p \\int _V gdy.\\end{equation}Repeating this argument with $ < 0$ and with $ g 0$ we obtain:\\begin{theorem}On the hypersurface S_j \\subset \\partial _* E\\begin{equation}\\Bigl |{\\rm {div}}\\Bigl ({{\\nabla \\varphi _j} \\over {\\sqrt{1 + |\\nabla \\varphi _j|^2}}}\\Bigr )\\Bigr | \\le \\lambda \\Vert K\\Vert _p.\\end{equation}when viewed as a distribution on (y,\\varphi _j)^{-1}(S_j).\\end{theorem}By (\\ref {eq(7.4)}) and Section 7.7 of \\cite {AmbrosioFuscoPallara}we see that $ j W2,2loc C1 + $ for any $ < 1.$Combining Theorem \\ref {Thm7.1} with Theorem \\ref {Thm6.4} we obtain:\\begin{theorem}Assume q =1 and u \\in {\\cal M}.", "Then for almost every t, E_t = \\lbrace u > t\\rbrace has finiteperimeter and \\Lambda _{d-1} almost everypoint of the measure theoretic boundary \\partial _*{E_t} lieson a C^{1+ \\alpha }, \\alpha < 1, surface having distributional mean curvature at most\\lambda \\Vert K\\Vert _p.\\end{theorem}We note that the ``distributional mean curvature\" defined by (\\ref {eq(7.4)})is the same as the generalized mean curvature defined by Allard in \\cite {Allard1},and thus Theorem \\ref {Thm7.2} complements Theorem 1.2 and Theorem 1.6 of \\cite {Allard1}.However, unlike the situation in Theorem 1, we cannot conclude that the $ C1 + $surface meeting $ *Et$ is real analytic because the leftside of (\\ref {eq(7.4)}) may not be Hölder continuous.$" ], [ "Radial Minimizers", "In this section we assume $q =1$ and $p =1$ .", "For convenience we assume the kernel $K(x) = e^{-\\pi |x|^2}$ , so that $K_t$ has the form $K_t(x) = t^{-d/2} K\\left({{x} \\over {\\sqrt{t}}}\\right)$ and $K_s * K_t = K_{s+t}.$ Note that (REF ) and (REF ) imply that $\\Vert K_t * f\\Vert _1 ~~~decreases~ in ~ t$ and for $f \\in L^1$ with compact support $\\lim _{t \\rightarrow \\infty }\\Vert K_t * f\\Vert _1 = \\left|\\int f dx\\right|.$ For fixed $\\lambda $ and $t$ we set $R(\\lambda ,t) = \\lbrace r > 0:$ $$ B(0,r) M}.", "$ By Theorem \\ref {Thm4.1} and Theorem \\ref {Thm6.4}, we have $ (,t) .$For $ t =0$ and $ K =I$ our problem (\\ref {(1.2)}) becomes the problem$$\\inf \\lbrace \\Vert u\\Vert _{\\dot{BV}} + \\lambda \\Vert f - u\\Vert _{L^1}\\rbrace $$ studied by Chan and Esedoglu in \\cite {ChanEsedoglu},and in that case Chan and Esedoglu showed $ R(,0) = [2 , ).$$ Theorem 4 There exists $r_0 = r_0(\\lambda ,t)$ such that $R(\\lambda ,t) = [r_0,\\infty ).$ Moreover $[0,\\infty ) \\ni t \\rightarrow r_0(t) ~~is~~ nondecreasing$ and $\\lim _{t \\rightarrow \\infty } r_0(t) = \\infty .$ Proof: Assume $r \\notin R(\\lambda ,t)$ and $0 < s < r.$ Write $\\alpha = {{r} \\over {s}} > 1$ and $f = $$$ B(0,r).$ By hypothesis there is $ g BV$ such that\\begin{equation}\\Vert g\\Vert _{\\dot{BV}} + \\lambda \\Vert K_t*(f-g)\\Vert _1 < \\Vert f\\Vert _{\\dot{BV}}.\\end{equation}We write $ g(x) = g(x), f(x) = f(x) = $\\chi $ B(0,s)(x),$ andchange variables carefully in (\\ref {EQ7.8}) to get$$\\alpha \\Vert \\tilde{g}\\Vert _{\\dot{BV}} + \\lambda \\Vert {{1} \\over {t^{d/2}}} \\int K\\bigl ({{x -y} \\over {\\sqrt{t}}}\\bigr )(\\tilde{f} - \\tilde{g})\\bigl ({{y} \\over {\\alpha }}\\bigr ) dy\\Vert _{L^1(x)} < \\alpha \\Vert \\tilde{f}\\Vert _{\\dot{BV}}$$so that$$\\alpha \\Vert \\tilde{g}\\Vert _{\\dot{BV}} + \\lambda \\Vert {{\\alpha ^d} \\over {t^{d/2}}} \\int K\\bigl ({{\\alpha x^{\\prime } - \\alpha y^{\\prime }}\\over {\\sqrt{t}}}\\bigr ) (\\tilde{f} - \\tilde{g})(y^{\\prime }) dy^{\\prime }\\Vert _{L^1(\\alpha x^{\\prime })} < \\alpha \\Vert \\tilde{f}\\Vert _{\\dot{BV}}$$and$$\\alpha \\Vert \\tilde{g}\\Vert _{\\dot{BV}} +\\lambda \\alpha ^d \\int \\Bigl |K_{{t} \\over {\\alpha ^2}} * (\\tilde{f} - \\tilde{g})(x^{\\prime })\\Bigr | dx^{\\prime }< \\alpha \\Vert \\tilde{f}\\Vert _{\\dot{BV}}.$$Since $ > 1$, this and (\\ref {EQ7.3}) show$$\\Vert \\tilde{g}\\Vert _{\\dot{BV}} + \\lambda \\Vert K_{t}* (\\tilde{f} - \\tilde{g})\\Vert _1< \\Vert \\tilde{f}\\Vert _{\\dot{BV}}$$so that $ s R(,t).$ That proves (\\ref {EQ7.5}), and (\\ref {EQ7.6}) now follows easilyfrom (\\ref {EQ7.3}).", "To prove (\\ref {EQ7.7}) take $ g = rd sd $\\chi $ B(0,s),  s >r$and use (\\ref {EQ7.4}).\\hfill $$\\bigskip $ We note that not all radial minimizers have the form $$$$ B(0,r)$.", "This is seen by considering separately, for large fixed $ t$ and $$, thefunction$$\\chi $ B(0,r2) + $\\chi $ B(0,r1) $ with $ r1$ and $ r2 - r1$ 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[ [ "Multiplicative excellent families of elliptic surfaces of type E_7 or\n E_8" ], [ "Abstract We describe explicit multiplicative excellent families of rational elliptic surfaces with Galois group isomorphic to the Weyl group of the root lattices E_7 or E_8.", "The Weierstrass coefficients of each family are related by an invertible polynomial transformation to the generators of the multiplicative invariant ring of the associated Weyl group, given by the fundamental characters of the corresponding Lie group.", "As an application, we give examples of elliptic surfaces with multiplicative reduction and all sections defined over Q for most of the entries of fiber configurations and Mordell-Weil lattices in [Oguiso-Shioda '91], as well as examples of explicit polynomials with Galois group W(E_7) or W(E_8)." ], [ "Introduction", "Given an elliptic curve $E$ over a field $K$ , the determination of its Mordell-Weil group is a fundamental question in algebraic geometry and number theory.", "When $K = k(t)$ is a rational function field in one variable, this question becomes a geometrical question of understanding sections of an elliptic surface with section.", "Lattice theoretic methods to attack this problem were described in [10].", "In particular, when ${\\mathcal {E}}\\rightarrow {\\mathbb {P}}^1_t$ is a rational elliptic surface given as a minimal proper model of $y^2 + a_1(t) xy + a_3(t) y = x^3 + a_2(t) x^2 + a_4(t) x + a_6(t)$ with $a_i(t) \\in k[t]$ of degree at most $i$ , the possible configurations (types) of bad fibers and Mordell-Weil groups were analyzed by Oguiso and Shioda [8].", "In [11], the second author studied sections for some families of elliptic surfaces with an additive fiber, by means of the specialization map, and obtained a relation between the coefficients of the Weierstrass equation and the fundamental invariants of the corresponding Weyl groups.", "This was expanded in [18], which studied families with a bad fiber of additive reduction more exhaustively.", "The formal notion of an excellent family was defined (see the next section), and the authors found excellent families for many of the “admissible” types.", "The analysis of rational elliptic surfaces of high Mordell-Weil rank, but with a fiber of multiplicative reduction, is much more challenging.", "However, understanding this situation is arguably more fundamental, since if we write down a “random” elliptic surface, then with probability close to 1 it will have Mordell-Weil lattice $E_8$ and twelve nodal fibers (i.e.", "of multiplicative reduction).", "To be more precise, if we choose Weierstrass coefficients $a_i(t)$ of degree $i$ , with coefficients chosen uniformly at random from among rational numbers (say) of height at most $N$ , then as $N \\rightarrow \\infty $ the surface will satisfy the above condition with probability approaching 1.", "One can make a similar statement for rational elliptic surfaces chosen to have Mordell-Weil lattice $E_7^*$ , $E_6^*$ , etc.", "In a recent work [13], this study was carried out for elliptic surfaces with a fiber of type $I_3$ and Mordell-Weil lattice isometric to $E_6^*$ , through a “multiplicative excellent family” of type $E_6$ .", "We will describe this case briefly in Section .", "The main result of this paper shows that two explicitly described families of rational elliptic surfaces with Mordell-Weil lattices $E_7^*$ or $E_8$ are multiplicative excellent.", "The proof involves a surprising connection with representation theory of the corresponding Lie groups, and in particular, their fundamental characters.", "In particular, we deduce that the Weierstrass coefficients give another natural set of generators for the multiplicative invariants of the respective Weyl groups, as a polynomial ring.", "We note that similar formulas were derived by Eguchi and Sakai [3] using calculations from string theory and mirror symmetry.", "The idea of an excellent family is quite useful and important in number theory.", "An excellent family of algebraic varieties leads to a Galois extension $F(\\mu )/F(\\lambda )$ of two purely transcendental extensions of a number field $F$ (say ${\\mathbb {Q}}$ ), with Galois group a desired finite group $G$ .", "This setup has an immediate number theoretic application, since one may specialize the parameters $\\lambda $ and apply Hilbert's irreducibility theorem to obtain Galois extensions over ${\\mathbb {Q}}$ with the same Galois group.", "Furthermore, we can make the construction effective if appropriate properties of the group $G$ are known (see examples REF and REF for the case $G=W(E_7)$ or $W(E_8)$ ).", "At the same time, an excellent family will give rise to a split situation very easily, by specializing the parameters $\\mu $ instead.", "For examples, in the situation considered in our paper, we obtain elliptic curves over ${\\mathbb {Q}}(t)$ with Mordell-Weil rank 7 or 8 together with explicit generators for the Mordell-Weil group (see examples REF and REF ).", "There are also applications to geometric specialization or degeneration of the family.", "Therefore, it is desirable (but quite nontrivial) to construct explicit excellent families of algebraic varieties.", "Such a situation is quite rare in general: theoretically, any finite reflection group is a candidate, but it is not generally neatly related to an algebraic geometric family.", "Hilbert treated the case of the symmetric group $S_n$ , corresponding to families of zero-dimensional varieties.", "Not many examples were known before the (additive) excellent families for the Weyl groups of the exceptional Lie groups $E_6$ , $E_7$ and $E_8$ were given in [11], using the theory of Mordell-Weil lattices.", "Here, we finish the story for the multiplicative excellent families for these Weyl groups." ], [ "Mordell-Weil lattices and excellent families", "Let $X \\stackrel{\\pi }{\\rightarrow } {\\mathbb {P}}^1$ be an elliptic surface with section $\\sigma : {\\mathbb {P}}^1 \\rightarrow X$ , i.e.", "a proper relatively minimal model of its generic fiber, which is an elliptic curve.", "We denote the image of $\\sigma $ by $O$ , which we take to be the zero section of the Néron model.", "We let $F$ be the class of a fiber in $\\operatorname{Pic}(X) \\cong \\operatorname{NS}(X)$ , and let the reducible fibers of $\\pi $ lie over $\\nu _1, \\dots , \\nu _k \\in {\\mathbb {P}}^1$ .", "The non-identity components of $\\pi ^{-1}(\\nu _i)$ give rise to a sublattice $T_i$ of $\\operatorname{NS}(X)$ , which is (the negative of) a root lattice (see [5], [19]).", "The trivial lattice $T$ is ${\\mathbb {Z}}O \\oplus {\\mathbb {Z}}F \\oplus (\\oplus T_i)$ , and we have the isomorphism $\\operatorname{MW}(X / {\\mathbb {P}}^1) \\cong \\operatorname{NS}(X)/T$ , which describes the Mordell-Weil group.", "In fact, one can induce a positive definite pairing on the Mordell-Weil group modulo torsion, by inducing it from the negative of the intersection pairing on $\\operatorname{NS}(X)$ .", "We refer the reader to [10] for more details.", "In this paper, we will call $\\oplus T_i$ the fibral lattice.", "Next, we recall the notion of an excellent family with Galois group $G$ from [18].", "Suppose $X \\rightarrow {\\mathbb {A}}^n$ is a family of algebraic varieties, varying with respect to $n$ parameters $\\lambda _1,\\dots , \\lambda _n$ .", "The generic member of this family $X_\\lambda $ is a variety over the rational function field $k_0 = {\\mathbb {Q}}(\\lambda )$ .", "Let $k =\\overline{k_0}$ be the algebraic closure, and suppose that ${\\mathcal {C}}(X_\\lambda )$ is a group of algebraic cycles on $X_\\lambda $ over the field $k$ (in other words, it is a group of algebraic cycles on $X_\\lambda \\times _{k_0} k$ ).", "Suppose in addition that there is an isomorphism $\\phi _\\lambda : {\\mathcal {C}}(X_\\lambda ) \\otimes {\\mathbb {Q}}\\cong V$ for a fixed vector space $V$ , and ${\\mathcal {C}}(X_\\lambda )$ is preserved by the Galois group $\\mathrm {Gal}(k/k_0)$ .", "Then we have the Galois representation $\\rho _\\lambda : \\mathrm {Gal}(k/k_0) \\rightarrow \\mathrm {Aut}({\\mathcal {C}}(X_\\lambda )) \\rightarrow \\mathrm {Aut}(V).$ We let $k_\\lambda $ be the fixed field of the kernel of $\\rho _\\lambda $ , i.e.", "it is the smallest extension of $k_0$ over which the cycles of ${\\mathcal {C}}(\\lambda )$ are defined.", "We call it the splitting field of ${\\mathcal {C}}(X_\\lambda )$ .", "Now let $G$ be a finite reflection group acting on the space $V$ .", "Definition 1 We say $\\lbrace X_\\lambda \\rbrace $ is an excellent family with Galois group $G$ if the following conditions hold: The image of $\\rho _\\lambda $ is equal to $G$ .", "There is a $\\mathrm {Gal}(k/k_0)$ -equivariant evaluation map $s:{\\mathcal {C}}(X_\\lambda ) \\rightarrow k$ .", "There exists a basis $\\lbrace Z_1, \\dots , Z_n \\rbrace $ of ${\\mathcal {C}}(X_\\lambda )$ such that if we set $u_i = s(Z_i)$ , then $u_1, \\dots , u_n$ are algebraically independent over ${\\mathbb {Q}}$ .", "${\\mathbb {Q}}[u_1, \\dots , u_n]^G = {\\mathbb {Q}}[\\lambda _1, \\dots , \\lambda _n]$ .", "As an example, for $G = W(E_8)$ , consider the following family of rational elliptic surfaces $y^2 = x^3 + x \\left( \\sum _{i=0}^3 p_{20 - 6i} t^i \\right) + \\left( \\sum _{j=0}^3 p_{30 - 6j} t^j + t^5 \\right)$ over $k_0 = {\\mathbb {Q}}(\\lambda )$ , with $\\lambda = (p_2, p_8, p_{12}, p_{14}, p_{18}, p_{20},p_{24}, p_{30})$ .", "It is shown in [11] that this is an excellent family with Galois group $G$ .", "The $p_i$ are related to the fundamental invariants of the Weyl group of $E_8$ , as is suggested by their degrees (subscripts).", "We now define the notion of a multiplicative excellent family for a group $G$ .", "As before, $X \\rightarrow {\\mathbb {A}}^n$ is a family of algebraic varieties, varying with respect to $\\lambda = (\\lambda _1, \\dots ,\\lambda _n)$ , and ${\\mathcal {C}}(X_\\lambda )$ is a group of algebraic cycles on $X_\\lambda $ , isomorphic (via a fixed isomorphism) to a fixed abelian group $M$ .", "The fields $k_0$ and $k$ are as before, and we have a Galois representation $\\rho _\\lambda : \\mathrm {Gal}(k/k_0) \\rightarrow \\mathrm {Aut}({\\mathcal {C}}(X_\\lambda )) \\rightarrow \\mathrm {Aut}(M).$ Suppose that $G$ is a group acting on $M$ .", "Definition 2 We say $\\lbrace X_\\lambda \\rbrace $ is a multiplicative excellent family with Galois group $G$ if the following conditions hold: The image of $\\rho _\\lambda $ is equal to $G$ .", "There is a $\\mathrm {Gal}(k/k_0)$ -equivariant evaluation map $s:{\\mathcal {C}}(X_\\lambda ) \\rightarrow k^*$ .", "There exists a basis $\\lbrace Z_1, \\dots , Z_n \\rbrace $ of ${\\mathcal {C}}(X_\\lambda )$ such that if we set $u_i = s(Z_i)$ , then $u_1, \\dots , u_n$ are algebraically independent over ${\\mathbb {Q}}$ .", "${\\mathbb {Q}}[u_1, \\dots , u_n, u_1^{-1}, \\dots , u_n^{-1}]^G = {\\mathbb {Q}}[\\lambda _1, \\dots , \\lambda _n]$ .", "Remark 3 Though we use similar notation, the specialization map $s$ and the $u_i$ in the multiplicative case are quite different from the ones in the additive case.", "Intuitively, one may think of these as exponentiated versions of the corresponding objects in the additive case.", "However, we want the specialization map to be an algebraic morphism, and so in general (additive) excellent families specified by Definition REF will be very different from multiplicative excellent families specified by Definition REF .", "In our examples, $G$ will be a finite reflection group acting on a lattice in Euclidean space, which will be our choice for $M$ .", "However, what is relevant here is not the ring of (additive) invariants of $G$ on the vector space spanned by $M$ .", "Instead, note that the action of $G$ on $M$ gives rise to a “multiplicative” or “monomial” action of $G$ on the group algebra ${\\mathbb {Q}}[M]$ , and we will be interested in the polynomials on this space which are invariant under $G$ .", "This is the subject of multiplicative invariant theory (see, for example, [7]).", "In the case when $G$ is the automorphism group of a root lattice or root system, multiplicative invariants were classically studied by using the terminology of “exponentiated” roots $e^{\\alpha }$ (for instance, see [1])." ], [ "The $E_6$ case", "We now briefly describe the construction of multiplicative excellent family in [13].", "Consider the family of rational elliptic surfaces $S_\\lambda $ with Weierstrass equation $y^2 + txy = x^3 + (p_0 + p_1 t + p_2 t^2) \\,x + q_0 + q_1 t + q_2 t^2 + t^3$ with parameter $\\lambda = (p_0, p_1, p_2, q_0, q_1, q_2)$ .", "The surface $S_\\lambda $ generically only has one reducible fiber at $t = \\infty $ , of type $I_3$ .", "Therefore, the Mordell-Weil lattice $M_\\lambda $ of $S_\\lambda $ is isomorphic to $E_6^*$ .", "There are 54 minimal sections of height $4/3$ , and exactly half of them have the property that $x$ and $y$ are linear in $t$ .", "If we have $x &= at + b \\\\y &= ct + d,$ then substituting these back in to the Weierstrass equation, we get a system of equations, and we may easily eliminate $b,c,d$ from the system to get a monic equation of degree 27 (subject to a genericity assumption), which we write as $\\Phi _\\lambda (a) = 0$ .", "Also, note that the specialization of a section of height $4/3$ to the fiber at $\\infty $ gives us a point on one of the two non-identity components of the special fiber of the Néron model (the same component for all the 27 sections).", "Identifying the smooth points of this component with ${\\mathbb {G}}_m \\times \\lbrace 1\\rbrace \\subset {\\mathbb {G}}_m \\times ({\\mathbb {Z}}/3{\\mathbb {Z}})$ , the specialization map $s$ takes the section to $(-1/a, 1)$ .", "Let the specializations be $s_i = -1/a_i$ for $1 \\le i \\le 27$ .", "We have $\\Phi _\\lambda (X) &= \\prod _{i=1}^{27} (X - a_i) \\\\&= \\prod _{i=1}^{27} (X + 1/s_i) \\\\&= X^{27} + \\epsilon _{-1} X^{26} + \\epsilon _{-2} X^{25} + \\dots + \\epsilon _4 X^4 + \\epsilon _3 X^3 + \\epsilon _2 X^2 + \\epsilon _1 X + 1.$ Here $\\epsilon _i$ is the $i$ 'th elementary symmetric polynomial of the $s_i$ and $\\epsilon _{-i}$ that of the $1/s_i$ .", "The coefficients of $\\Phi _\\lambda (X)$ are polynomials in the coordinates of $\\lambda $ , and we may use the equations for $\\epsilon _1, \\epsilon _2, \\epsilon _3, \\epsilon _4, \\epsilon _{-1}$ and $\\epsilon _{-2}$ to solve for $p_0, \\dots , q_3$ .", "However, the resulting solution has $\\epsilon _{-2}$ in the denominator.", "We may remedy this situation as follows: consider the construction of $E_6^*$ as described in [14]: let $v_1, \\dots , v_6$ be vectors in ${\\mathbb {R}}^6$ with $\\langle v_i, v_j \\rangle = \\delta _{ij} + 1/3$ , and let $u = (\\sum v_i)/3$ .", "The ${\\mathbb {Z}}$ -span of $v_1, \\dots , v_6, u$ is a lattice $L$ isometric to $E_6^*$ .", "It is clear that $v_1, \\dots ,v_5, u$ form a basis of $L$ .", "Here, we choose an isometry between the Mordell-Weil lattice and the lattice $L$ , and let the specializations of $v_1, \\dots , v_6, u$ be $s_1, \\dots , s_6, r$ respectively.", "These satisfy $s_1 s_2 \\dots s_6= r^3$ .", "The fifty-four nonzero minimal vectors of $E_6^*$ split up into two cosets (modulo $E_6$ ) of twenty-seven each, of which we have chosen one.", "The specializations of these twenty-seven special sections are given by $\\lbrace s_1, \\dots , s_{27} \\rbrace := \\lbrace s_i : 1 \\le i \\le 6 \\rbrace \\bigcup \\lbrace s_i/r : 1 \\le i \\le 6 \\rbrace \\bigcup \\lbrace r/(s_i s_j) : 1 \\le i < j \\le 6 \\rbrace .$ Let $\\delta _1 = r + \\frac{1}{r} + \\sum _{ i \\ne j} \\frac{s_i}{s_j} + \\sum _{i < j < k} \\left( \\frac{r}{s_i s_j s_k} + \\frac{s_i s_j s_k}{r} \\right).$ Then $\\delta _1$ belongs to the $G = W(E_6)$ -invariants of ${\\mathbb {Q}}[s_1,\\dots , s_5, r, s_1^{-1}, \\dots , s_5^{-1},r^{-1}]$ , and it is shown by explicit computation in [13] that ${\\mathbb {Q}}[s_1, \\dots , s_5, r, s_1^{-1}, \\dots , s_5^{-1}, r^{-1}]^G = {\\mathbb {Q}}[\\delta _1, \\epsilon _1, \\epsilon _2, \\epsilon _3, \\epsilon _{-1}, \\epsilon _{-2}] = {\\mathbb {Q}}[p_0, p_1, p_2, q_0, q_1, q_2].$ The explicit relation showing the second equality is as follows.", "$& \\delta _1 = -2 p_1 \\\\& \\epsilon _1 = 6 p_2 \\\\& \\epsilon _{-1} = p_2^2 - q_2 \\\\& \\epsilon _2 = 13 p_2^2 + p_0 - q_2 \\\\& \\epsilon _{-2} = -2p_1 p_2 + 6p_2 + q_1 \\\\& \\epsilon _3 = 8 p_2^3 + 2 p_0 p_2 + p_1^2 - 6 p_1 - q_0 + 9.$ We make the following additional observation.", "The six fundamental representations of the Lie algebra $E_6$ correspond to the fundamental weights in the following diagram, which displays the standard labeling of these representations.", "(0,0)–(4,0); (2,0)–(2,1); [white] (0,0) circle (0.1); [white] (1,0) circle (0.1); [white] (2,0) circle (0.1); [white] (3,0) circle (0.1); [white] (4,0) circle (0.1); [white] (2,1) circle (0.1); (0,0) circle (0.1); (1,0) circle (0.1); (2,0) circle (0.1); (3,0) circle (0.1); (4,0) circle (0.1); (2,1) circle (0.1); (0,-0.2) [below] node1; (1,-0.2) [below] node3; (2,-0.2) [below] node4; (3,-0.2) [below] node5; (4,-0.2) [below] node6; (2.2,1) [right] node2; The dimensions of these representations $V_1, \\dots , V_6$ are $27, 78,351, 2925, 351, 27$ respectively, and their characters are related to $\\epsilon _1, \\epsilon _2, \\epsilon _3, \\epsilon _{-1}, \\epsilon _{-2}, \\delta _1$ by the following nice transformation.", "$\\chi _1 &= \\epsilon _1 &\\chi _2 &= \\delta _1 + 6 &\\chi _3 &= \\epsilon _2 \\\\\\chi _4 &= \\epsilon _3 &\\chi _5 &= \\epsilon _{-2} &\\chi _6 &= \\epsilon _{-1}.$ This explains the reason for bringing in $\\delta _1$ into the picture, and also why there is a denominator when solving for $p_0, \\dots , q_2$ in terms of $\\epsilon _1, \\dots , \\epsilon _4, \\epsilon _{-1},\\epsilon _{-2}$ , as remarked in [13].", "The coefficients $\\epsilon _j$ are essentially the characters of $\\Lambda ^j V$ , where $V = V_1$ is the first fundamental representation, while $\\epsilon _{-j}$ are those of $\\Lambda ^j V^*$ , where $V_6 = V^*$ .", "Note that $\\Lambda ^3 V \\cong \\Lambda ^3 V^*$ .", "Therefore, from the expressions for $\\epsilon _1, \\epsilon _2, \\epsilon _3, \\epsilon _{-1}, \\epsilon _{-2}$ , we may obtain $p_2,q_2, p_0, q_1, q_0$ , in terms of the remaining variable $p_1$ , without introducing any denominators.", "However, representation $V_2$ cannot be obtained as a direct summand with multiplicity 1 from a tensor product of $\\Lambda ^j V$ (for $1 \\le j \\le 3$ ) and $\\Lambda ^k V^*$ (for $1\\le k \\le 2$ ).", "On the other hand, we do have the following isomorphism: $(V_2 \\otimes V_5) \\oplus V_5 \\oplus V_1 \\cong \\Lambda ^4 V_1 \\oplus (V_3 \\otimes V_6) \\oplus (V_6 \\otimes V_6).$ Therefore, we are able to solve for $p_1$ if we introduce a denominator of $\\epsilon _{-2}$ , which is the character of $V_5$ ." ], [ "Results", "Next, we exhibit a multiplicative excellent family for the Weyl group of $E_7$ .", "It is given by the Weierstrass equation $y^2 + txy = x^3 + (p_0 + p_1 t +p_2 t^2) \\, x + q_0 + q_1 t + q_2 t^2 + q_3 t^3 - t^4.$ For generic $\\lambda = (p_0, \\dots , p_2, q_0, \\dots , q_3)$ , this rational elliptic surface $X_\\lambda $ has a fiber of type $I_2$ at $t =\\infty $ , and no other reducible fibers.", "Hence, the Mordell-Weil group $M_\\lambda $ is $E_7^*$ .", "We note that any elliptic surface with a fiber of type $I_2$ can be put into this Weierstrass form (in general over a small degree algebraic extension of the ground field), after a fractional linear transformation of the parameter $t$ , and Weierstrass transformations of $x,y$ .", "Lemma 4 The smooth part of the special fiber is isomorphic to the group scheme ${\\mathbb {G}}_m \\times {\\mathbb {Z}}/2{\\mathbb {Z}}$ .", "The identity component is the non-singular part of the curve $y^2 + xy = x^3$ .", "A section of height 2 has $x$ - and $y$ -coordinates polynomials of degrees 2 and 3 respectively, and its specialization at $t = \\infty $ is $(\\lim _{t\\rightarrow \\infty } (y + tx)/y, 0) \\in k^* \\times \\lbrace 0,1\\rbrace $ .", "A section of height $3/2$ has $x$ and $y$ coordinates of the form $x &= at + b \\\\y &= ct^2 + dt + e.$ and specializes at $t = \\infty $ to $(c,1)$ .", "First, observe that to get a local chart for the elliptic surface near $t = \\infty $ , we set $x = \\widetilde{x}/u^2$ , $y =\\widetilde{y}/u^3$ and $t = 1/u$ , and look for $u$ near 0.", "Therefore, the special fiber (before blow-up) is given by $\\bar{y}^2 + \\bar{x} \\bar{y} = \\bar{x}^3$ , where $\\bar{x} =\\widetilde{x} |_{u = 0}$ and $\\bar{y} = \\widetilde{y}|_{u = 0}$ are the reductions of the coordinates at $u = 0$ respectively.", "It is an easy exercise to parametrize the smooth locus of this curve: it is given, for instance, by $\\bar{x} = s/(s-1)^2, \\bar{y} =s/(s-1)^3$ .", "We then check that $s = (\\bar{y}+\\bar{x})/\\bar{y}$ and the map from the smooth locus to ${\\mathbb {G}}_m$ which takes the point $(\\bar{x}, \\bar{y})$ to $s$ is a homomorphism from the secant group law to multiplication in $k^*$ .", "This proves the first half of the lemma.", "Note that we could just as well have taken $1/s$ to be the parameter on ${\\mathbb {G}}_m$ ; our choice is a matter of convention.", "To prove the specialization law for sections of height $3/2$ , we may, for instance, take the sum of such a section $Q$ with a section $P$ of height 2 with specialization $(s,0)$ .", "A direct calculation shows that the $y$ -coordinate of the sum has top (quadratic) coefficient $cs$ .", "Therefore the specialization of $Q$ must have the form $\\kappa c$ , where $\\kappa $ is a constant not depending on $Q$ .", "Finally, the sum of two sections $Q_1$ and $Q_2$ of height $3/2$ and having coefficients $c_1$ and $c_2$ for the $t^2$ term of their $y$ -coordinates can be checked to specialize to $(c_1 c_2, 0)$ .", "It follows that $\\kappa = \\pm 1$ , and we take the plus sign as a convention.", "(It is easy to see that both choices of sign are legitimate, since the sections of height 2 generate a copy of $E_7$ , whereas the sections of height $3/2$ lie in the nontrivial coset of $E_7$ in $E_7^*$ ).", "There are 56 sections of height $3/2$ , with $x$ and $y$ coordinates in the form above.", "Substituting the above formulas for $x$ and $y$ into the Weierstrass equation, we get the following system of equations.", "$c^2 + ac + 1 &= 0 \\\\q_3 + a p_2 + a^3 &= (2c + a) d + bc \\\\q_2 + b p_2 + 3 a^2 b &= (2c + a) e + (b+d) d \\\\q_1 + b p_1 + a p_0 + 3 a b^2 &= (2d + b) e \\\\q_0 + b p_0 + b^3 &= e^2.$ We solve for $a$ and $b$ from the first and second equations, and then $e$ from the third, assuming $c \\ne 1$ .", "Substituting these values back into the last two equations, we get two equations in the variables $c$ and $d$ .", "Taking the resultant of these two equations with respect to $d$ , and dividing by $c^{30} (c^2 - 1)^4$ , we obtain an equation of degree 56 in $c$ , which is monic, reciprocal and has coefficients in ${\\mathbb {Z}}[\\lambda ] = {\\mathbb {Z}}[p_0, \\dots , q_3]$ .", "We denote this polynomial by $\\Phi _\\lambda (X) = \\prod _{i=1}^{56} \\left(X - s(P) \\right) = X^{56} + \\epsilon _1 X^{55} + \\epsilon _2 X^{54}+ \\dots + \\epsilon _1 X + \\epsilon _0,$ where $P$ ranges over the 56 minimal sections of height $3/2$ .", "It is clear that $a,b,d,e$ are rational functions of $c$ with coefficients in $k_0$ .", "We have a Galois representation on the Mordell-Weil lattice $\\rho _\\lambda : \\mathrm {Gal}(k/k_0) \\rightarrow \\mathrm {Aut}(M_\\lambda ) \\cong \\mathrm {Aut}(E_7^*).$ Here $\\mathrm {Aut}(E_7^*) \\cong \\mathrm {Aut}(E_7) \\cong W(E_7)$ , the Weyl group of type $E_7$ .", "The splitting field of $M_\\lambda $ is the fixed field $k_\\lambda $ of $\\mathrm {Ker}(\\rho _\\lambda )$ .", "By definition, $\\mathrm {Gal}(k_\\lambda /k_0)\\cong \\textrm {Im}(\\rho _\\lambda )$ .", "The splitting field $k_\\lambda $ is equal to the splitting field of the polynomial $\\Phi _\\lambda (X)$ over $k_0$ , since the Mordell-Weil group is generated by the 56 sections of smallest height $P_i = (a_i t + b_i, c_i t^2 + d_i t + e_i)$ .", "We also have that $k_\\lambda = k_0(P_1, \\dots , P_{56}) = k_0(c_1, \\dots , c_{56}).$ We shall sometimes write $e^{\\alpha }$ , (for $\\alpha \\in E_7^*$ a minimal vector) to refer to the specializations of these sections $c(P_i)$ , for convenience.", "Theorem 5 Assume that $\\lambda $ is generic over ${\\mathbb {Q}}$ , i.e the coordinates $p_0, \\dots , q_3$ are algebraically independent over ${\\mathbb {Q}}$ .", "Then $\\rho _\\lambda $ induces an isomorphism $\\mathrm {Gal}(k_\\lambda / k_0) \\cong W(E_7)$ .", "The splitting field $k_\\lambda $ is a purely transcendental extension of ${\\mathbb {Q}}$ , isomorphic to the function field ${\\mathbb {Q}}(Y)$ of the toric hypersurface $Y \\subset {\\mathbb {G}}_m^8$ defined by $s_1 \\dots s_7 =r^3$ .", "There is an action of $W(E_7)$ on $Y$ such that ${\\mathbb {Q}}(Y)^{W(E_7)} = k_\\lambda ^{W(E_7)} = k_0$ .", "The ring of $W(E_7)$ -invariants in the affine coordinate ring ${\\mathbb {Q}}[Y] = \\frac{{\\mathbb {Q}}[s_i, r, 1/s_i, 1/r]}{(s_1\\dots s_7 - r^3)} \\cong {\\mathbb {Q}}[s_1, \\dots , s_6, r, s_1^{-1}, \\dots , s_6^{-1}, r^{-1}]$ is equal to the polynomial ring ${\\mathbb {Q}}[\\lambda ]$ : ${\\mathbb {Q}}[Y]^{W(E_7)} = {\\mathbb {Q}}[\\lambda ] = {\\mathbb {Q}}[p_0, p_1, p_2, q_0, q_1, q_2, q_3].$ In fact, we shall prove an explicit, invertible polynomial relation between the Weierstrass coefficients $\\lambda $ and the fundamental characters for $E_7$ .", "Let $V_1, \\dots , V_7$ be the fundamental representations of $E_7$ , and $\\chi _1, \\dots , \\chi _7$ their characters (on a maximal torus), as labelled below.", "For a description of the fundamental modules for the exceptional Lie groups see [2].", "(0,0)–(5,0); (2,0)–(2,1); [white] (0,0) circle (0.1); [white] (1,0) circle (0.1); [white] (2,0) circle (0.1); [white] (3,0) circle (0.1); [white] (4,0) circle (0.1); [white] (5,0) circle (0.1); [white] (2,1) circle (0.1); (0,0) circle (0.1); (1,0) circle (0.1); (2,0) circle (0.1); (3,0) circle (0.1); (4,0) circle (0.1); (5,0) circle (0.1); (2,1) circle (0.1); (0,-0.2) [below] node1; (1,-0.2) [below] node3; (2,-0.2) [below] node4; (3,-0.2) [below] node5; (4,-0.2) [below] node6; (5,-0.2) [below] node7; (2.2,1) [right] node2; Note that since the weight lattice $E_7^*$ has been equipped with a nice set of generators $(v_1, \\dots , v_7, u)$ with $\\sum v_i = 3 u$ (as in [14]), the characters $\\chi _1, \\dots , \\chi _7$ lie in the ring of Laurent polynomials ${\\mathbb {Q}}[s_i, r, 1/s_i, 1/r]$ where $s_i$ corresponds to $e^{v_i}$ and $r$ to $e^u$ , and are obviously invariant under the (multiplicative) action of the Weyl group on this ring of Laurent polynomials.", "Explicit formulae for the $\\chi _i$ are given in the auxiliary files.", "We also note that the roots of $\\Phi _\\lambda $ are given by $s_i, \\frac{1}{s_i} \\textrm { for } 1\\le i \\le 7 \\quad \\textrm { and } \\quad \\frac{r}{s_i s_j}, \\frac{s_i s_j}{r} \\textrm { for } 1 \\le i < j \\le 7.$ Theorem 6 For generic $\\lambda $ over ${\\mathbb {Q}}$ , we have ${\\mathbb {Q}}[\\chi _1, \\dots , \\chi _7] = {\\mathbb {Q}}[p_0, p_1, p_2, q_0, q_1, q_2, q_3].$ The transformation between these sets of generators is $\\chi _1 &= 6 p_2+25 \\\\\\chi _2 &= 6 q_3-2 p_1 \\\\\\chi _3 &= -q_2+13 p_2^2+108 p_2+p_0+221 \\\\\\chi _4 &= 9 q_3^2-6 p_1 q_3-q_2-q_0+8 p_2^3+85 p_2^2+(2 p_0+300) p_2+p_1^2+10 p_0+350 \\\\\\chi _5 &= (6 p_2+26) q_3+q_1-2 p_1 p_2-10 p_1 \\\\\\chi _6 &= -q_2+p_2^2+12 p_2+27 \\\\\\chi _7 &= q_3 \\\\& \\\\\\textrm {with inverse}\\\\& \\\\p_2 &= (\\chi _1-25)/6 \\\\p_1 &= (6 \\chi _7-\\chi _2)/2 \\\\p_0 &= -(3 \\chi _6-3 \\chi _3+\\chi _1^2-2 \\chi _1+7)/3 \\\\q_3 &= \\chi _7 \\\\q_2 &= -(36 \\chi _6-\\chi _1^2-22 \\chi _1+203)/36 \\\\q_1 &= (24 \\chi _7+6 \\chi _5+(-\\chi _1-5) \\chi _2)/6 \\\\q_0 &= (27 \\chi _2^2-8 \\chi _1^3-84 \\chi _1^2+120 \\chi _1-136)/108 \\\\& \\quad -(\\chi _1+2) \\chi _6/3 - \\chi _4 + (\\chi _1+ 5) \\chi _3/3.$ We note that our formulas agree with those of Eguchi and Sakai [3], who seem to derive these by using an ansatz.", "Next, we describe two examples through specialization, one of “small Galois” (in which all sections are defined over ${\\mathbb {Q}}[t]$ ) and one with “big Galois” (which has Galois group the full Weyl group).", "Example 7 The values $p_0 &= 244655370905444111/(3 \\mu ^2), \\quad p_1 = -4788369529481641525125/(16 \\mu ^2) \\\\q_3 &= 184185687325/(4 \\mu ), \\quad p_2 = 199937106590279644475038924955076599/(12 \\mu ^4) \\\\q_2 &= 57918534120411335989995011407800421/(9 \\mu ^3) \\\\q_1 &= -179880916617213624948875556502808560625/(4 \\mu ^4) \\\\q_0 &= 35316143754919755115952802080469762936626890880469201091/(1728 \\mu ^6) ,$ where $\\mu = 2 \\cdot 3 \\cdot 5 \\cdot 7 \\cdot 11 \\cdot 13 \\cdot 17 =102102$ , give rise to an elliptic surface for which we have $r = 2,s_1 = 3, s_2 = 5, s_3 = 7, s_4 = 11, s_5 = 13, s_6 = 17$ , the simplest choice of multiplicatively independent elements.", "The Mordell-Weil group has a basis of sections for which $c \\in \\lbrace 3,5,7,11,13,17,15/2\\rbrace $ .", "We write down their $x$ -coordinates below: $x(P_1) &= -(10/3)t - 707606695171055129/1563722760600 \\\\x(P_2) &= -(26/5)t - 611410735289928023/1563722760600 \\\\x(P_3) &= -(50/7)t - 513728975686763429/1563722760600 \\\\x(P_4) &= -(122/11)t - 316023939417997169/1563722760600 \\\\x(P_5) &= -(170/13)t - 216677827127591279/1563722760600 \\\\x(P_6) &= -(290/17)t - 17562556436754779/1563722760600 \\\\x(P_7) &= -(229/30)t - 140574879644393807/390930690150.$ In the auxiliary files the $x$ -and $y$ -coordinates are listed, and it is verified that they satisfy the Weierstrass equation.", "Example 8 The value $\\lambda = \\lambda _0 := (1,1,1,1,1,1,1)$ gives rise to an explicit polynomial $f(X) = \\Phi _{\\lambda _0}(X)$ , given by $f(X) &= X^{56} - X^{55} + 40X^{54} - 22X^{53} + 797X^{52} - 190X^{51} + 9878X^{50} - 1513X^{49} \\\\& \\quad + 82195X^{48} - 17689X^{47} + 496844X^{46} - 175584X^{45} + 2336237X^{44} \\\\& \\quad - 1196652X^{43} + 8957717X^{42} - 5726683X^{41} + 28574146X^{40} \\\\& \\quad - 20119954X^{39} + 75465618X^{38} - 53541106X^{37} + 163074206X^{36} \\\\& \\quad - 110505921X^{35} + 287854250X^{34} - 181247607X^{33} + 420186200X^{32} \\\\& \\quad - 243591901X^{31} + 518626022X^{30} - 278343633X^{29} + 554315411X^{28} \\\\& \\quad - 278343633X^{27} + 518626022X^{26} - 243591901X^{25} + 420186200X^{24} \\\\& \\quad - 181247607X^{23} + 287854250X^{22} - 110505921X^{21} + 163074206X^{20} \\\\& \\quad - 53541106X^{19} + 75465618X^{18} - 20119954X^{17} + 28574146X^{16} \\\\& \\quad - 5726683X^{15} + 8957717X^{14} - 1196652X^{13} + 2336237X^{12} \\\\& \\quad - 175584X^{11} + 496844X^{10} - 17689X^9 + 82195X^8 - 1513X^7 \\\\& \\quad + 9878X^6 - 190X^5 + 797X^4 - 22X^3 + 40X^2 - X + 1,$ for which we can show that the Galois group is the full group $W(E_7)$ , as follows.", "The reduction of $f(X)$ modulo 7 shows that $\\mathrm {Frob}_7$ has cycle decomposition of type $(7)^8$ , and similarly, $\\mathrm {Frob}_{101}$ has cycle decomposition of type $(3)^2(5)^4(15)^2$ .", "This implies, as in [12], that the Galois group is the entire Weyl group.", "We can also describe degenerations of this family of rational elliptic surfaces $X_\\lambda $ by the method of “vanishing roots”, where we drop the genericity assumption, and consider the situation where the elliptic fibration might have additional reducible fibers.", "Let $\\psi :Y \\rightarrow {\\mathbb {A}}^7$ be the finite surjective morphism associated to ${\\mathbb {Q}}[p_0, \\dots , q_3] \\hookrightarrow {\\mathbb {Q}}[Y] \\cong {\\mathbb {Q}}[s_1, \\dots , s_6, r, s_1^{-1}, \\dots , s_6^{-1}, r^{-1}].$ For $\\xi = (s_1, \\dots , s_7, r) \\in Y$ , let the multiset $\\Pi _\\xi $ consist of the 126 elements $s_i/r$ and $r/s_i$ (for $1 \\le i \\le 7$ ), $s_i/s_j$ (for $1 \\le i \\ne j \\le 7$ ) and $s_i s_j s_k/r$ and $r/(s_i s_j s_k)$ for $1 \\le i < j < k \\le 7$ , corresponding to the 126 roots of $E_7$ .", "Let $2\\nu (\\xi )$ be the number of times 1 appears in $\\Pi _\\xi $ , which is also the multiplicity of 1 as a root of $\\Psi _\\lambda (X)$ (to be defined in Section REF ), where $\\lambda = \\psi (\\xi )$ .", "We call the associated roots of $E_7$ the vanishing roots, in analogy with vanishing cycles in the deformation of singularities.", "By abuse of notation we label the rational elliptic surface $X_\\lambda $ as $X_\\xi $ .", "Theorem 9 The surface $X_\\xi $ has new reducible fibers (necessarily at $t \\ne \\infty $ ) if and only if $\\nu (\\xi ) > 0$ .", "The number of roots in the root lattice $T_{\\textrm {new}}$ is equal to $2\\nu (\\xi )$ , where $T_{\\textrm {new}} := \\oplus _{v \\ne \\infty } T_v$ is the new part of the trivial lattice.", "We may use this result to produce specializations with trivial lattice including $A_1$ , corresponding to the entries in the table of [8].", "Note that in earlier work [11], [18], examples of rational elliptic surfaces were produced with a fiber of additive type, for instance, a fiber of type $\\mathrm {III}$ (which contributes $A_1$ to the trivial lattice) or a fiber of type $\\mathrm {II}$ .", "Using our excellent family, we can produce examples with the $A_1$ fiber being of multiplicative type $I_2$ and all other irreducible singular fibers being nodal ($I_1$ ).", "We list below those types which are not already covered by [13].", "To produce these examples, we use an embedding of the new part $T_{\\textrm {new}}$ of the fibral lattice into $E_7$ , which gives us any extra conditions satisfied by $s_1,\\dots , s_7, r$ .", "The following multiplicative version of the labeling of simple roots of $E_7$ is useful (compare [14]).", "1.3 (0,0)–(5,0); (2,0)–(2,1); [white] (0,0) circle (0.1); [white] (1,0) circle (0.1); [white] (2,0) circle (0.1); [white] (3,0) circle (0.1); [white] (4,0) circle (0.1); [white] (5,0) circle (0.1); [white] (2,1) circle (0.1); (0,0) circle (0.1); (1,0) circle (0.1); (2,0) circle (0.1); (3,0) circle (0.1); (4,0) circle (0.1); (5,0) circle (0.1); (2,1) circle (0.1); (0,-0.2) [below] node$\\frac{s_1}{s_2}$ ; (1,-0.2) [below] node$\\frac{s_2}{s_3}$ ; (2,-0.2) [below] node$\\frac{s_3}{s_4}$ ; (3,-0.2) [below] node$\\frac{s_4}{s_5}$ ; (4,-0.2) [below] node$\\frac{s_5}{s_6}$ ; (5,-0.2) [below] node$\\frac{s_6}{s_7}$ ; (2.2,1) [right] node$\\frac{r}{s_1s_2s_3}$ ; [thick,dashed] (0.8, -0.2)–(3.2,-0.2)–(3.2,0.2)–(2.2,0.2)–(2.2,1.2)–(1.8,1.2)–(1.8,0.2)–(0.8,0.2)–(0.8,-0.2); For instance, to produce the example in line 18 of the table (i.e.", "with $T_{\\textrm {new}} = D_4$ ), we may use the embedding into $E_7$ indicated by embedding the $D_4$ Dynkin diagram within the dashed lines in the figure above.", "Thus, we must force $s_2 = s_3 = s_4= s_5$ and $r = s_1 s_2 s_3$ , and a simple solution with no extra coincidences is given in the rightmost column (note that $s_7 =18^3/(2\\cdot 3^4 \\cdot 5) = 36/5$ ).", "$\\begin{array}{cccl}\\textrm {Type in \\cite {OS}} & \\textrm {Fibral lattice} & \\textrm {MW group} & \\lbrace s_1, \\dots , s_6, r \\rbrace \\\\2 & A_1 & E_7^* & 3,5,7,11,13,17,2 \\\\4 & A_1^2 & D_6^* & 3,3,5,7,11,13,2 \\\\7 & A_1^3 & D_4^* \\oplus A_1^*& 3,3,5,5,7,11,2 \\\\10 & A_1 \\oplus A_3 & A_1^* \\oplus A_3^* & 3,3,3,3,5,7,2 \\\\13 & A_1^4 & D_4^* \\oplus {\\mathbb {Z}}/2{\\mathbb {Z}}& -1,2,3,5,7,49/30,7 \\\\14 & A_1^4 & A_1^{*4}& 3,3,5,5,7,7,2 \\\\17 & A_1 \\oplus A_4 & \\frac{1}{10} \\footnotesize { \\left( \\begin{array}{ccc} 3 & 1 & -1 \\\\ 1 & 7 & 3 \\\\ -1 & 3 & 7 \\end{array}\\right) } & 3,3,3,3,3,5,2 \\\\18 & A_1 \\oplus D_4 & A_1^{*3} & 2,3,3,3,3,5,18 \\\\21 & A_1^2 \\oplus A_3 & A_3^* \\oplus {\\mathbb {Z}}/2{\\mathbb {Z}}& 3,5,60,30,30,30,900 \\\\22 & A_1^2 \\oplus A_3 & A_1^{*2} \\oplus \\langle 1/4 \\rangle & 3,3,5,5,5,5,2 \\\\24 & A_1^5 & A_1^{*3} \\oplus {\\mathbb {Z}}/2{\\mathbb {Z}}& 15/4,2,2,3,3,5,15 \\\\28 & A_1 \\oplus A_5 & A_2^* \\oplus {\\mathbb {Z}}/2{\\mathbb {Z}}& 2,3,6,6,6,6,36 \\\\29 & A_1 \\oplus A_5 & A_1^* \\oplus \\langle 1/6 \\rangle & 2,2,2,2,2,2,3 \\\\30 & A_1 \\oplus D_5 & A_1^* \\oplus \\langle 1/4 \\rangle & 2,2,2,2,2,3,8 \\\\33 & A_1^2 \\oplus A_4 & \\frac{1}{10} \\footnotesize { \\left( \\begin{array}{cc} 2 & 1 \\\\ 1 & 3 \\end{array} \\right) } & 2,2,3,3,3,3,12 \\\\34 & A_1^2 \\oplus D_4 & A_1^{*2} \\oplus {\\mathbb {Z}}/2{\\mathbb {Z}}& 2,3,3,3,3,6,18\\end{array}$ $\\begin{array}{cccl}\\textrm {Type in \\cite {OS}} & \\textrm {Fibral lattice} & \\textrm {MW group} & \\lbrace s_1, \\dots , s_6, r \\rbrace \\\\38 & A_1^3 \\oplus A_3 & A_1^* \\oplus \\langle 1/4 \\rangle \\oplus {\\mathbb {Z}}/2{\\mathbb {Z}}& 2,2,3,3,3,4,12 \\\\42 & A_1^6 & A_1^{*2} \\oplus ({\\mathbb {Z}}/2{\\mathbb {Z}})^2 & 6,-1,-1,2,2,3,6 \\\\47 & A_1 \\oplus A_6 & \\langle 1/14 \\rangle & 8,8,8,8,8,8,128 \\\\48 & A_1 \\oplus D_6 & A_1^* \\oplus {\\mathbb {Z}}/2{\\mathbb {Z}}& 1,2,2,2,2,2,4 \\\\49 & A_1 \\oplus E_6 & \\langle 1/6 \\rangle & 2,2,2,2,2,2,8 \\\\52 & A_1^2 \\oplus D_5 & \\langle 1/4 \\rangle \\oplus {\\mathbb {Z}}/2{\\mathbb {Z}}& 2,2,2,2,2,4,8 \\\\53 & A_1^2 \\oplus A_5 & \\langle 1/6 \\rangle \\oplus {\\mathbb {Z}}/2{\\mathbb {Z}}& 2,2,4,4,4,4,16 \\\\57 & A_1^3 \\oplus D_4 & A_1^* \\oplus ({\\mathbb {Z}}/2{\\mathbb {Z}})^2 & -1,2,2,2,2,-2,-4 \\\\58 & A_1 \\oplus A_3^2 & A_1^* \\oplus {\\mathbb {Z}}/4{\\mathbb {Z}}& I,I,I,I,2,2,2 \\\\60 & A_1^4 \\oplus A_3 & \\langle 1/4 \\rangle \\oplus ({\\mathbb {Z}}/2{\\mathbb {Z}})^2 & 2,2,2,2,-1,-1,4 \\\\65 & A_1 \\oplus E_7 & {\\mathbb {Z}}/2{\\mathbb {Z}}& 1,1,1,1,1,1,1 \\\\70 & A_1 \\oplus A_7 & {\\mathbb {Z}}/4{\\mathbb {Z}}& I,I,I,I,I,I,I \\\\71 & A_1^2 \\oplus D_6 & ({\\mathbb {Z}}/2{\\mathbb {Z}})^2 & 1,1,1,1,1,1,-1 \\\\74 & A_1^2 \\oplus A_3^2 & ({\\mathbb {Z}}/2{\\mathbb {Z}}) \\oplus ({\\mathbb {Z}}/4{\\mathbb {Z}}) & I,I,I,I,-1,-1,-1\\end{array}$ Here $I = \\sqrt{-1}$ .", "Remark 10 For the examples in lines 58, 70 and 74 of the table, one can show that it is not possible to define a rational elliptic surface over ${\\mathbb {Q}}$ in the form we have assumed, such that all the specializations $s_i, r$ are rational.", "However, there do exist examples with all sections defined over ${\\mathbb {Q}}$ , not in the chosen Weierstrass form.", "The surface with Weierstrass equation $y^2 + xy + \\frac{(c^2-1)(t^2-1)}{16} y = x^3 + \\frac{(c^2-1)(t^2-1)}{16} x^2$ has a 4-torsion section $(0,0)$ and a non-torsion section $\\big ((c+1)(t^2-1)/8, (c+1)^2(t-1)^2(t+1)/32 \\big )$ of height $1/2$ , as well as two reducible fibers of type $I_4$ and a fiber of type $I_2$ .", "It is an example of type 58.", "The surface with Weierstrass equation $y^2 + xy + t^2 y = x^3 + t^2 x^2$ has a 4-torsion section $(0,0)$ , and reducible fibers of types $I_8$ and $I_2$ .", "It is an example of type 70.", "The surface with Weierstrass equation $y^2 + xy - \\left(t^2-\\frac{1}{16} \\right)y = x^3 - \\left( t^2 -\\frac{1}{16} \\right) x^2$ has two reducible fibers of type $I_4$ and two reducible fibers of type $I_2$ .", "It also has a 4-torsion section $(0,0)$ and a 2-torsion section $\\big ((4t-1)/8, (4t-1)^2/32\\big )$ , which generate the Mordell-Weil group.", "It is an example of type 74.", "This last example is the universal elliptic curve with ${\\mathbb {Z}}/4{\\mathbb {Z}}\\oplus {\\mathbb {Z}}/2{\\mathbb {Z}}$ torsion (compare [6])." ], [ "Proofs", "We start by considering the coefficients $\\epsilon _i$ of $\\Phi _\\lambda (X)$ ; we know that $(-1)^i \\epsilon _i$ is simply the $i$ 'th elementary symmetric polynomial in the 56 specializations $s(P_i)$ .", "One shows, either by explicit calculation with Laurent polynomials, or by calculating the decomposition of $\\Lambda ^i V$ (where $V = V_7$ is the 56-dimensional representation of $E_7$ ), and expressing its character as polynomials in the fundamental characters, the following formulae.", "Some more details are in Section and the auxiliary files.", "$\\epsilon _1 &= -\\chi _7 \\\\\\epsilon _2 &= \\chi _6 + 1 \\\\\\epsilon _3 &= -(\\chi _7+\\chi _5) \\\\\\epsilon _4 &= \\chi _6 + \\chi _4 + 1 \\\\\\epsilon _5 &= -(\\chi _6+\\chi _3-\\chi _1^2+\\chi _1+1) \\chi _7+(\\chi _1-1) \\chi _5-\\chi _2 \\chi _3 \\\\\\epsilon _6 &= \\chi _1 \\chi _7^2+(\\chi _5-(\\chi _1+1) \\chi _2) \\chi _7+\\chi _6^2 +2 (\\chi _3-\\chi _1^2+\\chi _1+1) \\chi _6 \\\\& \\quad -\\chi _2 \\chi _5 -(2 \\chi _1+1) \\chi _4 +\\chi _3^2+2 (2 \\chi _1+1) \\chi _3 \\\\& \\quad +\\chi _1 \\chi _2^2-2 \\chi _1^3+\\chi _1^2+2 \\chi _1+1 \\\\\\epsilon _7 &= (-(\\chi _1+1) \\chi _6+2 \\chi _4-2(\\chi _1+1) \\chi _3+\\chi _1^3-3 \\chi _1-1) \\chi _7 \\\\& \\quad -2 (\\chi _5-\\chi _1 \\chi _2) \\chi _6 -(\\chi _3-\\chi _1^2+\\chi _1+2) \\chi _5 +3 \\chi _2 \\chi _4 \\\\& \\quad -(\\chi _1+3) \\chi _2 \\chi _3-\\chi _2^3 +(2 \\chi _1-1) \\chi _1 \\chi _2 .$ On the other hand, we can explicitly calculate the first few coefficients $\\epsilon _i$ of $\\Phi _\\lambda (X)$ in terms of the Weierstrass coefficients, obtaining the following equations.", "Details for the method are in Section .", "$\\epsilon _1 &= -q_3 \\\\\\epsilon _2 &= p_2^2 + 12 p_2 - q_2 + 28 \\\\\\epsilon _3 &= -3 (2 p_2+9) q_3-q_1+2 p_1 (p_2+5) \\\\\\epsilon _4 &= 9 q_3^2-6 p_1 q_3-2 q_2-q_0+8 p_2^3+86 p_2^2+2 (p_0+156) p_2+p_1^2+10 p_0+378 \\\\\\epsilon _5 &= (8 q_2-20 p_2^2-174 p_2-7 p_0-351) q_3-2 p_1 q_2+6 (p_2+4) q_1 \\\\& \\quad +14 p_1 p_2^2 +108 p_1 p_2+2 (p_0+101) p_1 \\\\\\epsilon _6 &= 12 (4 p_2+15) q_3^2 -(5 q_1+38 p_1 p_2+140 p_1) q_3 + 4 q_2^2 \\\\& \\quad +(16 p_2^2+96 p_2-4 p_0+155) q_2 +2 p_1 q_1+3 (4 p_2+17) q_0 +28 p_2^4+360 p_2^3 \\\\& \\quad +(4 p_0+1765) p_2^2 +2 (4 p_1^2+21 p_0+1950) p_2+29 p_1^2+p_0^2+88 p_0+3276 \\\\\\epsilon _7 &= -36 q_3^3+42 p_1 q_3^2 +(4 q_2-20 q_0-56 p_2^3-628 p_2^2-14 (p_0+168) p_2-16 p_1^2 \\\\& \\quad -46 p_0-2925) q_3+(3 q_1+6 p_1 p_2+20 p_1) q_2+(21 p_2^2+162 p_2-p_0+323) q_1 \\\\& \\quad +6 p_1 q_0+42 p_1 p_2^3 +448 p_1 p_2^2+2 (p_0+799) p_1 p_2+2 p_1^3+6 (p_0+316) p_1 .$ Equating the two expressions we have obtained for each $\\epsilon _i$ , we get a system of seven equations, the first being $-\\chi _7 = -q_3.$ We label these equations $(1), \\dots , (7)$ .", "The last few of these polynomial equations are somewhat complicated, and so to obtain a few simpler ones, we may consider the 126 sections of height 2, which we analyze as follows.", "Substituting $x &= at^2 + bt + c \\\\y &= dt^3 + et^2 + ft + g$ in to the Weierstrass equation, we get another system of equations: $a^3 &= d^2 + ad \\\\3a^2b &= (2d + a) e + bd \\\\a(p_2 + 3 ac + 3b^2) &= (2d + a) f + e^2 + be + cd + 1 \\\\q_3 + b p_2 +a p_1 + 6abc+b^3 &= (2d + a) g + (2e + b) f + ce \\\\q_2 + c p_2 + b p_1 + a p_0 + 3 ac^2 + 3b^2 c &= (2e + b) g + f^2 + c f \\\\q_1 + c p_1 + b p_0 + 3 b c^2 &= (2f + c) g \\\\q_0 + c p_0 + c^3 &= g^2.$ The specialization of such a section at $t = \\infty $ is $1 +a/d$ .", "Setting $d = ar$ , we may as before eliminate other variables to obtain an equation of degree 126 for $r$ .", "Substituting $r =1/(u-1)$ , we get a monic polynomial $\\Psi _\\lambda (X) = 0$ of degree 126 for $u$ .", "Note that the roots are given by elements of the form $\\frac{s_i}{r} , \\frac{r}{s_i} \\textrm { for } 1 \\le i \\le 7, \\, \\frac{s_i}{s_j} \\textrm { for } 1 \\le i \\ne j \\le 7 \\, \\textrm { and } \\, \\frac{s_i s_j s_k}{r}, \\frac{r}{s_i s_j s_k} \\textrm { for } 1 \\le i < j < k \\le 7.$ As before, we can write the first few coefficients $\\eta _i$ of $\\Psi _\\lambda $ in terms of $\\lambda = (p_0, \\dots , q_3)$ , as well as in terms of the characters $\\chi _j$ , obtaining some more relations.", "We will only need the first two: $-\\chi _1 + 7 &= \\eta _1 = -18 - 6 p_2 \\\\-6 \\chi _1 + \\chi _3 + 28 &= \\eta _2 = p_0 + 72 p_2 + 13 p_2^2 - q_2 + 99$ which we call $(1^{\\prime })$ and $(2^{\\prime })$ respectively.", "Now we consider the system of six equations $(1)$ through $(4)$ , $(1^{\\prime })$ and $(2^{\\prime })$ .", "These may be solved for $(p_2, p_0, q_3, q_2, q_1,q_0)$ in terms of the $\\chi _j$ and $p_1$ .", "Substituting this solution into the other three relations $(5)$ , $(6)$ and $(7)$ , we obtain three equations for $p_1$ , of degrees 1, 2 and 3 respectively.", "These have a single common factor, linear in $p_1$ , which we then solve.", "This gives us the proof of Theorem REF .", "The proof of Theorem REF is now straightforward.", "Part (1) asserts that the image of $\\rho _\\lambda $ is surjective on to $W(E_7)$ : this follows from a standard Galois theoretic argument as follows.", "Let $F$ be the fixed field of $W(E_7)$ acting on $k_\\lambda = {\\mathbb {Q}}(\\lambda )(s_1,\\dots , s_6, r) = {\\mathbb {Q}}(s_1, \\dots , s_6,r)$ , where the last equality follows from the explicit expression of $\\lambda = (p_0, \\dots , q_3)$ in terms of the $\\chi _i$ , which are in ${\\mathbb {Q}}(s_1, \\dots , s_6, r)$ .", "Then we have that $k_0 \\subset F$ since $p_0, \\dots , q_3$ are polynomials in the $\\chi _i$ with rational coefficients, and the $\\chi _i$ are manifestly invariant under the Weyl group.", "Therefore $[k_\\lambda : k_0] \\ge [k_\\lambda :F] = |W(E_7)|$ , where the latter equality is from Galois theory.", "Finally, $[k_\\lambda : k_0] \\le |\\mathrm {Gal}(k_\\lambda /k_0)| \\le |W(E_7)|$ , since $\\mathrm {Gal}(k_\\lambda /k_0) \\hookrightarrow W(E_7)$ .", "Therefore, equality is forced.", "Another way to see that the Galois group is the full Weyl group is to demonstrate it for a specialization, such as Example REF , and use [9].", "Next, let $Y$ be the toric hypersurface given by $s_1 \\dots s_7 =r^3$ .", "Its function field is the splitting field of $\\Phi _\\lambda (X)$ , as we remarked above.", "We have seen that ${\\mathbb {Q}}(Y)^{W(E_7)} = k_0 ={\\mathbb {Q}}(\\lambda )$ .", "Since $\\Phi _\\lambda (X)$ is a monic polynomial with coefficients in ${\\mathbb {Q}}[\\lambda ]$ , we have that ${\\mathbb {Q}}[Y]$ is integral over ${\\mathbb {Q}}[\\lambda ]$ .", "Therefore ${\\mathbb {Q}}[Y]^{W(E_7)}$ is also integral over ${\\mathbb {Q}}[\\lambda ]$ , and contained in ${\\mathbb {Q}}(Y)^{W(E_7)} = k_0 = {\\mathbb {Q}}(\\lambda )$ .", "Since ${\\mathbb {Q}}[\\lambda ]$ is a polynomial ring, it is integrally closed in its ring of fractions.", "Therefore ${\\mathbb {Q}}[Y]^{W(E_7)} \\subset {\\mathbb {Q}}[\\lambda ]$ .", "We also know ${\\mathbb {Q}}[\\chi ] = {\\mathbb {Q}}[\\chi _1, \\dots , \\chi _7] \\subset {\\mathbb {Q}}[Y]^{W(E_7)}$ , since the $\\chi _j$ are invariant under the Weyl group.", "Therefore, we have ${\\mathbb {Q}}[\\chi ] \\subset {\\mathbb {Q}}[Y]^{W(E_7)} \\subset {\\mathbb {Q}}[\\lambda ]$ and Theorem REF , which says ${\\mathbb {Q}}[\\chi ] = {\\mathbb {Q}}[\\lambda ]$ , implies that all these three rings are equal.", "This completes the proof of Theorem REF .", "Remark 11 Note that this argument gives an independent proof of the fact that the ring of multiplicative invariants for $W(E_7)$ is a polynomial ring over $\\chi _1, \\dots , \\chi _7$ .", "See [1] or [7] for the classical proof that the Weyl-orbit sums of a set of fundamental weights are a set of algebraically independent generators of the multiplicative invariant ring; from there to the fundamental characters is an easy exercise.", "Remark 12 Now that we have found the explicit relation between the Weierstrass coefficients and the fundamental characters, we may go back and explore the “genericity condition” for this family to have Mordell-Weil lattice $E_7^*$ .", "To do this we compute the discriminant of the cubic in $x$ , after completing the square in $y$ , and take the discriminant with respect to $t$ of the resulting polynomial of degree 10.", "A computation shows that this discriminant factors as the cube of a polynomial $A(\\lambda )$ (which vanishes exactly when the family has a fiber of additive reduction, generically type $\\mathrm {II}$ ), times a polynomial $B(\\lambda )$ , whose zero locus corresponds to the occurrence of a reducible multiplicative fiber.", "In fact, we calculate (for instance, by evaluating the split case), that $B(\\lambda )$ is the product of $(e^{\\alpha } - 1)$ , where $\\alpha $ runs over 126 roots of $E_7$ .", "We deduce by further analyzing the type $\\mathrm {II}$ case that the condition to have Mordell-Weil lattice $E_7^*$ is that $\\prod (e^{\\alpha } - 1) = \\Psi _\\lambda (1) \\ne 0.$ Note that this is essentially the expression which occurs in Weyl's denominator formula.", "In addition, the condition for having only multiplicative fibers is that $\\Psi _\\lambda (1)$ and $A(\\lambda )$ both be non-zero.", "Finally, the proof of Theorem REF follows immediately from the discussion in [15], [16] (compare [16] for the additive reduction case)." ], [ "Results", "Finally, we show a multiplicative excellent family for the Weyl group of $E_8$ .", "It is given by the Weierstrass equation $y^2 = x^3 + t^2 \\, x^2 + (p_0 + p_1 t + p_2 t^2) \\,x + (q_0 + q_1 t + q_2 t^2 + q_3 t^3 + q_4 t^4 + t^5).$ For generic $\\lambda = (p_0, \\dots , p_2, q_0, \\dots , q_4)$ , this rational elliptic surface $X_\\lambda $ has no reducible fibers, only nodal $I_1$ fibers at twelve points, including $t = \\infty $ .", "We will use the specialization map at $\\infty $ .", "The Mordell-Weil lattice $M_\\lambda $ is isomorphic to the lattice $E_8$ .", "Any rational elliptic surface with a multiplicative fiber of type $I_1$ may be put in the above form (over a small degree algebraic extension of the base field), after a fractional linear transformation of $t$ and Weierstrass transformations of $x,y$ .", "Lemma 13 The smooth part of the special fiber is isomorphic to the group scheme ${\\mathbb {G}}_m$ .", "The identity component is the non-singular part of the curve $y^2 = x^3 + x^2$ .", "A section of height 2 has $x$ - and $y$ -coordinates polynomials of degrees 2 and 3 respectively, and its specialization at $t = \\infty $ may be taken as $\\lim _{t \\rightarrow \\infty } (y + tx)/(y-tx) \\in k^*$ .", "The proof of the lemma is similar to that in the $E_7$ case (and simpler!", "), and we omit it.", "There are 240 sections of minimal height 2, with $x$ and $y$ coordinates of the form $x &= gt^2 + at + b \\\\y &= ht^3 + ct^2 + dt + e.$ Under the identification with ${\\mathbb {G}}_m$ of the special fiber of the Néron model, this section goes to $(h+g)/(h-g)$ .", "Substituting the above formulas for $x$ and $y$ into the Weierstrass equation, we get the following system of equations.", "$h^2 &= g^3 + g^2 \\\\2ch &= 3ag^2+2ag+1 \\\\2dh+c^2 &= q_4+g p_2+3 b g^2+(2 b+3 a^2) g+a^2 \\\\2eh+2cd &= q_3+ap_2+gp_1+6abg+2ab+a^3 \\\\2ce+d^2 &= q_2+bp_2+ap_1+gp_0+3b^2g+b^2+3a^2b \\\\2de &= q_1+b p_1+a p_0+3 a b^2 \\\\e^2 &= q_0+b p_0+b^3.$ Setting $h = gu$ , we eliminate other variables to obtain an equation of degree 240 for $u$ .", "Finally, substituting in $u = (v + 1)/(v-1)$ , we get a monic reciprocal equation $\\Phi _\\lambda (X) = 0$ satisfied by $v$ , with coefficients in ${\\mathbb {Z}}[\\lambda ] = {\\mathbb {Z}}[p_0, \\dots , p_2, q_0, \\dots ,q_4]$ .", "We have $\\Phi _\\lambda (X) = \\prod _{i=1}^{240} (X - s(P)) = X^{240} + \\epsilon _1 X^{239} + \\dots + \\epsilon _1 X + \\epsilon _0,$ where $P$ ranges over the 240 minimal sections of height 2.", "It is clear that $a,\\dots ,h$ are rational functions of $v$ , with coefficients in $k_0$ .", "We have a Galois representation on the Mordell-Weil lattice $\\rho _\\lambda : \\mathrm {Gal}(k/k_0) \\rightarrow \\mathrm {Aut}(M_\\lambda ) \\cong \\mathrm {Aut}(E_8).$ Here $\\mathrm {Aut}(E_8) \\cong W(E_8)$ , the Weyl group of type $E_8$ .", "The splitting field of $M_\\lambda $ is the fixed field $k_\\lambda $ of $\\mathrm {Ker}(\\rho _\\lambda )$ .", "By definition, $\\mathrm {Gal}(k_\\lambda /k_0) \\cong \\textrm {Im}(\\rho _\\lambda )$ .", "The splitting field $k_\\lambda $ is equal to the splitting field of the polynomial $\\Phi _\\lambda (X)$ over $k_0$ , since the Mordell-Weil group is generated by the 240 sections of smallest height $P_i = (g_it^2 + a_i t + b_i, h_i t^3 + c_i t^2 + d_i t +e_i)$ .", "We also have that $k_\\lambda = k_0(P_1, \\dots , P_{240}) = k_0(v_1, \\dots , v_{240}).$ Theorem 14 Assume that $\\lambda $ is generic over ${\\mathbb {Q}}$ , i.e the coordinates $p_0, \\dots , q_4$ are algebraically independent over ${\\mathbb {Q}}$ .", "Then $\\rho _\\lambda $ induces an isomorphism $\\mathrm {Gal}(k_\\lambda / k_0) \\cong W(E_8)$ .", "The splitting field $k_\\lambda $ is a purely transcendental extension of ${\\mathbb {Q}}$ , isomorphic to the function field ${\\mathbb {Q}}(Y)$ of the toric hypersurface $Y \\subset {\\mathbb {G}}_m^9$ defined by $s_1 \\dots s_8 =r^3$ .", "There is an action of $W(E_8)$ on $Y$ such that ${\\mathbb {Q}}(Y)^{W(E_8)} = k_\\lambda ^{W(E_8)} = k_0$ .", "The ring of $W(E_8)$ -invariants in the affine coordinate ring ${\\mathbb {Q}}[Y] = {\\mathbb {Q}}[s_i, r, 1/s_i, 1/r]/(s_1 \\dots s_8 - r^3) \\cong {\\mathbb {Q}}[s_1, \\dots , s_7, r, s_1^{-1}, \\dots , s_7^{-1}, r^{-1}]$ is equal to the polynomial ring ${\\mathbb {Q}}[\\lambda ]$ : ${\\mathbb {Q}}[Y]^{W(E_8)} = {\\mathbb {Q}}[\\lambda ] = {\\mathbb {Q}}[p_0, p_1, p_2, q_0, q_1, q_2, q_3, q_4].$ As in the $E_7$ case, we prove an explicit, invertible polynomial relation between the Weierstrass coefficients $\\lambda $ and the fundamental characters for $E_8$ .", "Let $V_1, \\dots , V_8$ be the fundamental representations of $E_8$ , and $\\chi _1, \\dots , \\chi _8$ their characters as labelled below.", "(0,0)–(6,0); (2,0)–(2,1); [white] (0,0) circle (0.1); [white] (1,0) circle (0.1); [white] (2,0) circle (0.1); [white] (3,0) circle (0.1); [white] (4,0) circle (0.1); [white] (5,0) circle (0.1); [white] (6,0) circle (0.1); [white] (2,1) circle (0.1); (0,0) circle (0.1); (1,0) circle (0.1); (2,0) circle (0.1); (3,0) circle (0.1); (4,0) circle (0.1); (5,0) circle (0.1); (6,0) circle (0.1); (2,1) circle (0.1); (0,-0.2) [below] node1; (1,-0.2) [below] node3; (2,-0.2) [below] node4; (3,-0.2) [below] node5; (4,-0.2) [below] node6; (5,-0.2) [below] node7; (6,-0.2) [below] node8; (2.2,1) [right] node2; Again, for the set of generators of $E_8$ , we choose (as in [14]) vectors $v_1, \\dots , v_8, u$ with $\\sum v_i = 3 u$ and let $s_i$ correspond to $v_i$ and $r$ to $u$ , so that $\\prod s_i = r^3$ .", "The 240 roots of $\\Phi _\\lambda (X)$ are given by $s_i, \\frac{1}{s_i} \\textrm { for } 1 \\le i \\le 8, & \\quad \\frac{s_i}{s_j} \\textrm { for } 1 \\le i \\ne j \\le 8, \\\\\\frac{s_i s_j}{r}, \\frac{r}{s_i s_j} \\textrm { for } 1 \\le i < j \\le 8 & \\quad \\textrm { and } \\quad \\frac{s_i s_j s_k}{r}, \\frac{r}{s_i s_j s_k} \\textrm { for } 1 \\le i < j < k \\le 8.$ The characters $\\chi _1, \\dots , \\chi _7$ lie in the ring of Laurent polynomials ${\\mathbb {Q}}[s_i, r, 1/s_i, 1/r]$ , and are invariant under the multiplicative action of the Weyl group on this ring of Laurent polynomials.", "The $\\chi _i$ may be explicitly computed using the software LiE, as indicated in Section and the auxiliary files.", "Theorem 15 For generic $\\lambda $ over ${\\mathbb {Q}}$ , we have ${\\mathbb {Q}}[\\chi _1, \\dots , \\chi _8] = {\\mathbb {Q}}[p_0, p_1, p_2, q_0, q_1, q_2, q_3, q_4].$ The transformation between these sets of generators is $\\chi _1 &= -1600 q_4+1536 p_2+3875 \\\\\\chi _2 &= 2 (-45600 q_4+12288 q_3+40704 p_2-16384 p_1+73625) \\\\\\chi _3 &= 64 (14144 q_4^2-72 (384 p_2+1225) q_4+11200 q_3-4096 q_2+13312 p_2^2 \\\\& \\quad +87072 p_2-17920 p_1 +16384 p_0+104625 ) \\\\\\chi _4 &= -91750400 q_4^3+12288 (25600 p_2+222101) q_4^2 -256 (4530176 q_3-65536 q_2 \\\\& \\quad +1392640 p_2^2+21778944 p_2-8159232 p_1+2621440 p_0 +34773585) q_4 \\\\& \\quad + 32 ( 4718592 q_3^2+ 384 (80896 p_2-32768 p_1+225379) q_3-29589504 q_2 \\\\& \\quad +30408704 q_1 -33554432 q_0+4194304 p_2^3+88129536 p_2^2 \\\\& \\quad -64 (876544 p_1-262144 p_0-4399923) p_2+8388608 p_1^2-133996544 p_1 \\\\& \\quad +65175552 p_0+215596227 )$ $\\chi _5 &= 24760320 q_4^2-64 (106496 q_3+738816 p_2-163840 p_1+2360085) q_4 \\\\& \\quad +12288 (512 p_2+4797) q_3-30670848 q_2+16777216 q_1+20250624 p_2^2 \\\\&\\quad -512 (16384 p_1-235911) p_2-45154304 p_1+13631488 p_0+146325270 \\\\\\chi _6 &= 110592 q_4^2-1536 (128 p_2+1235) q_4+724992 q_3-262144 q_2+65536 p_2^2 \\\\& \\quad +1062912 p_2-229376 p_1+2450240 \\\\\\chi _7 &= -4 (3920 q_4-1024 q_3-1152 p_2-7595) \\\\\\chi _8 &= -8 (8 q_4-31).$ Remark 16 We omit the inverse for conciseness here; it is easily computed in a computer algebra system and is available in the auxiliary files.", "Remark 17 As before, our explicit formulas are compatible with those in [3].", "Also, the proof of Theorem REF gives another proof of the fact that the multiplicative invariants for $W(E_8)$ are freely generated by the fundamental characters (or by the orbit sums of the fundamental weights).", "Example 18 Let $\\mu = (2 \\cdot 3 \\cdot 5 \\cdot 7 \\cdot 11 \\cdot 13 \\cdot 17 \\cdot 19) = 9699690.$ $q_4 &= -2243374456559366834339/(2^5 \\cdot \\mu ^2) \\\\q_3 &= 430800343129403388346226518246078567/(2^{11} \\cdot \\mu ^3) \\\\q_2 &= 72555101947649011127391733034984158462573146409905769/(2^{22}\\cdot 3^2 \\cdot \\mu ^4) \\\\q_1 &= (-12881099305517291338207432378468368491584063772556981164919245 \\\\& \\qquad 30489)/(2^{29} \\cdot 3 \\cdot \\mu ^5) \\\\q_0 &= (8827176793323619929427303381485459401911918837196838709750423283 \\\\& \\qquad 443360357992650203)/(2^{42} \\cdot 3^3 \\cdot \\mu ^6) \\\\p_2 &= 146156773903879871001810589/(2^9 \\cdot 3 \\cdot \\mu ^2) \\\\p_1 &= -24909805041567866985469379779685360019313/(2^{20} \\cdot \\mu ^3) \\\\p_0 &= 14921071761102637668643191215755039801471771138867387/(2^{23} \\cdot 3 \\cdot \\mu ^4) \\\\$ These values give an elliptic surface for which we have $r = 2, s_1 =3, s_2 = 5, s_3 = 7, s_4 = 11, s_5 = 13, s_6 = 17, s_7 = 19$ , the simplest choice of multiplicatively independent elements.", "Here, the specializations of a basis are given by $v \\in \\lbrace 3,5,7,11,13,17,19,15/2\\rbrace $ .", "Once again, we list the $x$ -coordinates of the corresponding sections, and leave the remainder of the verification to the auxiliary files.", "$x(P_1) &= 3 t^2 - (99950606190359/620780160)t \\\\& \\quad + 4325327557647488120209649813/2642523476911718400 \\\\x(P_2) &= (5/4)t^2 - (153332163637781/1655413760)t \\\\& \\quad + 5414114237697608646836821/5138596941004800 \\\\x(P_3) &= (7/9)t^2 - (203120672689603/2793510720)t \\\\& \\quad + 6943164348569130636788638639/7927570430735155200 \\\\x(P_4) &= (11/25)t^2 - (8564057914757/147804800)t \\\\& \\quad + 115126372233675800396600989/155442557465395200$ $x(P_5) &= (13/36)t^2 - (347479008951469/6385167360)t \\\\& \\quad + 157133607680949617374030405417/221971972060584345600 \\\\x(P_6) &= (17/64)t^2 - (1327421017414859/26486620160)t \\\\& \\quad + 5942419292933021418457517303/8901131711702630400 \\\\x(P_7) &= (19/81)t^2 - (489830985359431/10056638592)t \\\\& \\quad + 46685137201743696441477454951/71348133876616396800 \\\\x(P_8) &= (120/169)t^2 - (30706596009257/440806080)t \\\\& \\quad + 76164443074828743662165466409/55823308449760051200.$ Example 19 The value $\\lambda = \\lambda _0 :=(1,1,1,1,1,1,1,1)$ gives rise to an explicit polynomial $g(X) =\\Phi _{\\lambda _0}(X)$ , for which we can show that the Galois group is $W(E_8)$ , as follows.", "The reduction of $g(X)$ modulo 79 shows that $\\mathrm {Frob}_{79}$ has cycle decomposition of type $(4)^2(8)^{29}$ , and similarly, $\\mathrm {Frob}_{179}$ has cycle decomposition of type $(15)^{16}$ .", "We deduce, as in [4] or [17], that the Galois group is the entire Weyl group.", "Since the coefficients of $g(X)$ are large, we do not display it here, but it is included in the auxiliary files.", "As in the case of $E_7$ , we can also describe degenerations of this family of rational elliptic surfaces $X_\\lambda $ by the method of “vanishing roots”, where we drop the genericity assumption, and consider the situation where the elliptic fibration might have additional reducible fibers.", "Let $\\psi : Y \\rightarrow {\\mathbb {A}}^8$ be the finite surjective morphism associated to ${\\mathbb {Q}}[p_0, \\dots , q_4] \\hookrightarrow {\\mathbb {Q}}[Y] \\cong {\\mathbb {Q}}[s_1, \\dots , s_7, r, s_1^{-1}, \\dots , s_7^{-1}, r^{-1}].$ For $\\xi = (s_1, \\dots , s_8, r) \\in Y$ , let the multiset $\\Pi _\\xi $ consist of the 240 elements $s_i$ and $1/s_i$ (for $1 \\le i \\le 8$ ), $s_i/s_j$ (for $1 \\le i \\ne j \\le 8$ ), $s_i s_j/r$ and $r/(s_is_j)$ (for $1 \\le i < j \\le 8$ ) and $s_i s_j s_k/r$ and $r/(s_i s_js_k)$ for $1 \\le i < j < k \\le 8$ , corresponding to the 240 roots of $E_6$ .", "Let $2\\nu (\\xi )$ be the number of times 1 appears in $\\Pi _\\xi $ , which is also the multiplicity of 1 as a root of $\\Phi _\\lambda (X)$ , with $\\lambda = \\psi (\\xi )$ .", "We call the associated roots of $E_8$ the vanishing roots, in analogy with vanishing cycles in the deformation of singularities.", "By abuse of notation we label the rational elliptic surface $X_\\lambda $ as $X_\\xi $ .", "Theorem 20 The surface $X_\\xi $ has new reducible fibers (necessarily at $t \\ne \\infty $ ) if and only if $\\nu (\\xi ) > 0$ .", "The number of roots in the root lattice $T_{\\textrm {new}}$ is equal to $2\\nu (\\xi )$ , where $T_{\\textrm {new}} := \\oplus _{v \\ne \\infty } T_v$ is the new part of the trivial lattice.", "We may use this result to produce specializations with trivial lattice corresponding to most of the entries of [8], and a nodal fiber.", "Below, we list those types which are not already covered by [11], [13] or our examples for the $E_7$ case, which have an $I_2$ fiber.", "$\\begin{array}{cccl}\\textrm {Type in \\cite {OS}} & \\textrm {Fibral lattice} & \\textrm {MW group} & \\lbrace s_1, \\dots , s_6, r \\rbrace \\\\1 & 0 & E_8 & 3,5,7,11,13,17,19,2 \\\\5 & A_3 & D_5^* & 2,2,2,2,5,7,11,3 \\\\8 & A_4 & A_4^* & 2,2,2,2,2,5,7,3 \\\\15 & A_5 & A_2^* \\oplus A_1^* & 2,2,2,2,2,2,5,3 \\\\16 & D_5 & A_3^* & 2,3,3,3,3,3,5,18 \\\\25 & A_6 & \\frac{1}{7} \\footnotesize { \\left( \\begin{array}{cc} 4 & -1 \\\\ -1 & 2 \\end{array} \\right) } & 2,2,2,2,2,2,2,3 \\\\26 & D_6 & A_1^{*2} & 2,3,3,3,3,3,3,18 \\\\35 & A_3^2 & A_1^{*2} \\oplus {\\mathbb {Z}}/2{\\mathbb {Z}}& 2,-1/2,3,3,3,1,1,-3\\end{array}$ $\\begin{array}{cccl}\\textrm {Type in \\cite {OS}} & \\textrm {Fibral lattice} & \\textrm {MW group} & \\lbrace s_1, \\dots , s_6, r \\rbrace \\\\36 & A_3^2 & \\langle 1/4 \\rangle & 8,8,8,8,27,27,27,1296 \\\\43 & E_7 & A_1^* & 2,2,2,2,2,2,2,8 \\\\44 & A_7 & A_1^* \\oplus {\\mathbb {Z}}/2{\\mathbb {Z}}& 2,2,2,2,2,2,2,-8 \\\\45 & A_7 & \\langle 1/8 \\rangle & 8,8,8,8,8,8,8,256 \\\\46 & D_7 & \\langle 1/4 \\rangle & 2,4,4,4,4,4,4,32 \\\\54 & A_3 \\oplus D_4 & \\langle 1/4 \\rangle \\oplus {\\mathbb {Z}}/2{\\mathbb {Z}}& 2,-1,-1,-1,-1,1,1,2 \\\\55 & A_3 \\oplus A_4 & \\langle 1/20 \\rangle & 16,16,16,16,32,32,32,4096 \\\\62 & E_8 & 0 & 1,1,1,1,1,1,1,1 \\\\63 & A_8 & {\\mathbb {Z}}/3{\\mathbb {Z}}& 1,1,1,1,1,1,1, \\zeta _3 \\\\64 & D_8 & {\\mathbb {Z}}/2{\\mathbb {Z}}& 1,1,1,1,1,1,1,-1 \\\\67 & A_4^2 & {\\mathbb {Z}}/5{\\mathbb {Z}}& 1,1,1,1,\\zeta _5,\\zeta _5,\\zeta _5,\\zeta _5^3 \\\\72 & A_3 \\oplus D_5 & {\\mathbb {Z}}/4{\\mathbb {Z}}& 1,1,1,I,I,I,I,-I\\end{array}$ Here $\\zeta _3$ , $I$ and $\\zeta _5$ are primitive third, fourth and fifth roots of unity.", "Remark 21 As before, for the examples in lines 63, 67 and 72 of the table, one can show that it is not possible to define a rational elliptic surface over ${\\mathbb {Q}}$ in the form we have assumed, such that all the specializations $s_i, r$ are rational.", "However, there do exist examples with all sections defined over ${\\mathbb {Q}}$ , not in the chosen Weierstrass form.", "The surface with Weierstrass equation $y^2 + xy + t^3 y = x^3$ has a 3-torsion point $(0, 0)$ and a fiber of type $I_9$ .", "It is an example of type 63.", "The surface with Weierstrass equation $y^2 + (t+1)xy + ty = x^3 + tx^2$ has a 5-torsion section $(0,0)$ and two fibers of type $I_5$ .", "It is an example of type 67.", "The surface with Weierstrass equation $y^2 + txy + \\frac{t^2(t - 1)}{16} y = x^3 + \\frac{t(t - 1)}{16}x^2$ has a 4-torsion section $(0,0)$ , and two fibers of types $I_4$ and $I_1^*$ .", "It is an example of type 72.", "Remark 22 Our tables and the one in [13] cover all the cases of [8], except lines 9, 27 and 73 of the table, with trivial lattice $D_4$ , $E_6$ and $D_4^2$ respectively.", "Since these have fibers with additive reduction, examples for them may be directly constructed using the families in [11].", "For instance, the elliptic surface $y^2 = x^3 - x t^2$ has two fibers of type $I_0^*$ and Mordell-Weil group $({\\mathbb {Z}}/2{\\mathbb {Z}})^2$ .", "This covers line 73 of the table.", "For the other two cases, we refer the reader to the original examples of additive reduction in Section 3 of [11]." ], [ "Proofs", "The proof proceeds analogously to the $E_7$ case: with two differences: we only have one polynomial $\\Phi _\\lambda (X)$ to work with (as opposed to having $\\Phi _\\lambda (X)$ and $\\Psi _\\lambda (X)$ ), and the equations are a good deal more complicated.", "We first write down the relation between the coefficients $\\epsilon _i$ , $1\\le i \\le 9$ , and the fundamental invariants $\\chi _j$ ; as before, we postpone the proofs to the auxiliary files and outline the idea in Section .", "Second, we write down the coefficients $\\epsilon _i$ in terms of $\\lambda = (p_0, \\dots , p_2, q_0, \\dots , q_4)$ ; see Section for an explanation of how this is carried out.", "In the interest of brevity, we do not write out either of these sets of equations, but relegate them to the auxiliary computer files.", "Finally, setting the corresponding expressions equal to each other, we obtain a set of equations $(1)$ through $(9)$ .", "To solve these equations, proceed as follows: first use $(1)$ through $(5)$ to solve for $q_0, \\dots , q_4$ in terms of $\\chi _j$ and $p_0,p_1, p_2$ .", "Substituting these in to the remaining equations, we obtain $(6^{\\prime })$ through $(9^{\\prime })$ .", "These have low degree in $p_0$ , which we eliminate, obtaining equations of relatively small degrees in $p_1$ and $p_2$ .", "Finally, we take resultants with respect to $p_1$ , obtaining two equations for $p_2$ , of which the only common root is the one listed above.", "Working back, we solve for all the other variables, obtaining the system above and completing the proof of Theorem REF .", "The deduction of Theorem REF now proceeds exactly as in the case of $E_7$ .", "Remark 23 As in the $E_7$ case, once we find the explicit relation between the Weierstrass coefficients and the fundamental characters, we may go back and explore the “genericity condition” for this family to have Mordell-Weil lattice isomorphic to $E_8$ .", "To do this we compute the discriminant of the cubic in $x$ , after completing the square in $y$ , and take the discriminant with respect to $t$ of the resulting polynomial of degree 11.", "A computation shows that this discriminant factors as the cube of a polynomial $A(\\lambda )$ (which vanishes exactly when the family has a fiber of additive reduction, generically type $\\mathrm {II}$ ), and the product of $(e^{\\alpha } - 1)$ , where $\\alpha $ runs over minimal vectors of $E_8$ .", "Again, the genericity condition to have Mordell-Weil lattice exactly $E_8$ is just the nonvanishing of $\\Phi _\\lambda (1) = \\prod (e^\\alpha - 1),$ the expression which occurs in the Weyl denominator formula.", "Furthermore, the condition to have only multiplicative fibers is that $\\Phi _\\lambda (1)A(\\lambda ) \\ne 0$ .", "As before, the proof of Theorem REF follows immediately from the results of [15], [16], by degeneration from a flat family." ], [ "Resultants, Interpolation and Computations", "We now explain a computational aid, used in obtaining the equations expressing the coefficients of $\\Phi _\\lambda $ (for $E_8$ ) or $\\Psi _\\lambda $ (for $E_7$ ) in terms of the Weierstrass coefficients of the associated family of rational elliptic surfaces.", "We illustrate this using the system of equations obtained for sections of the $E_8$ family: $h^2 &= g^3 + g^2 \\\\2ch &= 3ag^2+2ag+1 \\\\c^2 + 2dh &= q_4+gp_2+3bg^2+(2b+3a^2)g+a^2 \\\\2eh + 2cd &= q_3+ap_2+gp_1+6abg+2ab+a^3 \\\\2ce + d^2 &= q_2+bp_2+ap_1+gp_0+3b^2g+b^2+3a^2b \\\\2de &= q_1+bp_1+ap_0+3ab^2 \\\\e^2 &= q_0 + b p_0 + b^3.$ Setting $h = gu$ and solving the first equation for $g$ we have $g =u^2 - 1$ .", "We solve the next three equations for $c,d,e$ respectively.", "This leaves us with three equations $R_1(a,b,u) =R_2(a,b,u) = R_3(a,b,u) = 0$ .", "These have degrees $2,2,3$ respectively in $b$ .", "Taking the appropriate linear combination of $R_1$ and $R_2$ gives us an equation $S_1(a,b,u) = 0$ which is linear in $b$ .", "Similarly, we may use $R_1$ and $R_3$ to obtain another equation $S_2(a,b,u) = 0$ , linear in $b$ .", "We write $S_1(a,b,u) &= s_{11}(a,u) b + s_{10} (a,u) \\\\S_2(a,b,u) &= s_{21}(a,u) b + s_{20} (a,u) \\\\R_1(a,b,u) &= r_2 (a,u) b^2 + r_1 (a,u) b + r_0 (a,u).$ The resultant of the first two polynomials gives us an equation $T_1 (a,u) = s_{11} s_{20} - s_{10} s_{21} = 0$ while the resultant of the first and third gives us $T_2 (a,u) = r_2 s_{10}^2 - r_1 s_{10} s_{11} + r_0 s_{11}^2 = 0.$ Finally, we substitute $u =(v+1)/(v-1)$ throughout, obtaining two equations $\\tilde{T}_1(a,v) = 0$ and $\\tilde{T}_2(a,v) = 0$ .", "Next, we would like to compute the resultant of $\\tilde{T}_1(a,v)$ and $\\tilde{T}_2(a,v)$ , which have degrees 8 and 9 with respect to $a$ , to obtain a single equation satisfied by $v$ .", "However, the polynomials $\\tilde{T}_1$ and $\\tilde{T}_2$ are already fairly large (they take a few hundred kilobytes of memory), and since their degree in $a$ is not too small, it is beyond the current reach of computer algebra systems such as gp/PARI or Magma to compute their resultant.", "It would take too long to compute their resultant, and another issue is that the resultant would take too much memory to store, certainly more than is available on the authors' computer systems (for instance, it would take more than 16GB of memory).", "To circumvent this issue, what we shall do is to use several specializations of $\\lambda $ in ${\\mathbb {Q}}^8$ .", "Once we specialize, the polynomials take much less space to store, and the computations of the resultants becomes tremendously easier.", "Since the resultant can be computed via the Sylvester determinant $\\left|\\begin{array}{ccccccccccc}a_8 & \\dots & a_2 & a_1 & a_0 & 0 & 0 & \\dots & 0 \\\\0 & a_8 & \\dots & a_2 & a_1 & a_0 & 0 & \\dots & 0 \\\\\\vdots & \\ddots & \\ddots & & & \\ddots & \\ddots & \\ddots & \\vdots \\\\0 & \\dots & 0 & a_8 & \\dots & a_2 & a_1 & a_0 & 0 \\\\0 & \\dots & 0 & 0 & a_8 & \\dots & a_2 & a_1 & a_0 \\\\b_9 & b_8 & \\dots & b_2 & b_1 & b_0 & 0 & \\dots & 0 \\\\0 & b_9 & b_8 & \\dots & b_2 & b_1 & b_0 & \\ddots & \\vdots \\\\\\vdots & \\ddots & \\ddots & & & & \\ddots & \\ddots & 0 \\\\0 & \\dots & 0 & b_9 & b_8 & \\dots & b_2 & b_1 & b_0\\end{array}\\right|$ where $\\tilde{T}_1(a,v) = \\sum a_i(v) a^i$ and $\\tilde{T}_2(a,v) =\\sum b_i(v) a^i$ , we see that the resultant is a polynomial $Z(v) =\\sum z_i v^i$ with coefficients $z_i$ being polynomials in the coefficients of the $a_i$ and the $b_j$ , which happen to be elements of ${\\mathbb {Q}}[\\lambda ]$ (recall that $\\lambda = (p_0, \\dots , p_2, q_0, \\dots ,q_4)$ ).", "Furthermore, we can bound the degrees $m_i(j)$ of $z_i(v)$ with respect to the $j$ 'th coordinate of $\\lambda $ , by using explicit bounds on the multidegrees of the $a_i$ and $b_i$ .", "Therefore, by using Lagrange interpolation (with respect to the eight variables $\\lambda _j$ ) we can reconstruct $z_i(v)$ from its specializations for various values of $\\lambda $ .", "The same method lets us show that $Z(v)$ is divisible by $v^{22}$ (for instance, by showing that $z_0$ through $z_{21}$ are zero), and also by $(v+1)^{80}$ (by first shifting $v$ by 1 and then computing the Sylvester determinant, and proceeding as before), as well as by $(v^2 + v + 1)^8$ (this time, using cube roots of unity).", "Finally, it is clear that $Z(v)$ is divisible by the square of the resultant $G(v)$ of $s_{11}$ and $s_{10}$ with respect to $a$ .", "Removing these extraneous factors, we get a polynomial $\\Phi _\\lambda (v)$ which is monic and reciprocal of degree 240.", "We compute its top few coefficients by this interpolation method.", "Finally, we note the interpolation method above in fact is completely rigorous.", "Namely, let $\\epsilon _i(\\lambda )$ be the coefficient of $v^i$ in $\\Phi _\\lambda (v)$ , with bounds $(m_1, \\dots , m_8)$ for its multidegree, and $\\epsilon ^{\\prime }_i(\\lambda )$ is the putative polynomial we have computed using Lagrange interpolation on a set $L_1 \\times \\dots \\times L_8$ , where $L_i = \\lbrace \\ell _{i,0}, \\dots , \\ell _{i,m_i} \\rbrace $ for $1 \\le i \\le 8$ are sets of integers chosen generically enough to ensure that $G(v)$ has the correct degree and that $Z(v)$ is not divisible by any higher powers of $v$ , $v+1$ or $v^2 + v + 1$ than in the generic case.", "Then since $\\epsilon _j(\\ell _{1, i_1}, \\dots , \\ell _{8, i_8}) = \\epsilon ^{\\prime }_j(\\ell _{1,i_1}, \\dots , \\ell _{8, i_8})$ for all choices of $i_1, \\dots , i_8$ , we see that the difference of these polynomials must vanish." ], [ "Representation theory, and some identities in Laurent\npolynomials", "Finally, we demonstrate how to deduce the identities relating the coefficients of $\\Phi _{E_7, \\lambda }(X)$ or $\\Psi _{E_7, \\lambda }(X)$ to the fundamental characters for $E_7$ (and similarly, the coefficients of $\\Phi _{E_8, \\lambda (X)}$ to the fundamental characters of $E_8$ ).", "Conceptually, the simplest way to do this is to express the alternating powers of the 56-dimensional representation $V_7$ or the 133-dimensional representation $V_1$ in terms of the fundamental modules of $E_7$ and their tensor products.", "We know that the character $\\chi _1$ of $V_1$ is $7 + \\sum e^{\\alpha }$ , where the sum is over the 126 roots of $E_7$ .", "Therefore we have $(-1) \\eta _1 = \\chi _1 -7$ .", "For the next example, we consider $\\Lambda ^2 V_1 = V_3 \\oplus V_1$ .", "This gives rise to the equation $\\eta _2 + 7 \\cdot (-1)\\eta _1 + {7 \\atopwithdelims ()2} = \\chi _3 + \\chi _1$ which gives the relation $\\eta _2 = \\chi _3 - 6 \\chi _1 + 28$ .", "A similar analysis can be carried out to obtain all the other identities used in our proofs, using the software LiE, available from http://www-math.univ-poitiers.fr/maavl/LiE/.", "A more explicit method is to compute the expressions for the $\\chi _i$ as Laurent polynomials in $s_1, \\dots , s_6, r$ (note that $s_7 =r^3/(s_1 \\dots s_6)$ ), and then do the same for the $\\epsilon _i$ or $\\eta _i$ .", "The latter calculation is simplified by computing the power sums $\\sum (e^{\\alpha })^i$ (for $\\alpha $ running over the smallest vectors of $E_7^*$ or $E_7$ ), for $1 \\le i \\le 7$ and then using Newton's formulas to convert to the elementary symmetric polynomials, which are $(-1)^i \\epsilon _i$ or $(-1)^i \\eta _i$ .", "Finally, we check the identities by direct computation in the Laurent polynomial ring (it may be helpful to clear out denominators).", "This method has the advantage that we obtain explicit expressions for the $\\chi _i$ (and then for $\\lambda $ by Theorem REF ) in terms of $s_1,\\dots , s_6, r$ , which may then be used to generate examples such as Example REF ." ], [ "Acknowledgements", "We thank David Vogan for very helpful comments regarding computations in representation theory.", "The computer algebra systems gp/PARI, Magma, Maxima and the software LiE were used in the calculations for this paper.", "We thank Matthias Schütt and the referees for a careful reading of the paper and for many helpful comments.", "The auxiliary computer files for checking our calculations are available from the arXiv.org e-print archive, where it is file number arXiv.org:1204.1531.", "To access the auxiliary files, download the source file for the paper.", "This will produce not only the file for this paper, but also the computer code.", "The file README.txt gives an overview of the various computer files involved." ] ]

[ [ "A procedural framework and mathematical analysis for solid sweeps" ], [ "Abstract Sweeping is a powerful and versatile method of designing objects.", "Boundary of volumes (henceforth envelope) obtained by sweeping solids have been extensively investigated in the past, though, obtaining an accurate parametrization of the envelope remained computationally hard.", "The present work reports our approach to this problem as well as the important problem of identifying self-intersections within the envelope.", "Parametrization of the envelope is, of course, necessary for its use in most current CAD systems.", "We take the more interesting case when the solid is composed of several faces meeting smoothly.", "We show that the face structure of the envelope mimics locally that of the solid.", "We adopt the procedural approach at defining the geometry in this work which has the advantage of being accurate as well as computationally efficient.", "The problem of detecting local self-intersections is central to a robust implementation of the solid sweep.", "This has been addressed by computing a subtle mathematical invariant which detects self-intersections, and which is computationally benign and requires only point queries." ], [ "Introduction", "In this paper we focus on the problem of computing an accurate parametrization of the boundary of the volume obtained by sweeping a solid in $\\mathbb {R}^3$ along a trajectory and that of detecting local self-intersections in the envelope.", "Sweeping is an operation of fundamental importance in geometric design.", "It has applications like numerically controlled machining verification [3], [4], [17] and robot motion planning [23], [24].", "There have been several approaches to computing the boundary of swept volumes in the past.", "The works [6], [21] formulate the problem using the rank deficiency of the Jacobian, [3], [4], [5] compute the envelope by solving sweep differential equations, [20] uses inverse-trajectories for deriving a point membership test for a point to belong to the envelope.", "In [22] the authors give a close approximation of the envelope by restricting the trajectories to piecewise screw motions.", "Despite the extensive research done in the past in this area, computing an accurate parametrization of the envelope has remained an unsolved problem due to known mathematical and computational difficulties [15].", "In this work we attempt to arrive at an accurate parametrization of the envelope through the procedural approach, which is an abstract way of defining surfaces and curves when closed form formulae are not available.", "The procedural paradigm exploits the fact that from the users point of view, a parametric surface is just a map from $\\mathbb {R}^2$ to $\\mathbb {R}^3$ and hence can be represented in a computer by a procedure which takes as input $(u,v) \\in \\mathbb {R}^2$ and returns $(x,y,z) \\in \\mathbb {R}^3$ .", "Higher order derivatives of the surface can be returned similarly.", "The definition of splines through the De Casteljau's algorithm is an example of procedural parametrization.", "The authors in [14] compute the intersection curve of two parametric surfaces by procedural approach.", "The second problem that we tackle in this paper is that of detecting local self-intersections.", "Self-intersections cause anomalies in the envelope.", "There have been rather few attempts at solving this problem in the past.", "The paper [2] proposes an efficient and robust method of detecting global and local self-intersections by checking whether the inverse-trajectory of a point intersects the solid.", "In the paper [7] global and local self-intersections are detected by computing intersection of curves of contact at discrete time steps.", "This has the disadvantage of being computationally expensive.", "In the paper [18] self-intersections are accurately quantified and detected but their method is limited to sweeping tools for NC machining verification.", "The method employed by [25] for detecting local self-intersections is based on point set data and could be computationally expensive.", "In [9] the author solves the problem of detecting local self-intersections for sweeping planar profiles.", "In this paper we propose a novel test for detecting local self-intersections which is based on a subtle mathematical invariant of the envelope.", "It has the advantage of being computationally efficient and requires only point queries.", "The paper is organized as follows.", "In Section  we describe the input to the sweeping algorithm, in Section  we discuss the overall framework for the computation of the envelope, in Sections  and we study the mathematical structure of the envelope and quantify local self-intersections, giving a test for detecting them, in Section  we analyse the case when the envelope is free from self-intersections.", "In Section  we describe the algorithm for computing the procedural parametrization of the envelope.", "We conclude the paper in Section ." ], [ "Preliminaries", "This section outlines the basic representational structures associated with the problem.", "Subsection REF describes the boundary representation of a solid which is typical to many CAD systems and subsequent sections, the basic inputs and outputs of the sweep algorithm.", "In Subsection REF we define the trajectory in $\\mathbb {R}^3$ along which the solid is swept.", "Next, in subsection REF we define the various mathematical sub-entities which make up the envelope." ], [ "Boundary representation of a solid", "Boundary representation, also known as Brep, is a popular and standard method of representing a `closed' solid $M$ by its boundary $\\partial M$ .", "The boundary $\\partial M$ separates the interior of $M$ from the exterior of $M$ .", "$\\partial M$ is represented using a set of faces, edges and vertices.", "See figure REF for a Brep of a solid where different faces are coloured differently.", "Faces meet in edges and edges meet in vertices.", "The Brep of a solid consists of two interconnected pieces of information, viz.", "the geometric and the topological.", "Geometric information: This consists of geometric entities, namely, vertices, edges and faces.", "A vertex is simply a point in $\\mathbb {R}^3$ .", "An edge is obtained by restricting the underlying parametric curve by a pair of vertices.", "A parametric curve in $\\mathbb {R}^3$ is a continuous map $\\gamma : \\mathbb {R} \\rightarrow \\mathbb {R}^3$ .", "The curve $\\gamma $ is called regular at $s_0 \\in \\mathbb {R}$ if $\\gamma $ is differentiable and $\\frac{d \\gamma }{ds}|_{s_0} \\ne \\bar{0}$ .", "Here $s$ is the parameter of the curve.", "An edge is derived from the underlying curve by suitably restricting the parameter $s$ to an interval $[a,b]$ .", "Further, it is required that the edge (more precisely, the underlying curve) is regular at all points in the interval $[a,b]$ and devoid of self-intersections.", "Similarly, a face is obtained by restricting the underlying parametric surface by a set of edges.", "A parametric surface is a continuous map $S:\\mathbb {R}^2 \\rightarrow \\mathbb {R}^3$ .", "The surface $S$ is said to be regular at $(u_0,v_0) \\in \\mathbb {R}^2$ if $S$ is differentiable and $\\frac{\\partial S}{\\partial u}|_{(u_0,v_0)} \\in \\mathbb {R}^3$ and $\\frac{\\partial S}{\\partial v}|_{(u_0,v_0)} \\in \\mathbb {R}^3$ are linearly independent.", "Here $u$ and $v$ are the parameters of $S$ .", "A face is derived from a surface by suitably restricting the parameters $u$ and $v$ inside a `domain'.", "As expected, it is required that the face is regular at all points in the domain and devoid of self-intersection.", "Topological information: The topological/combinatorial information consists of spatial relationships between different geometric entities, i.e., the adjacency between faces, the incidence relationships between faces and edges and so on.", "In figure REF , for example, the orange and the green face are adjacent.", "Another important component is the orientation for each face, that is, a consistent choice of outward-normal for that face.", "The orientation of a regular face is a choice of a unit normal from amongst $\\frac{S_u \\times S_v}{\\Vert S_u \\times S_v \\Vert }$ and $-\\frac{S_u \\times S_v}{\\Vert S_u \\times S_v \\Vert }$ where $S$ is the underlying parametric surface.", "All the faces bounding the solid are oriented so that the unit normal at each point on each face is pointing towards the exterior of the solid.", "Conceptually, a Brep through its `global' topological information glues the `local' geometric entities which come equipped with associated mathematical parametrizations.", "Note that, the regularity assumptions on the geometric entities guarantee that the tangent space at every point on an edge or a face is of the right dimension.", "Typically, one also imposes higher-order `parametric' continuity requirements which are denoted by $C^k$ where $k$ refers to the order of continuity.", "For the sake of simplicity, throughout this paper, we will assume that the edges and faces bounding the solid are regular of class $C^k$ for some $k \\ge 2$ , i.e.", "the underlying parametrizations are twice differentiable with continuous second order derivatives.", "Note that, however, these do not rule out, e.g., adjacent faces meeting along sharp edges.", "Figure: A Brep of a solid" ], [ "A trajectory in $\\mathbb {R}^3$", "A trajectory in $\\mathbb {R}^3$ is a 1-parameter family of rigid motions in $\\mathbb {R}^3$ defined as follows.", "Definition 2.1 A trajectory in $\\mathbb {R}^3$ is specified by a map $h:[0,1] \\rightarrow (SO(3), \\mathbb {R}^3), h(t) = (A(t), b(t))$ where $ A(t) \\in SO(3) \\footnote {SO(3)=\\lbrace X \\mbox{ is a 3 $\\times $3 real matrix} |X^tX = I, det(X)=1 \\rbrace is the special orthogonal group, i.e.", "the group of rotational transforms.", "}, b(t) \\in \\mathbb {R}^3, A(0) = I, b(0) = 0$ .", "The parameter $t$ in this definition represents time.", "For technical convenience, we assume that $h$ is of class $C^k$ , for some $k \\ge 2$ ." ], [ "Boundary of the swept volume", "We begin by giving an intuitive description of the boundary of the swept volume.", "We will formalize these notions in Section .", "Let $M$ be a solid being swept along a given trajectory $h$ .", "By abuse of notation, a point in $M$ will mean a point in the interior of $M$ or on the boundary $\\partial M$ of $M$ .", "We denote by $M_t$ the position of $M$ at time $t \\in [0, 1]$ , i.e.", "$M_t = \\lbrace A(t)x + b(t) | x \\in M\\rbrace $ , and by $\\partial M_t$ , the boundary of $M_t$ .", "Then $\\displaystyle \\bigcup _{t \\in [0,1]} M_t$ is the volume swept by $M$ during this operation.", "Our goal is to compute the boundary of this swept volume as a Brep, which we will refer to as the envelope.", "For a fixed point $x \\in M$ , consider the trajectory of $x$ as the map $y:[0,1] \\rightarrow \\mathbb {R}^3$ given by $y(t)= A(t)x + b(t)$ .", "The trajectory of $x$ describes the motion $x$ in $\\mathbb {R}^3$ under the given trajectory $h$ .", "Clearly, if $x$ is in the interior of $M$ , no point in the image of the trajectory of $x$ can be on the envelope.", "Further, at a particular time instant $t_0$ , only a subset of points on $\\partial M_{t_0}$ will lie on the envelope.", "The union of such points for all $t_0 \\in [0,1]$ gives the final envelope.", "It is clear that, at a given time instant $t_0$ , only a part of $\\partial M_{t_0}$ is in `contact' with the envelope.", "To make this more clear, fix a point $x \\in \\partial M$ and the trajectory $y$ of $x$ .", "The derivative of the trajectory of $x$ at a given time instant $t_0$ , that is, $\\frac{dy}{dt}|_{t_0}$ gives the velocity of $x$ at $t_0$ .", "It is easy to show that (cf, Section ) $x$ (more precisely, $y(t_0)$ ) is in contact with the envelope at time $t_0$ only if the velocity of $x$ at $t_0$ is in the `tangent-space' of $\\partial M_{t_0}$ at $y(t_0)$ .", "In a generic situation, the set of points of $\\partial M_{t_0}$ which are in contact with the envelope will be the curve-of-contact.", "The union of these curves-of-contact is called the contact-set or the running envelope.", "Clearly, the total envelope and the contact-set are closely related.", "If all goes well, the envelope is obtained from the contact-set by `capping' it by appropriate parts of $M_0 $ and $M_1 $ , the object at times $t=0$ and $t=1$ .", "But all may not go well.", "The detection of anomalies is central to the use of the algorithm in industrial situations and is an important objective of this paper.", "Figure: A solid is swept along a helical trajectoryIn this section we briefly describe the overall framework for computing the sweep surface i.e.", "the envelope as a Brep.", "We continue using the notation from the previous section where $M$ denotes the Brep/solid and $h$ denotes the trajectory along which $M$ is swept.", "The naive approach to computing the envelope would be to discretize time, i.e, to construct a sequence $T=\\lbrace 0=t_1 ,\\ldots ,t_k =1 \\rbrace $ and constructing the approximate envelope as $E^{\\prime }=\\cup _i M_{t_i}$ , the union of the translates.", "The next step would be to construct a smooth version $E^{\\prime \\prime }$ of $E^{\\prime }$ above, by some fitting operation.", "However, this approach has several issues–(i) computation of $E^{\\prime }$ leads to unstable booleans of two very close-by objects, leading to sliver-faces, and (ii) the fit of $E^{\\prime \\prime }$ to the actual $E$ depends on a dense enough choice of $T$ which compounds problem (i) above.", "There are other options, but problems remain.", "In this work, we propose a novel approach based on the procedural paradigm (cf [11], [14]) which has gained ascendance in many numerical kernels, e.g., ACIS (cf [12]).", "We now describe the basic architecture for our algorithm.", "For this, we use a running example referred to in figure REF and figure REF .", "The object to be swept is $M$ as in figure REF , and the output contact-set is $\\mathcal {C}$ as shown in figure REF .", "The trajectory is roughly helical with a compounded rotation.", "A natural correspondence between the entities of $M$ and the entities of $\\mathcal {C}$ .", "Every point $p$ of the envelope comes from a curve of contact on $M_t $ , for some $t$ , and therefore belongs to some entity of $M$ , i.e., a vertex, edge or face.", "This sets up the correspondence between entities of $M$ and those of $\\mathcal {C}$ .", "The procedural approach attaches a common evaluation method to each such entity.", "Fig2 illustrates this correspondence.", "Faces of $\\mathcal {C}$ which are generated by a particular face of $M$ are shown in same colour.", "Curve-of-contact at time $t=0$ is shown imprinted on the solid in red.", "Along with the geometric definition of each entity, we must also construct the topological data to go with it.", "This data is constructed by observing that there is a local homeomorphism between a point on $\\mathcal {C}$ and a suitable point on $M$ .", "Accurate parametrizations of the geometric entities of $\\mathcal {C}$ with `time' as one of the central parameters.", "This is achieved through the procedural paradigm in which all key attributes/features of the geometric entities are made available through a set of associated procedures (cf [11], [14]).", "In our case, these procedures are based on Newton-Raphson solvers.", "This is the focus of Section .", "Topological and regularity analysis of $\\mathcal {C}$ .", "It is quite common to have a sweeping operation in which the resulting envelope/contact-set $\\mathcal {C}$ self-intersects.", "These self-intersections can be broadly classified into global and local self-intersections (see figure REF ).", "Once an accurate `local' parametrization (as in step 2) of the faces of $\\mathcal {C}$ is obtained, in principle, global self-intersections can be detected and dealt with by well-known (cf [14]) surface-surface intersection solvers.", "A more subtle mathematical issue is that of detecting singularities and local self-intersections.", "This is addressed in Sections  and .", "We are now in a position to define the scope of this paper.", "In this work we describe in detail tasks (ii) and (iii) described above, namely, detecting local self-intersections and obtaining a procedural parameterization of faces.", "The focus of Section  is task (i) in the interesting case when $M$ is composed of faces meeting smoothly.", "Other architectural aspects and solid-modelling implementation will be addressed in a later work." ], [ "Mathematical structure of the contact-set", "In this section we will study in detail the mathematical structure of the boundary of the volume obtained by sweeping the solid $M$ along the trajectory.", "For simplicity, we work with a single parametric surface patch $S$ and analyse the sweep of $S$ under the trajectory $h$ .", "As explained before, we assume that both $S$ and $h$ are regular of class $C^k$ for $k \\ge 2$ , and are devoid of self-intersections.", "In section , we lift the results of this section to the interesting case when $M$ is composed of several faces/surfaces meeting smoothly.", "For later use, we introduce the following notation: the tanget space to a manifold $X$ at a point $p \\in X$ will be denoted by $\\mathcal {T}_{X}(p)$ .", "We begin with the formal definition of the sweep map.", "Definition 4.1 Given $S$ and $h$ , the sweep is defined as a map $\\sigma : \\mathbb {R}^2 \\times [0,1] \\rightarrow \\mathbb {R}^3$ given by $\\sigma (u,v,t) = A(t)S(u,v) + b(t)$ .", "Here $u,v$ are the parameters of $S$ .", "The position of a point $S(u_0,v_0)$ on surface $S$ at time $t_0$ will be given by $\\sigma (u_0,v_0,t_0)=A(t_0)S(u_0,v_0)+b(t_0)$ and the velocity of the point $S(u_0,v_0)$ at time $t_0$ will be given by $V(u_0,v_0,t_0)=\\frac{\\partial \\sigma }{\\partial t} |_{(u_0,v_0,t_0)} = A^{\\prime }(t_0)S(u_0,v_0) + b^{\\prime }(t_0)$ , where $^{\\prime }$ denotes derivative with respect to $t$ .", "If $N(u_0,v_0)$ is the unit (outward) normal to $S$ at $(u_0,v_0)$ , then the unit normal to $S_{t_0}$ at $(u_0,v_0)$ is given by $\\hat{N} = A(t_0)N(u_0,v_0)$ where, $S_{t_0} = \\lbrace A(t_0)S(u,v) + b(t_0) | (u,v) \\in \\mathbb {R}^2 \\rbrace $ is the position of the surface at time instant $t_0$ .", "In order to formally define the contact-set, we look at the extended sweep in $\\mathbb {R}^4$ in which the fourth dimension is time [2].", "Definition 4.2 Given $S$ and $h$ , the extended sweep is defined as a map $\\hat{\\sigma } : \\mathbb {R}^2 \\times [0,1] \\rightarrow \\mathbb {R}^4$ given by $\\hat{\\sigma }(u,v,t) = (\\sigma (u,v,t), t)$ .", "Thus, the sweep $\\sigma $ is clearly the extended sweep $\\hat{\\sigma }$ composed with the projection map along the $t$ -dimension.", "Denoting partial derivatives using a subscript, we note that $\\hat{\\sigma }_u, \\hat{\\sigma }_v \\mbox{ and } \\hat{\\sigma }_t$ are linearly independent for all $(u,v,t) \\in \\mathbb {R}^2 \\times [0,1]$ and $\\hat{\\sigma }$ is injective.", "Hence the image of $\\hat{\\sigma }$ is a 3-dimensional manifold.", "We now define the contact-set.", "Definition 4.3 Given $S$ and $h$ , the contact-set is the set of points $\\sigma (u_0,v_0,t_0)$ such that the line $\\lbrace (x_0,y_0,z_0,t) \\in \\mathbb {R}^4| \\sigma (u_0,v_0,t_0) = (x_0,y_0,z_0), t \\in [0,1] \\rbrace $ is tangent to (the image of) $\\hat{\\sigma }$ at $\\hat{\\sigma }(u_0,v_0,t_0) = (x_0, y_0, z_0, t_0)$ .", "We will denote the contact-set by $\\mathcal {C}$ .", "We will refer to the domain of the map $\\sigma $ as the parameter space and the co-domain as the object space.", "Consider now the following function $f: \\mathbb {R}^2 \\times [0,1] \\rightarrow \\mathbb {R}$ given by $ f(u,v,t) = \\left< V(u,v,t), \\hat{N}(u,v,t) \\right>$ Recalling that $V(u,v,t)$ is the velocity of the point $S(u,v)$ at time $t$ and $\\hat{N}(u,v,t)$ is the normal to $S_t$ at $(u,v)$ , we look at the zero-set of this function in the parameter space.", "Definition 4.4 The funnel $\\mathcal {F}$ is defined as the zero-set of the function $f$ specified in Eq.", "REF , i.e., $\\mathcal {F} = \\lbrace (u,v,t) \\in \\mathbb {R}^2 \\times [0,1] |f(u,v,t) = 0 \\rbrace $ .", "In other words, if a point $p = (u_0,v_0,t_0) \\in \\mathcal {F}$ , then the velocity at the point $\\sigma (p)$ lies in the tangent space of $S_{t_0}$ at $\\sigma (p)$ .", "The following lemma shows that the contact-set is precisely the image of the funnel through the sweep map.", "Lemma 4.1 $\\sigma (\\mathcal {F}) = \\mathcal {C}$ Proof.", "Skipped here.", "$\\square $ Figure: Funnel and contact-setHence $S(u,v)$ `contributes' a point (namely, $A(t)S(u,v) + b(t)$ ) to the contact-set at time $t$ iff the triplet $(u,v,t)$ satisfies the following condition.", "$ f(u,v,t) = \\left< V(u,v,t), \\hat{N}(u,v,t) \\right> = 0$ For a fixed $t$ , Eq.", "REF is a system of one equation in two variables $u$ and $v$ , hence, in a generic situation, the solution will be a curve.", "Eq.", "REF can also be looked upon as the rank deficiency condition [6] of the Jacobian $J_{\\sigma }$ of the map $\\sigma $ defined in REF .", "To make this precise, let $ J_{\\sigma } =\\begin{bmatrix}\\sigma _u & \\sigma _v & \\sigma _t\\end{bmatrix}_{3\\times 3}$ where $\\sigma _u|_{(u,v,t)}= A(t)\\frac{\\partial S}{\\partial u}(u,v)$ and $\\sigma _v|_{(u,v,t)} = A(t)\\frac{\\partial S}{\\partial v} (u,v)$ and $\\sigma _t|_{(u,v,t)} = V(u,v,t)$ .", "Observe that regularity of $S$ ensures that $J_{\\sigma }$ has rank at least 2.", "Further, it is easy to show that $f(u,v,t)$ is a non-zero scalar multiple of the determinant of $J_{\\sigma }$ .", "Therefore, Eq.", "REF is precisely the rank deficiency condition of the Jacobian of $\\sigma $ .", "Note that, for a point $(u_0,v_0,t_0) \\in \\mathbb {R}^3$ , the Jacobian $J_{\\sigma }|_{(u_0,v_0,t_0)}$ is a map from the tangent space to the ambient parameter-space at $(u_0,v_0,t_0)$ to the tangent space to the ambient object space at $\\sigma (u_0,v_0,t_0)$ .", "As already noted, if $(u_0,v_0,t_0) \\in \\mathcal {F}$ , then $J_{\\sigma }|_{(u_0,v_0,t_0)}$ is rank-deficient and maps the 3-dimensional ambient tangent space at $(u_0,v_0,t_0)$ , (surjectively) onto, a 2-dimensional subspace of the ambient tangent space at $\\sigma (u_0,v_0,t_0)$ .", "The subset of $\\mathcal {F}$ for a fixed value of $t$ will in general be a curve and will be referred to as the pcurve-of-contact at time $t$ and its image through $\\sigma $ will be a subset of $\\mathcal {C}$ which will be referred to as the curve-of-contact since it is essentially the set of points on the surface $S$ where $S$ makes tangential contact with $\\mathcal {C}$ at time $t$ .", "The union of such curves-of-contact for all $t$ gives the contact-set $\\mathcal {C}$ .", "The curve-of-contact at $t$ will be denoted by $C_t$ and the pcurve-of-contact will be denoted by $c_t$ .", "Fig.", "REF schematically illustrates the funnel and the contact-set.", "Before proceeding further, we make the following the non-degeneracy assumptionExamples where this does not hold are (i) a cylinder being swept along its axis (ii) a planar face being swept in a direction orthogonal to its normal.", "Such cases can be separately and easily handled.", "that: $\\forall p \\in \\mathcal {F}, \\nabla f|_{p} \\ne (0, 0, 0) $ Further, for ease of discussion, we assume that (i) $\\mathcal {F}$ is connected, and (ii) $\\forall (u,v,t) \\in \\mathcal {F}, (f_u,f_v) \\ne (0,0)$ Our analysis can be easily extended to situations where these further assumptions do not hold.", "An important consequence of the assumption REF is that $\\mathcal {F}$ is a 2-dimensional manifold and, hence, $\\mathcal {T}_{\\mathcal {F}}(p)$ is 2-dimensional at all points $p \\in \\mathcal {F}$ .", "Observe that $\\mathcal {F}$ is also orientable as $\\nabla f$ provides a continuous non-vanishing normal.", "Thus, $\\mathcal {F}$ is topologically nice and regular, However, quite often, $\\mathcal {C}$ has `anomalies' which arise due to self-intersections.", "One of the main contributions of this paper is a subtle, efficiently computable mathematical function on $\\mathcal {F}$ which allows to identify points on $\\mathcal {F}$ which give rise to anomalies in $\\mathcal {C}$ .", "The key to our analysis ahead, is the restriction of the sweep map $\\sigma $ to $\\mathcal {F}$ .", "We will abuse the notation, denote this restriction, again by $\\sigma $ .", "So, $\\sigma : \\mathcal {F} \\rightarrow \\mathcal {C}$ .", "Now, fix a point $p=(u,v,t) \\in \\mathcal {F}$ and let $q=\\sigma (p) \\in \\mathcal {C}$ .", "Since $det(J_{\\sigma } (p))=0$ , $\\lbrace \\sigma _u(p), \\sigma _v(p), \\sigma _t(p)\\rbrace $ are linearly dependent.", "As $S$ is regular, the set $\\lbrace \\sigma _u(p), \\sigma _v(p)\\rbrace $ forms a basis for the tangent space to $S_t$ .", "Therefore, we must have $\\sigma _t = l\\sigma _u +m \\sigma _v $ where $l$ and $m$ are well-defined (unique) and are themselves continuous functions of $u, v$ and $t$ .", "Clearly, $(\\sigma _u(p), \\sigma _v(p))$ is a natural (ordered) 2-frame in the object space at point $q$ (recall that, $q=\\sigma (p)$ ).", "Further, let ${\\cal X}(p)$ be any ordered continuous 2-frame (basis) of the tangent space ${\\mathcal {T}}_{\\mathcal {F}}(p)$ .", "Note that, this 2-frame is in the parameter space and is associated to the point $p$ .", "Now, through $\\sigma $ , more precisely, $J_{\\sigma }(p)$ , the frame ${\\cal X}(p)$ can be transported to another natural 2-frame $\\sigma ({\\cal X}(p))$ in the object space at the point $q$ .", "The determinant of the linear transformation connecting these two natural frames at $q$ , namely (i) $(\\sigma _u(p) ,\\sigma _v(p))$ and (ii) $\\sigma ({\\cal X}(p))$ is the key to the subsequent analysis.", "As we show later, this determinant is a positive scalar multiple of the continuous function $\\theta : \\mathcal {F} \\rightarrow \\mathbb {R}$ defined as follows.", "$ \\theta (p)= l f_u +m f_v -f_t$ Here $p=(u,v,t)$ and $f_u, f_v$ and $f_t$ denote partial derivatives of the function $f$ w.r.t.", "$u,v$ and $t$ respectively at $p$ , and $l$ and $m$ are as defined before.", "Note that $\\theta $ is easily and robustly computed.", "We state an important result which we will prove in the coming sections: Theorem 4.1 The function $\\theta $ is such that (i) $\\theta (p)<0$ indicates that $p$ is a point of local self intersection as defined by most authors, (see  [2], [7]) and (ii) $\\theta (p)=0$ is where the rank of $J_{\\sigma } (\\mathcal {T}_\\mathcal {F} (p))<2$ , and finally (iii) excision of the region $\\lbrace p | \\theta (p)\\le 0 \\rbrace $ from the funnel $\\mathcal {F}$ simplifies the construction of the envelope." ], [ "A particular frame for $\\mathcal {T}_{\\mathcal {F}}$", "Let $p = (u,v,t) \\in \\mathcal {F}$ .", "In this section, we compute a natural 2-frame ${\\cal X}(p)$ in $\\mathcal {T}_{\\mathcal {F}}(p)$ .", "Note that, $\\mathcal {F}$ being the zero level-set of the function $f$ defined in Eq.", "REF , $\\nabla f|_p \\bot \\mathcal {T}_{\\mathcal {F}}(p)$ .", "We set $\\beta = (-f_v, f_u, 0) \\ne 0$ and note that $\\beta \\bot \\nabla f$ .", "It is easy to see that $\\beta $ is tangent to the pcurve-of-contact $c_t$ .", "Let $\\alpha = \\nabla f \\times \\beta = (-f_uf_t, -f_vf_t, f_u^2+f_v^2)$ .", "Here $\\times $ is the cross-product in $\\mathbb {R}^3$ .", "Clearly, the set $\\lbrace \\alpha , \\beta \\rbrace $ forms a basis of $\\mathcal {T}_{\\mathcal {F}}(p)$ .", "Figure REF illustrates the basis $\\lbrace \\alpha , \\beta \\rbrace $ schematically.", "Observe that $\\beta $ is tangent to the pcurve-of-contact at time $t$ and $\\alpha $ points towards the `next' pcurve-of-contact." ], [ "The determinant connecting the two frames", "We continue with the notation developed earlier.", "We have $\\alpha = (-f_tf_u, -f_tf_v, f_u^2+f_v^2 )$ and $\\beta = (-f_v, f_u, 0)$ .", "Hence, $J_{\\sigma }\\alpha &= -f_t f_u \\sigma _u - f_t f_v \\sigma _v + (f_u^2 + f_v^2) \\sigma _t \\\\&=(-f_t f_u + l(f_u^2+f_v^2)) \\sigma _u + (-f_t f_v + m(f_u^2+f_v^2))\\sigma _v \\\\J_{\\sigma }\\beta &= -f_v \\sigma _u + f_u \\sigma _v$ So, $\\lbrace J_{\\sigma }\\alpha , J_{\\sigma }\\beta \\rbrace $ can be expressed in terms of $\\lbrace \\sigma _u, \\sigma _v \\rbrace $ as follows $\\begin{bmatrix}J_{\\sigma }\\alpha & J_{\\sigma }\\beta \\end{bmatrix}=\\begin{bmatrix}\\sigma _u & \\sigma _v\\end{bmatrix}\\underbrace{\\begin{bmatrix}-f_t f_u + l(f_u^2+f_v^2) & -f_v \\\\-f_t f_v + m(f_u^2+f_v^2) & f_u\\end{bmatrix} }_{\\mathcal {D}(p)}$ Note that, $det(\\mathcal {D}(p)) &= (f_u^2 + f_v^2)(l f_u + m f_v - f_t) \\\\&=(f_u^2 +f_v^2 ) \\theta (p) $" ], [ "Singularities of $\\mathcal {C}$", "In this subsection, we propose an efficient test for detecting singularities on the contact-set.", "See Fig.", "REF for an example.", "Clearly, the detection of singularities is important in practice.", "We start with the following definition.", "Definition 4.5 We say that the sweep causes a singularity if the composite map $\\mathcal {F} \\stackrel{\\sigma }{\\rightarrow } \\mathcal {C} \\hookrightarrow \\mathbb {R}^3$ fails to be an immersion.", "In other words, the sweep causes a singularity if $\\exists p \\in \\mathcal {F}$ such that the rank of $J_{\\sigma }(p)(\\mathcal {T}_{\\mathcal {F}}(p))$ is less than 2.", "Following the standard usage [8], in this case we say that the point $p$ is a critical point.", "Lemma 4.2 A point $r \\in \\mathcal {F}$ is a critical point iff $\\theta (r)=0$ iff the rank of $J_{\\sigma }(r)(\\mathcal {T}_{\\mathcal {F}}(r))$ is less than 2.", "Proof.", "By equation , we have $\\det (\\mathcal {D}(r))=0$ iff $\\theta (r)=0$ .", "Recall that, as shown earlier, $\\lbrace \\alpha (r), \\beta (r)\\rbrace $ is a basis of $\\mathcal {T}_{\\mathcal {F}}(r)$ , and $\\lbrace \\sigma _u(r), \\sigma _v(r)\\rbrace $ is also a basis of the 2-frame associated at $\\sigma (r)$ .", "As $\\mathcal {D}(r)$ is the matrix expressing ${\\cal {X}} = \\lbrace J_{\\sigma }(r)(\\alpha (r)), J_{\\sigma }(r)(\\alpha (r))$ in terms of $\\lbrace \\sigma _u(r), \\sigma _v(r)\\rbrace $ , $\\det (\\mathcal {D}(r))=0$ iff rank of $J_{\\sigma }(r)(\\mathcal {T}_{\\mathcal {F}}(r))$ is less than 2.", "Thus, $r$ is a critical point iff $\\theta (r)=0$ iff the rank of $J_{\\sigma }(r)(\\mathcal {T}_{\\mathcal {F}}(r))$ is less than 2.", "$\\square $ Note that the above lemma proves part (ii) of theorem REF .", "Lemma 4.3 A sweep causes a singularity if there exists points $p$ and $q$ on $\\mathcal {F}$ such that $\\theta (p) \\le 0$ and $\\theta (q) \\ge 0$ .", "Proof.", "As $\\theta $ is a continuous function on $\\mathcal {F}$ , the existence of $p$ and $q$ on $\\mathcal {F}$ with the required properties implies existence of another point $r \\in \\mathcal {F}$ such that $\\theta (r)=0$ .", "By the previous lemma, this implies that the sweep causes a singularity.", "$\\square $ The above lemma leads to a computationally efficient test for detecting singularities: namely, evaluating $\\theta $ at sampled points on $\\mathcal {F}$ and checking if it changes sign on $\\mathcal {F}$ .", "The analysis done so far helps us detect singularities on the contact-set.", "In the next section we will perform a detailed analysis of local self-intersections which is topological in nature.", "Towards this, note that all points in the non-critical set may not lead to points on the envelope of the swept volume.", "For some $p=(u_0,v_0,t_0) \\in \\mathcal {F}$ , $\\sigma (p)$ may lie in the interior of the solid $M_t$ of which the surface patch $S_t$ is a part of, for some $t$ in neighbourhood of $t_0$ .", "In that case $\\sigma (p)$ will not be on the envelope.", "Fig.", "REF shows two sweeping examples with self-intersections.", "Curves-of-contact at a few time instances are shown.", "In the next section we focus on identifying such points.", "Figure: Examples of local self-intersection showing curves-of-contact at few time instances: (a) A cylinder with blended edges undergoing translation and rotation about xx-axis (b) An ellipsoid undergoing translation along a curvilinear path" ], [ "Topological and regularity analysis of ${\\mathcal {C}}$", "Quite often, the anomalies on $\\mathcal {C}$ arise due to self-intersections.", "If $\\mathcal {C}$ has self-intersections, it needs to be trimmed to obtain the envelope of the swept volume [2].", "Self-intersections can be broadly be classified into global and local.", "Fig.", "REF illustrates the difference between global and local self-intersections schematically.", "Figure: Global and local self-intersectionIf there are only global self-intersections occurring on $\\mathcal {C}$ , the composite map $\\mathcal {F} \\stackrel{\\sigma }{\\rightarrow } \\mathcal {C} \\hookrightarrow \\mathbb {R}^3$ fails to be an injection.", "However, it is an immersion (see [8]), i.e.", "$\\forall p \\in \\mathcal {F}$ , the rank of $J_{\\sigma }(p)({\\mathcal {T}_{\\mathcal {F}}(p)})$ is 2.", "In principle, global self-intersections can be detected by surface-surface intersection.", "(see [14]) The case of local self-intersection is more subtle as it leads to singularities on $\\mathcal {C}$ .", "Clearly, the detection of local self-intersections is also central to a robust implementation of the solid sweep in CAD systems.", "In literature [2], [7], local self-intersections have been quantified by looking at points in the contact-set which lie in the interior of the solid $M_t$ for some time instant $t$ .", "Clearly, such a point cannot be on the envelope of the swept volume.", "This approach was used in [2] for detecting local and global self-intersections, where the authors used implicit representation of the surface bounding the solid which is being swept.", "We adapt this concept to when the surface of the solid is represented parametrically.", "We will refer to this type of local self-intersection as type-2 L.S.I..", "It turns out that type-2 L.S.I.", "is intimately related to the analysis carried out in the previous section.", "To make this connection precise, we first introduce another type of local self-intersection.", "For lack of a better name, this is called as type-1 L.S.I." ], [ "Type-1 local self-intersections", "Definition 5.1 A type-1 L.S.I.", "is said to occur at a point $p \\in \\mathcal {F}$ if $\\theta (p) \\le 0$ .", "Thus, Type-1 L.S.I is our classification of a local self intersection.", "We will see in subsection REF that a for a Type-1 L.S.I point $p$ , the image $\\sigma (p)$ does not lie on the envelope of the swept volume." ], [ "Type-2 local self-intersection", "In order to define type-2 L.S.I.", "we first describe the inverse trajectory corresponding to a given trajectory [2], [20].", "Given a trajectory as in definition REF and a fixed point $x$ in object-space, we would like to compute the set of points in the object-space which get mapped to $x$ at some time instant.", "This set can be computed through the inverse trajectory defined as follows.", "Definition 5.2 Given a trajectory $h$ , the inverse trajectory $\\bar{h}$ is defined as the map $\\bar{h}:[0,1] \\rightarrow (SO(3), \\mathbb {R}^3)$ given by $\\bar{h}(t) = (A^t(t), -A^t(t)b(t))$ .", "Thus, for a fixed point $x \\in \\mathbb {R}^3$ , the inverse trajectory of $x$ is the map $\\bar{y}:[0,1] \\rightarrow \\mathbb {R}^3$ given by $\\bar{y}(t) = A^t(t)(x - b(t))$ .", "The range of $\\bar{y}$ is $\\lbrace A^t(t)x - A^t(t)b(t) | t \\in [0,1] \\rbrace = \\lbrace z \\in \\mathbb {R}^3 | \\exists t \\in [0,1], A(t)z + b(t) = x\\rbrace $ .", "We will denote the trajectory of $x$ by $y:[0,1] \\rightarrow \\mathbb {R}^3$ , $y(t) = A(t)x + b(t)$ .", "We now note a few useful facts about inverse trajectory of $x$ .", "We assume without loss of generality that $A(t_0) = I$ and $b(t_0)=0$ .", "Denoting the derivative with respect to $t$ by $\\dot{}$ , we have $\\dot{\\bar{y}}(t)=\\dot{A}^t(t)(x-b(t)) - A^t(t)\\dot{b}(t) $ Since $A \\in SO(3)$ we have, $A^t(t) A(t) &= I, \\forall t $ Differentiating Eq.", "REF w.r.t.", "$t$ we get $ \\dot{A}^t(t) A(t) + A^t(t)\\dot{A}(t) &= 0, \\forall t \\\\\\dot{A}^t(t_0) + \\dot{A}(t_0) &= 0 $ Differentiating Eq.", "REF w.r.t.", "$t$ we get $\\nonumber \\ddot{A}^t(t)A(t) + 2\\dot{A}^t(t)\\dot{A}(t) + A^t(t)\\ddot{A}(t) &=0, \\forall t \\\\\\ddot{A}^t(t_0) + 2\\dot{A}^t(t_0)\\dot{A}(t_0) + \\ddot{A}(t_0) &=0 $ Using Eq.", "REF and Eq.", "we get $\\dot{\\bar{y}}(t_0) &= -\\dot{A}(t_0)x - \\dot{b}(t_0) = -\\dot{y}(t_0) $ Differentiating Eq.", "REF w.r.t.", "time we get $\\ddot{\\bar{y}}(t) = \\ddot{A}^t(t)(x-b(t)) - 2\\dot{A}^t(t)\\dot{b}(t) - A^t(t)\\ddot{b}(t) $ Using Equations REF , and REF we get $\\ddot{\\bar{y}}(t_0) = -\\ddot{y}(t_0) + 2\\dot{A}(t_0)\\dot{y}(t_0) $ We now define type-2 L.S.I.", "The surface $S$ is the boundary of the solid $M$ being swept.", "We will refer to the interior of $M$ by $Int(M)$ and exterior of $M$ by $Ext(M)$ .", "Definition 5.3 A type-2 L.S.I is said to occur at a point $(u_0, v_0, t_0)$ if the inverse trajectory of the point $\\sigma (u_0, v_0, t_0)$ intersects $Int(M_{t_0})$ .", "(see [2]) Figure: Type-2 local self-intersectionFig.", "REF illustrates type-2 L.S.I.", "schematically where $\\bar{y}$ is the inverse trajectory of the point $x \\in S_{t_0}$ and $\\pi $ is the projection of $\\bar{y}$ on $S_{t_0}$ .", "Suppose the point $x = \\sigma (u_0,v_0,t_0) \\in \\mathcal {C}$ .", "Let $\\lambda (t)$ be the signed distance of $\\bar{y}(t)$ from surface $S_{t_0}$ .", "If the point $\\bar{y}(t)$ is in $Int(M_{t_0})$ , $Ext(M_{t_0})$ or on the surface $S_{t_0}$ , then $\\lambda (t)$ is negative, positive or zero respectively.", "Then we have $\\bar{y}(t) - \\pi (t) = \\lambda (t)N(t)$ , where $\\pi (t)$ is the projection of $\\bar{y}(t)$ on $S_{t_0}$ along the unit outward pointing normal $N(t)$ to $S_{t_0}$ at $\\pi (t)$ .", "Then, the following relation holds for $\\lambda $ .", "$ \\lambda (t) = \\left< \\bar{y}(t) - \\pi (t) , N(t) \\right>$ We now give a necessary and sufficient condition for type-2 L.S.I.", "to occur.", "Lemma 5.1 Type-2 L.S.I.", "occurs at a point $p=(u_0,v_0,t_0) \\in \\mathcal {F}$ if and only if either of the following conditions hold $\\ddot{\\lambda }(t_0)= \\left< -\\ddot{\\sigma }+2\\dot{A}V , N \\right> + \\kappa v^2 < 0$ where $\\kappa $ is the normal curvature of $S_{t_0}$ at $(u_0,v_0)$ along velocity $V(p)$ , $N$ is the unit length outward pointing normal to $S_{t_0}$ at $(u_0,v_0)$ and $v^2 = \\left< V(p), V(p) \\right>$ .", "$\\lambda (t)$ is negative for some $t$ in some nbd of $t_0$ .", "Remark: The statement of the above lemma (except for the insightful expression of $\\ddot{\\lambda }(t_0)$ ) is similar in spirit to the key Theorem-2 in [2] in the context of implicitly defined solids.", "Proof.", "Differentiating Eq.", "REF with respect to time and denoting derivative w.r.t.", "$t$ by $\\dot{}$ , we get $\\dot{ \\lambda }(t) &= \\left< \\dot{\\bar{y}}(t) - \\dot{\\pi }(t), N(t) \\right> + \\left< \\bar{y}(t) - \\pi (t) , \\dot{N}(t) \\right> \\\\\\nonumber \\ddot{\\lambda }(t) &= \\left< \\ddot{\\bar{y}}(t) - \\ddot{\\pi }(t), N(t) \\right> + 2\\left< \\dot{\\bar{y}}(t) - \\dot{\\pi }(t), \\dot{N}(t) \\right> \\\\&+ \\left< \\bar{y}(t) - \\pi (t), \\ddot{N}(t) \\right> $ At $t=t_0$ , $\\bar{y}(t_0) = \\pi (t_0)$ .", "Since $ \\dot{y}(t_0)=V(p) \\bot N(p)$ , it follows from Eq.", "REF that $\\dot{\\bar{y}}(t_0) \\bot N(p)$ .", "It is easy to verify that $\\dot{\\pi }(t_0) = \\dot{\\bar{y}}(t_0)$ .", "Hence, $\\lambda (t_0) = \\dot{\\lambda }(t_0) = 0 $ From Eq.", "REF and Eq.", "REF it follows that $\\nonumber \\ddot{\\lambda }(t_0) &= \\left< \\ddot{\\bar{y}}(t_0) - \\ddot{\\pi }(t_0), N(t_0) \\right>\\\\&= \\left< -\\ddot{y}(t_0) + 2\\dot{A}(t_0)\\dot{y}(t_0) - \\ddot{\\pi }(t_0), N(t_0) \\right> $ Since $\\pi (t) \\in S_{t_0} \\forall t$ in some neighbourhood $U$ of $t_0$ , we have that $\\left< \\dot{\\pi }(t), N(t) \\right> = 0, \\forall t \\in U$ .", "Hence $\\left< \\ddot{\\pi }(t), N(t) \\right> + \\left< \\dot{\\pi }(t), \\dot{N}(t) \\right> = 0, \\forall t \\in U$ .", "Hence $-\\left< \\ddot{\\pi }(t_0), N(t_0) \\right> = \\left< \\dot{\\pi }(t_0), \\dot{N}(t_0) \\right> = \\left< \\dot{\\pi }(t_0), \\mathcal {G}^*(\\dot{\\pi }(t_0)) \\right> = \\left< \\dot{y}(t_0), \\mathcal {G}^*(\\dot{y}(t_0)) \\right>$ = $\\left< V(p) , \\mathcal {G}^*(V(p)) \\right> =\\kappa v^2$ .", "Here $\\mathcal {G}^*(\\dot{y})$ is the differential of the Gauss map, i.e.", "the curvature tensor of $S_{t_0}$ at point $x$ .", "Using this in Eq.", "REF and the fact that $\\dot{y}(t_0) = \\dot{\\sigma }(p)$ , $\\ddot{y}(t_0) = \\ddot{\\sigma }(p)$ we get $\\ddot{\\lambda }(t_0) &= \\left< -\\ddot{\\sigma }(p) + 2\\dot{A}(t_0)V(p) , N(t_0) \\right> + \\kappa v^2 $ From Eq.", "REF and Eq.", "REF we conclude that if $\\ddot{\\lambda }(t_0) < 0$ the point $x=\\bar{y}(t_0) = \\sigma (u_0,v_0,t_0)$ is a local maxima of the function $\\lambda $ and the inverse trajectory of $x$ intersects with interior of the solid $M_{t_0}$ causing type-2 L.S.I.", "Similarly, if $\\ddot{\\lambda }(t_0) > 0$ we conclude that $x$ is a local minima of $\\lambda $ and the inverse trajectory of $x$ does not intersect with the interior of $M_{t_0}$ and there is no L.S.I.", "occurring at $x$ .", "However, if $\\ddot{\\lambda }(t_0)$ is zero, one needs to inspect a small neighbourhood of $t_0$ to see if $\\exists t$ such that $\\lambda (t)<0$ in order to check for type-2 L.S.I.", "$\\square $ If $\\ddot{\\lambda } = 0$ at a point, the structure of the contact-set $\\mathcal {C}$ is unknown at that point.", "We will see in the next subsection that at such a point, $\\mathcal {C}$ has singularity." ], [ "Relation between type-1 L.S.I. and type-2 L.S.I.", "In this subsection we will see that type-2 L.S.I.", "implies type-1 L.S.I.", "at any point $p = (u_0,v_0,t_0) \\in \\mathcal {F}$ Lemma 5.2 $\\theta (p) = \\ddot{\\lambda }(t_0)$ .", "Proof.", "Recalling definition of $\\theta (p)$ from Eq.", "REF $l f_u + m f_v - f_t &= \\left< l\\hat{N}_u + m\\hat{N}_v, V\\right> + \\left<\\hat{N}, l V_u + m V_v \\right>\\\\ &- \\left< \\hat{N}_t, V\\right> - \\left<\\hat{N} ,V_t \\right>$ Here $\\hat{N}_u = \\mathcal {G}^*(\\sigma _u)$ and $\\hat{N}_v = \\mathcal {G}^*(\\sigma _v)$ where $\\mathcal {G}^*$ is the shape operator (differential of the Gauss map) of $S_{t_0}$ at $(u_0,v_0)$ .", "Also, $V_u = A_t S_u$ and $V_v = A_t S_v$ .", "Assume without loss of generality that $A(t_0) = I$ and $b(t_0) = 0$ .", "Using Eq.", "and the fact that $\\sigma _t = l\\sigma _u+m\\sigma _v$ we get $\\nonumber l f_u + m f_v - f_t &= \\left< \\mathcal {G}^*V, V \\right> + 2\\left<A_t V ,N \\right> - \\left<V_t, N\\right> \\\\& = \\kappa v^2 + \\left< 2A_t V - V_t , N \\right> $ From Eqs.", "REF and REF and the fact that $\\frac{\\partial \\sigma }{\\partial t^2}=V_t$ we get $\\theta (p) = l f_u + m f_v - f_t = \\ddot{\\lambda }$ $\\square $ Figure: Region of local self-intersectionLet $p$ be such that $\\theta (p) < 0$ .", "By lemma REF , $\\ddot{\\lambda }(t_0)<0$ .", "Further, by lemma REF a type-2 L.S.I.", "occurs at $p$ .", "This proves parts (i) and (iii), and hence completes the proof of theorem REF .", "Fig.", "REF schematically illustrates the region on the funnel where local self-intersection occurs and the corresponding region on the contact-set.", "A curve-of-contact and the corresponding pcurve-of-contact is shown in red colour at a time instant $t_0$ when a local self-intersection occurs.", "The shaded region there corresponds to $\\theta < 0$ .", "Thus $sign(\\theta )$ changes(from $-$ ve to $+$ ve) as one moves from the interior to the exterior of the shaded region.", "Of course $\\theta =\\ddot{\\lambda } = 0$ on the boundary of the shaded region where $\\mathcal {C}$ has a singularity.", "Fig.", "REF shows an example of sweeping which clearly demonstrates the subtle difference between type-1 L.S.I.", "and type-2 L.S.I.", "In Fig.", "REF a sphere is rotated about an axis which is tangent to itself.", "The contact-set is shown in green and the curve-of-contacts on the contact-set at initial position $(t = 0)$ and final position $(t = 1)$ are also shown in blue.", "In Fig.", "REF there is type-1 L.S.I.", "but no type-2 L.S.I.", "since no point of the contact-set intersects with the interior of solid at any time.", "We now present two examples of sweeping which result in local self-intersections.", "We apply the test for type-1 L.S.I.", "on these examples to demonstrate its effectiveness.", "Example 5.1 Consider a solid cylinder along the $y$ -axis with height $2.5$ and radius 2, parameterized as $S(u, v) = (2\\cos v, u, -2\\sin v)$ , $u \\in [-1.25, 1.25]$ , $v \\in [-\\pi , \\pi ]$ being swept along the trajectory given by $h(t) = (A(t), b(t))$ where $A(t) = \\begin{bmatrix}1 & 0 & 0 \\\\0 & \\cos (0.1 \\pi t) & -\\sin (0.1 \\pi t) \\\\ 0 & \\sin (0.1 \\pi t) & \\cos (0.1\\pi t) \\end{bmatrix}$ and $b(t) = \\begin{bmatrix} 3 \\cos \\left(\\frac{\\pi }{2}t\\right) -3\\\\ 3 \\sin \\left(\\frac{\\pi }{2}t \\right) \\\\ 0 \\end{bmatrix}$ $t \\in [0,1]$ .", "The resulting envelope has local self-intersections as illustrated in Fig.", "REF (a) which shows the cylinder with blended edges and curves-of-contact at few time instances.", "Type-1 L.S.I is detected at time $t = 0.1$ by the test given in lemma REF at point $p = (u=0.18, v=1.53, t=0.1)$ .", "$\\theta (p) = -2.378$ .", "Example 5.2 Consider an ellipsoid with axes lengths 3, 1, and 1 parameterized as $S(u,v) = (-3\\cos (u) \\cos (v), \\sin (u),\\\\ \\cos (u) \\sin (v))$ , $u \\in [-\\frac{\\pi }{2}, \\frac{\\pi }{2}], v \\in [-\\pi , \\pi ]$ being swept along the curvilinear trajectory given by $h(t) = (A(t), b(t))$ where $A(t) = I \\forall t$ and $b(t) = \\begin{bmatrix} 3 \\cos \\left(\\frac{\\pi }{2}t\\right) -3\\\\ 3 \\sin \\left(\\frac{\\pi }{2}t \\right) \\\\ 0 \\end{bmatrix}$ $t \\in [0,1]$ .", "The curves-of-contact are shown in Fig.", "REF (b).", "Type-1 L.S.I.", "is detected at time $t=0.8$ .", "$\\theta (p) = -11.864$ at $(-0.791, -0.157, 0.8)$ ." ], [ "Mathematical structure of the smooth case", "In this section, we consider the smooth case where the solid $M$ is composed of faces meeting smoothly.", "As usual, each face (or the associated surface patch) is smooth (of class $C^k$ for $k \\ge 2$ ).", "Further, adjacent faces meet smoothly at the common edge.", "This is referred to as $G^1$ continuity [13] which formally means that the unit outward normals to the adjacent faces match on the common edge.", "Similarly, at a vertex, all the unit outward normals to faces incident on this vertex are identical.", "The solid shown in figure REF is such a solid.", "Consider a sweep of $M$ along a trajectory $h$ which causes no self-intersections/anomalies on the contact-set $\\mathcal {C}$ .", "As described in Section , every point $p$ on $\\mathcal {C}$ comes from a curve of contact on $M$ and therefore is associated to a point $q$ of $M$ .", "Let $\\pi : \\mathcal {C} \\rightarrow M$ ($p \\mapsto \\pi (p)=q$ ) denote this natural map.", "For every $p \\in \\mathcal {C}$ , $\\pi (p)$ belongs to some geometric entity of $M$ , i.e., a vertex, edge or face.", "This sets up the natural correspondence between geometric entities of $\\mathcal {C}$ and that of $M$ .", "For a face $F$ of $M$ , let $\\mathcal {C}_F$ denote the part of $\\mathcal {C}$ which corresponds to the face $F$ under this correspondence.", "For example, in figure REF , the green face on the solid $M$ corresponds to multiple green faces of $\\mathcal {C}$ .", "Clearly, the map $\\pi $ restricts naturally from $\\mathcal {C}_F \\rightarrow F$ .", "There are situations in which, for example, an edge (or a part of it) on $M$ remains on the boundary for a while and thus, `sweeps' a face on $\\mathcal {C}$ .", "For simplicity, we assume that such situations are ruled out.", "In other words, no lower dimensional geometric entity of $M$ gives rise to a higher dimensional geometric entity on $\\mathcal {C}$ .", "This is the case for the sweep operation illustrated in figure REF ." ], [ "Local similarity within a face", "Firstly, recall that each face $F$ of $M$ is derived from an underlying surface $S_F$ by restricting the parameters of $S_F$ to a suitable domain $D_F$ .", "Now, by applying the `local' analysis of Section  to the surface $S_F$ , we have Lemma 6.1 The set $\\mathcal {C}_F$ has no self-intersections and is a smooth manifold.", "Further, let $p \\in \\mathcal {C}_F$ correspond to $q \\in F$ at time $t$ .", "Then, the unit normal $N(p)$ to $\\mathcal {C}_F$ at $p$ is simply $A(t)N(q)$ where $N(q)$ is the unit normal to $F$ at $q$ .", "Further, we would like to show that `locally', $\\mathcal {C}_F$ has the same topology as that of $F$ .", "More precisely, the natural map $\\pi : \\mathcal {C}_F \\rightarrow F$ is a local homeomorphism onto its image.", "Thanks to the `local' nature, we may analyse this via the underlying surface $S_F$ .", "For ease of notation, we sometimes omit the reference to $F$ and freely use notations from Section .", "As the sweep $\\sigma $ is free of self-intersections, the key map $\\sigma : \\mathcal {F} \\rightarrow \\mathcal {C}_F$ is a bijective immersion (recall that, in Section , $\\mathcal {F}$ is the funnel defined via the function $f=0$ ).", "Thus, by the inverse function theorem, it is invertible via a continuous inverse.", "Therefore, in order to show that $\\pi : \\mathcal {C}_F \\rightarrow F$ is a local homeomorphism onto its image it suffices to prove the following lemma.", "Lemma 6.2 The natural map $\\pi ^{\\prime }: \\mathcal {F} \\rightarrow F$ defined as: for $p=(u,v,t) \\in \\mathcal {F}$ , $\\pi ^{\\prime }(p) = S(u,v)$ , is a local homeomorphism.", "Remark The map $\\pi ^{\\prime }$ is simply the composition of $\\sigma $ and $\\pi $ .", "Proof.", "Let $p=(u_0,v_0,t_0) \\in \\mathcal {F}$ .", "Here, the assumption that $f_t\\ne 0$ at $p$ is very crucial.", "Firstly, by the implicit function theorem, there exists a neighbourhood $O$ of $(u_0, v_0)$ and a continuous function $t=g(u,v)$ defined on $O$ such that $\\forall (u,v) \\in O$ , $f(u,v,g(u,v))=0$ .", "Further, the set $O(p) = \\lbrace (u,v,g(u,v)) \\mid (u,v) \\in O\\rbrace $ is a neighbourhood of $p$ in $\\mathcal {F}$ .", "On this neighbourhood of $p$ , the function $\\pi ^{\\prime }$ is a bijection and invertible.", "This easily follows from the fact that, for every $(u,v) \\in O$ , there is a unique time $t$ , namely, $t=g(u,v)$ , such that $(u,v,g(u,v)) \\in O(p)$ .", "$\\square $" ], [ "Local similarity across faces", "We begin by studying the variation of the unit normal across faces of $\\mathcal {C}$ .", "Let $C_t$ denote the curve of contact of $\\mathcal {C}$ at time $t$ .", "Let $p \\in C_t$ be such that $p$ is common to (only) $\\mathcal {C}_{F_1}$ and $\\mathcal {C}_{F_2}$ where $F_1$ and $F_2$ are two distinct faces of $M$ .", "Lemma 6.3 The faces $F_1$ and $F_2$ are adjacent in $M$ .", "Further, the normal to $\\mathcal {C}_{F_1}$ at $p$ is identical to the normal to $\\mathcal {C}_{F_2}$ at $p$ .", "Proof.", "Suppose $p$ corresponds to $q$ .", "Clearly, $q$ is common to (only) $F_1$ and $F_2$ .", "Thus $F_1$ and $F_2$ are adjacent in $M$ .", "Further, by $G^1$ continuity, the normal to $F_1$ at $q$ is identical to the normal to $F_2$ at $q$ .", "By lemma REF , it is clear that, the normals to $\\mathcal {C}_{F_1}$ , and to $\\mathcal {C}_{F_2}$ , at $p$ are identical.", "$\\square $ Thus, the adjacencies on $\\mathcal {C}$ are the `same' as the adjacencies on $M$ .", "See figure REF , which effectively illustrates this through colours.", "Further, the adjacent entities of $\\mathcal {C}$ meet smoothly across common lower-dimensional entities.", "In other words, $\\mathcal {C}$ is also of class $G^1$ .", "Recall that the overall envelope may be obtained from the contact-set $\\mathcal {C}$ by simply capping the appropriate parts of the solid $M$ at the initial and final position.", "Therefore, the topological structure of the contact-set and hence, that of the envelope, mimics that of the solid.", "The following theorem summarizes the analysis so far.", "Theorem 6.1 The map $\\pi $ from the contact-set/envelope to the solid is an adjacency-respecting local homeomorphism onto its image." ], [ "Curvature of $\\mathcal {C}$", "In the special case when the trajectory $h$ consists only of translations, i.e.", "$A(t)=I$ $\\forall t$ , the Gaussian curvature of $\\mathcal {C}$ can be expressed in terms of the Gaussian curvature of $S$ and the curvature of $h$ .", "Since $A(t) = I$ $\\forall t$ , $\\sigma _u = S_u$ and $\\sigma _v = S_v$ .", "Also, $V_u = V_v = 0$ .", "Consider a point $p = (u_0, v_0, t_0) \\in \\mathcal {F}$ .", "By definition of $\\mathcal {F}$ , we have that $f(u_0, v_0, t_0)=0$ .", "Given that $\\nabla f|_p \\ne 0$ suppose without loss of generality that $f_t|_p \\ne 0$ .", "Then by the implicit function theorem there exists a neighbourhood $\\mathcal {N}$ of $q=(u_0, v_0)$ such that $\\forall (u,v) \\in \\mathcal {N}$ , $f(u,v,t(u,v))=0$ .", "Hence, $t_u = -\\frac{f_u}{f_t}$ and $t_v = -\\frac{f_v}{f_t}$ and we get a local parameterization of $\\mathcal {C}$ in $\\mathcal {N}$ by $\\psi (u,v) = \\sigma (u,v,t(u,v))$ .", "$\\mathcal {T}_{\\mathcal {C}}(p)$ is spanned by $\\psi _u = \\sigma _u + \\sigma _t t_u$ and $\\psi _v = \\sigma _v + \\sigma _t t_v$ .", "Since $p \\in \\mathcal {F}$ by lemma REF , $\\sigma _t$ is in the space spanned by $\\sigma _u$ and $\\sigma _v$ .", "Let $\\sigma _t = l \\sigma _u + m \\sigma _v$ .", "Hence we express basis for $\\mathcal {T}_{\\mathcal {C}}(p)$ in terms of basis of $\\mathcal {T}_{S}(q)$ as follows $ \\begin{bmatrix}\\psi _u & \\psi _v\\end{bmatrix}=\\begin{bmatrix}\\sigma _u & \\sigma _v\\end{bmatrix}\\mathbf {M}$ where $\\mathbf {M} = \\begin{bmatrix} 1+l t_u & l t_v \\\\ m t_u & 1+m t_v \\end{bmatrix}$ .", "The unit normal to $\\mathcal {C}$ is given by $\\hat{N}(u,v) = A(t(u,v))N(u,v) = N(u,v)$ where $N$ is the unit normal to $S_t$ .", "Hence, $\\hat{N}_u = N_u$ and $\\hat{N}_v = N_v$ .", "Further, $ \\begin{bmatrix}N_u & N_v\\end{bmatrix}=\\begin{bmatrix}\\sigma _u & \\sigma _v\\end{bmatrix}\\mathbf {W}$ where $\\mathbf {W}$ is the Weingarten matrix of $S$ at point $q$ whose determinant gives the Gaussian curvature of $S$ at $q$ (see  [1]).", "From Eq.", "REF and Eq.", "REF we have $\\begin{bmatrix}\\hat{N}_u & \\hat{N}_v\\end{bmatrix}=\\begin{bmatrix}\\psi _u & \\psi _v\\end{bmatrix}\\mathbf {M}^{-1}\\mathbf {W}$ So the Weingarten matrix of $\\mathcal {C}$ with respect to parameterization $\\psi $ is given by $\\mathbf {M}^{-1}\\mathbf {W}$ and the Gaussian curvature is given by $det(\\mathbf {M}^{-1}\\mathbf {W})$ .", "From Eq.", "REF and the fact that $N_t = V_u = V_v = 0$ we note that $f = \\left< N, V \\right>$ , $f_u = \\left< N_u, V \\right>$ , $f_v = \\left< N_v, V \\right>$ and $f_t = \\left<N, V_t \\right>$ .", "So, $det(\\mathbf {M}^{-1}) &= \\frac{1}{1+l t_u + m t_v} = \\frac{f_t}{f_t - l f_u - m f_v} \\\\&= \\frac{\\left< N, V_t \\right>}{ \\left<N, V_t \\right> - \\left< l N_u + m N_v, V \\right>} \\\\&= \\frac{\\left< N, V_t \\right>}{ \\left<N, V_t \\right> - \\left< \\mathbf {W}(V), V \\right>}$ The Gaussian curvature of $\\mathcal {C}$ is computed as $det(\\mathbf {M}^{-1}\\mathbf {W}) &= \\frac{\\left< N, V_t \\right> }{ \\left<N, V_t \\right> - \\left< \\mathbf {W}(V), V \\right>} det(\\mathbf {W})$ where, $\\left< N, V_t \\right>$ is the curvature of the trajectory scaled by $\\Vert V_t \\Vert $ , $\\left< \\mathbf {W}(V), V \\right>$ is the normal curvature of $S$ at $p$ in direction $V$ scaled by $\\Vert V\\Vert ^2$ and $det(\\mathbf {W})$ is the Gaussian curvature of $S$ at $p$ ." ], [ "Envelope computation", "In this section we describe the construction of the envelope $\\mathcal {E}$ assuming that it is free from self-intersections and hence regular.", "We obtain a procedural parametrization of $\\mathcal {E}$ .", "The procedural paradigm is an abstract way of defining curves and surfaces.", "It relies on the fact that from the user's point of view, a parametric surface(curve) in $\\mathbb {R}^3$ is a map from $\\mathbb {R}^2 (\\mathbb {R})$ to $\\mathbb {R}^3$ and hence is merely a set of programs which allow the user to query the key attributes of the surface(curve), e.g.", "its domain and to evaluate the surface(curve) and its derivatives at the given parameter value.", "The procedural approach to defining geometry is especially useful when closed form formulae are not available for the parametrization map and one must resort to iterative numerical methods.", "We use the Newton-Raphson(NR) method for this purpose.", "As an example, the parametrization of the intersection curve of two surfaces is computed procedurally in [14].", "As we will see, this approach has the advantage of being computationally efficient as well as accurate.", "For a detailed discussion on the procedural framework, see [11].", "The computational framework is as follows.", "For the input parametric surface $S$ and trajectory $h$ , an approximate envelope is first computed, which we will refer to as the seed surface.", "Now, when the user wishes to evaluate the actual envelope or its derivative at some parameter value, a NR method will be started with seed obtained from the seed surface.", "The NR method will converge, upto the required tolerance, to the required point on the envelope, or to its derivative, as required.", "Here, the precision of the evaluation is only restricted by the finite precision of the computer and hence is accurate.", "It has the advantage that if a tighter degree of tolerance is required while evaluation of the surface or its derivative, the seed surface does not need to be recomputed.", "Thus, for the procedural definition of the envelope we need the following: a NR formulation for computing points on $\\mathcal {E}$ and its derivatives, which we describe in subsection  REF Seed surface for seeding the NR procedure, which we describe in subsection  REF Recall that by the non-degeneracy assumption, $\\mathcal {E}$ is the union of $C_t, \\forall t$ .", "This suggests a natural parametrization of $\\mathcal {E}$ in which one of the surface parameters is time $t$ .", "We will call the other parameter $p$ and denote the seed surface by $\\gamma $ which is a map from the parameter space of $\\mathcal {E}$ to the parameter space of $\\mathcal {\\sigma }$ , i.e.", "$\\gamma (p,t) = (\\bar{u}(p,t), \\bar{v}(p,t), t)$ and while the point $\\sigma (\\gamma (p,t))$ may not belong to $\\mathcal {E}$ , it is close to $\\mathcal {E}$ .", "In other words, $\\gamma (p,t)$ is close to $\\mathcal {F}$ .", "We call the image of the seed surface through the sweep map $\\sigma $ as the approximate envelope and denote it by $\\bar{\\mathcal {E}}$ , i.e.", "$\\bar{\\mathcal {E}}(p,t) = \\sigma (\\gamma (p,t))$ .", "We make the following assumption about $\\bar{\\mathcal {E}}$ .", "At every point on the iso-t curve of $\\bar{\\mathcal {E}}$ , the normal plane to the iso-t curve intersects the iso-t curve of $\\mathcal {E}$ in exactly one point.", "Note that this is not a very strong assumption and holds true in practice even with rather sparse sampling of points for the seed surface.", "We now describe the Newton-Raphson formulation for evaluating points on $\\mathcal {E}$ and its derivatives at a given parameter value." ], [ "NR formulation for faces of $\\mathcal {E}$", "Recall that the points on $\\mathcal {E}$ were characterized by the tangency condition given in Eq.", "REF .", "Introducing the parameters $(p,t)$ of $\\mathcal {E}$ , we rewrite Eq.", "REF $\\forall (p_0, t_0)$ : $ \\nonumber f(u(p_0,t_0), v(p_0,t_0), t_0 ) &= \\left< \\hat{N}(u(p_0,t_0), v(p_0,t_0), t_0), \\right.", "\\\\&\\left.", "V(u(p_0,t_0), v(p_0,t_0), t_0) \\right> = 0$ So, given $(p_0,t_0)$ , we have one equation in two unknowns, viz.", "$u(p_0,t_0)$ and $v(p_0, t_0)$ .", "$\\mathcal {E}(p_0,t_0)$ is defined as the intersection of the plane normal to the iso-$t$ (for $t=t_0$ ) curve of $\\bar{\\mathcal {E}}$ at $\\bar{\\mathcal {E}}(p_0,t_0)$ with the iso-$t$ (for $t=t_0$ ) curve of $\\mathcal {E}$ which is nothing but $C_{t_0}$ .", "Recall that $C_{t_0}$ is given by $\\sigma (u(p, t_0), v(p, t_0), t_0)$ where $u, v, t$ obey Eq.", "REF .", "Henceforth, we will suppress the notation that $u,v, \\bar{u}$ and $\\bar{v}$ are functions of $p$ and $t$ .", "Also, all the evaluations will be understood to be done at parameter values $(p_0,t_0)$ .", "The tangent to iso-$t$ curve of $\\bar{\\mathcal {E}}$ at $(p_0,t_0)$ is given by $\\frac{\\partial \\bar{\\mathcal {E}}}{\\partial p} =\\frac{\\partial \\sigma }{\\partial u} \\frac{\\partial \\bar{u}}{\\partial p} + \\frac{\\partial \\sigma }{\\partial v} \\frac{\\partial \\bar{v}}{\\partial p}$ Hence, $\\mathcal {E}(p_0,t_0)$ is the solution of simultaneous system of equations REF and  REF $ \\left< \\sigma (u, v ,t_0) - \\sigma (\\bar{u}, \\bar{v}, t_0) , \\frac{\\partial \\bar{\\mathcal {E}}}{\\partial p} \\right> = 0$ Eq.", "REF and Eq.", "REF give us a system of two equations in two unknowns, $u$ and $v$ and hence can be put into NR framework by computing their first order derivatives w.r.t $u$ and $v$ .", "For any given parameter value $(p_0,t_0)$ , we seed the NR method with the point $(\\bar{u}(p_0,t_0), \\bar{v}(p_0,t_0))$ and solve Eq.", "REF and Eq.", "REF for $(u(p_0,t_0), v(p_0,t_0))$ and compute $\\mathcal {E}(p_0,t_0)$ .", "Having computed $\\mathcal {E}(p,t)$ we now compute first order derivatives of $\\mathcal {E}$ assuming that they exist.", "In order to compute $\\frac{\\partial \\mathcal {E}}{\\partial p}$ , we differentiate Eq.", "REF and Eq.", "REF w.r.t.", "$p$ to obtain $&\\left< \\frac{\\partial \\hat{N}}{\\partial u} \\frac{\\partial u}{\\partial p} + \\frac{\\partial \\hat{N}}{\\partial v} \\frac{\\partial v}{\\partial p}, V \\right> + \\left< \\hat{N} , \\frac{\\partial V}{\\partial u} \\frac{\\partial u}{\\partial p} + \\frac{\\partial V}{\\partial v} \\frac{\\partial v}{\\partial p} \\right>=0 \\\\\\nonumber &\\left< \\frac{\\partial \\sigma }{\\partial u} \\frac{\\partial u}{\\partial p} + \\frac{\\partial \\sigma }{\\partial v} \\frac{\\partial v}{\\partial p} - \\frac{\\partial \\sigma }{\\partial u} \\frac{\\partial \\bar{u}}{\\partial p} + \\frac{\\partial \\sigma }{\\partial v} \\frac{\\partial \\bar{v}}{\\partial p}, \\frac{\\partial \\bar{\\mathcal {E}}}{\\partial p}\\right> \\\\&+ \\left< \\sigma (u, v ,t_0) - \\sigma (\\bar{u}, \\bar{v}, t_0) , \\frac{\\partial ^2 \\bar{\\mathcal {E}}}{\\partial p^2} \\right>= 0 $ Eq.", "REF and Eq.", "give a system of two equations in two unknowns, viz., $\\frac{\\partial u}{\\partial p}$ and $\\frac{\\partial v}{\\partial p}$ and can be put into NR framework by computing first order derivatives w.r.t.", "$\\frac{\\partial u}{\\partial p}$ and $\\frac{\\partial v}{\\partial p}$ .", "Note that Eq.", "REF and Eq.", "also involve $u$ and $v$ whose computation we have already described.", "After computing $\\frac{\\partial u}{\\partial p}$ and $\\frac{\\partial v}{\\partial p}$ , $\\frac{\\partial \\mathcal {E}}{\\partial p}$ can be computed as $\\frac{\\partial \\sigma }{\\partial u} \\frac{\\partial {u}}{\\partial p} + \\frac{\\partial \\sigma }{\\partial v} \\frac{\\partial {v}}{\\partial p}$ .", "$\\frac{\\partial \\mathcal {E}}{\\partial t}$ can similarly be computed by differentiating Eq.", "REF and Eq.", "REF w.r.t.", "$t$ .", "Higher order derivatives can be computed in a similar manner." ], [ "Computation of seed surface", "The seed surface is constructed by sampling a few points on the envelope and fitting a tensor product B-spline surface through these points.", "For this, we first sample a few time instants, say, $T =\\lbrace t_1, t_2, \\ldots , t_n \\rbrace $ from the time interval of the sweep.", "For each $t_i \\in T$ , we sample a few points on the curve-of-contact $C_{t_i}$ .", "For this, we begin with one point $p$ on $C_{t_i}$ and compute the tangent to $C_{t_i}$ at $p$ , call it $\\mathcal {T}_{C_{i}}(p)$ .", "$p+\\mathcal {T}_{C_{i}}(p)$ is used as a seed in Newton-Raphson method to obtain the next point on $C_{t_i}$ and this process is repeated.", "While we do not know of any structured way of choosing the number of sampled points, in practice even a small number of points suffice to ensure that the Assumption  is valid." ], [ "NR formulation for edges and vertices of $\\mathcal {E}$", "An NR formulation for edges and vertices of $\\mathcal {E}$ can be obtained in a manner similar to that for faces of $\\mathcal {E}$ which we described in subsection REF .", "In order to obtain a procedural parametrization for edges of $\\mathcal {E}$ , again seed curves need to be computed." ], [ "Discussion", "In this work, we have proposed a novel computationally efficient test for detecting anomalies on the envelope.", "This has been achieved through a delicate mathematical analysis of an `invariant'.", "We have provided a rich procedural framework for computing the Brep of the envelope along with its accurate parametrization.", "Another contribution is a natural correspondence between the geometric/topological entities of the Brep envelope and that of the Brep solid.", "This framework has been implemented using the ACIS kernel [12] and has been used to produce the running examples of this paper.", "Ongoing work includes, for example, extending the proposed procedural framework to handle (i) deeper topological information of the Brep envelope, (ii) swept edges (an edge on the solid sweeping a face on the envelope) and so on (iii) faces meeting with $G^0$ continuity (i.e.", "sharp edges).", "We also to plan to extend the detection of anomalies to above settings and further, trim the appropriate part to obtain the final envelope.", "Another exciting future direction would be to analyse sweeps in which some numerical invariant associated with a a curve of contact, varies over time.", "For example, one may imagine sweeping a torus along a trajectory where the number of components of the curve of contact changes over time.", "In such a case, one would like to efficiently compute deeper topological invariants, say genus, of the envelope.", "Our mathematical analysis coupled with a Morse-theoretic analysis appears promising." ] ]

[ [ "Black Topologies Production in Extra Dimensions" ], [ "Abstract The configuration resulting after a collision of gravitational sources in a higher dimensional space with extra dimensions is investigated.", "Evidence is found that as the energy increases, there is a phase transition in the topology of the black object that is being formed: from the Black Hole to the Black String topology.", "An intuitive mechanism for the way the transition takes place is being proposed.", "The transition occurs at a finite value of the energy where an upper and a lower bound is found.", "Furthermore, at low energies the compact dimension behaves as an extended one while at high energies the extra dimension seems to decouple.", "Finally, the implications about the Gregory- Laflamme instability, the implications to the accelerators as well as holographic implications are being discussed." ], [ "Context", "We analyze the problem of a black object formation in the presence of compact-extra dimensions in flat backgrounds when shock-waves collide.", "In particular, we attempt to follow the evolution of the topology [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11] of the resulting (black) object as the energy involved in the collision, changes.", "We also estimate the entropy associated with the corresponding trapped surface in the spirit of [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25].", "For concreteness we consider five extended and one compact dimensions where an analytic shock wave solution in closed form is possible.", "The literature is rich regarding the investigation of shock-waves in gravity.", "For instance, in flat spaces they have been studied from [26], [27], [28], [29], [30], [31], [32], [33], [12], [34], [35], [36], [37] while for more generalized backgrounds they have been studied in [17], [16], [38], [39], [40], [19], [41], [42], [32], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52].", "In this work it is argued that the there exists a finite energy of the shock where the trapped surface and as a consequence the produced Black Hole (BH) that will result immediately after, occupies all of the compact space.", "At that energy, a phase transition should appear: the localized apparent horizons should cover the compact dimension and merge yielding to a Black-String (BS) transition; that is the system seems to yield to a BH$\\rightarrow $ BS phase transition [4], [5], [6], [7], [8], [9], [53], [54], [55], [56], [57], [58], [59], [10], [11].", "The energy $E$ where this occurs, in the 5+1 (compact) dimensions that we work, should be found numerically.", "The analytical investigation of this work, provides the bound $0.8 \\lesssim \\mu _{mer}\\equiv \\frac{16 \\pi G_6}{L^3}E_{mer}\\lesssim 29=\\mbox{finite,} \\hspace{14.45377pt}L=2\\pi R$ where $G_6$ and $R$ are the Newton constants in six dimensions and the radius of the compact dimension respectively.", "The dimensionless parameter $\\mu _{mer}$ corresponds to the merging energy $E_{mer}$ where the topology transition takes place: the BH covers the whole compact dimension and becomes an (initially non-uniform) BS.", "The bound of (REF ) respects the numerical works of [60], [61], [62] for five space-time plus one compact dimensions.", "A detailed and pedagogical review on these topics is the work [63] which provides additional references while reference [64] examines the problem from a microscopical point of view (see also the related reviews [65], [66])." ], [ "Setting up the problem", "Trapped surfaces are created when two shocks collide.", "The one shock moves along $x^-$ and we call it $\\phi _{+}$ to distinguish it from the second that moves along $x^+$ and which we call $\\phi _{-}$ .", "In our case the shocks will be taken identical and hence the subscripts will be soon dropped A more complete set of notes on the theory of trapped surfaces may be found in the appendices of [14]..", "In terms of metrics before the collision, one then has $ds^2&= -2dx^+dx^-+ dx^idx^i+(R d\\theta )^2 \\\\ &+ \\lbrace \\phi _+(R\\theta ,r)\\delta (x^+)(dx^+)^2 +(+\\leftrightarrow -) \\rbrace , \\hspace{7.22743pt}x_{\\pm }<0$ where $i=1,2,3$ , $r=\\sqrt{(x^i)^2}$ , $x^{\\pm }=(x^0\\pm x^4)/\\sqrt{2}$ , $\\theta \\in [-\\pi ,\\pi ]$ and $\\phi _+&=\\frac{2^{3/2} E G_6}{\\pi R}\\frac{1}{r} \\left( 1+\\frac{1}{e^{\\frac{r}{R}+i \\theta }-1} +\\frac{1}{e^{\\frac{r}{R}-i \\theta }-1} \\right) \\\\&=\\frac{2^{3/2} E G_6}{\\pi R}\\frac{1}{r} \\frac{\\sinh (r)}{\\cosh (r)-\\cos (\\theta )}$ A few remarks on $\\phi _+$ of (REF ) follow: It is periodic with respect to the compact dimension $\\theta $ as should and has $\\theta \\leftrightarrow -\\theta $ symmetry.", "It satisfies $\\nabla _{\\perp }^2\\phi _+\\equiv (\\nabla _i^2+1/R^2\\partial ^2_{\\theta })\\phi _+=-16\\sqrt{2}\\pi G_6 E/R \\delta ^{(3)}($r$)\\delta (x^+)\\delta (\\theta )$ In order to verify it one has to note that the fraction that involves the trigonometric quantities, in the limit where $r\\rightarrow 0$ results to $2\\pi \\delta (\\theta )$ ..", "Taking into account that $R_{++}=-\\frac{1}{2}\\nabla _{\\perp }^2\\phi _+=8\\pi G_6 T_{++}$ it is deduced that the shock $\\phi _{+}$ is associated with a point-like stress-tensor $T_{++}=E\\sqrt{2}/R \\delta ^{(3)}($r$)\\delta (x^+)\\delta (\\theta )$ which moves with the speed of light along $-x^4$ axis.", "It happens that $\\phi _+$ satisfies the Poisson equation of gravity in 3 (non compact) + 1 (compact) dimensions.", "At large distances $r\\gg R$ it behaves as $\\sim 1/r$ , that is as in Newtons law in 3D.", "On the other hand, al small distances $r\\ll R$ and assuming that the measurement of the gravitational potential is restricted on some brane with $\\theta =0$ in the spirit of [67], then (REF ) behaves as $\\sim R/r^2$ .", "This behavior, coincides with the ideas of [68], [69], [70].", "However, $\\phi _+$ here will be used in a different context.", "The ordering of limits $\\lim _{r \\rightarrow 0}$ and $\\lim _{\\theta \\rightarrow 0}$ does not commute.", "Associated with the shock-wave $\\phi _{\\pm }$ in (REF ), we parametrize (half of the) trapped surface $S_{\\pm }$ by $x^{\\pm }=0 \\hspace{14.45377pt} x^{\\mp }+\\frac{1}{2}\\psi _{\\pm }(\\theta ,r)=0$ where $\\psi _+$ remains to be determined.", "The function $\\psi _+$ satisfies the following differential equation $\\nabla ^2_{\\perp } (\\psi _{\\pm }-\\phi _{\\pm })=0.$ It is pointed out that $(\\nabla ^2_{\\perp } ) \\phi _{\\pm }$ provides a source term for $(\\nabla ^2_{\\perp } ) \\psi _{\\pm }$ .", "The missing ingredient is the boundary conditions and are given by $\\psi _{\\pm } \\Big |_C=0 \\hspace{28.90755pt} \\sum _{i=R\\theta ,r}\\left[ \\nabla _i \\psi _+ \\nabla _i \\psi _- \\right] \\Big |_{C}=8$ for some curve $C$ which defines the boundary of the trapped surface and where both, $S_+$ and $S_-$ end.", "The produced entropy is then bounded below by the area of the surface obtained by adjoining the two pieces of the trapped surface associated with each of the shocks.", "It is given by $S_{prod} \\ge S_{trap}=2\\times \\frac{R}{4G_6} \\int _C d\\theta d^3 {\\bf x} = \\frac{2 \\pi R}{3G_6} \\int _{\\theta _1}^{\\theta _2} r^3(\\theta ,E)d\\theta $ where the (generalized) curve $C$ defines the boundary of the trapped surface $S_+$ and $S_-$ which are identical; thus the overall factor of 2.", "The integral with respect to the non compact direction gives $r^3$ when considering a head-on collision.", "Typically, $\\theta _1$ , $\\theta _2$ and $r$ carry the information of the shock $\\phi $ We drop the subscripts $+,-$ from $\\phi $ 's and $\\psi $ 's from now on..", "The boundary conditions define a curve $C$ $r(\\theta ,E)$ which is explicitly given below, in equation (REF ) in two extreme limits.", "Evidently, the dimensionless quantity $EG_6/R^3$ sets the high and the low energy limit of the process.", "The last step would be to (numerically) solve equation (REF ) for $r$ and integrate for several (but fixed) values of the parameter $x \\equiv R^3 \\pi / EG_6.$ In what follows, we will derive analytical results for the two extreme cases of high and low energies." ], [ "BH $\\rightarrow $ BS Phase Transition", "In this section it is argued that the resulting system covers the whole compact dimension at a finite energy and passes from a BH phase to a BS phase.", "This is done by analyzing the low and the high energy asymptotics.", "In the low energy asymptotics we find a trapped horizon with topologyIn what follows, we use the symbol $D^4$ (and $D^3$ ) for the 4-disk (and 3-disk) imbedded in (ignoring time) $\\mathbb {R}^{4+1}$ where plus one refers to the compact dimension.", "We prefer the symbol $D^n$ instead of the more traditional $B^n$ (unit ball) in order to highlight that these disks are imbedded in a higher dimensional space.", "For instance $D^4$ can be thought as the intersection of a 5-ball with the $x^1,x^2,x^3$ hyperplane.", "$D^4$ .", "After the horizon will obtain rapidity dependenceThat is after the horizon will expand along $x^4$ ; the direction where the shocks are moving initially., it will presumably evolve to an $ S^4$ horizon (likely to a Schwarzschild BH).", "At high energies on the other hand, we find $D^3\\times S^1$ to be evolved to $S^3\\times S^1$ ; hence the BH $\\rightarrow $ BS phase transition.", "These limiting geometries that we find below after the shock-waves collide, agree with the expected behavior in the literature [4].", "The starting point is the general solution to (REF ) which has the form $\\psi =\\phi +\\phi _h, \\hspace{7.22743pt}\\mbox{where} \\hspace{7.22743pt} \\nabla _{\\perp }^2\\phi _h=0.$ and where all the three functions generally depend on the energy $E$ .", "For the two extreme cases that we are interested, the solutions $\\phi _h$ of the homogeneous degenerate to two (different) constants respectively.", "In this case, the second (non-linear) boundary condition of (REF ), becomes $& (-2u + 2u \\cos (\\theta )\\cosh (u)-2 \\cos (\\theta )\\sinh (u)+\\sinh ^2(2u))^2\\\\ &-4 x^2 u^4(\\cos (\\theta )-\\cosh (u))^4 \\\\ &+4 u^2\\sin ^2(\\theta )\\sinh ^2(u)=0,\\hspace{7.22743pt}u=r/R.$ Figure: Trapped boundary for the two distinct boundary conditions (); red and blue curves correspond to the first and second boundary condition respectively.", "Top panel: φ h x≫1 =C x≫1 =-2 7/6 Rx -1/3 \\phi _h^{x\\gg 1}=C^{x\\gg 1}=-2^{7/6} R x^{-1/3} and x=1,8,10 2 ,10 3 x=1,8,10^2,10^3.", "The pairs of curves shrink as xx increases and soon they merge (third pair of curves for x=8x=8; low energy limit that results to a BH).", "Bottom panel: φ h x≪1 =C x≪1 =-2 3/2 Rx -1/2 \\phi _h^{x\\ll 1}=C^{x\\ll 1}=-2^{3/2} R x^{-1/2} and x=0.1x=0.1, 0.0350.035 (dashed plots), 0.020.02.", "The pairs of curves move upwards as xx decreases and soon they merge (top pair of curves with x=0.02x=0.02; high energy limit that results to a BS)." ], [ "Limiting Behaviours", "I.", "Low Energy Asymptotics: $x\\gg 1$ .", "The claim is that the choice $\\phi _h\\equiv C_x=-2^{7/6} R x^{-1/3}$ and the boundary $C: u^2+\\theta ^2=2^{4/3}x^{-2/3}$ satisfy both of the conditions of (REF ).", "Indeed, the second boundary condition for large $x$ yields $C: x^2(u^2+\\theta ^2)^3=16$ while the first one yields $u^2+\\theta ^2=-2^{5/2}R/xC_x$ from where one identifies that $\\phi _h\\equiv C_x=-2^{7/6} R x^{-1/3}$ .", "This configuration (BH) is shown in the top plot of fig.", "REF and REF .", "The (lower) entropy bound may now be found.", "Using (REF ) immediately yields $S_{trap} & = \\frac{2 \\pi R^4}{3G_6} 2\\int _{0}^{2^{2/3}x^{-1/3}} (2^{4/3}x^{ - 2/3}-\\theta ^2)^{3/2}d\\theta \\\\&=2^{2/3} \\pi ^2 \\frac{ R^4}{G_6} x^{-4/3}=(2 \\pi )^{2/3}(E^4G_6)^{1/3}.$ There are two related observations about this result.", "The first one is that the $S_{prod}|_{x \\gg 1}$ is independent from the radius of the compact dimension $R$ .", "The second observation is that the result agrees exactly with the result of Giddings and Eardley [12] for $D=6$ extended space-time dimensions and which states that in $D$ dimensions the trapped entropy is $S_{prod}^D \\sim \\frac{1}{G_D} \\left(\\frac{EG_D}{\\Omega _D}\\right)^{\\frac{D-2}{D-3}}$ with $\\Omega _D$ the solid angle.", "The physics behind this result is that at low energies, the dynamics are not capable to resolve the finiteness of the compact dimension.", "Therefore, the extra dimension behaves as being (also) extended.", "In other words, the resulting BH has size much smaller than $R$ .", "This statement, according to (REF ), is consistent with $x\\gg 1$ and the fact that the final BH will have a small (presumably) Schwarzschild horizon $r_h \\sim G_6E/R^2 \\ll R$ (see [4]).", "II.", "High Energy Asymptotics: $x\\ll 1$ .", "The claim is that the choice $\\phi _h\\equiv C_x=-2^{3/2} R x^{-1/2}$ and the boundary $C: u=x^{-1/2}$ satisfy both of the conditions of (REF ).", "Indeed, the second boundary condition for large $x$ and yields $C: x^2 u^4=1$ while the first one $2^{3/2}x^{-1} R u+C_x=0$ from where one identifies that $\\phi _h\\equiv C_x=-2^{3/2} R x^{-1/2}$ .", "The trapped boundary is evidently independent on $\\theta $ and yields to the BS configuration.", "This is depicted on the bottom plot of fig.", "REF and REF for several large values of $E$ .", "The $S_{prod}$ is then trivial to find and yields $&S_{trap}= \\frac{2\\pi R^3}{3G_6} \\left(x^{-1/2} \\right)^3 \\times R\\int _{-\\pi }^{\\pi }\\theta = \\\\ &\\frac{2\\pi }{3} \\frac{R^3}{G_6} x^{-3/2} L = \\frac{8}{3\\sqrt{2}} \\pi (E^3G_5)^{1/2}, \\hspace{7.22743pt} G_5\\equiv \\frac{G_6}{L} .$ The result agrees with the result of [12], equation (REF ), with $D=5$ and an effective Newton's constant $G_5=G_6/L$ and also the expectations of [4], [71]This work deals with 4 extended and one compact dimensions..", "The physics behind this result becomes clear once we make the logical hypothesis that the final state will be a Schwarzschild object in the sense that its size ($r_h$ ) will increase with the energy and occupy the whole compact dimension.", "In fact, for high enough energies, it will be true that $r_h\\gg R$ and hence $R$ may be neglected and can appear in the expression of the entropy only trivially.", "Indeed the compact dimension appears as a product space; this is the BS solution.", "The only trace of the compact dimension is in the effective Newton's constant which changes from $G_6$ to $G_5\\equiv \\frac{G_6}{2 \\pi R}$ ." ], [ "Possible Paths: BH$\\rightarrow $ BS", "The solution $\\phi _h=C_x=$ constant (see (REF )) yields to two generally distinct families (that are indexed by $x$ ) of trapped boundaries.", "The one family is given by the curves defined by the first and the other by the second boundary condition of (REF ).", "The two families coincide at large $x$ (small $E$ ; see top plots of fig.", "REF ) and also at small $x$ (large $E$ ; see bottom plots of fig.", "REF ); only in these two cases the trapped surfaces of fig.", "REF are the desired solutions to the boundary value problem defined by equations (REF ) and (REF ).", "Figure: As the energy increases, it is expected that the“nearby\" BHs will grow and approach each other.On the other hand, as $x$ decreases, the homogenous solution $\\phi _h^x$ which we explicitly index by $x$ , traces a path in the space of functions.", "It begins from $\\phi _h^{x\\gg 1}=C^{x \\gg 1}=-2^{7/6} R x^{-1/3}$ (low energies) and ends to $\\phi _h^{x\\ll 1}=C^{x\\ll 1}=-2^{3/2} R x^{-1/2}$ (high energies) corresponding to the top and bottom plots of fig.", "REF respectively.", "It is instructive, to plot the curve $C$ resulting from the first boundary condition for $\\phi _h^{x\\ll 1}=C^{x\\ll 1}$ (see fig.", "REF ) and the second boundary condition for $\\phi _h^x=C^{x\\gg 1}$ (see (REF ) and figs.", "REF and REF ) as $x$ varies.", "Although strictly speaking, this is correct only in the two extreme cases, we still may gain some intuition about the transition from BH to BS assuming is valid for all values of $x$ (energies).", "Figure: The curves result from equation ().", "The value of xx decreases from the top to the bottom plot.", "In the top plot, the trapped horizon which is not convex, covers the whole compact dimension because it extends up to values of θ=±π\\theta =\\pm \\pi .", "Increasing the energy further (two bottom plots), seems to yield to a phase transition: the resulting BH begins to wind the compact dimension changing the topology from D 4 D^4 to D 3 ×S 1 D^3 \\times S^1.", "This is a non-uniform BS topology.", "As the energy increases further (xx decreases), the curves move upwards ending up to straight lines independent on θ\\theta and matching those of fig.", "(uniform BS configuration).", "It is pointed out that the shaded areas consisting of the two (half) ellipses left and right of r=0r=0 are not a part of the trapped surface.", "There presence is to shield the image charges (see sec.", ").Figure: [t]Trapped boundary resulting from first condition of () with φ h x≪1 =C x≪1 =-2 3/2 Rx -1/2 \\phi _h^{x\\ll 1}=C^{x\\ll 1}=-2^{3/2} R x^{-1/2}.", "As energy increases (xx decreases), the curves move upwards ending up to straight lines and matching those of fig.", "(uniform BS configuration).", "During the xx evolution, the trapped area remains convex (in each subinterval: [(2n-1)π,(2n+1)π][(2n-1)\\pi ,(2n+1)\\pi ], n∈ℤn \\in \\mathbb {Z}) unlike the plots of fig.", ".In both, fig.", "REF and fig.", "REF the transition from BH$\\rightarrow $ BS is depicted and looks similar: the BH increases in size and merges with the nearby BHs and becomes a BS warping around the compact dimension.", "In both cases, it seems that there is a singularity on the topology of the trapped horizon which appears as a cusp (see [4]) at $\\theta =\\pm \\pi $ .", "Certainly, the information we have here is incomplete.", "It is likely that the truth should lay somewhere between these two possibilities.", "However, it is notable that in both cases, a cusp on the trapped horizon at the merging point seems to appear.", "Regarding the convexity or non-convexity of the trapped surface fig.", "REF shows convexity for the intermediate energies and seems to agree with the current intuition in the literature (see for instance fig.", "6 in [4]) about the way the transition to the BS takes place.", "Fig.", "REF on the other hand, shows non-convexity.", "In fact, after the BS stage, there seem to appear holes inside the surface (see shaded regions in fig.", "REF ); most likely due to the image charges (see section ).", "These holes are external to the trapped surface.", "We argue in the next section that most likely the transition has the (non-convex with holes) form of fig.", "REF ." ], [ "Towards the trapped boundary for all energies: finding $\\phi _h$", "The complete solution, if exists, should respect the $r$ , $\\theta \\leftrightarrow -\\theta $ and $\\theta \\leftrightarrow 2\\pi +\\theta $ symmetries.", "Thus, the only possible solution for $\\nabla _{\\perp }^2\\phi _h=0$ is $\\phi _h=$ constant which, according to the discussion of the previous section, does not work.", "Hence, we are forced to relax the condition $\\nabla _{\\perp }^2\\phi _h=0$ to $\\nabla _{\\perp }^2\\phi _h=$ “image charge\", provided that the “charge(s)\", lies outside the trapped surface and (also) respects the symmetries.", "In this case, the only possible solution consistent with the symmetries of the problem has the form $&\\psi =\\phi +C_x+\\frac{1}{x u}\\sum _i f_i \\\\&f_i \\equiv \\left[\\frac{C_i(x)\\sinh (u)}{\\cosh (u)-\\cos (\\theta -\\theta _i(x))}+(\\theta \\leftrightarrow -\\theta ) \\right], \\hspace{7.22743pt}u=r/R$ where all the $\\theta _i$ 's should lay outside $C$ .", "Two additional conditions are: (a) $\\lim _{x \\rightarrow \\infty }\\left[ C_x+2/x \\sum _i f_i/(1-\\cos (\\theta _i(x))\\right] \\rightarrow C^{x\\gg 1}$ (low energies; small trapped surface) and (b) $\\lim _{u, \\rightarrow \\infty ,x \\rightarrow 0}\\left[ C_x+1/(xu) \\sum _i f_i\\right] \\rightarrow C^{x\\ll 1}$ (high energies; large trapped surface).", "A few remarks about the possible form of the trapped surface follow.", "Let is consider the range $\\theta $ in $[-\\pi ,\\pi ]$ and take into account that the image charges are located at $r=0$ and $\\theta =\\pm \\theta _i$ .", "As $x$ decreases, the image charges should move to the right (left) if at a given $x$ , the charges are already located right (left) of the $r$ axis.", "In the extreme limit where $x\\rightarrow 0$ , the sources should move at $\\pm \\pi $ ; that is at $\\theta _i \\rightarrow \\pi $ .", "For instance, in fig.", "REF , the image charges at positive angles should lay (approximately) in the interval $\\theta \\in (2.25,\\pi )$ .", "Conditions (a) and (b) reproduce the low and the high energy limit behavior.", "Since $C^{x\\gg 1}<0$ and $C^{x\\ll 1}<0$ , conditions (a) and (b) imply that a subset of the image charges $C_i(x)$ (see (REF )) is possibly negative (a similar situation appears in [12]); i.e.", "the images correspond to negative energies.", "From an electrostatic analogy point of view, this is not a surprise as typically the image charges usually appear with opposite sign.", "Previous point, implies that at small $r$ there will be a repulsive force between the image charges and the actual charge (located left and right of say $\\theta =0$ ) causing the non-convexity of the trapped surface (see fig.", "REF ).", "At the same time, the image charges located left and right of (say) $\\theta =\\pi $ , will attract each other (being both negative!).", "These would create an (external to the trapped surface,) surface.", "This surface of the image charges isolates them from the the trapped surface once (see top plot of fig.", "REF ) the trapped surface reaches the whole compact dimension (see two bottom plots of fig.", "REF and in particular the shaded areas).", "The surface that shields the image charges, is centered at $\\theta =\\pm \\pi $ and shrinks to zero (see last plot of fig.", "REF ) as the energy tends to infinity.", "In other words, according to this scenario, there seem to exist holes inside the Black Holes.", "Concluding, we have argued that the topology should qualitatively change as in fig.", "REF contrary to the current intuition which is more compatible with fig.", "REF .", "The trapped boundary is certainly continuous but it appears a kink (cusp) at $\\theta =\\pi ,r \\ne 0$ and at the transition energy, it appears as not convex.", "The non-convexity is due to the repulsive action of the image charges.", "Certainly, a more thorough investigation is required to confirm or not our current intuition." ], [ "Conclusions", " In this work, we study the evolution of the topology of the black object that will be formed during a shock-wave collision in the presence of extra dimensions following the Penrose method of trapped surfaces.", "It is emphasized that this method provides a lower bound on the extend of the actual horizon (apparent horizon).", "This implies that the black objects that we have studied are at least as large as the Penrose method predicts.", "Consequently, our conclusions (see below) might apply for lower but certainly not higher energies.", "We find evidence that there will be a transition from the BH to the BS configuration as the energy increases.", "The transition occurs at a finite energy while the topology, based on our preliminary investigation, seems to exhibit a (cusp) singularity at the transition point.", "A mechanism of this transition is being proposed (see fig.", "REF ).", "In particular, we argue about the possibility that the non-uniform BSNon-uniform BS implies that the BS has energy greater but comparable with the one of the merging point.", "will contain a cavity around $\\theta =\\pi $ ; that is as far as possible from the position of the (localized) energy of the colliding shocks (at $r=\\theta =0$ ).", "This cavity surrounds the image charges and shrinks to zero as the energy increases; that is as the non-uniform BS tends to become a uniform BS.", "The entropyAnd possibly all the rest thermodynamical quantities.", "of the black object (BS) at high enough energies behaves as if the extra dimensions are absent.", "In other words the entropy depends only on the extended dimensions in this limit.", "This agrees with the expectations of [71]We would like to thank S. Giddings for his correspondence and for pointing out this particular issue.. A BH is being created at lower energies while a BS is created at higher energies while there exists a merger point at some critical value of the energy corresponding to some $x$ , called $x_m$ .", "According to the two top (coinciding) curves of the lower panel of fig.", "REF , $x_m>0.02$ .", "According to(REF )) $x_m>0.02$ implies that $16G_6E_m/L^3 \\lesssim 29$ is a lower bound for the merging energy (see (REF )).", "Similarly, using the upper panel of fig.", "REF , one observes that $x_h<8$ is a satisfactory bound where our analysis is (approximately) correct.", "This provides the higher bound on the energy that corresponds to the BH configuration.", "For $x=x_h=8$ , equation (REF ) implies $16G_6E_m/L^3\\gtrsim 0.8$ .", "These two (crude) bounds are the ones appearing in equation (REF ).", "This result is consistent with [60], [61], [62], [72], [10], [63] .", "On the other hand, according to the Gregory-Laflamme (GL) analysis [73], [74], [75] there is an energy $E_{GL}$ where for lower energies, the uniform BS becomes unstable (GL instability).", "In five extended plus one compact dimensions, the corresponding (defined analogously to (REF )) dimensionless parameter $\\mu _{GL}$ , results to an unstable uniform BS when $\\mu _{GL}\\lesssim 3$ .", "We argue that in physical processesReal life does not involve five extended dimensions but the argument still applies.", "this instability might not appear: a black object (BH or BS) will be generally formed through some scattering.", "Now, according to the present investigation, a uniform BS is being formed for energies that saturate the upper bound of (REF ) or even higher ($\\mu >29$ ) .", "For smaller energies, either a non-uniform BS or a BH is being formed.", "But this implies that there is no meaning in perturbing a uniform BS at energies (corresponding to) $\\mu _{GL}=3$ or lower as this geometry can never be created (through a scattering) at this low energy; the perturbation of a uniform BS at $\\mu _{GL}=3$ thus seems to be meaningless Formally, one may write a uniform BS solution for any small energy but it seems that dynamically, a low energy (almost) uniform BS can not be created.", "in this set up.", "The BS configuration obviously is larger and hence $S_{BS}>S_{BH}$ .", "There is no meaning to compare the two for fixed energies $E$ as they are created and exist for different collision energy $2E$ .", "Figures REF and REF suggest that there exist an intermediate region.", "This corresponds to a non-uniform BS creation.", "Once the energy is increased further, it becomes uniform.", "Hence, the entropy inequalities become $S_{BS}>S_{non\\hspace{0.72229pt}un.", "}>S_{BH}$ .", "Maybe a (Un.", "BS)$\\rightarrow $ (Non-Un.", "BS)$\\rightarrow $ (BH) transition is possible after the BS radiates enough energy but not the other way around.", "This does not contradict the maximum entropy principle as the total entropy, taking into account the entropy carried by the radiation, should increase in a BS$\\rightarrow $ BH process.", "If extra dimensions are present, then the produced entropy bound $S_{trap}$ will generally be different from $S_{trap} \\sim (\\sqrt{s})^2$ ($\\sqrt{s}$ is the c.m.", "energy; see (REF )) which applies for the real world, that is for $\\mathbb {R}^{1,3}$ space-time.", "Equation $S_{trap} \\sim (\\sqrt{s})^2$ applies only in the extreme (high energy) limit where $G_D E/(R_i)^{D-3}\\gg 1$ for all $i=1,2,...D-4$ and hence all the extra dimensions decouple.", "For lower energies, more extra dimensions will contribute to the energy dependence of $S_{trap}$ and hence to the produced entropy $S_{prod}$That is $D$ in (REF ) equals $4+n$ where $n$ the number of compact dimensions that satisfy $G_D E/(R_i)^{D-3}\\ll 1;i=1,...,n$ ..", "Hence, if the final entropy in a collision as a function of $\\sqrt{s}$ is estimated, it could yield to information about the presence of extra dimensions [76].", "The same reasoning applies to the AdS/CFT calculations which estimate total multiplicities at the LHC [14], [19], [20], [21], [22].", "Taking into account that the full string-theory is 10-dimensional as there exists a compact 5-dimensional compact manifold that surrounds the $AdS_5$ space, implies a possible change in the results of these works depending on the energy range of interest.", "It would be interesting to find the full $\\phi _h^x$ even numerically and trace the trajectory of the topology as the energy changes.", "In fact, the authors of [23], [24] which apply the numerical methods devised in [77], have already developed techniques that may solve the boundary value problem of finding the trapped surface.", "Then, with the full (trapped) solution at hand one could investigate the proposals of this work and also search for any possible discontinuities in the thermodynamical quantities.", "In particular one could search for a third order phase transition in the entropy analogously to [4], [78], [79].", "We leave this for a future investigation.", "We would like to thank Y. Constantinou, S. Giddings, E. Kiritsis, B. Kol, R. Meyer, K. Sfetsos, N. Toumbas and especially V. Niarchos for informative discussions.", "We also would like to thank The Un.", "of Cyprus, LPTENS and The Un.", "of Arizona for their warm hospitality during several stages of this effort.", "This work was partially supported by European Union grants FP7-REGPOT-2008-1-CreteHEP Cosmo-228644, and PERG07-GA-2010-268246 as well as EU program“Thalis\" ESF/NSRF 2007-2013 ." ] ]

[ [ "Flavor Physics in the LHC era: the role of the lattice" ], [ "Abstract We discuss the present status of global fits to the CKM unitary triangle using the latest experimental and theoretical constraints.", "For the required nonperturbative weak matrix elements, we use three-flavor lattice QCD averages from www.latticeaverages.org; these have been updated from Ref.", "[1] to reflect all available lattice calculations as of the \"End of 2011\".", "Because of the greater than 3 sigma disagreement between the extraction of |Vub| from inclusive and exclusive semileptonic b -> u l nu (l = e,mu) decays, particular emphasis is given to a clean fit in which we remove the information from these decays.", "Given current theoretical and experimental inputs, we observe an approximately 3 sigma tension in the CKM unitarity triangle that may indicate the presence of new physics in the quark-flavor sector.", "Using a model-independent parameterization of new-physics effects, we test the compatibility of the data with scenarios in which the new physics is in kaon mixing, in B-mixing, or in B -> tau nu decay.", "We find that scenarios with new physics in B -> tau nu decay or in B-mixing are approximately equally preferred.", "Finally, we interpret these results in terms contributions to Delta S = 2 and Delta B = 2 four-fermion operators.", "We find that the preferred scale of new physics (with Standard-Model like couplings) is in the few hundred GeV range." ], [ "Motivation", "The $B$ -factories and the Tevatron have produced a remarkable wealth of data needed to determine elements of the Cabibbo-Kobayashi-Maskawa (CKM) matrix and to search for new physics beyond the Standard Model CKM framework.", "Despite the great experimental success of the Standard Model, however, there is now considerable evidence for physics beyond the Standard Model, such as dark matter, dark energy, and neutrino masses.", "Generic new-physics models to explain such phenomena also lead to additional $CP$ -violating phases beyond the single one in the Standard Model; this would lead to apparent inconsistencies between independent determinations of the CKM matrix elements.", "Although there is presently reasonably good agreement with the Standard Model prediction of a single $CP$ -violating phase, as measured by global fits of the CKM unitarity triangle, some tensions have been observed [2], [3], [4], [5], [6], [7], [8].", "In this work we use the latest theoretical and experimental inputs to test the Standard-Model CKM framework in the quark flavor sector.", "We first use a global fit to the CKM unitarity triangle to quantify the tension with the Standard Model, and then identify within a largely model-independent framework the most likely sources of the new physics.", "The standard analysis of the unitarity triangle involves a simultaneous fit to several quantities: $\\varepsilon _K$ , $\\Delta M_{B_d}$ , $\\Delta M_{B_s}$ , time–dependent $CP$ asymmetry in $B\\rightarrow J/\\psi K_s$ ($S_{\\psi K} = \\sin (2\\beta )$ , where $\\beta $ is the phase of $V_{td}^*$ ),A discussion of penguin pollution in $S_{\\psi K}$ can be found in Ref.", "[9]; see also Refs.", "[10], [11], [12], [13], [14], [15], [16].", "direct $CP$ asymmetries in $B\\rightarrow D^{(*)} K^{(*)}$ ($\\gamma $ is the phase of $V_{ub}^*$ ) time dependent $CP$ asymmetries in $B\\rightarrow (\\pi \\pi , \\rho \\rho , \\rho \\pi )$ ($\\alpha = \\pi -\\beta -\\gamma $ ), ${\\rm BR} (B\\rightarrow \\tau \\nu )$ , $|V_{ub}|$ and $|V_{cb}|$ (from both inclusive and exclusive $b\\rightarrow (u,c)\\ell \\nu $ with $\\ell = e,\\mu $ ).", "We summarize the relevant inputs required for this analysis in Table REF .", "There are several choices for how to implement the constraints from $B\\rightarrow \\tau \\nu $ leptonic decay and $B_{d,s}$ mixing ($\\Delta M_{B_d}$ and $\\Delta M_{B_s}$ ) on the unitarity triangle because one can parameterize the nonperturbative weak matrix element contributions to these quantities in different ways.", "Certain combinations of lattice inputs are preferable, however, because they minimize correlations between the three different unitarity triangle constraints so that they can safely be neglected in the global fit.", "Let us now summarize the main considerations that lead to a reasonable choice of uncorrelated inputs.", "First of all it is important to include only one input with mass dimension 1 in order to eliminate correlations due to the determination of the lattice scale.", "Another important consideration is that the largest source of uncertainty in the $SU(3)$ –breaking ratios, $\\xi $ and $f_{B_s}/f_{B_d}$ , is the chiral extrapolation.", "Because the chiral logarithms are larger when the quark masses are lighter, the chiral extrapolation in the $SU(3)$ -breaking ratios is primarily due to the chiral extrapolation in the $B_d$ quantities (i.e.", "$f_{B_d}$ and $\\widehat{B}_d$ ).", "These ratios are, therefore, more correlated with $B_d$ rather than with $B_s$ quantities.", "Finally we note that the decay constants $f_{B_d}$ and $f_{B_s}$ have a stronger chiral extrapolation than the $B$ -parameters $\\widehat{B}_d$ and $\\widehat{B}_s$ .", "In view of these considerations we choose to describe $B_q$ mixing in terms of $f_{B_s}\\sqrt{\\widehat{B}_s}$ and $\\xi $ .", "We then need one additional input to describe ${\\rm BR} (B\\rightarrow \\tau \\nu )$ .", "Although in principle the choice of $\\widehat{B}_d$ is preferable because $f_{B_d}$ has mass dimension and is more correlated with $\\xi $ through statistics and the chiral extrapolation, we choose to use $f_{B_d}$ anyway.", "This is because two (mostly) independent 2+1 flavor determinations of $f_{B_d}$ are currently available while only one group has presented the calculation of the bag parameters $\\widehat{B}_d$ .", "Further, $f_{B_d}$ is known much more precisely and hence places a stronger constraint on the unitarity triangle.", "In our analysis, we write $\\Delta M_{B_d} \\propto \\left(\\frac{ f_{B_s} \\sqrt{\\widehat{B}_s}}{\\xi }\\right)^2 \\; , \\quad \\quad \\Delta M_{B_s} \\propto \\left(f_{B_s} \\textstyle \\sqrt{\\widehat{B}_s}\\right)^2 \\; ,\\quad \\quad {\\rm BR} (B\\rightarrow \\tau \\nu ) \\propto f_{B_d}^2 \\; .$ For completeness we point out that there is an alternative choice of inputs ($f_{B_s}/f_{B_d}$ , $\\widehat{B}_s/\\widehat{B}_d$ , $f_{B_s}$ and $\\widehat{B}_s$ ) for which correlations are again fairly small.", "Table: Lattice-QCD and other inputs to the unitarity triangle analysis.", "The determination of α\\alpha is obtained from a combined isospin analysis of B→(ππ,ρρ,ρπ)B\\rightarrow (\\pi \\pi ,\\; \\rho \\rho , \\; \\rho \\pi ) branching ratios and CPCP asymmetries .", "Details on the lattice-QCD averages are given in the \"Note on the correlations between the various lattice calculations\" at 𝚠𝚠𝚠.𝚕𝚊𝚝𝚝𝚒𝚌𝚎𝚊𝚟𝚎𝚛𝚊𝚐𝚎𝚜.𝚘𝚛𝚐\\texttt {www.latticeaverages.org}.", "Recent summaries of lattice-QCD progress on pion, kaon, charm, and bottom physics can be found in Refs.", ", , .Among all quantities required in the UT fit, the value of $|V_{ub}|$ is the most problematic.", "The determinations of $|V_{ub}|$ obtained from inclusive and exclusive semileptonic $b\\rightarrow u \\ell \\nu \\; (\\ell =e,\\mu )$ decays differ at the 3.3 $\\sigma $ level, and may indicate the presence of underestimated uncertainties.", "For the “standard fit\" presented in Sec.", "REF we use a weighted average of inclusive and exclusive $|V_{ub}|$ with an error rescaled by $\\sqrt{\\chi ^2/{\\rm dof}} = 3.3$ following the PDG prescription.", "In Sec.", "REF we also consider an alternative fit in which we omit the constraint from semileptonic $b\\rightarrow u \\ell \\nu \\; (\\ell =e,\\mu )$ decays.", "We consider this the “clean fit\" because all remaining inputs are on excellent theoretical ground.", "(Although there is also a slight disagreement between the determinations of $|V_{cb}|$ obtained from inclusive and exclusive semileptonic $b\\rightarrow c \\ell \\nu \\; (\\ell =e,\\mu )$ , the disagreement is less than $2\\sigma $ and we do not consider it a major concern.)" ], [ "Interpretation as New Physics", "Given the presence of a tension in the CKM unitarity triangle, we can use a model-independent approach to test the compatibility of the data with various new-physics scenarios.", "On general grounds, NP contributions to processes that appear at the 1-loop level in the SM ($K$ , $B_d$ and $B_s$ mixing) are expected to be sizable.", "On the other hand, tree–level SM processes do not usually receive large corrections.", "An exception to this statement are contributions to $B\\rightarrow \\tau \\nu $ (e.g.", "charged-Higgs exchange) and to exclusive $b\\rightarrow u \\ell \\nu \\; (\\ell =e,\\mu )$ decays (e.g.", "loop–induced effective right–handed $W-u_R-b_R$ interaction in the MSSM).", "New-physics contributions to $B_s$ mixing are mostly decoupled from the fit ($\\Delta M_{B_s}$ does not depend on $\\rho $ and $\\eta $ ) and will not be considered here.", "Given these considerations, in Sec.", "we consider scenarios in which the new physics is either in kaon mixing, $B_d$ -mixing, or $B\\rightarrow \\tau \\nu $ (details of the analysis method are given in Ref. [8]).", "In Sec.", "we consider the possibility of new physics effects in exclusive $b\\rightarrow u \\ell \\nu \\; (\\ell =e,\\mu )$ decays.", "We adopt a model-independent parametrization of new physics effects in the three observables: $|\\varepsilon _K^{\\rm NP}| &=& C_\\varepsilon \\; |\\varepsilon _K^{\\rm SM}| \\; , \\\\M_{12}^{d,{\\rm NP}} &=& r_d^2 \\; e^{2 i \\theta _d} \\; M_{12}^{d,{\\rm SM}} \\;, \\\\{\\rm BR} (B\\rightarrow \\tau \\nu )^{\\rm NP} &=& \\left(1- \\frac{\\tan ^2 \\beta \\; m_{B^+}^2}{m_{H^+}^2 (1+\\epsilon _0 \\tan \\beta )} \\right) {\\rm BR} (B\\rightarrow \\tau \\nu )^{\\rm SM} \\; \\\\&=&r_H \\; {\\rm BR} (B\\rightarrow \\tau \\nu )^{\\rm SM} \\; ,$ where in the Standard Model ($C_\\varepsilon , \\; r_H, \\; r_d )= 1$ and $\\theta _d = 0$ .", "In presence of non-vanishing contributions to $B_d$ mixing the following other observables are also affected: $S_{\\psi K_S} &=& \\sin 2 (\\beta + \\theta _d) \\; , \\\\\\sin (2 \\alpha _{\\rm eff}) &=& \\sin 2 (\\alpha - \\theta _d) \\; , \\\\X_{sd} &=& \\frac{\\Delta M_{B_s}}{\\Delta M_{B_d}} = X_{sd}^{\\rm SM} \\; r_d^{-2} \\; .$ When considering new physics in $B_d$ mixing we allow simultaneous variations of both $\\theta _d$ and $r_d$ .", "We find that new physics in $|M_{12}^d|$ has a limited effect on the tension between the direct and indirect determinations of $\\sin (2\\beta )$ ; as a consequence, our results for $r_d$ and $\\theta _d$ point to larger effects on the latter.", "Finally we interpret the constraints on the parameters $C_\\varepsilon , \\; r_d$ , and $\\theta _d$ in terms of generic new physics contributions to $\\Delta S = 2$ and $\\Delta B = 2$ four-fermion operators.", "The most general effective Hamiltonian for $B_d$ –mixing can be written as The Hamiltonians for $B_s$ – and $K$ –mixing are obtained by replacing $(d,b) \\rightarrow (s,b)$ and $(d,b) \\rightarrow (d,s)$ , respectively.", "${\\cal H}_{\\rm eff} = \\frac{G_F^2 m_W^2}{16 \\pi ^2} \\left( V_{tb}^{} V_{td}^*\\right)^2 \\left(\\sum _{i=1}^5 C_i O_i + \\sum _{i=1}^3 \\widetilde{C}_i \\widetilde{O}_i \\right) \\,,$ where $\\begin{tabular}{lcl}O_1 = ( \\overline{d}_L \\gamma _\\mu b_L) ( \\overline{d}_L \\gamma _\\mu b_L)& \\phantom{ciaciacia} &\\widetilde{O}_1 = ( \\overline{d}_R \\gamma _\\mu b_R) ( \\overline{d}_R \\gamma _\\mu b_R)\\cr O_2 = ( \\overline{d}_R b_L) ( \\overline{d}_R b_L) & &\\widetilde{O}_2 = ( \\overline{d}_L b_R) ( \\overline{d}_L b_R)\\cr O_3 = ( \\overline{d}^{\\alpha }_R b_L^\\beta ) ( \\overline{d}^{\\beta }_R b_L^\\alpha )& &\\widetilde{O}_3 = ( \\overline{d}^{\\alpha }_L b_R^\\beta ) ( \\overline{d}^{\\beta }_L b_R^\\alpha ) \\cr O_4 = ( \\overline{d}_R b_L) ( \\overline{d}_L b_R) & &O_5 = ( \\overline{d}^{\\alpha }_R b_L^\\beta ) ( \\overline{d}^{\\beta }_L b_R^\\alpha ) .\\cr \\end{tabular}$ Within the Standard Model, only the operator $O_1$ receives a non-vanishing contribution at a high scale $\\mu _H \\sim m_t$ .", "For our analysis we assume that all new physics effects can be effectively taken into account by a suitable contribution to $C_1$ : ${\\cal H}_{\\rm eff} = \\frac{G_F^2 m_W^4}{16 \\pi ^2} \\left( V_{tb}^{} V_{td}^*\\right)^2 C_1^{\\rm SM}\\left( \\frac{1}{m_W^2} - \\frac{e^{i\\varphi }}{ \\Lambda ^2} \\right) O_1 \\; ,$ where the minus sign has been introduced a posteriori (as we will see that the fit will point to new physics phases $\\varphi \\sim O(1)$ ).", "In this parametrization $\\Lambda $ is the scale of some new physics model whose interactions are identical to the Standard Model with the exception of an additional arbitrary $CP$ violating phase: $C_1 = C_1^{\\rm SM} \\left( 1 - e^{i\\varphi } \\frac{m_W^2}{\\Lambda ^2} \\right) \\;.$" ], [ "Unitarity Triangle Fit Results and Constraints on New Physics", "In this section we present the results obtained for the full fit and for the fits in which semileptonic decays (for the extraction of $|V_{ub}|$ and $|V_{cb}|$ ) are not used.", "For each set of constraints we present the fitted values of the CKM parameters $\\overline{\\rho }$ , $\\overline{\\eta }$ and $A$ .", "We also show the predictions for several interesting quantities (most importantly $S_{\\psi K}$ and ${\\rm BR} (B\\rightarrow \\tau \\nu )$ ) that we obtain after removing the corresponding direct determination from the fit.", "Finally we interpret the observed discrepancies in terms of new physics in $\\varepsilon _K$ , $B_d$ –mixing or $B\\rightarrow \\tau \\nu $ .", "In the upper panels of Figs.", "REF –REF , we show the global CKM unitarity triangle fit for the three sets of inputs that we consider (complete fit, no $V_{ub}$ fit, no $V_{qb}$ fit).", "In each figure, the black contours and $p$ –values in the top, middle and bottom panels correspond to the complete fit, the fit with a new phase in $B$ mixing (i.e.", "without using $S_{\\psi K}$ and $\\alpha $ ) and the fit with new physics in $B\\rightarrow \\tau \\nu $ (i.e.", "without using ${\\rm BR} (B\\rightarrow \\tau \\nu )$ ), respectively.", "For fits with a new phase in $B$ mixing we show the fit predictions for $\\sin (2\\beta )$ , while for the fits with new physics in $B\\rightarrow \\tau \\nu $ , we show the fit predictions for ${\\rm BR} (B\\rightarrow \\tau \\nu )$ .", "(Note that the individual contours in these figures never use the same constraint twice: in particular, the $B\\rightarrow \\tau \\nu $ allowed area is obtained by using $\\Delta M_{B_s}$ instead of the direct determination of $|V_{cb}|$ .)", "The lower panels of Figs.", "REF –REF show the interpretation of the tensions highlighted in the various fits in terms of possible new-physics scenarios.", "In the first panel of each figure we present the result of the two-dimensional fit in the $(\\theta _d,r_d)$ plane and in the second panel we map this allowed region onto the $(\\Lambda ,\\varphi )$ plane (see Eq.", "(REF )) under the assumption of new physics in $O_1$ only.", "We do not show the corresponding plots for new physics in $K$ mixing because these contributions cannot relieve the tension in the fit (as can be seen by the poor $p$ -values in Eqs.", "(REF ), (REF ) and (REF ))." ], [ "Standard Fit", "We include constraints from $\\varepsilon _K$ , $\\Delta M_{B_d}$ , $\\Delta M_{B_s}$ , $\\alpha $ , $S_{\\psi K}$ , $\\gamma $ , ${\\rm BR} (B\\rightarrow \\tau \\nu )$ , $|V_{cb}|$ and $|V_{ub}|$ .", "The overall $p$ -value of the Standard-Model fit is $ p = 6.2\\% $ and the results of the fit are $\\overline{\\rho }= 0.131 \\pm 0.018\\quad \\quad \\overline{\\eta }= 0.344 \\pm 0.013\\quad \\quad A = 0.818 \\pm 0.014\\; .$ The predictions from all other information when the direct determination of the quantity is removed from the fit are $& |V_{ub}| = (3.53 \\pm 0.13 ) \\; \\times 10^{-3} \\quad (0.4\\; \\sigma )\\\\& S_{\\psi K} = 0.827 \\pm 0.052 \\quad (2.7\\; \\sigma )\\\\& |V_{cb}| = (42.1 \\pm 1.0 ) \\; \\times 10^{-3} \\quad (0.8\\; \\sigma )\\\\& \\widehat{B}_K = 0.880 \\pm 0.11 \\quad (1.0\\; \\sigma )\\\\& f_{B_d} \\sqrt{\\widehat{B}_d} = (209.3 \\pm 4.8 ) \\; {\\rm MeV} \\quad (1.0\\; \\sigma )\\\\& {\\rm BR} (B\\rightarrow \\tau \\nu ) = (0.775 \\pm 0.067 ) \\; \\times 10^{-4} \\quad (2.8\\; \\sigma )\\\\& f_{B_d} = (280.", "\\pm 28. )", "\\; {\\rm MeV} \\quad (3.2\\; \\sigma )$ where we indicate the deviation from the corresponding direct determination in parentheses.", "The interpretation of the above discrepancies in terms of new physics in $K$ –mixing, $B_d$ –mixing and $B\\rightarrow \\tau \\nu $ yields $& \\hphantom{5\\;\\;} C_\\varepsilon = 1.15 \\pm 0.14 \\quad \\quad ( 1.0\\; \\sigma ,\\; p = 0.053) \\\\&{\\left\\lbrace \\begin{array}{ll}\\theta _d = - (6.7 \\pm 2.8)^{\\rm o}\\\\r_d = 0.97 \\pm 0.042\\\\\\end{array}\\right.}", "( 2.5\\; \\sigma ,\\; p = 0.33)\\\\&\\hphantom{5\\;\\;} r_H = 2.18 \\pm 0.44 \\quad \\quad ( 2.9\\; \\sigma ,\\; p = 0.57)\\; .$ In Figure REF we summarize these results and present the two dimensional $(\\theta _d,r_d)$ allowed regions and the interpretation of these results in terms of a minimal flavor violating new-physics scale." ], [ "Fit Without $|V_{ub}|$", "We include constraints from $\\varepsilon _K$ , $\\Delta M_{B_d}$ , $\\Delta M_{B_s}$ , $\\alpha $ , $S_{\\psi K}$ , $\\gamma $ , ${\\rm BR} (B\\rightarrow \\tau \\nu )$ and $|V_{cb}|$ .", "The overall $p$ -value of the Standard-Model fit is $ p = 3.6\\% $ and the results of the fit are $\\overline{\\rho }= 0.131 \\pm 0.018\\quad \\quad \\overline{\\eta }= 0.344 \\pm 0.013\\quad \\quad A = 0.818 \\pm 0.014\\; .$ The predictions from all other information when the direct determination of the quantity is removed from the fit are $& |V_{ub}| = (3.53 \\pm 0.13 ) \\; \\times 10^{-3} \\quad (0.4\\; \\sigma )\\\\& S_{\\psi K} = 0.878 \\pm 0.057 \\quad (3.0\\; \\sigma )\\\\& |V_{cb}| = (42.1 \\pm 1.0 ) \\; \\times 10^{-3} \\quad (0.8\\; \\sigma )\\\\& \\widehat{B}_K = 0.888 \\pm 0.11 \\quad (1.1\\; \\sigma )\\\\& f_{B_d} \\sqrt{\\widehat{B}_d} = (209.3 \\pm 4.8 ) \\; {\\rm MeV} \\quad (1.0\\; \\sigma )\\\\& {\\rm BR} (B\\rightarrow \\tau \\nu ) = (0.770 \\pm 0.067 ) \\; \\times 10^{-4} \\quad (2.9\\; \\sigma )\\\\& f_{B_d} = (281.", "\\pm 28. )", "\\; {\\rm MeV} \\quad (3.2\\; \\sigma )$ where we indicate the deviation from the corresponding direct determination in parentheses.", "The interpretation of the above discrepancies in terms of new physics in $K$ –mixing, $B_d$ –mixing and $B\\rightarrow \\tau \\nu $ yields $& \\hphantom{5\\;\\;} C_\\varepsilon = 1.16 \\pm 0.14 \\quad \\quad ( 1.1\\; \\sigma ,\\; p = 0.031) \\\\&{\\left\\lbrace \\begin{array}{ll}\\theta _d = - (9.3 \\pm 3.5)^{\\rm o}\\\\r_d = 0.98 \\pm 0.046\\\\\\end{array}\\right.}", "( 2.8\\; \\sigma ,\\; p = 0.39)\\\\&\\hphantom{5\\;\\;} r_H = 2.20 \\pm 0.45 \\quad \\quad ( 2.9\\; \\sigma ,\\; p = 0.45)\\; .$ In Figure REF we summarize these results and present the two dimensional $(\\theta _d,r_d)$ allowed regions and the interpretation of these results in terms of a minimal flavor violating new-physics scale." ], [ "Fit Without $|V_{ub}|$ and {{formula:2750de8f-3be9-41e4-bc5c-71ccb717c6bc}}", "The strategy for removing $|V_{cb}|$ by combining the constraints from $\\varepsilon _K$ , $\\Delta M_{B_s}$ , and ${\\rm BR} (B\\rightarrow \\tau \\nu )$ was proposed and is described in detail in Ref. [57].", "We include constraints from $\\varepsilon _K$ , $\\Delta M_{B_d}$ , $\\Delta M_{B_s}$ , $\\alpha $ , $S_{\\psi K}$ , $\\gamma $ and ${\\rm BR} (B\\rightarrow \\tau \\nu )$ .", "The overall $p$ -value of the Standard-Model fit is $ p = 2.4\\% $ and the results of the fit are $\\overline{\\rho }= 0.132 \\pm 0.018\\quad \\quad \\overline{\\eta }= 0.341 \\pm 0.013\\quad \\quad A = 0.829 \\pm 0.020\\; .$ The predictions from all other information when the direct determination of the quantity is removed from the fit are $& |V_{ub}| = (3.55 \\pm 0.13 ) \\; \\times 10^{-3} \\quad (0.3\\; \\sigma )\\\\& S_{\\psi K} = 0.914 \\pm 0.057 \\quad (2.9\\; \\sigma )\\\\& |V_{cb}| = (42.1 \\pm 1.0 ) \\; \\times 10^{-3} \\quad (0.8\\; \\sigma )\\\\& \\widehat{B}_K = 0.918 \\pm 0.18 \\quad (0.8\\; \\sigma )\\\\& f_{B_d} \\sqrt{\\widehat{B}_d} = (207.2 \\pm 5.4 ) \\; {\\rm MeV} \\quad (1.1\\; \\sigma )\\\\& {\\rm BR} (B\\rightarrow \\tau \\nu ) = (0.779 \\pm 0.070 ) \\; \\times 10^{-4} \\quad (2.8\\; \\sigma )\\\\& f_{B_d} = (280.", "\\pm 28. )", "\\; {\\rm MeV} \\quad (3.1\\; \\sigma )$ where we indicate the deviation from the corresponding direct determination in parentheses.", "The interpretation of the above discrepancies in terms of new physics in $K$ –mixing, $B_d$ –mixing and $B\\rightarrow \\tau \\nu $ yields $& \\hphantom{5\\;\\;} C_\\varepsilon = 1.20 \\pm 0.24 \\quad \\quad ( 0.78\\; \\sigma ,\\; p = 0.014)\\\\&{\\left\\lbrace \\begin{array}{ll}\\theta _d = - (12.", "\\pm 3.8)^{\\rm o}\\\\r_d = 0.95 \\pm 0.048\\\\\\end{array}\\right.}", "( 3.1\\; \\sigma ,\\; p = 0.73)\\\\&\\hphantom{5\\;\\;} r_H = 2.18 \\pm 0.44 \\quad \\quad ( 2.8\\; \\sigma ,\\; p = 0.36)\\; .$ In Figure REF we summarize these results and present the two dimensional $(\\theta _d,r_d)$ allowed regions and the interpretation of these results in terms of a minimal flavor violating new-physics scale." ], [ "Separate Treatment of Inclusive and Exclusive $|V_{ub}|$", "In this section we approach the full fit to the unitarity triangle from a different perspective: instead of averaging the extractions of $|V_{ub}|$ from inclusive and exclusive semileptonic $b\\rightarrow u\\ell \\nu \\; (\\ell = e,\\mu )$ decays (and inflating the resulting error), we take the $3.3\\sigma $ disagreement between the two determinations at face value.", "The fit therefore includes constraints from $\\varepsilon _K$ , $\\Delta M_{B_d}$ , $\\Delta M_{B_s}$ , $\\alpha $ , $S_{\\psi K}$ , $\\gamma $ , ${\\rm BR} (B\\rightarrow \\tau \\nu )$ , $|V_{cb}|$ and $|V_{ub}|^{\\rm excl}$ and $|V_{ub}|^{\\rm incl}$ .", "The overall $p$ -value of the fit is now very small ($p = 0.1\\% $ ) and the results we obtain are $\\overline{\\rho }= 0.136 \\pm 0.017\\quad \\quad \\overline{\\eta }= 0.349 \\pm 0.012\\quad \\quad A = 0.820 \\pm 0.014\\; .$ The predictions from all other information when the direct determination of the quantity is removed from the fit are $& |V_{ub}| = (3.53 \\pm 0.13 ) \\; \\times 10^{-3} \\quad (0.4\\; \\sigma )\\\\& S_{\\psi K} = 0.750 \\pm 0.029 \\quad (2.3\\; \\sigma )\\\\& |V_{cb}| = (42.25 \\pm 0.98 ) \\; \\times 10^{-3} \\quad (0.9\\; \\sigma )\\\\& \\widehat{B}_K = 0.838 \\pm 0.097 \\quad (0.7\\; \\sigma )\\\\& f_{B_d} \\sqrt{\\widehat{B}_d} = (209.3 \\pm 4.9 ) \\; {\\rm MeV} \\quad (1.0\\; \\sigma )\\\\& {\\rm BR} (B\\rightarrow \\tau \\nu ) = (0.813 \\pm 0.063 ) \\; \\times 10^{-4} \\quad (2.7\\; \\sigma )\\\\& f_{B_d} = (273.", "\\pm 27. )", "\\; {\\rm MeV} \\quad (3.1\\; \\sigma )$ where we indicate the deviation from the corresponding direct determination in parentheses.", "The interpretation of the above discrepancies in terms of new physics in $K$ –mixing, $B_d$ –mixing and $B\\rightarrow \\tau \\nu $ yields $& \\hphantom{5\\;\\;} C_\\varepsilon = 1.09 \\pm 0.13 \\quad \\quad ( 0.68\\; \\sigma ,\\; p = 0.00063) \\\\&{\\left\\lbrace \\begin{array}{ll}\\theta _d = - (3.4 \\pm 1.5)^{\\rm o}\\\\r_d = 0.96 \\pm 0.038\\\\\\end{array}\\right.}", "( 2.3\\; \\sigma ,\\; p = 0.0032)\\\\&\\hphantom{5\\;\\;} r_H = 2.08 \\pm 0.41 \\quad \\quad ( 2.7\\; \\sigma ,\\; p = 0.011)\\; .$ Figure REF summarizes these results.", "Note that none of the new physics scenarios we consider is able to lift the overall $p$ -value of the fit: the tension between the inclusive and exclusive determination of $|V_{ub}|$ , if taken at face value, can only be addressed by dedicated new-physics contributions.", "Alleviating the discrepancy between inclusive and exclusive $|V_{ub}|$ requires the introduction of interactions whose impact on exclusive $B\\rightarrow X_u \\ell \\nu $ decays is much larger than in inclusive ones.", "The introduction of a right–handed effective $\\overline{u}_R W\\hspace{-6.49994pt}/b_R$ coupling offers the most elegant solution of the “$V_{ub}$ puzzle” (see for instance Refs.", "[58], [59], [60], [61]).", "In this scenario we have: $V_{ub} \\; \\overline{u}_L W\\hspace{-6.49994pt}/b_L \\Longrightarrow V_{ub} \\; \\left( \\overline{u}_L W\\hspace{-6.49994pt}/b_L + \\xi _{ub}^R \\; \\overline{u}_R W\\hspace{-6.49994pt}/b_R \\right) \\; .$ The effective parameter $\\xi _{ub}^R$ affects all $b\\rightarrow u \\ell \\nu \\; (\\ell = e,\\mu ,\\tau )$ transitions: $\\left| V_{ub} \\right|_{\\rm incl} &\\Longrightarrow \\sqrt{1+ \\left| \\xi _{ub}^R\\right|^2} \\; \\left| V_{ub} \\right| \\; , \\\\\\left| V_{ub} \\right|_{\\rm excl} &\\Longrightarrow \\left| 1+ \\xi _{ub}^R \\right| \\; \\left| V_{ub} \\right| \\; ,\\\\{\\rm BR} (B\\rightarrow \\tau \\nu ) &\\Longrightarrow \\left| 1- \\xi _{ub}^R \\right|^2 \\; {\\rm BR} (B\\rightarrow \\tau \\nu ) \\; .", "$ The result of the fit to the unitarity triangle in which we allow $\\xi _{ub}^R$ , $\\theta _d$ and $r_d$ to vary simultaneouslyAlthough the introduction of right-handed $b\\rightarrow u \\ell \\nu $ currents can resolve the more than 3$\\sigma $ tension between $|V_{ub}|_{\\rm incl}$ , $|V_{ub}|_{\\rm excl}$ , and $B\\rightarrow \\tau \\nu $ , additional new physics in $B_d$ -mixing is needed to bring the global CKM unitarity-triangle fit into complete agreement.", "This is because, under the assumptions in Eqs.", "(REF )–(), inclusive $|V_{ub}|$ is affected minimally by new physics, so the tension between inclusive $|V_{ub}|$ and $\\sin (2\\beta )$ remains.", "The tenth entry in the upper panel of Fig.", "REF shows that even if exclusive $|V_{ub}|$ and $B\\rightarrow \\tau \\nu $ are removed from the fit, there is still a 3.4$\\sigma $ discrepancy.", "yields $\\xi _{ub}^R &= -0.245 \\pm 0.055 \\quad \\quad \\quad (4.0 \\sigma ) \\; ,\\\\\\theta _d &= - (4.8 \\pm 1.5)^{\\rm o} \\quad \\quad \\quad \\quad (3.2 \\sigma ) \\; ,\\\\r_d &= 0.978 \\pm 0.041 \\quad \\quad \\quad \\quad (0.5 \\sigma ) \\; .$ In the lower panel of Fig.", "REF we show the two–dimensional allowed regions in the $[\\xi _{ub}^R,\\theta _d]$ plane.", "From the direct inspection of Fig.", "REF , we see a clear correlation between $\\xi _{ub}^R$ and $\\theta _d$ (the allowed region is not a simple ellipsis with axes parallel to the coordinate axes).", "The origin of this correlation is two-fold.", "The inclusion of the rest of the fit allows for a more precise determination of $|V_{ub}|$ and, through Eqs.", "(REF )–(), of $\\xi _{ub}^R$ .", "The value of $\\xi _{ub}^R$ that comes from Eqs.", "(REF )–(), however, implies a value of $|V_{ub}|$ that is not optimal to lift the residual tension in $\\sin (2\\beta )$ .", "Finally we note that the impact of a $3\\%$ determination of ${\\rm BR} (B\\rightarrow \\tau \\nu )$ would be to reduce by a factor of two the longer axis in Fig.", "REF , while leaving the other axis unchanged." ], [ "Future Expectations", "In the near future, we expect improved lattice-QCD calculations to impact the CKM unitarity-triangle fit in several ways.", "Now that the uncertainty in the neutral kaon mixing matrix element $B_K$ is below 2%, the constraint on the UT from $\\epsilon _K$ is limited by the uncertainty in the Wolfenstein parameter $A$ (recall that $\\varepsilon _K \\propto A^4$ ).", "Currently $b\\rightarrow c\\ell \\nu $ decays offer the best determination of $A = |V_{cb}|/\\lambda ^2$ with an uncertainty of about $2\\%$ .", "Residual theoretical uncertainties on the extraction of $|V_{cb}|$ from inclusive $B \\rightarrow X_c \\ell \\nu $ decays will make it very hard to further reduce the error in $|V_{cb}|$ obtained from this approach.", "In the next few years, however, lattice-QCD calculations of the $B\\rightarrow D \\ell \\nu $ and $B \\rightarrow D^* \\ell \\nu $ form factors at nonzero recoil should allow the uncertainties in direct determinations of $|V_{cb}|$ to approach $\\mathrel {\\hbox{t}o 0pt{}\\hss }$$$ 1%$.$ An alternative extraction of $A$ is also possible via $\\Delta M_{B_s}$ ( $A \\propto \\Delta M_{B_s} / (f_{B_s} \\widehat{B}_s^{1/2})$ ).", "(In fact, for the first time there are now at least two $N_f = 2+1$ determinations of the quantities $\\xi $ , $f_{B_s} \\sqrt{\\widehat{B}_s}$ and $f_B$ .)", "Although this approach is presently limited by a lattice-QCD uncertainty of about $6\\%$ , future improvements in lattice-QCD calculations of will push the latter uncertainty to the $1\\%$ level, making this indirect determination of $A$ competitive with the direct extraction from semileptonic decays.", "In order to study the potential of this indirect method we consider the impact of a reduced uncertainty on $f_{B_s} \\widehat{B}_s^{1/2}$ , in conjunction with the expected Belle II/SuperB determinations of ${\\rm BR}(B\\rightarrow \\tau \\nu )$ with 50 ${\\rm ab}^{-1}$ , in our “no $V_{qb}$ ” scenario.", "The results are summarized in Table REF .", "Table: Impact of improved determinations of f B s B ^ s 1/2 f_{B_s} \\widehat{B}_s^{1/2} and BR (B→τν){\\rm BR} (B\\rightarrow \\tau \\nu ) on the no-V qb V_{qb} fit.", "We define δ s =δf B s B ^ s 1/2 \\delta _s = \\delta \\left[ f_{B_s} \\widehat{B}_s^{1/2}\\right] and δ τ =δ BR (B→τν)\\delta _\\tau = \\delta \\left[{\\rm BR} (B\\rightarrow \\tau \\nu )\\right].Finally, in Fig.", "REF we show the impact of a much improved determination of the angle $\\gamma $ .", "The super flavor factories (Belle II and SuperB) and BES III should be able to push the uncertainty on $\\gamma $ below $1^{\\rm o}$ ." ], [ "Discussion", "We collect the fit results obtained with different input selections and show the corresponding predictions for $\\sin (2\\beta )$ and ${\\rm BR} (B\\rightarrow \\tau \\nu )$ in Fig.", "REF .", "These tables are useful to get a clear picture of the stability of the observed tension against a variation of the inputs used.", "The main lessons to be learned from the fits described in the previous sections are: The CKM description of flavor and $CP$ violation displays a tension at the 3$\\sigma $ level, where most of the tension is driven by $B\\rightarrow \\tau \\nu $ and $S_{\\psi K}$ .", "The values of $|V_{ub}|$ obtained from inclusive and exclusive semileptonic $b\\rightarrow u \\ell \\nu \\; (\\ell = e,\\mu )$ differ by $3.3\\sigma $ .", "This disagreement is cause for serious concern, and may indicate the presence of underestimated theoretical uncertainties.", "Although the use of $|V_{ub}|$ in the UT fit is problematic, the tension persists even when the constraint from $|V_{ub}|$ is omitted.", "A possibly related problem is that the Standard-Model prediction for ${\\rm BR} (B\\rightarrow \\tau \\nu )$ from $f_B$ and $|V_{ub}|_{\\rm excl}$ (we obtain $(0.62 \\pm 0.11) \\times 10^{-4}$ ) is $3.2\\sigma $ below the direct experimental measurement (see Table REF ).", "These two $3\\sigma $ tensions in the $B\\rightarrow \\tau \\nu $ , $f_B$ , $|V_{ub}|$ system could be quite naturally resolved by a shift in the extraction of $|V_{ub}|$ , either due to new physics or to improved theoretical understanding of inclusive and exclusive semileptonic $b\\rightarrow u \\ell \\nu $ decays.", "Once $|V_{ub}|$ is removed from the fit, the tension is driven by the disagreement between $B\\rightarrow \\tau \\nu $ and $S_{\\psi K}$ , and can be alleviated by omitting either constraint.", "In fact, the tension in the $S_{\\psi K}$ prediction is reduced from $3.0\\sigma $ to $1.5\\sigma $ with the removal of $B\\rightarrow \\tau \\nu $ (see the first and ninth entries in the upper table in Fig.", "REF ).", "Similarly, the tension in the ${\\rm BR} (B\\rightarrow \\tau \\nu )$ prediction is reduced from $2.9\\sigma $ to $1.1\\sigma $ with the removal of $S_{\\psi K}$ (see the first and sixth entries in the lower table in Fig.", "REF ).", "Given the significant role played by BR($B\\rightarrow \\tau \\nu $ ) in causing the tension in the UT fit, the possibilities of statistical fluctuations and/or of underestimated systematic uncertainties in the theoretical and experimental inputs to the constraint from $B\\rightarrow \\tau \\nu $ should be given serious consideration.", "If we interpret the $3\\sigma $ tension in the UT fit in terms of new physics, the fit prefers new contributions to $B\\rightarrow \\tau \\nu $ and/or to $B_d$ mixing.", "Scenarios with new physics in kaon mixing are clearly disfavored (see the $p$ -values in Eqs.", "(REF -)).", "In terms of a new-physics model whose interactions mimic closely the SM (see Eq.", "(REF )), this tension points to a few hundred GeV mass scale.", "Even allowing for a generous model dependence in the couplings, it seems that such new particles, if the tension in the fit stands confirmed, cannot escape detection in direct production experiments." ], [ "Acknowledgements", "We would like to thank Christian Hoelbling and Urs Heller for their comments on this manuscript.", "We are especially grateful to Christine Davies for pointing out a missing renormalization factor in our interpretation of the $B^0_{(d,s)}$ -mixing matrix elements.", "Figure: Unitarity triangle fit with all constraints included.", "Quantities that are not used to generate the black contour are grayed out.Figure: Unitarity triangle fit without |V ub ||V_{ub}|.", "Quantities that are not used to generate the black contour are grayed out.Figure: Unitarity triangle fit without |V ub ||V_{ub}| and |V cb ||V_{cb}|.", "Quantities that are not used to generate the black contour are grayed out.Figure: Unitarity triangle fit with all constraints included (inclusive and exclusive V ub V_{ub} are included separately in the fit.", ").Figure: Unitarity triangle fit without |V ub ||V_{ub}|.", "We adopt a 1%1\\% uncertainty on the determination of γ\\gamma .Figure: Upper panel: Summary of sin(2β)\\sin (2\\beta ) determinations.", "The entry marked +++ (tenth from the top) corresponds to adding an hadronic uncertainty δΔS ψK =0.021\\delta \\Delta S_{\\psi K} = 0.021 to the relation between sin(2β)\\sin (2\\beta ) and S ψK S_{\\psi K}.", "Lower panel: Summary of BR (B→τν){\\rm BR} (B\\rightarrow \\tau \\nu ) determinations." ] ]

[ [ "Strongly magnetized cold electron degenerate gas: Mass-radius relation\n of the magnetized white dwarf" ], [ "Abstract We consider a relativistic, degenerate electron gas at zero-temperature under the influence of a strong, uniform, static magnetic field, neglecting any form of interactions.", "Since the density of states for the electrons changes due to the presence of the magnetic field (which gives rise to Landau quantization), the corresponding equation of state also gets modified.", "In order to investigate the effect of very strong magnetic field, we focus only on systems in which a maximum of either one, two or three Landau level(s) is/are occupied.", "This is important since, if a very large number of Landau levels are filled, it implies a very low magnetic field strength which yields back Chandrasekhar's celebrated non-magnetic results.", "The maximum number of occupied Landau levels is fixed by the correct choice of two parameters, namely the magnetic field strength and the maximum Fermi energy of the system.", "We study the equations of state of these one-level, two-level and three-level systems and compare them by taking three different maximum Fermi energies.", "We also find the effect of the strong magnetic field on the mass-radius relation of the underlying star composed of the gas stated above.", "We obtain an exciting result that, it is possible to have an electron degenerate static star, namely magnetized white dwarfs, with a mass significantly greater than the Chandrasekhar limit in the range 2.3-2.6M_Sun, provided it has an appropriate magnetic field strength and central density.", "In fact, recent observations of peculiar Type Ia supernovae - SN 2006gz, SN 2007if, SN 2009dc, SN 2003fg - seem to suggest super-Chandrasekhar-mass white dwarfs with masses up to 2.4-2.8M_Sun, as their most likely progenitors.", "Interestingly our results seem to lie within the observational limits." ], [ "Introduction", "Neutron stars are known to have high magnetic fields as large as $10^{12}$ G or more on their surfaces.", "Several magnetic white dwarfs have also been discovered with surface fields from about $10^{5}$ G to $10^{9}$ G [1], [2], [3], [4], [5] [6] and the physics of these objects have also been studied from a long time [7], [8], [9].", "It is likely that stronger fields exist in the centers of neutron stars or even white dwarfs, the limit to which is set by the scalar virial theorem [10]: $2T + W + 3\\Pi + \\mathcal {M} = 0,$ where $T$ is the total kinetic (rotational) energy, $W$ the gravitational potential energy, $\\Pi $ arises due to the internal energy and $\\mathcal {M}$ the magnetic energy.", "Since $T$ and $\\Pi $ are both positive, the maximum magnetic energy can be compared to, but can never exceed, the gravitational energy in an equilibrium configuration.", "For a star of mass $M$ and radius $R$ this gives $(4\\pi R^{3}/3)(B_{max}^{2}/8\\pi ) \\sim GM^{2}/R$ , or $B_{max} \\sim 2 \\times 10^{8} (M/M_{\\odot })(R/R_{\\odot })^{-2}$ G. For white dwarfs this limit is $10^{12}$ G. Ostriker and Hartwick [11] had constructed models of magnetic white dwarfs with magnetic field strength $B \\sim 10^{12}$ G at the center but with a much smaller field at the surface.", "Thus high interior magnetic fields in white dwarfs, although rarely observed in nature so far, are not completely implausible.", "It was proposed by Ginzburg [12] and Woltjer [13] that the magnetic flux $\\phi _{B} \\sim 4\\pi B R^{2}$ of a star is conserved during its evolution and subsequent collapse to form a remnant degenerate star (flux freezing phenomenon).", "Thus degenerate stars of small size and large magnetic fields are expected to be formed from parent stars which originally could have quite high magnetic fields of the order $\\sim 10^{8}$ G [14], [15].", "Thus the study of such highly magnetized degenerate stars will help us understand the origin and evolution of stellar magnetic fields.", "The mass-radius relation for (non-magnetic) white dwarfs was first determined by Chandrasekhar [16].", "He obtained a maximum mass for stable white dwarfs, known as the famous Chandrasekhar limit ($\\sim 1.44M_{\\odot }$ ), such that electron degeneracy pressure is just adequate to counteract gravitational collapse of the star.", "Suh et al.", "[17] obtained the mass-radius relation for white dwarfs with $B \\sim 4.4 \\times 10^{11-13}$ G. They, however, worked in the weak field limit (thus ignoring Landau quantization) and applied Euler-MacLaurin expansion to the equation of state of a fully degenerate electron gas in a strong magnetic field [10], in order to recover the usual equation of state in absence of a magnetic field.", "They found that both the mass and radius of magnetic white dwarfs increase compared to non-magnetic white dwarfs, having the same central density.", "In this paper, we consider a relativistic, degenerate electron gas at zero-temperature under the influence of a strong, uniform, static magnetic field.", "We neglect any form of interactions between the electrons.", "We study the effect of strong magnetic field on the equation of state of the degenerate matter and consequently obtain the mass-radius relation for a collapsed static star that might be composed of such matter.", "In order to highlight the effect of Landau quantization of electrons due to strong magnetic field, we restrict our systems to have at most one, two or three Landau level(s).", "We hypothesize the possibility of existence of purely electron degenerate stars with extremely high magnetic fields $\\sim 10^{15}-10^{17}$ G at the center, plausibly strongly magnetized white dwarfs.", "We also investigate the possibility of such stars having a mass greater than the Chandrasekhar limit, in the range $2.3-2.6M_{\\odot }$ .", "In a simple analytical framework existence of such stars has already been reported recently [18] and its astrophysical implications based on numerical analysis was also discussed [19].", "Interestingly recent observations of peculiar Type 1a supernovae - SN 2006gz, SN 2007if, SN 2009dc, SN 2003fg - seem to suggest super-Chandrasekhar-mass white dwarfs as their most likely progenitors [20], [21].", "These white dwarfs are believed to have masses up to $2.4-2.8M_{\\odot }$ .", "Proposed mechanism by which these white dwarfs exceed the Chandrasekhar limit is chiefly mass accretion from a binary companion accompanied by differential rotation [22].", "This is fundamentally different from what we are proposing here.", "However, these observations are quite stimulating as not only do they support the existence of super-Chandrasekhar mass white dwarfs, but also our results seem to lie within the observational limits.", "The paper is organized as follows.", "In the next section, we first recall how the equation of state of a cold electron degenerate gas gets modified due to the presence of a strong magnetic field and then state the numerical procedure followed to obtain results.", "Subsequently in §3 we discuss the numerical results describing the nature of the equations of state and the mass-radius relations.", "In §4 we elaborate on some of the key points of this work, for example, the timescale of magnetic field decay, comparison with Chandrasekhar's standard results, unstable branch of the mass-radius relations, the justifications for a constant magnetic field, the non-relativistic equation for hydrostatic equilibrium, neglecting Coulomb interactions and the anisotropy in pressure due to strong magnetic field.", "Finally, in §5 we summarize our findings with conclusions." ], [ "Basic equations", "The energy states of a free electron in a uniform magnetic field are quantized into what is known as Landau orbitals, which define the motion of the electron in a plane perpendicular to the magnetic field.", "On solving the Schrödinger equation in an external, uniform and static magnetic field directed along the $z$ -axis, one obtains the following dispersion relation [23] $E_{\\nu ,\\,p_{z}} = \\nu \\hbar \\omega _{c} + \\frac{p_{z}^{2}}{2m_{e}} ,$ where quantum number $\\nu $ denotes the Landau level and is given by $\\nu = j + \\frac{1}{2} + \\sigma ,$ when $j$ being the principal quantum number of the Landau level ($j$ = 0, 1, 2,...), $\\sigma = \\pm \\frac{1}{2}$ , the spin of the electron, $m_{e}$ the rest mass of the electron, $\\hbar $ the Planck's constant and $p_{z}$ the momentum of the electron along the $z$ -axis which may be treated as continuous (the motion along the field is not quantized).", "The cyclotron energy is $\\hbar \\omega _{c} = \\hbar (eB/m_{e}c)$ , where $e$ is the charge of the electron, $c$ the speed of light and $B$ the magnetic field.", "Now, the electrons can become relativistic in either of the two cases: (i) when the density is high enough such that the mean Fermi energy of an electron exceeds its rest-mass energy, (ii) when the cyclotron energy of the electron exceeds its rest-mass energy.", "We can define a critical magnetic field strength $B_{c}$ from the relation $\\hbar \\omega _{c} = m_{e}c^{2}$ , which gives $B_{c} = m_{e}^{2}c^{3}/\\hbar e = 4.414 \\times 10^{13}$ G. Thus in order to study the effect of a strong magnetic field ($B \\gtrsim B_{c}$ ) on the equation of state of a relativistic, degenerate electron gas we have to solve the relativistic Dirac equation.", "We mention here that for the present purpose, the magnetic field considered in the electron degenerate star originates due to the flux freezing phenomenon during the gravitational collapse of the parent star.", "The energy eigenstates in this case turn out to be [10] $E_{\\nu ,\\,p_{z}} = [p_{z}^{2}c^{2} + m_{e}^{2}c^{4}(1 + \\nu \\frac{2B}{B_{c}})]^{1/2}.$ The main effect of the magnetic field is to modify the available density of states for the electrons.", "The number of states per unit volume in an interval $\\Delta p_{z}$ for a given Landau level $\\nu $ is $g_{\\nu }(eB/h^{2}c)\\Delta p_{z}$ , where $g_{\\nu }$ is the degeneracy that arises due to the Landau level splitting, such that, $g_{\\nu } = 1$ for $\\nu = 0$ and $g_{\\nu } = 2$ for $\\nu \\ge 1$ .", "Therefore the electron state density in the absence of magnetic field $\\frac{2}{h^{3}}\\int d^{3}p ,$ has to be replaced with $\\sum _{\\nu } \\frac{2eB}{h^{2}c}\\,g_{\\nu } \\int dp_{z}$ in case of a non-zero magnetic field.", "Now in order to calculate the electron number density $n_{e}$ at zero temperature, we have to evaluate the integral in equation (REF ) from $p_{z} = 0$ to $p_{F}(\\nu )$ , which is the Fermi momentum of the electron for the Landau level $\\nu $ , to obtain [10] $n_{e} = \\sum _{\\nu =0}^{\\nu _{m}} \\frac{2eB}{h^{2}c}\\,g_{\\nu }\\,p_{F}(\\nu ).$ The Fermi energy $E_{F}$ of the electrons for the Landau level $\\nu $ is given by $E_{F}^{2} = p_{F}(\\nu )^{2}c^{2} + m_{e}^{2}c^{4}(1 + 2\\nu \\frac{B}{B_{c}}).$ The upper limit $\\nu _{m}$ of the summation in equation (REF ) is derived from the condition that $p_{F}(\\nu )^{2} \\ge 0$ , which implies $E_{F}^{2} \\ge m_{E}^{2}c^{4}(1 + 2\\nu \\frac{B}{B_{c}})$ and we obtain $\\nu \\le \\frac{\\epsilon _{F}^{2} - 1}{2B_{D}}$ or $\\nu _{m} = \\frac{\\epsilon _{Fmax}^{2} - 1}{2B_{D}},$ where $\\epsilon _{F} = E_{F}/m_{e}c^{2}$ , is the dimensionless Fermi energy, $B_{D} = B/B_{c}$ , the dimensionless magnetic field and $\\epsilon _{Fmax} = E_{Fmax}/m_{e}c^{2}$ , the dimensionless maximum Fermi energy of a system for a given $B_{D}$ and $\\nu _{m}$ .", "We note that $\\nu _{m}$ is taken be the nearest lowest integer in equation (REF ).", "For example, if $0 \\le \\nu _{m} < 1$ for a particular value of $\\epsilon _{Fmax}$ and $B_{D}$ , then the upper limit is taken to be $\\nu _{m}=0$ .", "If we define a dimensionless Fermi momentum $x_{F}(\\nu ) = p_{F}(\\nu )/m_{e}c$ , then equations (REF ) and (REF ) may be written as $n_{e} = \\frac{2B_{D}}{(2\\pi )^{2} \\lambda _{e}^{3}} \\sum _{\\nu =0}^{\\nu _{m}} g_{\\nu }x_{F}(\\nu )$ and $\\epsilon _{F} = [x_{F}(\\nu )^{2} + 1 + 2\\nu B_{D}]^{1/2}$ or $x_{F}(\\nu ) = [\\epsilon _{F}^{2} - (1 + 2\\nu B_{D})]^{1/2},$ where $\\lambda _{e} = \\hbar /m_{e}c$ , is the Compton wavelength of the electron.", "The matter density $\\rho $ can be written as $\\rho = \\mu _{e}m_{H}n_{e},$ where $\\mu _{e}$ is the mean molecular weight per electrons and $m_{H}$ the mass of hydrogen atom.", "The electron energy density at zero temperature is $\\varepsilon _{e} & = & \\frac{2B_{D}}{(2\\pi )^{2}\\lambda _{e}^{3}}\\sum _{\\nu =0}^{\\nu _{m}} g_{\\nu } \\int \\limits _{0}^{x_{F}(\\nu )} E_{\\nu ,\\,p_{z}}d \\left(\\frac{p_{z}}{m_{e}c} \\right) \\nonumber \\\\& = & \\frac{2B_{D}}{(2\\pi )^{2}\\lambda _{e}^{3}}m_{e}c^{2}\\sum _{\\nu =0}^{\\nu _{m}} g_{\\nu } (1 + 2\\nu B_{D}) {\\psi } \\left(\\frac{x_{F}(\\nu )}{(1 + 2\\nu B_{D})^{1/2}} \\right),$ where ${\\psi }(z) = \\int \\limits _{0}^{z} (1 + y^{2})^{1/2}dy = \\frac{1}{2}z \\sqrt{1 + z^{2}} + \\frac{1}{2}\\ln (z + \\sqrt{1 + z^{2}}).$ Then the pressure of an electron gas in a magnetic field is given by $P_{e} & = & n_{e}^{2}\\frac{d}{dn_{e}}\\left(\\frac{\\varepsilon _{e}}{n_{e}} \\right) = - \\varepsilon _{e} + n_{e}E_{F} \\nonumber \\\\& = & \\frac{2B_{D}}{(2\\pi )^{2}\\lambda _{e}^{3}}m_{e}c^{2}\\sum _{\\nu =0}^{\\nu _{m}} g_{\\nu } (1 + 2\\nu B_{D}) {\\eta } \\left(\\frac{x_{F}(\\nu )}{(1 + 2\\nu B_{D})^{1/2}} \\right),$ where ${\\eta }(z) = \\frac{1}{2}z \\sqrt{1 + z^{2}} - \\frac{1}{2}\\ln (z + \\sqrt{1 + z^{2}}).$" ], [ "Procedure", "Referring to the qualitative discussion by Lai and Shapiro [10], let $\\nu _{m}$ also denote the maximum number of Landau levels occupied by a cold gas of electrons in a magnetic field.", "In this case, from equation (REF ), $\\nu _{m}$ will be the nearest highest integer.", "Then if $\\nu _{m} \\gg 1$ then the Landau energy level spacing becomes a very small fraction of the Fermi energy and the discrete sum over $\\nu $ can be replaced by an integral and we get back the non-magnetic results.", "From equation (REF ) we see that if the magnetic field strength is high (for a fixed Fermi energy), i.e., $B_{D} \\gg 1$ , then $\\nu _{m}$ is small and the electrons are restricted to the lower Landau levels only.", "It is in this case that the magnetic field plays an important role in influencing the equation of state for the relativistic degenerate gas.", "Since we are investigating the effects of high magnetic field in this work, we fix $\\nu _{m}$ , such that it can only take values 1, 2 or 3, which we call as one-level, two-level and three-level system respectively.", "To clarify further, by one-level system we mean where only the ground Landau level, $\\nu = 0$ , is occupied, two-level means where both the ground and the first ($\\nu = 1$ ) Landau levels are occupied and three-level means where the ground, first and second ($\\nu = 2$ ) Landau levels are occupied.", "Now from equation (REF ) we see that once we fix $\\nu _{m}$ , we obtain a fixed $B_{D}$ on supplying a desired $E_{Fmax}$ , which corresponds to the maximum possible density (in a star which corresponds to its central density) for that $\\nu _m$ .", "Hence, in our framework, the magnetic field is in accordance with the density of the system.", "We choose $E_{Fmax} = 2\\, m_{e}c^{2}$ , $20\\, m_{e}c^{2}$ and $200\\, m_{e}c^{2}$ and for each we study the one-level, two-level and three-level systems, giving a total of 9 cases which are listed in Table 1.", "We mention here that for a given value of $E_{Fmax}$ , the value of $B_{D}$ listed here corresponds to a lower limit.", "For example, when $E_{Fmax} = 20\\,m_{e}c^{2}$ , $B_{D}$ with a value of 199.5 just results in a one-level system but, if we choose any $B_{D} >$ 199.5 that would also lead to a one-level system.", "For each of these cases we obtain the equation of state by simultaneously solving equations (REF ), (REF ) and (REF ) numerically from $E_{F} = m_{e}c^{2}$ to $E_{F} = E_{Fmax}$ , when each value of $E_{F}$ gives one point in the $P_{e}-\\rho $ plot.", "In this work we choose $\\mu _{e} = 2$ throughout.", "Figure 1 shows the equations of state for the different cases given in Table 1, which will be discussed in §3.", "Table: NO_CAPTIONIf we are to construct the model of a strongly magnetized star made out of electron degenerate matter, which is approximated to be spherical in presence of constant magnetic field, we require to solve the following differential equation which basically comes from the condition of hydrostatic equilibrium [24] $\\frac{1}{r^{2}}\\frac{d}{dr}\\left(\\frac{r^{2}}{\\rho }\\frac{dP}{dr} \\right) = -4\\pi G \\rho ,$ where we consider $P = P_{e}$ throughout this work.", "See, however, the Appendix in order to understand the effect of deviation from spherical symmetry due to the anisotropic effects of magnetic field, as discussed in §4.7.", "Since the pressure cannot be expressed as an analytical function of density, unlike that of Chandrasekhar's work [16], we fit the equation of state with the following polytropic relation: $P = K\\rho ^{\\Gamma },$ with different values of the adiabatic index $\\Gamma $ in different density ranges ($K$ being a dimensional constant).", "Thus the actual equation of state is reconstructed using multiple polytropic equations of state.", "One such fit is shown in Figure 1(d) and the parameters $K$ and $\\Gamma $ are stated in Table 2.", "The motivation behind doing such a fit is the following.", "First of all with this fitting we can determine the effect of magnetic field on the adiabatic index of the matter.", "More importantly, once we use an equation of state of the form (REF ), the problem essentially reduces to solving the Lane-Emden equation which arises in the non-magnetic case, except that in our case, $K$ and $\\Gamma $ also carry information about the magnetic field in the system.", "We briefly recall here the Lane-Emden equation, since we would be referring to some of its solutions in the next section.", "We start with equations (REF ) and (REF ), and write $\\Gamma = 1 + \\frac{1}{n}$ , where $n$ is the polytropic index.", "Next, two variable transformations are made as follows [24]: $\\rho = \\rho _{c}\\theta ^{n},$ where $\\rho _{c}$ is the central density of the star and $\\theta $ is a dimensionless variable and $r = a\\xi ,$ where $\\xi $ is another dimensionless variable and $a$ is defined as $a = \\left[\\frac{(n+1)K\\rho _{c}^{\\frac{1-n}{n}}}{4\\pi G} \\right]^{1/2},$ which has the dimension of length.", "Thus using equations (REF ), (REF ), (REF ) and (REF ), equation (REF ) reduces to the famous Lane-Emden equation: $\\frac{1}{\\xi ^{2}}\\frac{d}{d\\xi }\\left(\\xi ^{2} \\frac{d\\theta }{d\\xi } \\right) = - \\theta ^{n},$ which can be solved for a given $n$ , subjected to the following two boundary conditions: $\\theta (\\xi = 0) = 1$ and $\\left(\\frac{d\\theta }{d\\xi } \\right)_{\\xi =0} = 0.$ If $n < 5$ , then $\\theta $ falls to zero for a finite value of $\\xi $ , called as $\\xi _{1}$ , which basically denotes the surface of the star where the pressure goes to zero (and then density becomes zero too in the present context).", "The physical radius of the star is then given by $R = a\\xi _{1}.$ We note that the value of $n$ must be such that $n \\ge -1$ , so that $a$ is real in equation (REF ) and $R \\ge 0$ in equation (REF ).", "Each value of central density $\\rho _{c}$ corresponds to a particular value of radius $R$ and mass $M$ of a star.", "Substituting $a$ from equation (REF ) in equation (REF ) we find $R \\propto \\rho _{c}^{\\frac{1-n}{2n}} = \\rho _{c}^{\\frac{\\Gamma - 2}{2}}.$ The mass of the spherical star (see, however, the Appendix discussing the equations for an oblate spheroid) is obtained by integrating the following equation: $\\frac{dM}{dr} = 4\\pi r^{2}\\rho .$ Hence, $M = 4\\pi \\int \\limits _{0}^{R} r^{2}\\rho \\, dr = 4\\pi a^{3} \\rho _{c}\\int \\limits _{0}^{\\xi _{1}} \\xi ^{2}\\theta ^{n}\\, d\\xi .$ Again substituting $a$ from equation (REF ) in equation (REF ) we find $M \\propto \\rho _{c}^{\\frac{3-n}{2n}} = \\rho _{c}^{\\frac{3\\Gamma -4}{2}},$ and then combining equations (REF ) and (REF ) we obtain the following mass-radius relation: $R \\propto M^{\\frac{1-n}{3-n}} = M^{\\frac{\\Gamma -2}{3\\Gamma -4}}.$ In the present work we do not make the transformations (REF ) and (REF ), but directly solve equation (REF ) (in different density regions corresponding to the particular values of $\\Gamma $ or $n$ ) subjected to the same boundary conditions as in equations (REF ) and (REF ) which are written as $\\rho (r = 0) = \\rho _{c}$ and $\\left(\\frac{d\\rho }{dr} \\right)_{r=0} = 0.$ Hence the results in equations (REF ), (REF ) and (REF ) still remain applicable to our case.", "A plot of $R$ as a function of $M$ gives the mass-radius relation for the magnetic, degenerate static star.", "Figure 2 shows the mass-radius relations for the one-level, two-level and three-level systems with $E_{Fmax} = 20\\,m_{e}c^{2}$ (results for $E_{Fmax} = 2\\, m_{e}c^{2}$ and $200\\, m_{e}c^{2}$ also show the same trend).", "Figure 3 shows a comparison between the mass-radius relations for all the cases stated in Table 1.", "All these are discussed in detail in the next section." ], [ "Equations of State", "[subfigure]position=top Figure: Equations of state in a strong magnetic field (given in Table 1) for (a) E Fmax =2m e c 2 E_{Fmax} = 2\\,m_{e}c^{2}, (b) E Fmax =20m e c 2 E_{Fmax} = 20\\,m_{e}c^{2}, (c) E Fmax =200m e c 2 E_{Fmax} = 200\\,m_{e}c^{2}.", "In all three cases the solid line, the dotted line and the dashed lines indicate one-level, two-level and three-level systems respectively.", "In (d) the solid line is same as the dashed line in (b), but fitted with the dotted line by analytical formalism (see text for details).", "Here P D P_{D} is the pressure in units of 2.668×10 27 2.668 \\times 10^{27} erg/cc and ρ D \\rho _{D} is the density in units of 2×10 9 2 \\times 10^{9} gm/cc.Now we come to the discussions of the results obtained.", "We start with the equations of state shown in Figure 1.", "Let us consider the panel (b) which shows the cases with $E_{Fmax} = 20\\, m_{e}c^{2}$ .", "From Table 1 we see that the one-level system (the solid line) for $E_{Fmax} = 20\\, m_{e}c^{2}$ corresponds to a magnetic field strength $B_{D} = 199.5$ , and the two-level and three level systems (the dotted and dashed lines respectively) correspond to $B_{D} = 99.75$ and $ 66.5$ respectively.", "We notice that the solid curve is free of any kink, the dotted curve has one kink and the dashed curve has two kinks.", "The kinks appear when there is a transition from a lower Landau level to the next and they demarcate regions of the equation of state where the pressure becomes briefly independent of density.", "Let us consider the two-level systems.", "The portion of the equation of state below the kink represents the ground Landau level and the one above the kink represents the first Landau level.", "As the Fermi energy of the electrons increases, more and more electrons occupy the ground Landau level and both the density and pressure of the system keep increasing.", "Once the ground level is completely filled, one observes that, on increasing the Fermi energy of the electrons the density increases but, the pressure remains fairly constant for a while, after which the pressure again starts increasing with density.", "It is as if, the increase in Fermi energy during the transition is being used by the system to move to a higher Landau level instead of increasing the pressure.", "This situation seems analogous to that of phase transition in matter (where the temperature remains constant with respect to the input heat energy during the change of phase).", "Similar features are also seen in Figures 1(a) and (c) for $E_{Fmax} = 2\\, m_{e}c^{2}$ and $200\\, m_{e}c^{2}$ respectively.", "Thus looking at Figure 1, we observe that in the one-level systems (solid lines), all the electrons are in the ground Landau level and hence there is no kink.", "In the two-level systems (dotted lines), as the electrons start filling up the first Landau level, a kink develops in the equation of state.", "Finally in the three-level systems (dashed lines), there are two kinks $-$ the one at the lower density indicating transition to the first Landau level and another at the higher density indicating transition to the second Landau level.", "The value of $E_{Fmax}$ determines the maximum density of the system and hence the positions of the kinks shift accordingly in Figures 1(a), (b) and (c)." ], [ "Mass-radius relations", "Next we come to Figure 2.", "Here we show the mass-radius relations for the one-level, two-level and three level systems with $E_{Fmax} = 20\\,m_{e}c^{2}$ (the explanation that follows also holds true for the cases with $E_{Fmax} = 2\\,m_{e}c^{2} \\textrm { and } 200\\,m_{e}c^{2}$ ).", "Each point in the mass-radius curve corresponds to a star with a particular value of central density $\\rho _{c}$ which is supplied by us as a boundary condition ($R_D$ and $M_{D}$ are the dimensionless radius and mass of a star respectively as defined in Figure 2 caption).", "[subfigure]position=top Figure: Mass-radius relations with E Fmax =20m e c 2 E_{Fmax} = 20\\,m_{e}c^{2} for (a) one-level system, (b) two-level system, (c) three-level system.", "Here M D M_D is the mass of the star in units of M ⊙ M_{\\odot } and R D R_D is the radius of the star in units of 10 8 10^{8} cm (the solid, dotted and dashed lines have the same meaning as in Figure 1).Figure 2(a) shows the mass-radius relation for the one-level system ($B_{D} = 199.5$ ).", "We see that initially as $\\rho _{c}$ increases both the mass and radius increase and then at higher central densities the radius becomes nearly independent of the mass (we will be explaining later in this section as to why such a trend is observed).", "We note that the last point on this curve has a mass $\\sim $ 2.3 $M_{\\odot }$ and radius $\\sim $ $6.4 \\times 10^{7}$ cm, which corresponds to the maximum density point of the solid curve in Figure 1(b).", "This denotes the density ($ \\sim 1.16 \\times 10^{10}$ gm/cc) at which the ground Landau level is completely filled.", "Thus a star with this $\\rho _{c}$ and a magnetic field strength of $B = 199.5 B_{c} = 8.81 \\times 10^{15}$ G has a mass greater than the Chandrasekhar limit ($\\sim $ 1.44 $M_{\\odot }$ for $\\mu _{e} = 2$ ).", "Figure 2(b) shows the mass-radius relation for the two-level system ($B_{D} = 99.75$ ).", "In this case, a maximum mass $\\sim 2.3 M_{\\odot }$ is reached at a radius $\\sim 8.9 \\times 10^{7}$ cm for $\\rho _{c} \\sim 4.0 \\times 10^{9}$ gm/cc, in the same way as in Figure 2(a) when the ground Landau level is completely filled.", "After which there is a turning point in the curve, from where the mass starts decreasing.", "This turning point corresponds to the kink in the corresponding equation of state (dotted curve in Figure 1(b)).", "During the transition (when $ 4.0 \\times 10^{9} \\textrm { gm/cc} < \\rho _{c} \\lesssim 8.0 \\times 10^{9} $ gm/cc) from ground to first Landau level, the radius and mass both decrease with increasing $\\rho _{c}$ .", "Then there is a brief range of densities ($ 8.0 \\times 10^{9} \\textrm { gm/cc} < \\rho _{c} \\lesssim 1.2 \\times 10^{10} $ gm/cc) where the radius decreases as the mass remains fairly constant and ultimately at very high densities ($ 1.2 \\times 10^{9} \\textrm { gm/cc} < \\rho _{c} \\lesssim 1.5 \\times 10^{10} $ gm/cc) the the radius is again nearly independent of the mass, as in the uppermost branch.", "Figure 2(c) shows the mass-radius relation for the three-level system ($B_{D} = 66.5$ ).", "Here we see two turning points, denoted by the decrease of both mass and radius, which correspond to the two kinks in the corresponding equation of state (dashed curve in Figure 1(b)).", "The maximum mass at the first turning point $\\sim 2.3 M_{\\odot }$ is reached at a radius $\\sim 1.1 \\times 10^{8}$ cm for $\\rho _{c} \\sim 2.2 \\times 10^{9}$ gm/cc (the first kink in the equation of state).", "The mass at the second turning point is $\\sim 1.2 M_{\\odot }$ which has a radius $\\sim 6.3 \\times 10^{7}$ cm for $\\rho _{c} \\sim 7.6 \\times 10^{9}$ gm/cc (the second kink in the equation of state).", "Just after either of the turning points denoted by the decrease of radius and mass both, briefly the radius decreases as the mass remains almost constant and finally the radius becomes nearly independent of mass.", "The maximum mass $\\sim 2.3 M_{\\odot }$ occurs at the central density where the ground Landau level is completely filled and transition to the first Landau level is about to start.", "We observe that this density follows $\\rho _{c}(\\rm {three-level}) < \\rho _{c}(\\rm {two-level}) < \\rho _{c}(\\rm {one-level})$ as is also seen from the positions of the kinks in Figure 1.", "In order to explain this behavior in further detail, we resort to the Lane-Emden relations (REF ), (REF ) and (REF ).", "From (REF ) we note that if $\\Gamma > 2$ , then $R$ increases with $\\rho _{c}$ and if $\\Gamma = 2$ , then $R$ is independent of $\\rho _{c}$ .", "From (REF ) we note that if $\\Gamma > 4/3$ , then $M$ increases with $\\rho _{c}$ and if $\\Gamma = 4/3$ , then $M$ is independent of $\\rho _{c}$ .", "Finally from (REF ) we note that if $\\Gamma > 2$ , then $R$ increases with $M$ , if $\\Gamma = 2$ , then $R$ is independent of $M$ and if $\\Gamma = 4/3$ , then $M$ is independent of $R$ .", "This is exactly what is observed in Figure 2.", "Table: NO_CAPTIONLet us look again at the mass-radius relation corresponding to Figure 2(c) (also see Table 2).", "At very low densities $\\Gamma $ is $\\sim 3$ (which is the case for non-relativistic electrons in the ground Landau level; $P_{e}(= P) \\propto \\rho ^{3}$ ; see [25]) and the value of $\\Gamma $ keeps decreasing with increasing density.", "Up to the first turning point density $\\rho _{c} \\sim 2.2 \\times 10^{9}$ gm/cc, both the radius and mass keep increasing with $\\rho _{c}$ and then the radius becomes nearly independent of mass when $\\Gamma \\sim 2$ .", "Then $\\Gamma $ suddenly drops to the small value $\\sim 0.35$ which marks the onset of the transition from ground Landau level to first Landau level.", "In this region the pressure becomes independent of density, revealing an unstable zone in the equation of state (see detailed discussion in §4.5).", "As the density increases further, $\\Gamma $ approaches the relativistic value of $4/3$ .", "In this regime we see that the radius decreases slightly as the mass does not change significantly, as is also true for the mass-radius relation in the classical non-magnetic case for $\\Gamma =4/3$ (see Figure 4(b)).", "Next $\\Gamma $ takes up a value of 2 and the radius again becomes nearly independent of mass till it reaches the second turning point at density $\\rho _{c} \\sim 7.6 \\times 10^{9}$ gm/cc.", "Again during the transition from the first Landau level to the second, $\\Gamma $ drops to $0.35$ , followed by values of $4/3$ and 2, which have the same explanations as stated above.", "We also observe that for a certain range of masses in the two-level and three-level systems, it is possible to have multiple values of the radius for a given value of mass.", "For instance, looking at the two-level system in Figure 2(b) in the range $\\sim 0.6-1.0$ $M_{\\odot }$ , we observe that the same value of mass corresponds to three different values of the radius.", "Let us call them $R_{1}$ , $R_{2}$ and $R_{3}$ , such that $R_{1} > R_{2} > R_{3}$ and $\\rho _{c}(R_{1}) < \\rho _{c}(R_{2}) < \\rho _{c}(R_{3})$ .", "To explain why we observe such a behavior, we recall that $B_{D}$ of the system is such that $\\nu _{m}$ is fixed for a given value of $E_{Fmax}$ (see Table 1 for the values).", "In Figure 2(b), $B_{D}=99.75$ ensures that the system can have at the most two Landau levels (ground and first), but to what extent they will be filled depends on the Fermi energy of the electrons ($\\le E_{Fmax}$ ).", "For low central densities (i.e.", "low Fermi energy) the electrons occupy only the ground Landau level.", "In order for the electrons to start occupying the first Landau level, the central density of the star must be adequately high.", "Thus, based on previous discussion for the mass-radius curves in Figure 2, it is possible that the same mass of a star corresponds to more than one radius, depending on the Landau level occupancy.", "Now, $R_{1}$ lies on the branch of the mass-radius relation which corresponds to the stars in which the electrons occupy only the ground Landau level, while $R_{3}$ lies on the branch which corresponds to the electrons occupying the first Landau level (the ground level being already filled).", "$R_{2}$ lies on the unstable branch of the mass-radius relation (see discussion in §4.5) when the electrons are in the transition mode form the ground to the first Landau level.", "The three-level system in Figure 2(c) can also be explained likewise, except that it has two additional possible radii corresponding to the same mass due to the presence of the second Landau level.", "In the one-level system in Figure 2(a), multiple values of radius are not observed because the magnetic field ($B_{D} = 199.5$ ) is such that the electrons can occupy only the ground Landau level.", "[subfigure]position=top Figure: Comparison of the mass-radius relations, for (a) one-level system, (b) two-level system, (c) three-level system, when the solid, dotted and dashed lines represent E Fmax =2m e c 2 ,20m e c 2 E_{Fmax} = 2\\,m_{e}c^{2}, 20\\,m_{e}c^{2} and 200m e c 2 200\\,m_{e}c^{2} respectively.", "In each of the three panels the yy-axis is in log scale.", "See Table 1 for details.Finally we come to Figure 3.", "For each of the Figures (a), (b) and (c) the value of $E_{Fmax}$ increases from the top to the bottom curve and we see that the overall radius decreases from the top to the bottom curve.", "We also note that an increase in $E_{Fmax}$ corresponds to an increase in the magnetic field for a fixed $\\nu _{m}$ , which means that as the magnetic field strength increases the degenerate stars become more and more compact in size.", "The same trend has been observed in the case of neutron stars with a high magnetic field [26], [27], [28] (see discussion in §4.2.)", "Interestingly, as seen from Figure 3(a), for $E_{Fmax}=200\\,m_{e}c^{2}$ the maximum mass of the star is even higher than that of $E_{Fmax} = 20\\,m_{e}c^{2}$ , which is $\\sim 2.6M_{\\odot }$ .", "Also to be noted is the fact that for a given $\\nu _{m}$ , the curve corresponding to $E_{Fmax} = 20\\,m_{e}c^{2}$ covers almost the same range in mass as that of the $E_{Fmax} = 200\\,m_{e}c^{2}$ curve, while the curve for $E_{Fmax} = 2\\,m_{e}c^{2}$ covers a considerably smaller range in mass.", "The reason behind this saturation at higher $E_{Fmax}$ may be due to the fact that a Fermi energy $\\sim 2\\,m_{e}c^{2}$ corresponds to very low density such that the electrons are at the most only mildly relativistic.", "By the time Fermi energy reaches a value of about $20\\,m_{e}c^{2}$ (which corresponds to high density), the electrons have become highly relativistic, giving rise to denser, more massive stars and hence further increase of $E_{Fmax}$ could not bring any new effect in the system." ], [ "Timescale of decay of magnetic field", "Generally the magnetic fields inside an electron degenerate star undergo Ohmic decay.", "The timescale for this is given by $t_{ohm}=\\frac{4 \\pi \\sigma _{E} L^{2}}{c^{2}}$ , where $\\sigma _{E}$ is the electrical conductivity and $L$ the length-scale over which the magnetic field changes.", "Theoretical calculations of Ohmic decay in isolated, cooling white dwarfs, which are generally known in nature, show that the magnetic field changes little over their lifetime [29], [30], [31].", "Cumming [32] estimated a lowest order decay time as $t_{ohm} \\approx 10^{10} \\,\\rm yrs \\left(\\frac{\\rho _{c}}{3\\times 10^{6}\\, \\rm gm/cc}\\right)^{1/3}\\left(\\frac{\\rm R}{10^{9}\\, \\rm cm}\\right)^{1/2} \\left(\\frac{10<\\rho >}{\\rho _{c}}\\right),$ where $<\\rho >$ is the mean density of the star.", "For the three stars represented in Figure 5(d), above timescale turns out to be $\\gtrsim 10^{9}$ years.", "Thus we can say that the magnetic fields of these stars do not decay significantly via Ohmic dissipation in their lifetime.", "However, while for $B \\lesssim 10^{11}$ G, Ohmic decay is the dominant phenomenon, for $B \\sim 10^{12}-10^{13}$ G decay is supposed to take place via Hall drift and for $B \\gtrsim 10^{14}$ G, it undergoes ambipolar diffusion [33].", "Note that all the decay timescales stated above are valid only for normal or non-superfluid matter.", "Hence, while they might be unrealistic for neutron stars [34], these timescales are applicable to white dwarfs since they do not have a superconducting core [35], [36].", "Heyl & Kulkarni [33] examined the consequences of the magnetic field decay in magnetars having surface field $10^{14}$ G to $10^{16}$ G by using an appropriate cooling model and by solving the following decay equation: $\\frac{dB}{dt} = -B\\left(\\frac{1}{t_{ohmic}} + \\frac{1}{t_{ambip}} + \\frac{1}{t_{Hall}}\\right).$ The strongly magnetized white dwarfs considered in the present work have central magnetic field strengths $\\sim 10^{15}$ G, which is comparable to the surface field strengths of magnetars.", "It is then likely that the magnetic fields in both these cases would undergo similar decay mechanisms.", "The decay of such strong fields is dominated by ambipolar diffusion [33].", "However, for a typical initial field strength of $10^{15}$ G, the magnetic field remains nearly constant up to about $10^{5}$ years and in the next 100 years its value decreases by at most an order of magnitude [33].", "As discussed in §4.3 below, it is the central magnetic field which is crucial for the super-Chandrasekhar mass of the white dwarfs.", "Thus applying the above results to these magnetized white dwarfs, we can conclude that the central field will not decay appreciably for a long period of time.", "An alternate scenario could arise if the magnetized white dwarfs are accreting.", "In this case, the heat generated due to accretion decreases the electrical conductivity of the surface of the star, causing a faster decay of the (surface) magnetic field due to a reduced Ohmic decay timescale.", "However, as the mass of the star increases, it becomes more compact and the current carrying accreted material is pushed deeper into the star.", "Since conductivity is a steeply increasing function of density, the higher conductivity of the denser inner region of the star will again slow down further decay of the magnetic field [37].", "Since we are working with magnetic field strengths $B>B_{c}$ , one might be concerned about the process of electron-positron pair creation (via Schwinger process) at the expense of magnetic energy, which might lead to a reduction of the field strength.", "However, Canuto and Chiu [38], [39], [40] and also Daugherty et al.", "[41] showed that it is impossible to have spontaneous pair creation in a magnetic field alone, irrespective of its strength.", "Now, from Maxwell's equations in a steady state, we have $\\nabla \\times {\\bf B} = \\frac{4\\pi }{c}{\\bf j}$ and Ohm's law states ${\\bf E}=\\frac{\\bf j}{\\sigma _{E}},$ when ${\\bf E}$ is the electric field and ${\\bf j}$ the current density.", "Thus for an electric field to be generated, the magnetic field must vary with space as seen from equation (REF ).", "However, in this work we have chosen a constant magnetic field, which leads to ${\\bf j}=0$ and hence ${\\bf E}=0$ from equation (REF ).", "Now the main effect of the magnetic field, which is to give rise to a mass exceeding the Chandrasekhar limit, is restricted to the high density region, where the field remains essentially constant.", "Thus our choice of a constant magnetic field is justified (see §4.3 for detailed discussion).", "Moreover since the magnetic field in the degenerate star is generated due to the flux freezing phenomenon, it incorporates the fact that the conductivity $\\sigma _{E}$ is very large.", "Thus even if the magnetic field is highly inhomogeneous, from equation (REF ) we see that the electric field generated would again be negligible.", "Thus the electron degenerate stars in this work are magnetically dominated systems, i.e., both the magnetic field strength is very high ($B>B_{c}$ ) and the electric field strength is negligible.", "Recently, Jones [42] calculated the cross-section for photon-induced pair creation in very high magnetic fields and has arrived at the result that, there is a rapid decrease of the pair creation cross-section at $B>B_{c}$ .", "Hence one can ignore the effect of pair creation in reducing the magnetic field strength in these stars." ], [ "Comparison with Chandrasekhar's results", "[subfigure]position=top Figure: Comparison with Chandrasekhar's non-magnetic results for E Fmax =20m e c 2 E_{Fmax} = 20\\,m_{e}c^{2}.", "(a) Equations of state - the solid line represents Chandrasekhar's equation of state.", "The dot-dashed, dotted and dashed lines represent the one-level (ν m =1\\nu _{m}=1), two-level (ν m =2\\nu _{m}=2) and three-level (ν m =3\\nu _{m}=3) systems respectively.", "The equation of state for ν m =20\\nu _{m} = 20 is also shown, which appears as a series of kinks on top of the solid line.", "(b) Mass-radius relations - the vertical line marks the 1.44M ⊙ M_{\\odot } limit and the solid line represents Chandrasekhar's mass-radius relation.", "From top to bottom the other lines represent the cases for ν m =500,20,3,2\\nu _{m} = 500\\,, 20\\,, 3\\,, 2 and 1 respectively (the yy-axis is in log scale).", "(c) Density as a function of radius inside a non-magnetized star (solid line) and a star having ν m =1\\nu _{m}=1 (dot-dashed line), both having the same central density ρ c =5\\rho _{c}=5 in units of 2×10 9 2 \\times 10^{9} gm/cc.In Figure 4 we put together our results along with Chandrasekhar's result for non-magnetic white dwarfs to obtain a more complete picture.", "Figure 4(a) shows the equation of state obtained by Chandrasekhar and those corresponding to the one ($\\nu _{m}=1$ ), two ($\\nu _{m}=2$ ) and three ($\\nu _{m}=3$ ) level systems (same as in Figure 1(b)).", "Interestingly, the equation of state for $\\nu _{m}=20$ almost grazes Chandrasekhar's equation of state, except the appearance of a series of kinks.", "This clearly shows that as the magnetic field strength decreases, or equivalently as the maximum number of occupied Landau levels increases, the equation of state approaches Chandrasekhar's equation of state.", "Figure 4(b) represents the mass-radius relations corresponding to the equations of state shown in Figure 4(a).", "The mass-radius relation for $\\nu _{m}=500$ (500 Landau levels) is also shown.", "We observe that as the magnetic field strength decreases or as the maximum number of occupied Landau levels increases, the mass-radius relation approaches the non-magnetic relation and one recovers Chandrasekhar's mass limit.", "Figure 4(b) also shows that the stars of a given mass become more and more compact in size as the magnetic field strength increases.", "In order to explain this behavior we resort to Figures 4(a) and (c).", "Let us for simplicity look at Chandrasekhar's equation of state (solid line) and that of the one-level system (dot-dashed line), which corresponds to $B=199.5B_{c}$ , in Figure 4(a).", "We notice that at low densities, the dot-dashed line lies below the solid line and at higher densities, $\\rho _{D}\\gtrsim 2$ , the dot-dashed line lies above the solid line.", "In other words, the equation of state for the one-level system is softer than Chandrasekhar's equation of state at low densities, which means that the pressure does not rise with density as rapidly as that in Chandrasekhar's case.", "This trend reverses at higher densities and the equation of state for the one-level case becomes stiffer than that of Chandrasekhar's.", "Now, matter with a softer equation of state is less efficient in counteracting gravity and hence stars made out of such matter will be more compact in size.", "Keeping this in mind we now look at Figure 4(c).", "Figure 4(c) shows the variation of density with radius, within a non-magnetized star and a magnetized star with $\\nu _{m}=1$ (one-level), both having the same central density ($\\rho _{c}=5$ ).", "We mention here that for ease of explanation we have chosen a high central density such that the equations of state for both the stars cover almost the entire range of density.", "We observe that for a given radius, density for the magnetized star is higher than that of the non-magnetized star for a large range, from $\\rho _{D}=\\rho _{c}=5$ to $\\rho _{D} \\sim 0.3$ .", "But at very low densities, $\\rho _{D}<0.3$ , the density of the magnetized star sharply falls to zero due to smaller pressure, leading to a smaller star ($R=6.4\\times 10^{7}$ cm) than in the non-magnetized case ($R=1.3\\times 10^{8}$ cm).", "From the analysis of the equations of state for these two stars, we can say that if the equation of state of the magnetized star would have remained stiffer than the non-magnetized star throughout, then the magnetized star would have a larger radius, as in this case the pressure would be more efficient in counteracting the gravitational collapse of the star.", "But this is not true.", "At very low densities the equation of state suddenly becomes softer for the magnetized star and the pressure is not able to counteract gravity efficiently, causing the star to collapse rapidly.", "Hence the density goes to zero very rapidly, causing the star to have a (much) smaller radius.", "We also note from Figure 4(b) that as the magnetic field strength increases, the probability that the stars will have masses exceeding the Chandrasekhar limit increases.", "From Figure 4(c) we see that the density of the magnetized star is much higher than that of the non-magnetized star, except for a very small range of low densities.", "Thus calculating the total mass of the stars from equation (REF ), we obtain a much higher value for the magnetized star." ], [ "Choice of constant magnetic field", "[subfigure]position=top Figure: Mass as a function of density within an electron degenerate star with B D B_{D} (a) 199.5 (dashed line), (b) 99.75 (dotted line), (c) 66.5 (dot-dashed line), for E Fmax =20m e c 2 E_{Fmax} = 20\\,m_{e}c^{2}.", "(d) Variation of mass as a function of radius within the stars in (a), (b) and (c).", "The horizontal line indicates the 1.44M ⊙ M_{\\odot } limit.", "The radius r D r_{D} is in units of 10 8 10^{8} cm.", "See Table 1 for details.Figures 5(a), (b) and (c) show the variation of mass as a function of density within a magnetized electron degenerate star for three different magnetic field strengths.", "In all the cases we note that, by the time the density falls to about half the value of the central density, the mass has increased significantly, crossing the Chandrasekhar limit (indicated by the horizontal line) soon after.", "Hence, although we have considered a constant magnetic field, their effect is restricted to the high density regime, where the field remains essentially constant in reality.", "Hence we can also interpret this constant magnetic field as the central magnetic field of the star.", "This would be more clear from the description of the variation of magnetic field given by [26], [43], which show that an inhomogeneous magnetic profile in a compact star could be such that the magnetic field is nearly constant throughout most of the star and then gradually falls off close to the surface (see Figure 5(b) in [26]).", "Thus choosing an inhomogeneous magnetic profile would not affect our main finding that the Chandrasekhar mass limit can be exceeded for high magnetic field strengths." ], [ "Choice of non-relativistic equation of hydrostatic equilibrium", "Figure 5(d) shows the variation of mass as a function of radius within the stars represented in Figures 5(a), (b) and (c).", "All these stars have a total mass $\\sim 2.3M_{\\odot }$ and hence their Schwarzschild radius $R_{g}=\\frac{2GM}{c^{2}}=6.8\\times 10^{5}$ cm.", "Now, general relativistic effects usually start becoming important at a radius $\\lesssim 10 R_{g}$ , i.e.", "$r_{D} \\lesssim 0.068$ for the above mentioned stars.", "However, from Figure 5(d) we see that for all the three stars the contribution to the mass from a radius $< 10R_{g}$ , i.e.", "the central region, is negligible.", "Contribution to the mass rather effectively starts from a radius $r_{D} \\gtrsim 0.13$ ($\\sim 20 R_{g}$ ) and the Chandrasekhar limit is crossed at $R_{g} \\sim 66,\\, 95$ and 118, for the stars represented by the dashed, dotted and dot-dashed lines, respectively.", "Thus significant contribution to the total mass of the stars comes from a region well beyond the regime of general relativity.", "Hence one need not necessarily consider the Tolman-Oppenheimer-Volkoff equation and our choice of the non-relativistic equation of equilibrium, equation (REF ), is justified." ], [ "Unstable branch of the mass-radius relations", "Figure 6(a) shows the variation of density as a function of radius inside the degenerate stars having $E_{Fmax} = 20\\,m_{e}c^{2}$ and $B=99.75B_{c}$ .", "If we look at the equation of state for the two-level system with $E_{Fmax} = 20\\,m_{e}c^{2}$ , i.e.", "the dotted line in Figure 1(b), we see that both $\\rho _{D}=1.5$ and 2 lie below the density at which transition takes place from ground Landau level to first Landau level.", "For stars with these central densities, the pressure rises monotonically with density throughout and one forms a stable star.", "From the corresponding curves (solid and dashed) in Figure 6(b) also we note that the sound speed ($c_{s}$ ) varies smoothly with radius, reaching a maximum value at the center.", "Both the mass and radius of these stars increase with central density and hence they lie on the uppermost branch of the mass-radius relation in Figure 2(b).", "[subfigure]position=top Figure: (a) Density and (b) sound speed in units of cc, as a function of radius, inside the stars having E Fmax =20m e c 2 E_{Fmax} = 20\\,m_{e}c^{2} and B=99.75B c B=99.75B_{c}.", "The solid, dashed, dotted and dot-dashed lines represent the stars with central densities 1.5, 2, 2.5 and 7, in units of 2×10 9 2 \\times 10^{9} gm/cc, respectively.Now $\\rho _{D}=2.5$ lies on the plateau following the kink in the equation of state.", "A star having this central density will have within it a zone where pressure does not steadily increase with density, but is nearly constant.", "Thus such a star will tend to collapse faster under gravity and hence will have a smaller radius.", "From the dotted line in Figure 6(a) we see that the density of such a star falls steeply to zero compared to the $\\rho _{D}=2$ case indicated by the dashed line.", "The corresponding curve in Figure 6(b) shows a peak in $c_{s}$ at the radius where pressure starts becoming independent of density.", "Since mass of the star is calculated using equation (REF ), they have smaller masses too due to smaller size.", "These stars lie on the middle branch of the corresponding mass-radius relation.", "Since these stars consist of an unstable zone mentioned above, we argue that they constitute the somewhat unstable branch in the mass-radius relation.", "Finally, let us choose a larger density, say $\\rho _{D}=7$ , which lies on that portion of the equation of state above the kink, where pressure again rises monotonically with density, but not as steeply as that in the regions around $\\rho _{D}=1.5$ and 2.", "Although these stars also have an unstable zone inside them, but softer pressure in a large range of density, which together cause them to have a smaller radius, their central density is very high.", "As a result their mass starts increasing again, as can be inferred from the dot-dashed line in Figure 6(a).", "The $c_{s}$ again shows a sharp peak indicating the appearance of unstable zone within the star.", "These stars constitute the bottommost branch of the mass-radius relation." ], [ "Neglecting Coulomb interactions", "The distance between nuclei in highly magnetized electron degenerate stars could be as small as 50 Fermi.", "The Coulomb repulsion energy between electrons $\\frac{e^{2}}{r}$ , at the separation of 50 Fermi, is of the order of $4\\times 10^{-8}$ ergs which is quite less than the rest mass energy of an electron $m_{e}c^{2} \\sim 8\\times 10^{-7}$ ergs.", "There can be Coulomb interaction between electrons and the ions, which is given by $\\frac{Ze^{2}}{r}$ .", "Commonly electron degenerate stars consist of helium, carbon, oxygen, etcetera, so that $Z$ can have a value of 10 at the most.", "Hence, the Coulomb interaction energy would still be less than, or at most the same order as, the rest mass energy.", "Thus we can neglect the effects of Coulomb interaction for the present purpose." ], [ "Anisotropy in pressure due to strong magnetic field", "The strong magnetic field causes the pressure to become anisotropic [27], [44], [45].", "The total energy momentum tensor due to both matter and magnetic field is to be given by $T^{\\mu \\nu } = T^{\\mu \\nu }_{m} + T^{\\mu \\nu }_{f},$ where, $T^{\\mu \\nu }_{m} = \\epsilon _{m}u^{\\mu }u^{\\nu } - P_{m}(g^{\\mu \\nu } - u^{\\mu }u^{\\nu })$ and $T^{\\mu \\nu }_{f} = \\frac{B^{2}}{4\\pi }(u^{\\mu }u^{\\nu } - \\frac{1}{2}g^{\\mu \\nu }) - \\frac{B^{\\mu }B^{\\nu }}{4\\pi },$ when, $\\epsilon _{m} = \\epsilon _{e}$ is the matter energy density given by equation (REF ) and $P_{m} = P_{e}$ is the matter pressure given by equation (REF ).", "The first term in equation (REF ) is equivalent to magnetic pressure, while the second term gives rise to the magnetic tension.", "If $B$ is along the $z$ -axis, then we have $T^{\\mu \\nu }_{f} =\\begin{bmatrix}\\frac{B^{2}}{8\\pi } & 0 & 0 & 0 \\\\0 & \\frac{B^{2}}{8\\pi } & 0 & 0 \\\\0 & 0 & \\frac{B^{2}}{8\\pi } & 0 \\\\0 & 0 & 0 & -\\frac{B^{2}}{8\\pi }\\end{bmatrix}.$ Thus we see that pressure becomes anisotropic.", "The total pressure in the perpendicular direction to the magnetic field is given by $P_{\\bot } = P_{m} + \\frac{B^{2}}{8\\pi }$ and that in the parallel direction to the magnetic field is given by $P_{\\Vert } = P_{m} - \\frac{B^{2}}{8\\pi }.$ Now the parallel pressure becomes negative if the magnetic pressure exceeds the fluid pressure.", "In order to understand this effect, we write the component of $T^{\\mu \\nu }_{f}$ along the $z$ -axis as $T^{zz}_{f} = \\frac{B^{2}}{8\\pi } - \\frac{B^{2}}{4\\pi }.$ The second term $- B^{2}/4\\pi $ corresponds to an excess negative pressure or tension along the direction to the magnetic field.", "Thus the total energy momentum tensor can be written as $T^{\\mu \\nu } =\\begin{bmatrix}\\epsilon _{m} + \\frac{B^{2}}{8\\pi } & 0 & 0 & 0 \\\\0 & P_{m} + \\frac{B^{2}}{8\\pi } & 0 & 0 \\\\0 & 0 & P_{m} + \\frac{B^{2}}{8\\pi } & 0 \\\\0 & 0 & 0 & (P_{m} +\\frac{B^{2}}{8\\pi }) - \\frac{B^{2}}{4\\pi }\\end{bmatrix}.$ The strong magnetic field also reveals anisotropy due to magnetization pressure [27].", "Hence, actually the pressure in the perpendicular direction to the magnetic field is given by $P_{\\bot } = P_{m} + \\frac{B^{2}}{8\\pi } - \\cal M \\textit {B},$ where $\\cal M$ is the magnetization of the system, which is given by ${\\cal M} = - \\frac{\\partial \\epsilon _{m}}{\\partial \\textit {B}}.$ However, for the magnetic fields considered in the present work exhibiting super-Chandrasekhar masses, $B^2/8\\pi >>\\cal M \\textit {B}$ .", "Therefore, we do not include magnetization term in the pressure which does not affect the result practically for the present purpose.", "Here we refer to the work by Bocquet et al.", "[46], which models rotating neutron stars with magnetic fields, by using an extension of the electromagnetic code used by Bonazzola et al.", "[47].", "They observed that the component of the total energy momentum tensor along the symmetry axis becomes negative (equivalent to $T^{zz} < 0$ in our case), since the fluid pressure decreases more rapidly than the magnetic pressure away from the center of the star.", "This happens because the combined fluid-magnetic medium develops a tension.", "As a result of this magnetic tension, the star displays a pinch across the symmetry axis and assumes a flattened shape.", "A similar effect is expected to occur in our work, where the magnetic tension will be responsible for deforming the magnetized white dwarf along the direction to the magnetic field and turns it into a kind of oblate spheroid.", "Hence, one should be cautious before considering equation (REF ) which is applicable for a spherical star [45].", "However, in this work we consider a constant magnetic field, since our interest is to see the effect of the magnetic field of the central region of the white dwarf, where the field is supposed to be (almost) constant (see §4.3).", "Thus even if we use either the parallel or the perpendicular pressure in the hydrostatic equilibrium equation (REF ), $B$ does not appear explicitly in the equation ($dB/dr=0$ ) — only the gravitational field will be modified due to deformation.", "Hence, it is still possible to have super-Chandrasekhar mass white dwarfs - only they will be deformed in shape due to the strong field, which might even render a more massive white dwarf (as is discussed in Appendix A)." ], [ "Summary and Conclusions", "We have studied the effect of high magnetic field on the equation of state of purely electron degenerate matter at zero temperature.", "In the equation of state, we have considered only the electron degeneracy pressure modified by the strong magnetic field.", "We have focused on those Landau quantized systems in which the maximum number of Landau level(s) occupied is/are one, two or three, which we have named to be one-level, two-level and three-level system respectively.", "We have found that whenever a lower Landau level is completely filled and the next higher level is to be filled, a kink appears in the equation of state, followed by a plateau - which is a small region where the pressure becomes nearly independent of the density.", "The one-level system, which has only the ground Landau level filled, has no kink, the two-level system has one kink at the ground to first level transition and the three-level system has two kinks, one at the ground to first and the other at the first to second-level transition.", "We have studied each of these systems at three maximum Fermi energies $E_{Fmax} = 2\\,m_{e}c^{2}$ , $20\\,m_{e}c^{2}$ and $200\\,m_{e}c^{2}$ and obtained the mass-radius relations of the corresponding stars.", "The mass-radius relations show turning point(s), denoted by the decrease of both mass and radius, which correspond(s) to the kink(s) in the equation of state.", "The most interesting result obtained is that there are possible stars found on the mass-radius relations whose mass exceeds the Chandrasekhar limit.", "They could be potential magnetized white dwarfs.", "The maximum mass obtained is about $2.3-2.6 M_{\\odot }$ and is seen to occur for various combinations of central density, magnetic field strength and maximum number of occupied Landau levels.", "Interestingly such super-massive white dwarfs have been suggested to be the most likely progenitors of recently observed Type Ia supernovae [20], [21].", "Out of the equations of state considered in the present work, the system with the lowest magnetic field which gives rise to this mass is the three-level system with $E_{Fmax} = 20\\,m_{e}c^{2}$ , $B_{D} = 66.5$ (or $B = 2.94 \\times 10^{15}$ G) and the corresponding central density being $2.2 \\times 10^{9}$ gm/cc.", "The nature of the mass-radius relations is governed by the fact whether the system is one-level, two-level or three-level and is independent of the value of $E_{Fmax}$ .", "However, $E_{Fmax}$ determines how relativistic the system is.", "For instance, the Chandrasekhar mass limit is not exceeded for a low $E_{Fmax}$ (say = $2\\,m_{e}c^{2}$ ), no matter what the central density is.", "We have however observed that as $E_{Fmax}$ increases, which corresponds to an increase in the magnetic field strength, the degenerate stars become more compact in size.", "As discussed, the minimum magnetic field required to have a $2.3M_{\\odot }$ degenerate star is $B = 2.94 \\times 10^{15}$ G. The magnetic field of the original star of radius $R_{\\odot }$ , which collapses into the above degenerate star of radius $\\sim 10^{8}$ cm, turns out to be $\\sim 6 \\times 10^{9}$ G, based on the flux freezing theorem.", "Existence of such stars is not ruled out [15].", "However, the anisotropy in pressure due to strong magnetic field causes a deformation in the white dwarfs which adopt a flattened shape.", "This effect of flattening leads to more massive white dwarfs, even at relatively lower magnetic field strengths.", "One might wonder as to why highly magnetized white dwarfs have not yet been observed.", "A plausible reason could be that the surface magnetic field is being screened due to some physical processes.", "For instance, if the white dwarf is in a binary system and is accreting matter from its companion, as is proposed for Type Ia supernovae progenitors, then the plasma that is being deposited on the surface of the star could induce an opposite magnetic moment.", "This would result in a reduction of the surface field strength.", "However, the central magnetic field strength, which is presumably unaffected by the above processes, could be several orders of magnitude higher than the surface field.", "Indeed as seen in §4.3, it is the central field which is crucial for exceeding the Chandrasekhar mass limit." ], [ "Acknowledgments", "This work was partly supported by the ISRO grant ISRO/RES/2/367/10-11.", "We would like to thank the anonymous referees, Dipankar Bhattacharya and Efrain J. Ferrer for their useful comments, which have helped us greatly in improving this work." ], [ "Appendix", "In order to estimate the effect of deviation from spherical symmetry due to the magnetic field we have performed a few calculations.", "If the magnetic field is very strong, then the magnetic tension will flatten the star along the direction to the field (as discussed in §4.7).", "If we consider the white dwarf to be an oblate spheroid with the $z$ -axis being the symmetry axis, then its equation is given by $\\frac{r_{eq}^{2}}{a^{2}} + \\frac{z^{2}}{c^{2}} = 1,$ where, $r_{eq}^{2} = x^{2} + y^{2}$ , the semi-axis $a$ is the equatorial radius of the spheroid and $c$ is the distance from center to the pole along the symmetry axis ($c<a$ for an oblate spheroid).", "The equatorial force balance equation can be written as $\\frac{1}{\\rho } \\frac{dP}{dr_{eq}} = -\\frac{GM}{r^{2}}\\left(\\frac{r_{eq}}{r}\\right),$ where, $r^{2} = r_{eq}^{2} + z^{2}$ .", "The above equation for hydrostatic equilibrium has to be supplemented by an equation to determine the mass of the star.", "Now, $dM = \\rho dV$ and the volume element for an oblate spheroid is given by $dV = \\pi r_{eq}(z)^{2} dz.$ For a simpler visualization, one can also assume a cylindrical geometry and the equation for the mass can be given by $\\frac{dM}{dr_{eq}} = 2\\pi r_{eq} h \\rho ,$ where $h$ denotes an average height of the cylinder (assuming that the density does not change appreciably with $z$ ) and is a parameter that quantifies the degree of flattening.", "At high field strengths, the white dwarf will be more flattened and $h$ will be small.", "At low field strengths it is likely that the star will be less flattened and $h$ will have a larger value.", "Keeping this in mind we solved the above equations and obtained the mass-radius relations for some of the cases as shown in Figure REF .", "The one-level and two-level systems have strong magnetic fields and the corresponding mass-radius relations for the spherical case are denoted by the solid lines in Figures 7(a) and (b).", "From the dashed lines in Figures 7(a) and (b) we see that the flattened white dwarfs could have much higher masses than the perfectly spherical ones.", "Interestingly we note from the dashed lines of Figures 7(c) and (d) that even for much lower magnetic field strengths ($B = 4.4\\times 10^{14}$ G for $\\nu _{m}=20$ and $1.8\\times 10^{13}$ G for $\\nu _{m}=500$ ), the Chandrasekhar mass limit is exceeded even if one includes the corresponding reduced flattening effect - these white dwarfs were sub-Chandrasekhar for the spherical cases.", "One can also see that the radius of the stars represented by the dashed lines is larger than that represented by the solid lines.", "One must note here that for the dashed lines this radius is the equatorial radius, which will automatically be larger than the spherical radius as a consequence of the flattening effect.", "Hence, the effect of flattening leads to a more massive star.", "The effect is very similar to the flattening due to centrifugal force in rapidly rotating stars, which are known to be more massive than their slow-rotating counterparts (e.g.", "[46]).", "These are also shown to have larger mass in presence of high magnetic field.", "Thus the strong magnetic field is responsible for the increasing the mass of the white dwarfs, while the deformation (or flattening) of the white dwarf due to the field further adds on to the mass.", "Therefore, our estimate of the mass of the white dwarf, in fact just sets a lower bound.", "More interestingly, flattening effects due to magnetic field will render super-Chandrasekhar white dwarfs even at a smaller magnetic field - such relatively low magnetized white dwarfs are more probable in nature.", "[subfigure]position=top Figure: Mass-radius relations with E Fmax =20m e c 2 E_{Fmax} = 20\\,m_{e}c^{2} for (a) one-level system (ν m =1\\nu _{m}=1), (b) two-level system (ν m =2\\nu _{m}=2), (c) twenty-level system (ν m =20\\nu _{m}=20) and (d) five hundred-level system (ν m =500\\nu _{m}=500).", "Here M D M_D is the mass of the white dwarf in units of M ⊙ M_{\\odot } and R eq R_{eq} is the equatorial radius of the white dwarf in units of 10 8 10^{8} cm.", "All the solid lines represent the mass-radius relations for the cases, if the stars would have been spherical.", "The dashed lines in (a) and (b) represent the mass-radius relations for highly flattened (strongly magnetized) white dwarfs, while the dashed lines in (c) and (d) represent the mass-radius relations for less flattened (relatively weakly magnetized) white dwarfs.", "In all the four panels the yy-axis is in log scale." ] ]

[ [ "Prompt photon production and photon-hadron correlations at RHIC and the\n LHC from the Color Glass Condensate" ], [ "Abstract We investigate inclusive prompt photon and semi-inclusive prompt photon-hadron production in high energy proton-nucleus collisions using the Color Glass Condensate (CGC) formalism which incorporates non-linear dynamics of gluon saturation at small x via Balitsky-Kovchegov equation with running coupling.", "For inclusive prompt photon production, we rewrite the cross-section in terms of direct and fragmentation contributions and show that the direct photon (and isolated prompt photon) production is more sensitive to gluon saturation effects.", "We then analyze azimuthal correlations in photon-hadron production in high energy proton-nucleus collisions and obtain a strong suppression of the away-side peak in photon-hadron correlations at forward rapidities, similar to the observed mono-jet production in deuteron-gold collisions at forward rapidity at RHIC.", "We make predictions for the nuclear modification factor R_{p(d)A} and photon-hadron azimuthal correlations in proton(deuteron)-nucleus collisions at RHIC and the LHC at various rapidities." ], [ "Introduction", "The Color Glass Condensate (CGC) formalism has been successfully applied to many processes in high energy collisions involving at least one hadron or nucleus in the initial state.", "Examples are structure functions (inclusive and diffractive) in Deeply Inelastic Scattering of electrons on protons or nuclei, and particle production in proton-proton, proton-nucleus and nucleus-nucleus collisions, for a recent review see Ref. [1].", "The predicted suppression of $R_{dA}$ for the single inclusive hadron production in deuteron-nucleus (dA) collisions as well as the disappearance of the away side peak in di-hadron angular correlations in the forward rapidity region of RHIC [2], [3] are two of most robust predictions of the formalism which have been confirmed [4], see also Refs.", "[5], [6], [7].", "The CGC formalism has been also successful in providing predictions for the first LHC data [8] in proton-proton (pp) and nucleus-nucleus collisions [9], [10], [11].", "Nevertheless, there are more recent, alternative phenomenological approaches which combine nuclear shadowing, transverse momentum broadening and cold matter energy loss to describe the RHIC data [12], [13].", "Therefore, one needs to consider other observables which may help clarify the underlying dynamics of forward rapidity particle production at small $x$ .", "Inclusive prompt photon production [14] and prompt photon-hadron angular correlations [15] in the forward rapidity region are two such examples.", "Furthermore, there are advantages to studying prompt photon production as compared to hadron production.", "It is theoretically cleaner; one avoids the difficulties involved with description of hadronization of final state quarks and gluons, usually described by fragmentation functions valid at high transverse momentum.", "Also, one does not have to worry about possible initial state-final state interference effects which may be present for hadron production.", "In case of photon-hadron vs. di-hadron angular correlations, again the underlying theoretical understanding is more robust.", "Unlike di-hadron correlations which involve higher number of Wilson lines [16], photon-hadron correlations depend only on the dipole cross section properties of which are well understood.", "Both processes have been investigated previously, albeit not in detail and only in a limited kinematic range [14], [15], see also Refs.", "[17], [18].", "In this work, we extend the existing results for inclusive prompt photon production by clearly separating the contribution of direct and fragmentation photons.", "We show that direct photons are more sensitive to gluon saturation effects in the kinematics regions considered.", "We then investigate the dependence of prompt photon-hadron azimuthal angular correlations on high gluon density effects and show that gluon saturation effects lead to disappearance of the away side peak.", "The effect is very similar to the disappearance of the away side peak in di-hadron correlations observed in the forward rapidity region of RHIC in dA collisions  [19].", "Therefore, a measurement of this correlation at RHIC and the LHC would greatly help to clarify the role of CGC in the dynamics of particle production at high energy.", "The advantage of the CGC formalism over the more phenomenological models is that the cross section for many of these processes have the same common ingredient [1], [20], [21], the dipole total cross section; the imaginary part of the forward scattering amplitude of a quark-antiquark dipole on a proton or nucleus target.", "Its rapidity (energy) dependence is governed by the B-JIMWLK/BK evolutions equations [22], [23] and is pretty well-understood.", "The most recent advances in our understanding of the rapidity dependence of the dipole cross section include the running coupling constant corrections and the full Next-to-Leading Order corrections [24].", "The only input is the dipole profile (dependence on the dipole size $r_t$ ) at the initial rapidity $y_0$ which is modeled, usually motivated by the McLerran-Venugopalan (MV) model [25].", "The sensitivity to this initial condition is expected to go away at very large rapidities, see Sec.", "III and Ref. [6].", "This paper is organized as follows; we consider prompt photon-hadron production cross section in section IIA and inclusive prompt photon production in section IIB where we describe how to separate the contribution of direct and fragmentation photons.", "We then present our detailed numerical results and predictions at kinematics appropriate for RHIC and the LHC experiments in section III.", "We summarize our results in section IV.", "The cross section for production of a quark and a prompt photon with 4-momenta $l$ and $k$ respectively (both on-shell) in scattering of a on-shell quark with 4-momentum $p$ on a target (either proton or nucleus) in the CGC formalism has been calculated in [14] and is given by $d\\, \\sigma = \\frac{e^2\\,e_q^2}{2}\\frac{d^3 k}{(2\\pi )^3\\,2 k^-} \\frac{d^3 l}{(2\\pi )^3\\, 2l^-} \\frac{1}{2 p^-}(2\\pi )\\, \\delta (p^- - l^- -k^-) \\, {\\rm tr}_D\\,[\\cdots ]\\,d^2 \\vec{b_t}\\, d^2 \\vec{r_t}\\, e^{i (\\vec{l_t} + \\vec{k_t})\\cdot \\vec{r_t}} \\, N_F (b_t, r_t, x_g),$ where $Tr_D\\, [\\cdots ]$ is given by ${\\rm tr}_D\\, [\\cdots ] = 8\\, [(p^-)^2 + (l^-)^2] \\bigg [\\frac{p\\cdot l}{p\\cdot k\\, l\\cdot k} +\\frac{1}{l\\cdot k} - \\frac{1}{p\\cdot k} \\bigg ],$ which, after using the explicit forms of the momenta in the expression for the trace, can be written as, $&&{d\\sigma ^{q(p)\\, T \\rightarrow q(l)\\,\\gamma (k)\\, X}\\over d^2\\vec{b_t}\\, dk_t^2\\, dl_t^2\\, dy_{\\gamma }\\, dy_l\\, d\\theta } ={e_q^2\\, \\alpha _{em} \\over \\sqrt{2}(2\\pi )^3} \\,{k^-\\over k_t^2 \\sqrt{S}} \\,{1 + ({l^-\\over p^-})^2 \\over [k^- \\, \\vec{l_t} - l^- \\vec{k_t}]^2}\\nonumber \\\\&&\\delta [x_q - {l_t \\over \\sqrt{S}} e^{y_l} - {k_t \\over \\sqrt{S}} e^{y_{\\gamma }} ] \\,\\bigg [ 2 l^- k^-\\, \\vec{l_t} \\cdot \\vec{k_t} + k^- (p^- -k^-)\\, l_t^2 + l^- (p^- -l^-)\\, k_t^2 \\bigg ]\\nonumber \\\\&&\\int d^2 \\vec{r_t} \\, e^{i (\\vec{l_t} + \\vec{k_t})\\cdot \\vec{r_t}} \\, N_F (b_t, r_t, x_g) ,$ where the symbol $T$ stands for a proton $p$ or a nucleus $A$ target, $\\sqrt{S}$ is the nucleon-nucleon center of mass energy and $x_q$ is the ratio of the incoming quark to nucleon energies such that $p^-=x_q \\, \\sqrt{S/2}$ .", "The outgoing photon and quark rapidities are defined via $k^-={k_t \\over \\sqrt{2}} e^{y_{\\gamma }}$ and $l^-={l_t \\over \\sqrt{2}} e^{y_l}$ whereas $\\Delta \\theta $ is angle between the final state quark and photon, $cos(\\Delta \\theta ) \\equiv {\\vec{l}_t \\cdot \\vec{k}_t \\over l_t k_t}$ .", "We note that this cross section was first computed in [17] in coordinate space using the dipole formalism, the result of which agrees with the expression in Eq.", "(REF ) after Fourier transforming to momentum space.", "The imaginary part of of (quark-antiquark) dipole-target forward scattering amplitude $N_F (b_t, r_t, x_g)$ satisfies the B-JIMWLK equation and has all the multiple scattering and small $x$ evolution effects encoded.", "It is defined as $N_F(b_t,r_t,x_g) = {1\\over N_c} \\, < Tr [1 - V^{\\dagger } (x_t) V (y_t) ] >,$ where $N_c$ is the number of color.", "The vector $\\vec{b_t}\\equiv (\\vec{x_t} + \\vec{y_t})/2$ is the impact parameter of the dipole from the target and $\\vec{r_t}\\equiv \\vec{x_t} - \\vec{y_t}$ denotes the dipole transverse vector.", "The matrix $V (y_t)$ is a unitary matrix in fundamental representation of $SU(N_c)$ containing the interactions of a quark and the colored glass condensate target.", "The dipole scattering probability depends on Bjorken $x_g$ via the B-JIMWLK renormalization group equations.", "In the present case, it is related to the prompt photon and final state quark rapidities and transverse momenta via $x_g = {1\\over \\sqrt{S}}[k_t e^{-y_{\\gamma }} + l_t e^{-y_l}]\\, .$ In order to relate the above partonic production cross-section to proton (deuteron)-target collisions, one needs to convolute the partonic cross-section in Eq.", "(REF ) with the quark and antiquark distribution functions of a proton (deuteron) and the quark-hadron fragmentation function: $\\frac{d\\sigma ^{p\\, T \\rightarrow \\gamma (k)\\, h (q)\\, X}}{d^2\\vec{b_t} \\, dk^2_t\\, dq^2_t \\,d\\eta _{\\gamma }\\, d\\eta _{h}d\\theta }&=& \\int ^1_{z_{f}^{min}} \\frac{dz_f}{z_f^2} \\,\\int \\, dx_q\\,f (x_q,Q^2) \\frac{d\\sigma ^{q\\, T \\rightarrow \\gamma \\, q \\, X}}{d^2\\vec{b_t}\\, dk^2_t\\,dl^2_t\\,d\\eta _{\\gamma }\\, d\\eta _{h}\\, d\\theta } D_{h/q}(z_f,Q^2),$ where $q_t$ is the transverse momentum of the produced hadron, and $f(x_q,Q^2)$ is the parton distribution function (PDF) of the incoming proton (deuteron) which depends on the light-cone momentum fraction $x_q$ and the hard scale $Q$ .", "A summation over the quark and antiquark flavors in the above expression should be understood.", "The function $D_{h/q}(z_f,Q)$ is the quark-hadron fragmentation function (FF) where $z_f$ is the ratio of energies of the produced hadron and quark Since produced hadrons are assumed to be massless, we make no distinction between the rapidity of a quark and the hadron to which it fragments.", "Moreover, for massless hadrons, rapidity $y$ and pseudo-rapidity $\\eta $ is the same..", "Note that due to the assumption of collinear fragmentation of a quark into a hadron, the angle $\\Delta \\theta $ is now the angle between the produced photon and hadron.", "The light-cone momentum fraction $x_q, x_{\\bar{q}}, x_g$ are related to the transverse momenta and rapidities of the produced hadron and prompt photon via (details are given in the appendix) $x_q&=&x_{\\bar{q}}=\\frac{1}{\\sqrt{S}}\\left(k_t\\, e^{\\eta _{\\gamma }}+\\frac{q_t}{z_f}\\, e^{\\eta _{h}}\\right),\\nonumber \\\\x_g&=&\\frac{1}{\\sqrt{S}}\\left(k_t\\, e^{-\\eta _{\\gamma }}+ \\frac{q_t}{z_f}\\, e^{-\\eta _{h}}\\right),\\nonumber \\\\z_f&=&q_t/l_t \\hspace{28.45274pt} \\text{with}~~~~~ z_{f}^{min}=\\frac{q_t}{\\sqrt{S}}\\left(\\frac{e^{\\eta _h}}{1 - {k_t\\over \\sqrt{S}}\\, e^{\\eta _{\\gamma }}}\\,\\right).\\ $" ], [ "Single inclusive prompt photon production in proton-nuclear collisions", "The single inclusive prompt photon cross section can be readily obtained from Eq.", "(REF ) by integrating over the momenta of the final state quark.", "Integration over the quark energy $l^-$ is trivially done by using the delta function and leads to (after shifting $\\vec{l_t} \\rightarrow \\vec{l_t} - \\vec{k_t}$ ) , $\\frac{d\\sigma ^{q (p) T \\rightarrow \\gamma (k) \\, X}}{d^2\\vec{b_t} dk^2_t d\\eta _{\\gamma }}&=&\\frac{e_q^2 \\alpha _{em}}{(2\\pi )^3} z^2[1+(1-z)^2]\\frac{1}{k^2_t} \\int d^2 \\vec{r_t} \\,d^2 \\vec{l_t}\\frac{l_t^2}{[z\\, \\vec{l}_t - \\vec{k}_t ]^2}\\, e^{i \\vec{l_t}\\cdot \\vec{r_t}} \\, N_F (b_t, r_t, x_g),$ where $z \\equiv k^-/p^-$ denotes the fraction of the projectile quark energy $p^-$ carried by the photon and $d\\eta _\\gamma \\equiv \\frac{d z}{z}$ .", "Various limits of this expression have been studied in [14] where it was shown that in the limit where photon has a large transverse momentum $k_t \\gg z\\, l_t$ such that the collinear singularity is suppressed, one recovers the LO pQCD result for direct photon production process $q\\, g \\rightarrow q\\, \\gamma $ convoluted with the unintegrated gluon distribution function of the target.", "On the other hand, if one performs the $l_t$ integration above without any restriction, one recovers the LO pQCD expression for quark-photon fragmentation function convoluted with dipole scattering probability.", "In the limit where one can ignore multiple scattering of the quark on the target (\"leading twist\" kinematics), this expression reduces to the pQCD one describing LO production of fragmentation photons.", "It is therefore useful to explicitly separate the contribution of fragmentation photons from that of the direct photons.", "To this end, we rewrite Eq.", "(REF ) as $\\frac{d\\sigma ^{q (p)\\, T \\rightarrow \\gamma (k) \\, X}}{d^2\\vec{b_t} d^2\\vec{k_t} d\\eta _{\\gamma }}&=&\\frac{e_q^2 \\alpha _{em}}{\\pi (2\\pi )^3} z^2[1 + (1-z)^2]\\frac{1}{k^2_t}\\int d^2 \\vec{l_t}\\, l_t^2\\Big [\\frac{1}{[z \\vec{l}_t -\\vec{k}_t]^2}-\\frac{1}{k_t^2}\\Big ]N_F(x_g,b_t,l_t), \\nonumber \\\\&+& \\frac{e_q^2 \\alpha _{em}}{\\pi (2\\pi )^3} z^2 [1 + (1 - z)^2]\\frac{1}{k^4_t}\\int d^2 \\vec{l_t} \\, l_t^2N_F(x_g,b_t,l_t),$ where we have added and subtracted the second term.", "Notice that we use the same notation for coordinate representation of the forward dipole-target scattering amplitude $N_F$ and its two-dimensional Fourier transform.", "The second term in this expression describes production of direct photons whereas the first term gives the contribution of fragmentation photons.", "In order to see this more explicitly, we let $\\vec{l_t} \\rightarrow \\vec{l_t} + \\frac{\\vec{k_t}}{z}$ in the first term and keep the most divergent piece of the $l_t$ integral to get $\\frac{d\\sigma ^{q (p)\\, T \\rightarrow \\gamma (k) \\, X}}{d^2 \\vec{b_t} d^2 \\vec{k_t} d\\eta _{\\gamma }}&=&\\frac{d\\sigma ^{\\text{Fragmentation}}}{d^2 \\vec{b_t} d^2 \\vec{k_t} d\\eta _{\\gamma }}+\\frac{d\\sigma ^{\\text{Direct}}}{d^2 \\vec{b_t} d^2 \\vec{k_t} d\\eta _{\\gamma }}\\nonumber \\\\&=&\\frac{1}{(2\\pi )^2}\\frac{1}{z}\\, D_{\\gamma /q}(z,k_t^2)\\,N_F(x_g,b_t,k_t/z) +\\frac{e_q^2 \\alpha _{em}}{\\pi (2\\pi )^3}z^2[1+(1 - z)^2]\\frac{1}{k^4_t}\\int ^{k^2_t}d^2\\vec{l_t}\\,l_t^2\\,N_F(\\bar{x}_g,b_t,l_t)$ where $D_{\\gamma /q}(z,k_t^2)$ is the leading order quark-photon fragmentation function [26], $D_{\\gamma /q}(z,Q^2)=\\frac{e_q^2 \\alpha _{em}}{2\\pi }\\frac{1 + (1 - z)^2}{z}\\ln {Q^2/\\Lambda ^2}.$ Eq.", "(REF ) is new and is our main result for single inclusive prompt photon production which includes contribution of both fragmentation (first term) and direct (second term) photons.", "In order to ensure that the divergence present in Eq.", "(REF ) is properly removed, one needs to regulate it self-consistently.", "Here we have done this separation by imposing a hard cutoff which would result in a mismatch between the finite corrections to our results and those that are included in parameterizations of photon fragmentation function, for example, using the $\\overline{MS}$ scheme.", "However this mismatch is a higher order effect in the coupling constant and is therefore expected to be parametrically small.", "It should be noted that the separation between the direct and fragmentation contributions depends on the hard scale, chosen to be the photon transverse momentum, which is already well-known in pQCD.", "Eq.", "(REF ) exhibits some interesting features; the dipole scattering probability $N_F$ is probed at $k_t/z$ (where $k_t$ is the external momentum) in case of fragmentation photons whereas it depends on the internal momentum $l_t$ in case of direct photons.", "Furthermore, in case of direct photons, the integrand is peaked at values of transverse momenta $l_t \\sim Q_s$ .", "This means that fragmentation photons should be much less sensitive to high gluon density effects than direct photons since they probe the target structure at higher transverse momenta.", "This will be verified numerically in the following sections.", "In order to relate the partonic cross-section given by Eq.", "(REF ) to photon production in deuteron (proton)-nucleus collisions, we convolute Eq.", "(REF ) with quark and antiquark distribution functions of the projectile deuteron (or proton), $\\frac{d\\sigma ^{p\\, T \\rightarrow \\gamma (k) \\, X}}{d^2\\vec{b_t} d^2\\vec{k_t} d\\eta _{\\gamma }}=\\int _{x_q^{min}}^1 d x_q [f_q(x_q,k_t^2)+ f_{\\bar{q}}(x_{\\bar{q}},k_t^2)]\\frac{d\\sigma ^{q (p) \\, T \\rightarrow \\gamma (k) \\, X}}{d^2 \\vec{b_t} d^2\\vec{ k_t} d\\eta _{\\gamma }},$ where a summation over different flavors is understood.", "Equations (REF ,REF ) are our final results for the single inclusive prompt photon production.", "The light-cone fraction variables $x_g,\\bar{x}_g,z$ in Eq.", "(REF ,REF ) are defined as follows, $x_g&=& \\frac{k_t^2}{z^2\\, x_q\\, S} = x_q \\, e^{-2\\, \\eta _\\gamma } \\\\\\bar{x}_g &=& \\frac{1}{x_q\\, S} \\left[{k_t^2\\over z} + \\frac{(l_t-k_t)^2}{1-z}\\right]\\approx \\frac{1}{x_q\\, S} {k_t^2 \\over z (1-z)},\\\\z&\\equiv & \\frac{k^-}{p^-} = \\frac{k_t}{x_q\\, \\sqrt{S}}e^{\\eta _{\\gamma }} = \\frac{x_q^{min}}{x_q}\\hspace{28.45274pt} \\text{with}~~~~~ x_q^{min}=z_{min}=\\frac{k_t}{\\sqrt{S}}e^{\\eta _{\\gamma }}.", "\\ $ where in Eq.", "() the right hand-side approximation is valid if $l_t\\ll k_t$ .", "Notice that since now $\\bar{x}_g$ depends on the angle between $l_t$ and $k_t$ , the integral over the angle in Eq.", "(REF ) is not more trivial and can be done numerically.", "One should also note that the light-cone fraction variables defined above for the inclusive prompt photon cross-section Eqs.", "(REF ,REF ) are different from the corresponding semi-inclusive hadron-photon cross-section Eqs.", "(REF ,REF ) defined in Eqs.", "(), see the appendix for the derivation." ], [ "Numerical results and predictions", "The forward dipole-target scattering amplitude appears in both semi-inclusive photon-hadron and inclusive prompt photon cross-section Eq.", "(REF ,REF ) and incorporates small-x dynamics which can be computed via first principle non-linear B-JIMWLK equations [22] in the CGC formalism.", "In the large $N_c$ limit, the coupled B-JIMWLK equations are simplified to the Balitsky-Kovchegov (BK) equation [23], a closed-form equation for the evolution of the dipole amplitude in which both linear radiative processes and non-linear recombination effects are systematically incorporated.", "The running-coupling BK (rcBK) equation has the following simple form: $\\frac{\\partial N_{F}(r,x)}{\\partial \\ln (x_0/x)}=\\int d^2{\\vec{r}_1}\\ K^{{\\rm run}}({\\vec{r}},{\\vec{r}_1},{\\vec{r}_2})\\left[N_{F}(r_1,x)+N_{F}(r_2,x)-N_{F}(r,x)-N_{F}(r_1,x) N_{F}(r_2,x)\\right]\\,,$ where $\\vec{r}_2=\\vec{r}- \\vec{r}_1$ .", "The evolution kernel $K^{{\\rm run}}$ is given by Balitsky's prescription [27] with the running coupling.", "The explicit form of $K^{{\\rm run}}$ with details can be found in Refs.", "[27], [28].", "The only external input for the rcBK non-linear equation is the initial condition for the evolution which is taken to have the following form motivated by McLerran-Venugopalan (MV) model [25], $\\mathcal {N}(r,Y\\!=\\!0)=1-\\exp \\left[-\\frac{\\left(r^2\\,Q_{0s}^2\\right)^{\\gamma }}{4}\\,\\ln \\left(\\frac{1}{\\Lambda \\,r}+e\\right)\\right]\\ ,$ where $\\Lambda =0.241$ GeV [10], [29].", "The initial saturation scale $Q_{0s}$ (with $s=p,A$ for a proton and nuclear target) at starting point of evolution (at $x_0=0.01$ ) and the parameter $\\gamma $ , are free parameters which are determined from a fit to other experimental measurements at small-x.", "It was shown that inclusive single hadron data in pp collisions at RHIC can be described with a initial saturation scale within $Q_{0p}^2= 0.168\\div 0.336~\\text{GeV}^2$ [5], [6], [30].", "However, HERA data on proton structure functions prefers the lower value for the proton initial saturation scale $Q_{0p}^2\\approx 0.168~\\text{GeV}^2$ [29].", "One have also freedom to run $\\gamma $ as a free parameter in the $\\chi ^2$ minimization and obtain its preferred value in a fit to HERA data.", "In order to investigate the uncertainties due to initial condition of the rcBK equation, we will consider the following three parameter sets which all provide an excellent fit to the HERA data for proton targets [10], [29]: $ \\text{set I}:&&\\hspace{14.22636pt} Q_{0p}^2=0.2\\,\\text{GeV}^2\\hspace{22.76228pt}\\gamma =1 ,\\nonumber \\\\\\text{set II}:&&\\hspace{14.22636pt} Q_{0p}^2=0.168\\,\\text{GeV}^2\\hspace{14.22636pt}\\gamma =1.119,\\nonumber \\\\\\text{set III}:&&\\hspace{14.22636pt} Q_{0p}^2=0.157\\,\\text{GeV}^2\\hspace{14.22636pt}\\gamma =1.101.\\ $ In the MV model [25], the parameter $\\gamma $ in Eq.", "(REF ) is $\\gamma =1$ .", "However, it has been recently shown [31] that the effective value of $\\gamma $ can be larger than one when the sub-leading corrections to the MV model are included [32].", "The parameter $\\gamma $ appears to be also important in order to correctly reproduce the single inclusive particle spectra, and a larger value $\\gamma >1$ is apparently preferable at large-$k_t$ [10], [6].", "In our approach the difference between proton and nuclei originates from different initial saturation scales $Q_{0s}$ in the rcBK equation via Eq.", "(REF ).", "In the case of inclusive hadron production in proton-nucleus collisions, due to theoretical uncertainties and rather large errors of the experimental data, it is not possible to uniquely fix the initial value of $Q_{0A}$ .", "In the case of minimum-bias dAu collisions, the initial nuclear (gold) saturation scale within $Q_{0A}^2=3 \\div 4~Q_{0p}^2$ is consistent with the RHIC inclusive hadron production data [2], [5], [6], [30].", "The extracted value of $Q_{0A}$ is also consistent with the DIS data for nuclear targets [29], [30].", "Here, we will also consider the uncertainties due to the initial condition of the rcBK equation for a nuclear target.", "Note that $Q_{0A}$ should be considered as an impact-parameter averaged value since it was extracted from the minimum-bias data.", "For the minimum-bias collisions, one may assume that the initial saturation scale of a nuclei with atomic mass number A, scales linearly with $A^{1/3}$ , namely we have $Q_{0A}^2=cA^{1/3}~Q_{0p}^2$ where the parameter $c$ is fixed from a fit to data.", "In Ref.", "[30], it was shown that NMC data can be described with $c\\approx 0.5$ .", "We will use the NLO MSTW 2008 PDFs [33] and the NLO KKP FFs [34].", "For the photon fragmentation function, we will use the full leading log parametrization [26], [35].", "We assume the factorization scale $Q$ in the FFs and the PDFs to be equal and its value is taken to be $q_t$ and $k_t$ for the semi-inclusive and inclusive prompt photon production, respectively." ], [ "Direct and fragmentation prompt photon in pp and pA collisions at RHIC and the LHC", "We start by considering direct and fragmentation photon production in pA collisions at RHIC and the LHC.", "In nuclear collisions, nuclear effects on single particle production are usually evaluated in terms of ratios of particle yields in pA and pp collisions (scaled with a proper normalization), the so-called nuclear modification factor $R_{pA}$ .", "The nuclear modification factor $R_{p(d)A}$ is defined as $R_{dA}^{\\gamma }&=&\\frac{1}{2\\, N_{coll}}\\frac{dN^{d A \\rightarrow \\gamma X}}{d^2p_T d\\eta }/\\frac{dN^{p p \\rightarrow \\gamma X}}{d^2p_T d\\eta }, \\nonumber \\\\R_{pA}^{\\gamma }&=&\\frac{1}{N_{coll}}\\frac{dN^{p A \\rightarrow \\gamma X}}{d^2p_T d\\eta }/\\frac{dN^{p p \\rightarrow \\gamma X}}{d^2p_T d\\eta },\\ $ where the yield $\\frac{dN^{p(d) A(p) \\rightarrow \\gamma X }}{d^2p_T d\\eta }$ can be calculated from the invariant cross-section given in Eq.", "(REF ).", "The normalization constant $N_{coll}$ is the number of binary proton-nucleus collisions.", "We take $N_{coll}=3.6, 6.5$ and $7.4$ at $\\sqrt{s}=0.2, 4.4$ and $8.8$ TeV, respectively [36].", "In order to compare our predictions for $R_{pA}^{\\gamma }$ with the experimental value, one should take into account possible discrepancy between our assumed normalization $N_{coll}$ and the experimentally measured value for $N_{coll}$ by rescaling our curves.", "Again we expect that some of the theoretical uncertainties, such as sensitivity to $K$ factors (which effectively incorporates the missing higher order corrections), will drop out in $R_{p(d)A}^{\\gamma }$ .", "In Fig.", "REF we show the nuclear modification factor for both pA and dA collisions at RHIC.", "This is to facilitate a comparison of and to distinguish between the genuine saturation effects in the target nucleus from isospin effects in the projectile deuteron.", "Figure: Nuclear modification factor for direct photon production in minimum-bias pA (dashed line) and dA (solid line) collisions at RHIC (S=0.2\\sqrt{S}=0.2 TeV) at η=2,3\\eta = 2, 3.", "The curves are obtained from Eq.", "() using the solution to rcBK equation with the initial saturation scale Q 0p 2 =0.168GeV 2 Q_{0p}^2=0.168~\\text{GeV}^2 for proton and Q 0A 2 =3Q 0p 2 Q_{0A}^2=3Q_{0p}^2 for a nucleus (gold).Clearly there is a large difference between a proton and a deuteron projectile as far as prompt photon production is concerned.", "This difference is more pronounced in the forward rapidity region and at high transverse momentum where one probes the quark content of the projectile.", "This is due to difference between the up and down quark distributions of a proton (note that nuclear effects in the wave function of a deuteron are ignored as they are known to be small).", "This is a well known effect, nevertheless, for sake of clarity and to illustrate the difference between inclusive hadron production and QED probe namely prompt photon production, we illustrate this in some detail.", "In case of photon production, the production cross section is weighed by the the charged squared of a given quark flavor.", "For example (assuming only two flavor), for a proton projectile this is given by $e_q^2\\, f_{q/p} = (2/3)^2 \\, u_p + (1/3)^2\\, d_p$ where $u_p, d_p$ denote the distribution functions of up and down quarks in a proton.", "Ignoring nuclear effects in a deuteron, we assume a deuteron is a system of free proton and a neutron in which case the corresponding expression is $e_q^2\\, f_{q/d} = (2/3)^2 \\, u_p + (1/3)^2\\, d_p + (2/3)^2\\, u_n + (1/3)^2\\, d_n$ where $u_n, d_n$ denote the distribution functions of up and down quarks in a neutron.", "Assuming isospin symmetry gives $u_n = d_p$ and $d_n = u_p$ which leads to $(5/9)\\, [u_p + d_p]$ for a deuteron.", "Comparing this expression with two times that of a proton, the relative contribution of up quarks in a deuteron ($5/9$ ) is smaller than that of a twice a proton ($8/9$ ).", "Since there are more up quarks than down quarks (by a factor of $2\\div 3$ in this kinematics) in a proton, and their relative weight is smaller, this leads to a further reduction of $R_{dA}$ as compared with $R_{pA}$ in prompt photon production, see Fig.", "REF .", "We note that in the absence of the charged squared factor, which is the case for inclusive hadron production, one would get $d = p + n = 2\\, p$ as one should since possible nuclear effects in a deuteron are ignored here.", "At the LHC the isospin effect is absent since the same projectile is used for the pp and pA collisions.", "This helps to understand the physics of QCD saturation more clearly, as the suppression of the signal will not be contaminated with isospin effect.", "Figure: Nuclear modification factor for direct, fragmentation and inclusive prompt photon production in minimum-bias p(d)A collisions at RHIC S=0.2\\sqrt{S}=0.2 TeV (right) and the LHC S=4.4\\sqrt{S}=4.4 TeV (left) energy at various rapidities.", "The curves are the results obtained from Eq.", "() and the solution to rcBK equation with the initial saturation scale Q 0p 2 =0.168GeV 2 Q_{0p}^2=0.168~\\text{GeV}^2 for a proton and Q 0A 2 =3Q 0p 2 Q_{0A}^2=3Q_{0p}^2 for a nucleus (gold), corresponding to set II in Eq.", "().Figure: Nuclear modification factor for direct photon production in p(d)A collisions at various rapidities at RHIC S=0.2\\sqrt{S}=0.2 TeV (right) and the LHC S=4.4\\sqrt{S}=4.4 TeV energy (left).", "The curves are the results obtained from Eq.", "() and the solution to rcBK equation using different initial saturation scales for a proton Q 0p Q_{0p} and a nucleus Q 0A Q_{0A}.", "The band shows our theoretical uncertainties arising from allowing a variation of the initial saturation scale of the nucleus in a range consistent with previous studies of DIS structure functions as well as particleproduction in minimum-bias pp, pA and AA collisions in the CGC formalism, see the text for the details.Figure: Nuclear modification factor for direct photon (right) and inclusive prompt photon (left) production in pA collisions at various rapidities a the LHC S=8.8\\sqrt{S}=8.8 TeV energy.", "The band (CGC-rcBK-av) similar to Fig.", "corresponds to the results obtained from Eq.", "() and the solutions to the rcBK evolution equation using different initial saturation scales for a proton Q 0p Q_{0p} and a nucleus Q 0A Q_{0A}, see the text for the details.Figure: Right: Nuclear modification factor for direct photon production at η=3\\eta =3 in minimum-bias dA S=0.2\\sqrt{S}=0.2 TeV (RHIC) and pA S=4.4,8.8\\sqrt{S}=4.4, 8.8 TeV (LHC) collisions.", "The curves are the results obtained from Eq.", "() and the solution to rcBK equation with the initial saturation scale Q 0p 2 =0.168GeV 2 Q_{0p}^2=0.168~\\text{GeV}^2 for a proton and Q 0A 2 =3Q 0p 2 Q_{0A}^2=3Q_{0p}^2 for a nucleus.", "Left: Comparison of the inclusive prompt photon nuclear modification factor predictions from the CGC (in this paper) and the standard collinear factorization approach .", "The band CGC-rcBK-av is the same as in Fig.", ".In Fig.", "REF , we show the minimum-bias nuclear modification factor for the direct, fragmentation and the inclusive prompt photon production at RHIC and the LHC energies $\\sqrt{S}=0.2, 4.4$ TeV at various rapidities $\\eta $ obtained from Eqs.", "(REF , REF ) supplemented with rcBK solution Eq.", "(REF ) with the initial saturation scale for proton $Q_{0p}^2\\approx 0.168~\\text{GeV}^2$ and nuclei $Q_{0A}^2=3 Q_{0p}^2$ .", "It is seen that the nuclear modification $R_{p(d)A}^{\\gamma }$ for the fragmentation photon is bigger than the direct and inclusive prompt photon.", "This is what we expected in our picture since direct photon cross-section in Eq.", "(REF ) probes the target structure function at lower transverse momentum $k_t$ (and consequently lower x) than the fragmentation part with transverse momentum $k_t/z$ and therefore is more sensitive to the suppression of structure function and the saturation effect.", "However, as we increase the energy the enhancement of the fragmentation photon $R_{p(d)A}^{\\gamma }$ at RHIC will be also replaced with suppression at the LHC, see Fig.", "REF top panel.", "This is simply due to the fact that both the fragmentation and the direct part Eq.", "(REF ) depend on the color dipole forward amplitude which encodes the small-x dynamics and at higher energy, the small-x evolution leads to suppression of $R_{p(d)A}^{\\gamma }$ .", "In a collider experiment such as the LHC, the secondary photons coming from the decays of hadrons, overwhelm the inclusive prompt photon measurements with order of magnitudes.", "In order to reject the background, isolation cuts are imposed [37].", "Contribution of fragmentation prompt photon is reduced by imposing an isolation cutIf we assume that $p_c$ is the total transverse momentum of a fragmentation jet, the photon's energy is then $E_\\gamma =z p_c$ and the total hadronic energy within the jet is $E_h=(1-z)p_c$ .", "By isolation cut criterion, the hadronic energy does not have to be more than $\\epsilon E_\\gamma $ in the isolation cone.", "This gives the lower limit of $z$ (or $x_q$ convolution) in Eq.", "(REF ) integral, namely $z_c=1/(1+\\epsilon )<z$ .", "Given that the integrand of fragmented part is proportional to $1/z$ and dominated at lower limit of integrand, we expect that the isolation cut reduces the fragmentation contribution more severely than the direct one.. A proper incorporation of the isolation cut criterion in our framework is beyond the scope of this paper.", "However, from Fig.", "REF it is seen that at higher energy at forward collisions, $R_{p(d)A}^{\\gamma }$ for direct and single inclusive prompt photon becomes remarkably similar, indicating that to a good approximation, one may assume that the nuclear modification factor for direct and isolated prompt photon are equal.", "In Fig.", "REF , we show the minimum-bias nuclear modification factor for the direct photon production at RHIC and the LHC energies $\\sqrt{S}=0.2, 4.4$ TeV at various rapidities $\\eta $ obtained from rcBK solutions Eq.", "(REF ) with the initial proton saturation scale $Q_{0p}^2\\approx 0.168$ and $0.2\\,\\text{GeV}^2$ corresponding to parameter sets I and II in Eq.", "(REF ).", "For nuclear target in minimum-bias collisions, we take two initial saturation scales for nuclei (gold and lead) $Q_{0A}^2=3\\div 4 Q_{0p}^2$ which are extracted from a fit to other experimental data on heavy nuclear target [5], [6], [30].", "For a proton target, we have checked that parameter sets II and III give similar results for $R_{p(d)A}^{\\gamma }$ with better than $10\\% $ accuracy.", "Therefore, in Fig.", "REF we only show results obtained from two parameter sets I and II in Eq.", "(REF ).", "The band in Fig.", "REF shows our uncertainties arising from a variation of the initial saturation scale of the nucleus in a range consistent with previous studies of DIS structure functions as well as particle production in minimum-bias pp, pA and AA collisions in the CGC formalism.", "One may therefore expect that the possible effects of fluctuations (of nucleons in a nucleus) on particle production is effectively contained in our error band.", "From Fig.", "REF , it is seen that the nuclear modification for direct photon production is very sensitive to the initial saturation scale in proton and nuclei.", "However, this uncertainties will be reduced for more forward collisions at higher energy at the LHC.", "The same effect has been observed for the inclusive hadron production in pA collisions [6].", "This clearly indicates that the nuclear modification in p(d)A collisions is a sensitive probe of saturation effects and $R_{p(d)A}^{\\gamma }$ measurements for direct photon and inclusive hadron provide crucial complementary information about initial saturation scale and small-x evolution dynamics.", "In Fig.", "REF , it is seen that at a fixed rapidity and energy for a fixed initial saturation scale for proton $Q_{0p}$ , a bigger initial saturation scale for nuclei $Q_{0A}$ leads to a bigger broadening and consequently enhances the cross-section and $R_{p(d)A}^{\\gamma }$ if $N_{coll}$ is kept fixed.", "In Fig.", "REF , we show our predictions for $R_{pA}$ for direct photon (right) and inclusive prompt photon (left) production in pA collisions at various rapidities a the LHC $\\sqrt{S}=8.8$ TeV energy.", "The band (CGC-rcBK-av) similar to Fig.", "REF corresponds to the results obtained from Eq.", "(REF ) with the solutions of the rcBK evolution equation (REF ).", "In Fig.", "REF (right), we compare $R_{p(d)A}^{\\gamma }$ for direct photon at $\\eta =3$ for RHIC energy $\\sqrt{S}=0.2$ TeV and the LHC energies $4.4, 8.8$ TeV.", "It is seen that the suppression of $R_{pA}^{\\gamma }$ at the LHC is larger compared to $R_{dA}$ at RHIC and persists at higher transverse momentum.", "This larger suppression is even more impressive given that fact that a good amount of the observed suppression of $R_{dA}$ at RHIC is due to the projectile being a deuteron rather than a proton.", "In Fig.", "REF (left), we compare the CGC prediction (CGC-rcBK-av) obtained here with the collinear factorization result (EPS09) [12] for inclusive prompt photon $R_{pA}^{\\gamma }$ at $\\eta =3$ at the LHC.", "It is seen that the LHC measurements of the inclusive prompt photon at forward rapidities can discriminate between the collinear (standard parton model) and the CGC approach.", "Some words of caution are in order here.", "Strictly speaking our formalism is less reliable for collisions at around mid-rapidities and high transverse momenta.", "This is due to the fact that our formula is valid for asymmetric collisions like pA or pp collisions at forward rapidities when a projectile can be treated in the standard collinear approximation while for the target we systematically incorporated the small-x re-summation (at the leading log approximation) effects.", "Note however, for the case of pp collisions (our reference for $R_{dA}^{\\gamma }$ ) at RHIC, the saturation scale of target proton is rather small, and it is not clear that the CGC formulation will be applicable.", "Moreover, our parameter sets in Eq.", "(REF ) was obtained from a fit to HERA data at small-x $x<0.01$ and for virtualities $Q^2\\in [0.25,40] \\,\\text{GeV}^2$ [29].", "Therefore, our predictions are less reliable at high-$k_t$ ($k_t> 6\\div 7$ GeV).", "Figure: The relative azimuthal correlation P(Δθ)P(\\Delta \\theta ) defined in Eq.", "() for minimum-bias p(d)A and pp collisions at RHIC S=0.2\\sqrt{S}=0.2 TeV (upper) and the LHC S=4.4\\sqrt{S}=4.4 TeV energy (lower) obtained from the rcBK solutions with different initial saturation scales." ], [ "Prompt photon-hadron correlations at RHIC and the LHC; the signature of saturation", "We now focus on azimuthal angle $\\Delta \\theta $ correlations of the prompt photon-hadron spectrum, where the angle $\\Delta \\theta $ is the difference between the azimuthal angle of the measured hadron and single prompt photon.", "We present our predictions for semi-inclusive prompt photon-hadron (for hadron we consider only neutral pion here) production at RHIC and the LHC in pp and p(d)A collisions in terms of $P(\\Delta \\theta )$ defined as follows, $P(\\Delta \\theta )={d\\sigma ^{p(d)\\, T \\rightarrow h(q)\\,\\gamma (k)\\, X}\\over d^2\\vec{b_t}\\, dk_t^2\\, dq_t^2\\, dy_{\\gamma }\\, dy_l\\, d\\theta } [\\Delta \\theta ]/{d\\sigma ^{p(d)\\, T \\rightarrow h(q)\\,\\gamma (k)\\, X}\\over d^2\\vec{b_t}\\, dk_t^2\\, dq_t^2\\, dy_{\\gamma }\\, dy_l\\, d\\theta } [\\Delta \\theta = \\Delta \\theta _c],$ where the prompt photon-hadron cross-section in above expression is given in Eq.", "(REF ).", "This definition has a simple meaning of the probability of, the single semi-inclusive prompt photon-hadron production at a certain kinematics and angle $\\Delta \\theta $ given the production with the same kinematics at a fixed reference angle $\\Delta \\theta _c$ .", "We take $\\Delta \\theta _c=\\pi /2$ .", "As will show the $P(\\Delta \\theta )$ defined in this way has a non-trivial structure and can probe the physics of small-x and gluon saturation.", "In principle, one is free to chose a different reference angle $\\Delta \\theta _c$ , however any value $\\Delta \\theta _c<<\\pi $ will only change the normalization rather than the main picture.", "The advantage of the above definition for the azimuthal correlations is that it is experimentally easier to measure as it does not require a different experimental setup and run for the trigger or reference.", "Moreover, in dA collisions at RHIC, the isospin effect in $P(\\Delta \\theta )$ will drop out via normalizationWe checked that numerically the isospin effect brings less than $2\\%$ contribution to the azimuthal correlation defined via Eq.", "(REF ).", "Therefore, due to our particular definition of $P(\\Delta \\theta )$ in Eq.", "(REF ), the differences between a deuteron and a proton projectile are negligible unlike the prompt photon production case.", "and this facilitates to single out the importance of the saturation effect at forward rapidities in contrast to the nuclear modification factor $R_{dA}^{\\gamma }$ .", "Figure: The relative azimuthal correlation P(Δθ)P(\\Delta \\theta ) defined in Eq.", "() for minimum-bias dA collisions at RHIC S=0.2\\sqrt{S}=0.2 TeV for two different windows of kinematics of transverse momentum of produced prompt photon k t k_t and hadron (neutral pion) q t q_t at fixed rapidity η h =η γ =3\\eta _h=\\eta _{\\gamma }=3.", "The curves are the results obtained from the rcBK equation solution with the initial saturation scale Q 0p 2 =0.168GeV 2 Q_{0p}^2=0.168~\\text{GeV}^2 for proton and Q 0A 2 =3Q 0p 2 Q_{0A}^2=3Q_{0p}^2 for a nuclei.Figure: The relative azimuthal correlation P(Δθ)P(\\Delta \\theta ) for minimum-bias p(d)A collisions at different energies S=0.2,4.4,8.8\\sqrt{S}=0.2, 4.4, 8.8 TeV for a fixed rapidity η h =η γ =3\\eta _h=\\eta _{\\gamma }=3 and transverse momentum of the produced prompt photon k t =5k_t=5 GeV and hadron (neutral pion) q t =3q_t=3 GeV.", "The curves are the results obtained from the rcBK equation solution with the initial saturation scale Q 0p 2 =0.168GeV 2 Q_{0p}^2=0.168~\\text{GeV}^2 for proton and Q 0A 2 =3Q 0p 2 Q_{0A}^2=3Q_{0p}^2 for a nucleiIn this approach a fast valence quark from the projectile proton radiates a photon before and after multiply scattering on the color glass condensate target, see Fig.", "REF .", "In this picture, the projectile is treated in the collinear factorization, and therefore the photon radiation from quark at this level has the standard features of pQCD, including the back-to-back correlation in the transverse momentum.", "As a result of multiple scatterings, the quark acquires a transverse momentum comparable with the saturation scale, the only relevant scale in the system, and the intrinsic angular correlations are washed way.", "In Fig.", "REF , we show $P(\\Delta \\theta )$ at forward rapidity $\\eta _h=\\eta _{\\gamma }=3$ for $q_h=k_t=2$ GeV at RHIC and $q_h=k_t=6$ GeV at the LHC for two different initial saturation scale for proton $Q^2_{0p}=0.168, 0.2\\,\\text{GeV}^2$ and nuclei $Q_{0A}^2=3\\div 4 Q_{0p}^2$ .", "For such low $p_t$ 's we are most likely probing the saturation region of the nuclear wave function due to the small values of $x_g$ .", "It is clear that the away-side prompt photon-hadron cross-section (at $\\Delta \\theta \\approx \\pi $ ) is suppressed for the bigger saturation scale (corresponding to a denser system).", "It is also seen from Fig.", "REF that $P(\\Delta \\theta )$ is very sensitive to the initial saturation scale.", "Unfortunately, this bring rather large theoretical uncertainties.", "However, as we will show in the following the suppression of the way-side correlations seems to be a robust feature of our results and it less depends on our theoretical uncertainties.", "In Fig.", "REF we show the relative azimuthal correlations obtained from the rcBK solution for a fixed initial saturation scale $Q^2_{0s}=0.168\\,\\text{GeV}^2$ and $Q^2_{0A}=3 Q_{0p}^2$ at forward rapidities $\\eta _h=\\eta _{\\gamma }=3$ at RHIC $\\sqrt{S}=0.2$ TeV for two different kinematics windows of transverse momenta: We show in top panel, the results with a fix transverse momentum of prompt photon $q_t=5$ GeV at different transverse momentum of produced hadron, and in down panel, with a fixed transverse momentum of hadron $k_t=5$ GeV but at various transverse momentum of the produced prompt photon.", "It is seen, the relative azimuthal correlation is suppressed at $\\Delta \\theta =\\pi $ as the transverse momentum of the produced hadron or prompt photon decreases and becomes comparable to the actual saturation scale of the system.", "This is the case regardless which of two transverse momenta of the hadron or prompt photon decreases.", "In Fig.", "REF lower panel, it is seen that when the prompt photon transverse momentum becomes comparable with the saturation effect the away-side azimuthal angular correlation of photon-hadron completely washes away.", "The same effect happens at lower transverse momentum of the produced hadron.", "This is simply because of fragmentation effect namely the transverse momentum of the produced parton (that should be compared with the saturation scale) is higher than the transverse momentum of the fragmented hadron.", "Again, the suppression of away-side correlations is clearly due to the saturation effect since as we lower the transverse momentum of the produced particle, the system of hadron-photon become more sensitive to the small-x gluon saturation.", "Note that the hadron-photon cross-section in Eq.", "(REF ) has collinear singularity.", "Therefore for a proper investigation of the correlations at $\\Delta \\theta \\approx 0$ , in principle, one should first extract the collinear singularity in a same fashion as demonstrated in section II by introducing the quark-photon fragmentation function.", "Therefore, our results at near-side $\\Delta \\theta \\approx 0$ should be less reliable.", "Nevertheless, we expect that the sensitivity to the collinear singularity effect should drop out in the correlation defined in Eq.", "(REF ) via normalization.", "We checked that contrary to the away-side correlations, the near-side peak is not sensitive to the saturation physics as the correlations does not change with varying the density of the system, see also Fig.", "REF .", "In order to further understand the relative sensitivity of the away side peak to saturation dynamics, in Fig.", "REF , $P(\\Delta \\theta )$ we compare at various energies at RHIC and the LHC for a fixed transverse momentum of the produced prompt photon $k_t=5$ GeV and hadron $q_t=3$ GeV at rapidity $\\eta _h=\\eta _{\\gamma }=3$ .", "It is clear that the away side peak goes away as one increases the energy.", "Again this is due to the fact that as we increase the energy, the gluon density increases and non-linear gluon recombination or the saturation effect becomes important.", "From Fig.", "REF , it is obvious that at the LHC the away-side azimuthal correlations of photon-hadron will be strongly suppressed.", "We conclude that the suppression of the away-side azimuthal photon-hadron correlations defined via Eq.", "(REF ), with decreasing the transverse momentum of the produced prompt photon or hadron, or increasing the energy, or increasing the size/density of system, all uniquely can be explained within the universal picture of gluon saturation without invoking any new parameters or ingredients to our model." ], [ "Summary", "We have investigated prompt photon production and prompt photon-hadron azimuthal angular correlations in proton-proton and proton-nucleus collisions using the Color Glass Condensate formalism.", "We have provided predictions in the kinematic regions appropriate to RHIC and the LHC experiments.", "We have shown that single inclusive and direct prompt photon production cross section in p(d)A collisions at forward rapidities at both RHIC and the LHC is suppressed, as compared to normalized production cross section in proton-proton collisions.", "At RHIC, a good portion of the predicted suppression is due to the projectile being a deuteron rather than a proton.", "This suppression is larger at the LHC compared to RHIC which is even more impressive given that the projectile at the LHC is a proton.", "We showed that direct photon production is most affected by gluon saturation effects in the target nucleus than the fragmentation photons.", "However, at the LHC energies at forward rapidities the nuclear modification suppression for direct, fragmentation and inclusive prompt photon production is rather similar.", "We showed that the nuclear modification factor $R_{p(d)A}^{\\gamma }$ for inclusive prompt photon production at RHIC and the LHC is a sensitive probe of small-x dynamics.", "We note that our results based on gluon saturation dynamics and using the Color Glass Condensate formalism are different from those coming from the collinear factorization [12].", "Therefore, $R_{p(d)A}^{\\gamma }$ measurement at RHIC and the LHC is a crucial test of different factorization schemes, see also Refs.", "[6], [38], [39] for other observables.", "We have also studied prompt photon-hadron azimuthal angular correlations in kinematic regions which can be probed by RHIC and the LHC experiments.", "It is shown that the away side peak in photon-hadron angular correlation goes away as one lowers the final state particle's momenta, very similarly to the disappearance of the away side peak in di-hadron correlations in forward rapidity dA collisions at RHIC.", "At fixed transverse momenta, the suppression of the away side peak gets stronger as one goes to larger rapidities (more forward) or higher energy or denser system as expected, due to stronger saturation effects in the target nucleus.", "Presently, we are not aware of any alternative approach which leads to this novel phenomenon.", "Note that in contrast to the nuclear modification factor for prompt photon, the prompt photon-hadron azimuthal angular correlation defined via Eq.", "(REF ) is free from the isospin effect, and can be considered as a cleaner probe of saturation effect.", "Finally, we emphasize that prompt photon-hadron azimuthal angular correlations suffers from much less theoretical uncertainties as compared to di-hadron azimuthal angular correlations and thus a measurement of this correlation would go a long way toward establishing the dominance of gluon saturation effects at small $x_g$ .", "It will be interesting to see what the predictions of pQCD-motivated models [13] are for photon-hadron azimuthal angular correlations.", "In these models one usually needs to combine models of higher twist shadowing, the Cronin effect and cold matter energy loss in order to describe the data on single inclusive hadron production and di-hadron azimuthal angular correlations.", "The advantage of the CGC formalism is that the same framework can be used to describe nuclear shadowing of structure functions [40] at small $x$ and includes transverse momentum broadening (the Cronin effect) [41].", "It does not however include cold matter energy loss due to longitudinal momentum transfer between the projectile and the target which may be important at the very forward rapidities.", "It is not clear at the moment how to calculate this effect from first principles QCD.", "Even though this energy loss itself is small, due to steepness of the production cross section at forward rapidity, it can suppress the cross section significantly.", "Figure: The diagrams (at leading log approximation) contributing to the prompt photon+hadron production within the color glass condensate picture.", "The crossed white blob denotes the interaction of the projectile quark to all orders with the strong background field of the target nucleus A.", "The black blob represent the quark-hadron fragmentation process." ], [ "The purpose of this appendix is to define the kinematics and derive the needed relations between various light-cone energy fractions which appear in the production cross sections used.", "This is slightly different from the standard relations used in production cross sections based on collinear factorization theorems of pQCD.", "We first consider scattering of a quark on the target where a photon and a quark are produced, depicted in Fig.", "1, $q(p)\\, A (p_A)\\rightarrow \\gamma (k) \\, q(l) \\, X,$ where $A$ is a label for the multi-gluon state, described by a classical field representing a proton or nucleus target.", "In the standard pQCD (leading twist) kinematics, only one parton from the target interacts.", "This is not the case here since the target is described by a classical gluon field representing a multi-gluon state with intrinsic momentum rather than an individual gluon with a well defined energy fraction $x_g$ and zero transverse momentum.", "Nevertheless, since most of the gluons in the target wave function have momentum of order $Q_s$ , one can think of the state describing the target as being labeled by a (four) momentum $p_A$ .", "In this sense, the gluons in the target collectively carry fraction $x_g$ of the target energy and have intrinsic transverse momentum denoted $p_{A,t}$ .", "This also means that there is no integration over $x_g$ in our case unlike the collinearly factorized cross sections in pQCD (this basically corresponds to setting $x_g$ equal to the lower limit of $x_g$ integration in pQCD cross sections).", "We thus have $p^\\mu &=&\\left(p^-=x_q\\sqrt{S/2},~p^+=0,~p_t=0 \\right),\\nonumber \\\\P^\\mu &=&\\left(P^-=\\sqrt{S/2},~P^+=0,~P_t=0 \\right),\\nonumber \\\\p_A^\\mu &=&\\left(p^{-}_A=0,~p_A^{+}=x_g\\sqrt{S/2},~p_{A,t} \\right),\\nonumber \\\\P_A^\\mu &=&\\left(P^{-}_A=0,~P_A^{+}=\\sqrt{S/2},~P_{A,t} = 0 \\right),\\nonumber \\\\l^\\mu &=&\\left(l^-,~l^+=l_t^2/2l^-,~l_t \\right),\\nonumber \\\\q^\\mu &=&\\left(q^- = z_f\\, l^-,~q^+=q_t^2/2q^-,~q_t = z_f\\, l_t \\right),\\nonumber \\\\k^\\mu &=& \\left(k^-,~k^+=k_t^2/2k^-,~k_t \\right), \\ $ where $P^\\mu , P_A^\\mu , q^\\mu $ are the momenta of the incoming projectile, target and the produced hadron respectively.", "(Pseudo)-rapidities of the produced quark and photon are related to their energies via $l^-=\\frac{l_t}{\\sqrt{2}}e^{\\eta _h},\\hspace{56.9055pt} k^-=\\frac{k_t}{\\sqrt{2}}e^{\\eta _\\gamma },$ Imposing energy-momentum conservation at the partonic level via $\\delta ^4 (p + p_A - l -k)$ and using Eq.", "(REF ) leads to $p^-&=&k^- + l^-, \\\\p^{+}_A&=&k^+ + l^+,\\\\\\vec{p}_{A,t} &=& \\vec{k}_t + \\vec{l}_t.$ The above relations and Eq.", "(REF ) (and the on mass shell condition) can be used to derive the following expressions for the energy fractions $x_q,x_g$ .", "We obtain, $x_q&=&x_{\\bar{q}}=\\frac{1}{\\sqrt{S}}\\left(k_t\\, e^{\\eta _{\\gamma }}+\\frac{q_t}{z_f}\\, e^{\\eta _{h}}\\right),\\\\x_g&=&\\frac{1}{\\sqrt{S}}\\left(k_t\\,e^{-\\eta _{\\gamma }}+ \\frac{q_t}{z_f}\\, e^{-\\eta _{h}}\\right),\\ $ where the final hadron transverse momentum and rapidity are denoted by $q_t$ and $\\eta _h$ , and we used $z_f=q_t/l_t$ .", "Note that light-cone momentum fraction $x_g$ appears in the dipole forward scattering amplitude $N_F (b_t, r_t, x_g)$ whereas $x_q$ is the fraction of the projectile proton (deuteron) carried by the incident quark, see Eq.", "(REF ).", "To derive an expression for the lower limit of $z_f$ integration in Eq.", "(REF ), we note that $0 \\le \\, x_q \\, \\le \\, 1$ .", "Using the relation between the minus components of the four momenta given above, we get $x_q \\, \\sqrt{S/2} = k^- + \\frac{q^-}{z_f}.$ The minimum value of $z_f$ occurs when $x_q$ is maximum, i.e., $x_q = 1$ .", "We then have $z_{f}^{min} = {q^- \\over \\sqrt{S/2} - k^-},$ which can be written in terms of the transverse momenta and rapidities of the final state hadron and photon as $z_{f}^{min} = \\frac{q_t}{\\sqrt{S}} \\, {e^{\\eta _h} \\over 1 - \\frac{k_t}{\\sqrt{S}}\\, e^{\\eta _\\gamma }}.$ We now consider the kinematics of single inclusive photon production cross section.", "The cross section is obtained from Eq.", "(REF ) by integrating over the final state quark momenta.", "This requires some care as we have now explicitly separated direct and fragmentation photons in Eq.", "(REF ).", "Again using Eqs.", "(REF , REF , REF ), we obtain the following relation, $x_g=\\frac{1}{\\sqrt{S}}\\left(k_t\\,e^{-\\eta _{\\gamma }}+ \\frac{l^2_t}{\\bar{l}\\sqrt{2}}\\right), $ where opposite to Eq.", "(), we avoided to introduce $\\eta _h$ and $z_f$ .", "One can use the energy-momentum delta functions in Eq.", "(REF ) to obtain the following relation $\\bar{l}=x_q\\sqrt{S/2}-k_t/\\sqrt{2}e^{-\\eta _{\\gamma }}.$ Now using the above relation and Eq.", "(REF ), we obtain $x_g = \\frac{1}{x_q\\, S} \\left[{k_t^2\\over z} + {l_t^2\\over 1-z}\\right],$ where the parameter $z$ is the fraction of energy of parton carried away by photon and it is defined as follows, $z \\equiv \\frac{k^-}{p^-} = \\frac{k_t}{x_q\\, \\sqrt{S}}e^{\\eta _{\\gamma }}.$ In case of direct photons with transverse momentum $k_t$ , one should shift momentum $\\vec{l_t} \\rightarrow \\vec{l_t} - \\vec{k_t}$ in Eq.", "(REF ) (this is how we obtained the expersion Eq.", "(REF )).", "Assuming that $l_t<<k_t$ , we get $\\bar{x}_g = \\frac{1}{x_q\\, S} {k_t^2 \\over z (1-z)},$ where we have now used $\\bar{x}_g$ to denote the light cone momentum fraction of the target carried by gluons for production of direct photons so as to distinguish it from the momentum fraction involved in production of fragmentation photons.", "In the later case, the integration variable $l_t$ has been shifted twice.", "Implementing the shifts in Eq.", "(REF ) and noting that the $l_t$ integration in the fragmentation photon production cross section is dominated by its singularity at $l_t\\rightarrow 0$ we get, for fragmentation photons, $x_g=\\frac{k^2}{z^2 x_q S}.$ We are grateful to Javier Albacete for useful communications and for providing us with the latest tables for the numerical solution of rcBK equation.", "We would also like to thank William Brooks and Raju Venugopalan for useful discussions.", "We acknowledge and thank Barbara Jacak, John Lajoie and Richard Seto for helpful discussions about PHENIX detector capabilities.", "J.J-M. is supported in part by the DOE Office of Nuclear Physics through Grant No.", "DE-FG02-09ER41620, from the “Lab Directed Research and Development” grant LDRD 10-043 (Brookhaven National Laboratory), and from The City University of New York through the PSC-CUNY Research Program, grant 64554-00 42.", "The work of A.H.R is supported in part by Fondecyt grants 1110781." ] ]

[ [ "Weighted Plancherel estimates and sharp spectral multipliers for the\n Grushin operators" ], [ "Abstract We study the Grushin operators acting on $\\R^{d_1}_{x'}\\times \\R^{d_2}_{x\"}$ and defined by the formula \\[ L=-\\sum_{\\jone=1}^{d_1}\\partial_{x'_\\jone}^2 - (\\sum_{\\jone=1}^{d_1}|x'_\\jone|^2) \\sum_{\\jtwo=1}^{d_2}\\partial_{x\"_\\jtwo}^2.", "\\] We obtain weighted Plancherel estimates for the considered operators.", "As a consequence we prove $L^p$ spectral multiplier results and Bochner-Riesz summability for the Grushin operators.", "These multiplier results are sharp if $d_1 \\ge d_2$.", "We discuss also an interesting phenomenon for weighted Plancherel estimates for $d_1 <d_2$.", "The described spectral multiplier theorem is the analogue of the result for the sublaplacian on the Heisenberg group obtained by D. M\\\"uller and E.M. Stein and by W. Hebisch." ], [ "Introduction", "Let $(\\mathrm {X},\\mu )$ be a measure space and $L$ be a (possibly unbounded) self-adjoint operator on $L^2(\\mathrm {X})$ .", "If $E$ denotes the spectral resolution of $L$ on $\\mathbb {R}$ , then a functional calculus for $L$ can be defined via spectral integration and, for every Borel function $F : \\mathbb {R}\\rightarrow , the operator$$F(L) = \\int _\\mathbb {R}F(\\lambda ) \\,dE(\\lambda )$$is bounded on $ L2(X)$ if and only if the ``spectral multiplier^{\\prime \\prime } $ F$ is an ($ E$-essentially) bounded function.", "Characterizing, or at least giving (non-trivial) sufficient conditions for the $ Lp$-boundedness of the operator $ F(L)$, for some $ p 2$, in terms of properties of the multiplier $ F$, is a much more complicated issue, and a huge amount of literature is devoted to instances of this problem (we refer the reader to \\cite {cowling_spectral_2001, duong_singular_1999, duong_plancherel-type_2002, hebisch_multiplier_1993, hebisch_functional_1995, martini_multipliers_2010, mauceri_vectorvalued_1990, mller_spectral_1994, sikora_imaginary_2001} for a detailed discussion of the relevant literature).$ Here we are interested in the case where $\\mathrm {X}= \\mathbb {R}^{d_1} \\times \\mathbb {R}^{d_2}$ , with Lebesgue measure, and $L$ is the Grushin operator, that is, $L = -\\Delta _{x^{\\prime }} - |x^{\\prime }|^2 \\Delta _{x^{\\prime \\prime }},$ where $x^{\\prime },x^{\\prime \\prime }$ denote the two components of a point $x \\in \\mathbb {R}^{d_1} \\times \\mathbb {R}^{d_2}$ , while $\\Delta _{x^{\\prime }},\\Delta _{x^{\\prime \\prime }}$ are the corresponding partial Laplacians, and $|x^{\\prime }|$ is the Euclidean norm of $x^{\\prime }$ .", "Let $W_q^s(\\mathbb {R})$ denote the $L^q$ Sobolev space on $\\mathbb {R}$ of (fractional) order $s$ , and define a “local Sobolev norm” by the formula $\\Vert F\\Vert _{MW_q^s} = \\sup _{t>0} \\Vert \\eta \\, F_{(t)} \\Vert _{W_q^s},$ where $F_{(t)}(\\lambda ) =F(t \\lambda )$ , and $\\eta \\in C^\\infty _c(\\left]0,\\infty \\right[)$ is a not identically zero auxiliary function; note that different choices of $\\eta $ give rise to equivalent local norms.", "Next set $D = \\max \\lbrace d_1+d_2,2d_2\\rbrace $ .", "Then our main result reads as follows.", "Theorem 1 Suppose that a function $F : \\mathbb {R}\\rightarrow satisfies\\begin{equation}\\Vert F\\Vert _{MW_2^s} < \\infty \\end{equation}for some $ s > D/2$.", "Then the operator $ F(L)$ is of weak type $ (1,1)$ and bounded on $ Lp(X)$ for all $ p ]1,[$.", "In addition,\\begin{equation}\\Vert F(L)\\Vert _{L^1 \\rightarrow L^{1,w}} \\le C \\Vert F\\Vert _{MW_2^s}, \\qquad \\Vert F(L)\\Vert _{L^p \\rightarrow L^{p}} \\le C_p \\Vert F\\Vert _{MW_2^s}\\end{equation}for all $ r 0$.$ When $d_1=d_2=1$ , this result proves the conjecture stated on page 5 of [18], and in fact we obtain a far-going generalization of that statement.", "Note that, in the case $d_1 \\ge d_2$ , the lower bound for the order of differentiability $s$ of the multiplier $F$ required in Theorem REF is $(d_1+d_2)/2$ , that is, half the topological dimension of $\\mathrm {X}$ ; since the Grushin operator $L$ is elliptic in the region where $x^{\\prime }\\ne 0$ , the argument in [22] can be adapted to show that our result is sharp (see Section  below for more details).", "In the case $d_2 > d_1$ , instead, a gap of $(d_2-d_1)/2$ remains between half the topological dimension and the threshold on $s$ in Theorem REF .", "If one disregarded the constraint on $s$ , then the above result would follow from a general theorem [10], [8] proved in the context of a doubling metric-measure space $(\\mathrm {X},\\varrho ,\\mu )$ , with an operator $L$ satisfying Gaussian-type heat kernel bounds expressed in terms of the distance $\\varrho $ .", "In this general context, the mentioned weak type and $L^p$ -boundedness of $F(L)$ hold whenever $\\Vert F\\Vert _{MW_\\infty ^s} < \\infty $ for some $s > Q/2$ , where $Q$ denotes the “homogeneous dimension” of the metric-measure space.", "If $\\mathrm {X}$ is $\\mathbb {R}^{d_1} \\times \\mathbb {R}^{d_2}$ with Lebesgue measure and $L$ is the Grushin operator, a “control distance” $\\varrho $ associated to $L$ can be introduced, for which $Q = d_1 + 2d_2$ and $L$ satisfies Gaussian-type bounds [21], hence the general theorem applies in this case.", "Our Theorem REF gives a better result, with respect to both the order of differentiability required on the multiplier (since $D < Q$ ) and the type of Sobolev norm used ($L^2$ instead of $L^\\infty $ ).", "Our approach allows us to consider also the Bochner-Riesz means associated to the Grushin operator.", "Since these correspond to compactly supported multipliers, we can obtain a better result than the one given by direct application of Theorem REF .", "Theorem 2 Suppose that $\\kappa > (D-1)/2$ and $p \\in \\left[1,\\infty \\right]$ .", "Then the Bochner-Riesz means $(1-t L)_+^\\kappa $ are bounded on $L^p(\\mathrm {X})$ uniformly in $t \\in \\left[0,\\infty \\right[$ .", "As in many other works on the subject, the proof of our results is based on the analysis of the integral kernel $\\operatorname{\\mathcal {K}}_{F(L)} : \\mathrm {X}\\times \\mathrm {X}\\rightarrow of the operator $ F(L)$, defined by the identity\\begin{equation}F(L) f(x) = \\int _\\mathrm {X}\\operatorname{\\mathcal {K}}_{F(L)}(x,y) \\, f(y) \\,dy.\\end{equation}To be precise, if $ F$ is bounded and compactly supported, then there exists a Borel function $ KF(L)$ such that (\\ref {eq:integralkernel}) holds for all $ f L2(X)$ and for almost all $ x X$ (cf.\\ \\cite [Lemma 2.2]{duong_plancherel-type_2002}).", "However, a multiplier $ F$ satisfying (\\ref {eq:mhcond}) need not be compactly supported, and the integral kernel $ KF(L)$ in general exists only as a distribution; nevertheless the Calderón-Zygmund theory of singular integral operators allows one to derive the weak type $ (1,1)$ of $ F(L)$ from suitable estimates on the integral kernels corresponding to the compactly supported pieces in a dyadic decomposition of $ F$.$ As highlighted in [8], a crucial step in this approach is a “Plancherel estimate”, which in its basic form is the inequality $\\operatornamewithlimits{\\mathrm {ess\\,sup}}_{y \\in \\mathrm {X}} |B(y,R^{-1})|^{1/2} \\, \\Vert \\operatorname{\\mathcal {K}}_{F( L)}(\\cdot ,y)\\Vert _{L^2(\\mathrm {X})}\\le C \\Vert F_{(R^2)}\\Vert _{L^\\infty (\\mathbb {R})},$ for all $R > 0$ and all $F : \\mathbb {R}\\rightarrow supported in the interval $ [R2,4R2 ]$; here $ |B(x,r)|$ denotes the Lebesgue measure of the $$-ball of center $ x X$ and radius $ r$.Such an estimate holds, \\emph {mutatis mutandis}, for any operator $ L$ satisfying Gaussian heat kernel bounds, but usually it does not lead to optimal spectral multiplier results.In the present paper we obtain for the Grushin operator $ L$ an improvement of (\\ref {eq:basicplancherel}), that is, a ``weighted Plancherel estimate^{\\prime \\prime } of the form\\begin{equation}\\operatornamewithlimits{\\mathrm {ess\\,sup}}_{y \\in \\mathrm {X}} |B(y,R^{-1})|^{1/2} \\, \\Vert (1+w_R(\\cdot ,y))^\\gamma \\, \\operatorname{\\mathcal {K}}_{F( L)}(\\cdot ,y)\\Vert _{L^2(\\mathrm {X})} \\le C_\\gamma \\Vert F_{(R^2)}\\Vert _{L^2(\\mathbb {R})},\\end{equation}where $ [0, d2/2[$ and\\begin{equation}w_R(x,y) = \\min \\lbrace R,|y^{\\prime }|^{-1}\\rbrace |x^{\\prime }|.\\end{equation}The improvement of the Plancherel estimate yields, via the interpolation technique of \\cite {mauceri_vectorvalued_1990}, a sharp multiplier theorem, at least for $ d1 d2$.", "In the case $ d1 > d2$, an interesting phenomenon occurs: although (\\ref {eq:weightedplancherel}) holds for all $ [0, d2/2[$, we can exploit it only when $ < d1/2$; whence the gap between the threshold $ D/2$ in Theorem~\\ref {thm:maintheorem} and half the topological dimension.$ The use of weighted Plancherel estimates in multiplier theorems is not new [9], [20], [10], [4], and in particular they have been employed to obtain sharp results for homogeneous sublaplacians on Heisenberg and related groups.", "In the case $d_2 = 1$ , the Grushin operator $L$ corresponds, via a suitable quotient, to the homogeneous sublaplacian on the $(2d_1+1)$ -dimensional Heisenberg group.", "For $d_2 > 1$ , an analogous correspondence holds if we replace the Heisenberg group with a Heisenberg-Reiter group $\\mathrm {H}_{d_1,d_2}$ (see details below), and a multiplier theorem for the homogeneous sublaplacian on $\\mathrm {H}_{d_1,d_2}$ holds [15], giving the same gap between the threshold and half the topological dimension that appears in Theorem REF .", "We remark, however, that the weighted Plancherel estimate for the Grushin operator is not an immediate consequence of the analogous estimate on the Heisenberg-Reiter group: because of the absence of the group structure, here some careful estimates are needed, exploiting known asymptotics for the Hermite functions.", "In a recent work [12] the Grushin operator in the case $d_2 = 1$ is considered and results analogous to our Theorems REF and REF are obtained.", "However in the present paper the requirements on the order of differentiability $s$ and on the order of Bochner-Riesz means $\\kappa $ are essentially lower.", "Moreover the method used in [12] apparently does not yield the weak type $(1,1)$ in the multiplier theorem, nor the $L^1$ -boundedness of the Bochner-Riesz means." ], [ "Notation and preliminaries", "As above, let $\\mathrm {X}$ be $\\mathbb {R}^{d_1}\\times \\mathbb {R}^{d_2}$ with Lebesgue measure.", "In order to study the Grushin operator $L$ on $\\mathrm {X}$ , it is convenient to introduce at the same time a family of operators which commute with $L$ .", "Given a point $x = (x^{\\prime },x^{\\prime \\prime }) \\in \\mathrm {X}$ , we denote by $x^{\\prime }_{{j}}$ and $x^{\\prime \\prime }_{{k}}$ the ${j}$ -th component of $x^{\\prime }$ and the ${k}$ -th component of $x^{\\prime \\prime }$ .", "For all ${j}\\in \\lbrace 1,\\dots ,d_1\\rbrace $ and ${k}\\in \\lbrace 1,\\dots ,d_2\\rbrace $ , let then $L_{{j}}$ , $T_{{k}}$ , $P_{{j}}$ be the differential operators on $\\mathrm {X}$ given by $L_{{j}} = (-i\\partial _{x^{\\prime }_{{j}}})^2 + (x^{\\prime }_{{j}})^2 \\sum _{l=1}^{d_2} (-i\\partial _{x^{\\prime \\prime }_l})^2, \\qquad T_{{k}} = -i\\partial _{x^{\\prime \\prime }_{{k}}}, \\qquad P_{{j}} = x^{\\prime }_{{j}}.$ If $(\\delta _r)_{r > 0}$ is the family of dilations on $\\mathrm {X}$ defined by $\\delta _r(x^{\\prime },x^{\\prime \\prime }) = (rx^{\\prime },r^2 x^{\\prime \\prime }),$ then $\\Vert f \\circ \\delta _r\\Vert _2 = r^{-Q/2} \\Vert f\\Vert _2$ , where $Q = d_1 + 2d_2$ .", "We also note that $\\begin{split}P_{{j}} (f \\circ \\delta _r) = r^{-1} (P_{{j}} f) \\circ \\delta _r, &\\qquad L_{{j}}(f \\circ \\delta _r) = r^2 (L_{{j}} f) \\circ \\delta _r, \\\\ T_{{k}} (f \\circ \\delta _r) &= r^2 (T_{{k}} f) \\circ \\delta _r.\\end{split}$ The Grushin operator $L$ on $\\mathrm {X}$ is the sum $L_1 + \\dots + L_{d_1}$ .", "$L$ is a second-order subelliptic differential operator with smooth coefficients.", "For such operators, several ways of introducing a control distance $\\varrho $ are available in the literature, and we refer the reader to [11] for a survey.", "In particular, $L$ belongs to the class of operators studied in [21], where the following estimates are obtained.", "Proposition 3 The control distance $\\varrho $ of the Grushin operator $L$ on $\\mathrm {X}$ is homogeneous with respect to the dilations $\\delta _r$ , that is, $\\varrho (\\delta _r(x),\\delta _r(y)) = r \\varrho (x,y)$ for all $r > 0$ and $x,y \\in \\mathrm {X}$ , and moreover $\\varrho (x,y) \\sim |x^{\\prime } - y^{\\prime }| + {\\left\\lbrace \\begin{array}{ll}\\frac{|x^{\\prime \\prime }-y^{\\prime \\prime }|}{|x^{\\prime }| + |y^{\\prime }|} &\\text{if $|x^{\\prime \\prime }-y^{\\prime \\prime }|^{1/2} \\le |x^{\\prime }| + |y^{\\prime }|$,}\\\\|x^{\\prime \\prime }-y^{\\prime \\prime }|^{1/2} &\\text{if $|x^{\\prime \\prime }-y^{\\prime \\prime }|^{1/2} \\ge |x^{\\prime }| + |y^{\\prime }|$.}\\end{array}\\right.", "}$ Consequently, if $B(x,r)$ denotes the $\\varrho $ -ball of center $x \\in \\mathrm {X}$ and radius $r \\ge 0$ , then $|B(x,r)| \\sim r^{d_1+d_2} \\max \\lbrace r,|x^{\\prime }|\\rbrace ^{d_2},$ and in particular, for all $\\lambda \\ge 0$ , $|B(x,\\lambda r)| \\le C (1+\\lambda )^Q |B(x,r)|.$ Moreover, there exist constants $b,C > 0$ such that, for all $t > 0$ , the integral kernel $p_t$ of the operator $\\exp (-tL)$ is a function satisfying $|p_t(x,y)| \\le C |B(y,t^{1/2})|^{-1} e^{-b \\varrho (x,y)^2/t}$ for all $x,y \\in \\mathrm {X}$ .", "The homogeneity of $\\varrho $ follows immediately from its definition [21] and the homogeneity of $L$ .", "For the remaining estimates, see [21].", "The inequality (REF ) says that $\\mathrm {X}$ with the distance $\\varrho $ and the Lebesgue measure is a doubling metric-measure space of homogeneous dimension $Q$ (cf.", "[7] or [8]), whereas (REF ) expresses Gaussian-type heat kernel bounds for $L$ .", "Several properties of $L$ and the other operators introduced above can be easily recovered by considering $\\mathrm {X}$ as the quotient of a suitable stratified Lie group (cf.", "[2], [1]).", "Denote by $\\mathbb {R}^{d_1 \\times d_2}$ the set of $(d_1 \\times d_2)$ -matrices with real coefficients.", "Both $\\mathbb {R}^{d_1 \\times d_2}$ and $\\mathbb {R}^{d_1} \\times \\mathbb {R}^{d_2}$ are abelian Lie groups with respect to addition.", "Let $\\mathrm {H}_{d_1,d_2}$ be the semidirect product group $\\mathbb {R}^{d_1 \\times d_2} \\ltimes (\\mathbb {R}^{d_1} \\times \\mathbb {R}^{d_2})$ , with multiplication $(x,y,t) \\cdot (x_0, y_0, t_0) = (x+ x_0, y+ y_0, t+ t_0 - (x^T y_0 - x_0^T y)/2).$ This is a particular instance of Heisenberg-Reiter group (see [25] and references therein).", "If $\\lbrace \\tilde{X}_{1,1},\\dots ,\\tilde{X}_{d_1,d_2}, \\tilde{Y}_{1}, \\dots ,\\tilde{Y}_{d_1}, \\tilde{T}_{1}, \\dots , \\tilde{T}_{d_2}\\rbrace $ is the standard basis of the Lie algebra of $\\mathrm {H}_{d_1,d_2}$ (i.e., the set of the left-invariant vector fields extending the standard basis of $\\mathbb {R}^{d_1 \\times d_2} \\times \\mathbb {R}^{d_1} \\times \\mathbb {R}^{d_2}$ at the identity), then the only non-trivial Lie brackets among the elements of the basis are $[\\tilde{X}_{{j},{k}},\\tilde{Y}_{{j}}] = -[\\tilde{Y}_{{j}},\\tilde{X}_{{j},{k}}] = -\\tilde{T}_{{k}} \\qquad \\text{for all ${j}= 1,\\dots ,d_1$, ${k}= 1,\\dots ,d_2$.", "}$ $\\mathrm {H}_{d_1,d_2}$ is a 2-step stratified Lie group, with dilations $(\\tilde{\\delta }_r)_{r>0}$ defined by $\\tilde{\\delta }_r(\\tilde{X}_{{j},{k}}) = r\\tilde{X}_{{j},{k}}, \\quad \\tilde{\\delta }_r(\\tilde{Y}_{{j}}) = r\\tilde{Y}_{{j}}, \\quad \\tilde{\\delta }_r(\\tilde{T}_{{k}}) = r^2 \\tilde{T}_{{k}},$ and the homogeneous sublaplacian $\\tilde{L}$ on $\\mathrm {H}_{d_1,d_2}$ is given by $\\tilde{L} = -\\sum _{{j},{k}} \\tilde{X}_{{j},{k}}^2 - \\sum _{{j}} \\tilde{Y}_{{j}}^2.$ Note that, when $d_2=1$ , $\\mathrm {H}_{d_1,d_2}$ is the $(2d_1+1)$ -dimensional Heisenberg group.", "When $d_2 > 1$ , $\\mathrm {H}_{d_1,d_2}$ is not an H-type group (in the sense of Kaplan), nor a Métivier group.", "Nevertheless, in the terminology of [14], [15], $\\mathrm {H}_{d_1,d_2}$ is $h$ -capacious where $h= \\min \\lbrace d_1,d_2\\rbrace $ .", "In particular, the following multiplier theorem holds: the operator $F(\\tilde{L})$ is of weak type $(1,1)$ and bounded on $L^p(\\mathrm {H}_{d_1,d_1})$ for all $p \\in \\left]1,\\infty \\right[$ whenever $\\Vert F\\Vert _{MW_2^s} < \\infty $ for some $s > (\\dim \\mathrm {H}_{d_1,d_2} + (d_2-d_1)_+)/2$ , where $\\dim \\mathrm {H}_{d_1,d_2}$ is the topological dimension $d_1d_2 + d_1 +d_2$ [15].", "$\\mathrm {X}$ can be identified with the left quotient $\\mathbb {R}^{d_1 \\times d_2} \\backslash \\mathrm {H}_{d_1,d_2}$ via the projection map $(x,y,t) \\mapsto (y,t+x^T y/2)$ .", "Hence $\\mathrm {H}_{d_1,d_2}$ acts by right translations on $\\mathrm {X}$ , that is, $\\tau _{(x,y,t)} : \\mathrm {X}\\ni (z^{\\prime },z^{\\prime \\prime }) \\mapsto (z^{\\prime }-y,z^{\\prime \\prime }-x^T z^{\\prime } -t + x^T y/2) \\in \\mathrm {X}$ is a measure-preserving affine transformation of $\\mathrm {X}$ for all $(x,y,t) \\in \\mathrm {H}_{d_1,d_2}$ , and $\\tau _{g h} = \\tau _g \\tau _h$ .", "This in turn induces a unitary representation $\\sigma $ of $\\mathrm {H}_{d_1,d_2}$ on $L^2(\\mathrm {X})$ , given by $\\sigma (g) f = f \\circ \\tau _g^{-1}$ , and $\\begin{split}T_{{k}} = d\\sigma (-i\\tilde{T}_{{k}}), &\\qquad P_{{j}} T_{{k}} = d\\sigma (-i\\tilde{X}_{{j},{k}}), \\\\L_{{j}} = d\\sigma \\Biggl (-\\tilde{Y}_{{j}}^2 - &\\sum _{{k}} \\tilde{X}_{{j},{k}}^2 \\Biggr ), \\qquad L = d\\sigma (\\tilde{L}).\\end{split}$ This shows in particular that the operators $L_1,\\dots ,L_{d_1},T_1,\\dots ,T_{d_2}$ (and all the polynomials in $L_1,\\dots ,L_{d_1},T_1,\\dots ,T_{d_2}$ ) are essentially self-adjoint on $C^\\infty _c(\\mathrm {X})$ and commute strongly (that is, their spectral resolutions commute), so they admit a joint functional calculus on $L^2(\\mathrm {X})$ in the sense of the spectral theorem [16].", "Arguing analogously, by the use of the unitary representation $\\varpi $ of $\\mathbb {R}^{d_1} \\times \\mathbb {R}^{d_2}$ on $L^2(\\mathrm {X})$ given by $(\\varpi (u^{\\prime },u^{\\prime \\prime }) f)(x^{\\prime },x^{\\prime \\prime }) = e^{i \\langle x^{\\prime } , u^{\\prime } \\rangle } f(x^{\\prime }, x^{\\prime \\prime }+u^{\\prime \\prime }),$ one obtains that the operators $P_1,\\dots ,P_{d_1},T_1,\\dots ,T_{d_2}$ are essentially self-adjoint on $C^\\infty _c(\\mathrm {X})$ and commute strongly.", "Because of the mentioned commutation properties, it is convenient to introduce in our notation the following “vectors of operators”: $\\mathbf {L}= (L_1,\\dots ,L_{d_1}), \\qquad \\mathbf {T}= (T_1,\\dots ,T_{d_2}), \\qquad \\mathbf {P}= (P_1,\\dots ,P_{d_1}).$ Thus, for instance, $|\\mathbf {T}|$ stands for the operator $(|T_1|^2 + \\dots + |T_{d_2}|^2)^{1/2}$ , that is, the square root $(-\\Delta _{x^{\\prime \\prime }})^{1/2}$ of minus the second partial Laplacian on $\\mathbb {R}^{d_1} \\times \\mathbb {R}^{d_2}$ , while $|\\mathbf {P}|$ is the operator of multiplication by $|x^{\\prime }|$ .", "The subellipticity of $\\tilde{L}$ then yields the following estimate.", "Proposition 4 For all $\\gamma \\in \\left[0,\\infty \\right[$ and $f \\in L^2(\\mathrm {X})$ , $\\Vert \\, |\\mathbf {P}|^{\\gamma } f\\Vert _2 \\le C_\\gamma \\Vert L^{\\gamma /2} |\\mathbf {T}|^{-\\gamma } f\\Vert _2,$ where the $L^2$ norm on each side of (REF ) is understood to be $+\\infty $ when $f$ does not belong to the domain of the corresponding operator.", "We may assume $\\gamma > 0$ .", "Let $\\mathbf {P}\\mathbf {T}$ denote the double-indexed vector of operators $(P_{{j}} T_{{k}})_{{j},{k}}$ , and note that $|\\mathbf {P}\\mathbf {T}|^{\\gamma } = |\\mathbf {P}|^{\\gamma } |\\mathbf {T}|^{\\gamma }$ (modulo closures).", "Moreover the spectrum $\\left[0,+\\infty \\right[$ of $|\\mathbf {T}|^{\\gamma }$ is purely continuous, so $|\\mathbf {T}|^{\\gamma }$ is injective and its image is dense in $L^2(\\mathrm {X})$ .", "Therefore (REF ) is reduced to the proof of the inequality $\\Vert \\, |\\mathbf {P}\\mathbf {T}|^{\\gamma } g\\Vert _2 \\le C_\\gamma \\Vert L^{\\gamma /2} g\\Vert _2$ for all $g \\in L^2(\\mathrm {X})$ .", "By (REF ), the differential operator $\\tilde{W} = -\\sum _{{j},{k}} \\tilde{X}_{{j},{k}}^2$ on $\\mathrm {H}_{d_1,d_2}$ corresponds to the operator $|\\mathbf {P}\\mathbf {T}|^2$ on $\\mathrm {X}$ .", "Since $\\tilde{W}$ is $\\tilde{\\delta }_r$ -homogeneous, with the same homogeneity degree as the sublaplacian $\\tilde{L}$ , from (REF ) and [16] we deduce (REF ) for all $\\gamma \\in 2\\mathbb {N}$ and $g \\in L^2(\\mathrm {X})$ .", "We want now to extend (REF ) to all the real $\\gamma \\ge 0$ .", "For this, fix $m \\in \\mathbb {N}$ and let $A$ and $B$ be the closures of $|\\mathbf {P}\\mathbf {T}|^{2m}$ and $L^m$ on $L^2(\\mathrm {X})$ respectively.", "Since $A$ and $B$ are nonnegative self-adjoint operators on $L^2(\\mathrm {X})$ , by [13], for all $\\theta \\in \\left]0,1 \\right[$ , $(L^2(\\mathrm {X}),\\operatorname{\\mathrm {dom}}A)_{[\\theta ]} = \\operatorname{\\mathrm {dom}}A^\\theta , \\qquad (L^2(\\mathrm {X}),\\operatorname{\\mathrm {dom}}B)_{[\\theta ]} = \\operatorname{\\mathrm {dom}}B^\\theta ,$ with equivalent norms, where $(\\cdot ,\\cdot )_{[\\theta ]}$ denotes interpolation with respect to the complex method, and the domains of the various operators are endowed with the graph norms.", "On the other hand, (REF ) implies that $\\operatorname{\\mathrm {dom}}B \\subseteq \\operatorname{\\mathrm {dom}}A$ , with continuous inclusion.", "By interpolation [3], we conclude that $\\operatorname{\\mathrm {dom}}B^\\theta \\subseteq \\operatorname{\\mathrm {dom}}A^\\theta $ , with continuous inclusion.", "This implies that $\\Vert \\, |\\mathbf {P}\\mathbf {T}|^{\\gamma } g\\Vert _2 \\le C_\\gamma (\\Vert g\\Vert _2 + \\Vert L^{\\gamma /2} g\\Vert _2)$ for all $\\gamma \\in \\left[0,2m \\right]$ and $g \\in L^2(\\mathrm {X})$ .", "The bound (REF ) then follows by replacing $f$ with $f \\circ \\delta _r$ in the previous inequality, exploiting the homogeneity relations (REF ), and taking the limit for $r \\rightarrow \\infty $ ." ], [ "Plancherel estimates", "From the previous section we know that the operators $L_1,\\dots ,L_{d_1},T_1,\\dots ,T_{d_2}$ have a joint functional calculus.", "In fact one can obtain a quite explicit formula for the integral kernel $\\operatorname{\\mathcal {K}}_{G(\\mathbf {L},\\mathbf {T})}$ of an operator $G(\\mathbf {L},\\mathbf {T})$ in the functional calculus, in terms of the Hermite functions (cf.", "[18] for the case $d_1=d_2=1$ , and [23] for the analogue on the Heisenberg groups).", "Namely, for all $\\ell \\in \\mathbb {N}$ , let $h_\\ell $ denote the $\\ell $ -th Hermite function, that is, $h_\\ell (t) = (-1)^\\ell (\\ell !", "\\, 2^\\ell \\sqrt{\\pi })^{-1/2} e^{t^2/2} \\left(\\frac{d}{dt}\\right)^\\ell e^{-t^2},$ and set, for all $n \\in \\mathbb {N}^{d_1}$ , $u \\in \\mathbb {R}^{d_1}$ , $\\xi \\in \\mathbb {R}^{d_2}$ , $\\tilde{h}_n(u,\\xi ) = |\\xi |^{d_1/4} h_{n_1}(|\\xi |^{1/2} u_1) \\cdots h_{n_{d_1}}(|\\xi |^{1/2} u_{d_1}).$ Proposition 5 For all bounded Borel functions $G : \\mathbb {R}^{d_1} \\times \\mathbb {R}^{d_2} \\rightarrow compactly supported in $ Rd1 (Rd2 {0})$,\\begin{multline*}\\operatorname{\\mathcal {K}}_{G(\\mathbf {L},\\mathbf {T})}(x,y) \\\\= (2\\pi )^{-d_2} \\int _{\\mathbb {R}^{d_2}} \\sum _{n \\in \\mathbb {N}^{d_1}} G(|\\xi |(2n+\\tilde{1}), \\xi ) \\, \\tilde{h}_n(y^{\\prime },\\xi ) \\, \\tilde{h}_n(x^{\\prime },\\xi ) \\, e^{i \\langle \\xi , x^{\\prime \\prime }-y^{\\prime \\prime } \\rangle } \\,d\\xi \\end{multline*}for almost all $ x,y X$, where $ 1= (1,...,1) Nd1$.", "In particular\\begin{equation}\\Vert \\operatorname{\\mathcal {K}}_{G(\\mathbf {L},\\mathbf {T})}(\\cdot ,y)\\Vert _2^2 = (2\\pi )^{-d_2} \\int _{\\mathbb {R}^{d_2}} \\sum _{n \\in \\mathbb {N}^{d_1}} |G(|\\xi |(2n+\\tilde{1}),\\xi )|^2 \\, \\tilde{h}_n^2(y^{\\prime },\\xi ) \\,d\\xi \\end{equation}for almost all $ y X$.$ Let $\\mathcal {F} : L^2(\\mathbb {R}^{d_1} \\times \\mathbb {R}^{d_2}) \\rightarrow L^2(\\mathbb {R}^{d_1} \\times \\mathbb {R}^{d_2})$ be the isometry defined by $\\mathcal {F} \\phi (x^{\\prime },\\xi ) = (2\\pi )^{-d_2/2} \\int _{\\mathbb {R}^{d_2}} \\phi (x^{\\prime },x^{\\prime \\prime }) \\, e^{-i \\langle \\xi , x^{\\prime \\prime } \\rangle } \\, dx^{\\prime \\prime },$ i.e., the Fourier transform with respect to $x^{\\prime \\prime }$ .", "Then $\\mathcal {F} L_{{j}}\\phi (x^{\\prime },\\xi ) = L_{{j},\\xi } \\, \\mathcal {F} \\phi (x^{\\prime },\\xi ), \\qquad \\mathcal {F} T_{{k}}\\phi (x^{\\prime },\\xi ) = \\xi _{{k}} \\, \\mathcal {F} \\phi (x^{\\prime },\\xi ),$ at least for $\\phi $ in the Schwartz class, where $L_{{j},\\xi } = (-i\\partial _{x^{\\prime }_{{j}}})^2 + |\\xi |^2 (x^{\\prime }_{{j}})^2.$ For all $\\xi \\ne 0$ , $\\lbrace \\tilde{h}_n(\\cdot ,\\xi )\\rbrace _{n \\in \\mathbb {N}^{d_1}}$ is a complete orthonormal system for $L^2(\\mathbb {R}^{d_1})$ made of real-valued functions and $L_{{j},\\xi } \\, \\tilde{h}_n(x^{\\prime },\\xi ) = (2n_{{j}}+1) |\\xi | \\, \\tilde{h}_n(x^{\\prime },\\xi ).$ In particular, if $\\mathcal {G} : L^2(\\mathbb {R}^{d_1} \\times \\mathbb {R}^{d_2}) \\rightarrow L^2(\\mathbb {N}^{d_1} \\times \\mathbb {R}^{d_2})$ is the isometry defined by $\\mathcal {G} \\psi (n,\\xi ) = \\int _{\\mathbb {R}^{d_1}} \\psi (x^{\\prime },\\xi ) \\, \\tilde{h}_n(x^{\\prime },\\xi ) \\,dx^{\\prime },$ then $\\mathcal {G} \\mathcal {F} L_{{j}} \\phi (n,\\xi ) = (2n_{{j}}+1) |\\xi | \\, \\mathcal {G} \\mathcal {F} \\phi (n,\\xi ), \\qquad \\mathcal {G} \\mathcal {F} T_{{k}} \\phi (n,\\xi ) = \\xi _{{k}} \\, \\mathcal {G} \\mathcal {F} \\phi (n,\\xi ).$ The isometry $\\mathcal {G}\\mathcal {F}$ intertwines the operators $L_{{j}}$ and $T_{{k}}$ with some multiplication operators on $\\mathbb {N}^{d_1} \\times \\mathbb {R}^{d_2}$ , hence it intertwines the corresponding functional calculi: $\\mathcal {G} \\mathcal {F} \\, G(\\mathbf {L},\\mathbf {T}) \\, \\phi (n,\\xi ) = G(|\\xi |(2n+\\tilde{1}),\\xi ) \\, \\mathcal {G} \\mathcal {F} \\phi (n,\\xi ).$ The inversion formulae for $\\mathcal {F}$ and $\\mathcal {G}$ and some easy manipulations then give the above expression for $\\operatorname{\\mathcal {K}}_{G(\\mathbf {L},\\mathbf {T})}$ .", "Moreover, if we set $G_{y}(n,\\xi ) = (2\\pi )^{-d_2/2} G(|\\xi |(2n+\\tilde{1}),\\xi ) \\, \\tilde{h}_n(y^{\\prime },\\xi ) \\, e^{-i \\langle \\xi , y^{\\prime \\prime } \\rangle },$ then the formula for $\\operatorname{\\mathcal {K}}_{G(\\mathbf {L},\\mathbf {T})}$ can be rewritten as $\\operatorname{\\mathcal {K}}_{G(\\mathbf {L},\\mathbf {T})}(\\cdot ,y) = (\\mathcal {G}\\mathcal {F})^{-1} G_{y},$ and since $\\mathcal {G}\\mathcal {F} : L^2(\\mathrm {X}) \\rightarrow L^2(\\mathbb {N}^{d_1} \\times \\mathbb {R}^{d_2})$ is an isometry we obtain ().", "If we restrict to the joint functional calculus of $L,T_1,\\dots ,T_{d_2}$ , the formula () can be rewritten as follows.", "For all positive integers $d$ , set $\\mathbb {N}_{d} = 2\\mathbb {N}+ d$ , and define, for all $N \\in \\mathbb {N}_{d}$ and $u \\in \\mathbb {R}^{d}$ , $H_{d,N}(u) = \\sum _{\\begin{array}{c}n_1,\\dots ,n_{d} \\in \\mathbb {N}\\\\ 2n_1 + \\dots + 2 n_{d} + d = N\\end{array}} h^2_{n_1}(u_1) \\cdots h^2_{n_{d}}(u_{d}).$ Corollary 6 For all bounded Borel functions $G : \\mathbb {R}\\times \\mathbb {R}^{d_2} \\rightarrow compactly supported in $ R(Rd2 {0})$,\\begin{equation}\\Vert \\operatorname{\\mathcal {K}}_{G(L,\\mathbf {T})}(\\cdot ,y)\\Vert _2^2 = (2\\pi )^{-d_2} \\int _{\\mathbb {R}^{d_2}} \\sum _{N \\in \\mathbb {N}_{d_1}} |G(N|\\xi |,\\xi )|^2 \\, |\\xi |^{d_1/2} H_{d_1,N}(|\\xi |^{1/2} y^{\\prime }) \\,d\\xi \\end{equation}for almost all $ y X$.$ We can now combine (REF ) and () to get the following weighted inequalities.", "Proposition 7 For all $\\gamma \\ge 0$ and for all compactly supported bounded Borel functions $F : \\mathbb {R}\\rightarrow ,\\begin{equation}\\Vert \\,|\\mathbf {P}|^\\gamma \\operatorname{\\mathcal {K}}_{F(L)}(\\cdot ,y)\\Vert _2^2 \\le C_\\gamma \\int _0^\\infty |F(\\lambda )|^2 \\sum _{N \\in \\mathbb {N}_{d_1}} \\frac{\\lambda ^{Q/2-\\gamma }}{N^{Q/2-2\\gamma }} \\,H_{d_1,N}\\left(\\frac{\\lambda ^{1/2} y^{\\prime }}{N^{1/2}}\\right) \\,\\frac{d\\lambda }{\\lambda }\\end{equation}for almost all $ y X$.$ Let $G : \\mathbb {R}\\times \\mathbb {R}^{d_2} \\rightarrow be as in Corollary \\ref {cor:plancherel}.", "In particular $ KG(L,T)(,y) L2(X)$ for almost all $ y X$, and from \\cite [Theorem III.6.20]{dunford_linear_1958} and the definition of integral kernel one may deduce$$L^{\\gamma /2} |\\mathbf {T}|^{-\\gamma } \\left( \\operatorname{\\mathcal {K}}_{G(L,\\mathbf {T})}(\\cdot , y) \\right) = \\operatorname{\\mathcal {K}}_{L^{\\gamma /2} |\\mathbf {T}|^{-\\gamma } G(L,\\mathbf {T})}(\\cdot , y)$$for all $ 0$ and almost all $ y X$.", "This equality, together with (\\ref {eq:plancherel}) and (\\ref {eq:fractionalpowers2}), implies that$$\\Vert \\,|\\mathbf {P}|^\\gamma \\operatorname{\\mathcal {K}}_{G(L,\\mathbf {T})}(\\cdot ,y)\\Vert _2^2 \\le C_\\gamma \\sum _{N \\in \\mathbb {N}_{d_1}} \\int _{\\mathbb {R}^{d_2}} |G(N|\\xi |,\\xi )|^2 \\, N^\\gamma \\, |\\xi |^{d_1/2-\\gamma } \\, H_{d_1,N}(|\\xi |^{1/2} y^{\\prime }) \\,d\\xi .$$$ Choose now an increasing sequence $(\\zeta _n)_{n \\in \\mathbb {N}}$ of nonnegative Borel functions on $\\mathbb {R}$ , compactly supported in $\\mathbb {R}\\setminus \\lbrace 0\\rbrace $ and converging pointwise on $\\mathbb {R}\\setminus \\lbrace 0\\rbrace $ to the constant 1, and define $G_n(\\lambda ,\\xi ) = F(\\lambda ) \\,\\zeta _n(|\\xi |)$ .", "Note that $\\operatorname{\\mathcal {K}}_{F(L)}(\\cdot ,y) \\in L^2(\\mathrm {X})$ for almost all $y \\in \\mathrm {X}$ [8], hence $\\operatorname{\\mathcal {K}}_{G_n(L,\\mathbf {T})}(\\cdot ,y) = \\zeta _n(|\\mathbf {T}|) (\\operatorname{\\mathcal {K}}_{F(L)}(\\cdot ,y))$ for almost all $y \\in \\mathrm {X}$ , as before, and $\\operatorname{\\mathcal {K}}_{G_n(L,\\mathbf {T})}(\\cdot ,y) \\rightarrow \\operatorname{\\mathcal {K}}_{F(L)}(\\cdot ,y)$ in $L^2(\\mathrm {X})$ for almost all $y$ , because $|\\mathbf {T}|$ has trivial kernel.", "The conclusion then follows by applying the previous inequality when $G = G_n$ and letting $n$ tend to infinity.", "Now we recall some well-known estimates for the Hermite functions which we need in the sequel.", "Lemma 8 For all $N = 2n+1 \\in \\mathbb {N}_1$ , $H_{1,N}(u) = h_n^2(u) \\le {\\left\\lbrace \\begin{array}{ll}C(N^{1/3} + |u^2-N|)^{-1/2} &\\text{for all $u \\in \\mathbb {R}$.", "}\\\\C\\exp (-cu^2) &\\text{when $u^2 \\ge 2N$,}\\end{array}\\right.", "}$ Moreover, if $d \\ge 2$ , then, for all $N \\in \\mathbb {N}_d$ , $H_{d,N}(u) \\le {\\left\\lbrace \\begin{array}{ll}CN^{d/2-1} &\\text{for all $u \\in \\mathbb {R}^d$,}\\\\C\\exp (-c|u|_\\infty ^2) &\\text{when $|u|_\\infty ^2 \\ge 2N$,}\\end{array}\\right.", "}$ where $|u|_\\infty = \\max \\lbrace |u_1|,\\dots ,|u_d|\\rbrace $ .", "For the bounds (REF ), see [19] or [24].", "For the first inequality in (REF ), see [24]; the second inequality is an easy consequence of (REF ).", "These bounds allow us to obtain the following crucial estimate.", "Lemma 9 For all fixed $d \\in \\mathbb {N}\\setminus \\lbrace 0\\rbrace $ and $\\varepsilon \\in \\left]0,\\infty \\right[$ , the sum $\\sum _{N \\in \\mathbb {N}_d} \\frac{\\max \\lbrace 1,|u|\\rbrace ^{\\varepsilon }}{N^{d/2+\\varepsilon }} \\, H_{d,N} \\left( \\frac{u}{N^{1/2}}\\right)$ has a finite upper bound, independent of $u \\in \\mathbb {R}^d$ .", "We split the sum into several parts, and use the bounds (REF ), (REF ).", "The part where $N \\le |u|/2$ is empty unless $|u| \\ge 1$ ; in this case, moreover, $|N^{-1/2} u|^2 \\ge 4N$ , hence $H_{d,N}(N^{-1/2} u) \\le C \\exp (-c|u|^2/N)$ , and $\\sup _u \\sum _{N \\le |u|/2} |u|^\\varepsilon N^{-d/2-\\varepsilon } \\exp (-c|u|^2/N) \\le \\sum _{N \\in \\mathbb {N}_d} \\sup _{t \\ge 4N} t^{\\varepsilon /2} \\exp (-ct),$ which is finite.", "If $N \\ge |u|/2$ and $d \\ge 2$ , then $H_{d,N}(N^{-1/2} u) \\le C N^{d/2-1}$ , and $\\sup _u \\sum _{N \\ge |u|/2} \\max \\lbrace 1,|u|\\rbrace ^\\varepsilon N^{-1-\\varepsilon } < \\infty .$ When $d = 1$ , the same argument works for the part where $N \\ge 2|u|$ , because in this case $|N^{-1/2} u|^2 \\le N/4$ and the bound $H_{N}(N^{-1/2} u) \\le C N^{-1/2}$ holds.", "However the part where $|u|/2 < N < 2|u|$ requires a different estimate.", "Namely, the part of (REF ) where $|u|/2 < N \\le |u|-1$ is majorized by $C_\\varepsilon \\, |u|^{-1} \\sum _{|u|/2 < N < |u|-1} |1-N/u|^{-1/2} \\le C_\\varepsilon \\int _{1/2}^1 |1-t|^{-1/2} \\,dt,$ which is finite and independent of $u$ .", "Analogously one bounds the part of (REF ) where $|u|+1 \\le N < 2|u|$ .", "The remaining part, where $|u|-1 < N < |u|+1$ , contains at most one summand, which moreover is bounded by a constant.", "The previous inequality allows us to simplify () considerably and to obtain the weighted Plancherel estimates announced in the introduction.", "Recall that $w_R$ denotes the weight function defined by () for all $R > 0$ .", "Proposition 10 For all $\\gamma \\in \\left[0,d_2/2 \\right[$ and all bounded compactly supported Borel functions $F : \\mathbb {R}\\rightarrow ,$$\\Vert \\, |\\mathbf {P}|^\\gamma \\operatorname{\\mathcal {K}}_{F(L)}(\\cdot ,y)\\Vert _2^2 \\le C_\\gamma \\int _0^\\infty |F(\\lambda )|^2 \\, \\lambda ^{(d_1+d_2)/2} \\, \\min \\lbrace \\lambda ^{d_2/2-\\gamma },|y^{\\prime }|^{2\\gamma -d_2}\\rbrace \\,\\frac{d\\lambda }{\\lambda }$$for almost all $ y X$.", "In particular, for all $ R > 0$, if $ suppF [R2,4R2 ]$, then$$\\operatornamewithlimits{\\mathrm {ess\\,sup}}_{y \\in \\mathrm {X}} |B(y,R^{-1})|^{1/2} \\, \\Vert (1+w_R(\\cdot ,y))^\\gamma \\operatorname{\\mathcal {K}}_{F(L)}(\\cdot ,y)\\Vert _2 \\le C_{\\gamma } \\Vert F_{(R^2)}\\Vert _{L^2},$$where the constant $ C$ does not depend on $ R$.$ In view of (), the first inequality follows immediately from Lemma REF with $d = d_1$ and $\\varepsilon = d_2-2\\gamma $ .", "In the case $\\operatorname{\\mathrm {supp}}F \\subseteq \\left[R^2,4R^2 \\right]$ , a simple manipulation, together with () and (REF ), gives the second inequality." ], [ "The multiplier theorems", "We now show how the weighted Plancherel estimates obtained in the previous section can be used to improve the known multiplier theorems for the Grushin operator.", "First we recall the basic known estimates for operators satisfying Gaussian-type heat kernel bounds in a doubling metric-measure space.", "Proposition 11 For all $R > 0$ , $\\alpha \\ge 0$ , $\\beta > \\alpha $ , and for all functions $F : \\mathbb {R}\\rightarrow such that $ suppF [-4R2,4R2]$,\\begin{equation}\\operatornamewithlimits{\\mathrm {ess\\,sup}}_{y \\in \\mathrm {X}} |B(y,R^{-1})|^{1/2} \\Vert (1+ R\\varrho (\\cdot ,y))^\\alpha \\operatorname{\\mathcal {K}}_F(\\cdot ,y)\\Vert _2 \\le C_{\\alpha ,\\beta } \\Vert F_{(R^2)}\\Vert _{W_\\infty ^\\beta },\\end{equation}where the constant $ C,$ does not depend on $ R$.", "If in addition $ > +Q/2$, then\\begin{equation}\\operatornamewithlimits{\\mathrm {ess\\,sup}}_{y \\in \\mathrm {X}} \\Vert (1+ R\\varrho (\\cdot ,y))^\\alpha \\operatorname{\\mathcal {K}}_F(\\cdot ,y)\\Vert _1 \\le C_{\\alpha ,\\beta }\\Vert F_{(R^2)} \\Vert _{W_\\infty ^\\beta },\\end{equation}where again $ C,$ does not depend on $ R$.$ For the first inequality, see [10] or [8]; note that the statement in [8] seems to require that the multiplier $F$ is supported away from the origin, but its proof clarifies that this is not necessary, because here we do not perform the change of variable $\\lambda \\mapsto \\sqrt{\\lambda }$ in the multiplier function.", "The second inequality is an immediate consequence of the first, via Hölder's inequality and [8].", "These inequalities can be improved by means of the weighted Plancherel estimates.", "For this, some properties of the weight functions $w_R$ are needed.", "Lemma 12 Suppose that $0 \\le \\gamma < \\min \\lbrace d_1,d_2\\rbrace /2$ and $\\beta > Q/2 - \\alpha $ .", "For all $y \\in \\mathrm {X}$ and $R > 0$ , $\\int _\\mathrm {X}(1+w_R(x,y))^{-2\\gamma } (1 + R \\varrho (x,y))^{-2\\beta } \\,dx \\le C_{\\alpha ,\\beta } |B(y,R^{-1})|.$ Moreover, for all $x,y \\in \\mathrm {X}$ and $R > 0$ , $w_R(x,y) \\le C (1 + R \\varrho (x,y)).$ By exploiting the homogeneity properties of the distance $\\varrho $ and the weights $w_R$ , we may suppose that $R = 1$ .", "Then (REF ) immediately follows from the fact that $\\min \\lbrace 1,|y^{\\prime }|^{-1}\\rbrace |x^{\\prime }| \\le 1 + |x^{\\prime } - y^{\\prime }| \\le C (1 + \\varrho (x,y)),$ by (REF ).", "To show (REF ) we note that by translation-invariance we may also suppose that $y^{\\prime \\prime } = 0$ .", "By (REF ), we must then prove that $\\int _\\mathrm {X}\\left(1+\\frac{|x^{\\prime }-y^{\\prime }|}{1+|y^{\\prime }|}\\right)^{-2\\gamma } (1 + \\varrho (x,y))^{-2\\beta } \\,dx \\le C_{\\gamma ,\\beta } (1 + |y^{\\prime }|)^{d_2}.$ We split the integral into two parts, according to the asymptotics (REF ).", "In the region $\\mathrm {X}_1 = \\lbrace x \\in \\mathrm {X}\\,:\\,|x^{\\prime \\prime }|^{1/2} \\ge |x^{\\prime }| + |y^{\\prime }|\\rbrace $ , we decompose $\\beta = \\beta _1 + \\beta _2$ so that $\\beta _1 > d_1/2-\\gamma $ and $\\beta _2 > d_2$ , whence the integral on $\\mathrm {X}_1$ is at most $(1+|y^{\\prime }|)^{2\\gamma } \\int _{\\mathbb {R}^{d_1}} (1+|x^{\\prime }-y^{\\prime }|)^{-2(\\gamma +\\beta _1)} \\,dx^{\\prime } \\int _{\\mathbb {R}^{d_2}} (1+|x^{\\prime \\prime }|^{1/2})^{-2\\beta _2} \\,dx^{\\prime \\prime }.$ In the region $\\mathrm {X}_2 = \\lbrace x \\in \\mathrm {X}\\,:\\,|x^{\\prime \\prime }|^{1/2} < |x^{\\prime }| + |y^{\\prime }|\\rbrace $ , instead, we decompose $\\beta = \\tilde{\\beta }_1+\\tilde{\\beta }_2$ so that $\\tilde{\\beta }_1 > (d_1+d_2)/2-\\gamma $ and $\\tilde{\\beta }_2 > d_2/2$ , whence the integral on $\\mathrm {X}_2$ is at most $\\begin{split}\\int _\\mathrm {X}&\\left(1+\\frac{|x^{\\prime }-y^{\\prime }|}{1+|y^{\\prime }|}\\right)^{-2\\gamma } (1+|x^{\\prime }-y^{\\prime }|)^{-2\\tilde{\\beta }_1} \\left(1 + \\frac{|x^{\\prime \\prime }|}{|x^{\\prime }|+|y^{\\prime }|}\\right)^{-2\\tilde{\\beta }_2} \\,dx \\\\&\\le C_{\\gamma ,\\beta } \\int _{\\mathbb {R}^{d_1}} \\left(1+\\frac{|u|}{1+|y^{\\prime }|}\\right)^{-2\\gamma } (1+|u|)^{-2\\tilde{\\beta }_1} (|u+y^{\\prime }|+|y^{\\prime }|)^{d_2} \\,du \\\\&\\le C_{\\gamma ,\\beta } \\left( (1+|y^{\\prime }|)^{2\\gamma } \\int _{\\mathbb {R}^{d_1}} (1+|u|)^{-2\\nu } \\,du + |y^{\\prime }|^{d_2} \\int _{\\mathbb {R}^{d_1}} (1+|u|)^{-2\\tilde{\\beta }_1} \\,du \\right),\\end{split}$ where $\\nu = \\tilde{\\beta }_1 + \\gamma - d_2/2 > d_1/2$ .", "The conclusion follows.", "A strengthened weighted version of () can now be obtained using the Mauceri-Meda interpolation trick [17] (see also [15] and [8]).", "Proposition 13 For all $R > 0$ , $\\alpha \\ge 0$ , $\\beta > \\alpha $ , $\\gamma \\in \\left[0,d_2/2\\right[$ , and for all functions $F : \\mathbb {R}\\rightarrow such that $ suppF [R2,4R2 ]$,\\begin{multline}\\operatornamewithlimits{\\mathrm {ess\\,sup}}_{y \\in \\mathrm {X}} |B(y,R^{-1})|^{1/2} \\, \\Vert (1 + R\\varrho (\\cdot ,y))^\\alpha (1+w_R(\\cdot ,y))^\\gamma \\operatorname{\\mathcal {K}}_{F(L)}(\\cdot ,y)\\Vert _2 \\\\\\le C_{\\alpha ,\\beta ,\\gamma } \\Vert F_{(R^2)} \\Vert _{W_2^\\beta },\\end{multline}where the constant $ C,,$ does not depend on $ R$.$ The estimate (), together with (REF ) and a Sobolev embedding, immediately implies () in the case $\\beta > \\alpha + d_2/2 + 1/2$ .", "On the other hand, in the case $\\alpha = 0$ , () is given by Proposition REF for all $\\beta > 0$ .", "The conclusion then follows by interpolation (see, e.g., [3], [5]).", "An alternative proof of Proposition REF can be obtained using minor adjustments of the technique developed in [4].", "Let $D = Q - \\min \\lbrace d_1,d_2\\rbrace = \\max \\lbrace d_1+d_2,2d_2\\rbrace $ .", "Proposition REF , together with (REF ) and Hölder's inequality, then yields an improvement of ().", "Corollary 14 For all $R > 0$ , $\\alpha \\ge 0$ , $\\beta > \\alpha + D/2$ , and for all functions $F : \\mathbb {R}\\rightarrow such that $ suppF [R2,4R2 ]$,\\begin{equation}\\operatornamewithlimits{\\mathrm {ess\\,sup}}_{y \\in \\mathrm {X}} \\Vert (1 + R\\varrho (\\cdot ,y))^\\alpha \\operatorname{\\mathcal {K}}_{F(L)}(\\cdot ,y)\\Vert _1\\le C_{\\alpha ,\\beta } \\Vert F_{(R^2)}\\Vert _{W_2^\\beta },\\end{equation}where the constant $ C,$ does not depend on $ R$.", "In particular, under the same hypotheses,\\begin{equation}\\operatornamewithlimits{\\mathrm {ess\\,sup}}_{y \\in \\mathrm {X}} \\int _{\\mathrm {X}\\setminus B(y,r)} |\\operatorname{\\mathcal {K}}_{F(L)}(x,y)| \\,dx\\le C_{\\alpha ,\\beta } (1+rR)^{-\\alpha } \\Vert F_{(R^2)} \\Vert _{W_2^\\beta }.\\end{equation}$ We are finally able to prove our main results.", "We can follow the lines of the proof of [8], where the inequality (4.18) there is replaced by our ().", "Choose $\\beta \\in \\left]D/2, \\kappa +1/2 \\right[$ .", "Let $\\eta \\in C^\\infty _c(\\mathbb {R})$ be supported in $\\left[-1/2,1/2\\right]$ and equal to 1 in a neighborhood of the origin, and set $F(\\lambda ) = (1-|\\lambda |)_+^\\kappa $ .", "The function $\\eta F$ is smooth and compactly supported, while $(1-\\eta )F$ is compactly supported away from the origin and belongs to $W_2^\\beta $ .", "The inequalities () and () then imply that the operators $\\eta (tL) F(tL)$ and $(1-\\eta (tL)) F(tL)$ are bounded on $L^1(\\mathrm {X})$ , uniformly in $t > 0$ , and the same holds for their sum $(1-tL)_+^\\kappa $ .", "The conclusion follows by self-adjointness and interpolation." ], [ "Sharpness of the obtained results", "The aim of this section is to show that, if $d_1 \\ge d_2$ , then the result in Theorem REF is sharp.", "More precisely, if $d_1 \\ge d_2$ and $s<D/2=(d_1+d_2)/2$ , then the first inequality in () cannot hold.", "Indeed, if we consider the functions $H_t(\\lambda )=\\lambda ^{it}$ , then, for $t>1$ , $C \\Vert H_t \\Vert _{MW_2^s} \\sim t^s.$ On the other hand, we make the following observation.", "Proposition 15 Suppose that $L$ is the Grushin operator acting on $\\mathrm {X}= \\mathbb {R}^{d_1}\\times \\mathbb {R}^{d_2}$ .", "Then the following lower bounds holds: $\\Vert H_t(L)\\Vert _{L^1\\rightarrow L^{1,w}}=\\Vert L^{it}\\Vert _{L^1\\rightarrow L^{1,w}} \\ge C (1+|t|)^{(d_1+d_2)/2}$ for all $t>0$ .", "Because the Grushin operator is elliptic on $\\mathrm {X}_0 = \\lbrace x \\in \\mathrm {X}\\,:\\,x^{\\prime } \\ne 0\\rbrace $ , one can use the same argument as in [22] to prove that, for all $y \\in \\mathrm {X}_0$ , $|p_t(x,y) - |y^{\\prime }|^{- d_2}(4\\pi t)^{-(d_1+d_2)/2}{\\rm e}^{-\\varrho (x,y)^2/4t}|\\le Ct^{1/2}t^{-(d_1+d_2)/2}$ for all $x$ in a small neighborhood of $y$ and all $t \\in \\left]0,1 \\right[$ .", "Here $p_t=\\operatorname{\\mathcal {K}}_{\\exp (-tL)}$ is the heat kernel corresponding to the Grushin operator.", "The rest of the argument is the same as in [22], so we skip it here." ], [ "Notation and preliminaries", "As above, let $\\mathrm {X}$ be $\\mathbb {R}^{d_1}\\times \\mathbb {R}^{d_2}$ with Lebesgue measure.", "In order to study the Grushin operator $L$ on $\\mathrm {X}$ , it is convenient to introduce at the same time a family of operators which commute with $L$ .", "Given a point $x = (x^{\\prime },x^{\\prime \\prime }) \\in \\mathrm {X}$ , we denote by $x^{\\prime }_{{j}}$ and $x^{\\prime \\prime }_{{k}}$ the ${j}$ -th component of $x^{\\prime }$ and the ${k}$ -th component of $x^{\\prime \\prime }$ .", "For all ${j}\\in \\lbrace 1,\\dots ,d_1\\rbrace $ and ${k}\\in \\lbrace 1,\\dots ,d_2\\rbrace $ , let then $L_{{j}}$ , $T_{{k}}$ , $P_{{j}}$ be the differential operators on $\\mathrm {X}$ given by $L_{{j}} = (-i\\partial _{x^{\\prime }_{{j}}})^2 + (x^{\\prime }_{{j}})^2 \\sum _{l=1}^{d_2} (-i\\partial _{x^{\\prime \\prime }_l})^2, \\qquad T_{{k}} = -i\\partial _{x^{\\prime \\prime }_{{k}}}, \\qquad P_{{j}} = x^{\\prime }_{{j}}.$ If $(\\delta _r)_{r > 0}$ is the family of dilations on $\\mathrm {X}$ defined by $\\delta _r(x^{\\prime },x^{\\prime \\prime }) = (rx^{\\prime },r^2 x^{\\prime \\prime }),$ then $\\Vert f \\circ \\delta _r\\Vert _2 = r^{-Q/2} \\Vert f\\Vert _2$ , where $Q = d_1 + 2d_2$ .", "We also note that $\\begin{split}P_{{j}} (f \\circ \\delta _r) = r^{-1} (P_{{j}} f) \\circ \\delta _r, &\\qquad L_{{j}}(f \\circ \\delta _r) = r^2 (L_{{j}} f) \\circ \\delta _r, \\\\ T_{{k}} (f \\circ \\delta _r) &= r^2 (T_{{k}} f) \\circ \\delta _r.\\end{split}$ The Grushin operator $L$ on $\\mathrm {X}$ is the sum $L_1 + \\dots + L_{d_1}$ .", "$L$ is a second-order subelliptic differential operator with smooth coefficients.", "For such operators, several ways of introducing a control distance $\\varrho $ are available in the literature, and we refer the reader to [11] for a survey.", "In particular, $L$ belongs to the class of operators studied in [21], where the following estimates are obtained.", "Proposition 3 The control distance $\\varrho $ of the Grushin operator $L$ on $\\mathrm {X}$ is homogeneous with respect to the dilations $\\delta _r$ , that is, $\\varrho (\\delta _r(x),\\delta _r(y)) = r \\varrho (x,y)$ for all $r > 0$ and $x,y \\in \\mathrm {X}$ , and moreover $\\varrho (x,y) \\sim |x^{\\prime } - y^{\\prime }| + {\\left\\lbrace \\begin{array}{ll}\\frac{|x^{\\prime \\prime }-y^{\\prime \\prime }|}{|x^{\\prime }| + |y^{\\prime }|} &\\text{if $|x^{\\prime \\prime }-y^{\\prime \\prime }|^{1/2} \\le |x^{\\prime }| + |y^{\\prime }|$,}\\\\|x^{\\prime \\prime }-y^{\\prime \\prime }|^{1/2} &\\text{if $|x^{\\prime \\prime }-y^{\\prime \\prime }|^{1/2} \\ge |x^{\\prime }| + |y^{\\prime }|$.}\\end{array}\\right.", "}$ Consequently, if $B(x,r)$ denotes the $\\varrho $ -ball of center $x \\in \\mathrm {X}$ and radius $r \\ge 0$ , then $|B(x,r)| \\sim r^{d_1+d_2} \\max \\lbrace r,|x^{\\prime }|\\rbrace ^{d_2},$ and in particular, for all $\\lambda \\ge 0$ , $|B(x,\\lambda r)| \\le C (1+\\lambda )^Q |B(x,r)|.$ Moreover, there exist constants $b,C > 0$ such that, for all $t > 0$ , the integral kernel $p_t$ of the operator $\\exp (-tL)$ is a function satisfying $|p_t(x,y)| \\le C |B(y,t^{1/2})|^{-1} e^{-b \\varrho (x,y)^2/t}$ for all $x,y \\in \\mathrm {X}$ .", "The homogeneity of $\\varrho $ follows immediately from its definition [21] and the homogeneity of $L$ .", "For the remaining estimates, see [21].", "The inequality (REF ) says that $\\mathrm {X}$ with the distance $\\varrho $ and the Lebesgue measure is a doubling metric-measure space of homogeneous dimension $Q$ (cf.", "[7] or [8]), whereas (REF ) expresses Gaussian-type heat kernel bounds for $L$ .", "Several properties of $L$ and the other operators introduced above can be easily recovered by considering $\\mathrm {X}$ as the quotient of a suitable stratified Lie group (cf.", "[2], [1]).", "Denote by $\\mathbb {R}^{d_1 \\times d_2}$ the set of $(d_1 \\times d_2)$ -matrices with real coefficients.", "Both $\\mathbb {R}^{d_1 \\times d_2}$ and $\\mathbb {R}^{d_1} \\times \\mathbb {R}^{d_2}$ are abelian Lie groups with respect to addition.", "Let $\\mathrm {H}_{d_1,d_2}$ be the semidirect product group $\\mathbb {R}^{d_1 \\times d_2} \\ltimes (\\mathbb {R}^{d_1} \\times \\mathbb {R}^{d_2})$ , with multiplication $(x,y,t) \\cdot (x_0, y_0, t_0) = (x+ x_0, y+ y_0, t+ t_0 - (x^T y_0 - x_0^T y)/2).$ This is a particular instance of Heisenberg-Reiter group (see [25] and references therein).", "If $\\lbrace \\tilde{X}_{1,1},\\dots ,\\tilde{X}_{d_1,d_2}, \\tilde{Y}_{1}, \\dots ,\\tilde{Y}_{d_1}, \\tilde{T}_{1}, \\dots , \\tilde{T}_{d_2}\\rbrace $ is the standard basis of the Lie algebra of $\\mathrm {H}_{d_1,d_2}$ (i.e., the set of the left-invariant vector fields extending the standard basis of $\\mathbb {R}^{d_1 \\times d_2} \\times \\mathbb {R}^{d_1} \\times \\mathbb {R}^{d_2}$ at the identity), then the only non-trivial Lie brackets among the elements of the basis are $[\\tilde{X}_{{j},{k}},\\tilde{Y}_{{j}}] = -[\\tilde{Y}_{{j}},\\tilde{X}_{{j},{k}}] = -\\tilde{T}_{{k}} \\qquad \\text{for all ${j}= 1,\\dots ,d_1$, ${k}= 1,\\dots ,d_2$.", "}$ $\\mathrm {H}_{d_1,d_2}$ is a 2-step stratified Lie group, with dilations $(\\tilde{\\delta }_r)_{r>0}$ defined by $\\tilde{\\delta }_r(\\tilde{X}_{{j},{k}}) = r\\tilde{X}_{{j},{k}}, \\quad \\tilde{\\delta }_r(\\tilde{Y}_{{j}}) = r\\tilde{Y}_{{j}}, \\quad \\tilde{\\delta }_r(\\tilde{T}_{{k}}) = r^2 \\tilde{T}_{{k}},$ and the homogeneous sublaplacian $\\tilde{L}$ on $\\mathrm {H}_{d_1,d_2}$ is given by $\\tilde{L} = -\\sum _{{j},{k}} \\tilde{X}_{{j},{k}}^2 - \\sum _{{j}} \\tilde{Y}_{{j}}^2.$ Note that, when $d_2=1$ , $\\mathrm {H}_{d_1,d_2}$ is the $(2d_1+1)$ -dimensional Heisenberg group.", "When $d_2 > 1$ , $\\mathrm {H}_{d_1,d_2}$ is not an H-type group (in the sense of Kaplan), nor a Métivier group.", "Nevertheless, in the terminology of [14], [15], $\\mathrm {H}_{d_1,d_2}$ is $h$ -capacious where $h= \\min \\lbrace d_1,d_2\\rbrace $ .", "In particular, the following multiplier theorem holds: the operator $F(\\tilde{L})$ is of weak type $(1,1)$ and bounded on $L^p(\\mathrm {H}_{d_1,d_1})$ for all $p \\in \\left]1,\\infty \\right[$ whenever $\\Vert F\\Vert _{MW_2^s} < \\infty $ for some $s > (\\dim \\mathrm {H}_{d_1,d_2} + (d_2-d_1)_+)/2$ , where $\\dim \\mathrm {H}_{d_1,d_2}$ is the topological dimension $d_1d_2 + d_1 +d_2$ [15].", "$\\mathrm {X}$ can be identified with the left quotient $\\mathbb {R}^{d_1 \\times d_2} \\backslash \\mathrm {H}_{d_1,d_2}$ via the projection map $(x,y,t) \\mapsto (y,t+x^T y/2)$ .", "Hence $\\mathrm {H}_{d_1,d_2}$ acts by right translations on $\\mathrm {X}$ , that is, $\\tau _{(x,y,t)} : \\mathrm {X}\\ni (z^{\\prime },z^{\\prime \\prime }) \\mapsto (z^{\\prime }-y,z^{\\prime \\prime }-x^T z^{\\prime } -t + x^T y/2) \\in \\mathrm {X}$ is a measure-preserving affine transformation of $\\mathrm {X}$ for all $(x,y,t) \\in \\mathrm {H}_{d_1,d_2}$ , and $\\tau _{g h} = \\tau _g \\tau _h$ .", "This in turn induces a unitary representation $\\sigma $ of $\\mathrm {H}_{d_1,d_2}$ on $L^2(\\mathrm {X})$ , given by $\\sigma (g) f = f \\circ \\tau _g^{-1}$ , and $\\begin{split}T_{{k}} = d\\sigma (-i\\tilde{T}_{{k}}), &\\qquad P_{{j}} T_{{k}} = d\\sigma (-i\\tilde{X}_{{j},{k}}), \\\\L_{{j}} = d\\sigma \\Biggl (-\\tilde{Y}_{{j}}^2 - &\\sum _{{k}} \\tilde{X}_{{j},{k}}^2 \\Biggr ), \\qquad L = d\\sigma (\\tilde{L}).\\end{split}$ This shows in particular that the operators $L_1,\\dots ,L_{d_1},T_1,\\dots ,T_{d_2}$ (and all the polynomials in $L_1,\\dots ,L_{d_1},T_1,\\dots ,T_{d_2}$ ) are essentially self-adjoint on $C^\\infty _c(\\mathrm {X})$ and commute strongly (that is, their spectral resolutions commute), so they admit a joint functional calculus on $L^2(\\mathrm {X})$ in the sense of the spectral theorem [16].", "Arguing analogously, by the use of the unitary representation $\\varpi $ of $\\mathbb {R}^{d_1} \\times \\mathbb {R}^{d_2}$ on $L^2(\\mathrm {X})$ given by $(\\varpi (u^{\\prime },u^{\\prime \\prime }) f)(x^{\\prime },x^{\\prime \\prime }) = e^{i \\langle x^{\\prime } , u^{\\prime } \\rangle } f(x^{\\prime }, x^{\\prime \\prime }+u^{\\prime \\prime }),$ one obtains that the operators $P_1,\\dots ,P_{d_1},T_1,\\dots ,T_{d_2}$ are essentially self-adjoint on $C^\\infty _c(\\mathrm {X})$ and commute strongly.", "Because of the mentioned commutation properties, it is convenient to introduce in our notation the following “vectors of operators”: $\\mathbf {L}= (L_1,\\dots ,L_{d_1}), \\qquad \\mathbf {T}= (T_1,\\dots ,T_{d_2}), \\qquad \\mathbf {P}= (P_1,\\dots ,P_{d_1}).$ Thus, for instance, $|\\mathbf {T}|$ stands for the operator $(|T_1|^2 + \\dots + |T_{d_2}|^2)^{1/2}$ , that is, the square root $(-\\Delta _{x^{\\prime \\prime }})^{1/2}$ of minus the second partial Laplacian on $\\mathbb {R}^{d_1} \\times \\mathbb {R}^{d_2}$ , while $|\\mathbf {P}|$ is the operator of multiplication by $|x^{\\prime }|$ .", "The subellipticity of $\\tilde{L}$ then yields the following estimate.", "Proposition 4 For all $\\gamma \\in \\left[0,\\infty \\right[$ and $f \\in L^2(\\mathrm {X})$ , $\\Vert \\, |\\mathbf {P}|^{\\gamma } f\\Vert _2 \\le C_\\gamma \\Vert L^{\\gamma /2} |\\mathbf {T}|^{-\\gamma } f\\Vert _2,$ where the $L^2$ norm on each side of (REF ) is understood to be $+\\infty $ when $f$ does not belong to the domain of the corresponding operator.", "We may assume $\\gamma > 0$ .", "Let $\\mathbf {P}\\mathbf {T}$ denote the double-indexed vector of operators $(P_{{j}} T_{{k}})_{{j},{k}}$ , and note that $|\\mathbf {P}\\mathbf {T}|^{\\gamma } = |\\mathbf {P}|^{\\gamma } |\\mathbf {T}|^{\\gamma }$ (modulo closures).", "Moreover the spectrum $\\left[0,+\\infty \\right[$ of $|\\mathbf {T}|^{\\gamma }$ is purely continuous, so $|\\mathbf {T}|^{\\gamma }$ is injective and its image is dense in $L^2(\\mathrm {X})$ .", "Therefore (REF ) is reduced to the proof of the inequality $\\Vert \\, |\\mathbf {P}\\mathbf {T}|^{\\gamma } g\\Vert _2 \\le C_\\gamma \\Vert L^{\\gamma /2} g\\Vert _2$ for all $g \\in L^2(\\mathrm {X})$ .", "By (REF ), the differential operator $\\tilde{W} = -\\sum _{{j},{k}} \\tilde{X}_{{j},{k}}^2$ on $\\mathrm {H}_{d_1,d_2}$ corresponds to the operator $|\\mathbf {P}\\mathbf {T}|^2$ on $\\mathrm {X}$ .", "Since $\\tilde{W}$ is $\\tilde{\\delta }_r$ -homogeneous, with the same homogeneity degree as the sublaplacian $\\tilde{L}$ , from (REF ) and [16] we deduce (REF ) for all $\\gamma \\in 2\\mathbb {N}$ and $g \\in L^2(\\mathrm {X})$ .", "We want now to extend (REF ) to all the real $\\gamma \\ge 0$ .", "For this, fix $m \\in \\mathbb {N}$ and let $A$ and $B$ be the closures of $|\\mathbf {P}\\mathbf {T}|^{2m}$ and $L^m$ on $L^2(\\mathrm {X})$ respectively.", "Since $A$ and $B$ are nonnegative self-adjoint operators on $L^2(\\mathrm {X})$ , by [13], for all $\\theta \\in \\left]0,1 \\right[$ , $(L^2(\\mathrm {X}),\\operatorname{\\mathrm {dom}}A)_{[\\theta ]} = \\operatorname{\\mathrm {dom}}A^\\theta , \\qquad (L^2(\\mathrm {X}),\\operatorname{\\mathrm {dom}}B)_{[\\theta ]} = \\operatorname{\\mathrm {dom}}B^\\theta ,$ with equivalent norms, where $(\\cdot ,\\cdot )_{[\\theta ]}$ denotes interpolation with respect to the complex method, and the domains of the various operators are endowed with the graph norms.", "On the other hand, (REF ) implies that $\\operatorname{\\mathrm {dom}}B \\subseteq \\operatorname{\\mathrm {dom}}A$ , with continuous inclusion.", "By interpolation [3], we conclude that $\\operatorname{\\mathrm {dom}}B^\\theta \\subseteq \\operatorname{\\mathrm {dom}}A^\\theta $ , with continuous inclusion.", "This implies that $\\Vert \\, |\\mathbf {P}\\mathbf {T}|^{\\gamma } g\\Vert _2 \\le C_\\gamma (\\Vert g\\Vert _2 + \\Vert L^{\\gamma /2} g\\Vert _2)$ for all $\\gamma \\in \\left[0,2m \\right]$ and $g \\in L^2(\\mathrm {X})$ .", "The bound (REF ) then follows by replacing $f$ with $f \\circ \\delta _r$ in the previous inequality, exploiting the homogeneity relations (REF ), and taking the limit for $r \\rightarrow \\infty $ ." ], [ "Plancherel estimates", "From the previous section we know that the operators $L_1,\\dots ,L_{d_1},T_1,\\dots ,T_{d_2}$ have a joint functional calculus.", "In fact one can obtain a quite explicit formula for the integral kernel $\\operatorname{\\mathcal {K}}_{G(\\mathbf {L},\\mathbf {T})}$ of an operator $G(\\mathbf {L},\\mathbf {T})$ in the functional calculus, in terms of the Hermite functions (cf.", "[18] for the case $d_1=d_2=1$ , and [23] for the analogue on the Heisenberg groups).", "Namely, for all $\\ell \\in \\mathbb {N}$ , let $h_\\ell $ denote the $\\ell $ -th Hermite function, that is, $h_\\ell (t) = (-1)^\\ell (\\ell !", "\\, 2^\\ell \\sqrt{\\pi })^{-1/2} e^{t^2/2} \\left(\\frac{d}{dt}\\right)^\\ell e^{-t^2},$ and set, for all $n \\in \\mathbb {N}^{d_1}$ , $u \\in \\mathbb {R}^{d_1}$ , $\\xi \\in \\mathbb {R}^{d_2}$ , $\\tilde{h}_n(u,\\xi ) = |\\xi |^{d_1/4} h_{n_1}(|\\xi |^{1/2} u_1) \\cdots h_{n_{d_1}}(|\\xi |^{1/2} u_{d_1}).$ Proposition 5 For all bounded Borel functions $G : \\mathbb {R}^{d_1} \\times \\mathbb {R}^{d_2} \\rightarrow compactly supported in $ Rd1 (Rd2 {0})$,\\begin{multline*}\\operatorname{\\mathcal {K}}_{G(\\mathbf {L},\\mathbf {T})}(x,y) \\\\= (2\\pi )^{-d_2} \\int _{\\mathbb {R}^{d_2}} \\sum _{n \\in \\mathbb {N}^{d_1}} G(|\\xi |(2n+\\tilde{1}), \\xi ) \\, \\tilde{h}_n(y^{\\prime },\\xi ) \\, \\tilde{h}_n(x^{\\prime },\\xi ) \\, e^{i \\langle \\xi , x^{\\prime \\prime }-y^{\\prime \\prime } \\rangle } \\,d\\xi \\end{multline*}for almost all $ x,y X$, where $ 1= (1,...,1) Nd1$.", "In particular\\begin{equation}\\Vert \\operatorname{\\mathcal {K}}_{G(\\mathbf {L},\\mathbf {T})}(\\cdot ,y)\\Vert _2^2 = (2\\pi )^{-d_2} \\int _{\\mathbb {R}^{d_2}} \\sum _{n \\in \\mathbb {N}^{d_1}} |G(|\\xi |(2n+\\tilde{1}),\\xi )|^2 \\, \\tilde{h}_n^2(y^{\\prime },\\xi ) \\,d\\xi \\end{equation}for almost all $ y X$.$ Let $\\mathcal {F} : L^2(\\mathbb {R}^{d_1} \\times \\mathbb {R}^{d_2}) \\rightarrow L^2(\\mathbb {R}^{d_1} \\times \\mathbb {R}^{d_2})$ be the isometry defined by $\\mathcal {F} \\phi (x^{\\prime },\\xi ) = (2\\pi )^{-d_2/2} \\int _{\\mathbb {R}^{d_2}} \\phi (x^{\\prime },x^{\\prime \\prime }) \\, e^{-i \\langle \\xi , x^{\\prime \\prime } \\rangle } \\, dx^{\\prime \\prime },$ i.e., the Fourier transform with respect to $x^{\\prime \\prime }$ .", "Then $\\mathcal {F} L_{{j}}\\phi (x^{\\prime },\\xi ) = L_{{j},\\xi } \\, \\mathcal {F} \\phi (x^{\\prime },\\xi ), \\qquad \\mathcal {F} T_{{k}}\\phi (x^{\\prime },\\xi ) = \\xi _{{k}} \\, \\mathcal {F} \\phi (x^{\\prime },\\xi ),$ at least for $\\phi $ in the Schwartz class, where $L_{{j},\\xi } = (-i\\partial _{x^{\\prime }_{{j}}})^2 + |\\xi |^2 (x^{\\prime }_{{j}})^2.$ For all $\\xi \\ne 0$ , $\\lbrace \\tilde{h}_n(\\cdot ,\\xi )\\rbrace _{n \\in \\mathbb {N}^{d_1}}$ is a complete orthonormal system for $L^2(\\mathbb {R}^{d_1})$ made of real-valued functions and $L_{{j},\\xi } \\, \\tilde{h}_n(x^{\\prime },\\xi ) = (2n_{{j}}+1) |\\xi | \\, \\tilde{h}_n(x^{\\prime },\\xi ).$ In particular, if $\\mathcal {G} : L^2(\\mathbb {R}^{d_1} \\times \\mathbb {R}^{d_2}) \\rightarrow L^2(\\mathbb {N}^{d_1} \\times \\mathbb {R}^{d_2})$ is the isometry defined by $\\mathcal {G} \\psi (n,\\xi ) = \\int _{\\mathbb {R}^{d_1}} \\psi (x^{\\prime },\\xi ) \\, \\tilde{h}_n(x^{\\prime },\\xi ) \\,dx^{\\prime },$ then $\\mathcal {G} \\mathcal {F} L_{{j}} \\phi (n,\\xi ) = (2n_{{j}}+1) |\\xi | \\, \\mathcal {G} \\mathcal {F} \\phi (n,\\xi ), \\qquad \\mathcal {G} \\mathcal {F} T_{{k}} \\phi (n,\\xi ) = \\xi _{{k}} \\, \\mathcal {G} \\mathcal {F} \\phi (n,\\xi ).$ The isometry $\\mathcal {G}\\mathcal {F}$ intertwines the operators $L_{{j}}$ and $T_{{k}}$ with some multiplication operators on $\\mathbb {N}^{d_1} \\times \\mathbb {R}^{d_2}$ , hence it intertwines the corresponding functional calculi: $\\mathcal {G} \\mathcal {F} \\, G(\\mathbf {L},\\mathbf {T}) \\, \\phi (n,\\xi ) = G(|\\xi |(2n+\\tilde{1}),\\xi ) \\, \\mathcal {G} \\mathcal {F} \\phi (n,\\xi ).$ The inversion formulae for $\\mathcal {F}$ and $\\mathcal {G}$ and some easy manipulations then give the above expression for $\\operatorname{\\mathcal {K}}_{G(\\mathbf {L},\\mathbf {T})}$ .", "Moreover, if we set $G_{y}(n,\\xi ) = (2\\pi )^{-d_2/2} G(|\\xi |(2n+\\tilde{1}),\\xi ) \\, \\tilde{h}_n(y^{\\prime },\\xi ) \\, e^{-i \\langle \\xi , y^{\\prime \\prime } \\rangle },$ then the formula for $\\operatorname{\\mathcal {K}}_{G(\\mathbf {L},\\mathbf {T})}$ can be rewritten as $\\operatorname{\\mathcal {K}}_{G(\\mathbf {L},\\mathbf {T})}(\\cdot ,y) = (\\mathcal {G}\\mathcal {F})^{-1} G_{y},$ and since $\\mathcal {G}\\mathcal {F} : L^2(\\mathrm {X}) \\rightarrow L^2(\\mathbb {N}^{d_1} \\times \\mathbb {R}^{d_2})$ is an isometry we obtain ().", "If we restrict to the joint functional calculus of $L,T_1,\\dots ,T_{d_2}$ , the formula () can be rewritten as follows.", "For all positive integers $d$ , set $\\mathbb {N}_{d} = 2\\mathbb {N}+ d$ , and define, for all $N \\in \\mathbb {N}_{d}$ and $u \\in \\mathbb {R}^{d}$ , $H_{d,N}(u) = \\sum _{\\begin{array}{c}n_1,\\dots ,n_{d} \\in \\mathbb {N}\\\\ 2n_1 + \\dots + 2 n_{d} + d = N\\end{array}} h^2_{n_1}(u_1) \\cdots h^2_{n_{d}}(u_{d}).$ Corollary 6 For all bounded Borel functions $G : \\mathbb {R}\\times \\mathbb {R}^{d_2} \\rightarrow compactly supported in $ R(Rd2 {0})$,\\begin{equation}\\Vert \\operatorname{\\mathcal {K}}_{G(L,\\mathbf {T})}(\\cdot ,y)\\Vert _2^2 = (2\\pi )^{-d_2} \\int _{\\mathbb {R}^{d_2}} \\sum _{N \\in \\mathbb {N}_{d_1}} |G(N|\\xi |,\\xi )|^2 \\, |\\xi |^{d_1/2} H_{d_1,N}(|\\xi |^{1/2} y^{\\prime }) \\,d\\xi \\end{equation}for almost all $ y X$.$ We can now combine (REF ) and () to get the following weighted inequalities.", "Proposition 7 For all $\\gamma \\ge 0$ and for all compactly supported bounded Borel functions $F : \\mathbb {R}\\rightarrow ,\\begin{equation}\\Vert \\,|\\mathbf {P}|^\\gamma \\operatorname{\\mathcal {K}}_{F(L)}(\\cdot ,y)\\Vert _2^2 \\le C_\\gamma \\int _0^\\infty |F(\\lambda )|^2 \\sum _{N \\in \\mathbb {N}_{d_1}} \\frac{\\lambda ^{Q/2-\\gamma }}{N^{Q/2-2\\gamma }} \\,H_{d_1,N}\\left(\\frac{\\lambda ^{1/2} y^{\\prime }}{N^{1/2}}\\right) \\,\\frac{d\\lambda }{\\lambda }\\end{equation}for almost all $ y X$.$ Let $G : \\mathbb {R}\\times \\mathbb {R}^{d_2} \\rightarrow be as in Corollary \\ref {cor:plancherel}.", "In particular $ KG(L,T)(,y) L2(X)$ for almost all $ y X$, and from \\cite [Theorem III.6.20]{dunford_linear_1958} and the definition of integral kernel one may deduce$$L^{\\gamma /2} |\\mathbf {T}|^{-\\gamma } \\left( \\operatorname{\\mathcal {K}}_{G(L,\\mathbf {T})}(\\cdot , y) \\right) = \\operatorname{\\mathcal {K}}_{L^{\\gamma /2} |\\mathbf {T}|^{-\\gamma } G(L,\\mathbf {T})}(\\cdot , y)$$for all $ 0$ and almost all $ y X$.", "This equality, together with (\\ref {eq:plancherel}) and (\\ref {eq:fractionalpowers2}), implies that$$\\Vert \\,|\\mathbf {P}|^\\gamma \\operatorname{\\mathcal {K}}_{G(L,\\mathbf {T})}(\\cdot ,y)\\Vert _2^2 \\le C_\\gamma \\sum _{N \\in \\mathbb {N}_{d_1}} \\int _{\\mathbb {R}^{d_2}} |G(N|\\xi |,\\xi )|^2 \\, N^\\gamma \\, |\\xi |^{d_1/2-\\gamma } \\, H_{d_1,N}(|\\xi |^{1/2} y^{\\prime }) \\,d\\xi .$$$ Choose now an increasing sequence $(\\zeta _n)_{n \\in \\mathbb {N}}$ of nonnegative Borel functions on $\\mathbb {R}$ , compactly supported in $\\mathbb {R}\\setminus \\lbrace 0\\rbrace $ and converging pointwise on $\\mathbb {R}\\setminus \\lbrace 0\\rbrace $ to the constant 1, and define $G_n(\\lambda ,\\xi ) = F(\\lambda ) \\,\\zeta _n(|\\xi |)$ .", "Note that $\\operatorname{\\mathcal {K}}_{F(L)}(\\cdot ,y) \\in L^2(\\mathrm {X})$ for almost all $y \\in \\mathrm {X}$ [8], hence $\\operatorname{\\mathcal {K}}_{G_n(L,\\mathbf {T})}(\\cdot ,y) = \\zeta _n(|\\mathbf {T}|) (\\operatorname{\\mathcal {K}}_{F(L)}(\\cdot ,y))$ for almost all $y \\in \\mathrm {X}$ , as before, and $\\operatorname{\\mathcal {K}}_{G_n(L,\\mathbf {T})}(\\cdot ,y) \\rightarrow \\operatorname{\\mathcal {K}}_{F(L)}(\\cdot ,y)$ in $L^2(\\mathrm {X})$ for almost all $y$ , because $|\\mathbf {T}|$ has trivial kernel.", "The conclusion then follows by applying the previous inequality when $G = G_n$ and letting $n$ tend to infinity.", "Now we recall some well-known estimates for the Hermite functions which we need in the sequel.", "Lemma 8 For all $N = 2n+1 \\in \\mathbb {N}_1$ , $H_{1,N}(u) = h_n^2(u) \\le {\\left\\lbrace \\begin{array}{ll}C(N^{1/3} + |u^2-N|)^{-1/2} &\\text{for all $u \\in \\mathbb {R}$.", "}\\\\C\\exp (-cu^2) &\\text{when $u^2 \\ge 2N$,}\\end{array}\\right.", "}$ Moreover, if $d \\ge 2$ , then, for all $N \\in \\mathbb {N}_d$ , $H_{d,N}(u) \\le {\\left\\lbrace \\begin{array}{ll}CN^{d/2-1} &\\text{for all $u \\in \\mathbb {R}^d$,}\\\\C\\exp (-c|u|_\\infty ^2) &\\text{when $|u|_\\infty ^2 \\ge 2N$,}\\end{array}\\right.", "}$ where $|u|_\\infty = \\max \\lbrace |u_1|,\\dots ,|u_d|\\rbrace $ .", "For the bounds (REF ), see [19] or [24].", "For the first inequality in (REF ), see [24]; the second inequality is an easy consequence of (REF ).", "These bounds allow us to obtain the following crucial estimate.", "Lemma 9 For all fixed $d \\in \\mathbb {N}\\setminus \\lbrace 0\\rbrace $ and $\\varepsilon \\in \\left]0,\\infty \\right[$ , the sum $\\sum _{N \\in \\mathbb {N}_d} \\frac{\\max \\lbrace 1,|u|\\rbrace ^{\\varepsilon }}{N^{d/2+\\varepsilon }} \\, H_{d,N} \\left( \\frac{u}{N^{1/2}}\\right)$ has a finite upper bound, independent of $u \\in \\mathbb {R}^d$ .", "We split the sum into several parts, and use the bounds (REF ), (REF ).", "The part where $N \\le |u|/2$ is empty unless $|u| \\ge 1$ ; in this case, moreover, $|N^{-1/2} u|^2 \\ge 4N$ , hence $H_{d,N}(N^{-1/2} u) \\le C \\exp (-c|u|^2/N)$ , and $\\sup _u \\sum _{N \\le |u|/2} |u|^\\varepsilon N^{-d/2-\\varepsilon } \\exp (-c|u|^2/N) \\le \\sum _{N \\in \\mathbb {N}_d} \\sup _{t \\ge 4N} t^{\\varepsilon /2} \\exp (-ct),$ which is finite.", "If $N \\ge |u|/2$ and $d \\ge 2$ , then $H_{d,N}(N^{-1/2} u) \\le C N^{d/2-1}$ , and $\\sup _u \\sum _{N \\ge |u|/2} \\max \\lbrace 1,|u|\\rbrace ^\\varepsilon N^{-1-\\varepsilon } < \\infty .$ When $d = 1$ , the same argument works for the part where $N \\ge 2|u|$ , because in this case $|N^{-1/2} u|^2 \\le N/4$ and the bound $H_{N}(N^{-1/2} u) \\le C N^{-1/2}$ holds.", "However the part where $|u|/2 < N < 2|u|$ requires a different estimate.", "Namely, the part of (REF ) where $|u|/2 < N \\le |u|-1$ is majorized by $C_\\varepsilon \\, |u|^{-1} \\sum _{|u|/2 < N < |u|-1} |1-N/u|^{-1/2} \\le C_\\varepsilon \\int _{1/2}^1 |1-t|^{-1/2} \\,dt,$ which is finite and independent of $u$ .", "Analogously one bounds the part of (REF ) where $|u|+1 \\le N < 2|u|$ .", "The remaining part, where $|u|-1 < N < |u|+1$ , contains at most one summand, which moreover is bounded by a constant.", "The previous inequality allows us to simplify () considerably and to obtain the weighted Plancherel estimates announced in the introduction.", "Recall that $w_R$ denotes the weight function defined by () for all $R > 0$ .", "Proposition 10 For all $\\gamma \\in \\left[0,d_2/2 \\right[$ and all bounded compactly supported Borel functions $F : \\mathbb {R}\\rightarrow ,$$\\Vert \\, |\\mathbf {P}|^\\gamma \\operatorname{\\mathcal {K}}_{F(L)}(\\cdot ,y)\\Vert _2^2 \\le C_\\gamma \\int _0^\\infty |F(\\lambda )|^2 \\, \\lambda ^{(d_1+d_2)/2} \\, \\min \\lbrace \\lambda ^{d_2/2-\\gamma },|y^{\\prime }|^{2\\gamma -d_2}\\rbrace \\,\\frac{d\\lambda }{\\lambda }$$for almost all $ y X$.", "In particular, for all $ R > 0$, if $ suppF [R2,4R2 ]$, then$$\\operatornamewithlimits{\\mathrm {ess\\,sup}}_{y \\in \\mathrm {X}} |B(y,R^{-1})|^{1/2} \\, \\Vert (1+w_R(\\cdot ,y))^\\gamma \\operatorname{\\mathcal {K}}_{F(L)}(\\cdot ,y)\\Vert _2 \\le C_{\\gamma } \\Vert F_{(R^2)}\\Vert _{L^2},$$where the constant $ C$ does not depend on $ R$.$ In view of (), the first inequality follows immediately from Lemma REF with $d = d_1$ and $\\varepsilon = d_2-2\\gamma $ .", "In the case $\\operatorname{\\mathrm {supp}}F \\subseteq \\left[R^2,4R^2 \\right]$ , a simple manipulation, together with () and (REF ), gives the second inequality." ], [ "The multiplier theorems", "We now show how the weighted Plancherel estimates obtained in the previous section can be used to improve the known multiplier theorems for the Grushin operator.", "First we recall the basic known estimates for operators satisfying Gaussian-type heat kernel bounds in a doubling metric-measure space.", "Proposition 11 For all $R > 0$ , $\\alpha \\ge 0$ , $\\beta > \\alpha $ , and for all functions $F : \\mathbb {R}\\rightarrow such that $ suppF [-4R2,4R2]$,\\begin{equation}\\operatornamewithlimits{\\mathrm {ess\\,sup}}_{y \\in \\mathrm {X}} |B(y,R^{-1})|^{1/2} \\Vert (1+ R\\varrho (\\cdot ,y))^\\alpha \\operatorname{\\mathcal {K}}_F(\\cdot ,y)\\Vert _2 \\le C_{\\alpha ,\\beta } \\Vert F_{(R^2)}\\Vert _{W_\\infty ^\\beta },\\end{equation}where the constant $ C,$ does not depend on $ R$.", "If in addition $ > +Q/2$, then\\begin{equation}\\operatornamewithlimits{\\mathrm {ess\\,sup}}_{y \\in \\mathrm {X}} \\Vert (1+ R\\varrho (\\cdot ,y))^\\alpha \\operatorname{\\mathcal {K}}_F(\\cdot ,y)\\Vert _1 \\le C_{\\alpha ,\\beta }\\Vert F_{(R^2)} \\Vert _{W_\\infty ^\\beta },\\end{equation}where again $ C,$ does not depend on $ R$.$ For the first inequality, see [10] or [8]; note that the statement in [8] seems to require that the multiplier $F$ is supported away from the origin, but its proof clarifies that this is not necessary, because here we do not perform the change of variable $\\lambda \\mapsto \\sqrt{\\lambda }$ in the multiplier function.", "The second inequality is an immediate consequence of the first, via Hölder's inequality and [8].", "These inequalities can be improved by means of the weighted Plancherel estimates.", "For this, some properties of the weight functions $w_R$ are needed.", "Lemma 12 Suppose that $0 \\le \\gamma < \\min \\lbrace d_1,d_2\\rbrace /2$ and $\\beta > Q/2 - \\alpha $ .", "For all $y \\in \\mathrm {X}$ and $R > 0$ , $\\int _\\mathrm {X}(1+w_R(x,y))^{-2\\gamma } (1 + R \\varrho (x,y))^{-2\\beta } \\,dx \\le C_{\\alpha ,\\beta } |B(y,R^{-1})|.$ Moreover, for all $x,y \\in \\mathrm {X}$ and $R > 0$ , $w_R(x,y) \\le C (1 + R \\varrho (x,y)).$ By exploiting the homogeneity properties of the distance $\\varrho $ and the weights $w_R$ , we may suppose that $R = 1$ .", "Then (REF ) immediately follows from the fact that $\\min \\lbrace 1,|y^{\\prime }|^{-1}\\rbrace |x^{\\prime }| \\le 1 + |x^{\\prime } - y^{\\prime }| \\le C (1 + \\varrho (x,y)),$ by (REF ).", "To show (REF ) we note that by translation-invariance we may also suppose that $y^{\\prime \\prime } = 0$ .", "By (REF ), we must then prove that $\\int _\\mathrm {X}\\left(1+\\frac{|x^{\\prime }-y^{\\prime }|}{1+|y^{\\prime }|}\\right)^{-2\\gamma } (1 + \\varrho (x,y))^{-2\\beta } \\,dx \\le C_{\\gamma ,\\beta } (1 + |y^{\\prime }|)^{d_2}.$ We split the integral into two parts, according to the asymptotics (REF ).", "In the region $\\mathrm {X}_1 = \\lbrace x \\in \\mathrm {X}\\,:\\,|x^{\\prime \\prime }|^{1/2} \\ge |x^{\\prime }| + |y^{\\prime }|\\rbrace $ , we decompose $\\beta = \\beta _1 + \\beta _2$ so that $\\beta _1 > d_1/2-\\gamma $ and $\\beta _2 > d_2$ , whence the integral on $\\mathrm {X}_1$ is at most $(1+|y^{\\prime }|)^{2\\gamma } \\int _{\\mathbb {R}^{d_1}} (1+|x^{\\prime }-y^{\\prime }|)^{-2(\\gamma +\\beta _1)} \\,dx^{\\prime } \\int _{\\mathbb {R}^{d_2}} (1+|x^{\\prime \\prime }|^{1/2})^{-2\\beta _2} \\,dx^{\\prime \\prime }.$ In the region $\\mathrm {X}_2 = \\lbrace x \\in \\mathrm {X}\\,:\\,|x^{\\prime \\prime }|^{1/2} < |x^{\\prime }| + |y^{\\prime }|\\rbrace $ , instead, we decompose $\\beta = \\tilde{\\beta }_1+\\tilde{\\beta }_2$ so that $\\tilde{\\beta }_1 > (d_1+d_2)/2-\\gamma $ and $\\tilde{\\beta }_2 > d_2/2$ , whence the integral on $\\mathrm {X}_2$ is at most $\\begin{split}\\int _\\mathrm {X}&\\left(1+\\frac{|x^{\\prime }-y^{\\prime }|}{1+|y^{\\prime }|}\\right)^{-2\\gamma } (1+|x^{\\prime }-y^{\\prime }|)^{-2\\tilde{\\beta }_1} \\left(1 + \\frac{|x^{\\prime \\prime }|}{|x^{\\prime }|+|y^{\\prime }|}\\right)^{-2\\tilde{\\beta }_2} \\,dx \\\\&\\le C_{\\gamma ,\\beta } \\int _{\\mathbb {R}^{d_1}} \\left(1+\\frac{|u|}{1+|y^{\\prime }|}\\right)^{-2\\gamma } (1+|u|)^{-2\\tilde{\\beta }_1} (|u+y^{\\prime }|+|y^{\\prime }|)^{d_2} \\,du \\\\&\\le C_{\\gamma ,\\beta } \\left( (1+|y^{\\prime }|)^{2\\gamma } \\int _{\\mathbb {R}^{d_1}} (1+|u|)^{-2\\nu } \\,du + |y^{\\prime }|^{d_2} \\int _{\\mathbb {R}^{d_1}} (1+|u|)^{-2\\tilde{\\beta }_1} \\,du \\right),\\end{split}$ where $\\nu = \\tilde{\\beta }_1 + \\gamma - d_2/2 > d_1/2$ .", "The conclusion follows.", "A strengthened weighted version of () can now be obtained using the Mauceri-Meda interpolation trick [17] (see also [15] and [8]).", "Proposition 13 For all $R > 0$ , $\\alpha \\ge 0$ , $\\beta > \\alpha $ , $\\gamma \\in \\left[0,d_2/2\\right[$ , and for all functions $F : \\mathbb {R}\\rightarrow such that $ suppF [R2,4R2 ]$,\\begin{multline}\\operatornamewithlimits{\\mathrm {ess\\,sup}}_{y \\in \\mathrm {X}} |B(y,R^{-1})|^{1/2} \\, \\Vert (1 + R\\varrho (\\cdot ,y))^\\alpha (1+w_R(\\cdot ,y))^\\gamma \\operatorname{\\mathcal {K}}_{F(L)}(\\cdot ,y)\\Vert _2 \\\\\\le C_{\\alpha ,\\beta ,\\gamma } \\Vert F_{(R^2)} \\Vert _{W_2^\\beta },\\end{multline}where the constant $ C,,$ does not depend on $ R$.$ The estimate (), together with (REF ) and a Sobolev embedding, immediately implies () in the case $\\beta > \\alpha + d_2/2 + 1/2$ .", "On the other hand, in the case $\\alpha = 0$ , () is given by Proposition REF for all $\\beta > 0$ .", "The conclusion then follows by interpolation (see, e.g., [3], [5]).", "An alternative proof of Proposition REF can be obtained using minor adjustments of the technique developed in [4].", "Let $D = Q - \\min \\lbrace d_1,d_2\\rbrace = \\max \\lbrace d_1+d_2,2d_2\\rbrace $ .", "Proposition REF , together with (REF ) and Hölder's inequality, then yields an improvement of ().", "Corollary 14 For all $R > 0$ , $\\alpha \\ge 0$ , $\\beta > \\alpha + D/2$ , and for all functions $F : \\mathbb {R}\\rightarrow such that $ suppF [R2,4R2 ]$,\\begin{equation}\\operatornamewithlimits{\\mathrm {ess\\,sup}}_{y \\in \\mathrm {X}} \\Vert (1 + R\\varrho (\\cdot ,y))^\\alpha \\operatorname{\\mathcal {K}}_{F(L)}(\\cdot ,y)\\Vert _1\\le C_{\\alpha ,\\beta } \\Vert F_{(R^2)}\\Vert _{W_2^\\beta },\\end{equation}where the constant $ C,$ does not depend on $ R$.", "In particular, under the same hypotheses,\\begin{equation}\\operatornamewithlimits{\\mathrm {ess\\,sup}}_{y \\in \\mathrm {X}} \\int _{\\mathrm {X}\\setminus B(y,r)} |\\operatorname{\\mathcal {K}}_{F(L)}(x,y)| \\,dx\\le C_{\\alpha ,\\beta } (1+rR)^{-\\alpha } \\Vert F_{(R^2)} \\Vert _{W_2^\\beta }.\\end{equation}$ We are finally able to prove our main results.", "We can follow the lines of the proof of [8], where the inequality (4.18) there is replaced by our ().", "Choose $\\beta \\in \\left]D/2, \\kappa +1/2 \\right[$ .", "Let $\\eta \\in C^\\infty _c(\\mathbb {R})$ be supported in $\\left[-1/2,1/2\\right]$ and equal to 1 in a neighborhood of the origin, and set $F(\\lambda ) = (1-|\\lambda |)_+^\\kappa $ .", "The function $\\eta F$ is smooth and compactly supported, while $(1-\\eta )F$ is compactly supported away from the origin and belongs to $W_2^\\beta $ .", "The inequalities () and () then imply that the operators $\\eta (tL) F(tL)$ and $(1-\\eta (tL)) F(tL)$ are bounded on $L^1(\\mathrm {X})$ , uniformly in $t > 0$ , and the same holds for their sum $(1-tL)_+^\\kappa $ .", "The conclusion follows by self-adjointness and interpolation." ], [ "Sharpness of the obtained results", "The aim of this section is to show that, if $d_1 \\ge d_2$ , then the result in Theorem REF is sharp.", "More precisely, if $d_1 \\ge d_2$ and $s<D/2=(d_1+d_2)/2$ , then the first inequality in () cannot hold.", "Indeed, if we consider the functions $H_t(\\lambda )=\\lambda ^{it}$ , then, for $t>1$ , $C \\Vert H_t \\Vert _{MW_2^s} \\sim t^s.$ On the other hand, we make the following observation.", "Proposition 15 Suppose that $L$ is the Grushin operator acting on $\\mathrm {X}= \\mathbb {R}^{d_1}\\times \\mathbb {R}^{d_2}$ .", "Then the following lower bounds holds: $\\Vert H_t(L)\\Vert _{L^1\\rightarrow L^{1,w}}=\\Vert L^{it}\\Vert _{L^1\\rightarrow L^{1,w}} \\ge C (1+|t|)^{(d_1+d_2)/2}$ for all $t>0$ .", "Because the Grushin operator is elliptic on $\\mathrm {X}_0 = \\lbrace x \\in \\mathrm {X}\\,:\\,x^{\\prime } \\ne 0\\rbrace $ , one can use the same argument as in [22] to prove that, for all $y \\in \\mathrm {X}_0$ , $|p_t(x,y) - |y^{\\prime }|^{- d_2}(4\\pi t)^{-(d_1+d_2)/2}{\\rm e}^{-\\varrho (x,y)^2/4t}|\\le Ct^{1/2}t^{-(d_1+d_2)/2}$ for all $x$ in a small neighborhood of $y$ and all $t \\in \\left]0,1 \\right[$ .", "Here $p_t=\\operatorname{\\mathcal {K}}_{\\exp (-tL)}$ is the heat kernel corresponding to the Grushin operator.", "The rest of the argument is the same as in [22], so we skip it here." ], [ "The multiplier theorems", "We now show how the weighted Plancherel estimates obtained in the previous section can be used to improve the known multiplier theorems for the Grushin operator.", "First we recall the basic known estimates for operators satisfying Gaussian-type heat kernel bounds in a doubling metric-measure space.", "Proposition 11 For all $R > 0$ , $\\alpha \\ge 0$ , $\\beta > \\alpha $ , and for all functions $F : \\mathbb {R}\\rightarrow such that $ suppF [-4R2,4R2]$,\\begin{equation}\\operatornamewithlimits{\\mathrm {ess\\,sup}}_{y \\in \\mathrm {X}} |B(y,R^{-1})|^{1/2} \\Vert (1+ R\\varrho (\\cdot ,y))^\\alpha \\operatorname{\\mathcal {K}}_F(\\cdot ,y)\\Vert _2 \\le C_{\\alpha ,\\beta } \\Vert F_{(R^2)}\\Vert _{W_\\infty ^\\beta },\\end{equation}where the constant $ C,$ does not depend on $ R$.", "If in addition $ > +Q/2$, then\\begin{equation}\\operatornamewithlimits{\\mathrm {ess\\,sup}}_{y \\in \\mathrm {X}} \\Vert (1+ R\\varrho (\\cdot ,y))^\\alpha \\operatorname{\\mathcal {K}}_F(\\cdot ,y)\\Vert _1 \\le C_{\\alpha ,\\beta }\\Vert F_{(R^2)} \\Vert _{W_\\infty ^\\beta },\\end{equation}where again $ C,$ does not depend on $ R$.$ For the first inequality, see [10] or [8]; note that the statement in [8] seems to require that the multiplier $F$ is supported away from the origin, but its proof clarifies that this is not necessary, because here we do not perform the change of variable $\\lambda \\mapsto \\sqrt{\\lambda }$ in the multiplier function.", "The second inequality is an immediate consequence of the first, via Hölder's inequality and [8].", "These inequalities can be improved by means of the weighted Plancherel estimates.", "For this, some properties of the weight functions $w_R$ are needed.", "Lemma 12 Suppose that $0 \\le \\gamma < \\min \\lbrace d_1,d_2\\rbrace /2$ and $\\beta > Q/2 - \\alpha $ .", "For all $y \\in \\mathrm {X}$ and $R > 0$ , $\\int _\\mathrm {X}(1+w_R(x,y))^{-2\\gamma } (1 + R \\varrho (x,y))^{-2\\beta } \\,dx \\le C_{\\alpha ,\\beta } |B(y,R^{-1})|.$ Moreover, for all $x,y \\in \\mathrm {X}$ and $R > 0$ , $w_R(x,y) \\le C (1 + R \\varrho (x,y)).$ By exploiting the homogeneity properties of the distance $\\varrho $ and the weights $w_R$ , we may suppose that $R = 1$ .", "Then (REF ) immediately follows from the fact that $\\min \\lbrace 1,|y^{\\prime }|^{-1}\\rbrace |x^{\\prime }| \\le 1 + |x^{\\prime } - y^{\\prime }| \\le C (1 + \\varrho (x,y)),$ by (REF ).", "To show (REF ) we note that by translation-invariance we may also suppose that $y^{\\prime \\prime } = 0$ .", "By (REF ), we must then prove that $\\int _\\mathrm {X}\\left(1+\\frac{|x^{\\prime }-y^{\\prime }|}{1+|y^{\\prime }|}\\right)^{-2\\gamma } (1 + \\varrho (x,y))^{-2\\beta } \\,dx \\le C_{\\gamma ,\\beta } (1 + |y^{\\prime }|)^{d_2}.$ We split the integral into two parts, according to the asymptotics (REF ).", "In the region $\\mathrm {X}_1 = \\lbrace x \\in \\mathrm {X}\\,:\\,|x^{\\prime \\prime }|^{1/2} \\ge |x^{\\prime }| + |y^{\\prime }|\\rbrace $ , we decompose $\\beta = \\beta _1 + \\beta _2$ so that $\\beta _1 > d_1/2-\\gamma $ and $\\beta _2 > d_2$ , whence the integral on $\\mathrm {X}_1$ is at most $(1+|y^{\\prime }|)^{2\\gamma } \\int _{\\mathbb {R}^{d_1}} (1+|x^{\\prime }-y^{\\prime }|)^{-2(\\gamma +\\beta _1)} \\,dx^{\\prime } \\int _{\\mathbb {R}^{d_2}} (1+|x^{\\prime \\prime }|^{1/2})^{-2\\beta _2} \\,dx^{\\prime \\prime }.$ In the region $\\mathrm {X}_2 = \\lbrace x \\in \\mathrm {X}\\,:\\,|x^{\\prime \\prime }|^{1/2} < |x^{\\prime }| + |y^{\\prime }|\\rbrace $ , instead, we decompose $\\beta = \\tilde{\\beta }_1+\\tilde{\\beta }_2$ so that $\\tilde{\\beta }_1 > (d_1+d_2)/2-\\gamma $ and $\\tilde{\\beta }_2 > d_2/2$ , whence the integral on $\\mathrm {X}_2$ is at most $\\begin{split}\\int _\\mathrm {X}&\\left(1+\\frac{|x^{\\prime }-y^{\\prime }|}{1+|y^{\\prime }|}\\right)^{-2\\gamma } (1+|x^{\\prime }-y^{\\prime }|)^{-2\\tilde{\\beta }_1} \\left(1 + \\frac{|x^{\\prime \\prime }|}{|x^{\\prime }|+|y^{\\prime }|}\\right)^{-2\\tilde{\\beta }_2} \\,dx \\\\&\\le C_{\\gamma ,\\beta } \\int _{\\mathbb {R}^{d_1}} \\left(1+\\frac{|u|}{1+|y^{\\prime }|}\\right)^{-2\\gamma } (1+|u|)^{-2\\tilde{\\beta }_1} (|u+y^{\\prime }|+|y^{\\prime }|)^{d_2} \\,du \\\\&\\le C_{\\gamma ,\\beta } \\left( (1+|y^{\\prime }|)^{2\\gamma } \\int _{\\mathbb {R}^{d_1}} (1+|u|)^{-2\\nu } \\,du + |y^{\\prime }|^{d_2} \\int _{\\mathbb {R}^{d_1}} (1+|u|)^{-2\\tilde{\\beta }_1} \\,du \\right),\\end{split}$ where $\\nu = \\tilde{\\beta }_1 + \\gamma - d_2/2 > d_1/2$ .", "The conclusion follows.", "A strengthened weighted version of () can now be obtained using the Mauceri-Meda interpolation trick [17] (see also [15] and [8]).", "Proposition 13 For all $R > 0$ , $\\alpha \\ge 0$ , $\\beta > \\alpha $ , $\\gamma \\in \\left[0,d_2/2\\right[$ , and for all functions $F : \\mathbb {R}\\rightarrow such that $ suppF [R2,4R2 ]$,\\begin{multline}\\operatornamewithlimits{\\mathrm {ess\\,sup}}_{y \\in \\mathrm {X}} |B(y,R^{-1})|^{1/2} \\, \\Vert (1 + R\\varrho (\\cdot ,y))^\\alpha (1+w_R(\\cdot ,y))^\\gamma \\operatorname{\\mathcal {K}}_{F(L)}(\\cdot ,y)\\Vert _2 \\\\\\le C_{\\alpha ,\\beta ,\\gamma } \\Vert F_{(R^2)} \\Vert _{W_2^\\beta },\\end{multline}where the constant $ C,,$ does not depend on $ R$.$ The estimate (), together with (REF ) and a Sobolev embedding, immediately implies () in the case $\\beta > \\alpha + d_2/2 + 1/2$ .", "On the other hand, in the case $\\alpha = 0$ , () is given by Proposition REF for all $\\beta > 0$ .", "The conclusion then follows by interpolation (see, e.g., [3], [5]).", "An alternative proof of Proposition REF can be obtained using minor adjustments of the technique developed in [4].", "Let $D = Q - \\min \\lbrace d_1,d_2\\rbrace = \\max \\lbrace d_1+d_2,2d_2\\rbrace $ .", "Proposition REF , together with (REF ) and Hölder's inequality, then yields an improvement of ().", "Corollary 14 For all $R > 0$ , $\\alpha \\ge 0$ , $\\beta > \\alpha + D/2$ , and for all functions $F : \\mathbb {R}\\rightarrow such that $ suppF [R2,4R2 ]$,\\begin{equation}\\operatornamewithlimits{\\mathrm {ess\\,sup}}_{y \\in \\mathrm {X}} \\Vert (1 + R\\varrho (\\cdot ,y))^\\alpha \\operatorname{\\mathcal {K}}_{F(L)}(\\cdot ,y)\\Vert _1\\le C_{\\alpha ,\\beta } \\Vert F_{(R^2)}\\Vert _{W_2^\\beta },\\end{equation}where the constant $ C,$ does not depend on $ R$.", "In particular, under the same hypotheses,\\begin{equation}\\operatornamewithlimits{\\mathrm {ess\\,sup}}_{y \\in \\mathrm {X}} \\int _{\\mathrm {X}\\setminus B(y,r)} |\\operatorname{\\mathcal {K}}_{F(L)}(x,y)| \\,dx\\le C_{\\alpha ,\\beta } (1+rR)^{-\\alpha } \\Vert F_{(R^2)} \\Vert _{W_2^\\beta }.\\end{equation}$ We are finally able to prove our main results.", "We can follow the lines of the proof of [8], where the inequality (4.18) there is replaced by our ().", "Choose $\\beta \\in \\left]D/2, \\kappa +1/2 \\right[$ .", "Let $\\eta \\in C^\\infty _c(\\mathbb {R})$ be supported in $\\left[-1/2,1/2\\right]$ and equal to 1 in a neighborhood of the origin, and set $F(\\lambda ) = (1-|\\lambda |)_+^\\kappa $ .", "The function $\\eta F$ is smooth and compactly supported, while $(1-\\eta )F$ is compactly supported away from the origin and belongs to $W_2^\\beta $ .", "The inequalities () and () then imply that the operators $\\eta (tL) F(tL)$ and $(1-\\eta (tL)) F(tL)$ are bounded on $L^1(\\mathrm {X})$ , uniformly in $t > 0$ , and the same holds for their sum $(1-tL)_+^\\kappa $ .", "The conclusion follows by self-adjointness and interpolation." ], [ "Sharpness of the obtained results", "The aim of this section is to show that, if $d_1 \\ge d_2$ , then the result in Theorem REF is sharp.", "More precisely, if $d_1 \\ge d_2$ and $s<D/2=(d_1+d_2)/2$ , then the first inequality in () cannot hold.", "Indeed, if we consider the functions $H_t(\\lambda )=\\lambda ^{it}$ , then, for $t>1$ , $C \\Vert H_t \\Vert _{MW_2^s} \\sim t^s.$ On the other hand, we make the following observation.", "Proposition 15 Suppose that $L$ is the Grushin operator acting on $\\mathrm {X}= \\mathbb {R}^{d_1}\\times \\mathbb {R}^{d_2}$ .", "Then the following lower bounds holds: $\\Vert H_t(L)\\Vert _{L^1\\rightarrow L^{1,w}}=\\Vert L^{it}\\Vert _{L^1\\rightarrow L^{1,w}} \\ge C (1+|t|)^{(d_1+d_2)/2}$ for all $t>0$ .", "Because the Grushin operator is elliptic on $\\mathrm {X}_0 = \\lbrace x \\in \\mathrm {X}\\,:\\,x^{\\prime } \\ne 0\\rbrace $ , one can use the same argument as in [22] to prove that, for all $y \\in \\mathrm {X}_0$ , $|p_t(x,y) - |y^{\\prime }|^{- d_2}(4\\pi t)^{-(d_1+d_2)/2}{\\rm e}^{-\\varrho (x,y)^2/4t}|\\le Ct^{1/2}t^{-(d_1+d_2)/2}$ for all $x$ in a small neighborhood of $y$ and all $t \\in \\left]0,1 \\right[$ .", "Here $p_t=\\operatorname{\\mathcal {K}}_{\\exp (-tL)}$ is the heat kernel corresponding to the Grushin operator.", "The rest of the argument is the same as in [22], so we skip it here.", "The aim of this section is to show that, if $d_1 \\ge d_2$ , then the result in Theorem REF is sharp.", "More precisely, if $d_1 \\ge d_2$ and $s<D/2=(d_1+d_2)/2$ , then the first inequality in () cannot hold.", "Indeed, if we consider the functions $H_t(\\lambda )=\\lambda ^{it}$ , then, for $t>1$ , $C \\Vert H_t \\Vert _{MW_2^s} \\sim t^s.$ On the other hand, we make the following observation.", "Proposition 15 Suppose that $L$ is the Grushin operator acting on $\\mathrm {X}= \\mathbb {R}^{d_1}\\times \\mathbb {R}^{d_2}$ .", "Then the following lower bounds holds: $\\Vert H_t(L)\\Vert _{L^1\\rightarrow L^{1,w}}=\\Vert L^{it}\\Vert _{L^1\\rightarrow L^{1,w}} \\ge C (1+|t|)^{(d_1+d_2)/2}$ for all $t>0$ .", "Because the Grushin operator is elliptic on $\\mathrm {X}_0 = \\lbrace x \\in \\mathrm {X}\\,:\\,x^{\\prime } \\ne 0\\rbrace $ , one can use the same argument as in [22] to prove that, for all $y \\in \\mathrm {X}_0$ , $|p_t(x,y) - |y^{\\prime }|^{- d_2}(4\\pi t)^{-(d_1+d_2)/2}{\\rm e}^{-\\varrho (x,y)^2/4t}|\\le Ct^{1/2}t^{-(d_1+d_2)/2}$ for all $x$ in a small neighborhood of $y$ and all $t \\in \\left]0,1 \\right[$ .", "Here $p_t=\\operatorname{\\mathcal {K}}_{\\exp (-tL)}$ is the heat kernel corresponding to the Grushin operator.", "The rest of the argument is the same as in [22], so we skip it here." ] ]

[ [ "Dynamical magnetic anisotropy and quantum phase transitions in a\n vibrating spin-1 molecular junction" ], [ "Abstract We study the electronic transport through a spin-1 molecule in which mechanical stretching produces a magnetic anisotropy.", "In this type of device, a vibron mode along the stretching axis will couple naturally to the molecular spin.", "We consider a single molecular vibrational mode and find that the electron-vibron interaction induces an effective correction to the magnetic anisotropy that shifts the ground state of the device toward a non-Fermi liquid phase.", "A transition into a Fermi liquid phase could then be achieved, by means of mechanical stretching, passing through an underscreened spin-1 Kondo regime.", "We present numerical renormalization group results for the differential conductance, the spectral density, and the magnetic susceptibility across the transition." ], [ "Introduction", "The electronic properties of nanostructures depend critically on their symmetries.", "As a consequence, the ability to modify these microscopic symmetries makes it possible to drive the system through different physical regimes at will, possibly resulting in quantum phase transitions (QPTs) as the system visits different ground states.", "[1] Quantum dots[2], [3], [4] and molecular devices consisting of complex molecules deposited on metallic break junctions[5], [6], [7], [8] are good examples of systems where physical regimes can be explored by tuning their parameters.", "The electronic transport through molecular devices is very sensitive to the hybridization of the molecular energy levels with the bands of the metallic leads in the break junction, as well as to electron–electron (e-e) and electron–vibron interactions within the molecule.", "[9], [10], [11], [12], [13], [14], [15], , , [18], [19], [17], [20] Success has been achieved in the past few years in controlling the properties of molecular junctions through tuning of the molecular levels by different means.", "It has been shown, for example, that a gate voltage is capable of inducing a singlet–triplet transition in a $\\text{C}_{60}$ (buckyball) molecule trapped between metallic leads, due to distinct hybridizations of these states with the electronic states of the leads.", "[5] In recent experiments by Parks et al.,[6] it was shown that mechanical stretching of the spin–1 molecule $\\text{Co(tpy-SH)}_2$ (4'-mercapto-2, 2':6',2\"-terpyridine) along the transport axis in a break junction setup, can be used to control the magnetic properties of the molecule.", "The stretching induces a splitting of the spin–1 triplet ground state,[21], [22], [23], [24] raising the energy of the doublet with spin projection $S_z = \\pm 1$ by as much as 4 meV with respect to the state of $S_z = 0$ ,[6] and leaving the latter as the molecular ground state.", "This magnetic anisotropy is critical to the low–temperature transport through the molecular junction in this experiment.", "In the absence of anisotropy, the system exhibits an underscreened spin–1 Kondo effect, signaled by enhanced conductivity for temperatures below the Kondo temperature $T_K^0$ .", "[25] Any positive (hard–axis) anisotropy breaks the ground state degeneracy of the isolated molecule and drives the system into a Fermi liquid ground state with an associated low conductance.", "[26], , [28] The converse case of negative (easy–axis) anisotropy could be reached, in principle, by compressing the molecule.", "This situation would set the $S_z = \\pm 1$ doublet as the molecular ground state, yielding non–Fermi–liquid (NFL) behavior.", "[29], [30], [31], [26] In this context, visiting different ground states becomes a matter of adjusting the magnetic anisotropy.", "The system will go from an ordinary Fermi liquid (FL) in the hard-axis regime to a NFL in the easy-axis regime, passing through an underscreened Kondo ground state[31] —a singular Fermi liquid (SFL)— when the triplet is exactly degenerate.", "[32] These different regimes and the associated QPTs can be studied as function of a single parameter: the magnetic anisotropy.", "All of the effects described above can be caused by static deformations of the molecule.", "Moreover, it is to be expected that dynamical effects may arise via an analogous coupling between the spin and the mechanical degrees of freedom of the molecule.", "[33] As we show below, a coupling between molecular vibrations and spin will induce a deformation in the molecular ground state, opening access to an easy–axis regime.", "With this motivation, in this paper we study a model that encompasses the anisotropy regimes described above, and considers in addition the mechanical degrees of freedom of the molecule through a vibrational mode.", "In the same way as the static deformation, vibrations along the axis couple naturally to the spin projection of the molecule.", "Figure: (Color online) Phase diagram of a deformable spin–1 molecular break junction in the presence of a local vibrational mode.", "A rich variety of phases can be found at temperatures below T K T_K, at which Kondo physics dominates the transport properties of the system.", "The effective anisotropy A eff A_{\\text{eff}}, induced by both static and dynamic deformations of the molecule, drives the system through different quantum phases.In the case of an isolated molecule, we find that the coupling to the vibrational mode indeed opens access to the easy–axis regime, i.e., the spin–1 doublet of $S_z = \\pm 1$ becomes the ground state.", "When the molecule is coupled to leads, we find that the magnetic anisotropy is further renormalized.", "The resulting effective anisotropy could be tuned to explore a variety of ground states which we will discuss below.", "We studied the different highly–correlated regimes of the model by means of numerical renormalization group (NRG) calculations.", "[34], [35], [36] The results are summarized in the phase diagram shown in Fig.", "REF , to which we will come back later." ], [ "Model", "We model the molecular device shown in Fig.", "REF (a) as depicted in REF (b).", "The Hamiltonian of the spin–1 molecule can be represented by a 2–orbital model of the form $H_{0} = \\sum _{i=\\text{a, b}}\\left( \\varepsilon \\,n_{i} + U\\,n_{i\\uparrow }n_{i\\downarrow } \\right) - J\\,\\vec{S}_a\\cdot \\vec{S}_b.$ Here, $i=a,b$ are the two degenerate molecular orbitals with energy $\\varepsilon $ and intra-orbital Coulomb repulsion $U$ , $n_{i\\sigma } = d_{i\\sigma }^{\\dagger }d_{i\\sigma }$ and $d_{i\\sigma }^{\\dagger }$ ($d_{i\\sigma }$ ) the creation (annihilation) operator of the corresponding orbital; $n_{i} = n_{i\\uparrow } + n_{i\\downarrow }$ , and $\\vec{S}_{i}$ is the spin operator asociated to orbital $i$ .", "For simplicity, we consider the electron–hole (e-h) symmetric case where $\\varepsilon = -U/2$ and the Fermi level of the leads is $\\varepsilon _F=0$ .", "The ferromagnetic ($J>0$ ) coupling enforces Hund's rule, setting the spin–1 triplet as the ground state of the molecule.", "These states are defined as $\\begin{split}\\left| T,\\,+1 \\right> = \\left| \\uparrow _a\\,\\uparrow _b \\right>\\quad , \\quad \\left| T,\\,-1 \\right> = \\left| \\downarrow _a\\,\\downarrow _b \\right>&,\\\\\\left| T,\\,0 \\right> = \\frac{1}{\\sqrt{2}}\\Big ( \\left| \\uparrow _a\\,\\downarrow _b \\right> + \\left| \\downarrow _a\\,\\uparrow _b \\right> \\Big )&, \\\\\\end{split}$ where $\\left| \\sigma _a\\,\\sigma _b \\right>$ are the states with one electron on each orbital.", "Figure: (Color online) Spin–1, deformable molecular device.", "(a) A break junction supports a spin–1 molecule that can be stretched by mechanical means.", "(b) The molecule is modeled by two coupled orbitals, only one of them connected to metallic leads.", "The spin–1 regime is enforced by an inter–orbital ferromagnetic coupling.", "Magnetic anisotropy, induced by static and dynamic (vibrational) stretching of the molecule, is accounted for by the term A(z)S z 2 A(z)S_z^2 in the Hamiltonian.", "(c) Lowest lying level energies of the isolated molecule, obtained for different values of the effective static anisotropy.The deformation–induced anisotropy arising from the breaking of the octahedral symmetry can be written as a function of the elongation along the $\\hat{z}$ axis, as $A(z)S_z^2 $ .", "A purely static model [$A(z)=A_0$ ] of this kind was applied in reference [cornagliaepl2011] to describe the system of reference [parksscience2010].", "For small oscillations around the equilibrium position $z_0$ we take $A(z)= (A_0 + A_1\\,\\delta z)\\,S_z^2$ , where $\\delta z=z-z_0$ .", "The coupling $A_0$ represents the static deformation, whereas $A_1$ arises due to the e-ph coupling.", "We choose units such that $\\delta z=a+a^{\\dagger }$ , where $a$ and $a^\\dagger $ are phonon operators.", "The Hamiltonian of the molecule including the electron–vibron interaction becomes $H_{\\text{M}} = H_0 + A_0\\,S_z^2 + A_1\\,S_z^2\\left(a + a^{\\dagger } \\right) + \\omega _0\\,a^{\\dagger }a,$ with $\\omega _0$ the phonon frequency.", "By means of the unitary transformation $\\tilde{H}_{\\text{M}}=\\mathcal {U}\\,H_{\\text{M}}\\,\\mathcal {U}^{\\dagger } \\quad ;\\quad \\mathcal {U} = \\exp {\\left\\lbrace -\\frac{A_1}{\\omega _0}S_z^2\\left(a - a^{\\dagger } \\right) \\right\\rbrace },$ we diagonalize (REF ), which adopts the form $\\tilde{H}_{\\text{M}} = H_0 + \\tilde{A} S_z^2 + \\omega _0\\,b^{\\dagger }b,$ where $b=a+(A_1/\\omega _0)\\, S_z^2$ is a displaced phonon operator.", "Note that the magnetic anisotropy of the isolated molecule $\\tilde{A} = A_0 - A_1^2/\\omega _0,$ is negative in the absence of a static distortion ($A_0=0$ ).", "The lowest energy eigenstates of (REF ) for each molecular spin projection are $\\left| T,\\,\\pm 1;\\,\\tilde{0} \\right> = \\left| T,\\,\\pm 1 \\right>\\left| \\tilde{0} \\right),\\quad \\left| T,\\,0;\\,0 \\right> = \\left| T,\\,0 \\right>\\left| 0 \\right),$ where $\\left| n \\right)$ is an eigenvector of the operator $a^\\dagger a$ , and $\\left| \\tilde{n} \\right)=\\text{e}^{\\frac{A_1}{\\omega _0}(a - a^{\\dagger })}\\left| n \\right)$ is an eigenvector of $b^\\dagger b$ , with eigenvalue $n$ .", "We now leave the isolated molecule (molecular limit) to explore the consequences of coupling the metallic leads to the molecular orbital $a$ , through the hybridization term $H_{\\text{M-E}} = \\sqrt{2}\\,V\\sum _{\\vec{k},\\sigma }\\left( d_{a\\sigma }^{\\dagger }c_{\\vec{k}\\sigma } + \\text{H.c.} \\right).$ This “hanging–level” arrangement correctly describes the low–energy behavior of the system, and other configurations can be related to it by means of a level rotation, so there is no loss of generality.", "[23] Because we assume identical right (R) and left (L) leads and couplings, we have defined the lead–symmetric operators $c_{\\vec{k}\\sigma } \\equiv \\left( c_{\\text{L}\\vec{k}\\sigma } + c_{\\text{R}\\vec{k}\\sigma } \\right)/\\sqrt{2},$ which are the only combinations that couple to the molecule.", "Their anti–symmetric counterparts contribute only a constant energy term to the Hamiltonian.", "In the displaced basis the full transformed Hamiltonian reads $\\begin{split}\\tilde{H} = H_0 &+ \\tilde{A}\\,S_z^2 + \\omega _0\\,b^{\\dagger }b \\\\&+ \\sum _{\\vec{k},\\sigma }\\left( \\tilde{V}_{\\sigma }\\, c_{\\vec{k}\\sigma }^{\\dagger }d_{a\\sigma } + \\text{H.c.} \\right),\\end{split}$ with a hybridization operator $\\tilde{V}_{\\sigma } = \\sqrt{2}V\\,\\exp \\left\\lbrace \\frac{A_1}{4\\omega _0}\\left(a - a^{\\dagger } \\right)\\left[1 - 2n_{a\\bar{\\sigma }} +4\\sigma S_z^b \\right]\\right\\rbrace .$ Although the expression for the transformed Hamiltonain is more complicated than the starting one, we can appreciate an interesting feature.", "Namely, that it couples the $\\pm 1$ and 0 spin projections of the triplet to the electronic states of the band with different strengths.", "As we will see below, this asymmetry leads to a contribution to the total magnetic anisotropy." ], [ "Analytical results", "To expose the splitting of the triplet due to the spin dependent couplings to the band, we perform a Schrieffer–Wolff transformation[37] in the limit of large $U$ and $J$ .", "Here we set $\\tilde{A}=0$ so that the isolated molecule ground state is triply degenerate.", "We obtain an anisotropic Kondo Hamiltonian[38] $H_{K} = A_d S_z^2 + J_K^{\\parallel }s_z\\,S_z + J_K^{\\perp }\\left( s_x\\,S_x + s_y\\,S_y \\right),$ with exchange couplings $ J_K^{\\parallel } = \\sum _{n=0}^{\\infty }\\frac{4V^2\\left|\\left( \\tilde{0} \\right|\\left.", "n \\right) \\right|^2}{\\frac{U}{2} + \\frac{J}{4} + n\\omega _0},\\quad J_K^{\\perp } = \\text{e}^{-\\left(\\frac{A_1}{\\omega _0} \\right)^2/2} J_K^0,$ where $J_K^0=4V^2/(\\frac{U}{2} + \\frac{J}{4})$ is the Kondo coupling in the absence of e-ph coupling ($A_1=0$ ), and $A_d = 4V^2\\left(\\frac{1}{\\frac{U}{2} + \\frac{J}{4}} - \\sum _{n}\\frac{\\left|\\left( \\tilde{0} \\right|\\left.", "n \\right) \\right|^2}{\\frac{U}{2} + \\frac{J}{4} + n\\omega _0} \\right) > 0,$ where $\\left| \\left( \\tilde{0} \\right|\\left.", "n \\right) \\right|^2 = \\left(\\frac{A_1}{\\omega _0}\\right)^{2n}\\frac{\\text{e}^{-\\left(\\frac{A_1}{\\omega _0} \\right)^2}}{n!", "}.$ The anisotropy term $A_d$ is a consequence of the hybridization shifting the state $\\left| T,\\,0;\\,0 \\right>$ down in energy further than the states $\\left| T,\\,\\pm 1;\\,0 \\right>$ , for any state of the band.", "The e-ph interaction induces a reduction of both Kondo couplings that become anisotropic ($J_K^\\perp , J_K^\\parallel < J_K^0$ ).", "In the experimentaly relevant case $U\\gg \\omega _0$ we have $J_K^\\perp < J_K^\\parallel $ .", "For weak e-ph interaction ($A_1/\\omega _0\\ll 1$ ) and $U\\gg \\omega _0$ we obtain: $J_K^\\parallel /J_K^0&\\sim & 1-\\frac{\\omega _0}{\\frac{J}{4}+\\frac{U}{2}}\\left(\\frac{A_1}{\\omega _0}\\right)^2,\\\\J_K^\\perp /J_K^0&\\sim & 1-\\frac{1}{2}\\left(\\frac{A_1}{\\omega _0}\\right)^2,\\\\A_d/J_K^0&\\sim & \\frac{\\omega _0}{\\frac{J}{4}+\\frac{U}{2}}\\left(\\frac{A_1}{\\omega _0}\\right)^2.$ The expresions for $J_K^\\perp $ and $A_d$ are also valid in the strong e-ph coupling regime ($A_1 \\gg \\omega _0$ ), provided $A_1^2/\\omega _0\\ll U$ , while $J_K^\\perp $ is exponentially supressed [see Eq.", "(REF )] making the Kondo couplings strongly anisotropic.", "As a result of the correction $A_d$ , the degeneracy of the triplet is broken preventing the underscreened spin-1 Kondo effect from taking place at $\\tilde{A}=0$ .", "Therefore, a shift in the static anisotropy will be needed to restore the full spin-1 degeneracy.", "It is also important to point out that the anisotropy of the Kondo couplings leads to effects similar to those obtained by a static magnetic anisotropy term ($A_0$ ).", "In our case of $J_K^{\\parallel } > J_K^{\\perp } > 0$ , this anisotropy results in a shift toward the easy–axis regime.", "[39], [26] To summarize, the exact solution of the isolated spin–1 molecule, along with a perturbative analysis to second order in the hybridization, suggest that the main effect of the electron–phonon interaction considered in this break junction setup is to reduce the Kondo couplings, and to introduce corrections to the magnetic anisotropy.", "The different contributions to the magnetic anisotropy result in an effective anisotropy given by $A_{\\text{eff}}=A_0-A_1^2/\\omega _0+ A_d + \\delta A,$ where $\\delta A$ stems from the anisotropy of the Kondo couplings." ], [ "Numerical results", "We analyze the different magnetic anisotropy regimes of the system as a function of $A_0$ , $A_1$ and $\\omega _0$ , by means of NRG calculations.", "In this non–perturbative method, the continuum of electronic states in the leads is logarithmically discretized according to a parameter $\\Lambda > 1$ .", "It is then mapped onto a chain of states with exponentially decaying hopping terms that is diagonalized iteratively, defining a renormalization group transformation to the low energy spectrum.", "[35] All calculations shown below were obtained with discretization factor $\\Lambda = 2.5$ , with no fewer than 1200 states kept, and a constant hybridization $\\Gamma = \\pi \\left|V\\right|^2/2D=0.005\\,D $ , where $D$ is the half–bandwidth.", "We choose the parameters $\\varepsilon = -U/2 = -5\\,\\Gamma $ , with ferromagnetic coupling set to $J = 0.2\\,\\Gamma $ .", "Our conclusions, however, remain qualitatively unchanged for other parameters within the Kondo regime (such as $J > \\Gamma $ ).", "The spectral densities were calculated following Ref.", "[BullaCostiVollhardtPRB2001].", "For $A_0 = A_1 = 0$ , the system is expected to have an underscreened spin–1 Kondo ground state; we have verified this with NRG calculations.", "This regime serves as a reference for comparison with our results in other regions of parameter space.", "The infinite–dimensional boson sector of the Hilbert space has to be truncated for the numerical calculation.", "We analyze the regime of $\\omega _0 \\gtrsim T_K^0$ and $A_1/\\omega _0<1$ , so that relevant phonon excitations include transitions from the ground state with boson component given by $\\left| \\tilde{0} \\right)$ to states with occupation $n$ ; these have amplitudes given by Eq.", "(REF ).", "This Poisson distribution peaks at $n = A_1/\\omega _0<1$ and then falls to zero rapidly, suggesting a cutoff at phonon occupation $n \\sim 1$ .Within the explored range of $\\omega _0$ and $A_1$ , we found that a maximum occupation of 3 phonons in our NRG runs correctly describes the behavior of our system.", "This was explicitly verified in our numerical results, and is in accordance with the analysis of [hewsonjphyscondensmatter2002].", "Figure: (Color online) (a) Spectral density at orbital aa for A 0 =A 1 =0A_0 = A_1 = 0 (reference system).", "The screening of the molecular triplet ground state by the band is characterized by the sharp Kondo resonance of width T K 0 T_K^0 at the Fermi level (ω=0\\omega = 0), followed by shoulders at approximately the singlet–triplet (red dashed lines), and the charge (blue dashed lines in the inset) excitation energies of the isolated molecule.", "The Kondo effect yields unitary conductance at low (≪T K 0 \\ll T_K^0) temperatures.", "(b) Same as (a), but for finite magnetic anisotropy (A 0 =ω 0 =5T K 0 A_0 = \\omega _0 = 5\\,T_K^0, A 1 A_1 from 2T K 0 2\\,T_K^0 to 11T K 0 11\\,T_K^0 ).", "In the easy axis regime of A eff <0A_{\\text{eff}} < 0 (red curves), the system presents a finite amplitude of the spectral density at the Fermi level.", "The hard–axis regime (A eff >0A_{\\text{eff}} > 0; blue curves) presents a dip at the Fermi level, keeping the system in a Coulomb blockade.", "The underscreened Kondo state is recovered when A eff =0A_{\\text{eff}}=0 (black curve)." ], [ "Transport properties at low temperatures", "The low–temperature, linear transport properties of the system can be obtained in terms of the spectral density at orbital $a$ , $\\rho _a(\\omega )=\\sum _{\\sigma }\\rho _a^{\\sigma }(\\omega )$ , which relates to the conductance through[42], [43] $\\frac{G(T)}{G_0} = \\pi \\Gamma \\int _{-\\infty }^{\\infty } \\,\\mathrm {d}\\omega \\left(-\\frac{\\partial f(\\omega ,T)}{\\partial \\omega } \\right)\\,\\rho _{a}(\\omega ),$ where $G_0 = 2e^2/h$ , and $f(\\omega ,\\, T)$ is the Fermi distribution.", "In the reference system, where $A_0 = A_1 = 0$ , $\\rho _a(\\omega )$ shows a sharp Kondo resonance of width $T_K^0 \\sim 10^{-5}\\,D$ at the Fermi level[25], [44] ($\\varepsilon _F$ , set here to zero) as shown in Fig.", "REF (a).", "At low temperatures the conductance reaches the unitary limit.", "Shoulders in the reference spectral density appear at approximately the singlet–triplet excitation energy, $\\omega \\sim \\pm J$ , and peaks at the charge excitation energies $\\omega \\sim \\pm U/2$ .", "With this in mind we move on to analyze Fig.", "REF (b) in the hard–axis regime ($A_{\\text{eff}} > 0$ ).", "The spectral density has two peaks close to the Fermi level separated by a sharp dip, which according to (REF ) puts the device in a transport blockade with a vanishing zero–bias conductance at $T \\rightarrow 0$ .", "In the easy–axis regime ($A_{\\text{eff}} < 0$ ), conduction electrons will scatter off the molecule's ground state,[26] the doublet $\\left| T,\\pm 1 \\right>$ .", "At zero temperature $\\rho _a(\\varepsilon _F)$ presents a non–zero amplitude, which means that the system will conduct at zero bias, although not unitarily as in the case of zero anisotropy.", "Figure: (Color online) Conductance as a function of temperature, for different values of A eff A_{\\text{eff}}.", "In (a), varying A 1 A_1 from 2T K 0 2\\,T_K^0 to 12T K 0 12\\,T_K^0 for ω 0 =A 0 =5T K 0 \\omega _0 = \\,A_0 = 5\\,T_K^0, and in (b) by varying A 0 A_0 from 4T K 0 4\\,T_K^0 to 9T K 0 9\\,T_K^0 for A 1 =ω 0 =5T K 0 A_1 = \\omega _0 = 5\\,T_K^0.", "In (c), as a function of A 0 A_0 for different isotherms.", "The zero–temperature conductance vanishes for positive anisotropy, due to the lack of available states for transport in the molecule at the Fermi level (see Fig.", ").", "As A eff →0A_{\\text{eff}} \\rightarrow 0 the conductance curves vary as indicated by the blue arrows until the unitary limit is reached, signaled in the plot by a plateau that extends to lower temperatures.", "Unitarity is then lost due to many–body scattering of the leads' electrons off the S z =±1S_z = \\pm 1 degenerate ground state of the molecule at negative anisotropies.", "As A eff A_{\\text{eff}} goes from zero to negative the conductance curves follow the trend indicated by the red arrow.", "In (c) we show isotherms for the conductance as a function of A 0 A_0 (corresponding to the curves in (b)), which can be directly related to experiments in which the conductance is measured in a stretchable device, at constant temperature.", "The quantum critical point becomes apparent at very low temperatures, where a sudden drop in conductance signals the change of sign of A eff A_{\\text{eff}}.", "The Kondo temperatures of these cases are lower than the reference scale T K 0 T_K^0 by one order of magnitude, as can be seen by the onset of the conductance plateaus below T=T K 0 T = T_K^0.The different regimes described above are shown in Fig.", "REF (b) from NRG calculations in which $A_{\\text{eff}}$ was tuned by varying $A_1$ , at constant static anisotropy $A_0$ and phonon frequency $\\omega _0$ .", "Curves of conductance as a function of temperature, corresponding to the parameters of Fig.", "REF (b), are shown in Fig.", "REF (a), whereas in Fig.", "REF (b) the same regimes are explored by varying $A_0$ instead.", "As mentioned above, tuning $A_0$ in molecular devices has been achieved experimentally by Parks et al.,[6] through stretching.", "Assuming that the value of $A_0$ increases from zero by stretching the molecule, it could be used to change the sign of the effective magnetic anisotropy.", "This is depicted in Fig.", "REF (b), where we begin with a positive value of $A_0$ such that the effective anisotropy is negative, and then increase $A_0$ until we reach the hard–axis regime.", "Experiments such as those of reference [parksscience2010] may perhaps be more easily related to Fig.", "REF (c), where we show the zero–bias conductance at constant temperature, for different values of $A_0$ .", "The sudden drop in conductance is a clear signature of the transition from hard–axis to easy–axis behavior at zero temperature." ], [ "Effective anisotropy and underscreened Kondo effect restoration", "As we will see below, the numerical results confirm that the main effect of the e–ph coupling considered is to renormalize the magnetic anisotropy toward the hard–axis regime and reduce the Kondo exchange couplings.", "In fact, the static anisotropy term can be tuned to recover the underscreened spin–1 Kondo effect.", "We define $\\Delta = A_{\\text{eff}}-\\tilde{A}$ and calculate it numerically through an analysis of the effective magnetic–moment–squared, $\\mu ^2$ , at low temperature.", "[36] The onset of Kondo screening is signaled by a drop in the molecule's magnetic moment at temperatures below some $T_K(A_0,\\, A_1,\\, \\omega _0)$ .", "Notice that $T_K < T_K^0$ due to the reduced Kondo couplings [see Eqs.", "(REF )].", "An example of this is shown in Fig.", "REF (b).", "As the molecule is coupled to a single conduction electron channel, the electrons in the leads are able to screen one half of the molecule's spin, leaving an asymptotically free spin–$1/2$ object[31], [45] and producing a plateau $\\mu ^2(T<T_K) = 0.25$ , as in Fig.", "REF .", "Further screening associated to a second–stage Kondo effect is known to arise when a net positive anisotropy $0< A_{\\text{eff}}\\ll T_K$ is present.", "We have $\\mu ^2(T) \\sim 0$ for temperatures below a second stage Kondo temperature $T_K^*$ , given by[23] $T_K^* = c_1\\,T_K\\,\\text{e}^{-2\\,c_2\\sqrt{\\frac{T_K}{A_{\\text{eff}}}}},$ with constants $c_1,\\,c_2 \\sim 1$ that depend weakly on the parameters.", "On the easy–axis side, the Kondo screening is suppressed by the energy gap between the molecular ground state $\\left| T,\\,\\pm 1;\\,0 \\right>$ and the first excited state $\\left| T,\\,0;\\,0 \\right>$ .", "In other words, Kondo screening becomes increasingly less efficient as we go farther away from full degeneracy of the triplet, and we obtain $0.25 < \\mu ^2 < 1$ for all temperatures (not shown).", "In order to quantify $A_{\\text{eff}}$ and estimate $\\Delta $ , we carry out NRG calculations at fixed $A_0$ , varying $A_1$ and $\\omega _0$ .", "The transition from a hard to an easy axis is observed as a sudden jump (at zero temperature) from the hard–axis value of $\\mu ^2 = 0$ to $\\mu ^2 = 0.25$ in the spin–1 Kondo regime, as shown in Fig.", "REF (a).", "For the molecular limit where the magnetic anisotropy coefficient is exactly given by $\\tilde{A}$ , the change of regime is indicated by the red curve in Fig.", "REF (a) and (c), given by $\\tilde{A}=0$ .", "When the coupling to the band is taken into account, the transition occurs for $A_{\\text{eff}} = 0$ and is signaled by $\\mu ^2(T\\rightarrow 0) = 0.25$ .", "We indicate this transition with black dots in the figures, and find $\\Delta =A_{\\text{eff}}- \\tilde{A}>0$ as expected from our discussion above.", "The values of $\\Delta $ for every $\\omega _0$ are presented in Fig.", "REF (b) and (d) for their respective maps.", "Notice that the points at larger $\\omega _0$ correspond to smaller $A_1/\\omega _0$ , which explains the decreasing trend of $\\Delta $ with increasing $\\omega _0$ .", "Figure: (Color online) (a) and (c): Magnetic–moment–squared of the molecule at zero temperature (μ 2 (T→0)\\mu ^2(T\\rightarrow 0)) as a function of A 1 A_1 and ω 0 \\omega _0, for (a) A 0 =5T K 0 A_0 = 5\\,T_K^0 and (c) A 0 =6T K 0 A_0 = 6\\,T_K^0.", "In each case, the red curve (A 1 =ω 0 A 0 A_1 = \\sqrt{\\omega _0A_0}) indicates the parameters of full degeneracy of the spin–1 triplet in the molecular limit.", "The black dots indicate the parameters where the underscreened spin–1 Kondo effect is recovered (μ 2 =0.25\\mu ^2 = 0.25).", "(b) and (d): Correction to the magnetic anisotropy due to the coupling to the leads for the parameters of (a) and (c), respectively.The e–ph interactions suppress the Kondo couplings, reducing the spin–1 Kondo temperature $T_K$ of the device.", "This results, according to Eq.", "(REF ), in an enhancement of $T_K^*$ .", "This can be observed in Fig.", "REF , where we compare our device to one with only static anisotropy, and find that they behave similarly, differing only in the values of $T_K$ and $T_K^*$ .", "As further verification, we utilize the values of $\\Delta $ obtained from Fig.", "REF to compute values of $A_{\\text{eff}}$ for substitution into Eq.", "(REF ), and the degree of agreement is excellent, as can be appreciated in Fig.", "REF (c).", "The excellent fit to the theory reassures us of our picture, in which all the effects of the electron–phonon (e–ph) interactions can be absorbed into an effective anisotropy term, and a reduced hybridization due to polaronic effects.", "Figure: (Color online) Second–stage Kondo temperature, T K * T_K^*, for different values of A 1 A_1 and ω 0 \\omega _0.", "(a) A fast fall of T K * T_K^* as a function of A 1 /ω 0 A_1/\\omega _0 is observed, consistent with a vibronic induced shift of the magnetic anisotropy and Eq.", "().", "(b) Plotting T K * T_K^* against A 1 2 /ω 0 =A 0 -A ˜A_1^2/\\omega _0 = A_0 - \\tilde{A}, makes the presence of a correction Δ\\Delta to the anisotropy evident; the dotted lines indicate the value of A 1 A_1 where the molecular limit anisotropy A ˜\\tilde{A} changes sign, for the corresponding values of A 0 A_0.", "The numerical results are clearly shifted toward positive values of A eff A_{\\text{eff}}.", "(c) Verification of Eq.", "() within our picture of effective anisotropy and suppressed Kondo exchange couplings, for fixed A 1 A_1 and ω 0 \\omega _0, varying A 0 A_0.", "We substitute A eff A_{\\text{eff}} in the equation, with Δ\\Delta computed from numerical results such as those of Fig.", "(b) and (d).", "The slope of each curve is proportional to -2T K /T K 0 -2\\sqrt{T_K/T_K^0}, with T K T_K the Kondo temperature corresponding to the curve." ], [ "Conclusions", "We have studied the behavior of a stretchable spin–1 molecule deposited on a break junction, that presents a vibrational mode along the junction axis.", "We performed NRG calculations to study the Kondo physics of this system; its signatures in transport and thermodynamic quantities, as well as the nature of the system's ground state.", "We find that the vibrational degrees of freedom induce a negative magnetic anisotropy that shifts the ground state of the undeformed molecule toward an easy–axis regime, which would be accessible to the static molecule only through compression.", "Polaronic corrections arising from the electron–phonon interactions suppress the Kondo exchange couplings of the molecular spin with the leads' fermionic states, in a spin–asymmetric fashion.", "This reduces the Kondo temperature of the device and introduces an effective correction to the magnetic anisotropy.", "Static stretching of the molecule could then be used to explore a QPT from the non–Fermi–liquid ground state of the easy–axis regime, to the Fermi liquid of the hard–axis regime, visiting an underscreened Kondo critical point at zero anisotropy.", "The resulting phase diagram of the system is presented in Fig.", "REF .", "The different phases exhibit clear conductance signatures that can be explored experimentally.", "From our analysis, we may expect the un–stretched device in the experiment of Parks et al., to be in an easy–axis regime due to the coupling to molecular vibrations.", "In that case, a careful analysis of the low-temperature behavior of the conductance, such as that indicated inf Fig.", "REF (c), would make it possible to identify the signatures of this QPT in a well-controlled environment.", "DRT and SU thank E. Vernek for helpful discussions, and especially for the NRG code that served as an important first component of this work.", "DRT and SU acknowledge support from NSF PIRE and CIAM/MWN (US), and CONACyT (México).", "PSC and CAB acknowledge financial support from PIP 1821 of CONICET and PICT-Bicentenario 2010-1060 of the ANPCyT." ] ]

[ [ "Multimodality of rich clusters from the SDSS DR8 within the\n supercluster-void network" ], [ "Abstract We study the relations between the multimodality of galaxy clusters drawn from the SDSS DR8 and the environment where they reside.", "As cluster environment we consider the global luminosity density field, supercluster membership, and supercluster morphology.", "We use 3D normal mixture modelling, the Dressler-Shectman test, and the peculiar velocity of cluster main galaxies as signatures of multimodality of clusters.", "We calculate the luminosity density field to study the environmental densities around clusters, and to find superclusters where clusters reside.", "We determine the morphology of superclusters with the Minkowski functionals and compare the properties of clusters in superclusters of different morphology.", "We apply principal component analysis to study the relations between the multimodality parametres of clusters and their environment simultaneously.", "We find that multimodal clusters reside in higher density environment than unimodal clusters.", "Clusters in superclusters have higher probability to have substructure than isolated clusters.", "The superclusters can be divided into two main morphological types, spiders and filaments.", "Clusters in superclusters of spider morphology have higher probabilities to have substructure and larger peculiar velocities of their main galaxies than clusters in superclusters of filament morphology.", "The most luminous clusters are located in the high-density cores of rich superclusters.", "Five of seven most luminous clusters, and five of seven most multimodal clusters reside in spider-type superclusters; four of seven most unimodal clusters reside in filament-type superclusters.", "Our study shows the importance of the role of superclusters as high density environment which affects the properties of galaxy systems in them." ], [ "Introduction", "Most galaxies in the Universe are located in groups and clusters of galaxies, which themselves reside in larger systems – in superclusters of galaxies or in filaments crossing underdense regions between superclusters [44], [98], [61], [15].", "Cluster studies, in combination with the study of their environment are needed to understand the physics of clusters themselves, and the evolution of structure in the Universe.", "In the $\\Lambda $ CDM concordance cosmological model groups and clusters of galaxies and their filaments are created by density perturbations of scale up to 32 $h^{-1}$  Mpc, and superclusters of galaxies by larger perturbations, up to 100 $h^{-1}$  Mpc [18], [87].", "Still larger perturbations modulate the richness of galaxy systems.", "Superclusters of galaxies are the largest density enhancements in the cosmic web.", "Studies of their properties and galaxy and cluster content are needed to understand the formation, evolution, and properties of the large-scale structure and to compare cosmological models with observations [47], [16], [26], [40], [3], [25], [81], [51].", "The structures forming the cosmic web grow by hierarchical clustering driven by gravity [52], [53].", "Galaxy clusters form at intersections of filaments, through them galaxies and galaxy groups merge with clusters.", "An indicator of former or ongoing mergers in groups and clusters of galaxies is their multimodality: the presence of a substructure (several galaxy associations within clusters), a large peculiar velocities of their main galaxies, and non-Gaussian velocity distribution of their galaxies [6], [66], [86], [46], [11], [35], [9], [2], [41], [31].", "More references can be found in E12.", "Several studies have shown that richer and more luminous groups and clusters of galaxies from observations and simulations are located in a higher density environment [19], [22], [28], [14], [69].", "[68] and [67] showed that dynamically younger clusters are more strongly clustered than overall cluster population.", "In this study we analyse the relations between the multimodality of rich clusters from the SDSS DR8 and the environment where they reside.", "We calculate the luminosity density field to trace the supercluster-void network, to define the values of the environmental density around clusters, and to determine superclusters of galaxies.", "For each cluster we find whether the cluster is located in a supercluster and study the relations between the properties of superclusters and clusters.", "We compare the properties of isolated clusters, and clusters in superclusters, and compare the properties of clusters in superclusters of different morphology, to understand whether the morphology of superclusters is also an important environmental factor in shaping the properties of groups and clusters in superclusters.", "E12 analysed the substructure and velocity distributions of galaxies in the richest clusters from the SDSS DR8 with at least 50 member galaxies using a number of tests of different dimensionality.", "They showed that two most sensitive tests for the presence of substructure were 3D normal mixture modelling and the Dressler-Shectman (DS or $\\Delta $ ) test [66], [41].", "In this study we use the results of these two tests as an indicators of cluster substructure, and the peculiar velocities of the main galaxies in clusters.", "With principal component analysis we study the relation between the multimodality of clusters and their environment characterised by the values of the environmental density and supercluster luminosities.", "In Sect.", "we describe the data we used, in Sect.", "we give the results.", "We discuss the results and draw conclusions in Sect. .", "We assume the standard cosmological parametres: the Hubble parametre $H_0=100~h$ km s$^{-1}$ Mpc$^{-1}$ , the matter density $\\Omega _{\\rm m} = 0.27$ , and the dark energy density $\\Omega _{\\Lambda } = 0.73$ ." ], [ "Data", "We use the MAIN galaxy sample of the 8th data release of the Sloan Digital Sky Survey [1] with the apparent $r$ magnitudes $r \\le 17.77$ , and the redshifts $0.009 \\le z \\le 0.200$ , in total 576493 galaxies.", "We corrected the redshifts of galaxies for the motion relative to the CMB and computed the co-moving distances [57] of galaxies.", "The absolute magnitudes of galaxies were determined in the $r$ -band ($M_r$ ) with the $k$ - corrections for the SDSS galaxies, calculated using the KCORRECT algorithm [8].", "In addition, we applied evolution corrections, using the luminosity evolution model of [7].", "The magnitudes correspond to the rest-frame at the redshift $z=0$ .", "The details about data reduction and the description of the group catalogue can be found in [92].", "We determine groups of galaxies using the Friends-of-Friends cluster analysis method introduced in cosmology by [93], [98], [42].", "A galaxy belongs to a group of galaxies if this galaxy has at least one group member galaxy closer than a linking length.", "In a flux-limited sample the density of galaxies slowly decreases with distance.", "To take this selection effect into account properly when constructing a group catalogue from a flux-limited sample, we rescaled the linking length with distance, calibrating the scaling relation by observed groups [88], [89].", "As a result, the maximum sizes in the sky projection and the velocity dispersions of our groups are similar at all distances.", "We use in this study systems from the group catalogue with at least 50 member galaxies analysed for substructure in E12.", "These clusters are chosen from the distance interval 120 $h^{-1}$  Mpc $\\le D \\le $ 340 $h^{-1}$  Mpc (the redshift range $0.04 < z< 0.12$ ) where the selection effects are the smallest [89].", "This sample of 109 clusters includes all clusters from the SDSS DR8 with at least 50 member galaxies in this distance interval.", "E12 showed that more than 80% of clusters in this sample demonstrate a signs of multimodality according to several 3D, 2D, and 1D tests: the presence of multiple components, large probabilities to have a substructure, and the deviations of galaxy velocity distributions in clusters from Gaussianity.", "The larger the dimensionality of the test, the more sensitive it is to the presence of substructure in clusters (for details we refer to E12).", "In this study we use the results of two 3D test to characterise the multimodality in clusters: the 3D normal mixture modelling and the Dressler- Shectman (DS) test.", "We describe these tests in Appendix .", "In addition, we use the peculiar velocity of the main galaxies, $V_{\\mathrm {pec}}$ .", "In the group catalogue the main galaxy of a group is defined as the most luminous galaxy in the $r$ -band.", "We use this definition also in the present paper.", "We calculate the galaxy luminosity density field to reconstruct the underlying luminosity distribution, and to find the environmental density around clusters.", "Environmental densities are important for understanding the influence of the local and/or global environment on cluster properties.", "Three smoothing lengths are used for environmental densities around clusters, 4, 8, and 16 $h^{-1}$  Mpc to characterise environment at scales around clusters from cluster local surroundings to supercluster scales.", "For details we refer to Appendix  and to [92].", "To determine supercluster membership for clusters, we first found superclusters (extended systems of galaxies) in the luminosity density field at smoothing length 8 $h^{-1}$  Mpc.", "We created a set of density contours by choosing a density threshold and defined connected volumes above a certain density threshold as superclusters.", "In order to choose proper density levels to determine individual superclusters, we analysed the density field superclusters at a series of density levels.", "As a result we used the density level $D = 5.0$ (in units of mean density, $\\ell _{\\mathrm {mean}}$ = 1.65$\\cdot 10^{-2}$ $\\frac{10^{10} h^{-2}L_\\odot }{(h^{-1}\\mathrm {Mpc} )^3})$ to determine individual superclusters.", "At this density level superclusters in the richest chains of superclusters in the volume under study still form separate systems; at lower density levels they join into huge percolating systems.", "At higher threshold density levels superclusters are smaller and their number decreases.", "Superclusters are characterised by their total luminosity, richness, and morphology, determined with Minkowski functionals.", "The total luminosity of the superclusters $L_{\\mathrm {scl}}$ is calculated as the sum of weighted galaxy luminosities: $ L_{\\mathrm {scl}} = \\sum _{\\mathrm {gal} \\in \\mathrm {scl}} W_L (d_{\\mathrm {gal}}) L_{\\mathrm {gal}}.", "$ Here the $W_L(d_{\\mathrm {gal}})$ is the distance-dependent weight of a galaxy (the ratio of the expected total luminosity to the luminosity within the visibility window).", "The description of the supercluster catalogues is given in [50] and in Liivamägi et al.", "(in preparation, DR8 catalogue).", "Figure: Distribution of groups with at leastfour member galaxies in superclusters in x, y, and z coordinates (in h -1 h^{-1} Mpc,grey dots).Black filled circles denote clusters with at least 50 member galaxies insuperclusters, dark grey empty circles those clusters with at least 50 member galaxieswhich are not located in superclusters.Numbers are ID numbers of superclusters with at least 500 member galaxies.The supercluster morphology is fully characterised by the four Minkowski functionals $V_0$ –$V_3$.", "For a given surface the four Minkowski functionals (from the first to the fourth) are proportional to the enclosed volume $V$ , the area of the surface $S$ , the integrated mean curvature $C$ , and the integrated Gaussian curvature $\\chi $ [78], [57], [80], [77], [76].", "We give formulaes in Appendix .", "The overall morphology of a supercluster is described by the shapefinders $K_1$ (planarity) and $K_2$ (filamentarity), and their ratio, $K_1$ /$K_2$ (the shape parametre), calculated using the first three Minkowski functionals.", "They contain information both about the sizes of superclusters and about their outer shape.", "The smaller the shape parametre, the more elongated a supercluster is.", "The maximum value of the fourth Minkowski functional $V_3$ (the clumpiness) characterises the inner structure of the superclusters and gives the number of isolated clumps, the number of void bubbles, and the number of tunnels (voids open from both sides) in the region [77].", "The larger the value of $V_3$ , the more complicated the inner morphology of a supercluster is; superclusters may be clumpy, and they also may have holes or tunnels in them [26], [25].", "Superclusters show large morphological variety in which [24] determined four main morphological types: spiders, multispiders, filaments, and multibranching filaments.", "Spiders and multispiders are systems of one or several high-density clumps with a number of outgoing filaments connecting them.", "The Local Supercluster is an example of a typical poor spider.", "Filaments and multibranching filaments are superclusters with filament-like main body which connects clusters.", "An example of an exceptionally rich and dense multibranching filament is the richest supercluster in the Sloan Great Wall [26], [25].", "For simplicity, in this study we classify superclusters as spiders and filaments.", "Data about clusters and superclusters are given in online Tables REF and  REF .", "We cross-identify groups with Abell clusters (Table REF ).", "We consider a group identified with an Abell cluster, if the distance between their centres is smaller than at least the linear radius of one of the clusters, and the distance between their centres in the radial (line-of-sight) direction is less than 600 $km~s^{-1}$ (an empirical value).", "In some cases one group can be identified with more than one Abell cluster and vice versa (for details we refer to E12).", "In Table REF we give to superclusters the ID number from [20] catalogue if there is at least one Abell cluster in common between this catalogue and the present supercluster sample.", "A common cluster does not always mean that superclusters can be fully identified with each other.", "A number of superclusters from E01 are split between several superclusters in our present catalogue, an examples of such systems are SCl 019 and SCl 054, which both belong to SCl 111 in [20] catalogue." ], [ "The large-scale environment of clusters", "To study the distribution of clusters in the supercluster-void network we present in Figure REF the distribution of clusters with at least four member galaxies in superclusters, and the distribution of isolated clusters with at least 50 member galaxies in cartesian coordinates $x$ , $y$ , and $z$ defined as in [65] and in [50]: $\\begin{array}{l}x = -d \\sin \\lambda , \\nonumber \\\\[3pt]y = d \\cos \\lambda \\cos \\eta ,\\\\[3pt]z = d \\cos \\lambda \\sin \\eta ,\\nonumber \\end{array}$ where $d$ is the comoving distance, and $\\lambda $ and $\\eta $ are the SDSS survey coordinates.", "In Fig.", "REF we plot the values of the environmental density around groups with at least 4 member galaxies at smoothing length 8 $h^{-1}$  Mpc vs. the distance of groups.", "In this figure circles represent clusters with at least 50 member galaxies.", "The size of circles is proportional to the number of components in clusters determined with the 3D normal mixture modelling.", "Figure REF (left panel) shows that at the smallest distances from us (at low $y$ values, up to distances approximately 180 $h^{-1}$  Mpc) the sample crosses the void region.", "This is the void between the nearby rich superclusters [24].", "Groups and clusters form two filaments of poor superclusters and isolated clusters crossing this void.", "The richest superclusters in these filaments are SCl 352 and SCl 782 (we identify supercluster members among clusters in Sect.", "REF ).", "The density distribution in Fig.", "REF shows that even the maximal values of the environmental densities in this region are low, up to $D8 \\approx 5$ , in the density peaks in filaments at the locations of superclusters $D8 < 8$ (in units of the mean density).", "Figures REF and  REF shows that rich clusters in superclusters mark the peaks in the density distribution, isolated clusters are located at lower densities in filaments.", "The sizes of symbols in Fig.", "REF show that among these clusters there are both multicomponent and one-component clusters.", "At distances between 180 $h^{-1}$  Mpc and 270 $h^{-1}$  Mpc the SDSS survey crosses systems of rich superclusters.", "The richest superclusters in these systems are SCl 027 and SCl 019 in the Sloan Great Wall, SCl 211 (the Ursa Major supercluster) in another chain of superclusters, and SCl 099 (the Corona Borealis supercluster) and SCl 001 in the dominant supercluster plane at the intersection of the supercluster chains [29], [24].", "The values of the environmental densities $D8 < 8$ in the foreground of rich superclusters at distances less than 200 $h^{-1}$  Mpc.", "At larger distances, in rich superclusters the maximal values of the environmental densities are much higher than in the void region behind them.", "Again rich clusters mark the high density peaks in the density field (Fig.", "REF ).", "Some rich clusters are located in the cores of rich superclusters with the highest values of the environmental densities, $D8 > 10$ [21], [90], [91].", "The environmental density is the largest ($D8 = 21.3$ ) in the supercluster SCl 001, around rich cluster 29587 (Abell cluster A2142).", "At still larger distances the SDSS sample reaches the void region behind the Sloan Great Wall and other rich superclusters, and the values of the environmental densities are lower again.", "The farthest rich cluster in our sample belong to the rich supercluster SCl 003 behind this void at a distance of 336 $h^{-1}$  Mpc.", "Here the value of the environmental density is also very high, $D8 = 20.1$ .", "Figure REF shows that the lowest values of the environmental densities around rich clusters slightly increase with distance.", "This is due to the use of groups with 50 and more members.", "Due to the flux-limited sample, the groups with the same richness are also brighter further away.", "In E12 we showed that the richness of rich clusters in our sample does not depend on distance, therefore our sample of clusters is not strongly affected by this selection effects.", "When comparing the environmental densities around clusters in some cases we shall use two distance intervals, to analyse densities and the properties of clusters in void region and in supercluster region separately, and to minimise the influence of this selection effect.", "A visual inspection of Fig.", "REF shows that both multicomponent and one-component clusters are located in all density peaks.", "Next we analyse the values of the environmental densities around clusters in more detail.", "Figure: Global densities at smoothing length 8 h -1 h^{-1} Mpc (in units of mean density) around groupsand clusters vs. their distance.", "Black circles denote clusterswith at least 50 member galaxies, the size of circles is proportional to thenumber of components found by 3D normal mixture modelling.", "Grey dots denote groupswith 4–49 member galaxies.We plot cluster luminosities vs. environmental densities at three smoothing lengths in Fig.", "REF , and search for the pairwise correlations between the parametres of clusters and the environmental densities around them with the Pearson's correlation test (Table REF ).", "In Fig.", "REF we mark those clusters which are unimodal according to all the tests applied in E12 with filled circles, and those which are multimodal with stars (we discuss them in detail in Sect.", "REF ).", "We exclude from this analysis the cluster Gr1573 near the edge of the survey for which environmental densities cannot be determined reliably ($D = -999$ in Table REF ).", "Figure: From left to right: cluster luminosities (in 10 10 h -2 L 10^{10} h^{-2} L_{})vs. their environmental densities at the smoothing lengths4, 8, and 16 h -1 h^{-1} Mpc (in units of mean density) (grey circles).The size of circles is proportional to thenumber of components found by 3D normal mixture modelling.Stars denote most multimodal clusters,and filled circles denote most unimodal clusters as described in the text.At the smoothing length 4 $h^{-1}$  Mpc the luminosity density is determined by cluster members, and galaxies and galaxy systems in the close neighbourhood of clusters.", "Therefore the correlation between the luminosities of clusters and environmental densities is strong (the correlation coefficient is $r = 0.91$ with very high statistical significance, Table REF ).", "Figure REF shows that densities around clusters of the same luminosity may differ up to 1.5 - 2 times depending on the systems around clusters.", "At this smoothing length there is no statistically significant correlation between the number of components in clusters and the environmental density around clusters.", "The statistical significance to have substructure in clusters according to the DS test $p_{\\mathrm {\\Delta }}$ is weakly anticorrelated with the environmental density.", "Small $p_{\\mathrm {\\Delta }}$ values show higher significance of having substructure, therefore this test shows that there is a tendency that clusters with higher probabilities of having a substructure reside in a higher density environments.", "However, Table REF shows that the statistical significance of this result is low.", "Figure REF shows that one of the most luminous clusters have relatively low density local environment around it (Gr34727 in the supercluster SCl 7, Table REF ).", "At the smoothing length 8 $h^{-1}$  Mpc (Fig.", "REF , middle panel) the scatter of the relation between cluster luminosities and environmental densities increases – the difference between the environmental densities around clusters of the same luminosity increases and the correlations between the cluster luminosities and environmental density become weaker.", "The scatter is espacially large at densities $D > 8$ , in the cores of rich superclusters, where both high- and low-luminosity clusters reside.", "All most luminous clusters are located in supercluster cores.", "In poor superclusters environmental densities are lower.", "At the largest smoothing length, 16 $h^{-1}$  Mpc which characterises large scale supercluster environment around clusters the scatter of the relation between cluster luminosities and environmental densities increases and the correlations between the clusters luminosities and environmental density become weaker.", "The number of components in clusters at large smoothing lengths is not correlated with the environmental density around clusters, the correlation between the probability to have substructure in clusters and environmental density also becomes weaker.", "The correlations between the peculiar velocities of the main galaxies in clusters and environmental density are statistically highly significant – clusters in higher density environments have larger peculiar velocities of the main galaxies.", "The correlation coefficients are not large, from 0.24 at smoothing length 4 $h^{-1}$  Mpc to 0.18 at smoothing length 16 $h^{-1}$  Mpc.", "The correlations between the number of galaxies in clusters and the environmental density of clusters are statistically highly significant (Table REF ).", "The correlations are not very strong, with the correlation coefficient $r\\approx 0.5$ , being stronger at small smoothing length and weaker at large smoothing length.", "In addition, the larger the smoothing length, the stronger are the correlations between the luminosity of superclusters and environmental density – richer and more luminous superclusters have also higher environmental densities, as found earlier by [16].", "Table: Correlations between the environmental density around clusters,and cluster parametres.Figure REF shows the cumulative distributions of the values of the environmental densities at smoothing length 4 $h^{-1}$  Mpc, at which the correlations between the environmental density and multimodality parametres were the strongest, for two distance intervals: 120 $h^{-1}$  Mpc $\\le D \\le $ 180 $h^{-1}$  Mpc (upper row), and 180 $h^{-1}$  Mpc $\\le D \\le $ 300 $h^{-1}$  Mpc (lower row).", "There are 42 clusters in the closer distance interval, and 65 clusters in the farther distance interval.", "We show the cumulative distributions of densities around clusters divided into populations according to the different indicators of multimodality: multicomponent and one- component clusters according to the 3D normal mixture modelling, clusters with and without significant substructure according to the DS test ($p_{\\mathrm {\\Delta }} <= 0.05$ , and $p_{\\mathrm {\\Delta }} > 0.05$ , correspondingly), and clusters with small and large peculiar velocities of their main galaxies.", "E12 showed that, approximately, the peculiar velocity limit between these two populations is of about 250 $km~s^{-1}$ .", "Figure REF shows that densities around multicomponent clusters and clusters with significant substructure have higher values than densities around one-component clusters without significant substructure.", "The differences between the densities around clusters with small and large peculiar velocities of their main galaxies in the void region (120 $h^{-1}$  Mpc $\\le D \\le $ 180 $h^{-1}$  Mpc) are small, in the farther region in superclusters clusters with large peculiar velocities of their main galaxies have higher environmental densities around them than clusters with small values of the peculiar velocities of their main galaxies.", "Figure: Cumulative distributions of the values of the environmental densities around clustersfor the smoothing length 4 h -1 h^{-1} Mpc.", "Solid black line denote densities aroundmulticomponent clusters(left panel), clusters with significant substructure (middle panel), andclusters with the peculiar velocities of their main galaxies larger than 250 kms -1 km~s^{-1}(right panel).", "Dashed grey line denote densities aroundone-component clusters(left panel), clusters without significant substructure (middle panel), andclusters with the peculiar velocities of their main galaxies smaller than 250 kms -1 km~s^{-1}(right panel).", "Upper row – distance interval 120 h -1 h^{-1} Mpc ≤D≤\\le D \\le 180 h -1 h^{-1} Mpc,lower row – distance interval 180 h -1 h^{-1} Mpc ≤D≤\\le D \\le 300 h -1 h^{-1} Mpc.Table: Results of the principal component analysis for the multimodalityand environmental parametres of clusters.The relations between the parametres of clusters, the indicators of substructure, and the environmental parametres of clusters can be studied simultaneously with the principal component analysis (PCA).", "The PCA transforms the data to a new coordinate system, where the greatest variance by any projection of the data lies along the first coordinate (the first principal component), the second greatest variance – along the second coordinate, and so on.", "There are as many principal components as there are parametres, but often only the first few are needed to explain most of the total variation.", "The principal components PC$i$ ($i \\in \\mathbb {N}$ , $i \\le N_{\\mathrm {tot}}$ ) are linear combinations of the original parametres: $PCi = \\sum _{k=1}^{N_{\\mathrm {tot}}} a(k)_{i} V_{k},$ where $-1 \\le a(k)_i \\le 1$ are the coefficients of the linear transformation, $V_k$ are the original parametres and $N_{\\mathrm {tot}}$ is the number of the original parametres.", "In the analysis the parametres are standardised – they are centred on their means, $ V_{k} - \\overline{V_{k}}$ , and normalised, divided by their standard deviations, $\\sigma ( V_{k})$ .", "E12 used PCA to analyse the relations between the multimodality parametres of clusters and their physical properties.", "We refer to [23] for the references about applications of the PCA in astronomy.", "We include into the calculations the number of components as determined with the 3D normal mixture modelling, $p_{\\mathrm {\\Delta }}$ showing the probability to have substructure according to the DS test, the peculiar velocity of the main galaxy in clusters, $V_{\\mathrm {pec}}$ , the number of galaxies in clusters, $N_{\\mathrm {gal}}$ , the environmental density around clusters with smoothing length 8 $h^{-1}$  Mpc, $D8$ (environmental densities are correlated, therefore we include only one of them), and the luminosity of a supercluster where a cluster resides, $L_{\\mathrm {scl}}$ .", "We use $1 - p_{\\mathrm {\\Delta }}$ since larger values of $1 - p_{\\mathrm {\\Delta }}$ suggest a higher probability to have substructure, therefore the arrows in biplot corresponding to the number of the components point towards the same direction as the arrows corresponding to the DS test.", "We use logarithms of the peculiar velocities of main galaxies and environmental parametres.", "Figure REF and Table REF show the results of this analysis.", "Table REF shows that the coefficients of the first principal component are the largest for the environmental density around clusters, for the number of galaxies in clusters, and for the total luminosity of superclusters.", "This shows that richer clusters are located in a higher density environment, and richer superclusters have higher environmental densities in them [21], [16], as also shown with the analysis above.", "In the biplot showing the results of the PCA (Fig.", "REF ) the arrows corresponding to the tests about substructure and arrows corresponding to the other parametres of clusters are not pointed into the same direction.", "This suggests that the correlations between substructure parametres and the environment of clusters are not strong, as also the correlation calculations showed.", "In Fig.", "REF the arrow corresponding to the peculiar velocity of the main galaxies in clusters is pointed approximately into the same direction as the arrow for richness of clusters, showing that these velocities are larger in richer clusters.", "The length of the arrows and coefficients in Table REF show that the importance of the peculiar velocity of the main galaxies is smaller than the importance of the richness of clusters.", "The first principal component accounts for about 1/3 of the variance of parametres, the second principal component for about 1/4 of the variance.", "However, five principal components are needed to explain more than 90% of the variance of the parametres, thus clusters with their environment are complicated objects whose properties cannot be explained with a small number of parametres as found also for the dark matter haloes by [45].", "Figure: Biplot of the principal component analysis with the multimodality indicatorsand the environmental parametres of clusters, as described in the text.The locations of clusters in the PC1-PC2 plane shows that unimodal clusters in superclusters are located at upper lefthand part of the plot and have larger PC2 and smaller (larger negative) PC1 values (for example, clusters 608, 13408, 25078, and 28508).", "Rich multimodal clusters of high environmental density value around them populate lower and middle righthand area of the biplot (clusters 34276, 34727, 914, 29587).", "Multimodal clusters in low environmental density environment populate the lefthand lower area of the PC1-PC2 plane (clusters 11474, 11015).", "Unimodal clusters in very rich superclusters populate the upper righthand area of the plane (67116, 63361, and others).", "On the lefthand area of the plane are located isolated multimodal poor clusters (50657, 58323, and others).", "In Fig.", "REF we show for clusters in superclusters the number of galaxies in clusters vs. the total luminosity of the host supercluster.", "Here the supercluster of the highest luminosity is SCl 027, the richest system in the Sloan Great Wall.", "This figure shows that this supercluster, as well as other superclusters host both multicomponent and one-component clusters, as a result there is no correlation between the host supercluster luminosity and the number of components in clusters.", "Figure: The number of galaxies in clusters vs. the total luminosity of superclusterswhere they reside (in 10 10 h -2 L 10^{10}h^{-2} L_{}).", "The size of symbolsis proportional to the number of components in clusters." ], [ "Properties of clusters and supercluster morphology", "In this section we analyse the properties of clusters in superclusters of different morphology.", "At first we searched for the host superclusters for each cluster and found that 80 of our clusters lie in superclusters.", "Next we determined for these superclusters their morphological parametres and types using Minkowski functionals and shapefinders, and visual inspection.", "The physical and morphological parametres of superclusters (the values of the fourth Minkowski functional (the clumpiness) $V_3$ and the shapefinders $K_1$ (the planarity), $K_2$ (the filamentarity), and their ratio (the shape parametre) for each supercluster are given in Table REF .", "According to their overall shape superclusters are elongated with the value of the filamentarity $K_2$ being larger than the value of the planarity $K_1$ .", "There are only 4 systems with the shape parametre $K_1/K_2 > 1.0$ resembling pancakes.", "The superclusters with very small values of the shapefinders have large negative values of the shape parametre owing to noisiness in the data.", "There are 15 superclusters of filament morphology, and 35 of spider morphology in our sample.", "Figure REF shows the shapefinders plane for the superclusters with the size of symbols proportional to the clumpiness of superclusters takes together the morphological information about superclusters.", "Superclusters with higher values of planarity and filamentarity have also larger values of clumpiness and therefore more complicated inner morphology.", "Poor superclusters are mostly of spider morphology [24].", "Most of them are located close to us, they are members of the filaments crossing the void region in front of the Sloan Great Wall and other rich superclusters at distances larger than 180 $h^{-1}$  Mpc.", "Figure: Shapefinders K 1 K_1 (planarity) and K 2 K_2 (filamentarity) forthe superclusters.The symbol sizes are proportional to the fourthMinkowski functional V 3 V_3.Circles denote the superclusters of spider morphologyand squares denote the superclusters of filament morphology.In Fig.", "REF we show the examples of the fourth Minkowski functional $V_3$ vs. mass fraction $mf$ and the shapefinders $K_1$ and $K_2$ for two superclusters of filament morphology, SCl 027 and SCl 059, in Fig.REF for superclusters of spider morphology, SCl 019 and SCl 092.", "The superclusters SCl 027 and SCl 019 are the richest two superclusters in the Sloan Great Wall [25], [24].", "In middle panel of these figures we plot the clumpiness $V_3$ vs. the (excluded) mass fraction $mf$ .", "At small mass fractions the isodensity surface includes the whole supercluster and the value of the 4th Minkowski functional $V_3 = 1$ .", "As we move to higher mass fractions, the iso-surfaces move from the outer supercluster boundary towards the higher density parts of a supercluster, and some galaxies do not contribute to the supercluster any more.", "Individual high density regions in a supercluster begin to separate from each other, also the holes or tunnels may appear, therefore the value of the clumpiness increases.", "At a certain mass fraction $V_3$ has a maximum, showing the largest number of isolated clumps in a given supercluster.", "At still higher mass fraction only the high density peaks remain in the supercluster and the value of $V_3$ decreases again.", "When we increase the mass fraction, the changes in the morphological signature accompany the changes of the 4th Minkowski functional (right panels of the figures).", "As the mass fraction increases, at first the planarity $K_1$ almost does not change, while the filamentarity $K_2$ increases – at higher density levels superclusters become more filament-like than the whole supercluster.", "Then also the planarity starts to decrease, and at a mass fraction of about $m_f = 0.7$ the characteristic morphology of a supercluster changes.", "We see the crossover from the outskirts of a supercluster to the core of a supercluster [26].", "The figures of the fourth Minkowski functional and shapefinders for rich superclusters with at least 300 member galaxies from SDSS DR7 can be found in [24].", "Figure: Sky distribution of galaxies (left panel), thefourth Minkowski functional V 3 V_3 (middle panel)and the shapefinder's K 1 K_1-K 2 K_2 plane (right panel) for twosuperclusters of filament morphology.", "Upper row – the supercluster SCl 027,lower row – the supercluster SCl 059.", "In the left panel black filled circles denotegalaxies in clusters with at least 50 member galaxies, grey dots denote othergalaxies.", "On the right panel triangle corresponds to K 1 K_1 and K 2 K_2 values atthe mass fraction mf=0mf = 0 (the whole supercluster).", "Mass fraction increasesanti-clockwise along the K 1 K_1-K 2 K_2 curve (the morphological signature).Figure: Sky distribution of galaxies (left panel), thefourth Minkowski functional V 3 V_3 (middle panel)and the shapefinder's K 1 K_1-K 2 K_2 plane (right panel) for twosuperclusters of spider morphology.", "Upper row – the supercluster SCl 019,lower row – the supercluster SCl 092.", "Notations as in Fig.", ".Table: Properties of clusters in superclusters of spider and filament morphology.Table: Properties of clusters in superclusters and isolated clusters.Figure: Cumulative distributions of the numbers of components in clusters, N comp N_{\\mathrm {comp}},peculiar velocities of cluster main galaxies, V pec V_{\\mathrm {pec}}(in kms -1 km~s^{-1}), andp-value of the DS test, p Δ p_{\\mathrm {\\Delta }}for clusters in superclusters of filament morphology (F, black solid line),of spider morphology (S, grey dashed line), and for isolated clusters(I, thin grey solid line) in the distance interval180 h -1 h^{-1} Mpc ≤D≤\\le D \\le 300 h -1 h^{-1} Mpc (upper row), and for clusters in superclusters (Scl, black solid line) and for superclustersin low density regions (isolated clusters I, grey solid line)in the distance interval 120 h -1 h^{-1} Mpc ≤D≤\\le D \\le 180 h -1 h^{-1} Mpc (lower row).Next we compare the properties of clusters in superclusters of filament and of spider morphology, and isolated clusters in two distance intervals.", "At distances up to approximately 180 $h^{-1}$  Mpc most superclusters are poor, and of spider morphology.", "In this distance interval we compare the properties of isolated clusters and supercluster members, without dividing them according to the host supercluster morphology.", "At distances of 180 $h^{-1}$  Mpc $\\le D \\le $ 300 $h^{-1}$  Mpc there are only six isolated clusters, therefore we compare the properties of clusters in superclusters of different morphology.", "For a comparison we also show parametres of isolated clusters in this distance interval.", "We present cumulative distributions of the cluster substructure parametres in Figure REF and the median values of cluster parametres in Tables REF and REF .", "Table REF and Fig.", "REF (upper row) show that clusters in superclusters of spider morphology have higher probabilities to have substructure, and larger peculiar velocities of their main galaxies than clusters in filament-type superclusters.", "Clusters in spider-type superclusters are slightly richer than those in filament-type superclusters.", "Differences in a number of components found by 3D normal mixture modelling are small.", "The Kolmogorov-Smirnov test with substructure parametres centred on their means showed that in this case the differences between the probalitities to have substructure according to the DS test are statistically of very high significance ($p < 10^{-6}$ ), differences between other centred parametres are not significant ($p > 0.2$ ).", "Isolated clusters in this distance interval are poor, even poorer than nearby isolated clusters, with 52 median number of galaxies.", "The maximal values of the peculiar velocities of their main galaxies, and maximal number of components are smaller than those for clusters in superclusters of both types and smaller than those for nearby isolated clusters, but only one of them is a one-component cluster without significant substructure.", "Figure: Distribution of galaxies in R.A. vs. Dec.,and R.A. vs. velocity (in 10 2 kms -1 10^{2}~km~s^{-1}) plot(right panel) in the supercluster SCl 352.", "Filled circles denotegalaxies in Gr34726, empty circles galaxies in Gr9350.", "Greycrosses denote other galaxies in the supercluster.", "Black stars show the location ofthe main galaxies in both rich cluster.As an example of a nearby supercluster of spider morphology we show in Fig.", "REF the distribution of galaxies in SCl 352 in the filament crossing the void between nearby superclusters and the Sloan Great Wall (Fig.", "REF ).", "This supercluster contain two clusters with at least 50 member galaxies, Gr9350 and Gr34926.", "Both have four components and high probability to have substructure according to the DS test.", "The peculiar velocities of their main galaxies are small.", "In the Gr9350 the main galaxy is located in the main component of the cluster with large rms velocities of galaxies (the finger-of-god, seen in the right panel of the figure).", "In Gr34926 the main galaxy is located in another component with smaller rms velocities of galaxies.", "We see in this figure rich inner structure of the supercluster, where clusters and groups of galaxies are connected by galaxy filaments.", "Some components of rich clusters may be infalling, a hint that clusters are not virialised yet.", "Figure: Distribution of galaxies in R.A. vs. Dec.,and R.A. vs. velocity (in 10 2 kms -1 10^{2}~km~s^{-1}) plot(right panel) in the supercluster SCl 211 (the UrsaMajor supercluster).Filled circles denotegalaxies in Gr5217 (Abell cluster A 1436), empty circles galaxies inother clusters with at least 50member galaxies.", "Grey crosses denote other galaxies in the supercluster.Nearby clusters in superclusters are richer than more distant clusters in superclusters.", "They have larger probabilities to have substructure than clusters in more distant superclusters, but smaller peculiar velocities of their main galaxies.", "We use flux-limited sample of galaxies to define clusters, and nearby clusters contain galaxies of lower luminosity than distant clusters.", "It is possible that these clusters have substructure in their outer regions formed by fainter galaxies, absent in more distant sample, and this may explain the difference between them.", "They also have different global environment: nearby superclusters are located in poor filaments surrounded by voids.", "More distant superclusters are richer and form several chains of rich superclusters [24].", "Next we plan to analyse the galaxy content of clusters in more detail, and also a larger sample of clusters and superclusters, to understand better the difference between clusters.", "Isolated clusters are poorer than clusters in superclusters.", "The number of components in isolated clusters is close to that for supercluster members in the same distance interval.", "The peculiar velocities of the main galaxies in isolated clusters are larger than in supercluster members, but maximal values of the peculiar velocities of main galaxies in isolated clusters are smaller than those in supercluster members.", "The DS test shows that the probability to have substructure is larger among nearby supercluster member clusters than in isolated clusters.", "Among six distant isolated clusters only one has no significant substructure." ], [ "Examples of selected clusters", "Most luminous clusters.", "Our sample contains seven clusters of very high luminosity (Table REF ).", "All of them have been identified with Abell clusters.", "They are located in high density cores of superclusters (Sect.REF .", "Five of seven most luminous clusters are located in superclusters of spider morphology.", "Among them are the cluster Gr34727 and the cluster Gr39489 in superclusters of the dominant supercluster plane (SCl 007, and SCl 099, the Corona Borealis supercluster, correspondingly).", "The Corona Borealis supercluster and clusters in it have been studied by [84] and [63].", "One of the high- luminosity clusters in the filament-type supercluster is Gr914 (Abell cluster A 1750) in the richest supercluster of the Sloan Great Wall, SCl 027, another – Gr29587 (A2142) in SCl 001 in the dominant supercluster plane.", "The luminosity density around A2142 in SCl 001 is very high [24].", "The clusters A1750, A2028, A2029, A2065, A2069, and A2142 are (probably merging) X-ray clusters [55], [5], [12], [10], [62], [37].", "All high- luminosity clusters have multiple components, high probability to have substructure, and most of them have large peculiar velocities of their main galaxies.", "Most unimodal and most multimodal clusters.", "Some of our clusters are unimodal according to all the tests applied in E12, they are one-component systems with very low probability of substructure, the sky distribution of their member galaxies is symmetrical, and the galaxy velocity distribution is Gaussian.", "We list them in Table REF .", "There are also multimodal clusters according to all the tests with multiple components, asymmetrical galaxy distribution and non-Gaussian distribution of galaxy velocities (Table REF .", "We marked them in luminosity-density plots in Fig.", "REF .", "Tables REF and  REF show that unimodal clusters are, in average, poorer and of lower luminosity than multimodal clusters.", "Figure REF showed that unimodal clusters are typically located in a lower density environment, but in Fig.", "REF we saw that some of them reside also in high-density cores of rich superclusters.", "Of seven most unimodal clusters four are located in superclusters of filament morphology, two in spider-type superclusters and one is isolated.", "In contrary, of seven most multimodal clusters five are located in spider-type superclusers, and only two in filament-type superclusters.", "There are two superclusters in our sample which both hosts one most multimodal and one most unimodal cluster - SCl 027, and SCl 211.", "In Fig.", "REF we plot the distribution of galaxies in the supercluster SCl 211 (the Ursa Major supercluster).", "Here Gr5217 is located approximately at $R.A.", "= 180$ and $Dec.", "=56.2$ degrees, and Gr28387 at $R.A.", "= 170$ and $Dec.", "= 54.5$ degrees.", "Unimodal cluster Gr25078 (Abell cluster A1650) in the core of SCl 027 is a compact X-ray source, possibly located at a cold spot in the CMB [94].", "The cluster A1809 in the supercluster SCl 027 is also a X-ray cluster.", "Multimodal clusters A1291, A1983 and A671 are also X-ray clusters [72], [24], [64]." ], [ "Discussion and conclusions", "We studied the environment of rich clusters from SDSS DR8, defined by the environmental luminosity density around clusters, and by membership of clusters in superclusters of different morphology.", "We found a correlation between the environmental density around clusters and the presence of substructure in clusters.", "However, both multimodal and unimodal clusters can be found in regions of low and high environmental density, and correlations with the environmental density are not strong.", "In the study of the substructure of the richest clusters of the Sloan Great Wall [30] found clusters with substructure in both rich and poor superclusters of the Wall.", "In this study we showed using a larger sample of superclusters and clusters that superclusters of different richness may host both multicomponent and one-component clusters, and there is no correlation between the host supercluster richness (luminosity) and the multimodality of clusters.", "In higher density environment the peculiar velocities of the main galaxies in clusters are larger, a suggestion that by this measure also clusters in our sample are dynamically more active in high density environments.", "Isolated clusters are poorer and have smaller maximal number of components than cluster in superclusters in the same distance interval.", "[68], [67], [73] and [32] showed that dynamically younger clusters with more substructure are more strongly clustered than overall cluster population.", "They used cluster centroid shift and DS test results as indicators of the dynamical state of clusters.", "We only use the data about rich clusters in our sample, and majority of them (80%) show multimodality according to at least one test in E12 while other studies included data also about poorer systems.", "This is probably the reason why we found weaker correlations between the presence of substructure and cluster environment than in other studies.", "We also found clusters in our sample with almost no close galaxies or galaxy filaments (within the SDSS survey magnitude limits) in the sky distribution.", "One example of such a cluster is given in Fig.", "REF .", "This is the unimodal cluster Gr5217 (A1436), surrounded by an almost empty region without visible galaxies, seen also in Figure 1b by [48].", "Other rich clusters in this system are connected by galaxy filaments.", "This contradicts with understanding that rich clusters are located at the intersections of galaxy filaments.", "One possible reason for that may be that all brighter galaxies in this region have already merged into the cluster.", "[49] write that A1436 is probably relaxing after a recent merger, which agrees with our interpretation.", "We found about ten of such clusters in our sample, both one- component and multicomponent ones, and continue to study them to better understand the relation between rich clusters and their small and large scale environment.", "Cosmological simulations of the future evolution of the structure in the Universe predict that future superclusters (at $a = 100$ , $a$ is the expansion factor) are much more spherical than present-day superclusters.", "Clusters in superclusters will merge into a single cluster in the far future.", "In other words, the differences noted in this study between superclusters of different morphology, and clusters in them, may disappear in the distant future [3].", "In the framework of the hierarchical formation of the structure galaxy clusters are located at the intersections of filaments which form the cosmic web.", "Matter flows through filaments from lower density regions into clusters.", "The merging and growth of dark matter haloes have been studied with cosmological simulations [74], [59], [58], [33] which show that the late time formation of the main haloes and the number of recent major mergers can cause the late time subgrouping of haloes and the presence of substructure [85], [30], [71].", "In high- density regions groups and clusters of galaxies form early and could be more evolved dynamically [90], but in high-density regions the velocities of haloes in the vicinity of larger haloes are high [28] and the possibility of mergers is also high.", "As a result in high-density regions clusters have a larger amount of substructure and higher peculiar velocities of their main galaxies than in low-density regions.", "Superclusters of spider morphology have richer inner structure than superclusters of filament morphology with large number of filaments between clusters in them.", "This may lead to the differences noted in this study: clusters in superclusters of spider morphology are dynamically younger than clusters in superclusters of filament morphology with their higher probability to have substructure and larger peculiar velocities of main galaxies.", "Five of seven most multimodal clusters are located in superclusters of spider morphology, while four of six most unimodal clusters in superclusters reside in filament-type superclusters.", "One example of the high-luminosity multimodal cluster in the spider-type supercluster is Gr34727 (Abell cluster A2028).", "[37] suggest that the subclusters of A 2028 are probably merging to produce a more massive cluster.", "Merging X-ray clusters have been found also in the high-density cores of other superclusters [75], [4].", "[25] calculated the peculiar velocities of the main galaxies in groups from the two richest superclusters in the Sloan Great Wall and found that groups in the supercluster SCl 027 of filamentary morphology (see Table REF ) are dynamically more active with their larger peculiar velocities of main galaxies than groups in SCl 019 of spider morphology.", "In SCl 027 the environmental densities are very high, this may be the reason why in this supercluster of filament morphology clusters show higher dynamical activity than another supercluster of the Sloan Great Wall.", "[17], [18], and [87] showed how the syncronisation of phases of density waves of different scales affect the richness of galaxy systems: the larger the scale of density waves, where the maxima of waves of different scales have close positions, the larger are the masses of galaxy clusters.", "In lower density regions the formation of rich clusters is suppressed by the combined negative sections of medium- and large- scale density perturbations.", "This process makes voids empty of galaxies and their systems, clusters of galaxies can be found in filaments crossing the voids, and they are not so rich as galaxy clusters in higher density regions.", "This is what we found in this study.", "Richer clusters from our sample are located in regions of high environmental density, in agreement also with earlier results about observations and simulations [19], [22], [28], [14], where positive sections of medium- and large-scale density perturbations combine.", "The most luminous clusters are located in high-density cores of rich superclusters, five of seven most luminous clusters are in spider- type superclusters.", "From the other hand, [24] studied the morphology of superclusters from SDSS DR7 and found that among them there are no compact and very luminous superclusters.", "Poor superclusters have lower luminosities and they host clusters of lower luminosity.", "Correlations between the cluster parametres and the environmental density are stronger at small smoothing lengths and become weaker as we increase the smoothing length, in agreement with [19] and [19] who showed that the properties of galaxy groups depend on environmental density up to scales of about 15 – 20 $h^{-1}$  Mpc.", "Simulations show that also halo spin, overall shape and other properties depend on environment [28], [38], [96], [60], [95], [13] although [45] and [83] showed that dependence on environment is weaker than found in other studies.", "Summarising, our conclusions are as follows.", "1) Clusters from our sample are located in density peaks in filaments crossing voids and in superclusters.", "2) The values of the environmental densities around multimodal clusters (i.e.", "those with large number of components, high probabilities to have substructure, and large peculiar velocities of their main galaxies) are higher than those around unimodal clusters.", "3) We determined the values of the fourth Minkowski functional and shapefinders, and morphological types for each supercluster hosting clusters from our sample.", "Of 50 superclusters hosting rich clusters 35 are of spider type and 15 of filament type.", "4) Clusters in superclusters of spider morphology have higher probabilities to have substructure, and higher values of the peculiar velocities of their main galaxies than clusters in superclusters of filament morphology.", "5) High-luminosity clusters reside in the cores of rich superclusters.", "Five out of seven high-luminosity clusters belong to superclusters of spider morphology.", "The most multimodal clusters are preferentially located in spider-type superclusters, while four of seven most unimodal clusters reside in filament-type superclusters.", "6) Isolated clusters are poorer and they have smaller maximal number of components and lower maximal (but higher median) values of the peculiar velocities of their main galaxies than supercluster members.", "7) High luminosity superclusters have higher values of the environmental densities and peak densities than low luminosity superclusters.", "Our study shows the importance of the role of superclusters as high density environment which affects the properties of galaxies and galaxy systems in them [67], [97], [39], [26], [70], [90], [34], [91], [25].", "Earlier studies of galaxy superclusters revealed that while according to their overall properties superclusters can be described with a small number of parametres [23], the analysis of the morphology and galaxy and group content of the richest superclusters from the 2dF Galaxy Redshift Survey and from the Sloan Great Wall with SDSS data [27], [25] demonstrate a large variety of supercluster morphologies and differences in the distribution of galaxies from various populations, and groups of galaxies in superclusters.", "Observations have already revealed large differences between galaxy and group content in high- redshift superclusters [54], [79].", "Such a large variety of observational properties is not yet recovered by simulations [26], [25] and not well understood from observations.", "As a next step we plan to study the properties of a large sample of groups and clusters in superclusters of different morphology to better understand the differences and similarities between them." ], [ "Acknowledgments", "We thank our referee for very detailed comments which helped to improve the paper.", "We are pleased to thank the SDSS Team for the publicly available data releases.", "Funding for the Sloan Digital Sky Survey (SDSS) and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, and the Max Planck Society, and the Higher Education Funding Council for England.", "The SDSS Web site is http://www.sdss.org/.", "The SDSS is managed by the Astrophysical Research Consortium (ARC) for the Participating Institutions.", "The Participating Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, University of Cambridge, Case Western Reserve University, The University of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, The Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of Washington.", "The present study was supported by the Estonian Science Foundation grants No.", "8005, 7765, 9428, and MJD272, by the Estonian Ministry for Education and Science research project SF0060067s08, and by the European Structural Funds grant for the Centre of Excellence \"Dark Matter in (Astro)particle Physics and Cosmology\" TK120.", "This work has also been supported by ICRAnet through a professorship for Jaan Einasto.", "P.N.", "was supported by the Academy of Finland, P.H.", "by Turku University Foundation.", "V. M. was supported by the Spanish MICINN CONSOLIDER projects ATA2006-14056 and CSD2007-00060, including FEDER contributions, and by the Generalitat Valenciana project of excellence PROMETEO/2009/064." ], [ "Multimodality of clusters", "We employ two 3D methods to search for substructure in clusters.", "With Mclust package [36] from R, an open-source free statistical environment developed under the GNU GPL [43] we search for an optimal model for the clustering of the data among models with varying shape, orientation and volume, under assumption that the multivariate sample is a random sample from a mixture of multivariate normal distributions.", "Mclust finds the optimal number of components and calculates for every galaxy the probabilities to belong to any of the components which are used to calculate the uncertainties for galaxies to belong to a component.", "The mean uncertainty for the full sample is used as a statistical estimate of the reliability of the results.", "The calculations in E12 showed that uncertainties are small, their values can be found in online tables of E12.", "We tested how the possible errors in the line-of-sight positions of galaxies affect the results of Mclust, shifting randomly the peculiar velocities of galaxies 1000 times and searching each time for the components with Mclust.", "The random shifts were chosen from a Gaussian distribution with the dispersion equal to the sample velocity dispersion of galaxies in a cluster.", "The number of the components found by Mclust remained unchanged, demonstrating that the results of Mclust are not sensitive to such errors.", "The Dressler-Shectman (DS) test searches for deviations of the local velocity mean and dispersion from the cluster mean values.", "The algorithm starts by calculating the mean velocity ($v_\\mathrm {local}$ ) and the velocity dispersion ($\\sigma _\\mathrm {local}$ ) for each galaxy of the cluster, using its $n$ nearest neighbours.", "These values of local kinematics are compared with the mean velocity ($v_c$ ) and the velocity dispersion ($\\sigma _c$ ) determined for the entire cluster of $N_{gal}$ galaxies.", "The differences between the local and global kinematics are quantified by $ \\delta _i^2 = (n+1)/\\sigma _c^2[(v_\\mathrm {local}-v_c)^2 + (\\sigma _\\mathrm {local}-\\sigma _c)^2].", "$ The cumulative deviation $\\Delta = \\Sigma \\delta _i$ is used as a statistic for quantifying (the significance of) the substructure.", "The results of the DS-test depend on the number of local galaxies $n$ .", "We use $n = \\sqrt{N}_{gal}$ , as suggested by [66].", "The $\\Delta $ statistic for each cluster should be calibrated by Monte Carlo simulations.", "In Monte Carlo models the velocities of galaxies are randomly shuffled among the positions.", "We ran 25000 models for each cluster and calculated every time $\\Delta _\\mathrm {sim}$ .", "The significance of having substructure (the $p_\\Delta $ -value) can be quantified by the ratio $N(\\Delta _\\mathrm {sim} > \\Delta _{obs})/N_\\mathrm {sim}$ – the ratio of the number of simulations in which the value of $\\Delta $ is larger than the observed value, and the total number of simulations.", "The smaller the $p_\\Delta $ -value, the larger is the probability of substructure." ], [ "Luminosity density field and superclusters", "To calculate the luminosity density field, we calculate the luminosities of groups first.", "In flux-limited samples, galaxies outside the observational window remain unobserved.", "To take into account the luminosities of the galaxies that lie outside the sample limits also we multiply the observed galaxy luminosities by the weight $W_d$ .", "The distance-dependent weight factor $W_d$ was calculated as $W_d = {\\frac{\\int _0^\\infty L\\,n(L)\\mathrm {d}L}{\\int _{L_1}^{L_2} L\\,n(L)\\mathrm {d}L}} ,$ where $L_{1,2}=L_{} 10^{0.4(M_{}-M_{1,2})}$ are the luminosity limits of the observational window at a distance $d$ , corresponding to the absolute magnitude limits of the window $M_1$ and $M_2$ ; we took $M_{}=4.64$  mag in the $r$ -band [8].", "Due to their peculiar velocities, the distances of galaxies are somewhat uncertain; if the galaxy belongs to a group, we use the group distance to determine the weight factor.", "Details of the calculations of weights are given also in [91] and in [23].", "To calculate a luminosity density field, we convert the spatial positions of galaxies $\\mathbf {r}_i$ and their luminosities $L_i$ into spatial (luminosity) densities using kernel densities [82]: $\\rho (\\mathbf {r}) = \\sum _i K\\left( \\mathbf {r} - \\mathbf {r}_i; a\\right) L_i,$ where the sum is over all galaxies, and $K\\left(\\mathbf {r};a\\right)$ is a kernel function of a width $a$ .", "Good kernels for calculating densities on a spatial grid are generated by box splines $B_J$ .", "Box splines are local and they are interpolating on a grid: $\\sum _i B_J \\left(x-i \\right) = 1,$ for any $x$ and a small number of indices that give non-zero values for $B_J(x)$ .", "We use the popular $B_3$ spline function: $B_3(x)&=&\\left(|x-2|^3-4|x-1|^3+6|x|^3-\\right.\\nonumber \\\\&&\\left.-4|x+1|^3+|x+2|^3\\right)/12.$ The (one-dimensional) $B_3$ box spline kernel $K_B^{(1)}$ of the width $a$ is defined as $K_B^{(1)}(x;a,\\delta ) = B_3(x/a)(\\delta / a),$ where $\\delta $ is the grid step.", "This kernel differs from zero only in the interval $x\\in [-2a,2a]$ .", "It is close to a Gaussian with $\\sigma =0.6$ in the region $x\\in [-a,a]$ , so its effective width is $2a$ [76].", "The kernel preserves the interpolation property exactly for all values of $a$ and $\\delta $ , where the ratio $a/\\delta $ is an integer.", "(This kernel can be used also if this ratio is not an integer, and $a \\gg \\delta $ ; the kernel sums to 1 in this case, too, with a very small error.)", "This means that if we apply this kernel to $N$ points on a one-dimensional grid, the sum of the densities over the grid is exactly $N$ .", "The three-dimensional kernel $K_B^{(3)}$ is given by the direct product of three one-dimensional kernels: $K_B^{(3)}(\\mathbf {r};a,\\delta ) \\equiv K_3^{(1)}(x;a,\\delta ) K_3^{(1)}(y;a,\\delta ) K_3^{(1)}(z;a,\\delta ),$ where $\\mathbf {r} \\equiv \\lbrace x,y,z\\rbrace $ .", "Although this is a direct product, it is isotropic to a good degree [76].", "The densities were calculated on a cartesian grid based on the SDSS $\\eta $ , $\\lambda $ coordinate system.", "The grid coordinates are calculated according to Eq.REF .", "We used an 1 $h^{-1}$  Mpc step grid and chose the kernel width $a=8$  $h^{-1}$  Mpc.", "This kernel differs from zero within the radius 16 $h^{-1}$  Mpc, but significantly so only inside the 8 $h^{-1}$  Mpc radius.", "As a lower limit for the volume of superclusters we used the value $(a/2)$  $h^{-1}$  Mpc$^3$ (64 grid cells).", "We also used density field with the kernel widths $a=4$  $h^{-1}$  Mpc, $a=8$  $h^{-1}$  Mpc, and $a=16$  $h^{-1}$  Mpc to characterise the environmental density around clusters.", "Before extracting superclusters we apply the DR7 mask constructed by P. Arnalte-Mur [56], [50] to the density field and convert densities into units of mean density.", "The mean density is defined as the average over all pixel values inside the mask.", "The mask is designed to follow the edges of the survey and the galaxy distribution inside the mask is assumed to be homogeneous." ], [ "Minkowski functionals and shapefinders", "For a given surface the four Minkowski functionals (from the first to the fourth) are proportional to the enclosed volume $V$ , the area of the surface $S$ , the integrated mean curvature $C$ , and the integrated Gaussian curvature $\\chi $ .", "Consider an excursion set $F_{\\phi _0}$ of a field $\\phi (\\mathbf {x})$ (the set of all points where the density is higher than a given limit, $\\phi (\\mathbf {x}\\ge \\phi _0$ )).", "Then, the first Minkowski functional (the volume functional) is the volume of this region (the excursion set): $V_0(\\phi _0)=\\int _{F_{\\phi _0}}\\mathrm {d}^3x\\;.$ The second Minkowski functional is proportional to the surface area of the boundary $\\delta F_\\phi $ of the excursion set: $V_1(\\phi _0)=\\frac{1}{6}\\int _{\\delta F_{\\phi _0}}\\mathrm {d}S(\\mathbf {x})\\;,$ (but it is not the area itself, notice the constant).", "The third Minkowski functional is proportional to the integrated mean curvature $C$ of the boundary: $V_2(\\phi _0)=\\frac{1}{6\\pi }\\int _{\\delta F_{\\phi _0}}\\left(\\frac{1}{R_1(\\mathbf {x})}+\\frac{1}{R_2(\\mathbf {x})}\\right)\\mathrm {d}S(\\mathbf {x})\\;,$ where $R_1(\\mathbf {x})$ and $R_2(\\mathbf {x})$ are the principal radii of curvature of the boundary.", "The fourth Minkowski functional is proportional to the integrated Gaussian curvature (the Euler characteristic) of the boundary: $V_3(\\phi _0)=\\frac{1}{4\\pi }\\int _{\\delta F_{\\phi _0}}\\frac{1}{R_1(\\mathbf {x})R_2(\\mathbf {x})}\\mathrm {d}S(\\mathbf {x})\\;.$ At high (low) densities this functional gives us the number of isolated clumps (void bubbles) in the sample [57], [77]: $V_3=N_{\\mbox{clumps}} + N_{\\mbox{bubbles}} - N_{\\mbox{tunnels}}.$ As the argument labelling the isodensity surfaces, we chose the (excluded) mass fraction $mf$ – the ratio of the mass in the regions with the density lower than the density at the surface, to the total mass of the supercluster.", "When this ratio runs from 0 to 1, the iso-surfaces move from the outer limiting boundary into the centre of the supercluster, i.e., the fraction $mf=0$ corresponds to the whole supercluster, and $mf=1$ – to its highest density peak.", "The first three functionals were used to calculate the shapefinders [78], [80], [76].", "The shapefinders are defined as a set of combinations of Minkowski functionals: $H_1=3V/S$ (thickness), $H_2=S/C$ (width), and $H_3=C/4\\pi $ (length).", "The shapefinders have dimensions of length and are normalized to give $H_i=R$ for a sphere of radius $R$ .", "For smooth (ellipsoidal) surfaces, the shapefinders $H_i$ follow the inequalities $H_1\\le H_2\\le H_3$ .", "Oblate ellipsoids (pancakes) are characterised by $H_1 << H_2 \\approx H_3$ , while prolate ellipsoids (filaments) are described by $H_1 \\approx H_2 << H_3$ .", "[78] also defined two dimensionless shapefinders $K_1$ (planarity) and $K_2$ (filamentarity): $K_1 = (H_2 - H_1)/(H_2 + H_1)$ and $K_2 = (H_3 -H_2)/(H_3 + H_2)$ .", "In the $(K_1,K_2)$ -plane filaments are located near the $K_2$ -axis, pancakes near the $K_1$ -axis, and ribbons along the diameter, connecting the spheres at the origin with the ideal ribbon at $(1,1)$ .", "In [26] we calculated typical morphological signatures of a series of empirical models that serve as morphological templates to compare with the characteristic curves for superclusters in the $(K_1,K_2)$ -plane." ], [ "Data on selected clusters", "5 Table: Data on clusters5 Table: ...continued6 Table: Data on superclusters" ] ]

[ [ "Sodium Atoms in the Lunar Exotail: Observed Velocity and Spatial\n Distributions" ], [ "Abstract The lunar sodium tail extends long distances due to radiation pressure on sodium atoms in the lunar exosphere.", "Our earlier observations measured the average radial velocity of sodium atoms moving down the lunar tail beyond Earth (i.e., near the anti-lunar point) to be $\\sim 12.5$ km/s.", "Here we use the Wisconsin H-alpha Mapper to obtain the first kinematically resolved maps of the intensity and velocity distribution of this emission over a $15 \\times 15 \\deg$ region on the sky near the anti-lunar point.", "We present both spatially and spectrally resolved observations obtained over four nights bracketing new Moon in October 2007.", "The spatial distribution of the sodium atoms is elongated along the ecliptic with the location of the peak intensity drifting $3 \\deg$ east along the ecliptic per night.", "Preliminary modeling results suggest the spatial and velocity distributions in the sodium exotail are sensitive to the near surface lunar sodium velocity distribution.", "Future observations of this sort along with detailed modeling offer new opportunities to describe the time history of lunar surface sputtering over several days." ], [ "Introduction", "The Moon is known to have a trace atmosphere of helium (He), argon (Ar), sodium (Na), potassium (K) and other trace species [see e.g., Stern 1999]; however its tenuous nature makes remote observations of elements other than the alkalis difficult.", "Sodium “D-line” emission at 5895.924 Å (D1) and 5889.950 Å (D2) has been used since the late 1980s to observe and interpret the morphology of the lunar sodium atmosphere beginning with its detection by Potter and Morgan [1988] and Tyler et al.", "[1988].", "Likely source mechanisms are: thermal desorption, photo-desorption, ion sputtering and meteoric impact ablation.", "The relative importance of these mechanisms remains uncertain, both with regard to spatial and to temporal trends.", "Once released, sputtered gases in the lunar atmosphere can be pulled back to the regolith by gravity, escape to space, get pushed away by solar radiation pressure, or become photoionized and swept away by the solar wind.", "Mendillo et al.", "[1991] obtained the first broadband imaging observations (D1 + D2) of the extended lunar sodium atmosphere, observing emission out to $\\sim $ 5 lunar radii ($R_{m}$ ) on the dayside, and out to 15–20 $R_{m}$ in a “tail-like” structure on the nightside.", "The lunar sodium tail is now known to extend to great distances (many hundreds, and perhaps thousands, of lunar radii) due to the strong influence of the Sun's radiation pressure on Na atoms in the lunar exosphere [Wilson et al., 1999].", "Near new Moon phase, the extended lunar sodium tail can be observed as it sweeps over the Earth and is gravitationally-focused into a visible sodium “spot” in the anti-solar direction [Smith et al., 1999; Matta et al., 2009].", "Refer to Figure 1.", "Velocity resolved observations of the extended lunar sodium tail were first obtained by Mierkiewicz et al.", "[2006a] using a large aperture (15 cm) double etalon Fabry-Perot spectrometer with $\\sim 3.5$ km/s spectral resolution at 5890 Å; see Mierkiewicz et al.", "[2006b] for further instrument details.", "Observations were made within 2–14 hours of new Moon from the Pine Bluff Observatory (PBO), Wisconsin, on 29 March 2006, 27 April 2006 and 28 April 2006.", "The average observed radial velocity of the lunar sodium tail in the vicinity of the anti-solar/lunar point for the three nights was $\\sim 12.5$ km/s (from geocentric zero).", "This velocity is consistent with sodium atoms escaping from the Moon and being accelerated by radiation pressure for 2+ days.", "In some cases the line profile appeared asymmetric, with lunar sodium emission well in excess of 12.5 km/s.", "In this paper we report new results using the unique mapping capabilities of the Wisconsin H-alpha Mapper (WHAM), where we have traded spectral resolution in favor of increased sensitivity and spatial resolution." ], [ "Instrumentation", "WHAM was built to map the distribution and kinematics of ionized gas in our Galaxy [Haffner et al., 2003].", "Here we leverage WHAM's unique combination of high sensitivity, spectral resolution and automated pointing capabilities to map Na emission in the extended lunar sodium tail.", "At the time of these observations WHAM was located at Kitt Peak Observatory, AZ.", "Similar in design to the PBO Fabry-Perot used in our earlier work, WHAM is a large aperture (15 cm) double etalon Fabry-Perot coupled to a siderostat with a circular $1 \\deg $ field-of-view (FOV) (compared with $1.5 \\deg $ for PBO) .", "WHAM has a resolving power of 25,000, covering a 200 km/s spectral region with 12 km/s spectral resolution at 5890 Å [Haffner et al., 2003]." ], [ "Observations & Reduction", "Using WHAM we have mapped the intensity and velocity distribution of the extended lunar sodium tail over a $15 \\times 15 \\deg $ region near the anti-lunar point with $1 \\deg $ spatial resolution.", "Observations were made during 4 nights bracketing the 11 October 2007 (5:01 UT) new moon period.", "The automated pointing capability of WHAM was used to build a map of the lunar Na emission by rastering WHAM's $1 \\deg $ circular FOV in a “block” surrounding the anti-lunar point; see Figure 2.", "The number of $1 \\deg $ pointings per block was between 121 and 256.", "The exposure time for each observation was 120 s; a map was generated in approximately 6 hours.", "These observations provide the first kinematically resolved maps of the extended lunar sodium tail observed in the anti-lunar direction.", "Individual WHAM spectra were reduced with a four-component model: one Gaussian component for an atmospheric OH feature near -90 km/s from geocentric zero, a second Gaussian component for an unidentified feature near -70 km/s, a Voigt profile for the terrestrial sodium emission at 0 km/s and a Gaussian component for the lunar sodium emission near 13 km/s; refer to Figure 2a.", "Due to the partial blending of the terrestrial and lunar Na emission, the component fitting of the WHAM data required two steps.", "First, a constrained fit was applied to the data in which the Doppler shift of the lunar emission with respect to the terrestrial sodium sky glow line was fixed at 12.5 km/s based on our experience with the higher resolution PBO observations.", "Next, after a best-fit solution was obtained, the Doppler separation of the lunar emission was freed and the fitting routine was run again.", "In all cases, fit components were convolved with an instrumental profile and iterated, subject to the above constraints, to produce a least squares, best-fit to the data using the VoigtFit code of Woodward [private communication, 2012].", "Briefly, VoigtFit is a parameter estimation package for the analysis of spectral data.", "VoigtFit uses the Levenberg-Marquardt method of estimating parameters by minimizing chi-square using a hybrid of the steepest-decent and quadratic (Hessian) methods; the Levenberg-Marquardt method is described in Numerical Recipes [Press et al., 1986].", "Of particular importance here is VoigtFit's ability to: 1) analytically link parameters, which allows overlapping or faint lines to be fit without unnecessary free parameters, and 2) incorporate an empirical instrumental profile into the fitting process (obtained from a spectrum of a thorium emission line from a hollow cathode lamp) [Woodward, private communication, 2012].", "After fitting, we plot the lunar sodium emission spectra arranged according to their positions on the sky (Figure REF b).", "The intensity (i.e., the integrated area under the emission line, converted to Rayleighs, where 1R $= 10^{6}/4\\pi $ photons cm$^{-2}$ s$^{-1}$ str$^{-1}$ ) of the lunar sodium emission is then converted into colored beams representing WHAM's $1 \\deg $ FOV (Figure REF c).", "These beams are then smoothed (Figure REF d).", "We also generate a spatial map for the Doppler widths (full-width-at-half-maximum) of the lunar sodium emission lines.", "Intensity calibration is based on the surface brightness of NGC 7000, the “North American Nebula” (coordinates $\\alpha _{2000} = 20.97$ h, $\\delta _{2000} = 44.59 \\deg $ ).", "The NGC 7000 hydrogen Balmer $\\alpha $ (6563 Å) surface brightness is $\\sim 800$ R over WHAM's $1 \\deg $ FOV [Haffner et al., 2003].", "Sodium D2 line intensities are based on the assumption that WHAM's efficiency is unchanged between 6563 Å and 5890 Å, and a measured filter transmission ratio of T(5890 Å)/T(6563 Å) $\\sim 1$ .", "We estimate a 25% uncertainty in our sodium D2 absolute intensity calibration." ], [ "Results", "Spectra for the brightest beam for each night are given in Figure 3 and Table 1.", "Intensity and Doppler width maps for all pointings are given in Figure 4.", "In order to determine the sky background, we mapped a region of the sky $60 \\deg $ east of the anti-lunar point at $-20.15 \\deg $ ecliptic latitude and $57.82 \\deg $ ecliptic longitude.", "This off-direction dataset was processed with the Gaussian fitting procedure used in the analysis of the on-direction datasets as described in Section 3.", "We found intensities greater than 0.7 R to clearly be lunar Na emission (see Figure 4).", "In what follows we present a basic description of the observations for each night.", "10 October 2007: Observations were taken 22 to 16 hours before new Moon (first column of Figure 4), 169 one $\\deg $ pointings.", "The intensity distribution is elongated along the ecliptic with the location of the peak intensity to the southwest of the antisolar point.", "The Doppler width distribution appears to peak to the northeast of the brightest emission, and is also elongated along the ecliptic.", "The broadest emission is also the faintest emission for this night.", "The brightest emission occurs approximately $10.8 \\deg $ from the antisolar point, southwest along the ecliptic.", "11 October 2007: Observations were taken 1 hour before to 7 hours after new Moon (second column of Figure 4), 225 one $\\deg $ pointings.", "The intensity distribution is nearly axially symmetric; the Sun, Moon and Earth are nearly aligned, and we are looking almost directly down the lunar tail.", "As with the night of 10 October, the peak intensity is to the southwest of the antisolar point.", "The broadest emission for this night is near the brightest emission.", "The peak emission occurs $5.8 \\deg $ southwest of the anti-solar point.", "12 October 2007: Observations were taken 23 to 32 hours after new Moon (third column of Figure 4), 256 one $\\deg $ pointings.", "The intensity distribution again appears to be elongated along the ecliptic with the brightest emission occurring closer to the anti-solar point and the fainter emission occurring farther to the southwest.", "From the Doppler width map, there is a large region where the widths are roughly 15-20 km/s, with the broadest emission falling roughly in the same location as the brightest emission, $2.7 \\deg $ from the anti-solar point.", "13 October 2007: Observations were taken 48 to 52 hours after new Moon (fourth column of Figure 4), 121 one $\\deg $ pointings.", "The emission is extremely faint and remains elongated along the ecliptic.", "The broadest emission, upwards of 30 km/s, for this night appears to peak to the southwest of the brightest emission.", "The brightest emission is $1.8 \\deg $ to the southwest of the anti-solar point." ], [ "Sample Model Runs", "Here we present a sample set of data/model comparisons using the numerical Monte-Carlo lunar exospheric simulation model of Wilson et al.", "[1999; 2003].", "We include these sample model runs to illustrate the potential of future model/data analysis.", "The Wilson et al.", "[1999; 2003] Monte-Carlo model uses fourth-order Runga-Kutta integration to compute the accelerations and positions of $\\sim 10^{6}$ lunar exospheric sodium atoms due to gravitational effects and solar radiation pressure (a function of the atom's heliocentric distance and velocity); radiation pressure shadowing by the Moon and Earth is included in the model.", "The model accounts for the motion of the Moon around the Earth and the Earth around the Sun.", "Atom loss due to impact onto the Earth and by photoionization is also taken into account.", "A photoionization lifetime of 47 hours [Heubner et al., 1992] was used in all model runs presented here.", "In each model run, sodium atoms were randomly ejected from the Moon beginning 4 days before the desired simulation snapshot to ensure that the lunar exosphere/tail was well populated.", "The initial velocities at the Moon in the x, y and z directions were randomly determined from 0.1 km/s velocity bins between 2.1 and 2.6 km/s.", "The atom's ejection angle was set by the x, y and z components, resulting in an isotropic angular distribution.", "The relative fraction of particles in each velocity bin was a free model parameter (refer to Section 5.2).", "Because each of our maps was built over a $\\sim 6$ hour observing interval, a number of simulation snapshots (one per hour of observation) were computed for each observing interval and coadded to simulate the smearing inherent in our observations as the tail moves throughout the night.", "The sodium ejection rate was set to $1\\times 10^{22}$ atoms/s [Wilson et al., 1999].", "Note, although the ejection rate primarily controls the intensity of the sodium tail and therefore may be treated as a retrieved model parameter, at this time we are not attempting to derive the ejection rate from our data.", "Here we remain focused on the initial velocity distributions and their effect on the morphology of the extended lunar tail.", "In the sections that follow, we compare the model and data both spatially (Section 5.1) and spectrally (Section 5.2)." ], [ "Spatial Map Comparisons", "Figure 5 shows the tail geometry at the time of our observations for each night, along with the model intensity distribution projected onto the sky, and our data for comparison.", "Figure 6 shows the tail line-of-sight velocities of the atoms with respect to Earth.", "These model runs are for a flat initial velocity distribution (see Figure 7 & Section 5.2).", "The line-of-sight through the tail for the anti-solar direction (dotted line Figures 5 & 6) and the direction of the brightest emission (solid line in Figures 5 & 6) are shown to help visualize how the tail appears in projection onto the plane of the nightsky.", "The dense core of the tail (see Figure 5) is due to gravitational focusing of the sodium atoms by the Earth, and hence the brightest emission tends to occur while looking down this portion of the tail.", "As seen in the data, and with agreement in the model runs, the brightest emission always occurs to the west of the anti-solar point and moves eastward over the course of our four nights of observations.", "This eastward drift is also seen as a decrease in the angle between the anti-solar line-of-sight and the brightest emission line-of-sight over our four nights of observations (see Figure 5).", "When the Moon is to the east of the Earth-Sun line, as it is on the night of 10 October, the sodium atoms are deflected to the west.", "This deflection is responsible for the large westward location of the brightest emission observed on this night.", "As the Moon moves to the west of the Earth-Sun line (see Figure 5), the Earth deflects the lunar sodium atoms to the east.", "This explains why the brightest emission on the sky appears closer to the anti-solar point on the nights of 12 and 13 October.", "In both the data and the model images in Figure 5 a dark spot appears near the anti-solar point.", "It's unclear to what extent the Earth's gravitational deflection may contribute to this dark spot, but the Earth's shadow must play a role as it reduces/prevents illumination by the Sun." ], [ "Spectral Comparisons", "Spectral averaging is a useful metric to explore how the velocity distribution evolves from night-to-night.", "Both the data and model averages were normalized for these comparisons.", "Model calculations were binned to a $1 \\deg $ spatial resolution to match the $1 \\deg $ spatial resolution of the WHAM spectrometer.", "Model spectra are generated with significantly higher velocity resolution than WHAM's spectral resolution of 12 km/s.", "As such, model output was convolved with a WHAM instrumental function in order to facilitate direct data/model comparisons.", "We investigated how variations in the initial sodium particle velocity distribution at the Moon manifest themselves in our observed line profiles from the 11 October 2007 dataset.", "The solid line in Figure 7 is an average line profile for all WHAM lunar sodium observations obtained on 11 October 2007 after removal of the fitted terrestrial sodium emission from the original data.", "The line profile was normalized to the peak intensity, ignoring the residual noise spikes associated with sodium terrestrial line subtraction.", "The dashed line in Figure 7 is the corresponding model run average, normalized to its peak intensity.", "The model run average spectra were computed from single model simulation snapshots for the time of new Moon.", "Three different initial velocity distributions were used in our data/model comparison: slow (initial velocities falling between 2.1 and 2.15 km/s), fast (initial velocities falling between 2.56 and 2.6 km/s) and flat (an equal number of particles leaving the Moon with velocities of 2.1–2.6 km/s).", "These velocity distributions are based on the work of Wilson et al.", "[1999; 2003].", "Although the initial velocity distribution was not fully constrained by Wilson et al.", "[1999; 2003], the 2003 paper did show that there is a significant contribution to the extended lunar Na tail from source speeds in the range of 2.1–2.4 km/s.", "Even though more sophisticated velocity distributions, such as a Maxwell-Boltzman distribution, could be used, here we simply wanted to explore population extremes in order to determine the feasibility of this and future data to constrain the lunar surface velocity distribution.", "Inspection of Figure 7 indicates that the shape of the observed and modeled line profiles in the anti-lunar direction near the time of new Moon are sensitive to the initial velocity distribution of the sodium atoms leaving the Moon.", "A fast initial velocity distribution results in a broader average spectrum whereas a slow initial distribution produces a much narrower spectrum.", "It is important to emphasize that changes in the lunar source rates $\\sim 2$ days prior to the observations (when the observed Na atoms were being released from the lunar surface) could produce similar signatures, as would changes in the sodium photoionization lifetime due to solar activity.", "The flat velocity distribution produces a spectrum closest to that observed; see Figure 7, lower right.", "Figure 8 shows model runs for the other three nights of observations (together with 11 October), using the flat initial velocity distribution.", "As with Figure 7, the solid line in each panel of Figure 8 is an average line profile for all WHAM lunar sodium observations obtained on that night after removal of the fitted terrestrial sodium emission from the original data.", "The line profiles were normalized to the peak intensity, ignoring the residual noise spikes associated with sodium terrestrial line subtraction; the dashed line is the corresponding model run average, normalized to its peak intensity.", "In this case model spectra were computed and averaged over the observation time ranges for each night (refer to Figure 4).", "In both the data and model, the night of 13 October displays the broadest emission.", "Geometrically, as seen in Figure 6, our observations on this night have the best view down the tail, unobscured by the dense core, sampling the greatest range of radial velocities.", "This effect manifests itself in the Doppler width maps of Figure 4 (lower panel).", "The broadest emission observed on 13 October is $\\sim 30$ km/s; sodium atoms achieve these velocities $\\sim $ 1.5 million km (3.75 times the Earth-Moon distance) downwind from the Earth." ], [ "Conclusions", "Over the course of 4 nights ($\\sim $ 70 hours) of observations, the peak of the intensity distribution (see Figure 4 and Table 1) drifted east along the ecliptic a total $\\sim 11.5 \\deg $ (an average of $0.16 \\deg $ per hour), consistent with the 3–4 $\\deg $ eastward drift per day observed by Smith et al.", "[1999].", "For reference, the lunar motion is about $0.5 \\deg $ per hour eastward.", "The observed eastward drift of the brightest emission is due to a combination of the Moon's orbital motion and the gravitational deflection of sodium atoms by the Earth.", "The brightest emissions occurred on the nights of 11 and 12 October as the observation geometry on these nights presents the most direct look down the gravitationally focused part of the lunar tail (and hence the maximum column emission).", "The broadest line profiles were detected on the nights away from new Moon, occurring northeast of the peak intensity for the night preceding new Moon, and to the southwest following new Moon.", "Our preliminary modeling efforts suggest that the changes in the observed morphology are related to viewing geometry as the tail sweeps past the Earth (see Figures 5 & 6).", "At new Moon the nearby bright “core” of Na atoms, recently gravitationally focused by the Earth [Wilson et al., 1999], dominates the tailÕs appearance giving it a nearly axisymmetric core emission at 12.5 km/s, obscuring the dimmer signal of the older (and faster) more distant atoms.", "Before and after new Moon, however, the Earth's gravity more strongly influences atoms at one outer edge of the sodium tail and the tail is observed off-axis, leaving relatively more atoms in an extended diffuse “un-focused” tail away from the 12.5 km/s core emission and a correspondingly larger influence on the Doppler width observed in our data.", "Refer to Figures 4–6.", "Our sample model runs and recent work by Lee et al.", "[2011] confirm that velocity resolved observations and spatial mapping of the extended lunar tail offer new opportunities to describe the time history of lunar surface sputtering over several days, and set constraints for models of exospheric source mechanisms and their variabilities." ], [ "Acknowledgments", "The authors thank M. Mendillo and R. Reynolds for their valuable assistance as well as K. Nordsieck for providing the sodium filter.", "We thank all the members of the WHAM collaboration, in particular K. Jaehnig, A. Hill, G. Madsen and K. Barger.", "Finally, we thank the National Solar Observatory mountain support staff, C. Plymate and E. Galayda for their support and hosting us during the WHAM observations.", "M. Line's involvement as an undergraduate at Wisconsin was partially supported by a UW-Madison Hilldale Undergraduate Fellowship.", "This work was also funded by NASA award NNX11AE38G.", "WHAM construction and operations were primarily supported by the National Science Foundation; in particular, the use of WHAM described here was partially supported by awards AST-0607512 and AST-1108911." ] ]

[ [ "Mathematicians take a stand" ], [ "Abstract We survey the reasons for the ongoing boycott of the publisher Elsevier.", "We examine Elsevier's pricing and bundling policies, restrictions on dissemination by authors, and lapses in ethics and peer review, and we conclude with thoughts about the future of mathematical publishing." ], [ "=1 8.5in 11in [Mathematicians Take a Stand Douglas N. Arnold and Henry Cohn ] Mathematicians care deeply about the mathematical literature.", "We devote much of our lives to learning from it, expanding it, and guaranteeing its quality.", "We depend on it for our livelihoods, and our contributions to it will be our intellectual legacy.$\\!\\!\\!\\!$ Douglas N. Arnold is McKnight Presidential Professor of Mathematics at the University of Minnesota.", "His email address is [email protected].", "Henry Cohn is principal researcher at Microsoft Research New England and adjunct professor of mathematics at MIT.", "His e-mail address is [email protected].", "$\\!\\!\\!\\!$ Douglas N. Arnold is McKnight Presidential Professor of Mathematics at the University of Minnesota.", "His email address is [email protected].", "Henry Cohn is principal researcher at Microsoft Research New England and adjunct professor of mathematics at MIT.", "His e-mail address is [email protected].", "It has long been anticipated that technological advances will make the literature more affordable and accessible.", "Sadly, this potential is not being fully realized.", "The prices libraries pay for journals have been growing with no end in sight, even as the costs of publication and distribution have gone down, and many libraries are unable to maintain their subscriptions.For example, MIT's spending on serials increased by 426 percent over the period 1986–2009, while the number of serials purchased decreased by 16 percent, and the Consumer Price Index increased by only 96 percent.", "The normal market mechanisms we count on to keep prices in check have failed for a variety of reasons.", "For example, mathematicians have a professional obligation to follow the relevant literature, which leads to inelastic pricing.", "This situation is particularly perverse because we provide the content, editorial services, and peer review free of charge, implicitly subsidized by our institutions.", "The journal publishers then turn to the same institutions and demand prices that seem unjustifiable.", "Although the detailed situation is complex, the fundamental cause of this sad state of affairs is not hard to find.", "While libraries are being forced to cut acquisitions, a small number of commercial publishers have been making breathtaking profits year after year.", "The largest of these, Elsevier, made an adjusted operating profit of $1.12 billion in 2010 on $3.14 billion in revenue, for a profit margin of 36 percent, up from 35 percent in 2009 and 33 percent in 2008.Reed Elsevier Annual Report 2010, SEC form 20-F (based on data from p. 25 and average exchange rate from p. 6).", "Adding insult to injury, Elsevier has aggressively pushed bundling arrangements that result in libraries paying for journals they do not want and that obscure the actual costs.T.", "Bergstrom, Librarians and the terrible fix: economics of the Big Deal, Serials 23 (2010), 77–82.", "They have fought transparency of pricing, going so far as to seek a court injunction in an unsuccessful attempt to stop a state university from revealing the terms of their subscription contract.", "They have imposed unacceptable restrictions on dissemination by authors.", "And, while their best journals make important contributions to the mathematical literature, Elsevier also publishes many weaker journals, some of which have been caught in major lapses of peer review or ethical standards.", "These scandals have done harm to the integrity and reputation of mathematics.", "This situation has been extensively analyzed many times before, including in the Notices.", "There have been some high-profile actions, such as mass resignations of entire Elsevier editorial boards over pricing concerns: the Journal of Logic Programming in 1999, the Journal of Algorithms in 2003, and Topology in 2006.", "These boards have done a valuable service for the community by founding replacement journals, but there has been little relief from the overall trend.", "As Timothy Gowers wrote in his blog in January, “It might seem inexplicable that this situation has been allowed to continue.", "After all, mathematicians (and other scientists) have been complaining about it for a long time.", "Why can't we just tell Elsevier that we no longer wish to publish with them?” Gowers then revealed that he had been quietly boycotting Elsevier for years, and he suggested it would be valuable to create a website where like-minded researchers could publicly declare their unwillingness to contribute to Elsevier journals.", "Within days, Tyler Neylon responded to this need by creating http://thecostofknowledge.com.", "More than 2,000 people signed on in the first week, and participation has grown steadily since then, to over 8,000 as of early March.", "Each participant chooses whether to refrain from publishing papers in, refereeing for, or editing Elsevier journals.", "The boycott is ongoing, and it holds the promise of sparking real change.", "We urge you to consider adding your voice.", "The boycott is a true grassroots movement.", "No individual or group is in charge, beyond Gowers's symbolic position as the first boycotter.", "However, a group of thirty-four mathematiciansScott Aaronson, Douglas N. Arnold, Artur Avila, John Baez, Folkmar Bornemann, Danny Calegari, Henry Cohn, Ingrid Daubechies, Jordan Ellenberg, Matthew Emerton, Marie Farge, David Gabai, Timothy Gowers, Ben Green, Martin Grötschel, Michael Harris, Frédéric Hélein, Rob Kirby, Vincent Lafforgue, Gregory F. Lawler, Randall J. LeVeque, László Lovász, Peter J. Olver, Olof Sisask, Terence Tao, Richard Taylor, Bernard Teissier, Burt Totaro, Lloyd N. Trefethen, Takashi Tsuboi, Marie-France Vignéras, Wendelin Werner, Amie Wilkinson, and Günter M. Ziegler.", "(including Gowers and the authors of the present article) issued their best attempt at a consensus statement of purpose for the boycott.", "It is available online,See http://umn.edu/~arnold/sop.pdf or the March 2012 London Mathematical Society Newsletter.", "and we highly recommend it for reading.", "For reasons of space, we will not cover every aspect of that statement here.", "Before we proceed, we must address two pressing questions about the boycott.", "First, why is a boycott appropriate?", "After all, Elsevier employs many reasonable and thoughtful people, and many mathematicians volunteer their services, helping to produce journals of real value.", "Isn't a boycott overly confrontational?", "Could one not take a more collaborative approach?", "Unfortunately, such approaches have been tried time and again without success.", "Fifteen years of reasoned discussions have failed to sway Elsevier.R.", "Kirby, Comparative prices of math journals, 1997, http://math.berkeley.edu/~kirby/journals.html; J. Birman, Scientific publishing: a mathematician's viewpoint, Notices of the AMS 47 (2000), 770–774; R. Kirby, Fleeced?, Notices of the AMS 51 (2004), 181; W. Neumann, What we can do about journal pricing, 2005, http://www.math.columbia.edu/~neumann/journal.html; D. N. Arnold, Integrity under attack: the state of scholarly publishing, SIAM News 42 (2009), 2–3; P. Olver, Journals in flux, Notices of the AMS 58 (2011), 1124–1126.", "Elsevier's leadership seems to be driven only by their fiduciary responsibility to maximize profit for their shareholders.", "The one hope we see for change is to demonstrate that their business depends on us and that we will not cooperate with them unless they earn our respect and goodwill.", "Second, why is the focus solely on Elsevier?", "Some of the problems we discuss are common among large commercial publishers, and indeed we hope the boycott will help spur changes in the whole industry.", "But we must start somewhere, and we believe it is more effective to focus on one publisher whose behavior has been particularly egregious than to directly confront an entire industry at once.", "Many of the successful boycotts in history took the same tack.", "Table: Historical prices per page in constant 2012 dollars." ] ]

[ [ "X-Ray Emission from the Supergiant Shell in IC 2574" ], [ "Abstract The M81 group member dwarf galaxy IC 2574 hosts a supergiant shell of current and recent star-formation activity surrounding a 1000 x 500 pc hole in the ambient Hi gas distribution.", "Chandra X-ray Observatory imaging observations reveal a luminous, L_x ~ 6.5 x 10^{38} erg/s in the 0.3 - 8.0 keV band, point-like source within the hole but offset from its center and fainter diffuse emission extending throughout and beyond the hole.", "The star formation history at the location of the point source indicates a burst of star formation beginning ~25 Myr ago and currently weakening and there is a young nearby star cluster, at least 5 Myr old, bracketing the likely age of the X-ray source at between 5 and ~25 Myr.", "The source is thus likely a bright high-mass X-ray binary --- either a neutron star or black hole accreting from an early B star undergoing thermal-timescale mass transfer through Roche lobe overflow.", "The properties of the residual diffuse X-ray emission are consistent with those expected from hot gas associated with the recent star-formation activity in the region." ], [ "Introduction", "IC 2574 is a low metallicity [12], [10], [4], gas-rich, dwarf irregular galaxy located 4.0 Mpc distant [5] in the M81 group of galaxies.", "The galaxy hosts numerous Hi holes and shells [19], [18], one of which is located in the most prominent region of current and recent star-formation and is near the outskirts of the galaxy.", "This 1000$\\times $ 500 pc supergiant shell has been studied intensively at UV [17], optical [13], [21], mid-infrared [2], and radio wavelengths [19], [18].", "Recent analysis of resolved stars imaged with Hubble [21] shows that the most significant episodes of star formation began nearly 100 Myr ago and isolated bursts along the shell periphery are as young as 10 Myr.", "The ages of the younger star-formation events are consistent with those derived from broadband photometry [17], [2], [13] and only slightly younger than the estimated dynamical age of the Hi shell, 14 Myr [19], [18].", "[19] also report analysis of soft X-ray emission from the supergiant shell observed with the ROSAT PSPC.", "They found that the emission was extended beyond the $\\sim $ 28 FWHM instrumental point-spread function (PSF) with a surface brightness peak near the center of the Hi hole.", "Based on spectral hardness ratios, [19] compute a plasma temperature of $\\sim $ 0.54 keV, a luminosity of 1.6$\\times $ 10$^{38}$  erg s$^{-1}$ in the 0.3$-$ 2.4 keV range, and an electron density of 0.03 cm$^{-3}$ (assuming a metallicity $Z/Z_{\\odot }=0.15$ and a spherical emitting volume of 700 pc diameter).", "Here, we analyze higher resolution Chandra spectrophotometric imaging observations of the supergiant shell.", "We show that the bulk of the emission arises from a bright point source located within the Hi hole (§ ) with a mildly-absorbed hard power law spectrum (photon index 1.6) and intrinsic luminosity 6.5$\\times $ 10$^{38}$  erg s$^{-1}$ in the 0.3$-$ 8.0 keV range.", "There remains residual X-ray emission (§ ) which can be characterized as a diffuse thermal plasma (temperature 0.5 keV, 0.3$-$ 8.0 keV luminosity 2.7$\\times $ 10$^{37}$  erg s$^{-1}$ ).", "The emission extends throughout and beyond the Hi hole and is likely the result of the recent star formation activity.", "We then determine the mass, age, and extinction towards several young star clusters inside and on the supergiant shell (§ ) and use this to argue that the bright point source is a high-mass X-ray binary (HMXB) with an early B star companion (§ ).", "The residual X-ray emission accounts for no more than $\\sim $ 1% of the mechanical energy produced by massive stars and supernovae in the massive star cluster in the Hi hole." ], [ "X-ray Image Analysis", "IC 2574 was observed with the Chandra X-ray Observatory Advanced CCD Imaging Spectrometer (ACIS) on 2000 January 7 (ObsID 792) and again on 2008 June 30 (ObsID 9541).", "Both observations were of approximately 10 ks duration and are sensitive to on-axis point-like sources as faint as $\\sim $ 1.5$\\times $ 10$^{37}$  erg s$^{-1}$ .", "Here, the earlier observation is analyzed because the Hi hole was imaged close to the aimpoint and the back-illuminated S3 CCD used for this observation has higher effective area below 1 keV, where most of the X-ray emission is expected from a hot gas bubble, than the front-illuminated ACIS-I array used for the latter observation.", "We also note that the observation in 2000 was taken early enough in the mission that the soft response of the ACIS CCDs was not yet compromised by the buildup of material on their Optical Blocking Filters.", "The observation was made in the full-frame timed exposure mode using the standard 3.2 s frame time and the VFAINT telemetry format.", "The data were reprocessed beginning with the Level 1 event list.", "A time-dependent gain correction was applied and pixel randomization was removed using the Chandra X-ray Center's CIAO ver.", "4.3 tool acis_process_events.", "The standard ASCA event grades were selected and hot pixels, columns, and cosmic-ray afterglows were removed to create a Level 2 event list for analysis.", "An $\\sim $ 800 s period of high background near the end of the observation was omitted leaving a usable exposure time of $\\sim $ 9 ks." ], [ "CXOU J102843.0$+$ 682816", "There is a bright X-ray source located inside the supergiant shell.", "[19] judged it to be extended based on a ROSAT PSPC observation.", "However, it is clear in the Chandra image that most of the X-ray emission is contributed from a single point-like source.", "An elliptical Gaussian model (plus a constant term) was fit to the spatial distribution of X-ray events in the 0.3$-$ 8.0 keV range to determine the source centroid, Gaussian width, and approximate source asymmetry.", "The best-fitting source position is 10$^{\\rm h}$ 28$^{\\rm m}$ 43.08$^{\\rm s}$ , $+$ 682816.37.", "The source is symmetrical within measurement errors with an rms Gaussian width of 0.315$\\pm $ 0.004.", "A radial profile of the data was compared to a simulation of the PSF built using the Chandra Ray Tracer (ChaRT) and MARX suites of programs available from the Chandra X-ray Center.", "Fitting a Gaussian to the profile of the data and of the simulated PSF (plus a Lorentzian for the wings and a constant for the background) results in a best-fit width of the simulated PSF slightly larger than the observed source width.", "The source is clearly point-like.", "Events were extracted from a 3 radius region surrounding the source for spectral analysis.", "This encircles roughly 99% of the 0.3$-$ 8.0 keV flux from the source.", "A background spectrum was extracted from the surrounding region.", "An absorbed power law model, including a multiplicative pileup correction, was fit to the background-subtracted spectrum using the XSPEC ver.", "11.3.2ag analysis tool.", "The best-fitting model column density is (5.4$\\pm $ 5.8)$\\times $ 10$^{20}$  cm$^{-2}$ which is roughly the Galactic line-of-sight value of 2.2$\\times $ 10$^{20}$  cm$^{-2}$ , the best-fitting power law index is 1.6$\\pm $ 0.6 ($\\chi ^2=20.1$ for 31 dof), and the intrinsic X-ray luminosity is (6.5$\\pm $ 1.5)$\\times $ 10$^{38}$  erg s$^{-1}$ after correcting for $\\sim $ 10% pileup.", "A thermal plasma model also provides an acceptable fit to the point source spectrum but the resulting temperature, $kT_e \\sim 5$ to 18 keV, and X-ray luminosity are much too high to be representative of a hot gas phase associated with star formation.", "lcccrrr 7 0pt Properties of Star-forming Regions Region Age Mass $A_{V}$ $L_{\\rm W}$ Net Counts $L_{\\rm X}^{\\rm int}$ (Myr) ($10^{5}$  $M_{\\odot }$ ) (mag) (10$^{38}$  erg s$^{-1}$ ) (10$^{35}$  erg s$^{-1}$ ) C1 7 0.8 0.0 27 6.8$\\pm $ 2.8 38$\\pm $ 16 C2 17 1.5 0.0 29 9.3$\\pm $ 3.3 52$\\pm $ 19 C3 17 0.9 0.1 16 1.0$\\pm $ 1.4 6$\\pm $ 8 C4 7 0.2 0.1 4.7 $-$ 0.4$\\pm $ 1.0 $-$ 2$\\pm $ 6" ], [ "Diffuse X-ray Emission", "There remains an excess of X-ray events within and around the Hi hole (66$\\times $ 32) after removing CXOU J102843.0$+$ 682816 from the image.", "This soft X-ray diffuse emission has low surface brightness, which leads to a grainy, low S/N, image.", "The emission can be seen more easily if the image is artificially smoothed.", "The point source was removed using a 3 radius circle, which is large enough to avoid leaving residual emission from the wings of the point source.", "Then, the excluded region was filled using the local background level Poisson method of the CIAO tool dmfilth.", "The filled image was then divided by a 0.5 keV monochromatic exposure map, then smoothed with the CIAO tool aconvolve using a Gaussian smoothing function with a width of 10 pixels ($\\sim $ 5).", "The left panel of Figure REF shows the smoothed soft X-ray diffuse map of a 100$\\times $ 100 region around the supergiant shell.", "The Hi hole is denoted by an ellipse and the location of CXOU J102843.0$+$ 682816 by a small circle.", "There is a net of 28.1$\\pm $ 5.3 X-ray events in the energy range of 0.3$-$ 8.0 keV inside the Hi hole after subtracting background.", "(The background was defined as emission in the S3 chip outside the optical extent of IC 2574, defined by an ellipse approximating the 25 mag s$^{-1}$ blue light isophote, after excluding any detected point sources.)", "We note that 90% of the emission is found in the soft band (0.3$-$ 2.0 keV).", "The right panel of Figure REF displays the radial profile of events in the 0.3$-$ 2.0 keV energy range of the diffuse emission after deprojecting the image.", "The deprojection was made assuming that the 66$\\times $ 32 elliptical Hi hole is actually circular.", "The radial profile is centered on the Hi hole center.", "It clearly shows the excess emission within the Hi hole and perhaps extending slightly beyond the hole.", "Note that there is no emission enhancement within the current star-forming regions in the shell surrounding the hole.", "The broad energy distribution of the X-ray events within the Hi hole was compared to a grid of models of an absorbed thermal plasma.", "Specifically, the ratios of observed events in the 0.3$-$ 1.0 to 0.3$-$ 8.0 and in the 0.3$-$ 1.0 to 0.3$-$ 2.0 bands most closely match a model of temperature 0.5$\\pm $ 0.1 keV and intervening absorption column of (4$\\pm $ 2)$\\times $ 10$^{20}$  cm$^{-2}$ where the errors denote the model grid spacing.", "This model results in a luminosity of (2.7$\\pm $ 0.5)$\\times $ 10$^{37}$  erg s$^{-1}$ in the 0.3$-$ 8.0 keV range where the error is based on the net source counts." ], [ "Mass, Age, and Extinction in Supergiant Shell Star-Forming Regions", "As mentioned in the introduction, several previous studies have provided estimates of the mass and/or age of star-forming regions associated with the supergiant shell.", "We have performed our own multi-wavelength analysis which produces self-consistent age, mass, and extinction estimates for individual (assumed co-eval) star clusters.", "The method is presented in detail in [22].", "Briefly, the analysis uses the Starburst99 stellar synthesis program [8] to compute star cluster model spectra as a parameterized function of cluster age, star formation history, and metallicity from a combination of stellar evolution tracks and a grid of stellar model atmospheres weighted by an initial mass function (IMF).", "We assume a (single) instantaneous burst history, solar metallicity, and a Salpeter IMF [15].", "Transmission of the composite stellar spectra through the surrounding ISM is modeled using a standard starburst dust extinction model [1] that modifies the blue portion of the spectrum and we assume that the dust re-emits this radiation as a blackbody [2] in the IR band.", "Finally, the (dust-modified) emergent model spectra are convolved with instrumental response functions appropriate to the various instruments and the best-fitting solution to the observed broadband photometry is determined.", "(The observations are corrected for a line-of-sight Galactic extinction of $A_V$ = 0.120 [16] using the [3] dust model.)", "This method self-consistently overcomes the age-extinction degeneracy when fitting models to UV-to-IR broadband photometry.", "Here, we use calibrated GALEX  FUV and NUV observations available through the GalexView data interfacehttp://galex.stsci.edu/GalexView/ and Spitzer 3.6, 4.5, and 24 $\\mu $ m observations processed and made available as part of the Spitzer Infrared Nearby Galaxies Survey [6].", "Images of the supergiant shell at several passbands are displayed in Figure REF .", "A continuum-subtracted H$\\alpha $ image observed with the Kitt Peak 2.1 m telescope using the T2KB CCD Imager in 2008 March (1.3 seeing) is also shown.", "Figure: NO_CAPTIONOur cluster selection is to be both FUV and 24 $\\mu $ m bright sources.", "Three clusters around the shell were identified from both FUV and 24 $\\mu $ m images.", "The UV bright cluster located in the Hi hole has been well studied [17], [13] therefore it is also investigated further, although it lacks 24 $\\mu $ m emission.", "Overall, the total of four isolated star clusters were analyzed.", "These are marked in the figure along with the position of CXOU J102843.0$+$ 682816 and the elliptical boundary of the Hi hole.", "The best-fit age, mass, and extinction derived for these clusters are listed in Table .", "Also listed are the current mechanical luminosities from stellar winds and supernovae, $L_{\\rm W}$ , corresponding to the best-fitting Starburst99 model, X-ray background-subtracted net counts in circular apertures of size equal to the 99% FUV encircled energy region in the 0.5$-$ 2.0 keV energy range, and the corresponding X-ray luminosity, $L_{\\rm X}$ , in the 0.5$-$ 2.0 keV energy range assuming a 0.5 keV thermal plasma X-ray spectral shape.", "All clusters are young and the masses of the clusters are on the order of 10$^{4}$  $M_{\\odot }$ with the exception of C2.", "C2 is the central stellar cluster located within the Hi hole surrounded by the supergiant shell.", "The age of this cluster has been estimated as 11 Myr from the FUV observation [17] and 5 Myr from the optical data [13].", "We obtain a slightly older age.", "[17] also estimate the mass of the cluster as 1.4$\\times $ 10$^{5}$  $M_{\\odot }$ consistent with the present result of 1.5$\\times $ 10$^{5}$  $M_{\\odot }$ .", "Both authors also found that the Hii regions in the vicinity of the cluster are younger, having ages of 1$-$ 3 Myr, which is less than the present results.", "(We note that we obtain ages of 2$-$ 3 Myr if we assume a canonical dust temperature of 70 K.) This age gradient supports the premise of propagating star formation such that the mechanical energy from the central cluster compresses the surrounding ISM and triggers subsequent star formation at the rim.", "The mass of C2 is large enough to drive star formation in the surroundings: The derived mechanical luminosity for C2 is 2.9 $\\times $ 10$^{39}$  erg s$^{-1}$ .", "The cumulative energy released is 2$\\times $ 10$^{54}$  erg over the lifetime of the cluster, which is much more than needed to create the hole [19]." ], [ "Summary and Discussion", "High-angular resolution Chandra spectrophotometric imaging of the Hi hole and surrounding supergiant shell in IC 2574 reveal a bright point source, CXOU J102843.0$+$ 682816, near the center of the Hi hole and residual low surface brightness soft X-ray emission throughout the hole and in the surrounding shell.", "Analysis of the UV-to-IR spectral energy distributions of several representative star clusters in and around the Hi hole results in age and mass estimates consistent with previous studies.", "In particular, there is a massive, 1.5$\\times $ 10$^5$  $M_{\\odot }$ , intermediate-age, 5$-$ 17 Myr, cluster near the center of the hole that has released enough mechanical energy from massive stars and supernovae over its lifetime to have created the hole.", "Our analysis is consistent with previous analyses of the supergiant shell in IC 2574 [19], [18], [17], [13], [21] and supports the classical picture of bubble formation and evolution such that younger star clusters on the periphery of the hole likely formed as a consequence of this stellar feedback sweeping ambient material into dense clouds that collapse under self-gravity and subsequently form new stars [7], [20], [11], [9].", "We have shown that the ratio of X-ray to mechanical luminosity in these young star clusters in the region is $\\lesssim $ 0.2%.", "Observed ratios of 0.1% to 1% are typical of many star-forming regions [22] but are poorly constrained theoretically because of uncertainties in ambient conditions, starburst histories, and three-dimensional geometries.", "For example, we note that the central cluster alone could be responsible for all the X-ray diffuse emission within the Hi hole; the mechanical luminosity from the central cluster is 2.9$\\times $ 10$^{39}$  erg s$^{-1}$ and the diffuse X-ray luminosity from the hole is 2.7$\\times $ 10$^{37}$  erg s$^{-1}$ .", "[21] estimate that the cumulative energy released from the whole region in the past $\\sim $ 25 Myr is $\\sim $ 10$^{55}$  erg.", "Even at a constant (current) rate, less than 0.1% of the total energy from stars has been lost to the X-ray radiation.", "Thus, the central cluster in the Hi hole alone can account for the diffuse X-ray emission and the creation of the Hi hole.", "Perhaps we are witnessing powerful positive feedback from this central cluster in the form of subsequent star formation in the surrounding supergiant shell.", "Most of the X-ray emission in the Hi hole in IC 2574 comes from the single bright source CXOU J102843.0$+$ 682816.", "The presence of such a luminous X-ray point source in even a massive supergiant shell is uncommon.", "Recent supernovae can be this luminous but they, too, are rare and typically much softer than CXOU J102843.0$+$ 682816.", "The high luminosity, hard spectral shape, and large numbers of recently-formed stars in the vicinity suggest that CXOU J102843.0$+$ 682816 is an HMXB.", "We can estimate the age of the binary system at between 5 Myr, the youngest age of the nearby massive star cluster, and 25 Myr, the age of the peak in the star-formation rate at the location of CXOU J102843.0$+$ 682816 as determined by [21].", "The compact object must then have had a minimum initial mass between about 10 and 40 $M_{\\odot }$ to have already evolved to a supernova endpoint.", "The companion is likely also of roughly this mass in order to provide the required high mass accretion rate.", "This rate can be estimated assuming $L_{\\rm X} \\sim L_{\\rm bol} \\sim \\eta \\dot{m} c^2$ with an efficiency, $\\eta \\sim 0.1$ , giving $\\dot{m} \\sim 10^{-7}$  $M_{\\odot }$  yr$^{-1}$ .", "According to [14], this requires a companion star mass of at least 8$-$ 10 $M_{\\odot }$ and higher if the system is older, if the bolometric luminosity exceeds the X-ray luminosity, or if mass is lost from the system.", "[14] also estimate the rate of formation of systems of this type scaled to a fiducial supernova rate.", "Their rate can be converted to a total number of systems in the central star cluster as there are about one $M>8$  $M_{\\odot }$ star (i.e., stars that eventually undergo supernovae) per 50 $M_{\\odot }$ of star formation for a Salpeter IMF.", "Thus, the [14] rate of $<$ 10$^{-4}$  yr$^{-1}$ per 8 $M_{\\odot }$ star translates to 0.3 luminous HMXBs for the central star cluster of mass 1.5$\\times $ 10$^5$  $M_{\\odot }$ .", "This is a high estimate because not all the potential compact objects have yet been created by supernova events.", "Thus, objects like CXOU J102843.0$+$ 682816 are indeed rare even for massive star clusters.", "We gratefully acknowledge the referee, Leisa Townsley, for her expert critique and especially for alerting us to the possibility of pile-up of the point source.", "Support for this research was provided in part by NASA through an Astrophysics Data Analysis Program grant NNX08AJ49G." ] ]

[ [ "On the cosmology of Weyl's gauge invariant gravity" ], [ "Abstract Recently the vector inflation has been proposed as the alternative to inflationary models based on scalar bosons and quintessence scalar fields.", "In the vector inflationary model, the vector field non-minimally couples to gravity.", "We should, however, inquire if there exists a relevant fundamental theory which supports the inflationary scenario.", "We investigate the possibility that Weyl's gauge gravity theory could be such a fundamental theory.", "That is the reason why the Weyl's gauge invariant vector and scalar fields are naturally introduced.", "After rescaling the Weyl's gauge invariant Lagrangian to the Einstein frame, we find that in four dimensions the Lagrangian is equivalent to Einstein-Proca theory and does not have the vector field non-minimally coupled to gravity, but has the scalar boson with a polynomial potential which leads to the spontaneously symmetry breakdown." ], [ "Introduction", "Inflationary models are proposed as some resolutions for the cosmological problems, e.g., the flatness, horizon and monopole problems.", "These successful models, for example, chaotic inflation [1], $k$ -inflation [2], are based on models of scalar bosons.", "The chaotic inflationary model have at least a difficulty in which bosonic fields could condensate some domains, i.e., in the early stage, some domains successfully exit but others keep expanding.", "In $k$ -inflation and the modified modes [3], these inflation are driven by non-minimal and non-canonical kinetic terms, but need some adjustments of conditions of scalar fields and its potentials.", "In addition we have detected no such scalar bosons by experiments.", "From these reasons, recently the vector inflation has been proposed by Ford [4] and some authors have studied the model [5], [6], [7].", "They consider the following action: $S=\\int _{}^{} {d^4x}\\sqrt{-g}\\left\\lbrace {\\frac{R}{16\\pi }-\\frac{1}{4}F^2-\\frac{1}{2} ( m^2+\\frac{R}{6} ) A^2} \\right\\rbrace \\,, $ where $R$ is the scalar curvature, $F$ denotes the field strength of vector field $A$ and $m$ is the mass of the vector field.", "It is worth noting that the massive vector field non-minimally couples to gravity in Eq.", "(REF ).", "The isotropy and the stability of the inflationary model have been discussed [6].", "The isotropy of expansion is achieved by N randomly organized vector fields or by a triplet of orthogonal vector fields.", "However, these discussions have been made to solve the cosmological problems from a aspect of cosmological observations.", "These models are assumed the bosonic inflatons with potentials that are not completely supported by fundamental physics.", "It is, therefore, necessary to investigate how the fundamental physical theories support the inflationary models.", "In the very early stage of our universe, the gravitational theory is expected to be different from the ordinary Einstein gravity [8], e.g.", "higher derivative gravity, scalar-tensor gravity.", "Indeed, quantum gravity or string corrections would affect the cosmological evolution near the Planck scale.", "In particular, gravitational theory could be speculated to be a scale invariant in this stage as well as other fundamental physics.", "In this paper, we study the possibility that the Weyl's gauge gravity is a such fundamental theory.", "Weyl's gauge theory of gravity is an extension of the Einstein gravity [9-24].", "Especially the vector and scalar bosons are naturally introduced in this theory by the scale invariant symmetry.", "We consider that Weyl's gauge invariant scalar and vector field are expected to play cosmological important role in the very early stage of our universe.", "In Sec.", "2, Weyl's gauge transformation is introduced as the local scale transformation.", "Then we construct the minimal Weyl's gauge invariant Lagrangian in arbitrary dimensions in Sec. 3.", "In Sec.", "4, we discuss the cosmology by using the Lagrangian obtained in Sec.", "3." ], [ "Weyl's gauge gravity theory", "In this section, we review the Weyl's gauge transformation to construct the gauge invariant Lagrangian.", "Consider the scale transformation in $D$ -dimensions $ds\\rightarrow ds^{\\prime }=e^{\\Lambda (x)}ds\\,,$ where $\\Lambda (x)$ is an arbitrary function of the coordinates $x^\\mu $ ($\\mu =0,\\dots , D-1$ ).", "Then the transformation of metric is realized by $g_{\\mu \\nu }\\rightarrow g^{\\prime }_{\\mu \\nu }=e^{2\\Lambda (x)}g_{\\mu \\nu }\\,.$ Thus $g^{\\mu \\nu }\\rightarrow g^{\\prime \\mu \\nu }=e^{-2\\Lambda (x)}g^{\\mu \\nu }\\,,$ and $\\sqrt{-g}\\rightarrow \\sqrt{-g^\\prime }=e^{D\\Lambda (x)}\\sqrt{-g}\\,.$ We can define the field with weight $d=-\\frac{D-2}{2}$ which transforms as $\\Phi \\rightarrow \\Phi ^{\\prime }=e^{-\\frac{D-2}{2}\\Lambda (x)}\\Phi \\,.$ We consider the covariant derivative of the scalar field $\\partial _{\\mu }\\Phi \\Rightarrow \\tilde{\\partial }_\\mu \\Phi \\equiv \\partial _\\mu \\Phi -\\frac{D-2}{2}\\,A_\\mu \\Phi \\,,$ where $A_\\mu $ is a Weyl's gauge invariant vector meson and its field strength is given by $F_{\\mu \\nu }\\equiv \\partial _\\mu A_\\nu -\\partial _\\nu A_\\mu \\,.$ Under the Weyl's gauge field transformation $A_\\mu \\rightarrow A^{\\prime }_\\mu =A_\\mu -\\partial _\\mu \\Lambda (x)\\,,$ we obtain the transformation of the covariant derivative of the scalar field as $\\tilde{\\partial }_\\mu \\Phi \\rightarrow e^{-\\frac{D-2}{2}\\Lambda (x)}\\tilde{\\partial }_\\mu \\Phi \\,.$ Moreover it is easily seen that $F_{\\mu \\nu }\\rightarrow F^{\\prime }_{\\mu \\nu }=F_{\\mu \\nu }\\,.$ The modified Christoffel symbol is defined as $\\tilde{\\Gamma }^\\lambda _{\\mu \\nu }\\equiv \\frac{1}{2}g^{\\lambda \\sigma }\\left(\\tilde{\\partial }_\\mu g_{\\sigma \\nu }+\\tilde{\\partial }_\\nu g_{\\mu \\sigma }-\\tilde{\\partial }_\\sigma g_{\\mu \\nu }\\right)\\,,$ and the modified curvature is given as follows: $\\tilde{R}^\\mu {}_{\\nu \\rho \\sigma }\\equiv \\partial _\\rho \\tilde{\\Gamma }^\\mu _{\\nu \\sigma }-\\partial _\\sigma \\tilde{\\Gamma }^\\mu _{\\nu \\rho }+\\tilde{\\Gamma }^\\mu _{\\lambda \\rho }\\tilde{\\Gamma }^\\lambda _{\\nu \\sigma }-\\tilde{\\Gamma }^\\mu _{\\lambda \\sigma }\\tilde{\\Gamma }^\\lambda _{\\nu \\rho }\\,.$ In Weyl's gauge theory of gravity, the Lagrangian should be invariant under the scale transformation." ], [ "Weyl invariant Lagrangian", "First, we show the Weyl's gauge invariant sectors of the vector field, the scalar field, the curvature $R$ and $R^2$ in $D$ -dimensions: ${\\cal L}_A&=&-\\frac{1}{4e^2}\\sqrt{-g}\\,\\Phi ^{\\frac{2(D-4)}{D-2}}g^{\\mu \\rho }g^{\\nu \\sigma }F_{\\mu \\nu }F_{\\rho \\sigma }\\,,\\\\{\\cal L}_{\\Phi }&=&-\\sqrt{-g}\\left[\\frac{1}{2}g^{\\mu \\nu }{\\tilde{\\partial }}_{\\mu }\\Phi {\\tilde{\\partial }}_{\\nu }\\Phi +\\frac{1}{4}\\lambda \\Phi ^{\\frac{2D}{D-2}}\\right]\\,,\\\\{\\cal L}_{R}&=&\\frac{1}{2}\\sqrt{-g}\\,\\epsilon \\Phi ^{2}\\tilde{R}\\,,\\\\{\\cal L}_{R^2}&=&\\sqrt{-g}\\,\\alpha \\Phi ^{\\frac{2(D-4)}{D-2}}{\\tilde{R}}^2\\,,$ where $\\lambda $ , $\\epsilon $ , $e$ and $\\alpha $ are dimensionless constants and $\\tilde{R}=R-2(D-1)\\nabla _\\mu A^\\mu -(D-1)(D-2)A_\\mu A^\\mu .$ As seen from (REF ), ${\\cal L}_{R^2}$ seems to include the term of $R A_{\\mu }A^{\\mu }$ .", "The simple Lagrangian which consists of $A_\\mu $ ,$\\Phi $ , $\\tilde{R}$ and $\\tilde{R}^2$ is the combination of the above sectors.", "Kao investigated the cosmology of Weyl's gauge gravity in four dimensions [17].", "He focused on the higher derivative $R^{2}$ and introduced effective scalar potentials.", "Thus we take the more general higher derivative of $R$ into account, then in general we consider the following Lagrangian including higher order of the curvature $\\tilde{R}^{n}$ : ${\\cal L}/\\sqrt{-g}&=&-\\frac{1}{4e^2}\\,\\Phi ^{\\frac{2(D-4)}{D-2}}g^{\\mu \\rho }g^{\\nu \\sigma }F_{\\mu \\nu }F_{\\rho \\sigma }-\\frac{1}{2}g^{\\mu \\nu }{\\tilde{\\partial }}_{\\mu }\\Phi {\\tilde{\\partial }}_{\\nu }\\Phi -\\frac{1}{4}\\lambda \\Phi ^{\\frac{2D}{D-2}}\\nonumber \\\\&&+\\frac{1}{2}\\,\\epsilon \\Phi ^{\\frac{2D}{D-2}}(\\Phi ^{\\frac{-4}{D-2}}\\tilde{R})+\\,\\alpha \\Phi ^{\\frac{2D}{D-2}}(\\Phi ^{\\frac{-4}{D-2}}\\tilde{R})^n.$ Introducing an auxiliary field $\\chi $ , we get the equivalent Lagrangian as $& &{\\cal L}/\\sqrt{-g}=-\\frac{1}{4e^2}\\,\\Phi ^{\\frac{2(D-4)}{D-2}}g^{\\mu \\rho }g^{\\nu \\sigma }F_{\\mu \\nu }F_{\\rho \\sigma }-\\frac{1}{2}g^{\\mu \\nu }{\\tilde{\\partial }}_{\\mu }\\Phi {\\tilde{\\partial }}_{\\nu }\\Phi -\\frac{1}{4}\\lambda \\Phi ^{\\frac{2D}{D-2}}\\qquad \\quad \\quad \\nonumber \\\\ & &+\\frac{1}{2}\\,\\epsilon \\Phi ^{\\frac{2D}{D-2}}\\chi +\\,\\alpha \\Phi ^{\\frac{2D}{D-2}}\\chi ^n+\\left(\\frac{1}{2}\\,\\epsilon \\Phi ^{\\frac{2D}{D-2}}+\\,n\\alpha \\Phi ^{\\frac{2D}{D-2}}\\chi ^{n-1}\\right)\\left(\\Phi ^{\\frac{-4}{D-2}}\\tilde{R}-\\chi \\right).$ Furthermore the Lagrangian (REF ) can be rewritten by the new metric conformally related to the original one and new variables.", "Here we choose $\\hat{g}_{\\mu \\nu }\\equiv e^{2\\Lambda (x)}g_{\\mu \\nu }\\,,$ and $\\hat{A}_\\mu \\equiv A_\\mu -\\partial _\\mu \\Lambda (x)\\,,$ where $e^{-\\Lambda (x)}=f\\left(\\Phi ^{2}+\\,\\frac{2n\\alpha }{\\epsilon }\\Phi ^{2}\\chi \\right)^{-\\frac{1}{D-2}}\\,.$ Note that a mass scale $f$ was introduced here.", "Now we can rewrite Eq.", "(REF ) to the following Lagrangian ${\\cal L}/\\sqrt{-\\hat{g}}&=&-\\frac{1}{4e^2}\\,\\phi ^{2\\frac{D-4}{D-2}}\\hat{g}^{\\mu \\rho }\\hat{g}^{\\nu \\sigma }\\hat{F}_{\\mu \\nu }\\hat{F}_{\\rho \\sigma }-\\frac{1}{2}\\left(\\partial _\\mu \\phi -\\frac{D-2}{2}\\hat{A}_\\mu \\phi \\right)^2\\nonumber \\\\&&-\\frac{1}{4}\\lambda \\,\\phi ^{\\frac{2D}{D-2}}-(n-1)\\alpha \\left(\\frac{\\epsilon }{2n\\alpha }\\right)^{\\frac{n}{n-1}}\\phi ^{\\frac{2D}{D-2}-\\frac{2n}{n-1}}\\left({f^{D-2}}-\\phi ^2\\right)^{\\frac{n}{n-1}}\\nonumber \\\\&&+\\frac{1}{2}\\,\\epsilon f^{D-2}(\\hat{R}-2(D-1)\\hat{\\nabla }_\\mu \\hat{A}^\\mu -(D-1)(D-2)\\hat{A}_\\mu \\hat{A}^\\mu )\\,,$ where $\\phi \\equiv f^{\\frac{D-2}{2}} \\left(1+\\,\\frac{2n\\alpha }{\\epsilon }\\chi ^{n-1}\\right)^{-1/2}\\, ,$ and “ $\\hat{~}$ ” indicates the derived quantities from new variables.", "We should note that $\\hat{R}\\hat{A}_{\\mu }\\hat{A}^{\\mu }$ term and higher terms of the scalar curvature $\\hat{R}$ disappear in this expression." ], [ "Cosmology of Weyl's gauge Gravity", "Since we obtained the Weyl's gauge invariant Lagrangian in the Einstein frame, we can study the cosmology by using this Lagrangian.", "Also we consider in four dimensions: $D=4$ and the order of higher derivative of curvature as $n=2$ .", "The Lagrangian (REF ) reads ${\\cal L}/\\sqrt{-\\hat{g}}&=&\\frac{1}{2}\\,\\epsilon f^{2}(\\hat{R}-6\\hat{A}_\\mu \\hat{A}^\\mu )-\\frac{1}{2}\\left(\\partial _\\mu \\phi -\\hat{A}_\\mu \\phi \\right)^2 \\nonumber \\\\&&-\\frac{1}{4e^2}\\hat{F}^2-\\frac{\\epsilon ^2}{16\\alpha }\\left({f^{2}}-\\phi ^2\\right)^{2}-\\frac{1}{4}\\lambda \\,\\phi ^{4}\\,.$ This Lagrangian becomes markedly simple for $D=4$ , namely, it consists of a massive vector meson and a canonical scalar sector with a polynomial potential which leads to a spontaneously symmetry breakdown.", "Hence, the universe is expected to behave similarly to the well-known inflationary scenario for our minimal Lagrangian (REF ).", "From Eq.", "(REF ), the conformal vector field could not be candidate of inflaton but could affect the evolution of our universe.", "The cosmology of Weyl's gauge gravity has been investigated in four dimensions by Kao [17].", "He has focused on the higher curvature term $R^{2}$ and introduced an effective potential which the scalar field leads to symmetry-breakdown in the low energy region.", "Also the vector mesons are not taken into account in contrast to our model.", "As this result, his model missed the canonical form of the scalar sector." ], [ "Summary and Outlook", "In the early stage of the universe, vector inflations have been discussed as the alternative to the successfully inflationary scenarios based on scalar bosons.", "However, these inflaton has not been completely supported by relevant fundamental theory of physics.", "Also the gravity theory is expected to be different from the Einstein gravity in the very early universe.", "In particular, gravitational physics is speculated to have a symmetry of scale invariance near Planck scale like other particle physics.", "Thus, we study the possibility of the Weyl's gauge invariant theory as a fundamental theory in the early universe.", "One of the reasons is that the Weyl's gauge invariant scalar and vector field can be naturally introduced.", "We construct the Weyl's gauge invariant Lagrangian in arbitrary dimensions that includes an arbitrary higher order of the scalar curvature $R^n$ .", "This Lagrangian has the $RA^{\\mu }A_{\\mu }$ term.", "In order to investigate the cosmology, we rewrite this Lagrangian to the the Einstein-like form by using the Weyl's gauge transformation.", "Especially, for $D=4$ , the transformed Lagrangian is markedly simple.", "In this Lagrangian, $\\hat{R}$ is not minimally-coupled to the massive vector $\\hat{A}^{\\mu }$ .", "Therefore, the Weyl's gauge invariant vector field could not be an inflaton of the vector inflation.", "However, the Lagrangian has a scalar boson with polynomial potentials, namely, the canonical scalar sector with $\\phi ^4$ -potential.", "Hence the universe behaves similar to the ordinary one which has been discussed by many authors.", "While the massive vector field could not be inflaton of the vector inflation, nevertheless, it is expected that the vector meson relates to the dark matter and dark energy.", "In is worth noting that the study of the cosmology of Weyl's gravity by Kao [17].", "In the contrast to our model, he has focused on the higher derivative $R^2$ and introduced an effective scalar potentials but not taken the vector fields into account.", "From theas reasons, the effective action missed the canonical form of the scalar sector.", "From the Weyl's gauge gravity point of view, if the inflaton is the Weyl's gauge invariant scalar, the nature seems to select the polynomial potential instead of one in the new inflation.", "We need to analyze the behavior of vector field to obtain rigorous behavior of the inflaton.", "It will be studied in a separate publication.", "Also we should investigate the generalization to the case of higher and lower dimensions.", "This will be published in the forthcoming paper.", "This study is supported in part by the Grant-in-Aid of Nikaido Research Fund." ] ]

[ [ "Demagnetization Borne Microscale Skyrmions" ], [ "Abstract Magnetic systems are an exciting realm of study that is being explored on smaller and smaller scales.", "One extremely interesting magnetic state that has gained momentum in recent years is the skyrmionic state.", "It is characterized by a vortex where the edge magnetic moments point opposite to the core.", "Although skyrmions have many possible realizations, in practice, creating them in a lab is a difficult task to accomplish.", "In this work, new methods for skyrmion generation and customization are suggested.", "Skyrmionic behavior was numerically observed in minimally customized simulations of spheres, hemisphere, ellipsoids, and hemi-ellipsoids, for typ- ical Cobalt parameters, in a range from approximately 40 nm to 120 nm in diameter simply by applying a field." ], [ "Introduction", "A skyrmion, theorized first by Skyrme in 1962 [1], is a state with a vectorial order parameter which is aligned at the system boundary at an opposite direction to what the order parameter assumes at the origin.", "Skyrmions may appear in diverse arenas, such as elementary particles [1], [2], [3], [4], [5], liquid crystals [6], Bose-Einstein condensates [7], [8], [9], thin magnetic films [10], quantum Hall systems [11], [12], [13], [14], and potentially vortex lattices in type II superconductors [15], [16].", "Being able to experimentally observe or generate skyrmions is a current research thrust [1], [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19].", "In this work we demonstrate via micromagnetic simulations that achieving a skyrmion is as simple as creating a nanoparticle of many possible geometries, which is large enough to support a single vortex but small enough to prevent multiple vortices.", "The demagnetization energy allows for the formation of a vortex at zero-field.", "We find that as the field increases such that it lies in a direction opposite to the core, the magnetization at the edges may realign itself parallel to the field direction more readily than the magnetization next to the core.", "Immediately prior to annihilation of the vortex (i.e., the flipping of the magnetization at the system core to become parallel to the applied field direction), the skyrmionic state is most notable.", "We observed this, relatively ubiquitous, effect in systems with disparate geometries- spheres, hemispheres, ellipsoids, and hemi-ellipsoids.", "It may be possible to generalize this process so as to experimentally synthesize a skyrmion lattice by simply creating an array of nanoparticles with tunable size and spacing, such as by self-organzation [20], [21].", "Preliminary simulations of a two-by-two grid of Cobalt hemispheres of radius $20 \\: nm$ with varying inter-hemisphere separation indicate that beyond a threshold distance of twice the radius, an array of skyrmions is formed.", "As the center to center separation is steadily increased, the skyrmionic state becomes more lucid.", "For small separations, interactions partially thwart the creation of the individual skyrmions.", "As is well known, we can quantify a skyrmionic state by calculating the Pontryagin index (also known as a winding number) that is given by [22] $Q=\\frac{1}{8\\pi }\\int d^{2}x\\epsilon _{ij}\\hat{M}\\cdot (\\partial _{i}\\hat{M}\\times \\partial _{j}\\hat{M}).$ In this expression, $\\epsilon _{ij}$ is the two dimensional anti-symmetric tensor and $\\hat{M}$ is the normalized magnetization.", "For a single skyrmion, this winding number (or topological charge) is equal to unity.", "Skyrmions are characterized by the non-trivial homotopy class $\\pi _{2}(S^{2})$ .", "This homotopy class is characterized by an integer that, for this case, is the Pontryagin index.", "States with different integer skyrmion number (the Pontryagin index) cannot be continuously deformed into one another.", "In the current context, the skyrmionic state resides on a two dimensional plane.", "On each spatial point of the plane, there is a three dimensional order parameter which, in our case, is the magnetization $\\vec{M}$ .", "Topologically, a skyrmion is a magnetic state such that when it is mapped onto a sphere (via stereographic projection) resembles a monopole or hairy ball.", "This means that on mapping from a flat space to the surface of a sphere, the individual magnetic moments will always point perpendicular to the surface of the sphere, much like a magnetic monopole.", "The above topological classification is valid for an “ideal” skyrmion on an infinite two-dimensional plane or disk with the condition that the local moment $\\vec{M}(\\vec{r})$ at spatial infinity (irrespective of the location $\\vec{r}$ on the infinite disk) all orient in the same direction: $\\lim _{r\\rightarrow \\infty }\\hat{M}({\\vec{r}})=\\hat{M}_{0}$ .", "In such a case $\\hat{M}_{0}$ corresponds to the magnetization at the “point at infinity”.", "On applying a stereographic projection of the infinite plane onto a unit sphere, $\\hat{M}_{0}$ maps onto the magnetization at the north pole of the unit sphere while the oppositely oriented $\\hat{M}$ at the origin corresponds to the magnetization at the south pole.", "In such a case, the winding number is identically equal (in absolute value) to unity.", "In many physically pertinent geometries, including the systems simulated in this work, there are finite size limits which only allow the magnetization $\\vec{M}$ to exhibit the trend of approaching a uniform value $\\vec{M}_{0}$ as one moves away from the center of the system.", "In this case, the integral in Eq.", "REF is not an integer.", "However, it is clear that, in the limit of infinite planar size, these states would become ideal skyrmions and the winding number $Q$ would approach an integer value.", "The remainder of this article is organized as follows.", "In Section , we provide necessary background.", "We briefly describe the simulations employed in this work and discuss energetic considerations.", "Section reports on our central result- the numerical observation of skyrmions.", "We discuss a higher dimensional generalization and the possibility of generating skyrmion lattices.", "We conclude in section with a summary of our results." ], [ "Simulation Theory", "In this work of simulating magnetic states of nanoparticles, the Object Oriented Micromagnetic Framework (OOMMF) 1.2a distribution as provided from NIST was utilized [23].", "The OOMMF code numerically solves the Landau-Lifshitz Ordinary Differential Equation given by, $\\frac{d\\vec{M}}{dt}=-|\\bar{\\gamma }|\\vec{M}\\times \\vec{H}_{eff}-\\frac{|\\bar{\\gamma }|\\tilde{\\alpha }}{M_{s}}\\vec{M}\\times \\left(\\vec{M}\\times \\vec{H}_{eff}\\right)$ where $\\vec{M}$ is the magnetization, $\\bar{\\gamma }$ is the Landau-Lifshitz gyromagnetic ratio, $M_{s}$ is the saturation magnetization, $\\tilde{\\alpha }$ is the damping coefficient, and $H_{eff}$ is the effective field given by derivatives of the Gibbs free energy.", "The Gibbs free energy, in this case, is given by [24], $G=\\int (\\frac{1}{2}C\\left[\\left(\\vec{\\nabla }\\alpha \\right)^{2}+\\left(\\vec{\\nabla }\\beta \\right)^{2}+\\left(\\vec{\\nabla }\\gamma \\right)^{2}\\right]+w_{a}\\nonumber \\\\-\\frac{1}{2}\\vec{M}\\cdot \\vec{H}^{\\prime }-\\vec{M}\\cdot \\vec{H}_{0})d\\tau $ where $\\alpha $ , $\\beta $ , and $\\gamma $ are the directional cosines, $C$ is proportional to the exchange stiffness constant and depends on the crystal structure, $w_{a}$ is the crystalline anisotropy term, $\\vec{H}^{\\prime }$ is the demagnetization field, and $\\vec{H}_{0}$ is the external magnetic field.", "The crystalline anisotropy term can be expressed in terms of anisotropy constants, $K_{1}$ and $K_{2}$ , and directional cosines as, $w_{a}=K_{1}\\left(\\alpha ^{2}\\beta ^{2}+\\beta ^{2}\\gamma ^{2}+\\gamma ^{2}\\alpha ^{2}\\right)+K_{2}\\alpha ^{2}\\beta ^{2}\\gamma ^{2}.$ In the simulations, a metastable state was determined to have been reached when the maximum torque experienced by any one magnetic moment, measured in $\\frac{degrees}{ns}$ , dropped below $0.2$ .", "Once this level of torque was reached, the magnetic state data were saved to a file along with the other properties of the system, including but not limited to, the energies associated with each contribution, overall magnetization, and number of iterations.", "The magnetic field was then changed to the next value and the iterations continued until saturation of the magnetization was obtained.", "The magnetic field steps were chosen such that half the steps (typically, a few hundred) were during the increasing field portion and the other half in the decreasing field portion.", "The data stored in the file were used later to generate the hysteresis plots, track the energy changes associated with the field variations, and the spatial orientations of the magnetic moments.", "Unless specified otherwise, the parameters chosen in the simulations correspond to those for Cobalt, as shown in Table REF .", "Table: Table of parameters used in the simulations of particles in this work.The exchange stiffness constant, saturation magnetization, and crystallineanisotropy constant are material specific and are chosen for Cobalt.The damping constant, Landau-Lifshitz-Giblert gyromagnetic ratio,and stopping torque are material independent parameters." ], [ "Energy Considerations", "In our simulations, we considered field, demagnetization, and exchange energies.", "For simplicity, we neglected crystalline anisotropy effects.", "The field tries to align the local magnetic moments parallel to it while exchange effects favor an alignment of the magnetic moments with their nearest neighbors.", "The (universally geometry borne) demagnetization energy directly relates to dipole-dipole interactions [24].", "Demagnetization energy is often the dominant term for long range behaviors while exchange effects tend to dominate at short spatial scales.", "As is well known, the competition between the long range and the short range energy contributions leads to the creation of domain walls.", "The demagnetization favors oppositely oriented moments at the expense of exchange effects that favor slow variations amongst neighbors.", "Ultimately, this tradeoff gives rise to domain walls in micromagnetic systems.", "The potential energy from demagnetization of a system is given by $\\mathcal {E}_{M}=-\\frac{1}{2}\\sum _{i}\\vec{m}_{i}\\cdot \\vec{h}^{\\prime }_{i},$ where $\\vec{h}^{\\prime }_{i}$ is the effective field at position $i$ that originates from all other dipoles.", "This field can be written as $\\vec{h}^{\\prime }_{i}=\\vec{H}^{\\prime }+\\frac{4}{3}\\pi \\vec{M}+\\vec{h}^{\\prime \\prime }_{i},$ where $\\vec{H}^{\\prime }$ is the megascopic field from the poles due to $\\vec{M}$ outside of a physically small sphere around site $i$ .", "The second term subtracts the effective field inside an arbitrary small region (or sphere) centered about point $i$ , and $\\vec{h}^{\\prime \\prime }_{i}$ is the field at site $i$ created by dipoles inside this region.", "In general, $\\vec{h}^{\\prime \\prime }_{i}$ depends on the crystal lattice structure.", "In the continuum limit, the sum becomes an integral of the form, $\\mathcal {E}_{M}=-\\frac{1}{2}\\int \\vec{M}\\cdot (\\vec{H}^{\\prime }+\\frac{4}{3}\\pi \\vec{M}+\\Lambda \\cdot \\vec{M})dV.$ The tensor $\\Lambda $ in the third term depends only on the crystal structure and local magnetization and can grouped with crystalline anisotropy.", "This tensor also vanishes for cubic crystals identically.", "The second term in this expression is a constant proportional to $M_{s}^{2}$ and can be ignored.", "The $\\Lambda $ tensor also vanishes for cubic crystals identically leaving, $\\mathcal {E}_{M}=-\\frac{1}{2}\\int \\vec{M}\\cdot \\vec{H}^{\\prime }dV.$ The demagnetization field, $\\vec{H}^{\\prime }$ , can equivalently be derived from Maxwell's equations.", "It can be expressed as the negative gradient of a potential, $U$ that satisfies the equations, $\\nabla ^{2}U_{in}=\\gamma _{B}\\vec{\\nabla }\\cdot \\vec{M}\\\\\\nabla ^{2}U_{out}=0,$ with the surface boundary conditions, $U_{in}=U_{out}\\\\\\frac{\\partial U_{in}}{\\partial n}-\\frac{\\partial U_{out}}{\\partial n}=\\gamma _{B}\\vec{M}\\cdot \\vec{n}.$ where the constant $\\gamma _{B}$ is, in our units, $4 \\pi $ .", "Lastly, the potential needs to be regular at infinity, such that $|rU|$ and $|r^{2}U|$ are bounded as $r\\rightarrow \\infty $ .", "Our simulations directly capture the demagnetization field effects.", "From the standpoint of energy, for a skyrmion to be possible, the dimensions of the ellipsoid must be larger than the critical dimensions at which vortices can nucleate in a given system.", "For example, for the hemispherical geometry, with the typical values of Table REF , the critical radius was found to be $19 \\: nm$ .", "For larger radii, vortices are the preferred state before reaching zero field.", "The vortex will nucleate such that the core is parallel to the field and the remainder of the vortex lies in the plane perpendicular to the field.", "Once the field begins to oppose the direction of the moments at the core, the energy cost of eliminating the core is significantly higher than allowing the outer magnetic moments to align more with the field.", "When the exchange energy cost of the skyrmionic state becomes greater than the demagnetization energy for a uniform magnetization, the core flips, annihilating the skyrmion, and the magnetization saturates.", "Immediately, prior to this, though, a skyrmionic state can be achieved.", "Ezawa [25] raised the specter of a skyrmionic state in thin films via the computation of the energy of such assumed variational states within a field theoretic framework of a non-linear sigma model.", "Dipole-dipole interactions may stabilize such a state below a critical field.", "Our exact numerical calculations for the evolution of the magnetic states demonstrate that not only are skyrmionic states viable structures, but are actually the precise lowest energy state for slices of hemispheres and other general structures." ], [ "Observation of a Skyrmion", "As our numerical simulations vividly illustrate, just prior to the annihilation of the vortex, the magnetic moments at the edge of the system start to orient themselves in a direction opposite to that in the core.", "On increasing the radius of the simulated hemispheres and spheres, the configurations next to the basal plane better conformed to the full skyrmion topology (i.e., that on an infinite plane).t should be noted here, that as the radius of a hemisphere increases, the crossover to a double vortex state will eventually occur, but if one vortex is maintained, in the limit of large radii, a full skyrmion would be expected.", "This may be possible in materials with large exchange constant and small saturation magnetization.", "In what follows, we will employ the typical values appearing in Table REF .", "The skyrmion state for the bottom layer (basal plane) of a hemisphere of radius $24 \\: nm$ is shown in Figure REF .", "Figure: Vector plot of the skyrmion state for the bottom slice of a hemisphereof radius 24nm24 \\: nm.", "Not all local magnetic moments are shown for the sakeof clarity.A similar configuration was observed in simulation runs for nanospheres.", "For a sphere, symmetry does not favor any particular direction, but that symmetry is broken once a field is applied.", "Skyrmions were observed in runs of spheres large enough to support a vortex which corresponds to a radius of $\\approx 15 \\: nm$ .", "As the radius of the sphere increases, the edge magnetic moments and the core magnetic moments become more antiparallel.", "A skyrmion in a sphere of radius $59nm$ is shown in Figure REF .", "Figure: Vector plot of the skyrmion state in a sphere of radius 59nm59nm.The slice is along the equator of the sphere.", "Only a subset of localmagnetic moments is shown for clarity.Once skyrmions were observed in these systems, it begged the question, “Do these occur in ellipsoids and hemi-ellipsoids?\"", "Upon examining this, indeed skyrmions can be observed in oblate ellipsoids and hemi-ellipsoids as shown in Figures REF and REF .", "Figure: Vector plot of the skyrmion state in an ellipsoid with major axisof 20nm20 \\: nm and minor axis of 15nm15 \\: nm.", "The slice is along the equatorof the ellipsoid.", "Only a subset of local magnetic moments is shownfor clarity.Figure: Vector plot of the skyrmion state in a hemi-ellipsoid with major axisof 20nm20nm and minor axis of 15nm15 \\: nm.", "The slice is along the baseof the hemi-ellipsoid.", "Only a subset of local magnetic moments isshown for clarity.To verify that these are structures approach those of skyrmions and to quantitively monitor their deviations from an ideal skyrmionic state (for which the Pontryfin index is unity),we computed the Pontryagin index at different cross sections of the hemisphere.", "These cross sections were those of the hemisphere with planes parallel to the basal plane(i.e., that at the base of the hemisphere).", "For a hemisphere with radius $30 \\: nm$ , we calculated the skyrmion number Q for thirty individual parallel layers vertically separated by $1 \\: nm$ .", "We numerically evaluated the integral of Eq.", "REF for all of these layers and examined how it changes as the field increases from 0 to $0.8\\: T$ .", "These data are shown in Figure REF .", "Figure: Plot of the Pontryagin index versus the z-coordinate of the slicetaken from the hemisphere of radius 30nm30 \\: nm.", "These are shown for increasing field fromzero field (dark blue dot-dash line), 0.2T0.2 \\: T (green dotted line), 0.4T0.4 \\: T (red dashed line), and 0.6T0.6 \\: T(teal solid line).Visualizing this in the geometry of the hemisphere specifically, one can look at how the Pontryagin index varies along various planes of a hemisphere, starting from the equator and moving to the pole.", "It can be clearly seen that the skyrmionic behavior exists for most of the height of the hemisphere and only the cap deviates from the rest of the system.", "The size of this cap depends on the given field strength as can be seen in the case of 0 field (Figure REF ) and with a field of $0.6 \\: T$ (Figure REF ).", "At higher fields, prior to the annihilation of the vortex, the Pontryagin index approaches an integer value, as expected for an ideal skyrmionic state.", "Figure: Three dimensional plots of the Pontryagin index for a hemisphere of radius30nm30 \\: nm at (a) zero field and (b) 0.6T0.6 \\: TPerforming similar analysis on the hemi-ellipsoids and visualizing the Pontryagin index and its variance with height, it can be seen that the same behavior exists in a less extreme way than the hemispheres.", "This behavior can be seen in Figure REF for hemi-ellipsoids of fixed $30 \\: nm$ major axis and varying minor axis.", "Figure: Plots of the Pontryagin index and how it varies with height insidehemi-ellipsoids of 30nm30 \\: nm radius major axis as the minor axis varies from 15nm15 \\: nm (a) to 10nm10 \\: nm(c) to 5nm5 \\: nm (e).", "This is shown for (a) field equal to 0.2T0.2 \\: T pointing in the negative z-direction(perpendicular to the face of the hemi-ellipsoids).", "As will be noted, theexistence of skyrmionic behavior is not prevalent in the more flattenedhemiellipsoinds and vanishes at this field between minor axis 15nm15 \\: nm and 10nm10 \\: nm.The associated partial hysteresis loops for each of these hemi-ellipsoid runs are shown in Figs.", "(b), (d),and (f), respectively.In examining the hysteresis behavior of the hemi-ellipsoids, one can see a trend as the z-dimension goes from the hemisphere radius ($20 \\: nm$ ) to the minimum simulated size of $5 \\: nm$ .", "This trend shows a movement from extensive vortex and skyrmionic behavior in the more hemispheric geometries and less vortex and skyrmionic behavior in the more ellipsoidal geometries.", "Although it will not be considered in this work, crystalline anisotropy could influence the formation of skyrmions in a number of ways.", "In the case of a single crystal, the vortex state would be more difficult to nucleate and thus the skyrmionic state is less energetically favorable.", "When many crystalline grains are present, the results discussed here are valid as the large number or randomly oriented crystals will, on average, not favor any direction, and thus will not favor any one direction." ], [ "Generalization to a Hedgehog", "These results lead to the question of whether this can be generalized to more than two dimensions.", "The natural generalization from the two-dimensional skyrmion to a three-dimensional magnetic state would be the hedgehog.", "The hedgehog resides in three spatial dimensions coupled with a three dimensional order parameter.", "The canonical example of a hedgehog is$\\vec{M}=M_{s}\\hat{r}$ where the magnetization always points outwards.", "A skyrmion is related to a hedgehog via a stereographic projection from the sphere onto a plane where the south pole of the hedgehog projects to the core of the skyrmion on the plane and the north pole of the hedgehog projects to the points at infinity on the plane.", "Calculating the demagnetizing field for this state in a sphere gives rise to a potential and field equal to $U(r)=\\gamma _{B}M_{s}(r-R),\\\\\\vec{H}=-\\gamma _{B}M_{s}\\hat{r}.$ Plugging this into Equation REF , one finds the energy of the hedgehog to be $2\\pi M_{s}^{2}(4\\pi /3)R^{3}$ .", "Comparing this to the energy of the uniformly magnetized state, $(1/2)(4\\pi /3)^{2}M_{s}^{2}R^{3}$ , it can easily be seen that the hedgehog has three times the energy of the uniform state.", "This, combined with the fact that the exchange energy and the field energy will favor the uniform state, the hedgehog state will not be possible in a sphere.", "If one were to continuously deform the hedgehog by rotating the local magnetic moments by $\\pi /2$ such that $\\vec{M}=M_{s}f(z)\\hat{\\phi }$ where $f(z)$ is a function that goes to 0 as $z\\rightarrow 0$ such that the exchange energy does not diverge, one would find the demagnetization energy of that state to be identically 0.", "The field energy in this system is also 0 for a field that is applied along the z-axis.", "The exchange energy is given by $(4\\pi /3)RC$ where $C$ is the exchange stiffness constant.", "The total energy of this state is equal to the exchange energy, and comparing this to the uniform state, a hedgehog of this form is favorable for, $R\\ge \\sqrt{\\frac{C}{\\frac{2\\pi \\mu _{0}M^{2}}{3}-MH_{0}}}.$ For $C=2.5\\times 10^{-11}J/m$ and $M_{s}=1.4\\times 10^{6}A/m$ as it is for Cobalt, at 0 field, this radius works out to be $\\approx 3.5\\mu m$ ." ], [ "Skyrmion Array", "It is illuminating to consider the possibility of an array of skyrmions.", "As briefly discussed below, we find that effective particle interactions may thwart the creation of a skyrmion lattice when these particles are not far separated.", "However, for sufficiently large center to center separations, a Skryme lattice may be achieved.", "In preliminary simulations of arrays of nanoparticle arrays, simulations of a two-by-two grid of hemispheres of radius $20 \\: nm$ with a variable separation show that a center to center separation of four times the radius is close enough that the nanoparticles still interact magnetically and prevent the formation of an array of skyrmions.", "As expected, further separation should approach the the single particle result of skyrmions, as we briefly discuss next.", "The transition from the array of particles which support individual vortices to the array of particles that are clearly interacting with each other can be seen in Figure REF .", "In this figure, the annihilation of the vortices can be seen as the particles realign their magnetization to form a state where the local magnetization orients in the counterclockwise direction from particle to particle, yet within each particle, when moving in the counterclockwise direction, the local magnetization changes from oriented in the negative z-direction to the positive z-direction.", "In repeating these simulations for a 3x3 array of hemispherical nanoparticles, the same behavior was observed.", "This array was similar to the 2x2 array in that it had nanoparticles with diameters of $40 \\: nm$ and center to center separation of $80 \\: nm$ .", "The annihilation of the vortices occurred at a slightly larger field (0.08T rather than 0.1T) as shown in Fig REF ." ], [ "Conclusion", "We conclude with a brief synopsis of our findings.", "We carried a systematic numerical study of the magnetization of small nanoparticles in the presence of an external magnetic field.", "These systems were simulated for different sizes and geometry (sphere, hemisphere, ellipsoids).", "Our analysis ignored anisotropy (crystalline, shape, strain, etc.)", "effects.", "We find that, as has been widely reported in the literature [26], [27], beyond a critical diameter, the particles enter into a single vortex state under zero external field; multiple vortices are possible for much larger particles.", "Our key new result concerns the creation of skyrmions in the single vortex state.", "As the field is increased, vortex annihilation is accompanied by the formation of a skyrmionic state wherein the magnetization of the vortex core points to a direction opposite to that at the edge of the nanoparticle.", "Our result illustrates how geometry plays a pivotal role.", "Spheres and hemispheres more readily achieve skyrmionic states than higher eccentricity ellipsoids.", "Our preliminary results suggested that for center to center separations larger than twice the particle diameters, an array of skyrmions may be realized.", "More detailed studies of skyrmion lattices for such particle arrays are planned for the future.", "Acknowledgements.", "Work at Washington University was partially supported by NSF grants DMR-1106293 and DMR-0856707, and by the Center for Materials Innovation (CMI) of Washington University.", "Work at the university of Tennessee was partially supported by NSF DMR-0856707.", "Figure: Vector plot of a 2x2 array of hemispheres with radius 20nm20 \\: nm and center to center separation 80nm80 \\: nm at fields of 0.12T0.12 \\: T pointing in the negative z-direction (a) and 0.1T0.1\\: T pointing in the negative z-direction (b).", "Colorscale corresponds to z-component of the local magnetic moment in units of A/mA/m.Figure: Vector plot of a 3x3 array of hemispheres with radius 20nm20 \\: nm and center to center separation 80nm80 \\: nm at fields of 0.1T0.1 \\: T pointing in the negative z-direction (a) and 0.08T0.08\\: T pointing in the negative z-direction (b).", "Colorscale corresponds to z-component of the local magnetic moment in units of A/mA/m." ] ]

[ [ "Asymptotics of the $s$-perimeter as $s\\searrow 0$" ], [ "Abstract We deal with the asymptotic behavior of the $s$-perimeter of a set $E$ inside a domain $\\Omega$ as $s\\searrow0$.", "We prove necessary and sufficient conditions for the existence of such limit, by also providing an explicit formulation in terms of the Lebesgue measure of $E$ and $\\Omega$.", "Moreover, we construct examples of sets for which the limit does not exist." ], [ "Introduction", "Given $s\\in (0,1)$ and a bounded open set $\\Omega \\subset {R}^n$ with $C^{1,\\gamma }$ -boundary, the $s$ -perimeter of a (measurable) set $E\\subseteq {R}^n$ in $\\Omega $ is defined as $\\begin{split}&{\\text{\\rm Per}}_{s} (E;\\Omega ):=L(E\\cap \\Omega , ({C}E)\\cap \\Omega )\\\\ &\\qquad \\qquad \\qquad +L(E\\cap \\Omega , ({C}E)\\cap ({C}\\Omega ))+L(E\\cap ({C}\\Omega ), ({C}E)\\cap \\Omega ),\\end{split}$ where ${C}E = {R}^n\\setminus E$ denotes the complement of $E$ , and $L(A,B)$ denotes the following nonlocal interaction term $\\displaystyle L(A,B):=\\int _A \\int _B \\frac{1}{|x-y|^{n+s}}\\,dx\\,dy \\qquad \\forall \\, A, B\\subseteq {R}^n.$ Here we are using the standard convention for which $L(A,B)=0$ if either $A=\\varnothing $ or $B=\\varnothing $ .", "This notion of $s$ -perimeter and the corresponding minimization problem were introduced in [3] (see also the pioneering work [14], [15], where some functionals related to the one in (REF ) have been analyzed in connection with fractal dimensions).", "Recently, the $s$ -perimeter has inspired a variety of literature in different directions, both in the pure mathematical settings (for instance, as regards the regularity of surfaces with minimal $s$ -perimeter, see [2], [7], [6], [13]) and in view of concrete applications (such as phase transition problems with long range interactions, see [4], [11], [12]).", "In general, the nonlocal behavior of the functional is the source of major difficulties, conceptual differences, and challenging technical complications.", "We refer to [9] for an introductory review on this subject.", "The limits as $s\\searrow 0$ and $s\\nearrow 1$ are somehow the critical cases for the $s$ -perimeter, since the functional in (REF ) diverges as it is.", "Nevertheless, when appropriately rescaled, these limits seem to give meaningful information on the problem.", "In particular, it was shown in [5], [1] that $(1-s){\\text{\\rm Per}}_{s}$ approaches the classical perimeter functional as $s\\nearrow 1$ (up to normalizing multiplicative constants), and this implies that surfaces of minimal $s$ -perimeter inherit the regularity properties of the classical minimal surfaces for $s$ sufficiently close to 1 (see [6]).", "As far as we know, the asymptotic as $s\\searrow 0$ of $s{\\text{\\rm Per}}_{s}$ was not studied yet (see however [10] for some results in this direction), and this is the question that we would like to address in this paper.", "That is, we are interested in the quantity $\\mu (E):=\\lim _{s\\searrow 0}s{\\text{\\rm Per}}_{s}(E;\\Omega )$ whenever the limit exists.", "Of course, if it exists then $ \\mu (E)=\\mu ({C}E),$ since ${\\text{\\rm Per}}_{s}(E;\\Omega )={\\text{\\rm Per}}_{s}({C}E;\\Omega ).$ We will show that, though $\\mu $ is subadditive (see Proposition REF below), in general it is not a measure (see Proposition REF , and this is a major difference with respect to the setting in [10]).", "On the other hand, $\\mu $ is additive on bounded, separated sets, and it agrees with the Lebesgue measure of $E \\cap \\Omega $ (up to normalization) when $E$ is bounded (see Corollary REF ).", "As we will show below, a precise characterization of $\\mu (E)$ will be given in terms of the behavior of the set $E$ towards infinity, which is encoded in the quantity $ \\alpha (E):=\\lim _{s\\searrow 0} s\\int _{E\\cap ({C}B_1)}\\frac{1}{|y|^{n+s}}\\,dy,$ whenever it exists (see Theorem REF and Corollary REF ).", "In fact, the existence of the limit defining $\\alpha $ is in general equivalent to the one defining $\\mu $ (see Theorem REF (ii)).", "As a counterpart of these results, we will construct an explicit example of set $E$ for which both the limits $\\mu (E)$ and $\\alpha (E)$ do not exist (see Example REF ): this says that the assumptions we take cannot, in general, be removed.", "Also, notice that, in order to make sense of the limit in (REF ), it is necessary to assume thatIt is easily seen that if (REF ) holds, then $ {\\text{\\rm Per}}_{s}(E;\\Omega )<\\infty $ for any $s\\in (0,s_0)$ .", "Moreover, if $\\partial E$ is smooth, then (REF ) is always satisfied.", "${\\mbox{$ {\\text{\\rm Per}}_{s_0}(E;\\Omega )<\\infty $, for some $s_0 \\in (0,1)$.", "}}$ To stress that (REF ) cannot be dropped, we will construct a simple example in which such a condition is violated (see Example REF ).", "The paper is organized as follows.", "In the following section, we collect the precise statements of all the results we mentioned above.", "Section  is devoted to the proofs." ], [ "List of the main results", "We define ${{E}}$ to be the family of sets $E\\subseteq {R}^n$ for which the limit defining $\\mu (E)$ in (REF ) exists.", "We prove the following result: Proposition 2.1 $\\mu $ is subadditive on ${{E}}$ , i.e.", "$\\mu (E\\cup F)\\leqslant \\mu (E)+\\mu (F)$ for any $E$ , $F\\in {{E}}$ .", "First, it is convenient to consider the normalized Lebesgue measure ${{M}}$ , that is the standard Lebesgue measure scaled by the factor ${{H}}^{n-1}(S^{n-1})$ , namely $ {{M}}(E):={{H}}^{n-1}(S^{n-1})\\,|E|,$ where, as usual, we denote by $S^{n-1}$ the $(n\\!-\\!1)$ -dimensional sphere.", "Now, we recall the main result in [10]; that is, Theorem 2.2 (see [10]).", "Let $s\\in (0,1)$ .", "Then, for all $u\\in H^s({R}^n)$ , $\\displaystyle \\lim _{s \\searrow 0} \\,\\frac{s}{2}\\,\\int _{{R}^n}\\int _{{R}^n} \\frac{|u(x)-u(y)|^2}{|x-y|^{n+s}}\\,dx\\,dy\\, = \\, {H}^{n-1}({S}^{n-1}) \\int _{{R}^n} |u|^2\\, dx.$ An easy consequence of the result above is that when $E\\in {{E}}$ and $E\\subseteq \\Omega $ then $\\mu (E)$ agrees with ${{M}}(E)$ (in fact, we will generalize this statement in Theorem REF and Corollary REF ).", "Based on this property valid for subsets of $\\Omega $ , one may be tempted to infer that $\\mu $ is always related to the Lebesgue measure, up to normalization, or at least to some more general type of measures.", "The next result points out that this cannot be true: Proposition 2.3 $\\mu $ is not necessarily additive on separated sets in $ {{E}}$ , i.e.", "there exist $E,F\\in {{E}}$ such that $\\text{\\rm dist}(E,F)\\geqslant c>0$ , but $\\mu (E\\cup F)< \\mu (E)+\\mu (F)$ .", "Also, $\\mu $ is not necessarily monotone on ${{E}}$ , i.e.", "it is not true that $E\\subseteq F$ implies $\\mu (E)\\leqslant \\mu (F)$ .", "In particular, we deduce from Proposition REF that $\\mu $ is not a measure.", "On the other hand, in some circumstances the additivity property holds true: Proposition 2.4 $\\mu $ is additive on bounded, separated sets in ${{E}}$ , i.e.", "if $E$ , $F\\in {{E}}$ , $E$ and $F$ are bounded, disjoint and  $\\text{\\rm dist}(E,F)\\geqslant c>0$ , then $E\\cup F\\in {{E}}$ and $\\mu (E\\cup F)=\\mu (E)+\\mu (F)$ .", "There is a natural condition under which $\\mu (E)$ does exist, based on the weighted volume of $E$ towards infinity, as next result points out: Theorem 2.5 Suppose that  ${\\text{\\rm Per}}_{s_0}(E;\\Omega )<\\infty $ for some $s_0 \\in (0,1)$ , and that the following limit exists $\\alpha (E):=\\lim _{s\\searrow 0}s\\int _{E\\cap ({C}B_1)}\\frac{1}{|y|^{n+s}}\\,dy.$ Then $E\\in {{E}}$ and $\\mu (E)=\\big (1-\\widetilde{\\alpha }(E)\\big )\\,{{M}}(E\\cap \\Omega )+\\widetilde{\\alpha }(E)\\,{{M}}(\\Omega \\setminus E),$ where $\\widetilde{\\alpha }(E):=\\frac{\\alpha (E)}{{{H}}^{n-1}(S^{n-1})}.$ As a consequence of Theorem REF , one obtains the existence and the exact expression of $\\mu (E)$ for a bounded set $E$ , as described by the following result: Corollary 2.6 Let $E$ be a bounded set, and  ${\\text{\\rm Per}}_{s_0}(E;\\Omega )<\\infty $ for some $s_0 \\in (0,1)$ .", "Then $E\\in {{E}}$ and $\\mu (E)={{M}}(E\\cap \\Omega ).$ In particular, if $E\\subseteq \\Omega $ and ${\\text{\\rm Per}}_{s_0}(E;\\Omega )<\\infty $ for some $s_0 \\in (0,1)$ , then $\\mu (E)={{M}}(E)$ .", "Condition (REF ) is also in general necessary for the existence of the limit in (REF ).", "Indeed, next result shows that the existence of the limit in (REF ) is equivalent to the existence of the limit in (REF ), except in the special case in which the set $E$ occupies exactly half of the measure of $\\Omega $ (in this case the limit in (REF ) always exists, independently on the existence of the limit in (REF )).", "Theorem 2.7 Suppose that  ${\\text{\\rm Per}}_{s_0}(E;\\Omega )<\\infty $ , for some $s_0 \\in (0,1)$ .", "Then: If $|\\Omega \\setminus E| = |E \\cap \\Omega |$ , then $E\\in {{E}}$ and $\\mu (E)=M(E\\cap \\Omega )$ .", "If $|\\Omega \\setminus E| \\ne |E \\cap \\Omega |$ and $E\\in {{E}}$ , then the limit in (REF ) exists and $ \\alpha (E) = \\frac{\\mu (E) - {{M}}(E \\cap \\Omega )}{|\\Omega \\setminus E| - |E \\cap \\Omega |}.", "$ In the statements above we assumed the existence of the limits in (REF ) and (REF ).", "Such assumptions cannot be removed, since the limits in (REF ) and (REF ) may not exist, as we now point out: Example 2.8 There exists a set $E$ with $C^\\infty $ -boundary for which the limits in (REF ) and (REF ) do not exist.", "Example 2.9 There exists a set $E$ with $C^\\infty $ -boundary for which the limit in (REF ) exists and the limit in (REF ) does not exist.", "Notice that Examples REF and REF are provided by smooth sets, and therefore they have finite $s$ -perimeter for any $s\\in (0,1)$ (see, e.g., Lemma 11 in [5]).", "On the other hand, as regards condition (REF ), we point out that it cannot be dropped in general, since there are sets that do not satisfy it (and for them the limit in (REF ) does not make sense): Example 2.10 There exists a set $E$ for which  $\\text{\\rm Per}_s(E;\\Omega )=+\\infty $ for any $s\\in (0,1)$ ." ], [ "Proof of Proposition ", "We observe that ${\\mbox{the $s$-perimeter is subadditive.", "}}$ To check this, let $\\Omega _1$ , $\\Omega _2$ be open sets of ${R}^n$ .", "We remark that $&& \\!\\!\\!\\!\\!\\!L((E\\cup F)\\cap \\Omega _1,({C}(E\\cup F))\\cap \\Omega _2)\\\\[1ex]&&\\qquad \\quad =L((E\\cap \\Omega _1)\\cup (F\\cap \\Omega _1),({C}E)\\cap ({C}F)\\cap \\Omega _2)\\\\[1ex]&&\\qquad \\quad \\leqslant L( E\\cap \\Omega _1,({C}E)\\cap ({C}F)\\cap \\Omega _2)+L(F\\cap \\Omega _1,({C}E)\\cap ({C}F)\\cap \\Omega _2)\\\\[1ex]&&\\qquad \\quad \\leqslant L( E\\cap \\Omega _1,({C}E)\\cap \\Omega _2)+L(F\\cap \\Omega _1,({C}F)\\cap \\Omega _2).$ By taking $\\Omega _1:=\\Omega $ and $\\Omega _2:={R}^n$ we obtain $ L((E\\cup F)\\cap \\Omega ,{C}(E\\cup F))\\leqslant L( E\\cap \\Omega ,{C}E)+L(F\\cap \\Omega ,{C}F),$ while, by taking $\\Omega _1:={C}\\Omega $ and $\\Omega _2:=\\Omega $ , we conclude that $ L((E\\cup F)\\cap ({C}\\Omega ),({C}(E\\cup F))\\cap \\Omega )\\leqslant L( E\\cap ({C}\\Omega ),({C}E)\\cap \\Omega )+L(F\\cap ({C}\\Omega ),({C}F)\\cap \\Omega ).$ By summing up, we get $&& \\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!", "{\\text{\\rm Per}}_s(E\\cup F;\\Omega )\\\\&& \\quad = L( (E\\cup F)\\cap \\Omega , {C}(E\\cup F))+L((E\\cup F)\\cap ({C}\\Omega ),({C}(E\\cup F))\\cap \\Omega )\\\\[1ex]&& \\quad \\leqslant L( E\\cap \\Omega ,{C}E)+L(F\\cap \\Omega ,{C}F)\\\\&&\\quad \\quad +\\,L( E\\cap ({C}\\Omega ),({C}E)\\cap \\Omega )+L(F\\cap ({C}\\Omega ),({C}F)\\cap \\Omega )\\\\[1ex]&& \\quad =\\text{\\rm Per}_s(E;\\Omega )+ \\text{\\rm Per}_s(F;\\Omega ).$ This establishes (REF ) and then Proposition REF follows by taking the limit as $s\\searrow 0$ .", "$\\square $" ], [ "Proof of Proposition ", "First we show that $\\mu $ is not additive.", "Here and in the sequel, we denote by $B_R$ the open ball centered at $0\\in {R}^n$ of radius $R>0$ .", "We observe that if $x\\in B_1$ and $y\\in {C}B_2$ then $|x-y|\\,\\leqslant \\,|x|+|y|\\,\\leqslant \\, 2|y|$ , therefore $ s L(B_1,{C}B_2)\\,\\geqslant \\, c_1 s\\int _{B_1} dx \\int _{{C}B_2} dy \\frac{1}{|y|^{n+s}}\\,\\geqslant \\, c_2 s\\int _2^{+\\infty }\\frac{d\\rho }{\\rho ^{1+s}}\\,\\geqslant \\, c_3,$ for some positive constants $c_1$ , $c_2$ and $c_3$ .", "Now we take $E:={C}B_2$ , $F:=\\Omega :=B_1$ .", "Then $&& {\\text{\\rm Per}}_{s} (E;\\Omega )=L(B_1,{C}B_2),\\\\&& {\\text{\\rm Per}}_{s} (F;\\Omega )=L(B_1,{C}B_1)=L(B_1,{C}B_2)+L(B_1,B_2\\setminus B_1)\\\\\\,{\\mbox{ and }}\\,&& {\\text{\\rm Per}}_{s} (E\\cup F;\\Omega )=L(B_1,B_2\\setminus B_1).$ Therefore $s\\,{\\text{\\rm Per}}_{s} (E;\\Omega )+s\\,{\\text{\\rm Per}}_{s} (F;\\Omega ) &= & 2sL(B_1,{C}B_2)+sL(B_1,B_2\\setminus B_1) \\\\& \\geqslant &2c_3+s\\,L(B_1,B_2\\setminus B_1)\\\\& = & 2c_3 +s\\,{\\text{\\rm Per}}_{s} (E \\cup F;\\Omega ).$ By sending $s\\searrow 0$ , we conclude that $\\mu (E)+\\mu (F)\\geqslant 2c_3+\\mu (E\\cup F)$ , so $\\mu $ is not additive.", "Now we show that $\\mu $ is not monotone either.", "For this we take $E$ such that $\\mu (E)>0$ (for instance, one can take $E$ a small ball inside $\\Omega $ ; see Corollary REF ), and $F:={R}^n$ : with this choice, $E\\subset F$ and $\\text{\\rm Per}_s(F;\\Omega )=0$ , so $\\mu (E)>0=\\mu (F)$ .", "$\\square $" ], [ "Auxiliary observations", "Here we collect some observations, to be exploited in the subsequent proofs.", "Observation 1.", "First of all, we observe that $\\begin{split}&{\\mbox{if $A$ and $B$ are bounded, disjoints sets with~$\\mbox{dist}(A,B)\\geqslant c>0$, then }}\\\\&\\qquad \\displaystyle \\lim _{s\\searrow 0} s \\,L(A,B)=0.\\end{split}$ To check this, suppose that $A$ and $B$ lie in $B_R$ .", "Then $ \\int _A \\int _B \\frac{1}{|x-y|^{n+s}}\\,dx\\,dy\\,\\leqslant \\,\\int _{B_R} \\int _{B_R} \\frac{1}{c^{n+s}}\\,dx\\,dy\\,=\\,\\frac{({H}^{n-1}({S}^{n-1}))^2 R^{2n} }{n^2c^{n+s}}$ and this establishes (REF ).", "Observation 2.", "Now we would like to remark that the quantity $ \\lim _{s\\searrow 0}s\\int _{E\\cap ({C}B_R)}\\frac{1}{|y|^{n+s}}\\,dy$ is independent of $R$ , if the limit exists.", "More precisely, we show that for any $R\\geqslant r>0$ $\\lim _{s\\searrow 0}s\\left(\\int _{E\\cap ({C}B_R)}\\frac{1}{|y|^{n+s}}\\,dy-\\int _{E\\cap ({C}B_r)}\\frac{1}{|y|^{n+s}}\\,dy\\right)=0.$ To prove this, we notice that $s\\int _{E \\cap (B_R \\setminus B_r)}\\frac{1}{|y|^{n+s}}\\,dy\\leqslant s\\int _{B_R \\setminus B_r}\\frac{1}{|y|^{n+s}}\\,dy= s {{H}}^{n-1}(S^{n-1}) \\int _{r}^{R}\\frac{1}{\\rho ^{1+s}}\\,d\\rho \\nonumber \\\\= {{H}}^{n-1}(S^{n-1})\\left( \\frac{1}{r^s}-\\frac{1}{R^s}\\right)$ and so, by taking limit in $s$ , $\\lim _{s\\searrow 0} s\\int _{E \\cap (B_R \\setminus B_r)}\\frac{1}{|y|^{n+s}}\\,dy =0,$ which establishes (REF ).", "Observation 3.", "As a consequence of (REF ), it follows that if the limit in (REF ) exists then $\\alpha (E)= \\lim _{s\\searrow 0}s\\int _{E\\cap ({C}B_R)}\\frac{1}{|y|^{n+s}}\\,dy \\qquad \\forall \\,R>0.", "$ Observation 4.", "For any $s\\in (0,1)$ , we define $\\alpha _s(E) := s \\int _{E \\cap ({C}B_1)} \\frac{1}{|y|^{n+s}}\\,dy $ and we prove that, for any bounded set $F\\subset {R}^n$ , and any set $E\\subseteq {R}^n$ , $\\lim _{R\\rightarrow +\\infty }\\limsup _{s\\searrow 0}\\left|\\alpha _s(E)\\,|F| -s \\int _{F}\\int _{E\\cap ({C}B_R)}\\frac{1}{|x-y|^{n+s}}\\,dx\\,dy\\right|=0.$ To prove this, we take $r>0$ such that $F\\subset B_r$ and $R>1+2r$ (later on $R$ will be taken as large as we wish).", "We observe that, for any $z\\in B_r$ and $y\\in {C}B_R$ , $ |z-y|\\geqslant |y|-|z|=\\left(1-\\frac{r}{R}\\right)|y|+\\frac{r}{R}|y|-|z|\\geqslant \\frac{|y|}{2}.$ Therefore, if, for any fixed $y\\in {C}B_R$ we consider the map $ h(z):=\\frac{1}{|z-y|^{n+s}},\\qquad z\\in B_r,$ we have that $ |\\nabla h(z)|=\\frac{n+s}{|z-y|^{n+s+1}}\\leqslant \\frac{2^{n+s+1}(n+s)}{|y|^{n+s+1}},$ for any $z\\in B_r$ , which implies $ \\left| \\frac{1}{|x-y|^{n+s}}-\\frac{1}{|y|^{n+s}}\\right|=|h(x)-h(0)|\\leqslant \\frac{2^{n+s+1}(n+s)|x|}{|y|^{n+s+1}}\\qquad \\forall \\,x\\in B_r,\\,y\\in {C}B_R.$ Therefore $&&\\left|\\int _F\\left(\\int _{E\\cap ({C}B_R)}\\frac{1}{|y|^{n+s}}\\,dy\\right)\\,dx-\\int _{F}\\int _{E\\cap ({C}B_R)}\\frac{1}{|x-y|^{n+s}}\\,dx\\,dy\\right|\\\\&&\\qquad \\leqslant \\int _F\\left(\\int _{E\\cap ({C}B_R)}\\left|\\frac{1}{|y|^{n+s}}-\\frac{1}{|x-y|^{n+s}} \\right|\\,dy\\right)\\,dx\\\\&&\\qquad \\leqslant \\int _F\\left(\\int _{E\\cap ({C}B_R)}\\frac{2^{n+s+1}(n+s)|x|}{|y|^{n+s+1}}\\,dy\\right)\\,dx\\\\&&\\qquad \\leqslant 2^{n+s+1}(n+s) |F| r\\int _{{C}B_R}\\frac{1}{|y|^{n+s+1}}\\,dy\\,\\leqslant \\,C$ for some $C>0$ independent of $s$ .", "As a consequence $&& \\left|\\alpha _s(E)\\,|F| -s \\int _{F}\\int _{E\\cap ({C}B_R)}\\frac{1}{|x-y|^{n+s}}\\,dx\\,dy\\right|\\\\&&\\qquad \\qquad \\qquad \\qquad \\qquad \\leqslant |F|\\,\\left|\\alpha _s(E)-s\\int _{E\\cap ({C}B_R)}\\frac{1}{|y|^{n+s}}\\,dy\\right|+Cs.$ This and (REF ) (applied here with $r:=1$ ) imply (REF ).", "Observation 5.", "If the limit in (REF ) exists, then (REF ) boils down to $\\lim _{R\\rightarrow +\\infty }\\limsup _{s\\searrow 0}\\left|\\alpha (E)\\,|F| -s \\int _{F}\\int _{E\\cap ({C}B_R)}\\frac{1}{|x-y|^{n+s}}\\,dx\\,dy\\right|=0.$ Observation 6.", "Now we point out that, if $F\\subseteq \\Omega \\subset B_R$ for some $R>0$ , and $F$ has finite $s_0$ -perimeter in $\\Omega $ for some $s_0 \\in (0,1)$ , then $\\lim _{s\\searrow 0} s\\int _{F}\\int _{B_R\\setminus F}\\frac{1}{|x-y|^{n+s}}\\,dx\\,dy=0.$ Indeed, for any $s\\in (0,s_0)$ , $&& \\int _{F}\\int _{B_R\\setminus F}\\frac{1}{|x-y|^{n+s}}\\,dx\\,dy\\\\ &&\\qquad \\leqslant \\int _{F}\\int _{(B_R\\setminus F)\\cap \\lbrace |x-y|\\leqslant 1\\rbrace }\\frac{1}{|x-y|^{n+s_0}}\\,dx\\,dy+\\int _{F}\\int _{(B_R\\setminus F)\\cap \\lbrace |x-y|> 1\\rbrace }1\\,dx\\,dy\\\\ &&\\qquad \\leqslant {\\text{\\rm Per}}_{s_0}(F;\\Omega )+|B_R|^2,$ which implies (REF ).", "In particular, thanks to [1], the argument above also shows that if $F\\Subset \\Omega \\subset B_R$ and $\\chi _F \\in BV(\\Omega )$ , then $F$ has finite $s$ -perimeter in $\\Omega $ for any $s \\in (0,1)$ .", "Observation 7.", "Let $E_1:=E\\cap \\Omega $ and $E_2:=E\\setminus \\Omega $ .", "Then $\\begin{split}{\\text{\\rm Per}}_s(E;\\Omega )\\,&={\\text{\\rm Per}}_s(E_1\\cup E_2;\\Omega )\\\\ &= L(E_1,\\Omega \\setminus E_1)+L(E_1,({C}\\Omega )\\setminus E_2)+L(E_2,\\Omega \\setminus E_1)\\\\ &= L(E_1,{C}E_1)-L(E_1,E_2)+L(E_2,\\Omega \\setminus E_1)\\\\ &= {\\text{\\rm Per}}_s(E_1;\\Omega )-L(E_1,E_2)+L(E_2,\\Omega \\setminus E_1).\\end{split}$ With these observations in hand, we are ready to continue the proofs of the main results." ], [ "Proof of Proposition ", "We prove Proposition REF by suitably modifying the proof of Proposition REF .", "Given two open sets $\\Omega _1$ and $\\Omega _2$ , and two disjoint sets $E$ and $F$ , we have that $&&\\!\\!\\!\\!\\!\\!\\!\\!", "L((E\\cup F)\\cap \\Omega _1,({C}(E\\cup F))\\cap \\Omega _2)\\\\[1ex]&&\\qquad =L((E\\cap \\Omega _1)\\cup (F\\cap \\Omega _1),({C}E)\\cap ({C}F)\\cap \\Omega _2)\\\\[1ex]&&\\qquad =L(E\\cap \\Omega _1,({C}E)\\cap ({C}F)\\cap \\Omega _2)+L(F\\cap \\Omega _1,({C}E)\\cap ({C}F)\\cap \\Omega _2).$ By taking $\\Omega _1:=\\Omega $ and $\\Omega _2:={R}^n$ we obtain $ L((E\\cup F)\\cap \\Omega , {C}(E\\cup F) )=L(E\\cap \\Omega ,({C}E)\\cap ({C}F))+ L(F\\cap \\Omega ,({C}E)\\cap ({C}F) )$ while, by taking $\\Omega _1:={C}\\Omega $ and $\\Omega _2:=\\Omega $ , we conclude that $&& \\!\\!\\!\\!\\!\\!\\!\\!", "L((E\\cup F)\\cap ({C}\\Omega ),({C}(E\\cup F))\\cap \\Omega ) \\\\&& \\qquad = \\, L(E\\cap ({C}\\Omega ),({C}E)\\cap ({C}F)\\cap \\Omega )+L(F\\cap ({C}\\Omega ),({C}E)\\cap ({C}F)\\cap \\Omega ).$ As a consequence, $&& \\!\\!\\!\\!\\!\\!", "{\\text{\\rm Per}}_s(E\\cup F;\\Omega )\\\\ &&\\qquad = \\, L((E\\cup F)\\cap \\Omega , {C}(E\\cup F) )+L((E\\cup F)\\cap ({C}\\Omega ),({C}(E\\cup F))\\cap \\Omega )\\\\[1ex]&&\\qquad = \\, L(E\\cap \\Omega ,({C}E)\\cap ({C}F))+ L(F\\cap \\Omega ,({C}E)\\cap ({C}F) )\\\\&&\\qquad \\quad + \\,L(E\\cap ({C}\\Omega ),({C}E)\\cap ({C}F)\\cap \\Omega )+L(F\\cap ({C}\\Omega ),({C}E)\\cap ({C}F)\\cap \\Omega )\\\\[1ex]&&\\qquad = \\,{\\text{\\rm Per}}_s(E;\\Omega )+{\\text{\\rm Per}}_s(F;\\Omega )\\\\ &&\\qquad \\quad -\\, L(E\\cap \\Omega , ({C}E)\\cap F)-L(F\\cap \\Omega , E\\cap ({C}F))\\\\ &&\\qquad \\quad -\\,L(E\\cap ({C}\\Omega ), ({C}E)\\cap F\\cap \\Omega )-L(F\\cap ({C}\\Omega ), E\\cap ({C}F)\\cap \\Omega ).$ We remark that the last interactions involve only bounded, separated sets, since so are $E$ and $F$ , therefore, by (REF ), $ \\lim _{s\\searrow 0} s \\,{\\text{\\rm Per}}_s(E\\cup F;\\Omega )=\\lim _{s\\searrow 0} \\big ( s \\,{\\text{\\rm Per}}_s(E;\\Omega )+s \\,{\\text{\\rm Per}}_s(F;\\Omega )\\big ),$ which completes the proof of Proposition REF .", "$\\square $" ], [ "Proof of Theorem ", "We suppose that $\\Omega \\subset B_r$ , for some $r>0$ , and we take  $R>1+2r$ .", "Let $E_1:=E\\cap \\Omega $ and $E_2:=E\\setminus \\Omega $ .", "Notice that, for any $F\\subseteq \\Omega $ , which has finite $s_0$ -perimeter in $\\Omega $ for some $s_0\\in (0,1)$ , $E_2\\cap B_R\\subseteq B_R\\setminus \\Omega \\subseteq B_R\\setminus F$ and so (REF ) gives that $ \\lim _{s\\searrow 0} s\\int _{F}\\int _{E_2\\cap B_R}\\frac{1}{|x-y|^{n+s}}\\,dx\\,dy=0,$ provided that $F$ has finite $s_0$ -perimeter in $\\Omega $ .", "Using this and (REF ), we conclude that, for any $F\\subseteq \\Omega $ of finite $s_0$ -perimeter in $\\Omega $ , $\\begin{split}&\\lim _{s\\searrow 0}s\\int _{F}\\int _{E_2}\\frac{1}{|x-y|^{n+s}}\\,dx\\,dy\\\\&\\qquad \\qquad \\qquad \\qquad =\\,\\lim _{R\\rightarrow +\\infty }\\lim _{s\\searrow 0}s\\int _{F}\\int _{E_2}\\frac{1}{|x-y|^{n+s}}\\,dx\\,dy\\\\ &\\qquad \\qquad \\qquad \\qquad =\\,\\lim _{R\\rightarrow +\\infty }\\lim _{s\\searrow 0}s\\int _{F}\\int _{E_2\\cap ({C}B_R)}\\frac{1}{|x-y|^{n+s}}\\,dx\\,dy\\\\[1ex]&\\qquad \\qquad \\qquad \\qquad =\\,\\alpha (E)\\,|F|.\\end{split}$ In particularWe stress that both $E_1$ and $\\Omega \\setminus E_1$ have finite $s_0$ -perimeter in $\\Omega $ if so has $E$ , thanks to our smoothness assumption on $\\partial \\Omega $ .", "We check this claim for $E_1$ , the other being analogous.", "First of all, fixed $B_R\\supset B_r\\supset \\Omega $ , we have that $ L\\big (E_1,(E\\setminus \\Omega )\\cap ({C}B_R)\\big )\\leqslant L(B_r, {C}B_R)<+\\infty .$ Also $L\\big (\\Omega \\cap B_R, ({C}\\Omega )\\cap B_R\\big )<+\\infty $ (see, e.g., Lemma 11 in [5]), therefore $ && {\\text{\\rm Per}}_{s_0} (E_1;\\Omega )=L(E_1,{C}E_1)=L(E_1,{C}E)+L(E_1, E\\setminus \\Omega )\\\\ &&\\qquad \\leqslant {\\text{\\rm Per}}_{s_0} (E;\\Omega )+L\\big (E_1, (E\\setminus \\Omega )\\cap B_R\\big )+L\\big (E_1, (E\\setminus \\Omega )\\cap ({C}B_R)\\big )\\\\ &&\\qquad \\leqslant {\\text{\\rm Per}}_{s_0} (E;\\Omega )+L\\big (\\Omega , ({C}\\Omega )\\cap B_R\\big )+L\\big (E_1, (E\\setminus \\Omega )\\cap ({C}B_R)\\big ),$ that is finite.", ", by taking $F:=E_1$ and $F:=\\Omega \\setminus E_1$ , and recalling (REF ) and (REF ), $\\begin{split}&\\lim _{s\\searrow 0}s\\int _{E_1}\\int _{E_2}\\frac{1}{|x-y|^{n+s}}\\,dx\\,dy=\\alpha (E)\\,|E_1|=\\widetilde{\\alpha }(E)\\,{{M}}(E_1)\\\\{\\mbox{and }}\\;&\\lim _{s\\searrow 0}s\\int _{\\Omega \\setminus E_1}\\int _{E_2}\\frac{1}{|x-y|^{n+s}}\\,dx\\,dy=\\alpha (E)\\,|\\Omega \\setminus E_1|=\\widetilde{\\alpha }(E)\\,{{M}}(\\Omega \\setminus E_1).\\end{split}$ We now claim $\\lim _{s\\searrow 0}s\\,{\\text{\\rm Per}}_s(E_1;\\Omega )={{M}}(E_1).$ Indeed, since $E_1\\subseteq \\Omega $ , this is a plain consequence of Theorem REF (see also Remark 4.3 in [8] for another elementary proof) by simply choosing $u=\\chi _{E_1}$ there: $\\begin{split}\\lim _{s\\searrow 0} s \\,{\\text{\\rm Per}}_s(E_1;\\Omega )& = \\, \\lim _{s\\searrow 0} s \\,L(E_1, {C}E_1) \\\\& = \\, \\lim _{s\\searrow 0} \\frac{s}{2}\\int _{{R}^n}\\int _{{R}^n}\\frac{ |\\chi _{E_1}(x)-\\chi _{E_1}(y)|^2}{|x-y|^{n+s}}\\,dx\\,dy\\\\[1ex]& = \\,{{H}}^{n-1}(S^{n-1})\\,\\Vert \\chi _{E_1}\\Vert _{L^2({R}^n)}^2 \\ = \\ {{H}}^{n-1}(S^{n-1})\\,|E_1|,\\end{split}$ as desired.", "Thus, using (REF ), (REF ), and (REF ), we obtain $\\lim _{s\\searrow 0} s{\\text{\\rm Per}}_s(E;\\Omega )= {{M}}(E_1)-\\widetilde{\\alpha }(E) {{M}}(E_1)+\\widetilde{\\alpha }(E) {{M}}(\\Omega \\setminus E_1),$ which is the desired result.", "$\\square $" ], [ "Proof of Corollary ", "We fix $R$ large enough so that $E\\subset B_R$ , hence $E \\cap ({C}B_R) = \\varnothing $ .", "By the expression of $\\alpha (E)$ in $(\\ref {alfa})$ , we have that the limit in $(\\ref {Rj})$ exists and $\\alpha (E)=0$ .", "Then the result follows by Theorem REF .", "$\\square $" ], [ "Proof of Theorem ", "We suppose that $\\Omega \\subset B_r$ , for some $r>0$ , and we take  $R>1+2r$ .", "Let $E_1:=E\\cap \\Omega $ and $E_2:=E\\setminus \\Omega $ .", "By (REF ), $\\begin{split}&\\!\\!\\!\\!\\!", "s{\\text{\\rm Per}}_s(E;\\Omega ) -s {\\text{\\rm Per}}_s(E_1;\\Omega ) \\\\[1ex]& \\qquad =s L(E_2,\\Omega \\setminus E_1) - s L(E_1,E_2) \\\\[1ex]& \\qquad = s \\int _{\\Omega \\setminus E_1} \\int _{E_2 \\cap B_R}\\frac{1}{|x-y|^{n+s}}\\, dx \\, dy+ s \\int _{\\Omega \\setminus E_1} \\int _{E_2 \\cap ({C}B_R)}\\frac{1}{|x-y|^{n+s}}\\, dx \\, dy \\\\& \\qquad \\quad - s \\int _{ E_1} \\int _{E_2 \\cap B_R} \\frac{1}{|x-y|^{n+s}}\\, dx \\, dy- s \\int _{ E_1} \\int _{E_2 \\cap ({C}B_R)} \\frac{1}{|x-y|^{n+s}}\\, dx\\, dy.\\end{split}$ By rearranging the terms, we obtain $\\begin{split}& I(s,R):=s \\int _{\\Omega \\setminus E_1} \\int _{E_2 \\cap ({C}B_R)}\\frac{1}{|x-y|^{n+s}}\\, dx \\, dy- s \\int _{ E_1} \\int _{E_2 \\cap ({C}B_R)} \\frac{1}{|x-y|^{n+s}}\\, dx \\,dy\\\\[1ex]& \\qquad \\quad \\ = s{\\text{\\rm Per}}_s(E;\\Omega ) -s{\\text{\\rm Per}}_s(E_1;\\Omega )- s \\int _{\\Omega \\setminus E_1} \\int _{E_2 \\cap B_R}\\frac{1}{|x-y|^{n+s}}\\, dx \\, dy \\\\& \\qquad \\qquad \\ + s \\int _{ E_1} \\int _{E_2 \\cap B_R} \\frac{1}{|x-y|^{n+s}}\\, dx \\, dy.\\end{split}$ By using (REF ) with $F:=\\Omega \\setminus E_1$ and $F:= E_1$ (which have finite $s_0$ -perimeter in $\\Omega $ , recall the footnote on page REF ), we have that the last two terms in (REF ) converge to zero as $s\\searrow 0$ , thus $\\lim _{s\\searrow 0} I(s,R)=\\lim _{s\\searrow 0}\\Big (s{\\text{\\rm Per}}_s(E;\\Omega ) -s{\\text{\\rm Per}}_s(E_1;\\Omega )\\Big ).$ We now recall the notation in (REF ) and we write $\\alpha _s(E)\\,|\\Omega \\setminus E_1|&=& s \\int _{\\Omega \\setminus E_1}\\int _{E_2 \\cap ({C}B_R)} \\frac{1}{|x-y|^{n+s}}\\, dx \\, dy\\\\&&\\ +\\, \\alpha _s(E)\\,|\\Omega \\setminus E_1|-s \\int _{\\Omega \\setminus E_1} \\int _{E_2 \\cap ({C}B_R)}\\frac{1}{|x-y|^{n+s}} \\, dx \\, dy,$ and $\\alpha _s(E)\\,|E_1|&=& s \\int _{ E_1}\\int _{E_2 \\cap ({C}B_R)} \\frac{1}{|x-y|^{n+s}}\\, dx \\, dy\\\\ &&+\\,\\alpha _s(E)\\,|E_1|-s \\int _{E_1} \\int _{E_2 \\cap ({C}B_R)}\\frac{1}{|x-y|^{n+s}} \\, dx \\, dy.$ By subtracting term by term, we obtain that $&& \\alpha _s(E)\\,\\Big ( |\\Omega \\setminus E_1|-|E_1|\\Big )\\\\&& \\qquad \\qquad = I(s,R)+\\left(\\alpha _s(E)\\,|\\Omega \\setminus E_1|-s \\int _{\\Omega \\setminus E_1} \\int _{E_2 \\cap ({C}B_R)}\\frac{1}{|x-y|^{n+s}} \\, dx \\, dy\\right)\\\\ &&\\qquad \\qquad \\quad -\\,\\left(\\alpha _s(E)\\,|E_1|-s \\int _{E_1} \\int _{E_2 \\cap ({C}B_R)}\\frac{1}{|x-y|^{n+s}} \\, dx \\, dy\\right).$ As a consequence, by using (REF ) (applied here both with $F:=\\Omega \\setminus E_1$ and $F:=E_1$ ), $\\lim _{R\\rightarrow +\\infty }\\lim _{s\\searrow 0}\\left[ \\alpha _s(E)\\,\\Big ( |\\Omega \\setminus E_1|-|E_1|\\Big )-I(s,R)\\right]=0.$ Now, if $|\\Omega \\setminus E|=|E\\cap \\Omega |$ then $|\\Omega \\setminus E_1|-|E_1|=0$ , and from (REF ), (REF ), and Corollary REF we get $0=\\lim _{R\\rightarrow +\\infty }\\lim _{s\\searrow 0} I(s,R)=\\lim _{s\\searrow 0}s{\\text{\\rm Per}}_s(E;\\Omega ) -M(E\\cap \\Omega ),$ which proves that $E \\in E$ and $\\mu (E)=M(E\\cap \\Omega )$ .", "This establishes Theorem REF (i).", "On the other hand, if $|\\Omega \\setminus E|\\ne |E\\cap \\Omega |$ , then by (REF ), (REF ), and Corollary REF we obtain the existence of the limit $&&\\Big ( |\\Omega \\setminus E_1|-|E_1|\\Big )\\, \\lim _{s\\searrow 0}\\alpha _s(E) \\\\&& \\qquad \\qquad \\qquad =\\lim _{R\\rightarrow +\\infty }\\lim _{s\\searrow 0}\\alpha _s(E)\\,\\Big ( |\\Omega \\setminus E_1|-|E_1|\\Big )\\\\[1ex]&& \\qquad \\qquad \\qquad =\\lim _{R\\rightarrow +\\infty }\\lim _{s\\searrow 0}\\bigg \\lbrace \\left[ \\alpha _s(E)\\,\\Big ( |\\Omega \\setminus E_1|-|E_1|\\Big )-I(s,R)\\right]+\\, I(s,R)\\bigg \\rbrace \\\\[2ex]&& \\qquad \\qquad \\qquad = \\mu (E)-\\mu (E_1)= \\mu (E)-{M}(E\\cap \\Omega ),$ which completes the proof of Theorem REF (ii).", "$\\square $" ], [ "Construction of Example ", "We start with some preliminary computations.", "Let $a_k:=10^{k^2}$ , for any $k\\in {N}$ , and let $I_j:=\\bigcup _{k\\in {N}} \\big [a_{4k+j}, a_{4k+j+1}\\big ), \\quad \\text{for} \\ j=0,1,2,3.$ Notice that $[1,+\\infty )$ may be written as the disjoint union of the $I_j$ 's.", "Let $\\varphi \\in C^\\infty \\big ([0,+\\infty ),\\,[0,1]\\big )$ be such that $\\varphi =0$ in $[0,1]\\cup I_0$ , $\\varphi =1$ in $I_2$ , and then $\\varphi $ smoothly interpolates between 0 and 1 in $I_1\\cup I_3$ .", "We claim that there exist two sequences $\\nu _{0,k}\\rightarrow +\\infty $ and $\\nu _{1,k}\\rightarrow +\\infty $ such that $\\lim _{k\\rightarrow +\\infty }\\int _0^{+\\infty } \\varphi (\\nu _{0,k} x) e^{-x}\\,dx=0\\; \\ {\\mbox{ and }}\\;\\lim _{k\\rightarrow +\\infty }\\int _0^{+\\infty } \\varphi (\\nu _{1,k} x) e^{-x}\\,dx=1.$ To check (REF ), we take $\\nu _{0,k}:=a_{4k+1}/k$ and $\\nu _{1,k}:=a_{4k+3}/k$ .", "We observe that, by construction, $\\varphi =0$ in $\\big [a_{4k}, a_{4k+1}\\big )$ and $\\varphi =1$ in $\\big [a_{4k+2}, a_{4k+3}\\big )$ , so $\\varphi (\\nu _{0,k}x)=0$ for any $x\\in [kb_{0,k}, \\,k)$ and $\\varphi (\\nu _{1,k}x)=1$ in $[kb_{1,k},\\,k)$ , where $ b_{0,k}:=\\frac{a_{4k}}{a_{4k+1}}=10^{-(8k+1)}\\;{\\mbox{ and }}b_{1,k}:=\\frac{a_{4k+2}}{a_{4k+3}}=10^{-(8k+5)}.$ We deduce that $&& \\int _0^{+\\infty } \\varphi (\\nu _{0,k} x) e^{-x}\\,dx\\leqslant \\int _0^{kb_{0,k}} e^{-x}\\,dx+\\int _{k}^{+\\infty } e^{-x}\\,dx=1-e^{-kb_{0,k}}+e^{-k}\\\\ {\\mbox{and }}&&\\int _0^{+\\infty } \\varphi (\\nu _{1,k} x) e^{-x}\\,dx\\geqslant \\int _{kb_{1,k}}^{k} e^{-x}\\,dx=e^{-kb_{1,k}}-e^{-k}.$ This implies (REF ) by noticing that $\\lim _{k\\rightarrow +\\infty } k b_{0,k}=0=\\lim _{k\\rightarrow +\\infty } k b_{1,k}.$ Now we construct our example by using the above function $\\varphi $ and (REF ).", "We take $\\Omega := B_{1/2}$ and $E := \\big \\lbrace x =\\left( \\rho \\cos \\gamma , \\rho \\sin \\gamma \\right),\\rho >1, \\gamma \\in \\left[0,\\theta \\left(\\rho \\right) \\right] \\big \\rbrace \\subset {R}^2$ , where $\\theta \\left( \\rho \\right) := \\varphi \\left(\\log \\rho \\right)$ .", "First of all, since $\\Omega =B_{1/2}$ and $E \\subset {R}^n\\setminus B_1$ , it is easy to see that ${\\text{\\rm Per}}_s(E;\\Omega ) = \\int _\\Omega \\int _E\\frac{1}{|x-y|^{n+s}}\\,dx\\,dy\\leqslant |\\Omega | \\, \\int _{{R}^n\\setminus B_1} \\frac{2^{n+s}}{|z|^{n+s}}\\,dz<\\infty $ for any $s \\in (0,1)$ (notice that, since $E$ has smooth boundary, the fact that $E$ has finite $s$ -perimeter is also a consequence of Lemma 11 in [5]).", "Then, recalling (REF ) we have $ \\alpha _s(E) =s \\int _{1}^{+\\infty } \\int _{0}^{\\theta \\left(\\rho \\right)} \\frac{\\rho ^{n-1}}{\\rho ^{n+s}} \\, d\\theta \\, d\\rho = s \\int _{1}^{+\\infty } \\theta \\left(\\rho \\right) \\frac{1}{\\rho ^{1+s}} \\, d\\rho .$ Therefore, by the change of variable $\\log \\rho = r$ , we have $\\alpha _s(E) = s \\int _{0}^{+\\infty } \\varphi \\left( r\\right) e^{-rs}\\,dr,$ and, by the further change $rs=x$ , we have $\\alpha _s(E) = \\int _{0}^{+\\infty } \\varphi \\left( \\frac{x}{s}\\right) e^{-x}\\,dx.$ If we set $\\nu ={1}/{s}$ , the limit in (REF ) becomes the following: $\\alpha (E) = \\lim _{\\nu \\rightarrow \\infty } \\int _{0}^{+\\infty } \\varphi \\left( \\nu x\\right) e^{-x}\\,dx,$ and, by (REF ), we get that such a limit does not exist.", "This shows that the limit in (REF ) does not exist.", "Since $|\\Omega \\setminus E|=|B_{1/2}|>0=|E\\cap \\Omega |$ , by Theorem REF (ii), the limit in (REF ) does not exist either.", "$\\square $" ], [ "Construction of Example ", "It is sufficient to modify Example REF inside $\\Omega =B_{1/2}$ in such a way that $|\\Omega \\setminus E|=|E\\cap \\Omega |$ .", "Notice that, since the set $E$ has smooth boundary, then it has finite $s$ -perimeter for any $s \\in (0,1)$ (see Lemma 11 in [5]).", "Then (REF ) is not affected by this modification and so the limit in (REF ) does not exist in this case too.", "On the other hand, the limit in (REF ) exists, thanks to Theorem REF (i).", "$\\square $" ], [ "Construction of Example ", "We take a decreasing sequence $\\beta _k$ such that $\\beta _k>0$ for any $k\\geqslant 1$ , $ M:=\\sum _{k=1}^{+\\infty } \\beta _k<+\\infty $ but $\\sum _{k=1}^{+\\infty } \\beta _{2k}^{1-s}=+\\infty \\qquad \\forall \\, s\\in (0,1).$ For instance, one can take $\\beta _1:=\\displaystyle \\frac{1}{\\log ^2 2}$ and $\\displaystyle \\beta _k:=\\frac{1}{k\\log ^2 k}$ for any $k\\geqslant 2$ .", "Now, we define $&& \\Omega :=(0,M)\\subset {R},\\\\&& \\sigma _m:=\\sum _{k=1}^m \\beta _k,\\\\ && I_m:=(\\sigma _m,\\sigma _{m+1}),\\\\{\\mbox{and }}&&E:=\\bigcup _{j=1}^{+\\infty } I_{2j}.$ Notice that $E\\subset \\Omega $ and ${\\text{\\rm Per}}_s(E;\\Omega ) & = & L(E,{C}E) \\nonumber \\\\& \\geqslant & \\sum _{j=1}^{+\\infty }L(I_{2j},I_{2j+1})\\ =\\ \\sum _{j=1}^{+\\infty }\\int _{\\sigma _{2j}}^{\\sigma _{2j+1}} \\int _{\\sigma _{2j+1}}^{\\sigma _{2j+2}}\\frac{1}{|x-y|^{1+s}}\\,dx\\,dy.$ An integral computation shows that if $a<b<c$ then $ \\int _{a}^{b}\\int _{b}^{c}\\frac{1}{|x-y|^{1+s}}\\,dx\\,dy=\\frac{1}{s(1-s)}\\Big [(c-b)^{1-s}+(b-a)^{1-s}-(c-a)^{1-s}\\Big ].$ By plugging this into (REF ), we obtain $\\begin{split}& s(1-s){\\text{\\rm Per}}_s(E;\\Omega )\\\\ &\\qquad \\geqslant \\sum _{j=1}^{+\\infty } \\Big [(\\sigma _{2j+2}-\\sigma _{2j+1})^{1-s}+(\\sigma _{2j+1}-\\sigma _{2j})^{1-s}-(\\sigma _{2j+2}-\\sigma _{2j})^{1-s}\\Big ] \\\\& \\qquad = \\sum _{j=1}^{+\\infty }\\beta _{2j+2}^{1-s}+\\beta _{2j+1}^{1-s}-(\\beta _{2j+2}+\\beta _{2j+1})^{1-s}.\\end{split}$ Now we observe that the map $[0,1)\\ni t\\mapsto (1+t)^{1-s}$ is concave, therefore $ (1+t)^{1-s}\\leqslant 1+(1-s)t\\leqslant 1+(1-s) t^{1-s}$ for any $t\\in [0,1)$ , that is $ 1+t^{1-s}-(1+t)^{1-s}\\geqslant s t^{1-s}.$ By taking $t:=\\beta _{2j+2}/\\beta _{2j+1}$ and then multiplying by $\\beta _{2j+1}^{1-s}$ , we obtain $\\beta _{2j+1}^{1-s}+\\beta _{2j+2}^{1-s}-(\\beta _{2j+1}+\\beta _{2j+2})^{1-s}\\geqslant s \\beta _{2j+2}^{1-s}.$ By plugging this into (REF ) and using (REF ), we conclude that ${\\text{\\rm Per}}_s (E;\\Omega )\\, \\geqslant \\, \\frac{1}{1-s}\\sum _{j=1}^{+\\infty }\\beta _{2j+2}^{1-s}=+\\infty \\qquad \\forall \\, s\\in (0,1),$ as desired.$\\square $" ] ]

[ [ "Numerical Analysis of Parallel Replica Dynamics" ], [ "Abstract Parallel replica dynamics is a method for accelerating the computation of processes characterized by a sequence of infrequent events.", "In this work, the processes are governed by the overdamped Langevin equation.", "Such processes spend much of their time about the minima of the underlying potential, occasionally transitioning into different basins of attraction.", "The essential idea of parallel replica dynamics is that the exit time distribution from a given well for a single process can be approximated by the minimum of the exit time distributions of $N$ independent identical processes, each run for only 1/N-th the amount of time.", "While promising, this leads to a series of numerical analysis questions about the accuracy of the exit distributions.", "Building upon the recent work in Le Bris et al., we prove a unified error estimate on the exit distributions of the algorithm against an unaccelerated process.", "Furthermore, we study a dephasing mechanism, and prove that it will successfully complete." ], [ "Introduction", "Parallel replica dynamics (ParRep) is a numerical tool first introduced by Voter in [26] (see also[27], [22]) for accelerating the simulation of stochastic processes characterized by a sequence of infrequent, but rapid, transitions from one state to another.", "A standard and important problem in which such a separation of scales is present is the migration of defects through a crystalline lattice; see [22] and references therein for examples.", "Roughly, the idea behind parallel replica dynamics is as follows.", "Suppose a trajectory spends time $t$ in a particular state, before transitioning into another.", "Furthermore, assume $t$ is large, relative to the scale of the time step discretization.", "We wish to avoid directly simulating a single realization for time $t$ .", "We approximate the simulation of a single trajectory for time $t$ with $N$ independent copies, each simulated for time $t/N$ , and follow the particular trajectory that escapes first.", "This holds out the promise for a linear speedup with the number of independent realizations we are able to simulate.", "Of course, this is not exact, and error is introduced.", "A particular concern is error in the exit distributions of the system as it migrates from one state to another – does ParRep disrupt the state to state dynamics?", "Inspired by the tools proposed in [4], we prove an error estimate on the exit distributions over a single “cycle” of ParRep (the transition from one state to the next)." ], [ "The Algorithm", "We assume the system we wish to accelerate evolves according to the overdamped Langevin equation, $dX_t = - \\nabla V(X_t) dt + \\sqrt{2 \\beta ^{-1}} dB_t, \\quad X_t \\in \\mathbb {R}^{n},$ where $B_t$ is a Wiener process and $\\beta $ is proportional to inverse temperature.", "Though ParRep was originally developed for the Langevin equations, it is readily adapted to this problem.", "We next assume that our system is such that $V$ has a denumerable set of local minima, $x_j$ , $j=1,2,\\ldots $ For each minima, we associate a set $W_j\\subset \\mathbb {R}^{n}$ , the “well.” $W_j$ could be the basin of attraction of $x_j$ ; if $y(t)$ solves the ODE $\\dot{y}= - \\nabla V(y),\\quad y(0) = y_0\\in \\mathbb {R}^n,$ then $W_j = \\left\\lbrace y_0 : \\lim _{t\\rightarrow + \\infty } y(t) = x_j\\right\\rbrace .$ However, this definition is not essential; for the sake of our analysis, $W_j$ need only be a bounded set in $\\mathbb {R}^n$ with sufficiently regular boundary.", "This motivates defining the well selection function, $\\mathcal {S} :\\mathbb {R}^{n} \\rightarrow \\mathbb {N},$ which identifies the basin associated with a given position.", "Associated with this is the “coarse grained” trajectory, $\\mathcal {S}_t \\equiv \\mathcal {S}(X_t)$ which only identifies the present well.", "If the wells are “deep” with well-defined minima, then $X_t$ will infrequently transition from one to another.", "Such a well corresponds to a metastable state.", "Much of the simulation time will be spent waiting for a jump to occur.", "The goal of ParRep is to reduce this computational expense by providing a satisfactory approximation of the form $\\mathcal {S}_t \\approx \\mathcal {S}^{\\rm ParRep}_t.$ In other words, we are willing to sacrifice information about where the trajectory is within each well, for the sake of rapidly computing the sequence of wells the trajectory visits.", "We now describe the ParRep algorithm in the following steps: the decorrelation step; the dephasing step; and the parallel step.", "These steps are diagrammed in Figures REF and REF .", "We assume that the reference process $X_t^\\mathrm {ref}$ enters well $W_j$ at time $t_{\\mathrm {sim}}$ .", "Figure: An illustration of the decorrelation and parallel steps ofthe ParRep algorithm in the case that the reference walker neverleaves well W j W_j.", "X t k ☆ X^{k_\\star }_t is the first process to exitthe well, doing so at the computer time t corr +T ☆ t_{\\mathrm {corr}}+ T^{\\star }.This is then translated into the lab, or physical, time t sim +t corr +NT ☆ t_{\\mathrm {sim}}+t_{\\mathrm {corr}}+ N T^{\\star }.", "See Figure for anillustration of a dephasing step.Figure: An illustration of a dephasing step for the ParRepalgorithm.", "In this implementation, the replicas all start fromthe same position; μ phase 0 =δ x 1 \\mu _{\\rm phase}^0 = \\delta _{x_1}.", "WhenX t 2 X^2_t leaves before t phase t_{\\mathrm {phase}}, it is relaunched from the sameposition.", "Decorrelation Step: Let $X_t^{\\mathrm {ref}}$ evolve under (REF ) for $t_{\\mathrm {sim}}\\le t\\le t_{\\mathrm {sim}}+{t_{\\mathrm {corr}}}$ .", "If $\\mathcal {S}(X_t^{\\mathrm {ref}}) = \\mathcal {S}(X_{t_\\mathrm {sim}}^{\\mathrm {ref}})$ for all $t_{\\mathrm {sim}}\\le t\\le t_{\\mathrm {sim}} + {t_{\\mathrm {corr}}} $ , then time advances to $t_\\mathrm {sim}+ {t_{\\mathrm {corr}}}$ and proceed.", "Otherwise, denote the first exit time from the well, $T = \\inf \\left\\lbrace t\\mid \\mathcal {S}(X_{t_\\mathrm {sim}+ t}^{\\mathrm {ref}}) \\ne \\mathcal {S}(X_{t_\\mathrm {sim}}^{\\mathrm {ref}}) \\right\\rbrace $ and time advances to $t_\\mathrm {sim}+ T$ .", "Return to the beginning of the decorrelation step in the new well.", "Dephasing Step: In conjunction with the decorrelation step, we launch $N$ replicas with starting positions drawn from distribution $\\mu _{\\mathrm {phase}}^{0}$ .", "These are run for $t_{\\mathrm {phase}}$ amount of time, the dephasing time.", "If at any time before $t_{\\mathrm {phase}}$ a replica leaves the well, it is restarted.", "A replica has successfully dephased if it remains in the well for all of $t_{\\mathrm {phase}}$ .", "At the completion of the decorrelation and dephasing steps, assuming the reference walker has not exited, we have $N$ independent walkers with the same distribution.", "We discard the reference process.", "If at any time during the dephasing process the reference walker leaves the well, the dephasing process terminates and the replicas are discarded.", "Parallel Step: We now let the $N$ replicas evolve independently and define $k_\\star & = \\underset{k}{\\rm argmin}\\; T^k,\\\\X^\\star _t & = X^{k_\\star }_t,\\\\T^\\star &= T^{k_\\star }.$ The system advances to the next well: $t_\\mathrm {sim}\\mapsto t_\\mathrm {sim}+ t_{\\mathrm {corr}}+ N T^\\star \\\\X^\\mathrm {ref}_{ t_\\mathrm {sim}+ t_{\\mathrm {corr}}+ N T^\\star } = X^{\\star }_{T^\\star }.$ Finally, we return to the decorrelation step.", "This is a different dephasing algorithm than described in [4].", "There, after the decorrelation step, the replicas are initiated at the the position of the reference process and run for $t_{\\mathrm {phase}}$ .", "The simulation clock is not advanced, and replicas are replaced as need be should they exit the well.", "Our implementation has the advantage that no processor sits idle.", "The reader may wonder why we would want to have a distinguished reference process – why not relaunch the reference process, as we would a replica, should it exit?", "We retain this feature to allow for realizations where the process is in a well for a very short period, far less than the decorrelation time.", "These correlated events, such as recrossings, appear in serial simulations and should be preserved.", "One may also ask why we discard the reference process.", "This is to simplify the analysis, as it permits us to declare that the $N$ replicas are drawn from the same distribution when the parallel step begins.", "In addition to the choice of $t_{\\mathrm {corr}}$ and $t_{\\mathrm {phase}}$ , there is also the question of what $\\mu _{\\mathrm {phase}}^0$ should be.", "Again, there is significant flexibility.", "One possibility is to allow the reference process to evolve for some amount of time, and then the replicas could be launched from its position.", "A method used in practice is to find a local minima associated with the well, and initiate the replicas from that position, [21].", "We emphasize that the dephasing mechanism need not depend on any information associated with the reference process.", "In principle, ParRep offers a nearly linear speedup with the number of independent replicas, provided $t_{\\mathrm {corr}}$ is short relative to the typical exit time.", "With the explosion in the availability of distributed computing clusters, parallel replica dynamics is an attractive tool for studying infrequent event processes." ], [ "Main Results", "The essential aspects of a process undergoing infrequent events are How often does it transition from one state to another?", "What state does it transition to?", "These properties are captured in $\\mathcal {S}_t$ .", "To assess how well $\\mathcal {S}_t^{\\rm ParRep}$ approximates it, we are motivated to first consider the exit distribution of a process, and how well it is preserved.", "In [4], the authors proposed a rigorous framework in which to study ParRep.", "The purpose of this study is to unify those ideas and assess the total error, over a single cycle of ParRep, as a function of the parameters.", "Note: For brevity, we shall now take $t_\\mathrm {sim}= 0$ and $W_j =W$ .", "Throughout our paper, we shall assume: $W\\subset \\mathbb {R}^n$ is bounded; $\\partial W$ is sufficiently smooth; $V$ is sufficiently smooth on $\\overline{W}$ .", "Though $W$ need not correspond to a basin of attraction, we shall continue to call it a well.", "To motivate our results, we introduce some important objects.", "Let $\\mu _t$ denote the law of $X_t$ , conditioned on having not left the well: $\\mu _t(A) = \\mathbb {P}^{\\mu _0}\\left[X_t \\in A\\mid T> t\\right] =\\frac{\\mathbb {P}^{\\mu _0}\\left[X_t \\in A, T>t\\right]}{\\mathbb {P}^{\\mu _0}\\left[T> t\\right]}.$ The above expression is the probability of finding the processes, $X_t$ , in the set $A\\subset W$ , at time $t$ , conditioned on the exit time from the well, $T$ , being beyond $t$ , and $X_0$ being initially distributed by $\\mu _0$ .", "Additional details on our notation are given below, in Section REF .", "Under certain assumptions, the limit $\\lim _{t\\rightarrow \\infty } \\mu _t = \\nu ,$ exists.", "$\\nu $ is the quasistationary distribution (QSD) and characterizes the long term survivors of (REF ) in well $W$ .", "The properties of $\\nu $ are reviewed for the reader below in Section .", "In the following theorems, we shall refer to “admissible distributions.” This class is quite broad and includes the Dirac distribution.", "It is defined and explored in subsequent sections.", "First, we have the following result on the convergence of the exit distribution of $X_t.$ Theorem 1.1 (Convergence to the QSD) Assume $\\mu _0$ is admissible.", "There exist positive constants $\\lambda _2 > \\lambda _1$ , $C$ and $\\underline{t}$ , such that for all $t \\ge \\underline{t}$ and bounded and measurable $f(\\tau ,\\xi ):\\mathbb {R}^+\\times \\partial W\\rightarrow \\mathbb {R}$ we have $\\left| \\mathbb {E}^{\\mu _t}\\left[ f(T, X_{T})\\right] -\\mathbb {E}^{\\nu }\\left[f(T,X_{T})\\right] \\right| \\le C {\\left\\Vert f\\right\\Vert _{L^\\infty }} e^{-(\\lambda _2 - \\lambda _1) t}.$ The constant $C$ is independent of $t$ and $f$ .", "Taking $t$ sufficiently large so as to make this small corresponds to the satisfactory completion of the decorrelation step; this reflects (REF ).", "We give a more precise statement of this theorem at the beginning of Section , after introducing some additional notation in Section .", "This result also plays a role in studying the dephasing step.", "The constants $C$ and $\\underline{t}$ depend on $\\mu _0$ , $V,$ and the geometry of the well.", "We will use the notation $C_\\mathrm {phase}$ and $C_\\mathrm {corr}$ , and $\\underline{t}_\\mathrm {phase}$ and $\\underline{t}_\\mathrm {corr}$ to distinguish the constants induced by the dephasing and decorrelation steps.", "The next result ensures that the dephasing step terminates successfully: Theorem 1.2 (Dephasing Process) For an admissible distribution $\\mu ^0_{\\mathrm {phase}}$ and $t_\\mathrm {phase}\\ge \\underline{t}_\\mathrm {phase}$ : Dephasing produces $N$ independent replicas with distributions $\\mu _{\\mathrm {phase}}$ ; Given any $\\epsilon >0$ , by taking $t_{\\mathrm {phase}}\\ge \\underline{t}_\\mathrm {phase}$ , $\\left| \\mathbb {E}^{\\mu _\\mathrm {phase}}\\left[ f(T^k, X^k_{T^k})\\right] -\\mathbb {E}^{\\nu }\\left[f(T,X_{T})\\right] \\right| \\le C_\\mathrm {phase}e^{-(\\lambda _2 - \\lambda _1)t_{\\mathrm {phase}}}\\left\\Vert f\\right\\Vert _{L^\\infty }$ The expected number of times a replica is relaunched is finite.", "Next, the error in the parallel step cascading from the dephasing step can be controlled: Theorem 1.3 (Parallel Error) Given $t_{\\mathrm {phase}}\\ge \\underline{t}_{\\mathrm {phase}}$ , let $\\epsilon _\\mathrm {phase}\\equiv C_\\mathrm {phase}e^{-(\\lambda _2 - \\lambda _1)t_{\\mathrm {phase}}},$ and assume the dephasing step has produced $N$ i.i.d.", "replicas drawn from distribution $\\mu _\\mathrm {phase}$ .", "Then the exit time converges to an exponential law, with parameter $N\\lambda _1$ , $\\left| \\mathbb {P}^{\\mu _\\mathrm {phase}}\\left[T^\\star > t\\right] - e^{-N \\lambda _1 t} \\right|\\le N\\epsilon _\\mathrm {phase}(1+\\epsilon _\\mathrm {phase})^{N-1} e^{-N\\lambda _1 t} .$ If we additionally assume that $N\\epsilon _\\mathrm {phase}(1+\\epsilon _\\mathrm {phase})^{N-1}<1$ , then the hitting point distribution is asymptotically independent of the exit time $\\left| \\mathbb {P}^{\\mu _\\mathrm {phase}}\\left[X_{T^\\star }^\\star \\in A \\mid {T^\\star >t}\\right]- \\int _A d\\rho \\right| \\lesssim \\frac{ N^2\\epsilon _\\mathrm {phase}(1+\\epsilon _\\mathrm {phase})^{N-1}}{1-N\\epsilon _\\mathrm {phase}(1+\\epsilon _\\mathrm {phase})^{N-1}},$ where $\\rho $ is the hitting point density and $A\\subset \\partial W$ .", "Thus, for $t_{\\mathrm {phase}}$ large enough, we achieve the ideal factor of $N$ speedup and we do not disrupt the hitting point distribution too much.", "The reader may find the $N$ dependence in the error terms to be disconcerting, but it can easily be controlled by taking $t_{\\mathrm {phase}}\\gtrsim \\log N/(\\lambda _2 - \\lambda _1)$ .", "We will return to this in the discussion.", "We also note that there is a slight abuse of notation in the above expressions.", "The superscripts, $\\nu $ and $\\mu _{\\mathrm {phase}}$ , should be interpreted as $N$ -tensor products, with a distinct realization drawn for each replica.", "A more detailed statement of this theorem, with explicit constants, is given at the beginning of Section .", "The hitting point density $\\rho $ is defined by (REF ).", "However, Theorem REF is only a comparison between the parallel step and the QSD.", "Our final result is a comparison between the ParRep algorithm, including decorrelation, dephasing and parallel steps, with an unaccelerated, serial process: Theorem 1.4 (ParRep Error) Let $X^{\\rm s}_t$ denote the unaccelerated (serial) process and $X^{\\rm p}_t$ denote the ParRep process, and let both the serial process and the reference process be initially distributed under $\\mu _0$ , an admissible distribution.", "Furthermore, assume the replicas are initialized from $\\mu _\\mathrm {phase}^0$ , also an admissible distribution.", "Given $t_{\\mathrm {corr}}\\ge \\underline{t}_\\mathrm {corr}$ and $t_{\\mathrm {phase}}\\ge \\underline{t}_\\mathrm {phase}$ , let $\\epsilon _\\mathrm {corr}& = C_\\mathrm {corr}e^{-(\\lambda _2 - \\lambda _1)t_{\\mathrm {corr}}},\\\\\\epsilon _\\mathrm {phase}& = C_\\mathrm {phase}e^{-(\\lambda _2 - \\lambda _1)t_{\\mathrm {phase}}}.$ Letting $T^{\\rm s}$ and $T^{\\rm p}$ denote the physical exit times, we have $\\begin{split}&\\left| \\mathbb {P}^{\\mu _0}\\left[{ T^{\\rm s}>t}\\right]-\\mathbb {P}^{\\mu _0}\\left[{T^{\\rm p}>t}\\right]\\right|\\\\&\\quad \\lesssim \\left[\\epsilon _\\mathrm {corr}+N\\epsilon _\\mathrm {phase}(1+\\epsilon _\\mathrm {phase})^{N-1}\\right]e^{-\\lambda _1(t-t_{\\mathrm {corr}})_+}.\\end{split}$ If, in addition, $t_{\\mathrm {corr}}$ is sufficiently large that $\\epsilon _\\mathrm {corr}<1$ , then for $A\\subset \\partial W$ , $\\begin{split}&\\left| \\mathbb {P}^{\\mu _0}\\left[X_{T^{\\rm s}}^{\\rm s} \\in A \\mid {T^{\\rm s}>t}\\right]-\\mathbb {P}^{\\mu _0}\\left[X_{T^{\\rm p}}^{\\rm p}\\in A\\mid {T^{\\rm p}>t}\\right]\\right|\\\\&\\quad \\lesssim \\frac{\\epsilon _\\mathrm {corr}+N^2\\epsilon _\\mathrm {phase}(1+\\epsilon _\\mathrm {phase})^{N-1}}{1-\\epsilon _\\mathrm {corr}}\\end{split}$ Thus, over a single cycle, the error in ParRep can be approximately decomposed as $\\text{Error} = \\text{Decorrelation error} + \\text{Parallel error}(\\text{Dephasing error}),$ where we view the parellel error as a function of the dephasing error.", "The speedup can be seen when $T^\\mathrm {p}$ is given further consideration.", "When $T^\\mathrm {p}> t_{\\mathrm {corr}}$ , $T^\\mathrm {p}= NT^\\star + t_{\\mathrm {corr}}$ where $T^\\star $ is the exit time of the particular replica which escapes first.", "There will be no speedup if the exit is before $t_{\\mathrm {corr}}$ ." ], [ "Outline of the Paper", "In section , we review some important results for (REF ).", "Our main Theorems are proven in Sections , , and .", "We then discuss our results in Section .", "Some additional calculations appear in the appendix." ], [ "Notation", "Random variables, such as the position, $X_t$ , and the exit time from the well, $T$ , will appear in capital letters.", "Deterministic values, such as $x$ , $t$ , $t_{\\mathrm {corr}}$ , etc.", "will be lower case.", "We will frequently use indicator functions in our analysis, which we write as $1_A$ , with $A$ indicating the set on which the value is one.", "We are often interested in probabilities and expectations of solutions of $X_t$ solving (REF ), and its exit time $T$ from some region $W$ .", "When we write $\\mathbb {E}^x \\left[f(T, X_{T})\\right] \\text{ or }\\mathbb {P}^x \\left[T \\ge t\\right]=\\mathbb {E}^x \\left[1_{T\\ge t}\\right]$ the superscript $x$ indicates that $x$ is the initial condition of $X_t$ ; $X_0 = x$ , and the expectation and probability are then taken with respect to the underlying Wiener measure of $B_t$ .", "When $X_0$ is given by some distribution $\\mu _0$ over $W$ , we write $\\mathbb {E}^{\\mu _0} \\left[f(T, X_{T})\\right] \\equiv \\int _W \\mathbb {E}^x \\left[f(T, X_{T})\\right] d\\mu _0(x).$ When we write a conditional expectation with respect to distribution $\\mu _0$ , we mean $\\mathbb {E}^{\\mu _0} \\left[f(T, X_{T})\\mid T > t\\right]\\equiv \\frac{\\mathbb {E}^{\\mu _0} \\left[f(T, X_{T})1_{ T > t}\\right] }{\\mathbb {P}^{\\mu _0} \\left[ T > t\\right] }.$ For the reader more accustomed to the computational physics literature, $\\mathbb {E}^{\\mu _0}\\left[\\mathcal {O}(X_t)\\right] = \\left\\langle \\mathcal {O}(t) \\right\\rangle .$ It is helpful to explicitly include the starting distribution, $\\mu _0$ associated with the process $X_t$ , to avoid any ambiguity.", "When we write $f \\lesssim g$ , we mean that there exists a constant $C>0$ such that $f \\le C g$ , but that the constant is not noteworthy." ], [ "Acknowledgements", "The authors wish to thank D. Aristoff, K. Leder, S. Mayboroda, A. Shapeev, and O. Zeitouni for helpful conversations in developing these ideas.", "We also thank D. Perez and A.F.", "Voter for conversations at LANL that motivated important refinements of our estimates.", "This work was supported by the NSF PIRE grant OISE-0967140 and the DOE grant DE-SC0002085." ], [ "Preliminary Results", "Before proceeding to our main results on ParRep, we review some important results on the overdamped Langevin equation.", "These results are where our regularity assumptions on $V$ , $W,$ and $\\partial W$ are needed.", "Two essential tools in our study of (REF ) are the Feynman-Kac formula and the quasistationary distribution, which we briefly review here; see [4] for additional details.", "First, let us recall the Feynman-Kac formula which relates solutions of a parabolic equation with corresponding elliptic operator $L \\equiv - \\nabla V \\cdot \\nabla + \\beta ^{-1} \\Delta $ to solutions of (REF ).", "Proposition 2.1 (Proposition 1 of [4]) On the parabolic domain $W \\times \\mathbb {R}^+$ , let $v$ solve $\\partial _t v &= L v,\\\\v\\mid _{\\partial W} & = \\phi : \\partial W \\rightarrow \\mathbb {R}, \\\\v(t=0) & = v_0: W \\rightarrow \\mathbb {R}.$ Then, $v(t,x) = \\mathbb {E}^x\\left[1_{{T}\\le t } \\phi (X_{T}) \\right] + \\mathbb {E}^x\\left[1_{T > t} v_0(X_t) \\right].$ To say a bit more about the elliptic operator $L$ , recall the invariant measure of (REF ): $d\\mu \\equiv Z^{-1} \\exp \\left(- \\beta V(x)\\right)dx,$ where $Z$ is the appropriate normalization.", "We introduce the Hilbert space $L^2_\\mu $ , with inner product $\\left\\langle f,g\\right\\rangle _\\mu \\equiv \\int f g d\\mu .$ An elementary calculation shows that $L$ is self adjoint and negative definite with respect to this inner product when supplemented with homogeneous Dirichlet boundary conditions on $\\partial W$ .", "Standard functional analysis and elliptic theory tell us that $L$ has infinitely many eigenvalue/eigenfunction pairs $(\\lambda _k, u_k)$ ; the eigenvalues can be ordered $0 > - \\lambda _1 > - \\lambda _2 \\ge -\\lambda _3 \\ge \\ldots ;$ and the eigenfunctions form a complete orthonormal basis for $L^2_\\mu (W)$ .", "In addition, the ground state, $u_1$ , is unique and positive.", "For details, see, for example, [13], [14], [16].", "The $\\lambda _1$ and $\\lambda _2$ appearing in our theorems are precisely the first two eigenvalues.", "When solving (REF ) with $\\phi = 0$ , the solution can be expressed as $v(x,t) = \\sum _{k=1}^\\infty e^{-\\lambda _k t} \\left\\langle v_0,u_k\\right\\rangle _\\mu u_k.$ Out of this spectral problem, we build the norm $\\left\\Vert f\\right\\Vert _{H_\\mu ^{s}}^2 \\equiv \\sum _{k=1}^\\infty \\lambda _k^{s} \\left| \\left\\langle f,u_k\\right\\rangle _\\mu \\right|^2.$ This generalizes to measures $\\left\\Vert \\mu _0\\right\\Vert _{H_\\mu ^{s}}^2 \\equiv \\sum _{k=1}^\\infty \\lambda _k^{s}\\left| \\int u_k d\\mu _0\\right|^2,$ and to sequences, $\\mathbf {a} = (a_1, a_2,\\ldots )$ $\\left\\Vert \\mathbf {a}\\right\\Vert _{H_\\mu ^{s}}^2 \\equiv \\sum _{k=1}^\\infty \\lambda _k^{s}\\left| a_k\\right|^2.$ If $\\mu _0$ has an Radon-Nikodym derivative with respect to $\\mu $ , (REF ) and (REF ) agree.", "We then define the function spaces, $H^{s}_\\mu = \\left\\lbrace v \\in {S}(W)^{\\prime }\\mid \\left\\Vert v\\right\\Vert _{H^{s}_\\mu }<\\infty \\right\\rbrace ,$ where ${S}$ is the set of smooth functions with support in $W$ , and ${S}^{\\prime }$ is its dual.", "We also define the projection operator, $P_{\\mathcal {I}}$ , where $\\mathcal {I} \\subset \\mathbb {N}$ , $P_{\\mathcal {I}} f = \\sum _{k\\in \\mathcal {I}} \\left\\langle f,u_k\\right\\rangle _{\\mu } u_k.$ Having introduced these spaces and norms, we can now clarify what was meant by the term admissible distribution used in the introduction.", "In this work, a distribution will be admissible with respect to $W$ if $\\mathrm {supp}\\:\\mu _0 \\subset W$ , and for some $s\\ge 0$ , $\\left\\Vert \\mu _0\\right\\Vert _{H^{-s}_\\mu }<\\infty $ .", "The aforementioned quasistationary distribution of (REF ) associated with the set $W$ is closely related to the spectral structure of $L$ .", "The QSD, $\\nu $ , is a time independent probability measure satisfying, for all measurable $A\\subset W$ and $t>0$ : $\\nu (A) = \\frac{\\int _W \\mathbb {P}^x \\left[X_t \\in A,\\; t < T\\right] d\\nu }{\\int _W \\mathbb {P}^x \\left[ t < T\\right] d\\nu } = \\mathbb {P}^\\nu \\left[X_t \\in A\\mid t < T\\right].$ The QSD measure $\\nu $ exists and Proposition 2.2 (Proposition 2 of [4]) $d\\nu = \\frac{u_1 d\\mu }{\\int _W u_1 d\\mu } = \\frac{u_1 e^{-\\beta V}dx.", "}{\\int _W u_1 e^{-\\beta V} dx }$ We refer the reader to, amongst others, [5], [6], [7], [19], [20], [25] for additional details on the QSD.", "The utility of the QSD stems from the property that if $X_0$ is distributed according to $\\nu $ , then: Proposition 2.3 (Proposition 3 of [4]) Let $\\phi : \\partial W \\rightarrow \\mathbb {R}$ be smooth.", "Then for $t>0$ $\\begin{split}\\mathbb {E}^\\nu \\left[1_{T< t} \\phi (X_{T})\\right] =\\mathbb {P}^{\\nu }\\left[T <t\\right]E^{\\nu }\\left[\\phi (X_{T})\\right]= (1-e^{-\\lambda _1t})\\int _{\\partial _W} \\phi \\,d\\rho \\end{split}$ where the exit density is given by $d\\rho = -\\frac{1}{\\lambda _1\\beta } \\nabla \\frac{d\\nu }{dx} \\cdot {\\bf n}\\, dS_x = -\\frac{\\nabla (u_1 e^{-\\beta V} )\\cdot {\\bf n}}{\\lambda _1 \\beta \\int _W u_1 e^{-\\beta V} dx} \\, dS_x,$ with ${\\bf n}$ the outward pointing normal and $dS_x$ the surface measure.", "In words, $T$ is exponentially distributed with parameter $\\lambda _1$ , and the first hitting point is independent of the first hitting time.", "Being initially distributed according to $\\nu $ is, in a sense, ideal.", "As shown by Proposition 5 of [4], were this the case for the replicas, the parallel step of ParRep would be exact.", "In practice, $X_0$ is never distributed by $\\nu $ , and it is the propagation of this error that we explore.", "Many of these quantities can be reformulated in terms of the Fokker-Planck equation for density $p^x(t,y)$ , $x\\in W$ , $\\partial _t p^x= L^\\ast p^x = \\nabla _y \\cdot \\left(p^x\\nabla V +\\beta ^{-1} \\nabla p^x\\right), \\\\p^x|_{\\partial W} = 0, \\quad p_0^x = \\delta _x(y).$ Though we will not make use of this, the reader more accustomed to Fokker-Planck may find it helpful to re-express various quantities in terms of $p^x$ .", "With regard to exit distributions, $\\mathbb {E}^{x}\\left[\\phi (X_T)1_{T<t}\\right] &= \\int _0^t\\int _{\\partial W}-\\phi (y)\\beta ^{-1}\\nabla p^x\\cdot {\\bf n}\\,dS_y,\\\\\\mathbb {P}^{x}\\left[t< T\\right] & = \\int _t^\\infty \\int _{\\partial W}-\\beta ^{-1}\\nabla p^x\\cdot {\\bf n} dS_y = \\int _W p^x(t,y)\\,dy.$ These can be integrated against the density of the QSD, $\\tfrac{d\\nu }{dy}$ , which solves $L^\\ast \\tfrac{d\\nu }{dx} = -\\lambda _1\\tfrac{d\\nu }{dy}$ , to obtain $p^\\nu (y,t) = e^{-\\lambda _1t}\\frac{d\\nu }{dy}$ as a particular solution of the Fokker-Planck equation.", "This directly shows the independence of exit time and hitting point.", "Substituting into the above integrals reproduces Proposition REF ." ], [ "Convergence to the QSD – Proof of Theorem\n", "In this section we prove Theorem REF , which we first restate with more detail: Theorem 3.1 (Convergence to the QSD) Given $s \\ge 0$ , let $\\mu _0$ be a distribution with $\\mathrm {supp}\\:\\mu _0\\subset W$ and $\\left\\Vert \\mu _0\\right\\Vert _{H^{-s}_\\mu }<\\infty $ .", "There exists $\\underline{t} \\gtrsim \\left\\lbrace \\left\\Vert P_{[2,\\infty )}\\mu _0\\right\\Vert _{H^{-s}_\\mu } / \\int u_1d\\mu _0\\right\\rbrace ^{4/( n + 2s)}$ such that for all $t \\ge \\underline{t}$ and for all bounded and measurable $f(\\tau , \\xi ):\\mathbb {R}^+ \\times \\partial W \\rightarrow \\mathbb {R}$ $\\begin{split}&\\left| \\mathbb {E}^{\\mu _t}\\left[ f(T, X_{T})\\right] -\\mathbb {E}^{\\nu } \\left[f(T,X_{T})\\right] \\right| \\\\&\\ \\lesssim {\\left\\Vert f\\right\\Vert _{L^\\infty }}\\left(\\int u_1 d\\mu _0\\right)^{-1} {\\underline{t}}^{-n/4- s/2} e^{-(\\lambda _2 - \\lambda _1) (t-\\underline{t})} \\left\\Vert P_{[2,\\infty )}\\mu _0\\right\\Vert _{H_\\mu ^{-s}}.\\end{split}$ This is a refinement of Proposition 6 from [4], which now admits initial distributions which lack an $L^2$ Radon-Nikodym derivative.", "Indeed, for appropriate $s$ , $\\mu _0$ can be a Dirac distribution.", "Though this is a parabolic flow which will instantaneously regularize such rough data, it is essential to an analysis of ParRep as one often wants to use Dirac mass initial conditions.", "In addition to this result, we present an extension which is essential to obtaining the results in Section on the parallel step." ], [ "Proof of Theorem ", "We first write $\\mathbb {E}^{\\mu _t}\\left[f(T, X_{T})\\right] = \\int _W \\mathbb {E}^x\\left[ f(T,X_T)\\right] d\\mu _t = \\int _W F(x) d\\mu _t$ where we have defined $F(x) \\equiv \\mathbb {E}^x\\left[ f(T,X_{T})\\right]$ .", "Thus, $\\mathbb {E}^{\\mu _t}\\left[f(T, X_{T})\\right] = \\frac{\\int _W\\mathbb {E}^x\\left[F(X_t)1_{T> t}\\right]d\\mu _0}{\\int _W\\mathbb {E}^x \\left[1_{T> t}\\right] d\\mu _0}.$ Applying Feynman-Kac, (REF ), to this, $\\mathbb {E}^{\\mu _t}\\left[f(T, X_{T})\\right] = \\frac{\\int _W v(t,x)d\\mu _0}{\\int _W \\bar{v}(t,x) d\\mu _0}$ where $v$ solves (REF ) with $v_0 = F$ and $\\phi =0$ , while $\\bar{v}$ solves it with $v_0 = 1$ and $\\phi = 0$ .", "For brevity, let $\\hat{F}_k = \\int Fu_k d\\mu , \\quad \\hat{1}_k = \\int u_k d\\mu , \\quad \\hat{\\mu }_{0,k} = \\int u_k d\\mu _0.$ Expressing $v$ and $\\bar{v}$ as series solutions using (REF ), we have $v(t,x) = \\sum _{k=1}^\\infty e^{-\\lambda _k t} \\hat{F}_k u_k(x),\\quad \\bar{v}(t,x) = \\sum _{k=1}^\\infty e^{-\\lambda _k t} \\hat{1}_k u_k(x).$ After a bit of rearrangement, the error can be expressed as $\\begin{split}e(t) &\\equiv \\left| \\mathbb {E}^{\\mu _t}\\left[f(T, X_{T})\\right] - \\mathbb {E}^\\nu \\left[f(T, X_{T}) \\right]\\right|\\\\&= \\left|\\frac{\\sum _k e^{-(\\lambda _k-\\lambda _1) t}\\left(\\hat{F}_k- \\hat{1}_k\\int Fd\\nu \\right) \\hat{\\mu }_{0,k} }{\\hat{1}_1 \\hat{\\mu }_{0,1} + \\sum _k e^{-(\\lambda _k-\\lambda _1) t}\\hat{1}_k \\hat{\\mu }_{0,k} }\\right|\\end{split}$ where the sums are from $k=2$ to $\\infty $ since $\\hat{F}_1=\\hat{1}_1\\int _W F d\\nu $ .", "Noting that $\\begin{split}\\left| \\hat{F}_k - \\hat{1}_k \\int F d\\nu \\right| &\\le \\int \\left| F u_k\\right|d\\mu + \\int \\left| F\\right|d\\nu \\int \\left| u_k\\right| d\\mu \\\\&\\quad \\le 2\\left\\Vert f\\right\\Vert _{L^\\infty } \\int \\left| u_k\\right|d\\mu \\le 2\\left\\Vert f\\right\\Vert _{L^\\infty } \\sqrt{\\mu (W)},\\end{split}$ we can rewrite the numerator as $\\begin{split}&\\left| \\sum _{k=2}^\\infty e^{-(\\lambda _k - \\lambda _1)t} \\left(\\hat{F}_k -\\hat{1}_k \\int F d\\nu \\right)\\hat{\\mu }_{0,k}\\right|\\\\&\\quad \\le 2 \\sqrt{\\mu (W)} \\left\\Vert f\\right\\Vert _{L^\\infty } \\sum _{k=2}^\\infty e^{-(\\lambda _k - \\lambda _1)t} \\left| \\hat{\\mu }_{0,k}\\right|\\\\&\\quad \\le 2\\sqrt{\\mu (W)} \\left\\Vert f\\right\\Vert _{L^\\infty } e^{-(\\lambda _2 -\\lambda _1)(t-t_1)}\\sum _{k=2}^\\infty e^{-(\\lambda _k - \\lambda _1)t_1} \\left| \\hat{\\mu }_{0,k}\\right|\\\\&\\quad \\le 2 \\sqrt{\\mu (W)} \\left\\Vert f\\right\\Vert _{L^\\infty } e^{-(\\lambda _2 -\\lambda _1)(t-t_1)}\\sum _{k=2}^\\infty e^{-\\kappa \\lambda _k t_1}\\left| \\hat{\\mu }_{0,k}\\right|\\end{split}$ where $\\kappa = 1- \\lambda _1/ \\lambda _2$ and $t\\ge t_1 >0$ .", "Applying Proposition REF from the appendix to this, the numerator is bounded by $\\begin{split}&\\left| \\sum _{k=2}^\\infty e^{-(\\lambda _k - \\lambda _1)t} \\left(\\hat{F}_k -\\hat{1}_k \\int F d\\nu \\right)\\hat{\\mu }_{0,k}\\right|\\\\&\\quad \\lesssim \\left\\Vert P_{[2,\\infty )}\\mu _0\\right\\Vert _{H^{-s}_\\mu }\\left\\Vert f\\right\\Vert _{L^\\infty }e^{-(\\lambda _2 - \\lambda _1)(t-t_1)} t_1^{-n/4 - s/2}.\\end{split}$ The constant that has been absorbed into the $\\lesssim $ symbol is independent of $t$ , $f$ and $\\mu _0$ .", "To ensure the denominator is uniformly bounded away from zero, we use a similar treatment, $\\begin{split}\\sum _{k=2}^\\infty e^{-(\\lambda _k - \\lambda _1)t} \\hat{1}_k \\hat{\\mu }_{0,k} &\\le \\sqrt{\\mu (W)} e^{-(\\lambda _2 -\\lambda _1)(t-t_2)}\\sum _{k=2}^\\infty e^{-\\kappa \\lambda _k t_2} \\left| \\hat{\\mu }_{0,k}\\right| \\\\&\\lesssim \\left\\Vert P_{[2,\\infty )}\\mu _0\\right\\Vert _{H^{-s}_\\mu } e^{-(\\lambda _2- \\lambda _1)t}t_2^{-n/4 - s/2}\\end{split}$ for $t\\ge t_2 >0$ , which may differ from $t_1$ .", "Therefore, $\\begin{split}&\\hat{1}_1 \\hat{\\mu }_{0,1} + \\sum _{k=2}^\\infty e^{-(\\lambda _k -\\lambda _1)t} \\hat{1}_k \\hat{\\mu }_{0,k}\\\\&\\quad \\gtrsim \\hat{1}_1 \\hat{\\mu }_{0,1} - e^{-(\\lambda _2 -\\lambda _1) (t-t_2)} t_2^{-n/4 - s/2}\\left\\Vert P_{[2,\\infty )}\\mu _0\\right\\Vert _{H^{-s}_\\mu }.\\end{split}$ For a sufficiently large $t \\ge \\underline{t} \\ge t_2 > 0$ , the denominator is bounded from below by $\\hat{1}_1 \\hat{\\mu }_{0,1} + \\sum _{k=2}^\\infty e^{-(\\lambda _k - \\lambda _1)t}\\hat{1}_k\\hat{\\mu }_{0,k} \\ge \\frac{1}{2}\\hat{1}_1 \\hat{\\mu }_{0,1} = \\frac{1}{2}\\int u_1 d\\mu \\int u_1d\\mu _0 > 0.$ Roughly, $\\underline{t} \\gtrsim \\left\\lbrace \\left\\Vert P_{[2,\\infty )}\\mu _0\\right\\Vert _{H^{-s}_\\mu } / \\int u_1d\\mu _0\\right\\rbrace ^{4/( n + 2s)}.$ Taking $t_1 = t_2 = \\underline{t} $ in (REF ) and (REF ) we have that for $t\\ge \\underline{t} $ $\\begin{split}e(t) &\\lesssim \\left(\\int u_1 d\\mu _0\\right)^{-1} e^{-(\\lambda _2-\\lambda _1)(t-\\underline{t})} ( \\underline{t} )^{-n/4-s/2}\\\\&\\quad \\times \\left\\Vert f\\right\\Vert _{L^\\infty }\\left\\Vert P_{[2,\\infty )}\\mu _0\\right\\Vert _{H^{-s}_\\mu }.\\end{split}$ Finally, for this estimate to hold for general bounded and measurable $f$ , we apply a density argument with respect to the $L^\\infty $ norm.", "The inclusion of $\\int u_1 d\\mu _0$ in the preceding result is deliberate as $\\mu _0$ is, to a degree, a user specified parameter.", "Moreover, $\\int u_1 d\\mu _0$ could be quite small.", "Indeed, when a $X_t$ first enters $W$ , it is near $\\partial W$ and the support of $\\mu _0$ is in a neighborhood of $\\partial W$ ; we may have $\\mu _0 =\\delta _{x}$ where $x$ is close to $\\partial W$ .", "As $u_1$ is continuous and vanishes on $\\partial W$ , $\\int _W u_1\\delta _{x} = \\mathrm {O}\\left(\\mathrm {dist}(x,\\partial W )\\right).$ We also see that as $\\mu _0 \\rightarrow \\nu $ , $\\left\\Vert P_{[2,\\infty )}\\mu _0\\right\\Vert _{H^{-s}_\\mu } \\rightarrow 0$ , and the error vanishes.", "It remains to identify distributions and values of $s$ for which $\\left\\Vert \\mu _0\\right\\Vert _{H^{-s}_\\mu }<\\infty $ .", "In the case that $\\mu _0$ has an $L^2_\\mu $ Radon-Nikodym derivative, one readily sees that $\\left\\Vert \\mu _0\\right\\Vert _{H^{-s}_\\mu }<\\infty $ for $s\\le 0.$ Indeed, when $s=0$ , this results collapses onto the $L^2_\\mu $ estimate of [4].", "This extends to $\\mu _0$ possessing $L^p_\\mu $ densities for any $p \\ge 2$ .", "For the case $\\mu _0 = \\delta _{x}$ , a Dirac mass, we shall have that $\\mu _0 \\in H^{-s}_\\mu $ when $s$ is large enough to embed $H^s_\\mu $ into $L^\\infty $ .", "If the $\\partial W$ is sufficiently smooth, then by standard elliptic theory, $H^s_\\mu $ and $H^s$ will be equivalent for $s \\ge 0$ , and we have the embedding for $s>n/2$ , [14], [13], [1].", "Refined elliptic estimates may weaken such assumptions on the boundary." ], [ "Exit Times", "In the case that we are interested in exit times, we have a result closely related to Theorem REF .", "Theorem 3.2 Assume $\\mu _0$ satisfies the assumptions of Theorem REF and $t_0 \\ge \\underline{t}$ .", "Then for $t\\ge 0$ , $\\left| \\mathbb {P}^{\\mu _{t_0}}\\left[T>t\\right] - e^{-\\lambda _1 t}\\right| \\le Ce^{-\\lambda _1 t} e^{-(\\lambda _2 - \\lambda _1) t_0}$ where $C$ is the pre-exponential factor in (REF ) and is independent of $t$ and $t_0$ .", "As before, we rely on (REF ) and the series expansions (REF ) to write $\\mathbb {P}^{\\mu _{t_1}}\\left[T>t\\right] = \\frac{\\mathbb {P}^{\\mu _0}[T> t_1 +t]}{\\mathbb {P}^{\\mu _0}[T> t_1]} = \\frac{\\sum _{k=1}^\\infty e^{-\\lambda _k(t + t_1) }\\hat{1}_k \\hat{\\mu }_{0,k}}{\\sum _{k=1}^\\infty e^{-\\lambda _k t_1}\\hat{1}_k \\hat{\\mu }_{0,k}}.$ Comparing against the QSD, $\\begin{split}\\left| \\mathbb {P}^{\\mu _{t_1}}\\left[T>t\\right] - e^{-\\lambda _1t} \\right| &=\\left| \\frac{\\sum _{k=1}^\\infty e^{-\\lambda _k (t + t_1) }\\hat{1}_k\\hat{\\mu }_{0,k}}{\\sum _{k=1}^\\infty e^{-\\lambda _k t_1}\\hat{1}_k \\hat{\\mu }_{0,k}} - e^{-\\lambda _1 t}\\right|\\\\& = \\left| \\frac{\\sum _{k=1}^\\infty \\left(e^{-\\lambda _k (t + t_1)} - e^{-\\lambda _k t_1 - \\lambda _1 t}\\right)\\hat{1}_k\\hat{\\mu }_{0,k}}{\\sum _{k=1}^\\infty e^{-\\lambda _k t_1}\\hat{1}_k \\hat{\\mu }_{0,k}} \\right|\\\\\\end{split}$ In the numerator, the $k=1$ term vanishes, leaving $e^{-\\lambda _1 t} \\left| \\frac{\\sum _{k=2}^\\infty \\left(1- e^{-(\\lambda _k-\\lambda _1) t}\\right) e^{-\\lambda _k t_1}\\hat{1}_k \\hat{\\mu }_{0,k}}{\\sum _{k=1}^\\infty e^{-\\lambda _k t_1}\\hat{1}_k \\hat{\\mu }_{0,k}} \\right|\\le e^{-\\lambda _1 t}\\frac{\\sum _{k=2}^\\infty e^{-\\lambda _k t_0}\\left| \\hat{1}_k \\hat{\\mu }_{0,k}\\right|}{\\left| \\sum _{k=1}^\\infty e^{-\\lambda _k t_0}\\hat{1}_k \\hat{\\mu }_{0,k}\\right|}$ Using the same methods as in the proof of Theorem REF , $\\frac{\\sum _{k=2}^\\infty e^{-\\lambda _k t_1}\\left| \\hat{\\mu }_{0,k}\\right|}{\\left| \\sum _{k=1}^\\infty e^{-\\lambda _kt_1}\\hat{1}_k \\hat{\\mu }_{0,k}\\right|}\\lesssim \\left(\\int u_1d\\mu _0\\right)^{-1} {\\underline{t}}^{-n/4- s/2} e^{-(\\lambda _2 -\\lambda _1) (t_0-\\underline{t})}\\left\\Vert P_{[2,\\infty )}\\mu _0\\right\\Vert _{H_\\mu ^{-s}}$ This estimate plays an important role in our analysis of ParRep.", "Indeed, we will frequently confront terms of the form $\\mathbb {E}^{\\mu _{t_0}}\\left[f(X,T)1_{T>t}\\right]$ , and we will want to compare against the corresponding term for the QSD.", "One could naively apply Theorem REF to estimate such a term, with observable $g_t(\\xi ,\\tau ) = f(\\xi , \\tau )1_{\\tau >t}$ .", "However, this is wasteful, as the observable is going to be taken over realizations which not only have not left the well before $t_0$ , but remain in the well for at least an additional $t$ .", "We thus have the following identity.", "Lemma 3.1 Given $t, t_0 \\ge 0$ , $\\mathbb {E}^{\\mu _{t_0}}\\left[f(X_T, T) 1_{T>t}\\right] = \\mathbb {E}^{\\mu _{t_0 + t}}\\left[f(X_T, T+t)\\right]\\mathbb {P}^{\\mu _{t_0}}\\left[T>t\\right].$ This reflects the Markovian nature of the process.", "Writing out the lefthand side, $\\begin{split}\\mathbb {E}^{\\mu _{t_0}} \\left[f(X_T, T) 1_{T>t}\\right] &= \\int _W \\mathbb {E}^x\\left[f(X_T, T) 1_{T>t}\\right] \\mu _{t_0}(dx)\\\\& = \\frac{\\int _W \\mathbb {E}^x \\left[f(X_T, T) 1_{T>t}\\right]\\mathbb {P}^{\\mu _0}\\left[X_t \\in dx, T>t_0\\right]}{\\mathbb {P}^{\\mu _0}\\left[T>t_0\\right] }\\end{split}$ The numerator is $\\begin{split} \\int _W \\mathbb {E}^x \\left[f(X_T, T) 1_{T>t}\\right]\\mathbb {P}^{\\mu _0}\\left[X_t \\in dx, T>t_0\\right] &=\\mathbb {E}^{\\mu _0}\\left[f(X_T, T-t_0)1_{T-t_0>t}1_{T>t_0}\\right]\\\\&=\\mathbb {E}^{\\mu _0}\\left[f(X_T, T-t_0)1_{T>t_0+t}\\right],\\end{split}$ where $t_0$ is subtracted off to make the observable consistent.", "The same argument shows $\\mathbb {E}^{\\mu _{t_0 + t}}\\left[f(X_T, T+t)\\right] = \\frac{\\mathbb {E}^{\\mu _0}\\left[f(X_T, T-t_0)1_{T>t+t_0}\\right]}{\\mathbb {P}^{\\mu _0}\\left[T>t+t_0\\right] }.$ Combining these three expressions completes the proof.", "In principle, we can use this Lemma and Theorem REF to obtain refinements on Theorem REF for observables that include $1_{T>t}$ terms." ], [ "The Dephasing Step – Proof of Theorem\n", "We now examine our dephasing step, Theorem 4.1 Given $s\\ge 0$ , assume $\\mathrm {supp}\\:\\mu _{\\mathrm {phase}}^0 \\subset W$ and $\\left\\Vert \\mu ^0_{\\mathrm {phase}}\\right\\Vert _{H^{-s}_\\mu }<\\infty $ .", "Then The dephasing step produces $N$ independent replicas with distributions $\\mu _{\\mathrm {phase}}$ , $\\mu _{\\mathrm {phase}}(A) = \\mathbb {P}^{\\mu _\\mathrm {phase}^0}\\left[X_{t_{\\mathrm {phase}}} \\in A\\mid T>t_{\\mathrm {phase}}\\right];$ There exists $\\underline{t}_\\mathrm {phase}$ and $C_\\mathrm {phase}$ such that for $t_{\\mathrm {phase}}\\ge \\underline{t}_\\mathrm {phase}$ , $\\left| \\mathbb {E}^{\\mu _\\mathrm {phase}}\\left[ f(T^k, X^k_{T^k})\\right] -\\mathbb {E}^{\\nu }\\left[f(T,X_{T})\\right] \\right| \\le \\left\\Vert f\\right\\Vert _{L^\\infty } C_\\mathrm {phase}e^{-(\\lambda _2 - \\lambda _1) t_{\\mathrm {phase}}};$ for any bounded measurable $f:\\mathbb {R}^+\\times \\partial W\\rightarrow \\mathbb {R}$ and all $k=1,\\ldots , N$ .", "The expected number of times a replica is relaunched is finite.", "To prove Theorem REF , we must establish: The replicas are independent and have law $\\mu _\\mathrm {phase}$ ; The error of $\\mu _\\mathrm {phase}$ can be made small; The expected number of relaunches is finite.", "The first property is obvious as each of the replicas is driven by an independent Brownian motion, and we only retain realizations for which $T> t_{\\mathrm {phase}}$ .", "The second property follows from Theorem REF .", "To prove the third property, we must establish that replicas initiated from $\\mu _\\mathrm {phase}^0$ have a nonzero chance of surviving till $t_{\\mathrm {phase}}$ : Lemma 4.1 Assume that $\\mu _{\\mathrm {phase}}^0$ satisfies the hypotheses of Theorem REF , $\\mathbb {P}^{\\mu _{\\mathrm {phase}}^0}\\left[T^k \\ge t_{\\mathrm {phase}}\\right] \\equiv p > 0.$ Observe that we have the following monotonicity property for $t_2 >t_1$ , $0\\le \\mathbb {P}^{\\mu _{\\mathrm {phase}}^0}\\left[T\\ge t_2\\right] \\le \\mathbb {P}^{\\mu _{\\mathrm {phase}}^0}\\left[T\\ge t_1\\right].$ We now argue by contradiction.", "Assume that at some $t_1>0$ , $\\mathbb {P}^{\\mu _{\\mathrm {phase}}^0}\\left[T\\ge t_1\\right]=0$ .", "By the above monotonicity, $\\mathbb {P}^{\\mu _{\\mathrm {phase}}^0}\\left[T\\ge t_2\\right]=0$ for all $t_2\\ge t_1$ .", "Using a similar approach as in the proof of Theorem REF , we write $\\mathbb {P}^{\\mu _{\\mathrm {phase}}^0}\\left[T\\ge t\\right] = \\bar{v}(x,t) =\\sum _{k=1}^\\infty e^{-\\lambda _kt} \\int u_k d{\\mu _{\\mathrm {phase}}^0} \\int u_k d\\mu $ where $\\bar{v}$ solves (REF ) with $v_0 =1$ and $\\phi = 0$ .", "Therefore, $\\begin{split}\\bar{v}(x,t_2) &= \\sum _{k=1}^\\infty e^{-\\lambda _kt_2}\\hat{\\mu }_{\\mathrm {phase}, k}^0 \\int u_k d\\mu = \\sum _{k=1}^\\infty e^{-\\lambda _kt_2} \\hat{\\mu }_{\\mathrm {phase}, k}^0 \\hat{1}_k\\\\&\\ge e^{-\\lambda _1 t_2} \\left\\lbrace \\hat{\\mu }_{\\mathrm {phase}, 1}^0 \\hat{1}_1- e^{-(\\lambda _2-\\lambda _1)t_2}\\sum _{k=2}^\\infty e^{-\\kappa \\lambda _k t_2} \\left| \\hat{1}_k\\right| \\left| \\hat{\\mu }_{\\mathrm {phase}, k}^0 \\right|\\right\\rbrace \\\\& \\gtrsim e^{-\\lambda _1 t_2}\\left\\lbrace \\hat{\\mu }_{\\mathrm {phase}, 1}^0\\hat{1}_1 - e^{-(\\lambda _2-\\lambda _1)t_2} t_2^{-n/4-s/2}\\left\\Vert P_{[2, \\infty )}{\\mu }_{\\mathrm {phase}}^0 \\right\\Vert _{H^{-s}_\\mu }\\right\\rbrace \\end{split}$ Then taking $t_2$ sufficiently large, $\\begin{split}&\\hat{\\mu }_{\\mathrm {phase}, 1}^0 \\hat{1}_1 -e^{-(\\lambda _2-\\lambda _1)t_2} t_2^{-n/4-s/2}\\left\\Vert P_{[2,\\infty )}\\delta _x\\right\\Vert _{H^{-s}_\\mu }\\\\&\\ge \\frac{1}{2}\\hat{\\mu }_{\\mathrm {phase}, 1}^0\\hat{1}_1 =\\frac{1}{2}\\int u_1d\\mu _{\\mathrm {phase}}^0 \\int u_1 d\\mu >0,\\end{split}$ since $\\int u_1d\\mu _{\\mathrm {phase}}^0 >0$ .", "Thus, we have a contradiction.", "This calculation reveals a role played by the choice of $\\mu _{\\mathrm {phase}}^0$ .", "If concentrated near the well boundary, ${\\mathbb {P}^{\\mu _{\\mathrm {phase}}^0}\\left[T^k \\ge t_{\\mathrm {phase}}\\right]=p}$ could be quite small.", "This will induce the replicas to relaunch many times, as the next result shows.", "Thus, for computational efficiency, a distribution concentrated deep in the well's interior is desirable.", "Lemma 4.2 Assume that $\\mu _{\\mathrm {phase}}^0$ satisfies the hypotheses of Theorem REF , and that $\\mathbb {P}^{\\mu _{\\mathrm {phase}}^0}\\left[T^k \\ge t_{\\mathrm {phase}}\\right]=p>0$ .", "Then $\\mathbb {E}^{\\mu _{\\mathrm {phase}}^0}\\left[\\text{Number of relaunches}\\right] = (1-p)/p < \\infty $ The probability of relaunching $m$ times is the probability of exiting $m$ times and surviving on the $m+1$ -th time.", "Interpreting this in terms of $T^k$ and using the assumption, $\\mathbb {P}^{\\mu _{\\mathrm {phase}}^0}[\\text{$m$ relaunches}] = (1-p)^{m} p$ .", "Thus, $\\begin{split}\\mathbb {E}^{\\mu _{\\mathrm {phase}}^0}\\left[\\text{Number of relaunches}\\right]& = \\sum _{m=0}^\\infty m \\cdot \\mathbb {P}^{\\mu _{\\mathrm {phase}}^0}[\\text{$m$ relaunches}] \\\\& = \\sum _{m=0}^\\infty m (1-p)^m p= \\frac{1-p}{p}<\\infty .\\end{split}$" ], [ "The Parallel Step – Proofs of Theorems ", "First, we restate Theorem REF with additional detail: Theorem 5.1 (Parallel Error) Given $t_{\\mathrm {phase}}\\ge \\underline{t}_{\\mathrm {phase}}$ , let $\\epsilon _\\mathrm {phase}\\equiv C_\\mathrm {phase}e^{-(\\lambda _2 - \\lambda _1)t_{\\mathrm {phase}}},$ and assume the dephasing step has produced $N$ i.i.d.", "replicas drawn from distribution $\\mu _\\mathrm {phase}$ .", "Then the exit time distribution of the parallel step converges to an exponential, $\\left| \\mathbb {P}^{\\mu _\\mathrm {phase}}\\left[T^\\star > t\\right] - e^{-N \\lambda _1 t} \\right|\\le \\epsilon _\\mathrm {phase}N (1 + \\epsilon _\\mathrm {phase})^{N-1} e^{-N\\lambda _1 t}.$ If $\\phi : \\partial W \\rightarrow \\mathbb {R}$ is bounded and measurable, the exit distribution converges to one that is independent of exit time, $\\begin{split}&\\left| \\mathbb {E}^{\\mu _\\mathrm {phase}}\\left[1_{T^\\star >t}\\phi (X_{T^\\star }^\\star )\\right]- e^{-N \\lambda _1 t} \\int _{\\partial W} \\phi d\\rho \\right|\\\\&\\quad \\lesssim N^2 (1 + \\epsilon _\\mathrm {phase})^{N-1}\\epsilon _\\mathrm {phase}\\left\\Vert \\phi \\right\\Vert _{L^\\infty } e^{-N\\lambda _1 t}.\\end{split}$ If, in addition, $N \\epsilon _\\mathrm {phase}(1+\\epsilon _\\mathrm {phase})^{N-1} < 1$ , then $\\begin{split}& \\left| \\mathbb {E}^{\\mu _\\mathrm {phase}}\\left[\\phi (X_{T^\\star }^\\star )\\mid {T^\\star >t}\\right]- \\int _{\\partial W} \\phi d\\rho \\right|\\\\&\\quad \\lesssim \\frac{N^2 \\left\\Vert \\phi \\right\\Vert _{L^\\infty } \\epsilon _\\mathrm {phase}(1 +\\epsilon _\\mathrm {phase})^{N-1}}{1 - N \\epsilon _\\mathrm {phase}(1+\\epsilon _\\mathrm {phase})^{N-1}}.\\end{split}$ To prove (REF ), we begin by writing, $\\begin{split}&\\left| \\mathbb {P}^{\\mu _{\\mathrm {phase}}}\\left[ T^\\star > t\\right] -e^{-N\\lambda _1 t}\\right|\\\\&\\quad = \\left| \\Pi _{k=1}^N \\mathbb {P}^{\\mu _{\\mathrm {phase}}}\\left[T^k> t\\right]-\\Pi _{k=1}^N \\mathbb {P}^{\\nu }\\left[T^k> t\\right] \\right| \\\\&\\quad = \\left| \\mathbb {P}^{\\mu _{\\mathrm {phase}}}\\left[T^1> t\\right]^{N}-e^{-N\\lambda _1 t} \\right|\\\\&\\quad = \\left| \\mathbb {P}^{\\mu _{\\mathrm {phase}}}\\left[T^1> t\\right] -e^{-\\lambda _1t} \\right| \\left| \\sum _{k=0}^{N-1} \\mathbb {P}^{\\mu _{\\mathrm {phase}}}\\left[T^1>t\\right]^{k}e^{-(N-1-k)\\lambda _1 t} \\right|.\\end{split}$ From Theorem REF , we know $\\left| \\mathbb {P}^{\\mu _{\\mathrm {phase}}}\\left[T> t\\right]-e^{-\\lambda _1 t} \\right|\\le \\epsilon _{\\mathrm {phase}}e^{-\\lambda _1 t}.$ Therefore, $\\left| \\mathbb {P}^{\\mu _{\\mathrm {phase}}}\\left[ T^\\star \\ge t\\right] -e^{-N\\lambda _1 t}\\right|\\le \\epsilon _\\mathrm {phase}e^{-\\lambda _1 t} N (1+\\epsilon _\\mathrm {phase})^{N-1}e^{-(N-1)\\lambda _1 t}.$ To prove (REF ), we begin by writing the expectation as $\\begin{split}\\mathbb {E}^{\\mu _{\\mathrm {phase}}} \\left[1_{T^\\star > t} \\phi (X_{T^\\star }^\\star ) \\right] &= \\mathbb {E}^{\\mu _{\\mathrm {phase}}} \\left[1_{T^{k_\\star }> t}\\phi (X_{T^{k_\\star }}^{k_\\star }) \\right]\\\\& = \\sum _{k=1}^N \\mathbb {E}^{\\mu _{\\mathrm {phase}}} \\left[1_{T^k> t}\\phi (X_{T^{k}}^k) 1_{k = k_\\star } \\right]\\\\& = \\sum _{k=1}^N\\mathbb {E}^{\\mu _{\\mathrm {phase}}}\\left[1_{T^k> t}\\phi (X_{T^{k}}^k) \\Pi _{l \\ne k} 1_{T^l > T^k}1_{T^l>t} \\right].\\end{split}$ In the above expression, we have used that since $T^\\star >t$ , $T^l>t$ for each $l$ .", "Then, using Lemma REF on each of the processes, $\\begin{split}&\\mathbb {E}^{\\mu _{\\mathrm {phase}}}\\left[1_{T^k> t}\\phi (X_{T^{k}}^k) \\Pi _{l \\ne k} 1_{T^l > T^k}1_{T^l>t} \\right]\\\\& = \\mathbb {E}^{\\mu _{t_{\\mathrm {phase}}+t}}\\left[ \\phi (X_{T^{k}}^k) \\Pi _{l \\ne k}1_{T^l> T^k}\\right]\\Pi _{l=1}^N\\mathbb {P}^{\\mu _\\mathrm {phase}}[T^l>t]\\\\& = \\mathbb {E}^{\\mu _{t_{\\mathrm {phase}}+t}}\\left[ \\phi (X_{T^{k}}^k) 1_{k =k_\\star }\\right]\\mathbb {P}^{\\mu _\\mathrm {phase}}[T>t]^N.\\end{split}$ This leads to the expression $\\mathbb {E}^{\\mu _{\\mathrm {phase}}} \\left[1_{T^\\star > t} \\phi (X_{T^\\star }^\\star ) \\right] = \\mathbb {E}^{\\mu _{t_\\mathrm {phase}+t}} \\left[\\phi (X_{T^\\star }^\\star ) \\right]\\mathbb {P}^{\\mu _\\mathrm {phase}}[T^\\star >t].$ Comparing against the QSD, $\\begin{split}&\\left| \\mathbb {E}^{\\mu _{\\mathrm {phase}}} \\left[1_{T^\\star > t}\\phi (X_{T^\\star }^\\star ) \\right] - \\mathbb {E}^{\\nu } \\left[1_{T^\\star >t} \\phi (X_{T^\\star }^\\star ) \\right] \\right|\\\\&\\le \\left| \\mathbb {E}^{\\mu _{t_\\mathrm {phase}+t}} \\left[\\phi (X_{T^\\star }^\\star ) \\right] \\right|\\left| \\mathbb {P}^{\\mu _\\mathrm {phase}}[T^\\star >t]- e^{-N\\lambda _1 t}\\right|\\\\& \\quad + e^{-N\\lambda _1 t} \\left| \\mathbb {E}^{\\mu _{t_\\mathrm {phase}+t}}\\left[ \\phi (X_{T^\\star }^\\star ) \\right]- \\mathbb {E}^{\\nu } \\left[\\phi (X_{T^\\star }^\\star ) \\right] \\right|.\\end{split}$ The first difference can be treated by (REF ), but the second difference requires more care.", "Given an arbitrary distribution $\\eta $ for $X_0$ , we define $\\mathcal {P}^{\\eta }(t) \\equiv \\mathbb {P}^{\\eta }[T>t]=\\mathbb {P}^{\\eta }[T^k > t],\\quad k = 1\\ldots N.$ Consequently, $\\mathcal {P}^{\\nu }(t) = e^{-\\lambda _1 t}$ and $\\begin{split}\\mathbb {E}^{\\mu _{t_\\mathrm {phase}+t}} \\left[ \\phi (X_{T^\\star }^\\star ) \\right] &=\\sum _{k=1}^N \\mathbb {E}^{\\mu _{t_\\mathrm {phase}+t}} \\left[\\phi (X_{T}^k)\\Pi _{l\\ne k} 1_{T_l>T_k}\\right]\\\\& = \\sum _{k=1}^N \\mathbb {E}^{\\mu _{t_\\mathrm {phase}+t}} \\left[ \\phi (X_{T}^k)\\mathcal {P}^{\\mu _{t_\\mathrm {phase}+t}}(T^k)^{N-1} \\right].\\end{split}$ An analogous expansion can be made with $\\nu $ in place of ${\\mu _{t_\\mathrm {phase}+t}}$ .", "Taking the difference of the two sums, and comparing term by term, $\\begin{split}&\\left| \\mathbb {E}^{\\mu _{t_\\mathrm {phase}+t}} \\left[ \\phi (X_{T}^k)\\mathcal {P}^{\\mu _{t_\\mathrm {phase}+t}}(T^k)^{N-1} \\right] - \\mathbb {E}^\\nu \\left[\\phi (X_{T}^k) \\mathcal {P}^\\nu (T^k)^{N-1} \\right]\\right|\\\\& \\le \\left| \\mathbb {E}^{\\mu _{t_\\mathrm {phase}+t}} \\left[ \\phi (X_{T}^k)\\mathcal {P}^{\\mu _{t_\\mathrm {phase}+t}}(T^k)^{N-1} \\right]- \\mathbb {E}^\\nu \\left[\\phi (X_{T}^k) \\mathcal {P}^{\\mu _{t_\\mathrm {phase}+t}}(T^k)^{N-1} \\right]\\right|\\\\& \\quad + \\left| \\mathbb {E}^\\nu \\left[ \\phi (X_{T}^k)\\mathcal {P}^{\\mu _{t_\\mathrm {phase}+t}}(T^k)^{N-1} \\right]-\\mathbb {E}^\\nu \\left[\\phi (X_{T}^k) \\mathcal {P}^\\nu (T^k)^{N-1} \\right]\\right|.\\end{split}$ By Theorem REF the first difference in (REF ) is bounded by $\\begin{split}&\\left| \\mathbb {E}^{\\mu _{t_\\mathrm {phase}+t}} \\left[ \\phi (X_{T}^k)\\mathcal {P}^{\\mu _{t_\\mathrm {phase}+t}}(T^k)^{N-1} \\right]- \\mathbb {E}^\\nu \\left[\\phi (X_{T}^k) \\mathcal {P}^{\\mu _{t_\\mathrm {phase}+t}}(T^k)^{N-1} \\right]\\right|\\\\&\\quad \\le \\epsilon _\\mathrm {phase}e^{-(\\lambda _2-\\lambda _1)t}\\left\\Vert \\phi \\right\\Vert _{L^\\infty },\\end{split}$ since $\\mathcal {P}\\le 1$ .", "For the other difference in (REF ), we can replicate the proof of (REF ) to obtain, for any $\\tau \\ge 0$ , $\\begin{split}&\\left| \\mathcal {P}^{\\mu _{t_\\mathrm {phase}+t}}(\\tau )^{N-1} - \\mathcal {P}^\\nu (\\tau )^{N-1} \\right|\\\\& \\le (N-1)\\epsilon _\\mathrm {phase}e^{-(\\lambda _2-\\lambda _1)t}(1+\\epsilon _\\mathrm {phase}e^{-(\\lambda _2-\\lambda _1)t})^{N-2}e^{-(N-1)\\lambda _1 \\tau }\\\\&\\le (N-1)\\epsilon _\\mathrm {phase}(1+\\epsilon _\\mathrm {phase})^{N-2}.\\end{split}$ Therefore, $\\begin{split}&\\left| \\mathbb {E}^\\nu \\left[ \\phi (X_{T}^k)\\mathcal {P}^{\\mu _{t_\\mathrm {phase}+t}}(T^k)^{N-1} \\right]-\\mathbb {E}^\\nu \\left[\\phi (X_{T}^k) \\mathcal {P}^\\nu (T^k)^{N-1} \\right]\\right|\\\\& \\le \\left\\Vert \\phi \\right\\Vert _{L^\\infty } \\epsilon _\\mathrm {phase}(N-1) (1+\\epsilon _\\mathrm {phase})^{N-2}.\\end{split}$ So (REF ) can be bounded by $\\begin{split}&\\left| \\mathbb {E}^{\\mu _{t_\\mathrm {phase}+t}} \\left[ \\phi (X_{T}^k)\\mathcal {P}^{\\mu _{t_\\mathrm {phase}+t}}(T^k)^{N-1} \\right] - \\mathbb {E}^\\nu \\left[\\phi (X_{T}^k) \\mathcal {P}^\\nu (T^k)^{N-1} \\right]\\right|\\\\&\\le \\left\\Vert \\phi \\right\\Vert _{L^\\infty } \\epsilon _\\mathrm {phase}\\left[1 + (N-1) (1+\\epsilon _\\mathrm {phase})^{N-2}\\right].\\end{split}$ Returning to (REF ), using (REF ) to treat the first difference and the preceding calculation to treat the second, we have: $\\begin{split}&\\left| \\mathbb {E}^{\\mu _{\\mathrm {phase}}} \\left[1_{T^\\star > t}\\phi (X_{T^\\star }^\\star ) \\right] - \\mathbb {E}^{\\nu } \\left[1_{T^\\star >t} \\phi (X_{T^\\star }^\\star ) \\right] \\right|\\\\& \\le N\\left\\Vert \\phi \\right\\Vert _{L^\\infty }\\epsilon _\\mathrm {phase}e^{-N\\lambda _1 t} (1 +\\epsilon _\\mathrm {phase})^{N-1} \\\\&\\quad +N\\left\\Vert \\phi \\right\\Vert _{L^\\infty } \\epsilon _\\mathrm {phase}e^{-N\\lambda _1 t} \\left[1 + (N-1) (1+ \\epsilon _\\mathrm {phase})^{N-2}\\right]\\\\& \\lesssim N^2 \\left\\Vert \\phi \\right\\Vert _{L^\\infty } \\epsilon _\\mathrm {phase}e^{-N\\lambda _1t} (1 + \\epsilon _\\mathrm {phase})^{N-1}.\\end{split}$ Finally, to prove (REF ), $\\begin{split}&\\left| \\mathbb {E}^{\\mu _{\\mathrm {phase}}} \\left[\\phi (X_{T^\\star }^\\star ) \\mid {T^\\star > t} \\right] - \\mathbb {E}^{\\nu } \\left[\\phi (X_{T^\\star }^\\star ) \\mid {T^\\star > t} \\right] \\right|\\\\&=\\left| \\frac{\\mathbb {E}^{\\mu _{\\mathrm {phase}}} \\left[\\phi (X_{T^\\star }^\\star )1_{T^\\star > t} \\right]}{\\mathbb {P}^{\\mu _\\mathrm {phase}}\\left[T^\\star > t\\right] }-\\frac{\\mathbb {E}^{\\nu } \\left[\\phi (X_{T^\\star }^\\star ) 1_{T^\\star > t} \\right]}{\\mathbb {P}^\\nu \\left[T^\\star > t\\right] } \\right|\\\\&\\le \\left| \\mathbb {E}^{\\mu _{\\mathrm {phase}}} \\left[\\phi (X_{T^\\star }^\\star )1_{T^\\star > t} \\right]-\\mathbb {E}^\\nu \\left[\\phi (X_{T^\\star }^\\star )1_{T^\\star > t}\\right]\\right|\\frac{1}{{\\mathbb {P}^{\\mu _\\mathrm {phase}}\\left[T^\\star > t\\right] }} \\\\&\\quad + \\left| {\\mathbb {E}^{\\nu } \\left[\\phi (X_{T^\\star }^\\star )1_{T^\\star > t} \\right]}\\right|\\frac{\\left| \\mathbb {P}^{\\mu _\\mathrm {phase}}\\left[T^\\star > t\\right] -\\mathbb {P}^\\nu \\left[T^\\star > t\\right]\\right|}{\\mathbb {P}^{\\mu _\\mathrm {phase}}\\left[T^\\star >t\\right]\\mathbb {P}^\\nu \\left[T^\\star > t\\right] }.\\end{split}$ For the first difference, $\\begin{split}&\\left| \\mathbb {E}^{\\mu _{\\mathrm {phase}}} \\left[\\phi (X_{T^\\star }^\\star )1_{T^\\star > t} \\right]-\\mathbb {E}^\\nu \\left[\\phi (X_{T^\\star }^\\star )1_{T^\\star > t}\\right]\\right|\\frac{1}{{\\mathbb {P}^{\\mu _\\mathrm {phase}}\\left[T^\\star > t\\right] }}\\\\& \\lesssim \\frac{N^2 \\left\\Vert \\phi \\right\\Vert _{L^\\infty } \\epsilon _\\mathrm {phase}(1 +\\epsilon _\\mathrm {phase})^{N-1}}{1 - N \\epsilon _\\mathrm {phase}(1+\\epsilon _\\mathrm {phase})^{N-1}}.\\end{split}$ For the second difference, $\\begin{split}&\\left| {\\mathbb {E}^{\\nu } \\left[\\phi (X_{T^\\star }^\\star ) 1_{T^\\star > t}\\right]}\\right| \\frac{\\left| \\mathbb {P}^{\\mu _\\mathrm {phase}}\\left[T^\\star > t\\right] -\\mathbb {P}^\\nu \\left[T^\\star > t\\right]\\right|}{\\mathbb {P}^{\\mu _\\mathrm {phase}}\\left[T^\\star >t\\right]\\mathbb {P}^\\nu \\left[T^\\star > t\\right] }\\\\&\\le \\frac{N\\left\\Vert \\phi \\right\\Vert _{L^\\infty }\\epsilon _\\mathrm {phase}(1+\\epsilon _\\mathrm {phase})^{N-1}}{1 - N\\epsilon _\\mathrm {phase}(1+\\epsilon _\\mathrm {phase})^{N-1}}.\\end{split}$ Combining these estimates, we have our result.", "Lastly, we prove Theorem REF , which we first restate with additional detail: Theorem 5.2 (ParRep Error) Let $X^{\\rm s}_t$ denote the unaccelerated (serial) process and $X^{\\rm p}_t$ denote the ParRep process, and assume that both $X^\\mathrm {ref}_t$ and $X^{\\rm s}_t$ are initially distributed under $\\mu _0$ , an admissible distribution.", "Also assume that $\\mu _\\mathrm {phase}^0$ is admissible.", "Given $t_{\\mathrm {corr}}\\ge \\underline{t}_\\mathrm {corr}$ and $t_{\\mathrm {phase}}\\ge \\underline{t}_{\\mathrm {phase}}$ , let $\\epsilon _\\mathrm {corr}& =C_\\mathrm {corr}e^{-(\\lambda _2 - \\lambda _1)t_{\\mathrm {corr}}},\\\\\\epsilon _\\mathrm {phase}& = C_\\mathrm {phase}e^{-(\\lambda _2 - \\lambda _1)t_{\\mathrm {phase}}}.$ Letting $T^{\\rm s}$ and $T^{\\rm p}$ denote the physical times, we have $\\begin{split}&\\left| \\mathbb {P}^{\\mu _0}\\left[{ T^{\\rm s}>t}\\right]-\\mathbb {P}^{\\mu _0}\\left[{T^{\\rm p}>t}\\right]\\right|\\\\&\\quad \\le {\\epsilon _\\mathrm {corr}e^{-\\lambda _1 t} +\\epsilon _\\mathrm {phase}N (1 +\\epsilon _\\mathrm {phase})^{N-1}e^{-\\lambda _1(t -t_{\\mathrm {corr}})_+}},\\end{split}$ $\\begin{split}&\\left| \\mathbb {E}^{\\mu _0}\\left[\\phi (X_{T^{\\rm s}}^{\\rm s})1_{ T^{\\rm s}>t}\\right]-\\mathbb {E}^{\\mu _0}\\left[\\phi (X_{T^{\\rm p}}^{\\rm p}) 1_{T^{\\rm p}>t}\\right]\\right|\\\\&\\quad \\lesssim \\left[\\epsilon _\\mathrm {corr}+ \\epsilon _\\mathrm {phase}N^2 (1+\\epsilon _\\mathrm {phase})^{N-1}\\right]\\left\\Vert \\phi \\right\\Vert _{L^\\infty } e^{-\\lambda _1(t-t_{\\mathrm {corr}})_+}.\\end{split}$ If, in addition, $\\epsilon _\\mathrm {corr}<1$ , then $\\begin{split}&\\left| \\mathbb {E}^{\\mu _0}\\left[\\phi (X_{T^{\\rm s}}^{\\rm s})\\mid {T^{\\rm s}>t}\\right]-\\mathbb {E}^{\\mu _0}\\left[\\phi (X_{T^{\\rm p}}^{\\rm p})\\mid {T^{\\rm p}>t}\\right]\\right|\\\\&\\quad \\lesssim \\frac{\\epsilon _\\mathrm {corr}+ \\epsilon _\\mathrm {phase}N^2(1+\\epsilon _\\mathrm {phase})^{N-1}}{1-\\epsilon _\\mathrm {corr}}\\left\\Vert \\phi \\right\\Vert _{L^\\infty },\\end{split}$ We begin by decomposing $\\begin{split}\\mathbb {P}^{\\mu _0}\\left[T^\\mathrm {s}> t\\right] &=\\mathbb {P}^{\\mu _0}\\left[T^\\mathrm {s}>t\\mid T^\\mathrm {s}\\le t_{\\mathrm {corr}}\\right]\\mathbb {P}^{\\mu _0} \\left[T^\\mathrm {s}\\le t_{\\mathrm {corr}}\\right]\\\\&\\quad + \\mathbb {P}^{\\mu _0}\\left[T^\\mathrm {s}> t\\mid T^\\mathrm {s}>t_{\\mathrm {corr}}\\right]\\mathbb {P}^{\\mu _0} \\left[T^\\mathrm {s}> t_{\\mathrm {corr}}\\right].\\end{split}$ We analogously decompose $\\mathbb {P}^{\\mu _0}\\left[T^\\mathrm {p}> t\\right]$ .", "For $t\\le t_{\\mathrm {corr}}$ , the serial algorithm and the reference process of ParRep have the same law.", "Hence, $\\mathbb {P}^{\\mu _0}\\left[T^{\\rm s}\\le t_{\\mathrm {corr}}\\right] =\\mathbb {P}^{\\mu _0}\\left[T^{\\rm p}\\le t_{\\mathrm {corr}}\\right],\\\\\\mathbb {P}^{\\mu _0}\\left[T^\\mathrm {s}> t\\mid T^\\mathrm {s}\\le t_{\\mathrm {corr}}\\right] =\\mathbb {P}^{\\mu _0}\\left[T^\\mathrm {p}> t\\mid T^\\mathrm {p}\\le t_{\\mathrm {corr}}\\right].$ Consequently, error only manifests itself if the parallel step is engaged, $\\begin{split}&\\left| \\mathbb {P}^{\\mu _0}\\left[T^\\mathrm {s}> t\\right] -\\mathbb {P}^{\\mu _0}\\left[T^\\mathrm {p}> t\\right] \\right|\\\\&= \\mathbb {P}^{\\mu _0}\\left[T^{\\rm s}>t_{\\mathrm {corr}}\\right]\\left| \\mathbb {P}^{\\mu _0}\\left[T^\\mathrm {s}> t\\mid T^\\mathrm {s}>t_{\\mathrm {corr}}\\right] - \\mathbb {P}^{\\mu _0}\\left[T^\\mathrm {p}> t\\mid T^\\mathrm {p}>t_{\\mathrm {corr}}\\right]\\right|.\\end{split}$ Comparing against the QSD, $\\begin{split}&\\left| \\mathbb {P}^{\\mu _0}\\left[T^\\mathrm {s}> t\\right] -\\mathbb {P}^{\\mu _0}\\left[T^\\mathrm {p}> t\\right] \\right|\\\\&\\le \\mathbb {P}^{\\mu _0}\\left[T^{\\rm s}>t_{\\mathrm {corr}}\\right]\\left| \\mathbb {P}^{\\mu _0}\\left[T^\\mathrm {s}>t\\mid T^\\mathrm {s}>t_{\\mathrm {corr}}\\right] - \\mathbb {P}^{\\nu }\\left[T > (t-t_{\\mathrm {corr}})_+\\right] \\right| \\\\&\\quad + \\mathbb {P}^{\\mu _0}\\left[T^{\\rm s}>t_{\\mathrm {corr}}\\right]\\left| \\mathbb {P}^{\\mu _0}\\left[T^\\mathrm {p}> t\\mid T^\\mathrm {p}>t_{\\mathrm {corr}}\\right]- \\mathbb {P}^{\\nu }\\left[T > (t-t_{\\mathrm {corr}})_+\\right] \\right|.\\end{split}$ Examining the first term, $\\mathbb {P}^{\\mu _0}\\left[T^\\mathrm {s}>t\\mid T^\\mathrm {s}>t_{\\mathrm {corr}}\\right] = \\mathbb {P}^{\\mu _\\mathrm {corr}} \\left[T^\\mathrm {s}> (t-t_{\\mathrm {corr}})_+\\right].$ By assumption and Theorem REF $\\left| \\mathbb {P}^{\\mu _\\mathrm {corr}} \\left[T^\\mathrm {s}> (t-t_{\\mathrm {corr}})_+\\right]-\\mathbb {P}^{\\nu }\\left[T > (t-t_{\\mathrm {corr}})_+\\right] \\right| \\le \\epsilon _\\mathrm {corr}e^{-\\lambda _1(t-t_{\\mathrm {corr}})_+}.$ For the other term, since the exit time is beyond $t_{\\mathrm {corr}}$ the parallel step engages.", "The single reference process is replaced by the ensemble of $N$ replicas drawn from $\\mu _\\mathrm {phase}$ , and $T^\\mathrm {p}=N T^\\star + t_{\\mathrm {corr}}$ .", "Hence, $\\mathbb {P}^{\\mu _0}\\left[T^\\mathrm {p}>t\\mid T^\\mathrm {p}> t_{\\mathrm {corr}}\\right] = \\mathbb {P}^{\\mu _\\mathrm {phase}}\\left[T^\\star > \\tfrac{1}{N}(t-t_{\\mathrm {corr}})_+\\right].$ Therefore, by Theorem REF $\\begin{split}&\\left| \\mathbb {P}^{\\mu _\\mathrm {phase}}\\left[T^\\star >\\tfrac{1}{N}(t-t_{\\mathrm {corr}})_+\\right] -\\mathbb {P}^{\\nu }\\left[T > (t-t_{\\mathrm {corr}})_+\\right]\\right|\\\\&\\quad \\le \\epsilon _\\mathrm {phase}N (1+ \\epsilon _\\mathrm {phase})^{N-1}e^{-\\lambda _1(t-t_{\\mathrm {corr}})_+}\\end{split}$ Substituting (REF ) and (REF ) into (REF ), we obtain (REF ).", "To obtain (REF ), we again decompose as $\\begin{split}\\mathbb {E}^{\\mu _0}\\left[\\phi (X_{T^{\\rm s}}^{\\rm s})1_{T^\\mathrm {s}>t}\\right]& =\\mathbb {E}^{\\mu _0}\\left[\\phi (X_{T^{\\rm s}}^{\\rm s})1_{T^\\mathrm {s}>t}\\mid T^{\\rm s}\\le t_{\\mathrm {corr}}\\right]\\mathbb {P}^{\\mu _0}\\left[T^{\\rm s}\\le t_{\\mathrm {corr}}\\right] \\\\&\\quad + \\mathbb {E}^{\\mu _0}\\left[\\phi (X_{T^{\\rm s}}^{\\rm s})1_{T^{\\rm s}>t}\\mid T^{\\rm s}>t_{\\mathrm {corr}}\\right]\\mathbb {P}^{\\mu _0}\\left[T^{\\rm s}> t_{\\mathrm {corr}}\\right].\\end{split}$ and analogously decompose the ParRep expectation.", "Again, for $t\\le t_{\\mathrm {corr}}$ , the serial algorithm and the reference process of ParRep have the same law.", "Thus $\\mathbb {E}^{\\mu _0}\\left[\\phi (X_{T^{\\rm s}}^{\\rm s})1_{T^{\\rm s}>t}\\mid T^{\\rm s}\\le t_{\\mathrm {corr}}\\right] = \\mathbb {E}^{\\mu _0}\\left[\\phi (X_{T^{\\rm p}}^{\\rm p} )1_{T^{\\rm p}>t}\\mid T^{\\rm p}\\le t_{\\mathrm {corr}}\\right].$ Consequently, $\\begin{split}&\\left| \\mathbb {E}^{\\mu _0}\\left[\\phi (X_{T^\\mathrm {s}}^{\\mathrm {s}})1_{T^\\mathrm {s}>t}\\right]- \\mathbb {E}^{\\mu _0}\\left[\\phi (X_{T^\\mathrm {p}}^\\mathrm {p})1_{T^\\mathrm {p}>t}\\right] \\right|\\\\&= \\mathbb {P}^{\\mu _0}\\left[T^\\mathrm {s}> t_{\\mathrm {corr}}\\right]\\left| \\mathbb {E}^{\\mu _0}\\left[\\phi (X_{T^\\mathrm {s}}^{\\mathrm {s}})1_{T^\\mathrm {s}>t}\\mid T^\\mathrm {s}>t_{\\mathrm {corr}}\\right] -\\mathbb {E}^{\\mu _0}\\left[\\phi (X_{T^\\mathrm {p}}^\\mathrm {p})1_{T^\\mathrm {p}>t}\\mid T^\\mathrm {p}>t_{\\mathrm {corr}}\\right]\\right|.\\end{split}$ Using the QSD as an intermediary, $\\begin{split}&\\left| \\mathbb {E}^{\\mu _0}\\left[\\phi (X_{T^\\mathrm {s}}^{\\mathrm {s}})1_{T^\\mathrm {s}>t}\\mid T^\\mathrm {s}>t_{\\mathrm {corr}}\\right] -\\mathbb {E}^{\\mu _0}\\left[\\phi (X_{T^\\mathrm {p}}^\\mathrm {p})1_{T^\\mathrm {p}>t}\\mid T^\\mathrm {p}>t_{\\mathrm {corr}}\\right]\\right|\\\\&\\le \\left| \\mathbb {E}^{\\mu _0}\\left[\\phi (X_{T^\\mathrm {s}}^{\\mathrm {s}})1_{T^\\mathrm {s}>t}\\mid T^\\mathrm {s}>t_{\\mathrm {corr}}\\right] - \\mathbb {E}^{\\nu }\\left[\\phi (X_{T})1_{T>(t-t_{\\mathrm {corr}})_+} \\right]\\right|\\\\&\\quad + \\left| \\mathbb {E}^{\\nu }\\left[\\phi (X_{T})1_{T>(t-t_{\\mathrm {corr}})_+}\\right] -\\mathbb {E}^{\\mu _0}\\left[\\phi (X_{T^\\mathrm {p}}^\\mathrm {p})1_{T^\\mathrm {p}>t}\\mid T^\\mathrm {p}>t_{\\mathrm {corr}}\\right]\\right|.\\end{split}$ For the first term, $\\begin{split}\\mathbb {E}^{\\mu _0}\\left[\\phi (X_{T^\\mathrm {s}}^{\\mathrm {s}})1_{T^\\mathrm {s}>t}\\mid T^\\mathrm {s}>t_{\\mathrm {corr}}\\right] &=\\mathbb {E}^{\\mu _\\mathrm {corr}}\\left[\\phi (X_{T^\\mathrm {s}}^{\\mathrm {s}})1_{T^\\mathrm {s}>(t-t_{\\mathrm {corr}})_+}\\right]\\\\& = \\mathbb {E}^{\\mu _{t_{\\mathrm {corr}}+(t-t_{\\mathrm {corr}})_+}}\\left[\\phi (X_{T^\\mathrm {s}}^{\\mathrm {s}})\\right]\\mathbb {P}^{\\mu _\\mathrm {corr}}\\left[{T^\\mathrm {s}>(t-t_{\\mathrm {corr}})_+} \\right].\\end{split}$ Hence, $\\begin{split}&\\left| \\mathbb {E}^{\\mu _0}\\left[\\phi (X_{T^\\mathrm {s}}^{\\mathrm {s}})1_{T^\\mathrm {s}>t}\\mid T^\\mathrm {s}>t_{\\mathrm {corr}}\\right] -\\mathbb {E}^{\\nu }\\left[\\phi (X_{T})1_{T>(t-t_{\\mathrm {corr}})_+}\\right]\\right|\\\\&\\lesssim \\epsilon _{\\mathrm {corr}}\\left\\Vert \\phi \\right\\Vert _{L^\\infty }e^{-\\lambda _1(t-t_{\\mathrm {corr}})_+}.\\end{split}$ For the other term, since the parallel step has engaged, $\\mathbb {E}^{\\mu _0}\\left[\\phi (X_{T^\\mathrm {p}}^\\mathrm {p})1_{T^\\mathrm {p}>t}\\mid T^\\mathrm {p}>t_{\\mathrm {corr}}\\right] = \\mathbb {E}^{\\mu _\\mathrm {phase}}\\left[\\phi (X_{T^\\star }^\\star )1_{T^\\star >\\frac{1}{N}(t-t_{\\mathrm {corr}})_+}\\right].$ By Theorem REF , $\\begin{split}&\\left| \\mathbb {E}^{\\nu }\\left[\\phi (X_{T})1_{T>(t-t_{\\mathrm {corr}})_+}\\right] -\\mathbb {E}^{\\mu _0}\\left[\\phi (X_{T^\\mathrm {p}}^\\mathrm {p})1_{T^\\mathrm {p}>t}\\mid T^\\mathrm {p}>t_{\\mathrm {corr}}\\right]\\right|\\\\& = \\left| \\mathbb {E}^{\\nu }\\left[\\phi (X_{T})1_{T>(t-t_{\\mathrm {corr}})_+}\\right] -\\mathbb {E}^{\\mu _\\mathrm {phase}}\\left[\\phi (X_{T^\\star }^\\star )1_{T^\\star >\\frac{1}{N}(t-t_{\\mathrm {corr}})_+}\\right]\\right|\\\\&\\le \\epsilon _\\mathrm {phase}N^2 \\left\\Vert \\phi \\right\\Vert _{L^\\infty }(1+\\epsilon _\\mathrm {phase})^{N-1} e^{-\\lambda _1 (t-t_{\\mathrm {corr}})_+}.\\end{split}$ Using (REF ) and (REF ) in (REF ) gives (REF ).", "(REF ) is proved using the preceding estimates, $\\begin{split}&\\left| \\mathbb {E}^{\\mu _0}\\left[\\phi (X_{T^{\\rm s}}^{\\rm s})\\mid {T^{\\rm s}>t}\\right]-\\mathbb {E}^{\\mu _0}\\left[\\phi (X_{T^{\\rm p}}^{\\rm p})\\mid {T^{\\rm p}>t}\\right]\\right|\\\\&\\le \\left| \\frac{\\mathbb {E}^{\\mu _0}\\left[\\phi (X_{T^{\\rm s}}^{\\rm s})1_{ T^{\\rm s}>t}\\right]-\\mathbb {E}^{\\mu _0}\\left[\\phi (X_{T^{\\rm p}}^{\\rm p})1_{T^{\\rm p}>t}\\right]}{\\mathbb {P}^{\\mu _0}\\left[T^\\mathrm {s}>t\\right]}\\right|\\\\&\\quad + \\left| \\mathbb {E}^{\\mu _0}\\left[\\phi (X_{T^{\\rm p}}^{\\rm p})1_{T^{\\rm p}>t}\\right]\\right|\\left| \\frac{\\mathbb {P}^{\\mu _0}\\left[T^\\mathrm {s}>t\\right] -\\mathbb {P}^{\\mu _0}\\left[T^\\mathrm {p}>t\\right]}{\\mathbb {P}^{\\mu _0}\\left[T^\\mathrm {s}>t\\right]\\mathbb {P}^{\\mu _0}\\left[T^\\mathrm {p}>t\\right]}\\right|\\\\&\\lesssim \\left[\\epsilon _\\mathrm {corr}+ \\epsilon _\\mathrm {phase}N^2(1+\\epsilon _\\mathrm {phase})^{N-1}\\right]\\left\\Vert \\phi \\right\\Vert _{L^\\infty }\\frac{e^{-\\lambda _1(t-t_{\\mathrm {corr}})_+}\\mathbb {P}^{\\mu _0}\\left[T^\\mathrm {s}>t_{\\mathrm {corr}}\\right]}{\\mathbb {P}^{\\mu _0}\\left[T^\\mathrm {s}>t\\right]}\\\\&\\quad + \\left[ \\epsilon _\\mathrm {corr}+ \\epsilon _\\mathrm {phase}N (1 +\\epsilon _\\mathrm {phase})^{N-1}\\right]\\left\\Vert \\phi \\right\\Vert _{L^\\infty }\\frac{e^{-\\lambda _1(t-t_{\\mathrm {corr}})_+}\\mathbb {P}^{\\mu _0}\\left[T^\\mathrm {s}>t_{\\mathrm {corr}}\\right]}{\\mathbb {P}^{\\mu _0}\\left[T^\\mathrm {s}>t\\right]}\\\\&\\lesssim \\left[\\epsilon _\\mathrm {corr}+ \\epsilon _\\mathrm {phase}N^2(1+\\epsilon _\\mathrm {phase})^{N-1}\\right]\\left\\Vert \\phi \\right\\Vert _{L^\\infty }\\frac{e^{-\\lambda _1(t-t_{\\mathrm {corr}})_+}\\mathbb {P}^{\\mu _0}\\left[T^\\mathrm {s}>t_{\\mathrm {corr}}\\right]}{\\mathbb {P}^{\\mu _0}\\left[T^\\mathrm {s}>t\\right]}.\\end{split}$ Since $(t-t_{\\mathrm {corr}})_+ + t_{\\mathrm {corr}}\\ge t$ , $\\mathbb {P}^{\\mu _0}\\left[T^\\mathrm {s}> (t-t_{\\mathrm {corr}})_+ + t_{\\mathrm {corr}}\\right]\\le \\mathbb {P}^{\\mu _0}\\left[T^\\mathrm {s}> t\\right].$ Therefore, $\\frac{\\mathbb {P}^{\\mu _0}\\left[T^\\mathrm {s}>t_{\\mathrm {corr}}\\right]}{\\mathbb {P}^{\\mu _0}\\left[T^\\mathrm {s}>t\\right]}\\le \\frac{\\mathbb {P}^{\\mu _0}\\left[T^\\mathrm {s}>t_{\\mathrm {corr}}\\right]}{\\mathbb {P}^{\\mu _0}\\left[T^\\mathrm {s}> (t-t_{\\mathrm {corr}})_+ + t_{\\mathrm {corr}}\\right]} =\\frac{1}{\\mathbb {P}^{\\mu _\\mathrm {corr}}\\left[T^\\mathrm {s}> (t-t_{\\mathrm {corr}})_+\\right]}$ and $\\frac{e^{-\\lambda _1(t-t_{\\mathrm {corr}})_+}\\mathbb {P}^{\\mu _0}\\left[T^\\mathrm {s}>t_{\\mathrm {corr}}\\right]}{\\mathbb {P}^{\\mu _0}\\left[T^\\mathrm {s}>t\\right]}\\le \\frac{1}{1-\\epsilon _\\mathrm {corr}}.$ Substituting this estimate into (REF ) yields (REF )." ], [ "Discussion", "We have proven several theorems on the convergence of the exit distributions of parallel replica dynamics to the underlying unaccelerated problem.", "We have also demonstrated the effectiveness of a dephasing algorithm done in conjunction with the decorrelation step.", "However, there remain several problems associated with ParRep, both in fully justifying it as an algorithm, and implementing it in practice." ], [ "Error Estimates", "As we pointed out in the introduction, the error estimates in Theorem REF and Theorem REF include terms which grow as $N\\rightarrow \\infty $ .", "If we take $t_{\\mathrm {phase}}\\gtrsim k_\\mathrm {phase}\\frac{\\log N}{\\lambda _2 - \\lambda _1}$ for some multiplier, $k_\\mathrm {phase}$ , then the most egregious term in the estimates is bounded by $\\begin{split}\\lim _{N\\rightarrow \\infty } N^2 \\epsilon _\\mathrm {phase}(1 + \\epsilon _\\mathrm {phase})^{N-1} & \\le \\lim _{N\\rightarrow \\infty } C_\\mathrm {phase}e^{-k_\\mathrm {phase}/2} \\left(1 +e^{-k_\\mathrm {phase}}{C_\\mathrm {phase}}/{N}\\right)^{N-1}\\\\& \\quad = e^{-k_\\mathrm {phase}/2} e^{C_\\mathrm {phase}e^{-k_\\mathrm {phase}}}.\\end{split}$ Hence, taking $k_\\mathrm {phase}$ large enough, the error can be made arbitrarily small.", "In contrast, the decorrelation error is independent of $N$ , and reducing the decorrelation error will not correct for the error due to more replicas.", "The error estimate on the exit time in Theorem REF is a bit deceiving and merits additional comment.", "It would appear that when we consider this cumulative distribution function at any $t>0$ , then, sending $N\\rightarrow \\infty $ , the error vanishes.", "This is a reflection on the estimate being an absolute error.", "Dividing out by $e^{-N\\lambda _1 t}$ lets us evaluate the relative error, which we see is uniformly bounded in $t$ .", "We also remark that since $\\mathbb {E}[T] = \\int _0^\\infty \\mathbb {P}[T>t] dt,$ we can obtain error estimates on the expected exit time.", "Using the estimates in Theorem REF , we see that provided $N\\epsilon _\\mathrm {phase}(1+\\epsilon _\\mathrm {phase})^{N-1}<1$ , we have $\\left| \\mathbb {E}^{\\mu _\\mathrm {phase}}[T^\\star ]-\\frac{1}{N\\lambda _1}\\right|\\le N\\epsilon _\\mathrm {phase}(1+\\epsilon _\\mathrm {phase})^{N-1}.$ Similarly, using the estimates in Theorem REF , $\\left| \\mathbb {E}^{\\mu _0}[T^{\\rm s}]-\\mathbb {E}^{\\mu _0}[T^{\\rm p}] \\right|\\lesssim \\epsilon _\\mathrm {corr}+ N \\epsilon _\\mathrm {phase}(1+\\epsilon _\\mathrm {phase})^{N-1}.$ It remains to be determined whether our estimates are sharp – is the growth in $N$ real, or an artifact of our analysis?", "While we cannot yet address the sharpness, a simple numerical experiment indicates that there is growth in the error as $N$ increases.", "Consider the problem $dX_t = -4 X_t dt + \\sqrt{2} dB_t$ for the well $W=[-1,1]$ , and suppose we launch $N$ replicas from the Dirac distribution $X_0 = .1$ .", "By symmetry, we know that if we had perfect dephasing, then during the parallel step $\\mathbb {P}^{\\nu }[X^\\star _{T^\\star } = 1] = \\mathbb {P}^{\\nu }[X^\\star _{T^\\star } = -1]=\\tfrac{1}{2}.$ But if we incompletely dephase, then, because of our asymmetric initial condition, we expect a higher probability of escaping at 1 than $-1$ .", "For this problem, we can compute by spectral methods that $\\lambda _1 \\approx 0.971972 $ and $\\lambda _2\\approx 8.98262$ .", "To test our conjecture, that the error increases with $N$ , we ran 10000 realizations of the dephasing and parallel steps with values of $N=100,200, \\ldots 1000$ .", "We employed Euler-Maruyama time stepping with $\\Delta t = 10^{-4}$ .", "We then ran this with with $t_\\mathrm {phase}= .05$ , .1 and .2.", "The results appear in Figure REF .", "Figure: t phase =.2t_{\\mathrm {phase}}=.2As we predicted, the errors decrease as $t_{\\mathrm {phase}}$ increases.", "For the smallest dephasing time, we also see the error increase with $N$ .", "At $t_{\\mathrm {phase}}= .1$ , there is still some increase in the error as $N$ increases, though it is less dramatic.", "When $t_{\\mathrm {phase}}= .2$ , the trend appears to have been lost to numerical error and sampling variability." ], [ "Numerical Parameters & Eigenvalues", "An essential question is how to choose of the dephasing and decorrelation time parameters.", "Based on the arguments in the preceding section, roughly, if we desire the errors from decorrelation and dephasing to be of the same order, then, $2\\log (N)t_{\\mathrm {corr}}\\sim t_{\\mathrm {phase}}.$ So, while they should not be the same, if we can estimate one, we can infer the other.", "There will also be some mismatch due to different starting distributions for the reference process and the dephasing replicas.", "$t_{\\mathrm {corr}}$ must be large enough so as to be representative of the QSD while remaining computationally efficient.", "Taking too large a value of $t_{\\mathrm {corr}}$ will just replicate the serial implementation with no acceleration.", "Theorem REF provides some insight, already discussed in [4].", "The error of $\\mu _\\mathrm {corr}$ is controlled by the following quantities: The $\\mu _0$ initial distribution, $\\left\\Vert P_{[2,\\infty )}\\mu _0\\right\\Vert _{H^{-s}_\\mu }$ , the mismatch between the initial distribution of the reference process and the quasistationary distribution, $\\nu $ ; The value of $ \\underline{t} $ ; $\\int u_1 d \\mu _0$ ; $\\lambda _2 - \\lambda _1$ , the spectral gap between the first two eigenvalues.", "Based on these quantities, and how they relate to $t_{\\mathrm {corr}}$ , to make the decorrelation error small, we would certainly need $t_{\\mathrm {corr}}\\gtrsim \\frac{\\ln \\left[\\left(\\int u_1 d\\mu _0\\right)^{-1} \\left\\Vert P_{[2,\\infty )}\\mu _0\\right\\Vert _{H^{-s}_\\mu } \\right]}{\\lambda _2 -\\lambda _1} + \\underline{t} .$ The eigenvalues also play an important role in determining which problems would benefit from ParRep is an outstanding problem, which is an outstanding issue.", "For ParRep to be efficient, we need $t_{\\mathrm {corr}}\\ll \\mathbb {E}^{\\mu _{\\mathrm {corr}}}[T]\\sim \\mathbb {E}^{\\nu }[T] = \\frac{1}{\\lambda _1}.$ This is desirable because, in the event $X_t$ does not leave the well during the decorrelation step, it is will now take a comparatively long time to exit.", "In [4], the authors suggested $t_{\\mathrm {corr}}\\le \\mathbb {E}^{\\mu _0}[T].$ However, this can be problematic, depending on $\\mu _0$ .", "As previously discussed, if the replicas launch from a position too close to the boundary, $\\mathbb {E}^{\\mu _0}[T]$ might be rather small.", "This is mitigated as $t_{\\mathrm {corr}}$ becomes larger, leading to $\\mathbb {E}^{\\mu _{\\mathrm {corr}}}[T]$ approaching the escape time of the QSD, $\\lambda _1^{-1}$ .", "We can see from (REF ) and constraint (REF ) that ParRep will be most effective when $\\frac{1}{\\lambda _2 - \\lambda _1}\\ll \\frac{1}{\\lambda _1},$ or, alternatively, when $\\lambda _1 \\ll \\lambda _2$ .", "Under these conditions, $\\mu _\\mathrm {corr}$ converges to $\\nu $ much more rapidly than we expect $X_t$ to exit $W$ .", "(REF ) can also be viewed as a characterization of when $W$ corresponds to a metastable state for (REF ).", "Computing $\\lambda _1$ and $\\lambda _2$ directly from a discretization of the elliptic operator L is intractable for all but the lowest dimensional systems.", "Instead, one must use Monte Carlo methods, such as those found in [17], [18], [12], [11], [23].", "However, these studies, some of which use branching particles processes like Fleming-Viot (discussed below), only yield $\\lambda _1$ .", "In a forthcoming work, we explore a mechanism for computing $\\lambda _2 - \\lambda _1$ using observables.", "The idea stems from calculations in Theorem REF , that, for an observable $\\mathcal {O}(x)$ , as $t\\rightarrow \\infty $ , $\\mathbb {E}^{\\mu _0}\\left[\\mathcal {O}(X_t)\\mid T> t\\right] = \\int _W \\mathcal {O}(x)d\\nu (x) +C(\\mu _0, \\mathcal {O})e^{-(\\lambda _2-\\lambda _1)t} + \\ldots $ In principle, $\\lambda _2-\\lambda _1$ could be extracted from a time series of $\\mathbb {E}^{\\mu _0}\\left[\\mathcal {O}(X_t)\\mid T> t\\right]$ .", "This introduces a variety of questions, such as what observables to use and how to perform such a fitting.", "Thus, we will have a method for dynamically estimating $t_{\\mathrm {corr}}$ and $t_{\\mathrm {phase}}$ ." ], [ "Dephasing Mechanism", "The efficiency of our dephasing algorithm can be improved by the availability of multiple processors.", "For instance, assume we have $N$ processors available for the replicas and that $k$ replicas have successfully been run until $t_{\\mathrm {corr}}$ .", "We are still waiting for $N-k$ replicas to successfully dephase.", "Rather than let $k$ processors sit idle, they could record the successful replicas, and run independent realizations.", "As more replicas finish dephasing, more processors can be brought to bear on the outstanding replicas.", "In practice, as replicas are deemed to have been successfully dephased, they are promoted to the parallel step, [21].", "Thus, there is no bottleneck at the dephasing step from waiting to get $N$ realizations dephased.", "There are other approaches to dephasing too, such as Fleming-Viot or Moran branching interacting particle processes, [2], [3], [15], [9].", "These merit consideration for ParRep.", "These approaches, which randomly split a surviving process every time another process exits the well, can provide additional information, such as an estimate of $\\lambda _1$ .", "Moreover, no processor sits idle at anytime.", "However, two challenges are introduced.", "On a practical level, one needs to implement additional communication routines and synchronization across the processors to request and send configurations as trajectories are killed.", "The second challenge is analytical, as the dephased processes will now be only approximately independent.", "This complicates the analysis of the how the error in the dephasing step cascades through the parallel step." ], [ "Other Challenges", "Another task is to assess the cumulative error over many ParRep cycles.", "The hitting point distribution will be perturbed by the algorithm, meaning that the sequence in which the states are visited would also be perturbed.", "Quantifying the error across many steps, and showing that it may be made small, would complete the justification of ParRep over the lifetime of a simulation.", "But to begin such a study, one must decide how to measure $\\mathrm {dist}(\\mathcal {S}_t, \\mathcal {S}_t^{\\rm ParRep}).$ The problem is $\\mathcal {S}_t$ is not a Markovian process.", "A particle that sits near the edge of the well is likely to exit much sooner than one which is near the minima of the well.", "But that information is lost in the coarse graining.", "Knowing how long $X_t$ has been in the well provides some amount of information; it tells us the proximity to the QSD, from which we can get an exponential exit time.", "Despite the challenge of studying the coarse grained flow, we can report that ParRep appears to work as predicted over multiple wells.", "Consider the flow $dX_t = -2 \\pi \\sin (\\pi X_t) dt + \\sqrt{2} dB_t.$ For this equation, with initial condition $X_0 = 0$ , we examined the time it would take to reach the wells centered at $x = \\pm 10$ .", "In other words, we sought to compute $T_{\\pm 10} = \\inf \\left\\lbrace t\\mid \\left| X_t\\right| \\ge 9\\right\\rbrace .$ For this problem, we ran the full ParRep algorithm (decorrelation, dephasing and parallel steps) within each well.", "During dephasing, the replicas were initiated from the minima of the present well, $0, \\pm 2\\pi , \\pm 4\\pi , \\ldots $ We ran 10000 realizations of this experiment, varying $k_\\mathrm {corr}$ and $k_\\mathrm {phase}$ , where $t_{\\mathrm {corr}}= \\frac{k_\\mathrm {corr}}{\\lambda _2 - \\lambda _1}, \\quad t_{\\mathrm {phase}}=\\frac{k_\\mathrm {phase}}{\\lambda _2 - \\lambda _1}.$ Since the wells are periodic, we can use spectral methods to compute $\\lambda _1\\approx .202280$ and $\\lambda _2 \\approx 16.2588$ once, and we then have these values for all the wells.", "The results, with $\\Delta t = 10^{-4}$ and $N=100$ replicas, appear in Figure REF Figure: The cumulative distribution for the time for it takestrajectory () to reach the wells centered at ±10\\pm 10.", "10000 realizations of each case were run with time stepΔt=10 -4 \\Delta t = 10^{-4}.", "t corr t_{\\mathrm {corr}} and t phase t_{\\mathrm {phase}} relate to k corr k_\\mathrm {corr}and k phase k_\\mathrm {phase} via ().", "As expected, largervalues of t corr t_{\\mathrm {corr}} and t phase t_{\\mathrm {phase}} give better agreement with anunaccelerated process.As we expect, for sufficiently large values of $t_{\\mathrm {corr}}$ and $t_{\\mathrm {phase}}$ , the distributions agree with the serial process.", "Indeed, in the cases $k_\\mathrm {corr}= k_\\mathrm {phase}= 5$ and $k_\\mathrm {corr}=1$ , $k_\\mathrm {phase}=5$ , the exit times agree with the serial realization at 5% significance level under a Kolmogorov-Smirnov test.", "In addition, this experiment also supports our calculations that, through the dephasing error, the total error should be magnified by $N$ since increasing the dephasing time improves the fit much more than increasing the decorrelation time does.", "Finally, we remark that we have only analyzed the continuous in time problem, though we are ultimately interested in the associated discrete in time algorithm.", "Much of the analysis carries over to the discrete in time case.", "A discrete in time quasistationary distribution exists, and there are extensive results on using interacting particle algorithms for dephasing, [10], [9].", "As in the continuous in time case, there remains the subtlety of how to analyze the parallel step when the dephased ensemble is only approximately independent.", "However, the discrete time step introduces other subtleties.", "Assume one uses Euler-Maruyama time discretization with time step $\\Delta t$ , and define the exit time as $T^{\\Delta t} =\\inf \\left\\lbrace t_n\\mid X_{t_n}\\notin W\\right\\rbrace .$ For a uniform time step, we see that with no acceleration of the dynamics, the exit times are integer multiples of $\\Delta t$ .", "For ParRep, this remains true for exits that take place during the decorrelation step.", "But for exit times taking place during the parallel step, the exit times will be determined by multiples of $N \\Delta t$ .", "With a large number of processors, this effective time step could be quite large.", "When comparing against the continuous in time problem, the error of discretization could be magnified in ParRep.", "In the preceding experiment, $N\\Delta t = .01$ , which is small relative to the exit time scale ($1/\\lambda _1\\approx 4.9$ ) and the decorrelation time scale ($1/(\\lambda _2 - \\lambda _1) \\approx .062$ ).", "Clearly, the discrete in time case warrants a thorough investigation." ], [ "Summation Bounds", "Much of our analysis relies on bounding series solutions, (REF ), of (REF ), to obtain information about $X_t$ through the Feynman-Kac equation, (REF ).", "The key estimates needed in our work stem from Weyl's Law for $L$ : Proposition 1.1 (Weyl's Law for $L$ ) There exist positive constants $c_1$ and $c_2$ , independent of $k$ , such that the eigenvalues of (REF ) satisfy $c_1 k^{{2}/{n}} \\le \\lambda _k \\le c_2 k^{{2}/{n}}.$ Recall that $n$ denotes the dimension of the underlying problem; $X_t\\in \\mathbb {R}^n$ .", "We will not reproduce the proof here, which is accomplished by rewriting the eigenvalue problem as $- \\beta ^{-1} \\nabla \\cdot \\left(e^{-\\beta V} \\nabla u\\right) = \\lambda e^{-\\beta V} u.$ This is justified because $V$ is smooth and $W$ is bounded; thus $e^{-\\beta V}$ is smooth and nondegenerate.", "This is now in the form of Theorem 6.3.1 of [8] on Weyl's Law, yielding the result.", "Using Weyl's Law, we have our main summation result, Proposition 1.2 Given $s\\ge 0$ , let $\\mathbf {a} = (a_1, a_2, \\ldots )$ satisfy $\\left\\lbrace \\sum _{k=1}^\\infty \\lambda _k^{-s}\\left| a_k\\right|^2 \\right\\rbrace ^{1/2} =\\left\\Vert \\mathbf {a} \\right\\Vert _{H^{-s}_\\mu }<\\infty .$ Let $f$ be defined as $f(\\tau ) \\equiv \\sum _{k=1}^\\infty a_k \\lambda _k^\\alpha e^{-\\tau \\lambda _k}.$ For $ a > 0$ , we have: $ \\sup _{\\tau \\ge a} \\left| f(\\tau )\\right|\\lesssim a^{-n/4 - \\max \\left\\lbrace s/2 + \\alpha , 0\\right\\rbrace } \\left\\Vert \\mathbf {a}\\right\\Vert _{H^{-s}_\\mu } < \\infty ;$ The convergence of the series is uniform in $\\tau \\ge a$ ; $f$ is continuous.", "To prove Proposition REF , we first have the following lemma.", "Lemma 1.1 Let $\\lambda _k$ be the eigenvalues and eigenfunctions of $L$ , (REF ).", "There exists a constant $C>0$ , independent of $\\tau $ , such that for all $\\tau >0$ , $\\sum _{k=1}^\\infty \\lambda _k^{\\alpha } e^{-\\tau \\lambda _k}\\le C \\tau ^{-n/2 - \\max \\left\\lbrace \\alpha , 0\\right\\rbrace }.$ The reader should rightfully expect the lefthand side of (REF ) to grow as $\\alpha \\rightarrow \\infty $ .", "Indeed, the constant $C$ depends on $\\alpha $ and will grow.", "However, as $\\alpha $ is fixed, and we are interested in an estimate in $\\tau $ , this is suppressed.", "For $\\alpha \\le 0$ , $\\begin{split}\\sum _{k=1}^{\\infty } e^{-\\tau \\lambda _k} \\lambda _k^\\alpha &\\le \\sum _{k=1}^{\\infty } e^{-\\tau \\lambda _k} \\lambda _1^\\alpha \\le \\sum _{k=1}^{\\infty } e^{-c_1 \\tau k^{2/n}} \\lambda _1^\\alpha \\\\& \\le \\lambda _1^\\alpha \\int _0^\\infty e^{-c_1 \\tau k^{2/n}}dk =\\lambda _1^\\alpha (c_1 \\tau )^{-n/2} \\Gamma \\left[1 +\\frac{n}{2}\\right].\\end{split}$ In the above computation, we approximated the sum as the lower Riemann sum of the integral.", "For $\\alpha > 0$ , we begin by estimating $\\sum _{k=1}^{\\infty } e^{-\\tau \\lambda _k} \\lambda _k^\\alpha \\le \\sum _{k = 1}^{\\infty } e^{-c_1\\tau k^{2/n}}c_2^{\\alpha } k^{2\\alpha / n }.$ For sufficiently large $k$ , $k \\ge k_1 \\equiv \\left\\lceil {\\left(\\frac{\\alpha }{ c_1 \\tau }\\right)^{n/2}}\\right\\rceil ,$ the summand is monotonically decreasing, while for $k < k_1$ , it is monotonically increasing.", "Splitting the sum up, $\\begin{split}\\sum _{k = 1}^{\\infty } e^{-c_1\\tau k^{2/n}} k^{2\\alpha / n} &=\\sum _{k=1}^{k_1} e^{-c_1\\tau k^{2/n}} k^{2\\alpha / n + 1 } +\\sum _{k=k_1+1}^{\\infty } e^{-c_1\\tau k^{2/n}}k^{2\\alpha / n } \\\\& \\le e^{-c_1 \\tau } \\sum _{k=1}^{k_1} k^{2\\alpha / n }+\\sum _{k=k_1+1}^{\\infty } e^{-c_1\\tau k^{2/n}} k^{2\\alpha / n }.\\end{split}$ Crudely bounding the first sum in terms of a $\\max $ , and treating the latter sum as a lower Riemann approximations of an integral, $\\begin{split}&\\sum _{k = 1}^{\\infty } e^{-c_1\\tau k^{2/n}} k^{2\\alpha / n } \\le e^{-c_1 \\tau } k_1 \\cdot k_1 ^{2\\alpha / n }+ \\int _{k_1}^\\infty e^{-c_1\\tau k^{2/n}} k^{2\\alpha / n }dk \\\\&\\quad \\le e^{-c_1 \\tau } \\left[\\left(\\frac{\\alpha }{ c_1\\tau }\\right)^{n/2} +1\\right]^{2\\alpha / n +1 }+ \\int _{0}^\\infty e^{-c_1\\tau k^{2/n}} k^{2\\alpha / n }dk \\\\& \\quad \\le \\left(\\frac{c_1 \\tau }{ \\alpha }\\right)^{-n/2-\\alpha }e^{-c_1 \\tau }\\left[1 +\\left(\\frac{c_1 \\tau }{ \\alpha }\\right)^{n/2}\\right]^{2\\alpha /n +1}+\\frac{n}{2}(c_1\\tau )^{-n/2 - \\alpha } \\Gamma \\left[\\frac{n}{2}+\\alpha \\right]\\\\&\\quad \\lesssim \\tau ^{-n/2 -\\alpha }.\\end{split}$ The integrals were computed using Mathematica, with the commands Integrate[Exp[-c*t*k^(2/n)],{k,0,Infinity}] Integrate[Exp[-c*t*k^(2/n)]*k^(2*a/n+1),{k,0,Infinity}] Now we prove Proposition REF .", "We first observe that $f$ is well defined and bounded: $\\left| f(\\tau )\\right| \\le \\sum _{k=1}^\\infty \\left| a_k\\right| \\lambda _k^\\alpha e^{-\\tau \\lambda _k}\\le \\left\\lbrace \\sum _{k=1}^\\infty \\lambda ^{s+ 2\\alpha }e^{- 2\\lambda _k \\tau } \\right\\rbrace ^{1/2}\\left\\Vert \\mathbf {a}\\right\\Vert _{H^{-s}_\\mu }.$ Applying Lemma REF with $\\alpha \\mapsto s + 2\\alpha $ and $\\tau \\mapsto 2 a$ , $\\sum _{k=1}^\\infty \\lambda ^{s + 2\\alpha } e^{- 2 \\lambda _k a }\\lesssim (2a)^{-n/2 - \\max \\left\\lbrace s+ 2\\alpha , 0\\right\\rbrace }.$ To prove uniform convergence, let $f_m(\\tau ) \\equiv \\sum _{k=1}^{m} a_k \\lambda _k^\\alpha e^{- \\tau \\lambda _k}$ denote the partial sum.", "Obviously, each partial sum is continuous in $\\tau $ .", "Then $\\begin{split}\\left| f(\\tau )-f_m(\\tau )\\right| &\\le \\sum _{k={m+1}}^\\infty \\left| a_k\\right|\\lambda _k^\\alpha e^{-\\tau \\lambda _k}\\\\& \\le \\left\\lbrace \\sum _{k=m+1}^\\infty \\lambda _k^{s + 2\\alpha }e^{-2\\lambda _k a} \\right\\rbrace ^{1/2}\\left\\Vert P_{[m+1,\\infty )}\\mathbf {a}\\right\\Vert _{H_\\mu ^{-s}}\\\\&\\le \\left\\Vert \\mathbf {a}\\right\\Vert _{H_\\mu ^{-s}}\\left\\lbrace \\sum _{k=m+1}^\\infty \\lambda _k^{s+ 2\\alpha } e^{-2\\lambda _k a} \\right\\rbrace ^{1/2}.\\end{split}$ Examining the sum, $\\begin{split}\\sum _{k=m+1}^\\infty \\lambda _k^{s+ 2\\alpha } e^{-2\\lambda _k a} &\\lesssim \\sum _{k=m+1}^\\infty k^{2s/n + 4 \\alpha /n} e^{-2 c_1 ak^{2/n}}.\\end{split}$ Taking $m$ sufficiently large, the summand will be strictly decreasing in $k$ , so we can treat it as a lower Riemann sum for the integral $\\int _{m}^\\infty k^{2s/n + 4 \\alpha /n} e^{-2c_1 a k^{2/n}} dk.$ Changing variables by letting $k^{2/n} = l$ , $\\sum _{k=m+1}^\\infty \\lambda _k^{s+ 2\\alpha } e^{-2\\lambda _k a}\\lesssim \\int _{m^{2/n}}^\\infty l^{s+ 2\\alpha + n/2 -1} e^{- 2c_1 a l} dl.$ If ${s + 2\\alpha + n/2 -1} \\le 0$ , then $\\sum _{k=m+1}^\\infty \\lambda _k^{s + 2\\alpha } e^{-2\\lambda _ka}\\lesssim \\int _{m^{2/n}}^\\infty e^{- 2c_1 a l} dl = \\frac{1}{2c_1a}e^{-2 m^{2/n} c_1 a}.$ On the other hand, if ${s+ 2\\alpha + n/2 -1}> 0$ , we can trade some of the exponential decay to eliminate the algebraic term, $\\sum _{k=m+1}^\\infty \\lambda _k^{s + 2\\alpha } e^{-2\\lambda _ka}\\lesssim \\int _{m^{2/n}}^\\infty e^{- c_1 a l} dl = \\frac{1}{c_1a}e^{-m^{2/n} c_1 a}.$ In either case, we see that for any $a>0$ , $\\lim _{m\\rightarrow \\infty }\\sup _{\\tau \\ge a}\\left| f(\\tau ) - f_m(\\tau )\\right| = 0.$ Since the partial sums converge uniformly to $f$ , it is now a classical result to conclude that $f$ is continuous for $\\tau \\ge a> 0$ , [24]." ] ]

[ [ "Optimal bounds for monotonicity and Lipschitz testing over hypercubes\n and hypergrids" ], [ "Abstract The problem of monotonicity testing over the hypergrid and its special case, the hypercube, is a classic, well-studied, yet unsolved question in property testing.", "We are given query access to $f:[k]^n \\mapsto \\R$ (for some ordered range $\\R$).", "The hypergrid/cube has a natural partial order given by coordinate-wise ordering, denoted by $\\prec$.", "A function is \\emph{monotone} if for all pairs $x \\prec y$, $f(x) \\leq f(y)$.", "The distance to monotonicity, $\\eps_f$, is the minimum fraction of values of $f$ that need to be changed to make $f$ monotone.", "For $k=2$ (the boolean hypercube), the usual tester is the \\emph{edge tester}, which checks monotonicity on adjacent pairs of domain points.", "It is known that the edge tester using $O(\\eps^{-1}n\\log|\\R|)$ samples can distinguish a monotone function from one where $\\eps_f > \\eps$.", "On the other hand, the best lower bound for monotonicity testing over the hypercube is $\\min(|\\R|^2,n)$.", "This leaves a quadratic gap in our knowledge, since $|\\R|$ can be $2^n$.", "We resolve this long standing open problem and prove that $O(n/\\eps)$ samples suffice for the edge tester.", "For hypergrids, known testers require $O(\\eps^{-1}n\\log k\\log |\\R|)$ samples, while the best known (non-adaptive) lower bound is $\\Omega(\\eps^{-1} n\\log k)$.", "We give a (non-adaptive) monotonicity tester for hypergrids running in $O(\\eps^{-1} n\\log k)$ time.", "Our techniques lead to optimal property testers (with the same running time) for the natural \\emph{Lipschitz property} on hypercubes and hypergrids.", "(A $c$-Lipschitz function is one where $|f(x) - f(y)| \\leq c\\|x-y\\|_1$.)", "In fact, we give a general unified proof for $O(\\eps^{-1}n\\log k)$-query testers for a class of \"bounded-derivative\" properties, a class containing both monotonicity and Lipschitz." ], [ "Introduction", "Monotonicity testing over hypergrids [18] is a classic problem in property testing.", "We focus on functions $f:\\mathbf {D}\\mapsto \\mathbf {R}$ , where the domain, $\\mathbf {D}$ , is the hypergrid $[k]^n$ and the range, $\\mathbf {R}$ , is a total order.", "The hypergrid/hypercube defines the natural coordinate-wise partial order: $x \\preceq y$ , iff $\\forall i \\in [n], x_i \\le y_i$ .", "A function $f$ is monotone if $f(x) \\le f(y)$ whenever $x\\preceq y$ .", "The distance to monotonicity, denoted by $\\varepsilon _f$ , is the minimum fraction of places at which $f$ must be changed to have the property $\\mathcal {P}$ .", "Formally, if $\\mathcal {M}$ is the set of all monotone functions, $\\varepsilon _f ~\\triangleq ~\\min _{g \\in \\mathcal {M}}\\left(|\\lbrace x | f(x) \\ne g(x)\\rbrace |/|\\mathbf {D}|\\right).$ Given a parameter $\\varepsilon \\in (0,1)$ , the aim is to design a randomized algorithm for the following problem.", "If $\\varepsilon _f = 0$ (meaning $f$ is monotone), the algorithm must accept with probability $>2/3$ , and if $\\varepsilon _f > \\varepsilon $ , it must reject with probability $>2/3$ .", "If $\\varepsilon _f \\in (0,\\varepsilon )$ , then any answer is allowed.", "Such an algorithm is called a monotonicity tester.", "The quality of a tester is determined by the number of queries to $f$ .", "A one-sided tester accepts with probability 1 if the function is monotone.", "A non-adaptive tester decides all of its queries in advance, so the queries are independent of the answers it receives.", "Monotonicity testing has been studied extensively in the past decade [15], [18], [13], [21], [17], [1], [16], [19], [22], [2], [4], [6], [9], [7].", "Of special interest is the hypercube domain, $\\lbrace 0,1\\rbrace ^n$ .", "[18] introduced the edge tester.", "Let $\\mathbf {H}$ be the pairs that differ in precisely one coordinate (the edges of the hypercube).", "The edge tester picks a pair in $\\mathbf {H}$ uniformly at random and checks if monotonicity is satisfied by this pair.", "For boolean range,  [18] prove $O(n/\\varepsilon )$ samples suffice to give a bonafide montonicity tester.", "[13] subsequently showed that $O(\\varepsilon ^{-1}n\\log |\\mathbf {R}|)$ samples suffice for a general range $\\mathbf {R}$ .", "In the worst case, $|\\mathbf {R}| = 2^n$ , and so this gives a $O(n^2/\\varepsilon )$ -query tester.", "The best known general lower bound is $\\Omega (\\min (n,|\\mathbf {R}|^2))$  [7].", "It has been an outstanding open problem in property testing (see Question 5 in the Open Problems list from the Bertinoro Workshop ) to give an optimal bound for monotonicity testing over the hypercube.", "We resolve this by showing that the edge tester is indeed optimal (when $|\\mathbf {R}| \\ge \\sqrt{n}$ ).", "The edge tester is a $O(n/\\varepsilon )$ -query non-adaptive, one-sided monotonicity tester for functions $f:\\lbrace 0,1\\rbrace ^n\\mapsto \\mathbf {R}$ .", "For general hypergrids $[k]^n$ , [13] give a $O(\\varepsilon ^{-1}n\\log k\\log |\\mathbf {R}|)$ -query monotonicity tester.", "Since $|\\mathbf {R}|$ can be as large as $k^n$ , this gives a $O(\\varepsilon ^{-1}n^2\\log ^2k)$ -query tester.", "In this paper, we give a $O(\\varepsilon ^{-1}n\\log k)$ -query monotonicity tester on hypergrids that generalizes the edge tester.", "This tester is also a uniform pair tester, in the sense it defines a set $\\mathbf {H}$ of pairs, picks a pair uniformly at random from it, and checks for monotonicity among this pair.", "The pairs in $\\mathbf {H}$ also differ in exactly one coordinate, as in the edge tester.", "There exists a non-adaptive, one-sided $O(\\varepsilon ^{-1}n\\log k)$ -query monotonicity tester for functions $f:[k]^n\\mapsto \\mathbf {R}$ .", "Subsequent to the conference version of this work, the authors proved a $\\Omega (\\varepsilon ^{-1}n\\log k)$ -query lower bound for monotonicity testing on the hypergrid for any (adaptive, two-sided error) tester [11].", "Thus, both the above theorems are optimal.", "A property that has been studied recently is that of a function being Lipschitz: a function $f:[k]^n \\mapsto \\mathbf {R}$ is called $c$ -Lipschitz if for all $x,y\\in [k]^n, |f(x) - f(y)| \\le c\\Vert x-y\\Vert _1$ .", "The Lipschitz testing question was introduced by [20], who show that for the range $\\mathbf {R}= \\delta \\mathbb {Z}$ , $O((\\delta \\varepsilon )^{-1}n^2)$ queries suffice for Lipschitz testing.", "For general hypergrids, [3] recently give an $O((\\delta \\varepsilon )^{-1}n^2k\\log k)$ -query tester for the same range.", "[8] prove a lower bound of $\\Omega (n\\log k)$ queries for non-adaptive monotonicity testers (for sufficiently large $\\mathbf {R}$ ).", "We give a tester for the Lipschitz property that improves all known results and matches existing lower bounds.", "Observe that the following holds for arbitrary ranges.", "There exists a non-adaptive, one-sided $O(\\varepsilon ^{-1}n\\log k)$ -query $c$ -Lipschitz tester for functions $f:[k]^n\\mapsto \\mathbf {R}$ .", "Our techniques apply to a class of properties that contains monotonicity and Lipschitz.", "We call it the bounded derivative property, or more technically, the $(\\alpha ,\\beta )$ -Lipschitz property.", "Given parameters $\\alpha ,\\beta $ , with $\\alpha < \\beta $ , we say that a function $f:[k]^n \\mapsto \\mathbf {R}$ has the $(\\alpha ,\\beta )$ -Lipschitz property if for any $x\\in [k]^n$ , and $y$ obtained by increasing exactly one coordinate of $x$ by exactly 1, we have $\\alpha \\le f(y)-f(x) \\le \\beta $ .", "Note that when $(\\alpha =0,\\beta =\\infty )$If the reader is uncomfortable with the choice of $\\beta $ as $\\infty $ , $\\beta $ can be thought of as much larger than any value in $f$ ., we get monotonicity.", "When $(\\alpha =-c,\\beta =+c)$ , we get $c$ -Lipschitz.", "There exists a non-adaptive, one-sided $O(\\varepsilon ^{-1}n\\log k)$ -query $(\\alpha ,\\beta )$ -Lipschitz tester for functions $f:[k]^n\\mapsto \\mathbf {R}$ , for any $\\alpha <\\beta $ .", "There is no dependence in the running time on $\\alpha $ and $\\beta $ .", "Although [thm:main]Theorem thm:main implies all the other theorems stated above, we prove [thm:mono-hc]Theorem thm:mono-hc and [thm:mono-hg]Theorem thm:mono-hg before giving a whole proof of [thm:main]Theorem thm:main.", "The final proof is a little heavy on notation, and the proof of the monotonicity theorems illustrates the new techniques.", "We discuss some other previous work on monotonicity testers for hypergrids.", "For the total order (the case $n=1$ ), which has been called the monotonicity testing problem on the line, [15] give a $O(\\varepsilon ^{-1}\\log k)$ -query tester, and this is optimal [15], [16].", "Results for general posets were first obtained by [17].", "The elegant concept of 2-TC spanners introduced by [6] give a general class of monotonicity testers for various posets.", "It is known that such constructions give testers with polynomial dependence of $n$ for the hypergrid [5].", "For constant $n$ , [19], [1] give a $O(\\varepsilon ^{-1}\\log k)$ -query tester (although the dependency on $n$ is exponential).", "From the lower bound side, [17] first prove an $\\Omega (\\sqrt{n})$ (non-adaptive, one-sided) lower bound for hypercubes.", "[9] give an $\\Omega (n/\\varepsilon )$ -lower bound for non-adaptive, one-sided testers, and a breakthrough result of [7] prove a general $\\Omega (\\min (n,|\\mathbf {R}|^2)$ lower bound.", "Testing the Lipschitz property is a natural question that arises in many applications.", "For instance, given a computer program, one may like to test the robustness of the program's output to the input.", "This has been studied before, for instance in [12], however, the solution provided looks into the code to detect if the program satisfies Lipschitz or not.", "The property testing setting is a black-box approach to the problem.", "[20] also provide an application to differential privacy; a class of mechanisms known as Laplace mechanisms proposed by [14] achieve privacy in the process of outputting a function by adding a noise proportional to the Lipschitz constant of the function.", "[20] gave numerous results on Lipschitz testing over hypergrids.", "They give a $O(\\varepsilon ^{-1}\\log k)$ -query tester for the line, a general $\\Omega (n)$ -query lower bound for the Lipschitz testing question on the hypercube, and a non-adaptive, 1-sided $\\Omega (\\log k)$ -query lower bound on the line." ], [ "The Proof Roadmap", "The challenge of property testing is to relate the tester behavior to the distance of the function to the property.", "Consider monotonicity over the hypercube.", "To argue about the edge tester, we want to show that a large distance to monotonicity implies many violated edges.", "Most current analyses of the edge tester go via what we could call the contrapositive route.", "If there are few violated edges in $f$ , then they show the distance to monotonicity is small.", "This is done by modifying $f$ to make it monotone, and bounding the number of changes as a function of the number of violated edges.", "There is an inherently “constructive\" viewpoint to this: it specifies a method to convert non-monotone functions to monotone ones.", "Implementing this becomes difficult when the range is large, and existing bounds degrade with $\\mathbf {R}$ .", "For the Lipschitz property, this route becomes incredibly complex.", "A non-constructive approach may give more power, but how does one get a handle on the distance?", "The violation graph provides a method.", "The violation graph has $[k]^n$ as the vertex set and an edge between any pair of comparable domain vertices $(x,y)$ ($x \\prec y$ ) if $f(x) > f(y)$ .", "The following theorem can be found as Corollary 2 in [17].", "[[17]] The size of the minimum vertex cover of the violation graph is exactly $\\varepsilon _f|\\mathbf {D}|$ .", "As a corollary, the size of any maximal matching in the violation graph is at least $\\frac{1}{2}\\varepsilon _f|\\mathbf {D}|$ .", "Can a large matching in the violated graph imply there are many violated edges?", "[21] give an approach by reducing the monotonicity testing problem on the hypercube to routing problems.", "For any $k$ source-sink pairs on the directed hypercube, suppose $k\\mu (k)$ edges need to be deleted in order to pairwise separate them.", "Then $O(n/\\varepsilon \\mu (n))$ queries suffice for the edge tester.", "Therefore, if $\\mu (n)$ is at least a constant, one gets a linear query monotonicity tester on the cube.", "Lehman and Ron [21] explicitly ask for bounds on $\\mu (n)$ .", "[9] show that $\\mu (n)$ could be as small as $1/\\sqrt{n}$ , thereby putting an $\\Omega (n^{3/2}/\\varepsilon )$ bottleneck to the above approach.", "In the reduction above, the function values are altogether ignored.", "More precisely, once one moves to the combinatorial routing question on source-sink pairs, the fact that they are related by actual function values is lost.", "Our analysis crucially uses the value of the functions to argue about the structure of the maximal matching in the violation graph." ], [ "It's all about matchings", "The key insight is to move to a weighted violation graph.", "The weight of violation $(x,y)$ depends on the property at hand; for now it suffices to know that for monotonicity, the weight of $(x,y)$ ($x \\prec y$ ) is $f(x) - f(y)$ .", "This can be thought of as a measure of the magnitude of the violation.", "(Violation weights were also used for Lipschitz testers [20].)", "We now look at a maximum weighted matching $\\mathbf {M}$ in the violation graph.", "Naturally, this is maximal as well, so $|\\mathbf {M}|\\ge \\frac{1}{2}\\varepsilon _f |\\mathbf {D}|$ .", "All our algorithms pick a pair uniformly at random from a predefined set $\\mathbf {H}$ of pairs, and check the property on that pair.", "For the hypercube domain, $\\mathbf {H}$ is the set of all edges of the hypercube.", "Our analysis is based on the construction of a one-to-one mapping from pairs in $\\mathbf {M}$ to violating pairs in $\\mathbf {H}$ .", "This mapping implies the number of violated pairs in $\\mathbf {H}$ is at least $|\\mathbf {M}|$ , and thus the uniform pair tester succeeds with probability $\\Omega (\\varepsilon _f|\\mathbf {D}|/|\\mathbf {H}|)$ , implying $O(|\\mathbf {H}|/\\varepsilon _f|\\mathbf {D}|)$ queries suffice to test monotonicity.", "For the hypercube, $|\\mathbf {H}| = n2^{n-1}$ and $|\\mathbf {D}| = 2^n$ , giving the final bound of $O(n/\\varepsilon _f)$ .", "To obtain this mapping, we first decompose $\\mathbf {M}$ into sets $M_1,M_2,\\ldots ,M_t$ such that each pair in $\\mathbf {M}$ is in at least one $M_i$ .", "Furthermore, we partition $\\mathbf {H}$ into perfect matchings $H_1,H_2,\\ldots , H_t$ .", "In the hypercube case, $M_i$ is the collection of pairs in $\\mathbf {M}$ whose $i$ th coordinates differ, and $H_i$ is the collection of hypercube edges differing only in the $i$ th coordinate; for the hypergrid case, the partitions are more involved.", "We map each pair in $M_i$ to a unique violating pair in $H_i$ .", "For simplicity, let us ignore subscripts and call the matchings $M$ and $H$ .", "We will assume in this discussion that $M \\cap H = \\emptyset $ .", "Consider the alternating paths and cycles generated by the symmetric difference of $\\mathbf {M}\\setminus M$ and $H$ .", "Take a point $x$ involved in a pair of $M$ , and note that it can only be present as the endpoint of an alternating path, denoted by ${\\bf S}_x$ .", "Our main technical lemma shows that each such ${\\bf S}_x$ contains a violated $H$ -pair.", "Figure: The alternating path: the dotted lines connect pairs of MM, the solid curved lines connect pairs of 𝐌∖M\\mathbf {M}\\setminus M, and the dashed lines are HH-pairs." ], [ "Getting the violating $H$ -pairs", "Consider $M$ , the pairs of $\\mathbf {M}$ which differ on the $i$ th coordinate, and $H$ is the set of edges in the dimension cut along this coordinate.", "Let $(x,y)\\in M$ , and say $x[i] = 0$ giving us $x\\prec y$ .", "(We denote the $a$ th coordinate of $x$ by $x[a]$ .)", "Recall that the weight of this violation is $f(x) - f(y)$ .", "It is convenient to think of ${\\bf S}_x$ as follows.", "We begin from $x$ and take the incident $H$ -edge to reach $s_1$ (note that that $s_1 \\prec y$ ).", "Then we take the $(\\mathbf {M}\\setminus M)$ -pair containing $s_1$ to get $s_2$ .", "But what if no such pair existed?", "This can be possible in two ways: either $s_1$ was $\\mathbf {M}$ -unmatched or $s_1$ is $M$ -matched.", "If $s_1$ is $\\mathbf {M}$ -unmatched, then delete $(x,y)$ from $\\mathbf {M}$ and add $(s_1,y)$ to obtain a new matching.", "If $(x,s_1)$ was not a violation, and therefore $f(x) < f(s_1)$We are assuming here that all function values are distinct; as we show in [clm:pert]Claim clm:pert this is without loss of generality., we get $f(s_1) - f(y) > f(x) - f(y)$ .", "Thus the new matching has strictly larger weight, contradicting the choice of $\\mathbf {M}$ .", "If $s_1$ was $M$ -matched, then let $(s_1,s_2)\\in M$ .", "First observe that $s_1 \\succ s_2$ .", "This is because $s_1[i] = 1$ (since $s_1[i] \\ne x[i]$ ) and since $(s_1,s_2)\\in M$ they must differ on the $i$ th coordinate implying $s_2[i]=0$ .", "This implies $s_2 \\prec y$ , and so we could replace pairs $(x,y)$ and $(s_2,s_1)$ in $\\mathbf {M}$ with $(s_2,y)$ .", "Again, if $(x,s_1)$ is not a violation, then $f(s_2) - f(y) > [f(s_2) - f(s_1)] + [f(x) - f(y)]$ , contradicting the maximality of $\\mathbf {M}$ .", "Therefore, we can taje a $(\\mathbf {M}\\setminus M)$ -pair to reach $s_2$ .", "With care, this argument can be carried over till we find a violation, and a detailed description of this is given in [sec:mono-hc]§sec:mono-hc.", "Let us demonstrate a little further (refer to the left of [fig:two-step]Fig. fig:two-step).", "Start with $(x,y)\\in M$ , and $x[i] = 0$ .", "Following the sequence ${\\bf S}_x$ , the first term $s_1$ is $x$ projected “up\" dimension cut $H$ .", "The second term is obtained by following the $\\mathbf {M}\\setminus M$ -pair incident to $s_1$ to get $s_2$ .", "Now we claim that $s_2 \\succ s_1$ , for otherwise one can remove $(x,y)$ and $(s_1, s_2)$ and add $(x,s_1)$ and $(s_2, y)$ to increase the matching weight.", "(We just made the argument earlier; the interested reader may wish to verify.)", "In the next step, $s_2$ is projected “down\" along $H$ to get $s_3$ .", "By the nature of the dimension cut $H$ , $x \\prec s_3$ and $s_1 \\prec y$ .", "So, if $s_3$ is unmatched and $(s_2, s_3)$ is not a violation, we can again rearrange the matching to improve the weight.", "We alternately go “up\" and “down\" $H$ in traversing ${\\bf S}_x$ , because of which we can modify the pairs in $\\mathbf {M}$ and get other matchings in the violation graph.", "The maximality of $\\mathbf {M}$ imposes additional structure, which leads to violating edges in $H$ .", "In general, the spirit of all our arguments is as follows.", "Take an endpoint of $M$ and start walking along the sequence given by the alternating paths generated by $\\mathbf {M}\\setminus M$ and $H$ .", "Naturally, this sequence must terminate somewhere.", "If we never encounter a violating pair of $H$ during the entire sequence, then we can rewire the matching $\\mathbf {M}$ and increase the weight.", "Contradiction!", "Observe the crucial nature of alternating up and down movements along $H$ .", "This happens because the first coordinate of the points in ${\\bf S}_x$ switches between the two values of 0 and 1 (for $k=2$ ).", "Such a reasoning does not hold water in the hypergrid domain.", "The structure of $\\mathbf {H}$ needs to be more complex, and is not as simple as a partition of the edges of the hypergrid.", "Consider the extreme case of the line $[k]$ .", "Let $2^r$ be less than $k$ .", "We break $[k]$ into contiguous pieces of length $2^r$ .", "We can now match the first part to the second, the third to the fourth, etc.", "In other words, the pairs look like $(1,2^r+1)$ , $(2, 2^r+2)$ , $\\ldots $ , $(2^r, 2^{r+1})$ , then $(2^{r+1}+1,2^{r+1}+2^r+1)$ , $(2^{r+1}+2,2^{r+1}+2^r+2)$ , etc.", "We can construct such matchings for all powers of 2 less than $k$ , and these will be our $H_i$ 's.", "Those familiar with existing proofs for monotonicity on $[k]$ will not be surprised by this set of matchings.", "All methods need to cover all “scales\" from 1 to $k$ (achieved by making them all powers of 2 up to $k$ ).", "It can also be easily generalized to $[k]^n$ .", "What about the choice of $\\mathbf {M}$ ?", "Simply choosing $\\mathbf {M}$ to be a maximum weight matching and setting up the sequences ${\\bf S}_x$ does not seem to work.", "It suffices to look at $[k]^2$ and the matching $H$ along the first coordinate where $r=0$ , so the pairs are $\\lbrace (x,x^{\\prime }) | x[1] = 2i-1, x^{\\prime }[1] = 2i, x[2] = x^{\\prime }[2]\\rbrace $ .", "A good candidate for the corresponding $M$ is the set of pairs in $\\mathbf {M}$ that connect lower endpoints of $H$ to higher endpoints of $H$ .", "Let us now follow ${\\bf S}_x$ as before.", "Refer to the right part of [fig:two-step]Fig. fig:two-step.", "Take $(x,y) \\in M$ and let $x \\prec y$ .", "We get $s_1$ by following the $H$ -edge on $x$ , so $s_1 \\succ x$ .", "We follow the $\\mathbf {M}\\setminus M$ -pair incident to $s_1$ (suppose it exists) to get $s_2$ .", "It could be that $s_2 \\succ s_1$ .", "It is in $s_3$ that we see a change from the hypercube.", "We could get $s_3 \\succ s_2$ , because there is no guarantee that $s_2$ is at the higher end of an $H$ -pair.", "This could not happen in the hypercube.", "We could have a situation where $s_3$ is unmatched, we have not encountered a violation in $H$ , and yet we cannot rearrange $\\mathbf {M}$ to increase the weight.", "For a concrete example, consider the points as given in [fig:two-step]Fig.", "fig:two-step with function values $f(x) = f(s_1) = f(s_3) = 1$ , $f(y) = f(s_2) = 0$ .", "Some thought leads to the conclusion that $s_3$ must be less than $s_2$ for any such rearrangement argument to work.", "The road out of this impasse is suggested by the two observations.", "First, the difference in 1-coordinates between $s_1$ and $s_2$ must be odd.", "Next, we could rearrange and match $(x,s_2)$ and $(s_1,y)$ instead.", "The weight may not increase, but this matching might be more amenable to the alternating path approach.", "We could start from a maximum weight matching that also maximizes the number of pairs where coordinate differences are even.", "Indeed, the insight for hypergrids is the definition of a potential $\\Phi $ for $\\mathbf {M}$ .", "The potential $\\Phi $ is obtained by summing for every pair $(x,y) \\in \\mathbf {M}$ and every coordinate $a$ , the largest power of 2 dividing the difference $|x[a] - y[a]|$ .", "We can show that a maximum weight matching that also maximizes $\\Phi $ does not end up in the bad situation above.", "With some addition arguments, we can generalize the hypercube proof.", "We describe this in [sec:mono-hg]§sec:mono-hg." ], [ "Attacking the generalized Lipschitz property", "One of the challenges in dealing with the Lipschitz property is the lack of direction.", "The Lipschitz property, defined as $\\forall x,y, |f(x) - f(y)| \\le \\Vert x-y\\Vert _1$ , is an undirected property, as opposed to monotonicity.", "In monotonicity, a point $x$ only “interacts\" with the subcube above and below $x$ , while in Lipschitz, constraints are defined between all pairs of points.", "Previous results for Lipschitz testing require very technical and clever machinery to deal with this issue, since arguments analogous to monotonicity do not work.", "The alternating paths argument given above for monotonicity also exploits this directionality, as can be seen by heavy use of inequalities in the informal calculations.", "Observe that in the monotonicity example for hypergrids in [fig:two-step]Fig.", "fig:two-step, the fact that $s_3 \\succ s_2$ (as opposed to $s_3 \\prec s_2$ ) required the potential $\\Phi $ (and a whole new proof).", "A subtle point is that while the property of Lipschitz is undirected, violations to Lipschitz are “directed\".", "If $|f(x) - f(y)| > \\Vert x-y\\Vert _1$ , then either $f(x) - f(y) > \\Vert x-y\\Vert _1$ or $f(y) - f(x) > \\Vert x-y\\Vert _1$ , but never both.", "This can be interpreted as a direction for violations.", "In the alternating paths for monotonicity (especially for the hypercube), the partial order relation between successive terms follow a fixed pattern.", "This is crucial for performing the matching rewiring.", "As might be guessed, the weight of a violation $(x,y)$ becomes $\\max (f(x) - f(y) - \\Vert x-y\\Vert _1, f(y) - f(x) - \\Vert x-y\\Vert _1)$ .", "For the generalized Lipschitz problem, this is defined in terms of a pseudo-distance over the domain.", "We look at the maximum weight matching as before (and use the same potential function $\\Phi $ ).", "The notion of “direction\" takes the place of the partial order relation in monotonicity.", "The main technical arguments show that these directions follow a fixed pattern in the corresponding alternating paths.", "Once we have this pattern, we can perform the matching rewiring argument for the generalized Lipschitz problem.", "The framework of this section is applicable for all $(\\alpha ,\\beta )$ -Lipschitz properties over hypergrids.", "We begin with two objects: $\\mathbf {M}$ , the matching of violating pairs, and $H$ , a matching of $\\mathbf {D}$ .", "The pairs in $H$ will be aligned along a fixed dimension (denote it by $r$ ) with the same $\\ell _1$ distance, called the $H$ -distance.", "That is, each pair $(x,y)$ in $H$ will differ only in one coordinate and the difference will be the same for all pairs.", "We now give some definitions.", "$L(H), U(H)$ : Each pair $(x,y) \\in H$ has a “lower\" end $x$ and an “upper\" end $y$ depending on the value of the coordinate at which they differ.", "We use $L(H)$ (resp.", "$U(H)$ ) to denote the set of lower (resp.", "upper) endpoints.", "Note that $L(H) \\cap U(H) = \\emptyset $ .", "$H$ -straight pairs, $st_{H}(\\mathbf {M})$ : All pairs $(x,y) \\in \\mathbf {M}$ with both ends in $L(H)$ or both in $U(H)$ .", "$H$ -cross pairs, $cr_{H}(\\mathbf {M})$ : All pairs $(x,y) \\in \\mathbf {M}\\setminus H$ such that $x \\in L(H)$ , $y \\in U(H)$ , and the $H$ -distance divides $|y[r] - x[r]|$ .", "$H$ -skew pairs, $sk_{H}(\\mathbf {M})= \\mathbf {M}\\setminus (st_{H}(\\mathbf {M}) \\cup cr_{H}(\\mathbf {M}))$ .", "$X$ : A set of lower endpoints in $cr_{H}(\\mathbf {M}) \\setminus H$ .", "Consider the domain $\\lbrace 0,1\\rbrace ^n$ .", "We set $H$ to be (say) the first dimension cut.", "$st_{H}(\\mathbf {M})$ is the set of pairs in $(x,y) \\in \\mathbf {M}$ where $x[1] = y[1]$ .", "All other pairs $(x,y) \\in \\mathbf {M}$ ($x \\prec y$ ) are in $cr_{H}(\\mathbf {M})$ since $x[1] = 0$ and $y[1] = 1$ .", "There are no $H$ -skew pairs.", "The set $X$ will be chosen differently for the applications.", "We require the following technical definition of adequate matchings.", "This arises because we will use matchings that are not necessarily perfect.", "A perfect matching $H$ is always adequate.", "A matching $H$ is adequate if for every violation $(x,y)$ , both $x$ and $y$ participate in the matching $H$ .", "We will henceforth assume that $H$ is adequate.", "The symmetric difference of $st_{H}(\\mathbf {M})$ and $H$ is a collection of alternating paths and cycles.", "Because $H$ is adequate and $st_{H}(\\mathbf {M}) \\cap cr_{H}(\\mathbf {M}) = \\emptyset $ , any point in $x \\in X$ is the endpoint of some alternating path (denoted by ${\\bf S}_x$ ).", "Throughout the paper, $i$ denotes an even index, $j$ denotes an odd index, and $k$ is an arbitrary index.", "The first term ${\\bf S}_x(0)$ is $x$ .", "For even $i$ , ${\\bf S}_x(i+1) = H({\\bf S}_x(i))$ .", "For odd $j$ : if ${\\bf S}_x(j)$ is $st_{H}(\\mathbf {M})$ -matched, ${\\bf S}_x(j+1) = \\mathbf {M}({\\bf S}_x(j))$ .", "Otherwise, terminate.", "We start with a simple property of these alternating paths.", "For $k\\equiv 0,3 \\ (\\operatorname{mod} 4)$ , $s_k \\in L(H)$ .", "For non-negative $k \\equiv 1,2 \\ (\\operatorname{mod} 4)$ , $s_k \\in U(H)$ .", "If $k$ is even, then $(s_{k},s_{k+1})\\in H$ .", "Therefore, either $s_k \\in L(H)$ and $s_{k+1} \\in U(H)$ or vice versa.", "If $k$ is odd, $(s_k, s_{k+1})$ is a straight pair.", "So $s_k$ and $s_{k+1}$ lie in the same sets.", "Starting with $s_0 \\in L(H)$ , a trivial induction completes the proof.", "The following is a direct corollary of [prop:sub]Prop. prop:sub.", "If $i \\equiv 0 \\ (\\operatorname{mod} 4)$ , $s_{i} \\prec s_{i+1}$ .", "If $i \\equiv 2 \\ (\\operatorname{mod} 4)$ , $s_{i+1} \\prec s_{i}$ .", "We will prove that every ${\\bf S}_x$ contains a violated $H$ -pair.", "Henceforth, our focus is entirely on some fixed sequence ${\\bf S}_x$ .", "The sets $E_-(i)$ and $E_+(i)$ Our proofs are based on matching rearrangements, and this motivates the definitions in this subsection.", "For convenience, we denote ${\\bf S}_x$ by $x = s_0, s_1, s_2, \\ldots $ .", "We also set $s_{-1} = y$ .", "Consider the sequence $s_{-1}, s_0, s_1, \\ldots , s_i$ , for even $i > 1$ .", "We define $ E_-(i) = (s_{-1}, s_0), (s_1, s_2), (s_3, s_4), \\ldots , (s_{i-1}, s_i) = \\lbrace (s_j, s_{j+1}): \\textrm {$ $ odd}, -1 \\le j < i\\rbrace $$This is simply the set of $ M$-pairs in $Sx$ up to $ si$.", "We now define $ E+(i)$.Think of this as follows.", "We first pair up $ (s-1,s1)$.", "Then, we go in order of $Sx$ to pairup the rest.", "We pick the first unmatched $ sk$ and pair it to the first term of \\emph {opposite} parity.We follow this till $ si+1$ is paired.", "These sets are illustrated in [fig:lemma4point3]{Fig.\\,\\ref *{fig:lemma4point3}}.\\begin{eqnarray*}E_+(i) & = & (s_{-1}, s_1), (s_0, s_3), (s_2, s_5), \\ldots , (s_{i-4},s_{i-1}), (s_{i-2}, s_{i+1}) \\\\& = & \\lbrace (s_{-1}, s_1)\\rbrace \\cup \\lbrace (s_{i^{\\prime }}, s_{i^{\\prime }+3}): \\textrm {i^{\\prime } even}, 0 \\le i^{\\prime } \\le i-2\\rbrace \\end{eqnarray*}\\begin{figure}[h]\\includegraphics [trim=4cm 6cm 3cm 6cm,clip=true,scale=0.55]{lemma4point3-other}\\caption {Illustration for i=8.", "The light vertical edges are H-edges.", "The dark black ones are st_{H}(\\mathbf {M})-pairs.", "The green, double-lined one on the left is the starting M-pair.", "The dotted red pairs form E_+(8).All points alove the horizonatal line are in U(H), the ones below are in L(H).", "}\\end{figure}\\vspace{-20.0pt}\\begin{proposition} E_-(i) involves s_{-1}, s_0, \\ldots , s_i,while E_+(i) involves s_{-1}, s_0, \\ldots , s_{i-1}, s_{i+1}.\\end{proposition}\\section {The Structure of \\protect {\\bf S}_x for Monotonicity} We now focus on monotonicity, and show that $Sx$ is highly structured.", "(The proof for general Lipschitzwill also follow the same setup, but requires more definitions.", ")The weight of a pair $ (x,y)$ is defined to be $ f(x)-f(y)$ if $ xy$, and is $ -$ otherwise.We will assume that all function values are distinct.", "This is without loss of generality although we prove it formally later in [clm:pert]{Claim~\\ref *{clm:pert}}.Thus violating pairs have positive weight.We choose a maximum weight matching $ M$ of pairs.", "Note that every pair in $ M$ is a violating pair.We remind the readerthat for even $ k$, $ (sk,sk+1) H$ and for odd $ k$, $ (sk,sk+1)stH(M)$.\\subsection {Preliminary observations}\\begin{proposition} For all x,y \\in L(H) (or U(H)),x \\prec y iff H(x) \\prec H(y).Consider pair (x,y) \\in cr_{H}(\\mathbf {M}) such that x \\prec y.Then H(x) \\prec y and x \\prec H(y).\\end{proposition}\\begin{proof} For any point in x \\in L(H), H(x) is obtained by adding the H-distanceto a specific coordinate.", "This proves the first part.The H-distance divides |[y[r] - x[r]| (where H is aligned in dimension r) and (x,y), x \\prec y is a cross pair.Hence y[r] - x[r] is at least the H-distance.", "Note that H(x)is obtained by simply adding this distance to the r coordinate of x, so H(x) \\prec y.\\end{proof}\\begin{proposition} All pairs in E_-(i) and E_+(i) are comparable.Furthermore, s_{1} \\prec s_{-1} and for all even 0 \\le k \\le i-2, s_{k} \\prec s_{k+3} iff s_{k+1} \\prec s_{k+2}.\\end{proposition}\\begin{proof}All pairs in E_-(k) are in \\mathbf {M}, and hence comparable.", "Consider pair (s_{-1},s_1) \\in E_+(k).Since s_1 = H(s_0) and (s_0,s_1) is a cross-pair, by [prop:cross]{Prop.~\\ref *{prop:cross}}, s_1 \\prec s_{-1}.Consider pair (s_k, s_{k+3}),where k is even.", "(Refer to [fig:lemma4point3]{Fig.\\,\\ref *{fig:lemma4point3}}.", ")The pair (H(s_k), H(s_{k+3})) = (s_{k+1}, s_{k+2}) is in st_{H}(\\mathbf {M}).", "Hence, the pointsare comparable and both lie in L(H) or U(H).", "By [prop:cross]{Prop.~\\ref *{prop:cross}},s_k, s_{k+3} inherit their comparability from s_{k+1}, s_{k+2}.\\end{proof}For some even $ i$, suppose $ (si,si+1)$ is a not a violation.", "[cor:sub]{Corollary~\\ref *{cor:sub}} implies{\\begin{@align}{1}{-1}\\textrm {If} \\ i\\equiv 0\\ (\\operatorname{mod} 4), f(s_{i+1}) - f(s_{i}) > 0.\\nonumber \\\\\\textrm {If} \\ i\\equiv 2\\ (\\operatorname{mod} 4), f(s_{i}) - f(s_{i+1}) > 0.", "\\end{@align}}We will also state an \\emph {ordering} condition on the sequence.", "{\\begin{@align}{1}{-1}\\textrm {If} \\ i\\equiv 0\\ (\\operatorname{mod} 4), \\ s_i \\prec s_{i-1}.", "\\nonumber \\\\\\textrm {If} \\ i\\equiv 2\\ (\\operatorname{mod} 4), \\ s_i \\succ s_{i-1}.", "\\end{@align}}Remember these conditions and [cor:sub]{Corollary~\\ref *{cor:sub}} together as follows.", "If \\qquad \\mathrm {(*,**)}$ i 0 (mod 4)$,$ si$ is on smaller side, otherwise it is on the larger side.In other words, if $ i 0 (mod 4)$,$ si$ is smaller than its ``neighbors\" in $Sx$.", "For $ i 2 (mod 4)$, it is bigger.For condition (\\ref {eq:*}), if $ i 0 (mod 4)$, $ f(si) < f(si-1)$.\\subsection {The structure lemmas}We will prove a series of lemmas that prove structural properties of $Sx$ that are intimatelyconnected to conditions (\\ref {eq:*}) and (\\ref {eq:**}).", "These proofs are where much of the insight lies.\\begin{lemma}Consider some even index i such that s_i exists.Suppose conditions (\\ref {eq:*}) and (\\ref {eq:**}) held for all even indices \\le i.Then, s_{i+1} is \\mathbf {M}-matched.\\end{lemma}\\begin{proof} The proof is by contradiction, so assume that \\mathbf {M}(s_{i+1}) does not exist.Assume i\\equiv 0\\ (\\operatorname{mod} 4).", "(The proof for the case i\\equiv 2\\ (\\operatorname{mod} 4) is similar and omitted.", ")Consider sets E_-(i) and E_+(i).", "Note that s_{-1}, s_0, s_1, \\ldots , s_{i+1} are all distinct.By [prop:ek]{Prop.~\\ref *{prop:ek}}, \\mathbf {M}^{\\prime } = \\mathbf {M}- E_-(i) + E_+(i) is a valid matching.", "We willargue that w(\\mathbf {M}^{\\prime }) > w(\\mathbf {M}), a contradiction.By condition (\\ref {eq:**}),\\begin{eqnarray}w(E_-(i)) & = & [f(s_{0}) - f(s_{-1})] + [f(s_1) - f(s_2)] + [f(s_4) - f(s_3)] + \\cdots \\nonumber \\\\&& \\cdots + [f(s_{i-3}) - f(s_{i-2})] + [f(s_{i}) - f(s_{i-1})] \\end{eqnarray}By the second part of [prop:ek-comp]{Prop.~\\ref *{prop:ek-comp}} (for even k, s_{k} \\prec s_{k+3} iff s_{k+1} \\prec s_{k+2}) and condition (\\ref {eq:**}), we know the comparisons for all pairs in E_+(i).\\begin{eqnarray}w(E_+(i+2)) & = & [f(s_{1}) - f(s_{-1})] + [f(s_0) - f(s_3)] + [f(s_5) - f(s_2)] + \\cdots \\nonumber \\\\& & \\cdots + [f(s_{i-4}) - f(s_{i-1})] + [f(s_{i+1}) - f(s_{i-2})] \\end{eqnarray}Note that the coefficients of common terms in w(E_+(i)) and w(E_-(i)) are identical.The only terms not involves (by [prop:ek]{Prop.~\\ref *{prop:ek}}) are f(s_{i+1}) in w(E_+(i)) and f(s_i) in w(E_-(i)).The weight of the new matching is precisely w(\\mathbf {M}) - W_- + W_+ = w(\\mathbf {M}) + f(s_{i+1}) - f(s_{i}).By (\\ref {eq:*}) for i, this is strictly greater than w(\\mathbf {M}), contradicting the maximality of \\mathbf {M}.\\end{proof}So, under the condition of [lem:cons-mon]{Lemma\\,\\ref *{lem:cons-mon}}, $ si+1$ is $ M$-matched.", "We can also specifythe comparison relation of $ si+1$, $ M(si+1)$ (as condition (\\ref {eq:**})) using an almost identical argument.Abusing notation, we will denote $ M(si+1)$ as $ si+2$.", "(This is no abuse if $ (si+1, M(si+1))$is a straight pair.", ")\\begin{lemma}Consider some even index i such that s_i exists.Suppose conditions (\\ref {eq:*}) and (\\ref {eq:**}) held for all even indices \\le i.Then, condition (\\ref {eq:**}) holds for i+2.\\end{lemma}Before we prove this lemma, we need the following distinctness claim.\\begin{claim} Consider some odd j such that s_j and \\mathbf {M}(s_{j}) exist.Suppose condition (\\ref {eq:*}) and (\\ref {eq:**}) held for all even i < j.Then the sequence s_{-1}, s_0, s_1, \\ldots , s_j, \\mathbf {M}(s_{j}) are distinct.\\end{claim}\\begin{proof} (If (s_j, \\mathbf {M}(s_{j})) \\in st_{H}(\\mathbf {M}), this is obviously true.", "Thechallenge is when {\\bf S}_x terminates at s_j.)", "The sequence from s_0 to s_{j} is an alternating path, soall terms are distinct.", "If s_j \\ne y, then the claim holds.Suppose s_j = y.", "Note that j > 1, since (x,y) \\notin H. Since y \\in U(H), by [prop:sub]{Prop.~\\ref *{prop:sub}}, j \\equiv 1 \\ (\\operatorname{mod} 4).", "Condition (\\ref {eq:**})holds for j-1, so s_{j-1} \\prec s_{j} = y and by [cor:sub]{Corollary~\\ref *{cor:sub}}, s_{j-1} \\prec s_{j-2}.", "Note that (s_{j-1}, s_{j}) \\in Hand (x,s_{j}) is a cross pair.", "By [prop:cross]{Prop.~\\ref *{prop:cross}}, x \\prec s_{j-1} and thus x \\prec s_{j-2}.We replace pairs A = \\lbrace (x,y), (s_{j-2},s_{j-1})\\rbrace \\in \\mathbf {M} with (x,s_{j-2}), and argue that the weight has increased.We have w(A) = [f(x) - f(y)] + [f(s_{j-1}) - f(s_{j-2})] = [f(x) - f(s_{j-2})] - [f(y) - f(s_{j-1})].By condition (\\ref {eq:*}) on i, f(y) = f(s_{j}) > f(s_{j-1}), contradicting the maximality of \\mathbf {M}.\\end{proof}\\begin{proof} (of [lem:cons-mon2]{Lemma\\,\\ref *{lem:cons-mon2}}) By [lem:cons-mon]{Lemma\\,\\ref *{lem:cons-mon}}, \\mathbf {M}(s_{i+1}) exists.Assume i \\equiv 0 \\ (\\operatorname{mod} 4) (the other case is analogous and omitted).", "The proof is again by contradiction,so we assume condition (\\ref {eq:**}) does not hold for i+2.", "This means s_{i+2} = \\mathbf {M}(s_{i+1}) \\prec s_{i+1}.Consider sets E_-(i+2) and E^{\\prime } = E_+(i-2) \\cup (s_{i-2},s_{i+2}).", "By [clm:rep]{Claim~\\ref *{clm:rep}}, s_{-1}, s_0, s_1, \\ldots , s_{i+2} are distinct.So \\mathbf {M}^{\\prime } = \\mathbf {M}- E_-(i) + E^{\\prime } is a valid matching and we argue that w(\\mathbf {M}^{\\prime }) > w(\\mathbf {M}).By condition (\\ref {eq:**}) for even i^{\\prime } < i+2 and the assumption s_{i+2} \\prec s_{i+1}.\\begin{eqnarray}w(E_-(i+2)) & = & [f(s_{0}) - f(s_{-1})] + [f(s_1) - f(s_2)] + [f(s_4) - f(s_3)] + \\cdots \\nonumber \\\\&& \\cdots + [f(s_{i-3}) - f(s_{i-2})] + [f(s_{i}) - f(s_{i-1})] + [f(s_{i+2}) - f(s_{i+1})] \\nonumber \\end{eqnarray}Observe how the last term in the summation differs from the trend.All comparisons in E_+(i-2) are determined by [prop:ek]{Prop.~\\ref *{prop:ek}}, just as we arguedin the proof of [lem:cons-mon]{Lemma\\,\\ref *{lem:cons-mon}}.", "The expression for w(E_+(i-2)) is basicallygiven in (\\ref {eq:we+}).", "It remains to deal with (s_{i-2},s_{i+2}).By condition (\\ref {eq:**}) for i, s_i \\prec s_{i-1}.Thus, by [prop:ek]{Prop.~\\ref *{prop:ek}}, s_{i+1} \\prec s_{i-2}.Combining with the assumption of s_{i+2} \\prec s_{i+1}, we deduce s_{i+2} \\prec s_{i-2}.\\begin{eqnarray}w(E_+(i+2)) & = & [f(s_{1}) - f(s_{-1})] + [f(s_0) - f(s_3)] + [f(s_5) - f(s_2)] + \\cdots \\nonumber \\\\& & \\cdots + [f(s_{i-3}) - f(s_{i-6})] + [f(s_{i-4}) - f(s_{i-1})] + [f(s_{i+2}) - f(s_{i-2})] \\nonumber \\end{eqnarray}The coefficients are identical, except that f(s_i) and f(s_{i+1}) do not appear in w(E_+(i+2)).We get w(\\mathbf {M}) - W_- + W_+ = w(\\mathbf {M}) + f(s_{i+1}) - f(s_{i}).", "By (\\ref {eq:*}) for i, we contradict the maximality of \\mathbf {M}.\\end{proof}A direct combination of the above statements yields the main structure lemma.\\begin{lemma} Suppose {\\bf S}_x contains no violated H-pair.", "Let the lastterm by s_j (j is odd).", "For every even i \\le j+1, condition (\\ref {eq:**}) holds,and s_j belongs to a pair in sk_{H}(\\mathbf {M}).\\end{lemma}\\begin{proof} We prove the first statement by contradiction.", "Consider the smallest even i \\le j+1where condition (\\ref {eq:**}) does not hold.", "Note that for i=0, the condition does hold,so i \\ge 2.", "We can apply [lem:cons-mon2]{Lemma\\,\\ref *{lem:cons-mon2}} for i-2, since all even indices at most i-2satisfy (\\ref {eq:*}) and (\\ref {eq:**}).", "But condition (\\ref {eq:**}) holds for i, completing the proof.Now apply [lem:cons-mon]{Lemma\\,\\ref *{lem:cons-mon}} and [lem:cons-mon2]{Lemma\\,\\ref *{lem:cons-mon2}} for j-1.", "Conditions (\\ref {eq:*}) and (\\ref {eq:**}) hold for all relevant even indices.Hence, s_{j} must be \\mathbf {M}-matched and condition (\\ref {eq:**}) holds for j+1.", "Since {\\bf S}_x terminatesat s_j, s_j cannot be st_{H}(\\mathbf {M})-matched.", "Suppose s_j was cr_{H}(\\mathbf {M}) matched.", "Let j \\equiv 1 \\ (\\operatorname{mod} 4).", "By [prop:sub]{Prop.~\\ref *{prop:sub}}, s_j \\in U(H),so s_{j+1} = \\mathbf {M}(s_j) \\prec s_j, violating condition (\\ref {eq:**}).", "A similar argument holds when j \\equiv 3 \\ (\\operatorname{mod} 4).Hence, s_j must be sk_{H}(\\mathbf {M})-matched.\\end{proof}\\section {Monotonicity on Boolean Hypercube}We prove [thm:mono-hc]{Theorem\\,\\ref *{thm:mono-hc}}.", "Since $ M$ is also is a maximal family of disjoint violating pairs, and therefore, $ |M| 12f2n$.We denote the set of all edges of the hypercube as $ H$.", "We partition $ H$ into $ H1,...,Hn$ where $ Hr$ is the collection of hypercube edges which differ in the $ r$th coordinate.", "Each $ Hr$ is a perfect matching and is adequate.", "Note that $ stHr(M)$ is the set of $ M$-pairs which do not differ in the $ r$th coordinate.The $ H$-distance is trivially $ 1$, so $ crHr(M)$ is the set of $ M$-pairs that differ in the $ r$th coordinate.Importantly, $ skHr(M) = $.\\begin{lemma}For all 1\\le r\\le n, the number of violating H_r-edges is at least cr_{H_r}(\\mathbf {M})/2.\\end{lemma}\\begin{proof} Feed in \\mathbf {M} and H_r to the alternating path machinery.", "Set X to be the setof all lower endpoints of cr_{H_r}(\\mathbf {M}) \\setminus H_r, so |X| = |cr_{H_r}(\\mathbf {M}) \\setminus H_r|/2.", "Since sk_{H_r}(\\mathbf {M}) = \\emptyset ,by [lem:last]{Lemma\\,\\ref *{lem:last}}, all sequences {\\bf S}_x must contain a violated H_r-edge.", "The total numberof violated H_r-edges is at least |X| + |cr_{H_r}(\\mathbf {M}) \\cap H_r|.\\end{proof}The above lemma proves [thm:mono-hc]{Theorem\\,\\ref *{thm:mono-hc}}.Observe that every pair in $ M$ belongs to some set $ crHr(M)$.The edge tester only requires $ O(n/)$ queries, sincethe success probability of a single test is at least $$\\frac{1}{|\\mathbf {H}|}\\sum _{r=1}^n cr_{H_r}(\\mathbf {M})/2 \\ge |\\mathbf {M}|/(n2^{n-2}) \\ge \\varepsilon /2n.$$\\section {Setting up for Hypergrids} We setup the framework for hypergrid domains.", "The arguments here are property independent.Consider domain $ [k]n$ and set $ = k $.We define $ H$ to be pairs that differ in exactly one coordinate, and furthermore, the difference is a power of $ 2$.The tester chooses a pair in $ H$ uniformly at random, and checks the property on this pair.We partition $ H$ into $ n(+ 1)$ sets $ Ha,b$, $ 1an$, $ 0b$.$ Ha,b$ consists of pairs $ (x,y)$ which differ only in the $ a$th coordinate, and furthermore $ |y[a] - x[a]| = 2b$.Unfortunately, $ Ha,b$ is not a matching, since each point can participate in potentially twopairs in $ Ha,b$.To remedy this, we further partition $ Ha,b$ into $ H0a,b$ and $ H1a,b$.For any pair $ (x,y)Ha,b$, exactly one among $ x[a] (mod 2b+1)$\\footnote {We abuse notation and define p \\ (\\operatorname{mod} 2^{b+1}) to be 2^{b+1} (instead of 0) if 2^{b+1}\\mid p.} and $ y[a] (mod 2b+1)$ is $ >2b$ and one is $ 2b$.We put $ (x,y)Ha,b$ with $ xy$ in $ H0a,b$ if $ y[a] (mod 2b+1) > 2b$, and in the set $ H1a,b$ if $ 1y[a] (mod 2b+1) 2b$.For example, $ H1,0$ has all pairs that only differ by $ 20 = 1$ in the first coordinate.We partition these pairs depending on whether the higher endpoint has even or odd first coordinate.Note that each $ H0a,b$ and $ H1a,b$ are matchings.We have $ L(H0a,b) = {x | x[a] (mod 2b+1) 2b}$ and $ U(H0a,b) = {y | y[a] (mod 2b+1) > 2b}$.The sets are exactly switched for $ H1a,b$.Because of the matchings are not perfect, we are forced to introduce the notion of adequacy of matchings.A matching $ H$ is adequate if for every violation $ (x,y)$, both $ x$ and $ y$ participate in the matching $ H$ ([def:adeq]{Definition~\\ref *{def:adeq}}).We will eventually prove the following theorem.\\begin{theorem} Let k be a power of 2.", "Suppose for everyviolation (x,y) and every coordinate a, |y[a] - x[a]| \\le 2^c (for some c).Furthermore, suppose that for b \\le c, all matchings H^0_{a,b}, H^1_{a,b} are adequate.", "Then there exists a maximal matching \\mathbf {M} of the violationgraph such that the number of violating pairs in \\mathbf {H} is at least |\\mathbf {M}|/2.\\end{theorem}We reduce to this special case using a simple padding argument.The following theorem implies [thm:mono-hg]{Theorem\\,\\ref *{thm:mono-hg}}.\\begin{theorem} Consider any function f:[k]^n \\mapsto R.At least an \\varepsilon _f/(4n(\\lceil \\log k \\rceil + 1)-fraction of pairs in \\mathbf {H} are violations.\\end{theorem}\\begin{proof} Let \\hat{k} = 2^\\ell be the smallest power of 2 larger than 4k.Let us construct a function \\hat{f}:[\\hat{k}]^n \\mapsto \\mathbf {R}\\cup \\lbrace -\\infty , +\\infty \\rbrace .Let {\\bf 1} denote the n-dimensional vector all 1s vector.For x such that all x_i \\in [\\hat{k}/4 + 1, \\hat{k}/4 + k-1], we set \\hat{f}(x) = f(x - \\frac{\\hat{k}\\cdot {\\bf 1}}{4}).", "(We will refer to this region as the ``original domain\".", ")If any coordinate of x is less than \\hat{k}/4, we set \\hat{f}(x) = -\\infty .Otherwise, we set f(x) = +\\infty .All violations are contained in the original domain.For any violation (x,y) and coordinate a, |y[a] - x[a]| \\le k < 2^{\\ell -2}.Let \\hat{\\mathbf {H}} be the corresponding set of pairs in domain [\\hat{k}]^n.For b \\le \\ell -2 (and every a), every point in the original domain participates inall matchings in \\hat{\\mathbf {H}}.So, each of these matchings is adequate.", "Since every maximal matchingof the violation graph has size at least \\varepsilon _f k^n/2, by [thm:adeq]{Theorem\\,\\ref *{thm:adeq}},the number of violating pairs in \\hat{\\mathbf {H}} is at least \\varepsilon _f k^n/2.The matching \\mathbf {H} is exactly the set of pairs of \\hat{\\mathbf {H}}completely contained in the original domain.", "All violating pairs in \\hat{\\mathbf {H}}are contained in \\mathbf {H}.", "The total size of \\mathbf {H} is at most nk^n(\\lceil \\log k \\rceil + 1).The proof is completed by dividing \\varepsilon _f k^n/4 by the size of \\mathbf {H}.\\end{proof}Henceforth, we will assume that $ k = 2$ and thatall matchings $ H0a,b, H1a,b$ are adequate (for $ b c$, where$ 2c$ is an upper bound on the coordinate difference for any violation).\\subsection {The potential \\Phi } Define $ (a)$ of a non-negative integer $ a$ to be the largest power of $ 2$ which divides $ a$.", "That is, $ (a) = p$ implies $ 2pa$ but $ 2p+1a$.", "We define $ (0) := +1$.", "For any $ x Zn$, define $ (x) = c = 1n (|x[c]|)$.Now given a matching $ M$, define the following potential.\\begin{equation}\\Phi (\\mathbf {M}) := \\sum _{(x,y)\\in \\mathbf {M}} \\Phi (x-y) = \\sum _{(x,y) \\in \\mathbf {M}} \\sum _{c=1}^n (|y[c] - x[c]|).\\end{equation}We will choose maximum weighted matchings that also maximize $ (M)$.To give some intuition for the potential, note that it is aligned towards picking pairs which differ in as few coordinates as possible (since $ (0)$ is large).", "Furthermore, divisibility by powers of $ 2$ is favored.\\medskip \\section {Monotonicity on Hypergrids}In this section, we prove [thm:mono-hg]{Theorem\\,\\ref *{thm:mono-hg}}.As in the hypercube case, the weight of a pair $ (x,y)$ is defined to be $ f(x)-f(y)$ if $ xy$, and $ -$ otherwise.We set $ M$ to be a maximum weighted matching that maximizes $ (M)$.So $ |M| f kn/2$.Fix $ Hra,b$.It is instructive to explicitly see the pairs in $ stHra,b(M)$ and $ crHra,b(M)$.Consider a pair $ (x,y)$, $ x y$ in these sets.\\begin{asparaitem}\\item st_{H^r_{a,b}}(\\mathbf {M}): x[a], y[a] {\\ (\\operatorname{mod} 2^{b+1})} \\le 2^b, or x[a], y[a] {\\ (\\operatorname{mod} 2^{b+1})} > 2^b.\\item cr_{H^r_{a,b}}(\\mathbf {M}): (|y[a] - x[a]|) = b, x \\in L(H^r_{a,b}) (thus y \\in U(H^r_{a,b})).\\end{asparaitem}Now we do have skew pairs, and the potential $$ was designed specifically to handle such pairs.Note that every pair in $ M$ belongs to some $ crHra,b(M)$.There exists some $ a,b$ such that $ (|y[a] - x[a]|) = b$.", "If $ x[a] (mod 2b+1) 2b$,then $ (x,y) crH0a,b(M)$, otherwise $ (x,y) crH1a,b(M)$.Therefore, the following lemma directly implies [thm:adeq]{Theorem\\,\\ref *{thm:adeq}}.\\begin{lemma}For all r,a,b, the number of violated H^r_{a,b}-pairs is at least |cr_{H^r_{a,b}}(\\mathbf {M})|/2.\\end{lemma}\\begin{proof} We assume that H^r_{a,b} is adequate.Feed in H^r_{a,b} and \\mathbf {M} to the alternating paths machinery,with X as the set of lower endpoints in cr_{H^r_{a,b}}(\\mathbf {M}) \\setminus H^r_{a,b}.By [lem:last]{Lemma\\,\\ref *{lem:last}}, if a sequence {\\bf S}_x does not contain a violating H^r_{a,b}-pair,then the last term s_j must belong to sk_{H^r_{a,b}}(\\mathbf {M}).By [lem:skew]{Lemma\\,\\ref *{lem:skew}}, (|s_j[a] - \\mathbf {M}(s_j)[a]|) > b.But then both s_j and \\mathbf {M}(s_j) belongto L(H^r_{a,b}) or U(H^r_{a,b}), implying (s_j, \\mathbf {M}(s_j)) \\in st_{H}(\\mathbf {M}).", "Contradiction.Every sequence {\\bf S}_x contains a violating H^r_{a,b}-pair, and the calculationin [lem:mono-hc]{Lemma\\,\\ref *{lem:mono-hc}} completes the proof.\\end{proof}The main technical work is in the proof of [lem:skew]{Lemma\\,\\ref *{lem:skew}}.Fix $ a,b,r$.", "For convenience, we lose all superscripts and subscripts.\\begin{lemma} Suppose {\\bf S}_x contains no violated H-pair.", "Let the last term be s_j (j is odd).Then (|s_j[a] - \\mathbf {M}(s_j)[a]|) > b.\\end{lemma}\\begin{proof} For convenience, we denote s_{j+1} = \\mathbf {M}(s_j).We prove by contradiction, so (|s_{j}[a] - s_{j+1}[a]|) \\le b.By [lem:last]{Lemma\\,\\ref *{lem:last}}, for all even i \\le j+1, condition (\\ref {eq:**}) holds ands_j belongs to an H-skew pair.", "We will rewire \\mathbf {M} to \\mathbf {M}^{\\prime } such that weight remains the samebut the potential increases.", "We will remove the set E_-(j+1) from \\mathbf {M} and add the set\\hat{E} = E_+(j-1) \\cup (s_{j-1},s_{j+1}).Observe that both E_-(j+1) and \\hat{E} involve all terms in s_{-1},\\ldots ,s_{j+1}.We will assume that j \\equiv 1 \\ (\\operatorname{mod} 4) (the other case is analogous and omitted).By (\\ref {eq:**}), w(E_-(j+1)) = [f(s_{0}) - f(s_{-1})] + [f(s_1) - f(s_2)] + [f(s_4) - f(s_3)] + \\cdots + [f(s_{j-1}) - f(s_{j-2})] + [f(s_{j}) - f(s_{j+1})]Now for w(\\hat{E}), all pairs other than (s_{j-1},s_{j+1}) have their order decided by [prop:ek]{Prop.~\\ref *{prop:ek}}.By (\\ref {eq:**}) for j-1 and [cor:sub]{Corollary~\\ref *{cor:sub}} for j+1, s_{j-1} \\prec s_j \\prec s_{j+1}.", "w(\\hat{E}) = [f(s_{1}) - f(s_{-1})] + [f(s_0) - f(s_3)] + [f(s_5) - f(s_2)] + \\cdots + [f(s_{j}) - f(s_{j-3})] + [f(s_{j-1}) - f(s_{j+1})]We get w(E_-(j+1)) = w(\\hat{E}), so the weight stays the same.It remains the argue that the potential has increased, as argued in [clm:pot]{Claim~\\ref *{clm:pot}}\\end{proof}\\begin{claim} Suppose (|s_{j}[a] - s_{j+1}[a]|) \\le b.", "Then \\Phi (\\hat{E}) > \\Phi (E_-(j+1)).\\end{claim}\\begin{proof} Consider (s_{j^{\\prime }},s_{j^{\\prime }+1}) for odd -1 < j^{\\prime } < j.Both these terms are either in L(H) or U(H).", "Hence, \\Phi (s_{j^{\\prime }} - s_{j^{\\prime }+1}) = \\Phi (H(s_{j^{\\prime }}) - H(s_{j^{\\prime }+1}))= \\Phi (s_{j^{\\prime }-1} - s_{j^{\\prime }+2}).So most quantities in \\Phi (E_-(j+1)) and \\Phi (\\hat{E}) areidentical.", "\\Phi (\\hat{E}) - \\Phi (E_-(j+1)) = \\Phi (s_{-1} - s_1) + \\Phi (s_{j+1} - s_{j-1}) - [\\Phi (s_{-1} - s_0) + \\Phi (s_{j} - s_{j+1})] Since s_1 = H(s_0), the points s_{-1} - s_1 and s_{-1} - s_0 only differ in the ath coordinate.A similar argument works for the remaining terms.Using |\\cdot |_a to denote the absolute value of the ath coordinate, \\Phi (\\hat{E}) - \\Phi (E_-(j+1)) = (|s_{-1} - s_1|_a) + (|s_{j+1} - s_{j-1}|_a) - [(|s_{-1} - s_0|_a) + (|s_{j} - s_{j+1})|_a] Note that (|s_{-1} - s_0|_a) = b, by definition, since it lies in cr_{H^0_{a,b}}(\\mathbf {M}).Furthermore |s_{-1} - s_1|_a = |s_{-1} - H(s_0)|_a = |s_{-1} - s_0|_a - 2^b, so (|s_{-1} - s_{1}|_a) > b.", "(Note the strict inequality.", ")It suffices to show that (|s_{j+1} - s_{j-1}|_a) \\ge (|s_{j} - s_{j+1}|_a).Because s_{j-1} = H(s_j), |s_{j+1} - s_{j-1}|_a is either |2^b + |s_{j} - s_{j+1}|_a| or |2^b - |s_{j} - s_{j+1}|_a|.In either case, the assumption (|s_{j} - s_{j+1}|_a) \\le b implies (|s_{j+1} - s_{j-1}|_a) \\ge (|s_{j} - s_{j+1}|_a).\\end{proof}\\section {A pseudo-distance for (\\alpha ,\\beta )-Lipschitz}A key concept that unifies Lipschitz and monotonicity is apseudo-distance defined on $ D$.The challenge faced in the final proof is tweezing out allthe places in the previous argument where the distance function is ``hidden\".We define a weighted directed graph $ G= (D,E)$ where $ D$ is the hypergrid $ [k]n$.$ E$ contains directed edges of the form $ (x,y)$, where $ x-y1 = 1$.The length of edge $ (x,y)$ is gives as follows.", "If $ x y$, the length is $ -$.If $ x y$, the length is $$.\\begin{definition} The function {\\sf d}(x,y) between x,y \\in \\mathbf {D}is the shortest path length from x to y in {\\mathsf {G}}.\\end{definition}This function is asymmetric, meaning that $d(x,y)$ and $d(y,x)$ arepossibly different.", "Furthermore, $d(x,y)$ can be negative, so this isnot a distance in the usual parlance of metrics.Nonetheless, $d(x,y)$ has many useful properties, which can be proven byexpressing it in a more convenient form.Given any $ x,yD$, we define $ hcd(x,y)$ to be the $ zD$ maximizing $ ||z||1$ such that $ xz$ and $ yz$.Note that if $ xy$ then $ hcd(x,y) = y$.\\begin{claim} For any x,y\\in \\mathbf {D}, {\\sf d}(x,y) = \\beta {||x - \\mathsf {hcd}(x,y)||_1} - \\alpha {||y - \\mathsf {hcd}(x,y)||_1}.\\end{claim}\\begin{proof} Let us partition the coordinate set [n] = A \\sqcup B \\sqcup Cwith the following property.", "For all i \\in A, x_i > y_i.For all i \\in B, x_i < y_i, and for all i \\in C, x_i = y_i.Any path in {\\mathsf {G}} can be thought of as sequence of coordinate incrementsand decrements.Any path from x to y must finally decrement all coordinates in A,increment all coordinates in B, and preserve coordinates in C.Furthermore, increments add -\\alpha to the path length,and decrements add \\beta .Fix a path, and let I_i and D_i denote the number of increments and decrementsin dimension i.", "For i \\in A, D_i = I_i + |x_i-y_i|, for i \\in B, I_i = D_i + |x_i-y_i|,and for i \\in C, I_i = D_i.", "The path length is given by\\begin{eqnarray*} & & \\sum _{i \\in A} (\\beta D_i - \\alpha I_i) + \\sum _{i \\in B} (\\beta D_i - \\alpha I_i)+ \\sum _{i \\in C} (\\beta D_i - \\alpha I_i) \\\\& = & \\sum _{i \\in A} [\\beta |x_i-y_i| + I_i(\\beta -\\alpha )]+ \\sum _{i \\in B} [-\\alpha |x_i-y_i| + D_i(\\beta -\\alpha )] + \\sum _{i \\in C} I_i(\\beta -\\alpha )\\\\& \\ge & \\beta \\sum _{i\\in A} (x_i - y_i) - \\alpha \\sum _{i\\in B} (y_i - x_i)\\end{eqnarray*}For the inequality, we use \\beta \\ge \\alpha .Let z = \\mathsf {hcd}(x,y).", "Note that z_i = \\min (x_i,y_i).Consider the path from x that only decrements to reach z, and then only increments to reach y.The length of this path is exactly \\beta \\sum _{i\\in A}(x_i-y_i) - \\alpha \\sum _{i\\in B}(y_i-x_i).\\end{proof}It is instructive see the distance for monotonicity and Lipschitz.", "In the case of monotonicity (when $ =0, =$),$d(x,y) = 0$ if $ xy$ and $d(x,y) = $ otherwise.", "In the case of Lipschitz, $d(x,y) = ||x-y||1$.\\smallskip The next two claims establish some properties of the pseudo-distance.\\begin{claim}\\begin{asparaitem}\\item (Triangle equality) Fix x,y.", "Suppose z has the property that for all coordinates a, z[a] lies in [x[a],y[a]] or [y[a],x[a]] (whicheveris valid).", "Then, {\\sf d}(x,y) = {\\sf d}(x,z) + {\\sf d}(z,y).\\item (Triangle inequality) {\\sf d}(x,y) \\le {\\sf d}(x,z) + {\\sf d}(z,y).\\item (Projection)Let v be a vector with a single non-zero coordinate.Let x^{\\prime } = x + v and y^{\\prime } = y + v. Then {\\sf d}(x,y) = {\\sf d}(x^{\\prime },y^{\\prime }).\\item (Positivity) Consider a ``cycle\" of distinct points x_1, x_2, \\ldots , x_s, x_{s+1} = x_1Then \\sum _{c=1}^s {\\sf d}(x_c, x_{c+1}) > 0.\\end{asparaitem}\\end{claim}\\begin{proof}The triangle equality property follows from [clm:d]{Claim~\\ref *{clm:d}}.", "Suppose x \\succ z \\succ y.We have \\mathsf {hcd}(x,y) = y, \\mathsf {hcd}(x,z) = z, and \\mathsf {hcd}(y,z) = y.Hence, {\\sf d}(x,y) = \\beta {||x - y||_1} = \\beta ({||x - z||_1} + {||z - y||_1})= {\\sf d}(x,z) + {\\sf d}(z,y).", "The other case is analogous.The triangle inequality follows because {\\sf d}(x,y) is a shortest path length.For the projection property, let z = \\mathsf {hcd}(x,y) and let z^{\\prime } = \\mathsf {hcd}(x^{\\prime },y^{\\prime }).", "Note that z and z^{\\prime } also differ only in (say) the ath coordinate by the same amount v_a.", "Thus, {||x - z||_1} = {||x^{\\prime } - z^{\\prime }||_1} and {||y - z||_1} = {||y^{\\prime } - z^{\\prime }||_1}, implying {\\sf d}(x,y) = {\\sf d}(x^{\\prime },y^{\\prime }).For positivity, note that {\\sf d}(s_c,s_{c+1}) is the length of a path in {\\mathsf {G}}.So \\sum _{c=1}^s {\\sf d}(x_c, x_{c+1}) is length of a non-trivial cycle in {\\mathsf {G}}.Each coordinate increment adds -\\alpha to the length, and a decrement adds \\beta .The number of increments and decrements are the same, so the length is a strictly positive multiple of \\beta - \\alpha , a strictly positivequantity.\\end{proof}The following lemma connects the distance to the $ (,)$-Lipschitz property.\\begin{lemma}A function is (\\alpha ,\\beta )-Lipschitz iff for all x,y\\in {\\mathbf {D}}, f(x) - f(y) - {\\sf d}(x,y) \\le 0.\\end{lemma}\\begin{proof} Suppose the function satisfied the inequality for all x,y.If x and y differ in one-coordinate by 1 with x\\prec y, we get f(y) - f(x) \\le \\beta = {\\sf d}(y,x) and f(y) - f(x) \\ge \\alpha = -{\\sf d}(x,y) implying f is (\\alpha ,\\beta )-Lipschitz.", "Conversely, suppose f is (\\alpha ,\\beta )-Lipschitz.Setting z=\\mathsf {hcd}(x,y),f(x) - f(z) \\le \\beta {||x - z||_1} and \\alpha {||y - z||_1} \\le f(y) - f(z).", "Summing these,f(x) - f(y) \\le \\beta {||x - z||_1} - \\alpha {||y - z||_1} = {\\sf d}(x,y).\\end{proof}We give a simple, but important fact about distances related to the function values.\\begin{claim} \\min (f(x) - f(y) - {\\sf d}(x,y), f(y) - f(x) - {\\sf d}(y,x)) < 0.\\end{claim}\\begin{proof} Suppose not.", "Then f(x) - f(y) - {\\sf d}(x,y) + f(y) - f(x) - {\\sf d}(y,x) \\ge 0,implying {\\sf d}(x,y) + {\\sf d}(y,x) \\le 0.", "This violates the positivity of [clm:d-prop]{Claim~\\ref *{clm:d-prop}}.\\end{proof}The next lemma is a generalization of [thm:vc]{Theorem\\,\\ref *{thm:vc}}, which argued that the size of a minimum vertex coveris exactly $ f |D|$.We crucially use the triangleinequality for $d(x,y)$.We define an undirected weighted clique on $ D$.Given a function $ f$, we define the weight $ w(x,y)$ (for any $ x,y D$) as follows:\\begin{equation}w(x,y) ~:= ~~ \\max \\Big (f(x) - f(y) - {\\sf d}(x,y), ~~f(y) - f(x) - {\\sf d}(y,x)\\Big ) \\end{equation}Note that although the distance $d$ is asymmetric, the weight is symmetric.", "[lem:l]{Lemma\\,\\ref *{lem:l}} shows that a function is $ (,)$-Lipschitz iff all $ w(x,y)0$.Once again, consider the special cases of monotonicity and Lipschitz.", "For monotonicity,$ w(x,y) = f(x) - f(y)$ when $ xy$ and $ -$ otherwise.", "For Lipschitz,$ w(x,y) = |f(x) - f(y)| - ||x - y||1$.We define the unweighted {\\em violation graph} as$ VGf = (D,E)$ where $ E = {(x,y): w(x,y) > 0}$.The following lemma generalizes [thm:vc]{Theorem\\,\\ref *{thm:vc}} from \\cite {FLNRRS02}.\\begin{lemma} The size of a minimum vertex cover in VG_fis exactly \\varepsilon _f|\\mathbf {D}|.\\end{lemma}\\begin{proof} Let U be a minimum vertex cover in VG_f.Since each edge in VG_f is a violation, the points at which the function is modified must intersect all edges, and therefore should form a vertex cover.Thus, \\varepsilon _f |\\mathbf {D}| \\ge |U|.", "We show how to modify the function values at U to get a function f^{\\prime } with no violations.We invoke the following claim with V = \\mathbf {D}- U, and f^{\\prime }(x) = f(x), \\forall x \\in V.This gives a function f^{\\prime } such that \\Delta (f,f^{\\prime }) = |U|/|\\mathbf {D}|.", "By [lem:l]{Lemma\\,\\ref *{lem:l}}, f^{\\prime }is (\\alpha ,\\beta )-Lipschitz, and |U| \\ge \\varepsilon _f |\\mathbf {D}|.", "Hence, |U| = \\varepsilon _f|\\mathbf {D}|.\\begin{claim} Consider partial function f^{\\prime } defined on a subset V \\subseteq \\mathbf {D},such that for all \\forall x,y \\in V, f^{\\prime }(x) - f^{\\prime }(y) \\le {\\sf d}(x,y).It is possible to fill in the remaining values such that \\forall x,y \\in {\\mathbf {D}}, f^{\\prime }(x) - f^{\\prime }(y) \\le {\\sf d}(x,y).\\end{claim}\\begin{proof} We prove by backwards induction on the size of V. If |V| = |\\mathbf {D}|, this is triviallytrue.", "Now for the induction step.", "It suffices define f^{\\prime } for some u \\notin V.We need to define f^{\\prime }(u) so thatf^{\\prime }(u) - f^{\\prime }(y)\\le {\\sf d}(u,y) and f^{\\prime }(x) - f^{\\prime }(u) \\le {\\sf d}(x,u) for all x,y\\in V. It sufficesto argue that m := \\max _{x\\in V} \\left(f^{\\prime }(x) - {\\sf d}(x,u)\\right) \\ \\le \\ \\min _{y\\in V} \\left(f^{\\prime }(y) + {\\sf d}(u,y)\\right) =: MSuppose not, so for some x, y \\in V, f^{\\prime }(x) - {\\sf d}(x,u) > f^{\\prime }(y) + {\\sf d}(u,y).That implies that f^{\\prime }(x) - f^{\\prime }(y) > {\\sf d}(x,u) + {\\sf d}(u,y) \\ge {\\sf d}(x,y) (using triangle inequality).Contradiction, so m \\le M.\\end{proof}\\end{proof}The following is a simple corollary of the previous lemma.\\begin{corollary}The size of any maximal matching in VG_f is at least \\frac{1}{2}\\varepsilon _f|\\mathbf {D}|.\\end{corollary}By a perturbation argument, we can assume that $ w(x,y)$ is never exactly zero.This justifies the strict inequalities used in the monotonicity proofs.\\begin{claim} For any function f, there exists a function f^{\\prime }with the following properties.", "Both f and f^{\\prime } have the same set of violatedpairs, \\varepsilon _f = \\varepsilon _{f^{\\prime }}, and for all x,y \\in {\\mathbf {D}}, w_{f^{\\prime }}(x,y) \\ne 0.\\end{claim}\\begin{proof} We will construct a function f^{\\prime } such that w_{f^{\\prime }}(x,y)has the same sign as w_f(x,y).", "When w_f(x,y) = 0, then w_{f^{\\prime }}(x,y) < 0.Since exactly the same pairs have a strictly positive weight, their violationgraphs are identical.", "By [lem:vc]{Lemma\\,\\ref *{lem:vc}}, \\varepsilon _f = \\varepsilon _{f^{\\prime }}.Construct the following digraph T on \\mathbf {D}.", "For every x,y such that f(x) - f(y) - {\\sf d}(x,y) = 0, puta directed edge from y to x.", "Suppose there is a cycle x_1, x_2, \\ldots , x_s, x_{s+1} = x_1in this digraph.", "Then \\sum _{c=1}^s [f(s_c) - f(s_{c+1}) - {\\sf d}(s_c, s_{c+1})]= -\\sum _{c=1}^s {\\sf d}(s_c, s_{c+1}) = 0.", "This violates the positivity of [clm:d-prop]{Claim~\\ref *{clm:d-prop}},so T is a DAG.Pick a sink s. For any x, f(x) - f(s) - {\\sf d}(x,s) is non-zero.", "Infinitesimallydecrease f(s) (call the new function f^{\\prime }).", "For all x, w_{f^{\\prime }}(x,s) has the samesign as w_f(x,s) and is strictly negative if w_f(x,s) = 0.By iterating in this manner, we generate the desired function f^{\\prime }.", "\\end{proof}\\section {Generalized Lipschitz Testing on Hypergrids}In this section, we prove [thm:main]{Theorem\\,\\ref *{thm:main}}.", "With the distance $d(x,y)$ in place, the basic spirit of the monotonicity proofs canbe carried over.", "The final proof requires manipulations of the distance function.We do not explicitly have the ``directed\" behavior of monotonicity that allows for many of rewiringarguments.The matching $ H$ is the same as in [sec:setup]{§\\ref *{sec:setup}}.", "The generalized Lipschitz tester picks a pair $ (x,y)H$ at random.We choose $ M$ to be the maximum weight matching that also maximizes $ (M)$ (as defined by (\\ref {eq:phi})).We again set up the alternating paths as in [sec:altpaths]{§\\ref *{sec:altpaths}}, by fixing some matching $ Hra,b$ and takingalternating paths with $ stHra,b(M)$.We have a minor change that aids in some case analysis.By [clm:min]{Claim~\\ref *{clm:min}}, either $ f(x) - f(y) > d(x,y)$ or $ f(y) - f(x) > d(y,x)$, but not both.We will show that it suffices to consider only one of these cases.To that effect, define the set $ X$ as follows.$$ X = \\lbrace x | (x,y) \\in cr_{H^r_{a,b}}(\\mathbf {M}) \\setminus H^r_{a,b}, x \\in L(H^r_{a,b}), f(x) - f(y) > {\\sf d}(x,y)\\rbrace $$(For monotonicity, the last condition is redundant.)", "As before, the main lemma is the following.\\begin{lemma} For all x \\in X, {\\bf S}_x contains aviolated H^r_{a,b}-pair.\\end{lemma}We apply some symmetry arguments to show the next lemma, which proves [thm:adeq]{Theorem\\,\\ref *{thm:adeq}}.", "For convenience, we drop the sub/superscriptsin $ Hra,b$.", "(Note that we do not lose the $ 2$ factor here, as compared to [lem:mono-hc]{Lemma\\,\\ref *{lem:mono-hc}}.", ")\\begin{lemma} The number of violations in His at least cr_{H}(\\mathbf {M}).\\end{lemma}\\begin{proof} We can classify the endpoints of cr_{H}(\\mathbf {M}) \\setminus H^r_{a,b} intothe following sets.", "Consider a generic (x,y) \\in cr_{H}(\\mathbf {M}) where x \\in L(H).If f(x) - f(y) > {\\sf d}(x,y), we put x in X and y in Y.", "Otherwise, f(y) - f(x) > {\\sf d}(y,x),and we put x in X^{\\prime } and y in Y^{\\prime }.By [lem:sx-gen]{Lemma\\,\\ref *{lem:sx-gen}}, for x \\in X, {\\bf S}_x has a violated H-pair.", "Consider x^{\\prime } \\in X^{\\prime }.Take the function \\hat{f} = -f and the (-\\beta ,-\\alpha )-Lipschitz property.By [clm:d]{Claim~\\ref *{clm:d}}, the new distance satisfies \\hat{{\\sf d}}(u,v) = {\\sf d}(v,u).", "If f(u) - f(v) > {\\sf d}(u,v),then \\hat{f}(v) - \\hat{f}(u) > \\hat{{\\sf d}}(v,u) (and vice versa).", "Hence, the violation graphs, the weights, \\mathbf {M}, and thealternating paths are identical.Take x^{\\prime } \\in X^{\\prime }, so it belongs to some (x^{\\prime },y^{\\prime }) \\in cr_{H}(\\mathbf {M}).We have \\hat{f}(x) - \\hat{f}(y) > \\hat{{\\sf d}}(x,y).", "Applying [lem:sx-gen]{Lemma\\,\\ref *{lem:sx-gen}} to \\hat{f} for the (-\\beta ,-\\alpha )-Lipschitz property,{\\bf S}_{x^{\\prime }} has a violated H-pair.", "All in all, for any x \\in X \\cup X^{\\prime }, {\\bf S}_x containsa violated H-pair.To deal with Y \\cup Y^{\\prime }, we will first reverse the entire domain, by switching the directionof all edges in the hypergrid.", "(Represent this transformation by \\Psi :[k]^n \\rightarrow [k]^n, and notethat \\Psi ^{-1} = \\Psi .)", "By the shortest path definition of {\\sf d},the new distance satisfies \\hat{{\\sf d}}(u,v) = {\\sf d}(\\Psi (v),\\Psi (u)).", "Hence,we are looking at the (-\\beta , -\\alpha )-Lipschitz property.The matching H remains the same, but the identities of L(H) and U(H)have switched.", "Construct function \\hat{f}(x) = -f(\\Psi (x)).", "If in the original domain f(u) - f(v) > {\\sf d}(u,v),then \\hat{f}(\\Psi (v)) - \\hat{f}(\\Psi (u)) > \\hat{{\\sf d}}(\\Psi (v),\\Psi (u)) (and vice versa).", "Again, the alternating pathstructure is identical.", "Consider in the original domain (x,y) \\in cr_{H}(\\mathbf {M}) where x \\in L(H).In the new domain, \\Psi (y) \\in L(H).", "Hence, we can apply the conclusion of the previous paragraphfor all points in y \\in \\Psi (Y \\cup Y^{\\prime }), and deduce that {\\bf S}_y contains a violated H-pair.Finally, we conclude that every alternating path with an endpoint of cr_{H}(\\mathbf {M}) \\setminus H^r_{a,b}contains a violated pair.", "There are at least |cr_{H}(\\mathbf {M}) \\setminus H^r_{a,b}| such (disjoint) alternating paths.\\end{proof}\\subsection {Preliminary setup} All the propositions of [sec:altpaths]{§\\ref *{sec:altpaths}} hold, since they were independent of the property at hand.We start by generalizing the monotonicity-specific setup done in [sec:struct]{§\\ref *{sec:struct}}.", "We fix somematching $ Hra,b$, and drop all super/subscripts for ease of notation.\\begin{proposition} Consider the pairs in E_-(i) and E^+(i).", "For alleven 0 \\le j \\le i-2, {\\sf d}(s_j,s_{j+3}) = {\\sf d}(s_{j+1},s_{j+2}) and {\\sf d}(s_{j+3},s_j) = {\\sf d}(s_{j+2},s_{j+1}).\\end{proposition}\\begin{proof} By [prop:sub]{Prop.~\\ref *{prop:sub}}, s_j and s_{j+3} both lie in L(H) or U(H).", "Hence, s_{j+1} = H(s_j)and s_{j+2} = H(s_{j+3}) are both obtained by adding or subtracting 2^b from the ath coordinate.By the projection property, {\\sf d}(s_j,s_{j+3}) = {\\sf d}(s_{j+1},s_{j+2}) and {\\sf d}(s_{j+3},s_j) = {\\sf d}(s_{j+2},s_{j+1}).\\end{proof}Our aim is to generalize the conditions (\\ref {eq:*}) and (\\ref {eq:**}).", "The former condition is obtainedby assuming that $ (si,si+1)$ is not a violation.", "For monotonicity, this implies a single inequality,but here, there are two inequalities.", "It turns out that because we are in the setting where$ w(x,y) = f(x) - f(y) - d(x,y) > 0$, only one of these is necessary.", "Foreven $ i$, if $ (si,si+1)$ is not a violation, [cor:sub]{Corollary~\\ref *{cor:sub}} implies{\\begin{@align}{1}{-1}\\textrm {If} \\ i\\equiv 0\\ (\\operatorname{mod} 4), f(s_{i+1}) - f(s_{i}) > \\alpha 2^b.\\nonumber \\\\\\textrm {If} \\ i\\equiv 2\\ (\\operatorname{mod} 4), f(s_{i}) - f(s_{i+1}) > \\alpha 2^b.", "\\end{@align}}Nowe we generalize (\\ref {eq:**}).", "The pair \\qquad \\mathrm {(\\circ )}$ (si-1,si)$ is a violation, but we do notknow whether $ w(si-1,si)$ is $ f(si-1) - f(si) - d(si-1,si)$ or $ f(si) - f(si-1) - d(si,si-1)$.The following is the equivalentof the ordering condition of (\\ref {eq:**}).", "{\\begin{@align}{1}{-1}\\textrm {If} \\ i\\equiv 0\\ (\\operatorname{mod} 4), f(s_{i}) - f(s_{i-1}) > {\\sf d}(s_i,s_{i-1}).\\nonumber \\\\\\textrm {If} \\ i\\equiv 2\\ (\\operatorname{mod} 4), f(s_{i-1}) - f(s_{i}) > {\\sf d}(s_{i-1},s_i).", "\\end{@align}}\\subsection {The structure lemmas} This lemma is the direct analogue of [lem:cons-mon]{Lemma\\,\\ref *{lem:cons-mon}}.", "The proof is also along similar lines.\\begin{lemma}Consider some even index \\qquad \\mathrm {(\\circ \\circ )}i such that s_i exists.Suppose conditions (\\ref {eq:circ}) and (\\ref {eq:2-circ}) held for all even indices \\le i.Then, s_{i+1} is \\mathbf {M}-matched.\\end{lemma}\\begin{proof} The proof is by contradiction.Assume i\\equiv 0\\ (\\operatorname{mod} 4).", "(The proof for the case i\\equiv 2\\ (\\operatorname{mod} 4) is similar and omitted.", ")As in the proof of [lem:cons-mon]{Lemma\\,\\ref *{lem:cons-mon}}, we argue that w(\\mathbf {M}^{\\prime }) > w(\\mathbf {M}),where \\mathbf {M}^{\\prime } = \\mathbf {M}- E_-(i) + E_+(i).By condition (\\ref {eq:**}),\\begin{eqnarray}w(E_-(i)) & = & [f(s_{0}) - f(s_{-1}) - {\\sf d}(s_0, s_{-1})] + [f(s_1) - f(s_2) - {\\sf d}(s_1, s_2)] \\nonumber \\\\& & + [f(s_4) - f(s_3) - {\\sf d}(s_4,s_3)] + [f(s_5) - f(s_6) - {\\sf d}(s_5,s_6)] + \\cdots \\nonumber \\\\& & + [f(s_{i-3}) - f(s_{i-2}) - {\\sf d}(s_{i-3},s_{i-2})] + [f(s_{i}) - f(s_{i-1}) - {\\sf d}(s_i,s_{i-1})] \\end{eqnarray}For w(E_+(i)), it suffices to find a lower bound.", "Since (for any u,v \\in \\mathbf {D}) w(u,v) is the maximumof two expressions, we can choose the expression to match w(E_-(i)) as much as possible.For a pair (s_k,s_{k+3}) in E_+(i), we bound the weight byf(s_k) - f(s_{k+3}) - {\\sf d}(s_k, s_{k+3}) if j \\equiv 0 \\ (\\operatorname{mod} 4) and by f(s_{k+3}) - f(s_k) - {\\sf d}(s_{k+3},s_k) if j \\equiv 2 \\ (\\operatorname{mod} 4).This ensure that the coefficients of f(\\cdot ) are identical to those in [eq:w-lip]{(\\ref *{eq:w-lip})}.\\begin{eqnarray}w(E_+(i)) & \\ge & [f(s_{1}) - f(s_{-1}) - {\\sf d}(s_1, s_{-1})] + [f(s_0) - f(s_3) - {\\sf d}(s_0, s_3)] \\nonumber \\\\& & + [f(s_5) - f(s_2) - {\\sf d}(s_5,s_2)] + [f(s_4) - f(s_7) - {\\sf d}(s_4,s_7)] + \\cdots \\nonumber \\\\& & + [f(s_{i-4}) - f(s_{i-1}) - {\\sf d}(s_{i-4},s_{i-1})] + [f(s_{i+1}) - f(s_{i-2}) - {\\sf d}(s_{i+1},s_{i-2})] \\end{eqnarray}Note that only w(E_+(i)) involves f(s_{i+1}) and only w(E_-(i)) involves f(s_i), but allother f(\\cdot ) terms have identical coefficients.To deal with the difference of the distances, we use [prop:lip-comp]{Prop.~\\ref *{prop:lip-comp}}.", "All the distance termsin [eq:w+lip]{(\\ref *{eq:w+lip})} except for the first cancel out with an equivalent term in [eq:w-lip]{(\\ref *{eq:w-lip})}.\\begin{eqnarray*}w(E_+(i)) - w(E_-(i)) & \\ge & f(s_{i+1}) - f(s_i) - {\\sf d}(s_1,s_{-1}) + {\\sf d}(s_0, s_{-1})\\end{eqnarray*}Since (s_0,s_{-1}) is a cross pair and s_1 = H(s_0), we can use triangle equality to deduce that{\\sf d}(s_0, s_{-1}) - {\\sf d}(s_1,s_{-1}) = {\\sf d}(s_0,s_1) = - \\alpha 2^b.Combining, w(E_+(i)) - w(E_-(i)) \\ge f(s_{i+1}) - f(s_i) - \\alpha 2^b.", "By condition (\\ref {eq:circ})for i, the RHS is strictly positive.", "Contradiction.\\end{proof}Now for analogue of [lem:cons-mon2]{Lemma\\,\\ref *{lem:cons-mon2}} and [clm:rep]{Claim~\\ref *{clm:rep}}.", "We will prove the latter first.\\begin{lemma}Consider some even index i such that s_i exists.Suppose conditions (\\ref {eq:circ}) and (\\ref {eq:2-circ}) held for all even indices \\le i.Then, condition (\\ref {eq:2-circ}) holds for i+2.\\end{lemma}\\begin{claim} Let j be the last index of {\\bf S}_x.", "Supposeconditions (\\ref {eq:circ}) and (\\ref {eq:2-circ}) hold for all even i < j.", "Then the sequence s_{-1}, s_0 ,s_1, \\ldots , s_j, \\mathbf {M}(s_j) are distinct.\\end{claim}\\begin{proof} By the arguments in [clm:rep]{Claim~\\ref *{clm:rep}}, it suffices to get a contradiction assuming s_j = y.Since y \\in U(H), by [prop:sub]{Prop.~\\ref *{prop:sub}}, j \\equiv 1 \\ (\\operatorname{mod} 4).Note that s_{j-1} = H(y) and (x,y) is a cross pair.", "Therefore, we have the triangleequality {\\sf d}(x,y) = {\\sf d}(x,s_{j-1}) + {\\sf d}(s_{j-1},y) = {\\sf d}(x,s_{j-1}) - \\alpha 2^b.We will replace pairs A = \\lbrace (x,y), (s_{j-1},s_{j-2})\\rbrace \\in \\mathbf {M} with (x,s_{j-2}), and argue that the weight has increased.Applying condition (\\ref {eq:2-circ}) for j-1,\\begin{eqnarray*}w(A) & = & [f(x) - f(y) - {\\sf d}(x,y)] + [f(s_{j-1}) - f(s_{j-2}) - {\\sf d}(s_{j-1},s_{j-2})]\\\\& = & f(x) - f(y) + f(s_{j-1}) - f(s_{j-2}) - {\\sf d}(x,s_{j-1}) + \\alpha 2^b - {\\sf d}(s_{j-1},s_{j-2}) \\\\& \\le & f(x) - f(y) + f(s_{j-1}) - f(s_{j-2}) -{\\sf d}(x,s_{j-2}) + \\alpha 2^b \\ \\ \\ \\ \\textrm {(triangle inequality)}\\\\& = & [f(x) - f(s_{j-2}) - {\\sf d}(x,s_{j-2})] - [f(y) - f(s_{j-1}) - \\alpha 2^b] \\\\& \\le & w(x,s_{j-2}) - [f(y) - f(s_{j-1}) - \\alpha 2^b]\\end{eqnarray*}The second term is strictly positive (by condition (\\ref {eq:circ}) for j-1 \\equiv 0 \\ (\\operatorname{mod} 4)), contradicting the maximality of \\mathbf {M}.\\end{proof}\\begin{proof} (of [lem:lip-cons2]{Lemma\\,\\ref *{lem:lip-cons2}}) Assume i\\equiv 0\\ (\\operatorname{mod} 4).", "(The proof for the case i\\equiv 2\\ (\\operatorname{mod} 4) is similar and omitted.", ")By [lem:lip-cons]{Lemma\\,\\ref *{lem:lip-cons}}, \\mathbf {M}(s_{i+1}) exists, and is denoted by s_{i+2}.The proof is by contradiction, so assume condition (\\ref {eq:2-circ}) does not hold for i+2 \\equiv 2 \\ (\\operatorname{mod} 4).This means f(s_{i+1}) - f(s_{i+2}) \\le {\\sf d}(s_{i+1},s_{i+2}).", "Since (s_{i+1},s_{i+2}) is a violation,this implies w(s_{i+1},s_{i+2}) = f(s_{i+}) - f(s_{i+1}) - {\\sf d}(s_{i+2},s_{i+1}).We set E^{\\prime } = E_+(i-2) \\cup (s_{i-2},s_{i+2}).We argue that w(\\mathbf {M}^{\\prime }) > w(\\mathbf {M}),where \\mathbf {M}^{\\prime } = \\mathbf {M}- E_-(i+2) + E^{\\prime }.", "By [prop:ek]{Prop.~\\ref *{prop:ek}} and [clm:lip-rep]{Claim~\\ref *{clm:lip-rep}}, \\mathbf {M}^{\\prime } is a valid matching.By condition (\\ref {eq:2-circ}) for even k < i+2 and the above conclusion on w(s_{i+1},s_{i+2}),we get almost the same expression as (\\ref {eq:w-lip}).\\begin{eqnarray}w(E_-(i+2)) & = & [f(s_{0}) - f(s_{-1}) - {\\sf d}(s_0, s_{-1})] + [f(s_1) - f(s_2) - {\\sf d}(s_1, s_2)] \\nonumber \\\\& & + [f(s_4) - f(s_3) - {\\sf d}(s_4,s_3)] + [f(s_5) - f(s_6) - {\\sf d}(s_5,s_6)] + \\cdots \\nonumber \\\\& & + [f(s_{i-3}) - f(s_{i-2}) - {\\sf d}(s_{i-3},s_{i-2})] + [f(s_{i}) - f(s_{i-1}) - {\\sf d}(s_i,s_{i-1})] \\nonumber \\\\& & + [f(s_{i+2}) - f(s_{i+1}) - {\\sf d}(s_{i+2},s_{i+1})] \\end{eqnarray}For w(E^{\\prime }), we follow the same pattern in (\\ref {eq:w+lip}).\\begin{eqnarray}w(E^{\\prime }) & \\ge & [f(s_{1}) - f(s_{-1}) - {\\sf d}(s_1, s_{-1})] + [f(s_0) - f(s_3) - {\\sf d}(s_0, s_3)] \\nonumber \\\\& & + [f(s_5) - f(s_2) - {\\sf d}(s_5,s_2)] + [f(s_4) - f(s_7) - {\\sf d}(s_4,s_7)] + \\cdots \\nonumber \\\\& & + [f(s_{i-3}) - f(s_{i-6}) - {\\sf d}(s_{i-3},s_{i-6})] + [f(s_{i-4}) - f(s_{i-1}) - {\\sf d}(s_{i-4},s_{i-1})] \\nonumber \\\\& & + [f(s_{i+2}) - f(s_{i-2}) - {\\sf d}(s_{i+2},s_{i-2})] \\end{eqnarray}By [prop:lip-comp]{Prop.~\\ref *{prop:lip-comp}}, all distance termsin [eq:w+lip]{(\\ref *{eq:w+lip})} barring the first and last are identical to an equivalent term in [eq:w-lip]{(\\ref *{eq:w-lip})}.\\begin{eqnarray*}w(E_+(i+2)) - w(E_-(i+2)) & \\ge & f(s_{i+1}) - f(s_i) \\\\& & - {\\sf d}(s_1,s_{-1}) - {\\sf d}(s_{i+2},s_{i-2}) + {\\sf d}(s_0, s_{-1}) + {\\sf d}(s_i, s_{i-1}) + {\\sf d}(s_{i+2},s_{i+1})\\end{eqnarray*}As in the proof of [lem:lip-cons]{Lemma\\,\\ref *{lem:lip-cons}},{\\sf d}(s_0, s_{-1}) - {\\sf d}(s_1,s_{-1}) = {\\sf d}(s_0,s_1) = - \\alpha 2^b.", "Furthermore,\\begin{eqnarray*}- {\\sf d}(s_{i+2},s_{i-2}) + {\\sf d}(s_i, s_{i-1}) + {\\sf d}(s_{i+2},s_{i+1}) & \\ge & {\\sf d}(s_i,s_{i-1}) - {\\sf d}(s_{i+1},s_{i-2}) \\ \\ \\ \\textrm {(triangle inequality)} \\\\& = & 0 \\ \\ \\ \\textrm {([prop:lip-comp]{Prop.~\\ref *{prop:lip-comp}})}\\end{eqnarray*}Combining, w(E^{\\prime }) - w(E_-(i+2)) \\ge f(s_{i+1}) - f(s_i) - \\alpha 2^b.", "This is strictly positive, by condition (\\ref {eq:circ})for i. Contradiction.\\end{proof}We proceed to the analogue of [lem:last]{Lemma\\,\\ref *{lem:last}}.", "Because of the use of distances and potentials,we require a much simpler statement.\\begin{lemma} Suppose {\\bf S}_x contains no violated H-pair.", "Let the lastterm by s_j (j is odd).", "For every even i \\le j+1, condition (\\ref {eq:2-circ}) holds.Furthermore, s_j is \\mathbf {M}\\setminus st_{H}(\\mathbf {M})-matched.\\end{lemma}\\begin{proof} The first part is identical to that of [lem:last]{Lemma\\,\\ref *{lem:last}}.Condition (\\ref {eq:2-circ}) holds for i=0, and applications of [lem:lip-cons2]{Lemma\\,\\ref *{lem:lip-cons2}}complete the proof.", "By [lem:lip-cons]{Lemma\\,\\ref *{lem:lip-cons}} s_j is \\mathbf {M}-matched, but beingthe last term cannot be st_{H}(\\mathbf {M})-matched.\\end{proof}\\subsection {The existence of a violated edge in {\\bf S}_x}We show the existence of a violated $ H$-edge in $Sx$,proving [lem:sx-gen]{Lemma\\,\\ref *{lem:sx-gen}}.Suppose $Sx$ has no violated $ H$-pair.By [lem:lip-last]{Lemma\\,\\ref *{lem:lip-last}}, $ sj$ is $ MstH(M)$-matched.By the following lemma (analogue of [lem:skew]{Lemma\\,\\ref *{lem:skew}}) asserts $ (sj[a] - sj+1[a]) > b$, implying $ sj$ is $ stH(M)$-matched.\\begin{lemma} Suppose {\\bf S}_x contains no violated H-pair.", "Let the lastterm by s_j (j is odd).", "Then (s_j[a] - s_{j+1}[a]) > b.\\end{lemma}\\begin{proof} The proof is analogous to that of [lem:skew]{Lemma\\,\\ref *{lem:skew}}.By [lem:lip-last]{Lemma\\,\\ref *{lem:lip-last}}, for all even i \\le j+1, condition (\\ref {eq:2-circ}) holds.By [clm:lip-rep]{Claim~\\ref *{clm:lip-rep}}, s_{-1}, s_0, s_1, \\ldots , s_j, \\mathbf {M}(s_j) = s_{j+1} are all distinct.We rewire \\mathbf {M} to \\mathbf {M}^{\\prime } by removing E_-(j+1) from \\mathbf {M} and adding the set \\hat{E} = E_+(j-1) \\cup (s_{j-1},s_{j+1}).We will assume that j \\equiv 1 \\ (\\operatorname{mod} 4) (the other case is analogous and omitted).By (\\ref {eq:2-circ}), we can exactly express w(E_-(j+1)).\\begin{eqnarray*}w(E_-(j+1)) & = & [f(s_{0}) - f(s_{-1}) - {\\sf d}(s_0, s_{-1})] + [f(s_1) - f(s_2) - {\\sf d}(s_1, s_2)] \\nonumber \\\\& & + [f(s_4) - f(s_3) - {\\sf d}(s_4,s_3)] + [f(s_5) - f(s_6) - {\\sf d}(s_5,s_6)] + \\cdots \\nonumber \\\\& & + [f(s_{j-1}) - f(s_{j-2}) - {\\sf d}(s_{j-1},s_{j-2})] + [f(s_{j}) - f(s_{j+1}) - {\\sf d}(s_j,s_{j+1})]\\end{eqnarray*}We get a lower bound for w(\\hat{E}) that matches the f terms exactly.\\begin{eqnarray*}w(\\hat{E}) & \\ge & [f(s_{1}) - f(s_{-1}) - {\\sf d}(s_1, s_{-1})] + [f(s_0) - f(s_3) - {\\sf d}(s_0, s_3)] \\nonumber \\\\& & + [f(s_5) - f(s_2) - {\\sf d}(s_5,s_2)] + [f(s_4) - f(s_7) - {\\sf d}(s_4,s_7)] + \\cdots \\nonumber \\\\& & + [f(s_{j}) - f(s_{j-3}) - {\\sf d}(s_{j},s_{j-3})] + [f(s_{j-1}) - f(s_{j+1}) - {\\sf d}(s_{j-1},s_{j+1})]\\end{eqnarray*}By [prop:lip-comp]{Prop.~\\ref *{prop:lip-comp}}, the distance terms {\\sf d}(s_c,s_{c+3}) and {\\sf d}(s_{c+3},s_c) can bematched to equivalent terms.", "In the following, we use the equality {\\sf d}(s_0,s_{-1}) - {\\sf d}(s_1, s_{-1}) = -\\alpha 2^b.\\begin{eqnarray*}w(\\hat{E}) - w(E_-(j+1)) & \\ge & - {\\sf d}(s_1, s_{-1}) - {\\sf d}(s_{j-1},s_{j+1}) + {\\sf d}(s_0, s_{-1}) + {\\sf d}(s_{j},s_{j+1}) \\\\& \\ge & -\\alpha 2^b - {\\sf d}(s_{j-1},s_j) \\ \\ \\ \\textrm {(triangle inequality)}\\\\& = & -\\alpha 2^b - (-\\alpha 2^b) = 0 \\ \\ \\ \\textrm {(By [prop:sub]{Prop.~\\ref *{prop:sub}}, j \\equiv 1 \\ (\\operatorname{mod} 4), so s_j \\in U(H).", ")}\\end{eqnarray*}So \\mathbf {M}^{\\prime } is also a maximum weight matching.Observe that the potential \\Phi is independent of the property at hand.", "[clm:pot]{Claim~\\ref *{clm:pot}} only uses the basicstructure of the alternating paths and is applicable here.", "It asserts thatif (s_j[a] - s_{j+1}[a]) \\le b, then \\Phi (\\mathbf {M}^{\\prime }) > \\Phi (\\mathbf {M}), contradicting the choice of \\mathbf {M}.\\end{proof}\\section {Acknowledgements}Sandia National 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[ [ "Single-atom quantum memory with degenerate atomic levels" ], [ "Abstract The storage and retrieval of a single-photon polarization q-bit by means of STIRAP through the atoms with degenerate levels is studied theoretically for arbitrary polarization of the driving laser field and arbitrary values of the angular momenta of resonant atomic levels.", "The dependence of the probability of long-term photon storage on the polarization of the driving field and on the initial atomic state is examined." ], [ "Introduction", "The search for an efficient optical quantum memory and its implementation for quantum networks is an active area (see, e.g., the reviews [1], [2], [3], [4]).", "The role of carrier of information – flying q-bit – in such devices is played by a single photon and the most natural way to encode the q-bit is provided by its two polarization degrees of freedom (see [5], [6], [7], [8] for some recent proposals).", "While in the majority of the proposed schemes the photon quantum state is mapped to the quantum state of some atomic ensemble [6], [7], [8], in the experiment [5] the photon polarization q-bit was recorded in the superposition of magnetic substates of the long-lived degenerate level of a single rubidium atom.", "Such single-atom quantum memory offers an advantage of processing of the stored quantum information.", "Since at the storage stage the state of a single atom, in which the photon q-bit is encoded, is well determined, it may be altered in a controlled way, e.g., by some laser pulse, so that the state of the retrieved photon will differ from that of the recorded one in a desirable way implementing thus the performance of quantum gates.", "The setup employed in [5] for the photon storage and retrieval was based on the same technique of vacuum stimulated Raman scattering involving adiabatic passage (STIRAP), which was previously successfully employed for deterministic single photon emission [9], [10], [11].", "In these experiments the three-level $\\Lambda $ - type atom was trapped inside the high-finesse cavity, one branch of the atomic $\\Lambda $ - type transitions was coupled to the quantized cavity field, while the other one was coupled to the driving coherent laser field.", "The STIRAP with the three-level atom with non-degenerate levels is well described in the reviews and textbooks (see, e.g., [12], [13]).", "In case of non-degenerate levels there exists the only dark state – the superposition of the ground state $a$ and some metastable state $b$ – uncoupled to the excited level $c$ (Figure 1).", "In course of Raman scattering the atom is adiabatically transferred from the initial state $b$ to the target state $a$ .", "The STIRAP with degenerate atomic levels and with classical coherent resonant fields was studied in [14], were it was shown that unlike the non-degenerate case the population from the initial state $b$ is not always totally transferred to the target level $a$ .", "The objective of the present paper is to analyze the process of storage and retrieval of a single-photon polarization q-bit by means of STIRAP through the atoms with degenerate levels for arbitrary values of the angular momenta of resonant atomic levels and to study the dependence of the probability of long-term photon storage on the polarization of the driving field and on the initial atomic state.", "In section 2 the interaction model is described and the instantaneous eigenvectors of the interaction operator, which determine the evolution operator in the adiabatic approximation, are constructed.", "In case of degenerate levels there appear the new types of these eigenvectors, non-existing in case of non-degenerate levels, like the dark states, which atomic part belongs to only one of the lower levels $a$ or $b$ , and the bright states, which couple the excited level $c$ with only one of the lower levels $a$ or $b$ .", "In section 3 the formula for the probability of long-term storage of the photon polarization q-bit is obtained and the conditions for the photon storage with unit probability independent on its polarization are outlined.", "The transitions with the angular momenta $J_{b}=J_{c}=1$ , $J_{a}=2$ , corresponding to the transitions between the hyperfine structure components of the electronic levels $5^{2}S_{1/2}$ and $5^{2}P_{1/2}$ of the $^{87}Rb$ atom, which were employed in the experiments [5], are analyzed." ], [ "Basic equations", "We consider the three-level $\\Lambda $ - type atom in a high-finesse cavity, one branch – the transition $J_{a}\\rightarrow J_{c}$ with the frequency $\\omega _{c0}$ is coupled to the driving coherent laser field, while the other – the transition $J_{b}\\rightarrow J_{c}$ with the frequency $\\omega _{0}$ is coupled to the quantized cavity field (Figure 1), $J_{a}$ , $J_{b}$ and $J_{c}$ being the values of the angular momenta of the levels.", "The electric field strength of the coherent laser field may be put down as follows: $\\textbf {E}_{c}=e_{c}(t)\\textbf {l}_{c} e^{-i\\omega _{c}t}+ c.c.,$ while the quantized field of the cavity in the interaction representation is described by the operator: $\\hat{\\textbf {E}}=e(t)(\\hat{a}_{1}\\textbf {l}_{1} +\\hat{a}_{2}\\textbf {l}_{2}) e^{-i\\omega t}+h.c.,$ where the carrier frequencies $\\omega _{c}$ and $\\omega $ are in resonance with the frequencies $\\omega _{c0}$ and $\\omega _{0}$ , $e_{c}(t)$ and $\\textbf {l}_{c}$ are the slowly varying amplitude and the unit polarization vector of the driving field, $e(t)$ is the slowly varying amplitude of the photon field, $\\textbf {l}_{1}$ and $\\textbf {l}_{2}$ are the two unit orthogonal vectors of the two polarization modes of the photon field, $\\hat{a}_{1}$ and $\\hat{a}_{2}$ are the photon annihilation operators for this modes.", "The temporal dependence $e(t)$ of the cavity field amplitude appears due to some tailored alteration of cavity parameters.", "The equation for the slowly-varying density matrix $\\hat{\\rho }$ of the system, which consists of a single three-level atom and two-mode cavity field, in the rotating-wave approximation and in case of Raman resonance $\\omega _{c0}-\\omega _{c}=\\omega _{0}-\\omega =\\Delta $ is as follows: $\\frac{d}{dt} \\hat{\\rho } =\\frac{i}{2}\\left[\\hat{V}(t),\\hat{\\rho }\\right],$ $\\hat{V}(t) = -2\\Delta \\hat{P}_{c} + \\hat{G}(t) + \\hat{G}^{\\dag }(t),$ $\\hat{G}(t) = \\Omega _{a}(t)\\hat{g}_{a} + \\Omega _{b}(t)\\hat{g}_{b},$ $\\hat{g}_{b}=\\hat{g}_{b1}\\hat{a}_{1}^{\\dag }+\\hat{g}_{b2}\\hat{a}_{2}^{\\dag }.$ Here $\\hat{P}_{c}$ is the projector on the subspace of the atomic excited level $J_{c}$ , $\\Omega _{a}(t)=2|d_{a}|e_{c}(t)/\\hbar $ and $\\Omega _{b}(t)=2|d_{b}|e(t)/\\hbar $ are the reduced Rabi frequencies for the coherent laser field and for the cavity field, $d_{a}=d(J_{a}J_{c})$ and $d_{b}=d(J_{b}J_{c})$ are the reduced matrix elements of the electric dipole moment operator for the transitions $J_{a}\\rightarrow J_{c}$ and $J_{b}\\rightarrow J_{c}$ , while $\\hat{g}_{a}=\\hat{\\textbf {g}}_{a}\\textbf {l}_{c}^{*},~\\hat{g}_{bi}=\\hat{\\textbf {g}}_{b}\\textbf {l}^{*}_{i},~i=1,2,$ $\\hat{\\textbf {g}}_{a}$ and $\\hat{\\textbf {g}}_{b}$ are the dimensionless electric dipole moment operators for the transitions $J_{a}\\rightarrow J_{c}$ and $J_{b}\\rightarrow J_{c}$ .", "The matrix elements of the circular components $\\hat{g}_{\\alpha q}$ ($\\alpha =a,b$ ; $q=0,\\pm 1$ ) of these vector operators are expressed through Wigner 3J-symbols [15]: $(\\hat{g}_{\\alpha q})_{m_{\\alpha },m_{c}}=(-1)^{J_{\\alpha }-m_{\\alpha }}\\left({J_{\\alpha }&1&J_{c} \\cr -m_{\\alpha }&q&m_{c}}\\right),$ The solution of the equation (REF ) is expressed through the evolution operator $\\hat{S}(t)$ : $\\hat{\\rho }(t)=\\hat{S}(t)\\hat{\\rho }(0)\\hat{S}^{+}(t).$ In the adiabatic approximation the evolution operator $\\hat{S}(t)$ is defined by the instantaneous eigenvectors $|v_{k}(t)>$ and eigenvalues $\\lambda _{k}(t)$ of the interaction operator $\\hat{V}(t)$ in a following way: $\\hat{S}(t)=\\sum _{k}\\exp \\lbrace i\\phi _{k}(t)\\rbrace |v_{k}(t)><v_{k}(0)|,$ $\\phi _{k}(t)=\\frac{1}{2}\\int _{0}^{t}\\lambda _{k}(t^{\\prime })dt^{\\prime }.$ Since only single photon storage and retrieval are discussed in the present paper and no decay or dephasing processes are taken into account it is sufficient to limit the system space to the subspace with the basis vectors $|J_{a}m_{a}>|0,0>$ , $|J_{b}m_{b}>|1,0>$ , $|J_{b}m_{b}>|0,1>$ and $|J_{c}m_{c}>|0,0>$ , where $|J_{a}m_{a}>$ , $|J_{b}m_{b}>$ and $|J_{c}m_{c}>$ denote the atomic Zeeman states, while $|n_{1},n_{2}>$ ($n_{1,2}=0,1$ ) are the field number states with $n_{1}$ photons in the first polarization mode and $n_{2}$ – in the second.", "This subspace with the dimension $N=2(J_{a}+2J_{b}+J_{c}+2)$ constitutes the invariant subspace of the interaction operator $\\hat{V}(t)$ , so that its matrix represents itself a square hermitian $N\\times N$ matrix.", "The states $|a>|0,0>$ and $|c>|0,0>$ , which atomic part belongs to the level $a$ or $c$ , may be represented as columns with $2J_{a}+1$ or $2J_{c}+1$ elements correspondingly, while the states $|b_{1}>|1,0>+|b_{2}>|0,1>$ , which atomic part belongs to the level $b$ , may be represented as columns with $2(2J_{b}+1)$ elements: $|b_{1}>|1,0>+|b_{2}>|0,1> = \\left({|b_{1}> \\cr |b_{2}>}\\right).$ Then the operator $\\hat{g}_{a}$ will be represented by the $(2J_{a}+1)\\times (2J_{c}+1)$ matrix, while the operator $\\hat{g}_{b}$ will be represented by the $2(2J_{b}+1)\\times (2J_{c}+1)$ matrix $\\hat{g}_{b} = \\left( {\\hat{g}_{b1} \\cr \\hat{g}_{b2}} \\right),$ were each block $\\hat{g}_{bi}$ ($i=1,2$ ) represents itself a $(2J_{b}+1)\\times (2J_{c}+1)$ matrix.", "In order to find out the instantaneous eigenvectors $|v_{k}(t)>$ and eigenvalues $\\lambda _{k}(t)$ of the interaction operator $\\hat{V}(t)$ let us start with the operator $\\hat{G}^{\\dag }(t)\\hat{G}(t) = \\Omega _{a}^{2}(t)\\hat{g}_{a}^{\\dag }\\hat{g}_{a}+ \\Omega _{b}^{2}(t)\\hat{g}_{b}^{\\dag }\\hat{g}_{b},$ which acts at the subspace of the upper atomic level $c$ (note that $\\hat{g}_{a}^{\\dag }\\hat{g}_{b}=0$ , $\\hat{g}_{b}^{\\dag }\\hat{g}_{a}=0$ ).", "The eigenvectors of this operator form an orthonormal set, while its eigenvalues are real and non-negative.", "The eigenvectors $|D_{n}^{c}(t)>$ with zero eigenvalues: $\\hat{G}^{\\dag }(t)\\hat{G}(t)|D_{n}^{c}(t)>=0,~n=1,...,N_{c}^{d},$ uncoupled to the lower levels $a$ and $b$ , are at the same time the eigenvectors of the interaction operator $\\hat{V}(t)$ with the eigenvalues $\\lambda _{cn}=-2\\Delta $ .", "The eigenvectors $|C_{n}(t)>$ with positive eigenvalues $c_{n}^{2}(t)>0$ : $\\hat{G}^{\\dag }(t)\\hat{G}(t)|C_{n}(t)>=c_{n}^{2}(t)|C_{n}(t)>,~n=1,...,N_{c},$ are coupled by the electric dipole transitions to the states $|F_{n}(t)>=\\frac{1}{c_{n}(t)}\\hat{G}(t)|C_{n}(t)>.$ These states $|F_{n}(t)>$ also form an orthonormal subset at the subspace of the lower atomic levels $a$ and $b$ , as it follows from (REF )-(REF ).", "With the introduction of the states $|C_{n}(t)>$ and $|F_{n}(t)>$ all the eigenvectors of the interaction operator $\\hat{V}(t)$ with non-zero eigenvalues may be easily obtained.", "This operator has $2N_{c}$ eigenvectors $|V_{n}^{(+)}(t)> = \\sin \\theta _{n}(t) |F_{n}(t) + \\cos \\theta _{n}(t)|C_{n}(t)>,$ $|V_{n}^{(-)}(t)> = \\cos \\theta _{n}(t) |F_{n}(t)> - \\sin \\theta _{n}(t) |C_{n}(t)>,$ $\\tan 2\\theta _{n}(t) = - \\frac{c_{n}(t)}{\\Delta },$ with the eigenvalues $\\lambda _{n}^{(\\pm )}(t) = -\\Delta \\pm \\sqrt{\\Delta ^{2}+c_{n}^{2}(t)}.$ All the states $|D_{n}^{c}(t)>$ and $|V_{n}^{(\\pm )}(t)>$ constitute the orthonormal set of $N^{f}=2(2J_{c}+1)-N^{d}_{c}$ eigenvectors of the operator $\\hat{V}(t)$ with non-zero eigenvalues.", "The other $N^{d}=N-N^{f}$ eigenvectors $|D_{n}(t)>$ of the interaction operator $\\hat{V}(t)$ obtain zero eigenvalues.", "These states – dark states – with the atomic part belonging to the subspace of the lower atomic levels $a$ and $b$ remain uncoupled to the upper level $c$ : $\\hat{G}^{\\dag }(t)|D_{n}(t)> = 0,~ n=1,...,N^{d}.$ As it follows from (REF ) and (REF ), all the dark states $|D_{n}(t)>$ are orthogonal to all the bright states $|F_{n}(t)>$ .", "Among all the dark states let us distinguish first the dark states $|D_{n}^{a}>$ and $|D_{n}^{b}>$ , which atomic part belongs to the lower level $a$ or $b$ only.", "These states are time independent and may be obtained as the eigenvectors of operators $\\hat{g}_{a}\\hat{g}_{a}^{\\dag }$ or $\\hat{g}_{b}\\hat{g}_{b}^{\\dag }$ with zero eigenvalues: $\\hat{g}_{a}\\hat{g}_{a}^{\\dag }|D_{n}^{a}> = 0,~ k=1,...,N_{a}^{d},$ $\\hat{g}_{b}\\hat{g}_{b}^{\\dag }|D_{n}^{b}> = 0,~ k=1,...,N_{b}^{d}.$ The other $N^{d}_{ab}$ eigenvectors $|D^{ab}_{n}(t)>$ of the interaction operator $\\hat{V}(t)$ , which atomic part belongs to both atomic lower levels $a$ and $b$ , obtain zero eigenvalues and satisfy the equation (REF ), while $\\hat{g}_{a}^{\\dag }|D_{n}^{ab}(t)> \\ne 0,~\\hat{g}_{b}^{\\dag }|D_{n}^{ab}(t)> \\ne 0.$ The temporal dependence of these dark states may be immediately obtained from the equation (REF ): $|D^{ab}_{n}(t)> = Z_{n}(t)[\\Omega _{a}(t)|\\tilde{B}_{n}^{d}> -\\Omega _{b}(t)|A_{n}^{d}>],$ where $Z_{n}(t)$ is the normalization factor, while $|A_{n}^{d}>$ and $|\\tilde{B}_{n}^{d}>$ are temporally independent states, which atomic parts belong to the levels $a$ and $b$ correspondingly and which satisfy the equation $\\hat{g}_{a}^{\\dag }|A_{n}^{d}> =\\hat{g}_{b}^{\\dag }|\\tilde{B}_{n}^{d}>\\ne 0.$ Introducing the matrix $\\hat{D}_{b} = \\sum _{n}\\frac{1}{b_{n}^{2}}|B_{n}><B_{n}|,$ containing only the eigenvectors $|B_{n}>$ of matrix $\\hat{g}_{b}\\hat{g}_{b}^{\\dag }$ with non-zero eigenvalues $b_{n}^{2}$ , we may write the equation (REF ) as follows: $|\\tilde{B}_{n}^{d}> = \\hat{D}_{ba}|A_{n}^{d}>,~ \\hat{D}_{ba} =\\hat{D}_{b}\\hat{g}_{b}\\hat{g}_{a}^{\\dag }.$ Now we may define the orthonormal set of states $|A_{n}^{d}>$ as the eigenvectors of the hermitian matrix $\\hat{D}_{ba}^{\\dag }\\hat{D}_{ba}$ with non-zero eigenvalues: $\\hat{D}_{ba}^{\\dag }\\hat{D}_{ba}|A_{n}^{d}> = a_{dn}^{2}|A_{n}^{d}>,~a_{dn}>0.$ Then, as it follows from (REF ) and (REF ), the states $|B_{n}^{d}> = \\frac{1}{a_{dn}}\\hat{D}_{ba}|A_{n}^{d}>$ also constitute the orthonormal set, so that the orthonormal set of dark states $|D^{ab}_{n}(t)>$ may be expressed through these states $|A_{n}^{d}>$ and $|B_{n}^{d}>$ by the formula: $|D^{ab}_{n}(t)> = \\frac{a_{dn}\\Omega _{a}(t)|B_{n}^{d}> -\\Omega _{b}(t)|A_{n}^{d}>}{\\sqrt{\\Omega _{b}^{2}(t)+a_{dn}^{2}\\Omega _{a}^{2}(t)}}.$ All the eigenvectors of the interaction operator $\\hat{V}(t)$ , comprising the set of states $|D_{n}^{c}(t)>$ , $|V_{n}^{(\\pm )}(t)>$ , and the set of dark states $|D_{n}^{a}>$ , $|D_{n}^{b}>$ and $|D_{n}^{ab}(t)>$ constitute the complete orthonormal set of states, which determines the evolution operator (REF ) in the adiabatic approximation." ], [ "Photon storage and retrieval", "Initially the atom is at the lower level $b$ , its state being defined by the atomic density matrix $\\hat{\\rho }_{0}^{b}$ , while the state of the field is a pure single-photon state $|f>=\\xi _{1}|1,0>+\\xi _{2}|0,1>,~|\\xi _{1}|^{2}+|\\xi _{2}|^{2}=1,$ where the two complex numbers $\\xi _{1}$ and $\\xi _{2}$ define the photon polarization (the polarization q-bit), so that the initial density matrix $\\hat{\\rho }(0)$ of the atom+field system is: $\\hat{\\rho }(0) = \\hat{\\rho }_{0}^{b}\\cdot |f><f|.$ The classical coherent laser field is adiabatically switched off, while the interaction with the quantum field is adiabatically switched on in the time interval $T_{1}$ , so that: $\\Omega _{a}(0)=\\Omega _{a1},~ \\Omega _{a}(T_{1})=0,\\Omega _{b}(0)=0,~ \\Omega _{b}(T_{1})=\\Omega _{b1}.$ The atomic part of the initial bright states $|F_{n}(0)>$ , defined by the equation (REF ), with non-zero eigenvalues $c^{2}_{n}(0)\\ne 0$ belongs to the level $a$ only, so that the corresponding initial eigenvectors $|V^{\\pm }_{n}(0)>$ of the interaction operator, defined by (REF )-(REF ), will not contribute to the evolution.", "However the atomic part of the initial bright states $|F_{n}(0)>$ with zero eigenvalues $c^{2}_{n}(0)=0$ , obtained as the limit at $t\\rightarrow +0$ , belongs to the level $b$ .", "Since $c^{2}_{n}(0)=\\Omega _{a1}^{2}c^{2}_{an}$ , where $c^{2}_{an}$ are the eigenvalues of the operator $\\hat{g}_{a}^{\\dag }\\hat{g}_{a}$ , such states will be coupled to the eigenvectors $|C_{n}^{b}>$ of the operator $\\hat{g}_{a}^{\\dag }\\hat{g}_{a}$ with zero eigenvalues: $\\hat{g}_{a}^{\\dag }\\hat{g}_{a}|C_{n}^{b}>=0,~n=1,...,N_{c}^{b}.$ So, the evolution operator (REF ) will be determined by the normalized states $\\hat{g}_{b}|C_{n}^{b}>$ and the dark states $|D_{n}^{b}>$ and $|D_{n}^{ab}(t)>$ , defined by (REF ) and (REF ), which atomic part also belongs to the level $b$ .", "The states $|D_{n}^{b}>$ in the evolution operator are responsible for the photon passage through the cavity without interaction, while the states $\\hat{g}_{b}|C_{n}^{b}>$ are responsible for the process of photon absorption and the adiabatic transfer of the atom from the initial level $b$ to the excited level $c$ to some state, which is uncoupled to the lower level $a$ .", "In both cases the final system state is unstable and for purposes of long-term photon storage the presence of such states in the initial system state should be avoided by the proper choice of polarization of the driving field and by the preparation of the atomic state.", "With such states excluded the evolution is determined only by the states $|D_{n}^{ab}(0)>$ , which are responsible for the long-term recording of the photon polarization q-bit onto the superposition of Zeeman substates of the metastable level $a$ .", "The evolution operator at the instant of time $t=T_{1}$ after the recording process is finished is as follows: $\\hat{S}_{1}=-\\sum _{n}|A_{n}^{d}><B_{n}^{d}|,$ so that each state $|B_{n}^{d}>$ from the level $b$ , defined by (REF ), is adiabatically transferred to the corresponding state $|A_{n}^{d}>$ from the level $a$ , defined by (REF ).", "The total probability $w$ of the photon storage at the level $a$ generally depends on the photon polarization $\\xi _{1}$ and $\\xi _{2}$ : $w= tr\\lbrace \\hat{S}_{1}\\hat{\\rho }(0)\\hat{S}_{1}^{\\dag }\\rbrace =\\sum _{i,j=1}^{2}\\xi _{i}\\xi _{j}^{*}w_{ij},$ where $w_{ij}=\\sum _{n}<b_{n}^{di}|\\hat{\\rho }_{0}^{b}|b_{n}^{dj}>,$ while $|b_{n}^{d1}>=<1,0|B_{n}^{d}>$ and $|b_{n}^{d2}>=<0,1|B_{n}^{d}>$ are the pure atomic states at the level $b$ – the projections of the field+atom states $|B_{n}^{d}>$ on the two photon polarization states $|1,0>$ and $|0,1>$ .", "For purposes of q-bit storage the total probability must be unity and must not depend on photon polarization, so that the condition $w_{ij}=\\delta _{ij}$ must be satisfied.", "To retrieve the photon after the storage time $\\tau $ the classical coherent laser field with the same polarization $\\textbf {l}_{c}$ , as that of the field applied to store the photon, is adiabatically switched on, while the interaction with the quantum field is adiabatically switched off in the time interval $T_{2}$ , so that: $\\Omega _{a}(T_{1}+\\tau )=0,~\\Omega _{a}(T_{1}+\\tau +T_{2})=\\Omega _{a2},$ $\\Omega _{b}(T_{1}+\\tau )=\\Omega _{b2},~\\Omega _{b}(T_{1}+\\tau +T_{2})=0.$ If the photon was stored with unit probability, then at the start of storage process the atom was prepared at some state of the level $b$ , which contained only the parts $|B_{n}^{d}>$ of the \"dark\" states $|D_{n}^{ab}>$ , while the the initially unpopulated level $a$ was populated during the storage process by the adiabatic transfer of the parts $|B_{n}^{d}>$ of the \"dark\" states $|D_{n}^{ab}>$ to their counterparts $|A_{n}^{d}>$ at the level $a$ , so that at the start of the retrieval stage only the \"dark\" states $|D_{n}^{ab}>$ will be present in the initial atomic state, provided that the polarization of the retrieving field is the same as that of the recording field, ensuring that the \"dark\" states $|D_{n}^{ab}>$ are the same for both stages – the storage and the retrieval.", "In this case only the dark states $|D_{n}^{ab}(t)>$ will contribute to the evolution operator (REF ).", "Then, at the end of the retrieval process at the instant of time $t_{2}=T_{1}+\\tau +T_{2}$ the evolution operator $\\hat{S}_{2}=-\\sum _{n}|B_{n}^{d}><A_{n}^{d}|$ is determined by the same states $|A_{n}^{d}>$ and $|B_{n}^{d}>$ as the evolution operator $\\hat{S}_{1}$ (REF ) and the product of the evolution operators $\\hat{S}_{2}\\hat{S}_{1}$ is reduced to the projector on the subspace of states $|B_{n}^{d}>$ : $\\hat{S}_{2}\\hat{S}_{1}=\\sum _{n}|B_{n}^{d}><B_{n}^{d}|,$ and since only such states are present at the initial system state $\\hat{\\rho }(0)$ , then this state will be retrieved without distortion $\\hat{\\rho }_{2}=\\hat{S}_{2}\\hat{S}_{1} \\hat{\\rho }(0)\\hat{S}^{\\dag }_{2}\\hat{S}^{\\dag }_{1} = \\hat{\\rho }(0),$ reproducing a single photon $|f>$ with the recorded polarization and the atom at the initial state with the density matrix $\\hat{\\rho }_{0}^{b}$ .", "Let us choose the two unit polarization vectors $\\textbf {l}_{1}$ and $\\textbf {l}_{2}$ of the quantum field in the plane $XY$ as the two vectors with opposite circular components: $l_{1q} = \\delta _{q,-1},~ l_{2q} = \\delta _{q,1},$ while the polarization vector $\\textbf {l}_{c}$ of the classical field generally contains also $Z$ -projections.", "In the experiment [5] the angular momenta of the atomic levels were $J_{a}=2$ , $J_{b}=J_{c}=1$ , while the driving field was linearly polarized along the axis $Z$ : $l_{cq} =\\delta _{q,0}$ ($\\pi $ -polarized field).", "In this case there are three states $|A_{n}^{d}>$ at the level $a$ , defined by (REF ): $ |A_{1}^{d}> = \\left( {0\\cr 0\\cr 0\\cr 1\\cr 0}\\right),~|A_{2}^{d}> = \\left( {0\\cr 1\\cr 0\\cr 0\\cr 0}\\right),~|A_{3}^{d}> = \\left( {0\\cr 0\\cr 1\\cr 0\\cr 0}\\right),$ which are adiabatically coupled to the three states $|B_{n}^{d}>$ at the level $b$ , defined by (REF ): $ |B_{1}^{d}> = \\left( {0 \\cr 1 \\cr 0}\\right)|1,0>,~|B_{2}^{d}> = -\\left( {0 \\cr 1 \\cr 0}\\right)|0,1>,$ $|B_{3}^{d}> = \\frac{1}{\\sqrt{2}}\\left\\lbrace \\left( {1 \\cr 0 \\cr 0}\\right)|1,0>-\\left( {0 \\cr 0 \\cr 1}\\right)|0,1>\\right\\rbrace .$ The elements of matrix $w_{i,j}$ (REF ) determining the photon storage probability are expressed through the elements $\\rho ^{bb}_{m_{b},m^{\\prime }_{b}}$ of the initial atomic density matrix $\\hat{\\rho }_{0}^{b}$ in a following way: $w_{1,1} = \\rho ^{bb}_{0,0} + \\frac{1}{2}\\rho ^{bb}_{-1,-1},$ $w_{2,2} = \\rho ^{bb}_{0,0} + \\frac{1}{2}\\rho ^{bb}_{1,1},$ $w_{2,1} = w_{1,2}^{*} = - \\frac{1}{2}\\rho ^{bb}_{1,-1},$ so that the conditions (REF ) may be met only if the atom is initially in the pure state $|J_{b}=1,m_{b}=0>$ , as it was in the experiment.", "Then, as it follows from the expression (REF ) for the evolution operator, the initial system pure state $|J_{b}=1,m_{b}=0>(\\xi _{1}|1,0>+\\xi _{2}|0,1>)$ is adiabatically transferred to the pure state $(\\xi _{2}|J_{a}=2,m_{a}=-1> - \\xi _{1}|J_{a}=2,m_{a}=1>)|0,0>.$ The conditions (REF ) may be also met, as calculations show, with the driving field linearly polarized along the axis $X$ : $l_{cq}=\\frac{1}{\\sqrt{2}}\\left(\\delta _{q,-1}-\\delta _{q,1}\\right)$ and with the same initial atomic state $|J_{b}=1,m_{b}=0>$ , though in this case the photon polarization will be mapped to the superposition of states $|J_{a}=2,m_{a}=0>$ and $|J_{a}=2,m_{a}=\\pm 2>$ : $a^{0}|m_{a}=0> + a^{+}|m_{a}=2> + a^{-}|m_{a}=-2>,$ $a^{0}=\\frac{\\xi _{1}+\\xi _{2}}{2\\sqrt{2}},~a^{\\pm }=\\frac{\\sqrt{3}(\\xi _{1}+\\xi _{2})}{4} \\pm \\frac{(\\xi _{1}-\\xi _{2})}{2}.$ For the lower values $J_{b}=0$ and $J_{b}=1/2$ of the angular momenta of the initial atomic level $b$ the conditions (REF ) for the photon storage with unit probability, independent on photon polarization, may be easily met.", "With the driving field linearly polarized along the axis $Z$ : $l_{cq} =\\delta _{q,0}$ , these conditions will be satisfied, for example, for the transitions with $J_{b}=0$ , $J_{a}=J_{c}=1$ and with $J_{b}=1/2$ , $J_{a}=J_{c}=3/2$ for any initial atomic state, in particular, for the equilibrium state: $\\rho ^{bb}_{-1/2,-1/2} =\\rho ^{bb}_{1/2,1/2}=1/2$ , $\\rho ^{bb}_{-1/2,1/2} =\\rho ^{bb}_{1/2,-1/2}=0$ .", "However for larger values of angular momentum $J_{b}>1$ it becomes hard (if not impossible) to meet the conditions (REF ).", "Indeed, the memory efficiency (or better to say inefficiency) is determined by the number $N_{d}^{b}$ of the \"dark\" states of the type $|D_{n}^{b}>$ , defined by the equation (REF ), and by the weight of these states in the initial atomic state.", "The number of states $2(2J_{b}+1)$ with the atomic part at the level $b$ is doubled due to the two possible polarizations of the cavity field and only $2J_{c}+1$ of them are coupled to the excited level $c$ , while the other $N_{d}^{b}=2(2J_{b}-J_{c}+1)-1$ states – the \"dark\" states of the type $|D_{n}^{b}>$ – stay uncoupled, reducing the memory efficiency.", "The number of such states obtains its minimum $N_{d}^{b}=2J_{b}-1$ on the transitions $J_{b}\\rightarrow J_{c}=J_{b}+1$ , so that at $J_{b}\\ge 1$ the presence of these states becomes mandatory.", "The number $N_{d}^{b}$ of the states $|D_{n}^{b}>$ grows with $J_{b}$ and at $J_{b}>1$ it becomes highly improbable that each Zeeman state $|J_{b},m_{b}>$ will not be contained in these \"dark\" states $|D_{n}^{b}>$ .", "The general approach, developed in the present paper, enables one to analyze numerically the memory efficiency for any reasonable values of the angular momenta of resonant atomic levels.", "For example, let us consider the transitions $J_{b}=2\\rightarrow J_{c}=3\\rightarrow J_{a}=4$ and the $\\pi $ -polarized driving field ($l_{cq}=\\delta _{q,0}$ ).", "Then the elements of the probability matrix $w_{i,j}$ (REF ) are expressed through the elements $\\rho ^{bb}_{m_{b},m^{\\prime }_{b}}$ of the initial atomic density matrix $\\hat{\\rho }_{0}^{b}$ in a following way: $w_{1,1} = \\frac{1}{7}\\rho ^{bb}_{-2,-2} +\\frac{1}{2}\\rho ^{bb}_{-1,-1} + \\frac{6}{7}\\rho ^{bb}_{0,0} +\\rho ^{bb}_{1,1} + \\rho ^{bb}_{2,2},$ $w_{2,2} = \\frac{1}{7}\\rho ^{bb}_{2,2} + \\frac{1}{2}\\rho ^{bb}_{1,1} +\\frac{6}{7}\\rho ^{bb}_{0,0} + \\rho ^{bb}_{-1,-1} + \\rho ^{bb}_{-2,-2},$ while $w_{1,2}$ and $w_{2,1}$ are expressed only through the non-diagonal elements $\\rho ^{bb}_{m_{b},m^{\\prime }_{b}}$ ($m_{b}\\ne m^{\\prime }_{b}$ ) of the atomic density matrix.", "In addition the populations of Zeeman sublevels are non-negative $\\rho ^{bb}_{mm}\\ge 0$ and the total population of the level is unity $\\rho ^{bb}_{-2,-2} + \\rho ^{bb}_{-1,-1} + \\rho ^{bb}_{0,0} +\\rho ^{bb}_{1,1} + \\rho ^{bb}_{2,2} = 1.$ For a hundred per cent memory efficiency $w_{1,1}=w_{2,2}=1$ , then extracting (REF ) from (REF ) we obtain $\\frac{6}{7}\\rho ^{bb}_{-2,-2} + \\frac{1}{2}\\rho ^{bb}_{-1,-1} +\\frac{1}{7}\\rho ^{bb}_{0,0} = 0,$ which is true only if $\\rho ^{bb}_{-2,-2} = \\rho ^{bb}_{-1,-1} = \\rho ^{bb}_{0,0} = 0$ , and extracting (REF ) from (REF ) we also obtain that $\\rho ^{bb}_{2,2} = \\rho ^{bb}_{1,1} = 0$ , so that in this case the photon q-bit cannot be stored with unit probability.", "However, if the atom is initially prepared at the state $|J_{b}=2,m_{b}=0>$ , then $w_{ij}=(6/7)\\delta _{ij}$ , and the photon will be stored with the probability $w=6/7$ independent of its polarization." ], [ "Conclusions", "In the present paper we have obtained the general expression for the probability of long-term storage of a single-photon polarization q-bit at the magnetic substates of some long-lived degenerate atomic level by means of STIRAP.", "The dependence of this probability on the polarization of the driving field and on the initial atomic state was studied.", "It was shown that the conditions for the photon long-term storage with unit probability independent on the photon polarization may be satisfied with low values of angular momentum of the initial atomic level $J_{b}=0,1/2,1$ .", "At larger values of this angular momentum $J_{b}>1$ it becomes hard (if not impossible) to store the arbitrarily polarized photon with unit probability.", "The general approach, developed in the present paper, enables one to analyze numerically the memory efficiency for any reasonable values of the angular momenta of resonant atomic levels.", "It was also shown that, since the conditions of the photon storage with unit probability are satisfied, the driving field with the same polarization as that used to store the photon will retrieve the initial atomic state and a single photon with the recorded polarization without distortion.", "Acknowledgements Authors are indebted for financial support of this work to Russian Ministry of Science and Education (grant 2407)." ] ]

[ [ "Extending Gaussian hypergeometric series to the $p$-adic setting" ], [ "Abstract We define a function which extends Gaussian hypergeometric series to the $p$-adic setting.", "This new function allows results involving Gaussian hypergeometric series to be extended to a wider class of primes.", "We demonstrate this by providing various congruences between the function and truncated classical hypergeometric series.", "These congruences provide a framework for proving the supercongruence conjectures of Rodriguez-Villegas." ], [ "Introduction", "This work was supported by the UCD Ad Astra Research Scholarship program.", "In [8], Greene introduced hypergeometric series over finite fields or Gaussian hypergeometric series.", "Let $\\mathbb {F}_{p}$ denote the finite field with $p$ , a prime, elements.", "We extend the domain of all multiplicative characters $\\chi $ of $\\mathbb {F}^{*}_{p}$ to $\\mathbb {F}_{p}$ , by defining $\\chi (0):=0$ (including the trivial character $\\varepsilon $ ) and denote $\\overline{B}$ as the inverse of $B$ .", "For characters $A$ and $B$ of $\\mathbb {F}_{p}^*$ , define $\\binom{A}{B} :=\\frac{B(-1)}{p} \\sum _{x \\in \\mathbb {F}_{p}} A(x) \\overline{B}(1-x).$ For characters $A_0,A_1,\\cdots , A_n$ and $B_1, \\cdots , B_n$ of $\\mathbb {F}_{p}^*$ and $x \\in \\mathbb {F}_{p}$ , define the Gaussian hypergeometric series by ${_{n+1}F_n} {\\left( \\begin{array}{cccc} A_0, & A_1, & \\cdots , & A_n \\\\\\phantom{A_0} & B_1, & \\cdots , & B_n \\end{array}\\Big | \\; x \\right)}_{p}:= \\frac{p}{p-1} \\sum _{\\chi } \\binom{A_0 \\chi }{\\chi } \\binom{A_1 \\chi }{B_1 \\chi }\\cdots \\binom{A_n \\chi }{B_n \\chi } \\chi (x)$ where the sum is over all multiplicative characters $\\chi $ of $\\mathbb {F}_{p}^*$ .", "These series are analogous to classical hypergeometric series and have been used in character sum evaluations [12], finite field versions of the Lagrange inversion formula [9], the representation theory of SL($2, \\mathbb {R}$ ) [10], formula for traces of Hecke operators [6], [7], formulas for Ramanujan's $\\tau $ -function [6], [22], and evaluating the number of points over $\\mathbb {F}_{p}$ of certain algebraic varieties [2], [6], [24].", "They have also played an important role in the proof of many supercongruences [1], [2], [13], [15], [16], [17], [21].", "The main approach was originally established by Ahlgren and Ono in proving the Apéry number supercongruence [2].", "For $n\\ge 0$ , let $A(n)$ be the the numbers defined by $A(n):= \\sum _{j=0}^{n} \\left({\\genfrac{}{}{0.0pt}{}{n+j}{j}}\\right)^2 \\left({\\genfrac{}{}{0.0pt}{}{n}{j}}\\right)^2.$ These numbers were used by Apéry in his proof of the irrationality of $\\zeta (3)$ [3], [26] and are commonly known as the Apéry numbers.", "If we define integers $\\gamma (n)$ by $\\sum _{n=1}^{\\infty } \\gamma (n)q^n := q \\prod _{n=1}^{\\infty }(1-q^{2n})^4 (1-q^{4n})^4,$ then Beukers conjectured [4] that for $p\\ge 3$ a prime, $A\\left(\\tfrac{p-1}{2}\\right)\\equiv \\gamma (p) \\pmod {p^2}.$ For a complex number $a$ and a non-negative integer $n$ let ${\\left({a}\\right)}_{n}$ denote the rising factorial defined by ${\\left({a}\\right)}_{0}:=1 \\quad \\textup {and} \\quad {\\left({a}\\right)}_{n} := a(a+1)(a+2)\\cdots (a+n-1) \\textup { for } n>0.$ Then, for complex numbers $a_i$ , $b_j$ and $z$ , with none of the $b_j$ being negative integers or zero, we define the truncated generalized hypergeometric series ${{_rF_s} \\left[ \\begin{array}{ccccc} a_1, & a_2, & a_3, & \\cdots , & a_r \\vspace{3.61371pt}\\\\\\phantom{a_1} & b_1, & b_2, & \\cdots , & b_s \\end{array}\\Big | \\; z \\right]}_{m}:=\\sum ^{m}_{n=0}\\frac{{\\left({a_1}\\right)}_{n} {\\left({a_2}\\right)}_{n} {\\left({a_3}\\right)}_{n} \\cdots {\\left({a_r}\\right)}_{n}}{{\\left({b_1}\\right)}_{n} {\\left({b_2}\\right)}_{n} \\cdots {\\left({b_s}\\right)}_{n}}\\; \\frac{z^n}{{n!", "}}.$ If we first recognize that $A\\left(\\tfrac{p-1}{2}\\right) \\equiv {_{4}F_3} \\Biggl [ \\begin{array}{cccc} \\frac{1}{2}, & \\frac{1}{2}, & \\frac{1}{2}, & \\frac{1}{2}\\vspace{3.61371pt}\\\\\\phantom{\\frac{1}{2}} & 1, & 1, & 1 \\end{array}\\bigg | \\; 1 \\Biggr ]_{p-1}\\pmod {p^2}$ then we can summarize Ahlgren and Ono's proof of (REF ) in the following two results.", "Theorem 1.1 (Ahlgren and Ono [2]) If $p$ is an odd prime and $\\phi $ is the character of order 2 of $\\mathbb {F}_p^*$ , then ${_{4}F_3} \\Biggl [ \\begin{array}{cccc} \\frac{1}{2}, & \\frac{1}{2}, & \\frac{1}{2}, & \\frac{1}{2}\\vspace{3.61371pt}\\\\\\phantom{\\frac{1}{2}} & 1, & 1, & 1 \\end{array}\\bigg | \\; 1 \\Biggr ]_{p-1}\\equiv -p^3 \\: {_{4}F_3} \\Biggl ( \\begin{array}{cccc} \\phi , & \\phi , & \\phi , & \\phi \\\\\\phantom{\\phi } & \\varepsilon , & \\varepsilon , & \\varepsilon \\end{array}\\bigg | \\; 1 \\Biggr )_{p}-p \\pmod {p^2} .$ Theorem 1.2 (Ahlgren and Ono [2]) If $p$ is an odd prime and $\\phi $ is the character of order 2 of $\\mathbb {F}_p^*$ , then $-p^3 \\: {_{4}F_3} \\Biggl ( \\begin{array}{cccc} \\phi , & \\phi , & \\phi , & \\phi \\\\\\phantom{\\phi } & \\varepsilon , & \\varepsilon , & \\varepsilon \\end{array}\\bigg | \\; 1 \\Biggr )_{p}-p=\\gamma (p) .$ It is interesting to note the analogue between the parameters in the hypergeometric series in Theorem REF .", "“One over two” has been replaced with a character of order 2 and “one over one” has been replaced with a character of order one.", "This approach of using the Gaussian hypergeometric series as an intermediate step has since become the template for proving these types of supercongruences and the analogue between the parameters of the generalized and Gaussian hypergeometric series has also been in evidence in these cases.", "In [23] Rodriguez-Villegas examined the relationship between the number of points over $\\mathbb {F}_p$ on certain Calabi-Yau manifolds and truncated hypergeometric series which correspond to a particular period of the manifold.", "In doing so, he identified numerically 22 possible supercongruences.", "18 of these relate truncated generalized hypergeometric series to the $p$ -th Fourier coefficient of certain modular forms via modulo $p^2$ and $p^3$ congruences.", "Two of the 18 have been proven outright [1], [13] with three more established for primes in a particular congruence class and up to sign otherwise [17], all using Gaussian hypergeometric series as an intermediate step.", "(We note that the case proved in [1] had previously been established in [25] by other means.)", "We now consider one of the outstanding conjectures of Rodriguez-Villegas.", "Let $f(z):= f_1(z)+5f_2(z)+20f_3(z)+25f_4(z)+25f_5(z)=\\sum _{n=1}^{\\infty } c(n) q^n$ where $f_i(z):=\\eta ^{5-i}(z) \\hspace{2.0pt} \\eta ^4(5z) \\hspace{2.0pt} \\eta ^{i-1}(25z)$ , $\\eta (z):=q^{\\frac{1}{24}} \\prod _{n=1}^{\\infty }(1-q^n)$ is the Dedekind eta function and $q:=e^{2 \\pi i z}$ .", "Then $f$ is a cusp form of weight four on the congruence subgroup $\\Gamma _0(25)$ and we have the following conjecture.", "Conjecture 1.3 (Rodriguez-Villegas [23]) If $p \\ne 5$ is prime and $c(p)$ is as defined in (REF ), then ${_4F_3} \\Biggl [ \\begin{array}{cccc} \\frac{1}{5}, & \\frac{2}{5}, & \\frac{3}{5}, & \\frac{4}{5} \\\\\\phantom{\\frac{1}{5},} & 1, & 1, & 1\\end{array}\\bigg | \\; 1 \\Biggr ]_{p-1}\\equiv c(p) \\pmod {p^3}.$ Using the approach of Ahlgren and Ono we would expect to be able to relate the truncated hypergeometric series on the left hand side of (REF ) to the Gaussian hypergeometric series $_4F_3 \\Biggl ( \\begin{array}{cccc} \\chi _5, & \\chi _5^2, & \\chi _5^3, & \\chi _5^4 \\\\\\phantom{\\chi _5} & \\varepsilon , & \\varepsilon , & \\varepsilon \\end{array}\\bigg | \\; 1 \\Biggr )_p,$ where $\\chi _5$ is a character of order 5 of $\\mathbb {F}_p^*$ .", "However this series is only defined for $p \\equiv 1 \\pmod {5}$ which would restrict any eventual results.", "A similar restriction was also encountered by the author of [17].", "This issue did not affect the authors of [1], [13] as all top line parameters in their cases were characters of order 2 thus restricting the series to odd primes which was all that was required.", "Many of the other applications of these series also encounter these restrictions.", "For example the results in [6] are restricted to primes congruent to 1 modulo 12.", "We would like to develop some generalization of Gaussian hypergeometric series which does not have such restrictions implicit in its definition.", "We achieve this by reformulating the series as an expression in terms of the $p$ -adic gamma function which we can then extend in the $p$ -adic setting." ], [ "Statement of Results", "The Gaussian hypergeometric series which occur in the applications of the series referenced in Section 1 have all been of a certain form.", "All top line characters, $A_i$ , have been non-trivial, all bottom line characters, $B_j$ , have been trivial and the argument $x$ has equaled 1.", "The Gaussian hypergeometric series analogous to the classical series in the supercongruence conjectures of Rodriguez-Villegas would also be of this form.", "Therefore, our starting point for extending Gaussian hypergeometric series is to examine series of this type.", "In a similar manner to [16], [17], using the relationship between Jacobi and Gauss sums, and the Gross-Koblitz formula [11], we can express the series in terms of the $p$ -adic gamma function.", "For a positive integer $n$ , let $m_1, m_2, \\dots , m_{n+1}, d_1, d_2, \\dots , d_{n+1}$ also be positive integers, such that $0<\\frac{m_i}{d_i}< 1$ for $1 \\le i \\le n+1$ .", "Let $p$ be a prime such that $p \\equiv 1 \\pmod {d_i}$ for each $i$ and let $\\rho _i$ be the character of order $d_i$ of $\\mathbb {F}_p^*$ given by $\\overline{\\omega }^{\\frac{p-1}{d_i}}$ , where $\\omega $ is the Teichmüller character.", "Without loss of generality we can assume $0<\\frac{m_1}{d_1} \\le \\frac{m_2}{d_2} \\le \\dots \\le \\frac{m_{n+1}}{d_{n+1}}<1$ .", "We define $r_i:=\\frac{p-1}{d_i}$ for brevity.", "Then we also define $m_0:=-1$ , $m_{n+2}:=p-2$ and $d_0=d_{n+2}:=p-1$ so that $m_0r_0=-1$ and $m_{n+2}r_{n+2}=p-2$ .", "If $\\Gamma _p{\\left({\\cdot }\\right)}$ is the $p$ -adic gamma function, then we have $&(-1)^n \\: p^n \\: {_{n+1}F_n} {\\left( \\begin{array}{cccc} \\rho _1^{m_1}, & \\rho _2^{m_2}, & \\cdots , & \\rho _{n+1}^{m_{n+1}} \\\\\\phantom{\\rho _1^{m_1}} & \\varepsilon , & \\cdots , & \\varepsilon \\end{array}\\Big | \\; 1 \\right)}_{p}\\\\&= \\frac{-1}{p-1} \\sum _{k=0}^{n+1} (-p)^k \\sum _{j=m_kr_k+1}^{m_{k+1}r_{k+1}} \\omega ^{j(n+1)}(-1){\\Gamma _p{\\bigl ({\\tfrac{j}{p-1}}\\bigr )}}^{n+1}\\prod _{\\begin{array}{c}i=1\\\\i>k\\end{array}}^{n+1} \\frac{\\Gamma _p{\\bigl ({\\frac{m_i}{d_i}-\\frac{j}{p-1}}\\bigr )}}{\\Gamma _p{\\bigl ({\\frac{m_i}{d_i}}\\bigr )}}\\prod _{\\begin{array}{c}i=1\\\\i\\le k\\end{array}}^{n+1} \\frac{\\Gamma _p{\\bigl ({\\frac{d_i+m_i}{d_i}-\\frac{j}{p-1}}\\bigr )}}{\\Gamma _p{\\bigl ({\\frac{m_i}{d_i}}\\bigr )}}.$ Let $\\left\\lfloor x \\right\\rfloor $ denote the greatest integer less than or equal to $x$ and let $\\langle x \\rangle $ be the fractional part of $x$ , i.e.", "$x- \\left\\lfloor x \\right\\rfloor $ .", "We then define the following generalization.", "Definition 2.1 Let $p$ be an odd prime and let $n \\in \\mathbb {Z}^{+}$ .", "For $1 \\le i \\le n+1$ , let $\\frac{m_i}{d_i} \\in \\mathbb {Q} \\cap \\mathbb {Z}_p$ such that $0<\\frac{m_i}{d_i}<1$ .", "Then define ${_{n+1}G} \\left( \\tfrac{m_1}{d_1}, \\tfrac{m_2}{d_2}, \\cdots , \\tfrac{m_{n+1}}{d_{n+1}} \\right)_p:= \\frac{-1}{p-1} \\sum _{j=0}^{p-2}{\\left((-1)^j\\Gamma _p{\\bigl ({\\tfrac{j}{p-1}}\\bigr )}\\right)}^{n+1}\\prod _{i=1}^{n+1} \\frac{\\Gamma _p{\\bigl ({\\langle \\frac{m_i}{d_i}-\\frac{j}{p-1}\\rangle }\\bigr )}}{\\Gamma _p{\\bigl ({\\frac{m_i}{d_i}}\\bigr )}}(-p)^{-\\lfloor {\\frac{m_i}{d_i}-\\frac{j}{p-1}\\rfloor }}.$ We first note that ${_{n+1}G} \\in \\mathbb {Z}_p$ .", "We also note that the function is defined at all primes not dividing the $d_i$ .", "Combining (REF ) and Definition REF , one can easily see that we recover the Gaussian hypergeometric series via the following result.", "Proposition 2.2 Let $n \\in \\mathbb {Z}^{+}$ and, for $1 \\le i \\le n+1$ , let $\\frac{m_i}{d_i} \\in \\mathbb {Q}$ such that $0<\\frac{m_i}{d_i}<1$ .", "Let $p \\equiv 1 \\pmod {d_i}$ , for each $i$ , be prime and let $\\rho _i$ be the character of order $d_i$ of $\\mathbb {F}_p^*$ given by $\\overline{\\omega }^{\\frac{p-1}{d_i}}$ .", "Then ${_{n+1}G} \\left( \\tfrac{m_1}{d_1}, \\tfrac{m_2}{d_2}, \\cdots , \\tfrac{m_{n+1}}{d_{n+1}} \\right)_p=(-1)^n \\: p^n \\: {_{n+1}F_n} {\\Bigg ( \\begin{array}{cccc} \\rho _1^{m_1}, & \\rho _2^{m_2}, & \\cdots , & \\rho _{n+1}^{m_{n+1}} \\\\\\phantom{\\rho _1^{m_1}} & \\varepsilon , & \\cdots , & \\varepsilon \\end{array}\\bigg | \\; 1 \\Biggr )}_{p} \\; \\genfrac{}{}{0.0pt}{}{}{\\genfrac{}{}{0.0pt}{}{}{.", "}}$ Using this proposition, the $_{n+1}G$ function can be used in place of Greene's function in most applications and should allow results to be extended to a wider class of primes in many cases.", "We demonstrate this here by considering its relationship with the classical series.", "One of the main results offering congruences between generalized and Gaussian hypergeometric series has been Theorem 1 in [17].", "This theorem relates the same Gaussian hypergeometric series as appears on the right hand side of Proposition REF to a truncated generalized hypergeometric series via a modulo $p^2$ congruence.", "Therefore, applying Proposition REF to Theorem 1 in [17] we get the following result.", "Theorem 2.3 Let $n \\in \\mathbb {Z}^{+}$ and, for $1 \\le i \\le n+1$ , let $\\frac{m_i}{d_i} \\in \\mathbb {Q}$ such that $0<\\frac{m_i}{d_i}<1$ .", "Let $p \\equiv 1 \\pmod {d_i}$ , for each $i$ , be prime and let $\\rho _i$ be the character of order $d_i$ of $\\mathbb {F}_p^*$ given by $\\overline{\\omega }^{\\frac{p-1}{d_i}}$ .", "If $S:=\\sum _{i=1}^{n+1} \\frac{m_i}{d_i} \\ge n-1$ and $\\delta :=\\prod _{i=1}^{n+1} \\Gamma _p{\\bigl ({1-\\frac{m_i}{d_i}}\\bigr )}$ when $S=n-1$ and zero otherwise, then ${_{n+1}G} \\left( \\tfrac{m_1}{d_1}, \\tfrac{m_2}{d_2}, \\cdots , \\tfrac{m_{n+1}}{d_{n+1}} \\right)_p\\equiv {_{n+1}F_n} \\Biggl [ \\begin{array}{cccc} \\frac{m_1}{d_1}, & \\frac{m_2}{d_2}, & \\cdots , & \\frac{m_{n+1}}{d_{n+1}} \\vspace{3.61371pt}\\\\\\phantom{\\frac{m_1}{d_1}} & 1, & \\cdots , & 1 \\end{array}\\bigg | \\; 1 \\Biggr ]_{p-1}+ \\delta \\cdot p \\pmod {p^2}.$ The main results of this paper establish 4 families of congruences between the $_{n+1}G$ function and truncated generalized hypergeometric series which extend Theorem REF to primes in other congruence classes, as follows.", "Theorem 2.4 Let $2 \\le d \\in \\mathbb {Z}$ and let $p$ be an odd prime such that $p\\equiv \\pm 1 \\hspace{2.77771pt}({mod}\\,\\,d)$ .", "Then ${_{2}G} \\Bigl ( \\tfrac{1}{d}, 1-\\tfrac{1}{d} \\Bigr )_p&\\equiv {_{2}F_1} \\Biggl [ \\begin{array}{cc} \\frac{1}{d}, & 1-\\frac{1}{d}\\vspace{3.61371pt}\\\\\\phantom{\\frac{1}{d}} & 1 \\end{array}\\bigg | \\; 1 \\Biggr ]_{p-1}\\pmod {p^2} .$ Theorem 2.5 Let $2 \\le d \\in \\mathbb {Z}$ and let $p$ be an odd prime such that $p\\equiv \\pm 1 \\hspace{2.77771pt}({mod}\\,\\,d)$ .", "Then ${_{3}G} \\Bigl ( \\tfrac{1}{2}, \\tfrac{1}{d} , 1-\\tfrac{1}{d} \\Bigr )_p&\\equiv {_{3}F_2} \\Biggl [ \\begin{array}{ccc} \\frac{1}{2}, & \\frac{1}{d}, & 1-\\frac{1}{d}\\vspace{3.61371pt}\\\\\\phantom{\\frac{1}{d}} & 1, &1 \\end{array}\\bigg | \\; 1 \\Biggr ]_{p-1}\\pmod {p^2} .$ Theorem 2.6 Let $d_1, d_2 \\ge 2$ be integers and let $p$ be an odd prime such that $p\\equiv \\pm 1 \\hspace{2.77771pt}({mod}\\,\\,d_1)$ and $p\\equiv \\pm 1 \\hspace{2.77771pt}({mod}\\,\\,d_2)$ .", "If $s(p):=\\Gamma _p{\\bigl ({\\frac{1}{d_1}}\\bigr )}\\Gamma _p{\\bigl ({\\frac{d_1-1}{d_1}}\\bigr )}\\Gamma _p{\\bigl ({\\frac{1}{d_2}}\\bigr )}\\Gamma _p{\\bigl ({\\frac{d_2-1}{d_2}}\\bigr )}=(-1)^{\\left\\lfloor \\frac{p-1}{d_1} \\right\\rfloor +\\left\\lfloor \\frac{p-1}{d_2} \\right\\rfloor }$ , then ${_{4}G} \\left(\\tfrac{1}{d_1} , 1-\\tfrac{1}{d_1}, \\tfrac{1}{d_2} , 1-\\tfrac{1}{d_2}\\right)_p\\equiv {_{4}F_3} \\Biggl [ \\begin{array}{cccc} \\frac{1}{d_1}, & 1-\\frac{1}{d_1}, & \\frac{1}{d_2}, & 1-\\frac{1}{d_2}\\vspace{3.61371pt}\\\\\\phantom{\\frac{1}{d_1}} & 1, & 1, & 1 \\end{array}\\bigg | \\; 1 \\Biggr ]_{p-1}+s(p)\\hspace{1.0pt}p\\pmod {p^3}.$ Theorem 2.7 Let $r, d \\in \\mathbb {Z}$ such that $2 \\le r \\le d-2$ and $\\gcd (r,d)=1$ .", "Let $p$ be an odd prime such that $p\\equiv \\pm 1 \\hspace{2.77771pt}({mod}\\,\\,d)$ or $p\\equiv \\pm r \\hspace{2.77771pt}({mod}\\,\\,d)$ with $r^2 \\equiv \\pm 1 \\hspace{2.77771pt}({mod}\\,\\,d)$ .", "If $s(p) := \\Gamma _p{\\left({\\tfrac{1}{d}}\\right)} \\Gamma _p{\\left({\\tfrac{r}{d}}\\right)}\\Gamma _p{\\left({\\tfrac{d-r}{d}}\\right)}\\Gamma _p{\\left({\\tfrac{d-1}{d}}\\right)}$ , then ${_{4}G} \\Bigl (\\tfrac{1}{d} , \\tfrac{r}{d}, 1-\\tfrac{r}{d} , 1-\\tfrac{1}{d}\\Bigr )_p&\\equiv {_{4}F_3} \\Biggl [ \\begin{array}{cccc} \\frac{1}{d}, & \\frac{r}{d}, & 1-\\frac{r}{d}, & 1-\\frac{1}{d}\\vspace{3.61371pt}\\\\\\phantom{\\frac{1}{d_1}} & 1, & 1, & 1 \\end{array}\\bigg | \\; 1 \\Biggr ]_{p-1}+s(p)\\hspace{1.0pt} p\\pmod {p^3}.$ Theorems REF and REF extend Theorem REF , when $n=1,2$ , to primes in an additional congruence class.", "However, the price of this extension is a loss in some generality of the arguments of the series.", "In the case $n=3$ , Theorems REF and REF not only extend Theorem REF to primes in additional congruence classes but also to a modulo $p^3$ relation, which is a significant development, as we will see in the next paragraph.", "Again this extension is accompanied by a loss in generality of the arguments of the series.", "One of the consequences of the methods contained in this paper is that the sum of the arguments of $_{n+1}G$ in any extension will equal $\\frac{n+1}{2}$ .", "This means that the condition on $S$ in Theorem REF will only be satisfied if $n \\le 3$ .", "Therefore, extension of Theorem REF to primes beyond $p \\equiv 1 \\pmod {d_i}$ when $n>3$ is not possible using current methods.", "Numerical testing would also suggest that such a general formula does not exist.", "The truncated generalized hypergeometric series appearing in Theorems REF to REF include all 22 truncated hypergeometric series occurring in the supercongruence conjectures of Rodriguez-Villegas [23].", "Furthermore, in each of these cases, the congruences in Theorems REF to REF hold for all primes required in the conjectures, due to the particular parameters involved.", "Thus, these congruences provide a framework for proving all 22 cases.", "In fact, in [19] we use Theorem REF with $d=5$ to prove Conjecture REF .", "Using Proposition REF , it is easy to see the following corollaries of Theorems REF to REF .", "Corollary 2.8 Let $2 \\le d \\in \\mathbb {Z}$ and let $p$ be a prime such that $p\\equiv 1 \\pmod {d}$ .", "If $\\rho $ is the character of order $d$ of $\\mathbb {F}_p^*$ given by $\\overline{\\omega }^{\\frac{p-1}{d}}$ , then $-p \\: {_{2}F_1} {\\Biggl ( \\begin{array}{cc} \\rho , & \\overline{\\rho } \\\\\\phantom{\\rho } & \\varepsilon \\end{array}\\bigg | \\; 1 \\Biggr )}_{p}\\equiv {_{2}F_1} \\Biggl [ \\begin{array}{cc} \\frac{1}{d}, & 1-\\frac{1}{d}\\vspace{3.61371pt}\\\\\\phantom{\\frac{1}{d}} & 1 \\end{array}\\bigg | \\; 1 \\Biggr ]_{p-1}\\pmod {p^2}.$ Corollary 2.9 Let $2 \\le d \\in \\mathbb {Z}$ and let $p$ be a prime such that $p\\equiv 1 \\pmod {d}$ .", "If $\\psi $ is the character of order 2 and $\\rho $ is the character of order $d$ of $\\mathbb {F}_p^*$ given by $\\overline{\\omega }^{\\frac{p-1}{d}}$ , then $p^2 \\: {_{3}F_2} {\\Biggl ( \\begin{array}{ccc} \\psi , & \\rho , & \\overline{\\rho } \\\\\\phantom{\\psi } & \\varepsilon , & \\varepsilon \\end{array}\\bigg | \\; 1 \\Biggr )}_{p}\\equiv {_{3}F_2} \\Biggl [ \\begin{array}{ccc} \\frac{1}{2}, & \\frac{1}{d}, & 1-\\frac{1}{d}\\vspace{3.61371pt}\\\\\\phantom{\\frac{1}{d}} & 1, &1 \\end{array}\\bigg | \\; 1 \\Biggr ]_{p-1}\\pmod {p^2}.$ Corollary 2.10 Let $2 \\le d_1, d_2 \\in \\mathbb {Z}$ and let $p$ be a prime such that $p\\equiv ~1~{\\pmod {d_1}}$ and $p\\equiv 1 \\pmod {d_2}$ .", "If $s(p):=\\Gamma _p{\\bigl ({\\frac{1}{d_1}}\\bigr )}\\Gamma _p{\\bigl ({\\frac{d_1-1}{d_1}}\\bigr )}\\Gamma _p{\\bigl ({\\frac{1}{d_2}}\\bigr )}\\Gamma _p{\\bigl ({\\frac{d_2-1}{d_2}}\\bigr )}=(-1)^{\\left\\lfloor \\frac{p-1}{d_1} \\right\\rfloor +\\left\\lfloor \\frac{p-1}{d_2} \\right\\rfloor }$ and $\\rho _i$ is the character of order $d_i$ of $\\mathbb {F}_p^*$ given by $\\overline{\\omega }^{\\frac{p-1}{d_i}}$ , then $-p^3 \\: {_{4}F_3} {\\Biggl ( \\begin{array}{cccc} \\rho _1, & \\overline{\\rho _1}, & \\rho _2, & \\overline{\\rho _2} \\\\\\phantom{\\rho _1} & \\varepsilon , & \\varepsilon , & \\varepsilon \\end{array}\\bigg | \\; 1 \\Biggr )}_{p}\\equiv {_{4}F_3} \\Biggl [ \\begin{array}{cccc} \\frac{1}{d_1}, & 1-\\frac{1}{d_1}, & \\frac{1}{d_2}, & 1-\\frac{1}{d_2}\\vspace{3.61371pt}\\\\\\phantom{\\frac{1}{d_1}} & 1, & 1, & 1 \\end{array}\\bigg | \\; 1 \\Biggr ]_{p-1}+s(p)\\hspace{1.0pt}p\\pmod {p^3}.$ Corollary 2.11 Let $r, d \\in \\mathbb {Z}$ such that $2 \\le r \\le d-2$ and $\\gcd (r,d)=1$ .", "Let $p$ be a prime such that $p\\equiv 1 \\pmod {d}$ .", "If $s(p) := \\Gamma _p{\\left({\\tfrac{1}{d}}\\right)} \\Gamma _p{\\left({\\tfrac{r}{d}}\\right)}\\Gamma _p{\\left({\\tfrac{d-r}{d}}\\right)}\\Gamma _p{\\left({\\tfrac{d-1}{d}}\\right)}$ and $\\rho $ is the character of order $d$ of $\\mathbb {F}_p^*$ given by $\\overline{\\omega }^{\\frac{p-1}{d_i}}$ , then $-p^3 \\: {_{4}F_3} {\\Biggl ( \\begin{array}{cccc} \\rho , & \\overline{\\rho }, & \\rho ^r, & \\overline{\\rho }^r \\\\\\phantom{\\rho _1} & \\varepsilon , & \\varepsilon , & \\varepsilon \\end{array}\\bigg | \\; 1 \\Biggr )}_{p}\\equiv {_{4}F_3} \\Biggl [ \\begin{array}{cccc} \\frac{1}{d}, & \\frac{r}{d}, & 1-\\frac{r}{d}, & 1-\\frac{1}{d}\\vspace{3.61371pt}\\\\\\phantom{\\frac{1}{d_1}} & 1, & 1, & 1 \\end{array}\\bigg | \\; 1 \\Biggr ]_{p-1}+s(p)\\hspace{1.0pt} p\\pmod {p^3}.$ We note Corollaries REF and REF are special cases of Theorem 1 in [17].", "However Corollaries REF and REF are new and are the first general modulo $p^3$ results in this area.", "The remainder of the paper is organized as follows.", "In Section 3 we recall some properties of the $p$ -adic gamma function and its logarithmic derivatives and we also develop some preliminary results for later use.", "Section 4 deals with the proofs of Theorems REF to REF ." ], [ "$p$ -adic preliminaries", "We first recall the $p$ -adic gamma function.", "For further details, see [14].", "Let $p$ be an odd prime.", "For $n \\in \\mathbb {Z}^{+}$ we define the $p$ -adic gamma function as $\\Gamma _p{\\left({n}\\right)} &:= {(-1)}^n \\prod _{\\begin{array}{c}0<j<n\\\\p \\nmid j\\end{array}} j \\\\\\multicolumn{2}{l}{\\text{and extend to all $x \\in \\mathbb {Z}_p$ by setting $\\Gamma _p{\\left({0}\\right)}:=1$ and}}\\\\\\Gamma _p{\\left({x}\\right)} &:= \\lim _{n \\rightarrow x} \\Gamma _p{\\left({n}\\right)}$ for $x\\ne 0$ , where $n$ runs through any sequence of positive integers $p$ -adically approaching $x$ .", "This limit exists, is independent of how $n$ approaches $x$ , and determines a continuous function on $\\mathbb {Z}_p$ with values in $\\mathbb {Z}^{*}_p$ .", "We now state some basic properties of the $p$ -adic gamma function.", "Proposition 3.1 ([14] Chapter II.6) Let $x, y \\in \\mathbb {Z}_{p}$ and $n \\in \\mathbb {Z}^{+}$ .", "Then (1) $\\Gamma _p{\\left({x+1}\\right)}={\\left\\lbrace \\begin{array}{ll}-x \\hspace{1.0pt} {\\Gamma _p{\\left({x}\\right)}} & \\quad \\text{if } x \\in \\mathbb {Z}_p^* \\, ,\\\\- {\\Gamma _p{\\left({x}\\right)}} & \\quad \\text{otherwise}.\\end{array}\\right.", "}$ (2) $\\Gamma _p(x)\\Gamma _p(1-x) = {(-1)}^{x_0}$ , where $x_0 \\in \\lbrace 1,2, \\cdots , {p}\\rbrace $ satisfies $x_0 \\equiv x \\pmod {p}$ .", "(3) If $x \\equiv y \\pmod {p^n}$ , then $\\Gamma _p{\\left({x}\\right)} \\equiv \\Gamma _p{\\left({y}\\right)} \\pmod {p^n}$ .", "We also consider the logarithmic derivatives of $\\Gamma _p$ .", "For $x \\in \\mathbb {Z}_{p}$ , define $G_1(x):= \\dfrac{\\Gamma _p^{\\prime }(x)}{\\Gamma _p{\\left({x}\\right)}} \\qquad \\textup {and} \\qquad G_2(x):= \\dfrac{\\Gamma _p^{\\prime \\prime }(x)}{\\Gamma _p{\\left({x}\\right)}}.$ These also satisfy some basic properties which we state below.", "Some of these results can be found in [2], [5] and [13].", "If not, we include a short proof.", "Proposition 3.2 Let $x \\in \\mathbb {Z}_{p}$ .", "Then (1) $G_1(x+1) - G_1(x) ={\\left\\lbrace \\begin{array}{ll}1/ x & \\quad \\text{if } x \\in \\mathbb {Z}_p^* \\, ,\\\\0 & \\quad \\text{otherwise}.\\end{array}\\right.", "}$ (2) $G_1(x+1)^2 - G_2(x+1) - G_1(x)^2 + G_2(x)={\\left\\lbrace \\begin{array}{ll}1/ x^2 & \\quad \\text{if } x \\in \\mathbb {Z}_p^*\\, ,\\\\0 & \\quad \\text{otherwise}.\\end{array}\\right.", "}$ (3) $G_1(x) = G_1(1-x)$ .", "(4) $G_1(x)^2 - G_2(x) = - G_1(1-x)^2 + G_2(1-x)$ .", "(1) and (3) are obtained from differentiating the results in Proposition REF (1) and (2) respectively, while (2) and (4) follow from differentiating (1) and (3).", "We also have some congruence properties of the $p$ -adic gamma function and its logarithmic derivatives as follows.", "Proposition 3.3 Let $p \\ge 7$ be a prime, $x \\in \\mathbb {Z}_{p}$ and $z \\in p \\hspace{1.0pt} \\mathbb {Z}_{p}$ .", "Then (1) $G_1(x)$ , $G_2(x) \\in \\mathbb {Z}_p$ .", "(2) $\\Gamma _p(x+z) \\equiv \\Gamma _p(x) \\left(1+zG_1(x) +\\frac{z^2}{2} G_2(x) \\right) \\pmod {p^3}$ .", "(3) $\\Gamma _p^{\\prime }(x+z) \\equiv \\Gamma _p^{\\prime }(x) +z \\Gamma _p^{\\prime \\prime } (x) \\pmod {p^2}$ .", "See [13] Proposition 2.3.", "Corollary 3.4 Let $p \\ge 7$ be a prime, $x \\in \\mathbb {Z}_{p}$ and $z \\in p \\hspace{1.0pt} \\mathbb {Z}_{p}$ .", "Then (1) $\\Gamma _p^{\\prime }(x+z) \\equiv \\Gamma _p^{\\prime }(x) \\pmod {p}$ .", "(2) $\\Gamma _p^{\\prime \\prime }(x+z) \\equiv \\Gamma _p^{\\prime \\prime }(x) \\pmod {p}$ .", "(3) $G_1(x+z) \\equiv G_1(x) \\pmod {p}$ .", "(4) $G_2(x+z) \\equiv G_2(x) \\pmod {p}$ .", "By definition, $\\Gamma _p(x) \\in \\mathbb {Z}_p^{*}$ .", "Thus, by Proposition REF (1) and the definitions of $G_1(x)$ and $G_2(x)$ , we see that $\\Gamma _p^{\\prime }(x)$ and $\\Gamma _p^{\\prime \\prime }(x) \\in \\mathbb {Z}_p$ .", "Observe that (1) then follows from Proposition REF (3).", "For (2), one uses similar methods to Proposition 2.3 in [13].", "Finally, (3) and (4) follow from (1) and (2) and the definitions of $G_1(x)$ and $G_2(x)$ .", "Corollary 3.5 Let $p \\ge 7$ be a prime, $x \\in \\mathbb {Z}_{p}$ and $z \\in p \\hspace{1.0pt} \\mathbb {Z}_{p}$ .", "Then $G_1(x)\\equiv G_1(x+z)+z\\left(G_1(x+z)^2 - G_2(x+z)\\right)\\pmod {p^2}.$ By Proposition REF , we see that $G_1(x)= \\frac{\\Gamma _p^{\\prime }(x)}{\\Gamma _p{\\left({x}\\right)}}\\equiv \\frac{\\Gamma _p^{\\prime }(x+z) - z \\Gamma _p^{\\prime \\prime } (x)}{\\Gamma _p(x+z) - z \\Gamma _p^{\\prime }(x)}\\pmod {p^2}.$ Multiplying the numerator and denominator by ${\\Gamma _p(x+z) + z \\Gamma _p^{\\prime }(x)}$ we get that $G_1(x)&\\equiv \\frac{\\Gamma _p^{\\prime }(x+z) \\Gamma _p(x+z) + z\\left(\\Gamma _p^{\\prime }(x) \\Gamma _p^{\\prime }(x+z)-\\Gamma _p(x+z) \\Gamma _p^{\\prime \\prime } (x)\\right)}{\\Gamma _p(x+z)^2}\\\\&\\equiv \\frac{\\Gamma _p^{\\prime }(x+z) \\Gamma _p(x+z) + z\\left(\\Gamma _p^{\\prime }(x+z) \\Gamma _p^{\\prime }(x+z)-\\Gamma _p(x+z) \\Gamma _p^{\\prime \\prime } (x+z)\\right)}{\\Gamma _p(x+z)^2}\\\\&\\equiv G_1(x+z) +z \\left(G_1(x+z)^2-G_2(x+z)\\right)\\pmod {p^2}.$ We now introduce some notation for a $p$ -adic integer's basic representative modulo $p$ .", "Definition 3.6 For a prime $p$ and $x \\in \\mathbb {Z}_p$ we define $\\operatorname{rep}_p(x) \\in \\lbrace 1, 2, \\cdots , p\\rbrace $ via the congruence $\\operatorname{rep}_p(x) \\equiv x \\pmod {p}.$ We will drop the subscript $p$ when it is clear from the context.", "We have the following basic properties of $\\operatorname{rep}(\\cdot )$ .", "Proposition 3.7 Let $p$ be a prime and let $x \\in \\mathbb {Z}_p$ .", "Then $\\operatorname{rep}(1-x)=p+1-\\operatorname{rep}(x).$ We first note that $1-x \\in \\mathbb {Z}_p$ and that $p+1-\\operatorname{rep}(x) \\in \\lbrace 1, 2, \\cdots , p\\rbrace $ .", "We then see that $p+1-\\operatorname{rep}(x) \\equiv 1-\\operatorname{rep}(x) \\equiv 1- x \\pmod {p}.$ Lemma 3.8 Let $p$ be a prime and let $d \\in \\mathbb {Z}$ such that $p\\equiv a \\pmod {d}$ with ${0<a<d}$ .", "Then $\\operatorname{rep}(\\tfrac{a}{d})=p-\\lfloor \\tfrac{p-1}{d}\\rfloor $ and $\\operatorname{rep}(\\tfrac{d-a}{d})=\\lfloor \\tfrac{p-1}{d} \\rfloor +1 \\;.$ We first note that $\\tfrac{a}{d} \\in \\mathbb {Z}_p$ .", "It easy to see that $2 \\le p-\\lfloor \\tfrac{p-1}{d} \\rfloor \\le p$ and that $p-\\lfloor \\tfrac{p-1}{d} \\rfloor = p - \\tfrac{p-a}{d} \\equiv \\tfrac{a}{d} \\pmod {p}.$ Thus $\\operatorname{rep}(\\tfrac{a}{d})=p-\\lfloor \\tfrac{p-1}{d}\\rfloor $ .", "The second result follows from Proposition REF .", "We now use these properties to develop further results concerning the $p$ -adic gamma function.", "We recall the definition of the rising factorial $(a)_n$ in (REF ) and allow $a \\in \\mathbb {Z}_p$ .", "Proposition 3.9 Let $p$ be an odd prime and let $x \\in \\mathbb {Z}_p$ .", "If $0 \\le j \\le p\\in \\mathbb {Z}$ , then $\\Gamma _p{\\left({x+j}\\right)}={\\left\\lbrace \\begin{array}{ll}(-1)^j \\; \\Gamma _p{\\left({x}\\right)} \\, {\\left({x}\\right)}_{j} & \\qquad \\text{if } 0\\le j\\le p- \\operatorname{rep}(x),\\\\[6pt](-1)^j \\; \\Gamma _p{\\left({x}\\right)} \\, {\\left({x}\\right)}_{j} \\, {\\left({x+p-\\operatorname{rep}(x)}\\right)}^{-1} & \\qquad \\text{if } p- \\operatorname{rep}(x)+1\\le j\\le p.\\\\\\end{array}\\right.", "}$ For $j=0$ the result is trivial.", "Assume $j>0$ .", "For $0\\le k \\le j-1$ , $x +k \\in p \\hspace{1.0pt} \\mathbb {Z}_p \\Longleftrightarrow \\operatorname{rep}(x) +k \\in p \\hspace{1.0pt} \\mathbb {Z}_p \\Longleftrightarrow \\operatorname{rep}(x) +k = p \\Longleftrightarrow k = p -\\operatorname{rep}(x).$ Using Proposition REF (1) the result follows.", "For $i$ , $n \\in \\mathbb {Z}^{+}$ , we define the generalized harmonic sums, ${H}^{(i)}_{n}$ , by ${H}^{(i)}_{n}:= \\sum ^{n}_{j=1} \\frac{1}{j^i}$ and ${H}^{(i)}_{0}:=0$ .", "We can now use the above to develop some congruences for use in Section 4.", "Lemma 3.10 Let $p$ be an odd prime and let $x \\in \\mathbb {Z}_p$ .", "If $0 \\le j \\le p\\in \\mathbb {Z}$ , then $\\Gamma _p{\\left({x+j}\\right)}\\equiv \\left(\\operatorname{rep}(x)+j-1\\right)!", "{(-1)}^{\\operatorname{rep}(x)+j} \\cdot \\delta \\pmod {p}$ where $\\delta ={\\left\\lbrace \\begin{array}{ll}1 & \\text{if } 0\\le j\\le p- \\operatorname{rep}(x),\\\\[3pt]\\frac{1}{p} & \\text{if } p- \\operatorname{rep}(x)+1\\le j\\le p.\\\\[3pt]\\end{array}\\right.", "}$ By Proposition REF (3) we see that $\\Gamma _p{\\left({x+j}\\right)} \\equiv \\Gamma _p{\\left({\\operatorname{rep}(x)+j}\\right)} \\pmod {p}.$ Combining Proposition REF (1) and (REF ) yields the result.", "Lemma 3.11 Let $p\\ge 7$ be a prime and let $x \\in \\mathbb {Z}_p$ .", "If $0 \\le j \\le p-1\\in \\mathbb {Z}$ , then $G_1\\left(x+j\\right) -G_1(1+j)\\equiv H_{\\operatorname{rep}(x)-1+j}^{(1)}- H_{j}^{(1)}-\\delta \\\\[3pt]\\pmod {p}$ where $\\delta ={\\left\\lbrace \\begin{array}{ll}0 & \\text{if } 0\\le j\\le p- \\operatorname{rep}(x),\\\\[3pt]\\frac{1}{p} & \\text{if } p- \\operatorname{rep}(x)+1\\le j\\le p-1.\\\\[3pt]\\end{array}\\right.", "}$ By Corollary REF (3) we see that $G_1\\left(x+j\\right) -G_1(1+j) \\equiv G_1\\left(\\operatorname{rep}(x)+j\\right) -G_1(1+j) \\pmod {p}.$ Combining Proposition REF (1) and (REF ) yields the result.", "Lemma 3.12 Let $p\\ge 7$ be a prime and let $x \\in \\mathbb {Z}_p$ .", "If $0 \\le j \\le p-1\\in \\mathbb {Z}$ , then $G_1\\left(x+j\\right)^2-G_2\\left(x+j\\right) -G_1(1+j)^2+G_2(1+j)\\equiv H_{\\operatorname{rep}(x)-1+j}^{(2)}- H_{j}^{(2)}-\\delta \\\\[3pt]\\pmod {p}$ where $\\delta ={\\left\\lbrace \\begin{array}{ll}0 & \\text{if } 0\\le j\\le p- \\operatorname{rep}(x),\\\\[3pt]\\frac{1}{p^2} & \\text{if } p- \\operatorname{rep}(x)+1\\le j\\le p-1.\\\\[3pt]\\end{array}\\right.", "}$ By Corollary REF (3) and (4) we see that $G_1\\left(x+j\\right)^2-G_2\\left(x+j\\right) -G_1(1+j)^2+G_2(1+j)\\\\\\equiv G_1\\left(\\operatorname{rep}(x)+j\\right)^2-G_2\\left(\\operatorname{rep}(x)+j\\right) -G_1(1+j)^2+G_2(1+j)\\pmod {p}.$ Combining Proposition REF (2) and (REF ) yields the result.", "Lemma 3.13 Let $p\\ge 7$ be a prime and let $x \\in \\mathbb {Z}_p$ .", "Choose $m_1 \\in \\lbrace x, 1-x\\rbrace $ such that $\\operatorname{rep}(m_1) = \\max \\left(\\operatorname{rep}(x) ,\\operatorname{rep}(1-x)\\right)$ and set $m_2=1-m_1$ .", "Then for $0 \\le j<\\operatorname{rep}(m_1) \\in \\mathbb {Z}$ , $\\frac{\\Gamma _p{\\left({x+j}\\right)}\\Gamma _p{\\left({1-x+j}\\right)}}{\\Gamma _p{\\left({x}\\right)}\\Gamma _p{\\left({1-x}\\right)}{j!", "}^2} \\equiv (-1)^{j} \\left({\\genfrac{}{}{0.0pt}{}{\\operatorname{rep}(m_1)-1+j}{j}}\\right) \\left({\\genfrac{}{}{0.0pt}{}{\\operatorname{rep}(m_1)-1}{j}}\\right)\\cdot \\alpha \\\\\\cdot \\left[1-\\Bigl (\\operatorname{rep}(m_1)-m_1\\Bigr )\\left(H_{\\operatorname{rep}(m_1)-1+j}^{(1)}-H_{\\operatorname{rep}(m_2)-1+j}^{(1)}-\\beta \\right)\\right]\\pmod {p^2}$ where $\\alpha ={\\left\\lbrace \\begin{array}{ll}1 & \\text{if } 0\\le j \\le \\operatorname{rep}(m_2)-1,\\\\\\frac{1}{p} & \\text{if }\\operatorname{rep}(m_2) \\le j < \\operatorname{rep}(m_1),\\end{array}\\right.", "}& \\quad and \\quad \\beta ={\\left\\lbrace \\begin{array}{ll}0 & \\text{if } 0\\le j \\le \\operatorname{rep}(m_2)-1,\\\\\\frac{1}{p} & \\text{if }\\operatorname{rep}(m_2) \\le j < \\operatorname{rep}(m_1).\\end{array}\\right.", "}$ We first note that, by Proposition REF , $\\operatorname{rep}(m_2)-p-m_2=-\\left(\\operatorname{rep}(m_1)-m_1\\right) \\quad \\textup {and} \\quad \\operatorname{rep}(m_2)-p=1-\\operatorname{rep}(m_1).$ Then by Proposition REF (2) we get that $&\\Gamma _p{\\left({x+j}\\right)}\\Gamma _p{\\left({1-x+j}\\right)}=\\Gamma _p{\\left({m_1+j}\\right)}\\Gamma _p{\\left({m_2+j}\\right)}\\\\[6pt]& \\equiv \\left[\\Gamma _p{\\left({\\operatorname{rep}(m_1)+j}\\right)}-\\left(\\operatorname{rep}(m_1)-m_1\\right)\\Gamma _p^{\\prime }\\left(\\operatorname{rep}(m_1)+j\\right) \\right]\\\\[6pt]& \\quad \\; \\cdot \\left[\\Gamma _p{\\left({\\operatorname{rep}(m_2)-p+j}\\right)}-\\left(\\operatorname{rep}(m_2)-p-m_2\\right)\\Gamma _p^{\\prime }\\left(\\operatorname{rep}(m_2)+j\\right) \\right]\\\\[6pt]& \\equiv \\Gamma _p{\\left({\\operatorname{rep}(m_1)+j}\\right)} \\Gamma _p{\\left({1-\\operatorname{rep}(m_1)+j}\\right)} -\\left(\\operatorname{rep}(m_1)-m_1\\right)\\\\[6pt]& \\quad \\; \\cdot \\left[\\Gamma _p^{\\prime }\\left(\\operatorname{rep}(m_1)+j\\right) \\Gamma _p{\\left({\\operatorname{rep}(m_2)-p+j}\\right)}-\\Gamma _p^{\\prime }\\left(\\operatorname{rep}(m_2)+j\\right) \\Gamma _p{\\left({\\operatorname{rep}(m_1)+j}\\right)} \\right]\\pmod {p^2}.$ Using Proposition REF (3) we have $\\Gamma _p^{\\prime }&\\left(\\operatorname{rep}(m_1)+j\\right) \\Gamma _p{\\left({\\operatorname{rep}(m_2)-p+j}\\right)}-\\Gamma _p^{\\prime }\\left(\\operatorname{rep}(m_2)+j\\right) \\Gamma _p{\\left({\\operatorname{rep}(m_1)+j}\\right)}\\\\[6pt]&\\equiv \\Gamma _p{\\left({\\operatorname{rep}(m_1)+j}\\right)} \\Gamma _p{\\left({\\operatorname{rep}(m_2)+j}\\right)}\\left[G_1\\left(\\operatorname{rep}(m_1)+j\\right) -G_1\\left(\\operatorname{rep}(m_2)+j\\right)\\right]\\\\[6pt]&\\equiv \\Gamma _p{\\left({\\operatorname{rep}(m_1)+j}\\right)} \\Gamma _p{\\left({1-\\operatorname{rep}(m_1)+j}\\right)}\\left[G_1\\left(\\operatorname{rep}(m_1)+j\\right)-G_1\\left(\\operatorname{rep}(m_2)+j\\right)\\right]\\pmod {p}.$ So $\\Gamma _p{\\left({x+j}\\right)}\\Gamma _p{\\left({1-x+j}\\right)}\\equiv \\Gamma _p{\\left({\\operatorname{rep}(m_1)+j}\\right)} \\Gamma _p{\\left({1-\\operatorname{rep}(m_1)+j}\\right)} \\\\[6pt] \\cdot \\left[1-\\left(\\operatorname{rep}(m_1)-m_1\\right)\\left(G_1\\left(\\operatorname{rep}(m_1)+j\\right) -G_1\\left(\\operatorname{rep}(m_2)+j\\right)\\right)\\right]\\pmod {p^2}.$ By Proposition REF (1), $G_1\\left(\\operatorname{rep}(m_1)+j\\right) -G_1\\left(\\operatorname{rep}(m_2)+j\\right)= H_{\\operatorname{rep}(m_1)-1+j}^{(1)}-H_{\\operatorname{rep}(m_2)-1+j}^{(1)}-\\beta .$ For $j<\\operatorname{rep}(m_1)$ , Proposition REF gives us $\\Gamma _p{\\left({\\operatorname{rep}(m_1)+j}\\right)} \\Gamma _p{\\left({1-\\operatorname{rep}(m_1)+j}\\right)}&= (-1)^{\\operatorname{rep}(m_1)-j} \\frac{\\Gamma _p{\\left({\\operatorname{rep}(m_1)+j}\\right)}}{\\Gamma _p{\\left({\\operatorname{rep}(m_1)-j}\\right)}}\\\\[3pt]&=(-1)^{\\operatorname{rep}(m_1)-j} \\frac{\\left(\\operatorname{rep}(m_1)-1+j\\right)!", "\\, (\\alpha )}{\\left(\\operatorname{rep}(m_1)-1-j\\right)! }", "\\genfrac{}{}{0.0pt}{}{}{.", "}$ The result follows from noting that $\\frac{\\left(\\operatorname{rep}(m_1)-1+j\\right)!}{\\left(\\operatorname{rep}(m_1)-1-j\\right)!", "\\hspace{2.0pt} {j!", "}^2}= \\left({\\genfrac{}{}{0.0pt}{}{\\operatorname{rep}(m_1)-1+j}{j}}\\right) \\left({\\genfrac{}{}{0.0pt}{}{\\operatorname{rep}(m_1)-1}{j}}\\right) \\;$ and $\\Gamma _p{\\left({x}\\right)}\\Gamma _p{\\left({1-x}\\right)}=(-1)^{\\operatorname{rep}(m_1)}.$ Lemma 3.14 Let $p\\ge 7$ be a prime and let $x \\in \\mathbb {Z}_p$ .", "Choose $m_1 \\in \\lbrace x, 1-x\\rbrace $ such that $\\operatorname{rep}(m_1) = \\max \\left(\\operatorname{rep}(x) ,\\operatorname{rep}(1-x)\\right)$ and set $m_2=1-m_1$ .", "Then for $0 \\le j<\\operatorname{rep}(m_1) \\in \\mathbb {Z}$ , $G_1\\left(x+j\\right)+G_1\\left(1-x+j\\right) -2G_1(1+j)\\equiv H_{\\operatorname{rep}(m_1)-1+j}^{(1)}+H_{\\operatorname{rep}(m_1)-1-j}^{(1)}-2\\hspace{1.0pt} H_{j}^{(1)}-\\alpha \\\\[3pt]+\\left(\\operatorname{rep}(m_1)-m_1\\right)\\left(H_{\\operatorname{rep}(m_1)-1+j}^{(2)}-H_{\\operatorname{rep}(m_2)-1+j}^{(2)}-\\beta \\right)\\pmod {p^2}$ where $\\alpha ={\\left\\lbrace \\begin{array}{ll}0 & \\text{if }0\\le j \\le \\operatorname{rep}(m_2)-1,\\\\\\frac{1}{p} &\\text{if } \\operatorname{rep}(m_2) \\le j < \\operatorname{rep}(m_1),\\end{array}\\right.", "}& \\qquad and \\qquad \\beta ={\\left\\lbrace \\begin{array}{ll}0 &\\text{if } 0\\le j \\le \\operatorname{rep}(m_2)-1,\\\\\\frac{1}{p^2} &\\text{if } \\operatorname{rep}(m_2) \\le j < \\operatorname{rep}(m_1).\\end{array}\\right.", "}$ Using Corollary REF and Corollary REF we see that $G_1\\left(m_1+j\\right)\\equiv G_1\\left(\\operatorname{rep}(m_1)+j\\right)\\\\+\\left(\\operatorname{rep}(m_1)-m_1\\right)\\left(G_1\\bigl (\\operatorname{rep}(m_1)+j\\bigr )^2-G_2\\left(\\operatorname{rep}(m_1)+j\\right)\\right)\\pmod {p^2}$ and $G_1\\left(m_2+j\\right)\\equiv G_1\\left(\\operatorname{rep}(m_2)-p+j\\right)\\\\+\\left(\\operatorname{rep}(m_2)-p-m_2\\right)\\left(G_1\\bigl (\\operatorname{rep}(m_2)+j\\bigr )^2-G_2\\left(\\operatorname{rep}(m_2)+j\\right)\\right)\\pmod {p^2}.$ Therefore, using Proposition REF (3) and (REF ), $G_1\\left(x+j\\right)&+G_1\\left(1-x+j\\right)\\\\[4pt]&\\;=G_1\\left(m_1+j\\right)+G_1\\left(m_2+j\\right)\\\\[4pt]& \\; \\equiv G_1\\left(\\operatorname{rep}(m_1)+j\\right)+G_1\\left(\\operatorname{rep}(m_1)-j\\right)+\\left(\\operatorname{rep}(m_1)-m_1\\right) \\left[G_1\\bigl (\\operatorname{rep}(m_1)+j\\bigr )^2\\right.", "\\\\[4pt] & \\left.", "\\qquad -G_2\\left(\\operatorname{rep}(m_1)+j\\right)-G_1\\bigl (\\operatorname{rep}(m_2)+j\\bigr )^2+G_2\\bigl (\\operatorname{rep}(m_2)+j\\bigr )\\right]\\pmod {p^2}.$ By Proposition REF (1), (2) we get that $G_1\\left(\\operatorname{rep}(m_1)+j\\right)-G_1(1+j)=H_{\\operatorname{rep}(m_1)-1+j}^{(1)}- H_{j}^{(1)}-\\alpha ,$ $G_1\\bigl (\\operatorname{rep}(m_1)+j\\bigr )^2-G_2\\left(\\operatorname{rep}(m_1)+j\\right)-G_1\\bigl (\\operatorname{rep}(m_2)+j\\bigr )^2+G_2\\left(\\operatorname{rep}(m_2)+j\\right)\\\\[3pt]=H_{\\operatorname{rep}(m_1)-1+j}^{(2)}-H_{\\operatorname{rep}(m_2)-1+j}^{(2)}-\\beta $ and, for $0\\le j < \\operatorname{rep}(m_1)$ , $G_1\\left(\\operatorname{rep}(m_1)-j\\right)-G_1(1+j)=H_{\\operatorname{rep}(m_1)-1-j}^{(1)}- H_{j}^{(1)}.$ The result follows." ], [ "A Combinatorial Technique", "In this section we generalize a combinatorial technique from [17] to produce Lemmas REF and REF below.", "We will use both of these lemmas in proving Theorems REF to REF in Section 4.", "Lemma 3.15 Let $p$ be an odd prime and let $a_1, a_2, \\cdots , a_n \\in \\mathbb {Z}^{+}$ be such that ${T:=\\sum _{i=1}^{n} a_i \\le 2(p-1)}$ .", "Then $\\sum _{j=0}^{p-1}\\left[ \\prod _{i=1}^{n} {\\left({j+1}\\right)}_{a_i}\\right] \\left[1+j\\hspace{1.0pt} \\sum _{i=1}^{n} \\left( H_{a_i+j}^{(1)} - \\hspace{1.0pt} H_{j}^{(1)}\\right)\\right] \\equiv {\\left\\lbrace \\begin{array}{ll}0 & \\textup {if } T< 2(p-1),\\\\1 & \\textup {if } T=2(p-1),\\end{array}\\right.", "}\\pmod {p}.$ Let $P(j):= \\frac{d}{dj} \\left[ j \\prod _{i=1}^{n} {\\left({j+1}\\right)}_{a_i} \\right] = \\sum _{k=0}^{T} b_k j^k.$ Differentiating we see that $\\frac{d}{dj} \\left[ j \\prod _{i=1}^{n} {\\left({j+1}\\right)}_{a_i} \\right]&= \\left[ \\prod _{i=1}^{n} {\\left({j+1}\\right)}_{a_i} \\right] \\left[ 1 + j \\sum _{i=1}^{n} \\left( H_{a_i+j}^{(1)} - H_{j}^{(1)} \\right)\\right]\\genfrac{}{}{0.0pt}{}{}{.", "}$ So it suffices to show $\\sum _{j=0}^{p-1} P(j) \\equiv {\\left\\lbrace \\begin{array}{ll}0 & \\textup {if } T< 2(p-1),\\\\1 & \\textup {if } T=2(p-1),\\end{array}\\right.", "}\\pmod {p}.$ For a positive integer $k$ , we have $\\sum ^{p-1}_{j=1} j^k\\equiv {\\left\\lbrace \\begin{array}{ll}-1 \\pmod {p}& \\text{if $(p-1) \\vert k$} \\, ,\\\\\\phantom{-}0 \\pmod {p}& \\text{otherwise} \\, .\\end{array}\\right.", "}$ Consequently, $\\sum _{j=0}^{p-1} P(j)\\equiv \\sum _{k=1}^{T} b_k \\sum _{j=1}^{p-1} j^k\\equiv {\\left\\lbrace \\begin{array}{ll}0 & \\textup {if } T< p-1,\\\\-b_{p-1} & \\textup {if } p-1 \\le T< 2(p-1),\\\\-b_{p-1} -b_{2(p-1)} & \\textup {if } T=2(p-1),\\end{array}\\right.", "}\\pmod {p}.$ By definition of $P(j)$ we see that $\\prod _{i=1}^{n} {\\left({j+1}\\right)}_{a_i} = \\sum _{k=0}^{T} \\frac{b_k}{k+1} j^k \\;,$ which is monic with integer coefficients.", "Thus $p\\mid b_{p-1}$ for $T \\ge p-1$ and $b_{2(p-1)}=2p-1$ if $T=2(p-1)$ .", "Therefore $b_{p-1}\\equiv 0 \\pmod {p}$ and $b_{2(p-1)}\\equiv -1\\pmod {p}$ in these cases.", "Lemma 3.16 Let $p$ be an odd prime and let $a_1, a_2, \\cdots , a_n \\in \\mathbb {Z}^{+}$ be such that ${T:=\\sum _{i=1}^{n} a_i \\le 2(p-1)}$ .", "Then $\\sum _{j=0}^{p-1} \\Biggl [ \\prod _{i=1}^{n} {\\left({j+1}\\right)}_{a_i} \\Biggr ] \\Biggl [j\\hspace{1.0pt} \\sum _{i=1}^{n} \\left( H_{a_i+j}^{(1)} - \\hspace{1.0pt} H_{j}^{(1)}\\right)+\\frac{j^2}{2} \\Biggl \\lbrace \\left(\\sum _{i=1}^{n} \\left( H_{a_i+j}^{(1)} - \\hspace{1.0pt} H_{j}^{(1)}\\right)\\right)^2\\\\- \\sum _{i=1}^{n} \\left( H_{a_i+j}^{(2)} - \\hspace{1.0pt} H_{j}^{(2)}\\right) \\Biggr \\rbrace \\Biggr ] \\equiv {\\left\\lbrace \\begin{array}{ll}0 & \\textup {if } T< 2(p-1),\\\\-1 & \\textup {if } T=2(p-1),\\end{array}\\right.", "}\\pmod {p}.$ We first recognize that $\\Biggl [\\prod _{i=1}^{n} {\\left({j+1}\\right)}_{a_i} \\Biggr ] \\Biggl [j\\hspace{1.0pt} \\sum _{i=1}^{n} \\left( H_{a_i+j}^{(1)} - \\hspace{1.0pt} H_{j}^{(1)}\\right)+\\frac{j^2}{2} \\Biggl \\lbrace \\left(\\sum _{i=1}^{n} \\left( H_{a_i+j}^{(1)} - \\hspace{1.0pt} H_{j}^{(1)}\\right)\\right)^2- \\sum _{i=1}^{n} \\left( H_{a_i+j}^{(2)} - \\hspace{1.0pt} H_{j}^{(2)}\\right) \\Biggr \\rbrace \\Biggr ]\\\\=\\frac{j}{2} \\frac{d^2}{dj^2} \\left[ j \\prod _{i=1}^{n} {\\left({j+1}\\right)}_{a_i} \\right]= \\sum _{k=0}^{T} b_k j^k.$ Then applying (REF ) in a similar fashion to that used in the proof of Lemma REF yields the result." ], [ "Binomial Coefficient-Generalized Harmonic Sum Identities", "We now state two binomial coefficient–generalized harmonic sum identities from [20] which we will use in Section 4.", "Theorem 3.17 ([20] Thm.", "2) Let $m,n$ be positive integers with $m\\ge n$ .", "Then $\\sum _{k=0}^{n} \\biggl ({\\genfrac{}{}{0.0pt}{}{m+k}{k}}\\biggr ) \\biggl ({\\genfrac{}{}{0.0pt}{}{m}{k}}\\biggr ) \\biggl ({\\genfrac{}{}{0.0pt}{}{n+k}{k}}\\biggr ) \\biggl ({\\genfrac{}{}{0.0pt}{}{n}{k}}\\biggr )\\biggl [ 1+k \\left(H_{m+k}^{(1)} +H_{m-k}^{(1)} + H_{n+k}^{(1)}+ H_{n-k}^{(1)} -4H_k^{(1)}\\right) \\biggr ]\\\\[6pt]+\\sum _{k=n+1}^{m} (-1)^{k-n} \\biggl ({\\genfrac{}{}{0.0pt}{}{m+k}{k}}\\biggr ) \\biggl ({\\genfrac{}{}{0.0pt}{}{m}{k}}\\biggr ) \\biggl ({\\genfrac{}{}{0.0pt}{}{n+k}{k}}\\biggr ) \\Big / \\biggl ({\\genfrac{}{}{0.0pt}{}{k-1}{n}}\\biggr )=(-1)^{m+n}.$ Theorem 3.18 ([20] Thm.", "3) Let $l,m, n$ be positive integers with $l > m\\ge n\\ge \\frac{l}{2}$ and $c_1, c_2 \\in \\mathbb {Q}$ some constants.", "Then $\\sum _{k=0}^{n} \\biggl ({\\genfrac{}{}{0.0pt}{}{m+k}{k}}\\biggr ) \\biggl ({\\genfrac{}{}{0.0pt}{}{m}{k}}\\biggr ) \\biggl ({\\genfrac{}{}{0.0pt}{}{n+k}{k}}\\biggr ) \\biggl ({\\genfrac{}{}{0.0pt}{}{n}{k}}\\biggr )\\Biggl \\lbrace \\biggl [1+k \\Bigl (H_{m+k}^{(1)} +H_{m-k}^{(1)} + H_{n+k}^{(1)} + H_{n-k}^{(1)}-4H_k^{(1)} \\Bigr )\\biggr ]\\\\[5pt]\\cdot \\biggl [c_1\\left(H_{k+n}^{(1)} - H_{k+l-n-1}^{(1)}\\right) + c_2 \\Bigl (H_{k+m}^{(1)} - H_{k+l-m-1}^{(1)}\\Bigr )\\biggr ]-k\\biggl [c_1\\left(H_{k+n}^{(2)} - H_{k+l-n-1}^{(2)}\\right)\\\\[5pt]+ c_2 \\left(H_{k+m}^{(2)} - H_{k+l-m-1}^{(2)}\\right)\\biggr ] \\Biggr \\rbrace + \\sum _{k=n+1}^{m} (-1)^{k-n} \\biggl ({\\genfrac{}{}{0.0pt}{}{m+k}{k}}\\biggr ) \\biggl ({\\genfrac{}{}{0.0pt}{}{m}{k}}\\biggr ) \\biggl ({\\genfrac{}{}{0.0pt}{}{n+k}{k}}\\biggr ) \\Big / \\biggl ({\\genfrac{}{}{0.0pt}{}{k-1}{n}}\\biggr )\\\\[5pt]\\cdot \\biggl [c_1\\left(H_{k+n}^{(1)} - H_{k+l-n-1}^{(1)}\\right) + c_2 \\left(H_{k+m}^{(1)} - H_{k+l-m-1}^{(1)}\\right)\\biggr ] = 0.$" ], [ "Proofs", "The proofs of Theorems REF to REF proceed broadly along similar lines with the overall idea being to expand the relevant $_{n+1}G$ function using properties of the $p$ -adic gamma function and its logarithmic derivatives, identify the contribution of the truncated generalized hypergeometric series, and then use the results of Section 3.2 and 3.3 to simplify the remaining terms.", "In doing this we extend significantly the methods in [2], [13] and [17].", "For brevity we include only the proof of Theorem REF which is the most complex and uses all the additional methods developed for these results.", "Full details of all proofs can be found in [18].", "Assume without loss of generality that $r<d/2$ .", "We easily check the result for primes $p<7$ .", "Let $p\\ge 7$ be a prime which satisfies the conditions of the theorem.", "Let $p\\equiv a \\pmod {d}$ with $0<a<d$ .", "Then $a \\in \\lbrace 1,r,d-r,d-1\\rbrace $ .", "We define $s:=ar-d \\left\\lfloor \\frac{ar}{d} \\right\\rfloor = d \\, \\langle \\frac{ar}{d} \\rangle $ .", "It is easy to check that $\\lbrace 1, r, d-r, d-1\\rbrace = \\lbrace a, s, d-s, d-a\\rbrace $ for all possible $a$ .", "From Lemma REF we know that $\\operatorname{rep}(\\frac{a}{d}) = p- \\lfloor \\frac{p-1}{d} \\rfloor $ and $\\operatorname{rep}(\\frac{d-a}{d}) = \\lfloor \\frac{p-1}{d}\\rfloor +1$ .", "It is easy to check that $\\operatorname{rep}(\\frac{s}{d})=p-r \\lfloor \\tfrac{p-1}{d} \\rfloor - \\left\\lfloor \\tfrac{ar}{d} \\right\\rfloor $ .", "Then, by Proposition REF , $\\operatorname{rep}(\\frac{d-s}{d})= r \\lfloor \\tfrac{p-1}{d} \\rfloor +1 +\\left\\lfloor \\tfrac{ar}{d} \\right\\rfloor $ .", "Therefore $\\Bigl \\lbrace \\operatorname{rep}\\left(\\tfrac{1}{d}\\right),\\operatorname{rep}\\left(\\tfrac{r}{d}\\right),\\operatorname{rep}\\left(\\tfrac{d-r}{d}\\right),\\operatorname{rep}\\left(\\tfrac{d-1}{d}\\right)\\Bigr \\rbrace =\\Bigl \\lbrace \\lfloor \\tfrac{p-1}{d} \\rfloor +1,r \\lfloor \\tfrac{p-1}{d}\\rfloor +1 +\\lfloor \\tfrac{ar}{d} \\rfloor , p-r \\lfloor \\tfrac{p-1}{d} \\rfloor - \\lfloor \\tfrac{ar}{d} \\rfloor ,p-\\lfloor \\tfrac{p-1}{d} \\rfloor \\Bigr \\rbrace $ , where the exact correspondence between the elements of each set depends on the choice of $p$ .", "If we let $m_1:=\\frac{a}{d}$ , $m_2:=\\frac{s}{d}$ , $m_3:=\\frac{d-s}{d}$ and $m_4:=\\frac{d-a}{d}$ then $\\operatorname{rep}\\bigl (m_4\\bigr )< \\operatorname{rep}\\bigl (m_3\\bigr ) \\le \\operatorname{rep}\\bigl (m_2\\bigr ) < \\operatorname{rep}\\bigl (m_1\\bigr )$ .", "We reduce Definition REF modulo $p^3$ while using Proposition REF (2), (3) to expand the terms involved, noting that $t:=1+p+p^2 \\equiv \\frac{1}{1-p} \\pmod {p^3}$ , to get $&{_{4}G} \\left(\\tfrac{1}{d} , \\tfrac{r}{d}, 1-\\tfrac{r}{d} , 1-\\tfrac{1}{d}\\right)_p\\equiv \\sum _{j=0}^{\\left\\lfloor \\frac{p-1}{d} \\right\\rfloor }\\frac{\\Gamma _p{\\bigl ({\\frac{1}{d}+jt}\\bigr )}\\Gamma _p{\\bigl ({\\frac{r}{d}+jt}\\bigr )}\\Gamma _p{\\bigl ({\\frac{d-r}{d}+jt}\\bigr )}\\Gamma _p{\\bigl ({\\frac{d-1}{d}+jt}\\bigr )}}{\\Gamma _p{\\bigl ({\\frac{1}{d}}\\bigr )}\\Gamma _p{\\bigl ({\\frac{r}{d}}\\bigr )}\\Gamma _p{\\bigl ({\\frac{d-r}{d}}\\bigr )}\\Gamma _p{\\bigl ({\\frac{d-1}{d}}\\bigr )}{\\Gamma _p{\\bigl ({1+jt}\\bigr )}}^{4}}\\\\ &+p \\left\\lbrace \\sum _{j=0}^{\\left\\lfloor \\frac{p-1}{d} \\right\\rfloor }\\frac{\\Gamma _p{\\bigl ({\\frac{1}{d}+j+jp}\\bigr )}\\Gamma _p{\\bigl ({\\frac{r}{d}+j+jp}\\bigr )}\\Gamma _p{\\bigl ({\\frac{d-r}{d}+j+jp}\\bigr )}\\Gamma _p{\\bigl ({\\frac{d-1}{d}+j+jp}\\bigr )}}{\\Gamma _p{\\bigl ({\\frac{1}{d}}\\bigr )}\\Gamma _p{\\bigl ({\\frac{r}{d}}\\bigr )}\\Gamma _p{\\bigl ({\\frac{d-r}{d}}\\bigr )}\\Gamma _p{\\bigl ({\\frac{d-1}{d}}\\bigr )}{\\Gamma _p{\\bigl ({1+j+jp}\\bigr )}}^{4}}\\right.", "\\\\ &\\left.", "\\qquad - \\sum _{j=\\left\\lfloor \\frac{p-1}{d} \\right\\rfloor +1}^{\\left\\lfloor \\frac{r(p-1)}{d} \\right\\rfloor }\\frac{\\Gamma _p{\\bigl ({\\frac{d+1}{d}+j+jp}\\bigr )}\\Gamma _p{\\bigl ({\\frac{r}{d}+j+jp}\\bigr )}\\Gamma _p{\\bigl ({\\frac{d-r}{d}+j+jp}\\bigr )}\\Gamma _p{\\bigl ({\\frac{d-1}{d}+j+jp}\\bigr )}}{\\Gamma _p{\\bigl ({\\frac{1}{d}}\\bigr )}\\Gamma _p{\\bigl ({\\frac{r}{d}}\\bigr )}\\Gamma _p{\\bigl ({\\frac{d-r}{d}}\\bigr )}\\Gamma _p{\\bigl ({\\frac{d-1}{d}}\\bigr )}{\\Gamma _p{\\bigl ({1+j+jp}\\bigr )}}^{4}}\\right\\rbrace \\\\ &+p^2 \\left\\lbrace \\sum _{j=0}^{\\left\\lfloor \\frac{p-1}{d} \\right\\rfloor }\\frac{\\Gamma _p{\\bigl ({\\frac{1}{d}+j}\\bigr )}\\Gamma _p{\\bigl ({\\frac{r}{d}+j}\\bigr )}\\Gamma _p{\\bigl ({\\frac{d-r}{d}+j}\\bigr )}\\Gamma _p{\\bigl ({\\frac{d-1}{d}+j}\\bigr )}}{\\Gamma _p{\\bigl ({\\frac{1}{d}}\\bigr )}\\Gamma _p{\\bigl ({\\frac{r}{d}}\\bigr )}\\Gamma _p{\\bigl ({\\frac{d-r}{d}}\\bigr )}\\Gamma _p{\\bigl ({\\frac{d-1}{d}}\\bigr )}{\\Gamma _p{\\bigl ({1+j}\\bigr )}}^{4}}\\right.", "\\\\ &\\left.", "\\qquad \\; \\;- \\sum _{j=\\left\\lfloor \\frac{p-1}{d} \\right\\rfloor +1}^{\\left\\lfloor \\frac{r(p-1)}{d} \\right\\rfloor }\\frac{\\Gamma _p{\\bigl ({\\frac{d+1}{d}+j}\\bigr )}\\Gamma _p{\\bigl ({\\frac{r}{d}+j}\\bigr )}\\Gamma _p{\\bigl ({\\frac{d-r}{d}+j}\\bigr )}\\Gamma _p{\\bigl ({\\frac{d-1}{d}+j}\\bigr )}}{\\Gamma _p{\\bigl ({\\frac{1}{d}}\\bigr )}\\Gamma _p{\\bigl ({\\frac{r}{d}}\\bigr )}\\Gamma _p{\\bigl ({\\frac{d-r}{d}}\\bigr )}\\Gamma _p{\\bigl ({\\frac{d-1}{d}}\\bigr )}{\\Gamma _p{\\bigl ({1+j}\\bigr )}}^{4}}\\right.", "\\\\ &\\left.", "\\qquad \\; \\;+\\sum _{j=\\left\\lfloor \\frac{r(p-1)}{d} \\right\\rfloor +1}^{\\left\\lfloor (d-r)\\frac{p-1}{d} \\right\\rfloor }\\frac{\\Gamma _p{\\bigl ({\\frac{d+1}{d}+j}\\bigr )}\\Gamma _p{\\bigl ({\\frac{d+r}{d}+j}\\bigr )}\\Gamma _p{\\bigl ({\\frac{d-r}{d}+j}\\bigr )}\\Gamma _p{\\bigl ({\\frac{d-1}{d}+j}\\bigr )}}{\\Gamma _p{\\bigl ({\\frac{1}{d}}\\bigr )}\\Gamma _p{\\bigl ({\\frac{r}{d}}\\bigr )}\\Gamma _p{\\bigl ({\\frac{d-r}{d}}\\bigr )}\\Gamma _p{\\bigl ({\\frac{d-1}{d}}\\bigr )}{\\Gamma _p{\\bigl ({1+j}\\bigr )}}^{4}}\\right\\rbrace \\pmod {p^3}.$ Central to the proof will be the relationship between the $\\operatorname{rep}(m_k)$ , for $2 \\le k \\le 4$ , and the limits of summation of the individual sums in the expanded $_4G$ above.", "These relationships are outlined below and the reader may want to refer to them throughout the rest of the proof.", "$\\operatorname{rep}(m_4)= \\lfloor \\tfrac{p-1}{d} \\rfloor + 1$ .", "$\\operatorname{rep}(m_3)= \\bigl \\lfloor \\frac{r(p-1)}{d} \\bigr \\rfloor + 1 +{\\left\\lbrace \\begin{array}{ll}1 & \\textup {if } p \\equiv r \\pmod {d} \\textup { and } r^2 \\equiv 1 \\pmod {d},\\\\[3pt]1 & \\textup {if } p \\equiv d-r \\pmod {d} \\textup { and } r^2 \\equiv -1 \\pmod {d},\\\\[3pt]0 & \\textup {otherwise.}\\end{array}\\right.", "}$ $\\operatorname{rep}(m_2) = \\bigl \\lfloor (d-r)\\frac{(p-1)}{d} \\bigr \\rfloor + 1 +{\\left\\lbrace \\begin{array}{ll}1 & \\textup {if } p \\equiv r \\pmod {d} \\textup { and } r^2 \\equiv -1 \\pmod {d},\\\\[3pt]1 & \\textup {if } p \\equiv d-r \\pmod {d} \\textup { and } r^2 \\equiv 1 \\pmod {d},\\\\[3pt]1 & \\textup {if } p \\equiv d-1 \\pmod {d},\\\\[3pt]0 & \\textup {otherwise.}\\end{array}\\right.", "}$ We now consider $\\Gamma _p{\\left({\\frac{d+1}{d}+j+jp}\\right)}$ and $\\Gamma _p{\\left({\\frac{d+1}{d}+j}\\right)}$ .", "As $\\operatorname{rep}\\bigl (\\tfrac{d-1}{d}\\bigr )=p+1-\\operatorname{rep}\\bigl (\\tfrac{1}{d}\\bigr )$ by Proposition REF , we have that $\\tfrac{1}{d}+j +jp \\in p\\mathbb {Z}_p \\Longleftrightarrow \\tfrac{1}{d}+j \\in p\\mathbb {Z}_p\\Longleftrightarrow \\operatorname{rep}\\left(\\tfrac{1}{d}\\right)+j \\in p\\mathbb {Z}_p\\Longleftrightarrow \\operatorname{rep}\\left(\\tfrac{1}{d}\\right)+j =p\\Longleftrightarrow j= \\operatorname{rep}\\left(\\tfrac{d-1}{d}\\right) -1.$ Consequently, we see that the only time that $\\tfrac{1}{d}+j \\in p\\mathbb {Z}_p$ for $\\lfloor \\tfrac{p-1}{d} \\rfloor +1 \\le j \\le \\lfloor (d-r)\\tfrac{p-1}{d}\\rfloor $ is when $p\\equiv r \\pmod {d}$ with $r^2\\equiv 1 \\pmod {d}$ or $p\\equiv d-r \\pmod {d}$ with $r^2\\equiv -1 \\pmod {d}$ and in these cases $j=\\operatorname{rep}(m_3)-1=\\bigl \\lfloor \\tfrac{r(p-1)}{d} \\bigr \\rfloor +1$ .", "Therefore, using Proposition REF (1) we get that for $\\lfloor \\tfrac{p-1}{d}\\rfloor +1 \\le j \\le \\lfloor (d-r)\\tfrac{p-1}{d} \\rfloor $ , $\\Gamma _p{\\left({\\tfrac{d+1}{d}+j}\\right)}={\\left\\lbrace \\begin{array}{ll}-\\Gamma _p{\\left({\\tfrac{1}{d}+j}\\right)} & \\text{if } j=\\bigl \\lfloor \\tfrac{r(p-1)}{d} \\bigr \\rfloor +1,\\; p\\equiv r \\hspace{2.77771pt}({mod}\\,\\,d), \\;r^2\\equiv 1 \\hspace{2.77771pt}({mod}\\,\\,d),\\\\[3pt]-\\Gamma _p{\\left({\\tfrac{1}{d}+j}\\right)} & \\text{if }j=\\bigl \\lfloor \\tfrac{r(p-1)}{d} \\bigr \\rfloor +1, \\;p\\equiv d-r \\hspace{2.77771pt}({mod}\\,\\,d), \\; r^2\\equiv -1 \\hspace{2.77771pt}({mod}\\,\\,d),\\\\[3pt]-\\Gamma _p{\\left({\\tfrac{1}{d}+j}\\right)} \\left(\\tfrac{1}{d}+j\\right) & \\textup {otherwise,}\\end{array}\\right.", "}$ and that $\\Gamma _p{\\left({\\tfrac{d+1}{d}+j+jp}\\right)}=-\\left(\\tfrac{1}{d}+j+jp\\right)\\Gamma _p{\\left({\\tfrac{1}{d}+j+jp}\\right)}$ for $\\lfloor \\tfrac{p-1}{d} \\rfloor +1 \\le j \\le \\bigl \\lfloor \\tfrac{r(p-1)}{d} \\bigr \\rfloor $ .", "Considering $\\Gamma _p{\\left({\\frac{d+r}{d}+j}\\right)}$ in a similar fashion, we get that $\\Gamma _p{\\left({\\tfrac{d+r}{d}+j}\\right)}=-\\left(\\tfrac{r}{d}+j\\right)\\Gamma _p{\\left({\\tfrac{r}{d}+j}\\right)}$ for $\\bigl \\lfloor \\frac{r(p-1)}{d} \\bigr \\rfloor +1 \\le j \\le {\\lfloor (d-r)\\frac{p-1}{d} \\rfloor }$ .", "Applying these results and substituting $\\lbrace m_k\\rbrace $ for $\\lbrace \\frac{1}{d}, \\frac{r}{d}, \\frac{d-r}{d}, \\frac{d-1}{d}\\rbrace $ yields $&{_{4}G} \\left(\\tfrac{1}{d} , \\tfrac{r}{d}, 1-\\tfrac{r}{d} , 1-\\tfrac{1}{d}\\right)_p\\equiv \\sum _{j=0}^{\\left\\lfloor \\frac{p-1}{d} \\right\\rfloor }\\left[ \\, \\prod _{k=1}^{4} \\frac{\\Gamma _p{\\bigl ({m_k+j+jp+jp^2}\\bigr )}}{\\Gamma _p{\\bigl ({m_k}\\bigr )}{\\Gamma _p{\\bigl ({1+j+jp+jp^2}\\bigr )}}}\\right]\\\\ &+p \\left\\lbrace \\sum _{j=0}^{\\left\\lfloor \\frac{p-1}{d} \\right\\rfloor }\\left[ \\, \\prod _{k=1}^{4} \\frac{ \\Gamma _p{\\bigl ({m_k+j+jp}\\bigr )}}{ \\Gamma _p{\\bigl ({m_k}\\bigr )}{\\Gamma _p{\\bigl ({1+j+jp}\\bigr )}}}\\right]+ \\sum _{j=\\left\\lfloor \\frac{p-1}{d} \\right\\rfloor +1}^{\\left\\lfloor \\frac{r(p-1)}{d} \\right\\rfloor }\\left[ \\, \\prod _{k=1}^{4} \\frac{ \\Gamma _p{\\bigl ({m_k+j+jp}\\bigr )} }{ \\Gamma _p{\\bigl ({m_k}\\bigr )}{\\Gamma _p{\\bigl ({1+j+jp}\\bigr )}}}\\right]\\bigl (\\tfrac{1}{d}+j+jp\\bigr )\\right\\rbrace \\\\ &+p^2 \\left\\lbrace \\sum _{j=0}^{\\left\\lfloor \\frac{p-1}{d} \\right\\rfloor }\\left[ \\,\\prod _{k=1}^{4} \\frac{ \\Gamma _p{\\bigl ({m_k+j}\\bigr )}}{ \\Gamma _p{\\bigl ({m_k}\\bigr )}{\\Gamma _p{\\bigl ({1+j}\\bigr )}}} \\right]+ \\sum _{j=\\left\\lfloor \\frac{p-1}{d} \\right\\rfloor +1}^{\\left\\lfloor \\frac{r(p-1)}{d} \\right\\rfloor }\\left[\\,\\prod _{k=1}^{4} \\frac{ \\Gamma _p{\\bigl ({m_k+j}\\bigr )}}{ \\Gamma _p{\\bigl ({m_k}\\bigr )}{\\Gamma _p{\\bigl ({1+j}\\bigr )}}} \\right]\\bigl (\\tfrac{1}{d}+j\\bigr )\\right.", "\\\\ & \\left.", "\\qquad \\quad \\;+\\sum _{j=\\left\\lfloor \\frac{r(p-1)}{d} \\right\\rfloor +1}^{\\operatorname{rep}(m_3)-1}\\left[\\, \\prod _{k=1}^{4} \\frac{ \\Gamma _p{\\bigl ({m_k+j}\\bigr )} }{\\Gamma _p{\\bigl ({m_k}\\bigr )}{\\Gamma _p{\\bigl ({1+j}\\bigr )}}} \\right]\\bigl (\\tfrac{r}{d}+j\\bigr )\\right.", "\\\\ & \\left.", "\\qquad \\quad \\;+\\sum _{j=\\operatorname{rep}(m_3)}^{\\left\\lfloor (d-r)\\frac{p-1}{d} \\right\\rfloor }\\left[\\, \\prod _{k=1}^{4} \\frac{ \\Gamma _p{\\bigl ({m_k+j}\\bigr )} }{ \\Gamma _p{\\bigl ({m_k}\\bigr )}{\\Gamma _p{\\bigl ({1+j}\\bigr )}}} \\right]\\bigl (\\tfrac{1}{d}+j\\bigr )\\bigl (\\tfrac{r}{d}+j\\bigr )\\right\\rbrace \\pmod {p^3},$ where the second to last sum is vacuous unless $p\\equiv r \\pmod {d}$ and $r^2\\equiv 1 \\pmod {d}$ or $p\\equiv d-r \\pmod {d}$ and $r^2\\equiv -1 \\pmod {d}$ .", "By Proposition REF (2) we see that, for $1 \\le k \\le 4$ , $\\Gamma _p{\\bigl ({m_k+j+jp+jp^2}\\bigr )}\\\\\\equiv \\Gamma _p{\\bigl ({m_k+j}\\bigr )}\\left[1+(jp+jp^2)\\,G_1(m_k+j)+\\tfrac{j^2p^2}{2}\\,G_2(m_k+j)\\right]\\pmod {p^3},$ and also $\\Gamma _p{\\bigl ({1+j+jp+jp^2}\\bigr )}^{4}\\equiv \\Gamma _p{\\bigl ({1+j}\\bigr )}^4\\left[1+(jp+jp^2)\\, G_1(1+j)+\\tfrac{j^2p^2}{2}\\,G_2(1+j)\\right]^4\\pmod {p^3}.$ Multiplying the numerator and denominator by ${1-4(jp+jp^2)\\;G_1(1+j)-2j^2p^2\\, \\left(G_2(1+j)-5\\,G_1(1+j)^2\\right)}$ we get that $\\frac{ \\displaystyle \\prod _{k=1}^{4} \\left[ 1+(jp+jp^2)\\,G_1(m_k+j)+\\tfrac{j^2p^2}{2}\\,G_2(m_k+j)\\right] }{\\left[1+(jp+jp^2)\\,G_1(1+j)+\\frac{j^2p^2}{2}\\,G_2(1+j)\\right]^4}\\equiv 1+(jp+jp^2)A(j)+j^2p^2B(j)\\hspace{2.77771pt}({mod}\\,\\,p^3),$ where $A(j):=\\sum _{k=1}^{4} \\Bigl (G_1(m_k+j)-G_1(1+j)\\Bigr )$ and $B(j):=\\frac{1}{2}\\left[A(j)^2 - \\sum _{k=1}^{4} \\Bigl (G_1(m_k+j)^2-G_2(m_k+j)-G_1(1+j)^2 +G_2(1+j)\\Bigr )\\right]\\genfrac{}{}{0.0pt}{}{}{.", "}$ We note that both $A(j)$ and $B(j)$ $\\in \\mathbb {Z}_p$ by Proposition REF (1).", "Applying the above and noting that $\\Gamma _p{\\left({1+j}\\right)}=(-1)^{1+j} j!$ for $j<p$ , we get, after rearranging, ${_{4}G}& \\left(\\tfrac{1}{d} , \\tfrac{r}{d}, 1-\\tfrac{r}{d} , 1-\\tfrac{1}{d}\\right)_p\\equiv \\sum _{j=0}^{\\left\\lfloor \\frac{p-1}{d} \\right\\rfloor }\\left[ \\, \\prod _{k=1}^{4} \\frac{ \\Gamma _p{\\bigl ({m_k+j}\\bigr )}}{\\Gamma _p{\\bigl ({m_k}\\bigr )}\\, j!}", "\\right]\\\\[3pt] & \\: \\,+p \\left\\lbrace \\sum _{j=0}^{\\left\\lfloor \\frac{p-1}{d} \\right\\rfloor }\\left[ \\, \\prod _{k=1}^{4} \\frac{ \\Gamma _p{\\bigl ({m_k+j}\\bigr )}}{\\Gamma _p{\\bigl ({m_k}\\bigr )}\\, j!}", "\\right]\\Bigl [1+jA(j)\\Bigr ]+ \\sum _{j=\\left\\lfloor \\frac{p-1}{d} \\right\\rfloor +1}^{\\left\\lfloor \\frac{r(p-1)}{d} \\right\\rfloor }\\left[ \\, \\prod _{k=1}^{4} \\frac{ \\Gamma _p{\\bigl ({m_k+j}\\bigr )}}{\\Gamma _p{\\bigl ({m_k}\\bigr )}\\, j!}", "\\right]\\Bigl [\\tfrac{1}{d}+j\\Bigr ]\\right\\rbrace \\\\[3pt] &+p^2 \\left\\lbrace \\sum _{j=0}^{\\left\\lfloor \\frac{p-1}{d} \\right\\rfloor }\\left[ \\, \\prod _{k=1}^{4} \\frac{ \\Gamma _p{\\bigl ({m_k+j}\\bigr )}}{\\Gamma _p{\\bigl ({m_k}\\bigr )}\\, j!}", "\\right]\\Bigl [1+2jA(j)+j^2B(j)\\Bigr ]\\right.", "\\\\[3pt] & \\left.", "\\qquad \\quad \\;+\\sum _{j=\\left\\lfloor \\frac{p-1}{d} \\right\\rfloor +1}^{\\left\\lfloor \\frac{r(p-1)}{d} \\right\\rfloor }\\left[ \\, \\prod _{k=1}^{4} \\frac{ \\Gamma _p{\\bigl ({m_k+j}\\bigr )}}{\\Gamma _p{\\bigl ({m_k}\\bigr )}\\, j!}", "\\right]\\Bigl [\\Bigl (\\tfrac{1}{d}+j\\Bigr )\\Bigl (1+jA(j)\\Bigr )+j\\Bigr ]\\right.", "\\\\[3pt] &\\left.", "\\qquad \\quad \\;+\\sum _{j=\\left\\lfloor \\frac{r(p-1)}{d} \\right\\rfloor +1}^{\\operatorname{rep}(m_3)-1}\\left[ \\, \\prod _{k=1}^{4} \\frac{ \\Gamma _p{\\bigl ({m_k+j}\\bigr )}}{\\Gamma _p{\\bigl ({m_k}\\bigr )}\\, j!}", "\\right]\\Bigl [\\tfrac{r}{d}+j\\Bigr ]\\right.", "\\\\[3pt] &\\left.", "\\qquad \\quad \\;+\\sum _{j=\\operatorname{rep}(m_3)}^{\\left\\lfloor (d-r)\\frac{p-1}{d} \\right\\rfloor }\\left[ \\, \\prod _{k=1}^{4} \\frac{ \\Gamma _p{\\bigl ({m_k+j}\\bigr )}}{\\Gamma _p{\\bigl ({m_k}\\bigr )}\\, j!}", "\\right]\\Bigl [\\Bigl (\\tfrac{1}{d}+j\\Bigr )\\Bigl (\\tfrac{r}{d}+j\\Bigr )\\Bigr ]\\right\\rbrace \\pmod {p^3}.$ Proposition REF gives us ${_{4}F_3} & \\Biggl [ \\begin{array}{cccc} \\frac{1}{d}, & \\frac{r}{d}, & 1-\\frac{r}{d}, & 1-\\frac{1}{d}\\vspace{3.61371pt}\\\\\\phantom{\\frac{1}{d}} & 1, & 1, & 1 \\end{array}\\bigg | \\; 1 \\Biggr ]_{p-1} =\\sum _{j=0}^{p-1} \\,\\prod _{k=1}^{4} \\frac{{\\left({m_k}\\right)}_{j}}{{j!", "}}\\\\ &\\equiv \\sum _{j=0}^{\\operatorname{rep}(m_4)-1}\\left[ \\, \\prod _{k=1}^{4} \\frac{ \\Gamma _p{\\bigl ({m_k+j}\\bigr )}}{\\Gamma _p{\\bigl ({m_k}\\bigr )}\\, j!}", "\\right]+\\sum _{j=\\operatorname{rep}(m_4)}^{\\operatorname{rep}(m_3)-1}\\left[ \\, \\prod _{k=1}^{4} \\frac{ \\Gamma _p{\\bigl ({m_k+j}\\bigr )}}{\\Gamma _p{\\bigl ({m_k}\\bigr )}\\, j!}", "\\right]\\bigl (m_1+p-\\operatorname{rep}(m_1)\\bigr )\\\\ & \\qquad \\: \\,{+\\sum _{j=\\operatorname{rep}(m_3)}^{\\operatorname{rep}(m_2) - 1}\\left[ \\, \\prod _{k=1}^{4} \\frac{ \\Gamma _p{\\bigl ({m_k+j}\\bigr )}}{\\Gamma _p{\\bigl ({m_k}\\bigr )}\\, j!}", "\\right]\\bigl (m_1+p-\\operatorname{rep}(m_1)) \\bigl (m_2+p-\\operatorname{rep}(m_2))}\\\\ &\\equiv \\sum _{j=0}^{\\operatorname{rep}(m_4)-1}\\left[ \\, \\prod _{k=1}^{4} \\frac{ \\Gamma _p{\\bigl ({m_k+j}\\bigr )}}{\\Gamma _p{\\bigl ({m_k}\\bigr )}\\, j!}", "\\right]+p\\sum _{j=\\operatorname{rep}(m_4)}^{\\operatorname{rep}(m_3)-1}\\left[ \\, \\prod _{k=1}^{4} \\frac{ \\Gamma _p{\\bigl ({m_k+j}\\bigr )}}{\\Gamma _p{\\bigl ({m_k}\\bigr )}\\, j!}", "\\right]\\Bigl (\\tfrac{1}{d}\\Bigr )\\\\& \\qquad \\:\\,+p^2\\sum _{j=\\operatorname{rep}(m_3)}^{\\operatorname{rep}(m_2) - 1}\\left[ \\, \\prod _{k=1}^{4} \\frac{ \\Gamma _p{\\bigl ({m_k+j}\\bigr )}}{\\Gamma _p{\\bigl ({m_k}\\bigr )}\\, j!}", "\\right]\\Bigl (\\tfrac{1}{d}\\Bigr ) \\Bigl (\\tfrac{r}{d}\\Bigr )\\pmod {p^3}.$ This last step uses the fact that, by definition, $m_1+p - \\operatorname{rep}(m_1)=\\tfrac{a}{d}+p-\\left(p- \\lfloor \\tfrac{p-1}{d} \\rfloor \\right)=\\tfrac{a}{d}+ \\tfrac{p-a}{d} = \\tfrac{p}{d},\\\\[0pt]$ and $m_2+p - \\operatorname{rep}(m_2)=\\tfrac{s}{d}+ \\tfrac{r(p-a)}{d} +\\left\\lfloor \\tfrac{ar}{d} \\right\\rfloor =\\tfrac{ar}{d} - \\left\\lfloor \\tfrac{ar}{d} \\right\\rfloor + \\tfrac{rp}{d} -\\tfrac{ar}{d} + \\left\\lfloor \\tfrac{ar}{d} \\right\\rfloor = \\tfrac{rp}{d}.$ Therefore, combining (REF ) and (REF ), it suffices to prove $& \\sum _{j=0}^{\\operatorname{rep}(m_4)-1}\\left[ \\, \\prod _{k=1}^{4} \\frac{ \\Gamma _p{\\bigl ({m_k+j}\\bigr )}}{\\Gamma _p{\\bigl ({m_k}\\bigr )}\\, j!}", "\\right]\\Bigl [1+jA(j)\\Bigr ]+ \\sum _{j=\\operatorname{rep}(m_4)}^{\\left\\lfloor \\frac{r(p-1)}{d} \\right\\rfloor }\\left[ \\, \\prod _{k=1}^{4} \\frac{ \\Gamma _p{\\bigl ({m_k+j}\\bigr )}}{\\Gamma _p{\\bigl ({m_k}\\bigr )}\\, j!}", "\\right]\\Bigl [j\\Bigr ]\\\\ &{- \\sum _{j=\\left\\lfloor \\frac{r(p-1)}{d} \\right\\rfloor +1}^{\\operatorname{rep}(m_3)-1}\\left[ \\, \\prod _{k=1}^{4} \\frac{ \\Gamma _p{\\bigl ({m_k+j}\\bigr )}}{\\Gamma _p{\\bigl ({m_k}\\bigr )}\\, j!}", "\\right]\\Bigl [\\tfrac{1}{d}\\Bigr ]}+p \\left\\lbrace \\sum _{j=0}^{\\operatorname{rep}(m_4)-1}\\left[ \\, \\prod _{k=1}^{4} \\frac{ \\Gamma _p{\\bigl ({m_k+j}\\bigr )}}{\\Gamma _p{\\bigl ({m_k}\\bigr )}\\, j!}", "\\right]\\Bigl [1+2jA(j)+j^2B(j)\\Bigr ]\\right.", "\\\\ & \\left.", "{+\\sum _{j=\\operatorname{rep}(m_4)}^{\\left\\lfloor \\frac{r(p-1)}{d} \\right\\rfloor }\\left[ \\, \\prod _{k=1}^{4} \\frac{ \\Gamma _p{\\bigl ({m_k+j}\\bigr )}}{\\Gamma _p{\\bigl ({m_k}\\bigr )}\\, j!}", "\\right]\\Bigl [\\Bigl (\\tfrac{1}{d}+j\\Bigr )\\Bigl (1+jA(j)\\Bigr )+j\\Bigr ]}\\right.", "\\\\ & \\left.", "{+\\sum _{j=\\left\\lfloor \\frac{r(p-1)}{d} \\right\\rfloor +1}^{\\operatorname{rep}(m_3)-1}\\left[ \\, \\prod _{k=1}^{4} \\frac{ \\Gamma _p{\\bigl ({m_k+j}\\bigr )}}{\\Gamma _p{\\bigl ({m_k}\\bigr )}\\, j!}", "\\right]\\Bigl [\\tfrac{r}{d}+j\\Bigr ]}{+\\sum _{j=\\operatorname{rep}(m_3)}^{\\left\\lfloor (d-r)\\frac{p-1}{d} \\right\\rfloor }\\left[ \\, \\prod _{k=1}^{4} \\frac{ \\Gamma _p{\\bigl ({m_k+j}\\bigr )}}{\\Gamma _p{\\bigl ({m_k}\\bigr )}\\, j!}", "\\right]\\Bigl [j^2+j\\Bigl (\\tfrac{1}{d}+\\tfrac{r}{d}\\Bigr )\\Bigr ]}\\right.", "\\\\ & \\left.- \\sum _{j=\\left\\lfloor (d-r)\\frac{p-1}{d} \\right\\rfloor +1}^{\\operatorname{rep}(m_2) - 1}\\left[ \\, \\prod _{k=1}^{4} \\frac{ \\Gamma _p{\\bigl ({m_k+j}\\bigr )}}{\\Gamma _p{\\bigl ({m_k}\\bigr )}\\, j!}", "\\right]\\Bigl [\\Bigl (\\tfrac{1}{d}\\Bigr ) \\Bigl (\\tfrac{r}{d}\\Bigr )\\Bigr ]\\right\\rbrace \\equiv s(p)\\pmod {p^2}.$ We note that the terms inside the braces in (REF ) need only be considered modulo $p$ and can be rewritten as follows.", "$&\\sum _{j=0}^{\\operatorname{rep}(m_4)-1}\\left[ \\, \\prod _{k=1}^{4} \\frac{ \\Gamma _p{\\bigl ({m_k+j}\\bigr )}}{\\Gamma _p{\\bigl ({m_k}\\bigr )}\\, j!}", "\\right]\\Bigl [1+2jA(j)+j^2B(j)\\Bigr ]+\\sum _{j=\\operatorname{rep}(m_4)}^{\\left\\lfloor \\frac{r(p-1)}{d} \\right\\rfloor }\\left[ \\, \\prod _{k=1}^{4} \\frac{ \\Gamma _p{\\bigl ({m_k+j}\\bigr )}}{\\Gamma _p{\\bigl ({m_k}\\bigr )}\\, j!}", "\\right]\\Bigl [\\Bigl (\\tfrac{1}{d}\\Bigr )\\Bigl (1+jA(j)\\Bigr )\\Bigr ]\\\\ &+\\sum _{j=\\left\\lfloor \\frac{r(p-1)}{d} \\right\\rfloor +1}^{\\operatorname{rep}(m_3)-1}\\left[ \\, \\prod _{k=1}^{4} \\frac{ \\Gamma _p{\\bigl ({m_k+j}\\bigr )}}{\\Gamma _p{\\bigl ({m_k}\\bigr )}\\, j!}", "\\right]\\Bigl [\\tfrac{r}{d}-j-j^2A(j)\\Bigr ]+\\sum _{j=\\operatorname{rep}(m_4)}^{\\operatorname{rep}(m_3)-1}\\left[ \\, \\prod _{k=1}^{4} \\frac{ \\Gamma _p{\\bigl ({m_k+j}\\bigr )}}{\\Gamma _p{\\bigl ({m_k}\\bigr )}\\, j!}", "\\right]\\Bigl [2j+j^2A(j)\\Bigr ]\\\\ &+\\sum _{j=\\operatorname{rep}(m_3)}^{\\left\\lfloor (d-r)\\frac{p-1}{d} \\right\\rfloor }\\left[ \\, \\prod _{k=1}^{4} \\frac{ \\Gamma _p{\\bigl ({m_k+j}\\bigr )}}{\\Gamma _p{\\bigl ({m_k}\\bigr )}\\, j!}", "\\right]\\Bigl [j\\Bigl (\\tfrac{1}{d}+\\tfrac{r}{d}\\Bigr )\\Bigr ]- \\sum _{j=\\left\\lfloor (d-r)\\frac{p-1}{d} \\right\\rfloor +1}^{\\operatorname{rep}(m_2) - 1}\\left[ \\, \\prod _{k=1}^{4} \\frac{ \\Gamma _p{\\bigl ({m_k+j}\\bigr )}}{\\Gamma _p{\\bigl ({m_k}\\bigr )}\\, j!}", "\\right]\\Bigl [\\Bigl (\\tfrac{1}{d}\\Bigr ) \\Bigl (\\tfrac{r}{d}\\Bigr )+j^2\\Bigr ]\\\\ &+\\sum _{\\operatorname{rep}(m_3)}^{\\operatorname{rep}(m_2)-1}\\left[ \\, \\prod _{k=1}^{4} \\frac{ \\Gamma _p{\\bigl ({m_k+j}\\bigr )}}{\\Gamma _p{\\bigl ({m_k}\\bigr )}\\, j!}", "\\right]\\Bigl [j^2\\Bigr ].$ We now consider the first, fourth and last terms of (REF ).", "Define $X(j):=\\sum _{j=0}^{\\operatorname{rep}(m_4)-1}\\left[ \\, \\prod _{k=1}^{4} \\frac{ \\Gamma _p{\\bigl ({m_k+j}\\bigr )}}{\\Gamma _p{\\bigl ({m_k}\\bigr )}\\, j!}", "\\right]\\Bigl [1+2jA(j)+j^2B(j)\\Bigr ]\\\\+\\sum _{j=\\operatorname{rep}(m_4)}^{\\operatorname{rep}(m_3)-1}\\left[ \\, \\prod _{k=1}^{4} \\frac{ \\Gamma _p{\\bigl ({m_k+j}\\bigr )}}{\\Gamma _p{\\bigl ({m_k}\\bigr )}\\, j!}", "\\right]\\Bigl [2j+j^2A(j)\\Bigr ]+\\sum _{\\operatorname{rep}(m_3)}^{\\operatorname{rep}(m_2)-1}\\left[ \\, \\prod _{k=1}^{4} \\frac{ \\Gamma _p{\\bigl ({m_k+j}\\bigr )}}{\\Gamma _p{\\bigl ({m_k}\\bigr )}\\, j!}", "\\right]\\Bigl [j^2\\Bigr ].$ We will show $X(j)\\equiv 0 \\pmod {p}$ .", "We start by examining $A(j)$ , $B(j)$ , $\\prod _{k=1}^{4} \\Gamma _p{\\bigl ({m_k+j}\\bigr )}$ and $\\prod _{k=1}^{4} \\Gamma _p{\\bigl ({m_k}\\bigr )}$ modulo $p$ .", "Define, for $t\\in \\lbrace 1,2\\rbrace $ , $\\delta _t:={\\left\\lbrace \\begin{array}{ll}0 & \\text{if } 0\\le j\\le \\operatorname{rep}(m_4)-1, \\\\[6pt]\\frac{1}{p^t} &\\text{if } \\operatorname{rep}(m_4) \\le j\\le \\operatorname{rep}(m_3)-1,\\\\[6pt]\\frac{2}{p^t} & \\text{if } \\operatorname{rep}(m_3)\\le j\\le \\operatorname{rep}(m_2)-1,\\\\[6pt]\\end{array}\\right.", "}&& \\textup {and} &&\\gamma :={\\left\\lbrace \\begin{array}{ll}1 & \\text{if } 0\\le j\\le \\operatorname{rep}(m_4)-1, \\\\[6pt]\\frac{1}{p} & \\text{if } \\operatorname{rep}(m_4) \\le j\\le \\operatorname{rep}(m_3)-1,\\\\[6pt]\\frac{1}{p^2} & \\text{if } \\operatorname{rep}(m_3)\\le j\\le \\operatorname{rep}(m_2)-1.\\\\[6pt]\\end{array}\\right.", "}$ Then using Lemmas REF and REF we see that, for $j \\le \\operatorname{rep}(m_2)-1$ , $A(j)\\equiv \\sum _{k=1}^{4} \\Bigl (H_{\\operatorname{rep}(m_k)-1+j}^{(1)} - H_{j}^{(1)}\\Bigr )- \\delta _1\\pmod {p}$ and $\\sum _{k=1}^{4} \\Bigl (G_1\\left(m_k+j\\right)^2-G_2\\left(m_k+j\\right)-G_1\\left(1+j\\right)^2 +G_2\\left(1+j\\right)\\Bigr )\\\\\\equiv \\sum _{k=1}^{4} \\Biggl (H_{\\operatorname{rep}(m_k)+j-1}^{(2)}- \\hspace{1.0pt} H_{j}^{(2)}\\Biggr )-\\delta _2\\pmod {p}.$ Using Lemma REF we get that, for $j \\le \\operatorname{rep}(m_2)-1$ , $\\prod _{k=1}^{4} \\Gamma _p{\\bigl ({m_k+j}\\bigr )}\\\\\\equiv \\left[\\,\\prod _{k=1}^{4} \\left(\\operatorname{rep}(m_k)+j-1\\right)!", "\\,(-1)^{\\operatorname{rep}(m_k)+j}\\right] \\cdot \\gamma \\pmod {p}$ and by Proposition REF (2) we see that $\\prod _{k=1}^{4} \\Gamma _p{\\bigl ({m_k}\\bigr )} &= \\prod _{k=1}^{2} (-1)^{\\operatorname{rep}(m_k)}=(-1)^{\\operatorname{rep}(m_1)+\\operatorname{rep}(m_2)}=\\pm 1.$ Substituting for $A(j)$ , $B(j)$ , $\\prod _{k=1}^{4} \\Gamma _p{\\bigl ({m_k+j}\\bigr )}$ and $\\prod _{k=1}^{4} \\Gamma _p{\\bigl ({m_k}\\bigr )}$ modulo $p$ into (REF ) we have $\\pm X(j)\\equiv &\\sum _{j=0}^{\\operatorname{rep}(m_4)-1}\\Biggl [\\, \\prod _{k=1}^{4} {\\left({j+1}\\right)}_{\\operatorname{rep}(m_k)-1} \\Biggr ]\\Biggl [1+2j \\sum _{k=1}^{4} \\Bigl (H_{\\operatorname{rep}(m_k)-1+j}^{(1)}- H_{j}^{(1)}\\Bigr )\\\\ &+\\frac{j^2}{2} \\Biggl (\\left(\\sum _{k=1}^{4} \\Bigl (H_{\\operatorname{rep}(m_k)-1+j}^{(1)}- H_{j}^{(1)}\\Bigr )\\right)^2-\\sum _{k=1}^{4} \\Bigl (H_{\\operatorname{rep}(m_k)-1+j}^{(2)}- H_{j}^{(2)}\\Bigr )\\Biggr )\\Biggr ]\\\\ &+\\sum _{j=\\operatorname{rep}(m_4)}^{\\operatorname{rep}(m_3)-1}\\Biggl [\\, \\prod _{k=1}^{4} {\\left({j+1}\\right)}_{\\operatorname{rep}(m_k)-1} \\Biggr ]\\Biggl [ \\frac{1}{p} \\Biggr ]\\Biggl [2j+j^2 \\sum _{k=1}^{4} \\Bigl (H_{\\operatorname{rep}(m_k)-1+j}^{(1)}- H_{j}^{(1)}\\Bigr )-\\frac{j^2}{p} \\Biggr ]\\\\ &+\\sum _{\\operatorname{rep}(m_3)}^{\\operatorname{rep}(m_2)-1}\\Biggl [\\, \\prod _{k=1}^{4} {\\left({j+1}\\right)}_{\\operatorname{rep}(m_k)-1} \\Biggr ]\\Biggl [ \\frac{1}{p^2} \\Biggr ]\\Biggl [ j^2 \\Biggr ]\\pmod {p}.$ We can simplify this expression by combining the three terms into one single summation.", "For $\\operatorname{rep}(m_4) \\le j \\le \\operatorname{rep}(m_3) -1$ we note that $\\prod _{k=1}^{4} {\\left({j+1}\\right)}_{\\operatorname{rep}(m_k)-1} \\in p\\mathbb {Z}_p$ .", "Also, we see from (REF ) and (REF ) that $\\sum _{k=1}^{4} \\Bigl (H_{\\operatorname{rep}(m_k)-1+j}^{(1)}- H_{j}^{(1)}\\Bigr )-\\frac{1}{p} \\in \\mathbb {Z}_p$ and $\\sum _{k=1}^{4} \\Bigl (H_{\\operatorname{rep}(m_k)-1+j}^{(2)}- H_{j}^{(2)}\\Bigr )-\\frac{1}{p^2} \\in \\mathbb {Z}_p$ , for $j$ in the same range.", "Similarly, for $\\operatorname{rep}(m_3) \\le j \\le \\operatorname{rep}(m_2)- 1$ we have that $\\prod _{k=1}^{4} {\\left({j+1}\\right)}_{\\operatorname{rep}(m_k)-1} \\in p^2\\mathbb {Z}_p$ , $\\sum _{k=1}^{4} \\Bigl (H_{\\operatorname{rep}(m_k)-1+j}^{(1)}- H_{j}^{(1)}\\Bigr )-\\frac{2}{p} \\in \\mathbb {Z}_p$ and $\\sum _{k=1}^{4} \\Bigl (H_{\\operatorname{rep}(m_k)-1+j}^{(2)}- H_{j}^{(2)}\\Bigr )-\\frac{2}{p^2} \\in \\mathbb {Z}_p$ .", "Consequently, $\\pm X(j)\\equiv \\sum _{j=0}^{\\operatorname{rep}(m_2)-1}\\Biggl [\\, \\prod _{k=1}^{4} {\\left({j+1}\\right)}_{\\operatorname{rep}(m_k)-1} \\Biggr ]\\Biggl [1+2j\\sum _{k=1}^{4} \\Bigl (H_{\\operatorname{rep}(m_k)-1+j}^{(1)} -H_{j}^{(1)}\\Bigr )\\\\+\\frac{j^2}{2} \\Biggl (\\left(\\sum _{k=1}^{4} \\Bigl (H_{\\operatorname{rep}(m_k)-1+j}^{(1)} - H_{j}^{(1)}\\Bigr )\\right)^2-\\sum _{k=1}^{4} \\Bigl (H_{\\operatorname{rep}(m_k)-1+j}^{(2)} - H_{j}^{(2)}\\Bigr )\\Biggr )\\Biggr ]\\pmod {p}.$ Note we can extend the upper limit of this sum to $j=p-1$ as $\\prod _{k=1}^{4} {\\left({j+1}\\right)}_{\\operatorname{rep}(m_k)-1} \\in p^3\\mathbb {Z}_p$ for $\\operatorname{rep}(m_2) \\le j\\le p-1$ .", "By Lemmas REF and REF with $n=4$ and $a_i=\\operatorname{rep}(m_i)-1$ for $1 \\le i \\le 4$ we get that $X(j)\\equiv \\pm (1-1) \\equiv 0 \\pmod {p}.$ Accounting for (REF ) in (REF ), via (REF ) and (REF ), means we need only show $&\\sum _{j=0}^{\\operatorname{rep}(m_4)-1}\\left[ \\, \\prod _{k=1}^{4} \\frac{ \\Gamma _p{\\bigl ({m_k+j}\\bigr )}}{\\Gamma _p{\\bigl ({m_k}\\bigr )}\\, j!}", "\\right]\\Bigl [1+jA(j)\\Bigr ]+ \\sum _{j=\\operatorname{rep}(m_4)}^{\\left\\lfloor \\frac{r(p-1)}{d} \\right\\rfloor }\\left[ \\, \\prod _{k=1}^{4} \\frac{ \\Gamma _p{\\bigl ({m_k+j}\\bigr )}}{\\Gamma _p{\\bigl ({m_k}\\bigr )}\\, j!}", "\\right]\\Bigl [j\\Bigr ]\\\\[3pt] & - \\sum _{j=\\left\\lfloor \\frac{r(p-1)}{d} \\right\\rfloor +1}^{\\operatorname{rep}(m_3)-1}\\left[ \\, \\prod _{k=1}^{4} \\frac{ \\Gamma _p{\\bigl ({m_k+j}\\bigr )}}{\\Gamma _p{\\bigl ({m_k}\\bigr )}\\, j!}", "\\right]\\Bigl [\\tfrac{1}{d}\\Bigr ]+p \\left\\lbrace \\sum _{j=\\operatorname{rep}(m_4)}^{\\left\\lfloor \\frac{r(p-1)}{d} \\right\\rfloor }\\left[ \\, \\prod _{k=1}^{4} \\frac{ \\Gamma _p{\\bigl ({m_k+j}\\bigr )}}{\\Gamma _p{\\bigl ({m_k}\\bigr )}\\, j!}", "\\right]\\Bigl [\\Bigl (\\tfrac{1}{d}\\Bigr )\\Bigl (1+jA(j)\\Bigr )\\Bigr ]\\right.", "\\\\[3pt] &\\left.+\\sum _{j=\\left\\lfloor \\frac{r(p-1)}{d} \\right\\rfloor +1}^{\\operatorname{rep}(m_3)-1}\\left[ \\, \\prod _{k=1}^{4} \\frac{ \\Gamma _p{\\bigl ({m_k+j}\\bigr )}}{\\Gamma _p{\\bigl ({m_k}\\bigr )}\\, j!}", "\\right]\\Bigl [\\tfrac{r}{d}-j-j^2A(j)\\Bigr ]+\\sum _{j=\\operatorname{rep}(m_3)}^{\\left\\lfloor (d-r)\\frac{p-1}{d} \\right\\rfloor }\\left[ \\, \\prod _{k=1}^{4} \\frac{ \\Gamma _p{\\bigl ({m_k+j}\\bigr )}}{\\Gamma _p{\\bigl ({m_k}\\bigr )}\\, j!}", "\\right]\\Bigl [j\\Bigl (\\tfrac{1}{d}+\\tfrac{r}{d}\\Bigr )\\Bigr ]\\right.", "\\\\[3pt] &\\left.- \\sum _{j=\\left\\lfloor (d-r)\\frac{p-1}{d} \\right\\rfloor +1}^{\\operatorname{rep}(m_2) - 1}\\left[ \\, \\prod _{k=1}^{4} \\frac{ \\Gamma _p{\\bigl ({m_k+j}\\bigr )}}{\\Gamma _p{\\bigl ({m_k}\\bigr )}\\, j!}", "\\right]\\Bigl [\\Bigl (\\tfrac{1}{d}\\Bigr ) \\Bigl (\\tfrac{r}{d}\\Bigr )+j^2\\Bigr ]\\right\\rbrace \\equiv s(p)\\pmod {p^2}.$ We now convert these remaining terms to an expression involving binomial coefficients and harmonic sums and then use the results of Section 3.3 to simplify them.", "First we define $\\operatorname{Bin}\\hspace{1.0pt}(j) &:= \\left({\\genfrac{}{}{0.0pt}{}{\\operatorname{rep}(m_1)-1+j}{j}}\\right) \\left({\\genfrac{}{}{0.0pt}{}{\\operatorname{rep}(m_1)-1}{j}}\\right) \\left({\\genfrac{}{}{0.0pt}{}{\\operatorname{rep}(m_2)-1+j}{j}}\\right) \\left({\\genfrac{}{}{0.0pt}{}{\\operatorname{rep}(m_2)-1}{j}}\\right)\\genfrac{}{}{0.0pt}{}{}{,}\\\\[15pt]\\mathcal {H}(j)&:= H_{\\operatorname{rep}(m_1)-1+j}^{(1)}+H_{\\operatorname{rep}(m_1)-1-j}^{(1)}+H_{\\operatorname{rep}(m_2)-1+j}^{(1)}+H_{\\operatorname{rep}(m_2)-1-j}^{(1)}-4H_j^{(1)},\\\\[15pt]\\mathcal {A}(j)&:= \\Bigl (\\operatorname{rep}(m_1)-m_1\\Bigr )\\left(H_{\\operatorname{rep}(m_1)-1+j}^{(1)}-H_{\\operatorname{rep}(m_{4})-1+j}^{(1)}\\right)\\\\[6pt]& \\qquad \\qquad \\qquad \\qquad \\qquad \\quad + \\Bigl (\\operatorname{rep}(m_2)-m_2\\Bigr )\\left(H_{\\operatorname{rep}(m_2)-1+j}^{(1)}-H_{\\operatorname{rep}(m_{3})-1+j}^{(1)}\\right),\\multicolumn{2}{l}{\\text{and}}\\\\\\mathcal {B}(j)&:= \\Bigl (\\operatorname{rep}(m_1)-m_1\\Bigr )\\left(H_{\\operatorname{rep}(m_1)-1+j}^{(2)}-H_{\\operatorname{rep}(m_{4})-1+j}^{(2)}\\right)\\\\[6pt]& \\qquad \\qquad \\qquad \\qquad \\qquad \\quad + \\Bigl (\\operatorname{rep}(m_2)-m_2\\Bigr )\\left(H_{\\operatorname{rep}(m_2)-1+j}^{(2)}-H_{\\operatorname{rep}(m_{3})-1+j}^{(2)}\\right).$ By Lemma REF we see that for $j<\\operatorname{rep}({m_2})$ $\\prod _{k=1}^{4} \\frac{\\Gamma _p{\\bigl ({m_k+j}\\bigr )}}{\\Gamma _p{\\bigl ({m_k}\\bigr )} j!", "}&\\equiv \\Biggl [ \\prod _{i=1}^{2} \\left({\\genfrac{}{}{0.0pt}{}{\\operatorname{rep}(m_i)-1+j}{j}}\\right) \\left({\\genfrac{}{}{0.0pt}{}{\\operatorname{rep}(m_i)-1}{j}}\\right)\\Biggr ]\\cdot \\alpha \\\\[6pt]& \\qquad \\cdot \\left[1-\\sum _{i=1}^{2}\\Bigl (\\operatorname{rep}(m_i)-m_i\\Bigr )\\left(H_{\\operatorname{rep}(m_i)-1+j}^{(1)}-H_{\\operatorname{rep}(m_{5-i})-1+j}^{(1)}-\\beta _i\\right)\\right]\\\\[6pt]&\\equiv \\operatorname{Bin}\\hspace{1.0pt}(j) \\cdot \\alpha \\cdot \\left[1-\\mathcal {A}(j) + \\sum _{i=1}^{2}\\Bigl (\\operatorname{rep}(m_i)-m_i\\Bigr ) \\beta _i\\right]\\pmod {p^2},$ where $\\alpha ={\\left\\lbrace \\begin{array}{ll}1 & \\text{if } 0\\le j \\le \\operatorname{rep}(m_4)-1,\\\\\\frac{1}{p} & \\text{if } \\operatorname{rep}(m_4) \\le j \\le \\operatorname{rep}(m_3)-1,\\\\\\frac{1}{p^2} & \\text{if } \\operatorname{rep}(m_3) \\le j < \\operatorname{rep}(m_2),\\end{array}\\right.", "}&& \\textup {and} &&\\beta _i={\\left\\lbrace \\begin{array}{ll}0 & \\text{if } 0\\le j \\le \\operatorname{rep}(m_{5-i})-1,\\\\\\frac{1}{p} & \\text{if } \\operatorname{rep}(m_{5-i}) \\le j < \\operatorname{rep}(m_i).\\end{array}\\right.", "}$ Again for $j<\\operatorname{rep}({m_2})$ , Lemma REF gives us $A(j)&\\equiv \\sum _{i=1}^{2}\\left( H_{\\operatorname{rep}(m_i)-1+j}^{(1)}+H_{\\operatorname{rep}(m_i)-1-j}^{(1)}-2\\hspace{1.0pt} H_{j}^{(1)}\\right)-\\alpha ^{\\prime }\\\\&\\qquad \\qquad +\\sum _{i=1}^{2} \\left(\\operatorname{rep}(m_i)-m_i\\right)\\left(H_{\\operatorname{rep}(m_i)-1+j}^{(2)}-H_{\\operatorname{rep}(m_{5-i})-1+j}^{(2)}-\\beta _{i}^{\\prime }\\right)\\\\&\\equiv \\mathcal {H}(j) -\\alpha ^{\\prime } + \\mathcal {B}(j) - \\sum _{i=1}^{2} \\left(\\operatorname{rep}(m_i)-m_i\\right) \\beta _{i}^{\\prime }\\pmod {p^2}$ where $\\alpha ^{\\prime }={\\left\\lbrace \\begin{array}{ll}0 & \\text{if } 0\\le j \\le \\operatorname{rep}(m_4)-1,\\\\\\frac{1}{p} & \\text{if } \\operatorname{rep}(m_4) \\le j \\le \\operatorname{rep}(m_3)-1,\\\\\\frac{2}{p} & \\text{if } \\operatorname{rep}(m_3) \\le j < \\operatorname{rep}(m_2),\\;\\end{array}\\right.", "}&& \\textup {and} &&\\beta _{i}^{\\prime }={\\left\\lbrace \\begin{array}{ll}0 & \\text{if } 0\\le j \\le \\operatorname{rep}(m_{5-i})-1,\\\\\\frac{1}{p^2} & \\text{if } \\operatorname{rep}(m_{5-i}) \\le j < \\operatorname{rep}(m_i).\\end{array}\\right.", "}$ Reducing (REF ) and (REF ) modulo $p$ respectively we see that for $j<\\operatorname{rep}(m_2)$ , $\\prod _{k=1}^{4}\\frac{ \\Gamma _p{\\bigl ({m_k+j}\\bigr )}}{\\Gamma _p{\\bigl ({m_k}\\bigr )} j!", "}&\\equiv \\operatorname{Bin}\\hspace{1.0pt}(j) \\cdot \\alpha \\pmod {p}$ and $A(j) \\equiv \\mathcal {H}(j) -\\alpha ^{\\prime }\\pmod {p},$ as $\\operatorname{rep}(m_i)-m_i \\in p \\mathbb {Z}_p$ .", "Therefore, using (REF ), (REF ), (REF ) and (REF ), we see that (REF ) is equivalent to $\\sum _{j=0}^{\\operatorname{rep}(m_4)-1}\\operatorname{Bin}\\hspace{1.0pt}(j) \\Bigl [1-\\mathcal {A}(j) \\Bigr ]\\Bigl [1+j \\mathcal {H}(j) + j \\mathcal {B}(j) \\Bigr ]+ \\sum _{j=\\operatorname{rep}(m_4)}^{\\left\\lfloor \\frac{r(p-1)}{d} \\right\\rfloor }\\Bigl [\\tfrac{j \\operatorname{Bin}(j)}{p}\\Bigr ] \\Bigl [1-\\mathcal {A}(j) + \\tfrac{\\operatorname{rep}(m_1)-m_1}{p}\\Bigr ]\\\\- \\sum _{j=\\left\\lfloor \\frac{r(p-1)}{d} \\right\\rfloor +1}^{\\operatorname{rep}(m_3) -1}\\Bigl [\\tfrac{\\operatorname{Bin}(j)}{dp}\\Bigr ] \\Bigl [1-\\mathcal {A}(j) + \\tfrac{\\operatorname{rep}(m_1)-m_1}{p}\\Bigr ]+p \\left\\lbrace \\sum _{j=\\operatorname{rep}(m_4)}^{\\left\\lfloor \\frac{r(p-1)}{d} \\right\\rfloor }\\Bigl [\\tfrac{\\operatorname{Bin}(j)}{dp}\\Bigr ]\\Bigl [1+j\\mathcal {H}(j)-\\tfrac{j}{p}\\Bigr ]\\right.", "\\\\ \\multicolumn{1}{l}{\\left.+\\sum _{j=\\left\\lfloor \\frac{r(p-1)}{d} \\right\\rfloor +1}^{\\operatorname{rep}(m_3) -1}\\Bigl [\\tfrac{\\operatorname{Bin}(j)}{p}\\Bigr ] \\Bigl [\\tfrac{r}{d}-j-j^2 \\mathcal {H}(j)+\\tfrac{j^2}{p}\\Bigr ]+\\sum _{j=\\operatorname{rep}(m_3)}^{\\left\\lfloor (d-r)\\frac{p-1}{d} \\right\\rfloor }\\Bigl [\\tfrac{\\operatorname{Bin}(j)}{p^2}\\Bigr ]\\Bigl [j \\Bigl (\\tfrac{1}{d}+\\tfrac{r}{d}\\Bigr )\\Bigr ]\\right.}", "\\\\ \\left.-\\sum _{j=\\left\\lfloor (d-r)\\frac{p-1}{d} \\right\\rfloor +1}^{\\operatorname{rep}(m_2) - 1}\\Bigl [\\tfrac{\\operatorname{Bin}(j)}{p^2}\\Bigr ]\\Bigl [\\Bigl (\\tfrac{1}{d}\\Bigr )\\Bigl (\\tfrac{r}{d}\\Bigr )+j^2\\Bigr ]\\right\\rbrace \\equiv s(p)\\pmod {p^2}.$ We now consider $\\sum _{j=0}^{\\operatorname{rep}(m_2)- 1}\\operatorname{Bin}\\hspace{1.0pt}(j) \\Bigl [1+j \\mathcal {H}(j) \\Bigr ]+\\sum _{j=0}^{\\operatorname{rep}(m_2) - 1}\\operatorname{Bin}\\hspace{1.0pt}(j)\\Bigl [ j \\mathcal {B}(j) -\\mathcal {A}(j) - j \\mathcal {A}(j) \\mathcal {H}(j) \\Bigr ]\\pmod {p^2}.\\\\$ For $0\\le j \\le \\operatorname{rep}(m_4)-1$ we see that $\\operatorname{Bin}\\hspace{1.0pt}(j) \\Bigl [1-\\mathcal {A}(j) \\Bigr ]\\Bigl [1+j \\mathcal {H}(j) + j \\mathcal {B}(j) \\Bigr ]\\\\[6pt]\\equiv \\operatorname{Bin}\\hspace{1.0pt}(j) \\Bigl [1+j \\mathcal {H}(j)+ j \\mathcal {B}(j) -\\mathcal {A}(j) - j \\mathcal {A}(j) \\mathcal {H}(j) \\Bigr ]\\pmod {p^2},$ as $ \\mathcal {A}(j) \\mathcal {B}(j) \\in p^2 \\mathbb {Z}_p$ for such $j$ .", "For $\\operatorname{rep}(m_4) \\le j \\le \\operatorname{rep}(m_3)-1$ we note the following facts: $\\operatorname{Bin}\\hspace{1.0pt}(j) \\in p \\mathbb {Z}_p$ , which we can see from (REF ); $\\mathcal {H}(j) -\\frac{1}{p} \\in \\mathbb {Z}_p$ , from (REF ); $\\mathcal {A}(j)- (\\operatorname{rep}(m_1)-m_1)\\frac{1}{p} \\in p \\mathbb {Z}_p$ ; and $\\mathcal {B}(j)- (\\operatorname{rep}(m_1)-m_1)\\frac{1}{p^2} \\in p \\mathbb {Z}_p$ .", "The last two properties come directly from their definitions, noting that $\\operatorname{rep}(m_i)-m_i \\in p \\mathbb {Z}_p$ by definition.", "Therefore, for $j$ in this range, we have $\\operatorname{Bin}\\hspace{1.0pt}(&j) \\Bigl [1+j \\mathcal {H}(j)+ j \\mathcal {B}(j) -\\mathcal {A}(j) - j \\mathcal {A}(j) \\mathcal {H}(j) \\Bigr ] \\\\[9pt]&\\equiv \\operatorname{Bin}\\hspace{1.0pt}(j) \\Bigl \\lbrace 1+j \\mathcal {H}(j) +j \\tfrac{\\operatorname{rep}(m_1)-m_1}{p^2} - \\tfrac{\\operatorname{rep}(m_1)-m_1}{p}-j\\left[\\tfrac{1}{p} \\left(\\mathcal {A}(j)- \\tfrac{\\operatorname{rep}(m_1)-m_1}{p}\\right)\\right.\\\\[9pt] &\\left.", "\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\quad \\; \\; \\,+\\tfrac{\\operatorname{rep}(m_1)-m_1}{p} \\left(\\mathcal {H}(j)-\\tfrac{1}{p}\\right) +\\tfrac{\\operatorname{rep}(m_1)-m_1}{p^2}\\right]\\Bigr \\rbrace \\\\[9pt]&\\equiv \\operatorname{Bin}\\hspace{1.0pt}(j) \\Bigl [\\Bigl (1+j \\mathcal {H}(j)\\Bigr )\\left(1- \\tfrac{\\operatorname{rep}(m_1)-m_1}{p}\\right) +2j \\tfrac{\\operatorname{rep}(m_1)-m_1}{p^2} -\\tfrac{j}{p} \\mathcal {A}(j)\\Bigr ]\\pmod {p^2}$ Recall that $m_1:=\\frac{a}{d}$ , where $p \\equiv a \\pmod {d}$ with $a<d$ , and $\\operatorname{rep}(m_1)=p-\\lfloor \\frac{p-1}{d} \\rfloor $ .", "Hence $\\operatorname{rep}(m_1)-m_1 = p- \\lfloor \\tfrac{p-1}{d} \\rfloor - \\tfrac{a}{d}= p- \\tfrac{p-a}{d} - \\tfrac{a}{d} = p\\, \\bigl (1 - \\tfrac{1}{d}\\bigr ).$ Therefore, for $\\operatorname{rep}(m_4) \\le j \\le \\operatorname{rep}(m_3)-1$ , $\\operatorname{Bin}\\hspace{1.0pt}(j)& \\Bigl [1+j \\mathcal {H}(j)+ j \\mathcal {B}(j) -\\mathcal {A}(j) - j \\mathcal {A}(j) \\mathcal {H}(j) \\Bigr ] \\\\[9pt]&\\equiv \\operatorname{Bin}\\hspace{1.0pt}(j) \\Bigl [\\Bigl (\\tfrac{1}{d}\\Bigr )\\Bigl (1+j\\mathcal {H}(j)-\\tfrac{j}{p}\\Bigr )+\\Bigl (\\tfrac{j}{p}\\Bigr )\\Bigl (1-\\mathcal {A}(j)+\\tfrac{\\operatorname{rep}(m_1)-m_1}{p}\\Bigr )\\Bigr ]\\pmod {p^2}.$ Similarly, for $\\operatorname{rep}(m_3) \\le j \\le \\operatorname{rep}(m_2)-1$ , $\\operatorname{Bin}\\hspace{1.0pt}(j) &\\Bigl [1+j \\mathcal {H}(j)+ j \\mathcal {B}(j) -\\mathcal {A}(j) - j \\mathcal {A}(j) \\mathcal {H}(j) \\Bigr ]\\equiv \\operatorname{Bin}\\hspace{1.0pt}(j) \\Bigl [\\tfrac{1}{p}\\Bigr ]\\Bigl [j \\Bigl (\\tfrac{1}{d}+\\tfrac{r}{d}\\Bigr )\\Bigr ]\\pmod {p^2}.$ Accounting for (REF ), (REF ) and (REF ) in (REF ), it now suffices to show $\\sum _{j=0}^{\\operatorname{rep}(m_2) - 1}\\operatorname{Bin}\\hspace{1.0pt}(j) \\Bigl [1+j \\mathcal {H}(j) \\Bigr ]+\\sum _{j=0}^{\\operatorname{rep}(m_2) - 1}\\operatorname{Bin}\\hspace{1.0pt}(j)\\Bigl [ j \\mathcal {B}(j) -\\mathcal {A}(j) - j \\mathcal {A}(j) \\mathcal {H}(j) \\Bigr ]\\\\-\\sum _{j=\\left\\lfloor \\frac{r(p-1)}{d} \\right\\rfloor +1}^{\\operatorname{rep}(m_3)-1}\\operatorname{Bin}\\hspace{1.0pt}(j)\\biggl \\lbrace \\Bigl [\\tfrac{1}{d}+j\\Bigr ] \\Bigl [\\Bigl (\\tfrac{1}{p}\\Bigr ) \\left(1-\\mathcal {A}(j) + \\tfrac{\\operatorname{rep}(m_1)-m_1}{p}\\right) +1+j\\mathcal {H}(j)-\\tfrac{j}{p}\\Bigr ]-\\tfrac{r}{d}\\biggr \\rbrace \\\\-\\sum _{j=\\left\\lfloor (d-r)\\frac{p-1}{d} \\right\\rfloor +1}^{\\operatorname{rep}(m_2) - 1}\\operatorname{Bin}\\hspace{1.0pt}(j) \\Bigl [\\tfrac{1}{p}\\Bigr ]\\Bigl [j^2+\\Bigl (\\tfrac{1}{d}\\Bigr )\\Bigl (\\tfrac{r}{d}\\Bigr )+j \\Bigl (\\tfrac{1}{d}+\\tfrac{r}{d}\\Bigr )\\Bigr ]\\equiv s(p)\\pmod {p^2}.$ Recall that the third sum above is vacuous unless $p\\equiv r \\pmod {d}$ and $r^2\\equiv 1\\pmod {d}$ or $p\\equiv d-r \\pmod {d}$ and $r^2\\equiv -1\\pmod {d}$ .", "In these cases the sum is over one value of $j=\\operatorname{rep}(m_3)-1 = \\bigl \\lfloor \\tfrac{r(p-1)}{d} \\bigr \\rfloor +1$ and so $\\left(\\tfrac{1}{d} +j\\right) = \\tfrac{rp}{d}$ .", "Also $\\operatorname{rep}(m_1)-m_1=p\\left(1-\\tfrac{1}{d}\\right)$ .", "So we get $\\sum _{j=\\left\\lfloor \\frac{r(p-1)}{d} \\right\\rfloor +1}^{\\operatorname{rep}(m_3)-1}\\operatorname{Bin}\\hspace{1.0pt}(j)\\biggl [\\Bigl [\\tfrac{1}{d}+j\\Bigr ] \\Bigl [\\Bigl (\\tfrac{1}{p}\\Bigr ) \\left(1-\\mathcal {A}(j) + \\tfrac{\\operatorname{rep}(m_1)-m_1}{p}\\right) +1+j\\mathcal {H}(j)-\\tfrac{j}{p}\\Bigr ]-\\tfrac{r}{d}\\biggr ]\\\\=\\sum _{j=\\left\\lfloor \\frac{r(p-1)}{d} \\right\\rfloor +1}^{\\operatorname{rep}(m_3)-1}\\operatorname{Bin}\\hspace{1.0pt}(j)\\Bigl [\\tfrac{rpj}{d}\\mathcal {H}(j)-\\tfrac{r}{d}\\mathcal {A}(j)+\\tfrac{r}{d}\\tfrac{d-1}{d}+\\tfrac{rp}{d}-\\tfrac{rj}{d}\\Bigr ].$ Note $\\operatorname{Bin}\\hspace{1.0pt}(j) \\in p \\mathbb {Z}_p$ for $j=\\operatorname{rep}(m_3)-1$ .", "Thus $\\operatorname{Bin}\\hspace{1.0pt}(j) \\hspace{1.0pt}p\\hspace{1.0pt} \\mathcal {H}(j)\\equiv \\operatorname{Bin}\\hspace{1.0pt}(j) \\hspace{1.0pt}p \\left(\\frac{1}{p}\\right) \\pmod {p^2}$ and $\\operatorname{Bin}\\hspace{1.0pt}(j) \\mathcal {A}(j) \\equiv \\operatorname{Bin}\\hspace{1.0pt}(j) \\frac{\\operatorname{rep}(m_1)-m_1}{p} \\equiv \\operatorname{Bin}\\hspace{1.0pt}(j) \\frac{d-1}{d}\\pmod {p^2}$ for this value of $j$ , and $\\sum _{j=\\left\\lfloor \\frac{r(p-1)}{d} \\right\\rfloor +1}^{\\operatorname{rep}(m_3)-1}\\operatorname{Bin}\\hspace{1.0pt}(j)\\Bigl [\\tfrac{rpj}{d}\\mathcal {H}(j)-\\tfrac{r}{d}\\mathcal {A}(j)+\\tfrac{r}{d}\\tfrac{d-1}{d}+\\tfrac{rp}{d}-\\tfrac{rj}{d}\\Bigr ]\\equiv \\sum _{j=\\left\\lfloor \\frac{r(p-1)}{d} \\right\\rfloor +1}^{\\operatorname{rep}(m_3)-1}\\operatorname{Bin}\\hspace{1.0pt}(j)\\Bigl [\\tfrac{rp}{d}\\Bigr ]\\equiv 0\\pmod {p^2}.$ So the third term of (REF ) vanishes modulo $p^2$ .", "Next we examine the last term of (REF ), $\\sum _{j=\\left\\lfloor (d-r)\\frac{p-1}{d} \\right\\rfloor +1}^{\\operatorname{rep}(m_2) - 1}\\operatorname{Bin}\\hspace{1.0pt}(j) \\Bigl [\\tfrac{1}{p}\\Bigr ]\\Bigl [j^2+\\Bigl (\\tfrac{1}{d}\\Bigr )\\Bigl (\\tfrac{r}{d}\\Bigr )+j \\Bigl (\\tfrac{1}{d}+\\tfrac{r}{d}\\Bigr )\\Bigr ]=\\sum _{j=\\left\\lfloor (d-r)\\frac{p-1}{d} \\right\\rfloor +1}^{\\operatorname{rep}(m_2) - 1}\\operatorname{Bin}\\hspace{1.0pt}(j) \\Bigl [\\tfrac{1}{p}\\Bigr ]\\Bigl [\\Bigl (j+\\tfrac{1}{d}\\Bigr )\\Bigl (j+\\tfrac{r}{d}\\Bigr )\\Bigr ]$ modulo $p^2$ .", "Recall that this sum is vacuous unless $p\\equiv d-1\\pmod {d}$ , $p \\equiv r \\pmod {d}$ and $r^2\\equiv -1\\pmod {d}$ or $p \\equiv d-r \\pmod {d}$ and $r^2\\equiv 1\\pmod {d}$ .", "In these cases the limits of summation are equal and the sum is over one value of $j=\\operatorname{rep}(m_2)-1=p-r\\lfloor \\tfrac{p-1}{d} \\rfloor - \\left\\lfloor \\frac{ar}{d} \\right\\rfloor -1$ .", "Now $p-r\\lfloor \\tfrac{p-1}{d}\\rfloor - \\left\\lfloor \\tfrac{ar}{d} \\right\\rfloor -1=p-r\\left(\\tfrac{p-a}{d}\\right)- \\left\\lfloor \\tfrac{ar}{d} \\right\\rfloor -1=p\\left(1-\\tfrac{r}{d}\\right) + \\langle \\tfrac{ar}{d} \\rangle -1.$ Thus if $p\\equiv d-1 \\pmod {d}$ , $j=p\\left(1-\\tfrac{r}{d}\\right)+\\langle r-\\tfrac{r}{d}\\rangle -1\\\\=p\\left(1-\\tfrac{r}{d}\\right)+1-\\tfrac{r}{d}-1=p\\left(1-\\tfrac{r}{d}\\right)-\\tfrac{r}{d}.$ Then $\\left(j+\\tfrac{r}{d}\\right) = p\\left(1-\\tfrac{r}{d}\\right) \\in p \\mathbb {Z}_p$ and $\\left(j+\\tfrac{1}{d}\\right) = p\\left(1-\\tfrac{r}{d}\\right) -\\tfrac{r}{d} +\\tfrac{1}{d}\\in \\mathbb {Z}_p.$ If $p\\equiv r \\pmod {d}$ and $r^2 \\equiv -1 \\pmod {d}$ , $j=p\\left(1-\\tfrac{r}{d}\\right)-\\langle \\tfrac{r^2}{d} \\rangle -1=p\\left(1-\\tfrac{r}{d}\\right)+\\tfrac{d-1}{d}-1=p\\left(1-\\tfrac{r}{d}\\right)-\\tfrac{1}{d}.$ If $p\\equiv d-r \\pmod {d}$ and $r^2 \\equiv 1 \\pmod {d}$ , $j=p\\left(1-\\tfrac{r}{d}\\right)+\\langle \\tfrac{dr-r^2}{d} \\rangle -1=p\\left(1-\\tfrac{r}{d}\\right)+\\tfrac{d-1}{d}-1=p\\left(1-\\tfrac{r}{d}\\right)-\\tfrac{1}{d}.$ Then, in both cases, $\\left(j+\\tfrac{1}{d}\\right) = p\\left(1-\\tfrac{r}{d}\\right) \\in p \\mathbb {Z}_p$ and $\\left(j+\\tfrac{r}{d}\\right) = p\\left(1-\\tfrac{r}{d}\\right) -\\tfrac{1}{d} +\\tfrac{r}{d}\\in \\mathbb {Z}_p.$ Therefore, $\\sum _{j=\\left\\lfloor (d-r)\\frac{p-1}{d} \\right\\rfloor +1}^{\\operatorname{rep}(m_2) - 1}\\operatorname{Bin}\\hspace{1.0pt}(j) \\Bigl [\\tfrac{1}{p}\\Bigr ]\\Bigl [\\Bigl (j+\\tfrac{1}{d}\\Bigr )\\Bigl (j+\\tfrac{r}{d}\\Bigr )\\Bigr ]\\equiv 0\\pmod {p^2},\\vspace{12.0pt}$ as $\\operatorname{Bin}\\hspace{1.0pt}(j) \\in p^2\\mathbb {Z}_p$ for $j=\\operatorname{rep}(m_2)-1$ .", "Finally, we examine the two remaining terms of (REF ).", "Taking $m=\\operatorname{rep}(m_1)-1$ and $n=\\operatorname{rep}(m_2)-1$ in Corollary REF we get that $\\sum _{j=0}^{\\operatorname{rep}(m_2) - 1}\\operatorname{Bin}\\hspace{1.0pt}(j) \\Bigl [1+j \\mathcal {H}(j) \\Bigr ] \\equiv s(p)\\pmod {p^2},$ and taking $l=p$ , $m=\\operatorname{rep}(m_1)-1$ , $n=\\operatorname{rep}(m_2)-1$ , $c_2=\\operatorname{rep}(m_1)-m_1$ and $c_1=\\operatorname{rep}(m_2)-m_2$ in Corollary REF we get $\\sum _{j=0}^{p-\\left\\lfloor \\frac{p-1}{d_2} \\right\\rfloor - 1}\\operatorname{Bin}\\hspace{1.0pt}(j)\\Bigl [ j \\mathcal {B}(j) -\\mathcal {A}(j) - j \\mathcal {A}(j) \\mathcal {H}(j) \\Bigr ]\\equiv 0 \\pmod {p^2}$ as required." ] ]

[ [ "Faster Algorithms for Rectangular Matrix Multiplication" ], [ "Abstract Let {\\alpha} be the maximal value such that the product of an n x n^{\\alpha} matrix by an n^{\\alpha} x n matrix can be computed with n^{2+o(1)} arithmetic operations.", "In this paper we show that \\alpha>0.30298, which improves the previous record \\alpha>0.29462 by Coppersmith (Journal of Complexity, 1997).", "More generally, we construct a new algorithm for multiplying an n x n^k matrix by an n^k x n matrix, for any value k\\neq 1.", "The complexity of this algorithm is better than all known algorithms for rectangular matrix multiplication.", "In the case of square matrix multiplication (i.e., for k=1), we recover exactly the complexity of the algorithm by Coppersmith and Winograd (Journal of Symbolic Computation, 1990).", "These new upper bounds can be used to improve the time complexity of several known algorithms that rely on rectangular matrix multiplication.", "For example, we directly obtain a O(n^{2.5302})-time algorithm for the all-pairs shortest paths problem over directed graphs with small integer weights, improving over the O(n^{2.575})-time algorithm by Zwick (JACM 2002), and also improve the time complexity of sparse square matrix multiplication." ], [ "Background.", "Matrix multiplication is one of the most fundamental problems in computer science and mathematics.", "Besides the fact that several computational problems in linear algebra can be reduced to the computation of the product of two matrices, the complexity of matrix multiplication also arises as a bottleneck in a multitude of other computational tasks (e.g., graph algorithms).", "The standard method for multiplying two $n\\times n$ matrices uses $O(n^3)$ arithmetic operations.", "Strassen showed in 1969 that this trivial algorithm is not optimal, and gave a algorithm that uses only $O(n^{2.808})$ arithmetic operations.", "This has been the beginning of a long story of improvements that lead to the upper bound $O(n^{2.376})$ by Coppersmith and Winograd [10], which has been further improved to $O(n^{2.3727})$ very recently by Vassilevska Williams [27].", "Note that all the above complexities refer to the number of arithmetic operations involved, but naturally the same upper bounds hold for the time complexity as well when each arithmetic operation can be done in negligible time (e.g., in $\\mathrm {poly}(\\log n)$ time).", "Finding the optimal value of the exponent of square matrix multiplication is naturally one of the most important open problems in algebraic complexity.", "It is widely believed that the product of two $n\\times n$ matrices can be computed with $O(n^{2+\\epsilon })$ arithmetic operations for any constant $\\epsilon >0$ .", "Several conjectures, including conjectures about combinatorial structures [10] and about group theory [7], [6], would, if true, lead to this result (see also [1] for recent work on these conjectures).", "Another way to interpret this open problem is by considering the multiplication of an $n\\times m$ matrix by an $m\\times n$ matrix.", "Suppose that the matrices are defined over a field.", "For any $k>0$ , define the exponent of such a rectangular matrix multiplication as follows: $\\omega (1,1,k)=\\inf \\lbrace \\tau \\in \\mathbb {R}\\:|\\: C(n,n,\\lfloor n^k \\rfloor )=O(n^\\tau )\\rbrace ,$ where $C(n,n,\\lfloor n^k \\rfloor )$ denotes the minimum number of arithmetic operations needed to multiply an $n\\times \\lfloor n^k \\rfloor $ matrix by an $\\lfloor n^k \\rfloor \\times n$ matrix.", "Note that, while the value $\\omega (1,1,k)$ may depend on the field under consideration, it is known that it can depend only on the characteristic of the field [23].", "Define $\\omega =\\omega (1,1,1)$ and $\\alpha =\\sup \\lbrace k\\:|\\:\\omega (1,1,k)=2\\rbrace $ .", "The value $\\omega $ represents the exponent of square matrix multiplication, and the value $\\alpha $ essentially represents the largest value such that the product of an $n\\times n^\\alpha $ matrix by an $n^\\alpha \\times n$ matrix can be computed with $O(n^{2+\\epsilon })$ arithmetic operations for any constant $\\epsilon $ .", "Since $\\omega =2$ if and only if $\\alpha =1$ , one possible strategy towards showing that $\\omega =2$ is to give lower bounds on $\\alpha $ .", "Coppersmith [8] showed in 1982 that $\\alpha >0.172$ .", "Then, based on the techniques developed in [10], Coppersmith [9] improved this lower bound to $\\alpha >0.29462$ .", "This is the best lower bound on $\\alpha $ known so far.", "Excepting Coppersmith's works on the value $\\alpha $ , there have been relatively few algorithms that focused specifically on rectangular matrix multiplication.", "Since it is well known (see, e.g, [17]) that multiplying an $n\\times n$ matrix by an $n\\times m$ matrix, or an $m\\times n$ matrix by an $n\\times n$ matrix, can be done with the same number of arithmetic operations as multiplying an $n\\times m$ matrix by an $m\\times n$ matrix, the value $\\omega (1,1,k)$ represents the exponent of all these three types of rectangular matrix multiplications.", "Note that, by decomposing the product into smaller matrix products, it is easy to obtain (see, e.g, [17]) the following upper bound: $\\omega (1,1,k)=\\left\\lbrace \\begin{array}{ll}2 &\\textrm { if }0\\le k\\le \\alpha \\\\2+(\\omega -2)\\frac{k-\\alpha }{1-\\alpha } &\\textrm { if }\\alpha \\le k\\le 1.\\end{array}\\right.$ Lotti and Romani [17] obtained nontrivial upper bounds on $\\omega (1,1,k)$ based on the seminal result by Coppersmith [8] and on early works on square matrix multiplication.", "Huang and Pan [13] showed how to apply ideas from [10] to the rectangular setting and obtained the upper bound $\\omega (1,1,2)<3.333954$ , but this approach did not lead to any upper bound better than (REF ) for $k \\le 1$ .", "Ke, Zeng, Han and Pan [16] further improved Huang and Pan's result to $\\omega (1,1,2)<3.2699$ , by using again the approach from [10], and also reported the upper bounds $\\omega (1,1,0.8)<2.2356$ and $\\omega (1,1,0.5356)<2.0712$ , which are better than those obtained by (REF ).", "Their approach, nevertheless, did not give any improvement for the value of $\\alpha $ .", "Besides the fact that a better understanding of $\\omega (1,1,k)$ gives insights into the nature of matrix multiplication and ultimately may help showing that $\\omega =2$ , fast algorithms for multiplying an $n\\times n^k$ matrix by an $n^k\\times n$ with $k \\ne 1$ have also a multitude of applications.", "Typical examples not directly related to linear algebra include the construction of fast algorithms for the all-pairs shortest paths problem [2], [19], [29], [33], [34], the dynamic computation of the transitive closure [12], [22], finding ancestors [11], detecting directed cycles [31], or computing the diameter of a graph [30].", "Rectangular matrix multiplication has also been used in computational complexity [18], [28], and to speed-up sparse square matrix multiplication [3], [15], [32] or tasks in computational geometry [14], [15].", "Obtaining new upper bounds on $\\omega (1,1,k)$ would thus reduce the asymptotic time complexity of algorithms in a wide range of areas.", "We nevertheless stress that such improvements are only of theoretical interest, since the huge constants involved in the complexity of fast matrix multiplication usually make these algorithms impractical." ], [ "Short description of the approach by Coppersmith and Winograd.", "The results [9], [13], [16], [24], [27] mentioned above are all obtained by extending the approach by Coppersmith and Winograd [10].", "This approach is an illustration of a general methodology initiated in the 1970's based on the theory of bilinear and trilinear forms, through which most of the improvements for matrix multiplication have been obtained.", "Informally, the idea is to start with a basic construction (some small trilinear form), and then exploit general properties of matrix multiplication (in particular Schönhage's asymptotic sum inequality [23]) to derive an upper bound on the exponent $\\omega $ from this construction.", "The main contributions of [10] consist of two parts: the discovery of new basic constructions and the introduction of strong techniques to analyze them.", "In their paper, Coppersmith and Winograd actually present three algorithms, based on three different basic constructions.", "The first basic construction (Section 6 in [10]) is the simplest of the three and leads to the upper bound $\\omega <2.40364$ .", "The second basic construction (Section 7 in [10]), that we will refer in this paper as $F_q$ (here $q\\in \\mathbb {N}$ is a parameter), leads to the upper bound $\\omega <2.38719$ .", "The third basic construction (Section 8 in [10]) is $F_q\\otimes F_q$ , the tensor product of two instances of $F_q$ , and leads to the improved upper bound $\\omega <2.375477$ .", "In view of the last result, it was natural to ask if taking larger tensor powers of $F_q$ as the basic construction leads to better bounds on $\\omega $ .", "The case $r=3$ was explicitly mentioned as an open problem in [10] but did not seem to lead to any improvement.", "Stothers [24] and Vassilevska Williams [27] succeeded in analyzing the fourth tensor product $F_q^{\\otimes 4}$ and obtained a better upper bound on $\\omega $ , the first improvement in more that twenty years.", "Vassilevska Williams further presented a general framework that enables a systematic analysis for higher tensor products of the basis construction, and used this framework to show that $\\omega <2.3727$ , for the basic construction $F^{\\otimes 8}$ , the best upper bound obtained so far.", "The algorithms for rectangular matrix multiplication [9], [13], [16] already mentioned use a similar approach.", "Huang and Pan [13] obtained their improvement on $\\omega (1,1,2)$ by taking the easiest of the three construction in [10] and carefully modifying the analysis to evaluate the complexity of rectangular matrix multiplication.", "Ke, Zeng, Han and Pan [16] obtained their improvements similarly, but by using the second basic construction from [10] (the construction $F_q$ ) instead, which lead to better upper bounds.", "These approaches, while very natural, do not provide any nontrivial lower bounds on $\\alpha $ : the upper bounds on $\\omega (1,1,k)$ obtained are strictly larger than 2 even for small values of $k$ .", "In order to obtain the lower bound $\\alpha >0.29462$ , Coppersmith [9] relied on a more complex approach: the basic construction considered is still $F_q$ , but several instances for distinct values of $q$ are combined together in a subtle way in order to keep the complexity of the resulting algorithm small enough (i.e., not larger than $n^{2+o(1)}$ )." ], [ "Statement of our results and discussion.", "In this paper we construct new algorithms for rectangular matrix multiplication, by taking the tensor power $F_q\\otimes F_q$ as basic construction and analyzing this construction in the framework of rectangular matrix multiplication.", "We use these ideas to prove that $\\omega (1,1,k)=2$ for any $k\\le 0.3029805$ , as stated in the following theorem.", "Theorem 1.1 For any value $k<0.3029805... $ , the product of an $n\\times n^k$ matrix by an $n^k\\times n$ matrix can be computed with $O(n^{2+\\epsilon })$ arithmetic operations for any constant $\\epsilon >0$ .", "Theorem REF shows that $\\alpha >0.30298$ , which improves the previous record $\\alpha >0.29462$ by Coppersmith.", "More generally, in the present work we present an algorithm for multiplying an $n\\times n^k$ matrix by an $n^k\\times n$ matrix, for any value $k$ .", "We show that the complexity of this algorithm can be expressed as a (nonlinear) optimization problem, and use this formulation to derive upper bounds on $\\omega (1,1,k)$ .", "Table REF shows the bounds we obtain for several values of $k$ .", "The bounds obtained for $0\\le k\\le 1$ are represented in Figure REF as well.", "Table: Our upper bounds on the exponent of the multiplication of ann×n k n\\times n^k matrix by an n k ×nn^k\\times n matrix.Figure: Our upper bounds (in plain line) on ω(1,1,k)\\omega (1,1,k), for 0≤k≤10\\le k\\le 1.The dashed line represents the upper bounds on ω(1,1,k)\\omega (1,1,k) obtained by usingEquation () with thevalues α>0.30298\\alpha >0.30298 and ω<2.375477\\omega <2.375477.The results of this paper can be seen as a generalization of Coppersmith-Winograd's approach to the rectangular setting.", "In the case of square matrix multiplication (i.e., for $k=1$ ), we recover naturally the same upper bound $\\omega (1,1,1)<2.375477$ as the one obtained in [10].", "Let us mention that we can, in a rather straightforward way, combine our results with the upper bound $\\omega <2.3727$ by Vassilevska Williams [27] to obtain slightly improved bounds for $k\\approx 1$ .", "The idea is, very similarly to how Equation (REF ) was obtained, to exploit the convexity of the function $\\omega (1,1,k)$ .", "Concretely, for any fixed value $0\\le k_0<1$ , the inequality $\\omega (1,1,k)\\le \\omega (1,1,k_0)+(\\omega -\\omega (1,1,k_0))\\frac{k-k_0}{1-k_0}$ holds for any $k$ such that $k_0\\le k\\le 1$ .", "This enables us to combine an upper bound on $\\omega (1,1,k_0)$ , for instance one of the values in Table REF , with the improved upper bound $\\omega <2.3727$ by Vassilevska Williams.", "Since the improvement is small and concerns only the case $k\\approx 1$ , we will not discuss it further.", "For $k>0.29462$ and $k\\ne 1$ , the complexity of our algorithms is better than all known algorithms for rectangular matrix multiplication, including the algorithms [13], [16] mentioned above.", "Moreover, for $0.30298<k<1$ , our new bounds are significantly better than what can be obtained solely from the bound $\\alpha >0.30298$ and $\\omega <2.375477$ through Equation (REF ), as illustrated in Figure REF .", "This suggests that non-negligible improvements can be obtained for all applications of rectangular matrix multiplications that rely on this simple linear interpolation — we will elaborate on this subject in Subsection REF .", "Let us compare more precisely our results with those reported in [16].", "For $k=2$ , we obtain $\\omega (1,1,2)<3.256689$ while Ke et al.", "[16] obtained $\\omega (1,1,2)<3.2699$ by using the basic construction $F_q$ .", "Our improvements are of the same order for the other two values ($k=0.8$ and $k=0.5356$ ) analyzed in [16], as can be seen from Table REF .", "Note that the order of magnitude of the improvements here is similar to what was obtained in [10] by changing the basic construction from $F_q$ to $F_q\\otimes F_q$ for square matrix multiplication, which led to a improvement from $\\omega <2.38719$ to $\\omega <2.375477$ .", "A noteworthy point is that our algorithm directly leads to improved lower bounds on $\\alpha $ while, as already mentioned, to obtain a nontrivial lower bound on $\\alpha $ using the basic construction $F_q$ (as done in [9]) a specific methodology was needed.", "Our approach can then be considered as a general framework to study rectangular matrix multiplication, which leads to a unique optimization problem that gives upper bounds on $\\omega (1,1,k)$ for any value of $k$ ." ], [ "Applications", "As mentioned in the beginning of the introduction, improvements on the time complexity of rectangular matrix multiplication give faster algorithms for a multitude of computational problems.", "In this subsection we describe quantitatively the improvements that our new upper bounds imply for some of these problems: sparse square matrix multiplication, the all-pairs shortest paths problem and computing dynamically the transitive closure of a graph." ], [ "Sparse square matrix multiplication.", "Yuster and Zwick [32] have shown how fast algorithms for rectangular matrix multiplication can be used to construct fast algorithms for computing the product of two sparse square matrices (this result has been generalized to the product of sparse rectangular matrices in [15], and the case where the output matrix is also sparse has been studied in [3]).", "More precisely, let $M$ and $M^{\\prime }$ be two $n\\times n$ matrices such that each matrix has at most $m$ non-zero entries, where $0\\le m\\le n^2$ .", "Yuster and Zwick [32] showed that the product of $M$ and $M^{\\prime }$ can be computed in time $O\\left(\\min (nm,n^{\\omega (1,1,\\lambda _m)+o(1)},n^{\\omega +o(1)})\\right),$ where $\\lambda _m$ is the solution of the equation $\\lambda _m+\\omega (1,1,\\lambda _m)=2\\log _n (m)$ .", "Using the upper bounds on $\\omega (1,1,k)$ of Equation (REF ) with the values $\\alpha <0.294$ and $\\omega <2.376$ , this gives the complexity depicted in Figure REF .", "These upper bounds can be of course directly improved by using the new upper bound on $\\omega $ by Vassilevska Williams [27] and the new lower bound on $\\alpha $ given in the present work, but the improvement is small.", "A more significant improvement can be obtained by using directly the upper bounds on $\\omega (1,1,k)$ presented in Figure REF , which gives the new upper bounds on the complexity of sparse matrix multiplication depicted in Figure REF .", "For example, for $m=n^{4/3}$ , we obtain complexity $O(n^{2.087})$ , which is better than the original upper bound $O(n^{2.1293...})$ obtained from Equation (REF ) with $\\alpha >0.294$ and $\\omega <2.376$ .", "Note that replacing $\\omega <2.376$ with the the best known bound $\\omega <2.3727$ only decreases the latter bound to $O(n^{2.1287...})$ .", "Thus, even if the algorithms presented in the present paper do not give any improvement on $\\omega $ (i.e., for the product of dense square matrices), we do obtain improvements for computing the product of two sparse square matrices.", "Figure: Upper bounds on the exponent for the multiplication two n×nn\\times n matriceswith at most mm non-zero entries.", "The horizontal axis represents log n (m)\\log _n(m).The dashed line represents the results by Yuster and Zwick and showsthat the term n ω(1,1,λ m ) n^{\\omega (1,1,\\lambda _m)} dominates the complexitywhen 1≤log n (m)≤(1+ω)/21\\le \\log _n(m)\\le (1+\\omega )/2.The plain line represents the improvements we obtain." ], [ "Graph algorithms.", "Zwick [34] has shown how to use rectangular matrix multiplication to compute, with high probability, the all-pairs shortest paths in weighted direct graphs where the weights are bounded integers.", "The time complexity obtained is $O(n^{2+\\mu +\\epsilon })$ , for any constant $\\epsilon >0$ , where $\\mu $ is the solution of the equation $\\omega (1,1,\\mu )=1+2\\mu $ .", "Using the upper bounds on $\\omega (1,1,k)$ of Equation (REF ) with $\\alpha >0.294$ and $\\omega <2.376$ , this gives $\\mu <0.575$ and thus complexity $O(n^{2.575+\\epsilon })$ .", "This reduction to rectangular matrix multiplication is the asymptotically fastest known approach for weighted directed graphs with small integer weights.", "Our results (see Table REF ) show that $\\omega (1,1,0.5302)<2.0604$ , which gives the upper bound $\\mu <0.5302$ .", "We thus obtain the following result.", "Theorem 1.2 There exists a randomized algorithm that computes the shortest paths between all pairs of vertices in a weighted directed graph with bounded integer weights in time $O(n^{2.5302})$ , where $n$ is the number of vertices in the graph.", "Note that, even if $\\omega =2$ , the complexity of Zwick's algorithm is $O(n^{2.5+\\epsilon })$ .", "In this perspective, our improvements on the complexity of rectangular matrix multiplication offer a non-negligible speed-up for the all-pairs shortest paths problem in this setting.", "The same approach can be used to improve several other existing graph algorithms.", "Let us describe another example: algorithms for computing dynamically the transitive closure of a graph.", "Demetrescu and Italiano [12] presented a randomized algorithm for the dynamic transitive closure of directly acyclic graph with $n$ vertices that answers queries in $O(n^{\\mu })$ time, and performs updates in $O(n^{1+\\mu +\\epsilon })$ time, for any $\\epsilon >0$ .", "Here $\\mu $ is again the solution of the equation $\\omega (1,1,\\mu )=1+2\\mu $ .", "This was the first algorithm for this problem with subquadratic time complexity.", "This result have been generalized later to general graphs, with the same bounds, by Sankowski [21].", "Our new upper bounds thus show the existence of an algorithm for the dynamic transitive closure that answers queries in $O(n^{0.5302})$ time and performs updates in $O(n^{1.5302})$ time." ], [ "Overview of our techniques and organization of the paper", "Before presenting an overview of the techniques used in this paper, we will give an informal description of algebraic complexity theory (the contents of which will be superseded by the formal presentation of these notions in Section ).", "In this paper we will use, for any positive integer $n$ , the notation $[n]$ to represent the set $\\lbrace 1,\\ldots ,n\\rbrace $ ." ], [ "Trilinear forms and bilinear algorithms.", "The matrix multiplication of an $m\\times n$ matrix by an $n\\times p$ matrix can be represented by the following trilinear form, denoted as $\\langle m,n,p \\rangle $ : $\\langle m,n,p \\rangle =\\sum _{r=1}^m\\sum _{s=1}^n\\sum _{t=1}^p x_{rs}y_{st}z_{rt},$ where $x_{rs}$ , $y_{st}$ and $z_{rt}$ are formal variables.", "This form can be interpreted as follows: the $(r,t)$ -th entry of the product of an $m\\times n$ matrix $M$ by an $n\\times p$ matrix $M^{\\prime }$ can be obtained by setting $x_{ij}=M_{ij}$ for all $(i,j)\\in [m]\\times [n]$ and $y_{ij}=M^{\\prime }_{ij}$ for all $(i,j)\\in [n]\\times [p]$ , setting $z_{rt}=1$ and setting all the other $z$ -variables to zero.", "One can then think of the $z$ -variables as formal variables used to record the entries of the matrix product.", "More generally, a trilinear form $t$ is represented as $t=\\sum _{i\\in A}\\sum _{j\\in B}\\sum _{k\\in C} t_{ijk}x_{i}y_{j}z_{k}.$ where $A,B$ and $C$ are three sets, $x_{i}$ , $y_{j}$ and $z_{k}$ are formal variables and the $t_{ijk}$ 's are coefficients in a field $\\mathbb {F}$ .", "Note that the set of indexes for the trilinear form $\\langle m,n,p \\rangle $ are $A=[m]\\times [n]$ , $B=[n]\\times [p]$ and $C=[m]\\times [p]$ .", "An exact (bilinear) algorithm computing $t$ corresponds to an equality of the form $t=\\sum _{\\ell =1}^r \\left(\\sum _{i\\in A}\\alpha _{\\ell i}x_i\\right)\\left(\\sum _{j\\in B}\\beta _{\\ell j}y_j\\right)\\left(\\sum _{k\\in C}\\gamma _{\\ell k}z_k\\right)$ with coefficients $\\alpha _{\\ell i},\\beta _{\\ell j},\\gamma _{\\ell k}$ in $\\mathbb {F}$ .", "The minimum number $r$ such that such a decomposition exists is called the rank of the trilinear form $t$ , and denoted $R(t)$ .", "The rank of a trilinear form is an upper bound on the complexity of a (bilinear) algorithm computing the form: it precisely expresses the number of multiplications needed for the computation, and it can be shown that the number of additions or scalar multiplications affect the cost only in a negligible way.", "For any $k>0$ , the quantity $\\omega (1,1,k)$ can then be equivalently defined as follows: $\\omega (1,1,k)=\\inf \\lbrace \\tau \\in \\mathbb {R}\\:|\\: R(\\langle n,n,\\lfloor n^k \\rfloor \\rangle )=O(n^\\tau )\\rbrace .$ Approximate bilinear algorithms have been introduced to take advantage of the fact that the complexity of trilinear forms (and especially of matrix multiplication) may be reduced significantly by allowing small errors in the computation.", "Let $\\lambda $ be an indeterminate over $\\mathbb {F}$ , and let $\\mathbb {F}[\\lambda ]$ denote the set of all polynomials over $\\mathbb {F}$ in $\\lambda $ .", "Let $s$ be any nonnegative integer.", "A $\\lambda $ -approximate algorithm for $t$ is an equality of the form $\\lambda ^{s}t+\\lambda ^{s+1} \\left(\\sum _{i\\in A}\\sum _{j\\in B}\\sum _{k\\in C} d_{ijk}x_iy_jz_k\\right)=\\sum _{\\ell =1}^r \\left(\\sum _{i\\in A}\\alpha _{\\ell i}x_i\\right)\\left(\\sum _{j\\in B}\\beta _{\\ell j}y_j\\right)\\left(\\sum _{k\\in C}\\gamma _{\\ell k}z_k\\right)$ for coefficients $\\alpha _{\\ell i},\\beta _{\\ell j},\\gamma _{\\ell k},d_{ijk}$ in $\\mathbb {F}[\\lambda ]$ and some nonnegative integer $s$ .", "Informally, this means that the form $t$ can be computed by determining the coefficient of $\\lambda ^s$ in the right hand side.", "The minimum number $r$ such that such a decomposition exists is called the border rank of $t$ and denoted $\\underline{R}(t)$ .", "It is known that the border rank is an upper bound on the complexity of an algorithm that approximates the trilinear form, and that any such approximation algorithm can be converted into an exact algorithm with essentially the same complexity.", "A sum $\\sum _{i}t_i$ of trilinear forms is a direct sum if the $t_i$ 's do not share variables.", "Informally, Schönhage's asymptotic sum inequality [23] for rectangular matrix multiplication states that, if the form $t$ can be converted (in the $\\lambda $ -approximation sense) into a direct sum of $c$ trilinear forms, each form being isomorphic to $\\langle m,m,m^k \\rangle $ , then $c\\cdot m^{\\omega (1,1,k)}\\le \\underline{R}(t).$ This suggests that good bounds on $\\omega (1,1,k)$ can be obtained if the form $t$ can be used to derive many independent (i.e., not sharing any variables) matrix multiplications.", "This approach has been applied to derive almost all new bounds on matrix multiplication since its discovery in 1981 by Schönhage." ], [ "Overview of our techniques.", "Our algorithm uses, as its basic construction, the trilinear form $F_q\\otimes F_q$ from [10], which can be written as a sum of fifteen terms $T_{ijk}$ , for all fifteen nonnegative integers $i,j,k$ such that $i+j+k=4$ : $F_q\\otimes F_q=\\sum _{{{\\scriptstyle \\begin{matrix}0\\le i,j,k\\le 4\\\\ i+j+k=4\\end{matrix}}}}T_{ijk}.$ If the sum were direct, then, from Schönhage's asymptotic sum inequality and since an upper bound on the rank of $F_q\\otimes F_q$ is easy to obtain, this would reduce the problem to the analysis of each part $T_{ijk}$ .", "This is unfortunately not the case: the $T_{ijk}$ 's share variables.", "To solve this problem, the basic construction is manipulated in order to obtain a direct sum, similarly to [10].", "The first step is to take the $N$ -th tensor product of the basis construction, where $N$ is a large integer.", "This gives: $(F_q\\otimes F_q)^{\\otimes N}=\\sum _{IJK} T_{IJK}$ where the sum is over all triples of sequences $IJK$ with $I,J,K\\in \\lbrace 0,1,2,3,4\\rbrace ^N$ such that $I_\\ell +J_\\ell +K_\\ell =4$ for all $\\ell \\in \\lbrace 1,\\ldots ,N\\rbrace $ .", "Here each form $T_{IJK}$ is the tensor product of $N$ forms $T_{ijk}$ .", "The sum is nevertheless not yet direct.", "The key idea of the next step is to zero variables in order to remove some forms and obtain a sum where any two non-zero forms $T_{IJK}$ and $T_{I^{\\prime }J^{\\prime }K^{\\prime }}$ are such that $I\\ne I^{\\prime }$ , $J\\ne J^{\\prime }$ and $K\\ne K^{\\prime }$ , which will imply that the sum is direct.", "Moreover, in order to be able to apply Schönhage's asymptotic sum inequality, we would like all the remaining $T_{IJK}$ to be isomorphic to the same rectangular matrix product (i.e., there should exist values $m$ and $k$ such that each non-zero form $T_{IJK}$ is isomorphic to the matrix multiplication $\\langle m,m,m^k \\rangle $ ).", "From here our approach differs from Coppersmith and Winograd [10], since our concern is upper bounds for rectangular matrix multiplication.", "We will take fifteen integers $a_{ijk}$ and show how the form $(F_q\\otimes F_q)^{\\otimes N}$ can be transformed, by zeroing variables, into a direct sum of a large number of forms $T_{IJK}$ in which each $T_{IJK}$ is such that $T_{IJK}\\cong \\bigotimes _{{{\\scriptstyle \\begin{matrix}0\\le i,j,k\\le 4\\\\ i+j+k=4\\end{matrix}}}}T_{ijk}^{\\otimes a_{ijk}N}.$ The main difference here is that, in [10], the symmetry of square multiplication implied that the $a_{ijk}$ 's could be taken invariant under permutation of indices, with means that only four parameters ($a_{004}$ , $a_{013}$ , $a_{022}$ and $a_{112}$ ) needed to be considered.", "In our case, we still impose the condition $a_{ijk}=a_{ikj}$ , but not more.", "This reduces the number of parameters to nine: $a_{004}$ , $a_{400}$ , $a_{013}$ , $a_{103}$ , $a_{301}$ , $a_{022}$ , $a_{202}$ , $a_{112}$ and $a_{211}$ .", "Many nontrivial technical problems arise from this larger number of parameters.", "In particular, the equations that occur during the analysis do not have a unique solution and an optimization step is necessary.", "This is similar to the difficulties that appeared in the analysis of the basis construction $F_q^{\\otimes 4}$ (for square matrix multiplication) done in [24], [27].", "We will show that this further optimization step essentially imposes the additional (nonlinear) constraint $a_{013}a_{202}a_{112}=a_{103}a_{022}a_{211}$ .", "Showing that $(F_q\\otimes F_q)^{\\otimes N}$ can be transformed into a direct sum of many isomorphic forms, as claimed, was also used in previous works based on the approach by Coppersmith and Winograd.", "Our setting is nevertheless more general than in [10], for the following two reasons.", "First, our approach is asymmetric since our parameters $a_{ijk}$ are not invariant under permutations of the indices.", "Second, the technical problems due to the presence of many parameters require more precise arguments.", "The former complication was addressed implicitly in [9], and explicitly in [13], [16].", "The latter complication was addressed in [24], [27].", "Here we need to deal with these two complications simultaneously, which involves a careful analysis.", "Instead of approaching this task directly in the language of trilinear forms, we give a graph-theoretic interpretation of it, which will make the exposition more intuitive, and also simplify the analysis.", "Each form $T_{IJK}$ will correspond to one vertex in a graph, and an edge in the graph will represent the fact that two forms share one index.", "We will interpret the task of converting the sum $(F_q\\otimes F_q)^{\\otimes N}$ into a direct sum (i.e., a sum where the non-zero forms $T_{IJK}$ do not share any index) as the task of converting the graph, by using only simple graph operations, into an edgeless subgraph.", "We will present algorithms solving this latter graph-theoretic task and show that a large edgeless subgraph can be obtained, which means that many forms $T_{IJK}$ not sharing any index can be constructed from $(F_q\\otimes F_q)^{\\otimes N}$ .", "Now that we have a direct sum of many forms, each form being isomorphic to (REF ), the only thing to do before applying Schönhage's asymptotic sum inequality is to show that the form (REF ) is isomorphic to a direct sum of matrix products $\\langle m,m,m^k \\rangle $ .", "Some of the $T_{ijk}$ 's (more precisely, all the $T_{ijk}$ 's except $T_{112}$ , $T_{121}$ and $T_{211}$ ) can be analyzed in a straightforward way, since they correspond to matrix products, as originally observed in [10].", "The forms $T_{112}$ , $T_{121}$ and $T_{211}$ are delicate to analyze since they do not correspond to matrix products.", "In [10] (see also [24], [27]), they were analyzed individually through the concept of “value”, a quantity that evaluates the number of square matrix products (and their size) that can be created from the form under consideration.", "This is nevertheless useless for estimating $\\omega (1,1,k)$ , except when $k=1$ , because the value is intrinsically symmetric (in particular, the values of $T_{112}$ , $T_{121}$ and $T_{211}$ are identical), while here we are precisely interested in breaking the symmetry in order to obtain bounds for rectangular matrix multiplication.", "Instead, we will analyze the term $T_{112}^{\\otimes a_{112}N}\\otimes T_{121}^{\\otimes a_{121}N}\\otimes T_{211}^{\\otimes a_{211}N}$ globally.", "This is the key new idea leading to our new bounds on $\\alpha $ and more generally on $\\omega (1,1,k)$ for any $k\\ne 1$ .", "Note that this difficulty was not present in previous works on rectangular matrix multiplication [9], [13], [16]: for simpler basic constructions such as $F_q$ , all the smaller parts correspond to matrix products.", "We will show that $T_{112}$ , $T_{121}$ and $T_{211}$ can be converted into a large number of objects called “${C}$ -tensors” in Strassen's terminology [26].", "This will be done by relying on the graph-interpretation we introduced and showing how this conversion can be interpreted as finding large cliques in a graph (this is the main reason why we developed this graph-theoretic interpretation).", "While the fact that the form $T_{112}$ corresponds to a sum of ${C}$ -tensors was briefly mentioned in [10], and a proof sketched, we will need a complete analysis here.", "We will in particular rely on the fact that the ${C}$ -tensors obtained from $T_{112}$ , $T_{121}$ and $T_{211}$ are not identical.", "The success of our approach comes from the discovery that these ${C}$ -tensors are actually “complementary”: while the ${C}$ -tensors obtained individually from $T_{112}$ , $T_{121}$ and $T_{211}$ do not give any improvement for the exponent of rectangular matrix multiplication, their combination (i.e., the ${C}$ -tensors corresponding to the whole term (REF )) does lead to improvements when analyzed globally by the “laser method” developed by Strassen [26].", "This will show that the form (REF ), and thus the form (REF ) as well, is isomorphic to a direct sum of matrix products $\\langle m,m,m^k \\rangle $ .", "Finally, Schönhage's asymptotic sum inequality will give an inequality, depending on the parameters $a_{ijk}$ , that involves $\\omega (1,1,k)$ .", "Our new upper bounds on $\\omega (1,1,k)$ are obtained by optimizing these parameters.", "While this is done essentially though numerical calculations, the new lower bound on $\\alpha $ requires a more careful analysis where the optimal values of all but a few of the parameters are found analytically." ], [ "Higher powers.", "A natural question is whether our bounds can be improved by taking higher tensor powers of $F_q$ as the basic algorithm, i.e., taking $F_q^{\\otimes r}$ for $r>2$ .", "As can be expected from the analysis for $r=2$ we have outlined above, the analysis is much more difficult than in the square case, for two main reasons.", "The first reason is that the construction is not symmetric and thus more parameters have to be considered.", "This problem can be nevertheless addressed through a systematic framework similar to the one described in [27] — this is actually quite accessible without using a computer for $r=4$ .", "The second, and more fundamental, reason is that the analysis of the smaller parts is different from the square case since it does not use the concept of value.", "We solved this problem for $r=2$ by applying Strassen's laser method globally on the combination of $T_{112}$ , $T_{121}$ and $T_{211}$ , which is the key technical contribution of this paper.", "For larger values of $r$ the same approach can in principle be used, but other techniques seem to be necessary to convert these ideas into a systematic framework." ], [ "Organization of the paper.", "Section describes formally the notions of trilinear forms and Strassen's laser method.", "Section presents in details the basic construction $F_q\\otimes F_q$ from [10] and some of its properties.", "Section describes the graph-theoretic problems that will arise in the analysis of our trilinear forms.", "Section describes our algorithm, and Section shows how to use this algorithm to derive the optimization problem giving our new upper bounds on $\\omega (1,1,k)$ .", "Finally, the optimization is done in Section ." ], [ "Preliminaries", "In this section we present known results about trilinear forms, Strassen's laser method and Salem-Spencer sets that we will use in this paper.", "We refer to [5] for an extensive treatment of the first two topics." ], [ "Trilinear forms, degeneration of tensors and Schönhage's asymptotic sum inequality", "It will be convenient for us to use an abstract approach and represent trilinear forms as tensors.", "Our presentation is independent from what was informally defined and stated in Subsection REF , but describes essentially the same contents.", "We thus encourage the reader who encounter these notions for the first time to refer at Subsection REF for concrete illustrations.", "We assume that $\\mathbb {F}$ is an arbitrary field.", "Let $U=\\mathbb {F}^u$ , $V=\\mathbb {F}^v$ and $W=\\mathbb {F}^w$ be three vector spaces over $\\mathbb {F}$ , where $u,v$ and $w$ are three positive integers.", "A tensor $t$ of format $(u,v,w)$ is an element of $U\\otimes V\\otimes W=\\mathbb {F}^{u\\times v\\times w}$ , where $\\otimes $ denotes the tensor product.", "If we fix bases $\\lbrace x_i\\rbrace $ , $\\lbrace y_j\\rbrace $ and $\\lbrace z_k\\rbrace $ of $U$ , $V$ and $W$ , respectively, then we can express $t$ as $t=\\sum _{ijk}t_{ijk}\\:x_i\\otimes y_j\\otimes z_k$ for coefficients $t_{ijk}$ in $\\mathbb {F}$ .", "The tensor $t$ can then be represented by the 3-dimensional array $[t_{ijk}]$ .", "We will often write $x_i\\otimes y_j\\otimes z_j$ simply as $x_iy_jz_k$ .", "The tensor corresponding to the matrix multiplication of an $m\\times n$ matrix by an $n\\times p$ matrix is the tensor of format $(m\\times n,n\\times p,m\\times p)$ with coefficients $t_{ijk}=\\left\\lbrace \\begin{array}{ll}1&\\textrm { if } i=(r,s), j=(s,t) \\textrm { and } k=(r,t) \\textrm { for some integers } (r,s,t)\\in [m]\\times [n]\\times [p]\\\\0&\\textrm { otherwise }.\\end{array}\\right.$ This tensor will be denoted by $\\langle m,n,p \\rangle $ .", "Another important example is the tensor $\\sum _{\\ell =1}^n x_\\ell y_\\ell z_\\ell $ of format $(n,n,n)$ .", "This tensor is denoted $\\langle n \\rangle $ and corresponds to $n$ independent scalar products.", "Let $\\lambda $ be an indeterminate over $\\mathbb {F}$ and consider the extension $\\mathbb {F}[\\lambda ]$ of $\\mathbb {F}$ , i.e., the set of all polynomials over $\\mathbb {F}$ in $\\lambda $ .", "A triple of matrices $\\alpha \\in \\mathbb {F}[\\lambda ]^{u^{\\prime }\\times u}$ , $\\beta \\in \\mathbb {F}[\\lambda ]^{v^{\\prime }\\times v}$ and $\\gamma \\in \\mathbb {F}[\\lambda ]^{w^{\\prime }\\times w}$ transforms $t\\in \\mathbb {F}^{u\\times v\\times w}$ into the tensor $(\\alpha \\otimes \\beta \\otimes \\gamma )t\\in \\mathbb {F}[\\lambda ]^{u^{\\prime }\\times v^{\\prime }\\times w^{\\prime }}$ defined as $(\\alpha \\otimes \\beta \\otimes \\gamma )t=\\sum _{ijk}t_{ijk}\\:\\alpha (x_i)\\otimes \\beta (y_j)\\otimes \\gamma (z_j).$ This new tensor is called a restriction of $t$ .", "Intuitively, the fact that a tensor $t^{\\prime }$ is a restriction of $t$ means that an algorithm computing $t$ can be converted into an algorithm computing $t^{\\prime }$ that uses the same amount of multiplications (i.e., an algorithm with essentially the same complexity).", "We now give the definition of degeneration of tensors.", "Definition 2.1 Let $t\\in \\mathbb {F}^{u\\times v\\times w}$ and $t^{\\prime }\\in \\mathbb {F}^{u^{\\prime }\\times v^{\\prime }\\times w^{\\prime }}$ be two tensors.", "We say that $t^{\\prime }$ is a degeneration of $t$ , denoted $t^{\\prime }\\unlhd t$ , if there exists matrices $\\alpha \\in \\mathbb {F}[\\lambda ]^{u^{\\prime }\\times u}$ , $\\beta \\in \\mathbb {F}[\\lambda ]^{v^{\\prime }\\times v}$ and $\\gamma \\in \\mathbb {F}[\\lambda ]^{w^{\\prime }\\times w}$ such that $\\lambda ^st^{\\prime }+\\lambda ^{s+1} t^{\\prime \\prime }=(\\alpha \\otimes \\beta \\otimes \\gamma )t$ for some tensor $t^{\\prime \\prime }\\in \\mathbb {F}[\\lambda ]^{u^{\\prime }\\times v^{\\prime }\\times w^{\\prime }}$ and some nonnegative integer $s$ .", "Intuitively, the fact that a tensor $t^{\\prime }$ is a degeneration of a tensor $t$ means that an algorithm computing $t$ can be converted into an “approximate algorithm” computing $t^{\\prime }$ with essentially the same complexity.", "The notion of degeneration can be used to define the notion of border rank.", "Definition 2.2 Let $t$ be a tensor.", "Then $\\underline{R}(t)=\\min \\lbrace {r\\in \\mathbb {N}\\:|\\: t\\unlhd \\langle r \\rangle \\rbrace }$ .", "Let $t\\in U\\otimes V\\otimes W$ and $t^{\\prime }\\in U^{\\prime }\\otimes V^{\\prime }\\otimes W^{\\prime }$ be two tensors.", "We can naturally define the direct sum $t\\oplus t^{\\prime }$ , which is a tensor in $(U\\oplus U^{\\prime })\\otimes (V\\oplus V^{\\prime })\\otimes (W\\oplus W^{\\prime })$ , and the tensor product $t\\otimes t^{\\prime }$ , which is a tensor in $(U\\otimes U^{\\prime })\\otimes (V\\otimes V^{\\prime })\\otimes (W\\otimes W^{\\prime })$ .", "For any integer $c\\ge 1$ , we will denote the tensor $t\\oplus \\cdots \\oplus t$ (with $c$ occurrences of $t$ ) by $c\\cdot t$ and the tensor $t\\otimes \\cdots \\otimes t$ (with $c$ occurrences of $t$ ) by $t^{\\otimes c}$ .", "The degeneration of tensors has the following properties.", "Proposition 2.1 (Proposition 15.25 in [5]) Let $t_1,t_1^{\\prime },t_2$ and $t_2^{\\prime }$ be four tensors.", "Suppose that $t^{\\prime }_1\\unlhd t_1$ and $t^{\\prime }_2\\unlhd t_2$ .", "Then $t^{\\prime }_1\\oplus t^{\\prime }_2\\unlhd t_1\\oplus t_2$ and $t^{\\prime }_1\\otimes t^{\\prime }_2\\unlhd t_1\\otimes t_2$ .", "Schönhage's asymptotic sum inequality [23] will be one of the main tools used to prove our bounds.", "Its original statement is for estimating the exponent of square matrix multiplication, but it can be easily generalized to estimate the exponent of rectangular matrix multiplication as well.", "We will use the following form, which has been also used implicitly in [13], [16].", "A proof can be found in [17].", "Theorem 2.1 (Schönhage's asymptotic sum inequality) Let $k$ , $m$ and $c$ be three positive integers.", "Let $t$ be a tensor such that $c\\cdot \\langle m,m,m^k \\rangle \\unlhd t$ .", "Then $c\\cdot m^{\\omega (1,1,k)}\\le \\underline{R}(t).$ Theorem REF states that, if the form $t$ can be degenerated (i.e., approximately converted, in the sense of Definition REF ) into a direct sum of $c$ forms, each being isomorphic to $\\langle m,m,m^k \\rangle $ , then the inequality $c\\cdot m^{w(1,1,k)}\\le \\underline{R}(t)$ holds.", "Note that this is a powerful technique, since the concepts of degeneration and border rank refer to “approximate algorithms”, while $\\omega (1,1,k)$ refers to the complexity of exact algorithms for rectangular matrix multiplication." ], [ "Strassen's laser method and ${C}$ -tensors", "Strassen [25] introduced in 1986 a new approach, often referred as the laser method, to derive upper bounds on the exponent of matrix multiplication.", "To the best of our knowledge, all the applications of this method have focused so far on square matrix multiplication, in which case several simplifications can be done due to the symmetry of the problem.", "In this paper we will nevertheless need the full power of the laser method, and in particular the notion of ${C}$ -tensor introduced in [25], [26], to derive our new bounds on the exponent of rectangular matrix multiplication.", "The exposition below will mainly follow [5].", "Let $t\\in U\\otimes V\\otimes W$ be a tensor.", "Suppose that $U$ , $V$ and $W$ decompose as direct sums of subspaces as follows: $U=\\bigoplus _{i\\in S_U}U_i,\\hspace{8.53581pt}V=\\bigoplus _{j\\in S_V}V_j,\\hspace{8.53581pt}W=\\bigoplus _{k\\in S_W}W_k.$ Denote by $D$ this decomposition.", "We say that $t$ is a ${C}$ -tensor with respect to $D$ if $t$ can be written as $t=\\sum _{(i,j,k)\\in S_U\\times S_V\\times S_W} t_{ijk}$ where each $t_{ijk}$ is a tensor in $U_i\\otimes V_j\\otimes W_k$ .", "The support of $t$ is defined as $\\mathrm {supp}_D(t)=\\lbrace (i,j,k)\\in S_U\\times S_V\\times S_W\\:|\\: t_{ijk}\\ne 0\\rbrace ,$ and the nonzero $t_{ijk}$ 's are called the components of $t$ .", "We will usually omit the reference to $D$ when there is no ambiguity or when the decomposition does not matter.", "As a simple example, consider the complete decompositions of the spaces $U=\\mathbb {F}^{m\\times n}$ , $V=\\mathbb {F}^{n\\times p}$ and $W=\\mathbb {F}^{m\\times p}$ (i.e., their decomposition as direct sums of one-dimensional subspaces, each subspace being spanned by one element of their basis).", "With respect to this decomposition, the tensor of matrix multiplication $\\langle m,n,p \\rangle $ is a ${C}$ -tensor with support $\\mathrm {supp_c}(\\langle m,n,p \\rangle )=\\lbrace ((r,s),(s,t),(r,t))\\:|\\:(r,s,t)\\in [m]\\times [n]\\times [p]\\rbrace $ where each component is trivial (i.e., isomorphic to $\\langle 1,1,1 \\rangle $ ).", "In this paper the notation $\\mathrm {supp_c}(\\langle m,n,p \\rangle )$ will always refer to the support of $\\langle m,n,p \\rangle $ with respect to this complete decomposition.", "We now introduce the concept of combinatorial degeneration.", "A subset $\\Delta $ of $S_U\\times S_V\\times S_W$ is called diagonal if the three projections $\\Delta \\rightarrow S_U$ , $\\Delta \\rightarrow S_V$ and $\\Delta \\rightarrow S_W$ are injective.", "Let $\\Phi $ be a subset of $S_U\\times S_V\\times S_W$ .", "A set $\\Psi \\subseteq \\Phi $ is a combinatorial degeneration of $\\Phi $ if there exists tree functions $a\\colon S_U\\rightarrow \\mathbb {Z}$ , $b\\colon S_V\\rightarrow \\mathbb {Z}$ and $c\\colon S_W\\rightarrow \\mathbb {Z}$ such that for all $(i,j,k)\\in \\Psi $ , $a(i)+b(j)+c(k)=0$ ; for all $(i,j,k)\\in \\Phi \\backslash \\Psi $ , $a(i)+b(j)+c(k)>0$ .", "The most useful application of combinatorial degeneration will be the following result, which states that a sum, over indices in a diagonal combinatorial degeneration of $\\mathrm {supp}_D(t)$ , of the components $t_{ijk}$ is direct.", "Proposition 2.2 (Proposition 15.30 in [5]) Let $t$ be ${C}$ -tensor with support $\\mathrm {supp}_D(t)$ and components $t_{ijk}$ .", "Let $\\Delta \\subseteq \\mathrm {supp}_D(t)$ be a combinatorial degeneration of $\\mathrm {supp}_D(t)$ and assume that $\\Delta $ is diagonal.", "Then $\\bigoplus _{(i,j,k)\\in \\Delta }t_{ijk}\\unlhd t.$ In this work we will construct ${C}$ -tensors where all the components are isomorphic to $\\langle m,m,m^k \\rangle $ for some values $m$ and $k$ .", "Proposition REF then suggests that a good bound on the exponent of rectangular matrix multiplication can be derived, via Theorem REF , if the support of the ${C}$ -tensor contains a large diagonal combinatorial degeneration.", "When this support is isomorphic to $\\mathrm {supp_c}(\\langle e,h,\\ell \\rangle )$ for some positive integers $e,h$ and $\\ell $ , a powerful tool to construct large diagonal combinatorial degenerations is given by the following result by Strassen (Theorem 6.6 in [26]), restated in our terminology.", "Proposition 2.3 ([26]) Let $e_1, e_2$ and $e_3$ be three positive integers such that $e_1\\le e_2\\le e_3$ .", "For any permutation $\\sigma $ of $\\lbrace 1,2,3\\rbrace $ , there exists a diagonal set $\\Delta \\subseteq \\mathrm {supp_c}(\\langle e_{\\sigma (1)},e_{\\sigma (2)},e_{\\sigma (3)} \\rangle )$ with $|\\Delta |=\\left\\lbrace \\begin{array}{ll}e_1e_2-\\left\\lfloor \\frac{(e_1+e_2-e_3)^2}{4} \\right\\rfloor &\\textrm { if }e_1+e_2\\ge e_3\\\\e_1e_2&\\textrm { otherwise }\\end{array}\\right.$ that is a combinatorial degeneration of $\\mathrm {supp_c}(\\langle e_{\\sigma (1)},e_{\\sigma (2)},e_{\\sigma (3)} \\rangle )$ .", "In particular, $|\\Delta |\\ge \\left\\lceil 3e_1e_2/4 \\right\\rceil $ .", "Finally, we mention that the concept of ${C}$ -tensor is preserved by the tensor product.", "We will just state this property for the restricted class of ${C}$ -tensors that we will encounter in this paper (for which the precise decompositions do not matter), and refer to [5] or to Section 7 in [26] for a complete treatment.", "Proposition 2.4 Let $t$ be a ${C}$ -tensor with support isomorphic to $\\mathrm {supp_c}(\\langle e,h,\\ell \\rangle )$ in which each component is isomorphic to $\\langle m,n,p \\rangle $ .", "Let $t^{\\prime }$ be a ${C}$ -tensor with support isomorphic to $\\mathrm {supp_c}(\\langle e^{\\prime },h^{\\prime },\\ell ^{\\prime } \\rangle )$ in which each component is isomorphic to $\\langle m^{\\prime },n^{\\prime },p^{\\prime } \\rangle $ .", "Then $t\\otimes t^{\\prime }$ is a ${C}$ -tensor with support isomorphic to $\\mathrm {supp_c}(\\langle ee^{\\prime },hh^{\\prime },\\ell \\ell ^{\\prime } \\rangle )$ in which each component is isomorphic to $\\langle mm^{\\prime },nn^{\\prime },pp^{\\prime } \\rangle $ ." ], [ "Salem-Spencer sets", "Let $M$ be a positive integer and consider $\\mathbb {Z}_{M}=\\lbrace 0,1,\\ldots ,M-1\\rbrace $ .", "We say that a set $B\\subseteq \\mathbb {Z}_M$ has no length-3 arithmetic progression if, for any three elements $b_1,b_2$ and $b_3$ in $B$ , $b_1+b_2=2b_3 \\bmod M \\Longleftrightarrow b_1=b_2=b_3.$ Salem and Spenser have shown the existence of very dense sets with no length-3 arithmetic progression.", "Theorem 2.2 ([20]) For any $\\epsilon >0$ , there exists an integer $M_\\epsilon $ such that, for any integer $M>M_\\epsilon $ , there is a set $B\\subseteq \\mathbb {Z}_M$ of size $|B|>M^{1-\\epsilon }$ with no length-3 arithmetic progression.", "We refer to these sets as Salem-Spencer sets.", "They can be constructed in time polynomial in $M$ .", "Note that this construction has been improved by Behrend [4], but the above statement will be enough for our purpose." ], [ "Coppersmith-Winograd's construction", "In this section we describe the construction by Coppersmith and Winograd [10], which we will use as the basis of our algorithm, and several of its properties.", "We start with the simpler construction presented in Section 7 of [10].", "For any positive integer $q$ , let us define the following trilinear form $F_q$ , where $\\lambda $ is an indeterminate over $\\mathbb {F}$ .", "$F_{q}=&\\sum _{i=1}^q \\lambda ^{-2}(x_0+\\lambda x_i)(y_0+\\lambda y_i)(z_0+\\lambda z_i)\\:-\\\\&\\lambda ^{-3}(x_0+\\lambda ^2\\sum _{i=1}^q x_i)(y_0+\\lambda ^2\\sum _{i=1}^q y_i)(z_0+\\lambda ^2\\sum _{i=1}^q z_i)\\:+\\\\&(\\lambda ^{-3}-q\\lambda ^{-2})(x_0+\\lambda ^3x_{q+1})(y_0+\\lambda ^3y_{q+1})(z_0+\\lambda ^3z_{q+1})$ In this trilinear form the $x$ -variables are $x_0,x_1,\\ldots , x_{q+1}$ .", "Similarly, the number of $y$ -variables is $(q+2)$ and the number of $z$ -variables is $(q+2)$ as well.", "Define the form $F^{\\prime }_q=\\sum _{i=1}^q (x_0y_iz_i+x_iy_0z_i+x_iy_iz_0)+x_0y_0z_{q+1}+x_0y_{q+1}z_{0}+x_{q+1}y_0z_0.$ It is easy to check that the form $F_q$ can be written as $F_q=F^{\\prime }_q+\\lambda \\cdot F^{\\prime \\prime }_q$ , where $F^{\\prime \\prime }_q$ is a polynomial in $\\lambda $ and in the $x$ -variables, $y$ -variables and $z$ -variables.", "In the language of Section , this means that $F^{\\prime }_q\\unlhd F_q$ and, informally, this means that an algorithm computing $F_q$ can be converted into an algorithm computing $F^{\\prime }_q$ with the same complexity.", "Note that $\\underline{R}(F_q)\\le q+2$ since, by definition, $F_q$ is the sum of $q+2$ products.", "A more complex construction is proposed in Section 8 of [10].", "It is obtained by taking the tensor product of $F_q$ by itself.", "By Proposition REF we know that $F^{\\prime }_q\\otimes F^{\\prime }_q\\unlhd F_q\\otimes F_q$ .", "Consider the tensor product of $F^{\\prime }_q$ by itself: $F^{\\prime }_q\\otimes F^{\\prime }_q&=&T_{004}+T_{040}+T_{400}+T_{013}+T_{031}+T_{103}+T_{130}+T_{301}+T_{310}+\\\\&&T_{022}+T_{202}+T_{220}+T_{112}+T_{121}+T_{211}$ where $T_{004}&=&x_{0,0}^{0}y_{0,0}^{0}z_{q+1,q+1}^{4}\\\\T_{013}&=&\\sum _{i=1}^q x_{0,0}^{0}y_{i,0}^{1}z_{i,q+1}^{3}+\\sum _{k=1}^q x_{0,0}^{0}y_{0,k}^{1}z_{q+1,k}^{3}\\\\T_{022}&=&x_{0,0}^{0}y_{q+1,0}^{2}z_{0,q+1}^{2}+x_{0,0}^{0}y_{0,q+1}^{2}z_{q+1,0}^{2}+\\sum _{i,k=1}^q x_{0,0}^{0}y_{i,k}^{2}z_{i,k}^{2}\\\\T_{112}&=&\\sum _{i=1}^q x_{i,0}^{1}y_{i,0}^{1}z_{0,q+1}^{2}+\\sum _{k=1}^q x_{0,k}^{1}y_{0,k}^{1}z_{q+1,0}^{2}+\\sum _{i,k=1}^q x_{i,0}^{1}y_{0,k}^{1}z_{i,k}^{2}+\\sum _{i,k=1}^q x_{0,k}^{1}y_{i,0}^{1}z_{i,k}^{2}$ and the other eleven terms are obtained by permuting the indexes of the $x$ -variables, the $y$ -variables and $z$ -variables in the above expressions (e.g., $T_{040}=x_{0,0}^{0}y_{q+1,q+1}^{4}z_{0,0}^{0}$ and $T_{400}=x_{q+1,q+1}^{4}y_{0,0}^{0}z_{0,0}^{0}$ ).", "Let us describe in more details the notations used here.", "The number of $x$ -variables is $(q+2)^2$ .", "They are indexed as $x_{i,k}$ , for $i,k\\in \\lbrace 0,1,\\ldots , q+1\\rbrace $ .", "The superscript is assigned in the following way: the variable $x_{0,0}$ has superscript 0, the variables in $\\lbrace x_{i,0},x_{0,k}\\rbrace _{1\\le i,j\\le q}$ have superscript 1, the variables in $\\lbrace x_{q+1,0},x_{i,k},x_{0,q+1}\\rbrace _{1\\le i,j\\le q}$ have superscript 2, the variables in $\\lbrace x_{q+1,k},x_{i,q+1}\\rbrace _{1\\le i,j\\le q}$ have superscript 3 and the variable $x_{q+1,q+1}$ has superscript 4.", "Note that the superscript is completely determined by the subscript.", "Similarly, the number of $y$ -variables is $(q+2)^2$ , and the number of $z$ -variables is $(q+2)^2$ as well.", "The $y$ -variables and the $z$ -variables are assigned subscripts and superscripts exactly as for the $x$ -variables.", "Observe that any term $xyz$ that appears in $T_{ijk}$ is such that $x$ has superscript $i$ , $y$ has superscript $j$ and $z$ has superscript $k$ .", "We thus obtain $\\sum _{{{\\scriptstyle \\begin{matrix} 0\\le i,j,k\\le 4 \\\\ i+j+k=4 \\end{matrix}}}}T_{ijk} \\unlhd F_q\\otimes F_q.$ Moreover we know that $\\underline{R}(F_q\\otimes F_q)\\le (q+2)^2$ , since $F_q\\otimes F_q$ can be written using $(q+2)^2$ multiplications.", "We will later need to analyze all the forms $T_{ijk}$ .", "It happens, as observed in [10], that most of these forms (all the forms except $T_{112}$ , $T_{121}$ and $T_{211}$ ) can be analyzed in a straightforward way, since they are isomorphic to the following matrix products: $T_{004}\\cong T_{040}\\cong T_{400}&\\cong &\\langle 1,1,1\\rangle \\\\T_{013}\\cong T_{031}&\\cong &\\langle 1,1,2q\\rangle \\\\T_{103}\\cong T_{301}&\\cong & \\langle 2q,1,1\\rangle \\\\T_{130}\\cong T_{310}&\\cong & \\langle 1,2q,1\\rangle \\\\T_{022}&\\cong &\\langle 1,1,q^2+2\\rangle \\\\T_{202}&\\cong &\\langle q^2+2,1,1\\rangle \\\\T_{220}&\\cong &\\langle 1,q^2+2,1\\rangle .$ This can be seen from the definition of the trilinear form (or the tensor) corresponding to matrix multiplication described in Section .", "For example, the form $T_{013}$ is isomorphic to the tensor $\\sum _{\\ell =1}^{2q}x_0y_\\ell z_\\ell =\\langle 1,1,2q\\rangle $ , which represents the product of a $1\\times 1$ matrix (a scalar) by a $1\\times 2q$ matrix (a row)." ], [ "Graph-Theoretic Problems", "In this section we describe and solve several graph-theoretic problems that will arise in the analysis of our trilinear forms.", "While the presentation given here is independent from the remaining of the paper, the reader may prefer to read Section before going through this section." ], [ "Problem setting", "Let $\\tau $ be a fixed positive integer.", "Let $N$ be a large integer and define the set $\\Lambda = \\left\\lbrace (I,J,K)\\in [\\tau ]^N\\times [\\tau ]^N\\times [\\tau ]^N \\:|\\: I_\\ell +J_\\ell +K_\\ell =\\tau \\textrm { for all } \\ell \\in \\lbrace 1,\\ldots ,N\\rbrace \\right\\rbrace .$ Define the three coordinate functions $f_1,f_2,f_3\\colon [\\tau ]^N\\times [\\tau ]^N\\times [\\tau ]^N\\rightarrow [\\tau ]^N$ as follows.", "$f_1((I,J,K))&=&I\\\\f_2((I,J,K))&=&J\\\\f_3((I,J,K))&=&K$ From the definition of $\\Lambda $ , two distinct elements in $\\Lambda $ cannot agree on more than one coordinate.", "Since this simple observation will be crucial in our analysis, we state it explicitly as follows.", "Fact 1 Let $u$ and $v$ be two elements in $\\Lambda $ .", "If $f_i(u)=f_i(v)$ for more than one index $i\\in \\lbrace 1,2,3\\rbrace $ , then $u=v$ .", "Let $U$ be a subset of $\\Lambda $ such that there exist integers $\\mathcal {N}_1,\\mathcal {N}_2$ and $\\mathcal {N}_3$ for which the following property holds: for any $I\\in [\\tau ]^N$ , $|\\lbrace u\\in U \\:|\\: f_1(u)=I\\rbrace |&\\in &\\lbrace 0,\\mathcal {N}_1\\rbrace \\\\|\\lbrace u\\in U \\:|\\: f_2(u)=I\\rbrace |&\\in &\\lbrace 0,\\mathcal {N}_2\\rbrace \\\\|\\lbrace u\\in U \\:|\\: f_3(u)=I\\rbrace |&\\in &\\lbrace 0,\\mathcal {N}_3\\rbrace .$ This means that, for any $i\\in \\lbrace 1,2,3\\rbrace $ and any $I\\in [\\tau ]^N$ , the size of the set $f_i(I)^{-1}$ is either 0 or $\\mathcal {N}_i$ .", "Let us write $|f_1(U)|=T_1$ , $|f_2(U)|=T_2$ and $|f_3(U)|=T_3$ .", "It is easy to see that $|U|=T_1\\mathcal {N}_1=T_2\\mathcal {N}_2=T_3\\mathcal {N}_3.$ We are interested in stating asymptotic results holding when $N$ goes to infinity.", "Through this section the $\\mathcal {N}_i$ 's and the $T_i$ 's will be strictly increasing functions of $N$ .", "Let $G$ be the (simple and undirected) graph with vertex set $U$ in which two distinct vertices $u$ and $v$ are connected if and only there exists one index $i\\in \\lbrace 1,2,3\\rbrace $ such that $f_i(u)=f_i(v)$ .", "The goal will be to modify the graph $G$ to obtain a subgraph satisfying some specific properties.", "The only modification we allow is to remove all the vertices with a given sequence at a given position: given a sequence $I\\in [\\tau ]^N$ and a position $s\\in \\lbrace 1,2,3\\rbrace $ , remove all the vertices $u$ (if any) such that $f_i(u)=I$ .", "We call such an operation a removal operation.", "The reason why only such removal operations are allowed is that they will correspond, when considering trilinear forms, to setting to zero some variables, which is one of the only operations that can be performed on trilinear forms.", "While not stated explicitly in graph-theoretic terms, the key technical result by Coppersmith and Winograd [10] is a method to convert, when $\\mathcal {N}_1=\\mathcal {N}_2=\\mathcal {N}_3$ , the graph $G$ into an edgeless graph that still contains a non-negligible fraction of the vertices.", "We state this result in the following theorem.", "Theorem 4.1 ([10]) Suppose that $\\mathcal {N}_1=\\mathcal {N}_2=\\mathcal {N}_3$ .", "Then, for any constant $\\epsilon >0$ , the graph $G$ can be converted, with only removal operations, into an edgeless graph with $\\Omega \\left(\\frac{T_1}{\\mathcal {N}_1^{\\epsilon }}\\right)$ vertices.", "In this section we will give several generalizations of this result.", "In Subsection REF we state our generalizations.", "Then, in Subsections REF –REF , we describe the algorithms and prove the results.", "We stress that, in this section as in [10], the number of removal operations (i.e., the time complexity of the algorithms) is irrelevant.", "This is because, in the applications of these results to matrix multiplication, the parameter $N$ will be treated as a large constant independent of the size of the matrices considered." ], [ "Statement of our results", "The generalization we consider assume the existence of a known set $U^\\ast \\subseteq U$ such that $|f_i(U^\\ast )|=T_i$ for each $i\\in \\lbrace 1,2,3\\rbrace $ ; there exist integers $\\mathcal {N}^\\ast _1,\\mathcal {N}^\\ast _2$ and $\\mathcal {N}^\\ast _3$ such that $|\\lbrace u\\in U \\:|\\: f_i(u)=I\\rbrace |=\\mathcal {N}_i \\Leftrightarrow |\\lbrace u\\in U^\\ast \\:|\\: f_i(u)=I\\rbrace |=\\mathcal {N}_i^\\ast $ for each $I\\in [\\tau ]^N$ and each $i\\in \\lbrace 1,2,3\\rbrace $ .", "Note that we have necessarily $T_1\\mathcal {N}^\\ast _1=T_1\\mathcal {N}^\\ast _2=T_1\\mathcal {N}^\\ast _2$ .", "The first problem considered is again to convert the graph $G$ into an edgeless subgraph that contains a non-negligible fraction of the vertices using only removal operations, but we additionally require that all the remaining vertices are in $U^\\ast $ .", "Our first result is the following theorem.", "Theorem 4.2 For any constant $\\epsilon >0$ , the graph $G$ can be converted, with only removal operations, into an edgeless graph with $\\Omega \\left(\\frac{T_1\\mathcal {N}^\\ast _1}{(\\mathcal {N}_1+\\mathcal {N}_2+\\mathcal {N}_3)^{1+\\epsilon }}\\right)$ vertices, all of them being in $U^\\ast $ .", "Theorem REF is a special case of Theorem REF , for $U^\\ast =U$ .", "The case $\\mathcal {N}_1=\\mathcal {N}_2=\\mathcal {N}_3$ has been proved implicitly by Stothers [24] and Vassilevska Williams [27].", "Our second problem deals with another kind of conversion.", "Remember that, from the definition of the graph $G$ and Fact REF , for any edge connecting two vertices $u$ and $v$ in $G$ there exists exactly one index $i\\in \\lbrace 1,2,3\\rbrace $ such that $f_i(u)=f_i(v)$ .", "Let us define the concept of 1-clique as follows.", "Definition 4.1 A 1-clique in the graph $G$ is a set $U^{\\prime }\\subseteq U$ for which there exists a sequence $I\\in [\\tau ^N]$ such that $f_{1}(u)=I$ for all $u\\in U^{\\prime }$ .", "The size of the 1-clique is $|U^{\\prime }|$ .", "Our second result shows how to convert, using only removal operations, the graph $G$ into a graph with vertices in $U^\\ast $ that is the disjoint union of many large 1-cliques.", "The formal statement follows.", "Theorem 4.3 Suppose that $\\mathcal {N}_1\\ge \\mathcal {N}_2\\ge \\mathcal {N}_3$ .", "Assume that $\\frac{\\mathcal {N}_2T_1}{\\mathcal {N}_1^\\ast }+\\frac{\\mathcal {N}_2}{T_1}<\\frac{1}{1024}$ .", "Then, for any constant $\\epsilon >0$ , the graph $G$ can be converted, with only removal operations, into a graph satisfying the following conditions: all the vertices of the graph are in $U^\\ast $ ; each connected component is a 1-clique (i.e., the graph is a disjoint union of 1-cliques); among these 1-cliques, there are $\\Omega \\left(\\frac{T_1}{\\mathcal {N}_2^{\\epsilon }}\\right)$ 1-cliques that have size $\\Omega \\left(\\frac{\\mathcal {N}_1^\\ast }{\\mathcal {N}_2}\\right)$ .", "Note that, when only removal operations are allowed, the graph obtained is necessary a subgraph of $G$ induced be a subset of its vertices.", "Theorem REF thus states that there exist 1-cliques $U_r\\subseteq U^\\ast $ such that the graph obtained after the removal operations is the subgraph of $G$ induced by $\\cup _r U_r$ , at least $\\Omega \\left(T_1/ \\mathcal {N}_2^{\\epsilon }\\right)$ of these $U_r$ 's have size $\\Omega \\left(\\mathcal {N}_1^\\ast / \\mathcal {N}_2\\right)$ , and for any $r\\ne r^{\\prime }$ there is no edge with one extremity in $U_r$ and the other extremity in $U_{r^{\\prime }}$ .", "We mention that the constant $1/1024$ in the assumption of Theorem REF is chosen only for concreteness.", "The same theorem actually holds even for weaker conditions on $\\mathcal {N}^\\ast _1, T_1$ and $\\mathcal {N}_2$ , but this simpler version will be sufficient for our purpose." ], [ "Choice of the weight functions", "Let $M$ be a large prime number that will be chosen later.", "We take a Salem-Spencer set $\\Gamma $ with $|\\Gamma |\\ge M^{1-\\epsilon }$ , which existence is guaranteed by Theorem REF .", "Similarly to $\\cite {Coppersmith+90}$ , we take $N+2$ integers $\\omega _0$ , $\\omega _1$ ,$\\ldots $ , $\\omega _{N+1}$ uniformly at random in $\\mathbb {Z}_M=\\lbrace 0,1,\\ldots ,M-1\\rbrace $ and define three hash functions $b_1,b_2,b_3\\colon [\\tau ]^N\\rightarrow \\mathbb {Z}_M$ as follows.", "$b_1(I)&=&\\omega _0+\\sum _{j=1}^N I_j \\omega _j \\bmod M\\\\b_2(I)&=&\\omega _{N+1}+\\sum _{j=1}^N I_j \\omega _j \\bmod M\\\\b_3(I)&=&\\frac{1}{2}\\times \\left(\\omega _0+\\omega _{N+1}+\\sum _{j=1}^N (\\tau -I_j) \\omega _j\\right) \\bmod M$ A property of these functions is that, for a fixed sequence $I\\in [\\tau ]^N$ , the value $b_i(I)$ is uniformly distributed in $\\mathbb {Z}_M$ , for each $i\\in \\lbrace 1,2,3\\rbrace $ .", "Note that the term $\\omega _0$ is not used in [10], but we introduce it to obtain a uniform distribution even if $I$ is the all-zero sequence.", "Let us introduce the notion of a compatible vertex.", "Definition 4.2 A vertex $u\\in {U}$ is compatible if $b_i(f_i(u))\\in \\Gamma $ for each $i\\in \\lbrace 1,2,3\\rbrace $ .", "Let $V$ be the set of vertices in $U$ that are compatible, and denote $V^\\ast =V\\cap U^\\ast $ .", "We stress that the definition of compatibility (and thus the definitions of $V$ and $V^\\ast $ too) depend of the choice of $\\Gamma $ and of the weights $\\omega _i$ .", "Using the fact that $\\Gamma $ is a Salem-Spencer set we can give the following simple but very useful characterization of compatible vertices.", "Lemma 4.1 Let $i\\in \\lbrace 1,2,3\\rbrace $ and $i^{\\prime }\\in \\lbrace 1,2,3\\rbrace $ be any two distinct indexes.", "Then a vertex $u\\in U$ is compatible if and only if $b_i(f_i(u))=b_{i^{\\prime }}(f_{i^{\\prime }}(u))=b$ for some $b\\in \\Gamma $ .", "Let us take a vertex $u\\in U$ , and write $f_1(u)=I$ , $f_2(u)=J$ and $f_3(u)=K$ .", "Note that, since $I_\\ell +J_\\ell +K_\\ell =\\tau $ for any index $\\ell \\in \\lbrace 1,\\ldots ,N\\rbrace $ , the equality $b_1(I)+b_2(J)-2 b_3(K)=0\\bmod M$ always holds (i.e., holds for all choices of the weights $\\omega _j$ ).", "Suppose that $b_i(f_i(u))=b_{i^{\\prime }}(f_{i^{\\prime }}(u))=b$ for some $b\\in \\Gamma $ , where $i$ and $i^{\\prime }$ are distinct.", "For instance, suppose that $i=1$ and $i^{\\prime }=2$ .", "Then, since $M$ is a prime, the above property implies that $b_3(K)=b$ , which means that $u$ is compatible.", "The same conclusion is true for the other choices of $i$ and $i^{\\prime }$ .", "Now suppose that $u$ is compatible.", "From the definition of a Spencer-Salem set, we can conclude that there exists an element $b\\in \\Gamma $ such that $b_1(I)=b_2(J)=b_3(K)=b$ ." ], [ "The first pruning", "Similarly to [10], the first pruning simply eliminates all the nodes $u\\in U$ that are not compatible.", "Note that this can be done using removal operations: for each $i\\in \\lbrace 1,2,3\\rbrace $ , we remove all the vertices $w$ such that $f_i(w)\\notin b_i^{-1}(\\Gamma )$ .", "The vertices remaining are precisely those in $V$ .", "Among those remaining vertices, the vertices in $U^\\ast $ are precisely those in $V^\\ast =V\\cap U^\\ast $ .", "We now evaluate the expectation of $|V^\\ast |$ .", "The proof is very similar to what was shown in [10] for the case $U=U^\\ast $ , and to what was shown in [24], [27] for $\\mathcal {N}^\\ast _1=\\mathcal {N}^\\ast _2=\\mathcal {N}^\\ast _3$ .", "Lemma 4.2 $\\mathbb {E}[|V^\\ast |]=\\frac{T_1\\mathcal {N}_1^\\ast |\\Gamma |}{M^2}$ .", "We use Lemma REF with $i=1$ and $i^{\\prime }=2$ .", "Remember that $|U^\\ast |=T_1\\mathcal {N}_1^\\ast $ .", "For each vertex $u\\in U^\\ast $ and each value $b\\in \\mathbb {Z}_M$ , the probability that $b_1(f_1(u))=b_2(f_2(u))=b$ is $1/M^2$ .", "Note that the two events $b_1(f_1(u))=b$ and $b_2(f_2(u))=b$ are independent even when $f_1(u)=f_2(u)$ due to the terms $\\omega _0$ and $\\omega _{N+1}$ in the hash functions.", "Let $E$ be the edge set of the subgraph of $G$ induced by $V$ : it consists of all edges connecting two distinct vertices $u$ and $v$ in $V$ such that $f_i(u)= f_i(v)$ for some $i\\in \\lbrace 1,2,3\\rbrace $ .", "Let $E^{\\prime }\\subseteq E$ be the subset of edges in $E$ with (at least) one extremity in $V^\\ast $ .", "Let $E^{\\prime \\prime }\\subseteq E^{\\prime }$ be the subset of edges in $E^{\\prime }$ connecting two vertices $u$ and $v$ such that $f_1(u)\\ne f_1(v)$ (which means that either $f_2(u)= f_2(v)$ or $f_3(u)= f_3(v)$ ).", "The following lemma gives upper bounds on the expectations of $|E^{\\prime }|$ and $|E^{\\prime \\prime }|$ .", "The proof can be considered as a generalization of similar statements in [10], [24], [27].", "Lemma 4.3 $\\mathbb {E}[|E^{\\prime }|]\\le \\frac{T_1\\mathcal {N}_1^\\ast (\\mathcal {N}_1+\\mathcal {N}_2+\\mathcal {N}_3-3)|\\Gamma |}{M^3}$ and $\\:\\mathbb {E}[|E^{\\prime \\prime }|]\\le \\frac{T_1\\mathcal {N}_1^\\ast (\\mathcal {N}_2+\\mathcal {N}_3-2)|\\Gamma |}{M^3}$ .", "We show that, for any index $i\\in \\lbrace 1,2,3\\rbrace $ , the expected number of ordered pairs $(u,v)$ with $u\\in V^\\ast $ and $v\\in V\\backslash \\lbrace u\\rbrace $ such that $f_i(u)=f_i(v)$ is exactly $\\frac{T_1\\mathcal {N}_1^\\ast (\\mathcal {N}_i-1)|\\Gamma |}{M^3}.$ The expectation of the number of edges (i.e., unordered pairs) is necessarily smaller.", "The factor $T_1\\mathcal {N}_1^\\ast $ counts the number of vertices in $U^\\ast $ .", "For any vertex $u\\in U^\\ast $ , there are exactly $\\mathcal {N}_i-1$ vertices $v\\in U\\backslash \\lbrace u\\rbrace $ such that $f_i(v)=f_i(u)$ .", "Let $u$ be a vertex in $U^\\ast $ and $v$ be a vertex in $U\\backslash \\lbrace u\\rbrace $ such that $f_i(v)=f_i(u)$ .", "Take another index $i^{\\prime }\\in \\lbrace 1,2,3\\rbrace \\backslash \\lbrace i\\rbrace $ arbitrarily.", "From Lemma REF , both $u$ and $v$ are in $V$ if and only if $b_i(f_i(u))=b_{i^{\\prime }}(f_{i^{\\prime }}(u))=b_{i^{\\prime }}(f_{i^{\\prime }}(v))=b$ for some element $b\\in \\Gamma $ .", "This happens with probability $|\\Gamma |/M^3$ since the random variables $b_i(f_i(u))$ , $b_{i^{\\prime }}(f_{i^{\\prime }}(u))$ , and $b_{i^{\\prime }}(f_{i^{\\prime }}(v))$ are mutually independent.", "From the linearity of the expectation, the expected number of ordered pairs is $T_1\\mathcal {N}_1^\\ast (\\mathcal {N}_i-1)$ times this probability.", "Lemmas REF and REF focused on expected values and were proven, essentially, by the linearity of the expectation.", "This was sufficient for the applications to square matrix multiplication presented in [10], [24], [27], and this will be sufficient for proving Theorem REF as well.", "Since, in order to prove Theorem REF , we will need a more precise analysis of the behavior of the random variables considered, we now prove the following lemma, which gives a lower bound on the probability that the subgraph induced by $V^\\ast $ has many large 1-cliques.", "Lemma 4.4 With probability (on the choice of the weights $\\omega _i$ ) at least $1-\\frac{4MT_1}{\\mathcal {N}_1^\\ast }-\\frac{4M}{T_1|\\Gamma |}$ , there exists a set $R\\subseteq [\\tau ]^N$ satisfying the following two conditions: $|R|\\ge \\frac{T_1|\\Gamma |}{2M}$ ; for any $I\\in R$ , there exist at least $\\frac{\\mathcal {N}_1^\\ast }{2M}$ vertices $u\\in V^\\ast $ such that $f_1(u)=I$ .", "Let us consider the set $f_1(U^\\ast )=\\lbrace f_1(u)\\:|\\: u\\in U^\\ast \\rbrace $ .", "For each element $I\\in f_1(U^\\ast )$ , define the set $S_I=\\lbrace u\\in U^\\ast \\:|\\: f_1(u)=I\\rbrace .$ Note that $|f_1(U^\\ast )|=T_1$ , and $|S_I|=\\mathcal {N}_1^\\ast $ for each $I\\in f_1(U^\\ast )$ .", "Fix an element $I\\in f_1(U^\\ast )$ and define $X_I=|\\lbrace u\\in S_I\\:|\\: b_2(f_2(u))=b_1(I)\\rbrace |.$ This random variable represents the number of vertices from $S_I$ that are mapped by $b_2\\circ f_2$ into $b_1(I)$ .", "For any vertex $u=(I,J,K)\\in S_I$ , we have the equivalence $b_2(f_2(u))=b_1(I)\\Longleftrightarrow \\sum _{i=1}^N(J_i-I_i)\\omega _{i}=\\omega _{0}-\\omega _{N+1}.$ Thus the probability of the event $b_2(f_2(u))=b_1(I)$ is $1/M$ .", "Thus $\\mathbb {E}[X_I]=\\frac{\\mathcal {N}_1^\\ast }{M}$ .", "Note that, for any two distinct elements $u$ and $v$ in $S_I$ , the two events $b_2(f_2(u))=b_1(I)$ and $b_2(f_2(v))=b_1(I)$ are independent.", "From this pairwise independence, $\\mathrm {var}[X_I]=\\frac{\\mathcal {N}_1^\\ast }{M}(1-\\frac{1}{M})$ .", "This gives $\\Pr \\left[\\left|X_I-\\frac{\\mathcal {N}_1^\\ast }{M}\\right|\\ge \\frac{\\mathcal {N}_1^\\ast }{2M}\\right]\\le \\Pr \\left[\\left|X_I-\\frac{\\mathcal {N}_1^\\ast }{M}\\right|\\ge \\sqrt{\\frac{\\mathcal {N}_1^\\ast }{4M}}\\cdot \\sqrt{\\mathrm {var}(X_I)}\\right]\\le \\frac{4M}{\\mathcal {N}_1^\\ast },$ where the second inequality is obtained by Chebyshev's inequality.", "By the union bound we can conclude that $\\Pr \\left[|X_I|\\ge \\frac{\\mathcal {N}_1^\\ast }{2M} \\textrm { for all }I\\in f_1(U^\\ast )\\right]\\ge 1-\\frac{4MT_1}{\\mathcal {N}_1^\\ast }.$ Define $R=\\lbrace I\\in f_1(U^\\ast ) \\:|\\: b_1(I)\\in \\Gamma \\rbrace $ .", "For each element $I\\in f_1(U^\\ast )$ , the variable $b_1(I)$ is distributed uniformly at random in $\\mathbb {Z}_M$ .", "Moreover, the variables $b_1(I)$ 's are pairwise independent.", "Thus, by Chebyshev's inequality, we obtain $\\Pr \\left[\\left|R\\right|\\ge \\frac{T_1|\\Gamma |}{2M}\\right]\\ge 1-\\frac{4M}{T_1|\\Gamma |}.$ From the union bound we can conclude that, with probability at least $1-\\frac{4MT_1}{\\mathcal {N}_1^\\ast }-\\frac{4M}{T_1|\\Gamma |}$ , the inequality $\\left|R\\right|\\ge T_1|\\Gamma |/(2M)$ holds and simultaneously, for each element $I\\in R$ , there exist at least $\\mathcal {N}_1^\\ast /(2M)$ vertices $u\\in U^\\ast $ such that $f_1(u)=I$ and $b_2(f_2(u))=b_1(I)$ .", "These vertices are in $V^\\ast $ by Lemma REF ." ], [ "The second pruning: first version and proof of Theorem ", "In this subsection $M$ is an arbitrary prime number such that $2(\\mathcal {N}_1+\\mathcal {N}_2+\\mathcal {N}_3)<M<4(\\mathcal {N}_1+\\mathcal {N}_2+\\mathcal {N}_3)$ .", "The first pruning has transformed the graph $G$ into the subgraph induced by $V$ .", "The second pruning, similarly to [10], will further modify this subgraph by removing vertices in order to obtain a subgraph consisting of isolated vertices from $V^\\ast $ (i.e., an edgeless graph).", "This is done by constructing greedily a set $L\\subseteq V^\\ast $ of isolated vertices.", "Initially $L=\\emptyset $ and, at each iteration, either one remaining vertex in $V^\\ast \\backslash L$ will be added to $L$ or several vertices in $V$ will be removed.", "This will be repeated until there is no remaining vertex in $V^\\ast \\backslash L$ .", "Finally, all the remaining vertices not in $L$ will be removed.", "The detailed procedure is described in Figure REF , where $\\overline{V}$ represents the set of remaining vertices (initially $\\overline{V}=V$ ).", "Note that the procedure slightly differs from what was done in [10], [24], [27] since we need to take in consideration the asymmetry of the problem.", "Figure: The second pruning (first version)Let $\\overline{V_f}$ denote the contents of $\\overline{V}$ at the end of the procedure.", "The following proposition, shown using the same ideas as in [10], shows that what we obtain is a large set of isolated vertices from $V^\\ast $ .", "Proposition 4.1 The subgraph of $G$ induced by $\\overline{V_f}$ is an edgeless graph.", "Moreover, $\\overline{V_f}\\subseteq V^\\ast $ and the expectation of $|\\overline{V_f}|$ (over the choices of $\\omega _j$ ) is $\\Omega \\left(\\frac{T_1 \\mathcal {N}_1^\\ast }{(\\mathcal {N}_1+\\mathcal {N}_2+\\mathcal {N}_3)^{1+\\epsilon }}\\right).$ Let $L_f$ denote the contents of $L$ at the end of the procedure.", "First observe that $\\overline{V_f}\\subseteq L_f$ , due to Step 3.", "Note that any vertex added to $L$ cannot be later removed from $\\overline{V}$ , since it has no neighbor.", "Thus $L_f\\subseteq \\overline{V_f}$ , and we conclude that $\\overline{V_f}= L_f$ .", "This shows in particular that $\\overline{V_f}\\subseteq V^\\ast $ .", "Moreover, since each vertex in $L$ has no neighbor, the subgraph induced by $\\overline{V_f}$ is edgeless.", "To prove the second part, we will show an upper bound on the number of vertices in $V^\\ast $ removed from $\\overline{V}$ during the loop of Step 2.", "The bound will be obtained by considering the number of edges from $E^{\\prime }$ remaining in the subgraph induced by $\\overline{V}$ .", "Let us consider what happens during Step 2.2.", "Let $u\\in (\\overline{V}\\cap V^\\ast )\\backslash L$ be the vertex currently examined.", "Suppose that another vertex in $\\overline{V}$ sharing one index with $u$ is found.", "For example, suppose that we find another vertex $v\\in \\overline{V}$ with $f_1(v)=f_1(u)$ .", "Let $S=\\lbrace w\\in \\overline{V}\\cap V^\\ast \\:|\\:f_2(w)=f_2(u)\\rbrace $ be the set of vertices in $V^\\ast $ eliminated by the consequent removal operation.", "Observe that this removal operation will eliminate at least ${|S| \\atopwithdelims ()2}+1$ new edges from $E^{\\prime }$ : the edges between two vertices in $S$ , and the edge connecting $u$ and $v$ .", "Since ${|S| \\atopwithdelims ()2}+1\\ge |S|$ , the number of vertices in $V^\\ast $ removed during one execution of Step 2.2 is at most the number of edges eliminated from $E^{\\prime }$ .", "The total number of vertices from $V^\\ast $ that are removed by the procedure during the loop of Step 2 is thus at most $|E^{\\prime }|$ , which means that $|\\overline{V_f}\\cap V^\\ast |\\ge |V^\\ast |-|E^{\\prime }|$ .", "Since $\\overline{V}\\subseteq V^\\ast $ , Lemmas REF and REF imply that the expected number of vertices in $\\overline{V_f}$ is at least $\\frac{T_1\\mathcal {N}_1^\\ast |\\Gamma |}{M^2}\\left(1 - \\frac{(\\mathcal {N}_1+\\mathcal {N}_2+\\mathcal {N}_3-3)}{M}\\right)&\\ge &\\frac{T_1\\mathcal {N}_1^\\ast |\\Gamma |}{2M^2},$ where the inequality follows from the choice of $M$ .", "Since $M=O(\\mathcal {N}_1+\\mathcal {N}_2+\\mathcal {N}_3)$ and $|\\Gamma |\\ge M^{1-\\epsilon }$ , we conclude that $\\mathbb {E}[\\overline{|V_f}|]=\\Omega \\left(\\frac{T_1 \\mathcal {N}_1^\\ast }{(\\mathcal {N}_1+\\mathcal {N}_2+\\mathcal {N}_3)^{1+\\epsilon }}\\right)$ .", "Theorem REF follows from Proposition REF by fixing a choice of the weights $\\omega _0,\\omega _1,\\ldots ,\\omega _{N+1}$ for which $|\\overline{V_f}|\\ge \\mathbb {E}[|\\overline{V_f}|]$ (such a choice necessarily exists from the definition of the expectation)." ], [ "The second pruning: second version and proof of Theorem ", "In this subsection $M$ is an arbitrary prime such that $64\\mathcal {N}_2<M<128\\mathcal {N}_2$ .", "The first version of the pruning described in Subsection REF was designed to obtain an edgeless subgraph of $G$ .", "In this subsection we describe how to modify it to obtain a union of many large 1-cliques instead.", "The detailed procedure of the new pruning algorithm is described in Figure REF .", "The only difference is that, at Step 2.2, a vertex is not added in $L$ only if it is connected to another vertex with the same second or third index.", "Figure: The second pruning (second version)Let $\\overline{V_f}$ denote the contents of $\\overline{V}$ at the end of the procedure.", "By slightly modifying the arguments of the previous subsection, it is easy to see that the resulting graph has only vertices from $V^\\ast $ and is a disjoint union of 1-cliques (i.e., each connected component is a 1-clique).", "Proposition 4.2 The subgraph of $G$ induced by $\\overline{V_f}$ is a disjoint union of 1-cliques.", "Moreover, $\\overline{V_f}\\subseteq V^\\ast $ and $|\\overline{V_f}|\\ge |V^\\ast |-|E^{\\prime \\prime }|$ .", "Let $L_f$ denote the contents of $L$ at the end of the procedure.", "Due to Step 3, we know that $\\overline{V_f}\\subseteq L_f$ .", "Moreover, any vertex added to $L$ cannot be later removed from $\\overline{V}$ , since it has no neighbor with the same second or third index.", "Thus $L_f\\subseteq \\overline{V_f}$ , and we conclude that $\\overline{V_f}= L_f$ , which shows in particular that $\\overline{V_f}\\subseteq V^\\ast $ .", "Furthermore, since each vertex in $L$ has no neighbor with the same second or third index, the subgraph induced by $\\overline{V_f}$ is a disjoint union of 1-cliques.", "The inequality $|\\overline{V_f}|\\ge |V^\\ast |-|E^{\\prime \\prime }|$ is obtained as in the proof of Proposition REF , but replacing the edge set $E^{\\prime }$ by $E^{\\prime \\prime }$ .", "We now give the proof of Theorem REF , by using Lemmas REF and REF to evaluate the size and the numbers of 1-cliques in the resulting graph.", "[Proof of Theorem REF ] Using Lemma REF with the value $M>64\\mathcal {N}_2$ and Markov's bound, we conclude that $\\Pr \\left[|E^{\\prime \\prime }|\\le \\frac{T_1\\mathcal {N}_1^\\ast |\\Gamma |}{16M^2}\\right]\\ge \\frac{1}{2}.$ Note that the conditions $M<128\\mathcal {N}_2$ and $\\frac{\\mathcal {N}_2T_1}{\\mathcal {N}_1^\\ast }+\\frac{\\mathcal {N}_2}{T_1}<\\frac{1}{1024}$ imply that $1-\\frac{4MT_1}{\\mathcal {N}_1^\\ast }-\\frac{4M}{T_1|\\Gamma |}>1-512\\left(\\frac{\\mathcal {N}_2T_1}{\\mathcal {N}_1^\\ast }+\\frac{\\mathcal {N}_2}{T_1|\\Gamma |}\\right)>1-512\\left(\\frac{\\mathcal {N}_2T_1}{\\mathcal {N}_1^\\ast }+\\frac{\\mathcal {N}_2}{T_1}\\right)>\\frac{1}{2}.$ Thus the probability that a set $R$ as in Lemma REF exists and the inequality $|E^{\\prime \\prime }|\\le \\frac{T_1\\mathcal {N}_1^\\ast |\\Gamma |}{16M^2}$ simultaneously holds is positive.", "There then exists a choice of the weights $\\omega _0,\\omega _1,\\ldots ,\\omega _{N+1}$ such that this happens.", "Let us take such a choice.", "From Proposition REF we know that at most $|E^{\\prime \\prime }|$ vertices in $V^\\ast $ are removed by the second pruning.", "Then there exists a set $R^{\\prime }\\subseteq R$ of size $|R^{\\prime }|\\ge T_1|\\Gamma |/(4M)$ such that, for any $r^{\\prime }\\in R^{\\prime }$ , there are at least $\\mathcal {N}_1^\\ast /(4M)$ vertices $u$ with $f_1(u)=r^{\\prime }$ remaining after the second pruning.", "This is because, otherwise, from the properties of the set $R$ stated in Lemma REF it would be necessary to remove more than $T_1\\mathcal {N}_1^\\ast |\\Gamma |/(16M^2)$ vertices during the second pruning." ], [ "Algorithm for Rectangular Matrix Multiplication", "In this section we present our algorithm, which essentially consists in the two algorithmic steps described in Subsections REF and REF .", "We first start by explaining the construction we will use." ], [ "Our construction", "Let $a_{004},a_{400},a_{013},a_{103},a_{301},a_{022},a_{202}, a_{112},a_{211}$ be nine arbitrary positiveThe hypothesis that each $a_{ijk}$ is not zero is made only for convenience (all the bounds presented in this paper are obtained using positive values for these parameters).", "More specifically, this hypothesis is used only when approximating quantities like $(a_{ijk}N)!$ using Stirling's inequality.", "Without the hypothesis it would be necessary to treat the (trivial) case $a_{ijk}= 0$ separately.", "rational numbers such that $2a_{004}+a_{400}+2a_{013}+2a_{103}+2a_{301}+a_{022}+2a_{202}+2a_{112}+a_{211}=1$ and $a_{013}a_{202}a_{112}=a_{103}a_{022}a_{211}.$ Let us define rational numbers $A_0,A_1,A_2,A_3,A_4,B_0,B_1,B_2,B_3,B_4$ as follows: $A_0&=&2a_{004}+2a_{013}+a_{022}\\\\A_1&=&2a_{103}+2a_{112}\\\\A_2&=&2a_{202}+a_{211}\\\\A_3&=&2a_{301}\\\\A_4&=&a_{400}\\\\B_0&=&a_{004}+a_{400}+a_{103}+a_{301}+a_{202}\\\\B_1&=&a_{013}+a_{301}+a_{112}+a_{211}\\\\B_2&=&a_{022}+a_{202}+a_{112}\\\\B_3&=&a_{013}+a_{103}\\\\B_4&=&a_{004}.$ Note that $\\sum _{i=0}^4A_i=\\sum _{i=0}^4B_i=1.$ It will be convenient to define six additional numbers $a_{040}$ , $a_{031}$ , $a_{130}$ , $a_{310}$ , $a_{220}$ and $a_{121}$ as $a_{040}=a_{004}$ , $a_{031}=a_{013}$ , $a_{130}=a_{103}$ , $a_{310}=a_{301}$ , $a_{220}=a_{202}$ and $a_{121}=a_{112}$ .", "We can then rewrite concisely the $A_{i}$ 's and the $B_j$ 's as follows.", "$A_i&=&\\sum _{{{\\scriptstyle \\begin{matrix}0\\le j,k\\le 4\\\\ i+j+k=4\\end{matrix}}}} a_{ijk}\\hspace{8.53581pt}\\textrm { for } i=0,1,2,3,4\\\\B_j&=&\\sum _{{{\\scriptstyle \\begin{matrix}0\\le i,k\\le 4\\\\ i+j+k=4\\end{matrix}}}} a_{ijk}\\hspace{8.53581pt}\\textrm { for } j=0,1,2,3,4\\\\$ Let $N$ be a large enough positive integer such each $Na_{ijk}$ is an integer.", "We rise the construction $F_q\\otimes F_q$ described in Section to the $N$ -th power.", "Observe that $(F^{\\prime }_q\\otimes F^{\\prime }_q)^{\\otimes N} \\unlhd (F_q\\otimes F_q)^{\\otimes N}$ and $(F^{\\prime }_q\\otimes F^{\\prime }_q)^{\\otimes N}&=&\\sum _{{{\\scriptstyle \\begin{matrix} IJK\\end{matrix}}}}T_{IJK},$ where the sum is over all triples of sequences $IJK$ with $I,J,K\\in \\lbrace 0,1,2,3,4\\rbrace ^N$ such that $I_\\ell +J_\\ell +K_\\ell =4$ for all $\\ell \\in \\lbrace 1,\\ldots ,N\\rbrace $ .", "Here we use the notation $T_{IJK}=T_{I_1J_1K_1}\\otimes \\cdots \\otimes T_{I_NJ_NK_N}$ .", "Note that there are $15^N$ terms $T_{IJK}$ in the above sum.", "In the tensor product the number of $x$ -variables is $(q+2)^{2N}$ .", "The number of $y$ -variables and $z$ -variables is also $(q+2)^{2N}$ .", "Remember that in the original construction, each $x$ -variable was indexed by a superscript in $\\lbrace 0,1,2,3,4\\rbrace $ .", "Each $x$ -variable in the tensor product is thus indexed by a sequence of $N$ such superscripts, i.e., by an element $I\\in \\lbrace 0,1,2,3,4\\rbrace ^N$ .", "The same is true for the $y$ -variables and the $z$ -variables.", "Note that the $x$ -variables appearing in $T_{IJK}$ have superscript $I$ , the $y$ -variables appearing in $T_{IJK}$ have superscript $J$ , and the $z$ -variables appearing in $T_{IJK}$ have superscript $K$ .", "Let us introduce the following definition.", "Definition 5.1 Let $\\overline{a}_{004}$ , $ \\overline{a}_{040}$ , $\\overline{a}_{400}$ , $\\overline{a}_{013}$ , $\\overline{a}_{031}$ , $\\overline{a}_{103}$ , $\\overline{a}_{130}$ , $\\overline{a}_{301}$ , $\\overline{a}_{310}$ , $\\overline{a}_{022}$ , $\\overline{a}_{202}$ , $\\overline{a}_{220}$ , $\\overline{a}_{112}$ , $\\overline{a}_{121}$ , $\\overline{a}_{211}$ be fifteen nonnegative rational numbers.", "We say that a triple $IJK$ is of type $[\\overline{a}_{ijk}]$ if $|\\left\\lbrace \\ell \\in \\lbrace 1,\\ldots ,N\\rbrace \\:|\\: I_\\ell =i, J_\\ell =j \\textrm { and } K_\\ell =k\\right\\rbrace |=\\overline{a}_{ijk}N$ for all 15 combinations of positive $i,j,k$ with $i+j+k=4$ .", "With a slight abuse of notation, we will say that a form $T_{IJK}$ is of type $[\\overline{a}_{ijk}]$ if the triple $IJK$ is of type $[\\overline{a}_{ijk}]$ ." ], [ "The first step", "We set to zero all $x$ -variables except those satisfying the following condition: their superscript $I$ has exactly $A_0N$ coordinates with value 0, $A_1N$ coordinates with value 1, $A_2N$ coordinates with value 2, $A_3N$ coordinates with value 3 and $A_4N$ coordinates with value 4.", "We will say that such a sequence $I$ is of type $A$ .", "There are $T_X={N\\atopwithdelims (){A_0N,\\ldots ,A_4N}}=\\Theta \\left(\\frac{1}{N^{2}}\\left(\\frac{1}{A_0^{A_0}A_1^{A_1}A_2^{A_2}A_3^{A_3}A_4^{A_4}}\\right)^{N}\\right)$ sequences $I$ of type $A$ (the approximation is done using Stirling's formula).", "After the zeroing operation, all forms $T_{IJK}$ such that $I$ is not of type $A$ disappear (i.e., become zero).", "We process the $y$ -variables and the $z$ -variables slightly differently.", "We set to zero all $y$ -variables except those satisfying the following condition: their superscript $J$ has exactly $B_0N$ coordinates with value 0, $B_1N$ coordinates with value 1, $B_2N$ coordinates with value 2, $B_3N$ coordinates with value 3 and $B_4N$ coordinates with value 4.", "We will say that such a sequence is of type $B$ .", "There are $T_Y={N\\atopwithdelims (){B_0N,\\ldots ,B_4N}}=\\Theta \\left(\\frac{1}{N^{2}}\\left(\\frac{1}{B_0^{B_0}B_1^{B_1}B_2^{B_2}B_3^{B_3}B_4^{B_4}}\\right)^{N}\\right)$ sequences $J$ of type $B$ .", "Similarly, we set to zero all $z$ -variables except those such that their superscript $K$ is of type $B$ (there are $T_Y$ such sequences).", "After these three zeroing operations, the forms $T_{IJK}$ remaining are precisely those such that $I$ is of type $A$ , $J$ is of type $B$ , and $K$ is of type $B$ .", "Equivalently, the forms remaining are precisely the forms $T_{IJK}$ that are of type $[\\overline{a}_{ijk}]$ with fifteen numbers $\\overline{a}_{ijk}$ (for all fifteen combinations of positive $i,j,k$ such that $i+j+k=4$ ) satisfying the following four conditions: $\\hspace{14.22636pt}&\\overline{a}_{ijk}N\\in \\lbrace 0,1,\\ldots ,N\\rbrace \\:\\:\\:\\textrm {for all }i,j,k;\\\\\\hspace{14.22636pt}&A_i=\\sum _{j,k \\::\\: i+j+k=4} \\overline{a}_{ijk}\\hspace{8.53581pt}\\textrm { for } i=0,1,2,3,4;\\\\\\hspace{14.22636pt}&B_j=\\sum _{i,k \\::\\: i+j+k=4} \\overline{a}_{ijk}\\hspace{8.53581pt}\\textrm { for } j=0,1,2,3,4;\\\\\\hspace{14.22636pt}&B_k=\\sum _{i,j \\::\\: i+j+k=4} \\overline{a}_{ijk}\\hspace{8.53581pt}\\textrm { for } k=0,1,2,3,4.$ Let $I$ be a fixed sequence of type $A$ .", "The number of non-zeros forms $T_{IJK}$ with this sequence $I$ as its first index is thus precisely $\\mathcal {N}_X&=&\\sum _{[\\overline{a}_{ijk}]}{A_0N \\atopwithdelims ()\\overline{a}_{004}N,\\overline{a}_{040}N,\\overline{a}_{013}N,\\overline{a}_{031}N,\\overline{a}_{022}N}{A_1N \\atopwithdelims ()\\overline{a}_{103}N,\\overline{a}_{130}N,\\overline{a}_{112}N,\\overline{a}_{121}N}\\times \\\\&&\\hspace{136.57323pt}{A_2N \\atopwithdelims ()\\overline{a}_{202}N,\\overline{a}_{220}N,\\overline{a}_{211}N}{A_3N \\atopwithdelims ()\\overline{a}_{301}N,\\overline{a}_{310}N}{A_4N \\atopwithdelims ()\\overline{a}_{400}N},$ where the sum is over all the choices of fifteen parameters $\\overline{a}_{ijk}$ 's satisfying conditions (REF )–().", "For a fixed sequence $J$ of type $B$ , the number of non-zeros forms $T_{IJK}$ with this sequence $J$ as its second index is $\\mathcal {N}_Y&=&\\sum _{[\\overline{a}_{ijk}]}{B_0N \\atopwithdelims ()\\overline{a}_{004}N,\\overline{a}_{400}N,\\overline{a}_{103}N,\\overline{a}_{301}N,\\overline{a}_{202}N}{B_1N \\atopwithdelims ()\\overline{a}_{013},\\overline{a}_{310},\\overline{a}_{112},\\overline{a}_{211}N}\\times \\\\&&\\hspace{108.12047pt}{B_2N \\atopwithdelims ()\\overline{a}_{022}N,\\overline{a}_{220}N,\\overline{a}_{121}N}{B_3N \\atopwithdelims ()\\overline{a}_{031}N,\\overline{a}_{130}N}{B_4N \\atopwithdelims ()\\overline{a}_{040}N},$ where the sum is again over all the choices of fifteen parameters $\\overline{a}_{ijk}$ 's satisfying conditions ()–().", "Similarly, for a fixed sequence $K$ of type $B$ , the number of non-zeros forms $T_{IJK}$ with this sequence $K$ as its third index is $\\mathcal {N}_Z&=&\\sum _{[\\overline{a}_{ijk}]}{B_0N \\atopwithdelims ()\\overline{a}_{040}N,\\overline{a}_{400}N,\\overline{a}_{130}N,\\overline{a}_{310}N,\\overline{a}_{220}N}{B_1N \\atopwithdelims ()\\overline{a}_{031},\\overline{a}_{301},\\overline{a}_{121},\\overline{a}_{211}N}\\times \\\\&&\\hspace{108.12047pt}{B_2N \\atopwithdelims ()\\overline{a}_{022}N,\\overline{a}_{202}N,\\overline{a}_{112}N}{B_3N \\atopwithdelims ()\\overline{a}_{013}N,\\overline{a}_{103}N}{B_4N \\atopwithdelims ()\\overline{a}_{004}N}.$ The total number of remaining triples is $T_X\\mathcal {N}_X=T_Y\\mathcal {N}_Y=T_Y\\mathcal {N}_Z.$ Note that this implies that $\\mathcal {N}_Y=\\mathcal {N}_Z$ .", "We will also be interested in the number of remaining forms $T_{IJK}$ of type $[a_{ijk}]$ .", "For a fixed sequence $I$ of type $A$ , the number of non-zeros forms $T_{IJK}$ of type $[a_{ijk}]$ with this sequence $I$ as its first index is $\\mathcal {N}_X^\\ast &=&{A_0N \\atopwithdelims ()a_{004}N,a_{004}N,a_{013}N,a_{013}N,a_{022}N}{A_1N \\atopwithdelims ()a_{103}N,a_{103}N,a_{112}N,a_{112}N}\\times \\\\&&\\hspace{136.57323pt}{A_2N \\atopwithdelims ()a_{202}N,a_{202}N,a_{211}N}{A_3N \\atopwithdelims ()a_{301}N,a_{301}N}{A_4N \\atopwithdelims ()a_{400}N}.$ For a fixed sequence $J$ of type $B$ , the number of non-zeros forms $T_{IJK}$ of type $[a_{ijk}]$ with this sequence $J$ as its second index is $\\mathcal {N}_Y^\\ast &=&{B_0N \\atopwithdelims ()a_{004}N,a_{400}N,a_{103}N,a_{301}N,a_{202}N}{B_1N \\atopwithdelims ()a_{013},a_{301},a_{112},a_{211}N}\\times \\\\&&\\hspace{108.12047pt}{B_2N \\atopwithdelims ()a_{022}N,a_{202}N,a_{112}N}{B_3N \\atopwithdelims ()a_{013}N,a_{103}N}{B_4N \\atopwithdelims ()a_{004}N}.$ We have $T_X\\mathcal {N}^\\ast _X=T_Y\\mathcal {N}_Y^\\ast .$" ], [ "Approximation", "In this subsection we will use the notation $[\\overline{a}_{ijk}]$ to represent an arbitrary set of fifteen parameters $\\overline{a}_{ijk}$ such that $0\\le \\overline{a}_{ijk}\\le 1$ for each $i,j,k$ .", "Let $c([\\overline{a}_{ijk}])$ denote the number of nonzero elements among these fifteen parameters.", "Consider the following expression: $g([\\overline{a}_{ijk}])=\\left(\\overline{a}_{004}^{\\overline{a}_{004}}\\overline{a}_{040}^{\\overline{a}_{040}}\\overline{a}_{400}^{\\overline{a}_{400}}\\overline{a}_{013}^{\\overline{a}_{013}}\\overline{a}_{031}^{\\overline{a}_{031}}\\overline{a}_{103}^{\\overline{a}_{103}}\\overline{a}_{130}^{\\overline{a}_{130}}\\overline{a}_{301}^{\\overline{a}_{301}}\\overline{a}_{310}^{\\overline{a}_{310}}\\overline{a}_{022}^{\\overline{a}_{022}}\\overline{a}_{202}^{\\overline{a}_{202}}\\overline{a}_{220}^{\\overline{a}_{220}}\\overline{a}_{112}^{\\overline{a}_{112}}\\overline{a}_{121}^{\\overline{a}_{121}}\\overline{a}_{211}^{\\overline{a}_{211}}\\right)^{-1}.$ Using Stirling's formula, we can give the following approximations.", "$\\mathcal {N}_X&=&\\Theta \\left(\\sum _{[\\overline{a}_{ijk}]}\\frac{\\left[ A_0^{A_0}A_1^{A_1}A_2^{A_2}A_3^{A_3}A_4^{A_4}\\times g([\\overline{a}_{ijk}])\\right]^{N}}{N^{(c([\\overline{a}_{ijk}])-5)/2}}\\right)\\\\\\mathcal {N}_Y&=&\\Theta \\left(\\sum _{[\\overline{a}_{ijk}]}\\frac{\\left[ B_0^{B_0}B_1^{B_1}B_2^{B_2}B_3^{B_3}B_4^{B_4}\\times g([\\overline{a}_{ijk}])\\right]^{N}}{N^{(c([\\overline{a}_{ijk}])-5)/2}}\\right)\\\\\\mathcal {N}^\\ast _X&=&\\Theta \\left(\\frac{1}{N^5}\\left[ A_0^{A_0}A_1^{A_1}A_2^{A_2}A_3^{A_3}A_4^{A_4}\\times g([a_{ijk}])\\right]^{N}\\right)\\\\\\mathcal {N}^\\ast _Y&=&\\Theta \\left(\\frac{1}{N^5}\\left[ B_0^{B_0}B_1^{B_1}B_2^{B_2}B_3^{B_3}B_4^{B_4}\\times g([a_{ijk}])\\right]^{N}\\right)$ The first two sums are again over all the choices of fifteen parameters $\\overline{a}_{ijk}$ 's satisfying conditions (REF )–().", "Note that, for any $[\\overline{a}_{ijk}]$ satisfying these four conditions, $c([\\overline{a}_{ijk}])\\ge 5$ , since the $A_i$ 's are non-zero.", "We know that $\\mathcal {N}_X^\\ast \\le \\mathcal {N}_X$ and $\\mathcal {N}_Y^\\ast \\le \\mathcal {N}_Y$ , by definition.", "The following proposition shows that $\\mathcal {N}_X$ and $\\mathcal {N}_Y$ can actually be approximated by $\\mathcal {N}_X^\\ast $ and $\\mathcal {N}_Y^\\ast $ , respectively.", "Proposition 5.1 $\\mathcal {N}_X=O(N^8\\mathcal {N}_X^\\ast )\\:\\:\\textrm { and }\\:\\:\\mathcal {N}_Y=O(N^8\\mathcal {N}_Y^\\ast )$ .", "Any set of values $[\\overline{a}_{ijk}]$ that satisfies conditions (REF )–() is such that $\\overline{a}_{004}=a_{004}$ , $\\overline{a}_{040}=a_{040}$ and $\\overline{a}_{400}=a_{400}$ .", "Moreover, the other values $a_{ijk}$ depend only on $\\overline{a}_{103},\\overline{a}_{031}$ and $\\overline{a}_{301}$ : $\\overline{a}_{013}&=&B_3N-\\overline{a}_{103}\\\\\\overline{a}_{130}&=&B_3N-\\overline{a}_{031}\\\\\\overline{a}_{310}&=&A_3N-\\overline{a}_{301}\\\\\\overline{a}_{022}&=&(A_0-2B_4-B_3)N+\\overline{a}_{103}-\\overline{a}_{031}\\\\\\overline{a}_{202}&=&(-A_4+B_0-B_4)N-\\overline{a}_{103}-\\overline{a}_{301}\\\\\\overline{a}_{220}&=&(-A_3-A_4+B_0-B_3-B_4)N+\\overline{a}_{031}+\\overline{a}_{301}\\\\\\overline{a}_{112}&=&(-A_0+A_4-B_0+B_2+B_3+3B_4)N+\\overline{a}_{031}+\\overline{a}_{301}\\\\\\overline{a}_{121}&=&(-A_0+A_3+A_4-B_0+B_2+2B_3+3B_4)N-\\overline{a}_{103}-\\overline{a}_{301}\\\\\\overline{a}_{211}&=&(A_2+A_3+2A_4-2B_0+B_3+2B_4)N-\\overline{a}_{031}+\\overline{a}_{103}.$ Note that there are at most $(N+1)$ choices for each $\\overline{a}_{103},\\overline{a}_{031}$ and $\\overline{a}_{301}$ , from condition (REF ).", "There are thus at most $(N+1)^3$ choices for the values $[\\overline{a}_{ijk}]$ satisfying conditions (REF )–().", "We now show that the expression $g([\\overline{a}_{ijk}])$ is maximized for the values $[\\overline{a}_{ijk}]=[a_{ijk}]$ .", "Let us take the logarithm of the expression $g$ .", "Since this is a concave function on a convex domain, a local optimum of $\\log g$ is a global maximum of $g$ .", "The partial derivatives of $\\log g$ are as follows.", "$\\frac{\\partial \\log g}{\\partial \\overline{a}_{103}}&=&\\phantom{-}\\log (\\overline{a}_{013})-\\log (\\overline{a}_{103})-\\log (\\overline{a}_{022})+\\log (\\overline{a}_{202})+\\log (\\overline{a}_{121})-\\log (\\overline{a}_{211})\\\\\\frac{\\partial \\log g}{\\partial \\overline{a}_{031}}&=&-\\log (\\overline{a}_{031})+\\log (\\overline{a}_{130})+\\log (\\overline{a}_{022})-\\log (\\overline{a}_{220})-\\log (\\overline{a}_{112})+\\log (\\overline{a}_{211})\\\\\\frac{\\partial \\log g}{\\partial \\overline{a}_{301}}&=&-\\log (\\overline{a}_{301})+\\log (\\overline{a}_{310})+\\log (\\overline{a}_{202})-\\log (\\overline{a}_{220})-\\log (\\overline{a}_{112})+\\log (\\overline{a}_{121})\\\\$ The values $[\\overline{a}_{ijk}]=[a_{ijk}]$ satisfy $\\frac{\\partial \\log g}{\\partial \\overline{a}_{103}}=\\frac{\\partial \\log g}{\\partial \\overline{a}_{031}}=\\frac{\\partial \\log g}{\\partial \\overline{a}_{301}}=0$ since $a_{013}a_{202}a_{112}=a_{103}a_{022}a_{211}$ .", "We conclude that $\\mathcal {N}_X=O\\left((N+1)^3\\left[ A_0^{A_0}A_1^{A_1}A_2^{A_2}A_3^{A_3}A_4^{A_4}\\times g([a_{ijk}])\\right]^{N}\\right)=O\\left(N^8 \\mathcal {N}_X^\\ast \\right),$ and similarly $\\mathcal {N}_Y=O\\left(N^8 \\mathcal {N}_Y^\\ast \\right)$ ." ], [ "The second step", "We will now apply the results of Section , by associating to $U$ the set of all forms $T_{IJK}$ that are of type $[\\overline{a}_{ijk}]$ for values $\\overline{a}_{ijk}$ satisfying conditions (REF )–(), and to $U^\\ast $ the set of all forms $T_{IJK}$ of type $[a_{ijk}]$ .", "Note that all the conditions of Subsection REF are satisfied: we have $\\tau =4$ , and values $T_1=T_X$ , $T_2=T_3=T_Y$ , $\\mathcal {N}_1=\\mathcal {N}_X$ , $\\mathcal {N}_2=\\mathcal {N}_3=\\mathcal {N}_Y$ , $\\mathcal {N}_1^\\ast =\\mathcal {N}_X^\\ast $ and $\\mathcal {N}_2^\\ast =\\mathcal {N}_3^\\ast =\\mathcal {N}_Y^\\ast $ .", "With this association we can use the graph-theoretic framework developed in Section .", "The initial graph $G$ , as defined in Subsection REF , corresponds to the current sum of trilinear forms (each vertex corresponds to a form $T_{IJK}$ of type $[\\overline{a}_{ijk}]$ for values $\\overline{a}_{ijk}$ satisfying conditions (REF )–()).", "A removal operation on $G$ corresponds to zeroing variables with a given superscript.", "For instance, removing all vertices $u\\in G$ such that $f_1(u)=I$ corresponds to zeroing all the $x$ -variables with superscript $I$ .", "Our goal is to zero variables in order to obtain a sum of several forms $T_{IJK}$ satisfying the following two conditions.", "First, each form in the sum is of type $[a_{ijk}]$ .", "Then, the forms in the sum do not share any index (i.e., if $T_{IJK}$ and $T_{I^{\\prime }J^{\\prime }K^{\\prime }}$ are in the sum, then $I\\ne I^{\\prime }$ , $J\\ne J^{\\prime }$ and $K\\ne K^{\\prime }$ ), which implies that the same variable does not appear in more than one form, and thus that the sum is direct.", "This corresponds to constructing an edgeless subgraph of $G$ in which all the vertices are in $U^\\ast $ , so we are in a situation where Theorem REF can be applied.", "Suppose that the inequality $A_0^{A_0}A_1^{A_1}A_2^{A_2}A_3^{A_3}A_4^{A_4}\\ge B_0^{B_0}B_1^{B_1}B_2^{B_2}B_3^{B_3}B_4^{B_4}$ holds.", "Then $T_X=O(T_Y)$ , and Equalities (REF ) and (REF ) imply that $\\mathcal {N}_Y=O(\\mathcal {N}_X)$ and $\\mathcal {N}_Y^\\ast =O(\\mathcal {N}_X^\\ast )$ .", "From Proposition REF we then obtain the relation $\\mathcal {N}_1+\\mathcal {N}_2+\\mathcal {N}_3=O(N^8\\mathcal {N}_X^\\ast )$ .", "By the above discussion and Theorem REF , we can obtain a direct sum of $\\Omega \\left(\\frac{T_X\\mathcal {N}^\\ast _X}{(N^8\\mathcal {N}_X^\\ast )^{1+\\epsilon }}\\right)$ forms, all of type $[a_{ijk}]$ .", "By using the trivial upper bound $\\mathcal {N}_X^\\ast \\le 15^N$ , we obtain the following theorem.", "Theorem 5.1 Let $q$ be any positive integer and $a_{004}$ , $a_{400}$ , $a_{013}$ , $a_{103}$ , $a_{301}$ , $a_{022}$ , $a_{202}$ , $a_{112}$ and $a_{211}$ be any nine positive rational numbers satisfying the following three conditions: $2a_{004}+a_{400}+2a_{013}+2a_{103}+2a_{301}+a_{022}+2a_{202}+2a_{112}+a_{211}=1$ ; $a_{013}a_{202}a_{112}=a_{103}a_{022}a_{211}$ ; $A_0^{A_0}A_1^{A_1}A_2^{A_2}A_3^{A_3}A_4^{A_4}\\ge B_0^{B_0}B_1^{B_1}B_2^{B_2}B_3^{B_3}B_4^{B_4}$ .", "Then, for any constant $\\epsilon >0$ , the trilinear form $(F_q\\otimes F_q)^{\\otimes N}$ can be converted (i.e., degenerated in the sense of Definition REF ) into a direct sum of $\\Omega \\left(\\frac{1}{N^{10 +8\\epsilon }15^{N\\epsilon }}\\left[\\frac{1}{A_0^{A_0}A_1^{A_1}A_2^{A_2}A_3^{A_3}A_4^{A_4}}\\right]^{N}\\right)$ forms, each form being isomorphic to $\\bigotimes _{{{\\scriptstyle \\begin{matrix}0\\le i,j,k\\le 4\\\\ i+j+k=4\\end{matrix}}}}T_{ijk}^{\\otimes a_{ijk}N}.$" ], [ "Upper Bounds on the Exponent of Rectangular Matrix Multiplication", "Theorem REF showed how the form $(F_q\\otimes F_q)^{\\otimes N}$ can be used to obtain a direct sum of many forms $T_{IJK}$ such that $T_{IJK}\\cong \\bigotimes _{i,j,k:i+j+k=4}T_{ijk}^{\\otimes a_{ijk}N}.$ In order to apply Schönhage's asymptotic sum inequality (Theorem REF ), we need to analyze the smaller forms $T_{ijk}$ .", "All the forms except $T_{112}$ , $T_{121}$ and $T_{211}$ correspond to matrix multiplications, as described in Section .", "In Subsection REF we analyze the forms $T_{112}$ , $T_{121}$ and $T_{211}$ .", "Then, in Subsection REF , we put all our results together and prove our main result." ], [ "The forms $T_{112}$ , {{formula:efde4163-6c61-4872-84e0-94d4d9948431}} and {{formula:4f750395-9ba1-4395-839e-9677c13850bb}}", "Let us recall the definition of these three forms.", "$T_{112}&=&\\sum _{i=1}^q x_{i,0}^{1}y_{i,0}^{1}z_{0,q+1}^{2}+\\sum _{k=1}^q x_{0,k}^{1}y_{0,k}^{1}z_{q+1,0}^{2}+\\sum _{i,k=1}^q x_{i,0}^{1}y_{0,k}^{1}z_{i,k}^{2}+\\sum _{i,k=1}^q x_{0,k}^{1}y_{i,0}^{1}z_{i,k}^{2}\\\\T_{121}&=&\\sum _{i=1}^q x_{i,0}^{1}y_{0,q+1}^{2}z_{i,0}^{1}+\\sum _{k=1}^q x_{0,k}^{1}y_{q+1,0}^{2}z_{0,k}^{1}+\\sum _{i,k=1}^q x_{i,0}^{1}y_{i,k}^{2}z_{0,k}^{1}+\\sum _{i,k=1}^q x_{0,k}^{1}y_{i,k}^{2}z_{i,0}^{1}\\\\T_{211}&=&\\sum _{i=1}^q x_{0,q+1}^{2}y_{i,0}^{1}z_{i,0}^{1}+\\sum _{k=1}^q x_{q+1,0}^{2}y_{0,k}^{1}z_{0,k}^{1}+\\sum _{i,k=1}^q x_{i,k}^{2}y_{i,0}^{1}z_{0,k}^{1}+\\sum _{i,k=1}^q x_{i,k}^{2}y_{0,k}^{1}z_{i,0}^{1}$ We first focus on the form $T_{211}$ .", "It will be convenient to write $T_{211}=t_{011}+t_{101}+t_{110}+t_{200}$ , where $t_{011}&=&\\sum _{i=1}^q x_{0,q+1}^{0}y_{i,0}^{1}z_{i,0}^{1},\\\\t_{101}&=&\\sum _{i,k=1}^q x_{i,k}^{1}y_{0,k}^{0}z_{i,0}^{1},\\\\t_{110}&=&\\sum _{i,k=1}^q x_{i,k}^{1}y_{i,0}^{1}z_{0,k}^{0},\\\\t_{200}&=&\\sum _{k=1}^q x_{q+1,0}^{2}y_{0,k}^{0}z_{0,k}^{0}.$ Note that the superscripts in these forms differ from the original superscripts.", "They are nevertheless uniquely determined by the subscripts of the variables.", "Observe that $t_{011}\\cong t_{200}\\cong \\langle 1,1,q \\rangle $ , which corresponds to the product of a scalar by a row vector, and $t_{101}\\cong t_{110}\\cong \\langle q,q,1 \\rangle $ , which corresponds to the product of a $q\\times q$ matrix by a column vector.", "The following proposition states that tensor powers of $T_{211}$ can be used to construct a direct sum of several trilinear forms, each one being a ${C}$ -tensor in which the support and all the components are isomorphic to a rectangular matrix product.", "Proposition 6.1 Let $b$ be any constant such that $0.916027<b\\le 1$ .", "Then there exists a constant $c\\ge 1$ depending only on $b$ such that, for any $\\epsilon >0$ and any large enough integer $m$ , the form $T_{211}^{\\otimes 2m}$ can be converted into a direct sum of $\\Omega \\left(\\frac{1}{mc^{2\\epsilon m}}\\cdot \\left[\\frac{2}{(2b)^{b}(1-b)^{1-b}}\\right]^{2m}\\right)$ trilinear forms, each form being a ${C}$ -tensor in which: each component is isomorphic to $\\langle q^{2bm},q^{2bm},q^{2(1-b)m} \\rangle $ ; the support is isomorphic to $\\mathrm {supp_c}(\\langle 1,1,H \\rangle )$ , where $H=\\Omega \\left(\\frac{1}{\\sqrt{m}}\\cdot \\left[(2b)^b(1-b)^{(1-b)}\\right]^{2m}\\right).$ Remark.", "Proposition REF uses the convention $0^0=1$ .", "For the case $b=1$ , the proposition states that the form $T_{112}^{\\otimes 2m}$ can be used to construct at least one tensor with support isomorphic to $\\mathrm {supp_c}(\\langle 1,1,H \\rangle )$ for $H=\\Omega \\left(4^m/\\sqrt{m}\\right)$ , each component being isomorphic to $\\langle q^{2m},q^{2m},1 \\rangle $ .", "[Proof of Proposition REF ] For simplicity we suppose that $bm$ is an integer (otherwise, we can work with $\\left\\lfloor bm \\right\\rfloor $ , which gives the same asymptotic complexity).", "Let $S$ be the set of all triples $IJK$ with $I\\in \\lbrace 0,1,2\\rbrace ^{2m}$ and $J,K\\in \\lbrace 0,1\\rbrace ^{2m}$ such that $I_\\ell +J_\\ell +K_\\ell =2$ for all $\\ell \\in \\lbrace 1,\\ldots ,2m\\rbrace $ .", "We rise the form $T_{211}$ to the $2m$ -th power.", "This gives the form $\\sum _{IJK\\in S}t_{IJK}$ where $t_{IJK}=t_{I_1J_1K_1}\\otimes \\cdots \\otimes t_{I_{2m}J_{2m}K_{2m}}$ .", "Each $x$ -variable in the tensor product has for superscript a sequence in $\\lbrace 0,1,2\\rbrace ^{2m}$ , and each $y$ -variable or $z$ -variable has for superscript a sequence in $\\lbrace 0,1\\rbrace ^{2m}$ .", "Let us decompose the space of $x$ -variables as $\\bigoplus _{I\\in \\lbrace 0,1,2\\rbrace ^{2m}} X_I$ , where $X_I$ denotes the subspace of $x$ -variables with superscript $I$ .", "Similarly, decompose the space of $y$ -variables as $\\bigoplus _{J\\in \\lbrace 0,1\\rbrace ^{2m}} Y_J$ and the space of $z$ -variables as $\\bigoplus _{K\\in \\lbrace 0,1\\rbrace ^{2m}} Z_K$ .", "The form $\\sum t_{IJK}$ above is then a ${C}$ -tensor with respect to this decomposition, with support $S$ .", "We will now modify this form (by zeroing variables, which will modify its support) in order to obtain a simple expression for each of its components.", "We zero all the $x$ -variables except those for which the superscript $I$ has $(1-b)m$ coordinates with value 0, $(1-b)m$ coordinates with value 2 and $2bm$ coordinates with value 1.", "We zero all the $y$ -variables and $z$ -variables except those for which the superscript has $m$ coordinates with value 0 and $m$ coordinates with value 1.", "After these zeroing operations, the only forms remaining in the sum are those corresponding to indexes $IJK$ satisfying the following four conditions.", "$|\\left\\lbrace \\ell \\in \\lbrace 1,\\ldots ,2m\\rbrace \\:|\\: I_\\ell =0, J_\\ell =1 \\textrm { and } K_\\ell =1\\right\\rbrace |&=&(1-b)m\\\\|\\left\\lbrace \\ell \\in \\lbrace 1,\\ldots ,2m\\rbrace \\:|\\: I_\\ell =1, J_\\ell =0 \\textrm { and } K_\\ell =1\\right\\rbrace |&=&bm\\\\|\\left\\lbrace \\ell \\in \\lbrace 1,\\ldots ,2m\\rbrace \\:|\\: I_\\ell =1, J_\\ell =1 \\textrm { and } K_\\ell =0\\right\\rbrace |&=&bm\\\\|\\left\\lbrace \\ell \\in \\lbrace 1,\\ldots ,2m\\rbrace \\:|\\: I_\\ell =2, J_\\ell =0 \\textrm { and } K_\\ell =0\\right\\rbrace |&=&(1-b)m$ This means that each form $t_{IJK}$ in this new sum (i.e., each component in the corresponding ${C}$ -tensor) is isomorphic to $t_{011}^{\\otimes (1-b)m}\\otimes t_{101}^{\\otimes bm}\\otimes t_{110}^{\\otimes bm}\\otimes t_{200}^{\\otimes (1-b)m}\\cong \\langle q^{2bm},q^{2bm},q^{2(1-b)m} \\rangle .$ We now analyze the support of the new sum (the decomposition considered is unchanged).", "The case $b=1$ can be analyzed directly: there are ${2m \\atopwithdelims ()m}=\\Theta \\left(4^m/ \\sqrt{m}\\right)$ forms in the sum, all of them sharing the same first index (the all-one sequence $1=1\\cdots 1$ ).", "This sum is then $\\sum _{J}t_{1JK}$ , where for each $t_{1JK}$ the sequence $K$ is uniquely determined by $J$ .", "The support of the sum is thus isomorphic to $\\mathrm {supp_c}(\\langle 1,1,{2m \\atopwithdelims ()m} \\rangle )$ .", "To analyze the case $b<1$ , we will interpret the sum in the framework developed in Section , by letting $U$ be the set of triples $IJK$ satisfying the above four conditions.", "Indeed, all the requirements for $U$ are satisfied: we have $\\tau =2$ and $T_1&=&{2m \\atopwithdelims ()(1-b)m,(1-b)m,2mb}=\\Theta \\left(\\frac{1}{m}\\cdot \\left[\\frac{2}{(2b)^b(1-b)^{1-b}}\\right]^{2m}\\right)\\\\\\mathcal {N}_1&=&{2mb \\atopwithdelims ()mb}=\\Theta \\left(\\frac{1}{\\sqrt{m}}\\cdot \\left[2^b\\right]^{2m}\\right)\\\\T_2=T_3&=&{2m \\atopwithdelims ()m}=\\Theta \\left(\\frac{1}{\\sqrt{m}}\\cdot \\left[2\\right]^{2m}\\right)\\\\\\mathcal {N}_2=\\mathcal {N}_3&=&{m \\atopwithdelims ()m(1-b)}{m \\atopwithdelims ()m(1-b)}=\\Theta \\left(\\frac{1}{m}\\cdot \\left[\\frac{1}{b^b(1-b)^{1-b}}\\right]^{2m}\\right) .\\\\$ Note in particular that $N_1> N_2$ , since $(2b)^b(1-b)^{1-b}>1$ for any $b\\ge 0.773$ .", "Choose $U^\\ast =U$ (which means that $\\mathcal {N}_i=\\mathcal {N}_i^\\ast $ for each $i\\in \\lbrace 1,2,3\\rbrace $ ).", "The correspondence with the graph-theoretic interpretation of Section is as follows.", "Each vertex in the graph $G$ defined in Subsection REF corresponds to one form $T_{IJK}$ in the sum, which is isomorphic to $\\langle q^{2bm},q^{2bm},q^{2(1-b)m} \\rangle $ from the discussion above.", "One removal operation corresponds to zeroing variables.", "A 1-clique of length $n$ corresponds to a sum of $n$ forms sharing their first index, which is precisely a ${C}$ -tensor with support isomorphic to $\\mathrm {supp_c}(\\langle 1,1,n \\rangle )$ .", "For any value $b>0.916027$ we have $\\frac{2}{(2b)^{2b}(1-b)^{2(1-b)}}<1$ and thus $T_1\\mathcal {N}_2/\\mathcal {N}_1=o(1)$ .", "By Theorem REF , for any $\\epsilon >0$ we can then convert the sum into a direct sum of $\\Omega \\left(\\frac{T_1}{\\mathcal {N}_2^\\epsilon }\\right)=\\Omega \\left(\\frac{1}{m^{1-\\epsilon }}\\cdot \\left[\\frac{2}{2^b\\left(b^{b}(1-b)^{1-b}\\right)^{1-\\epsilon }}\\right]^{2m}\\right)$ ${C}$ -tensors, each tensor having support isomorphic to $\\mathrm {supp_c}(\\langle 1,1,n \\rangle )$ for $n=\\Omega \\left(\\frac{\\mathcal {N}_1}{\\mathcal {N}_2}\\right)=\\Omega \\left(\\sqrt{m}\\cdot \\left[(2b)^b(1-b)^{(1-b)}\\right]^{2m}\\right)$ and components isomorphic to $\\langle q^{2bm},q^{2bm},q^{2(1-b)m} \\rangle $ .", "Finally, note that $\\frac{1}{m^{1-\\epsilon }}\\cdot \\left[\\frac{2}{2^b\\left(b^{b}(1-b)^{1-b}\\right)^{1-\\epsilon }}\\right]^{2m}\\ge \\frac{1}{mc^{2\\epsilon m}}\\left[\\frac{2}{(2b)^{b}(1-b)^{1-b}}\\right]^{2m}$ for some constant $c\\ge 1$ depending only on $b$ , since $b^{b}(1-b)^{1-b}<1$ .", "The forms $T_{112}$ and $T_{121}$ can be analyzed in the same way as $T_{211}$ by permuting the roles of the $x$ -variables, the $y$ -variables and the $z$ -variables.", "Similarly to the statement of Proposition REF , the form $T_{112}^{\\otimes 2m}$ gives a direct sum of ${C}$ -tensors with support isomorphic to $\\langle 1,H,1 \\rangle $ , each component in the tensors being isomorphic to $\\langle q^{2bm},q^{2(1-b)m},q^{2bm} \\rangle $ .", "The form $T_{121}^{\\otimes 2m}$ gives a direct sum of ${C}$ -tensors with support isomorphic to $\\mathrm {supp_c}(\\langle H,1,1 \\rangle )$ , each component being isomorphic to $\\langle q^{2(1-b)m},q^{2bm},q^{2bm} \\rangle $ .", "Suppose that different constants are used to treat each of the three forms: the forms $T_{112}$ and $T_{121}$ are processed with some constant $b$ , while $T_{211}$ is processed with another constant $\\tilde{b}$ .", "For any fixed values $a_{112},a_{211}$ and any $\\epsilon >0$ , the form $T_{112}^{\\otimes a_{112}N}\\otimes T_{112}^{\\otimes a_{112}N}\\otimes T_{211}^{\\otimes a_{211}N}$ can then be used to construct a direct sum of $\\Omega \\left(\\frac{1}{N^3c^{\\prime N\\epsilon }}\\cdot \\left[\\frac{2}{(2b)^{b}(1-b)^{1-b}}\\right]^{2a_{112}N}\\times \\left[\\frac{2}{(2\\tilde{b})^{\\tilde{b}}(1-\\tilde{b})^{1-\\tilde{b}}}\\right]^{a_{211}N}\\right)$ ${C}$ -tensors, for some value $c^{\\prime }\\ge 1$ depending only on $b$ and $\\tilde{b}$ .", "Each of these ${C}$ -tensors has a support isomorphic to $\\mathrm {supp_c}(\\langle H_{112},H_{112},H_{211} \\rangle )$ , where $H_{112}&=&\\Omega \\left(\\frac{1}{\\sqrt{N}}\\cdot \\left[(2b)^b(1-b)^{(1-b)}\\right]^{a_{112}N}\\right)\\\\H_{211}&=&\\Omega \\left(\\frac{1}{\\sqrt{N}}\\cdot \\left[(2\\tilde{b})^{\\tilde{b}}(1-\\tilde{b})^{(1-\\tilde{b})}\\right]^{a_{211}N}\\right).$ In all these ${C}$ -tensors, each component is isomorphic to the rectangular matrix multiplication $\\langle q^{(a_{112}+ a_{211}\\tilde{b})N},q^{(a_{112}+ a_{211}\\tilde{b})N},q^{(2a_{112}b+ a_{211}(1-\\tilde{b}))N} \\rangle .$ We can then use Propositions REF and REF to convert each ${C}$ -tensor into a direct sum of at least $\\frac{3}{4}H_{112}\\times \\min (H_{112},H_{211})$ trilinear forms, each isomorphic to $\\langle q^{(a_{112}+ a_{211}\\tilde{b})N}\\!,q^{(a_{112}+ a_{211}\\tilde{b})N}\\!,q^{(2a_{112}b+ a_{211}(1-\\tilde{b}))N} \\rangle $ .", "We thus obtain the following result.", "Proposition 6.2 Let $a_{112}$ and $a_{211}$ be any two positive constants.", "Let $b$ and $\\tilde{b}$ be any two constants such that $0.916027<b,\\tilde{b}\\le 1$ .", "Then there exists a constant $c^{\\prime }\\ge 1$ such that, for any $\\epsilon >0$ , the form $T_{112}^{\\otimes a_{112}N}\\otimes T_{121}^{\\otimes a_{112}N}\\otimes T_{211}^{\\otimes a_{211}N}$ can be converted into a direct sum of $\\Omega \\left(\\frac{1}{N^4c^{\\prime N\\epsilon }}\\cdot \\left[\\frac{2^{2a_{112}+a_{211}}}{\\max \\left(\\left[(2b)^b(1-b)^{1-b}\\right]^{a_{112}},\\left[(2\\tilde{b})^{\\tilde{b}}(1-\\tilde{b})^{1-\\tilde{b}}\\right]^{a_{211}}\\right)}\\right]^N\\right)$ forms, each form being isomorphic to $\\langle q^{(a_{112}+ a_{211}\\tilde{b})N},q^{(a_{112}+ a_{211}\\tilde{b})N},q^{(2a_{112}b+a_{211}(1-\\tilde{b}) )N} \\rangle .$" ], [ "Main theorem", "Let us define the following three quantities.", "$Q&=&(2q)^{a_{103}+a_{301}}\\times (q^2+2)^{a_{202}}\\times q^{a_{112}+a_{211}\\tilde{b}}\\\\R&=&(2q)^{2a_{013}}\\times (q^2+2)^{a_{022}}\\times q^{2a_{112}b+(1-\\tilde{b})a_{211}}\\\\\\mathcal {M}&=&\\frac{2^{2a_{112}+a_{211}}}{A_0^{A_0}A_1^{A_1}A_2^{A_2}A_3^{A_3}A_4^{A_4}}\\times \\frac{1}{\\max \\left(\\left[(2b)^b(1-b)^{1-b}\\right]^{a_{112}},\\left[(2\\tilde{b})^{\\tilde{b}}(1-\\tilde{b})^{1-\\tilde{b}}\\right]^{a_{211}}\\right)}.\\\\$ Our main theorem gives an upper bound on $\\omega (1,1,k)$ that depends on these quantities.", "Theorem 6.1 Let $q$ be any positive integer and $b, \\tilde{b}$ be such that $0.916027<b,\\tilde{b}\\le 1$ .", "Let $a_{004}$ , $a_{400}$ , $a_{013}$ , $a_{103}$ , $a_{301}$ , $a_{022}$ , $a_{202}$ , $a_{112}$ and $a_{211}$ be any nine positive rational numbers satisfying the following three conditions: $2a_{004}+a_{400}+2a_{013}+2a_{103}+2a_{301}+a_{022}+2a_{202}+2a_{112}+a_{211}=1$ ; $a_{013}a_{202}a_{112}=a_{103}a_{022}a_{211}$ ; $A_0^{A_0}A_1^{A_1}A_2^{A_2}A_3^{A_3}A_4^{A_4}\\ge B_0^{B_0}B_1^{B_1}B_2^{B_2}B_3^{B_3}B_4^{B_4}$ .", "Then $\\mathcal {M}Q^{w(1,1,\\frac{\\log R}{\\log Q})}\\le (q+2)^2.$ Let $\\epsilon >0$ be an arbitrary positive value.", "Let $N$ be a large integer and consider the trilinear form $(F_q\\otimes F_q)^{\\otimes N}$ .", "Theorem REF shows that this form can be used to obtain a direct sum of $r_1=\\Omega \\left(\\frac{1}{N^{10+8\\epsilon }15^{N\\epsilon }}\\left[\\frac{1}{A_0^{A_0}A_1^{A_1}A_2^{A_2}A_3^{A_3}A_4^{A_4}}\\right]^{N}\\right)$ forms, each isomorphic to $\\bigotimes _{i,j,k:\\: i+j+k=4}T_{ijk}^{\\otimes a_{ijk}N}.$ All the terms $T_{ijk}$ in this form, except $T_{112}$ , $T_{121}$ and $T_{211}$ , correspond to matrix multiplications and have been analyzed in Section .", "By Proposition REF the part $T_{112}^{\\otimes a_{121}N}\\otimes T_{112}^{\\otimes a_{112}N}\\otimes T_{211}^{\\otimes a_{211}N}$ can be used to obtain a direct sum of $r_2=\\Omega \\left(\\frac{1}{N^4c^{\\prime N\\epsilon }}\\cdot \\left[\\frac{2^{2a_{112}+a_{211}}}{\\max \\left(\\left[(2b)^b(1-b)^{1-b}\\right]^{a_{112}},\\left[(2\\tilde{b})^{\\tilde{b}}(1-\\tilde{b})^{1-\\tilde{b}}\\right]^{a_{211}}\\right)}\\right]^N\\right)$ matrix multiplications $\\langle q^{(a_{112}+ a_{211}\\tilde{b})N},q^{(a_{112}+ a_{211}\\tilde{b})N},q^{(2a_{112}b+(1-\\tilde{b}) a_{211})N} \\rangle $ .", "This means that the trilinear form $(F_q\\otimes F_q)^{\\otimes N}$ can be converted into a direct sum of $r_1r_2$ matrix multiplications $\\langle Q^N,Q^N,R^N \\rangle $ .", "In other words: $r_1r_2 \\cdot \\langle Q^N,Q^N,R^N \\rangle \\unlhd (F_q\\otimes F_q)^{\\otimes N}.$ Since $\\underline{R}\\left(F_q\\otimes F_q\\right)\\le (q+2)^{2}$ , as mentioned in Section , we know that $\\underline{R}\\left((F_q\\otimes F_q)^{\\otimes N}\\right)\\le (q+2)^{2N}$ .", "By Schönhage's asymptotic sum inequality (Theorem REF ) we then conclude that $r_1r_2 \\times Q^{N\\omega (1,1,\\frac{\\log R}{\\log Q})}\\le (q+2)^{2N}.$ Taking the $N$ -th root, we obtain: $\\frac{1}{(15c^{\\prime })^\\epsilon N^{(14+8\\epsilon )/N}}\\times \\mathcal {M} Q^{\\omega (1,1,\\frac{\\log R}{\\log Q})}&\\le & (q+2)^{2}.$ For any $\\epsilon >0$ the above inequality holds for large enough integers $N$ .", "By letting $N$ grow to infinity, and then letting $\\epsilon $ decrease to zero, we conclude that $\\mathcal {M} Q^{\\omega (1,1,\\frac{\\log R}{\\log Q})}\\le (q+2)^{2}.$" ], [ "Optimization", "In this section we use Theorem REF to derive numerical upper bounds on the exponent of rectangular matrix multiplication, and prove Theorem REF ." ], [ "Square matrix multiplication", "In this subsection we briefly show that our results give, for the exponent of square matrix multiplication, the same upper bound as the bound obtained by Coppersmith and Winograd [10] .", "Due to the symmetry of square matrix multiplication, we take $b=\\tilde{b}$ , $a_{400}=a_{004}$ , $a_{103}=a_{013}=a_{301}$ , $a_{022}=a_{202}$ and $a_{112}=a_{211}$ .", "Then only six parameters remain, and the conditions $a_{013}a_{202}a_{112}=a_{103}a_{022}a_{211}$ and $A_0^{A_0}A_1^{A_1}A_2^{A_2}A_3^{A_3}A_4^{A_4}= B_0^{B_0}B_1^{B_1}B_2^{B_2}B_3^{B_3}B_4^{B_4}$ are immediately satisfied.", "Theorem REF shows that $\\frac{8^{a_{112}}\\times \\left[(2q)^{2a_{013}}\\times (q^2+2)^{a_{202}}\\times q^{(1+b)a_{112}}\\right]^{\\omega (1,1,1)}}{A_0^{A_0}A_1^{A_1}A_2^{A_2}A_3^{A_3}A_4^{A_4}\\times \\left[(2b)^b(1-b)^{(1-b)}\\right]^{a_{112}}}\\le (q+2)^2.$ By choosing $q=6$ , $b=0.9724317$ , $a_{103}=0.012506$ , $a_{202}=0.102546$ , $a_{112}=0.205542$ and $a_{004}=0.0007/3$ , we obtain the upper bound $\\omega (1,1,1)<2.375477$ .", "This upper bound on the exponent of square matrix multiplication is exactly the same value as in [10].", "This is not a coincidence.", "Indeed, by setting $b=\\frac{q^{\\omega (1,1,1)}}{q^{\\omega (1,1,1)}+2}$ , which is larger than 0.916027 for $q\\ge 5$ , we obtain $\\frac{q^{b\\omega (1,1,1)}}{(2b)^b(1-b)^{(1-b)}}=\\frac{q^{\\omega (1,1,1)}+2}{2}$ and our inequality becomes $\\frac{(2q)^{2a_{013}\\omega (1,1,1)}\\times (q^2+2)^{a_{202}\\omega (1,1,1)}\\times (4q^{\\omega (1,1,1)}(q^{\\omega (1,1,1)}+2))^{a_{112}}}{A_0^{A_0}A_1^{A_1}A_2^{A_2}A_3^{A_3}A_4^{A_4}}\\le (q+2)^2,$ which is exactly the same optimization problem as in Section 8 of [10]." ], [ "Rectangular matrix multiplication", "In this subsection we explain how to use Theorem REF to derive an upper bound on $\\omega (1,1,k)$ for an arbitrary value $k$ , and show how to obtain the results stated in Table REF and Figure REF .", "We use the following strategy.", "We take a positive integer $q$ , seven positive rational numbers $a_{400}$ , $a_{103}$ , $a_{301}$ , $a_{022}$ , $a_{202}$ , $a_{112}$ and $a_{211}$ , and two values $b,\\tilde{b}$ such that $0.916027<b,\\tilde{b}\\le 1$ .", "We then fix $a_{013}=\\frac{a_{103}a_{022}a_{211}}{a_{202}a_{112}}$ and $a_{004}=\\frac{1-(a_{400}+2a_{013}+2a_{103}+2a_{301}+a_{022}+2a_{202}+2a_{112}+a_{211})}{2}.$ The conditions that have to be satisfied are: $0<a_{004},a_{013}\\le 1$ ; $A_0^{A_0}A_1^{A_1}A_2^{A_2}A_3^{A_3}A_4^{A_4}\\ge B_0^{B_0}B_1^{B_1}B_2^{B_2}B_3^{B_3}B_4^{B_4}$ .", "If these conditions are satisfied, by Theorem REF this gives the upper bound $\\omega \\left(1,1,\\frac{\\log R}{\\log Q}\\right)\\le \\frac{2\\log (q+2)-\\log \\mathcal {M}}{\\log Q}.$ The above discussion reduces the problem of finding an upper bound on $\\omega (1,1,k)$ to solving a nonlinear optimization problem.", "The upper bounds presented in Table REF are obtained precisely by solving this optimization problem using Maple.", "We show exact values of the parameters proving that $\\omega (1,1,0.5302)<2.060396$ , $\\omega (1,1,0.75)< 2.190087$ and $\\omega (1,1,2)< 3.256689$ in Table REF .", "Table: Three solutions for our optimization problem.The first ten rows give (exact) values of the ten parameters.The numeral values of the next four rows show that the three conditions0<a 004 ,a 013 ≤10<a_{004},a_{013}\\le 1 and A 0 A 0 A 1 A 1 A 2 A 2 A 3 A 3 A 4 A 4 ≥B 0 B 0 B 1 B 1 B 2 B 2 B 3 B 3 B 4 B 4 A_0^{A_0}A_1^{A_1}A_2^{A_2}A_3^{A_3}A_4^{A_4}\\ge B_0^{B_0}B_1^{B_1}B_2^{B_2}B_3^{B_3}B_4^{B_4} are satisfied.The numerical values of the last two rows show that ω(1,1,0.5302)<2.060396\\omega (1,1,0.5302)<2.060396,ω(1,1,0.75)<2.190087\\omega (1,1,0.75)< 2.190087 and ω(1,1,2)<3.256689\\omega (1,1,2)< 3.256689." ], [ "The value $\\alpha $", "In this subsection we describe how to use Theorem REF to obtain a lower bound on the value $\\alpha $ , the largest value such that the product of an $n\\times n^\\alpha $ matrix by an $n^\\alpha \\times n$ matrix can be computed with $O(n^{2+\\epsilon })$ arithmetic operations for any $\\epsilon >0$ .", "The analysis is more delicate than in the previous subsection, since we will need to exhibit parameters such that $\\mathcal {M} Q^2=(q+2)^2$ , with an equality rather than an inequality, and is done by finding analytically the optimal values of all but a few parameters.", "Let $q$ be an integer such that $q\\ge 5$ .", "For convenience, we will write $\\kappa =1/(q+2)^2$ .", "Let $a_{112}$ and $a_{211}$ be any rational numbers such that $0<a_{112}< q\\kappa $ and $0< a_{211}< (q^2+2)\\kappa $ .", "We set the parameters $b$ , $\\tilde{b}$ , $a_{004}$ , $a_{103}$ , $a_{202}$ and $a_{301}$ as follows: $b&=&1\\\\\\tilde{b}&=& q^2/(q^2+2)\\\\a_{400}&=&\\kappa \\\\a_{103}&=&q\\kappa -a_{112}\\\\a_{202}&=&\\left((q^2+2)\\kappa -a_{211}\\right)/2\\\\a_{301}&=&q\\kappa .$ Putting these values in the formula for $Q$ , we obtain: $Q&=&\\left(2q\\right)^{q\\kappa +a_{103}}\\times \\left(q^2+2\\right)^{\\frac{(q^2+2)\\kappa -a_{211}}{2}}\\times q^{a_{112}+q^2a_{211}/(q^2+2)}\\\\&=&(2q)^{2q\\kappa }\\times (q^2+2)^\\frac{(q^2+2)\\kappa }{2}\\times 2^{-a_{112}}\\times \\left(\\frac{q^{q^2/(q^2+2)}}{\\sqrt{q^2+2}}\\right)^{a_{211}}.$ Observe that $A_1=A_3=2q\\kappa $ , $A_2=(q^2+2)\\kappa $ , $A_4=\\kappa $ and $A_0=1-(A_1+A_2+A_3+A_4)=\\kappa $ .", "Then we obtain the following equality.", "$\\frac{1}{A_0^{A_0}A_1^{A_1}A_2^{A_2}A_3^{A_3}A_4^{A_4}} &=&\\frac{(q+2)^2}{(2q)^{4q\\kappa }(q^2+2)^{(q^2+2)\\kappa }}$ The following lemma shows that, when $a_{112}$ is small enough, the condition $\\mathcal {M} Q^2=(q+2)^2$ is satisfied.", "Lemma 7.1 Suppose that $a_{112}\\le \\left(1+\\frac{2q^2}{q^2+2}\\log _2(q)-\\log _2(q^2+2)\\right)a_{211}.$ Then $\\mathcal {M} Q^2=(q+2)^2$ .", "Our choice for $b$ and $\\tilde{b}$ gives $\\left[(2b)^b(1-b)^{1-b}\\right]^{a_{112}}&=&2^{a_{112}}\\\\\\left[(2\\tilde{b})^{\\tilde{b}}(1-\\tilde{b})^{1-\\tilde{b}}\\right]^{a_{211}}&=&\\left[\\frac{2}{q^2+2}\\cdot q^{\\frac{2q^2}{q^2+2}}\\right]^{a_{211}}.$ Inequality (REF ) then implies that $\\left[(2b)^b(1-b)^{1-b}\\right]^{a_{112}}\\le \\left[(2\\tilde{b})^{\\tilde{b}}(1-\\tilde{b})^{1-\\tilde{b}}\\right]^{a_{211}}$ .", "In consequence, $\\mathcal {M}=\\frac{(q+2)^2}{(2q)^{4q\\kappa }(q^2+2)^{(q^2+2)\\kappa }}\\times 4^{a_{112}}\\times \\left[\\frac{q^2+2}{q^{2q^2/(q^2+2)}}\\right]^{a_{211}},$ which gives $\\mathcal {M} Q^2=(q+2)^2$ .", "We now explain how to determine the three remaining parameters $a_{004}$ , $a_{013}$ and $a_{022}$ .", "Remember that the parameters should satisfy the equalities $a_{013}=\\frac{a_{103}a_{211}}{a_{202}a_{112}}a_{022}$ and $2a_{004}+a_{400}+2a_{013}+2a_{103}+2a_{301}+a_{022}+2a_{202}+2a_{112}+a_{211}=1.$ From our choice of parameters, the second equality can be rewritten as $2a_{004}+2a_{013}+a_{022}=\\kappa $ .", "Since the parameter $a_{004}$ should be positive, we obtain the condition $\\left(\\frac{4(q\\kappa -a_{112})a_{211}}{((q^2+2)\\kappa -a_{211})a_{112}}+1\\right)a_{022}< \\kappa .$ If $a_{022}$ , $a_{112}$ and $a_{211}$ satisfy this inequality, then the parameter $a_{004}$ is fixed: $a_{004}=\\left(\\kappa -\\left(\\frac{4(q\\kappa -a_{112})a_{211}}{((q^2+2)\\kappa -a_{211})a_{112}}+1\\right)a_{022}\\right)/2.$ Note that Inequality (REF ) forces the value $a_{013}$ to be at most 1.", "All the values are thus determined by the choice of $q$ , $a_{022}$ , $a_{112}$ and $a_{211}$ .", "In particular, we obtain $R&=&\\left(2q\\right)^{\\frac{4(q\\kappa -a_{112})a_{211}}{((q^2+2)\\kappa -a_{211})a_{112}}}\\times \\left(q^2+2\\right)^{a_{022}}\\times q^{2a_{112}+\\frac{2}{q^2+2}a_{211}}.$ We can similarly express the values of $B_0$ , $B_1$ , $B_2$ , $B_3$ and $B_4$ in function of these four parameters.", "We then want to solve the following optimization problem.", "Table: NO_CAPTIONBy taking the values $q=5$ , $a_{022}=0.0174853$ , $a_{112}=0.0945442$ and $a_{211}=0.1773724$ , we obtain the value $\\alpha \\ge \\frac{\\log R}{\\log Q}>0.30298$ .", "These parameters satisfy all the constraints.", "We obtain in particular the following numerical values.", "$\\frac{(2q)^{4q\\kappa }(q^2+2)^{(q^2+2)\\kappa }}{(q+2)^2}&=&0.3211277...\\\\B_0^{B_0}B_1^{B_1}B_2^{B_2}B_3^{B_3}B_4^{B_4}&=&0.3211276....\\\\R&=&1.475744...\\\\Q&=&3.612672...$ A more precise lower bound on $\\alpha $ can be found using optimization software and high precision arithmetic.", "Using Maple and truncating the result of the optimization after the 25th digit, we find that for $q=5$ the values $a022 &=& 0.0174853267797595451457284\\\\a112 &=& 0.0945442542111395375830367\\\\a211 &=& 0.1773724081899825630904504$ give the lower bound $\\alpha > 0.3029805825293869820274449.$" ], [ "Acknowledgments", "The author is grateful to Virginia Vassilevska Williams and Ryan Williams for helpful correspondence about Andrew Stothers' work, and to Virginia Vassilevska Williams for suggesting that the next step is to use higher tensor powers of the basic construction to improve rectangular matrix multiplication.", "He also acknowledges support from the JSPS and the MEXT, under the grant-in-aids Nos.", "22800006 and 24700005." ] ]

[ [ "Ab initio molecular dynamics study of dissociation of water under an\n electric field" ], [ "Abstract The behavior of liquid water under an electric field is a crucial phenomenon in science and engineering.", "However, its detailed description at a microscopic level is difficult to achieve experimentally.", "Here we report on the first ab initio molecular-dynamics study on water under an electric field.", "We observe that the hydrogen-bond length and the molecular orientation are significantly modified at low-to-moderate field intensities.", "Fields beyond a threshold of about 0.35 V/\\AA are able to dissociate molecules and sustain an ionic current via a series of correlated proton jumps.", "Upon applying even more intense fields (1.0 V/\\AA), a 15-20% fraction of molecules are instantaneously dissociated and the resulting ionic flow yields a conductance of about 7.8 $\\Omega^{-1}cm^{-1}$, in good agreement with experimental values.", "This result paves the way to quantum-accurate microscopic studies of the effect of electric fields on aqueous solutions and, thus, to massive applications of ab initio molecular dynamics in neurobiology, electrochemistry and hydrogen economy." ], [ "Acknowledgements", "We acknowledge the IDRIS Supercomputing Facility for CPU time (CP9-101387)." ] ]

[ [ "Spin waves and Collisional Frequency Shifts of a Trapped-Atom Clock" ], [ "Abstract We excite spin-waves with spatially inhomogeneous pulses and study the resulting frequency shifts of a chip-scale atomic clock of trapped $^{87}$Rb.", "The density-dependent frequency shifts of the hyperfine transition simulate the s-wave collisional frequency shifts of fermions, including those of optical lattice clocks.", "As the spin polarizations oscillate in the trap, the frequency shift reverses and it depends on the area of the second Ramsey pulse, exhibiting a predicted beyond mean-field frequency shift.", "Numerical and analytic models illustrate the observed behaviors." ], [ "colorlinks=true, breaklinks=true, urlcolor= blue, linkcolor= red, bookmarksopen=false, Spin waves and Collisional Frequency Shifts of a Trapped-Atom Clock Wilfried Maineult LNE-SYRTE, Observatoire de Paris, CNRS, UPMC, 61 av de l'Observatoire, 75014 Paris, France Christian Deutsch Laboratoire Kastler Brossel, ENS, UPMC, CNRS, 24 rue Lhomond, 75005 Paris, France Kurt Gibble Department of Physics, The Pennsylvania State University, Pennsylvania 16802, USA Jakob Reichel Laboratoire Kastler Brossel, ENS, UPMC, CNRS, 24 rue Lhomond, 75005 Paris, France Peter [email protected] LNE-SYRTE, Observatoire de Paris, CNRS, UPMC, 61 av de l'Observatoire, 75014 Paris, France We excite spin-waves with spatially inhomogeneous pulses and study the resulting frequency shifts of a chip-scale atomic clock of trapped $^{87}$ Rb.", "The density-dependent frequency shifts of the hyperfine transition simulate the s-wave collisional frequency shifts of fermions, including those of optical lattice clocks.", "As the spin polarizations oscillate in the trap, the frequency shift reverses and it depends on the area of the second Ramsey pulse, exhibiting a predicted beyond mean-field frequency shift.", "Numerical and analytic models illustrate these observed behaviors.", "Quantum scattering in an ultracold gas of indistinguishable spin-1/2 atoms leads to rich and unexpected behaviors, even above the onset of quantum degeneracy.", "Among these, spin waves are a beautiful macroscopic manifestation of identical spin rotation (ISR) [1], [2], [3], [4].", "ISR also inhibits dephasing, which can dramatically increase the coherence time in a trapped ensemble of interacting atoms to tens of seconds [5], [6], with possible applications to compact atomic clocks and quantum memories.", "Another example is collisional interactions in optical lattice clocks [7], [8], [9], [10].", "Their detailed understanding is a prerequisite for optical lattice clocks to realize their full potential as future primary standards.", "At ultracold temperatures, scattering is purely s-wave, which is forbidden for indistinguishable fermions, suggesting that clocks using ultracold fermions are immune to collision shifts [11], [12].", "However, spatial inhomogeneities of the clock field, which are naturally larger for optical frequency fields than for radio-frequencies, allow fermions to become distinguishable and therefore can lead to s-wave clock shifts [8], [9], [10].", "A series of experiments attributed the collision shifts of Sr lattice clocks to these novel s-wave fermion collisions with inhomogeneous clock field excitations [7], [13], [14].", "However, subsequent work showed that p-waves dominate for Yb lattice clocks, and p-wave scattering is consistent with all the observed Sr collisional frequency shifts [15].", "Bosons with state-independent scattering lengths have fermion-like exchange interactions [16], [8], [17].", "This allows us to simulate the s-wave fermion collisional shift with a chip-scale clock that traps $^{87}$ Rb, a boson with nearly equal scattering lengths.", "We observe the distinguishing feature that the collisional shift in the presence of inhomogenous excitations depends on the area of the second Ramsey clock pulse.", "This dependence sets it apart from the well-known s-wave shift for homogeneous excitations, which is absent for fermions and, for bosons, depends only on the first pulse area, and hence the population dif- ference of the two clocks states [18], [8].", "Further, inhomogeneous excitations directly excite spin waves.", "We show an inextricable link between spin-waves and the s-wave fermion collisional shifts.", "Notably, we observe frequency shifts that change sign as spin polarizations oscillate in the trap.", "We perform Ramsey spectroscopy with two spatially inhomogeneous pulses to study spin waves and the collisional frequency shifts of trapped $^{87}$ Rb atoms.", "The first clock pulse creates an inhomogeneous spin polarization, which varies linearly in space.", "We directly observe a spatiotemporal oscillation of this spin polarization, which characterizes the strength of the atomic interactions.", "Driving a second Ramsey pulse, we measure frequency shifts of this clock.", "Here we vary the areas of each pulse and the interrogation time between the two pulses, to probe the unique behaviors of s-wave fermion clock shifts.", "We develop analytic and numerical models that describe the observed spin-waves and the novel dependence on the area of the second Ramsey pulse.", "Figure: Calculated microwave field amplitude of the coplanar waveguide on our atom chip, in arbitrary units.", "The rapidly decaying near-field causes a small vertical gradient of the spin-polarization across the trapped atom cloud.Our chip-scale atomic clock magnetically traps between $10^3$ and $10^5$ atoms at a distance $z_0 = 156\\,\\mu $ m below a microwave coplanar waveguide on our atom chip [19], [5].", "A microwave and radio-frequency, two-photon excitation drives the clock transition, $\\mathinner {|{\\downarrow }\\rangle }\\equiv \\mathinner {|{F=1,m_F=-1}\\rangle }$ to $\\mathinner {|{\\uparrow }\\rangle }\\equiv \\mathinner {|{F=2,m_F=1}\\rangle }$ .", "The near field of the microwave guide (Fig.", "REF ) creates a slightly inhomogeneous Rabi frequency in the vertical $z$ direction, $\\Omega (\\mathbf {r}) = \\Omega _0(1 + \\delta _1 z + \\delta _2 z^2+ \\ldots )$ .", "The trap frequencies are $(\\omega _x,\\omega _y,\\omega _z) =2\\pi (32(1),97(1),119.5(5))$ Hz.", "The temperature of the cloud is 175(6) nK, at least 30 nK above the onset of Bose Einstein condensation, with no measurable dependence on the atom number.", "At this temperature, the identical spin rotation rate $\\omega _{\\textit {ex}}$ , of order $ \\omega _{MF}\\equiv 4\\pi \\hbar |a_{\\uparrow \\downarrow }| \\bar{n}/m $ , dominates in our experiments.", "The lateral collision rate $\\gamma _{c}\\propto a_{\\uparrow \\downarrow }^2\\bar{n}v_T$ is always much lower than the trap frequencies, corresponding to the Knudsen regime.", "Here $a_{\\uparrow \\downarrow }$ is the inter-state scattering length, $\\bar{n}$ the mean density, $m$ the atomic mass and $v_T$ the thermal velocity.", "The magnetic field at the trap center is tuned to minimize the inhomogeneous spread of transition frequency [20] so that dephasing can be neglected on the timescales we consider The inhomogeneous spread of transition frequencies due to the magnetic potential and the spatially varying atom density, is less than 80 mHz..", "The variation of the Rabi frequency across the atom cloud is determined by fitting the resonant Rabi flopping using $\\Omega _0\\gg \\omega _z$ so that atomic motion during the pulse can be neglected (Fig.", "REF (a)).", "We find $\\delta _1\\approx 0.1~\\mathrm {\\xi _z}^{-1}$ and $\\delta _2\\approx 0$ which is reasonable since the r.m.s.", "cloud radius $\\xi _z= 4.1\\mathrm {\\mu m}\\ll z_0 $ .", "We checked experimentally that the inhomogeneity is predominantly vertical.", "Along $y$ there is no detectable variation of $\\Omega (\\mathbf {r})$ and its variation along $x$ is 5 times smaller than along $z$.", "Figure: (a) Inhomogeneous Rabi flopping with Ω(𝐫)=Ω 0 (1+δ 1 z)\\Omega (\\mathbf {r}) = \\Omega _0(1 + \\delta _1 z).", "The data are fitted to models of two trapped atoms: (red) an analytic model with a single motional sideband gives δ 1 =0.147(4)ξ z -1 \\delta _1 = 0.147(4)~\\xi _z^{-1}.", "(blue) numerical simulation for representative vibrational states η z \\eta _z = 30±830\\pm 8 yields δ 1 =0.090(4)ξ z -1 \\delta _1 = 0.090(4)~\\xi _z^{-1}.", "(b) Bloch sphere evolution of the spins of two representative atoms (blue and red), which are initially on opposite sides of the trap in state |↓〉\\mathinner {|{\\downarrow }\\rangle }.", "The first inhomogeneous Ramsey excitation pulse rotates the spins differently (θ red,1 =4π/5\\theta _{red,1}=4\\pi /5 and θ blue,1 =π/5\\theta _{blue,1}=\\pi /5).", "During the Ramsey interrogation time, the scattering produces an ISR rotation of the two spins around their sum (black arrow), here, by ω ex T R =π/2\\omega _{ex}T_R=\\pi /2.", "We take a second pulse with the same inhomogeneity as the first, but weaker, θ 2 ¯=π/8\\overline{\\theta _2}=\\pi /8.", "It barely moves the blue spin, whereas the red rotates to be more vertical.", "We detect the vertical projection of each spin S z S_z, corresponding to the points in (c).", "In (c) we trace S z S_z as a function of detuning, showing the resonance shifted to a negative detuning.", "For ω z T R =(2j+1)π\\omega _zT_R=(2j+1)\\pi , both atoms have switched sides of the trap so that the inhomogeneity of the 2nd pulse instead gives the blue spin a larger rotation, hence higher Ramsey fringe contrast.", "Thus the frequency shift changes sign as the spins oscillate in the trap.We initiate a spin-wave with a single $\\tau =1.05$  ms excitation pulse of area $\\Omega _0\\tau =2.5\\pi $ .", "We use a multiple of $\\pi /2$ to produce a larger spin inhomogeneity.", "This inhomogeneous spin population then oscillates in the trap and we observe the oscillation by holding the atoms in the trap for various times $t_h$ after the Rabi pulse, followed by 7 ms of time-of-flight and state-selective absorption imaging.", "Fig REF (a) shows the center of mass of the $\\mathinner {|{\\uparrow }\\rangle }$ component.", "The data for our lowest atom density exhibits a simple oscillation at $\\omega _z$ , and the $\\mathinner {|{\\downarrow }\\rangle }$ cloud (not shown) oscillates out of phase.", "The center of mass of the total population shows no measurable oscillation.", "Increasing the density, we observe a collapse and revival of the oscillation at shorter and shorter times.", "At $t_h=80$  ms, the oscillation for the highest density is out of phase with the lowest.", "As we show below, this is a signature of a spin wave driven by ISR.", "Figure: (a) Center of mass position of the |↑〉\\mathinner {|{\\uparrow }\\rangle } component of the cold atom cloud versus time after a single 5π/25\\pi /2 pulse for various atomic densities (in 10 12 10^{12} atoms/cm 3 ^{3}).", "Successive curves are offset vertically by 0.5ξ z 0.5\\xi _z.", "As the density increases, a beat appears between the trap frequency and the increasingly faster identical spin rotation rate ω 𝑒𝑥 \\omega _{\\textit {ex}}, characteristic for spin waves.", "(b) Fitting ω 𝑒𝑥 \\omega _{\\textit {ex}} versus density gives 2π1.4Hz/(10 12 atoms/cm 3 )×n ¯.2\\pi \\,1.4\\,\\text{Hz}/(10^{12} \\text{atoms}/\\text{cm}^3)\\times \\bar{n}.We can intuitively illustrate the spin dynamics by considering two localized atoms oscillating in a one-dimensional trap with frequency $\\omega _z$ (Fig.", "REF (b)).", "With the atoms initially on opposite sides of the trap center in state $\\mathinner {|{\\downarrow }\\rangle }$ , they are excited with a short Rabi pulse of mean area, typically $\\theta _1=\\Omega _0\\tau =\\pi /2$ .", "Since $\\Omega =\\Omega (z)$ , the atoms experience different Bloch vector rotations $\\theta _1\\pm \\Delta \\theta _1$ .", "After the pulse, the atoms oscillate in the trap during the Ramsey interrogation time $T_R$ as in Fig.", "REF (a).", "In the absence of interactions, each atom maintains its spin orientation, in a frame that rotates at the atomic transition frequency, and thus the spatial spin populations simply oscillate at $\\omega _z$ .", "Note that the phase of each atom's coherence is constant.", "However, if there are exchange interactions, the two spins will rotate with an ISR rate $\\omega _{\\textit {ex}}$ around their total spin as they repeatedly collide (Fig.", "REF (b)) [16], [5].", "In a time $\\pi /\\omega _{\\textit {ex}}$ , the two atoms exchange their spin polarizations, producing a beat between $\\omega _{\\textit {ex}}$ and $\\omega _z$ .", "This introduces the frequency $\\omega _{\\textit {ex}}$ into the spatial oscillation of the spins, producing a beat between $\\omega _{\\textit {ex}}$ and $\\omega _z$ .", "For each curve, we extract $\\omega _{\\textit {ex}}$ , which varies linearly with density (Fig.", "REF (b)), as $2\\pi \\,1.4\\,\\text{Hz}/(10^{12} \\text{at}/\\text{cm}^3)\\times \\bar{n}$ , within our uncertainty.", "This coefficient is five times smaller than $\\omega _{MF}$ [1].", "Small amplitude dipolar spin waves oscillate at $\\omega _{MF}/2$ [28] and models show that this frequency can decrease by a factor 2 for large amplitude spin waves.", "Additionally, we experimentally observe $\\omega _{\\textit {ex}}$ decreases by a factor of two as $\\theta _1=2.2 \\pi $ increases to $\\theta _1=2.8 \\pi $ The spatiotemporal spin oscillation has important consequences for Ramsey spectroscopy.", "The second Ramsey pulse reads-out the phase of the atomic coherences.", "Between the pulses, the exchange interaction modulates the phase of each atom's coherence as the spins rotate about one another (Fig.", "REF (b)).", "If we were to measure the transition probability of one of the atoms above [8], the apparent resonance frequency would depend on $T_R$ - it would be modulated at $\\omega _{\\textit {ex}}$ .", "In the usual case when both atoms are detected, the frequency excursion of this modulated collision shift is reduced (Fig.", "REF (c)).", "The shift averages to zero if the second Ramsey pulse is homogeneous.", "For an inhomogeneous second pulse, the two Bloch vectors experience different rotations $\\theta _2\\pm \\Delta \\theta _2=\\Omega (z_i(T_R))\\tau $ depending on their positions at the time of the pulse.", "This gives them different weights in the Ramsey measurement, making the cancellation incomplete, unless the second pulse is an odd multiple of $\\pi /2$ , which reads out the phases of both atoms with the same sensitivity [8].", "This simple model predicts a clock shift $\\delta \\nu = \\frac{\\Delta \\theta _1 \\Delta \\theta _2 \\sin \\left(\\omega _{\\textit {ex}}T_R\\right)\\cos \\left(\\omega _z T_R\\right) \\cos \\theta _2}{4\\pi T_R\\sin \\theta _1\\sin \\theta _2}.", "$ It extends the results in [8] to $\\omega _{\\textit {ex}}T_R \\ge 1$ and unresolved sidebands.", "Here we linearize the dependence on $\\Delta \\theta _i$ .", "A singlet-triplet basis provides helpful insight and also leads to Eq.", "(REF ).", "Before the first pulse the two atoms are in the triplet state $\\mathinner {|{S,m_s}\\rangle }=\\mathinner {|{1,-1}\\rangle }$ .", "The inhomogeneous excitation pulse makes them partially distinguishable and populates the singlet state $\\mathinner {|{0,0}\\rangle }$ Only the singlet state for fermions can have s-wave collisions.", "For bosons with equal inter and intra-state scattering lengths, the triplet states all have the same energy shift and the singlet state has none.", "Therefore, taking out the common shift of the triplet states, such bosons have the same interactions as fermions, albeit with the opposite $\\omega _{\\textit {ex}}$ [16], [8], [17]., which then accrues a collisional phase shift during the Ramsey interrogation time To derive Eq.", "REF in the singlet-triplet basis, we consider short pulses, $\\Omega _0\\gg \\omega _z,\\omega _{\\textit {ex}}$ , neglecting interactions during the pulses, and small inhomogeneities, $\\delta \\theta \\ll 1$ , exciting only a single trap sideband.", "We add the contributions for the upper and lower sidebands..", "When $\\Omega _0\\approx \\omega _z,\\omega _{\\textit {ex}}$ , we numerically calculate the evolution of the $S=1$ pseudo-spin system, coherently including all transitions up to the 5th sideband.", "We also treat the 5% scattering length difference $a_{\\uparrow \\uparrow }\\lesssim a_{\\uparrow \\downarrow }\\lesssim a_{\\downarrow \\downarrow }$ .", "To experimentally test eq.", "(REF ), we measure the shift of the clock's frequency with a Ramsey sequence for the same range of densities as in Fig.", "REF .", "Fig.", "REF (a) shows the measured shift as a function of density for two $\\tau _{1,2}=1.05$  ms pulses separated by a $T_R=100$  ms interrogation time, which is close to a multiple of the trap period.", "The first pulse area is $\\theta _1=5\\pi /2$ and the second is $\\theta _2=2.2 \\pi $ .", "Like above, a large pulse area is used to increase the inhomogeneity.", "The observed frequency shift indeed oscillates as a function of density, giving a frequency shift that is inconsistent with the often-used mean-field shift [18], [12], [7], [9].", "Moreover, the first zero of $\\delta \\nu $ indeed occurs for $\\omega _{\\textit {ex}}T_R=\\pi $ , with the value of $\\omega _{\\textit {ex}}$ being determined from the data in Fig.", "REF for this density.", "This confirms a distinguishing prediction of eq.", "(REF ).", "We also measure the shift as a function of $T_R$ (Fig.", "REF (b)).", "Again we use $\\theta _1= 2.5 \\pi $ and $\\theta _2=2.2 \\pi $ and determine the frequency shift $\\delta \\nu $ for each of the atomic densities, $\\overline{n}$ $\\approx $ $\\lbrace 0.4, 0.8, 1.3, 1.7 \\rbrace \\,10^{12}$ at/cm$^{3}$ .", "From a linear fit we extract the slope $\\alpha =d\\delta \\nu /dN_{at}$ .", "This suppresses potential density-independent frequency shifts that vary with $T_R$ .", "Whenever the spatial spin distribution is the same for the first and second Ramsey pulses, the shift has the same sign and has the opposite sign when the spatial spin distribution reverses.", "This emphasizes the importance of the correlation between the inhomogeneities of the first and second Ramsey pulses.", "In the low-density regime, ($\\overline{n}\\lesssim 10^{12}\\mbox{at}/\\mbox{cm}^{3}$ ), Eq.", "REF predicts that $\\alpha $ in Fig.", "REF (b) should oscillate at $\\omega _z$ .", "Here, the mean frequency shift is offset, as expected from the small scattering length differences.", "Figure: (a) Measured frequency shifts for a Ramsey sequence T R =100T_R=100 ms, θ 1 =2.5π\\theta _1=2.5\\pi and θ 2 =2.2π\\theta _2=2.2\\pi versus interaction strength, which is proportional to density.", "The shift is non-linear; in fact it oscillates as the density increases.", "For reference we plot the known shift for homogeneous excitations, δν=(a ↑↑ -a ↓↓ )/(2πa ↑↓ )ω 𝑒𝑥 \\delta \\nu =(a_{\\uparrow \\uparrow }-a_{\\downarrow \\downarrow })/(2\\pi a_{\\uparrow \\downarrow })\\omega _{\\textit {ex}} (black line).", "(b) Density shift per atom dδν/dN at d\\delta \\nu /dN_{at} in the low interaction regime versus T R T_R with θ 1 =2.5π\\theta _1=2.5\\pi and θ 2 =2.2π\\theta _2= 2.2\\pi .", "The shift oscillates as the spin polarizations oscillate in the trap.", "(c)Dependence on the first and second pulse areas.", "Black squares: dδν/dN at d\\delta \\nu / dN_{at} for T R =92T_R=92 ms with θ 1 =2.5π\\theta _1=2.5\\pi fixed and θ 2 \\theta _2 variable.", "The green horizontal line is a reference measurement with θ 2 =π/2\\theta _2=\\pi /2 giving -6μHz/atom-6\\,\\mu \\mbox{Hz}/\\mbox{atom}, the expected frequency shift in the absence of ISR and predicted from the difference of scattering lengths.", "A numerical calculation based on the singlet-triplet model for two atoms (blue line) reproduces the data with no free parameters.", "Eq.", "qualitatively reproduces the observed behaviours, but overestimates the shift by 60% (red line).", "Magenta dots: θ 1 \\theta _1 variable and θ 2 =2.5π\\theta _2=2.5\\pi fixed.", "This dependenceis not reproduced by either of the models.A distinguishing feature of the s-wave fermion collisions (eq.", "REF ) is that the frequency shift depends on the area of the second Ramsey pulse [8].", "For homogeneous excitations of ultracold bosons, the frequency shift depends on the area of the first Ramsey pulse, which determines the population difference of the clock states for the collisions during the Ramsey interrogation time.", "This unique feature was not demonstrated in the observations of collisional shifts of lattice clocks [7], [9], [10], [13], [14], [15].", "We apply $\\theta _1=2.5\\pi $ , $T_R=92$   ms and vary the second pulse area $\\theta _2$ via the microwave power, keeping the duration fixed.", "The resulting $\\alpha $ is shown in Fig.", "REF (c) (black squares).", "The $1/\\tan \\theta _2$ dependence predicted in [8] and (REF ) is clearly visible.", "To show the quantitative agreement between the data of Fig.", "REF and our models we use the experimental parameters, including $\\delta _1$ which is independently determined from the resonant Rabi flopping with the respective model (Fig.", "REF (a)).The numerical model reproduces the data To extract $\\alpha $ from the numerical model, we calculate the shift for the 4 densities and extract $\\alpha $ from a linear fit, in the same way that we analyze the data..", "The analytical model eq.", "REF reproduces the oscillations but overestimates the amplitude of the shift by 60%.", "Including many sidebands, the numerical model gives better agreement as expected.", "We also vary the area of the first Ramsey pulse $\\theta _1$ , keeping $\\theta _2=5\\pi /2$ fixed.", "Surprisingly, the shift is comparable to when $\\theta _2$ is varied.", "Eq.", "(REF ) has a small dependence on $\\theta _1$ when $\\theta _2$ is not exactly $5\\pi /2$ but the predicted shift is much smaller than observed and would not change sign near $\\theta _1=2.5\\pi $ .", "Similarly, shifts due to the small difference of scattering lengths $a_{\\uparrow \\uparrow }+a_{\\downarrow \\downarrow }-2a_{\\uparrow \\downarrow }$ are too small.", "However, we note that our first pulse motionally excites the spin components beyond a simple dipolar excitation.", "This leads to an oscillation of the cloud size, which unexpectedly varies with $\\theta _1$ .", "We speculate that this may produce some dephasing Such oscillations could create additional spatial inhomogeneities of the transition frequency through the trap, and add a new source of dephasing.", "Any dephasing directly couples pair-wise singlet and triplet states during the Ramsey time and generally leads to a dependence of the measured frequency on $\\theta _1$ .", "Since the contrast does not decrease significantly, the dephasing rate would have to be less than $\\omega _{\\textit {ex}}$, and thereby leads to a dependence on $\\theta _1$ .", "The striking connection between spin-waves and s-wave fermion collision shifts demonstrated here is very general.", "We can elucidate this connection by considering the resolved sideband regime, used in many ultraprecise atomic clocks, including optical lattice clocks.", "With resolved sidebands and weak interactions, even though the clock field cannot change the motional state of the atoms, we show that spin waves are excited.", "Here, the spatial inhomogeneity may give a low energy atom in vibrational state $\\mathinner {|{\\alpha }\\rangle }$ a large pulse area and a high energy atom in $\\mathinner {|{\\beta }\\rangle }$ a small pulse area, directly populating pair-wise singlet states $\\mathinner {|{0,0}\\rangle }$ [8].", "After the pulse, the two fermionic atoms evolve as $\\mathinner {|{\\Psi (T)}\\rangle }= s e^{i \\omega _{ex} T} \\mathinner {|{0,0}\\rangle } \\lbrace \\mathinner {|{\\alpha \\beta }\\rangle }\\rbrace ^++(t\\mathinner {|{1,0}\\rangle }+u\\mathinner {|{1,1}\\rangle }+d\\mathinner {|{1,-1}\\rangle }) \\lbrace \\mathinner {|{\\alpha \\beta }\\rangle }\\rbrace ^-$ , where $\\mathinner {|{1,m_S}\\rangle }$ are triplet states, $\\lbrace \\rbrace ^{(-)+}$ denotes (anti)symmetrization, and $u, d, t$ , and $s$ are the state amplitudes.", "Rewriting this two particle wavefunction as $\\mathinner {|{\\Psi (T)}\\rangle } = 1/\\sqrt{2}(t+s e^{i \\omega _{ex} T})\\lbrace \\mathinner {|{\\uparrow \\alpha }\\rangle }\\mathinner {|{\\downarrow \\beta }\\rangle }\\rbrace ^- +1/\\sqrt{2}(t-s e^{i \\omega _{ex} T})\\lbrace \\mathinner {|{\\downarrow \\alpha }\\rangle } \\mathinner {|{\\uparrow \\beta }\\rangle }\\rbrace ^- +(u\\mathinner {|{1,1}\\rangle }+d\\mathinner {|{1,-1}\\rangle })\\lbrace \\mathinner {|{\\alpha \\beta }\\rangle }\\rbrace ^-$ , we see that the $\\mathinner {|{\\uparrow }\\rangle }$ populations in state $\\mathinner {|{\\alpha }\\rangle }$ and $\\mathinner {|{\\beta }\\rangle }$ have an explicit oscillation at $\\omega _{ex}$ – at different times T, the $\\mathinner {|{\\uparrow }\\rangle }$ population in the vibrational states are different.", "Thus, whenever there is a fermion collision shift, spin waves must also exist, and the fermion collision shift will oscillate as the spin populations oscillate in the trap.", "In summary, we observe characteristic behaviors of the collisional frequency shifts due to inhomogeneous excitations in an atomic clock.", "The inhomogeneous excitations create spin-waves, which we show are inextricably connected to the s-wave frequency shifts of fermion clocks, including optical-frequency lattice clocks.", "We directly excite dipolar spin waves via an amplitude gradient of the excitation field.", "The spin populations oscillate, exhibiting a beat between the trap frequency and the frequency of spin rotation due to particle interactions.", "This leads to a collisional frequency shift that oscillates as the spin populations oscillate in the trap.", "We observe that the clock collision shift does not vary linearly with the atomic density and, in the spin-wave regime, varying the Ramsey interrogation time $T_R$ (Fig.", "REF (b)) could help to evaluate the accuracy of atomic clocks.", "The frequency shift exhibits the novel dependence on the area of the second Ramsey pulse, in stark contrast to the mean field expressions for frequency shifts with homogeneous excitations [8].", "While we intentionally exaggerate the spin wave excitations here, these frequency shifts can be minimized by using spatially homogenous fields, using sideband resolved pulses, and avoiding the Knudsen regime so that trap-state changing collisions further suppress the fermion shift.", "We acknowledge contribution of F. Reinhard.", "This work was supported by the Institut Francilien pour la Recherche sur les Atomes Froids (IFRAF), by the ANR (grant ANR-09-NANO-039) and by EU through the AQUTE Integrated Project (grant agreement 247687) and the project EMRP IND14, the NSF, and Penn State." ] ]

[ [ "The Jefferson Lab Frozen Spin Target" ], [ "Abstract A frozen spin polarized target, constructed at Jefferson Lab for use inside a large acceptance spectrometer, is described.", "The target has been utilized for photoproduction measurements with polarized tagged photons of both longitudinal and circular polarization.", "Protons in TEMPO-doped butanol were dynamically polarized to approximately 90% outside the spectrometer at 5 T and 200--300 mK.", "Photoproduction data were acquired with the target inside the spectrometer at a frozen-spin temperature of approximately 30 mK with the polarization maintained by a thin, superconducting coil installed inside the target cryostat.", "A 0.56 T solenoid was used for longitudinal target polarization and a 0.50 T dipole for transverse polarization.", "Spin-lattice relaxation times as high as 4000 hours were observed.", "We also report polarization results for deuterated propanediol doped with the trityl radical OX063." ], [ "Introduction", "One of the major research initiatives taking place in Hall B at Jefferson Lab is the NSTAR program, the experimental study of baryonic resonances.", "Despite decades of electron, meson, and photo-production studies, a complete and well-characterized spectrum of excited baryonic states remains missing.", "The parameters of many resonances (such as their mass, width, and decay couplings) are not well known, while other resonances, predicted by various QCD models, have yet to be experimentally verified.", "Experiments with both polarized beam and polarized target are critical to disentangling this complex spectrum of broad, overlapping resonances [1].", "Of particular importance are experiments that combine multiple combinations of beam and target polarization.", "In this article we describe a frozen spin polarized target explicitly constructed for such experiments inside a large acceptance spectrometer.", "The target can provide either longitudinal or transverse polarization, depending on the choice of magnet used to maintain the polarization.", "The centerpiece of the Hall B instrumentation package is the CEBAF Large Acceptance Spectrometer (CLAS), a multi-gap, high-acceptance magnetic spectrometer in which the field is generated by six superconducting coils in a toroidal configuration [2].", "This coil arrangement leaves a field-free region in the center of the detector that is well suited for the insertion of a polarized target.", "One such target, dynamically polarized by continuous microwave irradiation at 140 GHz, has been previously described [3] and has been used inside CLAS with electron beams up to 10 nA.", "However, this target features a 5 T superconducting magnet whose geometry limits the acceptance for scattered particles to $\\pm 55^\\circ $ in the forward direction, whereas the acceptance of CLAS spans $\\pm 135^\\circ $ .", "Furthermore, the magnetic field is by necessity parallel to the beam line and can therefore provide only longitudinal polarization.", "Figure: Schematic depicting the operation of the frozen spin target.To compliment the existing polarized target in Hall B, we have designed and constructed a frozen spin target, FROST, which permits the detection of scattered particles over an angular range of $\\pm 135^\\circ $ .", "The target has been utilized on two separate occasions, each lasting about six months.", "It was longitudinally polarized during the first set of experiments (g9a), and transversely polarized for the second (g9b).", "Both circularly and linearly polarized photon beams were used during g9a and g9b, so taken together, all four possible combinations of beam and target polarization were realized, resulting in a so-called “complete” experiment.", "The remainder of this article is organized as follows.", "A general overview of the target system and its operation is given in Section , with more detailed descriptions of the various components provided in Section .", "Its performance during both experiments is described in Section , and a summary is made in the final section." ], [ "System Overview", "The operation of the frozen spin target is depicted schematically in Fig.", "REF .", "The target sample is polarized with microwaves via Dynamic Nuclear Polarization (DNP) in the bore of a high field, high homogeneity magnet.", "The microwaves are then switched off, the target is cooled to a temperature below approximately 50 mK, and the polarization is maintained by a weaker magnetic field.", "In our case this field is produced by a small superconducting magnet installed inside the target cryostat.", "The scattering data is acquired while the target polarization decays in an exponential manner with a spin-lattice time constant $T_1$ that is determined in part by the temperature of the material and the strength of the magnetic field.", "The DNP process is repeated when the polarization has fallen to an unacceptably low value or to change the direction of the polarization.", "The major components of FROST, which we describe in the following section, are: a 5 T polarizing magnet with a horizontal warm bore; a bespoke, horizontal $^3$ He-$^4$ He dilution refrigerator (DR) with a high cooling capacity; a novel vacuum seal and insertion device for loading the target material directly into the mixing chamber of the DR; one of two internal superconducting holding coils; a 140 GHz microwave system for the DNP process; a continuous-wave, nuclear magnetic resonance (CW-NMR) system for measuring the polarization.", "The layout of the system is shown in Fig.", "REF .", "The target cryostat is suspended from the lower portion of a two-tiered insertion cart that is mounted on rails for travel into the center of the spectrometer.", "Most of the vacuum pumps for the cryostat are mounted to the upper tier, which is vibrationally isolated from the lower tier by four air springs.", "A control panel for the pumps, electronics for monitoring and controlling the target, and two gas panels for the DR are also mounted to the lower tier.", "Only a large chiller for water-cooled vacuum pumps and two tanks for storing the $^3$ He-$^4$ He mash are not mounted to the insertion cart.", "These items are located on Level 2 of the steel frame surrounding the spectrometer, and connect via flexible lines to the target on Level 1 below.", "The polarizing magnet is suspended at the entrance to the spectrometer from a set of rails perpendicular to the beamline, and can be moved approximately 1 m in the beam-left direction, allowing the target cart to be moved to the center of CLAS.", "The polarization process begins with the target cryostat inserted into the bore of the polarizing magnet (at 5 T) and the microwave tube energized.", "The time required to reach 80% proton polarization is about 2–3 hours, and another 3 hours are necessary to reach 90–95%.", "During this time the microwaves (50–100 mW) warm the target sample to about 0.3 K. The field of the polarizing magnet is parallel to the beamline, and so the target is longitudinally polarized.", "After the microwaves are switched off, 30–45 minutes are required for the target to cool to a temperature less than 50 mK.", "At that time the polarizing magnet is de-energized, while the internal holding magnet is simultaneously energized at a rate that maintains a net magnetic field of about 0.5 T (another 45 minutes).", "The direction of the polarization either remains longitudinal, or is rotated transverse to the beam, depending on which holding coil is installed inside the target cryostat.", "The cryostat is then retracted from the polarizing magnet, the latter is moved out of the way, and the cryostat is moved approximately 4 meters into the center of CLAS (about 2 minutes) where it continues to cool below 30 mK.", "The tagged photon beam is activated and photoproduction data is acquired for a period of 5–10 days, after which the polarization process is repeated, usually to reverse the target polarization.", "The photon beam deposits 10–20 $\\mu $ W to the refrigerator, warming it 2 mK or so.", "Even with beam on target, the polarization loss is only about 1% per day.", "Figure: Side view of the frozen spin target in the polarizing position.", "To acquire photoproductiondata the polarizing magnet is moved away, and the cryostat is rolled into the center of CLAS." ], [ "Polarizing Magnet", "A 5 Tesla solenoidCryomagnetics, Inc. was used during the DNP process to polarize the target material.", "The solenoid has a horizontal, room-temperature bore of 127 mm diameter and produces a maximum field of 5.1 T at 82.8 A.", "The inhomogeneity of the solenoid's central field is $\\le 5 \\times 10^{-5}$ over the volume of our sample, a 50 mm long, 15 mm diameter cylinder.", "It features a 45 l volume for liquid helium and two vapor-cooled heat shields, obviating the need for liquid nitrogen.", "Using a flexible transfer line, the magnet was automatically filled with LHe about every 4 days from a nearby 500 l dewar.", "Liquid in the dewar was continuously replenished from Jefferson Lab's End Station Refrigerator, or ESR." ], [ "Dilution Refrigerator", "To aid in the following discussion, a flow diagram for the frozen spin target is shown in Fig.", "REF , while a sectional view of the target is given in Fig.REF .", "There are two quasi-independent refrigeration systems inside the target cryostat: a $^3$ He-$^4$ He dilution refrigerator and a $^4$ He evaporation refrigerator, also called the precooler.", "To conserve space, the precooler is located inside the 180 mm diameter tube that is used for pumping $^3$ He from the dilution unit.", "It is sealed inside this pumping tube using an indium seal at one end and a rubber o-ring at the other.", "Both refrigeration units are constructed around a separate, thin-walled stainless tube that is 50 mm in diameter in the case of the precooler, and 40 mm for the dilution refrigerator.", "These bolt together using commercial knife-edge flanges, forming a single tube that extends from room temperature at one end to the mixing chamber at the other.", "This tube serves as both an evacuated beam pipe for the photon beam, and as a load-lock tube for inserting the polarizable target sample into the mixing chamber.", "At the room temperature end, the tube is sealed with a 0.13 mm thick polyimide beam-entrance window.", "The opposite end is sealed by a 0.13 mm thick aluminum window on the sample insertion device described in Section REF .", "Figure: Flow schematic of the target.", "Here HX indicates a heat exchanger and NV a needle valve.As its name implies, the precooler's purpose is to cool and condense $^3$ He before it circulates through the dilution unit.", "It consists of two counterflow gas-gas heat exchangers, and two 1 liter vessels, or “pots”, containing liquid helium at 4 K and 1 K, respectively.", "Both pots are welded around the central beam pipe and are instrumented with RuO temperature sensors and miniature superconducting level probes.", "The 4K pot is continuously filled with LHe from the same 500 l dewar that services the polarizing magnet.", "Vapor is pumped from this pot along two paths.", "The first is used to cool incoming $^3$ He via heat exchanger HX1, while the second cools both a 20 K heat shield surrounding the refrigerator as well as a copper plate that is located in the upstream end of the target and used to heat sink all tubes, wires, and cables as the enter the cryostat.", "The 1K pot receives liquid helium from the 4K pot via needle valve NV1.", "Vapor pumped from this pot is also used to cool $^3$ He gas using heat exchangers HX1 and HX2.", "The $^3$ He gas is condensed inside the 1K pot using HX3, the condenser.", "Figure: Sectional view of the frozen spin target.", "A: beam pipe, B: LHe inlet, C: 3 ^3He pump port,D: 4 K pot, E: 1 K pot, F: 1 K heat exchanger, G: still, H: vacuum chamber,I: sintered heat exchanger, J: mixing chamber, K: holding coil, L: target cup, M: target insert,N: 1 K heat shield, O: 20 K heat shield, P: beam pipe heat shield (one of three), Q: 3 ^3He pump tube,R: copper cold plate, S: waveguide, T: precool heat exchanger.The overall length of the cryostat is approximately 2 m.HX1 consists of three, thin-walled stainless tubes welded coaxially around the beam pipe.", "The tubes are approximately 80 cm long with annular gaps between one tube and the next of 0.25, 1, and 2 mm.", "Cold helium gas from the 1K and 4K pots is pumped through the inner and outermost gaps, while incoming $^3$ He flows in the central gap.", "In this manner the $^3$ He exchanges heat with gas from both the 4K and 1K pots and is cooled to about 3K.", "HX2 consists of a 19 mm diameter, 40 cm long tube for low pressure gas pumped from the 1K pot.", "A pair of 3 m long, 3 mm diameter tubes are coiled tightly inside and carry liquid helium from the 4K pot and $^3$ He from HX1.", "The condenser is a 10 cm$^3$ copper cup, filled with sintered 50 $\\mu $ m copper powder and located inside the 1K pot.", "The $^3$ He condensation pressure is set by needle valve NV2, located downstream of the condenser.", "A third needle valve, NV3, bypasses both HX2 and the condenser to deliver cold $^3$ He gas directly to the mixing chamber.", "This valve is used only for initially cooling the dilution unit from room temperature and is closed during normal operation.", "Two small diaphragm pumps are used to pump vapor from the 4K pot, while a 120 m$^3$  h$^{-1}$ roots system is used for the 1K pot.", "All gas pumped from the precooler is returned to the End Station Refrigerator for liquefaction.", "The dilution unit consist of the customary still, heat exchanger, and mixing chamber.", "It is located downstream of the 180 mm pumping tube and is surrounded by the 20 K heat shield.", "A second heat shield, at 1 K, is attached to the still and surrounds both the heat exchanger and mixer.", "Both heat shields are constructed of copper with 1 mm thick aluminum extensions around the mixing chamber.", "A photograph of the DR is shown in Fig.", "REF .", "Figure: Annotated photograph of the dilution refrigerator.", "The brass and PCTFE mixing chamberwas subject to leaks and was replaced with one fabricated entirely from PCTFE.The still is constructed from a 100 mm diameter stainless steel tube which seals with an indium o-ring against a circular flange welded around the central beam pipe.", "It is instrumented with thermometers, a heater, a capacitive liquid-level meter, and 1 m long, 1.5 mm diameter heat exchanger (HX4) for condensing any $^3$ He that may vaporize after expanding though NV2.", "The heater is a 10 m long coil of 0.6 mm NiChrome wire.", "The level probe consists of two copper-clad fiberglass plates with a separation of 0.5 mm, and the capacitance between the plates is measured using an AC bridge circuit.", "Gas from the still is pumped through a short, flexible bellows of 16 mm inner diameter connecting the top of the still to the 180 mm diameter pumping flange.", "All wiring for the DR, along with the $^3$ He bypass and $^3$ He condenser lines, pass through this bellows, or “umbilical”.", "$^3$ He is pumped from the still by two dry pumping systems operating in parallel.", "Each system consists of Alcatel RSV2000 and RSV600 roots pumps and an Edwards L70 dry pump, and the measured pumping speed for helium is 3300 m$^3$ /hr.", "The $^3$ He gas panel features two liquid nitrogen traps for filtering contaminates from the circulating gas (the second trap is a spare), but thanks to the all-dry pumping system, the trap has blocked only once, despite months of continuous use.", "The contaminates on that occasion were traced to a room-temperature gas fitting that had presumably vibrated loose.", "The main heat exchanger for the DR (HX5) is modeled after a design by Niinikoski [4] and is comprised of three sections.", "The first is a 2 m lengthLater shortened to 1 m, see Section  REF of cupronickel tube with an inner diameter of 0.5 mm.", "This serves as a secondary flow impedance for the circulating $^3$ He and was added after the refrigerator experienced flow and temperature instabilities during early tests.", "The second section of the heat exchanger is a 1.5 m long stainless steel tube of 1.0 mm inner diameter with copper fins brazed to the outside.Fin Tube Products, Inc.", "The final section of heat exchanger is comprised of six copper C122 tubes with copper powder sintered inside and out.", "Each tube is 28 cm long, with inner and outer diameters of 3.3 and 4.0 mm.", "The sinter on the outside is 1 mm thick, while the inside is filled with sinter except for a central, 1.5 mm diameter flow channel for the concentrated $^3$ He stream.", "The warmest two sections are sintered with 325 mesh copper powder (nominal size 20 $\\mu $ m), while 5 $\\mu $ m powder was used on the coldest four.", "The total surface area of the sintered heat exchanger is about 4 m$^2$ on the concentrated side and 7 m$^2$ on the dilute side, based on 77 K measurements of argon gas adsorption.", "All sections of the heat exchanger are wound in a spiral groove machined in a G10 fiberglass mandrel.", "The groove is 3.5 m long and has a 13 mm wide, 7 mm deep rectangular cross section.", "All wires to the mixing chamber and the 3 mm bypass tube are also wound in this groove.", "The G10 mandrel slides tightly over the central beam pipe and is covered by a second tightly-fitting stainless steel tube that attaches to the still at one end and to the mixing chamber at the other.", "The mixing chamber is a 1 mm thick PCTFE cup that seals against this tube using an indium o-ring.", "Concentrated $^3$ He enters the mixing chamber through a PTFE tube located in the lower half of the mixer, directly under the target sample.", "A series of holes are punctured along the length of the tube and distribute the concentrate evenly as it rises past the sample and collects at the top of the mixer.", "A series of three spring-energized PTFE radial seals constrains the excess concentrate to the top half of mixing chamber, while 3 mm holes under the seals allow $^3$ He to be removed from the dilute phase in the lower half.", "The mixing chamber is outfitted with a small nichrome heater for cooling power measurements and three RuO resistance thermometers.", "One thermometer,Lakeshore RX-202-AA has a room temperature resistance of 2 k$\\Omega $ and is calibrated 0.05–40 K. However, its resistance versus temperature dependence is too steep to permit accurate extrapolation to lower temperatures.", "The additional two resistorsDale 1 k$\\Omega $ RCW-575 were chosen because they are known to exhibit a well-behaved, log(T$^{n}$ ) response at temperatures as low as 25 mK [6] and are calibrated against the first thermometer.", "One of these is located in the center of the mixing chamber, near the polarized target sample, while the other is in the downstream end of the heat exchanger, in the dilute flow stream.", "All temperatures quotes in this article are based on this last sensor as it is believed to measure the average temperature of liquid exiting the mixing chamber.", "The thermometers in the mixing chamber and in the still are read by a Lakeshore Model 370 AC resistance bridge, which is also used to power both the still and mixing chamber heaters.", "Both the dilution and precooling refrigerators are housed inside a custom-built, stainless steel vacuum chamber.", "The downstream end of the chamber is made of closed-cell foam to reduce the energy loss of particles scattered from the polarized butanol sample.", "A thin layer of aluminum is glued to the foam's interior surface to decrease outgassing, and a 25 $\\mu $ m thick aluminum exit window is glued to its downstream end." ], [ "Target Material and Insert", "Frozen beads of butanol (C$_4$ H$_9$ OH) were used for the target material.", "The butanol was doped with the nitroxyl radical TEMPO2,2,6,6-Tetramethylpiperidinyloxy at a concentration of $2.0 \\times 10^{19}$  spins cm$^{-3}$ for dynamic polarization.", "Water (0.5% by weight) was added to the solution before freezing in order to avoid a crystalline solid.", "The 1–2 mm diameter beads were formed by dripping the solution through hypodermic needles into a bath of liquid nitrogen.", "The needles were held at an electrical potential of approximately 2 kV in order to control the bead size.", "This also provided the beads with a slight static charge which kept them from clumping together in the liquid nitrogen bath.", "Approximately 5 g of beads were loaded under liquid nitrogen into a 15 mm diameter, 50 mm long PCTFE target cup that attached to end of a 25 cm long stainless steel tube.", "The tube is sealed at the target end by a 0.13 mm thick aluminum vacuum window that also features a locking mechanism for the PCTFE cup.", "At the opposite end, a flange for a novel vacuum seal (described below) is welded to the tube.", "The cup has a number of 0.5 mm wide grooves machined into its underside which allows some of the $^3$ He concentrate entering the mixing chamber to penetrate directly into the cup.", "Since $^3$ He absorbs heat when it dilutes into liquid $^4$ He, we believe this may provide a more efficient cooling path to the target beads.", "We have developed a new superfluid-tight vacuum seal for quickly loading the target material into the mixing chamber of the dilution refrigerator while at cryogenic temperatures.", "Patterned after VCR face-seal fittings,Swagelok, Inc. it features a 0.6 mm high, toroidally-shaped ridge on the insert face for compressing a polyimide vacuum gasket.", "A threaded nut is used to screw the insert against the cryostat-side sealing face which has a polished, conical sealing surface and male threads.", "The conical design ensures that thermal contraction continues to tighten the seal after the warm insert (80 K) is screwed into the cold cryostat (10 K).", "A set of bearings between the insert's face seal and its threaded nut prevents the insert from rotating while the nut is turned.", "The seal is made by attaching the target insert, under liquid nitrogen, to a 2 m long pipe that acts as a long wrench.", "This pipe has a set of retractable pins that lock into matching holes in the insert.", "We load the insert into the empty mixing chamber at a temperature of about 10 K with a strong helium purge on the beam pipe.", "After making the seal, the pins are retracted and the wrench is removed.", "A set of three 13 $\\mu $ m thick aluminum heat shields is placed inside the beam pipe which is then evacuated and circulation of the $^3$ He-$^4$ He mixture through the dilution refrigerator is begun.", "The entire procedure requires about 90 minutes, during which the mixing chamber temperature rises to about 60–70 K. We have tested several gasket materials and have found that 180 $\\mu $ m thick polyimide gives the most reliable results.", "We have made the seal several dozen times with almost 100% success.", "Leaks are almost always due to contaminants such as moisture on the gasket or threaded surfaces.", "These contaminants are reduced by establishing a separate helium purge on the insert as soon as it is removed from the liquid nitrogen bath.", "In addition to the polarized butanol sample, two additional targets were installed in the cryostat and were used for background and dilution studies.", "A 1.5 mm thick carbon disk and a 3.5 mm thick CH$_2$ disk were mounted on the 1K and 20K heat shields, approximately 6 cm and 16 cm downstream of the butanol sample." ], [ "Holding Coils", "In the first generation of frozen spin targets, the holding field was generated by the fringe of the polarizing magnet or by a second, external magnet with a larger aperture.", "The field uniformity in these cases was not sufficient to resolve the NMR line of the target material, and so it was not possible to monitor the target polarization during the scattering experiment.", "Niinikoski first described a thin superconducting solenoid mounted inside the target cryostat for maintaining or rotating the target polarization [5], while the group at Bonn were the first to implement this type of solenoid in an actual experiment [7].", "With careful design, these coils can be thin enough for reaction products to pass through with acceptably low energy loss and uniform enough for precise NMR measurements.", "We have utilized two such coils for the g9a and g9b experiments, as shown in Fig.", "REF .", "For g9a, a 110 mm long solenoid was used to maintain longitudinal polarization.", "It consists of three layers of 0.14 mm copper-clad, multifiliment NbTi wireSupercon, Inc. type 54S43.", "wound on a 50 mm diameter, 1 mm thick aluminum former.", "Each layer has 785 windings, while an additional 162 turns are added at both ends to improve the field uniformity.", "Grooves machined in the former aid in the placement of the windings.", "Stycast epoxy 1365-65N was used to adhere the coil, which produces a 0.56 T central field at 22.0 A.", "The aluminum former was attached to the downstream end of the 1 K copper heat shield which was in turn thermally sunk to the still.", "The second coil, used for g9b, was wound as a four layer, saddle-shaped dipole.", "The two innermost layers each consist of 170 turns of wire, while 152 and 60 turns were used for the third and fourth layers, respectively.", "All eight coils were wound from a single, continuous length of 0.18 $\\mu $ m diameter superconducting wire using a custom-designed aluminum fixture coated with PTFE.", "The coils were glued to a 5 $\\mu $ m polyester backing to prevent the layers from separating as they were wrapped around and epoxied to an aluminum former similar to the one described above.", "In addition, a thin nylon thread was wrapped around the coils to further secure them to the former.", "This magnet has demonstrated a maximum field of 0.54 T, and was operated at 0.50 T (35.5 A) during g9b.", "The technique and equipment for winding this dipole were developed and built at Jefferson Lab for FROST, and later used to construct similar coils for a polarized target at the Hi$\\gamma $ s facility [8].", "Figure: Microwave system used during dynamic polarization of the frozen spin target.The variable attenuator was not used during g9a.Instead, the microwave power was adjusted with the EIO cathode voltage.Current leads for both coils consisted of a combination of copper wire, high temperature superconducting ribbon,American Superconductor HTS Cryoblock wire and 0.40 mm diameter NbTi wire.Supercon T48B-M The leads were heat sunk at both the 4K and 1K pots, and at the still, where they were soldered to excess wire from the holding coil.", "Both the solenoid and dipole magnets quenched at full field during the g9a and g9b runs.", "This happened on four occasions with the solenoid and once with the dipole.", "In the former case the quenches were eventually traced to a loose electrical connection at the room temperature feedthrough for the magnet leads.", "No quenches occurred after this was tightened, and the solenoid was at no time damaged.", "The dipole quenched when a water accident in the experimental hall shut off electrical power to the cryostat.", "In this instance one of the coil leads broke near the still heat sink.", "Fortunately enough excess wire remained to remake the solder joint at the still without having to splice in a new section of lead.", "From the widths of the NMR lines obtained with the two coils (see Fig.", "REF ), we estimate that $\\Delta B/B$ for the solenoid field was about 0.003 and about 0.008 for the dipole." ], [ "Microwaves", "Microwaves for dynamically polarizing the target were generated by an Extended Interaction OscillatorCommunications & Power Industries Canada, Inc. with a 140 GHz center frequency, a tuning range of approximately $\\pm $ 1 GHz, and a maximum power output of about 15 W. The frequency was adjusted by changing the size of the oscillator cavity using a small DC motor.", "Microwaves were transmitted to the target sample through a 2 m length of 4.3 mm diameter cupronickel waveguide that was sealed with a FEP window at the room temperature end and heat sunk at several locations inside the cryostat.", "It terminated outside the mixing chamber with a slight, upward bend that directed microwaves at the target sample.", "The aluminum former for the holding coil acted in some respect as a multimode cavity for the microwaves, but its efficiency is unknown and likely quite poor.", "Because of this, the best polarization was found to result from a substantial microwave power of about 50–100 mW to the target.", "A schematic diagram for the microwave system is shown in Fig.", "REF ." ], [ "NMR System", "The target polarization was measured using continuous-wave NMR circuits designed around the Liverpool Q-meter [9].", "As shown in Fig.REF , this consists of a series-tuned LRC circuit where the inductance takes the form of a small coil surrounding the target sample while a variable capacitor is used to adjust the resonance frequency.", "The RF field is swept at constant current through the nuclear resonance frequency, and the net energy absorbed or emitted by the target spins is observed as a change to the coil's impedance.", "Phase-sensitive detection (PSD) is used to measure the real part of the voltage across the circuit during the RF sweep.", "The polarization of the sample is proportional to the area under the resonance peak, with the constant of proportionality determined by calibrating the Q-meter system against a known sample polarization.", "Two NMR systems were utilized during the photoproduction measurements.", "The first system was tuned to resonate at 212.2 MHz and measured the target polarization at 5.0 T during the DNP process.", "This system was calibrated against the thermal equilibrium polarization of the target sample, which was measured at various temperatures between 0.9 and 1.8 K. The second NMR system was used to monitor the polarization of the target while in frozen spin mode.", "During g9a it was tuned at 23.8 MHz for use with the 0.56 T solenoid, and at 21.9 MHz for the 0.50 T dipole used during g9b.", "This second system was calibrated against the 212 MHz measurements each time the target was polarized.", "Comparison of the two systems is discussed in Section REF .", "Figure: NMR scans of the butanol sample.", "Clockwise from upper left: thermal equilibrium polarizationat 1 K and 5 T; approximately 85% polarization at 5.0 T; longitudinal holding coil at 0.56 T;transverse holding coil at 0.50 T.For the thermal equilibrium calibration measurements, the DR was operated as a simple $^4$ He evaporation refrigerator with a base temperature of approximately 0.75 K. This ensured a uniform sample temperature since the mixing chamber was filled with superfluid helium.", "The 2 k$\\Omega $ RuO sensor in the mixer was used to determine the sample temperature with an accuracy of 2%.", "A small ($\\sim $ 4%) correction was applied to account for the sensor's magnetoresistance at 5 T, previously measured against the $^3$ He vapor pressure curve.", "The NMR coils were cut from 25$\\mu $ m thick copper foil and were wrapped around the outside of the mixing chamber.", "They were held rigidly in place by a thin layer of FEP heat shrink tubing.", "Two separate coils were used for g9a, while a single coil was utilized for g9b.", "This is discussed in further detail in Section .", "The NMR coils were connected to the Q-meter circuit using resonant lengths ($n\\lambda / 2$ ) of semi-rigid coaxial cables specified for use at cryogenic temperatures.", "To first order, the $n\\lambda / 2$ cables mirror the coil inductance directly to the rest of the Q-meter circuit.", "However, they also generate a large background signal (Q-curve) which must be removed for precise signal analysis.", "This is accomplished by performing NMR sweeps with the magnetic field shifted slightly off resonance and subtracting the resultant Q-curve from subsequent, on-resonance scans.", "A typical NMR measurement consisted of sweeping through the resonance line multiple times and averaging the results in order to improve the signal-to-noise ratio.", "The sweep width was typically $\\pm $ 400 kHz for both the high and low field systems.", "The polarization was measured continuously during the DNP process, and twice per hour while in frozen spin mode.", "Typical NMR curves are shown in Fig.", "REF ." ], [ "Refrigerator Performance", "When mounted on a concrete floor during its initial tests, the dilution refrigerator achieved a base temperature of 26 mK.", "Vibration was more problematic in the experimental hall, resulting in a base temperature of 28 mK (during g9a) with a $^3$ He circulation rate of 1.0 mmol s$^{-1}$ .", "The maximum sustainable flow rate during g9a was about 16 mmol s$^{-1}$ and was limited by the circulation pressure.", "Exposed to a photon flux of about $5 \\times 10^{7}$  cm$^{-2}\\cdot $ s$^{-1}$ , the target warmed to approximately 30–32 mK at an optimal flow rate of 1.6 mmol s$^{-1}$ .", "The cooling power of the refrigerator was measured during g9a with flow rates up to 16 mmol s$^{-1}$ and is shown in Fig.", "REF .", "The cooling power was approximately 1 mW at 50 mK, 10 mW at 100 mK, and 60 mW at 300 mK.", "Two modifications were made to the DR between g9a and g9b.", "First, the secondary flow impedance was shortened from 2 m to 1 m, permitting the refrigerator to run with flow rates in excess of 30 mmol s$^{-1}$ .", "Second, a length of PTFE cord was placed in the G10 spiral, alongside the heat exchanger.", "This reduced the axial conduction of heat through the dilute solution and lowered the base temperature without beam to 24–25 mK and about 28 mK with beam.", "The cooling power was not measured following these modifications, but we estimate that the increased flow rate improved the cooling power to about 100 mW at 300 mK.", "As mentioned above, a 4 m long, flexible transfer line was used to supply liquid helium to the target cryostat from a 500 l dewar.", "During g9a the efficiency of this line was very poor, and the liquid helium consumption of the target was about 20 l h$^{-1}$ .", "A new transfer line with better thermal insulation was constructed for g9b, and the consumption dropped to 8–10 l h$^{-1}$ .", "The refrigerator operated continuously and without incident for the entire length of the g9a experiment, about 5 months.", "It ran six months for g9b, however its performance was compromised by a water accident in the experimental hall during the final weeks.", "This accident shut off power to the cryostat pumping system, damaging the transverse holding coil as noted above.", "The coil was repaired but a superfluid leak between the mixing chamber and beamline appeared on the subsequent cool down.", "Attempts to repair the leak in a timely fashion were unsuccessful, and the final two weeks of g9b were run with a base temperature of only 60 mK.", "Figure: Measured cooling power of the dilution refrigerator with 3 ^3He flow rates 1–16 mmol s -1 ^{-1}.The solid line is a guide to the eye." ], [ "Polarization Results", "The target was polarized a total of 21 times during g9a, with an average starting polarization of 84% in the positive spin state (9 times) and -86% in the negative.", "Typical relaxation times for positive polarization during g9a were about 2800 hours with beam on target and up to 3600 hours without beam.", "The target relaxed more quickly in the negative spin state, about 1400 hours with beam and 1900 hours without.", "The maximum polarization was -94%.", "The target was re-polarized (and the polarization reversed) about once per week.", "During g9b the target was polarized a total of 19 times, again about once per week.", "The target was most often polarized in the negative spin state (15 times) because it reached a higher starting polarization (-92%) than the positive spin state (83%).", "Therefore the orientation of the target spins with respect to the beam was usually determined by the direction of the transverse holding field.", "The relaxation time during g9b was somewhat higher, about 3400 hours for positive polarization with beam and 4000 hours without.", "The relaxation time for the negative spin state was once again about half that of the positive.", "The final two weeks of g9b were run with a superfluid leak between the mixing chamber and beam pipe.", "This reduced the starting polarization (three polarization cycles) to an average of only 69%, and the relaxation time decreased by about a factor of seven, necessitating bi-weekly polarizations.", "Figure: Spin-lattice relaxation times T 1 T_1 of polarized butanol as a function of sample temperature for two values of magnetic field.", "Circles: Bonn frozen spin target at 0.42 T. Squares: JLab frozen spin target at 0.56 T. The solid and dashed curves are fits to the data according to Eqs.", "and .Figure REF shows our measurements of the relaxation time for TEMPO-doped butanol at 0.56 T for temperatures 28–150 mK.", "Included are data for butanol doped with a similar spin density of porphyrexide measured with the Bonn frozen spin target at 0.42 T [10].", "We have fit both sets of data according to a semi-empirical relationship developed by de Boer [11] $T^{-1}_{1p} = [A T_{1e} B^2 \\cosh ^2(\\frac{h \\nu _e}{2kT})]^{-1} + [a \\frac{B^b}{T^c}]^{-1}$ where $AT_{1e} = 225[B^5 \\coth (\\frac{h \\nu _e}{2kT}) + 6.75 \\times 10^5 \\exp (\\frac{-1}{2T})]^{-1}$ In these equations $T_{1p}$ and $T_{1e}$ are the spin-lattice relaxation times (in hours) for the proton and electron respectively.", "The magnetic field strength (in kG) is $B$ , and $\\nu _e$ is the corresponding electron Larmor frequency.", "$A$ is a constant, and $a$ , $b$ and $c$ are fitting parameters.", "Reasonable agreement between both sets of data with the above equations could be found with $a=3.1\\times 10^{-4}$ , $b=4.3$ , and $c=2.5$ .", "Figure: NMR signal and polarization growth curve of deuterated propanediol doped with OXO63.Two separate NMR coils were used during g9a: a single-loop coil connected to a Q-meter tuned to 212 MHz for measuring the target polarization during DNP, and a two-loop coil and separate Q-meter at 24 MHz for monitoring the polarization in frozen-spin mode.", "The latter system was calibrated against the first each time the target was polarized, and the two were again compared after approximately seven days of beam time, just before the target was re-polarized.", "On average, the low frequency NMR system reported 4% more polarization loss than the high field system.", "We attribute this discrepancy to nonuniform heating induced by the photon beam.", "For geometrical reasons the two-loop coil was slightly more sensitive to material in the downstream end of the target sample, where heating and depolarization from forward-going charged particles was greatest.", "During g9b the same two-loop coil was utilized for both the high- and low-frequency NMR measurements, and no such discrepancy was observed." ], [ "Polarization of Deuterated Propanediol", "At the conclusion of g9b, the butanol target was replaced with a sample of fully-deuterated propanediol, again consisting of 1–2 mm frozen beads.", "This sample was doped with the trityl radical OX063Oxford Scientific Instruments with a spin concentration of $1.5 \\times 10^{19}$  cm$^{-3}$ .", "The newly synthesized trityl radicals have extremely narrow EPR linewidths [12], and deuteron polarizations as high as 80% have been observed in both d-butanol and d-propanediol [13].", "To our knowledge no attempt has been made to polarize trityl-doped samples at the operating conditions of FROST, 0.3 K and 5 T. The results of our test are shown in Figure REF .", "A maximum deuteron polarization of -87$\\pm 3$ % was obtained after approximately 16 hours of microwave irradiation, while -80% was obtained after about 7 hours.", "It should be stressed that only one attempt was made to polarize a trityl-doped sample, and we feel that even higher deuteron polarizations are possible.", "It is likely that the spin concentration, which was found to give the best results at 2.5 T [13], was not optimal for 5 T. Further studies of polarization with different spin densities will be performed in the future." ], [ "Summary", "We have described a frozen spin polarized target constructed for use inside the CEBAF Large Acceptance Spectrometer at Jefferson Lab.", "The primary components of the target are a horizontal dilution refrigerator with a high cooling power for dynamic polarization and a base temperature of about 25 mK, and internal, superconducting magnets for maintaining the target polarization while in frozen spin mode.", "Two such magnets have been used: a 0.56 T solenoid for longitudinal polarization, and a 0.50 T saddle coil for transverse polarization.", "The target has provided proton polarizations greater than 90%, and spin-lattice relaxation times up to 3400 hours have been observed with beam on target.", "The target has been used with both linearly- and circularly-polarized photons to perform the first “complete” photoproduction experiments at Jefferson Lab.", "The high reliability of the target system, along with the long relaxation times, resulted in an on-beam efficiency greater than 90% during these experiments.", "We have also demonstrated a deuteron polarization in this system of -87% in trityl-doped d-propanediol, with the expectation of even greater polarizations in the future." ], [ "Acknowledgements", "The authors gratefully acknowledge the expert support provided by the technical and engineering staffs of the Jefferson Lab Target Group and Jefferson Lab Experimental Hall B during the design, construction and operation of this target.", "Authored by Jefferson Science Associates, LLC under U.S. DOE Contract No.", "DE-AC05-06OR23177.", "The U.S. Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce this manuscript for U.S. Government purposes." ] ]

[ [ "Scale-Invariant Random Spatial Networks" ], [ "Abstract Real-world road networks have an approximate scale-invariance property; can one devise mathematical models of random networks whose distributions are {\\em exactly} invariant under Euclidean scaling?", "This requires working in the continuum plane.", "We introduce an axiomatization of a class of processes we call {\\em scale-invariant random spatial networks}, whose primitives are routes between each pair of points in the plane.", "We prove that one concrete model, based on minimum-time routes in a binary hierarchy of roads with different speed limits, satisfies the axioms, and note informally that two other constructions (based on Poisson line processes and on dynamic proximity graphs) are expected also to satisfy the axioms.", "We initiate study of structure theory and summary statistics for general processes in this class." ], [ "Introduction", "Familiar web sites such as Google maps provide road maps on adjustable scale (zoom in or out) and a suggested route between any two specified addresses.", "Given $k$ addresses in a country, one could find the route for each of the ${k 2}$ pairs, and call the union of these routes the subnetwork (of the country's entire road network) spanning the $k$ points.", "We abstract this idea by considering, for each pair of points $(z,z^\\prime )$ in the plane, a random route $\\mbox{${\\mathcal {R}}$}(z,z^\\prime ) = \\mbox{${\\mathcal {R}}$}(z^\\prime ,z)$ between $z$ and $z^\\prime $ .", "The collection of all routes (as $z$ and $z^\\prime $ vary) defines what one might call a continuum random spatial network, an idea we explain informally in this introduction (precise definitions will be given in section REF ).", "In particular, for each finite set $(z_1,\\ldots ,z_k)$ of points we get a random network $\\mathbf {span}(z_1,\\ldots ,z_k)$ , the spanning subnetwork linking the points, consisting of the union of the routes $\\mbox{${\\mathcal {R}}$}(z_i,z_j)$ .", "Mathematically natural structural properties we will impose on the distribution of such a process are (i) translation and rotation invariance (ii) scale-invariance.", "For $0<c<\\infty $ the scaling map $\\sigma _c: {\\mathbb {R}}^2 \\rightarrow {\\mathbb {R}}^2$ takes $z$ to $cz$ ; we emphasize that (ii) means “naive Euclidean scaling\", i.e.", "invariance under the action of $\\sigma _c$ , not any notion of “scaling exponent\".", "For instance, scale-invariance implies that the route-length $D_r$ between points at (Euclidean) distance $r$ apart must scale as $D_r \\ \\stackrel{d}{=} \\ r D_1$ , where of course $ 1 \\le D_1 \\le \\infty $ .", "The setup so far does not exclude the possibility that routes are fractal, with infinite length, and such cases do in fact arise naturally in the tree-like models of section REF .", "But, envisaging road networks rather than some other physical structure, we restrict attention to the case ${\\mathbb {E}}D_1 < \\infty $ .", "There is a rather trivial example, the complete network in which each $\\mbox{${\\mathcal {R}}$}(z_1,z_2)$ is the straight line segment from $z_1$ to $z_2$ , but the assumption “$\\ell < \\infty $ \" below will exclude this example.", "Much of our study involves sampled spanning subnetworks $\\mbox{${\\mathcal {S}}$}(\\lambda )$ , as follows.", "Write $\\Xi (\\lambda )$ for a Poisson point process in ${\\mathbb {R}}^2$ of intensity $\\lambda $ , independent of the network.", "Then the points $\\xi $ of $\\Xi (\\lambda )$ , together with the routes $\\mbox{${\\mathcal {R}}$}(\\xi ,\\xi ^\\prime )$ for each pair of such points, form a random subnetwork we denote by $\\mbox{${\\mathcal {S}}$}(\\lambda )$ .", "The distribution of $\\mbox{${\\mathcal {S}}$}(\\lambda )$ inherits the properties of translation- and rotation-invariance, and a form of scale-invariance described at ().", "In particular we can define a constant $0 < \\ell \\le \\infty $ by $ \\ell = \\mbox{mean length-per-unit-area of } \\mbox{${\\mathcal {S}}$}(1) $ (where “mean length-per-unit-area \" is formalized by edge-intensity at (REF )).", "In section REF we note a crude lower bound $\\ell \\ge \\frac{1}{4}$ .", "We impose the property $\\ell < \\infty .$ Regard $\\ell $ as “normalized network length\", for the purpose of comparing different networks.", "Everything mentioned so far makes sense when only finite-dimensional distributions $\\mbox{${\\mathcal {R}}$}(z_i,z_j)$ are specified.", "A first context in which we want to consider a process over the whole continuum concerns the following convenient abstraction of the notion of “major road\".", "Write $\\mbox{${\\mathcal {R}}$}_{(1)}(z_1,z_2)$ for the part of the route $\\mbox{${\\mathcal {R}}$}(z_1,z_2)$ that is at distance $\\ge 1$ from each of $z_1$ and $z_2$ .", "Conceptually, we want to study an edge-process $\\mbox{${\\mathcal {E}}$}$ viewed as the union of $\\mbox{${\\mathcal {R}}$}_{(1)}(z_1,z_2)$ over all pairs $(z_1,z_2)$ .", "To formalize this directly would require some notion of “regularity\" for a realization, for instance some notion of a.e.", "continuity of routes $\\mbox{${\\mathcal {R}}$}(z_1,z_2)$ as $z_1$ and $z_2$ vary.", "But we can avoid this complication by first considering only $z_1, z_2$ in $\\Xi (\\lambda )$ and then letting $\\lambda \\rightarrow \\infty $ .", "After defining $\\mbox{${\\mathcal {E}}$}$ in this way, we can define $p(1) := \\mbox{mean length-per-unit-area of } \\mbox{${\\mathcal {E}}$}$ and impose the requirement $p(1) < \\infty .$ If a process of random routes $\\mbox{${\\mathcal {R}}$}(z,z^\\prime )$ satisfies the properties we have described (as stated precisely in section REF ), then we will call it a scale-invariant random spatial network (SIRSN).", "As the choice of name suggests, it is the scale-invariance that makes such processes of mathematical interest; in section REF we briefly discuss its plausibility for real-world networks.", "We do not know any closely related previous work.", "We will discuss one related area of theory (discrete random spatial networks; section REF ) and one area of application (fast algorithms for shortest routes:; section REF ).", "Several more distantly related topics are mentioned in section REF ." ], [ "Outline of paper", "The purpose of this paper is to initiate study of SIRSNs, with three emphases.", "First, we give a careful formulation of an axiomatic setup for SIRSNs, with discussion of possible alternatives (section ).", "Second, it is not obvious that SIRSNs exist at all!", "We give details of one construction in section .", "That construction envisages a square lattice of freeways, with “speed level $j$ \" freeways spaced $2^j$ apart, and the routes are the minimum time paths.", "Being based on the discrete lattice makes some estimates technically straightforward, but completing the details of proof requires surprisingly intricate arguments.", "This construction is somewhat artificial in not naturally having all the desired invariance properties, so these need to be forced by external randomization.", "We briefly mention two other constructions (in section REF based on a weighted Poisson line process representing the different-level freeways, and in section REF based on a dynamic construction of random points and roads added accoding to a deterministic rule) which intuitively seem more natural but for which we have been unable to complete all the details of a proof.", "Third, in sections - we begin developing some general theory from the axiomatic setup.", "Of course scale-invariance is a rather weak assumption, loosely analogous to stationarity for a stochastic process, so one cannot expect sharp results holding throughout this general class of process.", "Our general results might be termed “structure theory\" and concern existence and uniqueness issues for singly- and doubly-infinite geodesics, continuity of routes $\\mbox{${\\mathcal {R}}$}(z_1,z_2)$ as a function of $(z_1,z_2)$ , numbers of routes connecting disjoint subsets, and bounds on the parameters ${\\mathbb {E}}D_1, \\ell , p(1)$ .", "One feature worth emphasis is that (very loosely analogous to entropy rate for a stationary process) the quantity $p(1)$ is a non-obvious statistic of a SIRSN, but turns out to be central in the foundational setup, in the structure theory, and in conceptual interpretation as a model for road networks.", "The latter is best illustrated by the “algorithms\" story in sections REF and REF .", "Being a new topic there are numerous open problems, both conceptual and technical, stated in a final discussion section .", "Before starting technical material, sections REF - REF give further verbal discussion of background to the topic." ], [ "Discrete spatial networks", "Traditional models of (deterministic or random) spatial networks start with a discrete set of points and then assign linking edges via some rule, e.g.", "the random geometric graph [20] or proximity graphs [15], surveyed in [5].", "One specific motivation for the present work was as a second attempt to resolve a paradox – more accurately, an unwelcome feature of a naive model – in the discrete setting, observed in [8].", "In studying the trade-off between total network length and the effectiveness of a network in providing short routes between discrete cities, one's first thought might be to measure the latter by the average, over all pairs $(x,y)$ , of the ratio $\\mbox{(route-length for $x$ to $y$)}/\\mbox{(Euclidean distance from $x$ to $y$)}$ instead of averaging over pairs at Euclidean distance $\\approx r$ to get our ${\\mathbb {E}}D_r$ .", "But it turns out that (in the $n \\rightarrow \\infty $ limit of a network on $n$ points) one can make this ratio tend to 1 for a network whose length is only $1 +o(1)$ times the length of the Steiner tree, by simply superimposing on the Steiner tree a sparse Poisson line process.", "Such “theoretically optimal\" networks are completely unrealistic, so there must be something wrong with the optimization criteria.", "What's wrong is that the networks are ineffective for small $r$ .", "One way to get a non-trivial tradeoff in the $n \\rightarrow \\infty $ limit was described in [5]: using the statistic $\\max _r r^{-1} {\\mathbb {E}}D_r$ as the measure of effectiveness leads to more realistic-looking networks.", "In the discrete setting a network model cannot be precisely scale-invariant, but such considerations prompted investigation of continuum models which are assumed to be scale-invariant, so that $r^{-1} {\\mathbb {E}}D_r$ is constant.", "We emphasize that our networks involve roads at definite positions in the plane.", "There is substantial recent literature, discussed in [10], involving quite different notions of random planar networks, based on identifying topologically equivalent networks." ], [ "Visualizing spanning subnetworks", "Both construction and analysis of general SIRSNs are based on studying subnetworks $\\mathbf {span}(z_1,\\ldots ,z_k)$ on fixed or (most often) random points.", "It is helpful to visualize what subnetworks look like – see Figure 1.", "Figure 1.", "Schematic for the subnetwork of a SIRSN on 7 points $\\bullet $ The qualitative appearance of Figure 1 is quite different from that of familiar spatial networks mentioned above, based on a discrete set of points, which could be viewed as abstractions of an inter-city road network, with cities as points.", "In contrast, we are abstracting the idea of the points $\\bullet $ being individual street addresses a long way apart.", "The real-world route between two such street addresses will typically consist, in the middle, of roughly straight freeway segments but, nearing an endpoint, of a more jagged trajectory of shorter segments of lower-capacity roads; our setup and Proposition REF imply the same behavior in our model." ], [ "Very fast shortest path algorithms", "There is an interesting connection with the “shortest path algorithms\" literature.", "Online mapping services and GPS devices require very quick computations of shortest routes.", "In this context, the U.S road network is represented as a graph on about 15 million street intersections (vertices) with edges (road segments) marked by distance (or typical driving time), and a given street address is recognized as being between two specific street intersections.", "Given a pair of (starting and destination) points, one wants to compute the shortest route.", "Neither of the two extremes – pre-compute and store the routes for all possible pairs; or use a classical Dijkstra-style algorithm for a given pair without any preprocessing – is practical.", "Bast et al (see [9] for an outline) find a set of about 10,000 intersections (which they call transit nodes) with the property that, unless the start and destination points are close, the shortest route goes via some transit node near the start and some transit node near the destination.", "Given such a set, one can pre-compute shortest routes and route-lengths between each pair of transit nodes; then answer a query by using the classical algorithm to calculate the route lengths from starting (and from destination) point to each nearby transit node, and finally minimizing over pairs of such transit nodes.", "This idea is actually used commercially (and patented).", "Mathematical discussion was initiated by Abraham et al [1], who introduced the notion of highway dimension, defined as the smallest integer $h$ such that for every $r$ and every ball of radius $4r$ , there exists a set of $h$ vertices such that every shortest route of length $>r$ within the ball passes through some vertex in the set.", "They discuss several algorithms whose performance can be analyzed in terms of highway dimension, and devise a particular model (a dynamic spanner construction on vertices given by an adversary) designed to have bounded highway dimension.", "Now saying one can find $h$ independent of $r$ is a form of approximate scale-invariance, so the empirical fact that one can find transit nodes in the real-world road networks is a weak form of empirical scale-invariance.", "Within our model where precise scale-invariance is assumed, we can derive quantitative estimates relating to transit nodes – see section REF .", "Incidently, the way we define edge-processes $\\mbox{${\\mathcal {E}}$}= \\mbox{${\\mathcal {E}}$}(\\lambda ,r)$ in terms of routes (mentioned in the Introduction and defined in section REF ) is closely related to the notion of reach in the algorithmic literature [14]." ], [ "Visualizing scale-invariance", "Visualizing a photo of a road, scale-invariance seems implausible, because it implies existence of roads of arbitrarily large and arbitrarily small “sizes\", however one interprets “size\".", "But scale-invariance is not referring to the physical roads but to the process of “shortest routes\", as in the discussion above.", "Figure 2 illustrates one aspect of scale-invariance.", "There is some number of crossing places (over the line) used by routes from one square to the other square.", "In our model, scale-invariance implies that the mean number of such crossings does not depend on the scale of the map.", "One could test this as a prediction about real-world road networks.", "Figure 2.", "Schematic for long-distance routes.", "As another empirical aspect of scale-invariance, [16] studied proportions of route-length, within distance-$r$ routes, spent on the $i$ 'th longest road segment in the route (identifying roads by their highway number designation) and observe that in the U.S. the averages of these ordered proportions are around $(0.40, 0.20, 0.13, 0.08, 0.05)$ as $r$ varies over a range of medium to large distances.", "Again, in our models (identifying roads as straight segments) scale-invariance implies there is some vector of expected proportions that is precisely independent of $r$ ." ], [ "Technical setup", "In formulating an axiomatic setup there are several alternative choices one could make.", "In section REF we state concisely the choices we made; section REF discusses alternatives, reasons for choices, and immediate consequences or non-consequences of the setup." ], [ "Stochastic geometry background", "We quote a fundamental identity from stochastic geometry (see [22] Chapter 8).", "Let $\\mbox{${\\mathcal {E}}$}$ be an edge process – for our purposes, a union of line segments – whose distribution is invariant under translation and rotation.", "Then $\\mbox{${\\mathcal {E}}$}$ has an edge-intensity, a constant $\\iota = \\mathrm {intensity}(\\mbox{${\\mathcal {E}}$}) \\in [0,\\infty ]$ such that $E (\\mbox{length of } \\mbox{${\\mathcal {E}}$}\\cap A) = \\iota \\times \\mathrm {area}(A), \\quad A \\subset {\\mathbb {R}}^2 .$ Moreover the positions and angles at which $\\mbox{${\\mathcal {E}}$}$ intersects the $x$ -axis (and hence any other line) are such that $\\mbox{mean number intersections per unit length} = 2 \\pi ^{-1} \\times \\mathrm {intensity}(\\mbox{${\\mathcal {E}}$})$ and the random angle $\\Theta \\in (0,\\pi )$ of a typical intersection has density $f_\\Theta (\\theta ) = {\\textstyle \\frac{1}{2}} \\sin \\theta .$" ], [ "Definitions", "Here we organize the setup via four aspects." ], [ "Some notation.", "$\\mathbf {0}$ denotes the origin; $\\mathrm {disc}(z, r)$ and $\\mathrm {circle}(z, r)$ denote the closed disc and the circle centered at $z$ ." ], [ "Aspect 1. Allowed routes and route-compatability.", "Define a jagged route between two points $z, z^\\prime $ of ${\\mathbb {R}}^2$ to consist of straight line segments between successive points $(z_i, - \\infty < i < \\infty ) $ with $\\lim _{i \\rightarrow - \\infty } z_i = z$ and $\\lim _{i \\rightarrow \\infty } z_i = z^\\prime $ , and such that the total length $\\sum _{i = -\\infty }^\\infty |z_i - z_{i-1}|$ is finite.", "A feasible route is either a jagged route or the variant with a finite or semi-infinite set of successive line segments; we further require that the route be non-self-intersecting.", "Write $\\mathsf {r}(z,z^\\prime )$ for a feasible route, which from now on we will just call route.", "We envisage a route $\\mbox{${\\mathsf {r}}$}(z,z^\\prime )$ as a one-dimensional subset of ${\\mathbb {R}}^2$ , equipped with a label indicating it is the route from $z$ to $z^\\prime $ .", "The route $\\mbox{${\\mathsf {r}}$}(z^\\prime ,z)$ is always the reversal of $\\mbox{${\\mathsf {r}}$}(z,z^\\prime )$ .", "When we have a collection of routes, we require the following pairwise compatability property.", "$ \\mbox{ If two routes $\\mathsf {r}(z_1,z_j), \\ \\mathsf {r}(z^\\prime _1,z^\\prime _2)$ meet at two points then the routes } $ $\\mbox{coincide on the subroute between the two meeting points.", "}$" ], [ "Aspect 2. Subnetworks on locally finite configurations.", "Given a locally finite configuration of points $(z_i)$ in the plane, and routes $\\mbox{${\\mathsf {r}}$}(z_i,z_j)$ satisfying the pairwise compatability property, write $\\mbox{${\\mathsf {s}}$}$ for the union of all these routes.", "If $\\mbox{${\\mathsf {s}}$}$ has the “finite length in bounded regions\" property $\\mathrm {len}(\\mbox{${\\mathsf {s}}$}\\cap \\mathrm {disc}(\\mathbf {0},r) ) < \\infty \\mbox{ for each } r < \\infty $ then call $\\mbox{${\\mathsf {s}}$}$ a feasible subnetwork.", "Here “$\\mathrm {len}$ ' denotes “length\".", "Formally $\\mbox{${\\mathsf {s}}$}$ consists of the vertex set $(z_i)$ , an edge set which is the union of the edge sets comprising each $\\mbox{${\\mathsf {r}}$}(z_i,z_j)$ , and marks on edges to indicate which routes they are in.", "Inclusion $\\mbox{${\\mathsf {s}}$}(1) \\subseteq \\mbox{${\\mathsf {s}}$}(2)$ means that $\\mbox{${\\mathsf {s}}$}(2)$ can be obtained from $\\mbox{${\\mathsf {s}}$}(1)$ by adding extra vertices and associated routes.", "As outlined in section REF there is a natural $\\sigma $ -field that makes the set of all feasible subnetworks into a measurable space, so it makes sense below to talk about random feasible subnetworks." ], [ "Aspect 3. Desired distributional properties of subnetworks.", "The precise definition of the class of processes we shall study uses “finite-dimensional distributions\" (FDDs), as follows.", "Given a finite set $z_1,\\ldots ,z_k$ let $\\mu _{z_1,\\ldots ,z_k}$ be the distribution of a random feasible subnetwork $\\mathbf {span}(z_1,\\ldots ,z_k)$ on $z_1,\\ldots ,z_k$ .", "Suppose a family (indexed by all finite sets) of FDDs satisfies $&&\\mbox{the natural consistency condition} \\\\&&\\mbox{invariance under translation and rotation} \\\\&&\\mbox{invariance under scaling.}", "$ To be precise about (), recall that the scaling map $\\sigma _c: {\\mathbb {R}}^2 \\rightarrow {\\mathbb {R}}^2$ takes $z$ to $cz$ .", "Then the action of $\\sigma _c$ on $\\mathbf {span}(z_1,\\ldots ,z_k)$ gives a random subnetwork whose distribution equals the distribution of $\\mathbf {span}(\\sigma _c z_1,\\ldots ,\\sigma _c z_k)$ .", "Appealing to the Kolmogorov extension theorem, we can associate with such a family a process of routes $\\mbox{${\\mathcal {R}}$}(z_1,z_2)$ , for each pair $z_1, z_2$ in ${\\mathbb {R}}^2$ , though for a process defined in that way we can only discuss properties determined by FDDs.", "As mentioned earlier, much of our study involves sampled spanning subnetworks, as follows.", "For each $0<\\lambda <\\infty $ let $\\Xi (\\lambda )$ be a Poisson point process of intensity $\\lambda $ (we sometimes call this point-intensity to distinguish from edge-intensity at (REF )).", "Make a process $(\\Xi (\\lambda ), \\ 0<\\lambda <\\infty )$ by coupling in the natural way (take a space-time Poisson point process and let $\\Xi (\\lambda )$ be the positions of points arriving during time $[0,\\lambda ]$ ).", "Taking Poisson points independent of the process of routes, we can define $\\mbox{${\\mathcal {S}}$}(\\lambda )$ as the subnetwork of routes $\\mbox{${\\mathcal {R}}$}(\\xi , \\xi ^\\prime ) $ for pairs $\\xi , \\xi ^\\prime $ in $\\Xi (\\lambda )$ .", "We want the resulting processes $\\mbox{${\\mathcal {S}}$}(\\lambda )$ to have the following properties.", "$&&\\mbox{for each $\\lambda $, $\\mbox{${\\mathcal {S}}$}(\\lambda )$ is a random feasible subnetwork on vertex-set $\\Xi (\\lambda )$} \\\\&&\\mbox{for each $\\lambda $, $\\mbox{${\\mathcal {S}}$}(\\lambda )$ has translation- and rotation-invariant distribution } \\\\&&\\mbox{$\\mbox{${\\mathcal {S}}$}(\\lambda _1) \\subseteq \\mbox{${\\mathcal {S}}$}(\\lambda _2)$ for $\\lambda _1 < \\lambda _2$} \\\\&&\\mbox{applying $\\sigma _c$ to $\\mbox{${\\mathcal {S}}$}(\\lambda )$ gives a network distributed as $\\mbox{${\\mathcal {S}}$}(c^{-2} \\lambda )$} .$ For (), recall that applying $\\sigma _c$ to $\\Xi (\\lambda )$ gives a point process distributed as $\\Xi (c^{-2} \\lambda )$ .", "We omit full measure-theoretic details of the construction of $\\mbox{${\\mathcal {S}}$}(\\lambda )$ , and just point out what extra conditions are needed to obtain properties (REF - ).", "First, we need to impose the technical condition $\\mbox{the map $(z_1,\\ldots ,z_k) \\rightarrow \\mu _{z_1,\\ldots ,z_k} $ is measurable}$ to ensure that $\\mbox{${\\mathcal {S}}$}(\\lambda )$ is measurable.", "Second, part of the “feasible\" assertion in (REF ) is the “finite length in bounded regions\" property (REF ), and this property for $\\mbox{${\\mathcal {S}}$}(\\lambda )$ cannot be a consequence of assumptions on FDDs only, so we need $\\mathrm {len}(\\mbox{${\\mathcal {S}}$}(\\lambda ) \\cap \\mathrm {disc}(\\mathbf {0},r) ) < \\infty \\ \\mbox{ a.s. for each } r < \\infty $ and this will follow from the stronger assumption (REF ) below." ], [ "Aspect 4. Final definition of a SIRSN.", "To summarize the above: given a process of routes $\\mbox{${\\mathcal {R}}$}(z_1,z_2)$ with FDDs satisfying (REF - , REF ), we can define the process of sampled subnetworks $(\\mbox{${\\mathcal {S}}$}(\\lambda ), 0 < \\lambda < \\infty )$ which, if (REF ) holds, will have properties (REF - ).", "Finally, we define a SIRSN as a process (denoted by the routes $\\mbox{${\\mathcal {R}}$}(z_1,z_2)$ or by the sampled subnetworks $(\\mbox{${\\mathcal {S}}$}(\\lambda ), 0 < \\lambda < \\infty )$ ) satisfying these assumptions (REF - , REF ) and also satisfying the extra conditions (REF ,REF ) below.", "These extra conditions merely repeat and formalize the requirements, stated in the introduction, that certain statistics be finite.", "As noted above, these assumptions imply that (REF - ) hold.", "Write $\\mathbf {1}= (1,0)$ and $D_1: = \\mathrm {len}\\ \\mbox{${\\mathcal {R}}$}(\\mathbf {0}, \\mathbf {1})$ .", "So $D_1$ represents route-length between points at distance 1 apart.", "Our definition of feasible route implies $1 \\le D_1 < \\infty $ a.s., and we impose the requirement $1 < {\\mathbb {E}}D_1 < \\infty .$ Next, our definition of feasible subnetwork implies that $\\mbox{${\\mathcal {S}}$}(1)$ must have a.s. finite length in a bounded region.", "We impose the stronger requirement of finite expected length.", "In terms of the edge-intensity (REF ), we require $\\ell := \\mathrm {intensity}( \\mbox{${\\mathcal {S}}$}(1)) < \\infty .$ Finally, we define $\\mbox{${\\mathcal {E}}$}(\\lambda , r) := \\bigcup _{\\xi , \\xi ^\\prime \\in \\Xi (\\lambda )} \\ \\ \\mbox{${\\mathcal {R}}$}(\\xi ,\\xi ^\\prime ) \\setminus (\\mathrm {disc}(\\xi ,r) \\cup \\mathrm {disc}(\\xi ^\\prime ,r))$ and edge-intensities $p(\\lambda , r) &:=& \\mathrm {intensity}(\\mbox{${\\mathcal {E}}$}(\\lambda ,r)) \\\\p(r) &:=& \\lim _{\\lambda \\rightarrow \\infty } p(\\lambda ,r) $ and impose the requirement $p(1) < \\infty $ whose significance is discussed in the next section.", "Lemma REF will show that (REF ) implies (REF ).", "If we do not require (REF ) but instead require (REF ), call the process a weak SIRSN." ], [ "Aspect 1. Allowed routes and route-compatability.", "Because we want routes to have a well-defined lengths, a minimum assumption would be that routes are rectifiable curves.", "We have assumed “feasible routes\" in order to simplify notation.", "We believe that the theory would be essentially unchanged if instead one allowed rectifiable curves, as in the (quite different) theory mentioned in section REF .", "It turns out (section REF ) that realizations of our models always have jagged routes.", "A consequence is that (as in Figure 1) a route $\\mbox{${\\mathcal {R}}$}(\\xi ,\\xi ^\\prime )$ between two points of $\\mbox{${\\mathcal {S}}$}(\\lambda )$ does not pass through any third point $\\xi ^{\\prime \\prime }$ of $\\mbox{${\\mathcal {S}}$}(\\lambda )$ .", "This prompts the precise definition of geodesic below.", "The route-compatability property is a property that would hold if routes were defined as minimum-cost paths, for some reasonable notion of “cost\".", "Note that our formal setup does not require routes to be minimum-cost in any explicit sense." ], [ "Aspect 2. Subnetworks on locally finite configurations.", "Here are some properties of a fixed feasible subnetwork.", "Lemma 1 Let $\\mbox{${\\mathsf {s}}$}$ be a feasible subnetwork on a locally finite, infinite configuration $(z_i)$ .", "(i) The set $ \\lbrace \\mbox{${\\mathsf {r}}$}(z_i,z_j) \\cap \\mathrm {disc}(z, r)\\rbrace _{i,j}$ of sub-routes appearing as intersections of some route with a fixed disc $\\mathrm {disc}(z, r)$ contains only finitely many distinct (non-identical) sub-routes.", "(ii) For each $i$ and each sequence $(z_j)$ with $|z_j| \\rightarrow \\infty $ there is a subsequence $z^\\prime _k = z_{j(k)}$ and a semi-infinite path $\\pi $ from $z_i$ in $\\mbox{${\\mathsf {s}}$}$ such that, for each $r > 0$ , $ \\mbox{${\\mathsf {r}}$}(z_i,z^\\prime _k) \\cap \\mathrm {disc}(z_i, r) = \\pi \\cap \\mathrm {disc}(z_i, r)\\mbox{ for all large } k .", "$ Proof.", "(ii) follows from (i) by a compactness argument.", "To outline (i), if false then (by the route-compatability property) the subroutes must meet the disc boundary at an infinite number of distinct points, and then (again by the route-compatability property) their extensions must meet the boundary of a slightly larger disc at an infinite number of distinct points, implying infinite length and contradicting the “finite length in bounded regions\" property (REF ) of $\\mbox{${\\mathsf {s}}$}$ .", "Note that Lemma REF is implicitly about compactness in a topology on the space of paths within a given subnetwork $\\mbox{${\\mathsf {s}}$}$ .", "This is quite different from the topology of the space of all subnetworks, mentioned later." ], [ "Terminology: paths, routes and geodesics.", "A path in $\\mbox{${\\mathsf {s}}$}$ has its usual network meaning.", "Typically there will be many paths between $z_i$ and $z_j$ , but (as part of the structure of a feasible subnetwork) one is distinguished as the route $\\mbox{${\\mathsf {r}}$}(z_i,z_j)$ .", "So a route is a path; and a path may or may not be part of one or more routes.", "A singly infinite geodesic in $\\mbox{${\\mathsf {s}}$}$ from $z_i$ is an infinite path, starting from $z_i$ , such that any finite portion of the path is a subroute of the route $\\mbox{${\\mathsf {r}}$}(z_i,z_k)$ for some $z_k$ .", "So Lemma REF (ii) says that there always exists at least one singly infinite geodesic from $z_i$ .", "A typical point $\\epsilon $ along a route $\\mbox{${\\mathsf {r}}$}(z_i,z_j)$ will sometimes be called a path element to distinguish it from the endpoints.", "Now write $\\mathfrak {S}$ for the set of all feasible subnetworks $\\mbox{${\\mathsf {s}}$}$ on all locally finite configurations ${\\mathbf {x}}= (x_j)$ .", "It is natural to want to regard $\\mbox{${\\mathcal {S}}$}(\\lambda )$ as a random element of $\\mathfrak {S}$ , which requires specifying a $\\sigma $ -field on $\\mathfrak {S}$ , and as traditional we can do this by specifying a complete separable metric space structure on $\\mathfrak {S}$ and using the Borel $\\sigma $ -field.", "We outline a “natural\" topology in an appendix.", "In this paper the topology plays no explicit role, but one can imagine developments where it does – one can imagine constructions using weak convergence, for instance, and compactness issues would be key to a proof of the existence part of Open Problem REF .", "However, it might be better to develop such theory within a framework where routes are allowed to be rectifiable curves." ], [ "Aspect 3. Desired distributional properties of subnetworks.", "The scale-invariance property () $ \\mbox{applying $\\sigma _c$ to $\\mbox{${\\mathcal {S}}$}(\\lambda )$ gives a network distributed as $\\mbox{${\\mathcal {S}}$}(c^{-2} \\lambda )$} $ is what gives SIRSNs a mathematically interesting structure, and almost all our general results in sections and rely on scale-invariance.", "To indicate how it is used, define $\\ell (\\lambda )$ analogously to (REF ): $\\ell (\\lambda ) := \\mathrm {intensity}( \\mbox{${\\mathcal {S}}$}(\\lambda ))$ so $\\ell (1) = \\ell $ .", "Then there is a scaling relation $\\ell (\\lambda ) = \\lambda ^{1/2} \\ell , \\quad 0 < \\lambda < \\infty .$ To derive this relation, consider the scaling map $\\sigma _{\\lambda ^{-1/2}}$ that takes $\\mbox{${\\mathcal {S}}$}(1)$ to $\\mbox{${\\mathcal {S}}$}(\\lambda )$ , and by considering the pre-image $A = [0,\\lambda ^{1/2}]^2$ of the unit square we see $ \\ell (\\lambda ) = \\lambda ^{-1/2} \\times \\mathrm {area}(A) \\times \\ell $ where the $\\lambda ^{-1/2}$ term is length rescaling.", "Similar relations, provable in the same way, will be stated later (REF ,REF , REF ) without repeating the proof." ], [ "Aspect 4. Final definition of a SIRSN.", "Starting from FDDs, a conceptual and technical issue is how to continue to understand a SIRSN as a process over the whole continuum.", "As an analogy, for continuous-time stochastic processes one typically seeks some sample path regularity property such as càdlàg.", "So one might seek some notion of “regularity\" for a realization, for instance a.e.", "continuity of routes $\\mbox{${\\mathcal {R}}$}(z_1,z_2)$ as $z_1$ and $z_2$ vary.", "A version of such continuity is proved, under extra assumptions, in section REF .", "But as we next explain, in the present context the assumption $p(1) < \\infty $ serves as an alternative regularity condition that enables us to study global properties of a SIRSN.", "There are several possible real-world measures of “size\" of a road segment, quantifying the minor road to major road spectrum – e.g.", "number of lanes; level in a highway classification system; traffic volume.", "What about within our model of a SIRSN?", "Recalling the definition (REF ) of $\\mbox{${\\mathcal {E}}$}(\\lambda , r)$ , the limit $ \\mbox{${\\mathcal {E}}$}(\\infty , r):= \\cup _{\\lambda <\\infty } \\mbox{${\\mathcal {E}}$}(\\lambda , r) $ has (because $\\cup _{\\lambda <\\infty } \\Xi (\\lambda )$ is dense) the interpretation of “the set of path elements $\\epsilon $ that are on some route $\\mbox{${\\mathcal {R}}$}(z_1,z_2)$ with both $z_1$ and $z_2$ at distance $> r$ from $\\epsilon $ \".", "As shown in section , assumption (REF ) implies that the edge-intensity $p(r)$ of $\\mbox{${\\mathcal {E}}$}(\\infty ,r)$ is finite and scales as $p(r) = p(1)/r$ .", "Moreover the random process $\\mbox{${\\mathcal {E}}$}(\\infty , r)$ is independent of the sampling process $(\\Xi (\\lambda ), 0 < \\lambda < \\infty )$ and is an intrinsic part of the global structure of the SIRSN.", "So if we intuitively interpret $\\mbox{${\\mathcal {E}}$}(\\infty ,r)$ as “the roads of size $\\ge r$ \", then we have a mathematically convenient notion of “size of a road segment\" emerging from our setup without explicit design.", "Intuitively, one could view the limit $\\mbox{${\\mathcal {E}}$}(\\infty , 0+) := \\cup _{r>0} \\ \\mbox{${\\mathcal {E}}$}(\\infty ,r)$ as the continuum network of interest.", "But at a technical level it is not clear what are the properties of a realization of $\\mbox{${\\mathcal {E}}$}(\\infty , 0+)$ , and we do not study it in this paper." ], [ "The binary hierarchy model", "The construction of this model, our basic example of a SIRSN, occupies all of section , in several steps.", "A construction on the integer lattice (sections REF - REF ) Extension to the plane (sections REF - REF ) Further randomization to obtain invariance properties (section REF )." ], [ "Routes on the lattice", "For an integer $x \\ne 0$ , write $\\mathrm {height}(x)$ for the largest $j \\in {\\mathbb {Z}}^+$ such that $2^j$ divides $x$ ; in other words the unique $j$ such that $x = (2k+1)2^j$ for some $k \\in {\\mathbb {Z}}$ .", "Set $\\mathrm {height}(0) = \\infty $ .", "For later use note that in one dimension, any integer interval $[m_1,m_2]$ contains a unique integer of maximal height, which we call $\\mathrm {peak}[m_1,m_2]$ .", "For instance $\\mathrm {peak}[67,99] = 96$ and $\\mathrm {peak}[34,59] = 48$ .", "Until section REF we will work on the integer lattice ${\\mathbb {Z}}^2$ , with vertices $z = (x,y)$ whose coordinates have heights $\\ge 0$ .", "While we are working on the lattice it is convenient to use $L^1$ distance $||z_2 - z_1||_1 := |x_2-x_1| + |y_2-y_1|$ .", "Note also that until section REF we work with deterministic constructions.", "Write $L^{(X)}_x$ and $L^{(Y)}_y$ for the lines through $\\lbrace (x,y), y \\in {\\mathbb {Z}}\\rbrace $ and $\\lbrace (x,y), x \\in {\\mathbb {Z}}\\rbrace $ .", "The height of a line $L^{(X)}_x$ is the height of $x$ .", "Fix a parameter $ 1/2 < \\gamma < 1$ .", "Associate with lines at height $h$ a cost-per-unit length equal to $\\gamma ^h$ .", "Now each path in the lattice has a cost, being the sum of the edge costs.", "Visualize a road network in which one can travel along a height-$h$ road at speed $1/\\gamma ^h$ ; so the cost equals time taken.", "Define the route $\\mbox{${\\mathsf {r}}$}(z_1,z_2)$ to be a minimum-cost path between $z_1$ and $z_2$ .", "There is a uniqueness issue: for instance, for any minimum-cost path from $(i,i)$ to $(j,j)$ there is an equal cost path obtained by reflection $(x,y) \\rightarrow (y,x)$ .", "However, the estimates from here through section REF hold when $\\mbox{${\\mathsf {r}}$}(z_1,z_2)$ is any choice of minimum-cost path.", "We will deal with uniqueness in section REF .", "A key point of the construction is that if we scale space by 2 then the scaled structure on the even lattice $(2{\\mathbb {Z}})^2$ agrees with the original substructure on the even lattice, up to a constant multiplicative factor in edge-costs, and so the route between two even points will be the same whether we work in ${\\mathbb {Z}}^2$ or $(2{\\mathbb {Z}})^2$ .", "So this “invariance under scaling by 2\" property is built into the model at the start.", "The fact that moving along the axes has zero cost may seem worrrying but actually causes no difficulty (we will later apply a random translation, and the original axes do not appear in the final process).", "Note that the cost associated with the line segment from $(2^h,2^h)$ to $(2^h,0)$ is $\\gamma ^h 2^h$ and the constraint $\\gamma > 1/2$ is needed to make this cost increase with $h$ .", "Intuitively, if $\\gamma $ is near 1 then the route $\\mbox{${\\mathsf {r}}$}(z_1,z_2)$ will stay inside or near the rectangle with opposite corners $z_1, z_2$ , whereas if $\\gamma $ is near $1/2$ then the route may go far away from the rectangle to exploit high-speed roads.", "For this model we will show a property stronger than (REF ); the ratio of route-length to distance is uniformly bounded.", "Proposition 2 There is a constant $K_\\gamma < \\infty $ such that $ \\mathrm {len}\\ \\mbox{${\\mathsf {r}}$}(z_1,z_2) \\le K_\\gamma ||z_2 - z_1||_1, \\quad \\forall z_1, z_2 \\in {\\mathbb {Z}}^2 .", "$ Some intuition about possible paths in this model is provided by Figure 3 (the reader should imagine the ratios of longer/shorter edge lengths as larger than drawn).", "We might have a route as shown in the figure, where the two long edges are very fast freeways.", "But such a route is not possible if the fast freeways are too far from the start and destination points.", "The latter assertion will follow from Lemma REF .", "Figure 3.", "Routes like this are possible.", "One might expect some explicit algorithmic description of routes $\\mbox{${\\mathsf {r}}$}(z_1,z_2)$ that one can use to prove the results in sections REF - REF , but we have been unable to do so.", "Instead our proofs rely on finding internal structural properties that routes must have." ], [ "Analysis of routes in the deterministic model", "Consider the route from $z_1 = (x_1,y_1)$ to $z_2 = (x_2,y_2)$ .", "The $x$ -values taken on the route form some interval $I_x \\supseteq [\\min (x_1,x_2), \\max (x_1,x_2)]$ , and similarly the $y$ -values form some interval $I_y$ .", "Consider the point $z^* = (x^*,y^*) = (\\mathrm {peak}(I_x),\\mathrm {peak}(I_y))$ and call this point $\\mathrm {peak}^{(2)}\\mbox{${\\mathsf {r}}$}(z_1,z_2)$ .", "The notation reminds us that $\\mathrm {peak}^{(2)}\\mbox{${\\mathsf {r}}$}(z_1,z_2)$ depends on the route $\\mbox{${\\mathsf {r}}$}(z_1,z_2)$ , which may not be unique.", "Lemma 3 Consider the route $\\mbox{${\\mathsf {r}}$}(z_1,z_2)$ from $z_1$ to $z_2$ .", "(i) The route passes through $z^* = \\mathrm {peak}^{(2)}\\mbox{${\\mathsf {r}}$}(z_1,z_2)$ .", "(ii) The route meets the line $L^{(X)}_{x^*}$ in either the single point $z^*$ or in one line segment containing $z^*$ (and similarly for $L^{(Y)}_{y^*}$ ).", "(iii) Suppose the route passes through a point $(x^*,y)$ (for some $y \\ne y^*$ ) and through a point $(x,y^*)$ (for some $x \\ne x^*$ ).", "Then the route between those points is the two-segment route via $z^*$ .", "(iv) Suppose $z^* = z_1$ .", "If $z_2$ is in a certain quadrant relative to $z_1$ , for instance the quadrant $[x_1,\\infty ) \\times [y_1,\\infty )$ , then the route from $z_1$ to $z_2$ stays in that quadrant.", "Proof.", "We first prove (iii).", "It is enough to prove that, amongst routes between $(x^*,y)$ and $(x,y^*)$ , the two-segment route via $z^*$ is the unique minimum-cost route.", "In order to get from $(x^*,y)$ to the line $L^{(Y)}_{y^*}$ the route must use at least $|y - y^*|$ vertical unit edges; by definition of $x^* = \\mathrm {peak}(I_x)$ , if these edges are not precisely the line segment from $(x^*,y)$ to $z^*$ then the cost of these edges will be strictly larger; and similarly for horizontal edges.", "This establishes the uniqueness assertion above, and hence (iii).", "For (i), if the hypothesis of (iii) fails then the route must go through $z^*$ , whereas if it holds then the conclusion of (iii) implies the route goes through $z^*$ .", "For (ii), if false then the the route passes through some two points $(x^*,y^\\prime )$ and $(x^*,y^{\\prime \\prime })$ but not the intervening points on that line.", "But (as in the argument for (iii)) the minimum cost path between those two points is the direct line between them.", "Finally, (iv) follows from (ii), because if (iv) fails then the route meets one boundary of the quadrant in more than one segment.", "Lemma 4 The route $\\mbox{${\\mathsf {r}}$}(z_1,z_2)$ from $z_1$ to $z_2$ stays within the square of side $K^\\prime _\\gamma \\ ||z_2-z_1||_1$ centered at $z_1$ , where $K^\\prime _\\gamma $ depends only on $\\gamma $ .", "Proof.", "Choose the integer $h$ such that $ 2^{h-1} < ||z_2 - z_1||_1 \\le 2^h .", "$ As illustrated in Figure 4, there is a square of the form $S = [(i-1)2^h, (i+1)2^h] \\times [(j-1)2^h,(j+1)2^h]$ containing both $z_1$ and $z_2$ (note here $i$ and $j$ may be even or odd).", "We may suppose the route does not stay within $S$ (otherwise the result is trivial).", "For any point $z$ outside $S$ , call the $L^\\infty $ distance from $z$ to $S$ , that is the number $d$ such that $z$ is on the boundary of the concentric square $S_d = [(i-1)2^h - d, (i+1)2^h + d] \\times [(j-1)2^h - d,(j+1)2^h + d]$ , the displacement of $z$ .", "Now let $d$ be the maximum displacement along the route $\\mbox{${\\mathsf {r}}$}(z_1,z_2)$ , and choose a point $z^\\prime $ along the route with displacement $d$ .", "So the route stays within $S_d$ .", "$z_1$$z_2$$d$$2^h$$S$$S_d$$z^\\prime $Figure 4.", "Construction for proof of Lemma REF .", "We may assume, as in Figure 4, that $z^\\prime $ is on the top edge of $S_d$ .", "The route needs to cover the vertical distance $d$ between the top edges of $S$ and $S_d$ twice (up and down) while staying within $S_d$ , which has side-length $2^{h+1} + 2d$ .", "Now within any integer interval of length $a$ the second-largest height $H$ satisfies $2^H \\le a$ .", "So the cost ($C$ , say) of the route outside $S$ is at least the cost associated with this second-largest height, which is given by $ d \\gamma ^H \\mbox{ where } 2^H \\le 2^{h+1} + 2d.", "$ Setting $d = b2^h$ , this inequality implies $ \\log _2 C \\ge \\log _2 b + h + (h+1 + \\log _2 (1+b)) \\log _2 \\gamma .", "$ But for this to be the minimum-cost path, the cost outside $S$ must be less than the cost of going round the boundary of $S$ , which is at most $\\gamma ^h \\times 2^{h+2}$ .", "So $ \\log _2 C \\le h \\log _2 \\gamma + h + 2 .", "$ This inequalities combine to show $ \\log _2 b + (1 + \\log _2 (1+b)) \\log _2 \\gamma \\le 2 $ which, because $\\gamma > 1/2$ , implies that $b$ is bounded by some constant $b_\\gamma $ .", "Lemma REF makes Proposition REF look very plausible, but to prove it we need to extend Lemma REF to develop internal structural properties that routes must have.", "Call a sequence of integers $i_1, i_2, \\ldots , i_m$ a height-monotone sequence from $i_1$ to $i_m$ if (i) $\\mathrm {height}(i_1) > \\mathrm {height}(i_2) > \\ldots > \\mathrm {height}(i_m) \\ge 0 $ ; (ii) $|i_{j+1} - i_j| < 2^{\\mathrm {height}(i_j)}, \\quad 1 \\le j < m$ .", "Suppose, for integers $m_1, m_2, m^*$ , we are given a height-monotone sequence $m^* = i_1, i_2, \\ldots , i_m = m_2$ and a height-monotone sequence $m^* = j_1, j_2, \\ldots , j_q = m_1$ .", "Then we can form the concatenation $m_1 = j_q, j_{q-1}, \\ldots , j_2, m^*, 1_2, \\ldots , i_m = m_2$ .", "Call a sequence that arises this way an admissable sequence from $m_1$ to $m_2$ .", "See Figure 5.", "Table: NO_CAPTIONFigure 5.", "An admissable path from 75 to 99.", "This path has range $100 - 72 = 28$ .", "Regard a height-monotone or admissable sequence as a path of steps where a step from $i$ to $j$ has length $|j-i|$ .", "It is clear from (ii) that the length of the path in (i) is at most twice the length of the first step.", "We deduce the following crude bound.", "(*) The total length of an admissable path is at most 4 times the range of the path, where the range is the difference between the maximum and minimum integer points visited by the path.", "Proposition REF follows immediately from Lemma REF , the bound (*) above and the following lemma.", "Lemma 5 The route from $z_1 = (x_1,y_1)$ to $z_2 = (x_2,y_2)$ consists of alternating horizontal and vertical segments, in which the successive distinct $x$ -values of the segment ends (the turning points) form an admissable sequence from $x_1$ to $x_2$ , and the successive distinct $y$ -values form an admissable sequence from $y_1$ to $y_2$ .", "Proof.", "In view of Lemma REF we can reduce to the case where $z_1 = \\mathrm {peak}^{(2)}\\mbox{${\\mathsf {r}}$}(z_1,z_2)$ , and we need to show that the successive distinct $x$ -values form a height-monotone sequence, as do the $y$ -values.", "Without loss of generality suppose that $x_1 \\le x_2$ , that $y_1 \\le y_2$ and that the first segment is horizontal.", "So the route is of the form $ (x_1,y_1) = (x_{(1)},y_{(1)}) \\rightarrow (x_{(2)},y_{(1)}) \\rightarrow (x_{(2)}, y_{(2)}) \\rightarrow (x_{(3)}, y_{(2)}) \\rightarrow \\ldots $ It suffices to show that for each edge of the route, say the edge $(x_{(i)}, y_{(i)}) \\rightarrow (x_{(i+1)}, y_{(i)})$ , and for each point (say $(x^*,y_{(i)})$ ) on the edge other than the starting point, we have $\\mathrm {height}(x^*) < \\mathrm {height}(x_{(i)})$ .", "This is true for the first two edges of the route by definition of $z_1$ as $ \\mathrm {peak}^{(2)}\\mbox{${\\mathsf {r}}$}(z_1,z_2)$ .", "Suppose it fails first at some point $(x^*,y_{(i)})$ .", "Then the route has proceeded $(x_{(i)},y_{(i-1)}) \\rightarrow (x_{(i)},y_{(i)}) \\rightarrow (x^*,y_{(i)})$ instead of the alternate path via $(x^*,y_{(i-1)})$ .", "Now inductively $\\mathrm {height}(y_{(i)}) < \\mathrm {height}(y_{(i-1)})$ , so the cost of the horizontal edge is less in the alternate path; so for the route to have smaller cost it must happen that the cost of its vertical edge is smaller than in the alternate path, that is $\\mathrm {height}(x_{(i)}) > \\mathrm {height}(x^*)$ , contradicting the supposed failure." ], [ "Further technical estimates", "The next lemma will be key to bounding network length, more specifically to showing $\\ell < \\infty $ later.", "Lemma 6 There exists an integer $b \\ge 1$ , depending only on $\\gamma $ , such that for all $h \\ge 0$ and all rectangles of the form $[i2^{h+b}, (i+1)2^{h+b}] \\times [j2^h, (j+1)2^h]$ , the route $\\mbox{${\\mathsf {r}}$}(z_1,z_2)$ between two points $z_1, z_2 \\in {\\mathbb {Z}}^2$ outside (or on the boundary of) the rectangle does not use any horizontal edge strictly inside the rectangle.", "Note there may be routes using a vertical line straight through the rectangle.", "Proof.", "Suppose false; then there are two points $z_1, z_2$ on the boundary of the rectangle such that the route between them lies strictly within the rectangle and contains a horizontal edge.", "Because the speed on an interior edge is less than the speed on a parallel boundary edge, this cannot happen when $z_1$ and $z_2$ are in the same or adjacent boundaries of the rectangle, because the path around the boundary is faster.", "Suppose they are on the top and the bottom boundaries.", "Then the height of the horizontal edge is less than the heights of the starting and ending $y$ -values, contradicting Lemma REF .", "The only remaining case is when $z_1$ and $z_2$ are on the left and right boundaries.", "Using Lemma REF again, the route cannot use a vertical edge inside the rectangle, so the only possibility is a single horizontal segment passing through the rectangle.", "Such a path has cost at least $2^{h+b} \\ \\gamma ^{h-1}$ , because the height of the line is at most $h-1$ , whereas the path around the boundary has cost at most $2^{h+b} \\ \\gamma ^{h} + 2^h \\ \\gamma ^{b+h}$ .", "So the potential route is impossible when $2^b + \\gamma ^b < 2^b \\gamma ^{-1}$ which holds for sufficiently large $b$ .", "Corollary 7 If a route $\\mbox{${\\mathsf {r}}$}(z_1,z_2)$ uses a height-$h$ segment through $z_0$ , then $\\min (||z_1-z_0||_1 , ||z_2-z_0||_1) \\le 2^h(2^b + 1) $ for $b$ as in Lemma REF .", "Proof.", "Consider a unit-length horizontal (without loss of generality) edge of height $h$ at $z_0$ .", "It is in the interior of some rectangle of the form $[i2^{h+1+b}, (i+1)2^{h+1+b}] \\times [j2^{h+1}, (j+1)2^{h+1}]$ .", "By Lemma REF applied with $h+1$ , either $z_1$ or $z_2$ must be within that rectangle.", "Perhaps surprisingly, we do not make much explicit use of the deterministic function $\\mbox{{\\bf cost}}(z_1,z_2)$ giving the cost of the minimum-cost route in the integer lattice, but will need the following bound.", "Lemma 8 There exists a constant $K^{\\prime \\prime }_\\gamma $ such that $ \\mbox{{\\bf cost}}(z_1,z_2) \\le K^{\\prime \\prime }_\\gamma ||z_2 - z_1||^\\beta $ where $\\beta := \\log (2 \\gamma ) / \\log 2$ .", "Proof.", "As in Figure 4 in the proof of Lemma REF , there is a square of the form $S = [(i-1)2^h, (i+1)2^h] \\times [(j-1)2^h,(j+1)2^h]$ containing both $z_1$ and $z_2$ , where $h$ is the integer such that $ 2^{h-1} < ||z_2 - z_1||_1 \\le 2^h $ .", "As observed there, the cost of going all around the boundary of $S$ is $O(\\gamma ^h 2^h)$ .", "By considering a path from $z_1$ using the “greedy\" rule of always switching to an orthogonal line of greater height, it is easy to check that the cost of this greedy path from $z_1$ to the boundary of $S$ is also $O(\\gamma ^h 2^h)$ .", "Hence $\\mbox{{\\bf cost}}(z_1,z_2) = O(\\gamma ^h 2^h)$ and the result follows." ], [ "Finessing uniqueness by secondary randomization", "As previously observed, minimum-cost paths are not always unique.", "We conjecture that, at least when $\\gamma $ is not algebraic, there is some simple classification of when and how non-uniqueness occurs.", "But instead of addressing that issue we can finesse it by introducing randomness (which we need later, anyway) at this stage.", "One possible way to do so would be to use the uniform distribution on minimum-cost paths.", "Instead we use what we will call secondary randomization to choose between non-unique minimum-cost paths.", "Place i.i.d.", "Normal$(0,1)$ random variables (“weights\") $\\zeta _e$ on the edges $e$ of ${\\mathbb {Z}}^2$ .", "Any path has a weight $\\sum _{e \\mbox{ in path}} \\zeta _e$ .", "Define the route $\\mbox{${\\mathcal {R}}$}_0(z_1,z_2)$ to be the minimum-weight path in the set of minimum-cost paths from $z_1$ to $z_2$ ." ], [ "Extension to the binary rational lattice", "The notion of height extends to binary rationals: if $x \\in {\\mathbb {R}}$ is a binary rational and $x \\ne 0$ , write $\\mathrm {height}(x)$ for the largest $j \\in {\\mathbb {Z}}$ such that $2^j$ divides $x$ ; in other words the unique $j$ such that $x = (2k+1)2^j$ for some $k \\in {\\mathbb {Z}}$ .", "For $- \\infty < H < \\infty $ let ${\\mathbb {Z}}^2_{H}$ be the lattice on vertex-set $\\lbrace 2^H z: \\ z \\in {\\mathbb {Z}}^2\\rbrace $ , in other words on the set of points in ${\\mathbb {R}}^2$ whose coordinates have height $\\ge H$ .", "So far we have been working on the integer lattice ${\\mathbb {Z}}^2$ , but now the results we have proved extend by (binary) scaling to analogous results on the lattices ${\\mathbb {Z}}^2_{H}$ .", "We will use such scaled results as needed.", "Note in particular the following consistency condition as $H$ varies.", "Take $H_1 < H_2$ .", "Consider the route, in ${\\mathbb {Z}}^2_{H_1}$ , between two vertices of ${\\mathbb {Z}}^2_{H_2}$ .", "By Lemma REF and the definition of admissable, any minimum-cost path stays within the lattice ${\\mathbb {Z}}^2_{H_2}$ .", "So the set of minimum-cost paths is the same whether we work in ${\\mathbb {Z}}^2_{H_1}$ or in ${\\mathbb {Z}}^2_{H_2}$ .", "Note also that each edge $e$ in ${\\mathbb {Z}}^2_H$ corresponds to two edges $e_1, e_2$ of ${\\mathbb {Z}}^2_{H-1}$ .", "So we can couple the edge-weights by making $\\zeta _e = \\zeta _{e_1} + \\zeta _{e_2}$ (only this infinite divisibility property of the Normal is relevant to the construction) and this gives a “consistency of secondary weights\" property, which implies that the random route $\\mbox{${\\mathcal {R}}$}_0(z_1,z_2)$ is also the same whether we work in ${\\mathbb {Z}}^2_{H_1}$ or in ${\\mathbb {Z}}^2_{H_2}$ .", "So we have now defined random routes $\\mbox{${\\mathcal {R}}$}_0(z_1,z_2)$ for all unordered pairs of vertices in ${\\mathbb {Z}}^2_{-\\infty }:= \\cup _{H>-\\infty } {\\mathbb {Z}}^2_{H}$ .", "From the “minimality\" in the construction it is clear that the routes satisfy the route-compatability properties (iii,iv) from section REF ." ], [ "Extension to the plane", "We want to define routes $\\mbox{${\\mathcal {R}}$}_0(z_1,z_2)$ between general points $z_1,z_2$ of ${\\mathbb {R}}^2$ as $H \\rightarrow - \\infty $ limits of the routes $\\mbox{${\\mathcal {R}}$}_0(z_1^H,z_2^H)$ between vertices such that $z_i^H \\in {\\mathbb {Z}}^2_H,\\ \\ z_i^H \\rightarrow z_i \\ \\ (i = 1,2)$ Proposition REF formalizes this idea.", "The proof in this section is the most intricate part of the construction, which can thereafter be completed (section REF ) by “soft\" arguments.", "As a first issue, what does it mean to say that, under (REF ), $\\mbox{routes$\\mbox{${\\mathsf {r}}$}(z_1^H,z_2^H)$ converge to a route $\\mbox{${\\mathsf {r}}$}(z_1,z_2)$?", "}$ We define this to mean: for each $H_0 > - \\infty $ , the subroute $\\mbox{${\\mathsf {r}}$}_{H_0}(z_1^H,z_2^H)$ consisting of path segments of $\\mbox{${\\mathsf {r}}$}(z_1^H,z_2^H)$ within lines of height $\\ge H_0$ is, for sufficiently large negative $H$ , a path not depending on $H$ – call this path $\\mbox{${\\mathsf {r}}$}_{H_0}(z_1,z_2)$ .", "When this property holds, Lemma REF implies that $\\mbox{${\\mathsf {r}}$}_{H_0}(z_1,z_2)$ is a connected path, consistent as $H_0$ decreases, and using Proposition REF and scaling we see that the closure of $\\cup _{H_0 > - \\infty } \\mbox{${\\mathsf {r}}$}_{H_0}(z_1,z_2)$ defines a route $\\mbox{${\\mathsf {r}}$}(z_1,z_2)$ satisfying the “jagged\" condition of section REF .", "Proposition 9 There exists a subset $A \\subset {\\mathbb {R}}^2$ of zero area such that, if $z_1$ and $z_2$ are outside $A$ , there exists a random jagged route $\\mbox{${\\mathcal {R}}$}_0(z_1,z_2)$ such that, whenever (REF ) holds, then (REF ) holds.", "The proof relies on the fact that, for particular configurations illustrated in Figure 6, routes from a certain neighborhood to distant destinations must pass through a particular point.", "In fact all that matters is the existence of such a configuration, not the particular one we now exhibit.", "Consider a square $G = [2^h ,2^h +2]^2$ and points $b = (2^h +1 , 2^h)$ and $d = (2^h + 1,2^h + 2^{-h})$ , illustrated in Figure 6.", "What is relevant is the heights of the lines involved, indicated in the figure.", "h1-h0h01$\\Sigma $GObd$\\varepsilon $$c^*$$b^*$NESENWSWEWFigure 6.", "The big square $G$ and the small square $\\Sigma $ .", "Marginal labels attached to lines are line-heights, not coordinates.", "Lemma 10 There exist large $h$ and small $\\varepsilon $ (depending on $\\gamma $ ) such that, in the configuration shown in Figure 6, every route from inside the small square $\\Sigma := d + [ - \\varepsilon ,0] \\times [0,\\varepsilon ]$ to the boundary of $G$ passes via $b$ .", "Proof.", "For each point $c$ on the boundary of $G$ there is a counter-clockwise path $\\pi _1(b,c)$ and a clockwise path $\\pi _2(b,c)$ along the boundary from $b$ to $c$ .", "These paths have equal cost for the point $c^* = (2^h + 2 - \\gamma ^{h-1}, 2^h + 2)$ , which is near the NE corner point of $G$ .", "We will need the following lemma.", "Lemma 11 There exists $h$ such that the following hold.", "(a) The paths $\\pi _1(b,c^*)$ and $\\pi _2(b,c^*)$ attain the minimum cost over all paths from $b$ to $c^*$ , and are the only paths to do so.", "(b) The only minimum-cost paths from $d$ to $c^*$ are the two paths consisting of the segment $[d,b]$ and the paths $\\pi _1(b,c^*)$ or $\\pi _2(b,c^*)$ .", "(c) There exists $\\eta > 0$ such that any path from $d$ to $c^*$ that avoids the segment $[d,b]$ has cost at least $\\eta $ greater that the minimum-cost paths.", "Note a technical point.", "We are working on ${\\mathbb {Z}}^2_{-\\infty }:= \\cup _{H>-\\infty } {\\mathbb {Z}}^2_{H}$ and $c^*$ may not be in ${\\mathbb {Z}}^2_{-\\infty }$ .", "To be precise we should replace $c^*$ in the arguments below by a sequence $c^*_H \\rightarrow c^*$ , but that requires awkward notation we prefer to avoid.", "Granted Lemma REF we deduce Lemma REF as follows.", "Consider a point $c$ on the counter-clockwise path from $b$ to $c^*$ (the clockwise case is similar).", "Then the following must hold, because any counter-example path to $c$ could be extended along the boundary from $c$ to $c^*$ and would give a counter-example to Lemma REF .", "(a) The path $\\pi _1(b,c)$ is the unique minimum-cost path from $b$ to $c$ .", "(b) The path consisting of the segment $[d,b]$ and the path $\\pi _1(b,c)$ is the unique minimum-cost path from $d$ to $c$ .", "(c) Any path from $d$ to $c$ that avoids the segment $[d,b]$ has cost at least $\\eta $ greater that the minimum-cost path.", "Lemma REF extends by scaling to $\\cup _{H>-\\infty } {\\mathbb {Z}}^2_{H}$ , and so the function $\\mbox{{\\bf cost}}(\\cdot ,\\cdot )$ extends to a continuous function on ${\\mathbb {R}}^2$ .", "So we can choose $H$ so that the square $\\Sigma = d + [-2^{-H},0]\\times [0,2^{-H}]$ satisfies $\\sup _{s \\in \\Sigma } \\mbox{{\\bf cost}}(s,d) \\le \\eta /3$ .", "It is easy to check that a minimum-cost path from $s \\in \\Sigma $ to $d$ does not meet $[d,b]$ except at $d$ .", "Consider $s \\in \\Sigma $ and a point $c$ as above.", "So $\\mbox{{\\bf cost}}(s,c) \\le \\mbox{{\\bf cost}}(d,c) + \\eta /3$ .", "Suppose a minimum-cost path from $s$ to $c$ does not meet the segment $[d,b]$ .", "Then the path from $d$ to $c$ via $s$ would have cost $\\le \\mbox{{\\bf cost}}(d,c) + 2\\eta /3$ and would not meet $[d,b]$ , contradicting (c).", "So a minimum-cost path from $s$ to $c$ must meet the segment $[d,b]$ .", "Then, by uniqueness in (b) (for the path from $d$ to $c$ ), it must continue via $b$ , establishing Lemma REF .", "Proof of Lemma REF .", "(I thank Justin Salez for completing the details of this proof.)", "In outline, we use the “structure of paths\" results in Lemmas REF and REF to reduce to comparing costs of a finite number of possible routes.", "We will make use of the following preliminary observations, which are straightforward to check : (i) the only minimum-cost path from $SE$ to $NW$ is $SE\\rightarrow SW \\rightarrow NW$ ; (ii) the only minimum-cost paths from $SW$ to $NE$ are $SW\\rightarrow NW \\rightarrow NE$ and $SW\\rightarrow SE \\rightarrow NE$ ; (iii) $O\\rightarrow b^* \\rightarrow NE$ is a minimum-cost path from $O$ to $NE$ .", "Consider assertion (a).", "The cost associated with paths $\\pi _1(b,c^*)$ and $\\pi _2(b,c^*)$ equals $2\\gamma ^h+2\\gamma $ , which (by choosing $h$ large) is less than 2.", "Now consider some minimum-cost path $\\pi $ from $b$ to $c^*$ .", "Since both end-points have their $y-$ coordinate at height $\\ge 1$ , all horizontal segments of $\\pi $ must have height $\\ge 1$ (Lemma REF ).", "In other words, the length of every vertical segment must be an even integer.", "If the last vertical segment of $\\pi $ were ending strictly between $NW$ and $NE$ , then its cost would be at least 2, contradicting optimality.", "Thus, $\\pi $ must pass through $NW$ or $NE$ , and observationss (i) or (ii) complete the proof.", "Now consider assertions (b) and (c).", "Let $\\pi $ be any path from $d$ to $c^*$ , and let $z$ be the point at which $\\pi $ first meets the boundary of the rectangle formed by $\\lbrace E,W,SW,SE\\rbrace $ .", "Let $\\pi ^{\\prime },\\pi ^{\\prime \\prime }$ denote the subpaths of $\\pi $ from $d$ to $z$ and from $z$ to $c^*$ , respectively.", "There are four possible cases : $z\\in (E,W)$ : since all segments in $\\pi ^{\\prime }$ have height $\\le 0$ , replacing $\\pi ^{\\prime }$ by $d\\rightarrow O\\rightarrow z$ cannot increase the overall cost.", "In the resulting path, one may further replace the subpath from $O$ to $c^*$ by $O\\rightarrow b^* \\rightarrow c^*$ without increasing the cost, by (iii).", "This shows : $\\mathrm {cost}(\\pi ) \\ge 2+\\gamma -2^{-h}-\\gamma ^h.$ $z\\in (W,SW)$ : all horizontal segments in $\\pi ^{\\prime }$ have height $\\le -1$ , so $\\mathrm {cost}(\\pi ^{\\prime })\\ge \\gamma ^{-1}.$ By (ii), one also has $\\mathrm {cost}(\\pi ^{\\prime \\prime })\\ge \\mathrm {cost}(z\\rightarrow NW\\rightarrow c^*).$ Combining these two facts yields $\\mathrm {cost}(\\pi ) \\ge 2\\gamma +\\gamma ^{-1}.$ $z\\in (SE,E)$ : replacing the subpath $\\pi ^{\\prime \\prime }$ by $z\\rightarrow NE \\rightarrow c^*$ cannot increase the overall cost, by part (a).", "In the resulting path, the subpath from $d$ to $E$ costs at least $\\gamma ^{-1}+\\gamma (1-2^{-h})$ , because the horizontal and vertical heights are $\\le -1$ and $\\le 1$ , respectively.", "Thus, $\\mathrm {cost}(\\pi ) \\ge 2\\gamma +\\gamma ^{-1}.$ $z\\in (SW,SE)$ : all segments of $\\pi ^{\\prime }$ have height $\\le -1$ except those included in $[O,b]$ , which have height 0.", "Thus, $\\mathrm {cost}(\\pi ^{\\prime })-\\mathrm {cost}(d\\rightarrow b\\rightarrow z) \\ge (\\gamma ^{-1}-1)\\mathrm {len}\\left([b,d]\\setminus \\pi ^{\\prime }\\right).$ Moreover, by part (a), $\\mathrm {cost}(b\\rightarrow z)+\\mathrm {cost}(\\pi ^{\\prime \\prime })\\ge \\mathrm {cost}(\\pi _1).$ Thus, $\\mathrm {cost}(\\pi )\\ge (\\gamma ^{-1}-1)\\mathrm {len}\\left( [b,d]\\setminus \\pi \\right)+\\mathrm {cost}(d\\rightarrow b)+\\mathrm {cost}(\\pi _1).$ Let us sum up: in the first three cases, the cost of $\\pi $ exceeds that of our two candidates by at least 1, for $h$ sufficiently large.", "In the fourth case, the excess is at least $(\\gamma ^{-1}-1)\\mathrm {len}\\left( [b,d]\\setminus \\pi \\right)$ .", "This proves both (b) and (c), with $\\eta = 2^{-h}(\\gamma ^{-1}-1)$ .", "Proof of Proposition REF .", "In each basic $2^{h+1} \\times 2^{h+1}$ square $G$ of ${\\mathbb {Z}}^2_{h+1}$ there is a copy of the Figure 6 configuration; let $\\Sigma _G$ be the corresponding small square.", "Let $B: = \\cup _G \\Sigma _G$ be the union of those squares, for the fixed $h$ given by Lemma REF .", "Then for $i \\ge 1$ let $B_i:= \\sigma _{2^{-i}}B$ be rescalings of $B$ .", "Each $B_i$ has the same density, which by Lemma REF is non-zero, and a straightforward use of the second Borel-Cantelli lemma (with sufficiently well-spaced values of $i$ ) shows that the set $ A:= \\lbrace z \\in {\\mathbb {R}}^2: \\ z \\mbox{ in only finitely many } B_i\\rbrace $ has area zero.", "Now consider $z_1 \\in A^c$ .", "Then there exists a sequence $i_j = i_j(z_1) \\rightarrow \\infty $ such that $z_1 \\in B_{i_j}$ and the associated $b_{i_j}(z_1) \\rightarrow z_1$ .", "Consider $z_2 \\ne z_1$ and $(z^H_1,z^H_2) \\rightarrow (z_1,z_2)$ as in (REF ).", "For $j$ larger than some $j_0(z_1,z_2)$ , Lemma REF implies that for all sufficiently large $H$ the route $\\mbox{${\\mathcal {R}}$}_0(z_1^H,z_2^H)$ passes through $b_{i_j}(z_1)$ .", "But the routes between the $b_{i_j}(z_1), j \\ge 1$ are specified by the construction on $\\cup _{H>-\\infty } {\\mathbb {Z}}^2_{H}$ .", "It follows that, when $z_1$ and $z_2$ are both in $A^c$ we have convergence in the sense of (REF ) to a route $\\mbox{${\\mathcal {R}}$}_0(z_1,z_2)$ .", "We digress to give the technical estimate that will show $\\ell < \\infty $ in this model.", "Lemma 12 For the routes $\\mbox{${\\mathcal {R}}$}_0$ in Proposition REF , take the union over points $\\xi , \\xi ^\\prime $ of a rate-1 Poisson point process $\\Xi (1)$ of the routes $\\mbox{${\\mathcal {R}}$}_0(\\xi ,\\xi ^\\prime )$ , and let $\\mbox{${\\mathcal {S}}$}^*$ be the intersection of that union with the interior of a unit square $U = [i,i+1] \\times [j,j+1]$ .", "Then the expected length of $\\mbox{${\\mathcal {S}}$}^*$ is at most $2^{b+2}$ , for $b$ as in Lemma REF .", "Proof.", "Lemma REF was stated for $h \\ge 0$ and vertices in ${\\mathbb {Z}}^2$ , but by scaling it holds for $h < 0$ and vertices in ${\\mathbb {R}}^2$ .", "Consider $h < 0$ .", "Within $U$ there are $2^{-h-1}$ horizontal unit-length line segments at height $h$ , and these can be split into $2^{-2h-1}$ segments of length $2^h$ .", "Consider such a line segment, $\\zeta $ say.", "It is in the interior of some rectangle of the form $[i2^{h+1+b}, (i+1)2^{h+1+b}] \\times [j2^{h+1}, (j+1)2^{h+1}]$ .", "By Lemma REF applied with $h+1$ , the only possible way that the segment $\\zeta $ can be in a route $\\mbox{${\\mathcal {R}}$}_0(\\xi ,\\xi ^\\prime )$ is if $\\xi $ or $\\xi ^\\prime $ is within the rectangle.", "(And the same holds for any piece of $\\zeta $ , by considering a sub-rectangle).", "The chance the Poisson process contains such a point is at most the area of the rectangle, which is $2^{2h+2+b}$ .", "So the contribution to mean length from a particular segment $\\zeta $ is at most $2^h \\times 2^{2h+2+b}$ , and then the contribution from height-$h$ horizontal lines is at most $2^h \\times 2^{2h+2+b} \\times 2^{-2h-1} = 2^{h+1+b}$ .", "Summing over $h \\le -1$ and adding the same contribution from vertical lines gives the bound $2^{2+b}$ ." ], [ "Completing the construction by forcing invariance", "Proposition REF gives paths $\\mbox{${\\mathcal {R}}$}_0(z_1,z_2)$ when $z_1, z_2 \\in A^c$ .", "The process $\\mbox{${\\mathcal {R}}$}_0$ cannot be translation- or rotation-invariant (in distribution), because the axes play a special role (infinite speed); though by construction the process is invariant under $\\sigma _2$ (scaling space by a factor 2).", "But there is a standard way of trying to make translation-invariant random processes out of deterministic processes, by taking weak limits of random translations of the original process.", "In our setting this can be done fairly explicitly as follows.", "For $u \\in {\\mathbb {R}}^2$ let $T_u$ be the translation map $T_u(z) = u + z, \\ z \\in {\\mathbb {R}}^2$ on points, and let $T_u$ act on routes in the natural way.", "Take $U_n$ uniform on the square $[0,2^n]^2$ , and couple the random variables $(U_n, n \\ge 1)$ by setting $U_n = U_{n+1} \\bmod 2^n$ coordinatewise.", "Define $\\mbox{${\\mathcal {R}}$}^{(n)}(z_1,z_2) = T_{- U_n} (\\mbox{${\\mathcal {R}}$}_0(z_1+U_n,z_2+U_n)) .$ In words, translate points by $U_n$ , use $\\mbox{${\\mathcal {R}}$}_0$ to define a route between the translated points, and then translate back to obtain a route between the original points.", "Now the only way that $\\mbox{${\\mathcal {R}}$}^{(n+1)}(z_1,z_2)$ could be different from $\\mbox{${\\mathcal {R}}$}^{(n)}(z_1,z_2)$ is if the route $\\mbox{${\\mathcal {R}}$}_0(z_1 + U_n, z_2+U_n)$ intersects the boundary of the square $[0,2^n]^2$ , which, using Lemma REF , has chance $O(2^{-n})$ .", "So we can define a random network $\\mbox{${\\mathcal {R}}$}_{\\mbox{{\\footnotesize t-i} }}$ via the a.s. limits $\\mbox{${\\mathcal {R}}$}_{\\mbox{{\\footnotesize t-i} }}(z_1,z_2) = \\mbox{${\\mathcal {R}}$}^{(n)}(z_1,z_2) \\mbox{ for all sufficiently large } n .$ This process is translation-invariant, because for fixed $z \\in {\\mathbb {R}}^2$ the variation distance between the distributions of $U_n$ and $U_n + z \\bmod 2^n$ tends to zero.", "For $0<c<\\infty $ write $\\sigma _c$ for the scaling map $z \\rightarrow cz$ on ${\\mathbb {R}}^2$ , and recall that $\\mbox{${\\mathcal {R}}$}_0$ is invariant under $\\sigma _{2}$ .", "Now for $\\mbox{${\\mathcal {R}}$}^{(n)}$ at (REF ), $\\sigma _2 \\mbox{${\\mathcal {R}}$}^{(n)}(z_1,z_2)&=& \\sigma _2 T_{-U_n} \\mbox{${\\mathcal {R}}$}_*(z_1 + U_n, z_2 + U_n) \\\\&=& T_{-2U_n} \\sigma _2 \\mbox{${\\mathcal {R}}$}_*(z_1 + U_n, z_2 + U_n) \\\\&\\ \\stackrel{d}{=} \\ & T_{-2U_n} \\mbox{${\\mathcal {R}}$}_*(2z_1 + 2U_n, 2z_2 + 2U_n) \\mbox{ by invariance of $\\mbox{${\\mathcal {R}}$}_0$ under $\\sigma _2$}\\\\&\\ \\stackrel{d}{=} \\ & T_{-U_{n+1}} \\mbox{${\\mathcal {R}}$}_*(2z_1 + U_{n+1}, 2z_2 + U_{n+1}) \\mbox{ because } U_{n+1}\\ \\stackrel{d}{=} \\ 2U_n \\\\&=& \\mbox{${\\mathcal {R}}$}^{(n+1)}(2z_1,2z_2).$ Hence the distribution of the limit $\\mbox{${\\mathcal {R}}$}_{\\mbox{{\\footnotesize t-i} }}$ is invariant under $\\sigma _2$ .", "Of course our construction so far is not rotationally invariant, but applying a uniform random rotation to $\\mbox{${\\mathcal {R}}$}_{\\mbox{{\\footnotesize t-i} }}$ gives a network $\\mbox{${\\mathcal {R}}$}_{\\mbox{{\\footnotesize r-i} }}$ whose distribution is invariant under rotation, as well as preserving distributional invariance under translation and under $\\sigma _2$ .", "Finally, we get a process $\\mbox{${\\mathcal {R}}$}$ with scale-invariant distribution by random rescaling via the scale-free distribution: $\\mbox{${\\mathcal {R}}$}(z_1,z_2) = \\sigma _{1/C} \\mbox{${\\mathcal {R}}$}_{\\mbox{{\\footnotesize r-i} }}(Cz_1,Cz_2) , \\quad \\mathbb {P}(C \\in dc) = {\\textstyle \\frac{1}{c \\ \\log 2}}, \\ 1 < c < 2 .$ This completes the construction of the binary hierarchy model $\\mbox{${\\mathcal {R}}$}$ .", "To check it satisfies the formal setup of a SIRSN in section REF , the only remaining issue is to check that the parameters ${\\mathbb {E}}D_1, \\ell $ and $p(1)$ are finite.", "For the former, Proposition REF implies the corresponding bound in terms of Euclidean distance $ \\mathrm {len}\\ \\mbox{${\\mathcal {R}}$}_0(z_1,z_2) \\le 2^{1/2} K_\\gamma ||z_2 - z_1||_2 $ and this bound is unaffected by the transformations taking $\\mbox{${\\mathcal {R}}$}_0$ to $\\mbox{${\\mathcal {R}}$}$ .", "So ${\\mathbb {E}}D_1 \\le 2^{1/2} K_\\gamma $ .", "For $\\ell $ , in the notation of Lemma REF , the edge-intensity of $\\cup _{\\xi , \\xi ^\\prime \\in \\Xi (1)} \\mbox{${\\mathcal {R}}$}_0(\\xi , \\xi ^\\prime )$ is at most $2^{b+2} + 2$ , the “$+2$ \" terms arising from the edges of ${\\mathbb {Z}}^2$ .", "This edge-intensity is unaffected by the transformations taking $\\mbox{${\\mathcal {R}}$}_0$ to $\\mbox{${\\mathcal {R}}$}_{\\mbox{{\\footnotesize r-i} }}$ .", "Scaling by $C$ in (REF ) multiplies edge-intensity by $C$ , so finally $\\ell \\le (2^{b+2} + 2) {\\mathbb {E}}C$ .", "To bound $p(1)$ , set $r(h) = 2^h(2^b +1)$ .", "Corollary REF implies that, for routes $\\mbox{${\\mathcal {R}}$}_0$ , if an edge element is in a route between some two points at distance $\\ge r(h)$ from the element, then the edge has height $\\ge h$ .", "The edge-intensity of edges with height $\\ge h$ equals $2^{1-h}$ .", "These quantities are unaffected by translation and rotation; and the scaling by $\\sigma _C$ can at most increase the edge-intensity by 4.", "So the edge-intensity in $\\mbox{${\\mathcal {R}}$}$ of $\\mbox{${\\mathcal {E}}$}(\\lambda , r(h))$ is $p(\\lambda , r(h)) \\le 4 \\cdot 2^{1-h}$ Choosing $h$ such that $r(h) < 1$ we deduce $p(1) < \\infty $ ." ], [ "Remarks on section ", "The “combinatorial\" arguments in sections REF - REF are obviously specific to this model.", "But the property implicit in Lemma REF (that there exist configurations in which all long routes from a small neighborhood exit the unit disc at the same point) is closely related to desirable structural properties of SIRSNs discussed in section .", "Lemma REF later shows that in general $\\ell \\le 2p(1)$ , so our argument above that $\\ell < \\infty $ could be omitted, though it is pleasant to have a self-contained construction." ], [ "Other possible constructions", "The model in section 3 has some very special features, in particular that in any realization we see a (scaled and rotated) square lattice of roads.", "Below we outline two other constructions which, we conjecture, produce SIRSNs, the technical dificulty being to prove a.s. uniqueness of routes defined as minimum-cost paths." ], [ "The Poisson line process model", "For each $m = 1,2,3,\\ldots $ take a rate-1 Poisson line process, and attach Uniform$(m-1,m)$ marks to the lines; the union of all these is a Poisson line process with “mark measure\" being Lebesgue measure on $(0,\\infty )$ .", "By a one-to-one mapping of marks one can transform to the mark measure with density $x^{- \\gamma }$ on $0<x<\\infty $ , where we take the parameter $2 < \\gamma < \\infty $ .", "So in any small disc, there is some finite largest mark amongst lines intersecting the disc.", "Picturing the lines as freeways and the marks as speeds, for any pair of points $z_1, z_2$ on the lines there is some finite minimum time $\\mathrm {t}(z_1,z_2)$ over all routes from $z_1$ to $z_2$ , and analogous to Lemma REF one can show (Wilf Kendall: personal communication) that this function extends to a random continuous function $\\mathrm {t}(z_1,z_2)$ on the plane.", "The technical difficulty is to show that for given $(z_1,z_2)$ there is an a.s. unique route attaining that time; if that were proved, establishing the remaining properties required of an SIRSN would be straightforward.", "In particular, scale-invariance would follow from the form $x^{-\\gamma }$ of the mark density." ], [ "A dynamic proximity graph model", "This potential construction of a SIRSN is based on a space-time Poisson point process $(\\Xi (\\lambda ), 0 < \\lambda < \\infty )$ .", "Note that to study such a SIRSN one would use an independent Poisson point process to define $\\mbox{${\\mathcal {S}}$}(\\lambda )$ .", "Note also that the corresponding “static\" model, called the Gabriel network, is a member of the family of proximity graphs described in [15], [5]; any family member could be used in the construction below.", "Here's the construction rule.", "When a point $\\xi $ arrives at time $\\lambda $ , consider in turn each existing point $\\xi ^\\prime \\in \\Xi (\\lambda -)$ , and create an edge $(\\xi ,\\xi ^\\prime )$ if the disc with diameter $(\\xi ,\\xi ^\\prime )$ contains no other point of $\\Xi (\\lambda -)$ .", "Write $\\mbox{${\\mathbb {G}}$}(\\lambda )$ for the time-$\\lambda $ network on points $\\Xi (\\lambda )$ .", "Note the automatic scale-invariance property the action of $\\sigma _c$ on $\\mbox{${\\mathbb {G}}$}(\\lambda )$ gives a network distributed as $\\mbox{${\\mathbb {G}}$}(c^{-2}\\lambda )$ .", "Now fix a parameter $0 \\le \\gamma < \\gamma _*$ for some sufficiently small $\\gamma _* > 0$ and view an edge created at time $\\lambda $ as a road with speed $\\lambda ^{-\\gamma }$ .", "Defining routes in $\\mbox{${\\mathbb {G}}$}(\\lambda )$ as minimum-time paths, it seems intuitively plausible, as in the Poisson line process model, that that we can extend the minimum-time function on $\\cup _\\lambda \\Xi (\\lambda )$ to a continuous function $\\mathrm {t}(z_1,z_2)$ and then prove there is an a.s. unique route attaining that time.", "Again, if that were proved, establishing the remaining properties required of an SIRSN would be straightforward.", "In particular, scale-invariance would follow from the fact that the construction rule is scale-invariant." ], [ "Properties of weak SIRSNs", "In this section we study properties that hold for any weak SIRSN, that is when we do not require (REF ) but instead require (REF ).", "These are essentially properties of the sampled subnetworks $\\mbox{${\\mathcal {S}}$}(\\lambda )$ for fixed $\\lambda $ – we cannot get $\\lambda \\rightarrow \\infty $ results." ], [ "No straight edges at typical points", "If a point $\\xi $ of $\\Xi (\\lambda )$ is the start of some straight line segment of length $\\ge r$ in $\\mbox{${\\mathcal {S}}$}(\\lambda )$ then consider the subroutes of length exactly $r$ from $\\xi $ .", "The edge process of such subroutes has some edge-intensity $\\iota (\\lambda ,r)$ .", "In independent copies of $\\Xi (1)$ these edge-processes cannot have any positive-length overlap.", "So by regarding $\\Xi (n)$ as the union of $n$ independent copies of $\\Xi (1)$ we have $\\iota (n,r) = n \\iota (1,r)$ .", "But by the general scaling property () $\\iota (\\lambda ,r) = \\lambda ^{1/2} \\iota (1,r \\lambda ^{1/2}).$ Since $\\iota (1,r) \\le \\ell < \\infty $ these two different scaling relations imply $\\iota (1,r) = 0$ for all $r>0$ .", "This proves (a) below; note the consequence (b), implied by the definition of feasible path in the section REF setup.", "Proposition 13 $S(\\lambda )$ has the following properties a.s. (a) $S(\\lambda )$ contains no line segment $[\\xi ,z]$ of positive length, for any $\\xi \\in \\Xi (\\lambda )$ .", "(b) The route $\\mbox{${\\mathcal {R}}$}(\\xi _1,\\xi _2)$ between two points of $\\Xi (\\lambda )$ does not pass through any third point $\\xi _3$ of $\\Xi (\\lambda )$ ." ], [ "Singly and doubly infinite geodesics", "Recall from section REF that a singly infinite geodesic from a point $\\xi _0$ in $\\mbox{${\\mathcal {S}}$}(\\lambda )$ is an infinite path, starting from $\\xi _0$ , such that any finite portion of the path is a subroute of some route $\\mbox{${\\mathcal {R}}$}(\\xi _0,\\xi )$ .", "Lemma REF showed $\\mbox{There is a.s. at least one singly infinite geodesic from each point of $\\mbox{${\\mathcal {S}}$}(\\lambda )$.", "}$ A doubly infinite geodesic in $\\mbox{${\\mathcal {S}}$}(\\lambda )$ is a path $\\pi $ which is an increasing union of segments $\\pi _k$ , where each $\\pi _k$ is a segment of some route $\\mbox{${\\mathcal {R}}$}(\\xi _k,\\xi ^\\prime _k)$ between two points of $\\Xi (\\lambda )$ , and both endpoints of $\\pi _k$ go to infinity.", "Previous work on very different (e.g.", "percolation-type [19]) networks suggests there may be a general principle: In natural models of random networks on ${\\mathbb {R}}^2$ or ${\\mathbb {Z}}^2$ , doubly infinite geodesics do not exist.", "Proposition REF proves this for weak SIRSNs based on a simple scaling argument.", "Note however this argument depends implicitly upon our assumption $\\ell < \\infty $ which seems rather special to our setting.", "Recall the setup of (REF , REF ).", "$\\mbox{${\\mathcal {E}}$}(\\lambda ,r) \\subset \\mbox{${\\mathcal {S}}$}(\\lambda )$ is the set of points $z$ in edges of $\\mbox{${\\mathcal {S}}$}(\\lambda )$ such that $z$ is in the route $\\mbox{${\\mathcal {R}}$}(\\xi ,\\xi ^\\prime )$ for some $\\xi , \\xi ^\\prime $ of $\\Xi (\\lambda )$ such that $\\min (|z-\\xi |, |z-\\xi ^\\prime |) \\ge r$ .", "And $p(\\lambda ,r)$ is the edge-intensity of $\\mbox{${\\mathcal {E}}$}(\\lambda ,r)$ .", "By scaling, $p(\\lambda ,r) = \\lambda ^{1/2} p(1,r\\lambda ^{1/2})$ Proposition 14 $p(\\lambda ,r) \\rightarrow 0$ as $r \\rightarrow \\infty $ .", "In particular, $\\mbox{${\\mathcal {S}}$}(\\lambda )$ has a.s. no doubly infinite geodesics.", "Proof.", "For fixed $\\lambda $ the edge-processes $\\mbox{${\\mathcal {E}}$}(\\lambda , r)$ can only decrease as $r$ increases, and the limit $\\mbox{${\\mathcal {E}}$}(\\lambda , \\infty ) := \\cap _r \\mbox{${\\mathcal {E}}$}(\\lambda , r)$ is by definition the set of path elements in doubly infinite geodesics.", "This limit has edge-intensity $p(\\lambda , \\infty ) = \\lim _{r \\rightarrow \\infty } p(\\lambda ,r) \\ge 0$ .", "So it is enough to prove $p(\\lambda ,\\infty ) = 0$ .", "Suppose not.", "Then by the scaling relation (REF ) $ p(\\lambda ,\\infty ) = \\lambda ^{1/2} p(1,\\infty ), \\quad 0 < \\lambda < \\infty .", "$ We claim that in fact $ \\mbox{$\\mbox{${\\mathcal {E}}$}(\\lambda , \\infty ) = \\mbox{${\\mathcal {E}}$}(1, \\infty )$ a.s. for $\\lambda < 1$,}$ which (because we know $p(1,\\infty ) < \\infty $ ) implies $p(1,\\infty ) = 0$ and completes the proof.", "To prove the claim, note that for any finite-length segment $\\pi _0$ of a doubly infinite geodesic in $\\mbox{${\\mathcal {S}}$}(1)$ , there are an infinite number of distinct pairs $\\xi _j, \\xi ^\\prime _j$ of $\\Xi (1)$ such that $\\mbox{${\\mathcal {R}}$}(\\xi _j,\\xi ^\\prime _j)$ contains $\\pi _0$ , and for each pair there is chance $\\lambda ^2$ that both points are in $\\Xi (\\lambda )$ .", "These events are independent (because $\\Xi (\\lambda )$ is obtained from $\\Xi (1)$ by independent sampling) so a.s. an infinite number of pairs $\\xi _j, \\xi ^\\prime _j$ are in $\\Xi (\\lambda )$ , implying that $\\pi _0$ is in a doubly infinite geodesic of $\\mbox{${\\mathcal {S}}$}(\\lambda )$ .", "Remark.", "The limit used here is different from the limit $p(r) := \\lim _{\\lambda \\rightarrow \\infty } p(\\lambda ,r)$ featuring in assumption (REF )." ], [ "Marginal interpretation of $\\ell $", "Recall $\\ell $ is defined as the edge-intensity of $\\mbox{${\\mathcal {S}}$}(1)$ , which is the subnetwork on a rate-1 Poisson point process $\\Xi (1)$ .", "Now augment the network $\\mbox{${\\mathcal {S}}$}(1)$ by including the point at the origin and the routes from the origin to each $\\xi \\in \\Xi (1)$ .", "The newly added edges have some random total length $L$ .", "Proposition 15 ${\\mathbb {E}}L = \\ell /2$ .", "Proof.", "Recall (REF ) the scaling relation $\\ell (\\lambda ) = \\lambda ^{1/2} \\ell $ , where $\\ell (\\lambda )$ is the edge-intensity of the subnetwork $ \\mbox{${\\mathcal {S}}$}(\\lambda )$ of a Poisson process of point-intensity $\\lambda $ .", "Differentiating with respect to $\\lambda $ , $ \\ell ^\\prime (1) = {\\textstyle \\frac{1}{2}} \\ell .", "$ So we need to show ${\\mathbb {E}}L = \\ell ^\\prime (1)$ .", "Consider the space-time Poisson point process $(\\Xi (\\lambda ), 0<\\lambda <\\infty )$ from section REF .", "Each arriving point creates some additional network length, say $\\tilde{L}(\\xi )$ , and for a point arriving at time $\\lambda $ , write $\\tilde{\\ell }(\\lambda )$ for the mean additional network length.", "Now $ \\ell (\\lambda _0) = {\\mathbb {E}}\\sum _{\\xi \\in \\Xi (\\lambda _0) \\cap [0,1]^2} \\tilde{L} (\\xi )= \\int _0^{\\lambda _0} \\tilde{\\ell }(\\lambda ) \\ d \\lambda $ and so $\\ell ^\\prime (1) = \\tilde{\\ell }(1)$ ." ], [ "A lower bound on network length", "Write $\\Delta $ for the parameter ${\\mathbb {E}}D_1$ of a SIRSN.", "Write $\\ell _*(\\Delta )$ for the minimum possible value of $\\ell $ in a SIRSN with a given value of $\\Delta $ .", "Proposition 16 $\\ell _*(\\Delta ) = \\Omega ((\\Delta -1)^{-1/2})$ as $\\Delta \\downarrow 1$ .", "The proof is based on a bound (Proposition REF ) involving the geometry of deterministic paths, somewhat similar to bounds used in [8] section 4.", "Figure 7 illustrates the argument to be used.", "0LU2L3L$z_1$$z_2$$\\beta $$\\xi $Figure 7.", "Proposition 17 Let $\\alpha , L, D$ and $\\theta _0$ be positive reals satisfying $\\theta _0< \\pi /2$ and $2 \\sqrt{(D-1)^2 + ({\\textstyle \\frac{3}{2}}L +1)^2} &=& (1+2\\alpha ) (3L+2) \\\\L\\left( {\\textstyle \\frac{1}{\\cos \\theta _0}} -1 \\right) &=& 4 \\alpha \\sqrt{(3L+2)^2 + 1} .", "$ Let $\\mbox{${\\mathcal {R}}$}$ be a route from some point $z_1$ in the unit square $[-1,0] \\times [0,1]$ to some point $z_2$ in the unit square $[3L, 3L+1] \\times [0,1]$ , and suppose $\\mathrm {len}(\\mbox{${\\mathcal {R}}$}) \\le (1 + 2\\alpha ) |z_2 - z_1| .$ Take $U$ uniform random on $[L,2L]$ .", "The route $\\mbox{${\\mathcal {R}}$}$ first crosses the vertical line $\\lbrace (U,y), -\\infty < y < \\infty \\rbrace $ at some random point $(U,\\xi (U))$ and at some angle $\\beta (U) \\in (- {\\textstyle \\frac{\\pi }{2}}, {\\textstyle \\frac{\\pi }{2}})$ relative to horizontal.", "Then (i) $|\\xi (U)| \\le D $ .", "(ii) $\\mathbb {P}(|\\beta (U)| \\le \\theta _0) \\ge {\\textstyle \\frac{1}{2}} $ .", "Proof.", "The maximum possible value of $\\xi (U)$ arises in the case where $z_1 = (-1,1), \\ z_2 = (3L+1,1), \\ U = {\\textstyle \\frac{3}{2}}L$ , the route consists of straight lines from $z_1$ to $(U,\\xi (U))$ to $z_2$ , and the route-length attains equality in (REF ).", "In this case the value of $\\xi (U)$ is the quantity $D$ satisfying (REF ), establishing (i).", "Writing $\\beta (u)$ for the angle (relative to horizontal) of the route at $x$ -coordinate $u$ , then the length ($\\Lambda $ , say) of the route between $x$ -coordinates $L$ and $2L$ equals $\\int _L^{2L} \\frac{1}{\\cos \\beta (u)} \\ du$ .", "This implies $ \\Lambda - L \\ge ({\\textstyle \\frac{1}{\\cos \\theta _0 }} - 1) \\times L \\mathbb {P}(\\beta (U) \\ge \\theta _0).", "$ But by considering excess length (relative to a horizontal route), (REF ) implies $ \\Lambda - L \\le 2 \\alpha |z_2 - z_1| \\le 2 \\alpha \\sqrt{(3L+2)^2+1} .", "$ Combining these inequalities gives a lower bound on $\\mathbb {P}(\\beta (U) \\ge \\theta _0)$ which equals $1/2$ when $\\theta _0$ satisfies (), establishing (ii)." ], [ "Proof of Proposition ", "Consider a SIRSN with parameters $\\ell $ and $\\Delta $ and with induced subnetwork $\\mbox{${\\mathcal {S}}$}$ on a Poisson point process $\\Xi $ .", "Set $\\alpha = \\Delta - 1$ .", "Suppose we can choose $L, D, \\theta _0$ to satisfy, along with the given $\\alpha $ , the equalities (REF ,) – note this leaves us one degree of freedom.", "With probability $(1 - e^{-1})^2$ there are points $z_1$ and $z_2$ of the Poisson process in the unit squares $[-1,0]\\times [0,1]$ and $[3L,3L+1] \\times [0,1]$ .", "By Markov's inequality and the definition of $\\Delta $ , with probability at least $1/2$ the route $\\mbox{${\\mathcal {R}}$}(z_1,z_2)$ has length at most $(1+ 2\\alpha )|z_2 - z_1|$ .", "Applying Proposition REF we deduce that, with probability $\\ge (1-e^{-1})^2/4$ , the network $\\mbox{${\\mathcal {S}}$}$ contains an edge that crosses the random vertical line $\\lbrace (U,y): \\ - \\infty < y < \\infty \\rbrace $ at some point $(U,\\xi (U))$ with $-D \\le \\xi (U) \\le D$ and crosses at some angle $\\beta (U) \\in (-\\theta _0, \\theta _0)$ relative to horizontal.", "If we translate vertically by $2D$ , to consider routes between the unit squares $[-1,0]\\times [2D,2D+1]$ and $[3L,3L+1] \\times [2D,2D+1]$ , then the potential crossing points (using the same r.v.", "$U$ ) for the translated and untranslated cases are distinct.", "Now by considering translates by all multiples of $2D$ , and noting that the distribution of crossings of the random vertical line $\\lbrace (U,y): \\ - \\infty < y < \\infty \\rbrace $ is the same as for the $y$ -axis, we have shown the mean intensity of crossings of the network $\\mbox{${\\mathcal {S}}$}$ over the $y$ -axis at angles $\\in (-\\theta _0, \\theta _0)$ relative to horizontal is at least $\\frac{(1-e^{-1})^2}{8D}$ .", "The stochastic geometry identities (REF , REF ) relates this mean intensity to the parameter $\\ell $ via $ \\mbox{this mean intensity} = {\\textstyle \\frac{\\ell }{\\pi }} \\int _{- \\theta _0}^{\\theta _0} \\cos \\theta \\ d \\theta \\le {\\textstyle \\frac{2 \\ell \\theta _0}{\\pi }} .", "$ Combining with the previous inequality we find $ \\ell \\ge \\frac{1}{21 D \\theta _0} .", "$ Now set $L = \\alpha ^{-1/2}$ and consider the solutions of (REF ,) in the limit as $\\alpha \\downarrow 0$ : we find that solutions exist with $ \\theta _0 \\sim \\sqrt{24 \\alpha }; \\quad D \\rightarrow 10 $ which establishes Proposition REF ." ], [ "The minimum value of $\\ell $ and the Steiner tree constant", "Take $k$ uniform random points $Z_1,\\ldots ,Z_k$ in a square of area $k$ and consider the length $L_{ST}(k)$ of the Steiner tree (the minimum-length connected network) on $Z_1,\\ldots ,Z_k$ .", "Well-known subadditivity arguments [21], [25] imply that ${\\mathbb {E}}L_{ST}(k) \\sim c_{\\mbox{{\\tiny ST}}}k$ for some constant $0< c_{\\mbox{{\\tiny ST}}}< \\infty $ .", "One can define $c_{\\mbox{{\\tiny ST}}}$ equivalently (see [4] for results of this kind) as the infimum of $c$ such that there exists a translation-invariant connected random network over $\\Xi (1)$ with edge-intensity $ c$ .", "From the latter description it is obvious that in any SIRSN we have $\\ell \\ge c_{\\mbox{{\\tiny ST}}}$ .", "So the overall infimum $\\ell _* := \\mbox{ infimum of $\\ell $ over all SIRSNs}$ satisfies $\\ell _* \\ge c_{\\mbox{{\\tiny ST}}}$ , and below we outline an argument that the inequality is strict.", "First we derive some simple lower bounds on $c_{\\mbox{{\\tiny ST}}}$ and $\\ell _*$ .", "(i) Write $b(\\xi )$ for the distance from $\\xi $ to its closest neighbor in $\\Xi (1)$ .", "The discs of center $\\xi $ and radius $b(\\xi )/2$ are disjoint as $\\xi $ varies and must contain network length at least $b(\\xi )/2$ , so $ c_{\\mbox{{\\tiny ST}}}\\ge {\\textstyle \\frac{1}{2}} {\\mathbb {E}}b(\\xi ) = {\\textstyle \\frac{1}{4}}.", "$ (ii) In a network of edge-intensity $c$ , (REF ) shows the mean number of edges crossing $\\mathrm {circle}(0,r)$ equals $2 \\pi r \\times 2 \\pi ^{-1} c = 4rc$ .", "If there is a point of $\\Xi (1)$ inside $\\mathrm {disc}(0,r)$ then there must be some such crossing edge, so $ 1 - \\exp (-\\pi r^2) \\le 4rc .", "$ So $ c_{\\mbox{{\\tiny ST}}}\\ge \\sup _r \\frac{1 - \\exp (-\\pi r^2)}{4r} \\approx 0.283 .", "$ (iii) We can get a better bound on $\\ell _*$ by using Proposition REF as follows.", "Using the intensity calculation above, in a network of edge-intensity $\\ell $ the probability that no edge crosses $\\mathrm {circle}(0,r)$ is at least $1 - 4r \\ell $ .", "When a new point arives at $\\xi $ in the $\\mbox{${\\mathcal {S}}$}(\\lambda )$ process at time $\\lambda = 1$ , if no existing edges cross $\\mathrm {circle}(\\xi ,r)$ then the added network length $L$ is at least $r$ .", "So $ {\\mathbb {E}}L \\ge \\sup _r r (1 - 4r \\ell ) = {\\textstyle \\frac{1}{16 \\ell }} .", "$ But Proposition REF says $\\ell = 2 {\\mathbb {E}}L$ and so we have shown $\\ell _* \\ge \\sqrt{1/8} \\approx 0.353 .$ One could no doubt obtain small improvements by similar arguments.", "Here is an outline argument that $\\ell _* > c_{\\mbox{{\\tiny ST}}}$ .", "(i) In the Steiner tree on the Posson point process $\\Xi (1)$ , vertices of degree $>1$ have non-zero density, and their edges meet at some varying angles, whereas at the Steiner points (non-vertex junctions) edges must meet at 120 degree angles.", "(ii) If there were a SIRSN with $\\ell \\approx c_{\\mbox{{\\tiny ST}}}$ , then $\\mbox{${\\mathcal {S}}$}(1)$ would have essentially the properties (i).", "But then in $\\mbox{${\\mathcal {S}}$}(1/2)$ , obtained by deleting half the vertices of $\\Xi (1)$ to get $\\Xi (1/2)$ , some of the deleted vertices would remain as junction points.", "The “varying angles\" property implies the edge-intensity $\\ell (1/2)$ of $\\mbox{${\\mathcal {S}}$}(1/2)$ is strictly larger than that of the Steiner tree on $\\Xi (1/2)$ , contradicting the scale-invariance property that the edge-intensities of $\\mbox{${\\mathcal {S}}$}(1)$ and the Steiner tree on $\\Xi (1)$ are essentially equal." ], [ " General SIRSNs and their properties", "In this section we study properties of $\\mbox{${\\mathcal {S}}$}(\\lambda )$ in the $\\lambda \\rightarrow \\infty $ limit, for a general SIRSN.", "Roughly speaking, this is studying “the whole SIRSN\" instead of sampled subnetworks, and such results depend on assumption (REF ).", "Recall again the setup from (REF ) - (REF ).", "So $p(\\lambda ,r)$ is the edge-intensity of $\\mbox{${\\mathcal {E}}$}(\\lambda ,r)$ , which is the process of points $z$ in edges of $\\mbox{${\\mathcal {S}}$}(\\lambda )$ such that $z$ is in the route $\\mbox{${\\mathcal {R}}$}(\\xi ,\\xi ^\\prime )$ for some $\\xi , \\xi ^\\prime $ in $\\Xi (\\lambda )$ such that $\\min (|z-\\xi |, |z-\\xi ^\\prime |) \\ge r$ .", "Recall also from (REF ) the scaling relation $p(\\lambda ,r) = \\lambda ^{1/2} p(1,r\\lambda ^{1/2})$ .", "Defining $p( r) := \\lim _{\\lambda \\rightarrow \\infty } p(\\lambda , r) < \\infty $ the assumption (REF ) that $p(1) < \\infty $ and scaling imply $p(r) = p(1) \\times r^{-1}, \\quad 0 < r < \\infty .$" ], [ "A connectivity bound", "Assumption (REF ) has a direct implication for the qualitative structure of a SIRSN: all the routes linking two regions, once they get away from a neighborhood of the regions, use only a finite number of different paths.", "We first give a version of this result in terms of discs.", "Figure 8.", "Schematic for routes from inside $\\mathrm {disc}(0,1/2)$ to outside $\\mathrm {disc}(0,3/2)$ crossing the unit circle.", "Proposition 18 Take $0<r<1$ and let $N(\\lambda , r)$ be the number of distinct points on the unit circle at which some route $R(\\xi ,\\xi ^\\prime )$ between some $\\xi \\in \\Xi (\\lambda ) \\cap \\mathrm {disc}(0,1-r)$ and some $\\xi ^\\prime \\in \\Xi (\\lambda ) \\cap ({\\mathbb {R}}^2 \\setminus \\mathrm {disc}(0,1+r))$ crosses the unit circle.", "Then $ {\\mathbb {E}}\\lim _{\\lambda \\rightarrow \\infty } N(\\lambda ,r) \\le 4 p(1) \\ r^{-1} .", "$ Proof.", "Any crossing point is in $\\mbox{${\\mathcal {E}}$}(\\lambda ,r)$ and so by identity (REF ) $ {\\mathbb {E}}N(\\lambda ,r) \\le 2\\pi \\times 2 \\pi ^{-1} p(\\lambda ,r) < \\infty $ and the result follows from (REF ).", "The following general version can be proved similarly.", "Proposition 19 Let $\\varepsilon > 0$ and let $K_1, K_2$ be compact sets whose $\\varepsilon $ -neighborhoods $K_1^\\varepsilon , K_2^\\varepsilon $ are disjoint.", "For $z_1 \\in K_1, z_2 \\in K_2$ let $\\mbox{${\\mathcal {R}}$}_\\varepsilon (z_1,z_2)$ be the subroute of $\\mbox{${\\mathcal {R}}$}(z_1,z_2)$ crossing from the boundary of $K_1^\\varepsilon $ to the boundary of $K_2^\\varepsilon $ .", "Let $N(\\lambda , \\varepsilon , K_1, K_2)$ be the number of distinct paths amongst the set $\\lbrace \\mbox{${\\mathcal {R}}$}_\\varepsilon (\\xi _1,\\xi _2)\\ : \\ \\xi _i \\in \\Xi (\\lambda ) \\cap K_i\\rbrace $ .", "Then ${\\mathbb {E}}\\lim _{\\lambda \\rightarrow \\infty } N(\\lambda , \\varepsilon , K_1, K_2)< \\infty .", "$" ], [ "A bound on normalized length", "Lemma 20 $\\ell \\le 2 p(1)$ .", "Proof.", "Define $ \\mbox{${\\mathcal {R}}$}_{\\delta }(\\xi ,\\xi ^\\prime ) =\\mbox{${\\mathcal {R}}$}(\\xi ,\\xi ^\\prime ) \\cap (\\mathrm {disc}(\\xi ,\\delta ) \\cup \\mathrm {disc}(\\xi ^\\prime ,\\delta )) $ in words, the part of the route that is within distance $\\delta $ from one or both endpoints.", "Then define $ \\widehat{\\mbox{${\\mathcal {S}}$}}_{\\delta }(\\lambda ) = \\cup _{\\xi , \\xi ^\\prime \\in \\Xi (\\lambda )}\\mbox{${\\mathcal {R}}$}_{\\delta }(\\xi ,\\xi ^\\prime ) .", "$ Note that clearly $S(\\lambda ) \\setminus \\widehat{\\mbox{${\\mathcal {S}}$}}_1(\\lambda ) \\subseteq \\mbox{${\\mathcal {E}}$}(\\lambda ,1) .$ By considering $\\lambda = 1$ , $ \\ell \\le p(1,1) + \\iota (\\widehat{\\mbox{${\\mathcal {S}}$}}_1(1) ) $ where $\\iota (\\cdot )$ denotes edge-intensity.", "Now write $ \\iota (\\widehat{\\mbox{${\\mathcal {S}}$}}_1(1) ) = \\sum _{k \\ge 1}\\iota (\\widehat{\\mbox{${\\mathcal {S}}$}}_{2^{1-k}}(1) \\setminus \\widehat{\\mbox{${\\mathcal {S}}$}}_{2^{-k}}(1) ) .", "$ For fixed $k \\ge 1$ , scaling by $2^k$ gives $\\iota (\\widehat{\\mbox{${\\mathcal {S}}$}}_{2^{1-k}}(1) \\setminus \\widehat{\\mbox{${\\mathcal {S}}$}}_{2^{-k}}(1) )&=& 2^{-k} \\iota (\\widehat{\\mbox{${\\mathcal {S}}$}}_{2}(2^{-2k}) \\setminus \\widehat{\\mbox{${\\mathcal {S}}$}}_{1}(2^{-2k}) ) \\\\&\\le & 2^{-k} p(2^{-2k}, 1) \\mbox{ by (\\ref {ShatS})} .$ So $ \\ell \\le \\sum _{k \\ge 0} 2^{-k} p(2^{-2k}, 1) \\le \\sum _{k \\ge 0} 2^{-k} p( 1).", "$" ], [ "The network $\\mbox{${\\mathcal {E}}$}(\\infty ,r)$ of major roads", "Intuitively, the point of assumption (REF ) and the scaling relation (REF ) is that we can define a proces $\\mbox{${\\mathcal {E}}$}(\\infty , r):= \\cup _{\\lambda < \\infty } \\mbox{${\\mathcal {E}}$}(\\lambda ,r)$ which must have edge-intensity $p(r) = p(1)/r$ , and that in results like Proposition REF we can replace $\\lim _{\\lambda \\rightarrow \\infty } N(\\lambda , r)$ by $N(\\infty ,r)$ .", "We don't want to give details of a completely rigorous treatment, but let us just suppose we can set up $\\mbox{${\\mathcal {E}}$}(\\infty ,r)$ as a random element of some suitable measurable space, as we did for $\\mbox{${\\mathcal {S}}$}(\\lambda )$ in section REF .", "The conceptual point is that $\\mbox{${\\mathcal {S}}$}(\\lambda )$ and $\\mbox{${\\mathcal {E}}$}(\\lambda ,r)$ depend on the external randomization, that is on the fact that we were studying a SIRSN via the random points $\\Xi (\\lambda )$ , but as outlined below $\\mbox{${\\mathcal {E}}$}(\\infty ,r)$ doesn't depend on such external randomization.", "Intuitively this is simply because $\\cup _\\lambda \\Xi (\\lambda )$ is dense in ${\\mathbb {R}}^2$ ; we outline a measure-theoretic argument below.", "Proposition 21 The FDDs $(\\mathbf {span}(z_1,\\ldots ,z_k))$ of a SIRSN can be extended to a joint distribution, of these FDDs jointly with a random process $\\mbox{${\\mathcal {E}}$}^*(\\infty ,r)$ , such that, for any space-time PPP $(\\Xi (\\lambda ), 0<\\lambda <\\infty )$ independent of the FDDs, we have $\\mbox{${\\mathcal {E}}$}^*(\\infty , r):= \\cup _{\\lambda < \\infty } \\mbox{${\\mathcal {E}}$}(\\lambda ,r)$ a.s. Outline proof.", "For a suitable formalization of “random subset of ${\\mathbb {R}}^2$ \" we have the implication if $\\mbox{${\\mathcal {A}}$}_1$ and $\\mbox{${\\mathcal {A}}$}_2$ are i.i.d.", "random subsets, and if $\\mbox{${\\mathcal {A}}$}_1 \\cup \\mbox{${\\mathcal {A}}$}_2 \\subseteq _{a.s.} \\mbox{${\\mathcal {A}}$}^\\prime \\ \\stackrel{d}{=} \\ \\mbox{${\\mathcal {A}}$}_1$ , then $\\mbox{${\\mathcal {A}}$}_1 = A$ a.s. for some non-random subset $A$ and then the corresponding “conditional\" implication if $Z$ is a random element of some space, if $\\mbox{${\\mathcal {A}}$}_1$ and $\\mbox{${\\mathcal {A}}$}_2$ are random subsets conditionally i.i.d.", "given $Z$ , and if $\\mbox{${\\mathcal {A}}$}_1 \\cup \\mbox{${\\mathcal {A}}$}_2 \\subseteq _{a.s.} \\mbox{${\\mathcal {A}}$}^\\prime $ where $(Z,\\mbox{${\\mathcal {A}}$}^\\prime ) \\ \\stackrel{d}{=} \\ (Z,\\mbox{${\\mathcal {A}}$}_1)$ , then $\\mbox{${\\mathcal {A}}$}_1 = \\mbox{${\\mathcal {A}}$}$ a.s. for some $Z$ -measurable random subset $A$ .", "So take two independent space-time PPPs $\\Xi ^1(\\lambda ), \\Xi ^2(\\lambda )$ and use a measure-preserving bijection $[0, \\infty ) \\cup [0,\\infty ) \\rightarrow [0,\\infty )$ to define another space-time PPP $\\Xi ^\\prime (\\lambda )$ in terms of $\\Xi ^1$ and $\\Xi ^2$ .", "The associated networks satisfy $ \\mbox{${\\mathcal {E}}$}^1(\\infty ,r) \\cup \\mbox{${\\mathcal {E}}$}^2(\\infty ,r) = \\mbox{${\\mathcal {E}}$}^\\prime (\\infty ,r) \\ \\stackrel{d}{=} \\ \\mbox{${\\mathcal {E}}$}^1(\\infty ,r) $ and this holds jointly with the FDDs of the SIRSN.", "Since $ \\mbox{${\\mathcal {E}}$}^1(\\infty ,r)$ and $\\mbox{${\\mathcal {E}}$}^2(\\infty ,r)$ are conditionally i.i.d.", "given the SIRSN.", "Proposition REF follows from the general “conditional implication\" above." ], [ "Transit nodes and shortest path algorithms", "Here we make a connection with the “shortest path algorithms\" literature mentioned in section REF .", "Fix $h$ and take the square grid of lines with inter-line spacing equal to $h$ .", "Define $\\mbox{${\\mathcal {T}}$}_h$ to be the set of points of intersection of $\\mbox{${\\mathcal {E}}$}(\\infty ,h)$ with that grid.", "Lemma 22 (i) $\\mbox{${\\mathcal {T}}$}_h$ has point-intensity $4 \\pi ^{-1} h^{-2} p(1)$ .", "(ii) For each $z \\in {\\mathbb {R}}^2$ there is a subset $T_z$ of $\\mbox{${\\mathcal {T}}$}_h$ , of mean size $24 \\pi ^{-1} p(1)$ , and with $|z^\\prime - z| \\le 2^{3/2} h$ for each $z^\\prime \\in T_z$ , such that for each pair $z_1, z_2$ with $|z_2 - z_1| > 3h$ the route $\\mbox{${\\mathcal {R}}$}(z_1,z_2)$ passes through some point of $T_{z_1}$ and some point of $T_{z_2}$ .", "Proof.", "The grid has edge-intensity $2h^{-1}$ , so from (REF ) the point-intensity of $\\mbox{${\\mathcal {T}}$}_h$ is $2\\pi ^{-1} \\times p( h) \\times 2h^{-1}$ , and (i) follows from scaling (REF ).", "For any starting point $z$ consider the closest grid intersection $(ih,jh)$ .", "Then $z$ is in some square with corner $(ih,jh)$ , say the square $[(i-1)h,ih] \\times [jh,(j+1)h]$ .", "Let $T_z$ be the set of points of intersection of $\\mbox{${\\mathcal {E}}$}(\\infty ,h)$ with the concentric square $S_z = [(i-2)h, (i+1)h] \\times [(j-1)h, (j+2)h]$ .", "This square has boundary length $12h$ and so the mean size of $T_z$ equals $2\\pi ^{-1} \\times p( h) \\times 12h = 24 \\pi ^{-1} p(1)$ .", "By construction $ {\\textstyle \\frac{3}{2}} h < |z^\\prime - z| \\le 2^{3/2} h \\mbox{ for each } z^\\prime \\mbox{ on the boundary of} S_z $ and in particular for each $z^\\prime \\in T_z$ .", "If $|z_2 - z_1| > 3h$ then the squares $S_{z_1}$ and $S_{z_2}$ do not overlap, and the points $z^\\prime _1$ and $z^\\prime _2$ at which the route crosses their boundaries are in $\\mbox{${\\mathcal {T}}$}_h$ ." ], [ "Informal algorithmic implications", "One cannot rigorously relate our “continuum\" setup to discrete algorithms, but in talks we present the following informal calculation.", "For the real-world road network in a country we have empirical statistics $A$ : area of country $\\eta $ : average number of road segments per unit area $M = \\eta A$ : total number of road segments in country $p(r)$ : “length per unit area\" of the subnetwork consisting of segments on routes with start/destination each at distance $>r$ from the segment.", "For a real-world network there is an inconsistency between scale-invariance and having a finite number $\\eta $ of road segments per unit area, but let us imagine approximate scale-invariance over scales of say 2 - 100 miles, and modify a scale-invariant model by deleting road segments of very short length.", "In what follows it is helpful to imagine the unit of length to be (say) 20 miles.", "Fix $r$ .", "Lemma REF (with $h = r$ ) suggests that in the real-world network we can find transit nodes such that there are $O(p(1))$ transit nodes within distance $O(r)$ of a typical point.", "If so then we can analyze the algorithmic procedure outlined in section REF .", "The local search involves a region of radius $r$ and hence with $O(\\eta r^2)$ edges.", "Regarding the time-cost of a single Dijkstra search as $c_1 \\times (\\mbox{number of edges})$ , the time-cost of finding the route to each local transit node is $O\\left(c_1 \\ (\\eta r^2) p(1) \\right)$ .", "Transit nodes have point-intensity $O(p(1)/r^2)$ , so the total number is $O(Ap(1)/r^2)$ .", "Regard the space-cost of storing a $k \\times k$ matrix of inter-transit-node routes as $c_2 k^2$ ; so this space-cost is $O \\left( c_2 \\ (p(1)A/r^2)^2 \\right)$ .", "Summing the two costs and optimizing over $r$ , the optimal cost is $O(c_1^{2/3}c_2^{1/3}\\eta ^{2/3}A^{2/3}p^{4/3}(1))= O(c_1^{2/3}c_2^{1/3}p^{4/3}(1)M^{2/3})$ and this $O(M^{2/3})$ scaling represents the improvement over the $O(M)$ scaling for Dijkstra.", "The corresponding optimal number of transit nodes is $O((c_1/c_2)^{1/3} p^{2/3}(1) M^{1/3})$ .", "The latter has a more interpretable formulation.", "If the only alternative algorithms were a Dijkstra search of cost $c_1 \\times (\\mbox{number of edges})$ or table look-up of cost $c_2 \\times (\\mbox{number of edges})^2$ , then there would be some critical number of edges at which one should switch between them, and this is just the solution $m_{\\mbox{{\\tiny crit}}}$ of $c_1 m_{\\mbox{{\\tiny crit}}}=c_2 m_{\\mbox{{\\tiny crit}}}^2$ .", "So the optimal number of transit nodes is $O(m_{\\mbox{{\\tiny crit}}}^{1/3}p^{2/3}(1) M^{1/3})$ ." ], [ "Number of singly infinite geodesics", "Write $\\mbox{${\\mathcal {S}}$}^*(\\lambda )$ for the spanning subnetwork obtained from $\\mbox{${\\mathcal {S}}$}(\\lambda )$ by adding a city at the origin $\\mathbf {0}$ .", "This process inherits the scaling-invariance property () of $\\mbox{${\\mathcal {S}}$}(\\lambda )$ .", "We know from (REF ) that at least one singly infinite geodesic from $\\mathbf {0}$ exists.", "The set of all singly infinite geodesics in $\\mbox{${\\mathcal {S}}$}^*(\\lambda )$ from $\\mathbf {0}$ forms a priori a tree, because two geodesics that branch cannot re-join, by route compatability property (iv) from section REF .", "So consider $ q(\\lambda ,r) := {\\mathbb {E}}\\mbox{(number of distinct points at which some singly infinite geodesic } $ $ \\mbox{ in $\\mbox{${\\mathcal {S}}$}^*(\\lambda )$ from $\\mathbf {0}$ first crosses the circle of radius $r$).", "}$ What we know in general is $ 1 \\le q(\\lambda , r) \\le \\infty ; \\quad r \\rightarrow q(\\lambda ,r) \\mbox{ is increasing;} \\quad \\lambda \\rightarrow q(\\lambda ,r) \\mbox{ is increasing} $ and the scaling property gives $q(\\lambda ,r) = q(1,r\\lambda ^{1/2}) .$ So the $\\lambda \\rightarrow \\infty $ limit $q(\\infty ,r) := \\lim _{\\lambda \\rightarrow \\infty } q(\\lambda ,r)$ exists (maybe infinite), and the scaling property implies $ q(\\infty ,r) = q(\\infty ,1) \\in [1,\\infty ], \\quad 0 < r < \\infty .", "$ So consider the property $q(\\infty ,1) < \\infty .$ By applying Proposition REF with $r \\approx 1$ we see $q(\\infty ,1) \\le 4 p(1) .$ So (REF ) implies (REF ).", "So we have shown the following.", "Corollary 23 As $\\lambda \\rightarrow \\infty $ the number of singly infinite geodesics in $\\mbox{${\\mathcal {S}}$}^*(\\lambda )$ from $\\mathbf {0}$ increases to a finite limit number (perhaps a random number with finite mean) $G$ .", "Moreover, if $G>1$ then these geodesics branch at $\\mathbf {0}$ ." ], [ "Unique singly-infinite geodesics and continuity", "For a SIRSN, let us call the property $G = 1$ a.s. (in the notation of Corollary REF above) the unique singly-infinite geodesics property.", "It is conceivable that this property always holds – we record this later in Open Problem REF .", "Uniqueness of geodesics is closely related to continuity of routes $\\mbox{${\\mathcal {R}}$}(z_1,z_2)$ as $(z_1,z_2)$ vary, as will be seen in section REF ." ], [ "Equivalent properties", "Here we show that several properties, the simplest being (REF ), are equivalent to the unique singly-infinite geodesics property.", "We will give definitions and proofs as we proceed, and then summarize as Proposition REF .", "Consider two independent uniform random points $U_1, U_2$ in $\\mathrm {disc}(\\mathbf {0}, 1)$ .", "By the route-compatability property, the intersection of $\\mbox{${\\mathcal {R}}$}(\\mathbf {0}, U_1)$ and $\\mbox{${\\mathcal {R}}$}(\\mathbf {0}, U_2)$ is a sub-route from $\\mathbf {0}$ to some branchpoint $B_{1,2}$ , where either $B_{1,2} \\ne \\mathbf {0}$ or the intersection consists of the single point $\\mathbf {0}$ (in which case, set $B_{1,2} = \\mathbf {0}$ ).", "So we can define a property $\\mathbb {P}(B_{1,2} = \\mathbf {0}) = 0 .$" ], [ "Unique singly-infinite geodesics imply (", "Suppose (REF ) fails.", "Then there exists $\\varepsilon > 0$ such that, for independent random points $U^1_1, U^1_2$ in $\\mathrm {disc}(\\mathbf {0}, 1) \\setminus \\mathrm {disc}(\\mathbf {0}, \\varepsilon ) $ , their branchpoint $B^1_{1,2}$ satisfies $\\mathbb {P}(B^1_{1,2} = \\mathbf {0}) \\ge \\varepsilon $ .", "Scaling by $\\varepsilon ^{-m}, m \\ge 1$ and using scale-invariance, for independent random points $U^m_1, U^m_2$ in $\\mathrm {disc}(\\mathbf {0}, \\varepsilon ^{-m}) \\setminus \\mathrm {disc}(\\mathbf {0}, \\varepsilon ^{1-m}) $ , their branchpoint $B^m_{1,2}$ satisfies $\\mathbb {P}(B^m_{1,2} = \\mathbf {0}) \\ge \\varepsilon $ .", "It follows that, with probability $\\ge \\varepsilon - o(1)$ as $m \\rightarrow \\infty $ , there exists points $\\xi ^m_1, \\xi ^m_2$ of $\\Xi (1) \\cap (\\mathrm {disc}(\\mathbf {0}, \\varepsilon ^{-m}) \\setminus \\mathrm {disc}(\\mathbf {0}, \\varepsilon ^{1-m}))$ such that $ \\mbox{ routes $\\mbox{${\\mathcal {R}}$}(\\mathbf {0}, \\xi ^m_1)$ and $\\mbox{${\\mathcal {R}}$}(\\mathbf {0}, \\xi ^m_2)$branch at $\\mathbf {0}$}.", "$ So on an event of probability $\\ge \\varepsilon $ this property holds for infinitely many $m$ .", "Then on that event we have $G > 1$ , by compactness within the spanning subnetwork $\\mbox{${\\mathcal {S}}$}^*(1)$ (Lemma REF ).", "Next consider the spanning subnetwork $\\mbox{${\\mathcal {S}}$}^*(\\lambda )$ on points $\\Xi (\\lambda ) \\cup \\lbrace \\mathbf {0}\\rbrace $ .", "The intersection of all routes $\\mbox{${\\mathcal {R}}$}(\\mathbf {0}, \\xi ), \\ \\xi \\in \\Xi (\\lambda )$ is a sub-route from $\\mathbf {0}$ to some branchpoint $B(\\lambda )$ .", "So we can define a property $\\mathbb {P}(B(1) = \\mathbf {0}) = 0 .$ Clearly (REF ) implies (REF ); we need to argue the converse." ], [ "(", "Suppose (REF ).", "For each $r< \\infty $ the intersection of routes $\\mbox{${\\mathcal {R}}$}(\\mathbf {0}, \\xi ), \\ \\xi \\in \\Xi (1) \\cap \\mathrm {disc}(\\mathbf {0}, r)$ is a subroute $\\pi (1,r)$ from $\\mathbf {0}$ to some branchpoint $B(1,r)$ , and by (REF ), scaling and the finiteness of $\\Xi (1) \\cap \\mathrm {disc}(\\mathbf {0}, r)$ we have $\\mathbb {P}( B(1,r) = \\mathbf {0}) = 0 , \\mbox{ each } r < \\infty .$ As $r$ increases the subroute $\\pi (1,r)$ can only shrink, and the quantity in (REF ) is the limit $B(1) = \\lim _{r \\rightarrow \\infty } B(1,r)$ .", "To prove (REF ) it suffices, by (REF ), to prove $B(1,r) \\mbox{ is constant for all large $r$, a.s. }$ We may suppose (otherwise the result is obvious) that for some $r_0 \\ge 4$ the subroute $\\pi (1,r_0)$ stays within $\\mathrm {disc}(\\mathbf {0}, 1)$ .", "As $r$ increases, the only way that $B(1,r)$ can change at $r$ is if there is a point $\\xi \\in \\Xi (1) \\cap \\mathrm {circle}(\\mathbf {0}, r)$ for which the route $\\mbox{${\\mathcal {R}}$}(\\mathbf {0},\\xi )$ diverges from the existing subroute $\\pi (1,r-)$ before the existing branchpoint $B(1,r-)$ .", "If this happens, at $r_1$ say, then consider the subroute $\\theta (r_1) = \\mbox{${\\mathcal {R}}$}(\\mathbf {0},\\xi ) \\cap (\\mathrm {disc}(\\mathbf {0}, 4) \\setminus \\mathrm {disc}(\\mathbf {0}, 1))$ which has length at least 3.", "Now suppose $B(1,r)$ again changes at some larger value $r_2$ .", "Then the corresponding subroute $\\theta (r_2)$ must be disjoint from $\\theta (r_1)$ , by the route-compatability property.", "Now the “finite length in bounded regions\" property (REF ) implies that $B(1,r)$ can change at only finitely many large values of $r$ , establishing (REF ).", "Now make a slight re-definition of $B(\\lambda )$ , by considering only points $\\xi $ outside the unit disc.", "That is, the intersection of all routes $\\mbox{${\\mathcal {R}}$}(\\mathbf {0}, \\xi ), \\ \\xi \\in \\Xi (\\lambda )\\setminus \\mathrm {disc}(\\mathbf {0}, 1) $ is a sub-route $\\tilde{\\pi }(\\lambda )$ from $\\mathbf {0}$ to some branchpoint $B_1(\\lambda )$ .", "Using scale-invariance it is easy to check that (REF ) is equivalent to $\\mathbb {P}(B_1(\\lambda ) = \\mathbf {0}) = 0 \\mbox{ for each } \\lambda < \\infty .$ As $\\lambda $ increases, the sub-routes $\\tilde{\\pi }(\\lambda )$ can only shrink, and the intersection of these subroutes over all $\\lambda < \\infty $ is again a subroute from $\\mathbf {0}$ to some point $B_1(\\infty )$ .", "So we can define a property $\\mathbb {P}(B_1(\\infty ) = \\mathbf {0}) = 0 .$ Clearly (REF ) implies (REF ); we need to argue the converse." ], [ "(", "Suppose (REF ).", "To prove (REF ) we essentially repeat the argument above, but use assumption (REF ) instead of (REF ).", "It is enough to show that, as $\\lambda $ increases, $B_1(\\lambda )$ can change at only finitely many large values of $\\lambda $ .", "And we may suppose that for large $\\lambda $ the subroute $\\tilde{\\pi }(\\lambda )$ stays within $\\mathrm {disc}(\\mathbf {0}, 1/4)$ .", "If $B_1(\\lambda )$ changes at $\\lambda _1$ then there is a point $\\xi $ appearing at “time\" $\\lambda _1$ for which $\\mbox{${\\mathcal {R}}$}(\\mathbf {0}, \\xi )$ diverges from the existing subroute $\\tilde{\\pi }(\\lambda _1 -)$ and so must cross $\\mathrm {circle}(\\mathbf {0}, 5/8)$ at some point $z(\\lambda _1) \\in \\mbox{${\\mathcal {E}}$}(\\lambda _1, 3/8) \\subset \\mbox{${\\mathcal {E}}$}(\\infty , 3/8)$ .", "By route-compatability the points $z(\\lambda _i)$ corresponding to different values $\\lambda _i$ where $B_1(\\lambda )$ changes must be distinct, and then (REF ) implies $\\mbox{${\\mathcal {E}}$}(\\infty , 3/8) \\cap \\mathrm {circle}(\\mathbf {0}, 5/8)$ is an a.s. finite set of points.", "Clearly (REF ) implies unique singly-infinite geodesics, by the final assertion of Corollary REF .", "We have now shown a cycle of equivalences.", "Finally, by scaling (REF ) is equivalent to the following property, where the notation is chosen to be consistent with notation in the next section.", "Define $Q(\\lambda , 0, B)$ to be the probability that the routes $\\mbox{${\\mathcal {R}}$}(\\mathbf {0},\\xi ^\\prime )$ to all points $\\xi ^\\prime \\in \\mbox{${\\mathcal {S}}$}(\\lambda ) \\cap (\\mathrm {disc}(\\mathbf {0},B))^c $ do not all first exit $\\mathrm {disc}(\\mathbf {0}, 1)$ at the same point.", "Then (REF ) is equivalent to $\\lim _{B \\uparrow \\infty } \\lim _{\\lambda \\rightarrow \\infty } Q(\\lambda , 0, B) = 0 .$ To summarize: Proposition 24 Properties (REF ), (REF ), (REF ), (REF ) and (REF ) are each equivalent to the unique singly-infinite geodesics property." ], [ "Continuity properties", "In the previous section we studied properties of long routes from a single point.", "We now consider long routes from nearby points, and in this context it seems harder to understand whether different properties are equivalent.", "Suppose, for this discussion, the unique singly-infinite geodesics property holds.", "Then the geodesics from $\\mathbf {0}$ and from $\\mathbf {1}= (1,0) \\in {\\mathbb {R}}^2$ are either disjoint or coalesce; we do not know (Open Problem REF ) whether the property $\\mbox{the geodesics from $\\mathbf {0}$ and from$\\mathbf {1}$ coalesce a.s }$ always holds or is stronger.", "There are several equivalent ways of saying (REF ) – see the end of this section – but what's relevant now is that it is equivalent to the property that, for each $\\lambda $ , the geodesics from each point of $\\mbox{${\\mathcal {S}}$}^*(\\lambda ) \\cap \\mathrm {disc}(\\mathbf {0},1)$ coincide outside a disc of random radius $R(\\lambda ) < \\infty $ a.s..", "So we can then ask whether the property $ R(\\infty ) := \\lim _{\\lambda \\rightarrow \\infty } R(\\lambda ) < \\infty \\mbox{ a.s. } $ is implied by property (REF ) or is stronger.", "We restate this latter property as (REF ) below.", "For $0 < \\varepsilon < 1 < B$ define $Q(\\lambda , \\varepsilon , B)$ to be the probability that the routes $\\mbox{${\\mathcal {R}}$}(\\xi ,\\xi ^\\prime )$ between points $\\xi \\in \\mbox{${\\mathcal {S}}$}(\\lambda ) \\cap \\mathrm {disc}(\\mathbf {0},\\varepsilon )$ and points $\\xi ^\\prime \\in \\mbox{${\\mathcal {S}}$}(\\lambda ) \\cap (\\mathrm {disc}(\\mathbf {0},B))^c $ do not all first exit $\\mathrm {disc}(\\mathbf {0}, 1)$ at the same point.", "Note $Q(\\lambda , \\varepsilon , B)$ is monotone increasing at $\\lambda $ increases, and decreasing as $B$ increases or $\\varepsilon $ decreases.", "So we can define $ Q(\\infty , \\varepsilon , B) := \\lim _{\\lambda \\rightarrow \\infty } Q(\\lambda , \\varepsilon , B) $ and then define a property of a SIRSN $\\lim _{\\varepsilon \\downarrow 0, B \\uparrow \\infty } Q(\\infty , \\varepsilon , B) = 0$ where the limit value is unaffected by the order of the double limit.", "In words, (REF ) says that (with high probability) every route from a small neighborhood of the origin to any distant point will first cross the unit circle at the same place.", "Property (REF ) implies (REF ) and implies form (REF ) of the unique singly-infinite geodesics property, which is the same assertion for routes from the origin only.", "The kinds of properties described above relate to questions about continuity of the routes $\\mbox{${\\mathcal {R}}$}(z_1,z_2)$ as $z_1, z_2$ vary, and we will give one such relation as Lemma REF below.", "Consider $0 < \\eta < \\delta < 1/2$ and for points $\\xi \\in \\mbox{${\\mathcal {S}}$}(\\lambda ) \\cap \\mathrm {disc}(\\mathbf {0},\\eta )$ and $\\xi ^\\prime \\in \\mbox{${\\mathcal {S}}$}(\\lambda ) \\cap \\mathrm {disc}(\\mathbf {1},\\eta )$ with route $\\mbox{${\\mathcal {R}}$}(\\xi ,\\xi ^\\prime )$ let $\\mbox{${\\mathcal {R}}$}_\\delta (\\xi ,\\xi ^\\prime )$ be the sub-route between the first exit from $\\mathrm {disc}(\\mathbf {0},\\delta )$ and the last entrance into $\\mathrm {disc}(\\mathbf {1},\\delta )$ .", "Let $\\Psi (\\lambda , \\eta , \\delta )$ be the probability that the sub-routes $\\mbox{${\\mathcal {R}}$}_\\delta (\\xi ,\\xi ^\\prime )$ for all $\\xi \\in \\mbox{${\\mathcal {S}}$}(\\lambda ) \\cap \\mathrm {disc}(\\mathbf {0},\\eta )$ and all $\\xi ^\\prime \\in \\mbox{${\\mathcal {S}}$}(\\lambda ) \\cap \\mathrm {disc}(\\mathbf {1},\\eta )$ are not all an identical sub-route.", "As above, by monotonicity we can define $ \\Psi (\\infty , \\eta , \\delta ) := \\lim _{\\lambda \\rightarrow \\infty } \\Psi (\\lambda , \\eta , \\delta ) $ and then define a property of a SIRSN $\\lim _{\\eta \\downarrow 0} \\Psi (\\infty , \\eta , \\delta ) = 0 \\quad \\forall \\delta .$ In words, (REF ) says that (with high probability) all routes from a very small neighborhood of the origin to a very small neighborhood of $\\mathbf {1}$ coincide outside of larger small neighborhoods.", "Lemma 25 Property (REF ) implies property (REF ).", "Proof.", "Choose $a$ such that $a \\eta < 1 < a \\delta $ .", "Take the definition of $\\Psi $ , scale by $a$ , and use scale-invariance to obtain the following.", "The probability that the sub-routes $\\mbox{${\\mathcal {R}}$}_{a \\delta }(\\xi ,\\xi ^\\prime )$ for all $\\xi \\in \\mbox{${\\mathcal {S}}$}(a^{-2} \\lambda ) \\cap \\mathrm {disc}(\\mathbf {0}, a \\eta )$ and all $\\xi ^\\prime \\in \\mbox{${\\mathcal {S}}$}(a^{-2} \\lambda ) \\cap \\mathrm {disc}((a,0), a \\eta )$ are not all an identical sub-route equals $\\Psi (\\lambda , \\eta , \\delta )$ .", "When this occurs there are two non-identical sub-routes between $\\mathrm {circle}(\\mathbf {0}, a \\delta )$ and $\\mathrm {circle}( (a,0), a \\delta )$ , which imply two non-identical sub-routes between $\\mathrm {circle}(\\mathbf {0}, 1)$ and $\\mathrm {circle}( (a,0), 1)$ .", "For this to happen, either the defining event for $Q(a^{-2} \\lambda , a \\eta , a/2)$ , or the analogous event with reference to $(a,0)$ instead of $\\mathbf {0}$ , must occur; otherwise all routes in question pass through the same points on $\\mathrm {circle}(\\mathbf {0}, 1)$ and $\\mathrm {circle}( (a,0), 1)$ , contradicting the route-compatability properties of section REF .", "So $ \\Psi (\\lambda , \\eta , \\delta ) \\le 2 Q(a^{-2} \\lambda , a \\eta , a/2) .", "$ Letting $\\lambda \\rightarrow \\infty $ $ \\Psi (\\infty , \\eta , \\delta ) \\le 2 Q(\\infty , a \\eta , a/2) .", "$ Choosing $a = \\eta ^{-1/2}$ establishes the lemma.", "Remark.", "Lemma REF is almost enough to prove that, under condition (REF ), we have the continuity property $\\mbox{ if } (z^n_1,z^n_2) \\rightarrow (z_1,z_2) \\mbox{ then } \\mbox{${\\mathcal {R}}$}(z^n_1,z^n_2) \\rightarrow \\mbox{${\\mathcal {R}}$}(z_1,z_2) \\mbox{ a.s. }$ where convergence of paths is in the sense of section REF .", "To deduce (REF ) one would need also to show that the lengths of $ \\mbox{${\\mathcal {R}}$}(z^n_1,z^n_2) \\cap (\\mathrm {disc}(z_1,\\varepsilon _n) \\cup \\mathrm {disc}(z_2,\\varepsilon _n))$ tend to 0 a.s. for all $\\varepsilon _n \\rightarrow 0$ .", "This is loosely related to Open Problem REF ." ], [ "Another property equivalent to (", "Because geodesics either colalesce or are disjoint, for any countable set of initial points there is some set of “geodesic ends\", where each such “end\" corresponds to a tree of coalescing geodesics from originating “leaves\".", "By a small modification of the proof of Corollary REF , the mean number of such ends from the points $\\Xi (\\lambda ) \\cap \\mathrm {disc}(\\mathbf {0}, 1)$ is at most $4 p(1)$ , so we can let $\\lambda \\rightarrow \\infty $ and deduce that the number $G^* \\ge 1$ of ends from initial points $\\Xi (\\infty ) \\cap \\mathrm {disc}(\\mathbf {0}, 1)$ satisfies ${\\mathbb {E}}G^* \\le 4 p(1)$ .", "Then by scale-invariance, for each $0<r<\\infty $ the number of ends from initial points $\\Xi (\\infty ) \\cap \\mathrm {disc}(\\mathbf {0}, r)$ equals $G^*$ .", "So the property $ G^* = 1 \\mbox{ a.s. } $ is clearly equivalent to property (REF ) (plus the unique singly-infinite geodesics property).", "Note that if $G^* > 1$ then there are a finite number of different “geodesic trees\" each of whose leaf-sets is dense in ${\\mathbb {R}}^2$ – behavior hard to visualize." ], [ "The binary hierarchy model", "Proposition 26 The binary hierarchy model has property (REF ) .", "Proof.", "Consider the last stages of construction of the model in section REF .", "Rotation and scaling do not affect the property of interest, so it will suffice to prove the property in the model $\\mbox{${\\mathcal {R}}$}_{\\mbox{{\\footnotesize t-i} }}$ .", "Consider the argument from “proof of Proposition REF \" in section REF but with large rescalings of $B$ instead of small rescalings.", "Combining this argument with the construction of $\\mbox{${\\mathcal {R}}$}_{\\mbox{{\\footnotesize t-i} }}$ at the start of section REF one can show (details omitted) that the set $ A^\\prime := \\lbrace z \\in {\\mathbb {R}}^2: \\ z \\mbox{ in only finitely many } B^\\prime _i\\rbrace $ has area zero; here $B^\\prime _i:= \\sigma _{2^{i}}B$ is the “large\" rescaling of the union $B: = \\cup _G \\Sigma _G$ of the translates $\\Sigma _G$ of the small subsquare $\\Sigma $ of the basic $2^{h+1} \\times 2^{h+1}$ square $G$ in the Figure 6 configuration.", "By translation-invariance, this implies that a.s. $\\mathbf {0}\\notin A^\\prime $ .", "For such a realization there is a random infinite sequence $i(j)$ with $\\mathbf {0}\\in \\sigma _{2^{i(j)}}B$ , and any singly-infinite geodesic from $\\mathbf {0}$ must pass through the corresponding infinite sequence $b_{i(j)}$ of points determined by Figure 6.", "This establishes the unique singly-infinite geodesic property.", "Moreover $\\mathbf {0}$ lies in some translated square $\\Sigma _{i(j)}$ of side $\\varepsilon 2^{i(j)}$ and for any other point in that square its geodesic must coalesce with the geodesic from $\\mathbf {0}$ at or before $b_{i(j)}$ .", "It is easy to check that the squares $\\Sigma _{i(j)}$ eventually cover any fixed disc, and this establishes property (REF )." ], [ "Other specific models?", "A major challenge is finding other explicit examples of SIRSN models.", "Let us pose the vague problems Open Problem 27 Give a construction of a SIRSN which is “mathematically natural\" in some sense, e.g.", "in the sense that there is an explicit formula for the distribution of subnetworks $\\mathbf {span}(z_1,\\ldots ,z_k)$ .", "Open Problem 28 Give a construction of a SIRSN which is “visually realistic\" in the sense of not looking very different from a real-world road network." ], [ "Quantitative bounds on statistics", "In designing a finite road network there is an obvious tradeoff between total length and the network's effectiveness in providing short routes, so in our context there is a tradeoff between $\\ell $ and $\\Delta := {\\mathbb {E}}D_1$ .", "More generally Open Problem 29 What can we say about the set of possible values, over all SIRSNs, of the triple $(\\Delta = {\\mathbb {E}}D_1, \\ell , p(1))$ of statistics of a SIRSN?", "This is a sensible question because each statistic is dimensionless, that is not dependent on choice of unit of length – a non-dimensionless statistic would take all values in $(0,\\infty )$ by scaling.", "We have given three results relating to this problem.", "Proposition REF gave a crude lower bound on the function $\\ell _*(\\Delta )$ defined as the infimum value of $\\ell $ over all SIRSNs with the given value of $\\Delta $ .", "Open Problem 30 (i) Give quantitative estimates of the function $\\ell _*(\\Delta )$ , improving Proposition REF .", "(ii) Do “optimal\" networks attaining the infimum exist, and (if so) can we say something about the structure of the associated optimal networks?", "One might make the (vague) conjecture that for some value of $\\Delta $ the optimal network exploits 4-fold symmetry in some way analogous to our section model, and that for some other value it exploits 6-fold symmetry.", "In section REF we showed (REF ) that the overall minimum normalized length $\\ell _*:= \\inf _\\Delta \\ell _*(\\Delta )$ satisfies $\\ell _*\\ge \\sqrt{1/8}$ .", "The third result was Lemma REF , showng $\\ell \\le 2p(1)$ ." ], [ "Traffic intensity", "As mentioned in section REF , the conceptual point of $\\mbox{${\\mathcal {E}}$}(\\infty ,r)$ is to capture the idea of the major road - minor road spectrum, and the particular definition of $\\mbox{${\\mathcal {E}}$}(\\infty ,r)$ is mathematically convenient because of the scaling property (REF ) of the edge-intensity $p(r)$ .", "But from a real-world perspective it seems more natural to use some notion of traffic intensity.", "Given any measure $\\psi $ on source-destination pairs $(z_1,z_2)$ , then length measure $\\mathrm {Leb}_1$ along the routes $\\mbox{${\\mathcal {R}}$}(z_1,z_2)$ in a SIRSN induces a “traffic intensity\" measure $\\widetilde{\\psi }$ on $\\cup _r \\mbox{${\\mathcal {E}}$}(\\infty ,r)$ .", "The natural measures $\\psi $ to consider are specified by (i) $z_1$ has Lebesgue measure $\\mathrm {Leb}_2$ on ${\\mathbb {R}}^2$ (ii) given $z_1$ , the measure on $z:= z_2 - z_1$ has density $|z|^{-\\beta }$ .", "The action of $\\sigma _c$ on $\\psi $ gives the measure specified by (i) $z_1$ has measure $c^{-2} \\mathrm {Leb}_2$ on ${\\mathbb {R}}^2$ (ii) $z:= z_2 - z_1$ has density $c^{\\beta -2} |z|^{- \\beta }$ (iii) the measure along paths is $c^{-1} \\mathrm {Leb}_1$ and this is the measure $\\widetilde{\\psi }$ scaled by $c^{\\beta - 5}$ .", "To make a rigorous treatment, the issue is to show that $\\widetilde{\\psi }$ is a locally finite measure on $\\mbox{${\\mathcal {E}}$}(\\infty ,1)$ .", "Heuristically one needs $\\beta > 2$ so that the contribution from large $|z_2 - z_1|$ is finite, and $\\beta < 4$ so that the contribution from small $|z_2 - z_1|$ is finite.", "Open Problem 31 Show that, perhaps under regularity assumptions on the SIRSN, for $2 < \\beta < 4$ the construction above gives a locally finite measure $\\widetilde{\\psi }$ on $\\mbox{${\\mathcal {E}}$}(\\infty ,1)$ and hence on $\\cup _r \\mbox{${\\mathcal {E}}$}(\\infty ,r)$ ." ], [ "Implications between different properties of a SIRSN", "We have given various results of the form “one property of a SIRSN implies another\" for which we conjecture the converse is false.", "In particular, we expect there are counter-examples to most of the following, though of course this requires constructing other examples of SIRSNs.", "Open Problem 32 Prove, or give a counter-example to: (i) (REF ) implies (REF ) (ii) the unique singly-infinite geodesics property implies (REF ) (iii) (REF ) implies (REF ) (iv) (REF ) implies (REF )." ], [ "Understanding the structure of $\\mbox{${\\mathcal {E}}$}(\\infty , 1)$ . ", "We envisage $\\mbox{${\\mathcal {E}}$}(\\infty , 1)$ as looking somewhat like a real-world network of major roads, but it is not clear what aspects of real networks appear automatically in our SIRSN model.", "For instance, a priori $\\mbox{${\\mathcal {E}}$}(\\infty , 1)$ need not be connected (it might contain a short segment in the middle of a route between two points at distance $2 + \\varepsilon $ apart) but it must contain an unbounded connected component (most of a singly-infinite geodesic).", "Open Problem 33 Does $\\mbox{${\\mathcal {E}}$}(\\infty , 1)$ have a.s. only a single unbounded connected component?" ], [ "Questions about lengths", "Even though we started the whole topic of SIRSNs by considering route-lengths, they have played a rather small role in our results, and many questions about route-lengths could be asked.", "Open Problem 34 Under what extra assumptions (if any) is it true that, for $U_1, U_2, \\ldots $ independent uniform on $\\mathrm {disc}(\\mathbf {0}, 1)$ , $ {\\mathbb {E}}\\sup _{i \\ge 1} \\mathrm {len}[\\mbox{${\\mathcal {R}}$}(\\mathbf {0}, U_i)] < \\infty ?", "$ The following (intuitively obvious) claim seems curiously hard to prove; the difficulty lies in showing that the spanning subnetwork does not have (necessarily with low probability) huge length a long way away from the square.", "Open Problem 35 Take $k$ uniform random points $Z_1,\\ldots ,Z_k$ in a square of area $k$ and consider the length $\\mathrm {len}[ \\mathbf {span}(Z_1,\\ldots ,Z_k) ]$ of the spanning subnetwork random network $\\mathbf {span}(Z_1,\\ldots ,Z_k)$ .", "Prove $ {\\mathbb {E}}\\ \\mathrm {len}[ \\mathbf {span}(Z_1,\\ldots ,Z_k) ] \\sim \\ell k \\mbox{ as } k \\rightarrow \\infty .", "$" ], [ "Alternative starting points for a setup", "We started the whole modeling process by assuming we are given routes between points, but one can imagine two different starting points.", "The first involves starting with a network of major roads and then adding successively more minor roads, so eventually the road network is dense in the plane.", "In other words, base a model on some explicit construction as $r$ decreases of some process $(\\mbox{${\\mathcal {E}}$}(r),\\ \\infty > r > 0)$ of “roads of size $\\ge r$ \" (in our setup this is achieved implicitly by the networks $\\mbox{${\\mathcal {E}}$}(\\infty ,r)$ ).", "Of course this corresponds to what we see when zooming in on an online map of the real-world road network; the maps are designed to show only the relatively major roads within the window, and hence to show progressively more minor roads as one zooms in.", "In talks we show such zooms along with the online “zooming in\" demonstration [24] of Brownian scaling to illustrate the concept of scale-invariance.", "The second, mathematically abstract, approach is to start with a random metric $d(z,z^\\prime )$ on the plane, and define routes as geodesics.", "But a technical difficulty with both of these approaches is that there seems no simple way to guarantee unique routes between a.a. pairs of points in the plane – in general one needs to add an assumption of uniqueness.", "The explicit models constructed in section and outlined in section do use the “random metric\" idea, but the hard part of the construction is proving the uniqueness of routes, even in these simplest models we can imagine.", "It is perhaps remarkable that our approach, taking routes as given with only the route-compatability property but with no explicit requirement that routes be minimum-cost in some sense, does lead to some non-obvious results." ], [ "Empirical evidence of scale-invariance?", "For real-world road networks, can scale-invariance be even roughly true over some range of distance?", "We mentioned one explicit piece of evidence (ordered segment lengths) in section REF ; there is also evidence that mean route length is indeed roughly proportional to distance, though this is also consistent with other (non scale-invariant) models [6].", "An interesting project would be to study the spanning subnetworks on (say) 4 real-world addresses, whose positions form roughly a square, randomly positioned, and find the empirical frequencies with which the various topologically different networks appear.", "Scale-invariance predicts these frequencies should not vary with the side-length of square; is this true?" ], [ "Hop count in spatial networks", "There has been study of spatial networks with respect to the trade-off between total network length and average graph distance (hop count), instead of route-length.", "See [23] for a recent literature survey and empirical analysis." ], [ "Continuum random trees in the plane", "Existence of continuum limits of discrete models of random trees has been conjectured, and studied non-rigorously in statistical physics, for a long time, and since 2000 spectacular progress has been made on rigorous proofs.", "For three models of random trees (uniform random spanning tree on ${\\mathbb {Z}}^2$ , minimal spanning tree on ${\\mathbb {Z}}^2$ (with random edge lengths), and the Euclidean minimal spanning tree on Poisson points), [2] established a rigorous “tightness\" result and gave sample properties of subsequential limits.", "A subsequent deep result [18] established the existence of a continuum limit in the first model.", "In these limits the paths have Hausdorff dimension greater than 1 so $D_1 = \\infty $ a.s..", "There should be a simple proof of the following, because our definition of SIRSN requires ${\\mathbb {E}}D_1 = \\infty $ .", "Open Problem 36 In a SIRSN, the subnetwork $\\mbox{${\\mathcal {S}}$}(1)$ cannot be a tree (with Steiner points)." ], [ "Geodesics in first-passage percolation", "Geodesics in particular models of first-passage percolation have been studied in [19].", "It is unclear whether there is any substantial connection between the behavior of geodesis in that setting and in our setting." ], [ "A Monge-Kantorovitch approach", "A completely different approach to continuum networks, starting from Monge-Kantorovitch optimal transport theory, is developed in the monograph by Buttazzo et al.", "[11].", "Their model assumes (i) some continuous distribution of sources and sinks (ii) an a priori arbitrary set $\\Sigma $ representing location of roads (iii) two different costs-per-unit-length for travel inside [resp.", "outside] $\\Sigma $ .", "An optimal network in one that minimizes total transportation cost for a given cost functional on $\\Sigma $ .", "It is shown that, under regularity conditions, the optimal network is covered by a finite number of Lipschitz curves of uniformly bounded length, although it may have even uncountably many connected components.", "But this theory does not seem to address statistics analogous to our $\\Delta $ and $\\ell $ in any quantitative way." ], [ "The method of exchangeable substructures", "The general methodology of studying complicated random structures by studying induced substructures on random points has many applications [7].", "In particular, the Brownian continuum random tree [3] provides an analogy for what we would like to see (Open Problem REF ) in some “mathematically natural\" SIRSN – see e.g.", "the formula (13) therein for the distribution of the induced subtree on random points – though that is in the “mean-field\" setting without any $d$ -dimensional geometry." ], [ "Urban road networks.", "There is scattered literature on models for urban road networks, mostly with a rather different focus, though [17] has some conceptual similarities with our work." ], [ "Dynamic random graphs.", "Conceptually, what we are doing with routes $\\mbox{${\\mathcal {R}}$}(z_1,z_2)$ and subnetworks $\\mbox{${\\mathcal {S}}$}(\\lambda )$ is exploring a given network.", "This is conceptually distinct from using sequential constructions of a network, a topic often called dynamic random graphs [13], even though the particular “dynamic Gabriel\" model outlined in section REF does fit the “dynamic\" category." ], [ "Acknowldgements", "My thanks to Justin Salez for details of the proof of Lemma REF , to Wilfrid Kendall for ongoing collaboration, and to Cliff Stein for references to the algorithmic literature." ], [ "Appendix: A topology on the space of feasible subnetworks.", "We first define convergence of routes.", "Recall a feasible route $\\mbox{${\\mathsf {r}}$}(z,z^\\prime )$ involves line segments between points $(z_i)$ which we will call turn points of the route.", "Given $\\varepsilon < |z^\\prime - z|/2$ there is (starting from $z$ ) a last turn point $z_{(\\varepsilon )}$ before the route $\\mbox{${\\mathsf {r}}$}(z,z^\\prime )$ first exits $\\mathrm {disc}(z,\\varepsilon )$ and there is (starting from $z^\\prime $ ) a last turn point $z^\\prime _{(\\varepsilon )}$ before the reverse route $\\mbox{${\\mathsf {r}}$}(z^\\prime ,z)$ first exits $\\mathrm {disc}(z^\\prime ,\\varepsilon )$ .", "Define $\\mbox{${\\mathsf {r}}$}(z(n),z^\\prime (n)) \\rightarrow \\mbox{${\\mathsf {r}}$}(z,z^\\prime )$ to mean (i) $z(n) \\rightarrow z, \\ z^\\prime (n) \\rightarrow z^\\prime \\ne z$ .", "(ii) For each $\\varepsilon < |z^\\prime - z|/2$ such that $\\mathrm {circle}(z,\\varepsilon )$ and $\\mathrm {circle}(z^\\prime ,\\varepsilon )$ do not contain any turn point of $\\mbox{${\\mathsf {r}}$}(z,z^\\prime )$ , writing the turn points of the subroutes $\\mbox{${\\mathsf {r}}$}(z_{(\\varepsilon )}(n),z^\\prime _{(\\varepsilon )}(n))$ as $(y_0(n), y_1(n),\\ldots , y_k(n))$ , we have $ (y_0(n), y_1(n),\\ldots , y_k(n)) \\rightarrow (y_0, y_1,\\ldots , y_k) $ the limit being the turn points of the subroute $\\mbox{${\\mathsf {r}}$}(z_{(\\varepsilon )},z^\\prime _{(\\varepsilon )})$ , where $k$ is finite from the definition of feasible route.", "(iii) The total lengths $L_{(\\varepsilon )}(n)$ of $\\mbox{${\\mathsf {r}}$}(z(n),z^\\prime (n)) \\cap (\\mathrm {disc}(z,\\varepsilon ) \\cup \\mathrm {disc}(z^\\prime ,\\varepsilon ))$ satisfy $ \\lim _{\\varepsilon \\rightarrow 0} \\limsup _n L_{(\\varepsilon )}(n) = 0 .", "$ Despite its inelegant formulation, this seems the “natural\" notion of convergence.", "Now we specify, in a way analogous to (i-iii) above, what it means for a sequence $\\mbox{${\\mathsf {s}}$}(n)$ of feasible subnetworks on locally finite sets ${\\mathbf {z}}(n) = \\lbrace z^i(n)\\rbrace $ to converge to a limit subnetwork $\\mbox{${\\mathsf {s}}$}$ on ${\\mathbf {z}}$ .", "(i) We need ${\\mathbf {z}}(n)$ to converge to ${\\mathbf {z}}$ in the usual sense of convergence of simple point processes [12].", "This is equivalent to saying that if we take any $R$ such that $\\mathrm {circle}(\\mathbf {0},R)$ contains no point of ${\\mathbf {z}}$ , then we can label the points of ${\\mathbf {z}}(n) \\cap \\mathrm {disc}(\\mathbf {0},R)$ as $(z^1(n), \\ldots , z^K(n))$ in such a way that $(z^1(n), \\ldots , z^K(n)) \\rightarrow (z^1, \\ldots , z^K)$ , the limit (here and in analogous assertions below) being the points of ${\\mathbf {z}}\\cap \\mathrm {disc}(\\mathbf {0},R)$ .", "(ii) Take $R$ and $(z^1, \\ldots , z^K)$ as above and take $\\varepsilon < {\\textstyle \\frac{1}{2}} \\min _{1\\le i < j \\le K} |z^i - z^j|$ such that $\\cup _{1 \\le i \\le K} \\mathrm {circle}(z^i,\\varepsilon )$ does not contain any turn point within $\\mbox{${\\mathsf {s}}$}$ .", "Then we can label the turn points of $\\cup _{1 \\le i \\le K} \\mbox{${\\mathsf {r}}$}(z^i_{(\\varepsilon )}(n), z^j_{(\\varepsilon )}(n))$ as $(y^u(n), 1 \\le u \\le L)$ in such a way that $(y^u(n), 1 \\le u \\le L)\\rightarrow (y^u, 1 \\le u \\le L)$ $(y^u(n),y^v(n))$ is an edge-segment of route $\\mbox{${\\mathsf {r}}$}(z^i(n),z^j(n))$ iff $(y^u,y^v)$ is an edge-segment of route $\\mbox{${\\mathsf {r}}$}(z^i,z^j)$ .", "(iii) For each $1 \\le i < j \\le K$ the routes $\\mbox{${\\mathsf {r}}$}(z^i(n),z^j(n))$ satisfy (iii) above.", "(iv) Lemma REF (i) implies that given $R$ , the following quantity (referring to the subnetwork $\\mbox{${\\mathsf {s}}$}$ ) is finite: $ R^* := \\min \\lbrace r: \\ \\bigcup _{z_i, z_j \\in \\mathrm {disc}(\\mathbf {0}, r)} \\mbox{${\\mathsf {r}}$}(z_i,z_j) \\cap \\mathrm {disc}(\\mathbf {0}, R) = \\mbox{${\\mathsf {s}}$}\\cap \\mathrm {disc}(\\mathbf {0}, R) \\ \\rbrace $ (that is, each edge of $\\mbox{${\\mathsf {s}}$}$ within $\\mathrm {disc}(\\mathbf {0}, R)$ is part of some route between endpoints in $\\mathrm {disc}(\\mathbf {0}, R^*)$ ).", "We require $ \\limsup _n R^*(n) < \\infty \\mbox{ for each } R < \\infty .", "$ We have described sequential convergence within the space of feasible subnetworks.", "It is routine to show this is convergence in some complete separable metric space, but we won't pursue such theory here." ] ]

[ [ "Combinatorics of the asymmetric exclusion process on a semi-infinite\n lattice" ], [ "Abstract We study two versions of the asymmetric exclusion process (ASEP) -- an ASEP on a semi-infinite lattice with an open left boundary, and an ASEP on a finite lattice with open left and right boundaries -- and we demonstrate a surprising relationship between their stationary measures.", "The semi-infinite ASEP was first studied by Liggett and then Grosskinsky, while the finite ASEP had been introduced earlier by Spitzer and Macdonald-Gibbs-Pipkin.", "We show that the finite correlation functions involving the first L sites for the stationary measures on the semi-infinite ASEP can be obtained as a nonphysical specialization of the stationary distribution of an ASEP on a finite one-dimensional lattice with L sites.", "Namely, if the output and input rates of particles at the right boundary of the finite ASEP are beta and delta, respectively, and we set delta=-beta, then this specialization corresponds to sending the right boundary of the lattice to infinity.", "Combining this observation with work of the second author and Corteel, we obtain a combinatorial formula for finite correlation functions of the ASEP on a semi-infinite lattice." ], [ "Introduction", "The asymmetric exclusion process (ASEP) is a model in which particles hop on a lattice, subject to the condition that there is at most one particle per site.", "It was first introduced by Spitzer [8] and also by Macdonal-Gibbs-Pipkin [7] in the context of protein synthesis, who studied this model on a finite lattice of $L$ sites.", "A version of the model where particles hop on the semi-infinite lattice $\\mathbb {Z}^+$ was studied by Liggett [6], and subsequently by Grosskinsky in his thesis [5].", "In the semi-infinite ASEP, particles may enter and exit at the left boundary at rates $\\alpha $ and $\\gamma $ , respectively, and in the bulk, particles may hop right and left to neighboring sites of the lattice at rates 1 and $q$ .", "Let $c$ be an additional parameter; it winds up determining the stationary current of particles.", "We denote states by vectors $\\eta = (\\eta _1, \\eta _2,\\dots )$ , where $\\eta _i \\in \\lbrace 0,1\\rbrace $ , and we denote the set of all states by $X$ .", "Figure: NO_CAPTIONIn the ASEP on a finite one-dimensional lattice of $L$ sites, particles may enter and exit at the left boundary at rates $\\alpha $ and $\\gamma $ , and may exit and enter at the right boundary at rates $\\beta $ and $\\delta $ .", "In the bulk, particles may hop right and left to neighboring sites of the lattice at rates 1 and $q$ .", "We refer to these two flavors of the ASEP as the semi-infinite ASEP and the finite ASEP.", "The two models are illustrated in Figure REF .", "In both models we assume that all parameters are non-negative.", "Given a measure $\\mu $ on $X$ , and a word $(\\eta _1,\\dots ,\\eta _L) \\in \\lbrace 0,1\\rbrace ^L$ , the correlation function $\\langle \\eta _1 \\dots \\eta _L \\rangle $ is the expected value with respect to $\\mu $ that the leftmost $L$ sites of a state in the semi-infinite ASEP will be $\\eta _1,\\dots ,\\eta _L$ .", "Our first result is the following.", "Theorem 1.1 The finite correlation functions involving the leftmost $L$ sites of the stationary measures of the semi-infinite ASEP can be obtained from the stationary distribution for the finite ASEP on a lattice of $L$ sites, after setting $\\beta =c$ and $\\delta =-c$ .", "More specifically, the correlation function $\\langle \\eta _1 \\dots \\eta _L \\rangle $ of the stationary measure for the semi-infinite ASEP corresponding to the stationary current $c$ is equal to $\\mu ^{fin}(\\alpha ,c,\\gamma , -c;q)(\\eta _1,\\dots ,\\eta _L)$ , the quantity one obtains by setting $\\beta =c$ and $\\delta = -c$ in the steady state probability of state $(\\eta _1,\\dots ,\\eta _L)$ in the finite ASEP.", "This theorem is illustrated in Figure REF .", "Figure: NO_CAPTIONBy combining Theorem REF with work of the second author and Corteel [2], [3] we can give a combinatorial formula for the finite correlation functions for the stationary measures for the semi-infinite ASEP.", "Before stating the result, we first introduce the staircase tableaux from [2], [3].", "Definition 1.2 A staircase tableau of size $L$ is a Young diagram of “staircase\" shape $(L, L-1, \\dots , 2, 1)$ such that boxes are either empty or labeled with $\\alpha , \\beta , \\gamma $ , or $\\delta $ , subject to the following conditions: no box along the diagonal is empty; all boxes in the same row and to the left of a $\\beta $ or a $\\delta $ are empty; all boxes in the same column and above an $\\alpha $ or a $\\gamma $ are empty.", "The type of a staircase tableau is a word in $\\lbrace \\circ , \\bullet \\rbrace ^L$ obtained by reading the diagonal boxes from northeast to southwest and writing a $\\bullet $ for each $\\alpha $ or $\\delta $ , and a $\\circ $ for each $\\beta $ or $\\gamma $ .", "See the left of Figure REF for an example of a staircase tableau.", "Figure: A staircase tableauof size 7 and type ∘∘•••∘∘\\circ \\circ \\bullet \\bullet \\bullet \\circ \\circ Definition 1.3 The weight $\\operatorname{wt}($ of a staircase tableau $is a monomial in$ , , , , q$, and $ u$, which we obtain as follows.Every blank box of $ is assigned a $q$ or $u$ , based on the label of the closest labeled box to its right in the same row and the label of the closest labeled box below it in the same column, such that: every blank box which sees a $\\beta $ to its right gets assigned a $u$ ; every blank box which sees a $\\delta $ to its right gets assigned a $q$ ; every blank box which sees an $\\alpha $ or $\\gamma $ to its right, and an $\\alpha $ or $\\delta $ below it, gets assigned a $u$ ; every blank box which sees an $\\alpha $ or $\\gamma $ to its right, and a $\\beta $ or $\\gamma $ below it, gets assigned a $q$ .", "After assigning a $q$ or $u$ to each blank box in this way, the weight of $ is then defined as the product of all labels in all boxes.$ The right of Figure REF shows that this staircase tableau has weight $\\alpha ^3 \\beta ^2 \\gamma ^3 \\delta ^3 q^9 u^8$ .", "Remark 1.4 The weight of a staircase tableau of size $L$ has degree $L(L+1)/2$ .", "We will typically set $u=1$ .", "Keeping $u$ general corresponds to particles in the bulk hopping right at rate $u$ instead of 1.", "The following result, concerning the stationary distribution of the finite ASEP, was announced in [2] and proved in [3].", "Theorem 1.5 [2], [3] Consider any state $\\tau $ of the finite ASEP with $L$ sites, where the parameters $\\alpha , \\beta , \\gamma , \\delta , q, u$ are general.", "Set $Z_L^{fin} = \\sum _{ \\operatorname{wt}(,where the sum is over all staircase tableaux of size L;we call Z_L^{fin} the \\emph {partition function} of the finite ASEP.Then the steady state probability that theASEP is at state \\tau isprecisely\\begin{equation*}\\frac{\\sum _{ \\operatorname{wt}(}{Z_L^{fin}},}{w}here the sum is over all staircase tableaux of type \\tau .\\end{equation*}}\\begin{example}Figure \\ref {tableaux} illustrates Theorem \\ref {OldThm}for the state \\bullet \\bullet of the ASEP.", "All staircase tableaux of type \\bullet \\bullet are shown.It follows that the steady state probability of \\bullet \\bullet is\\begin{equation*}\\frac{\\alpha ^2 u + \\delta ^2 q +\\alpha \\delta q +\\alpha \\delta u +\\alpha ^2 \\delta + \\alpha \\beta \\delta +\\alpha \\gamma \\delta +\\alpha \\delta ^2}{Z_2^{fin}}.\\end{equation*}\\end{example}\\begin{figure}[h]\\begin{picture}(0,0)\\includegraphics {Example.pstex}\\end{picture}\\begin{picture}(11744,1244)(279,-1883)\\put (1051,-1111){\\makebox{(}0,0)[lb]{\\smash{{\\normalfont \\fontfamily {}\\fontseries {}\\fontshape {}\\selectfont {\\color [rgb]{0,0,0}\\alpha }}}}}\\put (451,-1711){\\makebox{(}0,0)[lb]{\\smash{{\\normalfont \\fontfamily {}\\fontseries {}\\fontshape {}\\selectfont {\\color [rgb]{0,0,0}\\alpha }}}}}\\put (3451,-1711){\\makebox{(}0,0)[lb]{\\smash{{\\normalfont \\fontfamily {}\\fontseries {}\\fontshape {}\\selectfont {\\color [rgb]{0,0,0}\\alpha }}}}}\\put (5551,-1111){\\makebox{(}0,0)[lb]{\\smash{{\\normalfont \\fontfamily {}\\fontseries {}\\fontshape {}\\selectfont {\\color [rgb]{0,0,0}\\alpha }}}}}\\put (6451,-1111){\\makebox{(}0,0)[lb]{\\smash{{\\normalfont \\fontfamily {}\\fontseries {}\\fontshape {}\\selectfont {\\color [rgb]{0,0,0}\\alpha }}}}}\\put (7051,-1111){\\makebox{(}0,0)[lb]{\\smash{{\\normalfont \\fontfamily {}\\fontseries {}\\fontshape {}\\selectfont {\\color [rgb]{0,0,0}\\alpha }}}}}\\put (8551,-1111){\\makebox{(}0,0)[lb]{\\smash{{\\normalfont \\fontfamily {}\\fontseries {}\\fontshape {}\\selectfont {\\color [rgb]{0,0,0}\\alpha }}}}}\\put (10051,-1111){\\makebox{(}0,0)[lb]{\\smash{{\\normalfont \\fontfamily {}\\fontseries {}\\fontshape {}\\selectfont {\\color [rgb]{0,0,0}\\alpha }}}}}\\put (11551,-1111){\\makebox{(}0,0)[lb]{\\smash{{\\normalfont \\fontfamily {}\\fontseries {}\\fontshape {}\\selectfont {\\color [rgb]{0,0,0}\\alpha }}}}}\\put (10951,-1711){\\makebox{(}0,0)[lb]{\\smash{{\\normalfont \\fontfamily {}\\fontseries {}\\fontshape {}\\selectfont {\\color [rgb]{0,0,0}\\delta }}}}}\\put (10951,-1111){\\makebox{(}0,0)[lb]{\\smash{{\\normalfont \\fontfamily {}\\fontseries {}\\fontshape {}\\selectfont {\\color [rgb]{0,0,0}\\delta }}}}}\\put (9451,-1711){\\makebox{(}0,0)[lb]{\\smash{{\\normalfont \\fontfamily {}\\fontseries {}\\fontshape {}\\selectfont {\\color [rgb]{0,0,0}\\delta }}}}}\\put (9451,-1111){\\makebox{(}0,0)[lb]{\\smash{{\\normalfont \\fontfamily {}\\fontseries {}\\fontshape {}\\selectfont {\\color [rgb]{0,0,0}\\gamma }}}}}\\put (7951,-1111){\\makebox{(}0,0)[lb]{\\smash{{\\normalfont \\fontfamily {}\\fontseries {}\\fontshape {}\\selectfont {\\color [rgb]{0,0,0}\\beta }}}}}\\put (7951,-1711){\\makebox{(}0,0)[lb]{\\smash{{\\normalfont \\fontfamily {}\\fontseries {}\\fontshape {}\\selectfont {\\color [rgb]{0,0,0}\\delta }}}}}\\put (6451,-1711){\\makebox{(}0,0)[lb]{\\smash{{\\normalfont \\fontfamily {}\\fontseries {}\\fontshape {}\\selectfont {\\color [rgb]{0,0,0}\\delta }}}}}\\put (4951,-1711){\\makebox{(}0,0)[lb]{\\smash{{\\normalfont \\fontfamily {}\\fontseries {}\\fontshape {}\\selectfont {\\color [rgb]{0,0,0}\\delta }}}}}\\put (3451,-1111){\\makebox{(}0,0)[lb]{\\smash{{\\normalfont \\fontfamily {}\\fontseries {}\\fontshape {}\\selectfont {\\color [rgb]{0,0,0}q}}}}}\\put (2551,-1111){\\makebox{(}0,0)[lb]{\\smash{{\\normalfont \\fontfamily {}\\fontseries {}\\fontshape {}\\selectfont {\\color [rgb]{0,0,0}\\delta }}}}}\\put (1951,-1711){\\makebox{(}0,0)[lb]{\\smash{{\\normalfont \\fontfamily {}\\fontseries {}\\fontshape {}\\selectfont {\\color [rgb]{0,0,0}\\delta }}}}}\\put (1951,-1111){\\makebox{(}0,0)[lb]{\\smash{{\\normalfont \\fontfamily {}\\fontseries {}\\fontshape {}\\selectfont {\\color [rgb]{0,0,0}q}}}}}\\put (451,-1111){\\makebox{(}0,0)[lb]{\\smash{{\\normalfont \\fontfamily {}\\fontseries {}\\fontshape {}\\selectfont {\\color [rgb]{0,0,0}u}}}}}\\put (4951,-1111){\\makebox{(}0,0)[lb]{\\smash{{\\normalfont \\fontfamily {}\\fontseries {}\\fontshape {}\\selectfont {\\color [rgb]{0,0,0}u}}}}}\\put (4051,-1111){\\makebox{(}0,0)[lb]{\\smash{{\\normalfont \\fontfamily {}\\fontseries {}\\fontshape {}\\selectfont {\\color [rgb]{0,0,0}\\delta }}}}}\\end{picture} \\caption {The tableaux of type \\bullet \\bullet }\\end{figure}$ Remark 1.6 In [1], the second author together with Corteel, Stanley, and Stanton, studied staircase tableaux and their generating function $Z_L^{fin}(\\alpha ,\\beta ,\\gamma ,\\delta ;q)$ from a combinatorial point of view.", "In particular, Table 1 of [1] lists various specializations of $Z_L^{fin}(\\alpha ,\\beta ,\\gamma ,\\delta ;q)$ .", "The third row of the table shows that $Z_L^{fin}(\\alpha , \\beta , \\gamma , -\\beta ;q)=\\prod _{j=0}^{L-1}(\\alpha +q^j \\gamma ).$ Note that despite the fact that the specialization is nonphysical (we have made the hopping rate $\\delta $ a negative real number), the resulting quantity is positive.", "Also, the resulting quantity has no dependence on $\\beta $ and $\\delta $ .", "Corollary 1.7 Consider the semi-infinite ASEP, with parameters $\\alpha , \\gamma , q, c$ .", "Set $Z_L^{semi} = \\prod _{j=0}^{L-1} (\\alpha +q^j \\gamma )$ .", "Then the correlation function $\\langle \\eta _1 \\dots \\eta _L \\rangle $ for the stationary measure is precisely $\\frac{\\sum _{ \\operatorname{wt}(|_{u=1,\\beta =c,\\delta =-c}}{Z_L^{semi}},}{w}here the sum is over all staircase tableaux of type (\\eta _1,\\dots ,\\eta _L).$ Corollary REF follows from Theorem REF , Theorem REF , and Remark REF .", "Remark 1.8 For the finite ASEP these correlation functions could be written as polynomials in the parameters with all coefficients being positive (divided by a normalization factor).", "However, for the semi-infinite ASEP, since we use the substition $\\delta =-c$ in Corollary REF , this positivity property of the coefficients no longer holds.", "Nevertheless, in Theorem REF we will provide a sufficient condition for the quantities in (REF ) to be positive real numbers.", "Example 1.9 We can use Corollary REF and the tableaux of Figure to compute the correlation function $\\langle \\eta _1 \\eta _2 \\rangle =\\langle 1 1 \\rangle $ .", "Setting $u=1,\\beta =c,\\delta =-c$ gives $\\frac{\\alpha ^2 + c^2 q - \\alpha c q -\\alpha c -\\alpha ^2 c- \\alpha c^2 - \\alpha \\gamma c+\\alpha c^2}{(\\alpha +\\gamma )(\\alpha +q\\gamma )} =\\frac{\\alpha ^2 + c^2 q - \\alpha c q -\\alpha c -\\alpha ^2 c - \\alpha \\gamma c}{(\\alpha +\\gamma )(\\alpha +q\\gamma )}.$ Remark 1.10 The partition function $Z_L^{semi}$ for the finite correlation functions involving the first $L$ sites on the semi-infinite ASEP is $\\prod _{j=0}^{L-1}(\\alpha +q^j \\gamma )$ .", "In particular, this does not depend on $c$ .", "The structure of this paper is as follows.", "In Section we review some results on the finite ASEP, then in Section we define the ASEP on a semi-infinite lattice.", "In Section we state and prove a matrix ansatz describing the finite correlation functions of its (signed) stationary measures, and in Section we provide a sufficient condition for these signed stationary measures to be positive.", "In Section we prove Theorem REF .", "And in Section we summarize our results and end with some questions about “nonsensical\" specializations of Markov chains.", "Acknowledgments: The authors are grateful to A. Borodin, P. Deift, P.L.", "Ferrari, S. Grosskinsky, and H. Spohn for useful discussions." ], [ "Background on the finite ASEP and its matrix ansatz", "We start by recalling the Matrix Ansatz of Derrida, Evans, Hakim, and Pasquier [4] for the finite ASEP, as well as results of the first author together with Uchiyama and Wadati [9] on the current.", "Theorem 2.1 [4] Suppose that there are matrices $D$ , $E$ and vectors $\\langle W |$ , $|V\\rangle $ , which satisfy $DE-qED=D+E$ $\\alpha \\langle W|E-\\gamma \\langle W|D = \\langle W|$ $\\beta D|V\\rangle - \\delta E|V\\rangle = |V\\rangle .$ Let $\\eta = (\\eta _1,\\dots ,\\eta _L)$ be a state of the finite ASEP.", "Then the measure $\\mu $ defined by $\\mu (\\eta ) =\\frac{\\langle W| \\prod _{x=1}^L \\eta _x D+(1-\\eta _x) E|V\\rangle }{\\langle W| (D+E)^L | V\\rangle }$ is the unique stationary measure for the ASEP on a finite lattice of $L$ sites, where the rates of particles entering and exiting at the left are $\\alpha $ and $\\gamma $ , and the rates of particles exiting and entering at the right are $\\beta $ and $\\delta $ .", "Although Theorem REF was published in 1993, it was not until ten years later that a general solution to the ansatz was obtained.", "Theorem 2.2 [9] There is a solution $D$ , $E$ , $\\langle W|$ , $|V\\rangle $ which satisfies the relations of Theorem REF .", "The above solution was related to Askey-Wilson polynomials.", "Using properties of the Askey-Wilson integral, the authors calculated the current $J_L$ of the finite ASEP.", "Recall that $J_L = \\frac{Z_{L-1}}{Z_L}$ , where $Z_L = \\langle W| (D+E)^L|V\\rangle $ .", "Let $J = \\lim _{L \\rightarrow \\infty } J_L$ .", "Proposition 2.3 [9] Suppose that $q \\ne 1$ .", "Let $a&=\\frac{1-q-\\alpha +\\gamma +\\sqrt{(1-q-\\alpha +\\gamma )^2+4\\alpha \\gamma }}{2\\alpha }\\text{ and }\\\\b&=\\frac{1-q-\\beta +\\delta +\\sqrt{(1-q-\\beta +\\delta )^2+4\\beta \\delta }}{2\\beta }.$ If $a>1$ and $a>b$ then $J =(1-q) \\frac{a}{(1+a)^2}$ .", "If $a<1$ and $b<1$ then $J = \\frac{1-q}{4}$ .", "If $b>1$ and $b>a$ then $J = (1-q) \\frac{b}{(1+b)^2}.$" ], [ "Formal definition of the semi-infinite ASEP", "We now define the semi-infinite ASEP.", "Since this is a Markov process with infinitely many states, one must define it carefully; we give its Markov generator below.", "This Markov generator then determines a Markov semigroup and hence a Markov process, see [6] or [5] for details.", "Let $\\eta =(\\eta _1,\\eta _2,\\dots )$ be a state in $X$ .", "If $i$ is a positive integer, we define from $\\eta $ two new states $\\eta ^i$ and $\\eta ^{i,i+1}$ by $(\\eta ^i)_j ={\\left\\lbrace \\begin{array}{ll} 1-\\eta _i & \\text{ if $j=i$}\\\\\\eta _j &\\text{ if $j\\ne i$}\\end{array}\\right.}", "$ $\\text{ and }(\\eta ^{i,i+1})_j ={\\left\\lbrace \\begin{array}{ll}\\eta _{i+1} & \\text{ if $j=i$}\\\\\\eta _i & \\text{ if $j=i+1$}\\\\\\eta _j & \\text{ if $j\\ne i,i+1$}\\end{array}\\right.", "}$ Let $C_0(X)$ be the set of cylinder functions on $X$ , i.e.", "functions from $X$ to $\\mathbb {R}$ which depend on only finitely many sites.", "Definition 3.1 The Markov generator $\\mathcal {L}$ of the semi-infinite ASEP is defined as follows.", "Given any function $f\\in C_0(X)$ , $\\mathcal {L}f(\\eta ) &= \\alpha (1-\\eta _1)(f(\\eta ^1)-f(\\eta ))+\\gamma \\eta _1 (f(\\eta ^1)-f(\\eta ))\\\\&+\\sum _{x=1}^\\infty \\left( \\eta _x (1-\\eta _{x+1}) (f(\\eta ^{x,x+1})-f(\\eta ))+q(1-\\eta _x) \\eta _{x+1}(f(\\eta ^{x,x+1})-f(\\eta ))\\right).$ We are interested in stationary measures of the corresponding Markov process.", "A measure $\\mu $ is stationary if $\\mathbb {E}^{\\mu }(\\mathcal {L}f) = 0$ for all $f\\in C_0(X)$ .", "Here $\\mathbb {E}^{\\mu }$ is the expected value with respect to a measure $\\mu $ .", "Note that since the state space $X$ is infinite, the uniqueness of the stationary measure is no longer assured." ], [ "The matrix ansatz for the semi-infinite ASEP", "We first prove a matrix ansatz in the spirit of [4].", "The version which we shall state and prove for the semi-infinite ASEP is a generalization of a theorem of Grosskinsky [5]; his ansatz is the same as ours, except he set $\\gamma =0$ and $q=0$ .", "In what follows, we use the terminology signed measure for a measure which is not necessarily positive.", "We will first give a matrix ansatz which describes stationary signed measures (Theorem REF ), and then in the following section, we'll give a theorem (Theorem REF ) which provides conditions guaranteeing that the measures are positive.", "Theorem 4.1 Suppose there are matrices $D,E$ and vectors $\\langle W|,|V\\rangle $ , which satisfy $DE-qED=c(D+E)$ $\\alpha \\langle W|E-\\gamma \\langle W|D = c\\langle W|$ $(D+E)|V\\rangle = |V\\rangle .$ Let $\\eta = (\\eta _1, \\eta _2, \\dots , \\eta _L) \\in \\lbrace 0,1\\rbrace ^L$ .", "Then the signed measure $\\mu ^L$ defined by $ \\mu ^L(\\eta _1,\\dots ,\\eta _L) =\\frac{\\langle W| \\prod _{x=1}^L \\eta _x D+(1-\\eta _x) E|V\\rangle }{\\langle W| (D+E)^L | V\\rangle }$ is stationary for the process defined by $\\mathcal {L}$ .", "Here the parameter $c$ determines the stationary current, i.e.", "$\\mathbb {E}^{\\mu }( \\eta _x (1 - \\eta _{x+1})-q(1-\\eta _x)\\eta _{x+1}) = c$ for all $x\\in \\mathbb {Z}^+$ .", "Remark 4.2 The measure $\\mu ^L$ defined above does not depend on the choice of solution $D$ , $E$ , $\\langle W|$ , $|V\\rangle $ .", "Indeed, for any word $Y$ in $D$ and $E$ , by repeatedly applying relations (a.", "), (b.)", "and (c.), one can express $\\langle W|Y|V\\rangle $ in terms of $\\alpha $ , $\\gamma $ , $q$ , $c$ , and $\\langle W|V\\rangle $ .", "Suppose that $f\\in C_0(X)$ concentrates on sites $\\lbrace 1,2,\\dots ,L\\rbrace $ .", "Using Definition REF , the stationary condition which we must check becomes: $0 &=\\sum _{\\eta } \\alpha \\mu ^L(\\eta )(1-\\eta _1)(f(\\eta ^1)-f(\\eta ))+\\sum _{\\eta } \\gamma \\mu ^L(\\eta ) \\eta _1 (f(\\eta ^1)-f(\\eta )) +\\\\&\\hspace{14.22636pt}\\sum _{\\eta } \\sum _{x=1}^{L-1} \\left[ \\mu ^L(\\eta ) \\eta _x (1-\\eta _{x+1})(f(\\eta ^{x,x+1})-f(\\eta )) + q\\mu ^L(\\eta )(1-\\eta _x)\\eta _{x+1}(f(\\eta ^{x,x+1})-f(\\eta ))\\right] +\\\\&\\hspace{14.22636pt}\\sum _\\eta \\left[\\mu ^{L+1}(\\eta )\\eta _L(1-\\eta _{L+1})(f(\\eta ^{L,L+1})-f(\\eta )) +q \\mu ^{L+1}(\\eta )(1-\\eta _L)\\eta _{L+1}(f(\\eta ^{L,L+1})-f(\\eta ))\\right].$ Here the sum is over all $\\eta \\in \\lbrace 0,1\\rbrace ^L$ .", "Rewriting this equation gives $0 &=\\sum _{\\eta } f(\\eta ) \\bigg ( \\alpha \\mu ^L(\\eta ^1)(1-\\eta ^1_1)-\\alpha \\mu ^L(\\eta ) (1-\\eta _1)+\\gamma \\mu ^L(\\eta ^1) \\eta ^1_1-\\gamma \\mu ^L(\\eta ) \\eta _1) + \\\\&\\hspace{14.22636pt}\\sum _{x=1}^{L-1} [\\mu ^L(\\eta ^{x,x+1})\\eta _x^{x,x+1} (1-\\eta _{x+1}^{x,x+1})-\\mu ^L(\\eta ) \\eta _x (1-\\eta _{x+1})+q \\mu ^L(\\eta ^{x,x+1})(1-\\eta _x^{x,x+1})\\eta _{x+1}^{x,x+1}\\\\&\\hspace{28.45274pt}-q\\mu ^L(\\eta )(1-\\eta _x)\\eta _{x+1} ]\\\\&\\hspace{14.22636pt}+\\mu ^{L+1}(\\eta ^{L,L+1})\\eta _L^{L,L+1} (1-\\eta _{L+1}^{L,L+1}-\\mu ^{L+1}(\\eta ) \\eta _L (1-\\eta _{L+1})\\\\&\\hspace{14.22636pt}+q\\mu ^{L+1}(\\eta ^{L,L+1})(1-\\eta _L^{L,L+1})\\eta _{L+1}^{L,L+1}-q\\mu ^{L+1}(\\eta )(1-\\eta _L)\\eta _{L+1} \\bigg ).$ Note that $\\eta ^1_1 = 1-\\eta _1$ .", "The coefficient of $f(\\eta )$ in the above equation is $&\\sum _{x=1}^{L-1} [\\mu ^L(\\eta ^{x,x+1}) \\eta _{x+1}(1-\\eta _x)-\\mu ^L(\\eta ) \\eta _x (1-\\eta _{x+1})+q\\mu ^L(\\eta ^{x,x+1})(1-\\eta _{x+1})\\eta _x\\\\&\\hspace{14.22636pt} - q\\mu ^L(\\eta )(1-\\eta _x) \\eta _{x+1}]\\\\&+\\mu ^{L+1}(\\eta ^{L,L+1})\\eta _{L+1}(1-\\eta _L)-\\mu ^{L+1}(\\eta ) \\eta _L(1-\\eta _{L+1})\\\\&+q\\mu ^{L+1}(\\eta ^{L,L+1})(1-\\eta _{L+1})\\eta _L - q\\mu ^{L+1}(\\eta )(1-\\eta _L) \\eta _{L+1} \\\\&+\\alpha \\mu ^L(\\eta ^1)\\eta _1 - \\alpha \\mu ^L(\\eta ) (1-\\eta _1)+\\gamma \\mu ^L(\\eta ^1)(1-\\eta _1)-\\gamma \\mu ^L(\\eta ) \\eta _1.$ We aim to show that each coefficient is equal to 0.", "Rearranging terms gives $&\\sum _{x=1}^{L-1} \\bigg [\\mu ^L(\\eta ^{x,x+1}) \\eta _{x+1}(1-\\eta _x)- q\\mu ^L(\\eta )(1-\\eta _x) \\eta _{x+1}\\\\&\\hspace{19.91684pt}-(\\mu ^L(\\eta ) \\eta _x (1-\\eta _{x+1})- q\\mu ^L(\\eta ^{x,x+1})(1-\\eta _{x+1})\\eta _x)\\bigg ]\\\\&+\\mu ^{L+1}(\\eta ^{L,L+1})\\eta _{L+1}(1-\\eta _L) - q\\mu ^{L+1}(\\eta )(1-\\eta _L) \\eta _{L+1} \\\\&-[\\mu ^{L+1}(\\eta ) \\eta _L(1-\\eta _{L+1}) -q\\mu ^{L+1}(\\eta ^{L,L+1})(1-\\eta _{L+1})\\eta _L]\\\\&+[\\alpha \\mu ^L(\\eta ^1)\\eta _1 -\\gamma \\mu ^L(\\eta ) \\eta _1]- [\\alpha \\mu ^L(\\eta ) (1-\\eta _1)-\\gamma \\mu ^L(\\eta ^1)(1-\\eta _1)].$ Note that each configuration of particles can be seen as a sequence of empty and occupied blocks.", "Suppose that the first $L$ sites of $\\eta $ consists of $n$ such blocks $(\\circ \\dots \\circ ) (\\bullet \\dots \\bullet )(\\circ \\dots \\circ ) \\dots (\\bullet \\dots \\bullet )(\\circ \\dots \\circ )$ where there are $\\tau _1$ $\\circ $ 's in the first block, $\\tau _2$ $\\bullet $ 's in the second block, ..., and $\\tau _n$ $\\circ $ 's in the last block.", "Here we assume that all $\\tau _i$ 's are nonzero, so in particular, the first $L$ sites of $\\eta $ begin and end with $\\circ $ .", "Thinking of the configuration of particles as a sequence of empty and occupied blocks, we also use $\\tau $ to denote $\\eta $ .", "At a boundary between a full and empty block ($\\tau _i$ and $\\tau _{i+1}$ ) we can apply the bulk rule of the ansatz to get $\\tau -q\\tau ^{\\prime } = c(\\tau ^i + \\tau ^{i+1})$ .", "Here, $\\tau ^{\\prime }$ is the configuration obtained from $\\tau $ by swapping the adjacent $\\bullet $ and $\\circ $ in the $i$ th and $i+1$ st block, and $\\tau ^i$ is obtained from $\\tau $ by deleting one site in block $i$ .", "Noting that it has non-zero values only at the block boundaries, the sum over $x$ in (REF ) and () telescopes: $\\sum _{i=1, i \\text{ odd}}^{n-2}c [\\mu ^{L-1}(\\tau ^i)+\\mu ^{L-1}(\\tau ^{i+1}) -(\\mu ^{L-1}(\\tau ^{i+1})+\\mu ^{L-1}(\\tau ^{i+2}))]= c\\mu ^{L-1}(\\tau ^1)-c\\mu ^{L-1}(\\tau ^n).$ Since we have assumed that the first $L$ sites of $\\eta $ begin and end with a $\\circ $ , we have that $\\eta _1 = \\eta _L=0$ .", "Applying this and the relations of the ansatz allows us to simplify the quantities (), () and (): $+\\mu ^{L+1}(\\eta ^{L,L+1})\\eta _{L+1}(1-\\eta _L)-q\\mu ^{L+1}(\\eta )(1-\\eta _L)\\eta _{L+1} &= c(\\mu ^L(\\eta )+\\mu ^{L}(\\eta ^L)),\\\\-[\\mu ^{L+1}(\\eta )\\eta _L(1-\\eta _{L+1})-q\\mu ^{L+1}(\\eta ^{L,L+1})(1-\\eta _{L+1})\\eta _L] &= 0,\\\\+\\alpha \\mu ^L(\\eta ^1) \\eta _1 - \\gamma \\mu ^L(\\eta ) \\eta _1 &= 0,\\\\-[\\alpha \\mu ^L(\\eta )(1-\\eta _1) - \\gamma \\mu ^L(\\eta ^1)(1-\\eta _1)] &=-\\alpha \\mu ^L(\\eta ) + \\gamma \\mu ^{L}(\\eta ^1).$ Therefore the coefficient of $f(\\eta )$ , which is given by (REF ) through (), simplifies to $c\\mu ^{L-1}(\\tau ^1)-c\\mu ^{L-1}(\\tau ^n) + c\\mu ^L(\\eta ) + c\\mu ^L(\\eta ^L)-\\alpha \\mu ^L(\\eta )+\\gamma \\mu ^L(\\eta ^1).$ But now note that by relation (c.) of the ansatz, $ c\\mu ^L(\\eta ) + c\\mu ^L(\\eta ^L) =c\\mu ^{L-1}(\\tau ^n)$ , and by relation (b.)", "of the ansatz, $-\\alpha \\mu ^L(\\eta )+\\gamma \\mu ^L(\\eta ^1) = -c\\mu ^{L-1}(\\tau ^1).$ It follows that the coefficient of $f(\\eta )$ is 0.", "This completes the proof, when the first $L$ sites of $\\eta $ begin and end with $\\circ $ .", "The proof is analogous if the first $L$ sites begin or end with $\\bullet $ .", "Remark 4.3 In fact the above argument proves the following statement.", "Suppose that $g:\\lbrace D,E\\rbrace ^* \\rightarrow \\mathbb {R}$ is a function on words in $D$ and $E$ (extended linearly to linear combinations of such words) such that for any words $Y$ and $Y^{\\prime }$ in $D$ and $E$ , we have the following: $g(Y(DE-qED)Y^{\\prime })=cg(Y(D+E)Y^{\\prime })$ $g(\\alpha EY-\\gamma DY) = cg(Y)$ $g(Y(D+E)) = g(Y).$ Let $\\eta = (\\eta _1, \\eta _2, \\dots , \\eta _L) \\in \\lbrace 0,1\\rbrace ^L$ .", "Then the signed measure $\\mu ^L$ defined by $\\mu ^L(\\eta _1,\\dots ,\\eta _L) =\\frac{g(\\prod _{x=1}^L \\eta _x D+(1-\\eta _x) E)}{g((D+E)^L)}$ is stationary for the process defined by $\\mathcal {L}$ .", "Here the parameter $c$ determines the stationary current, i.e.", "$\\mathbb {E}^{\\mu }( \\eta _x (1 - \\eta _{x+1})-q(1-\\eta _x)\\eta _{x+1}) = c$ for all $x\\in \\mathbb {Z}^+$ ." ], [ "Positivity of the measures", "One would like to know when the signed measure defined in (REF ) or (REF ) is positive.", "Theorem 5.1 The signed measure defined in (REF ) (equivalently, (REF )) is positive provided that $q \\le 1$ and one of the inequalities below is satisfied: $a\\ge 1$ and $c \\le (1-q)a/(1+a)^2$ , or $a\\le 1$ and $c \\le (1-q)/4$ .", "Here $a$ is defined as in Proposition REF .", "We will prove Theorem REF by finding a solution to the semi-infinite matrix ansatz (the version in Remark REF ) which is obtained as a limit of a solution to the finite matrix ansatz.", "Proposition 5.2 Let $D$ , $E$ , $\\langle W|$ , $|V\\rangle $ denote the solution to the finite matrix ansatz from Theorem REF .", "Let $C = D+E$ .", "Then for any word $Y$ in $D$ and $E$ , the following limit exists: $\\lim _{m \\rightarrow \\infty } \\frac{\\langle W| YC^m |V\\rangle }{\\langle W|C^m|V\\rangle }.$ We will use relations (A.)", "and (B.)", "of the finite matrix ansatz together with the fact (Proposition REF ) that $\\lim _{m \\rightarrow \\infty } \\frac{\\langle W| C^{m-1} |V\\rangle }{\\langle W|C^m|V\\rangle }$ exists.", "Note that the latter fact implies that for any finite $\\ell $ , both $\\lim _{m \\rightarrow \\infty } \\frac{\\langle W| C^{\\ell } C^{m-1} |V\\rangle }{\\langle W|C^m|V\\rangle }$ and $\\lim _{m \\rightarrow \\infty } \\frac{\\langle W| C^{m-1} |V\\rangle }{\\langle W|C^{\\ell }C^m|V\\rangle }$ exist.", "We will prove the result by induction on the length of $Y$ .", "We start by considering the length 1 case, i.e.", "$Y=D$ or $Y=E$ .", "Let $x_m = \\frac{\\langle W|EC^m |V \\rangle }{\\langle W|C^{m+1}|V\\rangle }$ and $y_m = \\frac{\\langle W|DC^m |V \\rangle }{\\langle W|C^{m+1}|V\\rangle }$ .", "Then we have $x_m + y_m=1$ .", "But also, by relation (B.)", "of the ansatz, we have $\\alpha x_m - \\gamma y_m = \\frac{Z_m}{Z_{m+1}}=J_{m+1}$ .", "We can therefore solve for $x_m$ and $y_m$ in terms of $J_{m+1}$ ; since the limit of $J_{m+1}$ exists as $m \\rightarrow \\infty $ , so does the limit of $x_m$ and $y_m$ .", "It follows that for $Y = D$ or $Y=E$ , the limit $\\lim _{m \\rightarrow \\infty } \\frac{\\langle W| YC^m |V\\rangle }{\\langle W|C^m|V\\rangle }$ exists.", "More generally, for any word $Y^{\\prime }$ of length $\\ell (Y^{\\prime })>1$ , we will show that we can solve for $\\frac{\\langle W|Y^{\\prime } C^m |V\\rangle }{\\langle W|C^m |V\\rangle }$ in terms of quantities of the form $\\frac{\\langle W|Y C^m |V\\rangle }{\\langle W|C^m |V\\rangle }$ where the length $\\ell (Y)$ of $Y$ is at most $\\ell (Y^{\\prime })-1$ .", "This will complete the proof, since by the inductive hypothesis, we can write the latter quantities in terms of the parameters $\\alpha $ , $\\beta $ , $\\gamma $ , $\\delta $ , $q$ and $J_m$ 's, and hence can take the limit as $m$ goes to infinity.", "Note that any word $Y^{\\prime }$ of length greater than 1 can be written in the form $DY$ or $EY$ where the length of $Y$ is non-negative.", "Using relation (B.)", "of the finite matrix ansatz, for any word $Y$ in $D$ and $E$ , we have that $\\alpha \\langle W| EYC^m | V\\rangle - \\gamma \\langle W|DYC^m |V\\rangle = \\langle W|YC^m |V \\rangle .$ And by repeatedly using relation (A.)", "of the ansatz, we can write $q^{\\ell (Y)} \\langle W|EYC^m|V\\rangle &=q^{\\# E^{\\prime }s \\text{ in }Y} \\langle W|YEC^m|V\\rangle +\\text{ terms of shorter length.", "}\\\\\\langle W|DYC^m|V \\rangle &=q^{\\# E^{\\prime }s \\text{ in }Y} \\langle W|YDC^m|V\\rangle +\\text{ terms of shorter length.", "}$ Here a term of shorter length means a monomial in the parameters times a term of the form $\\langle W|Y^{\\prime \\prime }C^m|V \\rangle $ where $\\ell (Y^{\\prime \\prime }) < \\ell (YE) = \\ell (YD).$ Summing the last two equations gives $q^{\\ell (Y)} \\langle W|EYC^m|V\\rangle +\\langle W|DYC^m|V \\rangle =q^{\\# E^{\\prime }s \\text{ in }Y} \\langle W|YC^{m+1}|V\\rangle +\\text{ terms of shorter length.", "}$ But now since the right-hand sides of equations (REF ) and (REF ) are known quantities, and the determinant of the coefficient matrix is $\\alpha + \\gamma q^{\\ell (Y)}$ which is nonzero, we can solve for $ \\langle W|EYC^m|V\\rangle $ and $\\langle W|DYC^m|V \\rangle $ .", "This completes the proof.", "Proposition 5.3 Suppose that $q\\ne 1$ .", "Let $D$ , $E$ , $\\langle W|$ , and $|V\\rangle $ be as in Theorem REF , and set $C = D+E$ .", "Let $c=J$ (recall that $J$ is given by Proposition REF , depending on three cases).", "Denote the length of $Y$ by $\\ell (Y)$ .", "For each word $Y$ in $D$ and $E$ , define $g(Y) = c^{\\ell (Y)} \\lim _{m \\rightarrow \\infty }\\frac{\\langle W|YC^m |V\\rangle }{\\langle W| C^m |V\\rangle }.$ Then $g(Y)$ satisfies the relations of Remark REF .", "By Proposition REF , the definition of $g(Y)$ makes sense.", "Now note that the relations (a.)", "and (b.)", "of Remark REF follow directly from relations (A.)", "and (B.)", "of Theorem REF .", "To check relation (c.), note that $g(Y(D+E)) &= c^{\\ell (Y)+1} \\lim _{m\\rightarrow \\infty } \\frac{\\langle W|Y(D+E)C^m|V\\rangle }{\\langle W|C^m |V \\rangle } \\\\&= c^{\\ell (Y)+1} \\lim _{m\\rightarrow \\infty } \\frac{\\langle W|YC^{m+1}|V\\rangle }{\\langle W|C^m |V \\rangle } \\\\&= c^{\\ell (Y)+1} \\lim _{m\\rightarrow \\infty } \\frac{\\langle W|YC^{m+1}|V\\rangle }{\\langle W|C^{m+1} |V \\rangle } \\cdot \\frac{\\langle W|C^{m+1}|V\\rangle }{\\langle W|C^{m} |V \\rangle } \\\\&= c^{\\ell (Y)} \\lim _{m\\rightarrow \\infty } \\frac{\\langle W|YC^{m+1}|V\\rangle }{\\langle W|C^{m+1} |V \\rangle } \\\\&=g(Y).$ Finally we turn to the proof of Theorem REF .", "First note that if $q=1$ (hence $c=0$ ), it is easy to check that the measure defined by (REF ) is positive.", "This can be checked directly from the ansatz relations, which become very simple.", "We now consider the case that $q<1$ .", "By Proposition REF , we can define a function $g$ which satisfies the relations of Remark REF .", "Therefore the signed measure defined by (REF ) is stationary for the process defined by $\\mathcal {L}$ .", "Moreover, $\\frac{\\langle W|YC^m |V \\rangle }{\\langle W|C^{\\ell (Y)+m}|V \\rangle }$ is non-negative, because it is a correlation function in the finite ASEP.", "Since $c$ is positive, and $g(Y)$ is a limit of non-negative values, it follows that $g(Y)$ is non-negative.", "Therefore the signed measure defined by (REF ) using the function $g$ from Proposition REF is a positive measure.", "We now need to check that $c$ satisfies one of the inequalities in Theorem REF , and that indeed, any pair of $a$ and $c$ satisfying these inequalities can be obtained from Proposition REF using a suitable choice of $\\alpha $ , $\\beta $ , $\\gamma $ , and $\\delta $ .", "Recall that $J$ was computed in Proposition REF , and that we have set $c=J$ .", "Therefore in the first two cases, $c$ satisfies the inequalities of Theorem REF .", "We now consider the third case.", "If $a \\ge 1$ and $b>a$ then it follows that $c =J = \\frac{(1-q)b}{(1+b)^2} \\le \\frac{(1-q)a}{(1+a)^2}.$ While if $a \\le 1$ and $b>a$ then $c= J = \\frac{(1-q)b}{(1+b)^2} \\le \\frac{1-q}{4}.$ In both cases, it follows that $c=J$ satisfies the inequalities of Theorem REF .", "Moreoever, if we let $b$ tend to infinity (one may achieve this by sending $\\beta $ to 0), then $c = \\frac{(1-q)b}{(1+b)^2}$ tends to 0.", "Therefore it is possible to choose appropriate values $\\alpha $ , $\\beta $ , $\\gamma $ , $\\delta $ and $q$ so as to realize any pair $(a,c)$ satisfying the conditions of Theorem REF .", "By the previous arguments, the corresponding measure that we get in this case is positive." ], [ "Proof of Theorem ", "In this section we prove Theorem REF .", "Note that if we set $\\beta =c$ and $\\delta =-c$ in Theorem REF , then the relations (A.", "), (B.", "), and (C.) become: $DE-qED=D+E$ $\\alpha \\langle W|E-\\gamma \\langle W|D = \\langle W|$ $ (D+E)|V\\rangle = \\frac{1}{c} |V\\rangle .$ Note that these relations are nearly identical to the relations (a.", "), (b.)", "and (c.) from the Matrix Ansatz for the semi-infinite ASEP.", "Proposition 6.1 Suppose that $d, e$ , $\\langle w|$ and $|v \\rangle $ satisfy (a.", "), (b.", "), and (c.), and suppose that $D, E$ , $\\langle W|$ , and $|V\\rangle $ satisfy (A'.", "), (B'.", "), and (C'.).", "Let $y$ be an arbitrary word of length $\\ell $ in $d$ and $e$ , and let $Y$ be the corresponding word in $D$ and $E$ .", "Then $\\frac{\\langle w|y|v\\rangle }{\\langle w|(d+e)^{\\ell }|v\\rangle }= \\frac{\\langle W|Y|V\\rangle }{\\langle W|(D+E)^{\\ell }|V\\rangle }.$ Let $\\widetilde{D} = cD$ and $\\widetilde{E} = cE$ .", "Since $D, E$ , $\\langle W|$ , and $V \\rangle $ satisfy (A'.", "), (B'.", "), and (C'.", "), it is easy to verify that $\\widetilde{D}, \\widetilde{E}$ , $\\langle W|$ , and $V \\rangle $ satisfy (a.", "), (b.", "), and (c.).", "We also have that $d, e$ , $\\langle w|$ , and $|v \\rangle $ satisfy (a.", "), (b.", "), and (c.).", "Therefore both of them yield the same measure, as defined in Theorem REF .", "Letting $\\widetilde{Y}$ denote the word in $\\widetilde{D}$ and $\\widetilde{E}$ corresponding to $Y$ , we have that $\\frac{\\langle w|y|v\\rangle }{\\langle w|(d+e)^{\\ell }|v\\rangle }= \\frac{\\langle W|\\widetilde{Y}|V\\rangle }{\\langle W|(\\widetilde{D}+\\widetilde{E})^{\\ell }|V\\rangle }= \\frac{\\langle W|Y|V\\rangle }{\\langle W|(D+E)^{\\ell }|V\\rangle }.$ Theorem REF now follows from Proposition REF , and Theorems REF and REF ." ], [ "Conclusion", "In this paper we have given a combinatorial interpretation for the stationary measures of the semi-infinite ASEP.", "More specifically, one may compute the finite correlation functions of the stationary measures using sums over staircase tableaux, with the parameters $\\alpha $ , $\\beta =c$ , $\\gamma $ , $\\delta =-c$ , and $q$ .", "In particular, we have demonstrated that a rather nonsensical specialization of the stationary distribution of the finite ASEP – the specialization $\\delta = -\\beta $ – can be given a meaningful interpretation in terms of the ASEP on a semi-infinite lattice.", "One might ask more generally when this phenomenon can occur.", "For concreteness, in the discussion below, we will consider finite Markov chains.", "Consider a Markov chain $M$ whose transition matrix is written in terms of one or more parameters (e.g.", "hopping rates).", "Typically we don't consider $M$ to “make sense\" unless these parameters are non-negative.", "Recall that the stationary distribution $\\mu $ of a Markov chain is the unique left eigenvector of the transition matrix associated with eigenvalue 1.", "One may choose a specialization of the parameters and consider the corresponding specialization of $\\mu $ .", "If one makes one or more parameters negative (or even complex), when can one still give a probabilistic or physical meaning to the corresponding “stationary distribution,\" that is, the corresponding specialization of $\\mu $ ?" ], [ "Background on the finite ASEP and its matrix ansatz", "We start by recalling the Matrix Ansatz of Derrida, Evans, Hakim, and Pasquier [4] for the finite ASEP, as well as results of the first author together with Uchiyama and Wadati [9] on the current.", "Theorem 2.1 [4] Suppose that there are matrices $D$ , $E$ and vectors $\\langle W |$ , $|V\\rangle $ , which satisfy $DE-qED=D+E$ $\\alpha \\langle W|E-\\gamma \\langle W|D = \\langle W|$ $\\beta D|V\\rangle - \\delta E|V\\rangle = |V\\rangle .$ Let $\\eta = (\\eta _1,\\dots ,\\eta _L)$ be a state of the finite ASEP.", "Then the measure $\\mu $ defined by $\\mu (\\eta ) =\\frac{\\langle W| \\prod _{x=1}^L \\eta _x D+(1-\\eta _x) E|V\\rangle }{\\langle W| (D+E)^L | V\\rangle }$ is the unique stationary measure for the ASEP on a finite lattice of $L$ sites, where the rates of particles entering and exiting at the left are $\\alpha $ and $\\gamma $ , and the rates of particles exiting and entering at the right are $\\beta $ and $\\delta $ .", "Although Theorem REF was published in 1993, it was not until ten years later that a general solution to the ansatz was obtained.", "Theorem 2.2 [9] There is a solution $D$ , $E$ , $\\langle W|$ , $|V\\rangle $ which satisfies the relations of Theorem REF .", "The above solution was related to Askey-Wilson polynomials.", "Using properties of the Askey-Wilson integral, the authors calculated the current $J_L$ of the finite ASEP.", "Recall that $J_L = \\frac{Z_{L-1}}{Z_L}$ , where $Z_L = \\langle W| (D+E)^L|V\\rangle $ .", "Let $J = \\lim _{L \\rightarrow \\infty } J_L$ .", "Proposition 2.3 [9] Suppose that $q \\ne 1$ .", "Let $a&=\\frac{1-q-\\alpha +\\gamma +\\sqrt{(1-q-\\alpha +\\gamma )^2+4\\alpha \\gamma }}{2\\alpha }\\text{ and }\\\\b&=\\frac{1-q-\\beta +\\delta +\\sqrt{(1-q-\\beta +\\delta )^2+4\\beta \\delta }}{2\\beta }.$ If $a>1$ and $a>b$ then $J =(1-q) \\frac{a}{(1+a)^2}$ .", "If $a<1$ and $b<1$ then $J = \\frac{1-q}{4}$ .", "If $b>1$ and $b>a$ then $J = (1-q) \\frac{b}{(1+b)^2}.$" ], [ "Formal definition of the semi-infinite ASEP", "We now define the semi-infinite ASEP.", "Since this is a Markov process with infinitely many states, one must define it carefully; we give its Markov generator below.", "This Markov generator then determines a Markov semigroup and hence a Markov process, see [6] or [5] for details.", "Let $\\eta =(\\eta _1,\\eta _2,\\dots )$ be a state in $X$ .", "If $i$ is a positive integer, we define from $\\eta $ two new states $\\eta ^i$ and $\\eta ^{i,i+1}$ by $(\\eta ^i)_j ={\\left\\lbrace \\begin{array}{ll} 1-\\eta _i & \\text{ if $j=i$}\\\\\\eta _j &\\text{ if $j\\ne i$}\\end{array}\\right.}", "$ $\\text{ and }(\\eta ^{i,i+1})_j ={\\left\\lbrace \\begin{array}{ll}\\eta _{i+1} & \\text{ if $j=i$}\\\\\\eta _i & \\text{ if $j=i+1$}\\\\\\eta _j & \\text{ if $j\\ne i,i+1$}\\end{array}\\right.", "}$ Let $C_0(X)$ be the set of cylinder functions on $X$ , i.e.", "functions from $X$ to $\\mathbb {R}$ which depend on only finitely many sites.", "Definition 3.1 The Markov generator $\\mathcal {L}$ of the semi-infinite ASEP is defined as follows.", "Given any function $f\\in C_0(X)$ , $\\mathcal {L}f(\\eta ) &= \\alpha (1-\\eta _1)(f(\\eta ^1)-f(\\eta ))+\\gamma \\eta _1 (f(\\eta ^1)-f(\\eta ))\\\\&+\\sum _{x=1}^\\infty \\left( \\eta _x (1-\\eta _{x+1}) (f(\\eta ^{x,x+1})-f(\\eta ))+q(1-\\eta _x) \\eta _{x+1}(f(\\eta ^{x,x+1})-f(\\eta ))\\right).$ We are interested in stationary measures of the corresponding Markov process.", "A measure $\\mu $ is stationary if $\\mathbb {E}^{\\mu }(\\mathcal {L}f) = 0$ for all $f\\in C_0(X)$ .", "Here $\\mathbb {E}^{\\mu }$ is the expected value with respect to a measure $\\mu $ .", "Note that since the state space $X$ is infinite, the uniqueness of the stationary measure is no longer assured." ], [ "The matrix ansatz for the semi-infinite ASEP", "We first prove a matrix ansatz in the spirit of [4].", "The version which we shall state and prove for the semi-infinite ASEP is a generalization of a theorem of Grosskinsky [5]; his ansatz is the same as ours, except he set $\\gamma =0$ and $q=0$ .", "In what follows, we use the terminology signed measure for a measure which is not necessarily positive.", "We will first give a matrix ansatz which describes stationary signed measures (Theorem REF ), and then in the following section, we'll give a theorem (Theorem REF ) which provides conditions guaranteeing that the measures are positive.", "Theorem 4.1 Suppose there are matrices $D,E$ and vectors $\\langle W|,|V\\rangle $ , which satisfy $DE-qED=c(D+E)$ $\\alpha \\langle W|E-\\gamma \\langle W|D = c\\langle W|$ $(D+E)|V\\rangle = |V\\rangle .$ Let $\\eta = (\\eta _1, \\eta _2, \\dots , \\eta _L) \\in \\lbrace 0,1\\rbrace ^L$ .", "Then the signed measure $\\mu ^L$ defined by $ \\mu ^L(\\eta _1,\\dots ,\\eta _L) =\\frac{\\langle W| \\prod _{x=1}^L \\eta _x D+(1-\\eta _x) E|V\\rangle }{\\langle W| (D+E)^L | V\\rangle }$ is stationary for the process defined by $\\mathcal {L}$ .", "Here the parameter $c$ determines the stationary current, i.e.", "$\\mathbb {E}^{\\mu }( \\eta _x (1 - \\eta _{x+1})-q(1-\\eta _x)\\eta _{x+1}) = c$ for all $x\\in \\mathbb {Z}^+$ .", "Remark 4.2 The measure $\\mu ^L$ defined above does not depend on the choice of solution $D$ , $E$ , $\\langle W|$ , $|V\\rangle $ .", "Indeed, for any word $Y$ in $D$ and $E$ , by repeatedly applying relations (a.", "), (b.)", "and (c.), one can express $\\langle W|Y|V\\rangle $ in terms of $\\alpha $ , $\\gamma $ , $q$ , $c$ , and $\\langle W|V\\rangle $ .", "Suppose that $f\\in C_0(X)$ concentrates on sites $\\lbrace 1,2,\\dots ,L\\rbrace $ .", "Using Definition REF , the stationary condition which we must check becomes: $0 &=\\sum _{\\eta } \\alpha \\mu ^L(\\eta )(1-\\eta _1)(f(\\eta ^1)-f(\\eta ))+\\sum _{\\eta } \\gamma \\mu ^L(\\eta ) \\eta _1 (f(\\eta ^1)-f(\\eta )) +\\\\&\\hspace{14.22636pt}\\sum _{\\eta } \\sum _{x=1}^{L-1} \\left[ \\mu ^L(\\eta ) \\eta _x (1-\\eta _{x+1})(f(\\eta ^{x,x+1})-f(\\eta )) + q\\mu ^L(\\eta )(1-\\eta _x)\\eta _{x+1}(f(\\eta ^{x,x+1})-f(\\eta ))\\right] +\\\\&\\hspace{14.22636pt}\\sum _\\eta \\left[\\mu ^{L+1}(\\eta )\\eta _L(1-\\eta _{L+1})(f(\\eta ^{L,L+1})-f(\\eta )) +q \\mu ^{L+1}(\\eta )(1-\\eta _L)\\eta _{L+1}(f(\\eta ^{L,L+1})-f(\\eta ))\\right].$ Here the sum is over all $\\eta \\in \\lbrace 0,1\\rbrace ^L$ .", "Rewriting this equation gives $0 &=\\sum _{\\eta } f(\\eta ) \\bigg ( \\alpha \\mu ^L(\\eta ^1)(1-\\eta ^1_1)-\\alpha \\mu ^L(\\eta ) (1-\\eta _1)+\\gamma \\mu ^L(\\eta ^1) \\eta ^1_1-\\gamma \\mu ^L(\\eta ) \\eta _1) + \\\\&\\hspace{14.22636pt}\\sum _{x=1}^{L-1} [\\mu ^L(\\eta ^{x,x+1})\\eta _x^{x,x+1} (1-\\eta _{x+1}^{x,x+1})-\\mu ^L(\\eta ) \\eta _x (1-\\eta _{x+1})+q \\mu ^L(\\eta ^{x,x+1})(1-\\eta _x^{x,x+1})\\eta _{x+1}^{x,x+1}\\\\&\\hspace{28.45274pt}-q\\mu ^L(\\eta )(1-\\eta _x)\\eta _{x+1} ]\\\\&\\hspace{14.22636pt}+\\mu ^{L+1}(\\eta ^{L,L+1})\\eta _L^{L,L+1} (1-\\eta _{L+1}^{L,L+1}-\\mu ^{L+1}(\\eta ) \\eta _L (1-\\eta _{L+1})\\\\&\\hspace{14.22636pt}+q\\mu ^{L+1}(\\eta ^{L,L+1})(1-\\eta _L^{L,L+1})\\eta _{L+1}^{L,L+1}-q\\mu ^{L+1}(\\eta )(1-\\eta _L)\\eta _{L+1} \\bigg ).$ Note that $\\eta ^1_1 = 1-\\eta _1$ .", "The coefficient of $f(\\eta )$ in the above equation is $&\\sum _{x=1}^{L-1} [\\mu ^L(\\eta ^{x,x+1}) \\eta _{x+1}(1-\\eta _x)-\\mu ^L(\\eta ) \\eta _x (1-\\eta _{x+1})+q\\mu ^L(\\eta ^{x,x+1})(1-\\eta _{x+1})\\eta _x\\\\&\\hspace{14.22636pt} - q\\mu ^L(\\eta )(1-\\eta _x) \\eta _{x+1}]\\\\&+\\mu ^{L+1}(\\eta ^{L,L+1})\\eta _{L+1}(1-\\eta _L)-\\mu ^{L+1}(\\eta ) \\eta _L(1-\\eta _{L+1})\\\\&+q\\mu ^{L+1}(\\eta ^{L,L+1})(1-\\eta _{L+1})\\eta _L - q\\mu ^{L+1}(\\eta )(1-\\eta _L) \\eta _{L+1} \\\\&+\\alpha \\mu ^L(\\eta ^1)\\eta _1 - \\alpha \\mu ^L(\\eta ) (1-\\eta _1)+\\gamma \\mu ^L(\\eta ^1)(1-\\eta _1)-\\gamma \\mu ^L(\\eta ) \\eta _1.$ We aim to show that each coefficient is equal to 0.", "Rearranging terms gives $&\\sum _{x=1}^{L-1} \\bigg [\\mu ^L(\\eta ^{x,x+1}) \\eta _{x+1}(1-\\eta _x)- q\\mu ^L(\\eta )(1-\\eta _x) \\eta _{x+1}\\\\&\\hspace{19.91684pt}-(\\mu ^L(\\eta ) \\eta _x (1-\\eta _{x+1})- q\\mu ^L(\\eta ^{x,x+1})(1-\\eta _{x+1})\\eta _x)\\bigg ]\\\\&+\\mu ^{L+1}(\\eta ^{L,L+1})\\eta _{L+1}(1-\\eta _L) - q\\mu ^{L+1}(\\eta )(1-\\eta _L) \\eta _{L+1} \\\\&-[\\mu ^{L+1}(\\eta ) \\eta _L(1-\\eta _{L+1}) -q\\mu ^{L+1}(\\eta ^{L,L+1})(1-\\eta _{L+1})\\eta _L]\\\\&+[\\alpha \\mu ^L(\\eta ^1)\\eta _1 -\\gamma \\mu ^L(\\eta ) \\eta _1]- [\\alpha \\mu ^L(\\eta ) (1-\\eta _1)-\\gamma \\mu ^L(\\eta ^1)(1-\\eta _1)].$ Note that each configuration of particles can be seen as a sequence of empty and occupied blocks.", "Suppose that the first $L$ sites of $\\eta $ consists of $n$ such blocks $(\\circ \\dots \\circ ) (\\bullet \\dots \\bullet )(\\circ \\dots \\circ ) \\dots (\\bullet \\dots \\bullet )(\\circ \\dots \\circ )$ where there are $\\tau _1$ $\\circ $ 's in the first block, $\\tau _2$ $\\bullet $ 's in the second block, ..., and $\\tau _n$ $\\circ $ 's in the last block.", "Here we assume that all $\\tau _i$ 's are nonzero, so in particular, the first $L$ sites of $\\eta $ begin and end with $\\circ $ .", "Thinking of the configuration of particles as a sequence of empty and occupied blocks, we also use $\\tau $ to denote $\\eta $ .", "At a boundary between a full and empty block ($\\tau _i$ and $\\tau _{i+1}$ ) we can apply the bulk rule of the ansatz to get $\\tau -q\\tau ^{\\prime } = c(\\tau ^i + \\tau ^{i+1})$ .", "Here, $\\tau ^{\\prime }$ is the configuration obtained from $\\tau $ by swapping the adjacent $\\bullet $ and $\\circ $ in the $i$ th and $i+1$ st block, and $\\tau ^i$ is obtained from $\\tau $ by deleting one site in block $i$ .", "Noting that it has non-zero values only at the block boundaries, the sum over $x$ in (REF ) and () telescopes: $\\sum _{i=1, i \\text{ odd}}^{n-2}c [\\mu ^{L-1}(\\tau ^i)+\\mu ^{L-1}(\\tau ^{i+1}) -(\\mu ^{L-1}(\\tau ^{i+1})+\\mu ^{L-1}(\\tau ^{i+2}))]= c\\mu ^{L-1}(\\tau ^1)-c\\mu ^{L-1}(\\tau ^n).$ Since we have assumed that the first $L$ sites of $\\eta $ begin and end with a $\\circ $ , we have that $\\eta _1 = \\eta _L=0$ .", "Applying this and the relations of the ansatz allows us to simplify the quantities (), () and (): $+\\mu ^{L+1}(\\eta ^{L,L+1})\\eta _{L+1}(1-\\eta _L)-q\\mu ^{L+1}(\\eta )(1-\\eta _L)\\eta _{L+1} &= c(\\mu ^L(\\eta )+\\mu ^{L}(\\eta ^L)),\\\\-[\\mu ^{L+1}(\\eta )\\eta _L(1-\\eta _{L+1})-q\\mu ^{L+1}(\\eta ^{L,L+1})(1-\\eta _{L+1})\\eta _L] &= 0,\\\\+\\alpha \\mu ^L(\\eta ^1) \\eta _1 - \\gamma \\mu ^L(\\eta ) \\eta _1 &= 0,\\\\-[\\alpha \\mu ^L(\\eta )(1-\\eta _1) - \\gamma \\mu ^L(\\eta ^1)(1-\\eta _1)] &=-\\alpha \\mu ^L(\\eta ) + \\gamma \\mu ^{L}(\\eta ^1).$ Therefore the coefficient of $f(\\eta )$ , which is given by (REF ) through (), simplifies to $c\\mu ^{L-1}(\\tau ^1)-c\\mu ^{L-1}(\\tau ^n) + c\\mu ^L(\\eta ) + c\\mu ^L(\\eta ^L)-\\alpha \\mu ^L(\\eta )+\\gamma \\mu ^L(\\eta ^1).$ But now note that by relation (c.) of the ansatz, $ c\\mu ^L(\\eta ) + c\\mu ^L(\\eta ^L) =c\\mu ^{L-1}(\\tau ^n)$ , and by relation (b.)", "of the ansatz, $-\\alpha \\mu ^L(\\eta )+\\gamma \\mu ^L(\\eta ^1) = -c\\mu ^{L-1}(\\tau ^1).$ It follows that the coefficient of $f(\\eta )$ is 0.", "This completes the proof, when the first $L$ sites of $\\eta $ begin and end with $\\circ $ .", "The proof is analogous if the first $L$ sites begin or end with $\\bullet $ .", "Remark 4.3 In fact the above argument proves the following statement.", "Suppose that $g:\\lbrace D,E\\rbrace ^* \\rightarrow \\mathbb {R}$ is a function on words in $D$ and $E$ (extended linearly to linear combinations of such words) such that for any words $Y$ and $Y^{\\prime }$ in $D$ and $E$ , we have the following: $g(Y(DE-qED)Y^{\\prime })=cg(Y(D+E)Y^{\\prime })$ $g(\\alpha EY-\\gamma DY) = cg(Y)$ $g(Y(D+E)) = g(Y).$ Let $\\eta = (\\eta _1, \\eta _2, \\dots , \\eta _L) \\in \\lbrace 0,1\\rbrace ^L$ .", "Then the signed measure $\\mu ^L$ defined by $\\mu ^L(\\eta _1,\\dots ,\\eta _L) =\\frac{g(\\prod _{x=1}^L \\eta _x D+(1-\\eta _x) E)}{g((D+E)^L)}$ is stationary for the process defined by $\\mathcal {L}$ .", "Here the parameter $c$ determines the stationary current, i.e.", "$\\mathbb {E}^{\\mu }( \\eta _x (1 - \\eta _{x+1})-q(1-\\eta _x)\\eta _{x+1}) = c$ for all $x\\in \\mathbb {Z}^+$ ." ], [ "Positivity of the measures", "One would like to know when the signed measure defined in (REF ) or (REF ) is positive.", "Theorem 5.1 The signed measure defined in (REF ) (equivalently, (REF )) is positive provided that $q \\le 1$ and one of the inequalities below is satisfied: $a\\ge 1$ and $c \\le (1-q)a/(1+a)^2$ , or $a\\le 1$ and $c \\le (1-q)/4$ .", "Here $a$ is defined as in Proposition REF .", "We will prove Theorem REF by finding a solution to the semi-infinite matrix ansatz (the version in Remark REF ) which is obtained as a limit of a solution to the finite matrix ansatz.", "Proposition 5.2 Let $D$ , $E$ , $\\langle W|$ , $|V\\rangle $ denote the solution to the finite matrix ansatz from Theorem REF .", "Let $C = D+E$ .", "Then for any word $Y$ in $D$ and $E$ , the following limit exists: $\\lim _{m \\rightarrow \\infty } \\frac{\\langle W| YC^m |V\\rangle }{\\langle W|C^m|V\\rangle }.$ We will use relations (A.)", "and (B.)", "of the finite matrix ansatz together with the fact (Proposition REF ) that $\\lim _{m \\rightarrow \\infty } \\frac{\\langle W| C^{m-1} |V\\rangle }{\\langle W|C^m|V\\rangle }$ exists.", "Note that the latter fact implies that for any finite $\\ell $ , both $\\lim _{m \\rightarrow \\infty } \\frac{\\langle W| C^{\\ell } C^{m-1} |V\\rangle }{\\langle W|C^m|V\\rangle }$ and $\\lim _{m \\rightarrow \\infty } \\frac{\\langle W| C^{m-1} |V\\rangle }{\\langle W|C^{\\ell }C^m|V\\rangle }$ exist.", "We will prove the result by induction on the length of $Y$ .", "We start by considering the length 1 case, i.e.", "$Y=D$ or $Y=E$ .", "Let $x_m = \\frac{\\langle W|EC^m |V \\rangle }{\\langle W|C^{m+1}|V\\rangle }$ and $y_m = \\frac{\\langle W|DC^m |V \\rangle }{\\langle W|C^{m+1}|V\\rangle }$ .", "Then we have $x_m + y_m=1$ .", "But also, by relation (B.)", "of the ansatz, we have $\\alpha x_m - \\gamma y_m = \\frac{Z_m}{Z_{m+1}}=J_{m+1}$ .", "We can therefore solve for $x_m$ and $y_m$ in terms of $J_{m+1}$ ; since the limit of $J_{m+1}$ exists as $m \\rightarrow \\infty $ , so does the limit of $x_m$ and $y_m$ .", "It follows that for $Y = D$ or $Y=E$ , the limit $\\lim _{m \\rightarrow \\infty } \\frac{\\langle W| YC^m |V\\rangle }{\\langle W|C^m|V\\rangle }$ exists.", "More generally, for any word $Y^{\\prime }$ of length $\\ell (Y^{\\prime })>1$ , we will show that we can solve for $\\frac{\\langle W|Y^{\\prime } C^m |V\\rangle }{\\langle W|C^m |V\\rangle }$ in terms of quantities of the form $\\frac{\\langle W|Y C^m |V\\rangle }{\\langle W|C^m |V\\rangle }$ where the length $\\ell (Y)$ of $Y$ is at most $\\ell (Y^{\\prime })-1$ .", "This will complete the proof, since by the inductive hypothesis, we can write the latter quantities in terms of the parameters $\\alpha $ , $\\beta $ , $\\gamma $ , $\\delta $ , $q$ and $J_m$ 's, and hence can take the limit as $m$ goes to infinity.", "Note that any word $Y^{\\prime }$ of length greater than 1 can be written in the form $DY$ or $EY$ where the length of $Y$ is non-negative.", "Using relation (B.)", "of the finite matrix ansatz, for any word $Y$ in $D$ and $E$ , we have that $\\alpha \\langle W| EYC^m | V\\rangle - \\gamma \\langle W|DYC^m |V\\rangle = \\langle W|YC^m |V \\rangle .$ And by repeatedly using relation (A.)", "of the ansatz, we can write $q^{\\ell (Y)} \\langle W|EYC^m|V\\rangle &=q^{\\# E^{\\prime }s \\text{ in }Y} \\langle W|YEC^m|V\\rangle +\\text{ terms of shorter length.", "}\\\\\\langle W|DYC^m|V \\rangle &=q^{\\# E^{\\prime }s \\text{ in }Y} \\langle W|YDC^m|V\\rangle +\\text{ terms of shorter length.", "}$ Here a term of shorter length means a monomial in the parameters times a term of the form $\\langle W|Y^{\\prime \\prime }C^m|V \\rangle $ where $\\ell (Y^{\\prime \\prime }) < \\ell (YE) = \\ell (YD).$ Summing the last two equations gives $q^{\\ell (Y)} \\langle W|EYC^m|V\\rangle +\\langle W|DYC^m|V \\rangle =q^{\\# E^{\\prime }s \\text{ in }Y} \\langle W|YC^{m+1}|V\\rangle +\\text{ terms of shorter length.", "}$ But now since the right-hand sides of equations (REF ) and (REF ) are known quantities, and the determinant of the coefficient matrix is $\\alpha + \\gamma q^{\\ell (Y)}$ which is nonzero, we can solve for $ \\langle W|EYC^m|V\\rangle $ and $\\langle W|DYC^m|V \\rangle $ .", "This completes the proof.", "Proposition 5.3 Suppose that $q\\ne 1$ .", "Let $D$ , $E$ , $\\langle W|$ , and $|V\\rangle $ be as in Theorem REF , and set $C = D+E$ .", "Let $c=J$ (recall that $J$ is given by Proposition REF , depending on three cases).", "Denote the length of $Y$ by $\\ell (Y)$ .", "For each word $Y$ in $D$ and $E$ , define $g(Y) = c^{\\ell (Y)} \\lim _{m \\rightarrow \\infty }\\frac{\\langle W|YC^m |V\\rangle }{\\langle W| C^m |V\\rangle }.$ Then $g(Y)$ satisfies the relations of Remark REF .", "By Proposition REF , the definition of $g(Y)$ makes sense.", "Now note that the relations (a.)", "and (b.)", "of Remark REF follow directly from relations (A.)", "and (B.)", "of Theorem REF .", "To check relation (c.), note that $g(Y(D+E)) &= c^{\\ell (Y)+1} \\lim _{m\\rightarrow \\infty } \\frac{\\langle W|Y(D+E)C^m|V\\rangle }{\\langle W|C^m |V \\rangle } \\\\&= c^{\\ell (Y)+1} \\lim _{m\\rightarrow \\infty } \\frac{\\langle W|YC^{m+1}|V\\rangle }{\\langle W|C^m |V \\rangle } \\\\&= c^{\\ell (Y)+1} \\lim _{m\\rightarrow \\infty } \\frac{\\langle W|YC^{m+1}|V\\rangle }{\\langle W|C^{m+1} |V \\rangle } \\cdot \\frac{\\langle W|C^{m+1}|V\\rangle }{\\langle W|C^{m} |V \\rangle } \\\\&= c^{\\ell (Y)} \\lim _{m\\rightarrow \\infty } \\frac{\\langle W|YC^{m+1}|V\\rangle }{\\langle W|C^{m+1} |V \\rangle } \\\\&=g(Y).$ Finally we turn to the proof of Theorem REF .", "First note that if $q=1$ (hence $c=0$ ), it is easy to check that the measure defined by (REF ) is positive.", "This can be checked directly from the ansatz relations, which become very simple.", "We now consider the case that $q<1$ .", "By Proposition REF , we can define a function $g$ which satisfies the relations of Remark REF .", "Therefore the signed measure defined by (REF ) is stationary for the process defined by $\\mathcal {L}$ .", "Moreover, $\\frac{\\langle W|YC^m |V \\rangle }{\\langle W|C^{\\ell (Y)+m}|V \\rangle }$ is non-negative, because it is a correlation function in the finite ASEP.", "Since $c$ is positive, and $g(Y)$ is a limit of non-negative values, it follows that $g(Y)$ is non-negative.", "Therefore the signed measure defined by (REF ) using the function $g$ from Proposition REF is a positive measure.", "We now need to check that $c$ satisfies one of the inequalities in Theorem REF , and that indeed, any pair of $a$ and $c$ satisfying these inequalities can be obtained from Proposition REF using a suitable choice of $\\alpha $ , $\\beta $ , $\\gamma $ , and $\\delta $ .", "Recall that $J$ was computed in Proposition REF , and that we have set $c=J$ .", "Therefore in the first two cases, $c$ satisfies the inequalities of Theorem REF .", "We now consider the third case.", "If $a \\ge 1$ and $b>a$ then it follows that $c =J = \\frac{(1-q)b}{(1+b)^2} \\le \\frac{(1-q)a}{(1+a)^2}.$ While if $a \\le 1$ and $b>a$ then $c= J = \\frac{(1-q)b}{(1+b)^2} \\le \\frac{1-q}{4}.$ In both cases, it follows that $c=J$ satisfies the inequalities of Theorem REF .", "Moreoever, if we let $b$ tend to infinity (one may achieve this by sending $\\beta $ to 0), then $c = \\frac{(1-q)b}{(1+b)^2}$ tends to 0.", "Therefore it is possible to choose appropriate values $\\alpha $ , $\\beta $ , $\\gamma $ , $\\delta $ and $q$ so as to realize any pair $(a,c)$ satisfying the conditions of Theorem REF .", "By the previous arguments, the corresponding measure that we get in this case is positive." ], [ "Proof of Theorem ", "In this section we prove Theorem REF .", "Note that if we set $\\beta =c$ and $\\delta =-c$ in Theorem REF , then the relations (A.", "), (B.", "), and (C.) become: $DE-qED=D+E$ $\\alpha \\langle W|E-\\gamma \\langle W|D = \\langle W|$ $ (D+E)|V\\rangle = \\frac{1}{c} |V\\rangle .$ Note that these relations are nearly identical to the relations (a.", "), (b.)", "and (c.) from the Matrix Ansatz for the semi-infinite ASEP.", "Proposition 6.1 Suppose that $d, e$ , $\\langle w|$ and $|v \\rangle $ satisfy (a.", "), (b.", "), and (c.), and suppose that $D, E$ , $\\langle W|$ , and $|V\\rangle $ satisfy (A'.", "), (B'.", "), and (C'.).", "Let $y$ be an arbitrary word of length $\\ell $ in $d$ and $e$ , and let $Y$ be the corresponding word in $D$ and $E$ .", "Then $\\frac{\\langle w|y|v\\rangle }{\\langle w|(d+e)^{\\ell }|v\\rangle }= \\frac{\\langle W|Y|V\\rangle }{\\langle W|(D+E)^{\\ell }|V\\rangle }.$ Let $\\widetilde{D} = cD$ and $\\widetilde{E} = cE$ .", "Since $D, E$ , $\\langle W|$ , and $V \\rangle $ satisfy (A'.", "), (B'.", "), and (C'.", "), it is easy to verify that $\\widetilde{D}, \\widetilde{E}$ , $\\langle W|$ , and $V \\rangle $ satisfy (a.", "), (b.", "), and (c.).", "We also have that $d, e$ , $\\langle w|$ , and $|v \\rangle $ satisfy (a.", "), (b.", "), and (c.).", "Therefore both of them yield the same measure, as defined in Theorem REF .", "Letting $\\widetilde{Y}$ denote the word in $\\widetilde{D}$ and $\\widetilde{E}$ corresponding to $Y$ , we have that $\\frac{\\langle w|y|v\\rangle }{\\langle w|(d+e)^{\\ell }|v\\rangle }= \\frac{\\langle W|\\widetilde{Y}|V\\rangle }{\\langle W|(\\widetilde{D}+\\widetilde{E})^{\\ell }|V\\rangle }= \\frac{\\langle W|Y|V\\rangle }{\\langle W|(D+E)^{\\ell }|V\\rangle }.$ Theorem REF now follows from Proposition REF , and Theorems REF and REF ." ], [ "Conclusion", "In this paper we have given a combinatorial interpretation for the stationary measures of the semi-infinite ASEP.", "More specifically, one may compute the finite correlation functions of the stationary measures using sums over staircase tableaux, with the parameters $\\alpha $ , $\\beta =c$ , $\\gamma $ , $\\delta =-c$ , and $q$ .", "In particular, we have demonstrated that a rather nonsensical specialization of the stationary distribution of the finite ASEP – the specialization $\\delta = -\\beta $ – can be given a meaningful interpretation in terms of the ASEP on a semi-infinite lattice.", "One might ask more generally when this phenomenon can occur.", "For concreteness, in the discussion below, we will consider finite Markov chains.", "Consider a Markov chain $M$ whose transition matrix is written in terms of one or more parameters (e.g.", "hopping rates).", "Typically we don't consider $M$ to “make sense\" unless these parameters are non-negative.", "Recall that the stationary distribution $\\mu $ of a Markov chain is the unique left eigenvector of the transition matrix associated with eigenvalue 1.", "One may choose a specialization of the parameters and consider the corresponding specialization of $\\mu $ .", "If one makes one or more parameters negative (or even complex), when can one still give a probabilistic or physical meaning to the corresponding “stationary distribution,\" that is, the corresponding specialization of $\\mu $ ?" ] ]

[ [ "NLTE determination of the calcium abundance and 3D corrections in\n extremely metal-poor stars" ], [ "Abstract (Abridged) Extremely metal-poor stars contain the fossil records of the chemical composition of the early Galaxy.", "The NLTE profiles of the calcium lines were computed in a sample of 53 extremely metal-poor stars with a modified version of the program MULTI.", "With our new model atom we are able to reconcile the abundance of Ca deduced from the Ca I and Ca II lines in Procyon.", "-We find that [Ca/Fe] = 0.50 $\\pm$ 0.09 in the early Galaxy, a value slightly higher than the previous LTE estimations.", "-The scatter of the ratios [X/Ca] is generally smaller than the scatter of the ratio [X/Mg] where X is a \"light metal\" (O, Na, Mg, Al, S, and K) with the exception of Al.", "These scatters cannot be explained by error of measurements, except for oxygen.", "Surprisingly, the scatter of [X/Fe] is always equal to, or even smaller than, the scatter around the mean value of [X/Ca].", "-We note that at low metallicity, the wavelength of the Ca I resonance line is shifted relative to the (weaker) subordinate lines, a signature of the effect of convection.", "-The Ca abundance deduced from the Ca I resonance line (422.7 nm) is found to be systematically smaller at very low metallicity, than the abundance deduced from the subordinate lines." ], [ "Introduction", "An homogeneous sample of 53 metal-poor stars, most of them extremely metal-poor (EMP stars with $\\rm [Fe/H]<-2.9$ ), has been observed by Cayrel et al.", "([19]), and Bonifacio et al.", "([14], [15]).", "The aim of this paper is to determine more precisely the calcium abundance in these stars.", "120-125 F6.2 — Corr18 NLTE corrections, model (7000,2,-3.5,0.3) These low mass stars have been formed at the very early phases of the Galaxy and the chemical composition of their atmosphere reflects the yields of the first massive type II supernovae which have a very short life-time.", "These supernovae produce more “$\\alpha $ -elements” (O, Mg, Si, S, Ca) than “iron-peak” elements.", "In contrast, less massive type I supernovae, which have a much longer life-time and explode later, produce more iron-peak elements than $\\alpha $ -elements.", "As a consequence in the atmosphere of the EMP stars formed at the beginning of the Galaxy a relative enhancement of the $\\alpha $ -elements (compared to the Sun) is observed.", "The level of this overabundance is one of the fundamental parameters of the chemical evolution models of the Galaxy.", "On the other hand, the relative production of the $\\alpha $ -elements in a supernova depends on the mass of this supernova (e.g.", "Kobayashi et al., [30]): [Ca/Fe] tends to be lower for more massive supernovae.", "Therefore the level of [Ca/Fe] in the early Galaxy constrains the IMF in the early times.", "Moreover, Cayrel et al.", "([19]) and Bonifacio et al.", "([15]) have shown that for $\\rm [Fe/H]<-2.7$ the abundances of the $\\alpha $ -elements relative to iron ([Mg/Fe], [Si/Fe], [Ca/Fe], and [Ti/Fe]) are constant with a small scatter but that surprisingly the scatters of these elements relative to magnesium (another $\\alpha $ -element) are larger.", "This larger scatter cannot be explained by a lower precision of the abundance of magnesium.", "Cayrel et al.", "([19]) and Bonifacio et al.", "([15]) computed the abundances under the LTE hypothesis.", "The non-LTE (NLTE) abundances of two $\\alpha $ -elements Mg and S have been then computed in Andrievsky et al.", "([7]) and Spite et al.", "([57]).", "These papers confirm that the scatter of abundance ratios is generally larger when Mg replaces Fe as a reference element.", "120-125 F6.2 — Corr18 NLTE corrections, model (7000,2,-3.5,0.3) It is then interesting to check wether the link between the abundance of Ca and the abundance of the other $\\alpha $ -elements is closer.", "In this paper we have carried out a NLTE analysis of the calcium abundance in the atmosphere of these EMP stars and have tried to estimate the influence of convection (3D computations).", "For this new analysis we have used the resonance line of Ca I  and about 15 subordinate lines.", "We have also used the line of the Ca II  infrared triplet at 866.21nm.", "Unfortunately, the two other lines of this triplet (at 849.81 and 854.21nm) are outside the observed spectral range.", "Table: Program stars and their parameters.The stars marked with an asterisk are carbon-rich.Columns 2 to 5 give the main parameters of thestars.", "Column6 is the NLTE calcium abundance deduced from the subordinate Ca I lines with the number of lines and the standard deviation (columns 7and 8).", "Columns 9 and 10 list the abundance of Ca derived from aNLTE computation of the 422.67 and 866.22nm lines.", "The last twocolumns give [Ca/H] and [Ca/Fe] based on the calcium abundance deducedfrom the Ca I  subordinate lines." ], [ "Star sample and model parameters", "The spectra of the stars investigated here have been presented in detail in Cayrel et al.", "(2004) and Bonifacio et al.", "([14]).", "The observations were performed with the high-resolution spectrograph UVES at ESO-VLT (Dekker et al., [21]).", "The resolving power of the spectrograph is $R\\approx 45000$ , with about five pixels per resolution element and the S/N ratio per pixel is typically about 150.", "The fundamental parameters of the models (effective temperature $\\rm T_{eff}$ , logarithm of the gravity $\\log g$ , and metallicity) have been derived by Cayrel et al.", "([19]) for the giants and Bonifacio et al.", "([14]) for the turnoff stars.", "Briefly, temperatures of the giants are deduced from the colors with the calibration of Alonso et al.", "([3], [4]), and temperatures of the turnoff stars from the wings of the $\\rm H{\\alpha }$ line.", "Moreover we checked that these $\\rm H{\\alpha }$ temperatures agreed with the temperatures derived from the color V-K and the calibration of Alonso et al.", "([2]).", "The gravities are derived from the ionization equilibrium of iron (under the LTE approximation) and we note that they might be affected by NLTE effects.", "The parameters of the models are repeated in Table REF for the reader's convenience." ], [ "Determination of the calcium abundance", "The NLTE profiles of the calcium lines were computed with a modified version of the code MULTI (Carlsson, [18], Korotin et al., [33]), which allows a very good description of the radiation field.", "This version includes opacities from ATLAS9 (Kurucz, [34]), which modify the intensity distribution in the UV." ], [ "Atmospheric models", "For these computations we used Kurucz models without overshooting (Castelli et al., 1997).", "These models have been shown to provide LTE abundances very similar (within 0.05 dex) to those of the MARCS models used by Cayrel et al.", "([19]) and Bonifacio et al.", "([15]).", "The solar model was taken from Castellihttp://wwwuser.oats.inaf.it/castelli/sun/ap00t5777g44377k1asp.dat with a chromospheric contribution from the VAL-3C model of Vernazza et al.", "([63]) and the corresponding microturbulence distribution." ], [ "Atomic model", "Our model atom of calcium is similar to the one used by Mashonkina et al.", "([42]) but it includes some more levels and more recent atomic data.", "Seventy levels of Ca I, thirty-eight levels of Ca II, and the ground state of Ca III  were taken into account; in addition, more than 300 levels of Ca I  and Ca II  were included to keep the condition of the particle number conservation in LTE.", "The fine structure was taken into account for the levels $\\rm 3d^{2}D$ and $\\rm 4p^{2}P^{0}$ of Ca II.", "The energy levels were taken from the NIST atomic spectra database (Sugar & Corliss, [60]).", "The ionization cross-sections were taken from TOPBASE.", "The oscillator strengths of the Ca I  and Ca II  lines were taken from the most recent estimations: Wiese et al.", "([65]), Smith & Raggett ([54]), Smith ([53]), Theodosiou ([61]), Morton ([45]), Kurucz ([35]), and from TOPBASE for the lines occuring between non-splitted levels.", "For the forbidden transitions, we used TOPBASE and Hirata & Horaguchi ([27]).", "We considered 351 transitions in detail; for 375 weak transitions the radiative rates were fixed.", "Collisional rates between the ground level and the ten lower levels of Ca I  were taken from Samson & Berrington ([51]).", "For Ca II, collisional rates were taken from Meléndez et al.", "([43]) instead of those of Burgess et al.", "([16]) used by Mashonkina et al.", "([42]) for the lower seven terms.", "For the other transitions (without data) we used for allowed transitions, the Van Regemorter ([62]) formula, and for the forbidden transitions the Allen ([1]) formula.", "Electron impact ionization cross-sections were calculated by applying the formula of Seaton ([52]), with threshold photoionization cross-sections from the Opacity-Project data.", "Collisions with hydrogen atoms were computed using the Steenbock & Holweger ([58]) formula.", "The cross sections calculated with this formula were multiplied by a scaling factor $\\rm S_{H}$ : the “efficiency” of the hydrogenic collisions.", "This factor was constrained empirically by comparing the Ca abundance obtained from different lines in the Sun and in some reference stars (Procyon, HD 140283, HD 122563).", "The best agreement was obtained with $\\rm S_{H}= 0.1$ which agrees well with Ivanova et al.", "([29]) and Mashonkina et al.", "([42]).", "Figure: Profiles of Ca I and Ca II lines in Procyon.", "The wavelengths are in Å.", "The small crosses represent the observed spectrum and the (red) line the computed profile.", "With our model atom, the observed and computed profiles of the Ca I  and Ca II lines agree well.Figure: Profiles of Ca I and Ca II lines in HD 140283.", "The wavelengths are in Å.", "The symbols are the same as in Fig.", ".", "The observed spectrum and the synthetic profiles computed with logϵ\\log \\epsilon (Ca) =4.12 agree well." ], [ "Consistency check", "To test the Ca atom model, we computed the profiles of the calcium lines in the Sun, in Procyon, and in two classical metal-poor stars HD 122563, and HD 140283.", "The synthetic spectra were computed following the procedure described in Korotin ([32]): we calculated the departure coefficients factors “b” for the Ca lines with MULTI and then used these factors in the LTE synthetic spectrum code.", "$\\bullet $ For the Sun we used the solar atmosphere model computed by Castelli with a chromospheric contribution following Vernazza et al.", "([63]) (see section REF ).", "A micro turbulence velocity $\\rm \\xi _{t}=1.0$ $\\rm km s^{-1}$   was adopted in the atmosphere.", "We computed the profile of 49 lines of Ca I  and 17 lines of Ca II.", "The $\\log gf$ value of the lines and the broadening parameter due to collisions with hydrogen atoms, $\\log \\gamma _{VW}/N_{H}$ (for T=10 000K), are given in Table REF .", "We found $\\log \\epsilon $ (Ca) = $6.31 \\pm 0.05$ from the Ca I lines and $\\log \\epsilon $ (Ca) = $6.30 \\pm 0.07$ from the Ca II  lines.", "These values agree well with the meteoritic calcium abundance $\\log \\epsilon $ (Ca) = $6.31 \\pm 0.02$ (Lodders et al., [38]) and the photospheric abundances derived by Asplund et al.", "([9]) for the solar atmosphere: $\\log \\epsilon $ (Ca) = $6.34 \\pm 0.04$ (a value that includes 3D effects).", "For most of the lines in Table REF , the broadening parameter $\\log \\gamma _{VW}/N_{H}$ (for T=10 000K) due to collisions with hydrogen atoms, was taken from the precise calculations of Anstee & O'Mara ([8]), Barklem & O'Mara ([11], [12]), and Barklem et al.", "([13]).", "For the other lines, this parameter was derived from the fit of the solar atlas (Kurucz et al., [37]).", "These values are, for the high excitation lines of Ca II, higher than the values obtained from the Unsöld or Kurucz approximation.", "However, they agree well with the values obtained by Ivanova et al.", "([29]), who note that the use of the Unsöld or Kurucz approximation often leads to an underestimation of the broadening parameter.", "Table: Parameters of the Ca lines used in the solar spectrum.γ 1 =logγ VW /N H \\gamma _{1}=\\log \\gamma _{VW}/N_{H} for a temperature of 10 000K.$\\bullet $ We retrieved the spectrum of Procyon from the UVES POP (Bagnulo et al., [10]).", "Procyon has a solar-like chemical composition, and its surface gravity, derived from Hipparcos measurements (Perryman et al., [47]), is log g=3.96.", "We adopted a microturbulence $\\xi _{t}=$ 1.8 $\\rm km s^{-1}$ .", "For this star, Mashonkina et al.", "([42]) adopting $\\rm T_{eff}$ = 6510 K (Mashonkina et al., [41]) or $\\rm T_{eff}$ = 6590 K (Korn et al., [31]) found a subsolar abundance of Ca and a difference of about 0.2 dex between the NLTE calcium abundance derived from the Ca I  and Ca II  lines.", "With $\\rm T_{eff}$ =6510 K and our model atom, we derive for Procyon a near solar calcium abundance and a much better agreement between Ca I  and Ca II: $\\log \\epsilon $ (Ca) = 6.25 $\\pm 0.04$ from Ca I lines and 6.27 $\\pm 0.06$ from Ca II lines (Fig.", "REF ).", "$\\bullet $ We also tested our model on two classical metal-poor stars: HD 122563 (a star from our sample of giants) and HD 140283.", "The spectrum of HD 140283 was retrieved from the ESO POP (Bagnulo et al., [10]).", "For HD 122563 we adopted the parameters given in Table REF , and for HD 140283 the parameters given by Hosford et al.", "([28]): $\\rm T_{eff}$ =5750 K, $\\log g$ =3.4, $\\xi _{\\rm t}$ =1.5 $\\rm km s^{-1}$ .", "The agreement is good.", "From the Ca I lines we obtain $\\rm \\log \\epsilon (Ca) =4.12 \\pm 0.04$ and from the Ca II lines $\\rm \\log \\epsilon (Ca) =4.08 \\pm 0.05$ .", "In Fig.", "REF we show the fit between the observed spectrum of HD 140283 and the synthetic profiles computed with $\\rm \\log \\epsilon (Ca) =4.12$ .", "Figure: Profiles of three Ca lines computed for a giant starwith [ Ca /H]≈-2.5\\rm [Ca/H] \\approx -2.5 with LTE (thin blue line) and NLTE(thick red line) hypotheses.", "The wavelengths are given in Å.a)The NLTE profile of the Ca I  resonance line is narrower in thewings and deeper in the core.", "b) In a Ca I  subordinate line, forthe same abundance of calcium the equivalent width computed underthe NLTE hypothesis is slightly smaller.", "c) The NLTE correctionis important for the strong IR Ca II  line (note that the scale inwavelength is different for this line), but the wings are notaffected and a reliable calcium abundance can be deduced from thesewings via LTE analysis." ], [ "Comparison between the different systems", "In Fig.REF we show the influence of the NLTE effects on the profile of some typical calcium lines measurable in extremely metal-poor stars.", "Evidently, in particular the wings of the lines of the Ca II infrared triplet are not sensitive to NLTE effects until the wings are strong enough.", "Figure: NLTE corrections for the Ca I resonance line at 422.67 nmand for a typical subordinate line of Ca I  at 445.48 nm.In Fig.", "REF the correction NLTE-LTE is given as a funtion of the temperature, the gravity, and the calcium abundance for the Ca I  resonance line and a typical subordinate line of Ca I.", "The NLTE corrections for different temperatures, gravities, and abundances for all the Ca lines used in this analysis (Table REF ) are available in electronic formhttp://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.", "The influence of the complex NLTE effects as a function of the stellar parameters and of the characteristics of the lines has been discussed by Mashonkina et al.", "([42]) and Merle et al.", "([44]).", "In Table REF we give for each star the mean abundance derived from the subordinate lines of Ca I  (col. 6), the number of lines used for the analysis (col. 7), and the internal error (col. 8).", "Column 9 lists the abundance deduced from the resonance line of Ca I, and column 10 the abundance computed from the infra-red Ca II  line at 866.21nm (only this line of the red Ca II triplet is in the wavelength range of our spectra).", "Columns 11 and 12 list [Ca/H] and [Ca/Fe] deduced from the subordinate lines of Ca I.", "For an easier comparison to the previous “First Stars” papers, the solar abundance of calcium was taken from Grevesse & Sauval ([26]) as in Cayrel et al.", "([19]): $\\rm log\\epsilon _{(Ca)~\\odot }=6.36$ .", "The oscillator strengths of the calcium lines measured in our metal-poor stars were updated (to be the same as in section REF ).", "As a consequence, for some lines (Table REF ) they are slightly different from the $\\log ~gf$ values used in Cayrel et al.", "([19]) and Bonifacio et al.", "([15]).", "The abundance of Ca was deduced from the equivalent widths for the subordinate lines of Ca I.", "But for the resonance line, which is often strong and slightly blended with Fe I and CH lines, a fit of the profile was made.", "The Ca abundance was generally deduced from a fit of the wings (insensitive to NLTE effects) for the red Ca II  lines.", "However, in the turnoff stars and in the most metal-poor giants, the wings almost disappear (when the equivalent widths are less than 400 mÅ) and in this case, equivalent widths were used.", "Figure: Difference between [Ca/H] deduced from thesubordinate lines of Ca I  and [Ca/H] deduced from   a) the infrared triplet of Ca II  and    b) the resonance line ofCa I.", "The filled circles represent the turnoff stars and the opensymbols the giants.", "The error bar on [Ca/H], Δ Ca 866.21\\rm \\Delta ~Ca~866.21and Δ Ca 422.67\\rm \\Delta ~Ca~422.67 is generally less than 0.1dex.", "It is indicatedonly when it exceeds 0.1dex.a) The Ca abundance deduced from the line of the infrared tripletis, as a mean, 0.07 dex lower than the abundance derived from thesubordinate lines of Ca I.b) For [ Ca /H]≈-2\\rm [Ca/H]\\approx -2, the abundancededuced from the Ca I  resonance line agrees quite well with the abundances deduced fromsubordinate lines, but a discrepancy appears andincreases linearly when [Ca/H] decreases.It reaches about 0.4 dex for [Ca/H]= –3.5.In Fig.", "REF we compare the abundances of calcium derived from these different systems.", "The abundance deduced from the line of the infrared triplet of Ca II  at 862.21nm is, as a mean, 0.07 dex lower than the abundance deduced from the subordinate lines of Ca I  (Fig.", "REF a).", "This small difference is quite satisfactory since the abundance deduced from the Ca II  lines depends on the surface gravity of the model, and this gravity can be affected by a systematic error since it has been derived from the ionization equilibrium of iron under the LTE hypothesis (see Cayrel et al., [19]).", "In Fig REF b, the calcium abundance derived from the subordinate lines of Ca I  is compared to the abundance deduced from the Ca I resonance line at 422.67nm.", "It is well known that in very metal-poor stars, under the LTE hypothesis, this resonance line leads to an underestimation of the calcium abundance: e.g.", "Magain ([40]), Ryan et al.", "([50]).", "But it was expected that NLTE computations of the lines would remove this effect.", "For one typical metal-poor giant CS 22172-02 we show in Fig.", "REF the observed spectrum and theoretical spectra computed with abundances $\\log \\epsilon $ (Ca) = 2.52 (best fit), 2.8 and 3.1.", "The subordinate lines lead to a Ca abundance of 3.11; this value is well established: seven of the subordinate lines have an equivalent width stronger than 9 mÅ  and the scatter in abundance is only 0.09 dex.", "Figure: Observed profile of the resonance line of Ca I  (crosses) forone typical metal-poor giant compared to theoretical profilescomputed with logϵ( Ca )=2.52\\log ~\\epsilon \\rm (Ca)=2.52 (thick red line), 2.8 and 3.1(thin blue lines).", "The wavelengths are in Å.The (very small) difference between the observed spectrumand the profile computed with logϵ( Ca )=2.52\\log ~\\epsilon \\rm (Ca)=2.52is shown at the bottom of the figure (dots, shifted by 0.1).For this star, the subordinate lines lead to logϵ( Ca )=3.11\\log ~\\epsilon \\rm (Ca)=3.11(Table ).A similar discrepancy has been observed, after NLTE computations, in several metal-poor stars by Mashonkina et al.", "([42]) and the authors suggested that the explanation could lie with the 1D atmospheric models adopted for the computations, since the Ca I  resonance line is formed over a more extended range of atmospheric depths than the subordinate lines.", "The atmospheres of cool stars are not static.", "Velocity and intensity fluctuations caused by convection are observed in the Sun (Rutten et al., 2004) and in metal-poor stars.", "Ramírez et al.", "([48]) found that in HD 122563 (a star of our sample of giants) the cores of the Fe I  lines are shifted relative to the mean radial velocity of the star, this shift increases with the equivalent width of the line.", "Weaker lines form in deeper layers, where the granulation velocities and intensity contrast are higher.", "We also observe this phenomenon for the calcium lines in all stars of our sample.", "The radial velocity of the stars were determined from a constant set of iron lines, and relative to this “zero point”, the radial velocity derived from the Ca I  resonance line is about 0.4 $\\rm km s^{-1}$   higher in the giants and 0.2 $\\rm km s^{-1}$   higher in the dwarfs.", "In contrast, the radial velocity derived from the (weak) subordinate calcium lines is about 0.4 $\\rm km s^{-1}$   lower in the giants and about 0.2 $\\rm km s^{-1}$   lower in the dwarfs.", "(We were able to measure precisely the shift of the subordinate lines only on the “blue spectra” when these lines were larger than 20 mÅ.)", "At the resolution of our VLT/UVES spectra with $R \\approx 45000$ , the asymmetry of the lines cannot be reliably measured.", "The shift of the Ca I  resonance line does not clearly depend on [Ca/H], and therefore it does not seem that there is a clear correlation between the shift of the Ca lines and the discrepancy between the abundances deduced from the resonance or the subordinate lines." ], [ "Abundance correction", "The largest abundance corrections caused by granulation effects occur at low metallicities.", "This is mainly because the difference between the 1D and 3D predictions for the mean temperature of the outer layers of metal-poor stars is very large (see e.g.", "González Hernández et al., [25]).", "We tried to investigate the change in abundances caused by thermal inhomogeneities and differences in formation depth (3D corrections hereafter) for the Ca I resonance line at 422.67 nm and for a typical subordinate line of Ca I  at 445.48 nm.", "–For a representative turnoff star, we used a 3D-CO5BOLD model (Freytag et al., [23], [24]) from the CIFIST grid (see Ludwig et al., [39]) with parameters ($\\rm T_{eff}$ , $\\log g$ , [Fe/H]): 6270 K/4.0/–3.0).", "–For the giants we used two models (4488 K/2.0/–3.0 and 5020 K/2.5/–3.0) from the CIFIST grid.", "They have both a gravity slightly higher than the ones in our sample of giants, but no closer 3D model is available at the moment.", "From these computations we found that for [Fe/H]=–3, [Ca/H]=–2.6, the 3D correction for the subordinate lines of Ca I  in turnoff and in giant stars is very small.", "In giants, the 3D correction seems to be negligible also for the resonance line.", "But with the model of turnoff stars, we found for the resonance line a 3D correction of –0.44 dex, and therefore the 3D correction increases the discrepancy between the subordinate lines and the 422.67 resonance line.", "To date, it is not well understood why the abundance of calcium deduced from the resonance line of Ca I  is at very low metallicity, lower than the abundance deduced from the subordinate lines.", "It would be interesting to repeat the 3D computations with more metal-poor models (not available today).", "A fully consistent 3D NLTE model atmosphere and line formation scheme is currently beyond our reach." ], [ "Results and discussion", "In this section we adopt the Ca abundance deduced from the subordinate lines of Ca I  (Tab.", "REF , columns 11 and 12).", "In Fig.", "REF we present the new relation between [Ca/Fe] and [Fe/H].", "The error bar plotted in for the subordinate lines the figure is the quadratic sum of the error due to the uncertainty of the model and the random error of the Ca abundance derived from the subordinate lines.", "The agreement between the turnoff and the giant stars is excellent, the ratio [Ca/Fe] is constant in the interval $\\rm -4.5<[Fe/H]<-2.5$ .", "The slope of the regression line is –0.05.", "The scatter of [Ca/Fe] is a little smaller when NLTE effects are taken into account: 0.09 from NLTE computations compared to 0.10 from LTE computation .", "In Fig.", "REF we present the behavior of the abundances of O, Na, Mg, Al, S and K relative to Ca.", "The computation of these abundances take into account the NLTE effects (Andrievsky et al.", "[5], [6], [7] and Spite et al.", "[57]).", "Since the abundance of oxygen has been determined from the forbidden oxygen line at 630 nm (Cayrel et al.", "[19]), it is free of NLTE effects.", "In Fig.", "REF , different symbols are used for mixed and unmixed giants.", "After Spite et al.", "([55], [56]) we call “mixed giants”, those where, ownng to mixing with deep layers, carbon has been partially transformed into nitrogen ($\\rm [C/N]<-0.6$ ), part of $\\rm ^{12}C$ has been transformed into $\\rm ^{13}C$ ($\\rm ^{12}C/^{13}C < 10$ ), and lithium is not detected (lithium has been severely depleted by this mixing).", "In the HR diagram, these “mixed giants” are located above the “bump”.", "It has been found (Andrievsky et al., [5]) that some mixed giants are enriched in sodium, this is visible in Fig.", "REF b.", "This Na-enhancement reflects a deep internal mixing (or an AGB status, or a contamination by AGB stars), but it does not reflect an anomaly of the chemical composition of the cloud that formed the star.", "Therefore the mixed stars cannot be used to determine the ratio [Na/Ca] in the early galactic matter.", "Figure: Abundance ratios of O, Na, Mg, Al, S, and K relative to Cain the early Galaxy.The abundances of all these elements were computedtaking into account the NLTE effect.", "The black dots represent theturnoff stars, the open circles the giants (blue for the unmixedgiants and red for the mixed giants).Table: Mean value of the ratios [X/Fe], [X/Ca]and [X/Mg] in the interval -3.6<[ Fe /H]<-2.5\\rm -3.6 < [Fe/H] < -2.5 and scatteraround the meanSince the abundance ratios of O, Na, Al, S, and K relative to Ca are fairly flat vs. [Fe/H] in the central part of the diagram, we can define a mean value in this central interval, say $\\rm -3.6 < [Fe/H] < -2.5$ as has been done with Mg in Andrievsky et al.", "([7]).", "The mean values of [X/Fe], [X/Mg] and [X/Ca] are given in Table REF with the corresponding scatter.", "However, an anti-correlation seems to exist between [S/Ca] and [Fe/H] and also [S/Mg] and [Fe/H]: the Kendall $\\tau $ coefficient is 98.9% for [S/Ca] and 99.1% for [S/Mg].", "In these computations the weight of BD +$17^{\\circ }3248$ ([Fe/H]=–2.2) is important.", "This star is quite peculiar: according to For & Sneden ([22]) it is a red horizontal branch star strongly enriched in heavy elements (Cowan et al., [20]).", "However, if this star is removed from the computations, the Kendall $\\tau $ coefficient remains high: 97.6% for [S/Ca] and 97.9% for [S/Mg].", "One turnoff star, BS 16023-046, and one unmixed giant CS 22956-50, have abnormally strong and broad sodium D lines.", "Their radial velocity (–7.5 $\\rm km s^{-1}$  and –0.1 $\\rm km s^{-1}$ ) is low, and the stellar lines are very probably blended with interstellar lines.", "These stars have not been plotted on Fig.", "REF b and were not taken into account in computing the mean value and the scatter of [Na/Fe], [Na/Mg] and [Na/Ca] in Table REF .", "It is interesting to take advantage of the high quality of the data to compare in Table REF the scatter around the mean value when Fe, Mg, or Ca are taken as reference elements.", "The scatter of [O/Fe], [O/Mg] or [O/Ca] is fairly large: about 0.2 dex but this can be explained by the scatter of the oxygen abundance due to the reduced number of oxygen lines and the difficulty of the measurements.", "For the other ratios the scatter lies between 0.09 and 0.15 dex and we consider that this difference of 0.06 dex is significant.", "The scatter is always significantly smaller when Ca is taken as the reference element, with two exceptions: $\\bullet $ Oxygen: but in this case we have seen that the scatter is dominated by the error on the measurement of the very weak oxygen line.", "$\\bullet $ Aluminum: Al is better correlated with Mg than with Ca.", "But this correlation (opposite to the well known anti-correlation found in globular cluster stars) has at least one exception: CS 29516-024 is rather Al-poor but it is (relatively) Ca-rich and Mg-rich, and it does not seem possible to explain these differences by errors of measurements.", "A similar correlation between [Al/Fe] and [Mg/Fe] has been also suggested by Suda et al.", "([59]) even in a large but very inhomogeneous sample of metal-poor stars." ], [ "Concluding remarks", "We determined the calcium abundance in a homogeneous sample of of 53 metal-poor stars (31 of them with $\\rm [Fe/H]<-2.9$ ) taking into account departures from LTE.", "We have shown that the trend of the ratio [Ca/Fe] vs. [Fe/H], below [Fe/H]=–2.7 is almost flat and derived a new mean value of [Ca/Fe] in the early Galaxy: $\\rm [Ca/Fe]=0.5 \\pm 0.09$ .", "Generally speaking, our NLTE calculations agree quite well with those of Mashonkina et al.", "([42]).", "However, Mashonkina et al.", "had found different Ca abundances from the Ca I  and Ca II  lines in Procyon.", "With our new model atom of calcium, we are able to reconcile the calcium abundance deduced from neutral and ionized calcium lines in Procyon.", "We derived $\\log \\epsilon $ (Ca) = 6.25 $\\pm 0.04$ from Ca I lines and 6.27 $\\pm 0.06$ from Ca II lines (6.33 and 6.40 under the LTE hypothesis).", "Moreover, the calcium abundance is found to be solar, as expected.", "In metal-poor stars (below [Ca/H]=–2.5), a discrepancy clearly appears between the Ca abundances deduced from either the resonance Ca I  line, or the Ca I  subordinate lines.", "A rough estimation of the effect of convection on the profile of these lines does not explain this discrepancy.", "In the stars of our sample (giants and turnoff stars) the wavelengths of the calcium lines are shifted by convection as a function of the equivalent width of the lines as has been found for the iron lines by Ramírez et al.", "([48]) in HD 122563.", "The scatter around the mean value of [Ca/Fe] is small but since the NLTE correction is about the same for all the subordinate lines used in this analysis, the scatter is almost the same as it was in the LTE analyses (Cayrel et al., [19], Bonifacio et al., [15]).", "The abundance of the light metals O, Na, Al, S, K is well correlated with the calcium abundance.", "The correlation is generally a little better than it is with the magnesium abundance (with exception of the tight aluminium/magnesium correlation).", "However, it is striking that in Table REF , even when NLTE is taken into account, the correlation of the light elements O, Na, Al, S, K with Ca or Mg (all supposed to be formed mainly by hydrostatic fusion of C, Ne or O), is never better than the correlation with iron, providing some support to supernovae models predicting nucleosynthesis of all these elements predominantly in explosive modes (e.g.", "Nomoto et al.", "[46]).", "The authors thank the referee, who indicated new $\\log gf$ values for the high excitation Ca II  lines.", "This work has been supported in part by the \"Programme National de Physique Stellaire\" (CNRS).", "S.A. kindly thanks the Observatoire de Paris, the CNRS, and the laboratory GEPI for their hospitality and support during his stay in Meudon.", "He acknowledges the National Academy of Sciences of Ukraine and the franco-Ukrainian exchange program for its financial support under contract UKR CDIV N24008." ] ]

[ [ "Electron transport in a ferromagnetic/normal/ferromagnetic tunnel\n junction based on the surface of a topological insulator" ], [ "Abstract We theoretically study the electron transport properties in a ferromagnetic/normal/ferromagnetic tunnel junction, which is deposited on the top of a topological surface.", "The conductance at the parallel (\\textbf{P}) configuration can be much bigger than that at the antiparallel (\\textbf{AP}) configuration.", "Compared \\textbf{P} with \\textbf{AP} configuration, there exists a shift of phase which can be tuned by gate voltage.", "We find that the exchange field weakly affects the conductance of carriers for \\textbf{P} configuration but can dramatically suppress the conductance of carriers for \\textbf{AP} configuration.", "This controllable electron transport implies anomalous magnetoresistance in this topological spin valve, which may contribute to the development of spintronics .", "In addition, we find that there is a Fabry-Perot-like electron interference." ], [ "The concept of a topological insulator (TI) dates back to the work of Kane and Mele, who focused on two-dimensional (2D) systems $^{1}$ .", "There has been much recent interest in TIs, three-dimensional insulators with metallic surface states protected by time reversal invariance ${[1-25]}$ .", "Its theoretical ${[2]}$ and experimental ${[3]}$ discovery has accordingly generated a great deal of excitement in the condensed matter physics community.", "In particular, the surface of a three-dimensional (3D) TI, such as Bi$_{2}$ Se$_{3}$ or Bi$_{2}$ Te$_{3}$ ${[4]}$ , is a 2D metal, whose band structure consists of an odd number of Dirac cones, centered at time reversal invariant momenta in the surface Brillouin zone ${[5]}$ .", "This corresponds to the infinite mass Rashba model ${[6]}$ , where only one of the spin-split bands exists.", "This has been beautifully demonstrated by the spin- and angle-resolved photoemission spectroscopy ${[7,8]}$ .", "Surface sensitive experiments such as angle-resolved photoemission spectroscopy (ARPES) and scanning tunneling microscopy (STM) ${[9,10]}$ have confirmed the existence of this exotic surface metal, in its simplest form, which takes a single Dirac dispersion.", "Recent theoretical and experimental discovery of the two dimensional (2D) quantum spin Hall system ${[11-18]}$ and its generalization to the TI in three dimensions ${[19-21]}$ have established the state of matter in the time-reversal symmetric systems.", "The time-reversal invariant TI is a new state of matter, distinguished from a regular band insulator by a nontrivial topological invariant, which characterizes its band structure ${[11]}$ .", "Currently, most works focus on searching for TI materials and novel transport properties.", "To my knowledge, the fabrication of such TI-based nanostructure is still a challenging task.", "Usually such structures are fabricated by utilizing the split gate and etching technique ${[22]}$ .", "On the other hand, the 3D TIs are expected to show several unique properties when the time reversal symmetry is broken ${[23-25]}$ .", "This can be realized directly by a ferromagnetic insulating (FI) layer attached to the 3D TI surface.", "One remarkable feature of the Dirac fermions is that the Zeeman field acts like a vector potential: the Dirac Hamiltonian is transformed as $\\sigma \\cdot \\textbf {k}\\longrightarrow \\sigma \\cdot (\\textbf {k+H})$ by the Zeeman field $\\textbf {H}$ ${[26]}$ .", "This is in contrast to the Schr$\\ddot{o}$ dinger electrons in conventional semiconductor heterostructures modulated by nanomagnets ${[27-29]}$ .", "In this work, we study the electron transport properties in a ferromagnetic (F)/normal/N)/ ferromagnetic (F) tunnel junction, which is deposited on the top of a topological surface.", "Ferromagnetic Permalloy electrodes are formed by electron-beam lithography (EBL) followed by thermal evaporation; a second EBL step establishes contact to the Permalloy via Cr /Au electrodes ${[30]}$ .", "As shown in Fig.1, the FI is put on the top of the TI to induce an exchange field via the magnetic proximity effect.", "The easy axis of a FI stripe is usually along its length direction and thus either in parallel (P) or antiparallel (AP) with the $+y$ axis.", "We find that the conductance at the P configuration can be much bigger than that at the AP configuration.", "Compared P with AP configuration, there exists a shift of phase which can be tuned by gate voltage.", "We find that the exchange field weakly affects the conductance of carriers for P configuration but can dramatically suppress the conductance of carriers for AP configuration.", "This controllable electron transport implies anomalous magnetoresistance in this topological spin valve, which may contribute to the development of spintronics.", "Compared with the conventional F/N/F tunneling based on two dimensional electron gas (2DEG), the result implies the existence of Fabry-Perot-like electron interference in F/N/F based on the TI.", "In Sec.", "II , we introduce the model and method for our calculation.", "In Sec.", "III, the numerical analysis to our important issues is reported.", "Finally, a brief summary is given in sec.", "IV." ], [ " model and method", "Now, let us consider a F/N/ F tunnel junction which is deposited on the top of a topological surface where a gate electrode is attached to the ferromagnetic material.", "The ferromagnetism is induced due to the proximity effect by the ferromagnetic insulators deposited on the top as shown in Fig.", "1.", "We assume that the initial magnetization of FI stripes in the region I is aligned with the +y axis.", "In an actual experiment, one can use a magnet with very strong (soft) easy axis anisotropy to control the ferromagnetic material.", "Thus we focus on charge transport at the Fermi level of the surface of TIs, which is described by the 2D Dirac Hamiltonian $H = \\upsilon _{F} \\sigma \\cdot \\textbf {p} + \\sigma \\cdot \\textbf {M} +V{(x)},$ where $\\sigma $ is Pauli matrices , $\\textbf {M} = M_y(x)= M_0(\\Theta (-x)+ \\gamma \\Theta (x-L))$ is the effective exchange field and $V(x)=U_g \\Theta (x)\\Theta (L-x)+ V_g \\Theta (x-L)$ is the gate voltage, where $\\gamma =+1$ ($-1$ ) corresponds to the P (AP) configurations of magnetization and $\\Theta (x)$ is the Heaviside step function.", "Because of the translational invariance of the system along $y$ direction, the equation $H \\Psi (x, y)=E \\Psi (x, y)$ admits solutions of the form $\\Psi (x,y)=(\\Psi _{1}(x), \\Psi _{2}(x))^{T}\\exp (i k_y y)$ .", "We set $\\hbar = \\upsilon _F =1$ in the following.", "Then, with the above Hamiltonian, the wave function in the whole system is given by $\\Psi _{1}&=&\\left\\lbrace \\begin{array}{lllll } \\exp (i k_{x_1} x) + r\\exp (-i k_{x_1} x), &\\quad x<0, \\\\\\\\a \\exp (i q_{x} x) + b \\exp (-i q_{x} x), & \\quad 0<x<L,\\\\\\\\t \\exp (i k_{x_2} (x-L)) , & \\quad x>L,\\end{array} \\right.$ $\\Psi _{2}&=&\\left\\lbrace \\begin{array}{lllll } \\alpha ^{+} \\exp (i k_{x_1}x) + r \\alpha ^{-} \\exp (-i k_{x_1} x), &\\quad x<0, \\\\\\\\a \\beta ^{+} \\exp (i q_{x} x) + b \\beta ^{-} \\exp (-i q_{x} x), &\\quad 0<x<L,\\\\\\\\t \\alpha \\exp (i k_{x_2} (x-L)) , & \\quad x>L,\\end{array} \\right.$ where $k_{x_1}=E \\cos \\theta _{F_1}$ , $q_{x} = (E-U_{g})\\cos \\theta $ and $k_{x_2}=(E-V_{g})\\cos \\theta _{F_2}$ are wave vectors in region I, region II and region III, $\\alpha ^{\\pm }=\\pm \\exp (\\pm i \\theta _{F_1})$ , $\\beta ^{\\pm }=\\pm \\exp (\\pm i \\theta )$ and $\\alpha =\\exp (i \\theta _{F_2})$ .", "The momentum $k_y$ conservation should be satisfied everywhere such as $k_y = E \\sin \\theta _{F_1}-M_0 =(E-U_g)\\sin \\theta = (E-V_g) \\sin \\theta _{F_2}-\\gamma M_0$ .", "Also, $r$ and $t$ are reflection and transmission coefficients, respectively.", "Continuities of the wave function $\\Psi $ at $x=0$ and $x=L$ are $\\Psi (0^{-}) = \\Psi (0^{+})$ and $ \\Psi (L^{-}) = \\Psi (L^{+})$ , respectively.", "We find that the transmitted electron coefficient $t_\\gamma $ is given by $t_\\gamma =\\frac{2 \\cos \\theta _{F_1} \\cos \\theta \\exp (-ik_{x_2}L)}{s_{1, \\gamma } \\cos (q_x L) + i s_{2, \\gamma } \\sin (q_xL)},$ with $s_{1, \\gamma }= \\cos \\theta (\\exp (i \\theta _{F_2})+\\exp (-i\\theta _{F_1}))$ and $s_{2, \\gamma }= i \\sin \\theta (\\exp (-i\\theta _{F_1})-\\exp (i \\theta _{F_2}))-\\exp (i (\\theta _{F_2} -\\theta _{F_1}))-1$ .", "Then $T_\\gamma = |t_\\gamma |^2 \\Re (\\cos \\theta _{F_2}/\\cos \\theta _{F_1}),$ where the factor $\\Re (\\cos \\theta _{F_2}/\\cos \\theta _{F_1})$ is due to current conservation.", "In the linear transport regime and for low temperature, we can obtain the conductance $G$ by introducing it as the electron flow averaged over half the Fermi surface from the well-known Landauer-Buttiker formula ${[25, 31, 32]}$ $G_\\gamma \\sim 1/2\\int _{-\\pi /2}^{\\pi /2} T_\\gamma (E_{F},E_{F}\\cos \\theta _{F_1} ) \\cos \\theta _{F_1} d\\theta _{F_1}.$" ], [ "Results and Discussions", "For convenience we express all quantities in dimensionless units by means of the length of the basic unit $L$ and the energy $E_0=\\hbar v_F /L$ .", "For a typical value of $L=50$ nm and the Bi$_2$ Se$_3$ material $v_F=5\\times 10^5$ m/s, one has $E_0=6.6$ meV.", "We set the energy of electron $E=E_F$ and also define the value $\\eta $ with the form $\\eta =M_0 /E_F$ in our calculation.", "In Fig.2, we show gate voltage dependence of the conductances with a P ($\\gamma =1$ ) and AP ($\\gamma =-1$ ) configuration in the two cases: (a) $V_g /E_F =0$ and (b) $V_g /E_F=2$ .", "The value of the other parameter is $E_F =0.1$ and $\\eta =0.5$ .", "The presence of quantum modulation are seen in these two figures.", "We can see an oscillation of the electrical conductance with a period of $\\pi $ when the voltage $U_g$ is larger than $E_F$ .", "The conductance at the P configuration can be much bigger than that at the AP configuration.", "We find that a minimum of conductance at the P configuration corresponds to a maximum of conductance at the AP configuration [see in fig.2 (a)] when the voltage $U_g$ is larger than $E_F$ .", "In Fig.", "2(b), a similar tendency to Fig.", "2(a) is seen.", "In distinct contrast to Fig.2(a), a minimum of conductance at the P configuration here corresponds to a maximum of conductance at the AP configuration [see in fig.2 (b)].", "That is to say, there exists a shift of $\\pi $ -phase.", "To understand these results intuitively, we consider that the gate voltage $U_g$ is larger than the Fermi energy $E_F$ .", "For the given Fermi energy $E_F =0.1$ , the condition $U_g\\gg E_F$ is easily satisfied.", "In this limit we have $\\theta \\rightarrow 0$ and hence the transmission probability $T_\\gamma \\sim (2 \\cos ^2 \\theta _{F_1}/(1+ \\cos \\theta _{F_1}\\cos \\theta _{F_2} -\\cos (2 U_g L) \\sin \\theta _{F_1}\\sin \\theta _{F_2})\\Re (\\cos \\theta _{F_2}/\\cos \\theta _{F_1}) $ .", "For $\\gamma =1$ and $V_g/ E_F = 0$ (or 2), we find the $\\theta _{F_1}\\equiv \\theta _{F_2}$ (or $-\\theta _{F_2}$ ), and thus $T_\\gamma \\sim \\cos ^2\\theta _{F_1}/(1- \\cos ^2 ( U_g L+ \\delta ) \\sin ^2 \\theta _{F_1})$ where $\\delta =0$ (or $\\pi /2$ ) corresponds to $V_g / E_F = 0$ (or 2).", "Thus the phase difference between $V_g / E_F =0$ and $V_g / E_F =2$ is given by $ U_g L$ .", "We find $G_\\gamma \\propto \\cos ^2 (U_g L)$ for $V_g / E_F = 0$ but $G_\\gamma \\propto \\sin ^2 ( U_g L)$ for $V_g /E_F = 2$ .", "When $U_g L$ is equal to the half period of $\\pi $ , a minimum of conductance will appear for $V_g / E_F = 0$ but a maximum of conductance will appear for $V_g / E_F = 2$ .", "When $U_g L$ is equal to the period of $\\pi $ , a maximum of conductance will appear for $V_g / E_F = 0$ but a minimum of conductance will appear for $V_g / E_F = 2$ .", "Furthermore, we find that $G_\\gamma $ oscillates between $2/3$ and 1 for $\\gamma =1$ .", "For $\\gamma =-1$ and $V_g / E_F= 0$ (or 2), there is a similar tendency to the case of $\\gamma =1$ .", "We can see that $G_\\gamma $ is suppressed obviously by the strength of the effective exchange field .", "Nevertheless, there exists a shift of $\\pi $ -phase because of the factor $\\cos (2 U_gL)$ .", "Figure: Gate voltage dependence of the conductances with a P (γ=1\\gamma =1) and AP (γ=-1\\gamma =-1)configuration for four different values η=0,0.2,0.5\\eta =0, 0.2, 0.5, and0.80.8.", "The solid lines are for V g /E F =0V_g /E_F =0 while the dashed linesare for V g /E F =2V_g /E_F =2.", "The value of the other parameter is E F =0.1E_F=0.1.In order to observe the effect of the exchange field $\\eta $ on the conductance, in Fig.3 we show the gate voltage dependence of the conductances with a P ($\\gamma =1$ ) and AP ($\\gamma =-1$ ) configuration for four different values $\\eta =0, 0.2,0.5$ , and $0.8$ .", "The solid lines are for $V_g /E_F =0$ while the dashed lines are for $V_g /E_F =2$ .", "The value of the other parameter is $E_F =0.1$ .", "A similar tendency to Fig.", "2 is seen in Fig.", "3.", "It is easily seen that the exchange field $\\eta $ weakly affects the conductance of carriers for $\\gamma =1$ but profoundly influences the conductance of carriers for $\\gamma =-1$ .", "For $\\gamma =-1$ , $G_\\gamma $ is suppressed obviously by increasing the value $\\eta $ .", "Due to current conservation, the factor $\\Re (\\cos \\theta _{F_2}/\\cos \\theta _{F_1})$ must be real and then we have $\\sin \\theta _{F_1}=\\pm \\sin \\theta _{F_2}+ 2 \\eta $ where sign + (or -) corresponds to $V_g/E_F =0$ (or 2).", "We can see $2 \\eta -1\\le \\sin \\theta _{F_1} \\le 1$ and $2 \\eta -1\\le \\sin (\\mp \\theta _{F_2}) \\le 1$ where sign - (or +) corresponds to $V_g /E_F =0$ (or 2).", "Thus we find the ranges of the angle-allowable $\\theta _{F_1}$ and $\\theta _{F_2}$ depend on $\\eta $ .", "The transmission is nonzero only for $\\theta _{F_1}$ and $\\theta _{F_2}$ in these ranges and vanishes for $\\eta \\ge 1$ .", "The number of channels decreases with increasing of $\\eta $ , so we can see that $G_\\gamma $ dramatically decreases with the increase of $\\eta $ for $\\gamma =-1$ .", "Noting that the $\\eta \\ge 1$ for $\\gamma =-1$ , the conductance of carriers is forbidden, which implies anomalous magnetoresistance in this topological spin valve.", "Figure: Gate voltage dependence of the conductances with a P (γ=1\\gamma =1) and AP (γ=-1\\gamma =-1)configuration for three different values E F =0.1,1.0E_F = 0.1, 1.0, and5.05.0.", "In (a) and (b), the V g V_g is set as V g /E F =0V_g /E_F =0 while in(c) and (d) the V g V_g is set as V g /E F =2V_g /E_F =2.", "The value of theother parameter is η=0.5\\eta =0.5.In Fig.", "4, we show the gate voltage dependence of the conductances with a P ($\\gamma =1$ ) and AP ($\\gamma =-1$ ) configuration for three different values $E_F = 0.1, 1.0$ , and $5.0$ .", "In (a) and (b), the $V_g$ is set as $V_g /E_F =0$ while in (c) and (d) the $V_g$ is set as $V_g /E_F =2$ .", "The value of the other parameter is $\\eta =0.5$ .", "For $E_F =0.1$ , we can see that the $\\pi $ periodicity appears.", "However, the $\\pi $ periodicity is broken for $E_F =1$ (or 5) because the condition $U_g \\gg E_F$ is not satisfied for the smaller $U_g$ .", "Nevertheless, we get the $\\pi $ periodicity of conductance again by choosing a bigger $U_g$ for the bigger $E_F$ .", "Furthermore, we find that the minimum of the conductance will appear when the gate voltage arrives at a certain value.", "It is easily seen that the minimum of the conductance shifts to the right with increasing of the Fermi energy.", "The larger the Fermi energy is, the smaller the minimum of the conductance is.", "This phenomena is very obvious for the P ($\\gamma =1$ ) configuration [see in figs.4 (a) and (d)].", "From Figs.4 and 5, we find that the conductance at the parallel (P) configuration can be much bigger than that at the antiparallel (AP) configuration.", "However it may be not satisfied for the larger Fermi energy when the gate voltage is not bigger enough.", "We find that there is a Fabry-Perot-like electron interference in the F/N/F tunnel junction, which is deposited on the top of a topological surface.", "The two ferromagnetic electrodes and the barrier can compose a Fabry-Perot resonator ${[33,34]}$ .", "The transmitted electron waves in this resonator can be reflected by the two ferromagnetic electrodes.", "The electron waves undergo multiple reflections back and forth along the resonator between the two ferromagnetic electrodes.", "The conductance oscillations are caused by the interference of electron waves among the modes of the channel-allowable.", "When the gate voltage $U_g$ is larger than the Fermi energy $E_F$ , the round trip between the two ferromagnetic electrodes adds a further phase change $\\delta \\sim 4\\pi /\\lambda $ where the Fermi wavelength $\\lambda \\sim 2\\pi /U_g$ because of the value $\\theta \\sim 0$ .", "When the round trip between the two ferromagnetic electrodes is equal to the a multiple of wavelength, the quantum interference happens.", "This implies that the oscillation period is equal to $\\triangle U_g =\\pi $ .", "Figure: Gate voltage dependence of the conductances with a P (γ=1\\gamma =1) and AP (γ=-1\\gamma =-1)configuration for three different values E F =1.0,5.0E_F = 1.0, 5.0, and10.010.0.", "In (a) , the V g V_g is set as V g /E F =0V_g /E_F =0 while in (b)V g V_g is set as V g /E F =2V_g /E_F =2.", "The value of the other parameter isη=0.5\\eta =0.5.In order to investigate the solution of the standard electron described by the Schr$\\ddot{o}$ dinger equation with a parabolic band structure, we consider a 2DEG in (x,y) plane with a magnetic field B in the z direction as described in Refs.[26-28].", "Thus we can fabricate a F/N/F tunneling based on the 2DEG.", "We can apply all the relevant quantities in dimensionless units, which are the same with the Ref.", "28.", "So we can define the value $\\Delta = M_0= B$ where $M_0$ is the the effective exchange field corresponding to a F/N/F tunneling based on the TI and $B$ is the magnetic field corresponding to a F/N/F tunneling based on 2DEG.", "Nevertheless, we ignore the splitting of energy induced by the spin of electron.", "As described in Fig.1, we set the left electrode potential $V_1=0$ .", "For the electron with parabolic spectrum, ${E_F} \\sin {\\theta _{{F_1}}}$ in Eq.", "(6) should be replaced by $\\sqrt{2{E_F}}\\sin {\\theta _{{F_1}}}$ .", "Then the continuity of the wave function gives the transmission coefficient ${t_\\gamma } = \\frac{{2{\\hspace{1.0pt}} {k_1}{\\hspace{1.0pt}} {q_x}}}{{ - {q_x}{\\hspace{1.0pt}}({k_1} + {k_3})\\cos {\\hspace{1.0pt}} {\\hspace{1.0pt}} ({q_x}{\\hspace{1.0pt}} L){\\hspace{1.0pt}} +i{\\hspace{1.0pt}} {\\hspace{1.0pt}} ({k_1}{k_3} + q_x^2)s{\\hspace{1.0pt}} in{\\hspace{1.0pt}}({q_x}{\\hspace{1.0pt}} L){\\hspace{1.0pt}} {\\hspace{1.0pt}}}},$ and transmission probability ${T_\\gamma } = {\\left| {{t_\\gamma }}\\right|^2}{\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\mathop {\\rm \\Re }\\nolimits }({k_3}/{k_1})$ where ${k_1} = \\sqrt{2{\\hspace{1.0pt}} {\\hspace{1.0pt}}({E_F}{\\hspace{1.0pt}} - {V_1}){\\hspace{1.0pt}} {\\hspace{1.0pt}} } {\\hspace{1.0pt}}{\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} \\cos {\\hspace{1.0pt}} {\\hspace{1.0pt}}{\\hspace{1.0pt}} {\\theta _{{F_1}}}$ , ${q_x} = \\sqrt{2{\\hspace{1.0pt}} {\\hspace{1.0pt}} ({E_F}{\\hspace{1.0pt}} - {V_2}){\\hspace{1.0pt}} {\\hspace{1.0pt}} } \\cos {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} \\theta $ and ${k_3} = \\sqrt{2{\\hspace{1.0pt}} {\\hspace{1.0pt}} ({E_F}{\\hspace{1.0pt}} - {V_3}){\\hspace{1.0pt}}{\\hspace{1.0pt}} } {\\hspace{1.0pt}} {\\hspace{1.0pt}} {\\hspace{1.0pt}} \\cos {\\hspace{1.0pt}}{\\hspace{1.0pt}} {\\theta _{{F_3}}}$ .", "We can easily see that ${T_\\gamma }\\equiv 0$ when the Fermi energy is smaller than the right electrode voltage ${V_3}$ , which is different from the case of Dirac band structure.", "In Fig.6, we show the wave vector $k_y$ dependence of the transmission probability with a P ($\\gamma =1$ ) and AP ($\\gamma =-1$ ) configuration for Dirac electrons shown in (a) and the standard electrons shown in (b).", "The values of the other parameter are $\\Delta =0.5$ , $V_2= U_g=0$ , $V_3=V_g$ and $k_x = k_{x_1} = k_1 = 2$ .", "We can find that transmission is significantly more pronounced for Dirac electrons than for the usual electrons.", "Compared with the P ($\\gamma =1$ ) configuration , it can be seen that the channel of electron transporting from the left electrode to the right electron is suppressed especially for the standard electron.", "It is easily seen that the variety of the tunneling conductance of F/N/F with the change of the gate voltage is obviously different between the Dirac electron and standard electron [see in Fig.7].", "For the standard electron, the conductance will decrease monotonously with the increase of the gate voltage because of evanescent wave modes.", "However, we can see that the conductance with a $\\pi $ periodicity appears as the gate voltage is large enough for the Dirac electron.", "On the one hand, Dirac confined electron exhibits a jittering motion called ¡°Zitterbewegung¡±, originating from the interference of states with positive and negative energy.", "On the other hand, the transmitted Dirac electron waves in this resonator can be reflected by the two ferromagnetic electrodes one after another, so the phase interference will appear.", "As a result, this implies the existence of Fabry-Perot-like electron interference in a F/N/F tunneling based on the TI.", "Figure: Wave vector k y k_y dependence of the transmissionprobability with a P (γ=1\\gamma =1) and AP(γ=-1\\gamma =-1) configuration for the Dirac electrons shown in (a) andthe standard electrons shown in (b).", "The values of the otherparameter are Δ=0.5\\Delta =0.5, V 2 =U g =0V_2= U_g=0, V 3 =V g V_3=V_g andk x =k x 1 =k 1 =2k_x = k_{x_1} = k_1 = 2.Figure: Gate voltage U g U_g dependence of the conductance with a P (γ=1\\gamma =1) and AP (γ=-1\\gamma =-1)configuration for the Dirac electrons and the standard electrons.The values of the other parameter are η=0.5\\eta =0.5, V 3 =V g =0V_3=V_g=0 andE F =1.0E_F =1.0." ], [ "Conclusion", "In summary, we have theoretically investigated transport features of Dirac electrons on the surface of a three-dimensional TI under the modulation of a exchange field provided by an FI stripes.", "We find that the conductance at the P configuration can be much bigger than that at the AP configuration.", "Compared P with AP configuration, there exists a shift of phase which can be tuned by gate voltage.", "We find that the exchange field weakly affects the conductance of carriers for P configuration but can dramatically suppress the conductance of carriers for AP configuration.", "This controllable electron transport implies anomalous magnetoresistance in this topological spin valve, which may contribute to the development of spintronics .", "Acknowledgments: This work was supported by the National Natural Science Foundation of China under Grants No.", "10174024 and No.", "10474025." ] ]

[ [ "Effect of particle statistics in strongly correlated two-dimensional\n Hubbard models" ], [ "Abstract We study the onset of particle statistics effects as the temperature is lowered in strongly correlated two-dimensional Hubbard models.", "We utilize numerical linked-cluster expansions and focus on the properties of interacting lattice fermions and two-component hard-core bosons.", "In the weak-coupling regime, where the ground state of the bosonic system is a superfluid, the thermodynamic properties of the two systems at half filling exhibit very large differences even at high temperatures.", "In the strong-coupling regime, where the low-temperature behavior is governed by a Mott insulator for either particle statistics, the agreement between the thermodynamic properties of both systems extends to regions where the antiferromagnetic (iso)spin correlations are exponentially large.", "We analyze how particle statistics affects adiabatic cooling in those systems." ], [ "introduction", "The Fermi-Hubbard model has been the de facto playground for exploring the properties of high-temperature superconductors for more than two decades [1].", "Yet, still no analytical solution exists in more than one dimension, and state-of-the-art numerical calculations prove very difficult in regimes where the average number of fermions per site is different from 1.", "Unveiling the properties of the model in the latter regime, and addressing whether it supports superconductivity, may be crucial in understanding high-temperature superconductivity [2].", "More than a decade ago, it was proposed that one could “solve” strongly-interacting quantum lattice models by “simulating” them using ultracold atoms in optical lattices [3].", "More recently, the Mott insulator in the Bose-[4] and the Fermi-Hubbard [5] models were realized in these experiments.", "However, current accessible temperatures for fermions are still higher than one needs to observe even the relatively high temperature antiferromagnetic Neél transition in three dimensions.", "These experiments are done using atoms with internal degrees of freedom, which can be selected to emulate not only two-specie fermions (Fermi-Hubbard model) and single-specie bosons (Bose-Hubbard model), but also particles with exotic statistics and/or pseudo-spins [6], [7], [8], [9], [10].", "For instance, it has been shown that experiments with two-component (spin-1/2) bosons can lead to the realization of quantum spin models with tunable parameters [11], [12], [13], [14], [15].", "In the specific case where the intra-specie onsite repulsion is infinite, multiple occupancy of a single specie per site is forbidden.", "This case realizes an effective two-component hard-core boson (2HCB) model, which could be thought of as the bosonic equivalent of the Fermi-Hubbard model [11], [16], [17].", "Similar to fermions, in the strong-coupling regime (large inter-species interactions), the low-energy properties of this model can be described by a $t$ -$J$ model [18], [19].", "Despite the outward similarities between the Fermi-Hubbard model and the 2HCB-Hubbard model (in one dimension they share identical thermodynamic properties) in two dimensions, particle statistics plays a fundamental role in the properties of the system as the temperature is lowered, and in the selection of the ground state.", "At half filling, the ground state of fermions in two spatial dimensions is a Mott insulator with a long-range Neél order for any value of the interaction strength, $U$ .", "For 2HCBs, there is a quantum critical point at interaction $U_c/t\\sim 11$ , where $t$ is the hopping amplitude, which separates a phase with two miscible strongly-interacting superfluids (2SF) at small inter-specie interactions from a Mott insulator super-counter-fluid (SCF) [13] phase in the strong-coupling regime [16].", "The latter state corresponds to a superfluid of pairs of bosons from one specie and holes of the other specie, and can also be interpreted as a long-range $XY$ -ferromagnet in the iso-spin language.", "It is expected that such big contrasts in the nature of the ground states result in significant differences in the thermodynamic properties as well.", "Hence, for validating experiments with ultracold gases, which are performed at finite temperature, it is important to have access to exact quantitative results for the finite-temperature properties of the two systems.", "While there have been numerous finite-temperature numerical studies of fermions in optical lattices [20], [21], the same is not true for 2HCBs, in particular in two dimensions.", "In one dimension, calculations were done introducing a generalized Jordan-Wigner transformation [22].", "The three-dimensional model with attractive interaction was studied to describe the supersolid state of $^4$ He [23], [24].", "Recently, dynamical mean-field theory results have been reported for the two-component (soft-core) Bose-Hubbard model in two dimensions with an average of 1/2 particle of each specie per site and large intra-specie interactions [15], [25].", "It was found in those studies that, upon heating from zero temperature, the system quickly enters an unordered Mott insulator.", "These findings have been complemented by finite-temperature quantum Monte Carlo (QMC) simulations of magnetic phases [17], [26] as well as a field-theoretical treatment [27] in the hard-core limit.", "Here, we utilize numerical linked cluster expansions (NLCEs) to provide a comparative analysis between finite-temperature properties, such as the equation of state, entropy, specific heat, double occupancy, and spin correlations, of fermions and 2HCBs.", "We are particularly interested in identifying at what temperatures particle statistics become important in different interaction regimes, as well as what kind of qualitatively different behavior is produced by the statistics of the particles below those temperatures.", "We also discuss the implications of having bosonic vs fermionic statistics for adiabatic cooling protocols and for detecting short-range spin correlations in ultracold atoms experiments.", "We note that the lowest temperatures that are accessible with NLCEs are typically higher than the crossover temperatures to the $XY$ -ferromagnet phase [15].", "Therefore, the behavior of the strongly-correlated systems that we study at half filling is that of a Mott insulator with large spin correlations for either particle statistics.", "We show that, in the weak-coupling regime ($U<U_c$ ), where the bosonic system has a superfluid ground state, the disagreement between properties of fermions and 2HCBs is apparent at relatively high temperatures (of the order of the hopping).", "In contrast, by increasing $U$ in the strong-coupling regime, the agreement between the thermodynamic properties of the two systems extends to lower temperatures.", "For 2HCBs, although the $z$ -antiferromagnetic Neél ground state is known to win over the $XY$ -ferromagnet only in cases where the two species have different hopping amplitudes, the two phases are very close in energy for equal hopping amplitudes (the case considered here) as only terms beyond the second order perturbation in interaction determine the difference [11], [13].", "Consistent with this picture, we find that at strong interactions, short-range $z$ -antiferromagnetic correlations are large in the low-temperature Mott region and very close for the two particle statistics.", "Given that there are more efficient cooling techniques for bosons than for fermions, one could envision probing finite-temperature fermionic correlations using strongly correlated bosonic systems.", "Here, we present evidence that, through an adiabatic cooling mechanism that takes place in the bosonic system, the region with exponentially large antiferromagnetic (AF) correlations in two dimensions is more easily accessible with 2HCBs than with fermions.", "The exposition is organized as follows: In Sec.", ", we introduce the model and discuss the NLCEs used in this study.", "We present the results for the thermodynamic properties of the models as well as their implications for the optical lattice experiment in Sec. .", "Our findings are summarized in Sec.", "." ], [ "the model and NLCEs", "We consider the two-dimensional (2D) Hubbard Hamiltonian on the square lattice: $\\hat{H}&=&-t\\sum _{\\left<i,j\\right>\\sigma }(\\hat{a}^{\\dagger }_{i\\sigma }\\hat{a}_{j\\sigma } + \\text{H.c.})+U\\sum _i \\hat{n}_{i\\uparrow } \\hat{n}_{i\\downarrow }\\nonumber \\\\&&-\\mu \\sum _i(\\hat{n}_{i\\uparrow }+ \\hat{n}_{i\\downarrow })$ where $\\hat{a}^{\\dagger }_{i\\sigma }$ ($\\hat{a}_{i\\sigma }$ ) creates (annihilates) a particle with spin $\\sigma $ (for simplicity, we use spin instead of isospin for bosons too) on site $i$ , and $\\hat{n}_{i\\sigma }=\\hat{a}^{\\dagger }_{i\\sigma } \\hat{a}_{i\\sigma }$ is the number operator.", "$\\langle ..\\rangle $ denotes nearest neighbors (NN), and $U$ ($>0$ ) is the strength of the onsite repulsion.", "$t=1$ ($\\hbar =1$ and $k_B=1$ ) sets the unit of energy throughout this paper.", "We consider two different particle statistics: fermions ($\\hat{a}_{i\\sigma }=\\hat{f}_{i\\sigma }$ ) and 2HCBs ($\\hat{a}_{i\\sigma }=\\hat{b}_{i\\sigma }$ ).", "The 2HCB operators satisfy the following commutation relations and constraints: $[\\hat{b}^{}_{i\\sigma },\\hat{b}^{\\dagger }_{j\\sigma ^{\\prime }}]=\\delta _{ij}\\delta _{\\sigma \\sigma ^{\\prime }},\\qquad \\hat{b}^{\\dagger 2}_{i\\sigma }=\\hat{b}^{2}_{i\\sigma }=0.$" ], [ "Numerical linked-cluster expansions", "We solve the Hamiltonian (REF ) using the NLCEs introduced in Ref. [28].", "In NLCEs, an extended property of the lattice model per site in the thermodynamic limit, $P$ , is expanded in terms of contributions from all of the clusters, up to a certain size, that can be embedded in the lattice: $P=\\sum _c L(c)w_p(c),$ where $c$ represents the clusters.", "The contribution of each cluster with a particular topology is proportional to the number of ways it can be embedded in the lattice per site, $L(c)$ , and its weight for the property of interest, $w_p(c)$ .", "The weights are computed based on the inclusion-exclusion principle and given the property for each cluster, $\\mathcal {P}(c)$ , which is calculated using exact diagonalization [28]: $w_p(c)=\\mathcal {P}(c)-\\sum _{s\\subset c}w_p(s).$ Here, we carry out the calculations up to the ninth order in the site expansion (maximum cluster size of nine sites).", "NLCEs do not suffer from statistical or systematic errors, such as finite size effects and are not restricted to small or intermediate interaction strengths.", "For this reason, they are complementary to more commonly used methods, such as QMC simulations and dynamical mean-field theory, especially in the strong-coupling regime ($U$$\\gg $$t$ ) where computations in the latter approaches become more challenging.", "However, NLCE results are useful only in the temperature region in which the series converge, which has been shown to extend beyond the region accessible within high-temperature expansions [28], [29].", "The convergence of NLCEs can be accelerated by means of numerical resummations algorithms [28].", "Here, we use Euler and Wynn methods with different parameters and plot the resulting last two orders (or only the last order when the two are indistinguishable in the figures).", "The results from all of those algorithms agree with each other in the regions shown within the small fluctuations seen in some cases at the lowest temperatures.", "These fluctuations, which occur below the convergence temperature of NLCE direct sums, arise from numerical instabilities in the resummations routines.", "We work in the grand canonical ensemble [29], and so, the exact diagonalization for every finite cluster in the series is performed in all particle and spin sectors.", "For each $U$ , the partition function and all other observables are calculated in a dense grid of chemical potentials ($\\mu $ ) and temperatures ($T$ ).", "This allows us to study their behavior at constant density $n=\\langle \\hat{n}_{\\uparrow }+\\hat{n}_{\\downarrow }\\rangle =2\\langle \\hat{n}_{\\sigma }\\rangle $ [29], where $\\langle .. \\rangle $ denotes the expectation value.", "Since only NN hopping is considered, properties of the particle-doped system can be expressed in terms of those for the hole-doped system.", "Hence, in most cases away from half filling, we show results only for the hole-doped system ($n<1$ ).", "Generally, when using QMC-based methods, the specific heat ($C_v$ ) or the entropy ($S$ ) calculations involve numerical derivatives and/or integration by parts [30], [31], which can introduce systematic errors.", "Within NLCEs, these two quantities are computed directly from their definitions: $S=\\ln (Z)+\\frac{\\langle \\hat{H}\\rangle -\\mu \\langle \\hat{n}\\rangle }{T},$ where $Z$ is the partition function, and $C_v=\\left(\\frac{\\partial \\hat{\\langle H\\rangle }}{\\partial T}\\right)_n=\\left(\\frac{\\partial \\hat{\\langle H\\rangle }}{\\partial T}\\right)_{\\mu }+\\left(\\frac{\\partial \\hat{\\langle H\\rangle }}{\\partial \\mu }\\right)_T\\left(\\frac{\\partial \\mu }{\\partial T}\\right)_n.$ Since we work in the grand canonical ensemble, where the chemical potential and not the density is the control parameter, we have written this expression in a more suitable form for numerical evaluation.", "After straightforward mathematical derivations, using Maxwell equations, one can obtain the following closed form for the specific heat in terms of expectation values that can be computed directly in NLCEs: $C_v=\\frac{1}{T^2}\\left[\\langle \\Delta \\hat{H}^2\\rangle -\\frac{\\left(\\langle \\hat{H}\\hat{n}\\rangle -\\langle \\hat{H}\\rangle \\langle \\hat{n}\\rangle \\right)^2}{\\langle \\Delta \\hat{n}^2\\rangle }\\right],$ where $\\langle \\Delta \\hat{H}^2\\rangle = \\langle \\hat{H}^2\\rangle -\\langle \\hat{H}\\rangle ^2$ , and similarly $\\langle \\Delta \\hat{n}^2\\rangle =\\langle \\hat{n}^2\\rangle -\\langle \\hat{n}\\rangle ^2$ .", "Figure: (Color online) (a) Specific heat (C v C_v) of the Fermi-Hubbard modelat half filling vs temperature for values of the onsite interactionranging from U=2U=2 to three times the bandwidth (U=24U=24).", "(b) Difference between the specific heat of the Fermi-Hubbard modeland the 2HCB-Hubbard model at half filling vs temperature, for the sameinteractions as in (a).", "The left inset in (b) shows C v C_v forthe latter model vs TT for U=2,4,6U=2, 4, 6 and 8; and the right insetshows the difference between C v C_v of the Fermi-Hubbard modeland 2HCB-Hubbard model at half filling vs UU at three differenttemperatures.", "We have usedEuler sums for the last 6 terms.", "Thick (color) lines are theresults of the sums up to the 9th order and thin (black)lines up to the 8th order.We begin our study of the dependence of the observables on particle statistics as the temperature in the system is changed by showing, in Fig.", "REF , the specific heat for fermions and 2HCBs.", "Figure REF (a) shows the specific heat of the Fermi-Hubbard model from the weak-coupling regime to the strong-coupling regime with interactions up to three times the bandwidth ($U=24$ ).", "The trend in the deviation of the specific heat for 2HCBs from that of fermions for different interaction strengths is depicted in Fig.", "REF (b), where we show the difference between the two.", "It is clear that for $U\\lesssim 8$ , the difference is significant at temperatures greater than 1.", "For example, as shown in the left inset in Fig.", "REF (b), the specific heat of 2HCBs for $U=2$ is roughly twice as large as that of fermions around $T=1$ .", "Although the exact trend of the former at lower temperatures cannot be resolved within our method, such large high-temperature values in comparison to those for the fermionic case are suggestive of the lack of a second peak at lower temperatures in the 2HCB model.", "This is also supported by a fast drop in the entropy (not shown).", "We have found that for $U=2$ , at $T\\sim 0.8$ , already $\\sim 70\\%$ of the infinite-temperature entropy has been quenched.", "For 2HCBs, a double-peak structure in the specific heat becomes apparent for $U\\gtrsim 7$ .", "At the same time, the minimum temperature at which NLCEs converge, which for smaller $U$ is generally higher for 2HCBs in comparison to fermions, extends to roughly the location of the low-temperature peak.", "Figure: (Color online) (a) Temperature and (b) value of thehigh-temperature peak of the specific heat (C v P C_v^{P}) as functions ofUU.", "By decreasing UU in the strong-coupling regime, C v P C_v^{P} for thebosonic case deviates from the fermionic onearound U=12U=12.", "For U<8U<8, the two peaks in the specific heat of 2HCBsmerge, and the high-temperature peak is not well-definedanymore.", "(c) The location of the high-temperature crossingpoint between the specific heat curves for consecutive values of UU.", "(d) The value of C v C_v at that crossing point.For fermions, our NLCE calculations resolve the double-peak structure of the specific heat in the thermodynamic limit for $U\\ge 6$ .", "Since QMC simulations can access lower temperatures for $U\\lesssim 6$ , previous QMC studies of this model have established the existence of the low-$T$ peak for any finite value of the interaction strength [32], which signifies the crossover to the phase with exponentially large AF correlations.", "In the strong-coupling regime, the specific heat results for fermions and 2HCBs are very close to each other at high temperatures ($T>1$ ).", "In this regime, the difference between the $C_v$ of the two particle statistics decreases systematically for all accessible temperatures as $U$ is increased.", "As shown in the right inset in Fig.", "REF (b), the reduction of this difference is nearly exponential for large values of $U$ .", "The high-$T$ peak, which is associated with the freezing of charge degrees of freedom and moment formation, moves to higher $T$ as $U$ increases.", "In the atomic limit ($t\\rightarrow 0$ ), the location of the peak, $T^P$ , is determined by $\\alpha \\tanh (\\alpha )=1$ , where $\\alpha =U/16T^P$ , which can be approximated by $T^P=U/4.8$ .", "For large $U$ , and regardless of the particle statistics, we find a very good agreement with the atomic limit prediction for $T^P$ [see Fig.", "REF (a)].", "Note that there is no well-defined high-$T$ peak in the specific heat of 2HCBs for $U<7$ .", "Unlike its position, the value of this peak does not change monotonically with increasing $U$ and, as seen in Fig.", "REF (b), has a minimum around $U=7$ for fermions and around $U=9$ for 2HCBs.", "However, the relative change in the studied range of interactions is only about $20\\%$ .", "For large values of $U$ , the area under the high-$T$ peak of $C_v/T$ also approaches $\\ln (2)$ , consistent with results in the atomic limit.", "The low-temperature peak, which is associated with ordering of the moments, is expected to move to lower temperatures by increasing $U$ .", "This is because, for large values of $U$ ($\\gtrsim 12$ ), the low-$T$ system is essentially described by the antiferromagnetic Heisenberg model with a characteristic energy scale of $J\\propto t^2/U$  [21].", "Therefore, the position of the low-$T$ peak, which we do not report here, is expected to be inversely proportional to $U$ .", "An interesting feature discussed in the past for correlated fermionic systems is the near universal high-$T$ crossing of the specific heat curves for different values of the interaction [33], [34], [32].", "Due to its accuracy, NLCE provides an ideal tool for examining the precise behavior of the crossing point.", "We show its position, and the value of $C_v$ at the crossing, in Figs.", "REF (c) and REF (d), respectively.", "We find that, for fermions, the temperature and the specific heat value of the crossing point between two consecutive values of $U$ is nearly independent of $U$ for $U\\le 8$ ($T^\\text{cross.", "}\\sim 1.75$ , and $C_v^{\\text{cross.", "}}\\sim 0.36$ ), with changes of roughly $5\\%$ .", "These variations are slightly larger for 2HCBs in the same range of interactions.", "Figure: (Color online) Comparison of the entropy vs temperature forn=0.85n=0.85 (thick lines) and n=1.00n=1.00 (thin lines) between the Fermi-Hubbardand the 2HCB-Hubbard models for (a) U=4U=4, (b) U=8U=8, and (c) U=16U=16.Away from half filling, where the ground state of neither system is known, the trends in the deviation of finite-temperature properties of fermions and 2HCBs is different from the one reported so far.", "As an example, we show in Fig.", "REF the entropy (for $U=4$ , 8, and 16) vs $T$ for $n=0.85$ , and compare each curve with the one for $n=1.00$ .", "Interestingly, in the weak-coupling regime, the entropy does not change significantly at the accessible temperatures for either particle statistics as one dopes the system away from half filling.", "For $U=8$ [Fig.", "REF (b)], the entropy for the fermionic system with 15% doping is larger than the half-filled value for $0.3<T<2$ , while it does not change nearly as much with doping for 2HCBs for $T\\gtrsim 0.9$ .", "By increasing the interaction to $U=16$ , the agreement between the entropy of fermions and 2HCBs away from half filling improves, yet the deviations between the two for $n=0.85$ start at much higher temperatures in comparison to the half filled case [see Fig.", "REF (c)].", "These observations suggest that, even in the strong-coupling regime, the two systems away from half filling have fundamentally different phases.", "Further insight on the phases of the two systems at low temperatures can be gained by studying their equations of state and compressibilities.", "They are also of great interest to optical lattice experiments since those experiments are done in the presence of a confining potential that imposes a spatially varying chemical potential on the system.", "So, different regions in the trap correspond to different densities.", "In Fig.", "REF , we show $n$ and $\\partial n/\\partial \\mu $ vs $\\mu $ at $T=0.47$ for $U=9$ and $U=14$ , which are below and above the critical interaction value for 2HCBs.", "From QMC calculations it is known that, for $U<U_c$ , the ground state of 2HCBs does not have a charge density gap [16].", "One can also infer from Fig.", "REF that, regardless of the interaction strength, 2HCBs always have a smaller Mott gap at zero temperature than the fermions.", "This, in turn, implies that around half filling, the compressibility is always greater for bosons than for fermions, at any given temperature.", "A slight upturn in the double occupancy ($D$ ) as one decreases $T$ for the half-filled strongly-correlated fermionic system is a known phenomenon that has been attributed to the increase in virtual hoppings between allowed nearest neighbor sites due to the enhancement of short-range AF correlations [35].", "In fact, the onset of this increase, which can be measured in the experiments, may serve as a universal probe for large AF correlations [36].", "It can be shown to lead to adiabatic cooling by increasing the interaction strength [30], which is of great interest to optical lattice experiments.", "We have recently shown that such an increase in the double occupancy also occurs away from half-filling.", "Hence, in optical lattice experiments one also needs to make sure the density in most of the system is around half-filling in order for any increase in the double occupancy to be associated with the onset of antiferromagnetism [21].", "We find interesting trends in the double occupancy of 2HCBs when compared to fermions, especially for weak interactions.", "As shown in Fig.", "REF , and unlike in the fermionic case, the double occupancy of 2HCBs at and away from half filling increases sharply below $T\\sim 1$ for weak interactions, e.g., $U=4$ in Figs.", "REF (a) and REF (d), while such a large difference between the results for the two particle statistics is absent for large $U$ .", "The sharp low-$T$ rise in $D$ indicates that adiabatic cooling starting from the weakly interacting limit is efficient for 2HCBs [30].", "The double occupancy for 2HCBs even reaches the uncorrelated value of $1/4$ in the half-filled case for $U=4$ and $T\\sim 0.2$ , in stark contrast to $D$ of the fermionic system.", "This is consistent with the absence of a Mott insulator for 2HCBs in that parameter region.", "Figure: (Color online) Double occupancy vs temperature for differentvalues of UU at half filling (top panels) and for n=0.85n=0.85 (bottom panels).In the weak-coupling regime with U=4U=4 [(a) and (d)], DD for 2HCBsincreases significantly by lowering the temperature below 1.", "Thick(color) lines are results from the last order and thin (black) lines arethe results from the one-to-last order of the NLCEs after resummations.A related trend is also seen in the results for the NN spin correlations, $S^{zz}=|\\langle \\sum _{\\delta }S^z_i S^z_{i+\\delta }\\rangle |$ , where the sum runs over the four nearest neighbors of site $i$ .", "As depicted in Fig.", "REF (a), $S^{zz}$ for 2HCBs peaks around $T=0.5$ when $U=4$ before becoming vanishingly small at lower temperatures, which is again consistent with the superfluid nature of the ground state in this interaction region.", "On the other hand, it has been shown that for the fermionic case, $S^{zz}$ at half filling grows monotonically by decreasing the temperature [32], which is consistent with its antiferromagnetically ordered low-$T$ phase.", "It would be interesting to examine $S^{zz}$ at $T=0$ for 2HCBs and across the phase transition between the 2SF and the SCF phases, where $S^{zz}$ is presumably not small.", "Here, the difference between results for fermions and 2HCBs sets in around $T=1$ for $U=4$ and, like all other thermodynamic quantities at half filling, becomes smaller as $U$ is increased [see Figs.", "REF (a)-(c)].", "As shown in the inset of Figs.", "REF (a), at fixed temperatures, this difference becomes exponentially small with increasing $U$ in the strong-coupling regime.", "Figure: (Color online) Nearest-neighbor spin correlations vs temperatureat half filling for (a) U=4U=4, (b) U=8U=8, and (c) U=16U=16.", "(d)-(f)Uniform spin susceptibility at half filling vs temperature.", "Like forfermions, the spin susceptibility for 2HCBs peaks at a characteristictemperature T * T^*.", "The inset in (a) shows the exponential decrease ofthe difference between S zz S^{zz} of 2HCBs and fermions at fixed temperaturesby increasing UU in the strong-coupling regime.", "Lines are the same as inFig.", "Another thermodynamic quantity that highlights the difference between fermions and 2HCBs in the weak-coupling regime is the uniform spin susceptibility ($\\chi $ ).", "As seen in Fig.", "REF (d)-(f), we find that the deviation of $\\chi $ for 2HCBs from that of fermions is large at low temperature for $U=4$ and becomes smaller as $U$ increases to 8 and 16.", "More importantly, whereas a previous QMC study by Paiva et al.", "[20] has shown that the peak location in the fermionic case changes non-monotonically by increasing U (following the variations of the AF correlations in the system), for 2HCBs, the peak temperature decreases monotonically by increasing the interaction strength.", "This is because, unlike for fermions, the peak in $\\chi $ for 2HCBs in the weak-coupling regime does not signify moment ordering, but rather the disappearance of well-defined moments.", "This can be understood from the fact that NN spin correlations also decrease around the same temperature [see Fig.", "REF (a)].", "It is known for the fermionic Hubbard model that there exists the following unitary particle-hole transformation [37] that takes the repulsive Hubbard model to the attractive one: $\\hat{f}_{j\\uparrow } &=& \\hat{d}_{j\\uparrow }, \\nonumber \\\\\\hat{f}_{j\\downarrow } &=& e^{i(\\pi ,\\pi )\\cdot {\\bf R}_j}\\hat{d}^{\\dagger }_{j\\downarrow },$ where ${\\bf R}_j$ is the displacement vector of site $j$ .", "To see the effect of the transformation more clearly, it is easier to rewrite the Hamiltonian of Eq.", "(REF ) in the so-called particle-hole symmetric form: $\\hat{H}&=&-t\\sum _{\\left<i,j\\right>\\sigma }(\\hat{f}^{\\dagger }_{i\\sigma }\\hat{f}_{j\\sigma } + \\text{H.c.}) \\\\&&+U\\sum _i \\left(\\hat{n}_{i\\uparrow }-\\frac{1}{2}\\right)\\left(\\hat{n}_{i\\downarrow }-\\frac{1}{2}\\right)-\\mu ^{\\prime }\\sum _i (\\hat{n}_{i\\uparrow }+ \\hat{n}_{i\\downarrow }),\\nonumber $ where $\\mu ^{\\prime }=\\mu -\\frac{U}{2}$ .", "The transformation in Eq.", "(REF ) leaves the hopping term, as well as $n_{i\\uparrow }$ , invariant, but changes $n_{i\\downarrow }$ to $1-n_{i\\downarrow }$ .", "As a result, the Hamiltonian in Eq.", "(REF ) is transformed to: $\\hat{H}&=&-t\\sum _{\\left<i,j\\right>\\sigma }(\\hat{d}^{\\dagger }_{i\\sigma }\\hat{d}_{j\\sigma } + \\text{H.c.})\\\\&&-U\\sum _i \\left(\\hat{n}^{\\prime }_{i\\uparrow }-\\frac{1}{2}\\right)\\left(\\hat{n}^{\\prime }_{i\\downarrow }-\\frac{1}{2}\\right)-\\mu ^{\\prime }\\sum _i (\\hat{n}^{\\prime }_{i\\uparrow }- \\hat{n}^{\\prime }_{i\\downarrow })\\nonumber $ where $\\hat{n}^{\\prime }_{i\\sigma }=\\hat{d}^{\\dagger }_{i\\sigma }\\hat{d}_{i\\sigma }$ .", "At half filling ($\\mu ^{\\prime }=0$ ), the only change from Eq.", "(REF ) will be the sign of the interaction $U$ .", "Therefore, the energy spectral properties of the half-filled Fermi-Hubbard model, e.g., its specific heat, entropy, etc, are invariant under $U\\rightarrow -U$ .", "The nature of the ground state, however, is profoundly different in the repulsive and the attractive models since this transformation maps charge correlations to spin correlations and vice versa [38], which means that long-range AF order is mapped to a charge-density-wave one.", "A similar unitary transformation maps the repulsive Hubbard model for 2HCBs to an attractive one: $\\hat{b}_{j\\uparrow } &=& \\hat{d}_{j\\uparrow } \\nonumber \\\\\\hat{b}_{j\\downarrow } &=& \\hat{d}^{\\dagger }_{j\\downarrow }.$ Therefore, similar to the fermionic case, the Hamiltonian is invariant under the change of sign of the interaction at half filling.", "The same argument presented above for the nature of the ground state of the fermionic model also applies to 2HCBs for $U>U_c$ .", "In the regime $U<U_c$ , we expect the 2SF phase to be the ground even with attractive interactions since, like the repulsive case, the system presumably gains more energy through the condensation of each specie than by minimizing the interaction energy.", "As is clear from Eqs.", "(REF ) and (REF ), the attractive (repulsive) Hamiltonian away from half filling ($\\mu ^{\\prime }\\ne 0$ ) is equivalent to the repulsive (attractive) one in the presence of a magnetic field in the $z$ direction, $h$ , a role that is played by the chemical potential in the attractive (repulsive) case.", "In Fig.", "REF , we show how the specific heat of the repulsive Hubbard model for 2HCBs is modified in the presence of such a field, by plotting $C_v$ of the attractive model at $\\mu ^{\\prime }=0$ and $0.5$ ($h=0$ and 0.5 for the repulsive case).", "The low temperature region is not accessible to us at small $U$ and, in Fig.", "REF (a) for $U=8$ , only a small deviation around the high-temperature peak can be seen when the magnetic field is introduced.", "The results for $U=16$ in Fig.", "REF (b) show a suppression of the specific heat at $T<1$ , which are consistent with the fact that spin degrees of freedom emerge only in the latter temperature region.", "Results for the fermionic case show qualitatively the same behavior as for 2HCBs for those two values of $U$ .", "Figure: (Color online) Specific heat of the repulsive Hubbard modelfor 2HCBs with and without a magnetic field, h=0.5h = 0.5 and 0, for (a) U=8U=8 and (b)U=16U=16.", "Thick (thin) lines are the results from the last (one-to-last)order of the NLCEs after numerical resummation.Previously, we mapped out the isentropic paths of the 2D fermionic Hubbard model in the extended temperature-interaction space [21].", "In Fig.", "REF (a), we present a similar diagram for 2HCBs.", "As expected from the large negative slope in the low-$T$ double occupancy of 2HCBs for small $U$ vs $T$ (Fig.", "REF ), adiabatic cooling by increasing $U$ in the weak-coupling regime is much more efficient in comparison to the fermionic case.", "With the entropy per particle of 0.6, $T$ reduces roughly by a factor of 4 as the interaction increases from 1 to 24.", "However, the underlying physics in different regions of the diagram and the change in the strength of AF correlations is unlike that of fermions.", "As mentioned before, in the fermionic system, regardless of the value of the interaction, AF correlations are always enhanced by lowering the temperature.", "So, based on their dependence on $U$ and how rapidly the temperature falls in the adiabatic cooling process, one could drive the fermionic system into the region with exponentially large AF correlations.", "The latter region could be identified by the onset of a downturn in the uniform susceptibility ($T^*$ ) [21].", "On the contrary, for 2HCBs, the peak in the uniform susceptibility does not signify large AF correlations for weak interactions [see Figs.", "REF (a) and REF (d)].", "We emphasize this by plotting in the same figure the location of the peak in $S^{zz}$ ($T^\\text{sp}$ ) for a few values of $U$ .", "Beyond $U_c$ , where the Mott insulating ground state not only has long-range $xy$ -ferromagnetic order, but also very large $z$ -antiferromagnetic correlations, the maximum of $S^{zz}$ is likely at zero temperature.", "Figure: (Color online) (a) Isentropic dependence of temperature on theinteraction for different values of the entropy for the 2HCB-Hubbardmodel at half filling.", "Also shown in thispanel is the characteristic temperature, T * T^*, which is thelocation of the peak in the uniform spin susceptibility, andT sp T^\\text{sp}, which is the location of the peak in the NN spin correlations.The exact values of the latter are at lower temperatures than what isaccessible to us for U<4U<4 and U>6U>6.", "(b) Isothermic curves of S zz S^{zz}vs UU.", "(c) Isentropic curves of S zz S^{zz} vs UU for the entropy perparticle from 0.3 to ln(2)(2).Results in Fig.", "REF (b) and Fig.", "REF (c) provide further insight into the behavior of $S^{zz}$ for different interaction strengths.", "By comparing these figures to their fermionic counterparts [Figs.", "3(c) and 3(d) in Ref.", "Ekhatami11b], one can see that unlike in the weak-coupling regime, where the results are qualitatively different for the two particle statistics, in the strong-coupling regime they are very similar [as shown in Fig.", "REF (a), the difference decreases exponentially with $U$ ].", "As seen in Fig.", "REF (b), at fixed (and low) temperatures $S^{zz}$ for 2HCBs sharply drops to small values by decreasing $U$ .", "In Fig.", "REF (c), the isentropic curves of $S^{zz}$ are almost on top of each other for $U\\lesssim 7$ for the entropies shown, and are expected to become vanishingly small at lower entropies that are not accessible to us.", "This is, again, unlike the trends in the fermionic model in the weak-coupling regime.", "The opposite is true, however, for $U>U_c$ where, at the entropies accessible to us, $S^{zz}$ values of 2HCBs agree very well with their fermionic counterparts.", "So far, much lower temperatures have been achieved with bosons in optical lattices than with fermions.", "Employing novel cooling techniques, researchers have been able to access temperatures as low as 1 nK with bosons deep in the Mott insulating regime [39].", "It has also been shown in experiments with bosonic mixtures and that the inter- and intra-specie interactions can be tuned using Feshbach resonance and other techniques [7], [9], [10].", "Our results show that thermodynamic properties and short-range AF correlations of 2HCBs are very (exponentially) close to those of the fermions for strong interactions and up to intermediate to low temperatures.", "Therefore, by engineering the strong interaction regime studied here, optical lattice experiments with two-component bosons could also provide a tool for simulating intermediate to low temperature correlations in the Fermi-Hubbard model.", "Further studies need to be done to explore the properties of 2HCBs away from half filling and examine their relevance to the fermionic case." ], [ "Conclusions", "We have examined the particle statistics dependence of the thermodynamic properties of the 2D Hubbard model by means of an exact method (NLCEs) that yields properties such as entropy, specific heat, double occupancy, and spin correlations in the thermodynamic limit.", "The results are valid above a certain temperature that depends on the interaction and the filling factor.", "We considered two-component fermions and 2HCBs and compared their properties at various temperatures and interaction strengths, up to three times the bandwidth.", "We have shown that for weak interactions, in the regime where the ground state of the two models at half filling have fundamentally different natures, the results for observables at finite temperature differ significantly for the two particle statistics starting at relatively high temperatures.", "In contrast, in the strong-coupling regime (beyond a critical interaction that separates a superfluid from a Mott insulator phase for 2HCBs), the agreement between the thermodynamic quantities of the two systems, including short-range AF correlations, extends to much lower temperatures.", "We find that the trends in spin correlations in the $z$ direction are similar for both particle statistics and that the results differ only by exponentially small values for strong interactions.", "This provides an additional tool to probe correlations of fermionic systems in optical lattice experiments by emulating them using two-specie bosons, which can generally be cooled down to lower temperatures." ], [ "Acknowledgments", "This work was supported by the NSF under Grant No.", "OCI-0904597.", "We thank J. Carrasquilla for useful discussions." ] ]

[ [ "Colloids in active fluids: Anomalous micro-rheology and negative drag" ], [ "Abstract We simulate an experiment in which a colloidal probe is pulled through an active nematic fluid.", "We find that the drag on the particle is non-Stokesian (not proportional to its radius).", "Strikingly, a large enough particle in contractile fluid (such as an actomyosin gel) can show negative viscous drag in steady state: the particle moves in the opposite direction to the externally applied force.", "We explain this, and the qualitative trends seen in our simulations, in terms of the disruption of orientational order around the probe particle and the resulting modifications to the active stress." ], [ "Colloids in active fluids: Anomalous micro-rheology and negative drag G. Foffano$^1$ , J. S. Lintuvuori$^1$ , K. Stratford$^2$ , M. E. Cates$^1$ , D. Marenduzzo$^1$ $^1$ SUPA, School of Physics and Astronomy, University of Edinburgh, Mayfield Road, Edinburgh, EH9 3JZ, UK; $^2$ EPCC, School of Physics and Astronomy, University of Edinburgh, Mayfield Road, Edinburgh, EH9 3JZ, UK.", "We simulate an experiment in which a colloidal probe is pulled through an active nematic fluid.", "We find that the drag on the particle is non-Stokesian (not proportional to its radius).", "Strikingly, a large enough particle in contractile fluid (such as an actomyosin gel) can show negative viscous drag in steady state: the particle moves in the opposite direction to the externally applied force.", "We explain this, and the qualitative trends seen in our simulations, in terms of the disruption of orientational order around the probe particle and the resulting modifications to the active stress.", "87.10.−e, 47.50.−d, 47.63.mf, 83.60.Bc Active particles take energy from their surroundings and convert this into mechanical work.", "Active fluids are suspensions of such particles in a Newtonian solvent, and they represent an interesting class of nonequilibrium soft matter [1], [2].", "To lowest order, an active fluid may be modelled as a collection of force dipoles, exerted by the active particles, which quite often have orientational order (e.g., as a nematic phase).", "These force dipoles are either contractile, when the forces are exerted “inwards” towards the centre of mass of each particle, or extensile, in the opposite case.", "Suspensions of bacteria such as E.coli are extensile fluids, while a dispersion of Chlamydomonas (algae) is contractile, as is the actomyosin gel which constitutes the cytoskeleton of eukaryotic cells [1].", "The continuous dipolar forcing present in active fluids profoundly affects their macroscopic properties.", "For instance, active nematics flow spontaneously in steady state in the absence of any external force, provided the activity level (dipole density) is high enough [3], [4].", "Simulations [3], [5] show this flow to be chaotic, resembling the “bacterial turbulence” observed in concentrated films of B. subtilis [6].", "Furthermore, the bulk rheology of active fluids is strongly non-Newtonian [2], [7].", "Theory predicts an increase (reduction) in the effective viscosity for contractile (extensile) fluids [2].", "These predictions were confirmed by simulations [8], [9], and also by experiments on Chlamydomonas [10] and E. coli [11].", "The local flow properties of active fluids also differ from their passive counterparts.", "These can be addressed by monitoring the dynamics of a probe particle in a “micro-rheology\" experiment.", "Such studies [12] find marked violations of the fluctuation-dissipation theorem [13], [14], [15], which in near-equilibrium systems links the decay of random fluctuations to the linear force response.", "Local flow of active fluids is of strong biophysical relevance: for instance, the cytoskeleton in moving cells is mainly subjected to localized cues rather than global forces, and its response to these may be crucial to cell motility [16].", "Here we address a very basic issue of the local flow response in active fluids.", "We ask: what happens when a passive particle of radius $R$ is dragged through an active nematic with a force $F$ ?", "This represents perhaps the simplest micro-rheological experiment possible on an active fluid.", "We find by simulations that this simple experiment should lead to some very interesting results.", "First, the drag coefficient $\\xi =F/v$ (with $v$ a steady particle speed) exhibits a strongly nonlinear dependence on radius, in violation of Stokes' law.", "This is especially noticeable in the contractile case.", "We explain these violations in terms of the deformation of the orientational order in the active fluid around the probe particle, and present a simple scaling theory for the balance between active and viscous forces.", "Second, and strikingly, our theory and simulations show that in a contractile fluid the colloidal probe, if large enough, should move steadily in the direction opposite to the applied force.", "That is, we find a stable steady state with a negative drag coefficient.", "We also simulate a transient (force reversal) experiment that probes further the remarkable physics in this regime.", "Numerical model: In the continuum limit, the hydrodynamics of an active nematic fluid can be described by a set of continuum equations [1], [2] that govern the time evolution of the velocity field $u_\\alpha $ and of a (traceless, symmetric) tensor order parameter $Q_{\\alpha \\beta }$ .", "The latter describes the orientational order of the active particles (whether bacteria, algae, or cytoskeletal filaments) which usually have a rod-like shape and are thus capable of nematic alignment [1].", "Without activity, nematics are described by a Landau – de Gennes free-energy density ${\\cal F} = F(Q_{\\alpha \\beta }) + K(\\partial _{\\beta }Q_{\\alpha \\beta })^2/2$ , with $F(Q_{\\alpha \\beta }) = \\left(1-\\frac{\\gamma }{3}\\right)\\frac{Q_{\\alpha \\beta }^2}{2}-\\frac{\\gamma }{3}Q_{\\alpha \\beta }Q_{\\beta \\gamma }Q_{\\gamma \\alpha } + \\frac{\\gamma }{4}(Q_{\\alpha \\beta }^2)^2$ where indices denote Cartesian coordinates, summation over repeated indices is implied, $\\gamma $ controls the magnitude of nematic order, and $K$ is an elastic constant.", "The hydrodynamic equation for the evolution of the order parameter is: $D_t Q_{\\alpha \\beta } = \\Gamma H_{\\alpha \\beta }$ , with $D_t$ a material derivative describing advection by the fluid velocity $u_\\alpha $ , and rotation/stretch by flow gradients (see [3]).", "The molecular field is $H_{\\alpha \\beta }= -{\\delta {\\cal F} / \\delta Q_{\\alpha \\beta }} + (\\delta _{\\alpha \\beta }/3) {\\mbox{\\rm Tr}}({\\delta {\\cal F} / \\delta Q_{\\alpha \\beta }})$ and $\\Gamma $ is an inverse rotational friction.", "The fluid velocity obeys $\\partial _\\alpha u_\\alpha = 0$ , and also the Navier-Stokes equation, in which a passive $Q_{\\alpha \\beta }$ -dependent thermodynamic stress enters [3].", "The active force dipoles then create a further stress, $\\Pi _{\\alpha \\beta }=-\\zeta \\left(Q_{\\alpha \\beta }+\\frac{1}{3}\\delta _{\\alpha \\beta }\\right),$ where $\\zeta $ is the activity parameter that sets the dipolar force density [2].", "Note that $\\zeta <0$ for contractile fluids and $\\zeta >0$ for extensile ones.", "Within our hybrid numerical scheme, we solve the Navier-Stokes equation via lattice Boltzmann, and the equation for the order parameter via finite difference [17].", "Periodic boundary conditions are deployed throughout.", "We introduce a spherical colloidal probe by a standard method of bounce-back on links [18] which provides a no-slip boundary condition for the velocity field at the solid surface.", "Order parameter variations create additional elastic and active forces on the particle which are computed by integrating the total stress tensor over its surface.", "There we impose planar anchoring of the nematic director, which as usual is a headless unit vector $n_\\alpha $ oriented along the major principal axis of $Q_{\\alpha \\beta }$ .", "Planar anchoring is imposed via an additional quadratic term in the free energy (see [19], [20] for details).", "Below we give our results in simulation units [21].", "To convert them into physical ones, relevant for instance to a contractile actomyosin solution, we can assume a value of $K$ of 1.25 nN, and a rotational viscosity of 1.0 kPa/s.", "(These values hold for typical cytoskeletal gels [22].)", "In this way, the simulation units for force, length and time can be mapped onto 25 nN, 0.5 $\\mu $ m and 10 ms respectively.", "Note that the same equations also apply to an extensile bacterial fluid, but the mapping to physical units in that case leads to very different values [3].", "Our model also neglects motility which is important in bacterial fluids, where it naturally leads to density inhomogeneities [3], [23].", "Figure: (Color online) (a,b) Plots of v(F)v(F) curves for colloids of fixed radius R=11.3R=11.3, at various ζ\\zeta (see legend) in a contractile (a) and an extensile (b) active nematic, dragged respectively parallel and perpendicular to the far-field director.", "(c,d) Dependence of η eff \\eta _{\\rm eff}, measured according to Eq.", ", for both (c) contractile and (d) extensile cases.", "Filled symbols refer to an external force parallel to the far-field director, while empty symbols are used for orthogonal drag.", "Different symbols (colours) refer to different activities (see legend in (a) and (b)).", "In (a) and (b), insets, we sketch the director field and active force directions for a colloid pulled (a) in a contractile active fluid along the far-field director, and (b) in an extensile fluid perpendicular to it.Results: We now discuss the result of our micro-rheological simulation in which a colloid of radius $R$ is pulled through an active fluid, either contractile or extensile.", "The external force $F$ was directed either along or perpendicular to the far-field nematic director.", "For the values of $R$ chosen, plots of the steady state velocity $v$ (measured along $F$ ) versus $F$ (see Fig.", "1a,b) show a well-defined linear regime at small external force.", "In all our simulations $\\zeta $ is kept small enough such that no spontaneous flow arises in the absence of the probe particle [25].", "Figs.", "1a,b show that the linear drag coefficient $\\xi _0 = F/v|_{F\\rightarrow 0}$ increases with activity for a contractile fluid, and decreases for an extensile one (within the parameter range that we explored).", "Thus a contractile fluid opposes motion more strongly than its passive counterpart; an extensile fluid less.", "So far, this is in line with the respective increase (contractile) and decrease (extensile) of bulk fluid viscosities mentioned above [24].", "By analogy with the formula for Stokes drag on a probe of radius $R$ in a passive fluid of viscosity $\\eta $ , we can define an effective viscosity for the probe motion as $\\eta _{\\rm eff}(R)=\\xi _0/(6\\pi R).$ This is plotted as a function of $R$ in Figs.", "1c,d after correction for periodic boundary effects [26].", "If Stokes' law did hold, the effective viscosity should be independent of $R$ , and we see that this is indeed the case for a passive nematic ($\\zeta =0$ , open and filled circles in Fig.", "1c,d).", "When activity is switched on, this picture changes dramatically and the drag coefficient becomes strongly non-Stokesian.", "For a contractile fluid, this anomaly is most apparent when pulling along the director field, where the effective viscosity increases sharply with the radius.", "In the extensile case the most markedly anomalous response is instead obtained when the particle is pulled perpendicularly to the far-field director, and the effective viscosity this time drops with size.", "These results show that for active fluids microrheology experiments do not simply probe the fluid, but measure a scale-dependent property of the probe and surrounding fluid in combination.", "Although microrheology is thereby disconnected from bulk rheology, such experiments may offer essential insights into the physical response of (for instance) an actomyosin gel on the length scale relevant to transport of organelles or other subcellular objects [16].", "We can qualitatively explain the anomalous drag $\\xi _0(R,\\zeta )$ in Fig.", "1c via the following argument (sketched in Fig.", "1a, inset).", "As the colloidal probe is pulled through the contractile fluid, it deforms the director field.", "With the pulling force oriented parallel to the far-field director, planar anchoring requires an elastic splay of the director locally.", "Because of the fluid flow created by particle displacement, this splay will be stronger in front of the moving particle than behind it.", "Through its effect on the active stress, the splay will result in a large force opposing the motion at the front, and in a smaller force favoring it at the rear.", "The net effect is to slow the particle down: $\\xi _0$ is increased.", "The same argument holds for pulling perpendicular to the director, but here is much smaller.", "We attribute the different magnitude of the effect in the two directions to the fact that contractile activity enhances the splay response (leading at large wavelengths to an instability involving splay but not bend [1]).", "One can likewise understand the results in Fig.", "1d for an extensile fluid, by sketching the expected nematic deformation in front and at the rear of the particle (see inset in Fig. 1b).", "Now the incipient instability is towards bending, so pulling perpendicularly to the director gives the larger effect.", "Moreover, the active stress on the front of the particle now favors motion and the smaller one at the back opposes it: so drag is reduced.", "These arguments suggest that the $v(F)$ curves should strongly depend on the anchoring conditions at the particle surface, and we have confirmed this numerically (data not shown).", "Building more quantitatively on these arguments, we now estimate the extra force acting on a moving particle arising from the active contribution to the stress (integrated over the colloidal surface).", "This depends on the surrounding flow field and nematic deformation, details of which can only be computed numerically.", "Nonetheless, we can argue that activity leads to an additional force on the particle which, on dimensional grounds, equals $A(F) \\zeta R^2$ .", "To estimate $A$ , we note that for $F=0$ the particle is immobile, so that the extra force should also vanish; $A(0)=0$ .", "To first order, dimensional analysis now suggests $A\\sim c F/K$ with $c$ a positive dimensionless number of order unity [27].", "(Our previous arguments show that $A/F$ is positive for both signs of $\\zeta $ .)", "The force balance for the moving colloid in steady state is then given by $F+c \\zeta FR^2/K=6\\pi \\tilde{\\eta } R v$ , where $\\tilde{\\eta }$ is the passive nematic viscosity, which may depend on pulling direction.", "This equation together with Eq.", "REF leads to the following prediction for the effective viscosity $\\frac{1}{\\eta _{\\rm eff}} = \\frac{1}{\\tilde{\\eta }}\\left(1+\\frac{c \\zeta R^2}{K} \\right)$ This provides a surprisingly robust explanation for the size dependence of the drag coefficient: Fig.", "2a shows that for a colloid in a contractile fluid, plotting $1/{\\eta _{\\rm eff}}$ versus $R^2$ yields a straight line as predicted by Eq.", "REF .", "By fitting our numerical data, we also obtain $c\\sim 0.06$ for parallel pulling independent of $\\zeta $ , which further validates our approximate theory.", "However Eq.", "REF works less well in the extensile case.", "There the effect is smaller, and passive contributions to the stress tensor possibly play a more significant role.", "Figure: (Color online) (a) Plot of 1/η eff 1/\\eta _{\\rm eff} on R 2 R^2, yielding an approximately linear dependence in agreement with Eq. .", "(b) v(F)v(F) curve for a case with negative drag (R=30,ζ=-0.002R = 30,\\zeta = -0.002).", "Filled circles and open squares refer to simulations in which the force is applied to a quiescent and moving colloid respectively.The velocity fields in (c) and (d) correspond to the case of a quiescent particle (c) and of a particle moving to the left, opposite to the applied force (large arrow) (d).Our simplified theory in Eq.", "REF formally predicts that when the dimensionless quantity ${c\\zeta R^2}/{K}<-1$ , $\\eta _{\\rm eff}<0$ , giving a negative drag coefficient ($\\xi = F/v<0$ ).", "However, one might reasonably expect that before this regime is reached our leading-order expression $A\\sim cF/K$ would break down and some different physics would come into play, so that $\\xi $ remains positive.", "Remarkably though, this is not the case.", "Although positive $\\xi (F)$ appears to be restored at extremely small forces, for modest but finite $F$ , we robustly find that the particle moves in the opposite direction to the externally applied force.", "A steady-state $v(F)$ curve showing this effect for ${c\\zeta R^2}/{K} \\sim 2.2$ is in Fig. 2b.", "This reverse-sigmoidal curve shows bistability at small forces, with two stable branches each of negative drag, $\\xi = F/v<0$ , and also negative slope, $dF/dV<0$ .", "Interestingly, there seems to be no simple relation between negative local drag in contractile fluids and the homogeneous bulk rheology of active fluids.", "In the latter, a negative ratio of stress to strain rate can arise, but only in the extensile case [3].", "Moreover, the resulting flows are generically unstable [5].", "To shed more light on the nature of the negative drag state in contractile fluids, we plot in Figs.", "2c,d the active fluid flow around a stationary, unforced probe particle and that for one moving against the external force.", "For the static particle, a symmetric pattern of eddies is created by contractile forces arising from the deformation of the director (horizontal in far field), which splays around the sides of the probe and bends at its top and bottom.", "In the negative drag regime with the external force acting to the right, the eddies become asymmetric, creating a packet of left-moving fluid.", "The resulting leftward advective velocity of the particle exceeds its rightward speed relative to the local fluid packet, giving an overall leftward motion in the lab frame.", "We have examined how this remarkable steady-state flow pattern is reached dynamically.", "On introducing a rightward force, the particle first moves rightward (see Fig.", "3) creating strong elastic distortions on its leading (right-hand) side.", "Such deformations then establish the leftward moving fluid packet (as in Fig.", "2d), which sets up deformation in the surrounding director field to finally create the steady state travelling flow packet with the particle, as well as the fluid, moving leftward.", "We emphasize that this spontaneously moving state is quite distinct from the bulk spontaneous flows known to occur in active fluids when activity exceeds a system-size dependent threshold [4].", "Because we are below that threshold, in our simulations the fluid velocity is essentially localized in the neighborhood of the probe particle.", "Figure: (Color online.)", "Dependence of the external force (step functions, axis on the right) and of the particle velocity (axis on the left) on time for two different 'experiments'.", "In both cases a force F=0.8F=0.8 is initially applied to the particle and turned off.", "Then a smaller force is applied along the direction of particle motion: the blue dotted line and the red dot-dashed one give the values of the external force in the two cases.", "The correspondent particle velocities are given by the black solid and green dashed lines respectively.", "The dynamics is discussed in the text.", "In the sketches arrows on the fluid refer to the active force direction, the arrow on the colloid represents the external force, while that above the particle shows the actual direction of motion.A further exploration of the bistable negative-drag region at small $F$ is presented in Fig. 3.", "Here, we start with a static particle, of radius large enough to give negative drag ($c\\zeta R^2/K \\sim 2.2$ ) and first pull it to the right.", "The transient response was already described above, and leads finally to steady leftward motion.", "We now reduce $F$ to zero: the particle keeps moving to the left, but slows down.", "We next start pulling the particle with a tiny leftward force: the probe continues to move leftward, albeit even more slowly than before.", "If this last stage is repeated with a much larger leftward force, the active flow and deformation pattern around the colloid dynamically reconstructs itself with the opposite sense, leading to a rightward probe velocity and once more to a negative drag.", "This sequence of states can be viewed as a trajectory on the $v(F)$ curve of Fig. 2b.", "The initial large rightward force puts the particle on the lower branch of the curve to the right of the vertical axis.", "As the force is removed and then applied to the left, the system ascends this lower branch.", "Where that branch ends, the system must jump to the upper branch.", "Conclusion: we have simulated numerically a simple microrheological experiment, which should be realizable in the laboratory (with some caveats, noted below).", "In this experiment a colloidal probe is pulled through an active nematic fluid.", "We have shown that the activity leads to a non-Stokesian drag force that increases approximately quadratically with particle size.", "This behavior creates a new regime, arising in a contractile fluid such as an actomyosin gel, at large values of a dimensionless parameter ${c \\zeta R^2}/{K}$ .", "Strikingly, in this regime the colloidal probe is predicted to move against the driving force to create a stable steady state of negative drag.", "This contrasts with the bulk behavior of active fluids where negative ratios of stress to strain rate can arise in principle, but are unstable, and expected only for the opposite sign of activity (extensile rather than contractile) [3].", "Stable negative drag is counterintuitive but no physical law prevents it in active systems.", "Related phenomena have been reported for an ensemble of molecular motors with load-accelerated dissociation [28], in axon mechanics [29], for filament fluctuations in active media [7], and also in the upstream migration against a flow field of slime-mold cells [30] and bacteria [31].", "In our context, the anomalous (ultimately negative) drag stems from the active stress, Eq.", "REF , that emerges from a well-accepted coarse-grained description of activity in systems such as cytoskeletal gels [1].", "Finally, an important requirement for any experimental test of our predictions with actomyosin is that issues arising from macroscopic network clustering should be avoided [32].", "This might be feasible either through a suitable choice of parameters such as myosin concentration etc., or by working in metastable uniform networks.", "However, such experiments may well prove to be challenging.", "We acknowledge EU network ITN-COMPLOIDS, FP7-234810, and EPSRC EP/J007404/1, for funding.", "MEC is funded by the Royal Society." ] ]

[ [ "Hecke modules and supersingular representations of U(2,1)" ], [ "Abstract Let F be a nonarchimedean local field of odd residual characteristic p. We classify finite-dimensional simple right modules for the pro-p-Iwahori-Hecke algebra $\\mathcal{H}_C(G,I(1))$, where G is the unramified unitary group U(2,1)(E/F) in three variables.", "Using this description when C is the algebraic closure of $\\mathbb{F}_p$, we define supersingular Hecke modules and show that the functor of I(1)-invariants induces a bijection between irreducible nonsupersingular mod-p representations of G and nonsupersingular simple right $\\mathcal{H}_C(G,I(1))$-modules.", "We then use an argument of Paskunas to construct supersingular representations of G." ], [ "Introduction", "This article is set in the framework of the mod-$p$ representation theory of $p$ -adic reductive groups.", "Our motivation comes from the possibility of a mod-$p$ Local Langlands Correspondence, that is to say a matching between (packets of) smooth mod-$p$ representations of a $p$ -adic reductive group and certain Galois representations.", "The case of $\\textrm {GL}_2(\\mathbb {Q}_p)$ has been most extensively studied, and a mod-$p$ Local Langlands Correspondence has been established by Breuil ([5]) based on the explicit determination of the irreducible smooth mod-$p$ representations of $\\textrm {GL}_2(\\mathbb {Q}_p)$ .", "Moreover, this correspondence is compatible with the $p$ -adic Local Langlands Correspondence established by the work of several mathematicians: see [6], [7], [11], [14], [23], [24], [27], and the references therein.", "The case of $\\textrm {GL}_2(F)$ with $F\\ne \\mathbb {Q}_p$ is already much more complicated.", "For example, when $F$ is a nontrivial unramified extension of $\\mathbb {Q}_p$ , Breuil and Paškūnas ([8]) have shown that there exists an infinite family of supersingular $\\textrm {GL}_2(F)$ -representations associated to a “generic” Galois representation.", "Recently, Abdellatif has classified the irreducible smooth mod-$p$ representations of $\\textrm {SL}_2(\\mathbb {Q}_p)$ (cf.", "[1]) by restricting the irreducible representations of $\\textrm {GL}_2(\\mathbb {Q}_p)$ , allowing her to take the first steps towards a mod-$p$ Local Langlands Correspondence for $\\textrm {SL}_2(\\mathbb {Q}_p)$ .", "In addition, the results of [1] are the first to consider a mod-$p$ Local Langlands Correspondence with $L$ -packets.", "The explicit classification of mod-$p$ representations of $p$ -adic reductive groups other than $\\textrm {GL}_2(\\mathbb {Q}_p)$ and $\\textrm {SL}_2(\\mathbb {Q}_p)$ is not yet known, however.", "In the present article, we investigate the smooth mod-$p$ representations of the unitary group $G = \\textrm {U}(2,1)(E/F)$ , where $E/F$ is an unramified quadratic extension of nonarchimedean local fields, and where the residue field of $F$ is of size $q$ , a power of $p$ .", "The irreducible subquotients of parabolically induced representations have been classified by Abdellatif ([1]).", "We are interested in the smooth irreducible representations that do not appear in this fashion, which we call supersingular representations (we will comment on this terminology at the end of this introduction).", "These representations are the ones which are expected to play a crucial role in a potential Local Langlands Correspondence.", "The purpose of this article is to construct such representations.", "We now describe the ingredients in our method, inspired by the work of Vignéras and Paškūnas.", "Let $I(1)$ be the unique pro-$p$ -Sylow subgroup of the standard Iwahori subgroup $I$ of $G$ , and let $C$ denote an algebraically closed field.", "The pro-$p$ -Iwahori-Hecke algebra $\\mathcal {H}_C(G,I(1))$ is the convolution algebra of compactly supported, $C$ -valued functions on the double coset space $I(1)\\backslash G/I(1)$ .", "Under a mild assumption on the characteristic of $C$ , we determine explicitly the structure of the algebra $\\mathcal {H}_C(G,I(1))$ and describe its center.", "This allows us to classify all simple finite-dimensional right modules of $\\mathcal {H}_C(G,I(1))$ for any field $C$ satisfying Assumption REF (Section 3).", "The motivation for considering modules of the algebra $\\mathcal {H}_C(G,I(1))$ comes from the following observation.", "Attaching to a smooth representation $\\pi $ of $G$ its space of $I(1)$ -invariants $\\pi ^{I(1)}$ yields a functor with values in the category of $\\mathcal {H}_C(G,I(1))$ -modules.", "If $C$ is of characteristic $p$ , then $\\pi ^{I(1)}$ is nonzero provided $\\pi $ is nonzero; this suggests that the functor of $I(1)$ -invariants is likely to give information about representations generated by their $I(1)$ -invariants (though in general, we don't expect an equivalence of categories, given the $\\textrm {GL}_2(F)$ case, $F\\ne \\mathbb {Q}_p$ (cf.", "[25])).", "Using our explicit description of finite-dimensional simple $\\mathcal {H}_{\\overline{\\mathbb {F}}_p}(G,I(1))$ modules, we establish a bijection between irreducible smooth nonsupersingular representations of $G$ and certain simple modules (Corollary REF ).", "In particular, we show that the simple $\\mathcal {H}_{\\overline{\\mathbb {F}}_p}(G,I(1))$ -modules not arising in this fashion are precisely those with a “zero” central character.", "We call these modules supersingular (Definition REF ), and note that they are all one-dimensional (cf.", "Definition REF ).", "Our goal is to attach an irreducible smooth supersingular representation of $G$ to every supersingular $\\mathcal {H}_{\\overline{\\mathbb {F}}_p}(G,I(1))$ -module.", "We achieve this goal by showing the existence of such representations, and construct them explicitly in the case $q = p$ .", "The tool we will use is (homological) coefficient systems on the semisimple Bruhat-Tits building $X$ of $G$ (Section 6).", "In [28], Schneider and Stuhler introduced coefficient systems on the Bruhat-Tits building and used them to study complex representations of $p$ -adic reductive groups.", "Coefficient systems were later used in the mod-$p$ setting by Paškūnas to construct supersingular representations of $\\textrm {GL}_2(F)$ .", "The use of coefficient systems in this context has proved extremely useful (cf.", "[26]), but so far has only been considered for the group $\\textrm {GL}_2(F)$ .", "We adapt this method to representations of $G = \\textrm {U}(2,1)(E/F)$ .", "To this end, we define an analog of Paškūnas' diagrams, which are easier to handle than coefficient systems (Definition REF ).", "In particular, the category of diagrams is equivalent to the category of $G$ -equivariant coefficient systems on $X$ (this is the content of Section REF ).", "Next, we attach to every supersingular module $M$ a diagram ${D}_M$ .", "The 0-homology of the corresponding coefficient system $\\mathcal {D}_M$ is naturally a smooth $G$ -representation, and we show that the $I(1)$ -invariants of any of its nonzero irreducible quotients contain an $\\mathcal {H}_{\\overline{\\mathbb {F}}_p}(G,I(1))$ -module isomorphic to $M$ .", "This implies that any nonzero irreducible quotient is a supersingular representation of $G$ (Corollary REF ).", "We then want to produce such irreducible quotients, and for this purpose we construct an auxiliary coefficient system $\\mathcal {E}_M$ of a relatively simple form, along with a morphism $\\mathcal {D}_M\\rightarrow \\mathcal {E}_M$ .", "Constructing $\\mathcal {E}_M$ involves analyzing injective envelopes of irreducible representations of the finite groups $\\Gamma =\\mathbf {U}(2,1)(\\mathbb {F}_{q^2}/\\mathbb {F}_{q})$ and $\\Gamma ^{\\prime } = (\\mathbf {U}(1,1)\\times \\mathbf {U}(1))(\\mathbb {F}_{q^2}/\\mathbb {F}_{q})$ associated to $G$ .", "In the course of constructing $\\mathcal {E}_M$ , it will become necessary to have several descriptions of the irreducible mod-$p$ representations of $\\Gamma $ and $\\Gamma ^{\\prime }$ , and their injective envelopes.", "In Section 5, we recall two parametrizations of these representations: one in terms of the simple modules of the respective finite Hecke algebras, based on the work of Carter and Lusztig ([10]), and another in terms of highest weight modules.", "We provide a dictionary between these two descriptions, and prove a useful decomposition when $q = p$ , which is used in determining the decomposition of certain injective envelopes.", "We next specialize to the case $q = p$ .", "In this setting we are able to construct explicitly an auxiliary coefficient system $\\mathcal {E}_M$ along with a morphism $\\mathcal {D}_M\\rightarrow \\mathcal {E}_M$ (Section 7).", "This morphism induces a map on the 0-homology of the coefficient systems, and we consider the representation afforded by the image $\\pi _{\\mathcal {E}_M} = \\textrm {im}(H_0(X,\\mathcal {D}_M)\\rightarrow H_0(X,\\mathcal {E}_M)).$ The result here is the following: Theorem (Corollary REF ) Assume $q = p$ .", "The representation $\\pi _{\\mathcal {E}_M}$ is nonzero, irreducible, admissible, and supersingular.", "For nonisomorphic supersingular $\\mathcal {H}_{\\overline{\\mathbb {F}}_p}(G,I(1))$ -modules $M,M^{\\prime }$ , the representations $\\pi _{\\mathcal {E}_M},~\\pi _{\\mathcal {E}_{M^{\\prime }}}$ are nonisomorphic.", "We remark that while $\\mathcal {D}_M$ is uniquely determined, the choice of the coefficient system $\\mathcal {E}_M$ is in general not unique.", "Therefore, to every supersingular module $M$ we attach at least one supersingular representation; in this way, we construct at least $p^2(p+1)$ supersingular representations of $G$ .", "We next address the shortcomings of our method when $q\\ne p$ .", "As mentioned before, our method relies on the comparison of injective envelopes for representations of the finite groups $\\Gamma $ and $\\Gamma ^{\\prime }$ .", "For $q\\ne p$ , we demonstrate cases where the construction of Section 7 would produce a coefficient system $\\mathcal {E}_M$ which is “too big,” in the sense that we cannot guarantee irreducibility of the resulting representation.", "Our main tool will be Dordowsky's Diplomarbeit ([13]), in which the dimensions of injective envelopes of representations of $\\Gamma $ are computed.", "To conclude, we draw some comparisons between our results and the analogous results for the group $\\textrm {SL}_2(F)$ , drawing on the results of Abdellatif in [1].", "We first use the explicit classification of Hecke modules to determine by elimination which are supersingular.", "As is the case for $\\textrm {U}(2,1)(E/F)$ , the action of $\\textrm {SL}_2(F)$ on its Bruhat-Tits tree $X_S$ partitions the set of vertices into two disjoint orbits and acts transtitively on the edges.", "Therefore, the results of Section 6 hold equally well for $\\textrm {SL}_2(F)$ (the proofs carry over formally).", "When $q = p$ , we attach to every supersingular $\\mathcal {H}_{\\overline{\\mathbb {F}}_p}(\\textrm {SL}_2(F),I_S(1))$ -module $M_S$ two coefficient systems $\\mathcal {D}_{M_S}$ and $\\mathcal {E}_{M_S}$ .", "There is one striking difference between this case and the case of $\\textrm {U}(2,1)(E/F)$ , however: when $q = p$ , there is a canonical choice of auxiliary diagram $\\mathcal {E}_{M_S}$ .", "In this way, we construct $p$ supersingular representations of $\\textrm {SL}_2(F)$ : exactly one such representation for every supersingular $\\mathcal {H}_{\\overline{\\mathbb {F}}_p}(\\textrm {SL}_2(F),I_S(1))$ -module.", "We record the result here.", "Theorem (Theorems REF and REF ) Assume $q = p$ .", "For each of the $p$ nonisomorphic supersingular $\\mathcal {H}_{\\overline{\\mathbb {F}}_p}(\\textnormal {SL}_2(F),I_S(1))$ -modules $M_S$ there is a canonical pair of associated coefficient systems $(\\mathcal {D}_{M_S},\\mathcal {E}_{M_S})$ .", "The resulting $\\textnormal {SL}_2(F)$ -representation afforded by $\\pi _{{M_S}} = \\textnormal {im}(H_0(X_S,\\mathcal {D}_{M_S})\\rightarrow H_0(X_S,\\mathcal {E}_{M_S}))$ is nonzero, irreducible, admissible, and supersingular.", "For nonisomorphic $\\mathcal {H}_{\\overline{\\mathbb {F}}_p}(\\textnormal {SL}_2(F),I_S(1))$ -modules $M_S,M_S^{\\prime }$ , the representations $\\pi _{{M_S}},~\\pi _{{M_S^{\\prime }}}$ are nonisomorphic.", "In particular, when $F = \\mathbb {Q}_p,$ we recover in this way all $p$ nonisomorphic supersingular representations of $\\textnormal {SL}_2(\\mathbb {Q}_p)$ as classified in [1].", "Remark on Terminology.", "We briefly address our choice of nomenclature.", "The notion of supersingularity was introduced by Barthel and Livné ([3] and [4]) in their classification of smooth, irreducible, nonsupercuspidal mod-$p$ representations of $\\textrm {GL}_2(F)$ .", "A smooth representation of $\\textrm {GL}_2(F)$ is called supersingular if a certain operator of the spherical Hecke algebra acts by zero, while it is called supercuspidal if it is not a subquotient of a parabolically induced representation.", "Theorems 33, 34, and Corollary 36(1) of [3] show that a smooth representation of $\\textrm {GL}_2(F)$ admitting a central character is supercuspidal if and only if it is supersingular.", "In the present case, the study of spherical Hecke algebras for the group $\\textrm {U}(2,1)(E/F)$ has been initiated by Henniart and Vignéras in [16], and Abdellatif in [1], with an analogous notion of supersingularity defined in [1].", "The arguments in [3] and [4] may be extended to the group $\\textrm {U}(2,1)(E/F)$ ; these results, combined with those of [1], would show the equivalence of the notions of supercuspidality and supersingularity.", "Based on results contained in [17] and [18], it is very likely that these two notions coincide.", "In anticipation of these results, we shall assume that this is the case.", "We use the term supersingular representation for what might otherwise be referred to as a supercuspidal representation, and henceforth refer only to supersingular representations.", "Acknowledgements.", "The authors would like to thank their advisors, Professors Rachel Ollivier and Shaun Stevens, for suggesting this problem, as well as for many helpful and illuminating discussions throughout the course of working on this paper.", "Part of this work was done while the first author was a visitor at the University of East Anglia, and he wishes to thank the institution for their support.", "Additionally, the first author would like to warmly thank Professor James Humphreys for several enlightening discussions, and for providing a copy of Dordowsky's thesis.", "The first author was supported by NSF Grant DMS-0739400; the second author was supported by EPSRC Grant EP/H00534X/1." ], [ "General Notation", "Fix a prime number $p$ greater than 2, and let $F$ be a nonarchimedean local field of residual characteristic $p$ .", "Denote by $\\mathfrak {o}_F$ its ring of integers, and by $\\mathfrak {p}_F$ the unique maximal ideal of $\\mathfrak {o}_F$ .", "Fix a uniformizer $\\varpi _F$ and the normalized valuation $\\nu $ given by $\\nu (\\varpi _F) = 1$ .", "Let $k_F = \\mathfrak {o}_F/\\mathfrak {p}_F$ denote the (finite) residue field.", "The field $k_F$ is a finite extension of $\\mathbb {F}_p$ of size $q = p^f$ .", "We shall identify $k_F$ with $\\mathbb {F}_q$ in a fixed algebraic closure $\\overline{\\mathbb {F}}_p$ of $\\mathbb {F}_p$ .", "We fix also a separable closure $\\overline{F}$ of $F$ , compatible with the chosen algebraic closure of the residue field, and let $E$ denote the unique unramified extension of degree 2 in $\\overline{F}$ .", "We write $E = F(\\sqrt{\\epsilon })$ , where $\\epsilon \\in F$ is some fixed but arbitrary nonsquare unit.", "We let $x\\mapsto \\overline{x}$ denote the nontrivial Galois automorphism of $E$ fixing $F$ .", "The ring of integers of $E$ is denoted $\\mathfrak {o}_E$ , and $\\mathfrak {p}_E$ is its unique maximal ideal.", "Since $E$ is unramified, we may and do take $\\varpi _E = \\varpi _F =: \\varpi $ as our uniformizer.", "The residue field of $\\mathfrak {o}_E$ is denoted $k_E = \\mathbb {F}_{q^2}$ , and is a degree 2 extension of $k_F$ .", "Denote by $G$ the $F$ -rational points of the algebraic group $\\mathbf {U}(2,1)$ .", "We perform our computations using the following realization of $G$ : let $V$ denote a three-dimensional vector space over $E$ .", "We identify $V$ with $E^3$ by a choice of basis, and for $\\vec{x} = (x_1,x_2,x_3)^\\top , \\vec{y} = (y_1,y_2,y_3)^\\top \\in V$ we define a nondegenerate Hermitian form $\\langle \\cdot , \\cdot \\rangle $ by $\\langle \\vec{x}, \\vec{y}\\rangle = \\overline{x_1}{y_3} + \\overline{x_2}{y_2} + \\overline{x_3}{y_1}.$ Letting $s = \\begin{pmatrix}0 & 0 & 1\\\\ 0 & 1 & 0\\\\ 1 & 0 & 0\\end{pmatrix},$ our form is represented by $\\langle \\vec{x},\\vec{y}\\rangle = \\vec{x}^{\\phantom{^{\\prime }}*}s\\vec{y}$ , where $m^*$ denotes the conjugate transpose of a matrix $m$ .", "With this notation, we have $G = \\lbrace g\\in \\textrm {GL}_3(E): g^*sg = s\\rbrace $ .", "The group $G$ possesses, up to conjugacy, two maximal compact subgroups (cf.", "[32], Sections 2.10 and 3.2), given by $K := \\textrm {GL}_3(\\mathfrak {o}_E)\\cap G\\quad \\textrm {and}\\quad K^{\\prime } := \\begin{pmatrix}\\mathfrak {o}_E & \\mathfrak {o}_E & \\mathfrak {p}_E^{-1}\\\\ \\mathfrak {p}_E & \\mathfrak {o}_E & \\mathfrak {o}_E \\\\ \\mathfrak {p}_E & \\mathfrak {p}_E & \\mathfrak {o}_E \\end{pmatrix} \\cap G.$ Let $K_1, K^{\\prime }_1$ be the following subgroups of $G$ : $K_1 :=\\begin{pmatrix}1 + \\mathfrak {p}_E & \\mathfrak {p}_E & \\mathfrak {p}_E\\\\ \\mathfrak {p}_E & 1 + \\mathfrak {p}_E & \\mathfrak {p}_E \\\\ \\mathfrak {p}_E & \\mathfrak {p}_E & 1 + \\mathfrak {p}_E\\end{pmatrix}\\cap G,\\qquad K^{\\prime }_1 := \\begin{pmatrix}1 + \\mathfrak {p}_E & \\mathfrak {o}_E & \\mathfrak {o}_E \\\\ \\mathfrak {p}_E & 1 + \\mathfrak {p}_E & \\mathfrak {o}_E \\\\ \\mathfrak {p}_E^2 & \\mathfrak {p}_E & 1 + \\mathfrak {p}_E\\end{pmatrix}\\cap G.$ The group $K_1$ (resp.", "$K^{\\prime }_1$ ) is the maximal normal pro-$p$ subgroup of $K$ (resp.", "$K^{\\prime }$ ).", "We define $\\Gamma := K/K_1 \\cong \\mathbf {U}(2,1)(\\mathbb {F}_{q^2}/\\mathbb {F}_{q}),\\qquad \\Gamma ^{\\prime } := K^{\\prime }/K^{\\prime }_1 \\cong (\\mathbf {U}(1,1)\\times \\mathbf {U}(1))(\\mathbb {F}_{q^2}/\\mathbb {F}_{q}).$ We let $\\mathbb {B}$ denote the upper Borel subgroup of $\\Gamma $ , $\\mathbb {U}$ its unipotent radical, and $\\mathbb {U}^-$ the opposite unipotent; let $\\mathbb {B}^{\\prime }$ denote the lower Borel subgroup of $\\Gamma ^{\\prime }$ , $\\mathbb {U}^{\\prime }$ its unipotent radical, and $\\mathbb {U}^{\\prime -}$ the opposite unipotent.", "The groups $\\mathbb {U}$ and $\\mathbb {U}^{\\prime }$ are $p$ -Sylow subgroups of $\\Gamma $ and $\\Gamma ^{\\prime }$ , respectively.", "The intersection of $K$ and $K^{\\prime }$ is the Iwahori subgroup $I$ , which we may also think of as the preimage under the reduction-modulo-$\\varpi $ map of $\\mathbb {B}\\le \\mathbf {U}(2,1)(\\mathbb {F}_{q^2}/\\mathbb {F}_q)$ .", "We denote by $I(1)$ the unique pro-$p$ -Sylow subgroup of $I$ , which is the preimage of $\\mathbb {U}$ .", "We define the following distinguished elements of $G$ : Table: NO_CAPTION" ], [ "Weyl Groups", "The maximal torus $T$ of $G$ consists of all elements of the form $\\begin{pmatrix}a & 0 & 0 \\\\ 0 & \\delta & 0 \\\\ 0 & 0 & \\overline{a}^{-1}\\end{pmatrix},$ with $a\\in E^\\times , \\delta \\in \\mathbf {U}(1)(E/F)$ .", "Note that $T$ is not split over $F$ .", "Let $T_0 := T\\cap K = T\\cap K^{\\prime },\\qquad T_1:= T\\cap K_1 = T\\cap K^{\\prime }_1,$ $H := T_0/T_1 \\cong I/I(1) \\cong \\mathbb {F}_{q^2}^\\times \\times \\mathbf {U}(1)(\\mathbb {F}_{q^2}/\\mathbb {F}_q).$ We will identify the characters of $H$ and those of $I/I(1)$ .", "We will also identify $\\mathbb {F}_{q^2}$ with the image of the Teichmüller lifting map $[\\ \\cdot \\ ]:\\mathbb {F}_{q^2}\\rightarrow \\mathfrak {o}_E$ when convenient.", "Let $N$ denote the normalizer of $T$ in $G$ .", "Then the affine Weyl group $W_{\\textrm {aff}}$ is defined as $N/T_0$ , and the finite Weyl group $W$ is defined as $N/T$ .", "The group $W_{\\textrm {aff}}$ is a Coxeter group, generated by the classes of the two reflections $s$ and $s^{\\prime }$ .", "We have a decomposition $G = INI$ , where two cosets $InI$ and $In^{\\prime }I$ are equal if and only if $n$ and $n^{\\prime }$ have the same image in $W_{\\textrm {aff}}$ .", "This yields the Bruhat decomposition for the BN pair $(I,N)$ : $G = \\bigsqcup _{w\\in W_{\\textrm {aff}}}IwI;$ here we engage in the standard abuse of notation, letting $IwI$ denote $I\\dot{w}I$ for any preimage $\\dot{w}$ of $w$ in $N$ .", "We will take as our double coset representatives the elements $\\alpha ^n, n_s\\alpha ^n$ , for $n\\in \\mathbb {Z}$ .", "We let $\\ell $ denote the length of an element of $W_{\\textrm {aff}}$ , defined by $q^{\\ell (w)} = [IwI:I]$ (cf.", "Section 3.3.1 in [32]).", "In particular, we have $\\ell (n_s) = 3,\\ \\ell (n_{s^{\\prime }}) = 1$ .", "Let $U$ and $U^-$ denote the upper and lower unipotent elements of $G$ , respectively, and define $u(x,y):=\\begin{pmatrix}1 & x & y \\\\ 0 & 1 & -\\overline{x} \\\\ 0 & 0 & 1\\end{pmatrix},\\qquad u^-(x,y):=\\begin{pmatrix}1 & 0 & 0\\\\ x & 1 & 0\\\\ y & -\\overline{x} & 1\\end{pmatrix},$ where $x,y\\in E$ are such that $x\\overline{x} + y + \\overline{y} = 0$ .", "We have $u(x,y)^{-1} = u(-x,\\overline{y}), u^-(x,y)^{-1} = u^-(-x,\\overline{y})$ ." ], [ "Preliminaries", "Let $C$ denote an algebraically closed field.", "We shall be interested in the category $\\mathcal {REP}_C(G)$ of smooth representations of $G$ over $C$ .", "We briefly recall some preliminary results.", "Let $J$ be a closed subgroup of $G$ , and let $(\\sigma ,V)$ be a smooth $C$ -representation of $J$ (meaning that stabilizers are open).", "We denote by $\\textnormal {ind}_J^G(\\sigma )$ the space of functions $f:G\\rightarrow V$ such that $f(jg) = \\sigma (j)f(g)$ for $j\\in J, g\\in G$ , and such that the action of $G$ given by right translation is smooth (meaning that there exists some open subgroup $J^{\\prime }$ , depending on $f$ , such that $f(gj^{\\prime }) = f(g)$ for every $j^{\\prime }\\in J^{\\prime }, g\\in G$ ).", "We let $\\textnormal {c-ind}_J^G(\\sigma )$ denote the subspace of $\\textnormal {ind}_J^G(\\sigma )$ spanned by functions whose support in $J\\backslash G$ is compact.", "These functors are called induction and compact induction, respectively.", "We will mostly be concerned with the cases when $J$ is a parabolic subgroup of $G$ , or when $J$ is a compact open subgroup." ], [ "Pro-$p$ -Iwahori-Hecke Algebra", "Let $\\pi $ be a smooth $C$ -representation of $G$ .", "Frobenius Reciprocity for compact induction gives $\\pi ^{I(1)}\\cong \\textrm {Hom}_{I(1)}(1,\\pi |_{I(1)})\\cong \\textrm {Hom}_G(\\textrm {c-ind}_{I(1)}^G(1),\\pi ),$ where 1 denotes the trivial character of $I(1)$ .", "The pro-$p$ -Iwahori-Hecke algebra $\\mathcal {H}_C(G,I(1)) = \\textnormal {End}_G(\\textnormal {c-ind}_{I(1)}^G(1))$ is the algebra of $G$ -equivariant endomorphisms of the universal module $\\textnormal {c-ind}_{I(1)}^G(1)$ .", "This algebra has a natural right action on $\\textrm {Hom}_G(\\textrm {c-ind}_{I(1)}^G(1),\\pi )$ by pre-composition, which induces a right action on $\\pi ^{I(1)}$ .", "In this way, we obtain the functor of $I(1)$ -invariants, $\\pi \\mapsto \\pi ^{I(1)}$ , from the category of smooth $C$ -representations of $G$ to the category of right $\\mathcal {H}_C(G,I(1))$ -modules.", "By adjunction, we have a natural identification $\\mathcal {H}_C(G,I(1)) = \\textnormal {End}_G(\\textnormal {c-ind}_{I(1)}^G(1))\\cong \\textnormal {Hom}_{I(1)}(1,\\textnormal {c-ind}_{I(1)}^G(1)|_{I(1)})\\cong \\textnormal {c-ind}_{I(1)}^G(1)^{I(1)},$ so we may view endomorphisms of $\\textnormal {c-ind}_{I(1)}^G(1)$ as compactly supported functions on $G$ which are $I(1)$ -biinvariant.", "This leads to the following definition.", "Definition 3.1 Let $g\\in G$ .", "We let $g\\in \\mathcal {H}_C(G,I(1))$ denote the endomorphism of $\\textnormal {c-ind}_{I(1)}^G(1)$ corresponding by adjunction to the characteristic function of ${I(1)gI(1)}$ ; in particular, $g$ maps the characteristic function of $I(1)$ to the characteristic function of $I(1)gI(1)$ .", "From this definition it is clear that $g = {g^{\\prime }}$ if and only if $I(1)gI(1) = I(1)g^{\\prime }I(1)$ ; moreover, since $W_{\\textrm {aff}} = N/T_0$ is a set of representatives for the double coset space $I\\backslash G/I$ , the group $N/T_1$ gives a set of representatives for $I(1)\\backslash G/I(1)$ .", "We therefore only consider the operators $n$ , where $n$ is a representative of a coset in $N/T_1$ .", "These operators give a basis for $\\mathcal {H}_C(G,I(1))$ as a vector space over $C$ .", "Lemma 3.2 Let $n\\in N$ .", "If $\\pi $ is a smooth $C$ -representation of $G$ and $v\\in \\pi ^{I(1)}$ , then $v\\cdot n = \\sum _{u \\in I(1)\\backslash I(1)nI(1)}u^{-1}.v = \\sum _{u\\in I(1)/I(1)\\cap n^{-1}I(1)n} un^{-1}.v.$ See [3], Proposition 6.", "Lemma 3.3 Let $n,n^{\\prime }\\in N$ , and assume that $n$ normalizes $I(1)$ .", "We then have $n{n^{\\prime }} = {nn^{\\prime }}, {n^{\\prime }}n = {n^{\\prime }n}$ .", "This follows readily from the definition of $n$ ." ], [ "Decomposition of the pro-$p$ -Iwahori-Hecke Algebra", "Let $\\widehat{H}$ denote the group of all $C^\\times $ -valued characters of $H = T_0/T_1$ , and let $\\chi :H\\rightarrow C^\\times $ be an element of $\\widehat{H}$ .", "We define $\\zeta :\\mathbb {F}_{q^2}^\\times \\rightarrow C^\\times $ and $\\eta :\\mathbf {U}(1)(\\mathbb {F}_{q^2}/\\mathbb {F}_q)\\rightarrow C^\\times $ by $\\zeta (a) = \\chi \\begin{pmatrix}a & 0 & 0 \\\\ 0 & \\overline{a}a^{-1} & 0 \\\\ 0 & 0 & \\overline{a}^{-1}\\end{pmatrix},\\qquad \\eta (\\delta ) = \\chi \\begin{pmatrix}1 & 0 & 0 \\\\ 0 & \\delta & 0 \\\\ 0 & 0 & 1\\end{pmatrix},$ where $a\\in \\mathbb {F}_{q^2}^\\times , \\delta \\in \\mathbf {U}(1)(\\mathbb {F}_{q^2}/\\mathbb {F}_q)$ .", "We stress that the characters $\\zeta $ and $\\eta $ depend on $\\chi $, though we will supress this dependence from our notation, and write $\\chi = \\zeta \\otimes \\eta $ when convenient.", "The finite Weyl group $W$ acts on the characters $\\chi $ by conjugation; we denote by $\\chi ^s$ the character $\\chi ^s: h\\mapsto \\chi (n^{-1}hn),$ where $h\\in H$ and $n\\in N\\setminus T$ .", "Definition 3.4 Let $\\chi :H\\rightarrow C^\\times $ be a character.", "We say $\\chi $ is of trivial Iwahori type if $\\chi $ factors through the determinant, $\\chi $ is hybrid if $\\chi ^s = \\chi $ , but $\\chi $ does not factor through the determinant, and $\\chi $ is regular if $\\chi ^s \\ne \\chi $ .", "Note that $\\chi = \\zeta \\otimes \\eta $ factors through the determinant if and only if $\\zeta $ is trivial, and $\\chi ^s = \\chi $ if and only if $\\zeta ^{q+1}$ is trivial.", "For a character $\\chi $ , we let $\\gamma _\\chi $ denote the representation of $H$ defined by $\\gamma _\\chi = \\chi $ if $\\chi ^s = \\chi $ , and $\\gamma _\\chi = \\chi \\oplus \\chi ^s$ otherwise.", "From this point onwards, we make the following technical assumption: Assumption 3.5 The integers $\\textnormal {char}(C)$ and $|H|$ are relatively prime.", "With this hypothesis, we will now decompose $\\mathcal {H}_C(G,I(1))$ into blocks indexed by $W$ -orbits of $C$ -characters of $H$ .", "For $h\\in H$ , we define $h\\in \\mathcal {H}_{C}(G,I(1))$ to be the operator ${t_0}$ , for any preimage $t_0$ of $h$ in $T_0$ .", "Definition 3.6 For a $C$ -character $\\chi $ of $H$ , we define $e_\\chi = |H|^{-1}\\sum _{h\\in H}\\chi (h)h,$ $e_{\\gamma _\\chi } = {\\left\\lbrace \\begin{array}{ll}e_\\chi & \\textnormal {if}~\\chi ^s = \\chi ,\\\\ e_\\chi + e_{\\chi ^s} & \\textnormal {if}~\\chi ^s\\ne \\chi .\\end{array}\\right.", "}$ The operators $e_\\chi $ have the following properties: $e_\\chi e_\\chi = e_\\chi $ , $e_\\chi e_{\\chi ^{\\prime }} = 0$ for $\\chi \\ne \\chi ^{\\prime }$ , $\\textnormal {id}_{\\textnormal {c-ind}_{I(1)}^G(1)} = \\sum _{\\chi \\in \\widehat{H}}e_\\chi $ .", "These follow readily from the orthogonality relations of characters.", "Applying these relations to $\\pi ^{I(1)}$ gives the following Lemma.", "Lemma 3.7 Let $\\pi $ be a smooth $C$ -representation of $G$ .", "Then $(\\pi ^{I(1)})\\cdot e_\\chi = \\pi ^{I,\\chi }$ , and $\\pi ^{I(1)}\\cong \\bigoplus _{\\chi \\in \\widehat{H}} (\\pi ^{I(1)})\\cdot e_\\chi = \\bigoplus _{\\chi \\in \\widehat{H}} \\pi ^{I,\\chi }$ .", "Here $\\pi ^{I,\\chi } = \\lbrace v\\in \\pi : i.v = \\chi (i)v\\ \\textrm {for every}\\ i\\in I\\rbrace $ is the $\\chi $ -isotypic subspace of $\\pi $ .", "Since $I(1)$ is normal in $I$ and $I/I(1)\\cong H$ is abelian and of order prime to $\\textnormal {char}(C)$ , the action of $I$ on $\\pi ^{I(1)}$ is semisimple and decomposes as a sum of characters.", "Since (lifts of) elements of $H$ normalize $I(1)$ , Lemma REF implies that $(\\pi ^{I(1)})\\cdot e_\\chi = \\pi ^{I,\\chi }$ .", "The orthogonality properties above imply the direct sum decomposition.", "We now use the idempotents $e_{\\gamma _\\chi }$ to decompose the algebra $\\mathcal {H}_C(G,I(1))$ .", "Denote by $\\mathcal {H}_C(G,\\gamma _\\chi )$ the algebra $\\textrm {End}_G(\\textrm {c-ind}_I^G(\\gamma _\\chi ))$ .", "Using this notation, we obtain the following Proposition: Proposition 3.8 There is an isomorphism of $C$ -algebras $\\mathcal {H}_C(G,I(1)) \\cong \\bigoplus _{\\gamma _\\chi }\\mathcal {H}_C(G,\\gamma _\\chi )\\cong \\bigoplus _{\\gamma _\\chi }\\mathcal {H}_C(G,I(1))e_{\\gamma _\\chi },$ the sums taken over all $W$ -orbits of $C$ -characters of $H$ .", "Assumption REF guarantees that the regular representation of $I/I(1)$ is semisimple.", "Using this fact, the proof is nearly identical to that in [33], Proposition 3.1.", "Proposition 3.9 The operators ${n_s}, {n_{s^{\\prime }}}$ and $e_\\chi $ for all $C$ -characters $\\chi $ generate $\\mathcal {H}_C(G,I(1))$ as an algebra.", "We first claim that $e_\\chi (\\textrm {c-ind}_{I(1)}^G(1)) \\cong \\textrm {c-ind}_I^G(\\chi )$ .", "Indeed, since the characteristic function of $I(1)$ generates $\\textrm {c-ind}_{I(1)}^G(1)$ as a $G$ -representation, its image under $e_\\chi $ will generate $e_\\chi (\\textrm {c-ind}_{I(1)}^G(1))$ .", "Denote this image by $\\varphi _\\chi $ .", "By definition of the operators $e_\\chi $ , we have $\\textrm {supp}(\\varphi _\\chi ) = I$ and $\\varphi _\\chi (h) = |H|^{-1}\\chi (h)$ for $h\\in I$ (via the isomorphism $H\\cong I/I(1)$ ).", "The action of $G$ on $\\varphi _\\chi $ shows that the representation it generates is canonically isomorphic to $\\textrm {c-ind}_I^G(\\chi )$ .", "Let $\\mathcal {M}$ be the subalgebra of $\\mathcal {H}_C(G,I(1))$ generated by ${n_s}, {n_{s^{\\prime }}}$ and the operators $e_\\chi $ for every $\\chi \\in \\widehat{H}$ .", "Using the decomposition of Proposition REF , we have that $\\mathcal {M}e_{\\gamma _\\chi }$ is a subalgebra of $\\mathcal {H}_C(G,\\gamma _\\chi )$ .", "Assume first that $\\chi ^s = \\chi $ .", "The claim above shows that ${n_s}e_\\chi $ and ${n_{s^{\\prime }}}e_\\chi $ are elements of $\\mathcal {H}_C(G,\\chi )$ , and Propositions REF and REF imply that these elements generate $\\mathcal {H}_C(G,\\chi )$ .", "We therefore have $\\mathcal {M}e_{\\gamma _\\chi } \\cong \\mathcal {H}_C(G,\\chi )$ .", "Assume now that $\\chi ^s\\ne \\chi $ .", "The claim above shows that ${n_s}e_{\\gamma _\\chi }$ and ${n_{s^{\\prime }}}e_{\\gamma _\\chi }$ are elements of $\\mathcal {H}_C(G,\\gamma _\\chi )$ .", "The algebra $\\mathcal {M}e_{\\gamma _\\chi }$ also contains the elements $e_\\chi $ and $e_{\\chi ^s}$ , which implies that each of the elements ${n_s}e_\\chi , {n_{s^{\\prime }}}e_\\chi , {n_s}e_{\\chi ^s}, {n_{s^{\\prime }}}e_{\\chi ^s}$ are contained in $\\mathcal {M}e_{\\gamma _\\chi }$ .", "Propositions REF and REF show that these elements generate $\\mathcal {H}_C(G,\\gamma _\\chi )$ , so that $\\mathcal {M}e_{\\gamma _\\chi } \\cong \\mathcal {H}_C(G,\\gamma _\\chi )$ .", "Combining these results with the decomposition of Proposition REF shows that $\\mathcal {M} = \\mathcal {H}_C(G,I(1))$ .", "Theorem 3.10 The algebra $\\mathcal {H}_C(G,I(1))$ is a noncommutative algebra, generated by the elements ${n_s}, {n_{s^{\\prime }}}$ and $e_{\\chi }$ for every $\\chi \\in \\widehat{H}$ , subject to the following relations: ${n_s}e_\\chi = e_{\\chi ^s}{n_s},\\qquad {n_{s^{\\prime }}}e_\\chi = e_{\\chi ^s}{n_{s^{\\prime }}},$ $e_{\\chi }e_{\\chi ^{\\prime }} = {\\left\\lbrace \\begin{array}{ll}e_\\chi & \\textnormal {if}~ \\chi ^{\\prime } = \\chi ,\\\\ 0 & \\textnormal {if}~ \\chi ^{\\prime }\\ne \\chi .\\end{array}\\right.", "}$ If $\\chi $ factorizes through the determinant, then ${n_s}^2e_\\chi = (q^3 - 1){n_s}e_\\chi + q^3e_\\chi ,\\qquad {n_{s^{\\prime }}}^2e_\\chi = (q - 1){n_{s^{\\prime }}}e_\\chi + qe_\\chi .$ If $\\chi ^s = \\chi $ , but $\\chi $ does not factorize through the determinant, then ${n_s}^2e_\\chi = (q - q^2){n_s}e_\\chi + q^3e_\\chi ,\\qquad {n_{s^{\\prime }}}^2e_\\chi = (q - 1){n_{s^{\\prime }}}e_\\chi + qe_\\chi .$ If $\\chi ^s\\ne \\chi = \\zeta \\otimes \\eta $ , then ${n_s}^2e_\\chi = \\zeta (-1)q^3e_\\chi ,\\qquad {n_{s^{\\prime }}}^2e_\\chi = \\zeta (-1)qe_\\chi ,$ ${n_s}^2e_{\\gamma _\\chi } = \\zeta (-1)q^3e_{\\gamma _\\chi },\\qquad {n_{s^{\\prime }}}^2e_{\\gamma _\\chi } = \\zeta (-1)qe_{\\gamma _\\chi }.$ The center $\\mathcal {Z}$ of $\\mathcal {H}_C(G,I(1))$ is generated by the idempotents $e_{\\gamma _\\chi }$ , and the elements ${\\left\\lbrace \\begin{array}{ll} ({n_s}({n_{s^{\\prime }}} - (q - 1)) + {n_{s^{\\prime }}}({n_s} - (q^3 - 1)) + 1)e_\\chi & \\textit {for}~ \\chi = \\eta \\circ \\det ,\\\\({n_s}({n_{s^{\\prime }}} - (q - 1)) + {n_{s^{\\prime }}}({n_s} - (q - q^2)))e_\\chi & \\textit {for}~ \\chi ^s = \\chi , \\textit {but}~ \\chi \\ne \\eta \\circ \\det ,\\\\\\zeta (-1)({n_{s^{\\prime }}}{n_s}e_\\chi + {n_s}{n_{s^{\\prime }}}e_{\\chi ^s})~ \\textit {and} & \\\\\\qquad \\zeta (-1)({n_{s^{\\prime }}}{n_s}e_{\\chi ^s} + {n_s}{n_{s^{\\prime }}}e_\\chi ) & \\textit {for}~ \\chi ^s\\ne \\chi = \\zeta \\otimes \\eta .", "\\end{array}\\right.", "}$ Part (i) follows directly from the definitions and Lemma REF .", "To prove part (ii), we may either appeal to Propositions REF , REF , and REF below, or note that these results are a special case of [10], Proposition 3.18.", "One simply needs to use the fact that ${n_s}^2e_\\chi $ (resp.", "${n_{s^{\\prime }}}^2e_\\chi $ ) maps the characteristic function of $I(1)$ to a function with support contained in $K$ (resp.", "$K^{\\prime }$ ) and reduce the computations to those in the respective finite groups, as in [26], Lemma 2.11.", "Part (iii) follows from Propositions REF , REF , and Corollary REF .", "Remark Let $h_s:\\mathbb {F}_{q^2}^\\times \\rightarrow H$ be the homomorphism defined by $h_s(y) = \\begin{pmatrix}y & 0 & 0\\\\ 0 & \\overline{y}y^{-1} & 0 \\\\ 0 & 0 & \\overline{y}^{-1}\\end{pmatrix},$ and set $\\tau _s := (q + 1)\\sum _{y\\in \\mathbb {F}_{q^2}^\\times }{h_s(y)} - q\\sum _{y\\in \\mathbb {F}_q^\\times }{h_s(y)},\\qquad \\tau _{s^{\\prime }} := \\sum _{y\\in \\mathbb {F}_q^\\times } {h_s(y)}.$ These elements satisfy the relation $\\tau _s\\tau _{s^{\\prime }} = \\tau _{s^{\\prime }}\\tau _s = (q-1)\\tau _{s}$ .", "Using Fourier inversion and the theorem above, the quadratic relations take the form ${n_s}^2 & = & {n_s}\\tau _s + q^3{h_s(-1)}\\\\{n_{s^{\\prime }}}^2 & = & {n_{s^{\\prime }}}\\tau _{s^{\\prime }} + q{h_s(-1)}.$ Moreover, we see that the center $\\mathcal {Z}$ of $\\mathcal {H}_C(G,I(1))$ is generated by the central idempotents $e_{\\gamma _\\chi }$ and the elements ${n_{s^{\\prime }}}{n_s}\\vartheta _1 + {n_s}{n_{s^{\\prime }}}\\vartheta _2 - {n_s}\\tau _{s^{\\prime }} - {n_{s^{\\prime }}}\\tau _s + (q-1)\\tau _s,$ ${n_{s^{\\prime }}}{n_s}\\vartheta _2 + {n_s}{n_{s^{\\prime }}}\\vartheta _1 - {n_s}\\tau _{s^{\\prime }} - {n_{s^{\\prime }}}\\tau _s + (q-1)\\tau _s.$ Here $\\vartheta _1 = \\sum _{\\chi ^s = \\chi }e_\\chi + 2\\sum _{\\genfrac{}{}{0.0pt}{}{\\chi ^s\\ne \\chi }{\\chi \\in \\lbrace \\chi ,\\chi ^s\\rbrace }}e_\\chi ,\\quad \\vartheta _2 = \\sum _{\\chi ^s = \\chi }e_\\chi + 2\\sum _{\\genfrac{}{}{0.0pt}{}{\\chi ^s\\ne \\chi }{\\chi ^s\\in \\lbrace \\chi ,\\chi ^s\\rbrace }}e_{\\chi ^s},$ where the sums are taken over $W$ -orbits of $C$ -characters, such that $\\vartheta _1 + \\vartheta _2 = 2\\cdot \\textnormal {id}_{\\textnormal {c-ind}_{I(1)}^G(1)}$ .", "In light of Theorem REF , we make the following definition: Definition 3.11 Assume $\\textnormal {char}(C) = p$ , and let $M$ be a nonzero simple right $\\mathcal {H}_C(G,I(1))$ -module which admits a central character.", "We say $M$ is supersingular if every generator of the center $\\mathcal {Z}$ (as given in Theorem REF ) which is not a central idempotent $e_{\\gamma _\\chi }$ , acts by 0.", "In the subsequent sections, we describe the structures of the Hecke algebras $\\mathcal {H}_C(G,\\gamma _\\chi )$ .", "From the descriptions of these blocks, we obtain Proposition REF and Theorem REF , and identify the supersingular modules of $\\mathcal {H}_C(G,I(1))$ when $\\textnormal {char}(C) = p$ ." ], [ "The Trivial Case", "We first assume that $\\chi $ is “trivial,” meaning $\\chi $ factors through the determinant and $\\chi = \\eta \\circ \\det $ , for $\\eta $ a character of $\\mathbf {U}(1)(\\mathbb {F}_{q^2}/\\mathbb {F}_q)$ .", "In this case, we have $\\mathcal {H}_C(G,\\chi )\\cong \\mathcal {H}_C(G,1) = \\textnormal {End}_G(\\textnormal {c-ind}_I^G(1))$ ; this equivalence is induced by the isomorphism $\\textnormal {c-ind}_{I}^G(\\chi )\\cong \\eta \\circ \\det \\otimes \\textnormal {c-ind}_{I}^G(1)$ and Frobenius Reciprocity.", "Let $\\mathbf {1}_I\\in \\textrm {c-ind}_{I}^G(\\chi )$ denote the function with support in $I$ , taking the value 1 at the identity.", "We let $\\mathcal {T}_{n_s}$ (resp.", "$\\mathcal {T}_{n_{s^{\\prime }}}$ ) denote the endomorphism of $\\textrm {c-ind}_{I}^G(\\chi )$ sending $\\mathbf {1}_I$ to the function with support $In_sI$ (resp.", "$In_{s^{\\prime }}I$ ), taking the value 1 at $n_s$ (resp.", "$n_{s^{\\prime }}$ ), on which $I$ acts by $\\chi $ .", "In the notation of the previous subsection, we have $\\mathcal {T}_{n_s} = {n_s}e_\\chi ,\\qquad \\mathcal {T}_{n_{s^{\\prime }}} = {n_{s^{\\prime }}}e_\\chi .$ We now arrive at the following result on the structure of $\\mathcal {H}_C(G,\\chi )$ : Proposition 3.12 The algebra $\\mathcal {H}_C(G,\\chi )$ is a noncommutative algebra, generated by $\\mathcal {T}_{n_s}$ and $\\mathcal {T}_{n_{s^{\\prime }}}$ , subject to the relations $(\\mathcal {T}_{n_s} + 1)(\\mathcal {T}_{n_s} - q^3) & = & 0\\\\(\\mathcal {T}_{n_{s^{\\prime }}} + 1)(\\mathcal {T}_{n_{s^{\\prime }}} - q) & = & 0.$ The center $\\mathcal {Z}_\\chi $ is generated by $Z = \\mathcal {T}_{n_s}(\\mathcal {T}_{n_{s^{\\prime }}} - (q - 1)) + \\mathcal {T}_{n_{s^{\\prime }}}(\\mathcal {T}_{n_s} - (q^3 - 1)) + 1$ .", "We have an isomorphism of algebras $\\mathcal {H}_C(G,\\chi )\\cong C\\langle X,Y\\rangle /(X^2 +(1-q^3)X - q^3, Y^2 + (1-q)Y - q),$ sending $\\mathcal {T}_{n_s}$ to $X$ and $\\mathcal {T}_{n_{s^{\\prime }}}$ to $Y$ .", "Here $C\\langle X,Y \\rangle $ denotes the noncommutative polynomial algebra in two variables over $C$ .", "Remark Note that using the length function on $W_{\\textrm {aff}}$ , the Hecke relations take the simple form $(\\mathcal {T}_n + 1)(\\mathcal {T}_n - q^{\\ell (n)}) = 0$ , where $n = n_s$ or $n_{s^{\\prime }}$ .", "The verification of the Proposition is included in the proof of Proposition REF below.", "Given this result, we can quickly classify the finite-dimensional simple right $\\mathcal {H}_C(G,\\chi )$ -modules.", "Definition 3.13 Let $(\\theta ,\\theta ^{\\prime })\\in \\lbrace -1,q^3\\rbrace \\times \\lbrace -1,q\\rbrace $ .", "We define the characters $\\mu _{\\theta ,\\theta ^{\\prime }}:\\mathcal {H}_C(G,\\chi )\\rightarrow C$ by $\\mathcal {T}_{n_s} \\mapsto \\theta ,\\quad \\mathcal {T}_{n_{s^{\\prime }}} \\mapsto \\theta ^{\\prime }.$ The central element $Z$ maps to $\\theta (\\theta ^{\\prime } - q + 1) + \\theta ^{\\prime }(\\theta - q^3 + 1) + 1$ .", "Let $\\langle v_1, v_2\\rangle _C$ be a two-dimensional vector space over $C$ , and let $\\lambda \\in C$ .", "We define $M(\\lambda )$ to be the following right $\\mathcal {H}_C(G,\\chi )$ -module: Table: NO_CAPTION The central element $Z$ acts by $\\lambda $ .", "One may check directly that the action of $\\mathcal {H}_C(G,\\chi )$ on $M(\\lambda )$ is well-defined.", "This fact will also be made clear in the proof of Theorem REF .", "Proposition 3.14 Assume $q^3 + 1\\ne 0$ in $C$ .", "Then the module $M(\\lambda )$ is reducible if and only if $\\lambda = q^3 + q + 1$ or $\\lambda = -q^4$ .", "In these cases, we have the following exact sequences: $0\\rightarrow \\mu _{q^3,q}\\rightarrow M(q^3 + q + 1)\\rightarrow \\mu _{-1,-1}\\rightarrow 0$ $0\\rightarrow \\mu _{q^3,-1}\\rightarrow M(-q^4)\\rightarrow \\mu _{-1.q}\\rightarrow 0$ The sequences are not split.", "Assume that $M(\\lambda )$ is reducible, so that we have some character $\\mu \\subset M(\\lambda )$ .", "This means exactly that there is some vector $v\\in M(\\lambda )$ which is a common eigenvector for $\\mathcal {T}_{n_s}$ and $\\mathcal {T}_{n_{s^{\\prime }}}$ .", "The eigenvectors for $\\mathcal {T}_{n_s}$ are $(\\lambda - q)v_1 + (q^3+1)v_2$ and $v_1$ , with eigenvalues $q^3$ and $-1$ , respectively; the eigenvectors for $\\mathcal {T}_{n_{s^{\\prime }}}$ are $v_1 + v_2$ and $-qv_1 + v_2$ , with eigenvalues $q$ and $-1$ , respectively.", "We see that the only possibility for a common eigenvector is if $(\\lambda - q)v_1 + (q^3+1)v_2$ is a scalar multiple of $v_1 + v_2$ or of $-qv_1 + v_2$ .", "Assume the former.", "We then have $\\lambda - q = q^3 + 1$ , implying $\\lambda = q^3 + q + 1$ .", "Thus, $\\langle v_1 + v_2\\rangle _C\\cong \\mu _{q^3,q}\\subset M(q^3 + q + 1)\\quad \\textnormal {and}\\quad M(q^3 + q + 1)/\\mu _{q^3,q}\\cong \\mu _{-1,-1}.$ If the surjection split, then there would exist a $-1$ -eigenvector in $M(q^3 + q + 1)$ for both $\\mathcal {T}_{n_s}$ and $\\mathcal {T}_{n_{s^{\\prime }}}$ , which clearly cannot happen.", "Assume now that $(\\lambda - q)v_1 + (q^3+1)v_2$ is a scalar multiple of $-qv_1 + v_2$ .", "We then have $\\lambda - q = -q^4 - q$ , implying that $\\lambda = -q^4$ .", "Thus, we have $\\langle -qv_1 + v_2\\rangle _C\\cong \\mu _{q^3,-1}\\subset M(-q^4)\\quad \\textnormal {and}\\quad M(-q^4)/\\mu _{q^3,-1}\\cong \\mu _{-1,q}.$ By the same reasoning as before, the surjection cannot split.", "Proposition 3.15 Assume $q^3 + 1 = 0$ in $C$ .", "Then the module $M(\\lambda )$ is reducible if and only if $\\lambda = q$ .", "In this case the module decomposes as $M(q) \\cong \\mu _{-1,q}\\oplus \\mu _{-1,-1}$ .", "Assume that $M(\\lambda )$ is reducible, so that it contains either $\\mu _{-1,-1}$ or $\\mu _{-1,q}$ .", "In either case, the central element $\\mathcal {T}_{n_s}(\\mathcal {T}_{n_{s^{\\prime }}} - (q - 1)) + \\mathcal {T}_{n_{s^{\\prime }}}(\\mathcal {T}_{n_s} + 2) + 1$ acts by $q$ , so we must have $\\lambda = q$ .", "The action of $\\mathcal {T}_{n_s}$ and $\\mathcal {T}_{n_{s^{\\prime }}}$ on $M(q)$ shows that $\\langle v_1 + v_1\\rangle _C \\cong \\mu _{-1,q}\\quad \\textnormal {and}\\quad \\langle -qv_1 + v_2\\rangle _C \\cong \\mu _{-1,-1},$ so that $M(q) = \\mu _{-1,q}\\oplus \\mu _{-1,-1}.$ We now imitate the proof of Theorem 1.2 in [33] to classify simple right $\\mathcal {H}_C(G,\\chi )$ -modules.", "Theorem 3.16 Every finite-dimensional simple right $\\mathcal {H}_C(G,\\chi )$ -module is either a character $\\mu _{\\theta ,\\theta ^{\\prime }}, (\\theta ,\\theta ^{\\prime })\\in \\lbrace -1,q^3\\rbrace \\times \\lbrace -1,q\\rbrace $ , or a module of the form $M(\\lambda ), \\lambda \\ne q^3 + q + 1, -q^4$ .", "Assume $M$ is a nonzero simple right module which is not a character, and assume that $Z$ acts by $\\lambda $ .", "Consider the space $\\ker (\\mathcal {T}_{n_s} + 1)$ .", "We claim that this is a nontrivial proper subspace of $M$ .", "Indeed, if $\\ker (\\mathcal {T}_{n_s} + 1) = \\lbrace 0\\rbrace $ or $M$ , the element $\\mathcal {T}_{n_s}$ would act by a scalar, and any nonzero eigenvector for $\\mathcal {T}_{n_{s^{\\prime }}}$ would generate a one-dimensional submodule.", "This gives a contradiction, since $M$ was assumed simple of dimension greater than 1.", "The element $\\mathcal {T}_{n_{s^{\\prime }}}(\\mathcal {T}_{n_s} - q^3)$ maps $\\ker (\\mathcal {T}_{n_s} + 1)$ into itself, and therefore has an eigenvector $v$ in $\\ker (\\mathcal {T}_{n_s} + 1)$ .", "We have $v\\cdot \\mathcal {T}_{n_{s^{\\prime }}}(\\mathcal {T}_{n_s} - q^3) & = & v\\cdot (Z - (\\mathcal {T}_{n_s} + 1)\\mathcal {T}_{n_{s^{\\prime }}} + (q - 1)\\mathcal {T}_{n_s} - 1)\\\\& = & \\lambda v + (q - 1)v\\cdot \\mathcal {T}_{n_s} - v\\\\& = & (\\lambda - q)v.$ Consider now the subspace $V := \\langle v\\rangle _C + \\langle v\\cdot \\mathcal {T}_{n_{s^{\\prime }}}\\rangle _C$ .", "The quadratic relations and the computation above show that $V$ is stable under $\\mathcal {H}_C(G,\\chi )$ , and therefore must be all of $M$ by simplicity.", "Moreover, since $M$ was assumed to be of dimension greater than one, we have $v\\cdot \\mathcal {T}_{n_{s^{\\prime }}}\\ne 0$ , and the sum $\\langle v\\rangle _C + \\langle v\\cdot \\mathcal {T}_{n_s}\\rangle _C$ is direct.", "Writing out the actions of $\\mathcal {T}_{n_s}$ and $\\mathcal {T}_{n_{s^{\\prime }}}$ on the basis $\\lbrace v, v\\cdot \\mathcal {T}_{n_{s^{\\prime }}}\\rbrace $ shows that $M\\cong M(\\lambda )$ .", "We again use simplicity of $M$ to deduce that $\\lambda \\ne q^3 + q + 1, -q^4$ ." ], [ "The Hybrid Case", "We now assume that $\\chi ^s = \\chi = \\zeta \\otimes \\eta $ , but that $\\chi $ does not factor through the determinant.", "This condition implies that the character $\\zeta $ is nontrivial.", "In addition, we have $\\zeta (a) = \\zeta (\\overline{a}^{-1})$ ; since the map $a\\mapsto a^{q+1}$ maps $\\mathbb {F}_{q^2}^\\times $ onto $\\mathbb {F}_q^\\times $ , this implies $\\zeta $ is trivial on $\\mathbb {F}_q^\\times $ .", "As before, we let $\\mathbf {1}_I\\in \\textrm {c-ind}_I^G(\\chi )$ denote the function with support in $I$ , taking the value 1 at the identity.", "We let $\\mathcal {T}_{n_s}$ (resp.", "$\\mathcal {T}_{n_{s^{\\prime }}}$ ) denote the endomorphism of $\\textrm {c-ind}_{I}^G(\\chi )$ sending $\\mathbf {1}_I$ to the function with support $In_sI$ (resp.", "$In_{s^{\\prime }}I$ ), taking the value 1 at $n_s$ (resp.", "$n_{s^{\\prime }}$ ), on which $I$ acts by $\\chi $ .", "In the notation of Section REF , we have $\\mathcal {T}_{n_s} = {n_s}e_\\chi ,\\qquad \\mathcal {T}_{n_{s^{\\prime }}} = {n_{s^{\\prime }}}e_\\chi .$ Proposition 3.17 The algebra $\\mathcal {H}_C(G,\\chi )$ is a noncommutative algebra, generated by $\\mathcal {T}_{n_s}$ and $\\mathcal {T}_{n_{s^{\\prime }}}$ , subject to the relations $(\\mathcal {T}_{n_s} + q^2)(\\mathcal {T}_{n_s} - q) & = & 0\\\\(\\mathcal {T}_{n_{s^{\\prime }}} + 1)(\\mathcal {T}_{n_{s^{\\prime }}} - q) & = & 0.$ The center $\\mathcal {Z}_\\chi $ is generated by $Z = \\mathcal {T}_{n_s}(\\mathcal {T}_{n_{s^{\\prime }}} - (q - 1)) + \\mathcal {T}_{n_{s^{\\prime }}}(\\mathcal {T}_{n_s} - (q - q^2))$ .", "We have an ismorphism of algebras $\\mathcal {H}_C(G,\\chi )\\cong C\\langle X,Y\\rangle /(X^2 + (q^2 - q)X - q^3,Y^2 + (1-q)Y - q),$ sending $\\mathcal {T}_{n_s}$ to $X$ and $\\mathcal {T}_{n_{s^{\\prime }}}$ to $Y$ .", "Here $C\\langle X,Y \\rangle $ denotes the noncommutative polynomial algebra in two variables over $C$ .", "We shall prove Propositions REF and REF simultaneously.", "We begin with only the assumption that $\\chi ^s = \\chi $ .", "By Frobenius Reciprocity, we may view elements of $\\mathcal {H}_C(G,\\chi )$ as functions $\\varphi :G\\rightarrow C$ satisfying $\\varphi (igi^{\\prime }) = \\chi (i)\\varphi (g)\\chi (i^{\\prime })$ for $g\\in G, i,i^{\\prime }\\in I$ .", "If $\\mathcal {T}_{\\varphi _1}, \\mathcal {T}_{\\varphi _2}$ are the endomorphisms associated to $\\varphi _1,\\varphi _2$ , respectively, then the composition product on $\\mathcal {H}_C(G,\\chi )$ gives $\\mathcal {T}_{\\varphi _1}\\mathcal {T}_{\\varphi _2} = \\mathcal {T}_{\\varphi _1*\\varphi _2},$ where $\\varphi _1*\\varphi _2(g) = \\sum _{h\\in G/I}\\varphi _1(h)\\varphi _2(h^{-1}g).$ Assume that $\\varphi $ has support in $I\\dot{w}I$ , where $\\dot{w}$ is some representative of $w\\in W_{\\textrm {aff}}$ .", "Let $w = s_1s_2\\cdots s_k$ be a reduced word expression for $w$ , where $s_i\\in \\lbrace s,s^{\\prime }\\rbrace $ , and let $\\varphi _{n_s}$ (resp.", "$\\varphi _{n_{s^{\\prime }}}$ ) be the function with support in $In_sI$ (resp.", "$In_{s^{\\prime }}I$ ) taking the value 1 at $n_s$ (resp.", "$n_{s^{\\prime }}$ ).", "We claim that $\\varphi $ is a scalar multiple of $\\varphi _{n_{s_1}}*\\varphi _{n_{s_2}}*\\ldots *\\varphi _{n_{s_k}}$ .", "Indeed, the definition of the convolution product shows that $\\textrm {supp}(\\varphi _1*\\varphi _2)\\subset \\textrm {supp}(\\varphi _1)\\textrm {supp}(\\varphi _2)$ .", "By induction, we have that $\\textrm {supp}(\\varphi _{n_{s_1}}*\\varphi _{n_{s_2}}*\\ldots *\\varphi _{n_{s_k}}) & \\subset & \\textrm {supp}(\\varphi _{n_{s_1}})\\textrm {supp}(\\varphi _{n_{s_2}})\\cdots \\textrm {supp}(\\varphi _{n_{s_k}})\\\\& = & In_{s_1}In_{s_2}I\\cdots In_{s_k}I\\\\& = & I\\dot{w}I,$ where the last equality follows from [2], Prop.", "6.36(4).", "An elementary inductive argument shows that $\\varphi _{n_{s_1}}*\\varphi _{n_{s_2}}*\\ldots *\\varphi _{n_{s_k}}\\ne 0$ , which implies that $\\mathcal {H}_C(G,\\chi )$ is generated as an algebra by $\\mathcal {T}_{n_s}$ and $\\mathcal {T}_{n_{s^{\\prime }}}$ .", "It now suffices to determine relations for the operators $\\mathcal {T}_{n_s}$ and $\\mathcal {T}_{n_{s^{\\prime }}}$ .", "We again identify these operators with $\\varphi _{n_s}$ and $\\varphi _{n_{s^{\\prime }}}$ , respectively.", "We will make use of the following decompositions: $In_sI & = & \\bigsqcup _{\\genfrac{}{}{0.0pt}{}{x,y\\in \\mathbb {F}_{q^2}}{x\\overline{x} + y + \\overline{y} = 0}}u(x,y)n_sI\\\\In_{s^{\\prime }}I & = & \\bigsqcup _{\\genfrac{}{}{0.0pt}{}{y\\in \\mathbb {F}_{q^2}}{y + \\overline{y} = 0}}u^-(0,\\varpi y)n_{s^{\\prime }}I$ Here (and henceforth) we use the notational convention that $x$ and $y$ are representatives in $\\mathfrak {o}_E$ for the set $\\mathbb {F}_{q^2}$ , satisfying $x\\overline{x} + y + \\overline{y} = 0$ .", "As before, we have $\\textrm {supp}(\\varphi _{n_s}*\\varphi _{n_s})\\subset In_sIn_sI = In_sI \\sqcup I$ , and therefore it suffices to evaluate this function at 1 and $n_s$ .", "We shall also make use of the following identity (for $y\\ne 0$ ): $u^-(x,y) = u\\left(-\\overline{x}\\ \\overline{y}^{-1},y^{-1}\\right)n_s\\textrm {diag}(y\\sqrt{\\epsilon }^{-1},-\\overline{y}y^{-1}, -\\overline{y}^{-1}\\sqrt{\\epsilon })u(-\\overline{x}y^{-1},y^{-1})$ Note that if $u^-$ is a nonidentity lower unipotent element, then $u^-\\in Un_sTU$ .", "We compute: $\\varphi _{n_s}*\\varphi _{n_s}(1) & = & \\sum _{\\genfrac{}{}{0.0pt}{}{x,y\\in \\mathbb {F}_{q^2}}{x\\overline{x} + y + \\overline{y} = 0}}\\varphi _{n_s}(u(x,y)n_s)\\varphi _{n_s}(n_s^{-1}u(x,y)^{-1})\\\\& = & q^3\\\\\\varphi _{n_s}*\\varphi _{n_s}(n_s) & = & \\sum _{\\genfrac{}{}{0.0pt}{}{x,y\\in \\mathbb {F}_{q^2}}{x\\overline{x} + y + \\overline{y} = 0}}\\varphi _{n_s}(u(x,y)n_s)\\varphi _{n_s}(n_s^{-1}u(x,y)^{-1}n_{s})\\\\& = & \\sum _{\\genfrac{}{}{0.0pt}{}{x,y\\in \\mathbb {F}_{q^2}}{x\\overline{x} + y + \\overline{y} = 0}}\\varphi _{n_s}(u^-(\\overline{x}\\sqrt{\\epsilon }, -\\overline{y}\\epsilon ))\\\\& = & \\sum _{\\genfrac{}{}{0.0pt}{}{x,y\\in \\mathbb {F}_{q^2}}{x\\overline{x} + y + \\overline{y} = 0, y\\ne 0}}\\varphi _{n_s}(n_s\\textrm {diag}(-\\overline{y}\\sqrt{\\epsilon }, -y\\overline{y}^{-1}, y^{-1}\\sqrt{\\epsilon }^{-1}))\\\\& = & \\sum _{\\genfrac{}{}{0.0pt}{}{x,y\\in \\mathbb {F}_{q^2}}{x\\overline{x} + y + \\overline{y} = 0, y\\ne 0}}\\chi (\\textrm {diag}(-\\overline{y}\\sqrt{\\epsilon }, -y\\overline{y}^{-1}, y^{-1}\\sqrt{\\epsilon }^{-1})).$ If $\\chi $ factors through the determinant, then the last sum equals $q^3 - 1$ .", "Assume that $\\chi $ does not factor through the determinant.", "We then have $\\varphi _{n_s}*\\varphi _{n_s}(n_s) & = & \\sum _{\\genfrac{}{}{0.0pt}{}{y\\in \\mathbb {F}_{q^2}^\\times }{y + \\overline{y} = 0}}\\zeta (-\\overline{y}\\sqrt{\\epsilon }) + \\sum _{t\\in \\mathbb {F}_q^\\times }\\sum _{\\genfrac{}{}{0.0pt}{}{x\\in \\mathbb {F}_{q^2}^\\times }{x\\overline{x} = -t}}\\sum _{\\genfrac{}{}{0.0pt}{}{y\\in \\mathbb {F}_{q^2}^\\times }{y + \\overline{y} = t}}\\zeta (-\\overline{y}\\sqrt{\\epsilon })\\\\& \\stackrel{\\star }{=} & (q-1) + (q+1)\\sum _{\\genfrac{}{}{0.0pt}{}{y\\in \\mathbb {F}_{q^2}^\\times }{y + \\overline{y}\\ne 0}}\\zeta (-\\overline{y}\\sqrt{\\epsilon })\\\\& = & (q-1) + (q+1)\\left(\\sum _{y\\in \\mathbb {F}_{q^2}^\\times }\\zeta (-\\overline{y}\\sqrt{\\epsilon }) - \\sum _{\\genfrac{}{}{0.0pt}{}{y\\in \\mathbb {F}_{q^2}^\\times }{y + \\overline{y} = 0}}\\zeta (-\\overline{y}\\sqrt{\\epsilon })\\right)\\\\& \\stackrel{\\star \\star }{=} & (q-1) + (1 - q^2)\\\\& = & q - q^2.$ The equality $(\\star )$ follows from the fact that if $y + \\overline{y} = 0$ , then $-\\overline{y}\\sqrt{\\epsilon }\\in \\mathbb {F}_q$ , while $(\\star \\star )$ follows from the fact that $\\zeta $ is a nontrivial character.", "We now compute the Hecke relations for $\\varphi _{n_{s^{\\prime }}}$ .", "Again it suffices to evaluate $\\varphi _{n_{s^{\\prime }}}*\\varphi _{n_{s^{\\prime }}}$ at 1 and $n_{s^{\\prime }}$ .", "$\\varphi _{n_{s^{\\prime }}}*\\varphi _{n_{s^{\\prime }}}(1) & = & \\sum _{\\genfrac{}{}{0.0pt}{}{y\\in \\mathbb {F}_{q^2}}{y + \\overline{y} = 0}}\\varphi _{n_{s^{\\prime }}}(u^-(0,\\varpi y)n_{s^{\\prime }})\\varphi _{n_{s^{\\prime }}}(n_{s^{\\prime }}^{-1}u^-(0,\\varpi y)^{-1})\\\\& = & q\\\\\\varphi _{n_{s^{\\prime }}}*\\varphi _{n_{s^{\\prime }}}(n_{s^{\\prime }}) & = & \\sum _{\\genfrac{}{}{0.0pt}{}{y\\in \\mathbb {F}_{q^2}}{y + \\overline{y} = 0}}\\varphi _{n_{s^{\\prime }}}(u(0,y\\varpi ^{-1}\\epsilon ^{-1}))\\\\& = & \\sum _{\\genfrac{}{}{0.0pt}{}{y\\in \\mathbb {F}_{q^2}^\\times }{y + \\overline{y} = 0}}\\varphi _{n_{s^{\\prime }}}(\\textrm {diag}(-y\\sqrt{\\epsilon }^{-1}, 1, -y^{-1}\\sqrt{\\epsilon })n_{s^{\\prime }})\\\\& = & \\sum _{\\genfrac{}{}{0.0pt}{}{y\\in \\mathbb {F}_{q^2}^\\times }{y + \\overline{y} = 0}}\\chi (\\textrm {diag}(-y\\sqrt{\\epsilon }^{-1}, 1, -y^{-1}\\sqrt{\\epsilon }))\\\\& = & q - 1.$ Note that the last equality depends only on the fact that $\\chi ^s = \\chi $ .", "We again assume only that $\\chi ^s = \\chi $ .", "It is an elementary computation to check that $Z\\in \\mathcal {Z}_\\chi $ .", "To verify the claim about the centers $\\mathcal {Z}_\\chi $ of the algebras $\\mathcal {H}_C(G,\\chi )$ in general, we first note that any central element, when viewed as a polynomial in $\\mathcal {T}_{n_s}$ and $\\mathcal {T}_{n_{s^{\\prime }}}$ , must be of even degree (this follows from the Hecke relations).", "Moreover, the Hecke relations imply that the coefficients of the two highest even-degree terms must be equal.", "Let $\\mathcal {Y}\\in \\mathcal {Z}_\\chi $ be of degree $2k$ .", "Then there exists some $c\\in C$ such that $\\mathcal {Y} - cZ^k$ is of strictly smaller degree.", "Proceeding by induction, we see that $\\mathcal {Y}$ is a polynomial in $Z$ , and therefore $\\mathcal {Z}_\\chi = C[Z]$ .", "As before, we can now classify the finite-dimensional simple right $\\mathcal {H}_C(G,\\chi )$ -modules.", "Definition 3.18 Let $(\\theta ,\\theta ^{\\prime })\\in \\lbrace -q^2, q\\rbrace \\times \\lbrace -1,q\\rbrace $ .", "We define the characters $\\mu _{\\theta ,\\theta ^{\\prime }}:\\mathcal {H}_C(G,\\chi )\\rightarrow C$ by $\\mathcal {T}_{n_s} \\mapsto \\theta ,\\quad \\mathcal {T}_{n_{s^{\\prime }}} \\mapsto \\theta ^{\\prime }.$ The central element $Z$ maps to $\\theta (\\theta ^{\\prime } - q + 1) + \\theta ^{\\prime }(\\theta - q + q^2)$ .", "Let $\\langle v_1, v_2\\rangle _C$ be a two-dimensional vector space over $C$ , and let $\\lambda \\in C$ .", "We define $M(\\lambda )$ to be the following right $\\mathcal {H}_C(G,\\chi )$ -module: Table: NO_CAPTION The central element $Z$ acts by $\\lambda $ .", "Again, the proof of Proposition REF shows that the action of $\\mathcal {H}_C(G,\\chi )$ on $M(\\lambda )$ is well-defined.", "Proposition 3.19 Assume $\\textnormal {char}(C) \\ne p$ .", "Then $M(\\lambda )$ is reducible if and only if $\\lambda = q^3 + q$ or $\\lambda = -2q^2$ .", "In these cases, we have the following exact sequences: $0\\rightarrow \\mu _{q, -1}\\rightarrow M(-2q^2)\\rightarrow \\mu _{-q^2,q}\\rightarrow 0$ $0\\rightarrow \\mu _{q, q}\\rightarrow M(q^3 + q)\\rightarrow \\mu _{-q^2, -1}\\rightarrow 0$ The sequences are not split.", "Assume that $M(\\lambda )$ is reducible, so that it contains a character $\\mu $ .", "The operator $\\mathcal {T}_{n_s}$ has eigenvectors $v_1$ and $(\\lambda + q^2 - q^3)v_1 + (q + q^2)v_2$ , with eigenvalues $-q^2$ and $q$ , respectively; the operator $\\mathcal {T}_{n_{s^{\\prime }}}$ has eigenvectors $-qv_1 + v_2$ and $v_1 + v_2$ , with eigenvalues $-1$ and $q$ , respectively.", "We see that the only possibility for a common eigenvector of $\\mathcal {T}_{n_s}$ and $\\mathcal {T}_{n_{s^{\\prime }}}$ is if the vector $(\\lambda + q^2 - q^3)v_1 + (q + q^2)v_2$ is a scalar multiple of $-qv_1 + v_2$ or $v_1 + v_2$ .", "Assume the former.", "Then $\\lambda + q^2 - q^3 = -q(q + q^2)$ , which implies $\\lambda = -2q^2$ .", "It is then clear that $\\langle -qv_1 + v_2\\rangle _C\\cong \\mu _{q, -1}\\subset M(-2q^2)\\quad \\textnormal {and}\\quad M(-2q^2)/\\mu _{q, -1}\\cong \\mu _{-q^2, q}.$ Since $M(-2q^2)$ does not contain a $(-q^2,q)$ -eigenvector for $\\mathcal {T}_{n_s}, \\mathcal {T}_{n_{s^{\\prime }}}$ , the surjection cannot split.", "Assume now that $(\\lambda + q^2 - q^3)v_1 + (q + q^2)v_2$ is a scalar multiple of $v_1 + v_2$ .", "This implies $\\lambda = q^3 + q$ .", "It is clear that $\\langle v_1 + v_2\\rangle _C\\cong \\mu _{q, q}\\subset M(q^3 + q)\\quad \\textnormal {and}\\quad M(q^3 + q)/\\mu _{q, q}\\cong \\mu _{-q^2, -1}.$ By the same reasoning as above, the surjection doesn't split.", "Proposition 3.20 Assume $\\textnormal {char}(C) = p$ .", "Then $M(\\lambda )$ is reducible if and only if $\\lambda = 0$ .", "In this case the module decomposes as $M(0) \\cong \\mu _{0,0}\\oplus \\mu _{0,-1}$ .", "Assume that $M(\\lambda )$ is reducible, so that it contains either $\\mu _{0,0}$ or $\\mu _{0,-1}$ .", "In either case, the central element $Z = \\mathcal {T}_{n_s}(\\mathcal {T}_{n_{s^{\\prime }}} + 1) + \\mathcal {T}_{n_{s^{\\prime }}}\\mathcal {T}_{n_s}$ acts by 0, so we must have $\\lambda = 0$ .", "The action of $\\mathcal {T}_{n_s}$ and $\\mathcal {T}_{n_{s^{\\prime }}}$ on $M(0)$ shows that $\\langle v_1 + v_2\\rangle _C \\cong \\mu _{0,0}\\quad \\textnormal {and}\\quad \\langle v_2\\rangle _C\\cong \\mu _{0,-1},$ so that $M(0)\\cong \\mu _{0,0}\\oplus \\mu _{0,-1}.$ We may now classify the simple right $\\mathcal {H}_C(G,\\chi )$ -modules.", "Theorem 3.21 Every finite-dimensional simple right $\\mathcal {H}_C(G,\\chi )$ -module is either a character $\\mu _{\\theta ,\\theta ^{\\prime }}, (\\theta ,\\theta ^{\\prime })\\in \\lbrace -q^2, q\\rbrace \\times \\lbrace -1,q\\rbrace $ , or a module of the form $M(\\lambda ), \\lambda \\ne q^3 + q, -2q^2$ .", "The proof is virtually the same as the proof of REF , with only a few cosmetic changes.", "More precisely, we consider the space $\\ker (\\mathcal {T}_{n_s} + q^2)$ in $M$ , and compute the action of $\\mathcal {T}_{n_{s^{\\prime }}}(\\mathcal {T}_{n_s} - q)$ on an eigenvector $v$ in $\\ker (\\mathcal {T}_{n_s} + q^2)$ .", "The set $\\lbrace v, v\\cdot \\mathcal {T}_{n_{s^{\\prime }}}\\rbrace $ then forms a basis for $M$ ." ], [ "The Regular Case", "We assume now that $\\chi ^s \\ne \\chi = \\zeta \\otimes \\eta $ .", "In this case we have nontrivial intertwining maps between $\\textrm {c-ind}_I^G(\\chi )$ and $\\textrm {c-ind}_I^G(\\chi ^s)$ , and we are led to consider the algebra $\\mathcal {H}_C(G,\\gamma _\\chi ) & = & \\mathcal {H}_C(G,\\chi \\oplus \\chi ^s)\\\\& = & \\mathcal {H}_C(G,\\chi )\\oplus \\textrm {Hom}_G(\\textrm {c-ind}_I^G(\\chi ), \\textrm {c-ind}_I^G(\\chi ^s))\\\\& & \\qquad \\oplus \\ \\textrm {Hom}_G(\\textrm {c-ind}_I^G(\\chi ^s), \\textrm {c-ind}_I^G(\\chi ))\\oplus \\mathcal {H}_C(G,\\chi ^s).$ We first determine the algebra $\\mathcal {H}_C(G,\\chi )$ .", "For $n\\in N$ , we denote by $\\mathbf {1}_{InI}\\in \\textrm {c-ind}_I^G(\\chi )$ the function with support $InI$ , taking the value 1 at $n$ , on which $I$ acts by $\\chi $ or $\\chi ^s$ (depending on the class of $n$ in $W$ ).", "We let $\\mathcal {T}_{\\alpha ^{-1}}$ (resp.", "$\\mathcal {T}_\\alpha $ ) denote the endomorphism of $\\textrm {c-ind}_I^G(\\chi )$ sending $\\mathbf {1}_I$ to $\\mathbf {1}_{I\\alpha ^{-1}I}$ (resp.", "$\\mathbf {1}_{I\\alpha I}$ ).", "Proposition 3.22 The algebra $\\mathcal {H}_C(G,\\chi )$ is commutative, generated by $\\mathcal {T}_{\\alpha ^{-1}}$ and $\\mathcal {T}_{\\alpha }$ , with the relations $\\mathcal {T}_{\\alpha ^{-1}}\\mathcal {T}_\\alpha = \\mathcal {T}_\\alpha \\mathcal {T}_{\\alpha ^{-1}} = q^4.$ We have an isomorphism of algebras $\\mathcal {H}_C(G,\\chi )\\cong C[X,Y]/(XY - q^4)$ , sending $\\mathcal {T}_{\\alpha ^{-1}}$ to $X$ and $\\mathcal {T}_\\alpha $ to $Y$ .", "We adopt the same method as in the proof of REF , viewing elements of $\\mathcal {H}_C(G,\\chi )$ as functions $\\varphi $ on the double cosets $I\\backslash G/I$ .", "In this case, however, the relation $\\varphi (igi^{\\prime }) = \\chi (i)\\varphi (g)\\chi (i^{\\prime })$ shows that the functions $\\varphi $ associated to elements of $\\mathcal {H}_C(G,\\chi )$ are supported only on cosets of the form $I\\alpha ^nI$ , $n\\in \\mathbb {Z}$ .", "Once again using properties of the Bruhat decomposition for BN pairs (cf.", "[2]), if $\\varphi $ has support in $I\\alpha ^{-n}I$ (resp.", "$I\\alpha ^nI$ ) with $n>0$ , then $\\varphi $ is a scalar multiple of $\\varphi _{\\alpha ^{-1}}*\\varphi _{\\alpha ^{-1}}*\\ldots *\\varphi _{\\alpha ^{-1}}$ (resp.", "$\\varphi _\\alpha *\\varphi _\\alpha *\\ldots *\\varphi _\\alpha $ ), the convolution taken $n$ times.", "It therefore suffices to compute the products $\\varphi _{\\alpha ^{-1}}*\\varphi _\\alpha $ and $\\varphi _\\alpha *\\varphi _{\\alpha ^{-1}}$ .", "We compute the first of these; the method of computation for the second is the same.", "We have $\\textrm {supp}(\\varphi _{\\alpha ^{-1}}*\\varphi _\\alpha )\\subset I\\alpha ^{-1}I\\alpha I \\subset I\\sqcup In_sI\\sqcup I\\alpha ^{-1}n_sI$ , and since the convolution must have support on cosets of the form $I\\alpha ^nI$ , we actually have $\\textrm {supp}(\\varphi _{\\alpha ^{-1}}*\\varphi _\\alpha )\\subset I$ .", "Hence, we need only evaluate this function at 1, using the decomposition $I\\alpha ^{-1}I = \\bigsqcup _{\\genfrac{}{}{0.0pt}{}{x\\in \\mathbb {F}_{q^2},y\\in \\mathfrak {o}_E/\\mathfrak {p}_E^2}{x\\overline{x} + y + \\overline{y} = 0}}u(x,y)\\alpha ^{-1}I.$ We have: $\\varphi _{\\alpha ^{-1}}*\\varphi _\\alpha (1) & = & \\sum _{\\genfrac{}{}{0.0pt}{}{x\\in \\mathbb {F}_{q^2},y\\in \\mathfrak {o}_E/\\mathfrak {p}_E^2}{x\\overline{x} + y + \\overline{y} = 0}}\\varphi _{\\alpha ^{-1}}(u(x,y)\\alpha ^{-1})\\varphi _\\alpha (\\alpha u(x,y)^{-1})\\\\& = & q^4.$ We now turn our attention to $\\textrm {Hom}_G(\\textrm {c-ind}_I^G(\\chi ), \\textrm {c-ind}_I^G(\\chi ^s))$ .", "This has the structure of an $(\\mathcal {H}_C(G,\\chi ^s),\\mathcal {H}_C(G,\\chi ))$ -bimodule, with the action given by post-composition and pre-composition, respectively.", "By Frobenius Reciprocity we have $\\textrm {Hom}_G(\\textrm {c-ind}_I^G(\\chi ), \\textrm {c-ind}_I^G(\\chi ^s)) \\cong \\textrm {Hom}_I(\\chi ,\\textrm {c-ind}_I^G(\\chi ^s)|_I)\\cong \\textrm {c-ind}_I^G(\\chi ^s)^{I,\\chi }$ , which has a basis consisting of the functions $\\mathbf {1}_{In_s\\alpha ^nI}$ with support $In_s\\alpha ^nI$ and value 1 at $n_s\\alpha ^n$ , on which $I$ acts by $\\chi $ .", "We let $\\mathcal {S}_{n,\\chi }$ denote the homomorphism sending $\\mathbf {1}_I\\in \\textrm {c-ind}_I^G(\\chi )$ to $\\mathbf {1}_{In_s\\alpha ^nI}\\in \\textrm {c-ind}_I^G(\\chi ^s)$ , and append a $\\chi $ (or $\\chi ^s$ ) to the parameters for the operator $\\mathcal {T}_{\\alpha }$ (or $\\mathcal {T}_{\\alpha ^{-1}}$ ) to denote the Hecke algebra to which it corresponds.", "In the notation of Section REF , we have $\\mathcal {S}_{0,\\chi } = {n_s}e_\\chi ,\\qquad \\mathcal {S}_{0,\\chi ^s} = {n_s}e_{\\chi ^s},$ $\\mathcal {S}_{-1,\\chi } = {n_{s^{\\prime }}}e_\\chi ,\\qquad \\mathcal {S}_{-1,\\chi ^s} = {n_{s^{\\prime }}}e_{\\chi ^s}.$ We note that the set $\\lbrace \\textnormal {id}_{\\chi },~ \\mathcal {T}_{\\alpha ,\\chi }^m,~ \\mathcal {T}_{\\alpha ^{-1},\\chi }^m,~ \\textnormal {id}_{\\chi ^s},~ \\mathcal {T}_{\\alpha ,\\chi ^s}^m,~ \\mathcal {T}_{\\alpha ^{-1},\\chi ^s}^m,~ \\mathcal {S}_{n,\\chi },~ \\mathcal {S}_{n,\\chi ^s}\\rbrace _{m > 0, n\\in \\mathbb {Z}}$ forms a basis for $\\mathcal {H}_C(G,\\gamma _\\chi )$ , where $\\textnormal {id}_{\\chi }$ (resp.", "$\\textnormal {id}_{\\chi ^s}$ ) denotes the identity element of $\\mathcal {H}_C(G,\\chi )$ (resp.", "$\\mathcal {H}_C(G,\\chi ^s)$ ).", "Proposition 3.23 We have the following relations for the $(\\mathcal {H}_C(G,\\chi ^s),\\mathcal {H}_C(G,\\chi ))$ -bimodule $\\textnormal {Hom}_G(\\textnormal {c-ind}_I^G(\\chi ), \\textnormal {c-ind}_I^G(\\chi ^s))$ : $\\mathcal {T}_{\\alpha ^{-1},\\chi ^s}\\mathcal {S}_{n,\\chi } = \\mathcal {S}_{n,\\chi }\\mathcal {T}_{\\alpha ,\\chi } & = & {\\left\\lbrace \\begin{array}{ll}\\mathcal {S}_{n+1,\\chi } & n\\ge 0\\\\ q\\mathcal {S}_{n+1,\\chi } & n = -1\\\\ q^4\\mathcal {S}_{n+1,\\chi } & n\\le -2\\end{array}\\right.", "}\\\\\\mathcal {T}_{\\alpha ,\\chi ^s}\\mathcal {S}_{n,\\chi } = \\mathcal {S}_{n,\\chi }\\mathcal {T}_{\\alpha ^{-1},\\chi } & = & {\\left\\lbrace \\begin{array}{ll}q^4\\mathcal {S}_{n-1,\\chi } & n\\ge 1\\\\ q^3\\mathcal {S}_{n-1,\\chi } & n = 0\\\\ \\mathcal {S}_{n-1,\\chi } & n\\le -1 \\end{array}\\right.", "}$ In particular, $\\textnormal {Hom}_G(\\textnormal {c-ind}_I^G(\\chi ), \\textnormal {c-ind}_I^G(\\chi ^s))$ is generated as a module by $\\mathcal {S}_{0,\\chi }$ and $\\mathcal {S}_{-1,\\chi }$ .", "We will need the following coset decompositions: $I\\alpha I & = & \\bigsqcup _{\\genfrac{}{}{0.0pt}{}{x\\in \\mathbb {F}_{q^2},y\\in \\mathfrak {o}_E/\\mathfrak {p}_E^2}{x\\overline{x} + y + \\overline{y} = 0}}I\\alpha u(x,y)\\\\I\\alpha ^{-1}I & = & \\bigsqcup _{\\genfrac{}{}{0.0pt}{}{x\\in \\mathbb {F}_{q^2},y\\in \\mathfrak {o}_E/\\mathfrak {p}_E^2}{\\varpi x\\overline{x} + y + \\overline{y} = 0}}I\\alpha ^{-1}u^-(\\varpi x,\\varpi y)\\\\In_s\\alpha ^nI & = & \\bigsqcup _{\\genfrac{}{}{0.0pt}{}{x\\in \\mathfrak {o}_E/\\mathfrak {p}_E^{n+1},y\\in \\mathfrak {o}_E/\\mathfrak {p}_E^{2n+1}}{x\\overline{x} + y + \\overline{y} = 0}}In_s\\alpha ^nu(x,y)\\ \\ \\qquad \\qquad \\textrm {if}\\ n\\ge 0 \\\\In_s\\alpha ^nI & = & \\bigsqcup _{\\genfrac{}{}{0.0pt}{}{x\\in \\mathfrak {o}_E/\\mathfrak {p}_E^{-n-1},y\\in \\mathfrak {o}_E/\\mathfrak {p}_E^{-2n-1}}{\\varpi x\\overline{x} + y + \\overline{y} = 0}}In_s\\alpha ^nu^-(\\varpi x,\\varpi y)\\qquad \\textrm {if}\\ n < 0$ In order to compute $\\mathcal {S}_{n,\\chi }\\mathcal {T}_{\\alpha ,\\chi }$ , it suffices to know its action on the function $\\mathbf {1}_I\\in \\textrm {c-ind}_I^G(\\chi )$ .", "The definitions of $\\mathcal {S}_{n,\\chi }$ and $\\mathcal {T}_{\\alpha ,\\chi }$ show that the image will have support contained in $In_s\\alpha ^nI\\alpha I$ .", "Using Proposition 6.36 and Exercise 6.37 of [2], we see that this product of double cosets is equal to $In_s\\alpha ^{n+1}I$ (if $\\ell (n_s\\alpha ^{n+1}) = \\ell (n_s\\alpha ^n) + \\ell (\\alpha )$ ), or is contained in $In_s\\alpha ^{n+1}I \\sqcup I\\alpha ^{-n}I \\sqcup I\\alpha ^{-n-1}I$ (if $\\ell (n_s\\alpha ^{n+1}) \\ne \\ell (n_s\\alpha ^n) + \\ell (\\alpha )$ ).", "Since the support of $\\mathcal {S}_{n,\\chi }\\mathcal {T}_{\\alpha ,\\chi }(\\mathbf {1}_I)$ must be of the form $In_s\\alpha ^mI$ , we see that in both cases the support is contained in $In_s\\alpha ^{n+1}I$ , and therefore it suffices to evaluate the function at $n_s\\alpha ^{n+1}$ .", "This gives $\\mathcal {S}_{n,\\chi }\\mathcal {T}_{\\alpha ,\\chi }(\\mathbf {1}_I)(n_s\\alpha ^{n+1}) & = & \\mathcal {S}_{n,\\chi }\\left(\\sum _{\\genfrac{}{}{0.0pt}{}{x\\in \\mathbb {F}_{q^2},y\\in \\mathfrak {o}_E/\\mathfrak {p}_E^2}{x\\overline{x} + y + \\overline{y} = 0}} u(-x,\\overline{y})\\alpha ^{-1}.\\mathbf {1}_I\\right)(n_s\\alpha ^{n+1})\\\\& = & \\sum _{\\genfrac{}{}{0.0pt}{}{x\\in \\mathbb {F}_{q^2},y\\in \\mathfrak {o}_E/\\mathfrak {p}_E^2}{x\\overline{x} + y + \\overline{y} = 0}}u(-x,\\overline{y})\\alpha ^{-1}.\\mathbf {1}_{In_s\\alpha ^nI}(n_s\\alpha ^{n+1})\\\\& = & \\sum _{\\genfrac{}{}{0.0pt}{}{x\\in \\mathbb {F}_{q^2},y\\in \\mathfrak {o}_E/\\mathfrak {p}_E^2}{x\\overline{x} + y + \\overline{y} = 0}}\\mathbf {1}_{In_s\\alpha ^nI}(n_s\\alpha ^{n+1}u(-x,\\overline{y})\\alpha ^{-1})\\\\& = & {\\left\\lbrace \\begin{array}{ll}1 & n\\ge 0\\\\q & n = -1\\\\ q^4 & n\\le -2.\\end{array}\\right.", "}$ The last line follows from (the transpose of) equation (REF ).", "Using the same methods as above, we see that the support of $\\mathcal {S}_{n,\\chi }\\mathcal {T}_{\\alpha ^{-1},\\chi }(\\mathbf {1}_I)$ is contained in $In_s\\alpha ^{n-1}I$ .", "This gives $\\mathcal {S}_{n,\\chi }\\mathcal {T}_{\\alpha ^{-1},\\chi }(\\mathbf {1}_I)(n_s\\alpha ^{n-1}) & = & \\mathcal {S}_{n,\\chi }\\left(\\sum _{\\genfrac{}{}{0.0pt}{}{x\\in \\mathbb {F}_{q^2},y\\in \\mathfrak {o}_E/\\mathfrak {p}_E^2}{\\varpi x\\overline{x} + y + \\overline{y} = 0}} u^-(-\\varpi x,\\varpi \\overline{y})\\alpha .\\mathbf {1}_I\\right)(n_s\\alpha ^{n-1})\\\\& = & \\sum _{\\genfrac{}{}{0.0pt}{}{x\\in \\mathbb {F}_{q^2},y\\in \\mathfrak {o}_E/\\mathfrak {p}_E^2}{\\varpi x\\overline{x} + y + \\overline{y} = 0}}u^-(-\\varpi x,\\varpi \\overline{y})\\alpha .\\mathbf {1}_{In_s\\alpha ^nI}(n_s\\alpha ^{n-1})\\\\& = & \\sum _{\\genfrac{}{}{0.0pt}{}{x\\in \\mathbb {F}_{q^2},y\\in \\mathfrak {o}_E/\\mathfrak {p}_E^2}{\\varpi x\\overline{x} + y + \\overline{y} = 0}}\\mathbf {1}_{In_s\\alpha ^nI}(n_s\\alpha ^{n-1}u^-(-\\varpi x,\\varpi \\overline{y})\\alpha )\\\\& = & {\\left\\lbrace \\begin{array}{ll}q^4 & n\\ge 1\\\\q^3 & n = 0\\\\ 1 & n\\le -1.\\end{array}\\right.", "}$ Again, the last line follows from equation (REF ).", "The support of $\\mathcal {T}_{\\alpha ,\\chi ^s}\\mathcal {S}_{n,\\chi }(\\mathbf {1}_I)$ is contained in the coset $In_s\\alpha ^{n-1}I$ .", "In computing the value of $\\mathcal {T}_{\\alpha ,\\chi ^s}\\mathcal {S}_{n,\\chi }(\\mathbf {1}_I)(n_s\\alpha ^{n-1})$ , we must treat the cases $n\\ge 0$ and $n<0$ separately (based on the coset decomposition of $In_s\\alpha ^nI$ ).", "For $n\\ge 0$ , we have $\\mathcal {T}_{\\alpha ,\\chi ^s}\\mathcal {S}_{n,\\chi }(\\mathbf {1}_I)(n_s\\alpha ^{n-1}) & = & \\mathcal {T}_{\\alpha ,\\chi ^s}\\left(\\sum _{\\genfrac{}{}{0.0pt}{}{x\\in \\mathfrak {o}_E/\\mathfrak {p}_E^{n+1},y\\in \\mathfrak {o}_E/\\mathfrak {p}_E^{2n+1}}{x\\overline{x} + y + \\overline{y} = 0}} u(-x,\\overline{y})\\alpha ^{-n}n_s^{-1}.\\mathbf {1}_I\\right)(n_s\\alpha ^{n-1}) \\\\& = & \\sum _{\\genfrac{}{}{0.0pt}{}{x\\in \\mathfrak {o}_E/\\mathfrak {p}_E^{n+1},y\\in \\mathfrak {o}_E/\\mathfrak {p}_E^{2n+1}}{x\\overline{x} + y + \\overline{y} = 0}} u(-x,\\overline{y})\\alpha ^{-n}n_s^{-1}.\\mathbf {1}_{I\\alpha I}(n_s\\alpha ^{n-1})\\\\& = & \\sum _{\\genfrac{}{}{0.0pt}{}{x\\in \\mathfrak {o}_E/\\mathfrak {p}_E^{n+1},y\\in \\mathfrak {o}_E/\\mathfrak {p}_E^{2n+1}}{x\\overline{x} + y + \\overline{y} = 0}} \\mathbf {1}_{I\\alpha I}(n_s\\alpha ^{n-1}u(-x,\\overline{y})\\alpha ^{-n}n_s^{-1})\\\\& = & {\\left\\lbrace \\begin{array}{ll} q^4 & n \\ge 1\\\\ q^3 & n=0.", "\\end{array}\\right.", "}$ For $n<0$ , we have $\\mathcal {T}_{\\alpha ,\\chi ^s}\\mathcal {S}_{n,\\chi }(\\mathbf {1}_I)(n_s\\alpha ^{n-1}) & = & \\mathcal {T}_{\\alpha ,\\chi ^s}\\left(\\sum _{\\genfrac{}{}{0.0pt}{}{x\\in \\mathfrak {o}_E/\\mathfrak {p}_E^{-n-1},y\\in \\mathfrak {o}_E/\\mathfrak {p}_E^{-2n-1}}{\\varpi x\\overline{x} + y + \\overline{y} = 0}}u^-(-\\varpi x,\\varpi \\overline{y})\\alpha ^{-n}n_s^{-1}.\\mathbf {1}_I\\right)(n_s\\alpha ^{n-1})\\\\& = & \\sum _{\\genfrac{}{}{0.0pt}{}{x\\in \\mathfrak {o}_E/\\mathfrak {p}_E^{-n-1},y\\in \\mathfrak {o}_E/\\mathfrak {p}_E^{-2n-1}}{\\varpi x\\overline{x} + y + \\overline{y} = 0}}u^-(-\\varpi x, \\varpi \\overline{y})\\alpha ^{-n}n_s^{-1}.\\mathbf {1}_{I\\alpha I}(n_s\\alpha ^{n-1})\\\\& = & \\sum _{\\genfrac{}{}{0.0pt}{}{x\\in \\mathfrak {o}_E/\\mathfrak {p}_E^{-n-1},y\\in \\mathfrak {o}_E/\\mathfrak {p}_E^{-2n-1}}{\\varpi x\\overline{x} + y + \\overline{y} = 0}} \\mathbf {1}_{I\\alpha I}(n_s\\alpha ^{n-1}u^-(-\\varpi x,\\varpi \\overline{y})\\alpha ^{-n}n_s^{-1})\\\\& = & 1.$ Equation (REF ) once again shows that the element $n_s\\alpha ^{n-1}u^-(-\\varpi x,\\varpi \\overline{y})\\alpha ^{-n}n_s^{-1}$ is not in $I\\alpha I$ for $y\\ne 0$ .", "We omit the argument for $\\mathcal {T}_{\\alpha ^{-1},\\chi ^s}\\mathcal {S}_{n,\\chi }$ , as the computation is virtually the same as that for $\\mathcal {T}_{\\alpha ,\\chi ^s}\\mathcal {S}_{n,\\chi }$ .", "Corollary 3.24 As a right $\\mathcal {H}_C(G,\\chi )$ -module, we have $\\textnormal {Hom}_G(\\textnormal {c-ind}_I^G(\\chi ), \\textnormal {c-ind}_I^G(\\chi ^s)) \\cong (\\mathcal {H}_C(G,\\chi )\\oplus \\mathcal {H}_C(G,\\chi ))/((\\mathcal {T}_{\\alpha ^{-1},\\chi },-q^3),(-q,\\mathcal {T}_{\\alpha ,\\chi })),$ the isomorphism sending $\\mathcal {S}_{0,\\chi }$ to $(1,0)$ and $\\mathcal {S}_{-1,\\chi }$ to $(0,1)$ .", "Likewise, as a left $\\mathcal {H}_C(G,\\chi ^s)$ -module, we have $\\textnormal {Hom}_G(\\textnormal {c-ind}_I^G(\\chi ), \\textnormal {c-ind}_I^G(\\chi ^s)) \\cong (\\mathcal {H}_C(G,\\chi ^s)\\oplus \\mathcal {H}_C(G,\\chi ^s))/((\\mathcal {T}_{\\alpha ,\\chi ^s},-q^3),(-q,\\mathcal {T}_{\\alpha ^{-1},\\chi ^s})),$ the isomorphism sending $\\mathcal {S}_{0,\\chi }$ to $(1,0)$ and $\\mathcal {S}_{-1,\\chi }$ to $(0,1)$ .", "In addition to the bimodule structure on $\\textrm {Hom}_G(\\textrm {c-ind}_I^G(\\chi ), \\textrm {c-ind}_I^G(\\chi ^s))$ , we also have a composition product between elements $\\mathcal {S}_{n,\\chi ^s}\\in \\textrm {Hom}_G(\\textrm {c-ind}_I^G(\\chi ^s), \\textrm {c-ind}_I^G(\\chi ))$ and elements $\\mathcal {S}_{m,\\chi }\\in \\textrm {Hom}_G(\\textrm {c-ind}_I^G(\\chi ), \\textrm {c-ind}_I^G(\\chi ^s))$ .", "The product of two such homomorphisms will be an element of $\\mathcal {H}_C(G,\\chi )$ .", "We have the following result: Proposition 3.25 The composition $\\mathcal {S}_{n,\\chi ^s}\\mathcal {S}_{m,\\chi }$ has the following property: $\\mathcal {S}_{n,\\chi ^s}\\mathcal {S}_{m,\\chi } = {\\left\\lbrace \\begin{array}{ll}\\zeta (-1)q^{3 + 4\\min (n,m)}\\mathcal {T}_{\\alpha ,\\chi }^{\\max (0,m-n)}\\mathcal {T}_{\\alpha ^{-1},\\chi }^{\\max (n-m,0)} & n,m\\ge 0\\\\ \\zeta (-1)q^{1+4\\min (-n-1,-m-1)}\\mathcal {T}_{\\alpha ,\\chi }^{\\max (0,m-n)}\\mathcal {T}_{\\alpha ^{-1},\\chi }^{\\max (n-m,0)} & n,m<0\\\\ \\zeta (-1)\\mathcal {T}_{\\alpha ,\\chi }^{m-n} & n<0, m\\ge 0\\\\ \\zeta (-1)\\mathcal {T}_{\\alpha ^{-1},\\chi }^{n-m} & m<0, n\\ge 0.\\end{array}\\right.", "}$ By Proposition REF , it suffices to compute the four products $\\mathcal {S}_{0,\\chi ^s}\\mathcal {S}_{0,\\chi },\\ \\mathcal {S}_{-1,\\chi ^s}\\mathcal {S}_{-1,\\chi },$ $\\ \\mathcal {S}_{-1,\\chi ^s}\\mathcal {S}_{0,\\chi }$ and $\\mathcal {S}_{0,\\chi ^s}\\mathcal {S}_{-1,\\chi }$ .", "The method of proof is the same as in the proof of Proposition REF , this time using equations (REF ) and () for $n = 0$ and $n = -1$ .", "We give the proof for the first of these products.", "The definition of $\\mathcal {S}_{0,\\chi ^s}$ and $\\mathcal {S}_{0,\\chi }$ shows that the function $\\mathcal {S}_{0,\\chi ^s}\\mathcal {S}_{0,\\chi }(\\mathbf {1}_I)$ will have support contained in $I \\sqcup In_sI$ ; as $\\mathcal {S}_{0,\\chi ^s}\\mathcal {S}_{0,\\chi }\\in \\mathcal {H}_C(G,\\chi )$ , the support is actually contained in $I$ .", "This gives $\\mathcal {S}_{0,\\chi ^s}\\mathcal {S}_{0,\\chi }(\\mathbf {1}_I)(1) & = & \\mathcal {S}_{0,\\chi ^s}\\left(\\sum _{\\genfrac{}{}{0.0pt}{}{x,y\\in \\mathbb {F}_{q^2}}{x\\overline{x} + y + \\overline{y} = 0}}u(-x,\\overline{y})n_s^{-1}.\\mathbf {1}_I\\right)(1)\\\\& = & \\sum _{\\genfrac{}{}{0.0pt}{}{x,y\\in \\mathbb {F}_{q^2}}{x\\overline{x} + y + \\overline{y} = 0}} u(-x,\\overline{y})n_s^{-1}.\\mathbf {1}_{In_sI}(1)\\\\& = & \\sum _{\\genfrac{}{}{0.0pt}{}{x,y\\in \\mathbb {F}_{q^2}}{x\\overline{x} + y + \\overline{y} = 0}} \\mathbf {1}_{In_sI}(u(-x,\\overline{y})n_s^{-1})\\\\& = & \\zeta (-1)q^3.$ The other products follow similarly.", "Combining Propositions REF , REF , and REF , we now have a full description of the algebra structure of $\\mathcal {H}_C(G,\\gamma _\\chi )$ .", "When $\\textnormal {char}(C) = p$ , there is a more elegant presentation of $\\mathcal {H}_C(G,\\gamma _\\chi )$ .", "We record the result here.", "Corollary 3.26 Assume $\\textnormal {char}(C) = p$ .", "We then have $\\mathcal {H}_C(G,\\gamma _\\chi )\\cong \\begin{pmatrix}C[X,Y]/(XY) & C[X]\\oplus C[Y]\\\\ C[Y]\\oplus C[X] & C[X,Y]/(XY) \\end{pmatrix},$ where the algebra on the right is a “twisted matrix algebra.” If $(f(X),f^{\\prime }(Y))\\in C[X]\\oplus C[Y], (g^{\\prime }(Y),g(X))\\in C[Y]\\oplus C[X]$ , then we define their product to be $(f(X),f^{\\prime }(Y))(g^{\\prime }(Y),g(X)) & := & \\zeta (-1)Xf(X)g(X) + \\zeta (-1)Yf^{\\prime }(Y)g^{\\prime }(Y)\\\\& =: & (g^{\\prime }(Y),g(X))(f(X),f^{\\prime }(Y)).$ The isomorphism is given by Table: NO_CAPTIONThe center $\\mathcal {Z}_{\\gamma _\\chi }$ of $\\mathcal {H}_C(G,\\gamma _\\chi )$ consists of all elements of the form $h(\\mathcal {T}_{\\alpha ,\\chi },\\mathcal {T}_{\\alpha ^{-1},\\chi }) + h(\\mathcal {T}_{\\alpha ^{-1},\\chi ^s}, \\mathcal {T}_{\\alpha ,\\chi ^s}),$ where $h$ is a polynomial of two variables.", "It only remains to verify the claim about the center $\\mathcal {Z}_{\\gamma _\\chi }$ of $\\mathcal {H}_C(G,\\gamma _\\chi )$ .", "Let $\\mathcal {Y}$ be an arbitrary element of $\\mathcal {Z}_{\\gamma _\\chi }$ .", "Writing $\\mathcal {Y}$ as a linear combination of the basis elements, multiplying $\\mathcal {Y}$ on the left and right by the “diagonal” elements $\\mathcal {T}_{\\alpha ,\\chi }, \\mathcal {T}_{\\alpha ^{-1},\\chi }$ , etc., and using Proposition REF shows that the coefficients of $\\mathcal {S}_{n,\\chi }$ and $\\mathcal {S}_{n,\\chi ^s}$ in $\\mathcal {Y}$ must be zero.", "Subtracting an appropriate central element of the form $h(\\mathcal {T}_{\\alpha ,\\chi },\\mathcal {T}_{\\alpha ^{-1},\\chi }) + h(\\mathcal {T}_{\\alpha ^{-1},\\chi ^s}, \\mathcal {T}_{\\alpha ,\\chi ^s})$ , we may assume that $\\mathcal {Y}$ is a polynomial in $\\mathcal {T}_{\\alpha ,\\chi }$ and $\\mathcal {T}_{\\alpha ^{-1},\\chi }$ alone.", "Proposition REF again shows that $0 = \\mathcal {S}_{0,\\chi ^s}\\mathcal {Y} = \\mathcal {Y}\\mathcal {S}_{0,\\chi ^s},\\ 0 = \\mathcal {S}_{-1,\\chi ^s}\\mathcal {Y} = \\mathcal {Y}\\mathcal {S}_{-1,\\chi ^s}$ , which is enough to conclude that $\\mathcal {Y} = 0$ .", "Remark The characterization of the center $\\mathcal {Z}_{\\gamma _\\chi }$ of $\\mathcal {H}_C(G,\\gamma _\\chi )$ is the same for the case $\\textnormal {char}(C)\\ne p$ .", "The proof carries over without any essential change.", "We may now classify finite-dimensional simple modules for the algebra $\\mathcal {H}_C(G,\\gamma _\\chi )$ .", "We begin with the characters.", "Proposition 3.27 Assume $\\textnormal {char}(C)\\ne p$ .", "Then $\\mathcal {H}_C(G,\\gamma _\\chi )$ has no characters.", "Let $\\mu $ be a character of $\\mathcal {H}_C(G,\\gamma _\\chi )$ .", "Since $\\mathcal {S}_{n,\\chi }^2 = \\mathcal {S}_{n,\\chi ^s}^2 = 0$ , we must have $\\mu (\\mathcal {S}_{n,\\chi }) = \\mu (\\mathcal {S}_{n,\\chi ^s}) = 0$ for every $n\\in \\mathbb {Z}$ .", "Proposition REF now implies that all elements of $\\mathcal {H}_C(G,\\chi )$ and $\\mathcal {H}_C(G,\\chi ^s)$ map to 0.", "This gives a contradiction, since $1 = \\mu (\\textrm {id}_{\\textrm {c-ind}_I^G(\\gamma _\\chi )}) = \\mu (\\textnormal {id}_{\\chi } + \\textnormal {id}_{\\chi ^s}) = 0$ .", "Assume now that $\\textnormal {char}(C) = p$ .", "Definition 3.28 Let $i\\in \\lbrace 0,1\\rbrace $ .", "We define $\\mu _i: \\mathcal {H}_C(G,\\gamma _\\chi )\\rightarrow C$ to be the character for which $\\textnormal {id}_{\\chi ^{s^i}} \\mapsto 1$ and every other basis element maps to 0.", "Proposition 3.29 Assume $\\textnormal {char}(C)= p$ .", "Then the characters of $\\mathcal {H}_C(G,\\gamma _\\chi )$ are exactly $\\mu _0$ and $\\mu _1$ .", "As in the characteristic prime-to-$p$ case, we use Propositions REF and REF to conclude that every basis element besides $\\textnormal {id}_{\\chi }$ and $\\textnormal {id}_{\\chi ^s}$ must map to zero.", "Since $\\textnormal {id}_{\\chi } + \\textnormal {id}_{\\chi ^s} = \\textrm {id}_{\\textrm {c-ind}_I^G(\\gamma _\\chi )}$ and $\\textnormal {id}_{\\chi } \\textnormal {id}_{\\chi ^s} = 0$ , we see that the characters must be exactly those stated.", "We now turn our attention to modules of dimension greater than one.", "We first assume that $\\textnormal {char}(C) \\ne p$ .", "Let $\\sqrt{\\zeta (-1)}$ denote a fixed square root of $\\zeta (-1)$ , and let $\\mathcal {A} = \\sqrt{\\zeta (-1)}(\\mathcal {S}_{0,\\chi }+\\mathcal {S}_{-1,\\chi ^s}).$ We have that $\\mathcal {A}^2 = \\mathcal {T}_{\\alpha ,\\chi } + \\mathcal {T}_{\\alpha ^{-1},\\chi ^s}$ , and that $\\mathcal {H}_C(G,\\gamma _\\chi )$ is free of rank two over $\\mathcal {Z}_{\\gamma _\\chi }[\\mathcal {A}]$ , with basis $\\lbrace \\textnormal {id}_{\\chi }, \\textnormal {id}_{\\chi ^s}\\rbrace $ .", "Let $\\lambda \\in C^\\times $ , and fix a square root $\\sqrt{\\lambda }$ .", "Let $\\mu _{\\lambda ,\\sqrt{\\lambda }}$ denote the representation of $\\mathcal {Z}_{\\gamma _\\chi }[\\mathcal {A}]$ spanned by $v$ , with action given by $v\\cdot (\\mathcal {T}_{\\alpha ,\\chi } + \\mathcal {T}_{\\alpha ^{-1},\\chi ^s}) = \\lambda v,\\quad v\\cdot (\\mathcal {T}_{\\alpha ,\\chi ^s} + \\mathcal {T}_{\\alpha ^{-1},\\chi }) = q^4\\lambda ^{-1} v,$ $v\\cdot \\mathcal {A} = \\sqrt{\\lambda }v.$ We consider the induced representation $\\mu _{\\lambda ,\\sqrt{\\lambda }}\\otimes _{\\mathcal {Z}_{\\gamma _\\chi }[\\mathcal {A}]}\\mathcal {H}_C(G,\\gamma _\\chi )$ .", "Since the algebra $\\mathcal {H}_C(G,\\gamma _\\chi )$ admits no characters, this immediately implies that this module is simple.", "Lemma 3.30 The (isomorphism class of the) representation $\\mu _{\\lambda ,\\sqrt{\\lambda }}\\otimes _{\\mathcal {Z}_{\\gamma _\\chi }[\\mathcal {A}]}\\mathcal {H}_C(G,\\gamma _\\chi )$ is independent of the choice of square root $\\sqrt{\\lambda }$ .", "Let $\\langle v\\rangle _C$ denote the underlying space of $\\mu _{\\lambda ,\\sqrt{\\lambda }}$ .", "Then $\\mu _{\\lambda ,\\sqrt{\\lambda }}\\otimes _{\\mathcal {Z}_{\\gamma _\\chi }[\\mathcal {A}]}\\mathcal {H}_C(G,\\gamma _\\chi )$ is spanned by $\\lbrace v\\otimes \\textnormal {id}_{\\chi }, v\\otimes \\textnormal {id}_{\\chi ^s}\\rbrace $ .", "The action of $\\mathcal {Z}_{\\gamma _\\chi }[\\mathcal {A}]$ on the vector $v\\otimes (\\textnormal {id}_{\\chi } - \\textnormal {id}_{\\chi ^s})$ shows that $\\mu _{\\lambda ,-\\sqrt{\\lambda }}$ is contained in $\\mu _{\\lambda ,\\sqrt{\\lambda }}\\otimes _{\\mathcal {Z}_{\\gamma _\\chi }[\\mathcal {A}]}\\mathcal {H}_C(G,\\gamma _\\chi )|_{\\mathcal {Z}_{\\gamma _\\chi }[\\mathcal {A}]}$ .", "By Frobenius Reciprocity we have $\\lbrace 0\\rbrace & \\ne & \\textrm {Hom}_{\\mathcal {Z}_{\\gamma _\\chi }[\\mathcal {A}]}(\\mu _{\\lambda ,-\\sqrt{\\lambda }},~\\mu _{\\lambda ,\\sqrt{\\lambda }}\\otimes _{\\mathcal {Z}_{\\gamma _\\chi }[\\mathcal {A}]}\\mathcal {H}_C(G,\\gamma _\\chi )|_{\\mathcal {Z}_{\\gamma _\\chi }[\\mathcal {A}]})\\\\& \\cong & \\textrm {Hom}_{\\mathcal {H}_C(G,\\gamma _\\chi )}(\\mu _{\\lambda ,-\\sqrt{\\lambda }}\\otimes _{\\mathcal {Z}_{\\gamma _\\chi }[\\mathcal {A}]}\\mathcal {H}_C(G,\\gamma _\\chi ),~\\mu _{\\lambda ,\\sqrt{\\lambda }}\\otimes _{\\mathcal {Z}_{\\gamma _\\chi }[\\mathcal {A}]}\\mathcal {H}_C(G,\\gamma _\\chi )).$ As both modules are simple, the result follows.", "With this lemma, we may unambiguously define $M(\\lambda ) = \\mu _{\\lambda ,\\sqrt{\\lambda }}\\otimes _{\\mathcal {Z}_{\\gamma _\\chi }[\\mathcal {A}]}\\mathcal {H}_C(G,\\gamma _\\chi )$ .", "By considering central characters, we see that the modules $M(\\lambda )$ are pairwise nonisomorphic for distinct values of $\\lambda $ .", "Theorem 3.31 Assume $\\textnormal {char}(C) \\ne p$ .", "Every finite-dimensional simple right $\\mathcal {H}_C(G,\\gamma _\\chi )$ -module is of the form $M(\\lambda ), \\lambda \\in C^\\times $ .", "Assume $M$ is a nonzero simple right module, and assume that $M|_{\\mathcal {Z}_{\\gamma _\\chi }[\\mathcal {A}]}$ contains a character $\\mu _{\\lambda ,\\sqrt{\\lambda }}$ .", "Frobenius Reciprocity gives $\\lbrace 0\\rbrace \\ne \\textrm {Hom}_{\\mathcal {Z}_{\\gamma _\\chi }[\\mathcal {A}]}(\\mu _{\\lambda ,\\sqrt{\\lambda }},M|_{\\mathcal {Z}_{\\gamma _\\chi }[\\mathcal {A}]})\\cong \\textrm {Hom}_{\\mathcal {H}_C(G,\\gamma _\\chi )}(M(\\lambda ), M),$ which implies $M(\\lambda )\\cong M$ by simplicity of $M(\\lambda )$ and $M$ .", "Assume now that $\\textnormal {char}(C) = p$ .", "Let $\\sqrt{\\zeta (-1)}$ denote a fixed square root of $\\zeta (-1)$ , and let $\\mathcal {A}_1 = \\sqrt{\\zeta (-1)}(\\mathcal {S}_{0,\\chi } + \\mathcal {S}_{-1,\\chi ^s}),\\qquad \\mathcal {A}_2 = \\sqrt{\\zeta (-1)}(\\mathcal {S}_{0,\\chi ^s} + \\mathcal {S}_{-1,\\chi }).$ Note that $\\mathcal {A}_1\\mathcal {A}_2 = \\mathcal {A}_2\\mathcal {A}_1 = 0$ , $\\mathcal {A}_1^2 = \\mathcal {T}_{\\alpha ,\\chi } + \\mathcal {T}_{\\alpha ^{-1},\\chi ^s}$ , and $\\mathcal {A}_2^2 = \\mathcal {T}_{\\alpha ^{-1},\\chi } + \\mathcal {T}_{\\alpha ,\\chi ^s}$ .", "The algebra $\\mathcal {H}_C(G,\\gamma _\\chi )$ is free of rank two over $\\mathcal {Z}_{\\gamma _\\chi }[\\mathcal {A}_1,\\mathcal {A}_2]$ , with basis $\\lbrace \\textnormal {id}_{\\chi },\\textnormal {id}_{\\chi ^s}\\rbrace $ .", "Let $\\lambda ,\\lambda ^{\\prime }\\in C$ be such that $\\lambda \\lambda ^{\\prime } = 0$ , and fix square roots $\\sqrt{\\lambda }, \\sqrt{\\lambda ^{\\prime }}$ .", "We let $\\mu _{\\lambda ,\\lambda ^{\\prime },\\sqrt{\\lambda },\\sqrt{\\lambda ^{\\prime }}}$ denote the representation of $\\mathcal {Z}_{\\gamma _\\chi }[\\mathcal {A}_1,\\mathcal {A}_2]$ spanned by $v$ , with action given by Table: NO_CAPTIONWe consider the induced representation $\\mu _{\\lambda ,\\lambda ^{\\prime },\\sqrt{\\lambda },\\sqrt{\\lambda ^{\\prime }}}\\otimes _{\\mathcal {Z}_{\\gamma _\\chi }[\\mathcal {A}_1,\\mathcal {A}_2]}\\mathcal {H}_C(G,\\gamma _\\chi )$ .", "Proposition 3.32 The module $\\mu _{\\lambda ,\\lambda ^{\\prime },\\sqrt{\\lambda },\\sqrt{\\lambda ^{\\prime }}}\\otimes _{\\mathcal {Z}_{\\gamma _\\chi }[\\mathcal {A}_1,\\mathcal {A}_2]}\\mathcal {H}_C(G,\\gamma _\\chi )$ is reducible if and only if $(\\lambda ,\\lambda ^{\\prime }) = (0,0)$ .", "In this case, we have $\\mu _{0,0,0,0}\\otimes _{\\mathcal {Z}_{\\gamma _\\chi }[\\mathcal {A}_1,\\mathcal {A}_2]}\\mathcal {H}_C(G,\\gamma _\\chi )\\cong \\mu _0\\oplus \\mu _1.$ Assume that $\\mu _{\\lambda ,\\lambda ^{\\prime },\\sqrt{\\lambda },\\sqrt{\\lambda ^{\\prime }}}\\otimes _{\\mathcal {Z}_{\\gamma _\\chi }[\\mathcal {A}_1,\\mathcal {A}_2]}\\mathcal {H}_C(G,\\gamma _\\chi )$ is reducible, so that it contains either $\\mu _0$ or $\\mu _1$ .", "In either case, both $\\mathcal {A}_1$ and $\\mathcal {A}_2$ must act by 0, and therefore $(\\lambda ,\\lambda ^{\\prime }) = (0,0)$ .", "The action of $\\textnormal {id}_{\\chi }$ and $\\textnormal {id}_{\\chi ^s}$ show that if $\\langle v\\rangle _C\\cong \\mu _{0,0,0,0}$ as a $\\mathcal {Z}_{\\gamma _\\chi }[\\mathcal {A}_1,\\mathcal {A}_2]$ -module, then $\\langle v\\otimes \\textnormal {id}_{\\chi }\\rangle _C \\cong \\mu _0$ and $\\langle v\\otimes \\textnormal {id}_{\\chi ^s}\\rangle _C \\cong \\mu _1$ as $\\mathcal {H}_C(G,\\gamma _\\chi )$ -modules.", "Lemma 3.33 The (isomorphism class of the) representation $\\mu _{\\lambda ,\\lambda ^{\\prime },\\sqrt{\\lambda },\\sqrt{\\lambda ^{\\prime }}}\\otimes _{\\mathcal {Z}_{\\gamma _\\chi }[\\mathcal {A}_1,\\mathcal {A}_2]}\\mathcal {H}_C(G,\\gamma _\\chi )$ is independent of the choice of square roots $\\sqrt{\\lambda }, \\sqrt{\\lambda ^{\\prime }}$ .", "This is obvious if $\\lambda = \\lambda ^{\\prime } = 0$ , so assume that $\\lambda ^{\\prime } = 0, \\lambda \\ne 0$ .", "If we let $\\langle v\\rangle _C$ denote the underlying space of the character $\\mu _{\\lambda ,0,\\sqrt{\\lambda },0}$ , then $\\mu _{\\lambda ,0,\\sqrt{\\lambda },0}\\otimes _{\\mathcal {Z}_{\\gamma _\\chi }[\\mathcal {A}_1,\\mathcal {A}_2]}\\mathcal {H}_C(G,\\gamma _\\chi )$ is spanned by $\\lbrace v\\otimes \\textnormal {id}_{\\chi }, v\\otimes \\textnormal {id}_{\\chi ^s}\\rbrace $ .", "Considering the action of $\\mathcal {Z}_{\\gamma _\\chi }[\\mathcal {A}_1,\\mathcal {A}_2]$ on the vector $v\\otimes (\\textnormal {id}_{\\chi } - \\textnormal {id}_{\\chi ^s})$ , we see that $\\langle v\\otimes (\\textnormal {id}_{\\chi } - \\textnormal {id}_{\\chi ^s})\\rangle _C \\cong \\mu _{\\lambda , 0, -\\sqrt{\\lambda },0}$ as $\\mathcal {Z}_{\\gamma _\\chi }[\\mathcal {A}_1,\\mathcal {A}_2]$ -modules.", "By Frobenius Reciprocity we have $\\lbrace 0\\rbrace & \\ne & \\textrm {Hom}_{\\mathcal {Z}_{\\gamma _\\chi }[\\mathcal {A}_1,\\mathcal {A}_2]}(\\mu _{\\lambda ,0,-\\sqrt{\\lambda },0},~~ \\mu _{\\lambda ,0,\\sqrt{\\lambda },0}\\otimes _{\\mathcal {Z}_{\\gamma _\\chi }[\\mathcal {A}_1,\\mathcal {A}_2]}\\mathcal {H}_C(G,\\gamma _\\chi )|_{\\mathcal {Z}_{\\gamma _\\chi }[\\mathcal {A}_1,\\mathcal {A}_2]})\\\\& \\cong & \\textrm {Hom}_{\\mathcal {H}_C(G,\\gamma _\\chi )}(\\mu _{\\lambda ,0,-\\sqrt{\\lambda },0}\\otimes _{\\mathcal {Z}_{\\gamma _\\chi }[\\mathcal {A}_1,\\mathcal {A}_2]}\\mathcal {H}_C(G,\\gamma _\\chi ),~~ \\mu _{\\lambda ,0,\\sqrt{\\lambda },0}\\otimes _{\\mathcal {Z}_{\\gamma _\\chi }[\\mathcal {A}_1,\\mathcal {A}_2]}\\mathcal {H}_C(G,\\gamma _\\chi )).$ As both modules are simple, the result follows.", "The case $\\lambda = 0, \\lambda ^{\\prime }\\ne 0$ is similar.", "With this lemma, we can unambiguously define $M(\\lambda ,\\lambda ^{\\prime }) = \\mu _{\\lambda ,\\lambda ^{\\prime },\\sqrt{\\lambda },\\sqrt{\\lambda ^{\\prime }}}\\otimes _{\\mathcal {Z}_{\\gamma _\\chi }[\\mathcal {A}_1,\\mathcal {A}_2]}\\mathcal {H}_C(G,\\gamma _\\chi )$ .", "By examining central characters, we see that the modules $M(\\lambda ,\\lambda ^{\\prime })$ are pairwise nonisomorphic for distinct pairs $(\\lambda ,\\lambda ^{\\prime })$ .", "Theorem 3.34 Assume $\\textnormal {char}(C) = p$ .", "Every finite-dimensional simple right $\\mathcal {H}_C(G,\\gamma _\\chi )$ -module is either a character $\\mu _0$ or $\\mu _1$ , or a module of the form $M(\\lambda ,\\lambda ^{\\prime })$ with $\\lambda \\lambda ^{\\prime } = 0, (\\lambda ,\\lambda ^{\\prime })\\ne (0,0)$ .", "Assume $M$ is a nonzero simple right module which is not a character, and assume that $M|_{\\mathcal {Z}_{\\gamma _\\chi }[\\mathcal {A}_1,\\mathcal {A}_2]}$ contains a character $\\mu _{\\lambda ,\\lambda ^{\\prime },\\sqrt{\\lambda },\\sqrt{\\lambda ^{\\prime }}}$ .", "If $(\\lambda ,\\lambda ^{\\prime }) = (0,0)$ , then $M$ would contain either $\\mu _0$ or $\\mu _1$ , and by simplicity would be equal to a character, giving a contradiction.", "Frobenius Reciprocity now gives $\\lbrace 0\\rbrace \\ne \\textrm {Hom}_{\\mathcal {Z}_{\\gamma _\\chi }[\\mathcal {A}_1,\\mathcal {A}_2]}(\\mu _{\\lambda ,\\lambda ^{\\prime },\\sqrt{\\lambda },\\sqrt{\\lambda ^{\\prime }}}, M|_{\\mathcal {Z}_{\\gamma _\\chi }[\\mathcal {A}_1,\\mathcal {A}_2]})\\cong \\textrm {Hom}_{\\mathcal {H}_C(G,\\gamma _\\chi )}(M(\\lambda ,\\lambda ^{\\prime }),M),$ which implies $M(\\lambda ,\\lambda ^{\\prime })\\cong M$ by simplicity of $M(\\lambda ,\\lambda ^{\\prime })$ and $M$ .", "We conclude with one final definition.", "Definition 3.35 Let $\\chi :H\\rightarrow C^\\times $ be an arbitrary character, and let $M$ be a finite-dimensional simple module for $\\mathcal {H}_C(G,\\gamma _\\chi )$ .", "We append $\\chi $ to the list of parameters of $M$ , and use this notation to denote the corresponding module for $\\mathcal {H}_C(G,I(1))$ , via the decomposition of Proposition REF .", "Remark The isomorphism in Corollary REF depends on the ordered pair $(\\chi ,\\chi ^s)$ .", "There is an obvious isomorphism of algebras $\\mathcal {H}_C(G,\\chi \\oplus \\chi ^s)\\cong \\mathcal {H}_C(G,\\chi ^s\\oplus \\chi )$ , which identifies simple modules.", "In particular, the isomorphism gives $M(\\lambda ,\\chi )\\cong M(q^4\\lambda ^{-1},\\chi ^s)$ if $\\textnormal {char}(C)\\ne p$ , and $\\mu _{0,\\chi }\\cong \\mu _{1,\\chi ^s},\\ \\mu _{1,\\chi }\\cong \\mu _{0,\\chi ^s},\\ M(\\lambda ,\\lambda ^{\\prime },\\chi )\\cong M(\\lambda ^{\\prime },\\lambda ,\\chi ^s)$ if $\\textnormal {char}(C) = p$ ." ], [ "Principal Series", "We shall assume from this point onwards that $C = \\overline{\\mathbb {F}}_p$ , and that all representations are smooth $\\overline{\\mathbb {F}}_p$ -representations.", "We call such representations mod-$p$ or modular representations.", "We let $\\iota :\\mathbb {F}_{q^2}\\hookrightarrow \\overline{\\mathbb {F}}_p$ denote a fixed embedding, and assume that every $\\overline{\\mathbb {F}}_p^\\times $ -valued character of $H$ factors through $\\iota $ .", "In an attempt to understand supersingular representations of $G$ (cf.", "Introduction), we will make use of the functor sending a smooth representation $\\pi $ to $\\pi ^{I(1)}$ , called the functor of $I(1)$ -invariants.", "By Lemma 3(1) of [3], if $\\pi $ is a nonzero smooth representation of $G$ , then the module $\\pi ^{I(1)}$ will also be nonzero.", "Let $\\varepsilon = \\widetilde{\\zeta }\\otimes \\widetilde{\\eta }$ be a smooth character of the full torus $T$ of $G$ , and consider the principal series representation $\\textrm {ind}_B^G(\\varepsilon )$ , where $B$ is the standard upper Borel subgroup of $G$ , and $\\widetilde{\\zeta }$ and $\\widetilde{\\eta }$ are characters of $E^\\times $ and $\\mathbf {U}(1)(E/F)$ , respectively.", "In Proposition 4.4.9 of [1], Abdellatif has shown that the principal series representation is reducible if and only if $\\varepsilon = \\widetilde{\\eta }\\circ \\det $ , in which case it is of length 2.", "More precisely, we have a nonsplit short exact sequence $0 \\rightarrow \\widetilde{\\eta }\\circ \\det \\rightarrow \\textnormal {ind}_B^G(\\varepsilon )\\rightarrow \\widetilde{\\eta }\\circ \\det \\otimes \\textnormal {St}_G\\rightarrow 0,$ where $\\textnormal {St}_G = \\textnormal {ind}_B^G(1)/1$ is the Steinberg representation of $G$ .", "The Bruhat decomposition applied to $K$ and the Iwasawa decomposition together imply that $G = BI \\sqcup Bn_sI = BI(1) \\sqcup Bn_sI(1).$ Therefore, we may take as a basis for the space of $I(1)$ -invariants of $\\textrm {ind}_B^G(\\varepsilon )$ the functions $\\lbrace f_1,f_2\\rbrace $ , defined by Table: NO_CAPTIONthe function $f_1$ is the unique $I(1)$ -invariant function with support $BI(1)$ taking the value 1 at the identity (likewise for $f_2$ , supported in $Bn_sI(1)$ ).", "For a smooth character $\\varepsilon $ of $T$ , we recall that since $T_1$ is a pro-$p$ subgroup, the restriction of $\\varepsilon $ to $T_1$ must be trivial.", "Let $\\varepsilon ^*$ denote the representation of $H = T_0/T_1$ , given by restricting $\\varepsilon $ to $T_0$ .", "The action of $\\mathcal {H}_{\\overline{\\mathbb {F}}_p}(G,I(1))$ on $\\textnormal {ind}_B^G(\\varepsilon )^{I(1)}$ will depend on the character $\\varepsilon ^*$ .", "If $(\\varepsilon ^*)^s = \\varepsilon ^*$ , then in the notation of Lemma REF we have $\\textrm {ind}_B^G(\\varepsilon )^{I(1)} = \\textrm {ind}_B^G(\\varepsilon )^{I,\\varepsilon ^*},$ and the action of $\\mathcal {H}_{\\overline{\\mathbb {F}}_p}(G,I(1))$ factors through algebra $\\mathcal {H}_{\\overline{\\mathbb {F}}_p}(G,\\varepsilon ^*)$ (via the decomposition of Proposition REF ).", "Likewise, if $(\\varepsilon ^*)^s\\ne \\varepsilon ^*$ , then $\\textrm {ind}_B^G(\\varepsilon )^{I(1)} = \\textrm {ind}_B^G(\\varepsilon )^{I,\\varepsilon ^*\\oplus (\\varepsilon ^*)^s} = \\textrm {ind}_B^G(\\varepsilon )^{I,\\varepsilon ^*}\\oplus \\textrm {ind}_B^G(\\varepsilon )^{I,(\\varepsilon ^*)^s},$ and the action of $\\mathcal {H}_{\\overline{\\mathbb {F}}_p}(G,I(1))$ factors through $\\mathcal {H}_{\\overline{\\mathbb {F}}_p}(G,\\varepsilon ^*\\oplus (\\varepsilon ^*)^s)$ .", "Theorem 4.1 The algebra $\\mathcal {H}_{\\overline{\\mathbb {F}}_p}(G,I(1))$ acts on $\\langle f_1, f_2\\rangle _{\\overline{\\mathbb {F}}_p}$ in the following way: If $\\varepsilon ^*$ factors through the determinant, then Table: NO_CAPTION for $\\chi \\ne \\varepsilon ^*$ .", "If $(\\varepsilon ^*)^s = \\varepsilon ^*$ but $\\varepsilon ^*$ does not factor through the determinant, then Table: NO_CAPTION for $\\chi \\ne \\varepsilon ^*$ .", "If $(\\varepsilon ^*)^s\\ne \\varepsilon ^*$ , then Table: NO_CAPTION for $\\chi \\ne \\varepsilon ^*,~\\chi ^{\\prime }\\ne (\\varepsilon ^*)^s$ .", "See Appendix.", "Corollary 4.2 In the notation of Definition REF , the $\\mathcal {H}_{\\overline{\\mathbb {F}}_p}(G,I(1))$ -module $\\textnormal {ind}_B^G(\\varepsilon )^{I(1)}$ is given by the following: Assume $\\varepsilon ^*$ factors through the determinant.", "Then $\\textnormal {ind}_B^G(\\varepsilon )^{I(1)}\\cong M(\\varepsilon (\\alpha ),\\varepsilon ^*)$ as right $\\mathcal {H}_{\\overline{\\mathbb {F}}_p}(G,I(1))$ -modules.", "Assume $(\\varepsilon ^*)^s = \\varepsilon ^*$ , but $\\varepsilon ^*$ does not factor through the determinant.", "Then $\\textnormal {ind}_B^G(\\varepsilon )^{I(1)}\\cong M(\\varepsilon (\\alpha ),\\varepsilon ^*)$ as right $\\mathcal {H}_{\\overline{\\mathbb {F}}_p}(G,I(1))$ -modules.", "Assume $(\\varepsilon ^*)^s\\ne \\varepsilon ^*$ .", "Then $\\textnormal {ind}_B^G(\\varepsilon )^{I(1)}\\cong M(0,\\varepsilon (\\alpha ),\\varepsilon ^*)$ as right $\\mathcal {H}_{\\overline{\\mathbb {F}}_p}(G,I(1))$ -modules.", "Firstly, note that by Theorem REF , the central element of $\\mathcal {H}_{\\overline{\\mathbb {F}}_p}(G,\\varepsilon ^*)$ acts by $\\varepsilon (\\alpha )$ .", "Assume that $\\textrm {ind}_B^G(\\varepsilon )^{I(1)}$ is reducible as an $\\mathcal {H}_{\\overline{\\mathbb {F}}_p}(G,\\varepsilon ^*)$ -module, and let $c_1f_1 + c_2f_2$ , for $c_1,c_2\\in \\overline{\\mathbb {F}}_p$ , span a one-dimensional invariant subspace.", "The action of ${n_s}$ shows that either $c_1 = c_2$ , or $c_1 = 0$ .", "In the first case, the action of ${n_{s^{\\prime }}}$ implies $\\varepsilon (\\alpha ) = 1$ , while in the second case it implies $\\varepsilon (\\alpha ) = 0$ , an impossibility.", "If $\\varepsilon (\\alpha )\\ne 1$ , then $\\textrm {ind}_B^G(\\varepsilon )^{I(1)}$ is a simple module, with the center of $\\mathcal {H}_{\\overline{\\mathbb {F}}_p}(G,\\varepsilon ^*)$ acting by $\\varepsilon (\\alpha )$ .", "Therefore $\\textrm {ind}_B^G(\\varepsilon )^{I(1)}\\cong M(\\varepsilon (\\alpha ),\\varepsilon ^*)$ .", "If $\\varepsilon (\\alpha ) = 1$ , then the action of the operators ${n_s}$ and ${n_{s^{\\prime }}}$ on the basis $\\lbrace f_2,f_1\\rbrace $ shows that $\\textrm {ind}_B^G(\\varepsilon )^{I(1)}\\cong M(1,\\varepsilon ^*)$ .", "The condition $\\varepsilon (\\alpha ) = 1$ implies $\\varepsilon = \\widetilde{\\eta }\\circ \\det $ , in which case we have a short exact sequence 0[r] [r] indBG() [r] StG [r] 0.", "Using the same argument as in [33], we conclude that taking $I(1)$ -invariants is exact in this case, and we obtain: 0[r] [r][d] indBG()I(1)[r][d] StI(1)[r][d] 00 [r] 0,0,*[r] M(1,*)[r] -1,-1,* [r] 0 Applying the Five Lemma gives $\\widetilde{\\eta }\\circ \\det \\otimes \\textrm {St}^{I(1)}\\cong \\mu _{-1,-1,\\varepsilon ^*}$ .", "By Theorem REF , the central element of $\\mathcal {H}_{\\overline{\\mathbb {F}}_p}(G,\\varepsilon ^*)$ acts by $\\varepsilon (\\alpha )$ .", "Assume that $\\textrm {ind}_B^G(\\varepsilon )^{I(1)}$ is reducible as an $\\mathcal {H}_{\\overline{\\mathbb {F}}_p}(G,\\varepsilon ^*)$ -module, so that it contains either $\\mu _{0,0,\\varepsilon ^*}$ or $\\mu _{0,-1,\\varepsilon ^*}$ .", "In either case, the central element acts as 0, which implies $\\varepsilon (\\alpha ) = 0$ , an impossibility.", "Thus, the module $\\textrm {ind}_B^G(\\varepsilon )^{I(1)}$ is simple, and by Theorem REF , we must have $\\textrm {ind}_B^G(\\varepsilon )^{I(1)}\\cong M(\\varepsilon (\\alpha ),\\varepsilon ^*)$ as right $\\mathcal {H}_{\\overline{\\mathbb {F}}_p}(G,I(1))$ -modules.", "The action described in Theorem REF shows that $\\textrm {ind}_B^G(\\varepsilon )^{I(1)}$ does not contain a character.", "Thus $\\textrm {ind}_B^G(\\varepsilon )^{I(1)}$ is simple as a right $\\mathcal {H}_{\\overline{\\mathbb {F}}_p}(G,\\varepsilon ^*\\oplus (\\varepsilon ^*)^s)$ -module, and is therefore determined by the action of the center.", "Proposition REF and Corollary REF imply that the center is generated by $\\widetilde{\\zeta }(-1)(e_{\\varepsilon ^*}{n_{s^{\\prime }}}{n_s} + e_{(\\varepsilon ^*)^s}{n_s}{n_{s^{\\prime }}})\\quad \\textnormal {and}\\quad \\widetilde{\\zeta }(-1)(e_{(\\varepsilon ^*)^s}{n_{s^{\\prime }}}{n_s} + e_{\\varepsilon ^*}{n_s}{n_{s^{\\prime }}}).$ The first element acts by 0, while the second element acts by $\\varepsilon (\\alpha )$ .", "Therefore, we have $\\textrm {ind}_B^G(\\varepsilon )^{I(1)}\\cong M(0,\\varepsilon (\\alpha ),\\varepsilon ^*)$ as right $\\mathcal {H}_{\\overline{\\mathbb {F}}_p}(G,I(1))$ -modules." ], [ "Supersingular Modules", "In light of the results of the previous section, we make the following definition: Definition 4.3 Let $\\chi = \\zeta \\otimes \\eta $ be a character of the finite torus $H$ .", "We define the following characters of $\\mathcal {H}_{\\overline{\\mathbb {F}}_p}(G,I(1))$ : Assume that $\\chi = \\eta \\circ \\det $ .", "We set Table: NO_CAPTION for $\\chi ^{\\prime }\\ne \\chi $ .", "Assume that $\\chi ^s = \\chi $ , but $\\chi \\ne \\eta \\circ \\det $ .", "We set Table: NO_CAPTION for $\\chi ^{\\prime }\\ne \\chi $ .", "Assume that $\\chi ^s\\ne \\chi $ .", "We set Table: NO_CAPTION for $\\chi ^{\\prime }\\ne \\chi $ .", "The modules defined in this way are supersingular (as defined in Definition REF ).", "We will denote a generic supersingular module above by $M_{\\chi ,\\mathbf {J}}$ , where $\\mathbf {J} = (J,J^{\\prime })$ is an ordered pair as above with $J\\subset J_0(\\chi ), J^{\\prime }\\subset J_0^{\\prime }(\\chi )$ .", "This notation is motivated by the notation of Section (cf.", "Definitions REF and REF ).", "Note that we have $M_{\\chi ,\\mathbf {J}} \\cong M_{\\chi ^{\\prime },\\mathbf {J}^{\\prime }}$ if and only if $\\chi = \\chi ^{\\prime }$ and $\\mathbf {J}= \\mathbf {J}^{\\prime }$ .", "The computations of the previous sections lead to the following Corollary: Corollary 4.4 Let $M$ be a finite-dimensional supersingular $\\mathcal {H}_{\\overline{\\mathbb {F}}_p}(G,I(1))$ -module.", "Then $M\\cong M_{\\chi ,\\mathbf {J}}$ for some $\\chi $ and $\\mathbf {J}$ , where $M_{\\chi ,\\mathbf {J}}$ is a module as in Definition REF .", "The functor of $I(1)$ -invariants induces a bijection between irreducible nonsupersingular representations of $G$ and nonsupersingular finite-dimensional simple right $\\mathcal {H}_{\\overline{\\mathbb {F}}_p}(G,I(1))$ -modules.", "Moreover, if $M$ is a simple right $\\mathcal {H}_{\\overline{\\mathbb {F}}_p}(G,I(1))$ -module such that $M\\lnot \\cong \\pi ^{I(1)}$ for any nonsupersingular representation $\\pi $ , then $M$ is a supersingular module.", "This follows from Theorems REF , REF , REF , and Corollary REF .", "Corollary 4.5 Let $\\pi $ be a smooth irreducible representation of $G$ .", "If $\\pi ^{I(1)}$ contains a submodule isomorphic to a supersingular module, then $\\pi $ is supersingular." ], [ "Representations of the Finite Groups and Finite Hecke Algebras", "In this section, we recall results about mod-$p$ representations of the finite groups $\\Gamma =\\mathbf {U}(2,1)(\\mathbb {F}_{q^2}/\\mathbb {F}_{q})$ and $\\Gamma ^{\\prime } = (\\mathbf {U}(1,1)\\times \\mathbf {U}(1))(\\mathbb {F}_{q^2}/\\mathbb {F}_{q})$ .", "On one hand, we have a complete description of such representations in terms of characters $\\chi $ of $H$ and subsets of a certain set $J_0(\\chi )$ , due to Carter and Lusztig (cf.", "[10]); on the other hand, we have a more classical description in terms of highest weight modules.", "Our goal will be to provide a dictionary for matching the two sets of representations.", "For references on the highest weight classification, the reader is urged to consult the lecture notes of Steinberg ([30]) or Humphreys ([20]).", "We fix some notation.", "Let $S$ and $S^{\\prime }$ denote the sets of Coxeter generators for the Weyl groups associated to $\\Gamma $ and $\\Gamma ^{\\prime }$ , respectively.", "In both cases, the sets $S$ and $S^{\\prime }$ have size 1, consisting of the class of the elements $s$ and $s^{\\prime }$ , respectively." ], [ "Finite Hecke Algebras", "We first describe the Hecke algebras for the finite groups $\\Gamma $ and $\\Gamma ^{\\prime }$ , and their associated simple modules.", "Definition 5.1 We define $\\mathcal {H}_\\Gamma := \\textrm {End}_\\Gamma (\\textrm {ind}_\\mathbb {U}^\\Gamma (1)),\\qquad \\mathcal {H}_{\\Gamma ^{\\prime }} := \\textrm {End}_{\\Gamma ^{\\prime }}(\\textrm {ind}_{\\mathbb {U}^{\\prime }}^{\\Gamma ^{\\prime }}(1)),$ where $\\textnormal {ind}$ denotes induction in the category of representations of finite groups and 1 denotes the trivial character of $\\mathbb {U}$ or $\\mathbb {U}^{\\prime }$ .", "Extending functions by zero induces the injections $\\textrm {ind}_{\\mathbb {U}}^{\\Gamma }(1)\\cong \\textrm {ind}_{I(1)}^{K}(1)\\hookrightarrow \\textrm {c-ind}_{I(1)}^G(1)$ and $\\textrm {ind}_{\\mathbb {U}^{\\prime }}^{\\Gamma ^{\\prime }}(1)\\cong \\textrm {ind}_{I(1)}^{K^{\\prime }}(1)\\hookrightarrow \\textrm {c-ind}_{I(1)}^G(1)$ .", "Passing to $I(1)$ -invariants, we may view the algebras $\\mathcal {H}_\\Gamma $ and $\\mathcal {H}_{\\Gamma ^{\\prime }}$ as subalgebras of $\\mathcal {H}_{\\overline{\\mathbb {F}}_p}(G,I(1))$ by the morphisms $\\mathcal {H}_\\Gamma \\hookrightarrow \\textrm {Hom}_K(\\textrm {ind}_{I(1)}^{K}(1),\\textrm {c-ind}_{I(1)}^G(1)|_K) \\cong \\textrm {Hom}_G(\\textrm {c-ind}_{I(1)}^{G}(1),\\textrm {c-ind}_{I(1)}^G(1)) = \\mathcal {H}_{\\overline{\\mathbb {F}}_p}(G,I(1)),$ $\\mathcal {H}_{\\Gamma ^{\\prime }} \\hookrightarrow \\textrm {Hom}_{K^{\\prime }}(\\textrm {ind}_{I(1)}^{K^{\\prime }}(1),\\textrm {c-ind}_{I(1)}^G(1)|_{K^{\\prime }}) \\cong \\textrm {Hom}_G(\\textrm {c-ind}_{I(1)}^{G}(1),\\textrm {c-ind}_{I(1)}^G(1)) = \\mathcal {H}_{\\overline{\\mathbb {F}}_p}(G,I(1)).$ We deduce from these morphisms that the algebra $\\mathcal {H}_{\\Gamma }$ is generated by ${n_s}$ and $e_\\chi $ for all characters $\\chi $ of $H$ , while $\\mathcal {H}_{\\Gamma ^{\\prime }}$ is generated by ${n_{s^{\\prime }}}$ and $e_\\chi $ for all characters $\\chi $ of $H$ .", "Definition 5.2 We define Table: NO_CAPTIONDefinition 5.3 Let $\\chi :H\\rightarrow \\overline{\\mathbb {F}}_p^\\times $ be a character.", "Let $J\\subset J_0(\\chi )$ , and let $M_{\\chi ,J}$ denote the character of $\\mathcal {H}_\\Gamma $ given by Table: NO_CAPTION for $\\chi ^{\\prime }\\ne \\chi $ .", "Let $J^{\\prime }\\subset J_0^{\\prime }(\\chi )$ , and let $M_{\\chi ,J^{\\prime }}^{\\prime }$ denote the character of $\\mathcal {H}_{\\Gamma ^{\\prime }}$ given by Table: NO_CAPTION for $\\chi ^{\\prime }\\ne \\chi $ .", "With these definitions in place, we arrive at the following Proposition.", "Proposition 5.4 Let $\\chi :H\\rightarrow \\overline{\\mathbb {F}}_p^\\times $ be a character.", "Every simple right $\\mathcal {H}_\\Gamma $ -module is isomorphic to a character $M_{\\chi ,J}$ with $J\\subset J_0(\\chi )$ .", "Every simple right $\\mathcal {H}_{\\Gamma ^{\\prime }}$ -module is isomorphic to a character $M^{\\prime }_{\\chi ,J^{\\prime }}$ with $J^{\\prime }\\subset J_0^{\\prime }(\\chi )$ .", "The pairs $(\\mathbb {B},(N\\cap K)/(N\\cap K_1))$ and $(\\mathbb {B}^{\\prime },(N\\cap K^{\\prime })/(N\\cap K_1^{\\prime }))$ form “strongly split BN pairs of characteristic $p$ ” (cf.", "[9] Definition 2.20).", "The result then follows from Theorem 6.10(iii) of [9]." ], [ "Carter-Lusztig Theory", "Using the results of the previous section, we may begin classifying the mod-$p$ representations of the finite groups $\\Gamma $ and $\\Gamma ^{\\prime }$ .", "The starting point of this theory relies on Proposition 26 of [29]: if $\\rho $ is a nonzero mod-$p$ representation of $\\Gamma $ , then $\\rho ^\\mathbb {U}\\ne \\lbrace 0\\rbrace $ .", "The latter space has an action of the Hecke algebra $\\mathcal {H}_\\Gamma $ , so we obtain a functor from the category of mod-$p$ representations of $\\Gamma $ to right $\\mathcal {H}_\\Gamma $ -modules.", "We remark that this discussion holds equally well for $\\Gamma ^{\\prime }$ and $\\mathbb {U}^{\\prime }$ .", "The properties of this functor are made precise by the following Proposition: Proposition 5.5 The functor $\\rho \\mapsto \\rho ^{\\mathbb {U}}$ induces a bijection between irreducible representations of $\\Gamma $ and simple right $\\mathcal {H}_{\\Gamma }$ -modules.", "The functor $\\rho ^{\\prime }\\mapsto (\\rho ^{\\prime })^{\\mathbb {U}^{\\prime }}$ induces a bijection between irreducible representations of $\\Gamma ^{\\prime }$ and simple right $\\mathcal {H}_{\\Gamma ^{\\prime }}$ -modules.", "Since $\\mathcal {H}_\\Gamma $ and $\\mathcal {H}_{\\Gamma ^{\\prime }}$ are Frobenius algebras, the result follows from Proposition 1.25(ii) of [9].", "In light of this Proposition, we make the following definition: Definition 5.6 Let $\\chi :H\\rightarrow \\overline{\\mathbb {F}}_p^\\times $ be a character.", "For $J\\subset J_0(\\chi )$ , we define $\\rho _{\\chi ,J}$ to be the representation of $\\Gamma $ such that $\\rho _{\\chi ,J}^\\mathbb {U}\\cong M_{\\chi ,J}$ .", "For $J^{\\prime }\\subset J_0^{\\prime }(\\chi )$ , we define $\\rho _{\\chi ,J^{\\prime }}^{\\prime }$ to be the representation of $\\Gamma ^{\\prime }$ such that $(\\rho _{\\chi ,J^{\\prime }}^{\\prime })^{\\mathbb {U}^{\\prime }} \\cong M_{\\chi ,J^{\\prime }}^{\\prime }$ .", "The irreducible mod-$p$ representations of $\\Gamma $ (resp.", "$\\Gamma ^{\\prime }$ ) have been classified by Carter and Lusztig in terms of characters $\\chi $ of $H$ and certain subsets of $S$ (resp.", "$S^{\\prime }$ ).", "More precisely, given a nonzero irreducible mod-$p$ representation $\\rho $ of $\\Gamma $ , we have $\\rho ^\\mathbb {U}\\ne \\lbrace 0\\rbrace $ ; by Frobenius Reciprocity for finite groups, we obtain a surjection from $\\textnormal {ind}_\\mathbb {U}^\\Gamma (1)$ onto $\\rho $ , where 1 denotes the trivial character of $\\mathbb {U}$ .", "Since $\\textnormal {ind}_\\mathbb {U}^\\Gamma (1)$ decomposes as a direct sum of $\\textnormal {ind}_\\mathbb {B}^\\Gamma (\\chi )$ , we see that $\\rho $ is actually a quotient of a parabolically induced representation.", "In [10], Carter and Lusztig show how to construct irreducible quotients of parabolic inductions $\\textnormal {ind}_\\mathbb {B}^\\Gamma (\\chi )$ by using the Hecke operators $e_\\chi $ and ${n_s}$ (with analogous results holding for the group $\\Gamma ^{\\prime }$ ).", "Proposition 5.7 Let $\\chi :H\\rightarrow \\overline{\\mathbb {F}}_p^\\times $ be a character.", "If $\\chi $ factors through the determinant, then $\\rho _{\\chi ,S} \\cong \\textnormal {im}(1 + {n_s}:\\textnormal {ind}_\\mathbb {B}^\\Gamma (\\chi )\\rightarrow \\textnormal {ind}_\\mathbb {B}^\\Gamma (\\chi )),$ $\\rho _{\\chi ,\\emptyset } \\cong \\textnormal {im}({n_s}:\\textnormal {ind}_\\mathbb {B}^\\Gamma (\\chi )\\rightarrow \\textnormal {ind}_\\mathbb {B}^\\Gamma (\\chi )).$ If $\\chi $ does not factor through the determinant, then $\\rho _{\\chi ,\\emptyset } \\cong \\textnormal {im}({n_s}:\\textnormal {ind}_\\mathbb {B}^\\Gamma (\\chi )\\rightarrow \\textnormal {ind}_\\mathbb {B}^\\Gamma (\\chi ^s)).$ If $\\chi ^s = \\chi $ , then $\\rho ^{\\prime }_{\\chi ,S^{\\prime }} \\cong \\textnormal {im}(1 + {n_{s^{\\prime }}}:\\textnormal {ind}_{\\mathbb {B}^{\\prime }}^{\\Gamma ^{\\prime }}(\\chi )\\rightarrow \\textnormal {ind}_{\\mathbb {B}^{\\prime }}^{\\Gamma ^{\\prime }}(\\chi )),$ $\\rho ^{\\prime }_{\\chi ,\\emptyset } \\cong \\textnormal {im}({n_{s^{\\prime }}}:\\textnormal {ind}_{\\mathbb {B}^{\\prime }}^{\\Gamma ^{\\prime }}(\\chi )\\rightarrow \\textnormal {ind}_{\\mathbb {B}^{\\prime }}^{\\Gamma ^{\\prime }}(\\chi )).$ If $\\chi ^s \\ne \\chi $ , then $\\rho ^{\\prime }_{\\chi ,\\emptyset } \\cong \\textnormal {im}({n_{s^{\\prime }}}:\\textnormal {ind}_{\\mathbb {B}^{\\prime }}^{\\Gamma ^{\\prime }}(\\chi )\\rightarrow \\textnormal {ind}_{\\mathbb {B}^{\\prime }}^{\\Gamma ^{\\prime }}(\\chi ^s)).$ Theorem 7.1 and Corollary 7.5 of [10] imply that the images of the Hecke operators are irreducible and inequivalent; it therefore suffices to match the two sets of representations.", "Theorem 7.1 and Proposition 6.6 of [10] give the action of $\\mathcal {H}_\\Gamma $ and $\\mathcal {H}_{\\Gamma ^{\\prime }}$ on the $\\mathbb {U}$ - and $\\mathbb {U}^{\\prime }$ -invariants of the image representations.", "The claim then follows from Proposition REF and Definition REF .", "Lemma 5.8 Let $\\chi = \\zeta \\otimes \\eta :H\\rightarrow \\overline{\\mathbb {F}}_p^\\times $ be a character.", "Assume $\\chi = \\eta \\circ \\det $ .", "Then $e_\\chi (1 + {n_s})e_\\chi $ and $-e_\\chi {n_s}e_\\chi $ are orthogonal idempotents, and induce a splitting $\\textnormal {ind}_\\mathbb {B}^\\Gamma (\\chi )\\cong \\rho _{\\chi ,S}\\oplus \\rho _{\\chi ,\\emptyset }.$ Moreover, we have $\\rho _{\\chi ,S} \\cong \\eta \\circ \\det ,\\ \\rho _{\\chi ,\\emptyset }\\cong \\eta \\circ \\det \\otimes \\textnormal {St},$ where $\\textnormal {St} = \\textnormal {ind}_\\mathbb {B}^\\Gamma (1)/1$ is the Steinberg representation of $\\Gamma $ .", "Assume $\\chi ^s = \\chi $ .", "Then $e_\\chi (1 + {n_{s^{\\prime }}})e_\\chi $ and $-e_\\chi {n_{s^{\\prime }}}e_\\chi $ are orthogonal idempotents, and induce a splitting $\\textnormal {ind}_{\\mathbb {B}^{\\prime }}^{\\Gamma ^{\\prime }}(\\chi )\\cong \\rho ^{\\prime }_{\\chi ,S^{\\prime }}\\oplus \\rho ^{\\prime }_{\\chi ,\\emptyset }.$ Let $\\det ^\\star :\\mathbf {U}(1,1)(\\mathbb {F}_{q^2}/\\mathbb {F}_q)\\rightarrow \\mathbf {U}(1)(\\mathbb {F}_{q^2}/\\mathbb {F}_q)$ denote the determinant map of the group $\\mathbf {U}(1,1)(\\mathbb {F}_{q^2}/\\mathbb {F}_q)$ .", "Then there exists a unique character $\\zeta ^{\\prime }:\\mathbf {U}(1)(\\mathbb {F}_{q^2}/\\mathbb {F}_q)\\rightarrow \\overline{\\mathbb {F}}_p^\\times $ such that $\\chi \\cong ((\\zeta ^{\\prime }\\circ \\det ^\\star )\\boxtimes \\eta )|_H$ , and we have Table: NO_CAPTION where $\\textnormal {St}^{\\prime } = \\textnormal {ind}_{\\mathbb {B}^{\\prime }\\cap \\mathbf {U}(1,1)(\\mathbb {F}_{q^2}/\\mathbb {F}_q)}^{\\mathbf {U}(1,1)(\\mathbb {F}_{q^2}/\\mathbb {F}_q)}(1)/1$ is the Steinberg representation of $\\mathbf {U}(1,1)(\\mathbb {F}_{q^2}/\\mathbb {F}_q)$ .", "Remark We use the notation $\\boxtimes $ to denote the external tensor product of representations of $\\mathbf {U}(1,1)(\\mathbb {F}_{q^2}/\\mathbb {F}_q)$ and $\\mathbf {U}(1)(\\mathbb {F}_{q^2}/\\mathbb {F}_q)$ .", "The first claim of parts (i) and (ii) follow from Theorem REF .", "For a character $\\chi = \\eta \\circ \\det $ that factors through the determinant, we have $\\textrm {ind}_\\mathbb {B}^\\Gamma (\\chi )\\cong (\\eta \\circ \\det )\\otimes \\textrm {ind}_\\mathbb {B}^\\Gamma (1)$ , so it suffices to assume $\\chi = 1$ is the trivial character of $\\mathbb {B}$ .", "Theorem 7.1 of [10] and Proposition REF now imply that $\\rho _{1,S}$ is the trivial representation, which means $\\rho _{1,\\emptyset } = \\textrm {St}$ .", "This proof also applies mutatis mutandis for representations of $\\Gamma ^{\\prime }$ .", "We record one final result regarding the constituents of $\\textnormal {ind}_{\\mathbb {B}^{\\prime }}^{\\Gamma ^{\\prime }}(\\chi )$ , which will be of use later.", "Lemma 5.9 Let $\\chi :H\\rightarrow \\overline{\\mathbb {F}}_p^\\times $ be a character.", "Assume $\\chi ^s = \\chi $ .", "Then we have $\\textnormal {ind}_{\\mathbb {B}^{\\prime }}^{\\Gamma ^{\\prime }}(\\chi ) \\cong \\rho _{\\chi ,S^{\\prime }}^{\\prime }\\oplus \\rho _{\\chi ,\\emptyset }^{\\prime }.$ Assume $\\chi ^s \\ne \\chi $ .", "Then the sequence $0 \\rightarrow \\rho _{\\chi ^s,\\emptyset }^{\\prime } \\rightarrow \\textnormal {ind}_{\\mathbb {B}^{\\prime }}^{\\Gamma ^{\\prime }}(\\chi ) \\rightarrow \\rho _{\\chi ,\\emptyset }^{\\prime } \\rightarrow 0$ is exact if and only if $q = p$ .", "In this case, the sequence is nonsplit.", "Part (i) follows from Lemma REF .", "For part (ii), note that the representation $\\rho _{\\chi ,\\emptyset }^{\\prime }$ is defined by $\\rho _{\\chi ,\\emptyset }^{\\prime } \\cong {n_{s^{\\prime }}}(\\textrm {ind}_{\\mathbb {B}^{\\prime }}^{\\Gamma ^{\\prime }}(\\chi ))$ ; moreover, the endomorphism ${n_{s^{\\prime }}}$ maps $\\textrm {ind}_{\\mathbb {B}^{\\prime }}^{\\Gamma ^{\\prime }}(\\chi ^s)$ into $\\textrm {ind}_{\\mathbb {B}^{\\prime }}^{\\Gamma ^{\\prime }}(\\chi )$ , which implies $\\rho _{\\chi ^s,\\emptyset }^{\\prime }\\subset \\textrm {ind}_{\\mathbb {B}^{\\prime }}^{\\Gamma ^{\\prime }}(\\chi )$ .", "Since ${n_{s^{\\prime }}}^2 = 0$ on the space $\\textrm {ind}_{\\mathbb {B}^{\\prime }}^{\\Gamma ^{\\prime }}(\\chi ^s)$ , we have $\\rho _{\\chi ^s,\\emptyset }^{\\prime } \\subset \\ker ({n_{s^{\\prime }}})$ .", "If $\\rho _{\\chi ,\\emptyset }^{\\prime }$ is isomorphic to the representation $V_{j,k}^{\\prime }\\boxtimes \\omega ^c$ defined below, Proposition REF implies $\\dim _{\\overline{\\mathbb {F}}_p}(\\rho _{\\chi ,\\emptyset }^{\\prime }) + \\dim _{\\overline{\\mathbb {F}}_p}(\\rho _{\\chi ^s,\\emptyset }^{\\prime }) = \\prod _{i = 0}^{f-1}(j_i + 1) + \\prod _{i = 0}^{f-1}(p - j_i);$ this quantity is equal to $q + 1 = \\dim _{\\overline{\\mathbb {F}}_p}(\\textrm {ind}_{\\mathbb {B}^{\\prime }}^{\\Gamma ^{\\prime }}(\\chi ))$ if and only if $q = p$ .", "Theorem 7.4 of [10] implies that the sequence is nonsplit.", "Remark One can show, by computing dimensions and using Proposition REF below, that for $\\chi \\ne \\eta \\circ \\det $ the sequence $0 \\rightarrow \\rho _{\\chi ^s,\\emptyset } \\rightarrow \\textnormal {ind}_{\\mathbb {B}}^{\\Gamma }(\\chi ) \\rightarrow \\rho _{\\chi ,\\emptyset } \\rightarrow 0$ is never exact, even for $q = p$ ." ], [ "Highest Weight Modules: U(2,1)", "We now describe a classification of representations of $\\Gamma $ in terms of highest weight modules.", "We begin with the representations of $\\textrm {SL}_3(\\overline{\\mathbb {F}}_p)$ .", "Let $\\chi _{j,k}$ denote the character of the maximal torus of $\\textrm {SL}_3(\\overline{\\mathbb {F}}_p)$ given by $\\chi _{j,k}\\begin{pmatrix}a & 0 & 0\\\\ 0 & a^{-1}b & 0\\\\ 0 & 0 & b^{-1}\\end{pmatrix} = a^jb^k,$ where $a,b\\in \\overline{\\mathbb {F}}_p^\\times $ and $j,k\\in \\mathbb {Z}$ .", "The characters $\\chi _{j,k}$ with $j,k\\ge 0$ are called the dominant weights (with respect to the “standard” choice of upper Borel subgroup).", "The characters $\\chi _{1,0}$ and $\\chi _{0,1}$ are the fundamental dominant weights.", "Theorem 5.10 The irreducible finite-dimensional mod-$p$ representations of $\\textnormal {SL}_3(\\overline{\\mathbb {F}}_p)$ are parametrized by the set of dominant weights.", "For a weight $\\chi _{j,k}$ , we let $V_{j,k}$ denote the corresponding representation.", "If $\\mathcal {U}$ denotes the upper unipotent elements of $\\textnormal {SL}_3(\\overline{\\mathbb {F}}_p)$ , then $V_{j,k}^\\mathcal {U}$ is one-dimensional, and the upper Borel subgroup of $\\textnormal {SL}_3(\\overline{\\mathbb {F}}_p)$ acts on $V_{j,k}^{\\mathcal {U}}$ by the character $\\chi _{j,k}$ .", "This is $\\mathsection $ 12, Theorem 39 in [30].", "Given a representation $V$ of $\\textrm {SL}_3(\\overline{\\mathbb {F}}_p)$ , we form a new representation $V^{\\textrm {Fr}}$ : the underlying space of $V^{\\textrm {Fr}}$ is the same as that of $V$ , with the action given by first applying the map $x\\mapsto x^p$ to the entries of an element of $\\textrm {SL}_3(\\overline{\\mathbb {F}}_p)$ .", "In particular, we have $V_{j,k}^{\\textrm {Fr}}\\cong V_{pj,pk}$ .", "With this tool we can be more precise about the structure of the representations $V_{j,k}$ thanks to Steinberg's Tensor Product Theorem: Theorem 5.11 Let $j,k\\in \\mathbb {Z}_{\\ge 0}$ , and let $j = \\sum _{i\\ge 0}j_ip^i, k = \\sum _{i\\ge 0}k_ip^i$ be the $p$ -adic expansions of $j$ and $k$ .", "Then $\\displaystyle {V_{j,k}\\cong \\bigotimes _{i = 0}^\\infty V_{j_i,k_i}^{\\textnormal {Fr}^i}}$ as $\\textnormal {SL}_3(\\overline{\\mathbb {F}}_p)$ -representations.", "This is $\\mathsection $ 12, Theorem 41 in [30].", "Remark The theorem above shows that in order to classify the irreducible finite-dimensional representations of $\\textnormal {SL}_3(\\overline{\\mathbb {F}}_p)$ , it suffices to understand the representations $V_{j,k}$ with $0\\le j,k < p$ .", "The precise structure of these representations is governed by the Linkage Principle.", "We shall not need these results here, but for more information on this topic the reader may consult the book of Jantzen ([22]).", "Theorem 5.12 The representations $V_{j,k}$ of $\\textnormal {SL}_3(\\overline{\\mathbb {F}}_p)$ with $0\\le j,k < q$ remain irreducible upon restriction to $\\mathbf {SU}(2,1)(\\mathbb {F}_{q^2}/\\mathbb {F}_q)$ .", "Moreover, the given representations $V_{j,k}$ exhaust the irreducible mod-$p$ representations of $\\mathbf {SU}(2,1)(\\mathbb {F}_{q^2}/\\mathbb {F}_q)$ .", "This is $\\mathsection $ 13, Theorem 43 in [30].", "To obtain the irreducible representations of $\\Gamma = \\mathbf {U}(2,1)(\\mathbb {F}_{q^2}/\\mathbb {F}_q)$ , we proceed as follows.", "The subgroup $\\left\\lbrace \\begin{pmatrix}1 & 0 & 0\\\\ 0 & \\delta & 0 \\\\ 0 & 0 & 1\\end{pmatrix}: \\delta \\in \\mathbf {U}(1)(\\mathbb {F}_{q^2}/\\mathbb {F}_q)\\right\\rbrace $ gives a full set of coset representatives for $\\Gamma /\\mathbf {SU}(2,1)(\\mathbb {F}_{q^2}/\\mathbb {F}_q)$ .", "For an irreducible representation $V_{j,k}$ of $\\mathbf {SU}(2,1)(\\mathbb {F}_{q^2}/\\mathbb {F}_q)$ , we let $V_{j,k}^\\delta $ denote the representation with the same underlying space as $V_{j,k}$ , with the action given by first conjugating an element of $\\mathbf {SU}(2,1)(\\mathbb {F}_{q^2}/\\mathbb {F}_q)$ by $\\left({\\begin{matrix}1 & 0 & 0 \\\\ 0 & \\delta & 0 \\\\ 0 & 0 & 1\\end{matrix}}\\right)$ .", "Since these diagonal elements normalize $\\mathbb {U}\\le \\mathbf {SU}(2,1)(\\mathbb {F}_{q^2}/\\mathbb {F}_q)$ , we have that $(V_{j,k}^\\delta )^\\mathbb {U}= V_{j,k}^\\mathbb {U}$ (as vector spaces), for every $\\delta \\in \\mathbf {U}(1)(\\mathbb {F}_{q^2}/\\mathbb {F}_q)$ .", "Moreover, the action of the maximal torus of $\\mathbf {SU}(2,1)(\\mathbb {F}_{q^2}/\\mathbb {F}_q)$ on these spaces is identical.", "This implies that $V_{j,k}^\\delta \\cong V_{j,k}$ for every $\\delta \\in \\mathbf {U}(1)(\\mathbb {F}_{q^2}/\\mathbb {F}_q)$ , which means we may lift $V_{j,k}$ to a projective representation of $\\Gamma $ .", "Since $H^2(\\Gamma /\\mathbf {SU}(2,1)(\\mathbb {F}_{q^2}/\\mathbb {F}_q),\\overline{\\mathbb {F}}_p^\\times ) = \\lbrace 0\\rbrace ,$ this representation lifts to a genuine representation of $\\Gamma $ .", "After twisting by an appropriate power of the determinant, we can ensure that the element $\\left({\\begin{matrix}a & 0 & 0 \\\\ 0 & \\delta & 0 \\\\ 0 & 0 & \\overline{a}^{-1}\\end{matrix}}\\right)$ acts by the scalar $a^{j+qk}$ on $V_{j,k}^\\mathbb {U}$ .", "We continue to denote by $V_{j,k}$ this representation of $\\Gamma $ .", "Corollary 5.13 The irreducible mod-$p$ representations of $\\Gamma $ are given by $V_{j,k}\\otimes (\\det )^c$ , where $0\\le j,k < q$ and $0\\le c < q+1$ .", "Restricting $V_{j,k}\\otimes (\\det )^c$ to $\\mathbf {SU}(2,1)(\\mathbb {F}_{q^2}/\\mathbb {F}_q)$ verifies that each representation is irreducible, and by examining the action of $H$ (and dimensions of the $V_{j,k}$ ) we see that they are pairwise nonisomorphic.", "Since the number of $p$ -regular conjugacy classes of $\\Gamma $ is $q^2(q+1)$ , we conclude that these exhaust all irreduble representations.", "We can now provide a dictionary between the Carter-Lusztig description of representations and the description in terms of highest weight modules.", "Proposition 5.14 Let $\\chi = \\zeta \\otimes \\eta :H\\rightarrow \\overline{\\mathbb {F}}_p^\\times $ be a character.", "Let $0\\le r < q^2 - 1$ be the unique integer such that $\\zeta (a) = a^r$ for every $a\\in \\mathbb {F}_{q^2}^\\times $ , and let $0\\le c<q + 1$ be the unique integer such that $\\eta (\\delta ) = \\delta ^c$ for every $\\delta \\in \\mathbf {U}(1)(\\mathbb {F}_{q^2}/\\mathbb {F}_q)$ .", "Assume $r = 0$ .", "Then $\\chi = \\eta \\circ \\det $ factors through the determinant, and we have $\\rho _{\\chi ,S} & \\cong & V_{0,0}\\otimes V_{0,0}^{\\textnormal {Fr}}\\otimes \\cdots \\otimes V_{0,0}^{\\textnormal {Fr}^{f-1}}\\otimes (\\det )^c \\cong (\\det )^c\\\\\\rho _{\\chi ,\\emptyset } & \\cong & V_{p-1,p-1}\\otimes V_{p-1,p-1}^{\\textnormal {Fr}}\\otimes \\cdots \\otimes V_{p-1,p-1}^{\\textnormal {Fr}^{f-1}}\\otimes (\\det )^c \\cong \\textnormal {St}\\otimes (\\det )^c,$ where $\\textnormal {St}$ denotes the Steinberg representation.", "Assume $r\\ne 0$ .", "Then there exists a unique pair $(j,k)$ such that $0\\le j,k < q$ and $j+qk = r$ .", "Let $j = \\sum _{i=0}^{f-1}j_ip^i,\\ k = \\sum _{i=0}^{f-1} k_ip^i$ be the $p$ -adic expansions of $j$ and $k$ .", "We have $\\rho _{\\chi ,\\emptyset } & \\cong & V_{j_0,k_0}\\otimes V_{j_1,k_1}^{\\textnormal {Fr}}\\otimes \\cdots \\otimes V_{j_{f-1},k_{f-1}}^{\\textnormal {Fr}^{f-1}}\\otimes (\\det )^c.$ In each description of irreducibles, we have $q^2(q+1)$ representations; it therefore suffices to match these representations.", "Given a character $\\chi = \\zeta \\otimes \\eta :H\\rightarrow \\overline{\\mathbb {F}}_p^\\times $ with parameters $(r,c)$ , we have $\\chi \\begin{pmatrix}a & 0 & 0\\\\ 0 & \\delta & 0 \\\\ 0 & 0 & \\overline{a}^{-1}\\end{pmatrix} = a^r(a\\overline{a}^{-1}\\delta )^c.$ We see that $r = 0$ if and only if $\\chi = \\eta \\circ \\det $ .", "In this case, Lemma REF implies that $\\rho _{\\chi ,S}\\cong \\eta \\circ \\det = (\\det )^c\\quad \\textnormal {and}\\quad \\rho _{\\chi ,\\emptyset }\\cong \\textrm {St}\\otimes \\eta \\circ \\det = \\textrm {St}\\otimes (\\det )^c.$ Of the representations $V_{j,k}\\otimes (\\det )^c$ , the only representations on which $H$ acts by $(a\\overline{a}^{-1}\\delta )^c$ on the $\\mathbb {U}$ -invariants are $V_{0,0}\\otimes (\\det )^c$ and $V_{q-1,q-1}\\otimes (\\det )^c$ .", "We have that the dimension of $V_{0,0}\\otimes (\\det )^c$ is 1, while the representation $\\textrm {St}$ has dimension $|\\mathbb {U}| = q^3$ ; hence we must have $(\\det )^c\\cong V_{0,0}\\otimes (\\det )^c\\quad \\textnormal {and}\\quad \\textrm {St}\\otimes (\\det )^c\\cong V_{q-1,q-1}\\otimes (\\det )^c.$ As before, it suffices to match the action of $H$ on the $\\mathbb {U}$ -invariants of each representation.", "Let $(j,k)$ be integers such that $0\\le j,k < q$ and $r = j + qk$ .", "We see that $H$ acts on $(V_{j,k}\\otimes (\\det )^c)^{\\mathbb {U}}$ by $a^{j+qk}(a\\overline{a}^{-1}\\delta )^c = a^r(a\\overline{a}^{-1}\\delta )^c$ .", "Writing out the $p$ -adic expansions of $j$ and $k$ implies $\\rho _{\\chi ,\\emptyset }\\cong V_{j_0,k_0}\\otimes V_{j_1,k_1}^{\\textrm {Fr}}\\otimes \\cdots \\otimes V_{j_{f-1},k_{f-1}}^{\\textrm {Fr}^{f-1}}\\otimes (\\det )^c.$ Remark The results of [30] apply more generally to a split reductive algebraic group over a finite field with a semisimple derived subgroup, or one of their “twisted analogues.” In particular, we may match the representations $\\Theta _{w_0}^J{F}_\\chi $ of Carter-Lusztig with the highest weight modules as described in [30]." ], [ "Highest Weight Modules: U(1,1)$\\times $ U(1)", "We now describe representations of $\\Gamma ^{\\prime } = (\\mathbf {U}(1,1)\\times \\mathbf {U}(1))(\\mathbb {F}_{q^2}/\\mathbb {F}_q)$ in terms of highest weight modules.", "Every such representation is of the form $\\rho ^{\\prime }\\boxtimes \\eta $ , where $\\rho ^{\\prime }$ is a representation of $\\mathbf {U}(1,1)(\\mathbb {F}_{q^2}/\\mathbb {F}_q)$ and $\\eta $ is a character of $\\mathbf {U}(1)(\\mathbb {F}_{q^2}/\\mathbb {F}_q)$ .", "Though we may again use the results of [30], we instead proceed in a more explicit and ad hoc manner.", "Definition 5.15 Let $0\\le j < q, 0 \\le k < q + 1$ , and let $j = \\sum _{i = 0}^{f-1}j_ip^i$ be the $p$ -adic expansion of $j$ .", "We denote by $V_{j,k}^{\\prime }$ the representation of $\\mathbf {U}(1,1)(\\mathbb {F}_{q^2}/\\mathbb {F}_q)$ given by $\\textrm {Sym}^{j_0}(\\overline{\\mathbb {F}}_p^2)\\otimes \\textrm {Sym}^{j_1}(\\overline{\\mathbb {F}}_p^2)^{\\textrm {Fr}}\\otimes \\cdots \\otimes \\textrm {Sym}^{j_{f-1}}(\\overline{\\mathbb {F}}_p^2)^{\\textrm {Fr}^{f-1}}\\otimes (\\mathop {\\mmlmultiscripts{\\det {\\mmlnone }{\\star }}}\\limits )^k,$ where $\\det ^\\star $ denotes the determinant map of $\\mathbf {U}(1,1)(\\mathbb {F}_{q^2}/\\mathbb {F}_q)$ .", "Definition 5.16 We let $\\omega :\\mathbf {U}(1)(\\mathbb {F}_{q^2}/\\mathbb {F}_q)\\hookrightarrow \\mathbb {F}_{q^2}^\\times \\stackrel{\\iota }{\\rightarrow } \\overline{\\mathbb {F}}_p^\\times $ denote a fixed fundamental character of $\\mathbf {U}(1)(\\mathbb {F}_{q^2}/\\mathbb {F}_q)$ .", "Every $\\overline{\\mathbb {F}}_p^\\times $ -valued character of $\\mathbf {U}(1)(\\mathbb {F}_{q^2}/\\mathbb {F}_q)$ is of the form $\\omega ^c$ , $0\\le c < q+1$ .", "Theorem 5.17 The irreducible mod-$p$ representations of $\\Gamma ^{\\prime }$ are given by $V_{j,k}^{\\prime }\\boxtimes \\omega ^c$ , where $0\\le j < q$ and $0\\le k,c < q + 1$ .", "One may check that the $q(q+1)^2$ representations $V_{j,k}^{\\prime }\\boxtimes \\omega ^c$ are inequivalent and irreducible.", "Since the number of $p$ -regular conjugacy classes of $\\Gamma ^{\\prime }$ is $q(q+1)^2$ , we conclude that these exhaust all irreducible representations.", "Remark We remark that this theorem may be deduced from the fact that $\\mathbf {SU}(1,1)(\\mathbb {F}_{q^2}/\\mathbb {F}_q)$ is conjugate to $\\textrm {SL}_2(\\mathbb {F}_q)$ inside of $\\textrm {SL}_2(\\overline{\\mathbb {F}}_p)$ .", "Lemma 5.18 Let $0\\le j < q$ and $0\\le k,c < q + 1$ .", "Then the action of $H$ on $(V_{j,k}^{\\prime }\\boxtimes \\omega ^c)^{\\mathbb {U}^{\\prime }}$ is given by the character $\\left({\\begin{matrix}a & 0 & 0 \\\\ 0 & \\delta & 0 \\\\ 0 & 0 & \\overline{a}^{-1}\\end{matrix}}\\right)\\mapsto a^{-qj + (1-q)k}\\delta ^c$ .", "The previous theorem implies that $V_{j,k}^{\\prime }\\boxtimes \\omega ^c$ is irreducible, and Proposition REF implies that $(V_{j,k}^{\\prime }\\boxtimes \\omega ^c)^{\\mathbb {U}^{\\prime }}$ is one-dimensional.", "Let $j = \\sum _{i=0}^{f-1}j_ip^i$ be the $p$ -adic expansion of $j$ , and let $\\lbrace v_1,v_2\\rbrace $ be the standard basis of $\\overline{\\mathbb {F}}_p^2$ .", "The vector $v_2^{j_0}\\otimes v_2^{j_1}\\otimes \\cdots \\otimes v_2^{j_{f-1}}\\otimes 1\\boxtimes 1$ is fixed by $\\mathbb {U}^{\\prime }$ , and therefore spans $(V_{j,k}^{\\prime }\\boxtimes \\omega ^c)^{\\mathbb {U}^{\\prime }}$ .", "The action of $\\left({\\begin{matrix}a & 0 & 0 \\\\ 0 & \\delta & 0 \\\\ 0 & 0 & \\overline{a}^{-1}\\end{matrix}}\\right)$ on this vector is given by $(\\overline{a}^{-1})^ja^{(1-q)k}\\delta ^c = a^{-qj + (1-q)k}\\delta ^c$ .", "We may now provide a dictionary between the representations $\\rho _{\\chi ,J^{\\prime }}^{\\prime }$ and $V_{j,k}^{\\prime }\\boxtimes \\omega ^c$ .", "Proposition 5.19 Let $\\chi = \\zeta \\otimes \\eta :H\\rightarrow \\overline{\\mathbb {F}}_p^\\times $ be a character.", "Let $0\\le c<q + 1$ be the unique integer such that $\\eta (\\delta ) = \\delta ^c$ for every $\\delta \\in \\mathbf {U}(1)(\\mathbb {F}_{q^2}/\\mathbb {F}_q)$ , and let $0\\le r < q^2 - 1$ be the unique integer such that $\\zeta (a) = a^{r + c(q-1)}$ for every $a\\in \\mathbb {F}_{q^2}^\\times $ .", "Assume $r \\equiv 0\\ (\\textnormal {mod}\\ q - 1)$ .", "Then $\\chi ^s = \\chi $ , and there exists a unique integer $0\\le k < q + 1$ such that $(1-q)k \\equiv r\\ (\\textnormal {mod}\\ q^2 - 1)$ .", "We have $\\rho _{\\chi ,S^{\\prime }}^{\\prime } & \\cong & V_{0,k}^{\\prime }\\boxtimes \\omega ^c \\cong (\\mathop {\\mmlmultiscripts{\\det {\\mmlnone }{\\star }}}\\limits )^k\\boxtimes \\omega ^c\\\\\\rho _{\\chi ,\\emptyset }^{\\prime } & \\cong & V_{q-1,k+1}^{\\prime }\\boxtimes \\omega ^c \\cong (\\textnormal {St}^{\\prime }\\otimes (\\mathop {\\mmlmultiscripts{\\det {\\mmlnone }{\\star }}}\\limits )^k)\\boxtimes \\omega ^c,$ where $\\textnormal {St}^{\\prime }$ denotes the Steinberg representation of $\\mathbf {U}(1,1)(\\mathbb {F}_{q^2}/\\mathbb {F}_q)$ .", "Assume $r\\lnot \\equiv 0\\ (\\textnormal {mod}\\ q - 1)$ .", "Then $\\chi ^s\\ne \\chi $ , and there exists a unique pair $(j,k)$ such that $0< j<q, 0\\le k < q+1$ and $-qj + (1-q)k\\ \\equiv r\\ (\\textnormal {mod}\\ q^2 - 1)$ .", "Let $j = \\sum _{j=0}^{f-1}j_ip^i$ be the $p$ -adic expansion of $j$ .", "We have $\\rho _{\\chi ,\\emptyset }^{\\prime } & \\cong & V_{j,k}^{\\prime }\\boxtimes \\omega ^c \\cong (\\textnormal {Sym}^{j_0}(\\overline{\\mathbb {F}}_p^2)\\otimes \\textnormal {Sym}^{j_1}(\\overline{\\mathbb {F}}_p^2)^{\\textnormal {Fr}}\\otimes \\cdots \\otimes \\textnormal {Sym}^{j_{f-1}}(\\overline{\\mathbb {F}}_p^2)^{\\textnormal {Fr}^{f-1}}\\otimes (\\mathop {\\mmlmultiscripts{\\det {\\mmlnone }{\\star }}}\\limits )^k)\\boxtimes \\omega ^c.$ In each description of irreducibles, we have $q(q+1)^2$ representations; it suffices to match these representations.", "Given a character $\\chi = \\zeta \\otimes \\eta :H \\rightarrow \\overline{\\mathbb {F}}_p^\\times $ with parameters $(r,c)$ , we have $\\chi \\begin{pmatrix}a & 0 & 0 \\\\ 0 & \\delta & 0\\\\ 0 & 0 & \\overline{a}^{-1}\\end{pmatrix} = a^r\\delta ^c.$ We see that $r \\equiv 0\\ (\\textrm {mod}\\ q - 1)$ if and only if $\\chi ^s = \\chi $ .", "Let $0\\le k < q+1$ be the unique integer such that $(1-q)k \\equiv r\\ (\\textnormal {mod}\\ q^2 - 1)$ .", "In this case, Lemma REF implies that there exists a unique $\\zeta ^{\\prime }:\\mathbf {U}(1)(\\mathbb {F}_{q^2}/\\mathbb {F}_q)\\rightarrow \\overline{\\mathbb {F}}_p^\\times $ such that $\\rho _{\\chi ,S^{\\prime }}^{\\prime }\\cong (\\zeta ^{\\prime }\\circ \\mathop {\\mmlmultiscripts{\\det {\\mmlnone }{\\star }}}\\limits )\\boxtimes \\eta = (\\mathop {\\mmlmultiscripts{\\det {\\mmlnone }{\\star }}}\\limits )^k\\boxtimes \\omega ^c~~ \\textnormal {and}~~ \\rho _{\\chi ,\\emptyset }^{\\prime } \\cong (\\textrm {St}^{\\prime }\\otimes \\zeta ^{\\prime }\\circ \\mathop {\\mmlmultiscripts{\\det {\\mmlnone }{\\star }}}\\limits )\\boxtimes \\eta = (\\textrm {St}^{\\prime }\\otimes (\\mathop {\\mmlmultiscripts{\\det {\\mmlnone }{\\star }}}\\limits )^k)\\boxtimes \\omega ^c.$ Of the representations $V_{j,k}^{\\prime }\\boxtimes \\omega ^c$ , the only representations on which $H$ acts by $a^{(1-q)k}\\delta ^c$ on the $\\mathbb {U}^{\\prime }$ -invariants are $V_{0,k}^{\\prime }\\boxtimes \\omega ^c$ and $V_{q-1,k+1}^{\\prime }\\boxtimes \\omega ^c$ .", "Since $\\dim _{\\overline{\\mathbb {F}}_p}(V_{0,k}^{\\prime }\\boxtimes \\omega ^c) = 1$ and the representation $\\textrm {St}^{\\prime }$ has dimension $|\\mathbb {U}^{\\prime }| = q$ , we see that we must have $(\\zeta ^{\\prime }\\circ \\mathop {\\mmlmultiscripts{\\det {\\mmlnone }{\\star }}}\\limits )\\boxtimes \\eta \\cong V_{0,k}^{\\prime }\\boxtimes \\omega ^c\\quad \\textnormal {and}\\quad (\\textrm {St}^{\\prime }\\otimes (\\mathop {\\mmlmultiscripts{\\det {\\mmlnone }{\\star }}}\\limits )^k)\\boxtimes \\omega ^c \\cong V_{q-1,k+1}^{\\prime }\\boxtimes \\omega ^c.$ As before, it suffices to compute the action of $H$ on the $\\mathbb {U}^{\\prime }$ -invariants of each representation.", "Let $(j,k)$ be a pair of integers such that $0\\le j < q, 0\\le k < q+1$ and $r \\equiv -qj + (1-q)k\\ (\\textrm {mod}\\ q^2 - 1)$ ; the condition $r\\lnot \\equiv 0\\ (\\textrm {mod}\\ q - 1)$ implies $1\\le j\\le q-2$ and $\\chi ^s \\ne \\chi $ , which in turn implies that the pair $(j,k)$ is unique.", "Lemma 5.3.5 implies that $H$ acts on $(V_{j,k}^{\\prime }\\boxtimes \\omega ^c)^{\\mathbb {U}^{\\prime }}$ by $a^{-qj + (1-q)k}\\delta ^c = a^r\\delta ^c$ .", "Writing out the $p$ -adic expansion of $j$ implies $\\rho _{\\chi ,\\emptyset }\\cong V_{j,k}^{\\prime }\\boxtimes \\omega ^c \\cong (\\textrm {Sym}^{j_0}(\\overline{\\mathbb {F}}_p^2)\\otimes \\textrm {Sym}^{j_1}(\\overline{\\mathbb {F}}_p^2)^{\\textrm {Fr}}\\otimes \\cdots \\otimes \\textrm {Sym}^{j_{f-1}}(\\overline{\\mathbb {F}}_p^2)^{\\textrm {Fr}^{f-1}}\\otimes (\\det ^\\star )^k)\\boxtimes \\omega ^c.$" ], [ "Definitions and First Properties", "In this section, we follow [26] closely and translate the language of coefficient systems and diagrams to the group $G$ .", "In fact, our case is even easier to some extent, due mainly to the fact that the extended Bruhat-Tits building of $G$ coincides with the reduced Bruhat-Tits building, and therefore stabilizers of vertices are maximal compact subgroups of $G$ .", "Let $X$ be the reduced Bruhat-Tits building of $G$ (which is also the Bruhat-Tits building of $\\mathbf {SU}(2,1)(E/F)$ ).", "We refer the reader to [32] for an excellent exposition, particularly Sections 2.7 and 2.10.", "The building $X$ is a simplicial complex of dimension 1 (that is, a tree), with a natural action of $G$ .", "We let $X_0$ denote the set of all vertices on the tree, and let $X_1$ denote the set of all edges of $X$ .", "Given a simplex $\\sigma \\subset X$ , we let $\\mathfrak {K}(\\sigma )\\le G$ denote its stabilizer subgroup.", "We denote by $A$ the apartment corresponding to the maximal $F$ -split subtorus of $T$ .", "Since the group $K$ is hyperspecial, there exists a vertex $\\sigma _0$ in $A$ such that $\\mathfrak {K}(\\sigma _0) = K$ .", "Moreover, there exists a unique vertex $\\sigma _0^{\\prime }$ neighboring $\\sigma _0$ in $A$ such that $\\mathfrak {K}(\\sigma _0^{\\prime }) = K^{\\prime }$ (cf.", "[32] Section 3.1.1).", "The vertex $\\sigma _0^{\\prime }$ has $q + 1$ neighboring vertices in $X$ ; the vertex $\\sigma _0$ has $q^3 + 1$ neighboring vertices and is hyperspecial (these facts follow from Statement 3.5.4 in [32] and $|\\Gamma /\\mathbb {B}| = q^3 + 1, |\\Gamma ^{\\prime }/\\mathbb {B}^{\\prime }| = q+1$ ).", "Moreover, in any fixed apartment the vertices alternate valency (that is, the number of neighboring vertices in $X$ ) between $q^3 + 1$ and $q + 1$ .", "We let $\\tau _1$ denote the edge from $\\sigma _0$ to $\\sigma _0^{\\prime }$ ; we have $\\mathfrak {K}(\\tau _1) = I$ .", "We remark that we may alternatively define $X$ in terms of additive norms (cf.", "[32] Examples 2.10 and 3.11).", "The tree $X$ has a natural (combinatorial) $G$ -invariant distance function, and we denote by $X_0^e$ (resp.", "$X_0^o$ ) the set of vertices at an even (resp.", "odd) distance from $\\sigma _0$ .", "The group $N$ acts on $A$ , and under this action the points of $A\\cap X_0^e$ (resp.", "$A\\cap X_0^o$ ) are all conjugate.", "Given any vertex $\\sigma $ in $X_0^e$ (resp.", "$X_0^o$ ) and an apartment $A^{\\prime }$ containing $\\sigma $ and $\\sigma _0$ , there exists an element $g\\in G$ fixing $\\sigma _0$ such that $g.A^{\\prime } = A$ (cf.", "Section 2.2.1 in [32]).", "This implies that all vertices in $X_0^e$ (resp.", "$X_0^o$ ) are conjugate.", "Since the action of $G$ preserves valency, we conclude that $X_0^e$ and $X_0^o$ constitute two disjoint orbits for the action of $G$ on $X_0$ .", "Coefficient systems over $\\mathbb {C}$ were first introduced in [28] by Schneider and Stuhler, and used in the mod-$p$ setting by Paškūnas in [26].", "We recall the definition.", "Definition 6.1 A coefficient system $\\mathcal {V}=(V_{\\sigma })_{\\sigma }$ on $X$ consists of $\\overline{\\mathbb {F}}_p$ -vector spaces $V_\\sigma $ for every simplex $\\sigma \\subset X$ , along with linear restriction maps $r^\\tau _\\sigma :V_\\tau \\rightarrow V_\\sigma $ for every inclusion $\\sigma \\subset \\tau $ , such that $r^\\sigma _\\sigma = \\textnormal {id}_{V_\\sigma }$ for every $\\sigma $ .", "Definition 6.2 Let $\\mathcal {V}=(V_{\\sigma })_{\\sigma }$ be a coefficient system on $X$ .", "We say the group $G$ acts on $\\mathcal {V}$ if for every $g\\in G$ and every simplex $\\sigma \\subset X$ , we have linear maps $g_\\sigma :V_\\sigma \\rightarrow V_{g.\\sigma }$ satisfying the following properties: For every $g,h\\in G$ and every simplex $\\sigma \\subset X$ , we have $(gh)_\\sigma = g_{h.\\sigma }\\circ h_\\sigma $ .", "For every simplex $\\sigma \\subset X$ , we have $1_\\sigma = \\textnormal {id}_{V_\\sigma }$ .", "For every $g\\in G$ and every inclusion $\\sigma \\subset \\tau $ , the following diagram commutes:   V[r]g[d]r Vg.", "[d]rg.g.V[r]g Vg..", "Definition 6.3 Let $\\mathcal {V}=(V_{\\sigma })_{\\sigma }$ be a coefficient system on which $G$ acts.", "In particular, the definition above implies that $V_\\sigma $ is a representation of $\\mathfrak {K}(\\sigma )$ for every simplex $\\sigma \\subset X$ .", "If this action is smooth, we call $\\mathcal {V}$ a $G$ -equivariant coefficient system.", "We denote by $\\mathcal {COEF}_{G}$ the category of all $G$ -equivariant coefficient systems on $X$ , with the evident morphisms.", "Before going on to more details, we record the following useful fact: given a $G$ -equivariant coefficient system $\\mathcal {V}=(V_{\\sigma })_{\\sigma }$ , let $\\tau =\\lbrace \\sigma , \\sigma ^{\\prime }\\rbrace $ be an edge such that $\\sigma \\in X_0^e, \\sigma ^{\\prime }\\in X_0^o$ .", "There exists $g\\in G$ such that $\\tau = g.\\tau _1$ , meaning $\\sigma = g.\\sigma _0$ and $\\sigma ^{\\prime }=g.\\sigma ^{\\prime }_0$ .", "We see that this implies $V_{\\sigma }= g_{\\sigma _0}.V_{\\sigma _0},\\quad V_{\\sigma ^{\\prime }}= g_{\\sigma ^{\\prime }_0}.V_{\\sigma ^{\\prime }_0},\\quad V_{\\tau }=g_{\\tau _1}.V_{\\tau _1}.$ From these translation relations, we have the following relations on the restriction maps $r^\\tau _\\sigma $ : $r^{\\tau }_{\\sigma }= g_{\\sigma _0}\\circ r^{\\tau _1}_{\\sigma _0}\\circ (g^{-1})_{\\tau },\\quad r^{\\tau }_{\\sigma ^{\\prime }}= g_{\\sigma ^{\\prime }_0}\\circ r^{\\tau _1}_{\\sigma ^{\\prime }_0}\\circ (g^{-1})_{\\tau }.$ Definition 6.4 A diagram is a quintuple $D=(D_{0}, D^{\\prime }_{0}, D_1, r, r^{\\prime })$ , in which $(\\rho _0, D_0)$ is a smooth representation of $K$ , $(\\rho ^{\\prime }_0, D^{\\prime }_0)$ is a smooth representation of $K^{\\prime }$ , $(\\rho _1, D_1)$ is a smooth representation of $I$ , and $r\\in \\textnormal {Hom}_{I}(D_1, D_0|_I)$ , $r^{\\prime }\\in \\textnormal {Hom}_{I}(D_1, D^{\\prime }_0|_I)$ .", "We may represent a diagram pictorally as: D0 D1 [ur]r[dr]r' D0' Definition 6.5 A morphism $\\psi $ between two diagrams $D=(D_0, D^{\\prime }_{0}, D_1, r_D, r^{\\prime }_D)$ and $E=(E_0, E^{\\prime }_0, E_1, r_E, r^{\\prime }_E)$ is a triple $(\\psi _{0}, \\psi ^{\\prime }_{0}, \\eta _1)$ , where $\\psi _0\\in \\textnormal {Hom}_{K}(D_0,E_0)$ , $\\psi ^{\\prime }_0\\in \\textnormal {Hom}_{K^{\\prime }}(D^{\\prime }_0,E^{\\prime }_0)$ , and $\\eta _1\\in \\textnormal {Hom}_{I}(D_1,E_1)$ , such that the squares in the following diagram commute as $I$ -representations: D0 [rr]<<<<<<<<<0 E0D1 [ur]rD[dr]rD'[rr]1 E1[ur]rE[dr]rE' D0'[rr]<<<<<<<<<0' E0' We say $\\psi $ is an embedding if the maps $\\psi _0, \\psi ^{\\prime }_0,$ and $\\eta _1$ are injective.", "The set of diagrams with the morphisms defined above becomes a category, which we denote by $\\mathcal {DIAG}$ .", "The main result here is: Theorem 6.6 The categories $\\mathcal {DIAG}$ and $\\mathcal {COEF}_{G}$ are equivalent.", "The equivalence is induced by the functors Table: NO_CAPTIONwhere $\\mathcal {C}$ and $\\mathcal {D}$ are as in Definitions REF and REF .", "See Appendix." ], [ "Homology", "Let $\\mathcal {V}=(V_{\\tau })_{\\tau }$ be a $G$ -equivariant coefficient system.", "We denote by $C_{c}(X_{0}, \\mathcal {V})$ the $\\overline{\\mathbb {F}}_p$ -vector space of all maps: $\\omega : X_{0}\\rightarrow \\bigoplus _{\\sigma \\in X_0} V_{\\sigma },$ such that: $\\omega $ has finite support; $\\omega (\\sigma )\\in V_{\\sigma }$ for every vertex $\\sigma $ .", "We call such a map $\\omega $ a 0-chain.", "Let $X_{(1)}$ be the set of all oriented edges: if $\\lbrace \\sigma , \\sigma ^{\\prime }\\rbrace $ is an edge, we let $(\\sigma , \\sigma ^{\\prime })$ denote the oriented edge from $\\sigma $ to $\\sigma ^{\\prime }$ .", "Denote by $C_{c}(X_{(1)}, \\mathcal {V})$ the $\\overline{\\mathbb {F}}_p$ -vector space of all maps: $\\omega : X_{(1)}\\rightarrow \\bigoplus _{\\lbrace \\sigma , \\sigma ^{\\prime }\\rbrace \\in X_1} ~V_{\\lbrace \\sigma , \\sigma ^{\\prime }\\rbrace },$ such that $\\omega $ has finite support; $\\omega ((\\sigma , \\sigma ^{\\prime }))\\in V_{\\lbrace \\sigma , \\sigma ^{\\prime }\\rbrace }$ ; $\\omega ((\\sigma ^{\\prime }, \\sigma )) = -\\omega ((\\sigma , \\sigma ^{\\prime }))$ .", "We call such a map $\\omega $ a 1-chain.", "There is an action of $G$ on the two spaces above, induced from the action of $G$ on the tree $X$ and the coefficient system $\\mathcal {V}$ .", "Explicitly, for an element $g\\in G$ , we have Table: NO_CAPTIONThe action on both spaces is smooth.", "The boundary map $\\partial $ is defined as: Table: NO_CAPTIONwhere $\\sigma ^{\\prime }$ ranges over all neighbors of the vertex $\\sigma $ .", "One may easily verify that $\\partial $ is a $G$ -equivariant map.", "We define $H_{0}(X, \\mathcal {V})$ as the cokernel of $\\partial $ and $H_1(X, \\mathcal {V})$ as the kernel of $\\partial $ , both of which inherit a smooth action of $G$ ." ], [ "Properties of $H_0(X, \\mathcal {V})$ and {{formula:3bd1f347-6596-4a1d-b7df-22e577c1b3e9}}", "We fix a $G$ -equivariant coefficient system $\\mathcal {V}=(V_{\\tau })_{\\tau }$ .", "Lemma 6.7 Let $\\omega $ be a 1-chain, supported on a single edge $\\tau =\\lbrace \\sigma , \\sigma ^{\\prime }\\rbrace $ .", "Then $\\partial (\\omega )= \\omega _{\\sigma } - \\omega _{\\sigma ^{\\prime }}$ , where $\\omega _{\\sigma }$ and $\\omega _{\\sigma ^{\\prime }}$ are two 0-chains, supported respectively on $\\sigma $ and $\\sigma ^{\\prime }$ .", "More precisely, letting $v=\\omega ((\\sigma ,\\sigma ^{\\prime }))$ , then we have $\\omega _{\\sigma }(\\sigma )= r^{\\tau }_{\\sigma }(v), ~\\text{and}~~\\omega _{\\sigma ^{\\prime }}(\\sigma ^{\\prime })= r^{\\tau }_{\\sigma ^{\\prime }}(v).$ This follows directly from the definition of the boundary map $\\partial $ .", "Lemma 6.8 Let $\\omega $ be a 0-chain, supported on a single vertex $\\sigma $ .", "Suppose that the two restriction maps $r^{\\tau _1}_{\\sigma _0}$ and $r^{\\tau _1}_{\\sigma ^{\\prime }_0}$ are both injective.", "Then the image of $\\omega $ in $H_{0}(X, \\mathcal {V})$ is nonzero.", "From the assumption and equation (REF ) above, we see every restriction map is injective.", "The claim then follows from [26], Lemma 5.7.", "Lemma 6.9 Suppose that the two restriction maps $r^{\\tau _1}_{\\sigma _0}$ and $r^{\\tau _1}_{\\sigma ^{\\prime }_0}$ are both injective.", "Then $H_1(X, \\mathcal {V}) = \\lbrace 0\\rbrace $ .", "Let $\\omega \\in C_c(X_{(1)}, \\mathcal {V})$ be a nonzero 1-chain such that $\\partial (\\omega ) = 0$ , and let $\\sigma $ be a vertex which is contained in only one edge $\\tau = \\lbrace \\sigma ,\\sigma ^{\\prime }\\rbrace $ of the support of $\\omega $ .", "We then have $0 = \\partial (\\omega )(\\sigma ) = r^\\tau _\\sigma (\\omega ((\\sigma ,\\sigma ^{\\prime })))$ .", "Injectivity of the restriction maps implies $r^\\tau _\\sigma (\\omega ((\\sigma ,\\sigma ^{\\prime }))) \\ne 0$ , a contradiction.", "Lemma 6.10 Let $\\omega $ be a 0-chain.", "Suppose the two restriction maps $r^{\\tau _1}_{\\sigma _0}$ and $r^{\\tau _1}_{\\sigma ^{\\prime }_0}$ are both surjective.", "Then, for any vertex $\\sigma $ , there is a 0-chain $\\omega _{\\sigma }$ , supported on the single vertex $\\sigma $ , such that $\\omega +\\partial (C_{c}(X_{(1)}, \\mathcal {V})) = \\omega _{\\sigma }+\\partial (C_{c}(X_{(1)}, \\mathcal {V})).$ As $r^{\\tau _1}_{\\sigma _0}$ and $r^{\\tau _1}_{\\sigma ^{\\prime }_0}$ are both surjective, we see that every restriction map is surjective by equation (REF ).", "The claim then follows from [26], Lemma 5.8.", "Proposition 6.11 Suppose $r^{\\tau _1}_{\\sigma _0}$ and $r^{\\tau _1}_{\\sigma ^{\\prime }_0}$ are both isomorphisms of vector spaces.", "We then have $H_0 (X, \\mathcal {V})|_K \\cong V_{\\sigma _{0}},~H_0 (X, \\mathcal {V})|_{K^{\\prime }} \\cong V_{\\sigma ^{\\prime }_{0}}$ , and $H_0 (X, \\mathcal {V})|_{I}~\\cong V_{\\tau _1}$ .", "For $\\sigma =\\sigma _0 ~\\text{or} ~\\sigma ^{\\prime }_0$ , denote by $C_{c}(\\sigma , \\mathcal {V})$ the vector space of 0-chains with support contained in $\\lbrace \\sigma \\rbrace $ .", "We then have an evaluation map $ev_\\sigma $ , which is an isomorphism of $\\mathfrak {K}(\\sigma )$ -representations: Table: NO_CAPTION Let $\\jmath _\\sigma $ be the composition of the inclusion $C_{c}(\\sigma , \\mathcal {V})\\rightarrow C_{c}(X_0, \\mathcal {V})$ and the canonical map $C_{c}(X_0, \\mathcal {V})\\rightarrow H_0 (X, \\mathcal {V})$ .", "It is easily seen to be $\\mathfrak {K}(\\sigma )$ -equivariant.", "Moreover, Lemma REF and Lemma REF imply that $\\jmath _\\sigma $ is an isomorphism of vector spaces.", "Hence, we get a $\\mathfrak {K}(\\sigma )$ -equivariant isomorphism $\\jmath _\\sigma \\circ (ev_\\sigma )^{-1}:~V_{\\sigma }\\rightarrow H_0 (X, \\mathcal {V})|_{\\mathfrak {K}(\\sigma )}.$ Since the restriction maps $r^{\\tau _1}_{\\sigma }$ are isomorphisms of $I$ -representations, and $I\\subset \\mathfrak {K}(\\sigma )$ , we see that $\\imath _\\sigma =\\jmath _\\sigma \\circ (ev_\\sigma )^{-1}\\circ r^{\\tau _1}_{\\sigma }:~V_{\\tau _1}\\rightarrow H_0 (X, \\mathcal {V})|_{I}$ is an isomorphism of $I$ -representations.", "Corollary 6.12 Suppose $r_{\\sigma _0}^{\\tau _1}$ and $r_{\\sigma _0^{\\prime }}^{\\tau _1}$ are both isomorphisms of vector spaces, and let $\\sigma = \\sigma _0$ or $\\sigma _0^{\\prime }$ .", "Then the following diagram of $I$ -representations commutes: V1 [rr]>>>>>>>>[d]r1 H0 (X, V)[d]idV [rr](ev)-1 H0 (X, V) This follows readily from the previous Theorem." ], [ "Constant functor", "Let $\\pi $ be a smooth representation of $G$ , with underlying space $W$ .", "We define a constant coefficient system $\\mathcal {K}_\\pi $ as follows.", "Let $\\sigma $ be a simplex on the tree $X$ , and set $(\\mathcal {K}_\\pi )_{\\sigma }= W$ .", "If $\\sigma \\subset \\tau $ are two simplices, the restriction map $r_{\\sigma }^{\\tau }$ is defined as $\\textrm {id}_{W}$ .", "For every $g\\in G$ , and every simplex $\\sigma $ in $X$ , the linear map $g_{\\sigma }$ is defined by: Table: NO_CAPTIONLemma 6.13 Let $\\pi $ be a smooth representation of $G$ .", "Then $H_{0}(X, \\mathcal {K}_{\\pi })\\cong \\pi $ as $G$ -representations.", "Define an evaluation map $ev$ from $C_{c}(X_{0}, \\mathcal {K}_{\\pi })$ to $\\pi $ : Table: NO_CAPTION As the restriction maps are $\\text{id}_W$ , we see from Lemma REF that the image of the boundary map $\\partial $ is contained in $\\ker (ev)$ .", "Hence $ev$ induces a $G$ -equivariant map: $ev:~H_{0}(X, \\mathcal {K}_{\\pi })\\rightarrow \\pi $ .", "We need to show that this map is an isomorphism of vector spaces.", "Since $(\\mathcal {K}_{\\pi })_{\\sigma }= W$ , we have $ev|_{C_{c}(\\sigma , \\mathcal {K}_{\\pi })}= ev_\\sigma $ , i.e., $ev_{\\sigma }= ev\\circ \\jmath _\\sigma $ .", "As the restriction maps are all $\\text{id}_W$ , Proposition REF implies $\\jmath _\\sigma $ is an isomorphism, which gives $ev= ev_{\\sigma }\\circ \\jmath ^{-1} _\\sigma :~H_{0}(X, \\mathcal {K}_{\\pi })\\stackrel{\\sim }{\\rightarrow } (\\mathcal {K}_{\\pi })_{\\sigma }$ , as desired.", "Proposition 6.14 Let $\\mathcal {V}= (V_{\\sigma })_\\sigma $ be a $G$ -equivariant coefficient system with restriction maps $r^{\\tau }_{\\sigma }$ , and let $(\\pi , W)$ be a smooth representation of $G$ .", "Then $\\textnormal {Hom}_{\\mathcal {COEF}_{G}}(\\mathcal {V}, \\mathcal {K}_{\\pi })\\cong \\textnormal {Hom}_{G}(H_{0}(X, \\mathcal {V}), \\pi )$ By Lemma REF , $H_{0}(X, \\mathcal {K}_{\\pi })\\cong \\pi $ .", "Any morphism between $G$ -equivariant coefficient systems induces a homomorphism between the corresponding 0-homology which is compatible with the action of $G$ ; that is, there is a map $\\textnormal {Hom}_{\\mathcal {COEF}_{G}}(\\mathcal {V}, \\mathcal {K}_{\\pi })\\rightarrow \\textnormal {Hom}_{G}(H_{0}(X, \\mathcal {V}), \\pi ),$ and it suffices to construct an inverse to this map.", "Let $\\phi \\in \\textnormal {Hom}_{G}(H_{0}(X, \\mathcal {V}), \\pi )$ .", "Given a vertex $\\sigma $ , and a vector $v$ in $V_{\\sigma }$ , let $\\omega _{\\sigma , v}$ be the 0-chain such that supp($\\omega _{\\sigma , v})\\subset \\lbrace \\sigma \\rbrace , ~ \\omega _{\\sigma , v}(\\sigma )=v$ .", "For this vertex $\\sigma $ , we define Table: NO_CAPTION For an edge $\\tau $ in $X$ , with endpoints $\\sigma $ and $\\sigma ^{\\prime }$ , we define: Table: NO_CAPTION The independence of the choice of the vertex $\\sigma $ in the definition of $\\phi _{\\tau }$ results from Lemma REF .", "The linear maps $(\\phi _{\\sigma })_{\\sigma }$ consitute a morphism from $\\mathcal {V}$ to $\\mathcal {K}_{\\pi }$ , respecting the $G$ -action on both.", "One can check that $(\\phi _\\sigma )_\\sigma $ induces $\\phi $ on the 0-homology." ], [ "The Functors $\\mathcal {C}$ and {{formula:6857e853-fe86-4a6e-b3d5-eba51082e755}}", "To prove the equivalence of $\\mathcal {COEF}_G$ and $\\mathcal {DIAG}$ , we first observe that there is an obvious functor in one direction: Definition 6.15 Let $\\mathcal {D}$ be the functor from $\\mathcal {COEF}_{G}$ to $\\mathcal {DIAG}$ given by: Table: NO_CAPTIONWe will construct in the next several subsections a functor $\\mathcal {C}:\\mathcal {DIAG}\\rightarrow \\mathcal {COEF}_G$ , such that $\\mathcal {D}$ and $\\mathcal {C}$ induce an equivalence of categories.", "We continue to follow [26] closely." ], [ "Underlying vector spaces", "Let $D= (D_{0}, D^{\\prime }_{0}, D_1, r, r^{\\prime })$ be a fixed diagram in $\\mathcal {DIAG}$ .", "We consider the following compactly induced representations: $\\text{c-ind}_{K}^{G}(\\rho _0), ~\\text{c-ind}_{K^{\\prime }}^{G}(\\rho ^{\\prime }_0), ~\\text{c-ind}_{I}^{G}(\\rho _1)$ .", "For a vertex $\\sigma \\in X_0$ , there exists $g\\in G$ such that $\\sigma =g.\\sigma _0$ or $\\sigma = g.\\sigma ^{\\prime }_0$ , depending on whether $\\sigma \\in X_0^e$ or $\\sigma \\in X_0^o$ .", "We define $F_\\sigma =$ ${\\left\\lbrace \\begin{array}{ll}\\lbrace f\\in \\text{c-ind}_{K}^{G}(\\rho _0): ~\\text{supp}(f) \\subset Kg^{-1}\\rbrace & \\text{if}~\\sigma \\in X_0^e,\\\\\\end{array}\\lbrace f\\in \\text{c-ind}_{K^{\\prime }}^{G}(\\rho ^{\\prime }_0): ~\\text{supp}(f) \\subset K^{\\prime }g^{-1}\\rbrace & \\text{if}~\\sigma \\in X_0^o.\\right.", "}$ For an edge $\\tau \\in X_1$ , there exists $g\\in G$ such that $\\tau =g.\\tau _1$ .", "We define $F_{\\tau }= \\lbrace f\\in \\text{c-ind}_{I}^{G}(\\rho _1): ~\\text{supp}(f)\\subset Ig^{-1}\\rbrace .$ We note that these definitions are independent of the choice of $g$ ." ], [ "Restriction maps", "To define restriction maps of the coefficient system, we begin with the two given maps $r$ and $r^{\\prime }$ , and extend by translations.", "The evaluation map $ev_{\\sigma _0}: F_{\\sigma _0} \\rightarrow D_0$ is naturally an isomorphism of $K$ -representations.", "Explicitly, it is defined by Table: NO_CAPTIONwith inverse $ev_{\\sigma _0}^{-1}$ given by Table: NO_CAPTIONwhere $f_v: G\\rightarrow D_0$ has support in $K$ , and $f_{v}(k)=\\rho _{0}(k)v$ for $k\\in K$ (this is the function denoted $\\widehat{f}_v$ in [3]).", "One defines $ev_{\\sigma ^{\\prime }_0}$ and its inverse $ev_{\\sigma ^{\\prime }_0}^{-1}$ similarly.", "We also have isomorphisms $ev_{\\tau _1}$ and $ev_{\\tau _1}^{-1}$ of $I$ -representations, given by Table: NO_CAPTIONand Table: NO_CAPTIONwhere $f_v:G\\rightarrow D_1$ has support in $I$ , and $f_{v}(i)=\\rho _1(i)v$ for $i\\in I$ .", "Let $r_{\\sigma _0}^{\\tau _1} = ev_{\\sigma _0}^{-1}\\circ r\\circ ev_{\\tau _1}$ ; this is an $I$ -equivariant map from $F_{\\tau _1}$ to $F_{\\sigma _0}$ .", "Explicitly, it is given by $r_{\\sigma _0}^{\\tau _1}(f_v)= f_{r(v)},$ where $v\\in D_1$ .", "We define $r_{\\sigma ^{\\prime }_0}^{\\tau _1} = ev_{\\sigma _0^{\\prime }}^{-1}\\circ r^{\\prime }\\circ ev_{\\tau _1}$ ; it enjoys the same properties as $r_{\\sigma _0}^{\\tau _1}$ .", "To summarize, given a diagram $D = (D_0, D_0^{\\prime }, D_1, r, r^{\\prime })$ , we may construct a diagram $\\widetilde{D}= (F_{\\sigma _0}, F_{\\sigma ^{\\prime }_0}, F_{\\tau _1}, r_{\\sigma _0}^{\\tau _1}, r_{\\sigma ^{\\prime }_0}^{\\tau _1})$ ; the diagrams $\\widetilde{D}$ and $D$ are isomorphic via $\\mathbf {ev}=(ev_{\\sigma _0}, ev_{\\sigma ^{\\prime }_0}, ev_{\\tau _1})$ .", "Now let $\\tau $ be an edge, containing a vertex $\\sigma $ , and suppose $\\sigma \\in X_0^e$ .", "Then there exists $g\\in G$ such that $\\tau =g.\\tau _{1}$ , and $\\sigma =g.\\sigma _0$ , where the choice of $g$ is unique up to an element of $I = \\mathfrak {K}(\\sigma _0)\\cap \\mathfrak {K}(\\tau _1)$ .", "We define the restriction map $r^{\\tau }_{\\sigma }$ from $F_{\\tau }$ to $F_{\\sigma }$ by Table: NO_CAPTIONNote that this is independent of the choice of $g$ .", "In particular, we have $r^{\\tau }_{\\sigma }(f)=g.f_{r(v)}$ , where $v= f(g^{-1})$ .", "When $\\sigma \\in X_0^o$ , we define $r^{\\tau }_{\\sigma }(f) = g.r_{\\sigma _0^{\\prime }}^{\\tau _1}(g^{-1}.f)$ ; it enjoys the same properties as when $\\sigma \\in X_0^e$ .", "Finally, given any simplex $\\tau $ , we define $r^{\\tau }_{\\tau }=\\text{id}_{F_{\\tau }}$ ." ], [ "$G$ -action", "Let $\\tau $ be a simplex of $X$ , and let $f\\in F_\\tau $ .", "Since the space $F_\\tau $ is a subspace of either $\\text{c-ind}_{K}^{G}(\\rho _0), \\text{c-ind}_{K^{\\prime }}^{G}(\\rho ^{\\prime }_0)$ , or $\\text{c-ind}_{I}^{G}(\\rho _1)$ , the element $g.f$ is well-defined, and induces linear maps Table: NO_CAPTIONWe have $1_{\\tau } = \\textrm {id}_{F_{\\tau }}$ and $g_{h\\tau }\\circ h_{\\tau }=(gh)_{\\tau }$ for every $g,h\\in G$ .", "It only remains to check the linear maps above are compatible with the restriction maps in REF .", "In other words, for an edge $\\tau $ containing a vertex $\\sigma $ , we must verify the following diagram is commutative for all $g\\in G$ : F [r]g[d]r Fg.", "[d]rg.g.F [r]g Fg.", "Assume $\\sigma \\in X_0^e$ , and let $g^{\\prime }$ be such that $\\tau = g^{\\prime }.\\tau _1, ~\\sigma = g^{\\prime }.\\sigma _0$ .", "The previous subsection implies $g_{\\tau }\\circ r^{\\tau }_{\\sigma }(f)= gg^{\\prime }.f_{r(v)}$ , where $v= f(g^{\\prime -1})$ .", "On the other hand, $r_{g.\\sigma }^{g.\\tau }\\circ g_{\\tau }(f)=r_{g.\\sigma }^{g.\\tau }(g.f)= gg^{\\prime }.f_{r(v^{\\prime })}$ , where $v^{\\prime }=g.f((gg^{\\prime })^{-1})=v$ .", "The same argument applies mutatis mutandis to the case $\\sigma \\in X_0^o$ .", "Combining all of these results, we see that to each diagram $D\\in \\mathcal {DIAG}$ we may associate a $G$ -equivariant coefficient system $\\mathcal {F}=(F_\\sigma )_\\sigma \\in \\mathcal {COEF}_G$ ." ], [ "Morphisms", "Let $D=(D_0, D^{\\prime }_{0}, D_1, r_D, r^{\\prime }_D)$ and $E=(E_0, E^{\\prime }_0, E_1, r_E, r^{\\prime }_E)$ be two diagrams, and $\\psi =(\\psi _{0}, \\psi ^{\\prime }_{0}, \\eta _1)$ be a morphism between them.", "Let $\\mathcal {F}=(F_\\sigma )_\\sigma $ and $\\mathcal {F}^{\\prime }=(F^{\\prime }_\\sigma )_\\sigma $ be the coefficient systems associated to $D$ and $E$ , respectively.", "Let $\\sigma \\in X_0^e$ , and let $g\\in G$ be such that $\\sigma =g.\\sigma _0$ .", "For $f\\in F_{\\sigma }$ , we let $v=f(g^{-1})$ , and define Table: NO_CAPTIONwhere $f_{\\psi _0(v)}$ is the unique function in $F^{\\prime }_{\\sigma _0}$ such that $f_{\\psi _0(v)}(1)=\\psi _0(v).$ We define $\\psi _{\\sigma }(f) = g.f_{\\psi _0^{\\prime }(v)}$ if $\\sigma \\in X_0^o$ and $\\sigma = g.\\sigma _0^{\\prime }$ .", "Let $\\tau $ be an edge, and let $g\\in G$ be such that $\\tau =g.\\tau _1$ .", "For $f\\in F_{\\tau }$ , we let $v=f(g^{-1})$ , and define Table: NO_CAPTIONwhere $f_{\\eta _1(v)}$ is the unique function in $F^{\\prime }_{\\tau _1}$ such that $f_{\\eta _1(v)}(1)=\\eta _1(v).$ Note that the definitions of $\\psi _\\sigma $ and $\\psi _\\tau $ are both independent of the choice of $g$ .", "This process gives a collection of linear maps $(\\psi _{\\tau })_\\tau $ ; we need to verify they are compatible with the restriction maps and the $G$ -action.", "That is, we must check that the following two diagrams commute: F [r][d]r F'[d](r') F [r][d]h F'[d]hF [r] F' Fh.", "[r]h. F'h.", "In the first square, $\\tau $ is an edge containing a vertex $\\sigma $ , and in the second, $\\tau $ is any simplex, $h\\in G$ .", "We begin with the first square.", "Suppose $\\sigma \\in X_0^e$ with $\\sigma = g.\\sigma _0$ and $\\tau = g.\\tau _1$ for some $g\\in G$ , and let $f\\in F_{\\tau }$ .", "We have $\\psi _{\\sigma }\\circ r^{\\tau }_{\\sigma }(f)= \\psi _{\\sigma }(g.f_{r_D(v)})$ , with $v = f(g^{-1})$ .", "As $g.f_{r_D(v)}(g^{-1})=r_D(v)$ , we get $\\psi _{\\sigma }\\circ r^{\\tau }_{\\sigma }(f)=g.f_{\\psi _0\\circ r_D(v)}$ .", "On the other hand, we have $(r^{\\prime })_{\\sigma }^{\\tau }\\circ \\psi _{\\tau }(f)=(r^{\\prime })_{\\sigma }^{\\tau }(g.f_{\\eta _1(v)})$ .", "As $g.f_{\\eta _1 (v)}(g^{-1})=\\eta _1(v)$ , we see $(r^{\\prime })_{\\sigma }^{\\tau }(g.f_{\\eta _1(v)})=g.f_{r_E (\\eta _1(v))}$ .", "Since $\\psi $ is a morphism of diagrams, we have $\\psi _0\\circ r_D(v)=r_E\\circ \\eta _1(v)$ , and the result follows.", "The argument is identical for the case $\\sigma \\in X_0^o$ .", "In the second diagram, we note that given $f\\in F_\\tau $ , we have $h.f((hg)^{-1})=f(g^{-1})=v$ .", "The commutativity then follows directly from the definitions.", "We may now make the following definition: Definition 6.16 Let $\\mathcal {C}$ be the map: Table: NO_CAPTIONwhere $(F_\\tau )_\\tau $ is the coefficient system defined above.", "The results of the previous subsections imply that $\\mathcal {C}$ is a bona fide functor between the two categories." ], [ "Initial Diagrams", "We begin with some general remarks.", "Given any irreducible mod-$p$ representation $\\rho $ of $\\Gamma $ , we may view it as a representation of $K$ via the projection $K\\twoheadrightarrow K/K_1\\cong \\Gamma $ .", "Conversely, any smooth irreducible representation of $K$ must be of this form; this follows from Lemma 3(1) of [3] and the fact that $K_1$ is a normal pro-$p$ subgroup of $K$ .", "In light of this, we shall abuse notation and identify smooth irreducible representations of $K$ and those of $\\Gamma $ .", "The same statements hold for the groups $K^{\\prime }$ and $\\Gamma ^{\\prime }$ , and $I$ and $H$ .", "Using the functor $\\mathcal {C}$ , we may now construct coefficient systems by defining the appropriate diagrams.", "In particular, to each supersingular Hecke module in Definition REF , we associate a diagram as follows.", "Definition 7.1 Let $\\chi = \\zeta \\otimes \\eta :H\\rightarrow \\overline{\\mathbb {F}}_p^\\times $ be a character.", "Assume $\\chi =\\eta \\circ \\textnormal {det}$ .", "We associate to $M_{\\chi , (S,\\emptyset )}$ and $M_{\\chi , (\\emptyset , S^{\\prime })}$ the diagrams Table: NO_CAPTION where $j$ and $j^{\\prime }$ are the inclusion maps.", "Assume $\\chi ^s = \\chi $ , but $\\chi \\ne \\eta \\circ \\det $ .", "We associate to $M_{\\chi , (\\emptyset , S^{\\prime })}$ and $M_{\\chi , (\\emptyset , \\emptyset )}$ the diagrams Table: NO_CAPTION where $j$ and $j^{\\prime }$ are the inclusion maps.", "Assume $\\chi ^s\\ne \\chi $ .", "We associate to $M_{\\chi ,(\\emptyset ,\\emptyset )}$ the diagram Table: NO_CAPTION where $j$ and $j^{\\prime }$ are the inclusion maps.", "If ${D}_{\\chi , \\mathbf {J}} = (\\rho ,\\rho ^{\\prime },\\chi ,j,j^{\\prime })$ is a diagram as defined above, we let the underlying space of the $I$ -representation $\\chi $ be spanned by a fixed vector $v$ , and identify $v$ with its image in $\\rho ^{I(1)} = \\rho ^\\mathbb {U}$ and $(\\rho ^{\\prime })^{I(1)} = (\\rho ^{\\prime })^{\\mathbb {U}^{\\prime }}$ via $j$ and $j^{\\prime }$ .", "For a diagram $D_{\\chi , \\mathbf {J}}$ , we define $\\mathcal {D}_{\\chi , \\mathbf {J}} = \\mathcal {C}(D_{\\chi , \\mathbf {J}})$ to be the associated $G$ -equivariant coefficient system.", "Remark We note that if $M_{\\chi ,\\mathbf {J}}$ is a supersingular module and $D_{\\chi ,\\mathbf {J}} = (\\rho , \\rho ^{\\prime }, \\chi , j, j^{\\prime })$ is the associated diagram, we have $\\rho ^\\mathbb {U}\\cong M_{\\chi ,\\mathbf {J}}|_{\\mathcal {H}_\\Gamma }$ as $\\mathcal {H}_\\Gamma $ -modules and $(\\rho ^{\\prime })^{\\mathbb {U}^{\\prime }}\\cong M_{\\chi ,\\mathbf {J}}|_{\\mathcal {H}_{\\Gamma ^{\\prime }}}$ as $\\mathcal {H}_{\\Gamma ^{\\prime }}$ -modules.", "Proposition 7.2 Let $M_{\\chi , \\mathbf {J}}$ be a supersingular $\\mathcal {H}_{\\overline{\\mathbb {F}}_p}(G,I(1))$ -module, and let $\\pi $ be a nonzero irreducible quotient of $H_0(X,\\mathcal {D}_{\\chi ,\\mathbf {J}})$ .", "Then $\\pi ^{I(1)}$ contains $M_{\\chi ,\\mathbf {J}}$ , and $\\pi $ is supersingular as a $G$ -representation.", "In the notation of Subsection REF , we let $\\omega _{\\sigma _0, f_v}$ , be the 0-chain supported on $\\sigma _0$ , such that $\\omega _{\\sigma _0, f_v}(\\sigma _0) = f_v$ , and let $\\bar{\\omega }_{\\sigma _0,f_v}$ denote its image in $H_0(X,\\mathcal {D}_{\\chi , \\mathbf {J}})$ .", "Recall that $f_v\\in \\textnormal {c-ind}_K^G(\\rho _0)$ denotes the unique function such that $\\textnormal {supp}(f) = K$ and $f_v(1) = v$ , where $v$ is a fixed vector spanning the underlying space of $\\chi $ .", "By definition of the $G$ -action, $\\omega _{\\sigma _0, f_v}$ is $I(1)$ -invariant and the group $I$ acts by the character $\\chi $ .", "To proceed, we must show two things: The element $\\bar{\\omega }_{\\sigma _0, f_v}$ generates $H_0(X, \\mathcal {D}_{\\chi , \\mathbf {J}})$ as a $G$ -representation.", "The right action of $\\mathcal {H}(G, I(1))$ on $\\langle \\bar{\\omega }_{\\sigma _0, f_v} \\rangle _{\\overline{\\mathbb {F}}_p}$ yields an isomorphism onto $M_{\\chi , \\mathbf {J}}$ .", "Assuming these two results, we let $\\pi $ be a nonzero irreducible quotient of $H(X, \\mathcal {D}_{\\chi , \\mathbf {J}})$ .", "Since $\\bar{\\omega }_{\\sigma _0,f_v}$ generates $H_0(X,\\mathcal {D}_{\\chi , \\mathbf {J}})$ , the image of $\\bar{\\omega }_{\\sigma _0,f_v}$ in $\\pi $ will be nonzero.", "The second result above then shows that $\\pi ^{I(1)}$ contains the $\\mathcal {H}(G, I(1))$ -module $M_{\\chi , \\mathbf {J}}$ and the Proposition follows from Corollary REF .", "It remains to prove the two claims.", "For the first, we note that if $\\omega _{\\sigma _0^{\\prime }, f_v}$ denotes the 0-chain supported on $\\sigma _0^{\\prime }$ such that $\\omega _{\\sigma _0^{\\prime }, f_v}(\\sigma _0^{\\prime }) = f_v$ , then Lemma REF implies $\\bar{\\omega }_{\\sigma _0, f_v} = \\bar{\\omega }_{\\sigma _0^{\\prime }, f_v}$ in $H_0(X,\\mathcal {D}_{\\chi , \\mathbf {J}})$ .", "The Carter-Lusztig theory tells us that any irreducible representation of $K$ or $K^{\\prime }$ is generated by its $I(1)$ -invariants, and therefore $\\bar{\\omega }_{\\sigma _0, f_v}$ (resp.", "$\\bar{\\omega }_{\\sigma _0^{\\prime }, f_v}$ ) generates the image in $H_0(X,\\mathcal {D}_{\\chi , \\mathbf {J}})$ of the space $C_c(\\sigma _0,\\mathcal {D}_{\\chi , \\mathbf {J}})$ (resp.", "$C_c(\\sigma _0^{\\prime },\\mathcal {D}_{\\chi , \\mathbf {J}})$ ).", "This fact, combined with the observation that $G$ acts transitively on the sets $X_0^e$ and $X_0^o$ , verifies the claim.", "For the second claim, note that by Definition REF and our choice of irreducible $K$ - and $K^{\\prime }$ -representations, we have Table: NO_CAPTION where $\\mathbf {J}= (J,J^{\\prime })$ .", "We conclude from Proposition REF that $\\langle \\bar{\\omega }_{\\sigma _0, f_v} \\rangle _{\\overline{\\mathbb {F}}_p}$ is equivalent to $M_{\\chi , \\mathbf {J}}$ as a right $\\mathcal {H}(G, I(1))$ -module." ], [ "Injective Envelopes", "In this section we briefly recall some definitions and notation regarding socles and injective envelopes, which will be of use in subsequent sections.", "For more details, we refer to [29] and [26].", "Let $\\mathcal {K}$ be any finite or profinite group, and denote by $\\mathcal {REP}_{\\overline{\\mathbb {F}}_p}(\\mathcal {K})$ the category of smooth $\\overline{\\mathbb {F}}_p$ -representations of $\\mathcal {K}$ .", "Denote by $\\text{Irr}_{\\mathcal {K}}$ be the subcategory of smooth irreducible representations of $\\mathcal {K}$ .", "Definition 7.3 Let $\\pi \\in \\mathcal {REP}_{\\overline{\\mathbb {F}}_p}(\\mathcal {K})$ and let $\\rho $ be a subrepresentation of $\\pi $ .", "We say $\\pi $ is an essential extension of $\\rho $ if for every nonzero subrepresentation $\\pi ^{\\prime }$ of $\\pi $ , we have $\\pi ^{\\prime }\\cap \\rho \\ne \\lbrace 0\\rbrace $ .", "Let $\\rho \\in \\mathcal {REP}_{\\overline{\\mathbb {F}}_p}(\\mathcal {K})$ and let $\\mathfrak {I}$ be an injective object of $\\mathcal {REP}_{\\overline{\\mathbb {F}}_p}(\\mathcal {K})$ .", "We say $\\mathfrak {I}$ is an injective envelope of $\\rho $ if there exists an injection $\\rho \\hookrightarrow \\mathfrak {I}$ such that $\\mathfrak {I}$ is an essential extension of the image of $\\rho $ .", "We write $\\mathfrak {I} = \\textnormal {inj}_\\mathcal {K}(\\rho )$ .", "It is known that injective envelopes exist, and are unique up to isomorphism: for finite groups, see Chapter 14 of [29], and for profinite groups, see Section 3.1 of [31].", "Definition 7.4 Let $\\rho \\in \\mathcal {REP}_{\\overline{\\mathbb {F}}_p}(\\mathcal {K})$ .", "The socle of $\\rho $ , denoted $\\textnormal {soc}_{\\mathcal {K}}(\\rho )$ , is the maximal semisimple subrepresentation of $\\rho $ .", "We now use these constructions for the groups $K, K^{\\prime }$ and $I$ .", "Lemma 7.5 Let $\\rho $ be an irreducible representation of $K$ and let $\\rho \\hookrightarrow \\textnormal {inj}_K(\\rho )$ be an injective envelope of $\\rho $ .", "Then $\\textnormal {inj}_K(\\rho )|_I \\cong \\bigoplus _{\\chi \\in \\widehat{H}} \\textnormal {inj}_I(\\chi )^{\\oplus m_{\\rho ,\\chi }},$ where $m_{\\rho ,\\chi } = \\dim _{\\mathbb {F}_p}(\\textnormal {Hom}_H (\\chi , \\textnormal {inj}_\\Gamma (\\rho )^{\\mathbb {U}}))$ .", "Let $\\rho ^{\\prime }$ be an irreducible representation of $K^{\\prime }$ and let $\\rho ^{\\prime }\\hookrightarrow \\textnormal {inj}_{K^{\\prime }}(\\rho ^{\\prime })$ be an injective envelope of $\\rho ^{\\prime }$ .", "Then $\\textnormal {inj}_{K^{\\prime }}(\\rho ^{\\prime })|_I \\cong \\bigoplus _{\\chi \\in \\widehat{H}} \\textnormal {inj}_I(\\chi )^{\\oplus m_{\\rho ^{\\prime },\\chi }},$ where $m_{\\rho ^{\\prime },\\chi } = \\dim _{{\\mathbb {F}}_p}(\\textnormal {Hom}_H (\\chi , \\textnormal {inj}_{\\Gamma ^{\\prime }} (\\rho ^{\\prime })^{\\mathbb {U}^{\\prime }}))$ .", "In particular, the integers $m_{\\rho ,\\chi }$ and $m_{\\rho ^{\\prime },\\chi }$ are finite for every character $\\chi $ of $H$ .", "The proof is identical to the proof of Lemma 6.19 of [26].", "Remark Under the assumption $q=p$ , we will determine $(\\ref {decom for K^{\\prime }})$ explicitly, using a simple counting argument.", "Corollary 7.6 Let $\\mathcal {K}\\in \\lbrace K,~ K^{\\prime }\\rbrace $ , and let $\\rho \\in \\mathcal {REP}_{\\overline{\\mathbb {F}}_p}(\\mathcal {K})$ be a representation such that $\\textnormal {soc}_{\\mathcal {K}}(\\rho )$ is of finite length as a $\\mathcal {K}$ -representation.", "Then the space of $I(1)$ -invariants of $\\textnormal {inj}_{\\mathcal {K}}(\\rho )$ is finite-dimensional and $\\rho $ is admissible.", "From Lemma REF , we have $\\textnormal {inj}_{\\mathcal {K}}(\\rho )|_I\\cong \\textnormal {inj}_{\\mathcal {K}}(\\textnormal {soc}_{\\mathcal {K}}(\\rho ))|_I\\cong \\bigoplus _{\\chi \\in \\widehat{H}}\\textnormal {inj}_I(\\chi )^{\\oplus m_{\\chi }},$ where the integers $m_\\chi $ are finite.", "Hence, we see that $\\rho ^{I(1)} & \\hookrightarrow & \\textnormal {inj}_{\\mathcal {K}}(\\rho )^{I(1)}\\\\& \\cong & \\left(\\bigoplus _{\\chi \\in \\widehat{H}}\\textnormal {inj}_I(\\chi )^{\\oplus m_{\\chi }}\\right)^{I(1)}\\\\& \\cong & \\bigoplus _{\\chi \\in \\widehat{H}}\\left(\\textnormal {inj}_I(\\chi )^{I(1)}\\right)^{\\oplus m_{\\chi }}\\\\& \\cong & \\bigoplus _{\\chi \\in \\widehat{H}} \\chi ^{\\oplus m_{\\chi }};$ for the last isomorphism, we use the fact that $\\textnormal {inj}_I (\\chi )^{I(1)}\\cong \\textnormal {inj}_{I/I(1)}(\\chi )\\cong \\chi $ as representations of $H$ .", "Admissibility now follows from [26], Lemma 6.18." ], [ "Pure Diagrams", "In light of Proposition REF , it suffices to construct irreducible quotients of $H_0(X,\\mathcal {D}_{\\chi , \\mathbf {J}})$ to produce supersingular representations.", "With this in mind, we adapt the arguments of [26] into a more formal context: Definition 7.7 Let $M_{\\chi ,\\mathbf {J}}$ be a supersingular module, and let ${D}=(D_0, D^{\\prime }_0, D_1, r_D, r^{\\prime }_D)$ be a diagram.", "We say ${D}$ is essentially pure for $M_{\\chi , \\mathbf {J}}$ if it satisfies the following conditions: There exists an embedding of diagrams: $\\psi : {D}_{\\chi , \\mathbf {J}}\\rightarrow D$ .", "The maps $r_D$ and $r_{D}^{\\prime }$ induce isomorphisms $D_0|_I \\cong D^{\\prime }_0|_I \\cong D_1 $ .", "Moreover, we say ${D}$ is pure for $M_{\\chi , \\mathbf {J}}$ , if it also satisfies the following extra condition: Either $\\textnormal {soc}_K(D_0)$ or $\\textnormal {soc}_{K^{\\prime }}(D^{\\prime }_0)$ is irreducible.", "With these definitions, we are able to prove a formal result, whose proof is due to Paškūnas.", "Theorem 7.8 Let $M_{\\chi , \\mathbf {J}}$ be a supersingular module, and suppose that $D$ is a pure diagram for $M_{\\chi ,\\mathbf {J}}$ .", "Then the image of the induced $G$ -morphism between the 0-homology $\\pi _{{D}} =\\textnormal {im}(\\psi _*:H_0(X, \\mathcal {D}_{\\chi , \\mathbf {J}})\\rightarrow H_0(X, \\mathcal {C}({D})))$ is an irreducible admissible supersingular representation.", "Moreover, we have $\\pi _D \\cong \\textnormal {soc}_G(H_0(X,\\mathcal {C}(D)))$ .", "To verify the result, it suffices to show $\\pi _{{D}}$ is irreducible, admissible and nonzero, by Proposition REF .", "Let us assume that $\\textnormal {soc}_K(D_0)$ is irreducible; the case with $\\textnormal {soc}_{K^{\\prime }}(D_0^{\\prime })$ irreducible is the same.", "We first claim that the space $\\textnormal {soc}_K(H_0(X, \\mathcal {C}({D}))|_K)^{I(1)}$ generates $\\pi _{D}$ .", "Let $\\psi _*(\\bar{\\omega }_{\\sigma _0, f_v})$ denote the image of the homology class $\\bar{\\omega }_{\\sigma _0, f_v}$ .", "Claim (i) in the proof of Proposition REF shows that $\\pi _{{D}}= \\langle G.\\psi _*(\\bar{\\omega }_{\\sigma _0, f_v})\\rangle _{\\overline{\\mathbb {F}}_p}$ ; since $\\psi $ is an embedding, we have $\\psi _*(\\bar{\\omega }_{\\sigma _0, f_v})\\ne 0$ and hence $\\pi _D\\ne \\lbrace 0\\rbrace $ .", "From the definition of pure diagrams and Proposition REF , we know that $H_0 (X, \\mathcal {C}({D}))|_K \\cong D_0$ .", "The purity condition on ${D}$ also implies that the $K$ -representation of the diagram $D_{\\chi , \\mathbf {J}}$ is exactly $\\textnormal {soc}_K(D_0)$ , so that $\\psi _*(\\bar{\\omega }_{\\sigma _0, f_v})\\in \\textnormal {soc}_K(H_0(X,\\mathcal {C}({D}))|_K)$ .", "This shows $\\langle \\psi _*(\\bar{\\omega }_{\\sigma _0, f_v}) \\rangle _{\\overline{\\mathbb {F}}_p} = \\textnormal {soc}_K(H_0(X,\\mathcal {C}({D}))|_K)^{I(1)}$ , which combined with the previous observation verifies the claim.", "Moreover, this shows how to define an action of $\\mathcal {H}_{\\overline{\\mathbb {F}}_p}(G,I(1))$ on $\\textnormal {soc}_K(H_0(X,\\mathcal {C}({D}))|_K)^{I(1)}$ such that $\\textnormal {soc}_K(H_0(X,\\mathcal {C}({D}))|_K)^{I(1)} \\cong M_{\\chi , \\mathbf {J}}$ as right $\\mathcal {H}_{\\overline{\\mathbb {F}}_p}(G,I(1))$ -modules.", "Now let $\\pi ^{\\prime }$ a be nonzero $G$ -invariant subspace of $\\pi _{{D}}$ .", "Since $K_1$ is a pro-$p$ group we have $(\\pi ^{\\prime })^{K_1}\\ne \\lbrace 0\\rbrace $ and consequently $\\textnormal {soc}_K(\\pi ^{\\prime }|_K )\\ne \\lbrace 0\\rbrace $ .", "We also note that $\\textnormal {soc}_K(H_0 (X, \\mathcal {C}({D}))|_K)^{I(1)} \\cap (\\pi ^{\\prime })^{I(1)}\\ne \\lbrace 0\\rbrace $ , as $\\textnormal {soc}_K(\\pi ^{\\prime }|_K )$ is contained in $\\textnormal {soc}_K(H_0 (X, \\mathcal {C}({D}))|_K)$ .", "However, we have just shown that the space $\\textnormal {soc}_K(H_0 (X, \\mathcal {C}({D}))|_K)^{I(1)}$ is simple as a right $\\mathcal {H}(G, I(1))$ -module, which implies $\\textnormal {soc}_K(H_0 (X, \\mathcal {C}({D}))|_K)^{I(1)} \\subset (\\pi ^{\\prime })^{I(1)}$ .", "Note that we can deduce this simply from the fact that $\\dim _{\\overline{\\mathbb {F}}_p}(\\textnormal {soc}_K(H_0 (X, \\mathcal {C}({D}))|_K)^{I(1)}) = 1.$ Collecting these results, we conclude that $\\pi ^{\\prime }=\\pi _{{D}}$ .", "This argument also shows that the socle of $H_0(X,\\mathcal {C}(D))$ is exactly $\\pi _D$ .", "To show admissibility, we observe that $\\pi _D|_K\\subset H_0(X,\\mathcal {C}(D))|_K \\cong D_0,$ which implies that $\\textnormal {soc}_K(\\pi _D|_K)$ is of finite length.", "The claim then follows from Corollary REF .", "The definitions of pure and essentially pure diagrams do not make it clear that such diagrams exist in general.", "We take up this question when $q = p$ in the next section, and in general propose the following: Conjecture 7.9 Given a supersingular module $M_{\\chi , \\mathbf {J}}$ , an essentially pure diagram $D$ for $M_{\\chi , \\mathbf {J}}$ exists, and the image of $H_0(X,\\mathcal {D}_{\\chi , \\mathbf {J}})$ in $H_0(X,\\mathcal {C}(D))$ is a sum of supersingular representations." ], [ "Construction of Pure Diagrams when $q = p$", "We now give an application of the formalism developed in the previous section, using results of Section 5.", "Theorem 7.10 Suppose $q=p$ .", "Then for every supersingular module $M_{\\chi , \\mathbf {J}},$ there exists a pure diagram for $M_{\\chi , \\mathbf {J}}$ .", "More precisely, the corresponding initial diagram $D_{\\chi , \\mathbf {J}}=(\\rho ,~ \\rho ^{\\prime },~ \\chi , ~j, ~ j^{\\prime })$ can be embedded into a pure diagram ${E}_{\\chi , \\mathbf {J}}= (\\textnormal {inj}_K(\\textnormal {P}), ~\\textnormal {inj}_{K^{\\prime }}(\\textnormal {P}^{\\prime }), ~\\textnormal {inj}_I(\\textnormal {X}),~ j_p,~j^{\\prime }_p)$ where $\\textnormal {P} = \\rho $ , $\\textnormal {P}^{\\prime }$ is a semisimple representation of $K^{\\prime }$ having $\\rho ^{\\prime }$ as a summand, and $\\textnormal {X}$ is a semisimple representation of $I$ having $\\chi $ as a summand.", "Furthermore, the maps $j_p$ and $j^{\\prime }_p$ induce isomorphisms $\\textnormal {inj}_K(\\textnormal {P})|_I \\cong \\textnormal {inj}_I(\\textnormal {X}) \\cong \\textnormal {inj}_{K^{\\prime }}(\\textnormal {P}^{\\prime })|_I$ .", "Our main tool in proving this Theorem will be Lemma REF , which states that if $\\rho $ and $\\rho ^{\\prime }$ are smooth irreducible representations of $K$ and $K^{\\prime }$ , respectively, then we have $\\textnormal {inj}_K(\\rho )|_I \\cong \\bigoplus _{\\chi \\in \\widehat{H}} \\textnormal {inj}_I(\\chi )^{\\oplus m_{\\rho ,\\chi }},\\qquad \\textnormal {inj}_{K^{\\prime }}(\\rho ^{\\prime })|_I \\cong \\bigoplus _{\\chi \\in \\widehat{H}} \\textnormal {inj}_I(\\chi )^{\\oplus m_{\\rho ^{\\prime },\\chi }}.$ In general, it is not clear how the multiplicities in the above equations compare with each other.", "We record one result in this direction, which holds for general $q$ : Lemma 7.11 We have $m_{\\rho , \\chi }=m_{\\rho , \\chi ^s}$ , and $m_{\\rho ^{\\prime }, \\chi }=m_{\\rho ^{\\prime }, \\chi ^s}.$ The definition of the numbers $m_{\\rho ,\\chi }$ and Frobenius Reciprocity give $m_{\\rho ,\\chi } & = & \\dim _{\\overline{\\mathbb {F}}_p}(\\textnormal {Hom}_H (\\chi , \\textnormal {inj}_\\Gamma (\\rho )^{\\mathbb {U}}))\\\\& = & \\dim _{\\overline{\\mathbb {F}}_p}(\\textnormal {Hom}_\\mathbb {B}(\\chi ,\\textnormal {inj}_\\Gamma (\\rho )|_\\mathbb {B}))\\\\& = & \\dim _{\\overline{\\mathbb {F}}_p}(\\textnormal {Hom}_\\Gamma (\\textnormal {ind}_\\mathbb {B}^\\Gamma (\\chi ),\\textnormal {inj}_\\Gamma (\\rho ))).$ We note that given an arbitrary finite-dimensional mod-$p$ representation $V$ of $\\Gamma $ , the number $\\dim _{\\overline{\\mathbb {F}}_p}(\\textnormal {Hom}_\\Gamma (V,\\textnormal {inj}_\\Gamma (\\rho )))$ is precisely the multiplicity with which $\\rho $ occurs as a composition factor of $V$ .", "Given a $\\overline{\\mathbb {Q}}_p$ -representation $\\mathcal {W}$ of $\\Gamma $ , there exists a $\\Gamma $ -stable $\\overline{\\mathbb {Z}}_p$ -lattice in $\\mathcal {W}$ .", "Reducing this lattice modulo the maximal ideal gives a mod-$p$ representation of $\\Gamma $ , and Theorem 32 in [29] asserts that the semisimplification of this quotient is independent of the choice of $\\overline{\\mathbb {Z}}_p$ -lattice.", "In our case, it is straightforward to verify that (the semisimplification of) every mod-$p$ principal series representation comes from a characteristic 0 principal series representation in this fashion.", "The character tables of irreducible complex representations of $\\Gamma $ (and $\\Gamma ^{\\prime }$ ) have been determined by Ennola in [15].", "Considering the character tables with values in $\\overline{\\mathbb {Q}}_p$ , we obtain the Brauer characters of the principal series representations $\\textnormal {ind}_\\mathbb {B}^\\Gamma (\\chi )$ .", "In particular, the Brauer characters of the representations $\\textnormal {ind}_\\mathbb {B}^\\Gamma (\\chi )$ and $\\textnormal {ind}_\\mathbb {B}^\\Gamma (\\chi ^s)$ are identical.", "Since Brauer characters determine mod-$p$ representations up to semisimplification, we conclude that $m_{\\rho ,\\chi } = m_{\\rho ,\\chi ^s}$ .", "The same proof holds for the numbers $m_{\\rho ^{\\prime },\\chi }$ .", "Remark We note that this result holds in a more general context.", "In particular, if $\\mathcal {G}$ is a finite group with a “split BN pair $(\\mathcal {B},\\mathcal {N})$ of characteristic $p$ ,” $\\chi $ is a character of the maximal torus of $\\mathcal {G}$ , and $w$ is an element of the Weyl group of $(\\mathcal {B},\\mathcal {N})$ , then the induced representations $\\textnormal {ind}_{\\mathcal {B}}^{\\mathcal {G}}(\\chi )$ and $\\textnormal {ind}_{\\mathcal {B}}^{\\mathcal {G}}(\\chi ^w)$ have the same composition factors.", "The proof may be found in the Remarks of Section 7.2 and Section 9.7 of [19].", "Proposition 7.12 Assume $q = p$ .", "We then have Table: NO_CAPTIONRecall from Corollary 4.3 of [26] that we have a decomposition $\\overline{\\mathbb {F}}_p [\\Gamma ^{\\prime }]\\cong \\bigoplus _{\\tau \\in \\textnormal {Irr}_{\\Gamma ^{\\prime }}}\\textnormal {inj}_{\\Gamma ^{\\prime }}(\\tau )^{\\oplus \\dim _{\\overline{\\mathbb {F}}_p}(\\tau )},$ where $\\textnormal {Irr}_{\\Gamma ^{\\prime }}$ denotes the set of equivalence classes of irreducible mod-$p$ representations of $\\Gamma ^{\\prime }$ .", "Using Proposition REF , we translate the above decomposition into a sum over characters of $H$ : $\\overline{\\mathbb {F}}_p [\\Gamma ^{\\prime }] & \\cong &\\bigoplus _{\\chi =\\chi ^s}\\textnormal {inj}_{\\Gamma ^{\\prime }}(\\rho ^{\\prime }_{\\chi ,S^{\\prime }}) \\oplus (\\rho ^{\\prime }_{\\chi ,\\emptyset })^{\\oplus p}\\\\& &\\oplus \\bigoplus _{\\chi \\ne \\chi ^s}\\textnormal {inj}_{\\Gamma ^{\\prime }}(\\rho ^{\\prime }_{\\chi , \\emptyset })^{\\oplus \\dim _{\\overline{\\mathbb {F}}_p}(\\rho ^{\\prime }_{\\chi , \\emptyset })}\\oplus \\textnormal {inj}_{\\Gamma ^{\\prime }}(\\rho ^{\\prime }_{\\chi ^s, \\emptyset })^{\\oplus \\dim _{\\overline{\\mathbb {F}}_p}(\\rho ^{\\prime }_{\\chi ^s, \\emptyset })},$ where we use the fact that $\\rho ^{\\prime }_{\\chi ,\\emptyset }$ with $\\chi = \\chi ^s$ is a twist of the Steinberg representation, and therefore is injective of dimension $p$ .", "A similar argument as in the proof of Lemma 4.7 of [26] implies that $\\dim _{\\overline{\\mathbb {F}}_p}(\\textnormal {inj}_{\\Gamma ^{\\prime }}(\\rho ^{\\prime }_{\\chi , \\emptyset }))\\ge 2p$ for any character $\\chi $ satisfying $\\chi \\ne \\chi ^s$ .", "Additionally, by Lemma REF , we have $\\dim _{\\overline{\\mathbb {F}}_p}(\\rho ^{\\prime }_{\\chi , \\emptyset })+\\dim _{\\overline{\\mathbb {F}}_p}(\\rho ^{\\prime }_{\\chi ^s, \\emptyset })=p+1$ .", "These two facts allow us to evaluate the dimensions of both sides in the decomposition above: $p(p+1)^2 (p^2 -1)\\ge \\sum _{\\chi =\\chi ^s}(\\dim _{\\overline{\\mathbb {F}}_p}(\\textnormal {inj}_{\\Gamma ^{\\prime }}(\\rho ^{\\prime }_{\\chi ,S^{\\prime }})) + p^2) +\\sum _{\\chi \\ne \\chi ^s}~2p(p+1)$ The number of $\\chi $ satisfying $\\chi =\\chi ^s$ is $(p+1)^2$ , while the number of unordered pairs $\\lbrace \\chi , \\chi ^s\\rbrace $ such that $\\chi \\ne \\chi ^s$ is $\\frac{1}{2}(p+1)^2(p-2)$ .", "The above inequality now reduces to $(p+1)^2 p\\ge \\sum _{\\chi =\\chi ^s}\\dim _{\\overline{\\mathbb {F}}_p}(\\textnormal {inj}_{\\Gamma ^{\\prime }}(\\rho ^{\\prime }_{\\chi ,S^{\\prime }})).$ Since the order of $\\mathbb {U}^{\\prime }$ is $p$ , Corollary 4.6 of [26] implies that $\\dim _{\\overline{\\mathbb {F}}_p}(\\textnormal {inj}_{\\Gamma ^{\\prime }}(\\rho ^{\\prime }_{\\chi ,S^{\\prime }}))\\ge p$ .", "We therefore have $\\sum _{\\chi =\\chi ^s}\\dim _{\\overline{\\mathbb {F}}_p}(\\textnormal {inj}_{\\Gamma ^{\\prime }}(\\rho ^{\\prime }_{\\chi ,S^{\\prime }})) \\ge (p+1)^2 p.$ which forces every inequality above to be an equality.", "To sum up, we have Table: NO_CAPTION To proceed, note that the dimensions computed above tell us precisely the number of terms appearing on the right-hand side of equation (REF ).", "Taking $\\mathbb {U}^{\\prime }$ -invariants of the exact sequence $0\\rightarrow \\rho ^{\\prime }_{\\chi ,J^{\\prime }} \\rightarrow \\textnormal {inj}_{\\Gamma ^{\\prime }}(\\rho ^{\\prime }_{\\chi ,J^{\\prime }})$ yields $0\\rightarrow \\chi \\rightarrow \\textnormal {inj}_{\\Gamma ^{\\prime }}(\\rho ^{\\prime }_{\\chi ,J^{\\prime }})^{\\mathbb {U}^{\\prime }};$ combining this with Lemma REF gives the result.", "Remark We note that an alternate proof of this result may be obtained by explicitly computing the composition factors of the representations $\\textnormal {ind}_{\\mathbb {B}^{\\prime }}^{\\Gamma ^{\\prime }}(\\chi )$ using Lemma REF .", "Using Lemma REF , we now rewrite equation $(\\ref {decom for K})$ as $\\textnormal {inj}_{K}(\\rho )|_I = \\bigoplus _{\\mu =\\mu ^s} \\textnormal {inj}_I(\\mu )^{\\oplus m_{\\rho , \\mu }}\\oplus \\bigoplus _{\\mu \\ne \\mu ^s} (\\textnormal {inj}_I(\\mu )\\oplus \\textnormal {inj}_I(\\mu ^s))^{\\oplus m_{\\rho , \\mu }},$ the sums being taken over $W$ -orbits of characters.", "We let $\\textnormal {X}$ be the representation of $I$ defined by $\\textnormal {X} = \\bigoplus _{\\mu =\\mu ^s} \\mu ^{\\oplus m_{\\rho , \\mu }}\\oplus \\bigoplus _{\\mu \\ne \\mu ^s} (\\mu \\oplus \\mu ^s)^{\\oplus m_{\\rho , \\mu }},$ and let $\\textnormal {P}^{\\prime }$ be a representation of $K^{\\prime }$ of the form $\\textnormal {P}^{\\prime } = \\bigoplus _{\\mu =\\mu ^s} (\\rho ^{\\prime }_{\\gamma _\\mu })^{\\oplus m_{\\rho , \\mu }}\\oplus \\bigoplus _{\\mu \\ne \\mu ^s}(\\rho ^{\\prime }_{\\gamma _\\mu })^{\\oplus m_{\\rho , \\mu }}.$ Here we choose $\\rho ^{\\prime }_{\\gamma _\\mu }\\in \\lbrace \\rho ^{\\prime }_{\\mu ,S^{\\prime }},\\rho ^{\\prime }_{\\mu ,\\emptyset }\\rbrace $ if $\\mu =\\mu ^s$ and $\\rho ^{\\prime }_{\\gamma _\\mu }\\in \\lbrace \\rho ^{\\prime }_{\\mu ,\\emptyset },\\rho ^{\\prime }_{\\mu ^s,\\emptyset }\\rbrace $ if $\\mu \\ne \\mu ^s$ ; the only stipulation we make is that $\\rho ^{\\prime }$ be among the summands.", "By definition and Proposition REF , we have $\\textnormal {inj}_K(\\textnormal {P})|_I \\cong \\textnormal {inj}_{K^{\\prime }}(\\textnormal {P}^{\\prime })|_I \\cong \\textnormal {inj}_I(\\textnormal {X})$ .", "We now have natural injective maps from $\\rho $ to $\\textnormal {inj}_K(\\textnormal {P})$ , from $\\rho ^{\\prime }$ to $\\textnormal {inj}_{K^{\\prime }}(\\textnormal {P}^{\\prime })$ and from $\\chi $ to $\\textnormal {inj}_I(\\textnormal {X})$ : they are defined by first mapping each representation into its respective injective envelope, followed by the canonical inclusion into the direct sum.", "Moreover, one can choose the maps $j_p$ and $j_p^{\\prime }$ such that these injective maps induce a morphism of diagrams $\\psi :~D_{\\chi , \\mathbf {J}}\\rightarrow {E}_{\\chi , \\mathbf {J}}$ .", "It is evident that the diagram $E_{\\chi , \\mathbf {J}}$ is pure for $M_{\\chi , \\mathbf {J}}$ .", "Corollary 7.13 Assume $q = p$ , let $M_{\\chi , \\mathbf {J}}$ be a supersingular module, and let $E_{\\chi , \\mathbf {J}}$ be a pure diagram for $M_{\\chi , \\mathbf {J}}$ , constructed as in the proof of the previous theorem.", "Set $\\mathcal {E}_{\\chi , \\mathbf {J}} = \\mathcal {C}(E_{\\chi , \\mathbf {J}})$ .", "Then the image $\\pi _{E_{\\chi , \\mathbf {J}}}=\\textnormal {im}(\\psi _*:H_0(X, \\mathcal {D}_{\\chi , \\mathbf {J}})\\rightarrow H_0(X, \\mathcal {E}_{\\chi , \\mathbf {J}}))$ is an irreducible admissible supersingular representation.", "Moreover, for distinct modules $M_{\\chi , \\mathbf {J}}, M_{\\chi ^{\\prime }, \\mathbf {J}^{\\prime }}$ , the representations $\\pi _{E_{\\chi , \\mathbf {J}}}, \\pi _{E_{\\chi ^{\\prime }, \\mathbf {J}^{\\prime }}}$ are nonisomorphic.", "The first part of the Corollary follows from Theorems REF and REF .", "To prove the second part, let us assume $\\phi :\\pi _{E_{\\chi , \\mathbf {J}}}\\stackrel{\\sim }{\\rightarrow } \\pi _{E_{\\chi ^{\\prime }, \\mathbf {J}^{\\prime }}}$ is an isomorphism; we then obtain an induced isomorphism $\\overline{\\phi }:\\textnormal {soc}_K(\\pi _{E_{\\chi , \\mathbf {J}}}|_K)^{I(1)}\\stackrel{\\sim }{\\rightarrow } \\textnormal {soc}_K(\\pi _{E_{\\chi ^{\\prime }, \\mathbf {J}^{\\prime }}}|_K)^{I(1)}.$ The proof of Theorem REF shows how to equip these spaces with an action of $\\mathcal {H}_{\\overline{\\mathbb {F}}_p}(G,I(1))$ , which gives $M_{\\chi , \\mathbf {J}}\\cong \\textnormal {soc}_K(\\pi _{E_{\\chi , \\mathbf {J}}}|_K)^{I(1)}\\stackrel{\\overline{\\phi }}{\\rightarrow } \\textnormal {soc}_K(\\pi _{E_{\\chi ^{\\prime }, \\mathbf {J}^{\\prime }}}|_K)^{I(1)}\\cong M_{\\chi ^{\\prime }, \\mathbf {J}^{\\prime }}.$ The claim now follows from the comments following Definition REF .", "Remark Assume $q = p$ .", "Given a supersingular module $M_{\\chi , \\mathbf {J}}$ , our construction shows that there may be many choices of pure diagram $E_{\\chi , \\mathbf {J}}$ associated to $M_{\\chi , \\mathbf {J}}$ .", "As a consequence, if $E_{\\chi , \\mathbf {J}}$ and $E_{\\chi , \\mathbf {J}}^\\star $ are two such diagrams, we obtain two supersingular representations $\\pi _{E_{\\chi , \\mathbf {J}}}$ and $\\pi _{E_{\\chi , \\mathbf {J}}^\\star }$ whose $I(1)$ -invariants contain $M_{\\chi , \\mathbf {J}}$ .", "It is not clear, however, if these representations are isomorphic." ], [ "The Case $q \\ne p$", "In this section we point out the shortcomings of our method in the case when $q \\ne p$ .", "We assume that $q = p^2$ for the sake of simplicity.", "Let 1 denote the trivial character of $H$ (or, equivalently, of $I$ ), and consider the diagram $D_{1, (\\emptyset , S^{\\prime })} = (\\rho _{1,\\emptyset },~ \\rho ^{\\prime }_{1,S^{\\prime }},~ 1,~ j,~ j^{\\prime })$ .", "Here $\\rho _{1,\\emptyset }$ is the Steinberg representation of $K$ , and $\\rho ^{\\prime }_{1,S^{\\prime }}$ is the trivial character of $K^{\\prime }$ .", "We claim that there does not exist a pure diagram $D$ for $M_{1, (\\emptyset , S^{\\prime })}$ of the form $(\\textnormal {inj}_K(\\textnormal {P}),~ \\textnormal {inj}_{K^{\\prime }}(\\textnormal {P}^{\\prime }),~ \\textnormal {inj}_I(\\textnormal {X}),~ j_p,~ j_p^{\\prime })$ , where $\\textnormal {P}$ is a semisimple representation of $K$ , $\\textnormal {P}^{\\prime }$ is a semisimple representation of $K^{\\prime }$ , and $\\textnormal {X}$ is a semisimple representation of $I$ .", "We require some preparatory facts.", "Let $\\mu $ and $\\mu ^\\star $ be two $\\overline{\\mathbb {F}}_p$ -characters of $H$ defined by $\\mu \\begin{pmatrix}a & 0 & 0 \\\\ 0 & \\delta & 0 \\\\ 0 & 0 & \\overline{a}^{-1}\\end{pmatrix} = a^{(p^2 + 1)(p - 1)},\\quad \\mu ^\\star \\begin{pmatrix}a & 0 & 0 \\\\ 0 & \\delta & 0 \\\\ 0 & 0 & \\overline{a}^{-1}\\end{pmatrix} = a^{(p^2 + 1)(p + 1)}.$ A computation with Brauer characters verifies that $\\textnormal {ind}_{\\mathbb {B}^{\\prime }}^{\\Gamma ^{\\prime }}(\\mu )^{\\textnormal {ss}} & \\cong & V_{0,0}^{\\prime }\\boxtimes \\omega ^0~ \\oplus ~ V_{p^2 - 2p + 1, p}^{\\prime }\\boxtimes \\omega ^0~ \\oplus ~ V_{2p - 2, 1 - p}^{\\prime }\\boxtimes \\omega ^0~ \\oplus ~ V_{p^2 - 2p - 3, p + 2}^{\\prime }\\boxtimes \\omega ^0 \\\\& \\cong & \\rho ^{\\prime }_{1,S^{\\prime }}~\\oplus ~ \\rho ^{\\prime }_{\\mu ,\\emptyset }~\\oplus ~ \\rho ^{\\prime }_{\\mu ^s,\\emptyset }~\\oplus ~ \\rho ^{\\prime }_{\\mu ^\\star ,\\emptyset },$ where the superscript “ss” denotes semisimplification.", "Alternatively, we may obtain this decomposition from a slightly modified version of Proposition 1.1 in [12], along with the character tables computed in [15].", "Using the fact that $\\mathbf {SU}(1,1)(\\mathbb {F}_{p^4}/\\mathbb {F}_{p^2})$ is conjugate to $\\textnormal {SL}_2(\\mathbb {F}_{p^2})$ , and modifying the arguments in Section 4.2 of [26] shows that $\\dim _{\\overline{\\mathbb {F}}_p}(\\textnormal {inj}_{\\Gamma ^{\\prime }}(\\rho ^{\\prime }_{1,S^{\\prime }})) = 3p^2$ .", "Combining these two facts with Lemma REF shows that $\\textnormal {inj}_{K^{\\prime }}(\\rho ^{\\prime }_{1,S^{\\prime }})|_I \\cong \\textnormal {inj}_I(1)\\oplus \\textnormal {inj}_I(\\mu )\\oplus \\textnormal {inj}_I(\\mu ^s).$ Assume now that we have an embedding of diagrams $D_{1, (\\emptyset , S^{\\prime })}\\rightarrow D$ , with $D$ pure: 1,[rr] injK(P)1[ur]j[dr]j'[rr] injK'(X)[ur]jp[dr]jp' '1,S' [rr] injK'(P') Assume first that the $K$ -representation of $D$ has simple $K$ -socle, so that $\\textnormal {P} \\cong \\rho _{1,\\emptyset }$ .", "Since $\\rho _{1,\\emptyset }$ is injective as a representation of $\\Gamma $ , we have $\\textnormal {inj}_K(\\rho _{1,\\emptyset })|_I \\cong \\textnormal {inj}_I(1)$ .", "We have an injection $\\rho ^{\\prime }_{1,S^{\\prime }}\\hookrightarrow \\textnormal {inj}_{K^{\\prime }}(\\textnormal {P}^{\\prime })$ and the latter representation is injective, so Lemma 6.13 of [26] implies there exists an injection $\\textnormal {inj}_{K^{\\prime }}(\\rho ^{\\prime }_{1,S^{\\prime }})\\hookrightarrow \\textnormal {inj}_{K^{\\prime }}(\\textnormal {P}^{\\prime }).$ Restricting to $I$ and using the defintion of purity gives $ \\textnormal {inj}_I(1)\\oplus \\textnormal {inj}_I(\\mu )\\oplus \\textnormal {inj}_I(\\mu ^s) \\cong \\textnormal {inj}_{K^{\\prime }}(\\rho ^{\\prime }_{1,S^{\\prime }})|_I \\hookrightarrow \\textnormal {inj}_{K^{\\prime }}(\\textnormal {P}^{\\prime })|_I\\cong \\textnormal {inj}_I(1),$ which is absurd.", "We may therefore assume that the $K^{\\prime }$ -representation of $D$ has simple $K^{\\prime }$ -socle, so that $\\textnormal {P}^{\\prime } \\cong \\rho ^{\\prime }_{1,S^{\\prime }}$ and $\\textnormal {inj}_{K^{\\prime }}(\\textnormal {P}^{\\prime })|_I \\cong \\textnormal {inj}_I(1)\\oplus \\textnormal {inj}_I(\\mu )\\oplus \\textnormal {inj}_I(\\mu ^s) \\cong \\textnormal {inj}_K(\\textnormal {P})|_I,$ by the definition of purity.", "This implies that we must have $\\textnormal {inj}_K(\\textnormal {P}/\\rho _{1,\\emptyset })|_I \\cong \\textnormal {inj}_I(\\mu )\\oplus \\textnormal {inj}_I(\\mu ^s);$ the only representations for which this could potentially be true are $\\rho _{\\mu ,\\emptyset }$ and $\\rho _{\\mu ^s,\\emptyset }$ .", "The dimensions of the injective envelopes of $\\mathbf {SU}(2,1)(\\mathbb {F}_{p^4}/\\mathbb {F}_{p^2})$ have been computed explicitly by Dordowsky in his Diplomarbeit ([13]).", "In particular, his results show that $\\dim _{\\overline{\\mathbb {F}}_p}(\\textnormal {inj}_\\Gamma (\\rho _{\\mu ,\\emptyset })) = \\dim _{\\overline{\\mathbb {F}}_p}(\\textnormal {inj}_\\Gamma (\\rho _{\\mu ^s,\\emptyset })) = 12p^6$ , which implies that the number of summands in the decompositions of $\\textnormal {inj}_K(\\rho _{\\mu ,\\emptyset })|_I$ and $\\textnormal {inj}_K(\\rho _{\\mu ^s,\\emptyset })|_I$ is 12.", "This verifies our claim." ], [ "Comparison with $\\textnormal {SL}_2(F)$", "In the course of defining diagrams and coefficient systems for $\\textrm {U}(2,1)(E/F)$ , there are several parallels one can draw between the formalism we have used and the analogous formalism for the group $\\textnormal {SL}_2(F)$ .", "We hope to make this connection precise here, drawing on results of Abdellatif in [1].", "In this section only, the prime $p$ may be arbitrary.", "We let $G_S = \\textnormal {SL}_2(F),~ K_S = \\textnormal {SL}_2(\\mathfrak {o}_F),$ and $K_S^{\\prime } = \\alpha _SK_S\\alpha _S^{-1}$ , where $\\alpha _S = \\begin{pmatrix}1 & 0 \\\\ 0 & \\varpi _F\\end{pmatrix};$ the groups $K_S$ and $K_S^{\\prime }$ are representatives of the two conjugacy classes of maximal compact subgroups of $G_S$ .", "We note that our notation differs slightly from that of [1].", "Let $I_S = K_S\\cap K_S^{\\prime }$ be the Iwahori subgroup, and $I_S(1)\\le I_S$ its unique pro-$p$ -Sylow subgroup.", "Let $w_s = \\begin{pmatrix}0 & -1 \\\\ 1 & 0\\end{pmatrix}~~\\textnormal {and}~~ w_{s^{\\prime }} = \\begin{pmatrix}0 & -\\varpi _F^{-1}\\\\ \\varpi _F & 0\\end{pmatrix},$ and for $r\\in \\mathbb {Z}$ , let $\\omega ^r$ denote the $\\overline{\\mathbb {F}}_p$ -character of the finite torus $H_S$ defined by $\\omega ^r\\begin{pmatrix}a & 0 \\\\ 0 & a^{-1}\\end{pmatrix} = a^r,$ where $a\\in \\mathbb {F}_q^\\times $ .", "As in Section 3, we denote by $\\mathcal {H}_{\\overline{\\mathbb {F}}_p}(G_S,I_S(1)) = \\textnormal {End}_{G_S}(\\textnormal {c-ind}_{I_S(1)}^{G_S}(1))$ the pro-$p$ -Iwahori-Hecke algebra, and let ${w_s}$ (resp.", "${w_{s^{\\prime }}}$ ) be the endomorphism corresponding by adjunction to the characteristic function of $I_S(1)w_sI_S(1)$ (resp.", "$I_S(1)w_{s^{\\prime }}I_S(1)$ ).", "For $0\\le r < q-1$ , we define $e_{\\omega ^r} = |H_S|^{-1}\\sum _{h\\in H_S}\\omega ^r(h){h},$ where $h$ denotes the endomorphism corresponding by adjunction to the characteristic function of $I_S(1)hI_S(1)$ .", "A proof similar to that of Proposition REF shows that $\\mathcal {H}_{\\overline{\\mathbb {F}}_p}(G_S,I_S(1))$ is generated by ${w_s}, {w_{s^{\\prime }}}$ and $e_{\\omega ^r}$ for $0\\le r<q-1$ .", "The supersingular Hecke modules (as defined in [34]) have been classified in [1], Chapitre 6.", "They naturally divide into three classes, depending on the nature of the character $\\omega ^r$ (or equivalently, the parameter $r$ ).", "Proposition 8.1 The supersingular $\\mathcal {H}_{\\overline{\\mathbb {F}}_p}(G_S,I_S(1))$ -modules are all one-dimensional.", "They are given by: Table: NO_CAPTIONwhere $0 < r < q-1,r\\ne \\frac{q-1}{2}$ , and 1 denotes the trivial character of $H_S$ .", "The module $M_{(q-1)/2}$ is nonexistent if $q$ is even, while the modules $M_r$ are nonexistent if $q = 2$ or $q = 3$ .", "As is the case for $\\textrm {U}(2,1)(E/F)$ , the Bruhat-Tits building $X_S$ of $G_S$ is a tree.", "We let $\\sigma _0$ denote the hyperspecial vertex for which $\\mathfrak {K}(\\sigma _0) = K_S$ .", "The action of $G_S$ partitions the vertices into two orbits, those at an even distance from $\\sigma _0$ and those at an odd distance from $\\sigma _0$ .", "Since the action of $G_S$ on the set of (nonoriented) edges is transitive, the notion of a diagram is the same as in Definition REF .", "Moreover, the results of Section 6 do not rely on any other properties of the group $\\textrm {U}(2,1)(E/F)$ ; replacing $G$ by $G_S$ , $K$ by $K_S$ , etc., shows that every conclusion holds equally well for $G_S$ .", "In particular, the categories $\\mathcal {COEF}_{G_S}$ and $\\mathcal {DIAG}$ are equivalent.", "With this analogy in mind, we define the following diagrams.", "Definition 8.2 Let $\\rho _{\\omega ^r,J_S}$ and $\\rho _{\\omega ^r,J_S^{\\prime }}^{\\prime }$ denote the representations of $K_S$ and $K_S^{\\prime }$ obtained by inflation from $\\textnormal {SL}_2(\\mathbb {F}_q)$ .", "We set Table: NO_CAPTIONwhere $0 < r < q-1, r\\ne \\frac{q-1}{2}$ , and where $j$ and $j^{\\prime }$ are the inclusion maps.", "Using the same arguments as in Section 7, one can show that given a diagram $D_r$ of the form above, the $I_S(1)$ -invariants of every nonzero irreducible quotient of $H_0(X_S, \\mathcal {C}(D_r))$ contain $M_r$ , and therefore such a quotient must be supersingular.", "Moreover, if we specialize to the case $q = p$ , there exists a canonical pure diagram $E_r$ corresponding to an initial diagram $D_r$ : Proposition 8.3 Assume $q = p$ .", "Let $M_r$ be a supersingular $\\mathcal {H}_{\\overline{\\mathbb {F}}_p}(G_S,I_S(1))$ -module, and let $D_r = (\\rho ,~ \\rho ^{\\prime },~ \\omega ^r,~ j,~ j^{\\prime })$ be the associated diagram as in Definition REF .", "Then the diagram $E_r = (\\textnormal {inj}_{K_S}(\\rho ),~ \\textnormal {inj}_{K_S^{\\prime }}(\\rho ^{\\prime }),~ \\textnormal {inj}_{K_S}(\\rho )|_{I_S},~ j_p,~ j_p^{\\prime })$ is pure for $M_r$ , where $j_p$ and $j_p^{\\prime }$ are isomorphisms.", "Theorem 8.4 Assume $q = p$ .", "Let $M_r$ be a supersingular $\\mathcal {H}_{\\overline{\\mathbb {F}}_p}(G_S,I_S(1))$ -module, let $D_r$ and $E_r$ be the diagrams constructed above, and let $\\psi :D_r\\rightarrow E_r$ denote the canonical embedding.", "Then the representation afforded by $\\textnormal {im}(\\psi _*:H_0(X_S,\\mathcal {C}(D_r))\\rightarrow H_0(X_S,\\mathcal {C}(E_r)))$ is irreducible, admissible and supersingular.", "For distinct supersingular modules $M_r, M_{r^{\\prime }}$ , the resulting representations are nonisomorphic.", "The proofs of Propositions REF and REF hold equally well in the context of the group $G_S$ , which implies the first claim.", "The proof of the second claim follows in a manner similar to the proof of Corollary REF .", "In this way, we have constructed $p$ irreducible supersingular representations, corresponding to the supersingular $\\mathcal {H}_{\\overline{\\mathbb {F}}_p}(G_S,I_S(1))$ -modules.", "In particular, for $F = \\mathbb {Q}_p$ , we recover the following classification of supersingular representations: Theorem 8.5 Let $M_r$ be a supersingular Hecke module for $\\textnormal {SL}_2(\\mathbb {Q}_p)$ , and let $D_r$ and $E_r$ be the diagrams constructed above.", "We then have $\\textnormal {im}(\\psi _*:H_0(X_S,\\mathcal {C}(D_r))\\rightarrow H_0(X_S,\\mathcal {C}(E_r)))\\cong \\pi _r,$ where $\\pi _r$ is the supersingular representation of $\\textnormal {SL}_2(\\mathbb {Q}_p)$ defined in [1], Chapitre 3.", "By Théorème 3.6.13 of [1], there are precisely $p$ isomorphism classes of irreducible supersingular representations of $\\textnormal {SL}_2(\\mathbb {Q}_p)$ , given by the representations $\\pi _r, 0\\le r\\le p-1$ .", "Likewise, the representations $\\textnormal {im}(\\psi _*:H_0(X_S,\\mathcal {C}(D_r))\\rightarrow H_0(X_S,\\mathcal {C}(E_r))),$ for $0\\le r\\le p-1$ constitute $p$ pairwise nonisomorphic irreducible supersingular representations.", "It therefore suffices to match these.", "Since the $I_S(1)$ -invariants of the image of the induced map on homology contain a Hecke module isomorphic to $M_r$ , and since $\\pi _r^{I_S(1)}\\cong M_r$ as right $\\mathcal {H}_{\\overline{\\mathbb {F}}_p}(G_S, I_S(1))$ -modules, we conclude $\\textnormal {im}(\\psi _*:H_0(X_S,\\mathcal {C}(D_r))\\rightarrow H_0(X_S,\\mathcal {C}(E_r)))\\cong \\pi _r.$ Remark When $q \\ne p$ , the above construction fails in a manner similar to the construction for $\\textrm {U}(2,1)(E/F)$ , meaning that pure diagrams of the form $(\\textnormal {inj}_{K_S}(\\textnormal {P}),~ \\textnormal {inj}_{K_S^{\\prime }}(\\textnormal {P}^{\\prime }),~ \\textnormal {inj}_{I_S}(\\textnormal {X}),~ j_p,~ j_p^{\\prime })$ do not always exist.", "One may translate the example of the previous section to the case of $\\textnormal {SL}_2(F)$ to produce such an example explicitly." ], [ "Proof of Theorem ", "Here we carry out the computations for Theorem REF .", "Recall that $\\varepsilon = \\widetilde{\\zeta }\\otimes \\widetilde{\\eta }$ is a character of the torus $T$ , and the space of $I(1)$ -invariants $\\textnormal {ind}_B^G(\\varepsilon )^{I(1)}$ is two-dimensional, spanned by the functions $f_1$ and $f_2$ (cf.", "Section 4).", "The computations are split up according to the nature of $\\varepsilon ^*$ .", "Using Proposition 6 in [3], we can compute the action of ${n_s}$ and ${n_{s^{\\prime }}}$ on $v\\in \\textrm {ind}_B^G(\\varepsilon )^{I(1)}$ .", "Equation (REF ) implies $v\\cdot {n_s} = \\sum _{\\genfrac{}{}{0.0pt}{}{x,y\\in \\mathbb {F}_{q^2}}{x\\overline{x} + y + \\overline{y} = 0}}u(-x,\\overline{y})n_s^{-1}.v.$ Substituting $f_1$ for $v$ and evaluating at 1 and $n_s$ gives $f_1\\cdot {n_s}(1) & = & \\sum _{\\genfrac{}{}{0.0pt}{}{x,y\\in \\mathbb {F}_{q^2}}{x\\overline{x} + y + \\overline{y} = 0}}f_1(u(-x,\\overline{y})n_s^{-1})\\\\& = & 0\\\\f_1\\cdot {n_s}(n_s) & = & \\sum _{\\genfrac{}{}{0.0pt}{}{x,y\\in \\mathbb {F}_{q^2}}{x\\overline{x} + y + \\overline{y} = 0}}f_1(n_su(-x,\\overline{y})n_s^{-1})\\\\& = & \\sum _{\\genfrac{}{}{0.0pt}{}{x,y\\in \\mathbb {F}_{q^2}}{x\\overline{x} + y + \\overline{y} = 0}}f_1(u^-(-\\overline{x}\\sqrt{\\epsilon },-\\overline{y}\\epsilon ))\\\\& = & 1,$ where the last equality follows from equation (REF ).", "Similarly, $f_2\\cdot {n_s}(1) & = & \\sum _{\\genfrac{}{}{0.0pt}{}{x,y\\in \\mathbb {F}_{q^2}}{x\\overline{x} + y + \\overline{y} = 0}}f_2(u(-x,\\overline{y})n_s^{-1})\\\\& = & 0\\\\f_2\\cdot {n_s}(n_s) & = & \\sum _{\\genfrac{}{}{0.0pt}{}{x,y\\in \\mathbb {F}_{q^2}}{x\\overline{x} + y + \\overline{y} = 0}}f_2(n_su(-x,\\overline{y})n_s^{-1})\\\\& = & \\sum _{\\genfrac{}{}{0.0pt}{}{x,y\\in \\mathbb {F}_{q^2}}{x\\overline{x} + y + \\overline{y} = 0}}f_2(u^-(-\\overline{x}\\sqrt{\\epsilon },-\\overline{y}\\epsilon ))\\\\& = & \\sum _{\\genfrac{}{}{0.0pt}{}{x,y\\in \\mathbb {F}_{q^2},y\\ne 0}{x\\overline{x} + y + \\overline{y} = 0}}f_2(n_s\\textrm {diag}(-\\overline{y}\\sqrt{\\epsilon },-y\\overline{y}^{-1}, y^{-1}\\sqrt{\\epsilon }^{-1}))\\\\& = & \\sum _{\\genfrac{}{}{0.0pt}{}{x,y\\in \\mathbb {F}_{q^2},y\\ne 0}{x\\overline{x} + y + \\overline{y} = 0}}\\varepsilon (\\textrm {diag}(y^{-1}\\sqrt{\\epsilon }^{-1},-y\\overline{y}^{-1},-\\overline{y}\\sqrt{\\epsilon }))\\\\& = & {\\left\\lbrace \\begin{array}{ll}-1 & \\textrm {if}\\ \\varepsilon ^* = \\widetilde{\\eta }\\circ \\det \\\\ \\phantom{-}0 & \\textrm {if}\\ (\\varepsilon ^*)^s = \\varepsilon ^*, \\varepsilon ^*\\ne \\widetilde{\\eta }\\circ \\det \\\\ \\phantom{-}0 & \\textrm {if}\\ (\\varepsilon ^*)^s \\ne \\varepsilon ^*.\\end{array}\\right.", "}$ The last equality is obtained in precisely the same manner as in the proof of Proposition REF (cf.", "the computation of $\\varphi _{n_s}*\\varphi _{n_s}(n_s)$ ).", "Equation () implies $v\\cdot {n_{s^{\\prime }}} = \\sum _{\\genfrac{}{}{0.0pt}{}{y\\in \\mathbb {F}_{q^2}}{y + \\overline{y} = 0}}u^-(0,\\varpi \\overline{y})\\alpha n_s^{-1}.v$ for $v\\in \\textrm {ind}_B^G(\\varepsilon )^{I(1)}$ .", "Thus $f_1\\cdot {n_{s^{\\prime }}}(1) & = & \\sum _{\\genfrac{}{}{0.0pt}{}{y\\in \\mathbb {F}_{q^2}}{y + \\overline{y} = 0}}f_1(u^-(0,\\varpi \\overline{y})\\alpha n_s^{-1})\\\\& = & \\sum _{\\genfrac{}{}{0.0pt}{}{y\\in \\mathbb {F}_{q^2}, y\\ne 0}{y + \\overline{y} = 0}}f_1(u(0,\\varpi ^{-1}\\overline{y}^{-1})\\textrm {diag}(-y^{-1}\\sqrt{\\epsilon },1,-y\\sqrt{\\epsilon }^{-1})u^-(0,\\varpi \\epsilon y^{-1}))\\\\& = & \\sum _{\\genfrac{}{}{0.0pt}{}{y\\in \\mathbb {F}_{q^2}, y\\ne 0}{y + \\overline{y} = 0}} \\varepsilon (\\textrm {diag}(-y^{-1}\\sqrt{\\epsilon },1,-y\\sqrt{\\epsilon }^{-1}))\\\\& \\stackrel{\\star }{=} & {\\left\\lbrace \\begin{array}{ll}-1 & \\textrm {if}\\ \\varepsilon ^* = \\widetilde{\\eta }\\circ \\det \\\\ -1 & \\textrm {if}\\ (\\varepsilon ^*)^s = \\varepsilon ^*, \\varepsilon ^*\\ne \\widetilde{\\eta }\\circ \\det \\\\ \\phantom{-}0 & \\textrm {if}\\ (\\varepsilon ^*)^s\\ne \\varepsilon ^*\\end{array}\\right.", "}\\\\f_1\\cdot {n_{s^{\\prime }}}(n_s) & = & \\sum _{\\genfrac{}{}{0.0pt}{}{y\\in \\mathbb {F}_q}{y + \\overline{y} = 0}}f_1(n_su^-(0,\\varpi \\overline{y})\\alpha n_s^{-1})\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\\\& = & \\sum _{\\genfrac{}{}{0.0pt}{}{y\\in \\mathbb {F}_q}{y + \\overline{y} = 0}}f_1(u(0,-\\varpi \\epsilon ^{-1}\\overline{y})\\alpha ^{-1})\\\\& = & 0.$ The equality $(\\star )$ follows from the fact that $\\widetilde{\\zeta }$ is trivial on $\\mathbb {F}_q^\\times $ if and only if $\\widetilde{\\zeta }^{q+1}$ is trivial on $\\mathfrak {o}_E^\\times $ .", "Similarly, we have $f_2\\cdot {n_{s^{\\prime }}}(1) & = & \\sum _{\\genfrac{}{}{0.0pt}{}{y\\in \\mathbb {F}_{q^2}}{y + \\overline{y} = 0}}f_2(u^-(0,\\varpi \\overline{y})\\alpha n_s^{-1})\\\\& = & \\widetilde{\\zeta }(-1)\\varepsilon (\\alpha ) + \\sum _{\\genfrac{}{}{0.0pt}{}{y\\in \\mathbb {F}_{q^2}, y\\ne 0}{y + \\overline{y} = 0}}f_2(u(0,\\varpi ^{-1}\\overline{y}^{-1})\\textrm {diag}(-y^{-1}\\sqrt{\\epsilon },1,-y\\sqrt{\\epsilon }^{-1})u^-(0,\\varpi \\epsilon y^{-1}))\\\\& = & \\widetilde{\\zeta }(-1)\\varepsilon (\\alpha )\\\\f_2\\cdot {n_{s^{\\prime }}}(n_s) & = & \\sum _{\\genfrac{}{}{0.0pt}{}{y\\in \\mathbb {F}_{q^2}}{y + \\overline{y} = 0}}f_2(n_su^-(0,\\varpi \\overline{y})\\alpha n_s^{-1})\\\\& = & \\sum _{\\genfrac{}{}{0.0pt}{}{y\\in \\mathbb {F}_{q^2}}{y + \\overline{y} = 0}}f_2(u(0,-\\varpi \\epsilon ^{-1}\\overline{y})\\alpha ^{-1})\\\\& = & 0.$" ], [ "Proof of Theorem ", "In order to prove that the categories $\\mathcal {COEF}_G$ and $\\mathcal {DIAG}$ are equivalent, we must verify that there is a natural transformation from $\\mathcal {D}\\circ \\mathcal {C}$ (resp.", "$\\mathcal {C}\\circ \\mathcal {D}$ ) to $\\text{id}_{\\mathcal {DIAG}}$ (resp.", "$\\text{id}_{\\mathcal {COEF}_{G}}$ ).", "Given a diagram $D=(D_0, D^{\\prime }_0, D_1, r_D, r^{\\prime }_D)$ , the definition of the functors $\\mathcal {D}$ and $\\mathcal {C}$ gives $\\mathcal {D}\\circ \\mathcal {C}(D)=\\widetilde{D}=(F_{\\sigma _0}, F_{\\sigma ^{\\prime }_0}, F_{\\tau _1}, r_{\\sigma _0}^{\\tau _1}, r_{\\sigma ^{\\prime }_0}^{\\tau _1})$ , and we have already shown that $\\mathbf {ev}=(ev_{\\sigma _0}, ev_{\\sigma ^{\\prime }_0}, ev_{\\tau _1})$ is an isomorphism from $\\widetilde{D}$ to $D$ .", "To show it gives a natural transformation, we let $E=(E_0, E^{\\prime }_0, E_1, r_E, r^{\\prime }_E)$ be another diagram and $\\psi =(\\psi _0, \\psi ^{\\prime }_0, \\eta _1)$ a morphism from $D$ to $E$ .", "Let $\\mathcal {D}\\circ \\mathcal {C}(E)= \\widetilde{E} = (F^{\\prime }_{\\sigma _0}, F^{\\prime }_{\\sigma ^{\\prime }_0}, F^{\\prime }_{\\tau _1}, (r^{\\prime })_{\\sigma _0}^{\\tau _1}, (r^{\\prime })_{\\sigma ^{\\prime }_0}^{\\tau _1}).$ We must check that the following diagram is commutative: D [r]DC()[d]ev E[d]evD [r] E Writing this out explicitly, commutativity of this diagram is equivalent to commutativity of the following three diagrams: F0 [r]DC(0)[d]ev0 F'0[d]ev0' F'0 [r]DC('0)[d]ev0' F''0[d]ev0''D0 [r]0 E0 D'0 [r]'0 E'0 F1 [r]DC(1)[d]ev1 F'1[d]ev1'D1 [r]1 E1 We check the first of these assertions.", "Given $f\\in F_{\\sigma _0}$ , we see $\\psi _0\\circ ev_{\\sigma _0}(f)=\\psi _0(f(1))$ .", "By the definitions of $\\mathcal {C}(\\psi )$ and $\\mathcal {D}$ , we know that $\\mathcal {D}\\circ \\mathcal {C}(\\psi )(f)=f_{\\psi _0(f(1))}$ , therefore $ev^{\\prime }_{\\sigma _0}(\\mathcal {D}\\circ \\mathcal {C}(\\psi )(f))= \\psi _0(f(1))$ .", "The other two follow similarly, and hence $\\mathbf {ev}$ gives a natural transformation between $\\mathcal {D}\\circ \\mathcal {C}$ and $\\text{id}_{\\mathcal {DIAG}}$ .", "For the other direction, let $\\mathcal {V}=(V_{\\tau })_\\tau $ be a $G$ -equivariant coefficient system, with restriction maps $t^{\\tau }_{\\sigma }$ .", "Let $\\mathcal {F}=(F_{\\tau })_\\tau $ be the coefficient system $\\mathcal {C}\\circ \\mathcal {D}(\\mathcal {V})$ , with restriction maps $r^{\\tau }_{\\sigma }$ .", "Given an edge $\\tau $ containing a vertex $\\sigma \\in X_0^e$ , let $g\\in G$ be such that $\\tau =g.\\tau _1$ and $\\sigma =g.\\sigma _0$ .", "For this vertex $\\sigma $ , we define a map $ev_\\sigma $ by Table: NO_CAPTIONwhere $v=f(g^{-1})\\in V_{\\sigma _0}$ .", "One defines $ev_{\\sigma }$ similarly if $\\sigma \\in X_0^o$ .", "For the edge $\\tau $ , define $ev_\\tau $ by Table: NO_CAPTIONwhere $v= f(g^{-1})\\in V_{\\tau _1}$ .", "Note that both definitions are independent of the choice of $g$ , and that the maps $ev_\\sigma , ev_\\tau $ are isomorphisms of vector spaces.", "We must now show that the system $(ev_\\tau )_{\\tau }$ is compatible with the $G$ -action.", "Let $\\sigma \\in X_0^e$ be such that $\\sigma = g.\\sigma _0$ for some $g$ .", "For an element $h\\in G$ and $f\\in F_\\sigma $ , we have $ev_{h.\\sigma }(h_{\\sigma }(f)) = ev_{h.\\sigma }(h.f) = (hg)_{\\sigma _0}.v$ , where $v=f(g^{-1})$ .", "On the other hand, we have $h_{\\sigma }\\circ ev_{\\sigma }(f)=h_{\\sigma }(g_{\\sigma _0}v) = (hg)_{\\sigma _0}.v$ .", "The same argument applies for the case $\\sigma \\in X_0^o$ or $\\tau \\in X_1$ .", "To check that the system $(ev_\\tau )_\\tau $ is compatible with the restriction maps, we need to verify the commutativity of the following diagram (for $\\tau \\in X_1$ containing $\\sigma \\in X_0$ ): F [r]ev[d]r V[d]tF[r]ev V Let $\\tau \\in X_1$ contain $\\sigma \\in X_0^e$ , and let $g\\in G$ be such that $\\tau = g.\\tau _1, \\sigma = g.\\sigma _0$ .", "Given $f\\in F_\\tau $ , we have $t^{\\tau }_{\\sigma }\\circ ev_\\tau (f)=t^{\\tau }_{\\sigma }(g_{\\tau _1}.", "v)$ , where $v = f(g^{-1})$ .", "On the other hand, since $\\mathcal {F}$ comes from the diagram $(V_{\\sigma _0}, V_{\\sigma ^{\\prime }_0}, V_{\\tau _1}, t^{\\tau _1}_{\\sigma _0}, t^{\\tau _1}_{\\sigma ^{\\prime }_0})$ , we have $ev_{\\sigma }\\circ r^{\\tau }_{\\sigma }(f)= ev_{\\sigma }(g.f_{t^{\\tau _1}_{\\sigma _0}(v)})= g_{\\sigma _0}\\circ t^{\\tau _1}_{\\sigma _0}(v)$ .", "By definition of a $G$ -equivariant coefficient system, we have $t^{\\tau }_{\\sigma }\\circ g_{\\tau _1}= g_{\\sigma _0}\\circ t^{\\tau _1}_{\\sigma _0}$ , and therefore the diagram commutes.", "The argument for the case $\\sigma \\in X_0^o$ is the same.", "To show that the system $(ev_\\tau )_\\tau $ defines a natural transformation from $\\mathcal {C}\\circ \\mathcal {D}$ to $\\text{id}_{\\mathcal {COEF}_G}$ , it only remains to check the compatibility of $(ev_\\tau )_\\tau $ with morphisms in $\\mathcal {COEF}_G$ .", "The proof is virtually identical to the proof of $\\mathcal {D}\\circ \\mathcal {C}\\simeq \\text{id}_{\\mathcal {DIAG}}$ , so we omit it.", "Collecting these results shows that the two categories are equivalent." ] ]

[ [ "Natural selection. IV. The Price equation" ], [ "Abstract The Price equation partitions total evolutionary change into two components.", "The first component provides an abstract expression of natural selection.", "The second component subsumes all other evolutionary processes, including changes during transmission.", "The natural selection component is often used in applications.", "Those applications attract widespread interest for their simplicity of expression and ease of interpretation.", "Those same applications attract widespread criticism by dropping the second component of evolutionary change and by leaving unspecified the detailed assumptions needed for a complete study of dynamics.", "Controversies over approximation and dynamics have nothing to do with the Price equation itself, which is simply a mathematical equivalence relation for total evolutionary change expressed in an alternative form.", "Disagreements about approach have to do with the tension between the relative valuation of abstract versus concrete analyses.", "The Price equation's greatest value has been on the abstract side, particularly the invariance relations that illuminate the understanding of natural selection.", "Those abstract insights lay the foundation for applications in terms of kin selection, information theory interpretations of natural selection, and partitions of causes by path analysis.", "I discuss recent critiques of the Price equation by Nowak and van Veelen." ], [ "Introduction", "Evolutionary theory analyzes the change in phenotype over time.", "We may interpret phenotype broadly to include organismal characters, variances of characters, correlations between characters, gene frequency, DNA sequence—essentially anything we can measure.", "How does a phenotype influence its own change in frequency or the change in the frequencies of correlated phenotypes?", "Can we separate that phenotypic influence from other evolutionary forces that also cause change?", "The association of a phenotype with change in frequency, separated from other forces that change phenotype, is one abstract way to describe natural selection.", "The Price equation is that kind of abstract separation.", "Do we really need such abstraction, which may seem rather distant and vague?", "Instead of wasting time on such things as the abstract essence of natural selection, why not get down to business and analyze real problems?", "For example, we may wish to know how the evolutionary forces of mutation and selection interact to determine biological pattern.", "We could make a model with genes that have phenotypic effects, selection that acts on those phenotypes to change gene frequency, and mutation that changes one gene into another.", "We could do some calculations, make some predictions about, for example, the frequency of deleterious mutations that cause disease, Figure: NO_CAPTION\\newlabelbox:preface11 and compare those predictions to observations.", "All clear and concrete, without need of any discussion of the essence of things.", "However, we may ask the following.", "Is there some reorientation for the expression of natural selection that may provide subtle perspective, from which we can understand our subject more deeply and analyze our problems with greater ease and greater insight?", "My answer is, as I have mentioned, that the Price equation provides that sort of reorientation.", "To argue the point, I will have to keep at the distinction between the concrete and the abstract, and the relative roles of those two endpoints in mature theoretical understanding.", "Several decades have passed since Price's price70selection,price72extension original articles.", "During that span, published claims, counter-claims and misunderstandings have accumulated to the point that it seems worthwhile to revisit the subject.", "On the one hand, the Price equation has been applied to numerous practical problems, and has also been elevated by some to almost mythical status, as if it were the ultimate path to enlightenment for those devoted to evolutionary study (Box ).", "On the other hand, the opposition has been gaining adherents who boast the sort of disparaging anecdotes and slogans that accompany battle.", "In a recent book, nowak11supercooperators: counter The Price equation did not, however, prove as useful as [Price and Hamilton] had hoped.", "It turned out to be the mathematical equivalent of a tautology.", "$\\ldots $ If the Price equation is used instead of an actual model, then the arguments hang in the air like a tantalizing mirage.", "The meaning will always lie just out of the reach of the inquisitive biologist.", "This mirage can be seductive and misleading.", "The Price equation can fool people into believing that they have built a mathematical model of whatever system they are studying.", "But this is often not the case.", "Although answers do indeed seem to pop out of the equation, like rabbits from a magician's hat, nothing is achieved in reality.", "nowak11supercooperators: approvingly quote vanveelen12group with regard to calling the Price equation a mathematical tautology.", "vanveelen12group emphasize the point by saying that the Price equation is like soccer/football star Johan Cruyff's quip about the secret of success: “You always have to make sure that you score one goal more than your opponent.” The statement is always true, but provides no insight.", "nowak11supercooperators: and vanveelen12group believe their arguments demonstrate that the Price equation is true in the same trivial sense, and they call that trivial type of truth a mathematical tautology.", "Interestingly, magazines, online articles, and the scientific literature have for several years been using the phrase mathematical tautology for the Price equation, although nowak11supercooperators: and vanveelen12group do not provide citations to previous literature.", "As far as I know, the first description of the Price equation as a mathematical tautology was in frank95george.", "I used the phrase in the sense of the epigraph from Mazur, a formal equivalence between different expressions of the same object.", "Mathematics and much of statistics are about formal equivalences between different expressions of the same object.", "For example, the Laplace transform changes a mathematical expression into an alternative form with the same information, and analysis of variance decomposes the total variance into a sum of component variances.", "For any mathematical or statistical equivalence, value depends on enhanced analytical power that eases further derivations and calculations, and on the ways in which previously hidden relations are revealed.", "In light of the contradictory points of view, the main goal of this article is to sort out exactly what the Price equation is, how we should think about it, and its value and limitations in reasoning about evolution.", "Subsequent articles will show the Price equation in action, applied to kin selection, causal analysis in evolutionary models, and an information perspective of natural selection and Fisher's fundamental theorem.", "The first section derives the Price equation in its full and most abstract form.", "That derivation allows us to evaluate the logical status of the equation in relation to various claims of fundamental flaw.", "The equation survives scrutiny.", "It is a mathematical relation that expresses the total amount of evolutionary change in an alternative and mathematically equivalent way.", "That equivalence provides insight into aspects of natural selection and also provides a guide that, in particular applications, often leads to good approaches for analysis.", "The second section contrasts two perspectives of evolutionary analysis.", "In standard models of evolutionary change, one begins with the initial population state and the rules of change.", "The rules of change include the fitness of each phenotype and the change in phenotype between ancestor and descendant.", "Given the initial state and rules of change, one deduces the state of the changed population.", "Alternatively, one may have data on the initial population state, the changed population state, and the ancestor-descendant relations that map entities from one population to the other.", "Those data may be reduced to the evolutionary distance between two populations, providing inductive information about the underlying rules of change.", "Natural populations have no intrinsic notion of fitness or rules of change.", "Instead, they inductively accumulate information.", "The Price equation includes both the standard deductive model of evolutionary change and the inductive model by which information accumulates in relation to the evolutionary distance between populations.", "The third and fourth sections discuss the Price equation's abstract properties of invariance and recursion.", "The invariance properties include the information theory interpretation of natural selection.", "Recursion provides the basis for analyzing group selection and other models of multilevel selection.", "The fifth section relates the Price equation to various expressions that have been used throughout the history of evolutionary theory to analyze natural selection.", "The most common form describes natural selection by the covariance between phenotype and fitness or by the covariance between genetic breeding value and fitness.", "The covariance expression is one part of the Price equation that, when used alone, describes the natural selection component of total evolutionary change.", "The essence of those covariance forms arose in the early studies of population and quantitative genetics, have been used extensively during much of the modern history of animal breeding, and began to receive more mathematical development in the 1960s and 1970s.", "Recent critiques of the Price equation focus on the same covariance expression that has been widely used throughout the history of population and quantitative genetics to analyze natural selection and to approximate total evolutionary change.", "The sixth section returns to the full abstract form of the equation.", "I compare a few variant expressions that have been promoted as improvements on the original Price equation.", "Variant forms are indeed helpful with regard to particular abstract problems or particular applications.", "However, most variants are simply minor rearrangements of the mathematical equivalence for total evolutionary change given by the original Price equation.", "The recent extension by kerr09generalization does provide a slightly more general formulation by expanding the fundamental set mapping that defines Price's approach.", "The set mapping basis for the Price equation deserves more careful study and further mathematical work.", "The seventh section analyzes various flaws that have been ascribed to the Price equation.", "For example, the Price equation in its most abstract form does not contain enough information to follow evolutionary dynamics through multiple rounds of natural selection.", "By contrast, classical dynamic models of population genetics are sufficient to follow change through time.", "Much has been Figure: NO_CAPTION\\newlabelbox:literature21 made of this distinction with regard to dynamic sufficiency.", "The distinction arises from the fact that classical dynamics in population genetics makes more initial assumptions than the abstract Price equation.", "It must be true that all mathematical equivalences for total evolutionary change have the same dynamic status given the same initial assumptions.", "Each additional well-chosen assumption typically enhances the specificity and reduces the scope and generality of the analysis.", "The epigraph from Boulding emphasizes that the degree of specificity versus generality is an explicit choice of the analyst with respect to initial assumptions.", "The Discussion considers the value and limitations of the Price equation in relation to recent criticisms by Nowak and van Veelen.", "The critics confuse the distinct roles of general abstract theory and concrete dynamical models for particular cases.", "The enduring power of the Price equation arises from the discovery of essential invariances in natural selection.", "For example, kin selection theory expresses biological problems in terms of relatedness coefficients.", "Relatedness measures the association between social partners.", "The proper measure of relatedness identifies distinct biological scenarios with the same (invariant) evolutionary outcome.", "Invariance relations provide the deepest insights of scientific thought.", "The mathematics given here applies not only to genetical selection but to selection in general.", "It is intended mainly for use in deriving general relations and constructing theories, and to clarify understanding of selection phenomena, rather than for numerical calculation [p. 485]price72extension.", "I have emphasized that the Price equation is a mathematical equivalence.", "The equation focuses on separation of total evolutionary change into a part attributed to selection and a remainder term.", "That separation provides an abstraction of the nature of selection.", "As Price wrote sometime around 1970 but published posthumously in price95the-nature: “Despite the pervading importance of selection in science and life, there has been no abstraction and generalization from genetical selection to obtain a general selection theory and general selection mathematics.” It is useful first to consider the Price equation in this most abstract form.", "I follow my earlier derivations frank95george,frank97the-price,frank98foundations,frank09natural, which differ little from the derivation given by price72extension when interpreted in light of price95the-nature.", "The abstract expression can best be thought of in terms of mapping items between two sets frank95george,price95the-nature.", "In biology, we usually think of an ancestral population at some time and a descendant population at a later time.", "Although there is no need to have an ancestor-descendant relation, I will for convenience refer to the two sets as ancestor and descendant.", "What does matter is the relations between the two sets, as follows.", "The full abstract power of the Price equation requires adhering strictly to particular definitions.", "The definitions arise from the general expression of the relations between two sets.", "Let $q_i$ be the frequency of the $i$ th type in the ancestral population.", "The index $i$ may be used as a label for any sort of property of things in the set, such as allele, genotype, phenotype, group of individuals, and so on.", "Let $q^{\\prime }_i$ be the frequencies in the descendant population, defined as the fraction of the descendant population that is derived from members of the ancestral population that have the label $i$ .", "Thus, if $i=2$ specifies a particular phenotype, then $q^{\\prime }_2$ is not the frequency of the phenotype $i=2$ among the descendants.", "Rather, it is the fraction of the descendants derived from entities with the phenotype $i=2$ in the ancestors.", "One can have partial assignments, such that a descendant entity derives from more than one ancestor, in which case each ancestor gets a fractional assignment of the descendant.", "The key is that the $i$ indexing is always with respect to the properties of the ancestors, and descendant frequencies have to do with the fraction of descendants derived from particular ancestors.", "Given this particular mapping between sets, we can specify a particular definition for fitness.", "Let $q^{\\prime }_i = q_i(w_i/\\bar{w})$ , where $w_i$ is the fitness of the $i$ th type and $\\bar{w}= \\sum q_iw_i$ is average fitness.", "Here, $w_i/\\bar{w}$ is proportional to the fraction of the descendant population that derives from type $i$ entities in the ancestors.", "Usually, we are interested in how some measurement changes or evolves between sets or over time.", "Let the measurement for each $i$ be $z_i$ .", "The value $z$ may be the frequency of a gene, the squared deviation of some phenotypic value in relation to the mean, the value obtained by multiplying measurements of two different phenotypes of the same entity, and so on.", "In other words, $z_i$ can be a measurement of any property of an entity with label, $i$ .", "The average property value is $\\bar{z}=\\sum q_iz_i$ , where this is a population average.", "The value $z^{\\prime }_i$ has a peculiar definition that parallels the definition for $q^{\\prime }_i$ .", "In particular, $z^{\\prime }_i$ is the average measurement of the property associated with $z$ among the descendants derived from ancestors with index $i$ .", "The population average among descendants is $\\bar{z}^{\\prime }=\\sum q^{\\prime }_iz^{\\prime }_i$ .", "The Price equation expresses the total change in the average property value, $\\Delta \\bar{z}=\\bar{z}^{\\prime }-\\bar{z}$ , in terms of these special definitions of set relations.", "This way of expressing total evolutionary change and the part of total change that can be separated out as selection is very different from the usual ways of thinking about populations and evolutionary change.", "The derivation itself is very easy, but grasping the meaning and becoming adept at using the equation is not so easy.", "I will present the derivation in two stages.", "The first stage makes the separation into a part ascribed to selection and a part ascribed to property change that covers everything beyond selection.", "The second stage retains this separation, changing the notation into standard statistical expressions that provide the form of the Price equation commonly found in the literature.", "I follow with some examples to illustrate how particular set relations are separated into selection and property change components.", "The next section considers two distinct interpretations of the Price equation in relation to dynamics.", "We use $\\Delta q_i = q^{\\prime }_i-q_i$ for frequency change associated with selection, and $\\Delta z_i = z^{\\prime }_i-z_i$ for property value change.", "Both expressions for change depend on the special set relation definitions given above.", "We are after an alternative expression for total change, $\\Delta \\bar{z}$ .", "Thus, z= z'-z = q'iz'i - qizi = q'i(z'i-zi) +q'izi - qizi = q'i(zi) + (qi)zi.", "Switching the order of the terms on the right side of the last line yields $\\Delta \\bar{z}= \\sum (\\Delta q_i)z_i + \\sum q^{\\prime }_i(\\Delta z_i),$ a form emphasized by [eqn 1]frank97the-price.", "The first term separates the part of total change caused by changes in frequency.", "We call this the part caused by selection, because this is the part that arises directly from differential contribution by ancestors to the descendant population price95the-nature.", "Because the set mappings define all of the direct attributions of success for each $i$ with respect to the associated properties $z_i$ , it is reasonable to separate out this direct component as the abstraction of selection.", "It is of course possible to define other separations.", "I discuss one particular alternative later.", "However, it is hard to think of other separations that would describe selection in a better way at the most abstract and general level of the mappings between two sets.", "This first term has also been called the partial evolutionary change caused by natural selection (Eq.", "(REF )).", "The second term describes the part of total change caused by changes in property values.", "Recall that $\\Delta z_i=z^{\\prime }_i-z_i$ , and that $z^{\\prime }_i$ is the property value among entities that descend from $i$ .", "Many different processes may cause descendant property values to differ from ancestral values.", "In fact, the assignment of a descendant to an ancestor can be entirely arbitrary, so that there is no reason to assume that descendants should be like ancestors.", "Usually, we will work with systems in which descendants do resemble ancestors, but the degree of such associations can be arranged arbitrarily.", "This term for change in property value encompasses everything beyond selection.", "The idea is that selection affects the relative contribution of ancestors and thus the changes in frequencies of representation, but what actually gets represented among the descendants will be subject to a variety of processes that may alter the value expressed by descendants.", "The equation is exact and must apply to every evolutionary system that can be expressed as two sets with certain ancestor-descendant or mapping relations.", "It is in that sense that I first used the phrase mathematical tautology frank95george.", "The nature of separation and abstraction is well described by the epigraph from Mazur at the start of this article.", "price72extension used statistical notation to write Eq.", "(REF ).", "For the first term, by following prior definitions we have qi = q'i - qi =qiwiw - qi =qi(wiw-1), so that $\\sum (\\Delta q_i)z_i = \\sum q_i\\left(\\frac{w_i}{\\bar{w}}-1\\right)z_i={\\hbox{\\rm Cov}}(w,z)/\\bar{w},$ using the standard definition for population covariance.", "For the second term, we have $\\sum q^{\\prime }_i(\\Delta z_i) = \\sum q_i\\frac{w_i}{\\bar{w}}(\\Delta z_i) = \\operatorname{E}(w\\Delta z)/\\bar{w},$ where $\\operatorname{E}$ means expectation, or average over the full population.", "Putting these statistical forms into Eq.", "(REF ) and moving $\\bar{w}$ to the left side for notational convenience yields a commonly published form of the Price equation $\\bar{w}\\Delta \\bar{z}= {\\hbox{\\rm Cov}}(w,z) + \\operatorname{E}(w\\Delta z).$ price95the-nature and frank95george present examples of set mappings expressed in relation to the Price equation.", "The Price equation describes evolutionary change between two populations.", "Three factors express one iteration of dynamical change: initial state, rules of change, and next state.", "In the Price equation, the phenotypes, $z_i$ , and their frequencies, $q_i$ , describe the initial population state.", "Fitnesses, $w_i$ , and property changes, $\\Delta z_i$ , set the rules of change.", "Derived phenotypes, $z_i^{\\prime }$ , and their frequencies, $q_i^{\\prime }$ , express the next population state.", "Models of evolutionary change essentially always analyze forward or deductive dynamics.", "In that case, one starts with initial conditions and rules of change and calculates the next state.", "Most applications of the Price equation use this traditional deductive analysis.", "Such applications lead to predictions of evolutionary outcome given assumptions about evolutionary process, expressed by the fitness parameters and property changes.", "Alternatively, one can take the state of the initial population and the state of the changed population as given.", "If one also has the mappings between initial and changed populations that connect each entity, $i$ , in the initial population to entities in the changed population, then one can calculate (induce) the underlying rules of change.", "At first glance, this inductive view of dynamics may seem rather odd and not particularly useful.", "Why start with knowledge of the evolutionary sequence of population states and ancestor-descendant relations as given, and inductively calculate fitnesses and property changes?", "The inductive view takes the fitnesses, $w_i$ , to be derived from the data rather than an intrinsic property of each type.", "The Price equation itself does not distinguish between the deductive and inductive interpretations.", "One can specify initial state and rules of change and then deduce outcome.", "Or one can specify initial state and outcome along with ancestor-descendant mappings, and then induce the underlying rules of change.", "It is useful to understand the Price equation in its full mathematical generality, and to understand that any specific interpretation arises from additional assumptions that one brings to a particular problem.", "Much of the abstract power of the Price equation comes from understanding that, by itself, the equation is a minimal description of change between populations.", "The deductive interpretation of the Price equation is clear.", "What value derives from the inductive perspective?", "In observational studies of evolutionary change, we only have data on population states.", "From those data, we use the inductive perspective to make inferences about the underlying rules of change.", "Note that inductive estimates for evolutionary process derive from the amount of change, or distance, between ancestor and descendant populations.", "The Price equation includes that inductive, or retrospective view, by expressing the distance between populations in terms of $\\Delta \\bar{z}$ .", "I develop that distance interpretation in the following sections.", "Perhaps more importantly, natural selection itself is inherently an inductive process by which information accumulates in populations.", "Nature does not intrinsically “know” of fitness parameters.", "Instead, frequency changes and the mappings between ancestor and descendant are inherent in a population's response to the environment, leading to a sequence of population states, each separated by an evolutionary distance.", "That evolutionary distance provides information that populations accumulate inductively about the fitnesses of each phenotype frank09natural.", "The Price equation includes both the deductive and inductive perspectives.", "We may choose to interpret the equation in either way depending on our goals of analysis.", "The Price equation describes selection by the term $\\sum (\\Delta q_i)z_i= {\\hbox{\\rm Cov}}(w,z)/\\bar{w}$ .", "Any instance of evolutionary change that has the same value for this sum has the same amount of total selection.", "Put another way, for any particular value for total selection, there is an infinite number of different combinations of frequency changes and character measurements that will add up to the same total value for selection.", "All of those different combinations lead to the same value with respect to the amount of selection.", "We may say that all of those different combinations are invariant with respect to the total quantity of selection.", "The deepest insights of science come from understanding what does not matter, so that one can also say exactly what does matter—what is invariant feynman67the-character,weyl83symmetry.", "The invariance of selection with respect to transformations of the fitnesses, $w$ , and the phenotypes, $z$ , that have the same ${\\hbox{\\rm Cov}}(w,z)$ means that, to evaluate selection, it is sufficient to analyze this covariance.", "At first glance, it may seem contradictory that the covariance, commonly thought of as a linear measure of association, can be a complete description for selection, including nonlinear processes.", "Let us step through this issue, first looking at why the covariance is a sufficient expression of selection, and then at the limitations of this covariance expression in evolutionary analysis.", "Much of the confusion with respect to covariance and variance terms in selection equations arises from thinking only of the traditional statistical usage.", "In statistics, covariance typically measures the linear association between pairs of observations, and variance is a measure of the squared spread of observations.", "Alternatively, covariances and variances provide measures of distance, which ultimately can be understood as measures of information frank09natural.", "This section introduces the notation for the geometric interpretation of distance.", "The next section gives the main geometric result, and the following section presents some examples.", "The identity $\\sum (\\Delta q_i)z_i= {\\hbox{\\rm Cov}}(w,z)/\\bar{w}$ provides the key insight.", "It helps to write this identity in an alternative form.", "Note from the prior definition $q_i^{\\prime } = q_iw_i/\\bar{w}$ that $\\Delta q_i = q^{\\prime }_i - q_i = q_i(w_i/\\bar{w}- 1) = q_ia_i,$ where $a_i=w_i/\\bar{w}- 1$ is Fisher's average excess in fitness, a commonly used expression in population and quantitative genetics fisher30the-genetical,fisher41average,crow70an-introduction.", "A value of zero means that an entity has average fitness, and therefore fitness effects and selection do not change the frequency of that entity.", "Using the average excess in fitness, we can write the invariant expression for selection as $\\sum (\\Delta q_i)z_i= \\sum q_ia_iz_i = {\\hbox{\\rm Cov}}(w,z)/\\bar{w}.$ We can think of the state of the population as the listing of character states, $z_i$ .", "Thus we write the population state as $\\mathbf {z}= (z_1,z_2,\\ldots )$ .", "The subscripts run over every different entity in the population, so the vector $\\mathbf {z}$ is a complete description of the entire population.", "Similarly, for the frequency fluctuations, $\\Delta q_i = q_ia_i$ , we can write the listing of all fluctuations as a vector, $\\mathbf {\\Delta q}=(\\Delta q_1,\\Delta q_2,\\ldots )$ .", "It is often convenient to use the dot product notation $\\mathbf {\\Delta q}\\cdot \\mathbf {z}=\\sum (\\Delta q_i)z_i= {\\hbox{\\rm Cov}}(w,z)/\\bar{w}$ in which the dot specifies the sum obtained by multiplying each pair of items from two vectors.", "Before turning to some geometric examples in the following section, we need a definition for the length of a vector.", "Traditionally, one uses the definition $\\Vert \\mathbf {z}\\Vert = \\sqrt{\\sum z_i^2},$ in which the length is the square root of the sum of squares, which is the standard measure of length in Euclidean geometry.", "A simple identity relates a dot product to a measure of distance and to covariance selection $\\mathbf {\\Delta q}\\cdot \\mathbf {z}= \\Vert \\mathbf {\\Delta q}\\Vert \\Vert \\mathbf {z}\\Vert \\cos \\phi ={\\hbox{\\rm Cov}}(w,z)/\\bar{w},$ where $\\phi $ is the angle between the vectors $\\mathbf {\\Delta q}$ and $\\mathbf {z}$ (Fig.", "REF ).", "If we standardize the character vector ${-\\hspace{-6.94443pt}\\mathbf {z}}= \\mathbf {z}/\\Vert \\mathbf {z}\\Vert $ , then the standardized vector has a length of one, $\\Vert {-\\hspace{-6.94443pt}\\mathbf {z}}\\Vert =1$ , which simplifies the dot product expression of selection to $\\mathbf {\\Delta q}\\cdot {-\\hspace{-6.94443pt}\\mathbf {z}}= \\Vert \\mathbf {\\Delta q}\\Vert \\cos \\phi ,$ providing the geometric representation illustrated in Fig.", "REF .", "The covariance can be expressed as the product of a regression coefficient and a variance term ${\\hbox{\\rm Cov}}(w,z)/\\bar{w}=\\beta _{z w}{\\hbox{\\rm Var}}(w)/\\bar{w}=\\beta _{w z}{\\hbox{\\rm Var}}(z)/\\bar{w},$ where the notation $\\beta _{x y}$ describes the regression coefficient of $x$ on $y$ price70selection.", "This identity shows that the expression of selection in terms of a regression coefficient and a variance term is equivalent to the geometric expression of selection in terms of distance.", "I emphasize these identities for two reasons.", "First, as Mazur stated in the epigraph: “The heart and soul of much mathematics consists of the fact that the `same' object can be presented to us in different ways.” If an object is important, such as natural selection surely is, then it pays to study that object from different perspectives to gain deeper insight.", "Second, the appearance of statistical functions, such as the covariance and variance, in selection equations sometimes leads to mistaken conclusions.", "In the selection equations, it is better to think of the covariance and variance terms arising because they are identities with geometric or other interpretations of selection, rather than thinking of those terms as summary statistics of probability distributions.", "The problem with thinking of those terms as statistics of probability distributions is that the variance and covariance are not in general sufficient descriptions for probability distributions.", "That lack of sufficiency for probability may lead one to conclude that those terms are not sufficient for a general expression of selection.", "However, those covariance and variance terms are sufficient.", "That sufficiency can be understood by thinking of those terms as identities for distance or measures of information frank09natural.", "It is true that in certain particular applications of quantitative genetics or stochastic sampling processes, one does interpret the variances and covariances as summary statistics of probability distributions, usually the normal or Gaussian distribution.", "However, it is important to distinguish those special applications from the general selection equations.", "For the general selection expression in Eq.", "(REF ), any transformations that do not affect the net values are invariant with respect to selection.", "For example, transformations of the fitnesses and associated frequency changes, $\\mathbf {\\Delta q}$ , are invariant if they leave unchanged the distance expressed by $\\mathbf {\\Delta q}\\cdot \\mathbf {z}={\\hbox{\\rm Cov}}(w,z)/\\bar{w}$ .", "Similarly, changes in the pattern of phenotypes are invariant to the Figure: Geometric expression of selection.", "The plots show the equivalence of the dot product, the geometric expression and the covariance, as given in Eq. ().", "For both plots, 𝐳=(1,4)\\mathbf {z}=(1,4) and -𝐳=𝐳/∥𝐳∥=(0.24,0.97){-\\hspace{-6.94443pt}\\mathbf {z}}=\\mathbf {z}/\\Vert \\mathbf {z}\\Vert =(0.24,0.97).", "The dashed line shows the perpendicular between the pattern of frequency changes derived from fitnesses, Δ𝐪\\mathbf {\\Delta q}, and the phenotypic pattern, -𝐳{-\\hspace{-6.94443pt}\\mathbf {z}}.", "The vertex of the two vectors is at the origin (0,0)(0,0).", "The distance from the origin to the intersection of the perpendicular along -𝐳{-\\hspace{-6.94443pt}\\mathbf {z}} is the total amount of selection, ∥Δ𝐪∥cosφ\\Vert \\mathbf {\\Delta q}\\Vert \\cos \\phi .", "(a) The vector of frequency changes that summarize fitness is Δ𝐪=(-0.4,0.4)\\mathbf {\\Delta q}=(-0.4,0.4).", "The angle between the vector of frequency changes and the phenotypes is φ=arccos(Δ𝐪·-𝐳)/∥Δ𝐪∥\\phi =\\arccos \\left[(\\mathbf {\\Delta q}\\cdot {-\\hspace{-6.94443pt}\\mathbf {z}})/\\Vert \\mathbf {\\Delta q}\\Vert \\right] which, in this example, is 1.03 radians or 59 ∘ 59^{\\circ }.", "In this case, the total selection is ∥Δ𝐪∥cosφ=0.29\\Vert \\mathbf {\\Delta q}\\Vert \\cos \\phi =0.29.", "(b) In this plot, Δ𝐪=(0.4,-0.4)\\mathbf {\\Delta q}=(0.4,-0.4), yielding an angle φ\\phi of 121 ∘ 121^{\\circ }.", "The perpendicular intersects the negative projection of the phenotype vector, shown as a dashed line, associated with the negative change by selection of ∥Δ𝐪∥cosφ=-0.29\\Vert \\mathbf {\\Delta q}\\Vert \\cos \\phi =-0.29.extent that they leave $\\mathbf {\\Delta q}\\cdot \\mathbf {z}$ unchanged.", "These invariance properties of selection, measured as distance, may not appear very interesting at first glance.", "They seem to be saying that the outcome is the outcome.", "However, the history of science suggests that studying the invariant properties of key expressions can lead to insight.", "Few authors have developed an interest in the invariant qualities of selection.", "fisher30the-genetical initiated discussion with his fundamental theorem of natural selection, a special case of Eq.", "(REF ) frank97the-price.", "Although many authors commented on the fundamental theorem, most articles did not analyze the theorem with respect to its essential mathematical insights about selection.", "ewens92an-optimizing reviewed the few attempts to understand the mathematical basis of the theorem and its invariant quantities.", "frank09natural tied the theorem to Fisher information frieden01population,frieden04science, hinting at an information theory interpretation that arises from the fundamental selection equation of Eq.", "(REF ).", "In spite of the importance of selection in many fields of science, the potential interpretation of Eq.", "(REF ) with respect to invariants of information theory has hardly been developed.", "I briefly outline the potential connections here frank09natural.", "I develop this information perspective of selection in a later article, along with Fisher's fundamental theorem.", "To start, define the partial change in phenotype caused by natural selection as $\\Delta _\\text{\\tiny S}\\hspace{0.27771pt}\\bar{z}= \\mathbf {\\Delta q}\\cdot \\mathbf {z}= {\\hbox{\\rm Cov}}(w,z)/\\bar{w}.$ The concept of a partial change caused by natural selection arises from Fisher's fundamental theorem fisher30the-genetical,price72fishers,ewens89an-interpretation,frank92fishers.", "With this definition, we can use eqns REF and REF to write $\\Delta _\\text{\\tiny S}\\hspace{0.27771pt}\\bar{z}= \\beta _{z w}{\\hbox{\\rm Var}}(w)/\\bar{w}= \\bar{w}\\beta _{z w}{\\hbox{\\rm Var}}(w/\\bar{w}).$ From Eq.", "(REF ), we have the definition for the average excess in fitness $a_i=w_i/\\bar{w}-1$ .", "Thus, we can expand the expression for the variance in fitness as ${\\hbox{\\rm Var}}(w/\\bar{w}) = \\sum q_i\\left(\\frac{w_i}{\\bar{w}}-1\\right)^2 =\\sum q_ia_i^2.$ From Eq.", "(REF ), we also have the change in frequency in terms of the average excess, $\\Delta q_i = q_ia_i$ , and equivalently, $a_i = \\Delta q_i/q_i$ , thus Var(w/w) = qi(qiqi)2 = (qiqi)2 = qq, where $\\Delta \\hat{q}_i=\\Delta q_i/\\sqrt{q_i}$ is a standardized fluctuation in frequency, and $\\mathbf {\\Delta \\hat{q}}$ is the vector of standardized fluctuations.", "These alternative forms simply express the variance in fitness in different ways.", "The interesting result follows from the fact that ${\\hbox{\\rm Var}}(w/\\bar{w})=\\mathbf {\\Delta \\hat{q}}\\cdot \\mathbf {\\Delta \\hat{q}}= F(\\mathbf {\\Delta \\hat{q}})$ is the Fisher information, $F$ , in the frequency fluctuations, $\\mathbf {\\Delta \\hat{q}}$ .", "Fisher information is a fundamental quantity in information theory, Bayesian analysis, likelihood theory and the informational foundations of statistical inference.", "Fisher information is a variant form of the more familiar Shannon and Kullback-Leibler information measures, in which the Fisherian form expresses changes in information.", "Once again, we have a simple identity.", "Although it is true that Fisher information is just an algebraic rearrangement of the variance in fitness, some insight may be gained by relating selection to information.", "The variance form calls to mind a statistical description of selection or a partial description of a probability distribution.", "The Fisher information form suggests a relation between natural selection and the way in which populations accumulate information frank09natural.", "We may now write our fundamental expression for selection as $\\Delta _\\text{\\tiny S}\\hspace{0.27771pt}\\bar{z}= \\bar{w}\\beta _{zw}\\,F(\\mathbf {\\Delta \\hat{q}}).$ We may read this expression for selection as: the change in mean character value caused by natural selection, $\\Delta _\\text{\\tiny S}\\hspace{0.27771pt}\\bar{z}$ , is equal to the total Fisher information in the frequency fluctuations, $F$ , multiplied the scaling $\\beta $ that describes the amount of the potential information that the population captures when expressed in units of phenotypic change.", "In other words, the distance $\\Delta _\\text{\\tiny S}\\hspace{0.27771pt}\\bar{z}$ measures the informational gain by the population caused by natural selection.", "The invariances set by this expression may be viewed in different ways.", "For example, the distance of evolutionary change by selection, $\\Delta _\\text{\\tiny S}\\hspace{0.27771pt}\\bar{z}$ , is invariant with respect to many different combinations of frequency fluctuations, $\\mathbf {\\Delta \\hat{q}}$ , and scalings between phenotype and fitness.", "Similarly, any transformations of frequency fluctuations that leave the measure of information, $F(\\mathbf {\\Delta \\hat{q}})$ , invariant do not alter the scaled change in phenotype caused by natural selection.", "The full implications remain to be explored.", "NOTE ADDED AFTER PUBLICATION OF THE JOURNAL ARTICLE.", "I have oversimplified a bit in this section, with the aim to keep the presentation brief.", "The proper expression of Fisher information is $F(\\mathbf {\\Delta \\hat{q}})(\\Delta \\theta )^2$ , where $\\Delta \\theta $ is the scale of change over which population differences are measured, typically taken as $\\Delta \\theta \\rightarrow 0$ frank09natural.", "The various identities for the part of total evolutionary change caused by selection include Sz= Cov(w,z)/w = wzwVar(w/w) = qz = qz = wzw(qq) = wzw F(q).", "These forms show the equivalence of the statistical, geometrical and informational expressions for natural selection.", "These general abstract forms make no assumptions about the nature of phenotypes and the patterns of frequency fluctuations caused by differential fitness.", "The phenotypes may be squared deviations so that the average is actually a variance, or the product of measurements on different characters leading to measures of association, or any other nonlinear combination of measurements.", "Thus, there is nothing inherently linear or restrictive about these expressions.", "The previous sections discussed the part of evolutionary change caused by selection.", "The full Price equation (Eq.", "(REF )) gives a complete and exact expression of total change, repeated here as $\\Delta \\bar{z}= {\\hbox{\\rm Cov}}(w,z)/\\bar{w}+ \\operatorname{E}(w\\Delta z)/\\bar{w}$ or in terms of the dot product notation as $\\Delta \\bar{z}= \\mathbf {\\Delta q}\\cdot \\mathbf {z}\\hspace{1.66656pt}+ \\hspace{1.66656pt}\\mathbf {q^{\\prime }}\\cdot \\mathbf {\\Delta z}.$ The full change in the phenotype is the sum of the two terms, which we may express in symbols as $\\Delta \\bar{z}= \\Delta _\\text{\\tiny S}\\hspace{0.27771pt}\\bar{z}+ \\Delta _\\text{\\tiny E}\\hspace{0.27771pt}\\bar{z}.$ fisher30the-genetical called the term $\\Delta _\\text{\\tiny E}\\hspace{0.27771pt}\\bar{z}$ the change caused by the environment frank92fishers.", "However, the word environment often leads to confusion.", "The proper interpretation is that $\\Delta _\\text{\\tiny E}\\hspace{0.27771pt}\\bar{z}$ encompasses everything not included in the expression for selection.", "The term is environmental only in the sense that it includes all those forces external to the particular definition of the selective forces for a particular problem.", "The $\\Delta _\\text{\\tiny E}\\hspace{0.27771pt}$ term is sometimes associated with changes in transmission frank95george,frank97the-price,frank12natural,okasha06evolution.", "This interpretation arises because $\\operatorname{E}(w\\Delta z)$ is the fitness weighted changes in character value between ancestor and descendant.", "One may think of changes in character values as changes during transmission.", "It is important to realize that everything truly means every possible force that might arise and that is not accounted for by the particular expression for selection.", "Lightning may strike.", "New food sources may appear.", "The Price equation in its general and abstract form is a mathematical identity—what I previously called a mathematical tautology frank95george.", "In applications, one considers how to express $\\Delta _\\text{\\tiny E}\\hspace{0.27771pt}\\bar{z}$ , or one searches for ways to formulate the problem so that $\\Delta _\\text{\\tiny E}\\hspace{0.27771pt}\\bar{z}$ is zero or approximately zero.", "This article is not about particular applications.", "Here, I simply note that when one works with Fisher's breeding value as $z$ , then near equilibria (fixed points), one typically obtains $\\Delta z\\rightarrow 0$ and thus $\\operatorname{E}(w\\Delta z)\\rightarrow 0$ .", "In other cases, the search for a good way to express a problem means finding a form of character measurement that defines $z$ such that characters tend to remain stable over time, so that $\\Delta z\\rightarrow 0$ and thus $\\operatorname{E}(w\\Delta z)\\rightarrow 0$ .", "For applications that emphasize calculation of complex dynamics rather than a more abstract conceptual analysis of a problem, methods other than the Price equation often work better.", "To iterate is human, to recurse, divine coplien98to-iterate.", "Essentially all modern discussions of multilevel selection and group selection derive from price72extension, as developed by hamilton75innate.", "Price and Hamilton noted that the Price equation can be expanded recursively to represent nested levels of analysis, for example, individuals living in groups.", "Start with the basic Price equation as given in Eq.", "(REF ).", "The left side is the total change in average phenotype, $\\bar{z}$ .", "The second term on the right side includes the terms $\\Delta z_i$ in $\\operatorname{E}(w\\Delta z) = \\sum {q_i w_i\\Delta z_i}$ .", "Recall that in defining $z_i$ , we specified the meaning of the index $i$ to be any sort of labeling of set members, subject to minimal consistency requirements.", "We may, for example, label all members of a group by $i$ , and measure $z_i$ as some property of the group.", "If the index $i$ itself represents a set, then we may consider the members of that set.", "For example, $z_{ij}$ may be the $j$ th member of the $i$ th set, or we may say, the $i$ th group.", "In the abstract mathematical expression, there is no need to think of the $i$ th group as having any spatial or biological meaning.", "However, we may consider $i$ as a label for spatially defined groups if we wish to do so.", "With $i$ defining a group, we may analyze the selection and evolution of that $i$ th group.", "The term $\\Delta z_i$ becomes the average change in the $z$ measure for the $i$ th group, composed of members with values $z_{ij}$ .", "The terms $z^{\\prime }_{ij}$ are the average property values of the descendants of the $j$ th entity in the $i$ th group.", "The descendant entities that derive from the $i$ th group do not have to form any sort of group or other meaningful structuring, just as the original $i$ labeling does not have to refer to group structuring in the ancestors.", "However, we may if we wish consider descendants of $i$ as retaining some sense of the ancestral grouping.", "Because $z_i$ represents an averaging over the entities $j$ in the $i$ th group, we are assuming the notational equivalence $\\Delta z_i = \\Delta \\bar{z}_i$ .", "From that point of view, for each group $i$ we may from Eq.", "(REF ) express the change in the group mean by thinking of each group as a separate set or population, yielding for each $i$ the expression $\\Delta z_i = \\Delta \\bar{z}_i = {\\hbox{\\rm Cov}}(w_i,z_i)/\\bar{w}_i + \\operatorname{E}(w_i\\Delta z_i)/\\bar{w}_i.$ We may substitute this expression for each $i$ into the $\\operatorname{E}(w\\Delta z) = \\sum {q_i w_i\\Delta z_i}$ term on the right side of Eq.", "(REF ).", "That substitution recursively expands each change in property value, $\\Delta z_i$ , to itself be composed of a selection term and property value change term.", "For each group, $i$ , we now have expressions for selection within the group, ${\\hbox{\\rm Cov}}(w_i,z_i)/\\bar{w}_i$ , and average property value change within the group, $\\operatorname{E}(w_i\\Delta z_i)/\\bar{w}_i$ .", "If we write out the full expression for this last term, we obtain $\\operatorname{E}(w_i\\Delta z_i)/\\bar{w}_i = \\sum _j w_{ij}\\Delta z_{ij}/\\bar{w}_i.$ In the term $\\Delta z_{ij}$ , each labeling, $j$ , may itself be a subgroup within the larger grouping represented by $i$ .", "The recursive nature of the Price equation allows another expansion to the characters $z_{ijk}$ for the $k$ th entity in the $j$ th grouping that is nested in the $i$ th group, and so on.", "Once again, the indexing for levels $i$ , $j$ , and $k$ do not have to correspond to any particular structuring, but we may choose to use a structuring if we wish.", "One could analyze biological problems of group selection without using the Price equation.", "Because the Price equation is a mathematical identity, there are always other ways of expressing the same thing.", "However, in the 1970s, when group selection was a very confused subject, the Price equation's recursive nature and Hamilton's development provided the foundation for subsequent understanding of the topic.", "All modern conceptual insights about group selection derive from Price's recursive expansion of his abstract expression of selection.", "I have emphasized the general and abstract form of the Price equation.", "That abstract form was first presented rather cryptically by price72extension.", "In that article, Price described the recursive expansion to analyze group selection.", "Apart from the recursive aspect, the more general abstract properties were hardly mentioned in price72extension and not developed by others until 1995.", "While I was writing my history of Price's contributions to evolutionary genetics frank95george, I found Price's unpublished manuscript The nature of selection among W. D. Hamilton's papers.", "Price's unpublished manuscript gave a very general and abstract scheme for analyzing selection in terms of set relations.", "However, Price did not explicitly connect the abstract set relation scheme to the Price equation or to his earlier publications price70selection,price72extension.", "I had The nature of selection published posthumously as price95the-nature.", "In my own article, I explicitly developed the general interpretation of the Price equation as the formal abstract expression of the relation between two sets frank95george.", "price70selection wrote an earlier article in which he presented a covariance selection equation that emphasized the connection to classical models of population genetics and gene frequency change.", "That earlier covariance form lacks the abstract set interpretation and generally has narrower scope.", "Preceding Price, robertson66a-mathematical and li67fundamental also presented selection equations that are similar to Price's price70selection covariance expression.", "Robertson's covariance form itself arises from classical quantitative genetics and the breeder's equation, ultimately deriving from the foundations of quantitative genetics established by fisher18the-correlation.", "Li's form presents a covariance type of expression for classical population genetic models of gene frequency change.", "One cannot understand the current literature without a clear sense of this history.", "Almost all applications of the Price equation to kin and group selection, and to other problems of evolutionary analysis, derive from either the classical expressions of quantitative genetics robertson66a-mathematical or classical expressions of population genetics li67fundamental.", "In light of this history, criticisms can be confusing with regard to the ways in which the Price equation is commonly used.", "For example, in applications to kin or group selection, the Price equation mainly serves to package the notation for the Robertson form of quantitative genetic analysis or the Li form of population genetic analysis.", "The Price equation packaging brings no extra assumptions.", "In some applications, critics may believe that the particular analysis lacks enough assumptions to attain a desired level of specificity.", "One can, of course, easily add more assumptions, at the expense of reduced generality.", "The following sections briefly describe some alternative forms of the Price equation and the associated history.", "That history helps to place criticisms of the Price equation and its applications into clearer light.", "fisher18the-correlation established the modern theory of quantitative genetics, following the early work of Galton, Pearson, Weldon, Yule and others.", "The equations of selection in quantitative genetics and animal breeding arose from that foundation.", "Many modern applications of the Price equation to particular problems follow this tradition of quantitative genetics.", "A criticism of these Price equation applications is a criticism of the central approach of evolutionary quantitative genetics.", "Such criticisms may be valid for certain applications, but they must be evaluated in the broader context of quantitative genetics theory.", "This section shows the relation between quantitative genetics and a commonly applied form of the Price equation rice04evolutionary.", "Evolutionary aspects of quantitative genetics developed from the breeder's equation $R = Sh^2,$ in which the response to selection, $R$ , equals the selection differential, $S$ , multiplied by the heritability, $h^2$ .", "The separation of selection and transmission is the key to the breeder's equation and to quantitative genetics theory.", "The covariance term of the Price equation is equivalent to the selection differential, $S$ , when one interprets the meaning of fitness and descendants in a particular way.", "Suppose that we label each potential parent in the ancestral population of size $N$ with the index, $i$ .", "The initial weighting of each parent in the ancestral population is $q_i=1/N$ .", "Assign to each potential parent a weighting with respect to breeding contribution, $q^{\\prime }_i=q_iw_i$ , with fitnesses standardized so that $\\bar{w}=1$ and the $w_i$ are relative fitnesses.", "With this setup, ancestors are the initial population of potential parents, each weighted equally, and descendants are the same population of parents, weighted by their breeding contribution.", "The character value for each individual remains unchanged between the ancestor and descendant labelings.", "These assumptions lead to $\\Delta \\bar{z}^*={\\hbox{\\rm Cov}}(w,z)$ , the change in the average character value between the breeding population and the initial population.", "That difference is defined as $S$ , the selection differential.", "To analyze the fraction of the selection differential transmitted to offspring, classical quantitative genetics follows fisher18the-correlation to separate the character value as $z=g+\\epsilon $ , with a transmissible genetic component, $g$ , and a component that is not transmitted, which we may call the environmental or unexplained component, $\\epsilon $ .", "Following standard regression theory for this sort of expression, $\\bar{\\epsilon }=0$ .", "For a parent with $z=g+\\epsilon $ , the average character value contribution ascribed to the parent among its descendants is $z^{\\prime }=g$ , following the idea that $g$ represents the component of the parental character that is transmitted to offspring.", "If we assume that the only fluctuations of average character value in offspring are caused by the transmissible component that comes from parents, then the genetic component measured by $g$ is sufficient to explain expected offspring character values.", "Thus, $\\Delta z = z^{\\prime }-z=-\\epsilon $ , and $\\operatorname{E}(w\\Delta z)=-{\\hbox{\\rm Cov}}(w,\\epsilon )$ .", "Substituting into the full Price equation from Eq.", "(REF ) and assuming $\\bar{w}=1$ so that all fitnesses are normalized z= Cov(w,z) + E(wz) = Cov(w,g) + Cov(w,) - Cov(w,) = Cov(w,g).", "The expression $\\Delta \\bar{z}= {\\hbox{\\rm Cov}}(w,g)$ was first emphasized by robertson66a-mathematical, and is sometimes called Robertson's secondary theorem of natural selection.", "Robertson's expression summarizes the foundational principles of quantitative genetics, as conceived by fisher18the-correlation and developed over the past century falconer96introduction,lynch98genetics,hartl06principles.", "It is commonly noted that Robertson's theorem is related to the classic breeder's equation.", "In particular, $R=\\Delta \\bar{z}={\\hbox{\\rm Cov}}(w,g)={\\hbox{\\rm Cov}}(w,z)h^2=Sh^2,$ where $R$ is the response to selection, $S={\\hbox{\\rm Cov}}(w,z)$ is the selection differential, and $h^2={\\hbox{\\rm Var}}(g)/{\\hbox{\\rm Var}}(z)$ is a form of heritability, a measure of the transmissible genetic component.", "Additional details and assumptions can be found in several articles and texts crow76the-rate,frank97the-price,rice04evolutionary.", "price70selection expressed his original formulation in terms of gene frequency change and classical population genetics, rather than the abstract set relations that I have emphasized.", "At that time, it seems likely that Price already had the broader, more abstract theory in hand, and was presenting the population genetics form because of its potential applications.", "The article begins This is a preliminary communication describing applications to genetical selection of a new mathematical treatment of selection in general.", "Gene frequency change is the basic event in biological evolution.", "The following equation$\\ldots $ which gives frequency change under selection from one generation to the next for a single gene or for any linear function of any number of genes at any number of loci, holds for any sort of dominance or epistasis, for sexual or asexual reproduction, for random or nonrandom mating, for diploid, haploid or polyploid species, and even for imaginary species with more than two sexes$\\ldots $ Using my notation, Price writes the basic covariance form $\\Delta P = {\\hbox{\\rm Cov}}(w,p)/\\bar{w}= \\beta _{wp}{\\hbox{\\rm Var}}(p)/\\bar{w}.$ In a simple application, $p$ could be interpreted as gene frequency at a single diploid locus with two alleles.", "Then $P=\\bar{p}$ is the gene frequency in the population, and $\\beta _{wp}$ is the regression of individual fitness on individual gene frequency, in which the individual gene frequency is either 0, $1/2$ or 1 for an individual with 0, 1 or 2 copies of the allele of interest.", "li67fundamental gave an identical gene frequency expression in his eqn 4.", "In more general applications, one can study a $p$ -score that summarizes the number of copies of various alleles present in an individual, or in whatever entities are being tracked.", "In classical population genetics, the $p$ -score would be, in Price's words above, “any linear function of any number of genes at any number of loci.” Here, linearity means that $p$ is essentially a counting of presence versus absence of various things within the $i$ th entity.", "Such counting does not preclude nonlinear interactions between alleles or those things being counted with respect to phenotype, which is why Price said that the expression holds for any form of dominance or epistasis.", "hamilton70selfish used Price's gene frequency form in his first clear derivations of the direct and the inclusive fitness models of kin selection theory.", "Most early applications of the Price equation used this gene frequency interpretation.", "price70selection emphasized that the value of Eq.", "(REF ) arises from its benefits for qualitative reasoning rather than calculation.", "The necessary assumptions can be seen from the form given by Price, which is always exact, here written in my notation $\\Delta P = {\\hbox{\\rm Cov}}(w,p)/\\bar{w}+ \\operatorname{E}(w\\Delta p)/\\bar{w},$ where $\\Delta p$ is interpreted as the change in state between parental gene frequency for the $i$ th entity and the average gene frequency for the part of descendants derived from the $i$ th entity.", "In practice, $\\Delta p=0$ usually means Mendelian segregation, no biased mutation, and no sampling biases associated with drift.", "Most population genetics theory of traits such as social behavior typically make those assumptions, so that Eq.", "(REF ) is sufficient with respect to analyzing change in gene frequency or in $p$ -scores grafen84natural.", "However, the direction of change in gene frequency or $p$ -score is not sufficient to predict the direction of change in phenotype.", "To associate the direction of change in $p$ -score to the direction of change in phenotype, one must make the assumption that phenotype changes monotonically with $p$ -score.", "Such monotonicity is a strong assumption, which is not always met.", "For that reason, $p$ -score models sometimes buy simplicity at a rather high cost.", "In other applications, monotonicity is a reasonable assumption, and the $p$ -score models provide a very simple and powerful approach to understanding the direction of evolutionary change.", "The costs and benefits of the $p$ -score model are not particular to the Price equation.", "Any analysis based on the same assumptions has the same limitations.", "The Price equation provides a concise and elegant way to explore the consequences when certain simplifying assumptions can reasonably be applied to a particular problem.", "The full Price equation partitions total evolutionary change into components.", "Many alternative partitions exist.", "A partition provides value if it improves conceptual clarity or eases calculation.", "Which partitions are better than others?", "Better is always partly subjective.", "What may seem hard for me may appear easy to you.", "Nonetheless, it would be a mistake to suggest that all differences are purely subjective.", "Some forms are surely better than others for particular problems, even if better remains hard to quantify.", "As [p. 14]russell58the-abc-of-relativity said in another context, “All such conventions are equally legitimate, though not all are equally convenient.” Many partitions of evolutionary change include some aspect of selection and some aspect of property or transmission change.", "Most of those variants arise by minor rearrangements or extensions of the basic Price expression.", "A few examples follow.", "heisler87a-method introduced the phrase contextual analysis to the evolutionary literature.", "Contextual analysis is a form of path analysis, which partitions causes by statistical regression models.", "Path analysis has been used throughout the history of genetics li75path.", "It is a useful approach whenever one wishes to partition variation with respect to candidate causes.", "The widely used method of lande83the-measurement to analyze selection is a particular form of path analysis.", "okasha06evolution argued that contextual analysis is an alternative to the Price equation.", "To develop a simple example, let us work with just the selection part of the Price equation $\\bar{w}\\Delta \\bar{z}= {\\hbox{\\rm Cov}}(w,z).$ A path (contextual) analysis refines this expression by partitioning the causes of fitness with a regression equation.", "Suppose we express fitness as depending on two predictors: the focal character that we are studying, $z$ , and another character, $y$ .", "Then we can write fitness as $w = \\beta _{wz}z+\\beta _{wy}y + \\epsilon $ in which the $\\beta $ terms are partial regressions of fitness on each character, and $\\epsilon $ is the unexplained residual of fitness.", "Substituting into the Price equation, we get the sort of expression made popular by lande83the-measurement $\\bar{w}\\Delta \\bar{z}= \\beta _{wz}{\\hbox{\\rm Var}}(z)+\\beta _{wy}{\\hbox{\\rm Cov}}(y,z).$ If the partitioning of fitness into causes is done in a useful way, this type of path analysis can provide significant insight.", "I based my own studies of natural selection and social evolution on this approach frank97the-price,frank98foundations.", "Authors such as okasha06evolution consider the partitioning of fitness into distinct causes as an alternative to the Price equation.", "If one thinks of the character $z$ in ${\\hbox{\\rm Cov}}(w,z)$ as a complete causal explanation for fitness, then a partition into separate causes $y$ and $z$ does indeed lead to a different causal understanding of fitness.", "In that regard, the Price equation and path analysis lead to different causal perspectives.", "One can find articles that use the Price equation and interpret $z$ as a lone cause of fitness <see>okasha06evolution.", "Thus, if one equates those specific applications with the general notion of the Price equation, then one can say that path or contextual analysis provides a significantly different perspective from the Price equation.", "To me, that seems like a socially constructed notion of logic and mathematics.", "If someone has applied an abstract truth in a specific way, and one can find an alternative method for the same specific application that seems more appealing, then one can say that the alternative method is superior to the general abstract truth.", "The abstract Price equation does not compel one to interpret $z$ strictly as a single cause explanation.", "Rather, in the general expression, $z$ should always be interpreted as an abstract placeholder.", "Path (contextual) analysis follows as a natural extension of the Price equation, in which one makes specific models of fitness expressed by regression.", "It does not make sense to discuss the Price equation and path analysis as alternatives.", "In the standard form of the Price equation, the fitness term, $w$ , appears in both components $\\bar{w}\\Delta \\bar{z}= {\\hbox{\\rm Cov}}(w,z) + \\operatorname{E}(w\\Delta z).$ frank97the-price,frank98foundations derived an alternative expression z= qi'zi' - qi zi = qi(wi/w)zi' - qi zi = qi(wi/w)zi' - qizi' + qizi' - qizi = qi(wi/w-1)zi' + qi(zi'-zi) = Cov(w,z')/w+ E(z).", "This form sometimes provides an easier method to calculate effects.", "For example, the second term now expresses the average change in phenotype between parent and offspring without weighting by fitness effects.", "A biased mutational process would be easy to calculate with this expression—one only needs to know about the mutation process to calculate the outcome.", "The new covariance term can be partitioned into meaningful components with minor assumptions [p. 1721]frank97the-price, yielding ${\\hbox{\\rm Cov}}(w,z^{\\prime }) = {\\hbox{\\rm Cov}}(w,z)\\beta _{z^{\\prime }z},$ where $\\beta _{z^{\\prime }z}$ is usually interpreted as the offspring-parent regression, which is a type of heritability.", "Thus, we may combine selection with the heritability component of transmission into the covariance term, with the second term containing only a fitness-independent measure of change during transmission.", "okasha06evolution strongly favored the alternative partition for the Price equation in Eq.", "(REF ), because it separates all fitness effects in the first term from a pure transmission interpretation of the second term.", "In my view, there are costs and benefits for the standard Price equation expression compared with Eq.", "(REF ).", "One gains by having both, and using the particular form that fits a particular problem.", "For example, the term $\\operatorname{E}(\\Delta z)$ is useful when one has to calculate the effects of a biased mutational process that operates independently of fitness.", "Alternatively, suppose most individuals have unbiased transmission, such that $\\Delta z=0$ , whereas very sick individuals do not reproduce but, if they were to reproduce, would have a very biased transmission process.", "Then $\\operatorname{E}(\\Delta z)$ differs significantly from zero, because the sick, nonreproducing individuals appear in this term equally with the reproducing population.", "However, the actual transmission bias that occurs in the population would be zero, $\\operatorname{E}(w\\Delta z)=0$ , because all reproducing individuals have nonbiased transmission.", "Both the standard Price form and the alternative in Eq.", "(REF ) can be useful.", "Different scenarios favor different ways of expressing problems.", "I cannot understand why one would adopt an a priori position that unduly limits one's perspective.", "The Price equation's power arises from its abstraction of selection in terms of mapping relations between sets frank95george,price95the-nature.", "Although the Price equation is widely cited in the literature, almost no work has developed the set mapping formalism beyond the description given in the initial publications.", "I know of only one article.", "kerr09generalization noted that, in the original Price formulation, every descendant must derive from one or more ancestors.", "There is no natural way for novel entities to appear.", "In applications, new entities could arise by immigration from outside the system or, in a cultural interpretation, by de novo generation of an idea or behavior.", "kerr09generalization present an extended expression to handle unconnected descendants.", "Their formulation depends on making explicit the connection number between each individual ancestor and each individual descendant, rather than using the fitnesses of types.", "Some descendants may have zero connections.", "With an explicit description of connections, an extended Price equation follows.", "The two core components of covariance for selection and expected change for transmission occur, plus a new factor to account for novel descendants unconnected to ancestors.", "The notation in kerr09generalization is complex, so I do not repeat it here.", "Instead, I show a simplified version.", "Suppose that a fraction $p$ of the descendants are unconnected to ancestors.", "Then we can write the average trait value among descendants as z' = pj z*j + (1-p)qi'zi', where $z^*_j$ is the phenotype for the $j$ th member of the descendant population that is unconnected to ancestors, and $\\alpha _j$ is the frequency of each unconnected type, with $\\sum \\alpha _j=1$ .", "Given those definitions, we can proceed with the usual Price equation expression z= z' -z = pj z*j + (1-p)qi'zi' - (p+1-p)qizi = (1-p)(qi'zi' - qizi)+ p(j z*j - qizi).", "Note that the term weighted by $1-p$ leads to the standard form of the Price equation, so we can write z= (1-p)(Cov(w,z)+E(wz))/w+ p(j z*j - qizi) = (1-p)(Cov(w,z)+E(wz))/w+ p(z*-z).", "In the component weighted by $p$ , no connections exist between the descendant $z^*_j$ and a member of the ancestral population.", "Thus, we have no basis to relate those terms to fitness, transmission, or property change.", "kerr09generalization use an alternative notation that associates all entities with their number of connections, including those with zero.", "The outcome is an extended set mapping theory for evolutionary change.", "The main concepts and the value of the approach are best explained by the application presented in the next section.", "fox12analyzing analyze changes in ecosystem function by modifying the method of kerr09generalization.", "They measure ecosystem function by summing the functional contribution of each species present in an ecosystem.", "To compare ecosystems, they consider an initial site and a second site.", "When comparing ecosystems, the notion of ancestors and descendants may not make sense.", "Instead, one appeals to the more general set mapping relations of the Price equation.", "Assume that there is an initial site with total function $T=\\sum z_i$ , where $z_i$ is the function of the $i$ th species.", "At the initial site, there are $s$ different species, thus we may also express the total as $T=s\\bar{z}$ , where $\\bar{z}$ is the average function per species.", "At a second site, total function is $T^{\\prime }=\\sum z_j^{\\prime }$ , with $s^{\\prime }$ different species in the summation, and $T^{\\prime }=s^{\\prime }\\bar{z}^{\\prime }$ .", "Let the number of species in common between the sites be $s_c$ .", "Thus, the initial site has $S=s-s_c$ unique species, and the second site has $S^{\\prime }=s^{\\prime }-s_c$ unique species.", "fox12analyzing write the change in total ecosystem function as T = T' - T = s'z' - sz = (s'-sc)z' - (s-sc)z+ sc(z'-z) = S'z' - Sz+ sc(z).", "The term $S^{\\prime }\\bar{z}^{\\prime }$ represents the change in function caused the gain of an average species, in which $S^{\\prime }$ is the number of newly added species, and $\\bar{z}^{\\prime }$ is the average function per species.", "Fox & Kerr suggest that a randomly added species would be expected to function as an average species, and so interpret this term as the contribution of random species gain.", "The term $S\\bar{z}$ is interpreted similarly as random species loss with respect to the $S$ unique species in the first ecosystem not present in the second ecosystem.", "Fox & Kerr partition the term $s_c(\\Delta \\bar{z})$ into three components of species function: deviation from the average for species gained at the second site, deviation from the average for species lost from the first site, and the changes in function for those species in common between sites.", "The point here concerns the approach rather than the theory of ecosystem function.", "To analyze changes between two sets, one often benefits by an explicit decomposition of the relations between the two sets.", "The original Price equation is one sort of decomposition, based on tracing the ways in which descendants derive from and change with respect to ancestors.", "fox12analyzing extend the decomposition of change by set mapping to include specific components that make sense in the context of changes in ecosystem function.", "More work on the mathematics of set mapping and decomposition would be very valuable.", "The Price equation and the extensions by Kerr, Godfrey-Smith, and Fox show the potential for thinking carefully about the abstract components of change between sets, and how to apply that abstract understanding to particular problems.", "No clear guidelines determine what constitutes an extension to the Price equation.", "From a broad perspective, many different partitions of total change have similarities, because they separate something like selection from other forces that alter the similarity between populations.", "For example, the stochastic effects of sampling and drift create a distribution of descendant phenotypes around the ancestral mean.", "In the classical Price formulation, there is only the single realization of the actual descendants.", "A stochastic version analyzes a collection of possible descendant sets over some probability distribution, and a mapping from the ancestor set to each possible realization of the descendant set.", "In other cases, partitions will split components more finely or add new components not in Price's formulation.", "I do not have space to review every partition of total change and consider how each may be related to Price's formulation.", "I list a few examples here.", "grafen99formal and rice08a-stochastic developed stochastic approaches.", "grafen07the-formal based a long-term project on interpretations and extensions of the Price equation.", "page02unifying related the Price equation to various other evolutionary analyses, providing some minor extensions.", "wolf98evolutionary, bijma08the-joint, and many others developed extended partitions by splitting causes with regression or similar methods such as path analysis.", "Various forms of the Price equation have been applied in economic theory andersen04population.", "A reliable way to make people believe in falsehoods is frequent repetition, because familiarity is not easily distinguished from truth [p. 62]kahneman11thinking.", "One must distinguish the full, exact Price equation from various derived forms used in applications.", "The derived forms always make additional assumptions or express approximate relations frank97the-price.", "Each assumption increases specificity and reduces generality in relation to particular goals.", "Critiques of the Price equation rarely distinguish the costs and benefits of particular assumptions in relation to particular goals.", "I use van Veelen's recent series of papers as a proxy for those critiques.", "That series repeats some of the common misunderstandings and adds some new ones.", "Nowak recently repeated van Veelen's critique as the basis for his commentary on the Price equation veelen05on-the-use-of-the-price,veelen10call,van-veelen11a-rule,vanveelen12group,nowak10the-evolution,nowak11supercooperators:.", "The Price equation describes the change in some measurement, expressed as $\\Delta \\bar{z}$ .", "Change is calculated with respect to particular mapping relations between ancestor and descendant populations.", "We can think of the mappings and the beginning value of $\\bar{z}$ as the initial conditions or inputs, and $\\Delta \\bar{z}$ as the output.", "The output, $\\bar{z}^{\\prime }=\\bar{z}+\\Delta \\bar{z}$ , does not provide enough information to iterate the calculation of change in order to get another value of $\\Delta \\bar{z}$ starting with $\\bar{z}^{\\prime }$ .", "We would also need the mapping relations between the new descendant population and its subsequent descendants.", "That information is not part of the initial input.", "Thus, we cannot study the dynamics of change over time without additional information.", "This limitation with regard to repeated iteration is called a lack of dynamic sufficiency lewontin74the-genetic.", "Confusion about the nature of dynamic sufficiency in relation to the Price equation has been common in the literature.", "In [pp.", "378–379]frank95george, I wrote It is not true, however, that dynamic sufficiency is a property that can be ascribed to the Price Equation—this equation is simply a mathematical tautology for the relationship among certain quantities of populations.", "Instead, dynamic sufficiency is a property of the assumptions and information provided in a particular problem, or added by additional assumptions contained within numerical techniques such as diffusion analysis or applied quantitative genetics.", "$\\ldots $ What problems can the Price equation solve that cannot be solved by other methods?", "The answer is, of course, none, because the Price Equation is derived from, and is no more than, a set of notational conventions.", "It is a mathematical tautology.", "I showed how the Price equation helps to define the necessary conditions for dynamic sufficiency.", "Once again, the Price equation proves valuable for clarifying the abstract structure of evolutionary analysis.", "Compare my statement to vanveelen12group Dynamic insufficiency is regularly mentioned as a drawback of the Price equation (see for example Frank, 1995; Rice, 2004).", "We think that this is not an entirely accurate description of the problem.", "We would like to argue that the perception of dynamic insufficiency is a symptom of the fundamental problem with the Price equation, and not just a drawback of an otherwise fine way to describe evolution.", "To begin with, it is important to realize that the Price equation itself, by its very nature, cannot be dynamically sufficient or insufficient.", "The Price equation is just an identity.", "If we are given a list of numbers that represent a transition from one generation to the next, then we can fill in those numbers in both the right and the left hand side of the Price equation.", "The fact that it is an identity guarantees that the numbers that appear on both sides of the equality sign are the same.", "There is nothing dynamically sufficient or insufficient about that (this point is also made by Gardner et al., 2007, p. 209).", "A model, on the other hand, can be dynamically sufficient or insufficient.", "This quote from vanveelen12group demonstrates an interesting approach to scholarship.", "They first cite Frank as stating that dynamic insufficiency is a drawback of the Price equation.", "They then disagree with that point of view, and present as their own interpretation an argument that is nearly identical in concept and phrasing to my own statement in the very paper that they cited as the foundation for their disagreement.", "In this case, I think it is important to clarify the concepts and history, because influential and widely cited authors, such as Nowak, are using van Veelen's articles as the basis for their own critiques of the Price equation and approaches to fundamental issues of evolutionary analysis.", "With regard to dynamics, any analysis achieves the same dynamic status given the same underlying assumptions.", "The Price equation, when used with the same underlying assumptions as population genetics, has the same attributes of dynamic sufficiency as population genetics.", "vanveelen12group claim that Maybe the most unfortunate thing about the Price equation is that the term on the right hand side is denoted as a covariance, even though it is not.", "The equation thereby turns into something that can easily set us off in the wrong direction, because it now resembles equations as they feature in other sciences, where probabilistic models are used that do use actual covariances.", "One can see the covariance expression in the standard form of the Price equation given in Eq.", "(REF ).", "In the Price equation, the covariance is measured with respect to the total population, in other words, it expresses the association over all members of the population.", "In many statistical applications, one only has data on a subset of the full population, that subset forming a sample.", "It is important to distinguish between population measures and sample measures, because they refer to different things.", "[p. 485]price72extension made clear that his equation is about total change in entire populations, so the covariance is interpreted as a population measure [W]e will be concerned with population functions and make no use of sample functions, hence we will not observe notational conventions for distinguishing population and sample variables and functions.", "In additional to population and sample measures, covariance also arises in mathematical models of process.", "Suppose, for example, that I develop a model in which random processes influence fitness and random processes influence phenotype.", "If the random fluctuations in fitness and the random fluctuations in phenotype are associated, the random variables of fitness and phenotype would covary.", "All of these different interpretations of covariance are legitimate, they simply reflect different situations.", "In frank95george, I wrote: “What problems can the Price equation solve that cannot be solved by other methods?", "The answer is, of course, none, because the Price Equation is derived from, and is no more than, a set of notational conventions.", "It is a mathematical tautology.” nowak11supercooperators: and vanveelen12group emphasize the same point in their critique of the Price equation, although they present the argument as a novel insight without attribution.", "Given that the Price equation is a set of notational conventions, it cannot uniquely specify any predictions or insights.", "A particular set of assumptions leads to the same predictions, no matter what notational conventions one uses.", "The Price equation is a tool that sometimes helps in analysis or in seeing general connections between apparently disparate ideas.", "For many problems, the Price equation provides no value, because it is the wrong tool for the job.", "If the Price equation is just an equivalence, or tautology, then why am I enthusiastic about it?", "Mathematics is, in its essence, about equivalences, as expressed beautifully in the epigraph from Mazur.", "Not all equivalences are interesting or useful, but some are, just as not all mathematical expressions are interesting or useful, but some are.", "That leads us to the question of how we might know whether the Price equation is truly useful or a mere identity?", "It is not always easy to say exactly what makes an abstract mathematical equivalence interesting or useful.", "However, given the controversy over the Price equation, we should try.", "Because there is no single answer, or even a truly unique and unambiguous question, the problem remains open.", "I list a few potential factors.", "“[A] good notation has a subtlety and suggestiveness which at times make it seem almost like a live teacher” [pp. 17–18]russell22introduction.", "Much of creativity and understanding comes from seeing previously hidden associations.", "The tools and forms of expression that we use play a strong role in suggesting connections and are inseparable from cognition kahneman11thinking.", "Equivalences and alternative notations are important.", "The various forms of the covariance component from the Price equation given in Eq.", "(REF ) show the equivalence of the statistical, geometrical and informational expressions for natural selection.", "The recursive form of the full Price equation provides the foundation for all modern studies of group selection and multilevel analysis.", "The Price equation helped in discovering those various connections, although there are many other ways in which to derive the same relations.", "hardy67a-mathematicians also emphasized the importance of seeing new connections between apparently disparate ideas: We may say, roughly, that a mathematical idea is `significant' if it can be connected, in a natural and illuminating way, with a large complex of other mathematical ideas.", "Thus a serious mathematical theorem, a theorem which connects significant ideas, is likely to lead to important advances in mathematics itself and even in other sciences.", "What sort of connections?", "One type concerns the invariances discovered or illuminated by the Price equation.", "I discussed some of those invariances in an earlier section, particularly the information theory interpretation of natural selection through the measure of Fisher information frank09natural.", "Fisher's fundamental theorem of natural selection is a similar sort of invariance frank12wrights.", "Kin selection theory derives much of its power by identifying an invariant informational quantity sufficient to unify a wide variety of seemingly disparate processes [Chapter 6]frank98foundations.", "The interpretation of kin selection as an informational invariance has not been fully developed and remains an open problem.", "Invariances provide the foundation of scientific understanding: “It is only slightly overstating the case to say that physics is the study of symmetry” anderson72more.", "Invariance and symmetry mean the same thing weyl83symmetry.", "feynman67the-character emphasized that invariance is The Character of Physical Law.", "The commonly observed patterns of probability can be unified by the study of invariance and its association to measurement frank10measurement,frank11a-simple.", "There has been little effort in biology to pursue similar understanding of invariance and measurement frank11measurement,houle11measurement.", "Price argued for the great value of abstraction, in the sense of the epigraph from Mazur.", "In price95the-nature [D]espite the pervading importance of selection in science and life, there has been no abstraction and generalization from genetical selection to obtain a general selection theory and general selection mathematics.", "Instead, particular selection problems are treated in ways appropriate to particular fields of science.", "Thus one might say that `selection theory' is a theory waiting to be born—much as communication theory was 50 years ago.", "Probably the main lack that has been holding back any development of a general selection theory is lack of a clear concept of the general nature or meaning of `selection'.", "This article has been about the Price equation in relation to its abstract properties and its connections to various topics, such as information or fundamental invariances.", "Some readers may feel that those aspects of abstraction and invariance are nice, but far from daily work in biology.", "What of the many applications of the Price equation to kin or group selection?", "Do those applications hold up?", "How much value has been added?", "Because the Price equation is a tool, one can always arrive at the same result by other methods.", "How well the Price equation works depends partly on the goal and partly on the fit of the tool to the problem.", "There is inevitably a strongly subjective aspect to deciding about how well a tool works.", "Nonetheless, hammers truly are good for nails and bad for screws.", "For valuing tools, there is a certain component that should be open to agreement.", "For example, the robertson66a-mathematical form of the Price equation is widely regarded as the foundational method for analyzing models of evolutionary quantitative genetics.", "However, not all problems in quantitative genetics are best studied with the Robertson-Price equation.", "And not all problems in social evolution benefit from a Price equation approach.", "The Price equation or descendant methods have led to many useful models for kin selection frank98foundations.", "The most powerful follow a path analysis decomposition of causes or use a simple maximization method to analyze easily what would otherwise have been difficult.", "I will return to those applications in subsequent articles.", "I thank R. M. Bush and W. J. Ewens for helpful comments.", "My research is supported by National Science Foundation grant EF-0822399, National Institute of General Medical Sciences MIDAS Program grant U01-GM-76499, and a grant from the James S. McDonnell Foundation." ] ]

[ [ "Deformation of Hypersurfaces Preserving the Moebius Metric and a\n Reduction Theorem" ], [ "Abstract A hypersurface without umbilics in the n+1 dimensional Euclidean space is known to be determined by the Moebius metric and the Moebius second fundamental form up to a Moebius transformation when n>2.", "In this paper we consider Moebius rigidity for hypersurfaces and deformations of a hypersurface preserving the Moebius metric in the high dimensional case n>3.", "When the highest multiplicity of principal curvatures is less than n-2, the hypersurface is Moebius rigid.", "Deformable hypersurfaces and the possible deformations are also classified completely.", "In addition, we establish a Reduction Theorem characterizing the classical construction of cylinders, cones, and rotational hypersurfaces, which helps to find all the non-trivial deformable examples in our classification with wider application in the future." ], [ "Introduction", "In submanifold theory a fundamental problem is to investigate which data are sufficient to determine a submanifold $M$ up to the action of a certain transformation group $G$ on the ambient space.", "The deformable case means that there exists non-congruent (depending on $G$ ) immersions with the same given invariants at corresponding points, and such different immersions are called deformations to each other.", "In contrast, the rigid case indicates that such deformations do not exist (or just be congruent by the action of $G$ ).", "In this paper we consider deformations of hypersurfaces $M^n$ preserving the so-called Möbius metric in the framework of Möbius geometry ($G$ is the Möbius transformation group acting on $R^{n+1}\\cup \\lbrace \\infty \\rbrace $ ).", "As a background let us review some classical results.", "It is known that a generic immersed surface in Euclidean three-space $u:M^2\\rightarrow R^3$ is determined, up to a rigid motion of $R^3$ , by its induced metric $I$ and mean curvature function $H$ .", "All exceptional immersions are called Bonnet surfaces, which were classified by Bonnet[2], Cartan[7] and Chern[8] into three distinct classes: 1) CMC (constant mean curvature) surfaces with a 1-parameter deformations preserving $I$ and $H$ (known as the associated family); 2) Not CMC and admits a continuous 1-parameter deformations; 3) Surfaces that admit exactly one such deformation.", "In either case, two Bonnet surfaces forming deformation to each other is called a Bonnet pair.", "These notions are directly generalized to other space forms $S^3$ and $H^3$ .", "See [1], [16], [17], [23] for recent works on this topic.", "For a hypersurface $f:M^n\\rightarrow R^{n+1}(n\\ge 3)$ , the well-known Beez-Killing rigidity theorem says that $f$ is isometrically rigid if the rank of its second fundamental form (i.e.", "the number of non-zero principal curvatures) is greater than or equal to 3 everywhere.", "Compared to surface case this is a stronger rigidity result, mainly due to the Gauss equations which forms an over-determined system when there are many non-zero principal curvatures.", "On the other hand, all isometrically deformable hypersurfaces have rank 2 or less.", "They are locally classified by Sbrana [22] and Cartan [3].", "According to their results there are four classes of them.", "The first two classes (surface-like and ruled) are highly deformable.", "The third class admits precisely a continuous 1-parameter family of deformations, and the fourth class has a unique deformation.", "In Möbius geometry, let $f,\\bar{f}: M^n\\rightarrow R^{n+1}$ be two hypersurfaces in the $(n+1)$ -dimensional Euclidean space $R^{n+1}$ .", "We say $f$ is Möbius equivalent to $\\bar{f}$ (or $f$ is Möbius congruent to $\\bar{f}$ ) if there exists a Möbius transformation $\\Psi $ such that $f=\\Psi \\circ \\bar{f}$ .", "It is natural to consider deformations preserving certain conformal invariants.", "In [4] Cartan considered the problem of conformal deformation, i.e.", "deformation of any given hypersurface preserving the conformal class of the induced metric.", "Cartan has given the following conformal rigidity result: Theorem 1.1 [4] A hypersurface $f:M^n\\rightarrow R^{n+1}~~(n\\ge 5)$ is conformally rigid if each principal curvature has multiplicity less than $n-2$ everywhere.", "In [11] do Carmo and Dajczer generalized Cartan's rigidity theorem to submanifolds of dimension $n\\ge 5$ .", "Note that the multiplicity of a principal curvature is Möbius invariant.", "When the highest multiplicity is $n$ or $n-1$ , it is the conformally flat case well-known to be highly deformable.", "When $n\\ge 5$ and the highest multiplicity is $n-2$ , Cartan [4] gave a quite similar classification of conformally deformable hypersurfaces into four cases: I) Surface-like hypersurfaces (which are cylinders, cones or revolution hypersurfaces over surfaces in 3-dim space forms); II) Conformally ruled hypersurfaces; III) One of those having a continuous 1-parameter family of deformations; IV) One of those that admits a unique deformation.", "In [9] and [10] Dajczer et.al.", "gave a modern account of Sbrana and Cartan's classification.", "Following Dajczer, we call such conformally deformable hypersurfaces as Cartan hypersurfaces of class I, II, III, and IV.", "We observe that in the conformal class of a given immersed hypersurface in $R^{n+1}$ there is a distinguished metric called the Möbius metric $g$.", "Together with the Möbius second fundamental form $B$ they form a complete system of invariants in Möbius geometry (see [25] or Theorem REF in this paper).", "Based on our experience, the deformation preserving the Möbius metric $g$ seems to be a natural and new topic.", "Definition 1.2 A hypersurface $f:M^n\\rightarrow R^{n+1}$ is said to be Möbius rigid if any other immersion $\\bar{f}:M^n\\rightarrow R^{n+1}$ sharing the same Möbius metric $g$ as $f$ , is Möbius equivalent to $f$ .", "An immersion $\\bar{f}:M^n\\rightarrow R^{n+1}$ is said to be a Möbius deformation of $f$ if they induce the same Möbius metric $g$ at corresponding points and $\\bar{f}(M)$ is not congruent to $f(M)$ up to any Möbius transformation.", "We obtain the following Möbius Rigidity Theorem.", "Theorem 1.3 Let $f: M^n\\rightarrow R^{n+1}~~(n\\ge 4)$ be a hypersurface in the $(n+1)$ -dimensional Euclidean space.", "If every principal curvature of $f$ has multiplicity less than $n-2$ everywhere, then $f$ is Möbius rigid.", "Remark 1.4 Compared with Cartan's notion before, a conformally rigid hypersurface $f: M^n\\rightarrow R^{n+1}$ is Möbius rigid, but the converse may not be true.", "On the other hand, if $f$ is Möbius deformable with deformation $\\bar{f}$ , they are also conformal deformations to each other, but the converse may not be true.", "Thus when $n\\ge 5$ our rigidity theorem is a corollary of Cartan's conformal rigidity result.", "On the other hand, Cartan treated the special dimensions $n=4,3$ in [5], [6].", "In particular, in [5] Cartan has shown that, for $n=4$ , there exist hypersurfaces $f,\\bar{f}:M^4\\rightarrow R^5$ that have four distinct principal curvatures at each point $p\\in M^4$ and are conformal deformations to each other.", "In contrast, our Möbius rigidity result as above still holds true for dimension $n=4$ .", "Because of this interesting difference and for the purpose of self-containedness we give a proof to Theorem REF in Section 7.", "The main result of this paper is the following classification theorem of all Möbius deformable hypersurfaces.", "Theorem 1.5 Let $f: M^n\\rightarrow R^{n+1}~~(n\\ge 4)$ be an umbilic free hypersurface in the $(n+1)$ -dimensional Euclidean space, whose principal curvatures have constant multiplicities.", "Suppose $f$ is Möbius deformable.", "$1)$ When one principal curvature of $f$ has multiplicity $n-1$ everywhere, this deformable $f$ must have constant Möbius sectional curvature.", "They are either cones, cylinders or rotational hypersurfaces over the so-called curvature-spirals in 2-dimensional space-forms.", "(See [13] for the classification or Section 4 for an independent proof.)", "$2)$ When one principal curvature of $f$ has multiplicity $n-2$ everywhere, locally $f$ is Möbius equivalent to either of the three classes below: (a) $f(M^n)\\subset L^2\\times R^{n-2}$ , where $L^2$ is a Bonnet surface in $R^3$ ; (b) $f(M^n)\\subset CL^2\\times R^{n-3}$ , where $CL^2\\subset R^4$ is a cone over $L^2\\subset S^3$ , and $L^2$ is a Bonnet surface in $S^3$ ; (c) $f(M^n)$ is a rotational hypersurface over $L^2\\subset R^3_+$ , where $L^2$ is a Bonnet surface in the hyperbolic half space model $R^3_+$ .", "Moreover, the Möbius deformation to any of them belongs to the same class and comes from the deformation of the corresponding Bonnet surface $L^2$ .", "Remark 1.6 The hypothesis that the principal curvatures have constant multiplicities is necessary in this paper, because we need smooth frame of principal vectors.", "But the hypothesis is weak, for there always exists an open dense subset $U$ of $M^n$ on which the multiplicities of the principal curvatures are locally constant (see [21]).", "However, the hypothesis is not necessary only in Theorem REF .", "For dimension $n=4$ , the condition that any principal curvature has multiplicity less than $n-2$ means that the principal curvatures have constant multiplicities.", "For dimension $n\\ge 5$ , the frame of principal vectors used in Proposition REF is pointwise , so we do not need smooth frame of principal vector fields.", "Remark 1.7 According to our classification result, among Cartan hypersurfaces [10], only the first class (surface-like hypersurfaces) may share the same Möbius metric with their conformal deformations.", "The other three classes of conformally deformable hypersurfaces are Möbius rigid in our sense.", "Remark 1.8 In the definition above, it is noteworthy that the non-congruence between $\\bar{f}(M), f(M)$ (the images) is stronger than the non-congruence between $\\bar{f}, f$ (the mappings), because the same hypersurface $f(M)\\subset R^n$ might be given different parameterizations $f$ and $\\bar{f}$ which are NOT Möbius equivalent.", "In other words there might exist an (isometrical) diffeomorphism $\\phi : M^n\\rightarrow M^n$ and a Möbius transformation $\\Psi :R^{n+1}\\cup \\lbrace \\infty \\rbrace \\rightarrow R^{n+1}\\cup \\lbrace \\infty \\rbrace $ such that the following diagram commutes: ${M^n [d]_{f} [r]^{\\psi }& M^n [d]^{\\bar{f}} \\\\R^{n+1}\\cup \\lbrace \\infty \\rbrace [r]_{\\Psi }& R^{n+1}\\cup \\lbrace \\infty \\rbrace }$ A typical example is the Möbius isoparametric hypersurface (see [19], [15] or Section 9 for the definition) with three distinct constant Möbius principal curvatures $\\sqrt{\\frac{n-1}{2n}},-\\sqrt{\\frac{n-1}{2n}},0,\\cdots ,0.$ It is part of the cone over the Cartan minimal isoparametric hypersurface $y:N^3\\rightarrow S^4(1)\\hookrightarrow R^5\\subset R^{n+1}$ with three distinct principal curvatures.", "This $N^3$ is a tube of a specific constant radius over the Veronese embedding $RP^2\\hookrightarrow S^4$ .", "It is well-known that its induced metric has a 4-dimensional isometry group whose elements do NOT preserve the principal distributions in general.", "Any such isometry $\\phi $ extends to an isometry of the cone (with respect to its Möbius metric $g$ ) which is surely NOT a Möbius transformation of the ambient space.", "Any possible deformation $\\bar{f}$ to the cone $f$ preserving Möbius metric $g$ arises in this way, hence is excluded from our notion (as well as the classification list) of Möbius deformable hypersurfaces.", "See the discussion of this example in Section 9.", "Remark 1.9 For a hypersurface $f:M\\rightarrow R^{n+1}$ of constant Möbius curvature $c$ , generally we can map any neighborhood of a given point $p\\in M$ to a neighborhood of another point $q\\in M$ by an isometry (of $(M,g)$ ) which is not induced from a Möbius transformation of the ambient space.", "This is because any such hypersurface is conformally flat with a specific principal direction which is not preserved by a generic isometry of $(M,g)$ .", "So they provide the first class of deformable hypersurfaces.", "Circular cylinder and spiral cylinder (constructed from a circle or a logarithmic spiral, respectively) belong to this class, yet they are different.", "Each of them is homogeneous, namely invariant under a subgroup of the Möbius group (of dimension at least $n$ ) which acts transitively on $M^n$ ).", "On the other hand each of them have a bigger isometry group (with respect to $(M^n,g)$ ) which generally are not induced from Möbius transformations.", "So they resemble Cartan's example in the previous remark.", "Yet these two hypersurfaces still have non-trivial deformations.", "See final remarks in Section 4.", "Remark 1.10 Some comments on low dimensional case $n=3$ or 2.", "We do not have any Möbius rigidity result because our algebraic theorem REF fails in this case (see Remark REF ).", "But the construction of Möbius deformable hypersurfaces in Section 3 is still valid for $n=3$ .", "When $n=2$ , generally a surface with a given Möbius metric is highly deformable.", "So we would consider deformation problems under stronger restrictions.", "We just mention that any Willmore surface admits a one-parameter associated family of Willmore surfaces endowed with the same Möbius metric.", "For more on related topics see [12].", "Remark 1.11 It is very interesting that the non-trivial deformable examples all arise from the classical construction of cylinders, cones or rotational hypersurfaces over a given hypersurface in a low-dimensional Euclidean subspace, sphere or hyperbolic half-space, respectively.", "Such constructions appeared many times in various contexts and problems in Möbius geometry and Lie sphere geometry.", "We find that such examples have a nice characterization (Theorem REF ) in terms of its Möbius invariants introduced by the third author in [25].", "We believe that this Reduction Theorem is a valuable tool in simplifying discussions of many similar problems.", "We organize the paper as follows.", "In Section 2, we introduce Möbius invariants and the Möbius congruence theorem for hypersurfaces in $R^{n+1}~~(n\\ge 3)$ .", "Examples of Möbius deformable hypersurfaces are given in Section 3 and 4 (in particular, Section 4 gives a new proof to the classification theorem of hypersurfaces with constant Möbius sectional curvature).", "These examples are characterized by our Reduction Theorems REF (used in Section 9) and REF (used in Section 4) proved in Section 5.", "After these preparations, as a purely algebraic consequence of the Gauss equation we show in Section 6 that the (Möbius) second fundamental forms of $f$ and its deformation $\\bar{f}$ could be diagonalized almost simultaneously.", "Then we investigate our problem case by case.", "When the highest multiplicity is less than $n-2$ we establish the rigidity result (Theorem REF ) in Section 7.", "Section 8 treats the conformally flat case (i.e.", "the highest multiplicity is equal to $n-1$ ) where we show such deformable examples must have constant Möbius curvature, which have been classified in Section 4.", "In Section 9 all deformable hypersurfaces with one principal curvature of multiplicity $n-2$ are proved to be reducible to cylinders, cones or rotational hypersurfaces using the Reduction Theorem in Section 5.", "This finishes the proof to the Main Theorem REF ." ], [ "Möbius invariants for hypersurfaces in $R^{n+1}$", "In this section we briefly review the theory of hypersurfaces in Möbius geometry.", "For details we refer to $\\cite {w},\\cite {w2}$ .", "Let $R^{n+3}_1$ be the Lorentz space, i.e., $R^{n+3}$ with inner product $<\\cdot ,\\cdot >$ defined by $<x,y>=-x_0y_0+x_1y_1+\\cdots +x_{n+2}y_{n+2},$ for $x=(x_0,x_1,\\cdots ,x_{n+2}), y=(y_0,y_1,\\cdots ,y_{n+2})\\in R^{n+3}$ .", "Let $f:M^{n}\\rightarrow R^{n+1}$ be a hypersurface without umbilics and assume that $\\lbrace e_i\\rbrace $ is an orthonormal basis with respect to the induced metric $I=df\\cdot df$ with $\\lbrace \\theta _i\\rbrace $ the dual basis.", "Let $II=\\sum _{ij}h_{ij}\\theta _i\\theta _j$ and $H=\\sum _i\\frac{h_{ii}}{n}$ be the second fundamental form and the mean curvature of $f$ , respectively.", "We define the Möbius position vector $Y: M^n\\rightarrow R^{n+3}_1$ of $f$ by $Y=\\rho \\left(\\frac{1+|f|^2}{2},\\frac{1-|f|^2}{2},f\\right)~,~~\\rho ^2=\\frac{n}{n-1}(|II|^2-nH^2).$ Theorem 2.1 [25] Two hypersurfaces $f,\\bar{f}: M^n\\rightarrow R^{n+1}$ are Möbius equivalent if and only if there exists $T$ in the Lorentz group $O(n+2,1)$ in $R^{n+3}_1$ such that $\\bar{Y}=YT.$ It follows immediately from Theorem 2.1 that $g=<dY,dY>=\\rho ^2df\\cdot df$ is a Möbius invariant, called the Möbius metric of $f$ .", "Let $\\Delta $ be the Laplacian with respect to $g$ .", "Define $N=-\\frac{1}{n}\\Delta Y-\\frac{1}{2n^2}<\\Delta Y,\\Delta Y>Y,$ which satisfies $<Y,Y>=0=<N,N>, ~~<N,Y>=1~.$ Let $\\lbrace E_1,\\cdots ,E_n\\rbrace $ be a local orthonormal basis for $(M^n,g)$ with dual basis $\\lbrace \\omega _1,\\cdots ,\\omega _n\\rbrace $ .", "Write $Y_i=E_i(Y)$ .", "Then we have $<Y_i,Y>=<Y_i,N>=0, ~<Y_i,Y_j>=\\delta _{ij}, ~~1\\le i,j\\le n.$ Let $\\xi $ be the mean curvature sphere of $f$ written as $\\xi =\\left(\\frac{1+|f|^2}{2}H+f\\cdot e_{n+1},\\frac{1-|f|^2}{2}H-f\\cdot e_{n+1},Hf+e_{n+1}\\right),$ where $e_{n+1}$ is the unit normal vector field of $f$ in $R^{n+1}$ .", "Then $\\lbrace Y,N,Y_1,\\cdots ,Y_n,\\xi \\rbrace $ forms a moving frame in $R^{n+3}_1$ along $M^n$ .", "We will use the following range of indices in this section: $1\\le i,j,k\\le n$ .", "We can write the structure equations as following: $&&dY=\\sum _iY_i\\omega _i,\\\\&&dN=\\sum _{ij}A_{ij}\\omega _iY_j+\\sum _iC_i\\omega _i\\xi ,\\\\&&dY_i=-\\sum _jA_{ij}\\omega _jY-\\omega _iN+\\sum _j\\omega _{ij}Y_j+\\sum _jB_{ij}\\omega _j\\xi ,\\\\&&d\\xi =-\\sum _iC_i\\omega _iY-\\sum _{ij}\\omega _iB_{ij}Y_j,$ where $\\omega _{ij}$ is the connection form of the Möbius metric $g$ and $\\omega _{ij}+\\omega _{ji}=0$ .", "The tensors ${\\bf A}=\\sum _{ij}A_{ij}\\omega _i\\otimes \\omega _j,~~{\\bf B}=\\sum _{ij}B_{ij}\\omega _i\\otimes \\omega _j,~~\\Phi =\\sum _iC_i\\omega _i$ are called the Blaschke tensor, the Möbius second fundamental form and the Möbius form of $f$ , respectively.", "The covariant derivative of $C_i, A_{ij}, B_{ij}$ are defined by $&&\\sum _jC_{i,j}\\omega _j=dC_i+\\sum _jC_j\\omega _{ji},\\\\&&\\sum _kA_{ij,k}\\omega _k=dA_{ij}+\\sum _kA_{ik}\\omega _{kj}+\\sum _kA_{kj}\\omega _{ki},\\\\&&\\sum _kB_{ij,k}\\omega _k=dB_{ij}+\\sum _kB_{ik}\\omega _{kj}+\\sum _kB_{kj}\\omega _{ki}.$ The integrability conditions for the structure equations are given by $&&A_{ij,k}-A_{ik,j}=B_{ik}C_j-B_{ij}C_k,\\\\&&C_{i,j}-C_{j,i}=\\sum _k(B_{ik}A_{kj}-B_{jk}A_{ki}),\\\\&&B_{ij,k}-B_{ik,j}=\\delta _{ij}C_k-\\delta _{ik}C_j,\\\\&&R_{ijkl}=B_{ik}B_{jl}-B_{il}B_{jk}+\\delta _{ik}A_{jl}+\\delta _{jl}A_{ik}-\\delta _{il}A_{jk}-\\delta _{jk}A_{il},\\\\&&R_{ij}:=\\sum _kR_{ikjk}=-\\sum _kB_{ik}B_{kj}+(tr{\\bf A})\\delta _{ij}+(n-2)A_{ij},\\\\&&\\sum _iB_{ii}=0, \\sum _{ij}(B_{ij})^2=\\frac{n-1}{n}, tr{\\bf A}=\\sum _iA_{ii}=\\frac{1}{2n}(1+n^2\\kappa ),$ where $R_{ijkl}$ denote the curvature tensor of $g$ , $\\kappa =\\frac{1}{n(n-1)}\\sum _{ij}R_{ijij}$ is its normalized Möbius scalar curvature.", "We know that all coefficients in the structure equations are determined by $\\lbrace g, {\\bf B}\\rbrace $ and we have Theorem 2.2 $\\cite {w}$ Two hypersurfaces $f: M^n\\rightarrow R^{n+1}$ and $\\bar{f}:M^n\\rightarrow R^{n+1} (n\\ge 3)$ are Möbius equivalent if and only if there exists a diffeomorphism $\\varphi : M^n\\rightarrow M^n$ which preserves the Möbius metric and the Möbius second fundamental form.", "The second covariant derivative of $B_{ij}$ are defined by $dB_{ij,k}+\\sum _mB_{mj,k}\\omega _{mi}+\\sum _mB_{im,k}\\omega _{mj}+\\sum _mB_{ij,m}\\omega _{mk}=\\sum _mB_{ij,km}\\omega _m.$ We have the following Ricci identities $B_{ij,kl}-B_{ij,lk}=\\sum _mB_{mj}R_{mikl}+\\sum _mB_{im}R_{mjkl}.$ Coefficients of Möbius invariants and Euclidean invariants are related by [18] $\\begin{split}B_{ij}&=\\rho ^{-1}(h_{ij}-H\\delta _{ij}),\\\\C_i&=-\\rho ^{-2}[e_i(H)+\\sum _j(h_{ij}-H\\delta _{ij})e_j(\\log \\rho )],\\\\A_{ij}&=-\\rho ^{-2}[Hess_{ij}(\\log \\rho )-e_i(\\log \\rho )e_j(\\log \\rho )-Hh_{ij}]\\\\&-\\frac{1}{2}\\rho ^{-2}(|\\nabla \\log \\rho |^2+H^2)\\delta _{ij},\\end{split}$ where $Hess_{ij}$ and $\\nabla $ are the Hessian matrix and the gradient with respect to $I=df\\cdot df$ .", "Then $A=\\rho ^2\\sum _{ij}A_{ij}\\theta _i\\otimes \\theta _j,~B=\\rho ^2\\sum _{ij}B_{ij}\\theta _i\\otimes \\theta _j,~\\Phi =\\rho \\sum _iC_i\\theta _i.$ We call eigenvalues of $(B_{ij})$ as Möbius principal curvatures of $f$ .", "Clearly the number of distinct Möbius principal curvatures is the same as that of its distinct Euclidean principal curvatures.", "Let $k_1,\\cdots ,k_n$ be the principal curvatures of $f$ , and $\\lbrace \\lambda _1,\\cdots ,\\lambda _n\\rbrace $ the corresponding Möbius principal curvatures, then the curvature sphere of principal curvature $k_i$ is $\\xi _i=\\lambda _iY+\\xi =\\left(\\frac{1+|f|^2}{2}k_i+f\\cdot e_{n+1},\\frac{1-|f|^2}{2}k_i-f\\cdot e_{n+1},k_if+e_{n+1}\\right).$ Note that $k_i=0$ if, and only if, $<\\xi _i,(1,-1,0,\\cdots ,0)>=0.$ This means that the curvature sphere of principal curvature $k_i$ is a hyperplane in $R^{n+1}$ ." ], [ "Examples of Möbius deformable hypersurfaces", "This section describes the construction of Möbius deformable hypersurfaces $M^n$ whose highest multiplicity of principal curvatures is $n-2$ .", "Example 3.1 Let $u: L^m\\longrightarrow R^{m+1}$ be an immersed hypersurface.", "We define the cylinder over $u$ in $R^{n+1}$ as $f=(u,id):L^m\\times R^{n-m}\\longrightarrow R^{m+1} \\times R^{n-m}=R^{n+1},$ where $id:R^{n-m}\\longrightarrow R^{n-m}$ is the identity map.", "Proposition 3.2 Let $u,\\bar{u}: L^2\\longrightarrow R^3$ be a Bonnet pair.", "Then the cylinders $f=(u,id):L^2\\times R^{n-2}\\longrightarrow R^{n+1}$ and $\\bar{f}=(\\bar{u},id)$ are Möbius deformations to each other.", "Let $\\eta $ be the unit normal vector of surface $u$ .", "Then $e_{n+1}=(\\eta ,\\vec{0})\\in R^{n+1}$ is the unit normal vector of hypersurface $f$ .", "The first fundamental form $I$ and the second fundamental form $II$ of hypersurface $f$ are given by $I=I_u+I_{R^{n-2}}, \\;\\; II=II_u,$ where $I_u,II_u$ are the first and second fundamental forms of $u$ , respectively, and $I_{R^{n-2}}$ denotes the standard metric of $R^{n-2}$ .", "Let $k_1,k_2$ be principal curvatures of surface $u$ .", "The principal curvatures of hypersurface $f$ are obviously $k_1,k_2,0,\\cdots ,0.$ The Möbius metric $g$ of hypersurface $f$ is $g=\\rho ^2I=\\frac{n}{n-1}(|II|^2-nH^2)I=\\left(4H_u^2-\\frac{2n}{n-1}K_u\\right)(I_u+I_{R^{n-2}}),$ where $H_u,K_u$ are the mean curvature of $u$ and Gauss curvature of $u$ , respectively.", "Since $\\bar{u}:L^2\\longrightarrow R^3$ share the same metric $I_u$ and mean curvature $H_u$ as $u$ , the cylinder $\\bar{f}=(\\bar{u},id):L^2\\times R^{n-2}\\longrightarrow R^{n+1}$ share the same factor $\\rho $ and Möbius metric, i.e.", "$g=\\bar{g}.$ Note that the correspondence between the Bonnet pair $u,\\bar{u}$ preserves the principal curvatures, yet NOT the principal directions.", "By (REF ) this is also true between $f,\\bar{f}$ .", "So we conclude that $\\bar{f}$ is a non-trivial Möbius deformation to $f$ .", "This completes the proof to Proposition REF .", "Example 3.3 Let $u:L^m\\longrightarrow S^{m+1}\\subset R^{m+2}$ be an immersed hypersurface.", "We define the cone over $u$ in $R^{n+1}$ as $\\begin{split}&f:L^m\\times R^+\\times R^{n-m-1}\\longrightarrow R^{n+1},\\\\&~~~~~~f(u,t,y)=(tu,y),\\end{split}$ Proposition 3.4 Let $u,\\bar{u}:L^2\\longrightarrow S^3$ be a Bonnet pair in the standard 3-sphere.", "Then the cone hypersurfaces $f:L^2\\times R^+\\times R^{n-3}\\longrightarrow R^{n+1}$ and $\\bar{f}$ over them are Möbius deformations to each other.", "The first and second fundamental forms of hypersurface $f$ are, respectively, $I=t^2I_u+I_{R^{n-2}}, \\;\\; II=t~II_u,$ where $I_u,II_u,I_{R^{n-2}}$ are understood as before.", "Let $k_1,k_2$ be principal curvatures of surface $u$ .", "The principal curvatures of hypersurface $f$ are $\\frac{1}{t}k_1,\\frac{1}{t}k_2,0,\\cdots ,0.$ Thus the Möbius metric $g$ of hypersurface $f$ is $\\begin{split}g=\\rho ^2I&=\\frac{1}{t^2}\\left[4H_u^2-\\frac{2n}{n-1}(K_u-1)\\right](t^2I_u+I_{R^{n-2}})\\\\&=\\left[4H_u^2-\\frac{2n}{n-1}(K_u-1)\\right](I_u+I_{H^{n-2}}),\\end{split}$ where $H_u,K_u$ are the mean curvature and Gauss curvature of $u$ , respectively, $I_{H^{n-2}}$ is the standard hyperbolic of $R^{n-2}_+=R^+\\times R^{n-3}$ .", "Since $\\bar{u}:L^2\\longrightarrow S^3$ share the same metric $I_u$ and mean curvature $H_u$ as $u$ , the cone over $\\bar{u}$ $\\bar{f}:L^2\\times R^+\\times R^{n-3}\\longrightarrow R^{n+1}$ share the same Möbius metric, i.e.", "$g=\\bar{g}.$ By the same reason in the proof to Proposition REF , we know that their principal directions do NOT correspond.", "So they are genuine deformations to each other.", "This completes the proof to Proposition REF .", "Example 3.5 Let $R^{m+1}_+=\\lbrace (x_1,\\cdots ,x_m,x_{m+1})\\in R^{m+1}|x_{m+1}>0\\rbrace $ be the upper half-space endowed with the standard hyperbolic metric $ds^2=\\frac{1}{x_{m+1}^2}\\sum _{i=1}^m dx_i^2.$ Let $u=(x_1,\\cdots ,x_{m+1}):M^m\\longrightarrow R^{m+1}_+$ be an immersed hypersurface.", "We define rotational hypersurface over $u$ in $R^{n+1}$ as $\\begin{split}&f:L^m\\times S^{n-m}\\longrightarrow R^{n+1},\\\\&f(x_1,\\cdots ,x_{m+1},\\phi )=(x_1,\\cdots ,x_m,x_{m+1}\\phi ),\\end{split}$ where $\\phi :S^{n-m}\\longrightarrow R^{n-m+1}$ is the standard sphere.", "Proposition 3.6 Let $u,\\bar{u}:L^2\\longrightarrow R^3_+$ be a Bonnet pair in the hyperbolic 3-space.", "Then the rotational hypersurfaces $f=(x_1,x_2,x_3\\phi ):L^2\\times S^{n-2}\\longrightarrow R^{n+1}$ and $\\bar{f}=(\\bar{x}_1,\\bar{x}_2,\\bar{x}_3\\phi )$ are Möbius deformations to each other.", "Let $R^4_1$ be the Lorentz space with inner product $<y,y>=-y_1^2+y_2^2+y_3^2+y_4^2,\\;\\; y=(y_1,y_2,y_3,y_4).$ Let $H^3=\\lbrace y\\in R^4_1|<y,y>=-1,y_1>0\\rbrace $ be the hyperbolic space.", "Introduce isometry $\\tau :R^3_+\\longrightarrow H^3$ as below: $\\tau (x_1,x_2,x_3)=\\left(\\frac{1+x_1^2+x_2^2+x_3^2}{2x_3},\\frac{1-x_1^2-x_2^2-x_3^2}{2x_3},\\frac{x_1}{x_3},\\frac{x_2}{x_3}\\right).$ The inverse $\\tau ^{-1}:H^3\\longrightarrow R^3_+$ is $\\tau ^{-1}(y_1,y_2,y_3,y_4)=(\\frac{y_3}{y_1+y_2},\\frac{y_4}{y_1+y_2},\\frac{1}{y_1+y_2}).$ Let $\\eta $ be the unit normal vector of surface $u$ in $R^3_+$ .", "Write $\\eta =(\\eta _1,\\eta _2,\\eta _3).$ Since $\\eta $ is the unit normal vector,then $\\frac{\\eta _1^2+\\eta _2^2+\\eta _3^2}{x_3^2}=1.$ Thus the unit normal vector of hypersurface $f$ in $R^{n+1}$ is $\\xi =\\frac{1}{x_3}(\\eta _1,\\eta _2,\\eta _3\\phi ).$ The first fundamental form of $u$ is $I_u=\\frac{1}{x_3^2}(dx_1\\cdot dx_1+dx_2\\cdot dx_2+dx_3\\cdot dx_3).$ The second fundamental form of $u$ is $II_u=-<\\tau _*(du),\\tau _*(d\\eta )>=\\frac{1}{x_3^2}(dx_1\\cdot d\\eta _1+dx_2\\cdot d\\eta _2+dx_3\\cdot d\\eta _3)-\\frac{\\eta _3}{x_3}I_u.$ Now we can write out the first and the second fundamental forms of $f$ : $I=df\\cdot df=x_3^2(I_u+I_{S^{n-2}}),~~II=x_3II_u-\\eta _3I_u-\\eta _3I_{s^{n-2}},$ where $I_{S^{n-2}}$ is the standard metric of $S^{n-2}$ .", "Let $k_1,k_2$ be principal curvatures of $u$ .", "Then principal curvatures of hypersurface $f$ are $\\frac{k_1}{x_3}-\\frac{\\eta _3}{x_3^2}~,~\\frac{k_2}{x_3}-\\frac{\\eta _3}{x_3^2}~,~\\frac{-\\eta _3}{x_3^2}~,\\cdots ,\\frac{-\\eta _3}{x_3^2}~.$ Thus $\\rho ^2=\\frac{n}{n-1}(|II|^2-nH^2)=\\frac{1}{x^2_3}\\left[4H_u^2-\\frac{2n}{n-1}(K_u+1)\\right],$ where $H_u,K_u$ are the mean curvature and Gauss curvature of $u$ , respectively.", "So the Möbius metric of hypersurface $f$ is $g=\\rho ^2I=\\left[4H_u^2-\\frac{2n}{n-1}(K_u+1)\\right](I_u+I_{S^{n-2}}).$ Since $u$ and $\\bar{u}$ are a pair of Bonnet surfaces, $H_u=H_{\\bar{u}},K_u=K_{\\bar{u}},I_u=I_{\\bar{u}}$ , thus $\\bar{f}=(\\bar{x}_1,\\bar{x}_2,\\bar{x}_3\\phi ):L^2\\times S^{n-2}\\longrightarrow R^{n+1}$ , the rotational hypersurface over $\\bar{u}$ , is endowed with the same Möbius metric $g$ .", "Similar to previous discussions we know that they are NOT congruent.", "This completes the proof to Proposition REF .", "Remark 3.7 We note that the Möbius metric $g$ in these three cases (REF )(REF )(REF ) could be unified in a single formula: $g=\\left[4H_u^2-\\frac{2n}{n-1}(K_u+c)\\right](I_u+I_{N^{n-2}(c)}).$ Here $H_u,K_u,I_u$ are the mean curvature, the Gauss curvature and the first fundamental form of the surface $u:L^2\\rightarrow N^3(-c)$ in a three dimensional space form of constant curvature $-c$ ; $I_{N^{n-2}(c)}$ is the Riemannian metric of a $(n-2)-$ dimensional space form of constant curvature $c$ .", "This will be used in Section 9 to show that any Möbius deformation to any example in these three propositions arises in this way.", "In other words, the possible deformations are as many as that of the corresponding Bonnet surface." ], [ "Hypersurfaces with constant Möbius curvature: deformations and classification", "As pointed out in the introduction, hypersurfaces with constant Möbius sectional curvature form a new class of deformable hypersurfaces.", "In this section, we list hypersurfaces with constant Möbius curvature, i.e., constant sectional curvature with respect to the Möbius metric $g$ , and compute the Möbius invariants.", "Then we give a new proof to the classification of such hypersurfaces using a reduction theorem REF in Section 5.", "Example 4.1 The cylinder in $R^{n+1}$ over $\\gamma (s)\\subset R^2$ is defined by $f(s,id)=(\\gamma (s),id):~I\\times R^{n-1}\\longrightarrow R^{n+1},$ where $id:R^{n-1}\\longrightarrow R^{n-1}$ is the identity mapping.", "Remark 4.2 This is exactly Example REF when $m=1$ .", "The first fundamental form $I$ and the second fundamental form $II$ of hypersurface $f$ are, respectively, $ I=ds^2+I_{R^{n-1}}, \\;\\; II=\\kappa (s) ds^2,$ where $\\kappa (s)$ is the geodesic curvature of $\\gamma \\subset R^2$ , $s$ is the arc-length parameter, and $I_{R^{n-1}}$ is the standard Euclidean metric of $R^{n-1}$ .", "So we have $(h_{ij})=\\operatorname{diag}(\\kappa ,0,\\cdots ,0)~,~H=\\frac{\\kappa }{n}~,~\\rho =\\kappa ~.$ Thus the Möbius metric $g$ of hypersurface $f$ is $g=\\rho ^2 I=\\kappa (s)^2(ds^2+I_{R^{n-1}}).$ The Möbius invariants of $f$ under an orthonormal frame (consisting of principal directions) can be obtained as below using (REF ): $\\begin{split}&C_1=-\\frac{\\kappa _s}{\\kappa ^2}~, ~ C_2=\\cdots =C_n=0,\\\\&(B_{ij})=\\operatorname{diag}\\left(\\frac{n-1}{n},\\frac{-1}{n},\\cdots ,\\frac{-1}{n}\\right),\\\\&(A_{ij})=\\operatorname{diag}(a_1,a_2,\\cdots ,a_2),\\end{split}$ where $a_1=-\\dfrac{\\kappa _{ss}}{\\kappa ^3}+\\dfrac{3}{2}\\dfrac{(\\kappa _s)^2}{\\kappa ^4}+\\dfrac{2n-1}{2n^2}~,~a_2=-\\dfrac{1}{2}\\left[\\dfrac{(\\kappa _s)^2}{\\kappa ^4}+\\dfrac{1}{n^2}\\right].$ Example 4.3 The cone in $R^{n+1}$ over $\\gamma (s)\\subset S^2(1)\\subset R^3$ is defined by $f(s,t,id)=(t\\gamma (s),id):~I\\times R^{+}\\times R^{n-2}\\longrightarrow R^{n+1},$ where $id:R^{n-2}\\longrightarrow R^{n-2}$ is identity mapping and $R^{+}=\\lbrace t|~t>0\\rbrace $ .", "Remark 4.4 This is exactly Example REF when $m=1$ .", "The first and second fundamental forms of hypersurface $f$ are $I=t^2ds^2+I_{R^{n-1}}~, \\;\\; II=t\\kappa (s) ds^2.$ So we have $(h_{ij})=\\operatorname{diag}\\left(\\frac{\\kappa }{t},0,\\cdots ,0\\right)~,~H=\\frac{\\kappa }{nt}~,~\\rho =\\frac{\\kappa }{t}~.$ Thus the Möbius metric $g$ of hypersurface $f$ is $g=\\rho ^2I=\\frac{\\kappa (s)^2}{t^2}\\left(t^2ds^2+I_{R^{n-1}}\\right)=\\kappa (s)^2(ds^2+I_{H^{n-1}}),$ where $I_{H^{n-1}}$ is the standard hyperbolic metric of $H^{n-1}(-1)$ .", "The Möbius invariants of $f$ under an orthonormal frame (consisting of principal directions) can be obtained similarly: $\\begin{split}&C_1=-\\frac{\\kappa _s}{\\kappa ^2}~,~ C_2=\\cdots =C_n=0,\\\\&(B_{ij})=\\operatorname{diag}\\left(\\frac{n-1}{n},\\frac{-1}{n},\\cdots ,\\frac{-1}{n}\\right),\\\\&(A_{ij})=\\operatorname{diag}(a_1,a_2,\\cdots ,a_2),\\end{split}$ where $a_1=-\\dfrac{\\kappa _{ss}}{\\kappa ^3}+\\dfrac{3}{2}\\dfrac{(\\kappa _s)^2}{\\kappa ^4}+\\dfrac{1}{2\\kappa ^2}+\\dfrac{2n-1}{2n^2}~,~a_2=-\\dfrac{1}{2}\\left[\\dfrac{(\\kappa _s)^2}{\\kappa ^4}+\\dfrac{1}{\\kappa ^2}+\\dfrac{1}{n^2}\\right].$ Example 4.5 The rotational hypersurface in $R^{n+1}$ over $\\gamma (s)\\subset R^2_+=\\lbrace (x,y)\\in R^2|~y>0\\rbrace \\subset R^3$ is defined by $f(x,y,\\theta )=(x,y\\theta ):~I\\times S^{n-1}\\longrightarrow R^{n+1},$ where $\\theta :S^{n-1}\\longrightarrow R^{n}$ is the standard immersion of a round sphere, $R^2_+$ is regarded as the Poincare half plane with the hyperbolic metric $ds^2=\\frac{1}{y^2}(dx^2+dy^2)$ .", "Remark 4.6 This is exactly Example REF when $m=1$ .", "In the Poincare half plane, denote the covariant differentiation of the hyperbolic metric as $D$ .", "Choose orthonormal frames $e_1=y\\frac{\\partial }{\\partial x},e_2=y\\frac{\\partial }{\\partial y}$ .", "It is easy to find $D_{e_1}e_1=e_2~,~D_{e_1}e_2=-e_1~,~D_{e_2}e_1=D_{e_2}e_2=0.$ For $\\gamma (s)=((x(s),y(s))\\subset R^2_+$ let $x^{\\prime }$ denote derivative $\\partial x/\\partial s$ and so on.", "Choose the unit tangent vector $\\alpha =\\frac{1}{y}(x^{\\prime }(s)e_1+y^{\\prime }(s)e_2)$ and the unit normal vector $\\beta =\\frac{1}{y}(-y^{\\prime }(s)e_1+x^{\\prime }(s)e_2)$ .", "The geodesic curvature is computed via $\\kappa (s)=\\langle D_\\alpha \\alpha ,\\beta \\rangle =\\frac{x^{\\prime }y^{\\prime \\prime }-x^{\\prime \\prime }y^{\\prime }}{y^2}+\\frac{x^{\\prime }}{y}.$ After these preparation, we see that the rotational hypersurface $f(x,y,\\theta )=(x,y\\theta )$ has differential $df=(x^{\\prime }ds,y^{\\prime }\\theta ds+y d\\theta )$ and unit normal vector $\\eta =\\frac{1}{y}(-y^{\\prime },x^{\\prime }\\theta ).$ Thus the first and second fundamental forms of hypersurface $f$ are $I=df\\cdot df=y^2(ds^2+I_{S^{n-1}})~,~ II=-df\\cdot d\\eta =(y\\kappa -x^{\\prime })ds^2-x^{\\prime }I_{S^{n-1}},$ where $I_{S^{n-1}}$ is the standard metric of $S^{n-1}(1)$ .", "Thus principal curvatures are $\\frac{\\kappa y-x^{\\prime }}{y^2},\\frac{-x^{\\prime }}{y^2},\\cdots ,\\frac{-x^{\\prime }}{y^2}.$ So $\\rho =\\frac{\\kappa }{y}$ , and the Möbius metric of $f$ is $g=\\rho ^2I=\\kappa ^2(ds^2+I_{S^{n-1}}).$ The coefficients of Möbius invariants are: $\\begin{split}&C_1=-\\frac{\\kappa _s}{\\kappa ^2}~,~ C_2=\\cdots =C_n=0,\\\\&(B_{ij})=\\operatorname{diag}\\left(\\frac{n-1}{n},\\frac{-1}{n},\\cdots ,\\frac{-1}{n}\\right),\\\\&(A_{ij})=\\operatorname{diag}(a_1,a_2,\\cdots ,a_2),\\end{split}$ where $a_1=\\dfrac{\\kappa _{ss}}{\\kappa ^3}-\\dfrac{5}{2}\\dfrac{(\\kappa _s)^2}{\\kappa ^4}-\\dfrac{1}{2\\kappa ^2}+\\dfrac{2n-1}{2n^2}~,~a_2=-\\dfrac{1}{2}\\left[\\dfrac{(\\kappa _s)^2}{\\kappa ^4}-\\dfrac{1}{\\kappa ^2}+\\dfrac{1}{n^2}\\right].$ Lemma 4.7 The Möbius metric of those hypersurfaces in Examples (REF ), (REF ) and (REF ) are of the warped-product form $g=\\kappa ^2(s)\\left(ds^2+I_{-\\epsilon }^{n-1}\\right),$ where $I_{-\\epsilon }^{n-1}$ is the metric of $n-1$ dimensional space form of constant curvature $-\\epsilon $ .", "This metric (REF ) is of constant sectional curvature $c$ if, and only if, the function $\\kappa (s)$ satisfies $\\left[\\frac{d}{ds}\\frac{1}{\\kappa }\\right]^2+\\epsilon \\left[\\frac{1}{\\kappa }\\right]^2=-c.$ The proof is an easy exercise and we omit it at here.", "Definition 4.8 We call a curve $\\gamma $ the curvature-spiral in a $2-$ dimensional space form $N^2(\\epsilon )=R^2,S^2,H^2$ (of Gauss curvature $\\epsilon =0,1,-1$ respectively), if its geodesic curvature $\\kappa (s)$ is not constant and satisfies (REF ).", "Note that (REF ) is equivalent to the harmonic oscillator equation for the function $\\kappa (s)$ : $(1/\\kappa )^{\\prime \\prime }+\\epsilon /\\kappa =0.$ It is easy to see that for fixed $\\epsilon , c$ the solution curve is unique (because $N^2(\\epsilon )$ is a two-point homogeneous space).", "In particular, when $\\epsilon =0$ , $N^2(\\epsilon )=R^2$ , the corresponding $\\gamma $ is a circle or a logarithmic spiral, and the cylinder $\\gamma \\times R^{n-1}$ is called the circular cylinder and the spiral cylinder [24], respectively.", "Theorem 4.9 ([13]) Let $f:M^n\\rightarrow R^{n+1}$ $(n\\ge 3)$ be an umbilic free immersed hypersurface with constant Möbius curvature $c$ .", "If $n=3$ we assume that $f$ has two distinct principal curvatures.", "Then locally $f$ is Möbius equivalent to one of the following examples: $(i)$ the circular cylinder (where $c=0$ ) or the spiral cylinder (where $c<0$ ); $(ii)$ a cone over a curvature-spiral in a 2-sphere (where $c<0$ ); $(iii)$ a rotation hypersurface over a curvature-spiral in a hyperbolic 2-plane (the constant curvature $c$ could be positive, negative or zero).", "Choose an orthonormal frame with respect to $g$ so that $(B_{ij})$ is diagonal.", "According to the following Remark REF , $f$ has two distinct principal curvatures, one of which is simple.", "The assumption of constant curvature for $g$ implies the Ricci curvature $R_{ij}=0$ for $i\\ne j$ .", "From the integrability equation () we deduce that $(A_{ij})$ is also diagonal.", "Thus the second reduction theorem REF in the next section says that the Möbius form is closed and $f$ is reducible.", "Invoking Lemma REF we finish the proof.", "Remark 4.10 Clearly hypersurfaces with constant Möbius curvature are conformally flat.", "Equivalently, when the dimension $n\\ge 4$ there must be a principal curvature of multiplicity $n-1$ everywhere (and the hypersurface is the envelop of a one-parameter family of $(n-1)$ dimensional spheres).", "On the other hand, a 3-dimensional hypersurface $f:M^3\\rightarrow R^4$ with constant Möbius sectional curvature may have three distinct principal curvatures.", "We have finished a classification of such examples which will be published later [20].", "Let's see for fixed $c$ how many different (global) examples exist.", "If $\\epsilon =0,~N^2(\\epsilon )=R^2$ , without loss of generality the solution to (REF ) is written as $\\kappa =1/(\\sqrt{-c}s) .~~~~~~~~~~~~~~~~\\text{(logarithmic-spiral)}$ When $\\epsilon =1,~N^2(\\epsilon )=S^2$ , without loss of generality the solution to (REF ) is written as $\\kappa =1/(\\sqrt{-c}\\sin s) .~~~~~~~~~~~~~~~~\\text{(sin-spiral)}$ When $\\epsilon =-1,~N^2(\\epsilon )=H^2(-1)$ , there are three different possibilities: $\\kappa =1/(\\sqrt{-c}\\sinh s),~~~~~~~~&&\\text{(sinh-spiral)}\\\\\\kappa =1/(\\sqrt{c}\\cosh s),~~~~~~~~~~&&\\text{(cosh-spiral)}\\\\\\kappa =e^s.~~~~~~~~~~~~~~~~~~&&\\text{(exp-spiral)}$ When $c>0$ we have a unique example (cosh-spiral).", "Yet this example is not homogeneous and should not be viewed as Möbius rigid according to Remark REF .", "In contrast, for hypersurfaces of Möbius curvature $c<0$ we have three non-congruent hypersurfaces: the spiral cylinder, the cone hypersurface, and the rotational hypersurface over the sinh-spiral.", "We conclude that either of them (in particular, the spiral cylinder) is Möbius deformable.", "(See Remark REF and REF .)", "When $c=0$ , according to our theorem, there exist two non-congruent examples: the circular cylinder and the rotational hypersurfaces over the exp-spiral as in equation ().", "So either of them is deformable." ], [ "The Reduction Theorem", "In this section we establish a criterion in terms of Möbius invariants for a hypersurface to be cylinders, cones and rotational hypersurfaces (Examples (REF )(REF )(REF )).", "This is used in the previous and the final section.", "Theorem 5.1 (Reduction Theorem) Let $f:M^n\\rightarrow R^{n+1} (n\\ge 3)$ be an umbilic free immersed hypersurface, whose principal curvatures have constant multiplicities.", "We diagonalize the Möbius second fundamental form under an orthonormal frame $\\lbrace E_1,E_2,\\cdots ,E_n\\rbrace $ with respect to the Möbius metric $g$ : $B_{ij}=\\mathrm {diag}\\lbrace \\lambda _1,\\cdots ,\\lambda _m,\\mu ,\\cdots ,\\mu \\rbrace .$ Assume: $(1)$  $\\lambda _1,\\cdots ,\\lambda _m$ are distinct from $\\mu $ .", "$(2)$  $2\\le m\\le n-2$ .", "(So the multiplicity of $\\mu $ is $n-m$ and $2\\le n-m\\le n-2$ .)", "$(3)$  $B_{pq,\\alpha }=0,~C_{\\alpha }=0,~~ ~1\\le p,q \\le m,~m+1\\le \\alpha \\le n.$ Then $f$ is Möbius congruent to one of the examples (REF ),(REF ) and (REF ).", "Let $\\lbrace Y,N,Y_1,\\cdots ,Y_n,\\xi \\rbrace $ be a moving frame in $R^{n+3}_1$ (see Section 2).", "In the proof below we adopt the convention on the range of indices as below: $1 \\le p,q,r,s,t \\le m,~~m+1 \\le \\alpha ,\\beta ,\\gamma \\le n,~~1 \\le i,j,k,l\\le n.$ Without loss of generality we make a new choice of frame vectors such that $A_{\\alpha \\beta }=a_{\\alpha }\\delta _{\\alpha \\beta }.$ Applying $dB_{ij}+\\sum _kB_{kj}\\omega _{ki}+\\sum _kB_{ik}\\omega _{kj}=\\sum _kB_{ij,k}\\omega _k$ for off-diagonal element $B_{\\alpha \\beta }$ ($\\alpha \\ne \\beta $ ) and using the fact $B_{\\alpha \\alpha }=B_{\\beta \\beta }=\\mu ,B_{\\alpha \\beta }=0$ we get $B_{\\alpha \\beta ,k}=0=B_{k\\alpha ,\\beta },~~\\forall ~\\alpha \\ne \\beta ,~1\\le k\\le n.$ The second equality is by the integrability equation.", "Since $n-m\\ge 2$ , we can always choose indices $\\alpha \\ne \\beta $ .", "Then by integrability equation and the assumption $C_{\\beta }=0$ one has $E_{\\beta }(\\mu )=B_{\\alpha \\alpha ,\\beta }=B_{\\alpha \\beta ,\\alpha }+\\delta _{\\alpha \\alpha }C_{\\beta }-\\delta _{\\alpha \\beta }C_{\\alpha }=C_{\\beta }=0,~~\\forall \\beta .$ Here $B_{\\alpha \\beta ,\\alpha }=0$ due to (REF ).", "Similarly we have $B_{p\\alpha ,q}=B_{pq,\\alpha }+\\delta _{p\\alpha }C_{q}-\\delta _{pq}C_{\\alpha }=B_{pq,\\alpha }$ and $B_{p\\alpha ,\\alpha }=B_{\\alpha \\alpha ,p}-C_{p} =E_{p}(\\mu )-C_{p}$ .", "Together with the assumption $B_{pq,\\alpha }=0$ we summarize that $B_{pq,\\alpha }=B_{p\\alpha ,q}=0,~B_{p\\alpha ,\\alpha }=E_{p}(\\mu )-C_{p},~~\\forall ~p,q,\\alpha .$ Now with the help of (REF ) and (REF ) we compute the covariant derivatives of off-diagonal components $B_{p\\alpha }$ and find $\\omega _{p\\alpha }=\\frac{B_{p\\alpha ,\\alpha }}{\\lambda _p-\\mu }\\omega _{\\alpha },~~\\forall ~p,\\alpha .$ Differentiating once more we obtain the curvature tensor.", "Compare the coefficient of the component $\\omega _p\\wedge \\omega _q$ for any given $p\\ne q$ We find that $R_{p\\alpha pq}=0.$ (This is the only place where we use the assumption $m\\ge 2$ , to guarantee that there exist such $p\\ne q$ ).", "From the integrability equation () we get $A_{q\\alpha }=0~,~~~1\\le q\\le m,m+1\\le \\alpha \\le n.$ Similarly by comparing the component $\\omega _p\\wedge \\omega _{\\alpha }$ we observe that $R_{p\\alpha p\\alpha }$ is independent of $\\alpha $ (here we use (REF )).", "Equation () yields $R_{p\\alpha p\\alpha }=\\lambda _p\\mu +A_{pp}+A_{\\alpha \\alpha }$ and $A_{\\alpha \\alpha }=a,~\\forall ~\\alpha ~.$ Next we compute the covariant derivatives of tensor $A$ and $C$ .", "By the condition $C_{\\alpha }=0$ and the integrability equation (REF ) $A_{ij,k}-A_{ik,j}=B_{ik}C_{j}-B_{ij}C_{k}$ , $E_{\\alpha }(a)=E_{\\alpha }(A_{\\beta \\beta })=A_{\\beta \\beta ,\\alpha }=A_{\\alpha \\beta ,\\beta }=0,~\\forall ~\\alpha \\ne \\beta .$ As a consequence of (REF ) and $dC_i+\\sum _kC_k\\omega _{ki}=\\sum _kC_{i,k}\\omega _k$ we get that $E_{\\alpha }(C_p)=C_{p,\\alpha }=C_{\\alpha ,p}=0,~\\forall ~p,\\alpha .$ Let's look at the geometric meaning of these results.", "From the formula in (REF ) we know that distributions $D_1\\triangleq \\mathrm {Span}\\lbrace E_p|1\\le p\\le m\\rbrace ,~~D_2\\triangleq \\mathrm {Span}\\lbrace E_{\\alpha }|m+1\\le \\alpha \\le n\\rbrace ,$ are integrable.", "Any integral submanifold of distribution $D_1$ is a $m$ -dimensional submanifold.", "On the other hand, along any integral submanifold of $D_2$ the hypersurface $Y$ is tangent to $F\\triangleq \\mu Y+\\xi ,$ the principal curvature sphere of multiplicity $n-m$ .", "Using (REF ), $E_p(\\mu )=B_{\\alpha \\alpha ,p}=B_{p\\alpha ,\\alpha }+C_{p}$ and the structure equation it is easy to get that $E_{\\alpha }(F)=0,~E_p(F)=B_{p\\alpha ,\\alpha }Y+(\\mu -\\lambda _p)Y_p.$ Then principal curvature sphere $F$ induces a $m$ -dimensional submanifold in the de-Sitter space $S^{n+2}_1$ $F:\\widetilde{M}^m=M^n/L\\rightarrow S^{n+2}_1,$ where fibers $L$ are integral submanifolds of distribution $D_2$ .", "In other words, $F$ form a $m$ -parameter family of $n$ -spheres enveloped by the hypersurface $Y$ .", "The next crucial observation is that $F$ is located in a fixed $(m+2)$ -dimensional linear subspace of $R^{n+3}_1$ .", "To show that we compute the repeated derivatives of $F$ , which contains all information of the envelope $Y$ .", "Straightforward yet tedious computation shows that the frames of $V_1\\triangleq \\text{Span}\\lbrace F,E_1(F),\\cdots ,E_m(F),P\\rbrace ,$ $\\text{where}~~~~~P\\triangleq A_{\\alpha \\alpha }Y-N+\\sum _{p=1}^m\\frac{B_{p\\alpha ,\\alpha }}{(\\mu -\\lambda _p)^2}E_p(F) +\\mu F,$ satisfy a linear first order PDE system.", "Hence these vectors, including $F$ itself, are contained in a fixed $(m+2)$ -dimensional subspace $V_1$ endowed with degenerate, Lorentzian, or positive definite inner product.", "This agrees with the geometry of cylinders, cones, and rotational hypersurfaces (see examples (3.1),(3.3),(3.5)), where the principal curvature sphere $F$ is orthogonal to a $(n-m+1)$ -parameter family of hyperplanes/hyperspheres.", "Moreover, the orthogonal complement $V_1^{\\perp }$ of $dim=n-m+1$ contains all $Y_{\\alpha },~~ m+1\\le \\alpha \\le n$ .", "The final fact above inspires us to proceed in an alternative and easier way.", "Differentiate any given $Y_{\\alpha }$ and modulo components in the subspace $\\text{Span}\\lbrace Y_{\\gamma },~m+1\\le \\gamma \\le n\\rbrace $ .", "By (REF )(REF )(REF ) one finds $E_i(Y_{\\alpha })&=&-A_{\\alpha i}Y-\\delta _{\\alpha i}N+\\sum {_{j}}~\\omega _{\\alpha j}(E_i)Y_j+B_{\\alpha i}\\xi \\\\&=&\\left\\lbrace \\begin{array}{ll}-T~(\\text{mod}~ Y_{\\gamma }),~\\text{when}~i=\\alpha ~;\\\\0~(\\text{mod}~ Y_{\\gamma }),~~\\text{otherwise~.}", "\\\\\\end{array}\\right.$ where $T\\triangleq A_{\\alpha \\alpha }Y+N+\\sum _{p=1}^m\\frac{B_{p\\alpha ,\\alpha }}{\\lambda _p-\\mu }Y_p-\\mu \\xi $ is independent of $\\alpha $ by (REF )(REF ).", "Then we assert that the subspace $V_2\\triangleq \\text{Span}\\lbrace T,Y_{\\gamma }|m+1\\le \\gamma \\le n\\rbrace $ is parallel along $M$ .", "According to our previous computation, $E_i(Y_{\\alpha })=0~(\\text{mod}~V_2),~\\forall \\alpha ~.$ So we need only to consider $E_i(T)$ .", "Fix $i$ and choose $\\alpha \\ne i$ .", "(Such $\\alpha $ exists by the assumption $n-m\\ge 2$ , which is the third and final time that we use it.", "Recall that this condition has been used to derive (REF )(REF ), i.e.", "$E_{\\alpha }(\\mu )=0=E_{\\alpha }(a)$ .)", "Rewrite the first equality of (REF ) as $T=-E_{\\alpha }(Y_{\\alpha })+\\sum {_{\\gamma }}(\\cdots )Y_{\\gamma }.$ By this clever choice of index $\\alpha $ we may prove in a unified way that $E_i(T)&=&-E_i(E_{\\alpha }(Y_{\\alpha }))+\\sum {_{\\gamma }}(\\cdots )E_i(Y_{\\gamma })~~(\\text{mod}~Y_{\\gamma })\\\\&=&-E_{\\alpha }(E_i(Y_{\\alpha }))+[E_{\\alpha },E_i](Y_{\\alpha })+\\sum {_{\\gamma }}(\\cdots )E_i(Y_{\\gamma })~~(\\text{mod}~Y_{\\gamma })\\\\&=&-E_{\\alpha }(\\sum {_{\\beta }} (\\cdots )Y_{\\beta }))+[E_{\\alpha },E_i](Y_{\\alpha })+\\sum {_{\\gamma }}(\\cdots )E_i(Y_{\\gamma })~~(\\text{mod}~Y_{\\gamma })\\\\&=&0~~(\\text{mod}~V_2).$ This verifies our previous assertion.", "More precisely, we have $E_p(T)=\\frac{B_{p\\alpha ,\\alpha }}{\\lambda _p-\\mu }T,~~E_{\\alpha }(T)=QY_{\\alpha },~~\\forall ~p,\\alpha $ where $Q\\triangleq \\langle T,T\\rangle =2A_{\\alpha \\alpha }+\\mu ^2+\\sum _{p=1}^m\\frac{B_{p\\alpha ,\\alpha }^2}{(\\lambda _p-\\mu )^2},$ satisfies $E_p(Q)=\\frac{2B_{p\\alpha ,\\alpha }}{\\lambda _p-\\mu }Q,\\;E_{\\alpha }(Q)=0.$ One could verify (REF ) directly.", "But the easy way is using $\\langle T,Y_{\\alpha }\\rangle =0$ and (REF ) to get $\\langle E_i(T),Y_{\\alpha }\\rangle =-\\langle T,E_i(Y_{\\alpha })\\rangle =\\left\\lbrace \\begin{array}{ll}Q~,~\\text{when}~i=\\alpha ;\\\\0~,~\\text{otherwise~.", "}\\\\\\end{array}\\right.$ This implies $E_p(T)\\parallel T$ for any $1\\le p\\le m$ .", "Then $E_p(T)$ as in (REF ) is derived by differentiating (REF ) and comparing the $\\xi $ component with $T$ .", "The formula for $E_p(Q)$ in (REF ) follows directly.", "On the other hand, we know $\\langle E_{\\alpha }(T),T\\rangle =\\frac{1}{2}E_{\\alpha }(Q)=0,$ where we used (REF ) and its consequence $[E_p,E_{\\alpha }]\\in D_2$ together with (REF )(REF )(REF ).", "Combined with (REF ) we have $E_{\\alpha }(T)=QY_{\\alpha }$ .", "Regarding (REF ) as a linear first-order ODE for $Q$ we see that $Q\\equiv 0$ or $Q\\ne 0$ on the connected manifold $M^n$ .", "Thus there are three possibilities for the induced metric on the fixed subspace $V_2\\subset \\mathbb {R}^{n+3}_1$ .", "Case 1, $Q=0$ on $M^n$ ; $V_2$ is endowed with a degenerate inner product.", "In this case, $\\langle T,T\\rangle =0$ .", "By (REF ), $E_p(T)\\parallel T$ , so $T$ determines a fixed light-like direction in $\\mathbb {R}^{n+3}_1$ , which we may take to be $[T]=[1,-1,0,\\cdots ,0]\\in \\mathbb {R}^{n+3}_1.$ This corresponds to $\\infty $ , the point at infinity of $\\mathbb {R}^{n+1}$ .", "Choose space-like vectors $X_{m+1},\\cdots ,X_n$ so that $V_2=\\text{Span}\\lbrace T,X_{m+1},\\cdots ,X_n\\rbrace $ .", "We interpret the geometry of hypersurface $f:M^n\\rightarrow R^{n+1}$ as below: 1) Any $X_{\\alpha }$ determines a hyperplane in $\\mathbb {R}^{n+1}$ because $\\langle T,X_{\\alpha }\\rangle =0$ ; 2) $\\rm {Span}\\lbrace X_{\\alpha }, (m+1\\le \\alpha \\le n)\\rbrace $ corresponds to a (n-m)-dimensional plane $\\Sigma $ in $\\mathbb {R}^{n+1}$ .", "3) $F$ is a $m$ -parameter family of hyperplanes orthogonal to the fixed plane $\\Sigma $ .", "$f(M)$ , as the envelope of this family of hyperplanes $F$ , is clearly a cylinder over a hypersurface $\\widetilde{M}\\subset \\mathbb {R}^{m+1}$ .", "Case 2, $Q<0$ on $M^n$ ; $V_2$ is a Lorentz subspace in $\\mathbb {R}^{n+3}_1$ .", "Fix a basis $\\lbrace P_0,P_{\\infty },X_{m+2},\\cdots ,X_n\\rbrace $ of the $(n-m+1)$ -dimensional $V_2$ so that $P_0,P_{\\infty }$ are light-like.", "Without loss of generality we may assume $P_0=(1,1,0,\\cdots ,0),~~P_{\\infty }=(1,-1,0,\\cdots ,0).$ Using the stereographic projection $\\sigma $ they correspond to the origin $O$ and the point at infinity $\\infty $ of the flat $\\mathbb {R}^{n+1}$ , respectively.", "We interpret $F$ and $V_2$ in terms of the geometry of $\\mathbb {R}^{n+1}$ : 1) $\\rm {Span}\\lbrace X_{\\alpha }:$ $m+2\\le \\alpha \\le n\\rbrace $ corresponds to a coordinate plane $\\mathbb {R}^{n-m-1}\\subset \\mathbb {R}^{n+1}$ , because $X_{\\alpha }$ must be space-like and orthogonal to $P_0,P_{\\infty }$ .", "2) $F$ is a $m$ -parameter family of hyperplanes (passing $O$ and $\\infty $ ) and orthogonal to this fixed $\\mathbb {R}^{n-m-1}$ .", "Based on the fact 1), $f(M)$ , the envelope of $F$ , is a cylinder over a $(m+1)$ -dimensional hypersurface in $\\mathbb {R}^{m+2}$ (the orthogonal complement of the previous $\\mathbb {R}^{n-m-1}$ ); moreover, the fact 2) means that $f(M)$ is a cone (with vertex $O$ ) over a $m$ -dimensional hypersurface in $S^{m+1}$ .", "Case 3, $Q>0$ on $M^n$ ; $V_2$ is a space-like subspace.", "Without loss of generality we assume that $P_{\\infty }=(1,-1,0,\\cdots ,0)$ is contained in the orthogonal complement of $V_2$ .", "As before we make the following interpretation: 1) $V_2$ corresponds to a $m$ -dimensional plane $\\mathbb {R}^m\\subset \\mathbb {R}^{n+1}$ .", "2) $F$ is a $(n-m)$ -parameter family of hyper-spheres orthogonal to this fixed plane $\\mathbb {R}^m$ with centers locating on it.", "Thus $F$ envelops a rotational hypersurface $f(M)$ (over a hypersurface in half-space $\\mathbb {R}_+^{m+1}$ ).", "Sum together we complete the proof to the Reduction Theorem.", "Remark 5.2 It is noteworthy that we may introduce $P\\triangleq QY-T$ which satisfies $\\langle P,T\\rangle =0,\\langle P,Y_{\\alpha }\\rangle =0,\\langle P,P\\rangle =-Q$ .", "So $P\\bot V_2$ and $QY=T+P$ is an orthogonal decomposition.", "Hence a direct proof for case 2 and 3 is to define $\\bar{P}=\\frac{P}{\\sqrt{|Q|}}~,~~\\theta =\\frac{T}{\\sqrt{|Q|}}~,~~\\langle \\bar{P},\\bar{P}\\rangle =-\\langle \\theta ,\\theta \\rangle =\\pm 1~.$ Either of them gives a map into the sphere or the hyperbolic space.", "Then $M^n=L^m\\times N^{n-m}$ is mapped to the lightcone of $\\mathbb {R}^{n+3}_1$ by $Y=\\frac{-1}{\\sqrt{|Q|}}(\\bar{P},\\theta )~\\in \\mathbb {R}^{n+3}_1=V_2^{\\bot }\\oplus V_2$ as a warped product of these two maps ($Q$ depends only on the component of $\\widetilde{M}^m$ ).", "Clearly such hypersurfaces are cones or rotational hypersurfaces.", "The construction of cylinders, cones and rotational hypersurfaces exists for any index $1\\le m \\le n-1$ .", "From this viewpoint the condition $(2)$ that $2\\le n-m\\le n-2$ in our Reduction Theorem REF is unsatisfying, not only conceptually, but also in that it limits the possible application.", "Upon closer examination we find that when $m=n-1$ (the Möbius principal curvature $\\mu $ is simple) one could not find a satisfying version of the Reduction Theorem.", "In particular it seems unavoidable to assume that $\\lambda _1,\\cdots ,\\lambda _{n-1}$ (and $\\mu $ ) be distinct (which seems to be a quite unnatural condition), so that we can derive $\\omega _{pq}=\\sum _{r=1}^{n-1}\\frac{B_{pq,r}}{\\lambda _p-\\lambda _q}\\omega _r$ (similar to (REF )) and use it to compute $E_i(T)$ .", "(As pointed out before (REF ) in our previous proof of Theorem REF , the condition $m\\le n-2$ has been used several times, in particular to show $E_i(T)=0(\\text{mod}~V_2)$ before (REF ).)", "It seems preferable to verify whether the subspace $V_1$ or $V_2$ defined in (REF )(REF ) is invariant or not when the Reduction Theorem could not apply directly.", "On the other hand, our Reduction Theorem can be generalized to the case $m=1$ with some modification on the assumptions.", "Theorem 5.3 Let $f: M^n\\rightarrow R^{n+1}~~(n\\ge 3)$ be a hypersurface in $(n+1)-$ dimensional Euclidean space with a principal curvature of multiplicity $n-1$ .", "Below are equivalent: $(1)$ $f$ is Möbius congruent to a cylinder, or a cone, or a rotation hypersurface over a curve $\\gamma \\subset N^2(\\epsilon )$ .", "$(2)$ The Möbius form $\\Phi =\\sum _i C_i\\omega _i$ of $f$ is closed.", "Write out $\\Phi =\\sum _iC_i\\omega _i$ , the coefficient matrices $(B_{ij})$ of the Möbius second fundamental form and $(A_{ij})$ of the Blaschke tensor under any orthonormal basis $\\lbrace E_1,\\cdots ,E_n\\rbrace $ with respect to the Möbius metric $g$ and dual basis $\\lbrace \\omega _1,\\cdots ,\\omega _n\\rbrace $ .", "Notice $d\\Phi =\\sum _idC_i\\wedge \\omega _i+\\sum _iC_id\\omega _i=\\sum _{ij}C_{i,j}\\omega _j\\wedge \\omega _i$ and the integrability equation ().", "Then the following are obviously equivalent: 1) $\\Phi $ is a closed 1-form; 2) $C_{i,j}$ define a symmetric tensor; 3) matrices $(B_{ij})$ and $(A_{ij})$ commute; 4)$(B_{ij})$ and $(A_{ij})$ can be diagonalized simultaneously.", "Suppose $f$ has a principal curvature of multiplicity $n-1$ and $\\Phi $ is closed.", "Then we can choose $\\lbrace E_1,\\cdots ,E_n\\rbrace $ such that $(B_{ij})=\\text{diag}(\\lambda ,\\mu ,\\cdots ,\\mu ),~~ (A_{ij})=\\text{diag}(a_1,a_2,\\cdots ,a_n).$ We are almost in the same context as in the proof of Theorem REF with $m=1$ and here we still assume $1\\le i,j,k \\le n; 2\\le \\alpha ,\\beta ,\\gamma \\le n.$ In particular (REF ) still holds true and we have $B_{\\alpha \\beta ,\\alpha }=0$ for any $\\alpha \\ne \\beta $ .", "Using () we know $\\lambda =\\frac{n-1}{n},\\mu =\\frac{-1}{n}$ identically.", "Differentiate them.", "We get $\\begin{split}B_{11,\\alpha }&=0,~~\\forall \\alpha ,\\\\0=E_{\\beta }(\\mu )&=B_{\\alpha \\alpha ,\\beta }=B_{\\alpha \\beta ,\\alpha }+\\delta _{\\alpha \\alpha }C_{\\beta }-\\delta _{\\alpha \\beta }C_{\\alpha }=C_{\\beta },~~\\forall \\alpha \\ne \\beta .\\end{split}$ This looks like (REF ) and we also use (REF )().", "But the assumption is different.", "Anyway we find that the condition $(3)$ in the Reduction Theorem REF is satisfied.", "Although here $m=1$ violates the condition $(2)$ , we observe that $m\\ge 2$ is only used only once in that proof to derive (REF ): $A_{q\\alpha }=0,$ which is an established fact at here already.", "Thus the previous proof to Theorem REF after (REF ) is still valid.", "The same argument shows that $f$ is reducible.", "Conversely, if $f$ could be reduced to Example (REF ), (REF ), or (REF ), by the computations in the previous section we know that $(B_{ij})$ and $(A_{ij})$ can be diagonalized simultaneously, thus $C$ is closed.", "This finishes the proof to Theorem REF .", "Remark 5.4 In [14], Guo and Lin obtained a classification of hypersurfaces with two distinct principal curvatures and closed Möbius form $\\Phi $ , which included our Theorem REF .", "We give an alternative proof here not only to be self-contained, but also because this proof looks simpler and unified with the Reduction Theorem REF ." ], [ "Algebraic characteristics of second fundamental forms of deformable hypersurface pairs", "Let $f,\\bar{f}: M^n\\rightarrow R^{n+1}~~(n\\ge 4)$ be two hypersurfaces without umbilics.", "If they induce the same Möbius metric, i.e., $g=\\bar{g}$ , then the Möbius second fundamental forms $B$ of $f$ , and $\\bar{B}$ of $\\bar{f}$ , have specific algebraic characteristics.", "The algebraic result is as below: Theorem 6.1 Let $V$ be a $n$ -dimensional vector space $(n\\ge 4)$ , and $B,\\bar{B}: V\\times V\\rightarrow R$ be two bilinear symmetric functions.", "Let $\\lbrace e_1,\\cdots ,e_n\\rbrace $ be an orthonormal basis of $V$ , and write $B(e_i,e_j)=B_{ij}, \\bar{B}(e_i,e_j)=\\bar{B}_{ij}$ .", "Denote $\\begin{split}S_{ijkl}=&B_{ik}B_{jl}-B_{il}B_{jk}\\\\&+\\frac{1}{n-2}\\sum _m\\lbrace \\delta _{ik}B_{jm}B_{ml}+\\delta _{jl}B_{im}B_{mk}-\\delta _{il}B_{jm}B_{mk}-\\delta _{jk}B_{im}B_{ml}\\rbrace .\\end{split}$ Obviously this defines a tensor $S: V^4\\rightarrow R$ associated with $B$ .", "$\\bar{S}$ and $\\bar{S}_{ijkl}$ are defined similarly for $\\bar{B}$ .", "Assume $S=\\bar{S}$ , i.e.", "$S_{ijkl}=\\bar{S}_{ijkl},\\;\\;\\forall \\, 1\\le i,j,k,l\\le n.$ Then either $B$ and $\\bar{B}$ can be diagonalized simultaneously, or there exists an orthonormal basis $\\lbrace e_1,\\cdots ,e_n\\rbrace $ of $V$ such that $\\lbrace \\bar{B}_{ij}\\rbrace =\\text{diag}(\\bar{\\lambda }_1,\\bar{\\lambda }_2,\\bar{\\mu },\\cdots ,\\bar{\\mu }), \\lbrace B_{ij}\\rbrace =\\left(\\begin{array}{ccccc}B_{11} & B_{12}& 0 & \\cdots &0\\\\B_{21} &B_{22} & 0 & \\cdots & 0 \\\\0 & 0 & \\mu & \\cdots & 0 \\\\\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\0 & 0 & 0 & \\cdots & \\mu \\\\\\end{array}\\right),$ where $\\bar{\\lambda }_1\\ne \\bar{\\lambda }_2, \\mu =\\pm \\bar{\\mu }$ .", "In the last case there exist an eigenvalue of $\\bar{B}$ with multiplicity at least $n-2$ .", "To prove Theorem REF , we need the following two lemmas.", "Lemma 6.2 Given $n\\ge 4$ .", "Assumptions as in Theorem REF except that $dim(V)=l, 3\\le l\\le n$ .", "That means we still have the fraction $\\frac{1}{n-2}$ in the expression (REF ), yet the range of those indices is from 1 to $l$ .", "Then we can find an orthonormal basis of $V$ so that $\\lbrace \\bar{B}_{ij}\\rbrace =\\text{diag}(\\bar{\\lambda }_1,\\cdots ,\\bar{\\lambda }_l)$ and $B_{ij}=0$ for some $i\\ne j$ (i.e.", "there is at least one off-diagonal element of $\\lbrace B_{ij}\\rbrace $ equals to zero).", "Since $\\bar{B}$ is symmetric, we can always diagonalize it as $\\lbrace \\bar{B}_{ij}\\rbrace =\\text{diag}(\\bar{\\lambda }_1,\\cdots ,\\bar{\\lambda }_l)$ with respect to an orthonormal basis of $V$ .", "If there has been some $B_{ij}=0$ with $i\\ne j$ at the same time, we are done.", "Otherwise, suppose all the off-diagonal elements of $\\lbrace B_{ij}\\rbrace $ are non-zero.", "In this case we make the following Assertion: $~\\lbrace \\bar{\\lambda }_1,\\cdots ,\\bar{\\lambda }_l\\rbrace $ could not be all distinct.", "Hence there must exist two equal eigenvalues $\\bar{\\lambda }_\\alpha =\\bar{\\lambda }_\\beta $ , which enables us to rotate the basis vectors $\\lbrace e_\\alpha ,e_\\beta \\rbrace $ properly in the plane $span\\lbrace e_\\alpha ,e_\\beta \\rbrace $ and to obtain a new orthonormal basis of $V$ , so that $\\lbrace \\bar{B}_{ij}\\rbrace $ is still a diagonal matrix and $B_{\\alpha \\beta }=0$ .", "This completes the proof.", "To prove the assertion above (on condition that $B_{ij}\\ne 0,\\forall ~i\\ne j$ ), we substitute the expressions of $S,\\bar{S}$ and $\\lbrace \\bar{B}_{ij}\\rbrace =\\text{diag}(\\bar{\\lambda }_1,\\cdots ,\\bar{\\lambda }_l)$ into the equality $S_{\\alpha 1\\alpha 1}-S_{\\alpha 2\\alpha 2} =\\bar{S}_{\\alpha 1\\alpha 1}-\\bar{S}_{\\alpha 2\\alpha 2}, ~~~\\forall ~3\\le \\alpha \\le l.$ As the result we obtain $B_{\\alpha \\alpha }&(B_{11}-B_{22})-(B_{1\\alpha }^2-B_{2\\alpha }^2)+\\frac{1}{n-2}\\sum _{m=1}^l(B_{1m}^2-B_{2m}^2) \\\\&=(\\bar{\\lambda }_1-\\bar{\\lambda }_2)[\\bar{\\lambda }_{\\alpha }+\\frac{1}{n-2}(\\bar{\\lambda }_1+\\bar{\\lambda }_2)],~~~~~\\forall ~3\\le \\alpha \\le l. $ In the following let the range of the index $\\alpha $ be $3\\le \\alpha \\le l$ .", "We want to show that the left hand side of (REF ) vanishes.", "First note that $\\lbrace \\bar{B}_{ij}\\rbrace =\\text{diag}(\\bar{\\lambda }_1,\\cdots ,\\bar{\\lambda }_l)$ implies $\\bar{S}_{ijik}=0,$ when $i,j,k$ are distinct.", "It follows from the equality $S_{ijik}=\\bar{S}_{ijik}$ that $B_{ii}B_{jk}-B_{ij}B_{ik}+\\frac{1}{n-2} \\sum _{m=1}^lB_{jm}B_{km}=0,~~~~\\forall ~\\text{distinct}~i,j,k.$ Hence $\\frac{B_{11}}{B_{12}}-\\frac{B_{1\\alpha }}{B_{2\\alpha }}&=-\\frac{1}{n-2}\\cdot \\frac{1}{B_{12}B_{2\\alpha }}\\cdot \\sum _{m=1}^l B_{2m}B_{m\\alpha } \\\\&=-\\frac{1}{n-2}\\left[\\frac{B_{1\\alpha }}{B_{2\\alpha }}+\\frac{B_{22}}{B_{12}}+\\sum _{m=3}^l\\frac{B_{m\\alpha }}{B_{12}}\\cdot \\frac{B_{2m}}{B_{2\\alpha }}\\right].$ Similarly one can find $\\frac{B_{22}}{B_{12}}-\\frac{B_{2\\alpha }}{B_{1\\alpha }}&=-\\frac{1}{n-2}\\cdot \\frac{1}{B_{12}B_{1\\alpha }}\\cdot \\sum _{m=1}^l B_{1m}B_{m\\alpha } \\\\&=-\\frac{1}{n-2}\\left[\\frac{B_{2\\alpha }}{B_{1\\alpha }}+\\frac{B_{11}}{B_{12}}+\\sum _{m=3}^l\\frac{B_{m\\alpha }}{B_{12}}\\cdot \\frac{B_{1m}}{B_{1\\alpha }}\\right].$ Taking $(\\ref {s4})-(\\ref {s5})$ yields $\\left[ \\frac{B_{11}-B_{22}}{B_{12}}-\\frac{B_{1\\alpha }}{B_{2\\alpha }}+\\frac{B_{2\\alpha }}{B_{1\\alpha }}\\right]\\left(1-\\frac{1}{n-2}\\right) =-\\frac{1}{n-2}\\sum _{m=3}^l\\frac{B_{m\\alpha }}{B_{12}} \\left(\\frac{B_{2m}}{B_{2\\alpha }}-\\frac{B_{1m}}{B_{1\\alpha }}\\right) =0,$ due to $B_{2m}B_{1\\alpha }-B_{2\\alpha }B_{1m}=S_{21m\\alpha }=\\bar{S}_{21m\\alpha }=0$ when $\\bar{B}$ is diagonal and $m,\\alpha \\ge 3$ .", "We conclude $\\frac{B_{11}-B_{22}}{B_{12}}=\\frac{B_{1\\alpha }}{B_{2\\alpha }}-\\frac{B_{2\\alpha }}{B_{1\\alpha }} =b,$ for some constant $b$ .", "It follows that $&B_{\\alpha \\alpha }(B_{11}-B_{22})-(B_{1\\alpha }^2-B_{2\\alpha }^2)+\\frac{1}{n-2}\\sum _{m=1}^l(B_{1m}^2-B_{2m}^2)\\\\=~&B_{\\alpha \\alpha }(B_{11}-B_{22})-(B_{1\\alpha }^2-B_{2\\alpha }^2)+\\frac{1}{n-2}(B_{11}^2-B_{22}^2)+\\frac{1}{n-2}\\sum _{m=3}^l(B_{1m}^2-B_{2m}^2)\\\\=~&B_{\\alpha \\alpha }\\cdot bB_{12}-b\\cdot B_{1\\alpha }B_{2\\alpha }+\\frac{1}{n-2}(B_{11}+B_{22})\\cdot bB_{12}+\\frac{1}{n-2}\\sum _{m=3}^l(b\\cdot B_{1m}B_{2m})\\\\=~&b\\left[B_{\\alpha \\alpha }B_{12}-B_{1\\alpha }B_{2\\alpha }+\\frac{1}{n-2}\\sum _{m=1}^l B_{1m}B_{2m}\\right]=0,$ by (REF ).", "From (REF ) we have $(\\bar{\\lambda }_1-\\bar{\\lambda }_2)[\\bar{\\lambda }_{\\alpha }+\\frac{1}{n-2}(\\bar{\\lambda }_1+\\bar{\\lambda }_2)]=0.$ So either $\\bar{\\lambda }_1=\\bar{\\lambda }_2$ , or $\\bar{\\lambda }_{\\alpha }=-\\frac{1}{n-2}(\\bar{\\lambda }_1+\\bar{\\lambda }_2)],~\\forall ~3\\le \\alpha \\le l.$ This verifies the assertion when $l\\ge 4$ .", "The only case unsolved is when $l=3$ .", "This time (REF ) takes the form $(\\bar{\\lambda }_1-\\bar{\\lambda }_2)[\\bar{\\lambda }_3+\\frac{1}{n-2}(\\bar{\\lambda }_1+\\bar{\\lambda }_2) ]=0.$ Taking permutation of the indices $1,2,3$ yields two other similar formulas.", "Now it is easy to prove that $\\bar{\\lambda }_1,\\bar{\\lambda }_2,\\bar{\\lambda }_3$ can not be all distinct by contradiction.", "Hence the proof to Lemma REF is finished.", "Remark 6.3 Note that in the proof above we used the fact $\\frac{1}{n-2}\\ne 1$ at two places.", "Thus the condition $n\\ge 4$ is necessary.", "On the other hand, by the integrability equations () the Weyl conformal tenor associated with the Möbius metric $g$ can be expressed by the Möbius invariants as below: $\\begin{split}C_{ijkl}&=B_{ik}B_{jl}-B_{il}B_{jk}-\\frac{1}{n(n-2)}(\\delta _{ik}\\delta _{jl}-\\delta _{jk}\\delta _{il})\\\\&+\\frac{1}{n-2}\\sum _m\\lbrace \\delta _{ik}B_{jm}B_{ml}+\\delta _{jl}B_{im}B_{mk}-\\delta _{il}B_{jm}B_{mk}-\\delta _{jk}B_{im}B_{ml}\\rbrace \\\\&=S_{ijkl}-\\frac{1}{n(n-2)}(\\delta _{ik}\\delta _{jl}-\\delta _{jk}\\delta _{il}).\\end{split}$ It is well known that the Weyl conformal tenor vanishes on three dimensional Riemannian manifold.", "Therefore when $n=3$ , $S_{ijkl}=\\frac{1}{3}(\\delta _{ik}\\delta _{jl}-\\delta _{jk}\\delta _{il})=\\bar{S}_{ijkl}$ is a trivial identity.", "Lemma 6.4 Assumptions as in Lemma REF .", "By the conclusion above, without loss of generality we may suppose that for a given orthonormal basis of $V$ there are $\\lbrace \\bar{B}_{ij}\\rbrace =\\text{diag}(\\bar{\\lambda }_1,\\cdots ,\\bar{\\lambda }_l)$ and $B_{ij}=0$ for some $i\\ne j$ .", "Then there exists a properly chosen new orthonormal basis of $V$ , with respect to which $\\lbrace \\bar{B}_{ij}\\rbrace $ is still diagonal and $\\lbrace B_{ij}\\rbrace =\\left(\\begin{array}{cccc}B_{11} & \\cdots & B_{1,l-1} & 0\\\\\\vdots & \\ddots & \\vdots & \\vdots \\\\B_{l-1,1} & \\cdots & B_{l-1,l-1} & 0 \\\\0 & \\cdots & 0 & B_{ll}\\\\\\end{array}\\right)$ is a semi-diagonal matrix.", "For simplicity denote $k=l-1$ .", "Without loss of generality we may assume that the off-diagonal element $B_{kl}=0$ .", "First we consider the easy case $l=3$ .", "As in (REF ), we have $0=B_{11}B_{23}-B_{12}B_{13}+\\frac{1}{n-2} \\sum _{m=1}^3 B_{2m}B_{m3}=\\left(\\frac{1}{n-2}-1\\right) B_{12}B_{13},$ because $B_{23}=0$ as assumed.", "It follows that either $B_{12}=0$ or $B_{13}=0$ , and the conclusion is proved.", "In general, when $l\\ge 4$ , for any $i<j<k=l-1$ there is $0=\\bar{S}_{ikjl}=S_{ikjl}=B_{ij}B_{kl}-B_{il}B_{kj} =-B_{il}B_{kj}.$ If $B_{il}=0$ for any $i< k=l-1$ , then all the off-diagonal elements in the $l$ -th column and the $l$ -th row vanish, and we are done.", "Otherwise, suppose $B_{1l}\\ne 0$ without loss of generality.", "Then by (REF ), $B_{jk}=0,~\\forall ~1<j<k=l-1$ .", "Using this result and $B_{lk}=0, B_{1l}\\ne 0$ , we may prove $B_{1k}=0$ by (REF ): $0=B_{11}B_{kl}-B_{1k}B_{1l}+\\frac{1}{n-2} \\sum _{j=1}^l B_{jk}B_{jl}=\\left(\\frac{1}{n-2}-1\\right) B_{1k}B_{1l}.$ So $B_{jk}=0,~\\forall ~j\\ne k$ .", "That means all the off-diagonal elements in the $k$ -th column and the $k$ -th row vanish.", "Interchanging the basis vectors $e_k$ and $e_l$ gives the desired result.", "The proof to Lemma REF is finished.", "[Proof to Theorem REF ] From Lemma REF and Lemma REF and by induction it is easy to see that $\\lbrace B_{ij}\\rbrace ,\\lbrace \\bar{B}_{ij}\\rbrace $ can be diagonalized simultaneously except that $B_{12}$ might be non-zero.", "Denote $\\lbrace \\bar{B}_{ij}\\rbrace =\\text{diag}(\\bar{\\lambda }_1,\\cdots ,\\bar{\\lambda }_n)$ as before.", "When $B_{12}=0$ it is the first case in the conclusion.", "If $B_{12}\\ne 0$ yet $\\bar{\\lambda }_1=\\bar{\\lambda }_2$ , one might rotate the basis vectors $\\lbrace e_1,e_2\\rbrace $ properly in the plane $span\\lbrace e_1,e_2\\rbrace $ and obtain a new orthonormal basis of $V$ such that $\\lbrace \\bar{B}_{ij}\\rbrace $ is invariant and $B_{12}=0$ , hence we are also done.", "The final part of the proof is to show that when $B_{12}\\ne 0$ and $\\bar{\\lambda }_1\\ne \\bar{\\lambda }_2$ , $\\lbrace B_{ij}\\rbrace $ and $\\lbrace \\bar{B}_{ij}\\rbrace $ must have the desired multiplicities of their eigenvalues.", "Again by (REF ), $\\forall ~3\\le \\alpha \\le n$ , $0=B_{\\alpha \\alpha }B_{12}-B_{\\alpha 1}B_{\\alpha 2}+\\frac{1}{n-2}\\sum _{m=1}^n B_{1m}B_{m2} =B_{12}\\left[B_{\\alpha \\alpha }+\\frac{1}{n-2}(B_{11}+B_{22})\\right].$ Thus $B_{\\alpha \\alpha }=-\\frac{1}{n-2}(B_{11}+B_{22})=\\mu $ for any $\\alpha \\ge 3$ .", "So $\\lbrace B_{ij}\\rbrace $ has the desired form.", "As a by-product we find that $tr(B)=B_{11}+B_{22}+(n-2)\\mu =0.$ Taking use of the fact above and the equalities $S_{1\\alpha 1\\alpha }=\\bar{S}_{1\\alpha 1\\alpha }, S_{2\\alpha 2\\alpha }=\\bar{S}_{2\\alpha 2\\alpha }$ , we have $B_{11}\\mu +\\frac{1}{n-2}\\left( \\lambda ^2+B_{11}^2+B_{12}^2\\right)&= \\bar{\\lambda }_1\\bar{\\lambda }_{\\alpha }+\\frac{1}{n-2}\\left(\\bar{\\lambda }_{\\alpha }^2+\\bar{\\lambda }_1^2\\right), \\\\B_{22}\\mu +\\frac{1}{n-2}\\left( \\lambda ^2+B_{22}^2+B_{12}^2\\right)&= \\bar{\\lambda }_2\\bar{\\lambda }_{\\alpha }+\\frac{1}{n-2}\\left(\\bar{\\lambda }_{\\alpha }^2+\\bar{\\lambda }_2^2\\right), $ for any $\\alpha \\ge 3$ .", "Taking $(\\ref {b1})-(\\ref {b2})$ yields $(B_{11}-B_{22})\\left[B_{\\alpha \\alpha }+\\frac{1}{n-2}(B_{11}+B_{22})\\right]=(\\bar{\\lambda }_1-\\bar{\\lambda }_2)\\left[\\bar{\\lambda }_{\\alpha }+\\frac{1}{n-2}(\\bar{\\lambda }_1+\\bar{\\lambda }_2)\\right].$ The left hand side vanishes by (REF ).", "It follows that $\\bar{\\lambda }_{\\alpha }=-\\frac{1}{n-2}(\\bar{\\lambda }_1+\\bar{\\lambda }_2)=\\bar{\\lambda }$ for all $\\alpha \\ge 3$ (keep in mind that $\\bar{\\lambda }_1\\ne \\bar{\\lambda }_2$ at here) and $tr(\\bar{B})=0$ .", "Finally $S_{3434}=\\bar{S}_{3434}$ implies $\\mu ^2=\\bar{\\mu }^2$ .", "This finishes the proof to Theorem REF ." ], [ "Hypersurfaces with low multiplicities: rigidity", "Let $f,\\bar{f}: M^n\\rightarrow R^{n+1}(n\\ge 4)$ be two hypersurfaces without umbilics.", "In this section and the following two, the Möbius invariants of $f$ will be denoted by $\\lbrace A,B,C\\rbrace $ and those of $\\bar{f}$ by $\\lbrace \\bar{A},\\bar{B},\\bar{C}\\rbrace $ .", "Theorem 7.1 Let $f,\\bar{f}: M^n\\rightarrow R^{n+1}(n\\ge 4)$ be two immersed hypersurfaces without umbilics, whose principal curvatures have constant multiplicities.", "Assume that they induce the same Möbius metrics $g$ , and all principal curvatures of $B$ have multiplicity less than $n-2$ everywhere.", "Then $f$ is Möbius congruent to $\\bar{f}$ .", "We divide our proof into two parts.", "The case of dimension $n=4$ is different from higher dimensional case ($n\\ge 5$ ) and need to be discussed separately.", "Before that we make some preparation first.", "The same Möbius metric $g$ for $f,\\bar{f}$ determines the same curvature tensor $R_{ijkl}$ .", "By the integrability equations ()(), the conclusion of Theorem REF applies to the Möbius second fundamental forms $B,\\bar{B}$ .", "Since the multiplicities of all principal curvatures are less than $n-2$ at here by assumption, Theorem REF guarantees that locally we can choose an orthonormal basis $\\lbrace E_1,\\cdots ,E_n\\rbrace $ with respect to $g$ such that $\\lbrace B_{ij}\\rbrace =\\text{diag}(\\lambda _1,\\cdots ,\\lambda _n);\\lbrace \\bar{B}_{ij}\\rbrace =\\text{diag}(\\bar{\\lambda }_1,\\cdots ,\\bar{\\lambda }_n).$ Now ()() imply $\\lambda _i\\lambda _j+\\frac{1}{n-2}(\\lambda _i^2+\\lambda _j^2)=\\bar{\\lambda }_i\\bar{\\lambda }_j+\\frac{1}{n-2}(\\bar{\\lambda }_i^2+\\bar{\\lambda }_j^2),~~\\forall ~i\\ne j.$ Changing the subscript of (REF ) and taking difference, we get $(\\lambda _i-\\lambda _k)[\\lambda _j+\\frac{1}{n-2}(\\lambda _i+\\lambda _k)]=(\\bar{\\lambda }_i-\\bar{\\lambda }_k)[\\bar{\\lambda }_j+\\frac{1}{n-2}(\\bar{\\lambda }_i+\\bar{\\lambda }_k)],~~\\forall ~\\text{distinct}~i,j,k.$ To obtain the rigidity result we need only to show that $\\bar{B}=\\pm B$ ; reverse the direction of the normal vector field of $f$ if necessary we will have $\\bar{B}= B$ , which shows that $f$ is congruent to $\\bar{f}$ by the fundamental theorem REF .", "Proposition 7.2 The conclusion of Theorem REF is valid when the dimension $n\\ge 5$ .", "We assert that there is a linear relation between $\\lambda _j$ and $\\bar{\\lambda }_j$ , i.e.", "there exists constants $b,c$ such that $\\bar{\\lambda }_j=b\\lambda _j+c,~~\\forall ~1\\le j\\le n.$ In other words, regard $p_j=(\\lambda _j,\\bar{\\lambda }_j)$ as coordinates of $n$ points on a plane, then these $n$ points are collinear.", "Without loss of generality assume that $\\lambda _1\\ne \\lambda _2$ .", "We just show $p_3=(\\lambda _3,\\bar{\\lambda }_3)$ is collinear with $p_1=(\\lambda _1,\\bar{\\lambda }_1),p_2=(\\lambda _2,\\bar{\\lambda }_2)$ .", "(For any other index $j\\ne 1,2,3$ the proof is the same.)", "Now we need to consider two cases separately.", "In the first case, $n\\ge 5$ and the highest multiplicity of principal curvatures is less than $n-3$ .", "We can find $\\lambda _i\\ne \\lambda _k$ which are distinct from $\\lbrace 1,2,3\\rbrace $ .", "Fix $i,k$ in (REF ), we see that all other $(\\lambda _j,\\bar{\\lambda }_j)$ ($j\\ne i,k$ ) satisfies a non-trivial linear equation (REF ).", "In particular, $p_1,p_2,p_3$ are collinear.", "In the second case, $\\lambda _i$ might be a constant for any indices $i\\ne 1,2,3$ .", "(Note that $\\lambda _i\\ne \\lambda _1,\\lambda _2,\\lambda _3$ .", "Otherwise there will be a principal curvature of multiplicity at least $n-2$ , contradiction.", "Yet $\\lambda _3$ might be equal to either of $\\lambda _1,\\lambda _2$ .)", "Fix $i=1,k=5$ , we have $\\lambda _1\\ne \\lambda _5$ and by (REF ) we know $p_2=(\\lambda _2,\\bar{\\lambda }_2),p_3=(\\lambda _3,\\bar{\\lambda }_3),p_4(\\lambda _4,\\bar{\\lambda }_4)~ \\text{are collinear}.$ Similarly we know $\\lbrace p_1,p_2,p_4\\rbrace $ and $\\lbrace p_1,p_3,p_4\\rbrace $ are collinear triples.", "This guarantees that $\\lbrace p_1,p_2,p_3\\rbrace $ (and other $p_j$ 's) are collinear and finishes the proof to our assertion.", "Now we know $\\bar{\\lambda }_j=b\\lambda _j+c$ for constants $b,c$ and for any $j$ .", "The fact $\\sum _j \\lambda _j=0=\\sum _j \\bar{\\lambda }_j$ (the first identity in ()) implies $c=0$ .", "Using the second identity $\\sum _j \\lambda _j^2=\\frac{n-1}{n}=\\sum _j \\bar{\\lambda }_j^2$ in () we conclude that $b=\\pm 1$ .", "This completes the proof to Proposition REF .", "Proposition 7.3 The conclusion of Theorem REF is valid when dimension $n=4$ .", "First note that when $n=4$ and the highest multiplicity is less than $n-2=2$ , four principal curvatures of $f$ are distinct.", "Consider $\\lbrace B_{ij}\\rbrace =\\text{diag}(\\lambda _1,\\lambda _2,\\lambda _3,\\lambda _4);~\\lbrace \\bar{B}_{ij}\\rbrace =\\text{diag}(\\bar{\\lambda }_1,\\bar{\\lambda }_2,\\bar{\\lambda }_3,\\bar{\\lambda }_4).$ By (REF ) and $n=4$ we have $\\lambda _i+\\lambda _j=\\pm (\\bar{\\lambda }_i+\\bar{\\lambda }_j),~i\\ne j.$ We assert that there are four possibilities on each and every point of $M$ : $\\begin{split}&(1)\\;\\; B=\\pm \\bar{B};\\\\&(2)\\;\\;\\lbrace B_{ij}\\rbrace =\\text{diag}(\\lambda _1,\\lambda _2,\\lambda _3,\\lambda _4),\\lbrace \\bar{B}_{ij}\\rbrace =\\text{diag}(\\pm \\lambda _2,\\pm \\lambda _1,\\pm \\lambda _4,\\pm \\lambda _3);\\\\&(3)\\;\\;\\lbrace B_{ij}\\rbrace =\\text{diag}(\\lambda _1,\\lambda _2,\\lambda _3,\\lambda _4),\\lbrace \\bar{B}_{ij}\\rbrace =\\text{diag}(\\pm \\lambda _3,\\pm \\lambda _4,\\pm \\lambda _1,\\pm \\lambda _2);\\\\&(4)\\;\\;\\lbrace B_{ij}\\rbrace =\\text{diag}(\\lambda _1,\\lambda _2,\\lambda _3,\\lambda _4),\\lbrace \\bar{B}_{ij}\\rbrace =\\text{diag}(\\pm \\lambda _4,\\pm \\lambda _3,\\pm \\lambda _2,\\pm \\lambda _1).\\end{split}$ Suppose $B\\ne \\pm \\bar{B}$ .", "Consider a special case $ \\lambda _1+\\lambda _2=(\\bar{\\lambda }_1+\\bar{\\lambda }_2),~\\lambda _1+\\lambda _3=-(\\bar{\\lambda }_1+\\bar{\\lambda }_3),~\\lambda _1+\\lambda _4=-(\\bar{\\lambda }_2+\\bar{\\lambda }_3).$ Taking sum of these three equalities and using the fact $\\sum _j \\lambda _j=0=\\sum _j \\bar{\\lambda }_j$ we get $\\lambda _1=\\bar{\\lambda }_2$ .", "Substitute this back and use $\\sum _j \\lambda _j=0=\\sum _j \\bar{\\lambda }_j$ again.", "We conclude that this is case $(2)$ .", "Reversing either of the normal vector fields and taking permutations reduce other possibilities to this special case.", "This verifies our assertion.", "We need to exclude possibility $(2)$ by contradiction.", "Other cases are similar.", "This time $\\lambda _1+\\lambda _2=0$ automatically implies $B=\\pm \\bar{B}$ by $\\sum _j \\lambda _j=0=\\sum _j \\bar{\\lambda }_j$ .", "So we need only to find contradiction when $\\lambda _1\\ne \\pm \\lambda _2, \\lambda _3\\ne \\pm \\lambda _4$ and $\\lbrace B_{ij}\\rbrace =\\text{diag}(\\lambda _1,\\lambda _2,\\lambda _3,\\lambda _4),\\lbrace \\bar{B}_{ij}\\rbrace =\\text{diag}(\\pm \\lambda _2,\\pm \\lambda _1,\\pm \\lambda _4,\\pm \\lambda _3),$ under a locally orthonormal basis $\\lbrace E_1,\\cdots ,E_4\\rbrace $ with respect to $g$ .", "Using the covariant derivative of $B$ and $\\bar{B}$ , we get $(\\lambda _i-\\lambda _j)\\omega _{ij}=\\sum _kB_{ij,k}\\omega _k,~~(\\bar{\\lambda }_i-\\bar{\\lambda }_j)\\omega _{ij}=\\sum _k\\bar{B}_{ij,k}\\omega _k, ~~i\\ne j.$ So $(\\bar{\\lambda }_i-\\bar{\\lambda }_j)B_{ij,k}=(\\lambda _i-\\lambda _j)\\bar{B}_{ij,k},~~~i\\ne j.$ Consequently, there is $B_{12,k}=-\\bar{B}_{12,k},~~B_{34,k}=-\\bar{B}_{34,k},$ because $\\bar{\\lambda }_1-\\bar{\\lambda }_2=\\lambda _2-\\lambda _1\\ne 0,\\bar{\\lambda }_3-\\bar{\\lambda }_4=\\lambda _4-\\lambda _3\\ne 0$ .", "It follows that $B_{ij,k}=\\bar{B}_{ij,k}=0,~~\\text{when $i,j,k$ are distinct}.$ To verify (REF ), consider the case when $i=1,j=2,k=3$ .", "If $B_{12,3}\\ne 0$ , then $B_{13,2}=-\\bar{B}_{13,2}\\ne 0$ .", "(Note $B_{ij,k}=B_{ik,j}$ when $i,j,k$ are distinct by ().)", "Combined with (REF ), there should be $\\lambda _1-\\lambda _3=\\bar{\\lambda }_3-\\bar{\\lambda }_1=\\lambda _4-\\lambda _2$ .", "Yet this implies $\\lambda _1+\\lambda _2=0$ under our condition $\\lambda _1+\\lambda _2+\\lambda _3+\\lambda _4=0$ , which contradicts the assumption $\\lambda _1\\ne \\pm \\lambda _2$ .", "Other cases are verified similarly.", "As a corollary of (REF ) and (REF ), $(\\lambda _i-\\lambda _j)\\omega _{ij}=B_{ij,i}\\omega _i +B_{ij,j}\\omega _j, ~~(\\bar{\\lambda }_i-\\bar{\\lambda }_j)\\omega _{ij}=\\bar{B}_{ij,i}\\omega _i+\\bar{B}_{ij,j}\\omega _j, ~~i\\ne j.$ To derive the exact expressions of the connection forms $\\omega _{ij}$ , we shall compute out all the quantities like $B_{ii,j}$ in terms of $\\lambda _k$ 's and $C_k$ 's.", "By the symmetry in our situation, obviously there is $\\bar{B}_{33,1}=B_{44,1},\\bar{B}_{44,1}=B_{33,1}$ .", "Together with (REF ) and (), it follows $(\\bar{\\lambda }_3-\\bar{\\lambda }_1)(B_{33,1}-C_1)=(\\bar{\\lambda }_3-\\bar{\\lambda }_1)B_{31,3}=(\\lambda _3-\\lambda _1)\\bar{B}_{31,3}=(\\lambda _3-\\lambda _1)(\\bar{B}_{33,1}-\\bar{C}_1).$ So we get $\\begin{split}&(\\lambda _4-\\lambda _2)(B_{33,1}-C_1)=(\\lambda _3-\\lambda _1)(B_{44,1}-\\bar{C}_1),\\\\&(\\lambda _3-\\lambda _2)(B_{44,1}-C_1)=(\\lambda _4-\\lambda _1)(B_{33,1}-\\bar{C}_1).\\end{split}$ The second equation is obtained in the similar way.", "On the other hand, (REF ) tells us that $B_{22,1}-C_1=B_{12,2}=-\\bar{B}_{12,2}=\\bar{B}_{22,1}-\\bar{C}_1=B_{11,1}-\\bar{C}_1.$ Thus $B_{11,1}+B_{22,1}=C_1+\\bar{C}_1$ .", "Note that $\\sum _i\\lambda _i=\\sum _iB_{ii}=0$ and $\\sum _iB_{ii}^2=\\frac{n-1}{n}$ imply $\\sum _{i=1}^4 B_{ii,k}=0,~~\\sum _{i=1}^4 \\lambda _i B_{ii,k}=0,~~\\forall ~k.$ In particular, $B_{33,1}+B_{44,1}=-(B_{11,1}+B_{22,1})=-C_1-\\bar{C}_1.$ Eliminating $B_{33,1},B_{44,1}$ from (REF )(REF ) (keep in mind $\\sum _i\\lambda _i=0$ ) yields $\\lambda _2 C_1=\\lambda _1 \\bar{C}_1$ .", "Because $\\lambda _1,\\lambda _2$ could not be zero at the same time ($\\lambda _1\\ne \\pm \\lambda _2$ ), we may denote $\\Delta _1:=\\frac{C_1}{\\lambda _1}=\\frac{\\bar{C}_1}{\\lambda _2}.$ In case that $\\lambda _1=0\\ne \\lambda _2$ , there must be $C_1=0$ , and we need only to take $\\Delta _1=\\frac{\\bar{C}_1}{\\lambda _2}$ which is well-defined.", "Putting (REF ) into (REF )(REF ) solves $B_{33,1},B_{44,1}$ .", "Then by (REF ) we get the complete solution: $\\begin{split}&B_{11,1}=\\frac{\\lambda _2\\lambda _3+\\lambda _2\\lambda _4-\\lambda _3^2-\\lambda _4^2}{\\lambda _1-\\lambda _2}\\Delta _1,~B_{33,1}=\\lambda _3 \\Delta _1,\\\\&B_{22,1}=\\frac{\\lambda _1\\lambda _3+\\lambda _1\\lambda _4-\\lambda _3^2-\\lambda _4^2}{\\lambda _2-\\lambda _1}\\Delta _1,~B_{44,1}=\\lambda _4 \\Delta _1.\\end{split}$ Similarly there are: $\\begin{split}&B_{11,2}=\\frac{\\lambda _2\\lambda _3+\\lambda _2\\lambda _4-\\lambda _3^2-\\lambda _4^2}{\\lambda _1-\\lambda _2}\\Delta _2,~B_{33,2}=\\lambda _3 \\Delta _2,\\\\&B_{22,2}=\\frac{\\lambda _1\\lambda _3+\\lambda _1\\lambda _4-\\lambda _3^2-\\lambda _4^2}{\\lambda _2-\\lambda _1}\\Delta _2,~B_{44,1}=\\lambda _4 \\Delta _2,\\\\&B_{11,3}=\\lambda _1 \\Delta _3,~B_{33,3}=\\frac{\\lambda _4\\lambda _1+\\lambda _4\\lambda _2-\\lambda _1^2-\\lambda _2^2}{\\lambda _3-\\lambda _4}\\Delta _3,\\\\&B_{22,3}=\\lambda _2 \\Delta _3,~B_{44,3}=\\frac{\\lambda _3\\lambda _1+\\lambda _3\\lambda _2-\\lambda _1^2-\\lambda _2^2}{\\lambda _4-\\lambda _3}\\Delta _3,\\\\&B_{11,4}=\\lambda _1 \\Delta _4,~B_{33,4}=\\frac{\\lambda _4\\lambda _1+\\lambda _4\\lambda _2-\\lambda _1^2-\\lambda _2^2}{\\lambda _3-\\lambda _4}\\Delta _4,\\\\&B_{22,4}=\\lambda _2 \\Delta _4,~B_{44,4}=\\frac{\\lambda _3\\lambda _1+\\lambda _3\\lambda _2-\\lambda _1^2-\\lambda _2^2}{\\lambda _4-\\lambda _3}\\Delta _4,\\end{split}$ where $\\Delta _2:=\\frac{C_2}{\\lambda _2}=\\frac{\\bar{C}_2}{\\lambda _1},~\\Delta _3:=\\frac{C_3}{\\lambda _3}=\\frac{\\bar{C}_3}{\\lambda _4},~\\Delta _4:=\\frac{C_4}{\\lambda _4}=\\frac{\\bar{C}_4}{\\lambda _3}.$ Now the connection forms $\\omega _{ij}$ could be determined.", "Since $\\lambda _1-\\lambda _2\\ne 0$ , by (REF )() (REF )(REF ) and $C_1=\\lambda _1\\Delta _1,C_2=\\lambda _2\\Delta _2, \\sum _i\\lambda _i=0$ , we get $\\omega _{12}=\\frac{B_{11,2}-C_2}{\\lambda _1-\\lambda _2}\\omega _1+\\frac{B_{22,1}-C_1}{\\lambda _1-\\lambda _2}\\omega _2=I_{12}(\\Delta _2 \\omega _1-\\Delta _1 \\omega _2),~~I_{12}:=-\\frac{2\\lambda _1\\lambda _2+\\lambda _3^2+\\lambda _4^2}{(\\lambda _1-\\lambda _2)^2}.$ Similarly, there is $\\omega _{34}=I_{34}(\\Delta _4 \\omega _3-\\Delta _3 \\omega _4),~~I_{34}:=-\\frac{2\\lambda _3\\lambda _4+\\lambda _1^2+\\lambda _2^2}{(\\lambda _3-\\lambda _4)^2}.$ Other connection forms are found in the same way, yet much easier: $\\begin{split}&\\omega _{13}=\\Delta _3 \\omega _1-\\Delta _1 \\omega _3,~~\\omega _{24}=\\Delta _4 \\omega _2-\\Delta _2 \\omega _4, \\\\&\\omega _{14}=\\Delta _4 \\omega _1-\\Delta _1 \\omega _4,~~\\omega _{23}=\\Delta _3 \\omega _2-\\Delta _2 \\omega _3.\\end{split}$ Finally, by the formula $d\\omega _{ij}-\\sum _l\\omega _{il}\\wedge \\omega _{jl}=-\\frac{1}{2}R_{ijkl}~\\omega _k\\wedge \\omega _l,$ the sectional curvatures are computed out: $\\begin{split}&\\tfrac{1}{2}R_{1313}=-E_1(\\Delta _1)-E_3(\\Delta _3)+\\Delta _1^2+\\Delta _3^2+I_{12}\\Delta _2^2+I_{34}\\Delta _4^2,\\\\&\\tfrac{1}{2}R_{2424}=-E_2(\\Delta _2)-E_4(\\Delta _4)+\\Delta _2^2+\\Delta _4^2+I_{12}\\Delta _1^2+I_{34}\\Delta _3^2,\\\\&\\tfrac{1}{2}R_{1414}=-E_1(\\Delta _1)-E_4(\\Delta _4)+\\Delta _1^2+\\Delta _4^2+I_{12}\\Delta _2^2+I_{34}\\Delta _3^2,\\\\&\\tfrac{1}{2}R_{2323}=-E_2(\\Delta _2)-E_3(\\Delta _3)+\\Delta _2^2+\\Delta _3^2+I_{12}\\Delta _1^2+I_{34}\\Delta _4^2.\\end{split}$ Here $E_i(\\Delta _j)$ is understood as the action of tangent vector $E_i$ on the function $\\Delta _j$ , and $\\Delta _i^2$ is the square of $\\Delta _i$ .", "As a corollary, $R_{1313}+R_{2424}-R_{1414}-R_{2323}=0.$ But on the other hand, () implies $R_{ijij}=\\lambda _i\\lambda _j+A_{ii}+A_{jj}$ when $i\\ne j$ .", "Substitute this into the final result above, we find $R_{1313}+R_{2424}-R_{1414}-R_{2323}=(\\lambda _1-\\lambda _2)(\\lambda _3-\\lambda _4)=0.$ This contradicts our assumption $\\lambda _1\\ne \\pm \\lambda _2, \\lambda _3\\ne \\pm \\lambda _4$ .", "Thus we have proved that the possibilities other than $B=\\pm \\bar{B}$ could not happen.", "This completes the proof to Proposition REF ." ], [ "Deformable hypersurfaces with one principal curvature of multiplicity $n-1$", "In this section and the next one we make use of the following convention on the range of indices: $1\\le i,j,k \\le n;\\ 3\\le \\alpha ,\\beta ,\\gamma \\le n.$ Proposition 8.1 Let $f,\\bar{f}: M^n\\rightarrow R^{n+1}(n\\ge 4)$ be two hypersurfaces without umbilics.", "Suppose that their Möbius metrics are equal, and one principal curvature of $B$ has multiplicity $n-1$ everywhere (this means that $(M^n,g)$ is conformally flat).", "Then either $f(M^n)$ is Möbius congruent to $\\bar{f}(M^n)$ , or $f(M^n)$ has constant Möbius curvature.", "Since one of principal curvatures of $B$ has multiplicity $n-1$ everywhere, from the algebraic Theorem REF , locally we can choose an orthonormal basis $\\lbrace E_1,\\cdots ,E_n\\rbrace $ with respect to $g$ such that $\\lbrace \\bar{B}_{ij}\\rbrace =\\text{diag}\\bigl ( \\Big [{\\begin{matrix}\\bar{B}_{11}& \\bar{B}_{12}\\\\\\bar{B}_{21}& \\bar{B}_{22}\\end{matrix}}\\Big ],\\bar{\\mu },\\cdots ,\\bar{\\mu }\\bigr ), \\lbrace B_{ij}\\rbrace =\\text{diag}(\\lambda ,\\mu ,\\cdots ,\\mu ),$ .", "where $\\lambda \\ne \\mu $ .", "From () we have $\\lambda +(n-1)\\mu =0$ and $\\lambda ^2+(n-1)\\mu ^2=\\frac{n-1}{n}$ .", "So $\\lbrace B_{ij}\\rbrace =\\text{diag}\\left(\\frac{n-1}{n},\\frac{-1}{n}, \\cdots ,\\frac{-1}{n}\\right).$ Since $\\bar{\\mu }=\\pm \\mu $ , up to change of the normal direction we may assume $\\bar{\\mu }=\\mu =\\frac{-1}{n}$ .", "Apply formula () to $\\bar{B}_{ij}$ .", "We know that the sub-matrices $\\begin{pmatrix}\\bar{B}_{11}& \\bar{B}_{12}\\\\\\bar{B}_{21}& \\bar{B}_{22}\\end{pmatrix}\\quad \\text{and}\\quad \\begin{pmatrix}\\frac{n-1}{n} & 0\\\\0 & \\frac{-1}{n}\\end{pmatrix}$ have equal traces and equal norms, and $B,\\bar{B}$ share equal eigenvalues.", "At any point of $M^n$ there exists a suitable $P=\\begin{pmatrix}\\cos \\theta & \\sin \\theta \\\\-\\sin \\theta & \\cos \\theta \\end{pmatrix}\\in SO(2)$ such that $\\begin{pmatrix}\\bar{B}_{11}& \\bar{B}_{12}\\\\\\bar{B}_{21}& \\bar{B}_{22}\\end{pmatrix}= P^{-1}\\begin{pmatrix}\\frac{n-1}{n} & 0\\\\0 & \\frac{-1}{n}\\end{pmatrix}P=\\begin{pmatrix}\\cos ^2\\theta -\\frac{1}{n} & \\cos \\theta \\sin \\theta \\\\\\cos \\theta \\sin \\theta & \\sin ^2\\theta -\\frac{1}{n}\\end{pmatrix}.$ We summarize some intermediate results as Lemma 8.2 For hypersurface $f$ , the coefficients of tensor $B,\\nabla B,C$ under the orthonormal basis $\\lbrace E_1,\\cdots ,E_n\\rbrace $ satisfy $\\begin{split}&B_{1j,j}=-C_1, j>1; ~~~~\\text{otherwise,} B_{ij,k}=0.\\\\&C_k=0,k>1;~~~~~\\omega _{1j}=-C_1\\omega _j, j>1.\\\\&R_{1i1i}-C_{1,1}+C_1^2=0, j>1;~~~~R_{1iji}-C_{1,j}=0,j>1.\\end{split}$ where $\\lbrace \\omega _1,\\cdots ,\\omega _n\\rbrace $ are the dual basis, and $\\lbrace \\omega _{ij}\\rbrace $ are its connection forms.", "From $dB_{ij}+\\sum _kB_{kj}\\omega _{ki}+B_{ik}\\omega _{kj}=\\sum _kB_{ij,k}\\omega _k$ and () we get the first four equalities.", "From $d\\omega _{1i}-\\sum _k\\omega _{1k}\\wedge \\omega _{ki}=-\\frac{1}{2}\\sum _{kl}R_{1ikl}\\omega _k\\wedge \\omega _l$ and invoking the proved equalities we get the equalities on the curvature tensor.", "In order to prove Proposition (REF ) We have to consider the following two cases: Case I, $\\bar{B}_{12}\\equiv 0$ ; Case II, $\\bar{B}_{12}\\ne 0$ .", "First we consider Case I.", "Since $\\bar{B}_{12}=-\\cos \\theta \\sin \\theta \\equiv 0$ , so $\\sin \\theta =0$ or $\\cos \\theta =0$ .", "If $\\sin \\theta =0$ , then $\\bar{B}=B$ , thus $f$ is Möbius congruent to $\\bar{f}$ .", "Next we assume that $\\cos \\theta =0$ .", "From (REF ) we get $(\\bar{B}_{ij})=\\text{diag}(\\mu ,\\lambda ,\\mu ,\\cdots ,\\mu )=\\text{diag}\\left(\\frac{-1}{n},\\frac{n-1}{n},\\frac{-1}{n},\\cdots ,\\frac{-1}{n}\\right).$ For hypersurface $\\bar{f}:M^n\\rightarrow S^{n+1}$ , since $g=\\bar{g}$ , we may use the same dual basis $\\lbrace \\omega _i\\rbrace $ with the same connection forms.", "Similar to Lemma REF we get $\\begin{split}&\\bar{B}_{2i,i}=-\\bar{C}_2;~~\\text{otherwise} ~~\\bar{B}_{ij,k}=0;\\\\&\\omega _{1j}=-\\bar{C}_2\\omega _j,j\\ne 2;~~\\bar{C}_i=0,i\\ne 2.\\end{split}$ Compared with Lemma REF we see $-C_1\\omega _2=\\omega _{12}=-\\omega _{21}=\\bar{C}_2\\omega _1.$ Therefore, $C_1=\\bar{C}_2=0$ , and the Möbius forms of both $f$ and $\\bar{f}$ vanish: $C=0,~~~\\bar{C}=0.$ Thus both $f$ and $\\bar{f}$ are Möbius isoparametric hypersurfaces with two distinct principal curvatures.", "Since $\\omega _{1i}=\\omega _{2i}=0$ and $R_{1212}=0$ , from [19], $f$ and $\\bar{f}$ are Möbius equivalent to the circular cylinder $S^1(1)\\times R^{n-1}\\subset R^{n+1}$ , hence congruent to each other.", "This completes the proof to Proposition REF for Case I.", "(In particular this is not a Möbius deformable case.)", "Next we consider Case II, $\\bar{B}_{12}=-\\sin \\theta \\cos \\theta \\ne 0$ .", "Since $B_{ij}=\\frac{-1}{n}\\delta _{ij}, 2\\le i,j\\le n,\\bar{B}_{\\alpha \\beta }=\\frac{-1}{n}\\delta _{\\alpha \\beta }$ , We can rechoose $\\lbrace E_2,\\cdots ,E_n\\rbrace $ such that $(A_{ij})=\\left(\\begin{array}{ccccc}A_{11} & A_{12} & A_{13} & \\cdots & A_{1n} \\\\A_{21} & a_2 & 0 & \\cdots & 0 \\\\A_{31}& 0 & a_3& \\cdots & 0 \\\\\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\A_{n1} & 0 & 0 & \\cdots &a_n \\\\\\end{array}\\right),(\\bar{A}_{ij})=\\left(\\begin{array}{cccccc}\\bar{A}_{11} & \\bar{A}_{12} & \\bar{A}_{13}& \\bar{A}_{14} & \\cdots & \\bar{A}_{1n} \\\\\\bar{A}_{21} & \\bar{A}_{22} & \\bar{A}_{23} & \\bar{A}_{24} & \\cdots & \\bar{A}_{2n} \\\\\\bar{A}_{31}& \\bar{A}_{32} & \\bar{a}_3&0& \\cdots & 0 \\\\\\bar{A}_{41}&\\bar{A}_{42}& 0& \\bar{a}_4 &\\cdots & 0\\\\\\vdots & \\vdots & \\vdots &\\vdots & \\ddots & \\vdots \\\\\\bar{A}_{n1} &\\bar{A}_{n2}& 0 & 0 & \\cdots &\\bar{a}_n \\\\\\end{array}\\right).$ Noting that Lemma REF holds under this basis, using $R_{1i1i}-C_{1,1}+C_1^2=0$ and () we get that $a_2=a_3=\\cdots =a_n,\\bar{a}_3=\\cdots =\\bar{a}_n.$ In formula (), Let $i=2, k=\\alpha ,j=l=1$ and $i=k=\\alpha ,k=l=\\beta $ we get that $\\bar{A}_{\\alpha 2}=\\bar{A}_{2\\alpha }=0,a_2=a_3=\\cdots =a_n=\\bar{a}_3=\\cdots =\\bar{a}_n.$ Thus we have $(A_{ij})=\\left(\\begin{array}{ccccc}A_{11} & A_{12} & A_{13} & \\cdots & A_{1n} \\\\A_{21} & a_2 & 0 & \\cdots & 0 \\\\A_{31}& 0 & a_2& \\cdots & 0 \\\\\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\A_{n1} & 0 & 0 & \\cdots &a_2 \\\\\\end{array}~~\\right),\\bar{(A)}_{ij}=\\left(\\begin{array}{ccccc}\\bar{A}_{11} & \\bar{A}_{12} & \\bar{A}_{13} & \\cdots & \\bar{A}_{1n} \\\\\\bar{A}_{21} & \\bar{A}_{22} & 0 & \\cdots & 0 \\\\\\bar{A}_{31}& 0 & a_2& \\cdots & 0 \\\\\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\\bar{A}_{n1} & 0 & 0 & \\cdots &a_2 \\\\\\end{array}\\right).$ Since $\\bar{B}_{1\\alpha }=0,\\bar{B}_{2\\alpha }=0,\\bar{B}_{\\alpha \\beta }=\\frac{-1}{n}\\delta _{\\alpha \\beta }$ , we do covariant differentiation to find $\\begin{split}&(\\bar{B}_{11}+\\frac{1}{n})\\omega _{1\\alpha }+\\bar{B}_{12}\\omega _{2\\alpha }=\\sum {}_k\\bar{B}_{1\\alpha ,k}\\omega _k;\\\\&\\bar{B}_{12}\\omega _{1\\alpha }+(\\bar{B}_{22}+\\frac{1}{n})\\omega _{2\\alpha }=\\sum {}_k\\bar{B}_{2\\alpha ,k}\\omega _k;\\\\&\\bar{B}_{\\alpha \\beta ,k}=0,~~\\forall ~\\alpha ,\\beta ,k.\\end{split}$ Using $E_{\\beta }(\\bar{B}_{\\alpha \\alpha })=\\bar{B}_{\\alpha \\alpha ,\\beta }$ , () and (REF ), we get that $\\bar{C}_{\\alpha }=0,\\bar{B}_{1\\alpha ,\\alpha }=-\\bar{C}_1,\\bar{B}_{2\\alpha ,\\alpha }=-\\bar{C}_2,\\bar{B}_{1\\alpha ,\\beta }=\\bar{B}_{2\\alpha ,\\beta }=0,\\alpha \\ne \\beta .$ Thus from (REF ) and (REF ) we have $\\begin{split}&\\cos ^2\\theta \\omega _{1\\alpha }+\\sin \\theta \\cos \\theta \\omega _{2\\alpha }=\\bar{B}_{1\\alpha ,1}\\omega _1+\\bar{B}_{1\\alpha ,2}\\omega _2+\\bar{B}_{1\\alpha ,\\alpha }\\omega _{\\alpha };\\\\&\\sin \\theta \\cos \\theta \\omega _{1\\alpha }+\\sin ^2\\theta \\omega _{2\\alpha }=\\bar{B}_{2\\alpha ,1}\\omega _1+\\bar{B}_{2\\alpha ,2}\\omega _2+\\bar{B}_{2\\alpha ,\\alpha }\\omega _{\\alpha }.\\end{split}$ Using $E_{\\beta }(\\bar{B}_{11})=\\bar{B}_{11,\\alpha }$ , $\\bar{B}_{11}+\\bar{B}_{22}-\\frac{n-2}{n}=0$ and (REF ) we derive $\\begin{split}&\\bar{B}_{1\\alpha ,1}=\\bar{B}_{2\\alpha ,2}=\\bar{B}_{1\\alpha ,2}=0,\\sin \\theta \\bar{C}_1=\\cos \\theta \\bar{C}_2,\\\\&\\omega _{2\\alpha }=\\frac{\\cos ^2\\theta C_1-\\bar{C}_1}{\\cos \\theta \\sin \\theta }\\omega _{\\alpha }.\\end{split}$ Using $d\\bar{B}_{2\\alpha ,2}+\\sum _k\\bar{B}_{k\\alpha ,2}\\omega _{k2}+\\sum _k\\bar{B}_{2k,2}\\omega _{k\\alpha }+\\sum _k\\bar{B}_{2\\alpha ,k}\\omega _{k2}=\\sum _k\\bar{B}_{2\\alpha ,2k}\\omega _k$ ,(REF ) and (REF ), we get $\\bar{B}_{2\\alpha ,21}=0.$ Similarly we can get $\\bar{B}_{2\\alpha ,12}=0.$ Using Ricci identity $\\bar{B}_{2\\alpha ,21}-\\bar{B}_{2\\alpha ,12}=\\sum _k\\bar{B}_{k\\alpha }R_{k221}+\\sum _k\\bar{B}_{2k}R_{k\\alpha 21}$ and $R_{1\\alpha 12}=A_{2\\alpha }=\\bar{A}_{2\\alpha }=0$ we get $R_{2\\alpha 21}=0.$ By () this implies $A_{1\\alpha }=\\bar{A}_{1\\alpha }=0,\\\\\\lbrace A_{ij}\\rbrace =\\text{diag}\\bigl ( \\Big [{\\begin{matrix}A_{11}& A_{12}\\\\A_{21}& a_2\\end{matrix}}\\Big ],a_2,\\cdots ,a_2\\bigr ),~~\\lbrace \\bar{A}_{ij}\\rbrace =\\text{diag}\\bigl ( \\Big [{\\begin{matrix}\\bar{A}_{11}& \\bar{A}_{12}\\\\\\bar{A}_{21}& \\bar{A}_{22}\\end{matrix}}\\Big ],a_2,\\cdots ,a_2\\bigr ).$ From () we have $\\bar{B}_{11}+\\bar{B}_{22}=-\\frac{n-2}{n},\\bar{B}^2_{11}+\\bar{B}^2_{22}+2\\bar{B}^2_{12}=\\frac{n^2-2n+2}{n^2}.$ Using the above identity and $E_k(\\bar{B}_{ij})+\\sum _l\\bar{B}_{lj}\\omega _{li}(E_k)+\\sum _l\\bar{B}_{il}\\omega _{lj}(E_k)=\\bar{B}_{ij,k}$ , we get $\\begin{split}&\\bar{B}_{11,1}+\\bar{B}_{22,1}=0, \\bar{B}_{11,2}+\\bar{B}_{22,2}=0,\\\\&\\bar{B}_{11}\\bar{B}_{11,1}+\\bar{B}_{22}\\bar{B}_{22,1}+2\\bar{B}_{12}\\bar{B}_{12,1}=0,\\\\&\\bar{B}_{11}\\bar{B}_{11,2}+\\bar{B}_{22}\\bar{B}_{22,2}+2\\bar{B}_{12}\\bar{B}_{12,2}=0.\\end{split}$ From the above equation, (), $\\bar{B}_{11,2}=\\bar{B}_{12,1}+\\bar{C}_2,$ and $\\bar{B}_{22,1}=\\bar{B}_{12,2}+\\bar{C}_1,$ we derive $\\begin{split}&\\bar{B}_{11,2}=2\\cos \\theta \\sin \\theta \\bar{C}_1,\\bar{B}_{22,1}=2\\cos \\theta \\sin \\theta \\bar{C}_2,\\\\&\\bar{B}_{12,1}=(\\cos ^2\\theta -\\sin ^2\\theta )\\bar{C}_2,\\bar{B}_{12,2}=(\\sin ^2\\theta -\\cos ^2\\theta )\\bar{C}_1.\\end{split}$ Since $E_1(\\bar{B}_{12})=E_1(\\cos \\theta \\sin \\theta )=(\\cos ^2\\theta -\\sin ^2\\theta )E_1(\\theta )$ and $E_1(\\bar{B}_{12})=\\bar{B}_{12,1}$ , from (REF ) we get $E_1(\\theta )=\\bar{C}_2$ .", "Similarly we have $E_2(\\theta )=C_1-\\bar{C}_1$ , thus we have $E_1(\\theta )=\\bar{C}_2, E_2(\\theta )=C_1-\\bar{C}_1,E_{\\alpha }(\\theta )=0.$ Combining Lemma 5.4 and (REF ) we have $\\begin{split}&d\\omega _1=0, d\\omega _2=-C_1\\omega _1\\wedge \\omega _2,\\\\&d\\omega _{\\alpha }=C_1\\omega _1\\wedge \\omega _{\\alpha }+\\frac{\\cos ^2\\theta C_1-\\bar{C}_1}{\\cos \\theta \\sin \\theta }\\omega _2\\wedge \\omega _{\\alpha }+\\sum _{\\beta }\\omega _{\\beta }\\wedge \\omega _{\\beta \\alpha }.\\end{split}$ Therefore we have $[E_1,E_2]=C_1E_2.$ Using $d\\bar{C}_1+\\bar{C}_2\\omega _{21}=\\sum _k\\bar{C}_{1,k}\\omega _k$ and $d\\bar{C}_1+\\bar{C}_2\\omega _{21}=\\sum _k\\bar{C}_{1,k}\\omega _k$ , we have $\\begin{split}&E_1(\\bar{C}_1)=\\bar{C}_{1,1},E_2(\\bar{C}_1)=\\bar{C}_{1,2}-C_1\\bar{C}_2,\\\\&E_1(\\bar{C}_2)=\\bar{C}_{2,1}, E_2(\\bar{C}_2)=\\bar{C}_{2,2}+C_1\\bar{C}_1.\\end{split}$ Using $[E_1,E_2](\\theta )=C_1E_2(\\theta )$ , (REF ) and (REF ) we get that $\\bar{C}_{1,1}+\\bar{C}_{2,2}=C_{1,1}-C_1^2=R_{1\\alpha 1\\alpha }.$ Combining $\\sin \\theta \\bar{C}_1=\\cos \\theta \\bar{C}_2$ , (REF ) and (REF ), we obtain that $\\bar{C}_{1,2}=\\frac{\\cos \\theta }{\\sin \\theta }(\\bar{C}_1^2+\\bar{C}_2^2+\\bar{C}_{2,2}),\\bar{C}_{2,1}=\\frac{\\sin \\theta }{\\cos \\theta }(\\bar{C}_1^2+\\bar{C}_2^2+\\bar{C}_{1,1})$ From above formula and (), we have $R_{1\\alpha 2\\alpha }=\\frac{\\cos ^2\\theta -\\sin ^2\\theta }{\\sin \\theta \\cos \\theta }(\\bar{C}_1^2+\\bar{C}_2^2)+\\frac{\\cos \\theta }{\\sin \\theta }\\bar{C}_{2,2}-\\frac{\\sin \\theta }{\\cos \\theta }\\bar{C}_{1,1}.$ Compute the covariant differentiation of $\\bar{B}_{1\\alpha ,\\alpha },\\bar{B}_{1\\alpha ,1}$ .", "By (REF ) and the Ricci identity, $R_{1\\alpha 2\\alpha }+\\frac{\\sin \\theta }{\\cos \\theta }R_{2\\alpha 2\\alpha }+\\frac{\\sin ^2\\theta -\\cos ^2\\theta }{\\sin ^2\\theta \\cos ^2\\theta }\\bar{C}_1\\bar{C}_2=\\frac{\\bar{C}_{2,2}}{\\cos \\theta \\sin \\theta }.$ Similarly using Ricci identity $\\bar{B}_{2\\alpha ,2\\alpha }-\\bar{B}_{2\\alpha ,\\alpha 2}=\\sum _k\\bar{B}_{k\\alpha }R_{k22\\alpha }+\\sum _k\\bar{B}_{2k}R_{k\\alpha 2\\alpha }$ we get $R_{1\\alpha 2\\alpha }+\\frac{\\cos \\theta }{\\sin \\theta }R_{1\\alpha 1\\alpha }+\\frac{\\cos ^2\\theta -\\sin ^2\\theta }{\\sin ^2\\theta \\cos ^2\\theta }\\bar{C}_1\\bar{C}_2=\\frac{\\bar{C}_{1,1}}{\\cos \\theta \\sin \\theta }.$ Sum (REF ) and (REF ).", "Using (REF ) we get that $2R_{1\\alpha 2\\alpha }+\\frac{\\sin \\theta }{\\cos \\theta }(R_{2\\alpha 2\\alpha }-R_{1\\alpha 1\\alpha })=0.$ Note that $\\sin \\theta \\bar{C}_1=\\cos \\theta \\bar{C}_2$ , combining (REF ),(REF ) and (REF ), we obtain that $2R_{1\\alpha 2\\alpha }-\\frac{\\cos \\theta }{\\sin \\theta }(R_{2\\alpha 2\\alpha }-R_{1\\alpha 1\\alpha })=0.$ From (REF ) and (REF ), we get that $R_{1\\alpha 1\\alpha }-R_{2\\alpha 2\\alpha }=0,~R_{1\\alpha 2\\alpha }=0.$ Therefore from () we get $R_{2\\alpha 2\\alpha }=R_{\\alpha \\beta \\alpha \\beta }$ .", "Hence $R_{ijkl}=R_{2\\alpha 2\\alpha }(\\delta _{ik}\\delta _{jl}-\\delta _{il}\\delta _{ij}).$ By Schur's theorem $(M^n,g)$ is of constant curvature.", "This completes the proof." ], [ "Deformable hypersurfaces with one principal curvature\nof multiplicity $n-2$", "This section is devoted to the proof of the following Proposition 9.1 Let $f,\\bar{f}: M^n\\rightarrow R^{n+1}~~(n\\ge 4)$ be two immersed hypersurfaces without umbilics, whose principal curvatures have constant multiplicities.", "Suppose their Möbius metrics are equal, and one of principal curvatures of $B$ has multiplicity $n-2$ everywhere.", "Assume that $f(M^n)$ is NOT Möbius congruent to $\\bar{f}(M^n)$ .", "Then it must be either of the following three cases: (1) $f(M^n)$ is congruent to part of $L^2\\times R^{n-2}$ and $\\bar{f}(M^n)$ is congruent to part of $\\bar{L}^2\\times R^{n-2}$ , where $L^2$ and $\\bar{L}^2$ are a pair of isometric Bonnet surface in $R^3$ .", "(2) $f(M^n)$ is congruent to part of $CL^2\\times R^{n-3}$ where $CL^2\\subset R^4$ is a cone over $L^2\\subset S^3$ , and $\\bar{f}(M^n)$ is congruent to part of $C\\bar{L}^2\\times R^{n-3}$ .", "$L^2$ and $\\bar{L}^2$ form a Bonnet pair in $S^3$ .", "(3) $f(M^n)$ is a rotation hypersurfaces over $L^2\\subset R^3_+$ , and $\\bar{f}(M^n)$ is a rotation hypersurfaces over $\\bar{L}^2\\subset R^3_+$ , where $L^2$ and $\\bar{L}^2$ form a Bonnet pair in hyperbolic half-space $R^3_+$ .", "Recall that we have adopted the following convention on the range of indices as the last section: $1\\le i,j,k \\le n;\\ 3\\le \\alpha ,\\beta ,\\gamma \\le n.$ Since one of principal curvatures of $B$ has multiplicity $(n-2)$ everywhere, by Theorem REF we can assume without loss of generality that there exists a local orthonormal basis $\\lbrace E_1,\\cdots ,E_n\\rbrace $ for $(M^n,g)$ which is shared by $f,\\bar{f}$ , such that $\\lbrace B_{ij}\\rbrace =\\text{diag}(\\lambda _1,\\lambda _2,\\mu ,\\cdots ,\\mu );\\lbrace \\bar{B}_{ij}\\rbrace =\\text{diag}\\Bigl (\\left[{\\begin{matrix}\\bar{B}_{11} & \\bar{B}_{12}\\\\\\bar{B}_{21} &\\bar{B}_{22}\\end{matrix}}\\right],\\mu ,\\cdots ,\\mu \\Bigr ),$ where $\\lambda _1\\ne \\mu , \\lambda _2\\ne \\mu .$ By the identities () we know that the sub-matrices $\\left({\\begin{matrix}\\lambda _1 & 0\\\\0 & \\lambda _2\\end{matrix}}\\right)\\quad \\text{and}\\quad \\left({\\begin{matrix}\\bar{B}_{11}& \\bar{B}_{12}\\\\\\bar{B}_{21}& \\bar{B}_{22}\\end{matrix}}\\right)$ have equal traces and equal norms.", "Hence $B,\\bar{B}$ share the same eigenvalues.", "Therefore, the highest multiplicity of principal curvatures of $\\bar{f}$ is also $n-2$ .", "Denote $\\bar{B}_{11}=\\bar{\\lambda }_1,~\\bar{B}_{22}=\\bar{\\lambda }_2$ .", "We assert that $\\lambda _1\\ne \\lambda _2$ and $\\lambda _1\\ne \\bar{\\lambda }_1$ on an open dense subset of $M$ .", "Otherwise, suppose $\\lambda _1=\\lambda _2$ on an open subset.", "Then the 2 by 2 sub-matrices share the same eigenvalues $\\lambda _1=\\lambda _2$ , hence both be scalar matrix $\\text{diag}(\\lambda _1,\\lambda _1)$ .", "We get $B=\\bar{B}$ on an open subset.", "So these two hypersurfaces are Möbius equivalent.", "Contradiction.", "If $\\lambda _1=\\bar{\\lambda }_1$ on an open subset we will get a similar contradiction.", "From now on, without loss of generality we assume that on $M^n$ $\\lambda _1\\ne \\mu , ~\\lambda _2\\ne \\mu , ~\\lambda _1\\ne \\lambda _2,~\\lambda _1\\ne \\bar{\\lambda }_1.$ By () they satisfy $\\begin{split}&\\lambda _1+\\lambda _2+(n-2)\\mu =\\bar{\\lambda }_1+\\bar{\\lambda }_2+(n-2)\\mu =0,\\\\&\\lambda _1^2+\\lambda _2^2+(n-2)\\mu ^2=\\bar{\\lambda }_1^2+\\bar{\\lambda }_2^2+2\\bar{B}_{12}^2+(n-2)\\mu ^2=\\frac{n-1}{n}.\\end{split}$ Choose the dual basis $\\lbrace \\omega _i\\rbrace $ with connection forms $\\omega _{ij}$ satisfying $d\\omega _i=\\sum _k\\omega _{ik}\\wedge \\omega _k,~\\omega _{ij}=-\\omega _{ji}$ .", "It follows from $dB_{ij}+\\sum _kB_{kj}\\omega _{ki}+\\sum _kB_{ik}\\omega _{kj}=\\sum _kB_{ij,k}\\omega _k$ that $\\begin{split}&0=B_{\\alpha \\beta ,i}=B_{1\\alpha ,\\beta }=B_{2\\alpha ,\\beta }, ~~\\alpha \\ne \\beta ;\\\\&(\\lambda _1-\\mu )\\omega _{1\\alpha } =B_{1\\alpha ,1}\\omega _1+B_{1\\alpha ,2}\\omega _2+B_{1\\alpha ,\\alpha }\\omega _{\\alpha };\\\\&(\\lambda _2-\\mu )\\omega _{2\\alpha } =B_{2\\alpha ,1}\\omega _1+B_{2\\alpha ,2}\\omega _2+B_{2\\alpha ,\\alpha }\\omega _{\\alpha } ;\\\\&(\\lambda _1-\\lambda _2)\\omega _{12} =\\sum {}_kB_{12,k}\\omega _k.\\end{split}$ Using $d\\bar{B}_{ij}+\\sum _k\\bar{B}_{kj}\\omega _{ki}+\\sum _k\\bar{B}_{ik}\\omega _{kj}=\\sum _k\\bar{B}_{ij,k}\\omega _k$ in a similar way we obtain $\\begin{split}&0=\\bar{B}_{\\alpha \\beta ,i}=\\bar{B}_{1\\alpha ,\\beta }=\\bar{B}_{2\\alpha ,\\beta }, ~~\\alpha \\ne \\beta ;\\\\&(\\bar{\\lambda }_1-\\mu )\\omega _{1\\alpha }+\\bar{B}_{12}\\omega _{2\\alpha }=\\bar{B}_{1\\alpha ,1}\\omega _1+\\bar{B}_{1\\alpha ,2}\\omega _2+\\bar{B}_{1\\alpha ,\\alpha }\\omega _{\\alpha };\\\\&(\\bar{\\lambda }_2-\\mu )\\omega _{2\\alpha }+\\bar{B}_{12}\\omega _{1\\alpha }=\\bar{B}_{2\\alpha ,1}\\omega _1+\\bar{B}_{2\\alpha ,2}\\omega _2+\\bar{B}_{2\\alpha ,\\alpha }\\omega _{\\alpha };\\\\&d\\bar{B}_{12}+(\\bar{\\lambda }_1-\\bar{\\lambda }_2)\\omega _{12}=\\sum {}_k\\bar{B}_{12,k}\\omega _k.\\end{split}$ Comparing (REF ) with (REF ) yields $\\begin{split}&\\bar{B}_{1\\alpha ,1}=\\frac{\\bar{\\lambda }_1-\\mu }{\\lambda _1-\\mu }B_{1\\alpha ,1}+\\frac{\\bar{B}_{12}}{\\lambda _2-\\mu }B_{2\\alpha ,1};~\\bar{B}_{2\\alpha ,1}=\\frac{\\bar{\\lambda }_2-\\mu }{\\lambda _2-\\mu }B_{2\\alpha ,1}+\\frac{\\bar{B}_{12}}{\\lambda _1-\\mu }B_{1\\alpha ,1};\\\\&\\bar{B}_{1\\alpha ,2}=\\frac{\\bar{\\lambda }_1-\\mu }{\\lambda _1-\\mu }B_{1\\alpha ,2}+\\frac{\\bar{B}_{12}}{\\lambda _2-\\mu }B_{2\\alpha ,2};~\\bar{B}_{2\\alpha ,2}=\\frac{\\bar{\\lambda }_2-\\mu }{\\lambda _2-\\mu }B_{2\\alpha ,2}+\\frac{\\bar{B}_{12}}{\\lambda _1-\\mu }B_{1\\alpha ,2};\\\\&\\bar{B}_{1\\alpha ,\\alpha }=\\frac{\\bar{\\lambda }_1-\\mu }{\\lambda _1-\\mu }B_{1\\alpha ,\\alpha }+\\frac{\\bar{B}_{12}}{\\lambda _2-\\mu }B_{2\\alpha ,\\alpha };~\\bar{B}_{2\\alpha ,\\alpha }=\\frac{\\bar{\\lambda }_2-\\mu }{\\lambda _2-\\mu }B_{2\\alpha ,\\alpha }+\\frac{\\bar{B}_{12}}{\\lambda _1-\\mu }B_{1\\alpha ,\\alpha }.\\end{split}$ Another corollary of (REF ),(REF ) and () is $B_{\\alpha \\alpha ,\\beta }=C_{\\beta }=\\bar{C}_{\\beta }=\\bar{B}_{\\alpha \\alpha ,\\beta }=B_{\\beta \\beta ,\\beta }=\\bar{B}_{\\beta \\beta ,\\beta }~, ~~\\forall ~\\alpha \\ne \\beta .$ Taking the covariant derivatives for the identities (REF ) and invoking (REF ), we have $\\begin{split}&B_{11,\\alpha }+B_{22,\\alpha } =(2-n)C_{\\alpha },~\\lambda _1B_{11,\\alpha }+\\lambda _2B_{22,\\alpha } =(2-n)\\mu C_{\\alpha };\\\\&\\bar{B}_{11,\\alpha }+\\bar{B}_{22,\\alpha }=(2-n)\\bar{C}_{\\alpha },~\\bar{\\lambda }_1\\bar{B}_{11,\\alpha }+\\bar{\\lambda }_2\\bar{B}_{22,\\alpha } =(2-n)\\mu \\bar{C}_{\\alpha }-2\\bar{B}_{12}\\bar{B}_{12,\\alpha }.\\end{split}$ The two equations in the first line have solution $B_{11,\\alpha }= \\frac{\\mu -\\lambda _1}{\\lambda _2-\\lambda _1}(2-n)C_{\\alpha },~ B_{22,\\alpha }=\\frac{\\lambda _2-\\mu }{\\lambda _2-\\lambda _1} (2-n)C_{\\alpha }.$ Take the sum of the first and the fourth equations in (REF ) and insert (REF ) into it.", "By (), the first equation in (REF ), and the identities (REF ), the result is as below after simplification: $(\\lambda _1-\\lambda _2) \\mu \\bar{B}_{12}B_{12,\\alpha }=\\frac{n-1}{n}(\\bar{\\lambda }_1 -\\lambda _1)C_{\\alpha }.$ On the other hand, take the difference between the second and the third equations in (REF ).", "After simplification as before we get $(\\lambda _1-\\bar{\\lambda }_1 )(\\lambda _1-\\lambda _2) \\mu B_{12,\\alpha }=\\frac{n-1}{n}\\bar{B}_{12}C_{\\alpha }.$ It follows from (REF )(REF ) that $C_{\\alpha }=\\bar{C}_{\\alpha }=0,~\\forall ~\\alpha .$ Otherwise there will be $B_{12,\\alpha }^2=-(\\bar{\\lambda }_1-\\lambda _1)^2$ which is impossible.", "As a corollary of (REF )(REF ) and (REF ), $B_{11,\\alpha }=B_{22,\\alpha }=B_{\\beta \\beta ,\\alpha }=\\bar{B}_{\\beta \\beta ,\\alpha }=0,~~B_{1\\alpha ,1}=B_{2\\alpha ,2}=0, ~~\\forall ~\\alpha ,\\beta .$ We emphasize that (REF )(REF ) and $C_{\\alpha }=0$ implies $\\mu \\cdot B_{12,\\alpha }=0$ .", "Now we divide the proof into two cases.", "Case I, $B_{12,\\alpha }=0$ , for all $\\alpha .$ Case II, $B_{12,\\alpha }\\ne 0,$ for some $\\alpha .$ First we consider Case I.", "Since $C_{\\alpha }=0, B_{12,\\alpha }=0,$ from the Reduction Theorem REF we know that $f$ is Möbius equivalent to a hypersurface given by Example (REF ),(REF ) or (REF ) when $Q=0, Q<0$ or $Q>0$ , respectively.", "We define $\\omega _{ij}=\\sum _k\\Gamma ^i_{jk}\\omega _k$ .", "From (REF ) and the definition of $Q$ in the proof of Theorem REF , we get that $Q=2A_{\\alpha \\alpha }+\\mu ^2+(\\Gamma ^1_{\\alpha \\alpha })^2+(\\Gamma ^2_{\\alpha \\alpha })^2.$ On the other hand, $R_{\\alpha \\beta \\alpha \\beta }=\\mu ^2+2A_{\\alpha \\alpha }=\\mu ^2+2\\bar{A}_{\\alpha \\alpha }$ , so $A_{\\alpha \\alpha }=\\bar{A}_{\\alpha \\alpha }$ .", "Therefore $Q=\\bar{Q},$ and $f,\\bar{f}$ are congruent to two cylinders, or two cones, or two rotational hypersurfaces over some surfaces in a 3-dimensional space form.", "According to Remark REF and (REF ), in either case they share the same metric $g=\\left[4H_u^2-\\frac{2n}{n-1}(K_u+c)\\right](I_u+I_{N^{n-2}(c)}).$ So they must share the same surface metric $I_u$ , hence the same surface curvature $K_u$ , hence also the same mean curvature.", "Therefore they come from a Bonnet pair in the corresponding 3-space.", "This finishes our proof of Proposition REF in Case I.", "Next we consider Case II where $B_{12,\\alpha }\\ne 0$ for some $\\alpha $ .", "We have the following results.", "Proposition 9.2 Let $f,\\bar{f}: M^n\\rightarrow R^{n+1}~~(n\\ge 4)$ be two an immersed hypersurfaces without umbilics, whose principal curvatures have constant multiplicities.", "Suppose their Möbius metrics are equal, and one of principal curvatures of $B$ has multiplicity $n-2$ everywhere.", "We can assume that there exists a local orthonormal basis $\\lbrace E_1,\\cdots ,E_n\\rbrace $ for $(M^n,g)$ which is shared by $f,\\bar{f}$ , such that $\\lbrace B_{ij}\\rbrace =\\text{diag}(\\lambda _1,\\lambda _2,\\mu ,\\cdots ,\\mu );\\lbrace \\bar{B}_{ij}\\rbrace =\\text{diag}\\Bigl (\\left[{\\begin{matrix}\\bar{B}_{11} & \\bar{B}_{12}\\\\\\bar{B}_{21} &\\bar{B}_{22}\\end{matrix}}\\right],\\mu ,\\cdots ,\\mu \\Bigr ),$ where $\\lambda _1\\ne \\mu , \\lambda _2\\ne \\mu .$ If $B_{12,\\alpha }\\ne 0$ , for some $\\alpha $ .", "Then there exist an diffeomorphism $\\psi :M^n\\rightarrow M^n$ and a Möbius transformation $\\Phi $ such that $\\Phi \\circ f=\\bar{f}\\circ \\psi :M^n\\rightarrow R^{n+1}$ .", "Moreover, $f$ is Möbius equivalent to the minimal hypersurface defined by $x=(x_1,x_2):M^n=N^3\\times H^{n-3}(-\\frac{n-1}{6n})\\rightarrow S^{n+1},$ where $x_1=\\frac{y_1}{y_0},x_2=\\frac{y_2}{y_0},y_0\\in R^+,y_1\\in R^5, y_2\\in R^{n-3}.$ Here $y_1:N^3\\rightarrow S^4(\\sqrt{\\frac{6n}{n-1}})\\hookrightarrow R^5$ is Cartan's minimal isoparametric hypersurface in $S^4(\\sqrt{\\frac{6n}{n-1}})$ with three principal curvatures, and $(y_0,y_2):H^{n-3}(-\\frac{n-1}{6n})\\hookrightarrow R^{n-2}_1$ is the standard embedding of the hyperbolic space of sectional curvature $-\\frac{n-1}{6n}$ into the $(n-2)$ -dimensional Lorentz space with $-y_0^2+y_2^2=\\frac{6n}{n-1}$ .", "Since $B_{\\alpha \\beta }=\\bar{B}_{\\alpha \\beta }=0$ , We can assume that $B_{12,3}\\ne 0,~~B_{12,\\alpha }=0, \\alpha \\ne 3.$ From (REF ) we have $\\begin{split}&\\omega _{12}=\\frac{-C_2}{2\\lambda _1}\\omega _1-\\frac{C_1}{2\\lambda _1}\\omega _2+\\frac{B_{12,3}}{2\\lambda _1}\\omega _3;\\\\&\\omega _{13}=\\frac{B_{12,3}}{\\lambda _1}\\omega _2-\\frac{C_1}{\\lambda _1}\\omega _3,~\\omega _{23}=\\frac{B_{12,3}}{\\lambda _2}\\omega _2-\\frac{C_2}{\\lambda _2}\\omega _3;\\\\&\\omega _{1\\alpha }=\\frac{-C_1}{\\lambda _1}\\omega _{\\alpha },~~\\omega _{2\\alpha }=\\frac{-C_2}{\\lambda _2}\\omega _{\\alpha }, \\alpha >3.\\end{split}$ Since $C_{\\alpha }=0$ , using $dC_i+\\sum _mC_m\\omega _{mi}=\\sum _mC_{i,m}\\omega _m$ and (REF ), we get $\\begin{split}&C_{\\alpha ,\\alpha }=\\frac{C_2^2-C_1^2}{\\lambda _1},~~C_{\\alpha ,k}=0,k\\ne \\alpha ,\\alpha >3;\\\\&C_{3,3}=\\frac{C_2^2-C_1^2}{\\lambda _1},C_{3,1}=\\frac{B_{12,3}C_2}{\\lambda _2},C_{3,2}=\\frac{B_{12,3}C_1}{\\lambda _1},C_{3,\\alpha }=0,\\alpha >3.\\end{split}$ Differentiating the equations (REF ), we get $\\begin{split}\\frac{-1}{2}\\sum _{kl}R_{12kl}\\omega _k\\wedge \\omega _l=\\frac{-1}{2\\lambda _1}\\sum _m[C_{2,m}\\omega _m\\wedge \\omega _1+C_{1,m}\\omega _m\\wedge \\omega _2]-3\\frac{B_{12,3}C_1}{2\\lambda _1^2}\\omega _1\\wedge \\omega _3\\\\+[\\frac{C_1^2+C_2^2}{2\\lambda _1^2}+2\\frac{B_{12,3}^2}{\\lambda _1^2}]\\omega _1\\wedge \\omega _2+3\\frac{B_{12,3}C_2}{2\\lambda _1^2}\\omega _2\\wedge \\omega _3+\\frac{dB_{12,3}}{2\\lambda _1}\\wedge \\omega _3.\\end{split}$ $\\begin{split}\\frac{-1}{2}\\sum _{kl}R_{13kl}\\omega _k\\wedge \\omega _l&=\\frac{1}{\\lambda _1}dB_{12,3}\\wedge \\omega _2-\\frac{1}{\\lambda _1}\\sum _m C_{1,m}\\omega _m\\wedge \\omega _3-2\\frac{B_{12,3}C_1}{\\lambda _1^2}\\omega _1\\wedge \\omega _2\\\\&+[\\frac{C_1^2+C_2^2}{2\\lambda _1^2}-\\frac{B_{12,3}^2}{\\lambda _1^2}]\\omega _1\\wedge \\omega _3.\\end{split}$ $\\begin{split}\\frac{-1}{2}\\sum _{kl}R_{23kl}\\omega _k\\wedge \\omega _l&=\\frac{1}{\\lambda _1}dB_{12,3}\\wedge \\omega _1-\\frac{1}{\\lambda _2}\\sum _m C_{2,m}\\omega _m\\wedge \\omega _3+2\\frac{B_{12,3}C_2}{\\lambda _1^2}\\omega _1\\wedge \\omega _2\\\\&+[\\frac{C_1^2+C_2^2}{2\\lambda _1^2}-\\frac{B_{12,3}^2}{\\lambda _1^2}]\\omega _2\\wedge \\omega _3.\\end{split}$ $\\begin{split}\\frac{-1}{2}\\sum _{kl}R_{1\\alpha kl}\\omega _k\\wedge \\omega _l&=\\frac{-1}{\\lambda _1}\\sum _m C_{1,m}\\omega _m\\wedge \\omega _{\\alpha }+\\frac{C_1^2+C_2^2}{\\lambda _1^2}\\omega _1\\wedge \\omega _{\\alpha }\\\\&-\\frac{B_{12,3}C_2}{\\lambda _1^2}\\omega _3\\wedge \\omega _{\\alpha }-\\frac{B_{12,3}}{\\lambda _1}\\omega _2\\wedge \\omega _{3\\alpha }.\\end{split}$ $\\begin{split}\\frac{-1}{2}\\sum _{kl}R_{2\\alpha kl}\\omega _k\\wedge \\omega _l&=\\frac{-1}{\\lambda _1}\\sum _m C_{2,m}\\omega _m\\wedge \\omega _{\\alpha }+\\frac{C_1^2+C_2^2}{\\lambda _1^2}\\omega _2\\wedge \\omega _{\\alpha }\\\\&-\\frac{B_{12,3}C_1}{\\lambda _1^2}\\omega _3\\wedge \\omega _{\\alpha }+\\frac{B_{12,3}}{\\lambda _1}\\omega _1\\wedge \\omega _{3\\alpha }.\\end{split}$ Comparing the coefficients of $\\omega _3\\wedge \\omega _{\\alpha }$ on both sides of (REF ), and using () we obtain $A_{1\\alpha }=0,~~A_{3\\alpha }=0, ~E_{\\alpha }(B_{12,3})=0, ~\\alpha >3.$ Similarly from (REF ),(REF ), (REF ) and (REF ), we have $\\begin{split}&A_{2\\alpha }=0,~~ A_{\\alpha \\beta }=0, \\alpha ,\\beta >3, \\alpha \\ne \\beta ;\\\\&\\omega _{3\\alpha }(E_1)=0,~\\omega _{3\\alpha }(E_2)=0,~\\omega _{3\\alpha }(E_3)=0,~\\omega _{3\\alpha }(E_{\\beta })=0,~\\alpha ,\\beta >3, \\alpha \\ne \\beta ;\\\\&R_{1\\alpha 1\\alpha }=-\\frac{C_1^2+C_2^2}{\\lambda _1^2}+\\frac{C_{1,1}}{\\lambda _1},R_{2\\alpha 2\\alpha }=-\\frac{C_1^2+C_2^2}{\\lambda _1^2}+\\frac{C_{2,2}}{\\lambda _2},\\alpha >3;\\\\&R_{1313}=\\frac{B_{12,3}^2}{\\lambda _1^2}-\\frac{C_1^2+C_2^2}{\\lambda _1^2}+\\frac{C_{1,1}}{\\lambda _1},R_{2323}=\\frac{B_{12,3}^2}{\\lambda _1^2}-\\frac{C_1^2+C_2^2}{\\lambda _1^2}+\\frac{C_{2,2}}{\\lambda _2};\\\\&R_{1212}=\\frac{C_{1,1}-C_{2,2}}{2\\lambda _1}-2\\frac{B_{12,3}^2}{\\lambda _1^2}-\\frac{C_1^2+C_2^2}{2\\lambda _1^2}.\\end{split}$ $\\begin{split}&E_2(B_{12,3})=\\lambda _1A_{13}-2\\frac{B_{12,3}C_2}{\\lambda _1},~E_1(B_{12,3})=\\lambda _2A_{23}+2\\frac{B_{12,3}C12}{\\lambda _1};\\\\&A_{12}=\\frac{C_{1,2}}{\\lambda _1}+\\frac{B_{12,3}}{\\lambda _1}\\omega _{3\\alpha }(E_{\\alpha }),~A_{12}=\\frac{-C_{2,1}}{\\lambda _1}-\\frac{B_{12,3}}{\\lambda _1}\\omega _{3\\alpha }(E_{\\alpha });\\\\&A_{12}=\\frac{-C_{1,2}}{\\lambda _1}+\\frac{-1}{\\lambda _1}E_3(B_{12,3}),~A_{12}=\\frac{C_{2,1}}{\\lambda _1}+\\frac{-1}{\\lambda _1}E_3(B_{12,3}).\\end{split}$ From (REF ), (REF ) and (REF ), we have $E_3(B_{12,3})=B_{12,3}\\omega _{3\\alpha }(E_{\\alpha }).$ Define $\\phi :=\\omega _{3\\alpha }(E_{\\alpha })=\\frac{E_3(B_{12,3})}{B_{12,3}}$ .", "From (REF ), we have $\\omega _{3\\alpha }=\\phi \\omega _{\\alpha }.$ Differentiating the equations (REF ), we get $\\begin{split}\\frac{-1}{2}\\sum _{kl}R_{3\\alpha kl}\\omega _k\\wedge \\omega _l&=d\\phi \\wedge \\omega _{\\alpha }-\\frac{C_1}{\\lambda _1}\\phi \\omega _1\\wedge \\omega _{\\alpha }-\\frac{C_2}{\\lambda _2}\\phi \\omega _2\\wedge \\omega _{\\alpha }+\\phi ^2\\omega _3\\wedge \\omega _{\\alpha }\\\\&-\\frac{B_{12,3}C_1}{\\lambda _1^2}\\omega _2\\wedge \\omega _{\\alpha }+\\frac{C_1^2+C_2^2}{\\lambda _1^2}\\omega _3\\wedge \\omega _{\\alpha }-\\frac{B_{12,3}C_2}{\\lambda _1^2}\\omega _1\\wedge \\omega _{\\alpha }.\\end{split}$ Comparing the coefficients of $\\omega _1\\wedge \\omega _{\\alpha }$ and $\\omega _2\\wedge \\omega _{\\alpha }$ on both sides of (REF ), and using () we obtain $-A_{13}=E_1(\\phi )-\\frac{C_1}{\\lambda _1}\\phi -\\frac{B_{12,3}C_2}{\\lambda _1^2},-A_{23}=E_2(\\phi )-\\frac{C_2}{\\lambda _2}\\phi -\\frac{B_{12,3}C_2}{\\lambda _1^2}.$ Using $dA_{ij}+\\sum _mA_{mj}\\omega _{mi}+\\sum _mA_{im}\\omega _{mj}=\\sum _mA_{ij,m}\\omega _m$ and (REF ), we obtain $\\begin{split}A_{1\\alpha ,\\alpha }=(A_{\\alpha \\alpha }-A_{11})\\frac{C_1}{\\lambda _1}+A_{12}\\frac{C_2}{\\lambda _1}+A_{13}\\omega _{3\\alpha }(E_{\\alpha }),~~A_{1\\alpha ,k}=0,k\\ne \\alpha ;\\\\A_{2\\alpha ,\\alpha }=(A_{22}-A_{\\alpha \\alpha })\\frac{C_2}{\\lambda _1}-A_{12}\\frac{C_1}{\\lambda _1}+A_{23}\\omega _{3\\alpha }(E_{\\alpha }),~~A_{2\\alpha ,k}=0,k\\ne \\alpha ;\\\\A_{3\\alpha ,\\alpha }=(A_{33}-A_{\\alpha \\alpha })\\omega _{3\\alpha }(E_{\\alpha })-A_{13}\\frac{C_1}{\\lambda _1}+A_{23}\\frac{C_2}{\\lambda _1},~~A_{3\\alpha ,k}=0,k\\ne \\alpha .\\end{split}$ On the other hands, from (REF ) we have $A_{33}-A_{\\alpha \\alpha }=\\frac{B_{12,3}^2}{\\lambda _1^2}, \\alpha >3.$ Noting that $E_{\\alpha }(B_{12,3})=0$ and $A_{33,\\alpha }=A_{3\\alpha ,3}=0$ , we get $E_{\\alpha }(A_{\\alpha \\alpha })=E_{\\alpha }(A_{\\beta \\beta })=0,\\alpha \\ne \\beta ,\\alpha ,\\beta >3.$ Combining (REF ), (REF ), (REF ),(REF ) and (REF ), we get $\\begin{split}A_{1\\alpha ,\\alpha 1}&=(A_{\\alpha \\alpha ,1}-A_{11,1})\\frac{C_1}{\\lambda _1}+(A_{11}-A_{\\alpha \\alpha })[\\frac{C_2^2}{\\lambda _1^2}-\\frac{C_{1,1}}{\\lambda _1}]+A_{12}\\frac{C_1C_2}{\\lambda _1^2}\\\\&+A_{12,1}\\frac{C_2}{\\lambda _1}+A_{12}C_{2,1}+A_{13,1}\\phi -A_{13}^2+\\frac{A_{13}\\phi C_1}{\\lambda _1}-A_{12}\\frac{B_{12,3}\\phi }{\\lambda _1};\\\\A_{1\\alpha ,1\\alpha }&=(2A_{\\alpha \\alpha ,1}-A_{11,1})\\frac{C_1}{\\lambda _1}+A_{12,2}\\frac{C_2}{\\lambda _1}+A_{13,1}\\phi .\\end{split}$ Combining (REF ), (REF ) and Ricci identity $A_{1\\alpha ,1\\alpha }-A_{1\\alpha ,\\alpha 1}=\\sum _mA_{m\\alpha }R_{m11\\alpha }+\\sum _mA_{1m}R_{m\\alpha 1\\alpha }$ , we obtain $A_{13}=0.$ Similarly using Ricci identity $A_{2\\alpha ,2\\alpha }-A_{2\\alpha ,\\alpha 2}=\\sum _mA_{m\\alpha }R_{m22\\alpha }+\\sum _mA_{2m}R_{m\\alpha 2\\alpha }$ , we have $A_{23}=0.$ Using (REF ), (REF ) and $dA_{ij}+\\sum _mA_{mj}\\omega _{mi}+\\sum _mA_{im}\\omega _{mj}=\\sum _mA_{ij,m}\\omega _m$ and (REF ), we obtain $A_{13,2}=(A_{11}-A_{33})\\frac{B_{12,3}}{\\lambda _1},~A_{23,1}=(A_{22}-A_{33})\\frac{B_{12,3}}{\\lambda _2}.$ From (REF ), we know that $A_{13,2}=A_{23,1}$ , thus equations (REF ) mean that $A_{11}+A_{22}=2A_{33}.$ Combining (REF ) and (REF ), we obtain $2\\lambda _1^2=6\\frac{B_{12,3}^2}{\\lambda _1^2}-\\frac{C_1^2+C_2^2}{\\lambda _1^2}.$ Taking derivatives for (REF ) along $E_3$ and using () and (REF ), we have $E_3({B_{12,3}})=0.$ This means that $\\phi =0$ .", "From (REF ), (REF ) and (REF ), we get $B_{12,3}\\frac{C_1}{\\lambda _1^2}=0,~~B_{12,3}\\frac{C_2}{\\lambda _1^2}=0.$ One deduces the Möbius form $\\Phi =0.$ Since $\\mu =0$ , we get from () that $\\lambda _1=\\sqrt{\\frac{n-1}{2n}},\\lambda _2=-\\sqrt{\\frac{n-1}{2n}}.$ Thus $f$ is a Möbius isoparametric hypersurface with Möbius principal curvatures $\\sqrt{\\frac{n-1}{2n}},-\\sqrt{\\frac{n-1}{2n}},0,\\cdots ,0.$ It is then easy to show (or by the classification result in [15] of Möbius isoparametric hypersurfaces with three distinct principal curvatures) that $f$ is Möbius equivalent to the minimal hypersurface defined by $x=(x_1,x_2):M^n=N^3\\times H^{n-3}(-\\frac{n-1}{6n})\\rightarrow S^{n+1},$ where $x_1=\\frac{y_1}{y_0},x_2=\\frac{y_2}{y_0},y_0\\in R^+,y_1\\in R^5, y_2\\in R^{n-3}.$ Here $y_1:N^3\\rightarrow S^4(\\sqrt{\\frac{6n}{n-1}})\\hookrightarrow R^5$ is Cartan's minimal isoparametric hypersurface in $S^4(\\sqrt{\\frac{6n}{n-1}})$ with three principal curvatures, and $(y_0,y_2):H^{n-3}(-\\frac{n-1}{6n})\\hookrightarrow R^{n-2}_1$ is the standard embedding of the hyperbolic space of sectional curvature $-\\frac{n-1}{6n}$ into the $(n-2)$ -dimensional Lorentz space with $-y_0^2+y_2^2=\\frac{6n}{n-1}$ .", "On the other hand, Since $\\bar{\\mu }=\\mu =0$ , then $B_{\\alpha \\alpha ,1}=\\bar{B}_{\\alpha \\alpha ,1}=E_1(\\bar{\\mu })=0, B_{\\alpha \\alpha ,2}=\\bar{B}_{\\alpha \\alpha ,2}=E_2(\\bar{\\mu })=0$ .", "Using (), we have $ \\bar{B}_{1\\alpha ,\\alpha }=-\\bar{C}_1,\\bar{B}_{2\\alpha ,\\alpha }=-\\bar{C}_2,B_{1\\alpha ,\\alpha }=C_1=0,B_{2\\alpha ,\\alpha }=C_2=0.$ For the last two of equations (REF ), we get that $\\bar{\\Phi }=0.$ Since $B,\\bar{B}$ share equal eigenvalues, then $\\bar{f}$ determines the same Möbius isoparametric hypersurface as $f$ .", "So $f(M)$ is Möbius equivalent to $\\bar{f}(M)$ .", "In fact, at every point of $M^n$ there exist a suitable $P=\\begin{pmatrix}\\cos \\theta & \\sin \\theta \\\\-\\sin \\theta & \\cos \\theta \\end{pmatrix}\\in SO(2)$ such that $\\begin{pmatrix}\\bar{B}_{11}& \\bar{B}_{12}\\\\\\bar{B}_{21}& \\bar{B}_{22}\\end{pmatrix}= P^{-1}\\begin{pmatrix}\\sqrt{\\frac{n-1}{2n}} & 0\\\\0 & -\\sqrt{\\frac{n-1}{2n}}\\end{pmatrix}P.$ One can show by computation that $\\theta $ is a constant.", "Then one can construct a local diffeomorphism $\\psi :M^n\\rightarrow M^n$ such that $\\bar{f}\\circ \\psi $ not only shares the same metric as $f$ , but also shares the same Möbius principal curvatures and the same principal directions.", "Thus there exists Möbius transformation $\\Psi $ such that $\\bar{f}\\circ \\psi =\\Psi \\circ f.$ This is exactly the case as in Remark REF .", "Thus we do not get new Möbius deformable examples.", "Thus we have verified Proposition REF in all cases and completed the proof to the Main Theorem REF .", "Acknowledgements: The authors thank the referees for helpful suggestions." ] ]

[ [ "Magnetic moment non-conservation in magnetohydrodynamic turbulence\n models" ], [ "Abstract The fundamental assumptions of the adiabatic theory do not apply in presence of sharp field gradients as well as in presence of well developed magnetohydrodynamic turbulence.", "For this reason in such conditions the magnetic moment $\\mu$ is no longer expected to be constant.", "This can influence particle acceleration and have considerable implications in many astrophysical problems.", "Starting with the resonant interaction between ions and a single parallel propagating electromagnetic wave, we derive expressions for the magnetic moment trapping width $\\Delta \\mu$ (defined as the half peak-to-peak difference in the particle magnetic moment) and the bounce frequency $\\omega_b$.", "We perform test-particle simulations to investigate magnetic moment behavior when resonances overlapping occurs and during the interaction of a ring-beam particle distribution with a broad-band slab spectrum.", "We find that magnetic moment dynamics is strictly related to pitch angle $\\alpha$ for a low level of magnetic fluctuation, $\\delta B/B_0 = (10^{-3}, \\, 10^{-2})$, where $B_0$ is the constant and uniform background magnetic field.", "Stochasticity arises for intermediate fluctuation values and its effect on pitch angle is the isotropization of the distribution function $f(\\alpha)$.", "This is a transient regime during which magnetic moment distribution $f(\\mu)$ exhibits a characteristic one-sided long tail and starts to be influenced by the onset of spatial parallel diffusion, i.e., the variance $<(\\Delta z)^2 >$ grows linearly in time as in normal diffusion.", "With strong fluctuations $f(\\alpha)$ isotropizes completely, spatial diffusion sets in and $f(\\mu)$ behavior is closely related to the sampling of the varying magnetic field associated with that spatial diffusion." ], [ "Introduction", "In this paper we study magnetic moment $\\mu $ conservation for charged particles in presence of a single electromagnetic wave as well as in presence of turbulent magnetic fields having one dimensional spectra comparable to those measured in the solar wind.", "Magnetic moment conservation is an important topic in plasma physics.", "Indeed, some of the most commonly used theories that describe particle motion in perturbed magnetic fields are based on the assumption that particles magnetic moment is on average constant over a gyroperiod.", "When $\\mu $ is not conserved, this approximation is not allowed and its effects can have a bearing on several astrophysical phenomena such as coronal heating, cosmic ray transport, temperature anisotropies observed in the solar wind [1] and particle acceleration near reconnection sites [2].", "Furthermore this issue is strictly related to particle confinement in plasma machines and dynamically chaotic systems [3].", "Therefore we want to established the validity range of the adiabatic approximation and the key mechanisms that regulate magnetic moment non-conservation.", "The guiding center approximation [4] splits particle motion into the motion of the guiding center and the gyromotion around it.", "When analyzing charged particle motion in nonuniform electromagnetic fields, we would like to neglect the rapid and relatively uninteresting gyromotion, focusing instead on the far slower motion of the guiding center.", "Averaging the particle equation of motion over the gyrophase, we obtain a reduced equation that describes the guiding center motion.", "In the non-relativistic case the equation of motion of the guiding center in the direction parallel to the magnetic field reads $\\frac{dp_{\\parallel }}{dt} = -\\mu \\nabla _{\\parallel } B + qE_{\\parallel },$ where particle magnetic moment is defined as $\\mu = v_{\\perp }^2/B$ and $\\nabla _{\\parallel } = (\\mathbf {\\hat{B}} \\cdot \\nabla )$ is the spatial derivative along the field direction.", "In the perpendicular direction the guiding center drifts with the velocity $v_D = \\frac{{\\bf F} \\times {\\bf B}}{qB^2},$ where ${\\bf F} = [q{\\bf E} - \\mu \\nabla B - (mv^2_{\\parallel })\\nabla _{\\parallel }{\\bf B}]$ is the total force acting on the guiding center, averaged over a gyroperiod, in the (non-inertial) frame co-moving with the guiding center.", "Therefore, as long as a particle moves through slowly varying electric and magnetic fields, its guiding center behaves like a particle with a magnetic moment $\\mu $ conserved.", "This approximation is valid when the smallest length-scales of the electromagnetic fields are much larger than the particles Larmor radius, i.e., when particle magnetic moment is a constant of motion on average over the particle gyroperiod.", "This corresponds to the well-known Born-Oppenheimer approximation in quantum mechanics.", "This description for particle motion in a non-uniform magnetic field is also useful for numerical simulations.", "Indeed direct simulations of kinetic equations (Vlasov, Boltzmann) with a large magnetic field require the numerical resolution of small spatial and time scales induced by the gyration along the magnetic field.", "The guiding center approximation, as well as gyrokinetics, are approximate models describing particle motion in presence of a strong magnetic field.", "However, the assumption that the scale of variation of the magnetic field is much larger than the particle Larmor radius can break down in presence of turbulence.", "Turbulent magnetic fluctuations are observed in space plasmas in practically all environments and at all scales.", "Furthermore the presence of waves in collisionless plasma introduces through wave-particle interactions a finite dissipation.", "In this case it seems invalid to resort to a guiding center theory.", "When the amplitude of the magnetic fluctuations is lower than that of the mean magnetic field (averaged over the fluctuations time-scale), a perturbation approach called quasilinear approximation is applicable [5], [6], [7].", "In this case the resonant fluctuations make the dominant contribution to particle scattering.", "The resonance condition for wave-particle interaction is given by: $\\omega - k_{\\parallel }v_{\\parallel } = n\\Omega $ where $\\omega $ is the wave frequency, $k_{\\parallel }$ and $v_{\\parallel }$ are respectively the wavevector and the particle velocity along the mean magnetic field ${\\bf B}_0$ , and $\\Omega = qB/m$ is the particle gyrofrequency.", "Landau resonance [8] is found at $n=0$ , while $n= \\pm 1$ , $\\pm 2,\\, \\ldots $ are the cyclotron resonances.", "In linear theory these resonances are represented by delta functions.", "In presence of well-developed magnetohydrodynamic turbulence we expect that the discrete resonances to be significantly broadened due to the rapid decorrelation of the waves phases in strong turbulence [9].", "The particle reaction to the perturbation is always periodic except when condition (REF ) is satisfied.", "In this case the perpendicular electric force due to the wave remains in phase with the particle cyclotron motion and particle reaction is secular or resonant and, over short times, non-oscillatory.", "The secular electric force acting on a given particle is constant over a particle gyroperiod, so that the magnetic moment is no longer conserved.", "Charged particles are scattered by their interaction with the waves and undergo pitch angle diffusion.", "The pitch angle, $\\theta = \\arctan (v_{\\perp } / v_{\\parallel })$ , is the angle between the direction of the magnetic field and the particle's helical trajectory.", "Scattering from magnetic fluctuations causes the distribution of pitch angle cosine, $\\alpha = v_{\\parallel }/|v|$ , to become isotropic.", "Magnetic moment, $\\mu $ , is formally related to the time averages of the cosine of pitch angle by: $ \\mu \\sim \\frac{v^2_{\\perp }}{|B|} = \\frac{v^2}{|B|}(1 - \\alpha ^2)$ We therefore expect the behavior of magnetic moment to be strongly related to pitch angle behavior." ], [ "Stochastic motion, trapping width and resonance overlapping", "Wave-particle interactions usually involve multiple resonances.", "Particle motion is substantially different depending on when these resonances overlap or not.", "Numerical simulations show a complex behavior that cannot be approached analytically, e.g., it is not possible to write an equation for the evolution of particles distributions when two resonances overlap [10].", "Such motions in the presence of overlapping resonances are commonly labeled stochastic.", "It is important to distinguish between two different kinds of stochasticity.", "Wave-particle interaction in presence of of uncorrelated small amplitude electromagnetic waves or plasma turbulence is called extrinsically diffusive [11].", "In this case the regular phase space structure for a charged particle interacting resonantly with an electromagnetic wave is perturbed by neighboring uncorrelated waves.", "This leads to extrinsic stochasticity and diffusive behavior.", "On the other hand nonlinear systems, such as particle interacting resonantly with a large amplitude obliquely propagating (with respect to $\\mathbf {B}_0$ ) electromagnetic plasma wave, can exhibit intrinsic stochasticity.", "Indeed, when the wave amplitude is sufficiently large, the resonances at the gyrofrequency harmonics are sufficiently broadened that they overlap with adjacent primary resonances.", "Therefore particles interacting even with a single monochromatic wave may exhibit intrinsically stochastic and diffusive behavior [12].", "This is the regime of nonlinear diffusion and irreversible chaotic mixing of orbits.", "Because one of the main hypothesis of quasilinear theory is that particles dynamics is adequately modeled by their unperturbed trajectories, the quasilinear timescale $\\tau _{c}$ must be much smaller than the timescale for the onset of nonlinear orbit effects $\\tau _{nl}$  [14], [13]: $\\tau _{c} \\ll \\tau _{nl} \\sim \\frac{1}{\\omega _b},$ where ${\\omega _b}$ is the bounce frequency.", "This means that the turbulent spectrum should be broad enough so that the typical timescale for a charged particle to interact with a resonant wave-packet would be much less than its typical bounce time, $\\tau _b=2\\pi /\\omega _b$ , in a monochromatic wave at the characteristic wavenumber and frequency of the wave-packet.", "The bounce time, $\\tau _b$ , for a particle in resonance with an electromagnetic wave is proportional to its oscillation period in the pseudo-potential well governing the resonant wave-particle interaction [12].", "This interaction can be approximated by a Hamiltonian pendulum in the vicinity of the resonance point.", "Particles in resonance with a single finite amplitude fluctuation undergo a finite amplitude nonlinear oscillation.", "This is the so-called trapping width, $\\Delta v_{\\parallel }$ , given by the half peak-to-peak difference in the particle velocity parallel component.", "The trapping width and the bounce frequency for a nonrelativistic particle interacting resonantly with an electromagnetic wave are given by Equations (5a)–(5c) of Ref. [15].", "These approximate expressions for $\\Delta v_{\\parallel }$ and $\\omega _b$ yield considerable physical insight into the diffusion process [16] when used in conjunction with the quasilinear diffusion coefficient." ], [ "Magnetic moment trapping width", "From the trapping width, $\\Delta v_{\\parallel }$ , and bounce frequency, $\\omega _b$ , computed by Ref.", "[16] for the case of a circularly polarized electromagnetic wave (see Appendix), it is possible to derive the pitch angle trapping half width as: $\\Delta \\alpha = \\frac{\\Delta v_\\parallel }{v} = 2 \\left[(1-\\alpha ^2)^{1/2}|\\alpha |\\frac{\\delta B}{B_0}\\right]^{1/2}$ As magnetic moment $\\mu $ is related to $\\alpha $ by Eq.", "(REF ), we can write the trapping width for the magnetic moment as: $\\Delta \\mu = 2 \\alpha \\Delta \\alpha = 4\\alpha \\left[(1-\\alpha ^2)^{1/2}|\\alpha |\\frac{\\delta B}{B_0}\\right]^{1/2}$ These expressions apply to a circularly polarized wave.", "From Eq.", "(REF ) we expect that $\\mu $ continues to be a good adiabatic invariant when resonances are not present or when particle interacts with extremely small amplitude waves." ], [ "Model and governing equations", "We investigate magnetic moment behavior first during the resonant interaction between one ion and a circularly polarized magnetic wave, then when resonance overlapping occurs and finally during the interaction between a distribution of particles and a broad-band turbulent spectrum.", "Because some of our normalization quantities are expressed in terms of typical time and length scales of the turbulence slab model [5], [17], we first give a general summary of the slab model.", "For the general one dimensional (1D) slab description, turbulence is made up of a sum of right and left handed circularly polarized nondispersive plane Alfvén waves propagating in the parallel direction.", "The magnetic field fluctuations are perpendicular to both the wave vector and the mean field.", "The fields are assumed to be magnetostatic.", "This amounts to the auxiliary assumption that the average particle speed is well in excess of the phase speed of the underlying linear wave mode.", "We ignore nonlinear wave-wave couplings in the spirit of quasilinear theory , [19], [20].", "Table: Characteristic physical quantities.Considering Alfvén waves propagating with $\\omega /k = \\omega /k_\\parallel \\simeq \\pm v_A$ , the magnetostatic approximation implies $|{\\bf v}| \\gg v_A$ (strictly $|v_\\parallel | \\gg v_A$ ).", "Since particle energy is conserved in a frame moving at the parallel component of the phase velocity of the wave ($\\omega / k_\\parallel $ ), quasilinear theory [18] implies: $(v_\\parallel - \\omega / k_\\parallel )^2 + {v_\\perp }^2 = \\mbox{const}.$ Because of the magnetostatic assumption, particle energy is conserved, i.e., energy diffusion in forbidden and in velocity space the resonant interaction diffuses pitch angle and gyrophase only.", "Finally, we ignore all inter-particle correlations resulting from their mutual interaction through their microfields (e.g.,Coulomb collisions, Debye shielding, and polarization).", "Furthermore the feedback of the particles on the macroscopic fields is ignored, i.e., we consider only test particles in prescribed macroscopic magnetostatic fields.", "By virtue of the inequality $v_{\\parallel } \\gg v_A$ , the turbulent electric field of the order $(\\delta B/B_0) v_A B_0$ is negligible compared to the motional electric field of the particle, $v_{\\parallel }B_0$ .", "The dispersionless hypothesis rules out phase mixing and, hence, phase decorrelation due to this process.", "Consequently the only way for a particle to see a “wavepacket” phase-decorrelate is to traverse an autocorrelation length of the turbulence [21].", "The autocorrelation time in this case is given by $\\tau _{c} = \\frac{1}{|\\Delta (\\omega - k_\\parallel v_\\parallel )|} =\\frac{1}{|v_\\parallel \\Delta k_\\parallel |} \\simeq \\frac{\\lambda _c}{|v_\\parallel |},$ where $\\lambda _c$ is the turbulence correlation length.", "The behavior of a test particle is described by its time dependent position ${\\bf r}(t)$ and three-dimensional velocity ${\\bf v}(t)$ , that are advanced according to $d{\\bf r}/dt = {\\bf v}$ and the Lorentz force equation: $m\\frac{d{\\bf v}}{dt} = q\\left[{\\bf E} + \\frac{\\mathbf {v}}{c} \\times {\\bf B} \\right]$ In order to render the equations non-dimensional, we use the characteristic quantities listed in Table REF , where $\\tau _A$ is the Alfvén crossing time, $v_A$ is the Alfvén velocity, $\\lambda = l_z$ is the turbulence coherence length related to the turbulence correlation length $\\lambda _c$ ($\\lambda _c = 0.747l_z$ for our particular slab configuration [22]).", "For the static case also the light speed may be used as a characteristic quantity [23].", "The introduction of an Alfvén speed in our test particle model, where the waves are treated as static, may appear rather artificial.", "However, the magnetostatic assumption is valid here provided that $|v_\\parallel | \\gg v_A$ and we introduce $v_A$ in anticipation of future work where we will drop the magnetostatic hypothesis.", "With our choice for the characteristic quantities (Table REF ) the dimensionless equations of motion of our charged test particles are given by: $ \\frac{d{\\bf r}}{dt} & = & {\\bf v}\\\\\\frac{d{\\bf v}}{dt} & = & \\beta ({\\bf E} + {\\bf v} \\times {\\bf B})$ Here $\\beta =\\Omega \\tau _A$ $ parameter in Ref.\\endcsname {AmbrosianoEA88} couples particle andfield spatial and temporal scales and provides a particularly useful means to relate our numericalexperiments to space and astrophysical plasmas.In general in a turbulent collisionless plasma the bandwidth of the inertial range fluctuationsmay extend from large fluctuations at the correlation scale, $ c$, to small fluctuationsat the ion inertial scale.", "In this case $ 1 $ andthe turbulent time-scales are much slower than the typical particle gyroradius~\\cite {GoldsteinEA86}.$ The resonant condition for the static case in terms of $\\beta $ is given by $k_{res}\\lambda = \\frac{n\\beta }{\\alpha (v/v_A)} = \\frac{n\\beta }{(v_\\parallel /v_A)}$ Time is advanced through a fourth-order Runge-Kutta integration method with an adaptive time-step [26]." ], [ "Numerical simulations", "Particles are loaded randomly in space at $t = 0$ throughout a one-dimensional simulation box of length $L$ .", "The fields are described in the following sections.", "In spherical coordinates, with the polar axis along the $z$ -direction parallel to the mean magnetic field of strength $B_0$ , particle velocity components are: $v_x = v\\sin \\theta \\cos \\phi \\quad v_y = v\\sin \\theta \\sin \\phi \\quad v_z = v\\cos \\theta $ Particles initial velocities are randomly distributed in the gyrophase $\\phi $ between $[0:2\\pi ]$ , while the velocity magnitude $v$ and pitch angle $\\theta $ are determined by the particular numerical experiment.", "Typical particle velocities used in our simulations are $10v_A$ and $100v_A$ , satisfying the magnetostatic constraint.", "In our analysis magnetic moments are expressed in units of the characteristic quantity $\\mu _n= v^2/B_0$ .", "We also define $\\delta b = \\delta B/B_0$ .", "Figure: Gyroresonant interaction between a circularly polarized waveand a particle with v=100v A v=100 v_A and α=1/8\\alpha =1/8: cosine of pitch angle α\\alpha (top row), particle magneticmoment μ\\mu (middle row) and parallel component of the induced electric field bottom (row).Different columns correspond to different wave amplitude:δb=0.001\\delta b = 0.001 (first column), δb=0.01\\delta b = 0.01 (second column), δb=0.1\\delta b = 0.1 (thirdcolumn) and δb=1.0\\delta b = 1.0 (fourth column).The statistic analysis of particle magnetic moment involves averaging trajectories over the particle gyroperiod $\\tau _g = 2\\pi / \\Omega $ .", "For each simulation we compute the effective number of gyroperiods $N_{\\tau _g}$ that particles complete in a given magnetic field configuration as: $N_{\\tau _g} = \\int _0^t \\frac{dt}{2\\pi } \\frac{eB(t)}{mc}.$ where $B(t)$ is the intensity of the total magnetic field.", "When $\\delta b \\ll 1$ , $B(t) \\simeq B_0$ ; however increasing $\\delta b$ toward unity the waves contribution to the strength of the total magnetic field $B(t)$ is not negligible." ], [ "Single wave", "We start studying the ion motion in presence of a constant magnetic field ${\\bf B}_0$ and a perpendicular left-handed circularly polarized wave with $ {\\bf B} = \\delta B_x \\cos (k_0z)\\, \\mathbf {\\hat{e}}_x - \\delta B_y \\sin {(k_0z)}\\, \\mathbf {\\hat{e}}_y +B_0\\, \\mathbf {\\hat{e}}_z,$ where $\\delta B_x$ and $\\delta B_y$ are the amplitudes of the wave and $k_0$ is the wavevector.", "We assume $\\delta B_x = \\delta B_y = \\delta B$ for the rms average values.", "In these simulations $\\beta =10^3$ , $v = 100v_A$ and $\\alpha = 0.125$ ($\\theta = 82^\\circ $ ).", "We follow the test-particles until they complete $N_{\\tau _g} = 100$ gyroperiods.", "For the resonance condition, Eq.", "(REF ), we set $k_0=80 /\\lambda $ .", "Particles injected with a pitch angle cosine different to $\\alpha = 0.125$ will not be in resonance with this wave, exhibiting a different behavior.", "For a direct comparison we also inject non-resonant particles, i.e., with $\\alpha = 0.5$ ($\\theta = 60^{\\circ }$ ).", "Figure REF shows the time evolution of the cosine of pitch angle $\\alpha $ , particle magnetic moment $\\mu $ , and the parallel component of the induced electric field $E_z$ , for a resonant particle ($\\alpha = 0.125$ ).", "Different columns corresponds to different values of the wave amplitude: $\\delta b = 0.001$ (first column), $\\delta b = 0.01$ (second column), $\\delta b = 0.1$ (third column) and $\\delta b = 1.0$ (fourth column).", "When the parallel component of the induced electric field is almost constant and equal to $E_z \\sim -v_{\\perp }\\delta b$ , the resonant interaction produces variations that are secular over a gyroperiod.", "However an oscillation occurs over a longer time, the bounce period $\\tau _b=2\\pi / \\omega _b$ (where $\\omega _b$ is the bounce frequency discussed in Section ).", "This is the typical timescale over which the velocity, and hence the particle trajectory, exhibits significant deviations from the linear $v_{\\parallel } = \\text{const}$ and $v_{\\perp } = \\text{const}$ case.", "Table: Trapping width values for α\\alpha and μ\\mu :comparison between theoretical (subscript th) and numerical (subscript sim) values.In Section  we derived the analytical expression for the half trapping-width of magnetic moment for a particle interacting with a left or right handed circularly polarized wave (see Eq.", "REF ).", "We now compute the values of the half peak-to-peak difference in $\\alpha $ and $\\mu $ , $\\Delta \\alpha = (\\alpha _{max} - \\alpha _{min})/2$ and $\\Delta \\mu = (\\mu _{max} - \\mu _{min})/2$ , for the resonant interaction simulations.", "These values and those obtained from the theoretical expressions (REF )-(REF ) are listed in Table REF and are in good agreement, confirming the validity of equations (REF )-(REF ) and reinforcing the intuitively idea that magnetic moment and pitch angle behaviors are strictly related.", "To compare resonant and non-resonant dynamics, we show in Figure REF the time evolution of cosine of pitch angle $\\alpha $ (first row), magnetic moment $\\mu $ (third row), and their distribution functions $f(\\alpha )$ (second row) and $f(\\mu )$ (fourth row) at the end of the simulation, for a resonant particle with $\\alpha = 0.125$ (left column), and a non-resonant one with $\\alpha = 0.5$ (right column).", "In contrast with the resonant case in which $\\alpha $ and $\\mu $ exhibit well-known secular variations with typical period equal to $\\tau _b$ , the $\\alpha $ and $\\mu $ profiles for a non-resonant particle show a regular oscillating behavior, a distinctive signature of regular particle motion.", "The values of the half peak-to-peak difference in $\\alpha $ and $\\mu $ obtained from the simulation are $\\Delta \\alpha _{sim} =0.0025$ and $\\Delta \\mu _{sim} = 0.003$ .", "These are smaller than the theoretical values computed from equations (REF )-(REF ) with $\\delta b=0.01$ and $\\alpha = 0.5$ , for which we obtain $\\Delta \\alpha _{th} =0.1316$ and $\\Delta \\mu _{th} = 0.1316$ .", "The distribution functions $f(\\alpha )$ and $f(\\mu )$ (Figure REF ) for a resonant particle are more spread in $\\alpha $ and $\\mu $ and are centered around their initial values $\\alpha = 0.125$ and $\\mu = 0.98$ .", "In the non-resonant case, $f(\\mu )$ remains peaked at its initial value, i.e., its magnetic moment is constant during particle motion.", "The spread in $\\alpha $ of its distribution is $\\sim 10\\%$ , small compared to the resonant case spreading of $\\sim 40\\%$ .", "Figure: Time evolution of cosine of pitch angle α\\alpha (first row) and its distribution function f(α)f(\\alpha ) (second row),time evolution of magnetic moment μ\\mu (third row) and its distribution function f(μ)f(\\mu ) (fourth row) of resonant(α=0.125\\alpha = 0.125, left column) and non-resonant particle (α=0.5\\alpha = 0.5, right column).", "v=100v A v=100\\, v_A.Figure REF shows the distribution functions, $f(\\alpha )$ and $f(\\mu )$ , at the end of the simulation for 1000 resonant and non-resonant particles injected in the simulation box with random positions and phases.", "For non-resonant particles (right column) the distributions remain peaked around their initial values $\\alpha = 0.5$ and $\\mu /\\mu _n = 0.75$ with very little spreading.", "For the resonant particles (left column) $f(\\alpha )$ acquires a Gaussian shape centered around its initial value $\\alpha = 0.125$ .", "Furthermore it spreads of $\\sim 0.1$ , comparable to the trapping width for the single particle $2\\Delta \\alpha = 0.014$ (Figure REF ).", "The magnetic moment distribution for the resonant case has a characteristic shape found for $\\mu $ in the parameter range in which pitch angle exhibits a Gaussian distribution and the density distribution function is still isotropic (particle free-streaming regime).", "As for the pitch angle, the spread in the magnetic moment distribution of $\\sim 0.03$ is comparable to the trapping width for the single particle $2\\Delta \\mu = 0.00352$ (see Eq.", "REF and Figure REF ).", "Figure: f(α)f(\\alpha ) (first row) and f(μ)f(\\mu ) (second row) at the end of the simulationfor an initial distribution of 1000 resonant (left column) and non-resonant (right column) particlesrandomly distributed in the simulation box.", "δb=0.01\\delta b=0.01." ], [ "Overlapping resonances", "In order to understand the effect of overlapping resonances on particle magnetic moment, we perform a numerical experiment with four different particles in the simulation box with random initial positions, same initial velocity $v = 100\\, v_A$ , but different values for pitch angle cosine: $\\alpha _1=1/2, \\alpha _2=1/4,\\alpha _3=1/8, \\alpha _4=1/32$ .", "For $\\beta =10^3$ , making use of the resonance condition for the static case [Eq.", "(REF )], the cyclotron resonances $n=1$ for the different values of $\\alpha $ are expected for $k_1 \\lambda =20$ , $k_2\\lambda =40$ , $k_3\\lambda =80$ , and $k_4\\lambda =320$ .", "The total magnetic field is given by: ${\\bf B} =B_0{\\bf \\hat{e}}_z + \\sum _{i=1}^{4} \\delta b \\cos [k_iz + \\phi _i] {\\bf \\hat{e}}_x - \\sum _{i=1}^{4}\\delta b \\sin [k_iz + \\phi _i] {\\bf \\hat{e}}_y,$ where the $\\phi _i$ are random phases.", "Taking into account resonance broadening effects, all particles with parallel velocities in the range $v_{\\parallel } - \\Delta v_{\\parallel } < v_{\\parallel } < v_{\\parallel } + \\Delta v_{\\parallel }$ can potentially resonate with a wave, whose wave number is $k_{\\parallel }=\\Omega /v_{\\parallel }$ .", "As found by Ref.", "[27], the direct evidence of resonances overlapping is the disappearance of constants of motion, i.e., the onset of stochasticity in the Hamiltonian formalism.", "We make simulations with four different waves amplitudes $\\delta b$ = 0.001, 0.01, 0.1, and 1.0.", "The values of the trapping half-widths $\\Delta v_{\\parallel }$ computed for the different pitch angles with Eq.", "(REF ) are listed in Table REF for the different $\\delta b$ considered.", "Table: Values of Δv ∥ \\Delta v_{\\parallel }for α\\alpha =1/2,  1/4,  1/8,  and 1/32 resonances at different δb\\delta b.Figure: Transition from non-overlapping to overlapping resonances: α\\alpha (left column) andμ\\mu (right column) profiles varying the waves amplitude: δb=0.001\\delta b=0.001 (first row), δb=0.01\\delta b=0.01(second row), δb=0.1\\delta b=0.1 (third row), δb=1.0\\delta b=1.0 (fourth row).Figure REF shows time histories of pitch angle cosine $\\alpha $ (left column) and magnetic moment $\\mu $ (right column) profiles for various $\\delta b$ .", "Again similar behavior is seen for $\\alpha $ and $\\mu $ .", "For the smallest wave amplitude, $\\delta b=0.001$ , it is possible to recognize very well the four different resonances in the profiles of $\\alpha $ and $\\mu $ .", "For $\\delta b=0.01$ the resonance at $\\alpha _3=1/8$ is overlapping with the resonance at $\\alpha _4=1/32$ .", "Indeed, the initial parallel velocity of the particle injected at the smallest pitch angle, $v_{\\parallel ,4}=3.125v_A$ , lies in the range of velocities [see Eq.", "(REF )] in possible resonance with $k_{\\parallel }=k_3$ .", "For higher wave amplitudes, $\\delta b=0.1$ and $\\delta b=1.0$ , the condition (REF ) is satisfied by all particles velocities.", "Stochasticity arises and the different resonances are indistinguishable.", "Figure: Transition from non-overlapping to overlapping resonances with varying wave amplitude: δb=0.001\\delta b=0.001(first row), δb=0.01\\delta b=0.01 (second row), δb=0.1\\delta b=0.1 (third row),δb=1.0\\delta b=1.0 (fourth row).", "The distribution functions f(α)f(\\alpha ) (left column), f(μ)f(\\mu ) (central column) andf(δz)f(\\delta z) (right column) are averaged over time.The distribution functions $f(\\alpha )$ , $f(\\mu )$ and $f(\\delta z)$ (where $\\delta z=z - z_0$ is the displacement along $z$ relative to the particle initial position $z_0$ ) after $100 \\tau _g$ (Figure REF ) exhibit similar characteristics.", "For $\\delta b = 0.001$ , $f(\\alpha )$ and $f(\\mu )$ are peaked in correspondence of their four initial values because of the good resonances separation.", "$f(\\delta z)$ shows that the particles are simply free-streaming in the parallel direction and, depending on their initial parallel velocity, they cover shorter or longer distances along $z$ .", "For $\\delta b=0.01$ , $f(\\alpha )$ spreads around its initial four peaks because particle interact resonantly with waves of larger amplitude, and resonances overlap for $\\alpha < 1/4$ , as discussed previously.", "Similar effects are shown also by $f(\\mu )$ , confirming that for small $\\delta b$ the resonant interaction affects magnetic moment and pitch angle in similar ways.", "While for $\\delta b=0.01$ particles continue to free-stream in the z-direction, different profiles for $f(\\delta z)$ appear for $\\delta b=0.1$ .", "Pitch angle distribution begins to isotropize and magnetic moment exhibits a one-sided long tail distribution extending toward smaller $\\mu $ .", "This behavior is similar to the regime found previously in the single wave experiment when $f(\\alpha )$ is nearly isotropic, $f(\\delta z)$ still indicates particles free-streaming, and the magnetic moment distribution displays a long tail.", "For $\\delta b=1.0$ , by $t = 100 \\tau _g$ the pitch angle cosine distribution $f(\\alpha )$ has become completely isotropic, while $f(\\delta z)$ approaches a gaussian distribution indicative of spatial diffusion.", "In this regime $f(\\mu )$ loses its long-tail and starts to acquire a gaussian shape.", "In that way we have identified three distinct regimes of statistical magnetic moment behavior with increasing degree of turbulence." ], [ "Slab spectrum", "In this section we present the results of our numerical simulations of test-particles in presence of a broad-band slab spectrum [see Eq.", "(REF ) and Figure REF ].", "We have performed simulations for different particles velocities and amplitude of the magnetic field fluctuations.", "Simulations use a unidimensional computational box of length $L = 10000\\, l_z$ ($l_z = 1$ is the coherence scale for the slab spectrum) with $N_{z} = 2^{28} = 268,435,456$ grid points.", "The magnetic field in physical space is generated from a spectrum $P(k)$ in Fourier space, via inverse fast Fourier transform (FFT).", "The turbulent magnetic field is given by: ${\\bf B}(z) = B_0{\\bf e}_z + \\delta {\\bf B}(z),$ with $\\delta {\\bf B}(z) = \\delta B_x(z)\\, {\\bf \\hat{e}}_x + \\delta B_y(z)\\, {\\bf \\hat{e}}_y$ and the solenoidality condition is identically satisfied.", "The modes of the magnetic field components in k-space are given by: $\\delta B_x(k_n)= [P(k_n)]^{1/2} e^{i\\Phi _n} \\\\\\delta B_y(k_n) = [P(k_n)]^{1/2} e^{i\\Psi _n}$ where $k_n = 2\\pi n/L$ and $\\Phi _n$ and $\\Psi _n$ are random phases.", "The slab spectrum $P(k)$ is given by: $P(k_n) ={\\left\\lbrace \\begin{array}{ll}C_{slab} [1 + (k_n l_z)^2]^{-5/6}, & \\text{for $k_n < k_{diss}$} \\\\C_{diss} \\left( \\frac{k_n}{k_{diss}} \\right)^{-7/3}, & \\text{for $k_n \\ge k_{diss}$}\\end{array}\\right.", "}$ where $C_{slab} = 2\\lambda _c \\delta b^2_{x,slab}$ is a constant specific to this form of the slab model, $\\delta b^2_{x,slab}$ is the mean square fluctuation, $k_{diss}$ is the dissipation range wavenumber, $C_{diss} = C_{slab}[1 + (k_{diss} l_z)^2]^{-5/6}$ is the constant for the dissipation range (set by the continuity of the spectrum $P(k)$ at $k_{diss}$ ).", "The vectors of Fourier coefficients are zero-padded for $N_{max} + 1 \\le n \\le N_z$ providing an extra level of smoothness to the fields by an effective trigonometric interpolation.", "In all the simulations we use $N_{max} = 6.7 \\times 10^7$ and a simple linear interpolation to compute the fields at the test particle position.", "The resulting spectrum is shown in Figure REF .", "Figure: Power spectrum of the turbulent magnetic.", "kk is normalized to the coherence length l z l_z.Several important scales are present in the system.", "They are labeled as $k_{min}$ , $k_{l_z}$ , $k_{diss}$ , $k_{max}$ and $k_{N_z}$ .", "The discrete wavenumbers are obtained through $k_n = 2 \\pi n/ L$ as: $N_k = \\frac{L}{2 \\pi } k \\sim 1600\\, k.$ Table: Characteristic scales in the spectrum.We summarize the values for $k$ and $N_k$ used in our simulations in Table REF , where: - $k_{min} = 2\\pi /L$ is the minimum wave vector of the spectrum, corresponding to $N_k = N_{kmin} = 1$ .", "- $k_{l_z} = 2\\pi /l_z =2\\pi $ is the wave vector that marks the beginning of the inertial range.", "Three decades of energy containing range from $k_{min}$ to $k_{l_z}$ ensure turbulence homogeneity.", "$l_z$ or $\\lambda _c = 0.747l_z$ correspond to the typical lengths scales over which the particles attain diffusive behavior of the pitch angle.", "Three decades of inertial range with $P(k) \\propto k^{-5/3}$ well represent solar wind conditions.", "- $k_{diss}$ is the wave vector corresponding to the beginning of the dissipation range.", "In our model, the spectrum extends beyond $k_{diss}$ with $P(k) \\propto k^{-7/3}$ .", "- At two decades higher wavenumber, $k_{MAX} = \\sqrt{m_i/m_e} k_{diss}$ determines the end of the dissipation range.", "- Extending for two decades beyond $k_{MAX} = 4.2 \\times 10^4$ , the spectrum includes zero-padding up to $k_{MAX1} = 8.4 \\times 10^4$ .", "- Another important scale, not labeled in Figure REF because it depends on test-particle velocity, is the wave vector corresponding to $z_{max} = vT_{tot}$ , the distance covered by a charged test particle moving at speed $v$ in the simulation running time $T_{tot}$ .", "To avoid periodicity effects it is important that the box length $L$ is large enough so that particles trajectories are limited to a small fraction of the full length, i.e., $L \\gg z_{max}$ or $k_{min} \\ll 1/z_{max}$ .", "Periodicity might indeed give rise to artificial field lines diffusion.", "We fix the value of the $\\beta $ parameter equal to $10^4$ .", "This corresponds approximately to observed solar wind turbulence properties at 1 AU, as follows: $\\beta = \\Omega \\tau _A = \\left(\\frac{q}{m}\\right) \\frac{\\lambda _c \\sqrt{4\\pi \\rho }}{c} = \\frac{\\omega _{pi}\\lambda _c}{c} = \\frac{\\lambda _c}{\\lambda _{ii}},$ where $\\lambda _c$ is the turbulence correlation length $\\omega _{pi}= (4\\pi n_{0i}{q_i}^2/m_i)^{1/2}$ is the ion plasma frequency ($q_i$ and $m_i$ are respectively the ion charge and mass) and $\\lambda _{ii} = c/\\omega _{pi}=\\left( c^2 m_i / 4 \\pi n_i e^2 \\right)^{1/2}$ is the ion inertial length.", "For $n_i = n_e$ then $\\lambda _{ii} = (m_i/m_e)^{1/2}\\rho _{ie}$ .", "Because the solar wind density at 1 AU is approximately $n \\sim (1,10)\\, cm^{-3}$ on average $\\lambda _{ii} \\sim 1000\\, km$ .", "At the same distance the turbulence correlation length $\\lambda _c$ is approximately $10^6$  km [28] and $\\beta \\simeq 10^4$ .", "Typically 1000 particles are injected in the simulation with initial random positions.", "Particles are loaded from a cold ring beam [see equations (REF )] distribution with constant velocity magnitude, $\\sin \\theta $ is set equal to $(1 - \\alpha _0^2)^{1/2}$ , where $\\alpha _0$ is the initial pitch angle cosine respect to the background field $B_0$ .", "The initial gyrophase $\\phi $ is chosen randomly.", "For all the simulations $\\alpha _0=0.125$ ($\\theta \\simeq 82^{\\circ }$ ).", "Table: Typical values used in the simulations.From the previous section, we know that the behavior of magnetic moment is correlated to pitch angle behavior for a low level of magnetic fluctuation ($\\delta b =0.001, 0.01$ ).", "Pitch angle and magnetic moment exhibit Gaussian distribution functions typical of normal diffusion processes.", "Increasing the turbulence level, pitch angle distribution approaches isotropization and a transient regime is observed with the magnetic moment starting to be influenced by the onset of spatial parallel diffusion.", "When $f(\\alpha )$ completely isotropizes, spatial diffusion sets in and $f(\\mu )$ behavior is closely related to the sampling of the varying magnetic field strength associated with that spatial diffusion.", "From quasilinear theory we know that velocity and real space diffusion occur at two different time scales.", "Typically, velocity space diffusion takes place with the time scale $\\tau _c = \\lambda _c/v$ shorter than the typical time scale at which parallel diffusion occurs $\\tau _{\\parallel } = \\lambda _{\\parallel }/v$ , where $\\lambda _{\\parallel } = 3D_\\parallel /v$ is the parallel mean free path.", "For this reason we follow test particles in the simulation box for a time $T > \\tau _c$ , typically with $T = 20\\tau _c$ .", "Particles parameters used in the simulations are listed in Table REF .", "An important parameter in the description of energetic test particles is $\\epsilon = r_L/\\lambda _c$ , which is sometimes called the dimensionless particle rigidity.", "It can be related to the bend-over wavenumber of the turbulence, $k_{bo} = 1/\\lambda _c$ , and the minimum resonant wavenumber, $k^{r}_{min} = 1/r_L$ , as $\\epsilon = k_{bo}/k^{r}_{min}$ .", "For example when $r_L \\gg \\lambda _c$ particles experience all possible $k$ -modes in few gyroperiods resonating with the energy containing scale ($k^{r}_{min} \\ll k_{bo}$ ).", "For lower energies the test particles resonate in the inertial range.", "Those with $v = 10\\, v_A$ will resonate at the end of the inertial range ($1/r_1$ in Fig.", "REF ), while those with $v\\, = 100v_A$ at the middle of the inertial range ($1/r_2$ in Fig.", "REF ).", "Furthermore, as explained previously, the condition $k_{min} \\ll z_{max}$ is necessary to avoid artificial effects in particle transport associated with periodicity of the magnetic field.", "Figure: Distribution functions of cosine of pitch angle f(α)f(\\alpha ) (left column), magnetic moment f(μ)f(\\mu ) (central column) and particle displacements relative to the initial position f(δz)f(\\delta z) (right column) at 20τ c 20\\, \\tau _cfor different waves amplitude: δb=0.001\\delta b=0.001 (first row), δb=0.01\\delta b=0.01 (second row), δb=0.1\\delta b=0.1 (third row)and δb=1.0\\delta b=1.0 (fourth row).", "Particle parameters at injection: v=10v A v = 10\\, v_A and α 0 =0.125\\alpha _0=0.125.Figure: Statistics for v=10v A v = 10\\, v_A.", "Variances for cosine of pitch angle α\\alpha (left plot), magnetic moment μ\\mu (center plot) and particle displacements relative to the initial position δz\\delta z (right plot)at different values of δb\\delta b: δb=0.001\\delta b=0.001 (black line), δb=0.01\\delta b= 0.01 (purple line),δb=0.1\\delta b = 0.1(red line) and δb=1.0\\delta b = 1.0 (blue line).", "The ss values indicate the different slopes for thescalings 〈(Δα) 2 〉,〈(Δμ) 2 〉\\langle ( \\Delta \\alpha )^2 \\rangle , \\langle ( \\Delta \\mu )^2 \\rangle and〈(Δz) 2 〉∝t s \\langle ( \\Delta z )^2 \\rangle \\propto t^s.", "The variances are fitted with the gray lines:dotted line is s=2s=2, dotted-dashed line s=1s=1, dashed line s=0.8s=0.8, three dotted-dashed line s=0.7s=0.7.Figure REF shows $f(\\alpha )$ (left column), $f(\\mu )$ (central column) and $f(\\delta z)$ (right column) for a distribution of particles moving with an initial velocity $v = 10\\, v_A$ in presence of the slab spectrum [Eq.", "(REF ), Figure REF ].", "All the distribution functions are computed at the end of the simulation, i.e., after $20\\, \\tau _c$ .", "The blue line and the green line indicate the initial value and the mean value of each distribution.", "As particles are injected at different positions, it is convenient to define the quantity $\\delta z = z(j) - z(0)$ ($j$ is a temporal index).", "In this way it is possible to take out from the distribution function $f(\\delta z)$ both the drift effect ($v_Dt$ ) and particle diffusion relative to their own positions ($\\Delta z_i$ ).", "The general expression for the $z$ position of the $i$ -th particle is given by $z_i = z_i(0) + v_Dt + \\Delta z_i = z_i(0) + \\delta z_i.$ The primary diagnostic for studying particle diffusion is the variance $\\sigma ^2(t) \\propto t^s$ of particles cosine of pitch angle, magnetic moment and position parallel to the mean field direction.", "Figure REF illustrates the time evolution of the variances, $\\langle \\left( \\Delta \\alpha \\right)^2 \\rangle $ (left figure), $\\langle \\left( \\Delta \\mu \\right)^2 \\rangle $ (central figure) and $\\langle \\left( \\Delta z \\right)^2 \\rangle $ (left figure) for a particles distribution moving with initial velocity equal to $10\\, v_A$ .", "Different colors correspond to different $\\delta b$ values: $\\delta b=0.001$ black line, $\\delta b= 0.01$ purple line, $\\delta b = 0.1$ red line and $\\delta b = 1.0$ blue line.", "The variances are fitted with the gray lines: the dotted line is used for $s=2$ , the dotted-dashed line for $s=1$ , the dashed line for $s=0.8$ and the three dotted-dashed line for $s=0.7$ .", "For $\\delta b=0.001$ , $\\alpha $ and $\\mu $ display Gaussian distributions while particles free-stream in the $z$ -direction.", "Particles that cover greater distance in $z$ are more scattered in pitch angle and consequently in $\\mu $ .", "Figure REF shows superdiffusive behavior (black line, $s = 2$ ) with particles free streaming along $z$ , and later, variance characteristic of normal diffusion with $\\langle \\left( \\Delta \\alpha \\right)^2 \\rangle $ and $\\langle \\left( \\Delta \\mu \\right)^2 \\rangle $ scaling $\\propto t$ .", "For $\\delta b = 0.01$ , particles cover only one side of the $\\alpha $ hemisphere continuing to travel along $z$ (purple line in Figure REF ).", "This is the transient regime already observed in Figure REF when $f(\\mu )$ exhibits a one-sided long tail distribution toward smaller $\\mu $ .", "For $\\delta b = 0.1$ , pitch angle distribution becomes completely isotropic and spatial diffusion sets in, as shown by the slope $s=1$ of $\\langle \\left( \\Delta z \\right)^2 \\rangle $ in Figure REF (red line) at the end of the simulation.", "The deviation from purely free-streaming or ballistic behavior means that, while the system has not become fully diffusive along the mean field direction, there are signs that diffusive processes in velocity space are beginning to diminish the free-streaming.", "Although $f(\\mu )$ still exhibits a long-tail, the influence of spatial diffusion starts to appear.", "The well-pronounced peak observed in $f(\\mu )$ for $\\delta b = 0.01$ is substantially reduced and the mean value of magnetic moment decreases.", "Moreover $\\mu $ displays subdiffusive behavior up to $0.02 \\tau _c$ .", "After this time particles diffuse in space and $\\langle \\left( \\Delta \\mu \\right)^2 \\rangle $ attains a plateau.", "The Gaussian shape is not reached yet probably because spatial diffusion is just at the beginning.", "For $\\delta b= 0.5$ [see Figure (REF )] and $\\delta b=1.0$ , $f(\\alpha )$ is isotropic, particle motion is completely diffusive in real space [as the slope $s=1$ in $\\langle \\left( \\Delta z \\right)^2 \\rangle $ in Figure REF (blue line) shows], and $f(\\mu )$ behavior is closely related to the sampling of varying magnetic field strength associated with that spatial diffusion, displaying a Gaussian distribution centered at the middle of $\\mu $ -space.", "From a more detailed analysis of the case $\\delta b = 0.5$ [Figure (REF )] we notice that magnetic moment variance (first figure) scales according to $\\langle \\left( \\Delta \\mu \\right)^2 \\rangle \\propto t^{0.17}$ (red-dashed line) up to $0.002\\tau _c$ and after $0.005\\tau _c$ a plateau is attained (blue-dashed line).", "Instead particles motion (see time evolution of $\\langle \\left( \\Delta z \\right)^2 \\rangle $ , second plot) becomes fully diffusive (blue-dashed line) only after $0.007\\tau _c$ .", "Magnetic moment distribution $f(\\mu )$ in the first part of the evolution (third plot) is in the transient regime characterized by the long tail.", "In contrast, when particles diffuse in real space (fourth plot), $f(\\mu )$ reacquires the gaussian profile.", "Thus the final stage of magnetic moment variance evolution, i.e.", "the formation of the plateau, can be considered as a precursor for the onset of the parallel diffusion of particles in space.", "Of course this effect is present in pitch angle variance too, but in addition in $\\mu $ behavior we have a direct signature of the onset of the spatial diffusion, that is the reappearance of the gaussian shape in the distribution function, while pitch angle distribution remains completely isotropic.", "Figure: Statistics for v=10v A v = 10\\, v_A and δb=0.5\\delta b = 0.5.", "Magnetic moment variance 〈Δμ 2 〉\\langle \\left( \\Delta \\mu \\right)^2 \\rangle (first plot) and mean square displacement 〈Δz 2 〉\\langle \\left( \\Delta z \\right)^2 \\rangle (second plot) versus time and magnetic moment distribution function f(μ)f(\\mu ) after 0.01τ c 0.01\\, \\tau _c (third plot) and at the end of the simulation (fourth plot).", "The ss values indicate the different slopes for thescaling of the variances 〈(Δμ) 2 〉\\langle ( \\Delta \\mu )^2 \\rangle and 〈(Δz) 2 〉∝t s \\langle ( \\Delta z )^2 \\rangle \\propto t^s.", "The variances are fitted with the dashed lines.Thus these transitions in magnetic moment behavior are related not just to the variation of the turbulence level, but also to the different time scale at which magnetic moment conservation is studied.", "The magnetic moment distribution functions $f(\\mu )$ and variances $\\langle \\left( \\Delta \\mu \\right)^2 \\rangle $ in the case $v = 100\\, v_A$ (not shown) exhibit the same features observed for $v = 10 v_A$ .", "However, increasing particle speed the total number of gyroperiods, $N_{\\tau _g}$ , performed by each particle decreases; as a consequence, faster particles sample less variation in magnetic field strength.", "This leads to a slower spatial diffusion, i.e., for $100v_A$ spatial diffusion occurs on a time scale longer than $20 \\tau _c$ .", "Figure: Standard deviation (blue triangle) and variation in magnetic moment (red circle)versus δb\\delta b for at 20τ c 20 \\tau _c v=100v A v = 100v_A and α=0.125\\alpha = 0.125.For $v = 100v_A$ we show in Figure (REF ) magnetic moment standard deviation $\\sigma _\\mu /\\mu _{min}$ (blue triangle) and the changes in its mean $\\Delta \\mu /\\mu _{in} = (\\bar{\\mu } -\\mu _{in})/\\mu _{in}$ versus $\\delta b$ after $20 \\tau _c$ .", "As $\\delta b$ increases toward unity the changes in magnetic moment distribution start to increase faster." ], [ "Conclusions", "In this paper we have investigated the conservation of charged particle magnetic moment in presence of turbulent magnetic fields.", "For slow spatial and temporal variations of the magnetic field respect to the particle gyroradius and gyroperiod, the magnetic moment $\\mu $ is an adiabatic invariant of the particle motion.", "Non-conservation of magnetic moment can influence particle acceleration and have considerable implications in many astrophysical problems such as coronal heating, cosmic rays transport and temperature anisotropies in the solar wind.", "These applications motivate the present basic study of the degree to which magnetic moments are conserved in increasingly complex models of one dimensional spectra.", "While all the models considered here have been very oversimplified relative to the spectra observed for example in the solar wind [17], [29] or in simulations of MHD turbulence [31], [30], the present study is intended to contribute to the basic understanding of the conditions for the onset of magnetic moment nonconservation.", "We point the interested reader also to a recent study by Ref.", "[30] that addresses this issue from a somewhat different perspective.", "In order to reproduce and extend some of the result obtained by Ref.", "[15], we started to study the resonant interaction between ions and a single parallel propagating electromagnetic wave (see Section REF ).", "Using the specialized expression for the trapping width $\\Delta v_{\\parallel }$ found by Ref.", "[16] in the case of a single circularly polarized wave, we have been able to write a similar expression for magnetic moment (see Eq.", "REF ).", "In presence of a single finite amplitude fluctuation the magnetic moment of a resonant particle undergoes a finite amplitude nonlinear oscillation too.", "We have performed several simulations changing both particle velocity and the amplitude of the wave.", "For each of them we compare the values of $\\Delta \\mu $ and $\\Delta v_{\\parallel }$ with those obtained using our specialized expression and they are in good agreement.", "We designed a particular experiment to study the effects of resonances overlapping (see Section REF ).", "From the analysis of the distribution functions of particles pitch angle, $f(\\alpha )$ , magnetic moment, $f(\\mu )$ , and $z$ -position, $f(z)$ , we distinguish three different regimes.", "First, for a low level of magnetic fluctuation, i.e., $\\delta B/B_0=0.001, 0.01$ , the magnetic moment distribution half-width is directly related to pitch angle distribution.", "Second for $\\delta B/B_0=0.1$ stochasticity arises as a consequence of overlapping resonances and its effect on pitch angle is isotropization of the distribution function.", "This is a transient regime during which magnetic moment exhibits a one-sided long-tail distribution and starts to be influenced by the onset of spatial parallel diffusion.", "Finally, when $f(\\alpha )$ completely isotropizes spatial diffusion sets in ( $\\delta B/B_0=0.1$ ), $f(\\mu )$ behavior is closely related to the sampling of varying magnetic field strength associated with that spatial diffusion.", "Other studies regarding particles interaction with two electromagnetic waves as well as a flat turbulent spectrum (not shown) were also conducted and they confirmed this general picture.", "Motivated by these results we studied the behavior of many particles interacting with a broad-band slab spectrum, generated in order to mimic some of the major features of the solar wind (see Figure REF ): (a) three decades of the energy containing scale ensure turbulence homogeneity, (b) three decades of inertial range well-reproduce the observations and (c) two decades of dissipation range enable us to cross the “$\\alpha _{min}$ barrier” related with the “resonance gap” predicted by quasilinear theory [32], [21].", "After that there are almost other two decades of zero-padding, important for the trigonometric interpolations and for the smoothness of the field.", "This is implemented using a numerical grid with $N_{z} = 2^{28} = 268,435,456$ points corresponding to $134,217,728$ wavevectors for the spectrum.", "Apart from the obvious limitation that this spectrum is purely one dimensional, it is constructed to correspond roughly to features of solar wind spectra observed by single spacecraft, where the fully three dimensional spectrum is in effect reduced to a one dimensional form.", "Information is lost in the process .", "In order to gain insight on magnetic moment conservation we have performed simulations changing both particles velocity, $v = (10, 100)\\, v_A$ , and the amplitude of magnetic field fluctuations $\\delta B/B_0 = (0.001, 0.01, 0.1, 0.5, 1.0)$ .", "Particles injected at different velocities start to resonate at different points of the spectrum.", "We analyzed the distribution function [see Figure REF ] and the variance [see Figure (REF )] of pitch angle cosine $\\alpha $ , magnetic moment $\\mu $ and parallel position $z$ .", "From the experiment of resonances overlapping we know that the three different regimes of $\\mu $ statistical behavior are related with other two effects: diffusion in velocity space and spatial parallel diffusion.", "These take place at different characteristic times, $\\tau _c$ and $\\tau _{\\parallel }$ respectively.", "In order to investigate the effects of both processes on magnetic moment distributions, we followed test-particles in the simulation box for times $T > \\tau _c$ .", "For a low level of magnetic fluctuations particles free-stream in the $z$ -direction while $\\alpha $ and $\\mu $ exhibit gaussian distributions around their initial values.", "For $\\delta B/B_0 = 0.01$ particles cover completely one side of the $\\alpha $ hemisphere continuing to stream freely along $z$ .", "This is the transient regime during which $f(\\mu )$ exhibits a one-sided long tail distribution in the direction of smaller $\\mu $ that appears to be a typical feature of magnetic moment distribution.", "During this transient regime the distribution of particles nearly conserves its magnetic moment.", "Increasing the value of $\\delta B/B_0$ spatial diffusion starts to take place, $f(\\mu )$ recovers the typical Gaussian shape centered in the middle of $\\mu $ -space.", "These different regimes of magnetic moment statistical behavior are related not just to the variation of the turbulence level $\\delta B/B_0$ , but also to the different time scale at which magnetic moment conservation is studied [see Figure (REF )].", "In spite of the limitations of the present approach the results presented here provide a basic view of how magnetic moments are modified in simplified models, and in particular how magnetic moment changes are related to pitch angle changes and sampling of magnetic variations due to spatial diffusion.", "It is clear that additional study is required to understand more fully the influences of turbulence on magnetic moment statistics.", "For example realistic three dimensional models of the magnetic field turbulence, as well as incorporation of electric field fluctuation effects, are expected to have significant effects.", "It is also possible that nonGaussian features of magnetic field fluctuations, such as, are associated with intermittency effects, may also influence magnetic moment changes, much as they influence spatial transport due to trapping and related influences [33], [34].", "In this regard the present results, along with those of Ref.", "[30], may be considered as baseline or minimal quantification of nonconservation of magnetic moments of a distribution of test particles in turbulence.", "Planned future studies will investigate quantitatively how additional realism in the modeling might produce even more significant departures from magnetic moment conservation.", "This research supported in part by the NASA Heliophysics Theory program NNX11AJ44G, and by the NSF Solar Terrestrial and SHINE programs.", "(AGS-1063439 & AGS-1156094), by the NASA MMS and Solar probe PLus Projects, and by Marie Curie Project FP7 PIRSES-2010-269297 - Turboplasmas.", "*" ], [ "Derivation of trapping half width for a circularly polarized wave", "Using equations $(5a)$ and $(5b)$ of [15] it is possible to derive a simplified expression for the trapping half-width and the bounce frequency in the case of an Alfvén static wave .", "For this particular case $k_{\\perp }=0$ and $\\phi =0$ .", "We can rewrite equation $(5c)$ of [15] as $Z_n &= mc^2 \\left\\lbrace \\frac{v_{\\perp }}{2c} \\left[ \\left( \\epsilon _2 - \\frac{k_{\\parallel }}{k}\\sigma \\epsilon _1 \\right) J_{n-1}(k_{\\perp \\rho }) + \\right.", "\\right.", "\\nonumber \\\\&\\left.", "- \\left( \\epsilon _2 + \\frac{k_{\\parallel }}{k} \\sigma \\epsilon _1 \\right) J_{n+1}(k_{\\perp \\rho }) \\right] + \\nonumber \\\\&\\left.", "+ \\sigma \\left( \\frac{v_{\\parallel } k_{\\perp }}{ck} \\epsilon _1 + \\epsilon _3 \\right) J_n(k_{\\perp \\rho }) \\right\\rbrace ,$ with $\\cos \\alpha =1$ and $\\sin \\alpha =0$ .", "Because ${\\bf k} \\parallel {\\bf B}_0$ we can choose ${\\bf \\hat{e}}_z={\\bf B}_0/|{\\bf B}_0|$ , ${\\bf \\hat{e}}_y$ is any arbitrary direction perpendicular to ${\\bf \\hat{e}}_z$ and ${\\bf \\hat{e}}_x= {\\bf \\hat{e}}_u \\times {\\bf \\hat{e}}_z$ .", "The vector potential can be obtained from the magnetic field $\\nabla \\times {\\bf B}_{\\perp } = B_x {\\bf \\hat{e}}_x + B_y {\\bf \\hat{e}}_y$ .", "In Fourier space $\\nabla \\rightarrow ik_z {\\bf \\hat{e}}_z$ , so we have: $A_x & = & - \\frac{i}{k_{\\parallel }}B_y \\nonumber \\\\A_y & = & \\frac{i}{k_{\\parallel }}B_x$ Considering only a single circularly polarized wave in space, for the two different possible helicities we can write: ${\\bf B}_{\\pm } = (B_\\pm {\\bf \\hat{e}}_\\pm ) \\exp {[i(k_{\\parallel }z)]}$ where $B_\\pm = \\frac{1}{\\sqrt{2}}(B_x \\mp iBy) \\quad \\textrm {and} \\quad {\\bf \\hat{e}}_\\pm = \\frac{1}{\\sqrt{2}}({\\bf \\hat{e}_x} \\mp i{\\bf \\hat{e}}_y)$ are respectively the complex amplitudes and the orthogonal polarization unit vectors.", "The $+ (-)$ polarization state is the positive (negative) helicity, i.e., the vector ${\\bf B}$ is rotating counter-clockwise (clockwise).", "At first, let's consider only the left-handed polarized wave ${\\bf B}_+$ .", "Assuming $B_+ = \\sqrt{2} \\delta B e^{-i \\pi /2}$ we can write the $x$ and $y$ components of the wave magnetic field as $B_x & = & \\delta B \\exp {[i(k_\\parallel z - \\pi /2)]} \\nonumber \\\\B_y & = & \\delta B \\exp {(ik_\\parallel z)}$ Inserting this two expressions into Eq.", "REF we obtain: $A_x & = & \\frac{\\delta B}{k_\\parallel } \\exp {[i(k_\\parallel z - \\pi /2)]} \\nonumber \\\\A_y & = &\\frac{\\delta B}{k_\\parallel } \\exp {(ik_\\parallel z)}$ Comparing the real parts of these equations with equation $(1b)$ of [15] we obtain an expression for the coefficients $A_1$ and $A_2$ and for the normalized components of the wave polarization vector $\\epsilon _1$ , $\\epsilon _2$ and $\\epsilon _3$ : $A_1 = \\eta \\frac{\\delta B}{k_\\parallel }, \\quad A_2 = \\frac{\\delta B}{k_\\parallel },\\quad \\textrm {where} \\quad \\eta =\\frac{k_\\parallel }{|k_\\parallel |}$ $\\epsilon _1 = \\frac{|q|\\eta \\delta B}{mc^2 k_\\parallel }, \\quad \\epsilon _2 = \\frac{|q|\\delta B}{mc^2 k_\\parallel },\\quad \\epsilon _3=0.$ Similarly, for a right-handed circularly polarized wave ${\\bf B}_-$ we have: $A_1 = - \\eta \\frac{\\delta B}{k_\\parallel }, \\quad A_2 = \\frac{\\delta B}{k_\\parallel },\\quad \\textrm {where} \\quad \\eta =\\frac{k_\\parallel }{|k_\\parallel |}$ $\\epsilon _1 = - \\frac{|q|\\eta \\delta B}{mc^2 k_\\parallel }, \\quad \\epsilon _2 = \\frac{|q|\\delta B}{mc^2 k_\\parallel },\\quad \\epsilon _3=0.$ In case of a single circularly polarized wave propagating parallel (or antiparallel) to the magnetic field there is only one resonance present and particle motion is integrable [15]: indeed $J_n(0)= 0$ unless $n=0$ .", "Therefore depending on the polarization of the wave and on its direction of propagation $\\eta $ only $l=1$ or $l=-1$ resonances contribute to the trapping width, as shown in Table REF .", "Table: Wave polarization and resonance contribution to trapping width.Thus, considering equations $(5a)$ and $(5b)$ of [15], Eq.", "REF and Eq.", "REF -REF with $J_0(0)=1$ , find a specialized formula for the trapping half width and bounce frequency applied to the case of a circularly polarized wave propagating parallel $k_\\parallel > 0$ and $n=-1$ , or antiparallel, $k_\\parallel > 0$ and $n=1$ to ${\\bf B}_0$ : $\\Delta {v_\\parallel }^{(-1)} & = & 2v \\left[(1-\\alpha ^2)^{1/2}|\\alpha |\\frac{\\delta B}{B_0}\\right]^{1/2} \\nonumber \\\\{\\omega _b}^{(-1)} & = & \\Omega _0\\left[\\frac{(1-\\alpha ^2)^{1/2}}{|\\alpha |}\\frac{\\delta B}{B_0}\\right]^{1/2}$ if $k_\\parallel v_\\parallel > 0$ and zero otherwise, in which $\\alpha =\\cos \\theta $ is the cosine of pitch angle.", "Exactly the same set of equations holds for $\\Delta v_\\parallel ^{(+1)}$ and $\\omega _b^{(+1)}$ .", "However the condition for their being nonzero is reversed, i.e., $k_\\parallel v_\\parallel > 0$ .", "We omit the superscripts $(\\pm 1)$ because of this degeneracy." ] ]

[ [ "Reflection of Channel-Guided Solitons at Junctions in Two-Dimensional\n Nonlinear Schroedinger Equation" ], [ "Abstract Solitons confined in channels are studied in the two-dimensional nonlinear Schr\\\"odinger equation.", "We study the dynamics of two channel-guided solitons near the junction where two channels are merged.", "The two solitons merge into one soliton, when there is no phase shift.", "If a phase difference is given to the two solitons, the Josephson oscillation is induced.", "The Josephson oscillation is amplified near the junction.", "The two solitons are reflected when the initial velocity is below a critical value." ], [ "Reflection of Channel-Guided Solitons at Junctions in Two-Dimensional Nonlinear Schrödinger Equation Yusuke Kageyama and Hidetsugu Sakaguchi Department of Applied Science for Electronics and Materials, Interdisciplinary Graduate School of Engineering Sciences, Kyushu University, Kasuga, Fukuoka 816-8580, Japan Solitons confined in channels are studied in the two-dimensional nonlinear Schrödinger equation.", "We study the dynamics of two channel-guided solitons near the junction where two channels are merged.", "The two solitons merge into one soliton, when there is no phase shift.", "If a phase difference is given to the two solitons, the Josephson oscillation is induced.", "The Josephson oscillation is amplified near the junction.", "The two solitons are reflected when the initial velocity is below a critical value.", "The one-dimensional nonlinear Schrödinger equation has been intensively studied as a typical soliton equation for optical solitons in optical fibers [1] and matter-wave solitons in the Bose-Einstein condensates (BECs) [2],[3].", "However, solitons in two- or three-dimensional nonlinear Schrödinger equations were not intensively studied.", "Two-dimensional solitons exist in guided channels in the two-dimensional nonlinear Schrödinger equation.", "[4] The guiding channel can be constructed by modifying the profile of the refraction index in an optical planar waveguide.", "Cigar-shaped traps are used as a guiding channel to confine matter-wave solitons in BECs.", "Solitons can propagate along the guiding channel with an arbitrary velocity if the guiding channel is uniform and the norm of the solitons is below a critical value for collapse.", "In our previous study [5], we investigated the reflection of a channel-guided soliton in a tapered channel and the splitting at a branching point where one channel branches into two channels [6].", "In this study, we investigate the motion of solitons in curved channels and near the junction where two channels merge into a channel.", "The model equation is written as $i\\frac{\\partial \\phi }{\\partial t}=-\\frac{1}{2}\\nabla ^2\\phi -|\\phi |^2\\phi +U(x,y)\\phi ,$ where $U(x,y)$ denotes the potential for confinement, which is used to make a guiding channel.", "For a straight channel, $U(x,y)=-U_0$ for $-x_0\\le x \\le x_0$ and $U(x,y)=0$ for other regions.", "The width of the channel is denoted as $2x_0$ and the depth of the potential is denoted as $U_0$ .", "There are stationary soliton solutions $\\phi _0(x,y)$ in the straight channel.", "The stationary solution $\\phi _0(x,y)$ can be numerically obtained by the imaginary-time evolution of eq.", "(1) as was carried out in our previous study [5].", "The two-dimensional soliton can move with any velocity $k_y$ if the initial condition is set to be $\\phi _0(x,y-y_0)\\exp \\lbrace ik_y (y-y_0)\\rbrace $ .", "In the previous study, we investigated the splitting and reflection of a two-dimensional soliton in a branching channel and designed a branching channel where a soliton can split smoothly into two solitons without changing its velocity.", "If the branching channels merge into a straight channel at a junction point again, a waveguide system, such as the Mach-Zehnder interferometer, can be constructed, as shown in Fig. 1(a).", "The branching point is $y=-40$ and the junction point is $y=40$ in the channel system in Fig. 1(a).", "A phase shift $\\Delta \\varphi $ is given to the two split solitons when the center of the solitons goes through the line $y=0$ as $\\phi (x,y)\\rightarrow \\phi (x,y)\\exp (i\\Delta \\varphi )$ for $x<0$ and $\\phi (x,y)\\rightarrow \\phi (x,y)\\exp (-i\\Delta \\varphi )$ for $x>0$ .", "This procedure is an artificial one, which is simple and easy for numerical simulation.", "In an actual experiment, the phase shift can be effectively given to the two solitons by controlling the profiles of the refraction index and the forms of the two channels similar to the typical Mach-Zehnder interferometer.", "When $\\Delta \\varphi =0$ , the two solitons merge again into one soliton at the junction located at $y=40$ .", "We have performed numerical simulation with the split-step Fourier methods with $128\\times 1024$ modes for this rectangular system.", "For nonzero $\\Delta \\varphi $ , a zigzag oscillation of the center of gravity is observed, as shown in Fig. 1(b).", "Figure 1(b) shows the trajectories of the center of gravity of the solitons for $\\Delta \\varphi =0,\\pi /64,\\pi /32$ and $\\pi /16$ at $k_y=1$ .", "The norm $N$ of the soliton is 5.", "A zigzag oscillation appears in a region of $y>20$ .", "The amplitude of the zigzag oscillation increases with $\\Delta \\varphi $ , but the period of the oscillation is almost the same.", "This zigzag oscillation is interpreted as a remnant of the Josephson oscillation that appears before the junction point at $y=40$ .", "The Josephson oscillation occurs owing to the tunnel effect through the potential wall of height $U_0$ between the two channels.", "The Josephson oscillation is explained in detail in the following.", "When the initial velocity $k_y$ is decreased for a fixed phase shift value, the solitons are reflected before the junction point at $y=40$ .", "Figure 1(c) shows the time evolution of $|\\phi |$ along the center lines of the left and right channels at $k_y=0.37$ for $\\Delta \\varphi =\\pi /256$ .", "Before the reflection, the amplitudes of the two solitons in the left and right channels are almost the same, but the difference between the amplitudes of the two solitons increases after the reflection, and the soliton becomes localized in one of the two channels.", "The critical velocity $k_{yc}$ is 0.38 for $\\Delta \\varphi =\\pi /256$ .", "The relationship between $k_{yc}$ and the phase shift $\\Delta \\varphi $ is shown in Fig. 1(d).", "Note that the horizontal axis is plotted with a logarithmic scale.", "Figure 1(d) implies that the critical velocity $k_{yc}$ increases rapidly near $\\Delta \\varphi =0$ .", "If there is a phase difference between two solitons located on different channels, the Josephson oscillation occurs [7],[8], because the two solitons interact with each other by the tunnel effect through the potential barrier of height, $U_0$ .", "We can take an ansatz for the form of $\\phi $ as $\\phi (x,y)=u(y,t)\\exp [-\\lbrace x-\\eta (y)\\rbrace ^2/(2b^2)]\\exp (-i\\mu t)+v(y,t)\\exp [-\\lbrace x+\\eta (y)\\rbrace ^2/(2b^2)]\\exp (-i\\mu t)$ , where $\\pm \\eta (y)$ is the $x$ -coordinate of the central point in the right and left channels at $y$ .", "The substitution of the form into the Lagrangian (2) and the variational principle: $\\partial /\\partial t(\\delta L/\\delta u_t)=\\delta L/\\delta u$ , $\\partial /\\partial t(\\delta L/\\delta v_t)=\\delta L/\\delta v$ , yield approximately coupled equations for $u$ and $v$ : $i\\frac{\\partial u}{\\partial t}&=&-\\frac{1}{2}\\frac{\\partial ^2u}{\\partial y^2}-c(|u|^2+g|v|^2)u-d(v-u),\\nonumber \\\\i\\frac{\\partial v}{\\partial t}&=&-\\frac{1}{2}\\frac{\\partial ^2v}{\\partial y^2}-c(|v|^2+g|u|^2)u-d(u-v),$ where $g=2\\exp (-2\\eta (y)^2/b^2)$ and $d=2U_0\\exp (-2\\eta (y)^2/b^2)$ .", "Here, we have neglected some complex terms, such as $u^{*}v^2$ and $v^{*}u^2$ .", "The parameter $c$ is assumed to be $c=1/\\lbrace \\sqrt{2}(1+g)\\rbrace $ , because the symmetric solution $u=v$ propagates without changing the profile of the solitons.", "When $\\eta $ is large, mutual interaction is weak and the two solitons propagate independently.", "However, as $y$ is close to the junction point $y=40$ , the Josephson effect becomes strong because of a small $\\eta $ value.", "Figure: (a) Channel with branching and junction points.", "(b) Trajectories of the center of gravity of solitons for k y =1k_y=1.", "The phase shifts of Δϕ=0,π/64,π/32\\Delta \\varphi =0, \\pi /64,\\pi /32 and π/16\\pi /16 are given at y=0y=0.", "The straight line is the trajectory for Δϕ=0\\Delta \\varphi =0.", "The zigzag oscillation grows with Δϕ\\Delta \\varphi .The trajectory that exhibits the largest zigzag oscillation corresponds to Δϕ=π/16\\Delta \\varphi =\\pi /16.", "(c) Time evolution of |φ||\\phi | along the center lines of left and right channels at k y =0.37k_y=0.37 for Δϕ=π/256\\Delta \\varphi =\\pi /256.", "(d) Critical values k yc k_{yc} of the reflection as a function of Δϕ\\Delta \\varphi .Figure: (a) Time evolution of |u||u| (solid curve) and |v||v| (dashed curve) for k y =0.29k_y=0.29 and Δϕ=π/256\\Delta \\varphi =\\pi /256.", "(b) Time evolution of R=(N u -N v )/(N u +N v )R=(N_u-N_v)/(N_u+N_v).", "(c) Time evolution of ξ(t)\\xi (t) determined using eq.", "(7) for k y =0.29k_y=0.29 and Δθ=π/128\\Delta \\theta =\\pi /128.", "(d) Time evolution of RR determined using eq.", "(7).Figure 2(a) shows the time evolution of $|u|$ and $|v|$ determined using the coupled equations (2) at $k_y=0.2$ .", "The initial conditions of $u$ and $v$ are $u=A/{\\rm cosh}(y/W)\\exp (i\\Delta \\varphi )$ and $v=A/{\\rm cosh}(y/W)\\exp (-i\\Delta \\varphi )$ where $b=0.9,N_0=5/(\\sqrt{\\pi }b),W=4\\sqrt{2}/N_0,A=\\sqrt{N_0/(4W)}$ , and $\\Delta \\varphi =\\pi /256$ .", "The two solitons are reflected, and one of the solitons dominates after the reflection.", "The critical velocity $k_y=0.295$ is slightly different from that in the direct numerical simulation, but the qualitative behaviors are similar.", "The asymmetry of the two solitons is expressed as $R=(N_u-N_v)/(N_u+N_v)$ where $N_u=\\int |u|^2dy$ and $N_v=\\int |v|^2dy$ are the norms of $u$ and $v$ , respectively.", "The time evolution of the ratio $R$ is shown in Fig. 2(b).", "The ratio $R$ increases from 0 and reaches a positive constant value after the reflection.", "This implies that one soliton dominates in the $u$ -channel after the reflection.", "If $u(y,t)=A{\\rm sech}\\lbrace (y-\\xi )/W\\rbrace \\exp \\lbrace ip(y-\\xi )-i\\theta _1\\rbrace $ and $v(y,t)=B{\\rm sech}\\lbrace (y-\\xi )/W\\rbrace \\exp \\lbrace ip(y-\\xi )-i\\theta _2\\rbrace $ are further assumed, the effective Lagrangian is evaluated as $L_{eff}&=&\\frac{1}{2}\\int \\lbrace i(u_tu^*-uu_t^*+v_tv^*-vv_t^*)-|u_y|^2-|v_y|^2+c(|u|^4+|v|^4+2g|u|^2|v|^2)\\nonumber \\\\& &+2d(uv^*+vu^*)-2d(|u|^2+|v|^2)\\rbrace dy\\nonumber \\\\&=&N_0[p\\xi _t-p^2/2-1/(6W^2)+(\\theta _{1t}+\\theta _{2t})/2+R(\\theta _{1t}-\\theta _{2t})/2\\nonumber \\\\& &+R^2N_0e/(16W^2)+\\tilde{d}\\lbrace \\sqrt{1-R^2}\\cos (\\theta _2-\\theta _1)-1\\rbrace ]$ where $N_0=\\int \\lbrace |u|^2+|v|^2\\rbrace dy=2(A^2+B^2)W$ , $R=(A^2-B^2)/(A^2+B^2)$ , $e(\\xi )=\\int c(y)\\lbrace 1-g(y)\\rbrace {\\rm sech}^4\\lbrace (y-\\xi )/W\\rbrace dy$ , and $\\tilde{d}(\\xi )=\\int d(y){\\rm sech}^2\\lbrace (y-\\xi )/W\\rbrace dy/\\int {\\rm sech}^2\\lbrace (y-\\xi )/W\\rbrace dy$ .", "The variational principle yields $W&=&\\frac{4}{\\lbrace 1+R^2+(1-R^2)g\\rbrace N_0c},\\nonumber \\\\\\frac{d\\xi }{dt}&=&p,\\nonumber \\\\\\frac{dp}{dt}&=&\\frac{R^2N_0}{16W^2}\\frac{\\partial e}{\\partial \\xi }+\\lbrace \\sqrt{1-R^2}\\cos (\\theta _2-\\theta _1)-1\\rbrace \\frac{\\partial \\tilde{d}}{\\partial \\xi },\\nonumber \\\\\\frac{dR}{dt}&=&2\\tilde{d}\\sqrt{1-R^2}\\sin \\Delta \\theta ,\\nonumber \\\\\\frac{d\\Delta \\theta }{dt}&=&\\frac{RN_0e}{4W^2}-\\frac{2\\tilde{d}R}{\\sqrt{1-R^2}}\\cos \\Delta \\theta ,$ where $\\Delta \\theta =\\theta _2-\\theta _1$ , and $\\xi $ is the $y$ -coordinate of the center of gravity of the solitons.", "If $R=0$ and $\\theta _1=\\theta _2$ , $dp/dt=0$ and the two solitons propagate with a constant velocity.", "If $R<<1$ and $\\Delta \\theta <<1$ , $d^2\\Delta \\theta /dt^2=-2\\tilde{d}\\lbrace 2\\tilde{d}-N_0e/(4W^2)\\rbrace \\Delta \\theta $ is obtained using the last two equations in eq.", "(4), which describes the Josephson oscillation.", "If $N_0e/(4W^2)-2\\tilde{d}>0$ , the Josephson oscillation is amplified and the symmetric state $u=v$ becomes unstable.", "Because $e(\\xi )$ and $\\tilde{d(\\xi })$ are not uniform, $dp/dt$ can become negative owing to the amplified Josephson oscillation.", "Figures 2(c) and 2(d) show the time evolution of $\\xi $ and $R$ for $N_0=5/(\\sqrt{\\pi }b)$ with $b=0.9$ and $\\xi (0)=0,p(0)=k_y=0.2$ , and the initial phase difference $\\Delta \\theta (0)=2\\Delta \\varphi =\\pi /128$ .", "The reflection of the trajectory and the amplification of the Josephson oscillation are observed.", "The ratio $R$ takes a constant value for large $t$ values, although the sign of the constant value depends strongly on the initial velocity $p(0)$ because of the many oscillations of $R$ near the reflection point.", "The critical value $k_{yc}$ for $\\Delta \\theta (0)=\\pi /128$ determined using eq.", "(4) is 0.305, which is close to that by eq. (1).", "The variational approximation is fairly good.", "To summarize, we have performed some numerical simulations of channel-guided solitons in the two-dimensional nonlinear Schrödinger equation.", "We have found that reflection occurs near the junction induced by the amplification of the Josephson oscillation.", "The complex dynamical behaviors can be analyzed approximately on the basis of the variational principle using effective Lagrangians.", "Channel-guided solitons exhibit various complex dynamics, and we would like to investigate this further in the future.", "We would like to thank Prof. B. A.", "Malomed for valuable discussions." ] ]

[ [ "STM investigation of structural properties of Si layers deposited on\n Si(001) vicinal surfaces" ], [ "Abstract This communication covers investigation of the structural properties of surfaces of Si epitaxial layers deposited on different Si(001) vicinal substrates.", "We have shown processes of generation and growth of surface defects to depend on tilt direction of a Si(001) wafer and epilayer growth mode.", "We suppose these effects to be connected with mutual interaction of monoatomic steps." ], [ "STM investigation of structural properties of Si layers deposited on Si(001) vicinal surfacesL. V.", "Arapkina, V. A. Chapnin, K. V. Chizh, L. A. Krylova, V. A. Yuryev STM investigation of structural properties of Si layers deposited on Si(001) vicinal surfacesL. V.", "Arapkina$^{1}$ , V. A. Chapnin$^{1}$ , K. V. Chizh$^{1}$ , L. A. Krylova$^{1}$ , V. A. Yuryev$^{1,2}$ $^{1}$  Prokhorov General Physics Institute of the Russian Academy of Sciences, 38 Vavilov Street, Moscow, 119991, Russia $^{2}$  Technopark of GPI RAS, 38 Vavilov Street, Moscow, 119991, Russia This report covers investigation of the structural properties of surfaces of Si epitaxial layers deposited on Si(001) vicinal substrates with different miscuts.", "We have shown processes of generation and growth of surface defects to depend on tilt direction of a Si(001) wafer and epilayer growth mode.", "We suppose these effects to be connected with interaction of monoatomic steps.", "A structure of a Si(001) epitaxial layer surface, especially its defects, could affect the formation of nanostructures.", "Perfect, defectless Si epilayers grown on Si(001) vicinal substrates are of special importance for such industrially significant problem as controllable formation of Ge/Si(001) nanostructures for optoelectronic device applications.", "This report covers experimental investigation of structural properties of surfaces of Si epitaxial layers deposited on Si(001) vicinal surfaces of substrates with different miscuts.", "Experiments were carried out in UHV using GPI-300 STM coupled with Riber EVA 32 MBE chamber [1].", "Epitaxial layers were deposited by MBE on substrates cut from Si(001) vicinal wafers tilted $\\sim $  0.2° towards the [110] or [100] direction.", "Initial surfaces were treated by the RCA etchant.", "Before Si deposition, we cleaned the surfaces by the standard methods of preliminary annealing at 600℃ and decomposition of the SiO$_2$ film under a weak flux of the Si atoms at 800℃ [2].", "The substrate temperature during Si deposition was chosen in the range from 360 to 700℃.", "We have explored structural properties of Si epitaxial films deposited on Si(001) vicinal substrates depending on the growth temperature and the rate of Si deposition.", "Two modes of Si epitaxial growth have been observed.", "We have found that the step-flow growth goes on at the temperatures above 600℃ whereas the island growth takes place at the temperatures below 600℃.", "Samples grown at the step-flow growth mode have smooth surfaces composed of terraces bounded by monoatomic steps.", "STM data shown in Figs.", "1 and 2 are related to the Si epilayers deposited on the wafers tilted towards the [110] and [100] direction respectively.", "At first, we consider STM data for Si/Si(001) surfaces tilted towards the [110] direction.", "The surface is composed by S$_{\\rm A}$ and S$_{\\rm B}$ monoatomic steps [3]; S$_{\\rm B}$ steps are wider than S$_{\\rm A}$ ones.", "We have observed formation of such defects as faceted pits on these surfaces.", "In Fig.", "1, an initial stage of the defect formation is shown.", "There observed the local stoppage of growth of an S$_{\\rm B}$ step and appearance of two S$_{\\rm A}$ steps instead.", "In other words, we have observed a gap of the S$_{\\rm B}$ step.", "The S$_{\\rm A}$ steps repulse each other and the defect cannot be overgrown quickly.", "The bottom of the deep pit has a rectangular shape and a long side of it formed by the S$_{\\rm A}$ step.", "Figure: STM image of the surface of the 50-nm Si epilayer deposited at 650℃ on the Si(001) vicinal wafer tilted ∼\\sim  0.2° towards [110], the deposition rate was ∼\\sim  0.3 Å/c.Figure: STM image of the surface of the 50-nm Si epilayer deposited at 650℃ on the Si(001) vicinal wafer tilted ∼\\sim  0.2° towards [100], the deposition rate was ∼\\sim  0.3 Å/c.Figure: STM image of the surface of the 50-nm Si epilayer deposited at 650℃ on the Si(001) vicinal wafer tilted ∼\\sim  0.2° towards [100], the deposition rate was ∼\\sim  0.1 Å/c.STM data for the Si/Si(001) surface tilted towards the [100] direction are presented in Fig. 2.", "The surface is composed by bent monoatomic steps.", "In this case every monoatomic step consists of short parts of S$_{\\rm A}$ and S$_{\\rm B}$ steps and runs along the [100] direction.", "There are local disarrangements of the structure.", "We suppose this kind of defects to be connected with a process of transition from the mixed S$_{\\rm A}$  + S$_{\\rm B}$ monoatomic step to two single S$_{\\rm A}$ and S$_{\\rm B}$ steps instead of formation of the D$_{\\rm B}$ step.", "We have investigated the structural properties of the Si film surface depending on the rate of Si deposition.", "We have found that reduction of the Si deposition rate from  0.3 Å/c (Fig.", "2) to 0.1 Å/c (Fig.", "3) results in appearance of the structure formed by monoatomic steps running along the [110] direction and formation of shapeless pits on the surface instead of the structure formed by the bent monoatomic steps which run along the [100] direction.", "In Figs.", "4 and 5, we present STM data for surfaces of the epilayers deposited at the temperatures of 550 and 470℃.", "Surfaces of the samples grown at 550℃ consist of S$_{\\rm A}$ and S$_{\\rm B}$ monoatomic steps (Fig. 4).", "The mixed S$_{\\rm A}$  + S$_{\\rm B}$ monoatomic steps which are typical for the Si(001) surfaces tilted towards [100] and obtained at higher temperatures are not observed.", "At this temperature, the transitional mode of the epilayer growth, intermediate between the island growth and the step-flow one, is observed for both tilt directions.", "Further reduction of the temperature down to 470℃ results in the island growth mode (Fig. 5).", "The Si/Si(001) surface is composed by small islands.", "In both causes such defects as pits are present on the surface.", "The structural properties of the surfaces of Si epilayers grown at the island growth mode do not depend on the direction of the surface tilt.", "Figure: STM image of the surface of the 50 nm thick Si epilayer deposited at 550℃ on the Si(001) vicinal wafer (deposition rate is ∼\\sim  0.3 Å/c); tilt is ∼\\sim  0.2° towards (a) [110] and (b) [100].Figure: STM image of the surface of the 50 nm thick Si epilayer deposited at 470℃ on the Si(001) vicinal wafer (deposition rate is ∼\\sim  0.3 Å/c); tilt is ∼\\sim  0.2° towards (a) [110] and (b) [100].Summarizing the above we can conclude that processes of generation and growth of surface defects arising during epitaxial growth of Si films on Si(001) vicinal substrates depend on tilt direction of a Si(001) wafer and the epilayer growth conditions.", "We suppose the observed effects to be a consequence of mutual interactions of monoatomic steps.", "This research has been supported by the Ministry of Education and Science of Russian Federation through the contracts No.", "14.740.11.0069 and 16.513.11.3046.", "Facilities of Center of Collective Use of Scientific Equipment of GPI RAS were utilized in this research.", "We appreciate the financial and technological support." ] ]

[ [ "Scanning of Rich Web Applications for Parameter Tampering\n Vulnerabilities" ], [ "Abstract Web applications require exchanging parameters between a client and a server to function properly.", "In real-world systems such as online banking transfer, traversing multiple pages with parameters contributed by both the user and server is a must, and hence the applications have to enforce workflow and parameter dependency controls across multiple requests.", "An application that applies insufficient server-side input validations is however vulnerable to parameter tampering attacks, which manipulate the exchanged parameters.", "Existing fuzzing-based scanning approaches however neglected these important controls, and this caused their fuzzing requests to be dropped before they can reach any vulnerable code.", "In this paper, we propose a novel approach to identify the workflow and parameter dependent constraints, which are then maintained and leveraged for automatic detection of server acceptances during fuzzing.", "We realized the approach by building a generic blackbox parameter tampering scanner.", "It successfully uncovered a number of severe vulnerabilities, including one from the largest multi-national banking website, which other scanners miss." ], [ "Introduction", "Web applications typically require traversing multiple pages with parameters exchanged between a client and a server to complete even a single action.", "Figure REF depicts a simplified workflow of an online banking transfer application used by the HSBC bank, in which a user can transfer a certain amount of money (i.e., AMT) from his account FROM only to an authorized account TO.", "While user-supplied transaction details (i.e.,FROM, TO, AMT) are taken from input elements of a form, those server-generated ones such as session ID (SESSID) and the one-time use tokens working against Cross-Site Request Forgeries (CSRF$_{1\\&2}$ ) are respectively set in Cookies and hidden fields.", "In Step$_A$ , client-side validations are applied to restrict and instantly prompt the user for input corrections before the form can be submitted.", "Upon a valid submission, the ServHandler$_A$ verifies the CSRF$_1$ token, validates the user inputs, and responds with a review page for Step$_B$ .", "The user confirms the transaction by submitting Req$_B$ , and finally the ServHandler$_B$ executes the actual banking transfer and returns an acknowledgment page for user's reference at Step$_C$ .", "It is worth noting that ServHandler$_B$ enforces, among others, that the received token CSRF$_2$ and TO account respectively match with their previously stored values (i.e., sess.CSRF$_2$ and sess.TO).", "Such kind of web applications are however vulnerable to parameter tampering that is known to attack insecure direct object references, which is ranked No.", "4 in the OWASP Top 10 Web Application Security Risks [26].", "The vulnerability often arises from a misconception that the object references directly exposed as parameters to the client-side (and their validations) are assumed immutable, and thus improper or insufficient validations were applied at the server [10].", "As shown in the motivating example, the server does not validate whether the TO account is authorized (i.e., TO is missing at valid(FROM,AMT) in ServHandler$_A$ ).", "Hence, an attacker who has compromised a victim's session can tamper the TO parameter and bypass any associated client-side validations in order to commit unauthorized banking transfers.", "The parameter tampering vulnerability revealed above is however hardly discoverable despite many research efforts that were dedicated to web vulnerability scanning [8], [7], [11], [9], [32], [29], [13], [5].", "There are two fundamental reasons.", "First, blackbox fuzzing-based scanners such as [7], [1], [25], [20] cannot preserve the intended workflow.", "It is because they literally work in a “crawl-once-fuzz-many” manner, that captures a set of requests only once during crawling, and these requests will become the only bases to generate subsequent fuzzing requests.", "However, the one-time use tokens such as CSRF$_{1\\&2}$ expire as soon as they are accepted by the server during crawling.", "Regardless of how parameters are mutated, subsequent fuzzing attempts that reuse such expired tokens will all be rejected.", "Second, existing blackbox scanners neglect cross-request parameter dependencies.", "To uncover the vulnerbility in the banking example, the TO account from Req$_A$ and the one later sent in Req$_B$ must be equal.", "It is because ServHandler$_A$ stores the user-supplied parameter TO from Req$_A$ into sess.TO.", "When receiving Req$_B$ , ServHandler$_B$ enforces that TO in Req$_B$ must match with the stored sess.TO.", "If they are different, Req$_B$ will be rejected before any parameters can reach the vulnerable code.", "None of the blackbox fuzzing tools can identify this constraint and maintain the required parameter dependencies across the requests.", "While a state-aware fuzzer is recently proposed to observe the workflow control [11], its fuzzing requests are still insensitive towards cross-request dependencies.", "For other approaches that consider parameter relationships [33], [34], they are largely manual and protocol-specific.", "This paper presents Cross-Request Scanner (CRS), a novel approach that respects and leverages the intended workflow and parameter dependency controls while scanning for parameter tampering vulnerabilities.", "CRS consists of two phases: capturing and fuzzing.", "In the capturing phase, CRS records a set of valid user actions, identifies the one-time tokens, tracks the cross-request parameter dependencies, and learns key features (e.g., locations of submit buttons and reflected parameters) that indicate a server acceptance from rendered server responses.", "In the fuzzing phase, CRS sequentially replays the user actions to fetch new responses while keeping those confirmed one-time and dependent parameters intact so as to preserve the intended workflow and parameter dependency.", "Other parameters are mutated and placed back to the application itself for validations.", "Only those client-side rejected parameters are then forcefully submitted by bypassing client-side validations.", "Finally, it reports a vulnerability if the server accepts the mutated parameters and gives responses that are in line with those key features learned in the initial valid submissions (e.g., submit buttons reappeared, parameters reflected).", "While CRS takes a set of manually provided user actions, we argue that it is unavoidable in discovering parameter tampering vulnerabilities (i.e., unlike discovering XSS by simply asserting a pop-up dialog after an injection of what comprises alert(1)).", "Referring to the banking example, a parameter tampering vulnerability can be resulted by mutating only a digit in the TO account.", "Therefore, it is clear that the semantic meaning and underlying consequence of a mutation can be domain-specific, and thus known only to humans.", "Existing work also corroborate the needs of manual effort [12], [34].", "CRS requires the least amount of manual assistance among blackbox parameter tampering scanners [7], [1], [25], [20].", "The contributions of this paper are as follows: A field study on online banking applications to understand their workflow and implementations, and how they can be intercepted for vulnerability scanning.", "A novel approach that respects the intended workflow and cross-request parameter dependency while fuzzing, as well as to correlate the dependency in both requests and responses for automatic detection of server acceptances.", "An “in-context fuzzing” technique to build a blackbox vulnerability scanner that drives fuzzing in the application context allowing dynamic features to be preserved, and thus improving coverage and accuracy.", "The discovery of real-world vulnerabilities that are uncovered only by our scanner, which existing approaches miss.", "The rest of this paper is organized as follows: Section  describes the online banking transfer applications of different banks.", "Section  discusses the CRS approach.", "Section  provides some technical background on the web applications, followed by the implementation of the CRS scanner.", "Section  evaluates the scanner, and details the vulnerabilities uncovered.", "Section  presents related work.", "Finally, we conclude." ], [ "Motivating Examples", "This paper focuses on real-world applications, and that makes the testing intrinsically blackbox and more challenging owing to the lack of server-side source code.", "This section summarizes four representative banking transfer applications, which are among the most security-critical operations being carried out over the Internet.", "They include Citibank, HSBC, Bank of China (BOC), and Bank of East Asia (BEA), with their headquarters respectively located at the US, UK, China, and Hong Kong.", "The unauthorized transfers that are made possible in HSBC and BEA are discussed in Section REF ." ], [ "Workflow Design", "As outlined in Figure REF , all banks adopt a three-step workflow design to receive instructions of banking transfers.", "Step$_A$.", "A user specifies a source (i.e., FROM) and a destination (i.e., TO) account besides setting an amount (i.e., AMT) to transfer.", "The government mandates that the user can transfer his money only to third-party accounts that are authorized through an out-of-band channel.", "HSBC accepts unauthorized accounts to be specified in this step, and provides on-the-fly authorization in the next step, as shown in Figure REF .", "Citibank and BOC users can pre-authorize an account beforehand in a separate online form with an One-Time Password (OTP) that is received through SMS.", "BEA offers only offline authorization (i.e., must register an account by person in a physical branch).", "Finally, the user submits Req$_A$ by clicking “Go”.", "Step$_B$.", "The user reviews the transaction details returned by the server.", "If an HSBC user has specified an unauthorized account in Step$_A$ , he needs to enter an OTP token from his own hardware device for on-the-fly authorization, as detailed in Section REF .", "The user finally submits again by clicking “confirm” to send Req$_B$ .", "Step$_C$.", "The transfer instruction is acknowledged with a transaction number.", "The transaction details are shown for user's reference.", "Table: The website features that are essential for effective fuzzing by blackbox web vulnerability scanners" ], [ "Workflow Implementation", "Notably, HSBC, BOC and BEA implement each step of the workflow in a separate page, while Citibank integrates all steps in a single AJAX page with the use of XMLHttpRequest API.", "Their workflow implementations are outlined as follows: Step$_A$.", "Client-side code is implemented to restrict the user-supplied input parameters (i.e., FROM, TO, AMT) before they can be submitted (as Req$_A$ ) together with a CSRF$_1$ token that is server-generated and placed in a hidden field.", "All banks apply some client-side pre-processing (e.g., manipulating values of hidden fields) before validating and submitting the form.", "The submission approaches vary from one bank to another, as detailed in Section REF .", "ServHandler$_A$.", "The server verifies whether the CSRF$_1$ token matches with what was previously stored.", "Upon validating other parameters that represent the transaction details, the server stores them temporarily.", "Finally, it returns a review page in which a newly generated CSRF$_2$ token is embedded in a hidden field.", "Notice that besides defending against CSRF, both CSRF$_{1\\&2}$ tokens can also serve the purposes of enforcing the intended workflow and preventing duplicate transactions owing to their nature of one-time uses.", "Step$_B$.", "It echoes all transaction-related parameters visually for user's review, and that a confirming click will naturally include the hidden CSRF$_2$ token in the submission Req$_B$ .", "In HSBC, an OTP parameter may be solicited for account authorization.", "For HSBC and BOC, this step also serves to resubmit those transaction parameters gathered from Step$_A$ by embedding them in hidden fields.", "ServHandler$_B$.", "The server again verifies the CSRF$_2$ token beforehand.", "In HSBC, the OTP parameter, if provided, will then be verified.", "HSBC and BOC further ensure if some transaction-related parameters match with those stored at ServHandler$_A$ .", "The server finally executes the transfer with the transaction parameters.", "Step$_C$.", "The transaction details and a reference number are shown." ], [ "Intrinsic Limitations of Existing Scanners", "We verify that the banks enforce the workflow design and parameter dependency across requests by empirically and systematically running a series of experiments in their banking transfer applications.", "We first interact with an application through its user interface (UI), and capture a pair of submission requests from a valid transaction, which is referred to as {Req$_{A.o}$ , Req$_{B.o}$ }.", "Next, we prepare a mutated pair of requests {Req$_{A.m}$ , Req$_{B.m}$ }, in which all AMT parameters are incremented by “1”.", "Neglecting Intended Workflow and One-time Tokens.", "With this mutated set of requests, we however found that they were all rejected by the server through the following experiments.", "First of all, the requests were replayed according to a sequence of a few Req$_{A.m}$ s and then some Req$_{B.m}$ s, of which the sequence obviously violates the intended workflow.", "The requests were all rejected.", "We then proceed to sequentially replay {Req$_{A.m}$ , Req$_{B.m}$ , Req$_{A.m}$ , Req$_{B.m}$ }, but they were still rejected despite observing the workflow.", "The server actually responds differently to requests that are initiated by re-interacting with the application as usual.", "Therefore, when we interact with the UI and specify the same user-supplied values (including the incremented AMT) as in those mutated requests, the new requests are accepted as expected.", "We investigated all the requests generated through multiple times of UI interactions.", "We then found some distinct server-generated tokens were introduced to every request.", "Since replaying Req$_{A.o}$ that was once accepted by the server would result in a failure, we conclude that these tokens are of one-time uses.", "Breaking Cross-request Parameter Dependency.", "As discussed in Section REF , it is found that HSBC and BOC submit some transaction-related and other parameters twice in both Req$_A$ and Req$_B$ .", "To understand the relationship between the parameters, we thus run the remaining three possible pairs of request combinations {Req$_{A.o}$ , Req$_{B.m}$ }, {Req$_{A.m}$ , Req$_{B.o}$ }, and {Req$_{A.m}$ , Req$_{B.m}$ }, in which the one-time tokens are observed.", "Here, all transaction-related parameters are “slightly” mutated one by one.", "When running {Req$_{A.o}$ , Req$_{B.m}$ }, it is found that some parameters mutated only at Req$_{B.m}$ are actually disregarded, while their corresponding values given at Req$_{A.o}$ are instead honored by the server.", "This result shows that these parameters must be mutated at Req$_A$ during fuzzing.", "Mutating them at Req$_{B.m}$ can be ineffective.", "For {Req$_{A.m}$ , Req$_{B.o}$ }, server rejections are resulted since some parameter values are inconsistent between the request pair.", "Clearly, the server enforces certain parameters to be equal across the requests.", "Likewise, for {Req$_{A.m}$ , Req$_{B.m}$ }, server rejections depend on whether some parameters are mutated in such a way that the required dependency are broken across both requests.", "In a nutshell, existing scanners [7], [1], [25], [20], [2], [31], [11] all suffer from these limitations despite the prevalence of multi-request applications.", "The limitations are generic and intrinsic to all web vulnerability scanners.", "Hence, most attack vectors in their fuzzing attempts are incapable of reaching the vulnerable code.", "These findings are confirmed by running the publicly available scanners [1], [20], [25], [2], [31], and analyzing their fuzzing patterns against a banking transfer website mimicked by us.", "We also carefully read through the descriptions of those tools that are unavailable to us [11], [7].", "We attribute the main reason for such limitations to their fundamental scanner design.", "All blackbox scanners are common in crawling a website only once, and will rely on this static set of captured requests for mutation and fuzzing.", "Therefore, it is unsurprising that they do not even work well with applications that have applied token-based CSRF defenses.", "However, it is non-trivial as to how such a scanner design can be patched to identify, renew, and relate a token from a former response to a corresponding fuzzing request as well as handling the synchronization issues.", "We aim at addressing not only the problems concerned but also other dynamic characteristics that have made scanning of real-world web applications difficult.", "It is hard to resolve all these issues in the existing scanner design largely owing to their crawl-once-and-replay-many and stateless fuzzing approaches.", "Hence, this paper proposes a parameter tampering scanner called Cross-Request Scanner (CRS), which structurally changes the traditional fuzzing approach.", "Table REF provides a quick comparison of various scanners, which shows their support to those website features that are essential for effective fuzzing.", "Here we outline our approach and its design considerations, which has enabled the discovery of some previously unknown vulnerabilities.", "Figure: The architecture of Cross-Request Scanner (CRS)" ], [ "Overview of CRS", "Figure REF depicts the high-level architecture of CRS, which is a blackbox parameter tampering scanner designed to find parameters that are client-side rejected yet server-side accepted [10].", "It comprises the capturing and fuzzing phases: Capturing Phase.", "While running CRS as a browser add-on, an analyst provides an initial valid set of submissions by interacting with a testing application as usual in the browser.", "CRS captures the user actions for later replays.", "Meanwhile, it tracks the parameter dependency across the submission requests, and thus a parameter with its value being the same as in the last request is marked dependent.", "From the server responses, it identifies some candidates of key features such as locations of submit buttons and reflected parameters, that can represent an acceptance by the server.", "It then replays the original user actions to get a second valid set of submissions, and classifies those parameter values that differ from the initial submissions as token candidates.", "In these two valid submissions, those key feature candidates that cannot be reproduced will be dropped.", "Finally, it confirms that each token candidate is of one-time use if a rejection is resulted upon repeating the same request.", "Fuzzing Phase.", "To observe the intended workflow and one-time use tokens, CRS sequentially replays the user actions and keeps the tokens intact during fuzzing.", "Each parameter is mutated and placed back to the application to undergo its client-side validations.", "The dependent parameters are exempted from fuzzing after the candidates are mutated, submitted, and actually confirmed to result in server rejections.", "Next, those parameters that are rejected by the client-side are forcefully submitted with the validations bypassed.", "With such a rejected parameter, CRS reports a vulnerability if the server reacts in such a way that the expected workflow and key features can be reproduced as in the valid submissions." ], [ "Design Considerations", "Here we explain the underlying considerations of the approach.", "Preserving Intended Workflow.", "Sequentially fuzzing a whole set of requests, one after another, is a necessity to preserve the intended workflow.", "CRS achieves this by replaying user actions.", "Respecting One-time Use Tokens.", "One-time use tokens are by definition those server-generated parameters that differ across two requests even though an identical set of user actions are provided.", "Before any user inputs are entered, these tokens are readily present either in hidden fields or the query string of submission URLs.", "This important clue can help eliminate many false classifications.", "However, proxy-based scanners [1], [25], [20] which capture a request at proxy level (i.e., outside the browser context) obviously possess no such knowledge.", "To extract these tokens, complex state maintenance and even synchronization issues are anticipated if we are to modify existing scanners to fetch a previous response for token extraction before making every fuzzing request.", "In contrast, CRS proposes to identify and fuzz directly in the browser so that these tokens are inherently recognizable and well-preserved.", "After confirming that they are of one-time uses, CRS excludes them from being mutated during fuzzing.", "Respecting Parameter Dependency.", "Given the two valid sets of requests, CRS marks a parameter as dependent candidate if its value is equal to any parameters in the last request.", "This simple classification might be an overkill, so CRS still fuzzes each dependent candidate once for confirmation.", "Hence, only those that result in server rejections will be confirmed as dependent.", "To preserve the confirmed dependency, the fuzzer mutates such a parameter only in the first request it appeared but not in any successive ones.", "For instance, if a parameter value in Req$_B$ is confirmed to be dependent to one in Req$_A$ , then it is mutated only at Req$_A$ but kept intact at Req$_B$ .", "In fact, this algorithm is also good for locating session-based tokens of which the values are present in multiple requests and will expire only after the session is terminated.", "Preserving Client-side Preprocessing.", "Web applications implement their client-side logic using JavaScript.", "Without any page loading, modern applications may dynamically add or drop input fields in response to a user action.", "In the banking examples studied, HSBC, BOC, HSB and BEA even manipulate some hidden fields before performing form validations and submissions.", "AJAX-driven websites such as Citibank use the asynchronous XMLHttpRequest API.", "Existing scanners are however ignorant to all of them.", "It may look intuitive to preserve these features simply by running the applications in a full-blown browser during testing.", "But this actually poses a challenge in preserving these client-side executions while mutating parameters, to be detailed in Section REF .", "Detecting Client-side Rejection.", "Some existing approaches infer the validation rules by coding analysis[7], [8].", "Given a blackbox environment, it is however difficult to preserve complex and dynamic validations that may even depend on AJAX feedbacks.", "Instead, to be generic, CRS places a mutated parameter back into the application itself to determine whether a submission is forbidden, and if so, the parameter must be rejected by client-side validations and will be used for fuzzing.", "Concerning the mutation algorithm, it heuristically derives new values (e.g., $+1$ , $\\times -1$ ) from either a user-given or default value besides adopting some static values from [20].", "Although it may not be as exhaustive as what can be generated through code analysis, it suffices in the current study.", "Detecting Server-side Acceptance.", "Most existing blackbox scanners use edit distances for response classifications [7], [25], [1].", "However, this approach often fails in classifying JSON-formatted [19] feedbacks that are commonly used in AJAX applications.", "For responses such as {success:1} and {success:0}, the zero edit distance between them makes server acceptances hardly discernible from rejections.", "In particular, NoTamper [7] also requires an analyst to provide a baseline pair of valid and invalid submissions, such that a fuzzed response can be clustered to the one with closer edit distance.", "In contrast, CRS requires only one valid set of submissions, and is able to figure out the invalid responses which must fail to reproduce the expected submit buttons, workflow, and reflected parameters.", "The concept of button detection is similar to a proposal that clusters server responses into different states according to their outgoing hyperlinks [11].", "But here, CRS can support those responses that may contain no hyperlinks and those with indistinguishable edit distances.", "It is because the classification is applied to those DOM elements that receive visual changes [23] upon AJAX feedbacks, rather than tapping directly to the response data." ], [ "Implementation", "To realize our approach as a usable tool, this section first presents a background study on how parameters are gathered, validated, and submitted in web applications in general.", "Next, we detail the implementation of the core components, including the Capturer and Fuzzer, of the scanner." ], [ "Sources of Parameters and Validations", "An HTML form typically encloses a variety of input controls to take users' input, and each of which is associated with a name and optionally a default value (e.g., with the value attribute).", "Such a name/value pair then contributes a request parameter during form submission.", "On the other hand, server-generated values are typically embedded in hidden fields, or hardcoded at the query string of submission URLs.", "The choice of an input control can obviously impose a restriction on the input format that a user is supposed to follow.", "For instances, two radio boxes denoting genders with values defaulted to M and F can prevent users from entering unexpected values, whereas the use of a hidden field is a means of validation that it must be equal to the given (or default) value.", "In addition, applications can be programmed to apply explicit client-side validations with JavaScript and HTML 5 API [18].", "For instances, application typically enforces string-valued input to be composed of only alphanumeric characters, and that numeric input to be within a proper range (e.g., $age >= 18$ ).", "If a form fails the validations, the submission must be forbidden.", "These explicit validations and those imposed by input controls are all referred to as client-side validations.", "Notice that they are implemented only for improved usability.", "Input validations must also be implemented at the server for security purposes." ], [ "Form Submissions", "Client-side validations must be applied before an actual submission by registering an event handler to either the submit event of the form or the click event of a clickable element.", "In this paper, the former approach is referred to as the event-based form submission, and the latter one as programmatic form submission, whereas those submitted using the XmlHttpRequest API are referred to as the AJAX form submission.", "Their behaviors, as detailed below, essentially govern how they can be intercepted for fuzzing: Event-based Form Submission.", "When a user clicks a submit button (e.g., <input type=image|submit>), or hits “Enter” in a text field, the submit handler of the form is invoked.", "If its return value is set to true, the form proceeds to submit and will trigger a page load; otherwise, the submission can be forbidden.", "JavaScript validations are applied inside this handler, and thus the form is submitted only if it is properly validated (i.e., return validated(form)).", "Programmatic Form Submission.", "When employing an (click) event that is non-specific to form submission, the event handler must explicitly invoke the submit() method for a form submission.", "Clearly, this approach will trigger a page load but not the submit event.", "The invocation is typically restricted by the form validity (i.e., validated(form) && form.submit()).", "AJAX Form Submission.", "Modern applications submit a form at the background without reloading the page.", "To achieve this with the submit event handler, false is always returned to cancel the default submission and thus the page load.", "There is no page load problem when using other event handlers such as onclick.", "When the form passes the validations, the event handler will invoke a custom function (e.g., sendAJAX()), that is made to encode the name/value input pairs as request parameters, and finally submit a request using the well-known XMLHttpRequest API.", "When the add-on is being launched alongside the browser, the Capturer actually begins by recognizing the form submission approach that is being used.", "Even before any user actions, it has already interposed on the submit() method of every form, and registered an handler to the beforeunload event.", "If the submit() method is called, the form submission approach is a programmatic one.", "If the beforeunload handler is first fired (i.e., not initiated by the submit() method), then it is clearly an event-based form submission.", "Otherwise, the form submission is using AJAX.", "Here we present the implementation of the smaller modules including the Action Recorder, Action Player, Parameter Dependency Detector, One-time Token Detector, and Valid Features Detector." ], [ "Action Recorder", "The Action Recorder is responsible for capturing the URLs, HTTP requests and responses of an initial valid set of submissions.", "The request URL and parameters are obtained from the attributes and input elements of the corresponding form, while the responses are also readily accessible by similar DOM API calls.", "On the other hand, it listens to all keystrokes and clicking events, and takes the unique id attributes or otherwise XPath positions [22] as references to those target elements that receive the events.", "This is implemented by registering event handlers at the document level, so that they are recorded before propagating down to the target elements or possibly canceled by the intermediate ones." ], [ "Action Player", "With the captured user actions, the Action Player of the Capturer can replay them to obtain the second set of valid submissions.", "This is achieved by synthesizing and replaying the events with the initEvent() and initMouseEvent() APIs.", "In general, the implementations of the event recorder and player is similar to Selenium, which is an open-source recording and replaying tool [30]." ], [ "One-time Tokens Detector", "This detector concerns only server-generated parameters that are originated from hidden fields and query string of submission URLs.", "Working as a browser add-on, CRS can easily sort out these parameters by DOM inspection.", "The name/value pairs in the query string are then properly decoded using the decodeURIComponent() API.", "Among these server-generated parameters, it identifies those parameters that have the same name yet distinct values across each corresponding pair of requests between the initial and second valid sets of submissions.", "Each of these identified candidates is confirmed of one-time use if making an identical request will indeed lead to a server rejection.", "Given a non-AJAX form, it is relatively easy to tamper the corresponding hidden field or submission URL and reuse the token in order to spoof an identical request.", "A server rejection, which is literally a lack of any of the key features, can then be asserted.", "However, for forms submitted over AJAX, more complex techniques to be detailed in Section REF and REF are similarly used here to spoof an identical request as well as to find if any key features are missing.", "The names of these confirmed parameters are recorded, and they will be preserved when fuzzing other parameters." ], [ "Parameter Dependency Detector", "This detector is actually responsible only for identifying candidates of dependent parameters, which are subject to further confirmation by the Fuzzer.", "First, CRS applies regular expressions to determine the value types.", "For instance, /^[\\d,]+(?", ":\\.\\d+)?$/ is used to test for numbers.", "With any commas stripped, the numeric values are casted into float using the parseFloat() API, while all other values are trimmed.", "When comparing the parameters across the requests of the initial valid set of submissions, it neglects the names but considers only those transformed values.", "A candidate parameter is finally considered dependent to another parameter of the last request only if two values are identical.", "The references (i.e., id attribute or XPath location) to these candidates are recorded." ], [ "Valid Features Detector", "This module is to determine the key features that indicate reproducible acceptances by the server.", "The references to the candidate elements of key features are recorded throughout the initial valid set of submissions.", "In the second valid set of submissions, the extracted features are relocated in the rendered server responses.", "Hence, the final key features to be used are all verified to be reproducible in both valid set of submissions.", "First of all, the buttons that are clicked and captured in the Action Recorder automatically qualify as the key features since they are always needed to follow the intended workflow.", "Next, CRS extracts a number of other key features by locating user-supplied parameters that are visually reflected in the rendered responses.", "This is achieved by searching for the presence of a parameter value in the textContent and value attributes of every leaf node through DOM tree traversal.", "This technique is commonly used in search engine keyword highlighting [36].", "Nevertheless, CRS handles number matching differently in order to tolerate formatting issues that are introduced by the server.", "For instance, 12345 is first converted to a regular expression object, i.e., \\/1[^\\d]?2[^\\d]?3[^\\d]?4[^\\d]?5\\/, and can thus find its presence in a node content such as $12,345.00.", "If a value has resulted in multiple occurrences that exceed a configurable limitFor instance, a single-character value such as 1 or 0 will likely occur everywhere.", "While a maximum of 3 times is generally reasonable for applications that reflect parameters, here we do not quest for a magic number but leave it configurable., the candidate parameter is disqualified.", "Obviously, each parameter is mutated based on its specific features.", "In case no parameter reflections can be identified, CRS will heuristically search for some keywords that indicate sever acceptance with the same DOM traversal approach.", "The case insensitive keyword choices include “success, done, complete(d), execute(d), ok(ay), or update(d)” but must also exclude “not, n't, fail(ed), err(or) or sorry”.", "Since the extension to multi-lingual support is trivial, so a string is now tested only as follows: 1 /\\b(?:success(?:ful)?|done|completed?|executed?|ok(?:ay)?|updated?", ")\\b/i.test(string) 2 && !/\\b(?:not|sorry|fail(?:ed)?|err(?:or)?)\\b|(?", ":[a-z]n\\'t\\b)/i.test(string) The approximate string matching algorithms [24] may better perform in some applications, but they are not used here for less complexity.", "In case of no identifiable features, the analyst will be prompted to provide a regular expression statement for asserting the server acceptance.", "We so far did not encounter such a manual need in our experiments.", "The CRS approach focuses on the rendered responses to locate key features even for AJAX applications.", "It generally ignores the raw response data as replied from the server, but simply allows the testing application itself to react accordingly and make DOM changes.", "In this regard, CRS is so designed to leverage the application itself to generate the required XMLHttpRequests.", "So, the relevant onstatechange handlers that respond to the AJAX feedbacks can be triggered.", "On the other hand, CRS limits the scope of feature searching to accurately the part that changes after the AJAX form submission.", "To implement this, CRS registers a callback function to the MutationObserver API [23] right before the submit button is clicked.", "After the application has made some DOM changes, the callback function will be invoked with those DOM objects that receive the changes, where the feature extraction algorithm will be applied to those visible objects.", "Only if no visual changes are ever made, CRS will use the HTTP status code and response data to determine server acceptances.", "However, we did not encounter such a case in the websites we tested." ], [ "The Fuzzer of CRS", "The Fuzzer begins by confirming the list of dependent parameters.", "It replays the captured user actions up to the page where a candidate dependent parameter is found.", "The value of which is then mutated to ensure that it differs from that of the last request to make a dependency breaking attempt.", "Finally, the dependency is confirmed if a server rejection is encountered.", "To assert this, a fuzzing technique that is also applied to other non-candidate parameters is used and described as follows.", "while (action = userActionList.next()) { \tif (action.isSubmissionClick() \t\t\t&& param = paramList.nextCandidate() \t\t\t&& !param.isOneTimeToken() \t\t\t&& !param.isDependencyDetected()) { \t\tgetClientRejected(param) \t\tform.forceSubmission() \t\tdetectServerAcceptance() \t} else { \t\taction.replay() \t\tparamList.addNewlyAppearedFields() \t} } function getClientRejected(param) { \tif (mutated = param.nextMutated()) { \t\tform.interceptSubmission(function() { \t\t\tform.cancelSubmission() \t\t\tgetClientRejected(param) \t\t}) \t\tform.fillIn(mutated) \t\taction.replay() \t\tparamList.addNewlyAppearedFields() \t} } Listing  outlines the core algorithm of the Fuzzer.", "It sequentially replays the captured user actions (Line 1 & 10) and stops before a submission click (Line 2).", "Parameters that are confirmed of one-time use (Line 4) or dependent to a previous request (Line 5) are exempted from fuzzing.", "Other parameters are mutated one by one with respect to its default or user-given value (Line 15).", "The mutated parameter is then placed back to the form, and the corresponding events (e.g., onclick, onchange) are also triggered (Line 20).", "Any newly appeared fields will also be captured here for later mutation (Line 11 & 22).", "Upon replaying the submission click event (Line 21), the web application itself validates the form concerned.", "If a submission is attempted, that means the mutated parameter is accepted by the client-side validations (Line 16).", "The Fuzzer will however cancel the submission (Line 17), and repeatedly mutate the parameter until it is rejected by the client-side (Line 18).", "With that client-side rejected parameter, the Fuzzer bypasses the validations to force a form submission (Line 7).", "Finally, it asserts a server acceptance by locating all the valid key features (Line 8), and is able to report those parameters that are rejected by the client-side validation but accepted by the server as vulnerable to parameter tampering attacks.", "Those smaller modules including the Parameter Mutator and Vulnerability Detector are detailed as follows.", "The discussion on Action Player is omitted as it is the same as that in the Capturer." ], [ "Parameter Mutator", "This module is responsible for generating mutated values.", "The default values, or otherwise analyst-provided ones form the bases for mutations.", "First, CRS mutates the numeric portion of this existing value in order to derive some integral, decimal, negated, incremented, decremented, and multiplied versions (e.g., parseInt(), $+n/1000$ , $-n$ , $+n$ , $\\times -1$ , $\\times n$ , where $n$ comes from some hardcoded and randomized integers).", "CRS then determines the input types by inferring from the whole value (e.g., /[\\d,\\.", "]+/ for numbers), type attributes (e.g., date from HTML 5), class attributes (e.g., alphanumeric, datepicker; as often employed by JavaScript validation libraries [28]), name attributes, and field labels (e.g., tel, date).", "For boolean values that are likely represented in 1, 0 ,Y, N, T, F, true, or false, new values are derived by negating the boolean.", "For certain input types such as date, time and percentage, CRS adds some relevant hardcoded values (e.g., 2013-9-22, 23:59, $111\\%$ ).", "Optionally, the Mutator is configurable to also output test cases for violating the input types and other field restrictions such as required and maxlength (e.g., concatenating an existing value with itself by multiple times).", "Here some of the hardcoding values are borrowed from a tool called TamperData [20].", "It is certain that this heuristic input generation module is imperfect, and can be further enhanced by performing extra coding analysis [7], [8], [3].", "However, the current setting is found sufficient in our experiments." ], [ "Vulnerability Detector", "This subsection mainly enumerates how to intercept and cancel submission attempts (client-side accepted parameters) as well as force a submission (client-side rejected parameters) for each form submission approach.", "Interestingly, this involves flipping the original decision on whether to proceed a submission.", "With a fuzzing attempt that is client-side rejected, CRS can then discern server acceptances by detecting those valid features found by the Capturer.", "Intervening in Event-based Form Submission.", "To intercept the submission, CRS interposes on the onsubmit handler of the form to wrap the original one.", "Our handler disables HTML 5 automatic validations by setting form.noValidate to false.", "It then discerns the parameter validity by examining the return values of the original onsubmit handler and form.checkValidity() method that respectively reflect the results of JavaScript and HTML 5 validations.", "If an accepted parameter is encountered (i.e., both returned true), CRS will return false in our onsubmit handler to cancel the submission.", "For rejected parameters, CRS forces the submission by returning true in our handler.", "Intervening in Programmatic Form Submission.", "To intercept the programmatic submission, CPS overrides the submit() method of the form.", "To discern parameter validity, we cannot rely on the return value which is always void.", "Instead, we know that our method will be called only with client-side accepted parameters.", "We will then simply ignore the submissions for these parameters.", "For other (i.e., rejected) parameters, CPS deliberately calls the original submit() method to force submissions.", "Intervening in AJAX Form Submission.", "Intervening in AJAX submissions is the most challenging implementation for the following reasons.", "First, we do not assume web applications to use any particular JavaScript libraries such as jQuery [28].", "Second, in contrast to the previous approaches of which the event and method are directly associated with a form object, XMLHttpRequest multiplexes all kinds of HTTP requests that are non-specific to form submissions (i.e., without a function like form.ajaxSubmit() that can be intercepted similarly).", "Third, owing to the asynchronous nature, multiple AJAX requests can simultaneously exist, such as refreshing a placeholder for advertisements and dynamic validations that involve server feedbacks (e.g.", "nickname uniqueness).", "Hence, it is non-trivial to distinguish which request is responsible for the form submission concerned.", "We solve this problem by first intercepting the methods open() and send() at the prototype level of XMLHttpRequest, which are invoked before making a request.", "It is necessary to intercept both open() and send() methods, through which input parameters are provided respectively for GET and POST requests.", "To distinguish the particular request that corresponds to the form submission, CPS appends a nonce to the value of an existing (hidden) input controlAlternatively, we can inject a hidden field with a nonce value, so that it will be submitted along with other field values., of which its validity is unaffected.", "Hence, the AJAX request that consists of this nonce can be isolated in our intercepted methods.", "Here, an invocation of our intercepted send() method implies that the parameter is client-side accepted.", "CPS discards the submission by ignoring it.", "For client-side rejected parameters, the application will not even instantiate the XMLHttpRequest.", "Given a simple application, it may be feasible to reproduce the instantiation by program slicing.", "But we instead develop a more succinct approach.", "CPS achieves a forced submission by reusing the set of valid parameters to trigger a submission request.", "But then the parameters are substituted with the client-side rejected ones at the intercepted methods.", "The nonce is also removed during so before sending the requests." ], [ "Evaluations", "This section focuses on evaluating the proposed scanning approach in terms of its practicality and effectiveness.", "Practicality considers how well CRS can automate and intercept real-world applications, while effectiveness measures the vulnerability scanning capabilities.", "All experiments are conducted with an entry-level PC using Firefox.", "The lightweight implementation actually incurs only negligible performance overhead, and that the scanning time is dominated by the server responsiveness and a configurable delay deliberately introduced to avoid “DoS-ing” the servers.", "We thus omit here a benchmark on performance overhead.", "If needed, the fuzzing time can be further reduced by using a headless browser [27].", "This section also briefly discusses some insights behind the severe vulnerabilities uncovered with the scanner, and shares how they are responsibly reported and subsequently resolved [34]." ], [ "Ethical Concerns", "For the practicality evaluation which generates no tampered requests, there are no ethical issues.", "For the effectiveness evaluation, we turned tampering on in real-world web applications.", "In the worst case scenarios, the tampered banking transfer requests could possibly result in money transferred either from our account to an unknown account or from an unknown account to ours.", "We thus tested only the interbank transfer applications which all require at least two days to take effect, and hence we have sufficient time to manually cancel (or make instant transfer before) any unwanted transactions.", "To further limit the risk, all requests are capped to use only a small amount of money during the automated fuzzing.", "To confirm a vulnerability to be a working exploit, we only manually made online transfers from and to accounts that are owned by us at different banks.", "All findings were promptly disclosed to the affected parties.", "We were grateful for being formally recognized by the government and banks' officials as “good citizens”." ], [ "Empirical Results", "We tested a total of seven applications, including five online banking transfer applications including HSBC, BEA, Citibank, BOC, and Hang Seng Bank (HSB), followed by two traveling websites including Jetstar airline and an online travel agent Webjet.", "These banks are chosen since their branches are in close geographical proximity to our institution, so that we can minimize the overhead of visiting them for opening bank accounts.", "The traveling websites are chosen because of a plan to visit Australia, where the headquarters of Jetstar and Webjet are located.", "Practicality Evaluation.", "We found that CRS can practically drive all the testing applications to run successfully.", "In this regard, CRS outperforms the existing scanners, which fail to support many critical features that are required for effective scanning, as tabulated in Table REF .", "Some scanners cannot even run the applications due to their lack of JavaScript and AJAX support.", "We are unaware of any unsupported form designs except those with non-standard input controls and those built with plug-ins such as Adobe Flash.", "This limitation is however non-specific to CRS but also other parameter tampering scanners [7], [8], [1], [20].", "Effectiveness Evaluation.", "We evaluated that CRS is effective in uncovering some previously unknown vulnerabilities even in banking websites.", "Table REF summarizes the confusion matrix of the vulnerability scanning results, which is further discussed below.", "True Positives.", "The vulnerabilities uncovered in two banking websites are separately detailed in Section REF .", "CRS discovered another vulnerability in an AJAX page of the official Jetstar website.", "There is a restriction on tracking on-time performance only from the last 10 days, but it is found bypass-able.", "We confirmed manually that it allowed us to check the performance of a randomly selected flight, which departed even three years ago.", "Given that the information was once (when within last 10 days) publicly accessible, we did not approach the organization for a patch.", "This finding serves to demonstrate the support of AJAX applications by CRS.", "True Negatives.", "In our experiments, CRS found no vulnerabilities in BOC, Citibank and HSB.", "We manually analyzed their client-side source code and classified them as true negatives largely because the servers have mapped each account number to a unique index (rather than using the actual account number).", "Thus, a tampered index could not go beyond a pre-defined mapping of authorized accounts that is maintained at the server.", "It is indeed a proper parameter tampering defense, as recommended by OWASP [26].", "False Positives.", "We are unaware of any false positives so far.", "False Negatives.", "Finding false negatives requires a set of known and unpatched vulnerabilities, which is intrinsically difficult in our blackbox evaluation.", "We managed to manually discover only one vulnerability in a travel booking website Webjet.", "It was missed because the lack of domain knowledge on airport code.", "More specifically, only some Australia airports such as HBA would cause the vulnerability.", "This limitation is non-specific to CRS but also any other scanners.", "We believe that this vulnerability is discoverable only by chance even with a human (when planned to visit Australia).", "We contacted Webjet, which appreciated our finding and fixed the vulnerability.", "Table: The confusion matrix of the CRS scanning resultsFigure: The online banking transfer application of HSBC responds with a different review page based on the choice of radio buttons" ], [ "Case Studies and Vulnerability Disclosure", "Here we discuss the parameter tampering vulnerabilities uncovered in HSBC and BEA by CRS, as well as our encounters of responsible vulnerability disclosures.", "We once expected banks to be well scrutinized by existing scanners and security audit.", "For instance, the Acunetix web vulnerability scanner being used by HSBC is also adopted by NASA and even the Pentagon [2].", "Being one of the largest multinational banks, HSBC also won the “Best Information Security Initiatives” award in the world's best consumer and corporate Internet banks [15].", "However, their existing mitigation approaches have apparently failed.", "Their application designs may provide some insights on why the vulnerabilities would evade the detection of their existing approaches." ], [ "Bypassed OTP Requirement in HSBC", "Figure REF shows two screenshots of the interbank transfer application of HSBC, which correspond to Step$_A$ and Step$_B$ in Figure REF .", "Legitimate Use Cases.", "When transitioning from Step$_A$ to Step$_B$ , the server first verifies the one-time token and cross-request parameter dependency.", "The logic flow then depends on a radio button that indicates whether the account TO is authorized (registered).", "If the upper radio button is chosen, the server will make use of the corresponding “Account Holder's Name and Account Number” and “Bank Code and Name” fields for further format verification.", "Therefore, the account number format should match with a regular expression /^[a-zA-Z]{1,20}~~\\d{1,12}$/.", "It is however unlikely that checksum verification can be applied to the TO account number since its definition is totally up to another bank.", "Given an authorized account, the user still has to click “confirm” in the review page.", "If the lower radio button is chosen, the server will take an authorized account from the corresponding “Bank Code and Name”, “Account No.”, and “Account Holder's Name” fields.", "The review page will then ask the user for an OTP generated from a dedicated hardware device.", "This second factor authentication is to assure that the transfer is authorized by the legitimate user.", "In either case, the bank sends out an SMS to the user.", "The TO account number is however partially masked, and that neither the holder name nor registered status are provided.", "Vulnerability Details.", "CRS is first provided with an initial valid set of user actions that go through the complete workflow to transfer $1 from our account to an authorized account.", "CRS then automates the rest of the discovery, and reported several parameter tampering possibilities for the TO parameter.", "Among the scanning results, the simplest possibility was to increment the numeric portion of the original value by one (i.e., FUND RECEIPIENT~~290 123456883).", "We manually confirmed this vulnerability by transferring money out from our HSBC account, and were able to receive it in an unauthorized account of our own in another bank.", "It is likely that the mutated (i.e., unauthorized) account number was handled by the logic branch responsible for authorized accounts.", "Hence, the most-trusted OTP was completely bypassed!", "It is reasonable that even a transaction-signing device [17] could not help in this regard.", "As a consequence, an attacker transferring his money to an “unauthorized” account can dispute the transaction on the basis that it does not correspond to a use of OTP.", "Alternatively, he can steal money from those victims whose sessions can be hijacked or whose credentials can be purchased from underground communities.", "Existing scanners miss this vulnerability due to the enforcements of one-time use tokens, cross-request parameter dependency, and client-side pre-processing.", "Disclosure.", "It was threatening that two out of four banks (by the time we ran the first set of experiments) were confirmed as being vulnerable, in spite of the limited applications we scanned.", "This made us believe that the prevalence of parameter tampering vulnerabilities were heavily under-estimated in online banks.", "Having failed to contact the right representative of HSBC, we anonymized its name and reported our findings to the government authority responsible for maintaining monetary and banking stability.", "The authority took our findings seriously and met with us in person to avoid any discloses through a wire.", "Promptly after the meeting, the authority alerted all licensed banks to request them for a check and possible fix, which may explain that no further vulnerabilities were later found in other banks.", "It later forwarded to us a technical clarification request from HSBC, which rolled out a patch in two days.", "The authority and HSBC expressed their gratitude in writing." ], [ "Unauthorized Transfer in Bank of East Asia", "Here we outline only the findings that are specific to BEA.", "Legitimate Use Case.", "The interbank transfer application of BEA allows transfers only to registered accounts, which can be authorized only by visiting a physical branch.", "In other words, one can never make an online transfer to any unregistered accounts as a protection.", "No SMS and email notifications were implemented.", "Vulnerability Details.", "Similar to HSBC, CRS was able to report a vulnerability that result in unauthorized transfers owing to the lack of proper server-side validations.", "We manually confirmed the vulnerability by transferring $10,000 from our account to an unregistered account of our own.", "We noticed an interesting implementation that a hidden field called MACcode is set at the client-side before any form validations and submission.", "Its value is prepared by concatenating the transaction parameters (i.e., FROM, TO, AMT) and encoding the result by the Base64 scheme.", "The server was found rejecting those requests if the MACcode received does not align with the transaction parameters.", "Since CRS places a mutated parameter back to the application, the parameter can undergo the same MACcode algorithm at the client-side.", "So, it is clear that such an algorithm cannot protect message authenticity, as opposed to its confusing name.", "Existing scanners miss this vulnerability owing to the enforcements of client-side pre-processing and intended workflow.", "Disclosure.", "More than a week after our experiments, we disclosed the vulnerability directly to BEA.", "Apparently, no monitoring alarms were triggered regardless of the large transaction amount.", "They acknowledged our finding, and promptly fixed it within a day.", "Their system was later enhanced by dispatching email notifications." ], [ "Parameter Tampering Scanners", "Blackbox.", "Blackbox scanners are not tied to any specific frameworks and languages used on the server-side.", "This makes them often more generic than whitebox approaches.", "In general, they all work by first crawling an application to capture any requests encountered, then fuzzing it by replaying the requests with some mutated parameters.", "Proxy-based parameter tampering scanners include the HTTP fuzzer in Acunetix [1], WebScarab [25], and TamperData [20].", "They accept per-request fuzzing, and thus cannot automatically observe the intended workflow.", "They also require considerable amount of manual configurations such as specifying (a) a self-signed SSL certificate for interceptions of encrypted traffic [14], and; (b) even explicit value range of each parameter.", "NoTamper [7] outperforms proxy-based parameter tampering by reducing the manual efforts needed.", "Similar to CRS, it automates parameter generation based on the client-side validations.", "With static symbolic execution and constraint solving (that currently support the event-based form submissions only), it is able to generate an exhaustive set of parameters that are rejected by the client-side.", "This approach can complement the Parameter Mutator of CRS.", "It however requires an analyst to provide two (i.e., valid and invalid) initial sets of requests though, so as to discern whether the server accepts the mutated requests.", "To preserve the intended workflow during fuzzing, research work is found only for scanning other web vulnerabilities such as XSS.", "A state-aware fuzzer [11] is able to detect whether a fuzzing attempt has triggered a state change.", "If changed, it backtracks to the previous state by page traversal.", "It thus allows every state to be completely fuzzed by all candidate parameters before moving on to the next state.", "However, this stateful scanner can handle static web pages only, i.e., AJAX applications are not supported.", "Nonetheless, all these scanners are limited owing to their fundamental scanner design.", "It is very hard to enhance them in handling the one-time use tokens, cross-request parameter dependencies, and client-side pre-processing.", "Their fuzzing requests are thus likely barred from reaching vulnerable code.", "On the other hand, there are studies that evaluate the effectiveness of existing web vulnerability scanners.", "According to [6], the commercial scanners being evaluated are found not to be as comprehensive as they are claimed to be.", "This evaluation however covers only vulnerabilities like XSS, and it does not cover parameter tampering.", "According to [12], the study found that none of its evaluated scanners could find a parameter tampering vulnerability.", "It also suggests that high-level domain knowledge on the application is needed, but can hardly be known without a human.", "PAPAS [5] focuses only on a specific variant of parameter tampering, namely the parameter pollution vulnerabilities.", "Its success relies on the use of a server-side parameter precedence algorithm that allows malicious parameter to take over the precedence of an existing parameter.", "Its scanning approach is thus very similar to finding reflected vectors such as XSS, and that is different from finding more general parameter tampering vulnerabilities.", "Whitebox.", "When server-side source code is available, whitebox scanners can be used to compare the inconsistencies between client-side and server-side validations.", "WAPTEC [8] extends the NoTamper [7] approach by applying similar coding analysis to the server-side validations.", "The additional knowledge eliminates the manual effort required, and improves the selection of candidate parameter.", "But it still suffers from the limitations of NoTamper such as the negligence of the cross-request parameter dependency.", "ViewPoints [3] discovers validation inconsistencies between the client-side and server-side by differential string analysis.", "It supports only a specific framework of the Java web applications.", "Besides reporting those parameters that are client-rejected yet server-accepted, its goal is to also report parameters that are client-accepted yet server-rejected.", "Eliminating this latter class of inconsistencies from the application can improve in usability but not its security.", "In general, what can be validated from the server and client sides is intrinsically inconsistent since the server is always more knowledgeable (e.g., access to DB) than the client, and thus these approaches must report many false positives.", "It is also difficult to first annotate all the expected inconsistencies particularly in real-world complex applications before running these tools." ], [ "Parameter Tampering Mitigations", "InteGuard [35] is a whitebox protection approach that operates a server-side proxy to verify the parameters exchanged between the server and a client as well as those further exchanged with a third-party server.", "It relies on a human to provide some valid transactions for analyzing the parameter dependencies in its learning phase.", "Then its online defense is to drop subsequent requests that violate the expected dependencies.", "Its mitigation is limited only to those parameters triggered by human.", "CRS can automatically explore the client-side events to uncover new parameters, which can be hardly exhausted by a human in the training phase.", "Some development frameworks such as .NET [4] and Java [16] can replicate the server-side validations to become client-side code.", "However, they are often limited to simple validations that are purposely programmed for replications.", "Swift [9] proposes that developers should annotate those parameters and codes that are sensitive to run only on server-side or safe for both sides to facilitate replication.", "But again, it is demanding to annotate even a simple program.", "Server-side validations and sanitizations are still fundamental defenses against parameter tampering.", "For client-side validations aiming at delivering better user experience, HTML 5 Validations [18] and JavaScript libraries such as jQuery [28] can be deployed." ], [ "Others", "Compared to [33], [34] which also explored parameter relationships, CRS systematically analyzes all the parameter dependencies between different tuples of requests and responses, as shown in Figure REF .", "As a result, CRS can automatically detect server acceptances by analyzing Req vs Resp and Resp vs Resp', which makes a significant improvement over the manual and protocol-specific approaches as with [33], [34].", "The proposed model checking also requires human effort in accurately transforming a subset of logic replica from the original application [34].", "Some existing work also involve in replaying user actions, but none of which have identified the advantage of fuzzing inside the application context, that is, by intercepting the submission requests for fuzzing.", "Mugshot [21] deterministically captures and replays user actions and other JavaScript functions for failure and usability analysis.", "Kudzu [29] enhances its code coverage by exploring client-side events for the discovery of more client-side injection vulnerabilities.", "Ripley [32] maintains a server-side replica of the client-side logic in a shadow browser hosted in a trusted proxy, and attempts to automatically replicate and replay the client-side events in the replica for integrity check.", "Although it ensures the computation integrity of the client-side code, it does not eliminate the need for server-side validations.", "Acunetix Web Vulnerability Scanner [2] is able to generate some user action events while scanning for XSS vulnerabilities." ], [ "Conclusion", "We studied a number of real-world web applications to understand their implementations, and how they can be intercepted for vulnerability scanning.", "Existing parameter tampering scanners do not consider the enforcements of, among others, the intended workflow, one-time use tokens, and parameter dependency across requests, which are all common in multi-request applications.", "Their effectiveness of vulnerability scanning is thus severely limited.", "We proposed the novel CRS approach to respect all these enforcements.", "We realized the approach by building a generic blackbox parameter tampering scanner.", "In our evaluation, it is practical to drive real applications to run for fuzzing.", "The importance of this work is demonstrated by the vulnerabilities uncovered in real-world applications including banks, which existing scanners miss.", "The scanning approach can also be extended to the discovery of other web application vulnerabilities including XSS and CSRF." ], [ "Acknowledgements", "This work is partially supported by the Sir Edward Youde Memorial Fellowship.", "We appreciate and thank HSBC and Hong Kong Monetary Authority for their prompt follow-up.", "Special thanks are due to Chris Smith, Clara Ho, Thomas Yung, Esmond Lee, Shu-pui Li, and James Tam for their support.", "We thank for the valuable comments from Shuo Chen, Kehuan Zhang and anonymous reviewers." ] ]

[ [ "Submillimeter Polarization of Galactic Clouds: A Comparison of 350\n micron and 850 micron Data" ], [ "Abstract The Hertz and SCUBA polarimeters, working at 350 micron and 850 micron respectively, have measured the polarized emission in scores of Galactic clouds.", "Of the clouds in each dataset, 17 were mapped by both instruments with good polarization signal-to-noise ratios.", "We present maps of each of these 17 clouds comparing the dual-wavelength polarization amplitudes and position angles at the same spatial locations.", "In total number of clouds compared, this is a four-fold increase over previous work.", "Across the entire data-set real position angle differences are seen between wavelengths.", "While the distribution of \\phi(850)-\\phi(350) is centered near zero (near-equal angles), 64% of data points with high polarization signal-to-noise (P >= 3\\sigma_p) have |\\phi(850)-\\phi(350)| > 10 degrees.", "Of those data with small changes in position angle (<= 10 degrees) the median ratio of the polarization amplitudes is P(850)/P(350) = 1.7 +/- 0.6.", "This value is consistent with previous work performed on smaller samples and models which require mixtures of different grain properties and polarization efficiencies.", "Along with the polarization data we have also compiled the intensity data at both wavelengths; we find a trend of decreasing polarization with increasing 850-to-350 micron intensity ratio.", "All the polarization and intensity data presented here (1699 points in total) are available in electronic format." ], [ "lccccccc 0pt Polarization Ratio Distributions 3cdata satisfying $P \\ge 3\\sigma _p$ 3c...also satisfying $\\vert \\Delta \\phi \\vert < 10$ Source Peak MAD $\\chi ^2_r$ Peak MAD $\\chi ^2_r$ OMC-1 1.3 0.5 13.4 1.4 0.5 20.8 OMC-3 1.4 0.5 2.9 1.2 0.3 3.7 DR 21 1.4 0.4 2.5 1.8 0.3 5.2 DR 21(Main) 1.3 0.3 2.3 1.8 0.3 6.6 All 1.5 0.6 6.0 1.4 0.5 10.9 The peak value and the median absolute deviation (MAD) of the polarization ratio ($P[850]/P[350]$ ) distributions which minimize the MAD (eq. []).", "Also shown are $\\chi ^2_r$ values as calculated from equation (); see Section .", "The columns labeled “$P \\ge 3\\sigma _p$ ” and “also $\\vert \\Delta \\phi \\vert < 10$ ” are defined as in Table ." ] ]

[ [ "Efficient spin control in high-quality-factor planar micro-cavities" ], [ "Abstract A semiconductor microcavity embedding donor impurities and excited by a laser field is modelled.", "By including general decay and dephasing processes, and in particular cavity photon leakage, detailed simulations show that control over the spin dynamics is significally enhanced in high-quality-factor cavities, in which case picosecond laser pulses may produce spin-flip with high-fidelity final states." ], [ "Introduction", "Photons and excitons (X) can be made to strongly interact in high-quality cavities containing a semiconductor quantum well, leading to a repetitive coherent exchange of energy between the two particles.", "[1] When the energy exchange ocurrs faster than the decay time of the individual components, a combined state, the exciton-polariton, is said to have formed.", "Exciton-polaritons show a variety of features, that motivate studies in multiple directions.", "Current interest in exciton-polariton research in 2D micro-cavities mainly focus on its liquid state and non-equilibrium Bose-Einstein condensation (BEC) covering many aspects of the problem[2].", "The interaction of polariton fields with impurities or defects has been studied for the case of a polariton fluid scattered by centers acting on the photonic component of the field[3], a polariton gas scattered by spin-independent disorder potential acting on the exciton degree of freedom[4], and the scattering of polaritons from spinless impurities acting on the excitonic component of the field[5].", "However, to the best of our knowledge, there are no reported studies on the interaction of a polariton field with a single spin degree of freedom.", "Here we study the dynamics, including relaxation processes, of a diluted exciton/photon field interacting with a single impurity of spin $s=1/2$ .", "The system is depicted in Fig.", "REF .", "It consists of a 2D photon cavity embedding a quantum well (QW), which contains few donor impurities [6], [7].", "The whole system is assumed to be at low temperature and excited by a laser from outside.", "We show that the quantum control of a single spin is more efficient for high-quality-factor cavities.", "Thus, a spin-flip in a high-fidelity final state could be produced with a single laser pulse of a few picoseconds.", "Since typical decoherence times for impurity spins in semiconductors are in the $\\mu $ s time-scale, the system can act as a high speed quantum memory or qubit[9], [8], [10], [11].", "We believe that the present proposal and that for the implementation of two-qubit polariton-induced operations[12], [13] suggest that a complete quantum-computing scalable architecture based on a solid-state system is possible using polaritons in 2D-microcavities.", "Figure: Pictorial representation of the system.", "Two Distributed Bragg Mirror (DBR) structures, placed at the sides of a quantum well (QW), confine photons injected from outside by a laser.", "The photons produce QW excitons that interact with the impurity spin localized at position 𝐑\\mathbf {R}." ], [ "Coupling with the environment", "Real systems cannot be entirely isolated from their environment.", "This is specially true for solid-state systems where several particles co-exist.", "When control is in mind, undesired interactions make the evolution unpredictable with the possibility of partial or total failure of the control operation.", "In our case, excitons, photons, and spins suffer from the coupling with the environment.", "For highly pure samples, low temperatures and low exciton density, the relevant decoherence processes for excitons are those causing spin flip of electrons and holes, with conversion between bright and dark excitons [22], [18].", "In general hole spins loss coherence faster than electron spins; for instance, in CdTe the spin relaxation of electrons is 29ps[18], while that for holes is $<7$ ps[21].", "In addition, the annihilation of excitons must be also considered, with associated lifetime of hundred ps in GaInNAs/GaAs [19].", "While different processes, such as structural disorder[23] are responsible for the loss of photon population and coherence, the main process is photon leakage off the cavity due to its finite Q-factor, which leads to a lifetime of the order of $\\tau = 15$ ps [24].", "At extremely low concentration of impurities with densities $n_I\\simeq 10^{13}$ cm$^{-3}$ , electrons bound to different donors are well localized and do not interact among them [27].", "The interaction with the nuclei is dominant (due to the strong confinement of the localized state).", "At temperatures $T < 10 K$ the transverse relaxation time T2$^*$ is a few ns for the electron bound to a donor, and the spin relaxation time is of the order of $\\mu $ s for donors in GaAs [27]." ], [ "Hamiltonian", "In what follows, we work in the Heisenberg picture; thus, time-dependent operators shall be everywhere understood.", "The free Hamiltonian reads $H_0 = \\sum _{\\alpha \\, \\mathbf {k}}\\varepsilon _{\\mathbf {k}}\\, \\hat{b}_{\\alpha \\mathbf {k}}^\\dag \\hat{b}_{\\alpha \\mathbf {k}} +\\sum _{\\chi \\, \\mathbf {k}}\\hbar \\omega _{\\mathbf {k}}\\, \\hat{c}_{\\chi \\mathbf {k}}^\\dag \\hat{c}_{\\chi \\mathbf {k}}~,$ where the first and second terms correspond to excitons ($\\hat{b}_{\\alpha \\mathbf {k}}^\\dag /\\hat{b}_{\\alpha \\mathbf {k}}$ ) and cavity photons ($\\hat{c}_{\\chi \\mathbf {k}}^\\dag /\\hat{c}_{\\chi \\mathbf {k}}$ ).", "The QW quantum confinement splits the heavy- and light-hole electronic bands, forming excitons out of conduction-band electrons with total angular momentum $j_z=1/2$ and valence-band heavy holes with $j_z=3/2$ .", "Bright ($j_z=1$ ) and dark ($j_z=2$ ) excitons are included: $\\alpha =\\lbrace 1,2,3,4\\rbrace =\\lbrace \\uparrow \\Uparrow , \\downarrow \\Uparrow , \\uparrow \\Downarrow , \\downarrow \\Downarrow \\rbrace $ , where the single (double) arrow identifies an electron (hole) angular momentum.", "The respective dispersion relations for excitons and photons are $\\varepsilon _{\\mathbf {k}} = \\varepsilon _0 + (\\hbar \\mathbf {k})^2/2m^*$ and $ \\omega _{\\mathbf {k}} = c/n \\, (\\mathbf {k}^2+\\mathbf {k}_z^2)^{1/2}$ , where $ \\mathbf {k}$ is the in-plane momentum; the momentum $\\mathbf {k}_z$ in the growth direction is determined by parameters of the cavity, $n$ is the index of refraction, and $c$ the speed of light.", "The polarization of the photon is $\\chi =\\lbrace 1,2\\rbrace $ .", "The ground state energy of the donor is set to zero.", "The system is excited by a classical laser field producing photons that propagate inside the cavity.", "Using the quasi-mode approximation (useful for high Q-factor cavities)[20], the cavity-laser interaction reads $H_{LC}=\\hbar \\sqrt{A} \\sum _{\\chi \\, \\mathbf {k}} \\,{\\cal {V}}_{\\chi \\mathbf {k}} (t)\\, e^{i (\\Omega _{\\mathbf {k}} - \\bar{\\Omega }) t}\\, \\hat{c}_{\\chi \\mathbf {k}} + H.c.~,$ where ${\\cal {V}}_{\\chi \\mathbf {q}} (t)$ is the coupling constant, $A=L^2$ is the area of the system, $\\Omega _{\\mathbf {k}}$ is the laser frequency and $\\bar{\\Omega }$ a constant adequately chosen to ease the numerical solution, see below.", "Cavity photons interact with excitons according to $H_{L}&=& \\sum _{ \\chi \\, \\alpha \\, \\mathbf {k} }g_{\\alpha \\chi \\mathbf {k}}(\\omega _{\\mathbf {k}}) \\,\\hat{c}_{\\chi \\mathbf {k}} \\, \\hat{b}_{\\alpha \\mathbf {k}}^\\dag + H.c.\\,,$ where $g_{\\alpha \\chi \\mathbf {k}}(\\omega _{\\mathbf {k}}) = 0$ for $\\alpha = 1,4$ .", "The QW contains donor impuritites.", "The assumption is made that, at low temperature, each impurity has an electron bound to it that contributes a spin $s=1/2$ ; in addition, the concentration of donors is low enough to ensure that excitons/polaritons will interact only with one selected impurity, located at position $\\mathbf {R}$ , when the laser spot is small enough.", "[28] Via Coulomb exchange, the electrons belonging to the exciton and the donor impurity interact through $ H_{XS} = H_{XS}^{(+)} + H. c.$ $H_{XS}^{(+)} \\hspace{-2.84526pt}&=& \\hspace{-8.53581pt}\\sum _{\\scriptsize \\begin{array}{cc}\\mathbf {k} \\, \\mathbf {k}^{\\prime } \\\\\\end{array}}\\frac{ j_{ \\mathbf {k} \\mathbf {k}^{\\prime }}}{A} \\,e^{-i({\\bf {k - k^{\\prime }}})\\cdot \\mathbf {R}}\\nonumber \\\\&&\\times \\, \\hat{ \\mathbf {s}} \\cdot \\left[\\frac{\\hbar }{2} \\hspace{-5.69054pt}\\sum _{\\scriptsize \\begin{array}{cc}\\chi \\, \\chi ^{\\prime } \\, \\eta \\\\\\end{array}}\\hat{b}_{(\\chi \\eta ) \\mathbf {k}}^\\dag \\,\\sigma _{\\chi \\chi ^{\\prime }} \\,\\hat{b}_{(\\chi ^{\\prime } \\eta ) \\mathbf {k}^{\\prime }} \\right]$ where the vector spin operator $\\hat{\\mathbf {s}} = (\\hat{s}_x, \\hat{s}_y, \\hat{s}_z)$ , $j_{ \\mathbf {k} \\mathbf {k}^{\\prime }} = j_0 [1 + a_I^{*\\, 2} (\\mathbf {k}- \\mathbf {k}^{\\prime })^2]^{-1/2}$ , and $a_I$ is a measure of the impurity electron localization[29].", "Here we adopted a more detailed notation for the spin $\\alpha $ of the exciton: the electron (hole) spin has index $\\chi $ ($\\eta $ ), and $\\sigma $ is the vector of Pauli matrices.", "[9] X-X interaction is disregarded, because we will study the case of low exciton concentration, where $n_X a_B^{*\\,2}/A < 1$ , with $a_B^{*}$ the exciton Bohr radius." ], [ "Method", "Different theoretical tools are employed to solve problems in exciton-polariton research.", "Heisenberg equations of motion describe the dynamics of mean values of either exciton/photon operators or polariton operators [3], [30], [31].", "It is also common the use of the Gross-Pitaevskii equation[25], [26].", "Other methods have also been used, such as the Hartree-Fock-Popov[32].", "We make use of the Heisenberg equations of motion (HEM) $ \\hbar {d\\langle \\hat{{\\cal O}} \\rangle }/{dt} = i \\langle \\left[ H,\\hat{{\\cal O}} \\right]\\rangle $ for mean values ($ \\langle \\ldots \\rangle $ ) of operators describing separately excitons, photons, and the impurity spin.", "This allows us to treat the cases of weak —where no polaritons exist— and strong coupling, as well as to include easily spin-flip processes that cause a polariton to dissociate into a dark exciton and a photon.", "In general, the HEM comprise a set of infinitely coupled equations, that can be ordered in a heirarchy, much as the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy of classical statistical mechanics.", "One must then set equations for products or correlation of increasing number of operators.", "In order to close the system of equations, a truncation of the hierarchy is necessary.", "We use the truncation scheme $\\langle \\hat{{\\cal O}}_1 \\hat{{\\cal O}}_2 \\rangle = \\langle \\hat{{\\cal O}}_1 \\rangle \\langle \\hat{{\\cal O}}_2 \\rangle $ .", "It is important to note that the photon-exciton coupling is not affected by the truncation scheme; therefore, the formation of polaritons (strong-coupling regime) is accurately described.", "The system-bath coupling is properly introduced in the HEM by the formalism of Quantum Heisenberg-Langevin equations, which leads to additional terms in the equations: damping, Lamb shift, and stochastic force $\\cal F$ .", "A simpler way to deal with the environment is by introducing constants, taken from experiments or other theoretical works, directly in the HEM, taking into account the results of the detailed microscopic derivation.", "Here we follow the phenomenological procedure, by adding constants directly in the HEM [3], [33].", "Photons are coupled to the radiation field at zero temperature outside of the cavity, resulting in the addition of a term $- \\xi _q c_{\\chi \\mathbf {q}}$ ; the damping $\\xi _q$ becomes very large when $q$ is such that the normal component of the field exceeds the critical angle separating low and high DBR reflectivity.", "For excitons we introduced a term $ - \\beta _\\alpha b_{\\alpha \\mathbf {k}}$ , with a spin-dependent constant $\\beta _\\alpha $ , accounting for radiative recombination and spin flip (no scattering is considered).", "For the impurity spin, a general constant $\\gamma $ is used in all component, since no extenal magnetic field exists to distinguish among them.", "For long times, the spin relaxes, but does not vanish; thus, an equilibrium state is defined.", "We consider a circularly-polarized laser field that excites $\\hat{b}_{2 0}^\\dag $ , and an impurity located at $\\mathbf {R} = 0$ .", "To simplify the calculations, we eliminate fast oscillations by moving to a rotating reference frame, with frequency $\\bar{\\Omega }$ , setting $\\langle \\hat{{\\cal O}} \\rangle = e^{-i \\bar{\\Omega }t} \\, \\langle \\hat{{\\cal O}}^{\\prime } \\rangle $ , for ${\\cal O} = c, b$ .", "For the sake of simplicity, we hereafter denote the rotating frame version $ \\langle \\hat{{\\cal O}}^{\\prime } \\rangle $ , simply as ${\\cal O}$ .", "The equations of motion read $\\frac{ds_{z}}{dt}&=&\\hspace{-5.69054pt}-\\gamma \\bar{s}_{z}+\\frac{\\hbar }{A}\\sum _{\\scriptsize \\begin{array}{cc}\\mathbf {k} \\mathbf {k}^{\\prime } \\\\\\end{array}}j_{\\mathbf {k} \\mathbf {k}^{\\prime }}^{+}\\left(s_{y}{\\rho }_{1 \\mathbf {k}, 2 \\mathbf {k}^{\\prime }}^{x}+s_{x}{\\rho }_{1 \\mathbf {k}, 2 \\mathbf {k}^{\\prime }}^{y}\\right) \\\\\\frac{ds_{x}}{dt}&=&\\hspace{-5.69054pt}-\\gamma s_{x}-\\frac{\\hbar }{A}\\sum _{\\scriptsize \\begin{array}{cc}\\mathbf {k} \\mathbf {k}^{\\prime } \\\\\\end{array}}j_{\\mathbf {k} \\mathbf {k}^{\\prime }}^{+}\\left(s_{y}{\\rho }_{1 \\mathbf {k}, 1 \\mathbf {k}^{\\prime }}^{z}+s_{z}{\\rho }_{1 \\mathbf {k}, 2 \\mathbf {k}^{\\prime }}^{y}\\right)\\\\\\frac{ds_{y}}{dt}&=&\\hspace{-5.69054pt}-\\gamma s_{y}+\\frac{\\hbar }{A}\\sum _{\\scriptsize \\begin{array}{cc}\\mathbf {k} \\mathbf {k}^{\\prime } \\\\\\end{array}}j_{\\mathbf {k} \\mathbf {k}^{\\prime }}^{+}\\left(s_{x}{\\rho }_{1 \\mathbf {k}, 1 \\mathbf {k}^{\\prime }}^{z}-s_{z}{\\rho }_{1 \\mathbf {k}, 2 \\mathbf {k}^{\\prime }}^{x}\\right)$ where $ \\bar{s}_{z} = s_{z}-s_{z \\infty } $ , ${\\rho }_{n \\mathbf {k}, n \\mathbf {k}^{\\prime }}^{z} = ( {\\rho }_{n \\mathbf {k}, n \\mathbf {k}^{\\prime }} - {\\rho }_{n+1 \\mathbf {k}, n+1 \\mathbf {k}^{\\prime }})/2$ , ${\\rho }_{n \\mathbf {k}, m \\mathbf {k}^{\\prime }}^{x} = ({\\rho }_{n \\mathbf {k}, m \\mathbf {k}^{\\prime }} + {\\rho }_{m \\mathbf {k}, n \\mathbf {k}^{\\prime }})/2$ and ${\\rho }_{n \\mathbf {k}, m \\mathbf {k}^{\\prime }}^{y} = i ( {\\rho }_{n \\mathbf {k}, m \\mathbf {k}^{\\prime }} - {\\rho }_{m \\mathbf {k}, n \\mathbf {k}^{\\prime }})/2$ ,[34] with ${\\rho }_{n \\mathbf {k}, m \\mathbf {k}^{\\prime }} = b_{n \\mathbf {k}}^* b_{m \\mathbf {k}^{\\prime }}$ and $j_{\\mathbf {k} \\mathbf {k}^{\\prime }}^{+} = j_{\\mathbf {k} \\mathbf {k}^{\\prime }} + j_{\\mathbf {k}^{\\prime } \\mathbf {k}}$ .", "${d {b}_{1 \\mathbf {q}} \\over dt}&=&- \\left[\\beta _{1}+ i \\left( \\frac{\\varepsilon _{\\mathbf {q}}}{\\hbar } - \\bar{\\Omega }\\right)\\right] {b}_{1 \\mathbf {q}}+\\beta _{12} {b}_{2 \\mathbf {q}}\\\\&&\\hspace{-14.22636pt} - \\frac{i}{2 A}\\sum _{\\mathbf {k}}j_{\\mathbf {q} \\mathbf {k}}^{+}\\left(s_- \\, {b}_{2 \\mathbf {k}}+ s_z \\, {b}_{1 \\mathbf {k}}\\right)\\nonumber \\\\{d {b}_{2 \\mathbf {q}} \\over dt}&=&- \\left[\\beta _{2}+ i \\left( \\frac{\\varepsilon _{\\mathbf {q}}}{\\hbar } - \\bar{\\Omega }\\right)\\right] {b}_{2 \\mathbf {q}}+ \\beta _{12} {b}_{1 \\mathbf {q}}\\nonumber \\\\&&\\hspace{-19.91692pt} - \\frac{i}{\\hbar } \\sum _\\chi g_{2 \\chi \\mathbf {q}} \\,{c}_{\\chi \\mathbf {q}}- \\frac{i}{2 A} \\sum _{\\mathbf {k} }j_{\\mathbf {q} \\mathbf {k}}^{+}\\left(s_+ {b}_{1 \\mathbf {k}}- s_z {b}_{2 \\mathbf {k}}\\right) \\vspace{8.53581pt}$ where $s_{\\pm } = s_x \\pm i s_y$ .", "Similar equations hold between ${b}_{2 \\mathbf {q}} \\leftrightarrow {b}_{3 \\mathbf {q}}$ and ${b}_{1 \\mathbf {q}} \\leftrightarrow {b}_{4 \\mathbf {q}}$ .", "${d {c}_{\\chi \\mathbf {q}} \\over dt}&=&- \\left[ \\xi _{q} + i ( \\omega _{\\mathbf {q}} - \\bar{\\Omega })\\right] {c}_{\\chi \\mathbf {q}}- \\frac{i}{\\hbar } \\sum _{\\sigma } g_{\\sigma \\chi \\mathbf {q}}^* \\,{b}_{\\sigma \\mathbf {q}} \\nonumber \\\\&&\\hspace{-14.22636pt} - i \\sum _{\\sigma \\mathbf {k}} \\sqrt{A}{\\cal {V}}_{\\sigma \\mathbf {k}}^*(t) e^{- i (\\Omega _k - \\bar{\\Omega }) t }\\delta _{\\sigma \\chi } \\delta _{\\mathbf {k}\\,\\mathbf {q}}\\,.$" ], [ "Results", "Numerical solution of the HEM is obtained using a 4th-order Runge-Kutta method in a 2D grid of $N \\times N$ modes in momentum space.", "Basic units are $\\lbrace $ meV, ps, nm$\\rbrace $ , and we use data compatible with GaAs[35].", "The values of the different parameters are taken, in most cases, directly from experimental or theoretical work, only ${\\cal {V}}_{0}$ and $j_{0} $ are adjusted using our calculations.", "When $g_{\\alpha ^{\\prime } \\alpha \\mathbf {q}}=0$ , $b_{2 \\mathbf {q}}(0) \\ne 0$ and $s_z(0)=\\hbar $ , the system of equations becomes linear, and can be solved exactly.", "$j_0$ is then adjusted to yield a negative eigenvalue that matches the reported binding energy of excitons to donors (about 1meV).", "The value so obtained for $\\hbar ^2 j_0/A \\simeq 10^{-5}$ meV is in agreement with previous reports[8], [12].", "We fix the value of the coupling ${\\cal {V}}_{0}$ by demanding that the total exciton density $n_X = \\sum _{i \\mathbf {q}} b^\\dag _{i \\mathbf {q}}b_{i \\mathbf {q}}$ be low, i. e.: $r=n_X a_B^{*2}/A<1$ , so that the X-X interaction can be neglected.", "We studied the evolution of spin components, exciton and photon populations when the system, represented by $N = 50$ modes (larger $N$ s do not change the result significantly), is excited by a circularly polarized normal-incidence monochromatic laser-pulse ${\\cal V}_{\\sigma \\mathbf {k}} = {\\cal V}_0 \\exp \\lbrace -(t-t_p)^2/w^2\\rbrace $ .", "Cases with and without decoherence are considered.", "It is instructive to analyze first the (idealized) decoherence-free case —no plot presented.", "We find neither $b_{3\\mathbf {q}}^*b_{3\\mathbf {q}}$ nor $b_{4\\mathbf {q}}^*b_{4\\mathbf {q}}$ populations, while $s_z$ and $b_{1\\mathbf {q}}^*b_{1\\mathbf {q}}$ change little from their initial values.", "The small change in $s_z$ can be understood as follows: according to Eqs.", "(REF ) $ds_z/dt \\propto (\\hbar j_0/A) b_{1\\mathbf {q}}^* b_{2 \\mathbf {q}}$ and to Eqs.", "(REF ) $d{b_{1 \\mathbf {q}}}/dt \\propto (\\hbar j_0/A) b_{2 \\mathbf {q}}$ , that roughly yields $ds_z/dt \\propto (\\hbar j_0/A)^2 b_{2 \\mathbf {q}}^* b_{2\\mathbf {q} }$ .", "This, compared to $ds_x/dt \\propto (\\hbar j_0/A) b_{2 \\mathbf {q}}^* b_{2 \\mathbf {q}}$ , is a very small quantity given the choosen value of $\\hbar j_0/A \\simeq 10^{-5}$ ps$^{-1}$ .", "On the contrary, the spin projection in the $xy$ -plane can rotate several cycles depending on the temporal width and intensity of the pulse.", "As it is well known from quantum optics, once the laser is turned off, there is a remaining oscillating population of excitons and photons.", "For certain values of the pulse parameters, these populations are so small ($r \\rightarrow 0$ ) that cannot produce important changes in the spin.", "When decoherence is included, there is conversion to dark states $b_{4\\mathbf {q}}$ , due to hole spin-flip, and to a lesser extent due to electron spin-flip.", "Because of the long life-time of these dark states, the fraction $r$ remains finite (though very small compared to its peak value).", "Fig.", "REF presents the results for a simulation with parameters $\\lbrace \\Omega _ 0 =2270.$ ps$^{-1}, \\bar{\\Omega }=2301.2$ ps$^{-1}, \\varepsilon _0/\\hbar =2301.2$ ps$^{-1}, \\xi _0=6.6\\, 10^{-2}$ ps$^{-1}, \\beta _1=\\beta _4=0, \\beta _2=\\beta _3=10^{-2}$ ps$^{-1}, \\beta _{12}= \\beta _{34}= 3\\, 10^{-2}$ ps$^{-1}, \\beta _{13}= \\beta _{24}= 1$ ps$^{-1} \\rbrace $ .", "We find that if ${\\cal V}_0 < 32$ ps$^{-1}$ nm$^{-1}$ then $r<1$ and the neglect of the X-X interaction is justified.", "Under this condition, we see that a single inversion $s_x \\rightarrow -s_x$ can be realized in few picoseconds.", "Faster spin motion is observed when the laser intensity (and so the photon/exciton populations) increases.", "As predicted in the previous paragraph, during the whole evolution, the change in $s_z$ is very small, as seen in the lower panel of Fig.", "REF .", "In addition the population of dark excitons is also very small compared to that of bright excitons: with the definition $r_i = b_{i\\mathbf {q}}^*b_{i\\mathbf {q}} a_B^{*2}/A$ , we obtain at $t=10$ ps $\\lbrace r_1 \\simeq 0.3, r_2 \\simeq 1.5 \\times 10^{-6}\\rbrace $ and at $t=20$ ps $\\lbrace r_1 \\simeq 6 \\times 10^{-6}, r_2 \\simeq 10^{-6}\\rbrace $ .", "Figure: (color online) Evolution under the excitation by a laser pulse of width w≃4.5w \\simeq 4.5ps and 𝒱 0 =25{\\cal V}_0=25ps -1 ^{-1}nm -1 ^{-1}; the initial state is a spin having mean values {s x =3ℏ/4,s y =0,s z =-ℏ/4}\\lbrace s_x=\\sqrt{3}\\hbar /4, s_y=0,s_z=-\\hbar /4\\rbrace .Upper Panel: Spin components s x s_x (solid blue) and s y s_y (red).Lower Panel: Spin component s z s_z.Inset: fraction r=n X a B *2 /Ar=n_X a_B^{*2}/A." ], [ "Spin rotation for strong and weak coupling", "The addition of decoherence allows us to address the regimes of strong and weak coupling, and in particular to study the effect that cavity losses have in the spin control.", "Weak coupling is characterized by $|\\xi _0-\\beta _2|>2g/\\hbar $ (in our case $2g/\\hbar \\simeq 2.2$ ps$^{-1}$ ), and this regime can be simulated by increasing the photon losses of all modes (increasing $\\xi _0$ ), which amounts to considering different cavities with varying quality factor Q.", "Two notes of caution: First, we have treated the laser-photon coupling in the quasimode approximation, valid for high-quality-factor cavities.", "Therefore, we will refrain from studying cases with large values of $\\xi _0$ .", "Second, the laser-photon coupling ${\\cal V}_0$ is, in general, affected by changes in the photon losses $\\xi _0$ ; however, we can envisage situations where one can increase $\\xi _0$ without affecting ${\\cal V}_0$ , for example –but not exclusively– by reducing only the reflectivity of the left DBR in Fig.", "REF .", "Fig REF shows the effect that the increase in photon leakage, at fixed laser field intensity, has on the rotation of the impurity spin.", "For simplicity other sources of decoherence are disregarded.", "For all simulations, we used one set of laser parameters {$w\\simeq 4.5$ ps, ${\\cal V}_0=15$ ps$^{-1}$ nm$^{-1}$ } for a gaussian pulse which produces, without photon loss, a rotation from the initial state $s_x=\\sqrt{3}\\hbar /4$ to the final state $s_x=-\\sqrt{3}\\hbar /4$ at $t=15$ ps, i. e. a change in the angle $\\Delta \\theta = \\pi $ .", "Next we simulated situations of increasing $\\xi _0$ and plotted $\\Delta \\theta (\\xi _0)$ .", "In addition, we plotted the maximum photon population acchieved during the pulse.", "We observe that for $\\xi _0<2$ ps$^{-1}$ ($Q > 1500$ ) there is almost full rotation of $s_x$ , and that for lower quality-factor cavities (high $\\xi _0$ ) the spin changes little.", "The population of cavity photons and excitons follow this tendency.", "Figure: (color online) Degree of spin in-plane rotation Δθ\\Delta \\theta (dashed blue) as a function of the cavity photon loss, and maximum cavity-photon population (solid red).", "Inset: Zoom-in of rotation angle Δθ\\Delta \\theta for low ξ 0 \\xi _0.The effect of decoherence can be characterized with the fidelity $F$ .", "If the final state we wish to obtain is the pure spin state $-1/2\\,|{\\uparrow }\\rangle +\\sqrt{3}/2\\,|{\\downarrow }\\rangle $ , having mean values $\\lbrace s_x = -\\sqrt{3}\\hbar /4, s_y = 0, s_z = \\hbar /4\\rbrace $ , the formula for the fidelty reduces to $F=(-1/2\\hbar ) (\\sqrt{3}s_x-s_z-\\hbar )$ , see Jozsa[36].", "For the cases $\\xi _0=3.5$ ps$^{-1}$ and $\\xi _0=20.5$ ps$^{-1}$ the resulting fidelity is $F=0.9965$ and $F=0.607$ , respectively.", "We interpret the enhanced rotation in high-Q cavities in the following way.", "For high Q, as seen in Fig.", "REF , the photon density is larger, and a repetitive and longer interaction with excitons is possible.", "This leads concomitantly to the formation of polaritons, with the excitonic component causing impurity spin rotations.", "In contrast, for lower values of Q, photons tend to leave the cavity faster, and there is small conversion to excitons.", "As was mentioned before, it is perhaps easier to envisage a cavity, whose Q factor is lowered by degrading the left DBR in Fig.", "REF .", "Then, a naive picture tells us that the laser field produces photons inside the cavity at the same rate in the high and moderate Q cases.", "In the latter, photons are more prompt to leak out and produce less excitons.", "In addition, we can ask what the pulse width $w$ should be to ensure a full rotation of $s_x$ , for different values of cavity loss (see Fig.", "REF ).", "As expected from the previous analysis, we see that one requires longer pulses to produce the rotation, but in contrast to what happened before, the fidelity is almost unchanged.", "We attribute the behavior of $F$ to the fact that the only source of decoherence is photon loss in these simulations and that the final state is forced (by changing $w$ ) to be the closest possible to the ideal state.", "Figure: (color online) Pulse width ww (solid blue) required to produce full rotation of the s x s_x spin component, and corresponding fidelity FF (dashed red) as a function of cavity loss ξ 0 \\xi _0.Finally, it is worth mentioning that we have used a conservative value for $j_0$ .", "For example, Puri et al[13] reports a much higher value of $j_0$ for QDs replacing impurities.", "This would lead to even faster spin control, together with more efficient control on $s_z$ .", "However, fs laser pulses have a broad frequency spectrum, and may excite several polariton modes.", "This may lead to destructive interference effects, which may reduce the effectiveness of a large $j_0$ ." ], [ "Conclusions", "We studied the optical control of single spins in micro-cavities accounting for all sources of decoherence.", "When the system is in the strong-coupling regime, the spin manipulation is most efficient and can be done by a few-picoseconds laser pulse.", "This suggests that single spins embedded in high Q-factor planar cavities can act as quantum memories and as qubits, with the optical excitation being the mechanism to control the state of the memory or to perform one-qubit operations for quantum computing.", "This optical control produces high-fidelity final states in very short time: a single operation can be performed $10^6$ faster than the typical decoherence time of the impurity spin qubit (compared to other proposals using for example ion traps[37]) with a fidelity of $F>99.8\\%$ .", "We believe that the present proposal for one-qubit operations, together with a previous one for implementing two-qubit operations in the same system[12], [13] show that polaritons in 2D-microcavities is a promising system for the implementation of solid-state quantum-computing scalable architectures." ], [ "acknowledgments", "G. F. Quinteiro thanks financial assistance by CONICET and ANPCyT (PICT2007-873), Argentina, and the Fulbright Commission, USA.", "A.", "A. Aligia thanks financial assistance by CONICET (PIP No.", "11220080101821) and ANPCyT (PICT R1776)." ] ]

[ [ "Using Satellites to Probe Extrasolar Planet Formation" ], [ "Abstract Planetary satellites are an integral part of the heirarchy of planetary systems.", "Here we make two predictions concerning their formation.", "First, primordial satellites, which have an array of distinguishing characteristics, form only around giant planets.", "If true, the size and duration of a planetary system's protostellar nebula, as well as the location of its snow line, can be constrained by knowing which of its planets possess primordial satellites and which do not.", "Second, all satellites around terrestrial planets form by impacts.", "If true, this greatly enhances the constraints that can be placed on the history of terrestrial planets by their satellites' compositions, sizes, and dynamics." ], [ "$^{a}$ Center for Space Physics, Boston University, 725 Commonwealth Avenue, Boston, MA 02215, USA.", "$^{b}$ Department of Physics, University of Idaho, 875 Perimeter Drive, Moscow, ID 83844-0903, USA.", "$^{*}$ Corresponding author email address: [email protected] Editorial Correspondence to: Paul Withers Center for Space Physics, Boston University, 725 Commonwealth Avenue, Boston, MA 02215, USA.", "Email: [email protected] Phone: (617) 353 1531 Fax: (617) 353 6463 Planetary satellites are an integral part of the heirarchy of planetary systems.", "Here we make two predictions concerning their formation.", "First, primordial satellites, which have an array of distinguishing characteristics, form only around giant planets.", "If true, the size and duration of a planetary system's protostellar nebula, as well as the location of its snow line, can be constrained by knowing which of its planets possess primordial satellites and which do not.", "Second, all satellites around terrestrial planets form by impacts.", "If true, this greatly enhances the constraints that can be placed on the history of terrestrial planets by their satellites' compositions, sizes, and dynamics.", "The formation of stars from gas and dust is a fundamental astrophysical process, yet stars do not form alone.", "A large and rapidly growing number of stars are known to be orbited by debris disks or planets.", "A comprehensive understanding of the life cycle of protostellar nebulae requires consideration of all of the types of condensed objects that form within them, not just stars.", "This includes planets, satellites, and objects analogous to asteroids or Kuiper belt objects.", "Many studies have investigated how the properties of planets and belts of smaller objects can be used to explore the formation of the planetary system, yet satellites have been somewhat neglected.", "Here we explore how the existence and nature of a planet's satellite system can be used to constrain how the planet formed.", "We make two predictions concerning the formation of planetary satellites and investigate their consequences.", "First, we predict that primordial satellites, which only form in dense regions of a protostellar nebula, are found only around giant planets.", "The distribution of primordial satellites in a planetary system can be used to constrain the location of a protostellar nebula's snow line and thereby to empirically constrain whether close-in extrasolar planets formed in-situ or migrated to their present locations.", "Second, we predict that all satellites around terrestrial planets formed by impacts.", "The composition of all terrestrial planets and their satellites reflects mixing across a broad region of the protostellar nebula.", "Rapid advances in observational capabilities suggest that the first satellite of an extrasolar planet will soon be discovered [18], [11].", "Our predictions outline the potential significance of such discoveries for the histories and current states of these satellites, planets, and planetary systems.", "All planets in our solar system can be classified as either terrestrial planets (Mercury, Venus, Earth and Mars) or giant planets (Jupiter, Saturn, Uranus and Neptune).", "The key physical characteristic that defines the boundary between these two classes is composition, not size.", "The abundance of volatile species relative to refractory species is much greater for giant planets than terrestrial planets.", "This difference is attributed to differences in how the planets formed.", "The giant planets formed in $\\sim $ 10 Myr from protoplanetary disks with high surface densities of volatile species, whereas the terrestrial planets formed in $\\sim $ 100 Myr from the accretion of refractory planetesimals [13], [2].", "Although recent observations of extrasolar planetary systems have drastically reshaped ideas of planet formation, models still predict a bimodal distribution of volatile-rich planets analogous to giant planets and refractory-rich planets analogous to terrestrial planets [16].", "Since satellites are abundant in the solar system, we expect that satellites are likely to be present around many extrasolar planets (as long as they have not been lost due to tidal evolution; see [1]).", "Several theories of satellite formation have been proposed to explain the diverse satellites within the solar system, including in situ formation in a protoplanetary disk, gravitational capture, atmospheric capture, fission, and the impact between a planet and another body [21].", "While the basic mechanisms of satellite formation are still actively debated by planetary scientists, certain trends are apparent.", "In particular, large, prograde satellites orbiting in the equatorial plane of giant planets all seem to have formed in-situ within the planet-forming sub-nebula [7].", "Highly inclined (even retrograde) satellites around giant planets formed via a capture mechanism [22], [14].", "Satellites of large solid planets are all consistent with an impact origin [5], [4].", "Some asteroid satellites formed by fission [19].", "These mechanisms are candidates for how extrasolar satellites form.", "Prediction 1: Satellites that formed in the same place and at the same time as their parent planet exist only around planets analogous to giant planets.", "This prediction results from the understanding that the formation of such primordial satellites requires locally high surface densities of gas, ice or dust close to the growing planet.", "They only form beyond the nebula's snow line.", "These conditions occur in the protoplanetary disks within which giant planets, but not terrestrial planets, form [21], [2].", "Thus if primordial satellites were found around a close-in planet with an orbital semimajor axis inside its star's snow line, the existence of the satellite would imply that the planet formed further out and migrated inward rather than having formed in place.", "The distinguishing characteristics of such satellites are: low eccentricity, prograde orbits near the planet's equatorial plane and well within the planet's Hill sphere; same age of formation as the parent planet; and elemental and isotopic compositions that, although potentially modified by accretion and subsequent processes, are related to the environmental conditions at the location and time of the formation of the parent planet.", "Extrasolar planets with satellites that possess these characteristics are predicted to be volatile-rich.", "That provides a constraint on relationships between the planet's mass, density, size, temperature and spectrum.", "They are also predicted to have formed rapidly within a protoplanetary disk.", "Since their elemental and isotopic composition reflects conditions where they formed, trends in these compositions can be used to determine spatial variations in the protostellar nebula and to constrain planetary migration post-formation.", "Mapping the distribution of such planets within a planetary system places constraints on the size and duration of the protostellar nebula from which the star, planets and satellites formed.", "If this prediction withstands scrutiny, then consideration of a stronger version may be warranted.", "Specifically, that all planets that formed within protoplanetary disks originally possessed primordial satellites.", "This would imply that if such a planet no longer possesses primordial satellites, like Neptune, then they must have been removed somehow.", "The removal process is likely to have been a major event that affected the state and subsequent evolution of the planet, such as a strong gravitational interaction with another planet or a series of impacts.", "If this prediction is disproven, then the presence of a protoplanetary disk during planet formation is not a discriminant between volatile-rich giant planets and refractory-rich terrestrial planets, which has substantial implications for planet formation.", "Prediction 2: All satellites above a threshold mass around planets analogous to terrestrial planets formed by the accretion of ejecta from an impact.", "This prediction results from the elimination of other possible formation mechanisms.", "Prediction 1 implies that planets analogous to terrestrial planets do not possess primordial satellites.", "Capture of a body by a planet via any mechanism requires energy loss [3], which is impractical for bodies exceeding some threshold mass [9] that we do not quantify in this work.", "It is difficult for a large solid body to accumulate enough angular momentum to form satellites by fission without also accumulating so much energy that it is catastrophically disrupted [15], [20], [19].", "In addition, a rapidly rotating body is likely to lose angular momentum by shedding loose material before approaching the threshold for fission [17].", "Only mechanisms based upon satellite formation from impact ejecta remain viable.", "The distinguishing characteristics of satellites formed from the accretion of impact ejecta are [10], [20], [6]: low abundance of siderophile constituents relative to the parent planet; low relative abundance of volatiles; large satellite to planet mass ratio; large ratio of the orbital angular momentum of the satellite to the total angular momentum of the system; and isotopic composition that, although potentially modified by subsequent processes, suggests a common origin for the material in the planet and satellite.", "Extrasolar planets with satellites that possess these characteristics are predicted to be refractory-rich.", "This provides a constraint on relationships between the planet's mass, density, size, temperature and spectrum.", "They are also predicted to have formed slowly from the accretion of refractory planetesimals.", "Due to mixing of planetesimals, the elemental and isotopic composition of such planets reflects conditions across a broad region of the protostellar nebula.", "Mapping the distribution of such planets within a planetary system places constraints on the size and duration of the protostellar nebula from which the star, planets and satellites formed.", "If this prediction withstands scrutiny, then consideration of a stronger version may be warranted.", "Specifically, that all such satellites around a planet formed from a single impact.", "This would imply that any large impact which produces a new satellite disrupts any pre-existing satellites.", "The composition and dynamics of all of the planet's satellites would be linked by their common origin in a single impact event.", "If this prediction is disproven, then either at least one seemingly impractical mechanism for satellite formation must occur more easily than is currently thought or an unsuspected mechanism for satellite formation exists.", "Either case has substantial implications for satellite formation.", "In order to better constrain the formation of extrasolar planets, we have made two predictions concerning the formation of planetary satellites.", "Both are consistent with current knowledge of the solar system.", "The first prediction has broad consequences for how the current state of a planetary system can be used to constrain temporal and spatial variations of conditions in the protostellar nebula within which the star, planets, satellites and other components of the system formed.", "The second prediction highlights the importance of stochastic impacts as a process that affects not only the geophysical and geochemical states of the objects involved, but also the hierarchical structure of planetary systems.", "A meaningful planetary classification scheme should be based on currently observable characteristics, yet also be related to planetary formation and history.", "Planets in the mass range $1 \\mathrm {M_\\oplus } \\le M \\le 10 \\mathrm {M_\\oplus }$ could either be “mini-Neptunes\", like GJ1214b [8] or “super-Earths\" like CoRoT-7b [12].", "Future discoveries of planets in this mass range are likely to blur the currently clear boundary between terrestrial planets and giant planets.", "Consideration of a planet's retinue of satellites, particularly primordial satellites, may help fix the boundaries of planetary classification schemes.", "Acknowledgments JWB is supported by a grant from the NASA Exobiology program." ] ]

[ [ "From Atiyah Classes to Homotopy Leibniz Algebras" ], [ "Abstract A celebrated theorem of Kapranov states that the Atiyah class of the tangent bundle of a complex manifold $X$ makes $T_X[-1]$ into a Lie algebra object in $D^+(X)$, the bounded below derived category of coherent sheaves on $X$.", "Furthermore Kapranov proved that, for a K\\\"ahler manifold $X$, the Dolbeault resolution $\\Omega^{\\bullet-1}(T_X^{1,0})$ of $T_X[-1]$ is an $L_\\infty$ algebra.", "In this paper, we prove that Kapranov's theorem holds in much wider generality for vector bundles over Lie pairs.", "Given a Lie pair $(L,A)$, i.e.", "a Lie algebroid $L$ together with a Lie subalgebroid $A$, we define the Atiyah class $\\alpha_E$ of an $A$-module $E$ (relative to $L$) as the obstruction to the existence of an $A$-compatible $L$-connection on $E$.", "We prove that the Atiyah classes $\\alpha_{L/A}$ and $\\alpha_E$ respectively make $L/A[-1]$ and $E[-1]$ into a Lie algebra and a Lie algebra module in the bounded below derived category $D^+(\\mathcal{A})$, where $\\mathcal{A}$ is the abelian category of left $\\mathcal{U}(A)$-modules and $\\mathcal{U}(A)$ is the universal enveloping algebra of $A$.", "Moreover, we produce a homotopy Leibniz algebra and a homotopy Leibniz module stemming from the Atiyah classes of $L/A$ and $E$, and inducing the aforesaid Lie structures in $D^+(\\mathcal{A})$." ], [ "Introduction", "The Atiyah class of a holomorphic vector bundle $E$ over a complex manifold $X$ , as initially introduced by Atiyah [3], constitutes the obstruction to the existence of a holomorphic connection on said holomorphic vector bundle.", "It is constructed in the following way.", "The vector bundle ${}{E}$ of jets (of order 1) of holomorphic sections of $E\\rightarrow X$ fits into the canonical short exact sequence $0\\rightarrow T^*_X\\otimes E\\rightarrow {}{E}\\rightarrow E\\rightarrow 0$ of holomorphic vector bundles (over the complex manifold $X$ ).", "The Atiyah class of $E\\rightarrow X$ is the extension class $\\alpha _E\\in ^1_{X}(E,T^*_X\\otimes E) \\cong H^1(X;T^*_X\\otimes E)$ of this short exact sequence [3], [22].", "In the late 1990's, Rozansky and Witten proposed a construction of a family of new 3-dimensional topological quantum field theories, indexed by compact (or asymptotically flat) hyper-Kähler manifolds [40].", "The Rozansky-Witten procedure does thus associate topological invariants of 3-manifolds to each hyper-Kähler manifold.", "In subsequent work, Kapranov [22] and Kontsevich [24] revealed the crucial role played by Atiyah classes in the construction of the Rozansky-Witten invariants.", "They [22], [24] in particular showed that the hyper-Kähler restriction is unnecessary and that the theory devised by Rozansky and Witten works for all holomorphic symplectic manifolds.", "In Kapranov's work lies the key fact that the Atiyah class of the tangent bundle of a complex manifold $X$ yields a map ${T_X}{-1}\\otimes {T_X}{-1} \\rightarrow {T_X}{-1}$ in the derived category $D^+ (X)$ of bounded below complexes of sheaves of $\\mathcal {O}_X$ -modules with coherent cohomology, which makes ${T_X}{-1}$ into a Lie algebra object in $D^+ (X)$ .", "Therefore, Kapranov's approach shone light on many similarities between the Rozansky-Witten and Chern-Simons theories [4], [5] as shown by Roberts and Willerton [39].", "Atiyah classes have also enjoyed renewed vigor due to Kontsevich's seminal work on deformation quantization [25], [23].", "Kontsevich indicated the existence of deep ties between the Todd genus of complex manifolds and the Duflo element in Lie theory [25], [23], [41], [11].", "This discovery inspired several subsequent works on Hochschild (co)homology and the Hochschild-Kostant-Rosenberg isomorphism for complex manifolds, by Dolgushev, Tarmarkin and Tsygan [17], [16], Căldăraru [15], Markarian [33], Ramadoss [37], and Calaque and Van den Bergh [12] among many others.", "In particular, the work of Markarian [33] (see also Ramadoss [37]) led to an alternative proof of the Hirzebruch-Riemann-Roch theorem and its variations.", "In [35], [36], Molino introduced an Atiyah class for connections “transverse to a foliation”, which measures the obstruction to their “projectability”.", "Molino class has many applications in geometry, for instance, in the study of differential operators on a foliate manifold [44], and of deformation quantization theory [6].", "This paper is the first in a sequence of works which aims at developing a theory of Atiyah classes in a general setting and studying their applications.", "Our goal is to explore emerging connections between derived geometry and classical areas of mathematics such as complex geometry, foliation theory, Poisson geometry and Lie theory.", "The present paper develops a framework which encompasses both the original Atiyah class of holomorphic vector bundles and the Molino class of real vector bundles foliated over a foliation as special cases.", "Lie algebroids are the starting point of our approach.", "Indeed, holomorphic vector bundles and vector bundles foliated over a foliation may both be seen as instances of the concept of module over a Lie algebroid, a straightforward generalization of the well-known representations of Lie algebras.", "Given a Lie algebroid $L$ over a base manifold $M$ , an $L$ -connection on a vector bundle $E\\rightarrow M$ is a bilinear map $X\\otimes s \\mapsto \\nabla _{X}s$ from $\\Gamma (L)\\otimes \\Gamma (E)$ to $\\Gamma (E)$ satisfying the usual axioms $ \\nabla _{fX}s=f\\nabla _X s$ and $\\nabla _X (fs)=\\rho (X)f\\cdot s+f\\nabla _X s$ for any $f\\in {M}$ .", "If the connection is flat, $E$ is said to be a module over the Lie algebroid $L$ .", "When the base is the one-point space, the $L$ -modules are simply Lie algebra modules in the classical sense.", "When the base is a complex manifold $X$ , the holomorphic vector bundles over a complex manifold $X$ are known to be the modules of the complex Lie algebroid $T^{0, 1}_X$ stemming from the complex manifold $X$ .", "Molino's foliated bundles are modules over the Lie algebroid structure carried by the characteristic distribution of the foliation of their base.", "We introduce a general theory of Atiyah classes of vector bundles over Lie algebroid pairs.", "By a Lie algebroid pair $(L,A)$ , we mean a Lie algebroid $L$ (over a manifold $M$ ) together with a Lie subalgebroid $A$ (over the same base $M$ ) of $L$ .", "And by a vector bundle $E$ over the Lie algebroid pair $(L,A)$ , we mean a vector bundle $E$ (over $M$ ), which is a module over the Lie subalgebroid $A$ .", "Given such a Lie algebroid pair $(L,A)$ and $A$ -module $E$ , we consider the jet bundle ${{}}{E}$ (of order 1), whose smooth sections are the $L$ -connections on $E$ extending the (infinitesimal) $A$ -action on $E$ in a compatible way.", "We prove the following [A] The jet bundle ${{}}{E}$ is naturally an $A$ -module.", "It fits in a short exact sequence of $A$ -modules $0\\rightarrow A^\\perp \\otimes E\\rightarrow {{}}{E}\\rightarrow E\\rightarrow 0$ .", "Here $A^\\perp $ denotes the annihilator of $A$ in $L$ .", "We call the extension class $\\alpha _E\\in ^1_{{A}}(E,A^\\perp \\otimes E)\\cong H^1(A, A^\\perp \\otimes E)$ of this short exact sequence, the Atiyah class of $E$ because, when $L=T_X\\otimes $ and $A=T_X^{0,1}$ for a complex manifold $X$ , ${{}}{E}$ is the space of 1-jets of holomorphic sections of $E$ and $\\alpha _E$ is its (classical) Atiyah class; and, when $L$ is the tangent bundle of a smooth manifold $M$ and $A$ is an integrable distribution on $M$ , $\\alpha _E$ is the Molino class of the vector bundle $E$ foliated over $A$ .", "Geometrically, the Atiyah class can thus be interpreted as the obstruction to the existence of a compatible $L$ -connection on $E$ which extends the $A$ -action with which $E$ is endowed.", "It turns out that the Atiyah class introduced in our general context and the classical Atiyah class of holomorphic vector bundles enjoy similar rich algebraic properties.", "We denote the (abelian) category of modules over the universal enveloping algebra ${A}$ of the Lie algebroid $A$ by the symbol ${A}$ .", "Every vector bundle over $M$ endowed with an $A$ -action — more pecisely its space of smooth sections — is an object of ${A}$ .", "The bounded below derived category of ${A}$ will be denoted by $D^+({A})$ .", "Given a Lie algebroid pair $(L, A)$ , the quotient $L/A$ is naturally an $A$ -module.", "When $L$ is the tangent bundle to a manifold $M$ and $A$ is an integrable distribution on $M$ , the $A$ -action on $$ is given by the Bott connection [8].", "We consider $L/A$ as a complex concentrated in degree 1 and refer to it as $L/A[-1]$ .", "We show that the Atiyah class of $$ makes $[-1]$ into a Lie algebra object in the derived category $D^+({A})$ .", "Indeed, being an element of $^1_{{A}}(, A^\\perp \\otimes ) \\cong ^1_{{A}}(\\otimes , ) \\cong \\\\_{D^+({A})}({}{-1}\\otimes {}{-1},{}{-1}),$ the Atiyah class $_{}$ of the $A$ -module $$ defines a “Lie bracket” on ${}{-1}$ .", "If, moreover, $E$ is an $A$ -module, its Atiyah class $ _{E}\\in ^1_{{A}}(E, A^\\perp \\otimes E)\\cong ^1_{{A}}(\\otimes E,E)\\cong _{D^+({A})}({}{-1}\\otimes {E}{-1},{E}{-1}) $ defines a “representation” on ${E}{-1}$ of the “Lie algebra” ${}{-1}$ .", "In summary, we prove the following [B] Let $(L,A)$ be a Lie algebroid pair with quotient $L/A$ .", "Then the Atiyah class of $L/A$ makes ${}{-1}$ into a Lie algebra object in the derived category $D^+({A})$ .", "Moreover, if $E$ is an $A$ -module, then ${E}{-1}$ is a module object over the Lie algebra object ${}{-1}$ in the derived category $D^+({A})$ .", "The above result suggests that, on the cochain level, the Atiyah class should define some kind of Lie algebra up to a certain homotopy.", "But how does one obtain a cocycle representing the Atiyah class?", "Recall that a Dolbeault representative of the Atiyah class of a holomorphic vector bundle $E\\rightarrow X$ can be obtained in the following way.", "Considering $T_X$ as a complex Lie algebroid, choose a $T_X$ -connection $\\nabla $ on $E$ .", "Being a holomorphic vector bundle, $E$ carries a canonical flat $T_X$ -connection $$ .", "Adding $\\nabla $ and $$ , we obtain a $T_X\\otimes $ -connection $\\nabla $ on $E$ .", "The element $\\mathcal {R}\\in ^{1,1}(E)$ defined by $\\mathcal {R}(X^{0,1};Y^{1,0})s=\\nabla _{X^{0,1}}\\nabla _{Y^{1,0}}s-\\nabla _{Y^{1,0}}\\nabla _{X^{0,1}}s-\\nabla _{[X^{0,1},Y^{1,0}]}s$ is a Dolbeault 1-cocycle whose cohomology class (which is independent of the choice of $\\nabla $ ) is the Atiyah class $\\alpha _E\\in H^{1,1}(X,E)\\cong H^1(X,T^*_X\\otimes E)$ .", "In the more general setting of vector bundles over Lie algebroid pairs, the Atiyah class can be defined in terms of Lie algebroid connections as follows.", "Assume $(L,A)$ is a Lie pair, and $E$ is an $A$ -module.", "Let $\\nabla $ be any $L$ -connection on $E$ extending its $A$ -action.", "The curvature of $\\nabla $ is the bundle map $R^\\nabla :\\wedge ^2 L\\rightarrow E$ defined by $R^\\nabla (l_1,l_2)=\\nabla _{l_1}\\nabla _{l_2}-\\nabla _{l_2}\\nabla _{l_1}-\\nabla _{{l_1}{l_2}}$ , for all $l_1, l_2\\in {L}$ .", "Since $E$ is an $A$ -module, its restriction to $\\wedge ^2 A$ vanishes.", "Hence the curvature induces a section $_E\\in {A^*\\otimes A^\\perp \\otimes E}$ , which is a 1-cocycle for the Lie algebroid $A$ with values in the $A$ -module $A^\\perp \\otimes E$ .", "We prove that the cohomology class $_E\\in H^1(A;A^\\perp \\otimes E)$ of the 1-cocycle $_E$ is precisely the Atiyah class of the $A$ -module $E$ .", "When the $A$ -module $E$ is the quotient $$ of the Lie algebroid pair $(L,A)$ , by choosing an $L$ -connection $\\nabla $ on $$ extending the $A$ -action, we get the Atiyah cocycle $_{}\\in {A^*\\otimes ()^*\\otimes ()}$ , which may be regarded as a bundle map $:\\otimes \\rightarrow A^*\\otimes $ .", "Starting from $$ and a splitting of the short exact sequence of vector bundles $0\\rightarrow A\\rightarrow L\\rightarrow \\rightarrow 0$ , a sequence $(R_n)_{n=2}^{\\infty }$ of bundle maps $R_n:\\otimes ^n \\rightarrow (A,)$ can be defined inductively by the relation $R_{n+1}=\\partial ^\\nabla R_n$ , for $n\\ge 2$ .", "Alternatively, $R_n$ can be seen as a section of the vector bundle $A^*\\otimes (\\otimes ^n ()^*)\\otimes $ .", "Then the graded vector space $\\bigoplus _{n=0}^\\infty {\\wedge ^n A^*\\otimes }$ can be endowed with a sequence $(\\lambda _k)_{k=1}^\\infty $ of multibrackets $\\lambda _k:\\otimes ^kV\\rightarrow V$ : the unary bracket $\\lambda _1$ is chosen to be coboundary operator $\\partial ^A$ of exterior forms on the Lie algebroid $A$ taking values in the $A$ -module $$ , while all higher order brackets $\\lambda _k$ are defined by the relation $\\lambda _k(\\xi _1\\otimes b_1,\\cdots ,\\xi _k\\otimes b_k)={{\\xi _1}+\\cdots +{\\xi _k}}\\xi _1\\wedge \\cdots \\wedge \\xi _k\\wedge R_k(b_1,\\cdots , b_k),$ where $b_1,\\dots ,b_k\\in {}$ and $\\xi _1,\\dots ,\\xi _k$ are arbitrary homogeneous elements of ${\\wedge ^\\bullet A^*}$ .", "By an $A$ -algebra, we mean a bundle of associative algebras $$ over $M$ which is an $A$ -module, and on which $\\Gamma (A)$ acts by derivations.", "For a commutative $A$ -algebra $$ , $(\\lambda _k)_{k=1}^\\infty $ extends in a natural way to the graded subspace $\\bigoplus _{n=0}^\\infty {\\wedge ^n A^*\\otimes \\otimes }$ .", "We prove [C] Assume that $(L,A)$ is a Lie pair and $$ is a commutative $A$ -algebra.", "When endowed with the sequence of multibrackets $(\\lambda _k)_{k\\in }$ , the graded vector space ${\\wedge ^\\bullet A^*\\otimes L/A\\otimes }[-1]$ becomes a Leibniz$_\\infty $ algebra — a natural generalization of Stasheff's $L_\\infty $ algebras [28] first introduced by Loday [30] in which the requirement that the multibrackets be (skew-)symmetric is dropped.", "If $E$ is an $A$ -module, the graded vector space ${\\wedge ^\\bullet A^*\\otimes E \\otimes }[-1]$ becomes a Leibniz$_\\infty $ module over the Leibniz$_\\infty $ algebra ${\\wedge ^\\bullet A^*\\otimes L/A \\otimes }[-1]$ .", "As a consequence, $\\bigoplus _{i\\ge 1}H^{i-1} (A, {} \\otimes )$ is a graded Lie algebra and $\\bigoplus _{i\\ge 1}H^{i-1}(A, E \\otimes )$ a module over it.", "We also identify a simple criterion for detecting when this Leibniz algebra is actually an $L_\\infty $ algebra.", "This situation happens when $X$ is a Kähler manifold, $L=T_X\\otimes $ and $A=T_X^{0, 1}$ .", "Then we recover the $L_\\infty $ -structure on $\\Omega ^{0, \\bullet -1}(T^{1, 0})$ discovered by Kapranov [22].", "The intrinsic meaning of this homotopic algebraic structure arising from our construction of the Atiyah class, and its relation with the $L_\\infty $ -space of Costello [13], [14] will be explored somewhere else.", "Note that our definition of the Atiyah class could easily be generalized to complexes of $A$ -modules as in [33].", "We also refer the interested reader to [12] for the Atiyah class of a DG-module over a dDG-algebra.", "After the first draft of this paper was posted on arXiv, Calaque inferred that, for Lie algebra pairs $(\\mathfrak {d},\\mathfrak {g})$ , i.e.", "Lie algebroid pairs with the one-point space as base manifold, the Atiyah class of the quotient $\\mathfrak {d}/\\mathfrak {g}$ coincides with the class capturing the obstruction to the “PBW problem” studied earlier by Calaque-Căldăraru-Tu [10] (see also [18]).", "Bordemann gave a nice interpretation of the Calaque-Căldăraru-Tu class as the obstruction to the existence of invariant connections on homogeneous spaces [7].", "Another very recent development is Calaque's beautiful work [9] on the relation between the Atiyah class of the $A$ -module $L/A$ with respect to the Lie pair $(L,A)$ and the relative PBW problem previously solved by Căldăraru-Calaque-Tu [10].", "Calaque also pointed out to us that our results should be related to the obstruction to a relative HKR isomorphism for closed embeddings of algebraic varieties identified by Arinkin-Căldăraru [2].", "This certainly deserves further investigation.", "Finally, we would like to mention, in relation to the homotopy algebra results of the present paper, Yu's very interesting doctoral thesis on $L_\\infty $ -algebroids [45].", "Stasheff's work on constrained Poisson algebras [42] is another interesting result which could well be related to the present paper.", "We would like to express our gratitude to several institutions for their hospitality while we were working on this project: Penn State University (Chen), Université Paris 7 (Stiénon), Université du Luxembourg (Chen and Stiénon), Institut des Hautes Études Scientifiques (Xu), and Beijing International Center for Mathematical Research (Xu).", "We would also like to thank Martin Bordemann, Paul Bressler, Damien Calaque, Murray Gerstenhaber, Grégory Ginot, Camille Laurent-Gengoux, Dmitry Roytenberg, Boris Shoikhet, Jim Stasheff, Izu Vaisman and Alan Weinstein for fruitful discussions and useful comments.", "Special thanks go to Jim Stasheff for carefully reading the preliminary version of the manuscript." ], [ "Preliminaries: connections, modules, Lie pairs, and matched pairs", "Let $M$ be a smooth manifold, let $\\rightarrow M$ be a Lie algebroid, and let $E{\\pi }M$ be a vector bundle.", "The anchor map of $$ is denoted by $$ .", "Recall that the Lie algebroid differential $d:\\Gamma (\\wedge ^{\\bullet }L^*)\\rightarrow \\Gamma ( \\wedge ^{\\bullet +1}L^* )$ is given by $\\big (d\\mu \\big )(x_0,\\cdots ,x_n)= \\sum _{i=0}^n (-1)^i \\rho (x_i) \\big ( \\mu (x_0,\\cdots ,\\widehat{x_i},\\cdots ,x_n) \\big ) \\\\+ \\sum _{i<j} (-1)^{i+j} \\mu ({x_i}{x_j},x_0,\\cdots ,\\widehat{x_i},\\cdots ,\\widehat{x_j},\\cdots ,x_n).$ The traditional description of a (linear) $L$ -connection on $E$ is in terms of a covariant derivative $ {L}\\times {E}\\rightarrow {E}: (x,e)\\mapsto \\nabla _x e $ characterized by the following two properties: $\\nabla _{fx} e=f\\nabla _x e , \\\\\\nabla _x (fe)=\\rho (x)f\\cdot e+f\\cdot \\nabla _x e,$ for all $x\\in {L}$ , $e\\in {E}$ , and $f\\in {M}$ .", "Here, we give three equivalent descriptions of (linear) $$ -connections on $E$ : covariant differential, horizontal lifting, and horizontal distribution.", "A (linear) $L$ -connection on $E$ is a map ${E}{d^\\nabla }{^*\\otimes E}$ , called covariant differential, satisfying the Leibniz rule $d^\\nabla (fe)=\\rho ^* (df) \\otimes e + f \\cdot d^\\nabla e ,$ for all $f\\in {M}$ and $e\\in {E}$ .", "The covariant differential ${E}{d^\\nabla }{^*\\otimes E}$ extends uniquely to a degree 1 operator $ {\\wedge ^{\\bullet } ^*\\otimes E}{d^\\nabla }{\\wedge ^{\\bullet +1} ^*\\otimes E} $ satisfying the Leibniz rule $ d^\\nabla (\\beta \\otimes e)=d\\beta \\otimes e+ (-1)^{b}\\beta \\otimes d^\\nabla e ,$ for all $\\beta \\in {\\wedge ^{b}^*}$ and $e\\in {E}$ .", "A (linear) $L$ -connection on $E$ is a map $\\times _M E{h} T_E$ , called horizontal lifting, such that the diagram $ { & & [rr]^{\\rho } [rd] & & T_M [ld] \\\\\\times _M E [rr]^{h} [rru] [rd] & & T_E [rru]^{\\pi _*} [ld] & M & \\\\& E [rru]_{\\pi } & & & } $ commutes and its faces $ { \\times _M E [r]^h [d] & T_E [d] \\\\ E [r]_{} & E }\\qquad \\text{and} \\qquad { \\times _M E [r]^h [d] & T_E [d]^{\\pi _*} \\\\ [r]_{\\rho } &T_M } $ are vector bundle maps.", "A vector field $X$ on $E$ is said to be projectable onto $M$ if $\\pi (e_1)=\\pi (e_2)$ implies $\\pi _*(X_{e_1})=\\pi _*(X_{e_2})$ .", "By $_\\pi (E)$ , we denote the space of vector fields $X$ on $E$ which are projectable onto $M$ and whose flow $\\Phi ^X_t:E\\rightarrow E$ is a vector bundle automorphism over the flow $\\Phi ^{\\pi _* X}_t:M\\rightarrow M$ of the projected vector field $\\pi _* X$ on $M$ .", "The space $_\\pi (E)$ is obviously a module over the ring ${M}$ .", "A (linear) $L$ -connection on $E$ is a morphism of ${M}$ -modules ${}{H}_\\pi (E)$ , called horizontal distribution, such that the diagram $ { {} [rr]^H [rd]_\\rho && _\\pi (E) [ld]^{\\pi _*}\\\\ & (M) & } $ commutes.", "Covariant differential, covariant derivative, horizontal lift, and horizontal distribution are related to one another by the identities $\\nabla _l e={d^\\nabla e}{l} ; \\\\e_*\\rho (l_x)-h(l_x,e_x)=\\tau _{e_x} (\\nabla _l e)_x ; \\\\{H(l)}|_{e_x}=h(l_x,e_x) ,$ which hold for all $x\\in M$ , $l\\in {}$ , and $e\\in {E}$ .", "Here in the second equation, $\\tau _{e_x}$ denotes the canonical isomorphism between the fiber $E_x$ and its tangent space at the point $e_x$ .", "This second equation can be rewritten as $h(l_x,e_x)f_{\\nu }={\\nabla _{l_x}\\nu }{e_x},$ where $f_{\\nu }$ denotes the fiberwise linear function on $E$ determined by $\\nu \\in {E^*}$ .", "The following assertions are equivalent: $\\nabla _{l_1}\\nabla _{l_2}-\\nabla _{l_2}\\nabla _{l_1}=\\nabla _{{l_1}{l_2}} ;\\\\H({l_1}{l_2})={H(l_1)}{H(l_2)} .$ When they are satisfied for all $l_1,l_2\\in {L}$ , the connection is said to be flat.", "An $L$ -module is a vector bundle $E \\rightarrow M$ endowed with a flat (linear) $L$ -connection.", "A flat (linear) $L$ -connection will also be called an $L$ -action or $L$ -representation.", "When the $L$ -connection on $E$ is flat, $(d^\\nabla )^2=0$ and $({\\wedge ^\\bullet L\\otimes E}, d^\\nabla )$ is a cochain complex, whose cohomology groups $H(L;E)$ are the so-called Lie algebroid cohomology groups of $L$ with values in $E$ .", "By a Lie pair $(L,A)$ , we mean a Lie algebroid $L$ over a manifold $M$ and a subalgebroid $A$ of $L$ .", "The quotient $L/A$ of a Lie pair $(L,A)$ is an $A$ -module; the action of $A$ on $L/A$ is defined by $ \\nabla _a \\bigl ((l)\\bigr )=({a}{l}), \\quad \\forall a\\in {A}, l\\in {L} ,$ where $$ denotes the projection $L\\rightarrow L/A$ .", "Being dual to $L/A$ , the annihilator $A^\\perp $ of $A$ in $L^*$ is also an $A$ -module.", "Assume now that $A$ and $B$ are two Lie subalgebroids of a Lie algebroid $L$ such that $L$ and $A\\oplus B$ are isomorphic as vector bundles.", "Then $L/AB$ is naturally an $A$ -module while $L/BA$ is naturally a $B$ -module.", "The Lie algebroids $A$ and $B$ are said to form a matched pair.", "[[31], [34], [32]] Two (real or complex) Lie algebroids $A$ and $B$ over the same base manifold $M$ and with respective anchors $\\rho _A$ and $\\rho _B$ are said to form a matched pair if there exists an action $\\nabla $ of $A$ on $B$ and an action $$ of $B$ on $A$ such that the identities ${(X)}{(Y)} = -\\big (_Y X\\big )+\\big (\\nabla _X Y\\big ) , \\\\\\nabla _X{Y_1}{Y_2} = {\\nabla _X Y_1}{Y_2} +{Y_1}{\\nabla _X Y_2} + \\nabla _{_{Y_2} X}Y_1 -\\nabla _{_{Y_1} X} Y_2 , \\\\_Y{X_1}{X_2} = {_Y X_1}{X_2} +{X_1}{_Y X_2} + _{\\nabla _{X_2} Y} X_1 -_{\\nabla _{X_1}Y}X_2 ,$ hold for all $X_1,X_2,X\\in {A}$ and $Y_1,Y_2,Y\\in {B}$ .", "[[34], [32]] Given a matched pair $(A,B)$ of Lie algebroids, there is a Lie algebroid structure $A\\bowtie B$ on the direct sum vector bundle $A\\oplus B$ , with anchor $ X\\oplus Y\\mapsto \\rho _A(X)+\\rho _B(Y) $ and bracket ${X_1\\oplus Y_1}{X_2\\oplus Y_2}= \\big ({X_1}{X_2} + _{Y_1}X_2 - _{Y_2}X_1 \\big )\\oplus \\big ( {Y_1}{Y_2} + \\nabla _{X_1}Y_2 - \\nabla _{X_2}Y_1 \\big ) .$ Conversely, if $A\\oplus B$ has a Lie algebroid structure for which $A\\oplus 0$ and $0\\oplus B$ are Lie subalgebroids, then the representations $\\nabla $ and $$ defined by $ {X\\oplus 0}{0\\oplus Y} = -_Y X\\oplus \\nabla _X Y $ endow the couple $(A,B)$ with a structure of matched pair.", "A Lie algebra is a Lie algebroid whose base manifold is the one-point space.", "If the direct sum $\\oplus ^*$ of a vector space $$ and its dual $^*$ is endowed with a Lie algebra structure such that the direct summands $$ and $^*$ are Lie subalgebras and $ [X,\\alpha ]={X}\\alpha - {\\alpha }X,\\qquad \\forall X\\in , \\alpha \\in ^* ,$ the pair $(,^*)$ is said to be a Lie bialgebra.", "Lie bialgebras are instances of matched pairs of Lie algebroids.", "Let $X$ be a complex manifold.", "Then $(T^{0,1}_X,T^{1,0}_X)$ is a matched pair, where the actions are given by $ \\nabla _{X^{0,1}}X^{1,0}=^{1,0}{X^{0,1}}{X^{1,0}}\\qquad \\text{ and } \\qquad _{X^{1,0}}X^{0,1}=^{0,1}{X^{1,0}}{X^{0,1}}, $ for all $X^{0,1}\\in {T_X}$ and $X^{1,0}\\in {T_X}$ .", "Hence $T^{0,1}_X\\bowtie T^{1,0}_X$ and $T_X\\otimes $ are isomorphic as complex Lie algebroids.", "More generally, given a holomorphic Lie algebroid $A$ , the pair $(A^{0,1},A^{1,0})$ is a matched pair of Lie algebroids and $A^{0,1}\\bowtie A^{1,0}$ is isomorphic, as a complex Lie algebroid, to $A\\otimes $ .", "Let $D$ be an integrable distribution on a smooth manifold $M$ .", "Then $D$ is a Lie subalgebroid of $T_X$ , and the normal bundle $T_X/D$ is canonically a $D$ -module.", "The $D$ -action on $T_X/D$ is usually called Bott connection [8].", "Moreover, if $\\mathcal {F}_1$ and $\\mathcal {F}_2$ are two transversal foliations on a smooth manifold $M$ , the corresponding tangent distributions $T_{\\mathcal {F}_1}$ and $T_{\\mathcal {F}_2}$ constitute a matched pair of Lie algebroids with $T_{\\mathcal {F}_1}\\bowtie T_{\\mathcal {F}_2}T_X$ .", "Let $G$ be a Poisson Lie group and let $(P,\\pi )$ be a Poisson $G$ -space, i.e.", "a Poisson manifold together with a $G$ -action defined by a Poisson map $G\\times P\\rightarrow P$ .", "According to Lu [31], $A=(T^*P)_\\pi $ and $B=P\\rtimes $ form a matched pair of Lie algebroids.", "A matched pair of Lie algebroids $L=A\\bowtie B$ can be seen as a Lie pair $(L,A)$ together with a splitting $j:B\\rightarrow L$ of the short exact sequence $0\\rightarrow A \\rightarrow L \\rightarrow B \\rightarrow 0$ , whose image $j(B)$ happens to be a Lie subalgebroid of $L$ ." ], [ "Prelude: holomorphic connections", "The Atiyah class of a holomorphic vector bundle $E$ over a complex manifold $X$ is the obstruction class to the existence of a holomorphic (linear) connection.", "It is constructed in the following way.", "The vector bundle ${}{E}$ of jets (of order 1) of holomorphic sections of $E\\rightarrow X$ fits into the canonical short exact sequence of holomorphic vector bundles $ 0\\rightarrow T^*_X\\otimes E\\rightarrow {}{E}\\rightarrow E\\rightarrow 0 $ over the complex manifold $X$ .", "The Atiyah class of $E\\rightarrow X$ is the extension class $ \\alpha _E\\in ^1_{X}(E,T^*_X\\otimes E) $ of this short exact sequence [3], [22].", "There are canonical isomorphisms between the abelian groups $^1_{X}(E,T^*_X\\otimes E)$ and $_{D^b(X)}(T_X\\otimes E,E[1])$ , the sheaf cohomology group $H^1(X,T^*_X\\otimes E)$ and the Dolbeault cohomology group $H^{1, 1}(X,E)$ .", "A Dolbeault representative of the Atiyah class can be obtained in the following way.", "Considering $T_X$ as a complex Lie algebroid, choose a $T_X$ -connection $\\nabla $ on $E$ .", "Being a holomorphic vector bundle, $E$ carries a canonical flat $T_X$ -connection $$ .", "Adding $\\nabla $ and $$ , we obtain a $T_X\\otimes $ -connection $\\nabla $ on $E$ .", "The element $\\mathcal {R}\\in ^{1,1}(E)$ defined by $\\mathcal {R}(X^{0,1};Y^{1,0})s=\\nabla _{X^{0,1}}\\nabla _{Y^{1,0}}s-\\nabla _{Y^{1,0}}\\nabla _{X^{0,1}}s-\\nabla _{[X^{0,1},Y^{1,0}]}s $ is a Dolbeault 1-cocycle whose cohomology class (which is independent of the choice of $\\nabla $ ) is the Atiyah class $\\alpha _E\\in H^{1, 1}(X,E)$ ." ], [ "Extension of an $A$ -action to a compatible {{formula:559e48ac-9fc5-41b4-8ef6-60d4277407c0}} -connection", "Throughout this section, $(L,A)$ is a Lie pair and $E$ is an $A$ -module.", "The symbols ${E}$ and ${L}$ will denote the sheaves on $M$ defined by $ {E}(U)={e\\in {U;E}\\text{ s.t.", "}\\nabla _{a}e=0,\\forall a \\in {U;A} } ,$ and $ {L}(U)={l\\in {U;L}\\text{ s.t.", "}{a}{l}\\in {U;A},\\forall a\\in {U;A}} ,$ where $U$ denotes an arbitrary open subset of $M$ .", "Given an $A$ -module $E$ , there always exists an $L$ -connection on $E$ extending the given $A$ -connection.", "Moreover, if $\\nabla ^1$ and $\\nabla ^2$ are two such extensions, then $d^{\\nabla ^2}-d^{\\nabla ^1}\\in {A^\\perp \\otimes E}$ , where $A^\\perp $ denotes the annihilator of $A$ in $L^*$ .", "Choose a subbundle $B$ of $L$ such that $L=A\\oplus B$ and a $T_M$ -connection $\\nabla ^{(T_M)}$ on $E$ — this is always possible.", "Then extend $A$ -connection $\\nabla ^{(A)}$ to an $L$ -connection $\\nabla ^{(L)}$ on $E$ by setting $ \\nabla ^{(L)}_{a+b}= \\nabla ^{(A)}_a+\\nabla ^{(T_M)}_{\\rho (b)} ,$ where $\\rho $ denotes the anchor map $L\\rightarrow T_M$ .", "The difference $l \\mapsto \\nabla ^1_l-\\nabla ^2_l$ of two such extensions $\\nabla ^1$ and $\\nabla ^2$ is a bundle map $L\\rightarrow E$ , which vanishes on $A$ .", "An $L$ -connection $\\nabla $ on $E$ is said to be $A$ -compatible if (1) it extends the given $A$ -action on $E$ and (2) it satisfies $ \\nabla _a \\nabla _l-\\nabla _l\\nabla _a=\\nabla _{{a}{l}}, \\quad \\forall a\\in {A}, l\\in {L} .$ Let $\\nabla $ be an $L$ -connection on $E$ extending its $A$ -action.", "Provided that the sheaf of smooth sections of $E$ is isomorphic to $C^{\\infty }_M\\otimes _{}{E}$ and the sheaf of smooth sections of $L$ is isomorphic to $C^{\\infty }_M\\otimes _{}{L}$ , the $L$ -connection $\\nabla $ is $A$ -compatible if and only if $\\nabla _{{L}}{E}\\subset {E}$ .", "For any $a\\in {U;A}$ , $l\\in {L}(U)$ and $e\\in {E}(U)$ , we have ${a}{l}\\in {U;A}$ , $\\nabla _a e=0$ , and $\\nabla _{{a}{e}}=0$ so that, if $\\nabla $ is $A$ -compatible, we obtain $ \\nabla _a \\nabla _l e=\\nabla _a \\nabla _l e-\\nabla _l \\nabla _a e-\\nabla _{{a}{l}} e=0 .$ Hence $\\nabla _l e\\in {E}(U)$ and this proves that $\\nabla _{{L}}{E}\\subset {E}$ .", "Conversely, if $\\nabla _{{L}}{E}\\subset {E}$ , then $ \\bigl (\\nabla _a\\nabla _{f\\cdot l}-\\nabla _{f\\cdot l}\\nabla _a-\\nabla _{{a}{ {f\\cdot l}}}\\bigr )(g\\cdot e) = fg\\cdot \\nabla _{a}\\nabla _{l}e =0 ,$ for all $a\\in {U;A}$ , $f,g\\in C^{\\infty }_U$ , $l\\in {L}(U)$ and $e\\in {E}(U)$ .", "Since ${U;E}=C^{\\infty }_U\\otimes _{}{E}$ , it follows that $\\nabla $ is $A$ -compatible.", "Given a matched pair of Lie algebroids $(A,B)$ and an $A$ -module $E$ , consider the Lie algebroid $L=A\\bowtie B$ .", "An $A$ -compatible $L$ -connection on $E$ determines a $B$ -connection on $E$ satisfying $ \\nabla _a\\nabla _b e-\\nabla _b\\nabla _a e=\\nabla _{{a}{b}} e,\\quad \\forall a\\in {A}, b\\in {B}, e\\in {E} .$ The converse is also true." ], [ "Atiyah class: obstruction to compatibility", "Assume $(L,A)$ is a Lie pair, $E$ is an $A$ -module, and $\\nabla $ is an $L$ -connection on $E$ extending its $A$ -action.", "The curvature of $\\nabla $ is the bundle map $R^\\nabla :\\wedge ^2 L\\rightarrow E $ defined by $R^\\nabla (l_1,l_2)=\\nabla _{l_1}\\nabla _{l_2}-\\nabla _{l_2}\\nabla _{l_1}-\\nabla _{{l_1}{l_2}}, \\quad \\forall ~ l_1, l_2\\in {L}.$ Since $E$ is an $A$ -module, its restriction to $\\wedge ^2 A$ vanishes.", "Hence the curvature induces a section $_E\\in {A^*\\otimes A^\\perp \\otimes E}$ or, equivalently, a bundle map $_E:A\\otimes (L/A)\\rightarrow E$ given by $_\\big (a;q(l)\\big )=R^\\nabla (a,l)=\\nabla _{a}\\nabla _{l}-\\nabla _{l}\\nabla _{a}-\\nabla _{{a}{l}},\\quad \\forall a\\in {},l\\in {}.$ The $L$ -connection $\\nabla $ is compatible with the $A$ -module structure of $E$ if and only if $_E=0$ .", "The section $_E$ of $A^*\\otimes A^\\perp \\otimes E $ is a 1-cocycle for the Lie algebroid $A$ with values in the $A$ -module $A^\\perp \\otimes E $ .", "We call $_E$ the Atiyah cocycle associated with the $L$ -connection $\\nabla $ that extends the $A$ -module structure of $E$ .", "The cohomology class $_E\\in H^1(A;A^\\perp \\otimes E)$ of the cocycle $_E$ does not depend on the choice of $L$ -connection extending the $A$ -action and is called the Atiyah class of the $A$ -module $E$ .", "The Atiyah class $_E$ of $E$ vanishes if and only if there exists an $A$ -compatible $L$ -connection on $E$ .", "We use the symbol $\\partial ^A$ to denote the covariant differential associated to the action of the Lie algebroid $A$ on $A^\\perp \\otimes E$ .", "(a) The differential Bianchi identity states that $d^\\nabla R^\\nabla :\\wedge ^3 L\\rightarrow E$ is identically zero.", "Thus, for any $a_1,a_2\\in {A}$ , $l\\in {L}$ , we have $0 =& (d^\\nabla R^\\nabla )(a_1,a_2,l) \\\\=& \\nabla _{a_1}(R^\\nabla (a_2,l))-\\nabla _{a_2}(R^\\nabla (a_1,l))+\\nabla _{l}(R^\\nabla (a_1,a_2)) \\\\& -R^\\nabla ({a_1}{a_2},l)+R^\\nabla ({a_1}{l},a_2)-R^\\nabla ({a_2}{l},a_1) \\\\=& \\nabla _{a_1}\\big (_E(a_2;(l))\\big )-\\nabla _{a_2}\\big (_E(a_1;(l))\\big ) \\\\& -_E({a_1}{a_2};(l))-_E(a_2;\\nabla _{a_1}(l))+_E(a_1;\\nabla _{a_2}(l)) \\\\=& \\Big (\\nabla _{a_1}\\big (_E(a_2;(l))\\big )-_E\\big (a_2;\\nabla _{a_1}(l)\\big )\\Big ) \\\\& -\\Big (\\nabla _{a_2}\\big (_E(a_1;(l))\\big )-_E\\big (a_1;\\nabla _{a_2}(l)\\big )\\Big )-_E\\big ({a_1}{a_2};(l)\\big ) \\\\=& \\big (\\partial ^A_E\\big )(a_1,a_2;(l)).$ Therefore $\\partial ^A_E=0$ .", "(b) By Lemma REF , if $\\nabla ^1$ and $\\nabla ^2$ are two $L$ -connections that extend the $A$ -action, then $\\nabla ^1_l-\\nabla ^2_l=\\phi ( l )$ for some $\\phi \\in {A^\\perp \\otimes E}$ , and $& R^{\\nabla _1}_E(a;(l))\\cdot e-R^{\\nabla _2}_E(a;(l))\\cdot e \\\\&\\qquad = \\nabla _a(\\nabla ^1_l-\\nabla ^2_l)e-(\\nabla ^1_l-\\nabla ^2_l)\\nabla _a e-(\\nabla ^1_{{a}{l}}-\\nabla ^2_{{a}{l}})e \\\\&\\qquad = \\nabla _a\\big (\\phi (l)\\cdot e\\big )-\\phi (l)\\cdot (\\nabla _a e)-\\phi ({a}{l})e \\\\&\\qquad = \\big (\\partial ^A\\phi \\big )(a;l)\\cdot e.$ So $R^{\\nabla _1}_E-R^{\\nabla _2}_E=\\partial ^A\\phi $ .", "(c) It is clear that $_E$ vanishes if and only if $\\nabla $ is $A$ -compatible.", "Now, if $_E=\\partial ^A\\phi $ for some $\\phi \\in {A^\\perp \\otimes E}$ , set $\\nabla ^{\\prime }=\\nabla -\\phi $ .", "Then $R^{\\nabla ^{\\prime }}_E=0$ , which implies that $\\nabla ^{\\prime }$ is $A$ -compatible.", "When the Lie pair $(L, A)$ is a matched pair of Lie algebroids, i.e.", "$L=A\\bowtie B$ , our definition of Atiyah class is a special case of the Atiyah class of a dDG algebra developed by Calaque and Van den Bergh [12].", "Hence in the matched pair case, Theorem REF (a)-(b) is a consequence of Lemma 8.2.4 in [12].", "Let $X$ be a complex manifold and $E$ a holomorphic vector bundle over $X$ .", "Then $A=T_X^{0,1}$ and $B=T_X^{1, 0}$ form a matched pair of Lie algebroids and $L=A\\bowtie B$ is isomorphic to $T_X\\otimes $ .", "Moreover $E$ is an $A$ -module [29].", "It is simple to see that holomorphic $T_X$ -connections on $E$ are equivalent to $L$ -connections on $E$ compatible with the $A$ -action (as well as to $A$ -compatible $B$ -connections on $E$ — see Remark REF ).", "In this case, the Atiyah cocycle is exactly the Dolbeault 1-cocycle $\\mathcal {R}$ defined by Equation (REF ).", "A holomorphic Lie algebroid $K$ over a complex manifold $X$ yields a matched pair of complex Lie algebroids $(T_X,K)$ [29].", "The Atiyah class of the $T_X$ -module $K$ is the Atiyah class for $K$ studied extensively by Calaque and Van den Bergh in [12].", "In [35], Molino introduced an Atiyah class for connections “transversal to a foliation,” which measures the obstruction to their “projectability.” Although not phrased in the language of Lie algebroids, his construction is a special case of ours.", "Here $L$ is the tangent bundle $T_M$ , $A$ is the tangent bundle to a foliation $\\mathcal {F}$ of $M$ , and the $A$ -module $E$ is a vector bundle on $M$ foliated over $\\mathcal {F}$ .", "A transversal connection is an $L$ -connection on $E$ which extends the $A$ -action.", "It is said to be projectable precisely if it is $A$ -compatible, i.e.", "if it is preserved by parallel transport along any path tangent to $\\mathcal {F}$ .", "Let $\\mathfrak {g}$ be a Lie subalgebra of a Lie algebra $\\mathfrak {d}$ .", "Given an $\\mathfrak {g}$ -module $E$ (i.e.", "a Lie algebra morphism $A:\\mathfrak {g}\\rightarrow E$ ), and a $\\mathfrak {d}$ -connection on $E$ extending it (i.e.", "a linear map $L:\\mathfrak {d}\\rightarrow E$ whose restriction to $\\mathfrak {g}$ is $A$ ), the Atiyah class is the element in the Chevalley-Eilenberg cohomology group $H^1(\\mathfrak {g};\\mathfrak {g}^\\perp \\otimes (E))$ determined by $\\partial ^{\\mathfrak {g}} L$ .", "(The symbol $\\partial ^{\\mathfrak {g}}$ denotes the Chevalley-Eilenberg coboundary of $\\mathfrak {d}^*\\otimes (E)$ -valued $\\mathfrak {g}$ -cochains.)", "Here $L$ is considered as an element in $\\mathfrak {d}^*\\otimes (E)$ , which is, in general, not in $\\mathfrak {g}^\\perp \\otimes (E)$ .", "Hence, in general, $\\partial ^{\\mathfrak {g}} L$ does not vanish in $H^1(\\mathfrak {g};\\mathfrak {g}^\\perp \\otimes (E))$ .", "The following example is due to Calaque-Căldăraru-Tu [10].", "Consider the Lie algebra $\\mathfrak {sl}_2()$ and its standard basis $ h= \\begin{pmatrix} 1 & 0 \\\\ 0 & -1 \\end{pmatrix} ,\\qquad e= \\begin{pmatrix} 0 & 1 \\\\ 0 & 0 \\end{pmatrix} ,\\qquad f= \\begin{pmatrix} 0 & 0 \\\\ 1 & 0 \\end{pmatrix} .$ We have $ [e,f]=h, \\qquad [h,e]=2e, \\qquad [h,f]=-2f .$ Together, the matrices $h$ and $e$ generate the Lie subalgebra $\\mathfrak {g}$ of $2\\times 2$ traceless upper triangular matrices.", "We identify the quotient $\\mathfrak {sl}_2()/\\mathfrak {g}$ to the nilpotent Lie subalgebra $\\mathfrak {n}$ generated by $f$ .", "Note that $\\mathfrak {g}$ and $\\mathfrak {n}$ form a matched pair of Lie algebras with sum $\\mathfrak {g}\\oplus \\mathfrak {n}=\\mathfrak {sl}_2()$ .", "The bilinear map $\\theta :\\mathfrak {n}\\otimes \\mathfrak {n}\\rightarrow \\mathfrak {n}$ defined by $\\theta (f,f)=f$ is a generator of the one-dimensional $\\mathfrak {g}$ -module $\\mathfrak {g}^\\perp \\otimes (\\mathfrak {n})(\\mathfrak {n}\\otimes \\mathfrak {n},\\mathfrak {n})$ .", "The action of $\\mathfrak {g}$ on $(\\mathfrak {n}\\otimes \\mathfrak {n},\\mathfrak {n})$ is given by $h\\cdot \\theta =2\\theta $ and $e\\cdot \\theta =0$ .", "One checks that the degree 1 cohomology $H^1(\\mathfrak {g};\\mathfrak {g}^\\perp \\otimes (\\mathfrak {n}))$ is a one-dimensional vector space generated by the Atiyah class $\\alpha _{\\mathfrak {n}}$ of the $\\mathfrak {g}$ -module $\\mathfrak {n}$ ." ], [ "Functoriality", "Let $M$ and $N$ be smooth manifolds, $f:N\\rightarrow M$ be a smooth map, $A$ be a Lie algebroid over $M$ with anchor $\\rho :A\\rightarrow TM$ , and $E$ be a smooth vector bundle over $M$ .", "Let $f^* E$ denote the pullback of $E$ through $f$ , i.e.", "the fibered product of $N$ and $E$ over $M$ : $ { f^*E [d] [r] & E [d] \\\\ N [r]_{f} & M. } $ If the anchor $\\rho $ and the differential of $f$ are transversal (i.e.", "$f_*(TN)+\\rho (A)=TM|_N$ ), we can consider the fibered product $f^\\star A$ of $TN$ and $A$ over $TM$ : $ { f^\\star A [d] [r] & A [d]^{\\rho } \\\\ TN [r]_{f_*} & TM. }", "$ Note that $\\rho $ and $f_*$ are automatically transversal when $f$ is a surjective submersion or when the Lie algebroid $A$ is transitive.", "It is clear that $f^\\star A $ is a vector bundle over $N$ .", "However, note that $f^\\star A \\ne f^* A$ .", "The fiber of $f^\\star A$ over a point $n\\in N$ is $ (f^\\star A)_n=\\left\\lbrace (x,a)\\in T_nN\\oplus A_{f(n)}| f_*(x)=\\rho (a) \\right\\rbrace .$ The Lie algebroid structure on $A$ induces a Lie algebroid structure on $f^\\star A\\rightarrow N$ ; its anchor is the projection $f^\\star A\\rightarrow TN$ and its bracket is given by $ {(x_1,a_1)}{(x_2,a_2)}= ({x_1}{x_2},{a_1}{a_2}) ,$ for any $x_1,x_2\\in (N)$ and $a_1,a_2\\in {A}$ such that $f_*(x_1)=\\rho (a_1)$ and $f_*(x_2)=\\rho (a_2)$ (see [19] for details).", "Let $A$ be a Lie algebroid over $M$ and $f:N\\rightarrow M$ a smooth map whose differential $f_*:TN\\rightarrow TM$ is transversal to the anchor of $A$ .", "Then If $E$ is a module over $A$ , then $f^*E$ is a module over $f^\\star A$ .", "The map $f$ induces a natural homomorphism $ f^\\dagger :H^\\bullet \\big (A;E\\big )\\rightarrow H^\\bullet \\big (f^\\star A;f^*E\\big ) .$ The first assertion is easily proved if one thinks of Lie algebroid modules in terms of horizontal lifting.", "The second assertion follows from a direct verification.", "The following proposition is immediate.", "If $(L,A)$ is a Lie pair over $M$ , and $f:N\\rightarrow M$ a smooth map whose differential $f_*:TN\\rightarrow TM$ is transversal to the anchor of $A$ , then $(f^\\star L,f^\\star A)$ is a Lie pair over $N$ .", "Given a Lie pair $(L,A)$ over a smooth manifold $M$ and a smooth map $f:N\\rightarrow M$ whose differential $f_*:TN\\rightarrow TM$ is transversal to the anchor of $A$ (otherwise $f^\\star A$ and $f^\\star L$ could be singular), there is a canonical morphism of vector bundles $f^\\star L \\rightarrow f^* (L/A)$ over $N$ : $ f^\\star L \\ni (x_n,a_{f(n)})\\mapsto a_{f(n)}+A_{f(n)} \\in f^*\\left(\\frac{L}{A}\\right) ,$ whose kernel is exactly $f^\\star A$ .", "In other words, we have an exact sequence of vector bundles $ 0 \\rightarrow f^\\star A \\rightarrow f^\\star L \\rightarrow f^*(L/A) .$ Therefore $f^\\star L / f^\\star A$ can be seen as a vector subbundle of $f^*(L/A)$ .", "Under the hypothesis of Proposition REF , the inclusion $ I: \\frac{f^\\star L}{f^\\star A}\\rightarrow f^*\\left(\\frac{L}{A}\\right) $ intertwines the $ f^\\star A$ -module structures of $(f^\\star L)/(f^\\star A)$ and $f^* (L/A)$ .", "Dualizing it, as a consequence, we obtain the epimorphism of vector bundles $ I^\\dagger : f^*(A^\\perp )\\rightarrow (f^\\star A)^\\perp ,$ which is a morphism of $ f^\\star A$ -modules.", "Note that, when $f$ is a surjective submersion, $I$ is surjective and thus both $I$ and $I^\\dagger $ are isomorphisms of $ f^\\star A$ -modules.", "We are now ready to state the main result in this subsection.", "Let $(L,A)$ be a Lie pair over $M$ , and $f:N\\rightarrow M$ a smooth map whose differential $f_*:TN\\rightarrow TM$ is transversal to the anchor of $A$ .", "Assume that $E$ is an $A$ -module.", "Then the composition of homomorphisms $ H^1\\big (A;A^\\perp \\otimes E\\big ){f^\\dagger }H^1\\big (f^\\star A;f^*(A^\\perp \\otimes E)\\big ){I^\\dagger }H^1\\big (f^\\star A;(f^\\star A)^\\perp \\otimes (f^* E)\\big ) $ maps the Atiyah class of $E$ relative to the Lie pair $(L,A)$ onto the Atiyah class of $f^* E$ relative to the Lie pair $(f^\\star L,f^\\star A)$ : $ (I^\\dagger f^\\dagger ) (\\alpha _E)= \\alpha _{f^* E}.$" ], [ "Scalar Atiyah classes and Todd class", "Let $(L,A)$ be a Lie pair.", "We define the scalar Atiyah classes [3] of an $A$ -module $E$ by $ c_k (E):=\\frac{1}{k!}", "\\left(\\frac{i}{2\\pi }\\right)^k \\big (\\alpha _E^k\\big )\\in H^k(A;\\wedge ^k A^\\perp ) .$ Here $\\alpha _E^k$ denotes the image of $\\alpha _E\\otimes \\cdots \\otimes \\alpha _E$ under the natural map $ H^1(A;A^\\perp \\otimes E) \\times \\cdots \\times H^1(A;A^\\perp \\otimes E)\\rightarrow H^k(A;\\wedge ^k A^\\perp \\otimes E) $ induced by the composition in $E$ and the wedge product in $\\wedge ^\\bullet A^\\perp $ .", "If $E$ is a holomorphic vector bundle over a compact Kähler manifold $X$ , the natural inclusion of $H^k(X,\\Omega ^k)$ into $H^{2k}(X,)$ maps the scalar Atiyah classes of $E$ relative to the Lie pair $(L=T_X\\otimes ,A=T_X)$ to the Chern classes of $E$ .", "The Todd class of the $A$ -module $E$ relative to the Lie pair $(L,A)$ is the cohomology class $ (E) =\\det \\left(\\frac{\\alpha _E}{1-e^{-\\alpha _E}}\\right)\\in H^{\\bullet }(A;\\wedge ^\\bullet A^\\perp ) .$ The following propositions can be verified directly.", "Let $(L,A)$ be a Lie pair and let $E_1$ , $E_2$ be $A$ -modules.", "Then $(E_1\\oplus E_2)=E_1 \\cdot E_2$ Under the hypothesis of Theorem REF , we have $c_k(f^*E) = (I^\\dagger f^\\dagger ) (c_k E)\\in H^k\\big (f^\\star A; \\wedge ^k (f^\\star A)^\\perp \\big ) \\\\(f^*E)=(I^\\dagger f^\\dagger )(E)\\in H^\\bullet \\big (f^\\star A; \\wedge ^\\bullet (f^\\star A)^\\perp \\big )$" ], [ "The jet bundle ${}{E}$", "Let $M$ be a smooth manifold, let $L\\rightarrow M$ be a Lie algebroid, and let $E{\\pi }M$ be a vector bundle.", "An $L$ -jet (of order 1) on $E$ (at $e_x\\in E$ ) is a linear map $L_{\\pi (e_x)}{\\phi }T_{e_x}E$ such that the diagram $ { L_{\\pi ({e_x})} [rr]^\\phi [rd]_\\rho &&T_{e_x} E [ld]^{\\pi _{*{e_x}}} \\\\ & T_{\\pi ({e_x})}M & } $ commutes.", "The jet space ${L}{E}$ is the manifold whose points are $L$ -jets on $E$ .", "It is a vector bundle over $X$ : the projection ${L}{E}\\rightarrow M$ maps $L_{\\pi ({e_x})}{\\phi } T_{e_x} E$ to $\\pi ({e_x})$ .", "It fits into the short exact sequence of vector bundles over $M$ : ${ 0 [r] & L^*\\otimes E [r]^{\\hat{f}} & {L}{E}[r]^{\\hat{g}} & E[r] & 0 .", "}$ The surjection $\\hat{g}$ maps $L_{\\pi ({e_x})}{\\phi } T_{e_x} E$ to ${e_x}$ while the injection $\\hat{f}$ maps $L_x{\\psi } E_x$ to $L_x{\\rho \\oplus \\psi }T_x M\\oplus E_xT_{0_x}E$ .", "A splitting $s:E\\rightarrow {L}{E}$ of the short exact sequence of vector bundles (REF ) determines a (linear) $L$ -connection on $E$ .", "The converse is also true.", "In general, there is no canonical choice of splitting for (REF ).", "However, the induced short exact sequence ${ 0 [r] & {L^*\\otimes E} [r]^{\\hat{f}_\\sharp } &{{L}{E}} [r]^{\\hat{g}_\\sharp } & {E} [r] & 0 }$ at the level of spaces of smooth sections splits canonically: if $e$ is a section of $E$ , then ${e}:=e_*\\rho $ is a section of ${L}{E}$ such that $\\hat{g}_\\sharp ({e})=e$ .", "We note that the covariant differential $d^\\nabla :{E}\\rightarrow {L^*\\otimes E}$ associated to a splitting $s:E\\rightarrow {L}{E}$ of the short exact sequence (REF ) is given by $\\hat{f}_\\sharp (d^\\nabla e)={e}-s_\\sharp (e) ,\\quad \\forall e\\in {E}.$ Now assume $A$ is a Lie subalgebroid of $L$ and $E$ is an $A$ -module.", "The symbol $h$ will denote the horizontal lifting associated to the $A$ -action on $E$ .", "An $h$ -extending $L$ -jet (of order 1) on $E$ is a linear map $L_{\\pi ({e_x})}{\\phi } T_{e_x} E$ such that the diagram $ { & A_{\\pi ({e_x})} [dl] [dr]^{h(-,{e_x})} & \\\\L_{\\pi ({e_x})} [rr]^\\phi [rd]_\\rho & & T_{e_x} E [ld]^{\\pi _{*{e_x}}} \\\\& T_{\\pi ({e_x})}M & } $ commutes.", "The jet space ${}{E}$ is the manifold whose points are $h$ -extending $L$ -jets on $E$ .", "It is a vector bundle over $M$ : the projection ${{}}{E}\\rightarrow M$ maps $L_{\\pi ({e_x})}{\\phi } T_{e_x} E$ to $\\pi ({e_x})$ .", "When $E$ is a holomorphic vector bundle over a complex manifold $X$ , $A=T_X^{0,1}$ and $L=T_X\\otimes $ , the jet bundle ${{}}{E}$ is simply the bundle of jets (of order 1) of holomorphic sections of $E$ .", "Consider the surjective morphism of vector bundles $\\breve{g}:{{}}{E}\\rightarrow E$ , which maps $L_{\\pi ({e_x})}{\\phi } T_{e_x} E$ to ${e_x}$ .", "Since $T_{0_x}E$ is canonically isomorphic to $T_x M\\oplus E_x$ , the kernel of $\\breve{g}$ can be identified naturally to the subbundle $K$ of $L^*\\otimes E \\rightarrow M$ consisting of all linear maps $L_x{\\psi }E_x$ which satisfy $ h(a_x,0_x)=\\rho (a_x)+\\psi (a_x), \\quad \\forall x\\in M, a_x\\in A_x .$ Since the $A$ -connection $h$ on $E$ is linear, $h(a_x,0_x)$ must be the image of $\\rho (a_x)$ under the differential of the zero section $M{0}E$ .", "Therefore, a linear map $L_x{\\psi }E_x$ is an element of $K$ if and only if $\\psi (a_x)=0$ for all $a_x\\in A$ , so that $K=A^\\perp \\otimes E$ .", "Hence we obtain the short exact sequence of vector bundles ${ 0 [r] & A^\\perp \\otimes E [r]^{\\breve{f}}& {{}}{E} [r]^{\\breve{g}} & E [r] & 0 ,}$ where the injection $\\breve{f}$ maps $L_x{\\psi } E_x$ to the jet $ L_x\\rightarrow T_{0_x}ET_x M\\oplus E_x \\qquad l_x\\mapsto \\rho (l_x)+\\psi (l_x) .$ In general, there is no canonical choice of splitting for (REF ).", "A splitting $s:E\\rightarrow {{}}{E}$ of the short exact sequence of vector bundles (REF ) determines a (linear) $L$ -connection on $E$ extending the $A$ -action $h$ .", "The converse is also true.", "Obviously, we have the commutative diagram with exact rows $ {0 [r] & A^\\perp \\otimes E [r]^{\\breve{f}} [d] & {L/A}{E} [r]^{\\breve{g}} [d] &E [r] [d]^{} & 0 \\\\0 [r] & L^*\\otimes E [r]^{\\hat{f}} & {L}{E} [r]^{\\hat{g}} & E [r] & 0 ,} $ all of whose vertical arrows denote inclusions." ], [ "An equivalent description of the jet bundle", "An $L$ -jet (of order 1) on $E$ extending the $A$ -action $\\nabla $ is a pair $(D_x,e_x)$ consisting of a linear map $D_x:{E^*}\\rightarrow L_x^*$ and a point $e_x$ in the fiber of $E$ over $x\\in M$ , satisfying ${D_x(\\varepsilon )}{a_x}={\\nabla _{a_x}\\varepsilon }{e_x}\\quad (\\text{or equivalently}\\quad D_x(\\varepsilon )={d^\\nabla \\varepsilon }{e_x}); \\\\D_x(f\\varepsilon )=f(x)\\cdot D_x(\\varepsilon )+{\\varepsilon _x}{e_x}\\cdot \\rho ^*(df), $ for all $a_x\\in A_x$ , $\\varepsilon \\in {E^*}$ , and $f\\in {M}$ .", "Given such a pair $(D_x,e_x)$ , each $l_x\\in L_x$ determines uniquely a tangent vector $\\tau _x\\in T_{e_x}E$ through the relations $ \\tau _x(\\pi ^* f)=\\rho (l_x)f={\\rho ^*(df)}{l_x} \\qquad \\text{and} \\qquad \\tau _x (f_{\\varepsilon })={D_x(\\varepsilon )}{l_x} ,$ where $\\varepsilon \\in {E^*}$ , $f\\in {M}$ , and $f_{\\varepsilon }\\in {E}$ is the fiberwise linear function associated to $\\varepsilon $ : $f_{\\varepsilon }(e_x)={\\varepsilon _x}{e_x} .$ Let $\\phi _D:L_x\\rightarrow T_{e_x}E$ be the map $l_x\\mapsto \\tau _x$ .", "Clearly, $\\phi _D$ is linear and satisfies $\\pi _*\\circ \\phi _D=\\rho $ .", "Moreover, $\\phi _D$ is an extension of the $A$ -action since $\\big (\\phi _D(a_x)\\big )(f_\\varepsilon )={D_x(\\varepsilon )}{a_x}={\\nabla _{a_x}\\varepsilon }{e_x}=h(a_x,e_x)(f_\\varepsilon ).$ Here we have made use of Equation (REF ).", "Hence $\\phi _D\\in \\big ({{}}E\\big )_x$ and $\\breve{g}(\\phi _D)=e_x$ .", "Conversely, given an element $\\phi :L_x\\rightarrow T_{e_x}E$ of $\\big ({{}}E\\big )_x$ that projects to $e_x$ under $\\breve{g}$ , we can define a linear map $D^\\phi _x:{E^*}\\rightarrow L_x$ by the relation $ {D^\\phi _x(\\varepsilon )}{l_x}=\\big (\\phi (l_x)\\big )(f_{\\varepsilon }) .$ It is straightforward to check that $(D^\\phi _x,e_x)$ satisfies (REF ) and ().", "The surjection $\\breve{g}:{{}}{E}\\rightarrow E$ in (REF ) maps the 1-jet $(D_x,e_x)$ to $e_x$ .", "The injection $\\breve{f}$ in (REF ) maps $\\psi \\in (A^\\perp \\otimes E)_x$ to $(\\psi ^\\dagger _x,0_x)\\in \\big ({{}}{E}\\big )_x$ , where $0_x$ is the zero vector of $E_x$ and $\\psi ^\\dagger _x:{E^*}\\rightarrow L_x^*$ is the linear map defined by $ {\\psi ^\\dagger _x(\\varepsilon )}{l_x}={\\varepsilon _x}{\\psi (l_x)},\\quad \\forall l_x\\in L_x,\\varepsilon \\in {E^*} .$ Here $\\psi $ is considered as a linear map $L_x\\rightarrow E_x$ whose kernel contains $A_x$ ." ], [ "The jet bundle is an $A$ -module", "The jet bundle ${{}}E$ can be naturally endowed with an $A$ -action.", "In the language of Proposition REF , a section of ${{}}E\\rightarrow M$ consists of a section $e$ of $E\\rightarrow M$ and an $$ -linear map $D:{E^*}\\rightarrow {L^*}$ satisfying ${D(\\varepsilon )}{a}={\\nabla _a\\varepsilon }{e} ;\\\\D(f\\varepsilon )=f\\cdot D(\\varepsilon )+{\\varepsilon }{e}\\cdot \\rho ^*(df),$ for all $f\\in {M}$ , $\\varepsilon \\in {E^*}$ , and $a\\in {A}$ .", "The jet bundle ${{}}E$ is a module over $A$ ; the covariant derivative $ {A}\\times {{{}}E}\\rightarrow {{{}}E} $ is given by $\\nabla _a (D,e)=(\\nabla _a D,\\nabla _a e)$ , where the $$ -linear map $\\nabla _a D:{E^*}\\rightarrow {L^*}$ is defined as follows: ${\\big (\\nabla _a D\\big )(\\varepsilon )}{l}=\\rho (a){D(\\varepsilon )}{l}-{D(\\nabla _a\\varepsilon )}{l}-{D(\\varepsilon )}{{a}{l}}.$ Diagram (REF ) is a short exact sequence of $A$ -modules.", "(a) Checking that (REF ) determines a connection is straightforward.", "The flatness of the $A$ -connection on ${{}}E$ is a consequence of the flatness of the $A$ -connection on $E$ .", "(b) By definition, $\\breve{g}$ is a morphism of $A$ -modules.", "Let us check that $\\breve{f}$ is also a morphism of $A$ -modules.", "For any $\\psi \\in {A^\\perp \\otimes E}$ , we have ${(\\nabla _a \\psi ^\\dagger )(\\varepsilon )}{l}= a){\\psi ^\\dagger (\\varepsilon )}{l}-{\\psi ^\\dagger (\\nabla _a\\varepsilon )}{l}-{\\psi ^\\dagger (\\varepsilon )}{{a}{l}} \\\\= a){\\psi (l)}{\\varepsilon }-{\\psi (l)}{\\nabla _a\\varepsilon }-{\\psi ({a}{l})}{\\varepsilon }= {(\\nabla _a\\psi )(l)}{\\varepsilon }= {(\\nabla _a\\psi )^\\dagger (\\varepsilon )}{l}.$" ], [ "Alternative description of the $A$ -action on {{formula:2978ebda-e3c0-4c8d-a604-c3d5a8badcb0}}", "In this section, $B$ will denote the quotient $L/A$ of the Lie pair $(L,A)$ .", "The proof of the following lemma is a tedious computation, which we omit.", "The splitting ${}:{E}\\rightarrow {{L}{E}}$ of the short exact sequence (REF ) is not ${M}$ -linear.", "For every $e\\in {E}$ and $f\\in {M}$ , we have $ {(f\\cdot e)}-f\\cdot {e}=\\hat{f}_\\sharp \\big (\\rho ^*(df)\\otimes e\\big ) .$ In general, ${e}$ need not be a section of ${L/A}{E}$ .", "Nevertheless, fixing a splitting of the short exact sequence of vector bundles $ {0 [r] & A [r]^i & L [r]^q & B [r] & 0 } ,$ i.e.", "a pair of maps $j:B\\rightarrow L$ and $p:L\\rightarrow A$ such that $qj=_B$ , $pi=_A$ and $ip+jq=_L$ : $ {0 @<.5ex>[r] & A @<.5ex>[r]^i @<.5ex>[l] & L @<.5ex>[r]^q @<.5ex>[l]^p & B@<.5ex>[r] @<.5ex>[l]^j & 0 @<.5ex>[l] } ,$ naturally determines a splitting ${}:{E}\\rightarrow {{L/A}{E}}$ of the short exact sequence of spaces of smooth sections ${ 0 [r] & {A^\\perp \\otimes E} [r]^{\\breve{f}_\\sharp } &{{L/A}{E}} [r]^{\\breve{g}_\\sharp } & {E} [r] & 0 }$ induced by (REF ).", "The image of $x\\in M$ under the section ${e}$ of ${L/A}{E}$ associated to a section $e$ of $E$ by the splitting ${}$ is the 1-jet $ L_x\\ni l_x {({e})_x}h(p(l_x),e_x)+e_{*x}\\big (\\rho jq(l_x)\\big ) \\in T_{e_x}E .$ It is not difficult to see that $\\breve{g}_\\sharp ({e})=e$ .", "The proof of the following lemma is a tedious computation, which we omit.", "The splitting ${}:{E}\\rightarrow {{L/A}{E}}$ is not ${M}$ -linear.", "For every $e\\in {E}$ and $f\\in {M}$ , we have $ {(f\\cdot e)}-f\\cdot {e}=\\breve{f}_\\sharp \\Big (\\big (l\\mapsto \\rho (jq(l))f\\big )\\otimes e\\Big ) .$ Since both $E$ and $A^\\perp $ are modules over $A$ , so is $A^\\perp \\otimes E$ : $ {\\nabla _a (\\lambda \\otimes e)}{l}=\\rho (a)\\big (\\lambda (l)\\big )\\cdot e-\\lambda ({a}{l})\\cdot e+\\lambda (l)\\cdot \\nabla _a e ,$ where $a\\in {A}$ , $\\lambda \\in {A^\\perp }$ , $e\\in {E}$ , and $l\\in {L}$ .", "If ${}{E}$ was an $A$ -module and $\\breve{f}:A^\\perp \\otimes E\\rightarrow {}{E}$ was a morphism of $A$ -modules, then tedious computations would yield $ {(\\nabla _{f\\cdot a}e)-\\nabla _{f\\cdot a}({e})}= f\\cdot \\big ({(\\nabla _a e)}-\\nabla _a({e})\\big )+ \\breve{f}_\\sharp \\Big (\\big (l\\mapsto \\rho (jq(l))f\\big )\\otimes \\nabla _a e\\Big ) $ and $ {\\big (\\nabla _a (f\\cdot e)\\big )}-\\nabla _{a}\\big ({(f\\cdot e)}\\big )= f\\cdot \\big ({(\\nabla _a e)}-\\nabla _a({e})\\big )+ \\breve{f}_\\sharp \\Big (\\big (l\\mapsto \\rho (ip{jq(l)}{a})f\\big )\\otimes e\\Big ) ,$ for any $a\\in {A}$ , $e\\in {E}$ , and $f\\in {M}$ .", "Therefore, one cannot expect to define an $A$ -action on the jet bundle ${}{E}$ simply by setting $\\nabla _a\\big (\\breve{f}_\\sharp (\\lambda \\otimes e)\\big )=\\breve{f}_\\sharp \\big (\\nabla _a(\\lambda \\otimes e)\\big ) ;\\\\\\nabla _a({e})={(\\nabla _a e)}.$ The proof of the following Lemma is a tedious computation, which we omit.", "Given $a\\in {A}$ and $e\\in {E}$ , define $\\Theta (a,e)\\in {A^\\perp \\otimes E}$ by the relation $ {\\Theta (a,e)}{l}=\\nabla _{ip{jq(l)}{a}}e,\\qquad \\forall l\\in {L} .$ Then, for any $f\\in {M}$ , $\\Theta (f\\cdot a,e)-f\\cdot \\Theta (a,e)=\\big (l\\mapsto \\rho (jq(l))f\\big )\\otimes \\nabla _a e ;\\\\\\Theta (a,f\\cdot e)-f\\cdot \\Theta (a,e)=\\big (l\\mapsto \\rho (ip{jq(l)}{a})f\\big )\\otimes e.$ This suggests the following proposition.", "The jet bundle ${L/A}{E}\\rightarrow M$ is an $A$ -module: the flat $A$ -connection on ${L/A}{E}$ is given by $\\nabla _a \\big (\\breve{f}_\\sharp (\\lambda \\otimes e)\\big )=\\breve{f}_\\sharp \\big (\\nabla _a (\\lambda \\otimes e)\\big ) \\\\\\nabla _a ({e})={(\\nabla _a e)}-\\breve{f}_\\sharp \\big (\\Theta (a,e)\\big ) , $ for all $a\\in {A}$ , $\\lambda \\in {A^\\perp }$ , and $e\\in {E}$ .", "Here $\\Theta (a,e)\\in {A^\\perp \\otimes E}$ is defined by $ {\\Theta (a,e)}{l}=\\nabla _{p[jq(l),a]}e,\\qquad \\forall l\\in {L} .$ Diagram (REF ) is a short exact sequence of $A$ -modules.", "(a) It follows from Lemmas REF and REF that what we have defined is indeed a covariant derivative.", "Flatness follows from the Jacobi identity in ${L}$ and the flatness of the $A$ -connections on $E$ and $A^\\perp \\otimes E$ .", "(b) It suffices to check that $\\breve{g}$ is a morphism of $A$ -modules.", "We have $ \\breve{g}_\\sharp \\big (\\nabla _a ({e})\\big )=\\breve{g}_\\sharp \\big ({(\\nabla _a e)}\\big ) - \\breve{g}_\\sharp \\breve{f}_\\sharp \\big (\\Theta (a,e)\\big )= \\nabla _a e = \\nabla _a \\big ( \\breve{g}_\\sharp ({e})\\big ) .$ For a matched pair $L=A\\bowtie B$ , observe that $\\Theta (a,e)\\in {B^*\\otimes E}$ is given by the simple formula ${\\Theta (a,e)}{b}=\\nabla _{_b a}e$ .", "The $A$ -actions on ${}E$ defined in Propositions REF and REF are identical.", "The $$ -linear map $D^{{e}}:{E^*}\\rightarrow {L^*}$ determined by the section ${e}$ of ${}E\\rightarrow M$ (as explained in the proof of Proposition REF ) satisfies ${D^{{e}}(\\varepsilon )}{a}={\\nabla _a\\varepsilon }{e} ;\\\\{D^{{e}}(\\varepsilon )}{j(b)}=\\Big ({e}\\big (j(b)\\big )\\Big )(f_{\\varepsilon })=\\Big (e_*\\rho \\big (j(b)\\big )\\Big )(f_{\\varepsilon })=\\rho \\big (j(b)\\big ){\\varepsilon }{e},$ for all $e\\in {E}$ , $\\varepsilon \\in {E^*}$ , $a\\in {A}$ , and $b\\in {B}$ .", "The image of $D^{{e}}$ under the action of $a\\in {A}$ is the $$ -linear map $\\nabla _a D^{{e}}:{E^*}\\rightarrow {L^*}$ defined by (REF ).", "We have $& {\\big (\\nabla _{a^{\\prime }} D^{{e}}\\big )(\\varepsilon )}{a} \\\\&\\qquad = \\rho (a^{\\prime }){D^{{e}}(\\varepsilon )}{a}-{D^{{e}}(\\nabla _{a^{\\prime }}\\varepsilon )}{a}-{D^{{e}}(\\varepsilon )}{{a^{\\prime }}{a}} \\\\&\\qquad = \\rho (a^{\\prime }){\\nabla _a \\varepsilon }{e}-{\\nabla _a\\nabla _{a^{\\prime }}\\varepsilon }{e}-{\\nabla _{{a^{\\prime }}{a}}\\varepsilon }{e} \\\\&\\qquad = \\rho (a^{\\prime }){\\nabla _a \\varepsilon }{e}-{\\nabla _{a^{\\prime }}\\nabla _a\\varepsilon }{e} \\\\&\\qquad = {\\nabla _a\\varepsilon }{\\nabla _{a^{\\prime }}e} \\\\&\\qquad = {D^{{(\\nabla _{a^{\\prime }}e)}}(\\varepsilon )}{a}$ and $& {\\big (\\nabla _{a^{\\prime }} D^{{e}}\\big )(\\varepsilon )}{j(b)} \\\\&\\qquad = \\rho (a^{\\prime }){D^{{e}}(\\varepsilon )}{j(b)}-{D^{{e}}(\\nabla _{a^{\\prime }}\\varepsilon )}{j(b)}-{D^{{e}}(\\varepsilon )}{{a^{\\prime }}{j(b)}} \\\\&\\qquad = \\rho (a^{\\prime })\\rho \\big (j(b)\\big ){\\varepsilon }{e} - \\rho \\big (j(b)\\big ){\\nabla _{a^{\\prime }}\\varepsilon }{e} \\\\&\\qquad \\quad -{D^{{e}}(\\varepsilon )}{-p{j(b)}{a^{\\prime }}+j(\\nabla _{a^{\\prime }}b)} \\\\&\\qquad = \\rho (a^{\\prime })\\rho \\big (j(b)\\big ){\\varepsilon }{e} - \\rho \\big (j(b)\\big )\\rho (a^{\\prime }){\\varepsilon }{e}+\\rho \\big (j(b)\\big ){\\varepsilon }{\\nabla _{a^{\\prime }}e} \\\\&\\qquad \\quad +{\\nabla _{p{j(b)}{a^{\\prime }}}\\varepsilon }{e}-\\rho \\big (j(\\nabla _{a^{\\prime }}b)\\big ){\\varepsilon }{e} \\\\&\\qquad = \\rho ({a^{\\prime }}{j(b)}){\\varepsilon }{e}+\\rho \\big (j(b)\\big ){\\varepsilon }{\\nabla _{a^{\\prime }}e}+\\rho (p{j(b)}{a^{\\prime }}){\\varepsilon }{e} \\\\&\\qquad \\quad -{\\varepsilon }{\\nabla _{p{j(b)}{a^{\\prime }}}e}-\\rho \\big (j(\\nabla _{a^{\\prime }}b)\\big ){\\varepsilon }{e} \\\\&\\qquad = \\rho \\big (j(b)\\big ){\\varepsilon }{\\nabla _{a^{\\prime }}e}-{\\varepsilon }{\\nabla _{p{j(b)}{a^{\\prime }}}e} \\\\&\\qquad = {D^{{(\\nabla _{a^{\\prime }}e)}}(\\varepsilon )}{j(b)}-{\\big (\\Theta (a^{\\prime },e)\\big )^\\dagger (\\varepsilon )}{j(b)}.$ Therefore, we obtain $ \\nabla _{a^{\\prime }}D^{{e}}=D^{{(\\nabla _{a^{\\prime }}e)}}-\\big (\\Theta (a^{\\prime },e)\\big )^\\dagger ,$ which is equivalent to Equation ()." ], [ "The abelian category ${A}$", "It is classical a result that the space ${A}$ of smooth sections of a Lie algebroid $A$ over a smooth manifold $M$ is a Lie-Rinehart algebra over the commutative ring $C^\\infty (M)$  [20], [38].", "We denote the (abelian) category of modules over this Lie-Rinehart algebra by the symbol ${A}$ .", "Alternatively, the objects of ${A}$ can be seen as left modules over the universal enveloping algebra ${A}$ of the Lie algebroid $A$ .", "The space of smooth sections of an $A$ -module, i.e.", "a vector bundle over $M$ endowed with an $A$ -action, is an object of ${A}$ .", "We note that the derived category of ${A}$ , which we denote by $D({A})$ , is a symmetric monoidal category [21].", "The interchange isomorphism $\\tau :X\\otimes Y\\rightarrow Y\\otimes X$ of a pair of objects $X$ and $Y$ of $D({A})$ is given by $\\tau (x\\otimes y)=(-1)^{{x}{y}}y\\otimes x.$" ], [ "Extension class of the jet sequence", "A short exact sequence of $A$ -modules ${0 [r] & P [r]^\\alpha & Q [r]^\\beta & R [r] & 0}$ determines an extension class in the group $\\text{Ext}^1_{A} (R, P)$ , which is naturally isomorphic to the Lie algebroid cohomology group $H^1(A;R^*\\otimes P)$ [20].", "Indeed, given a homomorphism of vector bundles $s:R\\rightarrow Q$ such that $\\beta s=_R$ , we have $ s_\\sharp (\\nabla _a r)-\\nabla _a \\big (s_\\sharp (r)\\big )\\in \\ker \\beta ,\\qquad \\forall a\\in {A}, r\\in {R} $ so that the equation $s_\\sharp (\\nabla _a r)-\\nabla _a \\big (s_\\sharp (r)\\big )=\\alpha _\\sharp \\big (\\xi _s(a)\\cdot r\\big )$ defines a vector bundle map $\\xi _s:A\\rightarrow (R,P).$ Rewriting (REF ) as $ (_{R^*}\\otimes \\alpha )\\xi _s=\\partial ^A s $ and recalling that $\\alpha :P\\rightarrow Q$ is a morphism of $A$ -modules, we immediately see that $ (_{R^*}\\otimes \\alpha )\\partial ^A\\xi _s=\\partial ^A\\big ((_{R^*}\\otimes \\alpha )\\xi _s\\big )=\\partial ^A(\\partial ^A s)=0 .$ Therefore $\\partial ^A\\xi _s=0$ , i.e.", "$\\xi _s$ is a 1-cocycle for the Lie algebroid $A$ with values in the $A$ -module $R^*\\otimes P$ .", "It follows from Equation (REF ) that the cohomology class $[\\xi _s]\\in H^1(A;R^*\\otimes P)$ of the 1-cocycle $\\xi _s$ defined by (REF ) is independent of the choice of the section $s:R\\rightarrow Q$ .", "In fact, $[\\xi _s]$ is the extension class in $^1_{A}(R, P)H^1(A;R^*\\otimes P)$ of the short exact sequence of $A$ -modules (REF ).", "Given a Lie pair $(L,A)$ and an $A$ -module $E$ , let $\\nabla $ denote the $L$ -connection on $E$ determined by a section $s:E\\rightarrow {{}}{E}$ of the short exact sequence (REF ).", "When considered as sections of $A^*\\otimes A^\\perp \\otimes E$ , the bundle maps $\\xi _s:A\\rightarrow (E,A^\\perp \\otimes E)$ and $R^\\nabla _E:A\\otimes (L/A)\\rightarrow E$ (respectively defined by (REF ) and (REF )) are one and the same.", "Define $\\breve{d}^\\nabla :{E}\\rightarrow {A^\\perp \\otimes E}$ by $\\breve{f}_\\sharp (\\breve{d}^\\nabla e)={e}-s_\\sharp (e).$ Then, for all $b\\in B$ , we have ${\\breve{d}^\\nabla e}{j(b)}=\\nabla _{j(b)}e$ .", "For all $a\\in {A}$ , and $e\\in {E}$ , we have $& \\breve{f}_\\sharp \\big (\\xi _s(a)\\cdot e\\big ) \\\\&\\qquad = s_\\sharp (\\nabla _a e)-\\nabla _a (s_\\sharp e) && \\text{by~(\\ref {threestars})}, \\\\&\\qquad = \\big ({(\\nabla _a e)}-\\breve{f}_\\sharp \\breve{d}^{\\nabla }(\\nabla _a e)\\big )-\\big (\\nabla _a({e})-\\nabla _a \\breve{f}_\\sharp (\\breve{d}^\\nabla e)\\big )&& \\text{by~(\\ref {shoe})}, \\\\&\\qquad = \\breve{f}_\\sharp \\big (\\Theta (a,e)+\\nabla _a (\\breve{d}^\\nabla e)-\\breve{d}^{\\nabla } (\\nabla _a e) \\big )&& \\text{by~(\\ref {gather:tp1}) and (\\ref {gather:tp2})}.$ Hence, for all $b\\in {B}$ , we get $& {\\xi _s(a)\\cdot e}{j(b)} \\\\&\\qquad = {\\Theta (a,e)+\\nabla _a (\\breve{d}^\\nabla e)-\\breve{d}^{\\nabla } (\\nabla _a e)}{j(b)} \\\\&\\qquad = \\nabla _{p{j(b)}{a}}e+\\nabla _a{\\breve{d}^\\nabla e}{j(b)}-{\\breve{d}^\\nabla e}{j(\\nabla _a b)} -{\\breve{d}^\\nabla (\\nabla _a e)}{j(b)} \\\\&\\qquad = \\nabla _{p{j(b)}{a}}e+\\nabla _a\\nabla _{j(b)}e-\\nabla _{j(\\nabla _a b)}e-\\nabla _{j(b)}\\nabla _a e \\\\&\\qquad = \\nabla _a\\nabla _{j(b)}e-\\nabla _{j(b)}\\nabla _a e-\\nabla _{j(\\nabla _a b)+p{a}{j(b)}}e \\\\&\\qquad = \\nabla _a\\nabla _{j(b)}e-\\nabla _{j(b)}\\nabla _a e-\\nabla _{{a}{j(b)}}e \\\\&\\qquad = R^\\nabla \\big (a,j(b)\\big )e \\\\&\\qquad = R^\\nabla _E(a;b)\\cdot e.$ This proves that $\\xi _s=R^\\nabla _E$ .", "Let $(L,A)$ be a Lie pair, and $E$ an $A$ -module.", "A section $s:E\\rightarrow {{}}{E}$ of the short exact sequence (REF ) is a morphism of $A$ -modules if and only if the $L$ -connection it induces on $E$ is compatible with the $A$ -action on $E$ .", "The short exact sequence of $A$ -modules (REF ) splits if and only if the Atiyah class $_E$ vanishes.", "Let $(L,A)$ be a Lie pair, and $E$ an $A$ -module.", "The natural isomorphism $ ^1_{A}(E,A^\\perp \\otimes E) {\\cong } H^1(A;A^\\perp \\otimes E) $ maps the extension class of the short exact sequence of $A$ -modules (REF ) to the Atiyah class of $E$ .", "We refer the reader to [12] for a related result regarding the Atiyah class of dDG algebras, which correspond to the matched pair case as pointed out in Remark REF ." ], [ " algebras", "In this section, we will explore the rich algebraic structures underlying the Atiyah class of a Lie pair.", "As we will see in the subsequent discussion, the adequate framework is the notion of algebras.", "Loday's algebras [30] are a natural generalization of Stasheff's $L_\\infty $ algebras [28], [27], where the (skew-)symmetry requirement is dropped.", "Throughout this section, we implicitly identify objects of ${A}$ to complexes in ${A}$ concentrated in degree 0.", "Moreover, we make frequent use of the shifting functor: the shift $V[k]$ of a graded vector space $V=\\bigoplus _n V_n$ is determined by the rule $\\big (V[k]\\big )_n=V_{k+n}$ .", "We defer most proofs to Section REF ." ], [ "Atiyah class as Lie algebra object", "Recall that a Lie algebra object in a monoidal category ${C}$ is an object $\\Lambda $ of ${C}$ together with a morphism $\\lambda \\in _{{C}}(\\Lambda \\otimes \\Lambda ,\\Lambda )$ such that $\\lambda \\tau =-\\lambda $ (skew-symmetry) and $ \\lambda (\\otimes \\lambda )= \\lambda (\\lambda \\otimes )+\\lambda (\\otimes \\lambda )(\\tau \\otimes )\\qquad \\text{(Jacobi identity)},$ where $\\tau :\\Lambda \\otimes \\Lambda \\rightarrow \\Lambda \\otimes \\Lambda $ is the braiding isomorphism.", "Let $(L,A)$ be a Lie pair with quotient $B=L/A$ .", "Note that $ _{D^b({A})}(B\\otimes B,{B}{1}) \\cong _{D^b({A})}({B}{-1}\\otimes {B}{-1},{B}{-1}) ,$ since a chain map $B\\otimes B\\rightarrow B[1]$ is equivalent to a map $B[-1]\\otimes B[-1]\\rightarrow B[-1]$ modulo a shift.", "Being an element of $^1_{{A}}(B,B^*\\otimes B) \\cong ^1_{{A}}(B\\otimes B,B) \\cong _{D^b({A})}(B\\otimes B,{B}{1}) \\cong \\\\_{D^b({A})}({B}{-1}\\otimes {B}{-1},{B}{-1}),$ the Atiyah class $_{B}$ of the $A$ -module $B$ defines a “Lie bracket” on ${B}{-1}$ .", "If, moreover, $E$ is an $A$ -module, its Atiyah class $ _{E}\\in ^1_{{A}}(E,B^*\\otimes E)\\cong ^1_{{A}}(B\\otimes E,E)\\cong _{D^b({A})}({B}{-1}\\otimes {E}{-1},{E}{-1}) $ defines a “representation” on ${E}{-1}$ of the “Lie algebra” ${B}{-1}$ .", "Let $(L,A)$ be a Lie pair with quotient $B=L/A$ .", "Then ${B}{-1}$ is a Lie algebra object in the derived category $D^+({A})$ .", "Moreover, if $E$ is an $A$ -module, then ${E}{-1}$ is a module object over the Lie algebra object ${B}{-1}$ in the derived category $D^+({A})$ .", "From the skew-symmetric property of a Lie algebra, it follows that the Atiyah class $\\alpha _{B}$ can indeed be considered as an element in $H^1(A, S^2 B^*\\otimes B)$ , or more precisely, in the image of the map $H^1(A, B^*\\otimes B) \\rightarrow H^1(A, S^2 B^*\\otimes B)$ induced by the $A$ -modules morphism $S^2 B^*\\otimes B\\rightarrow B^*\\otimes B^* \\otimes B (\\cong B^*\\otimes B)$ .", "It is implicitly stated in [22] (see also [39], [37]) that, if $X$ is a complex manifold, then $T_X[-1]$ is a Lie algebra object in the derived category $D^+(X)$ of bounded below complexes of sheaves of $\\mathcal {O}_X$ -modules with coherent cohomology.", "This is simply Theorem REF in the special case where $L=T_X\\otimes $ and $A=T_X$ ." ], [ "Jacobi identity up to homotopy", "Let $(L,A)$ be a Lie pair and $E$ an $A$ -module.", "The quotient $B={}$ is naturally an $A$ -module (see Proposition ).", "Consider the graded vector spaces $V=\\bigoplus _{n=0}^\\infty {\\wedge ^n A^*\\otimes B}$ and $W=\\bigoplus _{n=0}^\\infty {\\wedge ^n A^*\\otimes E}$ , and the covariant differentials $:\\Gamma \\big (\\wedge ^{\\bullet }A^*\\otimes B\\big )\\rightarrow \\Gamma \\big (\\wedge ^{\\bullet +1}A^*\\otimes B\\big ) \\\\:\\Gamma \\big (\\wedge ^{\\bullet }A^*\\otimes E\\big )\\rightarrow \\Gamma \\big (\\wedge ^{\\bullet +1}A^*\\otimes E\\big )$ associated to the $A$ -actions on $B$ and $E$ , respectively.", "Choosing an $L$ -connection $\\nabla $ on $$ extending the $A$ -action, we obtain the bundle maps $R_2:B\\otimes B\\rightarrow (A,B)$ and $S_2:B\\otimes E\\rightarrow (A,E)$ given by $A\\ni a{R_2(b_1,b_2)}R^\\nabla _B(a;b_1)b_2 \\in B , \\\\A\\ni a{S_2(b,e)}R^\\nabla _E(a;b)e \\in E , $ where $R^\\nabla _B:A\\otimes B\\rightarrow B$ and $R^\\nabla _E:A\\otimes B\\rightarrow E$ denote the Atiyah cocycles of $B$ and $E$ .", "Up to homotopies, the complex $(V[-1],\\partial ^A)$ is a differential graded Lie algebra and the complex $(W[-1],\\partial ^A)$ is a differential graded module over it.", "The Lie algebra bracket $ V[-1]\\otimes V[-1]{\\lambda } V[-1] $ and the representation $ V[-1]\\otimes W[-1]{\\mu } W[-1] $ are given by $\\lambda \\big ( (\\xi _1\\otimes b_1)\\otimes (\\xi _2\\otimes b_2) \\big )={k_2} \\xi _1\\wedge \\xi _2\\wedge R_2(b_1,b_2)$ and $\\mu \\big ( (\\xi _1\\otimes b)\\otimes (\\xi _2\\otimes e) \\big )={k_2} \\xi _1\\wedge \\xi _2\\wedge S_2(b,e),$ where $\\xi _1\\in {\\wedge ^{k_1}A^*}$ , $\\xi _2\\in {\\wedge ^{k_2}A^*}$ , $b_1,b_2,b\\in {B}$ , and $e\\in {E}$ .", "Consequently, the cohomology $\\bigoplus _{i\\ge 1}H^{i-1}(A;E)=H^\\bullet (W[-1],\\partial ^A)$ is a module over the (graded) Lie algebra $\\bigoplus _{i\\ge 1}H^{i-1}(A;B)=H^\\bullet (V[-1],\\partial ^A)$ .", "In Section REF , we will describe a result which keeps track of higher homotopies." ], [ " algebras", "Recall that a graded Leibniz algebra is a $$ -graded vector space $V=\\bigoplus _{k\\in }V_k$ equipped with a bilinear bracket $V\\otimes V{{-}{-}}V$ of degree 0 satisfying the graded Leibniz rule ${x}{{y}{z}}={{x}{y}}{z}+(-1)^{{x}{y}}{y}{{x}{z}},$ for all homogeneous elements $x,y,z\\in V$ .", "If, moreover, $V$ is endowed with a differential $\\delta $ of degree 1 satisfying $ \\delta {x}{y}={\\delta x}{y}+{{x}+1}{x}{\\delta y} $ for all homogeneous elements $x,y\\in V$ , then we say that $(V,{-}{-},\\delta )$ is a differential graded Leibniz algebra.", "Here ${V}{n}$ denotes the shifted graded vector space: $({V}{n})_k=V_{n+k}$ .", "A algebra is a $$ -graded vector space $V=\\bigoplus _{n\\in }V_n$ endowed with a sequence $(\\lambda _k)_{k=1}^\\infty $ of multilinear maps $\\lambda _k: \\otimes ^k V\\rightarrow V$ of degree 1 satisfying the identity $\\sum _{1\\le j\\le k\\le n}\\sum _{\\sigma \\in {k-j}{j-1}}{\\sigma ;v_1,\\cdots ,v_{k-1}}(-1)^{{v_{\\sigma (1)}}+{v_{\\sigma (2)}}+\\cdots +{v_{\\sigma (k-j)}}} \\\\\\lambda _{n-j+1}\\big (v_{\\sigma (1)},\\cdots ,v_{\\sigma (k-j)},\\lambda _j(v_{\\sigma (k+1-j)},\\cdots ,v_{\\sigma (k-1)},v_k),v_{k+1},\\cdots , v_{n}\\big )=0$ for each $n\\in $ and for any homogeneous vectors $v_1,v_2,\\dots ,v_n\\in V$ .", "Here ${k-j}{j-1}$ denotes the set of $(k-j,j-1)$ -shufflesA $(k-j, j-1)$ -shuffle is a permutation $\\sigma $ of the set $\\lbrace 1,2,\\cdots ,k-1\\rbrace $ such that $\\sigma (1)\\le \\sigma (2)\\le \\cdots \\le \\sigma (k-j)$ and $\\sigma (k-j+1)\\le \\sigma (k-j+2)\\le \\cdots \\le \\sigma (k-1)$ ., and ${\\sigma ; v_1, \\cdots , v_{k-1}}$ denotes the Koszul signThe Koszul sign of a permutation $\\sigma $ of the (homogeneous) vectors $v_1,v_2,\\dots ,v_n$ is determined by the relation $v_{\\sigma (1)}\\odot v_{\\sigma (2)}\\odot \\cdots \\odot v_{\\sigma (n)}= {\\sigma ; v_1, \\cdots , v_n} \\cdot v_1\\odot v_2\\odot \\cdots \\odot v_n$ .", "of the permutation $\\sigma $ of the (homogeneous) vectors $v_1,v_2,\\dots ,v_{k-1}$ .", "If all $\\lambda _k$ are zero except for $\\lambda _1$ , $(V,\\lambda _1)$ is simply a cochain complex.", "If $\\lambda _k=0$ ($k\\ge 3$ ), then $({V}{-1},{-}{-}, d)$ is a graded differential Leibniz algebra, where ${x}{y}={{x}}\\lambda _2(x,y)$ , and $d=\\lambda _1$ .", "A graded vector space $V$ is a algebra if and only if the shifted graded vector space $V[-1]$ is a algebra in the sense of Loday [1], [43].", "Working with algebras rather than algebras is convenient as all maps in the sequence $(\\lambda _k)_{k=1}^n$ have the same degree in this setting.", "A module over a algebra $V$ is a $$ -graded vector space $W=\\bigoplus _{n\\in }W_n$ together with a sequence $(\\mu _k)_{k=1}^\\infty $ of multilinear maps $ \\mu _k: (\\otimes ^{k-1} V)\\otimes W\\rightarrow W $ of degree 1 satisfying the identity $\\sum _{1\\le j\\le k\\le n-1}\\sum _{\\sigma \\in {k-j}{j-1}}{\\sigma ;v_1,\\cdots ,v_{k-1}}(-1)^{{v_{\\sigma (1)}}+{v_{\\sigma (2)}}+\\cdots +{v_{\\sigma (k-j)}}} \\\\\\mu _{n-j+1}\\big (v_{\\sigma (1)},\\cdots ,v_{\\sigma (k-j)},\\lambda _j(v_{\\sigma (k+1-j)},\\cdots ,v_{\\sigma (k-1)},v_k),v_{k+1},\\cdots , v_{n-1},w\\big ) \\\\+ \\sum _{1\\le j\\le n}\\sum _{\\sigma \\in {n-j}{j-1}}{\\sigma ;v_1,\\cdots ,v_{n-1}}(-1)^{{v_{\\sigma (1)}}+{v_{\\sigma (2)}}+\\cdots +{v_{\\sigma (n-j)}}} \\\\\\mu _{n-j+1}\\big (v_{\\sigma (1)},\\cdots ,v_{\\sigma (n-j)},\\mu _j(v_{\\sigma (n+1-j)},\\cdots ,v_{\\sigma (n-1)},w)\\big ) =0$ for each $n\\in $ and for any homogeneous vectors $v_1,v_2,\\dots ,v_{n-1}\\in V$ and $w\\in W$ .", "A graded vector space $W$ is a module over a algebra $V$ if and only if $V\\oplus W$ is a algebra such that $V$ is a subalgebra [26].", "The proof of the next proposition is a direct verification, which we omit.", "If $\\big (V,(\\lambda _k)_{k=1}^\\infty \\big )$ is a algebra, then $(V, \\lambda _1)$ is a cochain complex and its cohomology $H(V)[-1]$ is a graded Leibniz algebra with bracket $H(\\lambda _2)$ , the image of $\\lambda _2$ (seen as a chain map) under the cohomology functor.", "Moreover, if $\\big (W,(\\mu _k)_{k=1}^\\infty \\big )$ is a module over $\\big (V,(\\lambda _k)_{k=1}^\\infty \\big )$ , then $(W,\\mu _1)$ is a cochain complex and $H(\\mu _2)$ is a representation of $H(V)[-1]$ on the cohomology $H(W)[-1]$ of $(W,\\mu _1)$ ." ], [ "Main theorem", "Unless we state otherwise, we assume throughout this section that $(L,A)$ is a Lie pair and $E$ is an $A$ -module.", "The quotient $B={}$ is naturally an $A$ -module (see Proposition ).", "We use the symbol $\\partial ^A$ to denote the covariant differential $ :\\Gamma \\big (\\wedge ^{\\bullet }A^*\\otimes (\\otimes ^{\\star }B^*)\\otimes E\\big )\\rightarrow \\Gamma \\big (\\wedge ^{\\bullet +1}A^* \\otimes (\\otimes ^{\\star }B^*)\\otimes E\\big ) $ associated to the $A$ -action on $(\\otimes ^{\\star }B^*)\\otimes E$ .", "In particular, for any bundle map $\\mu :(\\wedge ^k A)\\otimes (\\otimes ^l B)\\rightarrow B$ , we have $\\big (\\partial ^A\\mu \\big )(a_0\\wedge \\cdots \\wedge a_k;b_1\\otimes \\cdots \\otimes b_l) = \\\\\\sum _{i=0}^k (-1)^i \\left\\lbrace \\nabla _{a_i} \\big ( \\mu (a_{\\widehat{i}};b_1\\otimes \\cdots \\otimes b_l)\\big )-\\mu \\big (a_{\\widehat{i}};\\nabla _{a_i} (b_1\\otimes \\cdots \\otimes b_l) \\big ) \\right\\rbrace \\\\+ \\sum _{i<j} (-1)^{i+j} \\mu ({a_i}{a_j}\\wedge a_{\\widehat{i,j}};b_1\\otimes \\cdots \\otimes b_l),$ where $a_{\\widehat{i}}$ stands for $a_0\\wedge \\cdots \\wedge \\widehat{a_i}\\wedge \\cdots \\wedge a_k$ and $a_{\\widehat{i,j}}$ for $a_0\\wedge \\cdots \\wedge \\widehat{a_i}\\wedge \\cdots \\wedge \\widehat{a_j}\\wedge \\cdots \\wedge a_k$ , and $\\nabla _{a_i}(b_1\\otimes \\cdots \\otimes b_l)$ for $\\sum _{j=1}^l b_1\\otimes \\cdots \\otimes \\nabla _{a_i}b_j\\otimes \\cdots \\otimes b_l$ ." ], [ "The operator $\\partial ^\\nabla $", "Now choose an extension of the $A$ -action on $E$ to an $L$ -connection $\\nabla $ on $E$ , an extension of the $A$ -action on $B$ to an $L$ -connection $\\nabla $ on $B$ , and a splitting of the short exact sequence of vector bundles ${0 [r] & A [r]^i & L [r]^q & B [r] & 0 },$ i.e.", "a pair of maps $j:B\\rightarrow L$ and $p:L\\rightarrow A$ such that $qj=_B$ , $pi=_A$ and $ip+jq=_L$ : $ {0 @<.5ex>[r] & A @<.5ex>[r]^i @<.5ex>[l] & L @<.5ex>[r]^q @<.5ex>[l]^p& B @<.5ex>[r] @<.5ex>[l]^j & 0 @<.5ex>[l] } .$ This splitting determines the map $ {B}\\times {A}\\rightarrow {A}: (b,a) \\mapsto p{j(b)}{i(a)} ,$ which we will denote by $$ since it satisfies the relations $ _{f b}a=f _{b}a\\qquad \\text{and}\\qquad _{b}(fa)={\\rho ^* df}{j(b)} a+f_b a ,$ for all $f\\in {M}$ , $b\\in {B}$ , and $a\\in {A}$ .", "In some sense, $B$ “acts” on $A$ .", "Identifying sections of $\\wedge ^\\bullet A^*\\otimes (\\otimes ^\\star B^*)\\otimes E$ with bundle maps $\\wedge ^\\bullet A\\otimes (\\otimes ^\\star B)\\rightarrow E$ , we define a differential operator $ \\partial ^\\nabla :{\\wedge ^\\bullet A^* \\otimes (\\otimes ^\\star B^*) \\otimes E}\\rightarrow {\\wedge ^\\bullet A^* \\otimes (\\otimes ^{\\star +1} B^*) \\otimes E} $ by $ (-1)^k b_0(\\partial ^\\nabla \\omega )=\\nabla _{j(b_0)}\\omega $ or, more precisely, ${k} \\big (\\partial ^\\nabla \\omega \\big )(a_1,\\cdots ,a_k;b_0,\\cdots ,b_l)= \\nabla _{j(b_0)}\\big (\\omega (a_1,\\cdots ,a_k;b_1,\\cdots ,b_l)\\big ) \\\\-\\omega (_{b_0}a_1,\\cdots ,a_k;b_1,\\cdots ,b_l) -\\cdots -\\omega (a_1,\\cdots ,_{b_0}a_k;b_1,\\cdots ,b_l) \\\\-\\omega (a_1,\\cdots ,a_k;\\nabla _{j(b_0)}b_1,\\cdots ,b_l) -\\cdots -\\omega (a_1,\\cdots ,a_k;b_1,\\cdots ,\\nabla _{j(b_0)}b_l),$ where $a_1,\\cdots ,a_k\\in {A}$ , $b_0,\\cdots ,b_l\\in {B}$ , and $\\omega :\\wedge ^k A\\otimes (\\otimes ^l B)\\rightarrow E$ .", "Note that $\\partial ^\\nabla $ depends on the choice of the $L$ -connections extending the $A$ -actions and the splitting $j:B\\rightarrow L$ of the short exact sequence (REF ), while $\\partial ^A$ does not.", "The chosen splitting of (REF ) does also determine three vector bundle maps $ \\alpha :\\wedge ^2 B\\rightarrow A, \\qquad \\beta :\\wedge ^2 B\\rightarrow B, \\quad \\text{and} \\quad \\Omega :\\wedge ^2 B\\rightarrow B $ given by $\\alpha (b_1,b_2)=p{j(b_1)}{j(b_2)} \\\\\\beta (b_1,b_2)=\\nabla _{j(b_1)}b_2-\\nabla _{j(b_2)}b_1-q{j(b_1)}{j(b_2)} \\\\\\multicolumn{2}{l}{\\text{and}}\\\\\\Omega (b_1,b_2)=\\nabla _{j(b_1)}\\nabla _{j(b_2)}-\\nabla _{j(b_2)}\\nabla _{j(b_1)}-\\nabla _{{j(b_1)}{j(b_2)}}.$ For any $a\\in {A}$ and $b_1,b_2\\in {B}$ , we have $ R^\\nabla _B(a;b_1)b_2-R^\\nabla _B(a;b_2)b_1=\\big (\\nabla _a\\beta \\big )(b_1,b_2) $ or, equivalently, $ R_2(b_1,b_2)-R_2(b_2,b_1)=\\big (\\partial ^A \\beta \\big )(b_1,b_2) .$ For convenience, we set $\\widetilde{b}=j(b)$ , $\\forall b\\in \\Gamma ( B)$ .", "Hence ${a}{\\widetilde{b}}=-_b a+\\widetilde{\\nabla _a b}$ and $ {\\widetilde{b_1}}{\\widetilde{b_2}}=\\alpha (b_1,b_2)+\\widetilde{\\nabla _{\\widetilde{b_1}} b_2}-\\widetilde{\\nabla _{\\widetilde{b_2}} b_1}-\\widetilde{\\beta (b_1,b_2)} .$ A straightforward computation yields the equality $ q\\big ({{a}{\\widetilde{b_1}}}{\\widetilde{b_2}}+{{\\widetilde{b_1}}{\\widetilde{b_2}}}{a}+{{\\widetilde{b_2}}{a}}{\\widetilde{b_1}}\\big ) \\\\=R^\\nabla _B(a;b_2)b_1-R^\\nabla _B(a;b_1)b_2+\\nabla _a\\big (\\beta (b_1,b_2)\\big )-\\beta (\\nabla _a b_1,b_2)-\\beta (b_1,\\nabla _a b_2).$ The result follows from the Jacobi identity in the Lie algebroid $L$ .", "Note that, since $R^\\nabla _B$ is (by its very definition) independent of the choice of the splitting, Proposition REF asserts that, unlike $\\beta $ , $\\partial ^A\\beta $ does not depend on the choice of splitting." ], [ "The maps $R_n$", "Recall the bundle map $R_2:B\\otimes B\\rightarrow (A,B)$ associated to the Atiyah cocycle of $B$ given by (REF ).", "Since $B$ is an $A$ -module, we can substitute $B$ for $E$ in the definitions of $\\partial ^A$ and $\\partial ^\\nabla $ above and define a sequence of bundle maps $ R_n:\\otimes ^n B\\rightarrow (A,B) $ inductively by the relation $ R_{n+1}=\\partial ^\\nabla R_n, \\qquad \\text{for } n\\ge 2 .$ Hence, we have $ R_{n+1}(b_0\\otimes b_1\\otimes \\cdots \\otimes b_n)=R_n\\big (\\nabla _{j(b_0)}(b_1\\otimes \\cdots \\otimes b_n)\\big )-\\nabla _{j(b_0)}\\big (R_n(b_1\\otimes \\cdots \\otimes b_n)\\big ) .$ Let $L=A\\bowtie B$ be a matched pair of Lie algebras.", "Any bilinear map $\\gamma :B\\otimes B\\rightarrow B$ determines an $L$ -connection $\\nabla $ on $B$ extending its $A$ -module structure (and conversely): $\\nabla _{b_1}b_2=\\gamma (b_1,b_2)$ .", "Taking $\\gamma =0$ , the Atiyah cocycle reads $_{B}(a;b_1)b_2=\\nabla _{\\nabla _{b_1}a}b_2$ .", "Hence $ (b_1,b_2,b_3,\\cdots ,b_n)= \\nabla _{\\nabla _{b_{n-1}}\\nabla _{b_{n-2}}\\cdots \\nabla _{b_{1}}()}b_n .$" ], [ " algebra (and modules) arising from a Lie pair", "Consider the sequence of $k$ -ary operations $\\lambda _k:\\otimes ^kV\\rightarrow V$ ($k\\in $ ) on the graded vector space $V=\\bigoplus _{n=0}^\\infty {\\wedge ^n A^*\\otimes B}$ defined by $\\lambda _1=\\partial ^A$ and, for $k\\ge 2$ , $\\lambda _k(\\xi _1\\otimes b_1,\\cdots ,\\xi _k\\otimes b_k)={{\\xi _1}+\\cdots +{\\xi _k}}\\xi _1\\wedge \\cdots \\wedge \\xi _k\\wedge R_k(b_1,\\cdots , b_k),$ where $b_1,\\dots ,b_k\\in {B}$ and $\\xi _1,\\dots ,\\xi _k$ are homogeneous elements of ${\\wedge ^\\bullet A^*}$ .", "When endowed with the sequence of multibrackets $(\\lambda _k)_{k\\in }$ defined above, the graded vector space $V=\\bigoplus _{n=0}^\\infty {\\wedge ^n A^*\\otimes B}$ becomes a algebra.", "Similarly, we can introduce the bundle map $S_2:B\\otimes E\\rightarrow (A,E)$ given by $ A\\ni a{S_2(b;e)}R^\\nabla _E(a;b)\\cdot e\\in E,$ where $R^\\nabla _E:A\\otimes B\\rightarrow E$ denotes the Atiyah cocycle of the $A$ -module $E$ , and then define a sequence of bundle maps $ S_n:(\\otimes ^{n-1} B)\\otimes E\\rightarrow (A,E) $ inductively by the relation $ S_{n+1}=\\partial ^\\nabla S_n, \\qquad \\text{for } n\\ge 2 .$ This leads to the graded vector space $W=\\bigoplus _{n=0}^\\infty {\\wedge ^n A^*\\otimes E}$ and the sequence of $k$ -ary brackets $\\mu _k:(\\otimes ^{k-1}V) \\otimes W \\rightarrow W$ ($k\\in $ ) defined by $\\mu _1=\\partial ^A$ and, for $k\\ge 2$ , $\\mu _k(\\xi _1\\otimes b_1,\\cdots ,\\xi _{k-1}\\otimes b_{k-1};\\xi _k\\otimes e)={{\\xi _1}+\\cdots +{\\xi _{k}}}\\xi _1\\wedge \\cdots \\wedge \\xi _k\\wedge S_k(b_1,\\cdots ,b_{k-1};e),$ where $b_1,\\dots ,b_{k-1}\\in {B}$ , $e\\in {E}$ , and $\\xi _1,\\dots ,\\xi _k$ are homogeneous elements of ${\\wedge ^\\bullet A^*}$ .", "When endowed with the sequence of multibrackets $(\\mu _k)_{k\\in }$ defined above, the graded vector space $W=\\bigoplus _{n=0}^\\infty {\\wedge ^n A^*\\otimes E}$ becomes a module over the algebra $\\big (V,(\\lambda _k)_{k\\in }\\big )$ .", "A Lie bialgebra $(,^*)$ is a matched pair of Lie algebras.", "Therefore, it induces two Lie pairs: $(\\bowtie ^*,)$ and $(\\bowtie ^*,^*)$ .", "It follows from Example REF and Theorem REF that both $\\bigoplus _{n\\ge 0}\\wedge ^n^*\\otimes ^*$ and $\\bigoplus _{n\\ge 0}\\wedge ^n\\otimes $ are algebras.", "Let $A$ be a Lie algebroid over a manifold $M$ .", "By an $A$ -algebra, we mean a bundle (of finite or infinite rank) of associative algebras $$ over $M$ , which is an $A$ -module, and on which $\\Gamma (A)$ acts by derivations.", "For a commutative $A$ -algebra $$ , the sequence of maps $(\\lambda _k)_{k\\in }$ extends in a natural way to the graded space $\\bigoplus _{n=0}^\\infty {\\wedge ^n A^*\\otimes B\\otimes }$ .", "Similarly, the sequence of maps $(\\mu _k)_{k\\in }$ extends to the graded space $\\bigoplus _{n=0}^\\infty {\\wedge ^n A^*\\otimes E\\otimes }$ .", "Let $(L,A)$ be a Lie pair with quotient $B$ , and let $$ be a commutative $A$ -algebra.", "When endowed with the sequence of multibrackets $(\\lambda _k)_{k\\in }$ , the graded vector space ${\\wedge ^\\bullet A^*\\otimes B\\otimes }$ becomes a Leibniz$_\\infty [1]$ algebra.", "Moreover, if $E$ is an $A$ -module, the graded vector space ${\\wedge ^\\bullet A^*\\otimes E \\otimes }[-1]$ becomes a Leibniz$_\\infty [1]$ module over the Leibniz$_\\infty [1]$ algebra ${\\wedge ^\\bullet A^*\\otimes B \\otimes }[-1]$ .", "As an immediate consequence, we have the following Under the same hypothesis as in Theorem REF , $\\bigoplus _{i\\ge 1}H^{i-1} (A,B\\otimes )$ is a graded Lie algebra and $\\bigoplus _{i\\ge 1}H^{i-1}(A, E \\otimes )$ a module over it.", "Let $$ be a Lie subalgebra of a Lie algebra $$ as in Example REF .", "Assume that $$ is a commutative $$ -algebra.", "Every linear map $L: \\rightarrow (/)$ that extends the $$ -module structure $\\rightarrow (/)$ induces a 2-ary bracket on $ \\wedge ^{\\bullet -1}^* \\otimes /\\otimes $ : $\\left[{\\xi _1\\otimes b_1\\otimes c_1},{\\xi _2\\otimes b_2 \\otimes c_2}\\right]={{\\xi _2}}{\\xi _1\\wedge \\xi _2\\otimes \\big (\\partial ^{}L\\big )(;b_1)\\cdot b_2} \\otimes c_1c_2,$ which in turn induces a (graded) Lie algebra bracket on the Chevalley-Eilenberg cohomology $\\bigoplus H^{\\bullet -1}(;/\\otimes )$ .", "Here $\\xi _i\\otimes b_i\\otimes c_i$ ($i=1,2$ ) are cocycles with $\\xi _1,\\xi _2 \\in \\wedge ^\\bullet ^*$ , $b_1,b_2\\in /$ and $c_1, c_2 \\in $ .", "Moreover, if $E$ is a $$ -module, every linear map $M:\\rightarrow E$ that extends the $$ -module structure $\\rightarrow E$ gives rise to a bilinear map $ {(\\xi _1\\otimes b\\otimes c_1)}{(\\xi _2\\otimes e \\otimes c_2)}={{\\xi _2}}{\\xi _1\\wedge \\xi _2\\otimes \\big (\\partial ^{}M\\big )(;b)\\cdot e} \\otimes c_1c_2, $ which induces a representation on $\\bigoplus H^{\\bullet -1}(;E \\otimes )$ of the graded Lie algebra $\\bigoplus H^{\\bullet -1}(;/\\otimes )$ .", "Here $\\xi _1\\otimes b\\otimes c_1$ and $\\xi _2\\otimes e \\otimes c_2$ are cocycles with $\\xi _1,\\xi _2 \\in \\wedge ^\\bullet ^*$ , $b\\in /$ , $e\\in E$ and $c_1, c_2 \\in $ .", "Take a complement $\\mathfrak {h}$ of $\\mathfrak {g}$ in $\\mathfrak {d}$ so that we can write $\\mathfrak {d}=\\mathfrak {g}\\oplus \\mathfrak {h}$ .", "Then $\\mathfrak {d}/\\mathfrak {g}$ can be identified with $\\mathfrak {h}$ , on which the $\\mathfrak {h}$ -action is given by $a\\cdot h= [a, h]$ .", "Take $L: \\rightarrow \\mathfrak {h}$ to be the trivial extension of the $\\mathfrak {g}$ -module structure $\\mathfrak {g} \\rightarrow \\mathfrak {h}$ , i.e.", "set $L|_\\mathfrak {h}=0$ .", "Then the 2-ary bracket in Equation (REF ) is given by $ [f\\otimes c_1, \\ g\\otimes c_2]=[f,g] \\otimes c_1c_2 ,$ where $f\\in \\wedge ^p\\mathfrak {g}\\rightarrow \\mathfrak {h}$ , $g\\in \\wedge ^q\\mathfrak {g}\\rightarrow \\mathfrak {h}$ , $c_1, c_2\\in $ , and $[f,g]\\in \\wedge ^{p+q+1}\\mathfrak {g}\\rightarrow \\mathfrak {h}$ is given by $ [f,g](a_0,a_1,\\cdots ,a_{p+q})= \\\\-\\sum _{\\sigma \\in \\mathfrak {S}_{1,p,q}}(\\sigma ){{a_{\\sigma (0)}}{f(a_{\\sigma (1)},\\cdots ,a_{\\sigma (p)})}}{g(a_{\\sigma (p+1)},\\cdots ,a_{\\sigma (p+q)})} ,$ where $\\mathfrak {S}_{1,p,q}$ is the set of all permutations $\\sigma $ of $\\lbrace 0,1,\\cdots ,p+q\\rbrace $ satisfying $\\sigma (1)<\\cdots <\\sigma (p)$ and $\\sigma (p+1)<\\cdots <\\sigma (p+q)$ .", "It is natural to ask how the algebra structure obtained in Theorem REF and the module structure in Theorem REF depend on the choice of connections and the splitting data.", "This question will be investigated somewhere else." ], [ "$$ rather than ", "Let $(A,B)$ be a matched pair of Lie algebroids with direct sum $L=A\\bowtie B$ .", "Assume there exists a flat torsion free $B$ -connection on $B$ .", "Then the maps $R_n$ defined above are totally symmetric, the multibrackets $\\lambda _k:\\otimes ^k V\\rightarrow V$ ($k\\in $ ) defined earlier on the graded vector space $V=\\bigoplus _{n=0}^{\\infty }{\\wedge ^n A^*\\otimes B}$ are graded symmetric, and $V[-1]$ is actually an $$ algebra.", "The following example is due to Camille Laurent-Gengoux.", "The general Lie algebra $\\mathfrak {gl}_n()$ decomposes as the direct sum of the unitary Lie algebra $\\mathfrak {u}_n$ and the Lie algebra $\\mathfrak {t}_n$ of upper triangular matrices with real diagonal coefficients.", "Both $\\mathfrak {u}_n$ and $\\mathfrak {t}_n$ are isotropic with respect to the natural nondegenerate ad-invariant inner product $X\\otimes Y\\mapsto \\big ((XY)\\big )$ on $\\mathfrak {gl}_n()$ .", "Hence $(\\mathfrak {u}_n,\\mathfrak {t}_n)$ is a matched pair of Lie algebras as well as a Lie bialgebra.", "Matrix multiplication being associative, setting $\\nabla _X Y=XY$ for any $X,Y\\in \\mathfrak {t}_n$ defines a flat torsion free $\\mathfrak {t}_n$ -connection on $\\mathfrak {t}_n$ .", "It follows from Theorem REF that ${\\wedge ^\\bullet \\mathfrak {u}_n^*\\otimes \\mathfrak {t}_n}[-1]{\\wedge ^\\bullet \\mathfrak {t}_n\\otimes \\mathfrak {t}_n}[-1]$ is an $L_{\\infty }$ algebra.", "[Kapranov] Suppose $X$ is a Kähler manifold.", "The complexification $\\nabla ^$ of its Levi-Civita connection is a $T_X\\otimes $ -connection on $T_X\\otimes $ .", "Set $A=T_X$ and $B=T_X$ .", "Then $(A,B)$ is a matched pair of Lie algebroids, whose direct sum $A\\bowtie B$ is isomorphic, as a Lie algebroid, to $T_X\\otimes $ .", "It is easy to see that $\\nabla ^$ induces a flat torsion free $B$ -connection on $B$ .", "In this context, the tensors $R_n\\in ^{0,1}\\big ((\\otimes ^n T,T)\\big )$ (where $T$ stands for $T_X$ ) are the curvature $R_2\\in ^{1,1}(T)^{0,2}\\big ((T\\otimes T,T)\\big )$ and its higher covariant derivatives: $R_{i+1}=\\partial ^\\nabla R_i$ .", "Applying Theorem REF , we recover a result of Kapranov [22]: [Kapranov] The shifted Dolbeault complex $^{0,\\bullet -1}(T)$ of a Kähler manifold is an $$ algebra.", "The $n$ -th multibracket $ \\lambda _n:^{0,j_1}(T)\\otimes \\cdots \\otimes ^{0,j_n}(T)\\rightarrow ^{0,j_1+\\cdots +j_n+1}(T) $ is the composition of the wedge product $ ^{0,j_1}(T)\\otimes \\cdots \\otimes ^{0,j_n}(T)\\rightarrow ^{0,j_1+\\cdots +j_n}(\\otimes ^n T) $ with the map $ ^{0,j_1+\\cdots +j_n}(\\otimes ^n T) \\rightarrow ^{0,j_1+\\cdots +j_n+1}(T) $ associated to $R_n\\in ^{0,1}\\big ((\\otimes ^n T,T)\\big )$ in the obvious way." ], [ "Proofs", "This section is devoted to the proofs of the theorems claimed in Section .", "For convenience, we set $\\widetilde{b}=j(b)$ , $\\forall b\\in \\Gamma (B) $ ." ], [ "Atiyah class as Lie algebra object", "For any $a_1,a_2\\in {A}$ and $b\\in {B}$ , we have $ {_b a_1}{a_2}+{a_1}{_b a_2}-_b{a_1}{a_2}=_{\\nabla _{a_1}b}a_2-_{\\nabla _{a_2}b}a_1 .$ We have $& p\\big ( {{}{a_1}}{a_2}+{{a_1}{a_2}}{}+{{a_2}{}}{a_1} \\big ) \\\\=& p{p{}{a_1}+\\widetilde{q{}{a_1}}}{a_2}-p{}{{a_1}{a_2}}+p{p{a_2}{}+\\widetilde{q{a_2}{}}}{a_1} \\\\=& {p{}{a_1}}{a_2}-p{\\widetilde{q{a_1}{}}}{a_2}-p{}{{a_1}{a_2}}+{a_1}{p{}{a_2}}+p{\\widetilde{q{a_2}{}}}{a_1} \\\\=& {_b a_1}{a_2}-_{\\nabla _{a_1}b}a_2-_b{a_1}{a_2}+{a_1}{_b a_2}+_{\\nabla _{a_2}b}a_1.$ The result follows from the Jacobi identity in the Lie algebroid $L$ .", "Note that a bundle map $\\omega :(\\wedge ^k A)\\otimes (\\otimes ^l B)\\rightarrow E$ determines a bundle map ${\\omega }:\\wedge ^k A\\rightarrow (\\otimes ^l B^*)\\otimes E$ and vice versa.", "For any bundle map $\\omega :(\\wedge ^k A)\\otimes (\\otimes ^l B)\\rightarrow E$ , any $a_0,\\dots ,a_k\\in {A}$ and any $b_0,\\dots ,b_l\\in {B}$ , we have $& (-1)^{k} \\big (\\partial ^A\\omega +\\partial ^A\\omega \\big )(a_0,\\cdots ,a_k;b_0,\\dots ,b_l) \\\\& \\quad = \\sum _{i=0}^k (-1)^{i} \\left\\langle \\nabla _{a_i}\\nabla _{\\widetilde{b_0}}\\big ({\\omega }(\\widehat{a_i})\\big )-\\nabla _{\\widetilde{b_0}}\\nabla _{a_i}\\big ({\\omega }(\\widehat{a_i})\\big )-\\nabla _{{a_i}{\\widetilde{b_0}}}\\big ({\\omega }(\\widehat{a_i})\\big ) | b_1\\otimes \\cdots \\otimes b_l \\right\\rangle \\\\& \\quad = \\sum _{i=0}^k (-1)^{i} \\left\\lbrace R^\\nabla _E(a_i;b_0)\\cdot \\omega (\\widehat{a_i};\\widehat{b_0})- \\sum _{j=1}^l \\omega \\big (\\widehat{a_i};b_1,\\cdots ,R^\\nabla _B(a_i;b_0)\\cdot b_j,\\cdots ,b_l\\big ) \\right\\rbrace ,$ where $\\widehat{a_i}$ stands for $a_0\\wedge \\cdots \\wedge a_{i-1}\\wedge a_{i+1}\\wedge \\cdots \\wedge a_k$ and $\\widehat{b_0}$ for $b_1\\otimes \\cdots \\otimes b_l$ .", "The first equality follows from a cumbersome computation at the last step of which use is made of Lemma REF .", "The second equality is immediate.", "Given $\\mu \\in {(\\wedge ^{k_1}A^*)\\otimes (\\otimes ^{l_1}B^*)\\otimes B}$ , $\\nu \\in {(\\wedge ^{k_2}A^*)\\otimes (\\otimes ^{l_2}B^*)\\otimes B}$ , and arbitrary sections $b_1,\\dots ,b_{l_1},b^{\\prime }_1,\\dots ,b^{\\prime }_{l_1}$ of $B$ , we can consider the bundle map $ {\\mu \\big (b_1,\\cdots ,b_{i-1},\\nu (b^{\\prime }_1,\\cdots ,b^{\\prime }_{l_2}),b_{i+1},\\cdots ,b_{l_1}\\big )}:\\wedge ^{k_1+k_2}A\\rightarrow B ,$ which maps $a_1\\wedge \\cdots \\wedge a_{k_1+k_2}$ to $ \\sum _{\\sigma \\in {k_1}{k_2}} (\\sigma )\\mu \\big (a_{\\sigma (1)},\\cdots ,a_{\\sigma (k_1)};b_1,\\cdots ,b_{i-1}, \\\\\\nu ( a_{\\sigma (k_1+1)},\\cdots ,a_{\\sigma (k_1+k_2)};b^{\\prime }_1,\\cdots ,b^{\\prime }_{l_2}),b_{i+1},\\cdots ,b_{l_1}\\big ) .$ In particular, if $\\mu =\\alpha _1\\otimes \\beta _1\\otimes u$ and $\\nu =\\alpha _2\\otimes \\beta _2\\otimes v$ with $\\alpha _1\\in {\\wedge ^{k_1}A^*}$ , $\\alpha _2\\in {\\wedge ^{k_2}A^*}$ , $\\beta _1\\in {\\otimes ^{l_1}B^*}$ , $\\beta _2\\in {\\otimes ^{l_2}B^*}$ , and $u,v\\in {B}$ , then ${\\mu \\big (b_1,\\cdots ,b_{i-1},\\nu (b^{\\prime }_1,\\cdots ,b^{\\prime }_{l_2}),b_{i+1},\\cdots ,b_{l_1}\\big )} = \\\\\\beta _1(b_1,\\cdots ,b_{i-1},v,b_{i+1},\\cdots ,b_{l_1})\\beta _2(b^{\\prime }_1,\\cdots ,b^{\\prime }_{l_2}) \\cdot (\\alpha _1\\wedge \\alpha _2)\\otimes u.$ For any $n\\ge 2$ and $b_0,\\dots ,b_n\\in {B}$ , we have $- \\big ((\\partial ^A+\\partial ^A)R_n\\big )(b_0,\\cdots ,b_n) = {R_2\\big (b_0,R_n(b_1,\\cdots ,b_n)\\big )} \\\\+\\sum _{j=1}^n {R_n\\big (b_1,\\cdots ,R_2(b_0,b_j),\\cdots ,b_n\\big )}.$ Apply Lemma REF to $\\omega =R_n$ .", "For any $b_0,b_1,b_2\\in {B}$ , we have $- \\big (\\partial ^A R_3\\big )(b_0,b_1,b_2) = {R_2\\big (b_0,R_2(b_1,b_2)\\big )}+{R_2\\big (R_2(b_0,b_1),b_2\\big )} \\\\ +{R_2\\big (b_1,R_2(b_0,b_2)\\big )}.$ Since $\\partial ^A R_2=0$ and $R_2=R_3$ , taking $n=2$ in Corollary REF yields the result.", "The interchange isomorphism $\\tau :B[-1]\\otimes B[-1]\\rightarrow B[-1]\\otimes B[-1]$ is the image in $D({A})$ of the chain map $\\tau :B[-1]\\otimes B[-1]\\rightarrow B[-1]\\otimes B[-1]$ given by $\\tau (b_1\\otimes b_2)=- b_2\\otimes b_1$ , $\\forall b_1,b_2\\in B$ — the negative sign is due to $B[-1]$ being a complex concentrated in degree 1, see Equation (REF ).", "Recall that $R_2$ is a cocycle (w.r.t.", "$\\partial ^A$ ).", "Its cohomology class $\\alpha _B$ , the Atiyah class of $B$ , can be seen as an element of $_{D^+({A})}(B[-1]\\otimes B[-1],B[-1])$ .", "Proposition REF implies the equality $\\alpha _B\\tau =-\\alpha _B$ in $_{D^+({A})}(B[-1]\\otimes B[-1],B[-1])$ .", "Corollary REF implies that the Jacobi identity $\\alpha _B(\\otimes \\alpha _B)=\\alpha _B(\\alpha _B\\otimes )+\\alpha _B(\\otimes \\alpha _B)(\\tau \\otimes )$ holds in $D^+({A})$ .", "Indeed, each of the terms ${\\cdots }$ can be interpreted as a Yoneda product, a composition of morphisms in the derived category." ], [ "Jacobi identity up to homotopy", "Consider the cochain complex $(V[-1],\\partial ^A)$ , where the graded vector space $V=\\bigoplus _{k=0}^{\\infty } V_k$ is given by $V_k={\\wedge ^k A^*\\otimes B}$ so that, if $\\xi \\in {\\wedge ^k A^*}$ and $b\\in {B}$ , then $\\xi \\otimes b\\in \\big (V[-1]\\big )_{k+1}$ .", "The graded linear map $\\lambda :V[-1]\\otimes V[-1]\\rightarrow V[-1]$ given by $ \\lambda (v_1\\otimes v_2) = (-1)^{k_2} \\xi _1\\wedge \\xi _2\\wedge R_2(b_1,b_2) $ for any $v_1=\\xi _1\\otimes b_1\\in (V[-1])_{k_1+1}$ and $v_2=\\xi _2\\otimes b_2\\in (V[-1])_{k_2+1}$ is a chain map.", "A straightforward computation yields $ \\big (\\partial ^A\\lambda -\\lambda \\partial ^A\\big )(v_1\\otimes v_2)=(-1)^{k_1}\\xi _1\\wedge \\xi _2\\wedge \\big (\\partial ^A R_2\\big )(b_1,b_2) .$ The result follows from $\\partial ^A R_2=0$ (see Theorem REF and the definition (REF ) of $R_2$ ).", "Now, consider the interchange isomorphism $\\tau :V[-1]\\otimes V[-1]\\rightarrow V[-1]\\otimes V[-1]$ given by $\\tau (v_1\\otimes v_2)=(-1)^{{v_1}{v_2}}v_2\\otimes v_1$ .", "The chain map $\\lambda $ is skew-symmetric up to a homotopy: $ \\lambda +\\lambda \\tau =\\partial ^A\\Theta +\\Theta \\partial _A ,$ where the graded map $\\Theta :V[-1]\\otimes V[-1]\\rightarrow V[-2]$ is given by $ \\Theta (v_1\\otimes v_2) = (-1)^{k_1} \\xi _1\\wedge \\xi _2\\otimes \\beta (b_1,b_2) $ for any $v_1=\\xi _1\\otimes b_1\\in (V[-1])_{k_1+1}$ and $v_2=\\xi _2\\otimes b_2\\in (V[-1])_{k_2+1}$ .", "Straightforward computations yield $ \\big (\\lambda +\\lambda \\tau \\big )(v_1\\otimes v_2)=(-1)^{k_2}\\xi _1\\wedge \\xi _2\\wedge \\left\\lbrace R_2(b_1,b_2)-R_2(b_2,b_1)\\right\\rbrace $ and $ \\big (\\partial ^A\\Theta +\\Theta \\partial _A\\big )(v_1\\otimes v_2)=(-1)^{k_2}\\xi _1\\wedge \\xi _2\\wedge \\left\\lbrace \\big (\\partial ^A\\beta \\big )(b_1,b_2)\\right\\rbrace $ The result follows from Proposition REF .", "The chain map $\\lambda $ satisfies the Jacobi identity up to a homotopy: $ -\\lambda (\\otimes \\lambda )+\\lambda (\\lambda \\otimes )+\\lambda (\\otimes \\lambda )(\\tau \\otimes )= \\partial ^A\\Xi +\\Xi \\partial _A ,$ where the graded map $\\Xi :V[-1]\\otimes V[-1]\\otimes V[-1]\\rightarrow V[-2]$ is given by $ \\Xi (v_0\\otimes v_1\\otimes v_2) = (-1)^{k_0+k_2} \\xi _0\\wedge \\xi _1\\wedge \\xi _2\\wedge R_3(b_0,b_1,b_2) $ for any $v_i=\\xi _i\\otimes b_i\\in (V[-1])_{k_i+1}$ with $i\\in \\lbrace 0,1,2\\rbrace $ .", "Straightforward computations yield $\\big ( \\lambda (\\otimes \\lambda ) \\big )(v_0\\otimes v_1\\otimes v_2)= (-1)^{k_1}\\xi _0\\wedge \\xi _1\\wedge \\xi _2\\wedge {R_2(b_0,R_2(b_1,b_2))} ,\\\\\\big ( \\lambda (\\lambda \\otimes ) \\big )(v_0\\otimes v_1\\otimes v_2)= -(-1)^{k_1}\\xi _0\\wedge \\xi _1\\wedge \\xi _2\\wedge {R_2(R_2(b_0,b_1),b_2)} ,\\\\\\big ( \\lambda (\\otimes \\lambda )(\\tau \\otimes ) \\big )(v_0\\otimes v_1\\otimes v_2)= -(-1)^{k_1}\\xi _0\\wedge \\xi _1\\wedge \\xi _2\\wedge {R_2(b_1,R_2(b_0,b_2))} ,\\\\\\multicolumn{2}{l}{\\text{and}}\\\\\\big ( \\partial ^A\\Theta +\\Theta \\partial _A \\big )(v_0\\otimes v_1\\otimes v_2)= (-1)^{k_1}\\xi _0\\wedge \\xi _1\\wedge \\xi _2\\wedge \\left\\lbrace \\big (\\partial ^A R_3\\big )(b_0,b_1,b_2)\\right\\rbrace .$ The result follows from Corollary REF .", "Theorem REF immediately follows from Lemmas REF , REF , and REF .", "Note that Theorem REF could also be seen as a corollary of Theorems REF and REF ." ], [ " algebra (and modules) arising from a Lie pair", "For any $n\\ge 3$ and $b_1,\\dots ,b_n$ are arbitrary sections of $B$ , we have $-\\big (\\partial ^A R_n\\big )(b_1,\\dots ,b_n)=\\sum _{\\begin{array}{c}i+j=n+1 \\\\ i\\ge 2 \\\\ j\\ge 2\\end{array}}\\sum _{k=j}^n\\sum _{\\sigma \\in {k-j}{j-1}} \\\\{R_i\\big (b_{\\sigma (1)},\\cdots ,b_{\\sigma (k-j)},R_j(b_{\\sigma (k+1-j)},\\cdots ,b_{\\sigma (k-1)},b_k),b_{k+1},\\cdots ,b_n\\big )}.$ We reason by induction.", "The formula holds for $n=3$ by Corollary REF .", "Assuming the formula holds for $n=N$ , we get $\\big (\\partial ^A R_N\\big )(b_0,\\dots ,b_N)=\\big (\\nabla _{b_0}(\\partial ^A R_N)\\big )(b_1,\\dots ,b_N)= \\sum _{\\begin{array}{c}i+j=N \\\\ i\\ge 2 \\\\ j\\ge 2\\end{array}}\\sum _{k=j}^N\\sum _{\\sigma \\in {k-j}{j-1}} \\\\\\Big \\lbrace {R_{i+1}\\big (b_0,b_{\\sigma (1)},\\cdots ,b_{\\sigma (k-j)},R_j(b_{\\sigma (k+1-j)},\\cdots ,b_{\\sigma (k-1)},b_k),b_{k+1},\\cdots ,b_n\\big )} \\\\+ {R_i\\big (b_{\\sigma (1)},\\cdots ,b_{\\sigma (k-j)},R_{j+1}(b_0,b_{\\sigma (k+1-j)},\\cdots ,b_{\\sigma (k-1)},b_k),b_{k+1},\\cdots ,b_n\\big )} \\Big \\rbrace .$ Observing that $ \\partial ^A R_{N+1}=(\\partial ^A+\\partial ^A) R_N + \\partial ^A R_N $ and recalling Corollary REF , it is easy to check that the desired formula holds for $n=N+1$ as well.", "For any bundle map $\\omega :(\\wedge ^k A)\\otimes (\\otimes ^l B)\\rightarrow B$ and any $b_1,\\dots ,b_l\\in {B}$ , we have $ \\partial ^A\\big ({\\omega }(b_1,\\dots ,b_l)\\big )-\\big ({\\partial ^A\\omega }\\big )(b_1,\\dots ,b_l)=(-1)^k\\sum _{j=0}^l {\\omega (b_1,\\cdots ,\\partial ^A b_j,\\cdots ,b_l)} .$ For any $a_0,\\dots ,a_k\\in {A}$ , we have $&\\left\\langle \\partial ^A\\big ({\\omega } (b_1,\\dots ,b_l)\\big )-\\big ({\\partial ^A\\omega }\\big )(b_1,\\dots ,b_l)| a_0\\wedge \\cdots \\wedge a_k \\right\\rangle \\\\=\\; & \\sum _{j=0}^l \\sum _{i=0}^k (-1)^i \\omega ( a_0,\\cdots ,a_{i-1},a_{i+1},\\cdots ,a_k ;b_1,\\cdots ,\\nabla _{a_i}b_j,\\cdots ,b_l) \\\\=\\; & (-1)^k \\sum _{j=0}^l \\sum _{\\sigma \\in {k}{1}}(\\sigma )\\omega ( a_{\\sigma (0)},\\cdots ,a_{\\sigma (k-1)}; b_1,\\cdots ,\\nabla _{a_{\\sigma (k)}}b_j,\\cdots ,b_l) \\\\=\\; & (-1)^k \\sum _{j=0}^l \\left\\langle {\\omega (b_1,\\cdots ,\\partial ^A b_j,\\cdots ,b_l)} |a_0\\wedge \\cdots \\wedge a_k \\right\\rangle .", "$ We only need to check that the generalized Leibniz identity (REF ) holds.", "Since $\\lambda _1=\\partial ^A$ and $(\\partial ^A)^2=0$ , Equation (REF ) is obviously true for $n=1$ .", "Let $n\\ge 2$ and $v_i=\\xi _i\\otimes b_i\\in {\\wedge ^{p_i}A^*\\otimes B}$ for all $i\\in \\lbrace 1,\\dots ,n\\rbrace $ .", "The l.h.s.", "of (REF ) is $\\sum _{1\\le j\\le k\\le n}\\sum _{\\sigma \\in {k-j}{j-1}}{\\sigma ;v_1,\\cdots ,v_{k-1}}(-1)^{{v_{\\sigma (1)}}+{v_{\\sigma (2)}}+\\cdots +{v_{\\sigma (k-j)}}} \\\\\\lambda _{n-j+1}\\big (v_{\\sigma (1)},\\cdots ,v_{\\sigma (k-j)},\\lambda _j(v_{\\sigma (k+1-j)},\\cdots ,v_{\\sigma (k-1)},v_k),v_{k+1},\\cdots , v_{n}\\big ) .$ Separating the terms involving $\\lambda _1$ (aka $\\partial ^A$ ) from the others, it can be rewritten as $\\partial ^A\\big (\\lambda _n(v_1,\\cdots ,v_n)\\big )+ \\sum _{\\begin{array}{c}i+j=n+1 \\\\ i\\ge 2 \\\\ j\\ge 2\\end{array}}\\sum _{k=j}^n\\sum _{\\sigma \\in {k-j}{j-1}}{\\sigma ;\\xi _1,\\cdots ,\\xi _{k-1}}(-1)^{p_{\\sigma (1)}+\\cdots +p_{\\sigma (k-j)}} \\\\\\lambda _{i}\\big (v_{\\sigma (1)},\\cdots ,v_{\\sigma (k-j)},\\lambda _j(v_{\\sigma (k+1-j)},\\cdots ,v_{\\sigma (k-1)},v_k),v_{k+1},\\cdots , v_{n}\\big ) \\\\+\\sum _{k=1}^n (-1)^{p_1+\\cdots +p_{k-1}} \\lambda _n\\big (v_1,\\cdots ,v_{k-1},(\\partial ^A \\xi _k)\\otimes b_k+(-1)^{p_k}\\xi _k\\otimes (\\partial ^A b_k) ,v_{k+1},\\cdots ,v_n\\big )$ and then, using the definition (REF ) of each $\\lambda _k$ in terms of the corresponding $R_k$ , as $\\partial ^A\\big ((-1)^{p_1+\\cdots +p_n}\\xi _1\\wedge \\cdots \\wedge \\xi _n\\wedge R_n(b_1,\\cdots ,b_n)\\big ) \\\\+ \\sum _{\\begin{array}{c}i+j=n+1 \\\\ i\\ge 2 \\\\ j\\ge 2\\end{array}}\\sum _{k=j}^n\\sum _{\\sigma \\in {k-j}{j-1}}{\\sigma ;\\xi _1,\\cdots ,\\xi _{k-1}}\\xi _{\\sigma (1)}\\wedge \\cdots \\wedge \\xi _{\\sigma (k-1)}\\wedge \\xi _k\\wedge \\cdots \\wedge \\xi _n\\wedge \\\\{R_i\\big (b_{\\sigma (1)},\\cdots ,b_{\\sigma (k-j)},R_j(b_{\\sigma (k+1-j)},\\cdots ,b_{\\sigma (k-1)},b_k),b_{k+1},\\cdots ,b_{n}\\big ) } \\\\+ \\sum _{k=1}^n (-1)^{1+p_k+p_{k+1}+\\cdots +p_n}\\xi _1\\wedge \\cdots \\wedge \\xi _{k-1}\\wedge \\partial ^A\\xi _k\\wedge \\xi _{k+1}\\wedge \\cdots \\wedge \\xi _n\\wedge R_n(b_1,\\cdots ,b_n) \\\\+ \\sum _{k=1}^n \\xi _1\\wedge \\cdots \\wedge \\xi _n\\wedge {R_n(b_1,\\cdots ,\\partial ^A b_k,\\cdots ,b_n)},$ which simplifies to $\\xi _1\\wedge \\cdots \\wedge \\xi _n\\wedge \\Big \\lbrace \\partial ^A\\big (R_n(b_1,\\cdots ,b_n)\\big )+\\sum _{\\begin{array}{c}i+j=n+1 \\\\ i\\ge 2 \\\\ j\\ge 2\\end{array}}\\sum _{k=j}^n\\sum _{\\sigma \\in {k-j}{j-1}} \\\\{R_i\\big (b_{\\sigma (1)},\\cdots ,b_{\\sigma (k-j)},R_j(b_{\\sigma (k+1-j)},\\cdots ,b_{\\sigma (k-1)},b_k),b_{k+1},\\cdots ,b_{n}\\big ) } \\\\+ \\sum _{k=1}^n {R_n(b_1,\\cdots ,\\partial ^A b_k,\\cdots ,b_n)} \\Big \\rbrace .$ The result now follows from Lemmas REF and REF .", "The proof of Theorem REF , Theorem REF and Corollary REF goes along the same line mutatis mutandis." ], [ "$L_\\infty $ rather than ", "For any $b_0,b_1,b_2\\in {B}$ , we have $ R_3(b_0,b_1,b_2)-R_3(b_1,b_0,b_2)=R_2\\big (\\beta (b_0,b_1),b_2\\big )-\\big (\\partial ^A\\Omega \\big )(b_0,b_1)\\cdot b_2 .$ The differential Bianchi identity $d^\\nabla R^\\nabla =0$ holds for the curvature $R^\\nabla :\\wedge ^2 L\\rightarrow B$ of the $L$ -connection $\\nabla $ on $B$ extending the $A$ -action.", "Hence, for any $a\\in {A}$ and $b_0,b_1,b_2\\in {B}$ , we have $0=& \\big (d^\\nabla R^\\nabla \\big )(a,_0,_1) \\\\=& \\nabla _a\\big ( R^\\nabla (_0,_1) \\big ) - \\nabla _{_0}\\big ( R^\\nabla (a,_1) \\big )+ \\nabla _{_1}\\big ( R^\\nabla (a,_0) \\big ) \\\\& - R^\\nabla ({a}{_0},_1) + R^\\nabla ({a}{_1},_0)- R^\\nabla ({_0}{_1},a) \\\\=& \\nabla _a\\big ( R^\\nabla (_0,_1) \\big ) - \\nabla _{_0}\\big ( R^\\nabla (a,_1) \\big )+ \\nabla _{_1}\\big ( R^\\nabla (a,_0) \\big ) \\\\& - R^\\nabla (\\widetilde{\\nabla _a b_0},_1) + R^\\nabla (_{b_0}a,_1)+ R^\\nabla (\\widetilde{\\nabla _a b_1},_0) - R^\\nabla (_{b_1}a,_0) \\\\& - R^\\nabla \\big (\\alpha (b_0,b_1),a\\big ) - R^\\nabla (\\widetilde{\\nabla _{_0}b_1},a)+ R^\\nabla (\\widetilde{\\nabla _{_1}b_0},a) +R^\\nabla \\big (\\widetilde{\\beta (b_0,b_1)},a\\big )$ and thus $0=& \\big (d^\\nabla R^\\nabla \\big )(a,_0,_1) \\cdot b_2 \\\\=& \\big (R^\\nabla _B\\big )(a;b_0,b_1) \\cdot b_2- \\big (R^\\nabla _B\\big )(a;b_1,b_0) \\cdot b_2- R^\\nabla _B\\big (a,\\beta (b_0,b_1)\\big ) \\cdot b_2 \\\\& + \\nabla _a\\big (R^\\nabla (_0,_1)\\big )\\cdot b_2- R^\\nabla \\big (\\widetilde{\\nabla _a b_0},_1\\big ) \\cdot b_2- R^\\nabla \\big (_0,\\widetilde{\\nabla _a b_1}\\big ) \\cdot b_2$ or, equivalently, $ 0=R_3(b_0,b_1,b_2)-R_3(b_1,b_0,b_2)-R_2\\big (\\beta (b_0,b_1),b_2\\big )+\\big (\\partial ^A\\Omega \\big )(b_0,b_1)\\cdot b_2 .$ For any $a\\in {A}$ and $b_0,b_1\\in {B}$ , we have $ {\\alpha (b_0,b_1)}{a}+\\alpha (\\nabla _a b_0,b_1)+\\alpha (b_0,\\nabla _a b_1)= _{b_0}_{b_1}a-_{b_1}_{b_0}a-_{\\widetilde{q{_0}{_1}}} a .$ We have $& p\\big ( {_1}{{_0}{a}}+{_0}{{a}{_1}}+{a}{{_1}{_0}} \\big ) \\\\=& p{_1}{p{_0}{a}}+p{_0}{p{a}{_1}}+p{a}{p{_1}{_0}} \\\\& +p{_1}{\\widetilde{q{_0}{a}}}+p{_0}{\\widetilde{q{a}{_1}}}+p{a}{\\widetilde{q{_1}{_0}}} \\\\=& _{b_1}_{b_0}a-_{b_0}_{b_1}a+ p{a}{\\alpha (b_1,b_0)}+p{\\widetilde{\\nabla _a b_0}}{_1}+p{_0}{\\widetilde{\\nabla _a b_1}}+p{\\widetilde{q{_0}{_1}}}{a} \\\\=& _{b_1}_{b_0}a-_{b_0}_{b_1}a+ {\\alpha (b_0,b_1)}{a} +\\alpha (\\nabla _a b_0,b_1)+\\alpha (b_0,\\nabla _a b_1)+_{\\widetilde{q{_0}{_1}}}a.$ The result follows from the Jacobi identity in $L$ .", "For any $n\\ge 2$ , $a\\in {A}$ and $b_0,b_1,\\dots ,b_n\\in {B}$ , we have $R_{n+1}(a;b_0,b_1,b_2,\\cdots ,b_n)-R_{n+1}(a;b_1,b_0,b_2,\\cdots ,b_n) = \\\\\\Omega (b_0,b_1)\\cdot R_{n-1}(a;b_2,\\cdots ,b_n)-\\sum _{j=2}^n R_{n-1}\\big (a;b_2,\\cdots ,\\Omega (b_0,b_1)\\cdot b_j,\\cdots ,b_n\\big ) \\\\+\\nabla _{\\alpha (b_0,b_1)}\\big (R_{n-1}(a;b_2,\\cdots ,b_n)\\big )-\\sum _{j=2}^n R_{n-1}\\big (a;b_2,\\cdots ,\\nabla _{\\alpha (b_0,b_1)} b_j,\\cdots ,b_n\\big ) \\\\-R_{n-1}\\big ( {\\alpha (b_0,b_1)}{a}+\\alpha (\\nabla _a b_0,b_1)+\\alpha (b_0,\\nabla _a b_1);b_2,\\cdots ,b_n \\big ) \\\\+R_n\\big (a;\\beta (b_0,b_1),b_2,\\cdots ,b_n\\big ).$ Straightforward computation at the last step of which use is made of Lemma REF .", "Let $L=A\\bowtie B$ be a matched pair of Lie algebroids endowed with a flat torsion free $B$ -connection on $B$ .", "(These data determine a splitting of the short exact sequence of vector bundles $0\\rightarrow A\\rightarrow L\\rightarrow B\\rightarrow 0$ and an $L$ -connection on $B$ extending the $A$ -action such that the three associated bundle maps $\\alpha $ , $\\beta $ , and $\\Omega $ are trivial.)", "Then each $R_n:\\otimes ^n B\\rightarrow (A,B)$ is totally symmetric in its $n$ arguments.", "It follows from Proposition REF and Lemma REF that $R_2$ and $R_3$ are invariant under the permutation of their first two arguments.", "By Lemma REF , the same property holds for all higher $R_n$ .", "Moreover, it is easy to see that, if $R_n$ is symmetric in its $n$ arguments, then $R_{n+1}$ is symmetric in its last $n$ arguments since $R_{n+1}=R_n$ .", "The result follows by induction.", "Theorem REF , which says that $V[-1]$ is an $L_\\infty $ -algebra when the assumptions of Proposition REF are satisfied, is an immediate consequence of Proposition REF and Theorem REF ." ] ]

[ [ "SDSS J184037.78+642312.3: The First Pulsating Extremely Low Mass White\n Dwarf" ], [ "Abstract We report the discovery of the first pulsating extremely low mass (ELM) white dwarf (WD), SDSS J184037.78+642312.3 (hereafter J1840).", "This DA (hydrogen-atmosphere) WD is by far the coolest and the lowest-mass pulsating WD, with Teff = 9100 \\pm 170 K and log g = 6.22 \\pm 0.06, which corresponds to a mass ~ 0.17 Msun.", "This low-mass pulsating WD greatly extends the DAV (or ZZ Ceti) instability strip, effectively bridging the log g gap between WDs and main sequence stars.", "We detect high-amplitude variability in J1840 on timescales exceeding 4000 s, with a non-sinusoidal pulse shape.", "Our observations also suggest that the variability is multi-periodic.", "The star is in a 4.6 hr binary with another compact object, most likely another WD.", "Future, more extensive time-series photometry of this ELM WD offers the first opportunity to probe the interior of a low-mass, presumably He-core WD using the tools of asteroseismology." ], [ "Introduction", "Asteroseismology allows us to probe below the photosphere and into the interiors of stars.", "There are many pulsational instability strips on the Hertzsprung-Russell diagram, including the DAV (or ZZ Ceti) instability strip, driven by a hydrogen partial ionization zone in the hydrogen atmosphere (DA) WDs.", "Seismology using the non-radial $g$ -mode pulsations of DAVs enables us to constrain the mass, core and envelope composition, rotation rate, and the behavior of convection in these objects (see reviews by [38] and [9]).", "The mass distribution of DA WDs in the SDSS shows a strong peak at 0.6 $M_{\\odot }$ with tails toward higher and lower masses [36]; masses of individual WDs range from about 0.2 $M_{\\odot }$ to 1.3 $M_{\\odot }$ .", "The roughly 150 DAVs known to date have masses $0.5-1.1$ $M_{\\odot }$ , implying they all likely contain C/O-cores.", "Lower mass WDs are likely to pulsate as well.", "However, previous searches have failed to detect such pulsations [33].", "The galaxy is not old enough to produce low mass ($<0.5$ $M_{\\odot }$ ) WDs through single-star evolution; these WDs are believed to be the product of binary evolution.", "Indeed, radial velocity surveys of low-mass WDs indicate that most form in binary systems [20], [4].", "Many short-period binaries go through one or two common-envelope phases, which may effectively remove enough mass to prevent ignition of He to C/O.", "However, there is little direct evidence that low-mass WDs have He-cores.", "But if they pulsate, as do their C/O-core brethren, we may differentiate their interior structure.", "We have been engaged in an ongoing search for low-mass DAVs for many years at McDonald Observatory.", "The benefits of a search for a low-mass (and putatively He-core) DAV were recently emphasized by [32].", "Should they pulsate in $g$ -modes like the C/O-core DAVs, the eigenfunctions of ELM WDs would globally sample the interior, making the pulsations sensitive to core composition.", "Seismology may also allow us to constrain the hydrogen layer mass; this is vitally important since hydrogen burning is expected to be a major or even dominant component of the luminosity of these stars [30].", "[33] outlined the null results of a search for pulsations in 12 low-mass WDs.", "We have extended a similar, systematic search for variable He-core WDs, armed with the many dozens of new extremely low-mass (ELM, $\\sim 0.2$ $M_{\\odot }$ ) WDs catalogued by the ELM Survey [3], [15], [6], [17].", "That search has yielded its first success.", "In this Letter, we report the discovery of pulsations in the ELM WD SDSS J184037.78+642312.3, and show that the photometric variations are likely multi-periodic.", "We have also included our null results for another eight low-mass WDs that were observed not to vary, to various detection limits." ], [ "Observations", "[6] present the spectroscopic discovery data for J1840 from the Blue Channel spectrograph on the 6.5m MMT.", "They use 37 separate spectra over more than a year to determine the system parameters.", "[6] find that J1840 is in a $4.5912\\pm 0.0012$ hr ($16528.32\\pm 4.32$ s) orbital period binary with a $K=272\\pm 2$ km s$^{-1}$ radial velocity semi-amplitude.", "However, a significant alias exists at 3.85 hr.", "Model fits to the co-added spectra for J1840 yield ${T}_{\\mathrm {eff}}$ $=9140\\pm 170$ K and $\\log {g}$ $= 6.16\\pm 0.06$ , which correspond to a mass of roughly 0.17 $M_{\\odot }$ .", "Given the mass function of the system, $f = 0.399\\pm 0.009$ $M_{\\odot }$ , the minimum mass of the unseen companion is 0.64 $M_{\\odot }$ ; if the orbital inclination is random, there is a 70% probability that the companion is a WD with $<1.4$ $M_{\\odot }$ .", "We note, however, that the nature of the unseen companion has no bearing on the impact of this Letter.", "We obtained high speed photometric observations of J1840 at the McDonald Observatory over three consecutive nights in 2011 October, for a total of more than 5.5 hr of coverage.", "We used the Argos instrument, a frame-transfer CCD mounted at the prime focus of the 2.1m Otto Struve telescope [26], to obtain many 15 s exposures on this $g_0=18.8$ mag WD.", "The seeing averaged 1.5 and transparency variations were low, although our second and third nights were cut short by clouds.", "Observations were obtained through a 1mm BG40 filter to reduce sky noise.", "We performed weighted aperture photometry on the calibrated frames using the external IRAF package $\\textit {ccd\\_hsp}$ written by Antonio Kanaan (the reduction method is outlined in [12]).", "We divided the sky-subtracted light curves by five brighter comparison stars in the field to allow for fluctuations in seeing and cloud cover, and applied a timing correction to each observation to account for the motion of the Earth around the barycenter of the solar system [34], [35].", "Figure REF shows all 1365 Argos light curve points obtained for J1840 from 25 Oct 2011 to 27 Oct 2011.", "We also include the light curve of the brightest comparison star in the field, SDSS J184043.21+642351.8, for reference." ], [ "Analysis", "Our photometric data set is relatively short, as we caught J1840 just before it went behind the Sun.", "This limits the significance of the detected periods, and we eagerly anticipate further observations.", "Still, we have sufficient data to show convincingly that this low-mass WD is a multi-periodic variable star.", "cccc 4 0.45 Frequency solutions for SDSS J1840+6423 Period Frequency Amplitude S/N (s) ($\\mu $ Hz) (%) 4cMulti-mode solution I 4445.9 $\\pm $ 1.4 224.926 $\\pm $ 0.070 6.59 $\\pm $ 0.19 8.0 2376 $\\pm $ 57 420 $\\pm $ 10 4.883 $\\pm $ 0.83 6.3 1578.56 $\\pm $ 0.37 633.49 $\\pm $ 0.15 2.95 $\\pm $ 0.25 4.4 4cMulti-mode solution II 4445.3 $\\pm $ 2.4 224.96 $\\pm $ 0.12 7.6 $\\pm $ 1.6 9.4 2376.07 $\\pm $ 0.74 420.86 $\\pm $ 0.13 4.817 $\\pm $ 0.46 6.3 1578.70 $\\pm $ 0.65 633.43 $\\pm $ 0.26 2.831 $\\pm $ 0.41 4.3 3930 $\\pm $ 300 254 $\\pm $ 19 2.7 $\\pm $ 2.0 3.4 1164.15 $\\pm $ 0.38 859.00 $\\pm $ 0.29 1.78 $\\pm $ 0.29 3.3 4cSingle-mode harmonic solution 4443.77 $\\pm $ 0.80 225.034 $\\pm $ 0.041 7.35 $\\pm $ 0.19 5.7 2221.89 $\\pm $ 0.40 450.068 $\\pm $ 0.081 4.03 $\\pm $ 0.19 3.5 1481.26 $\\pm $ 0.27 675.10 $\\pm $ 0.12 1.53 $\\pm $ 0.18 1.6 The high-amplitude variability is easy to distinguish in the raw light curve (Figure REF ), with more than 25% peak-to-peak variability.", "The highest peak in a Fourier transform (FT) of the brightest companion star in the field yields only a small signal (0.24% amplitude) at $7340\\pm 15$  s, consistent with low-frequency noise from atmospheric variability.", "Our apertures and sky annuli have been chosen to ensure there is no significant contamination from the nearby star SDSS J184038.73+642315.6, which is 7.0 away from our target.", "Thus the signal we are observing is intrinsic to the WD.", "Without evidence for a companion star or accretion from the spectra, the light curves, or existing broadband photometry, we are confident that the photometric variability results from pulsations on the surface of the ELM WD.", "The pulse shape appears non-sinusoidal, with a steep rise and decline.", "We first test whether a single mode ($f_1$ ) and its harmonics ($2f_1 + 3f_1 + ...$ ) could reproduce the observed light curve.", "A nonlinear least squares fit with the highest peak in the FT and its fixed harmonics converges on 225.03 $\\mu $ Hz (4444 s) as the best parent mode (see the bottom of Table  for a full solution).", "However, our data from multiple nights rule out this scenario.", "The single-mode harmonic solution fits the two peaks during the first night well, but it fails to match the peak in our second night of data, predicting a maximum in the light curve more than 500 s too soon.", "Figure: Fourier transforms of the light curve.", "The top panel shows the original data, the second panel the data after prewhitening by the peak at 224.926 μ\\mu Hz (4445.9 s), the third panel after prewhitening by the peak at 420 μ\\mu Hz (2376 s), and the fourth panel after prewhitening by the peak at 633.49 μ\\mu Hz (1578.56 s).", "See Table  for our frequency solution.", "The dashed blue line shows 4 times the noise level, as described in the text.", "The bottom panel shows the spectral window folded around 633.49 μ\\mu Hz.A multi-mode solution (see the top panels of Table ) where the frequencies are determined from the highest amplitudes in an FT of the entire dataset (see Figure REF ) improves the residuals by more than 20%.", "For more realistic estimates, the cited errors are not formal least-squares errors to the data but rather the product of $10^5$ Monte Carlo simulations of perturbed data using the software package Period04 [19].", "The signal-to-noise calculation is based on the average amplitude of a 1000 $\\mu $ Hz box after pre-whitening by the three significant, highest-amplitude frequencies.", "A small but possibly significant amount of power remains around 254 and 859 $\\mu $ Hz after prewhitening by the three significant periodicities (with S/N $> 4$ ), so we have included a frequency solution with these additional periods in Table .", "This set of period solutions is by no means exhaustive, but it establishes that this ELM WD is variable and multi-periodic." ], [ "Discussion", "We announce the discovery of the first pulsating extremely low mass white dwarf, J1840, which is both the coolest and lowest-mass WD known to pulsate.", "The object offers, for the first time, an opportunity to explore the interior of a putative He-core WD using asteroseismology.", "Asteroseismology of ELM WDs will help constrain the thickness of the surface hydrogen layers in these low-mass WDs.", "There are several millisecond pulsars with ELM WD companions, and the cooling ages of such companions can be used to calibrate the spin-down ages of these pulsars.", "However, current evolutionary models for ELM WDs are relatively unconstrained.", "For masses $M>0.17~M_\\odot $ , diffusion-induced hydrogen-shell flashes take place, which yield small hydrogen envelopes [1], [30], [14].", "The models with $M \\le 0.17~M_\\odot $ do not experience thermonuclear flashes.", "As a result, they have massive hydrogen envelopes, larger radii, lower surface gravities, and they are predicted to evolve much more slowly compared to more massive WDs.", "If enough modes are excited to observability, we hope to directly constrain the hydrogen layer mass.", "A plethora of excited modes would also allow for measuring the mean period spacing, which is a sensitive function of the mass of the star, and is also slightly dependent on the core mass fraction.", "The models of [32] found a mean period spacing of $\\sim 89$ s for $\\ell =1$ $g$ -modes of a 0.17 $M_{\\odot }$ ELM WD, about a factor of two larger than the observed $47\\pm 12$ s period spacing for $\\ell =1$ $g$ -modes in the cool C/O-core counterpart G29-38 [18].", "We do expect this non-radial pulsator to be multi-periodic even if just one pulsation mode and its non-linear harmonics are amplified to observability: J1840 is in a relatively close binary with an unseen compact object.", "This companion will influence the light curve in many ways.", "Although we lack the sensitivity and full phase coverage needed for a detection, we expect a $\\sim 0.3$ % amplitude Doppler beaming signal at the orbital period given the effective temperature of J1840 and the radial-velocity amplitude [31].", "The companion will also induce tidal distortions on the primary, as seen in many other ELM WDs [16], [5], [11].", "However, assuming that J1840 has a radius of roughly 0.054 $R_{\\odot }$ [30], ellipsoidal variations should be smaller than 0.1% for even the highest possible inclinations [24].", "The companion's effect on the rotation period is perhaps more significant.", "The $\\sim 1.7$ Gyr cooling age of this ELM WD [30] may be longer than the synchronization timescale for such a short-period binary [7].", "If so, and the ELM WD is rotating at the orbital period of 4.6 hr, then non-radial pulsations will be subsequently subject to rotational splittings determined by their modal degree.", "For example, if the 225 $\\mu $ Hz mode is an $\\ell = 1$ mode, a 4.6 hr rotation rate would cause it to be split by about 30 $\\mu $ Hz, assuming solid-body rotation [37].", "However, synchronization is not a guarantee: Recent Kepler observations of a close sdB+dM binary found the primary rotating much slower than the $\\sim 9.5$ hr orbital period [29].", "An analysis of the rotational splittings in the pulsations of J1840 will test tidal synchronization in this system, thereby probing the rigidness of this ELM WD.", "Figure: Search parameter space for pulsations in low-mass WDs.", "Known DAVs are included as purple dots, and our new pulsator is labeled and marked as a magenta dot.", "DA WDs observed not to vary to at least 1% are marked with an X; those from are marked in gray and those from our search (see Table ) in black.", "To guide the eye, we have marked the rough boundaries of the observed DAV instability strips as dotted lines; the short, blue dotted line marks the blue-edge of the theoretical instability strip.We also hope to use the nonlinearities in this non-sinusoidal light curve to constrain the size of the convection zone of this WD [22], [23].", "Since ELM WDs exist in a completely new regime of parameter space than do typical C/O core WDs, this analysis will provide important independent constraints on the structure of their outer layers, as well as providing a measure of the convective efficiency in this new regime.", "We discovered the pulsations in J1840 as part of a systematic search for pulsations in low-mass WDs, recently energized by the work of [32], [33].", "We have explored a vast area of the $\\log {g}$ — ${T}_{\\mathrm {eff}}$ parameter space for variability (see Figure REF ), and compiled a list of our first eight null results in Table .", "These null observations were reduced and analyzed in an identical manner to those outlined in Section , and the atmospheric parameters and their formal errors were determined by the references cited.", "lcccrc 6 1.0 Observed Low-Mass DAV Candidates and Null Results Object $g^\\prime $ -SDSS ${T}_{\\mathrm {eff}}$ $\\log {g}$ Reference Det.", "Limit (mag) (K) (cm s$^{-1}$ ) % SDSS J011210.25+183503.74 17.3 $9690\\pm 150$ $5.63\\pm 0.06$ [6] 0.1 SDSS J082212.57+275307.4 18.3 $8880\\pm \\phantom{1}60$ $6.44\\pm 0.11$ [14] 0.2 SDSS J091709.55+463821.8 18.7 $11850\\pm 170\\phantom{1}$ $5.55\\pm 0.05$ [13] 0.3 SDSS J122822.84+542752.92 19.6 $9921\\pm 143$ $7.25\\pm 0.08$ [8] 0.7 SDSS J123316.20+160204.6 19.8 $10920\\pm 160\\phantom{1}$ $5.12\\pm 0.07$ [3] 0.9 SDSS J174140.49+652638.7 18.4 $9790\\pm 240$ $5.19\\pm 0.06$ [6] 0.3 SDSS J210308.80-002748.89 18.5 $9788\\pm \\phantom{1}59$ $5.72\\pm 0.07$ [17] 0.2 SDSS J211921.96-001825.8 20.0 $10360\\pm 230\\phantom{1}$ $5.36\\pm 0.07$ [3] 0.6 Our newfound pulsator J1840 occupies a new, cooler region of the DAV instability strip, which may be an extension of the C/O-core region.", "The highest-amplitude period observed ($> 4400$  s) is the longest, to date, of any period observed in a DAV [25].", "This makes sense qualitatively, as we would expect the periods of pulsation modes to roughly scale with the dynamical timescale for the whole star, $P \\propto \\rho ^{-1/2}$ .", "[27] find longer pulsation periods in lower temperature and surface gravity subdwarf B stars.", "A similar trend would explain the relatively long pulsation period of J1840.", "The period observed is a factor of 4 longer than the models of [32] predict for an $\\ell =1$ , $k=10$ mode of a 0.17 $M_{\\odot }$ ELM WD; in the context of these models, this mode likely has an extremely high radial overtone, higher than that of any dominant mode observed in a normal-mass DAV.", "We note, however, that the driving mechanism for pulsation is largely unknown for this star.", "It is natural to assume that the same mechanism of convective driving that operates in the C/O-core DAVs [2], [39], [10] is also responsible for the pulsations of low-mass WDs.", "This mechanism is based on the assumption that the convective turnover timescale for a fluid element, $t_{\\rm to}$ , is much smaller than the oscillation periods, $P_i$ .", "For the $\\log g \\sim 8$ DAVs, $t_{\\rm to}\\sim 0.1$ –1 sec and $P_i \\sim 100$ –1000 sec, so $t_{\\rm to} \\ll P_i$ is satisfied.", "To estimate how this timescale scales with $g$ we note that $t_{\\rm to} \\sim l/v$ , where $l$ is the mixing length and $v$ is the velocity of the convective fluid elements.", "Employing a simplified version of mixing length theory (ignoring radiative losses) we find that $v \\propto (g F/\\rho )^{1/3}$ , where $F$ is the stellar flux and $\\rho $ is the mass density at the base of the star's surface convection zone [21].", "Taking the mixing length as a factor of order one times a pressure scale height, we find $l \\sim c_s^2/g \\propto T/g$ .", "Putting these results together we find $t_{\\rm to} \\propto T (\\rho /F)^{1/3} g^{-4/3} \\sim g^{-4/3}$ .", "Thus, a $\\log g \\sim 6$ object would have $t_{\\rm to} \\sim 50$ –500 sec.", "In order for convective driving to operate, $P_i \\gg t_{\\rm to}$ is required.", "Perhaps this is at least a partial explanation for the very long period ($> 4000$  s) seen in this pulsator.", "It may also set a lower limit to periods in this DAV of $\\sim 500$ –1000 s, although more detailed models will be needed to confirm this.", "Another potential source of driving is the $\\epsilon $ mechanism, i.e., driving due to the modulation of H burning in the envelope.", "We will address this additional mechanism after more extensive observations.", "We look forward to a coordinated effort for extensive follow-up observations and analysis of this exciting new pulsating WD, and to the discovery of additional pulsating ELM WDs in order to better understand this new (or extended) DAV instability strip.", "J.J.H., M.H.M.", "and D.E.W.", "gratefully acknowledge the support of the NSF under grant AST-0909107 and the Norman Hackerman Advanced Research Program under grant 003658-0252-2009.", "The authors are grateful to the McDonald Observatory support staff, F. Mullally for developing much of the data analysis pipeline used here, and to J. Pelletier of the UT Freshman Research Initiative for some of the analysis used in Table ." ] ]

[ [ "Parity oscillations of Kondo temperature in a single molecule break\n junction" ], [ "Abstract We study the Kondo temperature ($T_K$) of a single molecule break junction.", "By employing a numerical renormalization group calculations we have found that $T_K$ depends dramatically upon the position of the molecule in the wire formed between the contacts.", "We show that $T_K$ exhibits strong \\emph{oscillations} when the parity of the left {and/or} right number of atomic sites ($N_L,N_R$) is changed.", "For a given set of parameters, the maximum value of $T_K$ occurs for ($odd,odd$) combination, while its minimum values is observed for ($even,even$).", "These oscillations are fully understood in terms of the effective hybridization function." ], [ "Parity oscillations of Kondo temperature in a single molecule break junction B. M. F. Resende Instituto de Fí­sica - Universidade Federal de Uberlândia - Uberlândia, MG 38400-902 - Brazil E. Vernek [Corresponding author:][email protected] Instituto de Fí­sica - Universidade Federal de Uberlândia - Uberlândia, MG 38400-902 - Brazil 72.10.Fk, 72.15.Qm, 73.21.Ac, 73.21.Hb, 73.21.La, 73.63.Kv, 73.63.Nm, 73.21.La Single molecule break junction, Kondo effect, Kondo temperature, Quantum wire We study the Kondo temperature ($T_K$ ) of a single molecule break junction.", "By employing a numerical renormalization group calculations we have found that $T_K$ depends dramatically upon the position of the molecule in the wire formed between the contacts.", "We show that $T_K$ exhibits strong oscillations when the parity of the left and/or right number of atomic sites ($N_L,N_R$ ) is changed.", "For a given set of parameters, the maximum value of $T_K$ occurs for ($odd,odd$ ) combination, while its minimum values is observed for ($even,even$ ).", "These oscillations are fully understood in terms of the effective hybridization function.", "Kondo effect (KE) is one of the most intriguing phenomena of strong correlated systems,[1] which was beautifully explained by J. Kondo in the 60's in the seminal theoretical work on the minimal resistance in magnetic alloys.", "[2] KE has revived in the later 90's with the advent of the scanning tunneling microscope (STM) that has facilitated the manipulation of the matter at atomic scale.", "For instance, STM has allowed observation of interesting facets of the KE in quantum dots (QD)[3], and in single atom or molecule on metallic surfaces,[5], [6], [7] which have motivated a huge number of experimental[3], [4], [8], and theoretical[10], [11] investigations.", "One of the experimentally accessible signatures of the KE in nanoscopic system such as QD or magnetic molecules attached to metallic contacts is the strong modification in the conductance across the system, observable when the system is cooled down below the so-called Kondo temperature ($T_K$ ).", "In QD, for instance, $T_K$ is found to be in the sub Kelvin region whereas for large molecules attached to metallic leads $T_K$ can be much larger.", "[7] In both cases, in the limit of very strong Coulomb interaction, $T_K$ depends strongly upon the effective hybridization ($\\Delta $ ) that connects the localized magnetic moments to the conduction electrons[12], [7] as[1] $T_K\\sim \\exp ({\\pi \\varepsilon _d/\\Delta })$ , where $\\varepsilon _d$ ($<0$ ) is the energy of the localized orbital respect to the Fermi level.", "Controlling $\\Delta $ or $\\varepsilon _d$ is therefore crucial for obtaining higher $T_K$ , which is fundamental for possible technological application of KE.", "While tuning $\\varepsilon _d$ is relatively simple in QDs by mean of gate voltages, in molecules, on the other hand, it becomes a more complicated task.", "Conversely, geometrical parameters are more suitably modified in molecules than in QDs and has proven to produce important modifications in $T_K$ via hybridization function.", "[13] A suitable experimental arrangement to study the KE is the break junction (BJ) molecular structures, in which a metallic wire (gold wire, for instance) is stretched until a few-atom 1D chain bridges the gap between the electrodes before the complete break up of the wire.", "[14], [15], [16], [17], [18] Owing to the dependence of $T_K$ upon $\\Delta $ it has been shown that $T_K$ can be mechanically modulated in BJ experiments[13] by changing the distance between the electrodes.", "Motivated by this experiment, in the present work we study the Kondo temperature of a spin-$1/2$ magnetic impurity coupled to metallic contacts through two finite (left and right) quantum wires (QW), as illustrated in Fig.", "REF (a).", "By employing a numerical renormalization group[19], [20] (NRG) calculation we find a strong dependence of $T_K$ upon the parity of the number of sites ($N_R,N_L$ ) of each QW as well as their length.", "The dependence upon the ($N_L,N_R$ ) parity combination results in an oscillating behavior of $T_K$ as function of $N_L$ or $N_R$ , akin to what has been observed in Manganese phthalocyanine (MnPc) molecules on top of Pb islands, reported in Ref. PhysRevLett.99.256601.", "Although the system under investigation here is quite different from the one studied in Ref.", "PhysRevLett.99.256601, the origin of the oscillation of $T_K$ can be interpreted likewise.", "While in their case the enhancement of $\\Delta $ originates from the formation of multiple quantum well states between the Pb atomic layers, here the enhancement of $\\Delta $ results from the localized states of the atomic sites of the QW.", "Figure: (Color online) (a) Illustration of a Au quantumwire coupled to metallic contacts with a embedded C 60 _{60} molecule.", "(b) Pictorialrepresentation of the model.", "The site labeled as “0” represents the impurity site with strongon-site coulomb repulsion.Our system model is schematically represented in Fig.", "REF (b) and is modeled by the Anderson-type Hamiltonian that can be split into five terms as $H=H_{imp}+H_{cb}+H_{wires}+H_{imp-wires} +H_{cbs-wires},$ where $H_{imp}$ , $H_{cb}$ and $H_{wires}$ describe, respectively, the interacting impurity, the free electrons in the conduction bands and the electrons in the wires, $H_{imp-wires}$ couples the impurity to the two wires and $H_{cb-wires}$ couples the wires the their respective conduction bands.", "In terms of creation and annihilation fermion operators the Hamiltonians read $H_{imp}&=&\\sum _{\\sigma }\\varepsilon _dc^\\dagger _{d\\sigma }c_{d\\sigma }+Un_{d\\uparrow }n_{d\\downarrow },\\\\H_{cb}&=&\\sum _{\\ell =R,L}\\sum _{k\\sigma }\\varepsilon _kc^\\dagger _{\\ell k\\sigma }c_{\\ell k\\sigma }\\\\H_{wires}&=&\\sum _{\\ell =R,L}\\left[\\leavevmode {\\color {black}\\varepsilon _0\\sum _{i_\\ell =1\\atop \\sigma }^{N_\\ell }n_{i_\\ell \\sigma }+}t\\sum _{i_\\ell =1\\atop \\sigma }^{N_\\ell -1}\\left(c^\\dagger _{i_\\ell \\sigma }c_{i_\\ell +1} +H.c.\\right)\\right],\\\\H_{cb-wires}&=&\\sum _{\\ell =R,L}\\sum _{\\ell k\\sigma } \\left(V_{\\ell k}c^\\dagger _ { N_{\\ell }}c_{\\ell k\\sigma }+V^*_{\\ell k} c^\\dagger _{\\ell k\\sigma }c_{ N_\\ell }\\right)\\\\H_{imp-wires}&=&t^\\prime \\sum _{\\ell =R,L}\\sum _{\\ell \\sigma }\\left(c^\\dagger _{d\\sigma }c_{1_\\ell }+c^\\dagger _{1_\\ell \\sigma }c_{d\\sigma } \\right).$ In Eqs.", "REF -, the operators $c^\\dagger _{d\\sigma }$ ($c_{d\\sigma }$ ) creates (annihilates) an electron in the orbital $d$ with energy $\\varepsilon _d$ , $c^\\dagger _{\\ell k\\sigma }$ ($c_{\\ell k\\sigma }$ ) creates (annihilates) an electron in the $\\ell $ th conduction band with energy $\\varepsilon _k$ , $c^\\dagger _{i_\\ell \\sigma }$ ($c_{i_\\ell \\sigma }$ ) creates (annihilates) and electron in the $i$ th site of the $\\ell $ th QW with energy $\\varepsilon _0$ spin $\\sigma $ .", "Finally, $t$ is the hopping between two adjacent sites in the wires and $V_{\\ell k}$ and $t^\\prime $ couple the QWs to their conduction bands and to the impurity, respectively.", "The conduction bands are characterized by a flat density of states, $\\rho (\\omega )=(1/2D)\\Theta (D-|\\omega |)$ , where $D$ is their half bandwidth and $\\Theta (x)$ is the Heaviside step function.", "It is worth emphasizing that although the motivating experiment was realized using C$_{60}$ molecule coupled to Au metallic contacts, this model is rather general.", "In the particular context of molecular BJ, vibrations may be important in certain range of parameters, but this aspect is beyond the scope of the present work.", "In order to properly address the Kondo physics of the system, the full Hamiltonian is approached by using the numerical renormalization method, with which we can calculate the thermodynamical properties.", "Within the NRG approach we discretize the effective conduction band “seen” by the interacting impurity.", "The effective conduction band can be determined by exact calculation of the local non-interacting ($U=0$ ) Green's function (suppressing the spin index), $g_{dd}(\\omega )=[\\omega -\\epsilon _d+\\Sigma (\\omega )]^{-1}$ , where $\\Sigma =\\Sigma _R(\\omega )+\\Sigma _L(\\omega )$ , in which the $\\ell $ th self-energy is given by $\\Sigma _\\ell (\\omega )=-\\cfrac{t^{\\prime 2}}{\\omega -\\cfrac{t^2}{\\omega -\\cfrac{ t^2}{ \\omega -\\cfrac{t^2}{ \\cfrac{\\ddots t^2}{ \\omega -V_\\ell ^2\\tilde{g}(\\omega )}}}}},$ with $\\tilde{g}(\\omega )=-\\frac{1}{2D}\\ln \\Big |\\frac{\\omega -D}{\\omega +D}\\Big |-\\frac{i\\pi }{2D}\\Theta (D-|\\omega |)$ being the diagonal GF associated to the unperturbed conduction electrons in the leads.", "The fraction in Eq.", "REF is continued until all the sites of the of the $\\ell $ th wire and the $\\ell $ th conduction band are taken into account.", "The hybridization of the localized orbital “$d$ ” with the effective band is give by $\\Delta (\\omega )={\\tt Im}[\\Sigma (\\omega )]=\\Delta _L(\\omega )+\\Delta _R(\\omega )$ [Hereafter we will refers to $\\Delta (0)$ just as $\\Delta $ ].", "The hybridization function is logarithmically discretized[20], [22] to map the system in a Wilson's chain form,[19] $H=H_{imp}+t^\\prime \\sum _{\\sigma }\\left(c^\\dagger _{d\\sigma }c_{0\\sigma }+c^\\dagger _{0\\sigma }c_{d\\sigma }\\right)+\\sum _{i=0\\atop \\sigma }^\\infty \\varepsilon _{i\\sigma }c^\\dagger _{i\\sigma }c_{i\\sigma }\\nonumber \\\\+\\sum _{i=0\\atop \\sigma }^\\infty t_i\\left(c^\\dagger _{i\\sigma }c_{i+1\\sigma }+c^\\dagger _{i+1\\sigma }c_{i\\sigma }\\right),$ where $t_i$ 's are calculated via $\\Delta (\\omega )$ , following the recipes described in Ref. PhysRevB.52.14436.", "Once we have mapped the system in the Wilson's form, we proceed the NRG calculation, which is based in the iterative diagonalization of the effective Hamiltonian.", "[20] After reaching the strong coupling fixed point we can calculate the magnetic moment within the canonical ensemble as $k_BT\\chi (T)=\\frac{1}{Z(T)}\\sum _{\\nu }\\left[\\langle \\nu |S^2_z|\\nu \\rangle -(\\nu |S_z|\\nu \\rangle )^2\\right]e^{-E_\\nu /k_BT},$ where $k_B$ is the Boltzmann constant, $Z(T)=\\sum _\\nu \\exp (-E_\\nu /k_BT)$ is the canonical partition function, $S_z$ is the spin operator, $|\\nu \\rangle $ and $E_\\nu $ are, respectively, the eigenvector and its corresponding eigenvalue of the full interacting Hamiltonian, which are naturally calculated in the NRG procedure.All the results were obtained using the conventional NRG discretization parameter $\\Lambda =2.5$ and keeping typically 1500 states at each iteration.", "Following Wilson's criterion, we define $T_K$ from the “impurity” magnetic moment as $k_BT_K\\chi _{imp}(T_K)=0.0707(g\\mu _B)^2$ , (that is the magnetic moment of the full system subtracted by the contribution of the effective conduction band), $g$ is the electron $g$ -factor and $\\mu _B$ is the Bohr magneton.", "Before starting the presentation of our numerical results, lets analyze the hybridization function at the Fermi level, which is the most relevant parameter to determine the behavior of $T_K$ in our calculations.", "It is straightforward to show from Eq.", "REF that $\\Delta $ possesses only three distinct vales, $\\Delta =\\left\\lbrace \\begin{array}{cc}\\Delta _{min}=\\Delta _0 &\\quad \\mbox{for (even,even)}\\\\\\Delta _{int}=\\Delta _0\\left(\\frac{1}{2}+ \\alpha \\right) & \\quad \\mbox{for (even,odd) or(odd,even)}\\\\\\Delta _{max}=2\\alpha \\Delta _0 & \\quad \\mbox{for (odd,odd)}\\end{array}\\right.,$ where we have defined $\\Delta _0=\\pi t^{\\prime 2}/D$ and denoted $\\Delta _{min}$ , $\\Delta _{int}$ and $\\Delta _{max}$ , the minimum, intermediate and maximum value of $\\Delta $ , respectively, and $\\alpha =2(D/\\pi t)^2$ is a dimensionless parameter that can be modified, for instance, by stretching the QW as is was done in the Ref. PhysRevLett.99.026601.", "To obtain our numerical results lets set $D=1$ as our energy scale.", "With that we choose hereafter (unless otherwise stated) $U=0.5$ , $\\epsilon _{d}=-0.25$ , $\\varepsilon _0=0$ (at the particle-hole symmetric point), $V_R=V_L=t=0.15$ and $t^\\prime =0.1$ .", "Figure: (Color online) Hybridization function vs energy for various values ofN L N_{L} and N L N_L [denoted in the figure as (N L ,N R N_L,N_R) using V L =V R =t=0.15V_{L}=V_{R}=t=0.15 andt ' =0.1t^\\prime =0.1.", "Notice thatΔ\\Delta possesses three different values, depending on the parity of N L N_L and N R N_R.", "Theminimum (Δ min \\Delta _{min}) and maximum (Δ max \\Delta _{max}) value of Δ\\Delta is obtained for(even,even)(even,even) (b) and (odd,odd)(odd,odd) (e), respectively, while for all the other combinations Δ\\Delta has an intermediate value, Δ int \\Delta _{int}.In Fig.", "REF we show the hybridization function vs energy for various values of $N_L$ and $N_R$ .", "In Figs.", "REF (a), REF (b), and REF (c) we fix $N_L=0$ and show $\\Delta (\\omega )$ for $N_R=1$ , $N_R=2$ and $N_R=3$ , while in Figs.", "REF (d), REF (e), and REF (f) we keep $N_L=1$ fixed and show $\\Delta (\\omega )$ for $N_R=2$ , $N_R=3$ and $N_R=4$ .", "The number of peaks of $\\Delta (\\omega )$ is given by $\\max (N_L,N_R)$ for equal parity and $N_R+N_L$ for different parities.", "Although the structure of $\\Delta (\\omega )$ away from the Fermi level has some effect on $T_K$ , the most relevant contribution comes from the structures at or very close the the Fermi level.", "For the parameters set above, we obtain $\\Delta _{min}\\approx 0.0314$ , $\\Delta _{max}\\approx 0.566$ and $\\Delta _{int}\\approx 0.299$ .", "These distinct values of $\\Delta $ are crucial for determining the Kondo temperature of the system, which in our case can be roughly estimated as[24] $k_BT_K(\\Delta )\\sim \\exp [-\\pi U/8\\Delta ]$ .", "It is clear that $T_K$ increases as $\\Delta $ increases.", "Figure: (Color online) Magnetic moment as function of temperature for variousvalues of N R =N L =NN_R=N_L=N.", "Panel (a) and (b) correspond to NN even and odd,respectively (see values of NN in the legends).In Fig.", "REF (a) and Fig.", "REF (b) we show the magnetic moment as function temperature for various values of $N=N_L=N_R$ (the symmetric case) even and odd, respectively.", "The case of $N=0$ [Fig.", "REF (a), $$ (black)] corresponds to the single impurity coupled to two conduction band.", "The low temperature suppression in the magnetic moment results from the Kondo screening of the local spin [these curves are used to $T_K$ , as discussed above].", "On the other hand, in the high temperature limit the $k_BT\\chi \\rightarrow (g\\mu _B)^2/8$ , as expected.", "Notice in Fig.", "REF (a) that $T_K$ increases when $N$ (even) increases.", "Conversely, $T_K$ decreases when $N$ (odd) increases as seen in Fig.", "REF (b).", "Notice also that $T_K$ can be at least two order of magnitude larger for $N$ odd than for $N$ even.", "This huge difference will be analyze below.", "The result for $N=1$ [Fig.", "REF (b), $$ (black)] is equivalent to those reported in Ref. PhysRevLett.97.096603.", "The negative values of $k_BT\\chi _{imp}$ within a small range of $T$ results from the subtraction of the effective conduction band contribution.", "Figure: (Color online) Kondo temperature as function ofnumber of sites (N R N_{R}) for a fixed N L =1N_{L}=1.", "(black) and □\\square (red)correspond to even (odd) N R N_R, respectively.", "The zig-zag (green) line shows the (even,oddeven,odd)oscillation, similar to those observed in Ref.", "PhysRevLett.99.256601.In order to show the behavior of the Kondo temperature for larger and different values of $N_L$ and $N_R$ we show in Fig.", "REF $T_K$ as function of $N_R$ for a fixed number $N_L$ even (left) and $N_L$ odd (right).", "The $$ (black) curves correspond to $N_R$ even, while $\\square $ (red) curves corresponds to $N_R$ odd.", "Notice that for $N_L$ even [(Fig.", "REF (a), REF (b) and REF (c)] $T_K$ increases with $N_R$ even [$$ (black)] while it decreases for $N_R$ odd [$\\square $ (red)].", "Observe again that for small $N_R$ $T_K$ is almost two order of magnitude larger for $N_R$ odd than for $N_R$ even (keeping $N_L$ even).", "This difference decreases asymptotically for large $N_R$ and vanishes asymptotically as $N\\rightarrow \\infty $ .", "This results from the fact that in this situation the conduction electrons near the Fermi level are more strongly coupled to the impurity, reflecting the fact that $\\Delta _{int}$ is larger than $\\Delta _{min}$ as clearly shown in Fig.", "REF .", "For $N_L$ odd (Fig.", "REF (d), REF (e) and REF (f) we observe a similar behavior ($T_K$ increases as $N_R$ even increases and decreases as $N_R$ even increases) but in this case the curves collapse onto each other very quickly (typically for $N_R=10$ ) to a larger value, when compared to the case of $N_L$ even.", "At least for small $N_L$ and $N_R$ we can roughly estimate the ratio between $T_K$ 's for the three distinct values of $\\Delta $ as $\\frac{T_K(\\Delta _{a})}{T_K(\\Delta _{b})}=e^{\\frac{\\pi U}{8}\\left(\\frac{\\Delta _a-\\Delta _b}{\\Delta _a\\Delta _b}\\right)},$ where $a$ and $b$ stand for $min$ , $int$ and $max$ .", "Using the parameters chosen above we obtain $T_K(\\Delta _{int})/T_K(\\Delta _{min})\\approx 2.7\\times 10^2$ , while $T_K(\\Delta _{max})/T_K(\\Delta _{int})\\approx 1.36$ .", "These values are consistent with the huge difference between the values shown in $\\square $ (red) and $$ (black) curves of Figs.", "REF (a), REF (b) and REF (c) and small difference in the related curves of Figs.", "REF (d), REF (e) and REF (f).", "The behavior of $T_K$ with increasing $N_L$ and $N_R$ for the same parity combination cannot be explained in terms of $\\Delta $ .", "This can be reasonably understood in terms of the formation of a small sub-band inside the conduction band, due to a large number of atomic sites and the energy dependence of hybridization function near the Fermi level.", "In the limit of $N_L,N_R\\rightarrow \\infty $ the sub-band becomes a smooth curve, resulting in a $T_K$ independent of the lengths of the wires.", "The zig-zag (green) line in Fig.", "REF (b) shows the even-odd oscillations in $T_K$ , very similar to what was observed in Ref. PhysRevLett.99.256601.", "Finally, in Fig.", "REF we show the robustness of these results against particle-hole symmetry breaking.", "In Fig.", "REF (a) show $T_K$ as function of $\\varepsilon _d$ for $\\varepsilon _0=0$ .", "Notice that, although more pronounced for the $(even,even)$ case, $T_K$ increases as $\\varepsilon _d$ is shifted upward from $-U/2$ for all parities [(0,2), (1,2) and (1,3)].", "Same behavior is obtained for the other side (not shown).", "These are consistent with the general expression,[25] $T_K\\sim Exp{\\left[-\\pi |\\varepsilon _d|(\\varepsilon _d+U)/(2\\Delta U)\\right]}$ for constant hybridization function.", "When we keep $\\varepsilon _d$ and vary $\\varepsilon _0$ about the Fermi level [Fig.", "REF (b)] we see that $T_K$ increases for $(0,2)$ but decrease slightly for $(1,2)$ and $(1,3)$ .", "This results from the fact that, as $\\varepsilon _0$ deviates from the Fermi level, $\\Delta (0)$ decreases if $\\Delta (\\omega )$ possesses a peak at the Fermi level as in the $(even,odd)$ , $(odd,even)$ or $(odd,odd)$ cases, but increases when $\\Delta (\\omega )$ exhibits a dip at the Fermi level as in the $(even,even)$ configuration.", "Figure: (Color online) Kondo temperature as function ofε d \\varepsilon _d (ε 0 =0\\varepsilon _0=0) (a) and ε 0 \\varepsilon _0 (ε d =-U/2\\varepsilon _d=-U/2) (b) forvarious configuration of (N L ,N R )(N_L,N_R) as shown in the legend.", "The other parameter are the same as inthe previous figures.In conclusion, we have presented a detailed study of the Kondo temperature of a single molecule break junction.", "By employing a numerical renormalization group we show that $T_K$ is strongly dependent of the parity of the number of atomic sites in each piece of QW connecting the molecule to the contacts.", "More interesting, we show that the $T_K$ oscillates when the parity of the number of sites of the wires changes.", "These oscillations are interpreted in terms of the effective hybridization function $\\Delta (\\omega )$ .", "For $(even,even)$ and $(odd,odd)$ configurations the effective coupling $\\Delta $ is minimum and maximum, respectively, while for $(even,odd)$ or $(odd,even)$ configurations $\\Delta $ possesses an intermediate value.", "Within this picture, the huge variation of $T_K$ is readily estimated by a simple analytical calculation, which can vary up to a factor of $10^2$ [in the case of changing from ($even,even$ ) to ($even,odd$ )].", "Our results provide a very clear picture of the main ingredient responsible for the dramatic dependence of $T_K$ on geometrical configuration of single molecule break junctions as well as of magnetic molecule on atomic layer surfaces.", "Moreover, we believe our results can be used to guide experimental realizations of high-$T_K$ experiments.", "We would like to thank CNPq (under grant No.", "493299/2010-3) and FAPEMIG (under grant No.", "CEX-APQ-02371-10) for financial support.", "We also wish to acknowledge valuable discussions with F. M. Souza." ] ]

[ [ "Central Lyapunov exponent of partially hyperbolic diffeomorphisms of\n $\\mathbb{T}^3$" ], [ "Abstract In this paper we construct some \"pathological\" volume preserving partially hyperbolic diffeomorphisms on $\\toro{3}$ such that their behaviour in small scales in the central direction (Lyapunov exponent) is opposite to the behavior of their linearization.", "These examples are isotopic to Anosov.", "We also get partially hyperbolic diffeomorphisms isotopic to Anosov (consequently with non-compact central leaves) with zero central Lyapunov exponent at almost every point." ], [ "Introduction and statement of the result", "The ergodic theory of “beyond uniformly hyperbolic dynamics\" is an extensive research area and has connection to many other topics.", "Partial hyperbolicity is a form of relaxing the uniform hyperbolicity condition with natural interesting examples (see [2], [6]).", "One of the amazing issues raising in the study of ergodic properties of partially hyperbolic dynamics is the existence of invariant foliations and their topological and metric properties.", "A complete comprehension of invariant foliations by partially hyperbolic dynamics is also an important tool for the classification of these dynamics and the manifolds which support them.", "In this paper we introduce some new “pathological\" examples of partially hyperbolic diffeomorphisms.", "We study the relationship between central Lyapunov exponents and topology of leaves of central foliation of a partially hyperbolic diffeomorphism and its linearization.", "More precisely, we find an open set of partially hyperbolic diffeomorphisms $f: \\mathbb {T}^3 \\rightarrow \\mathbb {T}^3$ isotopic to linear Anosov diffeomorphisms $A$ such that the central Lyapunov exponent of $f$ is positive almost everywhere while the central bundle of $A$ is contracting.", "This opposite behavior in the asymptotic growth (manifested by the sign of Lyapunov exponent) which is a local issue contrasts with the compatible behavior in the large scale between $f$ and $A.$ (See Preliminaries Section.)", "We also obtain examples of partially hyperbolic diffeomorphisms isotopic to Anosov with non compact central leaves and zero central Lyapunov exponent almost everywhere.", "This is also a contrast between the global topology of central leaves of non linear and linearization of a partially hyperbolic diffeomorphism.", ".", "Theorem 1.1 There exists an open set of volume preserving partially hyperbolic diffeomorphisms $U$ such that for any $f \\in U$ almost every point has positive Lyapunov exponent and the linearization of $f$ is an Anosov diffeomorphism with splitting $E^u \\oplus E^s \\oplus E^{ss}.$ Theorem 1.2 There exist volume preserving partially hyperbolic diffeomorphism $f:\\mathbb {T}^{3} \\rightarrow \\mathbb {T}^{3}$ (isotopic to linear Anosov automorphism) with zero central Lyapunov exponent for Lebesgue almost every point of $\\mathbb {T}^{3}$ and non-compact central leaves.", "There are important questions on our pathological examples which are not known by us for the moment.", "We mention that by a recent result of A. Hammerlindl and R. Ures [10], a non-ergodic volume preserving isotopic to Anosov diffeomorphism on $\\mathbb {T}^3$ , if it exists, should have zero central Lyapunov exponent almost everywhere.", "The ergodicity of the diffeomorphisms with zero central exponent in Theorem REF is an open problem.", "Another interesting issue is related to absolute continuity of central foliation.", "We do not know whether the central foliation of diffeomorphisms with zero central exponent in Theorem REF is absolutely continuous (see [1] for a survey about absolute continuity) or not.", "We believe that it is not the case and formulate the following question.", "Question 1: Let $f:M \\rightarrow M$ be a partially hyperbolic diffeomorphism of $M=\\mathbb {T}^{3}$ with absolutely continuous central foliation $\\mathcal {F}^c$ and central Lyapunov exponent equal to zero at Lebesgue almost every point ($\\lambda ^c = 0$ Lebesgue-a.e).", "Is it true that the leaves of $\\mathcal {F}^c$ are compact?", "We remark that A. Tahzibi and F. Micena [11] recently gave an affirmative answer to this question assuming $\\mathcal {F}^c$ satisfies a uniformly bounded density condition which is a regularity condition stronger than leafwise absolute continuity.", "The idea of the proof of Theorems REF and REF is to take a family of hyperbolic linear automorphisms $f_k: \\mathbb {T}^{3} \\rightarrow \\mathbb {T}^{3}$ with eigenvalues $\\lambda ^s_k < \\lambda ^c_k < 1 < \\lambda ^u_k$ in such a way that $\\lambda ^s_k \\rightarrow 0, \\lambda ^c_k \\rightarrow 1, \\lambda ^u_k \\rightarrow \\infty $ as $k \\rightarrow \\infty $ and moreover the corresponding unitary eigenvectors converge to a fixed orthonormal basis.", "Then we apply Baraviera-Bonatti [3] method of local perturbations and by the choice of the Anosov automorphisms, we are able to show that this local perturbation yields a new partially hyperbolic diffeomorphism with positive central Lyapunov exponent in average.", "By continuity argument we find some isotopic to Anosov and partially hyperbolic diffeomorphism with vanishing integral of central Lyapunov exponent.", "Finally, by [10] the diffeomorphisms obtained in Theorem REF are ergodic and so almost every point have the same Lyapunov exponent.", "For diffeomorphisms obtained in Theorem REF , with vanishing average of central exponent, without proving the ergodicity we obtain vanishing Lyapunov exponent almost everywhere." ], [ "Preliminaries", "Let $M$ be a compact smooth manifold.", "A diffeomorphism $f: M \\rightarrow M$ is partially hyperbolic if there exists a $Df$ -invariant splitting of the tangent bundle $TM=E^s \\oplus E^c \\oplus E^u$ such that $Df$ uniformly expands all vectors in $E^u$ and uniformly contracts all vectors in $E^s.$ While vectors in $E^c$ are neither contracted as strongly as any nonzero vector in $E^s$ nor expanded as strongly as any nonzero vector in $E^u.$ $f$ is called absolutely partially hyperbolic if the domination property between the three mentioned sub bundles is uniform on the whole manifold, i.e.", "there are constants $a, b > 0$ such that for all $x \\in M$ and any unit vectors $v^*, * \\in \\lbrace s, c, u\\rbrace $ in $T_x M$ $\\Vert D_xf(v^s)\\Vert < a < \\Vert D_xf (v^c)\\Vert < b < \\Vert D_xf(v^u)\\Vert .$ Absolutely partial hyperbolicity conditions can be expressed equivalently in terms of invariant cone families.", "In the appendix we make some precise statements.", "In this paper we always deal with absolutely partially hyperbolic diffeomorphisms.", "For all partially hyperbolic diffeomorphisms, there are foliations $\\mathcal {F}^{\\tau }, \\tau =s,u$ tangent to the sub-bundles $E^{\\tau }, \\tau =s,u$ called stable and unstable foliation respectively.", "On the other hand, the integrability of the central sub-bundle $E^c$ is a subtle issue and is not the case in general partially hyperbolic setting (see [9]).", "By M. Brin, D. Burago, S. Ivanov [4] all absolutely partially hyperbolic diffeomorphisms on $\\mathbb {T}^{3}$ admit a central foliation tangent to $E^c.$ Let $f: \\mathbb {T}^{3} \\rightarrow \\mathbb {T}^{3}$ be a partially hyperbolic diffeomorphism.", "Consider $f_* : \\mathbb {Z}^3 \\rightarrow \\mathbb {Z}^3$ the action of $f$ on the fundamental group of $\\mathbb {T}^{3}.$ $f_*$ can be extended to $\\mathbb {R}^3$ and the extension is the lift of a unique linear automorphism $A : \\mathbb {T}^{3} \\rightarrow \\mathbb {T}^{3}$ which is called the linearization of $f.$ It can be proved that $A$ is a partially hyperbolic automorphism of torus ([4]).", "It is not difficult to see that in large scale $f$ and $A$ behaves similarly (see [7], corollary $2.2$ ).", "More precisely, for each $k \\in \\mathbb {Z}$ and $C > 1$ there is an $M > 0 $ such that for all $x, y \\in \\mathbb {R}^3$ , $\\Vert x - y\\Vert > M \\Rightarrow \\frac{1}{C} < \\frac{\\Vert \\tilde{f}^k(x) - \\tilde{f}^k(y)\\Vert }{\\Vert A^k(x) - A^k(y)\\Vert } < C.$ where $\\tilde{f}: \\mathbb {R}^3 \\rightarrow \\mathbb {R}^3$ is the lift of $f$ to $\\mathbb {R}^3.$ The examples in the open set $U$ of Theorem REF shows that in the infinitesimal scale opposite behaviors can occur.", "A. Hammerlindl proves that any absolutely partially hyperbolic diffeomorphism $f$ on $\\mathbb {T}^{3}$ is leaf conjugated to its linearization (for higher dimensions see [8]).", "In particular the central leaves of $f$ are all homeomorphic.", "It is easy to see that a linear partially hyperbolic diffeomorphism of $\\mathbb {T}^3$ is either Anosov or all of the leaves of $\\mathcal {F}^c$ are compact, i.e, homeomorphic to $\\mathbb {S}^1.$ In the latter case the central eigenvalue is equal to one.", "In theorem REF we give example of partially hyperbolic diffeomorphisms with zero central Lyapunov exponent almost everywhere and non compact leaves.", "Acknowledgement.", "We would like to thank C. Bonatti for very helpful suggestions (during the conference Beyond Uniform Hyperbolicity 2011- Marseille) for the conjecture of the second named author's talk in that conference.", "We are gratefull to A. Hammerlindl for elegant mathematical suggestions for the presentation of our results.", "We also would like to thank R. Varão for carefully reading the manuscript of this article." ], [ "Local Perturbation", "In this section we describe briefly a local perturbation process introduced by A. Baraviera, C. Bonatti [3].", "For any partially hyperbolic diffeomorphism $f_0$ they construct a $C^1$ -arc of diffeomorphisms $\\lbrace f_r\\rbrace $ for which the integral of central Lyapunov exponent of $f_r$ is strictly bigger than integral of the central exponent of $f_0$ .", "In [3], this perturbation procedure is made in a general case.", "Here we will use the perturbation argument just for the linear case.", "Let $f:\\mathbb {T}^{3} \\rightarrow \\mathbb {T}^{3}$ be a volume preserving, linear partially hyperbolic diffeomorphism.", "Denote by $\\lambda ^s < \\lambda ^c < \\lambda ^u$ the eigenvalues of $f$ and its unitary eigenvectors by $e_s,e_c,e_u$ respectively.", "Thus, the directions of $e_s,e_c,e_u$ are the directions of the subbundles $E^s,E^c,E^u$ .", "Let $p$ be a non-fixed point.", "We take a $C^1-$ local coordinate system on a neighborhood $V$ centered at $p$ such that $\\lbrace e_s, e_c, e_u\\rbrace $ are directed by $\\frac{\\partial }{\\partial x}, \\frac{\\partial }{\\partial y}, \\frac{\\partial }{\\partial z}.$ Moreover, the expression of volume form on $\\mathbb {T}^{3}$ coincides with the Lebesgue measure on $\\mathbb {R}^3.$ Let $B_1(0)$ be the unit ball of $\\mathbb {R}^3.$ Given any ball $B_r(p)$ inside $V$ we denote by $\\varphi _{r} : B_{r}(p) \\rightarrow B_1(0)$ the diffeomorphism which in local coordinates is a homothety of ratio $\\frac{1}{r}.$ More precisely, if $\\pi : V \\rightarrow \\mathbb {R}^3$ is the mentioned coordinate system $\\varphi _r(x):= \\frac{\\pi (x)}{r}.$ Let $h:B_1(0) \\rightarrow B_1(0)$ , $h \\ne Id$ , a volume preserving diffeomorphism which preserves the $x$ - direction, and equal to the identity on a neighborhood of the boundary of $B_1(0)$ .", "We define the diffeomorphism $h_{r}: \\mathbb {T}^{3} \\rightarrow \\mathbb {T}^{3}$ by: $ h_{r}(w) = \\left\\lbrace \\begin{array}{ll}w & \\mbox{, if } w \\notin B_{r}(p); \\\\ \\varphi _{r} ^{-1} \\circ h \\circ \\varphi _{r} (w) & \\mbox{, if } w \\in B_{r}(p).", "\\end{array}\\right.$ Finally, we define the arc of diffeomorphisms $\\lbrace f_r\\rbrace _{r\\in [0,1]}$ by: $ f_{r} := f \\circ h_{r}.$ Also, we take $h$ to satisfy $||h-Id||_{C^1} < 1.$ Since $h$ preserves the direction of $e_s$ we can write $ Dh(p)e_u = h^u(p)e_u + h^c(p)e_c.$ Lemma 3.1 [3] Let $h$ be as above, then $I(h) := \\int _{B_1(0)} \\log h^u(p) dm(p) <0.$ Consider $n_{r}$ the least positive integer such that $f^{n_{r}}(B_r) \\cap B_r \\ne \\emptyset .$ Denote by $\\lambda ^u_{r}(p)$ the unstable Lyapunov exponent of $f_r$ at $p$ and define: $\\sigma ^u_{f_{r}} = \\int \\log J^u_{f_{r}}(p) dm(p), \\hspace{8.5359pt} \\sigma ^c_{f_{r}} = \\int \\log J^c_{f_{r}}(p) dm(p)$ where $J^{\\tau }_{f_r}( p)$ denotes the Jacobian of $f_r$ on $E^{\\tau }_{f_r}( p )$ , that is, the modulus of the determinant of the restriction of $Df_r (p )$ to $E^{\\tau }_{f_r}( p )$ , $\\tau =s, c,u$ .", "Lemma 3.2 [3] Let $\\sigma ^u_{f_{r}}$ and $\\sigma ^c_{f_{r}}$ be as above, then $\\log \\lambda ^u - \\sigma ^u_{f_r} \\ge vol(B_r)(-I(h) - C \\alpha ^{n_r})$ where $\\alpha = \\lambda ^c/\\lambda ^u$ , and $C = \\max _{x\\in B_r} \\frac{h^c_r}{h^u_r} \\cdot \\max _{x \\in B_r}{|| \\text{Proj} _{u}(e_c)||}$ with $Proj_{u}(e_c)$ denoting the projection of $e_c$ over $E^u$ parallel to the new center bundle.", "Observe that by (REF ) the perturbation $h_r$ preserves the center unstable bundle and the center-unstable jacobian of $h_r$ is equal to one.", "So the integral of the logarithm of the jacobian of $f_r$ in the center-unstable direction is the same of the one for $f$ (see [3], pg.1664).", "Thus, we have the following corollary.", "Corollary 3.3 With the previous notations, the difference between $\\sigma ^c_{f_r}$ and $\\log (\\lambda ^c)$ is bounded from below as follows: $\\sigma ^c_{f_{r}} - \\log \\lambda ^c = \\log \\lambda ^u - \\sigma ^u_{f_{r}} \\ge vol (B_{r}(p)) \\cdot \\left(-I(h) - C \\cdot \\left(\\frac{\\lambda ^c}{\\lambda ^u}\\right)^{n_r}\\right).$" ], [ "Family of Anosov Linear Automorphisms", "In order to realize a perturbation that changes the sign of the central Lyapunov exponent, it is reasonable to take a diffeomorphim with central exponent close to 0 and big unstable exponent (so that we can borrow some hyperbolicity from the unstable direction).", "For each $k \\in \\mathbb {Z}^{}$ define the linear automorphism $f_k: \\mathbb {T}^{3} \\rightarrow \\mathbb {T}^{3}$ induced by the integer matrix: $A_k=\\left( \\begin{array}{ccc}0 & 0 & 1 \\\\0 & 1 & -1 \\\\-1 & -1 & k \\end{array} \\right).", "$ The characteristic polynomial of $A_k$ is $p_{k}(x)= x^3 -(k+1)x^2 + kx -1.$ Lemma 4.1 For all $k \\ge 5$ , $A_k$ has real eigenvalues $0<\\lambda ^s_k < \\lambda ^c_k < 1<\\lambda ^u_k$ and $\\lambda ^s_k \\rightarrow 0, \\lambda ^c_k \\rightarrow 1, \\lambda ^u_k \\rightarrow \\infty $ as $k \\rightarrow \\infty $ .", "First of all note that : $p_k(1/2) = \\frac{k}{4}-\\frac{9}{8} > 0$ , $\\forall k \\ge 5$ ; $p_k(1) = p_k(k) = -1$ , $\\forall k$ ; $p_k(k+1) = k(k+1)-1 \\ge 1$ , $\\forall k \\ge 1.$ So, for all $k \\ge 5$ , $p_k$ has a root $\\lambda ^u_k \\in (k,k+1)$ and a root $\\lambda ^c_k \\in (1/2,1)$ .", "Denoting by $\\lambda ^s_k$ the other root we have: $0<\\lambda ^s_k = \\frac{1}{\\lambda ^c_k \\cdot \\lambda ^u_k} < \\lambda ^c_k < 1< k <\\lambda ^u_k.$ Now, given any $0<\\varepsilon <1$ we have $p_k(1-\\varepsilon ) = k(1-\\varepsilon )\\varepsilon -\\varepsilon (1-\\varepsilon )^2 -1$ which is trivially positive for large values of $k$ .", "That is, $\\lambda ^c_k \\rightarrow 1$ when $k \\rightarrow \\infty $ .", "Also, since $k <\\lambda ^u_k$ and $\\lambda ^s_k \\cdot \\lambda ^c_k \\cdot \\lambda ^u_k =1$ we conclude that $\\lambda ^c_k \\rightarrow 1, \\lambda ^u_k \\rightarrow \\infty , \\text{ and } \\lambda ^s_k \\rightarrow 0$ as $k \\rightarrow \\infty .$ Next, we evaluate the stable, central and unstable directions of $f_k$ .", "$A_k \\cdot \\left(\\begin{array}{c}a \\\\b \\\\c \\end{array}\\right) =\\left(\\begin{array}{c}\\lambda a \\\\\\lambda b \\\\\\lambda c \\end{array}\\right) \\Rightarrow \\left(\\begin{array}{c}a \\\\b \\\\c \\end{array}\\right) = \\left(\\begin{array}{c}a \\\\\\frac{\\lambda a}{1-\\lambda } \\\\\\lambda a \\end{array}\\right).", "$ So the directions are $v^{\\tau }_k:=(1, \\lambda _k^{\\tau }/(1-\\lambda _k^{\\tau }), \\lambda _k^{\\tau })$ where $\\tau =s,c,u$ .", "Let ${\\displaystyle e^k_{\\tau }:=\\frac{v_k^{\\tau }}{||v_k^{\\tau }||}}$ , $\\tau =s,c,u$ .", "Then we have: $e^k_s \\rightarrow \\left(\\begin{array}{c}1 \\\\0 \\\\0 \\end{array}\\right),e^k_c \\rightarrow \\left(\\begin{array}{c}0 \\\\1 \\\\0 \\end{array}\\right),e^k_u \\rightarrow \\left(\\begin{array}{c}0 \\\\0 \\\\1 \\end{array}\\right).$ The following step is to apply local perturbations to each $f_k$ .", "The aim of the next section is to define a family of functions $h_k$ that we will use to do the perturbation." ], [ " Proof of Theorems ", "For a linear partially hyperbolic automorphim with eigenvalues $\\lambda ^s < \\lambda ^c < \\lambda ^u$ , corollary REF implies that the following quantities are relevant to the amount of change of the central Lyapunov exponent after local perturbation method of Baraviera-Bonatti: $\\alpha = \\lambda ^c / \\lambda ^u$ ; ${\\displaystyle C = \\max _{x \\in B_r} \\frac{h^c_r}{h^u_r} \\cdot \\max _{x \\in B_r}{|| \\text{Proj} _{u}(e_c)||}}$ ; $vol(B_r)$ and the return time $n_r$ ; $I(h).$ We consider the family $f_k$ constructed in the previous section.", "Recall that for each $k$ there exists an adapted inner product (which gives adapted metric) where $e^s_k, e^c_k, e^u_k$ form an orthonormal set.", "As these eigenspaces are converging to the canonical basis the adapted metrics are close to the euclidean metric when $k$ is large enough.", "By euclidean metric we mean the usual inner product coming from euclidean inner product of $\\mathbb {R}^3.$ We will take a non-fixed point $p$ ($f_k(p) \\ne p.$ ) and $0 < r < 1$ we take a local coordinate $\\pi _k: B_r^k(p) \\rightarrow B_r(0) \\subset \\mathbb {R}^3$ such that the adapted inner product is the pullback by $D \\pi _k$ of the euclidean inner product.", "Here $B_r^k$ is the ball of radius $r$ with respect to the adapted metric.", "Take an arbitrary point $p \\in F:= P( (0,2/3) \\times (0,1) \\times (5/6,1))$ where $P: \\mathbb {R}^3 \\rightarrow \\mathbb {T}^{3}$ is the usual canonical projection.", "Take fixed small $r$ such that $B^k_{r} (p) \\in F$ for large $k.$ It is possible to find such $r > 0$ , because for large $k$ all the adapted metrics are close to the euclidean metric.", "It is easy to see that $f_k^{-1}(F) \\cap F=\\emptyset $ so $ f_k(F) \\cap F =\\emptyset $ and $ f_k(B^k_{r_0}(p )) \\cap B^k_{r_0}( p) = \\emptyset $ , $\\forall k \\ge k_0$ .", "We take $h: B_1(0) \\rightarrow B_1(0)$ such that $ 0 \\ne \\Vert h - Id\\Vert _{C^1} < \\eta $ ($\\eta $ will be defined later).", "Let $\\pi _k : B^k_r(p) \\rightarrow B_r(0) \\subset \\mathbb {R}^3$ be a local coordinate which is isometry and the derivative of $\\pi _k$ sends $e^s_k, e^c_k, e^u_k$ to the canonical basis of $\\mathbb {R}^3$ .", "Let $\\xi : \\mathbb {R}^{3} \\rightarrow \\mathbb {R}^{3}$ , $\\xi _r(x):= \\frac{1}{r}x$ be the homothety of ratio $\\frac{1}{r}$ , then we define $\\varphi _{k,r}(x) := \\xi _r \\circ \\pi _k (x)$ and like in section () we construct $h_{k, r} : \\mathbb {T}^{3} \\rightarrow \\mathbb {T}^{3}$ as follows: $ h_{k,r}(w) = \\left\\lbrace \\begin{array}{ll}w & \\mbox{, if } w \\notin B^k_{r}(p); \\\\ \\varphi _{k,r} ^{-1} \\circ h \\circ \\varphi _{k,r} (w) & \\mbox{, if } w \\in B^k_{r}(p).", "\\end{array}\\right.$ Now the idea is to consider the arcs of diffeomorphisms $f_{k,r} := f_k \\circ h_{k,r}$ and show that for some positive $r$ and all large $k$ , $h_{k, r}$ is partially hyperbolic with positive Lyapunov exponent almost everywhere.", "We need to guarantee that these arcs are composed of partially hyperbolic diffeomorphisms.", "Even knowing that the set of partially hyperbolic diffeomorphisms is open in $C^1$ - topology, we do not know the “size\" of this set.", "To construct our desired examples in Theorems REF , REF we need the sequence $\\lbrace h_k\\rbrace $ to be “far from\" $Id$ , so it is not obvious that the composition $f_{k, r} = f_k \\circ h_{k,r}$ is partially hyperbolic.", "However, in our case this is not a serious issue.", "As $k$ grows, the domination between invariant sub bundles of $f_k$ is getting better and and the expansion and contraction of respectively expanding and contracting bundles increase.", "We observe that when the domination between bundles is bigger, one can take wider invariant cones in the definition of partial hyperbolicity by cones (see Appendix).", "So it is reasonable to expect that we can do bigger perturbations of $f_k$ and still remain in the partially hyperbolic diffeomorphisms set.", "Lemma 5.1 Let $\\lbrace f_k\\rbrace $ be the sequence of linear partially hyperbolic automorphisms defined as before and $0 < r < 1.$ There exist $\\eta >0$ and $K_0$ such that if $h: B_1(0) \\rightarrow B_1(0)$ is a diffeomorphism satisfying $||h-Id||_{C^1} < \\eta $ and equal to identity on a neighborhood of the boundary of $B_1(0)$ then, for $k \\ge k_0$ , $f_k \\circ h_{k, r}$ is absolutely partially hyperbolic.", "In the Appendix, for any linear partially hyperbolic diffeomorphism $f$ we estimate the size of the $C^1$ -neighborhood of $f$ inside absolutely partially hyperbolic diffeomorphisms.", "Here we are dealing with a sequence $f_k$ and the claim is that a same estimate for the size of neighborhood works for all large enough $k.$ Now by remark REF the size of permitted perturbation (i.e the number $\\varepsilon $ ) in the lemma REF depends increasingly on the ratio $\\Theta _k:= \\min \\left\\lbrace \\frac{|\\lambda ^u_k|}{|\\lambda ^c_k|}, \\frac{|\\lambda ^c_k|}{|\\lambda ^s_k|} \\right\\rbrace $ .", "When $k$ grows this ratio also grows.", "So we take the same $\\varepsilon $ for all $f_k.$ We should emphasize that the size of permitted perturbation is measured in the distance corresponding to the adapted metric of $f_k.$ So let $\\varepsilon $ be as above and take any $\\eta \\le \\varepsilon .$ We have $\\Vert \\xi _r^{-1}\\circ h\\circ \\xi _r - Id\\Vert _{C^1} \\le r \\Vert h - Id\\Vert _{C^1} \\le \\Vert h - Id\\Vert _{C^1} \\le \\varepsilon .$ By definition of adapted metric for each $f_k$ we have $D \\pi _k$ preserves norms and angles and consequently the distance (adapted norm corresponding to $f_k$ ) between $ (\\pi _k \\circ \\xi _r)^{-1} \\circ h \\circ (\\pi _k \\circ \\xi _r)$ and identity is also less than $\\varepsilon $ and we can apply lemma REF taking $g:=h_{k, r}.$ By the above lemma it follows that $f_{k,r}$ is partially hyperbolic for large enough $k$ .", "Also, since the same family of invariants cones works for both $f_{k,r}$ and $f_k$ , the angle between the new center bundle and $E^u_{f_{k,r}}$ is uniformly bounded.", "That is, the norm of the projection of $E^c_{f_{k,r}}$ over $E^u_{f_{k,r}}$ parallel to the new center bundle is uniformly bounded.", "Now let $n_r(k)$ be the least positive integer for which $(f_k)^{n_r(k)}(B^k_r(p)) \\cap B^k_r(p) \\ne \\emptyset $ .", "Then we have $\\sigma ^c_{f_{k,r}}- \\log \\lambda ^c _k\\ge vol(B^k_r(p)) \\cdot (-I(h) - C_k \\alpha _k^{n_r(k)})$ where ${\\displaystyle \\alpha _k = \\frac{\\lambda ^c}{\\lambda ^u}, \\text{ and }C_k= \\max _{x\\in B_r^k} \\frac{h^c_{r, k}}{h^u_{r, k}} \\cdot \\max _{x \\in B_r^k}{|| \\text{Proj} _{u}(e_c)||} }.$ As $\\max _{x \\in B_r^k}{|| \\text{Proj} _{u}(e_c)||} $ is uniformly bounded, it follows that $C_k$ is uniformly bounded, say $C_k<D, \\forall k$ .", "Thus, since $n_r(k) \\ge 2 $ for all $k$ , we get: $\\sigma ^c_{f_{k,r}} - \\log \\lambda ^c_k \\ge vol(B^k_r(p)) \\cdot (-I(h) - D \\alpha _k^2).$ Observe that $\\alpha _k \\rightarrow 0$ when $k \\rightarrow \\infty $ .", "So, for large values of $k$ we have $-I(h_k) - D \\alpha _k^2 \\ge \\frac{-I(h)}{2}$ which implies $\\sigma ^c_{f_{k, r}} - \\log \\lambda ^c_k \\ge -vol(B^k_r(p)) \\cdot \\frac{I(h)}{2} \\rightarrow -vol(B(r,p))\\frac{I(h)}{2} > 0.$ Observe that the volume appearing in the above equations is the fixed euclidean volume on the torus and as $k$ is large enough the volume of $B_r^k(p)$ is close to $B_r(0).$ Thus since $\\log \\lambda ^c_k \\rightarrow 0$ , for large values of $k$ we get $\\sigma ^c_{f_{k, r}}- \\log \\lambda ^c_k > -\\log \\lambda ^c_k \\Rightarrow \\sigma ^c_{f_{k,r}} > 0.$ Here we conclude the proof of Theorem REF .", "We have obtained $f_{k, r}$ isotopic to Anosov such that the average of central Lyapunov exponent is positive.", "Indeed, $H_k:[0,1] \\times \\mathbb {T}^{3} \\rightarrow \\mathbb {T}^{3}$ , $H_k(s,x) =f_k \\circ h_{k, sr} $ is an isotopy between $f_k$ and $f_{k,r}$ .", "By continuity argument we conclude that there exists an open subset of volume preserving diffeomorphisms $U$ containing $f_{k, r}$ such that for any $g \\in U$ we have $\\sigma ^c_g > 0.$ By the following result of Hammerlindl-Ures we conclude that all $g \\in U$ are ergodic and so the central Lyapunov exponent of almost every point is positive.", "Theorem 5.2 [10] Let $f: \\mathbb {T}^{3} \\rightarrow \\mathbb {T}^{3}$ be a $C^{1+\\alpha }$ volume preserving partially hyperbolic diffeomorphism, homotopic to a hyperbolic automorphism $A$ .", "Assume $f$ is not ergodic.", "Then, $E^s \\oplus E^u$ integrates to a minimal foliation; $f$ is topologically conjugate to $A$ and the conjugacy carries strong leaves of $f$ to the correspondent strong leaves of $A$ ; the central Lyapunov exponent of $f$ is 0 almost everywhere.", "Now we prove Theorem REF .", "Again from the continuity of $\\sigma ^c$ , there is some $0<r_0 < r$ for which $\\sigma ^c_{f_{k, r_0}} = 0$ .", "That is, we got a partially hyperbolic diffeomorphism $g := f_{k,r_0}: \\mathbb {T}^{3} \\rightarrow \\mathbb {T}^{3}$ , isotopic to an Anosov diffeomorphism and with $\\sigma ^c_g = 0$ .", "However, using again the above Theorem we obtain the following: Corollary 5.3 The diffeomorphism $g$ obtained above has zero central Lyapunov exponent almost everywhere.", "Indeed, if $g$ is ergodic then the Lyapunov exponents are constant almost everywhere.", "So we have $\\lambda ^c_g =\\sigma ^c_g = 0$ almost everywhere.", "On the other case, that is, if $g$ is not ergodic then by the previous theorem (third item) we have that $\\lambda ^c_g = 0$ almost everywhere.", "To finish the proof we note that, since $g$ is isotopic to a linear Anosov automorphism, by [8] all central leaves are homeomorphic to $\\mathbb {R}.$ $\\blacksquare $" ], [ "Appendix: Cone constructions and absolutely partially hyperbolic diffeomorphisms", "Definition 6.1 Given an orthogonal splitting of the tangent bundle of $M$ $E \\oplus F = T M$ and a real constant $\\beta >0$ , for each $x\\in M$ we define the cone centered in $E(x)$ with angle $\\beta $ as $C(E,x,\\beta ) = \\lbrace v \\in T_xM : ||v_F || \\le \\beta ||v_E|| , \\text{ where } v = v_E + v_F , v_E \\in E(x), v_F\\in F(x).\\rbrace $ Given a partially hyperbolic diffeomorphism $f: \\mathbb {T}^{3} \\rightarrow \\mathbb {T}^{3}$ with invariant splitting $T M = E^s \\oplus E^c \\oplus E^u$ there is an adapted inner product (and then an adapted norm) with respect to which the splitting is orthogonal (see [5]).", "Thus, given $\\beta >0$ we can define standard families of cones centered on the fiber bundles $E^{\\tau }(x)$ with angle $\\beta $ , $C^{\\tau }(x,\\beta )$ , $\\tau = s,c,u,cs,cu$ .", "Consider $f:M \\rightarrow M$ a (absolutely) partially hyperbolic diffeomorphism.", "By using an adapted norm $|| \\cdot ||$ , we can consider the invariant splitting $T M = E^s \\oplus E^c \\oplus E^u$ as being an orthogonal splitting, and there exist numbers $0 < \\lambda _1 \\le \\mu _1 < \\lambda _2 \\le \\mu _2 <\\lambda _3 \\le \\mu _3, \\hspace{8.5359pt} \\mu _1<1, \\hspace{8.5359pt} \\lambda _3>1$ for which $\\lambda _1 \\le ||Df(x)|E^s(x)|| \\le \\mu _1$ $\\lambda _2 \\le ||Df(x)|E^c(x)|| \\le \\mu _2$ $\\lambda _3 \\le ||Df(x)|E^u(x)|| \\le \\mu _3.$ Partial hyperbolicity can be described in terms of invariant cone families (see [5], pg.15).", "More specifically, let $f: \\mathbb {T}^{3} \\rightarrow \\mathbb {T}^{3}$ be a partially hyperbolic diffeomorphism and $T _x\\mathbb {T}^{3} = E^s(x) \\oplus E^c(x) \\oplus E^u(x)$ a continuous orthogonal splitting of $T \\mathbb {T}^{3}$ .", "Given a real number $\\beta >0$ define the families of cones $C^s(x,\\beta ) = C(x,E^s(x),\\beta ), C^u(x,\\beta ) = C(x,E^u(x),\\beta )$ $C^{cs}(x,\\beta ) = C(x,E^{cs}(x),\\beta ),C^{cu}(x,\\beta ) = C(x,E^{cu}(x),\\beta )$ where $E^{cs}(x) = E^c(x) \\oplus E^s(x), E^{cu}(x) = E^c(x) \\oplus E^u(x).$ Then, $f$ is absolutely partially hyperbolic if, and only if, there is $0 < \\beta < 1$ and constants $ 0<\\mu _1< \\lambda _2 \\le \\mu _2 < \\lambda _3 , \\hspace{8.5359pt} \\mu _1<1, \\hspace{8.5359pt} \\lambda _3>1$ for which $\\begin{split}Df^{-1}(x) (\\mathcal {C}^{\\tau }_{}(x,\\beta )) & \\subset \\mathcal {C}^{\\tau }_{}(f^{-1}(x),\\beta ), \\tau =s,cs ; \\\\Df(x) (\\mathcal {C}^{\\Psi }_{}(x,\\beta )) & \\subset \\mathcal {C}^{\\Psi }_{}(f(x),\\beta ) ,\\Psi =u,cu;\\end{split}$ and $\\begin{split}||Df^{-1}(x) v|| & > \\mu _1^{-1}||v||, v \\in C^{s}(x,\\beta ) ;\\\\||Df^{-1}(x) v|| & > \\mu _2^{-1}||v||, v \\in C^{cs}(x,\\beta ); \\\\||Df(x) v|| & > \\lambda _3 ||v||, v \\in C^{u}(x,\\beta ) ;\\\\||Df(x) v|| & > \\lambda _2||v||, v \\in C^{cu}(x,\\beta ).\\end{split}$ For the linear case, we can get some more precise and interesting conclusions regarding the relation of the constants of (REF ) and (REF ) and the Lyapunov exponents of the function.", "In what follows, for a linear partially hyperbolic diffeomorphism, we find an explicit relation between the angle of the invariant cones families and the ratio of domination between unstable, stable and central bundles.", "Consider $f: \\mathbb {T}^{3} \\rightarrow \\mathbb {T}^{3}$ a linear, volume preserving, partially hyperbolic diffeomorphism of $\\mathbb {T}^{3}$ .", "Denote by $\\lambda ^s,\\lambda ^c,\\lambda ^u$ its eigenvalues, where $|\\lambda ^s| < |\\lambda ^c| < |\\lambda ^u| , \\hspace{8.5359pt} |\\lambda ^s| <1 < |\\lambda ^u|.$ Put $ \\Theta := \\min \\left\\lbrace \\frac{|\\lambda ^c|}{|\\lambda ^s|} , \\frac{|\\lambda ^u|}{|\\lambda ^c|} \\right\\rbrace $ Then we can choose a constant $\\beta > 0$ such that $1< (1+\\beta )^2 < \\Theta .$ Therefore, by the definition of $\\beta $ , we have $ (1+\\beta )|\\lambda ^s| < \\frac{| \\lambda ^c|}{1+\\beta } < (1+\\beta ) |\\lambda ^c| , \\hspace{8.5359pt} (1+\\beta )|\\lambda ^s| <1<\\frac{|\\lambda ^u|}{1+\\beta }.$ Consequently we can find constants $\\mu _1, \\lambda _2, \\mu _2, \\lambda _3$ such that $ (1+\\beta )|\\lambda ^s| < \\mu _1 < \\lambda _2 < \\frac{| \\lambda ^c|}{1+\\beta } < (1+\\beta ) |\\lambda ^c| < \\mu _2 < \\lambda _3 < \\frac{|\\lambda ^u|}{1+\\beta } , \\hspace{8.5359pt} \\mu _1 <1<\\lambda _3.$ Now, it is straightforward to verify that with the constants defined by (REF ) and (REF ), the families of stable, unstable, center-stable and center-unstable cones satisfies (REF ) and (REF ).", "For example, if we take $v =v_s +v_{cu }\\in C^{cu}(x,\\beta )$ then $||Df(x)v_s|| = |\\lambda ^s| ||v_{s}|| \\le \\beta |\\lambda ^s| ||v_{cu}|| < \\beta |\\lambda ^c| ||v_{cu}|| \\le \\beta ||Df(x)v_{cu}||$ that is $Df(x) (C^{cu}(x,\\beta )) \\subset C^{cu}(f(x),\\beta ).$ Furthermore, $||Df(x)v||^2 \\ge ||Df(x)v_{cu}||^2 \\ge |\\lambda ^c|^2 ||v_{cu}||^2.$ But by (REF ) we know that $|\\lambda ^c| > (1+\\beta )\\lambda _2$ .", "So, $||Df(x)v||^2 > (1+\\beta )^2 (\\lambda _2)^2||v_{cu}||^2 \\ge (\\lambda _2)^2 (||v_{cu}||^2 + \\beta ^2 ||v_{cu}||^2) \\ge \\lambda _2^2 (\\Vert v_{cu}\\Vert ^2 + \\Vert v_s\\Vert ^2 )= (\\lambda _2||v|| )^2$ $ \\Rightarrow ||Df(x)v|| > \\lambda _2 ||v||.$ The argument for the other cones is similar." ], [ " Size of perturbation among absolutely partially hyperbolic diffeomorphisms", "Here we find an estimative for the size of the $C^1$ -neighborhood of a linear partially hyperbolic automorphism inside absolutely partially hyperbolic diffeomorphisms.", "Lemma 6.2 Let $f: \\mathbb {T}^{3} \\rightarrow \\mathbb {T}^{3}$ be a linear partially hyperbolic diffeomorphism, volume preserving, with eigenvalues $\\lambda ^s, \\lambda ^c, \\lambda ^u$ , where $|\\lambda ^s| < |\\lambda ^c| < |\\lambda ^u|$ .", "Then, there is a constant $\\varepsilon >0$ such that, for every diffeomorphism $g:\\mathbb {T}^{3} \\rightarrow \\mathbb {T}^{3}$ with $||g-Id ||_{C^1} < \\varepsilon $ (adapted norm corresponding to $f$ ), the composition $f \\circ g$ is an absolutely partially hyperbolic diffeomorphism.", "The constant $\\varepsilon $ depends only on $\\Theta := \\min \\left\\lbrace \\frac{|\\lambda ^u|}{|\\lambda ^c|}, \\frac{|\\lambda ^c|}{|\\lambda ^s|} \\right\\rbrace $ .", "Remark 6.3 It is easy to see from the proof that $\\varepsilon $ depends increasingly on $\\Theta .$ Bigger $\\Theta $ permits bigger perturbations.", "However we emphasize that the distance between $g$ and identity is considered in the adapted metric.", "Let $f$ be as in the statement and denote the invariant splitting of $f$ by $T_xM = E^s(x) \\oplus E^c(x) \\oplus E^u(x).$ Consider the adapted norm $||\\cdot ||$ with respect to which the invariant splitting is orthogonal.", "Now, since $f$ is a linear partially hyperbolic diffeomorphisms we can choose constants $0 < \\lambda _1 \\le \\mu _1 < \\lambda _2 \\le \\mu _2 < \\lambda _3 \\le \\mu _3$ $ \\mu _1 < 1 < \\lambda _3$ and a real value $\\beta >0$ as in (REF ) and (REF ).", "For $v \\in M$ we can write $v = v_s + v_c + v_u$ with $v_{\\tau } \\in E^{\\tau }(x)$ , $\\tau =s,c,u$ .", "If $v \\in C^{u}(x,\\beta )$ then $||v_{cs}|| \\le \\beta ||v_u||$ , where $v_{cs} = v_s + v_c$ .", "Thus, $||Df(x)v_{cs}|| < \\mu _2 || Df^{-1} \\circ Df(x) v_{cs}|| = \\mu _2 ||v_{cs}|| \\le \\mu _2 \\beta ||v_u|| < \\mu _2 \\beta (\\lambda _3)^{-1}||Df(x)v_u||.$ $\\Rightarrow Df(x)v \\in C^u(f(x),(\\mu _2 / \\lambda _3) \\cdot \\beta ).$ If $v \\in C^{cu}(x,\\beta )$ then $||v_s|| \\le \\beta ||v_{cu}||$ where $v_{cu} = v_c + v_u$ .", "Thus, $||Df(x)v_{s}|| < \\mu _1 || Df^{-1} \\circ Df(x) v_{s}|| = \\mu _1 ||v_{s}|| \\le \\mu _1 \\beta ||v_{cu}|| < \\mu _1 \\beta (\\lambda _2)^{-1}||Df(x)v_{cu}||$ $ \\Rightarrow Df(x) C^{cu}(x,\\beta ) \\subset C^u(f(x),( \\mu _1/\\lambda _2 )\\cdot \\beta ).$ If $v \\in E^s(x)$ then $||v_{cu}|| \\le \\beta || v_s|| $ .", "Thus, $ ||Df^{-1}(x)v_{cu}|| < \\lambda _2^{-1} || Df\\circ Df^{-1}(x) v_{cu}|| = \\lambda _2^{-1} ||v_{cu}|| \\le \\lambda _2^{-1} \\beta ||v_s|| < \\lambda _2^{-1} \\beta \\mu _1||Df^{-1}(x)v_s||$ $ \\Rightarrow Df^{-1}(x) C^s(x,\\beta ) \\subset C^{s}(f^{-1}(x),(\\mu _1 / \\lambda _2) \\cdot \\beta ).$ If $v\\in E^{cs}(x)$ then $||v_u||\\le \\beta ||v_{cs}||$ .", "Thus, $ ||Df^{-1}(x)v_{u}|| < \\lambda _3^{-1} || Df\\circ Df^{-1}(x) v_{u}|| = \\lambda _3^{-1} ||v_{u}|| \\le \\lambda _3^{-1} \\beta ||v_{cs}|| < \\lambda _3^{-1} \\beta \\mu _2 ||Df^{-1}(x)v_{cs}||$ $\\Rightarrow Df^{-1}(x) C^{cs}(x,\\beta ) \\subset C^{cs}(f^{-1}(x),(\\mu _2 / \\lambda _3) \\cdot \\beta ).$ Define $\\gamma := \\max \\left\\lbrace \\frac{\\mu _2}{\\lambda _3}, \\frac{\\mu _1}{\\lambda _2} \\right\\rbrace < 1.$ So we have $Df^{-1}(x) (\\mathcal {C}^{\\tau }_{}(x,\\beta )) & \\subset & \\mathcal {C}^{\\tau }_{}(f^{-1}(x),\\gamma \\cdot \\beta ), \\tau =s,cs ; \\\\Df(x) (\\mathcal {C}^{\\Psi }_{}(x,\\beta )) & \\subset & \\mathcal {C}^{\\Psi }_{}(f(x),\\gamma \\cdot \\beta ) ,\\Psi =u,cu;$ Observe that by (REF ) and (REF ), $\\beta $ and $\\gamma $ depends only on the ratios $|\\lambda ^u| / |\\lambda ^c| , |\\lambda ^c| / |\\lambda ^s|$ .", "Now, since the invariant splitting is constant, we can take an $\\varepsilon > 0$ depending only on the ratios $|\\lambda ^u| / |\\lambda ^c| , |\\lambda ^c| / |\\lambda ^s|$ such that, if $|| g -Id ||_{C^1} < \\varepsilon $ then $\\begin{split}Dg(x) C^{\\tau }\\left(x,\\gamma \\cdot \\beta \\right) & \\subset C^{\\tau }\\left(g(x),\\beta \\right), \\tau =u,cu \\\\Dg^{-1}(x) C^{\\Psi }\\left(x,\\gamma \\cdot \\beta \\right) & \\subset C^{\\Psi }\\left(g^{-1}(x), \\beta \\right), \\Psi =s,cs\\end{split}$ and $\\frac{l}{L} > \\gamma , \\hspace{8.5359pt} L < \\frac{1}{\\mu _1},\\hspace{8.5359pt} \\frac{1}{\\lambda _3} < l$ with $l ||v|| \\le ||Dg(x)v|| \\le L||v||.$ Figure: The grey cones denotes the cones with angle γ·β\\gamma \\cdot \\beta and the wider ones are the cones with angle β\\beta .Thus we have (see figure 1) $D(f\\circ g)(x) (C^{\\tau }(x,\\gamma \\cdot \\beta )) \\subset C^{\\tau }(f(g(x)),\\gamma \\cdot \\beta ), \\tau =u,cu ;$ $D(f \\circ g)^{-1}(x) (C^{\\Psi }(x,\\beta )) \\subset C^{\\Psi }(g^{-1}(f^{-1}(x)),\\beta ), \\Psi = s,cs.$ Now, we need to show uniform contraction and expansion on these families.", "If $v \\in C^{u}(x,\\gamma \\cdot \\beta )$ then $|| D(f \\circ g ) (x) v|| \\ge \\lambda _3 ||Dg(x)v|| \\ge \\lambda _3 \\cdot l \\cdot ||v||.", "$ If $v \\in C^{cs}(x,\\beta )$ then $||D(f\\circ g)^{-1}(x) v|| \\ge L^{-1} || Df^{-1}(x)v|| > L^{-1} \\cdot \\mu _2^{-1} ||v|| .$ If $v \\in C^{cu}(x,\\beta \\cdot \\gamma )$ then $|| D(f \\circ g ) (x) v|| \\ge \\lambda _2 ||Dg(x)v|| \\ge \\lambda _2 \\cdot l \\cdot ||v||.", "$ If $v \\in C^{s}(x,\\beta )$ then $||D(f\\circ g)^{-1}(x) v|| \\ge L^{-1} || Df^{-1}(x)v|| > L^{-1} \\cdot \\mu _1^{-1} ||v|| .$ Furthermore $0 < L \\cdot \\mu _1 < l \\cdot \\lambda _2 \\le L \\cdot \\mu _2 < l \\cdot \\lambda _3.$ and $L \\cdot \\mu _1 < 1 \\hspace{8.5359pt} , \\hspace{8.5359pt} l \\cdot \\lambda _3 >1,$ so that $f \\circ g$ is absolutely partially hyperbolic as we claimed." ] ]

[ [ "Extended supersymmetry and its applications in quantum mechanical models\n associated with self-dual gauge fields" ], [ "Abstract We study certain new models of supersymmetric quantum mechanics.", "The explicit form of the corresponding superfield and component actions, as well as of the quantum Hamiltonians and supercharges is given.", "It is shown that the Hamiltonian H=D*D, where D is flat four-dimensional Dirac operator in an external self-dual gauge background, Abelian or non-Abelian, is supersymmetric with N=4 supersymmetry.", "A generalization of this Hamiltonian to the motion on a curved conformally flat four-dimensional manifold exists.", "For an Abelian self-dual background, the corresponding Lagrangian can be derived from certain harmonic superspace expressions.", "If the Hamiltonian involves a non-Abelian self-dual gauge field, one can construct the Lagrangian formulation of it by introducing auxiliary bosonic variables with Wess-Zumino type action.", "For a special class of such Lagrangians when the gauge group is SU(2) and the gauge field is expressed in the `t Hooft ansatz form, it is possible to give a superfield description using the harmonic superspace formalism.", "As a new explicit example, the N=4 mechanics with Yang monopole in R^5 (= instanton on S^4) is considered.", "Independently, a similar system with N=4 supersymmetry in three dimensions also admits the superfield description.", "Although the three-dimensional system involves different superfields, its component Lagrangian and Hamiltonian appear to be the three-dimensional reduction of the mentioned four-dimensional system.", "The off-shell N=4 supersymmetry requires the gauge field to be a static form of the 't Hooft ansatz for the four-dimensional self-dual SU(2) gauge fields, that is a particular solution of Bogomolny equations for BPS monopoles." ], [ "a4paper,textwidth=16cm,right=2.5cm,top=3.5cm,textheight=23cm,headsep=0.8cm 1 UNIVERSITÉ DE NANTES FACULTÉ DES SCIENCES ET TECHNIQUES ———— ÉCOLE DOCTORALE MATÉRIAUX MATIÈRE MOLÉCULE EN PAYS DE LOIRE Année : 2012 N$^{\\circ }$ attribué par la bibliothèque Table: NO_CAPTION                    Année : 2012 N$^{\\circ }$ attribué par la bibliothèque Table: NO_CAPTION 2 Extended supersymmetry and its applications in quantum mechanical models associated with self-dual gauge fields 1.2 THÈSE DE DOCTORAT Discipline : Physique Spécialité : Physique Subatomique Présentée et soutenue publiquement par Maxim KONYUSHIKHIN Le 29 Mars 2012, devant le jury ci-dessous 0.25 Table: NO_CAPTION Directeur de thèse : Andrei SMILGA, Professeur, Subatech, Université de Nantes Andrei SMILGA, Professeur, Subatech, Université de Nantes N$^\\circ $ ED 500                   Acknowledgment Throughout the various stages of completion of this thesis and during my graduate education as a whole, I have greatly benefited from discussions with Andrei Smilga, my research advisor.", "I warmly thank him for sharing his insight, offering advice, maintaining interest and providing encouragement.", "The study involved in this thesis was done in an amiable and pleasant collaboration with Evgeny Ivanov.", "I deeply appreciate his participation in my education, his help and his advices during the writing of the manuscript.", "Let me also express my deep gratitude to my wife Olga Driga.", "I would not have completed my thesis without her unconditional support.", "I am also indebted to L. Alvarez-Gaume, F. Delduc, S. Fedoruk, A. Gorsky, O. Lechtenfeld, M. Shifman and A. Wipf for their illuminating discussions.", "Supersymmetric quantum mechanics (SQM) provides a proper venue for exploring and modeling salient features of supersymmetric field theories in diverse dimensions [1].", "Some SQM models represent one-dimensional reductions of higher-dimensional supersymmetric theories.", "At the same time, many interesting models of this kind can be constructed directly in (0+1) dimensions, without any reference to the dimensional reduction procedure.", "They exhibit some surprising properties related to peculiarities of one-dimensional supersymmetry.", "For any SQM model (like for any supersymmetric field theory), it is desirable, besides the component Hamiltonian and Lagrangian description, to have the appropriate superfield Lagrangian formulation.", "The latter makes supersymmetry manifest, prompts possible generalizations of the model and allows one to reveal relationships with other cognate theories.", "This study is devoted to the Hamiltonian and the Lagrangian as well as the superfield Lagrangian formulation for a certain class of ${\\cal N}=4$ SQM models Hereafter, in quantum mechanics, ${\\cal N}$ counts the number of real supercharges.", "with self-dual or anti-self-dual Abelian or non-Abelian gauge field backgrounds [2], [3], [4].", "Surprisingly, such systems did not attracted much attention so far.", "A natural framework for this formulation proves to be the harmonic superspace (HSS) approach [5] adapted to the one-dimensional case [6].", "The models of supersymmetric quantum mechanics with background gauge fields are of obvious interest for several reasons.", "One of them is a close relation of these systems to the Landau problem (motion of a charged particle in an external magnetic field) and its generalizations (see e.g. [7]).", "The Landau-type models constitute a basis of the theoretical description of quantum Hall effect (QHE), and it is natural to expect that their supersymmetric extensions, with extra fermionic variables added, may be relevant to spin versions of QHE.", "Also, these systems can provide quantum-mechanical realizations of various Hopf maps closely related to higher-dimensional QHE (see e.g.", "[8] and references therein).", "The first type of SQM models considered in this work represents a subclass of well-known systems which describe the motion of a fermion on an even-dimensional manifold with an arbitrary gauge background.", "It was observed many years ago that one can treat these systems as supersymmetric ones such that, e.g., the Atiyah-Singer index of the massless Dirac operator $\\, /\\!\\!\\!\\!", "can be interpreted as theWitten index of a certainsupersymmetric Hamiltonian \\cite {Gaume}.The corresponding supercharges and the Hamiltonian are\\begin{equation*}Q = /\\!\\!\\!\\!", "1 + \\gamma _5) ,\\quad \\quad \\bar{Q} = /\\!\\!\\!\\!", "1 - \\gamma _5) ,\\quad \\quad H = /\\!\\!\\!\\!", "2 ,\\end{equation*}where $ 5$ is the appropriate ``fifth gamma matrix^{\\prime \\prime } obeying $ 52=1$ and anticommuting with the Dirac operator,$ {5,    / } = 0$.Indeed, for any eigenstate $$ of the massless Dirac operator $   / with a nonzero eigenvalue $\\lambda $ , the state $\\gamma ^5 \\Psi $ is also an eigenstate of $\\, /\\!\\!\\!\\!", "with the eigenvalue$ -$.", "Thus, all excited states of $ H$ are doubly degenerate.$ It turns out that for a four-dimensional flat manifold and self-dual or anti-self-dual gauge field, Abelian or non-Abelian, the spectrum of $H$ is 4-fold degenerate implying the extended ${\\cal N} = 4$ supersymmetry.", "For a flat Dirac operator in the instanton background, this can be traced back to Ref. [10].", "${\\cal N}=4$ SQM models with the background Abelian gauge fields were treated in the pioneer papers [11], [12] and, more recently, e.g.", "in [13], [6], [14], [2].", "In particular, in [6] an off-shell Lagrangian superfield formulation of the general models associated with the multiplets $({\\bf 4, 4, 0})$ and $({\\bf 3, 4, 1})$ was given in the ${\\cal N}=4$ , $d=1$ harmonic superspace The first superfield formulation of general $({\\bf 3, 4, 1})$ SQM (without background gauge field couplings) was given in [15]..", "It was found that ${\\cal N}=4$ supersymmetry requires the gauge field to be (anti)self-dual in the four-dimensional $({\\bf 4, 4, 0})$ case, or to obey a “static” version of the (anti)self-duality condition in the three-dimensional $({\\bf 3, 4, 1})$ case.", "In the papers [14], [2], it was observed (in a Hamiltonian approach) that the Abelian $({\\bf 4, 4, 0})$ ${\\cal N}=4$ SQM admits a simple generalization to arbitrary self-dual non-Abelian background.", "In [3], an off-shell Lagrangian formulation was shown to exist for a particular class of such non-Abelian ${\\cal N}=4$ SQM models, with ${\\rm SU}(2)$ gauge group and 't Hooft ansatz [16] for the self-dual ${\\rm SU}(2)$ gauge field (see also [17]).", "As in the Abelian case, it was the use of ${\\cal N}=4$ , $d=1$ harmonic superspace that allowed us to construct such an off-shell formulation.", "A new non-trivial feature of the construction of [3] is the involvement of an auxiliary “semi-dynamical” $({\\bf 4, 4, 0})$ multiplet with the Wess-Zumino type action possessing an extra gauged ${\\rm U}(1)$ symmetry.", "After quantization, the corresponding bosonic fields become a sort of spin ${\\rm SU}(2)$ variables to which the background gauge field naturally couples The use of such auxiliary bosonic variables for setting up coupling of a particle to Yang-Mills fields can be traced back to [18].", "In the context of ${\\cal N}=4$ SQM, they were employed in [19], [20] and [8], [21]..", "The second class of SQM models that we consider can be obtained at the component level by the Hamiltonian reduction of the systems discussed above from four to three dimensions.", "Their superfield description is nontrivial and consists in coupling the coordinate supermultiplet $({\\bf 3, 4, 1})$ to an external non-Abelian gauge field through the introduction of the auxiliary $({\\bf 4, 4, 0})$ superfield.", "The off-shell ${\\cal N}=4$ supersymmetry restricts the external gauge field to be represented by a “static” version of the 't Hooft ansatz for four-dimensional (anti)self-dual ${\\rm SU}(2)$ gauge fields, i.e.", "to a particular solution of the general monopole Bogomolny equations [22] Some BPS monopole backgrounds in the framework of ${\\cal N}=2$ SQM were considered, e.g., in [23].. A new feature of the three-dimensional case is the appearance of “induced” potential term in the action as a result of eliminating the auxiliary field of the coordinate $({\\bf 3, 4, 1})$ supermultiplet.", "This term is bilinear in the ${\\rm SU}(2)$ gauge group generators.", "As a particular “spherically symmetric” case of the construction (with the exact ${\\rm SU}(2)$ R-symmetry) we recover the ${\\cal N}=4$ mechanics with Wu-Yang monopole [24] (recently considered in [21] with an essentially different treatment of the spin variables).", "The chapter REF is devoted to the introduction to supersymmetry in four-dimensional relativistic field theories.", "We discuss the motivation and the properties of supersymmetric theories as well as the their practical realization through the superfield approach.", "As an illustration, we consider the simplest example of the Wess-Zumino model – a complex scalar field coupled to a Weyl spinor field.", "In chapter REF , we discuss supersymmetry in quantum mechanics.", "The ordinary superspace and harmonic superspace formalisms are given.", "In particular, the structure of the supermultiplets $({\\bf 4, 4, 0})$ and $({\\bf 3, 4, 1})$ is explained.", "Additionally, we introduce necessary notations which will be used in the chapter REF .", "The chapter REF presents the original results of this study.", "We give the component and the superfield description of the four-dimensional and the three-dimensional models discussed above.", "In particular, the Hamiltonians and the corresponding supercharges are written.", "This chapter is purely introductory and is devoted to supersymmetric field theories in four-dimensional Minkowski space.", "We explain what is supersymmetry and why it is the only possible nontrivial extension of the Poincaré symmetry.", "We discuss main properties of any supersymmetric field theory and motivate why supersymmetric theories are interesting.", "Finally, we show how to work with such theories and explain the superfield formalism.", "As an illustration, we consider a simplest possible supersymmetric example – the Wess-Zumino model which describes supersymmetric dynamics of a complex scalar field.", "In a supersymmetric field theory, interactions between particles are fine-tuned in a special way, so that an additional continuous symmetry – supersymmetry – emerges.", "This symmetry mixes bosons and fermions (particles with different statistics) between each other.", "Supersymmetry admits natural resolution of certain inconsistency problems of field theories.", "For instance, a vacuum in a field theory usually has infinite energy density.", "In a theory with unbroken supersymmetry, however, the energy of a vacuum is exactly zero.", "This subject is discussed in details in Section REF .", "The infinite vacuum energy density in a field theory does not produce a problem by itself.", "Being coupled to gravity, however, such a theory becomes inconsistent.", "Thus, every known consistent field theory with gravity must be supersymmetric.", "In a similar manner, supersymmetry is included into every consistent string theory.", "Another notable property of supersymmetric field theories is the equality in the number of bosonic and fermionic particles.", "It is not what we observe experimentally.", "This does not mean, however, that the idea of supersymmetry is altogether unreasonable.", "Indeed, supersymmetry may describe particle interactions at very small distances and very high energies and be broken at our energy scale.", "Assuming that it is the case, the extra predicted particles may acquire large masses, which explains the fact that they have not been observed so far.", "In addition, supersymmetric gauge theories provide the lightest supersymmetric particle as a natural candidate for dark matter.", "All physical phenomena up to the TeV scale are well described by the Standard Model.", "Despite its success, however, it is conceivable that a new theory has to exist beyond the TeV scale.", "One reason is that we need a Higgs boson to break the electroweak symmetry.", "The radiative corrections to the mass of the Higgs boson are quadratically divergent and thus give an unacceptable large contribution if the cutoff scale is not of the TeV scale.", "This is called the naturalness problem.", "One way to address it is to consider a supersymmetric extension of the Standard Model.", "In the presence of supersymmetry, the mass of the Higgs boson is the same as the mass of its fermionic partner, while the fermion mass obtains only a logarithmic divergence due to the fact that an additional chiral symmetry appears in the absence of the mass term.", "Of course, supersymmetry should be broken below the TeV scale to be able to describe our non-supersymmetric world.", "One can introduce supersymmetry breaking terms by hand.", "They should break supersymmetry softly in the sense that quadratic divergences should be absent.", "Alternatively, supersymmetry can be broken dynamically, so that the soft terms are generated in the a energy effective theory.", "There is another aspect which makes supersymmetry attractive for a theorist.", "In particle physics, symmetries restrict particle dynamics and allow one to make theoretical predictions on kinematical grounds without actually doing any concrete dynamical calculation.", "The introduction of the extra symmetry on top of the Poincaré symmetry imposes more constraints on the amplitudes in a theory and makes it more accessible for theoretical studies.", "In it even believed that in some cases supersymmetry makes a theory exactly solvable.", "In fact, supersymmetry is a powerful instrument to study strong coupling dynamics and non-perturbative effects analytically.", "In addition, theoretically appealing property of supersymmetry is that it offers the only “loophole” to the Coleman-Mandula theorem (see Section REF ) which prohibits any nontrivial extension of the Poincaré symmetry in a field theory.", "It is possible to make a supersymmetric theory from almost any field theory.", "There exist en effective superfield technique for this.", "It is discussed in details in this chapter.", "In a certain way, supersymmetric extension of a theory can be compared with the extension of real numbers $\\mathbb {R}$ to complex numbers $\\mathbb {C}$ .", "Indeed, analytical functions on the complex plane are much more constrained than functions of real argument.", "It is well known that an analytical function of a complex argument, given in some region of the complex plane, can be uniquely analytically continued to other regions or even to the whole complex plane.", "It it also well known that an analytical function can be reconstructed from the knowledge of its zeros and poles, which is not the case for functions of real argument.", "These central properties of complex functions are of great importance in supersymmetric theories: the introduction of supersymmetry renders physical observables (e.g.", "amplitudes) depend on the parameters of the theory analytically.", "This allows one to calculate these quantities in one region of the theory (for example, in the region of the weak coupling, where the calculations can be carried out perturbatively) and then analytically continue the results to the strong coupling regime.", "As an example of such analysis, let us mention the paper by Seiberg and Witten [25] who studied the supersymmetric extension of quantum chromodynamics (QCD) and showed analytically that this theory is confining.", "Supersymmetry, however, is reacher than just the complex analysis.", "Whereas the complex extension of real numbers is unique, several supersymmetric extensions of a quantum theory may be possible.", "A theory may have an ordinary or an extended supersymmetry.", "A quantum field theory in four space-time dimensions may have as much as four independent supersymmetries.", "The four-dimensional field theories with gravity may have as much as eight independent supersymmetries.", "The number of independent supersymmetries is usually counted by the ${\\cal N}$ symbol, e.g.", "${\\cal N}=2$ means that a field theory has two independent supersymmetries.", "The more is the number of supersymmetries, the more a theory is constrained.", "This generally makes the theory more amenable for theoretical studies.", "The best known example is ${\\cal N} =4$ super-Yang-Mills theory – a theory similar to QCD, but extended to have four different supersymmetries.", "It is believed that this theory is exactly solvable.", "Still, this theory is very different from QCD, having zero $\\beta $ -function, no dimensional transmutation and no confinement.", "The simplest supersymmetric extension of QCD (with one supersymmetry) is still too complicated to be understood analytically.", "The ${\\cal N}=2$ super-Yang-Mills theory (which was studied by Seiberg and Witten) is intermediate between ${\\cal N}=1$ and ${\\cal N}=4$ theories.", "Being invented as a form of mathematical construction, supersymmetry produced the deepest impact on theoretical physics over the last several decades and became an essential part of modern high-energy physics.", "We denote the coordinates in four-dimensional Minkowski space as $x^\\mu $ with the Lorentz indices taken from the middle of the Greek alphabet, $\\mu ,\\, \\nu ,\\, \\rho ,\\,\\dots = 0,1,2,3.$ Thereby, the coordinate $x^0$ is associated with time.", "As for the three space components, $x^i$ , we use the indices from the middle of the Latin alphabet, $i,\\, j,\\, k,\\, \\dots = 1,2,3.$ Also, it is convenient to use the vectorial notation $\\vec{x}$ for the space components of the four-vector $x^\\mu $ .", "The Minkowski space metric tensor is $g_{\\mu \\nu } = \\left(\\begin{array}{cccc}1&0&0&0\\\\0&-1&0&0\\\\0&0&-1&0\\\\0&0&0&-1\\end{array}\\right)\\equiv {\\rm diag}\\left(1,-1,-1,-1\\right).$ As usual, it is used for raising and lowering Lorentz indices.", "For instance, one has for a tensor $A_{\\mu \\nu }$ with two Lorentz indices: $A^{\\mu }_{\\,\\,\\nu } = g^{\\mu \\rho } A_{\\rho \\nu },\\quad \\quad A_{\\mu \\nu } = g_{\\mu \\rho } A^{\\rho }_{\\,\\,\\nu },$ where $g^{\\mu \\nu }$ is the inverse metric tensor, which is equal to $g_{\\mu \\nu }$ .", "As usual, summing over the repeated indices is assumed, as in the formulas above.", "Supersymmetry involves fermions.", "Consequently, spinors and spinor notations are extensively exploited in all the chapters.", "The spinorial indices are denoted with undotted and dotted Greek letters from the beginning of the alphabet: $\\alpha ,\\beta = 1,2\\quad \\quad \\mbox{and}\\quad \\quad \\dot{\\alpha }, \\dot{\\beta }= 1,2.$ Throughout this chapter the following four-dimensional matrices are used: $\\left(\\sigma ^\\mu \\right)_{\\alpha \\dot{\\alpha }} = \\left\\lbrace 1,\\, \\vec{\\sigma }\\right\\rbrace _{\\alpha \\dot{\\alpha }},\\quad \\quad \\left(\\bar{\\sigma }^\\mu \\right)^{\\dot{\\alpha }\\alpha } = \\left\\lbrace 1,\\, -\\vec{\\sigma }\\right\\rbrace ^{\\dot{\\alpha }\\alpha },$ where $\\vec{\\sigma }$ are ordinary Pauli matrices.", "Note that starting from the next chapter, where a quantum-mechanical formalism is involved, the Euclidean version of these matrices will be used, see Eq.", "(REF ).", "The Poincaré group in Minkowski space parametrized by the coordinates $x^\\mu $ can be realized by linear transformations $x^{\\prime \\mu }= \\Lambda ^\\mu _\\nu x^\\nu + c^\\mu $ which preserve the space-time interval $ds^2 = g_{\\mu \\nu } dx^\\mu dx^\\nu .$ The subgroup of homogeneous transformations (i.e.", "those with parameters $\\Lambda ^\\mu _\\nu $ ) form the Lorentz group ${\\rm O}(1,3)$ .", "The invariance of $ds^2$ implies $g_{\\mu \\nu }\\Lambda ^\\mu _\\rho \\Lambda ^\\nu _\\sigma = g_{\\rho \\sigma }$ and, as a consequence, ${\\rm det} \\,\\Lambda = \\pm 1$ .", "Here we skip the consideration of the discrete Poincaré transformations (i.e.", "space-time reflections) some of which are related to the $\\det \\Lambda = -1$ branch of solutions of Eq.", "(REF ).", "Instead, we take the proper subgroup in the Lorentz group with ${\\rm det}\\, \\Lambda = 1$ .", "The infinitesimal (infinitely small) Lorentz transformation can be written as $\\Lambda ^\\mu _\\nu = \\delta ^\\mu _\\nu + \\omega ^\\mu _\\nu ,\\quad \\quad \\omega _{\\mu \\nu } = - \\omega _{\\nu \\mu },$ where, as usual, $\\omega _{\\mu \\nu } = g_{\\mu \\rho }\\omega ^\\rho _\\nu $ .", "In this way, the infinitesimal form of the transformations (REF ) is given by $\\delta x^\\mu = -i\\left[c^\\nu \\hat{P}_\\nu + \\frac{1}{2} \\omega ^{\\nu \\rho }\\hat{M}_{\\nu \\rho }\\right]x^\\mu ,$ where the differential operators $\\hat{P}_\\mu = i\\,\\partial _\\mu ,\\quad \\quad \\hat{M}_{\\mu \\nu } = -i\\left(x_\\mu \\partial _\\nu - x_\\nu \\partial _\\mu \\right)$ are the generators of the Poincaré algebra.", "The infinitesimal form for the action of the Poincaré group on functions $f(x)$ on the Minkowski space is $f^{\\prime }\\left(x\\right) =f(x) - i\\left[c^\\nu \\hat{P}_\\nu + \\frac{1}{2} \\omega ^{\\nu \\rho }\\hat{M}_{\\nu \\rho }\\right] f(x).$ The Poincaré algebra generators – four translations $P_\\mu $ and six space-time rotations $M_{\\mu \\nu }$ – form the Poincaré algebra Note that here and below we omit “hats” on the operators $P_\\mu $ and $M_{\\mu \\nu }$ .", "$\\begin{array}{l}[ P_\\mu \\,,\\, P_\\nu ] = 0,\\\\[2mm][ M_{\\mu \\nu }\\,,\\, P_\\lambda ]= i\\left( g_{\\mu \\lambda }\\, P_\\nu - g_{\\nu \\lambda }\\, P_\\mu \\right),\\\\[2mm][ M_{\\mu \\nu }\\,,\\, M_{\\rho \\sigma }]= i\\left( g_{\\mu \\rho }\\, M_{\\nu \\sigma }+g_{\\nu \\sigma }\\, M_{\\mu \\rho }-g_{\\nu \\rho }\\, M_{\\mu \\sigma } - g_{\\mu \\sigma }\\, M_{\\nu \\rho }\\right).\\end{array}$ Thus, the Poincaré algebra in Minkowski space in four dimensions has 10 independent generators: four space-time shifts, three space rotations and three boosts (transformations to other inertial reference frames).", "Supersymmetry unifies bosons and fermions and thus extensively uses the spinorial formalism.", "Here we recall the basic properties of this formalism in four-dimensional Minkowski space.", "Four-dimensional spinors realize irreducible representation of the Lorentz group (which has six generators: three spatial rotations and three Lorentz boosts).", "There are two types of spinors: left-handed and right-handed, which are marked by undotted and dotted indices, respectively, in the following way: $\\begin{array}{lll}\\mbox{left-handed:}\\quad \\quad & \\xi _\\alpha ,\\quad & \\alpha = 1,2,\\\\\\mbox{right-handed:}\\quad \\quad & \\bar{\\eta }_{\\dot{\\alpha }},\\quad & \\dot{\\alpha }= 1,2.\\end{array}$ It is possible to lower and raise spinor indices with the invariant Levi-Civita tensor from the left.", "For instance, $\\chi ^\\alpha = \\varepsilon ^{\\alpha \\beta }\\chi _\\beta ,\\quad \\quad \\chi _\\alpha = \\varepsilon _{\\alpha \\beta }\\chi ^\\beta $ and similar for spinors with dotted indices.", "The two-index antisymmetric Lorentz-invariant Levi-Civita tensors $\\varepsilon ^{\\alpha \\beta }$ , $\\varepsilon ^{\\dot{\\alpha }\\dot{\\beta }}$ , $\\varepsilon _{\\alpha \\beta }$ , and $\\varepsilon _{\\dot{\\alpha }\\dot{\\beta }}$ are defined as $\\begin{array}{l}\\varepsilon ^{\\alpha \\beta } = -\\varepsilon ^{\\beta \\alpha },\\quad \\varepsilon _{\\alpha \\beta } = -\\varepsilon _{\\beta \\alpha },\\quad \\varepsilon _{12}=-\\varepsilon ^{12} = 1,\\\\\\varepsilon ^{\\dot{\\alpha }\\dot{\\beta }}=-\\varepsilon ^{\\dot{\\beta }\\dot{\\alpha }},\\quad \\varepsilon _{\\dot{\\alpha }\\dot{\\beta }}=-\\varepsilon _{\\dot{\\beta }\\dot{\\alpha }},\\quad \\varepsilon _{\\dot{1}\\dot{2}}=-\\varepsilon ^{\\dot{1}\\dot{2}} = 1.\\end{array}$ The Lorentz transformation law for the undotted (left) spinors can be written as $\\xi ^{\\prime }_\\alpha = U_\\alpha ^\\beta \\xi _\\beta ,$ where the matrix $U$ has the form $U=\\exp \\left(-\\frac{i}{2} \\omega _{\\mu \\nu } \\sigma ^{\\mu \\nu }\\right)$ with $\\omega _{\\mu \\nu }$ being the same as in Eqs.", "(REF ), (REF ).", "The matrices $\\sigma ^{\\mu \\nu }$ give a particular matrix realization of the Lorentz rotations $M^{\\mu \\nu }$ and satisfy the last line in Eqs.", "(REF ).", "To be more specific, $\\sigma ^{\\mu \\nu } =\\frac{i}{4}\\left(\\sigma ^\\mu \\bar{\\sigma }^\\nu - \\sigma ^\\nu \\bar{\\sigma }^\\mu \\right)$ with the matrices $\\sigma ^\\mu $ and $\\bar{\\sigma }^\\mu $ being introduced in Eq.", "(REF ).", "Let us consider a spatial rotation.", "The matrix from Eq.", "(REF ) takes the following form: $U_{\\rm rot} = \\exp \\left(-i\\frac{\\theta }{2} \\vec{n}\\vec{\\sigma }\\right),\\quad \\quad \\theta \\, n^i = \\frac{1}{2}\\varepsilon ^{ijk} \\omega ^{jk},$ where $\\varepsilon ^{ijk}$ is antisymmetric Levi-Civita tensor ($\\varepsilon ^{123}=1$ ), $\\theta $ is the rotation angle and $\\vec{n}$ – the unit vector denoting the axis of rotation.", "Analogously for a Lorentz boost, the matrix from Eq.", "(REF ) has the form $U_{\\rm boost} = \\exp \\left(\\frac{\\phi }{2} \\vec{n}^{\\prime }\\vec{\\sigma }\\right),\\quad \\quad \\phi \\, n^i = \\omega ^{oi}.$ Here $\\tanh \\phi = v$ , where $v$ is the velocity in the units of speed of light of the first inertial reference frame with respect to the second, $\\vec{n}^{\\prime }$ denotes the velocity direction.", "Note that in the case of the spatial rotation the matrix $U_{\\rm rot}$ is unitary, $U^\\dagger _{\\rm rot} U_{\\rm rot} = 1$ , whereas in the case of the Lorentz boost the matrix $U_{\\rm boost}$ is not.", "This reflects the fact that the Lorentz group is the noncompact ${\\rm O}(1,3)$ group rather than the compact ${\\rm O}(4)$ group.", "Dotted spinors transform as complex conjugates of undotted spinors: $\\bar{\\eta }_{\\dot{\\alpha }} \\sim \\left(\\eta _\\alpha \\right)^*,$ where the sign $\\sim $ means “is transformed as”.", "Therefore, for dotted spinors the Lorentz transformation goes with the complex conjugated matrix, $\\bar{\\eta }^{\\prime }_{\\dot{\\alpha }} = \\left(U^*\\right)_{\\dot{\\alpha }}^{\\dot{\\beta }} \\bar{\\eta }_{\\dot{\\beta }},$ where $U^*=\\exp \\left(\\frac{i}{2} \\omega _{\\mu \\nu } \\bar{\\sigma }^{\\mu \\nu }\\right),$ and the matrices $\\bar{\\sigma }^{\\mu \\nu } =\\frac{i}{4}\\left(\\bar{\\sigma }^\\mu \\sigma ^\\nu - \\bar{\\sigma }^\\nu \\sigma ^\\mu \\right)$ give another matrix realization of the Lorentz rotations $M^{\\mu \\nu }$ and satisfy the last equation in Eqs.", "(REF ).", "Similarly for dotted spinors, one has for particular cases of spatial rotations and Lorentz boosts: $\\bar{\\eta }^{\\prime \\dot{\\alpha }}=\\left\\lbrace \\begin{array}{ll}\\left(U_{\\rm rot}\\right)_{\\dot{\\beta }}^{\\dot{\\alpha }}\\, \\bar{\\eta }^{\\dot{\\beta }}, \\quad & \\mbox{for rotations},\\\\[2mm]\\left(U_{\\rm boost}^{-1}\\right)_{\\dot{\\beta }}^{\\dot{\\alpha }}\\, \\bar{\\eta }^{\\dot{\\beta }}, \\quad & \\mbox{for boosts},\\end{array}\\right.$ where for convenience the index for the spinor $\\bar{\\eta }_{\\dot{\\alpha }}$ is raised with the antisymmetric Levi-Civita tensor.", "Note that under spatial rotations the undotted spinor $\\xi _\\alpha $ and the dotted spinor $\\bar{\\eta }^{\\dot{\\alpha }}$ transform under one and the same matrix $U_{\\rm rot}$ .", "The spinors $\\xi _\\alpha $ and $\\bar{\\eta }_{\\dot{\\alpha }}$ are referred to as Weyl spinors.", "In Minkowski space in four dimensions one undotted and one dotted Weyl spinor comprise one Dirac spinor (see, for example, the textbook [26] for a more detailed description).", "In order to be Lorentz-invariant, an equation which involves spinors must have the same number of undotted and dotted indices on each side, otherwise the equation becomes invalid under a change of reference frame.", "One should also remember, however, that complex conjugation implies the interchange of dotted and undotted indices.", "For instance, the relation $\\left(\\xi _{\\alpha \\beta }\\right)^* = \\bar{\\eta }_{\\dot{\\alpha }\\dot{\\beta }}$ is Lorentz-invariant.", "Lorentz scalars can be built by convolution of either undotted or dotted spinor indices.", "For example, the products $\\chi ^{\\alpha }\\xi _\\alpha \\quad \\quad \\mbox{and}\\quad \\quad \\bar{\\psi }_{\\dot{\\beta }}\\bar{\\eta }^{\\dot{\\beta }}$ are invariant under the Lorentz transformations.", "The Poincaré algebra (REF ) forms basis of geometric symmetries of a relativistic field theory.", "Other symmetries like flavour symmetry, isospin symmetry, etc.", "commute with the Poincaré group and are internal in the sense that they have nothing to do with Minkowski space.", "A natural question arises: is it possible to extend the Poincaré group with an additional symmetry which affects space-time coordinates?", "It was believed for a long time that this is not possible.", "In 1967 Coleman and Mandula formulated a theorem which states that, in a dynamically nontrivial relativistic quantum field theories of space-time dimension $d\\ge 3$ with interactions and with asymptotic states (particles), no geometric extension of the Poincaré group is possible [27].", "In other words, besides already known conserved generators carrying Lorentz indices (the energy-momentum operator $P_\\mu $ and the Lorentz transformations $M_{\\mu \\nu }$ ) no such new conserved charges (algebra generators) can appear.", "According to the theorem, the only allowed additional conserved charges must be Lorentz scalars, such as the electromagnetic charge.", "However, in 1970 Golfand and Likhtman found a loophole in this theorem [28] which, together with the Coleman-Mandula theorem, singles out supersymmetry as the only possible geometric extension of the Poincaré invariance in a relativistic field theory.", "A reason for which this statement is not valid in one and in two space-time dimensions will become clear shortly.", "The essence of the proof of the Coleman-Mandula theorem is the following.", "Let us take an interacting field theory and consider two-particle scattering process.", "The energy and momentum conservation laws present in every Poincaré-invariant field theory.", "Particularly, in our case $p_1^\\mu + p_2^\\mu = p_3^\\mu + p_4^\\mu ,\\quad \\quad \\mu = 0,1,2,3,$ where $p^\\mu _{1}$ , $p^\\mu _2$ are the particle 4-momenta before the interaction while $p^\\mu _{3}$ , $p^\\mu _{4}$ are the particle 4-momenta after the interaction.", "The kinematic constraints above leave only one essential free parameter – the scattering angle $\\theta $ , see Fig.", "REF .", "This angle cannot be determined on kinematical grounds and is defined by particular dynamics in the theory.", "Figure: Two-particle scattering.", "Only the scattering angle θ\\theta is undefined from energy and momentum conservation laws.Imagine now that there is an additional symmetry generator of space-time with some Lorentz indices.", "An additional exotic conservation law which have the same Lorentz indices corresponds to this symmetry and involves the particles 4-momenta.", "The presence of this conservation law would completely fix the scattering angle $\\theta $ (or at most would leave only a discrete set of possible angles).", "Since the scattering amplitude is an analytic function of the angle, it then must vanish for all angles.", "In other words, the theory has trivial $S$ -matrix, i.e.", "it is non-interacting.", "Consequently, the Coleman-Mandula theorem is not applicable in one and in two space-time dimensions, where there is no scattering angle between the two particles.", "The details of the proof and also its generalization to non-identical particles, particles with spin, etc.", "can be found in Refs.", "[27], [29], [30].", "Thus, no geometric extension of the Poincaré symmetry is possible on asymptotic states in a nontrivial field theory.", "Saying this differently, either the theory dynamics is trivial or the theory has no asymptotic states (particles), or extra geometric symmetries in the theory are broken on asymptotic states.", "The latter two statements can be illustrated on an example of a conformal field theory.", "The conformal symmetry adds scale invariance to a theory.", "For instance, any conformal field theory possesses an extra space-time symmetry $x^{\\prime \\mu }= \\lambda x^\\mu ,$ where $\\lambda $ is an arbitrary positive number.", "In its infinitesimal form $\\lambda =1+\\epsilon $ , $|\\epsilon |\\ll 1$ and $x^{\\prime \\mu }= x^\\mu + i\\epsilon \\hat{D} x^\\mu ,$ where $\\hat{D}$ is the so called dilatational operator, $\\hat{D} = -i x^\\mu \\partial _\\mu $ Together with certain additional conformal operators (special conformal transformations usually denoted as $\\hat{K}_\\mu $ ), the Poincaré algebra (REF ) extends in a nontrivial way.", "However, the asymptotic states (particles) in the theory would break conformal symmetry down to the Poincaré symmetry, in full accordance with Coleman-Mandula theorem.", "The second possibility – the theory has no asymptotic states at all.", "Such a theory is scale invariant so that the distance between two points in space is undetermined.", "For instance, ${\\cal N}= 4$ super-Yang-Mills theory is of this kind.", "The Coleman-Mandula theorem assumes that all symmetry generators in a relativistic field theory are operators which possibly have some Lorentz indices (or, equivalently, even number of spinor indices) and thus are bosonic operators, i.e.", "they form a Lie algebra with certain commutation relations.", "Meanwhile, generators with odd number of spinor indices are of fermionic nature and are not considered in the proof of the theorem since they cannot participate in commutation relations.", "The loophole in the theorem consists in the possibility to introduce the operators $Q_\\alpha $ and ${\\bar{Q}}_{\\dot{\\alpha }}$ with spinor indices.", "The algebra which now includes such operators must involve not only commutators, e.g.", "$[B_1, \\, B_2]$ and $[Q_\\alpha ,\\, B_3]$ (with $B_{1,2,3}$ being the bosonic operators), but also anticommutators, e.g.", "$\\lbrace Q_\\alpha , \\, \\bar{Q}_{\\dot{\\alpha }}\\rbrace $ .", "Hence, the spinor operators $Q_\\alpha $ , $\\bar{Q}_{\\dot{\\alpha }}$ , due to their nature, cannot produce additional restrictions on particle momenta in scattering processes so that no new conservation laws appear.", "Nevertheless, they relate to each other various scattering amplitudes which greatly constrains the $S$ -matrix in a quantum field theory.", "The complex operators $Q_\\alpha $ and ${\\bar{Q}}_{\\dot{\\alpha }}$ are Hermitian conjugated, ${\\bar{Q}}_{\\dot{\\alpha }} = \\left(Q_\\alpha \\right)^\\dagger ,$ and transform as ordinary Weyl spinors under the action of the Poincaré algebra, namely they satisfy the following commutation relations with the Poincaré algebra generators: $\\begin{array}{l}[P_\\mu ,\\, Q_\\alpha ] = [P_\\mu ,\\, {\\bar{Q}}^{\\dot{\\alpha }}] =0,\\\\[2mm][M^{\\mu \\nu },\\, Q_\\alpha ] = i \\left(\\sigma ^{\\mu \\nu }\\right)_\\alpha ^{\\,\\,\\,\\beta }\\,Q_\\beta ,\\\\[2mm][M^{\\mu \\nu },\\, {\\bar{Q}}^{\\dot{\\alpha }}] = i \\left(\\bar{\\sigma }^{\\mu \\nu }\\right)^{\\dot{\\alpha }}_{\\,\\,\\,\\dot{\\beta }}\\, {\\bar{Q}}^{\\dot{\\beta }},\\end{array}$ where the index for the supercharge $\\bar{Q}_{\\dot{\\alpha }}$ is raised with the antisymmetric Levi-Civita tensor, $\\bar{Q}^{\\dot{\\alpha }}=\\varepsilon ^{{\\dot{\\alpha }}{\\dot{\\beta }}}\\bar{Q}_{{\\dot{\\beta }}}$ , and the matrices $\\sigma ^{\\mu \\nu }$ and $\\bar{\\sigma }^{\\mu \\nu }$ were introduced in Eqs.", "(REF ) and (REF ).", "The operators $Q_\\alpha $ and ${\\bar{Q}}_{\\dot{\\alpha }}$ are also referred to as supercharges or supergenerators.", "According to their indices, the minimum number of such supergenerators in four-dimensional Minkowski space is four.", "To close the algebra (REF ), (REF ), one needs to specify the anticommutators $\\lbrace Q_\\alpha , {\\bar{Q}}_{\\dot{\\alpha }}\\rbrace $ , $\\lbrace Q_\\alpha , Q_\\beta \\rbrace $ and $\\lbrace \\bar{Q}_{\\dot{\\alpha }}, \\bar{Q}_{\\dot{\\beta }}\\rbrace $ .", "The first anticommutator can only be proportional to $P_\\mu \\left(\\sigma ^\\mu \\right)_{\\alpha \\dot{\\beta }}$ since it is the only operator with the appropriate Lorentz indices.", "The standard normalization is $\\lbrace Q_\\alpha , {\\bar{Q}}_{\\dot{\\alpha }}\\rbrace = 2 P_\\mu \\left( \\sigma ^\\mu \\right)_{\\alpha \\dot{\\alpha }}.$ The simplest choice for the other two anticommutators allowed by the Jacobi identities is $\\lbrace Q_\\alpha , Q_{\\beta }\\rbrace = \\lbrace {\\bar{Q}}_{\\dot{\\alpha }}, {\\bar{Q}}_{\\dot{\\beta }}\\rbrace = 0.$ Thereby, Eqs.", "(REF ), (REF ), (REF ), (REF ) form the super-Poincaré algebra first obtained by Golfand and Likhtman [28].", "This minimal super-Poincaré algebra with four supercharges can be further extended with additional supercharges.", "As was demonstrated in Ref.", "[31], one can construct extended supersymmetries, with up to sixteen supercharges in four dimensions.", "The minimal supersymmetry is referred to as ${\\mathcal {N}}=1$ .", "Correspondingly, one can consider ${\\mathcal {N}}=2$ (eight supercharges) or ${\\mathcal {N}}=4$ (sixteen supercharges).", "The extended supersymmetry in a four-dimensional field theory is discussed in Section REF .", "As was also demonstrated in Ref.", "[31], it is possible to modify the super-Poincaré algebra by an introduction of central charges in it.", "Such superalgebras are referred to as centrally extended.", "A central charge is an element of the superalgebra which commutes with other generators.", "It acts as a number with numerical value being dependent on a sector of a theory under consideration.", "The presence of central charges reflect possible existence of conserved topological currents and topological charges [32].", "For instance, if a theory under consideration supports topologically stable domain walls, the right-hand side of (REF ) can be modified in the following way: $\\lbrace Q_\\alpha , Q_{\\beta }\\rbrace = C_{\\alpha \\beta },\\quad \\quad \\lbrace \\bar{Q}_{\\dot{\\alpha }}, \\bar{Q}_{{\\dot{\\beta }}}\\rbrace = \\left(C_{\\alpha \\beta }\\right)^\\dagger .$ Here $C_{\\alpha \\beta }=C_{\\beta \\alpha }$ are the central charges.", "Let us remark that they have spinor indices and thus transform under Lorentz rotations.", "This is why $C_{\\alpha \\beta }$ are also called tensor central charges to distinguish them from “standard” central charges which commute with all superalgebra generators.", "(See also the footnote in Section REF , where it is shown how the central charges $Z^{IJ}$ , which are Lorentz scalars, can be introduced for the case of extended supersymmetry.)", "For any supersymmetric field theory, the following fundamental properties hold: a state in a supersymmetric field theory cannot have negative energy; if supersymmetry is unbroken, the vacuum has exactly zero energy; if there is a boson with mass $m$ , there must exist a fermion with exactly the same mass $m$ , and vice versa (Bose-Fermi degeneracy); any supersymmetric field theory has equal number of bosonic and fermionic degrees of freedom in every supermultiplet.", "a state in a supersymmetric field theory cannot have negative energy; if supersymmetry is unbroken, the vacuum has exactly zero energy; if there is a boson with mass $m$ , there must exist a fermion with exactly the same mass $m$ , and vice versa (Bose-Fermi degeneracy); any supersymmetric field theory has equal number of bosonic and fermionic degrees of freedom in every supermultiplet.", "Let us discuss these statements in details.", "The first consequence which follows from the super-Poincaré algebra is the fact that a state in a quantum field theory cannot have negative energy.", "This straightforwardly follows from Eqs.", "(REF ) and (REF ) if one takes the sum $P^0 =\\frac{1}{4} \\sum _{\\alpha =1}^2 \\left[Q_\\alpha \\left(Q_\\alpha \\right)^\\dagger + \\left(Q_\\alpha \\right)^\\dagger Q_\\alpha \\right]$ and calculates an average of the left and the right hand sides for a normalized eigenstate $\\left| \\Psi \\right>$ with the energy $E$ .", "Indeed, $\\left<\\Psi \\left| P^0\\right|\\Psi \\right> = E=\\frac{1}{4}\\sum \\limits _{\\alpha =1}^2\\left< \\Psi \\left| Q_\\alpha \\left(Q_\\alpha \\right)^\\dagger + \\left(Q_\\alpha \\right)^\\dagger Q_\\alpha \\right| \\Psi \\right>\\\\[2mm]=\\frac{1}{4}\\sum \\limits _{\\alpha =1}^2\\big < \\left(Q_\\alpha \\right)^\\dagger \\Psi \\big |\\left(Q_\\alpha \\right)^\\dagger \\Psi \\big >^*+\\frac{1}{4}\\sum \\limits _{\\alpha =1}^2\\big <Q_\\alpha \\Psi \\big | Q_\\alpha \\Psi \\big >^*$ The second line in this equality is always non-negative.", "Thus, for any quantum eigenstate its energy $E \\ge 0$ .", "The minimum $E=0$ is achieved on a vacuum state $\\left| 0 \\right>$ which is annihilated by the supercharges, $Q_\\alpha |0\\rangle = \\left(Q_\\alpha \\right)^\\dagger |0\\rangle =0.$ A field theory may have one or several vacua with zero energy.", "If a theory have no states with zero energy, i.e.", "$E_{\\rm vac} > 0$ , the supersymmetry is spontaneously broken.", "In fact, the vanishing of the vacuum energy is the necessary and sufficient condition for supersymmetry to be left unbroken.", "In a supersymmetric theory, if there is a boson with the mass $m$ , a fermion with the very same mass $m$ must exist too, and vice versa.", "To elaborate more on this point, let us introduce a bosonic state $\\left| B \\right>$ with the mass $m$ and associate with it one of the following fermionic states: $Q_\\alpha \\left| B \\right>$ , $\\left(Q_\\alpha \\right)^\\dagger \\left| B \\right>$ , where $\\alpha =1,2$ .", "At least one of these four states is nonzero.", "Indeed, the sum of the norms of these four states in positive: $\\sum \\limits _{\\alpha =1}^2\\big < \\left(Q_\\alpha \\right)^\\dagger B \\big |\\left(Q_\\alpha \\right)^\\dagger B\\big >+\\sum \\limits _{\\alpha =1}^2\\big <Q_\\alpha B \\big | Q_\\alpha B\\big >\\\\[2mm]=\\sum \\limits _{\\alpha =1}^2\\left< B \\left| Q_\\alpha \\left(Q_\\alpha \\right)^\\dagger + \\left(Q_\\alpha \\right)^\\dagger Q_\\alpha \\right| B\\right>^*=4\\left<B \\left| P^0\\right|B\\right>^* = 4 E_B,$ where $E_B > 0$ is the state $\\left| B \\right>$ energy.", "Note also that $P^2= P_\\mu P^\\mu $ which is a Casimir operator of the Poincaré algebra (it commutes with all the Poincaré algebra generators) is also a Casimir operator of the super-Poincaré algebra, because $[P^2,\\, Q_\\alpha ] = [P^2,\\, {\\bar{Q}}_{\\dot{\\alpha }}] = 0.$ Thus, from $P^2 \\left| B \\right>=m^2 \\left| B \\right>$ follows that $P^2 \\big |Q_\\alpha B\\big > = m^2 \\big |{Q_\\alpha B}\\big >\\quad \\quad \\mbox{and}\\quad \\quad P^2\\left| \\left(Q_\\alpha \\right)^\\dagger B \\right> = m^2 \\left| \\left(Q_\\alpha \\right)^\\dagger B \\right>.$ Combining the two observations above, one arrives to the statement of this section.", "In a similar manner, one can prove the reverse statement: for a fermion with the mass $m$ there exist a boson with the very same mass $m$ .", "The Poincaré group is not a compact group.", "That is why all its unitary representations (except for the trivial representation) are infinite-dimensional.", "This infinite dimensionality reveals itself in a widely known fact that particle states are labeled by the continuous parameters – particle 4-momentum $p_\\mu $ .", "The Poincaré algebra has two Casimir operators: $P^2 = P_\\mu P^\\mu $ and $W^2 = W_\\mu W^\\mu $ , where $W^\\mu $ is Pauli–Lubanski vector, $W^\\mu = \\frac{1}{2}\\, \\varepsilon ^{\\mu \\nu \\rho \\sigma }\\, P_\\nu \\, M_{\\rho \\sigma }$ ($\\varepsilon ^{\\mu \\nu \\rho \\sigma }$ is antisymmetric Levi-Civita tensor).", "The eigenvalues of the operator $P^2$ fix particle mass squared, $p_\\mu p^\\mu = m^2$ , while the eigenvalues of the operator $W^2$ are responsible for particle spin if the particle mass is not zero.", "To understand the latter statement, let us boost to a reference frame where the particle is at rest: $p_\\mu = (m,\\,0,\\,0,\\,0)$ .", "One can check that in this reference frame $W^2 = W_\\mu W^\\mu = - m^2 s(s+1),$ where $s$ is the particle spin.", "For massless particles $P^2=0$ and $W^2=0$ .", "Then, instead of spin, one must consider particle helicity.", "One can boost to the reference frame where the particle 4-momentum is $p_\\mu = (E,\\,0,\\,0,\\,E)$ with $E$ being the particle energy.", "Then the eigenvalues of the operator $M_{12}$ are $\\pm \\lambda $ with $\\lambda $ being the helicity.", "Hence, besides the particle 4-momentum $p_\\mu $ , a unitary irreducible representation of the Poincaré algebra is identified by the particle mass $m$ and the particle spin or helicity $s$ , if the particle has zero mass.", "In contrast with $P^2$ , the operator $W^2$ does not commute with the supercharges, i.e.", "$[W^2 ,\\, Q_\\alpha ]\\ne 0$ , as follows from Eq.", "(REF ).", "The same is true for the operator $M_{12}$ .", "Thus, massive irreducible superalgebra representations must contain particles with different spins, while massless irreducible superalgebra representations must contain particles with different helicities.", "Due to the property, $Q_\\alpha ^2 = \\bar{Q}_{\\dot{\\alpha }}^2 = 0$ , the supercharges may change the particle into another particle with different spin/helicity a finite number of times.", "The corresponding set of particles, all with the same mass $m$ , but with different spins/helicities is called a supermultiplet.", "In the simplest case a supermultiplet consists of two particles with spins $s$ , $s+1/2$ or helicities $\\lambda $ , $\\lambda +1/2$ .", "Further details on building the supermultiplets can be found e.g.", "in [33], [30], [29], [34].", "We omit here a formal proof of the equality of the number of bosonic and fermionic states in a supermultiplet.", "It will be given in the case of supersymmetric quantum mechanics in Section REF .", "Instead, let us discuss how this fact follows from the vanishing of the vacuum energy.", "Consider a free field theory.", "It is well known that bosons and fermions contribute to the vacuum energy due to zero-point oscillations.", "The bosonic contribution is $\\sum _B \\sum _{\\vec{p}}\\,\\sqrt{m_B^2+\\vec{p}^{\\,2}},$ where the (divergent) sum runs over all bosonic degrees of freedom and over all spatial momenta.", "The fermionic contribution is $- \\sum _F \\sum _{\\vec{p}}\\,\\sqrt{m_F^2+\\vec{p}^{\\,2}},$ where the sum runs over all fermionic degrees of freedom.", "The extra minus sign is due to $-1$ associated with the fermion loop in the corresponding Feynman diagram which describes the vacuum energy density.", "The vanishing of the vacuum energy density requires the cancellation of two contributions which is possible only if the following equations hold inside each supermultiplet: $n_B=n_F\\quad \\quad \\left[\\begin{array}{c}\\mbox{equal number of bosons and fermions}\\\\\\mbox{in a supermultiplet}\\end{array}\\right]$ and $m_B = m_F\\quad \\quad \\left[\\begin{array}{c}\\mbox{equal masses of bosons and fermions}\\\\\\mbox{ in a supermultiplet}\\end{array}\\right].$ Note that the latter property was already proven in Section REF from algebraic considerations.", "Note also that by bosons and fermions we mean physical (positive norm) degrees of freedom.", "For instance, a photon has two degrees of freedom corresponding to the two transverse polarizations (the two helicities $\\pm 1$ ).", "In a relativistic field theory, fields are functions (probably, with some vector or spinor indices) which locally depend on the space-time point $x^\\mu $ and transform in a certain way under the action of the Poincaré group.", "With introduction of supersymmetry which is the geometric extension of the Poincaré symmetry, it is very natural to expand the space-time by an addition of appropriate extra dimensions.", "By doing so, one expands the concept of space-time to the concept of superspace.", "In the superspace, the supercharges are realized as differential operators which generate supertranslations in a way similar to the energy-momentum operator which generates translations in four-dimensional space-time.", "Due to the anticommuting nature of the supercharges, the extra dimensions in the superspace are described by coordinates of Grassmann (anticommuting) nature.", "Finally, the concept of fields is extended to the concept of superfields which are functions of the coordinates on the superspace.", "This breakthrough idea was pioneered by Salam and Strathdee [35].", "The immediate advantage of this formalism is that it gives simple and explicit description of the action of supersymmetry on component fields (see below) and provides a very efficient method for constructing manifestly supersymmetric Lagrangians.", "In an ordinary ${\\cal N}=1$ supersymmetry, the superspace $\\lbrace x^\\mu ,\\, \\theta ^\\alpha ,\\,\\bar{\\theta }^{\\dot{\\alpha }}\\rbrace ,\\quad \\quad \\bar{\\theta }^{\\dot{\\alpha }} \\equiv \\left(\\theta ^{\\alpha }\\right)^*$ includes four complex Grassmann (anticommuting) variables $\\theta ^\\alpha $ and $\\bar{\\theta }^{\\dot{\\alpha }}$ which represent “quantum” or “fermionic” dimensions of the superspace.", "They are complex conjugated and anticommute between each other, $\\lbrace \\theta ^\\alpha ,\\, \\theta ^\\beta \\rbrace = \\lbrace \\bar{\\theta }^{\\dot{\\alpha }} ,\\, \\bar{\\theta }^{\\dot{\\beta }}\\rbrace =\\lbrace \\theta ^\\alpha ,\\, \\bar{\\theta }^{\\dot{\\beta }}\\rbrace = 0.$ Note also the peculiarity of the Leibniz rule for Grassmann derivatives, e.g.", "$\\frac{\\partial }{\\partial \\theta ^\\alpha } \\left(\\theta ^\\beta \\theta ^\\gamma \\right) =\\left(\\frac{\\partial }{\\partial \\theta ^\\alpha } \\theta ^\\beta \\right)\\theta ^\\gamma -\\theta ^\\beta \\left(\\frac{\\partial }{\\partial \\theta ^\\alpha }\\theta ^\\gamma \\right).$ In addition, Hermitian conjugation changes the order of anticommuting numbers: $\\left(\\theta ^1 \\theta ^2\\right)^\\dagger = (\\theta ^2)^\\dagger (\\theta ^1)^\\dagger = \\bar{\\theta }^{\\dot{2}}\\bar{\\theta }^{\\dot{1}}.$ A superfield is a function of the coordinates (REF ) [35], [36].", "One can expand it in power series of the Grassmann variables $\\theta ^\\alpha $ and $\\bar{\\theta }^{\\dot{\\alpha }}$ .", "This expansion has finite number of terms since the square of a given Grassmann parameter vanishes.", "Thus, the highest term in this expansion is $\\theta ^2\\bar{\\theta }^2$ , where $\\theta ^2 = \\theta ^\\alpha \\theta _\\alpha $ , $\\bar{\\theta }^2 = \\bar{\\theta }_{\\dot{\\alpha }}\\bar{\\theta }^{\\dot{\\alpha }}$ .", "The most general superfield with no external indices has the following form: $S(x, \\theta , \\bar{\\theta }) = \\phi + \\theta ^\\alpha \\psi _\\alpha + \\bar{\\theta }_{\\dot{\\alpha }}\\bar{\\chi }^{\\dot{\\alpha }}+ \\theta ^2 F + \\bar{\\theta }^2 G+\\theta ^\\alpha A_{\\alpha \\dot{\\alpha }} \\bar{\\theta }^{\\dot{\\alpha }}+\\theta ^2 (\\bar{\\theta }_{\\dot{\\alpha }}\\bar{\\lambda }^{\\dot{\\alpha }})+\\bar{\\theta }^2 (\\theta ^\\alpha \\rho _\\alpha ) +\\theta ^2 \\bar{\\theta }^2 D,$ where $\\phi $ , $\\psi _\\alpha $ , $\\bar{\\chi }^{\\dot{\\alpha }}$ , $\\dots $ , $D$ depend only on $x^\\mu $ and are referred to as the component fields.", "In what follows we will use shorthand notations for contraction of spinor indices: $A B = A^\\alpha B_\\alpha ,\\quad \\quad \\bar{A} \\bar{B} = \\bar{A}_{\\dot{\\alpha }} \\bar{B}^{\\dot{\\alpha }}.$ In particular, if $A$ and $B$ are anticommuting variables, $A B = B A,\\quad \\quad \\bar{A} \\bar{B} = \\bar{B}\\bar{A},\\quad \\quad \\left(A B\\right)^\\dagger = \\bar{A} \\bar{B}.$ The representation of the supercharges $Q_\\alpha $ and $\\bar{Q}_{\\dot{\\alpha }}$ as differential operators on the superspace (REF ) can be derived in the following standard way.", "Let us associate with each point of the superspace (REF ) an element of the group corresponding to the ${\\mathcal {N}}=1$ superalgebra (REF ), (REF ), (REF ), (REF ) as $G(x^\\mu , \\theta ,\\bar{\\theta }) =e^{i\\left(-x^\\mu P_\\mu +\\theta ^\\alpha Q_\\alpha +\\bar{\\theta }_{\\dot{\\alpha }}{\\bar{Q}}^{\\dot{\\alpha }}\\right)}.$ Then the product of two elements $G(0, \\epsilon ,\\bar{\\epsilon })$ and $G(x^\\mu , \\theta ,\\bar{\\theta })$ is All the elements (REF ) form an invariant space under the action of the Poincaré group: the result of the action of any Poincaré group element on (REF ) is of the same type.", "In fact, this space is invariant under the action of the whole super-Poincaré group.", "This statement partially follows from the equality (REF ).", "Let us also remark that the superspace (REF ) can be obtained as the factor of the super-Poincaré group over the Lorentz group much in the same way this can be done for the four-dimensional Minkowski space: $\\frac{\\mbox{Poincaré group}}{\\mbox{Lorentz group}}\\quad \\quad \\longrightarrow \\quad \\quad \\frac{\\mbox{super-Poincaré group}}{\\mbox{Lorentz group}}.$ The points in this factor space are orbits obtained by the action of the Lorentz group on the super-Poincaré group space.", "If we choose a certain point as the origin, then the superspace can be parametrized by (REF ).", "$G(0, \\epsilon ,\\bar{\\epsilon }) \\,\\, G(x^\\mu , \\theta ,\\bar{\\theta })=G(x^\\mu + i\\theta ^\\alpha \\sigma ^\\mu _{\\alpha \\dot{\\alpha }} \\bar{\\epsilon }^{\\dot{\\alpha }}- i\\epsilon ^\\alpha \\sigma ^\\mu _{\\alpha \\dot{\\alpha }} \\bar{\\theta }^{\\dot{\\alpha }}\\,,\\, \\theta +\\epsilon \\,,\\,\\bar{\\theta }+\\bar{\\epsilon }).$ All the elements (REF ) form an invariant space under the action of the Poincaré group: the result of the action of any Poincaré group element on (REF ) is of the same type.", "In fact, this space is invariant under the action of the whole super-Poincaré group.", "This statement partially follows from the equality (REF ).", "Let us also remark that the superspace (REF ) can be obtained as the factor of the super-Poincaré group over the Lorentz group much in the same way this can be done for the four-dimensional Minkowski space: $\\frac{\\mbox{Poincaré group}}{\\mbox{Lorentz group}}\\quad \\quad \\longrightarrow \\quad \\quad \\frac{\\mbox{super-Poincaré group}}{\\mbox{Lorentz group}}.$ The points in this factor space are orbits obtained by the action of the Lorentz group on the super-Poincaré group space.", "If we choose a certain point as the origin, then the superspace can be parametrized by (REF ).", "This equality can be proven by using the Hausdorff formula $e^Ae^B =e^{A+B +\\frac{1}{2} [A,B] + ...}$ (where the ellipsis corresponds to infinite series of multi-commutator terms) and taking into account the fact that the series on the right-hand side terminate at the first commutator for the algebra elements considered here.", "While doing this calculation, one should also remember that the parameters $\\epsilon $ , $\\bar{\\epsilon }$ , $\\theta $ , $\\bar{\\theta }$ as well as the supercharges $Q$ , $\\bar{Q}$ all anticommute between each other.", "Thereby, the action of the group element $G(0, \\epsilon ,\\bar{\\epsilon })$ on $G(x^\\mu , \\theta ,\\bar{\\theta })$ induces the following motion in the parameter space (REF ): $\\begin{array}{l}x^\\mu \\rightarrow x^\\mu + i\\theta ^\\alpha \\sigma ^\\mu _{\\alpha \\dot{\\alpha }} \\bar{\\epsilon }^{\\dot{\\alpha }}- i\\epsilon ^\\alpha \\sigma ^\\mu _{\\alpha \\dot{\\alpha }} \\bar{\\theta }^{\\dot{\\alpha }},\\\\[2mm]\\theta _\\alpha \\rightarrow \\theta _\\alpha + \\epsilon _\\alpha ,\\\\[2mm]\\bar{\\theta }_{\\dot{\\alpha }}\\rightarrow \\bar{\\theta }_{\\dot{\\alpha }}+ \\bar{\\epsilon }_{\\dot{\\alpha }}.\\end{array}$ This motion is generated by the operator $i\\left(\\epsilon ^\\alpha Q_\\alpha + \\bar{\\epsilon }_{\\dot{\\alpha }}\\bar{Q}^{\\dot{\\alpha }}\\right)$ , where the Hermitian-conjugated supercharges are $Q_\\alpha = -i\\frac{\\partial }{\\partial \\theta ^\\alpha } -\\sigma ^\\mu _{\\alpha {\\dot{\\alpha }}}\\,\\bar{\\theta }^{\\dot{\\alpha }}\\,\\partial _\\mu ,\\quad \\quad {\\bar{Q}}_{\\dot{\\alpha }}=i\\frac{\\partial }{\\partial \\bar{\\theta }^{\\dot{\\alpha }}}+\\theta ^{\\alpha }\\sigma ^\\mu _{\\alpha {\\dot{\\alpha }}}\\,\\partial _\\mu .$ They satisfy the anticommutation relations $\\begin{array}{lll}\\big \\lbrace Q_\\alpha ,\\,Q_\\beta \\big \\rbrace &=& \\left\\lbrace \\bar{Q}_{\\dot{\\alpha }},\\, \\bar{Q}_{\\dot{\\beta }}\\right\\rbrace \\,\\,=\\,\\, 0,\\\\[2mm]\\left\\lbrace Q_\\alpha ,\\, {\\bar{Q}}_{\\dot{\\alpha }}\\right\\rbrace &=& 2i\\sigma ^\\mu _{\\alpha \\dot{\\alpha }}\\,\\partial _\\mu \\end{array}$ and hence, together with $P_\\mu = i\\partial _\\mu $ and an appropriate expression for $M_{\\mu \\nu }$  If $M_{\\mu \\nu }$ is acting on a scalar, its expression is given in Eq.", "(REF ).", "One must add certain (matrix) extra terms in this expression, if $M_{\\mu \\nu }$ is acting on a field with some vector or spinor indices.", "give an explicit realization of the supersymmetry algebra, Eqs.", "(REF ), (REF ), (REF ), (REF ).", "One could study right multiplication instead of left multiplication in (REF ) and would found that the induced motion is generated by a different operator $\\epsilon ^\\alpha D_\\alpha + \\bar{\\epsilon }_{\\dot{\\alpha }}\\bar{D}^{\\dot{\\alpha }}$ , with the operators $D_\\alpha $ and $\\bar{D}_{\\dot{\\alpha }}$ defined as $D_\\alpha =\\frac{\\partial }{\\partial \\theta ^\\alpha }+ i \\sigma ^\\mu _{\\alpha {\\dot{\\alpha }}}\\,\\bar{\\theta }^{\\dot{\\alpha }}\\,\\partial _\\mu ,\\quad \\quad \\bar{D}_{\\dot{\\alpha }}=-\\frac{\\partial }{\\partial \\bar{\\theta }^{\\dot{\\alpha }}}-i\\theta ^{\\alpha }\\sigma ^\\mu _{\\alpha {\\dot{\\alpha }}}\\,\\partial _\\mu .$ with $\\bar{D}_{\\dot{\\alpha }}= -\\left(D_\\alpha \\right)^\\dagger $ .", "Note that we have used a different convention of multipliers in the operators above on purpose: this will be convenient in subsequent sections.", "The operators $D_\\alpha $ and $\\bar{D}_{\\dot{\\alpha }}$ are called superderivatives.", "By their very definition, they satisfy the following anticommutation relations: $\\big \\lbrace D_\\alpha ,\\,D_\\beta \\big \\rbrace = \\left\\lbrace \\bar{D}_{\\dot{\\alpha }},\\, \\bar{D}_{\\dot{\\beta }}\\right\\rbrace = 0,\\quad \\quad \\left\\lbrace D_\\alpha ,\\, {\\bar{D}}_{\\dot{\\alpha }}\\right\\rbrace = -2i\\sigma ^\\mu _{\\alpha \\dot{\\alpha }}\\,\\partial _\\mu .$ In addition, the superderivatives and the supercharges anticommute: $\\big \\lbrace D_\\alpha ,\\,Q_\\beta \\big \\rbrace =\\left\\lbrace D_\\alpha ,\\,\\bar{Q}_{\\dot{\\beta }}\\right\\rbrace =\\left\\lbrace \\bar{D}_{\\dot{\\alpha }},\\,Q_\\beta \\right\\rbrace =\\left\\lbrace \\bar{D}_{\\dot{\\alpha }},\\,\\bar{Q}_{\\dot{\\beta }}\\right\\rbrace = 0.$ This will allow us to reduce the number of independent components in superfields by imposing covariant (consistent with supersymmetry) constraints on them (see Section REF ).", "Superfields form linear representations of superalgebra.", "In general, however, the representations are highly reducible.", "Extra components in superfields can be eliminated by imposing covariant constraints which (anti)commute with the supersymmetry algebra.", "One could say that superfield formalism shifts the problem of finding supersymmetry representations to that of finding appropriate constraints.", "Note that one should constrain superfields without restricting their $x^\\mu $ dependence (i.e., for instance, by virtue of differential equations in the $x$ space).", "Let us remark first that the superspace (REF ) has two invariant subspaces in it: $\\lbrace x^\\mu _{\\rm L},\\,\\theta ^\\alpha \\rbrace ,\\quad \\quad x_{\\rm L}^\\mu = {x}^\\mu + i\\theta ^\\alpha \\sigma ^\\mu _{\\alpha \\dot{\\alpha }} \\bar{\\theta }^{\\dot{\\alpha }},$ and $\\lbrace x^\\mu _{\\rm R},\\,\\bar{\\theta }^{\\dot{\\alpha }}\\rbrace ,\\quad \\quad x_{\\rm R}^\\mu = {x}^\\mu - i\\theta ^\\alpha \\sigma ^\\mu _{\\alpha \\dot{\\alpha }} \\bar{\\theta }^{\\dot{\\alpha }}.$ Indeed, the supertransformations (REF ) give the following supertransformations in the subspaces: $\\begin{array}{l}x_{\\rm L}^\\mu \\rightarrow x_{\\rm L}^\\mu + 2i\\theta ^\\alpha \\sigma ^\\mu _{\\alpha \\dot{\\alpha }} \\bar{\\epsilon }^{\\dot{\\alpha }},\\\\[2mm]\\theta ^\\alpha \\rightarrow \\theta ^\\alpha + \\epsilon ^\\alpha ,\\end{array}$ and $\\begin{array}{l}x_{\\rm R}^\\mu \\rightarrow x_{\\rm R}^\\mu - 2i\\epsilon ^\\alpha \\sigma ^\\mu _{\\alpha \\dot{\\alpha }} \\bar{\\theta }^{\\dot{\\alpha }},\\\\[2mm]\\bar{\\theta }^{\\dot{\\alpha }}\\rightarrow \\bar{\\theta }^{\\dot{\\alpha }}+ \\bar{\\epsilon }^{\\dot{\\alpha }}.\\end{array}$ These two subspaces are referred to as chiral (or left) and antichiral (or right) respectively.", "Each of them is spanned by half of the Grassmann coordinates.", "Due to this, a function which is defined on the chiral or the antichiral subspace have much shorter component expansion.", "Consider, for instance, a chiral superfield $\\Phi (x_{\\rm L},\\theta )$ .", "Its component expansion ${\\Phi (x_{\\rm L},\\theta )} = \\phi ({x}_{\\rm L}) + \\sqrt{2}\\,\\theta ^\\alpha \\psi _\\alpha ({x}_{\\rm L}) + \\theta ^2 F({x}_{\\rm L})$ includes one complex scalar field $\\phi (x)$ (two bosonic states) and one complex Weyl spinor $\\psi _\\alpha (x)$ (two fermionic states) as well as the auxiliary $F$ term which is non-propagating: as we will see shortly, this field will appear in Lagrangian without a kinetic term.", "Chiral superfields are used for constructing matter sectors of various theories.", "Note that the chiral superfield $\\Phi (x_{\\rm L}, \\theta )$ , being expressed as a function of $x^\\mu $ , depends on $\\theta ^\\alpha $ as well as on $\\bar{\\theta }^{\\dot{\\alpha }}$ variables.", "In fact, the chiral superfield $\\Phi $ or the antichiral superfield $\\bar{\\Phi }$ can be obtained as solutions of the covariant constraints [37] $\\bar{D}_{\\dot{\\alpha }} \\Phi = 0\\quad \\quad {\\rm and}\\quad \\quad D_{\\alpha } \\bar{\\Phi }= 0$ which follow from $\\bar{D}_{\\dot{\\alpha }}\\, x_{\\rm L}^\\mu =0\\quad \\quad \\mbox{and}\\quad \\quad D_{\\alpha }\\, x_{\\rm R}^\\mu =0.$ Moreover, the covariant superderivatives (REF ) in the chiral basis $\\lbrace x_{\\rm L}^\\mu ,\\, \\theta ^\\alpha ,\\, \\bar{\\theta }^{\\dot{\\alpha }}\\rbrace $ are realized as $D_\\alpha =\\frac{\\partial }{\\partial \\theta ^\\alpha }+ 2i \\sigma ^\\mu _{\\alpha {\\dot{\\alpha }}}\\,\\bar{\\theta }^{\\dot{\\alpha }}\\frac{\\partial }{\\partial x^\\mu _{\\rm L}},\\quad \\quad \\bar{D}_{\\dot{\\alpha }}=-\\frac{\\partial }{\\partial \\bar{\\theta }^{\\dot{\\alpha }}},$ while in the antichiral basis $\\lbrace x_{\\rm R}^\\mu ,\\, \\theta ^\\alpha ,\\, \\bar{\\theta }^{\\dot{\\alpha }}\\rbrace $ they are realized as $D_\\alpha =\\frac{\\partial }{\\partial \\theta ^\\alpha },\\quad \\quad \\bar{D}_{\\dot{\\alpha }}=-\\frac{\\partial }{\\partial \\bar{\\theta }^{\\dot{\\alpha }}}-2i\\theta ^{\\alpha }\\sigma ^\\mu _{\\alpha {\\dot{\\alpha }}}\\frac{\\partial }{\\partial x^\\mu _{\\rm R}} .$ Thus, for instance, the chirality condition $\\bar{D}_{\\dot{\\alpha }}\\Phi = 0$ translates itself into $\\frac{\\partial }{\\partial \\bar{\\theta }^{\\dot{\\alpha }}}\\Phi = 0,$ from which the solution (REF ) is obvious.", "Let us write also the induced transformations of the component fields in the chiral superfield (REF ) under the infinitesimal supertransformations (REF ) or, equivalently, (REF ): $\\begin{array}{l}\\phi \\rightarrow \\phi + \\sqrt{2}\\, \\epsilon ^\\alpha \\psi _\\alpha ,\\\\[2mm]\\psi _\\alpha \\rightarrow \\psi _\\alpha + i\\sqrt{2}\\,\\partial _\\mu \\phi \\, \\sigma ^\\mu _{\\alpha {\\dot{\\alpha }}}\\,\\bar{\\epsilon }^{\\dot{\\alpha }}+ \\sqrt{2} F\\epsilon _\\alpha ,\\\\[2mm]F\\rightarrow F -i\\sqrt{2} \\,\\partial _\\mu \\left(\\psi ^\\alpha \\sigma ^\\mu _{\\alpha {\\dot{\\alpha }}}\\bar{\\epsilon }^{\\dot{\\alpha }}\\right) .\\end{array}$ Note that the $F$ term transforms through a total space-time derivative.", "Let us inspect Eq.", "(REF ).", "It gives a reducible representation of the supersymmetry algebra.", "We can simply constrain it with the reality condition $S^\\dagger = S$ .", "This gives what is called vector superfield: $V(x,\\theta ,\\bar{\\theta })= \\phi +\\theta \\psi + \\bar{\\theta }\\bar{\\psi }+ \\theta ^2 F + \\bar{\\theta }^2 {\\bar{F}}+\\theta ^\\alpha A_{\\alpha \\dot{\\alpha }} \\bar{\\theta }^{\\dot{\\alpha }}+\\theta ^2 (\\bar{\\theta }\\bar{\\lambda })+\\bar{\\theta }^2 (\\theta \\lambda )+ \\theta ^2\\bar{\\theta }^2 D.$ The superfield $V$ is real, $V=V^\\dagger $ , implying that the bosonic fields $\\phi $ , $D$ and $A^\\mu =\\frac{1}{2}\\left(\\bar{\\sigma }^\\mu \\right)^{\\dot{\\alpha }\\alpha }A_{\\alpha \\dot{\\alpha }}$ are real.", "The fields $\\psi _\\alpha $ , $\\lambda _\\alpha $ , $F$ are complex, with $\\bar{\\psi }_{\\dot{\\alpha }}= (\\psi _\\alpha )^\\dagger $ , $\\bar{\\lambda }_{\\dot{\\alpha }}=(\\lambda _\\alpha )^\\dagger $ , $\\bar{F} = F^*$ .", "The real superfield $V$ is used in construction of supersymmetric gauge theories.", "In fact, the (super)gauge freedom (which for the Abelian gauge superfield is $V\\rightarrow V+(\\Lambda +\\bar{\\Lambda })$ , where $\\Lambda $ is an arbitrary chiral superfield) allows one to eliminate the unwanted components $\\phi $ , $\\psi _\\alpha $ , $\\bar{\\psi }_{\\dot{\\alpha }}$ , $F$ , and $\\bar{F}$ , reducing the physical content of $V$ to $V\\rightarrow \\theta ^\\alpha A_{\\alpha {\\dot{\\alpha }}} \\bar{\\theta }^{\\dot{\\alpha }}+\\left\\lbrace \\theta ^2 (\\bar{\\theta }\\bar{\\lambda })+\\bar{\\theta }^2 (\\theta \\lambda )\\right\\rbrace + \\theta ^2\\bar{\\theta }^2 D$ so that $\\lambda _\\alpha $ is a fermionic superpartner of the gauge field $A_{\\alpha {\\dot{\\alpha }}}$ , while $D$ is an auxiliary field which is not dynamical and can be excluded from a corresponding Lagrangian by algebraic equations.", "The supersymmetry transformations (REF ) induce the transformations of the component fields in $V$ .", "Here, for later purposes, we quote only the corresponding transformation for the auxiliary field $D$ : $D\\rightarrow D + \\frac{i}{2} \\partial _\\mu \\left(\\epsilon ^\\alpha \\sigma ^\\mu _{\\alpha {\\dot{\\alpha }}}\\bar{\\lambda }^{\\dot{\\alpha }}-\\lambda ^\\alpha \\sigma ^\\mu _{\\alpha {\\dot{\\alpha }}} \\bar{\\epsilon }^{\\dot{\\alpha }}\\right).$ Note that, like in the chiral superfield case, it transforms through a total space-time derivative.", "This property is of a paramount importance for construction of supersymmetric theories.", "Let us enumerate the main properties of superfields which are used in construction of supersymmetric Lagrangians.", "Linear combinations of superfields as well as products of superfields are again superfields.", "In general, a function of superfields is a superfield.", "If $\\Phi $ is a chiral superfield, then $\\bar{\\Phi }=\\Phi ^\\dagger $ is an antichiral superfield and vice versa.", "Given a superfield, one can use the space-time derivatives $\\partial /\\partial x^\\mu $ to generate a new one.", "At the same time, the Grassmann derivatives $\\partial /\\partial \\theta ^\\alpha $ and $\\partial /\\partial \\bar{\\theta }^{\\dot{\\alpha }}$ , being applied to a superfield, do not produce a superfield.", "One can use the covariant superderivatives $D_\\alpha $ and $\\bar{D}_{\\dot{\\alpha }}$ instead since they anticommute with the supercharges.", "The squares of the covariant superderivatives $\\bar{D}^2$ , $D^2$ , being applied on a generic superfield, produce a chiral or antichiral superfield, respectively.", "This immediately follows from the expression for $\\bar{D}_{\\dot{\\alpha }}$ in (REF ) and from the expression for $D_\\alpha $ in (REF ).", "For instance, $ \\bar{D}^2 \\,D_\\alpha S$ (with arbitrary $S$ ) is a chiral superfield while $D_\\alpha \\Phi $ (with $\\Phi $ chiral) is not a chiral superfield.", "Indeed, $\\bar{D}_{\\dot{\\alpha }}\\left(D_\\alpha \\Phi \\right) = \\left\\lbrace \\bar{D}_{\\dot{\\alpha }} , \\, D_\\alpha \\right\\rbrace \\Phi = -2i \\sigma ^\\mu _{\\alpha \\dot{\\alpha }}\\,\\partial _\\mu \\Phi \\ne 0.$ At the same time, $D^2\\Phi $ is an antichiral superfield.", "Linear combinations of superfields as well as products of superfields are again superfields.", "In general, a function of superfields is a superfield.", "If $\\Phi $ is a chiral superfield, then $\\bar{\\Phi }=\\Phi ^\\dagger $ is an antichiral superfield and vice versa.", "Given a superfield, one can use the space-time derivatives $\\partial /\\partial x^\\mu $ to generate a new one.", "At the same time, the Grassmann derivatives $\\partial /\\partial \\theta ^\\alpha $ and $\\partial /\\partial \\bar{\\theta }^{\\dot{\\alpha }}$ , being applied to a superfield, do not produce a superfield.", "One can use the covariant superderivatives $D_\\alpha $ and $\\bar{D}_{\\dot{\\alpha }}$ instead since they anticommute with the supercharges.", "The squares of the covariant superderivatives $\\bar{D}^2$ , $D^2$ , being applied on a generic superfield, produce a chiral or antichiral superfield, respectively.", "This immediately follows from the expression for $\\bar{D}_{\\dot{\\alpha }}$ in (REF ) and from the expression for $D_\\alpha $ in (REF ).", "For instance, $ \\bar{D}^2 \\,D_\\alpha S$ (with arbitrary $S$ ) is a chiral superfield while $D_\\alpha \\Phi $ (with $\\Phi $ chiral) is not a chiral superfield.", "Indeed, $\\bar{D}_{\\dot{\\alpha }}\\left(D_\\alpha \\Phi \\right) = \\left\\lbrace \\bar{D}_{\\dot{\\alpha }} , \\, D_\\alpha \\right\\rbrace \\Phi = -2i \\sigma ^\\mu _{\\alpha \\dot{\\alpha }}\\,\\partial _\\mu \\Phi \\ne 0.$ At the same time, $D^2\\Phi $ is an antichiral superfield.", "With the extension of space-time to the superspace (REF ), the space-time integral $\\int d^4x$ must also include the integration over the new coordinates $\\theta ^\\alpha $ and $\\bar{\\theta }^{\\dot{\\alpha }}$ .", "As it is shown below, one can covariantly integrate superfields in the following three ways: $\\int d^4x \\,d^4\\theta \\quad \\mbox{or}\\quad \\int d^2\\theta \\, d^4x_{\\rm L},\\quad \\mbox{or}\\quad \\int d^2\\bar{\\theta }\\, d^4x_{\\rm R}.$ Do do so, one needs to define the rules of Grassmann integration.", "After that, manifestly supersymmetric Lagrangians can be straightforwardly constructed.", "We will consider the simplest example of a superfield action involving a single chiral superfield.", "The rules of integration over the Grassmann variables, also known as Berezin integrals [38], are the following.", "One-dimensional integrals are defined as $\\int d\\theta _\\alpha = 0,\\qquad \\int \\theta _\\alpha \\,d\\theta _\\beta = \\delta _{\\alpha \\beta }$ (and similarly for the Grassmann variables with dotted indices), while multi-dimensional integrals involving two and more Grassmann variables are to be understood as product of one-dimensional integrals.", "Note that the Grassmann variables $\\theta $ and $\\bar{\\theta }$ have the dimension of $\\mbox{[length]}^{1/2}$ , while the differentials $d\\theta $ and $d\\bar{\\theta }$ have the dimension of $\\mbox{[length]}^{-1/2}$ .", "If $c$ is a number, then $d(c\\,\\theta ) =c^{-1} d\\theta $ .", "This follows from the right equation in (REF ).", "We normalize the integral over all four Grassmann dimensions of the superspace, $\\int \\, d^4\\theta \\equiv \\int d^2\\theta \\, d^2\\bar{\\theta }\\sim \\int \\,d\\theta _1\\, d\\theta _2\\,d\\bar{\\theta }_{\\dot{1}}\\, d\\bar{\\theta }_{\\dot{2}},$ in such a way that $\\int d^2\\theta \\, d^2\\bar{\\theta }\\,\\,\\theta ^2\\,\\bar{\\theta }^2= 1.$ Respectively, the integrals over the chiral and the antichiral subspaces are normalized as $\\int d^2\\theta \\,\\,\\theta ^2 = 1,\\quad \\quad \\int d^2\\bar{\\theta }\\,\\,\\bar{\\theta }^2= 1.$ Consider the integral $\\int \\, d^4x\\,d^4\\theta \\, V(\\dots )$ with $V$ being the real superfield (REF ) which can be a function of other superfields.", "The rules of Grassmann integration (REF ), (REF ) imply that $D=\\int d^4\\,\\theta V$ , there $D$ is the coefficient in front of $\\theta ^2\\bar{\\theta }^2$ in $V$ .", "Since the supertransformations change the $D$ term through a total space-time derivative, Eq.", "(REF ), then the expression $\\int \\, d^4\\theta \\, V(\\dots )$ is superinvariant up to a total derivative.", "Let us exploit the above idea and construct kinetic term $S_{\\rm kin} =\\int d^4x\\, d^4\\theta \\, \\bar{\\Phi }\\Phi $ for a chiral superfield $\\Phi (x_{\\rm L},\\theta )$ .", "The component expansion of $\\Phi $ is given in Eq.", "(REF ).", "The product $\\Phi \\bar{\\Phi }$ , where $\\bar{\\Phi }=\\Phi ^\\dagger $ , is a real superfield.", "The calculation of the highest term in $\\theta ^\\alpha $ and $\\bar{\\theta }^{\\dot{\\alpha }}$ gives $\\bar{\\Phi }\\,\\Phi =\\dots +\\theta ^2\\bar{\\theta }^2\\left\\lbrace \\frac{1}{2} \\partial _\\mu \\bar{\\phi }\\,\\partial ^\\mu \\phi -\\frac{1}{4}\\bar{\\phi }\\,\\partial ^2\\phi -\\frac{1}{4}\\phi \\,\\partial ^2\\bar{\\phi }\\right.\\\\[2mm]\\left.-\\frac{i}{2}\\,\\psi ^{\\alpha }\\sigma ^\\mu _{\\alpha \\dot{\\alpha }}\\,\\partial _\\mu \\bar{\\psi }^{\\dot{\\alpha }}+\\frac{i}{2}\\,\\sigma ^\\mu _{\\alpha \\dot{\\alpha }}\\partial _\\mu \\psi ^{\\alpha }\\,\\bar{\\psi }^{\\dot{\\alpha }}+\\bar{F}F\\right\\rbrace ,$ where all the component fields depend on the space-time point $x^\\mu $ , and the ellipsis denotes all lower-order terms in $\\theta ^\\alpha $ and $\\bar{\\theta }^{\\dot{\\alpha }}$ .", "Substituting the last expression into (REF ), integrating over the Grassmann variables and integrating by parts reads $S_{\\rm kin}=\\int d^4x \\left\\lbrace \\partial _\\mu \\bar{\\phi }\\,\\partial ^\\mu \\phi - {i} \\psi ^\\alpha \\sigma ^\\mu _{\\alpha {\\dot{\\alpha }}}\\,\\partial _\\mu \\bar{\\psi }^{\\dot{\\alpha }}+\\bar{F}F\\right\\rbrace .$ Thus, this expression presents the kinetic term for the complex field $\\phi (x)$ and the kinetic term for the Weyl spinor $\\psi _\\alpha (x)$ .", "As one can see, the field $F(x)$ appears in the Lagrangian with no derivatives and does not represent any physical (propagating) degrees of freedom.", "It can be eliminated from the action by virtue of the equations of motion.", "Similarly to the previous section, the integral $\\int d^2\\theta d^4x_{\\rm L}\\, K(\\dots ),$ where $K$ is a chiral superfield (see Eq.", "(REF ) for its component expansion), which can be a function of other superfields, is invariant under the action of the supersymmetry group.", "To prove this statement, note that $\\int d^2\\theta \\,K(\\dots ) = F_K(x)- \\frac{i}{\\sqrt{2}} \\partial _\\mu \\left(\\psi ^\\alpha _K(x)\\,\\sigma ^\\mu _{\\alpha {\\dot{\\alpha }}}\\bar{\\theta }^{\\dot{\\alpha }}\\right),$ with $F_K$ being the coefficient in front of $\\theta ^2$ , while $\\psi _K^\\alpha $ being the coefficient in front of $\\sqrt{2}\\theta ^\\alpha $ in the component expansion of the chiral superfield $K$ .", "In addition, the last term in (REF ) is a full space-time derivative and thus vanishes if one integrates it over space-time.", "Thereby, we see that $\\int d^2\\theta d^4x_{\\rm L}\\, K(\\dots )=\\int d^4x\\, d^2\\theta \\, K(\\dots )=\\int d^4x \\, F_K(x).$ Finally, according to Eq.", "(REF ), the $F_K$ term transforms as a full space-time derivative under the action of the supersymmetry group, and thus the integral above is indeed superinvariant.", "It is also clear now that the integral $\\int d^2\\bar{\\theta }d^4x_{\\rm R}\\, \\bar{K}(\\dots ),$ where $\\bar{K}$ is an antichiral superfield, is superinvariant.", "Potential terms for component fields in the Lagrangian are those which enter with no space-time derivatives.", "They can be produced for the chiral superfield $\\Phi $ from the previous section by the following superfield action: $S_{\\rm int} =\\int d^2\\theta \\, d^4 x_{\\rm L}\\, {\\mathcal {W}}(\\Phi )+\\int d^2\\bar{\\theta }\\, d^4 x_{\\rm R}\\, \\bar{\\mathcal {W}}(\\bar{\\Phi }).$ Here ${\\mathcal {W}} (\\Phi )$ is a function of the chiral superfield termed superpotential.", "The second integral which involves $\\bar{\\mathcal {W}}(\\bar{\\Phi }) = \\left({\\mathcal {W}}(\\Phi )\\right)^\\dagger $ ensures that the expression above is real.", "After taking the integrals over the Grassmann variables, one obtains the following component action: $S_{\\rm int} = \\int d^4x \\left\\lbrace F\\,{\\cal W}^{\\prime }(\\phi ) - \\frac{1}{2} {\\cal W}^{\\prime \\prime }(\\phi )\\,\\psi ^2+ \\mbox{h.c.}\\right\\rbrace ,$ where $\\psi ^2=\\psi ^\\alpha \\psi _\\alpha $ , the prime denotes the derivative with respect to $\\phi $ and $\\mbox{h.c.}$ means the Hermitian conjugated expression.", "Let us combine the results of the previous two sections and write the full superfield action involving a single chiral superfield $\\Phi $ and its Hermitian conjugated superfield $\\bar{\\Phi }$ : $S_{\\rm WZ} = \\int \\!", "{ d}^4 x \\,{ d}^4\\theta \\,\\Phi \\bar{\\Phi } +\\int { d}^2 \\theta \\,d^4 x_{\\rm L}{\\cal W}(\\Phi ) +\\int { d}^2 \\bar{\\theta }\\,d^4x_{\\rm R}\\bar{\\cal W}(\\bar{\\Phi }) .$ This model of supersymmetric field theory was invented by Wess and Zumino [37] and bears their name.", "Let us remark that the first term in the superfield action is the integral over the full superspace, while the second and the third terms run over the chiral and antichiral subspaces, respectively.", "In components, the corresponding Lagrangian is ${\\mathcal {L}}_{\\rm WZ} =\\partial _\\mu \\bar{\\phi }\\,\\partial ^\\mu \\phi - {i} \\psi ^\\alpha \\sigma ^\\mu _{\\alpha {\\dot{\\alpha }}}\\,\\partial _\\mu \\bar{\\psi }^{\\dot{\\alpha }}+\\bar{F}F+\\left(F\\,{\\cal W}^{\\prime }(\\phi ) - \\frac{1}{2} {\\cal W}^{\\prime \\prime }(\\phi )\\,\\psi ^2\\right)+ \\mbox{h.c.}$ It is supersymmetry-invariant up to a total space-time derivative.", "The auxiliary field $F$ is non-dynamical and can be eliminated by virtue of its classical equation of motion, $\\bar{F} = -{\\mathcal {W}}^{\\prime }(\\phi ).$ The final expression for the component Lagrangian is ${\\mathcal {L}}_{\\rm WZ} =\\partial _\\mu \\bar{\\phi }\\,\\partial ^\\mu \\phi - {i} \\psi ^\\alpha \\sigma ^\\mu _{\\alpha {\\dot{\\alpha }}}\\,\\partial _\\mu \\bar{\\psi }^{\\dot{\\alpha }}-\\left|{\\cal W}^{\\prime }(\\phi )\\right|^2- \\frac{1}{2} {\\cal W}^{\\prime \\prime }(\\phi )\\,\\psi ^\\alpha \\psi _\\alpha - \\frac{1}{2} \\bar{\\cal W}^{\\prime \\prime }(\\bar{\\phi })\\,\\bar{\\psi }_{\\dot{\\alpha }}\\bar{\\psi }^{\\dot{\\alpha }}.$ It contains the scalar potential $\\left| {\\cal W}^{\\prime }(\\phi )\\right|^2$ describing the self-interaction of the complex field $\\phi $ .", "For a renormalizable field theory in four dimensions, the superpotential ${\\mathcal {W}} (\\Phi )$ must be a polynomial function of $\\Phi $ of power not higher than three.", "Then, it can be always reduced to the form ${\\cal W}(\\Phi ) = \\frac{m}{2} \\,\\Phi ^2 - \\frac{\\lambda }{3}\\Phi ^3$ with two complex constants $m$ and $\\lambda $ .", "In fact, one can always choose the phases of the constants $m$ and $\\lambda $ at will.", "As a simplest example, let us consider the case $\\lambda =0$ .", "Then the last three terms in the Lagrangian (REF ), $- |m|^2\\phi \\bar{\\phi }-\\frac{m}{2} \\psi ^2-\\frac{\\bar{m}}{2} \\bar{\\psi }^2,$ are the mass terms for the fields $\\phi $ and $\\psi _\\alpha $ .", "As expected, the masses of scalar and spinor particles are equal and are given by one and the same parameter $|m|$ .", "The Coleman–Mandula theorem states that all global bosonic symmetries must commute with the Poincaré group (i.e.", "be Lorentz scalars).", "However, it is not necessary for them to commute with all generators of the super-Poincaré group.", "Indeed, the form of the super-Poincaré algebra (REF ), (REF ), (REF ), (REF ) allows one to multiply the supercharges by a constant phase in such a way that the superalgebra itself stays unchanged: $Q_\\alpha \\rightarrow e^{-i\\varphi } Q_\\alpha ,\\quad \\quad \\bar{Q}_{\\dot{\\alpha }}\\rightarrow e^{i\\varphi } \\bar{Q}_{\\dot{\\alpha }}.$ The corresponding Hermitian ${\\rm U}(1)$ generator $R$ has the following commutation relations with the supercharges, $[R,\\,Q_\\alpha ] = -Q_\\alpha ,\\quad \\quad [R,\\, \\bar{Q}_{\\dot{\\alpha }} ] = \\,\\bar{Q}_{\\dot{\\alpha }},$ and it can be realized as a differential operator in the superspace (REF ), $R= \\theta ^\\alpha \\frac{\\partial }{\\partial \\theta ^\\alpha }- \\bar{\\theta }^{\\dot{\\alpha }}\\frac{\\partial }{\\partial \\bar{\\theta }^{\\dot{\\alpha }}}.$ This additional ${\\rm U}(1)$ symmetry is called the R-symmetry.", "It transforms the Grassmann parameters, $\\theta ^\\alpha \\rightarrow e^{i\\varphi }\\theta ^\\alpha ,\\quad \\quad \\bar{\\theta }^{\\dot{\\alpha }}\\rightarrow e^{-i\\varphi }\\,\\bar{\\theta }^{\\dot{\\alpha }},$ and the measure of the Grassmann integration, $d^2\\theta \\rightarrow e^{-2i\\varphi }d^2\\theta ,\\quad \\quad d^2\\bar{\\theta }\\rightarrow e^{2i\\varphi }d^2\\bar{\\theta },\\quad \\quad d^4\\theta \\rightarrow d^4\\theta .$ Thus, we assign the R-charge $+1$ to $\\theta ^\\alpha $ ($R\\,\\theta ^\\alpha = +\\theta ^\\alpha $ ) and the R-charge $-1$ to $\\bar{\\theta }^{\\dot{\\alpha }}$ ($R\\,\\bar{\\theta }^{\\dot{\\alpha }}=-\\bar{\\theta }^{\\dot{\\alpha }}$ ), while the R-charges of $d^2\\theta $ and $d^2\\bar{\\theta }$ are $-2$ and $+2$ , respectively.", "The R-charge of $d^4\\theta $ is zero.", "The R-symmetry can be a symmetry of a given system.", "To see how it works, let us consider the Wess-Zumino model (REF ) of a chiral superfield $\\Phi $ with the superpotential ${\\mathcal {W}}= -\\frac{\\lambda }{3}\\Phi ^3.$ Then, the action (REF ) is invariant under the R-symmetry, if the R-charge of the superfield $\\Phi $ is 2/3, $\\Phi \\rightarrow e^{2i\\varphi /3}\\Phi .$ Indeed, the first term in the action, $\\int d^4\\theta \\,\\Phi \\bar{\\Phi }$ , is invariant since the antichiral superfield $\\bar{\\Phi }$ transforms with the complex conjugated phase factor.", "The potential term is also invariant: $\\int d^2\\theta \\,{\\mathcal {W}}(\\Phi )\\longrightarrow \\int \\left( e^{-2i\\varphi }\\, d^2\\theta \\right) \\left( {\\mathcal {W}}(\\Phi ) \\, e^{2i\\varphi }\\right)=\\int d^2\\theta \\,{\\mathcal {W}}(\\Phi ).$ Since the R-symmetry does not commute with supersymmetry, the component fields (REF ) of the superfield $\\Phi $ do not all carry the same R-charge.", "It follows from Eqs.", "(REF ) and (REF ) that, under the R-symmetry, they are transformed as $\\begin{array}{l}\\phi (x) \\rightarrow e^{\\frac{2}{3} i\\varphi } \\,\\phi (x),\\\\[2mm]\\psi (x) \\rightarrow e^{\\left(\\frac{2}{3}-1\\right)i\\varphi }\\, \\psi (x),\\\\[2mm]F(x) \\rightarrow e^{\\left(\\frac{2}{3}-2\\right)i\\varphi } \\, F(x).\\end{array}$ As one can see, the R-charge of the lowest component $\\phi (x)$ coincides with the R-charge of the superfield $\\Phi $ itself.", "The above formulas can be used to explicitly check that the component Lagrangian (REF ) of the Wess-Zumino model is R-symmetry invariant provided $m=0$ .", "The R-symmetries play very important role in harmonic superspace approach, which is discussed in the case of supersymmetric quantum mechanics in the next chapter.", "As we already know, the Coleman-Mandula theorem can be circumvented by the introduction to the Poincaré algebra four complex supercharges $Q_\\alpha $ , $\\bar{Q}_{\\dot{\\alpha }}$ of fermionic nature.", "The supersymmetry algebra (REF ), (REF ), (REF ) and (REF ) is usually referred to as minimal supersymmetry or as ${\\mathcal {N}}=1$ supersymmetry.", "In fact, it can be extended even more, with more generators of fermionic nature, $Q_\\alpha ^I\\quad \\quad \\mbox{and}\\quad \\quad \\bar{Q}_{\\dot{\\alpha }}^J,$ where new indices $I,J = 1, 2,\\dots , {\\cal N}$ numerate the “flavours” of supercharges.", "Evidently, the defining relations (REF ) and (REF ) can be generalized with The relations (REF ) do not include possible central charges.", "One can, for instance, introduce them as $\\lbrace Q_\\alpha ^I,\\, Q_{\\beta }^J\\rbrace = \\varepsilon _{\\alpha \\beta } Z^{IJ}$ with $Z^{IJ}=-Z^{JI}$ .", "See also the discussion at the end of Section REF .", "$\\lbrace Q_\\alpha ^I,\\, \\bar{Q}_{\\dot{\\alpha }}^J\\rbrace &=&2 P_\\mu \\left( \\sigma ^\\mu \\right)_{\\alpha \\dot{\\alpha }}\\,\\delta ^{IJ}\\,,\\\\[3mm]\\lbrace Q_\\alpha ^I,\\, Q_{\\beta }^J\\rbrace &=&\\lbrace Q_{\\dot{\\alpha }}^I,\\, Q_{\\dot{\\beta }}^J\\rbrace \\,\\,=\\,\\, 0,$ The relations (REF ) do not include possible central charges.", "One can, for instance, introduce them as $\\lbrace Q_\\alpha ^I,\\, Q_{\\beta }^J\\rbrace = \\varepsilon _{\\alpha \\beta } Z^{IJ}$ with $Z^{IJ}=-Z^{JI}$ .", "See also the discussion at the end of Section REF .", "while the relation (REF ) stays untouched since it defines how spinors transform under the Poincaré symmetry.", "In ${\\cal N}=1$ supersymmetric renormalizable four-dimensional field theory which do now include gravity, the supermultiplets involve particles with spins (0, 1/2) or (1/2, 1), since the supercharges change a particle spin by 1/2.", "Similarly, with the introduction of extended supersymmetry, one can transform a particle with spin zero to a particle with spin one by using two supercharges of different flavours.", "Thus, such a theory must include particles with spins (0, 1/2, 1) in a supermultiplet, i.e.", "it must be a gauge theory.", "Gauge theories of this type are ${\\mathcal {N}}=2$ and ${\\mathcal {N}}=4$ super-Yang–Mills theories.", "They are obtained by dimensional reduction from the minimal super-Yang–Mills theories in six and ten dimensions, respectively.", "These theories are unsuitable for phenomenology, because all fermion fields they contain are nonchiral.", "Nevertheless, they have rich dynamics the study of which provides deep insights into a large number of problems in mathematical physics.", "All the properties of a supersymmetric field theory such as vanishing of vacuum energy and equal number of bosons and fermions in a supermultiplet, which were discussed in Section REF , remain intact.", "Moreover, the Abelian ${\\rm U}(1)$ R-symmetry from Eq.", "(REF ) is extended to be non-Abelian ${\\rm U}({\\cal N})$ symmetry.", "Simply speaking, the non-Abelian R-symmetry group just mixes the flavours of the supercharges (i.e.", "it mixes the supercharge flavour indices).", "In addition, the superspace (REF ) is trivially extended to $\\lbrace x^\\mu ,\\, \\theta ^{I\\,\\alpha },\\, \\bar{\\theta }^{J\\,\\dot{\\alpha }}\\rbrace .$ Note, however, that the usefulness of this ${\\cal N}=2$ or ${\\cal N}=4$ superspace is limited due to the fact that there are no chiral subspaces in it (which span over half of Grassmann coordinates).", "There exist a different superspace called harmonic superspace (HSS) which has these invariant subspaces [5].", "Such an approach gives more adequate superfield description.", "The harmonic superspace is obtained from the superspace (REF ) by extending it with bosonic coordinates which parametrize the R-symmetry group ${\\rm U}({\\cal N})$ space or a certain factor space of it.", "The harmonic superspace approach is discussed in supersymmetric quantum mechanics in the next chapter.", "This chapter is devoted to basic introduction to supersymmetry in quantum mechanics.", "We consider the main properties of any supersymmetric system, explain what is superspace and how the superfield formalism can be used in order to construct genuine supersymmetric Lagrangians.", "A simplest example of a supersymmetric system is considered.", "It involves a particle with spin moving in a potential field in one-dimensional space.", "The second part of the chapter is devoted to harmonic superspace approach in supersymmetric quantum mechanics.", "The essential definitions and notations are introduced.", "They will be extensively used in the next chapter.", "In particular, the supermultiplets ${\\bf (4,4,0)}$ and ${\\bf (3,4,1)}$ are described.", "The former is relevant in the context of four-dimensional quantum mechanics, while the latter is used in construction of three-dimensional systems.", "A quantum-mechanical system with traditional commutation relations between coordinates and momenta and a traditional Hilbert space of states is described by its Hamiltonian function $H$ .", "We introduce a set of complex operators $Q_i$ together with their Hermitian conjugated operators, $\\bar{Q}^i = \\left(Q_i\\right)^\\dagger .$ The system with the Hamiltonian $H$ and the supercharges $Q_i$ , $\\bar{Q}^j$ is supersymmetric, by definition, if $\\left\\lbrace Q_i,\\bar{Q}^j\\right\\rbrace &=& 2 \\delta _i^j H \\\\[2mm] \\big \\lbrace Q_i, Q_j\\big \\rbrace &=& \\left\\lbrace \\bar{Q}^i, \\bar{Q}^j\\right\\rbrace = 0,$ where, as usual, curly brackets denote the anticommutator.", "In particular, note the important property $Q_i^2 = \\left(\\bar{Q}^i\\right)^2 = 0$ for any $i$ .", "Another important consequence is that the supercharges commute with the Hamiltonian: $\\big [H,\\, Q_i\\big ] = \\left[ H,\\, \\bar{Q}^i\\right] = 0,$ which can be proven by direct computation.", "In this way, the supercharges $Q_i$ and $\\bar{Q}^j$ are considered as conserved spinorial operators in the system.", "Several comments are relevant here.", "The Latin indices $i$ , $j$ denote supercharge numbers and vary in the following region: $i,\\, j = 1,\\, 2, \\, \\dots , \\, {\\cal N}/2$ with number ${\\cal N}$ being even integer, see below.", "We distinguish the position of indices for the supercharge $Q_i$ and its Hermitian conjugated supercharge $\\bar{Q}^i$ .", "This is done because the ${\\rm SU}(\\frac{{\\cal N}}{2})$ subgroup of the R-symmetry group (in the ${\\cal N}\\ge 4$ case) acts on these indices differently, see Section REF for details.", "This subgroup will be of special importance for us later, when we limit ourselves to ${\\cal N}=4$ case and deploy harmonic superspace approach.", "Spinor products and their complex conjugates are conveniently written in such notations.", "For instance, for any two anticommuting fields $\\bar{\\psi }^i$ and $\\xi _j$ $\\bar{\\psi }\\,\\xi \\equiv \\bar{\\psi }^i \\,\\xi _i,\\quad \\quad \\mbox{while}\\quad \\quad \\left(\\bar{\\psi }\\,\\xi \\right)^*=\\bar{\\xi }\\,\\psi \\equiv \\bar{\\xi }^i \\,\\psi _i,$ where $\\bar{\\psi }^i = \\left(\\psi _i\\right)^*$ and $\\bar{\\xi }^i = \\left(\\xi _i\\right)^*$ .", "Finally, one can always pass to a real basis in the vector space of supercharges, for instance, by using the definitions $S_A =\\left\\lbrace \\begin{array}{l}Q_A+\\bar{Q}^A, \\quad \\quad \\quad \\,\\,\\,\\,\\mbox{for}\\quad A=1,\\, 2,\\, \\dots ,\\, {\\cal N}/2,\\\\[2mm]i\\left(Q_A-\\bar{Q}^A\\right),\\quad \\quad \\mbox{for}\\quad A = {\\cal N}/2+1,\\, \\dots ,\\, {\\cal N}\\end{array}\\right.$ which give commutation relations $\\left\\lbrace S_A,\\,S_B\\right\\rbrace = 4\\,\\delta _{AB} \\,H.$ The new indices $A$ , $B$ vary from 1 to ${\\cal N}$ .", "Thereby, ${\\cal N}$ counts the number of linearly independent real supercharges in the system at hand.", "For instance, ${\\cal N}=2$ corresponds to an ordinary supersymmetric quantum mechanics, while a system with ${\\cal N}=4$ is endowed with extended supersymmetry.", "Let us remark that there is also a different convention in quantum mechanics.", "According to it, ${\\cal N}$ counts the number of linearly independent complex supercharges which is, of course, twice smaller than the number of real supercharges.", "Note that, in a four-dimensional field theory context, the supercharges represent complex Weyl doublets, and ${\\cal N}$ counts the number of those doublets.", "We already saw in the previous chapter that a supersymmetric system is endowed with additional common properties, namely, vanishing of the vacuum energy, energy positiveness of any definite-energy state, equal number of bosons and fermions in a supermultiplet, their equal masses.", "Any supersymmetric quantum-mechanical system has similar features.", "Let us describe them in detail.", "To this end, let us denote a normalized vacuum state as $\\psi _{\\rm vac}$ , $\\left<{\\psi _{\\rm vac}}|{\\psi _{\\rm vac}}\\right> =1,$ while any other similarly normalized state as $\\psi $ .", "For simplicity, all the states will be treated as normalizable.", "Also, we may denote a state either as $\\psi $ or as $\\left| \\psi \\right>$ according to our convenience.", "This statement can be proven by sandwiching Eq.", "(REF ) (with $i=j$ being fixed) with an eigenstate $\\psi $ : $2 E = \\left<{\\psi }\\left| Q_i\\left(Q_i\\right)^\\dagger \\ \\right|{\\psi }\\right>+\\left<{\\psi }\\left| \\left(Q_i\\right)^\\dagger Q_i\\right| {\\psi }\\right>\\\\[3mm]=\\left<\\left(Q_i\\right)^\\dagger \\psi \\left| \\left(Q_i\\right)^\\dagger \\psi \\right.\\right>^*+\\Big <Q_i \\psi \\Big | Q_i \\psi \\Big >^* .$ The second line in this equality is always non-negative.", "Thus, $E\\ge 0 .$ If supersymmetry is unbroken, the vacuum $\\psi _{\\rm vac}$ has exactly zero energy.", "This statement straightforwardly follows from Eq.", "(REF ) if one changes $E\\rightarrow E_{\\rm vac}$ and $\\psi \\rightarrow \\psi _{\\rm vac}$ .", "Indeed, the minimum $E_{\\rm vac}=0$ is achieved when $Q_i\\left| \\psi _{\\rm vac} \\right> = \\bar{Q}^i\\left| \\psi _{\\rm vac} \\right> = 0,\\quad \\quad \\mbox{for all $i$.", "}$ In fact, the conditions (REF ) exactly correspond to the case of unbroken supersymmetry.", "The simplest example of supersymmetry breaking in quantum mechanics was constructed in [39] (see [40] for a good pedagogical review).", "The eigenstates of a supersymmetric Hamiltonian can be divided into supermultiplets.", "To this end, take a state $\\psi $ with positive energy $E$ .", "One can act on this state with supercharges $Q_i$ and $ \\left(Q_j\\right)^\\dagger $ and obtain new states with exactly the same energy $E$ .", "Indeed, using Eq.", "(REF ), we conclude that $H\\left| Q_i\\,\\psi \\right> = Q_i\\, H\\left| \\psi \\right> = E\\left| Q_i\\psi \\right>,$ and similarly for $\\left(Q_i\\right)^\\dagger $ .", "A supermultiplet is defined as a vector space of states of the same energy $E>0$ obtained by all possible actions of supercharges on a chosen reference state $\\psi $ .", "The basis in the supermultiplet can be explicitly constructed.", "To do so, consider the following states: $Q_{i_1}\\,Q_{i_2}\\,Q_{i_3}\\,\\dots \\left| \\psi \\right> .$ Due to the property $Q_i^2=0$ there exists a state $\\left| \\psi _{\\rm low} \\right> = Q_{k_1}\\,Q_{k_2}\\,\\dots Q_{k_n}\\left| \\psi \\right>,\\quad \\quad n\\le {\\cal N}/2$ which is not zero, $\\left<\\psi _{\\rm low}|\\psi _{\\rm low}\\right>\\sim 1$ , but all the supercharges $Q_i$ annihilate it: $Q_i\\left| \\psi _{\\rm low} \\right> = 0 \\quad \\quad \\mbox{for all $i$}.$ The initial reference state $\\psi $ can be restored from the state $\\psi _{\\rm low}$ in the following way: $\\left(Q_{k_1}\\right)^\\dagger \\left(Q_{k_2}\\right)^\\dagger \\dots \\left(Q_{k_n}\\right)^\\dagger \\left| \\psi _{\\rm low} \\right>= \\left(2E\\right)^n\\left| \\psi \\right>,$ where Eqs.", "(REF ) and (REF ) were used.", "Let us now show that the states $\\left(Q_{i_1}\\right)^\\dagger \\left(Q_{i_2}\\right)^\\dagger \\dots \\left(Q_{i_f}\\right)^\\dagger \\left| \\psi _{\\rm low} \\right>,\\quad \\quad i_1 < i_2 < \\dots < i_f, \\quad \\quad f \\le {\\cal N}/2$ form the basis in the supermultiplet.", "To this end, consider a state of type $Q_{i_1}\\left(Q_{j_1}\\right)^\\dagger Q_{i_2}\\, Q_{i_3} \\left(Q_{j_2}\\right)^\\dagger \\dots \\left| \\psi \\right>$ and plug into it the expression for $\\psi $ from Eq.", "(REF ).", "One can use the relations (REF ), () and order the supercharges in such a way that any $\\left(Q_i\\right)^\\dagger $ stays to the left from any $Q_j$ .", "After that, the identity (REF ) allows one to eliminate any terms which contain $Q_i$ .", "For instance, $Q_1 \\left(Q_2\\right)^\\dagger \\left(Q_1\\right)^\\dagger \\left| \\psi _{\\rm low} \\right>=\\left(Q_2\\right)^\\dagger Q_1 \\left(Q_1\\right)^\\dagger \\left| \\psi _{\\rm low} \\right>\\\\[2mm]=-\\left(Q_2\\right)^\\dagger \\left(Q_1\\right)^\\dagger Q_1\\left| \\psi _{\\rm low} \\right>+\\left(Q_2\\right)^\\dagger \\left\\lbrace Q_1,\\, \\left(Q_1\\right)^\\dagger \\right\\rbrace \\left| \\psi _{\\rm low} \\right>\\\\[2mm]=2E\\left(Q_2\\right)^\\dagger \\left| \\psi _{\\rm low} \\right>.$ Thus, any state in the supermultiplet adds up to a linear combination of the states (REF ).", "The states (REF ) are linearly independent, because their scalar products are zero.", "For instance, one has for the two states $\\left(Q_1\\right)^\\dagger \\left(Q_2\\right)^\\dagger \\left| \\psi _{\\rm low} \\right>$ and $\\left(Q_1\\right)^\\dagger \\left(Q_3\\right)^\\dagger \\left| \\psi _{\\rm low} \\right>$ : $\\left<\\left(Q_1\\right)^\\dagger \\left(Q_2\\right)^\\dagger \\psi _{\\rm low}\\left|\\left(Q_1\\right)^\\dagger \\left(Q_3\\right)^\\dagger \\psi _{\\rm low}\\right.\\right>=\\left<\\psi _{\\rm low}\\left| Q_2\\, Q_1 \\left(Q_1\\right)^\\dagger \\left(Q_3\\right)^\\dagger \\right|\\psi _{\\rm low}\\right>^*\\\\[2mm]=-\\left<\\psi _{\\rm low}\\left| Q_2 \\left(Q_1\\right)^\\dagger Q_1\\left(Q_3\\right)^\\dagger \\right|\\psi _{\\rm low}\\right>^*+\\left<\\psi _{\\rm low}\\left| Q_2 \\left\\lbrace Q_1,\\, \\left(Q_1\\right)^\\dagger \\right\\rbrace \\left(Q_3\\right)^\\dagger \\right|\\psi _{\\rm low}\\right>^*\\\\[2mm]=2E \\left<\\psi _{\\rm low}\\left| \\left\\lbrace Q_2,\\, \\left(Q_3\\right)^\\dagger \\right\\rbrace \\right|\\psi _{\\rm low}\\right>^* = 0$ Evidently, the states (REF ) themselves have positive normalization which follows from the calculations similar to above.", "Let us remark that we assume that the energy $E$ is positive.", "If not, Eq.", "(REF ) does not allow us to express $\\psi $ through $\\psi _{\\rm low}$ , and thus the considerations above cannot be applied.", "As it follows from Eq.", "(REF ), the dimension of the supermultiplet with $E>0$ is $2^{{\\cal N}/2}$ .", "In this way, in an ${\\cal N}=2$ system each positive-energy state is doubly degenerate, while for a system with extended ${\\cal N}=4$ supersymmetry each state with nonzero energy is four-times degenerate.", "The states of zero energy – vacuums – are of special interest.", "Their number cannot be found from general considerations, and Eqs.", "(REF ) have to be solved for that.", "If there are no vacuums in a theory, the supersymmetry is spontaneously broken (see e.g.", "Refs.", "[39], [40] for an example).", "In a field theory, supermultiplets involve bosonic and fermionic states.", "The same concerns supermultiplets in quantum mechanics.", "Let us consider the states (REF ).", "Each of them is characterized by the number of supercharges $f$ acting on $\\psi _{\\rm low}$ .", "We can introduce then the “fermionic number operator” $\\hat{N}_F$ with eigenvalues $f$ , $\\hat{N}_F\\left(Q_{i_1}\\right)^\\dagger \\left(Q_{i_2}\\right)^\\dagger \\dots \\left(Q_{i_m}\\right)^\\dagger \\left| \\psi _{\\rm low} \\right>=f\\left(Q_{i_1}\\right)^\\dagger \\left(Q_{i_2}\\right)^\\dagger \\dots \\left(Q_{i_m}\\right)^\\dagger \\left| \\psi _{\\rm low} \\right>.$ By definition, the states for which $\\hat{N}_F$ is even are called “bosonic”, while the states for which $\\hat{N}_F$ is odd – “fermionic” In our general consideration this is a matter of mere convention: we could equally well call the states for which $\\hat{N}_F$ is even fermionic.", "For example, consider ${\\cal N}=2$ supersymmetric quantum-mechanical system described by the Hamiltonian (REF ).", "It describes a particle with spin 1/2.", "The corresponding supermultiplets involve two basis states of different spin directions, and each of them can be equally well called bosonic or fermionic..", "In particular, the state $\\psi _{\\rm low}$ is bosonic.", "We have the following equalities for bosonic and fermionic states respectively: $\\left(-1\\right)^{\\hat{N}_F} \\left| \\psi _{\\rm B} \\right> = + \\left| \\psi _{\\rm B} \\right>,\\quad \\quad \\left(-1\\right)^{\\hat{N}_F} \\left| \\psi _{\\rm F} \\right> = - \\left| \\psi _{\\rm F} \\right>.$ Using the definitions above, one can check that $\\left\\lbrace \\left(-1\\right)^{\\hat{N}_F},\\, Q_i\\right\\rbrace = \\left\\lbrace \\left(-1\\right)^{\\hat{N}_F},\\, \\bar{Q}^i\\right\\rbrace = 0 .$ In this way, one obtains (the index $i$ is fixed) ${\\rm Tr}\\left(\\left(-1\\right)^{\\hat{N}_F}\\right)=\\\\[2mm]=\\frac{1}{2E}{\\rm Tr} \\left( \\left(-1\\right)^{\\hat{N}_F} \\left\\lbrace Q_i,\\,\\bar{Q}^i \\right\\rbrace \\right) =\\frac{1}{2E}{\\rm Tr} \\left( - Q_i \\left(-1\\right)^{\\hat{N}_F}\\bar{Q}^i +\\left(-1\\right)^{\\hat{N}_F}\\bar{Q}^i\\, Q_i\\right)=0,$ where the cyclic property of the trace was used.", "The trace is taken among the states of the (finite-dimensional) supermultiplet.", "For instance, it is the sum over the averages of the states (REF ).", "Summarizing all above, one concludes that any supermultiplet with positive energy has equal number of bosonic and fermionic degrees of freedom.", "Let us remark that the supercharges $Q_i$ and $\\bar{Q}^j$ can be linearly transformed in such a way that the form of Eqs.", "(REF ), () does not change.", "The corresponding group of automorphisms of the supersymmetry algebra is called R-symmetry group.", "To deduce the most general form of such linear transformations, one takes the ansatz $Q^{\\prime }_i = U_{\\!i}^{\\,\\,j} \\, Q_j,\\quad \\quad \\bar{Q}^{\\prime i} = \\bar{Q}^j \\, (U^\\dagger )_{\\!j}^{\\,\\,i}$ involving matrix $U_{\\!i}^{\\,\\,j}$ and its Hermitian-conjugated matrix $(U^\\dagger )_{\\!j}^{\\,\\,i}$ and substitutes it back into Eqs.", "(REF ), ().", "It appears that the matrix $U$ must be unitary, $U\\,U^\\dagger =1$ .", "Thus, the R-symmetry group is ${\\rm U}(\\frac{{\\cal N}}{2})$ .", "It is clear now why the supercharge $Q_i$ carries the superscript index, while the supercharge $\\bar{Q}^j$ carries the subscript index: they belong to complex-conjugated representations of the R-symmetry group.", "In particular case of the ${\\cal N}=2$ supersymmetry, the R-symmetry group is no more than just ${\\rm U}(1)$ group of multiplications of the supercharges by a phase factor: $Q^{\\prime }_i = {\\rm e}^{i\\varphi } Q_i,\\quad \\quad \\bar{Q}^{\\prime j} = {\\rm e}^{-i\\varphi } \\bar{Q}^j$ with $\\varphi $ being an arbitrary real number.", "In the ${\\cal N}\\ge 4$ case, the R-symmetry group is ${\\rm U}(\\frac{{\\cal N}}{2}) = {\\rm U}(1)\\times {\\rm SU}(\\frac{{\\cal N}}{2})$ , i.e.", "it contains the same ${\\rm U}(1)$ subgroup of multiplications by phase factor and also the non-Abelian subgroup ${\\rm SU}(\\frac{{\\cal N}}{2})$ .", "This section is devoted to superfield approach in supersymmetric quantum mechanics.", "It is shown how to construct systems endowed with supersymmetry in a rather universal way.", "As we already know from the previous chapter, supersymmetry can be realized as a geometrical symmetry which acts on the coordinates on certain extended space.", "In quantum mechanics, this space includes the time coordinate $t$ .", "The other space “dimensions” are of Grassmann nature, i.e.", "the corresponding coordinates anticommute among each other.", "Superfield technique is very general.", "It allows one to construct genuine supersymmetric systems.", "It also serves as an efficient instrument for generalizing already known supersymmetric systems.", "The extended space in quantum mechanics – superspace – is described by the following coordinates: $\\left\\lbrace t,\\, \\theta _i,\\, \\bar{\\theta }^j\\right\\rbrace ,\\quad \\quad i,\\, j = 1,\\, 2, \\, \\dots , \\, {\\cal N}/2$ with the identification $\\bar{\\theta }^i=\\left(\\theta _i\\right)^*$ : the Grassmann coordinates $\\theta _i$ and $\\bar{\\theta }^j$ are complex.", "They anticommute with each other: $\\big \\lbrace \\theta _i,\\, \\theta _j\\big \\rbrace =\\left\\lbrace \\theta _i,\\, \\bar{\\theta }^j\\right\\rbrace =\\left\\lbrace \\bar{\\theta }^i,\\, \\bar{\\theta }^j\\right\\rbrace = 0.$ Let us remind that the upper index on the conjugated Grassmann variable reflects the way how it transforms under the action of the R-symmetry group.", "Having thus defined the superspace, one introduces superfields depending on the superspace coordinates.", "The property $\\theta ^2=0$ for any Grassmann variable $\\theta $ limits the number of terms in the expansion of a general superfield $\\Phi (t,\\, \\theta _i,\\,\\bar{\\theta }^j)$ as series in Grassmann variables.", "Take, for instance, ${\\cal N}=2$ case.", "Omitting the indices on $\\theta _1$ and $\\bar{\\theta }^1$ , one obtains $\\Phi (t,\\,\\theta ,\\,\\bar{\\theta })= \\phi (t) + \\bar{\\psi }(t)\\theta + \\xi (t) \\bar{\\theta }+ D(t)\\theta \\bar{\\theta }.$ Two complex fields $\\phi (t)$ , $D(t)$ and two complex Grassmann fields $\\bar{\\psi }(t)$ , $\\xi (t)$ depend only on the time variable and may represent physical degrees of freedom in supersymmetric quantum mechanics.", "However, usually this field content is excessive, i.e.", "it is possible to reduce the number of fields by additional constraints on the superfield $\\Phi $ .", "Such constraints must be covariant with respect to supersymmetry transformations.", "For instance, such a constraint can be reality condition for the superfield $\\Phi $ , see Section REF for details.", "Infinitesimal supersymmetry transformations in quantum mechanics are realized as shifts on the superspace (REF ), $\\begin{array}{l}t\\rightarrow t +i\\left(\\epsilon _i \\bar{\\theta }^i + \\bar{\\epsilon }^i\\theta _i\\right),\\\\[2mm]\\theta _i\\rightarrow \\theta _i + \\epsilon _i,\\\\[2mm]\\bar{\\theta }^i\\rightarrow \\bar{\\theta }^i + \\bar{\\epsilon }^i\\end{array}$ with Grassmann parameter $\\epsilon _i$ , $\\bar{\\epsilon }^i=\\left(\\epsilon _i\\right)^*$ .", "Such transformations are induced by the operator $i\\left(\\bar{\\epsilon }^i Q_i - \\epsilon _i \\bar{Q}^i\\right)$ , where the supercharges $Q_i=-i\\frac{\\partial }{\\partial \\bar{\\theta }^i}+\\theta _i \\frac{\\partial }{\\partial t},\\quad \\quad \\bar{Q}^i=i\\frac{\\partial }{\\partial \\theta _i}-\\bar{\\theta }^i\\frac{\\partial }{\\partial t}$ satisfy the relations $\\big \\lbrace Q_i,\\, Q_j\\big \\rbrace & = & \\left\\lbrace \\bar{Q}^i,\\, \\bar{Q}^j\\right\\rbrace \\,\\,=\\,\\, 0,\\\\[2mm] \\left\\lbrace Q_i,\\, \\bar{Q}^j\\right\\rbrace &=& 2\\delta ^j_i\\, i\\partial _t$ with $\\partial _t =\\frac{\\partial }{\\partial t}.$ The supercharges (REF ) together with the operator $H=i\\partial _t$ realize a particular representation of the supersymmetry algebra (REF ), () on the superspace (REF ).", "Let us also introduce covariant superderivatives $D^i$ and $\\bar{D}_i$ defined as $D^i=\\frac{\\partial }{\\partial \\theta _i}-i\\bar{\\theta }^i\\frac{\\partial }{\\partial t} , \\quad \\quad \\bar{D}_i=\\frac{\\partial }{\\partial \\bar{\\theta }^i}-i\\theta _i \\frac{\\partial }{\\partial t}.$ These operators are of special interest, because they anticommute with the supercharges from above, $\\left\\lbrace D^i,\\, Q_j\\right\\rbrace =\\left\\lbrace D^i,\\, \\bar{Q}^j\\right\\rbrace =\\left\\lbrace \\bar{D}_i,\\, Q_j\\right\\rbrace =\\left\\lbrace \\bar{D}_i,\\, \\bar{Q}^j\\right\\rbrace = 0 ,$ meaning that they are covariant with respect to supertransformations (REF ).", "Thereby, they can be used in covariant constraints on superfields to reduce their number of independent components.", "The anticommutation relations for the superderivatives are $\\left\\lbrace D^i,\\, D^j\\right\\rbrace =\\left\\lbrace \\bar{D}_i,\\, \\bar{D}_j\\right\\rbrace = 0,\\quad \\quad \\left\\lbrace D^i,\\, \\bar{D}_j\\right\\rbrace = -2\\delta ^i_j\\, i\\partial _t .$ Note that, for later convenience, we intentionally use a different convention for the superderivatives (REF ) as compared with the supercharges (REF ).", "Like in supersymmetric field theory case, the ${\\cal N}=2$ supersymmetric quantum-mechanical superspace $\\left\\lbrace t,\\,\\theta ,\\,\\bar{\\theta }\\right\\rbrace $ is endowed with two invariant subspaces — chiral, $\\left\\lbrace t_{\\rm L},\\,\\theta \\right\\rbrace ,\\quad \\quad t_{\\rm L} = t - i\\theta \\bar{\\theta },$ and antichiral, $\\left\\lbrace t_{\\rm R},\\,\\bar{\\theta }\\right\\rbrace ,\\quad \\quad t_{\\rm R} = t + i\\theta \\bar{\\theta }.$ These two subspaces are invariant with respect to supersymmetry transformations (REF ): $\\begin{array}{ll}t_{\\rm L} \\rightarrow t_{\\rm L} + 2i\\bar{\\epsilon }\\,\\theta ,\\quad \\quad &\\theta \\rightarrow \\theta +\\epsilon ,\\\\[2mm]t_{\\rm R} \\rightarrow t_{\\rm R} + 2i\\epsilon \\,\\bar{\\theta },\\quad \\quad &\\bar{\\theta }\\rightarrow \\bar{\\theta }+\\bar{\\epsilon }.\\end{array}$ Practically this means that superfields which depend only on chiral (or antichiral) coordinates have twice as less Grassmann coordinates and thus their component expansion in Grassmann variables is shorter and contains smaller number of component fields.", "This feature of ${\\cal N}=2$ superspace is of primary importance in construction of supersymmetric quantum-mechanical systems.", "Consider, for instance, a superfield $q\\left(t_{\\rm L},\\,\\theta \\right)$ which depends only on the coordinates of the chiral subspace.", "It satisfies a covariant constraint $\\bar{D}\\,q\\left(t_{\\rm L},\\,\\theta \\right) =\\left(\\frac{\\partial }{\\partial \\bar{\\theta }}-i\\theta \\frac{\\partial }{\\partial t}\\right) q\\left(t_{\\rm L},\\,\\theta \\right)=0$ which is due to $\\bar{D}\\,t_{\\rm L}=0$ .", "Its component expansion $q\\left(t_{\\rm L},\\,\\theta \\right) = z(t) + \\psi (t)\\theta - \\dot{z}(t)\\,i\\theta \\bar{\\theta }$ contains one complex variable $z(t)$ and its superpartner – one complex Grassmann variable $\\psi (t)$ .", "This is indeed the minimum number of degrees of freedom one may have for such a superfield.", "The supertransformations (REF ) of the chiral superspace induce the supertransformations of the chiral superfield $q\\left(t_{\\rm L},\\,\\theta \\right)$ and its components: $q\\left(t_{\\rm L},\\,\\theta \\right)\\rightarrow q\\left(t_{\\rm L},\\,\\theta \\right) + \\delta q\\left(t_{\\rm L},\\,\\theta \\right)$ .", "Direct calculation yields: $\\begin{array}{l}z(t)\\rightarrow z(t) + \\psi (t)\\epsilon ,\\\\[2mm]\\psi (t)\\rightarrow \\psi (t) + 2i\\,\\dot{z}(t)\\bar{\\epsilon }\\end{array}$ Note that the field $\\psi $ transformation involves full time derivative.", "Along with the (anti)chiral superfields, real superfields are widely used in the ${\\cal N}=2$ supersymmetric quantum mechanics.", "Let us list their properties.", "We will use a real superfield below for the construction of an example of simple supersymmetric system.", "The real superfield $v(t,\\,\\theta ,\\,\\bar{\\theta })$ is defined by the reality condition, $v=v^\\dagger .$ Its component expansion reads: $v(t,\\,\\theta ,\\,\\bar{\\theta }) = x(t) +\\bar{\\psi }(t)\\theta +\\bar{\\theta }\\psi (t) + D(t)\\, \\theta \\bar{\\theta },$ where $\\bar{\\psi }=\\psi ^*$ and the fields $x(t)$ and $D(t)$ are real.", "Under the supertransformations (REF ) the component fields transform as $\\begin{array}{l}x\\rightarrow x + \\bar{\\psi }\\epsilon + \\bar{\\epsilon }\\psi ,\\\\[2mm]\\psi \\rightarrow \\psi - i\\dot{x}\\, \\epsilon ,\\\\[2mm]\\bar{\\psi }\\rightarrow \\bar{\\psi }+ i\\dot{x}\\, \\bar{\\epsilon },\\\\[2mm]D\\rightarrow D - i\\left(\\dot{\\bar{\\psi }}\\epsilon +\\dot{\\psi }\\bar{\\epsilon }\\right)\\end{array}$ Note that the $D$ -term transforms with full time derivative.", "It is this fact which allows one to build a supersymmetric system to which we now proceed.", "Let us illustrate the material above with an example of the simplest supersymmetric quantum-mechanical system of one real bosonic variable $x(t)$ and one complex Grassmann variable $\\psi (t)$ [39].", "To this end, we take the real superfield (REF ) and write the following action for it: $S=\\int dt\\, d\\bar{\\theta }d\\theta \\left\\lbrace \\frac{1}{2}\\bar{D} v \\, D v + \\Lambda (v)\\right\\rbrace $ which involves the covariant derivatives from Eq.", "(REF ) and an arbitrary real function $\\Lambda (v)$ .", "The rules of Grassmann integration were discussed in Section REF .", "The integration over the Grassmann variables leaves only the $D$ -term of the real superfield which stays under the integral in Eq.", "(REF ).", "Consequently, the action (REF ) is automatically supersymmetric.", "Indeed, under the supertransformations, the $D$ -term transforms as a full time derivative, see Eq.", "(REF ).", "As one may guess, the first term in the action (REF ) describes the kinetic term of the system, while the second term is the interaction term.", "The integration over the Grassmann variables can be straightforwardly performed.", "Using the definition $\\int d\\bar{\\theta }d\\theta \\, \\theta \\bar{\\theta }= 1$ , one obtains $S=\\int dt\\left\\lbrace \\frac{1}{2}\\dot{x}^2+ \\frac{i}{2}\\left(\\bar{\\psi }\\dot{\\psi }-\\dot{\\bar{\\psi }}\\psi \\right)+ \\frac{1}{2} D^2- D\\, W(x) - W^{\\prime }(x)\\, \\bar{\\psi }\\psi \\right\\rbrace ,$ where we have introduced $W(x)=\\Lambda ^{\\prime }(x)$ , and the prime denotes the derivative with respect to $x$ .", "Only the last two terms come from the function $\\Lambda (v)$ in the action.", "We now see that the variable $D(t)$ does not have kinetic term and, thus, it is not dynamical.", "One can integrate it out with its classical equations of motion, $D=W(x).$ Putting Eq.", "(REF ) back into the component action (REF ), one finally obtains the final Lagrangian of the system: $L =\\frac{1}{2}\\dot{x}^2- \\frac{1}{2} W^2(x)+ \\frac{i}{2}\\left(\\bar{\\psi }\\dot{\\psi }-\\dot{\\bar{\\psi }}\\psi \\right)- W^{\\prime }(x)\\, \\bar{\\psi }\\psi .$ This Lagrangian defines a traditional one-dimensional system with the usual kinetic term $\\frac{1}{2}\\dot{x}^2$ and the usual potential term $\\frac{1}{2} W^2(x)$ .", "The coordinate $x$ , however, is coupled to a fermion $\\psi $ .", "Note that here one has one bosonic degree of freedom and two fermionic degrees of freedom.", "Let us remark that the statement about the equality of the number of bosons and fermions concerns the number of states in a multiplet, not the number of fields in the system.", "To understand better the nature of the fermionic degree of freedom, let us perform the Legendre transform of the Lagrangian to the Hamiltonian and quantize the system.", "The third term in the Lagrangian (REF ) involves only one time derivative on $\\psi $ and $\\bar{\\psi }$ and will not enter the Hamiltonian.", "Nevertheless, it does provide the anticommutation relations of $\\psi $ and $\\bar{\\psi }$ .", "Indeed, if one considers $\\psi $ as a coordinate, then $\\bar{\\psi }$ would be the corresponding momentum and vice versa.", "Thus, the canonical anticommutation relations are $\\left\\lbrace \\psi ,\\,\\bar{\\psi }\\right\\rbrace =1 ,$ and the Hamiltonian reads: $H_{\\rm cl} =\\frac{1}{2} {p^2}+ \\frac{1}{2} W^2(x)+ W^{\\prime }(x)\\, \\bar{\\psi }\\psi ,$ where $p=\\dot{x}$ .", "This expression corresponds to the “classical” Hamiltonian, because the term with $\\bar{\\psi }\\psi $ has ordering ambiguity.", "The recipe to solve this issue is known [41]: one must take the classical supercharges (which can be obtained with Nöether theorem from the Lagrangian (REF )) and order the problematic terms with $\\bar{\\psi }\\psi $ , if any, in certain way.", "After that, one must apply the commutation relations (REF ) to obtain the “quantum” Hamiltonian which is indeed enjoys supersymmetry algebra.", "In fact, the supercharges do not have order ambiguity problem.", "Their expressions are as simple as $Q= \\frac{1}{\\sqrt{2}} \\left(p+iW\\right)\\psi ,\\quad \\quad \\bar{Q}= \\frac{1}{\\sqrt{2}} \\left(p-iW\\right)\\bar{\\psi }.$ The anticommutator of supercharges shows the difference between the classical and the quantum Hamiltonian, namely, the quantum Hamiltonian is obtained from the classical Hamiltonian (REF ) by the replacement $\\bar{\\psi }\\psi \\rightarrow \\frac{1}{2} \\left(\\bar{\\psi }\\psi - \\psi \\bar{\\psi }\\right) .$ One can realize the coordinate/momentum pair $\\hat{\\psi }$ , $\\hat{\\bar{\\psi }}$ as differential operators acting on the space of functions $f(\\psi )$ of the argument $\\psi $ in the following way We put “hats” on the operators for clearness.", ": $\\hat{\\psi }= \\psi ,\\quad \\quad \\hat{\\bar{\\psi }} = \\frac{\\partial }{\\partial \\psi }.$ Alternatively, these operators can be realized as matrices.", "Indeed, the space of functions $f(\\psi )$ is two-dimensional: $f(\\psi )\\equiv a + b\\psi $ .", "Let us introduce the basis functions $1\\equiv \\left(\\begin{array}{c}1 \\\\ 0\\end{array}\\right)\\quad \\quad \\mbox{and}\\quad \\quad \\psi \\equiv \\left(\\begin{array}{c}0 \\\\ 1\\end{array}\\right),$ so that $f(\\psi )=a+b\\psi \\equiv \\left(\\begin{array}{c}a \\\\ b\\end{array}\\right).$ Thus, the operators (REF ) have the following matrix form: $\\hat{\\psi }=\\left(\\begin{array}{cc}0 & 0\\\\ 1 &0\\end{array}\\right)\\quad \\quad \\mbox{and}\\quad \\quad \\hat{\\bar{\\psi }} =\\left(\\begin{array}{cc}0 & 1\\\\ 0 &0\\end{array}\\right).$ Thereby, $\\hat{\\bar{\\psi }}\\hat{\\psi }-\\hat{\\psi }\\hat{\\bar{\\psi }}=\\left(\\begin{array}{cc}1 & 0\\\\ 0 &-1\\end{array}\\right) = \\sigma _3 ,$ where $\\sigma _3$ denotes the third Pauli matrix.", "Combining Eqs.", "(REF ), (REF ) and (REF ), one finally obtains the quantum Hamiltonian $H =\\frac{1}{2} {p^2}+ \\frac{1}{2} W^2(x)+ \\frac{1}{2} W^{\\prime }(x)\\sigma _3 ,$ which describes a particle with spin moving in one-dimensional space.", "This Hamiltonian is supersymmetric with supercharges introduced above.", "As a simple exercise, one can take $W(x) = \\omega x$ which gives the system composed with non-interacting one-dimensional oscillator and one spin degree of freedom.", "The reader is referred to Ref.", "[40] for further details on this system, where the study of energy spectrum and quantum states is performed and where the illustration of spontaneous breaking of supersymmetry is presented.", "Let us emphasize that the existence of (anti)chiral subspace in the superspace (REF ) which leads to the reduction of the component expansion of any (anti)chiral superfield is inherent only to the ${\\cal N}=2$ case.", "The superspace (REF ) in the ${\\cal N}\\ge 4$ quantum mechanics does not have this feature.", "One, however, may still use covariant constraints (e.g.", "reality condition or equations involving superderivatives) to reduce number of components in superfields.", "Still, this is sometimes not enough to reduce the number to the expected minimum.", "Moreover, the constraints may be rather sophisticated.", "Some successful examples of manipulation with superfields and their covariant constraints are described in Ref. [42].", "It appeared that the case of ${\\cal N}=4$ is special: it also admits a superspace which has two invariant subspaces and allows one to reduce the number of Grassmann variables in superfields by a factor of two.", "This is the so called harmonic superspace (HSS) approach [5] invented by Galperin, Ivanov, Ogievetsky and Sokatchev.", "The key idea here is that the standard superspace (REF ) should be supplemented with additional coordinates of bosonic nature.", "Let us limit ourselves to the case of ${\\cal N}=4$ supersymmetric quantum mechanics.", "The harmonic superspace can be seen as one of the superspaces on which the supersymmetry group acts.", "The superspace (REF ) is one of possible choices.", "All the conceivable spaces on which supersymmetry acts can be described as a factor $\\mbox{superspace} = \\frac{\\mbox{supersymmetry group}}{\\mbox{one of its certain subgroups}}.$ We remind that ${\\cal N}=4$ supersymmetry algebra in quantum mechanics is invariant under the ${\\rm SU}(2)$ R-symmetry group (cf.", "Section REF ).", "Till this moment, the latter was factored out from the considerations.", "However, while searching for conceivable superspaces, the R-symmetry group space can be added to the superspace (REF ).", "It appears that the superfields on the extended superspace which deliver minimal component field content will be functions on a two-sphere ${\\rm S}^2 = {\\rm SU}(2)/{\\rm U}(1)$ – a factor of the R-symmetry group with respect to one of its ${\\rm U}(1)$ subgroups.", "The harmonic superspace (HSS) approach in quantum mechanics was developed in Ref. [6].", "The convention in this manuscript follows the convention of Ref.", "[2] and differs from the convention of Ref.", "[6] by the change of time direction $t\\rightarrow -t$ .", "With this, one reproduces the correct sign in the kinetic term for the spinor field in Eq.", "(REF ).", "From now and below, in ${\\cal N}=4$ supersymmetry, we use a different notation for spinor indices: the indices from the beginning of the Greek alphabet $\\alpha ,\\, \\beta = 1, 2$ are used instead of the indices $i$ , $j$ .", "For instance, the ordinary ${\\cal N}=4$ superspace is $\\left\\lbrace t,\\,\\theta _\\alpha ,\\,\\bar{\\theta }^\\beta \\right\\rbrace ,\\quad \\quad \\bar{\\theta }^\\beta = (\\theta _\\beta )^* .$ For later references, let us repeat here the expression for the supercharges of Eq.", "(REF ): $Q_\\alpha =-i\\frac{\\partial }{\\partial \\bar{\\theta }^\\alpha }+\\theta _\\alpha \\frac{\\partial }{\\partial t} ,\\quad \\quad \\bar{Q}^\\alpha =i\\frac{\\partial }{\\partial \\theta _\\alpha }-\\bar{\\theta }^\\alpha \\frac{\\partial }{\\partial t} ,$ and the superderivatives of Eq.", "(REF ): $D^\\alpha =\\frac{\\partial }{\\partial \\theta _\\alpha }-i\\bar{\\theta }^\\alpha \\frac{\\partial }{\\partial t} , \\quad \\quad \\bar{D}_\\alpha =\\frac{\\partial }{\\partial \\bar{\\theta }^\\alpha }-i\\theta _\\alpha \\frac{\\partial }{\\partial t}.$ The supersymmetry algebra (REF ), () is $ \\begin{array}{l}\\left\\lbrace Q_\\alpha ,\\bar{Q}^\\beta \\right\\rbrace = 2 \\delta _\\alpha ^\\beta H\\\\[3mm]\\big \\lbrace Q_\\alpha , Q_\\beta \\big \\rbrace = \\left\\lbrace \\bar{Q}^\\alpha , \\bar{Q}^\\beta \\right\\rbrace = 0,\\end{array}$ The ${\\rm SU}(2)$ R-symmetry group admits the possibility of raising and lowering spinor indices with the invariant antisymmetric Levi-Civita tensors $\\varepsilon _{\\alpha \\beta }$ and $\\varepsilon ^{\\alpha \\beta }$ .", "By definition, $\\begin{array}{ll}\\varepsilon _{\\alpha \\beta }=-\\varepsilon _{\\beta \\alpha }, \\quad \\quad &\\varepsilon _{12} = 1,\\\\[2mm]\\varepsilon ^{\\alpha \\beta }=-\\varepsilon ^{\\beta \\alpha }, \\quad \\quad &\\varepsilon ^{12} = -1,\\end{array}$ so that, for example, one has for a spinor $v_\\alpha $ : $v^\\alpha = \\varepsilon ^{\\alpha \\beta }v_\\beta ,\\quad \\quad v_\\alpha = \\varepsilon _{\\alpha \\beta }v^\\beta .$ Due to the invariance of the Levi-Civita tensors with respect to the action of ${\\rm SU}(2)$ R-symmetry group, the equations involving them are also invariant.", "Below, we will also introduce ${\\rm SU}(2)$ Pauli-Gürsey group [5] and dotted indices for it.", "Analogously, one can introduce Levi-Civita tensors with dotted indices: $\\begin{array}{ll}\\varepsilon _{\\dot{\\alpha }\\dot{\\beta }}=-\\varepsilon _{\\dot{\\beta }\\dot{\\alpha }}, \\quad \\quad &\\varepsilon _{\\dot{1} \\dot{2}} = 1,\\\\[2mm]\\varepsilon ^{\\dot{\\alpha }\\dot{\\beta }}=-\\varepsilon ^{\\dot{\\beta }\\dot{\\alpha }}, \\quad \\quad &\\varepsilon ^{\\dot{1}\\dot{2}} = -1.\\end{array}$ Before discussing the harmonic superspace as a whole it is instructive to study the coordinates on the ${\\rm SU}(2)/{\\rm U}(1)$ space which is a two-sphere.", "One could choose, for example, polar or stereographic coordinates on ${\\rm S}^2$ .", "However, it turns out much more convenient to deal with the homogeneous coordinates on the ${\\rm SU}(2)$ group space and constrain functions on it to live on the ${\\rm S}^2$ space.", "To elaborate more on this point, introduce the homogeneous complex coordinates $u^\\pm _\\alpha $ , $\\alpha =1,2$ called harmonics on a three-sphere ${\\rm SU}(2)$ .", "They satisfy the defining relations $u^{+\\alpha } u^-_\\alpha =1 ,\\quad \\quad u^-_\\alpha = (u^{+\\alpha })^* .$ (We are using raised indices in the above, see the previous subsection; for instance, $u^{+\\alpha }=\\varepsilon ^{\\alpha \\beta }u^+_\\beta $ .)", "Note also an important identity $u^+_\\alpha u^-_\\beta - u^-_\\alpha u^+_\\beta = \\varepsilon _{\\alpha \\beta }.$ We are interested in considering functions on the ${\\rm SU}(2)/{\\rm U}(1)$ space.", "Consider functions on a three-sphere which have definitive ${\\rm U}(1)$ charge.", "For illustration, take a function $f^+(u)$ of charge +1.", "In can be expanded into series in harmonics $u^\\pm _\\alpha $ : $f^+(u) = f^\\alpha u^+_\\alpha + f^{\\alpha \\beta \\gamma } u^+_\\alpha u^+_\\beta u^-_\\gamma + \\dots ,$ where the constants $f^\\alpha $ , $f^{\\alpha \\beta \\gamma }$ , $\\dots $ can always be taken symmetric in their indices.", "Indeed, using the relation (REF ), one can transform any product of harmonics to symmetric combinations plus products of harmonics of smaller orders.", "For example, $u^+_\\alpha u^+_\\beta u^-_\\gamma &= &\\frac{1}{3}\\left(u^+_\\alpha u^+_\\beta u^-_\\gamma + u^-_\\alpha u^+_\\beta u^+_\\gamma + u^+_\\alpha u^-_\\beta u^+_\\gamma \\right) +\\nonumber \\\\[2mm]&&+\\ \\frac{1}{3}\\left(u^+_\\alpha u^+_\\beta u^-_\\gamma - u^-_\\alpha u^+_\\beta u^+_\\gamma \\right) +\\nonumber \\\\[2mm]&&+\\ \\frac{1}{3} \\left(u^+_\\alpha u^+_\\beta u^-_\\gamma - u^+_\\alpha u^-_\\beta u^+_\\gamma \\right) =\\nonumber \\\\[2mm]&=& \\frac{1}{3}\\left(u^+_\\alpha u^+_\\beta u^-_\\gamma + u^-_\\alpha u^+_\\beta u^+_\\gamma + u^+_\\alpha u^-_\\beta u^+_\\gamma \\right) +\\frac{1}{3} \\varepsilon _{\\alpha \\gamma }u^+_\\beta + \\frac{1}{3} \\varepsilon _{\\beta \\gamma } u^+_\\alpha .$ With respect to the action of R-symmetry group, the function $f^+(u)$ undergo homogeneous ${\\rm U}(1)$ phase transformations (according to its overall charge) and thus is well defined on a two-sphere ${\\rm SU}(2)/{\\rm U}(1)$ .", "The same is also true for any function of harmonics with fixed ${\\rm U}(1)$ charge.", "In fact, the harmonics $u^\\pm _\\alpha $ are the fundamental spin 1/2 spherical harmonics familiar from quantum mechanics.", "This is why they are called harmonic variables.", "The harmonics can be used for projection of spinor indices onto harmonic space.", "For instance, $f^+=u^+_\\alpha f^\\alpha $ and $f^-= u^-_\\alpha f^\\alpha $ .", "The original spinor can be restored using Eq.", "(REF ): $f^\\alpha = u^{+\\alpha }f^- - u^{-\\alpha }f^+.$ The differential operators $D^{++}=u_\\alpha ^+\\frac{\\partial }{\\partial u_\\alpha ^-},\\quad \\quad D^{--} = u_\\alpha ^-\\frac{\\partial }{\\partial u_\\alpha ^+}\\ ,\\quad \\quad D^0 = u_\\alpha ^+\\frac{\\partial }{\\partial u_\\alpha ^+} - u_\\alpha ^-\\frac{\\partial }{\\partial u_\\alpha ^-}$ are called harmonic derivatives.", "The operator $D^0$ plays a role of the ${\\rm U}(1)$ charge operator.", "One has for a function $f^{+q}(u)$ of definite ${\\rm U}(1)$ charge $+q$ : $D^0 f^{+q}(u) = q\\, f^{+q}(u) .$ The coordinates $u^+_\\alpha $ have charge 1, while the coordinates $u^-_\\alpha $ have charge -1.", "The ${\\cal N}=4$ harmonic superspace formalism in quantum mechanics was developed in Ref. [6].", "In this formalism, the superfields depend on time $t$ and on harmonics $u^{\\pm \\alpha }$ which parametrize the R-symmetry group ${\\rm SU}(2)$ of the ${\\cal N} = 4$ superalgebra, and on Grassmann variables $\\theta _\\alpha $ , $\\bar{\\theta }^\\beta $ .", "The superspace is $\\left\\lbrace t,\\,\\theta _\\alpha ,\\,\\bar{\\theta }^\\beta ,\\,u^\\pm _\\gamma \\right\\rbrace ,\\quad \\quad \\bar{\\theta }^\\beta = (\\theta _\\beta )^* .$ This is the so called standard basis in harmonic superspace.", "Usually, instead of spinors $\\theta _\\alpha $ and $\\bar{\\theta }^\\beta $ it is preferable to use the following harmonic projections: $\\theta ^\\pm = u_\\alpha ^\\pm \\theta ^\\alpha ,\\quad \\quad \\bar{\\theta }^\\pm = u_\\alpha ^\\pm \\bar{\\theta }^\\alpha .$ One can also define harmonic projections of superderivatives, $D^\\pm =u^\\pm _\\alpha D^\\alpha $ , $\\bar{D}^\\pm =u^\\pm _\\alpha \\bar{D}^\\alpha $ .", "One can check that $D^+ = \\frac{\\partial }{\\partial \\theta ^-} - i\\bar{\\theta }^+ \\frac{\\partial }{\\partial t},&\\quad \\quad &D^- = -\\frac{\\partial }{\\partial \\theta ^+} - i\\bar{\\theta }^- \\frac{\\partial }{\\partial t},\\\\[2mm]\\bar{D}^+ = -\\frac{\\partial }{\\partial \\bar{\\theta }^-} - i\\theta ^+ \\frac{\\partial }{\\partial t},&\\quad \\quad &\\bar{D}^- = \\frac{\\partial }{\\partial \\bar{\\theta }^+} - i\\theta ^- \\frac{\\partial }{\\partial t},$ in the standard basis (REF ).", "The most striking feature of harmonic superspace is the presence of an analytic subspace $\\left\\lbrace t_{\\rm A},\\,\\theta ^+,\\, \\bar{\\theta }^+,\\, u^{\\pm \\alpha }\\right\\rbrace $ in it (an analog of ${\\cal N} = 2$ chiral superspace) involving the “analytic time” $t_A = t + i(\\theta ^+ \\bar{\\theta }^- + \\theta ^- \\bar{\\theta }^+)$ and containing twice as less fermionic coordinates.", "Let us elaborate more on this point.", "It is convenient to go over to the analytic basis in harmonic superspace, $\\left\\lbrace t_{\\rm A},\\, \\theta ^\\pm ,\\,\\bar{\\theta }^\\pm ,\\,u^\\pm _\\alpha \\right\\rbrace .$ In this basis, the covariant spinor derivatives $D^+,\\ \\bar{D}^+$ are as simple as $D^+=\\frac{\\partial }{\\partial \\theta ^-},\\quad \\quad \\bar{D}^+=-\\frac{\\partial }{\\partial \\bar{\\theta }^-} .$ It is this fact which allows one to translate superfield constraints $D^+ f= \\bar{D}^+ f$ for some superfield $f$ to be independent of $\\theta ^-$ and $\\bar{\\theta }^-$ : $f=f(t_A,\\,\\theta ^+,\\,\\bar{\\theta }^+,\\, u^\\pm _\\alpha )$ .", "One can check directly that the subspace (REF ) is invariant with respect to ${\\cal N}=4$ supersymmetry transformations.", "Indeed, using Eq.", "(REF ), one obtains $\\begin{array}{l}t_A\\rightarrow t_A + 2i\\left(\\epsilon ^- \\bar{\\theta }^+ - \\bar{\\epsilon }^-\\theta ^+ \\right),\\\\[2mm]\\theta ^\\pm \\rightarrow \\theta ^\\pm + \\epsilon ^\\pm ,\\\\[2mm]\\bar{\\theta }^\\pm \\rightarrow \\bar{\\theta }^\\pm + \\bar{\\epsilon }^\\pm ,\\\\[2mm]u^\\pm _\\alpha \\rightarrow u^\\pm _\\alpha ,\\end{array}$ where $\\bar{\\epsilon }^\\alpha = (\\epsilon _\\alpha )^*,\\quad \\quad \\epsilon ^\\pm = u^\\pm _\\alpha \\epsilon ^\\alpha ,\\quad \\quad \\bar{\\epsilon }^\\pm = u^\\pm _\\alpha \\bar{\\epsilon }^\\alpha .$ Finally, let us also write the form of harmonic derivatives $D^{++}$ and $D^{--}$ from Eq.", "(REF ) in the analytic basis: $D^{++}=u^+_\\alpha \\frac{\\partial }{\\partial u^-_\\alpha }+\\theta ^+\\frac{\\partial }{\\partial \\theta ^-}+\\bar{\\theta }^+\\frac{\\partial }{\\partial \\bar{\\theta }^-}+2i\\theta ^+\\bar{\\theta }^+ \\frac{\\partial }{\\partial t_{\\rm A}},\\\\[2mm]D^{--}=u^-_\\alpha \\frac{\\partial }{\\partial u^+_\\alpha }+\\theta ^-\\frac{\\partial }{\\partial \\theta ^+}+\\bar{\\theta }^-\\frac{\\partial }{\\partial \\bar{\\theta }^+}+2i\\theta ^-\\bar{\\theta }^- \\frac{\\partial }{\\partial t_{\\rm A}} .$ Note that in the subspace (REF ) the second and the third terms vanish.", "The superspace (REF ) admits an involution symmetry which commutes with supersymmetry transformations [6], [5].", "We denote the involution with a sign $\\widetilde{\\ }$ , e.g.", "its action is $f\\rightarrow \\widetilde{f}$ for an arbitrary superfield $f(t_A,\\,\\theta ^\\pm ,\\, \\bar{\\theta }^\\pm ,\\, u^\\pm _\\alpha )$ .", "By definition, the involution transformation acts just as the ordinary complex conjugation except its action on the harmonics $u^\\pm _\\alpha $ for which it is $\\widetilde{u^\\pm _\\alpha }=u^{\\pm \\alpha },\\quad \\quad \\widetilde{u^{\\pm \\alpha }}=-u^\\pm _\\alpha .$ This gives $\\widetilde{t_{\\rm A}}=t_{\\rm A},\\quad \\quad \\widetilde{\\theta ^\\pm }=\\bar{\\theta }^\\pm ,\\quad \\quad \\widetilde{\\bar{\\theta }^\\pm }=-\\theta ^\\pm .$ The action of the involution transformation on harmonics can be seen as a composition of complex conjugation and point inversion on the sphere ${\\rm S}^2$ .", "In general, the involution symmetry is very similar to the operation of complex conjugation, but is does not change ${\\rm U}(1)$ charges of superfields.", "It allows one to put additional constraints on superfields (e.g.", "reality condition) and is used in construction of supersymmetric Lagrangians.", "Let us now discuss possible minimal supermultiplets In Section REF , we discussed supermultiplets of quantum states (characterized by their wave functions), whereas here by a supermultiplet we mean a superfield with certain “minimal” superfield content.", "In particular, the statement about the equality of the number of bosons and fermions (Section REF ) is not applicable to the number of physical bosonic and fermionic fields in a system.", "This property was already observed in the example of the simplest quantum mechanics, see Section REF .", "Let us also remark that the above property is specific only to quantum mechanics.", "which are commonly involved in the superfield description.", "All such ${\\cal N}=4$ supermultiplets are usually referred to with three numbers, $({\\bf b, f, a})$ , where In Section REF , we discussed supermultiplets of quantum states (characterized by their wave functions), whereas here by a supermultiplet we mean a superfield with certain “minimal” superfield content.", "In particular, the statement about the equality of the number of bosons and fermions (Section REF ) is not applicable to the number of physical bosonic and fermionic fields in a system.", "This property was already observed in the example of the simplest quantum mechanics, see Section REF .", "Let us also remark that the above property is specific only to quantum mechanics.", "${\\bf b}$ is the number of physical bosonic degrees of freedom; ${\\bf f}$ is the number of physical fermionic degrees of freedom; ${\\bf a}$ is the number of auxiliary nondynamical bosonic degrees of freedom, which are integrated out of final Lagrangians.", "${\\bf b}$ is the number of physical bosonic degrees of freedom; ${\\bf f}$ is the number of physical fermionic degrees of freedom; ${\\bf a}$ is the number of auxiliary nondynamical bosonic degrees of freedom, which are integrated out of final Lagrangians.", "The widely used supermultiplets are $({\\bf 4, 4, 0})$ and $({\\bf 3, 4, 1})$ [43], [6], [42], [15] which usually describe four- and three-dimensional dynamics respectively.", "We discuss them in detail below.", "Also, the common way of obtaining ${\\cal N}=8$ supersymmetry or higher dimensional theories with ${\\cal N}=4$ supersymmetry is to take several superfields of such types.", "Other supermultiplets include $({\\bf 2, 4, 2})$ and $({\\bf 1, 4, 3})$ [44], [45] which are usually used to build systems of many particles in two and one dimensions.", "These multiplets are not discussed in this manuscript.", "In general, the number of physical fermions in all such multiplets is four, while the sum of physical and auxiliary bosonic degrees of freedom is also four.", "It is even possible to introduce the $({\\bf 0, 4, 4})$ supermultiplet [6].", "The derivative operators $D^+$ , $\\bar{D}^+$ , $D^{++}$ (anti)commute with each other and with supercharges.", "Because of this, it is possible to consider a superfield $q^+$ with ${\\rm U}(1)$ charge +1 satisfying $D^+ q^+=0,\\quad \\quad \\bar{D}^+ q^+=0,\\quad \\quad D^{++}q^+=0.$ In the analytic superspace coordinates, the first and the second equations mean that $q^+$ depend only on $\\theta ^+$ and $\\bar{\\theta }^+$ , but not on $\\theta ^-$ and $\\bar{\\theta }^-$ , see Eq.", "(REF ).", "In this way, the first and the second equations form the so-called superfield analyticity conditions.", "When expanding the superfield $q^+(t_A,\\, \\theta ^+,\\, \\bar{\\theta }^+,\\, u^\\pm _\\alpha )$ over spinor coordinates and the harmonics, one obtains an infinite set of physical fields.", "However, imposing also the condition $D^{++} q^+ = 0$ drastically reduces the number of such fields, making it finite.", "In the analytic basis (REF ), the solution of the constraints (REF ) reads $q^+=x^\\alpha (t_{\\rm A})u^+_\\alpha -2\\theta ^+\\chi (t_{\\rm A})-2\\bar{\\theta }^+\\bar{\\chi }^{\\prime }(t_{\\rm A})-2i\\theta ^+\\bar{\\theta }^+\\partial _{\\rm A}x^\\alpha (t_{\\rm A}) u^-_\\alpha $ with $\\partial _{\\rm A}\\equiv \\frac{\\partial }{\\partial t_A}$ and the factors $-2$ introduced for convenience.", "Thus, the $({\\bf 4,4,0})$ superfield $q^+$ involves two complex bosonic coordinates $x^\\alpha $ and two complex fermions – $\\chi $ , $\\bar{\\chi }^{\\prime }$ .", "The constraints $D^+ q^+ = \\bar{D}^+ q^+ = 0$ are akin to the discussed previously chirality constraints in ${\\cal N} =1$ four-dimensional supersymmetric field theories.", "Such constraints appear naturally in the HSS formalism and are common also in four-dimensional field theories.", "A possibility to impose the extra constraint $D^{++} q^+ = 0$ is specific for quantum mechanics only, where it has a pure kinematic nature.", "In ${\\cal N}=2$ supersymmetric field theories, the relation $D^{++} q^+ = 0$ is not a kinematic constraint, it is the equation of motion for the free hypermultiplet derived from the action $S = \\int d^4x \\, du \\, d^4\\theta ^+ \\, \\widetilde{q^+} D^{++} q^+$ [5].", "The constraints (REF ) admit an involution symmetry $q^+\\rightarrow \\widetilde{q^+}$ which commutes with supersymmetry transformations [6], [5]: $\\widetilde{q^+}=\\left[x_\\alpha (t_{\\rm A})\\right]^*u^+_\\alpha -2\\theta ^+\\bar{\\chi }^{\\prime *}(t_{\\rm A})+2\\bar{\\theta }^+\\chi ^*(t_{\\rm A})-2i \\theta ^+\\bar{\\theta }^+\\partial _{\\rm A}\\left[x_\\alpha (t_{\\rm A})\\right]^* u^-_\\alpha .$ It is straightforward to see that the field $\\widetilde{q^+}$ satisfies the same constraints (REF ) as the field $q^+$ .", "As we will use the $({\\bf 4,4,0})$ supermultiplet to construct supersymmetric quantum-mechanical system with four space dimensions, it will be more convenient for us to use the $q^+$ supermultiplet in different form.", "Namely, let us introduce the supermultiplet $q^{+\\dot{\\alpha }} = \\left\\lbrace q^+,\\, \\widetilde{q^+}\\right\\rbrace ,\\quad \\quad \\dot{\\alpha }= 1, 2 .$ The involution symmetry can used to impose the pseudoreality condition on the field $q^{+\\dot{\\alpha }}$ , $q^{+{\\dot{\\alpha }}}=\\varepsilon ^{{\\dot{\\alpha }}{\\dot{\\beta }}} \\widetilde{\\left(q^{+{\\dot{\\beta }}}\\right)} ,$ which is in fact equivalent to Eq.", "(REF ).", "In components, $q^{+\\dot{\\alpha }} = x^{\\alpha \\dot{\\alpha }}(t_{\\rm A})u^+_\\alpha -2\\theta ^+\\chi ^{\\dot{\\alpha }}(t_{\\rm A})-2\\bar{\\theta }^+\\bar{\\chi }^{\\dot{\\alpha }}(t_{\\rm A})-2i\\theta ^+\\bar{\\theta }^+\\partial _{\\rm A}x^{\\alpha \\dot{\\alpha }}(t_{\\rm A}) u^-_\\alpha .$ The constraint (REF ) implies $x^{\\alpha {\\dot{\\alpha }}}=-\\left(x_{\\alpha {\\dot{\\alpha }}}\\right)^* ,\\quad \\quad \\bar{\\chi }^{\\dot{\\alpha }}=\\left(\\chi _{\\dot{\\alpha }}\\right)^* .$ The form of the supermultiplet $q^{+\\dot{\\alpha }}$ suggests that one can associate the ${\\rm SU}(2)$ group related to the dotted index $\\dot{\\alpha }$ .", "This Pauli-Gürsey group [5] is also realized on the $q^+$ supermultiplet, but not manifestly.", "Consequently, the quantum-mechanical system which will be discussed in the next chapter, inherits the ${\\rm SO}(4)={\\rm SU}(2)_{\\rm R}\\times {\\rm SU}(2)_{\\rm PG}$ group composed from the R-symmetry group and the Pauli-Gürsey group.", "In general, this rotational ${\\rm SO}(4)$ group is completely broken by the presence of a four-dimensional gauge field, see next chapter.", "Instead of the coordinate superfield $q^{+\\dot{\\alpha }}$ one can deal with the analytic superfield $L^{++}$ of charge +2 which encompass the supermultiplet $(\\bf {3,4,1})$ and is subjected to the constraints $\\begin{array}{c}D^{+}L^{++} = \\bar{D}^{+}L^{++} = 0,\\\\[3mm]D^{++}L^{++} = 0,\\quad \\quad \\widetilde{(L^{++})} = -L^{++},\\quad \\quad \\quad \\quad \\,\\end{array}$ They restrict the analytic superfield $L^{++}$ to have the appropriate off-shell component field content, namely $({\\bf 3, 4, 1})$ : $L^{++} = \\ell ^{\\alpha \\beta }u^+_\\alpha u^+_\\beta + 2i\\theta ^+ \\chi ^\\alpha u^+_\\alpha +2i\\bar{\\theta }^+ \\bar{\\chi }^\\alpha u^+_\\alpha +\\theta ^+\\bar{\\theta }^+ [F - 2 i\\dot{\\ell }^{\\alpha \\beta } u^+_{\\alpha } u^-_{\\beta }]$ with $\\left(\\ell _{\\alpha \\beta }\\right)^* = -\\ell ^{\\alpha \\beta },\\quad \\quad (\\chi ^\\alpha )^* = \\bar{\\chi }_\\alpha .$ The multiplet $L^{++}$ involves the 3-dimensional target space coordinates $\\ell ^{\\alpha \\beta } = \\ell ^{\\beta \\alpha }$ , their fermionic partners $\\chi ^\\alpha $ , $\\bar{\\chi }^\\alpha $ and a real auxiliary field $F$ .", "The invariant actions involve the harmonic integral $\\int du$ .", "To find such integral of any function $f(u^\\pm _\\alpha )$ , one should expand $f$ in the harmonic Taylor series and, for each term, do the integrals using the rules $\\int du\\, 1=1,\\quad \\quad \\int du\\, u^+_{\\lbrace \\alpha _1}\\dots u^+_{\\alpha _k}u^-_{\\alpha _{k+1}}\\dots u^-_{\\alpha _{k+\\ell }\\rbrace }=0 \\ ,$ where the integrand in the right equation is symmetrized over all indices.", "The values of the integrals of all other harmonic monoms (for example, $\\int du \\, u^+_\\alpha u^-_\\beta = \\frac{1}{2} \\varepsilon _{\\alpha \\beta }$ ) follow from (REF ) and the definitions (REF ), (REF ).", "We keep the notation of the previous chapter for the four-dimensional Euclidean space vector indices, $\\mu ,\\nu = 0,1,2,3.$ The Euclidean four-dimensional sigma-matrices, however, are different; we use the following SO(4) notation (compare with Eq.", "(REF )): $(\\sigma _\\mu )_{\\alpha \\dot{\\alpha }} = \\left\\lbrace i, \\vec{\\sigma }\\right\\rbrace _{\\alpha \\dot{\\alpha }},\\quad \\quad \\left(\\sigma _\\mu ^\\dagger \\right)^{\\dot{\\alpha }\\alpha }=\\left\\lbrace -i,\\vec{\\sigma }\\right\\rbrace ^{\\dot{\\alpha }\\alpha },$ where $\\vec{\\sigma }$ are ordinary Pauli matrices.", "(These are more or less the conventions of [33] rotated to Euclidean space.)", "The matrix $\\sigma _\\mu ^\\dagger $ is obtained from the matrix $\\sigma _\\mu $ by the operation of raising of indices: $\\left(\\sigma _\\mu ^\\dagger \\right)^{\\dot{\\alpha }\\alpha }=-\\varepsilon ^{\\dot{\\alpha }\\dot{\\gamma }}\\varepsilon ^{\\alpha \\gamma }(\\sigma _\\mu )_{\\gamma \\dot{\\gamma }}\\,.$ The matrices $\\sigma _\\mu ,\\ \\sigma ^\\dagger _\\mu $ satisfy the identities $\\begin{array}{c}\\sigma _\\mu \\sigma ^\\dagger _\\nu +\\sigma _\\nu \\sigma ^\\dagger _\\mu =\\sigma ^\\dagger _\\mu \\sigma _\\nu +\\sigma ^\\dagger _\\nu \\sigma _\\mu =2\\delta _{\\mu \\nu }, \\\\\\sigma ^\\dagger _{\\mu }\\sigma _{\\nu } - \\sigma ^\\dagger _{\\nu }\\sigma _{\\mu }=2i\\,\\eta _{\\mu \\nu }^a \\sigma _a, \\\\\\sigma _{\\mu }\\sigma ^\\dagger _{\\nu } - \\sigma _{\\nu }\\sigma ^\\dagger _{\\mu } =2i\\,\\bar{\\eta }_{\\mu \\nu }^a \\sigma _a,\\end{array}$ where $\\eta _{\\mu \\nu }^a$ , $\\bar{\\eta }_{\\mu \\nu }^a$ are the 't Hooft symbols, $\\eta ^a_{ij} = \\bar{\\eta }^a_{ij} = \\varepsilon _{aij},\\ \\ \\eta ^a_{i0} = - \\eta ^a_{0i} = \\bar{\\eta }^a_{0i} = -\\bar{\\eta }^a_{i0} = \\delta _{ai}$ ($\\sigma _a$ – Pauli matrices, indices $a$ , $i$ , $j$ run from 1 to 3).", "They are self-dual and anti-self-dual respectively, $\\eta _{\\mu \\nu }^a=\\frac{1}{2}\\varepsilon _{\\mu \\nu \\rho \\lambda }\\eta _{\\rho \\lambda }^a,\\quad \\quad \\bar{\\eta }_{\\mu \\nu }^a=-\\frac{1}{2}\\varepsilon _{\\mu \\nu \\rho \\lambda }\\bar{\\eta }_{\\rho \\lambda }^a,$ with the convention $\\varepsilon _{0123} = -1.$ Another useful identity is $\\sigma _2 \\sigma _\\mu ^T \\sigma _2 = \\ -\\sigma ^\\dagger _\\mu .$ The $({\\bf 4,4,0})$ supermultiplet (REF ) involves the bosonic field $x^{\\alpha \\dot{\\alpha }}(t)$ .", "In fact, such bosonic field is equivalent to a real four-vector in the Euclidean space.", "Let us describe the transformation between the spinor notation $v^{\\alpha \\dot{\\alpha }}$ and the corresponding vector notation $v^\\mu $ for an arbitrary field $v$ : $ \\begin{array}{l}v_{\\alpha \\dot{\\alpha }} = v_\\mu (\\sigma _\\mu )_{\\alpha \\dot{\\alpha }} ,\\\\[2mm]v_\\mu = \\frac{1}{2} v_{\\alpha \\dot{\\alpha }}(\\sigma _\\mu ^\\dagger )^{\\dot{\\alpha }\\alpha }\\, =\\, -\\frac{1}{2} v^{\\alpha \\dot{\\alpha }}(\\sigma _\\mu )_{\\alpha \\dot{\\alpha }},\\\\[2mm]v^{\\alpha \\dot{\\alpha }} = \\varepsilon ^{\\alpha \\beta }\\varepsilon ^{\\dot{\\alpha }\\dot{\\beta }} v_{\\beta \\dot{\\beta }}\\, =\\, -v_\\mu (\\sigma _\\mu ^\\dagger )^{\\dot{\\alpha }\\alpha }.\\end{array}$ Particularly, it is straightforward to check that for the field $x^{\\alpha \\dot{\\alpha }}$ with the constraint (REF ) the corresponding vector field $x^\\mu $ is real.", "Here we give a list of relations for the 't Hooft symbols $\\eta ^a_{\\mu \\nu }$ and $\\bar{\\eta }^a_{\\mu \\nu }$ , defined by Eqs.", "(REF ), (REF ): $\\begin{array}{l}\\eta ^a_{\\mu \\nu }=-\\eta ^a_{\\nu \\mu },\\\\[2mm]\\eta ^a_{\\mu \\nu }\\eta ^a_{\\mu \\lambda }=3\\delta _{\\nu \\lambda },\\\\[2mm]\\eta ^a_{\\mu \\nu }\\eta ^b_{\\mu \\nu }=4\\delta ^{ab},\\\\[2mm]\\eta ^a_{\\mu \\nu }\\eta ^a_{\\gamma \\lambda }=\\delta _{\\mu \\gamma }\\delta _{\\nu \\lambda }-\\delta _{\\mu \\lambda }\\delta _{\\nu \\gamma }+\\varepsilon _{\\mu \\nu \\gamma \\lambda },\\\\[2mm]\\varepsilon _{\\mu \\nu \\lambda \\sigma }\\eta ^a_{\\gamma \\sigma }=\\delta _{\\gamma \\mu }\\eta ^a_{\\nu \\lambda }-\\delta _{\\gamma \\nu }\\eta ^a_{\\mu \\lambda }+\\delta _{\\gamma \\lambda }\\eta ^a_{\\mu \\nu },\\\\[2mm]\\eta ^a_{\\mu \\nu }\\eta ^b_{\\mu \\lambda }=\\delta ^{ab}\\delta _{\\nu \\lambda }+\\varepsilon ^{abc}\\eta ^c_{\\nu \\lambda },\\\\[2mm]\\varepsilon ^{abc}\\eta ^b_{\\mu \\nu }\\eta ^c_{\\gamma \\lambda }=\\delta _{\\mu \\gamma }\\eta ^a_{\\nu \\lambda }-\\delta _{\\mu \\lambda }\\eta ^a_{\\nu \\gamma }-\\delta _{\\nu \\gamma }\\eta ^a_{\\mu \\lambda }+\\delta _{\\nu \\lambda }\\eta ^a_{\\mu \\gamma },\\\\[2mm]\\eta ^a_{\\mu \\nu }\\bar{\\eta }^b_{\\mu \\nu }=0,\\\\[2mm]\\eta ^a_{\\gamma \\mu }\\bar{\\eta }^b_{\\gamma \\lambda }=\\eta ^a_{\\gamma \\lambda }\\bar{\\eta }^b_{\\gamma \\mu }.\\end{array}$ To pass from the relations for $\\eta ^a_{\\mu \\nu }$ , to those for $\\bar{\\eta }^a_{\\mu \\nu }$ it is necessary to make the substitution $\\eta ^a_{\\mu \\nu }\\rightarrow \\bar{\\eta }^a_{\\mu \\nu },\\quad \\varepsilon _{\\mu \\nu \\gamma \\delta }\\rightarrow -\\varepsilon _{\\mu \\nu \\gamma \\delta }.$   This is the central chapter of the manuscript describing certain new SQM models discussed and studied in the papers [2], [3], [4].", "The explicit form of the corresponding superfield and component actions, as well as of the quantum Hamiltonians and supercharges is given.", "The brief summary of the results is the following.", "It is shown that the Hamiltonian $H =\\, /\\!\\!\\!\\!", "{ ^2, where \\,/\\!\\!\\!\\!", "{ is flat four-dimensional Dirac operator in an external {\\em self-dual} gauge background, Abelian or non-Abelian, is supersymmetric with {\\cal N}=4 supersymmetry.A generalization of this Hamiltonian to the motion on a curved conformally flat four-dimensional manifold exists.For an {\\em Abelian} self-dual background, the corresponding Lagrangian can be derived from certain harmonic superspace expressions.", "}\\hspace{8.53581pt}If the Hamiltonian involves a {\\em non-Abelian} self-dual gauge field, one can construct the Lagrangian formulation by introducing auxiliary bosonic variables with Wess-Zumino type action.For a special class of such Lagrangians when the gauge group is {\\rm SU}(2) and the gauge field is expressed in the `t~Hooft ansatz form, it is possible to give a superfield description using the harmonic superspace formalism.", "As a new explicit example, the {\\cal N}=4 mechanics with {\\em Yang monopole} in {\\mathbb {R}}^5 (which coincides with an instanton on {\\rm S}^4) is considered.", "}\\hspace{8.53581pt}Independently, a similar system with $ N=4$ supersymmetry in {\\em three dimensions} also admits the superfield description.Although the three-dimensional system involves different superfields as compared with the four-dimensional case, its component Lagrangian and Hamiltonian appear to be the three-dimensional reduction of the mentioned four-dimensional system.The off-shell $ N=4$ supersymmetry requires the gauge field to be astatic form of the ^{\\prime }t Hooft ansatz for the four-dimensional self-dual SU(2)gauge fields, that is a particular solution of Bogomolny equationsfor {\\em BPS monopoles}.$ Consider the Dirac operator in flat four-dimensional Euclidean space $/\\!\\!\\!", "\\!=\\ \\sum _{\\mu =0,1,2,3} \\mu \\gamma _\\mu \\ ,$ where $\\mu = \\partial _\\mu - i {\\cal A}_\\mu $ with ${\\cal A}_\\mu $ being a gauge field and $\\gamma _\\mu $ are Euclidean anti-Hermitian gamma–matrices, $\\gamma _\\mu = \\left( \\begin{array}{cc} 0 & -\\sigma ^\\dagger _\\mu \\\\\\sigma _\\mu & 0\\end{array} \\right),\\quad \\quad \\left\\lbrace \\gamma _\\mu ,\\gamma _\\nu \\right\\rbrace =-2\\delta _{\\mu \\nu }.$ The matrices $\\sigma _\\mu $ and $\\sigma ^\\dagger _\\mu $ were introduced in Eq.", "(REF ).", "The Hamiltonians we are going to construct enjoy ${\\rm SO}(4) = {\\rm SU}(2) \\times {\\rm SU}(2)$ covariance such that the undotted spinor index refers to the first ${\\rm SU}(2)$ factor, while the dotted one to the second.", "Consider the operator $H = \\frac{1}{2}\\, /\\!\\!\\!\\!2 \\ =\\ - \\frac{1}{2} 2 - \\frac{i}{4} {\\cal F}_{\\mu \\nu } \\gamma _{\\mu } \\gamma _{\\nu }\\ ,$ where ${\\cal F}_{\\mu \\nu }=\\partial _\\mu {\\cal A}_\\nu -\\partial _\\nu {\\cal A}_\\mu -i\\left[{\\cal A}_\\mu , {\\cal A}_\\nu \\right]$ is the gauge field strength.", "It is well known that nonzero eigenvalues of the Euclidean Dirac operator come in pairs $(-\\lambda , \\lambda )$ and hence the spectrum of the Hamiltonian $H$ is double-degenerate for all excited states.", "This means that, for any external field ${\\cal A}_\\mu $ , this Hamiltonian is supersymmetric [9] admitting two different anticommuting real supercharges: $/\\!\\!\\!\\!", "and $ i/ 5$($ 5 = 0 1 2 3$).Suppose now that the background field is self-dual,\\begin{equation}{\\cal F}_{\\mu \\nu } = \\frac{1}{2} \\varepsilon _{\\mu \\nu \\rho \\delta } {\\cal F}_{\\rho \\delta } \\ \\ \\longleftrightarrow \\ \\ {\\cal F}_{\\mu \\nu } = \\eta ^a_{\\mu \\nu } B_a\\ ,\\end{equation}where $ a$ are the ^{\\prime }t~Hooft symbols defined in Eq.~(\\ref {eq_sigma}).One can be easily convinced that in this case the Hamiltonian admits {\\it four} different Hermitian square roots $ SA$ that satisfythe extended supersymmetry algebra~(\\ref {c3SAB}); we repeat these relations here:\\begin{equation}\\lbrace S_A, S_B\\rbrace = 4\\delta _{AB} H.\\end{equation}One of the choices is\\begin{equation}\\begin{array}{l}S_1 = /\\!\\!\\!\\!", "\\gamma _0 0 + \\gamma _1 1 + \\gamma _2 2 + \\gamma _3 3,\\\\S_2 = \\gamma _0 3 + \\gamma _1 2 - \\gamma _2 1 - \\gamma _3 0, \\\\S_3 = \\gamma _0 2 - \\gamma _1 3 - \\gamma _2 0 + \\gamma _3 1 , \\\\S_4 = \\gamma _0 1 -\\gamma _1 0 + \\gamma _2 3 - \\gamma _3 2 .\\end{array}\\end{equation}Introducing the complex supercharges\\begin{equation}\\begin{array}{c}Q_1 = (S_1 - iS_2)/2,\\quad \\quad Q_2 = (S_3 - iS_4)/2, \\\\\\bar{Q}^1 = (S_1 + iS_2)/2,\\quad \\quad \\bar{Q}^2 = (S_3 + iS_4)/2,\\end{array}\\end{equation}we obtain the standard$ N = 4$ supersymmetry algebra~(\\ref {c3e5x}).Correspondingly, the excited spectrum of $ H$ is four-fold degenerate, while the spectrum of $ / consists of the quartets involving two degenerate positive and two degenerate negative eigenvalues.", "Note that, in contrast to $/\\!\\!\\!\\!", ", the operator $ / 5$ is not expressed into a linear combination of $ SA$.In other words, the $ N=2$ supersymmetry algebra with the operators $ / 1 5)$ is not a subalgebra of the$ N=4$ algebra~(\\ref {c3e5x}).$ The algebra () with supercharges () holds for any self-dual field, irrespectively of whether it is Abelian or non-Abelian.", "Thus, the additional 2-fold degeneracy of the spectrum of the Dirac operator mentioned above should be there for a generic self-dual field.", "One particular example of a non-Abelian self-dual field is the instanton solution, where this degeneracy was observed back in [10] (see Eqs.", "(4.15) there).", "The generalization of the Dirac operator and (anti)self-duality conditions to higher-dimensional manifolds was considered in Ref. [14].", "To make contact with the Lagrangian (and, especially, superfield) description, it is convenient to introduce holomorphic fermion variables $\\psi _{\\dot{\\alpha }}$ and $\\bar{\\psi }^{\\dot{\\alpha }}$ , which satisfy the standard anticommutation relations $\\lbrace \\psi _{\\dot{\\alpha }}, \\psi _{\\dot{\\beta }} \\rbrace \\ =\\ \\lbrace \\bar{\\psi }^{\\dot{\\alpha }}, \\bar{\\psi }^{\\dot{\\beta }} \\rbrace \\ =\\ 0, \\quad \\quad \\lbrace \\bar{\\psi }^{\\dot{\\alpha }}, \\psi _{\\dot{\\beta }} \\rbrace = \\delta ^{\\dot{\\alpha }}_{\\dot{\\beta }}$ and are realized as matrices in the following way: $\\psi _{\\dot{1}} = \\frac{-\\gamma _0 + i\\gamma _3}{2},\\quad \\quad \\bar{\\psi }^{\\dot{1}} = \\frac{\\gamma _0 + i\\gamma _3}{2}, \\nonumber \\ \\\\\\psi _{\\dot{2}} = \\frac{\\gamma _2 + i\\gamma _1}{2},\\quad \\quad \\bar{\\psi }^{\\dot{2}} = \\frac{-\\gamma _2 + i\\gamma _1}{2}.\\ $ Then two complex supercharges () are expressed in a very simple way, namely $\\begin{array}{c}Q_\\alpha = \\left(\\sigma _\\mu \\bar{\\psi }\\right)_\\alpha \\left(\\hat{p}_\\mu -{\\cal A}_\\mu \\right),\\\\[3mm]\\bar{Q}^\\alpha = \\left(\\psi \\sigma ^\\dagger _\\mu \\right)^\\alpha \\left(\\hat{p}_\\mu -{\\cal A}_\\mu \\right),\\end{array}$ with $\\hat{p}_\\mu = -i\\partial _\\mu $ .", "The Hamiltonian (REF ) is expressed in these terms as $H = \\frac{1}{2} \\left(\\hat{p}_\\mu -{\\cal A}_\\mu \\right)^2+ \\frac{i}{4}\\, {\\cal F}_{\\mu \\nu }\\,\\psi \\sigma ^\\dagger _{\\mu } \\sigma _{\\nu }\\bar{\\psi }\\ .$ It is clear now why the spinor indices in Eqs.", "(REF ) are undotted, while in Eq.", "(REF ) they are dotted: the supercharges are rotated by the first ${\\rm SU}(2)$ and the variables $\\psi _{\\dot{\\alpha }}$ by the second Note that complex conjugation leaves the spinors in the same representation, the symmetry group here is ${\\rm SO}(4)$ rather than ${\\rm SO}(1,3)$ .. A careful distinction between two different ${\\rm SU}(2)$ factors allows one to understand better the reason why the supercharges (REF ) satisfy the simple algebra (REF ) in a self-dual background.", "The self-dual field density ${\\cal F}$ carries in the spinor notation only dotted indices.", "Therefore any expression involving ${\\cal F}, \\psi , \\bar{\\psi }$ is a scalar with respect to undotted ${\\rm SU}(2)$ .", "The only such scalar that can appear in the right hand side of the anticommutators of the supercharges $\\lbrace Q_\\alpha , \\bar{Q}^\\beta \\rbrace $ is the structure which is proportional to $\\delta _\\alpha ^\\beta $ , i.e.", "the Hamiltonian.", "No other operator is allowed.", "In the Abelian case, the supercharges (REF ) and the Hamiltonian (REF ) are scalar operators not carrying matrix indices anymore.", "This allows one to derive the Lagrangian, $L =\\frac{1}{2}\\, \\dot{x}_\\mu \\dot{x}_\\mu +{\\cal A}_\\mu (x)\\dot{x}_\\mu +i{\\bar{\\psi }}^{\\dot{\\alpha }} \\dot{\\psi }_{\\dot{\\alpha }}-\\frac{i}{4} {\\cal F}_{\\mu \\nu }\\,\\psi \\sigma ^\\dagger _{\\mu }\\sigma _{\\nu }\\bar{\\psi }\\ .$ In the non-Abelian case, the expressions (REF ) and (REF ) still keep their color matrix structure, and one cannot derive the Lagrangian in a so straightforward way.", "One of the ways to handle the matrix structure is to introduce a set of color fermion variables (say, in the fundamental representation of the group) and impose the extra constraint considering only the sector with unit fermion charge [9].", "An alternative non-Abelian construction of the Lagrangian is presented in Section REF .", "In this section, we limit ourselves only to Lagrangians for Abelian fields.", "As will be demonstrated explicitly in Section REF , the component Lagrangian (REF ) follows from the superfield action written earlier by Ivanov and Lechtenfeld in the framework of harmonic superspace approach [6].", "We will see that one can naturally derive in this way a $\\sigma $ -model type generalization of the Lagrangian (REF ) describing the motion over the manifold with nontrivial conformally flat metric $ds^2 = \\left\\lbrace f(x)\\right\\rbrace ^{-2} dx_\\mu dx_\\mu $ .", "It is written as follows: $L \\ =\\ \\frac{1}{2}f^{-2}\\, \\dot{x}_\\mu \\dot{x}_\\mu +i{\\bar{\\psi }}^{\\dot{\\alpha }} \\dot{\\psi }_{\\dot{\\alpha }}+\\frac{1}{4} \\left\\lbrace 3\\left(\\partial _\\mu f\\right)^2- f\\partial ^2 f \\right\\rbrace \\psi ^4+\\frac{i}{2} f^{-1}\\partial _\\mu f \\,\\dot{x}_\\nu \\,\\psi \\sigma ^\\dagger _{[\\mu }\\sigma _{\\nu ]} \\bar{\\psi }\\\\[3mm]+{\\cal A}_\\mu (x)\\dot{x}^\\mu - \\frac{i}{4}f^2 {\\cal F}_{\\mu \\nu }\\,\\psi \\sigma ^\\dagger _{\\mu }\\sigma _{\\nu }\\bar{\\psi },$ The corresponding quantum Nöether supercharges and the Hamiltonian are (see also Section REF for details) $\\begin{array}{l}Q_\\alpha = f \\left(\\sigma _\\mu \\bar{\\psi }\\right)_\\alpha \\left(\\hat{p}_\\mu -{\\cal A}_\\mu \\right)-\\psi _{\\dot{\\gamma }} \\bar{\\psi }^{\\dot{\\gamma }} \\left(\\sigma _\\mu \\bar{\\psi }\\right)_\\alpha i\\partial _\\mu f,\\\\[2mm]\\bar{Q}^\\alpha = \\left(\\psi \\sigma ^\\dagger _\\mu \\right)^\\alpha \\left(\\hat{p}_\\mu -{\\cal A}_\\mu \\right)f+i\\partial _\\mu f \\left(\\psi \\sigma ^\\dagger _\\mu \\right)^\\alpha \\cdot \\psi _{\\dot{\\gamma }}\\bar{\\psi }^{\\dot{\\gamma }},\\end{array}$ $H=\\frac{1}{2}f \\left(\\hat{p}_\\mu -{\\cal A}_\\mu \\right)^2 f+\\frac{i}{4}f^2\\, {\\cal F}_{\\mu \\nu }\\,\\psi \\sigma ^\\dagger _{\\mu }\\sigma _{\\nu }\\bar{\\psi }\\\\[2mm]- \\frac{1}{2} f i\\partial _\\mu f \\,(\\hat{p}_\\nu -{\\cal A}_\\nu )\\, \\psi \\sigma ^\\dagger _{[\\mu } \\sigma _{\\nu ]}\\bar{\\psi }+ f\\partial ^2 f\\left\\lbrace \\psi _{\\dot{\\gamma }}\\bar{\\psi }^{\\dot{\\gamma }}-\\frac{1}{2}\\left(\\psi _{\\dot{\\gamma }} \\bar{\\psi }^{\\dot{\\gamma }}\\right)^2\\right\\rbrace .$ On the other hand, one can explicitly calculate the anticommutators of the supercharges (REF ) for any self-dual Anti-self-duality conditions are obtained when one interchanges $\\sigma _\\mu $ and $\\sigma ^\\dagger _\\mu $ in all the formulas.", "This is equivalent to the interchange of two spinor representations of SO(4).", "field ${\\cal A}_\\mu (x)$ , Abelian or non-Abelian, and verify that the algebra (REF ) holds.", "While doing this, the use of the following Fierz identity $\\big (\\bar{\\psi }\\sigma ^\\dagger _\\mu \\big )^\\beta \\big (\\sigma _{\\nu }\\psi \\big )_\\alpha - \\big (\\sigma _{\\mu }\\bar{\\psi }\\big )_\\alpha \\big (\\psi \\sigma ^\\dagger _{\\nu }\\big )^\\beta =\\delta ^\\beta _\\alpha \\,\\bar{\\psi }\\sigma ^\\dagger _\\mu \\sigma _\\nu \\psi \\ ,$ which can be proven using (REF ), is convenient.", "Note that, with a nontrivial factor $f(x)$ , the Dirac operator $\\,/\\!\\!\\!\\!", "{ in a conformally flat background can be expressed as alinear combination of Q_\\alpha and \\bar{Q}^\\alpha {\\em only} if one also adds a certain {\\em torsion} proportional to the derivatives of f(x) \\cite {arXiv:1107.1429}.", "The Hamiltonian (\\ref {eq_susyham}) would also coincide with \\,/\\!\\!\\!\\!", "{ ^2/2 in this case.In fact, generalization of the system with the Hamiltonian~(\\ref {Hflatcherezpsi}) to the conformally flat case, which preserves {\\cal N}=4 supersymmetry, as in Eq.~(\\ref {eq_susyham}) always involves the torsion field.", "The Hamiltonian~(\\ref {Hflatcherezpsi}) also admits another type of {\\cal N}=4 extension to a curved space endowed with Hyper-Kähler metric and without a torsion \\cite {Kirchberg:2004za}.", "In its most generality, the Hamiltonian involves weak HKT geometry -- the generalization of Hyper-Kähler metric obtained by introduction of conformal factor and torsion~\\cite {arXiv:1107.1429}.", "}}The model (\\ref {eq_action}), (\\ref {eq_Qf}), (\\ref {eq_susyham}) is a close relative to the model constructed inRef.~\\cite {Smilga:1986rb} (see Eqs.~(30) and~(31) there), which describes the motion on a {\\it three}-dimensional conformallyflat manifold in external magnetic field and a scalar potential.", "In fact, the latter modelcan be obtained from the former, if assuming that the metricand the vector potential $ A(, A)$ depend only on three spatial coordinates $ xi$.If assuming further that the metric is flat, one is led to the Hamiltonian \\cite {de Crombrugghe:1982un}\\begin{equation}H \\ = \\ \\frac{1}{2}\\left(\\hat{\\vec{p}}-\\vec{{\\cal A}}\\right)^2+\\frac{1}{2}U^2+\\vec{\\nabla }U \\,\\psi \\vec{\\sigma }\\bar{\\psi },\\end{equation}which is supersymmetric under the condition $ Fij = ijk k U $ (the 3-dimensional reduction of the four-dimensional self-duality condition).", "It was noticed in Ref.~\\cite {Smilga:1986rb}that the effective Hamiltonian of a chiral supersymmetric electrodynamicsin finite spatial volume belongs to this class with $ U 1/|A|$.The vector potential $ A(A)$ describesin this case a Dirac magneticmonopole such that the Berry phase appears.", "The three dynamical variables $ A$ (do not confuse with curly $ A$ !", ")have in this case the meaning ofthe zero Fourier harmonic of the vectorpotential in the original field theory.", "In the leading order, the metric is flat.When higher loop corrections are included, a (conformally flat !)", "metric on the moduli space $ {A}$ appears.$ Performing the Hamiltonian reduction of Eq.", "(REF ) with non-Abelian ${\\cal A}_\\mu $ , a non-Abelian generalization of Eq.", "() can easily be derived.", "It keeps the gauge structure of Eq.", "() with matrix-valued $\\vec{{\\cal A}}$ and $U$ satisfying the condition ${\\cal F}_{ij}=\\varepsilon _{ijk}k U$ .", "Note that such Hamiltonian does not coincide with the non-Abelian 3-dimensional Hamiltonian derived in Ref. [21].", "As an illustration, consider the system described by the Hamiltonian (REF ) in a constant self-dual Abelian background.", "The constant self-dual field strength ${\\cal F}_{\\mu \\nu }= \\eta _{\\mu \\nu }^a B_a $ is parametrized by three independent components.", "Let us direct $B^a$ along the third axis, $B_a=(0,0,B)$ , and choose the gauge ${\\cal A}_0=Bx_3,\\quad \\quad {\\cal A}_2=Bx_1,\\quad \\quad {\\cal A}_1={\\cal A}_3=0.$ The Hamiltonian (REF ) acquires the form $H=\\left\\lbrace \\frac{1}{2} \\left(\\hat{p}_0-Bx_3\\right)^2+\\frac{1}{2} \\hat{p}_3^2+B \\left(\\chi _1\\bar{\\chi }^1-\\frac{1}{2}\\right)\\right\\rbrace \\\\+\\left\\lbrace \\frac{1}{2} \\left(\\hat{p}_2-Bx_1\\right)^2+\\frac{1}{2} \\hat{p}_1^2+B \\left(\\chi _2\\bar{\\chi }^2-\\frac{1}{2}\\right)\\right\\rbrace .$ For convenience, we have introduced notations $\\chi _1=\\bar{\\psi }^{\\dot{1}}$ , $\\bar{\\chi }^1=\\psi _{\\dot{1}}$ , $\\chi _2=\\psi _{\\dot{2}}$ , $\\bar{\\chi }^2=\\bar{\\psi }^{\\dot{2}}$ .", "The Hamiltonian is thus reduced to the sum $H_1 + H_2$ of two independent (acting in different Hilbert spaces) supersymmetric Hamiltonians, each describing the 2-dimensional motion of an electron in homogeneous orthogonal to the plane magnetic field $\\vec{B}$ .", "The bosonic sector of each such Hamiltonian corresponds to the spin projection $\\vec{s} \\vec{B}/|\\vec{B}| = -1/2$ , and the fermionic sector to the spin projection $\\vec{s} \\vec{B}/|\\vec{B}| = 1/2$ .", "This is the first and the simplest supersymmetric quantum problem ever considered [47].", "The energy levels for each Hamiltonian are $\\varepsilon _i=B\\left(n_i+\\frac{1}{2}+s_i\\right)$ , $n_i\\ge 0$ – integers, $s_i=\\pm \\frac{1}{2}$ .", "Each level of $H_i$ is doubly degenerate.", "Besides, there is an infinite degeneracy associated with the positions of the center of the orbit along the axes 1 and 3 that are proportional to the integrals of motion $p_2$ and $p_0$ .", "The full spectrum $E=B\\left(n_1+n_2+1+s_1+s_2\\right)$ is thus 4-fold degenerate at each level (except for the singlet state with $E=0$ ) for given $p_0$ , $p_2$ .", "It might be instructive to explicitly associate this degeneracy with the action of supercharges (REF ).", "Let us assume for definiteness $B > 0$ .", "One can represent $Q_\\alpha $ as $Q_1=\\sqrt{2B}\\left(b\\chi _1+a^\\dagger \\bar{\\chi }^2\\right),\\hspace{28.45274pt}Q_2=\\sqrt{2B}\\left(a\\chi _1-b^\\dagger \\bar{\\chi }^2\\right)\\ ,$ where $a^\\dagger $ , $b^\\dagger $ and $a$ , $b$ are the creation and annihilation operators, $a=\\frac{1}{\\sqrt{2B}}\\left(\\hat{p}_1 - iBx_1 \\ + i p_2 \\right),\\quad \\quad b=\\frac{1}{\\sqrt{2B}}\\left(\\hat{p}_3 - iBx_3 \\ + i p_0\\right)\\ ,$ $\\left[a,a^\\dagger \\right]=1,\\quad \\quad \\left[b,b^\\dagger \\right]=1.$ In these notations, the Hamiltonian (REF ) takes a very simple form $H=B\\left\\lbrace a^\\dagger a+b^\\dagger b+\\chi _1\\bar{\\chi }^1+\\chi _2\\bar{\\chi }^2\\right\\rbrace .$ Obviously, the energy levels of the Hamiltonian (REF ) are defined by two integrals of motion $p_{2,0}$ , two oscillator excitation numbers $n_{1,2}$ and two spins $s_{1,2}$ , as in Eq.", "(REF ).", "For each $p_2, p_0$ , there is a unique ground zero energy state $|0 \\rangle $ annihilated by all supercharges.", "A quartet of excited states can be represented as $\\left| n_1,n_2 \\right>,\\quad Q_1^\\dagger \\left| n_1,n_2 \\right>,\\quad Q_2^\\dagger \\left| n_1,n_2 \\right>,\\quad Q_1^\\dagger Q_2^\\dagger \\left| n_1,n_2 \\right> \\ ,$ where the state $ \\left| n_1,n_2 \\right>\\equiv \\chi _1\\cdot \\left(a^\\dagger \\right)^{n_1}\\left(b^\\dagger \\right)^{n_2}\\left| 0 \\right>$ of energy $E=B(n_1+n_2+1)$ is annihilated by both $Q_1$ and $Q_2$ .", "For each $p_2, p_0$ , there are $N$ such quartets at the energy level $E = BN$ .", "In this section, we derive the Hamiltonian (REF ) from the harmonic superspace approach.", "The introduction to the harmonic superspace and its salient features and definitions in application to quantum mechanical problems were already discussed in the previous chapter.", "The relevant superfield action was written in [6].", "Let us show here that the corresponding component Lagrangian coincides with (REF ).", "The corresponding supercharges (REF ) and the Hamiltonian (REF ) involve an Abelian self-dual gauge field ${\\cal A}_\\mu (x)$ .", "The non-Abelian field case is discussed later in this chapter.", "Let us introduce a doublet of superfields $q^{+{\\dot{\\alpha }}}$ with charge +1 ($D^0 q^{+\\dot{\\alpha }} = q^{+\\dot{\\alpha }}$ ) satisfying the constraints (REF ), (REF ).", "The index ${\\dot{\\alpha }}$ is the fundamental representation index of an additional external (Pauli-Gürsey) group ${\\rm SU}(2)$ .", "The solution for these constraints in the analytical basis was written in Eqs.", "(REF ), (REF ).", "It can be presented in the central basis (REF ) as $q^{+{\\dot{\\alpha }}}=u^+_\\alpha q^{\\alpha {\\dot{\\alpha }}}$ , where $q^{\\alpha {\\dot{\\alpha }}}$ does not depend on $u^\\pm _\\alpha $ (the latter follows from the constraint $D^{++} q^{+{\\dot{\\alpha }}} = 0$ and the definition $D^{++} = u^+_\\alpha \\frac{\\partial }{\\partial u^-_\\alpha }$ ).", "It is convenient to go over to the four-dimensional vector notation (REF ), introducing $q_\\mu = -\\frac{1}{2}\\left(\\sigma _\\mu \\right)_{\\alpha {\\dot{\\alpha }}} \\, q^{\\alpha {\\dot{\\alpha }}},\\quad \\quad q^{+{\\dot{\\alpha }}}=-q_\\mu \\left(\\sigma ^\\dagger _\\mu \\right)^{{\\dot{\\alpha }} \\alpha }u_\\alpha ^+.$ Now, $q_\\mu $ is a vector with respect to the group ${\\rm SO}(4)={\\rm SU_{\\rm R}}(2)\\times {\\rm SU_{\\rm PG}}(2)$ , with the first factor representing the ${\\cal N}=4$ R-symmetry group and the second one being the Pauli-Gürsey global ${\\rm SU}(2)$ group which rotates the dotted “flavor” indices.", "The pseudoreality condition (REF ) implies that the superfield $q_\\mu $ is real.", "The latter is expressed in components as follows: $q_\\mu =x_\\mu +\\theta \\sigma _\\mu \\chi +\\bar{\\theta }\\sigma _\\mu \\bar{\\chi }-\\frac{i}{2}\\dot{x}_\\nu \\, \\bar{\\theta }\\sigma _{[\\mu }\\sigma ^\\dagger _{\\nu ]}\\theta +\\frac{i}{2}\\bar{\\theta }\\sigma _\\mu \\dot{\\chi }\\,\\theta ^2-\\frac{i}{2}\\theta \\sigma _\\mu \\dot{\\bar{\\chi }} \\,\\bar{\\theta }^2-\\frac{1}{4}\\ddot{x}_\\mu \\,\\theta ^4 \\ ,$ where $\\theta ^2 \\equiv \\theta ^\\alpha \\theta _\\alpha $ , $\\bar{\\theta }^2 \\equiv \\bar{\\theta }^\\alpha \\bar{\\theta }_\\alpha $ , $\\theta ^4 \\equiv \\theta ^2\\bar{\\theta }^2$ .", "Moreover, the first equality in Eq.", "(REF ) implies that $x_\\mu = -\\frac{1}{2}x^{\\alpha \\dot{\\alpha }} (\\sigma _\\mu )_{\\alpha \\dot{\\alpha }}$ is also real, and we are left with four dynamic bosonic variables.", "The classical ${\\cal N}=2$ supersymmetric action for the superfield $q_\\mu $ can now be written.", "It consists of two parts, $S=S_{\\rm kin}+S_{\\rm int}$ .", "The kinetic part, $S_{\\rm kin}=\\int dt\\, d^4\\theta du\\, R^{\\prime }_{\\rm kin}(q^{+{\\dot{\\alpha }}}, q^{-{\\dot{\\beta }}}, u^\\pm _\\gamma )=\\int dt\\,d^4\\theta \\, R_{\\rm kin}(q_\\mu ),$ depends on an arbitrary function $R_{\\rm kin}(q_\\mu )$ .", "Note that one can forget the harmonic superspace coordinates here and work in an ordinary superspace.", "In other words, the kinetic term $S_{\\rm kin}$ does not require the additional coordinates $u^\\pm _\\alpha $ of the harmonic superspace.", "Plugging (REF ) into (REF ) and adding/subtracting proper total derivatives, one obtains $S_{\\rm kin}=\\int dt\\left\\lbrace \\frac{1}{2}g(x)\\, \\dot{x}_\\mu \\dot{x}_\\mu +\\frac{i}{2}g(x)\\left(\\bar{\\chi }^{\\dot{\\alpha }}\\dot{\\chi }_{\\dot{\\alpha }}-\\dot{\\bar{\\chi }}^{\\dot{\\alpha }}\\chi _{\\dot{\\alpha }}\\right)\\right.\\\\[3mm]\\left.+\\frac{1}{8}\\partial ^2 g(x) \\,\\chi ^4-\\frac{i}{4}\\partial _\\mu g(x) \\,\\dot{x}_\\nu \\,\\chi \\sigma ^\\dagger _{[\\mu }\\sigma _{\\nu ]}\\bar{\\chi }\\right\\rbrace ,$ where $g(x)=\\frac{1}{2}\\partial ^2_x R_{\\rm kin}(x)$ and $\\chi ^4=\\chi ^{\\dot{\\alpha }}\\chi _{\\dot{\\alpha }}\\,\\bar{\\chi }^{\\dot{\\beta }}\\bar{\\chi }_{\\dot{\\beta }}$ .", "To couple $x_\\mu $ to an external gauge field, one should add the interaction term $S_{\\rm int}$ that represents an integral over analytic superspace in the harmonic superspace, $S_{\\rm int}=\\int dt\\, du\\, d\\bar{\\theta }^+ d\\theta ^+ R_{\\rm int}^{++}\\left(q^{+{\\dot{\\alpha }}}(t_{\\rm A},\\theta ^+,\\bar{\\theta }^+),u^\\pm _\\gamma \\right).$ We choose $R_{\\rm int}^{++}$ (it carries the charge +2, $D^0 R_{\\rm int}^{++}=2R_{\\rm int}^{++}$ ) satisfying the condition $\\widetilde{R_{\\rm int}^{++}} = - R_{\\rm int}^{++}$ (the involution operation $\\widetilde{X}$ was defined in Section REF ) such that the action (REF ) is real.", "In contrast to the kinetic term, the interaction term involves the dependence on harmonics $u^\\pm _\\gamma $ and thus cannot be written in terms of superfields of ordinary superspace (REF ).", "To do the integral over $\\theta ^+$ and $\\bar{\\theta }^+$ , we substitute Eq.", "(REF ) into (REF ) and expand the latter in Taylor series over $\\theta ^+$ , $\\bar{\\theta }^+$ , keeping only terms $\\sim \\theta ^+\\bar{\\theta }^+$ : $R_{\\rm int}^{++}(q^{+\\dot{\\alpha }}, u^\\pm _\\gamma )=\\partial _{+\\dot{\\alpha }} R_{\\rm int}^{++}\\cdot \\left(-2i\\theta ^+\\bar{\\theta }^+ u^-_\\alpha \\dot{x}^{\\alpha \\dot{\\alpha }}\\right)\\\\[2mm]+2\\partial _{+\\dot{\\alpha }}\\partial _{+\\dot{\\beta }}R_{\\rm int}^{++}\\cdot \\theta ^+\\bar{\\theta }^+\\left(\\chi ^{\\dot{\\alpha }}\\bar{\\chi }^{\\dot{\\beta }}+\\chi ^{\\dot{\\beta }}\\bar{\\chi }^{\\dot{\\alpha }}\\right)+\\dots $ (ellipsis denote terms not proportional to $\\theta ^+\\bar{\\theta }^+$ ) with $\\partial _{+\\dot{\\alpha }} R_{\\rm int}^{++} (x,u)\\equiv \\frac{\\partial R_{\\rm int}^{++} (x^{+{\\dot{\\gamma }}}, u^\\pm _\\gamma )}{\\partial x^{+{\\dot{\\alpha }}}}$ and similarly for $\\partial _{+\\dot{\\alpha }}\\partial _{+\\dot{\\beta }}R_{\\rm int}^{++}$ .", "Let us also pass to vector notation $x_\\mu $ for the coordinates $x^{\\alpha {\\dot{\\alpha }}}$ , see Eq.", "(REF ).", "Consequently, $x^{+{\\dot{\\alpha }}}\\equiv x^{\\alpha \\dot{\\alpha }}u^+_\\alpha =-x_\\mu \\left(\\sigma ^\\dagger _\\mu \\right)^{{\\dot{\\alpha }} \\alpha }u^+_\\alpha .$ Then $S_{\\rm int}=\\int dt\\,du\\left\\lbrace 2i\\left(\\sigma ^\\dagger _\\mu \\right)^{{\\dot{\\alpha }} \\alpha } \\partial _{+\\dot{\\alpha }} R_{\\rm int}^{++}\\, u_\\alpha ^- \\cdot \\dot{x}_\\mu -4\\chi ^{\\dot{\\alpha }}\\bar{\\chi }^{\\dot{\\beta }}\\,\\partial _{+\\dot{\\alpha }}\\partial _{+\\dot{\\beta }} R_{\\rm int}^{++}\\right\\rbrace .$ Now, define the gauge field, ${\\cal A}_\\mu (x) \\equiv \\int du\\left\\lbrace 2i\\left(\\sigma ^\\dagger _\\mu \\right)^{{\\dot{\\alpha }} \\alpha } \\partial _{+\\dot{\\alpha }} R_{\\rm int}^{++}\\, u_\\alpha ^-\\right\\rbrace .$ As the action (REF ) is real, the field ${\\cal A}_\\mu (x)$ is also real.", "It automatically has zero divergence, $\\partial _\\mu {\\cal A}_\\mu =0.$ The field strength is expressed as ${\\cal F}_{\\mu \\nu } = \\partial _\\mu {\\cal A}_\\nu -\\partial _\\nu {\\cal A}_\\mu = -2 \\eta _{\\mu \\nu }^a \\int du\\, \\partial _{+\\dot{\\alpha }}\\partial _{+\\dot{\\beta }} R_{\\rm int}^{++}\\,\\varepsilon ^{{\\dot{\\alpha }}{\\dot{\\gamma }}}\\left( \\sigma _a \\right)^{\\!", "{\\dot{\\beta }}}_{\\,\\,{\\dot{\\gamma }}}$ (the identities (REF ) were used).", "It is obviously self-dual because the 't Hooft symbols are self-dual, see Eq.", "(REF ).", "With the definitions (REF ) and (REF ) in hand, one can represent the interaction term (REF ) simply as $S_{\\rm int}=\\int dt\\left\\lbrace {\\cal A}_\\mu (x)\\dot{x}_\\mu -\\frac{i}{4}{\\cal F}_{\\mu \\nu }\\,\\chi \\sigma ^\\dagger _{\\mu }\\sigma _{\\nu }\\bar{\\chi }\\right\\rbrace .$ Finally, one can get rid of the factor $g(x)$ in the fermion kinetic term (REF ) by introducing canonically conjugated $\\psi _{\\dot{\\alpha }}=f^{-1}(x)\\chi _{\\dot{\\alpha }},\\quad \\quad \\bar{\\psi }^{\\dot{\\alpha }} = f^{-1}(x) \\bar{\\chi }^{\\dot{\\alpha }}$ with $f(x) = g^{-1/2}(x)\\equiv \\left[\\frac{1}{2}\\,\\partial ^2_\\mu R_{\\rm kin}(x)\\right]^{-1/2}.$ Adding the kinetic term in (REF ) to the interaction term (REF ), one can explicitly check that the Lagrangian $L = L_{\\rm kin}+L_{\\rm int}$ coincides, up to a total derivative, with (REF ).", "As was noticed, the field $A_\\mu $ naturally obtained in the HSS framework satisfies the constraint $\\partial _\\mu {\\cal A}_\\mu =0$ [6].", "This does not really impose a restriction, however, because gauge transformations of $A_\\mu $ that shift it by the gradient of an arbitrary function amount to adding a total derivative in the Lagrangian (REF ).", "By construction, the action with the Lagrangian (REF ) is invariant under the following supersymmetry transformations: $\\begin{array}{c}x_\\mu \\rightarrow x_\\mu +f\\epsilon \\sigma _\\mu \\psi +f\\bar{\\epsilon }\\sigma _\\mu \\bar{\\psi },\\\\[3mm]f\\psi _{\\dot{\\alpha }}\\rightarrow f\\psi _{\\dot{\\alpha }}+i\\dot{x}_\\mu \\left(\\bar{\\epsilon }\\sigma _\\mu \\right)_{{\\dot{\\alpha }}},\\\\[3mm]f\\bar{\\psi }^{\\dot{\\alpha }}\\rightarrow f\\bar{\\psi }^{\\dot{\\alpha }}-i\\dot{x}_\\mu \\left(\\sigma ^\\dagger _\\mu \\epsilon \\right)^{{\\dot{\\alpha }}}.\\end{array}$ The Nöether classical supercharges expressed in terms of $\\psi _{\\dot{\\alpha }}$ and $\\bar{\\psi }^{\\dot{\\alpha }}$ , $x_\\mu $ and their canonical momenta, $p_\\mu \\ =\\ f^{-2}\\dot{x}_\\mu + {\\cal A}_\\mu - \\frac{i}{2} f^{-1} \\partial _\\nu f\\,\\psi \\sigma ^\\dagger _{[\\mu } \\sigma _{\\nu ]} \\bar{\\psi }\\ ,$ are $\\begin{array}{ccl}Q_\\alpha &=& f \\left(\\sigma _\\mu \\bar{\\psi }\\right)_\\alpha \\left( p_\\mu -{\\cal A}_\\mu \\right)-i \\partial _\\mu f \\psi _{\\dot{\\gamma }} \\bar{\\psi }^{\\dot{\\gamma }} \\left(\\sigma _\\mu \\bar{\\psi }\\right)_\\alpha ,\\\\[2mm]\\bar{Q}^\\alpha &=& \\mbox{[complex conjugate]} .\\end{array}$ The quantization procedure of the corresponding classical Hamiltonian has order ambiguity problem for the bosonic operators $\\hat{p}_\\mu = -i\\partial _\\mu $ and $x^\\mu $ as well as for the fermionic operators $\\hat{\\psi }_{\\dot{\\alpha }}$ and $\\hat{\\bar{\\psi }}^{\\dot{\\alpha }}$ with anticommutation relations (REF ).", "(We temporary restore “hats” on fermionic operators in order to distinguish them from the corresponding classical anticommuting variables $\\psi _{\\dot{\\alpha }}$ and $\\bar{\\psi }^{\\dot{\\alpha }}$ .)", "One must thus define an ordering procedure in such a way that the supersymmetry algebra (REF ) would hold.", "The solution to this problem is known [41] and it prescribes to order the supercharges in a certain way, while the quantum Hamiltonian should be obtained from the anticommutator $\\left\\lbrace Q_\\alpha ,\\,\\bar{Q}^\\alpha \\right\\rbrace $ .", "It is prescribed by Ref.", "[41] to order the operators in the classical supercharges (REF ) with the so called Weyl ordering procedure: any product of operators must be substituted with its totally symmetrized expression, taking into account the commuting/anticommuting nature of the operators.", "For instance, the expression $x^1 x^2 p_3$ , upon quantization, becomes $x^1 x^2p_3 \\quad \\longrightarrow \\quad \\frac{1}{6}\\left(x^1 x^2 \\hat{p}_3+x^2 x^1 \\hat{p}_3 + x^1 \\hat{p}_3 x^2 + x^2 \\hat{p}_3 x^1+ \\hat{p}_3 x^1 x^2 + \\hat{p}_3 x^2 x^1\\right),$ while the expression $\\psi _{\\dot{\\gamma }}\\bar{\\psi }^{\\dot{\\gamma }}\\, \\bar{\\psi }^{\\dot{\\beta }}$ becomes $\\psi _{\\dot{\\gamma }}\\bar{\\psi }^{\\dot{\\gamma }}\\, \\bar{\\psi }^{\\dot{\\beta }}\\quad \\longrightarrow \\quad \\frac{1}{6} \\left(\\hat{\\psi }_{\\dot{\\gamma }}\\hat{\\bar{\\psi }}^{\\dot{\\gamma }}\\hat{\\bar{\\psi }}^{\\dot{\\beta }}+ (-1)\\hat{\\psi }_{\\dot{\\gamma }}\\hat{\\bar{\\psi }}^{\\dot{\\beta }}\\hat{\\bar{\\psi }}^{\\dot{\\gamma }}+(-1)\\hat{\\bar{\\psi }}^{\\dot{\\gamma }}\\hat{\\psi }_{\\dot{\\gamma }}\\hat{\\bar{\\psi }}^{\\dot{\\beta }}\\right.\\\\[2mm]\\left.+\\hat{\\bar{\\psi }}^{\\dot{\\beta }}\\hat{\\psi }_{\\dot{\\gamma }}\\hat{\\bar{\\psi }}^{\\dot{\\gamma }}+\\hat{\\bar{\\psi }}^{\\dot{\\gamma }}\\hat{\\bar{\\psi }}^{\\dot{\\beta }}\\hat{\\psi }_{\\dot{\\gamma }}+(-1)\\hat{\\bar{\\psi }}^{\\dot{\\beta }}\\hat{\\bar{\\psi }}^{\\dot{\\gamma }}\\hat{\\psi }_{\\dot{\\gamma }}\\right).$ Let us elaborate more on the ordering of the expression $f(x) p_\\mu $ .", "For this, consider Weyl ordering of the expression $x^n p$ , where $x$ is any of the coordinates $x^\\mu $ , while $p$ is the corresponding conjugated momentum which, upon quantization, becomes $\\hat{p} =-i\\partial /\\partial x$ .", "One has: $x^n p \\quad \\longrightarrow \\quad \\frac{1}{n+1}\\left(x^n \\hat{p} + x^{n-1} \\hat{p} x + x^{n-2} \\hat{p} x^2+\\dots + \\hat{p} x^n\\right)\\\\[2mm]=x^n\\hat{p} +\\frac{1}{n+1}x^{n-1} \\left(1+2+\\dots +n\\right)\\cdot \\left[\\hat{p},\\,x\\right]\\\\[2mm]=x^n \\hat{p} +\\frac{1}{2} n x^{n-1}\\left[\\hat{p},\\,\\hat{x}\\right]$ Thus, in fact, Weyl ordering of the product $f(x)p_\\mu $ gives a simple formula: $f(x)p_\\mu \\quad \\longrightarrow \\quad \\frac{1}{2} \\big [f(x)\\hat{p}_\\mu + \\hat{p}_\\mu \\, f(x)\\big ]=f(x)\\hat{p}_\\mu - \\frac{1}{2} i\\,\\partial _\\mu f(x).$ In the same way, the expression in Eq.", "(REF ) can be simplified which gives $\\psi _{\\dot{\\gamma }}\\bar{\\psi }^{\\dot{\\gamma }}\\, \\bar{\\psi }^{\\dot{\\beta }}\\quad \\longrightarrow \\quad \\hat{\\psi }_{\\dot{\\gamma }}\\hat{\\bar{\\psi }}^{\\dot{\\gamma }}\\,\\hat{\\bar{\\psi }}^{\\dot{\\beta }}-\\frac{1}{2} \\hat{\\bar{\\psi }}^{\\dot{\\beta }}.$ Finally, combining Eqs.", "(REF ), (REF ), one obtains: $fp_\\mu \\bar{\\psi }^{\\dot{\\beta }}- i\\partial _\\mu f\\, \\psi _{\\dot{\\gamma }}\\bar{\\psi }^{\\dot{\\gamma }}\\,\\bar{\\psi }^{\\dot{\\beta }}\\quad \\longrightarrow \\quad f\\hat{p}_\\mu \\hat{\\bar{\\psi }}^{\\dot{\\beta }}-i\\partial _\\mu f \\,\\hat{\\psi }_{\\dot{\\gamma }}\\hat{\\bar{\\psi }}^{\\dot{\\gamma }}\\,\\hat{\\bar{\\psi }}^{\\dot{\\beta }}.$ This gives the quantum supercharges (REF ).", "One can check that the anticommutator $\\lbrace Q_\\alpha , \\bar{Q}^\\alpha \\rbrace $ gives the quantum Hamiltonian (REF ).", "For a matrix-valued non-Abelian self-dual field ${\\cal A}_\\mu $ , the (scalar) Lagrangian cannot be straightforwardly derived from the Hamiltonian (REF ) by a Legendre transformation as it was done in the Abelian case, Eq.", "(REF ).", "Nevertheless, this can be done in the case of ${\\rm SU}(N)$ gauge group by introducing extra “semi-dynamical” fields $\\varphi _i$ in the fundamental representation of ${\\rm SU}(N)$ and the auxiliary ${\\rm U}(1)$ gauge field $B(t)$ .", "The second line in (REF ) is then generalized to $L_{\\rm int}^{{\\rm SU}(N)}=i\\bar{\\varphi }^i\\left(\\dot{\\varphi }_i+iB\\varphi _i\\right)+kB+{\\cal A}_\\mu ^a T^a \\,\\dot{x}_\\mu -\\frac{i}{4}f^2{\\cal F}_{\\mu \\nu }^a T^a\\,\\psi \\sigma _\\mu ^\\dagger \\sigma _\\nu \\bar{\\psi }$ with integer $k$ and $T^a=\\bar{\\varphi }^i\\left(t^a\\right)^{\\,\\, j}_{\\!", "i} \\varphi _j,$ $t^a$ being standard ${\\rm SU}(N)$ algebra generators.", "The interaction Lagrangian (REF ) possesses ${\\cal N}=4$ supersymmetry.", "The corresponding supersymmetry transformations are written below in Eqs.", "(REF ).", "It is not difficult to check that it is invariant with respect to the non-Abelian gauge transformations of the target space: $\\begin{array}{c}{\\cal A}_\\mu ^a\\, t^a\\rightarrow U^\\dagger {\\cal A}_\\mu ^a\\, t^a U+iU^\\dagger \\partial _\\mu U\\\\[2mm]\\varphi _i\\rightarrow \\left(U^\\dagger \\varphi \\right)_i,\\,\\,\\,\\,\\bar{\\varphi }^i\\rightarrow \\left(\\bar{\\varphi }U\\right)^i ,\\end{array}$ where $U(x)\\in {\\rm SU}(N)$ .", "In addition, the expression (REF ) is also invariant with respect to the following gauge transformations of auxiliary fields $B(t)$ and $\\varphi _i$ : $B(t) \\rightarrow \\ B(t) + \\frac{d \\alpha (t)}{dt}, \\quad \\quad \\varphi _i(t) \\rightarrow e^{-i\\alpha (t)} \\varphi _i (t) .$ It is not immediately clear how to extend the Abelian superfield description to a general non-Abelian case, i.e.", "to the gauge group ${\\rm SU}(N)\\,$ .", "We succeeded in constructing such a description for the particular case of ${\\rm SU}(2)$ self-dual or anti-self-dual gauge fields expressed in the form $ {\\cal A}^a_\\mu \\ =\\ -\\bar{\\eta }^a_{\\mu \\nu }\\partial _\\nu \\ln h(x) \\quad {\\rm or}\\quad {\\cal A}^a_\\mu \\ =\\ -\\eta ^a_{\\mu \\nu } \\partial _\\nu \\ln h(x)$ respectively, with harmonic function $h(x)$ , $\\partial _\\mu ^2 h(x) = {\\rm a\\ finite\\ sum\\ of \\ delta\\ functions}.$ This is the so called 't Hooft ansatz for a multi-instanton ${\\rm SU}(2)$ solution [16] with the 't Hooft symbols $\\eta ^a_{\\mu \\nu }$ defined previously in Eq.", "(REF ).", "If one takes the function $h(x)$ to be vanishing at $|x|\\rightarrow \\infty $ , then this function can be presented as the following sum over instantons: $h(x)=1+\\sum \\limits _I \\frac{c_I}{\\left(x^\\mu -b^{\\mu }_I\\right)^2}.$ It involves particular instanton positions $b_I^\\mu $ as well as the numbers $c_I$ associated with each instanton.", "Let us understand how the interaction Lagrangian (REF ) gives rise to the matrix Hamiltonian (REF ).", "This is achieved upon quantization of the auxiliary variables $\\varphi _\\alpha $ and $\\bar{\\varphi }^\\beta $ .", "We consider only the particular case of ${\\rm SU}(2)$ gauge group when the indices $i$ , $j$ for the auxiliary variables take only two values, 1 and 2, and are denoted as $\\alpha $ , $\\beta $ .", "See Section REF for a general discussion of ${\\rm SU}(N)$ gauge group.", "Observe that the variables $\\varphi _\\alpha $ enter the Lagrangian with only one time derivative.", "Thus, they are not full-fledged dynamic variables (like $x_\\mu $ ) and not auxiliary fields (like $B(t)$ field or $\\omega _{1,2}$ fields in Eq.", "(REF ), see below).", "They have a kind of intermediate nature.", "In the context of ${\\cal N}=4$ SQM models, such variables (together with their analytic superfield carriers $v^+, \\widetilde{v^+}$ , see Eqs.", "(REF ), (REF ) below) were introduced in [19], [20].", "See also [21] for a recent application.", "To understand better the nature of the auxiliary fields, perform the quantization.", "The canonical commutation relations following from the action (REF ) through the standard Dirac prescription are $[ \\varphi _\\alpha , \\bar{\\varphi }^\\beta ] = \\delta _\\alpha ^\\beta \\ , \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ [\\varphi _\\alpha , \\varphi _\\beta ] = [\\bar{\\varphi }^\\alpha , \\bar{\\varphi }^\\beta ] = 0 .$ The fact that $k$ must be integer leads to the finite representations of the operator algebra $\\varphi _\\alpha $ , $\\bar{\\varphi }^\\alpha $ .", "Indeed, consider the constraint $\\bar{\\varphi }^\\alpha \\varphi _\\alpha = k\\,, $ which follows from (REF ) by varying with respect to $B$ .", "All real positive values of $k$ are classically allowed.", "As we will shortly see, in the quantum theory, $k$ must be integer, but not necessary positive.", "Consider the case of positive values of the integer $k$ .", "In quantum theory, one can choose $\\varphi _\\alpha \\equiv \\partial /\\partial \\bar{\\varphi }^\\alpha $ and impose (REF ) on the wave functions: $\\bar{\\varphi }^\\alpha \\varphi _\\alpha \\Psi = \\bar{\\varphi }^\\alpha \\frac{\\partial }{\\partial \\bar{\\varphi }^\\alpha } \\Psi = k\\Psi \\ .$ In other words, the wave functions represent homogeneous polynomials of $\\bar{\\varphi }^\\alpha $ of (an integer) degree $k\\,$ .", "In the case $k<0$ the algebra (REF ) is the same, but one must choose $\\bar{\\varphi }^\\alpha = - \\partial /\\partial \\varphi _\\alpha $ and consider polynomials of $\\varphi _\\alpha $ of degree $|k|$ .", "The number of such (linearly independent) polynomials is $|k|+1$ .", "Moreover, it is also easy to see that the operators (REF ) (which enter the interaction Lagrangian (REF )) satisfy the following algebra: $[T^a, T^b] \\ =\\ i\\varepsilon ^{abc} T^c .$ In addition, assuming $k>0$ and taking into account (REF ), one derives $T^a T^a \\ =\\ \\frac{1}{4} \\left[ (\\bar{\\varphi }^\\alpha \\varphi _\\alpha )^2 + 2 (\\bar{\\varphi }^\\alpha \\varphi _\\alpha ) \\right] \\ =\\ \\frac{k}{2} \\left(\\frac{k}{2} +1\\right) .$ In other words, $T^a$ can be treated as the generators of ${\\rm SU}(2)$ in the representation of spin $k/2$ .", "This way of quantizing semi-dynamical variables $\\varphi _\\alpha , \\bar{\\varphi }^\\alpha $ was employed in Ref. [20].", "Alternatively, one could interpret $\\varphi _\\alpha , \\bar{\\varphi }^\\alpha $ with the constraint (REF ) as a kind of the target harmonic variables representing a sphere ${\\rm S}^2$ , solve (REF ) in terms of the stereographic projection coordinates and quantize the system (see, for example, Ref. [48]).", "A nice feature is that this gauge ${\\rm SU}(2)$ group is in fact the R-symmetry group of ${\\cal N}=4$ supersymmetry algebra.", "The crucial role of the constraint (REF ) is to restrict the space of quantum states of the considered model to the finite set of irreducible ${\\rm SU}(2)$ multiplets of fixed spins (e.g., of the spin $k/2$ in the bosonic sector).", "This is an essential difference of this approach from that employed, e.g., in [18] (and later in [21], [17]) where no analog of the constraints (REF ) and (REF ) was imposed, thus allowing for the space of states to involve an infinite number of ${\\rm SU}(2)$ multiplets of all spins.", "The quantization scheme which we follow here was earlier used in the SQM context in [19], [20] and can be traced back to the work [49].", "Let us restrict ourselves by the first two terms in the interaction Lagrangian (REF ).", "The action $S \\ =\\ \\int dt\\, \\Big [ i \\bar{\\varphi }^i (\\dot{\\varphi }_i + iB \\varphi _i) \\ + \\ kB \\Big ]$ much resembles the three-dimensional Chern-Simons action, $S_{{\\rm CS}} = \\kappa \\int \\left( A \\wedge dA - \\frac{2i}{3} A\\wedge A \\wedge A \\right) .$ In both systems, the canonical Hamiltonian is zero, the canonical momenta are algebraically expressed through coordinates, and the quantization consists in imposing certain second class constraints (for a nice review of the classical and quantum aspects of the Chern-Simons theory, see [50]).", "Another well-known feature of CS theory is the quantization of the coupling, $k_{{\\rm CS}} = 4\\pi \\kappa $ = integer.", "This follows from the requirement for the Euclidean path integral to be invariant with respect to large (topologically nontrivial) gauge transformations.", "As was mentioned above, in our case the coefficient $k$ is also quantized.", "This can be derived following a similar reasoning.", "Consider the Euclidean version of the action (REF ), where one changes the time $t$ to the Euclidean time $\\tau $ by $t=-i\\tau $ and regularize it in the infrared by putting it on a finite Euclidean interval $\\tau \\in (0,\\beta )$ and imposing the periodic boundary conditions.", "This is of course equivalent to do calculations at finite temperature $T = 1/\\beta $ .", "Notice first that the action (REF ) is invariant with respect to gauge transformations (REF ) which, in the Euclidean version of the theory, become $B(\\tau ) \\rightarrow \\ B(\\tau ) + i \\frac{d\\alpha (\\tau )}{d\\tau } ,\\quad \\quad \\varphi _i(\\tau ) \\rightarrow e^{-i\\alpha (\\tau )} \\varphi _i (\\tau ) .$ Let us discover topologically nontrivial gauge transformation in the Euclidean version of this theory with the periodic boundary conditions $B(\\beta ) = B(0), \\quad \\quad \\varphi _i(\\beta ) = \\varphi _i(0).$ The only admissible gauge transformations (REF ) are those which do not break these periodicity conditions.", "We see that the transformation with $\\alpha (\\tau ) = 2\\pi \\tau /\\beta $ is topologically nontrivial: it cannot be reduced to a chain of infinitesimal transformations.", "This transformation shifts the Euclidean version of the last term in the action (REF ) by an imaginary constant, $\\Delta S_{\\rm FI} = -2\\pi i k$ .", "The requirement that the Euclidean path integrals (involving the factor $e^{-S_{\\rm FI}}$ ) are not changed leads [49] to the quantization condition $k \\ =\\ {\\rm integer}$ Thus, a benign quantum theory can only be defined if this requirement is fulfilled.", "To construct the action involving non-Abelian gauge fields, introduce, as earlier, a doublet of superfields $q^{+{\\dot{\\alpha }}}$ with charge +1 satisfying the constraints (REF ), (REF ).", "On top of that, we introduce an analytic gauge superfield $V^{++}$ of charge +2 satisfying the constraints $D^+ V^{++} = \\bar{D}^+ V^{++} = 0 \\ ,\\quad \\quad V^{++} = \\widetilde{V^{++}}$ and the “matter” superfield $v^+$ of charge +1.", "The constraints it satisfies, $D^+ v^{+} =0,\\quad \\quad \\bar{D}^+ v^{+} = 0,\\quad \\quad (D^{++} +i V^{++} ) v^{ +} =0 \\, ,$ differ from (REF ) by the presence of the covariant harmonic derivative ${\\cal D}^{++} = D^{++} + iV^{++}$ [5].", "The constraint ${\\cal D}^{++} v^+ = 0$ is covariant with respect to gauge transformations $V^{++} \\ \\rightarrow \\ V^{++} + D^{++} \\Lambda ,\\quad \\quad v^+ \\ \\rightarrow \\ e^{-i\\Lambda } v^+ ,\\quad \\quad D^+\\Lambda =\\bar{D}^+\\Lambda =0\\ .$ We can use this gauge freedom to eliminate almost all components from $V^{++}$ and to present it as $V^{++} \\ =\\ 2i\\, \\theta ^+ \\bar{\\theta }^+ B,$ where the gauge field $B(t)$ is real.", "This is a one-dimensional counterpart of the familiar Wess-Zumino gauge in four-dimensional theories.", "Observe also that Eq.", "(REF ) is a remnant of gauge transformations (REF ), which survives in the Wess-Zumino gauge (REF ).", "Then the superfield $v^+$ is expressed in the analytical basis as $v^+=\\phi ^\\alpha u^+_\\alpha -2\\theta ^+\\omega _1-2\\bar{\\theta }^+\\bar{\\omega }_2-2i\\theta ^+\\bar{\\theta }^+ (\\dot{\\phi }^\\alpha + i B\\phi ^\\alpha ) u^-_\\alpha \\ ,$ from which it follows that $\\widetilde{v^+}= \\bar{\\phi }^\\alpha u^+_\\alpha -2\\theta ^+\\omega _2 + 2\\bar{\\theta }^+ \\bar{\\omega }_1-2i\\theta ^+\\bar{\\theta }^+ ({\\dot{\\bar{\\phi }}^\\alpha } - iB \\bar{\\phi }^\\alpha ) u^-_\\alpha \\, \\ $ with $\\bar{\\phi }^\\alpha = (\\phi _\\alpha )^*$ .", "Thus, the fields $\\phi _\\alpha $ and $\\bar{\\phi }^\\alpha $ carry nonzero opposite ${\\rm U}(1)$ charges associated with the auxiliary gauge field $B$ .", "The ${\\cal N}=4$ supersymmetry-invariant action consists of three parts, $S=S_{\\rm kin}+S_{\\rm int} + S_{\\rm FI}$ .", "The kinetic part is more convenient to express in the central basis $\\lbrace t,\\, \\theta _\\alpha ,\\, \\bar{\\theta }^\\beta \\rbrace $ .", "It has the same form as in Eq.", "(REF ) and its component expansion coincides with the first line in Eq.", "(REF ), where the same change of variables (REF ) and (REF ) is performed as in the Abelian case.", "The interaction part is taken as $S_{\\rm int}= -\\frac{1}{2} \\int \\, dt \\, du\\, d\\bar{\\theta }^+d\\theta ^+\\,K\\left(q^{+\\dot{\\alpha }}, u^\\pm _\\beta \\right) v^+ \\widetilde{v^+} ,$ where the condition $\\widetilde{K}=K$ is imposed to ensure the action to be real.", "Finally, we add the Fayet-Iliopoulos term $S_{\\rm FI} \\ =\\ -\\frac{i k}{2} \\int \\, dt \\, du \\, d\\bar{\\theta }^+ d\\theta ^+ \\, V^{++}\\ =\\ k\\int dt\\, B ,$ which is invariant under gauge transformations (REF ).", "Let us concentrate on the interaction part.", "It is convenient to introduce new variables $\\varphi _\\alpha \\ =\\ \\phi _\\alpha \\sqrt{h(x) }\\ ,$ where $h(x) = \\int du\\, K(x^{+\\dot{\\alpha }}, u^\\pm _\\beta )\\quad \\quad \\quad \\big (x^{+\\dot{\\alpha }}=x^{\\alpha \\dot{\\alpha }}u_\\alpha ^+\\big ),$ is a harmonic function We assume here that $h(x)>0$ .", "The case $h(x)<0$ is treated similarly, if one redefines $h(x)\\rightarrow -h(x).$.", "Indeed, $\\partial _\\mu ^2 h(x) = 4\\,\\varepsilon ^{\\dot{\\alpha }\\dot{\\beta }}\\int du\\,\\partial _{+\\dot{\\alpha }}\\partial _{-\\dot{\\beta }}K(x^{+\\dot{\\gamma }}, u^\\pm _\\beta ) = 0\\,.$ Substituting (REF ), (REF ) and (REF ) into (REF ) and eliminating the auxiliary fermionic degrees of freedom $\\omega _{1,2}$ , $\\bar{\\omega }_{1,2}$ by their algebraic equations of motion, we derive after some algebra $L_{\\rm int} \\ =\\ i \\bar{\\varphi }^\\alpha ({\\dot{\\varphi }}_\\alpha + iB \\varphi _\\alpha )- \\frac{1}{2} \\bar{\\varphi }^\\beta \\varphi _\\gamma \\left({\\cal A}_{\\alpha {\\dot{\\alpha }}} \\right)_{\\!\\beta }^{\\,\\,\\gamma } {\\dot{x}}^{\\alpha {\\dot{\\alpha }}}- \\frac{i}{4} \\left( {\\cal F}_{{\\dot{\\alpha }}{\\dot{\\beta }}}\\right)^{\\,\\,\\gamma }_{\\!\\beta }\\chi ^{\\dot{\\alpha }}\\bar{\\chi }^{\\dot{\\beta }} \\bar{\\varphi }^\\beta \\varphi _\\gamma \\,.", "$ Here $\\left({\\cal A}_{\\alpha {\\dot{\\alpha }}} \\right)_{\\!\\beta }^{\\,\\,\\gamma } = -\\frac{2i}{\\int du \\, K} \\int du \\, \\partial _{+\\dot{\\alpha }}K \\left( u^{+\\gamma }\\varepsilon _{\\alpha \\beta }- \\frac{1}{2}u^+_\\alpha \\delta ^\\gamma _\\beta \\right)=\\frac{i}{h}\\left(\\varepsilon _{\\alpha \\beta } \\, \\partial ^\\gamma _{\\ \\dot{\\alpha }} h - \\frac{1}{2}\\delta ^\\gamma _\\beta \\,\\partial _{\\alpha {\\dot{\\alpha }}}h \\right)$ ($\\partial _{\\alpha \\dot{\\alpha }}\\equiv \\left(\\sigma _\\mu \\right)_{\\alpha \\dot{\\alpha }}\\partial _\\mu =-2\\partial /\\partial x^{\\alpha \\dot{\\alpha }}$ ) is a Hermitian traceless matrix – the gauge field, and $\\left( {\\cal F}_{{\\dot{\\alpha }}{\\dot{\\beta }}} \\right)^{\\,\\,\\gamma }_{\\!\\beta }= \\left({\\cal F}_{\\mu \\nu }\\right)^{\\,\\,\\gamma }_{\\!\\beta }\\left(\\sigma _\\mu ^\\dagger \\sigma _\\nu \\right)_{\\dot{\\alpha }\\dot{\\beta }}=\\partial _{\\delta \\dot{\\alpha }}( {\\cal A}^{\\delta }_{\\ {\\dot{\\beta }}} )^{\\,\\,\\gamma }_{\\!\\beta }-i ( {\\cal A}_{\\delta {\\dot{\\alpha }} } )^{\\,\\,\\lambda }_{\\!\\beta }( {\\cal A}^{\\delta }_{\\ {\\dot{\\beta }} } )^{\\,\\,\\gamma }_{\\!\\lambda } \\ +({\\dot{\\alpha }} \\leftrightarrow {\\dot{\\beta }} )$ is its self-dual part.", "It is easy to check explicitly, that the anti-self-dual part of the gauge field ${\\cal A}_\\mu $ vanishes, $\\left({\\cal F}_{\\alpha \\beta }\\right)_{\\!\\gamma }^{\\,\\,\\delta }=\\left({\\cal F}_{\\mu \\nu }\\right)_{\\!\\gamma }^{\\,\\,\\delta } \\left(\\sigma _\\mu \\sigma _\\nu ^\\dagger \\right)_{\\alpha \\beta }=-\\partial _{\\alpha \\dot{\\alpha }}({\\cal A}_\\beta ^{\\ \\dot{\\alpha }})^{\\,\\,\\delta }_{\\!\\gamma }+i({\\cal A}_{\\alpha \\dot{\\alpha }})^{\\,\\,\\lambda }_{\\!\\gamma }({\\cal A}_\\beta ^{\\ \\dot{\\alpha }})^{\\,\\,\\delta }_{\\!\\lambda }+\\left(\\alpha \\leftrightarrow \\beta \\right)=0.$ Thus, the field strength ${\\cal F}_{\\mu \\nu }^a$ is self-dual and belongs to the representation $(0,1)$ of ${\\rm SO}(4)={\\rm SU}(2)\\times {\\rm SU}(2)$ .", "Passing to ${\\cal A}_\\mu ^a$ such that $\\left({\\cal A}_\\mu \\right)^{\\,\\,\\gamma }_{\\!\\beta }={\\cal A}_\\mu ^a \\left(\\sigma _a\\right)^{\\,\\,\\gamma }_{\\!\\beta }\\!\\!/2$ , we find that the representation (REF ) precisely amounts to the self-dual 't Hooft ansatz (REF ), left equation.", "The anti-self-dual expression from the right equation arises if one interchanges dotted and undotted indices, i.e.", "effectively interchanges $\\sigma _\\mu $ and $\\sigma ^\\dagger _\\mu $ .", "This also implies passing to the harmonics $u^\\pm _{\\dot{\\alpha }}$ and in fact to another ${\\cal N}=4$ supersymmetry, with the second SU(2) (acting on dotted indices) as the R-symmetry group.", "Finally, substituting $\\bar{\\varphi }^\\beta \\varphi _\\gamma =T^a\\left(\\sigma _a\\right)_{\\!\\gamma }^{\\,\\,\\beta }$ and $\\chi _{\\dot{\\alpha }}=f\\psi _{\\dot{\\alpha }}$ into (REF ), where $T^a$ is defined in (REF ) with $t^a=\\frac{1}{2}\\sigma _a$ , one convinces himself that the interaction term together with the FI term (REF ) yields just (REF ) for the SU(2) gauge group case, and the quantum Hamiltonian derived from the Lagrangian $L_{\\rm kin} + L_{\\rm int} + L_{\\rm FI}$ has the form (REF ) with ${\\cal A}_\\mu \\equiv {\\cal A}_\\mu ^a T^a$ and ${\\cal F}_{\\mu \\nu } \\equiv {\\cal F}_{\\mu \\nu }^a T^a $ .", "The full Lagrangian in the non-Abelian case representing the sum of Eq.", "(REF ) and the first line of Eq.", "(REF ) is invariant, up to a total derivative, with respect to ${\\cal N}=4$ supersymmetry transformations (in the infinitesimal form) of Eqs.", "(REF ) supplemented with transformations for the auxiliary fields: $\\begin{array}{c}\\varphi _i\\rightarrow \\varphi _i+i f\\left(t^a\\varphi \\right)_i {\\cal A}_\\mu ^a\\left(\\epsilon \\sigma _\\mu \\psi +\\bar{\\epsilon }\\sigma _\\mu \\bar{\\psi }\\right),\\\\[3mm]\\bar{\\varphi }^i\\rightarrow \\bar{\\varphi }^i-i f\\left(\\bar{\\varphi }t^a\\right)^i {\\cal A}_\\mu ^a\\left(\\epsilon \\sigma _\\mu \\psi +\\bar{\\epsilon }\\sigma _\\mu \\bar{\\psi }\\right).\\end{array}$ Note that the above formulas are written for the case of the gauge group ${\\rm SU}(N)$ when the semi-dynamical fields $\\varphi _i$ , $\\bar{\\varphi }^j$ belong to a fundamental representation of ${\\rm SU}(N)$ .", "We have constructed the superfield action for the ${\\cal N}=4$ supersymmetric quantum mechanics corresponding to the Hamiltonian (REF ) with a non-Abelian ${\\rm SU}(2)$ gauge field ${\\cal A}_\\mu $ which lives on a conformally flat 4-manifold and is representable in the 't Hooft ansatz form (REF ).", "As an example of such a field, let us quote the instanton solution on ${\\rm S}^4$ .", "Generically, it depends on the radius $R$ of the sphere and the instanton size $\\rho $ .", "The configurations of maximal size, $\\rho = R$ , present a particular interest.", "In the stereographic coordinates on ${\\rm S}^4$ , $ds^2 \\ =\\ \\frac{4R^4 dx_\\mu ^2}{(x^2 + R^2)^2}\\ ,$ they are expressed by the same formulas as the flat instantons in the singular gauge, ${\\cal A}_\\mu ^a \\ =\\ \\frac{2R^2 \\bar{\\eta }^a_{\\mu \\nu } x_\\nu }{x^2(x^2+ R^2)} \\quad \\mbox{or} \\quad ({\\cal A}_{\\alpha \\dot{\\alpha }})^{\\,\\,\\gamma }_{\\!\\beta } = - \\frac{2i \\,R^2}{x^2(x^2 + R^2)}\\left(\\varepsilon _{\\alpha \\beta }x^\\gamma _{\\dot{\\alpha }}- \\frac{1}{2}\\,\\delta ^\\gamma _\\beta \\,x_{\\alpha \\dot{\\alpha }} \\right),$ and $({\\cal F}_{\\dot{\\alpha }\\dot{\\beta }})^{\\,\\,\\gamma }_{\\!\\beta } = \\frac{8i \\,R^2}{x^2(x^2 + R^2)^2}\\left(x^\\gamma _{\\dot{\\beta }}x_{\\beta \\dot{\\alpha }}+ x^\\gamma _{\\dot{\\alpha }}x_{\\beta \\dot{\\beta }}\\right).", "$ The corresponding functions in Eq.", "(REF ) are taken in the form $K(x^{+\\dot{\\alpha }},u^\\pm _\\beta )=1+\\frac{1}{\\left(c^-_{\\dot{\\alpha }}x^{+\\dot{\\alpha }}\\right)^2}\\ ,\\quad \\quad h(x)\\equiv \\int du\\, K(x^{+\\dot{\\alpha }},u^\\pm _\\beta )=1+\\frac{R^2}{ x_\\mu ^2},$ where $c^-_{\\dot{\\alpha }}=c_{\\ \\dot{\\alpha }}^\\alpha u^-_\\alpha $ , $c^{\\alpha \\dot{\\alpha }}$ – constant vector and $R^2=1/c^2_\\mu $ .", "The integral on the right hand side of Eq.", "(REF ) can be calculated as the power series in $c^-_{\\dot{\\alpha }}c^{+\\dot{\\alpha }}=-c_\\mu ^2$ or directly after noting that the form of this integral is ${\\rm SO}(4)$ invariant and putting $c_\\mu =(c,0,0,0)$ , $x_\\mu =(x_1,x_2,0,0)$  Let us describe the latter possibility in more detail.", "If $c_\\mu $ and $x_\\mu $ are chosen as mentioned above, this gives $c_{\\dot{\\alpha }}^- x^{+{\\dot{\\alpha }}} = -c\\, x^0 + i c\\, x^1 \\left(u_1^+ u_1^- - u_2^+ u_2^-\\right).$ To calculate the integral in Eq.", "(REF ), one realizes the harmonics $u^\\pm _\\alpha $ in the familiar stereographic parametrization [5]: $\\left(\\begin{array}{ll}u_1^+ & u_1^-\\\\u_2^+ & u_2^-\\end{array}\\right)=\\frac{1}{\\sqrt{1+t \\bar{t}}}\\left(\\begin{array}{ll}e^{i\\psi } & -\\bar{t}\\, e^{-i\\psi }\\\\t\\, e^{i\\psi } & e^{-i\\psi }\\end{array}\\right),$ where $t\\in {\\mathbb {C}}$ , $0 \\le \\psi < 2\\pi $ .", "It is also necessary to define the measure of integration, $\\int du\\quad \\longrightarrow \\quad \\frac{i}{4\\pi ^2}\\int _0^{2\\pi } d\\psi \\int \\frac{dt\\, d\\bar{t}}{(1+t\\bar{t})^2} .$ After this, the computation of the integral $\\frac{i}{2\\pi }\\int \\frac{dt\\,d\\bar{t}}{(1+t\\bar{t})^2}\\left[1 + \\frac{(1+t\\bar{t})^2}{\\left(c\\,x^0 (1+t\\bar{t})+ic\\, x^1(t+\\bar{t})\\right)^2}\\right]$ gives (REF ).", ".", "Let us describe the latter possibility in more detail.", "If $c_\\mu $ and $x_\\mu $ are chosen as mentioned above, this gives $c_{\\dot{\\alpha }}^- x^{+{\\dot{\\alpha }}} = -c\\, x^0 + i c\\, x^1 \\left(u_1^+ u_1^- - u_2^+ u_2^-\\right).$ To calculate the integral in Eq.", "(REF ), one realizes the harmonics $u^\\pm _\\alpha $ in the familiar stereographic parametrization [5]: $\\left(\\begin{array}{ll}u_1^+ & u_1^-\\\\u_2^+ & u_2^-\\end{array}\\right)=\\frac{1}{\\sqrt{1+t \\bar{t}}}\\left(\\begin{array}{ll}e^{i\\psi } & -\\bar{t}\\, e^{-i\\psi }\\\\t\\, e^{i\\psi } & e^{-i\\psi }\\end{array}\\right),$ where $t\\in {\\mathbb {C}}$ , $0 \\le \\psi < 2\\pi $ .", "It is also necessary to define the measure of integration, $\\int du\\quad \\longrightarrow \\quad \\frac{i}{4\\pi ^2}\\int _0^{2\\pi } d\\psi \\int \\frac{dt\\, d\\bar{t}}{(1+t\\bar{t})^2} .$ After this, the computation of the integral $\\frac{i}{2\\pi }\\int \\frac{dt\\,d\\bar{t}}{(1+t\\bar{t})^2}\\left[1 + \\frac{(1+t\\bar{t})^2}{\\left(c\\,x^0 (1+t\\bar{t})+ic\\, x^1(t+\\bar{t})\\right)^2}\\right]$ gives (REF ).", "The field ${\\cal A}_\\mu ^a$ can be brought to the nonsingular gauge ${\\cal A}_\\mu ^a=\\frac{2\\eta _{\\mu \\nu }^a x_\\nu }{x^2+R^2}\\,, \\qquad {\\cal F}^a_{\\mu \\nu }\\ =\\ - \\frac{4R^2\\eta ^a_{\\mu \\nu }}{(x^2 + R^2)^2}\\,,$ by the gauge transformations (REF ) with $U(x)=-i\\sigma _\\mu x_\\mu /\\sqrt{x^2}$ (this particular $U(x)$ form is prompted by the form of the field strength (REF )).", "The action density $\\sim {\\cal F}_{\\mu \\nu }{\\cal F}^{\\mu \\nu }$ is the same in this case at all points of ${\\rm S}^4$ .", "It is worth noting that the singular gauge transformation converts the undotted gauge group indices into the dotted ones: the self-dual gauge potential and the field strength in the spinorial notation become $({\\cal A}_{\\alpha \\dot{\\alpha }})^{\\!\\dot{\\gamma }}_{\\,\\,\\dot{\\beta }} = \\frac{2i}{x^2 + R^2}\\left(\\varepsilon _{\\dot{\\alpha }\\dot{\\beta }}x^{\\dot{\\gamma }}_{\\alpha }- \\frac{1}{2}\\,\\delta ^{\\dot{\\gamma }}_{\\dot{\\beta }} \\,x_{\\alpha \\dot{\\alpha }} \\right),\\quad ({\\cal F}_{\\dot{\\alpha }\\dot{\\beta }})^{\\!\\dot{\\gamma }}_{\\,\\,\\dot{\\delta }} = -\\frac{8i \\,R^2}{(x^2 + R^2)^2}\\left(\\varepsilon _{\\dot{\\alpha }\\dot{\\delta }}\\delta ^{\\dot{\\gamma }}_{\\dot{\\beta }} + \\varepsilon _{\\dot{\\beta }\\dot{\\delta }}\\delta ^{\\dot{\\gamma }}_{\\dot{\\alpha }}\\right)$ and, also, $\\varphi _\\alpha \\rightarrow \\varphi ^{\\dot{\\alpha }} = -i\\varphi _\\alpha \\,x^{\\alpha \\dot{\\alpha }}/\\sqrt{x^2}$ , $\\bar{\\varphi }^\\alpha \\rightarrow \\bar{\\varphi }_{\\dot{\\alpha }} = -i\\bar{\\varphi }^\\alpha \\,x_{\\alpha \\dot{\\alpha }}/\\sqrt{x^2} \\,$ .", "Note that, the field (REF ), (REF ) describes the Yang monopole living in ${\\mathbb {R}}^5$ [51].", "The potential (REF ) has a nice group-theoretical meaning as one of the two SU(2) connections on the coset manifold ${\\rm SO}(5)/[{\\rm SU}(2)\\times {\\rm SU}(2)] \\sim {\\rm S}^4$ (see e.g.", "[52]).", "It coincides with the flat self-dual instanton only in the conformally flat parametrization of ${\\rm S}^4$ as in (REF ).", "When coupled to the world-line through our semi-dynamical variables $\\varphi _\\alpha , \\bar{\\varphi }^\\alpha $ , the 5-dimensional Yang monopole is reduced to this SU(2) connection defined on ${\\rm S}^4$ .", "Let us elaborate on this point in more detail, choosing, without loss of generality, $R =1$ in the above formulas.", "Consider the following $d=1$ bosonic Lagrangian with the ${\\mathbb {R}}^5$ target space and an additional coupling to Yang monopole $L_{\\,{\\mathbb {R}}^5} = \\frac{1}{2}\\left(\\dot{y}_5 \\dot{y}_5 + \\dot{y}_\\mu \\dot{y}_\\mu \\right) + {\\cal B}_\\mu ^{\\, a} (y) T^a \\,\\dot{y}_\\mu \\,.$ Here, ${\\cal B}_\\mu ^{\\,a} $ is the standard form of the Yang monopole in the ${\\mathbb {R}}^5$ coordinates, ${\\cal B}_\\mu ^{\\,a} = \\frac{\\eta _{\\mu \\nu }^a y_\\nu }{r(r + y_5)}\\,, \\quad r = \\sqrt{y_5^2 + y^2_\\mu }\\,,$ $T^a$ are defined as in (REF ) with $t^a = {\\textstyle \\frac{1}{2}}\\sigma ^a\\,$ , and the action for the semi-dynamical variables $\\varphi _\\alpha , \\bar{\\varphi }^{\\alpha }$ is omitted.", "Now one passes to the polar decomposition of ${\\mathbb {R}}^5$ into a radius $r$ and the angular part ${\\rm S}^4\\,$ , $(y_5, y_\\mu ) \\;\\rightarrow \\; (r, \\tilde{y}_5, \\tilde{y}_\\mu )\\,, \\; \\tilde{y}_5 = \\sqrt{1 - \\tilde{y}^2_\\mu }\\,$ , and rewrites (REF ) as $L_{\\,{\\mathbb {R}}^5} =\\frac{1}{2}\\dot{r}{}^2 + \\frac{1}{2} r^2\\left(\\dot{\\tilde{y}}_5 \\dot{\\tilde{y}}_5 + \\dot{\\tilde{y}}_\\mu \\dot{\\tilde{y}}_\\mu \\right)+ \\frac{\\eta _{\\mu \\nu }^a \\tilde{y}_\\nu \\dot{\\tilde{y}}_\\mu \\,T^a }{1 + \\sqrt{1 - \\tilde{y}^2_\\mu }} \\,.$ The coordinates $\\tilde{y}_\\mu $ give a particular parametrization of ${\\rm S}^4$ .", "Passing to the stereographic coordinates is accomplished by the redefinition $\\tilde{y}_\\mu = 2\\, \\frac{x_\\mu }{ 1 + x^2}\\,,$ which casts (REF ) into the form $L_{\\,{\\mathbb {R}}^5} =\\frac{1}{2}\\left\\lbrace \\dot{r}{}^2 + 4 r^2\\frac{\\dot{x}_\\mu \\dot{x}_\\mu }{(1 + x^2)^2}\\right\\rbrace + \\frac{2 \\eta _{\\mu \\nu }^a x_\\nu \\dot{x}_\\mu \\,T^a }{1 + x^2} \\,.$ One sees that the ${\\rm S}^4$ metric (REF ) (with $R =1$ ) and the instanton vector potential (REF ) appear.", "Thus, current approach, as a by-product, provides a solution to the long-standing problem of constructing ${\\cal N}=4$ supersymmetric quantum mechanics with Yang monopole (see e.g.", "[8] and references therein).", "Obviously, the component Lagrangian (REF ) (with the relevant function $f(x)$ ) is just the ${\\rm S}^4$ part of the Lagrangian (REF ) with the “frozen” radial variable $r = 1$ .", "Presumably, one can restore the full 5-dimensional kinetic part in (REF ) by adding a coupling to the appropriately constrained scalar ${\\cal N}=4$ zero-charge superfield $X(t,\\theta , \\bar{\\theta })$ which describes an off-shell multiplet $({\\bf 1, 4, 3})$ with one physical bosonic field [44], such that $X|_{\\theta = \\bar{\\theta }= 0} = r\\,$ .", "The problem of finding a superfield formulation for a generic ${\\rm SU}(N)$ self-dual field is more complicated and is still an open question.", "However, by introducing extra variables $\\varphi _i$ , it is always possible to write a component Lagrangian (REF ) (together with the first line in (REF )) corresponding to the matrix Hamiltonian (REF ).", "This observation has actually nothing to do with supersymmetry.", "It boils down to the following.", "Consider the eigenvalue problem for a usual Hermitian matrix $H_{jk}$ .", "It can be treated as a Schrödinger problem $\\hat{H} \\Psi (\\varphi _j) = \\lambda \\Psi (\\varphi _j)$ with the constraint $\\hat{G} \\Psi = 0$ , where $\\hat{H} \\ =\\ \\varphi _j H_{jk} \\frac{\\partial }{\\partial \\varphi _k} \\ , \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\hat{G} = \\varphi _j\\frac{\\partial }{\\partial \\varphi _j} - 1 .$ The corresponding Lagrangian is $L \\ =\\ i\\bar{\\varphi }_j \\dot{\\varphi }_j - B(\\bar{\\varphi }_j \\varphi _j - 1) - \\bar{\\varphi }_j H_{jk} \\varphi _k ,$ where $\\bar{\\varphi }_j=(\\varphi _j)^*$ .", "This easily generalizes to the case where $H$ is an operator depending on a set of canonically conjugated variables $\\lbrace p_\\mu , x_\\mu \\rbrace $ .", "The only difference is that $-H_{jk}$ is now replaced by the matrix $L_{jk}$ obtained from $H_{jk}$ by the appropriate Legendre transformation This elementary observation should be well known, for example, in matrix models.", "Surprisingly, it is not found it in such a “chemically pure” form in the literature, but similar constructions were discussed, e.g., in Refs.", "[49], [53]..", "The initial goal was to find a Lagrangian representation for the Hamiltonian (REF ) with matrix-valued ${\\cal A}_\\mu $ , ${\\cal F}_{\\mu \\nu }$ .", "The construction just described, with $\\varphi _i$ in the fundamental representation of ${\\rm SU}(N)$ , leads to the $N\\times N$ matrix Hamiltonian.", "The Lagrangian (REF ) coincides in this case with the Lagrangian (REF ) with the choice $k=1\\,$ , to which the first line from Eq.", "(REF ) is also added.", "Obviously, one can describe the Hamiltonians in higher representations of ${\\rm SU}(N)$ in a similar way, by choosing the number of components $\\varphi _i$ equal to the dimension of the representation.", "We have seen, however, that in the ${\\rm SU}(2)$ case one can be more economic, introducing only a couple of dynamic variables $\\varphi _\\alpha $ and multiplying the term proportional to $B$ in the Lagrangian by an arbitrary integer $k$ .", "This leads to the Hamiltonian in the representation of spin $|k|/2$ .", "Certain ${\\rm SU}(N)$ representations (namely, the symmetric products of $|k|$ fundamental or $|k|$ antifundamental representations) can also be attained in this way.", "One can also construct in this way a ${\\cal N} =2 $ supersymmetric Lagrangian for the Hamiltonian (REF ) with generic (not necessarily self-dual) ${\\cal A}_\\mu $ .", "A similar construction (but with extra fermionic rather than bosonic variables) was in fact discussed in Ref. [9].", "A beauty of the harmonic superspace approach explored here is, however, that such extra variables and the constraint (REF ) are not introduced by hand, but arise naturally from the manifestly off-shell supersymmetric superfield actions.", "The Hamiltonian (REF ) presents the generalization of the Hamiltonian (REF ) to the conformally flat metric case in four dimensions.", "We succeeded in the construction of this generalization using the superfield formalism.", "In this section, we employ similar construction and generalize the three-dimensional system described by with the Hamiltonian () to the conformally flat case.", "Although the resulting Hamiltonian, the supercharges and the component Lagrangian appear to be just the three-dimensional reduction of the four-dimensional counterpart, the superfield formalism in the three-dimensional case involves a different superfield for the space coordinates $x^i$ and is thus implemented differently.", "In this section, instead of the coordinate superfield $q^{+\\dot{\\alpha }}$ one deals with the analytic superfield $L^{++}$ which encompass the multiplet $(\\bf {3,4,1})$ and is subjected to the constraints (REF ).", "The superfield $V^{++}$ and the auxiliary superfields $v^+$ and $\\widetilde{v^+}$ are defined by the same Eqs.", "(REF ), (REF ).", "The superfield $V^{++}$ in the Wess-Zumino gauge (REF ) is expressed through one independent component $B(t)$ .", "We remind that $B(t)$ is a real one-dimensional “gauge field” which transforms as $B \\rightarrow B + \\dot{\\lambda }\\,$ , with $\\lambda (t)$ being the parameter of the residual gauge U(1) symmetry.", "The explicit expressions for the superfields $q^{+\\dot{\\alpha }}$ , $v^+$ , $\\widetilde{v^+}$ and $V^{++}$ are written in Eqs.", "(REF ), (REF ), (REF ) and (REF ) respectively.", "The component expansion of the analytic superfield $L^{++}$ can be found in Eqs.", "(REF ), (REF ).", "The multiplet $L^{++}$ involves the three-dimensional target space coordinates $\\ell ^{\\alpha \\beta } = \\ell ^{\\beta \\alpha }$ , their fermionic partners $\\chi ^\\alpha $ , $\\bar{\\chi }^\\alpha $ and a real auxiliary field $F$ .", "Let us remark that the three-dimensional case involves only one ${\\rm SU}(2)$ (R-symmetry) group and thus no dotted indices present in the description.", "Note also that the constraint $\\bar{\\chi }_\\alpha = \\left(\\chi ^\\alpha \\right)^*$ involves different position of spinor indices compared to Eq.", "(REF ) in the four-dimensional case (see Section REF below).", "The transition from the spinor notation $\\ell ^{\\alpha \\beta }$ to the vector notation $\\ell ^i$ , $\\ell _\\alpha ^\\beta =\\ell _i\\left(\\sigma _i\\right)_{\\!\\alpha }^{\\,\\,\\beta },\\quad \\quad \\ell _i=\\frac{1}{2} \\ell ^\\alpha _\\beta \\left(\\sigma _i\\right)_{\\!\\alpha }^{\\,\\,\\beta },\\quad \\quad i=1,2,3$ ($\\sigma _i$ are Pauli matrices and, as usual, spinor indices are raised and lowered with antisymmetric Levi-Civita tensors $\\varepsilon _{\\alpha \\beta }$ and $\\varepsilon ^{\\alpha \\beta }$ ), for the three-dimensional coordinates is considered in Section REF .", "The condition (REF ) ensures that the coordinates $\\ell _i$ are real.", "The full Lagrangian ${\\cal L}$ entering the ${\\cal N}=4$ invariant off-shell action $S= \\int dt {\\cal L}$ consists of the three pieces The first superfield formulation of general $({\\bf 3, 4, 1})$ SQM without background gauge field couplings was given in [15].", "${\\cal L} = {\\cal L}_{\\rm kin} + {\\cal L}_{\\rm int} + {\\cal L}_{\\rm FI}= \\int du\\, d^4\\theta \\, R_{\\rm kin}(L^{++}, L^{+-}, L^{--}, u)\\\\[3mm]- \\frac{1}{2} \\int du\\, d\\bar{\\theta }^+ d\\theta ^+\\, K(L^{++}, u) v^+\\widetilde{v^+}-\\frac{i k}{2} \\int \\, du \\, d\\bar{\\theta }^+ d\\theta ^+ \\, V^{++} , $ where $L^{+-}=\\frac{1}{2}D^{--}L^{++}$ and $L^{--}=D^{--}L^{+-}$ .", "The superfield functions $R_{\\rm kin}$ and $K$ bear an arbitrary dependence on their arguments The superfield Lagrangian (REF ) is written in the non-Abelian case.", "In the Abelian case, the superfield Lagrangian is simpler as it does not involve the auxiliary superfields $V^{++}$ , $v^+$ , $\\widetilde{v^+}$ : ${\\cal L} = {\\cal L}_{\\rm kin} + {\\cal L}_{\\rm int}= \\int du\\, d^4\\theta \\, R_{\\rm kin}(L^{++}, L^{+-}, L^{--}, u)+\\int du\\, d\\bar{\\theta }^+ d\\theta ^+\\, K^{++}(L^{++}, u).$ Here the interaction term is defined by a function $K^{++}(L^{++},\\,u)$ of charge +2.", "One can check that although the corresponding component Lagrangian as well as the expression for the Abelian gauge field differ from that in the non-Abelian case, the quantum Hamiltonian and the supercharges in the Abelian case have exactly the same form as in the non-Abelian case.", ".", "The meaning of three terms in (REF ) is explained below.", "The superfield Lagrangian (REF ) is written in the non-Abelian case.", "In the Abelian case, the superfield Lagrangian is simpler as it does not involve the auxiliary superfields $V^{++}$ , $v^+$ , $\\widetilde{v^+}$ : ${\\cal L} = {\\cal L}_{\\rm kin} + {\\cal L}_{\\rm int}= \\int du\\, d^4\\theta \\, R_{\\rm kin}(L^{++}, L^{+-}, L^{--}, u)+\\int du\\, d\\bar{\\theta }^+ d\\theta ^+\\, K^{++}(L^{++}, u).$ Here the interaction term is defined by a function $K^{++}(L^{++},\\,u)$ of charge +2.", "One can check that although the corresponding component Lagrangian as well as the expression for the Abelian gauge field differ from that in the non-Abelian case, the quantum Hamiltonian and the supercharges in the Abelian case have exactly the same form as in the non-Abelian case.", "The first, sigma-model-type term in Eq.", "(REF ), after integrating over Grassmann and harmonic variables, yields the generalized kinetic terms for $\\ell ^{\\alpha \\beta }, \\chi ^\\alpha , \\bar{\\chi }_\\alpha $ : ${\\cal L}_{\\rm kin}=\\frac{1}{8}f^{-2}\\left(-2\\dot{\\ell }_{\\alpha \\beta }\\dot{\\ell }^{\\alpha \\beta }+F^2\\right)+\\frac{i}{2} f^{-2}\\left(\\bar{\\chi }_\\alpha \\dot{\\chi }^\\alpha -{\\dot{\\bar{\\chi }}}_\\alpha \\chi ^\\alpha \\right)+\\frac{1}{4}\\left(\\partial _{\\alpha \\beta }\\partial ^{\\alpha \\beta }f^{-2}\\right)\\chi ^4\\\\[2mm]+\\frac{i}{f^3}\\dot{\\ell }^{\\alpha \\beta }\\big \\lbrace \\partial _{\\alpha \\gamma }f\\chi _\\beta \\bar{\\chi }^\\gamma +\\partial _{\\beta \\gamma }f\\chi ^\\gamma \\bar{\\chi }_\\alpha \\big \\rbrace -\\frac{1}{f^3}F\\chi ^\\alpha \\bar{\\chi }^\\beta \\partial _{\\alpha \\beta }f,$ where $\\chi ^4=\\chi ^\\alpha \\chi _\\alpha \\, \\bar{\\chi }^\\beta \\bar{\\chi }_\\beta $ , $\\partial _{\\alpha \\beta } \\equiv \\frac{\\partial }{\\partial \\ell ^{\\alpha \\beta }}$ and $f(\\ell )$ is a conformal factor.", "The calculations are most easily performed in the central basis, where $L^{++}=u^+_\\alpha u^+_\\beta L^{\\alpha \\beta }\\left(t,\\theta _\\gamma ,\\bar{\\theta }^\\delta \\right)$ .", "Then $f^{-2}(\\ell )=-\\partial _{\\alpha \\beta }\\partial ^{\\alpha \\beta }\\int R_{\\rm kin}\\left(\\ell ^{\\alpha \\beta }u^+_\\alpha u^+_\\beta ,\\ell ^{\\alpha \\beta }u^+_\\alpha u^-_\\beta ,\\ell ^{\\alpha \\beta }u^-_\\alpha u^-_\\beta \\right)\\,du.$ The fermionic kinetic term can be brought to the canonical form by the change of variables $\\chi ^\\alpha =f \\psi ^\\alpha ,\\quad \\quad \\bar{\\chi }_\\alpha =f\\bar{\\psi }_\\alpha .$ It is worth pointing out that the R-symmetry ${\\rm SU}(2)$ group amounts to the rotational ${\\rm SO(3)}$ group in the $\\mathbb {R}^3$ target space parametrized by $\\ell ^{i}$ from Eq.", "(REF ).", "The conformal factor $f(\\ell )$ can bear an arbitrary dependence on $\\ell ^{\\alpha \\beta }$ , so this ${\\rm SO(3)}$ can be totally broken in the Lagrangian (REF ).", "The second piece in Eq.", "(REF ) describes the coupling to an external non-Abelian gauge field.", "Performing the integration over $\\theta ^+$ , $\\bar{\\theta }^+$ and $u^\\pm _\\alpha $ , eliminating the auxiliary fermionic fields $\\omega _{1,2}$ and, finally, rescaling the bosonic doublet variables as $\\varphi _\\alpha \\ =\\ \\phi _\\alpha \\sqrt{h(\\ell ) }$ , where $h(\\ell )=\\int du\\, K\\left(\\ell ^{\\alpha \\beta }u^+_\\alpha u^+_\\beta ,u^\\pm _\\gamma \\right), $ after some algebra one obtains ${\\cal L}_{\\rm int}=i\\bar{\\varphi }^\\alpha \\left(\\dot{\\varphi }_\\alpha +iB\\varphi _\\alpha \\right)+\\bar{\\varphi }^\\gamma \\varphi ^\\delta \\frac{1}{2}\\left({\\cal A}_{\\alpha \\beta }\\right)_{\\gamma \\delta } \\dot{\\ell }^{\\alpha \\beta }-\\frac{1}{2} F\\,\\bar{\\varphi }^\\gamma \\varphi ^\\delta \\,U_{\\gamma \\delta }+ \\chi ^\\alpha \\bar{\\chi }^\\beta \\bar{\\varphi }^\\gamma \\varphi ^\\delta \\nabla _{\\alpha \\beta }U_{\\gamma \\delta }.$ Here the non-Abelian background gauge field and the scalar (matrix) potential are fully specified by the function $h$ : $\\left({\\cal A}_{\\alpha \\beta }\\right)_{\\gamma \\delta }=\\frac{i}{2h} \\Big \\lbrace \\varepsilon _{\\gamma \\beta }\\partial _{\\alpha \\delta }h+\\varepsilon _{\\gamma \\alpha }\\partial _{\\beta \\delta }h+\\varepsilon _{\\delta \\beta }\\partial _{\\alpha \\gamma }h+\\varepsilon _{\\delta \\alpha }\\partial _{\\beta \\gamma }h\\Big \\rbrace ,\\quad \\quad U_{\\gamma \\delta }=\\frac{1}{h}\\partial _{\\gamma \\delta }h\\,.", "$ By its definition, the function $h$ obeys the 3-dimensional Laplace equation, $\\partial ^{\\alpha \\beta }\\partial _{\\alpha \\beta }\\,h = 0\\,.", "$ Using the explicit expressions (REF ), it is straightforward to check the relation $\\left({\\cal F}_{\\alpha \\beta }\\right)_{\\gamma \\delta }=2i\\nabla _{\\alpha \\beta }U_{\\gamma \\delta },$ where $\\left({\\cal F}_{\\alpha \\beta }\\right)_{\\gamma \\delta }=-2\\partial _{\\alpha }^{\\;\\lambda }\\left({\\cal A}_{\\lambda \\beta }\\right)_{\\gamma \\delta }+i\\left({\\cal A}_{\\alpha }^{\\,\\,\\,\\lambda }\\right)_{\\!\\gamma \\sigma }\\left({\\cal A}_{\\lambda \\beta }\\right)^{\\,\\,\\sigma }_{\\delta }+\\left(\\alpha \\leftrightarrow \\beta \\right),$ $\\nabla _{\\alpha \\beta }U_{\\gamma \\delta }=-2\\partial _{\\alpha \\beta }U_{\\gamma \\delta }+i\\left({\\cal A}_{\\alpha \\beta }\\right)_{\\gamma \\lambda }U^{\\,\\,\\lambda }_{\\!\\delta }+i\\left({\\cal A}_{\\alpha \\beta }\\right)_{\\delta \\lambda }U^{\\,\\,\\lambda }_{\\!\\gamma },$ and $\\left({\\cal F}_{\\alpha \\beta }\\right)_{\\gamma \\delta }$ is related to the standard gauge field strength in the vector notation, see below.", "As we shall see soon, the condition (REF ) is none other than the static form of the general self-duality condition for the ${\\rm SU}(2)$ Yang-Mills field on $\\mathbb {R}^4\\,$ (see Eq.", "(REF )), i.e.", "the Bogomolny equations for BPS monopoles [22], while (REF ) provides a particular solution to these equations, being a static form of the 't Hooft ansatz [16].", "Note that the relation (REF ) is covariant and the Lagrangian (REF ) is form-invariant under the “target space” ${\\rm SU}(2)$ gauge transformations written in Eq.", "(REF ).", "This is not a genuine symmetry; rather, it is a reparametrization of the Lagrangian which allows one to cast the background potentials (REF ) in some different equivalent forms.", "It is worth noting that the gauge group indices coincide with those of the R-symmetry group, like in the four-dimensional case.", "Nevertheless, the “gauge” reparametrizations (REF ) do not affect the doublet indices of the target space coordinates $\\ell ^{\\alpha \\beta }$ and their superpartners present in the superfield $L^{++}$ .", "They act only on the semi-dynamical spin variables $\\varphi _\\alpha , \\bar{\\varphi }^\\alpha $ and gauge and scalar potentials (REF ).", "Finally, the last piece in Eq.", "(REF ) yields the Fayet-Iliopoulos term, ${\\cal L}_{\\rm FI}=k B\\,.$ In the quantum case, the coefficient $k$ is quantized, $k \\in \\mathbb {Z}\\,$ , on the same grounds as in the 4-dimensional case, see Section REF .", "It is instructive to rewrite the above relations and expressions, including the full Lagrangian (REF ), in the vector notations.", "To this end, let us pass from $\\ell ^{\\alpha \\beta }$ to $\\ell _i$ as in Eq.", "(REF ) and associate the vector ${\\cal A}_i$ to the gauge field ${\\cal A}_{\\alpha \\beta }$ (with additional matrix indices which are omitted here for simplicity) by the rule ${\\cal A}_\\alpha ^\\beta ={\\cal A}_i\\left(\\sigma _i\\right)_{\\!\\alpha }^{\\,\\,\\beta },\\quad \\quad {\\cal A}_i=\\frac{1}{2} {\\cal A}^\\alpha _\\beta \\left(\\sigma _i\\right)_{\\!\\alpha }^{\\,\\,\\beta },\\quad \\quad i=1,2,3.", "$ One can check that the coordinates $\\ell _i$ are real while the matrix $\\left({\\cal A}_i\\right)_{\\!\\gamma }^{\\,\\,\\delta }$ is Hermitian.", "Note also the relation between the partial derivatives $\\partial _{\\alpha \\beta } = \\partial /\\partial \\ell ^{\\alpha \\beta }$ and $\\partial _i = \\partial /\\partial \\ell _i$ : $\\partial _{\\alpha \\beta }= -\\frac{1}{2}\\left(\\sigma _i\\right)_{\\!\\alpha \\beta }\\partial _i,\\quad \\quad \\partial _i= -\\left(\\sigma _i\\right)_{\\!\\alpha }^{\\,\\,\\beta }\\partial ^\\alpha _\\beta \\,.$ Let us also make a similar conversion of the gauge group indices, $M_{\\!\\gamma }^{\\,\\,\\delta }=\\frac{1}{2} M^a \\left(\\sigma _a\\right)_{\\!\\gamma }^{\\,\\,\\delta },\\quad \\quad M^a=M_{\\!\\delta }^{\\,\\,\\gamma }\\left(\\sigma _a\\right)_{\\!\\gamma }^{\\,\\,\\delta },\\quad \\quad a=1,2,3\\,,$ for any Hermitian traceless $2\\times 2$ matrix $M$ .", "In the new notations, the total Lagrangian (REF ) takes the following form: ${\\cal L}=\\frac{1}{2} f^{-2}\\dot{\\ell }_i^2+{\\cal A}_i^a T^a\\dot{\\ell }_i+i\\bar{\\varphi }^\\alpha \\left(\\dot{\\varphi }_\\alpha +iB\\varphi _\\alpha \\right)+kB+i\\bar{\\psi }_\\alpha \\dot{\\psi }^\\alpha +f^2\\nabla _i U^a T^a\\,\\psi \\sigma _i\\bar{\\psi }\\\\[2mm]+\\frac{1}{4}\\left\\lbrace f\\partial _i^2 f-3\\left(\\partial _i f\\right)^2\\right\\rbrace \\psi ^4+2 f^{-1}\\varepsilon _{ijk}\\partial _i f\\,\\dot{\\ell }_j\\, \\psi \\sigma _k\\bar{\\psi }\\\\[2mm]+\\frac{1}{8}f^{-2}F^2+\\frac{1}{2} F\\left( U^a T^a- f^{-1}\\partial _i f\\,\\psi \\sigma _i\\bar{\\psi }\\right).$ where $T^a$ defined in Eq.", "(REF ).", "Here $\\nabla _i U^a=\\partial _i U^a +\\varepsilon ^{abc}{\\cal A}_i^b U^c$ and the Bogomolny equations (REF ) relating ${\\cal A}_i^a$ and $U^a$ are equivalently rewritten in the more familiar form, ${\\cal F}_{ij}^a = \\varepsilon _{ijk}\\nabla _k U^a\\,, $ where ${\\cal F}_{ij}^a = \\partial _i{\\cal A}_j^a - \\partial _j{\\cal A}_i^a+\\varepsilon ^{abc}{\\cal A}_i^b{\\cal A}_j^c$ .", "Finally, the gauge field and the matrix potential defined in (REF ) are rewritten as ${\\cal A}_i^a=-\\varepsilon _{ija} \\partial _j \\ln h,\\quad \\quad U^a=-\\partial _a \\ln h\\,, \\qquad \\Delta \\,h = 0\\,.$ After eliminating the auxiliary field $F$ by its equation of motion, $F = 2 f^2\\left( f^{-1}\\partial _i f \\,\\psi \\sigma _i \\bar{\\psi }- U^aT^a\\right)\\,, $ the Lagrangian (REF ) takes the form ${\\cal L} =\\frac{1}{2} f^{-2}\\dot{\\ell }_i^2+{\\cal A}_i^a T^a\\dot{\\ell }_i+i\\bar{\\varphi }^\\alpha \\left(\\dot{\\varphi }_\\alpha +iB\\varphi _\\alpha \\right)+kB+i\\bar{\\psi }_\\alpha \\dot{\\psi }^\\alpha +f^2\\psi \\sigma _i\\bar{\\psi }\\left(\\nabla _i +f^{-1}\\partial _i f\\right) U^aT^a\\\\[3mm]+\\,\\frac{1}{4}\\left\\lbrace f\\partial _i^2 f- 4\\left(\\partial _i f\\right)^2\\right\\rbrace \\psi ^4+2 f^{-1}\\varepsilon _{ijk}\\partial _i f\\,\\dot{\\ell }_j\\, \\psi \\sigma _k\\bar{\\psi }-\\frac{1}{2} f^2 ( U^a T^a)^2\\,.$ We see that this Lagrangian involves three physical bosonic fields $\\ell _i$ and four physical fermionic fields $\\psi _\\alpha \\,$ .", "It is fully specified by two independent functions: the metric conformal factor $f(\\ell )$ which can bear an arbitrary dependence on $\\ell _i$ and the function $h(\\ell )$ which satisfies the 3-dimensional Laplace equation and determines the background non-Abelian gauge and scalar potentials.", "The representation (REF ) for $h$ in terms of the analytic function $K(\\ell ^{++}, u)$ yields in fact a general solution of the 3-dimensional Laplace equation [54].", "If one takes the function $h(\\ell )$ to be vanishing at $|\\vec{\\ell }|\\rightarrow \\infty $ , then this function can be presented as the following sum over monopoles: $h(\\ell )=1+\\sum \\limits _M \\frac{c_M}{\\left|\\vec{\\ell }-\\vec{b}_M\\right|}.$ It involves particular monopole positions $\\vec{b}_M$ as well as the numbers $c_M$ associated with each monopole.", "The Lagrangian (REF ) also contains the “semi-dynamical” spin variables $\\varphi _\\alpha , \\bar{\\varphi }^\\alpha \\,$ , the role of which is the same as in the four-dimensional case: after quantization they ensure that $T^a$ defined in (REF ) become matrix SU(2) generators corresponding to the spin $|k|/2$ representation.", "The component action corresponding to the Lagrangian (REF ) is partly on shell since we have already eliminated the fermionic fields of the auxiliary $v^+$ multiplet by their algebraic equations of motion.", "The fields of the coordinate multiplet $L^{++}$ are still off shell.", "The ${\\cal N}=4$ transformations leaving invariant the action $S = \\int dt \\,{\\cal L}$ look most transparent being expressed in terms of the component fields $\\ell _i, F, \\chi ^\\alpha $ , $\\bar{\\chi }^\\alpha $ , $\\phi ^\\beta $ , $\\bar{\\phi }^\\beta $ : $\\begin{array}{l}\\ell _i\\rightarrow \\ell _i +i\\epsilon \\sigma _i\\chi + i\\bar{\\epsilon }\\sigma _i\\bar{\\chi },\\\\[2mm]F\\rightarrow F - 2\\epsilon ^\\alpha \\dot{\\chi }_\\alpha - 2\\bar{\\epsilon }^\\alpha \\dot{\\bar{\\chi }}_\\alpha ,\\\\[2mm]\\chi ^\\alpha \\rightarrow \\chi ^\\alpha - \\frac{1}{2} iF\\bar{\\epsilon }^\\alpha - (\\bar{\\epsilon }\\sigma _i)^\\alpha \\,\\dot{\\ell }_i,\\\\[2mm]\\bar{\\chi }^\\alpha \\rightarrow \\bar{\\chi }^\\alpha +\\frac{1}{2} iF\\epsilon ^\\alpha + (\\epsilon \\sigma _i)^\\alpha \\,\\dot{\\ell }_i,\\\\[2mm]\\phi ^\\alpha \\rightarrow \\phi ^\\alpha - {i}\\big (\\epsilon ^\\alpha \\chi \\sigma _i\\phi + \\bar{\\epsilon }^\\alpha \\bar{\\chi }\\sigma _i\\phi \\big )\\partial _i\\ln h,\\\\[2mm]\\bar{\\phi }^\\alpha \\rightarrow \\bar{\\phi }^\\alpha - {i}\\left(\\epsilon ^\\alpha \\chi \\sigma _i\\bar{\\phi }+ \\bar{\\epsilon }^\\alpha \\bar{\\chi }\\sigma _i\\bar{\\phi }\\right)\\partial _i\\ln h.\\end{array}$ These transformations can be deduced from the analytic subspace realization of ${\\cal N}=4$ supersymmetry (REF ), with taking into account the compensating ${\\rm U}(1)$ gauge transformations of the superfields $v^+, \\widetilde{v}^+$ and $V^{++}$ needed to preserve the WZ gauge (REF ).", "Note that $\\delta B = 0$ under ${\\cal N}=4$ supersymmetry.", "This transformation law matches with the ${\\cal N}=4$ superalgebra in WZ gauge, taking into account that the translation of $B$ looks as a particular U(1) gauge transformation of the latter.", "The Lagrangian (REF ) is invariant, modulo a total time derivative, under the transformations (REF ) in which $F$ is expressed from (REF ).", "The Lagrangian (REF ) is the point of departure for setting up the Hamiltonian formulation of the model under consideration and quantizing the latter.", "After substitution of ${\\rm SU}(2)$ spin-$k/2$ generators instead of $T^a$ , the quantum Hamiltonian of this system takes the form $H=\\frac{1}{2}f \\left(\\hat{p}_i-{\\cal A}_i\\right)^2 f+\\frac{1}{2} f^2 U^2-f^2\\nabla _i U\\psi \\sigma _i\\bar{\\psi }\\\\[2mm]+\\Big \\lbrace \\varepsilon _{ijk}f\\partial _i f\\left(\\hat{p}_j-{\\cal A}_j\\right)-f\\partial _k f U\\Big \\rbrace \\psi \\sigma _k\\bar{\\psi }+ f\\partial ^2 f\\left\\lbrace \\psi ^{\\gamma }\\bar{\\psi }_{\\gamma }-\\frac{1}{2}\\left(\\psi ^{\\gamma } \\bar{\\psi }_{\\gamma }\\right)^2\\right\\rbrace ,$ which is just a static 3-dimensional reduction of the 4-dimensional Hamiltonian given by Eq.", "(REF ).", "In this expression, the gauge field ${\\cal A}_i = {\\cal A}_i^aT^a$ and the scalar potential $U = U^aT^a$ are ${\\rm SU}(2)$ matrices subjected to the constraint (REF ).", "It is also easy to find the supercharges $Q_\\alpha , \\bar{Q}^\\beta $ : $\\begin{array}{l}Q_\\alpha = f \\left(\\sigma _i \\bar{\\psi }\\right)_\\alpha \\left(\\hat{p}_i-{\\cal A}_i\\right)-\\psi ^{\\gamma } \\bar{\\psi }_{\\gamma } \\left(\\sigma _i\\bar{\\psi }\\right)_\\alpha i\\partial _i f-ifU\\bar{\\psi }_\\alpha ,\\\\[3mm]\\bar{Q}^\\alpha = \\left(\\psi \\sigma _i\\right)^\\alpha \\left(\\hat{p}_i-{\\cal A}_i\\right)f+i\\partial _i f \\left(\\psi \\sigma _i\\right)^\\alpha \\psi ^{\\gamma }\\bar{\\psi }_{\\gamma }+ifU\\psi ^\\alpha ,\\end{array}$ The ordering ambiguity arising in the case of the general conformal factor $f(\\ell )$ can be fixed, as usual, by Weyl ordering procedure [41], see Section REF for details.", "Let us emphasize that the only condition required from the background matrix fields ${\\cal A}_i$ and $U$ for the generators $Q_\\alpha $ and $\\bar{Q}^\\beta $ to form ${\\cal N}=4$ superalgebra (REF ) is that these fields satisfy the Bogomolny equations (REF ).", "Thus the expressions (REF ) and (REF ) define the ${\\cal N}=4$ SQM model in the field of arbitrary BPS monopole, not necessarily restricted to the ansatz (REF ).", "Also, one can extend the gauge group ${\\rm SU}(2)$ to ${\\rm SU}(N)$ in (REF ) and (REF ).", "Let us remark that the three-dimensional Hamiltonian (REF ) and the supercharges (REF ) were considered for the first time in Ref.", "[12] (in the Abelian case).", "Finally, as a simple example of the monopole background consistent with the off- and on-shell ${\\cal N}=4$ supersymmetry, let us consider a particular 3-dimensional spherically symmetric case.", "It corresponds to the most general ${\\rm SO}(3)$ invariant solution of the Laplace equation for the function $h$ , $h_{{\\rm so}(3)}(\\ell ) = c_0 + c_1\\,\\frac{1}{\\sqrt{\\ell ^2}}\\,.", "$ The corresponding potentials calculated according to Eqs.", "(REF ) read ${\\cal A}^a_i= \\varepsilon _{ija}\\frac{\\ell _j}{\\ell ^2} \\,\\frac{c_1}{c_1 + c_0\\sqrt{\\ell ^2}}, \\quad \\quad U^a =\\frac{\\ell _a}{\\ell ^2} \\,\\frac{c_1}{c_1 + c_0\\sqrt{\\ell ^2}}.", "$ This configuration becomes the Wu-Yang monopole [24] for the choice $c_0 = 0\\,$ .", "It is easy to find the analytic function $K(\\ell ^{++},u)$ which generates the solution (REF ) (see [6]): $h_{{\\rm so}(3)}(\\ell ) = \\int du\\, K_{{\\rm so}(3)}(\\ell ^{++}, u)\\,, \\quad K_{so(3)}(\\ell ^{++}, u)= c_0 + c_1 \\left(1 + a^{--}\\hat{\\ell }^{++}\\right)^{-\\frac{3}{2}}\\,, \\\\[3mm]\\ell ^{++} \\equiv \\hat{\\ell }^{++} + a^{++}\\,, \\quad a^{\\pm \\pm } = a^{\\alpha \\beta }u^\\pm _\\alpha u^\\pm _\\beta \\,, \\quad a^{\\alpha }_\\beta a^\\beta _\\alpha = 2\\,.$ One could equally choose as $h(\\ell )$ , e.g., the well-known multi-center solution to the Laplace equation, with the broken ${\\rm SO}(3)$ .", "Note that the ${\\cal N}=4$ mechanics with coupling to Wu-Yang monopole was recently constructed in [21], proceeding from a different approach, with the built-in ${\\rm SO}(3)$ invariance and the treatment of spin variables in the spirit of Ref. [18].", "Our general consideration shows, in particular, that the demand of ${\\rm SO}(3)$ symmetry is not necessary for the existence of ${\\cal N}=4$ SQM models with non-Abelian monopole backgrounds.", "It is instructive to show that (REF ) can indeed be viewed as a 3-dimensional reduction of the 't Hooft ansatz for the solution of general self-duality equation in $\\mathbb {R}^4$ for the gauge group ${\\rm SU}(2)$ , with the identification $U^a = {\\cal A}_0^a\\,$ , while the condition (REF ) is the 3-dimensional reduction of this equation.", "To establish this relationship, we use the following rules of correspondence between the ${\\rm SO}(4)= {\\rm SU}(2)\\times {\\rm SU}(2)$ spinor formalism and its ${\\rm SU}(2)$ reduction: $ \\begin{array}{l}\\left(\\sigma _\\mu \\right)_{\\alpha \\dot{\\beta }}\\rightarrow \\left\\lbrace i\\delta _\\alpha ^{\\beta },\\left(\\sigma _i\\right)_{\\!\\alpha }^{\\,\\,\\beta }\\right\\rbrace ,\\\\[2mm]\\varepsilon ^{\\dot{\\alpha }\\dot{\\beta }} \\rightarrow - \\varepsilon _{\\alpha \\beta },\\quad \\quad \\varepsilon _{\\dot{\\alpha }\\dot{\\beta }} \\rightarrow - \\varepsilon ^{\\alpha \\beta },\\\\[2mm]x_{\\alpha \\dot{\\beta }}\\rightarrow \\ell _\\alpha ^{\\beta },\\quad \\quad \\quad x^{\\alpha \\dot{\\beta }}\\rightarrow -\\ell ^\\alpha _{\\beta },\\\\[2mm]\\psi _{\\dot{\\alpha }}\\rightarrow \\psi ^{\\alpha }.\\end{array}$ This reflects the fact that the R-symmetry ${\\rm SU}(2)$ in the $({\\bf 3, 4, 1})$ models can be treated as a diagonal subgroup in the symmetry group ${\\rm SO}(4)= {\\rm SU}(2)\\times {\\rm SU}(2)$ of the $({\\bf 4, 4, 0})$ models, with the ${\\rm SU}(2)$ factors acting, respectively, on the undotted and dotted indices.", "The self-dual $\\mathbb {R}^4$ SU(2) gauge field in the 't Hooft ansatz is written in the spinor notation in Eq.", "(REF ).", "Then, using the rules (REF ), one performs the reduction $\\mathbb {R}^4 \\rightarrow \\mathbb {R}^3$ as $\\begin{array}{l}({\\cal A}_{\\alpha \\dot{\\beta }})_{\\!\\gamma }^{\\,\\,\\delta }\\; \\rightarrow \\; iU_{\\!\\gamma }^{\\,\\,\\delta } \\delta _\\alpha ^\\beta + ({\\cal A}_{\\alpha }^{\\beta })_{\\!\\gamma }^{\\,\\,\\delta },\\quad \\quad ({\\cal A}_{\\alpha }^{\\alpha })_{\\!\\gamma }^{\\,\\,\\delta } = 0,\\\\[3mm]h(x)\\; \\rightarrow \\;h(\\ell ), \\quad \\quad \\partial _\\beta ^\\alpha \\partial ^\\beta _\\alpha \\,h = 0.\\end{array}$ Upon this reduction, the four-dimensional ansatz (REF ) yields precisely (REF ), while the general self-duality condition (REF ) goes over into the Bogomolny equations (REF ).", "Of course, the same reduction can be performed in the vector notation, with ${\\cal F}_{\\mu \\nu }\\rightarrow \\left\\lbrace {\\cal F}_{ij}, {\\cal F}_{0k}=\\nabla _k U\\right\\rbrace $ , and Eqs.", "(REF ), (REF ) as an output.", "Thus, the general gauge field background prescribed by the off-shell ${\\cal N}=4$ supersymmetry in this $({\\bf 3,4,1})$ system is a static form of the 't Hooft ansatz for the self-dual ${\\rm SU}(2)$ gauge field in $\\mathbb {R}^4\\,$ .", "This suggests that the above bosonic target space reduction has its superfield counterpart relating the four-dimensional system described in Section REF to the one considered here.", "Indeed, the superfield $({\\bf 3,4,1})$ action (REF ) can be obtained from the $({\\bf 4,4,0})$ multiplet action composed from Eqs.", "(REF ), (REF ), (REF ) via the “automorphic duality” [55] by considering a restricted class of the $({\\bf 4,4,0})$ actions with ${\\rm U}(1)$ isometry and performing a superfield gauging of this isometry by an extra gauge superfield $V^{++}{}^{\\prime }$ along the general line of Ref. [56].", "Actually, the bosonic target space reduction we have just described corresponds to the shift isometry of the analytic superfield $q^{+\\dot{\\alpha }}$ accommodating the $({\\bf 4,4,0})$ multiplet, namely, to $q^{+\\dot{\\alpha }} \\rightarrow q^{+\\dot{\\alpha }} + \\omega u^{+ \\dot{\\alpha }}\\,$ .", "It is the invariant projection $q^{+\\dot{\\alpha }}u^+_{\\dot{\\alpha }}$ which is going to become the $({\\bf 3,4,1})$ superfield $L^{++}$ upon gauging this isometry and choosing the appropriate manifestly ${\\cal N}=4$ supersymmetric gauge.", "An important impact of this superfield reduction on the structure of the component action is the appearance of the new induced potential bilinear in the gauge group generators $\\sim U^2 = U^a U^b T^a T^b\\,$ .", "It comes out as a result of eliminating the auxiliary field $F$ in the off-shell $({\\bf 3, 4, 1})$ multiplet, and so is necessarily prescribed by ${\\cal N}=4$ supersymmetry.", "It is interesting that analogous potential terms were introduced in [57] at the bosonic level for ensuring the existence of some hidden symmetries in the models of 3-dimensional particle in a non-Abelian monopole background.", "The same reduction ${\\mathbb {R}}^4\\rightarrow {\\mathbb {R}}^3$ can be performed at the level of Hamiltonian and supercharges.", "In particular, the reduction of the Hamiltonian of the four-dimensional system of Eq.", "(REF ) yields the 3-dimensional Hamiltonian (REF ).", "Conclusion We studied some rather general off-shell ${\\cal N}=4$ supersymmetric coupling of the $d=1$ coordinate supermultiplets ${\\bf (4,4,0)}$ and ${\\bf (3,4,1)}$ to an external self-dual (or anti-self-dual) Abelian gauge field and discussed the ${\\bf (4,4,0)}$ case in details.", "Our main framework was the harmonic superspace approach.", "The use of an analytic “semi-dynamical” multiplet $({\\bf 4,4,0})$ with the Wess-Zumino type action allowed us to make coupling of the coordinate multiplets ${\\bf (4,4,0)}$ and ${\\bf (3,4,1)}$ to an external ${\\rm SU}(2)$ gauge field.", "This auxiliary multiplet incorporates ${\\rm SU}(2)$ doublet of bosonic spin variables which are crucial for arranging couplings to non-Abelian gauge fields.", "In the four-dimensional case, the off-shell ${\\cal N}=4$ supersymmetry restricts the non-Abelian gauge field to be self-dual (or anti-self-dual) and in a form of the 't Hooft ansatz for ${\\rm SU}(2)$ gauge field.", "In the three-dimensional case, the non-Abelian gauge field is a three-dimensional reduction of this 't Hooft ansatz, i.e.", "a particular solution of the Bogomolny monopole equations.", "Additionally, in three dimensions, at the component level, the coupling to a gauge field is necessarily accompanied by an induced potential which is bilinear in the ${\\rm SU}(2)$ generators and arises as a result of eliminating the auxiliary field in the coordinate ${\\bf (3,4,1)}$ multiplet.", "The explicit form of the Hamiltonians and the supercharges were presented.", "The corresponding expressions respect ${\\cal N}=4$ on-shell supersymmetry for any self-dual or anti-self-dual, Abelian or non-Abelian gauge field in four-dimensions, not necessarily in the 't Hooft ansatz form.", "In three dimensions, an arbitrary BPS monopole background can be used in the non-Abelian case.", "It is worthwhile to note that similar constraints (Bogomolny equations) on the external non-Abelian three-dimensional gauge field were found in [58], while considering an ${\\cal N}=4$ extension of Berry phase in quantum mechanics.", "However, no invariant actions and/or the explicit expressions for the Hamiltonian and ${\\cal N}=4$ supercharges were presented there.", "The nonlinear counterpart of $q^{+{\\dot{\\alpha }}}$ multiplet is discussed in [46].", "In this case, the bosonic target geometry is more general as compared to the conformally-flat geometry associated with the linear $({\\bf 4, 4, 0})$ multiplet.", "Among the possible directions of further study, we mention the construction of higher ${\\cal N}$ SQM models with non-Abelian gauge field backgrounds, e.g.", "${\\cal N}=8$ ones, as well as studying various supersymmetry-preserving reductions of these models to lower-dimensional target bosonic manifolds by the gauging procedure of [56].", "Actually, the method of the auxiliary “semi-dynamical” $({\\bf 4,4,0})$ multiplet with the Wess-Zumino type action, which was successfully applied in our construction here, could work with the equal efficiency for constructing a Lagrangian description of other supersymmetric quantum-mechanics problems involving the coupling to an external non-Abelian gauge field.", "Besides the obvious examples of quantum Hall effect (or Landau problem) in higher dimensions (see e.g.", "the discussion in [8]), let us also mention supersymmetric Wilson loop functionals which can be interpreted in terms of a non-Abelian version of Chern-Simons (super)quantum mechanics [59], with the parameter along the loop as an evolution parameter.", "We hope that the quantized semi-dynamical variables could provide a new efficient tool to study this class of problems.", "The 't Hooft type ansatz (REF ) and the choice of ${\\rm SU}(2)$ as the gauge group are required for the existence of the off-shell superfield formulation of the discussed SQM systems.", "It is not known whether the most general system can be derived from some off-shell superfield formalism, with general instanton/monopole backgrounds obtained from the ADHM construction [60] or its three-dimensional reduction.", "Additionally, there remains a problem of extending the models to a generic ${\\rm SU}(N)$ gauge group.", "Possibly, the above issues are related to the generalization of the interaction term (REF ) to $S_{\\rm int}= \\int \\, dt \\, du\\, d\\bar{\\theta }^+d\\theta ^+\\,K^{++}\\left(q^{+\\dot{\\alpha }},\\, u^\\pm _\\beta ,\\, v^+ \\widetilde{v^+} \\right) .$ It would be also interesting to study SQM models with nonlinear counterpart of the semi-dynamical multiplet $({\\bf 4, 4, 0})$ [56].", "tocchapterBibliography" ] ]

[ [ "N=1 Non-Abelian Tensor Multiplet in Four Dimensions" ], [ "Abstract We carry out the N=1 supersymmetrization of a physical non-Abelian tensor with non-trivial consistent couplings in four dimensions.", "Our system has three multiplets: (i) The usual non-Abelian vector multiplet (VM) (A_\\mu{}^I, \\lambda^I), (ii) A non-Abelian tensor multiplet (TM) (B_{\\mu\\nu}{}^I, \\chi^I, \\varphi^I), and (iii) A compensator vector multiplet (CVM) (C_\\mu{}^I, \\rho^I).", "All of these multiplets are in the adjoint representation of a non-Abelian group G. Unlike topological theory, all of our fields are propagating with kinetic terms.", "The C_\\mu{}^I-field plays the role of a Stueckelberg compensator absorbed into the longitudinal component of B_{\\mu\\nu}{}^I.", "We give not only the component lagrangian, but also a corresponding superspace reformulation, reconfirming the total consistency of the system.", "The adjoint representation of the TM and CVM is further generalized to an arbitrary real representation of general SO(N) gauge group.", "We also couple the globally N=1 supersymmetric system to supergravity, as an additional non-trivial confirmation." ], [ "[12pt]article =15ex CSULB–PA–11–3 (Revised Version) N$\\,$ =$\\,$ 1$\\,$ Non-Abelian Tensor Multiplet in Four Dimensions Hitoshi  NISHINO$^)$$^)$  E-Mail: [email protected]  and  Subhash  RAJPOOT$^)$$^)$  E-Mail: [email protected] Department of Physics & Astronomy California State University 1250 Bellflower Boulevard Long Beach, CA 90840 Abstract    We carry out the $~N=1$   supersymmetrization of a physical non-Abelian tensor with non-trivial consistent couplings in four dimensions.", "Our system has three multiplets: (i) The usual non-Abelian vector multiplet (VM) $~(A_{\\mu }{}^{I}, \\lambda ^I)$ ,  (ii) A non-Abelian tensor multiplet (TM) $~(B_{\\mu \\nu }{}^{I} , \\chi ^I, \\varphi ^I)$ , and (iii) A compensator vector multiplet (CVM) $\\,(C_{\\mu }{}^{I}, \\rho ^I)$ .", "All of these multiplets are in the adjoint representation of a non-Abelian group $\\, G$ .", "Unlike topological theory, all of our fields are propagating with kinetic terms.", "The $C_{\\mu }{}^{I}{\\hspace{1.5pt}}\\hbox{-}$ field plays the role of a Stueckelberg compensator absorbed into the longitudinal component of $~B_{\\mu \\nu }{}^{I}$ .", "We give not only the component lagrangian, but also a corresponding superspace reformulation, reconfirming the total consistency of the system.", "The adjoint representation of the TM and CVM is further generalized to an arbitrary real representation of general $~SO(N)$   gauge group.", "We also couple the globally $~N=1$   supersymmetric system to supergravity, as an additional non-trivial confirmation.", "PACS: 11.15.-q, 11.30.Pb, 12.60.Jv Key Words: Non-Abelian Tensor, $~N=1$   Supersymmetry, Tensor Multiplet, Vector Field                in Non-Trivial Representation, Consistency of Field Equations and Couplings.", "1.", "Introduction Recently, the long-standing problem with non-Abelian tensors [1] has been solved by de Wit, Samtleben, and Nicolai [2][3].", "The original motivation in [2] was to generalize the tensor and vector field interactions in manifestly $~E_{6(+6)}{\\hspace{1.5pt}}\\hbox{-}$ covariant formulation of five-dimensional (5D) maximal supergravity by gauging non-Abelian sub-groups.", "In [3], this work was further related to M-theory [4] by confirming the representation assignments under the duality group of the gauge charges.", "The underlying hierarchies of these tensor and vector gauge fields are presented with the consistency of general gaugings.", "The hierarchy in [2][3] has been further applied to the conformal supergravity in 6D [5].", "In ref.", "[5], the `minimal tensor hierarchy' as a special case of the more general hierarchy in [2][3] has been discussed.", "This hierarchy consists of $~A_{\\mu }{}^{r} $   and two-form gauge potentials $B_{\\mu \\nu }{}^{I}$ , with two labels $~{\\scriptstyle r}$   and $~{\\scriptstyle I}$ .", "Also introduced is the 3-form gauge potentials $~C_{\\mu \\nu \\rho \\, r}$   with the index $~_r$   is dual to $~^r$   of $~A_{\\mu }{}^{r}$ .", "The field strengths of vector and two-form gauge potentials are defined by [5] $ 2=0ptto{\\hfil \\displaystyle {#}&\\displaystyle {{}#}\\hfil &\\unknown.", "{#}\\crcr {\\cal F}_{\\mu \\nu }{}^{r} & \\equiv 2 \\partial _{\\lfloor {\\hspace{0.35pt}}\\!\\!\\!\\lceil \\mu } A_{\\nu \\rfloor {\\hspace{0.35pt}}\\!\\!\\!\\rceil }{}^{r}+ h_{I}{}^{r} B_{\\mu \\nu }{}^{I} ~~,&(1.1{\\rm a}) \\cr {\\cal H}_{\\mu \\nu \\rho }{}^{I} & \\equiv 3 D_{\\lfloor {\\hspace{0.35pt}}\\!\\!\\!\\lceil \\mu } B_{\\nu \\rho \\rfloor {\\hspace{0.35pt}}\\!\\!\\!\\rceil }{}^{I}+ 6 d_{r s}{}^{I} A_{\\lfloor {\\hspace{0.35pt}}\\!\\!\\!\\lceil \\mu }{}^{r} \\partial _\\nu A_{\\rho \\rfloor {\\hspace{0.35pt}}\\!\\!\\!\\rceil }{}^{s}- 2 f_{p q}{}^{s} d_{r s}{}^{I} A_{\\lfloor {\\hspace{0.35pt}}\\!\\!\\!\\lceil \\mu }{}^{r} A_{\\nu }{}^{p} A_{\\rho \\rfloor {\\hspace{0.35pt}}\\!\\!\\!\\rceil }{}^{q}+ g^{I r} C_{\\mu \\nu \\rho r} ~~.", "~~~~~&(1.1{\\rm b}) \\cr \\crcr } $ The prescription for tensor-vector system, which we will be based upon, is described with eq.$\\,$ (3.22) in [5].", "To be more specific, we consider in the present paper the product of two identical gauge groups $~G \\times G$   [6], whose adjoint indices are respectively $~{\\scriptstyle r,~s,~\\cdots }$   and $~{\\scriptstyle r^{\\prime }, ~s^{\\prime },~\\cdots }$ .", "Accordingly, we use the coefficients $ 2=0ptto{\\hfil \\displaystyle {#}&\\displaystyle {{}#}\\hfil &\\unknown.", "{#}\\crcr f_{r s}{}^{t} & = {\\rm f}_{r s}{}^{t} ~~, ~~~~ f_{r s^{\\prime }}{}^{t^{\\prime }} = - f_{s^{\\prime } r}{}^{t^{\\prime }}= +{\\textstyle {1\\over \\vphantom{2}\\smash{\\scriptstyle {2}}}}} {\\rm f}_{r s^{\\prime }}{}^{t^{\\prime }} ~~,&(1.2{\\rm a}) \\cr d_{r s^{\\prime }}^t & = d_{s^{\\prime } r}^t = - {\\textstyle {1\\over \\vphantom{2}\\smash{\\scriptstyle {2}}}}$ fr s't   ,      hr's = s r'   , (1.2b) $where $ fr st $~ is the structure constant of a non-Abelian gauge group.We use the same field content arising by this prescription.$ Since the outstanding paper [5] gives the extensive details of how to get our system from [2][3][6], there is nothing new to explain, except for our notational preparation.", "In our notation, the field strengths of the $~B$   and $~C{\\hspace{1.5pt}}\\hbox{-}$ fields are respectively $~G$   and $~H$   defined by $2=0ptto{\\hfil \\displaystyle {#}&\\displaystyle {{}#}\\hfil &\\unknown.", "{#}\\crcr G_{\\mu \\nu \\rho }{}^{I} & \\equiv + 3 D_{\\lfloor {\\hspace{0.35pt}}\\!\\!\\!\\lceil \\mu } B_{\\nu \\rho \\rfloor {\\hspace{0.35pt}}\\!\\!\\!\\rceil }{}^{I}- 3 f^{I J K} C_{\\lfloor {\\hspace{0.35pt}}\\!\\!\\!\\lceil \\mu }{}^{J} F_{\\nu \\rho \\rfloor {\\hspace{0.35pt}}\\!\\!\\!\\rceil }{}^{K} ~~,&(1.3{\\rm a}) \\cr H_{\\mu \\nu }{}^{I} & \\equiv + 2 D_{\\lfloor {\\hspace{0.35pt}}\\!\\!\\!\\lceil \\mu } C_{\\nu \\rfloor {\\hspace{0.35pt}}\\!\\!\\!\\rceil }{}^{I} + g B_{\\mu \\nu }{}^{I} ~~.&(1.3{\\rm b}) \\cr \\crcr } $ The gauge transformations for $~B,~C$   and $~A{\\hspace{1.5pt}}\\hbox{-}$ fields are $2=0ptto{\\hfil \\displaystyle {#}&\\displaystyle {{}#}\\hfil &\\unknown.", "{#}\\crcr \\delta _\\alpha (B_{\\mu \\nu }{}^{I} , C_{\\mu }{}^{I}, A_{\\mu }{}^{I} )& = ( \\, - f^{I J K}\\alpha ^J B_{\\mu \\nu }{}^{K} , ~- f^{I J K}\\alpha ^J C_{\\mu }{}^{K} ,~ + D_\\mu \\alpha ^I ) ~~,&(1.4{\\rm a}) \\cr \\delta _\\beta ( B_{\\mu \\nu }{}^{I}, C_{\\mu }{}^{I} ,A_{\\mu }{}^{I} )& = ( \\, + 2 D_{\\lfloor {\\hspace{0.35pt}}\\!\\!\\!\\lceil \\mu } \\beta _{\\nu \\rfloor {\\hspace{0.35pt}}\\!\\!\\!\\rceil }{}^{I} , ~ - g \\beta _{\\mu }{}^{I} , ~ 0) ~~,&(1.4{\\rm b}) \\cr \\delta _\\gamma ( B_{\\mu \\nu }{}^{I}, C_{\\mu }{}^{I} ,A_{\\mu }{}^{I} )& = ( \\, - f^{I J K}F_{\\mu \\nu }{}^{J} \\gamma ^K , ~D_\\mu \\gamma ^I , ~0) ~~.&(1.4{\\rm c}) \\cr \\crcr } $ As (1.3b) or (1.4b) shows, $~C_{\\mu }{}^{I}$   is a vectorial Stueckelberg field, absorbed into the longitudinal component of $~B_{\\mu \\nu }{}^{I}$ .", "Due to the general hierarchy [2][3], all field strengths are invariant: $2=0ptto{\\hfil \\displaystyle {#}&\\displaystyle {{}#}\\hfil &\\unknown.", "{#}\\crcr & \\delta _\\alpha (G_{\\mu \\nu \\rho }{}^{I} , ~H_{\\mu \\nu }{}^{I}, ~F_{\\mu \\nu }{}^{I} )= - f^{I J K}\\alpha ^J ( G_{\\mu \\nu \\rho }{}^{K} ,~ H_{\\mu \\nu }{}^{K} , ~F_{\\mu \\nu }{}^{K} ) ~~,&(1.5{\\rm a}) \\cr & \\delta _\\beta (G_{\\mu \\nu \\rho }{}^{I} , ~H_{\\mu \\nu }{}^{I} , ~F_{\\mu \\nu }{}^{I}) = 0 ~~,~~~~ \\delta _\\gamma (G_{\\mu \\nu \\rho }{}^{I} , ~H_{\\mu \\nu }{}^{I} ,~ F_{\\mu \\nu }{}^{I}) = 0 ~~.&(1.5{\\rm b}) \\cr \\crcr } $ Since the hierarchy given in [2][3] guarantees the gauge invariance of all field strengths, the construction of purely bosonic lagrangian is straightforward.", "Consider the action $~I_1 \\equiv \\int d^4 x \\, g^2 {\\cal L}_1$ $^)$$^)$  The reason we need the factor $~g^2$   in the action is due to the mass-dimension assignments of our fields.", "with $2=0ptto{\\hfil \\displaystyle {#}&\\displaystyle {{}#}\\hfil &\\unknown.", "{#}\\crcr & {\\cal L}_1 \\equiv -{\\textstyle {1\\over \\vphantom{2}\\smash{\\scriptstyle {12}}}}} ( G_{\\mu \\nu \\rho }{}^{I} )^2- {\\textstyle {1\\over \\vphantom{2}\\smash{\\scriptstyle {4}}}}$ ( HI)2 - 12$\\scriptstyle {4}$ ( FI)2   .", "(1.6) $The gauge invariances of all field strength also guarantee theconsistency of the $ A, B$~ and $  C-$field equations,such as the divergence $  D(L1 / BI )  .", "=  0$.", "{\\hspace{1.0pt}}$ )${^)~{We use the symbol ~{\\hspace{4.0pt}}{{\\hbox{\\LARGE .}}", "\\over =}\\,\\,{}~ for a field equation to be distinguished from analgebraic equation.}}", "Since we will do similar confirmation for supersymmetricsystem later, we skip the details for the purely bosonic system.$ The purpose of our present paper is to supersymmetrize this system.", "The rest of our paper is organized as follows.", "In section 2, we give the component formulation of $~N=1$   tensor multiplet (TM).", "In section 3, we give the superspace re-formulation of component result.", "In section 4, we give the generalization to non-adjoint representation of $~G=SO(N)$   case.", "In section 5, we give the supergravity coupling to non-Abelian TM, as supporting evidence for the consistency of the global case.", "Section 6 is for concluding remarks.", "Appendix A is devoted to purely bosonic systems of non-Abelian tensors with much simpler structures than has been presented in arbitrary space-time dimensions with arbitrary signature.", "An example of tensor-vector duality $~G = F^*$   in $~D=2+4$   dimensions, and its dimensional reduction (DR) into the self-dual YM $~F = F^*$   in $~D=2+2$   is also presented.", "2.", "Component Formulation of $\\,$ N=1$\\,$ TM The supersymmetrization of the purely bosonic system (1.6) is rather straightforward, except for subtlety to be mentioned later.", "Our system has three multiplets: (i) A TM $(B_{\\mu \\nu }{}^{I}, \\chi ^I, \\varphi ^I)$ , (ii) A compensating vector multiplet (CVM) $(C_{\\mu }{}^{I}, \\rho ^I)$ , and (iii) A Yang-Mills vector multiplet (YMVM) $(A_{\\mu }{}^{I}, \\lambda ^I)$ .", "Our total action $~I \\equiv \\int d^4 x\\, {\\cal L}$   has the lagrangian $2=0ptto{\\hfil \\displaystyle {#}&\\displaystyle {{}#}\\hfil &\\unknown.", "{#}\\crcr {\\cal L}= & - {\\textstyle {1\\over \\vphantom{2}\\smash{\\scriptstyle {12}}}}} ( G_{\\mu \\nu \\rho }{}^{I})^2+ {\\textstyle {1\\over \\vphantom{2}\\smash{\\scriptstyle {2}}}}$ (I D / I) - 12$\\scriptstyle {2}$ (DI)2 - 12$\\scriptstyle {2}$ g2 ( I)2 - g (I I) - 12$\\scriptstyle {4}$ (HI)2 + 12$\\scriptstyle {2}$ (I D / I) - 12$\\scriptstyle {4}$ (FI)2 + 12$\\scriptstyle {2}$ (I D / I) -12$\\scriptstyle {2}$ g fI J K (I J) K +12$\\scriptstyle {2}$ fI J K (I J) DK +12$\\scriptstyle {12}$ fI J K (I J ) GK          + 12$\\scriptstyle {4}$ fI J K (I J) FK -12$\\scriptstyle {4}$ fI J K (I J) HK -12$\\scriptstyle {2}$ fI J K FI H  J K   , (2.1) $up to quartic-order terms $ O(4)$.$ It is clear that the scalar $~\\varphi ^I$   has its mass $~g$ , while there is a mixture between $~\\chi ^I$   and $~\\rho ^I$ , again with the asme mass $~g$ .", "As has been mentioned after (1.4), $~C_{\\mu }{}^{I} $   plays the role of Stueckelberg field [7], being absorbed into the longitudinal component of $~B_{\\mu \\nu }{}^{I}$ .", "Eventually, the kinetic term of the $~C{\\hspace{1.5pt}}\\hbox{-}$ field becomes the mass term of $~B_{\\mu \\nu }{}^{I}$ .", "Accordingly, the degrees of freedom (DOF) for the massive TM fields are $~B_{\\mu \\nu }{}^{I} ~(3), ~\\rho ^I ~(4)$   and $~\\varphi ^I (1)$ , up to the adjoint index $~{\\scriptstyle I}$ .", "Our action $~I$   is invariant under global $~N=1$   supersymmetry $2=0ptto{\\hfil \\displaystyle {#}&\\displaystyle {{}#}\\hfil &\\unknown.", "{#}\\crcr \\delta _Q B_{\\mu \\nu }{}^{I} = & + (\\overline{\\epsilon }\\gamma _{\\mu \\nu } \\chi ^I)- 2 f^{I J K}C_{\\lfloor {\\hspace{0.35pt}}\\!\\!\\!\\lceil \\mu | }{}^{J} (\\delta _Q A_{| \\nu \\rfloor {\\hspace{0.35pt}}\\!\\!\\!\\rceil }{}^{K}) ~~,&(2.2{\\rm a}) \\cr \\delta _Q \\chi ^I = & + {\\textstyle {1\\over \\vphantom{2}\\smash{\\scriptstyle {6}}}}} (\\gamma ^{\\mu \\nu \\rho } \\epsilon ) G_{\\mu \\nu \\rho }{}^I- (\\gamma ^\\mu \\epsilon ) D_\\mu \\varphi ^I \\cr & + {\\textstyle {1\\over \\vphantom{2}\\smash{\\scriptstyle {2}}}}$ fI J K[ + (JK) - (5) (J 5K) - (5 ) (J 5K)]   , (2.2b) Q I = + (I)   , (2.2c) Q CI = +(I) + fI J K (J) K   , (2.2d) Q I = + 12$\\scriptstyle {2}$ () HI - g I - 12$\\scriptstyle {2}$ fI J K () FJK + 12$\\scriptstyle {4}$ fI J K [ + (J K) - () (J K ) + 12$\\scriptstyle {2}$ () (J K )                  - (5) (J 5 K ) - (5 ) (J 5K ) ]   , (2.2e) Q AI = + (I)   , (2.2f) Q I = + 12$\\scriptstyle {2}$ () FI + 12$\\scriptstyle {2}$ fI J K(5) (J5 K)   , (2.2g) $up to cubic terms $ O(3)$~ in fields.The fermionic quadratic terms in (2.2b), (2.2e) and (2.2g) are fixedin superspace formulation, as will be explained later.In the {\\it conventional} dimensionswith all the bosonic (or fermionic) fields with $  1$~ (or $  3/2$) massdimensions,{\\hspace{1.0pt}}$ )${^)~{Our bosonic (or fermionic) fields have dimensions~0 (or ~1/2), in contrast to the conventional dimensions~1 (or ~3/2).", "}}these terms lead to non-renormalizability.", "For example, the l.h.s.~of (2.2b) has dimension ~$ 3/2$, while its r.h.s.~for the $  ()$~term has $  (-1/2) + (3/2) + (3/2) = 5/2$.", "In other words, there is an implicitcoupling constant $  $~ with the dimension of length in front offermionic quadratic terms.", "This feature is also related to the existence of Pauli-terms which are non-renormalizable, already at a {\\it globally} supersymmetricsystem.", "These features are similar to supergravity[8],even though our system so far has only {\\it global} supersymmetry.$ The usual non-Abelian gauge transformation $~\\delta _\\alpha $   and our tensorial gauge transformation $~\\delta _\\beta $ , and $~\\delta _\\gamma {\\hspace{1.5pt}}\\hbox{-}$ transformation are exactly the same as (1.4), while all the fermionic fields are transforming only under $~\\delta _\\alpha $ , as the $~B$   and $~C{\\hspace{1.5pt}}\\hbox{-}$ fields do, so that there arises no problem with the $~\\delta _\\beta $   and $~\\delta _\\gamma {\\hspace{1.5pt}}\\hbox{-}$ invariances of the field strengths as in (1.5).", "These immediately lead to the invariances of our action $~\\delta _\\alpha I =0, ~\\delta _\\beta I = 0$   and $~\\delta _\\gamma I = 0$ .", "The Bianchi identities (BIds) for our field strengths $~G, ~H$   and $~F$   are: $ 2=0ptto{\\hfil \\displaystyle {#}&\\displaystyle {{}#}\\hfil &\\unknown.", "{#}\\crcr D_{\\lfloor {\\hspace{0.35pt}}\\!\\!\\!\\lceil \\mu } G_{\\nu \\rho \\sigma \\rfloor {\\hspace{0.35pt}}\\!\\!\\!\\rceil }{}^{I}- \\,\\hbox{\\large {${{\\textstyle {{3}\\over \\vphantom{2}\\smash{\\scriptstyle {{2}}}}}}$$$}\\, f^{I J K} F_{\\lfloor {\\hspace{0.35pt}}\\!\\!\\!\\lceil \\mu \\nu }{}^{J} H_{\\rho \\sigma \\rfloor {\\hspace{0.35pt}}\\!\\!\\!\\rceil }{}^{K} & \\equiv 0 ~~,&(2.3{\\rm a}) \\cr D_{\\lfloor {\\hspace{0.35pt}}\\!\\!\\!\\lceil \\mu } H_{ \\nu \\rho \\rfloor {\\hspace{0.35pt}}\\!\\!\\!\\rceil }{}^{I} - \\,\\hbox{\\large {${{\\textstyle {{1}\\over \\vphantom{2}\\smash{\\scriptstyle {{3}}}}}}$$$}\\, g\\, G_{\\mu \\nu \\rho }{}^I & \\equiv 0 ~~,&(2.3{\\rm b}) \\cr D_{\\lfloor {\\hspace{0.35pt}}\\!\\!\\!\\lceil \\mu } F_{ \\nu \\rho \\rfloor {\\hspace{0.35pt}}\\!\\!\\!\\rceil }{}^{I} & \\equiv 0 ~~.&(2.3{\\rm c}) \\cr \\crcr } $$Relevantly, the non-trivial $~\\delta _Q{\\hspace{1.5pt}}\\hbox{-}$transformationsof the field strengths are$$ 2=0ptto{\\hfil $\\displaystyle {#}$&$\\displaystyle {{}#}$\\hfil &\\unknown.", "{$#$}\\crcr \\delta _ Q G_{\\mu \\nu \\rho }{}^{I} = & + 3(\\overline{\\epsilon }\\gamma _{\\lfloor {\\hspace{0.35pt}}\\!\\!\\!\\lceil \\mu \\nu } D_{\\rho \\rfloor {\\hspace{0.35pt}}\\!\\!\\!\\rceil } \\chi ^I)+ 3 f^{I J K}(\\delta _Q A_{\\lfloor {\\hspace{0.35pt}}\\!\\!\\!\\lceil \\mu }{}^{J}) H_{\\nu \\rho \\rfloor {\\hspace{0.35pt}}\\!\\!\\!\\rceil }{}^{K}- 3 f^{I J K}(\\delta _Q C_{\\lfloor {\\hspace{0.35pt}}\\!\\!\\!\\lceil \\mu }{}^{J}) F_{\\nu \\rho \\rfloor {\\hspace{0.35pt}}\\!\\!\\!\\rceil }{}^{K} ~~, ~~~~~&(2.4{\\rm a}) \\cr \\delta _Q H_{\\mu \\nu }{}^{I} = & - 2 (\\overline{\\epsilon }\\gamma _{\\lfloor {\\hspace{0.35pt}}\\!\\!\\!\\lceil \\mu } D_{\\nu \\rfloor {\\hspace{0.35pt}}\\!\\!\\!\\rceil } \\rho ^I)+ g (\\overline{\\epsilon }\\gamma _{\\mu \\nu } \\chi ^I )+ 2 f^{I J K}D_{\\lfloor {\\hspace{0.35pt}}\\!\\!\\!\\lceil \\mu |}\\left[ (\\delta _Q A_{| \\nu \\rfloor {\\hspace{0.35pt}}\\!\\!\\!\\rceil }{}^{J} ) \\varphi ^K \\right] ~~,&(2.4{\\rm b}) \\cr \\delta _Q F_{\\mu \\nu }{}^{I} = & - 2 (\\overline{\\epsilon }\\gamma _{\\lfloor {\\hspace{0.35pt}}\\!\\!\\!\\lceil \\mu } D_{\\nu \\rfloor {\\hspace{0.35pt}}\\!\\!\\!\\rceil } \\lambda ^I) ~~,&(2.4{\\rm c}) \\cr \\crcr } $$reflecting the presence of CS terms.", "}}$ Note that our YMVM and CVM has on-shell DOF 2+2, while off-shell DOF 3+4, because we have not added the $~D{\\hspace{1.5pt}}\\hbox{-}$ auxiliary field.", "On the other hand, our TM is in the off-shell formulation, because the total off-shell DOF is $~4+4$ , because the off-shell DOF of each field are $~[(4-1)\\cdot (4-2)] / 2 = 3$   for $~B_{\\mu \\nu }$ , $~4$   for $~\\chi $   and $~1$   for $~\\varphi $ .", "The field equations for $~\\lambda ^I, ~\\chi ^I,~\\rho ^I, ~A_{\\mu }{}^{I}, ~B_{\\mu \\nu }{}^{I}, ~\\varphi ^I$   and $~C_{\\mu }{}^{I}$   are respectively$^)$$^)$  These equations are fixed up to $~{\\cal O}({\\phi }^{3})\\,$ -terms, due to the quartic fermion terms in the lagrangian.", "$2=0ptto{\\hfil \\displaystyle {#}&\\displaystyle {{}#}\\hfil &\\unknown.", "{#}\\crcr & + {}D \\!\\!\\!\\!", "/{\\,}\\lambda ^I - {\\textstyle {1\\over \\vphantom{2}\\smash{\\scriptstyle {2}}}}} g f^{I J K}\\chi ^J \\varphi ^K+ {\\textstyle {1\\over \\vphantom{2}\\smash{\\scriptstyle {2}}}}$ fI J K(J) DK                                     - 12$\\scriptstyle {4}$ fI J K( J) H K + 12$\\scriptstyle {12}$ fI J K( J) GK  .", "=  0   ,           (2.5a) + D / I - g I + 12$\\scriptstyle {2}$ g fI J KH K - 12$\\scriptstyle {4}$ fI J K( J) HK + 12$\\scriptstyle {4}$ fI J K( J) FK  .", "=  0   ,           (2.5b) + D / I - g I + 12$\\scriptstyle {2}$ fI J K(J) DK                      - 12$\\scriptstyle {12}$ fI J K( J) GK + 12$\\scriptstyle {4}$ fI J K( J) FK  .", "=  0    ,             (2.5c) + DF  I + g fI J KJ DK + 12$\\scriptstyle {2}$ g fI J K(J K) + fI J KHJ DK                   - 12$\\scriptstyle {2}$ fI J KGJ H  K + 12$\\scriptstyle {2}$ fI J K(J DK) + 12$\\scriptstyle {2}$ fI J K(J DK)  .", "=  0   , (2.5d) + DG  I - g H  I - 12$\\scriptstyle {2}$ fI J KD(J K)                                     + g fI J KF  J K - 12$\\scriptstyle {2}$ g fI J K(J K)  .", "=  0   , (2.5e) + D2 I - g fI J K(JK) - g2 I - 12$\\scriptstyle {2}$ fI J KFJ H  K  .", "=  0  , (2.5f) + DH  I - 12$\\scriptstyle {2}$ fI J KFJ G  K - 12$\\scriptstyle {2}$ fI J K(J DK) - 12$\\scriptstyle {2}$ fI J K(J DK)                                                   + 12$\\scriptstyle {2}$ g fI J K(J K) - fI J KF  J DK  .", "=  0   .", "(2.5g) $In the derivation of these field equations, we have also usedother field equations, in order to simply their final expressions, asa conventional prescription.$ In the above computation, we do not attempt to fix the $~{\\cal O}({\\phi }^{3}){\\hspace{1.5pt}}\\hbox{-}$ terms in field equations, or equivalently the fermionic $~{\\cal O}({\\phi }^{4}) {\\hspace{1.5pt}}\\hbox{-}$ terms in the lagrangian.", "There are several remarks about these terms.", "First, our system is non-renormalizable as supergravity theory [8], as has been mentioned after eq.$\\,$ (2.2).", "Accordingly, the $\\,(\\hbox{fermion})^2{\\hspace{1.5pt}}\\hbox{-}$ terms in the fermionic transformations such as (2.2b), (2.2e) and (2.2g) are accompanied by the implicit constant  $\\ell $   carrying the dimension of $~(\\hbox{legnth})$ .", "In supergravity theory [8], this is the gravitational coupling $~\\kappa $ .", "In our lagrangian, all the quartic-fermion terms carry $~\\ell ^2$ , so that the lagrangian has the mass dimension $+4$ .", "Accordingly, a typical Noether-term has the structure $~\\ell \\,\\Psi ^2 \\,\\partial \\,\\Phi $ , that produces the terms of the form $~\\ell ^2 \\, \\epsilon \\,\\Psi ^3 \\,\\partial \\,\\Phi $   via $~\\delta _Q \\,\\Psi \\approx \\,\\ell \\, \\epsilon \\, \\Psi ^2$ .", "Here $~\\Psi $   (or $~\\Phi $ ) is a general fermionic (or bosonic) fundamental field.", "These $~\\ell ^2 \\,\\epsilon \\,\\Psi ^3 \\,\\partial \\,\\Phi {\\hspace{1.5pt}}\\hbox{-}$ terms are cancelled by the variation of the fermionic quartic terms $~\\ell ^2\\, \\Psi ^4$ , via $~\\delta _Q \\Psi \\approx \\epsilon \\, \\partial \\Phi $ .", "In other words, the structure of these cancellations associated with quartic-fermion terms is parallel to supergravity [8], since $~\\ell $   is analogous to $~\\kappa $ .", "However, in our peculiar system, this cancellation mechanism may be not simply parallel to conventional supergravity [8].", "For example, there may be $~\\ell ^2\\Psi ^2 \\Phi \\partial \\Psi {\\hspace{1.5pt}}\\hbox{-}$ type terms in the action, while $~\\ell ^2 \\epsilon \\Psi ^2\\Phi {\\hspace{1.5pt}}\\hbox{-}$ type terms in the transformation rules may exist, because both of them yield $~\\ell ^2\\epsilon \\Psi ^3 \\partial \\Phi {\\hspace{1.5pt}}\\hbox{-}$ type terms, canceling each other in $~\\delta _Q I$ .", "At the present time, we do not know, if such terms arise, because the $~\\ell ^2 \\epsilon \\Psi ^2\\Phi {\\hspace{1.5pt}}\\hbox{-}$ type terms in transformations are at $~{\\cal O}({\\phi }^{3})$ , while $~\\ell ^2\\Psi ^2 \\Phi \\partial \\Psi {\\hspace{1.5pt}}\\hbox{-}$ type terms in the action are at $~{\\cal O}({\\phi }^{4})$ .", "In fact, even in the superspace re-confirmation in the next section, we have fixed only the $~{\\cal O}({\\phi }^{1})$   and $~{\\cal O}({\\phi }^{2}){\\hspace{1.5pt}}\\hbox{-}$ terms in the transformation rules for fermions, such as (3.2d), (3.2e) and (3.2f), but not cubic terms $~{\\cal O}({\\phi }^{3})$ .", "Our consistent principle in this paper is to fix only $~{\\cal O}({\\phi }^{1}), ~{\\cal O}({\\phi }^{2})$  and $~{\\cal O}({\\phi }^{3}){\\hspace{1.5pt}}\\hbox{-}$ terms in the lagrangian, $~{\\cal O}({\\phi }^{1})$   and $~{\\cal O}({\\phi }^{2}){\\hspace{1.5pt}}\\hbox{-}$ terms in all transformation rules, while $~{\\cal O}({\\phi }^{1})$   and $~{\\cal O}({\\phi }^{2}){\\hspace{1.5pt}}\\hbox{-}$ terms in all field equations.", "However, we try to fix neither $~{\\cal O}({\\phi }^{4}){\\hspace{1.5pt}}\\hbox{-}$ terms in the lagrangian, nor $~{\\cal O}({\\phi }^{3}){\\hspace{1.5pt}}\\hbox{-}$ term in all transformation rules, nor $~{\\cal O}({\\phi }^{3}){\\hspace{1.5pt}}\\hbox{-}$ terms in all field equations.", "We do not specify each field meant by $~\\phi $   is fermionic or bosonic in this paper, either.", "Second, as an additional difference from supergravity [8], the fermionic quartic terms do not contain any gravitino.", "This implies that we can not use the conventional technique of `supercovariantizing' fermionic field equations.", "Due to this feature, as well as the above-mentioned possible non-purely-fermionnic $~\\ell ^2\\Psi ^2 \\Phi \\partial \\Psi {\\hspace{1.5pt}}\\hbox{-}$ type terms, the quartic terms $~{\\cal O}({\\phi }^{4})$   at $~{\\cal O}({\\ell }^{2})$   will be more involved than conventional supergravity [8] which are tedious.", "For these reasons, we do not attempt to fix them in this paper.", "Third, according to the past experience in supergravity theory [8], it is understood that the series in terms of $~\\kappa $   in a lagrangian will stop at a finite order, such as the quartic-fermion terms at $~{\\cal O}({\\kappa }^{2})$ [8].", "However, at the present time, we do not know, whether this is also the case with our globally supersymmetric system.", "This is because of the above-mentioned differences of our system from supergravity [8], and therefore the analogy with supergravity might be not valid in our system.", "Fourth, since we have already fixed the cubic terms in the lagrangian, they seem sufficient for non-trivial and consistent couplings as a supersymmetric system.", "3.", "Superspace Reformulation of $\\,$ N=1$\\,$ TM As a reconfirmation of the total consistency of our system, we re-formulate our theory in terms of superspace language.", "Our basic superspace BIds for the superfield strengths $~F_{A B}{}^{I},~G_{A B C}{}^{I}$   and $~H_{A B}{}^{I}$   are$^)$$^)$  Only in this superspace section, we use the indices $\\,{\\scriptstyle A~=~(a,\\alpha ),~B~=~(b,\\beta ),~\\cdots }\\,$ for superspace coordinates, where $\\,{\\scriptstyle a,~b,~\\cdots ~=~0,\\,1,\\,2,\\,3}\\,$ (or $~{\\scriptstyle \\alpha ,~\\beta ,~\\cdots ~=~1,\\,2,\\,3,\\,4}$ ) are for bosonic (or fermionic) coordinates.", "In superspace, the (anti)symmetrization convention, e.g., $X_{\\lfloor {\\hspace{0.35pt}}\\!\\!\\!\\lceil A B)}\\equiv X_{A B} -(-1)^{A B} X_{B A}$   is different from our component notation.", "$ 2=0ptto{\\hfil \\displaystyle {#}&\\displaystyle {{}#}\\hfil &\\unknown.", "{#}\\crcr + \\,\\hbox{\\large {${{\\textstyle {{1}\\over \\vphantom{2}\\smash{\\scriptstyle {{6}}}}}}$$$}\\, \\nabla _{\\lfloor {\\hspace{0.35pt}}\\!\\!\\!\\lceil A} G_{B C D)}{}^{I}- \\,\\hbox{\\large {${{\\textstyle {{1}\\over \\vphantom{2}\\smash{\\scriptstyle {{4}}}}}}$$$}\\, T_{\\lfloor {\\hspace{0.35pt}}\\!\\!\\!\\lceil A B|}{}^{E} G_{E| C D)}-\\,\\hbox{\\large {${{\\textstyle {{1}\\over \\vphantom{2}\\smash{\\scriptstyle {{4}}}}}}$$$}\\, f^{I J K} F_{\\lfloor {\\hspace{0.35pt}}\\!\\!\\!\\lceil A B}{}^{J} H_{C D)}{}^{K} & \\equiv 0 ~~,& (3.1{\\rm a}) \\cr + \\,\\hbox{\\large {${{\\textstyle {{1}\\over \\vphantom{2}\\smash{\\scriptstyle {{2}}}}}}$$$}\\, \\nabla _{\\lfloor {\\hspace{0.35pt}}\\!\\!\\!\\lceil A} H_{ B C)}{}^{I}- \\,\\hbox{\\large {${{\\textstyle {{1}\\over \\vphantom{2}\\smash{\\scriptstyle {{2}}}}}}$$$}\\, T_{\\lfloor {\\hspace{0.35pt}}\\!\\!\\!\\lceil A B| }{}^{D} H_{ D| C)}{}^{I} - g \\, G_{A B C}{}^{I}& \\equiv 0 ~~,& (3.1{\\rm b}) \\cr + \\,\\hbox{\\large {${{\\textstyle {{1}\\over \\vphantom{2}\\smash{\\scriptstyle {{2}}}}}}$$$}\\, \\nabla _{\\lfloor {\\hspace{0.35pt}}\\!\\!\\!\\lceil A} F_{ B C)}{}^{I}- \\,\\hbox{\\large {${{\\textstyle {{1}\\over \\vphantom{2}\\smash{\\scriptstyle {{2}}}}}}$$$}\\, T_{\\lfloor {\\hspace{0.35pt}}\\!\\!\\!\\lceil A B| }{}^{D} F_{ D| C)}{}^{I} & \\equiv 0 ~~.& (3.1{\\rm b}) \\cr \\crcr } $$These BIds are the superspace generalizations of the component BIds(2.3), with the supertorsion terms added for local Lorentz indices, as usualin superspace.", "}Our basic superspace constraints at mass dimensions $~0 \\le d \\le 1$~ are$$2=0ptto{\\hfil $\\displaystyle {#}$&$\\displaystyle {{}#}$\\hfil &\\unknown.", "{$#$}\\crcr T_{\\alpha \\beta }{}^{c} = & + 2 (\\gamma ^c)_{\\alpha \\beta }~~, ~~~~G_{\\alpha \\beta c}{}^{I} = + 2 (\\gamma _c)_{\\alpha \\beta } \\, \\varphi ^I~~,&(3.2{\\rm a}) \\cr G_{\\alpha b c}{}^{I} = & - (\\gamma _{b c}\\chi ^I )_\\alpha ~~, ~~~~H_{\\alpha b}{}^{I} = - (\\gamma _b\\rho ^I)_\\alpha - f^{I J K} (\\gamma _b\\lambda ^J)_\\alpha \\, \\varphi ^K ~~,&(3.2{\\rm b}) \\cr F_{\\alpha b}{}^{I} = & - (\\gamma _b \\lambda ^I)_\\alpha ~~, ~~~~\\nabla _\\alpha \\varphi ^I = - \\chi _{\\alpha }{}^{I} ~~,&(3.2{\\rm c}) \\cr \\nabla _\\alpha \\chi _{\\beta }{}^{I} = & - \\,\\hbox{\\large {${{\\textstyle {{1}\\over \\vphantom{2}\\smash{\\scriptstyle {{6}}}}}}$$$}\\, (\\gamma ^{c d e})_{\\alpha \\beta } G_{c d e}{}^{I}- (\\gamma ^c)_{\\alpha \\beta } \\nabla _c \\varphi ^I \\cr & - {\\textstyle {1\\over \\vphantom{2}\\smash{\\scriptstyle {2}}}}} f^{I J K} \\Big [ + C_{\\alpha \\beta } (\\overline{\\lambda }{}^J \\rho ^K)- (\\gamma _5\\gamma ^c)_{\\alpha \\beta } (\\overline{\\lambda }{}^J \\gamma _5\\gamma _c\\rho ^K)- (\\gamma _5)_{\\alpha \\beta } (\\overline{\\lambda }{}^J \\gamma _5 \\rho ^K) \\Big ]~~, ~~~~~&(3.2{\\rm d}) \\cr \\nabla _\\alpha \\rho _{\\beta }{}^{I} = & + {\\textstyle {1\\over \\vphantom{2}\\smash{\\scriptstyle {2}}}}} (\\gamma ^{c d})_{\\alpha \\beta } H_{c d}{}^{I}+ g \\, C_{\\alpha \\beta } \\, \\varphi ^I- {\\textstyle {1\\over \\vphantom{2}\\smash{\\scriptstyle {2}}}}} f^{I J K} (\\gamma ^{c d})_{\\alpha \\beta } F_{c d}{}^{J} \\varphi ^K \\cr & - {\\textstyle {1\\over \\vphantom{2}\\smash{\\scriptstyle {4}}}}} f^{I J K} \\Big [ + C_{\\alpha \\beta } (\\overline{\\lambda }{}^J \\chi ^K)+ (\\gamma ^c)_{\\alpha \\beta } \\, (\\overline{\\lambda }{}^J \\gamma _c \\chi ^K)- {\\textstyle {1\\over \\vphantom{2}\\smash{\\scriptstyle {2}}}}} (\\gamma ^{c d})_{\\alpha \\beta } (\\overline{\\lambda }{}^J \\gamma _{c d}\\chi ^K) \\cr & ~~~~~ ~~~~~ ~~ \\,\\, - (\\gamma _5\\gamma ^c)_{\\alpha \\beta } (\\overline{\\lambda }{}^J \\gamma _5\\gamma _c\\chi ^K)- (\\gamma _5)_{\\alpha \\beta } (\\overline{\\lambda }{}^J \\gamma _5 \\chi ^K) ~~,&(3.2{\\rm e}) \\cr \\nabla _\\alpha \\lambda _{\\beta }{}^{I} = & + {\\textstyle {1\\over \\vphantom{2}\\smash{\\scriptstyle {2}}}}}(\\gamma ^{c d})_{\\alpha \\beta } F_{c d}{}^{I}- {\\textstyle {1\\over \\vphantom{2}\\smash{\\scriptstyle {2}}}}} (\\gamma _5)_{\\alpha \\beta } \\, f^{I J K} (\\overline{\\rho }{}^J\\gamma _5\\chi ^K) ~~.&(3.2{\\rm f}) \\cr \\crcr } $ All other components, such as $~G_{\\alpha \\beta \\gamma }{}^I, ~T_{\\alpha \\beta }{}^{\\gamma }, ~T_{a b}{}^{c}, ~H_{\\alpha \\beta }{}^{I} $   etc.", "at $~d\\le 1$   are zero.", "Note that $~\\hbox{(fermion)}^2{\\hspace{1.5pt}}\\hbox{-}$ terms in (3.2d) through (3.2f) have been determined in superspace by satisfying BIds at  $d=1$ .", "Note that these results are valid up to $~{\\cal O}({\\phi }^{3}){\\hspace{1.5pt}}\\hbox{-}$ terms, which we do not attempt to fix these terms in this paper.", "However, all the $~{\\cal O}({\\phi }^{2}){\\hspace{1.5pt}}\\hbox{-}$ terms have been included, as has been also mentioned at the end of last section.", "There are also useful relationships obtained from $~d = + 3/2$   BIds: $2=0ptto{\\hfil \\displaystyle {#}&\\displaystyle {{}#}\\hfil &\\unknown.", "{#}\\crcr \\nabla _\\alpha G_{b c d} = & - {\\textstyle {1\\over \\vphantom{2}\\smash{\\scriptstyle {2}}}}} (\\gamma _{\\lfloor {\\hspace{0.35pt}}\\!\\!\\!\\lceil b c} \\nabla _{d\\rfloor {\\hspace{0.35pt}}\\!\\!\\!\\rceil } \\chi ^I )_\\alpha - {\\textstyle {1\\over \\vphantom{2}\\smash{\\scriptstyle {2}}}}$ fI J K (b | J)H| c dK + 12$\\scriptstyle {2}$ fI J K (b | J)F| c dK   ,           (3.3a) Hb cI = + ( b c I)- g (b c I)- fI J K b [ (c J)K]   , (3.3b) Fb cI = + ( b c I)  , (3.3c) $up to $ O(3)-$terms.Note the existence of the $  O(2)-$terms in (3.3a) and (3.3b),reflecting the corresponding terms in the component results (2.4a)and (2.4b).$ As usual, the satisfaction of all the BIds in superspace by the constraints (3.2) and (3.3) is straightforward to perform, from the dimension $~d=0$   to $~d=3/2$ , as usual.", "In particular, the $\\,(\\hbox{Fermions})^2{\\hspace{1.5pt}}\\hbox{-}$ terms in (3.2d) through (3.2f) are the results of our superspace re-formulation.", "The fermionic $\\,\\lambda $   and $\\,\\rho {\\hspace{1.5pt}}\\hbox{-}$ field equations (2.5a) and (2.5c) are obtained as usual by computing $~\\lbrace \\nabla _\\alpha , \\nabla _\\beta \\rbrace \\, \\lambda ^{\\beta I}$   and $~\\lbrace \\nabla _\\alpha , \\nabla _\\beta \\rbrace \\, \\rho ^{\\beta I}$ , while the $~\\chi {\\hspace{1.5pt}}\\hbox{-}$ field equation is shown to be consistent with the component lagrangian.", "As has been mentioned, since the TM is off-shell multiplet, we can not get the $\\,\\chi {\\hspace{1.5pt}}\\hbox{-}$ field equation (2.5b) in superspace directly, but we can show that (2.5b) is consistent in superspace.", "The bosonic field equations (2.5d) - (2.5g) are obtained by applying another fermionic derivative on the fermionic field equations (2.5a) - (2.5c).", "4.", "Generalization to Non-Adjoint Representations of $\\,$ G = SO(N) We have so far considered the case for the TM and CVM both carrying only the adjoint representation.", "We can generalize this result to other more general representations, such as an arbitrary real representation of a $~SO(N){\\hspace{1.5pt}}\\hbox{-}$ type gauge group.$^)$$^)$  We can also consider the complex representation for $~SU(N)\\, $ -type gauge groups.", "To be more specific, we consider the TM $~(B_{\\mu \\nu }{}^{i}, \\chi ^i, \\varphi ^i)$   and the CVM $~(C_{\\mu }{}^{i}, \\rho ^i)$ , where the index $~{\\scriptstyle i}$   is for any real representation of a gauge group $~G = SO(N)$ .", "Let $~(T^I)^{j k}$   be the generator of the group $~G$ .", "Then our action $~I^{\\prime } \\equiv \\int d^4 x \\, {\\cal L}^{\\prime }$   has the lagrangian$^)$$^)$  Since the metric for the gauge group $~G = SO(N)$   is positive definite, we do not distinguish the upper or lower indices for $~{\\scriptstyle i,~j,~\\cdots ~=~1,~2,~\\cdots ,~\\hbox{dim} \\, R}$ , where $~{\\bf R}$   is a real representation of $~G$ .", "$2=0ptto{\\hfil \\displaystyle {#}&\\displaystyle {{}#}\\hfil &\\unknown.", "{#}\\crcr {\\cal L}^{\\prime } = & - {\\textstyle {1\\over \\vphantom{2}\\smash{\\scriptstyle {12}}}}} ( G_{\\mu \\nu \\rho }{}^{i})^2+ {\\textstyle {1\\over \\vphantom{2}\\smash{\\scriptstyle {2}}}}$ (i D / i) - 12$\\scriptstyle {2}$ (Di)2 - 12$\\scriptstyle {2}$ g2 ( i)2 - g (i i) - 12$\\scriptstyle {4}$ (Hi)2 + 12$\\scriptstyle {2}$ (i D / i) - 12$\\scriptstyle {4}$ (FI)2 + 12$\\scriptstyle {2}$ (I D / I) - 12$\\scriptstyle {2}$ g (TI)j k (I j)   k + 12$\\scriptstyle {2}$ (TI)j k (I j) Dk + 12$\\scriptstyle {12}$ (TI)j k (I j )   Gk          + 12$\\scriptstyle {4}$ (TI)j k(j k) FI - 12$\\scriptstyle {4}$ (TI)j k (I j) Hk - 12$\\scriptstyle {2}$ (TI)j k FI H  j k   , (4.1) $up to quartic terms $ O(4)$.Our action $  I'$~ is invariant under global $  N=1$~ supersymmetry$$2=0ptto{\\hfil \\displaystyle {#}&\\displaystyle {{}#}\\hfil &\\unknown.", "{#}\\crcr \\delta _Q B_{\\mu \\nu }{}^{i} = & + (\\overline{\\epsilon }\\gamma _{\\mu \\nu } \\chi ^i)- 2 (T^J)^{i k} C_{\\lfloor {\\hspace{0.35pt}}\\!\\!\\!\\lceil \\mu |}{}^{k} (\\delta _Q A_{| \\nu \\rfloor {\\hspace{0.35pt}}\\!\\!\\!\\rceil }{}^{J} ) ~~,&(4.2{\\rm a}) \\cr \\delta _Q \\chi ^i = & + {\\textstyle {1\\over \\vphantom{2}\\smash{\\scriptstyle {6}}}}} (\\gamma ^{\\mu \\nu \\rho } \\epsilon ) G_{\\mu \\nu \\rho }{}^i- (\\gamma ^\\mu \\epsilon ) D_\\mu \\varphi ^i \\cr & - {\\textstyle {1\\over \\vphantom{2}\\smash{\\scriptstyle {2}}}}$ (TJ)i k [ + (J k) - (5) (J 5 k) - (5 ) (J 5k)]   , (4.2b) Q i = + (i)   , (4.2c) Q Ci = + (i) - (TJ)i k (J) k   , (4.2d) Q i = + 12$\\scriptstyle {2}$ () Hi - g i + 12$\\scriptstyle {2}$ (TJ)i k () FJ k - 12$\\scriptstyle {4}$ (TJ)i k [ + (J k) - () (J k) + 12$\\scriptstyle {2}$ () (J k )                   - (5) (J 5 k ) - (5 ) (J 5k ) ]   , (4.2e) Q AI = + (I)   , (4.2f) Q I = + 12$\\scriptstyle {2}$ () FI - 12$\\scriptstyle {2}$ (TI)j k (5) (j5 k)   .", "(4.2g) $$ The essential point is that all the cubic-order terms contain one component field $~A_{\\mu }{}^{I}$   or $~\\lambda ^I$   with the index $~{\\scriptstyle I}$ , and the remaining two component fields out of either TM or CVM carry the indices $~{\\scriptstyle j}$   and $~{\\scriptstyle k}$ .", "So the cancellation structure is parallel to the adjoint-representation case, e.g., with the structure constant $~f^{I J K}$   replaced by the matrix $~- (T^J)^{i k}$   in $~D_\\mu \\chi ^I = \\partial _\\mu \\chi ^I + g f^{I J K}A_{\\mu }{}^{J} \\chi ^K ~~\\Longrightarrow ~~D_\\mu \\chi ^i = \\partial _\\mu \\chi ^i - g (T^J)^{i k} A_{\\mu }{}^{J} \\chi ^k$ .", "Accordingly, the Stueckelberg mechanism [7] works in a parallel fashion, because $~C_{\\mu }{}^{i}$   is absorbed into the longitudinal component of $~B_{\\mu \\nu }{}^{i}$ , both in the same representation $~{\\bf R}$ .", "5.", "Coupling to $~N=1$   Supergravity Once we have established the $~N=1$   global system of non-Abelian TM with non-trivial and consistent interactions, the next natural step is to make $~N=1$   supersymmetry local, coupling to $~N=1$   supergravity.", "This coupling is rather straightforward, because most of the basic structure is parallel to the usual matter coupling to supergravity, except for certain couplings to be mentioned later.", "Our result for the lagrangian $~{\\widetilde{{\\cal L}}}\\hspace{1.084pt}$   of our action is $~ {\\widetilde{I}}\\hspace{1.084pt}\\equiv \\int d^4 x\\, {\\widetilde{{\\cal L}}}\\hspace{1.084pt}$ : $ 2=0ptto{\\hfil \\displaystyle {#}&\\displaystyle {{}#}\\hfil &\\unknown.", "{#}\\crcr e^{-1} {\\widetilde{{\\cal L}}}\\hspace{1.084pt}= & - {\\textstyle {1\\over \\vphantom{2}\\smash{\\scriptstyle {4}}}}} R (\\omega ) - \\left[ \\overline{\\psi }_\\mu \\gamma ^{\\mu \\nu \\rho }D_\\nu (\\omega ) \\psi _\\rho \\right]- {\\textstyle {1\\over \\vphantom{2}\\smash{\\scriptstyle {12}}}}$ ( GI)2 + 12$\\scriptstyle {2}$ [   I D / ()I ] -" ] ]

[ [ "Bounds for Fisher information and its production under flow" ], [ "Abstract We prove that two well-known measures of information are interrelated in interesting and useful ways when applied to nonequilibrium circumstances.", "A nontrivial form of the lower bound for the Fisher information measure is derived in presence of a flux vector, which satisfies the continuity equation.", "We also establish a novel upper bound on the time derivative (production) in terms of the arrow of time and derive a lower bound by the logarithmic Sobolev inequality.", "These serve as the revealing dynamics of the information content and its limitations pertaining to nonequilibrium processes." ], [ "Introduction", "A fundamental aspect of physical systems being out of equilibrium lies in the existence of limits in information measures that the systems possess.", "It is known that a few measures of information only determined by probability distributions are closely related, but studies on their bounds had not been done until relatively recently when the flow vectors exist.", "The Fisher information measure, hereafter denoted as $I$ , is an intriguing measure behind physical laws [1] and it appears as the basic ingredient in bounding entropy production.", "Specifically, the time derivative of the Shannon entropy $dS/dt$ is bounded by $I$ [2], [3].", "In this sense, there is a fundamental motivation in probing the interplay between physical entities and information.", "Fisher information is defined using the time-dependent probability distribution $f(\\vec{x},t)$ as $I(t)=\\left\\langle \\left(\\frac{\\nabla f}{f}\\right)^2\\right\\rangle _f = \\sum _{k=1}^d\\left\\Vert \\frac{\\nabla _k f}{\\sqrt{f}}\\right\\Vert _2^2,$ where $\\langle \\cdot \\rangle _f$ denotes the average with $f$ over the domain of phase space coordinate $\\vec{x}$ , and the $\\Vert \\cdot \\Vert _2$ is $L^2$ norm.", "In the second expression, we use $\\nabla _k=\\partial /\\partial x_k, (k=1,\\ldots , d)$ with dimension $d$ .", "At the moment of research, two different types of lower bounds are obtained through the upper bound on the temporal change of entropy under the Dirichlet boundary conditions.", "More specifically, the out-flux vector $\\vec{j}(\\vec{x},t)$ and the quantity $\\vec{j}(\\vec{x},t)\\ln f$ are required to vanish at the boundary in combination with the continuity equation for probability density $\\frac{\\partial f(\\vec{x},t)}{\\partial t}+\\nabla \\cdot \\vec{j}(\\vec{x},t)=0.$ One of these expressions was given by Nikolov and Frieden [2] as $\\frac{dS}{dt}\\leqslant \\frac{1}{2d}I(t) \\frac{d\\langle r^2\\rangle }{dt},$ where $d$ is the spatial dimension and $\\langle r^2 \\rangle $ the mean square displacement of the particle investigated.", "The fact that the entropy increase rate never reaches infinity but is appropriately suppressed by two characteristic quantities offers deep insight.", "That is, it bridges between the information theoretical aspect through the shape of probability distribution and the kinematic one through the speed of particle.", "The other one was expressed by Brody and Meister [3] as $\\frac{dS}{dt}\\leqslant \\gamma \\sqrt{I},$ where coefficient $\\gamma $ is given as $\\langle (\\vec{j}/f)^2\\rangle _f$ .", "Moreover, this bound represents the collaborative effect of $I$ and the flow of transmitted matter.", "We can consider this formula as more versatile than Eq.", "(REF ), since it gives the bound if we specify a physical model (i.e., the form of the flux).", "On the other hand, a lower bound on $I$ itself was alternatively obtained as $\\langle \\nabla \\cdot (\\vec{j}/f)\\rangle _f^2/\\gamma \\leqslant I$ [4].", "Furthermore, according to the Fickian diffusion law, $\\vec{j}=-D\\nabla f$ with the diffusion constant $D$ , we can easily obtain $dS/dt=D\\cdot I \\geqslant 0$ .", "In either case, the quantity $I$ is found to play a vital role in limiting the entropy production.", "The present consideration is deeply rooted in the unidirectional characteristics of time (the arrow of time).", "Arrow of time is commonly understood as the consequence of the transformation of local information into nonlocal correlations, i.e., the production of information measure.", "In this sense, we focus on the rate of change of the Fisher information with time, which is termed as ”Fisher information production” $dI/dt$ in analogy to the entropy production $dS/dt$ , and we will present the upper bound for it in a general setting.", "The present consideration is of high importance not only in terms of the notion of the arrow of time, based on $I$ , but also from the viewpoint of the second law of thermodynamics.", "In Sec.", "II, we derive the lower bounds for $I$ with and without the flow and determine the form of the upper bound for $dI/dt$ .", "We present examples for these bounds in Sec.", "III.", "The lower bounds on the Fisher information production are derived in terms of Shannon entropy production via the logarithmic Sobolev inequality in Sec.", "IV.", "Finaly, we discuss the conclusions.", "Since the continuity of $\\nabla f$ is not obvious from the onset for a given $f$ in general, we first need to remark that we assume Leibniz's rule for differentiation under the integral sign; i.e., $dI/dt=\\int _{\\mathbb {R}^1} \\frac{\\partial }{\\partial t}((\\nabla f)^2/f) dx$ is assumed to be guaranteed.", "This implies that, in case $x$ and $t$ are in closed intervals, $(\\nabla f)^2/f$ is continuous for $x$ and $t$ .", "Then, $\\frac{\\partial }{\\partial t}((\\nabla f)^2/f)$ is also continuous.", "In case the integral involves an infinite domain, we require $\\int _{w}^{\\infty } \\frac{\\partial }{\\partial t}((\\nabla f)^2/f) dx$ to uniformly converge with respect to $t$ , where $w$ irrelevant to $t$ , such that $|\\int _{w}^{\\infty } \\frac{\\partial }{\\partial t}((\\nabla f)^2/f) dx| <\\epsilon $ , $\\forall \\epsilon $ .", "In this context, we obtain the time derivative of the one-dimensional Fisher information, which yields $\\frac{dI}{dt}=\\int _{\\mathbb {R}^1} \\frac{\\partial }{\\partial t}\\left( \\frac{(\\nabla f)^2}{f}\\right)dx=\\int _{\\mathbb {R}^1} \\left[ \\left(\\frac{\\nabla f}{f}\\right)^2\\nabla \\cdot \\vec{j}-2\\frac{\\nabla f}{f}\\cdot \\nabla (\\nabla \\cdot \\vec{j})\\right]dx, $ in which we have used the interchangeability $f_{xt}=f_{tx}$ , since we are assuming that both $f_{xt}$ and $f_{tx}$ are continuous.", "In addition, we have substituted the continuity equation $\\partial _t f=-\\nabla \\cdot \\vec{j}$ .", "Here we continue using the symbol $\\nabla $ instead of $\\partial /\\partial x$ because we want to include the high-dimensional case for consideration.", "The fact that $I$ decreases with time and attains its minimum has been shown in several contexts.", "Indeed, the never-increasing property $dI/dt\\leqslant 0$ holds true for solutions of the Fokker-Planck equation [5], indicating that the minimum Fisher information (MFI) principle [6] induces the existence of time asymmetry in terms of $I$ [7].", "Accordingly, it is apparently legitimate to expect that in a wider class of time evolution, possessing the continuity equation, the Fisher information production is negative.", "In Sec.IV, however, we note a situation, in which this qualification is not satisfied.", "Now, imposing the decreasing $I$ , we obtain the following inequality for the terms specified in Eq.", "(REF ) $\\int _{\\mathbb {R}^1} \\left[(\\nabla \\ln f)^2\\nabla \\cdot \\vec{j}\\right]dx \\leqslant 2\\int _{\\mathbb {R}^1} \\frac{\\nabla f}{\\sqrt{f}}\\cdot \\frac{\\nabla (\\nabla \\cdot \\vec{j})}{\\sqrt{f}} dx\\leqslant 2 \\sqrt{\\int _{\\mathbb {R}^1} \\frac{(\\nabla f)^2}{f}dx}\\sqrt{\\int _{\\mathbb {R}^1} \\frac{\\left[\\nabla (\\nabla \\cdot \\vec{j})\\right]^2}{f}dx}.$ The last inequality follows from the Schwarz inequality.", "Therefore, we obtain a lower bound when the leftmost term of the above inequality is a positive quantity $I \\geqslant \\frac{1}{4}\\frac{\\left( \\int _{\\mathbb {R}^1} dx(\\nabla \\ln f)^2\\nabla \\cdot \\vec{j}\\right)^2}{\\left\\langle \\left[\\frac{\\nabla (\\nabla \\cdot \\vec{j})}{f}\\right]^2\\right\\rangle }.$ This is equivalent to considering only systems with positive divergence flows.", "We note also that the inequality becomes valid when the absolute value of the leftmost part is less than or equal to the right hand side, i.e., $|\\int _{\\mathbb {R}^1} dx(\\nabla \\ln f)^2\\nabla \\cdot \\vec{j}|\\leqslant 2I\\sqrt{\\int _{\\mathbb {R}^1} dx\\left[\\nabla (\\nabla \\cdot \\vec{j})\\right]^2/f}$ .", "This condition limits the scope of applicability of the analysed distribution functions, and as we shall mention later in Sec.", "III, the Gaussian distribution lies within this applicable class.", "Consequently, the derived inequality Eq.", "(REF ) is yet another expression of a lower bound for $I$ , which is different from the previously reported one in [4].", "In the $d$ -dimensional case, the Schwarz inequality gives $\\int _{\\mathbb {R}^d}\\left[\\frac{\\nabla f}{f}\\cdot \\nabla (\\nabla \\cdot \\vec{j})\\right]d\\tau \\leqslant \\sum ^{d}_{k=1}\\left\\Vert \\frac{\\nabla _k f}{\\sqrt{f}}\\right\\Vert _2 \\left\\Vert \\frac{\\nabla _k(\\nabla \\cdot \\vec{j})}{\\sqrt{f}}\\right\\Vert _2,$ where $d\\tau =dx_1\\cdots dx_d$ .", "Then, we have the following inequality: $d\\mathbf {I}\\cdot d\\mathbf {J}\\geqslant \\frac{1}{2}\\int _{\\mathbb {R}^d}\\left[\\left(\\frac{\\nabla f}{f}\\right)^2\\nabla \\cdot \\vec{j}\\right]d\\tau ,$ in which we denote the vector $d\\mathbf {I}=((d\\mathbf {I})_1,\\ldots ,(d\\mathbf {I})_d)$ , whose components $\\Vert \\nabla _k f/\\sqrt{f}\\Vert _2$ ($k=1,\\ldots ,d$ ) constitute the Fisher information, such that $|d\\mathbf {I}|^2=I$ .", "It reflects the shape (geometry) property of the distribution function.", "On the other hand, the vector $d\\mathbf {J}$ associated with the $L^2$ -norm of the quantity $\\nabla _k(\\nabla \\cdot \\vec{j})/\\sqrt{f}$ provides information on the (phase) spatial change of the flow.", "It is interesting to note that the bound for $dS/dt$ by Brody and Meister [3] essentially stems from the application of the Schwarz inequality.", "Now, let us consider the one-dimensional case and evaluate the magnitude $|dI/dt|$ , which is our second objective.", "Let us put the absolute values of the rightmost integrations in Eq.", "(REF ) respectively as $C_1$ and $C_2$ , $C_1:=\\Big | \\int _{\\mathbb {R}^1} \\frac{\\nabla f}{\\sqrt{f}}\\frac{\\nabla f}{\\sqrt{f}}\\frac{\\nabla \\cdot \\vec{j}}{f}dx\\Big |, \\quad C_2:=\\Big | 2\\int _{\\mathbb {R}^1}\\frac{\\nabla f}{f}\\cdot \\nabla (\\nabla \\cdot \\vec{j})dx\\Big |.$ We apply the Hölder inequality for the three functions $f_1$ , $f_2$ , and $f_3$ $\\Big |\\int _{\\mathbb {R}^1} dx f_1f_2f_3\\Big | \\leqslant \\Vert f_1\\Vert _{p_1}\\Vert f_2\\Vert _{p_2}\\Vert f_3|_{p_3}\\quad {\\rm with} \\quad \\frac{1}{p_1}+\\frac{1}{p_2}+\\frac{1}{p_3}=1,$ where $\\Vert f\\Vert _p(t)=(\\int _{\\mathbb {R}^1} |f(x,t)|^p dx)^{1/p}$ ($1\\leqslant p <\\infty , t>0$ ) is the $L^p$ norm.", "Then, by setting $p_1=p_2=2$ and $p_3=\\infty $ , the limit for $C_1$ can be expressed as follows $C_1 \\leqslant \\left\\Vert \\frac{\\nabla f}{\\sqrt{f}}\\right\\Vert _2^2\\left\\Vert \\frac{\\nabla \\cdot \\vec{j}}{f}\\right\\Vert _{L^\\infty },$ where $\\Vert \\cdot \\Vert _{L^\\infty }=ess.\\sup |\\cdot |$ .", "In conjunction with the triangle inequality for the last expression of Eq.", "(REF ), we obtain the upper bound $\\Big |\\frac{dI}{dt}\\Big | \\leqslant C_1+C_2 = \\alpha I + \\beta \\sqrt{I}, $ where we have put the coefficients respectively as $\\alpha =\\mathop {ess.\\sup }\\limits _{x\\in \\mathbb {R}^1}\\Big | \\frac{\\nabla \\cdot \\vec{j}}{f}\\Big | \\quad {\\rm and} \\quad \\beta = 2\\sqrt{\\int _{\\mathbb {R}^1}\\frac{[\\nabla (\\nabla \\cdot \\vec{j})]^2}{f}dx}.$" ], [ "Lower bound expressions in terms of the Sobolev and logarithmic Sobolev inequalities", "In this section, we provide the lower bounds for $I$ depending on dimensions, when we do not consider the flux.", "Since the Fisher information captures the coarse-grained inclination of the distribution function, it is useful to find the operative limit of the average gradient in terms of the spreading of the function.", "For a function $g$ that vanishes at infinity with its gradient in $L^2(\\mathbb {R}^n)$ , i.e., $g\\in D^1(\\mathbb {R}^n)$ , the Sobolev inequality with dimension $n\\geqslant 3$ (e.g.", "[8]) reads $\\left\\Vert \\nabla g\\right\\Vert _2^2 \\geqslant \\frac{n(n-2)}{4}|\\mathbb {S}^n|^\\frac{2}{n}\\left\\Vert g\\right\\Vert _{\\frac{2n}{n-2}}^2,$ where $|\\mathbb {S}^{n-1}|=2\\pi ^{\\frac{n}{2}}/\\Gamma (n/2)$ is a sphere of radius 1 in $\\mathbb {R}^n$ .", "By substituting $g=\\sqrt{f}$ we obtain $I(f)\\geqslant \\frac{n(n-2)2^{\\frac{2}{n}}\\pi ^{1+\\frac{1}{n}}}{\\Gamma \\left(\\frac{n+1}{2}\\right)^\\frac{2}{n}}\\left\\Vert f\\right\\Vert _{\\frac{n}{n-2}}.$ In case of $n=2$ , for function $g$ and its gradient in $L^2(\\mathbb {R}^2)$ , the following inequality holds $\\left\\Vert \\nabla g\\right\\Vert _{H^1(\\mathbb {R}^2)}^2 \\geqslant C\\left\\Vert g\\right\\Vert _{L^q(\\mathbb {R}^2)}^2, \\quad (2\\leqslant q < \\infty )$ where the norm of the Sobolev space $H^1(\\mathbb {R}^2)$ is defined as $\\left\\Vert \\nabla g\\right\\Vert _{H^1(\\mathbb {R}^2)}:=\\left( \\int _{\\mathbb {R}^2}|g(x)|^2 dx +\\int _{\\mathbb {R}^2}|\\nabla g(x)|^2 dx \\right)^{1/2},$ and constant $C$ depends only on $q$ .", "Then, noting that $\\left\\Vert f\\right\\Vert _2^2=1$ by the normalization of the probability function, we obtain the lower bound $I(f)\\geqslant 4(C\\left\\Vert f\\right\\Vert _{\\frac{q}{2}}-1).$ In case of $n=1$ , for any $g\\in H^1(\\mathbb {R}^1)$ , $\\left\\Vert \\frac{dg}{dx}\\right\\Vert _2^2+\\left\\Vert g \\right\\Vert _2^2 \\geqslant 2\\left\\Vert g \\right\\Vert _\\infty ^2$ holds.", "By setting $g=\\sqrt{f}$ , we have the inequality $I(f)\\geqslant 4(2\\left\\Vert \\sqrt{f} \\right\\Vert _\\infty ^2-1).$ We use this inequality in Sec.", "III." ], [ "Normal diffusion", "As the benchmark evaluation of the bounds derived here, we examine the one-dimensional Wiener process, where the Gaussian probability distribution function with the time-dependent dispersion $\\sigma (t)$ and with the associated flow satisfying the continuity equation govern the system.", "This example has also been previously considered in the literature [3], [4].", "The flow and the distribution function are related as $\\vec{j}(x,t)=xf\\dot{\\sigma }(t)/\\sigma (t)$ according to the continuity equation and the relation $j=-\\partial f/\\partial x$ .", "By straightforward calculations, we obtain $\\left\\langle \\left(\\frac{j_{xx}}{f}\\right)^2\\right\\rangle =\\frac{6\\dot{\\sigma }^2(t)}{\\sigma ^4(t)},\\quad \\int ^{\\infty }_{-\\infty }(\\partial _x\\ln f)^2 j_xdx=-\\frac{2\\dot{\\sigma }(t)}{\\sigma ^3(t)}.$ Since the Fisher information is calculated to be $I=1/\\sigma ^2(t)$ , we can corroborate that the applicability condition of Eq.", "(REF ) is indeed satisfied, $|-2\\dot{\\sigma }(t)/\\sigma ^3(t)|<2\\sqrt{I}\\sqrt{\\langle \\left(j_{xx}/f\\right)^2\\rangle }$ .", "Therefore, it is well-justified to insert these into Eq.", "(REF ), and thus, we obtain the lower bound $I\\geqslant \\frac{1}{6\\sigma ^2(t)}.$ This provides a tighter bound compared to $\\sigma ^{-2}(t)$ as the lower bound obtained in [4] from the derived inequality (i.e., $I \\geqslant \\langle \\nabla \\cdot (\\vec{j}/f)\\rangle _f^2/\\gamma $ ) therein.", "A possible interpretation of the origin of tightness is that the occurrence of flow reduces the information in a system compared to the case without it.", "In this sense, the fact that the value of the lower bound $\\sigma ^{-2}(t)$ [4] coincides with that of the Fisher information $I$ , determined only from the form of the distribution, does not convey limitation, at least for one-dimensional normal diffusion, and the present lower bound may be a preferable alternative.", "The coefficient $1/6$ is for the one-dimensional case and it differs for the other dimensions.", "Moreover, we note that the lower bound Eq.", "(REF ), derived from the Sobolev inequality, for one dimension is indeed found to be satisfied, by considering $\\left\\Vert \\sqrt{f} \\right\\Vert _\\infty ^2=(\\mathop {ess.\\sup }\\limits _{x\\in \\mathbb {R}^1}|\\sqrt{f}|)^2=\\frac{1}{\\sqrt{2\\pi }\\sigma }$ and by the fact $(\\sigma -1/\\sqrt{2\\pi })^2+1/4-1/2\\pi \\geqslant 0$ ." ], [ "Truncated Gaussian", "The use of truncated Gaussian distributions is common in statistics, econometrics, and in many other areas of science, where the probability density function has a cutoff from below or above (or both) while keeping the Gaussian form (e.g., [9]).", "The density function of the truncated normal distribution defined in $x\\in [a,b]$ is $\\frac{\\frac{1}{\\sigma }\\phi (\\frac{x-\\mu }{\\sigma })}{\\Phi (\\frac{b-\\mu }{\\sigma })-\\Phi (\\frac{a-\\mu }{\\sigma })},$ where $\\phi (x)$ is the standard normal distribution with mean $\\mu $ and variance $\\sigma ^2$ .", "$\\Phi (x)$ denotes its cumulative distribution.", "Below, we use a time-dependent finite interval $x\\in [-\\sigma (t), \\sigma (t)]$ by setting $\\mu =0$ and proceed without the scaling factor $\\sigma (t)(\\Phi (1)-\\Phi (-1))$ , derived from the normalization.", "Then, the Fisher information is calculated to be $I=\\int ^{\\sigma (t)}_{-\\sigma (t)}\\frac{(f_x)^2}{f}dx =\\frac{1}{\\sigma ^2(t)}\\left\\lbrace \\rm {Erf}\\left( \\frac{1}{\\sqrt{2}}\\right)-\\sqrt{\\frac{2}{\\pi e}}\\right\\rbrace ,$ where the error function $\\rm {Erf}(1/\\sqrt{2})=0.6826...$ and the positivity $I>0$ is accordingly maintained.", "The use of the truncated Gaussian in our consideration is equivalent to incorporation of the flow form, same as that given in the previous example, because during the process, the distribution keeps the Gaussian within finite support.", "For the coefficients $\\alpha $ and $\\beta $ defined in Eq.", "(REF ), we calculate respectively as $\\alpha =\\mathop {ess.\\sup }\\limits _{x\\in [-\\sigma (t),\\sigma (t)]}\\Big | \\frac{j_x}{f}\\Big |=\\frac{\\dot{\\sigma }(t)}{\\sigma (t)},\\quad \\beta = 2\\sqrt{\\int ^{\\sigma (t)}_{-\\sigma (t)}\\frac{(j_{xx})^2}{f}dx}=\\frac{12\\dot{\\sigma }^2(t)}{\\sigma ^4(t)}\\left\\lbrace \\rm {Erf}\\left( \\frac{1}{\\sqrt{2}}\\right)-\\sqrt{\\frac{2}{\\pi e}}\\right\\rbrace .$ Therefore, we find that the absolute value of the time derivative is bounded from above as $\\Big |\\frac{dI}{dt}\\Big | \\leqslant \\frac{\\dot{\\sigma }(t)}{\\sigma (t)} I+12\\frac{\\dot{\\sigma }^2(t)}{\\sigma ^4(t)} I^{3/2}.$ Since we are dealing with moving boundaries (truncation positions) here, the time derivative of the integral of a bivariable function, whose limits depend on time has additional terms (e.g., [10]) as $\\frac{d}{dt}\\int ^{u(t)}_{v(t)}\\rho (x,t)dx=\\dot{u}(t)\\rho (u(t),t)-\\dot{v}(t)\\rho (v(t),t)+\\int ^{u(t)}_{v(t)}\\frac{\\partial }{\\partial t}\\rho (x,t)dx,$ where $\\rho (x,t)=-f(x,t)\\log f(x,t)$ in our case and $\\int ^{u(t)}_{v(t)}\\rho (x,t)dx$ is assumed to be continuous within the interval of interest in both $x$ and $t$ .", "We note that the first two extra terms calculated as $\\dot{\\sigma }(t)\\rho (\\sigma (t),t)-(-\\dot{\\sigma }(t))\\rho (-\\sigma (t),t)$ contribute only in shifting the upper bound, obtained in Eq.", "(REF ).", "Substituting the Gaussian form of $f$ into $\\rho $ , we have $\\rho (\\sigma (t),t)=\\sigma ^{-1}(t)(1/2+\\log \\sqrt{2\\pi }+\\log \\sigma (t))/\\sqrt{2\\pi e}$ .", "Then, the final upper bound becomes $\\Big |\\frac{dI}{dt}\\Big |\\leqslant \\frac{\\dot{\\sigma }(t)}{\\sigma (t)} I+12\\frac{\\dot{\\sigma }^2(t)}{\\sigma ^4(t)} I^{3/2}+ \\frac{\\dot{\\sigma }(t)}{\\sigma (t)\\sqrt{2\\pi e}}\\left( 1+\\log (2\\pi )+2\\log \\sigma (t)\\right).$ Note that by comparing Eq.", "(REF ) with Eq.", "(REF ), the effect of moving boundary appears as an increment in the bound." ], [ "A bound for $dI/dt$ in terms of entropy production", "In Eqs.", "(REF ) and (REF ), viewed conversely, $I$ and its square root are respectively bounded from below by entropy production $dS/dt$ .", "Therefore, we can conceive the available bound in any form to generate change in Fisher information in terms of flow.", "Specifically, we develop an interest in obtaining a general expression of the lower bound for the Fisher information production $dI/dt$ in terms of the entropy production when the systems follow the continuity equation $\\partial _t f=-\\nabla \\cdot \\vec{j}$ with the aid of the interdependence between $S$ and $I$ .", "One may think that finding the lower bound on $dI/dt$ contradicts the arrow of time employed in Sec.", "II.", "However, this characterization can only be true under well-organized circumstances such as systems in which the model follows the heat equation, although it is widely expected to hold as mentioned in Sec.", "II A.", "In fact, Fisher information does not necessarily decrease in time in certain cases.", "Indeed, within properly developing living cells, $I$ is locally maximized, i.e., within the cell delimited by the membrane [13].", "This can occur at the expense of transferring increased waste and disorder outside the cell.", "The latter ensures that the second law of thermodynamics is overall obeyed.", "Therefore, we proceed as follows.", "Recalling that the logarithmic Sobolev inequality for the function $g$ (e.g., [8]) $\\frac{1}{\\pi }\\int _{\\mathbb {R}^n} |\\nabla g(x)|^2 dx\\geqslant \\int _{\\mathbb {R}^n} |g(x)|^2\\ln \\left( \\frac{|g(x)|^2}{\\left\\Vert g \\right\\Vert _2^2}\\right)dx + n\\left\\Vert g \\right\\Vert _2^2.$ Setting $g=\\sqrt{f}$ and by normalization of the probability density, we find $I(f)\\geqslant 4\\pi (n-S(f)),$ where the Shannon entropy is $S(f)=-\\int _{\\mathbb {R}^n}f(x)\\ln f(x)dx$ .", "Unless the function $I(f)+4\\pi S(f)$ is a monotonically decreasing function with respect to time, there should exist at least a time region where its derivative becomes positive.", "Recalling that the identity $dS/dt=\\langle {\\rm div}(\\vec{j}/f)\\rangle _f$ holds true [4] under the setting same as ours, we can rewrite it further as $\\frac{dS}{dt}= \\left\\langle \\frac{{\\rm div}\\vec{j}}{f}\\right\\rangle _f-\\left\\langle \\vec{j}\\cdot \\frac{\\nabla f}{f^2}\\right\\rangle _f,$ where we have used the formula ${\\rm div}(\\phi \\vec{A})=\\vec{A}\\cdot \\nabla \\phi +\\phi \\nabla \\cdot \\vec{A}$ for a scalar function $\\phi $ and a vector $\\vec{A}$ .", "Therefore, by differentiating Eq.", "(REF ) we obtain the following lower bound $\\frac{dI}{dt} \\geqslant 4\\pi \\left( \\left\\langle \\vec{j}\\cdot \\frac{\\nabla f}{f^2}\\right\\rangle _f- \\left\\langle \\frac{{\\rm div}\\vec{j}}{f}\\right\\rangle _f \\right).$ A direct evidence for the contradiction between the property $dI/dt\\leqslant 0$ and the above-derived inequality can be checked in case of one-dimensional normal diffusion.", "That is, we obtain $dI/dt\\geqslant 4\\pi \\dot{\\sigma }/\\sigma $ , which is finite as long as particles diffuse (i.e., $\\dot{\\sigma }>0$ ).", "This fact shows the scope of the applicability." ], [ "Discussion", "If we specify the relation between flux vector $\\vec{j}$ and distribution $f$ , we obtain a physical model.", "In general, we can write the process as $P(L)f(x,t)=j(x,t)$ in the one-dimensional case, where $P(L)$ is a linear operator, represented by a polynomial of the differential operator $L=\\partial /\\partial x$ with constant coefficients.", "Fick's law is a special case, which follows from the choice $P(L)=-DL$ .", "Since the Fourier transform of the both sides becomes $P(\\xi )\\hat{f}(\\xi )=\\hat{j}(\\xi )$ when $P(\\xi )\\ne 0, (\\xi \\in \\mathbb {R}^1)$ , we can have the expression of the original distribution $f(x,t)=(\\sqrt{2\\pi })^{-1}\\int _{\\mathbb {R}^1}d\\xi e^{ix\\xi }\\hat{j}(\\xi )/P(\\xi )$ by inverse transform.", "Alternatively, in terms of the inverse operator $P^{-1}(L)$ , we have the expression $f(x,t)=P^{-1}(L)j(x,t)$ .", "Together with the definition of the Fisher information in Eq.", "(REF ), we regard $I$ as the result of the averaging function $\\chi (f)$ with $f$ determined, such that $I(t)=\\langle \\chi (f)\\rangle _f$ holds.", "If we specify the functional form of $\\chi (f)$ as $\\chi (f)=(j/f)^2$ , it is equivalent to choose the form of flux as Fick's law $j=-\\partial f /\\partial x$ .", "Other models are realized by fixing the form $\\chi $ as a function of flux and distribution.", "Assuming that $\\partial \\chi /\\partial t$ is continuous, we have $\\frac{dI}{dt}=\\int _{\\mathbb {R}^1}\\left( \\frac{\\partial f}{\\partial t}\\chi + f \\frac{\\partial \\chi }{\\partial t}\\right)dx.$ The upper bound of the above equation relies on the supremum, which is derived from the two competing terms $(\\partial _t{f})\\chi $ and $f\\partial _t{\\chi }$ , but the Schwarz' inequality at least provides the maxima for each term.", "Then, we have $\\Big |\\frac{dI}{dt}\\Big | \\leqslant \\left\\Vert \\frac{\\partial f}{\\partial t}\\right\\Vert _2 \\left\\Vert \\chi \\right\\Vert _2+\\left\\Vert f\\right\\Vert _2 \\left\\Vert \\frac{\\partial \\chi }{\\partial t}\\right\\Vert _2,$ which is a desirable expression for a bound with $L^2$ norms because we normally require square integrability of distributions in physics." ], [ "Conclusion", "Apart from the well-known Cramér-Rao bound [11], which asserts that $I$ cannot exceed the inverse of the mean square error of a measured quantity, the fact that the bounds for $I$ , $|dI/dt|$ , and $dI/dt$ , derived from the distribution functions, obey the laws of physics definitely links physics and the information contained in a physical system.", "The former (Cramér-Rao) originates from repeated active measurements (i.e., statistics), but in contrast, the latter originates from the flux $\\vec{j}$ of a physical entity.", "In this context, we have presented new alternative upper and lower bounds for the time derivative of $I$ .", "However, we neither have nor established a relation between the information flux and information production for $I$ , whereas in nonequilibrium thermodynamics [12] based on Shannon entropy, there is a familiar local formulation, i.e., $dS/dt= -\\nabla \\cdot \\mathbf {J}+\\sigma $ , where $\\mathbf {J}$ and $\\sigma $ denote the entropy flux and the entropy production, respectively.", "In irreversible processes, $\\sigma \\geqslant 0$ implies the Boltzmann's H-theorem.", "A search for the counterpart may lead to a deeper understanding of the informational structures, inherent in physical systems." ], [ "Acknowledgement", "The author wishes to thank the reviewer and the editor for valuable comments and suggestions to improve the presentation of the paper." ] ]

[ [ "Interacting Fibonacci anyons in a Rydberg gas" ], [ "Abstract A defining property of particles is their behavior under exchange.", "In two dimensions anyons can exist which, opposed to fermions and bosons, gain arbitrary relative phase factors or even undergo a change of their type.", "In the latter case one speaks of non-Abelian anyons - a particularly simple and aesthetic example of which are Fibonacci anyons.", "They have been studied in the context of fractional quantum Hall physics where they occur as quasiparticles in the $k=3$ Read-Rezayi state, which is conjectured to describe a fractional quantum Hall state at filling fraction $\\nu=12/5$.", "Here we show that the physics of interacting Fibonacci anyons can be studied with strongly interacting Rydberg atoms in a lattice, when due to the dipole blockade the simultaneous laser excitation of adjacent atoms is forbidden.", "The Hilbert space maps then directly on the fusion space of Fibonacci anyons and a proper tuning of the laser parameters renders the system into an interacting topological liquid of non-Abelian anyons.", "We discuss the low-energy properties of this system and show how to experimentally measure anyonic observables." ], [ "Interacting Fibonacci anyons in a Rydberg gas Igor Lesanovsky Midlands Ultracold Atom Research Centre (MUARC), School of Physics and Astronomy, The University of Nottingham, Nottingham NG7 2RD, United Kingdom Hosho Katsura Department of Physics, Gakushuin University, Mejiro, Toshima-ku, Tokyo 171-8588, Japan A defining property of particles is their behavior under exchange.", "In two dimensions anyons can exist which, opposed to fermions and bosons, gain arbitrary relative phase factors [1] or even undergo a change of their type.", "In the latter case one speaks of non-Abelian anyons - a particularly simple and aesthetic example of which are Fibonacci anyons [2].", "They have been studied in the context of fractional quantum Hall physics where they occur as quasiparticles in the $k=3$ Read-Rezayi state [3], which is conjectured to describe a fractional quantum Hall state at filling fraction $\\nu =12/5$ [4].", "Here we show that the physics of interacting Fibonacci anyons [5] can be studied with strongly interacting Rydberg atoms in a lattice, when due to the dipole blockade [6] the simultaneous laser excitation of adjacent atoms is forbidden.", "The Hilbert space maps then directly on the fusion space of Fibonacci anyons and a proper tuning of the laser parameters renders the system into an interacting topological liquid of non-Abelian anyons.", "We discuss the low-energy properties of this system and show how to experimentally measure anyonic observables.", "There is great interest in studying many-body systems of interacting anyons as they often exhibit exotic quantum phases [5], [7], [8], [9].", "A further motivation is that the exchange of anyons - the braiding - permits the implementation of robust protocols for quantum information processing [10], [11], [12].", "Physically, anyons emerge as quasi particles on the ground state of an interacting many-body system and recently there has been put much effort in implementing and exploring anyonic models with cold atoms [13], [14], [15], polar molecules [16] and trapped ions [17].", "A model which has been much studied - in particular in the context of quantum information processing - is Kitaev's toric code [18].", "Excitations on the ground state are Abelian anyons, which merely acquire a phase when braided.", "A more complex and exotic scenario is encountered in the non-Abelian case, i.e.", "when anyons can undergo an actual change of their type under braiding.", "Figure: (a) Fusion space.", "The Hilbert space of a system of NN Fibonacci anyons of type τ\\tau is spanned by all possible fusion paths.", "An example of such path is given by the symbols contained in the boxes.", "A fusion path translates into configurations of atoms which are either in the electronic ground state or excited to a Rydberg state.", "The Rydberg state (ground state) is identified with the fusion outcome 1 (τ\\tau ).", "(b) Level scheme and interaction energy.", "The excitation of a Rydberg atom on site kk shifts the energy of Rydberg states of atoms in the neighborhood.", "The strong interaction between neighboring atoms (here kk and k+1k+1) excludes their simultaneous excitation to Rydberg states and effectively constrains the Hilbert space of the atomic system to the fusion space of Fibonacci τ\\tau -anyons.A particularly simple example of non-Abelian anyons are Fibonacci anyons which occur in two types: A trivial particle referred to as 1 and a non-trivial particle denoted by $\\tau $ .", "A convenient way to define an anyonic system is through fusion rules for the particle types (see e.g.", "Refs.", "[12], [2]), which for Fibonacci anyons read $1 \\times 1 = 1,\\,\\,1 \\times \\tau = \\tau ,\\,\\,\\tau \\times 1 = \\tau ,\\,\\,\\tau \\times \\tau = 1 + \\tau .$ The first three rules are a consequence of the trivial nature of a 1-anyon.", "The fourth rule states that two anyons of type $\\tau $ can fuse such that the result is either the trivial particle or a $\\tau $ -anyon.", "This can be thought of as being analogous to the merging of two spin $1/2$ particles ($\\frac{1}{2}\\otimes \\frac{1}{2}=0\\oplus 1 $ ), which can yield either a singlet (total spin 0) or a triplet (total spin 1).", "For a set of $N$ anyons of type $\\tau $ it can be shown that a basis of the underlying Hilbert space is spanned by all possible fusion paths [2], i.e.", "the number of ways in which these anyons can be successively fused.", "For large $N$ , this number grows as $\\varphi ^N$ with $\\varphi =(1+\\sqrt{5})/2$ being the golden ratio.", "An example of a fusion path is illustrated in Fig.", "REF a and it is constructed as follows: Starting from the left the first $\\tau $ -anyon fuses with the $\\tau $ that is set by the left boundary condition.", "There are two possible fusion outcomes, 1 or $\\tau $ .", "In this particular realization we choose the outcome 1, which is written into the first box.", "This trivial particle then fuses with the second $\\tau $ -anyon.", "The fusion rules dictate that the outcome must be a $\\tau $ anyon (second box).", "Continuing this procedure by obeying the fusion rules results in a possible fusion path.", "It is important to note that since a trivial particle fusing with a $\\tau $ always results in another $\\tau $ the occurrence of two consecutive 1's is excluded.", "In the following we will show in detail that a Rydberg lattice gas constitutes an analog quantum simulator for Fibonacci anyons.", "The link between these two systems is the aforementioned exclusion principle.", "While for anyons this is a consequence of the underlying mathematical rules the exclusion in a Rydberg system is physically rooted in the dipole blockade [6] which prevents the simultaneous excitation of neighboring atoms to Rydberg states.", "The blockade originates from large electrostatic energy shifts between atoms in Rydberg states and has recently been demonstrated experimentally for atoms trapped in individual optical traps [19], [20].", "Also first experiments with Rydberg atoms in a lattice [21] have been conducted.", "We will discuss which anyonic interactions can be realized in such a system and how anyonic degrees of freedom can be measured experimentally.", "We expect that this comparatively simple quantum simulator platform for non-Abelian anyons will highlight a new route towards the study of exotic forms of quantum matter.", "Formally, the Rydberg lattice gas is described by a set of atoms with the electronic ground state $\\left|g\\right>\\equiv \\left|\\downarrow \\right>$ that is coupled by a laser with Rabi frequency $\\Omega $ to a high lying Rydberg state denoted as $\\left|r\\right>\\equiv \\left|\\uparrow \\right>$ .", "Excited atoms with position labels $k$ and $m$ interact with a van-der-Waals potential $V_{|k-m|}=(C_6/a^6)/|k-m|^{6}$ where $C_6$ is the interaction dispersion coefficient and $a$ the lattice spacing.", "Within the rotating-wave approximation the Hamiltonian of an ensemble of $N-1$ atoms reads $H=\\Omega \\sum _{k=1}^{N-1} \\sigma ^x_k+\\Delta \\sum _{k=1}^{N-1} n_k+\\sum _{k=1,m\\ne k}^{N-1} V_{|k-m|} n_k n_m.", "$ Here $\\Delta $ is the detuning of the laser frequency with respect to the frequency of the atomic transition $\\left|g\\right>\\leftrightarrow \\left|r\\right>$ , $n_k=(1+\\sigma ^z_k)/2$ is the local number operator and $\\sigma ^{x,z}_k$ are Pauli spin matrices referring to the internal degrees of freedom of each atom.", "Hamiltonian (REF ) has been studied in a number of theoretical works [22], [23], [24], [25] and recent experiments have shown that it accurately reflects the physics of laser driven Rydberg gases [26], [21].", "We are here interested in a parameter regime in which the nearest neighbor interaction is the largest energy scale, i.e.", "$|\\Omega |,|\\Delta |\\ll |V_1|$ .", "Fig.", "REF b depicts a sketch of this situation where we take $V_1$ ($C_6$ ) [24] negative.", "However, in this work we will discuss both positive as well as negative $V_1$ .", "It was shown in Refs.", "[23], [25] that when $|\\Omega |,|\\Delta |\\ll |V_1|$ the dynamics of the Rydberg gas is confined to a Hilbert space $\\mathcal {H}_\\mathrm {blockade}$ that is spanned by atomic configurations in which adjacent atoms are not simultaneously excited.", "This Hilbert space is equivalent to the fusion space of an ensemble of $N$ Fibonacci anyons of type $\\tau $ .", "To see this, one should think of the atoms as being located in between the $\\tau $ -anyons.", "A fusion path (see Fig.", "REF a for an example) is then encoded in the internal state of the atoms: the $k$ -th atom is in the state $\\left|r\\right>$ ($\\left|g\\right>$ ) if the outcome of the fusion of the $k$ -th anyon with the previous fusion outcome is 1($\\tau $ ).", "The state of each atom can thus be interpreted as the combined topological charge of the anyons located to its left [5].", "Having established a formal equivalence between the Hilbert spaces of a Rydberg lattice gas and the fusion space, we will now turn to the question concerning which anyonic interactions are actually realized by the atomic system.", "To this end, it is more convenient to work with an explicit projection of Hamiltonian (REF ) on the constrained Hilbert space $\\mathcal {H}_\\mathrm {blockade}$ .", "This effective Hamiltonian reads $H_\\mathrm {eff}=H_0+H_\\mathrm {vdW}+H_2$ with $H_0&=&\\Omega \\left[\\sigma _1^x P_2 +P_{N-2}\\sigma ^x_{N-1} \\right]+\\Omega \\sum _{k=2}^{N-2} P_{k-1}\\sigma ^x_k P_{k+1}\\nonumber \\\\&&+\\Delta \\sum _{k=1}^{N-1} n_k+V_2\\sum _{k=1}^{N-3} n_k n_{k+2}\\\\H_\\mathrm {vdW}&=&\\sum _{|k-m|>2} \\!\\!\\!\\!V_{|k-m|} n_k n_m\\\\H_2&=&-\\frac{\\Omega ^2}{V_1}\\left[2\\sum ^{N-1}_{k=1} n_k -\\frac{3}{2} n_k n_{k+2}\\right.\\nonumber \\\\&&+\\left.\\sum ^{N-4}_{k=1} P_{k}\\left(\\sigma ^+_{k+1} \\sigma ^-_{k+2}+\\sigma ^-_{k+1} \\sigma ^+_{k+2}\\right)P_{k+3}\\right],$ where $P_k=1-n_k$ .", "Here $H_0+H_\\mathrm {vdW}$ is the actual projection of Hamiltonian (REF ) onto $\\mathcal {H}_\\mathrm {blockade}$ .", "The term $H_2$ contains second order corrections that arise from a non-perfect blockade, i.e.", "due to a finite nearest-neighbor interaction.", "Since the relative strength of $H_2$ scales with $(\\Omega /V_1)^2$ it can be regarded as a perturbation to $H_0$ .", "Moreover, due to the quickly decaying tail of the van-der-Waals interaction also the term $H_\\mathrm {vdW}$ can be considered perturbatively.", "We will later discuss the influence of these terms on the spectrum, but let us first analyze the principal Hamiltonian $H_0$ .", "We start by rewriting it as $H_0=\\sum _{k=2}^{N-2}h_k+H_\\mathrm {b}$ where the $h_k$ are local three body Hamiltonians and $H_\\mathrm {b}$ defines the boundary term.", "Using the basis (centered around the $k$ -th site) $\\lbrace \\left|\\uparrow \\downarrow \\uparrow \\right>, \\left|\\downarrow \\downarrow \\uparrow \\right>, \\left|\\uparrow \\downarrow \\downarrow \\right>,\\left|\\downarrow \\uparrow \\downarrow \\right>,\\left|\\downarrow \\downarrow \\downarrow \\right>\\rbrace $ and taking into account that these site-triples overlap, we find that $h_k=\\left(\\begin{array}{ccccc}f_1 & & & & \\\\& f_2 & & & \\\\& & \\alpha & & \\\\& & & \\beta & \\Omega \\\\& & & \\Omega & \\gamma \\\\\\end{array}\\right)_{k}$ where $f_1=\\Delta +V_2-\\beta +2\\gamma $ , $f_2=\\Delta -\\alpha -\\beta +3\\gamma $ and $\\alpha $ , $\\beta $ and $\\gamma $ are constants.", "These constants are arbitrary, showing that there is a large class of local three-body Hamiltonians whose sum adds up to Hamiltonian $H_0$ .", "The boundary term reads $H^\\mathrm {o}_\\mathrm {b}=(\\Delta -\\alpha +\\gamma )n_1+(f_2-\\gamma ) n_2+(\\alpha -\\gamma ) n_{N-2}+(\\Delta -f_2-\\gamma )n_{N-1}+\\Omega \\left[\\sigma _1^x P_2 +P_{N-2}\\sigma ^x_{N-1}\\right]$ for open and $H^\\mathrm {c}_\\mathrm {b}=h_1+h_{N-1}$ for closed boundaries (with relabeling the indices $N\\rightarrow 1$ , $0\\rightarrow N-1$ ).", "An anyonic interaction which has been extensively discussed in the literature [5], [2] is the anyonic equivalent of the Heisenberg interaction between spins.", "As mentioned earlier two spin $1/2$ particles fuse according to the rule $\\frac{1}{2}\\otimes \\frac{1}{2}=0\\oplus 1$ , where 0 and 1 are the total spins (singlet and triplet) and the Heisenberg interaction, $\\propto \\mathbf {S}_1\\cdot \\mathbf {S}_2$ , introduces an energy difference between these fusion outcomes.", "According to $\\tau \\times \\tau =1 +\\tau $ an anyonic analogue thus would assign different energies to the two fusion outcomes 1 and $\\tau $ .", "The construction of this Hamiltonian is discussed in Refs.", "[5], [2], but we will briefly summarize here how an explicit representation of the interaction in fusion space and hence also in $\\mathcal {H}_\\mathrm {blockade}$ is obtained.", "Figure: (a) Changing the order of the fusion path.", "The FF-matrix is performing a basis change in fusion space such that the fusion outcome uu of the two neighboring τ\\tau -anyons becomes explicit.", "(b) Determination of the fusion outcome via a measurement of the three-point correlation function Π k \\left<\\Pi _k\\right>.", "We start by applying a strong laser pulse with Rabi frequency Ω rs ≫|V 2 |\\Omega _\\mathrm {rs}\\gg |V_2| on a transition from the Rydberg state r\\left|r\\right> to a stable hyperfine state s\\left|s\\right>.", "This state transfer is necessary for switching off the Rydberg-Rydberg interaction.", "We then measure the state of the atoms on sites k±1k\\pm 1 by monitoring fluorescence from a closed transition, i.e.", "we shine in a laser that resonantly couples the state g\\left|g\\right> to a short lived state a\\left|a\\right> that decays under the emission of a photon (with rate Γ\\Gamma ) back to g\\left|g\\right>.", "If the atom is in state g\\left|g\\right> this results in the cyclic emission of photons while in the opposite case (atom in s\\left|s\\right>) no photons are emitted.", "The presence/absence of these scattered photons thus allows directly to infer the atomic state.", "(c) List of measurement results and the corresponding fusion outcome.", "If one finds that the atoms located at sites k-1k-1 and k+1k+1 are in different internal states we directly conclude from the matrix representation () that Π k =0\\left<\\Pi _k\\right>=0 and hence the fusion outcome was τ\\tau .", "In turn, if the atoms are in the state s k-1 s k+1 \\left|s\\right>_{k-1}\\left|s\\right>_{k+1}, we can conclude that Π k =1\\left<\\Pi _k\\right>=1 (fusion of the two τ\\tau 's to a trivial anyon).", "The most involved scenario is encountered when the two outer atoms are in the state g k-1 g k+1 \\left|g\\right>_{k-1}\\left|g\\right>_{k+1}.", "In this case we have to measure the expectation value of the operator that corresponds to the lower right 2×22\\times 2-block of Π k \\Pi _k given in Eq.", "().", "This block can be written as a rotation of the number operator n=(1+σ z )/2n=(1+\\sigma ^z)/2: R 𝐧 † nR 𝐧 R^\\dagger _\\mathbf {n}\\,n\\,R_\\mathbf {n}.Here R 𝐧 =e iπ𝐧·σ/2 R_\\mathbf {n}=e^{i\\pi \\mathbf {n}\\cdot \\mathbf {\\sigma }/2} with σ=(σ x ,σ y ,σ z )\\mathbf {\\sigma }=(\\sigma ^x,\\sigma ^y,\\sigma ^z) conducts a rotation around the axis 𝐧=(ϕ -1/2 ,0,ϕ -1 )\\mathbf {n}=(\\varphi ^{-1/2},0,\\varphi ^{-1}) by an angle π\\pi .", "Hence, if we rotate the state of the kk-th atom on the Bloch sphere by applying R 𝐧 R_\\mathbf {n} and subsequently perform a projective measurement we find the fusion outcomes to be 1(τ\\tau ) if the atom is in the state s\\left|s\\right>(g\\left|g\\right>).When considering two $\\tau $ anyons there are different orders in which anyons can be fused, two of which are depicted in Fig.", "REF a.", "The upper panel represents a “conventional\" fusion path according to the rules discussed earlier: Here the left anyon of type $x$ merges with the first $\\tau $ -anyon.", "The outcome of this fusion is an anyon of type $y$ which fuses with the second $\\tau $ to yield an anyon of type $z$ .", "This is not a convenient basis for the discussion of anyonic interactions, as we cannot directly read off the fusion outcome of the two $\\tau $ 's.", "We thus perform a basis change from the “conventional\" fusion space $\\left|xyz\\right>\\,\\in \\,\\lbrace \\left|1\\tau 1\\right>, \\left|\\tau \\tau 1\\right>,\\left|1\\tau \\tau \\right>,\\left|\\tau 1 \\tau \\right>,\\left|\\tau \\tau \\tau \\right>\\rbrace $ to a basis where the fusion outcome (denoted as $u$ ) of the two $\\tau $ -anyons becomes explicit: $\\left|xuz\\right>\\,\\in \\,\\lbrace \\left|111\\right>, \\left|\\tau \\tau 1\\right>,\\left|1\\tau \\tau \\right>,\\left|\\tau 1 \\tau \\right>,\\left|\\tau \\tau \\tau \\right>\\rbrace $ .", "The change to this new basis, as depicted in Fig.", "REF a, is conducted by the so-called $F$ -matrix [5], [2] $F_k=\\left(\\begin{array}{ccccc}1 & & & & \\\\& 1 & & & \\\\& & 1 & & \\\\& & & \\varphi ^{-1} & \\varphi ^{-1/2} \\\\& & & \\varphi ^{-1/2} & -\\varphi ^{-1}\\end{array}\\right)_k.$ With this we are now equipped to construct the Heisenberg interaction.", "An operator that discriminates the two fusion results is a projector on the trivial particle.", "In the basis where the fusion outcome of the two $\\tau $ 's is explicit (bottom panel of Fig.", "REF a), this is a diagonal matrix with entries one, where $u=1$ and zero otherwise, $\\bar{\\Pi }_k=\\mathrm {diag}(1,0,0,1,0)$ .", "Upon transforming $\\bar{\\Pi }_k$ back into the “conventional\" fusion space we can thus write the anyonic Heisenberg Hamiltonian as $H_\\mathrm {aH}=-J \\sum _{k=1}^N \\Pi _k$ with $\\Pi _k=F_k\\,\\bar{\\Pi }_k\\,F_k=\\left(\\begin{array}{ccccc}1 & & & & \\\\& 0 & & & \\\\& & 0 & & \\\\& & & \\varphi ^{-2} & \\varphi ^{-3/2} \\\\& & & \\varphi ^{-3/2} & \\varphi ^{-1}\\end{array}\\right)_k.$ Comparing the coefficients of $\\Pi _k$ with those of the three-body Hamiltonians $h_k$ in Eq.", "(REF ) we find that this interaction is in fact naturally present in a system of interacting Rydberg atoms when $\\Omega =-J\\,\\varphi ^{-3/2},\\,\\Delta =-J\\,(\\varphi ^{-2}-3\\varphi ^{-1}),\\,V_2=-J\\,\\varphi .$ With this choice of parameters the Rydberg lattice gas behaves just like a system of Fibonacci anyons that interact with a Heisenberg interaction.", "For $J>0$ ($V_2<0$ ), it is energetically favorable for anyons to fuse to the trivial particle.", "In the opposite case, $J<0$ ($V_2>0$ ), the fusion to a $\\tau $ -anyon is favored.", "Before turning to the analysis of the spectral properties and the influence of imperfections such as a non-perfect Rydberg blockade or the van-der-Waals tail let us show how one can experimentally measure properties of the anyonic system.", "We have mentioned earlier that the state of an atom encodes the combined topological charge of the anyons located to its left.", "This property can hence be readily inferred from a site-resolved projective measurement.", "Another important experimental quantity - in particular in the context of the discussed Heisenberg interaction - is the result of a fusion of two $\\tau $ -anyons.", "To determine the fusion outcome of the anyons with label $k$ and $k+1$ one needs to determine the expectation value of the projection operator $\\Pi _k$ .", "This amounts to a measurement of a three-point correlation function on the atomic degrees of freedom.", "If the outcome is 1(0) we know that the particles have fused to an anyon of type 1($\\tau $ ).", "$\\left<\\Pi _k\\right>$ can be inferred from three separate projective measurements each of which concern only a single site.", "Such site resolved addressing in conjunction with projective measurements has been experimentally demonstrated in the context of interacting Rydberg atoms [19], [20].", "A schematics of the procedure and involved levels is given in Fig.", "REF b, and a comprehensive list of all measurement outcomes together with the corresponding fusion result is provided in Fig.", "REF c. Figure: Low energy excitation spectra.", "Energy spectra for H 0 H_0 (left column) and H eff H_{\\rm eff} (right column) with positive and negative JJ (referred to as case (i) and case (ii) in the text).", "The spectra have been shifted such that the lowest energy eigenvalues are zero.", "The dashed lines sketch low-energy dispersion relations and serve as a guide to the eye.", "The different symbols (green circles, red squares, blue diamonds) indicate different system sizes NN: top panels (19,21,23), bottom panels (16,19,22).Let us finally analyze the ground state and the low-energy properties of the model.", "The ground state is in general a strongly correlated many-body quantum state whose preparation typically requires an adiabatic passage protocol: Here, one starts in a simple initial (product) state with no Rydberg atoms present and slowly tunes the laser parameters until one reaches the point (REF ) in parameter space.", "This has to be achieved in a time much smaller than the radiative lifetime of Rydberg atoms, which is typically (for principal quantum numbers in the range $n=40\\ldots 60$ ) on the order of $100\\,\\mu $ s. The experimental feasibility of such protocol has recently been shown [27].", "Beyond that it is moreover possible to infer properties of the ground state such as critical behavior from the time evolution and the scaling properties of certain quantum mechanical observables as demonstrated in Ref.", "[26].", "The ground state phase diagram of the principal Hamiltonian $H_0$ has been studied in [28].", "It has a number of interesting features, e.g.", "for certain parameter choices the ground state becomes a simple matrix product state [23] and also the excitation gap can be calculated analytically [25].", "There are furthermore integrable lines in parameter space that also include the point (REF ) which represents the anyonic Heisenberg interaction [5].", "Here one can map the model onto the restricted-solid-on-solid model [29] and the critical properties of the ground state of $H_0$ are inferred from the exact solution.", "We consider the two cases (i) antiferromagnetic interaction between anyons ($J>0$ , $V_2<0$ ) and (ii) ferromagnetic interaction ($J<0$ , $V_2>0$ ), separately.", "For case (i), the low-energy excitation is gapless and the continuum limit of the model is described by the minimal conformal field theory (CFT) with central charge $c=7/10$ [30].", "For case (ii), the low-energy excitation is again gapless but the continuum limit is described by another CFT with $c=4/5$ [30].", "The left column of Fig.", "REF shows the low-energy spectra obtained by exact diagonalization up to $N=23$ with periodic boundary conditions.", "For (i), two gapless modes at momenta $K=0$ and $K=\\pi $ are clearly visible.", "On the other hand, for (ii), the gapless modes are located at $K=0$ , $2\\pi /3$ , and $4\\pi /3$ .", "These gapless modes suggest the existence of the quasi-long-range order.", "The difference of modulation period between (i) and (ii) can be understood as follows: In the original physical system, $J>0$ ($J<0$ ) and hence $V_2<0$ ($V_2>0$ ) means the attractive (repulsive) interaction between atoms in Rydberg states.", "But since there is a hard-core constraint caused by the Rydberg blockade, the density wave order of period 2 is favored when $J>0$ , while that of period 3 is favored when $J<0$ .", "We finally discuss the effect of the hitherto neglected perturbations, i.e., $H_{\\rm vdW}$ and $H_2$ .", "Fig.", "REF shows in the right column how the low-energy spectra are modified by these perturbations.", "The spectrum for case (i) with $H_{\\rm vdW}$ and $H_2$ suggests that the gap is small and the model still remains in the vicinity of the critical point.", "As demonstrated in Ref.", "[5], the gaplessness of the antiferromagnetic anyonic chain is protected by translation symmetry and topological ($Y$ ) symmetry.", "Since the first and the second terms in $H_2$ can be absorbed into $H_0$ by fine-tuning $\\Delta $ and $V_2$ , we expect that the criticality of the model is robust against the longer-range interactions stemming from the tail of the van-der-Waals potential and the correction due to the non-perfect blockade.", "On the other hand the spectrum for case (ii) clearly indicates the existence of a gap.", "The numerical results do not show any evidence of the quasi-degeneracy of the ground state caused by the density wave order.", "We thus conclude that the perturbations $H_{\\rm vdW}$ and $H_2$ lead to a disordered gapped state that does not break any symmetries.", "In conclusion, we have shown that a Rydberg lattice gas rather naturally constitutes an analog quantum simulator platform for Fibonacci anyons with tuneable interactions.", "Anyonic observables can be monitored by measuring atomic correlation functions which opens up the possibility for probing correlated quantum states of non-Abelian anyons in current experiments.", "We acknowledge C. Ates, M. Müller, B. Olmos and S. Furukawa for fruitful discussions.", "H.K.", "was supported in part by Grant-in-Aid for Young Scientists (B) (23740298).", "I.L.", "acknowledges support by EPSRC and through the Leverhulme Trust." ] ]

[ [ "Critical two-point functions for long-range statistical-mechanical\n models in high dimensions" ], [ "Abstract We consider long-range self-avoiding walk, percolation and the Ising model on $\\mathbb{Z}^d$ that are defined by power-law decaying pair potentials of the form $D(x)\\asymp|x|^{-d-\\alpha}$ with $\\alpha>0$.", "The upper-critical dimension $d_{\\mathrm{c}}$ is $2(\\alpha\\wedge2)$ for self-avoiding walk and the Ising model, and $3(\\alpha\\wedge2)$ for percolation.", "Let $\\alpha\\ne2$ and assume certain heat-kernel bounds on the $n$-step distribution of the underlying random walk.", "We prove that, for $d>d_{\\mathrm{c}}$ (and the spread-out parameter sufficiently large), the critical two-point function $G_{p_{\\mathrm{c}}}(x)$ for each model is asymptotically $C|x|^{\\alpha\\wedge2-d}$, where the constant $C\\in(0,\\infty)$ is expressed in terms of the model-dependent lace-expansion coefficients and exhibits crossover between $\\alpha<2$ and $\\alpha>2$.", "We also provide a class of random walks that satisfy those heat-kernel bounds." ], [ "Introduction", "The two-point function is one of the key observables to understand phase transitions and critical behavior.", "For example, the two-point function for the Ising model indicates how likely the spins located at those two sites point in the same direction.", "If it decays fast enough to be summable, then there is no macroscopic order.", "The summability of the two-point function is lost as soon as the model parameter (e.g., temperature) is above the critical point, and therefore it is naturally hard to investigate critical behavior.", "The lace expansion is a powerful tool to rigorously prove mean-field behavior above the model-dependent critical dimension.", "The mean-field behavior here is for the two-point function at the critical point to exhibit similar behavior to the underlying random walk.", "It has been successful to prove such behavior for various statistical-mechanical models, such as self-avoiding walk, percolation, lattice trees/animals and the Ising model.", "The best lace-expansion result obtained so far is to identify an asymptotic expression (= the Newtonian potential times a model-dependent constant) of the critical two-point function for finite-range models, such as the nearest-neighbor model.", "However, this ultimate goal has not been achieved, before this paper, for long-range models, especially when the 1-step distribution for the underlying random walk decays in powers of distance; only the infrared bound on the Fourier transform of the two-point function was available.", "This was partly because of our poor understanding of the long-range models in the $x$ -space, not in the Fourier space.", "For example, the random-walk Green's function is known to be asymptotically Newtonian/Riesz depending on the power of the aforementioned power-law decaying 1-step distribution, but we were unable to find optimal error estimates in the literature.", "Also, the subcritical two-point function is known to decay exponentially for the finite-range models, but this is not the case for the power-law decaying long-range models; as is shown in this paper, the decay rate of the subcritical two-point function is the same as the 1-step distribution of the underlying random walk.", "Therefore, the goal of this paper is to overcome those difficulties and derive an asymptotic expression of the critical two-point function for the power-law decaying long-range models above the critical dimension, using the lace expansion.", "We would also like to investigate crossover in the asymptotic expression when the power of the 1-step distribution of the underlying random walk changes." ], [ "Models and known results", "Self-avoiding walk (SAW) is a model for linear polymers.", "We define the two-point function for SAW on ${\\mathbb {Z}}^d$ as $G_p^{\\scriptscriptstyle \\rm SAW}(x)=\\sum _{\\omega :o\\rightarrow x}p^{|\\omega |}\\prod _{j=1}^{|\\omega |}D(\\omega _j-\\omega _{j-1})\\prod _{s<t}(1-\\delta _{\\omega _s,\\omega _t}),$ where $p\\ge 0$ is the fugacity, $|\\omega |$ is the length of a path $\\omega =(\\omega _0,\\omega _1,\\dots ,\\omega _{|\\omega |})$ and $D:{\\mathbb {Z}}^d\\rightarrow [0,1]$ is the ${\\mathbb {Z}}^d$ -symmetric non-degenerate (i.e., $D(o)\\ne 1$ ) 1-step distribution for the underlying random walk (RW); the contribution from the 0-step walk is considered to be $\\delta _{o,x}$ by convention.", "If the indicator function $\\prod _{s<t}(1-\\delta _{\\omega _s,\\omega _t})$ is replaced by 1, then $G_p^{\\scriptscriptstyle \\rm SAW}(x)$ turns into the RW Green's function $G_p^{\\scriptscriptstyle \\rm RW}(x)$ , whose radius of convergence $p_{\\rm c}^{\\scriptscriptstyle \\rm RW}$ is 1, as $\\chi _p^{\\scriptscriptstyle \\rm RW}\\equiv \\sum _{x\\in {\\mathbb {Z}}^d}G_p^{\\scriptscriptstyle \\rm RW}(x)=(1-p)^{-1}$ for $p<1$ and $\\chi _p^{\\scriptscriptstyle \\rm RW}=\\infty $ for $p\\ge 1$ .", "Therefore, the radius of convergence $p_{\\rm c}^{\\scriptscriptstyle \\rm SAW}$ for $G_p^{\\scriptscriptstyle \\rm SAW}(x)$ is not less than 1.", "It is known that $\\chi _p^{\\scriptscriptstyle \\rm SAW}\\equiv \\sum _{x\\in {\\mathbb {Z}}^d}G_p^{\\scriptscriptstyle \\rm SAW}(x)<\\infty $ if and only if $p<p_{\\rm c}^{\\scriptscriptstyle \\rm SAW}$ and diverges as $p\\uparrow p_{\\rm c}^{\\scriptscriptstyle \\rm SAW}$ .", "Here, and in the remainder of the paper, we often use “$\\equiv $ \" for definition.", "Percolation is a model for random media.", "Each bond $\\lbrace u,v\\rbrace $ , which is a pair of vertices in ${\\mathbb {Z}}^d$ , is either occupied or vacant independently of the other bonds.", "The probability that $\\lbrace u,v\\rbrace $ is occupied is defined to be $pD(v-u)$ , where $p\\ge 0$ is the percolation parameter.", "Since $D$ is a probability distribution, the expected number of occupied bonds per vertex equals $p\\sum _{x\\ne o}D(x)=p(1-D(o))$ .", "The percolation two-point function $G_p^{\\scriptscriptstyle \\rm perc}(x)$ is defined to be the probability that there is a self-avoiding path of occupied bonds from $o$ to $x$ ; again by convention, $G_p^{\\scriptscriptstyle \\rm perc}(o)=1$ .", "The Ising model is a model for magnets.", "For $\\Lambda \\subset {\\mathbb {Z}}^d$ and $\\varphi =\\lbrace \\varphi _v\\rbrace _{v\\in \\Lambda }\\in \\lbrace \\pm 1\\rbrace ^\\Lambda $ , we define the Hamiltonian (under the free-boundary condition) as $H_\\Lambda (\\varphi )=-\\sum _{\\lbrace u,v\\rbrace \\subset \\Lambda }J_{u,v}\\varphi _u\\varphi _v,$ where $J_{u,v}=J_{o,v-u}\\ge 0$ is the ferromagnetic pair potential and inherits the properties of the given $D$ , as explained below.", "The finite-volume two-point function at the inverse temperature $\\beta \\ge 0$ is defined as $\\langle \\varphi _o\\varphi _x\\rangle _{\\beta ,\\Lambda }=\\sum _{\\varphi \\in \\lbrace \\pm 1\\rbrace ^\\Lambda }\\varphi _o\\varphi _x\\;e^{-\\beta H_\\Lambda (\\varphi )}\\Bigg /\\sum _{\\varphi \\in \\lbrace \\pm 1\\rbrace ^\\Lambda }e^{-\\beta H_\\Lambda (\\varphi )}.$ It is known that $\\langle \\varphi _o\\varphi _x\\rangle _{\\beta ,\\Lambda }$ is increasing in $\\Lambda \\uparrow {\\mathbb {Z}}^d$ .", "Let $p=\\sum _{x\\in {\\mathbb {Z}}^d}\\tanh (\\beta J_{o,x})$ .", "The Ising two-point function $G_p^{\\scriptscriptstyle \\rm Ising}(x)$ is defined to be the increasing-volume limit of $\\langle \\varphi _o\\varphi _x\\rangle _{\\beta ,\\Lambda }$ : $G_p^{\\scriptscriptstyle \\rm Ising}(x)=\\lim _{\\Lambda \\uparrow {\\mathbb {Z}}^d}\\langle \\varphi _o\\varphi _x\\rangle _{\\beta ,\\Lambda }.$ Let $D(x)=p^{-1}\\tanh (\\beta J_{o,x})$ .", "For percolation and the Ising model, there is a model-dependent critical point $p_{\\rm c}\\ge 1$ (from now on, we omit the superscript, unless it causes any confusion) such that $\\chi _p\\equiv \\sum _{x\\in {\\mathbb {Z}}^d}G_p(x){\\left\\lbrace \\begin{array}{ll}<\\infty &[p<p_{\\rm c}],\\\\=\\infty &[p\\ge p_{\\rm c}],\\end{array}\\right.", "}&&\\theta _p\\equiv \\sqrt{\\lim _{|x|\\rightarrow \\infty }G_p(x)}{\\left\\lbrace \\begin{array}{ll}=0&[p<p_{\\rm c}],\\\\>0&[p>p_{\\rm c}].\\end{array}\\right.", "}$ The order parameter $\\theta _p^{\\scriptscriptstyle \\rm perc}$ is the probability that the occupied cluster of the origin is unbounded, while $\\theta _p^{\\scriptscriptstyle \\rm Ising}$ is the spontaneous magnetization, which is the infinite-volume limit of the finite-volume single-spin expectation $\\langle \\varphi _o\\rangle _{\\beta ,\\Lambda }^+$ under the plus-boundary condition.", "The continuity of $\\theta _p$ at $p=p_{\\rm c}$ in a general setting is still a remaining issue.", "We are interested in asymptotic behavior of $G_{p_{\\rm c}}(x)$ as $|x|\\rightarrow \\infty $ .", "For the “uniformly spread-out\" finite-range models, e.g., $D(x)={\\mathbb {1}}_{\\lbrace |x|=1\\rbrace }/(2d)$ or $D(x)={\\mathbb {1}}_{\\lbrace \\Vert x\\Vert _\\infty \\le L\\rbrace }/(2L+1)^d$ for some $L\\in [1,\\infty )$ , it has been proved [16], [18], [26] that, if $d>4$ for SAW and the Ising model and $d>6$ for percolation, and if $d$ or $L$ is sufficiently large (depending on the models), then there is a model-dependent constant $A$ ($=1$ for RW) such that $G_{p_{\\rm c}}(x)\\underset{|x|\\rightarrow \\infty }{\\sim }\\frac{a_d/\\sigma ^2}{A|x|^{d-2}},$ where “$\\sim $ \" means that the asymptotic ratio of the left-hand side to the right-hand side is 1, and $a_d=\\frac{d\\Gamma (\\frac{d-2}{2})}{2\\pi ^{d/2}},&&\\sigma ^2\\equiv \\sum _{x\\in {\\mathbb {Z}}^d}|x|^2D(x)=O(L^2).$ This is a sufficient condition for the following mean-field behavior [1], [2], [3], [5], [24]: $\\chi _p\\underset{p\\uparrow p_{\\rm c}}{\\asymp }(p_{\\rm c}-p)^{-1},&&\\theta _p\\underset{p\\downarrow p_{\\rm c}}{\\asymp }{\\left\\lbrace \\begin{array}{ll}\\sqrt{p-p_{\\rm c}}\\quad &[\\text{Ising}],\\\\~p-p_{\\rm c}&[\\text{percolation}],\\end{array}\\right.", "}$ where “$\\asymp $ \" means that the asymptotic ratio of the left-hand side to the right-hand side is bounded away from zero and infinity.", "The proof of the above result is based on the lace expansion (e.g., [19], [24], [26]).", "The core concept of the lace expansion is to systematically isolate interaction among individuals (e.g., mutual avoidance between distinct vertices for SAW or between distinct occupied pivotal bonds for percolation) and derive macroscopic recursive structure that yields the random-walk like behavior (REF ).", "When $d>d_{\\rm c}$ and $d\\vee L\\gg 1$ (i.e., $d$ or $L$ sufficiently large depending on the models), there is enough room for those individuals to be away from each other, and the lace expansion converges [19], [24], [26].", "The resultant recursion equation for $G_p$ is the following: $G_p(x)={\\left\\lbrace \\begin{array}{ll}\\displaystyle \\delta _{o,x}+\\sum _{v\\in {\\mathbb {Z}}^d}pD(v)\\,G_p(x-v)&[\\text{RW}],\\\\[1.5pc]\\displaystyle \\delta _{o,x}+\\sum _{v\\in {\\mathbb {Z}}^d}\\big (pD(v)+\\pi _p(v)\\big )\\,G_p(x-v)&[\\text{SAW}],\\\\[1.5pc]\\displaystyle \\pi _p(x)+\\sum _{\\begin{array}{c}u,v\\in {\\mathbb {Z}}^d\\\\ (u\\ne v)\\end{array}}\\pi _p(u)\\,pD(v-u)\\,G_p(x-v)\\quad &[\\text{Ising \\& percolation}],\\end{array}\\right.", "}$ where $\\pi _p$ is the lace-expansion coefficient.", "To treat all models simultaneously, we introduce the notation $f*g$ to denote the convolution of functions $f$ and $g$ in ${\\mathbb {Z}}^d$ : $(f*g)(x)=\\sum _{v\\in {\\mathbb {Z}}^d}f(v)\\,g(x-v).$ Then the above identities can be simplified as (the spatial variables are omitted) $G_p={\\left\\lbrace \\begin{array}{ll}\\delta +pD*G_p&[\\text{RW}],\\\\\\delta +(pD+\\pi _p)*G_p&[\\text{SAW}],\\\\\\pi _p+\\pi _p*p\\big (D-D(o)\\delta )*G_p\\quad &[\\text{Ising \\& percolation}].\\end{array}\\right.", "}$ Repeated use of these identities yieldsFor SAW, since $\\Vert \\pi _p\\Vert _1=o(1)$ as $d\\vee L\\rightarrow \\infty $ and $\\Vert G_p\\Vert _\\infty <\\infty $ for every $p\\le p_{\\rm c}$ [16], [18], $G_p=\\delta +pD*G_p+\\pi _p*\\!\\!\\underbrace{G_p}_\\text{replace}\\!\\!&=\\delta +pD*G_p+\\pi _p*\\big (\\delta +pD*G_p+\\pi _p*G_p\\big )\\\\&=(\\delta +\\pi _p)+(\\delta +\\pi _p)*pD*G_p+\\pi _p^{*2}*\\!\\!\\underbrace{G_p}_\\text{replace}\\!\\!=\\cdots \\rightarrow (\\ref {eq:lace-exp}).$ For percolation and the Ising model, since $D(o)=o(1)$ and $p\\Vert \\pi _p\\Vert _1=1+o(1)$ as $d\\vee L\\rightarrow \\infty $ and $\\Vert G_p\\Vert _\\infty \\le 1$ for every $p\\le p_{\\rm c}$ [16], [18], [26], $G_p&=\\pi _p+\\pi _p*pD*G_p-pD(o)\\pi _p*\\!\\!\\underbrace{G_p}_\\text{replace}\\!\\!\\\\&=\\pi _p+\\pi _p*pD*G_p-pD(o)\\pi _p*\\big (\\pi _p+\\pi _p*pD*G_p-pD(o)\\pi _p*G_p\\big )\\\\&=\\big (\\pi _p-pD(o)\\pi _p^{*2}\\big )+\\big (\\pi _p-pD(o)\\pi _p^{*2}\\big )*pD*G_p+\\big (-pD(o)\\big )^2\\pi _p^{*2}*\\!\\!\\underbrace{G_p}_\\text{replace}\\!\\!=\\cdots \\rightarrow (\\ref {eq:lace-exp}).$ $G_p=_p+_p*pD*G_p,$ where $_p(x)={\\left\\lbrace \\begin{array}{ll}\\delta _{o,x}&[\\text{RW}],\\\\[5pt]\\displaystyle \\sum _{n=0}^\\infty \\pi _p^{*n}(x)\\equiv \\sum _{n=0}^\\infty (\\underbrace{\\pi _p*\\cdots *\\pi _p}_\\text{$n$-fold})(x)\\quad &[\\text{SAW}],\\\\\\displaystyle \\sum _{n=1}^\\infty \\big (-pD(o)\\big )^{n-1}\\pi _p^{*n}(x)&[\\text{Ising \\& percolation}],\\end{array}\\right.", "}$ with the convention $f^{*0}(x)\\equiv \\delta _{o,x}$ for general $f$ .", "When $d>d_{\\rm c}$ and $d\\vee L\\gg 1$ , there is a $\\rho >0$ such that $|_{p_{\\rm c}}(x)|$ is summable and decays as $|x|^{-d-2-\\rho }$ [16], [18], [26].", "The multiplicative constant $A$ in (REF ) and $p_{\\rm c}$ can be represented in terms of $_{p_{\\rm c}}(x)$ as $p_{\\rm c}=\\bigg (\\sum _{x\\in {\\mathbb {Z}}^d}_{p_{\\rm c}}(x)\\bigg )^{-1},&&A=p_{\\rm c}\\bigg (1+\\frac{p_{\\rm c}}{\\sigma ^2}\\sum _{x\\in {\\mathbb {Z}}^d}|x|^2\\,_{p_{\\rm c}}(x)\\bigg ).$ In this paper, we investigate long-range SAW, percolation and the Ising model on ${\\mathbb {Z}}^d$ defined by power-law decaying pair potentials of the form $D(x)\\asymp |x|^{-d-\\alpha }$ with $\\alpha >0$ .", "For example, as in [10], we can consider the following uniformly spread-out long-range $D$ with parameter $L\\in [1,\\infty )$ : $D(x)=\\frac{|\\!|\\!|\\frac{x}{L}|\\!|\\!|_1^{-d-\\alpha }}{\\sum _{y\\in {\\mathbb {Z}}^d}|\\!|\\!|\\frac{y}{L}|\\!|\\!|_1^{-d-\\alpha }},$ where $|\\!|\\!|x|\\!|\\!|_\\ell =|x|\\vee \\ell $ .", "As a result, $D(x)=O(L^\\alpha )|\\!|\\!|x|\\!|\\!|_L^{-d-\\alpha },$ which we require throughout the paper (cf., Assumption REF below).", "The goal is to see how the asymptotic expression (REF ) of $G_{p_{\\rm c}}(x)$ changes depending on the value of $\\alpha $ .", "We note that (REF ) and (REF ) are invalid for $\\alpha \\le 2$ because then $\\sigma ^2=\\infty $ .", "Let $d_{\\rm c}={\\left\\lbrace \\begin{array}{ll}2(\\alpha \\wedge 2)&[\\text{SAW \\& Ising}],\\\\3(\\alpha \\wedge 2)&[\\text{percolation}].\\end{array}\\right.", "}$ It has been proved [20] that, for $d>d_{\\rm c}$ and $L\\gg 1$ , the Fourier transform $\\hat{G}_p(k)\\equiv \\sum _{x\\in {\\mathbb {Z}}^d}e^{ik\\cdot x}G_p(x)$ for the long-range models is bounded above and below by a multiple of $\\hat{G}_{\\hat{p}}^{\\scriptscriptstyle \\rm RW}(k)\\equiv (1-\\hat{p}\\hat{D}(k))^{-1}$ with $\\hat{p}=p/p_{\\rm c}$ , uniformly in $p<p_{\\rm c}$ .", "Although this gives an impression of the similarity between $G_{p_{\\rm c}}(x)$ and $G_1^{\\scriptscriptstyle \\rm RW}(x)$ , it is still too weak to identify the asymptotic expression of $G_{p_{\\rm c}}(x)$ .", "The proof of the above Fourier-space result makes use of the following properties of $D$ that we make use of here as well: there are $v_\\alpha =O(L^{\\alpha \\wedge 2})$ and $\\epsilon >0$ such that $\\hat{D}(k)\\equiv \\sum _{x\\in {\\mathbb {Z}}^d}e^{ik\\cdot x}D(x)=1-v_\\alpha |k|^{\\alpha \\wedge 2}\\times {\\left\\lbrace \\begin{array}{ll}1+O((L|k|)^\\epsilon )&[\\alpha \\ne 2],\\\\\\log \\frac{1}{L|k|}+O(1)&[\\alpha =2].\\end{array}\\right.", "}$ If $\\alpha >2$ , then $v_\\alpha =\\sigma ^2/(2d)$ .", "Moreover, if $L\\gg 1$ , then there is a constant $\\Delta \\in (0,1)$ such thatIn the proof of the bound on $\\Vert D^{*n}\\Vert _\\infty $ , we simply bounded the factor $\\log \\frac{\\pi }{2r}$ in [10] by a positive constant.", "If we make the most of that factor instead, we can readily improve the bound for $\\alpha =2$ as $\\Vert D^{*n}\\Vert _\\infty \\le O(L^{-d})\\,(n\\log n)^{-d/2}.$ $\\Vert D^{*n}\\Vert _\\infty \\le O(L^{-d})\\,n^{-\\frac{d}{\\alpha \\wedge 2}}\\quad [n\\ge 1],&&&&&&1-\\hat{D}(k){\\left\\lbrace \\begin{array}{ll}<2-\\Delta &[k\\in [-\\pi ,\\pi ]^d],\\\\>\\Delta &[\\Vert k\\Vert _\\infty \\ge L^{-1}].\\end{array}\\right.", "}$ All those properties hold for $D$ in (REF ) (cf., [10], [12])." ], [ "Main result", "In addition to the above properties, the $n$ -step transition probability obeys the following bound: $D^{*n}(x)\\le \\frac{O(L^{\\alpha \\wedge 2})}{|\\!|\\!|x|\\!|\\!|_L^{d+\\alpha \\wedge 2}}\\,n\\times {\\left\\lbrace \\begin{array}{ll}1&[\\alpha \\ne 2],\\\\\\log |\\!|\\!|x|\\!|\\!|_L\\quad &[\\alpha =2].\\end{array}\\right.", "}$ This is due to the following two facts: (i) the contribution from the walks that have at least one step which is longer than $c|\\!|\\!|x|\\!|\\!|_L$ for a given $c>0$ is bounded by $O(L^\\alpha )n/|\\!|\\!|x|\\!|\\!|_L^{d+\\alpha }$ ; (ii) the contribution from the walks whose $n$ steps are all shorter than $c|\\!|\\!|x|\\!|\\!|_L$ is bounded, due to the local CLT, by $O(\\tilde{v}n)^{-d/2}e^{-|x|^2/O(\\tilde{v}n)}\\le O(\\tilde{v}n)/|\\!|\\!|x|\\!|\\!|_L^{d+2}$ (times an exponentially small normalization constant), where $\\tilde{v}$ is the variance of the truncated 1-step distribution $\\tilde{D}(y)\\equiv D(y){\\mathbb {1}}_{\\lbrace |y|\\le c|x|\\rbrace }$ and equals $\\tilde{v}=\\sum _{y\\in {\\mathbb {Z}}^d}|y|^2\\tilde{D}(y)\\le O(L^{\\alpha \\wedge 2})\\times {\\left\\lbrace \\begin{array}{ll}|\\!|\\!|x|\\!|\\!|_L^{2-\\alpha }&[\\alpha <2],\\\\\\log |\\!|\\!|x|\\!|\\!|_L\\quad &[\\alpha =2],\\\\1&[\\alpha >2].\\end{array}\\right.", "}$ For $\\alpha \\ne 2$ , the inequality (REF ) is a discrete space-time version of the heat-kernel bound on the transition density $p_s(x)$ of an $\\alpha $ -stable/Gaussian process: $p_s(x)\\equiv \\int _{{\\mathbb {R}}^d}\\frac{\\text{d}^dk}{(2\\pi )^d}\\,e^{-ik\\cdot x-s|k|^{\\alpha \\wedge 2}}\\le \\frac{O(s)}{|x|^{d+\\alpha \\wedge 2}}.$ In Section REF , we will show that the properties (REF ), (REF ) and (REF ) are sufficient to obtain an asymptotic expression of $G_1^{\\scriptscriptstyle \\rm RW}(x)$ .", "However, these properties are not good enough to fully control error terms arising from convolutions of $D^{*n}(x)$ and $_p(x)$ in (REF ).", "To overcome this difficulty, we assume the following bound on the discrete derivative of the $n$ -step transition probability: $\\bigg |D^{*n}(x)-\\frac{D^{*n}(x+y)+D^{*n}(x-y)}{2}\\bigg |\\le \\frac{O(L^{\\alpha \\wedge 2})|\\!|\\!|y|\\!|\\!|_L^2}{|\\!|\\!|x|\\!|\\!|_L^{d+\\alpha \\wedge 2+2}}\\,n\\qquad [|y|\\le \\tfrac{1}{3}|x|].$ Here is the summary of the properties of $D$ that we use throughout the paper.", "Assumption 1.1 The ${\\mathbb {Z}}^d$ -symmetric 1-step distribution $D$ satisfies the properties (REF ), (REF ), (REF ), (REF ) and (REF ).", "In Appendix , we will show that the following $D$ satisfies all properties in the above assumption: $D(x)=\\sum _{t\\in \\mathbb {N}}U_L^{*t}(x)\\,T_\\alpha (t),$ where $U_L$ is in a class of ${\\mathbb {Z}}^d$ -symmetric distributions on ${\\mathbb {Z}}^d\\cap [-L,L]^d$ , and $T_\\alpha $ is the stable distribution on $\\mathbb {N}$ with parameter $\\alpha /2\\ne 1$ .", "Under the above assumption on $D$ , we can prove the following theorem: Theorem 1.2 Let $\\alpha >0$ , $\\alpha \\ne 2$ and $\\gamma _\\alpha =\\frac{\\Gamma (\\frac{d-\\alpha \\wedge 2}{2})}{2^{\\alpha \\wedge 2}\\pi ^{d/2}\\Gamma (\\frac{\\alpha \\wedge 2}{2})},$ and assume all properties of $D$ in Assumption REF .", "Then, for RW with $d>\\alpha \\wedge 2$ and any $L\\ge 1$ , and for SAW, percolation and the Ising model with $d>d_{\\rm c}$ and $L\\gg 1$ , there are $\\mu \\in (0,\\alpha \\wedge 2)$ and $A=A(\\alpha ,d,L)\\in (0,\\infty )$ ($A\\equiv 1$ for random walk) such that, as $|x|\\rightarrow \\infty $ , $G_{p_{\\rm c}}(x)=\\frac{\\gamma _\\alpha /v_\\alpha }{A|x|^{d-\\alpha \\wedge 2}}+\\frac{O(L^{-\\alpha \\wedge 2+\\mu })}{|x|^{d-\\alpha \\wedge 2+\\mu }}.$ As a result, by [20], $\\chi _p$ and $\\theta _p$ exhibit the mean-field behavior (REF ).", "Moreover, $p_{\\rm c}$ and $A$ can be expressed in term of $_p$ in (REF ) as $p_{\\rm c}=\\hat{}_{p_{\\rm c}}(0)^{-1},&&A=p_{\\rm c}+{\\left\\lbrace \\begin{array}{ll}0&[\\alpha <2],\\\\[5pt]\\displaystyle \\frac{p_{\\rm c}^2}{\\sigma ^2}\\sum _x|x|^2_{p_{\\rm c}}(x)\\quad &[\\alpha >2].\\end{array}\\right.", "}$ Remark 1.3 The finite-range models are formally considered as the $\\alpha =\\infty $ model.", "Indeed, the leading term in (REF ) for $\\alpha >2$ is identical to (REF ).", "Following the argument in [16], [26], we can “almost\" prove Theorem REF for $\\alpha >2$ without assuming the bounds on $D^{*n}(x)$ .", "The shortcoming is the restriction $d>10$ , not $d>6$ , for percolation.", "This is due to the peculiar diagrammatic estimate in [16], which we do not use in this paper.", "The asymptotic behavior of $G_{p_{\\rm c}}(x)$ in (REF ) or (REF ) is a key element for the so-called 1-arm exponent to take on its mean-field value [17], [21], [23], [25].", "For finite-range critical percolation, for example, the probability that $o\\in {\\mathbb {Z}}^d$ is connected to the surface of the $d$ -dimensional ball of radius $r$ centered at $o$ is bounded above and below by a multiple of $r^{-2}$ in high dimensions [23].", "The value of the exponent may change in a peculiar way depending on the value of $\\alpha $ [21].", "As described in (REF ), the constant $A$ exhibits crossover between $\\alpha <2$ and $\\alpha >2$ ; in particular, $A=p_{\\rm c}$ for $\\alpha <2$ (cf., () below).", "According to some rough computations, it seems that the asymptotic expression of $G_{p_{\\rm c}}(x)$ for $\\alpha =2$ is a mixture of those for $\\alpha <2$ and $\\alpha >2$ , with a logarithmic correction: $G_{p_{\\rm c}}(x)\\underset{|x|\\rightarrow \\infty }{\\sim }\\frac{\\gamma _2/v_2}{p_{\\rm c}|x|^{d-2}\\log |x|}.$ One of the obstacles to prove this conjecture is a lack of good control on convolutions of the RW Green's function and the lace-expansion coefficients for $\\alpha =2$ .", "As hinted in the above expression, we may have to deal with logarithmic factors more actively than ever.", "We are currently working in this direction." ], [ "Notation and the organization", "From now on, we distinguish $G_p^{\\scriptscriptstyle \\rm RW}$ from $G_p$ for the other three models, and define $S_p=G_p^{\\scriptscriptstyle \\rm RW}.$ Here, and in the remainder of the paper, the spatial variables are sometimes omitted.", "For example, $S_p=\\delta +pD*S_p$ is the abbreviated version of the convolution equation $S_p(x)=\\delta _{o,x}+(pD*S_p)(x)=\\delta _{o,x}+\\sum _{y\\in {\\mathbb {Z}}^d}pD(y)\\,S_p(x-y).$ We also recall the notation $|\\!|\\!|x|\\!|\\!|_\\ell =|x|\\vee \\ell .$ The remainder of the paper is organized as follows.", "In Section , we prove the asymptotic expression (REF ) for $S_1$ , as well as bounds on $S_p$ for $p\\le 1$ and some basic properties of $G_p$ for $p\\le p_{\\rm c}$ .", "Then, by using these facts and the diagrammatic bounds on the lace-expansion coefficients in [18], [26], we prove (REF ) for $G_{p_{\\rm c}}$ in Section ." ], [ "Preliminaries", "In this section, we derive the asymptotic expression (REF ) for $S_1$ , which will be restated as Proposition REF , and prove some properties of $G_p$ that will be used to prove Theorem REF in Section ." ], [ "Asymptotics of $S_p$", "Proposition 2.1 Let $\\alpha >0$ , $\\alpha \\ne 2$ and $d>\\alpha \\wedge 2$ , and assume all properties but (REF ) in Assumption REF .", "Then there is a $\\mu \\in (0,\\alpha \\wedge 2)$ such that, for any $L\\ge 1$ , $p\\le 1$ and $\\kappa >0$ , $\\delta _{o,x}\\le S_p(x)&\\le \\delta _{o,x}+\\frac{O(L^{-\\alpha \\wedge 2})}{|\\!|\\!|x|\\!|\\!|_L^{d-\\alpha \\wedge 2}}\\qquad [\\forall x\\in {\\mathbb {Z}}^d],\\\\S_1(x)&=\\frac{\\gamma _\\alpha /v_\\alpha }{|x|^{d-\\alpha \\wedge 2}}+\\frac{O(L^{-\\alpha \\wedge 2+\\mu })}{|x|^{d-\\alpha \\wedge 2+\\mu }}\\qquad [|x|>L^{1+\\kappa }],$ where a constant in the $O(L^{-\\alpha \\wedge 2+\\mu })$ term depends on $\\kappa $ ." ], [ "The inequality (REF ) is an immediate result of (REF ), $p\\le 1$ and (REF )–(REF ) asFor $\\alpha =2$ , we can readily bound $S_p(x)-\\delta _{o,x}$ by using (REF ) for $n\\ge N_x\\equiv |\\!|\\!|x|\\!|\\!|_L^2/(L^2\\log |\\!|\\!|x|\\!|\\!|_L)$ and (REF ) for $n<N_x$ as $S_p(x)-\\delta _{o,x}\\le \\sum _{n=1}^{N_x-1}D^{*n}(x)+\\sum _{n=N_x}^\\infty D^{*n}(x)\\le \\frac{O(L^{-2})}{|\\!|\\!|x|\\!|\\!|_L^{d-2}\\log |\\!|\\!|x|\\!|\\!|_L}.$ $0\\le S_p(x)-\\delta _{o,x}\\le \\sum _{n=1}^\\infty D^{*n}(x)&\\le \\frac{O(L^{\\alpha \\wedge 2})}{|\\!|\\!|x|\\!|\\!|_L^{d+\\alpha \\wedge 2}}\\sum _{n=1}^{(|\\!|\\!|x|\\!|\\!|_L/L)^{\\alpha \\wedge 2}}n+O(L^{-d})\\sum _{n=(|\\!|\\!|x|\\!|\\!|_L/L)^{\\alpha \\wedge 2}}^\\infty n^{-d/(\\alpha \\wedge 2)}\\nonumber \\\\&=\\frac{O(L^{-\\alpha \\wedge 2})}{|\\!|\\!|x|\\!|\\!|_L^{d-\\alpha \\wedge 2}}.$ To prove the asymptotic expression (), we first rewrite $S_1(x)$ for $d>\\alpha \\wedge 2$ as $S_1(x)=\\int _{[-\\pi ,\\pi ]^d}\\frac{\\text{d}^dk}{(2\\pi )^d}\\,\\frac{e^{-ik\\cdot x}}{1-\\hat{D}(k)}&=\\int _0^\\infty \\text{d}t\\int _{[-\\pi ,\\pi ]^d}\\frac{\\text{d}^dk}{(2\\pi )^d}\\,e^{-ik\\cdot x-t(1-\\hat{D}(k))}\\nonumber \\\\&=\\int _T^\\infty \\text{d}t\\int _{[-\\pi ,\\pi ]^d}\\frac{\\text{d}^dk}{(2\\pi )^d}\\,e^{-ik\\cdot x-t(1-\\hat{D}(k))}\\text{d}t+I_1,$ for any $T\\in (0,\\infty )$ , where $I_1=\\int _0^T\\text{d}t\\int _{[-\\pi ,\\pi ]^d}\\frac{\\text{d}^dk}{(2\\pi )^d}\\,e^{-ik\\cdot x-t(1-\\hat{D}(k))}\\text{d}t=\\int _0^T\\text{d}t\\,e^{-t}\\sum _{n=0}^\\infty \\frac{t^n}{n!", "}D^{*n}(x).$ Next we rewrite the large-$t$ integral as $\\int _T^\\infty \\text{d}t\\int _{[-\\pi ,\\pi ]^d}\\frac{\\text{d}^dk}{(2\\pi )^d}\\,e^{-ik\\cdot x-t(1-\\hat{D}(k))}=\\int _0^\\infty \\text{d}t\\,p_{v_\\alpha t}(x)+\\sum _{j=2}^5I_j,$ where $p_s(x)$ is the transition density of an $\\alpha $ -stable/Gaussian process (cf., (REF )), and for any $R\\in (0,\\pi )$ , $I_2&=-\\int _0^T\\text{d}t\\,p_{v_\\alpha t}(x)\\equiv -\\int _0^T\\text{d}t\\int _{{\\mathbb {R}}^d}\\frac{\\text{d}^dk}{(2\\pi )^d}\\,e^{-ik\\cdot x-v_\\alpha t|k|^{\\alpha \\wedge 2}},\\\\I_3&=\\int _T^\\infty \\text{d}t\\int _{|k|\\le R}\\frac{\\text{d}^dk}{(2\\pi )^d}\\,e^{-ik\\cdot x}\\Big (e^{-t(1-\\hat{D}(k))}-e^{-v_\\alpha t|k|^{\\alpha \\wedge 2}}\\Big ),\\\\I_4&=\\int _T^\\infty \\text{d}t\\int _{[-\\pi ,\\pi ]^d}\\frac{\\text{d}^dk}{(2\\pi )^d}\\,e^{-ik\\cdot x-t(1-\\hat{D}(k))}{\\mathbb {1}}_{\\lbrace |k|>R\\rbrace },\\\\I_5&=-\\int _T^\\infty \\text{d}t\\int _{|k|>R}\\frac{\\text{d}^dk}{(2\\pi )^d}\\,e^{-ik\\cdot x-v_\\alpha t|k|^{\\alpha \\wedge 2}}.$ By using the identity $\\int _0^\\infty \\text{d}t\\,e^{-v_\\alpha t|k|^{\\alpha \\wedge 2}}=\\frac{1}{v_\\alpha |k|^{\\alpha \\wedge 2}}=\\frac{1}{v_\\alpha \\Gamma (\\frac{\\alpha \\wedge 2}{2})}\\int _0^\\infty \\text{d}t~t^{\\frac{\\alpha \\wedge 2}{2}-1}e^{-|k|^2t},$ we obtain $\\int _0^\\infty \\text{d}t\\,p_{v_\\alpha t}(x)=\\frac{1}{v_\\alpha \\Gamma (\\frac{\\alpha \\wedge 2}{2})}\\int _0^\\infty \\text{d}t~t^{\\frac{\\alpha \\wedge 2}{2}-1}\\int _{{\\mathbb {R}}^d}\\frac{\\text{d}^dk}{(2\\pi )^d}\\,e^{-|k|^2t-ik\\cdot x}=\\frac{\\gamma _\\alpha /v_\\alpha }{|x|^{d-\\alpha \\wedge 2}}.$ As a result, we arrive at $S_1(x)=\\frac{\\gamma _\\alpha /v_\\alpha }{|x|^{d-\\alpha \\wedge 2}}+\\sum _{j=1}^5I_j.$ It remains to estimate $\\sum _{j=1}^5I_j$ .", "First, by (REF ) and (REF ), we can estimate $I_1+I_2$ for $|x|>L$ as $|I_1+I_2|\\le \\frac{O(L^{\\alpha \\wedge 2})}{|x|^{d+\\alpha \\wedge 2}}\\int _0^T\\text{d}t~t\\le \\frac{O(L^{\\alpha \\wedge 2})T^2}{|x|^{d+\\alpha \\wedge 2}}.$ Let $\\mu =\\frac{2(\\alpha \\wedge 2)\\epsilon }{d+\\alpha \\wedge 2+\\epsilon },&&T=\\bigg (\\frac{|x|}{L}\\bigg )^{\\alpha \\wedge 2-\\mu /2}.$ Then, we obtain $|I_1+I_2|\\le \\frac{O(L^{-\\alpha \\wedge 2+\\mu })}{|x|^{d-\\alpha \\wedge 2+\\mu }}.$ Next we estimate $I_3$ .", "For small $R$ , whose value will be determined shortly, we use (REF ) to obtain $\\Big |e^{-t(1-\\hat{D}(k))}-e^{-v_\\alpha t|k|^{\\alpha \\wedge 2}}\\Big |\\le O(L^{\\alpha \\wedge 2+\\epsilon })t|k|^{\\alpha \\wedge 2+\\epsilon }e^{-v_\\alpha t|k|^{\\alpha \\wedge 2}}.$ Therefore, by (REF ), $|I_3|&\\le O(L^{\\alpha \\wedge 2+\\epsilon })\\int _T^\\infty \\text{d}t~t\\int _{|k|\\le R}\\text{d}^dk~|k|^{\\alpha \\wedge 2+\\epsilon }e^{-v_\\alpha t|k|^{\\alpha \\wedge 2}}\\nonumber \\\\&=O(L^{\\alpha \\wedge 2+\\epsilon })\\int _T^\\infty \\text{d}t~t\\int _0^{v_\\alpha tR^{\\alpha \\wedge 2}}\\frac{\\text{d}r}{r}~\\bigg (\\frac{r}{v_\\alpha t}\\bigg )^{\\frac{d+\\alpha \\wedge 2+\\epsilon }{\\alpha \\wedge 2}}e^{-r}\\nonumber \\\\&\\le O(L^{\\alpha \\wedge 2+\\epsilon })\\int _T^\\infty \\text{d}t~t(v_\\alpha t)^{-\\frac{d+\\alpha \\wedge 2+\\epsilon }{\\alpha \\wedge 2}}\\nonumber \\\\&\\le O(L^{-d})T^{1-\\frac{d+\\epsilon }{\\alpha \\wedge 2}}=\\frac{O(L^{-\\alpha \\wedge 2+\\mu })}{|x|^{d-\\alpha \\wedge 2+\\mu }}.$ Finally we estimate $I_4+I_5$ and determine the value of $R$ during the course.", "First, by (REF )–(REF ), we have $|I_4|&\\le \\int _T^\\infty \\text{d}t\\int _{[-\\pi ,\\pi ]^d}\\frac{\\text{d}^dk}{(2\\pi )^d}\\,e^{-t(1-\\hat{D}(k))}{\\mathbb {1}}_{\\lbrace |k|>R\\rbrace }\\Big ({\\mathbb {1}}_{\\lbrace \\Vert k\\Vert _\\infty <L^{-1}\\rbrace }+{\\mathbb {1}}_{\\lbrace \\Vert k\\Vert _\\infty \\ge L^{-1}\\rbrace }\\Big )\\nonumber \\\\&\\le \\int _T^\\infty \\text{d}t~\\bigg (\\int _{|k|>R}\\frac{\\text{d}^dk}{(2\\pi )^d}\\,e^{-tc(L|k|)^{\\alpha \\wedge 2}}+O(1)e^{-t\\Delta }\\bigg )\\nonumber \\\\&\\le O(L^{-d})\\int _T^\\infty \\text{d}t~t^{-\\frac{d}{\\alpha \\wedge 2}}\\Gamma (\\tfrac{d}{\\alpha \\wedge 2};tc(LR)^{\\alpha \\wedge 2})+O(1)e^{-T\\Delta },$ where $\\Gamma (a;x)\\equiv \\int _x^\\infty \\text{d}t\\,t^{a-1}e^{-t}$ is the incomplete gamma function, which is bounded by $O(x^{a-1})e^{-x}$ for large $x$ .", "Here we choose $R$ to satisfy $tc(LR)^{\\alpha \\wedge 2}=\\frac{2\\epsilon }{\\alpha \\wedge 2}\\log t.$ Then, for large $t$ , $\\Gamma (\\tfrac{d}{\\alpha \\wedge 2};tc(LR)^{\\alpha \\wedge 2})&\\le O\\big ((tc(LR)^{\\alpha \\wedge 2})^{\\frac{d}{\\alpha \\wedge 2}-1}\\big )e^{-tc(LR)^{\\alpha \\wedge 2}}\\nonumber \\\\&=O\\big ((\\log t)^{\\frac{d}{\\alpha \\wedge 2}-1}\\big )t^{-\\frac{2\\epsilon }{\\alpha \\wedge 2}}\\le O(t^{-\\frac{\\epsilon }{\\alpha \\wedge 2}}).$ Therefore, again by (REF ) (cf., (REF )), $O(L^{-d})\\int _T^\\infty \\text{d}t~t^{-\\frac{d}{\\alpha \\wedge 2}}\\Gamma (\\tfrac{d}{\\alpha \\wedge 2};tc(LR)^{\\alpha \\wedge 2})&\\le O(L^{-d})T^{1-\\frac{d+\\epsilon }{\\alpha \\wedge 2}}=\\frac{O(L^{-\\alpha \\wedge 2+\\mu })}{|x|^{d-\\alpha \\wedge 2+\\mu }}.$ We can estimate $I_5$ in exactly the same way.", "The exponentially decaying term in (REF ) obeys the same bound, since, for sufficiently large $N$ (depending on $\\kappa $ ), $e^{-T\\Delta }&\\le \\frac{{}^\\exists c_N}{T^N}=c_NL^{-d}T^{1-\\frac{d+\\epsilon }{\\alpha \\wedge 2}}L^dT^{-(N+1-\\frac{d+\\epsilon }{\\alpha \\wedge 2})}\\nonumber \\\\&\\le c_NL^{-d}T^{1-\\frac{d+\\epsilon }{\\alpha \\wedge 2}}L^{d-(N+1-\\frac{d+\\epsilon }{\\alpha \\wedge 2})(\\alpha \\wedge 2-\\mu /2)\\kappa }\\qquad (\\because ~|x|>L^{1+\\kappa }\\Rightarrow T>L^{(\\alpha \\wedge 2-\\mu /2)\\kappa })\\nonumber \\\\&\\le c_NL^{-d}T^{1-\\frac{d+\\epsilon }{\\alpha \\wedge 2}}=\\frac{O(L^{-\\alpha \\wedge 2+\\mu })}{|x|^{d-\\alpha \\wedge 2+\\mu }}.$ Summarizing the above, we obtain that, for $|x|>L^{1+\\kappa }$ , $\\bigg |\\sum _{j=1}^5I_j\\bigg |\\le \\frac{O(L^{-\\alpha \\wedge 2+\\mu })}{|x|^{d-\\alpha \\wedge 2+\\mu }}.$ This together with (REF ) completes the proof of Proposition REF ." ], [ "Basic properties of $G_p$", "In this subsection, we summarize some basic properties of $G_p$ .", "Roughly speaking, those properties are the continuity up to $p=p_{\\rm c}$ (Lemma REF ), the RW bound that is optimal for $p\\le 1$ (Lemma REF ) and the a priori bound that is not sharp but finite as long as $p<p_{\\rm c}$ (Lemma REF ).", "We will use them in the next section (especially in Section REF ) to prove Theorem REF .", "Lemma 2.2 For every $x\\in {\\mathbb {Z}}^d$ , $G_p(x)$ is nondecreasing and continuous in $p<p_{\\rm c}$ for SAW, and in $p\\le p_{\\rm c}$ for percolation and the Ising model.", "The continuity up to $p=p_{\\rm c}^{\\scriptscriptstyle \\rm SAW}$ for SAW is also valid if $G_p^{\\scriptscriptstyle \\rm SAW}(x)$ is uniformly bounded in $p<p_{\\rm c}^{\\scriptscriptstyle \\rm SAW}$ ." ], [ "For SAW, since $G_p^{\\scriptscriptstyle \\rm SAW}(x)$ is a power series of $p\\ge 0$ with nonnegative coefficients, it is nondecreasing and continuous in $p<p_{\\rm c}^{\\scriptscriptstyle \\rm SAW}$ .", "The continuity up to $p=p_{\\rm c}^{\\scriptscriptstyle \\rm SAW}$ under the hypothesis is due to monotone convergence.", "For the Ising model, we first note that, by the Griffiths inequality [13], $\\langle \\varphi _o\\varphi _x\\rangle _{\\beta ,\\Lambda }$ is nondecreasing and continuous in $\\beta \\ge 0$ and nondecreasing in $\\Lambda \\subset {\\mathbb {Z}}^d$ .", "Therefore, the infinite-volume limit $G_p^{\\scriptscriptstyle \\rm Ising}(x)$ is nondecreasing and left-continuous in $p\\ge 0$ .", "The continuity in $p\\le p_{\\rm c}^{\\scriptscriptstyle \\rm Ising}$ follows from the fact that, for $p<p_{\\rm c}^{\\scriptscriptstyle \\rm Ising}$ , $G_p^{\\scriptscriptstyle \\rm Ising}(x)$ coincides with the decreasing limit of the finite-volume two-point function under the “plus-boundary\" condition, which is right-continuous in $p\\ge 0$ .", "For percolation, $G_p^{\\scriptscriptstyle \\rm perc}(x)$ is nondecreasing in $p\\ge 0$ because the event that there is a path of occupied bonds from $o$ to $x$ is an increasing event.", "The continuity in $p\\ge 0$ is obtained by following the same strategy as explained above for the Ising model and using the fact that there is at most one infinite occupied cluster for all $p\\ge 0$ .", "This completes the proof of Lemma REF .", "Lemma 2.3 For every $p<p_{\\rm c}$ and $x\\in {\\mathbb {Z}}^d$ , $G_p(x)&\\le S_p(x),&&pD(x)(1-\\delta _{o,x})\\le G_p(x)-\\delta _{o,x}\\le (pD*G_p)(x).$" ], [ "The first inequality for $p>1\\equiv p_{\\rm c}^{\\scriptscriptstyle \\rm RW}$ is trivial since $S_p(x)=\\infty $ for every $x\\in {\\mathbb {Z}}^d$ .", "On the other hand, the first inequality for $p\\le 1$ is obtained by using the second inequality $N$ times and then using (REF ), as $G_p(x)\\le \\sum _{n=0}^{N-1}p^nD^{*n}(x)+p^N(D^{*N}*G_p)(x)\\le S_p(x)+\\Vert D^{*N}\\Vert _\\infty \\,\\chi _p\\underset{N\\uparrow \\infty }{\\rightarrow }S_p(x).$ It remains to prove the second inequality in (REF ).", "In fact, it suffices to prove the inequality only for $x\\ne o$ , since $G_p(o)=1$ for all three models and therefore the inequality is trivial for $x=o$ .", "For SAW and percolation, the inequality is obtained by specifying the first step $pD$ and then using subadditivity for SAW or the BK inequality for percolation [6].", "For the Ising model, we use the following random-current representation [1], [14] (see also [26]): $\\langle \\varphi _o\\varphi _x\\rangle _{\\beta ,\\Lambda }=\\frac{\\displaystyle \\sum _{\\partial n=\\lbrace o\\rbrace \\triangle \\lbrace x\\rbrace }w_\\Lambda (n)}{\\displaystyle \\sum _{\\partial n=\\varnothing }w_\\Lambda (n)},&&w_\\Lambda (n)=\\prod _{\\lbrace u,v\\rbrace \\subset \\Lambda }\\frac{(\\beta J_{u,v})^{n_{u,v}}}{n_{u,v}!", "},$ where $n\\equiv \\lbrace n_{u,v}\\rbrace $ is a collection of ${\\mathbb {Z}}_+$ -valued undirected bond variables (i.e., $n_{u,v}=n_{v,u}\\in {\\mathbb {Z}}_+\\equiv \\lbrace 0\\rbrace \\cup \\mathbb {N}$ for each bond $\\lbrace u,v\\rbrace \\subset \\Lambda $ ), $\\partial n$ is the set of vertices $y$ such that $\\sum _{z\\in \\Lambda }n_{y,z}$ is an odd number, and “$\\triangle $ \" represents symmetric difference (i.e., $\\lbrace o\\rbrace \\triangle \\lbrace x\\rbrace =\\varnothing $ if $x=o$ , otherwise $\\lbrace o\\rbrace \\triangle \\lbrace x\\rbrace =\\lbrace o,x\\rbrace $ ).", "Using this representation, we prove below that, for $x\\ne o$ , $pD(x)\\le \\langle \\varphi _o\\varphi _x\\rangle _{\\beta ,\\Lambda }\\le \\sum _{y\\in \\Lambda }pD(y)\\langle \\varphi _y\\varphi _x\\rangle _{\\beta ,\\Lambda },$ where $pD(x)=\\tanh (\\beta J_{o,x})$ .", "The second inequality in (REF ) for the Ising model is the infinite-volume limit of the above inequality.", "To prove the lower bound of (REF ), we first specify the parity of $n_{o,x}$ to obtain that, for $x\\ne o$ (so that $\\lbrace o\\rbrace \\triangle \\lbrace x\\rbrace =\\lbrace o,x\\rbrace $ ), $\\langle \\varphi _o\\varphi _x\\rangle _{\\beta ,\\Lambda }=\\frac{\\displaystyle \\sum _{\\begin{array}{c}\\partial n=\\lbrace o,x\\rbrace \\\\ (n_{o,x}\\text{ odd})\\end{array}}w_\\Lambda (n)+\\sum _{\\begin{array}{c}\\partial n=\\lbrace o,x\\rbrace \\\\ (n_{o,x}\\text{ even})\\end{array}}w_\\Lambda (n)}{\\displaystyle \\sum _{\\begin{array}{c}\\partial n=\\varnothing \\\\ (n_{o,x}\\text{ odd})\\end{array}}w_\\Lambda (n)+\\sum _{\\begin{array}{c}\\partial n=\\varnothing \\\\ (n_{o,x}\\text{ even})\\end{array}}w_\\Lambda (n)}.$ Let $\\tilde{Y}_y(z,x)\\equiv \\sum _{\\begin{array}{c}\\partial n=\\lbrace z\\rbrace \\triangle \\lbrace x\\rbrace \\\\(n_{o,y}\\text{ even})\\end{array}}w_\\Lambda (n),&&\\tilde{Z}_y\\equiv \\sum _{\\begin{array}{c}\\partial n=\\varnothing \\\\ (n_{o,y}\\text{ even})\\end{array}}w_\\Lambda (n).$ Then, by changing the parity of $n_{o,x}$ (and the constraint on $\\partial n$ accordingly) and recalling $\\tanh (\\beta J_{o,x})=pD(x)$ , we obtain $\\sum _{\\begin{array}{c}\\partial n=\\lbrace o,x\\rbrace \\\\ (n_{o,x}\\text{ odd})\\end{array}}w_\\Lambda (n)&=pD(x)\\sum _{\\begin{array}{c}\\partial n=\\varnothing \\\\ (n_{o,x}\\text{ even})\\end{array}}w_\\Lambda (n)=pD(x)\\tilde{Z}_x,\\\\\\sum _{\\begin{array}{c}\\partial n=\\varnothing \\\\ (n_{o,x}\\text{ odd})\\end{array}}w_\\Lambda (n)&=pD(x)\\sum _{\\begin{array}{c}\\partial n=\\lbrace o,x\\rbrace \\\\ (n_{o,x}\\text{ even})\\end{array}}w_\\Lambda (n)=pD(x)\\tilde{Y}_x(o,x),$ hence $\\langle \\varphi _o\\varphi _x\\rangle _{\\beta ,\\Lambda }=\\frac{pD(x)\\tilde{Z}_x+\\tilde{Y}_x(o,x)}{pD(x)\\tilde{Y}_x(o,x)+\\tilde{Z}_x}=pD(x)+\\frac{(1-p^2D(x)^2)\\tilde{Y}_x(o,x)}{pD(x)\\tilde{Y}_x(o,x)+\\tilde{Z}_x}\\ge pD(x).$ To prove the upper bound in (REF ), we first note that, if $\\partial n=\\lbrace o,x\\rbrace $ , then there must be at least one $y\\in \\Lambda $ such that $n_{o,y}$ is an odd number.", "By similar computation to (REF ), we obtain that, for $x\\ne o$ , $\\sum _{\\partial n=\\lbrace o,x\\rbrace }w_\\Lambda (n)\\le \\sum _{y\\in \\Lambda }\\sum _{\\begin{array}{c}\\partial n=\\lbrace o,x\\rbrace \\\\ (n_{o,y}\\text{ odd})\\end{array}}w_\\Lambda (n)&=\\sum _{y\\in \\Lambda }pD(y)\\underbrace{\\sum _{\\begin{array}{c}\\partial n=\\lbrace y\\rbrace \\triangle \\lbrace x\\rbrace \\\\ (n_{o,y}\\text{ even})\\end{array}}w_\\Lambda (n)}_{\\tilde{Y}_y(y,x)}.$ On the other hand, $\\sum _{\\partial n=\\varnothing }w_\\Lambda (n)\\ge \\tilde{Z}_y$ for any $y\\in \\Lambda $ .", "Therefore, for $x\\ne o$ , $\\langle \\varphi _o\\varphi _x\\rangle _{\\beta ,\\Lambda }\\equiv \\frac{\\displaystyle \\sum _{\\partial n=\\lbrace o,x\\rbrace }w_\\Lambda (n)}{\\displaystyle \\sum _{\\partial n=\\varnothing }w_\\Lambda (n)}\\le \\sum _{y\\in \\Lambda }\\frac{pD(y)\\tilde{Y}_y(y,x)}{\\tilde{Z}_y}\\le \\sum _{y\\in \\Lambda }pD(y)\\langle \\varphi _y\\varphi _x\\rangle _{\\beta ,\\Lambda },$ where we have used the fact that $\\tilde{Y}_y(y,x)/\\tilde{Z}_y$ is equivalent to the finite-volume two-point function under the restriction $J_{o,y}=0$ and therefore, by using the Griffiths inequality [13], it is bounded above by $\\langle \\varphi _y\\varphi _x\\rangle _{\\beta ,\\Lambda }$ .", "This completes the proof of (REF ), hence the proof of Lemma REF .", "Lemma 2.4 Assume the property (REF ) in Assumption REF .", "Then, for every $\\alpha >0$ and $p<p_{\\rm c}$ , there is a $K_p=K_p(\\alpha ,d,L)<\\infty $ such that, for any $x\\in {\\mathbb {Z}}^d$ , $G_p(x)\\le K_p|\\!|\\!|x|\\!|\\!|_L^{-d-\\alpha }.$ Remark 2.5 This together with the lower bound in (REF ) implies that, for every $p<p_{\\rm c}$ , $G_p(x)$ is bounded above and below by a $p$ -dependent multiple of $|\\!|\\!|x|\\!|\\!|_L^{-d-\\alpha }$ .", "This shows sharp contrast to the exponential decay of $G_p(x)$ for the finite-range models." ], [ "Since $G_p(o)\\le \\chi _p<\\infty $ for $p<p_{\\rm c}$ , it suffices to prove (REF ) for $x\\ne o$ .", "We follow the idea of the proof of [4] for one-dimensional long-range percolation and extend it to those three models in general dimensions.", "The key ingredient is the following Simon-Lieb type inequality: for $0<\\ell <|x|$ , $G_p(x)\\le \\sum _{\\begin{array}{c}\\lbrace u,v\\rbrace \\subset {\\mathbb {Z}}^d\\\\ (|u|\\le \\ell <|v|)\\end{array}}G_p(u)\\,pD(v-u)\\,G_p(x-v).$ For SAW and percolation, this is a result of subadditivity or the BK inequality (cf., e.g., [15], [24]).", "For the Ising model, this is obtained by using the random-current representation (REF ) and a restricted version of the source-switching lemma [26], as follows.", "Let $Z_\\Lambda =\\sum _{\\partial n=\\varnothing }w_\\Lambda (n)$ such that, for $x\\ne o$ , $\\langle \\varphi _o\\varphi _x\\rangle _{\\beta ,\\Lambda }=\\sum _{\\partial n=\\lbrace o,x\\rbrace }\\frac{w_\\Lambda (n)}{Z_\\Lambda }.$ We note that, if $\\partial n=\\lbrace o,x\\rbrace $ , then there is a path $\\omega =(\\omega _0,\\omega _1,\\dots ,\\omega _t)\\subset \\Lambda $ from $\\omega _0=o$ to $\\omega _t=x$ such that $n_{\\omega _{s-1},\\omega _s}$ is odd for every $s\\in \\lbrace 1,\\dots ,t\\rbrace $ ; moreover, there is a unique $\\tau \\in \\lbrace 1,\\dots ,t\\rbrace $ such that $|\\omega _{\\tau -1}|\\le \\ell <|\\omega _\\tau |$ (i.e., $\\tau $ is the first time when $\\omega $ crosses the surface of the ball $B_\\ell $ of radius $\\ell $ centered at the origin).", "This can be restated as follows: if $\\partial n=\\lbrace o,x\\rbrace $ , then there is a bond $\\lbrace u,v\\rbrace \\subset \\Lambda $ such that $n_{u,v}$ is odd and that $u$ is connected from $o$ with a path of bonds $\\subset B_\\ell $ with odd numbers.", "Therefore, $\\langle \\varphi _o\\varphi _x\\rangle _{\\beta ,\\Lambda }\\le \\sum _{\\begin{array}{c}\\lbrace u,v\\rbrace \\subset \\Lambda \\\\ (|u|\\le \\ell <|v|)\\end{array}}\\sum _{\\partial n=\\lbrace o,x\\rbrace }\\frac{w_\\Lambda (n)}{Z_\\Lambda }\\,{\\mathbb {1}}_{\\lbrace n_{u,v}\\text{ odd}\\rbrace }\\,{\\mathbb {1}}_{\\lbrace o\\underset{n}{\\longleftrightarrow }u\\text{ in }B_\\ell \\rbrace },$ where $\\lbrace o\\underset{n}{\\longleftrightarrow }u$ in $B_\\ell \\rbrace $ is the event that $o$ is connected to $u$ with a path of bonds $b\\subset B_\\ell $ satisfying $n_b>0$ .", "Multiplying $Z_{B_\\ell }/Z_{B_\\ell }\\equiv 1$ to both sides of (REF ) and using the identity $Z_{B_\\ell }=\\sum _{\\partial m=\\varnothing }w_{B_\\ell }(m)$ , we obtain $\\langle \\varphi _o\\varphi _x\\rangle _{\\beta ,\\Lambda }&\\le \\sum _{\\begin{array}{c}\\lbrace u,v\\rbrace \\subset \\Lambda \\\\ (|u|\\le \\ell <|v|)\\end{array}}\\sum _{\\begin{array}{c}\\partial m=\\varnothing \\\\ \\partial n=\\lbrace o,x\\rbrace \\end{array}}\\frac{w_{B_\\ell }(m)}{Z_{B_\\ell }}\\,\\frac{w_\\Lambda (n)}{Z_\\Lambda }\\,{\\mathbb {1}}_{\\lbrace n_{u,v}\\text{ odd}\\rbrace }\\,{\\mathbb {1}}_{\\lbrace o\\underset{m+n}{\\longleftrightarrow }u\\text{ in }B_\\ell \\rbrace },$ where we have used the trivial inequality ${\\mathbb {1}}_{\\lbrace o\\underset{n}{\\longleftrightarrow }u\\text{ in }B_\\ell \\rbrace }\\le {\\mathbb {1}}_{\\lbrace o\\underset{m+n}{\\longleftrightarrow }u\\text{ in }B_\\ell \\rbrace }$ .", "Then, by using the source-switching lemma [26], we obtain $\\langle \\varphi _o\\varphi _x\\rangle _{\\beta ,\\Lambda }&\\le \\sum _{\\begin{array}{c}\\lbrace u,v\\rbrace \\subset \\Lambda \\\\ (|u|\\le \\ell <|v|)\\end{array}}\\sum _{\\begin{array}{c}\\partial m=\\lbrace o\\rbrace \\triangle \\lbrace u\\rbrace \\\\ \\partial n=\\lbrace u,x\\rbrace \\end{array}}\\frac{w_{B_\\ell }(m)}{Z_{B_\\ell }}\\,\\frac{w_\\Lambda (n)}{Z_\\Lambda }\\,{\\mathbb {1}}_{\\lbrace n_{u,v}\\text{ odd}\\rbrace }\\,{\\mathbb {1}}_{\\lbrace o\\underset{m+n}{\\longleftrightarrow }u\\text{ in }B_\\ell \\rbrace }\\nonumber \\\\&=\\sum _{\\begin{array}{c}\\lbrace u,v\\rbrace \\subset \\Lambda \\\\ (|u|\\le \\ell <|v|)\\end{array}}\\langle \\varphi _o\\varphi _u\\rangle _{\\beta ,B_\\ell }\\sum _{\\begin{array}{c}\\partial n=\\lbrace u,x\\rbrace \\\\ (n_{u,v}\\text{ odd})\\end{array}}\\frac{w_\\Lambda (n)}{Z_\\Lambda },$ where we have used the identity ${\\mathbb {1}}_{\\lbrace o\\underset{m+n}{\\longleftrightarrow }u\\text{ in }B_\\ell \\rbrace }=1$ given $\\partial m=\\lbrace o\\rbrace \\triangle \\lbrace u\\rbrace $ and then used (REF ).", "Finally, by following the same argument as in (REF )–(REF ) and then taking the infinite-volume limit, we obtain (REF ) for the Ising model.", "Now we prove (REF ) by using (REF ) with $\\ell =\\frac{1}{3}|x|$ (the factor $\\frac{1}{3}$ is unimportant as long as it is less than $\\frac{1}{2}$ ).", "Let $c_x=\\sum _{\\begin{array}{c}\\lbrace u,v\\rbrace \\subset {\\mathbb {Z}}^d\\\\ (|u|\\le \\frac{1}{3}|x|<|v|)\\end{array}}G_p(u)\\,D(v-u).$ We note that $c_x\\rightarrow 0$ as $|x|\\rightarrow \\infty $ , because $c_x&=\\sum _{\\begin{array}{c}\\lbrace u,v\\rbrace \\subset {\\mathbb {Z}}^d\\\\ (|u|\\le \\frac{1}{4}|x|,~\\frac{1}{3}|x|<|v|)\\end{array}}G_p(u)\\,pD(v-u)+\\sum _{\\begin{array}{c}\\lbrace u,v\\rbrace \\subset {\\mathbb {Z}}^d\\\\ (\\frac{1}{4}|x|<|u|\\le \\frac{1}{3}|x|<|v|)\\end{array}}G_p(u)\\,pD(v-u)\\nonumber \\\\&\\le \\chi _p\\,p\\underbrace{\\sup _{u:|u|\\le \\frac{1}{4}|x|}\\sum _{v:|v|>\\frac{1}{3}|x|}D(v-u)}_{O(|x|^{-\\alpha })}+p\\underbrace{\\sum _{u:|u|>\\frac{1}{4}|x|}G_p(u)}_{\\text{Tail of }\\chi _p<\\infty }.$ Therefore, for any $\\epsilon \\in (0,1)$ , there is an $\\tilde{\\ell }\\in [L,\\infty )$ such that $2^{d+\\alpha }c_xp\\le \\epsilon $ for all $|x|\\ge \\tilde{\\ell }$ .", "Then, for $|x|\\ge \\tilde{\\ell }$ , (REF ) implies $G_p(x)&\\le \\sum _{\\begin{array}{c}\\lbrace u,v\\rbrace \\subset {\\mathbb {Z}}^d\\\\ (|u|\\le \\frac{1}{3}|x|<|v|\\le \\frac{1}{2}|x|)\\end{array}}G_p(u)\\,pD(v-u)\\,G_p(x-v)\\nonumber \\\\&\\quad +\\sum _{\\begin{array}{c}\\lbrace u,v\\rbrace \\subset {\\mathbb {Z}}^d\\\\ (|u|\\le \\frac{1}{3}|x|,~|v|>\\frac{1}{2}|x|)\\end{array}}G_p(u)\\,pD(v-u)\\,G_p(x-v)\\nonumber \\\\&\\le c_xp\\sup _{v:|v|\\le \\frac{1}{2}|x|}G_p(x-v)+\\chi _p^2\\,p\\sup _{\\begin{array}{c}\\lbrace u,v\\rbrace \\subset {\\mathbb {Z}}^d\\\\ (|u|\\le \\frac{1}{3}|x|,~|v|>\\frac{1}{2}|x|)\\end{array}}D(v-u)\\nonumber \\\\&\\le 2^{-d-\\alpha }\\epsilon \\sup _{v:|v|>\\frac{1}{2}|x|}G_p(v)+\\frac{C_p}{|\\!|\\!|x|\\!|\\!|_L^{d+\\alpha }},$ for some $C_p=O(\\chi _p^2)$ .", "If $2\\tilde{\\ell }\\le |x|<4\\tilde{\\ell }$ , then we use (REF ) twice to obtain $G_p(x)&\\le (2^{-d-\\alpha }\\epsilon )^2\\sup _{v:|v|>\\frac{1}{4}|x|}G_p(v)+2^{-d-\\alpha }\\epsilon \\frac{C_p}{|\\!|\\!|x/2|\\!|\\!|_L^{d+\\alpha }}+\\frac{C_p}{|\\!|\\!|x|\\!|\\!|_L^{d+\\alpha }}\\nonumber \\\\&=(2^{-d-\\alpha }\\epsilon )^2\\sup _{v:|v|>\\frac{1}{4}|x|}G_p(v)+(1+\\epsilon )\\frac{C_p}{|\\!|\\!|x|\\!|\\!|_L^{d+\\alpha }}.$ In general, if $2^{n-1}\\tilde{\\ell }\\le |x|<2^n\\tilde{\\ell }$ for some $n\\in \\mathbb {N}$ , then we repeatedly use (REF ) to obtain $G_p(x)&\\le (2^{-d-\\alpha }\\epsilon )^n\\sup _{v:|v|>\\frac{1}{2^n}|x|}G_p(v)+(1+\\epsilon \\cdots +\\epsilon ^{n-1})\\frac{C_p}{|\\!|\\!|x|\\!|\\!|_L^{d+\\alpha }}\\nonumber \\\\&\\le \\frac{\\tilde{\\ell }^{d+\\alpha }}{|\\!|\\!|x|\\!|\\!|_L^{d+\\alpha }}\\chi _p+\\frac{C_p}{(1-\\epsilon )|\\!|\\!|x|\\!|\\!|_L^{d+\\alpha }}.$ For $|x|<\\tilde{\\ell }$ , we use the trivial inequality $G_p(x)\\le \\chi _p\\le \\tilde{\\ell }^{d+\\alpha }\\chi _p/|\\!|\\!|x|\\!|\\!|_L^{d+\\alpha }$ .", "This completes the proof of (REF ), where $K_p=\\tilde{\\ell }^{d+\\alpha }\\chi _p+C_p/(1-\\epsilon )$ ." ], [ "Proof of the main result", "In this section, we prove the asymptotic behavior (REF ) of $G_{p_{\\rm c}}$ in high dimensions.", "To do so, we show in Section REF that, if $d>d_{\\rm c}$ and $L\\gg 1$ , then $G_p$ for $p\\le p_{\\rm c}$ obeys the same bound as in (REF ) on $S_p$ for $p\\le 1$ .", "Then, in Section REF , we show that the obtained infrared bound on $G_{p_{\\rm c}}$ implies its asymptotic expression (REF ).", "The proofs rely on the lace expansion (REF ) for $G_p$ ." ], [ "Bounds on $_p$ assuming the infrared bound on {{formula:f4a92b95-c945-410f-bd7a-ec58bd18a8ba}}", "In this subsection, we assume the infrared bound on $G_p$ and prove bounds on $_p$ and related quantities, such as its sum $\\hat{}_p(0)\\equiv \\sum _x_p(x)$ , in high dimensions.", "Before stating this more precisely, we need introduce the following parameter for $\\alpha >0$ , $\\alpha \\ne 2$ and $d>\\alpha \\wedge 2$ (cf., (REF )): $\\lambda =\\sup _{x\\ne o}\\frac{S_1(x)}{|\\!|\\!|x|\\!|\\!|_L^{\\alpha \\wedge 2-d}}=O(L^{-\\alpha \\wedge 2}).$ Proposition 3.1 Let $\\alpha >0$ , $\\alpha \\ne 2$ and $d>d_{\\rm c}$ , and assume the properties (REF ) and (REF ) in Assumption REF .", "Suppose that $p\\le 3,&&G_p(x)\\le 3\\lambda |\\!|\\!|x|\\!|\\!|_L^{\\alpha \\wedge 2-d}\\qquad [x\\ne o].$ If $\\lambda \\ll 1$ (i.e., $L\\gg 1$ ), then, for any $x\\in {\\mathbb {Z}}^d$ , $(pD*G_p)(x)&\\le O(\\lambda )|\\!|\\!|x|\\!|\\!|_L^{\\alpha \\wedge 2-d},\\\\|_p(x)-\\delta _{o,x}|&\\le O(L^{-d})\\delta _{o,x}+O(\\lambda ^\\ell )|\\!|\\!|x|\\!|\\!|_L^{(\\alpha \\wedge 2-d)\\ell },$ where $\\ell =2$ for percolation and $\\ell =3$ for SAW and the Ising model.", "As a result, $\\hat{}_p(0)&=1+O(L^{-d}),\\\\\\bar{\\nabla }^{\\alpha \\wedge 2}\\hat{}_p(0)&\\equiv \\lim _{|k|\\rightarrow 0}\\frac{\\hat{}_p(0)-\\hat{}_p(k)}{1-\\hat{D}(k)}\\nonumber \\\\&={\\left\\lbrace \\begin{array}{ll}0&[\\alpha <2],\\\\\\frac{1}{\\sigma ^2}\\sum _x|x|^2_p(x)=O(L^{-d(\\ell -1)})&[\\alpha >2].\\end{array}\\right.", "}$ We prove this proposition by using the following lemma, which is an improved version of [18].", "Lemma 3.2 For any $a\\ge b>0$ with $a+b>d$ , there is an $L$ -independent constant $C=C(a,b,d)<\\infty $ such that $\\sum _{y\\in {\\mathbb {Z}}^d}|\\!|\\!|x-y|\\!|\\!|_L^{-a}\\,|\\!|\\!|y|\\!|\\!|_L^{-b}\\le {\\left\\lbrace \\begin{array}{ll}CL^{d-a}|\\!|\\!|x|\\!|\\!|_L^{-b}\\quad &[a>d],\\\\C|\\!|\\!|x|\\!|\\!|_L^{d-a-b}&[a<d].\\end{array}\\right.", "}$ Let $f$ and $g$ be functions on ${\\mathbb {Z}}^d$ , with $g$ being ${\\mathbb {Z}}^d$ -symmetric.", "Suppose that there are $C_1,C_2,C_3>0$ and $\\rho >0$ such that $f(x)=C_1|\\!|\\!|x|\\!|\\!|_L^{\\alpha \\wedge 2-d},&&|g(x)|\\le C_2\\delta _{o,x}+C_3|\\!|\\!|x|\\!|\\!|_L^{-d-\\rho }.$ Then there is a $\\rho ^{\\prime }\\in (0,\\rho \\wedge 2)$ such that, for $d>\\alpha \\wedge 2$ , $(f*g)(x)=\\frac{C_1\\Vert g\\Vert _1}{|\\!|\\!|x|\\!|\\!|_L^{d-\\alpha \\wedge 2}}+\\frac{O(C_1C_3)}{|\\!|\\!|x|\\!|\\!|_L^{d-\\alpha \\wedge 2+\\rho ^{\\prime }}}.$" ], [ "First we note that $D(x)=\\frac{O(L^\\alpha )}{|\\!|\\!|x|\\!|\\!|_L^{d+\\alpha }}=\\frac{O(L^\\alpha )|\\!|\\!|x|\\!|\\!|_L^{-\\alpha -\\alpha \\wedge 2}}{|\\!|\\!|x|\\!|\\!|_L^{d-\\alpha \\wedge 2}}\\le \\frac{O(\\lambda )}{|\\!|\\!|x|\\!|\\!|_L^{d-\\alpha \\wedge 2}}.$ We also note that the identity $G_p(y)=\\delta _{o,y}+G_p(y){\\mathbb {1}}_{\\lbrace y\\ne o\\rbrace }$ holds for all three models.", "Therefore, by using the assumed bound (REF ) and Lemma REF (i), we obtain (REF ) as $(D*G_p)(x)&=D(x)+\\sum _{y\\ne o}D(x-y)\\,G_p(y)\\nonumber \\\\&\\le \\frac{O(\\lambda )}{|\\!|\\!|x|\\!|\\!|_L^{d-\\alpha \\wedge 2}}+\\sum _{y\\in {\\mathbb {Z}}^d}\\frac{O(L^\\alpha )}{|\\!|\\!|x-y|\\!|\\!|_L^{d+\\alpha }}\\,\\frac{3\\lambda }{|\\!|\\!|y|\\!|\\!|_L^{d-\\alpha \\wedge 2}}\\le \\frac{O(\\lambda )}{|\\!|\\!|x|\\!|\\!|_L^{d-\\alpha \\wedge 2}}.$ The inequality () is obtained by repeatedly applying (REF )–(REF ) and Lemma REF (i) to the diagrammatic bounds on $_p(x)$ in [18], [26] ($_p(x)$ in this paper equals $\\delta _{o,x}+\\Pi _z(x)$ in [18]), where $\\ell $ is the number of disjoint paths in the diagrams from $o$ to $x$ (cf., Figure REF ).", "Figure: The left figure is an example of the lace-expansion diagrams for percolation,and the right one is for SAW and the Ising model.", "The number ℓ\\ell ofdisjoint paths from oo to xx using different sets of line segments is 2 inthe left figure and 3 in the right figure.The proof is quite similar to [18] and [26]; the only difference is the use of $|\\!|\\!|\\cdot |\\!|\\!|_L$ instead of $|\\!|\\!|\\cdot |\\!|\\!|_1$ and Lemma REF (i).", "Because of this, we gain the factor $O(L^{-d})~(=O(\\lambda )|\\!|\\!|o|\\!|\\!|_L^{\\alpha \\wedge 2-d})$ in (), which is much smaller than $O(\\lambda )$ as claimed in [18], [26].", "It remains to prove (REF )–().", "By (), we readily obtain (REF ) as $\\hat{}_p(0)\\equiv \\sum _{x\\in {\\mathbb {Z}}^d}_p(x)=1+O(L^{-d})+O(L^{-d(\\ell -1)})=1+O(L^{-d}).$ Moreover, $|\\hat{}_p(0)-\\hat{}_p(k)|\\equiv \\bigg |\\sum _{x\\in {\\mathbb {Z}}^d}\\big (1-\\cos (k\\cdot x)\\big )_p(x)\\bigg |\\le O(\\lambda ^\\ell )\\sum _{x\\in {\\mathbb {Z}}^d}\\frac{1-\\cos (k\\cdot x)}{|\\!|\\!|x|\\!|\\!|_L^{(d-\\alpha \\wedge 2)\\ell }}.$ If $\\alpha <2$ , then there is a $\\delta \\in (0,(2-\\alpha )\\wedge ((\\ell -1)(d-d_{\\rm c})))$ such that $1-\\cos (k\\cdot x)\\le O(|k\\cdot x|^{\\alpha +\\delta })$ , hence $|\\hat{}_p(0)-\\hat{}_p(k)|&\\le O(|k|^{\\alpha +\\delta })\\bigg (L^{-d\\ell }\\sum _{x:|x|\\le L}|x|^{\\alpha +\\delta }+L^{-\\alpha \\ell }\\sum _{x:|x|>L}\\frac{|x|^{\\alpha +\\delta }}{|x|^{(d-\\alpha )\\ell }}\\bigg )\\nonumber \\\\&=O(L^{-d(\\ell -1)+\\alpha +\\delta })|k|^{\\alpha +\\delta }.$ If $\\alpha >2$ , then there is a $\\delta \\in (0,2\\wedge ((\\ell -1)(d-d_{\\rm c})))$ such that $1-\\cos (k\\cdot x)=\\frac{1}{2}|k\\cdot x|^2+O(|k\\cdot x|^{2+\\delta })$ and therefore $\\hat{}_p(0)-\\hat{}_p(k)&=\\frac{1}{2}\\sum _{x\\in {\\mathbb {Z}}^d}|k\\cdot x|^2_p(x)+O(L^{-2\\ell })|k|^{2+\\delta }\\sum _{x\\in {\\mathbb {Z}}^d}\\frac{|x|^{2+\\delta }}{|\\!|\\!|x|\\!|\\!|_L^{(d-2)\\ell }}\\nonumber \\\\&=\\frac{|k|^2}{2d}\\sum _{x\\in {\\mathbb {Z}}^d}|x|^2_p(x)+O(L^{-d(\\ell -1)+2+\\delta })|k|^{2+\\delta }.$ Then, by the above estimates and (REF ), we obtain $\\frac{\\hat{}_p(0)-\\hat{}_p(k)}{1-\\hat{D}(k)}&={\\left\\lbrace \\begin{array}{ll}O(L^{-d(\\ell -1)+\\delta })|k|^\\delta &[\\alpha <2],\\\\\\frac{1}{\\sigma ^2}\\sum _x|x|^2_p(x)+O(L^{-d(\\ell -1)+\\delta })|k|^\\delta &[\\alpha >2],\\end{array}\\right.", "}$ hence () by taking $|k|\\rightarrow 0$ .", "This completes the proof of Proposition REF ." ], [ "The proof of (REF ) is almost identical to that of [18].", "However, since we are using $|\\!|\\!|\\cdot |\\!|\\!|_L$ rather than $|\\!|\\!|\\cdot |\\!|\\!|_1$ as in [18], we can gain the extra factor $L^{d-a}$ for $a>d$ in (REF ).", "To clarify this, we include the proof here.", "First of all, since $a\\ge b$ , we have $\\sum _{y\\in {\\mathbb {Z}}^d}|\\!|\\!|x-y|\\!|\\!|_L^{-a}\\,|\\!|\\!|y|\\!|\\!|_L^{-b}&\\le \\sum _{y:|x-y|\\le |y|}|\\!|\\!|x-y|\\!|\\!|_L^{-a}\\,|\\!|\\!|y|\\!|\\!|_L^{-b}+\\sum _{y:|x-y|>|y|}|\\!|\\!|x-y|\\!|\\!|_L^{-a}\\,|\\!|\\!|y|\\!|\\!|_L^{-b}\\nonumber \\\\&\\le 2\\sum _{y:|x-y|\\le |y|}|\\!|\\!|x-y|\\!|\\!|_L^{-a}\\,|\\!|\\!|y|\\!|\\!|_L^{-b}.$ Since $|x-y|\\le |y|$ implies $|y|\\ge \\frac{1}{2}|x|$ , we obtain that, for $a>d$ , $\\sum _{y:|x-y|\\le |y|}|\\!|\\!|x-y|\\!|\\!|_L^{-a}\\,|\\!|\\!|y|\\!|\\!|_L^{-b}\\le 2^b|\\!|\\!|x|\\!|\\!|_L^{-b}\\sum _{y\\in {\\mathbb {Z}}^d}|\\!|\\!|x-y|\\!|\\!|_L^{-a}=\\frac{C}{2}L^{d-a}|\\!|\\!|x|\\!|\\!|_L^{-b}.$ For $a<d$ , on the other hand, we use the identity $1={\\mathbb {1}}_{\\lbrace |y|\\le \\frac{3}{2}|x|\\rbrace }+{\\mathbb {1}}_{\\lbrace |y|>\\frac{3}{2}|x|\\rbrace }$ and the fact that $|y|>\\frac{3}{2}|x|$ implies $|x-y|\\ge \\frac{1}{3}|y|$ .", "Then, we obtain $&\\sum _{y:|x-y|\\le |y|}|\\!|\\!|x-y|\\!|\\!|_L^{-a}\\,|\\!|\\!|y|\\!|\\!|_L^{-b}\\nonumber \\\\&\\quad \\le 2^b|\\!|\\!|x|\\!|\\!|_L^{-b}\\sum _{y:|x-y|\\le |y|}|\\!|\\!|x-y|\\!|\\!|_L^{-a}{\\mathbb {1}}_{\\lbrace |y|\\le \\frac{3}{2}|x|\\rbrace }+3^a\\sum _{y:|x-y|\\le |y|}|\\!|\\!|y|\\!|\\!|_L^{-a-b}{\\mathbb {1}}_{\\lbrace |y|>\\frac{3}{2}|x|\\rbrace }\\nonumber \\\\&\\quad \\le 2^b|\\!|\\!|x|\\!|\\!|_L^{-b}\\sum _{y:|x-y|\\le \\frac{3}{2}|x|}|\\!|\\!|x-y|\\!|\\!|_L^{-a}+3^a\\sum _{y:|y|>\\frac{3}{2}|x|}|\\!|\\!|y|\\!|\\!|_L^{-a-b}~\\le \\frac{C}{2}|\\!|\\!|x|\\!|\\!|_L^{d-a-b}.$ This completes the proof of (REF ).", "The proof of (REF ) is also quite similar to that of [18], where [18] is used.", "However, [18] is valid only for $d>4$ , not $d>2$ as claimed in [18].", "In fact, it is not difficult to avoid this problem, and we include the proof here to clarify this.", "First we note that $(f*g)(x)=\\Vert g\\Vert _1f(x)+\\sum _{y\\in {\\mathbb {Z}}^d}g(y)\\big (f(x-y)-f(x)\\big ).$ To prove (REF ), it suffices to show that the sum in the right-hand side is the error term in (REF ).", "For that, we split the sum into the following three sums: $\\sum _{y\\in {\\mathbb {Z}}^d}=\\sum _{y:|y|\\le \\frac{1}{3}|x|}+\\sum _{y:|x-y|\\le \\frac{1}{3}|x|}+\\sum _{y:|y|\\wedge |x-y|>\\frac{1}{3}|x|}\\equiv {\\sum _y}^{\\prime }+{\\sum _y}^{\\prime \\prime }+{\\sum _y}^{\\prime \\prime \\prime }.$ It is not difficult to estimate the last two sums, as $\\bigg |{\\sum _y}^{\\prime \\prime }g(y)\\big (f(x-y)-f(x)\\big )\\bigg |\\le \\frac{O(C_3)}{|\\!|\\!|x|\\!|\\!|_L^{d+\\rho }}\\sum _{y:|x-y|\\le \\frac{1}{3}|x|}\\big (f(x-y)+f(x)\\big )\\le \\frac{O(C_1C_3)}{|\\!|\\!|x|\\!|\\!|_L^{d-\\alpha \\wedge 2+\\rho }},$ and $\\bigg |{\\sum _y}^{\\prime \\prime \\prime }g(y)\\big (f(x-y)-f(x)\\big )\\bigg |\\le \\frac{O(C_1)}{|\\!|\\!|x|\\!|\\!|_L^{d-\\alpha \\wedge 2}}\\sum _{y:|y|>\\frac{1}{3}|x|}\\frac{C_3}{|\\!|\\!|y|\\!|\\!|_L^{d+\\rho }}\\le \\frac{O(C_1C_3)}{|\\!|\\!|x|\\!|\\!|_L^{d-\\alpha \\wedge 2+\\rho }}.$ To estimate the sum $\\sum _y^{\\prime }$ , we use the ${\\mathbb {Z}}^d$ -symmetry of $g$ to obtain ${\\sum _y}^{\\prime }g(y)\\big (f(x-y)-f(x)\\big )=\\sum _{y:0<|y|\\le \\frac{1}{3}|x|}g(y)\\bigg (\\frac{f(x+y)+f(x-y)}{2}-f(x)\\bigg ).$ Notice that $\\bigg |\\frac{f(x+y)+f(x-y)}{2}-f(x)\\bigg |&\\le \\frac{O(C_1)}{|\\!|\\!|x|\\!|\\!|_L^{d-\\alpha \\wedge 2}}\\times {\\left\\lbrace \\begin{array}{ll}1&[|x|\\le \\tfrac{3}{2}L],\\\\|y|^2/|x|^2\\quad &[|x|\\ge \\tfrac{3}{2}L].\\end{array}\\right.", "}$ To verify this for $|x|\\le \\tfrac{3}{2}L$ , we simply bound each $f$ by $O(C_1)|\\!|\\!|x|\\!|\\!|_L^{\\alpha \\wedge 2-d}$ .", "For $|x|\\ge \\tfrac{3}{2}L$ , since $|x\\pm y|\\ge |x|-|y|\\ge \\frac{2}{3}|x|\\ge L$ , we have $f(x\\pm y)=C_1|x\\pm y|^{\\alpha \\wedge 2-d}$ .", "Then, by Taylor's theorem, since $\\big |\\pm 2\\frac{x\\cdot y}{|x|^2}+\\frac{|y|^4}{|x|^4}\\big |\\le \\frac{7}{9}<1$ , we have $|x\\pm y|^{\\alpha \\wedge 2-d}&=|x|^{\\alpha \\wedge 2-d}\\bigg (1\\pm 2\\frac{x\\cdot y}{|x|^2}+\\frac{|y|^4}{|x|^4}\\bigg )^{(\\alpha \\wedge 2-d)/2}\\nonumber \\\\&=|x|^{\\alpha \\wedge 2-d}\\Bigg (1\\mp (d-\\alpha \\wedge 2)\\frac{x\\cdot y}{|x|^2}+O\\bigg (\\frac{|y|^2}{|x|^2}\\bigg )\\Bigg ),$ and (REF ) follows.", "Therefore, if $|x|\\le \\frac{3}{2}L$ , then $|y|\\le \\frac{1}{2}L$ and we obtain $\\bigg |{\\sum _y}^{\\prime }g(y)\\big (f(x-y)-f(x)\\big )\\bigg |\\le \\frac{O(C_1)}{|\\!|\\!|x|\\!|\\!|_L^{d-\\alpha \\wedge 2}}\\sum _{y:0<|y|\\le \\frac{1}{2}L}\\frac{C_3}{L^{d+\\rho }}\\le \\frac{O(C_1C_3)}{|\\!|\\!|x|\\!|\\!|_L^{d-\\alpha \\wedge 2+\\rho }}.$ If $|x|\\ge \\frac{3}{2}L$ , then $|\\!|\\!|x|\\!|\\!|_L=|x|$ and we obtain $\\bigg |{\\sum _y}^{\\prime }g(y)\\big (f(x-y)-f(x)\\big )\\bigg |&\\le \\frac{O(C_1)}{|\\!|\\!|x|\\!|\\!|_L^{d-\\alpha \\wedge 2+2}}\\sum _{y:0<|y|\\le \\frac{1}{3}|x|}|y|^2\\bigg (\\frac{C_3{\\mathbb {1}}_{\\lbrace |y|\\le L\\rbrace }}{L^{d+\\rho }}+\\frac{C_3{\\mathbb {1}}_{\\lbrace |y|>L\\rbrace }}{|y|^{d+\\rho }}\\bigg )\\nonumber \\\\&\\le \\frac{O(C_1C_3)}{|\\!|\\!|x|\\!|\\!|_L^{d-\\alpha \\wedge 2+2}}\\times {\\left\\lbrace \\begin{array}{ll}L^{-\\rho +2}\\quad &[\\rho >2],\\\\\\log |x|&[\\rho =2],\\\\|x|^{2-\\rho }&[\\rho <2].\\end{array}\\right.", "}$ Summarizing the above yields (REF ).", "This completes the proof of Lemma REF ." ], [ "Proof of the infrared bound on $G_p$", "In this subsection, we prove that the hypothesis of Proposition REF indeed holds for $p\\le p_{\\rm c}$ in high dimensions.", "The precise statement is the following: Theorem 3.3 Let $\\alpha >0$ , $\\alpha \\ne 2$ and $d>d_{\\rm c}$ , and assume the properties (REF ), (REF ) and (REF ) in Assumption REF .", "Then, for $L\\gg 1$ and $p\\le p_{\\rm c}$ , $G_p(x)\\le O(L^{-\\alpha \\wedge 2})|\\!|\\!|x|\\!|\\!|_L^{\\alpha \\wedge 2-d}\\qquad [x\\ne o].$" ], [ "Let $g_p=p\\vee \\sup _{x\\ne o}\\frac{G_p(x)}{\\lambda |\\!|\\!|x|\\!|\\!|_L^{\\alpha \\wedge 2-d}},$ where we recall the definition (REF ) of $\\lambda $ .", "Suppose that the following properties hold: $g_p$ is continuous (and nondecreasing) in $p\\in [1,p_{\\rm c})$ .", "$g_1\\le 1$ .", "If $\\lambda \\ll 1$ (i.e., $L\\gg 1$ ), then $g_p\\le 3$ implies $g_p\\le 2$ for every $p\\in (1,p_{\\rm c})$ .", "If the above properties hold, then in fact $g_p\\le 2$ for all $p<p_{\\rm c}$ , as long as $d>d_{\\rm c}$ and $\\lambda \\ll 1$ .", "In particular, $G_p(x)\\le 2\\lambda |\\!|\\!|x|\\!|\\!|_L^{\\alpha \\wedge 2-d}$ for all $x\\ne o$ and $p<p_{\\rm c}\\,(\\le 2)$ .", "By Lemma REF , we can extend this bound up to $p=p_{\\rm c}$ , hence the proof completed.", "Now we verify those properties (i)–(iii)." ], [ "It suffices to show that, for every $p_0\\in (1,p_{\\rm c})$ , $\\sup _{x\\ne o}G_p(x)/|\\!|\\!|x|\\!|\\!|_L^{\\alpha \\wedge 2-d}$ is continuous in $p\\in [1,p_0]$ .", "By the monotonicity of $G_p(x)$ in $p\\le p_0$ and using Lemma REF , we have $\\frac{G_p(x)}{|\\!|\\!|x|\\!|\\!|_L^{\\alpha \\wedge 2-d}}\\le \\frac{G_{p_0}(x)}{|\\!|\\!|x|\\!|\\!|_L^{\\alpha \\wedge 2-d}}\\le \\frac{K_{p_0}|\\!|\\!|x|\\!|\\!|_L^{-d-\\alpha }}{|\\!|\\!|x|\\!|\\!|_L^{\\alpha \\wedge 2-d}}=\\frac{K_{p_0}}{|\\!|\\!|x|\\!|\\!|_L^{\\alpha +\\alpha \\wedge 2}}.$ On the other hand, for any $x_0\\ne o$ with $D(x_0)>0$ , there exists an $R=R(p_0,x_0)<\\infty $ such that, for all $|x|\\ge R$ , $\\frac{K_{p_0}}{|\\!|\\!|x|\\!|\\!|_L^{\\alpha +\\alpha \\wedge 2}}\\le \\frac{D(x_0)}{|\\!|\\!|x_0|\\!|\\!|_L^{\\alpha \\wedge 2-d}}.$ Moreover, by using $p\\ge 1$ and the lower bound of the second inequality in (REF ), we have $\\frac{D(x_0)}{|\\!|\\!|x_0|\\!|\\!|_L^{\\alpha \\wedge 2-d}}\\le \\frac{pD(x_0)}{|\\!|\\!|x_0|\\!|\\!|_L^{\\alpha \\wedge 2-d}}\\le \\frac{G_p(x_0)}{|\\!|\\!|x_0|\\!|\\!|_L^{\\alpha \\wedge 2-d}}.$ As a result, for any $p\\in [1,p_0]$ , we obtain $\\sup _{x\\ne o}\\frac{G_p(x)}{|\\!|\\!|x|\\!|\\!|_L^{\\alpha \\wedge 2-d}}=\\frac{G_p(x_0)}{|\\!|\\!|x_0|\\!|\\!|_L^{\\alpha \\wedge 2-d}}\\vee \\max _{x:0<|x|<R}\\frac{G_p(x)}{|\\!|\\!|x|\\!|\\!|_L^{\\alpha \\wedge 2-d}}.$ Since $G_p(x)$ is continuous in $p$ (cf., Lemma REF ) and the maximum of finitely many continuous functions is continuous, we can conclude that $g_p$ is continuous in $p\\in [1,p_0]$ , as required." ], [ "By the first inequality in (REF ) and the definition (REF ) of $\\lambda $ , we readily obtain $g_1=1\\vee \\sup _{x\\ne o}\\frac{G_1(x)}{\\lambda |\\!|\\!|x|\\!|\\!|_L^{\\alpha \\wedge 2-d}}\\le 1\\vee \\sup _{x\\ne o}\\frac{S_1(x)}{\\lambda |\\!|\\!|x|\\!|\\!|_L^{\\alpha \\wedge 2-d}}=1,$ as required." ], [ "If $d>d_{\\rm c}$ , $\\lambda \\ll 1$ and $g_p\\le 3$ , then, by Proposition REF , $_p$ satisfies ()–() as well as (REF ).", "We use these estimates and the lace expansion to prove $g_p\\le 2$ as follows.", "First we recall (REF ) and (REF ): $G_p=_p+_p*pD*G_p,&&S_p=\\delta +pD*S_p,$ or equivalently $_p=G_p*(\\delta -_p*pD),&&\\delta =(\\delta -pD)*S_p.$ Inspired by the similarity of the above identities, we approximate $G_p$ to $r_p*S_q$ with some constant $r\\in (0,\\infty )$ and the parameter change $q\\in [0,1]$ .", "Rewrite $G_p$ as follows: $G_p&=r_p*S_q+G_p*\\delta -r_p*S_q\\nonumber \\\\&=r_p*S_q+G_p*(\\delta -qD)*S_q-rG_p*(\\delta -_p*pD)*S_q\\nonumber \\\\&=r_p*S_q+G_p*E_{p,q,r}*S_q,$ where $E_{p,q,r}=\\delta -qD-r(\\delta -_p*pD).$ We choose $q,r$ to satisfy $\\hat{E}_{p,q,r}(0)=\\bar{\\nabla }^{\\alpha \\wedge 2}\\hat{E}_{p,q,r}(0)=0,$ or equivalently ${\\left\\lbrace \\begin{array}{ll}1-q-r\\big (1-\\hat{}_p(0)p\\big )=0,\\\\[5pt]-q+r\\big (\\hat{}_p(0)+\\bar{\\nabla }^{\\alpha \\wedge 2}\\hat{}_p(0)\\big )p=0.\\end{array}\\right.", "}$ Solving these simultaneous equations for $r$ and using (), we obtain $r=\\big (1+p\\bar{\\nabla }^{\\alpha \\wedge 2}\\hat{}_p(0)\\big )^{-1}=1+{\\left\\lbrace \\begin{array}{ll}0&[\\alpha <2],\\\\O(L^{-d(\\ell -1)})&[\\alpha >2].\\end{array}\\right.", "}$ On the other hand, by taking the Fourier transform of (REF ) and setting $k=0$ , we obtain $\\chi _p=\\hat{}_p(0)+\\hat{}_p(0)p\\chi _p,$ or equivalently $\\hat{}_p(0)=\\chi _p/(1+p\\chi _p)$ , and therefore $q=1-r\\big (1-\\hat{}_p(0)p\\big )=1-\\frac{r}{1+p\\chi _p}\\in (0,1],$ where we have used $p\\ge 1$ , $\\chi _p\\ge 1$ and (REF ) to guarantee the positivity (by taking $L\\gg 1$ if $\\alpha >2$ ).", "In addition, by solving (REF ) for $\\chi _p$ and using (REF ), we have $\\chi _p=\\frac{\\hat{}_p(0)}{1-\\hat{}_p(0)p}=\\frac{1+O(L^{-d})}{1-\\hat{}_p(0)p},$ hence $1-\\hat{}_p(0)p\\ge 0$ .", "In particular, $p\\le \\hat{}_p(0)^{-1}=1+O(L^{-d})\\le 2$ , as required.", "It remains to prove $G_p(x)\\le 2\\lambda |\\!|\\!|x|\\!|\\!|_L^{\\alpha \\wedge 2-d}$ .", "To do so, we use the following property of $E_{p,q,r}$ : Proposition 3.4 Let $q,r$ be defined as in (REF )–(REF ).", "Under the hypothesis of Proposition REF , there is a $\\rho \\in (0,\\alpha \\wedge 2)$ such that $|(E_{p,q,r}*S_q)(x)|\\le O(L^{-d(\\ell -1)})\\bigg ({\\mathbb {1}}_{\\lbrace \\alpha >2\\rbrace }\\delta _{o,x}+\\frac{L^\\rho }{|\\!|\\!|x|\\!|\\!|_L^{d+\\rho }}\\bigg ).$ For now we assume this proposition and complete verifying the property (iii).", "First, by rearranging (REF ) and using $S_q\\le S_1$ as well as (REF ) and (REF ) for $L\\gg 1$ , we obtain $G_p&=r_p*S_q+G_p*E_{p,q,r}*S_q\\nonumber \\\\&=r\\hat{}_p(0)S_q-r\\big (\\hat{}_p(0)\\delta -_p\\big )*S_q+G_p*E_{p,q,r}*S_q\\nonumber \\\\&\\le \\big (1+O(L^{-d})\\big )S_1-r\\big (\\hat{}_p(0)\\delta -_p\\big )*S_q+G_p*E_{p,q,r}*S_q.$ Then, by Proposition REF and Lemma REF (i), the third term is bounded as $|(G_p*E_{p,q,r}*S_q)(x)|&\\le O(L^{-d(\\ell -1)})\\sum _{y\\in {\\mathbb {Z}}^d}\\frac{3\\lambda }{|\\!|\\!|y|\\!|\\!|_L^{d-\\alpha \\wedge 2}}\\bigg (\\delta _{y,x}+\\frac{L^\\rho }{|\\!|\\!|x-y|\\!|\\!|_L^{d+\\rho }}\\bigg )\\nonumber \\\\&\\le \\frac{O(L^{-d(\\ell -1)})\\lambda }{|\\!|\\!|x|\\!|\\!|_L^{d-\\alpha \\wedge 2}}.$ Also, by () and Lemma REF (i), the second term in (REF ) is bounded as $\\Big |\\Big (\\big (\\hat{}_p(0)\\delta -_p\\big )*S_q\\Big )(x)\\Big |&=\\bigg |\\sum _{y\\ne o}_p(y)\\big (S_q(x)-S_q(x-y)\\big )\\bigg |\\nonumber \\\\&\\le \\sum _{y\\ne o}|_p(y)|\\,S_q(x)+\\sum _{y\\ne o}|_p(y)|\\,S_q(x-y)\\nonumber \\\\&\\le \\frac{O(L^{-d(\\ell -1)})\\lambda }{|\\!|\\!|x|\\!|\\!|_L^{d-\\alpha \\wedge 2}}.$ Putting these estimates back into (REF ), we obtain that, for $L\\gg 1$ , $G_p(x)\\le \\big (1+O(L^{-d})\\big )\\frac{\\lambda }{|\\!|\\!|x|\\!|\\!|_L^{d-\\alpha \\wedge 2}}+\\frac{O(L^{-d(\\ell -1)})\\lambda }{|\\!|\\!|x|\\!|\\!|_L^{d-\\alpha \\wedge 2}}\\le \\frac{2\\lambda }{|\\!|\\!|x|\\!|\\!|_L^{d-\\alpha \\wedge 2}},$ as required.", "This completes the proof of Theorem REF assuming Proposition REF ." ], [ "First, by substituting $q=1-r(1-\\hat{}_p(0)p)$ (cf., (REF )) into (REF ) and using $1-r=pr\\bar{\\nabla }^{\\alpha \\wedge 2}\\hat{}_p(0)$ (cf., (REF )), we obtain $E_{p,q,r}=pr\\Big (\\bar{\\nabla }^{\\alpha \\wedge 2}\\hat{}_p(0)(\\delta -D)-\\big (\\hat{}_p(0)\\delta -_p\\big )*D\\Big ).$ Using this representation, we prove (REF ) for $|x|\\le 2L$ and $|x|>2L$ , separately.", "For $|x|\\le 2L$ , we simply use (REF ) to bound $|(E_{p,q,r}*S_q)(x)|$ by $|(E_{p,q,r}*S_q)(x)|\\le |E_{p,q,r}(x)|+O(L^{-d})\\sum _{y\\in {\\mathbb {Z}}^d}|E_{p,q,r}(y)|.$ By (REF ), we have $\\frac{|E_{p,q,r}(x)|}{pr}&\\le |\\bar{\\nabla }^{\\alpha \\wedge 2}\\hat{}_p(0)|\\big (\\delta _{o,x}+D(x)\\big )+\\Big |\\Big (\\big (\\hat{}_p(0)\\delta -_p\\big )*D\\Big )(x)\\Big |\\nonumber \\\\&=|\\bar{\\nabla }^{\\alpha \\wedge 2}\\hat{}_p(0)|\\big (\\delta _{o,x}+D(x)\\big )+\\bigg |\\sum _{z\\ne o}_p(z)\\big (D(x)-D(x-z)\\big )\\bigg |\\nonumber \\\\&\\le |\\bar{\\nabla }^{\\alpha \\wedge 2}\\hat{}_p(0)|\\big (\\delta _{o,x}+D(x)\\big )+\\sum _{z\\ne o}|_p(z)|\\big (D(x)+D(x-z)\\big ).$ Using ()–() and (REF ), we obtain that $|E_{p,q,r}(x)|&\\le O(L^{-d(\\ell -1)}){\\mathbb {1}}_{\\lbrace \\alpha >2\\rbrace }\\big (\\delta _{o,x}+O(L^{-d})\\big )+O(L^{-d})\\sum _{z\\ne o}|_p(z)|\\\\&\\le O(L^{-d(\\ell -1)}){\\mathbb {1}}_{\\lbrace \\alpha >2\\rbrace }\\delta _{o,x}+O(L^{-d\\ell }),$ and that, by summing (REF ) over $x\\in {\\mathbb {Z}}^d$ , $O(L^{-d})\\sum _{x\\in {\\mathbb {Z}}^d}|E_{p,q,r}(x)|\\le O(L^{-d})\\bigg (2|\\bar{\\nabla }^{\\alpha \\wedge 2}\\hat{}_p(0)|+2\\sum _{z\\ne o}|_p(z)|\\bigg )\\le O(L^{-d\\ell }).$ Therefore, for $|x|\\le 2L$ , $|(E_{p,q,r}*S_q)(x)|&\\le O(L^{-d(\\ell -1)}){\\mathbb {1}}_{\\lbrace \\alpha >2\\rbrace }\\delta _{o,x}+O(L^{-d\\ell })\\nonumber \\\\&\\le O(L^{-d(\\ell -1)}){\\mathbb {1}}_{\\lbrace \\alpha >2\\rbrace }\\delta _{o,x}+\\frac{O(L^{-d(\\ell -1)+\\rho })}{|\\!|\\!|x|\\!|\\!|_L^{d+\\rho }}.$ It remains to prove (REF ) for $|x|>2L$ .", "To do so, we first rewrite $(E_{p,q,r}*S_q)(x)$ as $(E_{p,q,r}*S_q)(x)&=\\int _{[-\\pi ,\\pi ]^d}\\frac{\\text{d}^dk}{(2\\pi )^d}~\\hat{E}_{p,q,r}(k)\\,\\frac{e^{-ik\\cdot x}}{1-q\\hat{D}(k)}\\nonumber \\\\&=\\int _0^\\infty \\text{d}t\\int _{[-\\pi ,\\pi ]^d}\\frac{\\text{d}^dk}{(2\\pi )^d}~\\hat{E}_{p,q,r}(k)\\,e^{-t(1-q\\hat{D}(k))-ik\\cdot x}.$ Then we split the integral with respect to $t$ into $\\int _0^T$ and $\\int _T^\\infty $ , where $T$ is arbitrary for now, but it will be determined shortly.", "For the latter integral, we use the Fourier transform of (REF ), which is $\\hat{E}_{p,q,r}(k)=pr\\big (1-\\hat{D}(k)\\big )\\bigg (\\bar{\\nabla }^{\\alpha \\wedge 2}\\hat{}_p(0)-\\frac{\\hat{}_p(0)-\\hat{}_p(k)}{1-\\hat{D}(k)}\\hat{D}(k)\\bigg ).$ Because of (REF ), () and (REF ), there is a $\\delta >0$ such that $\\hat{E}_{p,q,r}(k)=O(L^{-d(\\ell -1)+\\alpha \\wedge 2+\\delta })|k|^{\\alpha \\wedge 2+\\delta }\\qquad [\\alpha \\ne 2].$ Since $1-q\\hat{D}(k)\\ge q(1-\\hat{D}(k))$ , the contribution to (REF ) from the large-$t$ integral is bounded as $&\\bigg |\\int _T^\\infty \\text{d}t\\int _{[-\\pi ,\\pi ]^d}\\frac{\\text{d}^dk}{(2\\pi )^d}~\\hat{E}_{p,q,r}(k)\\,e^{-t(1-q\\hat{D}(k))-ik\\cdot x}\\bigg |\\nonumber \\\\&\\quad \\le O(L^{-d(\\ell -1)+\\alpha \\wedge 2+\\delta })\\int _T^\\infty \\text{d}t\\int _{[-\\pi ,\\pi ]^d}\\frac{\\text{d}^dk}{(2\\pi )^d}~|k|^{\\alpha \\wedge 2+\\delta }e^{-tq(1-\\hat{D}(k))}.$ Since $p\\ge 1$ , we have $q\\ge 1-r/(1+\\chi _1)\\ge 1-r/2$ (cf., (REF )), which is bounded away from zero when $L\\gg 1$ .", "Therefore, by using (REF ), we obtain $\\int _{[-\\pi ,\\pi ]^d}\\frac{\\text{d}^dk}{(2\\pi )^d}~|k|^{\\alpha \\wedge 2+\\delta }e^{-tq(1-\\hat{D}(k))}=O(L^{\\alpha \\wedge 2}t)^{-1-\\frac{d+\\delta }{\\alpha \\wedge 2}},$ hence $\\bigg |\\int _T^\\infty \\text{d}t\\int _{[-\\pi ,\\pi ]^d}\\frac{\\text{d}^dk}{(2\\pi )^d}~\\hat{E}_{p,q,r}(k)\\,e^{-t(1-q\\hat{D}(k))-ik\\cdot x}\\bigg |\\le O(L^{-d\\ell })T^{-\\frac{d+\\delta }{\\alpha \\wedge 2}}.$ Let $\\rho =\\frac{(\\alpha \\wedge 2)\\delta }{d+\\alpha \\wedge 2+\\delta },&&T=\\bigg (\\frac{|x|}{L}\\bigg )^{\\alpha \\wedge 2-\\rho }.$ Then, since $|x|>2L$ , $O(L^{-d\\ell })T^{-\\frac{d+\\delta }{\\alpha \\wedge 2}}=\\frac{O(L^{-d(\\ell -1)+\\rho })}{|\\!|\\!|x|\\!|\\!|_L^{d+\\rho }}.$ To estimate the contribution to (REF ) from the small-$t$ integral, we use the identity $\\int _0^T\\text{d}t\\int _{[-\\pi ,\\pi ]^d}\\frac{\\text{d}^dk}{(2\\pi )^d}~\\hat{E}_{p,q,r}(k)\\,e^{-t(1-q\\hat{D}(k))-ik\\cdot x}=\\int _0^T\\text{d}t~e^{-t}\\sum _{n=0}^\\infty \\frac{(tq)^n}{n!", "}\\,(E_{p,q,r}*D^{*n})(x),$ where, by (REF ) and (), $(E_{p,q,r}*D^{*n})(x)&=\\underbrace{pr\\bar{\\nabla }^{\\alpha \\wedge 2}\\hat{}_p(0)}_{O(L^{-d(\\ell -1)}){\\mathbb {1}}_{\\lbrace \\alpha >2\\rbrace }}\\sum _{y\\in {\\mathbb {Z}}^d}D(y)\\big (D^{*n}(x)-D^{*n}(x-y)\\big )\\nonumber \\\\&\\quad -pr\\sum _{y\\in {\\mathbb {Z}}^d}_p(y)\\big (D^{*(n+1)}(x)-D^{*(n+1)}(x-y)\\big ).$ In the following, we use the decomposition (REF ) of $\\sum _y$ and estimate the contribution to (REF ) from $\\sum _y^{\\prime }$ , $\\sum _y^{\\prime \\prime }$ and $\\sum _y^{\\prime \\prime \\prime }$ , separately.", "First we estimate the contribution from $\\sum _y^{\\prime \\prime }\\equiv \\sum _{y:|x-y|\\le \\frac{1}{3}|x|}$ .", "Since $|y|\\ge |x|-|x-y|\\ge \\frac{2}{3}|x|$ in this domain of summation, we bound $|_p(y)|$ by $O(\\lambda ^\\ell )|\\!|\\!|x|\\!|\\!|_L^{(\\alpha \\wedge 2-d)\\ell }$ (cf., ()) and then use (REF ), $\\sum _y^{\\prime \\prime }1\\le O(|\\!|\\!|x|\\!|\\!|_L^d)$ and $\\sum _y^{\\prime \\prime }D^{*(n+1)}(x-y)\\le 1$ .", "As a result, $\\bigg |{\\sum _y}^{\\prime \\prime }_p(y)\\big (D^{*(n+1)}(x)-D^{*(n+1)}(x-y)\\big )\\bigg |&\\le \\frac{O(\\lambda ^\\ell )}{|\\!|\\!|x|\\!|\\!|_L^{(d-\\alpha \\wedge 2)\\ell }}\\bigg (\\frac{O(L^{\\alpha \\wedge 2})\\,n}{|\\!|\\!|x|\\!|\\!|_L^{\\alpha \\wedge 2}}+1\\bigg )\\nonumber \\\\&\\le \\frac{O(L^{-d(\\ell -1)+\\alpha \\wedge 2})}{|\\!|\\!|x|\\!|\\!|_L^{d+\\alpha \\wedge 2}}\\bigg (\\frac{O(L^{\\alpha \\wedge 2})\\,n}{|\\!|\\!|x|\\!|\\!|_L^{\\alpha \\wedge 2}}+1\\bigg ).$ Similarly, for $\\alpha >2$ , $O(L^{-d(\\ell -1)})\\bigg |{\\sum _y}^{\\prime \\prime }D(y)\\big (D^{*n}(x)-D^{*n}(x-y)\\big )\\bigg |&\\le \\frac{O(L^{-d(\\ell -1)+\\alpha })}{|\\!|\\!|x|\\!|\\!|_L^{d+\\alpha }}\\bigg (\\frac{O(L^2)\\,n}{|\\!|\\!|x|\\!|\\!|_L^2}+1\\bigg )\\nonumber \\\\&\\le \\frac{O(L^{-d(\\ell -1)+2})}{|\\!|\\!|x|\\!|\\!|_L^{d+2}}\\bigg (\\frac{O(L^2)\\,n}{|\\!|\\!|x|\\!|\\!|_L^2}+1\\bigg ).$ To estimate the contribution to (REF ) from $\\sum _y^{\\prime \\prime \\prime }\\equiv \\sum _{y:|y|\\wedge |x-y|>\\frac{1}{3}|x|}$ in (REF ), we bound $D^{*(n+1)}(x)$ and $D^{*(n+1)}(x-y)$ by $O(L^{\\alpha \\wedge 2})n/|\\!|\\!|x|\\!|\\!|_L^{d+\\alpha \\wedge 2}$ and then use () to bound $|_p(y)|$ .", "The result is $\\bigg |{\\sum _y}^{\\prime \\prime \\prime }_p(y)\\big (D^{*(n+1)}(x)-D^{*(n+1)}(x-y)\\big )\\bigg |&\\le \\frac{O(L^{\\alpha \\wedge 2})\\,n}{|\\!|\\!|x|\\!|\\!|_L^{d+\\alpha \\wedge 2}}\\sum _{y:|y|>\\frac{1}{3}|x|}|_p(y)|\\nonumber \\\\&\\le \\frac{O(L^{-(\\ell -1)(\\alpha \\wedge 2)})\\,n}{|\\!|\\!|x|\\!|\\!|_L^{d+2(\\alpha \\wedge 2)+(\\ell -1)(d-d_{\\rm c})}}\\nonumber \\\\&\\le \\frac{O(L^{-d(\\ell -1)+2(\\alpha \\wedge 2)})\\,n}{|\\!|\\!|x|\\!|\\!|_L^{d+2(\\alpha \\wedge 2)}}.$ Similarly, for $\\alpha >2$ , $O(L^{-d(\\ell -1)})\\bigg |{\\sum _y}^{\\prime \\prime \\prime }D(y)\\big (D^{*n}(x)-D^{*n}(x-y)\\big )\\bigg |&\\le \\frac{O(L^{-d(\\ell -1)+2})\\,n}{|\\!|\\!|x|\\!|\\!|_L^{d+2}}\\sum _{y:|y|>\\frac{1}{3}|x|}D(y)\\nonumber \\\\&\\le \\frac{O(L^{-d(\\ell -1)+4})\\,n}{|\\!|\\!|x|\\!|\\!|_L^{d+4}}.$ Finally we estimate the contribution to (REF ) from $\\sum _y^{\\prime }\\equiv \\sum _{y:|y|\\le \\frac{1}{3}|x|}$ in (REF ).", "By the ${\\mathbb {Z}}^d$ -symmetry of $_p$ and using () and the assumption (REF ), we obtain $&\\bigg |{\\sum _y}^{\\prime }_p(y)\\big (D^{*(n+1)}(x)-D^{*(n+1)}(x-y)\\big )\\bigg |\\nonumber \\\\&\\quad =\\bigg |{\\sum _y}^{\\prime }_p(y)\\bigg (D^{*(n+1)}(x)-\\frac{D^{*(n+1)}(x+y)+D^{*(n+1)}(x-y)}{2}\\bigg )\\bigg |\\nonumber \\\\&\\quad \\le \\frac{O(L^{\\alpha \\wedge 2})\\,n}{|\\!|\\!|x|\\!|\\!|_L^{d+\\alpha \\wedge 2+2}}\\sum _{y:|y|\\le \\frac{1}{3}|x|}\\frac{O(\\lambda ^\\ell )|\\!|\\!|y|\\!|\\!|_L^2}{|\\!|\\!|y|\\!|\\!|_L^{(d-\\alpha \\wedge 2)\\ell }}\\nonumber \\\\&\\quad \\le \\frac{O(L^{-(\\ell -1)(\\alpha \\wedge 2)})\\,n}{|\\!|\\!|x|\\!|\\!|_L^{d+\\alpha \\wedge 2+2}}\\times {\\left\\lbrace \\begin{array}{ll}|\\!|\\!|x|\\!|\\!|_L^{d+2-(d-\\alpha \\wedge 2)\\ell }&[d+2>(d-\\alpha \\wedge 2)\\ell ]\\\\1+\\log (|\\!|\\!|x|\\!|\\!|_L/L)\\quad &[d+2=(d-\\alpha \\wedge 2)\\ell ]\\\\L^{d+2-(d-\\alpha \\wedge 2)\\ell }&[d+2<(d-\\alpha \\wedge 2)\\ell ]\\end{array}\\right.", "}\\nonumber \\\\&\\quad \\le \\frac{O(L^{-d(\\ell -1)+2(\\alpha \\wedge 2)})\\,n}{|\\!|\\!|x|\\!|\\!|_L^{d+2(\\alpha \\wedge 2)}},$ where, to obtain the last inequality for $d+2=(d-\\alpha \\wedge 2)\\ell $ , which implies $\\alpha <2$ , we have used fact that $(|\\!|\\!|x|\\!|\\!|_L/L)^{\\alpha -2}(1+\\log (|\\!|\\!|x|\\!|\\!|_L/L))$ is bounded.", "Similarly, for $\\alpha >2$ , $&O(L^{-d(\\ell -1)})\\bigg |{\\sum _y}^{\\prime }D(y)\\big (D^{*n}(x)-D^{*n}(x-y)\\big )\\bigg |\\nonumber \\\\&\\quad =O(L^{-d(\\ell -1)})\\bigg |{\\sum _y}^{\\prime }D(y)\\bigg (D^{*n}(x)-\\frac{D^{*n}(x+y)+D^{*n}(x-y)}{2}\\bigg )\\bigg |\\nonumber \\\\&\\quad \\le \\frac{O(L^{-d(\\ell -1)+2})\\,n}{|\\!|\\!|x|\\!|\\!|_L^{d+4}}\\underbrace{\\sum _{y:|y|\\le \\frac{1}{3}|x|}\\frac{O(L^\\alpha )|\\!|\\!|y|\\!|\\!|_L^2}{|\\!|\\!|y|\\!|\\!|_L^{d+\\alpha }}}_{O(L^2)}=\\frac{O(L^{-d(\\ell -1)+4})\\,n}{|\\!|\\!|x|\\!|\\!|_L^{d+4}}.$ Now, by putting these estimates back into (REF ), we obtain $|(E_{p,q,r}*D^{*n})(x)|\\le \\frac{O(L^{-d(\\ell -1)+\\alpha \\wedge 2})}{|\\!|\\!|x|\\!|\\!|_L^{d+\\alpha \\wedge 2}}\\bigg (\\frac{O(L^{\\alpha \\wedge 2})}{|\\!|\\!|x|\\!|\\!|_L^{\\alpha \\wedge 2}}n+1\\bigg ),$ hence, by (REF ), $\\bigg |\\int _0^T\\text{d}t~e^{-t}\\sum _{n=0}^\\infty \\frac{(tq)^n}{n!", "}\\,(E_{p,q,r}*D^{*n})(x)\\bigg |&\\le \\frac{O(L^{-d(\\ell -1)+\\alpha \\wedge 2})}{|\\!|\\!|x|\\!|\\!|_L^{d+\\alpha \\wedge 2}}\\bigg (\\frac{O(L^{\\alpha \\wedge 2})}{|\\!|\\!|x|\\!|\\!|_L^{\\alpha \\wedge 2}}T^2+T\\bigg )\\nonumber \\\\&\\le \\frac{O(L^{-d(\\ell -1)+\\alpha \\wedge 2})}{|\\!|\\!|x|\\!|\\!|_L^{d+\\alpha \\wedge 2}}T\\nonumber \\\\&=\\frac{O(L^{-d(\\ell -1)+\\rho })}{|\\!|\\!|x|\\!|\\!|_L^{d+\\rho }}.$ This completes the proof of Proposition REF ." ], [ "Derivation of the asymptotics of $G_{p_{\\rm c}}$", "Finally we derive the asymptotic expression (REF ) for $G_{p_{\\rm c}}$ .", "First, by repeatedly applying (REF ), we obtain $G_p&=r_p*S_q+G_p*E_{p,q,r}*S_q\\nonumber \\\\&=r_p*S_q+(r_p*S_q+G_p*E_{p,q,r}*S_q)*E_{p,q,r}*S_q\\nonumber \\\\&=r_p*S_q*(\\delta +E_{p,q,r}*S_q)+G_p*(E_{p,q,r}*S_q)^{*2}\\nonumber \\\\&~\\:\\vdots \\nonumber \\\\&=r_p*S_q*\\sum _{n=0}^{N-1}(E_{p,q,r}*S_q)^{*n}+G_p*(E_{p,q,r}*S_q)^{*N}.$ By Proposition REF and Lemma REF (i), we have that, for $p\\le p_{\\rm c}$ , $|(E_{p,q,r}*S_q)^{*n}(x)|\\le O(L^{-d(\\ell -1)n})\\bigg ({\\mathbb {1}}_{\\lbrace \\alpha >2\\rbrace }\\delta _{o,x}+\\frac{L^\\rho }{|\\!|\\!|x|\\!|\\!|_L^{d+\\rho }}\\bigg ),$ hence, for any $N\\in \\mathbb {N}$ , $\\sum _{n=0}^{N-1}|(E_{p,q,r}*S_q)^{*n}(x)|\\le \\big (1+O(L^{-d(\\ell -1)})\\big )\\delta _{o,x}+\\frac{O(L^{-d(\\ell -1)+\\rho })}{|\\!|\\!|x|\\!|\\!|_L^{d+\\rho }}.$ Therefore, we can take $N\\rightarrow \\infty $ to obtain that, for $p\\le p_{\\rm c}$ , $G_p=r_p*S_q*\\sum _{n=0}^\\infty (E_{p,q,r}*S_q)^{*n}\\equiv H_p*S_q,$ where, by () and (REF ), $H_p(x)&=r\\bigg (_p*\\sum _{n=0}^\\infty (E_{p,q,r}*S_q)^{*n}\\bigg )(x)\\nonumber \\\\&=r\\sum _{y\\in {\\mathbb {Z}}^d}\\bigg (\\big (1+O(L^{-d})\\big )\\delta _{o,y}+\\frac{O(L^{-(\\alpha \\wedge 2)\\ell })}{|\\!|\\!|y|\\!|\\!|_L^{(d-\\alpha \\wedge 2)\\ell }}\\bigg )\\nonumber \\\\&\\qquad \\qquad \\times \\bigg (\\big (1+O(L^{-d(\\ell -1)})\\big )\\delta _{y,x}+\\frac{O(L^{-d(\\ell -1)+\\rho })}{|\\!|\\!|x-y|\\!|\\!|_L^{d+\\rho }}\\bigg ).$ Notice that, by Lemma REF (i) and using (REF ) and $d+\\rho <(d-\\alpha \\wedge 2)\\ell $ , $H_p(x)=\\big (r+O(L^{-d})\\big )\\delta _{o,x}+\\frac{O(L^{-d(\\ell -1)+\\rho })}{|\\!|\\!|x|\\!|\\!|_L^{d+\\rho }}.$ Now we set $p=p_{\\rm c}$ , so, by (REF ), $q=1$ .", "By Proposition REF and Lemma REF (ii), we obtain the asymptotic expression $G_{p_{\\rm c}}(x)=\\hat{H}_{p_{\\rm c}}(0)\\frac{\\gamma _\\alpha /v_\\alpha }{|\\!|\\!|x|\\!|\\!|_L^{d-\\alpha \\wedge 2}}+\\frac{O(L^{-\\alpha \\wedge 2+\\mu })}{|x|^{d-\\alpha \\wedge 2+\\mu }}+\\frac{O(L^{-d(\\ell -1)-\\alpha \\wedge 2+\\rho })}{|\\!|\\!|x|\\!|\\!|_L^{d-\\alpha \\wedge 2+\\rho ^{\\prime }}}.$ Since $H_{p_{\\rm c}}$ is absolutely summable, we can change the order of the limit and the sum as $\\hat{H}_{p_{\\rm c}}(0)=\\lim _{|k|\\rightarrow 0}\\hat{H}_{p_{\\rm c}}(k)&=\\lim _{|k|\\rightarrow 0}r\\hat{}_{p_{\\rm c}}(k)\\sum _{n=0}^\\infty \\big (\\hat{E}_{p_{\\rm c},1,r}(k)\\,\\hat{S}_1(k)\\big )^n\\nonumber \\\\&=r\\hat{}_{p_{\\rm c}}(0)+r\\hat{}_{p_{\\rm c}}(0)\\sum _{n=1}^\\infty \\Big (\\lim _{|k|\\rightarrow 0}\\hat{E}_{p_{\\rm c},1,r}(k)\\,\\hat{S}_1(k)\\Big )^n.$ By (REF ) and the fact that $\\chi _p$ diverges as $p\\uparrow p_{\\rm c}$ , we have $\\hat{}_{p_{\\rm c}}(0)=p_{\\rm c}^{-1}$ .", "Moreover, by (REF ) and (REF ), $\\hat{E}_{p_{\\rm c},1,r}(k)\\,\\hat{S}_1(k)=\\hat{E}_{p_{\\rm c},1,r}(k)\\,\\big (1-\\hat{D}(k)\\big )^{-1}\\underset{|k|\\rightarrow 0}{\\rightarrow }0.$ Therefore, $A=\\hat{H}_{p_{\\rm c}}(0)^{-1}=\\frac{p_{\\rm c}}{r}\\equiv p_{\\rm c}\\big (1+p_{\\rm c}\\bar{\\nabla }^{\\alpha \\wedge 2}\\hat{}_{p_{\\rm c}}(0)\\big ).$ This completes the proof of Theorem REF ." ], [ "Verification of Assumption ", "In this appendix, we show that the ${\\mathbb {Z}}^d$ -symmetric 1-step distribution $D$ in (REF ), defined more precisely below, satisfies the properties (REF ), (REF ), (REF ), (REF ) and (REF ) in Assumption REF .", "First, for $\\alpha >0$ and $\\alpha \\ne 2$ , we define $T_\\alpha (t)=\\frac{t^{-1-\\alpha /2}}{\\sum _{s\\in \\mathbb {N}}s^{-1-\\alpha /2}}\\qquad [t\\in \\mathbb {N}].$ Next, let $h$ be a nonnegative bounded function on ${\\mathbb {R}}^d$ that is piecewise continuous, ${\\mathbb {Z}}^d$ -symmetric, supported in $[-1,1]^d$ and normalized (i.e., $\\int _{[-1,1]^d}h(x)\\,\\text{d}^dx=1$ ); e.g., $h(x)=2^{-d}{\\mathbb {1}}_{\\lbrace \\Vert x\\Vert _\\infty \\le 1\\rbrace }$ .", "Then, for large $L$ (to ensure positivity of the denominator), we define $U_L(x)=\\frac{h(x/L)}{\\sum _{y\\in {\\mathbb {Z}}^d}h(y/L)}\\qquad [x\\in {\\mathbb {Z}}^d],$ where (cf., [12], [22]) $&\\sigma _L^2\\equiv \\sum _{x\\in {\\mathbb {Z}}^d}|x|^2U_L(x)=O(L^2),\\\\&\\hat{U}_L(k){\\left\\lbrace \\begin{array}{ll}\\in (-1+\\Delta ,1-\\Delta )&[|k|\\ge \\sigma _L^{-1}],\\\\=1-\\frac{\\sigma _L^2}{2d}|k|^2+O\\big ((L|k|)^{2+\\zeta }\\big )&[|k|\\rightarrow 0],\\end{array}\\right.", "}$ for some $\\Delta \\in (0,1)$ and $\\zeta \\in (0,2)$ .", "Combining these distributions, we define $D$ as $D(x)=\\sum _{t\\in \\mathbb {N}}U_L^{*t}(x)\\,T_\\alpha (t).$ We note that the above definition is a discrete version of the transition kernel for the so-called subordinate process (e.g., [8]).", "Just like (REF ), the transition kernel for the subordinate process is given by an integral of the Gaussian density with respect to the 1-dimensional $\\alpha /2$ -stable distribution.", "Bogdan and Jakubowski [9] make the most of this integral representation to estimate derivatives of the transition kernel.", "This is close to what we want: to prove (REF ).", "However, under the current discrete space-time setting, we cannot simply adopt their proof to show (REF ).", "To overcome this difficulty, we will approximate the lattice distribution $U_L^{*t}$ in (REF ) to a Gaussian density (multiplied by a polynomial) by using a discrete version of the Cramér-Edgeworth expansion [7].", "Before doing so, we first show that the above $D$ satisfies (REF ) and (REF )." ], [ "Due to the above definition of $U_L$ , we can follow the same argument as in [22] to verify the bound on $1-\\hat{D}$ in (REF ).", "Moreover, if (REF ) is also verified, then we can follow the same argument as in [10] to confirm the bound on $\\Vert D^{*n}\\Vert _\\infty $ in (REF ) as well.", "It remains to verify (REF ) for small $k$ .", "First we note that $1-\\hat{D}(k)=\\sum _{t\\in \\mathbb {N}}(1-\\hat{U}^t)\\,T_\\alpha (t)&=(1-\\hat{U})\\sum _{t\\in \\mathbb {N}}T_\\alpha (t)\\sum _{s=1}^t\\hat{U}^{s-1},$ where $\\hat{U}$ is an abbreviation for $\\hat{U}_L(k)$ .", "If $\\alpha >2$ , we can take any $\\xi \\in (0,\\alpha /2-1)$ to obtain $1-\\hat{D}(k)&=(1-\\hat{U})\\sum _{t\\in \\mathbb {N}}T_\\alpha (t)\\sum _{s=1}^t1-(1-\\hat{U})\\sum _{t\\in \\mathbb {N}}T_\\alpha (t)\\sum _{s=1}^t(1-\\hat{U}^{s-1})\\nonumber \\\\&=(1-\\hat{U})\\sum _{t\\in \\mathbb {N}}t\\,T_\\alpha (t)+O\\big ((1-\\hat{U})^{1+\\xi }\\big ),$ where we have used the inequality $\\sum _{t\\in \\mathbb {N}}T_\\alpha (t)\\sum _{s=1}^t(1-\\hat{U}^{s-1})&=(1-\\hat{U})^\\xi \\sum _{t\\in \\mathbb {N}}T_\\alpha (t)\\sum _{s=1}^t\\bigg (\\frac{1-\\hat{U}^{s-1}}{1-\\hat{U}}\\bigg )^\\xi (1-\\hat{U}^{s-1})^{1-\\xi }\\nonumber \\\\&\\le 2^{1-\\xi }(1-\\hat{U})^\\xi \\sum _{t\\in \\mathbb {N}}t^{1+\\xi }\\,T_\\alpha (t)=O\\big ((1-\\hat{U})^\\xi \\big ).$ This together with (REF )–() implies (REF ) for $\\alpha >2$ , with $\\epsilon =\\zeta \\wedge (2\\xi )$ and $v_\\alpha =\\frac{\\sigma _L^2}{2d}\\sum _{t\\in \\mathbb {N}}t\\,T_\\alpha (t)=O(L^2).$ If $\\alpha \\in (0,2)$ , on the other hand, we first rewrite (REF ) for small $k$ by setting $\\hat{u}\\equiv \\log 1/\\hat{U}$ and changing the order of summations as $1-\\hat{D}(k)=\\frac{1-\\hat{U}}{\\hat{U}}\\sum _{t\\in \\mathbb {N}}T_\\alpha (t)\\sum _{s=1}^te^{-\\hat{u}s}=\\frac{1-\\hat{U}}{\\hat{U}}\\frac{\\sum _{s\\in \\mathbb {N}}e^{-\\hat{u}s}\\sum _{t=s}^\\infty t^{-1-\\alpha /2}}{\\sum _{s\\in \\mathbb {N}}s^{-1-\\alpha /2}}.$ We note that, for small $k$ , $\\frac{1-\\hat{U}}{\\hat{U}}=1-\\hat{U}+O\\big ((1-\\hat{U})^2\\big ),&&\\hat{u}=1-\\hat{U}+O\\big ((1-\\hat{U})^2\\big ).$ Therefore, by a Riemann-sum approximation, we can estimate the numerator in (REF ) as $\\sum _{s\\in \\mathbb {N}}e^{-\\hat{u}s}\\sum _{t=s}^\\infty t^{-1-\\alpha /2}&=\\sum _{s\\in \\hat{u}\\mathbb {N}}e^{-s}\\sum _{\\begin{array}{c}t\\in \\hat{u}\\mathbb {N}\\\\ (t\\ge s)\\end{array}}\\bigg (\\frac{t}{\\hat{u}}\\bigg )^{-1-\\alpha /2}\\nonumber \\\\&=\\hat{u}^{\\alpha /2-1}\\big (1+O(\\hat{u})\\big )\\int _0^\\infty \\text{d}s~e^{-s}\\int _s^\\infty \\text{d}t~t^{-1-\\alpha /2}\\nonumber \\\\&=\\frac{2}{\\alpha }\\Gamma (1-\\alpha /2)\\,\\hat{u}^{\\alpha /2-1}\\big (1+O(\\hat{u})\\big ).$ This together with (REF )–() and (REF )–(REF ) implies (REF ) for $\\alpha \\in (0,2)$ , with $\\epsilon =\\zeta $ and $v_\\alpha =\\frac{2}{\\alpha }\\frac{\\Gamma (1-\\alpha /2)}{\\sum _{s\\in \\mathbb {N}}s^{-1-\\alpha /2}}\\bigg (\\frac{\\sigma _L^2}{2d}\\bigg )^{\\alpha /2}=O(L^\\alpha ).$ This verifies that $D$ in (REF ) satisfies both (REF ) and (REF )." ], [ "To verify these $x$ -space bounds on the transition probability $D^{*n}$ and its discrete derivative, we use the Cramér-Edgeworth expansion to approximate the lattice distribution $U_L^{*t}(x)$ in (REF ) to the Gaussian density $\\nu _{\\sigma _L^2t}(x)$ (multiplied by a polynomial of $x/\\sqrt{\\sigma _L^2t}$ ), where $\\nu _c(x)=\\bigg (\\frac{d}{2\\pi c}\\bigg )^{d/2}\\exp \\bigg (-\\frac{d|x|^2}{2c}\\bigg ).$ Before showing a precise statement (cf., Theorem REF below), we explain heuristic derivation of the formal expansion (REF ).", "First, we recall the expansion in terms of cumulants $Q_{\\vec{n}}$ for $\\vec{n}\\in {\\mathbb {Z}}_+^d$ : $\\log \\hat{U}_L(k)=\\sum _{\\begin{array}{c}\\vec{n}\\in {\\mathbb {Z}}_+^d\\\\ (\\Vert \\vec{n}\\Vert _1\\ge 1)\\end{array}}Q_{\\vec{n}}\\prod _{s=1}^d\\frac{(ik_s)^{n_s}}{n_s!", "}.$ Since $U_L$ is ${\\mathbb {Z}}^d$ -symmetric, we have $Q_{\\vec{n}}=0$ if $\\Vert \\vec{n}\\Vert _1$ is odd, and $Q_{(2,0,\\dots ,0)}=\\cdots =Q_{(0,\\dots ,0,2)}=\\sigma _L^2/d$ .", "Therefore, $\\log \\hat{U}_L(k)=-\\frac{\\sigma _L^2}{2d}|k|^2+\\sum _{l=4}^\\infty \\,\\sum _{\\begin{array}{c}\\vec{n}\\in {\\mathbb {Z}}_+^d\\\\(\\Vert \\vec{n}\\Vert _1=l)\\end{array}}Q_{\\vec{n}}\\prod _{s=1}^d\\frac{(ik_s)^{n_s}}{n_s!", "}.$ By the Fourier inversion, we may rewrite $U_L^{*t}(x)$ as $U_L^{*t}(x)&=\\int _{[-\\pi ,\\pi ]^d}\\frac{\\text{d}^dk}{(2\\pi )^d}~\\hat{U}_L(k)^te^{-ik\\cdot x}\\nonumber \\\\&=\\int _{[-\\pi ,\\pi ]^d}\\frac{\\text{d}^dk}{(2\\pi )^d}~e^{-\\frac{\\sigma _L^2}{2d}t|k|^2-ik\\cdot x}\\exp \\bigg (t\\sum _{l=4}^\\infty \\,\\sum _{\\begin{array}{c}\\vec{n}\\in {\\mathbb {Z}}_+^d\\\\(\\Vert \\vec{n}\\Vert _1=l)\\end{array}}Q_{\\vec{n}}\\prod _{s=1}^d\\frac{(ik_s)^{n_s}}{n_s!", "}\\bigg )\\nonumber \\\\&=(\\sigma _L^2t)^{-d/2}\\int _{\\sqrt{\\sigma _L^2t}[-\\pi ,\\pi ]^d}\\frac{\\text{d}^dk}{(2\\pi )^d}~e^{-\\frac{1}{2d}|k|^2-ik\\cdot \\tilde{x}}\\exp \\bigg (\\sum _{l=4}^\\infty t^{1-l/2}\\tilde{Q}_l(ik)\\bigg ),$ where, in the third equality, we have replaced $k$ by $k/\\sqrt{\\sigma _L^2t}$ and used the abbreviations $\\tilde{x}=\\frac{x}{\\sqrt{\\sigma _L^2t}},&&\\tilde{Q}_l(ik)=\\sum _{\\begin{array}{c}\\vec{n}\\in {\\mathbb {Z}}_+^d\\\\ (\\Vert \\vec{n}\\Vert _1=l)\\end{array}}\\frac{Q_{\\vec{n}}}{\\sigma _L^l}\\prod _{s=1}^d\\frac{(ik_s)^{n_s}}{n_s!", "}.$ Notice that the coefficients $Q_{\\vec{n}}/\\sigma _L^l$ for $\\Vert \\vec{n}\\Vert _1=l$ are uniformly bounded in $L$ .", "Then, the exponential factor involving higher-order cumulants in (REF ) may be expanded as $\\exp \\bigg (\\sum _{l=2}^\\infty t^{-l/2}\\tilde{Q}_{l+2}(ik)\\bigg )&=1+\\sum _{m=1}^\\infty \\frac{1}{m!", "}\\sum _{l_1,\\dots ,l_m\\ge 2}\\prod _{r=1}^m\\Big (t^{-l_r/2}\\tilde{Q}_{l_r+2}(ik)\\Big )\\nonumber \\\\&=1+\\sum _{j=2}^\\infty t^{-j/2}\\sum _{m=1}^{\\lfloor j/2\\rfloor }\\frac{1}{m!", "}\\sum _{\\begin{array}{c}l_1,\\dots ,l_m\\ge 2\\\\ (l_1+\\cdots +l_m=j)\\end{array}}\\prod _{r=1}^m\\tilde{Q}_{l_r+2}(ik).$ Let $P_0(ik)=1,\\qquad P_1(ik)=0,\\qquad P_j(ik)=\\sum _{m=1}^{\\lfloor j/2\\rfloor }\\frac{1}{m!", "}\\sum _{\\begin{array}{c}l_1,\\dots ,l_m\\ge 2\\\\ (l_1+\\cdots +l_m=j)\\end{array}}\\prod _{r=1}^m\\tilde{Q}_{l_r+2}(ik)\\quad [j\\ge 2].$ Then, by (REF ) and (REF ), we arrive at the formal Cramér-Edgeworth expansion $U_L^{*t}(x)&=(\\sigma _L^2t)^{-d/2}\\int _{\\sqrt{\\sigma _L^2t}[-\\pi ,\\pi ]^d}\\frac{\\text{d}^dk}{(2\\pi )^d}~e^{-\\frac{1}{2d}|k|^2-ik\\cdot \\tilde{x}}\\sum _{j=0}^\\infty t^{-j/2}P_j(ik).$ Now we note that, if $\\sqrt{\\sigma _L^2t}[-\\pi ,\\pi ]^d$ is replaced by ${\\mathbb {R}}^d$ , if $\\sum _{j=0}^\\infty $ is replaced by $\\sum _{j=0}^\\ell $ for some $\\ell <\\infty $ , and if $x$ is considered to be an element of ${\\mathbb {R}}^d$ instead of ${\\mathbb {Z}}^d$ , then we obtain $&(\\sigma _L^2t)^{-d/2}\\int _{{\\mathbb {R}}^d}\\frac{\\text{d}^dk}{(2\\pi )^d}~e^{-\\frac{1}{2d}|k|^2-ik\\cdot \\tilde{x}}\\sum _{j=0}^\\ell t^{-j/2}P_j(ik)\\nonumber \\\\&=(\\sigma _L^2t)^{-d/2}\\sum _{j=0}^\\ell t^{-j/2}\\tilde{P}_j\\int _{{\\mathbb {R}}^d}\\frac{\\text{d}^dk}{(2\\pi )^d}~e^{-\\frac{1}{2d}|k|^2-ik\\cdot \\tilde{x}}\\nonumber \\\\&=(\\sigma _L^2t)^{-d/2}\\sum _{j=0}^\\ell t^{-j/2}\\tilde{P}_j\\nu _1(\\tilde{x}),$ where $\\tilde{P}_j$ is the differential operator defined by replacing each $ik_s$ of $P_j(ik)$ in (REF ) by $-\\partial /\\partial \\tilde{x}_s$ : $\\tilde{P}_0=1,&& \\tilde{P}_1=0,&&\\tilde{P}_j=P_j\\bigg (\\frac{-\\partial }{\\partial \\tilde{x}_1},\\dots ,\\frac{-\\partial }{\\partial \\tilde{x}_d}\\bigg )\\quad [j\\ge 2].$ Notice that, by (REF ) and (REF ), $(\\sigma _L^2t)^{-d/2}\\tilde{P}_j\\nu _1(\\tilde{x})\\text{ in (\\ref {eq:U*t-simp3})}=H_{j+2}^{2j}\\bigg (\\frac{x}{\\sqrt{\\sigma _L^2t}}\\bigg )\\,\\nu _{\\sigma _L^2t}(x),$ where $H_{j+2}^{2j}$ is a polynomial of degree at least $j+2$ and at most $2j$ (due to the symmetry of $U_L$ ).", "The coefficients of the polynomial are uniformly bounded in $L$ , as explained below (REF ).", "The following theorem is a version of [7] for symmetric distributions, which gives a bound on the difference between $U_L^{*t}(x)$ and (REF ).", "Theorem A.1 For any $x\\in {\\mathbb {Z}}^d$ , $t\\in \\mathbb {N}$ and $\\ell \\in {\\mathbb {Z}}_+$ , $(1+|\\tilde{x}|^{\\ell +2})\\bigg |U_L^{*t}(x)-(\\sigma _L^2t)^{-d/2}\\sum _{j=0}^\\ell t^{-j/2}\\tilde{P}_j\\nu _1(\\tilde{x})\\bigg |\\le O(L^{-d})t^{-(d+\\ell )/2},$ where $\\tilde{x}$ and $\\tilde{P}_j$ are defined in (REF ) and (REF ), respectively.", "Before using this theorem to verify (REF ), (REF ) and (REF ), we briefly explain how to prove that the contribution which comes from 1 on the left-hand side of (REF ) is bounded by $O(L^{-d})t^{-(d+\\ell )/2}$ , as in (REF )To investigate the effect of the factor $|\\tilde{x}|^{\\ell +2}$ on the left-hand side of (REF ), we also use identities such as $\\tilde{x}_1^{\\ell +2}U_L^{*t}(x)&=(\\sigma _L^2t)^{-d/2}\\int _{\\sqrt{\\sigma _L^2t}[-\\pi ,\\pi ]^d}\\frac{\\text{d}^dk}{(2\\pi )^d}\\,e^{-ik\\cdot \\tilde{x}}\\,\\frac{\\partial ^{\\ell +2}}{\\partial (ik_1)^{\\ell +2}}\\hat{U}_L\\bigg (\\frac{k}{\\sqrt{\\sigma _L^2t}}\\bigg )^t,$ which is a result of integration by parts.", ".", "First, we split the domain of integration in the Fourier space into $E_1=\\lbrace k\\in {\\mathbb {R}}^d:|k|\\le \\sqrt{t}\\rbrace $ , $E_2=\\sqrt{\\sigma _L^2t}[-\\pi ,\\pi ]^d\\setminus E_1$ and $E_3={\\mathbb {R}}^d\\setminus E_1$ .", "Then, the difference between $U_L^{*t}(x)$ and (REF ) is equal to ${\\cal I}_1+{\\cal I}_2-{\\cal I}_3$ , where ${\\cal I}_1&=(\\sigma _L^2t)^{-d/2}\\int _{E_1}\\frac{\\text{d}^dk}{(2\\pi )^d}\\,e^{-ik\\cdot \\tilde{x}}\\Bigg (\\hat{U}_L\\bigg (\\frac{k}{\\sqrt{\\sigma _L^2t}}\\bigg )^t-e^{-\\frac{1}{2d}|k|^2}\\sum _{j=0}^\\ell t^{-j/2}P_j(ik)\\Bigg ),\\\\{\\cal I}_2&=(\\sigma _L^2t)^{-d/2}\\int _{E_2}\\frac{\\text{d}^dk}{(2\\pi )^d}\\,e^{-ik\\cdot \\tilde{x}}\\,\\hat{U}_L\\bigg (\\frac{k}{\\sqrt{\\sigma _L^2t}}\\bigg )^t,\\\\{\\cal I}_3&=(\\sigma _L^2t)^{-d/2}\\int _{E_3}\\frac{\\text{d}^dk}{(2\\pi )^d}\\,e^{-\\frac{1}{2d}|k|^2-ik\\cdot \\tilde{x}}\\sum _{j=0}^\\ell t^{-j/2}P_j(ik).$ Since (REF ) for $t=1$ is trivial, we can assume $t\\ge 2$ with no loss of generality.", "Then, it is not difficult to prove that ${\\cal I}_2$ and ${\\cal I}_3$ are both bounded by $O(L^{-d})t^{-(d+\\ell )/2}$ , due to direct computation for ${\\cal I}_3$ , and due to () and similar computation to [10] for ${\\cal I}_2$ .", "For ${\\cal I}_1$ , we can bound the integrand by $Ct^{-\\ell /2}(|k|^{\\ell +2}+|k|^{2\\ell })e^{-c|k|^2}$ for some $L$ -independent constants $C,c\\in (0,\\infty )$ , due to a version of [7] for symmetric distributions.", "Then, by direct computation, we can prove that ${\\cal I}_1$ is also bounded by $O(L^{-d})t^{-(d+\\ell )/2}$ .", "Now we apply (REF ) to verify the $x$ -space bounds (REF ), (REF ) and (REF ).", "In particular, by (REF ) and (REF )–(REF ), $D(x)=\\sum _{t=1}^\\infty \\nu _{\\sigma _L^2t}(x)\\,T_\\alpha (t)&+\\sum _{t=1}^\\infty \\sum _{j=2}^\\ell t^{-j/2}H_{j+2}^{2j}\\bigg (\\frac{x}{\\sqrt{\\sigma _L^2t}}\\bigg )\\,\\nu _{\\sigma _L^2t}(x)\\,T_\\alpha (t)\\nonumber \\\\&+\\sum _{t=1}^\\infty \\frac{O(L^{-d})}{t^{(d+\\ell )/2}}\\Bigg (1\\wedge \\bigg (\\frac{\\sqrt{\\sigma _L^2t}}{|x|}\\bigg )^{\\ell +2}\\Bigg )\\,T_\\alpha (t).$ The leading term is bounded as $\\sum _{t=1}^\\infty \\nu _{\\sigma _L^2t}(x)\\,T_\\alpha (t)&\\le O(L^{-d})\\Bigg (\\sum _{1\\le t<|\\!|\\!|x/\\sigma _L|\\!|\\!|_1^2}\\frac{\\exp \\big (-\\frac{d|x|^2}{2\\sigma _L^2t}\\big )}{t^{1+(d+\\alpha )/2}}+\\sum _{t\\ge |\\!|\\!|x/\\sigma _L|\\!|\\!|_1^2}t^{-1-(d+\\alpha )/2}\\Bigg )\\nonumber \\\\&\\le O(L^{-d})\\Bigg (\\sum _{1\\le t<|\\!|\\!|x/\\sigma _L|\\!|\\!|_1^2}\\frac{O\\big (\\big (\\frac{|x|^2}{\\sigma _L^2t}\\big )^{-1-(d+\\alpha )/2}\\big )}{t^{1+(d+\\alpha )/2}}+O\\big (|\\!|\\!|x/\\sigma _L|\\!|\\!|_1^{-(d+\\alpha )}\\big )\\Bigg )\\nonumber \\\\&=O(L^\\alpha )|\\!|\\!|x|\\!|\\!|_L^{-d-\\alpha }.$ The second term on the right-hand side of (REF ) is bounded, due to (REF ), as follows: for any $j\\in \\lbrace 2,\\dots ,\\ell \\rbrace $ and $h\\in \\lbrace j+2,\\dots ,2j\\rbrace $ , $&\\sum _{t=1}^\\infty t^{-j/2}\\bigg (\\frac{|x|}{\\sqrt{\\sigma _L^2t}}\\bigg )^h\\nu _{\\sigma _L^2t}(x)\\,T_\\alpha (t)\\nonumber \\\\&\\le O(L^{-d-h})|x|^h\\Bigg (\\sum _{1\\le t<|\\!|\\!|x/\\sigma _L|\\!|\\!|_1^2}\\frac{\\exp \\big (-\\frac{d|x|^2}{2\\sigma _L^2t}\\big )}{t^{1+(d+h+j+\\alpha )/2}}+\\sum _{t\\ge |\\!|\\!|x/\\sigma _L|\\!|\\!|_1^2}t^{-1-(d+h+j+\\alpha )/2}\\Bigg )\\nonumber \\\\&\\le O(L^{-d-h})|x|^h\\frac{O(L^{d+h+j+\\alpha })}{|\\!|\\!|x|\\!|\\!|_L^{d+h+j+\\alpha }}=\\frac{O(L^{j+\\alpha })}{|\\!|\\!|x|\\!|\\!|_L^{d+j+\\alpha }}\\le \\frac{O(L^{2+\\alpha })}{|\\!|\\!|x|\\!|\\!|_L^{d+2+\\alpha }}.$ Therefore, $\\sum _{t=1}^\\infty \\sum _{j=2}^\\ell t^{-j/2}H_{j+2}^{2j}\\bigg (\\frac{x}{\\sqrt{\\sigma _L^2t}}\\bigg )\\,\\nu _{\\sigma _L^2t}(x)\\,T_\\alpha (t)\\le \\frac{O(L^{\\alpha +2})}{|\\!|\\!|x|\\!|\\!|_L^{d+\\alpha +2}}.$ Similarly, the third term on the right-hand side of (REF ) is bounded as $&\\sum _{t=1}^\\infty \\frac{O(L^{-d})}{t^{(d+\\ell )/2}}\\Bigg (1\\wedge \\bigg (\\frac{\\sqrt{\\sigma _L^2t}}{|x|}\\bigg )^{\\ell +2}\\Bigg )\\,T_\\alpha (t)\\nonumber \\\\&\\le O(L^{-d})\\Bigg (\\bigg (\\frac{\\sigma _L}{|x|}\\bigg )^{\\ell +2}\\sum _{1\\le t<|\\!|\\!|x/\\sigma _L|\\!|\\!|_1^2}t^{-(d+\\alpha )/2}+\\sum _{t\\ge |\\!|\\!|x/\\sigma _L|\\!|\\!|_1^2}t^{-1-(d+\\ell +\\alpha )/2}\\Bigg )\\nonumber \\\\&={\\left\\lbrace \\begin{array}{ll}O(L^{-d+\\ell +2})|\\!|\\!|x|\\!|\\!|_L^{-\\ell -2}&[d+\\alpha >2],\\\\O(L^{-d+\\ell +2})|\\!|\\!|x|\\!|\\!|_L^{-\\ell -2}\\log |\\!|\\!|x/\\sigma _L|\\!|\\!|_1\\quad &[d+\\alpha =2],\\\\O(L^{\\alpha +\\ell })|\\!|\\!|x|\\!|\\!|_L^{-d-\\alpha -\\ell }&[d+\\alpha <2],\\end{array}\\right.", "}$ which is further bounded by $O(L^{\\alpha +2})|\\!|\\!|x|\\!|\\!|_L^{-d-\\alpha -2}$ for sufficiently large $\\ell $ .", "Summarizing the above estimates, we can conclude (REF ): $D(x)=\\sum _{t=1}^\\infty \\nu _{\\sigma _L^2t}(x)\\,T_\\alpha (t)+\\frac{O(L^{\\alpha +2})}{|\\!|\\!|x|\\!|\\!|_L^{d+\\alpha +2}}\\le \\frac{O(L^\\alpha )}{|\\!|\\!|x|\\!|\\!|_L^{d+\\alpha }}.$ The bound (REF ) on the $n$ -step transition probability is then automatically verified, due to the argument below (REF ).", "Heuristically, since $D^{*n}(x)&=\\sum _{t=n}^\\infty U_L^{*t}(x)\\,T_\\alpha ^{*n}(t),$ this suggests that $T_\\alpha ^{*n}(t)\\le O(n)\\,T_{\\alpha \\wedge 2}(t).$ In fact, we can verify this (or a stronger version) by following the same argument as given below (REF ), but we omit the details here.", "Finally we verify (REF ) by using (REF ) with sufficiently large $\\ell $ and (REF )–(REF ).", "For $|y|\\le \\frac{1}{3}|x|$ (so that $|x\\pm y|\\ge \\frac{2}{3}|x|$ ), we obtain $&D^{*n}(x)-\\frac{D^{*n}(x+y)+D^{*n}(x-y)}{2}\\nonumber \\\\&=\\sum _{t=1}^\\infty \\bigg (\\frac{\\eta }{\\pi t}\\bigg )^{d/2}\\bigg (e^{-\\eta |x|^2/t}-\\frac{e^{-\\eta |x+y|^2/t}+e^{-\\eta |x-y|^2/t}}{2}\\bigg )T_\\alpha ^{*n}(t)+\\frac{O(L^{\\alpha \\wedge 2+2})}{|\\!|\\!|x|\\!|\\!|_L^{d+\\alpha \\wedge 2+2}}\\,n,$ where we have set $\\eta =d/(2\\sigma _L^2)=O(L^{-2})$ for convenience.", "By a Taylor expansion, $e^{-\\eta |x|^2/t}-\\frac{e^{-\\eta |x+y|^2/t}+e^{-\\eta |x-y|^2/t}}{2}=\\frac{O(\\eta |y|^2)}{t}\\,e^{-\\eta |x|^2/t}.$ Using this and (REF ) and following the same analysis as in (REF )–(REF ), we can bound the sum in (REF ) by $O(\\eta ^{1+d/2})|y|^2n\\sum _{t=1}^\\infty \\frac{e^{-\\eta |x|^2/t}}{t^{2+(d+\\alpha \\wedge 2)/2}}=\\frac{O(\\eta ^{-(\\alpha \\wedge 2)/2})|y|^2}{|\\!|\\!|x|\\!|\\!|_{1/\\sqrt{\\eta }}^{d+\\alpha \\wedge 2+2}}\\,n.$ This together with (REF ) and $|\\!|\\!|y|\\!|\\!|_L=|y|\\vee L$ yields (REF )." ], [ "Acknowledgements", "The work of the first-named author was supported in part by the NSC grants, and the work of the second-named author was supported in part by the JSPS Grant-in-Aid for Young Scientists (B) and in part by the JSPS Grant-in-Aid for Scientific Research (C).", "The second-named author is grateful to Remco van der Hofstad for encouraging conversations, and to Panki Kim, Takashi Kumagai and Kôhei Uchiyama for pointing him to the relevant literature.", "We would like to thank the anonymous referees for many useful suggestions to improve presentation of the manuscript." ] ]

[ [ "The Mass-Loss Induced Eccentric Kozai Mechanism: A New Channel for the\n Production of Close Compact Object-Stellar Binaries" ], [ "Abstract Over a broad range of initial inclinations and eccentricities an appreciable fraction of hierarchical triple star systems with similar masses are essentially unaffected by the Kozai-Lidov mechanism (KM) until the primary in the central binary evolves into a compact object.", "Once it does, it may be much less massive than the other components in the ternary, enabling the \"eccentric Kozai mechanism (EKM):\" the mutual inclination between the inner and outer binary can flip signs driving the inner binary to very high eccentricity, leading to a close binary or collision.", "We demonstrate this \"Mass-loss Induced Eccentric Kozai\" (MIEK) mechanism by considering an example system and defining an ad-hoc minimal separation between the inner two members at which tidal affects become important.", "For fixed initial masses and semi-major axes, but uniform distributions of eccentricity and cosine of the mutual inclination, ~10% of systems interact tidally or collide while the primary is on the MS due to the KM or EKM.", "Those affected by the EKM are not captured by earlier quadrupole-order secular calculations.", "We show that fully ~30% of systems interact tidally or collide for the first time as the primary swells to AU scales, mostly as a result of the KM.", "Finally, ~2% of systems interact tidally or collide for the first time after the primary sheds most of its mass and becomes a WD, mostly as a result of the MIEK mechanism.", "These findings motivate a more detailed study of mass-loss in triple systems and the formation of close NS/WD-MS and NS/WD-NS/WD binaries without an initial common envelope phase." ], [ "Introduction", "[22] and [24] showed that a hierarchical triple system with an inner binary of masses $m_{\\textrm {0}}$ and $m_{\\textrm {1}}$ , and a tertiary of mass $m_{\\textrm {2}}$ , can exchange angular momentum between the inner and outer orbits periodically.", "These oscillations drive the eccentricity of the inner binary ($e_{\\textrm {1}}$ ) to high values if the tertiary is inclined with $39.2 ^\\circ \\lesssim i\\lesssim 141.8 ^\\circ $ .", "For nearly circular orbits, in the test-particle approximation $(m_{\\textrm {1}}\\ll m_{\\textrm {0}}, m_{\\textrm {2}})$ , the maximum eccentricity is given by [15] $e_{\\rm {in, max}} = \\left( 1 - \\frac{5}{3} \\rm {cos}^{2}i \\right)^{1/2},$ when the three-body Hamiltonian is expanded to quadrupole order in the ratio of the inner to outer semi-major axes ($a_{\\textrm {1}}/a_{\\textrm {2}}$ ), and neglecting tidal forces and general relativity [3], [33], [8].", "Tidal friction in Kozai-affected triples can cause $a_{\\textrm {1}}$ to decrease.", "As the inner binary is driven to high eccentricity, the periastron of the secondary approaches the primary, and tidal friction tends to circularize the orbit [31].", "This mechanism has been proposed as an explanation for the prevalence of “hot” Jupiters with few-day orbits around their host stars [55], [8], [56], [35], [36], the very high triple fraction of close solar-type binaries with periods less than $\\sim 5$ days [48], [8], and the formation of blue stragglers in globular clusters [38].", "The combination of Kozai cycles and tidal friction has also been explored generally for triple star systems in [21] and [6].", "As discussed by [55], for the case of Jupiter-mass planets around Sun-like stars, by equating the tidal and Kozai forcing terms in the equation for the secular evolution of $e_{\\textrm {1}}$ , one estimates the circularization semi-major axis of the inner binary to be $\\sim 3 \\rm {R}_\\odot $ .", "[25], [19], and [35], [36] have recently emphasized that the commonly used quadrupole-order expansion of the three-body Hamiltonian is insufficient to capture the secular dynamics of triple systems in the test particle approximation when the outer eccentricity ($e_{\\textrm {2}}$ ) is non-zero.", "The parameter $\\epsilon _{\\textrm {oct}}= \\left( \\frac{m_{\\textrm {0}}- m_{\\textrm {1}}}{m_{\\textrm {0}}+ m_{\\textrm {1}}} \\right) \\left(\\frac{a_{\\textrm {1}}}{a_{\\textrm {2}}} \\right) \\frac{e_{\\textrm {2}}}{1 - e_{\\textrm {2}}^{2}}$ measures the importance of the octupole-order terms in the doubly-averaged three-body Hamiltonian relative to the quadrupole-order terms.", "[35] first showed that when $(m_{\\textrm {1}}\\ll m_{\\textrm {0}},m_{\\textrm {2}})$ and $e_{\\textrm {2}}\\ne 0$ , it is possible for the triple to “flip”: the system exhibits quasi-periodic cycles in $\\textrm {cos} \\ \\textit {i}$ through 0, and the tertiary passes from prograde to retrograde and vice versa.", "These flips occur even for $\\epsilon _{\\textrm {oct}}{}$ as small as $10^{-3}$ in the test particle approximation, provided that the system is sufficiently inclined initially and the arguments of periastron of the inner orbit ($g_{\\textrm {1}}$ ) and the outer orbit ($g_{\\textrm {2}}$ ) are chosen judiciously (see Figures 7 and 3 of [25] and [19], respectively).", "During these flips, the inner binary is driven to extremely high eccentricities ($1-e_{\\textrm {1}}\\sim 10^{-5}$ ).", "This behavior occurs for a broad range of parameters in octupole-order calculations, but it is not present in the quadrupole-order calculations, and has been referred to as the “eccentric Kozai mechanism.” An important piece of physics missing from the test particle approximation of the eccentric Kozai mechanism is the $(m_{\\textrm {0}}- m_{\\textrm {1}})$ term in $\\epsilon _{\\textrm {oct}}$ [23], [9], [3], [36].", "Because the three-body Hamiltonian in Jacobi coordinates, expanded to all orders in $a_{\\textrm {2}}/ a_{\\textrm {1}}$ has a sum over $(m_{\\textrm {0}}^{j-1} - (-m_{\\textrm {1}})^{j-1})$ for $j=2,3,\\cdots $ , when $m_{\\textrm {0}}= m_{\\textrm {1}}$ all $j=\\rm {odd}$ terms are zero [11].", "Importantly, the flip phenomenon present in the test-particle calculations at modest $e_{\\textrm {2}}$ and $i$ is strongly suppressed as $m_{\\textrm {0}}\\rightarrow m_{\\textrm {1}}$ because the octupole-order terms become negligibleIntuitively, any odd-order terms in the multipole expansion of the three-body Hamiltonian are odd functions of $r_{01}$ , the separation vector between members of the inner binary, and thus averages to zero for symmetric inner masses.. Our own experiments with the octupole-order code based on [3] and [47] shows that in order for flips to occur over a broad range in $g_{\\textrm {1}}$ and $g_{\\textrm {2}}$ , $m_{\\textrm {1}}$ must be less than $m_{\\textrm {0}}$ by a factor of $\\sim 2$ for a given $a_{\\textrm {1}}/ a_{\\textrm {2}}$ and $e_{\\textrm {2}}$ .", "The dependence of the flip phenomenon on $(m_{\\textrm {0}}- m_{\\textrm {1}})$ implies that many systems will be affected by mass-loss as the primary becomes a WD or a NS.", "Indeed, many MS triple star systems will have $m_{\\textrm {0}}\\sim m_{\\textrm {1}}$ , because the inner binaries of solar-like triples typically have a flat mass distribution [30], [42] and may even have a preference for “twins” [40], [42].", "The similar masses of the inner binary suppresses the eccentric Kozai mechanism since $\\epsilon _{\\textrm {oct}}$ is small.", "Furthermore, a significant fraction of triple systems (those with modest mutual inclination) will have inner binaries that are wide enough to avoid tidal contact and collisions on the MS, even if the system undergoes normal Kozai-Lidov oscillations with $e_{\\textrm {max}}$ given approximately by Equation (REF ).", "However, once the primary of the inner binary evolves off the MS and loses the majority of its mass, the system effectively enters the test-particle approximation in many cases, particularly for intermediate- and high-mass primaries where the change in mass as the MS star becomes a compact object is large.", "After mass loss, $\\epsilon _{\\textrm {oct}}$ can increase dramatically, allowing the systems to exhibit the eccentric Kozai mechanism, and driving the inner binary to extremely high eccentricities and tidal contact or collision.", "The likely result of the MIEK mechanism is a close compact-object(WD/NS)-MS binary with separation $\\sim \\rm {R}_\\odot $ .", "Such a configuration would produce a variety interesting astrophysical systems.", "In this paper we demonstrate this \"Mass-Loss Induced Eccentric Kozai\" (MIEK) mechanism and provide a preliminary exploration of its dependence on the initial eccentricities and mutual inclination of the triple system, saving a detailed treatment for a future paper.", "In Section , we describe our method for integrating the orbits of triple systems with mass-loss, and we give two example systems with different mass-loss time-scales to illustrate the MIEK mechanism.", "By directly integrating triple systems with an $N$ -body code we circumvent the need to pick a limiting order when expanding the three-body Hamiltonian and solving the secular dynamics.", "In Section , we explore parameter space by investigating the effects of eccentricity and inclination on the MIEK mechanism for an example ternary.", "We find that a fraction of widely separated triple systems are brought to tidal contact for the first time only after the primary evolves off the MS.", "In Section , we discuss the possible outcomes and the implications of systems whose inner binary comes to tidal contact or collides at each evolutionary phase.", "In Section , we provide a brief conclusion." ], [ "Simulating Mass-Loss in Triples", "To simulate the dynamical effects of mass-loss on hierarchical triple systems, we modified the $N$ -body code FEWBODYFEWBODY is now available at http://fewbody.sourceforge.net/.", "[10] to create triple systems and evolve them.", "We create a triples with specified inner and outer semi-major axess ($a_{\\textrm {1}}$ and $a_{\\textrm {2}}$ ), eccentricities ($e_{\\textrm {1}}$ and $e_{\\textrm {2}}$ ), arguments of periastron ($g_{\\textrm {1}}$ and $g_{\\textrm {2}}$ ), primary mass in the central binary ($m_{\\textrm {0}}$ ), secondary mass in the central binary ($m_{\\textrm {1}}$ ), tertiary mass ($m_{\\textrm {2}}$ ), random phase angles, and mutual inclination between the two orbits ($\\textrm {cos} \\ \\textit {i}$ ).", "During integration of the triple, we check that the triple remains bound and stable.", "If the triple becomes unbound we continue integrating the system until a conservative minimum tidal perturbation on the binary is met ($\\delta < 10^{-7}$ ; see [10]), and we thus include any possible resonant interactions which could result when the triple system becomes unstableThis allows our calculations to include systems affected by the “triple evolution dynamical instability” (TEDI, [39])..", "Integration is also stopped if the time steps become exceedingly small.", "This occurs for a handful of triples where the separation between the inner binary approaches 0 because the stars are on nearly radial orbits.", "However, in the current study these cases only occur when the inner two stars are separated by $\\ll \\rm {R}_\\odot $ .", "To check our modified FEWBODY integrator we tested a sample of triple systems against the N-body integrator MERCURY [4] and found good agreement.", "We then tested our code against the octupole-order code of [3] with the general relativistic terms removed.", "An example triple system is shown in Figure REF .", "As can be seen in the figure there is reasonable agreement with the secular octupole-order calculation.", "There is, however, a slow drift in $g_{\\textrm {1}}- g_{\\textrm {2}}$ between the N-body codes and the secular code.", "Figure: Comparison between our modified FEWBODY code (black), and an octupole-order secular code (red, ).", "The triple system hasm 0 =7.0M ⊙ m_{\\textrm {0}}= 7.0 \\ \\rm {M}_\\odot , m 1 =6.5M ⊙ m_{\\textrm {1}}= 6.5 \\ \\rm {M}_\\odot , m 2 =6M ⊙ m_{\\textrm {2}}= 6 \\ \\rm {M}_\\odot , a 1 =10 AU a_{\\textrm {1}}= 10 \\ \\rm {AU}, a 2 =250 AU a_{\\textrm {2}}= 250 \\ \\rm {AU}, e 1 =0.3e_{\\textrm {1}}=0.3, e 2 =0.1e_{\\textrm {2}}= 0.1, g 1 =0 ∘ g_{\\textrm {1}}= 0^\\circ , g 2 =0 ∘ g_{\\textrm {2}}= 0^\\circ , and cos 𝑖=0.4\\textrm {cos} \\ \\textit {i}= 0.4.", "There is a good agreement between the calculations but there is a slow drift in g 1 -g 2 g_{\\textrm {1}}- g_{\\textrm {2}}.", "Figure discussed in §." ], [ "Evolutionary Phases", "When we evolve a triple system, we break its evolutionary phases into four distinct stages: (1) primary on the MS, (2) primary evolved into a giant but prior to significant mass loss, (3) during a period of significant mass loss, and (4) after the end of mass loss when the primary has become a compact object.", "However, because of the prohibitive computational time required to integrate close and highly eccentric triples for the full MS lifetime of the primary, we truncate our integrations after $10^{3}$ Kozai-Lidov times ($\\textit {t}_{\\textrm {K}}$ ) defined by [15], [13]: $\\textit {t}_{\\textrm {K}}&=&\\frac{4}{3}\\left(\\frac{a_{\\textrm {1}}^3(m_{\\textrm {0}}+ m_{\\textrm {1}})}{Gm_{\\textrm {2}}^2}\\right)^{1/2}\\left(\\frac{b_{\\textrm {2}}}{a_{\\textrm {1}}}\\right)^3\\nonumber \\\\&\\simeq & 6.4 \\times 10^{4} \\textrm {yr} \\left(\\frac{a_{\\textrm {1}}}{10\\textrm {AU}}\\right)^{3/2} \\left(\\frac{m_{\\textrm {0}}+ m_{\\textrm {1}}}{13.5\\rm {M}_\\odot }\\right)^{1/2} \\left(\\frac{6\\rm {M}_\\odot }{m_{\\textrm {2}}}\\right)\\left(\\frac{1}{25}\\frac{b_{\\textrm {2}}}{a_{\\textrm {1}}}\\right)^{3} \\nonumber \\\\$ where $b_{\\textrm {2}}=a_{\\textrm {2}}(1 - e_{\\textrm {2}})^{1/2}$ .", "Thus, we implicitly assume that the triple systems have no interesting behavior after this time.", "We account for the MS lifetime of the primary by only integrating the triple systems for a time ($\\textit {t}_{\\textrm {MS}}$ ) defined by: $\\textit {t}_{\\textrm {MS}}= \\textrm {min}(\\textit {t}_{\\textrm {ms, 0}}, 10^{3} \\textit {t}_{\\textrm {K}}) \\times (0.95 + 0.1 \\xi ),$ where $\\xi $ is a uniformly distributed random number on the interval $[0,1)$ , $\\textit {t}_{\\textrm {ms, 0}}$ is the MS lifetime of the primary.", "We determine the MS lifetimes by interpolating the logarithmic lifetimes on a grid of stellar models run with the Yale Rotating Evolution Code with input physics described in [50].", "We multiply by the second factor in Equation (REF ) in order to randomize exactly when the primary evolves off the main sequence to avoid any correlations of the start of mass loss between different triple systems in our sample.", "To simulate the giant phase of the primary, we continue to integrate the triple for a time ($\\textit {t}_{\\textrm {RG}}$ ) defined by: $\\textit {t}_{\\textrm {RG}}= \\rm {min}(0.1 \\textit {t}_{\\textrm {ms, 0}}, 10^{3} \\textit {t}_{\\textrm {K}}).$ Then, to simulate mass-loss, we linearly decrease $m_{\\textrm {0}}$ over a mass-loss time scale ($\\textit {t}_{\\textrm {ml}}$ ) to the WD mass as specified by the initial-final mass relation of [16] given by: $m_{0,\\textrm {f}} = (0.109 \\pm 0.007) \\, m_{\\textrm {0}}+ (0.394 \\pm 0.025) \\, \\rm {M}_\\odot ,$ where $m_{0,\\textrm {f}}$ is final mass of the primary.", "To test our mass-loss prescription, we integrated binaries where one member underwent mass loss and tested that the change in the semi-major axis and eccentricity of the binary matched that for instantaneous mass loss and adiabatic mass loss (i.e., Equations 4–6 in [1]) when $\\textit {t}_{\\textrm {ml}}\\ll P$ and when $\\textit {t}_{\\textrm {ml}}\\gg P$ , respectively, where $P$ is the orbital period of the orbit.", "We found good agreement.", "After mass-loss has finished we integrate the evolved triple for a time ($\\textit {t}_{\\textrm {after}}$ ) given by: $\\textit {t}_{\\textrm {after}}= \\rm {min}(\\textit {t}_{\\textrm {ms, 1}}- 1.1 \\textit {t}_{\\textrm {ms, 0}}- \\textit {t}_{\\textrm {ml}}, 10^{3} \\textit {t}_{\\textrm {K}}),$ where $\\textit {t}_{\\textrm {ms, 1}}$ is the MS lifetime of the next most massive star in the triple system and $\\textit {t}_{\\textrm {K}}$ is redefined by Equation (REF ) with the orbital parameters of the triple right after the conclusion of mass-loss." ], [ "Tidal Contact and/or Collisions", "A detailed treatment of stellar tides and tidal dissipation is beyond the scope of this paper.", "However, to determine when tidal effects are likely to be important, we define an ad-hoc minimal separation between the inner two binary members, $r_{\\textrm {tide}}$ , below which strong tidal contact or a collision is assumed to have occurred.", "That is, if the separation between the primary and secondary is $r < r_{\\textrm {tide}}$ , we assume either (1) that tidal dissipation will remove energy from the inner orbit decreasing $a_{\\textrm {1}}$ or (2) if the separation between the inner binary members changes dramatically in a single inner orbital period ($P_{\\textrm {1}}$ ), a physical collision may occur before tides become important, as in the recent work of [18].", "In either case, our simulations are not valid for $r < r_{\\textrm {tide}}$ .", "When choosing a tidal criterion we considered two tidal effects which will alter the evolution of the system.", "The first is the nondissipative contribution that tidal bulges on the members of the inner binary have on apsidal motion [46].", "This additional apsidal motion may detune the Kozai mechanism (e.g.", "[55]; [8]).", "We find that the timescale of this apsidal motion becomes comparable to $\\textit {t}_{\\textrm {K}}$ when $r_{\\textrm {peri}}\\sim 4 R_{\\textrm {ms, 0}}$ , assuming the typical tidal Love number valid for $n=3$ polytropes, $k=0.028$ , [6] and the system parameters for the fiducial system presented in §.", "The second effect is eccentricity damping due to dissipative tides [31], [56].", "For our fiducial system, we derive a critical inner seperation by equating the tidal and Kozai forcing terms in the equation for the secular evolution of $e_{\\textrm {1}}$ .", "We find a critial inner separation of: $r_{\\textrm {crit}} \\sim 3 R_{\\textrm {ms, 0}}\\left( \\frac{10^4}{Q} \\ \\frac{k}{0.028} \\right)^{2/13},$ where $Q$ is the tidal dissipation factor.", "As can be seen in Equation (REF ), $r_{\\textrm {crit}}$ is very insensitive to the exact values chosen for $k$ and $Q$ .", "For simplicity, and to allow a broad, albeit incomplete, survey of parameter space, we take $r_{\\textrm {tide}}= 3 \\times \\rm {max}(R_{\\textrm {ms, 0}}, R_{\\textrm {ms, 1}})$ for the bulk of this study, but we investigate the effect of changing $r_{\\textrm {tide}}$ on our results in §REF ." ], [ "General Relativistic Precession", "General relativistic (GR) periastron precession can “detune” the normal Kozai mechanism, and truncate the maximum eccentricity attainable for a triple system [3], [33], [47].", "FEWBODY does not contain post-Newtonian corrections, so we must check if it is permissible to ignore GR precession.", "The GR precession timescale (e.g., Equation 23 of [8]) is given by: $\\textit {t}_{\\textrm {GRp}}&=&\\frac{1}{3}\\frac{a_{\\textrm {1}}}{c}\\left(\\frac{a_{\\textrm {1}}c^2}{G(m_{\\textrm {0}}+ m_{\\textrm {1}})}\\right)^{3/2}\\hspace*{-5.69046pt}(1-e_{\\textrm {1}}^2) \\nonumber \\\\&\\simeq &3.4\\times 10^7{\\rm \\,\\,yr}\\,\\,\\left(\\frac{13.5 \\rm {M}_\\odot }{(m_{\\textrm {0}}+m_{\\textrm {1}})}\\right)^{3/2}\\left(\\frac{a_1}{10{\\rm AU}}\\right)^{5/2}(1-e_1^2).\\nonumber \\\\$ As discussed in [3], the Kozai mechanism only operates if $\\textit {t}_{\\textrm {K}}\\lesssim \\textit {t}_{\\textrm {GRp}}$ .", "We see that for the masses and semi-major axes of our fiducial case presented in §, $\\textit {t}_{\\textrm {K}}\\lesssim \\textit {t}_{\\textrm {GRp}}$ for any choice of $e_{\\textrm {2}}$ until the inner eccentricity is increased such that $r_{\\textrm {peri}}{} < R_{\\textrm {ms, 1}}$ .", "Thus the normal Kozai mechanism is free to operate in our triple integrations.", "However, we also see in § that the eccentric Kozai timescale ($\\textit {t}_{\\textrm {EK}}$ ), the timescale for \"flips\" and extreme eccentricity maxima, is much longer then $\\textit {t}_{\\textrm {K}}$ , $\\textit {t}_{\\textrm {EK}}\\sim 10^6$ years.", "Furthermore, we see that $\\textit {t}_{\\textrm {EK}}> \\textit {t}_{\\textrm {GRp}}$ for values of $e_{\\textrm {1}}$ where $r_{\\textrm {peri}}> \\rm {R}_\\odot $ , which would seem to imply that GR precession might detune the eccentric Kozai mechanism before tidal effects become important.", "This is not clear however, because the majority of the GR precession occurs when $e_{\\textrm {1}}$ is large, which is only a small fraction of the eccentric Kozai cycle.", "To test if GR precession will suppress the maximum $e_{\\textrm {1}}$ reached, we take the fiducial triple system after mass loss that exhibits the eccentric Kozai flip, and compare this to the equivalent octupole-order calculation that includes GR precession using the code of [47] [3].", "We found good agreement between the two calculations, implying that GR precession does not shut off the eccentric Kozai mechanism, and is thus not important for the current study.", "However, these effects should be more carefully considered in future studies where $a_{\\textrm {1}}$ is much smaller, and when the periastron distance can be significantly smaller than $R_\\odot $ if both the primary and secondary are compact objects.", "Figure REF presents two example systems with the left and right panels having mass-loss time scales of $\\textit {t}_{\\textrm {ml}}= 10^{4}$ yr and $\\textit {t}_{\\textrm {ml}}= 10^{6}$ yr, respectively.", "Both have $m_{\\textrm {0}}= 7.0 \\ \\rm {M}_\\odot $ , $m_{\\textrm {1}}= 6.5\\ \\rm {M}_\\odot $ , $m_{\\textrm {2}}= 6 \\ \\rm {M}_\\odot $ , $a_{\\textrm {1}}= 10 \\ \\rm {AU}$ , $a_{\\textrm {2}}= 250 \\ \\rm {AU}$ , $e_{\\textrm {1}}=0.1$ , $e_{\\textrm {2}}= 0.7$ , $g_{\\textrm {1}}= 0^\\circ $ , $g_{\\textrm {2}}= 180^\\circ $ , and $\\textrm {cos} \\ \\textit {i}= 0.5$ ($i=60 ^\\circ $ ).", "The top panels present the evolution of $a_{\\textrm {1}}$ , $a_{\\textrm {2}}$ , $e_{\\textrm {2}}$ , and $m_{\\textrm {0}}$ .", "The middle and bottom panels show the evolution of $\\textrm {cos} \\ \\textit {i}$ and $1 -e_{\\textrm {1}}$ , respectively.", "In the bottom panel, the red line shows where $e_{\\textrm {1}}$ is high enough such that the radius of periastron of the inner binary ($r_{\\textrm {peri}}$ ) is equal to $\\rm {R}_\\odot $ .", "The example systems go through normal Kozai-Lidov cycles before mass-loss begins, reaching a maximum $e_{\\textrm {1}}\\sim 0.75$ as predicted by Equation (REF ), but $r_{\\textrm {peri}}$ is still more than two orders of magnitude too large for tidal interactions to be significant.", "After adiabatic mass-loss $m_{\\textrm {0}}\\rightarrow 1.15 \\rm {M}_\\odot $ , and $\\epsilon _{\\textrm {oct}}$ increases (Equation REF ).", "To calculate this increase analytically, we first note that the increase in the semi-major axis and change in eccentricity of a binary that underwent adiabatic mass-loss is given by: $\\frac{a_{\\textrm {f}}}{a_{0}} = \\frac{M_{0}}{M_{\\textrm {f}}}, \\; \\; \\textrm {and} \\; \\; e_{\\textrm {f}} = e_{0},$ where $M_{0}$ and $M_{\\textrm {f}}$ are the initial total mass and final total mass of the binary, respectively.", "From Equation (REF ) we see that the change in the ratio of the semi-axes of a triple system that underwent adiabatic mass-loss is given by: $\\frac{a_{\\textrm {2, f}}}{a_{\\textrm {1, f}}} =\\frac{a_{\\textrm {2, i}}}{a_{\\textrm {1, i}}} \\left( \\frac{m_{\\textrm {0}}+m_{\\textrm {1}}+ m_{\\textrm {2}}}{m_{\\textrm {0, f}}+ m_{\\textrm {1}}+ m_{\\textrm {2}}} \\right)\\left( \\frac{m_{\\textrm {0, f}}+ m_{\\textrm {1}}}{m_{\\textrm {0}}+ m_{\\textrm {1}}} \\right),$ where $a_{\\textrm {2, i}}$ and $a_{\\textrm {2, f}}$ are the initial and final outer semi-major axes, respectively, and $a_{\\textrm {2, i}}$ and $a_{\\textrm {2, f}}$ are the initial and final inner semi-major axes, respectively.", "From Equation (REF ) one sees that the outer to inner semi-major axis ratio decreases for our fiducial system during adiabatic mass loss by $a_{\\textrm {2, f}}/ a_{\\textrm {1, f}}\\simeq 0.8 \\ a_{\\textrm {2, i}}/ a_{\\textrm {1, i}}$ .", "Then with Equation (REF ) we see that the fractional increaseWhile $\\epsilon _{\\textrm {oct, i}}$ and $\\epsilon _{\\textrm {oct, f}}$ will vary between systems in § depending on $e_{\\textrm {2}}$ , the fractional increase of $\\epsilon _{\\textrm {oct}}$ will, however, be the same because the mass loss is adiabatic so $e_{\\textrm {2}}$ drops out.", "in $\\epsilon _{\\textrm {oct}}$ due to adiabatic mass loss of the primary is given by: $\\frac{\\epsilon _{\\textrm {oct, f}}}{\\epsilon _{\\textrm {oct, i}}} =\\left|\\left( \\frac{m_{\\textrm {0, f}}- m_{\\textrm {1}}}{m_{\\textrm {0}}- m_{\\textrm {1}}} \\right)\\left( \\frac{m_{\\textrm {0, f}}+ m_{\\textrm {1}}+ m_{\\textrm {2}}}{m_{\\textrm {0}}+ m_{\\textrm {1}}+ m_{\\textrm {2}}} \\right)\\left( \\frac{m_{\\textrm {0}}+ m_{\\textrm {1}}}{m_{\\textrm {0, f}}+ m_{\\textrm {1}}} \\right)^{2}\\right|,$ where $\\epsilon _{\\textrm {oct, i}}$ and $\\epsilon _{\\textrm {oct, f}}$ are the initial and final values of $\\epsilon _{\\textrm {oct}}$ , respectively.", "The absolute value is due to the ambiguity as to which of the binary stars will be the most massive in the final system.", "After mass loss our fiducial triple system is in the near-test-particle limit and $\\epsilon _{\\textrm {oct}}$ increases by a factor of $23.3$ , from $0.002$ to $0.047$ .", "These factors enable the eccentric Kozai mechanism, the system then flips, and as $\\textrm {cos} \\ \\textit {i}$ passes through 0, $(1 - e_{\\textrm {1}})$ approaches $10^{-4}$ .", "Both systems would be affected by tidal interactions during their first flip as $r_{\\textrm {peri}}< \\rm {R}_\\odot $ .", "This example illustrates the essence of the MIEK mechanism." ], [ "Inclination and Eccentricity Dependence", "The importance of the MIEK mechanism for a full population of triple systems will depend on the adopted mass function of the stars, the eccentricity and semi-major axis distributions of the inner binary and outer tertiary, as well as the mutual inclination distribution.", "We save a full exploration of parameter space for a future paper, but we explore the effects of inclination and eccentricity on the dynamics of the example system from §.", "This system is an interesting test case because all the masses are approximately equal, so the octupole-order terms mostly become important during and after mass loss." ], [ "Parameters Space Exploration", "To explore the effects of inclination and eccentricity, we integrate $\\sim 10^{5}$ triple systems over an equally spaced grid with 41, 10, 20, and 4 values of $\\textrm {cos} \\ \\textit {i}$ , $e_{\\textrm {1}}$ , $e_{\\textrm {2}}$ , and $g_{\\textrm {1}}$ , respectively, while keeping the initial masses and initial semi-major axes from our example system.", "Because the outcomes of triple systems effected by MIEK may differ depending on the initial phase angles, we integrate three triple systems at each grid point in $\\textrm {cos} \\ \\textit {i}$ , $e_{\\textrm {1}}$ , $e_{\\textrm {2}}$ , and $g_{\\textrm {1}}$ with randomized initial phase angles.", "In order to determine if the inner binary of each system becomes tidally affected or collides by our ad-hoc tidal criterion (Section REF ), we record the minimum inner separation ($r_{\\textrm {min}}$ ) in each of the four evolutionary phases described in §.", "We integrate these triple systems assuming $\\textit {t}_{\\textrm {ml}}= 10^{4}$ years and posit that tides will affect the inner binary if $r_{\\textrm {min}}$ is smaller than some ad-hoc tidal radius ($r_{\\textrm {tide}}$ ).", "We define this ad-hoc tidal radius differently for the different evolutionary phases of the primary, in an attempt to capture the different strength of tidal forces due to the varying physical size and mass of the primary.", "Before the primary evolves off the MS we take $r_{\\textrm {tide}}= 3 \\ R_{\\textrm {ms, 0}}$ , where $R_{\\textrm {ms, 0}}$ in the MS radii of the primary.", "When the primary is a giant and during the mass-loss phase, we take $r_{\\textrm {tide}}= 1 \\textrm {AU}$ .", "Finally, when the primary becomes a WD we take $r_{\\textrm {tide}}= 3 \\ R_{\\textrm {ms, 1}}$ , where $R_{\\textrm {ms, 1}}$ in the MS radii of the secondary in the inner binary.", "We determine $R_{\\textrm {ms, 0}}$ and $R_{\\textrm {ms, 1}}$ from the [49] sample by taking the median of the observed radii for stars with similar masses ($\\left| \\log {M} - \\log {M^{\\prime }} \\right| < 0.1$ ).", "Additionally, we would like to determine when the eccentric Kozai mechanism is operating.", "Because orbital flips (i.e., a change of sign in $\\textrm {cos} \\ \\textit {i}$ ) only occur during the eccentric Kozai mechanism, we monitor each triple systems for flips during each of the aforementioned evolutionary phases.", "We then use the two above criteria to loosely define systems that are strongly affected by the MIEK mechanism, as systems that come into tidal contact and exhibit a flip after the onset of mass loss." ], [ "Results", "The results of the parameter space search in inclination and eccentricity are presented in Table REF , Figure REF , and Figure REF .", "In Table REF we break down the fraction of the triple systems that flip and become tidally affected or collide during the various evolutionary stages presented in §.", "We highlight any asymmetries about $\\textrm {cos} \\ \\textit {i}_{0}= 0$ by presenting the fractions for prograde and retrograde systems separately.", "We now describe each of the evolutionary phases of the primary in turn.", "lllll5 0pc Triple Outcomes Contact Flip Contact & Flip Unbound Combined $ \\ \\ \\ \\ $ Primary MS $0.13$ $0.064$ $0.060$ $0.11$ $ \\ \\ \\ \\ $ Primary Giant $0.28$ $0.000041$ $0.000030$ $0.00037$ $ \\ \\ \\ \\ $ During Mass-Loss $0.00016$ $0.0015$ $0.000030$ $0.0027$ $ \\ \\ \\ \\ $ After Mass-Loss $0.022$ $0.021$ $0.019$ $0.011$ Prograde $ \\ \\ \\ \\ $ Primary MS $0.039$ $0.0076$ $0.0059$ $0.15$ $ \\ \\ \\ \\ $ Primary Giant $0.30$ $0.000063$ $0.000042$ $0.00042$ $ \\ \\ \\ \\ $ During Mass-Loss $0.00013$ $0.0020$ $0.000021$ $0.0037$ $ \\ \\ \\ \\ $ After Mass-Loss $0.030$ $0.029$ $0.028$ $0.0083$ Retrograde $ \\ \\ \\ \\ $ Primary MS $0.22$ $0.12$ $0.11$ $0.082$ $ \\ \\ \\ \\ $ Primary Giant $0.26$ $0.000020$ $0.000020$ $0.00032$ $ \\ \\ \\ \\ $ During Mass-Loss $0.00020$ $0.00097$ $0.000040$ $0.0017$ $ \\ \\ \\ \\ $ After Mass-Loss $0.015$ $0.013$ $0.010$ $0.014$ Fractions of triple systems that become tidally affected, exhibit a flip, or are unbound at various evolutionary stages without being tidally affected or unbound in a previous evolutionary stage.", "Prograde and retrograde divide the sample in half where combined are the fractions of the whole sample.", "The rows Primary MS, Primary Giant, During Mass-Loss and After Mass-Loss present the fractions of triple systems with the primary star is on the main sequence, a giant that has evolved off the main sequence, an evolved star undergoing an $10^4$ year phase of significant mass-loss phase and a WD, respectively.", "The columns are as follows.", "Contact — Systems that satisfy our ad-hoc tidal criteria.", "Flip — Systems whose $\\textrm {cos} \\ \\textit {i}$ changes sign from its initial value.", "Contact & Flip — Systems common to the previous two columns.", "Unbound — Systems that become unbound without being tidally affected prior.", "Table discussed in §.", "Figure: Initial parameters for triple systems that were tidallyaffected or collided on the MS, r peri <3R ms ,0 r_{\\textrm {peri}}< 3 R_{\\textrm {ms, 0}} (left), and triplesystems that were tidally affected or collided after the primary evolved off the mainsequence but before the onset of significant mass-loss, r peri <1 AU r_{\\textrm {peri}}< 1 \\textrm {AU} (right).In the histograms, black shows the fraction of systems in each bin that aretidally affected or collide and red shows the fraction of systems that also exhibited anorbital flip.", "Contours present the fraction of systems at each grid point thatwould be tidally affected by our criteria.", "The contours are in steps of 20%starting at 10 % (black) Red dots show the grid points where systemswhere integrated.", "The teal line on the contours show where cos 𝑖=0\\textrm {cos} \\ \\textit {i}= 0.", "Notethat there is a pronounced asymmetry between prograde and retrograde triples.", "Figure discussed in§.Figure: Initial parameters for triple systems that were tidallyaffected or collide after the end of mass loss, r peri <3R ms ,1 r_{\\textrm {peri}}< 3 R_{\\textrm {ms, 1}}.", "In the histograms, black shows thefraction of systems in each bin that are tidally affected or collide and red shows thefraction of systems that also exhibited an orbital flip.", "Contours present thefraction of systems at each grid point that would be tidally affected by ourcriteria.", "The contours are in steps of 20% starting at 10% (black).", "Red dotsshow the grid points where systems where integrated.", "The teal line on thecontours show where cos 𝑖=0\\textrm {cos} \\ \\textit {i}= 0.", "Note there is a pronounced asymmetry between prograde andretrograde triples.", "Figure discussed in §.When the primary is on the MS (“Primary MS\" in Table REF ), $13\\% $ of triple systems become tidally affected or collide ($r_{\\textrm {min}}\\le r_{\\textrm {tide}}$ ).", "We show the initial inclinations and eccentricities of these triple systems in the left panel of Figure REF .", "Each contour plot displays the parameter space projected onto a two parameter plane.", "Each histogram marginalizes over all parameters except one, and then displays the fraction of systems in each bin that become tidally affected or collide.", "From the left panel of Figure REF , we see that when the primary is on the MS the majority of systems which become tidally affected or collide are initially highly inclined.", "These systems undergo normal Kozai-Lidov oscillations, which bring the inner binary to tidal contact with a maximum pericenter distance given by: $r_{\\textrm {peri}}= a_{\\textrm {1}}(1-e_{\\rm {1, max}})=a_{\\textrm {1}}\\left[1- \\left( 1 - \\frac{5}{3} \\cos ^{2}i_0 \\right)^{1/2} \\right]$ with $e_{\\textrm {1, max}}$ given by Equation (REF ) and $i_0$ the initial eccentricity.", "Then, setting the periastron distance equal to $r_{\\rm tide}$ , $r_{\\textrm {peri}}= r_{\\textrm {tide}}= 3 R_{\\textrm {ms, 0}}\\sim 12 \\rm {R}_\\odot \\sim 0.05 \\textrm {AU} \\sim 0.005 \\ a_{\\textrm {1}},$ we see that the required critical initial inclination to interact tidally when the primary is on the MS is $\\cos i_{0,\\rm crit}\\sim 0.08,$ which is in good agreement with the prograde orbits in the left panel of Figure REF .", "Note, however, that the distribution is asymmetric about $\\textrm {cos} \\ \\textit {i}_{0}= 0$ , with $22\\% $ of the retrograde systems becoming tidally affected or colliding while only $3.9\\% $ of the prograde systems do.", "This asymmetry is not captured by Equation (REF ) and Equation (REF ), because of the approximations under which Equation (REF ) is derived.", "A more complete treatment of the quadrupole-order Hamiltonian shows that $e_{1,\\rm max}$ does not occur at $\\cos i_0=0$ , but instead at retrograde mutual inclination (see Fig.", "1 of [33]).", "Importantly, we find that $6.0\\% $ of the triple systems, with $\\textrm {cos} \\ \\textit {i}_{0}{} \\simeq 0$ , become tidally affected and exhibit an orbital flip, which is characteristic of the eccentric Kozai mechanism.", "These systems comprise almost half of the total number of systems that become tidally affected or collide when the primary is on the MS, showing that the eccentric Kozai mechanism is also important.", "The implications that the eccentric Kozai has for previous studies involving the Kozai-Lidov mechanism and tidal friction is discussed further in §, but here we note that the systems that tidally interact or collide on the MS are analogous to the systems investigated by [8], except that these authors only included quadrupole-order terms in the secular Hamiltonian, and thus may have missed the systems that undergo the eccentric Kozai mechanism and execute a \"flip\" with $e_1\\rightarrow 1$ [35], [36].", "Additionally, $11\\% $ of triple systems become unbound when the primary is on the MS.", "The systems that are unbound have large $e_{\\textrm {2}}$ and do not satisfy the standard, empirically derived, stability criterion for triple systems given by [27]: $\\frac{a_{\\textrm {2}}}{a_{\\textrm {1}}} \\ge C \\: f \\left[ \\left( 1 + \\frac{m_{\\textrm {2}}}{m_{\\textrm {0}}+ m_{\\textrm {1}}} \\right) \\frac{1 + e_{\\textrm {2}}}{(1 - e_{\\textrm {2}})^{3}} \\right] ^{0.4},$ where $C=2.8$ and $f=1-\\frac{0.3}{\\pi }i$ is an ad-hoc inclination term.", "These systems are present in our study because we uniformly sample the eccentricities without regard for the initial dynamical stability of the system.", "We see from Equation (REF ) and our initial assumed semi-major axes, that if $\\textrm {cos} \\ \\textit {i}= 0$ then $e_{\\textrm {2}}\\le 0.83$ must hold for the system to satisfy the stability criterion.", "It is interesting to note that in the $\\textrm {cos} \\ \\textit {i}_{0}$ vs. $e_{\\textrm {2, 0}}$ contour plot in the left panel of Figure REF , many triple systems which are unstable given the stability criteria become tidally affected or collide before they are unbound.", "It is possible that tidal friction will quickly shrink $a_{\\textrm {1}}$ of these systems, causing them to become stable as the ratio $a_{\\textrm {2}}/a_{\\textrm {1}}$ increases.", "After the primary has evolved off the MS but before significant mass-loss has begun (“Primary Giant\" in Table REF ), $28\\% $ of triple systems become tidally affected or collide ($r_{\\textrm {min}}\\le 1 \\rm {AU}$ ) for the first time.", "We show the initial inclinations and eccentricities of these triple systems in the right panel of Figure REF .", "Such a large fraction of the triple systems become tidally affected or collide during this stage of evolution because the physical size of the primary dramatically increases, so we impose an ad-hoc increase to $r_{\\textrm {tide}}= 1 \\rm {AU}$ .", "Then from Equation (REF ) we find that $\\cos i_{0,\\,\\rm crit} \\simeq 0.34,$ encompassing about a third of our parameter space.", "Additionally, we see that only a small fraction of the systems exhibit the orbital flip characteristic of the eccentric Kozai mechanism when the primary is a giant, showing that the eccentric Kozai mechanism is not important during this phase of the primary's evolution.", "Lastly, only $0.037\\% $ of the systems are unbound during this evolutionary phase, emphasizing that initially dynamically unstable systems are either quickly unbound, when the primary is still on the MS, collide, or become tidally affected.", "During the $\\textit {t}_{\\textrm {ml}}=10^{4}$ year mass-loss phase (“During Mass-Loss\" in Table REF ) only a small fraction, $0.016\\% $ , of the triple systems become tidally affected or collide for the first time.", "This is because previous stages of evolution have removed the majority of triple systems that would tidally interact or collide with $r_{\\textrm {tide}}= 1 \\rm {AU}$ , and because $\\textit {t}_{\\textrm {ml}}< t_{\\rm {K}}$ , so the systems do not have time to complete a Kozai cycle during the primary's mass loss episode.", "We find that the fraction of systems which are unbound increases, to $0.27\\% $ , from the previous phase of evolution.", "More systems become unstable because, as shown by Equation (REF ), the outer to inner semi-major axis ratio decreases during mass loss.", "Then, as a result of Equation (REF ), this decrease in the semi-major axis ratio causes more triple systems to be unstable.", "Finally, after the end of mass-loss (“After Mass-Loss\" in Table REF ) we see that $2.2\\% $ of triple systems spanning a broad range in orbital parameters become tidally affected or collide for the first time.", "We note that the lion's share, $1.9\\% $ , of these systems exhibit an orbital flip characteristic of the eccentric Kozai mechanism and thus the MIEK mechanism because it is induced by mass loss of the primary.", "We now discuss the distribution of tidally affected or colliding systems after mass loss for the inclination and eccentricities individually.", "The distribution of these tidally affected or colliding systems is asymmetric about $\\textrm {cos} \\ \\textit {i}_{0}=0$ , with $3.0\\% $ of prograde systems and only $1.5\\% $ of retrograde systems becoming tidally affect of colliding.", "At least some of this asymmetry arises because, like previous stages of evolution of the primary, systems with $\\textrm {cos} \\ \\textit {i}_{0}$ near 0 are strongly preferred.", "However, earlier stages of evolution have removed a disproportionate fraction of the prograde systems, $33\\% $ , compared to the fraction of retrograde systems, $49\\% $ .", "This causes the reverse asymmetry, more prograde than retrograde systems, in the distribution of systems which become tidally affected or collide after the end of mass loss.", "The distribution in $e_{\\textrm {2, 0}}$ for the systems that become tidally affected or collide after the end of mass loss shows a strong preference for larger values, peaking at $e_{\\textrm {2, 0}}{} = 0.75$ .", "This preference is easily understood when considering Equation (REF ), which also shows a strong dependence on $e_{\\textrm {2, 0}}$ .", "Thus, the octupole order terms and consequently the eccentric Kozai mechanism are expected to become more important with larger $e_{\\textrm {2, 0}}$ .", "Even though there is a strong preference for large $e_{\\textrm {2, 0}}$ , the MIEK mechanism continues to operate for systems with modest $e_{\\textrm {2, 0}}\\sim 0.4$ .", "We also note that the $e_{\\textrm {2, 0}}$ distribution is truncated at very large values, which arises because most of these systems became tidally affected, collided, or were unbound when the primary was on the MS. Lastly, the distribution of $e_{\\textrm {1, 0}}$ for the systems that become tidally affected or collide shows a preference for both large and small values.", "The minimum in the distribution occurs at $e_{\\textrm {1, 0}}\\sim 0.5$ , with $\\sim 3$ times fewer systems becoming tidally affected or colliding then at $e_{\\textrm {1, 0}}= 0$ .", "The preference for large values of $e_{\\textrm {1, 0}}$ is easily understood because these systems do not need to increase their inner eccentricity as much to collide or come into tidal contact.", "The preference for smaller values of $e_{\\textrm {1, 0}}$ , however, is not easily understood.", "We also note that the majority of systems with $e_{\\textrm {1, 0}}= 0.9$ become tidally affected or collide in earlier stages of the primary's evolution, so these systems are absent in the current $e_{\\textrm {1, 0}}$ distribution." ], [ "Sensitivity to $r_{\\textrm {tide}}$", "We now briefly evaluate the sensitivity of the results presented in §REF on the ad-hoc tidal criterion chosen (Section REF ).", "First, we evaluate the dependence of the fraction of systems that become tidally affected or collide on our choice of tidal criterion.", "We both decreased and increased the tidal criterion from $ r_{\\textrm {tide}}= 3 \\ R_{*}$ to $ r_{\\textrm {tide}}= 2.5 \\ R_{*}$ , $ r_{\\textrm {tide}}= 3.5 \\ R_{*}$ , $ r_{\\textrm {tide}}= 5 \\ R_{*}$ , and $ r_{\\textrm {tide}}= 10.0 \\ R_{*}$ .", "$R_{*} = R_{\\textrm {ms, 0}}{}$ when the primary is on the MS and $R_{*} = R_{\\textrm {ms, 1}}{}$ when the primary is a WD.", "The fraction of systems that become tidally affected or collide after the end of mass-loss with these new tidal criteria are $0.022$ , $0.023$ , $0.023$ , and $0.025$ , respectively.", "When these fractions are compared to that presented in Table REF , we see that the fraction of systems affected by the MIEK mechanism is relatively insensitive to the exact choice of tidal criterion.", "This insensitivity results because an increase in the tidal radius on the MS affects systems that will already be affected by the large tidal radius when the primary becomes a Giant, and does not affect regions of parameter space shown in Figure REF where the MIEK mechanism operates.", "In fact, increasing the tidal radius marginally increases the fraction of systems affected by MIEK.", "Additionally, we emphasize that by eliminating all systems that collide or were tidally affected at a prior stage of evolution, $1.9\\% $ is a conservative estimate of the fraction of triple systems that will undergo the MIEK mechanism.", "This is because many of the systems we remove from the sample as the evolutionary stages progress may still be viable candidates for the MIEK mechanism.", "One possibility is that tides on the giant primary only detune the normal Kozai-Lidov mechanism, and do not allow $e_{\\textrm {1}}$ to increase so dramatically.", "Then, our removal of these systems from the sample would unrealistically decrease the number of systems that experience MIEK.", "These systems would survive in our sample to the stage after mass loss is complete.", "Another possibility is that the combination of the Kozai-Lidov mechanism and tides on a giant will work to shrink $a_{\\textrm {1}}$ to a few AU.", "These systems will then have $a_{\\textrm {2}}/a_{\\textrm {1}}\\sim 100$ , but they can still be affected by the eccentric Kozai mechanism after mass loss, which will shrink this semi-major axis ratio according to Equation (REF ).", "Evaluating these effects would involve a realistic calculation of the 3-body dynamics with tides since $\\cos i$ will evolve secularly during the tidal interaction [8].", "We save this for a future work.", "Here, to roughly quantify the number of systems that could potentially be affected by the MIEK mechanism, we changed the tidal criterion when the primary star is a giant and during mass loss to be the same as it was when the primary star was on the MS, $r_{\\textrm {tide}}= 3 R_{\\textrm {ms, 0}}$ .", "Under this new tidal criterion, the analogue of Figure REF is presented in Figure REF .", "As expected, we see a sharp decrease, from $28\\% $ to $0.011\\% $ , in the fraction of triple systems that are tidally affected or collide when the primary is a giant.", "There are then many additional systems available after mass loss to increase the fraction of systems that are tidally affected or collide after the primary becomes a WD.", "The total change in the fraction is from $2.2\\% $ to $8.7\\% $ .", "Similarly, there is a significant increase in the fraction of MIEK systems, systems that collide or become tidally affected and flip the sign of $\\textrm {cos} \\ \\textit {i}$ , from $1.9\\% $ to $6.7\\% $ .", "For the example system considered, and for a uniform distribution in eccentricities and mutual inclination, we consider $1.9\\% $ and $6.7\\% $ to be lower and upper limits, respectively, to the fraction of all systems that undergo MIEK.", "Figure: Same as Figure , except assuming theprimary's tidal radius remains unchanged as it evolves until it becomes aWD.", "This is taken to be anupper bound on the fraction of systems that become tidally affected or collide after theend of mass loss.", "Figure discussed in §." ], [ "Discussion", "It is interesting to speculate on the outcome of triple systems that become tidally affected at each of the various points in their evolution.", "In what follows, we discuss the possible outcomes for tidally interacting triple systems during the MS, giant, and WD evolutionary phases of the primary in §REF , §REF , and §REF , respectively.", "We then discuss the MIEK mechanism for massive star triples during NS formation of the primary in §REF .", "Lastly, we discuss extensions of the current study in §REF ." ], [ "Main Sequence Phase", "We have shown in our fiducial parameter search, that systems that interact tidally while the primary is on the MS undergo both normal Kozai-Lidov oscillations and the eccentric Kozai mechanism in approximately equal proportion.", "Those systems that undergo normal Kozai-Lidov oscillations are analogous to the systems considered by [8], who calculated the long-term evolution of a distribution of triple systems using the quadrupole-order secular equations of [6].", "[8] found that the combination of normal Kozai-Lidov cycles and tidal friction produces a substantial population of close binaries, and that their calculations matched the observations of [48], which show that essentially all close spectroscopic binaries are actually triple systems.", "Given these results, we anticipate that our systems with $\\cos i_0\\sim 0$ will also tidally circularize with small $a_{\\textrm {1}}$ and $P_{\\textrm {1}}$ , with the caveat that the MS lifetime of the primary we consider here is much shorter than for the systems considered by [8], and thus a fraction of our systems may not circularize before WD formation.", "Importantly, though, we find that half of the systems that tidally interact or collide on the MS do so as a result of the eccentric Kozai mechanism, which, as [25], [19], and [35], [36] have shown cannot be captured by the quadrupole-order calculations of [8].", "What is the fate of these systems?", "Unfortunately, no study has yet been performed to investigate the eccentric Kozai mechanism with the inclusion of tidal dissipation for triple stellar systems.", "However, [35] demonstrate with an octupole-order calculation that the eccentric Kozai mechanism and tidal friction can possibly explain the occurrence of retrograde hot Jupiters.", "They show that the extremely high eccentricities obtained by the inner binary when $\\textrm {cos} \\ \\textit {i}$ flips signs can lead to a rapid capture of a planet into a short period retrograde orbit (“Kozai capture.”).", "We suppose a similar mechanism operates for stellar triple systems, and that those that are tidally affected on the MS due to the eccentric Kozai mechanism rapidly dissipate energy from the inner binary and circularize with a semi-major axis of $a_1 \\approx 2 r_{\\textrm {peri}}\\sim 2r_{\\textrm {tide}}$ .", "If so, then the [8] study may be missing of order half of the systems that become close binaries.", "The action of the eccentric Kozai mechanism for MS binaries may also affect the distribution of binary eccentricities as a function of inner period as the population is circularized [43], [5].", "An example observed (probable) intermediate mass triple star system that has likely already tidally circularized is MT429 in the Cygnus OB2 association, where the inner $P_1\\simeq 3$  d B3V+B6V binary is orbited by a B0V tertiary at 138 AU [20].", "An example of an even more massive system that might give rise to a NS-MS triple system as discussion in Section REF is HD 150136.", "It consists of O3-3.5V+O5.5-6V ($\\sim 64+40$  M$_\\odot $ ) inner binary with $P_1\\simeq 2.7$  d and an O6.5-7V ($\\sim 35$  M$_\\odot $ ) tertiary with $P_2\\sim 3000-5000$  d [26].", "Finally, after the inner binary orbit has shrunk due to either normal Kozai-Lidov oscillations or the eccentric Kozai mechanism, the binary evolves like other close binaries, potentially leading to the progenitors of single degenerate (SD) supernovae [53], double degenerate (DD) supernovae [52], [14], barium stars (BA; [32]), and asymmetric planetary nebulae (PN) [34]." ], [ "Giant Phase", "We find that $\\sim 30$  % of the triple systems we consider interact tidally or collide as the primary evolves off the MS, but before mass loss, mostly as a result of normal Kozai-Lidov oscillations, which bring the pericenter of the secondary orbit inside $\\sim $  AU (see right panel, Fig.", "REF and Equation REF ).", "Since the post-MS phase is short (particularly for the intermediate and high mass stars we consider here), and since the efficiency of tidal friction is uncertain, it is unclear if the system will circularize before mass loss, or whether the Kozai-Lidov oscillation will simply be de-tuned such that pericenter $\\sim $  AU for the duration of the giant phase.", "The interaction of Kozai-Lidov oscillations as the primary evolves to the giant phase may also initiate mass transfer and complex secular dynamics, perhaps analogous to the system considered by [41].", "Another possibility is that this phase initiates common envelope evolution, perhaps circularizing the inner orbit at smaller semi-major axis.", "If the inner binary maintains significant eccentricity, or if $e_{\\textrm {1}}$ is pumped after the giant phase and during mass loss via normal Kozai-Lidov oscillations or MIEK, such systems may form off-center or bi-polar planetary nebulae (PNe) (e.g., [45], [44], [54]).", "On the other hand, if the inner binary is circularized, it might also participate in the shaping of PNe (e.g., [28], [29]).", "An example of some of these mechanisms in action may be ring planetary nebula SuWt 2 (Exter et al. 2010).", "Since a very large fraction of triple stars will be first affected by Kozai-Lidov cycles when the MS star evolves, this phase should be investigated in detail.", "Figure: MIEK for massive stars, similar toFigure , but here m 0 =11m_0=11 M ⊙ _\\odot ,m 1 =10m_1=10 M ⊙ _\\odot , m 2 =10m_2=10 M ⊙ _\\odot , e 1 =0.7e_1=0.7, e 2 =0e_2=0, cosi=0.25\\cos i=0.25,g 1 =0 ∘ g_1=0^\\circ , g 2 =270 ∘ g_2=270^\\circ , a 1 =15a_1=15 AU, and a 2 =300a_2=300 AU.", "At ∼7\\sim 7 Myr,m 0 m_0 decreases to 1.4 M ⊙ _\\odot , cosi\\cos i flips, and the system attains1-e 1 ∼10 -3 1-e_1\\sim 10^{-3} immediately." ], [ "After Mass Loss & MIEK", "We find that $\\sim 2$  % of our triple systems undergo the MIEK mechanism after WD formation (see Fig.", "REF ).", "We again emphasize that by removing all systems from our sample that were tidally affected during the giant phase, the fraction of triples that go through MIEK presented in Table REF (“After Mass-Loss\") is conservative since many of these systems may be available to go through MIEK.", "The reason for this is that as the primary enters its post-MS evolution and begins to interact tidally with the secondary, the orbits will be affected by processes that we do not model (e.g., [41]).", "We attempt to provide an upper limit on the fraction of triple systems that could be tidally affected or collide after mass-loss due to the MIEK mechanism in Section REF by assuming a small value for $r_{\\rm tide}$ during the giant phase (compare Figs.", "REF and REF ).", "This increases the fraction of triple systems that undergo MIEK from $1.9\\% $ to $6.7\\% $ .", "A calculation of the 3-body dynamics with tides appropriate to a MS star interacting with a red giant is clearly needed to address this issue in more detail and to address the issues raised in Section REF .", "Systems that undergo the MIEK mechanism also have many possible outcomes.", "One possibility is that the extreme eccentricities obtained during a “flip” might lead to physical collisions (Baoz et al.", "in prep.).", "This may occur if, in a single orbit of the inner binary, $r_{\\textrm {peri}}$ changes from $r_{\\textrm {peri}}> r_{\\textrm {tide}}$ to $r_{\\textrm {peri}}< R_{*}$ .", "This is possible because the angular momentum of the inner binary at such large $e_{\\textrm {1}}$ is small.", "Thus, the change in angular momentum required for $r_{\\textrm {peri}}{} \\sim 0$ is also small.", "Another possible outcome for the system is a tidal Kozai capture as presented in [35].", "The inner semi-major axis $a_{\\textrm {1}}$ will strongly decrease because of tides, leaving a close MS-WD binary.", "When the secondary subsequently evolves off the MS, the WD will either accrete or go through common envelope evolution.", "Both of these scenarios could lead to single- or double-degenerate (SD or DD) SNe Ia, cataclysmic variables, or other types of WD accretors such as AM CVn stars [51], which may produce faint “.Ia” supernovae [2].", "One interesting consequence of this evolutionary scheme for producing close WD-MS binaries is that it skips the initial common envelope evolutionary phase usually required in binary evolution to produce a close WD-MS pair.", "In the MIEK mechanism the close binary is produced without common envelope, and thus the WD produced should have a mass appropriate to a single star with the mass of the primary (e.g., [16]).", "This stands in contrast to the conventional picture for the production of WD-MS and WD-WD binaries relevant for SD and DD Ia progenitors, where the WD growth is truncated by common envelope interaction with the close secondary and the mass of the resulting WD is smaller than one would naively estimate from the single-star initial-final mass relation [14].", "For SD SN Ia models, this means that less mass needs to be accreted onto the WD from the secondary, leading to a shorter delay time between production of the system and explosion.", "For DD SN models, if the model presented in [37] is accurate, then the luminosity of the SN resulting from the merger of a WD-WD binary is primarily determined by the mass of the primary.", "Thus, all else being equal, if a WD-WD binary results from a MIEK triple system then its primary WD mass should be larger and thus result in a more luminous DD SN than the analogous close binary system that is produced through normal common envelope evolution." ], [ "Massive Star Triples & Neutron Star Formation", "The MIEK mechanism may also play an important role in triple systems after NS formation, particularly since the NS mass, $\\simeq 1.4$  M$_\\odot $ , is so much less than its progenitor massive star, leading to a large increase in $\\epsilon _{\\rm oct}$ at the time of the associated supernova (Equation REF ).", "However, several effects differ in the NS case versus the case of WD formation considered throughout this paper.", "First, NSs receive a “kick” at birth that may unbind the inner binary or the outer tertiary, depending on the masses of the constituents, their eccentricities, and their semi-major axes ([12]; [17]).", "Second, mass loss on the MS can affect the mass ratios as a function of time (e.g., [7]).", "Third, even in the absence of a NS kick, the large and instantaneous mass loss in the supernova explosion from the primary provides a kick to the center of mass of the inner binary which may unbind the tertiary, and shift the mutual inclination.", "Even so, we expect a population of NS-MS binaries to survive with massive tertiaries and large $e_{\\textrm {2}}$ due to the aforementioned kicks.", "Thus, $\\epsilon _{\\textrm {oct}}$ will be large for the surviving triple systems which will cause many to exhibit the MIEK mechanism.", "This is a qualitatively new way to produce close NS-MS binaries.", "As a test, we integrated the orbit of a system with $m_0=11$  M$_\\odot $ , $m_1=10$  M$_\\odot $ , $m_2=10$  M$_\\odot $ , $e_1=0.7$ , $e_2=0$ , $\\cos i=0.25$ , $g_1=0^\\circ $ , $g_2=270^\\circ $ , $a_1=15$  AU, and $a_2=300$  AU.", "At a time $\\sim 7$ Myr after the start of the simulation we instantaneously decrease the mass of the primary to $1.4$  M$_\\odot $ and apply no kick to the NS.", "Figure REF shows the resulting evolution.", "The system flips immediately after the supernova and evolves to an inner eccentricity of $1-e_1 > 10^{-3}$ , bringing the NS to tidal contact with the secondary.", "Note the large increase in $e_2$ from $\\sim 0$ to $\\sim 0.3$ immediately after the SN.", "This calculation is meant only to be illustrative since it does not include a NS kick.", "A full population study, with kicks and tides is clearly needed to assess the viability of this mechanism for forming close NS-MS binaries.", "A further aspect of this type of evolution is that the ratio of $a_{\\textrm {2}}/ a_{\\textrm {1}}$ can decrease during the strong MS mass loss and the supernova explosion, which can cause the triple to become dynamically unstable and eventually disrupt, or potentially initiate collisions (the TEDI mechanism of [39])." ], [ "Further Studies", "In this study we have explored a single set of $m_{\\textrm {0}}$ and $m_{\\textrm {1}}$ , however, different combinations of masses will affect the fraction of systems which become tidally affected or collide during each of the evolutionary phases.", "For example, if $m_{\\textrm {0}}\\approx m_{\\textrm {1}}$ the system will have little or no time after the primary becomes a compact object before the secondary enters its giant phase.", "This will reduce the fraction of systems that will flip and become tidally affected or collide due to the MIEK mechanism.", "However, if the difference between the masses of the inner binary members is significantly greater, then a larger fraction of systems will flip before the primary goes through mass-loss.", "Additionally, the difference in the masses after the primary has become a compact object will be smaller, further decreasing the fraction of systems that will be affected by the MIEK mechanism.", "Determining the optimal balance between these two effects is beyond the scope of the current study.", "In a future study we will present a more general study of mass loss in a full population of triples, quantifying the number of systems affected by Kozai-Lidov oscillations and the eccentric Kozai mechanism, an accounting of those that are unbound and those that are tidally affected or collide at each evolutionary stage, and those that undergo the MIEK mechanism.", "Any study which evolves a full population of triple systems with distributions in $a_{\\textrm {1}}$ and $a_{\\textrm {2}}$ will have a significant fraction of systems that will be affected by the TEDI.", "[39] demonstrate that systems that start with a small $a_{\\textrm {2}}/ a_{\\textrm {1}}$ ratio become unstable during mass-loss and evolve chaotically, leading to close encounters, collisions, and exchanges between the stellar components.", "Additionally, [39] also raise the issue of Kozai cycles and stellar evolution as important.", "Future studies will be needed to quantitatively evaluate the importance of both the TEDI and the MIEK mechanism." ], [ "Conclusion", "We have presented a preliminary investigation of stellar evolution and mass loss in triple systems and demonstrated the MIEK mechanism, a novel channel through which triple systems produce compact object – MS binaries that interact tidally or collide only after the primary has evolved off the MS and become a compact object.", "For a broad range of parameters, hierarchical triple star systems are unaffected by Kozai-Lidov oscillations until the primary in the central binary evolves off the MS and begins mass loss.", "Subsequently, the primary becomes a WD or a NS, and may then be much less massive than the other components in the ternary, enabling the “eccentric Kozai mechanism”.", "In this near-test-particle limit, the mutual inclination between the inner and outer binary can flip signs, driving the inner binary to very high eccentricity and tidal contact or collision.", "In this study, we define tidal contact by an ad-hoc minimal separation between the member of the inner binary, below which tidal affects are deemed important (Section REF ).", "Even distant binaries with initial semi-major axes larger then tens of AU can be strongly affected.", "We consider an example triple system with masses 7, 6.5 and 6 $\\rm {M}_\\odot $ as proof -of-principle (see Fig.", "REF ), and explore the MIEK mechanism's dependence on the initial eccentricities and inclination for this system.", "For a flat distribution of eccentricities and $\\cos i$ , we find that $13\\% $ , $28\\% $ , and $2.2\\% $ of systems are tidally affected or collide before the primary evolves off the MS, after the primary becomes a giant, and after the primary becomes a WD, respectively.", "On the MS, we find that roughly half of our systems interact tidally or collide as a result of the eccentric Kozai mechanism, and hence their dynamics would not have been captured by quadrupole-order secular calculations.", "In the giant phase, most of the systems are brought to tidal contact by normal Kozai-Lidov oscillations and the fact that pericenter passage occurs at semi-major axes less than $\\sim $  AU even for very modest inclinations.", "The large fraction of such systems motivates a detailed study of the dynamics and tidal interaction of triple systems with a giant primary.", "Finally, the MIEK mechanism dominates the last stage, causing the WD produced by the initial primary to come to tidal contact or collide with the initial secondary which reduces $a_{\\textrm {1}}$ without a common envelope phase.", "As a last application, we showed that some massive star triple systems will also undergo the MIEK mechanism after the primary's supernova explosion and the associated rapid mass loss (Figure REF ).", "We save a detailed study of the parameter space of triples affected by MIEK for a future paper, but here note that for a thermal distribution of eccentricities, as is commonly adopted for studies of binary stars, we expect more systems to be affected by MIEK since, in general, $\\epsilon _{\\rm oct}$ increases as $e_2$ increases (Equation REF ).", "We thank Chris Kochanek, Boaz Katz, Jennifer van Saders, Joe Antognini, Krzysztof Stanek, Smadar Naoz, Yoram Lithwick, Norm Murray, and Andy Gould for discussions and encouragement.", "We also thank J. Fregeau for discussions and for making the code FEWBODY publicly available.", "This work is supported in part by an Alfred P. Sloan Foundation Fellowship and NSF grant AST-0908816.", "B.J.S.", "was supported by a Graduate Research Fellowship from the National Science Foundation." ] ]

[ [ "An Approximate Newton Method for Markov Decision Processes" ], [ "Abstract Gradient-based algorithms are one of the methods of choice for the optimisation of Markov Decision Processes.", "In this article we will present a novel approximate Newton algorithm for the optimisation of such models.", "The algorithm has various desirable properties over the naive application of Newton's method.", "Firstly the approximate Hessian is guaranteed to be negative-semidefinite over the entire parameter space in the case where the controller is log-concave in the control parameters.", "Additionally the inference required for our approximate Newton method is often the same as that required for first order methods, such as steepest gradient ascent.", "The approximate Hessian also has many nice sparsity properties that are not present in the Hessian and that make its inversion efficient in many situations of interest.", "We also provide an analysis that highlights a relationship between our approximate Newton method and both Expectation Maximisation and natural gradient ascent.", "Empirical results suggest that the algorithm has excellent convergence and robustness properties." ], [ "Markov Decision Processes", "Markov Decision Processes (MDPs) are the most commonly used model for the description of sequential decision making processes in a fully observable environment, see e.g.", "[5].", "A MDP is described by the tuple $\\lbrace \\mathcal {S}, \\mathcal {A}, H, p_1, p, \\pi , R \\rbrace $ , where $\\mathcal {S}$ and $\\mathcal {A}$ are sets known respectively as the state and action space, $H \\in \\mathbb {N}$ is the planning horizon, which can be either finite or infinite, and $\\lbrace p_1, p, \\pi , R \\rbrace $ are functions that are referred as the initial state distribution, transition dynamics, policy (or controller) and the reward function.", "In general the state and action spaces can be arbitrary sets, but we restrict our attention to either discrete sets or subsets of $\\mathbb {R}^n$ , where $n \\in \\mathbb {N}$ .", "We use boldface notation to represent a vector and also use the notation $\\mathbf {z} = (\\mathbf {s}, \\mathbf {a})$ to denote a state-action pair.", "Given a MDP the trajectory of the agent is determined by the following recursive procedure: Given the agent's state, $\\mathbf {s}_t$ , at a given time-point, $t \\in \\mathbb {N}_{H}$ , an action is selected according to the policy, $\\mathbf {a}_t \\sim \\pi (\\cdot |\\mathbf {s}_t)$ ; The agent will then transition to a new state according to the transition dynamics, $\\mathbf {s}_{t+1} \\sim p(\\cdot | \\mathbf {a}_t, \\mathbf {s}_t)$ ; this process is iterated sequentially through all of the time-points in the planning horizon, where the state of the initial time-point is determined by the initial state distribution $\\mathbf {s}_1 \\sim p_1(\\cdot )$ .", "At each time-point the agent receives a (scalar) reward that is determined by the reward function, where this function depends on the current action and state of the environment.", "Typically the reward function is assumed to be bounded, but as the objective is linear in the reward function we assume w.l.o.g that it is non-negative.", "The most widely used objective in the MDP framework is to maximise the total expected reward of the agent over the course of the planning horizon.", "This objective can take various forms, including an infinite planning horizon, with either discounted or average rewards, or a finite planning horizon.", "The theoretical contributions of this paper are applicable to all three frameworks, but for notational ease and for reasons of space we concern ourselves with the infinite horizon framework with discounted rewards.", "In this framework the boundedness of the objective function is ensured by the introduction of a discount factor, $\\gamma \\in [0,1)$ , which scales the rewards of the various time-points in a geometric manner.", "Writing the objective function and trajectory distribution directly in terms of the parameter vector then, for any $\\mathbf {w} \\in \\mathcal {W}$ the objective function takes the form $U(\\mathbf {w}) = \\sum _{t=1}^{\\infty } \\mathbb {E}_{p_t(\\mathbf {a}, \\mathbf {s}; \\mathbf {w})} \\bigg [ \\gamma ^{t-1} R(\\mathbf {a},\\mathbf {s}) \\bigg ], $ where we have denoted the parameter space by $\\mathcal {W}$ and have used the notation $p_t(\\mathbf {a}, \\mathbf {s};\\mathbf {w})$ to represent the marginal $p(\\mathbf {s}_t \\!", "= \\!\\mathbf {s}, \\mathbf {a}_t \\!", "= \\!\\mathbf {a};\\mathbf {w})$ of the joint state-action trajectory distribution $p(\\mathbf {a}_{1:H}, \\mathbf {s}_{1:H};\\mathbf {w}) = \\pi (\\mathbf {a}_H|\\mathbf {s}_H;\\mathbf {w}) \\bigg \\lbrace \\prod _{t=1}^{H-1} p(\\mathbf {s}_{t+1}|\\mathbf {a}_t,\\mathbf {s}_t) \\pi (\\mathbf {a}_t|\\mathbf {s}_t;\\mathbf {w}) \\bigg \\rbrace p_1(\\mathbf {s}_1), \\quad H \\in \\mathbb {N}.", "$ Note that the policy is now written in terms of its parametric representation, $\\pi (\\mathbf {a}|\\mathbf {s}; \\mathbf {w})$ .", "It is well known that the global optimum of (REF ) can be obtained through dynamic programming, see e.g.", "[5].", "However, due to various issues, such as prohibitively large state-action spaces or highly non-linear transition dynamics, it is not possible to find the global optimum of (REF ) in most real-world problems of interest.", "Instead most research in this area focuses on obtaining approximate solutions, for which there exist numerous techniques, such as approximate dynamic programming methods [6], Monte-Carlo tree search methods [19] and policy search methods, both parametric [27], [21], [16], [18] and non-parametric [2], [25].", "This work is focused solely on parametric policy search methods, by which we mean gradient-based methods, such as steepest and natural gradient ascent [23], [1], along with Expectation Maximisation [11], which is a bound optimisation technique from the statistics literature.", "Since their introduction [14], [31], [10], [16] these methods have been the centre of a large amount of research, with much of it focusing on gradient estimation [21], [4], variance reduction techniques [30], [15], function approximation techniques [27], [8], [20] and real-world applications [18], [26].", "While steepest gradient ascent has enjoyed some success it is known to suffer from some substantial issues that often make it unattractive in practice, such as slow convergence and poor scaling [23].", "Various optimisation methods have been introduced as an alternative, most notably natural gradient ascent [16], [24], [3] and Expectation Maximisation [18], [28], which are currently considered the current state of the art in the field of parametric policy search algorithms.", "In this paper our primary focus is on the search-direction (in the parameter space) of these two methods." ], [ "Search Direction Analysis", "In this section we will perform a novel analysis of the search-direction of both natural gradient ascent and Expectation Maximisation.", "In gradient-based algorithms of Markov Decision Processes the update of the policy parameters take the form $\\mathbf {w}^{\\textnormal {new}} = \\mathbf {w} + \\alpha \\mathcal {M}(\\mathbf {w}) \\nabla _{\\mathbf {w}} U(\\mathbf {w}), $ where $\\alpha \\in \\mathbb {R}^+$ is the step-size parameter and $\\mathcal {M}(\\mathbf {w})$ is some positive-definite matrix that possibly depends on $\\mathbf {w}$ .", "It is well-known that such an update will increase the total expected reward, provided that $\\alpha $ is sufficiently small, and this process will converge to a local optimum of (REF ) provided the step-size sequence is appropriately selected.", "While EM doesn't have an update of the form given in (REF ) we shall see that the algorithm is closely related to such an update.", "It is convenient for later reference to note that the gradient $\\nabla _{\\mathbf {w}} U(\\mathbf {w})$ can be written in the following form $\\nabla _{\\mathbf {w}} U(\\mathbf {w}) = \\mathbb {E}_{p_{\\gamma }(\\mathbf {z}; \\mathbf {w}) Q(\\mathbf {z}; \\mathbf {w})} \\bigg [ \\nabla _{\\mathbf {w}} \\log \\pi (\\mathbf {a}|\\mathbf {s}; \\mathbf {w}) \\bigg ], $ where we use the expectation notation $\\mathbb {E}[\\cdot ]$ to denote the integral/summation w.r.t.", "a non-negative function.", "The term $p_{\\gamma }(\\mathbf {z}; \\mathbf {w})$ is a geometric weighted average of state-action occupancy marginals given by $p_{\\gamma }(\\mathbf {z}; \\mathbf {w}) = \\sum _{t=1}^{\\infty } \\gamma ^{t-1} p_t(\\mathbf {z}; \\mathbf {w}),$ while the term $Q(\\mathbf {z}; \\mathbf {w})$ is referred to as the state-action value function and is equal to the total expected future reward from the current time-point onwards, given the current state-action pair, $\\mathbf {z}$ , and parameter vector, $\\mathbf {w}$ , i.e.", "$Q(\\mathbf {z}; \\mathbf {w}) = \\sum _{t=1}^{\\infty } \\mathbb {E}_{p_t(\\mathbf {z}^{\\prime }; \\mathbf {w})} \\bigg [ \\gamma ^{t-1} R(\\mathbf {z}^{\\prime }) \\bigg | \\mathbf {z}_1 = \\mathbf {z} \\bigg ].$ This is a standard result and due to reasons of space we have omitted the details, but see e.g.", "[27] or section(REF ) of the supplementary material for more details.", "An immediate issue concerning updates of the form (REF ) is in the selection of the `optimal' choice of the matrix $\\mathcal {M}(\\mathbf {w})$ , which clearly depends on the sense in which optimality is defined.", "There are numerous reasonable properties that are desirable of such an update, including the numerical stability and computational complexity of the parameter update, as well as the rate of convergence of the overall algorithm resulting from these updates.", "While all reasonable criteria the rate of convergence is of such importance in an optimisation algorithm that it is a logical starting point in our analysis.", "For this reason we concern ourselves with relating these two parametric policy search algorithms to the Newton method, which has the highly desirable property of having a quadratic rate of convergence in the vicinity of a local optimum.", "The Newton method is well-known to suffer from problems that make it either infeasible or unattractive in practice, but in terms of forming a basis for theoretical comparisons it is a logical starting point.", "We shall discuss some of the issues with the Newton method in more detail in section().", "In the Newton method the matrix $\\mathcal {M}(\\mathbf {w})$ is set to the negative inverse Hessian, i.e.", "$\\mathcal {M}(\\mathbf {w}) = -\\mathcal {H}^{-1}(\\mathbf {w}), \\quad \\textnormal { where } \\textnormal { } \\mathcal {H}(\\mathbf {w}) = \\nabla _{\\mathbf {w}} \\nabla ^T_{\\mathbf {w}} U(\\mathbf {w}).$ where we have denoted the Hessian by $\\mathcal {H} (\\mathbf {w})$ .", "Using methods similar to those used to calculate the gradient, it can be shown that the Hessian takes the form $\\mathcal {H} (\\mathbf {w}) = \\mathcal {H}_1(\\mathbf {w}) + \\mathcal {H}_2(\\mathbf {w}), $ where $\\mathcal {H}_1(\\mathbf {w}) &= \\sum _{t=1}^{\\infty } \\mathbb {E}_{p(\\mathbf {z}_{1:t}; \\mathbf {w})} \\bigg [ \\gamma ^{t-1} R(\\mathbf {z}_t) \\nabla _{\\mathbf {w}} \\log p(\\mathbf {z}_{1:t}; \\mathbf {w}) \\nabla ^T_{\\mathbf {w}} \\log p(\\mathbf {z}_{1:t}; \\mathbf {w}) \\bigg ], \\\\\\mathcal {H}_2(\\mathbf {w}) &= \\sum _{t=1}^{\\infty } \\mathbb {E}_{p(\\mathbf {z}_{1:t}; \\mathbf {w})} \\bigg [ \\gamma ^{t-1} R(\\mathbf {z}_t) \\nabla _{\\mathbf {w}} \\nabla ^T_{\\mathbf {w}} \\log p(\\mathbf {z}_{1:t}; \\mathbf {w}) \\bigg ].", "$ We have omitted the details of the derivation, but these can be found in section(REF ) of the supplementary material, with a similar derivation of a sample-based estimate of the Hessian given in [4]." ], [ "Natural Gradient Ascent", "To overcome some of the issues that can hinder steepest gradient ascent an alternative, natural gradient, was introduced in [16].", "Natural gradient ascent techniques originated in the neural network and blind source separation literature, see e.g.", "[1], and take the perspective that the parameter space has a Riemannian manifold structure, as opposed to a Euclidean structure.", "Deriving the steepest ascent direction of $U(\\mathbf {w})$ w.r.t.", "a local norm defined on this parameter manifold (as opposed to w.r.t.", "the Euclidean norm, which is the case in steepest gradient ascent) results in natural gradient ascent.", "We denote the quadratic form that induces this local norm on the parameter manifold by $G(\\mathbf {w})$ , i.e.", "$d(\\mathbf {w})^2 = \\mathbf {w}^T G(\\mathbf {w}) \\mathbf {w}$ .", "The derivation for natural gradient ascent is well-known, see e.g.", "[1], and its application to the objective (REF ) results in a parameter update of the form $\\mathbf {w}_{k+1} = \\mathbf {w}_k + \\alpha _k G^{-1}(\\mathbf {w}_k) \\nabla _{\\mathbf {w}} U(\\mathbf {w}_k).$ In terms of (REF ) this corresponds to $\\mathcal {M}(\\mathbf {w}) = G^{-1}(\\mathbf {w})$ .", "In the case of MDPs the most commonly used local norm is given by the Fisher information matrix of the trajectory distribution, see e.g.", "[3], [24], and due to the Markovian structure of the dynamics it is given by $G(\\mathbf {w}) = - \\mathbb {E}_{p_{\\gamma }(\\mathbf {z}; \\mathbf {w})} \\bigg [ \\nabla _{\\mathbf {w}} \\nabla _{\\mathbf {w}}^T \\log \\pi (\\mathbf {a}| \\mathbf {s}; \\mathbf {w}) \\bigg ].", "$ We note that there is an alternate, but equivalent, form of writing the Fisher information matrix, see e.g.", "[24], but we do not use it in this work.", "In order to relate natural gradient ascent to the Newton method we first rewrite the matrix (REF ) into the following form $\\mathcal {H}_2(\\mathbf {w}) = \\mathbb {E}_{p_{\\gamma }(\\mathbf {z}; \\mathbf {w}) Q(\\mathbf {z}; \\mathbf {w})} \\bigg [ \\nabla _{\\mathbf {w}} \\nabla ^T_{\\mathbf {w}} \\log \\pi (\\mathbf {a}|\\mathbf {s}; \\mathbf {w}) \\bigg ].", "$ For reasons of space the details of this reformulation of (REF ) are left to section(REF ) of the supplementary material.", "Comparing the Fisher information matrix (REF ) with the form of $\\mathcal {H}_2(\\mathbf {w})$ given in (REF ) it is clear that natural gradient ascent has a relationship with the approximate Newton method that uses $\\mathcal {H}_2(\\mathbf {w})$ in place of $\\mathcal {H}(\\mathbf {w})$ .", "In terms of (REF ) this approximate Newton method corresponds to setting $\\mathcal {M}(\\mathbf {w}) = -\\mathcal {H}^{-1}_2(\\mathbf {w})$ .", "In particular it can be seen that the difference between the two methods lies in the non-negative function w.r.t.", "which the expectation is taken in (REF ) and (REF ).", "(It also appears that there is a difference in sign, but observing the form of $\\mathcal {M}(\\mathbf {w})$ for each algorithm shows that this is not the case.)", "In the Fisher information matrix the expectation is taken w.r.t.", "to the geometrically weighted summation of state-action occupancy marginals of the trajectory distribution, while in $\\mathcal {H}_2(\\mathbf {w})$ there is an additional weighting from the state-action value function.", "Hence, $\\mathcal {H}_2(\\mathbf {w})$ incorporates information about the reward structure of the objective function, whereas the Fisher information matrix does not, and so it will generally contain more information about the curvature of the objective function." ], [ "Expectation Maximisation", "The Expectation Maximisation algorithm, or EM-algorithm, is a powerful optimisation technique from the statistics literature, see e.g.", "[11], that has recently been the centre of much research in the planning and reinforcement learning communities, see e.g.", "[10], [28], [18].", "A quantity of central importance in the EM-algorithm for MDPs is the following lower-bound on the $\\log $ -objective $\\log U(\\mathbf {w}) \\ge \\mathcal {H}_{\\textnormal {entropy}}(q(\\mathbf {z}_{1:t}, t)) + \\mathbb {E}_{q(\\mathbf {z}_{1:t}, t)} \\bigg [ \\log \\gamma ^{t-1} R(\\mathbf {z}_t) p(\\mathbf {z}_{1:t}; \\mathbf {w}) \\bigg ], $ where $\\mathcal {H}_{\\textnormal {entropy}}$ is the entropy function and $q(\\mathbf {z}_{1:t}, t)$ is known as the `variational distribution'.", "Further details of the EM-algorithm for MDPs and a derivation of (REF ) are given in section(REF ) of the supplementary material or can be found in e.g.", "[18], [28].", "The parameter update of the EM-algorithm is given by the maximum (w.r.t.", "$\\mathbf {w}$ ) of the `energy' term, $\\mathcal {Q}(\\mathbf {w}, \\mathbf {w}_k) = \\mathbb {E}_{p_{\\gamma }(\\mathbf {z}; \\mathbf {w}_k) Q(\\mathbf {z}; \\mathbf {w}_k)} \\bigg [ \\log \\pi (\\mathbf {a} | \\mathbf {s}; \\mathbf {w}) \\bigg ].", "$ Note that $\\mathcal {Q}$ is a two-parameter function, where the first parameter occurs inside the expectation and the second parameter defines the non-negative function w.r.t.", "the expectation is taken.", "This decoupling allows the maximisation over $\\mathbf {w}$ to be performed explicitly in many cases of interest.", "For example, when the $\\log $ -policy is quadratic in $\\mathbf {w}$ the maximisation problems is equivalent to a least-squares problem and the optimum can be found through solving a linear system of equations.", "It has previously been noted, again see e.g.", "[18], that the parameter update of steepest gradient ascent and the EM-algorithm can be related through this `energy' term.", "In particular the gradient (REF ) evaluated at $\\mathbf {w}_k$ can also be written as follows $\\nabla _{\\mathbf {w}|\\mathbf {w} = \\mathbf {w}_k} U(\\mathbf {w}) = \\nabla ^{10}_{\\mathbf {w}|\\mathbf {w} = \\mathbf {w}_k} \\mathcal {Q}(\\mathbf {w}, \\mathbf {w}_k)$ , where we use the notation $\\nabla ^{10}_{\\mathbf {w}}$ to denote the first derivative w.r.t.", "the first parameter, while the update of the EM-algorithm is given by $\\mathbf {w}_{k+1} = \\operatorname{argmax}_{\\mathbf {w}\\in \\mathcal {W}}\\hspace{2.84526pt} \\mathcal {Q}(\\mathbf {w}, \\mathbf {w}_k)$ .", "In other words, steepest gradient ascent moves in the direction that most rapidly increases $\\mathcal {Q}(\\mathbf {w}, \\mathbf {w}_k)$ w.r.t.", "the first variable, while the EM-algorithm maximises $\\mathcal {Q}(\\mathbf {w}, \\mathbf {w}_k)$ w.r.t.", "the first variable.", "While this relationship is true, it is also quite a negative result.", "It states that in situations where it is not possible to explicitly perform the maximisation over $\\mathbf {w}$ in (REF ) then the alternative, in terms of the EM-algorithm, is this generalised EM-algorithm, which is equivalent to steepest gradient ascent.", "Considering that algorithms such as EM are typically considered because of the negative aspects related to steepest gradient ascent this is an undesirable alternative.", "It is possible to find the optimum of (REF ) numerically, but this is also undesirable as it results in a double-loop algorithm that could be computationally expensive.", "Finally, this result provides no insight into the behaviour of the EM-algorithm, in terms of the direction of its parameter update, when the maximisation over $\\mathbf {w}$ in (REF ) can be performed explicitly.", "Instead we provide the following result, which shows that the step-direction of the EM-algorithm has an underlying relationship with the Newton method.", "In particular we show that, under suitable regularity conditions, the direction of the EM-update, i.e.", "$\\mathbf {w}_{k+1} - \\mathbf {w}_k$ , is the same, up to first order, as the direction of an approximate Newton method that uses $\\mathcal {H}_2(\\mathbf {w})$ in place of $\\mathcal {H}(\\mathbf {w})$ .", "Theorem 1 Suppose we are given a Markov Decision Process with objective (REF ) and Markovian trajectory distribution (REF ).", "Consider the update of the parameter through Expectation Maximisation at the $k^{\\textnormal {th}}$ iteration of the algorithm, i.e.", "$\\mathbf {w}_{k+1} = \\operatorname{argmax}_{\\mathbf {w}\\in \\mathcal {W}}\\hspace{2.84526pt} \\mathcal {Q}(\\mathbf {w}, \\mathbf {w}_k).$ Provided that $\\mathcal {Q}(\\mathbf {w}, \\mathbf {w}_k)$ is twice continuously differentiable in the first parameter we have that $\\mathbf {w}_{k+1} - \\mathbf {w}_k = - \\mathcal {H}_2^{-1}(\\mathbf {w}_k) \\nabla _{\\mathbf {w}|\\mathbf {w} = \\mathbf {w}_k} U(\\mathbf {w}) + \\mathcal {O}(\\Vert \\mathbf {w}_{k+1} - \\mathbf {w}_k\\Vert ^2).", "$ Additionally, in the case where the $\\log $ -policy is quadratic the relation to the approximate Newton method is exact, i.e.", "the second term on the r.h.s.", "(REF ) is zero.", "The idea of the proof is simple and only involves performing a Taylor expansion of $\\nabla _{\\mathbf {w}}^{10} \\mathcal {Q}(\\mathbf {w}, \\mathbf {w}_k)$ .", "As $\\mathcal {Q}$ is assumed to be twice continuously differentiable in the first component this Taylor expansion is possible and gives $\\nabla ^{10}_{\\mathbf {w}} \\mathcal {Q}(\\mathbf {w}_{k+1}, \\mathbf {w}_k) = \\nabla ^{10}_{\\mathbf {w}} \\mathcal {Q}(\\mathbf {w}_k, \\mathbf {w}_k) + \\nabla ^{20}_{\\mathbf {w}} \\mathcal {Q}(\\mathbf {w}_k, \\mathbf {w}_k) (\\mathbf {w}_{k+1} - \\mathbf {w}_k) + \\mathcal {O}(\\Vert \\mathbf {w}_{k+1} - \\mathbf {w}_k\\Vert ^2).", "$ As $\\mathbf {w}_{k+1} = \\operatorname{argmax}_{\\mathbf {w} \\in \\mathcal {W}}\\hspace{2.84526pt} \\mathcal {Q}(\\mathbf {w}, \\mathbf {w}_k)$ it follows that $\\nabla ^{10}_{\\mathbf {w}}\\mathcal {Q}(\\mathbf {w}_{k+1}, \\mathbf {w}_k) = 0$ .", "This means that, upon ignoring higher order terms in $\\mathbf {w}_{k+1} - \\mathbf {w}_k$ , the Taylor expansion (REF ) can be rewritten into the form $\\mathbf {w}_{k+1} - \\mathbf {w}_k = - \\nabla ^{20}_{\\mathbf {w}} \\mathcal {Q}(\\mathbf {w}_k, \\mathbf {w}_k)^{-1} \\nabla ^{10}_{\\mathbf {w}} \\mathcal {Q}(\\mathbf {w}_k, \\mathbf {w}_k).$ The proof is completed by observing that $\\nabla ^{10}_{\\mathbf {w}} \\mathcal {Q}(\\mathbf {w}_k, \\mathbf {w}_k) = \\nabla _{\\mathbf {w}|\\mathbf {w} = \\mathbf {w}_k} U(\\mathbf {w})$ and $\\nabla ^{20}_{\\mathbf {w}}\\mathcal {Q}(\\mathbf {w}_k, \\mathbf {w}_k) = \\mathcal {H}_2(\\mathbf {w}_k)$ .", "The second statement follows because in the case where the $log$ -policy is quadratic the higher order terms in the Taylor expansion vanish." ], [ "Summary", "In this section we have provided a novel analysis of both natural gradient ascent and Expectation Maximisation when applied to the MDP framework.", "Previously, while both of these algorithms have proved popular methods for MDP optimisation, there has been little understanding of them in terms of their search-direction in the parameter space or their relation to the Newton method.", "Firstly, our analysis shows that the Fisher information matrix, which is used in natural gradient ascent, is similar to $\\mathcal {H}_2(\\mathbf {w})$ in (REF ) with the exception that the information about the reward structure of the problem is not contained in the Fisher information matrix, while such information is contained in $\\mathcal {H}_2(\\mathbf {w})$ .", "Additionally we have shown that the step-direction of the EM-algorithm is, up to first order, an approximate Newton method that uses $\\mathcal {H}_2(\\mathbf {w})$ in place of $\\mathcal {H}(\\mathbf {w})$ and employs a constant step-size of one." ], [ "An Approximate Newton Method", "A natural follow on from the analysis in section() is the consideration of using $\\mathcal {M}(\\mathbf {w}) = -\\mathcal {H}^{-1}_2(\\mathbf {w})$ in (REF ), a method we call the full approximate Newton method from this point onwards.", "In this section we show that this method has many desirable properties that make it an attractive alternative to other parametric policy search methods.", "Additionally, denoting the diagonal matrix formed from the diagonal elements of $\\mathcal {H}_2(\\mathbf {w})$ by $\\mathcal {D}_2(\\mathbf {w})$ , we shall also consider the method that uses $\\mathcal {M}(\\mathbf {w}) = -\\mathcal {D}^{-1}_2(\\mathbf {w})$ in (REF ).", "We call this second method the diagonal approximate Newton method.", "Recall that in (REF ) it is necessary that $\\mathcal {M}(\\mathbf {w})$ is positive-definite (in the Newton method this corresponds to requiring the Hessian to be negative-definite) to ensure an increase of the objective.", "In general the objective (REF ) is not concave, which means that the Hessian will not be negative-definite over the entire parameter space.", "In such cases the Newton method can actually lower the objective and this is an undesirable aspect of the Newton method.", "An attractive property of the approximate Hessian, $\\mathcal {H}_2(\\mathbf {w})$ , is that it is always negative-definite when the policy is $\\log $ –concave in the policy parameters.", "This fact follows from the observation that in such cases $\\mathcal {H}_2(\\mathbf {w})$ is a non-negative mixture of negative-definite matrices, which again is negative-definite [9].", "Additionally, the diagonal terms of a negative-definite matrix are negative and so $\\mathcal {D}_2(\\mathbf {w})$ is also negative-definite when the controller is $\\log $ -concave.", "To motivate this result we now briefly consider some widely used policies that are either $\\log $ -concave or blockwise $\\log $ -concave.", "Firstly, consider the Gibb's policy, $\\pi (\\mathbf {a}|\\mathbf {s}; \\mathbf {w}) \\propto \\exp \\mathbf {w}^T \\mathbf {\\phi }(\\mathbf {a}, \\mathbf {s}) $ , where $\\mathbf {\\phi }(\\mathbf {a}, \\mathbf {s}) \\in \\mathbb {R}^{n_{\\mathbf {w}}}$ is a feature vector.", "This policy is widely used in discrete systems and is $\\log $ -concave in $\\mathbf {w}$ , which can be seen from the fact that $\\log \\pi (\\mathbf {a}|\\mathbf {s}; \\mathbf {w})$ is the sum of a linear term and a negative log-sum-exp term, both of which are concave [9].", "In systems with a continuous state-action space a common choice of controller is $\\pi (\\mathbf {a}| \\mathbf {s}; \\mathbf {w}_{\\textnormal {mean}}, \\Sigma ) = \\mathcal {N}(\\mathbf {a}| K \\mathbf {\\phi }(\\mathbf {s}) + \\mathbf {m}, \\Sigma (\\mathbf {s}))$ , where $\\mathbf {w}_{\\textnormal {mean}} = \\lbrace K, \\mathbf {m}\\rbrace $ and $\\mathbf {\\phi }(\\mathbf {s}) \\in \\mathbb {R}^{n_{\\mathbf {w}}}$ is a feature vector.", "The notation $\\Sigma (\\mathbf {s})$ is used because there are cases where is it beneficial to have state dependent noise in the controller.", "This controller is not jointly $\\log $ -concave in $\\mathbf {w}_{\\textnormal {mean}}$ and $\\Sigma $ , but it is blockwise $\\log $ -concave in $\\mathbf {w}_{\\textnormal {mean}}$ and $\\Sigma ^{-1}$ .", "In terms of $\\mathbf {w}_{\\textnormal {mean}}$ the $\\log $ -policy is quadratic and the coefficient matrix of the quadratic term is negative-definite.", "In terms of $\\Sigma ^{-1}$ the $\\log $ -policy consists of a linear term and a $\\log $ -determinant term, both of which are concave.", "In terms of evaluating the search direction it is clear from the forms of $\\mathcal {D}_2(\\mathbf {w})$ and $\\mathcal {H}_2(\\mathbf {w})$ that many of the pre-existing gradient evaluation techniques can be extended to the approximate Newton framework with little difficulty.", "In particular, gradient evaluation requires calculating the expectation of the derivative of the $\\log $ -policy w.r.t.", "$p_{\\gamma }(\\mathbf {z};\\mathbf {w})Q(\\mathbf {z};\\mathbf {w})$ .", "In terms of inference the only additional calculation necessary to implement either the full or diagonal approximate Newton methods is the calculation of the expectation (w.r.t.", "to the same function) of the Hessian of the $\\log $ -policy, or its diagonal terms.", "As an example in section(REF ) of the supplementary material we detail the extension of the recurrent state formulation of gradient evaluation in the average reward framework, see e.g.", "[31], to the approximate Newton method.", "We use this extension in the Tetris experiment that we consider in section().", "Given $n_{s}$ samples and $n_{\\mathbf {w}}$ parameters the complexity of these extensions scale as $\\mathcal {O}(n_{s} n_{\\mathbf {w} })$ for the diagonal approximate Newton method, while it scales as $\\mathcal {O}(n_{s} n_{\\mathbf {w}}^2)$ for the full approximate Newton method.", "An issue with the Newton method is the inversion of the Hessian matrix, which scales with $\\mathcal {O}(n_{\\mathbf {w}}^3)$ in the worst case.", "In the standard application of the Newton method this inversion has to be performed at each iteration and in large parameter systems this becomes prohibitively costly.", "In general $\\mathcal {H}(\\mathbf {w})$ will be dense and no computational savings will be possible when performing this matrix inversion.", "The same is not true, however, of the matrices $\\mathcal {D}_2(\\mathbf {w})$ and $\\mathcal {H}_2(\\mathbf {w})$ .", "Firstly, as $\\mathcal {D}_2(\\mathbf {w})$ is a diagonal matrix it is trivial to invert.", "Secondly, there is an immediate source of sparsity that comes from taking the second derivative of the $\\log $ -trajectory distribution in (REF ).", "This property ensures that any (product) sparsity over the control parameters in the $\\log $ -trajectory distribution will correspond to sparsity in $\\mathcal {H}_2(\\mathbf {w})$ .", "For example, in a partially observable Markov Decision Processes where the policy is modeled through a finite state controller, see e.g.", "[22], there are three functions to be optimised, namely the initial belief distribution, the belief transition dynamics and the policy.", "When the parameters of these three functions are independent $\\mathcal {H}_2(\\mathbf {w})$ will be block-diagonal (across the parameters of the three functions) and the matrix inversion can be performed more efficiently by inverting each of the block matrices individually.", "The reason that $\\mathcal {H}(\\mathbf {w})$ does not exhibit any such sparsity properties is due to the term $\\mathcal {H}_1(\\mathbf {w})$ in (REF ), which consists of the non-negative mixture of outer-product matrices.", "The vector in these outer-products is the derivative of the $\\log $ -trajectory distribution and this typically produces a dense matrix.", "A undesirable aspect of steepest gradient ascent is that its performance is affected by the choice of basis used to represent the parameter space.", "An important and desirable property of the Newton method is that it is invariant to non-singular linear (affine) transformations of the parameter space, see e.g.", "[9].", "This means that given a non-singular linear (affine) mapping $\\mathcal {T} \\in \\mathbb {R}^{n_{\\mathbf {w}} \\times n_{\\mathbf {w}}}$ , the Newton update of the objective $\\tilde{U}(\\mathbf {w}) = U(\\mathcal {T}\\mathbf {w})$ is related to the Newton update of the original objective through the same linear (affine) mapping, i.e.", "$\\mathbf {v} + \\Delta \\mathbf {v}_{\\textnormal {nt}} = \\mathcal {T} \\big ( \\mathbf {w} + \\Delta \\mathbf {w}_{\\textnormal {nt}} \\big )$ , where $\\mathbf {v} = \\mathcal {T}\\mathbf {w}$ and $\\Delta \\mathbf {v}_{\\textnormal {nt}}$ and $\\Delta \\mathbf {w}_{\\textnormal {nt}}$ denote the respective Newton steps.", "In other words running the Newton method on $U(\\mathbf {w})$ and $\\tilde{U}(\\mathcal {T}^{-1}\\mathbf {w})$ will give identical results.", "An important point to note is that this desirable property is maintained when using $\\mathcal {H}_2(\\mathbf {w})$ or $\\mathcal {D}_2(\\mathbf {w})$ in an approximate Newton method.", "This can be seen by using the linearity of the expectation operator and a proof of this statement is provided in section(REF ) of the supplementary material." ], [ "Experiments", "The first experiment we performed was an empirical illustration that the approximate Newton method is invariant to linear transformations of the parameter space.", "We considered the simple two state example of [16] as it allows us to plot the trace of the policy during training, since the policy has only two parameters.", "The policy was trained using both steepest gradient ascent and the approximate Newton method and in both the original and linearly transformed parameter space.", "The policy traces of the two algorithms are plotted in figure(REF .", "*fig:policytrace).", "As expected steepest gradient ascent is affected by such mappings, whilst the approximate Newton method is invariant to them.", "Figure: (a) An empirical illustration of the affine invariance of the approximate Newton method, performed on the two state MDP of .", "The plot shows the trace of the policy during training for the two different parameter spaces, where the results of the latter have been mapped back into the original parameter space for comparison.", "The plot shows the two steepest gradient ascent traces (blue cross and blue circle) and the two traces of the approximate Newton method (red cross and red circle).", "(b) Results of the tetris problem for steepest gradient ascent (black), natural gradient ascent (green), the diagonal approximate Newton method (blue) and the approximate Newton method (red).The second experiment was aimed at demonstrating the scalability of the full and diagonal approximate Newton methods to problems with a large state space.", "We considered the tetris domain, which is a popular computer game designed by Alexey Pajitnov in 1985.", "See [12] for more details.", "Firstly, we compared the performance of the full and diagonal approximate Newton methods to other parametric policy search methods.", "Tetris is typically played on a $20\\times 10$ grid, but due to computational costs we considered a $10 \\times 10$ grid in the experiment.", "This results in a state space with roughly $7 \\times 2^{100}$ states.", "We modelled the policy through a Gibb's distribution, where we considered a feature vector with the following features: the heights of each column, the difference in heights between adjacent columns, the maximum height and the number of `holes'.", "Under this policy it is not possible to obtain the explicit maximum over $\\mathbf {w}$ in (REF ) and so a straightforward application of EM is not possible in this problem.", "We therefore compared the diagonal and full approximate Newton methods with steepest and natural gradient ascent.", "Due to reasons of space the exact implementation of the experiment is detailed in section(REF ) of the supplementary material.", "We ran 100 repetitions of the experiment, each consisting of 100 training iterations, and the mean and standard error of the results are given in figure(REF .", "*fig:tetris).", "It can be seen that the full approximate Newton method outperforms all of the other methods, while the performance of the diagonal approximate Newton method is comparable to natural gradient ascent.", "We also ran several training runs of the full approximate Newton method on the full-sized $20\\times 10$ board and were able to obtain a score in the region of $14,000$ completed lines, which was obtained after roughly 40 training iterations.", "An approximate dynamic programming based method has previously been applied to the Tetris domain in [7].", "The same set of features were used and a score of roughly $4,500$ completed lines was obtained after around 6 training iterations, after which the solution then deteriorated.", "In the third experiment we considered a finite horizon (controlled) linear dynamical system.", "This allowed the search-directions of the various algorithms to be computed exactly using [13] and removed any issues of approximate inference from the comparison.", "In particular we considered a 3-link rigid manipulator, linearized through feedback linearisation, see e.g.", "[17].", "This system has a 6-dimensional state space, 3-dimensional action space and a 22-dimensional parameter space.", "Further details of the system can be found in section(REF ) of the supplementary material.", "We ran the experiment 100 times and the mean and standard error of the results plotted in figure(REF .", "*fig:modelbased).", "In this experiment the approximate Newton method found substantially better solutions than either steepest gradient ascent, natural gradient ascent or Expectation Maximisation.", "The superiority of the results in comparison to either steepest or natural gradient ascent can be explained by the fact that $\\mathcal {H}_2(\\mathbf {w})$ gives a better estimate of the curvature of the objective function.", "Expectation Maximisation performed poorly in this experiment, exhibiting sub-linear convergence.", "Steepest gradient ascent performed $3684\\pm 314$ training iterations in this experiment which, in comparison to the $203\\pm 34$ and $310\\pm 40$ iterations of natural gradient ascent and the approximate Newton method respectively, illustrates the susceptibility of this method to poor scaling.", "In the final experiment we considered the synthetic non-linear system considered in [29].", "Full details of the system and the experiment can be found in section(REF ) of the supplementary material.", "We ran the experiment 100 times and the mean and standard error of the results are plotted in figure(REF .", "*fig:nonlinearmodelfree).", "Again the approximate Newton method outperforms both steepest and natural gradient ascent.", "In this example only the mean parameters of the Gaussian controller are optimised, while the parameters of the noise are held fixed, which means that the $\\log $ -policy is quadratic in the policy parameters.", "Hence, in this example the EM-algorithm is a particular (less general) version of the approximate Newton method, where a fixed step-size of one is used throughout.", "The marked difference in performance between the EM-algorithm and the approximate Newton method shows the benefit of being able to tune the step-size sequence.", "In this experiment we considered five different step-size sequences for the approximate Newton method and all of them obtained superior results than the EM-algorithm.", "In contrast only one of the seven step-size sequences considered for steepest and natural gradient ascent outperformed the EM-algorithm.", "Figure: (a) The normalised total expected reward plotted against training time, in seconds, for the 3-link rigid manipulator.", "The plot shows the results for steepest gradient ascent (black), EM (blue), natural gradient ascent (green) and the approximate Newton method (red), where the plot shows the mean and standard error of the results.", "(b) The normalised total expected reward plotted against training iterations for the synthetic non-linear system of .", "The plot shows the results for EM (blue), steepest gradient ascent (black), natural gradient ascent (green) and the approximate Newton method (red), where the plot shows the mean and standard error of the results." ], [ "Conclusion", "The contributions of this paper are twofold: Firstly we have given a novel analysis of Expectation Maximisation and natural gradient ascent when applied to the MDP framework, showing that both have close connections to an approximate Newton method; Secondly, prompted by this analysis we have considered the direct application of this approximate Newton method to the optimisation of MDPs, showing that it has numerous desirable properties that are not present in the naive application of the Newton method.", "In terms of empirical performance we have found the approximate Newton method to perform consistently well in comparison to EM and natural gradient ascent, highlighting its viability as an alternative to either of these methods.", "At present we have only considered actor type implementations of the approximate Newton method and the extension to actor-critic methods is a point of future research." ], [ "Gradient Derivation", "For ease of reference in this section we give a brief outline of the derivation for the derivative of (REF ).", "As in the rest of the paper we focus on the case of an infinite planning horizon with discounted rewards, where other frameworks follow similarly.", "The first point to note is that for any $t \\in \\mathbb {N}$ we have the following identity, often referred to as the `$\\log $ -trick', $\\nabla _{\\mathbf {w}} p(\\mathbf {z}_{1:t};\\mathbf {w}) = p(\\mathbf {z}_{1:t}; \\mathbf {w}) \\nabla _{\\mathbf {w}} \\log p(\\mathbf {z}_{1:t}; \\mathbf {w}),$ where the usual limit arguments are made in the case $p(\\mathbf {z}_{1:t}; \\mathbf {w})=0$ .", "Upon interchanging the order of integration and differentiation the gradient takes the form $\\nabla _{\\mathbf {w}} U(\\mathbf {w}) = \\sum _{t = 1}^{\\infty } \\mathbb {E}_{p(\\mathbf {z}_{1:t}; \\mathbf {w})} \\bigg [ \\gamma ^{t-1} R(\\mathbf {z}_t) \\nabla _{\\mathbf {w}} \\log p(\\mathbf {z}_{1:t}; \\mathbf {w}) \\bigg ].", "$ Due to the Markovian structure of the trajectory distribution (REF ) this derivative can be written in the equivalent form $\\nabla _{\\mathbf {w}} U(\\mathbf {w}) &= \\sum _{t=1}^{\\infty } \\sum _{\\tau =1}^t \\mathbb {E}_{p(\\mathbf {z}_{\\tau }, \\mathbf {z}_{t}; \\mathbf {w})} \\bigg [ \\gamma ^{t-1} R(\\mathbf {z}_t) \\nabla _{\\mathbf {w}} \\log p(\\mathbf {a}_{\\tau }|\\mathbf {s}_{\\tau }; \\mathbf {w}) \\bigg ], \\\\&= \\sum _{\\tau =1}^{\\infty } \\sum _{t = \\tau }^\\infty \\mathbb {E}_{p(\\mathbf {z}_{\\tau }, \\mathbf {z}_{t}; \\mathbf {w})} \\bigg [ \\gamma ^{t-1} R(\\mathbf {z}_t) \\nabla _{\\mathbf {w}} \\log p(\\mathbf {a}_{\\tau }|\\mathbf {s}_{\\tau }; \\mathbf {w}) \\bigg ],$ where the second line follows from the first through an interchange of the summations.", "The chain structure of the trajectory distribution allows the expectation over the marginals of the two time-points of the trajectory distribution to be written as follows $\\nabla _{\\mathbf {w}} U(\\mathbf {w}) = \\sum _{\\tau =1}^{\\infty } \\mathbb {E}_{p_{\\tau }(\\mathbf {z}; \\mathbf {w})} \\bigg [ \\sum _{t = \\tau }^\\infty \\mathbb {E}_{p_t(\\mathbf {z}^{\\prime }; \\mathbf {w})} \\bigg [ \\gamma ^{t-1} R(\\mathbf {z}^{\\prime }) \\big | \\mathbf {z}_{\\tau } = \\mathbf {z} \\bigg ] \\nabla _{\\mathbf {w}} \\log p(\\mathbf {a}|\\mathbf {s}; \\mathbf {w}) \\bigg ], $ where we have used the notation $p_{\\tau }(\\mathbf {z}; \\mathbf {w}) \\equiv p(\\mathbf {z}_{\\tau } = \\mathbf {z}; \\mathbf {w})$ , for $\\tau \\in \\mathbb {N}$ .", "The summation over the inner expectation in (REF ) can be seen to be equal to the state-action value function scaled by $\\gamma ^{\\tau -1}$ , i.e.", "$\\gamma ^{\\tau -1} Q(\\mathbf {z};\\mathbf {w}) = \\sum _{t = \\tau }^\\infty \\mathbb {E}_{p_t(\\mathbf {z}^{\\prime }; \\mathbf {w})} \\bigg [ \\gamma ^{t-1} R(\\mathbf {z}^{\\prime }) \\big | \\mathbf {z}_{\\tau } = \\mathbf {z} \\bigg ].$ Inserting this form for this inner expectation into (REF ) gives $\\nabla _{\\mathbf {w}} U(\\mathbf {w}) &= \\sum _{\\tau =1}^{\\infty } \\mathbb {E}_{p_{\\tau }(\\mathbf {z}; \\mathbf {w})} \\bigg [ \\gamma ^{\\tau -1} Q(\\mathbf {z};\\mathbf {w}) \\nabla _{\\mathbf {w}} \\log p(\\mathbf {a}|\\mathbf {s}; \\mathbf {w}) \\bigg ], \\\\&= \\mathbb {E}_{p_{\\gamma }(\\mathbf {z};\\mathbf {w}) Q(\\mathbf {z};\\mathbf {w})} \\bigg [ \\nabla _{\\mathbf {w}} \\log p(\\mathbf {a}|\\mathbf {s}; \\mathbf {w}) \\bigg ],$ where the second line follows from the definition of $p_{\\gamma }(\\mathbf {z};\\mathbf {w})$ .", "This completes the derivation of (REF ).", "This derivation is different, but equivalent, to the standard derivation in [27]." ], [ "Hessian Derivation", "Following similar calculations to those used section(REF ) it is simple to see that the Hessian takes the form $\\mathcal {H} (\\mathbf {w}) = \\sum _{t=1}^\\infty \\mathbb {E}_{p(\\mathbf {z}_{1:t}; \\mathbf {w})} \\bigg [ \\gamma ^{t-1} R(\\mathbf {z}_t) \\bigg ( \\nabla _{\\mathbf {w}} \\log p(\\mathbf {z}_{1:t}; \\mathbf {w}) \\nabla _{\\mathbf {w}}^T \\log p(\\mathbf {z}_{1:t}; \\mathbf {w}) + \\nabla _{\\mathbf {w}} \\nabla _{\\mathbf {w}}^T \\log p(\\mathbf {z}_{1:t}; \\mathbf {w}) \\bigg ) \\bigg ],$ where the first term is obtained by a second application of the `$\\log $ -trick'.", "This is the form of the Hessian given in (REF ).", "The form of $\\mathcal {H}_2(\\mathbf {w})$ given in (REF ) follows from analogous calculations to those performed in section(REF ) of the supplementary material." ], [ "Expectation Maximisation Derivation", "For ease of reference in this section we give a brief derivation of the application of Expectation Maximisation to Markov Decision Processes.", "As in the rest of the paper we focus on the case of an infinite planning horizon with discounted rewards, where other frameworks follow similarly.", "A quantity of central importance in this derivation is the following non-negative function $\\tilde{p}(\\mathbf {z}_{1:t},t;\\mathbf {w}) = \\gamma ^{t-1} R(\\mathbf {z}_t) p(\\mathbf {z}_{1:t}; \\mathbf {w}).$ This function is non-negative because the reward function is considered as non-negative.", "Note that this function is trans-dimensional, i.e.", "it is a function of both $t \\in \\mathbb {N}$ and the state-action variables up until the $t^{\\textnormal {th}}$ time-point.", "It can be seen from (REF ) and (REF ) that the `normalisation constant' of $\\tilde{p}(\\mathbf {z}_{1:t}, t; \\mathbf {w})$ is equal to $U(\\mathbf {w})$ , i.e.", "$U(\\mathbf {w}) = \\sum _{t=1}^{\\infty } \\sum _{\\mathbf {z}_{1:t}} \\tilde{p}(\\mathbf {z}_{1:t}, t; \\mathbf {w})$ .", "We denote the normalised version of this function by $\\hat{p}(\\mathbf {z}_{1:t}, t; \\mathbf {w})$ .", "To obtain the lower-bound on the $\\log $ -objective we note that the Kullback-Leibler divergence between any two distributions is always non-negative and so calculating the Kullback-Leibler divergence between the variational distribution, $q(\\mathbf {z}_{1:t}, t)$ , and $\\hat{p}(\\mathbf {z}_{1:t}, t; \\mathbf {w})$ gives $KL(q||\\hat{p}) = \\log U(\\mathbf {w}) - \\mathcal {H}_{\\textnormal {entropy}}(q(\\mathbf {z}_{1:t}, t)) - \\mathbb {E}_{q(\\mathbf {z}_{1:t}, t)} \\bigg [ \\log \\tilde{p}(\\mathbf {z}_{1:t}, t; \\mathbf {w}) \\bigg ] \\ge 0.$ The lower-bound (REF ) is now immediately obtained from the definition of $\\tilde{p}(\\mathbf {z}_{1:t}, t; \\mathbf {w})$ .", "An EM-algorithm is obtained through coordinate-wise optimisation of the bound (REF ) w.r.t.", "the variational distribution (the E-step) and the policy parameters (the M-step).", "In the E-step the lower-bound (REF ) is optimised when the Kullback-Leibler divergence is zero, namely when $q(\\mathbf {z}_{1:t}, t) \\equiv \\hat{p}(\\mathbf {z}_{1:t}, t; \\mathbf {w}_k)$ , where $\\mathbf {w}_k$ are the current policy parameters.", "In the E-step the lower-bound is optimised w.r.t.", "$\\mathbf {w}$ , which is equivalent to optimising the quantity $\\mathbb {E}_{\\tilde{p}(\\mathbf {z}_{1:t}, t;\\mathbf {w}_k)} \\bigg [ \\log \\tilde{p}(\\mathbf {z}_{1:t}, t; \\mathbf {w}) \\bigg ] = \\mathbb {E}_{\\tilde{p}(\\mathbf {z}_{1:t}, t;\\mathbf {w}_k)} \\bigg [ \\sum _{\\tau =1}^t \\log \\pi (\\mathbf {a}_{\\tau }|\\mathbf {s}_{\\tau }; \\mathbf {w}) \\bigg ] + \\textnormal {terms independent of } \\mathbf {w}.$ Using the definition of $\\tilde{p}(\\mathbf {z}_{1:t}, t;\\mathbf {w}_k)$ we have $\\mathbb {E}_{\\tilde{p}(\\mathbf {z}_{1:t}, t;\\mathbf {w}_k)} \\bigg [ \\sum _{\\tau =1}^t \\log \\pi (\\mathbf {a}_{\\tau }|\\mathbf {s}_{\\tau }; \\mathbf {w}) \\bigg ] = \\sum _{t=1}^{\\infty } \\sum _{\\tau =1}^t \\mathbb {E}_{p(\\mathbf {z}_{\\tau }, \\mathbf {z}_t;\\mathbf {w}_k)} \\bigg [ \\gamma ^{t-1} R(\\mathbf {z}_t) \\log \\pi (\\mathbf {a}_{\\tau }|\\mathbf {s}_{\\tau }; \\mathbf {w}) \\bigg ],$ so that using the same manipulations as those used in section(REF ) of the supplementary material we have that the E-step is equivalent to maximising the function $\\mathcal {Q}(\\mathbf {w},\\mathbf {w}_k) = \\mathbb {E}_{p_{\\gamma }(\\mathbf {z};\\mathbf {w}_k) Q(\\mathbf {z};\\mathbf {w}_k)} \\bigg [ \\log \\pi (\\mathbf {a}|\\mathbf {s}; \\mathbf {w}) \\bigg ],$ w.r.t.", "to the first parameter, $\\mathbf {w}$ .", "This completes the derivation of the EM-algorithm for Markov Decision Processes with an infinite planning horizon and discounted rewards." ], [ "Affine Invariance of Approximate Newton Method", "To show that the use of $\\mathcal {H}_2(\\mathbf {w})$ in place of the true Hessian maintains the linear (affine) invariance of the Newton method we use the following formulae $\\nabla _{\\mathbf {w}} \\tilde{f}(\\mathbf {w}) = \\mathcal {T}^T \\nabla _{\\mathbf {v}} f(\\mathbf {v}), \\qquad \\qquad \\qquad \\nabla _{\\mathbf {w}} \\nabla ^T_{\\mathbf {w}} \\tilde{f}(\\mathbf {w}) = \\mathcal {T}^T \\nabla _{\\mathbf {v}} \\nabla _{\\mathbf {v}}^T f(\\mathbf {v}) \\mathcal {T},$ where $f$ is some twice differentiable function of $\\mathbf {w}$ , $\\tilde{f}(\\mathbf {w}) = f(\\mathcal {T}\\mathbf {w})$ and $\\mathbf {v} = \\mathcal {T} \\mathbf {w}$ .", "Using these formulae we have the following two identities $\\nabla _{\\mathbf {w}} \\log p(\\mathbf {a}|\\mathbf {s}; \\mathcal {T} \\mathbf {w}) &= \\mathcal {T}^T \\nabla _{\\mathbf {v}} \\log p(\\mathbf {a}|\\mathbf {s}; \\mathbf {v}), \\\\\\nabla _{\\mathbf {w}} \\nabla ^T_{\\mathbf {w}} \\log p(\\mathbf {a}|\\mathbf {s}; \\mathcal {T} \\mathbf {w}) &= \\mathcal {T}^T \\nabla _{\\mathbf {v}} \\nabla ^T_{\\mathbf {v}} \\log p(\\mathbf {a}|\\mathbf {s}; \\mathbf {v}) \\mathcal {T},$ which hold for each $(\\mathbf {s}, \\mathbf {a}) \\in \\mathcal {S} \\times \\mathcal {A}$ .", "Defining $\\tilde{U}(\\mathbf {w}) = U(\\mathcal {T} \\mathbf {w})$ , we have $\\nabla _{\\mathbf {w}} \\tilde{U}(\\mathbf {w}) = \\mathcal {T}^T U(\\mathbf {v})$ .", "Following calculations almost identical to those in section(REF ) it can be shown that $\\tilde{\\mathcal {H}}_2(\\mathbf {w})$ takes the form $\\tilde{\\mathcal {H}}_2(\\mathbf {w}) = \\mathbb {E}_{p_{\\gamma }(\\mathbf {z}; \\mathcal {T} \\mathbf {w}) Q(\\mathbf {z};\\mathcal {T} \\mathbf {w})} \\bigg [ \\nabla _{\\mathbf {w}} \\nabla ^T_{\\mathbf {w}} \\log p(\\mathbf {a}|\\mathbf {s}; \\mathcal {T} \\mathbf {w}) \\bigg ], \\\\$ which gives the following $\\tilde{\\mathcal {H}}_2(\\mathbf {w}) &= \\mathbb {E}_{p_{\\gamma }(\\mathbf {z}; \\mathcal {T} \\mathbf {w})Q(\\mathbf {z};\\mathcal {T} \\mathbf {w})} \\bigg [ \\nabla _{\\mathbf {w}} \\nabla ^T_{\\mathbf {w}} \\log p(\\mathbf {a}|\\mathbf {s}; \\mathcal {T} \\mathbf {w}) \\bigg ], \\\\&= \\mathbb {E}_{p_{\\gamma }(\\mathbf {z}; \\mathcal {T} \\mathbf {w})Q(\\mathbf {z};\\mathcal {T} \\mathbf {w})} \\bigg [ \\mathcal {T}^T \\nabla _{\\mathbf {v}} \\nabla ^T_{\\mathbf {v}} \\log p(\\mathbf {a}|\\mathbf {s}; \\mathbf {v}) \\mathcal {T} \\bigg ], \\\\&= \\mathcal {T}^T \\mathbb {E}_{p_{\\gamma }(\\mathbf {z}; \\mathcal {T} \\mathbf {w})Q(\\mathbf {z};\\mathcal {T} \\mathbf {w})} \\bigg [ \\nabla _{\\mathbf {v}} \\nabla ^T_{\\mathbf {v}} \\log p(\\mathbf {a}|\\mathbf {s}; \\mathbf {v}) \\bigg ] \\mathcal {T}, \\\\& = \\mathcal {T}^T \\mathcal {H}_2(\\mathbf {v}) \\mathcal {T}.$ Using these two expressions we have that the parameter updates, under the approximate Newton method, of the objective functions $U$ and $\\tilde{U}$ are related as follows $\\mathbf {v}_{\\textnormal {new}} &= \\mathbf {v} + \\alpha \\mathcal {H}_2(\\mathbf {v})^{-1} \\nabla _{\\mathbf {v}} U(\\mathbf {v}), \\\\&= \\mathcal {T} \\big ( \\mathbf {w} + \\alpha \\tilde{\\mathcal {H}}_2(\\mathbf {w})^{-1} \\nabla _{\\mathbf {v}} \\tilde{U}(\\mathbf {w}) \\big ),$ where $\\alpha $ is some step-size parameter.", "This shows that the approximate Newton method is affine invariant.", "The affine invariance of the diagonal approximate Newton method follows similarly." ], [ "Recurrent State Search Direction Evaluation for Approximate Newton Method", "In [31] a sampling algorithm was provided for estimating the gradient of an infinite horizon MDP with average rewards.", "This algorithm makes use of a recurrent state, which we denote by $\\mathbf {s}^*$ .", "In algorithm(REF ) we detail a straightforward extension of this algorithm to the estimation the approximate Hessian, $\\mathcal {H}_2(\\mathbf {w})$ , in this MDP framework.", "The analogous algorithm for the estimation of the diagonal matrix, $\\mathcal {D}_2(\\mathbf {w})$ , follows similarly.", "In algorithm(REF ) we make use of an eligibility trace for both the gradient and the approximate Hessian, which we denote by $\\mathbf {\\Phi }^1$ and $\\mathbf {\\Phi }^2$ respectively.", "The estimates (up to a positive scalar) of the gradient and the approximate Hessian are denoted by $\\mathbf {\\Delta }^1$ and $\\mathbf {\\Delta }^2$ respectively." ], [ "Tetris Experiment Specification", "[tb] Recurrent state sampling algorithm to estimate the search direction of the approximate Newton method when applied to a MDP with an infinite planning horizon with average rewards.", "Sample a state from the initial state distribution: $\\mathbf {s}_1 \\sim p_1(\\cdot ).$ $t= 1, ....., N$ , for some $N \\in \\mathbb {N}$ , Given the current state, sample an action from the policy: $\\mathbf {a}_t \\sim \\pi (\\cdot |\\mathbf {s}_t;\\mathbf {w}).$ $\\mathbf {s}_t \\ne \\mathbf {s}^*$ Update the eligibility traces: $\\mathbf {\\Phi }^1 \\leftarrow \\mathbf {\\Phi }^1 + \\nabla _{\\mathbf {w}} \\log \\pi (\\mathbf {a}_t|\\mathbf {s}_t;\\mathbf {w}) \\quad \\quad \\quad \\quad \\mathbf {\\Phi }^2 \\leftarrow \\mathbf {\\Phi }^2 + \\nabla _{\\mathbf {w}} \\nabla ^T_{\\mathbf {w}} \\log \\pi (\\mathbf {a}_t|\\mathbf {s}_t;\\mathbf {w})$ reset the eligibility traces: $\\mathbf {\\Phi }^1 = \\mathbf {0}, \\quad \\quad \\quad \\quad \\quad \\mathbf {\\Phi }^2 = \\mathbf {0}.$ Update the estimates of the gradient and the approximate Hessian: $\\mathbf {\\Delta }^1 \\leftarrow \\mathbf {\\Delta }^1 + R(\\mathbf {a}_t, \\mathbf {s}_t) \\mathbf {\\Phi }^1, \\quad \\quad \\quad \\quad \\mathbf {\\Delta }^2 \\leftarrow \\mathbf {\\Delta }^2 + R(\\mathbf {a}_t, \\mathbf {s}_t) \\mathbf {\\Phi }^2.$ Sample state from the transition dynamics: $\\mathbf {s}_{t+1} \\sim p(\\cdot |\\mathbf {a}_t, \\mathbf {s}_t).$ Return the estimated gradient and approximate Hessian, which up to a positive scaling are given by $\\mathbf {\\Delta }^1$ and $\\mathbf {\\Delta }^2$ respectively.", "In this section we give a detailed specification of the procedure used for each of the algorithms in the tetris experiment.", "The same general procedure was used for all the algorithms considered in the experiments.", "We modelled the environment through with an infinite planning horizon with average rewards.", "The reward at each time-point is equal to the number of lines deleted.", "We used a recurrent state formulation [31] of the gradient of the average reward framework to perform the gradient evaluation.", "We used analogous versions of this recurrent state formulation for natural gradient ascent, the diagonal approximate Newton method and the full approximate Newton method.", "See section(REF ) of the supplementary material for more details.", "As in [16] we used the sample trajectories obtained during the gradient evaluation to estimate the Fisher information matrix.", "We used the empty board as a recurrent state.", "A new game starts with an empty board and so this too is a recurrent state.", "Irrespective of the policy a game of tetris is guaranteed to terminate after a finite number of turns, see e.g.", "[7], and this guarantees the recurrence property of this state.", "During each training iteration the (approximation of the) search direction was obtained by sampling 1000 games, where these games were sampled using the current policy parameters.", "Given the current approximate search direction the following basic line search method was used to obtain a step-size: For every step-size in a given finite set of step-sizes sample a set number of games and then return the step-size with the maximal score over these games.", "In practice, in order to reduce the susceptibility to random noise, we used the same simulator seed for each possible step-size in the set.", "To avoid over-fitting a different simulator seed was used during each training iteration.", "In this line search procedure we sampled 1000 games for each of the possible step-sizes.", "The same set of step-sizes was used in all of the different training algorithms considered in section(), where we used the set $\\big \\lbrace 0.1, 0.5, 1.0, 2.0, 4.0, 8.0, 16.0, 32.0, 64.0, 128.0 \\big \\rbrace .$ To reduce the amount of noise in the results we used the same set of simulator seeds in the search direction evaluation for each of the algorithms considered in section().", "In particular, we generated a $n_{\\textnormal {experiments}} \\times n_{\\textnormal {iterations}}$ matrix of simulator seeds, where $n_{\\textnormal {experiments}}$ was the number of repetitions of the experiment and $n_{\\textnormal {iterations}}$ was the number of training iterations in each experiment.", "We then used this one matrix in all of the different training algorithms, where the element in the $j^{\\textnormal {th}}$ column and $i^{\\textnormal {th}}$ row corresponds to the simulator seed used in the $j^{\\textnormal {th}}$ training iteration of the $i^{\\textnormal {th}}$ experiment.", "In a similar manner the set of simulator seeds used for the line search procedure was the same for all of the different training algorithms.", "Finally, to make the line search consistent among all of the different training algorithms the search direction was normlised and the resulting unit vector was the vector used in the line search procedure." ], [ "Linear System Specification", "The $N$ -link rigid robot arm manipulator is a standard continuous MDP model, consisting of an end effector connected to an $N$ -linked rigid body [17].", "A typical continuous control problem for such systems is to apply appropriate torque forces to the joints of the manipulator so as to move the end effector into a desired position.", "The state of the system is given by $\\mathbf {q}$ , $\\dot{\\mathbf {q}}$ , $\\ddot{\\mathbf {q}} \\in \\mathbb {R}^N$ , where $\\mathbf {q}$ , $\\dot{\\mathbf {q}}$ and $\\ddot{\\mathbf {q}}$ denote the angles, velocities and accelerations of the joints respectively, while the control variables are the torques applied to the joints $\\mathbf {\\tau } \\in \\mathbb {R}^N$ .", "The nonlinear state equations of the system are given by, see e.g.", "[17], $M(\\mathbf {q})\\ddot{\\mathbf {q}} + C(\\dot{\\mathbf {q}}, \\mathbf {q}) \\dot{\\mathbf {q}} + g(\\mathbf {q}) = \\mathbf {\\tau }$ where $M(\\mathbf {q})$ is the inertia matrix, $C(\\dot{\\mathbf {q}}, \\mathbf {q})$ denotes the Coriolis and centripetal forces and $g(\\mathbf {q})$ is the gravitational force.", "While this system is highly nonlinear it is possible to define an appropriate control function $\\hat{\\mathbf {\\tau }}(\\mathbf {q}, \\dot{\\mathbf {q}})$ that results in linear dynamics in a different state-action space.", "This process is called feedback linearisation, see e.g.", "[17], and in the case of an $N$ -link rigid manipulator recasts the torque action space into the acceleration action space.", "This means that the state of the system is now given by $\\mathbf {q}$ and $\\dot{\\mathbf {q}}$ , while the control is $\\mathbf {a} = \\ddot{\\mathbf {q}}$ .", "Ordinarily in such problems the reward would be a function of the generalised co-ordinates of the end effector, which results in a non-trivial reward function in terms of $\\mathbf {q}$ , $\\dot{\\mathbf {q}}$ and $\\ddot{\\mathbf {q}}$ .", "While this reward function can be modelled as a mixture of Gaussians for simplicity we consider the simpler problem where the reward is a function of $\\mathbf {q}$ , $\\dot{\\mathbf {q}}$ and $\\ddot{\\mathbf {q}}$ directly.", "In the experiments we considered a 3-link rigid manipulator, which results in a 9-dimensional state-action space and a 22-dimensional policy.", "In the experiment we discretised the continuous time dynamics into time-steps of $\\Delta _t = 0.1$ and considered a finite planning horizon of $H = 100$ .", "The mean of the initial state distribution was set to zero.", "The elements of the policy parameters $m$ and $\\pi _\\sigma $ were initialised randomly from the interval $[-2,2]$ and $[1,2]$ respectively.", "The matrix $K$ was initialised to be zero on inter-link entries, while intra-link entries were initialised using rejection sampling.", "We sampled the parameters for each link independently from the set $[-400, 40] \\times [-50, 10]$ and rejected the sample if the corresponding link was unstable.", "In the reward function the desired angle of each joint was randomly sampled from the interval $[\\pi /4, 3\\pi /4]$ .", "The covariance matrices of the initial state distribution and state transition dynamics were set to diagonals, where the diagonal elements were initialised randomly from the interval $[0,0.05]$ .", "The covariance matrix of the reward function was set to be a diagonal with all entries equal to $0.1$ .", "We used the minFuncThis software library is freely available at http://www.di.ens.fr/~mschmidt/Software/minFunc.html.", "optimisation library in all of the gradient-based algorithms.", "We found that both the line search algorithm and the step-size initialisation had an effect on the performance of all the algorithms.", "We therefore tried various combinations of these settings for each algorithm and selected the one that gave the best performance.", "We tried bracketing line search algorithms with: step-size halving; quadratic/cubic interpolation from new function values; cubic interpolation from new function and gradient values; step-size doubling and bisection; cubic interpolation/extrapolation with function and gradient values.", "We tried the following step-size initialisations: quadratic initialization using previous function value and new function value/gradient; twice the previous step-size.", "To handle situations where the initial policy parameterisation was in a `flat' area of the parameter space far from any optima we set the function and point toleration of minFunc to zero for all algorithms.", "To handle the large number of training iterations required by steepest gradient ascent we increased the maximum number of function evaluations and training iterations to $10,000$ ." ], [ "Non-Linear System Specification", "In section we detail the implementation of the experiment on the synthetic two-dimensional non-linear MDP considered in [29].", "The state-space of the problem is two-dimensional, $\\mathbf {s} = (s^1, s^2)$ , where $s^1$ is the agent's position and $s^2$ is the agent's velocity.", "The control is one-dimensional and the dynamics of the system is given as follows $s^1_{t+1} &= s^1_t + \\frac{1}{1 + e^{-u_t}} - 0.5 + \\kappa , \\\\s^2_{t+1} &= s^2_t - 0.1 s^1_{t+1} + \\kappa ,$ where $\\kappa $ is a zero-mean Gaussian random variable with standard deviation $\\sigma _{\\kappa } = 0.02$ .", "The agent starts in the state $\\mathbf {s} = (0,1)$ , with the addition of Gaussian noise with standard deviation $0.001$ , and the objective is to transport the agent to the state $(0,0)$ .", "We use the same policy as in [29], which is given by $a_t = (\\mathbf {w} + \\mathbf {\\epsilon }_t)^T \\mathbf {s}_t$ , where $\\mathbf {w}$ are the control parameters and $\\mathbf {\\epsilon }_t \\sim \\mathcal {N}(\\mathbf {\\epsilon }_t; \\mathbf {0}, \\sigma _{\\epsilon }^2 I)$ .", "The objective function is non-trivial for $\\mathbf {w} \\in [0, 60] \\times [-8, 0]$ .", "In the experiment the initial control parameters were sampled from the region $\\mathbf {w}_0 \\in [0, 60] \\times [-8, 0]$ .", "We considered a finite planning horizon with a planning horizon of $H = 80$ .", "We used forward sampling to perform the inference, where in all algorithms 50 trajectories were sampled during each training iteration.", "We also detail the procedure used for the step-size tuning for the approximate Newton method and natural gradient ascent.", "The tuning of the step-size sequence in steepest gradient ascent was similar in nature to natural gradient ascent and so is omitted from the discussion.", "Using the intuition that the approximate Newton method has a natural step-size of one, which corresponds to an EM-step in this problem because the $\\log $ -policy is quadratic in the control parameters, we considered step-size sequences of the form $\\alpha _k = (1 - \\frac{k}{N})\\alpha + \\frac{k}{N}$ , where $N$ is the total number of training iterations considered and $\\alpha \\in \\mathbb {R}^{+}$ .", "In the experiment we considered the values $\\alpha = 1$ , 6, 12, 18 and 24.", "The intuition used in this selection was that, provided that the steps were not so large as to cause overshooting in the parameter space, larger steps will increase performance.", "In steepest and natural gradient ascent we used step-size sequences that satisfied the Robbins-Munro conditions.", "In both of these methods it was necessary to obtain a gauge of a reasonable scale of a good step-size sequence.", "For this reason in the experiment we considered step-size sequences of the form $\\alpha _{k} = \\frac{\\alpha }{\\sqrt{k}}$ with $\\alpha = 0.0001$ , $0.001$ , $0.01$ , $0.1$ , 1, 2 and 4.", "It was found that the sequence $\\alpha = 18$ gave the best results for the approximate Newton method, while the sequence $\\alpha = 0.001$ gave the best results for natural gradient ascent." ] ]