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In this paper, we introduce a weighted $\ell_2/\ell_1$ minimization to recover block sparse signals with arbitrary prior support information. When partial prior support information is available, a sufficient condition based on the high order block RIP is derived to guarantee stable and robust recovery of block sparse signals via the weighted $\ell_2/\ell_1$ minimization. We then show if the accuracy of arbitrary prior block support estimate is at least $50\%$, the sufficient recovery condition by the weighted $\ell_2/\ell_{1}$ minimization is weaker than that by the $\ell_2/\ell_{1}$ minimization, and the weighted $\ell_2/\ell_{1}$ minimization provides better upper bounds on the recovery error in terms of the measurement noise and the compressibility of the signal. Moreover, we illustrate the advantages of the weighted $\ell_2/\ell_1$ minimization approach in the recovery performance of block sparse signals under uniform and nonuniform prior information by extensive numerical experim
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We start from a variational model for nematic elastomers that involves two energies: mechanical and nematic. The first one consists of a nonlinear elastic energy which is influenced by the orientation of the molecules of the nematic elastomer. The nematic energy is an OseenFrank energy in the deformed configuration. The constraint of the positivity of the determinant of the deformation gradient is imposed. The functionals are not assumed to have the usual polyconvexity or quasiconvexity assumptions to be lower semicontinuous. We instead compute its relaxation, that is, the lower semicontinuous envelope, which turns out to be the quasiconvexification of the mechanical term plus the tangential quasiconvexification of the nematic term. The main assumptions are that the quasiconvexification of the mechanical term is polyconvex and that the deformation is in the Sobolev space $W^{1,p}$ (with $p>n1$ and $n$ the dimension of the space) and does not present cavitation.
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We propose here a new discretization method for a class continuum gauge theories which action functionnals are polynomials of the curvature. Based on the notion of holonomy, this discretization procedure appears gaugeinvariant for discretized analogs of YangMills theories, and hence gaugefixing is fully rigorous for these discretized action functionnals. Heuristic parts are forwarded to the quantization procedure via Feynman integrals and the meaning of the heuristic infinite dimensional Lebesgue integral is questionned.
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The CartanHartogs domains are defined as a class of Hartogs type domains over irreducible bounded symmetric domains. For a CartanHartogs domain $\Omega^{B}(\mu)$ endowed with the natural K\"{a}hler metric $g(\mu),$ Zedda conjectured that the coefficient $a_2$ of the Rawnsley's $\varepsilon$function expansion for the CartanHartogs domain $(\Omega^{B}(\mu), g(\mu))$ is constant on $\Omega^{B}(\mu)$ if and only if $(\Omega^{B}(\mu), g(\mu))$ is biholomorphically isometric to the complex hyperbolic space. In this paper, following Zedda's argument, we give a geometric proof of the Zedda's conjecture by computing the curvature tensors of the CartanHartogs domain $(\Omega^{B}(\mu), g(\mu))$.
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The quantum Renyi relative entropies play a prominent role in quantum information theory, finding applications in characterizing error exponents and strong converse exponents for quantum hypothesis testing and quantum communication theory. On a different thread, quantum Gaussian states have been intensely investigated theoretically, motivated by the fact that they are more readily accessible in the laboratory than are other, more exotic quantum states. In this paper, we derive formulas for the quantum Renyi relative entropies of quantum Gaussian states. We consider both the traditional (Petz) Renyi relative entropy as well as the more recent sandwiched Renyi relative entropy, finding formulas that are expressed solely in terms of the mean vectors and covariance matrices of the underlying quantum Gaussian states. Our development handles the hitherto elusive case for the PetzRenyi relative entropy when the Renyi parameter is larger than one. Finally, we also derive a formula for the ma
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We will give general sufficient conditions under which a controller achieves robust regulation for a boundary control and observation system. Utilizing these conditions we construct a minimal order robust controller for an arbitrary order impedance passive linear portHamiltonian system. The theoretical results are illustrated with a numerical example where we implement a controller for a onedimensional EulerBernoulli beam with boundary controls and boundary observations.
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We derive a description of the family of canonical selfadjoint extensions of the operator of multiplication in a de Branges space in terms of singular rankone perturbations using distinguished elements from the set of functions associated with a de Branges space. The scale of rigged Hilbert spaces associated with this construction is also studied from the viewpoint of de Branges spaces.
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In the excitonpolariton system, a linear dispersive photon field is coupled to a nonlinear exciton field. Shorttime analysis of the lossless system shows that, when the photon field is excited, the time required for that field to exhibit nonlinear effects is longer than the time required for the nonlinear Schr\"odinger equation, in which the photon field itself is nonlinear. When the initial condition is scaled by $\epsilon^\alpha$, it is found that the relative error committed by omitting the nonlinear term in the excitonpolariton system remains within $\epsilon$ for all times up to $t=C\epsilon^\beta$, where $\beta=(1\alpha(p1))/(p+2)$. This is in contrast to $\beta=1\alpha(p1)$ for the nonlinear Schr\"odinger equation.
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We present the concept of Simultaneous Lightwave Information and Power Transfer (SLIPT) for indoor InternetofThings (IoT) applications. Specifically, we propose novel and fundamental SLIPT strategies, which can be implemented through Visible Light or Infrared communication systems, equipped with a simple solar panelbased receiver. These strategies are performed at the transmitter or at the receiver, or at both sides, named Adjusting transmission, Adjusting reception and Coordinated adjustment of transmission and reception, correspondingly. Furthermore, we deal with the fundamental tradeoff between harvested energy and qualityofservice (QoS), by maximizing the harvested energy, while achieving the required user's QoS. To this end, two optimization problems are formulated and optimally solved. Computer simulations validate the optimum solutions and reveal that the proposed strategies considerably increase the harvested energy, compared to SLIPT with fixed policies.
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We describe a "slow" version of the hierarchy of uniform reflection principles over Peano Arithmetic ($\mathbf{PA}$). These principles are unprovable in Peano Arithmetic (even when extended by usual reflection principles of lower complexity) and introduce a new provably total function. At the same time the consistency of $\mathbf{PA}$ plus slow reflection is provable in $\mathbf{PA}+\operatorname{Con}(\mathbf{PA})$. We deduce a conjecture of S.D. Friedman, Rathjen and Weiermann: Transfinite iterations of slow consistency generate a hierarchy of precisely $\varepsilon_0$ stages between $\mathbf{PA}$ and $\mathbf{PA}+\operatorname{Con}(\mathbf{PA})$ (where $\operatorname{Con}(\mathbf{PA})$ refers to the usual consistency statement).
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Answering a question of P\'alfy and Pyber, we first prove the following extension of the k(GV)Problem: Let G be a finite group and A\le Aut(G) such that (G,A)=1. Then the number of conjugacy classes of the semidirect product GA is at most G. As a consequence we verify Brauer's k(B)Conjecture for piblocks of piseparable groups which was proposed by Y. Liu. On the other hand, we construct a counterexample to a version of Olsson's Conjecture for piblocks which was also introduced by Liu.
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We consider the problem of managing a hydroelectric power plant system. The system consists of N hydropower dams, which all have some maximum production capacity. The inflow to the system is some stochastic process, representing the precipitation to each dam. The manager can control how much water to release from each dam at each time. She would like to choose this in a way which maximizes the total revenue from the initial time 0 to some terminal time T. The total revenue of the hydropower dam system depends on the price of electricity, which is also a stochastic process. The manager must take this price process into account when controlling the draining process. However, we assume that the manager only has partial information of how the price process is formed. She can observe the price, but not the underlying processes determining it. By using the conjugate duality framework of Rockafellar, we derive a dual problem to the problem of the manager. This dual problem turns out to be sim
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In this article we study Whitney (B) regular stratified spaces with the action of a compact Lie group $G$ which preserves the strata. We prove an equivariant submersion theorem and use it to show that such a $G$stratified space carries a system of $G$equivariant control data. As an application, we show that if $A \subset X$ is a closed $G$stratified subspace which is a union of strata of $X$, then the inclusion $i : A \hookrightarrow X$ is a $G$equivariant cofibration. In particular, this theorem applies whenever $X$ is a $G$invariant analytic subspace of an analytic $G$manifold $M$ and $A \hookrightarrow X$ is a closed $G$invariant analytic subspace of $X$.
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This paper studies cohomogeneity one Ricci solitons. If the isotropy representation of the principal orbit $G/K$ consists of two inequivalent $Ad_K$invariant irreducible summands, the existence of parameter families of nonhomothetic complete steady and expanding Ricci solitons on nontrivial bundles is shown. These examples were detected numerically by BuzanoDancerGallaugherWang. The analysis of the corresponding Ricci flat trajectories is used to reconstruct Einstein metrics of positive scalar curvature due to B\"ohm. The techniques also yield unifying proofs for the existence of $m$quasiEinstein metrics.
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A family of lines through the origin in Euclidean space is called equiangular if any pair of lines defines the same angle. The problem of estimating the maximum cardinality of such a family in $\mathbb{R}^n$ was extensively studied for the last 70 years. Motivated by a question of Lemmens and Seidel from 1973, in this paper we prove that for every fixed angle $\theta$ and sufficiently large $n$ there are at most $2n2$ lines in $\mathbb{R}^n$ with common angle $\theta$. Moreover, this is achievable only for $\theta = \arccos(1/3)$. We also show that for any set of $k$ fixed angles, one can find at most $O(n^k)$ lines in $\mathbb{R}^n$ having these angles. This bound, conjectured by Bukh, substantially improves the estimate of Delsarte, Goethals and Seidel from 1975. Various extensions of these results to the more general setting of spherical codes will be discussed as well.
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We compute the singleinterval Renyi entropy (replica partition function) for free fermions in 1+1d at finite temperature and finite spatial size by two methods: (i) using the highergenus partition function on the replica Riemann surface, and (ii) using twist operators on the torus. We compare the two answers for a restricted set of spin structures, leading to a nontrivial proposed equivalence between highergenus Siegel $\Theta$functions and Jacobi $\theta$functions. We exhibit this proposal and provide substantial evidence for it. The resulting expressions can be elegantly written in terms of Jacobi forms. Thereafter we argue that the correct Renyi entropy for modularinvariant freefermion theories, such as the Ising model and the Dirac CFT, is given by the highergenus computation summed over all spin structures. The result satisfies the physical checks of modular covariance, the thermal entropy relation, and BoseFermi equivalence.
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We show that the theory of derivators (or, more generally, of fibered multiderivators) on all small categories is equivalent to this theory on partially ordered sets, in the following sense: Every derivator (more generally, every fibered multiderivator) defined on partially ordered sets has an enlargement to all small categories that is unique up to equivalence of derivators. Furthermore, extending a theorem of Cisinski, we show that every bifibration of multimodel categories (basically a collection of model categories, and Quillen adjunctions in several variables between them) gives rise to a left and right fibered multiderivator on all small categories.
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We prove local existence of solutions to the Einsteinnull dust system under polarized $\mathbb U(1)$ symmetry in an elliptic gauge. Using in particular the previous work of the first author on the constraint equations, we show that one can identify freely prescribable data, solve the constraints equations, and construct a unique local in time solution in an elliptic gauge. Our main motivation for this work, in addition to merely constructing solutions in an elliptic gauge, is to provide a setup for our companion paper in which we study high frequency backreaction for the Einstein equations. In that work, the elliptic gauge we consider here plays a crucial role to handle high frequency terms in the equations. The main technical difficulty in the present paper, in view of the application in our companion paper, is that we need to build a framework consistent with the solution being high frequency, and therefore having large higher order norms. This difficulty is handled by exploiting a
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Based on the method of hydrodynamic projections we derive a concise formula for the Drude weight of the repulsive LiebLiniger $\delta$Bose gas. Our formula contains only quantities which are obtainable from the thermodynamic Bethe ansatz. The Drude weight is an infinitedimensional matrix, or bilinear functional: it is bilinear in the currents, and each current may refer to a general linear combination of the conserved charges of the model. As a byproduct we obtain the dynamical twopoint correlation functions involving charge and current densities at small wavelengths and long times, and in addition the scaled covariance matrix of charge transfer. We expect that our formulas extend to other integrable quantum models.
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Let $A$ and $B$ be unital $C^*$algebras and let $H$ be a finite dimensional $C^*$Hopf algebra. Let $H^0$ be its dual $C^*$Hopf algebra. Let $(\rho, u)$ and $(\sigma, v)$ be twisted coactions of $H^0$ on $A$ and $B$, respectively. In this paper, we shall show the following theorem: We suppose that the unital inclusions $A\subset A\rtimes_{\rho, u}H$ and $B\subset B\rtimes_{\sigma, v}H$ are strongly Morita equivalent. If $A'\cap (A\rtimes_{\rho, u}H)=\BC1$, then there is a $C^*$Hopf algebra automorphism $\lambda^0$ of $H^0$ such that the twisted coaction $(\rho, u)$ is strongly Morita equivalent to the twisted coaction $((\id_B \otimes\lambda^0 )\circ\sigma \, , \, (\id_B \otimes\lambda^0 \otimes\lambda^0 )(v))$ induced by $(\sigma, v)$ and $\lambda^0$.
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Let $F_n$ denote the distribution function of the normalized sum $Z_n = (X_1 + \dots + X_n)/\sigma\sqrt{n}$ of i.i.d. random variables with finite fourth absolute moment. In this paper, polynomial rates of convergence of $F_n$ to the normal law with respect to the Kolmogorov distance, as well as polynomial approximations of $F_n$ by the Edgeworth corrections (modulo logarithmically growing factors in $n$) are given in terms of the characteristic function of $X_1$. Particular cases of the problem are discussed in connection with Diophantine approximations.
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Finite element approximations of minimal surface are not always precise. They can even sometimes completely collapse. In this paper, we provide a simple and inexpensive method, in terms of computational cost, to improve finite element approximations of minimal surfaces by local boundary mesh refinements. By highlighting the fact that a collapse is simply the limit case of a locally bad approximation, we show that our method can also be used to avoid the collapse of finite element approximations. We also extend the study of such approximations to partially free boundary problems and give a theorem for their convergence. Numerical examples showing improvements induced by the method are given throughout the paper.
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A finite element method for the evolution of a twophase membrane in a sharp interface formulation is introduced. The evolution equations are given as an $L^2$gradient flow of an energy involving an elastic bending energy and a line energy. In the two phases Helfrichtype evolution equations are prescribed, and on the interface, an evolving curve on an evolving surface, highly nonlinear boundary conditions have to hold. Here we consider both $C^0$ and $C^1$matching conditions for the surface at the interface. A new weak formulation is introduced, allowing for a stable semidiscrete parametric finite element approximation of the governing equations. In addition, we show existence and uniqueness for a fully discrete version of the scheme. Numerical simulations demonstrate that the approach can deal with a multitude of geometries. In particular, the paper shows the first computations based on a sharp interface description, which are not restricted to the axisymmetric case.
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We perform an asymptotic study of the performance of filter bank multicarrier (FBMC) in the context of massive multiinput multioutput (MIMO). We show that the effects of channel distortions, i.e., intersymbol interference and intercarrier interference, do not vanish as the base station (BS) array size increases. As a result, the signaltointerferenceplusnoise ratio (SINR) cannot grow unboundedly by increasing the number of BS antennas, and is upper bounded by a certain deterministic value. We show that this phenomenon is a result of the correlation between the multiantenna combining tap values and the channel impulse responses between the mobile terminals and the BS antennas. To resolve this problem, we introduce an efficient equalization method that removes this correlation, enabling us to achieve arbitrarily large SINR values by increasing the number of BS antennas. We perform a thorough analysis of the proposed system and find analytical expressions for both equalizer coeffici
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"Bounds on information combining" are entropic inequalities that determine how the information (entropy) of a set of random variables can change when these are combined in certain prescribed ways. Such bounds play an important role in classical information theory, particularly in coding and Shannon theory; entropy power inequalities are special instances of them. The arguably most elementary kind of information combining is the addition of two binary random variables (a CNOT gate), and the resulting quantities play an important role in Belief propagation and Polar coding. We investigate this problem in the setting where quantum side information is available, which has been recognized as a hard setting for entropy power inequalities. Our main technical result is a nontrivial, and close to optimal, lower bound on the combined entropy, which can be seen as an almost optimal "quantum Mrs. Gerber's Lemma". Our proof uses three main ingredients: (1) a new bound on the concavity of von Neuma
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We have recently presented an extension of the standard variational calculus to include the presence of deformed derivatives in the Lagrangian of a system of particles and in the Lagrangian density of fieldtheoretic models. Classical EulerLagrange equations and the Hamiltonian formalism have been reassessed in this approach. Whenever applied to a number of physical systems, the resulting dynamical equations come out to be the correct ones found in the literature, specially with massdependent and with nonlinear equations for classical and quantummechanical systems. In the present contribution, we extend the variational approach with the intervalar form of deformed derivatives to study higherorder dissipative systems, with application to concrete situations, such as an accelerated point charge  this is the problem of the AbrahamLorentzDirac force  to stochastic dynamics like the Langevin, the advectionconvectionreaction and FokkerPlanck equations, Kortewegde Vries equation
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We prove a central limit theorem for the linear statistics of onedimensional loggases, or $\beta$ensembles. We use a method based on a change of variables which allows to treat fairly general situations, including multicut and, for the first time, critical cases, and generalizes the previously known results of Johansson, BorotGuionnet and Shcherbina. In the onecut regular case, our approach also allows to retrieve a rate of convergence as well as previously known expansions of the free energy to arbitrary order.
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In a recent work, the degenerate Stirling polynomials of the second kind were studied by T. Kim. In this paper, we investigate the extended degenerate Stirling numbers of the second kind and the extended degenerate Bell polynomials associated with them. As results, we give some expressions, identities and properties about the extended degener ate Stirling numbers of the second kind and the extended degenerate Bell polynomials.
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If D is a category and k is a commutative ring, the functors from D to kMod can be thought of as representations of D. By definition, D is dimension zero over k if its finitely generated representations have finite length. We characterize categories of dimension zero in terms of the existence of a "homological modulus" (Definition 1.4) which is combinatorial and linearalgebraic in nature.
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Since its very beginnings, topology has forged strong links with physics and the last Nobel prize in physics, awarded in 2016 to Thouless, Haldane and Kosterlitz " for theoretical discoveries of topological phase transitions and topological phases of matter", confirmed that these connections have been maintained up to contemporary physics. To give some (very) selected illustrations of what is, and still will be, a cross fertilization between topology and physics, hydrodynamics provides a natural domain through the common theme offered by the notion of vortex, relevant both in classical (\S 2) and in quantum fluids (\S 3). Before getting into the details, I will sketch in \S 1 a general perspective from which this intertwining between topology and physics can be appreciated: the old dichotomy between discreteness and continuity, first dealing with antithetic thesis, eventually appears to be made of two complementary sides of a single coin.
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This paper concerns the problem of lifting a KZ doctrine P to the 2category of pseudo Talgebras for some pseudomonad T. Here we show that this problem is equivalent to giving a pseudodistributive law (meaning that the lifted pseudomonad is automatically KZ), and that such distributive laws may be simply described algebraically and are essentially unique (as known to be the case in the (co)KZ over KZ setting). Moreover, we give a simple description of these distributive laws using Bunge and Funk's notion of admissible morphisms for a KZ doctrine (the principal goal of this paper). We then go on to show that the 2category of KZ doctrines on a 2category is biequivalent to a poset. We will also discuss here the case of lifting a locally fully faithful KZ doctrine, which we noted earlier enjoys most of the axioms of a Yoneda structure, and show that an oplaxlax bijection is exhibited on the lifted 'Yoneda structure' similar to Kelly's doctrinal adjunction. We also briefly discuss how
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The paper presents a distributed model predictive control (DMPC) scheme for continuoustime nonlinear systems based on the alternating direction method of multipliers (ADMM). A stopping criterion in the ADMM algorithm limits the iterations and therefore the required communication effort during the distributed MPC solution at the expense of a suboptimal solution. Stability results are presented for the suboptimal DMPC scheme under two different ADMM convergence assumptions. In particular, it is shown that the required iterations in each ADMM step are bounded, which is also confirmed in simulation studies.
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For a positive integer $n$ let $\mathfrak{P}_n=\prod_{s_p(n)\ge p} p,$ where $p$ runs over all primes and $s_p(n)$ is the sum of the base $p$ digits of $n$. For all $n$ we prove that $\mathfrak{P}_n$ is divisible by all "small" primes with at most one exception. We also show that $\mathfrak{P}_n$ is large, has many prime factors exceeding $\sqrt{n}$, with the largest one exceeding $n^{20/37}$. We establish Kellner's conjecture, which says that the number of prime factors exceeding $\sqrt{n}$ grows asymptotically as $\kappa \sqrt{n}/\log n$ for some constant $\kappa$ with $\kappa=2$. Further, we compare the sizes of $\mathfrak{P}_n$ and $\mathfrak{P}_{n+1}$, leading to the somewhat surprising conclusion that although $\mathfrak{P}_n$ tends to infinity with $n$, the inequality $\mathfrak{P}_n>\mathfrak{P}_{n+1}$ is more frequent than its reverse.
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We establish counting formulas and bijections for deformations of the braid arrangement. Precisely, we consider real hyperplane arrangements such that all the hyperplanes are of the form $x\_ix\_j=s$ for some integer $s$. Classical examples include the braid, Catalan, Shi, semiorder and Linial arrangements, as well as graphical arrangements. We express the number of regions of any such arrangement as a signed count of decorated plane trees. The characteristic and coboundary polynomials of these arrangements also have simple expressions in terms of these trees. We then focus on certain "wellbehaved" deformations of the braid arrangement that we call transitive. This includes the Catalan, Shi, semiorder and Linial arrangements, as well as many other arrangements appearing in the literature. For any transitive deformation of the braid arrangement we establish a simple bijection between regions of the arrangement and a set of plane trees defined by local conditions. This answers a questi
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Time dependent quantum systems have become indispensable in science and its applications, particularly at the atomic and molecular levels. Here, we discuss the approximation of closed time dependent quantum systems on bounded domains, via iterative methods in Sobolev space based upon evolution operators. Recently, existence and uniqueness of weak solutions were demonstrated by a contractive fixed point mapping defined by the evolution operators. Convergent successive approximation is then guaranteed. This article uses the same mapping to define quadratically convergent Newton and approximate Newton methods. Estimates for the constants used in the convergence estimates are provided. The evolution operators are ideally suited to serve as the framework for this operator approximation theory, since the Hamiltonian is time dependent. In addition, the hypotheses required to guarantee quadratic convergence of the Newton iteration build naturally upon the hypotheses used for the existence/uniq
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Slotted ALOHA (SA) algorithms with Successive Interference Cancellation (SIC) decoding have received significant attention lately due to their ability to dramatically increase the throughput of traditional SA. Motivated by increased density of cellular radio access networks due to the introduction of small cells, and dramatic increase of user density in MachinetoMachine (M2M) communications, SA algorithms with SIC operating cooperatively in multi base station (BS) scenario are recently considered. In this paper, we generalize our previous work on Slotted ALOHA with multipleBS (SAMBS) by considering users that use directional antennas. In particular, we focus on a simple randomized beamforming strategy where, for every packet transmission, a user orients its main beam in a randomly selected direction. We are interested in the total achievable system throughput for two decoding scenarios: i) noncooperative scenario in which traditional SA operates at each BS independently, and ii) c
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Let $F_p$ be the prime field with $p$ elements. We derive the homogeneous weight on the Frobenius matrix ring $M_2(F_p)$ in terms of the generating character. We also give a generalization of the Lee weight on the finite chain ring $F_{p^2}+uF_{p^2}$ where $u^2=0$. A noncommutative ring, denoted by $\mathcal{F}_{p^2}+\mathbf{v}_p \mathcal{F}_{p^2}$, $\mathbf{v}_p$ an involution in $M_2(F_p)$, that is isomorphic to $M_2(F_p)$ and is a left $F_{p^2}$vector space, is constructed through a unital embedding $\tau$ from $F_{p^2}$ to $M_2(F_p)$. The elements of $\mathcal{F}_{p^2}$ come from $M_2(F_p)$ such that $\tau(F_{p^2})=\mathcal{F}_{p^2}$. The irreducible polynomial $f(x)=x^2+x+(p1) \in F_p[x]$ required in $\tau$ restricts our study of cyclic codes over $M_2(F_p)$ endowed with the Bachoc weight to the case $p\equiv$ $2$ or $3$ mod $5$. The images of these codes via a left $F_p$module isometry are additive cyclic codes over $F_{p^2}+uF_{p^2}$ endowed with the Lee weight. New examples
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We prove the existence of solutions of the mass critical generalized Kortewegde Vries equation $\partial_t u + \partial_x(\partial_{xx} u + u^5) = 0$ containing an arbitrary number $K\geq 2$ of blow up bubbles, for any choice of sign and scaling parameters: for any $\ell_1>\ell_2>\cdots>\ell_K>0$ and $\epsilon_1,\ldots,\epsilon_K\in\{\pm1\}$, there exists an $H^1$ solution $u$ of the equation such that \[ u(t)  \sum_{k=1}^K \frac {\epsilon_k}{\lambda_k^\frac12(t)} Q\left( \frac {\cdot  x_k(t)}{\lambda_k(t)} \right) \longrightarrow 0 \quad\mbox{ in }\ H^1 \mbox{ as }\ t\downarrow 0, \] with $\lambda_k(t)\sim \ell_k t$ and $x_k(t)\sim \ell_k^{2}t^{1}$ as $t\downarrow 0$. The construction uses and extends techniques developed mainly by Martel, Merle and Rapha\"el. Due to strong interactions between the bubbles, it also relies decisively on the sharp properties of the minimal mass blow up solution (single bubble case) proved by the authors in arXiv:1602.03519.
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In this paper, we study moduli spaces of representations of certain quivers with relations. For quivers without relations and other categories of homological dimension one, a lot of information is known about the cohomology of their moduli spaces of objects. On the other hand, categories of higher homological dimension remain more mysterious from this point of view, with few general methods. In this paper, we will see how some of the methods used to study quivers can be extended to work for representations of any (noncommutative) monomial algebra with relations of length two. In particular, we will give an algorithm to calculate in many cases the classes of these moduli spaces in the Grothendieck ring of varieties.
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Assuming three strongly compact cardinals, it is consistent that \[ \aleph_1 < \mathrm{add}(\mathrm{null}) < \mathrm{cov}(\mathrm{null}) < \mathfrak{b} < \mathfrak{d} < \mathrm{non}(\mathrm{null}) < \mathrm{cof}(\mathrm{null}) < 2^{\aleph_0}.\] Under the same assumption, it is consistent that \[ \aleph_1 < \mathrm{add}(\mathrm{null}) < \mathrm{cov}(\mathrm{null}) < \mathrm{non}(\mathrm{meager}) < \mathrm{cov}(\mathrm{meager}) < \mathrm{non}(\mathrm{null}) < \mathrm{cof}(\mathrm{null}) < 2^{\aleph_0}.\]
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Solomon showed that the Poincar\'e polynomial of a Coxeter group $W$ satisfies a product decomposition depending on the exponents of $W$. This polynomial coincides with the rankgenerating function of the poset of regions of the underlying Coxeter arrangement. In this note we determine all instances when the analogous factorization property of the rankgenerating function of the poset of regions holds for a restriction of a Coxeter arrangement. It turns out that this is always the case with the exception of some instances in type $E_8$.
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In the past three decades, many theoretical measures of complexity have been proposed to help understand complex systems. In this work, for the first time, we place these measures on a level playing field, to explore the qualitative similarities and differences between them, and their shortcomings. Specifically, using the Boltzmann machine architecture (a fully connected recurrent neural network) with uniformly distributed weights as our model of study, we numerically measure how complexity changes as a function of network dynamics and network parameters. We apply an extension of one such informationtheoretic measure of complexity to understand incremental Hebbian learning in Hopfield networks, a fully recurrent architecture model of autoassociative memory. In the course of Hebbian learning, the total information flow reflects a natural upward trend in complexity as the network attempts to learn more and more patterns.
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In this work we develop arbitraryorder Discontinuous Skeletal Gradient Discretisations (DSGD) on general polytopal meshes. Discontinuous Skeletal refers to the fact that the globally coupled unknowns are broken polynomial on the mesh skeleton. The key ingredient is a highorder gradient reconstruction composed of two terms: (i) a consistent contribution obtained mimicking an integration by parts formula inside each element and (ii) a stabilising term for which sufficient design conditions are provided. An example of stabilisation that satisfies the design conditions is proposed based on a local lifting of highorder residuals on a RaviartThomasN\'ed\'elec subspace. We prove that the novel DSGDs satisfy coercivity, consistency, limitconformity, and compactness requirements that ensure convergence for a variety of elliptic and parabolic problems. Links with Hybrid HighOrder, nonconforming Mimetic Finite Difference and nonconforming Virtual Element methods are also studied. Numeric
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In classical coding theory, Gray isometries are usually defined as mappings between finite Frobenius rings, which include the ring $Z_m$ of integers modulo $m$, and the finite fields. In this paper, we derive an isometric mapping from $Z_8$ to $Z_4^2$ from the composition of the Gray isometries on $Z_8$ and on $Z_4^2$. The image under this composition of a $Z_8$linear block code of length $n$ with homogeneous distance $d$ is a (not necessarily linear) quaternary block code of length $2n$ with Lee distance $d$.
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We show that for almost any vector $v$ in $\mathbb{R}^n$, for any $\epsilon>0$ there exists $\delta>0$ such that the dimension of the set of vectors $w$ satisfying $\liminf_{k\to\infty} k^{1/n}<kvw> \ge \epsilon$ (where $<\cdot>$ denotes the distance from the nearest integer), is bounded above by $n\delta$. This result is obtained as a corollary of a discussion in homogeneous dynamics and the main tool in the proof is a relative version of the principle of uniqueness of measures with maximal entropy.
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This paper presents a novel method for assessing multifault diagnosability and detectability of non linear parametrized dynamical models. This method is based on computer algebra algorithms which return precomputed values of algebraic expressions characterizing the presence of some multifault(s). Estimations of these expressions, obtained from inputs and outputs measurements, permit then the detection and the isolation of multifaults possibly acting on the system. This method applied on a coupled watertank model attests the relevance of the suggested approach.
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Reliable estimation (or measurement) of vehicle states has always been an active topic of research in the automotive industry and academia. Among the vehicle states, vehicle speed has a priority due to its critical importance in traction and stability control. Moreover, the emergence of new generation of communication technologies has brought a new avenue to traditional studies on vehicle estimation and control. To this end, this paper introduces a set of distributed function calculation algorithms for vehicle networks, robust to communication failures. The introduced algorithms enable each vehicle to gather information from other vehicles in the network in a distributed manner. A procedure to use such a bank of information for a single vehicle to diagnose and correct a possible fault in its own speed estimation/measurement is discussed. The functionality and performance of the proposed algorithms are verified via illustrative examples and simulation results.
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The main goal of this paper is to study a stochastic game connected to a system of forward backward stochastic differential equations (FBSDEs) involving delay and socalled noisy memory. We derive suffcient and necessary maximum principles for a set of controls for the players to be a Nash equilibrium in such a game. Furthermore, we study a corresponding FBSDE involving Malliavin derivatives, which (to the best of our knowledge) is a kind of equation which has not been studied before. The maximum principles give conditions for determining the Nash equilibrium of the game. We use this to derive a closed form Nash equilibrium for a specifc model in economics where the players aim to maximize their consumption with respect recursive utility.
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We construct finitely generated simple algebras with prescribed growth types, which can be arbitrarily taken from a large variety of (superpolynomial) growth types. This (partially) answers a question raised by the author in a recent paper. Our construction goes through a construction of finitely generated justinfinite, primitive monomial algebras with prescribed growth type, from which we construct uniformly recurrent infinite words with subword complexity having the same growth type. We also discuss the connection between entropy of algebras and their homomorphic images, as well as the degrees of their generators of free subalgebras.
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We use Ricci flow to obtain a local biHolder correspondence between Ricci limit spaces in three dimensions and smooth manifolds. This is more than a complete resolution of the threedimensional case of the conjecture of AndersonCheegerColdingTian, describing how Ricci limit spaces in three dimensions must be homeomorphic to manifolds, and we obtain this in the most general, locally noncollapsed case. The proofs build on results and ideas from recent papers of Hochard and the current authors.
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Let $K$ be a finite extension of $\mathbf{Q}_p$. We use the theory of $(\varphi,\Gamma)$modules in the LubinTate setting to construct some corestrictioncompatible families of classes in the cohomology of $V$, for certain representations $V$ of $\mathrm{Gal}(\overline{\mathbf{Q}}_p/K)$. If in addition $V$ is crystalline, we describe these classes explicitly using BlochKato's exponential maps. This allows us to generalize PerrinRiou's period map to the LubinTate setting.
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A representation for a solution $u(\omega,x)$ of the equation $u"+q(x)u=\omega^2 u$, satisfying the initial conditions $u(\omega,0)=1$, $u'(\omega,0)=i\omega$ is derived in the form \[ u(\omega,x)=e^{i\omega x}\left( 1+\frac{u_1(x)}{\omega}+ \frac{u_2(x)}{\omega^2}\right) +\frac{e^{i\omega x}u_3(x)}{\omega^2}\frac{1}{\omega^2}\sum_{n=0}^{\infty} i^{n}\alpha_n(x)j_n(\omega x), \] where $u_m(x)$, $m=1,2,3$ are given in a closed form, $j_n$ stands for a spherical Bessel function of order $n$ and the coefficients $\alpha_n$ are calculated by a recurrent integration procedure. The following estimate is proved $\vert u(\omega,x) u_N(\omega,x)\vert \leq \frac{1}{\vert \omega \vert^2}\varepsilon_N(x)\sqrt{\frac{\sinh(2\mathop{\rm Im}\omega\,x)}{\mathop{\rm Im}\omega}}$ for any $\omega\in\mathbb{C}\backslash \{0\}$, where $u_N(\omega,x)$ is an approximate solution given by truncating the series in the representation for $u(\omega,x)$ and $\varepsilon_N(x)$ is a nonnegative function tending
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Let $X=(X_1,\ldots,X_n)$ be a vector of i.i.d. random variables where $X_i$'s take their values over $\mathbb{N}$. The purpose of this paper is to study the number of weakly increasing subsequences of $X$ of a given length $k$, and the number of all weakly increasing subsequences of $X$. For the former of these two, it is shown that a central limit theorem holds. Also, the first two moments of each of these two random variables are analyzed, their asymptotics are investigated, and results are related to the case of similar statistics in uniformly random permutations. We conclude the paper with applications on a similarity measure of Steele, and on increasing subsequences of riffle shuffles.
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Demand response aims to stimulate electricity consumers to modify their loads at critical time periods. In this paper, we consider signals in demand response programs as a binary treatment to the customers and estimate the average treatment effect, which is the average change in consumption under the demand response signals. More specifically, we propose to estimate this effect by linear regression models and derive several estimators based on the different models. From both synthetic and real data, we show that including more information about the customers does not always improve estimation accuracy: the interaction between the side information and the demand response signal must be carefully modeled. In addition, we compare the traditional linear regression model with the modified covariate method which models the interaction between treatment effect and covariates. We analyze the variances of these estimators and discuss different cases where each respective estimator works the bes
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This book provides a rather selfcontained survey of the construction of Hadamard states for scalar field theories in a large class of notable spacetimes, possessing a (conformal) lightlike boundary. The first two sections focus on explaining a few introductory aspects of this topic and on providing the relevant geometric background material. The notions of asymptotically flat spacetimes and of expanding universes with a cosmological horizon are analysed in detail, devoting special attention to the characterization of asymptotic symmetries. In the central part of the book, the quantization of a real scalar field theory on such class of backgrounds is discussed within the framework of algebraic quantum field theory. Subsequently it is explained how it is possible to encode the information of the observables of the theory in a second, ancillary counterpart, which is built directly on the conformal (null) boundary. This procedure, dubbed bulktoboundary correspondence, has the net advan
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A local description of the nonflat infinitesimally bendable Euclidean hypersurfaces was recently given by Dajczer and Vlachos \cite{DaVl}. From their classification, it follows that there is an abundance of infinitesimally bendable hypersurfaces that are not isometrically bendable. In this paper we consider the case of complete hypersurfaces $f\colon M^n\to\mathbb{R}^{n+1}$, $n\geq 4$. If there is no open subset where $f$ is either totally geodesic or a cylinder over an unbounded hypersurface of $\mathbb{R}^4$, we prove that $f$ is infinitesimally bendable only along ruled strips. In particular, if the hypersurface is simply connected, this implies that any infinitesimal bending of $f$ is the variational field of an isometric bending.
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We generalize the following univariate characterization of Kummer and Gamma distributions to the cone of symmetric positive definite matrices: let $X$ and $Y$ be independent, nondegenerate random variables valued in $(0, \infty)$, then $U= Y/(1+X)$ and $V = X(1+U)$ are independent if and only if $X$ follows the Kummer distribution and $Y$ follows the the Gamma distribution with appropriate parameters. We solve a related functional equation in the cone of symmetric positive definite matrices, which is our first main result and apply its solution to prove the characterization theorem, which is our second main result.
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We consider importance sampling (IS), pseudomarginal (PM), and delayed acceptance (DA) approaches to reversible Markov chain Monte Carlo, and compare the asymptotic variances. Despite their similarity in terms of using an approximation or unbiased estimators, not much has been said about the relative efficiency of IS and PM/DA. Simple examples demonstrate that the answer is setting specific. We show that the IS asymptotic variance is strictly less than a constant times the PM/DA asymptotic variance, where the constant is twice the essential supremum of the importance weight, and that the inequality becomes an equality as the weight approaches unity in the uniform norm. A version of the inequality also holds for the case of unbounded weight estimators, as long as the estimators are square integrable. The result, together with robustness and computational cost considerations in the context of parallel computing, lends weight to the suggestion of using IS over that of PM/DA when reasonabl
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A homogeneous Riemannian manifold $(M=G/K, g)$ is called a space with homogeneous geodesics or a $G$g.o. space if every geodesic $\gamma (t)$ of $M$ is an orbit of a oneparameter subgroup of $G$, that is $\gamma(t) = \exp(tX)\cdot o$, for some non zero vector $X$ in the Lie algebra of $G$. We give an exposition on the subject, by presenting techniques that have been used so far and a wide selection of previous and recent results. We also present some open problems.
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We consider submanifolds into Riemannian manifold with metallic structures. We obtain some new results for hypersurfaces in these spaces and we express the fundamental theorem of submanifolds into products spaces in terms of metallic structures. Moreover, we define new structures called complex metallic structures. We show that these structures are linked with complex structures. Then, we consider submanifolds into Riemannian manifold with such structures with a focus on invariant submanifolds and hypersurfaces. We also express in particular the fundamental theorem of submanifolds of complex space form in terms of complex metallic structures.
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Known examples in plane symmetry or Gowdy symmetry show that given a $1$parameter family of solutions to the vacuum Einstein equations, it may have a weak limit which does not satisfy the vacuum equations, but instead has a nontrivial stressenergymomentum tensor. We consider this phenomenon under polarized $\mathbb U(1)$ symmetry  a much weaker symmetry than most of the known examples  such that the stressenergymomentum tensor can be identified with that of multiple families of null dust propagating in distinct directions. We prove that any generic localintime smalldata polarized$\mathbb U(1)$symmetric solution to the Einsteinmultiple null dust system can be achieved as a weak limit of vacuum solutions. Our construction allows the number of families to be arbitrarily large, and appears to be the first construction of such examples with more than two families.
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We describe a class of coadjoint orbits of the group of Hamiltonian diffeomorphisms of a symplectic manifold $(M,\omega)$ by implementing symplectic reduction for the dual pair associated to the Hamiltonian description of ideal fluids. The description is given in terms of nonlinear Grassmannians (manifolds of submanifolds) with additional geometric structures. Reduction at zero momentum yields the identification of coadjoint orbits with Grassmannians of isotropic volume submanifolds, slightly generalizing the results in Weinstein [1990] and Lee [2009]. At the other extreme, the case of a nondegenerate momentum recovers the identification of connected components of the nonlinear symplectic Grassmannian with coadjoint orbits, thereby recovering the result of Haller and Vizman [2004]. We also comment on the intermediate cases which correspond to new classes of coadjoint orbits. The description of these coadjoint orbits as well as their orbit symplectic form is obtained in a systematic way
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The structure of a certain subgroup $S$ of the automorphism group of a partially commutative group (RAAG) $G$ is described in detail: namely the subgroup generated by inversions and elementary transvections. We define admissible subsets of the generators of $G$, and show that $S$ is the subgroup of automorphisms which fix all subgroups $\langle Y\rangle$ of $G$, for all admissible subsets $Y$. A decomposition of $S$ as an iterated tower of semidirect products in given and the structure of the factors of this decomposition described. The construction allows a presentation of $S$ to be computed, from the commutation graph of $G$.
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In a low energy approximation of the massless Yukawa theory (Nelson model) we derive a FaddeevKulish type formula for the scattering matrix of $N$ electrons and reformulate it in LSZ terms. To this end, we perform a decomposition of the infrared finite Dollard modifier into clouds of real and virtual photons, whose infrared divergencies mutually cancel. We point out that in the original work of Faddeev and Kulish the clouds of real photons are omitted, and consequently their scattering matrix is illdefined on the Fock space of free electrons. To support our observations, we compare our final LSZ expression for $N=1$ with a rigorous nonperturbative construction due to Pizzo. While our discussion contains some heuristic steps, they can be formulated as clearcut mathematical conjectures.
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We study the random conductance model on the lattice $\mathbb{Z}^d$, i.e.\ we consider a linear, finitedifference, divergenceform operator with random coefficients and the associated random walk under random conductances. We allow the conductances to be unbounded and degenerate elliptic, but they need to satisfy a strong moment condition and a quantified ergodicity assumption in form of a spectral gap estimate. As a main result we obtain in dimension $d\geq 3$ quantitative central limit theorems for the random walk in form of a BerryEsseen estimate with speed $t^{\frac 1 5+\varepsilon}$ for $d\geq 4$ and $t^{\frac{1}{10}+\varepsilon}$ for $d=3$. In addition, for $d\geq 3$ we show nearoptimal decay estimates on the semigroup associated with the environment process, which plays a central role in quantitative stochastic homogenization. This extends some recent results by Gloria, Otto and the second author to the degenerate elliptic case.
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We use the Mallows permutation model to construct a new family of stationary finitely dependent proper colorings of the integers. We prove that these colorings can be expressed as finitary factors of i.i.d. processes with finite mean coding radii. They are the first colorings known to have these properties. Moreover, we prove that the coding radii have exponential tails, and that the colorings can also be expressed as functions of countablestate Markov chains. We deduce analogous existence statements concerning shifts of finite type and higherdimensional colorings.
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A noncooperative differential (dynamic) game model of opinion dynamics, where the agents' motives are shaped by how susceptible they are to get influenced by others, how stubborn they are, and how quick they are willing to change their opinions on a set of issues in a prescribed time interval is considered. We prove that a unique Nash equilibrium exists in the game if there is a harmony of views among the agents of the network. The harmony may be in the form of similarity in pairwise conceptions about the issues but may also be a collective agreement on the status of a "leader" in the network. The existence of a Nash equilibrium can be interpreted as an emergent collective behavior out of the local interaction rules and individual motives.
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