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In this article we study the stochastic six vertex model under the scaling proposed by Borodin and Gorin (2018), where the weights of cornershape vertices are tuned to zero, and prove Conjecture 6.1 therein: that the height fluctuation converges in finite dimensional distributions to the solution of stochastic telegraph equation.
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Twitter will begin removing tens of millions of suspicious accounts from users' followers on Thursday, signaling a major new effort to restore trust on the popular but embattled platform. From a report: The reform takes aim at a pervasive form of social media fraud. Many users have inflated their followers on Twitter or other services with automated or fake accounts, buying the appearance of social influence to bolster their political activism, business endeavors or entertainment careers. Twitter's decision will have an immediate impact: Beginning on Thursday, many users, including those who have bought fake followers and any others who are followed by suspicious accounts, will see their follower numbers fall. While Twitter declined to provide an exact number of affected users, the company said it would strip tens of millions of questionable accounts from users' followers.
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We study the statistical properties of an estimator derived by applying a gradient ascent method with multiple initializations to a multimodal likelihood function. We derive the population quantity that is the target of this estimator and study the properties of confidence intervals (CIs) constructed from asymptotic normality and the bootstrap approach. In particular, we analyze the coverage deficiency due to finite number of random initializations. We also investigate the CIs by inverting the likelihood ratio test, the score test, and the Wald test, and we show that the resulting CIs may be very different. We provide a summary of the uncertainties that we need to consider while making inference about the population. Note that we do not provide a solution to the problem of multiple local maxima; instead, our goal is to investigate the effect from local maxima on the behavior of our estimator. In addition, we analyze the performance of the EM algorithm under random initializations and
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We introduce and describe the class of split regular $Hom$Leibniz color $3$algebras as the natural extension of the class of split Lie algebras, split Leibniz algebras, split Lie $3$algebras, split Lie triple systems, split Leibniz $3$algebras, and some other algebras. More precisely, we show that any of such split regular $Hom$Leibniz color $3$algebras $T$ is of the form ${T}={\mathcal U} +\sum\limits_{j}I_{j}$, with $\mathcal U$ a subspace of the $0$root space ${T}_0$, and $I_{j}$ an ideal of $T$ satisfying {for} $j\neq k:$ \[[{ T},I_j,I_k]+[I_j,{ T},I_k]+[I_j,I_k,T]=0.\] Moreover, if $T$ is of maximal length, we characterize the simplicity of $T$ in terms of a connectivity property in its set of nonzero roots.
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Broadcom on Wednesday announced plans to buy IT management software company CA for $18.9 billion in cash, just months after U.S. regulators blocked Broadcom's deal to buy fellow chipmaker Qualcomm. Some history of CA, via CNBC reporter Ari Levy: 14 years ago CA was called Computer Associates. The former CEO was charged with securities fraud, conspiracy and obstruction of justice. The lead prosecutor was a Deputy Attorney General by the name James Comey. "The investigators in this case went up against highly sophisticated and allegedly corrupt corporate executives who used every means at their disposal to delay, deceive and derail the government's investigation," Comey said. "The Computer Associates story also includes a failed coverup, replete with lies to government investigators, lies under oath, and the use of attorneys to obstruct and impede the government's investigation of this fraud," he said.
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We prove that splendid Morita equivalences between principal blocks of finite groups with dihedral Sylow $2$subgroups realised by Scott modules can be lifted to splendid Morita equivalences between principal blocks of finite groups with generalised quaternion Sylow $2$subgroups realised by Scott modules.
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In this paper, we discuss the variation of the numbers of the isomorphic classes of stable lattices when the weight and the level varies in a Hida deformation by using the KubotaLeopoldt $p$adic $L$function. As a corollary, we give a sufficient condition for the numbers of the isomorphic classes of stable lattices in Hida deformation to be infinite.
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lkcl writes: Phoronix and The Register have an insightful look into an effort by ARM that is reminiscent of Microsoft's "Get The Facts" campaign. RISCV's design is a revamp of the RISC concept that is intended from the ground up to fix the mistakes and learn from the lessons of the past 30 years. Power efficiency is 40% better than ARM or Intel. Compressed instructions reduce Icache misses by 2025%, which is roughly comparable to the same performance that would be achieved by doubling the Instruction Cache size. Yet despite El Reg's insightful analysis,all is not as it seems: on further investigation, some of ARM's criticism has merit, whilst some of it is clear outandout FUD from ARM that, being so critically dependent on free software, had its own employees complain so much that the site was pulled. Also we cannot help but wonder which "Big Chip" company offered sevenfigure salaries to try to shut down the IIT Madras Shakti Project. Most interesting however is the fact that ARM
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We study the relationship between symmetric nonnegative forms and symmetric sums of squares. Our particular emphasis is on the asymptotic behavior when the degree 2d is fixed and the number of variables $n$ grows. We show that in sharp contrast to the general case the difference between symmetric forms and sums of squares does not grow arbitrarily large for any fixed degree 2d. For degree 4 we show that the difference between symmetric nonnegative forms and sums of squares asymptotically goes to 0. More precisely we relate nonnegative symmetric forms to symmetric mean inequalities, valid independent of the number of variables. Given a symmetric quartic we show that the related symmetric mean inequality holds for all $n\geq 4$, if and only if the symmetric mean inequality can be written as a sum of squares. We conjecture that this is true for arbitrary degree 2d.
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FCC boss Ajit Pai says the agency will finally take steps to shore up the security of the FCC's public comment system after being widely criticized for turning a blind eye to routine fraud and abuse. From a report: If you'll recall, more than 22 million Americans voiced their thoughts on the Trump FCC's attack on net neutrality last fall via the agency's website. The vast majority of comments opposed the move, closely reflecting surveys that show widespread, bipartisan support for the rules. [...] Not a single one of your comments was cited in the FCC's 218 page justification for its decision. [...] Back in May, Senators Senators Jeff Merkley (DOR) and Pat Toomey (RPA) fired off a letter to Pai demanding he actually do something about the abuse of FCC systems. [...] In a response letter this week provided to the Wall Street Journal, Pai says the agency is finally taking steps to address the problem, while acknowledging his own identity was hijacked during the comment process. "It is
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The stable throughput region of the twouser interference channel is investigated here. First, the stability region for the general case is characterized. Second, we study the cases where the receivers treat interference as noise or perform successive interference cancellation. Furthermore, we provide conditions for the convexity of the stability region and for which a certain interference management strategy leads to broader stability region. Finally, we study the effect of random access on the stability region of the twouser interference channel.
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According to a note shared by reliable Apple analyst MingChi Kuo, Apple is planning to refresh a number of its computing product lineups later this year. Via MacRumors: iPhone: There are three iPhones in the works, two OLED models in 5.8 and 6.5inch sizes and one LED model that will be available in a 6.1inch size. iPad: Apple is working on two new 11 and 12.9inch models that are equipped with a fullscreen design and no Home button, with Apple to replace Touch ID with Face ID. Mac mini: Processor upgrades expected. MacBook Pro: Processor upgrades expected. MacBook: Processor upgrades expected. New LowPriced Notebook: Kuo believes Apple is designing a new lowpriced notebook. He originally said that this would be in the MacBook Air family, but now has changed his mind. Previous rumors have suggested this machine could be a 12inch MacBook. iMac: Significant display performance upgrade alongside a processor upgrade. Apple Watch: Two new models in sizes that include 1.57 inches (39.9
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In this paper we determine the motivic classin particular, the weight polynomial and conjecturally the Poincar\'e polynomialof the open de Rham space, defined and studied by Boalch, of certain moduli of irregular meromorphic connections on the trivial bundle on $\mathbb{P}^1$. The computation is by motivic Fourier transform. We show that the result satisfies the purity conjecture, that is, it agrees with the pure part of the conjectured mixed Hodge polynomial of the corresponding wild character variety. We also identify the open de Rham spaces with quiver varieties with multiplicities of Yamakawa and GeissLeclercSchr\"oer. We finish with constructing natural complete hyperk\"ahler metrics on them, which in the $4$dimensional cases are expected to be of type ALF.
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We consider the stochastic heat equation on $\mathbb R^d$ with multiplicative spacetime white noise noise smoothed in space. For $d\geq 3$ and small noise intensity, the solution is known to converge to a strictly positive random variable as the smoothing parameter vanishes. In this regime, we study the rate of convergence and show that the pointwise fluctuations of the smoothened solutions as well as that of the underlying martingale of the Brownian directed polymer converge to a Gaussian limit.
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We make use of the metric version of the conformal Einstein field equations to construct antide Sitterlike spacetimes by means of a suitably posed initialboundary value problem. The evolution system associated to this initialboundary value problem consists of a set of conformal wave equations for a number of conformal fields and the conformal metric. This formulation makes use of generalised wave coordinates and allows the free specification of the Ricci scalar of the conformal metric via a conformal gauge source function. We consider Dirichlet boundary conditions for the evolution equations at the conformal boundary and show that these boundary conditions can, in turn, be constructed from the 3dimensional Lorentzian metric of the conformal boundary and a linear combination of the incoming and outgoing radiation as measured by certain components of the Weyl tensor. To show that a solution to the conformal evolution equations implies a solution to the Einstein field equations we al
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We use a Hamiltonian interacting particle system to derive a stochastic mean field system whose McKeanVlasov equation yields the incompressible Navier Stokes equation. Since the system is Hamiltonian, the particle relabeling symmetry implies a Kelvin Circulation Theorem along stochastic Lagrangian paths. Moreover, issues of energy dissipation are discussed and the model is connected to other approaches in the literature.
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We study a number of natural linear systems carried by any polarized Nikulin surface of genus g. We determine their positivity and establish their BrillNoether theory. Relying upon recent work of Farkas and Rim\'{a}nyi, we compute the class of some natural effective divisors associated to these linear systems on the moduli space of Nikulin surfaces, in this way obtaining for any genus g roughly $\sqrt{g/2}$ relations in its tautological ring.
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Recent advances in open source interiorpoint optimization methods and power system related software have provided researchers and educators with the necessary platform for simulating and optimizing power networks with unprecedented convenience. Within the Matpower software platform a combination of several different interior point optimization methods are provided and four different optimal power flow (OPF) formulations are recently available: the PolarPower, PolarCurrent, CartesianPower, and CartesianCurrent. The robustness and reliability of interiorpoint methods for different OPF formulations for minimizing the generation cost starting from different initial guesses, for a wide range of networks provided in the Matpower library ranging from 1951 buses to 193000 buses, will be investigated. Performance profiles are presented for iteration counts, overall time, and memory consumption, revealing the most reliable optimization method for the particular metric.
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Given any admissible $k$dimensional family of immersions of a given closed oriented surface into an arbitrary closed Riemannian manifold, we prove that the corresponding minmax width for the area is achieved by a smooth (possibly branched) immersed minimal surface with multiplicity one and Morse index bounded by $k$.
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Finding roots of equations is at the heart of most computational science. A wellknown and widely used iterative algorithm is the Newton's method. However, its convergence depends heavily on the initial guess, with poor choices often leading to slow convergence or even divergence. In this paper, we present a new class of methods that improve upon the classical Newton's method. The key idea behind the new approach is to develop a relatively simple multiplicative transformation of the original equations, which leads to a significant reduction in nonlinearities, thereby alleviating the limitations of the Newton's method. Based on this idea, we propose two novel classes of methods and present their application to several mathematical functions (real, complex, and vector). Across all examples, our numerical experiments suggest that the new methods converge for a significantly wider range of initial guesses with minimal increase in computational cost. Given the ubiquity of Newton's method, a
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A linear constrained switching system is a discretetime linear switched system whose switching sequences are constrained by a deterministic finite automaton. As a characterization of the asymptotic stability of a constrained switching system, the constrained joint spectral radius is difficult to compute or approximate. Using the semitensor product of matrices, we express dynamics of a deterministic finite automaton, an arbitrary switching system and a constrained switching system into their matrix forms, respectively, where the matrix expression of a constrained switching system can be seen as the matrix expression of a lifted arbitrary switching system. Inspired by this, we propose a lifting method for the constrained switching system, and prove that the constrained joint/generalized spectral radius of the constrained switching system is equivalent to the joint/generalized spectral radius of the lifted arbitrary switching system. Examples are provided to show the advantages of the p
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In this study, a pairwise comparison matrix is generalized to the case when coefficients create Lie group $G$, non necessarily abelian. A necessary and sufficient criterion for pairwise comparisons matrices to be consistent is provided. Basic criteria for finding a nearest consistent pairwise comparisons matrix (extended to the class of group $G$) are proposed. A geometric interpretation of pairwise comparisons matrices in terms of connections to a simplex is given. Approximate reasoning is more effective when inconsistency in data is reduced.
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The problem of finding an explicit formula for the probability density function of two zero mean correlated normal random variables dates back to 1936. Perhaps surprisingly, this problem was not resolved until 2016. This is all the more surprising given that a very simple proof is available, which is the subject of this note; we identify the product of two zero mean correlated normal random variables as a variancegamma random variable, from which an explicit formula for probability density function is immediate.
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We prove an area law for the entanglement entropy in gapped one dimensional quantum systems. The bound on the entropy grows surprisingly rapidly with the correlation length; we discuss this in terms of properties of quantum expanders and present a conjecture on completely positive maps which may provide an alternate way of arriving at an area law. We also show that, for gapped, local systems, the bound on Von Neumann entropy implies a bound on R\'{e}nyi entropy for sufficiently large $\alpha<1$ and implies the ability to approximate the ground state by a matrix product state.
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ReedMuller (RM) and polar codes are a class of capacityachieving channel coding schemes with the same factor graph representation. Lowcomplexity decoding algorithms fall short in providing a good errorcorrection performance for RM and polar codes. Using the symmetric group of RM and polar codes, the specific decoding algorithm can be carried out on multiple permutations of the factor graph to boost the errorcorrection performance. However, this approach results in high decoding complexity. In this paper, we first derive the total number of factor graph permutations on which the decoding can be performed. We further propose a successive permutation (SP) scheme which finds the permutations on the fly, thus the decoding always progresses on a single factor graph permutation. We show that SP can be used to improve the errorcorrection performance of RM and polar codes under successivecancellation (SC) and SC list (SCL) decoding, while keeping the memory requirements of the decoders u
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We study conditions for the abstract linear functional differential equation $\dot{x}=Ax+F(t)x_t+f(t), t\ge 0$ to have asymptotic almost periodic solutions, where $F(\cdot )$ is periodic, $f$ is asymptotic almost periodic. The main conditions are stated in terms of the spectrum of the monodromy operator associated with the equation and the circular spectrum of the forcing term $f$. The obtained results extend recent results on the subject.
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In this note, we discuss the cobordism maps on periodic Floer homology(PFH) induced by Lefschetz fibration. In the first part of the note, we define the cobordism maps on PFH induced by Lefschetz fibration via Seiberg Witten theory and the isomorphism between PFH and Seiberg Witten cohomology. The second part is to define the cobordism maps induced by Lefschetz fibration provided that the cobordism satisfies certain conditions. Under certain monotone assumptions, we show that these two definitions in fact are equivalent.
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Recently, Shahaf Nitzan and JanFredrik Olsen showed that BalianLow type theorems exist for discrete Gabor systems defined on $\mathbb{Z}_d$. Here, we extend these results to higher dimensional analogs of these systems on $\mathbb{Z}_d^{l}$, and show a variety of applications of both the discrete and continuous verisons of the socalled Quantitative BalianLow Theorem, also of Nitzan and Olsen. In particular, we prove nonsymmetric versions of the finite and continuous BalianLow Theorems holding for $\ell_2(\mathbb{Z}_d^l)$ and $L^2(\mathbb{R}^l)$, respectively.
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The present paper is devoted to quasiParaSasakian manifolds. Basic properties of such manifolds are obtained and general curvature identities are investigated. Next it is proved that if $M$ is quasiParaSasakian manifold of constant curvature $K$. Then $K$ $\leq 0$ and $(i)~$if $K=0$, the manifold is paracosymplectic, $(ii)$ if $K<0$, the quasiparaSasakian structure of $M$ is obtained by a homothetic deformation of a paraSasakian structure.
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In this paper, we consider the characterization of normparallelism problem in some classical Banach spaces. In particular, for two continuous functions $f, g$ on a compact Hausdorff space $K$, we show that $f$ is normparallel to $g$ if and only if there exists a probability measure (i.e. positive and of full measure equal to $1$) $\mu$ with its support contained in the norm attaining set $\{x\in K: \, f(x) = \f\\}$ such that $\big\int_K \overline{f(x)}g(x)d\mu(x)\big = \f\\,\g\$.
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The main objective of the present paper is to establish a new uncertainty principle (UP) for the twosided quaternion Fourier transform (QFT). This result is an extension of a result of Benedicks, Amrein and Berthier, which states that a nonzero function in $L^1\left({\mathbb{R}}^2, {\mathbb{H}}\right)$ and its twosided QFT cannot both have support of finite measure.
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We consider a onedimensional lattice system of unbounded, realvalued spins with arbitrary strong, quadratic, finiterange interaction. We show the equivalence of cor relations of the grand canonical (gce) and the canonical ensemble (ce). As a corollary we obtain that the correlations of the ce decay exponentially plus a volume correction term. Then, we use the decay of correlation to verify a conjecture that the infinitevolume Gibbs measure of the ce is unique on a onedimensional lattice. For the equivalence of correlations, we modify a method that was recently used to show the equivalence of the ce and the gce on the level of thermodynamic functions. In this article we also show that the equivalence of the ce and the gce holds on the level of observables. One should be able to extend the methods and results to graphs with bounded degree as long as the gce has a sufficient strong decay of correlations.
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A. Takahashi suggested a conjectural method to find mirror symmetric pairs consisting of invertible polynomials and symmetry groups generated by some diagonal symmetries and some permutations of variables. Here we generalize the Saito duality between Burnside rings to a case of nonabelian groups and prove a "nonabelian" generalization of the statement about the equivariant Saito duality property for invertible polynomials. It turns out that the statement holds only under a special condition on the action of the subgroup of the permutation group called here PC ("parity condition"). An inspection of data on CalabiYau threefolds obtained from quotients by nonabelian groups shows that the pairs found on the basis of the method of Takahashi have symmetric pairs of Hodge numbers if and only if they satisfy PC.
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The canonical tensor rank approximation problem (TAP) consists of approximating a realvalued tensor by one of low canonical rank, which is a challenging nonlinear, nonconvex, constrained optimization problem, where the constraint set forms a nonsmooth semialgebraic set. We introduce a Riemannian GaussNewton method with trust region for solving smallscale, dense TAPs. The novelty of our approach is threefold. First, we parametrize the constraint set as the Cartesian product of Segre manifolds, hereby formulating the TAP as a Riemannian optimization problem, and we argue why this parametrization is among the theoretically best possible. Second, an original STHOSVDbased retraction operator is proposed. Third, we introduce a hot restart mechanism that efficiently detects when the optimization process is tending to an illconditioned tensor rank decomposition and which often yields a quick escape path from such spurious decompositions. Numerical experiments show improvements of up
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We show that symbolic finitetoone extensions of the type constructed by O. Sarig for surface diffeomorphisms induce H\"oldercontinuous conjugacies on large sets, sometimes preserving transitivity. We deduce this from their Bowen property. This notion, introduced in a joint work with M. Boyle, generalizes a fact first observed by R. Bowen for Markov partitions. We use the notion of degree from finite equivalence theory and magic word isomorphisms. As an application, we improve Sarig's lower bound on the number of periodic points for surface diffeomorphisms. Finally we characterize surface diffeomorphisms admitting a H\"oldercontinuous coding of all their aperiodic hyperbolic measures.
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The Coordinate Bethe Ansatz (CBA) expresses, as a sum over permutations, the matrix element of an XXX Heisenberg spin chain Hamiltonian eigenstate with a state with fixed spins. These matrix elements comprise the wave functions of the Hamiltonian eigenstates. However, as the complexity of the sum grows rapidly with the length N of the spin chain, the exact wave function in the continuum limit is too cumbersome to be exploited. In this note we provide an approximation to the CBA whose complexity does not directly depend upon N. This consists of two steps. First, we add an anchor to the argument of the exponential in the CBA. The anchor is a permutationdependent integral multiple of 2 pi. Once anchored, the distribution of these arguments simplifies, becoming approximately Gaussian. The wave function is given by the Fourier transform of this distribution and so the calculation of the wave function reduces to the calculation of the moments of the distribution. Second, we parametrize the
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We give a new proof of the cutandjoin equation for the monotone Hurwitz numbers, derived first by Goulden, GuayPaquet, and Novak. Our proof in particular uses a combinatorial technique developed by Han. The main interest in this particular equation is its close relation to the quadratic loop equation in the theory of spectral curve topological recursion, and we recall this motivation giving a new proof of the topological recursion for monotone Hurwitz numbers, obtained first by Do, Dyer, and Mathews.
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In this paper, we propose and analyze an additive domain decomposition method (DDM) for solving the highfrequency Helmholtz equation with the Sommerfeld radiation condition. In the proposed method, the computational domain is partitioned into structured subdomains along all spatial directions, and each subdomain contains an overlapping region for source transferring. At each iteration all subdomain PML problems are solved completely in parallel, then all horizontal, vertical and corner directional residuals on each subdomain are passed to its corresponding neighbor subdomains as the source for the next iteration. This DDM method is highly scalable in nature and theoretically shown to produce the exact solution for the PML problem defined in ${\mathbb{R}}^2$ in the constant medium case. A slightly modified version of the method for bounded truncated domains is also developed for its use in practice and an error estimate is rigorously proved. Various numerical experiments in two and thr
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Given a positive integer $M$ and $q\in(1,M+1]$, let $\mathcal U_q$ be the set of $x\in[0, M/(q1)]$ having a unique $q$expansion: there exists a unique sequence $(x_i)=x_1x_2\ldots$ with each $x_i\in\{0,1,\ldots, M\}$ such that \[ x=\frac{x_1}{q}+\frac{x_2}{q^2}+\frac{x_3}{q^3}+\cdots. \] Denote by $\mathbf U_q$ the set of corresponding sequences of all points in $\mathcal U_q$. It is wellknown that the function $H: q\mapsto h(\mathbf U_q)$ is a Devil's staircase, where $h(\mathbf U_q)$ denotes the topological entropy of $\mathbf U_q$. In this paper we {give several characterizations of} the bifurcation set \[ \mathcal B:=\{q\in(1,M+1]: H(p)\ne H(q)\textrm{ for any }p\ne q\}. \] Note that $\mathcal B$ is contained in the set $\mathcal{U}^R$ of bases $q\in(1,M+1]$ such that $1\in\mathcal U_q$. By using a transversality technique we also calculate the Hausdorff dimension of the difference $\mathcal B\backslash\mathcal{U}^R$. Interestingly this quantity is always strictly between $0$ an
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We present a conjecture (and a proof for G=SL(2)) generalizing a result of J. Arthur which expresses a character value of a cuspidal representation of a $p$adic group as a weighted orbital integral of its matrix coefficient. It also generalizes a conjecture by the second author proved by SchneiderStuhler and (independently) the first author. The latter statement expresses an elliptic character value as an orbital integral of a pseudomatrix coefficient defined via the Chern character map taking value in zeroth Hochschild homology of the Hecke algebra. The present conjecture generalizes the construction of pseudomatrix coefficient using compactly supported Hochschild homology, as well as a modification of the category of smooth representations, the so called compactified category of smooth $G$modules. This newly defined "compactified pseudomatrix coefficient" lies in a certain space on which the weighted orbital integral is a conjugation invariant linear functional, our conjecture
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Two widely studied models of multiplenode repair in distributed storage systems are centralized repair and cooperative repair. The centralized model assumes that all the failed nodes are recreated in one location, while the cooperative one stipulates that the failed nodes may communicate but are distinct, and the amount of data exchanged between them is included in the repair bandwidth. As our first result, we prove a lower bound on the minimum bandwidth of cooperative repair. We also show that the cooperative model is stronger than the centralized one, in the sense that any MDS code with optimal repair bandwidth under the former model also has optimal bandwidth under the latter one. These results were previously known under the additional "uniform download" assumption, which is removed in our proofs. As our main result, we give explicit constructions of MDS codes with optimal cooperative repair for all possible parameters. More precisely, given any $n,k,h,d$ such that $2\le h \le nd
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This paper studies the dynamical behavior near one kind of singularity solutions (selfsimilar solutions) for the classcial BornInfeld equation in $1+1$ dimension. This model arises in nonlinear electrodynamics, as well as string theory and geometric minimal surfaces theory in Minkowski space. Although the quasilinear term of it satisfies null condition, we show that this model admits a family of explicit timelike selfsimilar solutions, which are also explicit selfsimilar solutions of linear wave equation in one dimension. Meanwhile, Lynapunov nonlinear stability of those selfsimilar solutions are given inside a strictly proper subset of the backward light cone.
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We consider the unnormalized Yamabe flow on manifolds with conical singularities. Under certain geometric assumption on the initial crosssection we show well posedness of the short time solution in the $L^q$setting. Moreover, we give a picture of the deformation of the conical tips under the flow by providing an asymptotic expansion of the evolving metric close to the boundary in terms of the initial local geometry. Due to the blow up of the scalar curvature close to the singularities we use maximal $L^q$regularity theory for conically degenerate operators.
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Given a Hamiltonian system $ (M,\omega, G,\mu) $ where $(M,\omega)$ is a symplectic manifold, $G$ is a compact connected Lie group acting on $(M,\omega)$ with moment map $ \mu:M \rightarrow\mathfrak{g}^{*}$, then one may construct the symplectic quotient $(M//G, \omega_{red})$ where $M//G := \mu^{1}(0)/G$. Kirwan used the normsquare of the moment map, $\mu^2$, as a Gequivariant Morse function on $M$ to derive formulas for the rational Betti numbers of $M//G$. A real Hamiltonian system $(M,\omega, G,\mu, \sigma, \phi) $ is a Hamiltonian system along with a pair of involutions $(\sigma:M \rightarrow M, \phi:G \rightarrow G) $ satisfying certain compatibility conditions. These imply that the fixed point set $M^{\sigma}$ is a Lagrangian submanifold of $(M,\omega)$ and that $M^{\sigma}//G^{\phi} := (\mu^{1}(0) \cap M^{\sigma})/G^{\phi}$ is a Lagrangian submanifold of $(M//G, \omega_{red})$. In this paper we prove analogues of Kirwan's Theorems that can be used to calculate the $\mathb
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It is well known that every positive integer can be expressed as a sum of nonconsecutive Fibonacci numbers provided the Fibonacci numbers satisfy $F_n =F_{n1}+F_{n2}$ for $n\geq 3$, $F_1 =1$ and $F_2 =2$. In this paper, for any $n,m\in\mathbb{N}$ we create a sequence called the $(n,m)$bin sequence with which we can define a notion of a legal decomposition for every positive integer. These sequences are not always positive linear recurrences, which have been studied in the literature, yet we prove, that like positive linear recurrences, these decompositions exist and are unique. Moreover, our main result proves that the distribution of the number of summands used in the $(n,m)$bin legal decompositions displays Gaussian behavior.
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The Gaussian stochastic process (GaSP) is a useful technique for predicting nonlinear outcomes. The estimated mean function in a GaSP, however, can be far from the reality in terms of the $L_2$ distance. This problem was widely observed in calibrating imperfect mathematical models using experimental data, when the discrepancy function is modeled as a GaSP. In this work, we study the theoretical properties of the scaled Gaussian stochastic process (SGaSP), a new stochastic process to address the identifiability problem of the mean function in the GaSP model. The GaSP is a special case of the SGaSP with the scaling parameter being zero. We establish the explicit connection between the GaSP and SGaSP through the orthogonal series representation. We show the predictive mean estimator in the SGaSP calibration model converges to the reality at the same rate as the GaSP with the suitable choice of the regularization parameter and scaling parameter. We also show the calibrated mathematical
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We study conditions for the abstract periodic linear functional differential equation $\dot{x}=Ax+F(t)x_t+f(t)$ to have almost periodic with the same structure of frequencies as $f$. The main conditions are stated in terms of the spectrum of the monodromy operator associated with the equation and the frequencies of the forcing term $f$. The obtained results extend recent results on the subject. A discussion on how the results could be extended to the case when $A$ depends on $t$ is given.
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We construct a family of twisted generalized Weyl algebras which includes WeylClifford superalgebras and quotients of the enveloping algebras of $\mathfrak{gl}(mn)$ and $\mathfrak{osp}(m2n)$. We give a condition for when a canonical representation by differential operators is faithful. Lastly, we give a description of the graded support of these algebras in terms of patternavoiding vector compositions.
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We consider a hyperbolic Diractype operator with growing potential on a a spatially noncompact globally hyperbolic manifold. We show that the AtiyahPatodiSinger boundary value problem for such operator is Fredholm and obtain a formula for this index in terms of the local integrals and the relative etainvariant introduced by Braverman and Shi. This extends recent results of B\"ar and Strohmaier, who studied the index of a hyperbolic Dirac operator on a spatially compact globally hyperbolic manifold.
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Entropyconservative numerical flux functions can be used to construct highorder, entropystable discretizations of the Euler and NavierStokes equations. The purpose of this short communication is to present a novel family of such entropyconservative flux functions. The proposed flux functions are solutions to quadratic optimization problems and admit closedform, computationally affordable expressions. We establish the properties of the flux functions including their continuous differentiability, which is necessary for highorder discretizations.
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Differential privacy provides a rigorous framework for privacypreserving data analysis. This paper proposes the first differentially private procedure for controlling the false discovery rate (FDR) in multiple hypothesis testing. Inspired by the Benjamini Hochberg procedure (BHq), our approach is to first repeatedly add noise to the logarithms of the pvalues to ensure differential privacy and to select an approximately smallest pvalue serving as a promising candidate at each iteration; the selected pvalues are further supplied to the BHq and our private procedure releases only the rejected ones. Apart from the privacy considerations, we develop a new technique that is based on a backward submartingale for proving FDR control of a broad class of multiple testing procedures, including our private procedure, and both the BHq stepup and stepdown procedures. As a novel aspect, the proof works for arbitrary dependence between the true null and false null test statistics, while FDR con
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We improve a recent result by giving the optimal conclusion possible both to the frequent universality criterion and the frequent hypercyclicity criterion using the notion of Adensities, where A refers to some weighted densities sharper than the natural lower density. Moreover we construct an operator which is logarithmicallyfrequently hypercyclic but not frequently hypercyclic.
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Using Hecke characters, we construct two infinite families of newforms with complex multiplication, one by $\mathbb{Q}(\sqrt{3})$ and the other by $\mathbb{Q}(\sqrt{2})$. The values of the $p$th Fourier coefficients of all the forms in each family can be described by a single formula, which we provide explicitly. This allows us to establish a formula relating the $p$th Fourier coefficients of forms of different weights, within each family. We then prove congruence relations between the $p$th Fourier coefficients of these newforms at all odd weights and values coming from two of Zagier's sporadic Ap\'erylike sequences.
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The elliptical range theorem asserts that the field of values (or numerical range) of a twobytwo matrix with complex entries is an elliptical disk, the foci of which are the eigenvalues of the given matrix. Many proofs of this result are available in the literature, but most, with one exception, are computational and quite involved. In this note, it is shown that the elliptical range theorem follows from the properties of plane algebraic curves and a straightforward application of a wellknown result due to Kippenhahn.
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Let $\nu$ be either the Ozsv\'athSzab\'o $\tau$invariant or the Rasmussen $s$invariant, suitably normalized. For a knot $K$, Livingston and Naik defined the invariant $t_\nu(K)$ to be the minimum of $k$ for which $\nu$ of the $k$twisted positive Whitehead double of $K$ vanishes. They proved that $t_\nu(K)$ is bounded above by $TB(K)$, where $TB$ is the maximal ThurstonBennequin number. We use a blowing up process to find a crossing change formula and a new upper bound for $t_\nu$ in terms of the unknotting number. As an application, we present infinitely many knots $K$ such that the difference between LivingstonNaik's upper bound $TB(K)$ and $t_\nu(K)$ can be arbitrarily large.
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The geometric flow theory and its applications turned into one of the most intensively developing branches of modern geometry. Here, a brief introduction to Finslerian Ricci flow and their selfsimilar solutions known as Ricci solitons are given and some recent results are presented. They are a generalization of Einstein metrics and are previously developed by the present authors for Finsler manifolds. In the present work, it is shown that a complete shrinking Ricci soliton Finsler manifold has a finite fundamental group.
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An (r,alpha)bounded excess flow ((r,alpha)flow) in an orientation of a graph G=(V,E) is an assignment of a real "flow value" between 1 and r1 to every edge. Rather than 0 as in an actual flow, some flow excess, which does not exceed alpha may accumulate in any vertex. Bounded excess flows suggest a generalization of Circular nowhere zero flows, which can be regarded as (r,0)flows. We define (r,alpha) as Stronger or equivalent to (s,beta) If the existence of an (r,alpha)flow in a cubic graph always implies the existence of an (s,beta)flow in the same graph. Then we study the structure of the twodimensional flow strength poset. A major role is played by the "Trace" parameter: tr(r,alpha)=(r2alpha) divided by (1alpha). Among points with the same trace the stronger is the one with the larger r (an rcnzf is of trace r). About one half of the article is devoted to proving the main result: Every cubic graph admits a (3.5,0.5)flow. tr(3.5,0.5)=5 so it can be considered a step in the
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We consider random set partitions of size $n$ with exactly $k$ blocks, chosen uniformly from all such, as counted by $S(n,k)$, the Stirling number of the second kind, and random permutations of size $n$ with exactly $k$ cycles, chosen uniformly from all such, as counted by $s(n,k)$, the unsigned Stirling number of the first kind, under the regime where $r \equiv r(n,k) := nk \sim t\sqrt{n}$. In this regime, there is a simple approximation for the entire process of component counts; in particular the number of components of size 3 converges in distribution to Poisson with mean $\frac{2}{3}t^2$ for set partitions, and mean $\frac{4}{3}t^2$ for permutations, and with high probability, all other components have size one or two. These approximations are proved, with quantitative error bounds, using combinatorial bijections for placements of $r$ rooks on a triangular half of an $n\times n$ chess board, together with the ChenStein method for processes of indicator random variables.
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In this paper, we use the method of Thue and Siegel, based on explicit Pade approximations to algebraic functions, to completely solve a family of quartic Thue equations. From this result, we can also solve the diophantine equation in the title. We prove that this equation has at most one solution in positive integers when $d \geq 3$. Moreover, when such a solution exists, it is of the form $(u,\sqrt{v})$ where $(u,v)$ is the fundamental solution of $X^{2}+1=dY^{2}$.
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We consider d independent walkers on Z, m of them performing simple symmetric random walk and r = d  m of them performing recurrent RWRE (Sinai walk), in I independent random environments. We show that the product is recurrent, almost surely, if and only if m $\le$ 1 or m = d = 2. In the transient case with r $\ge$ 1, we prove that the walkers meet infinitely often, almost surely, if and only if m = 2 and r $\ge$ I = 1. In particular, while I does not have an influence for the recurrence or transience, it does play a role for the probability to have infinitely many meetings. To obtain these statements, we prove two subtle localization results for a single walker in a recurrent random environment, which are of independent interest.
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To compute generators for the MordellWeil group of an elliptic curve over a number field, one needs to bound the difference between the naive and the canonical height from above. We give an elementary and fast method to compute an upper bound for the local contribution to this difference at an archimedean place, which sometimes gives better results than previous algorithms.
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In this paper we answer several questions raised by Sidorov on the set $\mathcal B_2$ of bases in which there exist numbers with exactly two expansions. In particular, we prove that the set $\mathcal B_2$ is closed, and it contains both infinitely many isolated and accumulation points in $(1, q_{KL})$, where $q_{KL}\approx 1.78723$ is the KomornikLoreti constant. Consequently we show that the second smallest element of $\mathcal B_2$ is the smallest accumulation point of $\mathcal B_2$. We also investigate the higher order derived sets of $\mathcal B_2$. Finally, we prove that there exists a $\delta>0$ such that \begin{equation*} \dim_H(\mathcal B_2\cap(q_{KL}, q_{KL}+\delta))<1, \end{equation*} where $\dim_H$ denotes the Hausdorff dimension.
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