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It is known that fuzzy set theory can be viewed as taking place within a topos. There are several equivalent ways to construct this topos, one is as the topos of \'{e}tal\'{e} spaces over the topological space $Y=[0,1)$ with lower topology. In this topos, the fuzzy subsets of a set $X$ are the subobjects of the constant \'{e}tal\'{e} $X\times Y$ where $X$ has the discrete topology. Here we show that the type2 fuzzy truth value algebra is isomorphic to the complex algebra formed from the subobjects of the constant relational \'{e}tal\'{e} given by the type1 fuzzy truth value algebra $\mathfrak{I}=([0,1],\wedge,\vee,\neg,0,1)$. More generally, we show that if $L$ is the lattice of open sets of a topological space $Y$ and $\mathfrak{X}$ is a relational structure, then the convolution algebra $L^\mathfrak{X}$ is isomorphic to the complex algebra formed from the subobjects of the constant relational \'{e}tal\'{e} given by $\mathfrak{X}$ in the topos of \'{e}tal\'{e} spaces over $Y$.
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For families of smooth complex projective varieties we show that normal functions arising from algebraically trivial cycle classes are algebraic, and defined over the field of definition of the family. As a consequence, we prove a conjecture of Charles and KerrPearlstein, that zero loci of normal functions arising from algebraically trivial cycle classes are algebraic, and defined over the field of definition of the family. In particular, this gives a short proof of a special, algebraically motivated case of a result of Saito, BrosnanPearlstein, and Schnell, conjectured by GreenGriffiths, on zero loci of admissible normal functions.
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Emerging applications of sensor networks for detection sometimes suggest that classical problems ought be revisited under new assumptions. This is the case of binary hypothesis testing with independent  but not necessarily identically distributed  observations under the two hypotheses, a formalism so orthodox that it is used as an opening example in many detection classes. However, let us insert a new element, and address an issue perhaps with impact on strategies to deal with "big data" applications: What would happen if the structure were streamlined such that data flowed freely throughout the system without provenance? How much information (for detection) is contained in the sample values, and how much in their labels? How should decisionmaking proceed in this case? The theoretical contribution of this work is to answer these questions by establishing the fundamental limits, in terms of error exponents, of the aforementioned binary hypothesis test with unlabeled observations draw
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The performance of alloptical dualhop relayed freespace optical communication systems is analytically studied and evaluated. We consider the case when the total received signal undergoes turbulenceinduced channel fading, modeled by the versatile mixtureGamma distribution. Also, the misalignmentinduced fading due to the presence of pointing errors is jointly considered in the enclosed analysis. The performance of both amplifyandforward and decodeandforward relaying transmission is studied, when heterodyne detection is applied. New closedform expressions are derived regarding some key performance metrics of the considered system; namely, the system outage probability and average biterror rate.
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This paper presents a modified quasireversibility method for computing the exponentially unstable solution of a nonlocal terminalboundary value parabolic problem with noisy data. Based on data measurements, we perturb the problem by the socalled filter regularized operator to design an approximate problem. Different from recently developed approaches that consist in the conventional spectral methods, we analyze this new approximation in a variational framework, where the finite element method can be applied. To see the whole skeleton of this method, our main results lie in the analysis of a semilinear case and we discuss some generalizations where this analysis can be adapted. As is omnipresent in many physical processes, there is likely a myriad of models derived from this simpler case, such as source localization problems for brain tumors and heat conduction problems with nonlinear sinks in nuclear science. With respect to each noise level, we benefit from the FaedoGalerkin meth
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We prove existence, uniqueness and regularity of solutions of nonlocal heat equations associated to anisotropic stable diffusion operators. The main features are that the righthand side has very few regularity and that the spectral measure can be singular in some directions. The proofs require having good enough estimates for the corresponding heat kernels and their derivatives.
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We exhibit a new construction of edgeregular graphs with regular cliques that are not strongly regular. The infinite family of graphs resulting from this construction includes an edgeregular graph with parameters $(24,8,2)$. We also show that edgeregular graphs with $1$regular cliques that are not strongly regular must have at least $24$ vertices.
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In this paper we introduce a particular class of matrices. We study the concept of a matrix to be balanced. We study some properties of this concept in the context of matrix operations. We examine the behaviour of various matrix statistics in this setting. The crux will be to understanding the determinants and the eigenvalues of balanced matrices. It turns out that there does exist a direct communication among the leading entry, the trace, determinants and, hence, the eigenvalues of these matrices of order $2\times 2$. These matrices have an interesting property that enables us to predict their quadratic forms, even without knowing their entries but given their spectrum.
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Brinde [Approximating fixed points of weak contractions using the Picard itration, Nonlinear Anal. Forum 9 (2004), 4353] introduced almost contraction mappings and proved Banach contraction principle for such mappings. The aim of this paper is to introduce the notion of multivalued almost $\Theta$ contraction mappings and present some best proximity point results for this new class of mappings. As applications, best proximity point and fixed point results for weak single valued $\Theta$contraction mappings are obtained. An example is presented to support the results presented herein. An application to a nonlinear differential equation is also provided.
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We study the family of Bethe subalgebras in the Yangian $Y(\mathfrak{g})$ parameterized by the corresponding adjoint Lie group $G$. We describe their classical limits as subalgebras in the algebra of polynomial functions on the formal Lie group $G_1[[t^{1}]]$. In particular we show that, for regular values of the parameter, these subalgebras are free polynomial algebras with the same Poincare series as the Cartan subalgebra of the Yangian. Next, we extend the family of Bethe subalgebras to the De ConciniProcesi wonderful compactification $\overline{G}\supset G$ and describe the subalgebras corresponding to generic points of any stratum in $\overline{G}$ as Bethe subalgebras in the Yangian of the corresponding Levi subalgebra in $\mathfrak{g}$.
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For the focusing cubic wave equation, we find an explicit, nontrivial selfsimilar blowup solution $u^*_T$, which is defined on the whole space and exists in all supercritical dimensions $d \geq 5$. For $d=7$, we analyze its stability properties without any symmetry assumptions and prove the existence of a codimension one Lipschitz manifold consisting of initial data whose solutions blowup in finite time and converge asymptotically to $u^*_T$ (modulo spacetime shifts and Lorentz boosts) in the backward lightcone of the blowup point. The underlying topology is strictly above scaling.
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We provide a modulitheoretic framework for the collapsing of Ricciflat Kahler metrics via compactification of moduli varieties of MorganShalen and Satake type. In patricular, we use it to study the GromovHausdorff limits of hyperKahler metrics with fixed diameters, especially for K3 surfaces.
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In this work, we show that along a particular choice of Hermitian curvature flow, the nonpositivity of ChernRicci curvature will be preserved if the initial metric has nonpositive bisectional curvature. As a corollary, we show that the canonical line bundle of a compact Hermitian manifold with nonpositive bisectional curvature and quasinegative ChernRicci curvature is ample.
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A sequence is called $r$sparse if every contiguous subsequence of length $r$ has no repeated letters. A $DS(n, s)$sequence is a $2$sparse sequence with $n$ distinct letters that avoids alternations of length $s+2$. Pettie and Wellman (2018) asked whether there exist $r$sparse $DS(n, s)$sequences of length $\Omega(s n^{2})$ for $s \geq n$ and $r > 2$, which would generalize a result of Roselle and Stanton (1971) for the case $r = 2$. We construct $r$sparse $DS(n, s)$sequences of length $\Omega(s n^{2})$ for $s \geq n$ and $r > 2$. Our construction uses linear hypergraph edgecoloring bounds. We also use the construction to generalize a result of Pettie and Wellman by proving that if $s = \Omega(n^{1/t} (t1)!)$, then there are $r$sparse $DS(n, s)$sequences of length $\Omega(n^{2} s / (t1)!)$ for all $r \geq 2$. In addition, we find related results about the lengths of sequences avoiding $(r, s)$formations.
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This article is based on a talk at the RIEMain in Contact conference in Cagliari, Italy in honor of the 78th birthday of David Blair one of the founders of modern Riemannian contact geometry. The present article is a survey of a special type of Riemannian contact structure known as Sasakian geometry. An ultimate goal of this survey is to understand the moduli of classes of Sasakian structures as well as the moduli of extremal and constant scalar curvature Sasaki metrics, and in particular the moduli of SasakiEinstein metrics.
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Under the assumption that sequences of graphs equipped with resistances, associated measures, walks and local times converge in a suitable GromovHausdorff topology, we establish asymptotic bounds on the distribution of the $\varepsilon$blanket times of the random walks in the sequence. The precise nature of these bounds ensures convergence of the $\varepsilon$blanket times of the random walks if the $\varepsilon$blanket time of the limiting diffusion is continuous with probability one at $\varepsilon$. This result enables us to prove annealed convergence in various examples of critical random graphs, including critical GaltonWatson trees, the Erd\H{o}sR\'enyi random graph in the critical window and the configuration model in the scaling critical window. We highlight that proving continuity of the $\varepsilon$blanket time of the limiting diffusion relies on the scale invariance of a finite measure that gives rise to realizations of the limiting compact random metric space, and t
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It is shown that the solution of the Cauchy problem for the BBMKP equation converges to the solution of the Cauchy problem for the BBM equation in a suitable function space whenever the initial data for both equations are close as the transverse variable $y \rightarrow \pm \infty$.
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LQG mean field game systems consisting of a major agent and a large population of minor agents have been addressed in the literature. In this paper, a novel convex analysis approach is utilized to retrieve the best response strategies for the major agent and each individual minor agent which collectively yield an $\epsilon$Nash equilibrium for the entire system.
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The covert capacity is characterized for a noncoherent fast Rayleighfading wireless channel, in which a legitimate user wishes to communicate reliably with a legitimate receiver while escaping detection from a warden. It is shown that the covert capacity is achieved with an amplitudeconstrained input distribution that consists of a finite number of mass points including one at zero and numerically tractable bounds are provided. It is also conjectured that distributions with two mass points in fixed locations are optimal.
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This paper deals with the analysis of qualitative properties involved in the dynamics of KellerSegel type systems in which the diffusion mechanisms of the cells are driven by porousmedia fluxsaturated phenomena. We study the regularization inside the support of a solution with jump discontinuity at the boundary of the support. We analyze the behavior of the size of the support and blowup of the solution, and the possible convergence in finite time towards a Dirac mass in terms of the three constants of the system: the mass, the fluxsaturated characteristic speed, and the chemoattractant sensitivity constant. These constants of motion also characterize the dynamics of regular and singular traveling waves.
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For a polynomial $F(t,A_1,\ldots,A_n)\in\mathbf{F}_p[t,A_1,\ldots,A_n]$ ($p$ being a prime number) we study the decomposition statistics of its specializations $$F(t,a_1,\ldots,a_n)\in\mathbf{F}_p[t]$$ with $(a_1,\ldots,a_n)\in S$, where $S\subset\mathbf{F}_p^n$ is a subset, in the limit $p\to\infty$ and $\deg F$ fixed. We show that for a sufficiently large and regular subset $S\subset\mathbf{F}_p^n$, e.g. a product of $n$ intervals of length $H_1,\ldots,H_n$ with $\prod_{i=1}^nH_n>p^{n1/2+\epsilon}$, the decomposition statistics is the same as for unrestricted specializations (i.e. $S=\mathbf{F}_p^n$) up to a small error. This is a generalization of the wellknown P\'olyaVinogradov estimate of the number of quadratic residues modulo $p$ in an interval.
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In this paper, wireless video transmission to multiple users under total transmission power and minimum required video quality constraints is studied. In order to provide the desired performance levels to the endusers in realtime video transmissions while using the energy resources efficiently, we assume that power control is employed. Due to the presence of interference, determining the optimal power control is a nonconvex problem but can be solved via monotonic optimization framework. However, monotonic optimization is an iterative algorithm and can often entail considerable computational complexity, making it not suitable for realtime applications. To address this, we propose a learningbased approach that treats the input and output of a resource allocation algorithm as an unknown nonlinear mapping and a deep neural network (DNN) is employed to learn this mapping. This learned mapping via DNN can provide the optimal power level quickly for given channel conditions.
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Recent explorations of Deep Learning in the physical layer (PHY) of wireless communication have shown the capabilities of Deep Neuron Networks in tasks like channel coding, modulation, and parametric estimation. However, it is unclear if Deep Neuron Networks could also learn the advanced waveforms of current and nextgeneration wireless networks, and potentially create new ones. In this paper, a Deep Complex Convolutional Network (DCCN) without explicit Discrete Fourier Transform (DFT) is developed as an Orthogonal FrequencyDivision Multiplexing (OFDM) receiver. Compared to existing deep neuron network receivers composed of fullyconnected layers followed by nonlinear activations, the developed DCCN not only contains convolutional layers but is also almost (and could be fully) linear. Moreover, the developed DCCN not only learns to convert OFDM waveform with Quadrature Amplitude Modulation (QAM) into bits under noisy and Rayleigh channels, but also outperforms expert OFDM receiver ba
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In this paper, we introduce asymptotically periodic functions and study these functions from the point of view of operator algebras and dynamical systems. We show that the M\"{o}bius function is disjoint from any strongly asymptotically periodic functions. As a consequence, Sarnak's M\"{o}bius Disjointness Conjecture holds for all countable compact spaces. Whenever Sarnak's conjecture holds, we show that the M\"{o}bius function is disjoint from all asymptotically periodic functions.
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We introduce a tool for analysing models of $\textnormal{CT}^$, the compositional truth theory over Peano Arithmetic. We present a new proof of Lachlan's theorem that arithmetical part of models of $\textnormal{PA}$ are recursively saturated. We use this tool to provide a new proof that all models of $\textnormal{CT}^$ carry a partial inductive truth predicate. Finally, we construct a partial truth predicate defined for formulae from a nonstandard cut which cannot be extended to a full truth predicate satisfying $\textnormal{CT}^$.
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The aim of this work is an analysis of distal and nondistal behavior in dense pairs of ominimal structures. A characterization of distal types is given through orthogonality to a generic type in $M^{\operatorname{eq}}$, nondistality is geometrically analyzed through Keisler measures, and a distal expansion for the case of pairs of ordered vector spaces is computed.
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We introduce two new measures for the dependence of $n \ge 2$ random variables: distance multivariance and total distance multivariance. Both measures are based on the weighted $L^2$distance of quantities related to the characteristic functions of the underlying random variables. These extend distance covariance (introduced by Sz\'ekely, Rizzo and Bakirov) from pairs of random variables to $n$tuplets of random variables. We show that total distance multivariance can be used to detect the independence of $n$ random variables and has a simple finitesample representation in terms of distance matrices of the sample points, where distance is measured by a continuous negative definite function. Under some mild moment conditions, this leads to a test for independence of multiple random vectors which is consistent against all alternatives.
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In 2016, Yuri Zarhin gave formulas for "dividing a point on a hyperelliptic curve by 2." Given a point $P$ on a hyperelliptic curve $\mathcal{C}$, Zarhin gives the Mumford's representation of every degree $g$ divisor $D$ such that $2(D  g \infty) \sim P  \infty$. The aim of this paper is to generalize Zarhin's result to the superelliptic situation; instead of dividing by 2, we divide by $1  \zeta$. Even though there is no Mumford's representation for superelliptic curves, we give a formula for functions which cut out $D$.
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We study Nash equilibria for a twoplayer zerosum optimal stopping game with incomplete and asymmetric information. In our setup, the drift of the underlying diffusion process is unknown to one player (incomplete information feature), but known to the other one (asymmetric information feature). We formulate the problem and reduce it to a fully Markovian setup where the uninformed player optimises over stopping times and the informed one uses randomised stopping times in order to hide their informational advantage. Then we provide a general verification result which allows us to find Nash equilibria by solving suitable quasivariational inequalities with some nonstandard constraints. Finally, we study an example with linear payoffs, in which an explicit solution of the corresponding quasivariational inequalities can be obtained.
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We prove that a general complete intersection of dimension $n$, codimension $c$ and type $d_1, \dots, d_c$ in $\mathbb{P}^N$ has ample cotangent bundle if $c \geq 2n2$ and the $d_i$'s are all greater than a bound that is $O(1)$ in $N$ and quadratic in $n$. This degree bound substantially improves the currently bestknown superexponential bound in $N$ by Deng, although our result does not address the case $n \leq c < 2n2$.
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Lowrank inducing unitarily invariant norms have been introduced to convexify problems with lowrank/sparsity constraint. They are the convex envelope of a unitary invariant norm and the indicator function of an upper bounding rank constraint. The most wellknown member of this family is the socalled nuclear norm. To solve optimization problems involving such norms with proximal splitting methods, efficient ways of evaluating the proximal mapping of the lowrank inducing norms are needed. This is known for the nuclear norm, but not for most other members of the lowrank inducing family. This work supplies a framework that reduces the proximal mapping evaluation into a nested binary search, in which each iteration requires the solution of a much simpler problem. This simpler problem can often be solved analytically as it is demonstrated for the socalled lowrank inducing Frobenius and spectral norms. Moreover, the framework allows to compute the proximal mapping of compositions of the
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In this paper we fully characterize the sequentially weakly lower semicontinuity of the parameterdepending energy functional associated with the critical Kirchhoff problem. We also establish sufficient criteria with respect to the parameters for the convexity and validity of the PalaisSmale condition of the same energy functional. We then apply these regularity properties in the study of some elliptic problems involving the critical Kirchhoff term.
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The enhanced power graph $\mathcal G_e(\mathbf G)$ of a group $\mathbf G$ is the graph with vertex set $G$ such that two vertices $x$ and $y$ are adjacent if they are contained in a same cyclic subgroup. We prove that finite groups with isomorphic enhanced power graphs have isomorphic directed power graphs. We show that any isomorphism between power graphs of finite groups is an isomorhism between enhanced power graphs of these groups, and we find all finite groups $\mathbf G$ for which $\mathrm{Aut}(\mathcal G_e(\mathbf G)$ is abelian, all finite groups $\mathbf G$ with $\lvert\mathrm{Aut}(\mathcal G_e(\mathbf G)\rvert$ being prime power, and all finite groups $\mathbf G$ with $\lvert\mathrm{Aut}(\mathcal G_e(\mathbf G)\rvert$ being square free. Also we describe enhanced power graphs of finite abelian groups. Finally, we give a characterization of finite nilpotent groups whose enhanced power graphs are perfect, and we present a sufficient condition for a finite group to have weakly pe
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We exploit the so called \emph{atomic condition}, recently defined by De~Philippis, De~Rosa, and Ghiraldin in [Comm. Pure Appl. Math.] and proved to be necessary and sufficient for the validity of the anisotropic counterpart of the Allard rectifiability theorem. In~particular, we address an open question of this seminal work, showing that the atomic condition implies the strict Almgren geometric ellipticity condition.
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Suppose $G$ is a finite group and $p$ is either a prime number or $0$. For $p$ positive, we say that $G$ is weakly tame at $p$ if $G$ has no nontrivial normal $p$subgroups. By convention we say that every finite group is weakly tame at $0$. Now suppose that $G$ is a finite group which is weakly tame at the residue characteristic of a discrete valuation ring $R$. Our main result shows that the essential dimension of $G$ over the fraction field $K$ of $R$ is at least as large as the essential dimension of $G$ over the residue field $k$. We also prove a more general statement of this type for a class of \'etale gerbes over $R$. As a corollary, we show that, if $G$ is weakly tame at $p$ and $k$ is any field of characteristic $p >0$ containing the algebraic closure of $\mathbb{F}_p$, then the essential dimension of $G$ over $k$ is less than or equal to the essential dimension of $G$ over any characteristic $0$ field. A conjecture of A. Ledet asserts that the essential dimension, $\math
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Eta quotients on $\Gamma_0(6)$ yield evaluations of sunrise integrals at 2, 3, 4 and 6 loops. At 2 and 3 loops, they provide modular parametrizations of inhomogeneous differential equations whose solutions are readily obtained by expanding in the nome $q$. AtkinLehner transformations that permute cusps ensure fast convergence for all external momenta. At 4 and 6 loops, onshell integrals are periods of modular forms of weights 4 and 6 given by Eichler integrals of eta quotients. Weakly holomorphic eta quotients determine quasiperiods. A Rademacher sum formula is given for Fourier coefficients of an eta quotient that is a Hauptmodul for $\Gamma_0(6)$ and its generalization is found for all levels with genus 0, namely for $N = 1,2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 18, 25$. There are elliptic obstructions at $N = 11, 14, 15, 17, 19, 20, 21, 24, 27, 32, 36, 49,$ with genus 1. We surmount these, finding explicit formulas for Fourier coefficients of eta quotients in thousands of cases.
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We consider positive solution to the weighted elliptic problem \begin{equation*} \left \{ \begin{array}{ll} {\rm div} (x^\theta \nabla u)=x^\ell u^p \;\;\; \mbox{in $\mathbb{R}^N \backslash {\overline B}$},\\ u=0 \;\;\; \mbox{on $\partial B$}, \end{array} \right. \end{equation*} where $B$ is the standard unit ball of $\mathbb{R}^N$. We give a complete answer for the existence question when $N':=N+\theta>2$. In particular, for $N'> 2$ and $\tau:=\ell\theta >2$, it is shown that the problem admits a unique positive radial solution for $p>p_s:=\frac{N'+2+2\tau}{N'2}$, while for any $ 0<p \leq p_s$, the only nonnegative solution is $u \equiv 0$.
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We study existence of densities for solutions to stochastic differential equations with H\"older continuous coefficients and driven by a $d$dimensional L\'evy process $Z=(Z_{t})_{t\geq 0}$, where, for $t>0$, the density function $f_{t}$ of $Z_{t}$ exists and satisfies, for some $(\alpha_{i})_{i=1,\dots,d}\subset(0,2)$ and $C>0$, \begin{align*} \limsup\limits _{t \to 0}t^{1/\alpha_{i}}\int\limits _{\mathbb{R}^{d}}f_{t}(z+e_{i}h)f_{t}(z)dz\leq Ch,\ \ h\in \mathbb{R},\ \ i=1,\dots,d. \end{align*} Here $e_{1},\dots,e_{d}$ denote the canonical basis vectors in $\mathbb{R}^{d}$. The latter condition covers anisotropic $(\alpha_{1},\dots,\alpha_{d})$stable laws but also particular cases of subordinate Brownian motion. To prove our result we use some ideas taken from \citep{DF13}.
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With $\Fq$ the finite field of $q$ elements, we investigate the following question. If $\gamma$ generates $\Fqn$ over $\Fq$ and $\beta$ is a nonzero element of $\Fqn$, is there always an $a \in \Fq$ such that $\beta(\gamma + a)$ is a primitive element? We resolve this case when $n=3$, thereby proving a conjecture by Cohen. We also improve substantially on what is known when $n=4$.
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In this paper, we investigate exterior and symmetric (co)homology of groups. We give a new approach to symmetric cohomology and also introduce symmetric homology of groups. We compute symmetric homology and exterior (co)homology of some finite groups. Further, we compare the classical, exterior and symmetric (co)homology and introduce some new (co)homologies of groups.
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Given a 3manifold $M$ fibering over the circle, we investigate how the asymptotic translation lengths of pseudoAnosov monodromies in the arc complex vary as we vary the fibration. We formalize this problem by defining normalized asymptotic translation length functions $\mu_d$ for every integer $d \ge 1$ on the rational points of a fibered face of the unit ball of the Thurston norm on $H^1(M;\mathbb{R})$. We show that even though the functions $\mu_d$ themselves are typically nowhere continuous, the sets of accumulation points of their graphs on $d$dimensional slices of the fibered face are rather nice and in a way reminiscent of Fried's convex and continuous normalized stretch factor function. We also show that these sets of accumulation points depend only on the shape of the corresponding slice. We obtain a particularly concrete description of these sets when the slice is a simplex. We also compute $\mu_1$ at infinitely many points for the mapping torus of the simplest hyperbolic b
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In this paper we first discuss a TemperleyLieb algebra associated to the Coxeter group of type $\mathtt{B}$ which is the natural extension of the classical case, in the sense that it can be expressed as a quotient of the Hecke algebra of type B over an appropriate twosided ideal. We then give the necessary and sufficient conditions so that the Markov trace defined on the Hecke algebra of type $\mathtt{B}$ factors through to the quotient algebra and we construct the corresponding knot invariants. Next, following the results recently obtained for groups of type $\mathtt{A}$, we define a framization of such a TemperleyLieb algebra as a proper quotient of the YokonumaHecke algebra of type $\mathtt{B}$. The main theorem provides necessary and sufficient conditions for the Markov trace defined on the YokonumaHecke algebra of type $\mathtt{B}$ to pass through to the framization quotient algebra. Finally, we present the derived invariants for framed and classical knots and links inside th
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The second eigenvalue of the Robin Laplacian is shown to be maximal for the ball among domains of fixed volume, for negative values of the Robin parameter $\alpha$ in the regime connecting the first nontrivial Neumann and Steklov eigenvalues, and even somewhat beyond the Steklov regime. The result is close to optimal, since the ball is not maximal when $\alpha$ is sufficiently large negative, and the problem admits no maximiser when $\alpha$ is positive.
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Let $A$ be a set and $f:A\rightarrow A$ a bijective function. Necessary and sufficient conditions on $f$ are determined which makes it possible to endow $A$ with a binary operation $*$ such that $(A,*)$ is a cyclic group and $f\in \mbox{Aut}(A)$. This result is extended to all abelian groups in case $A=p^2, \ p$ a prime. Finally, in case $A$ is countably infinite, those $f$ for which it is possible to turn $A$ into a group $(A,*)$ isomorphic to ${\Bbb Z}^n$ for some $n\ge 1$, and with $f\in \mbox{Aut} (A)$, are completely characterised.
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In this paper we propose a combinatorial approach to generalized mathematical derangements and anagrams without fixed letters. In sections 1 and 2 we introduce the functions $P$  the number of generalized derangements of a set, and $P'$  the number of anagrams without fixed letters of a given word. The preliminary observations in these chapters provide the toolbox for developing two recursive algorithms in section 3 for computing $P$ and $P'$. The second algorithm leads to several different inequalities. They allow us to roughly estimate the values of $P$ and $P'$ and partially order them. The final section of this paper is dedicated to some number theoretical properties of $P'.$ The focus is on divisibility and the main technique is partitioning the anagrams into classes of equivalence in different ways. The article ends with a conjecture, which generalizes one of the theorems in the last chapter.
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Let $\mathfrak{g}(A)$ be the KacMoody algebra with respect to a symmetrizable generalized Cartan matrix $A$. We give an explicit presentation of the fixpoint Lie subalgebra $\mathfrak{k}(A)$ of $\mathfrak{g}(A)$ with respect to the Chevalley involution. It is a presentation of $\mathfrak{k}(A)$ involving inhomogeneous versions of the Serre relations, or, from a different perspective, a presentation generalizing the DolanGrady presentation of the Onsager algebra. In the finite and untwisted affine case we explicitly compute the structure constants of $\mathfrak{k}(A)$ in terms of a Chevalley type basis of $\mathfrak{k}(A)$. For the symplectic Lie algebra and its untwisted affine extension we explicitly describe the onedimensional representations of $\mathfrak{k}(A)$.
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In this paper, we study the global dynamics of a general $2\times 2$ competition models with nonsymmetric nonlocal dispersal operators. Our results indicate that local stability implies global stability provided that one of the diffusion rates is sufficiently small. This paper continues the work in \cite{BaiLi2017}, where competition models with symmetric nonlocal operators are considered.
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At its annual Adobe Max conference, Adobe announced plans to bring a complete version of Photoshop to the iPad in 2019. Photoshop CC for iPad will feature a revamped interface designed specifically for a touch experience, but it will bring the power and functionality people are accustomed to on the desktop. This is the real, full photoshop  the same codebase as the regular Photoshop, but running on the iPad with a touch UI. The Verge's Dami Lee and artist colleagues at The Verge got to test this new version of Photoshop, and they are very clear to stress that the biggest news here isn't even having the "real" Photoshop on the iPad, but the plans Adobe has for the PSD file format. But the biggest change of all is a total rethinking of the classic .psd file for the cloud, which will turn using Photoshop into something much more like Google Docs. Photoshop for the iPad is a big deal, but Cloud PSD is the change that will let Adobe bring Photoshop everywhere. This does se
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An anonymous reader quotes a report from The Hill: An Amazon employee is seeking to put new pressure on the company to stop selling its facial recognition technology to law enforcement. An anonymous worker, whose employment at Amazon was verified by Medium, published an oped on that platform on Tuesday criticizing the company's facial recognition work and urging the company to respond to an open letter delivered by a group of employees. The employee wrote that the government has used surveillance tools in a way that disproportionately hurts "communities of color, immigrants, and people exercising their First Amendment rights." "Ignoring these urgent concerns while deploying powerful technologies to government and law enforcement agencies is dangerous and irresponsible," the person wrote. "That's why we were disappointed when Teresa Carlson, vice president of the worldwide public sector of Amazon Web Services, recently said that Amazon 'unwaveringly supports' law enforcement, defense,
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The Bro network security monitoring project has announced a name change to "Zeek". "On the Leadership Team of the Bro Project, we heard clear concerns from the Bro community that the name 'Bro' has taken on strongly negative connotations, such as 'Bro culture'. These send a sharp, antiinclusive  and wholly unintended and undesirable  message to those who might use Bro. The problems were significant enough that during BroCon community sessions, several people have mentioned substantial difficulties in getting their upper management to even consider using opensource software with such a seemingly illchosen, offputting name."
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Krystalo quotes a report from VentureBeat: Google today launched Chrome 70 for Windows, Mac, and Linux. The release includes an option to disable linking Google site and Chrome signins, Progressive Web Apps on Windows, the ability for users to restrict extensions' access to a custom list of sites, an AV1 decoder, and plenty more. You can update to the latest version now using Chrome's builtin updater or download it directly from google.com/chrome. An anonymous Slashdot reader adds: "The most anticipated addition to today's release is a new Chrome setting panel option that allows users to control how the browser behaves when they log into a Google account," reports ZDNet. "Google added this new setting after the company was accused last month of secretly logging users into their Chrome browser accounts whenever they logged into a Google website." Chrome 70 also comes with support for the AV1 video format, TLS 1.3 final, persite Chrome extension permissions, TouchID and fingerprint s
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An anonymous reader shares a report: Earlier this year, Google made a seemingly crowdpleasing tweak to its Chrome browser and created a crisis for web game developers. Its May release of Chrome 66 muted sites that played sound automatically, saving internet users from the plague of annoying autoplaying videos. But the new system also broke the audio of games and web art designed for the old audio standard  including hugely popular games like QWOP, clever experiments like the Infinite Jukebox, and even projects officially showcased by Google. After a backlash over the summer, Google kept blocking autoplay for basic video and audio, but it pushed the change for games and web applications to a later version. That browser version, Chrome 70, is on the verge of full release  but the new, autoplayblocking Web Audio API isn't part of it yet. Google communications manager Ivy Choi tells The Verge that Chrome will start learning the sites where users commonly play audio, so it can tailor
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An anonymous reader quotes a report from Recode: Facebook announced Portal last week, its take on the inhome, voiceactivated speaker to rival competitors from Amazon, Google and Apple. Last Monday, we wrote: "No data collected through Portal  even call log data or app usage data, like the fact that you listened to Spotify  will be used to target users with ads on Facebook." We wrote that because that's what we were told by Facebook executives. But Facebook has since reached out to change its answer: Portal doesn't have ads, but data about who you call and data about which apps you use on Portal can be used to target you with ads on other Facebookowned properties. "Portal voice calling is built on the Messenger infrastructure, so when you make a video call on Portal, we collect the same types of information (i.e. usage data such as length of calls, frequency of calls) that we collect on other Messengerenabled devices. We may use this information to inform the ads we show you acr
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Facebook is developing hardware for the TV, news outlet Cheddar reported Tuesday. From the report: The world's largest social network is building a cameraequipped device that sits atop a TV and allows video calling along with entertainment services like Facebook's YouTube competitor, according to people familiar with the matter. The project, internally codenamed "Ripley," uses the same core technology as Facebook's recently announced Portal video chat device for the home. Portal begins shipping next month and uses A.I. to automatically detect and follow people as they move throughout the frame during a video call. Facebook currently plans to announce project Ripley in the spring of 2019, according to a person with direct knowledge of the project. But the device is still in development and the date could be changed.
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Let $\{T(t)\}_{t\ge 0}$ be a $C_0$semigroup on a separable Hilbert space $H$. We characterize that $T(t)$ is an $m$isometry for every $t$ in terms that the mapping $t\in \Bbb R^+ \rightarrow \T(t)x\^2$ is a polynomial of degree less than $m$ for each $x\in H$. This fact is used to study $m$isometric right translation semigroup on weighted $L^p$spaces. We characterize the above property in terms of conditions on the infinitesimal generator operator or in terms of the cogenerator operator of $\{ T(t)\}_{t\geq 0}$. Moreover, we prove that a nonunitary $2$isometry on a Hilbert space satisfying the kernel condition, that is, $$ T^*T(KerT^*)\subset KerT^*\;, $$ then $T$ can be embedded into a $C_0$semigroup if and only if $dim (KerT^*)=\infty$.
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In this paper we study the contribution of monopole bubbling to the expectation value of supersymmetric 't Hooft defects in Lagrangian theories of class $\mathcal{S}$ on $\mathbb{R}^3\times S^1$. This can be understood as the Witten index of an SQM living on the world volume of the 't Hooft defect that couples to the bulk 4D theory. The computation of this Witten index has many subtleties originating from a continuous spectrum of scattering states along the noncompact vacuum branches. We find that even after properly dealing with the spectral asymmetry, the standard localization result for the 't Hooft defect does not agree with the result obtained from the AGT correspondence. In this paper we will explicitly show that one must correct the localization result by adding an extra term to the standard JeffreyKirwan residue formula. This extra term accounts for the contribution of ground states localized along the noncompact branches. This extra term restores both the expected symmetry
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