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Millions of dollars in funding and billions of dollars in valuations have made scooters the next big thing since the last big thing. From a report: When Michael Ramsey, an analyst for technology research firm Gartner, started in February to put together his 2018 "hype cycle" report for the future of transportation, he had plenty of topics to choose from: electric vehicles, flying cars, 5G, blockchain, and, of course, autonomous vehicles. But one type of transportation is conspicuously absent from the results of the report: electric scooters. "At the time, outside of California, these scooters were really not that common," Ramsey said. "That's how much has happened." As for autonomous vehicles, which have enjoyed years of hype as the next big thing, Ramsey labeled them sliding into "the trough of disillusionment," which Ramsey described as "when expectations don't meet the truth." In a matter of months, electric scooter startups have gone from tech oddity to global phenomenon. In some c
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Essential Products, a startup founded in 2015 by Android creator Andy Rubin, was started to create a smartphone with highend design features that wasn't associated with a particular operatingsystem maker. Unfortunately, reaching that goal has been harder than anticipated as the company has laid off about 30 percent of its staff. Fortune reports: Cuts were particularly deep in hardware and marketing. The company's website indicates it has about 120 employees. A company spokesperson didn't confirm the extent of layoffs, but said that the decision was difficult for the firm to make and, "We are confident that our sharpened product focus will help us deliver a truly game changing consumer product." The firm was Rubin's first startup after leaving Google in 2014, which had acquired his cofounded firm, Android, in 2005. Essential's first phone came out in August 2017, a few weeks later than initially promised. It received mixed reviews, with most critics citing its lower quality and missi
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Zorro shares a report from The Mercury News: Not only did Facebook inflate adwatching metrics by up to 900 percent (Warning: source may be paywalled, alternative source), it knew for more than a year that its averageviewership estimates were wrong and kept quiet about it, a new legal filing claims. A group of small advertisers suing the Menlo Park social media titan alleged in the filing that Facebook "induced" advertisers to buy video ads on its platform because advertisers believed Facebook users were watching video ads for longer than they actually were. That "unethical, unscrupulous" behavior by Facebook constituted fraud because it was "likely to deceive" advertisers, the filing alleged. The latest allegations arose out of a lawsuit that the advertisers filed against Mark Zuckerbergled Facebook in federal court in 2016 over alleged inflation of adwatching metrics. "Suggestions that we in any way tried to hide this issue from our partners are false," the company told The Wall S
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People's Facebook posts might predict whether they are suffering from depression, researchers reported this week. From a report: The researchers found that the words people used seemed to indicate whether they would later be diagnosed with depression. The findings offer a way to flag people who may be in need of help, but they also raise important questions about people's health privacy, the team reported in the Proceedings of the National Academy of Sciences. People who were later clinically diagnosed with depression used more "I" language, according to Johannes Eichstaedt of the University of Pennsylvania and his colleagues. They also used more words reflecting loneliness, sadness and hostility. "We observed that users who ultimately had a diagnosis of depression used more firstperson singular pronouns, suggesting a preoccupation with the self," they wrote. That is an indicator of depression in some people. The team recruited 683 people who visited an emergency room for their study
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Three state treasurers and a top official from New York have joined a shareholders' motion to install an independent chairman at Facebook, claiming the move would improve governance and accountability. [...] The move comes as Facebook was presented with a new legal challenge. The technology company has been accused of misleading advertisers by inflating the viewing figures for videos on its site. A group of US advertisers launched a fraud claim against the social media giant on Tuesday, stating that it had overstated the average viewing time of advertising videos on the site by between 100 and 900pc before reporting them in 2016. All tech companies are pretty terrible as far as companies go, but Facebook really seems to be going out of its way to lead the pack. As far as I'm concerned, we shut it down. Would anyone really miss it?
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In the early days of what ultimately became Waymo, Google's selfdriving car division (known at the time as "Project Chauffeur"), there were "more than a dozen accidents, at least three of which were serious," according to a new article in The New Yorker . From a report: The magazine profiled Anthony Levandowski, the former Google engineer who was at the center of the Waymo v. Uber trade secrets lawsuit. According to the article, back in 2011, Levandowski also modified the autonomous software to take the prototype Priuses on "otherwise forbidden routes." Citing an anonymous source, The New Yorker reports that Levandowski sat behind the wheel as the safety driver, along with Isaac Taylor, a Google executive. But while they were in the car, the Prius "accidentally boxed in another vehicle," a Camry. As The New Yorker wrote: "A human driver could easily have handled the situation by slowing down and letting the Camry merge into traffic, but Google's software wasn't prepared for this scena
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An anonymous reader shares a report: Sellers have been modifying lower end NVIDIA graphics cards and selling them more powerful cards online. In a recent version of the GPUZ graphics card information utility, TechPowerUp has added the ability to now detect these fake NVIDIA cards. This new feature allows buyers of cards to detect if the card is actually a relabled NVIDIA G84, G86, G92, G94, G96, GT215, GT216, GT218, GF108, GF106, GF114, GF116, GF119, or GK106 GPU by displaying an exclamation point where the NVIDIA logo would normally appear and also prepends the string "[FAKE]" before the card's name.
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Gabrielle Hamilton, April Bloomfield, and the Problem with Leaving Women to
1018 THE NEW YORKER 4660 
In this paper, we prove a new cohomology theory that is an invariant of a planar trivalent graph with a given perfect matching. This bigraded cohomology theory appears to be very powerful: the graded Euler characteristic of the cohomology is a one variable polynomial (called the 2factor polynomial) that, if nonzero when evaluated at one, implies that the perfect matching is even. This polynomial can be used to construct a polynomial invariant of the graph called the even matching polynomial. We conjecture that the even matching polynomial is positive when evaluated at one for all bridgeless planar trivalent graphs. This conjecture, if true, implies the existence of an even perfect matching for the graph, and thus the trivalent planar graph is 3edgecolorable. This is equivalent to the four color theorema famous conjecture in mathematics that was proven using a computer program in 1970s. While these polynomial invariants may not have enough strength as invariants to prove such a co
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In this paper, we extend a class of globally convergent evolution strategies to handle general constrained optimization problems. The proposed framework handles relaxable constraints using a merit function approach combined with a specific restoration procedure. The unrelaxable constraints in our framework, when present, are treated either by using the extreme barrier function or through a projection approach. The introduced extension guaranties to the regarded class of evolution strategies global convergence properties for first order stationary constraints. Preliminary numerical experiments are carried out on a set of known test problems as well as on a multidisciplinary design optimization problem
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In this work we present an algorithm to construct sparsepaving matroids over finite set $S$. From this algorithm we derive some useful bounds on the cardinality of the set of circuits of any SparsePaving matroids which allow us to prove in a simple way an asymptotic relation between the class of Sparsepaving matroids and the whole class of matroids. Additionally we introduce a matrix based method which render an explicit partition of the $r$subsets of $S$, $\binom{S}{r}=\sqcup_{i=1}^{\gamma }\mathcal{U}_{i}$ such that each $\mathcal{U}_{i}$ defines a sparsepaving matroid of rank $r$.
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Recently, Misanantenaina and Wagner characterized the set of induced $N$free and bowtiefree posets as a certain class of recursively defined subposets which they term "$\mathcal{V}$posets". Here we offer a new characterization of $\mathcal{V}$posets by introducing a property we refer to as \emph{autonomy}. A poset $\mathcal{P}$ is said to be \emph{autonomous} if there exists a directed acyclic graph $D$ (with adjacency matrix $U$) whose transitive closure is $\mathcal{P}$, with the property that any total ordering of the vertices of $D$ so that Gaussian elimination of $U^TU$ proceeds without row swaps is a linear extension of $\mathcal{P}$. Autonomous posets arise from the theory of pressing sequences in graphs, a problem with origins in phylogenetics. The pressing sequences of a graph can be partitioned into families corresponding to posets; because of the interest in enumerating pressing sequences, we investigate when this partition has only one block, that is, when the pressing
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In many contemporary optimization problems, such as hyperparameter tuning for deep learning architectures, it is computationally challenging or even infeasible to evaluate an entire function or its derivatives. This necessitates the use of stochastic algorithms that sample problem data, which can jeopardize the guarantees classically obtained through globalization techniques via a trust region or a line search. Using subsampled function values is particularly challenging for the latter strategy, that relies upon multiple evaluations. On top of that all, there has been an increasing interest for nonconvex formulations of datarelated problems. For such instances, one aims at developing methods that converge to secondorder stationary points, which is particularly delicate to ensure when one only accesses subsampled approximations of the objective and its derivatives. This paper contributes to this rapidly expanding field by presenting a stochastic algorithm based on negative curvature a
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The set of all perfect matchings of a plane (weakly) elementary bipartite graph equipped with a partial order is a poset, moreover the poset is a finite distributive lattice and its Hasse diagram is isomorphic to $Z$transformation directed graph of the graph. A finite distributive lattice is matchable if its Hasse diagram is isomorphic to a $Z$transformation directed graph of a plane weakly elementary bipartite graph, otherwise nonmatchable. We introduce the meetirreducible cell with respect to a perfect matching of a plane (weakly) elementary bipartite graph and give its equivalent characterizations. Using these, we extend a result on nonmatchable distributive lattices, and obtain a class of new nonmatchable distributive lattices.
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The spectral properties of two special classes of Jacobi operators are studied. For the first class represented by the $2M$dimensional real Jacobi matrices whose entries are symmetric with respect to the secondary diagonal, a new polynomial identity relating the eigenvalues of such matrices with their matrix { entries} is obtained. In the limit $M\to\infty$ this identity induces some requirements, which should satisfy the scattering data of the resulting infinitedimensional Jacobi operator in the halfline, which super and subdiagonal matrix elements are equal to 1. We obtain such requirements in the simplest case of the discrete Schr\"odinger operator acting in ${l}^2( \mathbb{N})$, which does not have bound and semibound states, and which potential has a compact support.
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The characterization of local regularity is a fundamental issue in signal and image processing, since it contains relevant information about the underlying systems. The 2microlocal frontier, a monotone concave downward curve in $\mathbb {R}^2$, provides a complete and profound classification of pointwise singularity. In \cite{Meyer1998}, \cite{GuiJaffardLevy1998} and \cite{LevySeuret2004} the authors show the following: given a monotone concave downward curve in the plane it is possible to exhibit one function (or distribution) such that its 2microlocal frontier al $x_0$ is the given curve. In this work we are able to unify the previous results, by obtaining a large class of functions (or distributions), that includes the three examples mentioned above, for which the 2microlocal frontier is the given curve. The three examples above are in this class. Further, if the curve is a line, we characterize all the functions whose 2microlocal frontier at $x_0$ is the given line.
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A tetravalent $2$arctransitive graph of order $728$ is either the known $7$arctransitive incidence graph of the classical generalized hexagon $GH(3,3)$ or a normal cover of a $2$transitive graph of order $182$ denoted $A[182,1]$ or $A[182,2]$ in the $2009$ list of Poto\v{c}nik.
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We propose an index for gapped quantum lattice systems that conserve a $\mathrm{U}(1)$charge. This index takes integer values and it is therefore stable under perturbations. Our formulation is general, but we show that the index reduces to (i) an index of projections in the noninteracting case, (ii) the filling factor for translational invariant systems, (iii) the quantum Hall conductance in the twodimensional setting without any additional symmetry. Example (ii) recovers the LiebSchultzMattis theorem, (iii) provides a new and short proof of quantization of Hall conductance in interacting manybody systems. Additionally, we provide a new proof of Bloch's theorem on the vanishing of ground state currents.
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Let $R=K[x_1,...,x_n]$ be the polynomial ring in $n$ variables over a field $K$ and $I$ be a monomial ideal generated in degree $d$. Bandari and Herzog conjectured that a monomial ideal $I$ is polymatroidal if and only if all its monomial localizations have a linear resolution. In this paper we give an affirmative answer to the conjecture in the following cases: $(i)$ ${\rm height}(I)=n1$; $(ii)$ $I$ contains at least $n3$ pure powers of the variables $x_1^d,...,x_{n3}^d$; $(iii)$ $I$ is a monomial ideal in at most four variables.
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We look at the number $L(n)$ of $O$sequences of length $n$. This interesting and naturallydefined sequence $L(n)$ was first investigated in a recent paper by commutative algebraists Enkosky and Stone, inspired by Huneke. In this note, we significantly improve both of their upper and lower bounds, by means of a very short partitiontheoretic argument. In particular, it turns out that, for suitable positive constants $c_1$ and $c_2$ and all $n\ge 1$, $$e^{c_1\sqrt{n}}\le L(n)\le e^{c_2\sqrt{n}\log n}.$$ It remains an open problem to determine an exact asymptotic estimate for $L(n)$.
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We study the continuoustime evolution of the recombination equation of population genetics. This evolution is given by a differential equation that acts on a product probability space, and its solution can be described by a Markov chain on a set of partitions that converges to the finest partition. We study an explicit form of the law of this process by using a family of trees. We also describe the geometric decay rate to the finest partition and the quasistationary behavior of the Markov chain when conditioned on the event that the chain does not hit the limit.
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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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