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We present families of large undirected and directed Cayley graphs whose construction is related to butterfly networks. One approach yields, for every large $k$ and for values of $d$ taken from a large interval, the largest known Cayley graphs and digraphs of diameter $k$ and degree $d$. Another method yields, for sufficiently large $k$ and infinitely many values of $d$, Cayley graphs and digraphs of diameter $k$ and degree $d$ whose order is exponentially larger in $k$ than any previously constructed. In the directed case, these are within a linear factor in $k$ of the Moore bound.
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We classify finite dimensional $H_{m^2}(\zeta)$simple $H_{m^2}(\zeta)$module Lie algebras $L$ over an algebraically closed field of characteristic $0$ where $H_{m^2}(\zeta)$ is the $m$th Taft algebra. As an application, we show that despite the fact that $L$ can be nonsemisimple in ordinary sense, $\lim_{n\to\infty}\sqrt[n]{c_n^{H_{m^2}(\zeta)}(L)} = \dim L$ where $c_n^{H_{m^2}(\zeta)}(L)$ is the codimension sequence of polynomial $H_{m^2}(\zeta)$identities of $L$. In particular, the analog of Amitsur's conjecture holds for $c_n^{H_{m^2}(\zeta)}(L)$.
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In this paper additive bifree convolution is defined for general Borel probability measures, and the limiting distributions for sums of bifree pairs of selfadjoint commuting random variables in an infinitesimal triangular array are determined. These distributions are characterized by their bifreely infinite divisibility, and moreover, a transfer principle is established for limit theorems in classical probability theory and Voiculescu's bifree probability theory. Complete descriptions of bifree stability and fullness of planar probability distributions are also set down. All these results reveal one important feature about the theory of bifree probability that it parallels the classical theory perfectly well. The emphasis in the whole work is not on the tool of bifree combinatorics but only on the analytic machinery.
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Conjectural results for cohomological invariants of wild character varieties are obtained by counting curves in degenerate CalabiYau threefolds. A conjectural formula for Epolynomials is derived from the GromovWitten theory of local CalabiYau threefolds with normal crossing singularities. A refinement is also conjectured, generalizing existing results of Hausel, Mereb and Wong as well as recent joint work of Donagi, Pantev and the author for weighted Poincar\'e polynomials of wild character varieties.
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We consider the flow in direction $\theta$ on a translation surface and we study the asymptotic behavior for $r\to 0$ of the time needed by orbits to hit the $r$neighborhood of a prescribed point, or more precisely the exponent of the corresponding power law, which is known as \emph{hitting time}. For flat tori the limsup of hitting time is equal to the diophantine type of the direction $\theta$. In higher genus, we consider an extended geometric notion of diophantine type of a direction $\theta$ and we seek for relations with hitting time. For genus two surfaces with just one conical singularity we prove that the limsup of hitting time is always less or equal to the square of the diophantine type. For any squaretiled surface with the same topology, the diophantine type itself is a lower bound. Moreover both bounds are sharp for big sets of directions. Our results apply to Lshaped billiards.
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The capacity of symmetric, neighboring and consecutive sideinformation single unicast index coding problems (SNCSUICP) with number of messages equal to the number of receivers was given by Maleki, Cadambe and Jafar. For these index coding problems, an optimal index code construction by using Vandermonde matrices was proposed. This construction requires all the sideinformation at the receivers to decode their wanted messages and also requires large field size. In an earlier work, we constructed binary matrices of size $m \times n (m\geq n)$ such that any $n$ adjacent rows of the matrix are linearly independent over every field. Calling these matrices as Adjacent Independent Row (AIR) matrices using which we gave an optimal scalar linear index code for the onesided SNCSUICP for any given number of messages and onesided sideinformation. By using Vandermonde matrices or AIR matrices, every receiver needs to solve $KD$ equations with $KD$ unknowns to obtain its wanted message, wher
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On the basis of loop group decompositions (Birkhoff decompositions), we give a discrete version of the nonlinear d'Alembert formula, a method of separation of variables of difference equations, for discrete constant negative Gauss curvature (pseudospherical) surfaces in Euclidean three space. We also compute two examples by this formula in detail.
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A rapid transformation is derived between spherical harmonic expansions and their analogues in a bivariate Fourier series. The change of basis is described in two steps: firstly, expansions in normalized associated Legendre functions of all orders are converted to those of order zero and one; then, these intermediate expressions are reexpanded in trigonometric form. The first step proceeds with a butterfly factorization of the wellconditioned matrices of connection coefficients. The second step proceeds with fast orthogonal polynomial transforms via hierarchically offdiagonal lowrank matrix decompositions. Total precomputation requires at best $\mathcal{O}(n^3\log n)$ flops; and, asymptotically optimal execution time of $\mathcal{O}(n^2\log^2 n)$ is rigorously proved via connection to Fourier integral operators.
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It is proved that for $k\geq 4$, if the points of $k$dimensional Euclidean space are coloured in red and blue, then there are either two red points distance one apart or $k+3$ blue collinear points with distance one between any two consecutive points. This result is new for $4\leq k\leq 10$.
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Twosided group digraphs, introduced by Iradmusa and Praeger, provide a generalization of Cayley digraphs in which arcs are determined by left and right multiplying by elements of two subsets of the group. We characterize when twosided group digraphs are connected and count connected components, using both an explicit elementary perspective and group actions. Our results and examples address four open problems posed by Iradmusa and Praeger that concern connectedness and valency.
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These notes contain an introduction to the theory of complex semisimple quantum groups. Our main aim is to discuss the classification of irreducible HarishChandra modules for these quantum groups, following Joseph and Letzter. Along the way we cover extensive background material on quantized universal enveloping algebras and explain connections to the analytical theory in the setting of locally compact quantum groups.
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The Cluster Value Theorem is known for being a weak version of the classical Corona Theorem. Given a Banach space $X$, we study the Cluster Value Problem for the ball algebra $A_u(B_X)$, the Banach algebra of all uniformly continuous holomorphic functions on the unit ball $B_X$; and also for the Fr\'echet algebra $H_b(X)$ of holomorphic functions of bounded type on $X$ (more generally, for $H_b(U)$, the algebra of holomorphic functions of bounded type on a given balanced open subset $U \subset X$). We show that Cluster Value Theorems hold for all of these algebras whenever the dual of $X$ has the bounded approximation property. These results are an important advance in this problem, since the validity of these theorems was known only for trivial cases (where the spectrum is formed only by evaluation functionals) and for the infinite dimensional Hilbert space. As a consequence , we obtain weak analytic Nullstellensatz theorems and several structural results for the spectrum of these alg
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The theory of free MajoranaWeyl spinors is the prototype of conformal field theory in two dimensions in which the gravitational anomaly and the Weyl anomaly obstruct extending the flat spacetime results to curved backgrounds. In this paper, we investigate a quantization scheme in which the short distance singularity in the twopoint function of chiral fermions on a two dimensional curved spacetime is given by the Green's function corresponding to the classical field equation. We compute the singular term in the Green's function explicitly and observe that the short distance limit is not welldefined in general. We identify constraints on the geometry which are necessary to resolve this problem. On such special backgrounds the theory has locally $c=\frac{1}{2}$ conformal symmetry.
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The purpose of this article is to investigate the properties of the category of mixed plectic Hodge structures defined by Nekov\'a\v{r} and Scholl. We give an equivalent description of mixed plectic Hodge structures in terms of the weight and partial Hodge filtrations. We also construct an explicit complex calculating the extension groups in this category.
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In this paper we give a necessary and suffcient conditions for the existence and uniqueness of periodic solutions of functional differential equations with n delay d dt x(t) = Ax(t) + n j=1 Bx(t  r j) + f (t). The conditions are obtained in terms of Rboundedness of operator valued Fourier multipliers.
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A general class of strongly coupled elliptic systems with quadratic growth in gradients is considered and the existence of their strong solutions is established. The results greatly improve those in a recent paper \cite{dleJFA} as the systems can be either degenerate or singular when their solutions become unbounded. A unified proof for both cases is presented. Most importantly, the VMO assumption in \cite{dleJFA} will be replaced by a much versatile one thanks to a new local weighted GagliardoNirenberg involving BMO norms. Examples in physical models will be provided.
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In this expository paper we survey some recent results on Dirichlet problems of the form $Lu=f(x,u)$ in $\Omega$, $u\equiv0$ in $\mathbb R^n\backslash\Omega$. We first discuss in detail the boundary regularity of solutions, stating the main known results of Grubb and of the author and Serra. We also give a simplified proof of one of such results, focusing on the main ideas and on the blowup techniques that we developed in \cite{RSK,RSstable}. After this, we present the Pohozaev identities established in \cite{RSPoh,RSV,GrubbPoh} and give a sketch of their proofs, which use strongly the fine boundary regularity results discussed previously. Finally, we show how these Pohozaev identities can be used to deduce nonexistence of solutions or unique continuation properties. The operators $L$ under consideration are integrodifferential operator of order $2s$, $s\in(0,1)$, the model case being the fractional Laplacian $L=(\Delta)^s$.
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In blind detection, a set of candidates has to be decoded within a strict time constraint, to identify which transmissions are directed at the user equipment. Blind detection is an operation required by the 3GPP LTE/LTEAdvanced standard, and it will be required in the 5th generation wireless communication standard (5G) as well. We propose a blind detection scheme based on polar codes, where the radio network temporary identifier (RNTI) is transmitted instead of some of the frozen bits. A lowcomplexity decoding stage decodes all candidates, selecting a subset that is decoded by a highperformance algorithm. Simulations results show good missed detection and false alarm rates, that meet the system specifications. We also propose an early stopping criterion for the second decoding stage that can reduce the number of operations performed, improving both average latency and energy consumption. The detection speed is analyzed and different system parameter combinations are shown to meet th
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The study of higherdimensional black holes is a subject which has recently attracted a vast interest. Perhaps one of the most surprising discoveries is a realization that the properties of higherdimensional black holes with the spherical horizon topology are very similar to the properties of the well known fourdimensional Kerr metric. This remarkable result stems from the existence of a single object called the principal tensor. In our review we discuss explicit and hidden symmetries of higherdimensional black holes. We start with the overview of the Liouville theory of completely integrable systems and introduce Killing and KillingYano objects representing explicit and hidden symmetries. We demonstrate that the principal tensor can be used as a 'seed object' which generates all these symmetries. It determines the form of the black hole geometry, as well as guarantees its remarkable properties, such as special algebraic type of the spacetime, complete integrability of geodesic mot
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We consider a sensor network focused on target localization, where sensors measure the signal strength emitted from the target. Each measurement is quantized to one bit and sent to the fusion center. A general attack is considered at some sensors that attempts to cause the fusion center to produce an inaccurate estimation of the target location with a large meansquareerror. The attack is a combination of maninthemiddle, hacking, and spoofing attacks that can effectively change both signals going into and coming out of the sensor nodes in a realistic manner. We show that the essential effect of attacks is to alter the estimated distance between the target and each attacked sensor to a different extent, giving rise to a geometric inconsistency among the attacked and unattacked sensors. Hence, with the help of two secure sensors, a class of detectors are proposed to detect the attacked sensors by scrutinizing the existence of the geometric inconsistency. We show that the false alarm
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The Ardour audio editor project has announced the 5.9 release. "Ardour 5.9 is now available, representing several months of development that spans some new features and many improvements and fixes. Among other things, some significant optimizations were made to redraw performance on OS X/macOS that may be apparent if you are using Ardour on that platform. There were further improvements to tempo and MIDI related features and lots of small improvements to state serialization. Support for the Presonus Faderport 8 control surface was added"
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In our recent work, the sampling and reconstruction of nondecaying signals, modeled as members of weighted$L_p$ spaces, were shown to be stable with an appropriate choice of the generating kernel for the shiftinvariant reconstruction space. In this paper, we extend the StrangFix theory to show that, for $d$dimensional signals whose derivatives up to order $L$ are all in some weighted$L_p$ space, the weighted norm of the approximation error can be made to go down as $O(h^L)$ when the sampling step $h$ tends to $0$. The sufficient condition for this decay rate is that the generating kernel belongs to a particular hybridnorm space and satisfies the StrangFix conditions of order $L$. We show that the $O(h^L)$ behavior of the error is attainable for both approximation schemes using projection (when the signal is prefiltered with the dual kernel) and interpolation (when a prefilter is unavailable). The requirement on the signal for the interpolation method, however, is slightly more
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Ranking items to be recommended to users is one of the main problems in large scale social media applications. This problem can be set up as a multiobjective optimization problem to allow for trading off multiple, potentially conflicting objectives (that are driven by those items) against each other. Most previous approaches to this problem optimize for a single slot without considering the interaction effect of these items on one another. In this paper, we develop a constrained multislot optimization formulation, which allows for modeling interactions among the items on the different slots. We characterize the solution in terms of problem parameters and identify conditions under which an efficient solution is possible. The problem formulation results in a quadratically constrained quadratic program (QCQP). We provide an algorithm that gives us an efficient solution by relaxing the constraints of the QCQP minimally. Through simulated experiments, we show the benefits of modeling inte
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Random input/output (RIO) code is a coding scheme that enables reading of one logical page using a single read threshold in multilevel flash memory. The construction of RIO codes is equivalent to the construction of WOM codes. Parallel RIO (PRIO) code is an RIO code that encodes all pages in parallel. In this paper, we utilize coset coding with Hamming codes in order to construct PRIO codes. Coset coding is a technique that constructs WOM codes using linear binary codes. We leverage the information on the data of all pages to encode each page. Our constructed codes store more pages than RIO codes constructed via coset coding.
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We study two closely related problems stemming from the random wave conjecture for Maa\ss{} forms. The first problem is bounding the $L^4$norm of a Maa\ss{} form in the large eigenvalue limit; we complete the work of Spinu to show that the $L^4$norm of an Eisenstein series $E(z,1/2+it_g)$ restricted to compact sets is bounded by $\sqrt{\log t_g}$. The second problem is quantum unique ergodicity in shrinking sets; we show that by averaging over the centre of hyperbolic balls in $\Gamma \backslash \mathbb{H}$, quantum unique ergodicity holds for almost every shrinking ball whose radius is larger than the Planck scale. This result is conditional on the generalised Lindel\"{o}f hypothesis for Maa\ss{} eigenforms but is unconditional for Eisenstein series. We also show that equidistribution for Maa\ss{} eigenforms need not hold at or below the Planck scale. Finally, we prove similar equidistribution results in shrinking sets for Heegner points and closed geodesics associated to ideal clas
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Polar codes are a family of capacityachieving errorcorrecting codes, and they have been selected as part of the next generation wireless communication standard. Each polar code bitchannel is assigned a reliability value, used to determine which bits transmit information and which parity. Relative reliabilities need to be known by both encoders and decoders: in case of multimode systems, where multiple code lengths and code rates are supported, the storage of relative reliabilities can lead to high implementation complexity. In this work, observe patterns among code reliabilities. We propose an approximate computation technique to easily represent the reliabilities of multiple codes, through a limited set of variables and update rules. The proposed method allows to tune the tradeoff between reliability accuracy and implementation complexity. An approximate computation architecture for encoders and decoders is designed and implemented, showing 50.7% less area occupation than storage
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Edgecaching has received much attention as an efficient technique to reduce delivery latency and network congestion during peaktraffic times by bringing data closer to end users. Existing works usually design caching algorithms separately from physical layer design. In this paper, we analyse edgecaching wireless networks by taking into account the caching capability when designing the signal transmission. Particularly, we investigate multilayer caching where both base station (BS) and users are capable of storing content data in their local cache and analyse the performance of edgecaching wireless networks under two notable uncoded and coded caching strategies. Firstly, we propose a coded caching strategy that is applied to arbitrary values of cache size. The required backhaul and access rates are derived as a function of the BS and user cache size. Secondly, closedform expressions for the system energy efficiency (EE) corresponding to the two caching methods are derived. Based o
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Let $\hat{\mathfrak{g}}$ be an untwisted affine KacMoody algebra, with its SklyaninDrinfel'd structure of Lie bialgebra, and let $\hat{\mathfrak{h}}$ be the dual Lie bialgebra. By dualizing the quantum double construction  via formal Hopf algebras  we construct a new quantum group $U_q(\hat{\mathfrak{h}})$, dual of $U_q(\hat{\mathfrak{g}})$. Studying its restricted and unrestricted integer forms and their specializations at roots of 1 (in particular, their classical limits), we prove that $U_q(\hat{\mathfrak{h}})$ yields quantizations of $\hat{\mathfrak{h}}$ and $\hat{G}^\infty$ (the formal group attached to $\hat{\mathfrak{g}}$), and we construct new quantum Frobenius morphisms. The whole picture extends to the untwisted affine case the results known for quantum groups of finite type.
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Troy Hunt hits some nails on their heads: If you had any version of Windows since Vista running the default Windows Update, you would have had the critical Microsoft Security Bulletin known as "MS17010" pushed down to your PC and automatically installed. Without doing a thing, when WannaCry came along almost 2 months later, the machine was protected because the exploit it targeted had already been patched. It's because of this essential protection provided by automatic updates that those advocating for disabling the process are being labelled the IT equivalents of antivaxxers and whilst I don't fully agree with real world analogies like this, you can certainly see where they're coming from. As with vaccinations, patches protect the host from nasty things that the vast majority of people simply don't understand. Great article, which also goes into Windows Update itself for a bit.
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We consider $d\times d$ tensors $A(x)$ that are symmetric, positive semidefinite, and whose rowdivergence vanishes identically. We establish sharp inequalities for the integral of $(\det A)^{\frac1{d1}}$. We apply them to models of compressible inviscid fluids: Euler equations, EulerFourier, relativistic Euler, Boltzman, BGK, etc... We deduce an {\em a priori} estimate for a new quantity, namely the spacetime integral of $\rho^{\frac1n}p$, where $\rho$ is the mass density, $p$ the pressure and $n$ the space dimension. For kinetic models, the corresponding quantity generalizes Bony's functional.
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This paper considers continuoustime coordination algorithms for networks of agents that seek to collectively solve a general class of nonsmooth convex optimization problems with an inherent distributed structure. Our algorithm design builds on the characterization of the solutions of the nonsmooth convex program as saddle points of an augmented Lagrangian. We show that the associated saddlepoint dynamics are asymptotically correct but, in general, not distributed because of the presence of a global penalty parameter. This motivates the design of a discontinuous saddlepointlike algorithm that enjoys the same convergence properties and is fully amenable to distributed implementation. Our convergence proofs rely on the identification of a novel global Lyapunov function for saddlepoint dynamics. This novelty also allows us to identify mild convexity and regularity conditions on the objective function that guarantee the exponential convergence rate of the proposed algorithms for convex
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We present a new algorithm for the discretization of the VlasovMaxwell system of equations for the study of plasmas in the kinetic regime. Using the discontinuous Galerkin finite element method for the spatial discretization, we obtain a high order accurate solution for the plasma's distribution function. Time stepping for the distribution function is done explicitly with a third order strongstability preserving RungeKutta method. Since the Vlasov equation in the VlasovMaxwell system is a high dimensional transport equation, up to six dimensions plus time, we take special care to note various features we have implemented to reduce the cost while maintaining the integrity of the solution, including the use of a reduced highorder basis set. A series of benchmarks, from simple wave and shock calculations, to a five dimensional turbulence simulation, are presented to verify the efficacy of our set of numerical methods, as well as demonstrate the power of the implemented features.
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We prove an ifandonlyif criterion for direct sum decomposability of a smooth homogeneous form in terms of the factorization properties of the Macaulay inverse system of its Milnor algebra. This criterion leads to an algorithm for computing direct sum decompositions over any field either of characteristic zero, or of sufficiently large positive characteristic, for which polynomial factorization algorithms exist.
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A $2n$dimensional Poisson manifold $(M ,\Pi)$ is said to be $b^m$symplectic if it is symplectic on the complement of a hypersurface $Z$ and has a simple Darboux canonical form at points of $Z$ which we will describe below. In this paper we will discuss a desingularization procedure which, for $m$ even, converts $\Pi$ into a family of symplectic forms $\omega_{\epsilon}$ having the property that $\omega_{\epsilon}$ is equal to the $b^m$symplectic form dual to $\Pi$ outside an $\epsilon$neighborhood of $Z$ and, in addition, converges to this form as $\epsilon$ tends to zero in a sense that will be made precise in the theorem below. We will then use this construction to show that a number of somewhat mysterious properties of $b^m$manifolds can be more clearly understood by viewing them as limits of analogous properties of the $\omega_{\epsilon}$'s. We will also prove versions of these results for $m$ odd; however, in the odd case the family $\omega_{\epsilon}$ has to be replaced by a
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We examine the problem of exactly or approximately counting all perfect matchings in hereditary classes of nonbipartite graphs. In particular, we consider the switch Markov chain of Diaconis, Graham and Holmes. We determine the largest hereditary class for which the chain is ergodic, and define a large new hereditary class of graphs for which it is rapidly mixing. We go on to show that the chain has exponential mixing time for a slightly larger class. We also examine the question of ergodicity of the switch chain in a arbitrary graph. Finally, we give exact counting algorithms for three classes.
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The elliptic integral and its various generalizations are playing very important and basic role in different branches of modern mathematics. It is well known that they cannot be represented by the elementary transcendental functions. Therefore, there is a need for sharp computable bounds for the family of integrals. In this paper, by virtue of two new tools, we study monotonicity and convexity of certain combinations of the complete elliptic integrals of the first kind, and obtain new sharp bounds and inequalities for them. In particular, we prove that the function $\mathcal{K}\left( \sqrt{% x}\right) /\ln \left( c/\sqrt{1x}\right) $ is concave on $\left( 0,1\right) $ if and only if $c=e^{4/3}$, where $\mathcal{K}$ denotes the complete elliptic integrals of the first kind.
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This text was published in the book "Penser les math{\'e}matiques: s\'eminaire de philosophie et math\'ematiques de l'\'Ecole normale sup\'erieure (J. Dieudonn\'e, M. Loi, R. Thom)" edited by F. Gu\'enard and G. Leli\`evre, Paris, \'editions du Seuil, 1982, pp. 5872. It is reproduced with the kind authorisation of Fran\c{c}ois Ap\'ery.
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An antimatroid is a combinatorial structure abstracting the convexity in geometry. In this paper, we explore novel connections between antimatroids and matchings in a bipartite graph. In particular, we prove that a combinatorial structure induced by stable matchings or maximumweight matchings is an antimatroid. Moreover, we demonstrate that every antimatroid admits such a representation by stable matchings and maximumweight matchings.
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This paper presents an efficient method to perform Structured Matrix Approximation by Separation and Hierarchy (SMASH), when the original dense matrix is associated with a kernel function. Given points in a domain, a tree structure is first constructed based on an adaptive partitioning of the computational domain to facilitate subsequent approximation procedures. In contrast to existing schemes based on either analytic or purely algebraic approximations, SMASH takes advantage of both approaches and greatly improves the efficiency. The algorithm follows a bottomup traversal of the tree and is able to perform the operations associated with each node on the same level in parallel. A strong rankrevealing factorization is applied to the initial analytic approximation in the separation regime so that a special structure is incorporated into the final nested bases. As a consequence, the storage is significantly reduced on one hand and a hierarchy of the original grid is constructed on the o
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We establish a coarse version of the CartanHadamard theorem, which states that proper coarsely convex spaces are coarsely homotopy equivalent to the open cones of their ideal boundaries. As an application, we show that such spaces satisfy the coarse BaumConnes conjecture. Combined with the result of OsajdaPrzytycki, it implies that systolic groups and locally finite systolic complexes satisfy the coarse BaumConnes conjecture.
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Let K be a field equipped with a valuation. Tropical varieties over K can be defined with a theory of Gr{\"o}bner bases taking into account the valuation of K. While generalizing the classical theory of Gr{\"o}bner bases, it is not clear how modern algorithms for computing Gr{\"o}bner bases can be adapted to the tropical case. Among them, one of the most efficient is the celebrated F5 Algorithm of Faug{\`e}re. In this article, we prove that, for homogeneous ideals, it can be adapted to the tropical case. We prove termination and correctness. Because of the use of the valuation, the theory of tropical Gr{\"o}bner bases is promising for stable computations over polynomial rings over a padic field. We provide numerical examples to illustrate timecomplexity and padic stability of this tropical F5 algorithm.
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In this work, the set of quasiprimary ideals of a commutative ring with identity is equipped with a topology and is called quasiprimary spectrum. Some topological properties of this space are examined. Further, a sheaf of rings on the quasiprimary spectrum is constructed and it is shown that this sheaf is the direct image sheaf with respect to the inclusion map from the prime spectrum of a ring to the quasiprimary spectrum of the same ring.
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We propose a onedimensional SaintVenant (also known as open channel or shallow water) equation model for overland flows including a water inputoutput source term. We derive the model from the twodimensional NavierStokes equations under the shallow water assumption, with boundary conditions including recharge via ground infiltration and runoff. We show that the energyconsistency of the resulting SaintVenant model is strictly dependent upon the assumed level of rain or rechargeinduced friction. The proposed model extends most extant models by adding more scope depending on friction terms that depend on the rate of water entering or exiting the flow via recharge and infiltration. The obtained entropy relation for our model validate it both mathematically and physically. We compare both models computationally based on a kinetic finite volume scheme; in particular, we provide numerical evidence that the two models may show drastically different results, where the model conditione
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In this paper, we consider the semilinear heat equations under Dirichlet boundary condition \[ u_{t}\left(x,t\right)=\Delta u\left(x,t\right)+f(u(x,t)), & \left(x,t\right)\in \Omega\times\left(0,+\infty\right), u\left(x,t\right)=0, & \left(x,t\right)\in\partial \Omega\times\left[0,+\infty\right), u\left(x,0\right)=u_{0}\geq0, & x\in\overline{\Omega}, \] where $\Omega$ is a bounded domain of $\mathbb{R}^{N}$ $(N\geq1)$ with smooth boundary $\partial\Omega$. The main contribution of our work is to introduce a new condition \[ (C) \alpha \int_{0}^{u}f(s)ds \leq uf(u)+\beta u^{2}+\gamma,\,\,u>0 \] for some $\alpha, \beta, \gamma>0$ with $0<\beta\leq\frac{\left(\alpha2\right)\lambda_{0}}{2}$, where $\lambda_{0}$ is the first eigenvalue of Laplacian $\Delta$, and we use the concavity method to obtain the blowup solutions to the semilinear heat equations. In fact, it will be seen that the condition (C) improves the conditions known so far.
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Extended welded links are a generalization of Fenn, Rim\'{a}nyi, and Rourke's welded links. Their braided counterpart are extended welded braids, which are closely related to ribbon braids and loop braids. In this paper we prove versions of Alexander and Markov's theorems for extended welded braids and links, following Kamada's approach to the case of welded objects.
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In this work we obtain a Liouville theorem for positive, bounded solutions of the equation $$ (\Delta)^s u= h(x_N)f(u) \quad \hbox{in }\mathbb{R}^{N} $$ where $(\Delta)^s$ stands for the fractional Laplacian with $s\in (0,1)$, and the functions $h$ and $f$ are nondecreasing. The main feature is that the function $h$ changes sign in $\mathbb{R}$, therefore the problem is sometimes termed as indefinite. As an application we obtain a priori bounds for positive solutions of some boundary value problems, which give existence of such solutions by means of bifurcation methods.
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Power distribution system's security is often jeopardized by network uncertainties, usually caused by the recent increase in intermittent power injections from renewable energy sources, and nontraditional energy demands. Naturally, there is an ever increasing need for a tool for assessing distribution system security. This paper presents a fast and reliable tool for constructing \emph{inner approximations} of steady state voltage stability regions in multidimensional injection space such that every point in our constructed region is guaranteed to be solvable. Numerical simulations demonstrate that our approach outperforms all existing inner approximation methods in most cases. Furthermore, the constructed regions are shown to cover substantial fractions of the true voltage stability region. The paper will later discuss a number of important applications of the proposed technique, including fast screening for viable injection changes, constructing an effective solvability index and rig
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We show that the $G$equivariant coherent derived category of $D$modules on $\mathfrak{g}$ admits an orthogonal decomposition in to blocks indexed by cuspidal data (in the sense of Lusztig). Each block admits a monadic description in terms a certain differential graded algebra related to the homology of Steinberg varieties, which resembles a "triple affine" Hecke algebra. Our results generalize the work of Rider and RiderRussell on constructible complexes on the nilpotent cone, and the earlier work of the author on the abelian category of equivariant $D$modules on $\mathfrak{g}$. However, the algebra controlling the entire derived category of $D$modules appears to be substantially more complicated than either of these special cases, as evidenced by the nonsplitting of the Mackey filtration on the monad controlling each block. This paper is a sequel to arXiv:1510.02452.
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A modular semilattice is a semilattice generalization of a modular lattice. We establish a Birkhofftype representation theorem for modular semilattices, which says that every modular semilattice is isomorphic to the family of ideals in a certain poset with additional relations.This new poset structure, which we axiomatize in this paper, is called a PPIP (projective poset with inconsistent pairs). A PPIP is a common generalization of a PIP (poset with inconsistent pairs) and a projective ordered space. The former was introduced by Barth\'elemy and Constantin for establishing Birkhofftype theorem for median semilattices, and the latter by Herrmann, Pickering, and Roddy for modular lattices. We show the $\Theta (n)$ representation complexityand a construction algorithm for PPIPrepresentations of $(\wedge, \vee)$closed sets in the product $L^n$ of modular semilattice $L$. This generalizes the results of Hirai and Oki for a special median semilattice $S_k$. We also investigate implicati
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A randomisation of the Berele insertion algorithm is proposed, where the insertion of a letter to a symplectic Young tableau leads to a distribution over the set of symplectic Young tableaux. Berele's algorithm provides a bijection between words from an alphabet and a symplectic Young tableau along with a recording oscillating tableau. The randomised version of the algorithm is achieved by introducing a parameter $0 < q < 1$. The classic Berele algorithm corresponds to letting the parameter $q \to 0$. The new version provides a probabilistic framework that allows to prove Littlewoodtype identities for a $q$deformation of the symplectic Schur functions. These functions correspond to multilevel extensions of the continuous $q$Hermite polynomials. Finally, we show that when both the original and the $q$modified insertion algorithms are applied to a random word then the shape of the symplectic Young tableau evolves as a Markov chain on the set of partitions.
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Friday saw the largest global ransomware attack in internet history, and the world did not handle it well. We're only beginning to calculate the damage inflicted by the WannaCry program  in both dollars and lives lost from hospital downtime  but at the same time, we're also calculating blame. There's a long list of parties responsible, including the criminals, the NSA, and the victims themselves  but the most controversial has been Microsoft itself. The attack exploited a Windows networking protocol to spread within networks, and while Microsoft released a patch nearly two months ago, itâs become painfully clear that patch didnât reach all users. Microsoft was following the best practices for security and still left hundreds of thousands of computers vulnerable, with dire consequences. Was it good enough? If you're still running Windows XP today and you do not pay for Microsoft's extended support, the blame for this whole thing rests solely on your shoulders  whether that be
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The rational Calogero model based on an arbitrary rank$n$ Coxeter root system is spherically reduced to a superintegrable angular model of a particle moving on $S^{n1}$ subject to a very particular potential singular at the reflection hyperplanes. It is outlined how to find conserved charges and to construct intertwining operators. We deform these models in a ${\cal PT}$symmetric manner by judicious complex coordinate transformations, which render the potential less singular. The ${\cal PT}$ deformation does not change the energy eigenvalues but in some cases adds a previously unphysical tower of states. For integral couplings the new and old energy levels coincide, which roughly doubles the previous degeneracy and allows for a conserved nonlinear supersymmetry charge. We present the details for the generic ranktwo ($A_2$, $G_2$) and all rankthree Coxeter systems ($AD_3$, $BC_3$ and $H_3$), including a reducible case ($A_1^{\otimes 3}$).
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We consider the family of dehomogenized Loud's centers $X_{\mu}=y(x1)\partial_x+(x+Dx^2+Fy^2)\partial_y,$ where $\mu=(D,F)\in\mathbb{R}^2,$ and we study the number of critical periodic orbits that emerge or dissapear from the polycycle at the boundary of the period annulus. This number is defined exactly the same way as the wellknown notion of cyclicity of a limit periodic set and we call it criticality. The previous results on the issue for the family $\{X_{\mu},\mu\in\mathbb{R}^2\}$ distinguish between parameters with criticality equal to zero (regular parameters) and those with criticality greater than zero (bifurcation parameters). A challenging problem not tackled so far is the computation of the criticality of the bifurcation parameters, which form a set $\Gamma_{B}$ of codimension 1 in $\mathbb{R}^2$. In the present paper we succeed in proving that a subset of $\Gamma_{B}$ has criticality equal to one.
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We introduce and analyze a discontinuous Galerkin method for a timeharmonic eddy current problem formulated in terms of the magnetic field. The scheme is obtained by putting together a DG method for the approximation of the vector field variable representing the magnetic field in the conductor and a DG method for the Laplace equation whose solution is a scalar magnetic potential in the insulator. The transmission conditions linking the two problems are taken into account weakly in the global discontinuous Galerkin scheme. We prove that the numerical method is uniformly stable and obtain quasioptimal error estimates in the DGenergy norm.
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In 1987 Harris proved (Proc. Amer. Math. Soc., 101)  among others that for each $1\le p<2$ there exists a twodimensional function $f\in L^p$ such that its triangular WalshFourier series diverges almost everywhere. In this paper we investigate the Fej\'er (or $(C,1)$) means of the triangle two variable WalshFourier series of $L^1$ functions. Namely, we prove the a.e. convergence $\sigma_n^{\bigtriangleup}f = \frac{1}{n}\sum_{k=0}^{n1}S_{k, nk}f\to f$ ($n\to\infty$) for each integrable twovariable function $f$.
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This paper introduces an Algebraic MultiScale method for simulation of flow in heterogeneous porous media with embedded discrete Fractures (FAMS). First, multiscale coarse grids are independently constructed for both porous matrix and fracture networks. Then, a map between coarse and finescale is obtained by algebraically computing basis functions with local support. In order to extend the localization assumption to the fractured media, four types of basis functions are investigated: (1) DecoupledAMS, in which the two media are completely decoupled, (2) FracAMS and (3) RockAMS, which take into account only oneway transmissibilities, and (4) CoupledAMS, in which the matrix and fracture interpolators are fully coupled. In order to ensure scalability, the FAMS framework permits full flexibility in terms of the resolution of the fracture coarse grids. Numerical results are presented for two and threedimensional heterogeneous test cases. During these experiments, the performance
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This paper presents the development of an Adaptive Algebraic Multiscale Solver for Compressible flow (CAMS) in heterogeneous porous media. Similar to the recently developed AMS for incompressible (linear) flows [Wang et al., JCP, 2014], CAMS operates by defining primal and dualcoarse blocks on top of the finescale grid. These coarse grids facilitate the construction of a conservative (finite volume) coarsescale system and the computation of local basis functions, respectively. However, unlike the incompressible (elliptic) case, the choice of equations to solve for basis functions in compressible problems is not trivial. Therefore, several basis function formulations (incompressible and compressible, with and without accumulation) are considered in order to construct an efficient multiscale prolongation operator. As for the restriction operator, CAMS allows for both multiscale finite volume (MSFV) and finite element (MSFE) methods. Finally, in order to resolve highfrequency error
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The numerical invariants (global) cohomological length, (global) cohomological width, and (global) cohomological range of complexes (algebras) are introduced. Cohomological range leads to the concepts of derived bounded algebras and strongly derived unbounded algebras naturally. The first and second BrauerThrall type theorems for the bounded derived category of a finitedimensional algebra over an algebraically closed field are obtained. The first BrauerThrall type theorem says that derived bounded algebras are just derived finite algebras. The second BrauerThrall type theorem says that an algebra is either derived discrete or strongly derived unbounded, but not both. Moreover, piecewise hereditary algebras and derived discrete algebras are characterized as the algebras of finite global cohomological width and finite global cohomological length respectively.
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The GNOME project has, after a period of contemplation, put forward a proposal to move to a GitLab installation on GNOME's infrastructure. "We are confident that GitLab is a good choice for GNOME, and we can’t wait for GNOME to modernise our developer experience with it. It will provide us with vastly more effective tools, an easier landing for newcomers, and lots of opportunities to improve the way that we work. We're ready to start working on the migration."This wiki page describes the idea in detail.
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In the previous report [Phys. Rev. B {\bf{62}} 13812 (2000)], by proposing the mechanism under which electric conductivity is caused by the activational hopping conduction with the Wigner surmise of the level statistics, the temperaturedependent of electronic conductivity of a highly disordered carbon system was evaluated including apparent metalinsulator transition. Since the system consists of small pieces of graphite, it was assumed that the reason why the level statistics appears is due to the behavior of the quantum chaos in each granular graphite. In this article, we revise the assumption and show another origin of the Wigner surmise, which is more natural for the carbon system based on a recent investigation of graph zeta function in graph theory.
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We definitively solve the old problem of finding a minimal integrity basis of polynomial invariants of the fourthorder elasticity tensor C. Decomposing C into its SO(3)irreducible components we reduce this problem to finding joint invariants of a triplet (a, b, D), where a and b are secondorder harmonic tensors, and D is a fourthorder harmonic tensor. Combining theorems of classical invariant theory and formal computations, a minimal integrity basis of 297 polynomial invariants for the elasticity tensor is obtained for the first time.
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Conceptually, a rough number is a positive integer with no small prime factors. Formally, for real numbers $x$ and $y$, let $\Phi(x,y)$ denote the number of positive integers at most $x$ with no prime factors less than $y$. In this paper we establish the lower bound $\Phi(n,p)\geq \lfloor 2n/p \rfloor +1$ when $p\geq 11$ is prime and $n\geq 2p$.
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In this paper we study both the Cauchy problem and the initial boundary value problem for the equation $\partial_tu+\mbox{div}\left(\nabla\Delta u{\bf g}(\nabla u)\right)=0$. This equation has been proposed as a continuum model for kinetic roughening and coarsening in thin films. In the Cauchy problem, we obtain that local existence of a weak solution is guaranteed as long as the vectorvalued function ${\bf g}$ is continuous and the initial datum $u_0$ lies in $C^1(\mathbb{R}^N)$ with $\max_{\mathbb{R}^N}\nabla u_0<\infty$, and that the global existence assertion also holds true if we further assume that ${\bf g}$ satisfies the growth condition ${\bf g}(\xi) \leq c\xi^\alpha$ for some $c>0, \alpha\in (2, 3)$ and the norm of $u_0$ in the space $L^{\frac{(\alpha1)N}{3\alpha}}(\mathbb{R}^N) $ is sufficiently small. This is done by exploring various properties of the biharmonic heat kernel. In the initial boundary value problem, we assume that ${\bf g}$ is continuous and sa
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Quartz looks at recent developments in the Artifex v. Hancom case. Artifex makes Ghostscript, an opensource (GPL) PDF interpreter. Hancom used Ghostscript in its Hancom Office product and did not abide by the license, so Artifex sued Hancom. "The enforceability of open source licenses like the GNU GPL has long been an open legal question. The Federal Circuit Court of Appeals held in a 2006 case, Jacobsen v. Katzer, that violations of open source licenses could be treated like copyright claims. But whether they could legally considered breaches of contract had yet to be determined, until the issue came up in Artifex v. Hancom. That happened when Hancom issued a motion to dismiss the case on the grounds that the company didn’t sign anything, so the license wasn’t a real contract." Judge Jacqueline Scott Corley disagreed with Hancom and said: "These allegations sufficiently plead the existence of a contract." (Thanks to Paul Wise)
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We prove a dynamical Shafarevich theorem on the finiteness of the set of isomorphism classes of rational maps with fixed degeneracies. More precisely, fix an integer d at least 2 and let K be either a number field or the function field of a curve X over a field k, where k is of characteristic zero or p>2d2 that is either algebraically closed or finite. Let S be a finite set of places of K. We prove the finiteness of the set of isomorphism classes of rational maps over K with a natural kind of good reduction outside of S. We also prove auxiliary results on finiteness of reduced effective divisors in $\mathbb{P}^1_K$ with good reduction outside of S and on the existence of global models for rational maps.
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The distance standard deviation, which arises in distance correlation analysis of multivariate data, is studied as a measure of spread. New representations for the distance standard deviation are obtained in terms of Gini's mean difference and in terms of the moments of spacings of order statistics. Inequalities for the distance variance are derived, proving that the distance standard deviation is bounded above by the classical standard deviation and by Gini's mean difference. Further, it is shown that the distance standard deviation satisfies the axiomatic properties of a measure of spread. Explicit closedform expressions for the distance variance are obtained for a broad class of parametric distributions. The asymptotic distribution of the sample distance variance is derived.
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