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We present recent progress in theory of local conformal nets which is an operator algebraic approach to study chiral conformal field theory. We emphasize representation theoretic aspects and relations to theory of vertex operator algebras which gives a different and algebraic formulation of chiral conformal field theory.
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We show that, assuming Vojta's height conjecture, the height of a rational point on an algebraically hyperbolic variety can be bounded "uniformly" in families. This generalizes a result of SuIon Ih for curves of genus at least two to higherdimensional varieties. As an application, we show that, assuming Vojta's height conjecture, the height of a rational point on a curve of general type is uniformly bounded. Finally, we prove a similar result for smooth hyperbolic surfaces with $c_1^2 > c_2$.
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In this note we prove that the Borel class of representations of 3manifold groups to PGL(n,C) is preserved under Cartan involution up to sign. For representations to PGL(3,C) this is implied by a more general result of E. Falbel and Q. Wang, however our proof appears to be much shorter for that special case.
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The notion of weakly monotone functions extends the classical definition of monotone function, that can be traced back to H.Lebesgue. It was introduced, in the setting of Sobolev spaces, by J.Manfredi, and thoroughly investigated in the more general framework of OrliczSobolev spaces by diverse authors, including T.Iwaniec, J.Kauhanen, P.Koskela, J.Maly, J.Onninen, X.Zhong. The present paper complements and augments the available theory of pointwise regularity properties of weakly monotone functions in OrliczSobolev spaces. In particular, a variant is proposed in a customary condition ensuring the continuity of functions from these spaces which avoids a technical additional assumption, and applies to certain situations when the latter is not fulfilled. The continuity outside sets of zero Orlicz capacity, and outside sets of (generalized) zero Hausdorff measure, will are also established when everywhere continuity fails.
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We consider the problem of finding confidence intervals for the risk of forecasting the future of a stationary, ergodic stochastic process, using a model estimated from the past of the process. We show that a bootstrap procedure provides valid confidence intervals for the risk, when the data source is sufficiently mixing, and the loss function and the estimator are suitably smooth. Autoregressive (AR(d)) models estimated by least squares obey the necessary regularity conditions, even when misspecified, and simulations show that the finite sample coverage of our bounds quickly converges to the theoretical, asymptotic level. As an intermediate step, we derive sufficient conditions for asymptotic independence between empirical distribution functions formed by splitting a realization of a stochastic process, of independent interest.
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A linear operator $T$ between two latticenormed locally solid Riesz spaces is said to be $p_\tau$continuous if, for any $p_\tau$null net $(x_\alpha)$, the net $(Tx_\alpha)$ is $p_\tau$null, and $T$ is also said to be $p_\tau$bounded operator if it sends $p_\tau$bounded subsets to $p_\tau$bounded subsets. They are generalize several known classes of operators such as continuous, order continuous, $p$continuous, order bounded, $p$bounded operators, etc. We also study $up_\tau$continuous operators between latticenormed locally solids Riesz spaces.
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Lowrank tensor approximations are plagued by a wellknown problem  a tensor may fail to have a best rank$r$ approximation. Over $\mathbb{R}$, it is known that such failures can occur with positive probability, sometimes with certainty. We will show that while such failures still occur over $\mathbb{C}$, they happen with zero probability. In fact we establish a more general result with useful implications on recent scientific and engineering applications that rely on sparse and/or lowrank approximations: Let $V$ be a complex vector space with a Hermitian inner product, and $X$ be a closed irreducible complex analytic variety in $V$. Given any complex analytic subvariety $Z \subseteq X$ with $\dim Z < \dim X$, we prove that a general $p \in V$ has a unique best $X$approximation $\pi_X (p)$ that does not lie in $Z$. In particular, it implies that over $\mathbb{C}$, any tensor almost always has a unique best rank$r$ approximation when $r$ is less than the generic rank. Our result
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Let $R$ be a ring, let $\mathfrak{a}\subseteq R$ be an ideal, and let $M$ be an $R$module. Let $\Gamma_{\mathfrak{a}}$ denote the $\mathfrak{a}$torsion functor. Conditions are given for the (weakly) associated primes of $\Gamma_{\mathfrak{a}}(M)$ to be the (weakly) associated primes of $M$ containing $\mathfrak{a}$, and for the (weakly) associated primes of $M/\Gamma_{\mathfrak{a}}(M)$ to be the (weakly) associated primes of $M$ not containing $\mathfrak{a}$.
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For a graph $G$ and a nonnegative integral weight function $w$ on the vertex set of $G$, a set $S$ of vertices of $G$ is $w$safe if $w(C)\geq w(D)$ for every component $C$ of the subgraph of $G$ induced by $S$ and every component $D$ of the subgraph of $G$ induced by the complement of $S$ such that some vertex in $C$ is adjacent to some vertex of $D$. The minimum weight $w(S)$ of a $w$safe set $S$ is the safe number $s(G,w)$ of the weighted graph $(G,w)$, and the minimum weight of a $w$safe set that induces a connected subgraph of $G$ is its connected safe number $cs(G,w)$. Bapat et al. showed that computing $cs(G,w)$ is NPhard even when $G$ is a star. For a given weighted tree $(T,w)$, they described an efficient $2$approximation algorithm for $cs(T,w)$ as well as an efficient $4$approximation algorithm for $s(T,w)$. Addressing a problem they posed, we present a PTAS for the connected safe number of a weighted tree. Our PTAS partly relies on an exact pseudopolynomial time algor
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We propose a generative graph model for electrical infrastructure networks that accounts for heterogeneity in both node and edge type. To inform the design of this model, we analyze the properties of power grid graphs derived from the U.S. Eastern Interconnection, Texas Interconnection, and Poland transmission system power grids. Across these datasets, we find subgraphs induced by nodes of the same voltage level exhibit shared structural properties atypical to smallworld networks, including low local clustering, large diameter and large average distance. On the other hand, we find subgraphs induced by transformer edges linking nodes of different voltage types contain a more limited structure, consisting mainly of small, disjoint star graphs. The goal of our proposed model is to match both these inter and intranetwork properties by proceeding in two phases: we first generate subgraphs for each voltage level and then generate transformer edges that connect these subgraphs. The first ph
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We prove a generalized Dade's Lemma for quotients of local rings by ideals generated by regular sequences. That is, given a pair of finitely generated modules over such a ring with algebraically closed residue field, we prove a sufficient (and necessary) condition for the vanishing of all higher Ext or Tor of the modules. This condition involves the vanishing of all higher Ext or Tor of the modules over all quotients by a minimal generator of the ideal generated by the regular sequence.
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A fully discrete approximation of the onedimensional stochastic heat equation driven by multiplicative spacetime white noise is presented. The standard finite difference approximation is used in space and a stochastic exponential method is used for the temporal approximation. Observe that the proposed exponential scheme does not suffer from any kind of CFLtype step size restriction. When the drift term and the diffusion coefficient are assumed to be globally Lipschitz, this explicit time integrator allows for error bounds in $L^q(\Omega)$, for all $q\geq2$, improving some existing results in the literature. On top of this, we also prove almost sure convergence of the numerical scheme. In the case of nonglobally Lipschitz coefficients, we provide sufficient conditions under which the numerical solution converges in probability to the exact solution. Numerical experiments are presented to illustrate the theoretical results.
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We give a new expression for the law of the eigenvalues of the discrete Anderson model on the finite interval $[0,N]$, in terms of two random processes starting at both ends of the interval. Using this formula, we deduce that the tail of the eigenvectors behaves approximatelylike $\exp(\sigma B\_{nk}\gamma\frac{nk}{4})$ where $B\_{s}$ is the Brownian motion and $k$ is uniformly chosen in $[0,N]$ independentlyof $B\_{s}$. A similar result has recently been shown by B. Rifkind and B. Virag in the critical case, that is, when the random potential is multiplied by a factor $\frac{1}{\sqrt{N}}$
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We present exponential generating function analogues to two classical identities involving the ordinary generating function of the complete homogeneous symmetric functions. After a suitable specialization the new identities reduce to identities involving the first and second order Eulerian polynomials. The study of these identities led us to consider a family of symmetric functions associated with a class of permutations introduced by Gessel and Stanley, known in the literature as Stirling permutations. In particular, we define certain type statistics on Stirling permutations that refine the statistics of descents, ascents and plateaux and we show that their refined versions are equidistributed, generalizing a result of B\'ona. The definition of this family of symmetric functions extends to the generality of $r$Stirling permutations. We discuss some occurrences of these symmetric functions in the cases of $r=1$ and $r=2$.
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The simple length spectrum of a Riemannian manifold is the set of lengths of its simple closed geodesics. We prove that in any Riemannian 2sphere whose simple length spectrum consists of only one element L, any geodesic is simple closed with length L. We also show that, if the simple length spectrum of a Riemannian 2sphere contains at most two elements, for at least one such element L every point of the 2sphere lies on a simple closed geodesic of length L.
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During early development, waves of activity propagate across the retina and play a key role in the proper wiring of the early visual system. During a particular phase of the retina development (stage II) these waves are triggered by a transient network of neurons called Starburst Amacrine Cells (SACs) showing a bursting activity which disappears upon further maturation. The underlying mechanisms of the spontaneous bursting and the transient excitability of immature SACs are not completely clear yet. While several models have tried to reproduce retinal waves, none of them is able to mimic the rhythmic autonomous bursting of individual SACs and understand how these cells change their intrinsic properties during development. Here, we introduce a mathematical model, grounded on biophysics, which enables us to reproduce the bursting activity of SACs and to propose a plausible, generic and robust, mechanism that generates it. Based on a bifurcation analysis we exhibit a few biophysical param
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Smoothed particle hydrodynamics (SPH) has been extensively used to model high and low Reynolds number flows, free surface flows and collapse of dams, study porescale flow and dispersion, elasticity, and thermal problems. In different applications, it is required to have a stable and accurate discretization of the elliptic operator with homogeneous and heterogeneous coefficients. In this paper, the stability and approximation analysis of different SPH discretization schemes (traditional and new) of the diagonal elliptic operator for homogeneous and heterogeneous media are presented. The optimum and new discretization scheme is also proposed. This scheme enhances the Laplace approximation (Brookshaw's scheme (1985) and Schwaiger's scheme (2008)) used in the SPH community for thermal, viscous, and pressure projection problems with an isotropic elliptic operator. The numerical results are illustrated by numerical examples, where the comparison between different versions of the meshless di
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A theorem of Erdos asserts that every infinite subset of Euclidean nspace R^n has a subset of the same cardinality having no repeated distances. This theorem is generalized here as follows: If (R^n,E) is an algebraic hypergraph that does not have an infinite, complete subset, then every infinite subset of it has an independent subset of the same cardinality.
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The free sum is a basic geometric operation among convex polytopes. This note focuses on the relationship between the normalized volume of the free sum and that of the summands. In particular, we show that the normalized volume of the free sum of full dimensional polytopes is precisely the product of the normalized volumes of the summands.
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In this work we develop an effective Monte Carlo method for estimating sensitivities, or gradients of expectations of sufficiently smooth functionals, of a reflected diffusion in a convex polyhedral domain with respect to its defining parameters  namely, its initial condition, drift and diffusion coefficients, and directions of reflection. Our method, which falls into the class of infinitesimal perturbation analysis (IPA) methods, uses a probabilistic representation for such sensitivities as the expectation of a functional of the reflected diffusion and its associated derivative process. The latter process is the unique solution to a constrained linear stochastic differential equation with jumps whose coefficients, domain and directions of reflection are modulated by the reflected diffusion. We propose an asymptotically unbiased estimator for such sensitivities using an Euler approximation of the reflected diffusion and its associated derivative process. Proving that the Euler appro
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We formulate a stochastic game of mean field type where the agents solve optimal stopping problems and interact through the proportion of players that have already stopped. Working with a continuum of agents, typical equilibria become functions of the common noise that all agents are exposed to, whereas idiosyncratic randomness can be eliminated by an Exact Law of Large Numbers. Under a structural monotonicity assumption, we can identify equilibria with solutions of a simple equation involving the distribution function of the idiosyncratic noise. Solvable examples allow us to gain insight into the uniqueness of equilibria and the dynamics in the population.
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The axial anomaly of the massless free fermion is given by the index of the corresponding Dirac operator. We use the BV formalism and the methods of equivariant quantization of Costello and Gwilliam to produce a new, mathematical derivation of this result. We formalize through these methods the conventional wisdom that the anomaly measures the failure to construct a welldefined partition function.
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We prove several classification results for $p$Laplacian problems on bounded and unbounded domains, and deal with qualitative properties of signchanging solutions to $p$Laplacian equations on $\mathbb R^N$ involving critical nonlinearities. Moreover, on radial domains we characterise the compactness of possibly signchanging PalaisSmale sequences.
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Motivated by considerations of euclidean quantum gravity, we investigate a central question of spectral geometry, namely the question of reconstructability of compact Riemannian manifolds from the spectra of their Laplace operators. To this end, we study analytic paths of metrics that induce isospectral LaplaceBeltrami operators over oriented compact surfaces without boundary. Applying perturbation theory, we show that sets of conformally equivalent metrics on such surfaces contain no nontrivial convex subsets. This indicates that cases where the manifolds cannot be reconstructed from their spectra are highly constrained.
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We give a global geometric decomposition of continuously differentiable vector fields on $\mathbb{R}^n$. More precisely, given a vector field of class $\mathcal{C}^{1}$ on $\mathbb{R}^{n}$, and a geometric structure on $\mathbb{R}^n$, we provide a unique global decomposition of the vector field as the sum of a left (right) gradientlike vector field (naturally associated to the geometric structure) with potential function vanishing at the origin, and a vector field which is left (right) orthogonal to the identity, with respect to the geometric structure. As application, we provide a criterion to decide topological conjugacy of complete vector fields of class $\mathcal{C}^1$ on $\mathbb{R}^{n}$ based on topological conjugacy of the corresponding parts given by the associated geometric decompositions.
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We consider the cost of general orthogonal range queries in random quadtrees. The cost of a given query is encoded into a (random) function of four variables which characterize the coordinates of two opposite corners of the query rectangle. We prove that, when suitably shifted and rescaled, the random cost function converges uniformly in probability towards a random field that is characterized as the unique solution to a distributional fixedpoint equation. Our results imply for instance that the worst case query satisfies the same asymptotic estimates as a typical query, and thereby resolve an old question of Chanzy, Devroye and ZamoraCura [\emph{Acta Inf.}, 37:355383, 2000]
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Amazon has announced the release of FreeRTOS kernel version 10, with a new license: "FreeRTOS was created in 2003 by Richard Barry. It rapidly became popular, consistently ranking very high in EETimes surveys on embedded operating systems. After 15 years of maintaining this critical piece of software infrastructure with very limited human resources, last year Richard joined Amazon. Today we are releasing the core open source code as FreeRTOS kernel version 10, now under the MIT license (instead of its previous modified GPLv2 license). Simplified licensing has long been requested by the FreeRTOS community. The specific choice of the MIT license was based on the needs of the embedded systems community: the MIT license is commonly used in open hardware projects, and is generally whitelisted for enterprise use." While the modified GPLv2 was removed, it was replaced with a slightly modified MIT license that adds: "If you wish to use our Amazon FreeRTOS name, please do so in a fair
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This paper analyzes an emerging architecture of cellular network utilizing both planar base stations uniformly distributed in Euclidean plane and base stations located on roads. An example of this architecture is that where, in addition to conventional planar cellular base stations and users, vehicles also play the role of both base stations and users. A Poisson line process is used to model the road network and, conditionally on the lines, linear Poisson point processes are used to model the vehicles on the roads. The conventional planar base stations and users are modeled by independent planar Poisson point processes. The joint stationarity of the elements in this model allows one to use Palm calculus to investigate statistical properties of such a network. Specifically, this paper discusses two different Palm distributions, with respect to the user point processes depending on its type: planar or vehicular. We derive the distance to the nearest base station, the association of the t
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Due to its excellent shockcapturing capability and high resolution, the WENO scheme family has been widely used in varieties of compressive flow simulation. However, for problems containing strong shocks and contact discontinuities, such as the Lax shock tube problem, the WENO scheme still produces numerical oscillations. To avoid such numerical oscillations, the characteristicwise construction method should be applied. Compared to componentwise reconstruction, characteristicwise reconstruction leads to much more computational cost and thus is not suite for large scale simulation such as direct numeric simulation of turbulence. In this paper, an adaptive characteristicwise reconstruction WENO scheme, i.e. the AdaWENO scheme, is proposed to improve the computational efficiency of the characteristicwise reconstruction method. The new scheme performs characteristicwise reconstruction near discontinuities while switching to componentwise reconstruction for smooth regions. Meanwhile
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In this paper, we discuss the implementation of a cell based smoothed finite element method (CSFEM) within the commercial finite element software Abaqus. The salient feature of the CSFEM is that it does not require an explicit form of the derivative of the shape functions and there is no isoparametric mapping. This implementation is accomplished by employing the user element subroutine (UEL) feature of the software. The details on the input data format together with the proposed user element subroutine, which forms the core of the finite element analysis are given. A few benchmark problems from linear elastostatics in both two and three dimensions are solved to validate the proposed implementation. The developed UELs and the associated input files can be downloaded from Github repository link: https://github.com/nsundar/SFEM\_in\_Abaqus.
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We provide a natural answer to Lewis Carroll's pillow problem of what is the probability that a triangle is obtuse, Prob(Obtuse). This arises by straightforward combination of a) Kendall's Theorem  that the space of all triangles is a sphere  and b) the natural map sending triangles in space to points in this shape sphere. The answer is 3/4. Our method moreover readily generalizes to a wider class of problems, since a) and b) both have many applications and admit large generalizations: Shape Theory. An elementary and thus widely accessible prototype for Shape Theory is thereby desirable, and extending Kendall's alreadynotable prototype a) by demonstrating that b) readily solves Lewis Carroll's wellknown pillow problem indeed provides a memorable and considerably stronger prototype. This is a prototype of, namely, mapping flat geometry problems directly realized in a space to shape space, where differentialgeometric tools are readily available to solve the problem and then finally
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Let $G$ be a finitely generated group of isometries of $\HH^m$, hyperbolic $m$space, for some positive integer $m$. %or equivalently elements of $PSL(2,\CC)$. The discreteness problem is to determine whether or not $G$ is discrete. Even in the case of a two generator nonelementary subgroup of $\HH^2$ (equivalently $PSL(2,\mathbb{R})$) the problem requires an algorithm \cite{GM,JGtwo}. If $G$ is discrete, one can ask when adjoining an $n$th root of a generator results in a discrete group. In this paper we address the issue for pairs of hyperbolic generators in $PSL(2, \RR)$ with disjoint axes and obtain necessary and sufficient conditions for adjoining roots for the case when the two hyperbolics have a hyperbolic product and are what as known as {\sl stopping generators} for the GilmanMaskit algorithm \cite{GM}. We give an algorithmic solution in other cases. It applies to all other types of pair of generators that arise in what is known as the {\sl intertwining case}. The results ar
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Providing a cutfree sequent calculus for S5 has a long history. The efforts in this direction are numerous, and each of them presents some difficulties. In this paper, we present a sequent calculus system for modal logic S5. It has in a sense subformula property. We show that the cut rule is admissible in this system.
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We introduce a variational theory for processes adapted to the multidimensional Brownian motion filtration that provides a differential structure allowing to describe infinitesimal evolution of Wiener functionals at very small scales. The main novel idea is to compute the "sensitivities" of processes, namely derivatives of martingale components and a weak notion of infinitesimal generators, via a finitedimensional approximation procedure based on controlled interarrival times and approximating martingales. The theory comes with convergence results that allow to interpret a large class of Wiener functionals beyond semimartingales as limiting objects of differential forms which can be computed path wisely over finitedimensional spaces. The theory reveals that solutions of BSDEs are minimizers of energy functionals w.r.t Brownian motion driving noise.
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We prove a robust version of Freiman's $3k  4$ theorem on the restricted sumset $A+_{\Gamma}B$, which applies when the doubling constant is at most $\tfrac{3+\sqrt{5}}{2}$ in general and at most $3$ in the special case when $A = B$. As applications, we derive robust results with other types of assumptions on popular sums, and structure theorems for sets satisfying almost equalities in discrete and continuous versions of the RieszSobolev inequality.
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The present paper aims at providing a numerical strategy to deal with PDEconstrained optimization problems solved with the adjoint method. It is done through out a unified formulation of the constraint PDE and the adjoint model. The resulting model is a nonconservative hyperbolic system and thus a finite volume scheme is proposed to solve it. In this form, the scheme sets in a single frame both constraint PDE and adjoint model. The forward and backward evolutions are controlled by a single parameter $\eta$ and a stable time step is obtained only once at each optimization iteration. The methodology requires the complete eigenstructure of the system as well as the gradient of the cost functional. Numerical tests evidence the applicability of the present technique
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Let $D$ be a strongly connected digraph. An arc set $S$ of $D$ is a \emph{restricted arccut} of $D$ if $DS$ has a nontrivial strong component $D_{1}$ such that $DV(D_{1})$ contains an arc. The \emph{restricted arcconnectivity} $\lambda'(D)$ of a digraph $D$ is the minimum cardinality over all restricted arccuts of $D$. A strongly connected digraph $D$ is \emph{$\lambda'$connected} when $\lambda'(D)$ exists. This paper presents a family $\cal{F}$ of strong digraphs of girth four that are not $\lambda'$connected and for every strong digraph $D\notin \cal{F}$ with girth four it follows that it is $\lambda'$connected. Also, an upper and lower bound for $\lambda'(D)$ are given.
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In 1991, Baker and Harman proved, under the assumption of the generalized Riemann hypothesis, that $\max_{ \theta \in [0,1) }\left\sum_{ n \leq x } \mu(n) e(n \theta) \right \ll_\epsilon x^{3/4 + \epsilon}$. The purpose of this note is to deduce an analogous bound in the context of polynomials over a finite field using Weil's Riemann Hypothesis for curves over a finite field. Our approach is based on the work of Hayes who studied exponential sums over irreducible polynomials.
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We give an new proof of the wellknown competitive exclusion principle in the chemostat model with $n$ species competing for a single resource, for any set of increasing growth functions. The proof is constructed by induction on the number of the species, after being ordered. It uses elementary analysis and comparisons of solutions of ordinary differential equations.
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We set up a correspondence between solutions of the YangMills equations on ${\mathbb R}\times S^3$ and in Minkowski spacetime via de Sitter space. Some known Abelian and nonAbelian exact solutions are rederived. For the Maxwell case we present a straightforward algorithm to generate an infinite number of explicit solutions, with fields and potentials in Minkowski coordinates given by rational functions of increasing complexity. We illustrate our method with a nontrivial example.
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A new grid system on a sphere is proposed that allows for straightforward implementation of both sphericalharmonicsbased spectral methods and gridpointbased multigrid methods. The latitudinal gridpoints in the new grid are equidistant and spectral transforms in the latitudinal direction are performed using ClenshawCurtis quadrature. The spectral transforms with this new grid and quadrature are shown to be exact within the machine precision provided that the grid truncation is such that there are at least 2N + 1 latitudinal gridpoints for the total truncation wavenumber of N. The new grid and quadrature is implemented and tested on a shallowwater equations model and the hydrostatic dry dynamical core of the global NWP model JMAGSM. The integration results obtained with the new quadrature are shown to be almost identical to those obtained with the conventional Gaussian quadrature on Gaussian grid.
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This paper is dedicated to the global existence and optimal decay estimates of strong solutions to the compressible viscoelastic flows in the whole space $\mathbb{R}^n$ with any $n\geq2$. We aim at extending those works by Qian \& Zhang and Hu \& Wang to the critical $L^p$ Besov space, which is not related to the usual energy space. With aid of intrinsic properties of viscoelastic fluids as in \cite{QZ1}, we consider a more complicated hyperbolicparabolic system than usual NavierStokes equations. We define "\emph{two effective velocities}", which allows us to cancel out the coupling among the density, the velocity and the deformation tensor. Consequently, the global existence of strong solutions is constructed by using elementary energy approaches only. Besides, the optimal timedecay estimates of strong solutions will be shown in the general $L^p$ critical framework, which improves those decay results due to Hu \& Wu such that initial velocity could be \textit{large high
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Polar codes are a class of linear block codes that provably achieves channel capacity, and have been selected as a coding scheme for $5^{\rm th}$ generation wireless communication standards. Successivecancellation (SC) decoding of polar codes has mediocre errorcorrection performance on short to moderate codeword lengths: the SCFlip decoding algorithm is one of the solutions that have been proposed to overcome this issue. On the other hand, SCFlip has a higher implementation complexity compared to SC due to the required loglikelihood ratio (LLR) selection and sorting process. Moreover, it requires a high number of iterations to reach good errorcorrection performance. In this work, we propose two techniques to improve the SCFlip decoding algorithm for lowrate codes, based on the observation of channelinduced error distributions. The first one is a fixed index selection (FIS) scheme to avoid the substantial implementation cost of LLR selection and sorting with no cost on errorco
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This is a continuation of our previous work with Botvinnik on the nontriviality of the secondary index invariant on spaces of metrics of positive scalar curvature, in which we take the fundamental group of the manifolds into account. We show that the secondary index invariant associated to the vanishing of the Rosenberg index can be highly nontrivial, for positive scalar curvature Spin manifolds with torsionfree fundamental groups which satisfy the BaumConnes conjecture. For example, we produce a compact Spin 6manifold such that its space of positive scalar curvature metrics has each rational homotopy group infinite dimensional. At a more technical level, we introduce the notion of "stable metrics" and prove a basic existence theorem for them, which generalises the GromovLawson surgery technique, and we also give a method for rounding corners of manifold with positive scalar curvature metrics.
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In this paper, we propose and study the iteration complexity of an inexact DouglasRachford splitting (DRS) method and a DouglasRachfordTseng's forwardbackward (FB) splitting method for solving twooperator and fouroperator monotone inclusions, respectively. The former method (although based on a slightly different mechanism of iteration) is motivated by the recent work of J. Eckstein and W. Yao, in which an inexact DRS method is derived from a special instance of the hybrid proximal extragradient (HPE) method of Solodov and Svaiter, while the latter one combines the proposed inexact DRS method (used as an outer iteration) with a Tseng's FB splitting type method (used as an inner iteration) for solving the corresponding subproblems. We prove iteration complexity bounds for both algorithms in the pointwise (nonergodic) as well as in the ergodic sense by showing that they admit two different iterations: one that can be embedded into the HPE method, for which the iteration complexi
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We prove that constant scalar curvature K\"ahler (cscK) manifolds with transcendental cohomology class are Ksemistable, naturally generalising the situation for polarised manifolds. Relying on a very recent result by R. Berman, T. Darvas and C. Lu regarding properness of the Kenergy, it moreover follows that cscK manifolds with finite automorphism group are uniformly Kstable. As a main step of the proof we establish, in the general K\"ahler setting, a formula relating the (generalised) DonaldsonFutaki invariant to the asymptotic slope of the Kenergy along weak geodesic rays.
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In this paper, a new online scheme is presented to design the optimal coordination control for the consensus problem of multiagent differential games by fuzzy adaptive dynamic programming (FADP), which brings together game theory, generalized fuzzy hyperbolic model (GFHM) and adaptive dynamic programming. In general, the optimal coordination control for multiagent differential games is the solution of the coupled HamiltonJacobi (HJ) equations. Here, for the first time, GFHMs are used to approximate the solution (value functions) of the coupled HJ equations, based on policy iteration (PI) algorithm. Namely, for each agent, GFHM is used to capture the mapping between the local consensus error and local value function. Since our scheme uses the singlenetwork rchitecture for each agent (which eliminates the action network model compared with dualnetwork architecture), it is a more reasonable architecture for multiagent systems. Furthermore, the approximation solution is utilized to
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Let $K$ be the fraction field of a complete discrete valuation ring, with algebraically closed residue field of characteristic $p > 0$. This paper studies the index of a smooth, proper $K$variety $X$ with logarithmic good reduction. We prove that it is prime to $p$ in `most' cases, for example if the Euler number of $X$ does not vanish, but (perhaps surprisingly) not always. We also fully characterise curves of genus $1$ with logarithmic good reduction, thereby completing classical results of T. Saito and Stix valid for curves of genus at least $2$.
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Recent research has shown the potential utility of probability distributions designed through hierarchical constructions which are conditionally Gaussian. This body of work is placed in a common framework and, through recursion, several classes of deep Gaussian processes are defined. The resulting samples have a Markovian structure with respect to the depth parameter and the effective depth of the process is interpreted in terms of the ergodicity, or nonergodicity, of the resulting Markov chain.
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We prove that the Tate conjecture is invariant under Homological Projective Duality (=HPD). As an application, we prove the Tate conjecture in the new cases of linear sections of determinantal varieties, and also in the cases of complete intersections of two quadrics. Furthermore, we extend the Tate conjecture from schemes to stacks and prove it for certain global orbifolds
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Under appropriate assumptions on the $N(\Omega)$fucntion, the De Giorgi process is presented in the framework of MusielakOrliczSobolev space to prove the H\"{o}lder continuity of fully nonlinear elliptic problems. As the applications, the H\"{o}lder continuity of the minimizers for a class of the energy functionals in MusielakOrliczSobolev spaces is proved; and furthermore, the H\"{o}lder continuity of the weak solutions for a class of fully nonlinear elliptic equations is provided.
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We establish heat kernel upper bounds for a continuoustime random walk under unbounded conductances satisfying an integrability assumption, where we correct and extend recent results by the authors to a general class of speed measures. The resulting heat kernel estimates are governed by the intrinsic metric induced by the speed measure. We also provide a comparison result of this metric with the usual graph distance, which is optimal in the context of the random conductance model with ergodic conductances.
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We develop a novel Hybrid HighOrder method for the simulation of Darcy flows in fractured porous media. The discretization hinges on a mixed formulation in the bulk region and on a primal formulation inside the fracture. Salient features of the method include a seamless treatment of nonconforming discretizations of the fracture, as well as the support of arbitrary approximation orders on fairly general meshes. For the version of the method corresponding to a polynomial degree k \v{e} 0, we prove convergence in h^{k+1} of the discretization error measured in an energylike norm. In the error estimate, we explicitly track the dependence of the constants on the problem data, showing that the method is fully robust with respect to the heterogeneity of the permeability coefficients, and it exhibits only a mild dependence on the square root of the local anisotropy of the bulk permeability. The numerical validation on a comprehensive set of test cases confirms the theoretical results.
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Using recently introduced DebordSkandalis Blup's groupoids we study index theory for a compact foliated manifold with boundary inducing a foliation in its boundary. For this we consider first a blup groupoid whose Lie algebroid has sections consisting of vector fields tangent to the leaves in the interior and tangent to the leaves of the foliation in the boundary. In particular the holonomy $b$groupoid allows us to consider the appropriate pseudodifferential calculus and the appropriate index problems. In this paper we further use the blup groupoids, and in particular its functoriality properties, to actually get index theorems. For the previous geometric situtation we have two index morphisms, one related to ellipticity and a second one related to fully ellipticity (generalized Fredholmness). For the first we are able to extend the longitudinal ConnesSkandalis index theorem and use it to get that every $b$longitudinal elliptic operator can be perturbed (up to stable homotopy) with
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Massive MIMO is a variant of multiuser MIMO in which the number of antennas at the base station (BS) $M$ is very large and typically much larger than the number of served users (data streams) $K$. Recent research has illustrated the systemlevel advantages of such a system and in particular the beneficial effect of increasing the number of antennas $M$. These benefits, however, come at the cost of dramatic increase in hardware and computational complexity. This is partly due to the fact that the BS needs to compute suitable beamforming vectors in order to coherently transmit/receive data to/from each user, where the resulting complexity grows proportionally to the number of antennas $M$ and the number of served users $K$. Recently, different algorithms based on tools from random matrix theory in the asymptotic regime of $M,K \to \infty$ with $\frac{K}{M} \to \rho \in (0,1)$ have been proposed to reduce the complexity. The underlying assumption in all these techniques, however, is that
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The ability to dynamically and efficiently allocate resources to meet the need of growing diversity in services and user behavior marks the future of wireless networks, giving rise to intelligent processing, which aims at enabling the system to perceive and assess the available resources, to autonomously learn to adapt to the perceived wireless environment, and to reconfigure its operating mode to maximize the utility of the available resources. The perception capability and reconfigurability are the essential features of cognitive technology while modern machine learning techniques project effectiveness in system adaptation. In this paper, we discuss the development of the cognitive technology and machine learning techniques and emphasize their roles in improving both spectrum and energy efficiency of the future wireless networks. We describe in detail the stateoftheart of cognitive technology, covering spectrum sensing and access approaches that may enhance spectrum utilization an
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Boyer, Gordon, and Watson have conjectured that an irreducible rational homology 3sphere is an Lspace if and only if its fundamental group is not leftorderable. Since Dehn surgeries on knots in $S^3$ can produce large families of Lspaces, it is natural to examine the conjecture on these 3manifolds. Greene, Lewallen, and Vafaee have proved that all 1bridge braids are Lspace knots. In this paper, we consider three families of 1bridge braids. First we calculate the knot groups and peripheral subgroups. We then verify the conjecture on the three cases by applying the criterion developed by Christianson, Goluboff, Hamann, and Varadaraj, when they verified the same conjecture for certain twisted torus knots and generalized the criteria of Clay and Watson and of Ichihara and Temma.
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Let $K\subset \Bbb R^d$ be a set with positive and finite Lebesgue measure. Let $\Lambda=M(\Bbb Z^{2d})$ be a lattice in $\Bbb R^{2d}$ with density dens$(\Lambda)=1$. It is wellknown that if $M$ is a diagonal block matrix with diagonal matrices $A$ and $B$, then $\mathcal G(K^{1/2}\chi_K, \Lambda)$ is an orthonormal basis for $L^2(\Bbb R^d)$ if and only if $K$ tiles both by $A(\Bbb Z^d)$ and $B^{t}(\Bbb Z^d)$. However, there has not been any intensive study when $M$ is not a diagonal matrix. We investigate this problem for a large class of important cases of $M$. In particular, if $M$ is any lower block triangular matrix with diagonal matrices $A$ and $B$, we prove that if $\mathcal G(K^{1/2}\chi_K, \Lambda)$ is an orthonormal basis, then $K$ can be written as a finite union of fundamental domains of $A({\mathbb Z}^d)$ and at the same time, as a finite union of fundamental domains of $B^{t}({\mathbb Z}^d)$. If $A^tB$ is an integer matrix, then there is only one common fundamen
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A physical model of a onedimensional flow of a viscous bubbly fluid in an intermediate regime between bubble formation and breakage is presented. The model takes into account multiple physical effects including gravity, viscosity, and surface tension, and is based on mechanics and thermal exchange that occurs at the boundary of a single bubble, coupled to the dynamics of the viscous fluid as a whole. The dimensionless versions of the resulting nonlinear model are obtained, and typical values of dimensionless parameters are estimated for typical magma flows in horizontal subaerial lava fields and vertical volcanic conduits. Exact solutions of the resulting system of nonlinear equations corresponding to equilibrium flows and traveling waves are analyzed. Generalized SuGardnertype perturbation analysis is employed to study approximate solutions of the model in the the longwave ansatz. Simplified nonlinear partial differential equations (PDE) satisfied by the leading terms of the pertu
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This is a note on MacPherson's local Euler obstruction, which plays an important role recently in DonaldsonThomas theory by the work of Behrend. We introduce MacPherson's original definition, and prove that it is equivalent to the algebraic definition used by Behrend, following the method of GonzalezSprinberg. We also give a formula of the local Euler obstruction in terms of Lagrangian intersections. As an application, we consider a scheme or DM stack $X$ admitting a symmetric obstruction theory. Furthermore we assume that there is a $\CC^*$ action on $X$, which makes the obstruction theory $\CC^*$equivariant. The $\CC^*$action on the obstruction theory naturally gives rise to a cosection map in the sense of KiemLi. We prove that Behrend's weighted Euler characteristic of $X$ is the same as KiemLi localized invariant of $X$ by the $\CC^*$action.
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Let $\alpha\in]0,1]$ and let $Q_j$ $(1\leqslant j\leqslant r)$ denote distinct irreducible polynomials with integer coefficients. We show that, for vectors with coordinates not exceeding a constant multiple of their mean, the joint local distribution of the number of prime factors of the $Q_j(n)$ for $x<n\leqslant x+x^\alpha$ is majorized by a constant multiple of the pairwise independency model, and we provide an upper bound for the constant in terms of the coefficients of the $Q_j$.
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The article discusses the gradient discretisation method (GDM) for distributed optimal control problems governed by diffusion equation with pure Neumann boundary condition. Using the GDM framework enables to develop an analysis that directly applies to a wide range of numerical schemes, from conforming and nonconforming finite elements, to mixed finite elements, to finite volumes and mimetic finite differences methods. Optimal order error estimates for state, adjoint and control variables for low order schemes are derived under standard regularity assumptions. A novel projection relation between the optimal control and the adjoint variable allows the proof of a superconvergence result for postprocessed control. Numerical experiments performed using a modified active set strategy algorithm for conforming, nonconforming and mimetic finite difference methods confirm the theoretical rates of convergence.
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This work explores the tradeoff between the number of samples required to accurately build models of dynamical systems and the degradation of performance in various control objectives due to a coarse approximation. In particular, we show that simple models can be easily fit from input/output data and are sufficient for achieving various control objectives. We derive bounds on the number of noisy input/output samples from a stable linear timeinvariant system that are sufficient to guarantee that the corresponding finite impulse response approximation is close to the true system in the $\mathcal{H}_\infty$norm. We demonstrate that these demands are lower than those derived in prior art which aimed to accurately identify dynamical models. We also explore how different physical input constraints, such as power constraints, affect the sample complexity. Finally, we show how our analysis fits within the established framework of robust control, by demonstrating how a controller designed fo
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