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Included in a new book by Robyn Lea, titled “Dinner With Georgia O’Keeffe,” is a fullpage photograph of the artist standing at her kitchen table, in New Mexico, slicing vegetables. She wears a starched white apron over a dark work shirt. Behind her is a modest, fourburner white stove, the same workmanlike model you used to find in nine out of ten American kitchens. O’Keeffe made dinner for me on that stove one night, in the fall of 1973, when I visited her at the Ghost Ranch.
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In this paper, we begin with the LehmanWalsh formula counting oneface maps and construct two involutions on pairs of permutations to obtain a new formula for the number $A(n,g)$ of oneface maps of genus $g$. Our new formula is in the form of a convolution of the Stirling numbers of the first kind which immediately implies a formula for the generating function $A_n(x)=\sum_{g\geq 0}A(n,g)x^{n+12g}$ other than the wellknown HarerZagier formula. By reformulating our expression for $A_n(x)$ in terms of the backward shift operator $E: f(x)\rightarrow f(x1)$ and proving a property satisfied by polynomials of the form $p(E)f(x)$, we easily establish the recursion obtained by Chapuy for $A(n,g)$. Moreover, we give a simple combinatorial interpretation for the HarerZagier recurrence.
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We show that functions definable in power bounded $T$convex fields have the (multidimensional) Jacobian property. Building on work of I. Halupczok, this implies that a certain notion of nonarchimedean stratifications is available in such valued fields. From the existence of these stratifications, we derive some applications in an archimedean ominimal setting. As a minor result, we also show that if $T$ is power bounded, the theory of $T$convex valued fields is $b$minimal with centres.
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Strategic decisions to develop a mineral deposit are subject to geological uncertainty, due to the sparsity of drill core samples. The selection of metallurgical equipment is especially critical, since it restricts the processing options that are available to different ore blocks, even as the nature of the deposit is still highly uncertain. Current approaches for longterm mine planning are successful at addressing geological uncertainty, but do not adequately represent alternate modes of operation for the mineral processing plant, nor do they provide sufficient guidance for developing processing options. Nonetheless, recent developments in stochastic optimization and computer data structures have resulted in a framework that can integrate operational modes into strategic mine planning algorithms. A logical next step is to incorporate geometallurgical models that relate mineralogical features to plant performance, as described in this paper.
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Hadwiger's Conjecture asserts that every $K_t$minorfree graph has a proper $(t1)$colouring. We relax the conclusion in Hadwiger's Conjecture via improper colourings. We prove that every $K_t$minorfree graph is $(2t2)$colourable with monochromatic components of order at most $\lceil\frac{t2}{2}\rceil$. This result has less colours and much smaller monochromatic components than all previous results in this direction. We then prove that every $K_t$minorfree graph is $(t1)$colourable with monochromatic degree at most $t2$. This is the best known degree bound for such a result. Both these theorems are based on a decomposition result of independent interest. We also give analogous results for $K_{s,t}$minorfree graphs, which also leads to improved bounds on generalised colouring numbers for these classes.
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We introduce a class of nonlinear least square error precoders with a general penalty function for multiuser massive MIMO systems. The generality of the penalty function allows us to consider several hardware limitations including transmitters with a predefined constellation and restricted number of active antennas. The largesystem performance is then investigated via the replica method under the assumption of replica symmetry. It is shown that the least square precoders exhibit the "marginal decoupling property" meaning that the marginal distributions of all precoded symbols converge to a deterministic distribution. As a result, the asymptotic performance of the precoders is described by an equivalent singleuser system. To address some applications of the results, we further study the asymptotic performance of the precoders when both the peaktoaverage power ratio and number of active transmit antennas are constrained. Our numerical investigations show that for a desired distortion
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Recently, there has been a strong ambition to translate models and algorithms from traditional image processing to nonEuclidean domains, e.g., to manifoldvalued data. While the task of denoising has been extensively studied in the last years, there was rarely an attempt to perform image inpainting on manifoldvalued data. In this paper we present a nonlocal inpainting method for manifoldvalued data given on a finite weighted graph. We introduce a new graph infinityLaplace operator based on the idea of discrete minimizing Lipschitz extensions, which we use to formulate the inpainting problem as PDE on the graph. Furthermore, we derive an explicit numerical solving scheme, which we evaluate on two classes of synthetic manifoldvalued images.
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In this article we give an approach to continuous functional calculus for right linear densely defined closed operators defined in a quaternionic Hilbert space. As a consequence, we obtain the spectral theorem for normal operator (not necessarily bounded) in quaternionic Hilbert spaces.
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We consider the nonlinear, inverse problem of identifying the stored energy function of a hyperelastic material from full knowledge of the displacement field as well as from surface sensor measurements. The displacement field is represented as a solution of Cauchy's equation of motion, which is a nonlinear, elastic wave equation. Hyperelasticity means that the first PiolaKirchhoff stress tensor is given as the gradient of the stored energy function. We assume that a dictionary of suitable functions is available and the aim is to recover the stored energy with respect to this dictionary. The considered inverse problem is of vital interest for the development of structural health monitoring systems which are constructed to detect defects in elastic materials from boundary measurements of the displacement field, since the stored energy encodes the mechanical peroperties of the underlying structure. In this article we develope a numerical solver for both settings using the attenuated Land
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We complete the mathematical analysis of the fine structure of harmonic measure on SLE curves that was initiated by Beliaev and Smirnov, as described by the averaged integral means spectrum. For the unbounded version of wholeplane SLE as studied by Duplantier, Nguyen, Nguyen and Zinsmeister, and Loutsenko and Yermolayeva, a phase transition has been shown to occur for high enough moments from the bulk spectrum towards a novel spectrum related to the point at infinity. For the bounded version of wholeplane SLE studied here, a similar transition phenomenon, now associated with the SLE origin, is proved to exist for low enough moments, but we show that it is superseded by the earlier occurrence of the transition to the SLE tip spectrum.
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The word problem of a finitely generated group is the formal language of words over the generators which are equal to the identity in the group. If this language happens to be contextfree, then the group is called contextfree. Finitely generated virtually free groups are contextfree. In a seminal paper Muller and Schupp showed the converse: A contextfree group is virtually free. Over the past decades a wide range of other characterizations of contextfree groups have been found. The present notes survey most of these characterizations. Our aim is to show how the different characterizations of contextfree groups are interconnected. Moreover, we present a selfcontained access to the MullerSchupp theorem without using Stallings' structure theorem or a separate accessibility result. We also give an introduction to some classical results linking groups with formal language theory.
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Stochastic dynamical systems with continuous symmetries arise commonly in nature and often give rise to coherent spatiotemporal patterns. However, because of their random locations, these patterns are not well captured by current order reduction techniques and a large number of modes is typically necessary for an accurate solution. In this work, we introduce a new methodology for efficient order reduction of such systems by combining (i) the method of slices, a symmetry reduction tool, with (ii) any standard order reduction technique, resulting in efficient mixed symmetrydimensionality reduction schemes. In particular, using the Dynamically Orthogonal (DO) equations in the second step, we obtain a novel nonlinear Symmetryreduced Dynamically Orthogonal (SDO) scheme. We demonstrate the performance of the SDO scheme on stochastic solutions of the 1D KdV and 2D NavierStokes equations.
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Quantized massive multipleinputmultipleoutput (MIMO) systems are gaining more interest due to their power efficiency. We present a new precoding technique to mitigate the multiuser interference and the quantization distortions in a downlink multiuser (MU) multipleinputsingleoutput (MISO) system with 1bit quantization at the transmitter. This work is restricted to PSK modulation schemes. The transmit signal vector is optimized for every desired received vector taking into account the 1bit quantization. The optimization is based on maximizing the safety margin to the decision thresholds of the PSK modulation. Simulation results show a significant gain in terms of the uncoded biterrorratio (BER) compared to the existing linear precoding techniques.
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It is wellknown that odddimensional manifolds have Euler characteristic zero and orientable manifolds have an even Euler characteristic unless the dimension is a multiple of $4$. We here prove a generalisation of these statements: a $k$orientable manifold (or more generally Poincar\'e complex) has even Euler characteristic unless the dimension is a multiple of $2^{k+1}$, where we call a manifold $k$orientable if the $i^{th}$ StiefelWhitney class vanishes for all $i< 2^k$ ($k\geq 0$). This theorem is strict for $k=0,1,2,3$, but whether there exist 4orientable manifolds with an odd Euler characteristic is an open question.
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In this paper we prove the existence of infinitely many nontrivial solutions of the following equations driven by a nonlocal integrodifferential operator $L_K$ with concaveconvex nonlinearities and homogeneous Dirichlet boundary conditions \begin{eqnarray*} \mathcal{L}_{K} u + \mu\, u^{q1}u + \lambda\,u^{p1}u &=& 0 \quad\text{in}\quad \Omega, \\[2mm] u&=&0 \quad\mbox{in}\quad\mathbb{R}^N\setminus\Omega, \end{eqnarray*} where $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$, $N>2s$, $s\in(0, 1)$, $0<q<1<p\leq \frac{N+2s}{N2s}$. Moreover, when $L_K$ reduces to the fractional laplacian operator $(\Delta)^s $, $p=\frac{N+2s}{N2s}$, $\frac{1}{2}(\frac{N+2s}{N2s})<q<1$, $N>6s$, $\lambda=1$, we find $\mu^*>0$ such that for any $\mu\in(0,\mu^*)$, there exists at least one sign changing solution.
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We consider the wholeplane SLE conformal map f from the unit disk to the slit plane, and show that its mixed moments, involving a power p of the derivative modulus f' and a power q of the map f itself, have closed forms along some integrability curves in the (p,q) moment plane, which depend continuously on the SLE parameter kappa. The generalization of this integrability property to the mfold transform of f is also given. We define a generalized integral means spectrum corresponding to the singular behavior of the mixed moments above. By inversion, it allows for a unified description of the unbounded interior and bounded exterior versions of wholeplane SLE, and of their mfold generalizations. The average generalized spectrum of wholeplane SLE takes four possible forms, separated by five phase transition lines in the moment plane, whereas the average generalized spectrum of the mfold wholeplane SLE is directly obtained from a linear map acting in that plane. We also conjectur
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For a discrete subgroup of an indefinite unitary group $U(1,n+1)$, $n\geq 1$, consider the attached modular variety. Using local Borcherds products, we study Heegner divisors in the local Picard group over a boundary component the compactified variety. We obtain a criterion for local Heegner divisors to be torsion elements in the local Picard group. As an application, we find that the obstructions for a local Heegner divisor to be a torsion element can be described through spaces of vector valued elliptic cusp forms spanned by certain thetaseries.
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Let $f\colon I\to I$ be a unimodal map with topological entropy $h(f)>\frac12\log2$, and let $\widehat{f}\colon\widehat{I}\to\widehat{I}$ be its natural extension, where $\widehat{I}=\varprojlim(I,f)$. Subject to some regularity conditions, which are satisfied for tent maps and quadratic maps, we give a complete description of the prime ends of the BargeMartin embedding of $\widehat{I}$ into the disk, and identify the prime ends rotation number with the height of $f$. We also show that $\widehat{f}$ is semiconjugate to a sphere homeomorphism by a semiconjugacy for which all fibers except one contain at most three points. In the case where $f$ is a postcritically finite tent map, we show that the corresponding sphere homeomorphism is a generalized pseudoAnosov map.
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We study a $(2+1)$dimensional stochastic interface growth model, that is believed to belong to the socalled Anisotropic KPZ (AKPZ) universality class [Borodin and Ferrari, 2014]. It can be seen either as a twodimensional interacting particle process with drift, that generalizes the onedimensional Hammersley process [Aldous and Diaconis 1995, Seppalainen 1996], or as an irreversible dynamics of lozenge tilings of the plane [Borodin and Ferrari 2014, Toninelli 2015]. Our main result is a hydrodynamic limit: the interface height profile converges, after a hyperbolic scaling of space and time, to the solution of a nonlinear first order PDE of HamiltonJacobi type with nonconvex Hamiltonian (nonconvexity of the Hamiltonian is a distinguishing feature of the AKPZ class). We prove the result in two situations: (i) for smooth initial profiles and times smaller than the time $T_{shock}$ when singularities (shocks) appear or (ii) for all times, including $t>T_{shock}$, if the initial p
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We show in this paper that a small subset of agents of a formation of n agents in Euclidean space can control the position and orientation of the entire formation. We consider here formations tasked with maintaining interagent distances at prescribed values. It is known that when the interagent distances specified can be realized as the edges of a rigid graph, there is a finite number of possible configurations of the agents that satisfy the distance constraints, up to rotations and translations of the entire formation. We show here that under mild conditions on the type of control used by the agents, a subset of them forming a clique can work together to control the position and orientation of the formation as a whole. Mathematically, we investigate the effect of certain allowable perturbations of a nominal dynamics of the formation system. In particular, we show that any such perturbation leads to a rigid motion of the entire formation. Furthermore, we show that the map which assig
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We study the potential employment of improper Gaussian signaling (IGS) in fullduplex relaying (FDR) with nonnegligible residual selfinterference (RSI) under Nakagamim fading. IGS is recently shown to outperform traditional proper Gaussian signaling (PGS) in several interferencelimited settings. In this work, IGS is employed as an attempt to alleviate RSI. We use two performance metrics, namely, the outage probability and ergodic rate. First, we provide upper and lower bounds for the system performance in terms of the relay transmit power and circularity coefficient, a measure of the signal impropriety. Then, we numerically optimize the relay signal parameters based only on the channel statistics to improve the system performance. Based on the analysis, IGS allows FDR to operate even with high RSI. The results show that IGS can leverage higher power budgets to enhance the performance, meanwhile it relieves RSI impact via tuning the signal impropriety. Interestingly, onedimensional
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We prove in this paper a classicality result for overconvergent modular forms on PEL Shimura varieties of type (A) or (C), without any ramification hypothesis. We use an analytic continuation method, which generalizes previous results in the unramified setting. We work with the rational model of the Shimura variety, and use an embedding into the Siegel variety to define the integral structures on the rigid space.
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KolmogorovSinai entropy is an invariant of measurepreserving actions of the group of integers that is central to classification theory. There are two recently developed invariants, sofic entropy and Rokhlin entropy, that generalize classical entropy to actions of countable groups. These new theories have counterintuitive properties such as factor maps that increase entropy. This survey article focusses on examples, many of which have not appeared before, that highlight the differences and similarities with classical theory.
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We consider the problem of minimising an inhomogeneous anisotropic elliptic functional in a class of closed $m$~dimensional subsets of~$\mathbf{R}^n$ which is stable under taking smooth deformations homotopic to identity and under local Hausdorff limits. We~prove that the minimiser exists inside the class and is an $(\mathscr H^m,m)$ rectifiable set in the sense of Federer. The class of competitors encodes a notion of spanning a~boundary. We admit unrectifiable and noncompact competitors and boundaries, and we make no restrictions on dimension~$m$ and codimension $nm$ other than $1 \le m < n$. An important tool for the proof is a novel smooth deformation theorem. The skeleton of the proof and the main ideas originate from Almgren's 1968 paper. In the end we show that classes of sets spanning some closed set $B$ in homological and cohomological sense satisfy our axioms.
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For classical onedimensional dynamical systems, such as circle diffeomorphisms, unimodal interval maps at the boundary of chaos, critical circle maps, the topological classes coincide with the rigidity classes. That is, the topological conjugacies are differentiable on the attractors. This phenomenon is known as rigidity. A natural question is now to investigate the rigidity phenomenon for more general dynamical systems. The situation is clearly more intricate. We cannot expect that the topological classes will always coincide with the rigidity classes, like in the classical context. See for example H\'enon maps at the boundary of chaos. Actually already for $C^3$ circle maps with a flat interval and Fibonacci rotation number the topological class is a $C^1$ co dimension $1$ manifold which is foliated by co dimension $3$ rigidity classes. The rigidity paradigm breaks, but in a very organized way. The foliations by rigidity classes will be an integral part of rigidity theory in general
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The nonnegative and positive semidefinite (PSD) ranks are closely connected to the nonnegative and positive semidefinite extension complexities of a polytope, which are the minimal dimensions of linear and SDP programs which represent this polytope. Though some exponential lower bounds on the nonnegative and PSD ranks has recently been proved for the slack matrices of some particular polytopes, there are still no tight bounds for these quantities. We explore some existing bounds on the PSDrank and prove that they cannot give exponential lower bounds on the extension complexity. Our approach consists in proving that the existing bounds are upper bounded by the polynomials of the regular rank of the matrix, which is equal to the dimension of the polytope (up to an additive constant). As one of the implications, we also retrieve an upper bound on the mutual information of an arbitrary matrix of a joint distribution, based on its regular rank.
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In this paper, we study the problem of finding the Euclidean distance to a convex cone generated by a set of discrete points in $\mathbb{R}^n_+$. In particular, we are interested in problems where the discrete points are the set of feasible solutions of some binary linear programming constraints. This problem has applications in manufacturing, machine learning, clustering, pattern recognition, and statistics. Our problem is a highdimensional constrained optimization problem. We propose a FrankWolfe based algorithm to solve this nonconvex optimization problem with a convexnoncompact feasible set. Our approach consists of two major steps: presenting an equivalent convex optimization problem with a noncompact domain, and finding a compactconvex set that includes the iterates of the algorithm. We discuss the convergence property of the proposed approach. Our numerical work shows the effectiveness of this approach.
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For any graph $G$, let $t(G)$ be the number of spanning trees of $G$, $L(G)$ be the line graph of $G$ and for any nonnegative integer $r$, $S_r(G)$ be the graph obtained from $G$ by replacing each edge $e$ by a path of length $r+1$ connecting the two ends of $e$. In this paper we obtain an expression for $t(L(S_r(G)))$ in terms of spanning trees of $G$ by a combinatorial approach. This result generalizes some known results on the relation between $t(L(S_r(G)))$ and $t(G)$ and gives an explicit expression $t(L(S_r(G)))=k^{m+sn1}(rk+2)^{mn+1}t(G)$ if $G$ is of order $n+s$ and size $m+s$ in which $s$ vertices are of degree $1$ and the others are of degree $k$. Thus we prove a conjecture on $t(L(S_1(G)))$ for such a graph $G$.
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A set of m distinct positive integers {a_{1},...a_{m}} is called a Diophantine mtuple if a_{i}a_{j}+n is a square for each 1\leqi<j\leqm . The aim of this study is to show that some P_{k} sets can not be extendible to a Diophantine quadruple when k=2 and k=3 and also to give some properties about P_{k} sets.
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This article improves the existing proven rates of regret decay in optimal policy estimation. We give a marginfree result showing that the regret decay for estimating a withinclass optimal policy is secondorder for empirical risk minimizers over Donsker classes, with regret decaying at a faster rate than the standard error of an efficient estimator of the value of an optimal policy. We also give a result from the classification literature that shows that faster regret decay is possible via plugin estimation provided a margin condition holds. Four examples are considered. In these examples, the regret is expressed in terms of either the mean value or the median value; the number of possible actions is either two or finitely many; and the sampling scheme is either independent and identically distributed or sequential, where the latter represents a contextual bandit sampling scheme.
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We study the boundary structure of closed convex cones, with a focus on facially dual complete (nice) cones. These cones form a proper subset of facially exposed convex cones, and they behave well in the context of duality theory for convex optimization. Using the wellknown and very commonly used concept of tangent cones in nonlinear optimization, we introduce some new notions for exposure of faces of convex sets. Based on these new notions, we obtain some necessary conditions and some sufficient conditions for a cone to be facially dual complete using tangent cones and a new notion of lexicographic tangent cones (these are a family of cones obtained from a recursive application of the tangent cone concept). Lexicographic tangent cones are related to Nesterov's lexicographic derivatives.
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An extensive empirical literature documents a generally negative correlation, named the "leverage effect" between asset returns and changes of volatility. It is more challenging to establish such a returnvolatility relationship for jumps in highfrequency data. We propose new nonparametric methods to assess and test for a discontinuous leverage effect  that is, a relation between contemporaneous jumps in prices and volatility  in highfrequency data with market microstructure noise. We present local tests and estimators for price jumps and volatility jumps. Five years of transaction data from 320 NASDAQ firms display no negative relation between price and volatility cojumps. We show, however, that there is a strong relation between pricevolatility cojumps if one conditions on the sign of price jumps and whether the price jumps are marketwide or idiosyncratic.
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Antenna current optimization is often used to analyze the optimal performance of antennas. Antenna performance can be quantified in e.g., minimum Qfactor and maximal gain over Qfactor ratio. The performance of MIMO antennas is more involved and a single parameter is, in general, not sufficient to quantify it. Here, the capacity of an idealized channel is used as the main performance quantity. An optimization problem in the current distribution for optimal capacity, measured in spectral efficiency, given a fixed Qfactor and efficiency is formulated as a semidefinite optimization problem. A model order reduction based on characteristic and energy modes is employed to improve the computational efficiency. The bound is illustrated by solving the optimization problem numerically for a rectangular plate.
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We prove general results about completeness of cotorsion theories and existence of covers and envelopes in locally presentable abelian categories, extending the wellestablished theory for module categories and Grothendieck categories. These results are then applied to the categories of contramodules over topological rings, which provide examples and counterexamples.
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For any geodesic current we associated a quasimetric space. For a subclass of geodesic currents, called filling, it defines a metric and we study the critical exponent associated to this space. We show that is is equal to the exponential growth rate of the intersection function for closed curves.
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There has been a growing effort in studying the distributed optimization problem over a network. The objective is to optimize a global function formed by a sum of local functions, using only local computation and communication. Literature has developed consensusbased distributed (sub)gradient descent (DGD) methods and has shown that they have the same convergence rate $O(\frac{\log t}{\sqrt{t}})$ as the centralized (sub)gradient methods (CGD) when the function is convex but possibly nonsmooth. However, when the function is convex and smooth, under the framework of DGD, it is unclear how to harness the smoothness to obtain a faster convergence rate comparable to CGD's convergence rate. In this paper, we propose a distributed algorithm that, despite using the same amount of communication per iteration as DGD, can effectively harnesses the function smoothness and converge to the optimum with a rate of $O(\frac{1}{t})$. If the objective function is further strongly convex, our algorithm h
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In this paper we use Morawetz estimates with geometric energy estimates the socalled vector field method to prove decay results for the Maxwell field in the static exterior region of the ReissnerNordstr{\o}mde Sitter black hole. We prove two types of decay: The first is a uniform decay of the energy of the Maxwell field on achronal hypersurfaces as the hypersurfaces approach timelike infinities. The second decay result is a pointwise decay in time with a rate of $t^{1}$ which follows from local energy decay by Sobolev estimates. Both results are consequences of bounds on the conformal energy defined by the Morawetz conformal vector field. These bounds are obtained through wave analysis on the middle spin component of the field. The results hold for a more general class of spherically symmetric spacetimes with the same arguments used in this paper.
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We study the probability of $k$hop connection between two nodes in a wireless multihop network, addressing the difficulty of providing an exact formula for the scaling of hop counts with Euclidean distance without first making a sort of mean field approximation, which in this case assumes all nodes in the network have uncorrelated degrees. We therefore study the mean and variance of the number of $k$hop paths between two vertices $x,y$ in the random connection model, which is a random geometric graph where nodes connect probabilistically rather than according to a law of intersecting spheres. In the example case where Rayleigh fading is modelled, the variance of the number of three hop paths is in fact composed of four separate decaying exponentials, one of which is the mean, which decays slowest as $\lVert xy \rVert \to \infty$. These terms each correspond to one of exactly four distinct substructures with can form when pairs of paths intersect in a specific way, for example at e
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Feynman diagrams in $\phi^4$ theory have as their underlying structure 4regular graphs. In particular, any 4point $\phi^4$ graph can be uniquely derived from a 4regular graph by deleting a vertex. The Feynman period is a simplified version of the Feynman integral, and is of special interest, as it maintains much of the important number theoretic information from the Feynman integral. It is also of structural interest, as it is known to be preserved by a number of graph theoretic operations. In particular, the vertex deleted in constructing the 4point graph does not affect the Feynman period, and it is invariant under planar duality and the Schnetz twist, an operation that redirects edges incident to a 4vertex cut. Further, a 4regular graph may be produced by a 3sum operation on triangles in two 4regular graphs. The Feynman period of this graph with a vertex deleted is equal to the product of the Feynman periods of the two smaller graphs with one vertex deleted each. These opera
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Let $G$ be a finite group with centre $Z(G)$. The commuting graph of a nonAbelian group $G$, denoted by $\Gamma_G$, is a simple undirected graph whose vertex set is $G\setminus Z(G)$, and two vertices $x$ and $y$ are adjacent if and only if $xy = yx$. In this article we aim to compute the ordinary energy of $\Gamma_G$ for groups $G$ whose centralizers are Abelian.
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In this article, we continue the study of the problem of $L^p$boundedness of the maximal operator $M$ associated to averages along isotropic dilates of a given, smooth hypersurface $S$ of finite type in 3dimensional Euclidean space. An essentially complete answer to this problem had been given about seven years ago by the last named two authors in joint work with M. Kempe for the case where the height h of the given surface is at least two. In the present article, we turn to the case $h<2.$ More precisely, in this Part I, we study the case where $h<2,$ assuming that $S$ is contained in a sufficiently small neighborhood of a given point $x^0\in S$ at which both principal curvatures of $S$ vanish. Under these assumptions and a natural transversality assumption, we show that, as in the case where $h\ge 2,$ the critical Lebesgue exponent for the boundedness of $M$ remains to be $p_c=h,$ even though the proof of this result turns out to require new methods, some of which are inspire
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We develop a theory which allows making qualitative conclusions about the dynamics of both monotone and nonmonotone Moreau sweeping processes. Specifically, we first prove that any sweeping processes with almost periodic monotone righthandsides admits a globally exponentially stable almost periodic solution. And then we describe the extent to which such a globally stable solution persists under nonmonotone perturbations.
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With the recent surge of interest in using robotics and automation for civil purposes, providing safety and performance guarantees has become extremely important. In the past, differential games have been successfully used for the analysis of safetycritical systems. In particular, the HamiltonJacobi (HJ) formulation of differential games provides a flexible way to compute the reachable set, which can characterize the set of states which lead to either desirable or undesirable configurations, depending on the application. While HJ reachability is applicable to many small practical systems, the curse of dimensionality prevents the direct application of HJ reachability to many larger systems. To address computation complexity issues, various efficient computation methods in the literature have been developed for approximating or exactly computing the solution to HJ partial differential equations, but only when the system dynamics are of specific forms. In this paper, we propose a flexib
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We consider \L ojasiewicz inequalities for a nondegenerate holomorphic function with an isolated singularity at the origin. We give an explicit estimation of the \L ojasiewicz exponent in a slightly weaker form than the assertion in Fukui.For a weighted homogeneous polynomial, we give a better estimation in the form which is conjectured by Brzostowski, Krasinski and Oleksik under under some condition (the \L ojasiewicz nondegeneracy). We also introduce \L ojasiewicz inequality for strongly nondegenerate mixed functions and generalize this estimation for mixed functions.
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We give an account of the "gravitational memory effect" in the presence of an exact plane wave solution of Einstein's vacuum equations. This allows an elementary but exact description of the soft gravitons and how their presence may be detected by observing the motion of freely falling particles.
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In this article we discuss the Mass Transference Principle due to Beresnevich and Velani and survey several generalisations and variants, both deterministic and random. Using a Hausdorff measure analogue of the inhomogeneous KhintchineGroshev Theorem, proved recently via an extension of the Mass Transference Principle to systems of linear forms, we give an alternative proof of a general inhomogeneous Jarn\'{\i}kBesicovitch Theorem which was originally proved by Levesley. We additionally show that without monotonicity Levesley's theorem no longer holds in general. Thereafter, we discuss recent advances by Wang, Wu and Xu towards mass transference principles where one transitions from $\limsup$ sets defined by balls to $\limsup$ sets defined by rectangles (rather than from "balls to balls" as is the case in the original Mass Transference Principle). Furthermore, we consider mass transference principles for transitioning from rectangles to rectangles and extend known results using a sli
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We study the problem of approximating the partition function of the ferromagnetic Ising model in graphs and hypergraphs. Our first result is a deterministic approximation scheme (an FPTAS) for the partition function in bounded degree graphs that is valid over the entire range of parameters $\beta$ (the interaction) and $\lambda$ (the external field), except for the case $\vert{\lambda}\vert=1$ (the "zerofield" case). A randomized algorithm (FPRAS) for all graphs, and all $\beta,\lambda$, has long been known. Unlike most other deterministic approximation algorithms for problems in statistical physics and counting, our algorithm does not rely on the "decay of correlations" property. Rather, we exploit and extend machinery developed recently by Barvinok, and Patel and Regts, based on the location of the complex zeros of the partition function, which can be seen as an algorithmic realization of the classical LeeYang approach to phase transitions. Our approach extends to the more general
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Latent Block Model (LBM) is a modelbased method to cluster simultaneously the $d$ columns and $n$ rows of a data matrix. Parameter estimation in LBM is a difficult and multifaceted problem. Although various estimation strategies have been proposed and are now well understood empirically, theoretical guarantees about their asymptotic behavior is rather sparse. We show here that under some mild conditions on the parameter space, and in an asymptotic regime where $\log(d)/n$ and $\log(n)/d$ tend to $0$ when $n$ and $d$ tend to $+\infty$, (1) the maximumlikelihood estimate of the complete model (with known labels) is consistent and (2) the loglikelihood ratios are equivalent under the complete and observed (with unknown labels) models. This equivalence allows us to transfer the asymptotic consistency to the maximum likelihood estimate under the observed model. Moreover, the variational estimator is also consistent.
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New contributions are offered to the theory and practice of the Discrete Empirical Interpolation Method (DEIM). These include a detailed characterization of the canonical structure; a substantial tightening of the error bound for the DEIM oblique projection, based on index selection via a strong rank revealing QR factorization; and an extension of the DEIM approximation to weighted inner products defined by a real symmetric positivedefinite matrix $W$. The weighted DEIM ($W$DEIM) can be deployed in the more general framework where the POD Galerkin projection is formulated in a discretization of a suitable energy inner product such that the Galerkin projection preserves important physical properties such as e.g. stability. Also, a special case of $W$DEIM is introduced, which is DGEIM, a discrete version of the Generalized Empirical Interpolation Method that allows generalization of the interpolation via a dictionary of linear functionals.
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We establish the decidability of the $\Sigma_2$ theory of $\mathscr{D}_h(\leq_h \mathcal{O})$, the hyperarithmetic degrees below Kleene's $\mathcal{O}$, in the language of uppersemilattices with least and greatest element. This requires a new kind of initialsegment result and a new extension of embeddings result both in the hyperarithmetic setting.
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Consider the classical Gaussian unitary ensemble of size $N$ and the real Wishart ensemble $W_N(n,I)$. In the limits as $N \to \infty$ and $N/n \to \gamma > 0$, the expected number of eigenvalues that exit the upper bulk edge is less than one, 0.031 and 0.170 respectively, the latter number being independent of $\gamma$. These statements are consequences of quantitative bounds on tail sums of eigenvalues outside the bulk which are established here for applications in high dimensional covariance matrix estimation.
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Compared with the twocomponent CamassaHolm system, the modified twocomponent CamassaHolm system introduces a regularized density which makes possible the existence of solutions of lower regularity, and in particular of multipeakon solutions. In this paper, we derive a new pointwise invariant for the modified twocomponent CamassaHolm system. The derivation of the invariant uses directly the symmetry of the system, following the classical argument of Noether's theorem. The existence of the multipeakon solutions can be directly inferred from this pointwise invariant. This derivation shows the strong connection between symmetries and the existence of special solutions. The observation also holds for the scalar CamassaHolm equation and, for comparison, we have also included the corresponding derivation. Finally, we compute explicitly the solutions obtained for the peakonantipeakon case. We observe the existence of a periodic solution which has not been reported in the literature pre
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In this paper, we discuss stochastic comparisons of parallel systems with independent heterogeneous exponentiated NadarajahHaghighi (ENH) components in terms of the usual stochastic order, dispersive order, convex transform order and the likelihood ratio order. In the presence of the Archimedean copula, we study stochastic comparison of series dependent systems in terms of the usual stochastic order.
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In this paper, we consider a diffusive mobile molecular communication (MC) system consisting of a pair of mobile transmitter and receiver nanomachines suspended in a fluid medium, where we model the mobility of the nanomachines by Brownian motion. The transmitter and receiver nanomachines exchange information via diffusive signaling molecules. Due to the random movements of the transmitter and receiver nanomachines, the statistics of the channel impulse response (CIR) change over time. We introduce a statistical framework for characterization of the impulse response of timevariant MC channels. In particular, we derive closedform analytical expressions for the mean and the autocorrelation function of the impulse response of the channel. Given the autocorrelation function, we define the coherence time of the timevariant MC channel as a metric that characterizes the variations of the impulse response. Furthermore, we derive an analytical expression for evaluation of the expected er
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A simple analytical solution is proposed for the stationary loss system of two parallel queues with finite capacity $K$, in which new customers join the shortest queue, or one of the two with equal probability if their lengths are equal. The arrival process is Poisson, service times at each queue have exponential distribution with the same parameter, and both queues have equal capacity. Using standard generating function arguments, a simple expression of the blocking probability is derived, which as far as we know is original. Then using coupling arguments and explicit formulas, comparisons with related loss systems are provided, as well as bounds for the average total number of customers. Furthermore, from the balance equations, all stationary probabilities are obtained as explicit combinations of their values at states $(0,k)$ for $0 \le k \le K$. These expressions extend to the infinite capacity and asymmetric cases, i.e., when the queues have different service rates. For the initia
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Consider the thinfilm equation $h_t + \left(h h_{yyy}\right)_y = 0$ with a zero contact angle at the free boundary, that is, at the triple junction where liquid, gas, and solid meet. Previous results on stability and wellposedness of this equation have focused on perturbations of equilibriumstationary or selfsimilar profiles, the latter eventually wetting the whole surface. These solutions have their counterparts for the secondorder porousmedium equation $h_t  (h^m)_{yy} = 0$, where $m > 1$ is a free parameter. Both porousmedium and thinfilm equation degenerate as $h \searrow 0$, but the porousmedium equation additionally fulfills a comparison principle while the thinfilm equation does not. In this note, we consider traveling waves $h = \frac V 6 x^3 + \nu x^2$ for $x \ge 0$, where $x = yV t$ and $V, \nu \ge 0$ are free parameters. These traveling waves are receding and therefore describe dewetting, a phenomenon genuinely linked to the fourthorder nature of the thinfi
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In spoken languages, speakers divide up the space of phonetic possibilities into different regions, corresponding to different phonemes. We consider a simple exemplar model of how this division of phonetic space varies over time among a population of language users. In the particular model we consider, we show that, once the system is initialized with a given set of phonemes, that phonemes do not become extinct: all phonemes will be maintained in the system for all time. This is in contrast to what is observed in more complex models. Furthermore, we show that the boundaries between phonemes fluctuate and we quantitatively study the fluctuations in a simple instance of our model. These results prepare the ground for more sophisticated models in which some phonemes go extinct or new phonemes emerge through other processes.
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A host algebra of a (possibly infinite dimensional) Lie group $G$ is a $C^*$algebra whose representations are in onetoone correspondence with certain continuous unitary representations $\pi \colon G \to \U(\cH)$. In this paper we present a new approach to host algebras for infinite dimensional Lie groups which is based on smoothing operators, i.e., operators whose range is contained in the space $\cH^\infty$ of smooth vectors. Our first major result is a characterization of smoothing operators $A$ that in particular implies smoothness of the maps $\pi^A \colon G \to B(\cH), g \mapsto \pi(g)A$. The concept of a smoothing operator is particularly powerful for representations $(\pi,\cH)$ which are semibounded, i.e., there exists an element $x_0 \in\g$ for which all operators $i\dd\pi(x)$, $x \in \g$, from the derived representation are uniformly bounded from above in some neighborhood of $x_0$. Our second main result asserts that this implies that $\cH^\infty$ coincides with the space
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We prove that digital sequences modulo $m$ along squares are normal; which covers some prominent sequences like the sum of digits in base $q$ modulo $m$, the RudinShapiro sequence and some generalizations. This gives, for any base, a class of explicit normal numbers that can be efficiently generated.
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We consider qdeformations of Witt rings, based on geometric operations on zeta functions of motives over finite fields, and we use these deformations to construct qanalogs of the BostConnes quantum statistical mechanical system. We show that the qdeformations obtained in this way can be related to Habiro ring constructions of analytic functions over the field with one element and to categorifications of BostConnes systems.
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Many hardness results in computational social choice make use of the fact that every directed graph may be induced as the pairwise majority relation of some preference profile. However, this fact requires a number of voters that is almost linear in the number of alternatives. It is therefore unclear whether these results remain intact when the number of voters is bounded, as is, for example, typically the case in search engine aggregation settings. In this paper, we provide a systematic study of majority digraphs for a constant number of voters resulting in analytical, experimental, and complexitytheoretic insights. First, we characterize the set of digraphs that can be induced by two and three voters, respectively, and give sufficient conditions for larger numbers of voters. Second, we present a surprisingly efficient implementation via SAT solving for computing the minimal number of voters that is required to induce a given digraph and experimentally evaluate how many voters are req
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Visibility algorithms are a family of geometric and ordering criteria by which a realvalued time series of N data is mapped into a graph of N nodes. This graph has been shown to often inherit in its topology nontrivial properties of the series structure, and can thus be seen as a combinatorial representation of a dynamical system. Here we explore in some detail the relation between visibility graphs and symbolic dynamics. To do that, we consider the degree sequence of horizontal visibility graphs generated by the oneparameter logistic map, for a range of values of the parameter for which the map shows chaotic behaviour. Numerically, we observe that in the chaotic region the block entropies of these sequences systematically converge to the Lyapunov exponent of the system. Via Pesin identity, this in turn suggests that these block entropies are converging to the Kolmogorov Sinai entropy of the map, which ultimately suggests that the algorithm is implicitly and adaptively constructing
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A matching $M$ in a graph $G$ is said to be uniquely restricted if $M$ is the only perfect matching in the subgraph of $G$ induced by vertices saturated by $M$. For any connected multigraph $G=(V,E)$ and a fixed vertex $x_0$ in $G$, there is a bijection from the set of spanning trees of $G$ to the set of uniquely restricted matchings of size $V1$ in the bipartite graph $S(G)x_0$, where $S(G)$ is obtained from $G$ by subdividing each edge in $G$. Motivated by this observation, we extend the concept of Gparking functions of graphs to Bparking functions $f:X\rightarrow N_0$ for any bipartite graph $H=(X,Y)$, and establish a bijection $\psi$ from the set of uniquely restricted matchings in $H$ to the set of Bparking functions of $H$. If $M$ is a uniquely restricted matching of $H$ of size $X$ and $f=\psi(M)$, then for any $x\in X$, $f(x)$ is interpreted by the number of some elements $y\in Y$ which are not saturated by $M$ and are not externally Bactive with respect to $M$ in $H$
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We study a model introduced by Perthame and Vauchelet that describes the growth of a tumor governed by Brinkman's Law, which takes into account friction between the tumor cells. We adopt the viscosity solution approach to establish an optimal uniform convergence result of the tumor density as well as the pressure in the incompressible limit. The system lacks standard maximum principle, and thus modification of the usual approach is necessary.
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A net $(x_\alpha)$ in a vector lattice $X$ is said to be {unbounded order convergent} (or uoconvergent, for short) to $x\in X$ if the net $(\abs{x_\alphax}\wedge y)$ converges to 0 in order for all $y\in X_+$. In this paper, we study unbounded order convergence in dual spaces of Banach lattices. Let $X$ be a Banach lattice. We prove that every norm bounded uoconvergent net in $X^*$ is $w^*$convergent iff $X$ has order continuous norm, and that every $w^*$convergent net in $X^*$ is uoconvergent iff $X$ is atomic with order continuous norm. We also characterize among $\sigma$order complete Banach lattices the spaces in whose dual space every simultaneously uo and $w^*$convergent sequence converges weakly/in norm.
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We analyze the distribution of the distance between two nodes, sampled uniformly at random, in digraphs generated via the directed configuration model, in the supercritical regime. Under the assumption that the covariance between the indegree and outdegree is finite, we show that the distance grows logarithmically in the size of the graph. In contrast with the undirected case, this can happen even when the variance of the degrees is infinite. The main tool in the analysis is a new coupling between a breadthfirst graph exploration process and a suitable branching process based on the KantorovichRubinstein metric. This coupling holds uniformly for a much larger number of steps in the exploration process than existing ones, and is therefore of independent interest.
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Trace classes of Sobolevtype functions in metric spaces are subject of this paper. In particular, functions on domains whose boundary has an upper codimension$\theta$ bound are considered. Based on a Poincar\'e inequality, existence of a Borel measurable trace is proven whenever the power of integrability of the "gradient" exceeds $\theta$. The trace $T$ is shown to be a compact operator mapping a Sobolevtype space on a domain into a Besov space on the boundary. Sufficient conditions for $T$ to be surjective are found and counterexamples showing that surjectivity may fail are also provided. The case when the exponent of integrability of the "gradient" is equal to $\theta$, i.e., the codimension of the boundary, is also discussed. Under some additional assumptions, the trace lies in $L^\theta$ on the boundary then. Essential sharpness of these extra assumptions is illustrated by an example.
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The topic of this paper is the asymptotic distribution of random orthogonal matrices distributed according to Haar measure. We examine the total variation distance between the joint distribution of the entries of $Z_n$, the $p_n \times q_n$ upperleft block of a Haardistributed matrix, and that of $p_nq_n$ independent standard Gaussian random variables. We show that the total variation distance converges to zero when $p_nq_n = o(n)$.
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