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A new study claims 44.7 million metric tons (49.3 million tons) of TV sets, refrigerators, cellphones and other electrical good were discarded last year, with only a fifth recycled to recover the valuable raw materials inside. From a report: The U.N.backed study published Wednesday calculates that the amount of ewaste thrown away in 2016 included a million tons of chargers alone. The U.S. accounted for 6.3 million metric tons, partly due to the fact that the American market for heavy goods is saturated. The original study can be found here (PDF; Google Drive link).
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AT&T has started trials to deliver highspeed internet over power lines. The company announced the news on Wednesday and said that trials have started in Georgia state and a nonU.S. location. Reuters reports: AT&T aims to eventually deliver speeds faster than the 1 gigabit per second consumers can currently get through fiber internet service using highfrequency airwaves that travel along power lines. While the Georgia trial is in a rural area, the service could potentially be deployed in suburbs and cities, the company said in a statement. AT&T said it had no timeline for commercial deployment and that it would look to expand trials as it develops the technology. "We think this product is eventually one that could actually serve anywhere near a power line," said Marachel Knight, AT&T's senior vice president of wireless network architecture and design, in an interview. She added that AT&T chose an international trial location in part because the market opportunity
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Ahead of the Vulkan 1.0 debut nearly two years ago, we heard that for AMD's Vulkan Linux driver it was initially going to be closedsource and would then be opensourced once ready. At the time it sounded like something that would be opened up six months or so, but finally that milestone is being reached! Ahead of Christmas, AMD is publishing the source code to their official Vulkan Linux driver. There's some minor caveats noted in the linked article, but this is looking like great news.
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We demonstrate that a large class of firstorder quantum phase transitions can be described as a condensation in the space of states. Given a system having Hamiltonian $H=K+gV$, where $K$ and $V$ are hopping and potential operators acting on the space of states $\mathbb{F}$, we may always write $\mathbb{F}=\mathbb{F}_\mathrm{cond} \oplus \mathbb{F}_\mathrm{norm}$ where $\mathbb{F}_\mathrm{cond}$ is the subspace which spans the eigenstates of $V$ with minimal eigenvalue and $\mathbb{F}_\mathrm{norm}=\mathbb{F}_\mathrm{cond}^\perp$. If, in the thermodynamic limit, $M_\mathrm{cond}/M \to 0$, where $M$ and $M_\mathrm{cond}$ are, respectively, the dimensions of $\mathbb{F}$ and $\mathbb{F}_\mathrm{cond}$, the above decomposition of $\mathbb{F}$ becomes effective, in the sense that the ground state energy per particle of the system, $\epsilon$, coincides with the smaller between $\epsilon_\mathrm{cond}$ and $\epsilon_\mathrm{norm}$, the ground state energies per particle of the system restri
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Some viruses, such as human immunodeficiency virus, can infect several types of cell populations. The age of infection can also affect the dynamics of infected cells and production of viral particles. In this work, we study a virus model with infectionage and different types of target cells which takes into account the saturation effect in antibody immune response and a general nonlinear infection rate. We construct suitable Lyapunov functionals to show that the global dynamics of the model is completely determined by two critical values: the basic reproduction number of virus and the reproductive number of antibody response.
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In this work we consider the problem of global existence of small regular solutions to a type nonlinear waveKleinGordon system with semilinear interactions in two spatial dimension. We develop some new techniques on both wave equations and KleinGordon equations in order to get sufficient decay rates when energies are not uniformly bounded. These techniques are compatible with those introduced in previous work on two spatialdimensional quasilinear waveKleinGordon systems, and can be applied in much general cases.
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We study the Fluctuation Theorem (FT) for entropy production in chaotic discretetime dynamical systems on compact metric spaces, and extend it to empirical measures, all continuous potentials, and all weak Gibbs states. In particular, we establish the FT in the phase transition regime. These results hold under minimal chaoticity assumptions (expansiveness and specification) and require no ergodicity conditions. They are also valid for systems that are not necessarily invertible and involutions other than time reversal. Further extensions involve asymptotically additive potential sequences and the corresponding weak Gibbs measures. The generality of these results allows to view the FT as a structural facet of the thermodynamic formalism of dynamical systems.
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In this paper we obtain a detailed description of the global and cocycle attractors for the skewproduct semiflows induced by the mild solutions of a family of scalar lineardissipative parabolic problems over a minimal and uniquely ergodic flow. We consider the case of null upper Lyapunov exponent for the linear part of the problem. Then, two different types of attractors can appear, depending on whether the linear equations have a bounded or an unbounded associated real cocycle. In the first case (e.g.~in periodic equations), the structure of the attractor is simple, whereas in the second case (which occurs in aperiodic equations), the attractor is a pinched set with a complicated structure. We describe situations when the attractor is chaotic in measure in the sense of LiYorke. Besides, we obtain a nonautonomous discontinuous pitchfork bifurcation scenario for concave equations, applicable for instance to a lineardissipative version of the ChafeeInfante equation.
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In this paper we construct two groupoids from morphisms of groupoids, with one from a categorical viewpoint and the other from a geometric viewpoint. We show that for each pair of groupoids, the two kinds of groupoids of morphisms are equivalent. Then we study the automorphism groupoid of a groupoid.
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The sample frequency spectrum (SFS), which describes the distribution of mutant alleles in a sample of DNA sequences, is a widely used summary statistic in population genetics. The expected SFS has a strong dependence on the historical population demography and this property is exploited by popular statistical methods to infer complex demographic histories from DNA sequence data. Most, if not all, of these inference methods exhibit pathological behavior, however. Specifically, they often display runaway behavior in optimization, where the inferred population sizes and epoch durations can degenerate to 0 or diverge to infinity, and show undesirable sensitivity of the inferred demography to perturbations in the data. The goal of this paper is to provide theoretical insights into why such problems arise. To this end, we characterize the geometry of the expected SFS for piecewiseconstant demographic histories and use our results to show that the aforementioned pathological behavior of pop
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We consider a wireless network in which $K$ transmitters, each equipped with a single antenna, fully cooperate to serve $K$ single antenna receivers, each equipped with a cache memory. The transmitters have access to partial knowledge of the channel state information. For a symmetric setting, in terms of channel strength levels, partial channel knowledge levels and cache sizes, we characterize the generalized degrees of freedom (GDoF) up to a constant multiplicative factor. The achievability scheme exploits the interplay between spatial multiplexing gains and codedmulticasting gain. On the other hand, a cutset argument in conjunction with a new application of the aligned image sets approach are used to derive the outer bound. We further show that the characterized orderoptimal GDoF is also attained in a decentralized setting, where no coordination is required for content placement in the caches.
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A notion of parabolic Csubsolutions is introduced for parabolic equations, extending the theory of Csubsolutions recently developed by B. Guan and more specifically G. Sz\'ekelyhidi for elliptic equations. The resulting parabolic theory provides a convenient unified approach for the study of many geometric flows.
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Fixed two positive integers m and e, some algorithms for computing the minimal Frobenius number and minimal genus of the set of numerical semigroups with multiplicity m and embedding dimension e are provided. Besides, the semigroups where these minimal values are achieved are computed too.
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We show that the properties of Lagrangian mean curvature flow are a special case of a more general phenomenon, concerning couplings between geometric flows of the ambient space and of totally real submanifolds. Both flows are driven by ambient Ricci curvature or, in the nonK\"ahler case, by its analogues. To this end we explore the geometry of totally real submanifolds, defining (i) a new geometric flow in terms of the ambient canonical bundle, (ii) a modified volume functional which takes into account the totally real condition. We discuss shorttime existence for our flow and show it couples well with the StreetsTian symplectic curvature flow for almost K\"ahler manifolds. We also discuss possible applications to Lagrangian submanifolds and calibrated geometry.
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In this paper, also motivated by evolutionary biology and evolutionary computation, we study a variation of the accessibility percolation model. Consider a tree whose vertices are labeled with random numbers. We study the probability of having a monotone subsequence of a path from the root to a leave, where any $k$ consecutive vertices in the path contain at least one vertex of the subsequence. An $n$ary tree, with height $h$, is a tree whose vertices at distance at most $h1$ to the root have $n$ children. For the case of $n$ary trees, we proof that, as $h$ tends to infinity the probability of having such subsequence: tends to 1, if $n(h)\geq c\sqrt[k]{h/(ek)} $ and $c>1$; and tends to 0, if $n(h)\leq c\sqrt[k]{h/(ek)} $ and $c<1$.
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We consider a class of twodimensional Schr\"odinger operator with a singular interaction of the $\delta$ type and a fixed strength $\beta$ supported by an infinite family of concentric, equidistantly spaced circles, and discuss what happens below the essential spectrum when the system is amended by an AharonovBohm flux $\alpha\in [0,\frac12]$ in the center. It is shown that if $\beta\ne 0$, there is a critical value $\alpha_\mathrm{crit} \in(0,\frac12)$ such that the discrete spectrum has an accumulation point when $\alpha<\alpha_\mathrm{crit} $, while for $\alpha\ge\alpha_\mathrm{crit} $ the number of eigenvalues is at most finite, in particular, the discrete spectrum is empty for any fixed $\alpha\in (0,\frac12)$ and $\beta$ small enough.
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The limiting distribution function for the Mobius function is found in the paper. It is proved also the relation: $\lim_{n \to \infty} {P(S_n/\sqrt {2pn}<y)}=G(y)$, where $S_n$ is the sum of random variables having the distribution of the Mobius function, $G(y)$ is a function of the standard normal distribution and $p=3/\pi^2$. It is shown that the law of the iterated logarithm is fulfilled for the sum of random variables having the distribution of the Mobius function.
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For noncritical almost Mathieu operators with Diophantine frequency, we establish exponential asymptotics on the size of spectral gaps, and show that the spectrum is homogeneous. We also prove the homogeneity of the spectrum for Sch\"odinger operators with (measuretheoretically) typical quasiperiodic analytic potentials and fixed strong Diophantine frequency. As applications, we show the discrete version of Deift's conjecture \cite{Deift, Deift17} for subcritical analytic quasiperiodic initial data and solve a series of open problems of DamanikGoldstein et al \cite{BDGL, DGL1, dgsv, Go} and Kotani \cite{Kot97}.
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We use a weighted variant of the frequency functions introduced by Almgren to prove sharp asymptotic estimates for almost eigenfunctions of the drift Laplacian associated to the Gaussian weight on an asymptotically conical end. As a consequence, we obtain a purely elliptic proof of a result of L. Wang on the uniqueness of selfshrinkers of the mean curvature flow asymptotic to a given cone. Another consequence is a unique continuation property for selfexpanders of the mean curvature flow that flow from a cone.
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We consider a supercritical GaltonWatson tree whose nondegenerate offspring distribution has finite mean. We consider the random trees $\tau$n distributed as $\tau$ conditioned on the nth generation, Zn, to be of size an $\in$ N. We identify the possible local limits of $\tau$n as n goes to infinity according to the growth rate of an. In the low regime, the local limit $\tau$ 0 is the Kesten tree, in the moderate regime the family of local limits, $\tau$ $\theta$ for $\theta$ $\in$ (0, +$\infty$), is distributed as $\tau$ conditionally on {W = $\theta$}, where W is the (nontrivial) limit of the renormalization of Zn. In the high regime, we prove the local convergence towards $\tau$ $\infty$ in the Harris case (finite support of the offspring distribution) and we give a conjecture for the possible limit when the offspring distribution has some exponential moments. When the offspring distribution has a fat tail, the problem is open. The proof relies on the strong ratio theorem for G
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We investigate the tail asymptotic behavior of the sojourn time for a large class of centered Gaussian processes $X$, in both continuous and discretetime framework. All results obtained here are new for the discretetime case. In the continuoustime case, we complement the investigations of [1,2] for nonstationary $X$. A byproduct of our investigation is a new representation of Pickands constant which is important for MonteCarlo simulations and yields a sharp lower bound for Pickands constant.
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A computationally efficient method is presented for approximate optimization of cutting pattern of framesupported and pneumatic membrane structures. The plane cutting sheet is generated by minimizing the error from the shape obtained by reducing the stress from the desired curved shape. The equilibrium shape is obtained solving a minimization problem of total strain energy. The external work done by the pressure is also incorporated for analysis of pneumatic membrane. An approximate method is also proposed for analysis of an Ethylene TetraFluoroEthylene (ETFE) film, where elastoplastic behavior is modeled as a nonlinear elastic material under monotonic loading condition. Efficiency of the proposed method is demonstrated through examples of a framesupported PolyVinyl Chloride (PVC) membrane structure and an air pressured square ETFE film.
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Nous pr\'esentons dans cet article une approche constructive, dans le style de Bishop, de la th\'eorie des diviseurs et des anneaux de Krull. Nous accordons une place centrale aux "anneaux \`a diviseurs," appel\'es PvMD dans la litt\'erature anglaise. Les r\'esultats classiques sont obtenus comme r\'esultats d'algorithmes explicites sans faire appel aux hypoth\`eses de factorisation compl\`ete. We give give an elementary and constructive version of the theory of "Pr\"ufer vMultiplication Domains" (which we call "anneaux \`a diviseurs" in the paper) and Krull Domains. The main results of these theories are revisited from a constructive point of view, following the Bishop style, and without assuming properties of complete factorizations.
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Let $M$ be a real hypersurface in complex Grassmannians of rank two. Denote by $\mathfrak J$ the quaternionic K\"{a}hler structure of the ambient space, $TM^\perp$ the normal bundle over $M$ and $\mathfrak D^\perp=\mathfrak JTM^\perp$. The real hypersurface $M$ is said to be $\mathfrak D^\perp$invariant if $\mathfrak D^\perp$ is invariant under the shape operator of $M$. We showed that if $M$ is $\mathfrak D^\perp$invariant, then $M$ is Hopf. This improves the results of Berndt and Suh in [{Int. J. Math.} \textbf{23}(2012) 1250103] and [{Monatsh. Math.} \textbf{127}(1999), 114]. We also classified $\mathfrak D^\perp$ real hypersurface in complex Grassmannians of rank two with constant principal curvatures.
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Mixture models are a natural choice in many applications, but it can be difficult to place an apriori upper bound on the number of components. To circumvent this, investigators are turning increasingly to Dirichlet process mixture models (DPMMs) and, more generally, PitmanYor mixtures. These models are well suited to Bayesian density estimation. An interesting question is whether they can be turned to the problem of {\em classification} or {\em clustering}, which involves allocating observations to clusters. This is becoming increasingly widely used among investigators. This article considers the MAP (maximal posterior partition) clustering for the GaussGauss DPM (where the cluster means have Gaussian distribution and, for each cluster, the observations within the cluster have Gaussian distribution; the number and sizes of the clusters generated according to a Chinese Restaurant Process). It is proved that the convex hulls of the clusters created by the MAP are pairwise `almost disjo
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We show how the onemode pseudobosonic ladder operators provide concrete examples of nilpotent Lie algebras of dimension five. It is the first time that an algebraicgeometric structure of this kind is observed in the context of pseudobosonic operators. Indeed we don't find the well known Heisenberg algebras, which are involved in several quantum dynamical systems, but different Lie algebras which may be decomposed in the sum of two abelian Lie algebras in a prescribed way. We introduce the notion of semidirect sum (of Lie algebras) for this scope and find that it describes very well the behaviour of pseudobosonic operators in many quantum models.
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The edges of a graph are assigned weights and passage times which are assumed to be positive integers. We present a parallel algorithm for finding the shortest path whose total weight is smaller than a predetermined value. In each step the processing elements are not analyzing the entire graph. Instead they are focusing on a subset of vertices called {\em active vertices}. The set of active vertices at time $t$ is related to the boundary of the ball $B_t$ of radius $t$ in the first passage percolation metric. Although it is believed that the number of active vertices is an order of magnitude smaller than the size of the graph, we prove that this need not be the case with an example of a graph for which the active vertices form a large fractal. We analyze an OpenCL implementation of the algorithm on GPU for cubes in $\mathbb Z^d$.
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We provide a degree condition on a regular $n$vertex graph $G$ which ensures the existence of a near optimal packing of any family $\mathcal H$ of bounded degree $n$vertex $k$chromatic separable graphs into $G$. In general, this degree condition is best possible. Here a graph is separable if it has a sublinear separator whose removal results in a set of components of sublinear size. Equivalently, the separability condition can be replaced by that of having small bandwidth. Thus our result can be viewed as a version of the bandwidth theorem of B\"ottcher, Taraz and Schacht in the setting of approximate decompositions. More precisely, let $\delta_k$ be the infimum over all $\delta\ge 1/2$ ensuring an approximate $K_k$decomposition of any sufficiently large regular $n$vertex graph $G$ of degree at least $\delta n$. Now suppose that $G$ is an $n$vertex graph which is close to $r$regular for some $r \ge (\delta_k+o(1))n$ and suppose that $H_1,\dots,H_t$ is a sequence of bounded degre
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In this paper, we propose a network nonorthogonal multiple access (NNOMA) technique for the downlink coordinated multipoint (CoMP) communication scenario of a cellular network, with randomly deployed users. In the considered NNOMA scheme, superposition coding (SC) is employed to serve celledge users as well as users close to base stations (BSs) simultaneously, and distributed analog beamforming by the BSs to meet the celledge user's quality of service (QoS) requirements. The combination of SC and distributed analog beamforming significantly complicates the expressions for the signaltointerferenceplusnoise ratio (SINR) at the reveiver, which makes the performance analysis particularly challenging. However, by using rational approximations, insightful analytical results are obtained in order to characterize the outage performance of the considered NNOMA scheme. Computer simulation results are provided to show the superior performance of the proposed scheme as well as to demonst
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In this paper we will present a homological model for Coloured Jones Polynomials. For each color $N \in \N$, we will describe the invariant $J_N(L,q)$ as a graded intersection pairing of certain homological classes in a covering of the configuration space on the punctured disk. This construction is based on the Lawrence representation and a result due to Kohno that relates quantum representations and homological representations of the braid groups.
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In this paper, we propose an optimizationbased sparse learning approach to identify the set of most influential reactions in a chemical reaction network. This reduced set of reactions is then employed to construct a reduced chemical reaction mechanism, which is relevant to chemical interaction network modeling. The problem of identifying influential reactions is first formulated as a mixedinteger quadratic program, and then a relaxation method is leveraged to reduce the computational complexity of our approach. Qualitative and quantitative validation of the sparse encoding approach demonstrates that the model captures important network structural properties with moderate computational load.
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Given a smooth manifold $M$ equipped with a properly and discontinuous smooth action of a discrete group $G$, the nerve $M_{\bullet}G$ is a simplicial manifold and its vector space of differential forms $\operatorname{Tot}_{N}\left(A_{DR}(M_{\bullet}G)\right)$ carry a $C_{\infty}$algebra structure $m_{\bullet}$. We show that each $C_{\infty}$algebra $1$minimal model $g_{\bullet}\: : \: \left(W, {m'}_{\bullet} \right) \to \left( \operatorname{Tot}_{N}\left(A_{DR}(M_{\bullet}G)\right),m_{\bullet}\right) $ gives a flat connection $\nabla$ on a smooth trivial bundle $E$ on $M$ where the fiber is the Malcev Lie algebra of $\pi_{1}(M/G)$ and its monodromy representation is the Malcev completion of $\pi_{1}(M/G)$. This connection is unique in the sense that different $1$models give isomorphic connections. In particular, the resulting connections are isomorphic to Chen's flat connection on $M/G$. If the action is holomorphic and $g_{\bullet}$ has holomorphic image (with logarithmic singula
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In this paper we study the tensor powers of a $4$dimensional representation of the quantum superalgebra $U_q(sl(21)$, focusing on the rings of its algebra endomorphisms so called centralizer algebras, denoted by $LG_n$. Their dimensions were conjectured by I. Marin and E. Wagner \cite{MW}. We will prove this conjecture, describing the intertwiners spaces from a semisimple decomposition as sets consisting in certain paths in a planar lattice with integer coordinates.
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The smooth particle mesh Ewald (SPME) method is an FFT based method for the fast evaluation of electrostatic interactions under periodic boundary conditions. A highly optimized implementation of this method is available in GROMACS, a widely used software for molecular dynamics simulations. In this article, we compare a more recent method from the same family of methods, the spectral Ewald (SE) method, to the SPME method in terms of performance and efficiency. We consider serial and parallel implementations of both methods for single and multiple core computations on a desktop machine as well as the Beskow supercomputer at KTH Royal Institute of Technology. The implementation of the SE method has been well optimized, however not yet comparable to the level of the SPME implementation that has been improved upon for many years. We show that the SE method is very efficient whenever used to achieve high accuracy and that it already at this level of optimization can be competitive for low ac
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We introduce a framework for calculating sparse approximations to signals based on elements of continuous wavelet systems. The method is based on an extension of the continuous wavelet theory. In the new theory, the signal space is embedded in larger "abstract" signal space, which we call the windowsignal space. There is a canonical extension of the wavelet transform on the windowsignal space, which is an isometric isomorphism from the windowsignal space to a space of functions on phase space. Hence, the new framework is called a waveletPlancherel theory, and the extended wavelet transform is called the waveletPlancherel transform. Since the waveletPlancherel transform is an isometric isomorphism, any operation on phase space can be pulledback to an operation in the windowsignal space. Using this pull back property, it is possible to pull back a search for big wavelet coefficients to the windowsignal space. We can thus avoid inefficient calculations on phase space, performing
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We represent the Lebesgue measure on the unit interval as a boundary measure of the Farey tree and show that this representation has a certain symmetry related to the tree automorphism induced by Dyer's outer automorphism of the group PGL(2,Z). Our approach gives rise to three new measures on the unit interval which are possibly of arithmetic significance.
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As a simplified model for subsurface flows elliptic equations may be utilized. Insufficient measurements or uncertainty in those are commonly modeled by a random coefficient, which then accounts for the uncertain permeability of a given medium. As an extension of this methodology to flows in heterogeneous\fractured\porous media, we incorporate jumps in the diffusion coefficient. These discontinuities then represent transitions in the media. More precisely, we consider a second order elliptic problem where the random coefficient is given by the sum of a (continuous) Gaussian random field and a (discontinuous) jump part. To estimate moments of the solution to the resulting random partial differential equation, we use a pathwise numerical approximation combined with multilevel Monte Carlo sampling. In order to account for the discontinuities and improve the convergence of the pathwise approximation, the spatial domain is decomposed with respect to the jump positions in each sample, leadin
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We prove characterization theorems for relative entropy (also known as KullbackLeibler divergence), qlogarithmic entropy (also known as Tsallis entropy), and qlogarithmic relative entropy. All three have been characterized axiomatically before, but we show that earlier proofs can be simplified considerably, at the same time relaxing some of the hypotheses.
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The logarithmic strain measures $\lVert\log U\rVert^2$, where $\log U$ is the principal matrix logarithm of the stretch tensor $U=\sqrt{F^TF}$ corresponding to the deformation gradient $F$ and $\lVert\,.\,\rVert$ denotes the Frobenius matrix norm, arises naturally via the geodesic distance of $F$ to the special orthogonal group $\operatorname{SO}(n)$. This purely geometric characterization of this strain measure suggests that a viable constitutive law of nonlinear elasticity may be derived from an elastic energy potential which depends solely on this intrinsic property of the deformation, i.e. that an energy function $W\colon\operatorname{GL^+}(n)\to\mathbb{R}$ of the form \begin{equation} W(F)=\Psi(\lVert\log U\rVert^2) \tag{1} \end{equation} with a suitable function $\Psi\colon[0,\infty)\to\mathbb{R}$ should be used to describe finite elastic deformations. However, while such energy functions enjoy a number of favorable properties, we show that it is not possible to find a strictly m
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This article is a follow up of our submitted paper [11] in which a decomposition of the Richards equation along two soil layers was discussed. A decomposed problem was formulated and a decoupling and linearisation technique was presented to solve the problem in each time step in a fixed point type iteration. This article extends these ideas to the case of twophase in porous media and the convergence of the proposed domain decomposition method is rigorously shown.
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This article develops a duality principle for a semilinear model in micromagnetism. The results are obtained through standard tools of convex analysis and the Legendre transform concept. We emphasize the dual variational formulation presented is concave and suitable for numerical computations. Moreover, sufficient conditions of optimality are also established.
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Given a hypergraph $H = (V,E)$ and an integer parameter $k$, a coloring of $V$ is said to be $k$conflictfree ($k$CF in short) if for every hyperedge $S \in E$, there exists a color with multiplicity at most $k$ in $S$. A $k$CF coloring of a graph is a $k$CF coloring of the hypergraph induced by the (closed or punctured) neighborhoods of its vertices. The special case of $1$CF coloring of general graphs and hypergraphs has been studied extensively. In this paper we study $k$CF coloring of graphs and hypergraphs. First, we study the nongeometric case and prove that any hypergraph with $n$ vertices and $m$ hyperedges can be $k$CF colored with $\tilde{O}(m^{\frac{1}{k+1}})$ colors. This bound, which extends theorems of Cheilaris and of Pach and Tardos (2009), is tight, up to a logarithmic factor. Next, we study {\em string graphs}. We consider several families of string graphs on $n$ vertices for which the $1$CF chromatic number w.r.t. punctured neighborhoods is $\Omega(\sqrt{n})
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A supervised learning algorithm searches over a set of functions $A \to B$ parametrised by a space $P$ to find the best approximation to some ideal function $f\colon A \to B$. It does this by taking examples $(a,f(a)) \in A\times B$, and updating the parameter according to some rule. We define a category where these update rules may be composed, and show that gradient descentwith respect to a fixed step size and an error function satisfying a certain propertydefines a monoidal functor from a category of parametrised functions to this category of update rules. This provides a structural perspective on backpropagation, as well as a broad generalisation of neural networks.
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This is a course of lectures given for students of the Regional Mathematical Center of the Novosibirsk State University from October 20 to November 3, 2017. The course is devoted to some geometric problems of ramified coverings of the Riemann sphere. A special attention is payed to compact surfaces of genus one (complex tori). In the first section we give a short introduction to the theory of elliptic functions. Section 2 is devoted to oneparametric families of holomorphic and meromorphic functions. We recall the role of such families on Loewner's equation in solving some problems of the theory of univalent functions. Further we deduce a system of ODEs expressing dependence of critical points of a family of rational functions from their critical values. This gives an approximate method to find a conformal mapping of the Riemann sphere onto a given simplyconnected compact Riemann surface over the sphere. Thereafter a similar problem is solved for elliptic functions uniformizing comple
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In this paper we present explicit estimate for Lipschitz constant of solution to a problem of calculus of variations. The approach we use is due to Gamkrelidze and is based on the equivalence of the problem of calculus of variations and a timeoptimal control problem. The obtained estimate is used to compute complexity bounds for a pathfollowing method applied to a convex problem of calculus of variations with polyhedral endpoint constraints.
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We present exact analytical results for the Caputo fractional derivative of a wide class of elementary functions, including trigonometric and inverse trigonometric, hyperbolic and inverse hyperbolic, Gaussian, quartic Gaussian, and Lorentzian functions. These results are especially important for multiscale physical systems, such as porous materials, disordered media, and turbulent fluids, in which transport is described by fractional partial differential equations. The exact results for the Caputo fractional derivative are obtained from a single generalized Euler's integral transform of the generalized hypergeometric function with a powerlaw argument. We present a proof of the generalized Euler's integral transform and directly apply it to the exact evaluation of the Caputo fractional derivative of a broad spectrum of functions, provided that these functions can be expressed in terms of a generalized hypergeometric function with a powerlaw argument. We determine that the Caputo fr
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We are dealing with the question whether every group or semigroup action (with some additional property) on a continuum (with some additional property) has a fixed point. One of such results was given in 2009 by Shi and Sun. They proved that every nilpotent group action on a uniquely arcwise connected continuum has a fixed point. We are seeking for this type of results with e.g. commutative, compact or torsion groups and semigroups acting on dendrites, dendroids, $\lambda$dendroids and uniquely arcwise connected continua. We prove that every continuous action of a compact or torsion group on a uniquely arcwise connected continuum has a fixed point. We also prove that every continuous action of a compact and commutative semigroup on a uniquely arcwise connected continuum has a fixed point.
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This paper studies a twohop decodeandforward underlay cognitive radio system with interference alignment technique. An energyconstrained relay node harvests the energy from the interference signals through a powersplitting (PS) relaying protocol. Firstly, the beamforming matrices design for the primary and secondary networks is demonstrated. Then, a bit error rate (BER) performance of the system under perfect and imperfect channel state information (CSI) scenarios for PS protocol is calculated. Finally, the impact of the CSI mismatch parameters on the BER performance is simulated.
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In this paper, we consider a twohop amplifyandforward (AF) relaying system, where the relay node is energyconstrained and harvests energy from the source node. In the literature, there are three main energyharvesting (EH) protocols, namely, timeswitching relaying (TSR), powersplitting (PS) relaying (PSR) and ideal relaying receiver (IRR). Unlike the existing studies, in this paper, we consider $\alpha$$\mu$ fading channels. In this respect, we derive accurate unified analytical expressions for the ergodic capacity for the aforementioned protocols over independent but not identically distributed (i.n.i.d) $\alpha$$\mu$ fading channels. Three special cases of the $\alpha$$\mu$ model, namely, Rayleigh, NakagamimandWeibull fading channels were investigated. Our analysis is verified through numerical and simulation results. It is shown that finding the optimal value of the PS factor for the PSR protocol and the EH time fraction for the TSR protocol is a crucial step in achieving
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In this paper, we establish equivariant mirror symmetry for the weighted projective line. This extends the results by B. Fang, C.C. Liu and Z. Zong, where the projective line was considered [13]. More precisely, we prove the equivalence of the Rmatrices for Amodel and Bmodel for large radius limit, and establish isomorphism for $R$matrices for general radius. We further demonstrate the graph sum of higher genus cases for both models to be the same, hence establish equivariant mirror symmetry for the weighted projective line.
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This paper investigates an energyefficient nonorthogonal transmission design problem for two downlink receivers that have strict reliability and finite blocklength (latency) constraints. The Shannon capacity formula widely used in traditional designs needs the assumption of infinite blocklength and thus is no longer appropriate. We adopt the newly finite blocklength coding capacity formula for explicitly specifying the tradeoff between reliability and code blocklength. However, conventional successive interference cancellation (SIC) may become infeasible due to heterogeneous blocklengths. We thus consider several scenarios with different channel conditions and with/without SIC. By carefully examining the problem structure, we present in closedform the optimal power and code blocklength for energyefficient transmissions. Simulation results provide interesting insights into conditions for which nonorthogonal transmission is more energy efficient than the orthogonal transmission suc
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Recently, Yamanaka and Yamashita (2017) proposed the socalled positively homogeneous optimization problems, which generalize many important problems, in particular the absolutevalue and the gauge optimizations. They presented a closed dual formulation for these problems, proving weak duality results, and showing that it is equivalent to the Lagrangian dual under some conditions. In this work, we focus particularly in optimization problems whose objective functions and constraints consist of some gauge and linear functions. Through the positively homogeneous framework, we prove that both weak and strong duality results hold. We also discuss necessary and sufficient optimality conditions associated to these problems. Finally, we show that it is possible to recover primal solutions from KarushKuhnTucker points of the dual formulation.
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A graph $G$ is said to be distance magic if there exists a bijection $f:V\rightarrow \{1,2, \ldots , v\}$ and a constant {\sf k} such that for any vertex $x$, $\sum_{y\in N(x)} f(y) ={\sf k}$, where $N_(x)$ is the set of all neighbours of $x$. In this paper we shall study distance magic labelings of graphs obtained from four graph products: cartesian, strong, lexicographic, and cronecker. We shall utilise magic rectangle sets and magic column rectangles to construct the labelings.
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A unified treatment for fast and spectrally accurate evaluation of electrostatic potentials subject to periodic boundary conditions in any or none of the three space dimensions is presented. Ewald decomposition is used to split the problem into a real space and a Fourier space part, and the FFT based Spectral Ewald (SE) method is used to accelerate the computation of the latter. A key component in the unified treatment is an FFT based solution technique for the freespace Poisson problem in three, two or one dimensions, depending on the number of nonperiodic directions. The cost of calculations is furthermore reduced by employing an adaptive FFT for the doubly and singly periodic cases, allowing for different local upsampling rates. The SE method will always be most efficient for the triply periodic case as the cost for computing FFTs will be the smallest, whereas the computational cost for the rest of the algorithm is essentially independent of the periodicity. We show that the cost
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Given a $C^k$smooth closed embedded manifold $\mathcal N\subset{\mathbb R}^m$, with $k\ge 2$, and a compact connected smooth Riemannian surface $(S,g)$ with $\partial S\neq\emptyset$, we consider $\frac 12$harmonic maps $u\in H^{1/2}(\partial S,\mathcal N)$. These maps are critical points of the nonlocal energy \begin{equation}E(f;g):=\int_S\big\nabla\widetilde u\big^2\,d\text{vol}_g,\end{equation} where $\widetilde u$ is the harmonic extension of $u$ in $S$. We express the energy as a sum of the $\frac 12$energies at each boundary component of $\partial S$ (suitably identified with the circle $\mathcal S^1$), plus a quadratic term which is continuous in the $H^s(\mathcal S^1)$ topology, for any $s\in\mathbb R$. We show the $C^{k1,\delta}$ regularity of $\frac 12$harmonic maps. We also establish a connection between free boundary minimal surfaces and critical points of $E$ with respect to variations of the pair $(f,g)$, in terms of the Teichm\"uller space of $S$.
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Maxstable processes are very relevant for the modelling of spatial extremes. In this paper, we focus on some processes belonging to the class of spacetime maxstable models introduced in Embrechts et al. (2016). The mentioned processes are Markov chains with state space the space of continuous functions from the unit sphere of $\mathbb{R}^3$ to $(0, \infty)$. We show that these Markov chains are geometrically ergodic. An interesting feature lies in the fact that the previously mentioned state space is not locally compact, making the classical methodology to be found, e.g., in Meyn and Tweedie (2009), inapplicable. Instead, we use the fact that the state space is Polish and apply results on Markov chains with Polish state spaces presented in Hairer (2010).
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We introduce new Langevintype equations describing the rotational and translational motion of rigid bodies interacting through conservative and nonconservative forces, and hydrodynamic coupling. In the absence of nonconservative forces the Langevintype equations sample from the canonical ensemble. The rotational degrees of freedom are described using quaternions, the lengths of which are exactly preserved by the stochastic dynamics. For the proposed Langevintype equations, we construct a weak 2nd order geometric integrator which preserves the main geometric features of the continuous dynamics. The integrator uses Verlettype splitting for the deterministic part of Langevin equations appropriately combined with an exactly integrated OrnsteinUhlenbeck process. Numerical experiments are presented to illustrate both the new Langevin model and the numerical method for it, as well as to demonstrate how inertia and the coupling of rotational and translational motion can introduce qualit
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We provide an algorithm to solve the word problem in all fundamental groups of closed 3manifolds; in particular, we show that these groups are autostackable. This provides a common framework for a solution to the word problem in any closed 3manifold group using finite state automata. We also introduce the notion of a group which is autostackable respecting a subgroup, and show that a fundamental group of a graph of groups whose vertex groups are autostackable respecting any edge group is autostackable. A group that is strongly coset automatic over an autostackable subgroup, using a prefixclosed transversal, is also shown to be autostackable respecting that subgroup. Building on work by Antolin and Ciobanu, we show that a finitely generated group that is hyperbolic relative to a collection of abelian subgroups is also strongly coset automatic relative to each subgroup in the collection. Finally, we show that fundamental groups of compact geometric 3manifolds, with boundary consistin
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We propose strategies to estimate and make inference on key features of heterogeneous effects in randomized experiments. These key features include best linear predictors of the effects using machine learning proxies, average effects sorted by impact groups, and average characteristics of most and least impacted units. The approach is valid in high dimensional settings, where the effects are proxied by machine learning methods. We postprocess these proxies into the estimates of the key features. Our approach is agnostic about the properties of the machine learning estimators used to produce proxies, and it completely avoids making any strong assumption. Estimation and inference relies on repeated data splitting to avoid overfitting and achieve validity. Our variational inference method is shown to be uniformly valid and quantifies the uncertainty coming from both parameter estimation and data splitting. In essence, we take medians of pvalues and medians of confidence intervals, resul
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