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We construct a 6D ${\cal N}=(1,0)$ superconformal field theory by coupling an ${\cal N}=(1, 0)$ tensor multiplet to an ${\cal N}=(1, 0)$ hypermultiplet. While the ${\cal N}=(1, 0)$ tensor multiplet is in the adjoint representation of the gauge group, the hypermultiplet can be in the fundamental representation or any other representation. If the hypermultiplet is also in the adjoint representation of the gauge group, the supersymmetry is enhanced to ${\cal N}=(2, 0)$, and the theory is identical to the $(2,0)$ theory of Lambert and Papageorgakis (LP). Upon dimension reduction, the $(1, 0)$ theory can be reduced to a general ${\cal N}=1$ supersymmetric YangMills theory in 5D. We discuss briefly the possible applications of the theories to multi M5branes.
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A new definition of random sets is proposed. It is based on the distance in measurable space and uses negative definite kernels for continuation from initial space to that of random sets. This approach has no connection to Hausdorff distance between sets. Key words: random sets; measurable space; negative definite kernels; Hilbert space isometries.
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The (driven) Rabi model, together with its twomode, twophoton, and asymmetric generalizations, are exotic examples of quasiexactly solvable models in that a corresponding 2nd order ordinary differential equation (ODE) ${\cal L}\psi=0$ with polynomial coefficients (i) is not Fuchsian one and (ii) the differential operator ${\cal L}$ comprises energy E dependent terms $\sim Ez d_z$, $Ez$, $E^2$. When recast into a Schr\"odinger equation (SE) form with the first derivative term being eliminated and the coefficient of $d_x^2$ set to one, such an equation is characterized by a nontrivially energy dependent potential. The concept of a gradation slicing is introduced to analyze polynomial solutions of such equations. It is shown that the ODE of all the above Rabi models are characterized by the same unique set of grading parameters. General necessary and sufficient conditions for the existence of a polynomial solution are formulated. Unlike standard eigenvalue problems, the condition that
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Nesterov's accelerated gradient (AG) method for minimizing a smooth strongly convex function $f$ is known to reduce $f({\bf x}_k)f({\bf x}^*)$ by a factor of $\epsilon\in(0,1)$ after $k=O(\sqrt{L/\ell}\log(1/\epsilon))$ iterations, where $\ell,L$ are the two parameters of smooth strong convexity. Furthermore, it is known that this is the best possible complexity in the functiongradient oracle model of computation. Modulo a line search, the geometric descent (GD) method of Bubeck, Lee and Singh has the same bound for this class of functions. The method of linear conjugate gradients (CG) also satisfies the same complexity bound in the special case of strongly convex quadratic functions, but in this special case it can be faster than the AG and GD methods. Despite similarities in the algorithms and their asymptotic convergence rates, the conventional analysis of the running time of CG is mostly disjoint from that of AG and GD. The analyses of the AG and GD methods are also rather distin
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We give a de Finetti type representation for exchangeable random coalescent trees (formally described as semiultrametrics) in terms of sampling iid sequences from marked metric measure spaces. We apply this representation to define versions of treevalued FlemingViot processes from a $\Xi$lookdown model. As state spaces for these processes, we use, besides the space of isomorphy classes of metric measure spaces, also the space of isomorphy classes of marked metric measure spaces and a space of distance matrix distributions. This allows to include the case with dust in which the genealogical trees have isolated leaves.
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We propose a refinement of the RobertsonSchrodinger uncertainty principle (RSUP) using Wigner distributions. This new principle is stronger than the RSUP. In particular, and unlike the RSUP, which can be saturated by many phase space functions, the refined RSUP can be saturated by pure Gaussian Wigner functions only. Moreover, the new principle is technically as simple as the standard RSUP. In addition, it makes a direct connection with modern harmonic analysis, since it involves the Wigner transform and its symplectic Fourier transform, which is the radar ambiguity function. As a byproduct of the refined RSUP, we derive inequalities involving the entropy and the covariance matrix of Wigner distributions. These inequalities refine the Shanon and the Hirschman inequalities for the Wigner distribution of a mixed quantum state $\rho$. We prove sharp estimates which critically depend on the purity of $\rho$ and which are saturated in the Gaussian case.
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For an $n$dimensional local analytic differential system $\dot x=Ax+f(x)$ with $f(x)=O(x^2)$, the Poincar\'e nonintegrability theorem states that if the eigenvalues of $A$ are not resonant, the system does not have an analytic or a formal first integral in a neighborhood of the origin. This result was extended in 2003 to the case when $A$ admits one zero eigenvalue and the other are nonresonant: for $n=2$ the system has an analytic first integral at the origin if and only if the origin is a nonisolated singular point; for $n>2$ the system has a formal first integral at the origin if and only if the origin is not an isolated singular point. However, the question of \emph{whether the system has an analytic first integral at the origin provided that the origin is not an isolated singular point} remains open.
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We consider a probit model without covariates, but the latent Gaussian variables having compound symmetry covariance structure with a single parameter characterizing the common correlation. We study the parameter estimation problem under such oneparameter probit models. As a surprise, we demonstrate that the likelihood function does not yield consistent estimates for the correlation. We then formally prove the parameter's nonestimability by deriving a nonvanishing minimax lower bound. This counterintuitive phenomenon provides an interesting insight that one bit information of the latent Gaussian variables is not sufficient to consistently recover their correlation. On the other hand, we further show that trinary data generated from the Gaussian variables can consistently estimate the correlation with parametric convergence rate. Hence we reveal a phase transition phenomenon regarding the discretization of latent Gaussian variables while preserving the estimability of the correlation
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In this paper, we will study the following PDE in $\mathbb{R}^N$ involving multiple HardySobolev critical exponents: $$ \begin{cases} \Delta u+\sum_{i=1}^{l}\lambda_i \frac{u^{2^*(s_i)1}}{x^{s_i}}+u^{2^*1}=0\;\hbox{in}\;\mathbb{R}^N, u\in D_{0}^{1,2}(\mathbb{R}^N), \end{cases} $$ where $0<s_1<s_2<\cdots<s_l<2, 2^\ast:=\frac{2N}{N2}, \; 2^\ast(s):=\frac{2(Ns)}{N2}$ and there exists some $k\in [1, l]$ such that $\lambda_i>0$ for $1\leq i\leq k$; $\lambda_i<0$ for $k+1\leq i\leq l$. We develop an interesting way to study this class of equations involving mixed sign parameters. We prove the existence and nonexistence of the positive ground state solution. The regularity of the leastenergy solution are also investigated.
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In this work we consider a class of nonlocal nonautonomous evolution problems, which arise in neuronal activity, \[ \begin{cases} \partial_t u(t,x) = a(t)u(t,x) + b(t) \displaystyle\int_{\mathbb{R}^N} J(x,y)f(t,u(t,y))dy h +S(t,x) ,\ t\geq\tau \in \mathbb{R},\ x \in \Omega, u(\tau,x)=u_\tau(x),\ x\in\Omega u(t,x)= 0,\ t\geq\tau,\ x \in\mathbb{R}^N\backslash\Omega. \end{cases} \] Under suitable assumptions on the nonlinearity $f: \mathbb{R} \times \mathbb{R} \to\mathbb{R}$ and constraints on the functions $J: \mathbb{R}^N \times \mathbb{R}^{N}\to \mathbb{R}$;\, $S: \mathbb{R} \times \mathbb{R}^{N}\to\mathbb{R}$ and $a,b:\mathbb{R} \to\mathbb{R}$, we study the assimptotic behavior of the evolution process, generated by this problem, in an appropriated Banach space.% and we present a brief discussion on the model with a biological interpretation. We prove results on existence, uniqueness and smoothness of the solutions and the existence of pullback attracts for the evolution process as
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We revisit the scattering result of Duyckaerts, Holmer, and Roudenko for the nonradial $\dot H^{1/2}$critical focusing NLS. By proving an interaction Morawetz inequality, we give a simple proof of scattering below the ground state in dimensions $d\geq 3$ that avoids the use of concentration compactness.
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This work is a study of $p$adic multiple zeta values at roots of unity ($p$MZV$\mu_{N}$'s), the $p$adic periods of the crystalline prounipotent fundamental groupoid of $(\mathbb{P}^{1}  \{0,\mu_{N},\infty\})/ \mathbb{F}_{q}$. The main tool is new objects which we call $p$adic prounipotent harmonic actions. In this part IV we define and study $p$adic analogues of some elementary complex analytic functions which interpolate multiple zeta values at roots of unity such as the multiple zeta functions. The indices of $p$MZV$\mu_{N}$'s involve sequences of positive integers ; in this IV1, by considering an operation which we call localization (inverting certain integration operators) in the prounipotent fundamental groupoid of $\mathbb{P}^{1}  \{0,\mu_{N},\infty\}$, and by using $p$adic prounipotent harmonic actions, we extend the definition of $p$MZV$\mu_{N}$'s to indices for which these integers can be negative, and we study these generalized $p$MZV$\mu_{N}$'s.
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In this paper we extend analytic signal method to the functions in many dimensions. First it is shown how to obtain separate phaseshifted components and how combine them to obtain signal's envelope, instantaneous frequencies and phases in many dimensions. Second, we show that phaseshifted components may be obtained by positive frequency restriction of the Fourier transform defined in the algebra of commutative elliptic hypercomplex numbers. Finally we prove that for $d>2$ there is no corresponding CliffordFourier transform that allows to recover phaseshifted components correctly.
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An optimal control problem is considered for a stochastic differential equation containing a statedependent regime switching, with a recursive cost functional. Due to the nonexponential discounting in the cost functional, the problem is timeinconsistent in general. Therefore, instead of finding a global optimal control (which is not possible), we look for a timeconsistent (approximately) locally optimal equilibrium strategy. Such a strategy can be represented through the solution to a system of partial differential equations, called an equilibrium HamiltonJacobBellman (HJB, for short) equation which is constructed via a sequence of multiperson differential games. A verification theorem is proved and, under proper conditions, the wellposedness of the equilibrium HJB equation is established as well.
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This paper investigates reverse auctions that involve continuous values of different types of goods, general nonconvex constraints, and second stage costs. Our analysis seeks to design the payment rules and conditions under which coalitions of participants cannot influence the auction outcome in order to obtain higher collective utility. Under incentivecompatible bidding in the VickreyClarkeGroves mechanism, coalitionproof outcomes are achieved if the submitted bids are convex and the constraint sets are of polymatroidtype. Unfortunately, these conditions do not capture the complexity of the general class of reverse auctions under consideration. By relaxing the property of incentivecompatibility, we investigate further payment rules that are coalitionproof, but without any extra conditions. Among coalitionproof mechanisms, we select the mechanism that minimizes the participants' abilities to benefit from strategic manipulations, in order to incentivize truthful bidding from the
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Identifying coordinate transformations that make strongly nonlinear dynamics approximately linear is a central challenge in modern dynamical systems. These transformations have the potential to enable prediction, estimation, and control of nonlinear systems using standard linear theory. The Koopman operator has emerged as a leading datadriven embedding, as eigenfunctions of this operator provide intrinsic coordinates that globally linearize the dynamics. However, identifying and representing these eigenfunctions has proven to be mathematically and computationally challenging. This work leverages the power of deep learning to discover representations of Koopman eigenfunctions from trajectory data of dynamical systems. Our network is parsimonious and interpretable by construction, embedding the dynamics on a lowdimensional manifold that is of the intrinsic rank of the dynamics and parameterized by the Koopman eigenfunctions. In particular, we identify nonlinear coordinates on which the
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The paper studies the locus in the rank 2 Higgs bundle moduli space corresponding to points which are critical for d of the Poisson commuting functions. These correspond to the Higgs field vanishing on a divisor of degree D. The degree D critical locus has an induced integrable system related to K(D)twisted Higgs bundles. Topological and differentialgeometric properties of the critical loci are addressed.
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Let $\mathcal{T}$ be the group of smooth concordance classes of topologically slice knots, and $\{0\}\subset\cdots\subset \mathcal{T}_{n+1}\subset\mathcal{T}_{n}\subset \cdots\subset \mathcal{T}_{0}\subset \mathcal{T}$ be the bipolar filtration. In this paper, we show that a proper collection of the knots employed by Hedden, Kim, and Livingston to prove $\mathbb{Z}_2^{\infty} < \mathcal{T}$ can be used to see $\mathbb{Z}_2^{\infty} < \mathcal{T}_0/\mathcal{T}_1$.
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In this work we prove convergence of the finite difference scheme for equations of stationary states of a general class of the spatial segregation of reactiondiffusion systems with $m\geq 2$ components. More precisely, we show that the numerical solution $u_h^l$, given by the difference scheme, converges to the $l^{th}$ component $u_l,$ when the mesh size $h$ tends to zero, provided $u_l\in C^2(\Omega),$ for every $l=1,2,\dots,m.$ In particular, our proof provides convergence of a difference scheme for the multiphase obstacle problem.
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This paper is devoted to a description of a general approach introduced by Agrachev and Sarychev in 2005 for studying some control problems for NavierStokes equations. The example of a 1D Burgers equation is used to illustrate the main ideas. We begin with a short discussion of the Cauchy problem and establish a continuity property for the resolving operator. We next turn to the property of approximate controllability and prove that it can be achieved by a twodimensional external force. Finally, we investigate a stronger property, when the approximate controllability and the exact controllability of finitedimensional functionals are proved simultaneously.
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Given a polynomial basis $\Psi_i$ which spans the polynomial vector space $\mathcal{P}$, this paper addresses the construction and use of the algebraic dual space $\mathcal{P}'$ and its canonical basis. Differentiation of dual variables will be discussed. The method will be applied to a Dirichlet and Neumann problem presented in \cite{CarstensenDemkowiczGopalakrishnan} and it is shown that the finite dimensional approximations satisfy $\phi^h = \mbox{div}\, \mathbf{q}^h$ on any grid. The dual method is also applied to a constrained minimization problem, which leads to a mixed finite element formulation. The discretization of the constraint and the Lagrange multiplier will be independent of the grid size, grid shape and the polynomial degree of the basis functions.
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A line field on a manifold is a smooth map which assigns a tangent line to all but a finite number of points of the manifold. As such, it can be seen as a generalization of vector fields. They model a number of geometric and physical properties, e.g. the principal curvature directions dynamics on surfaces or the stress flux in elasticity. We propose a discretization of a MorseSmale line field on surfaces, extending Forman's construction for discrete vector fields. More general critical elements and their indices are defined from local matchings, for which Euler theorem and the characterization of homotopy type in terms of critical cells still hold.
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Let $S$ be the set of subsequences $(x_{n_k})$ of a given real sequence $(x_n)$ which preserve the set of statistical cluster points. It has been recently shown that $S$ is a set of full (Lebesgue) measure. Here, on the other hand, we prove that $S$ is meager if and only if there exists an ordinary limit point of $(x_n)$ which is not a statistical cluster point of $(x_n)$. This provides a nonanalogue between measure and category.
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In this work, we propose constructions that correct duplications of multiple consecutive symbols. These errors are known as tandem duplications, where a sequence of symbols is repeated; respectively as palindromic duplications, where a sequence is repeated in reversed order. We compare the redundancies of these constructions with code size upper bounds that are obtained from sphere packing arguments. Proving that an upper bound on the code cardinality for tandem deletions is also an upper bound for inserting tandem duplications, we derive the bounds based on this special tandem deletion error as this results in tighter bounds. Our upper bounds on the cardinality directly imply lower bounds on the redundancy which we compare with the redundancy of the best known construction correcting arbitrary burst insertions. Our results indicate that the correction of palindromic duplications requires more redundancy than the correction of tandem duplications and both significantly less than arbitr
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We establish the limiting distribution of certain subsets of Farey sequences, i.e., sequences of primitive rational points, on expanding horospheres in covers $\Delta\backslash\mathrm{SL}(n+1,\mathbb{R})$ of $\mathrm{SL}(n+1,\mathbb{Z})\backslash\mathrm{SL}(n+1,\mathbb{R})$, where $\Delta$ is a finite index subgroup of $\mathrm{SL}(n+1,\mathbb{Z})$. These subsets can be obtained by projecting to the hyperplane $\{(x_1,\ldots,x_{n+1})\in\mathbb{R}^{n+1}:x_{n+1}=1\}$ sets of the form $\mathbf{A}=\bigcup_{j=1}^J\boldsymbol{a}_j\Delta$, where for all $j$, $\boldsymbol{a}_j$ is a primitive lattice point in $\mathbb{Z}^{n+1}$. Our method involves applying the equidistribution of expanding horospheres in quotients of $\mathrm{SL}(n+1,\mathbb{R})$ developed by Marklof and Str\"{o}mbergsson, and more precisely understanding how the full Farey sequence distributes in $\Delta\backslash\mathrm{SL}(n+1,\mathbb{R})$ when embedded on expanding horospheres as done in previous work by Marklof. For each
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In this note we observe that one can contact embed all contact 3manifolds into a Stein fillable contact structure on the twisted $S^3$bundle over $S^2$ and also into a unique overtwisted contact structure on $S^3\times S^2$. These results are proven using "spun embeddings" and Lefschetz fibrations.
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In this paper, we investigate the problem of beam alignment in millimeter wave (mmWave) systems, and design an optimal algorithm to reduce the overhead. Specifically, due to directional communications, the transmitter and receiver beams need to be aligned, which incurs high delay overhead since without a priori knowledge of the transmitter/receiver location, the search space spans the entire angular domain. This is further exacerbated under dynamic conditions (e.g., moving vehicles) where the access to the base station (access point) is highly dynamic with intermittent onoff periods, requiring more frequent beam alignment and signal training. To mitigate this issue, we consider an online stochastic optimization formulation where the goal is to maximize the directivity gain (i.e., received energy) of the beam alignment policy within a time period. We exploit the inherent correlation and unimodality properties of the model, and demonstrate that contextual information improves the perfor
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Physicallayer security for wireless networks has become an effective approach and recently drawn significant attention in the literature. In particular, the deployment and allocation of resources such as relays to assist the transmission have gained significant interest due to their ability to improve the secrecy rate of wireless networks. In this work, we examine relay selection criteria with arbitrary knowledge of the channels of the users and the eavesdroppers. We present alternative optimization criteria based on the signaltointerference and the secrecy rate criteria that can be used for resource allocation and that do not require knowledge of the channels of the eavesdroppers and the interference. We then develop effective relay selection algorithms that can achieve a high secrecy rate performance without the need for the knowledge of the channels of the eavesdroppers and the interference. Simulation results show that the proposed criteria and algorithms achieve excellent perfo
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An economic interpretation of the Caputo derivatives of noninteger orders is proposed. The suggested economic interpretation of the fractional derivatives is based on a generalization of average and marginal values of economic indicators. We formulate an economic interpretation by using the concept of the Tindicator that allows us to describe economic processes with memory. The standard average and marginal values of indicator are special cases of the proposed Tindicator, when the order is equal to zero and one, respectively. The fractional derivatives are interpreted as economic characteristics (indicators) that are intermediate between the standard average and marginal values of indicators.
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This paper proposes a dynamic gamebased maintenance scheduling mechanism for the asset owners of the natural gas grid and the power grid by using a bilevel approach. In the upper level, the asset owners of the natural gas grid and the power grid schedule maintenance to maximize their own revenues. This level is modeled as a dynamic game problem, which is solved by the backward induction algorithm. In the lower level, the independent system operator (ISO) dispatches the system to minimize the loss of power load and natural gas load in consideration of the system operating conditions under maintenance plans from the asset owners in the upper level. This is modeled as a mixed integer linear programming problem. For the model of the natural gas grid, a piecewise linear approximation associated with the bigM approach is used to transform the original nonlinear model into the mixed integer linear model. Numerical tests on a 6bus system with a 4node gas grid show the effectiveness of the
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An edgecolored graph $G$ is \emph{conflictfree connected} if, between each pair of distinct vertices, there exists a path containing a color used on exactly one of its edges. The \emph{conflictfree connection number} of a connected graph $G$, denoted by $cfc(G)$, is defined as the smallest number of colors that are required in order to make $G$ conflictfree connected. A coloring of vertices of a hypergraph $H=(\mathcal{V},\mathcal{E})$ is called \emph{conflictfree} if each hyperedge $e$ of $H$ has a vertex of unique color that does not get repeated in $e$. The smallest number of colors required for such a coloring is called the \emph{conflictfree chromatic number} of $H$, and is denoted by \emph{$\chi_{cf}(H)$}. In this paper, we study the conflictfree connection coloring of trees, which is also the conflictfree coloring of edgepath hypergraphs of trees. We first prove that for a tree $T$ of order $n$, $cfc(T)\geq cfc(P_n)=\lceil \log_{2} n\rceil$, and this completely confirms
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In this paper, we compare the radio propagation channels characteristics between 28 and 140 GHz bands based on the wideband (several GHz) and directional channel sounding in a shopping mall environment. The measurements and data processing are conducted in such a way to meet requirements for a fair comparison of large and small scale channel parameters between the two bands. Our results reveal that there is high spatialtemporal correlation between 28 and 140 GHz channels, similar numbers of strong multipath components, and only small variations in the largescale parameters between the two bands. Furthermore, when including the weak paths there are higher total numbers of clusters and paths in 28 GHz as compared to those in 140 GHz bands. With these similarities, it would be very interesting to investigate the potentials of using 140 GHz band in the future mobile radio communications.
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We prove that the asymptotic of the bulk local statistics in models of random lozenge tilings is universal in the vicinity of straight boundaries of the tiled domains. The result applies to uniformly random lozenge tilings of large polygonal domains on triangular lattice and to the probability measures describing the decomposition in GelfandTsetlin bases of tensor products of representations of unitary groups. In a weaker form our theorem also applies to random domino tilings.
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The average nearest neighbor degree (ANND) of a node of degree $k$ is widely used to measure dependencies between degrees of neighbor nodes in a network. We formally analyze ANND in undirected random graphs when the graph size tends to infinity. The limiting behavior of ANND depends on the variance of the degree distribution. When the variance is finite, the ANND has a deterministic limit. When the variance is infinite, the ANND scales with the size of the graph, and we prove a corresponding central limit theorem in the configuration model (CM, a network with random connections). As ANND proved uninformative in the infinite variance scenario, we propose an alternative measure, the average nearest neighbor rank (ANNR). We prove that ANNR converges to a deterministic function whenever the degree distribution has finite mean. We then consider the erased configuration model (ECM), where selfloops and multiple edges are removed, and investigate the wellknown `structural negative correlati
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We prove almost sharp decay estimates for the small data solutions and their derivatives of the VlasovMaxwell system in dimension $n \geq 4$. The smallness assumption concerns only certains weighted $L^1$ or $L^2$ norms of the initial data. In particular, no compact support assumption is required on the Vlasov or the Maxwell fields. The main ingredients of the proof are vector field methods for both the kinetic and the wave equations, null properties of the VlasovMaxwell system to control high velocities and a new decay estimate for the velocity average of the solution of the relativistic massive transport equation. We also consider the massless VlasovMaxwell system under a lower bound on the velocity support of the Vlasov field. As we prove in this paper, the velocity support of the Vlasov field needs to be initially bounded away from $0$. We compensate the weaker decay estimate on the velocity averages of the massless Vlasov field near the light cone by an extra null decomposition
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In multicast scenario, all desired users are divided into $K$ groups. Each group receives its own individual confidential message stream. Eavesdropper group aims to intercept $K$ confidentialmessage streams. To achieve a secure transmission, two secure schemes are proposed: maximum group receive power plus nullspace (NS) projection (MaxGRP plus NSP) and leakage. The former obtains its precoding vector per group by maximizing its own group receive power subject to the orthogonal constraint, and its AN projection matrix consist of all bases of NS of all desired steering vectors from all groups. The latter attains its desired precoding vector per group by driving the current confidential message power to its group steering space and reducing its power leakage to eavesdropper group and other $K1$ desired ones by maximizing signal to leakage and noise ratio (MaxSLNR). And its AN projection matrix is designed by forcing AN power into the eavesdropper steering space by viewing AN as a u
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Android applications, running on either Android itself or on Chrome OS, pause whenever they're not in focus. While this makes sense on a phone, this doesn't make a whole lot of sense on desktop machines such as Chromebooks. As such, Google is addressing this shortcoming with Parallel Tasks. With that in mind, the expected behavior of an open app is that it would remain active and running even when the user clicks to another window. Coming from Windows, Linux, or Mac OS, this is what users expect and it is a bit confusing unless you understand what is happening. Parallel tasks on Android allow the OS to keep everything running and open until you pause the activity or close the app down. Again, with Chrome OS, this is much easier to manage. Just click the "X" on the app and it is closed. Simple. Nothing groundbreaking in and of itself, obviously, but a hugely important 'feature' to have on a laptop or desktop.
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In this article, we prove that for any probability distribution $\mu$ on $\mathbb{N}$ one can construct a stationary version of the infinitebin model an interacting particle system introduced by Foss and Konstantopoulos with move distribution $\mu$. Using this result, we obtain a new formula for the speed of the front of infinitebin models, as a series of positive terms. This implies that the growth rate $C(p)$ of the longest path in a BarakErd\H{o}s graph of parameter $p$ is analytic on $(0,1]$.
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In this paper, the $mn$dimensional space of tensorproduct polynomials of two variables, of degree at most $(m1)+(n1)$, is considered. A theory of twovariate polynomials is developed by establishing the algebra and basic algebraic properties with respect to the usual addition, scalar multiplication, and a newly defined algebraic operation in the considered space. Further, the existence of the considered space is established with respect to the matrix interpolation problem (MIP), $P(i,j)=a_{ij}$ for all $1 \leq i \leq m$, $1 \leq j \leq n$, corresponds to a given matrix $(a_{ij})_{m \times n}$ in the space of $m \times n$ order real matrices. The poisedness of the MIP is proved and three formulae are presented to construct the respective polynomial in the considered space. After that, using construction formulae, a polynomial map from the space of $m \times n$ order real matrices to the considered space is defined. Some properties of the polynomial map are investigated and some isom
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ButtScratching Caterpillars, Whispering Whales, and Eight Other Sounds Tha
1229 THE NEW YORKER 999 
In this paper, we propose a novel joint caching and nonorthogonal multiple access (NOMA) scheme to facilitate advanced downlink transmission for next generation cellular networks. In addition to reaping the conventional advantages of caching and NOMA transmission, the proposed cacheaided NOMA scheme also exploits cached data for interference cancellation which is not possible with separate caching and NOMA transmission designs. Furthermore, as caching can help to reduce the residual interference power, several decoding orders are feasible at the receivers, and these decoding orders can be flexibly selected for performance optimization. We characterize the achievable rate region of cacheaided NOMA and investigate its benefits for minimizing the time required to complete video file delivery. Our simulation results reveal that, compared to several baseline schemes, the proposed cacheaided NOMA scheme significantly expands the achievable rate region for downlink transmission, which tra
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We consider marginal loglinear models for parameterizing distributions on multidimensional contingency tables. These models generalize ordinary loglinear and multivariate logistic models, besides several others. First, we obtain some characteristic properties of marginal loglinear parameters. Then we define collapsibility and strict collapsibility of these parameters in a general sense. Several necessary and sufficient conditions for collapsibility and strict collapsibility are derived using the technique of M\"{o}bius inversion. These include results for an arbitrary set of marginal loglinear parameters having some common effects. The connections of collapsibility and strict collapsibility to various forms of independence of the variables are discussed. Finally, we establish a result on the relationship between parameters with the same effect but different margins, and use it to demonstrate smoothness of marginal loglinear models under collapsibility conditions thereby obtaining
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A virtual link diagram is called normal if the associated abstract link diagram is checkerboard colorable, and a virtual link is normal if it has a normal diagram as a representative. Normal virtual links have some properties similar to classical links.In this paper, we introduce a method of converting a virtual link diagram to a normal virtual link diagram. We show that the normal virtual link diagrams obtained by this method from two equivalent virtual link diagrams are equivalent. We discuss the relationship between this method and some invariants of virtual links.
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Performances of cellular networks over {\kappa}{\mu} shadowed fading with longterm shadowing has been studied in the existing literature. However, the impact of {\kappa}{\mu} shadowed fading with instantaneous shadowing on performances of cellular networks is unknown. Therefore, this letter analyzes the downlink coverage probability of a Poisson network with double shadowed fading which is composed of a largescale fading of lognormal distribution and {\kappa}{\mu} shadowed fading with integer fading parameters. The closest base station association rule without shadowing is considered. For analytical tractability, the double shadowed fading is approximated as a weighted sum of {\kappa}{\mu} shadowed distributions based on the GaussianHermit quadrature. As a main theoretical result, a closedform expression for the downlink coverage probability of a Poisson network under double shadowed fading for the desired signal and arbitrary fading for the interfering signals is successfully
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We investigate the existence of leftinvariant closed G$_2$structures on sevendimensional nonsolvable Lie groups, providing the first examples of this type. When the Lie algebra has trivial Levi decomposition, we show that such a structure exists only when the semisimple part is isomorphic to $\mathfrak{sl}(2,\mathbb{R})$ and the radical is unimodular and centerless. Moreover, we classify unimodular Lie algebras with nontrivial Levi decomposition admitting closed G$_2$structures.
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In this paper, we aim to provide a systematic study of the relationship between Chernoff information and topological, as well as algebraic properties of the corresponding Gaussian tree graphs for the underlying graphical testing problems. We first show the relationship between Chernoff information and generalized eigenvalues of the associated covariance matrices. It is then proved that Chernoff information between two Gaussian trees sharing certain local subtree structures can be transformed into that of two smaller trees. Under our proposed grafting operations, bottleneck Gaussian trees, namely, Gaussian trees connected by one such operation, can thus be simplified into two 3node Gaussian trees, whose topologies and edge weights are subject to the specifics of the operation. Thereafter, we provide a thorough study about how Chernoff information changes when small differences are accumulated into bigger ones via concatenated grafting operations. It is shown that the two Gaussian trees
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In 1885, Fedorov discovered that a convex domain can form a lattice tiling of the Euclidean plane if and only if it is a parallelogram or a centrally symmetric hexagon. It is known that there is no other convex domain which can form two, three or fourfold translative tiling in the Euclidean plane, but there are centrally symmetric convex octagons and decagons which can form fivefold translative tilings. This paper characterizes all the convex domains which can form fivefold translative tilings of the Euclidean plane.
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Following Semmes and Zinsmeister, we continue the study of Carleson measures and their invariance under pullback and pushforward operators. We also study the analogous statements for vanishing Carleson measures. As an application, we show that some quotient space of the space of chordarc curves has a natural complex structure.
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Apple will be releasing the code of the operating system and applications of the Apple Lisa. Just wanted to let everyone know the sources to the OS and applications were recovered, I converted them to Unix end of line conventions and space for Pascal tabs after the files using Disk Image Chef, and they are with Apple for review. After that's done, the Computer History Museum will do a CHM blog post about the historical significance of the software and the code that is cleared for release by Apple will be made available in 2018. The only thing I saw that probably won't be able to be released is the American Heritage dictionary for the spell checker in LisaWrite. Merry Christmas everybody.
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An anonymous reader quotes Kotaku: He's been this way for over an hour, and as word's gotten out the audience has swelled to over 30,000... The Twitch stream opened a couple hours ago on an empty chair. A few minutes later Kaplan walked in and sat down. He's been there ever since, sometimes crossing his legs, sometimes uncrossing them, and always looking, watching, waiting. And lest anyone think the stream is somehow a small segment of footage on loop, there have been a few weird moments sprinkled throughout, including one where Jeff gets booped by an off camera boom mic. In the other, less action filled parts, you can feel time passing as the rate of Jeff blinking changes. Three different blinking speeds, we'll call them long stare, short stare, and turbo eye lash flicking, have taken shape in the stream like the ghosts of Christmas past, present, and future... It's boring to the point of being impossible to look away. It's actually the opposite of what this time of year's supposed t
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2017 began with an AI named "Libratus" defeating four of the world's best poker players. Now the AI's creators reveal how exactly they did it. An anonymous reader quotes the Pittsburgh TribuneReview: First, the AI made the game easier to understand. There are 10**161 potential outcomes in the game of poker  that's a one followed by 161 zeros, potential outcomes in a game of poker. Libratus grouped similar hands, like a Kinghigh flush and a Queenhigh flush, and similar bet sizes to cut down that number. Libratus then created a detailed strategy for how it would play the early rounds of the game and a lessrefined strategy for the final rounds. As the game nears the end, Libratus refined the second strategy based on how the game had gone. A third strategy was at work as well. In realtime, Libratus created another model based on how its play stacked up against the play of the humans. If the humans did something unexpected to Libratus, the AI accounted for it and built it into the s
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An anonymous reader quotes Fortune: Call it a Christmas miracle  albeit of a rather perverse sort. Theranos, the digraced medicaltechnology startup that infamously inflated the capabilities of its devices, has secured $100 million in new funding in the form of a loan. The loan, reported by the Wall Street Journal, will come from Fortress Investment Group. Fortress, whose other underdog bets include a private passenger rail line under construction in Florida, is set to be acquired by Japan's SoftBank. Theranos was reportedly on the verge of bankrutpcy... By the end of 2016, the company reportedly still had $200 million in cash on hand, but had sharply limited prospects for attracting more capital. It has since settled a major lawsuit with Walgreens, a former client, for an undisclosed but likely substantial sum. According to the Journal, the Fortress loan is expected to keep Theranos solvent through 2018. That will give the company more time for its ongoing effort to reboot as a
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Longtime Slashdot reader b0s0z0ku writes: China has turned Xinjiang, the Northwestern part of the country surrounding Urumqi, into one of the most advanced surveillance states in the world. Officially, the purpose is to prevent terrorism and control resistance to the government in one of the few parts of China where ethnic Chinese are a minority. From routine use of facial recognition cameras, to police checkpoints where people's cell phones randomly are checked for unauthorized software, to needing to swipe an ID card and be photographed to buy gasoline and other necessities, the level of technology — and control — is frightening and aweinspiring.
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Suren Enfiajyan writes: Red Hat worker and GNOME blogger Christian F.K. Schaller wrote why GNU/Linux failed to become a mainstream desktop OS... "My thesis is that there really isn't one reason, but rather a range of issues that all have contributed to holding the Linux Desktop back from reaching a bigger market. Also to put this into context, success here in my mind would be having something like 10% market share of desktop systems. That to me means we reached critical mass." He named the following reasons:  A fragmented market  Lack of special applications  Lack of big name applications  Lack of API and ABI stability  Apple's resurgence  Microsoft's aggressive response  Windows piracy  Red Hat mostly stayed away  Canonical's business model not working out  Lack of original device manufacturer support Then he ended with some optimism: "So anyone who has read my blog posts probably knows I am an optimist by nature. This isn't just some kind of genetic dispositio
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Many scientific and engineering challenges can be formulated as optimization problems which are constrained by partial differential equations (PDEs). These include inverse problems, control problems, and design problems. As a major challenge, the associated optimization procedures are inherently largescale. To ensure computational tractability, the design of efficient and robust iterative methods becomes imperative. To meet this challenge, this paper introduces a fast and memoryefficient preconditioned iterative scheme for a class of distributed optimal control problems governed by convectiondiffusionreaction (CDR) equations. As an alternative to loworder discretizations and Schurcomplement block preconditioners, the scheme combines a highorder spectral Galerkin method with an efficient preconditioner tailored specifically for the CDR application. The preconditioner is matrixfree and can be applied within linear complexity where the proportionality constant is small. Numerical
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Answering a question of Benjamini, we present an isometryinvariant random partition of the euclidean space R^3 into infinite connected indinstinguishable pieces, such that the adjacency graph defined on the pieces is the 3regular infinite tree. Along the way, it is proved that any finitely generated amenable Cayley graph (or more generally, amenable unimodular random graph) can be represented in R^3 as an isometryinvariant random collection of polyhedral domains (tiles). A new technique is developed to prove indistinguishability for certain constructions, connecting this notion to factor of iid's.
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The main purpose of this paper is to establish, over Henselian valued fields $K$ with analytic structure, the closedness theorem to the effect that every projection with a projective fiber is a definably closed map. Here the ground field $K$ is assumed to be of equicharacteristic zero, possibly non algebraically closed.
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Presented is a method for efficient computation of the HamiltonJacobi (HJ) equation for timeoptimal control problems using the generalized Hopf formula. Typically, numerical methods to solve the HJ equation rely on a discrete grid of the solution space and exhibit exponential scaling with dimension. The generalized Hopf formula avoids the use of grids and numerical gradients by formulating an unconstrained convex optimization problem. The solution at each point is completely independent, and allows a massively parallel implementation if solutions at multiple points are desired. This work presents a primaldual method for efficient numeric solution and presents how the resulting optimal trajectory can be generated directly from the solution of the Hopf formula, without further optimization. Examples presented have execution times on the order of milliseconds and experiments show computation scales approximately polynomial in dimension with very small highorder coefficients.
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