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Millimeter wave (mmWave) communication has attracted increasing attention as a promising technology for 5G networks. One of the key architectural features of mmWave is the use of massive antenna arrays at both the transmitter and the receiver sides. Therefore, by employing directional beamforming (BF), both mmWave base stations (MBSs) and mmWave users (MUEs) are capable of supporting multibeam simultaneous transmissions. However, most researches have only considered a single beam, which means that they do not make full potential of mmWave. In this context, in order to improve the performance of shortrange indoor mmWave networks with multiple reflections, we investigate the challenges and potential solutions of downlink multiuser multibeam transmission, which can be described as a highdimensional (i.e., beamspace) multiuser multipleinput multipleoutput (MUMIMO) technique, including multiuser BF training, simultaneous users' grouping, and multiuser multibeam power allocation.
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This paper focuses on a stochastic formulation of Bayesian attitude estimation on the special orthogonal group. In particular, an exponential probability density model for random matrices, referred to as the matrix Fisher distribution is used to represent the uncertainties of attitude estimates and measurements in a global fashion. Various stochastic properties of the matrix Fisher distribution are derived on the special orthogonal group, and based on these, two types of intrinsic frameworks for Bayesian attitude estimation are constructed. These avoid complexities or singularities of the attitude estimators developed in terms of quaternions. The proposed approaches are particularly useful to deal with large estimation errors or large uncertainties for complex maneuvers to obtain accurate estimates of the attitude.
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This paper describes wideband (1 GHz) base station diversity and coordinated multipoint (CoMP)style largescale measurements at 73 GHz in an urban microcell open square scenario in downtown Brooklyn, New York on the NYU campus. The measurements consisted of ten random receiver locations at pedestrian level (1.4 meters) and ten random transmitter locations at lamppost level (4.0 meters) that provided 36 individual transmitterreceiver (TXRX) combinations. For each of the 36 radio links, extensive directional measurements were made to give insights into smallcell base station diversity at millimeterwave (mmWave) bands. Highgain steerable horn antennas with 7degree and 15degree halfpower beamwidths (HPBW) were used at the transmitter (TX) and receiver (RX), respectively. For each TXRX combination, the TX antenna was scanned over a 120degree sector and the RX antenna was scanned over the entire azimuth plane at the strongest RX elevation plane and two other elevation planes on bo
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This paper studies model order reduction of multiagent systems consisting of identical linear passive subsystems, where the interconnection topology is characterized by an undirected weighted graph. Balanced truncation based on a pair of specifically selected generalized Gramians is implemented on the asymptotically stable part of the fullorder networked model, which leads to a reducedorder system preserving the passivity of each subsystem. To restore the network structure, we then apply a coordinate transformation to convert the resulting reducedorder model to a statespace model of Laplacian dynamics. The proposed method simultaneously reduces the complexity of the network structure and individual agent dynamics. Moreover, it preserves the passivity of the subsystems and allows for the a priori computation of a bound on the approximation error. Finally, the feasibility of the method is demonstrated by an example.
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A classical result of Wente, motivated by the study of sessile capillarity droplets, demonstrates the axial symmetry of every hypersurface which meets a hyperplane at a constant angle and has mean curvature dependent only on the distance from that hyperplane. An analogous result is proven here for the fractional mean curvature operator.
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The AtiyahPatodiSinger(APS) index theorem attracts attention for understanding physics on the surface of materials in topological phases. The mathematical setup for this theorem is, however, not directly related to the physical fermion system, as it imposes on the fermion fields a nonlocal boundary condition known as the "APS boundary condition" by hand, which is unlikely to be realized in the materials. In this work, we attempt to reformulate the APS index in a "physicistfriendly" way for a simple setup with $U(1)$ or $SU(N)$ gauge group on a flat fourdimensional Euclidean space. We find that the same index as APS is obtained from the domainwall fermion Dirac operator with a local boundary condition, which is naturally given by the kink structure in the mass term. As the boundary condition does not depend on the gauge fields, our new definition of the index is easy to compute with the standard Fujikawa method.
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We established and estimate the full asymptotic expansion in integer powers of 1 N of the [ $\sqrt$ N ] first marginals of Nbody evolutions lying in a general paradigm containing Kac models and nonrelativistic quantum evolution. We prove that the coefficients of the expansion are, at any time, explicitly computable given the knowledge of the linearization on the onebody meanfield kinetic limit equation. Instead of working directly with the corresponding BBGKYtype hierarchy, we follows a method developed in [22] for the meanfield limit, dealing with error terms analogue to the vfunctions used in previous works. As a byproduct we get that the rate of convergence to the meanfield limit in 1 N is optimal.
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We establish necessary and sufficient conditions for boundedness of composition operators on the most general class of Hilbert spaces of entire Dirichlet series with real frequencies. Depending on whether or not the space contains any nonzero constant function, different criteria for boundedness are developed. Thus, we complete the characterization of bounded composition operators on all known Hilbert spaces of entire Dirichlet series of one variable.
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In applications that involve interactive curve and surface modeling, the intuitive manipulation of shapes is crucial. For instance, user interaction is facilitated if a geometrical object can be manipulated through control points that interpolate the shape itself. Additionally, models for shape representation often need to provide local shape control and they need to be able to reproduce common shape primitives such as ellipsoids, spheres, cylinders, or tori. We present a general framework to construct families of compactlysupported interpolators that are piecewiseexponential polynomial. They can be designed to satisfy regularity constraints of any order and they enable one to build parametric deformable shape models by suitable linear combinations of interpolators. They allow to change the resolution of shapes based on the refinability of Bsplines. We illustrate their use on examples to construct shape models that involve curves and surfaces with applications to interactive modelin
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Motivated by the theory of weighted shifts on directed trees and its multivariable counterpart, we address the question of identifying commutant and reflexivity of the multiplication $d$tuple $\mathscr M_z$ on a reproducing kernel Hilbert space $\mathscr H$ of $E$valued holomorphic functions on $\Omega$, where $E$ is a separable Hilbert space and $\Omega$ is a bounded starshaped domain in $\mathbb C^d$ with polynomially convex closure. In case $E$ is a finite dimensional cyclic subspace for $\mathscr M_z$, under some natural conditions on the $B(E)$valued kernel associated with $\mathscr H$, the commutant of $\mathscr M_z$ is shown to be the algebra $H^{\infty}_{_{B(E)}}(\Omega)$ of bounded holomorphic $B(E)$valued functions on $\Omega$, provided $\mathscr M_z$ satisfies the matrixvalued von Neumann's inequality. This generalizes a classical result of Shields and Wallen (the case of $\dim E=1$ and $d=1$). As an application, we determine the commutant of a Bergman shift on a leafl
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In 2009, Joselli et al introduced the Neighborhood Grid data structure for fast computation of neighborhood estimates in point clouds. Even though the data structure has been used in several applications and shown to be practically relevant, it is theoretically not yet well understood. The purpose of this paper is to present a polynomialtime algorithm to build the data structure. Furthermore, it is investigated whether the presented algorithm is optimal. This investigations leads to several combinatorial questions for which partial results are given.
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Let $k$ be a field of characteristic $\neq 2$. In this paper, we show that computing $n^{\rm th}$ root of an element of the group $SL_2(k)$ is equivalent to finding solutions of certain polynomial equations over the base field $k$. These polynomials are in two variables, and their description involves generalised Fibonacci polynomials. As an application, we prove some results on surjectivity of word maps over $SL_2(k)$. We prove that the word maps $X_1^2X_2^2$ and $X_1^4X_2^4X_3^4$ are surjective on $SL_2(k)$ and, with additional assumption that characteristic $\neq 3$, the word map $X_1^3X_2^3$ is surjective. Further, over finite field $\mathbb F_q$, $q$ odd, we show that the proportion of squares and, similarly, the proportion of conjugacy classes which are square in $SL_2(\mathbb F_q)$, is asymptotically $\frac{1}{2}$. More generally, for $n\geq 3$, a prime not dividing $q$ but dividing the order of $SL_2(\mathbb F_q)$, we show that the proportion of $n^{th}$ powers, and, similarly
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In any dimension $D$, the Euclidean EinsteinHilbert action, which describes gravity in the absence of matter, can be discretized over random discrete spaces obtained by gluing families of polytopes together in all possible ways. In the physical limit of small Newton constant, only the spaces which maximize the mean curvature survive. In two dimensions, this results in a theory of random discrete spheres, which converge in the continuum limit towards the Brownian sphere, a random fractal space interpreted as a quantum random spacetime. In this limit, the continuous Liouville theory of $D=2$ quantum gravity is recovered. Previous results in higher dimension regarded triangulations  gluings of tetrahedra or $D$dimensional generalizations, leading to the continuum random tree, or gluings of simple colored building blocks of small sizes, for which multitrace matrix model results are recovered. This work aims at providing combinatorial tools which would allow a systematic study of riche
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In this paper, we study the following fractional Schr\"{o}dingerPoisson system \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{2s}(\Delta)^su+V(x)u+\phi u=g(u) & \hbox{in $\mathbb{R}^3$,} \varepsilon^{2t}(\Delta)^t\phi=u^2,\,\, u>0& \hbox{in $\mathbb{R}^3$,} \end{array} \right. \end{equation*} where $s,t\in(0,1)$, $\varepsilon>0$ is a small parameter. Under some local assumptions on $V(x)$ and suitable assumptions on the nonlinearity $g$, we construct a family of positive solutions $u_{\varepsilon}\in H_{\varepsilon}$ which concentrates around the global minima of $V(x)$ as $\varepsilon\rightarrow0$.
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We formalize the intuition that cohomologically rigid overconvergent isocrystals on dense affine open subsets of the projective line over a perfect field of positive characteristic are the ones with no nontrivial infinitesimal deformations that preserve the Robba fibers. En route, we describe a general result showing that the Hochschild cochain complex governs deformations of modules over arbitrary associative algebras in characteristic 0, and we explain a relationship between the Hochschild cochain complex and the de Rham complex.
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We give an introduction to generalisations of conjectures of Brumer and Stark on the annihilator of the class group of a number field. We review the relation to the equivariant Tamagawa number conjecture, the main conjecture of Iwasawa theory for totally real fields, and a conjecture of Gross on the behaviour of $p$adic Artin $L$functions at zero.
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In this paper, we study a full irreducible complete isoparametric submanifold of codimension greater than one in a symmetric space of noncompact type. First we prove that, if such an isoparametric submanifold admits a reflective focal submanifold, then it is curvatureadapted, where the ambient symmetric space may be of compact type. Next we prove that, if such an isoparametric submanifold admits a reflective focal submanifold and if it is of real analytic, then it is a principal orbit of a Hermann type action.
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Let S be a compact connected surface and let f be an element of the group Homeo\_0(S) of homeomorphisms of S isotopic to the identity. Denote by \tilde{f} a lift of f to the universal cover of S. Fix a fundamental domain D of this universal cover. The homeomorphism f is said to be nonspreading if the sequence (d\_{n}/n) converges to 0, where d\_{n} is the diameter of \tilde{f}^{n}(D). Let us suppose now that the surface S is orientable with a nonempty boundary. We prove that, if S is different from the annulus and from the disc, a homeomorphism is nonspreading if and only if it has conjugates in Homeo\_{0}(S) arbitrarily close to the identity. In the case where the surface S is the annulus, we prove that a homeomorphism is nonspreading if and only if it has conjugates in Homeo\_{0}(S) arbitrarily close to a rotation (this was already known in most cases by a theorem by B{\'e}guin, Crovisier, Le Roux and Patou). We deduce that, for such surfaces S, an element of Homeo\_{0}(S) is dist
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We obtain exact solutions to the twodimensional (2D) Dirac equation for the onedimensional P\"oschlTeller potential which contains an asymmetry term. The eigenfunctions are expressed in terms of Heun confluent functions, while the eigenvalues are determined via the solutions of a simple transcendental equation. For the symmetric case, the eigenfunctions of the supercritical states are expressed as spheroidal wave functions, and approximate analytical expressions are obtained for the corresponding eigenvalues. A universal condition for any square integrable symmetric potential is obtained for the minimum strength of the potential required to hold a bound state of zero energy. Applications for smooth electron waveguides in 2D DiracWeyl systems are discussed.
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We obtain exact solutions to the twodimensional (2D) Dirac equation for the onedimensional P\"oschlTeller potential which contains an asymmetry term. The eigenfunctions are expressed in terms of Heun confluent functions, while the eigenvalues are determined via the solutions of a simple transcendental equation. For the symmetric case, the eigenfunctions of the supercritical states are expressed as spheroidal wave functions, and approximate analytical expressions are obtained for the corresponding eigenvalues. A universal condition for any square integrable symmetric potential is obtained for the minimum strength of the potential required to hold a bound state of zero energy. Applications for smooth electron waveguides in 2D DiracWeyl systems are discussed.
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In this note we study graphs $G_r$ with the property that every colouring of $E(G_r)$ with $r+1$ colours admits a copy of some graph $H$ using at most $r$ colours. Such graphs occur naturally at intermediate steps in the synthesis of a $2$colour Ramsey graph $G_1\longrightarrow H$. For $H=K_n$ we prove a result on building a $G_{r}$ from a $G_{r+1}$ and establish Ramseyinfiniteness. From the structural point of view, we characterise the class of the minimal $G_r$ in the case when $H$ is relaxed to be the graph property of containing a cycle; we then use it to progress towards a constructive description of that class by proving both a reduction and an extension theorem.
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In a previous paper we constructed a new class of Iwasawa modules as $\ell$adic realizations of what we called abstract $\ell$adic $1$motives in the number field setting. We proved in loc. cit. that the new Iwasawa modules satisfy an equivariant main conjecture. In this paper we link the new modules to the $\ell$adified Tate canonical class, defined by Tate in 1960 and give an explicit construction of (the minus part of) $\ell$adic Tate sequences for any Galois CM extension $K/k$ of an arbitrary totally real number field $k$. These explicit constructions are significant and useful in their own right but also due to their applications (via our previous results on the Equivariant Main Conjecture in Iwasawa theory) to a proof of the minus part of the far reaching Equivariant Tamagawa Number Conjecture for the Artin motive associated to the Galois extension $K/k$.
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In hierarchical searches for continuous gravitational waves, clustering of candidates is an important postprocessing step because it reduces the number of noise candidates that are followedup at successive stages [1][7][12]. Previous clustering procedures bundled together nearby candidates ascribing them to the same root cause (be it a signal or a disturbance), based on a predefined cluster volume. In this paper, we present a procedure that adapts the cluster volume to the data itself and checks for consistency of such volume with what is expected from a signal. This significantly improves the noise rejection capabilities at fixed detection threshold, and at fixed computing resources for the followup stages, this results in an overall more sensitive search. This new procedure was employed in the first Einstein@Home search on data from the first science run of the advanced LIGO detectors (O1) [11].
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This paper investigates estimation of the mean vector under invariant quadratic loss for a spherically symmetric location family with a residual vector with density of the form $ f(x,u)=\eta^{(p+n)/2}f(\eta\{\x\theta\^2+\u\^2\}) $, where $\eta$ is unknown. We show that the natural estimator $x$ is admissible for $p=1,2$. Also, for $p\geq 3$, we find classes of generalized Bayes estimators that are admissible within the class of equivariant estimators of the form $\{1\xi(x/\u\)\}x$. In the Gaussian case, a variant of the JamesStein estimator, $[1\{(p2)/(n+2)\}/\{\x\^2/\u\^2+(p2)/(n+2)+1\}]x$, which dominates the natural estimator $x$, is also admissible within this class. We also study the related regression model.
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We give an exact formula of the average of adjoint $L$functions of holomorphic Hilbert cusp forms with a fixed weight and a squarefree level, which is a generalization of Zagier's formula known for the case of elliptic cusp forms on ${\rm SL}_2(\mathbb{Z})$. As an application, we prove that the Satake parameters of Hilbert cusp forms with a fixed weight and with growing squarefree levels are equidistributed in an ensemble constructed by values of the adjoint $L$functions.
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We give an explicit weak solution to the Schottky problem, in the spirit of Riemann and Schottky. For any genus $g$, we write down a collection of polynomials in genus $g$ theta constants, such that their common zero locus contains the locus of Jacobians of genus $g$ curves as an irreducible component. These polynomials arise by applying a specific SchottkyJung proportionality to an explicit collection of quartic identities for theta constants in genus $g1$, which are suitable linear combinations of Riemann's quartic relations.
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We propose a novel method to find Nash equilibria in games with binary decision variables by including compensation payments and incentivecompatibility constraints from noncooperative game theory directly into an optimization framework in lieu of using first order conditions of a linearization, or relaxation of integrality conditions. The reformulation offers a new approach to obtain and interpret dual variables to binary constraints using the benefit or loss from deviation rather than marginal relaxations. The method endogenizes the tradeoff between overall (societal) efficiency and compensation payments necessary to align incentives of individual players. We provide existence results and conditions under which this problem can be solved as a mixedbinary linear program. We apply the solution approach to a stylized nodal powermarket equilibrium problem with binary onoff decisions. This illustrative example shows that our approach yields an exact solution to the binary Nash game w
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In this work, an accurate regularization technique based on the Meyer wavelet method is developed to solve the illposed backward heat conduction problem with timedependent thermal diffusivity factor in an infinite "strip". In principle, the extremely illposedness of the considered problem is caused by the amplified infinitely growth in the frequency components which lead to a blowup in the representation of the solution. Using the Meyer wavelet technique, some new stable estimates are proposed in the H\"older and Logarithmic types which are optimal in the sense of given by Tautenhahn. The stability and convergence rate of the proposed regularization technique are proved. The good performance and the highaccuracy of this technique is demonstrated through various one and two dimensional examples. Numerical simulations and some comparative results are presented.
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In this paper, we obtain a nonabelian analogue of Lubkin's embedding theorem for abelian categories. Our theorem faithfully embeds any small regular Mal'tsev category $\mathbb{C}$ in an $n$th power of a particular locally finitely presentable regular Mal'tsev category. The embedding preserves and reflects finite limits, isomorphisms and regular epimorphisms, as in the case of Barr's embedding theorem for regular categories. Furthermore, we show that we can take $n$ to be the (cardinal) number of subobjects of the terminal object in $\mathbb{C}$.
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A sparse random block matrix model suggested by the Hessian matrix used in the study of elastic vibrational modes of amorphous solids is presented and analyzed. By evaluating some moments, benchmarked against numerics, differences in the eigenvalue spectrum of this model in different limits of space dimension $d$, and for arbitrary values of the lattice coordination number $Z$, are shown and discussed. As a function of these two parameters (and their ratio $Z/d$), the most studied models in random matrix theory (ErdosRenyi graphs, effective medium, replicas) can be reproduced in the various limits of block dimensionality $d$. Remarkably, the MarchenkoPastur spectral density (which is recovered by replica calculations for the Laplacian matrix) is reproduced exactly in the limit of infinite size of the blocks, or $d\rightarrow\infty$, which for the first time clarifies the physical meaning of space dimension in these models. The approximate results for $d=3$ provided by our method have
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Published in 1999, Christodoulou proved that the naked singularities of a selfgravitating scalar field are not stable in spherical symmetry and therefore the cosmic censorship conjecture is true in this context. The original proof is by contradiction and sharp estimates are obtained strictly depending on spherical symmetry. In this paper, appropriate a priori estimates for the solution are obtained. These estimates are more relaxed but sufficient for giving another robust argument in proving the instability, in particular not by contradiction. In another related paper, we are able to prove instability theorems of the spherical symmetric naked singularities under certain isotropic gravitational perturbations without symmetries. The argument given in this paper plays a central role.
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We generalize the notion of selfsimilar groups of infinite tree automorphisms to allow for groups which are defined on a tree but do not act faithfully on it. The elements of such a group correspond to labeled trees which may be recognized by a tree automaton (e.g. Rabin, B\"{u}chi, etc.), or considered as elements of a tree shift (e.g. of finite type, sofic) as in symbolic dynamics. We give examples to show that the various classes of selfsimilar groups defined in this way do not coincide. As the main result, extending the classical result of Kitchens on onedimensional group shifts, we provide a sufficient condition for a selfsimilar group whose elements form a sofic tree shift to be a tree shift of finite type. As an application, we show that the closure of certain selfsimilar groups of tree automorphisms are not Rabinrecognizable. \end{abstract}
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This paper presents a Laguerre homotopy method for optimal control problems in semiinfinite intervals (LaHOC), with particular interests given to nonlinear interconnected largescale dynamic systems. In LaHOC, spectral homotopy analysis method is used to derive an iterative solver for the nonlinear twopoint boundary value problem derived from Pontryagins maximum principle. A proof of local convergence of the LaHOC is provided. Numerical comparisons are made between the LaHOC, Matlab BVP5C generated results and results from literature for two nonlinear optimal control problems. The results show that LaHOC is superior in both accuracy and efficiency.
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A radial probability measure is a probability measure with a density (with respect to the Lebesgue measure) which depends only on the distances to the origin. Consider the Euclidean space enhanced with a radial probability measure. A correlation problem concerns showing whether the radial measure of the intersection of two symmetric convex bodies is greater than the product of the radial measures of the two convex bodies. A radial measure satisfying this property is said to satisfy the correlation property. A major question in this field is about the correlation property of the (standard) Gaussian measure. The main result in this paper is a theorem suggesting a sufficient condition for a radial measure to satisfy the correlation property. A consequence of the main theorem will be a proof of the correlation property of the Gaussian measure.
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Throughout, let $R$ be a commutative Noetherian ring. A ring $R$ satisfies Serre's condition $(S_{\ell})$ if for all $P \in \textrm{ Spec }R,$ $\textrm{ depth } R_P \geq \min \{ \ell , \dim R_P \}$. Serre's condition has been a topic of expanding interest. In this paper, we examine a generalization of Serre's condition $(S_{\ell}^j)$. We say a ring satisfies $(S_{\ell}^j)$ when $\textrm{ depth } R_P \geq \min \{ \ell , \dim R_P j \}$ for all $P \in \textrm{ Spec }R$. We prove generalizations of results for rings satisfying Serre's condition.
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In this paper, we prove a theorem that gives a simple criterion for generating commuting pairs of generalized almost complex structures on spaces that are the product of two generalized almost contact metric spaces. We examine the implications of this theorem with regard to the definition of generalized Sasakian and generalized coK\"ahler geometry.
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As more attention is paid to security in the context of control systems and as attacks occur to real control systems throughout the world, it has become clear that some of the most nefarious attacks are those that evade detection. The term stealthy has come to encompass a variety of techniques that attackers can employ to avoid detection. Here we show how the states of the system (in particular, the reachable set corresponding to the attack) can be manipulated under two important types of stealthy attacks. We employ the chisquared fault detection method and demonstrate how this imposes a constraint on the attack sequence either to generate no alarms (zeroalarm attack) or to generate alarms at a rate indistinguishable from normal operation (hidden attack).
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The El Ni\~noSouthern Oscillation (ENSO) is a mode of interannual variability in the coupled equatorial Pacific coupled atmosphere/ocean system. El Ni\~no describes a state in which sea surface temperatures in the eastern Pacific increase and upwelling of colder, deep waters diminishes. El Ni\~no events typically peak in boreal winter, but their strength varies irregularly on decadal time scales. There were exceptionally strong El Ni\~no events in 198283, 199798 and 201516 that affected weather on a global scale. Widely publicized forecasts in 2014 predicted that the 201516 event would occur a year earlier. Predicting the strength of El Ni\~no is a matter of practical concern due to its effects on hydroclimate and agriculture around the world. This paper discusses the frequency and regularity of strong El Ni\~no events in the context of chaotic dynamical systems. We discover a mechanism that limits their predictability in a conceptual "recharge oscillator" model of ENSO. Weak seas
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In this paper, we introduce and investigate a novel class of analytic and univalent functions of negative coefficients in the open unit disk. For this function class, we obtain characterization and distortion theorems as well as the radii of closetoconvexity, starlikeness and convexity by using fractional calculus techniques.
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Given a triangulation of a point set in the plane, a \emph{flip} deletes an edge $e$ whose removal leaves a convex quadrilateral, and replaces $e$ by the opposite diagonal of the quadrilateral. It is well known that any triangulation of a point set can be reconfigured to any other triangulation by some sequence of flips. We explore this question in the setting where each edge of a triangulation has a label, and a flip transfers the label of the removed edge to the new edge. It is not true that every labelled triangulation of a point set can be reconfigured to every other labelled triangulation via a sequence of flips. We characterize when this is possible by proving the \emph{Orbit Conjecture} of Bose, Lubiw, Pathak and Verdonschot which states that \emph{all} labels can be simultaneously mapped to their destination if and only if \emph{each} label individually can be mapped to its destination. Furthermore, we give a polynomialtime algorithm to find a sequence of flips to reconfigure
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We construct a countably infinite simple rank $3$ matroid $M_*$ which $\wedge$embeds every finite simple rank $3$ matroid, and such that every isomorphism between finite $\wedge$subgeometries of $M_*$ extends to an automorphism of $M_*$. We prove that $M_*$ is not $\aleph_0$categorical, it has the independence property, it admits a stationary independence relation, and that $Aut(M_*)$ embeds the symmetric group $Sym(\omega)$. Finally, we use the free projective extension of $M_*$ to conclude the existence of a countably infinite projective plane embedding all the finite simple rank $3$ matroids and whose automorphism group contains $Sym(\omega)$.
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We deal with hypersurfaces in the framework of the relative differential geometry in $\mathbb{R}^4$. We consider a hypersurface $\varPhi$ in $\mathbb{R}^4$ with position vector field $\vect{x}$ which is relatively normalized by a relative normalization $\vect{y}$. Then $\vect{y}$ is also a relative normalization of every member of the oneparameter family $\mathcal{F}$ of hypersurfaces $\varPhi_\mu$ with position vector field $\vect{x}_\mu = \vect{x} + \mu \, \vect{y}$, where $\mu$ is a real constant. We call every hypersurface $\varPhi_\mu \in \mathcal{F}$ relatively parallel to $\varPhi$. This consideration includes both Euclidean and Blaschke hypersurfaces of the affine differential geometry. In this paper we express the relative mean curvature's functions of a hypersurface $\varPhi_\mu$ relatively parallel to $\varPhi$ by means of the ones of $\varPhi$ and the "relative distance" $\mu$. Then we prove several Bonnet's type theorems. More precisely, we show that if two relative mean
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Cao & Yuan obtained a Blichfeldttype result for the vertex set of the edgetoedge tiling of the plane by regular hexagons. Observing that every Archimedean tiling is the union of translates of a fixed lattice, we take a more general viewpoint and investigate basic questions for such point sets about the homogeneous and inhomogeneous problem in the Geometry of Numbers. The Archimedean tilings nicely exemplify our results.
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We consider a saddle point formulation for a sixth order partial differential equation and its finite element approximation, for two sets of boundary conditions. We follow the CiarletRaviart formulation for the biharmonic problem to formulate our saddle point problem and the finite element method. The new formulation allows us to use the $H^1$conforming Lagrange finite element spaces to approximate the solution. We prove a priori error estimates for our approach. Numerical results are presented for linear and quadratic finite element methods.
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An example of a graph that admits balanced fractional revival between antipodes is presented. It is obtained by establishing the correspondence between the quantum walk on a hypercube where the opposite vertices across the diagonals of each face are connected and, the coherent transport of single excitations in the extension of the Krawtchouk spin chain with nexttonearest neighbour interactions.
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In this paper we address a $n+1$body gravitational problem governed by the Newton's laws, where $n$ primary bodies orbit on a plane $\Pi$ and an additional massless particle moves on the perpendicular line to $\Pi$ passing through the center of mass of the primary bodies. We find a condition for that the configuration described be possible. In the case that the primaries are in a rigid motion we classify all the motions of the massless particle. We study the situation when the massless particle has a periodic motion with the same minimal period than primary bodies. We show that this fact is related with the existence of certain pyramidal central configuration.
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We construct a class of noncommutative, noncocommutative, semisimple Hopf algebras of dimension $2n^2$ and present conditions to define an inner faithful action of these Hopf algebras on quantum polynomial algebras, providing, in this way, more examples of semisimple Hopf actions which do not factor through group actions. Also, under certain condition, we classify the inner faithful Hopf actions of the KacPaljutkin Hopf algebra of dimension $8$, $H_8$, on the quantum plane.
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We construct qhypergeometric solutions of the equivariant quantum differential equations of the cotangent bundle of a partial flag variety. These qhypergeometric solutions manifest a LandauGinzburg mirror symmetry for the cotangent bundle. We formulate and prove Pieri rules for quantum equivariant cohomology of the cotangent bundle.
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This paper presents a general theory and isogeometric finite element implementation of phase fields on deforming surfaces. The problem is governed by two coupled fourth order partial differential equations (PDEs) that live on an evolving manifold. For the phase field, the PDE is the CahnHilliard equation for curved surfaces, which can be derived from surface mass balance. For the surface deformation, the PDE is the thin shell equation following from KirchhoffLove kinematics. Both PDEs can be efficiently discretized using $C^1$continous interpolation free of derivative dofs (degreesoffreedom) such as rotations. Structured NURBS and unstructured spline spaces with pointwise $C^1$continuity are considered for this. The resulting finite element formulation is discretized in time by the generalized$\alpha$ scheme with timestep size adaption, and it is fully linearized within a monolithic NewtonRaphson approach. A curvilinear surface parameterization is used throughout the formulati
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For a CohenMacaulay ideal of holomorphic functions, we construct by elementary means residue currents whose annihilator is precisely the given ideal. We give two proofs that the currents have the prescribed annihilator, one using the theory of linkage, and another using an explicit division formula involving these residue currents to express the ideal membership.
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For an arbitrary quiver Q and dimension vector d we prove that the dimension of the space of cuspidal functions on the moduli stack of representations of Q of dimension d over a finite field F_q is given by a polynomial in q. We define a variant of this polynomial counting absolutely cuspidal functions and conjecture that it has positive integral coefficients. In the case of totally negative quivers (such as the gloop quiver for g >1) we provide a closed formula for these polynomials in terms of Kac polynomials.
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We construct an example of a bounded degree, nonamenable, unimodular random rooted graph with $p_c=p_u$ for Bernoulli bond percolation, as well as an example of a bounded degree, unimodular random rooted graph with $p_c<1$ but with an infinite cluster at criticality. These examples show that two wellknown conjectures of Benjamini and Schramm are false when generalised from transitive graphs to unimodular random rooted graphs.
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Recent years have witnessed the success of employing convex relaxations of the AC optimal power flow (OPF) problem to find global or nearglobal optimal solutions. The majority of the effort has focused on solving problem formulations where variables live in continuous spaces. Our focus here is in the extension of these results to the cooptimization of network topology and the OPF problem. We employ binary variables to model topology reconfiguration in the standard semidefinite programming (SDP) formulation of the OPF problem. This makes the problem nonconvex, not only because the variables are binary, but also because of the presence of bilinear products between the binary and other continuous variables. Our proposed convex relaxation to this problem incorporates the bilinear terms in a novel way that improves over the commonly used McCormick approximation. We also address the exponential complexity associated with the discrete variables by partitioning the network graph in a way tha
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In this paper, we introduce a model describing the dynamic of vesicle membranes within an incompressible viscous fluid in $3D$ domains. The system consists of the NavierStokes equations, with an extra stress tensor depending on the membrane, coupled with a CahnHilliard phasefield equation associated to a bending energy plus a penalization term related to the area conservation. This problem has a dissipative in time freeenergy which leads, in particular, to prove the existence of global in time weak solutions. We analyze the largetime behavior of the weak solutions. By using a modified LojasiewiczSimon's result, we prove the convergence as time goes to infinity of each (whole) trajectory to a single equilibrium. Finally, the convergence of the trajectory of the phase is improved by imposing more regularity on the domain and initial phase.
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Given a control system on a manifold that is embedded in Euclidean space, it is sometimes convenient to use a single global coordinate system in the ambient Euclidean space for controller design rather than to use multiple local charts on the manifold or coordinatefree tools from differential geometry. In this paper, we develop a theory about this and apply it to the fully actuated rigid body system for stabilization and tracking. A noteworthy point in this theory is that we legitimately modify the system dynamics outside its statespace manifold before controller design so as to add attractiveness to the manifold in the resulting dynamics.
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We consider ferromagnetic Dyson models which display phase transitions. They are longrange onedimensional Ising ferromagnets, in which the interaction is given by $J_{x,y} = J(xy)\equiv \frac{1}{xy^{2\alpha}}$ with $\alpha \in [0, 1)$, in particular, $J(1)=1$. For this class of models one way in which one can prove the phase transition is via a kind of Peierls contour argument, using the adaptation of the Fr\"ohlichSpencer contours for $\alpha \neq 0$, proposed by Cassandro, Ferrari, Merola and Presutti. As proved by Fr\"ohlich and Spencer for $\alpha=0$ and conjectured by Cassandro et al for the region they could treat, $\alpha \in (0,\alpha_{+})$ for $\alpha_+=\log(3)/\log(2)1$, although in the literature dealing with contour methods for these models it is generally assumed that $J(1)\gg1$, we can show that this condition can be removed in the contour analysis. In addition, combining our theorem with a recent result of Littin and Picco we prove the persistence of the conto
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We continue our systematic search for symmetric Hadamard matrices based on the so called propus construction. In a previous paper this search covered the orders $4v$ with odd $v\le41$. In this paper we cover the cases $v=43,45,47,49,51$. The odd integers $v<120$ for which no symmetric Hadamard matrices of order $4v$ are known are the following: $$47,59,65,67,73,81,89,93,101,103,107,109,113,119.$$ By using the propus construction, we found several symmetric Hadamard matrices of order $4v$ for $v=47,73,113$.
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For LTI control systems, we provide mathematical tools  in terms of Linear Matrix Inequalities  for computing outer ellipsoidal bounds on the reachable sets that attacks can induce in the system when they are subject to the physical limits of the actuators. Next, for a given set of dangerous states, states that (if reached) compromise the integrity or safe operation of the system, we provide tools for designing new artificial limits on the actuators (smaller than their physical bounds) such that the new ellipsoidal bounds (and thus the new reachable sets) are as large as possible (in terms of volume) while guaranteeing that the dangerous states are not reachable. This guarantees that the new bounds cut as little as possible from the original reachable set to minimize the loss of system performance. Computer simulations using a platoon of vehicles are presented to illustrate the performance of our tools.
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Nonlinear generalizations of integrable equations in one dimension, such as the KdV and Boussinesq equations with $p$power nonlinearities, arise in many physical applications and are interesting in analysis due to critical behaviour. This paper studies analogous nonlinear $p$power generalizations of the integrable KP equation and the Boussinesq equation in two dimensions. Several results are obtained. First, for all $p\neq 0$, a Hamiltonian formulation of both generalized equations is given. Second, all Lie symmetries are derived, including any that exist for special powers $p\neq0$. Third, Noether's theorem is applied to obtain the conservation laws arising from the Lie symmetries that are variational. Finally, explicit line soliton solutions are derived for all powers $p>0$, and some of their properties are discussed.
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Given a simple Lie group $H$ of real rank at least $2$ we show that the maximum cardinality of a set of isospectral nonisometric $H$locally symmetric spaces of volume at most $x$ grows at least as fast as $x^{c\log x/ (\log\log x)^2}$ where $c = c(H)$ is a positive constant. In contrast with the real rank $1$ case, this bound comes surprisingly close to the total number of such spaces as estimated in a previous work of Belolipetsky and Lubotzky [BL]. Our proof uses Sunada's method, results of [BL], and some deep results from number theory. We also discuss an open numbertheoretical problem which would imply an even faster growth estimate.
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The bounded cohomology $H^n_b(F, \mathbb{R})$ of a nonabelian free group $F$ has uncountable dimension for $n=2,3$ but it is unkown for $n \geq 4$. The aim of this paper is to show that the cup product between many known bounded $2$cocycles does not yield nontrivial classes in degree $4$. Those classes include the counting quasimorphisms of Brooks.
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The nearby space surrounding the Earth is densely populated by artificial satellites and instruments, whose orbits are distributed within the LowEarthOrbit region (LEO), ranging between 90 and 2 000 $km$ of altitude. As a consequence of collisions and fragmentations, many space debris of different sizes are left in the LEO region. Given the threat raised by the possible damages which a collision of debris can provoke with operational or manned satellites, the study of their dynamics is nowadays mandatory. This work is focused on the existence of equilibria and the dynamics of resonances in LEO. We base our results on a simplified model which includes the geopotential and the atmospheric drag. Using such model, we make a qualitative study of the resonances and the equilibrium positions, including their location and stability. The dissipative effect due to the atmosphere provokes a tidal decay, but we give examples of different behaviors, precisely a straightforward passage through the
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The widespread application of wireless services and dense devices access have triggered huge energy consumption. Because of the environmental and financial considerations, energyefficient design in wireless networks becomes an inevitable trend. To the best of the authors' knowledge, energyefficient orthogonal frequency division multiple access heterogeneous small cell optimization comprehensively considering energy efficiency maximization, power allocation, wireless backhaul bandwidth allocation, and user Quality of Service is a novel approach and research direction, and it has not been investigated. In this paper, we study the energyefficient power allocation and wireless backhaul bandwidth allocation in orthogonal frequency division multiple access heterogeneous small cell networks. Different from the existing resource allocation schemes that maximize the throughput, the studied scheme maximizes energy efficiency by allocating both transmit power of each small cell base station to
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We present nonoverlapping Domain Decomposition Methods (DDM) based on quasioptimal transmission operators for the solution of Helmholtz transmission problems with piecewise constant material properties. The quasioptimal transmission boundary conditions incorporate readily available approximations of DirichlettoNeumann operators. These approximations consist of either complexified hypersingular boundary integral operators for the Helmholtz equation or square root Fourier multipliers with complex wavenumbers. We show that under certain regularity assumptions on the closed interface of material discontinuity, the DDM with quasioptimal transmission conditions are wellposed. We present a DDM framework based on RobintoRobin (RtR) operators that can be computed robustly via boundary integral formulations. More importantly, the use of quasioptimal transmission operators results in DDM that converge in small numbers of iterations even in the challenging highcontrast, highfrequency
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We introduce and study a dimensionallike characteristic of an uniformly almost periodic function, which we call the Diophantine dimension. By definition, it is the exponent in the asymptotic behavior of the inclusio length. Diophantine dimension is connected with recurrent and ergodic properties of an almost periodic function. We get some estimates of the Diophantine dimension for certain quasiperiodic functions and present methods to investigate such a characteristic for almost periodic trajectories of evolution equations. Also we discuss the link between the presented approach and the so called effective versions of the Kronecker theorem.
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