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For an integer $r$, the graph $P_6+rP_3$ has $r+1$ components, one of which is a path on $6$ vertices, and each of the others is a path on $3$ vertices. In this paper we provide a polynomialtime algorithm to test if a graph with no induced subgraph isomorphic to $P_6+rP_3$ is threecolorable. We also solve the list version of this problem, where each vertex is assigned a list of possible colors, which is a subset of $\{1,2,3\}$.
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An anonymous reader writes: On Tuesday, Wikipedia Italy set all of its pages to redirect to a statement raising awareness for the upcoming vote that (barring some legislative wrangling) would make the copyright directive law. The statement reads, in part (emphasis theirs): On July 5, 2018, The Plenary of the European Parliament will vote whether to proceed with a copyright directive proposal which, if approved, will significantly harm the openness of the Internet . The directive instead of updating the copyright laws in Europe and promoting the participation of all the citizens to the society of information, threatens online freedom and creates obstacles to accessing the Web, imposing new barriers, filters and restrictions. If the proposal would be approved in its current form, it could be impossible to share a news article on social networks, or find it through a search engine; Wikipedia itself would be at risk.
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There are many things to love about the Linux Plumbers Conference (LPC), but the event's web site has not often been considered one of them. This year, your editor took on the task of finding a new system to handle proposal submission, review, and scheduling, despite his own poor track record when it comes to creating attractive web sites. The search finally settled on a system called Indico; read on for some impressions of this interesting free eventmanagement system.
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We show that, given an infinite cardinal $\mu$, a graph has colouring number at most $\mu$ if and only if it contains neither of two types of subgraph. We also show that every graph with infinite colouring number has a wellordering of its vertices that simultaneously witnesses its colouring number and its cardinality.
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Vfsafe deltamatroids have the desirable property of behaving well under certain duality operations. Several important classes of deltamatroids are known to be vfsafe, including the class of ribbongraphic deltamatroids, which is related to the class of ribbon graphs or embedded graphs in the same way that graphic matroids correspond to graphs. In this paper, we characterize vfsafe deltamatroids and ribbongraphic deltamatroids by finding the minimal obstructions, called 3minors, to belonging to the class. We find the unique (up to twisted duality) excluded 3minor within the class of set systems for the class of vfsafe deltamatroids. In "Circle graph obstructions under pivoting", Geelen and Oum found the 166 (up to twists) excluded minors for ribbongraphic deltamatroids. By translating Bouchet's characterization of circle graphs to the language of 3minors, we show that this class can also be characterized amongst deltamatroids by a set of three excluded 3minors up to tw
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In this manuscript we introduce and study an extended version of the minimal dispersion of point sets, which has recently attracted considerable attention. Given a set $\mathscr P_n=\{x_1,\dots,x_n\}\subset [0,1]^d$ and $k\in\{0,1,\dots,n\}$, we define the $k$dispersion to be the volume of the largest box amidst a point set containing at most $k$ points. The minimal $k$dispersion is then given by the infimum over all possible point sets of cardinality $n$. We provide both upper and lower bounds for the minimal $k$dispersion that coincide with the known bounds for the classical minimal dispersion for a surprisingly large range of $k$'s.
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Let $T$ be an adjointable operator between two Hilbert $C^*$modules and $T^*$ be the adjoint operator of $T$. The polar decomposition of $T$ is characterized as $T=U(T^*T)^\frac12$ and $\mathcal{R}(U^*)=\overline{\mathcal{R}(T^*)}$, where $U$ is a partial isometry, $\mathcal{R}(U^*)$ and $\overline{\mathcal{R}(T^*)}$ denote the range of $U^*$ and the norm closure of the range of $T^*$, respectively. Based on this new characterization of the polar decomposition, an application to the study of centered operators is carried out.
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Coded caching can create coded multicasting thus significantly accelerates content delivery in broadcast channels with receiver caches. While the original delivery scheme in coded caching multicasts each coded message sequentially, it is not optimal for multipleinput multipleoutput (MIMO) broadcast channels. This work aims to investigate the full spatial multiplexing gain in multiantenna coded caching by transmitting all coded messages concurrently. In specific, we propose to treat the content delivery as the transmission problem with general message sets where all possible messages are present, each with different length and intended for different user set. We first obtain inner and outer bounds of the degrees of freedom (DoF) region of a $K$user $(M,N)$ broadcast channel with general message sets, with $M$ and $N$ being the number of transmit and receive antennas, respectively. Then for any given set of coded messages, we find its minimum normalized delivery time (NDT) by searchi
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We study the Hausdorff distance between a random polytope, defined as the convex hull of i.i.d. random points, and the convex hull of the support of their distribution. As particular examples, we consider uniform distributions on convex bodies, densities that decay at a certain rate when approaching the boundary of a convex body, projections of uniform distributions on higher dimensional convex bodies and uniform distributions on the boundary of convex bodies. We essentially distinguish two types of convex bodies: those with a smooth boundary and polytopes. In the case of uniform distributions, we prove that, in some sense, the random polytope achieves its best statistical accuracy under the Hausdorff metric when the support has a smooth boundary and its worst statistical accuracy when the support is a polytope. This is somewhat surprising, since the exact opposite is true under the Nikodym metric. We prove rate optimality of most our results in a minimax sense. In the case of uniform
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In this paper we deal with infinite horizon optimal control problems. Basing on weak variations in an extremal problem in weighted function spaces we prove necessary conditions in form of the adjoint equation and a variational inequality. The obtained optimality conditions imply several transversality conditions and the validity of Mangasarian type sufficiency conditions. Arising difficulties in the presence of state constraints are discussed.
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We introduce the notion of a weak (homotopy) moment map associated to a Lie group action on a multisymplectic manifold. We show that the existence/uniqueness theory governing these maps is a direct generalization from symplectic geometry. We use weak moment maps to extend Noether's theorem from Hamiltonian mechanics by exhibiting a correspondence between multisymplectic conserved quantities and continuous symmetries on a multiHamiltonian system. We find that a weak moment map interacts with this correspondence in a way analogous to the moment map in symplectic geometry. We define a multisymplectic analog of the classical momentum and position functions on the phase space of a physical system by introducing momentum and position forms. We show that these differential forms satisfy generalized Poisson bracket relations extending the classical bracket relations from Hamiltonian mechanics. We also apply our theory to derive some identities on manifolds with a closed $G_2$ structure.
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Weakly stable torsion classes were introduced by the author and Yekutieli to provide a torsion theoretic characterisation of the notion of weak proregularity from commutative algebra. In this paper we investigate weakly stable torsion classes, with a focus on aspects related to localisation and completion. We characterise when torsion classes arising from left denominator sets and idempotent ideals are weakly stable. We show that every weakly stable torsion class $\operatorname{\mathsf{T}}$ can be associated with a dg ring $A_{\operatorname{\mathsf{T}}}$; in well behaved situations there is a homological epimorphism $A\to A_{\operatorname{\mathsf{T}}}$. We end by studying torsion and completion with respect to a single regular and normal element.
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We propose a lowcomplexity subbanded DSP architecture for digital backpropagation where the walkoff effect is compensated using simple delay elements. For a simulated 96Gbaud signal and 2500 km optical link, our method achieves a 2.8 dB SNR improvement over linear equalization.
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Nash flows over time describe the behavior of selfish users eager to reach their destination as early as possible while traveling along the arcs of a network with capacities and transit times. Throughout the past decade, they have been thoroughly studied in singlesource singlesink networks for the deterministic queuing model, which is of particular relevance and frequently used in the context of traffic and transport networks. In this setting there exist Nash flows over time that can be described by a sequence of static flows featuring special properties, socalled `thin flows with resetting'. This insight can also be used algorithmically to compute Nash flows over time. We present an extension of these result to networks with multiple sources and sinks which are much more relevant in practical applications. In particular, we come up with a subtle generalization of thin flows with resetting which yields a compact description as well as an algorithmic approach for computing multiterm
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We use nonsmooth critical point theory and the theory of geodesics with obstacle to show a multiplicity result about orthogonal geodesic chords in a Riemannian manifold (with boundary) which is homeomorphic to an $N$disk. This applies to brake orbits in a potential well of a natural Hamiltonian system, providing a further step towards the proof of a celebrated conjecture by Seifert.
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One can define the notion of length spectrum for a simple regular periodic graph via counting the orbits of closed reduced cycles under an action of a discrete group of automorphisms. We prove that this length spectrum satisfies an analogue of the Multiplicity one property. We show that if all but finitely many cycles in two simple regular periodic graphs have equal lengths, then all the cycles have equal lengths. This is a graphtheoretic analogue of a similar theorem in the context of geodesics on hyperbolic spaces. We also prove, in the context of actions of finitely generated abelian groups on a graph, that if the adjacency operators for two actions of such a group on a graph are similar, then corresponding periodic graphs are length isospectral.
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Recently, Nunge studied Eulerian polynomials on segmented permutations, namely \emph{generalized Eulerian polynomials}, and further asked whether their coefficients form unimodal sequences. In this paper, we proved the stability of the generalized Eulerian poynomials and hence confirm Nunge's conjecture. Our proof is based on Br\"and\'en's stable multivariate Eulerian polynomials. By acting a stabilitypreserving linear operator on Br\"and\'en's polynomials, we get a multivariate refinement of the generalized Eulerian poynomials. To prove Nunge's conjecture, we also develop a general approach to obtain generalized Sturm sequences from bivariate stable polynomials.
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This paper is concerned with the design of a distributed cooperative synchronization controller for a class of higherorder nonlinear multiagent systems. The objective is to achieve synchronization and satisfy a predefined timebased performance. Dynamics of the agents (also called the nodes) are assumed to be unknown to the controller and are estimated using Neural Networks. The proposed robust neuroadaptive controller drives different states of nodes systematically to synchronize with the state of the leader node within the constraints of the prescribed performance. The nodes are connected through a weighted directed graph with a timeinvariant topology. Only few nodes have access to the leader. Lyapunovbased stability proofs demonstrate that the multiagent system is uniformly ultimately bounded stable. Highly nonlinear heterogeneous networked systems with uncertain parameters and external disturbances were used to validate the robustness and performance of the new novel approach
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Recently, Lin introduced two new partition functions $\textup{PD}_\textup{t}(n)$ and $\textup{PDO}_\textup{t}(n)$, which count the total number of tagged parts over all partitions of $n$ with designated summands and the total number of tagged parts over all partitions of $n$ with designated summands in which all parts are odd. Lin also proved some congruences modulo 3 and 9 for $\textup{PD}_\textup{t}(n)$ and $\textup{PDO}_\textup{t}(n)$, and conjectured some congruences modulo 8. Very recently, Adansie, Chern, and Xia found two new infinite families of congruences modulo 9 for $\textup{PD}_\textup{t}(n)$. In this paper, we prove the congruences modulo 8 conjectured by Lin and also find many new congruences and infinite families of congruences modulo some small powers of 2.
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We propose a non uniform web spline based finite element analysis for elliptic partial differential equation with the gradient type nonlinearity in their principal coefficients like plaplacian equation and QuasiNewtonian fluid flow equations. We discuss the wellposednes of the problems and also derive the apriori error estimates for the proposed finite element analysis and obtain convergence rate of $\mathcal{O}(h^{\alpha})$ for $\alpha > 0$.
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In our previous paper, viewing $D^b(K(l))$ as a noncommutative curve, we observed that it is reasonable to count noncommutative curves in certain triangulated categories, where $K(l)$ is the Kronecker quiver with $l$arrows. We gave a general definition, which specializes to the noncommutative curvecounting invariants. Roughly it defines the set of subcategories of a fixed type in a given category modulo conditions. Here, after recalling the definition, we focus on examples. We compute the noncommutative curvecounting invariants in $D^b(Q)$ for two affine quivers, in $ D^b(A_n)$, and in $D^b(D_4)$. We estimate these numbers for $D^b({\mathbb P}^2)$, in particular we prove finiteness and that the exact determining of these numbers for $D^b({\mathbb P}^2)$ leads to proving (or disproving) of Markov conjecture. Via homological mirror symmetry this gives a new approach to this conjecture. In the present paper are derived also formulas for counting of the subcategories of type $D^b(A_
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Two different types of generalized solutions, namely viscosity and variational solutions, were introduced to solve the firstorder evolutionary HamiltonJacobi equation. They coincide if the Hamiltonian is convex in the momentum variable. In this paper we prove that there exists no other class of integrable Hamiltonians sharing this property. To do so, we build for any nonconvex nonconcave integrable Hamiltonian a smooth initial condition such that the graph of the viscosity solution is not contained in the wavefront associated with the Cauchy problem. The construction is based on a new example for a saddle Hamiltonian and a precise analysis of the onedimensional case, coupled with reduction and approximation arguments.
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We propose nonlinear Dirac equations where the conformal degree of the selfinteraction terms are equal to that of the Dirac operator and the coupling parameters are dimensionless. As such, the massless equation is conformally invariant and preserves the conformal degree for both the linear and nonlinear components. In 2+1 spacetime, we use these features to propose three and fourparameter nonlinear Dirac equation models that might be useful for applications in 2D systems such as graphene sheets, ribbons and nanotubes.
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We introduce an alternative description of coarse proximities. We define a coarse normality condition for connected coarse spaces and show that this definition agrees with large scale normality defined in [3] and asymptotic normality defined in [7]. We utilize the alternative definition of coarse proximities to show that a connected coarse space naturally induces a coarse proximity if and only if the connected coarse space is coarsely normal. We conclude with showing that every connected asymptotic resemblance space induces a coarse proximity if and only if the connected asymptotic resemblance space is asymptotically normal.
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We develop an obstruction theory for the existence and uniqueness of a solution to the gluing problem for a destriction functor and apply it to some wellknown biset functors. The obstruction groups for this theory are reduced cohomology groups of a category ${\mathcal D}_G$, whose objects are the sections $(U,V)$ of $G$ with $V\neq 1$, and whose morphisms are defined as a generalization of morphisms in the orbit category. Using this obstruction theory, we calculate the obstruction group for the Dade group of a $p$group when $p$ is odd.
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Let $G=(V(G),E(G))$ be a simple, finite and undirected graph of order $p$ and size $q$. For $k\ge 1$, a bijection $f: V(G)\cup E(G) \to \{k, k+1, k+2, \ldots, k+p+q1\}$ such that $f(uv)= f(u)  f(v)$ for every edge $uv\in E(G)$ is said to be a $k$super graceful labeling of $G$. We say $G$ is $k$super graceful if it admits a $k$super graceful labeling. In this paper, we study the $k$super gracefulness of some standard graphs. Some general properties are obtained. Particularly, we found many sufficient conditions on $k$super gracefulness for many families of (complete) bipartite and tripartite graphs. We show that some of the conditions are also necessary.
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We outline a new method of construction of globalintime weak solutions of the Liouville equation  and also of the associated BBGKY hierarchy  corresponding to the hard sphere singular Hamiltonian. Our method makes use only of geometric reflection arguments on phase space. As a consequence of our method, in the case of $N=2$ hard spheres, we show for any chaotic initial data, the unique globalintime weak solution $F$ of the Liouville equation is realised as $F=\mathsf{R}(f\otimes f)$ in the sense of tempered distributions on $T\mathbb{R}^{6}\times (\infty, \infty)$, where $\mathsf{R}$ is a 'reflectiontype' operator on Schwartz space, and $f$ is a globalintime classical solution of the 1particle free transport Liouville equation on $T\mathbb{R}^{3}$.
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It is important to classify covering subgroups of the fundamental group of a topological space using their topological properties in the topologized fundamental group. In this paper, we introduce and study some topologies on the fundamental group and use them to classify coverings, semicoverings, and generalized coverings of a topological space. To do this, we use the concept of subgroup topology on a group and discuss their properties. In particular, we explore which of these topologies make the fundamental group a topological group. Moreover, we provide some examples of topological spaces to compare topologies of fundamental groups.
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Let $X$ be a complex, irreducible, quasiprojective variety, and $\pi:\widetilde X\to X$ a resolution of singularities of $X$. Assume that the singular locus ${\text{Sing}}(X)$ of $X$ is smooth, that the induced map $\pi^{1}({\text{Sing}}(X))\to {\text{Sing}}(X)$ is a smooth fibration admitting a cohomology extension of the fiber, and that $\pi^{1}({\text{Sing}}(X))$ has a negative normal bundle in $\widetilde X$. We present a very short and explicit proof of the Decomposition Theorem for $\pi$, providing a way to compute the intersection cohomology of $X$ by means of the cohomology of $\widetilde X$ and of $\pi^{1}({\text{Sing}}(X))$. Our result applies to special Schubert varieties with two strata, even if $\pi$ is nonsmall. And to certain hypersurfaces of $\mathbb P^5$ with onedimensional singular locus.
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We show that the analyticity of semigroups $(T_t)_{t \geq 0}$ of (not necessarily positive) selfadjoint contractive Fourier multipliers on $\mathrm{L}^p$spaces of some abelian locally compact groups is preserved by the tensorisation of the identity operator $\mathrm{Id}_X$ of a Banach space $X$ for a large class of $\mathrm{K}$convex Banach spaces, answering partially a conjecture of Pisier. The result is even new for semigroups of Fourier multipliers acting on $\mathrm{L}^p(\mathbb{R}^n)$ and relies on more general results for semigroups of Fourier multipliers acting on noncommutative $\mathrm{L}^p$spaces. Finally, we also give some result in the discrete case, i.e. for Ritt operators.
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Let $G=(V(G),E(G))$ be a simple, finite and undirected graph of order $p$ and size $q$. For $k\ge 1$, a bijection $f: V(G)\cup E(G) \to \{k, k+1, k+2, \ldots, k+p+q1\}$ such that $f(uv)= f(u)  f(v)$ for every edge $uv\in E(G)$ is said to be a $k$super graceful labeling of $G$. We say $G$ is $k$super graceful if it admits a $k$super graceful labeling. In this paper, we study the $k$super gracefulness of some regular and biregular graphs.
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Two planar embedded circle patterns with the same combinatorics and the same intersection angles can be considered to define a discrete conformal map. We show that two locally finite circle patterns covering the unit disc are related by a hyperbolic isometry. Furthermore, we prove an analogous rigidity statement for the complex plane if all exterior intersection angles of neighboring circles are uniformly bounded away from $0$. Finally, we study a sequence of two circle patterns with the same combinatorics each of which approximates a given simply connected domain. Assume that all kites are convex and all angles in the kites are uniformly bounded and the radii of one circle pattern converge to $0$. Then a subsequence of the corresponding discrete conformal maps converges to a Riemann map between the given domains.
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In this paper we study sums and products in a field. Let $F$ be a field with ${\rm ch}(F)\not=2$, where ${\rm ch}(F)$ is the characteristic of $F$. For any integer $k\ge4$, we show that each $x\in F$ can be written as $a_1+\ldots+a_k$ with $a_1,\ldots,a_k\in F$ and $a_1\ldots a_k=1$ if ${\rm ch}(F)\not=3$, and that for any $\alpha\in F\setminus\{0\}$ we can write each $x\in F$ as $a_1\ldots a_k$ with $a_1,\ldots,a_k\in F$ and $a_1+\ldots+a_k=\alpha$. We also prove that for any $x\in F$ and $k\in\{2,3,\ldots\}$ there are $a_1,\ldots,a_{2k}\in F$ such that $a_1+\ldots+a_{2k}=x=a_1\ldots a_{2k}$.
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Let $c$ and $c'$ be edge or vertex colourings of a graph $G$. We say that $c'$ is less symmetric than $c$ if the stabiliser (in $\operatorname{Aut} G$) of $c'$ is contained in the stabiliser of $c$. We show that if $G$ is not a bicentred tree, then for every vertex colouring of $G$ there is a less symmetric edge colouring with the same number of colours. On the other hand, if $T$ is a tree, then for every edge colouring there is a less symmetric vertex colouring with the same number of edges. Our results can be used to characterise those graphs whose distinguishing index is larger than their distinguishing number.
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A central question in Quantum Computing is how matrices in $SU(2)$ can be approximated by products over a small set of "generators". A topology will be defined on $SU(2)$ so as to introduce the notion of a covering exponent \cite{letter}, which compares the length of products required to covering $SU(2)$ with $\varepsilon$ balls against the Haar measure of $\varepsilon$ balls. An efficient universal set over $PSU(2)$ will be constructed using the Pauli matrices, using the metric of the covering exponent. Then, the relationship between $SU(2)$ and $S^3$ will be manipulated to correlate angles between points on $S^3$ and give a conjecture on the maximum of angles between points on a lattice. It will be shown how this conjecture can be used to compute the covering exponent, and how it can be generalized to universal sets in $SU(2)$.
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In this short paper, we are considering the connection between the \emph{Residual Distribution Schemes} (RD) and the \emph{Flux Reconstruction} (FR) approach. We demonstrate that flux reconstruction can be recast into the RD framework and vice versa. Because of this close connection we are able to apply known results from RD schemes to FR methods. In this context we propose a first demonstration of entropy stability for the FR schemes under consideration and show how to construct entropy stable numerical schemes based on our FR methods. Simultaneously, we do not restrict the mesh to tensor structures or triangle elements, but rather allow polygons. The key of our analysis is a proper choice of the correction functions for which we present an approach here.
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We study the local solvability of a class of operators with multiple characteristics. The class considered here complements and extends the one studied in [9], in that in this paper we consider some cases of operators with complex coefficients that were not present in [9]. The class of operators considered here ideally encompasses classes of degenerate parabolic and Schr\"odinger type operators. We will give local solvability theorems. In general, one has $L^2$ local solvability, but also cases of local solvability with better Sobolev regularity will be presented.
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Let $\mathbb{T}$ be the three dimensional torus acting on $\mathbb{P}^{3}$ and $\mathcal{M}^{\mathbb{T}}_{\mathbb{P}^{3}}(c)$ be the fixed locus of the corresponding action on the moduli space of rank $2$ framed instanton sheaves on $\mathbb{P}^{3}.$ In this work, we prove that $\mathcal{M}^{\mathbb{T}}_{\mathbb{P}^{3}}(c)$ consist only of non locallyfree instanton sheaves whose double dual is the trivial bundle $\mathcal{O}_{\mathbb{P}^{3}}^{\oplus 2}$. Moreover, We relate these instantons to PandharepandeThomas stable pairs and give a classification of their support. This allows to compute a lower bound on the number of components of $\mathcal{M}^{\mathbb{T}}_{\mathbb{P}^{3}}(c).$
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We consider a class of fourth order uniformly elliptic operators in planar Euclidean domains and study the associated heat kernel. For operators with $L^{\infty}$ coefficients we obtain Gaussian estimates with best constants, while for operators with constant coefficients we obtain short time asymptotic estimates. The novelty of this work is that we do not assume that the associated symbol is strongly convex. The short time asymptotics reveal a behavior which is qualitatively different from that of the strongly convex case.
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We give superexponential lower and upper bounds on the number of coloured $d$dimensional triangulations whose underlying space is a manifold, when the number of simplices goes to infinity and $d\geq 3$ is fixed. In the special case of dimension $3$, the lower and upper bounds match up to exponential factors, and we show that there are $2^{\Theta(n)} n^{\frac{n}{6}}$ coloured triangulations of $3$manifolds with $n$ tetrahedra. Our results also imply that random coloured triangulations of $3$manifolds have a sublinear number of vertices. Our upper bounds apply in particular to coloured $d$spheres for which they seem to be the best known bounds in any dimension $d\geq 3$, even though it is often conjectured that exponential bounds hold in this case. We also ask a related question on regular edgecoloured graphs having the property that each $3$coloured component is planar, which is of independent interest.
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We study the distribution of the discrete spectrum, embedded eigenvalues and the existence of limiting absorption principles for some nonselfadjoint perturbations of the onedimensional discrete Schr\"odinger operator. Our results are based on a suitable combination of complex scaling techniques, positive commutators methods, and resonance theory. These strategies can be applied fruitfully to many other models.
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This paper investigates the dynamic stability of an adaptive learning procedure in a traffic game. Using the RouthHurwitz criterion we study the stability of the rest points of the corresponding mean field dynamics. In the special case with two routes and two players we provide a full description of the number and nature of these rest points as well as the global asymptotic behavior of the dynamics. Depending on the parameters of the model, we find that there are either one, two or three equilibria and we show that in all cases the mean field trajectories converge towards a rest point for almost all initial conditions.
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A central problem of random matrix theory is to understand the eigenvalues of spiked random matrix models, introduced by Johnstone, in which a prominent eigenvector (or "spike") is planted into a random matrix. These distributions form natural statistical models for principal component analysis (PCA) problems throughout the sciences. Baik, Ben Arous and Peche showed that the spiked Wishart ensemble exhibits a sharp phase transition asymptotically: when the spike strength is above a critical threshold, it is possible to detect the presence of a spike based on the top eigenvalue, and below the threshold the top eigenvalue provides no information. Such results form the basis of our understanding of when PCA can detect a lowrank signal in the presence of noise. However, under structural assumptions on the spike, not all information is necessarily contained in the spectrum. We study the statistical limits of tests for the presence of a spike, including nonspectral tests. Our results lever
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In this paper, we build up an output feedback law to stabilize the heat equation, with a potential, in a bonded domain $\Omega$. The feedback law abides the following principles: We divide equally the time interval $[0,+\infty)$ into infinitely many disjoint time periods. Each time period is divided into three disjoint subintervals. In the first subinterval, we observe a solution over an open subset of $\Omega$ and sample the output at one time point; in the second subinterval, we add a timeinvariant output feedback control over another open subset of $\Omega$; in the last subinterval, we let the equation evolute free. Thus, the corresponding feedback control is of sampleddata. The advantages of our feedback law are as: the sampling period (which is the length of the above time period) can be arbitrarily given; it has an explicit expression in terms of the sampling period; its norm depends on the sampling period continuously; the behaviours of its norm, when the sampling period goes
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For a long time it is wellknown that highdimensional linear parabolic partial differential equations (PDEs) can be approximated by Monte Carlo methods with a computational effort which grows polynomially both in the dimension and in the reciprocal of the prescribed accuracy. In other words, linear PDEs do not suffer from the curse of dimensionality. For general semilinear PDEs with Lipschitz coefficients, however, it remained an open question whether these suffer from the curse of dimensionality. In this paper we partially solve this open problem. More precisely, we prove in the case of semilinear heat equations with gradientindependent and globally Lipschitz continuous nonlinearities that the computational effort of a variant of the recently introduced multilevel Picard approximations grows polynomially both in the dimension and in the reciprocal of the required accuracy.
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The multilevel hprefinement scheme is a powerful extension of the finite element method that allows local mesh adaptation without the trouble of constraining hanging nodes. This is achieved through hierarchical highorder overlay meshes, a hpscheme based on spatial refinement by superposition. An efficient parallelization of this method using standard domain decomposition approaches in combination with ghost elements faces the challenge of a large basis function support resulting from the overlay structure and is in many cases not feasible. In this contribution, a parallelization strategy for the multilevel hpscheme is presented that is adapted to the scheme's simple hierarchical structure. By distributing the computational domain among processes on the granularity of the active leaf elements and utilizing shared mesh data structures, good parallel performance is achieved, as redundant computations on ghost elements are avoided. We show the scheme's parallel scalability for proble
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The paper studies the partial infimal convolution of the generalized matrixfractional function with a closed, proper, convex function on the space of real symmetric matrices. Particular attention is given to support functions and convex indicator functions. It is shown that this process yields approximate smoothings of all KyFan norms, weighted nuclear norms, and much more. In particular, it is proven that all variational Gram functions are the square of a weighted gauge. A complete variational calculus is given for this class of functions including formulas for their conjugates and subdifferentials.
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In this paper, we present some criteria for the $2$$q$logconvexity and $3$$q$logconvexity of combinatorial sequences, which can be regarded as the first column of certain infinite triangular array $[A_{n,k}(q)]_{n,k\geq0}$ of polynomials in $q$ with nonnegative coefficients satisfying the recurrence relation $$A_{n,k}(q)=A_{n1,k1}(q)+g_k(q)A_{n1,k}(q)+h_{k+1}(q)A_{n1,k+1}(q).$$ Those criterions can also be presented by continued fractions and generating functions. These allow a unified treatment of the $2$$q$logconvexity of alternating Eulerian polynomials, $2$logconvexity of Euler numbers, and $3$$q$logconvexity of many classical polynomials, including the Bell polynomials, the Eulerian polynomials of Types $A$ and $B$, the $q$Schr\"{o}der numbers, $q$central Delannoy numbers, the Narayana polynomials of Types $A$ and $B$, the generating functions of rows in the Catalan triangles of Aigner and Shapiro, the generating functions of rows in the large Schr\"oder triang
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We establish a characterization of complex linear canonical transformations that are positive with respect to a pair of strictly plurisubharmonic quadratic weights. As an application, we show that the boundedness of a class of Toeplitz operators on the Bargmann space is implied by the boundedness of their Weyl symbols.
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We detect the deviation of the grid frequency from the nominal value (i.e., 50 Hz), which itself is an indicator of the power imbalance (i.e., mismatch between power generation and load demand). We first pass the noisy estimates of grid frequency through a hypothesis test which decides whether there is no deviation, positive deviation, or negative deviation from the nominal value. The hypothesis testing incurs missclassification errorsfalse alarms (i.e., there is no deviation but we declare a positive/negative deviation), and missed detections (i.e., there is a positive/negative deviation but we declare no deviation). Therefore, to improve further upon the performance of the hypothesis test, we represent the grid frequency's fluctuations over time as a discretetime hidden Markov model (HMM). We note that the outcomes of the hypothesis test are actually the emitted symbols, which are related to the true states via emission probability matrix. We then estimate the hidden Markov sequ
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We firstly consider fully asynchronous coded computation for matrix multiplication and combining this asynchronous coded computation with private information retrieval (PIR). Our scheme based on polynomial code, which has optimal recovery threshold. First, we propose the polynomial code for asynchronous coded computation. We also propose the polynomial code for PIR. By combining two polynomial codes, we propose asynchronous polynomial code for PIR. We compare the runtime performance of proposed scheme with conventional robust PIR (RPIR) scheme for coded computation.
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We consider the regional enlarged observability problem for fractional evolution differential equations involving Caputo derivatives. Using the Hilbert Uniqueness Method, we show that it is possible to rebuild the initial state between two prescribed functions only in an internal subregion of the whole domain. Finally, an example is provided to illustrate the theory.
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Among all affine, flat, finitely presented group schemes, we focus on those that are pure; this includes all groups which are extensions of a finite locally free group by a group with connected fibres. We prove that over an arbitrary base ring, pure group schemes have a classifying space satisfying the resolution property, an embedding into some GLn, a tensor generator for their category of finite type representations, and can be reconstructed from their category of projective finite type representations. In the case of an Artinian base ring, the same is true for all affine, flat, finitely presented group schemes; this answers a question of Conrad. We also prove that quotients of pure groups by closed pure subgroups over an arbitrary base scheme are Zariskilocally quasiprojective. This answers a question of Raynaud, in the case of affine groups. We give various applications.
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The aim of twodimensional line spectral estimation is to superresolve the spectral point sources of the signal from time samples. In many associated applications such as radar and sonar, due to cutoff and saturation regions in electronic devices, some of the numbers of samples are corrupted by spiky noise. To overcome this problem, we present a new convex program to simultaneously estimate spectral point sources and spiky noise in two dimensions. To prove uniqueness of the solution, it is sufficient to show that a dual certificate exists. Construction of the dual certificate imposes a mild condition on the separation of the spectral point sources. Also, the number of spikes and detectable sparse sources are shown to be a logarithmic function of the number of time samples. Simulation results confirm the conclusions of our general theory.
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We study the eigenvectors of Laplacian matrices of trees. The Laplacian matrix is reduced to a tridiagonal matrix using the Schur complement. This preserves the eigenvectors and allows us to provide fomulas for the ratio of eigenvector entries. We also obtain bounds on the ratio of eigenvector entries along a path in terms of the eigenvalue and Perron values. The results are then applied to the Fiedler vector. Here we locate the extremal entries of the Fiedler vector and study classes of graphs such that the extremal entries can be found at the end points of the longest path.
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With nonorthogonal multiple access(NOMA), we tackle the maximization of the secrecy rate for the strong user subject to a maximum allowable secrecy outage probability, while guaranteeing a constraint on the transmission rate to the weak user. For the first time, the dependence between the eavesdropper's ability to conduct successive interference cancellation and her channel quality is considered. We determine the optimal power allocation and the redundancy rate, based on which the cost of security in terms of the reduction in the strong user's secrecy rate is examined and the benefits of NOMA for secure transmissions are explicitly revealed.
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We initiate the study of simultaneous core multipartitions, generalising simultaneous core partitions, which have been studied extensively in the recent literature. Given a multipartition datum (sc), which consists of a nonnegative integer s and an ltuple c of integers, we introduce the notion of an (sc)core multipartition. Given an arbitrary set of multicore data, we give necessary and sufficient conditions for the corresponding set of simultaneous core multipartitions to be finite. We then study the special case of simultaneous core bipartitions, giving exact enumerative results in some special subcases.
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We consider a one dimensional random walk in random environment that is uniformly biased to one direction. In addition to the transition probability, the jump rate of the random walk is assumed to be spatially inhomogeneous and random. We study the probability that the random walk travels slower than its typical speed and determine its decay rate asymptotic.
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Repair of multiple partially failed cache nodes is studied in a distributed wireless content caching system, where $r$ out of a total of $n$ cache nodes lose part of their cached data. Broadcast repair of failed cache contents at the network edge is studied; that is, the surviving cache nodes transmit broadcast messages to the failed ones, which are then used, together with the surviving data in their local cache memories, to recover the lost content. The tradeoff between the storage capacity and the repair bandwidth is derived. It is shown that utilizing the broadcast nature of the wireless medium and the surviving cache contents at partially failed nodes significantly reduces the required repair bandwidth per node.
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