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We provide an atomic decomposition of the product Hardy spaces $H^p(\widetilde{X})$ which were recently developed by Han, Li, and Ward in the setting of product spaces of homogeneous type $\widetilde{X} = X_1 \times X_2$. Here each factor $(X_i,d_i,\mu_i)$, for $i = 1$, $2$, is a space of homogeneous type in the sense of Coifman and Weiss. These Hardy spaces make use of the orthogonal wavelet bases of Auscher and Hyt\"onen and their underlying reference dyadic grids. However, no additional assumptions on the quasimetric or on the doubling measure for each factor space are made. To carry out this program, we introduce product $(p,q)$atoms on $\widetilde{X}$ and product atomic Hardy spaces $H^{p,q}_{{\rm at}}(\widetilde{X})$. As consequences of the atomic decomposition of $H^p(\widetilde{X})$, we show that for all $q > 1$ the product atomic Hardy spaces coincide with the product Hardy spaces, and we show that the product Hardy spaces are independent of the particular choices of both
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In this article, we study the enumeration by length of several walk models on the square lattice. We obtain bijections between walks in the upper halfplane returning to the $x$axis and walks in the quarter plane. A recent work by Bostan, Chyzak, and Mahboubi has given a bijection for models using small north, west, and southeast steps. We adapt and generalize it to a bijection between halfplane walks using those three steps in two colours and a quarterplane model over the symmetrized step set consisting of north, northwest, west, south, southeast, and east. We then generalize our bijections to certain models with large steps: for given $p\geq1$, a bijection is given between the halfplane and quarterplane models obtained by keeping the small southeast step and replacing the two steps north and west of length 1 by the $p+1$ steps of length $p$ in directions between north and west. This model is close to, but distinct from, the model of generalized tandem walks studied by Bousqu
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We show that if $h(x,y)=ax^2+bxy+cy^2\in \mathbb{Z}[x,y]$ satisfies $b^2\neq 4ac$, then any subset of $\{1,2,\dots,N\}$ with no nonzero differences in the image of $h$ has size at most a constant depending on $h$ times $N\exp(c\sqrt{\log N})$, where $c=c(h)>0$. We achieve this goal by adapting an $L^2$ density increment strategy previously used to establish analogous results for sums of one or more singlevariable polynomials. Our exposition is thorough and selfcontained, in order to serve as an accessible gateway for readers who are unfamiliar with previous implementations of these techniques.
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The cycle joining method and the crossjoin pairing are two main construction techniques for de Bruijn sequences. This work shows how to combine Zech's logarithms and each of the two techniques to efficiently construct binary de Bruijn sequences of large orders. A basic implementation is supplied as a proofofconcept. In the cycle joining method, the cycles are generated by an LFSR with a chosen period. We prove that determining Zech's logarithms is equivalent to identifying conjugate pairs shared by any pair of cycles. The approach quickly finds enough number of conjugate pairs between any two cycles to ensure the existence of trees containing all vertices in the adjacency graph of the LFSR. When the characteristic polynomial of the LFSR is a product of distinct irreducible polynomials, the approach via Zech's logarithms combines nicely with a recently proposed method to determine the conjugate pairs. This allows us to efficiently generate de Bruijn sequences with larger orders. Alon
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We show that fixed dimensional klt weak Fano pairs with alphainvariants and volumes bounded away from $0$ and the coefficients of the boundaries belong to the set of hyperstandard multiplicities $\Phi(\mathscr{R})$ associated to a fixed finite set $\mathscr{R}$ form a bounded family. We also show $\alpha(X,B)^{d1}\mathrm{vol}((K_X+B))$ are bounded from above for all klt weak Fano pairs $(X,B)$ of a fixed dimension $d$.
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A paper of U. First & Z. Reichstein proves that if $R$ is a commutative ring of dimension $d$, then any Azumaya algebra $A$ over $R$ can be generated as an algebra by $d+2$ elements, by constructing such a generating set, but they do not prove that this number of generators is required, or even that for an arbitrarily large $r$ that there exists an Azumaya algebra requiring $r$ generators. In this paper, for any given fixed $n\ge 2$, we produce examples of a base ring $R$ of dimension $d$ and an Azumaya algebra of degree $n$ over $R$ that requires $r(d,n) = \lfloor \frac{d}{2n2} \rfloor + 2$ generators. While $r(d,n) < d+2$ in general, we at least show that there is no uniform upper bound on the number of generators required for Azumaya algebras. The method of proof is to consider certain varieties $B^r_n$ that are universal varieties for degree$n$ Azumaya algebras equipped with a set of $r$ generators, and specifically we show that a natural map on Chow group $CH^{(r1)(n1)}
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In 1968, H. Brezis introduced a notion of operator pseudomonotonicity which provides a unified approach to monotone and nonmonotone variational inequalities (VIs). A closely related notion is that of KyFan hemicontinuity, a continuity property which arises if the famous KyFan minimax inequality is applied to the VI framework. It is clear from the corresponding definitions that KyFan hemicontinuity implies Brezis pseudomonotonicity, but quite surprisingly, a recent publication by Sadeqi and Paydar (J. Optim. Theory Appl., 165(2):344358, 2015) claims the equivalence of the two properties. The purpose of the present note is to show that this equivalence is false; this is achieved by providing a concrete example of a nonlinear operator which is Brezis pseudomonotone but not KyFan hemicontinuous.
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We study Stochastic Gradient Descent (SGD) with diminishing step sizes for convex objective functions. We introduce a definitional framework and theory that defines and characterizes a core property, called curvature, of convex objective functions. In terms of curvature we can derive a new inequality that can be used to compute an optimal sequence of diminishing step sizes by solving a differential equation. Our exact solutions confirm known results in literature and allows us to fully characterize a new regularizer with its corresponding expected convergence rates.
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Millimeterwave communications rely on narrowbeam transmissions to cope with the strong signal attenuation at these frequencies, thus demanding precise alignment between transmitter and receiver. However, the beamalignment procedure may entail a huge overhead and its performance may be degraded by detection errors. This paper proposes a coded energyefficient beamalignment scheme, robust against detection errors. Specifically, the beamalignment sequence is designed such that the errorfree feedback sequences are generated from a codebook with the desired error correction capabilities. Therefore, in the presence of detection errors, the errorfree feedback sequences can be recovered with high probability. The assignment of beams to codewords is designed to optimize energy efficiency, and a waterfilling solution is proved. The numerical results with analog beams depict up to 4dB and 8dB gains over exhaustive and uncoded beamalignment schemes, respectively.
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We derive a nonlinear integrodifferential transport equation describing collective evolution of weights under gradient descent in largewidth neuralnetworklike models. We characterize stationary points of the evolution and analyze several scenarios where the transport equation can be solved approximately. We test our general method in the special case of linear freeknot splines, and find good agreement between theory and experiment in observations of global optima, stability of stationary points, and convergence rates.
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Recently, Kawarabayashi and Thorup presented the first deterministic edgeconnectivity recognition algorithm in nearlinear time. A crucial step in their algorithm uses the existence of vertex subsets of a simple graph $G$ on $n$ vertices whose contractions leave a multigraph with $\tilde{O}(n/\delta)$ vertices and $\tilde{O}(n)$ edges that preserves all nontrivial mincuts of $G$. We show a very simple argument that improves this contractionbased sparsifier by eliminating the polylogarithmic factors, that is, we show a contractionbased sparsification that leaves $O(n/\delta)$ vertices and $O(n)$ edges, preserves all nontrivial mincuts and can be computed in nearlinear time $\tilde{O}(E(G))$. As consequence, every simple graph has $O((n/\delta)^2)$ nontrivial mincuts. Our approach allows to represent all nontrivial mincuts of a graph by a cactus representation, whose cactus graph has $O(n/\delta)$ vertices. Moreover, this cactus representation can be derived directly from
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Full Approximation Scheme (FAS) is a widely used multigrid method for nonlinear problems. In this paper, a new framework to analyze FAS for convex optimization problems is developed. FAS can be recast as an inexact version of nonlinear multigrid methods based on space decomposition and subspace correction. The local problem in each subspace can be simplified to be linear and one gradient decent iteration is enough to ensure a linear convergence of FAS.
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In classical information theory, the information bottleneck method (IBM) can be regarded as a method of lossy data compression which focusses on preserving meaningful (or relevant) information. As such it has recently gained a lot of attention, primarily for its applications in machine learning and neural networks. A quantum analogue of the IBM has recently been defined, and an attempt at providing an operational interpretation of the socalled quantum IB function as an optimal rate of an informationtheoretic task, has recently been made by Salek et al. However, the interpretation given in that paper has a couple of drawbacks; firstly its proof is based on a conjecture that the quantum IB function is convex, and secondly, the expression for the rate function involves certain entropic quantities which occur explicitly in the very definition of the underlying informationtheoretic task, thus making the latter somewhat contrived. We overcome both of these drawbacks by first proving the c
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We establish that for every function $u \in L^1_\mathrm{loc}(\Omega)$ whose distributional Laplacian $\Delta u$ is a signed Borel measure in an open set $\Omega$ in $\mathbb{R}^{N}$, the distributional gradient $\nabla u$ is differentiable almost everywhere in $\Omega$ with respect to the weak$L^{\frac{N}{N1}}$ Marcinkiewicz norm. We show in addition that the absolutely continuous part of $\Delta u$ with respect to the Lebesgue measure equals zero almost everywhere on the level sets $\{u = \alpha\}$ and $\{\nabla u = e\}$, for every $\alpha \in \mathbb{R}$ and $e \in \mathbb{R}^N$. Our proofs rely on an adaptation of Calder\'on and Zygmund's singularintegral estimates inspired by subsequent work by Haj\l asz.
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Momentum is a popular technique to accelerate the convergence in practical training, and its impact on convergence guarantee has been wellstudied for firstorder algorithms. However, such a successful acceleration technique has not yet been proposed for secondorder algorithms in nonconvex optimization.In this paper, we apply the momentum scheme to cubic regularized (CR) Newton's method and explore the potential for acceleration. Our numerical experiments on various nonconvex optimization problems demonstrate that the momentum scheme can substantially facilitate the convergence of cubic regularization, and perform even better than the Nesterov's acceleration scheme for CR. Theoretically, we prove that CR under momentum achieves the best possible convergence rate to a secondorder stationary point for nonconvex optimization. Moreover, we study the proposed algorithm for solving problems satisfying an error bound condition and establish a local quadratic convergence rate. Then, particul
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We examine a class of CalabiYau varieties of the determinantal type in Grassmannians and clarify what kind of examples can be constructed explicitly. We also demonstrate how to compute their genus0 GromovWitten invariants from the analysis of the Givental $I$functions. By constructing $I$functions from the supersymmetric localization formula for the two dimensional gauged linear sigma models, we describe an algorithm to evaluate the genus0 Amodel correlation functions appropriately. We also check that our results for the GromovWitten invariants are consistent with previous results for known examples included in our construction.
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Call a colouring of a graph distinguishing, if the only colour preserving automorphism is the identity. A conjecture of Tucker states that if every automorphism of a graph $G$ moves infinitely many vertices, then there is a distinguishing $2$colouring. We confirm this conjecture for graphs with maximum degree $\Delta \leq 5$. Furthermore, using similar techniques we show that if an infinite graph has maximum degree $\Delta \geq 3$, then it admits a distinguishing colouring with $\Delta  1$ colours. This bound is sharp.
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The distributionally robust Markov Decision Process (MDP) approach asks for a distributionally robust policy that achieves the maximal expected total reward under the most adversarial distribution of uncertain parameters. In this paper, we study distributionally robust MDPs where ambiguity sets for the uncertain parameters are of a format that can easily incorporate in its description the uncertainty's generalized moment as well as statistical distance information. In this way, we generalize existing works on distributionally robust MDP with generalizedmomentbased and statisticaldistancebased ambiguity sets to incorporate information from the former class such as moments and dispersions to the latter class that critically depends on empirical observations of the uncertain parameters. We show that, under this format of ambiguity sets, the resulting distributionally robust MDP remains tractable under mild technical conditions. To be more specific, a distributionally robust policy can
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We present a novel direct transcription method to solve optimization problems subject to nonlinear differential and inequality constraints. In order to provide numerical convergence guarantees, it is sufficient for the functions that define the problem to satisfy boundedness and Lipschitz conditions. Our assumptions are the most general to date; we do not require uniqueness, differentiability or constraint qualifications to hold and we avoid the use of Lagrange multipliers. Our approach differs fundamentally from stateoftheart methods based on collocation. We follow a leastsquares approach to finding approximate solutions to the differential equations. The objective is augmented with the integral of a quadratic penalty on the differential equation residual and a logarithmic barrier for the inequality constraints, as well as a quadratic penalty on the point constraint residual. The resulting unconstrained infinitedimensional optimization problem is discretized using finite elements
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\'Elie Cartan's "g\'en\'eralisation de la notion de courbure" (1922) arose from a creative evaluation of the geometrical structures underlying both, Einstein's theory of gravity and the Cosserat brothers generalized theory of elasticity. In both theories groups operating in the infinitesimal played a crucial role. To judge from his publications in 192224, Cartan developed his concept of generalized spaces with the dual context of general relativity and nonstandard elasticity in mind. In this context it seemed natural to express the translational curvature of his new spaces by a rotational quantity (via a kind of Grassmann dualization). So Cartan called his translational curvature "torsion" and coupled it to a hypothetical rotational momentum of matter several years before spin was encountered in quantum mechanics.
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Recently it was obtained in [Tarzia, Thermal Sci. 21A (2017) 111] for the classical twophase Lam\'eClapeyronStefan problem an equivalence between the temperature and convective boundary conditions at the fixed face under a certain restriction. Motivated by this article we study the twophase Stefan problem for a semiinfinite material with a latent heat defined as a power function of the position and a convective boundary condition at the fixed face. An exact solution is constructed using Kummer functions in case that an inequality for the convective transfer coefficient is satisfied generalizing recent works for the corresponding onephase free boundary problem. We also consider the limit to our problem when that coefficient goes to infinity obtaining a new free boundary problem, which has been recently studied in [ZhouShiZhou, J. Engng. Math. (2017) DOI 10.1007/s106650179921y].
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In this paper, the pgeneralized modified error function is defined as the solution to a nonlinear ordinary differential problem of second order with a Robin type condition at $x=0$. Existence and uniqueness of a nonnegative analytic solution is proved by using a fixed point strategy. It is shown that the pgeneralized modified error function converges to the pmodified error function defined as the solution to a similar problem with a Dirichlet condition at x=0. In both problems, for p=1, the generalized modified error function and the modified error function, studied recently in literature, are recovered
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In this paper we study the Cauchy problem for the generalized Boussinesq equation with initial data in modulation spaces $M^{s}_{p^\prime,q}(\mathbb{R}^n),$ $n\geq 1.$ After a decomposition of the Boussinesq equation in a $2\times 2$nonlinear system, we obtain the existence of global and local solutions in several classes of functions with values in $ M^s_{p,q}\times D^{1}JM^s_{p,q}$ spaces for suitable $p,q$ and $s,$ including the special case $p=2,q=1$ and $s=0.$ Finally, we prove some results of scattering and asymptotic stability in the framework of modulation spaces.
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We study the geometry and topology of exotic Springer fibers for orbits corresponding to onerow bipartitions from an explicit, combinatorial point of view. This includes a detailed analysis of the structure of the irreducible components and their intersections as well as the construction of an explicit affine paving. Moreover, we compute the ring structure of cohomology by constructing a CWcomplex homotopy equivalent to the exotic Springer fiber. This homotopy equivalent space admits an action of the type C Weyl group inducing Kato's original exotic Springer representation on cohomology. Our results are described in terms of the diagrammatics of the twoboundary TemperleyLieb algebra. This provides a first step in generalizing the geometric versions of Khovanov's arc algebra to the exotic setting.
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Locally repairable codes (LRCs) have received significant recent attention as a method of designing data storage systems robust to server failure. Optimal LRCs offer the ideal tradeoff between minimum distance and locality, a measure of the cost of repairing a single codeword symbol. For optimal LRCs with minimum distance greater than or equal to 5, block length is bounded by a polynomial function of alphabet size. In this paper, we give explicit constructions of optimallength (in terms of alphabet size), optimal LRCs with minimum distance equal to 5.
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In this article, we introduce a notion of an exponential matrix, which is a polynomial matrix with exponential properties, and a notion of an equivalence relation of two exponential matrices, and then we initiate to study classifying exponential matrices in positive characteristic, up to equivalence. We classify exponential matrices of Heisenberg groups in positive characteristic, up to equivalence. We also classify exponential matrices of size fourbyfour in positive characteristic, up to equivalence. From these classifications, we obtain a classification of modular representations of elementary abelian $p$groups into Heisenberg groups, up to equivalence, and a classification of fourdimensional modular representations of elementary abelian $p$groups, up to equivalence.
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We use genus zero free energy functions of Hermitian matrix models to define spectral curves and their special deformations. They are special plane curves defined by formal power series with integral coefficients generalizing the Catalan numbers. This is done in two different versions, depending on two different genus expansions, and these two versions are in some sense dual to each other.
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We introduce the tools of intersection theory to the study of Feynman integrals, which allows for a new way of projecting integrals onto a basis. In order to illustrate this technique, we consider Baikov representation of maximal cuts in arbitrary spacetime dimension. We introduce a minimal basis of differential forms with logarithmic singularities on the boundaries of the corresponding integration cycles. We give an algorithm for computing a basis decomposition of an arbitrary maximal cut using the socalled intersection numbers and describe two alternative ways of computing them. Furthermore, we show how to obtain Pfaffian systems of differential equations for the basis integrals using the same technique. All the steps are illustrated on the example of a twoloop nonplanar triangle diagram with a massive loop.
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High dimensional data and systems with many degrees of freedom are often characterized by covariance matrices. In this paper, we consider the problem of simultaneously estimating the dimension of the principal (dominant) subspace of these covariance matrices and obtaining an approximation to the subspace. This problem arises in the popular principal component analysis (PCA), and in many applications of machine learning, data analysis, signal and image processing, and others. We first present a novel method for estimating the dimension of the principal subspace. We then show how this method can be coupled with a Krylov subspace method to simultaneously estimate the dimension and obtain an approximation to the subspace. The dimension estimation is achieved at no additional cost. The proposed method operates on a model selection framework, where the novel selection criterion is derived based on random matrix perturbation theory ideas. We present theoretical analyses which (a) show that th
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For a prime number $q\neq 2$ and $r>0$ we study, whether there exists an isometry of order $q^r$ acting on a free $\mathbb{Z}_{p^k}$module equipped with a scalar product. We investigate, whether there exists such an isometry with no nonzero fixed points. Both questions are completely answered in this paper if $p\neq 2,q$. As an application we refine Naik's criterion for periodicity of links in $S^3$. The periodicity criterion we obtain is effectively computable and gives concrete restrictions for periodicity of lowcrossing knots.
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The finite volume correction for a meanfield monomerdimer system with an attractive interaction are computed for the pressure density, the monomer density and the susceptibility. The results are obtained by introducing a twodimensional integral representation for the partition function decoupling both the hardcore interaction and the attractive one. The nexttoleading terms for each of the mentioned quantities is explicitly derived as well as the value of their sign that is related to their monotonic convergence in the thermodynamic limit.
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We prove $\sqrt{\log n}$ lower bounds on the order of growth fluctuations in three planar growth models (firstpassage percolation, lastpassage percolation, and directed polymers) under no assumptions on distribution of vertex or edge weights other than the minimum conditions required for avoiding pathologies. Such bounds were previously known only for certain restrictive classes of distributions. In addition, the firstpassage shape fluctuation exponent is shown to be at least $1/8$, extending previous results to more general distributions.
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In locally compact, separable metric measure spaces, heat kernels can be classified as either local (having exponential decay) or nonlocal (having polynomial decay). This dichotomy of heat kernels gives rise to operators that include (but are not restricted to) the generators of the classical Laplacian associated to Brownian processes as well as the fractional Laplacian associated with $\beta$stable L\'evy processes. Given embedded data that lie on or close to a compact Riemannian manifold, there is a practical difficulty in realizing this theory directly since these kernels are defined as functions of geodesic distance which is not directly accessible unless if the manifold (i.e., the embedding function or the Riemannian metric) is completely specified. This paper develops numerical methods to estimate the semigroups and generators corresponding to these heat kernels using embedded data that lie on or close to a compact Riemannian manifold (the estimators of the local kernels are res
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We consider unitary simple vertex operator algebras whose vertex operators satisfy certain energy bounds and a strong form of locality and call them strongly local. We present a general procedure which associates to every strongly local vertex operator algebra V a conformal net A_V acting on the Hilbert space completion of V and prove that the isomorphism class of A_V does not depend on the choice of the scalar product on V. We show that the class of strongly local vertex operator algebras is closed under taking tensor products and unitary subalgebras and that, for every strongly local vertex operator algebra V, the map W\mapsto A_W gives a onetoone correspondence between the unitary subalgebras W of V and the covariant subnets of A_V. Many known examples of vertex operator algebras such as the unitary Virasoro vertex operator algebras, the unitary affine Lie algebras vertex operator algebras, the known c=1 unitary vertex operator algebras, the moonshine vertex operator algebra, toge
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The global structure of the minimal spanning tree (MST) is expected to be universal for a large class of underlying random discrete structures. But very little is known about the intrinsic geometry of MSTs of most standard models, and so far the scaling limit of the MST viewed as a metric measure space has only been identified in the case of the complete graph [4]. In this work, we show that the MST constructed by assigning i.i.d. continuous edgeweights to either the random (simple) $3$regular graph or the $3$regular configuration model on $n$ vertices, endowed with the tree distance scaled by $n^{1/3}$ and the uniform probability measure on the vertices, converges in distribution with respect to GromovHausdorffProkhorov topology to a random compact metric measure space. Further, this limiting space has the same law as the scaling limit of the MST of the complete graph identified in [4] up to a scaling factor of $6^{1/3}$. Our proof relies on a novel argument that uses a coupling
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In this paper we consider rough differential equations on a smooth manifold $\left( M\right) .$ The main result of this paper gives sufficient conditions on the driving vectorfields so that the rough ODE's have global (in time) solutions. The sufficient conditions involve the existence of a complete Riemannian metric $\left( g\right) $ on $M$ such that the covariant derivatives of the driving fields and their commutators to a certain order (depending on the roughness of the driving path) are bounded. Many of the results of this paper are generalizations to manifolds of the fundamental results in \cite{Bailleul2015a}.
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We study the Cauchy problem for fractional Schr\"odinger equation with cubic convolution nonlinearity ($i\partial_t u  (\Delta)^{\frac{\alpha}{2}}u\pm (K\ast u^2) u =0$) with Cauchy data in the modulation spaces $M^{p,q}(\mathbb R^{d}).$ For $K(x)= x^{\gamma}$ $ (0< \gamma< \text{min} \{\alpha, d/2\})$, we establish global wellposedness results in $M^{p,q}(\mathbb R^{d}) (1\leq p \leq 2, 1\leq q < 2d/ (d+\gamma))$ when $\alpha =2, d\geq 1$, and with radial Cauchy data when $d\geq 2, \frac{2d}{2d1}< \alpha < 2. $ Similar results are proven in Fourier algebra $\mathcal{F}L^1(\mathbb R^d) \cap L^2(\mathbb R^d).$
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Dotsenko and Vallette discovered an extension to nonsymmetric operads of Buchberger's algorithm for Gr\"obner bases of polynomial ideals. In the free nonsymmetric operad with one ternary operation $({\ast}{\ast}{\ast})$, we compute a Gr\"obner basis for the ideal generated by partial associativity $((abc)de) + (a(bcd)e) + (ab(cde)$. In the category of $\mathbb{Z}$graded vector spaces with Koszul signs, the (homological) degree of $({\ast}{\ast}{\ast})$ may be even or odd. We use the Gr\"obner bases to calculate the dimension formulas for these operads.
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This is the first of a series of papers devoted to a thorough analysis of the class of gradient flows in a metric space $(X,\mathsf{d})$ that can be characterized by Evolution Variational Inequalities. We present new results concerning the structural properties of solutions to the $\mathrm{EVI}$ formulation, such as contraction, regularity, asymptotic expansion, precise energy identity, stability, asymptotic behaviour and their link with the geodesic convexity of the driving functional. Under the crucial assumption of the existence of an $\mathrm{EVI}$ gradient flow, we will also prove two main results: the equivalence with the De Giorgi variational characterization of curves of maximal slope and the convergence of the Minimizing MovementJKO scheme to the $\mathrm{EVI}$ gradient flow, with an explicit and uniform error estimate of order $1/2$ with respect to the step size, independent of any geometric hypothesis (such as upper or lower curvature bounds) on $\mathsf{d}$. In order to av
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We describe the effect of ramified morphisms on Harbourne constants of reduced effective divisors. With this goal, we introduce the pullback of a weighted cluster of infinitely near points under a dominant morphism between surfaces, and describe some of its basic properties. As an application, we describe configurations of curves with transversal intersections and $H$index arbitarily close to \(25/7\simeq 3.571\), smaller than any previously known result.
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Hardy's inequality for Laguerre expansions of Hermite type with the index $\al\in(\{1/2\}\cup[1/2,\infty))^d$ is proved in the multidimensional setting with the exponent $3d/4$. We also obtain the sharp analogue of Hardy's inequality with $L^1$ norm replacing $H^1$ norm at the expense of increasing the exponent by an arbitrarily small value.
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In this paper we study arithmetic properties of a oneparameter family ${\mathbf H}$ of H\'enon maps over the affine line. Given a family of initial points ${\mathbf P}$ satisfying a natural condition, we show the height function $h_{{\mathbf P}}$ associated to ${\mathbf H}$ and ${\mathbf P}$ is the restriction of the height function associated to a semipositive adelically metrized line bundle on projective line. We then show various local properties of $h_{{\mathbf P}}$. Next we consider the set $\Sigma({\mathbf P})$ consisting of periodic parameter values, and study when $\Sigma({\mathbf P})$ is an infinite set or not. We also study unlikely intersections of periodic parameter values.
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We describe the Hochschild cohomology ring for a family of selfinjective algebras of tree class $E_6$ in terms of generators and relations. Together with the results of the previous paper, this gives a complete description of the Hochschild cohomology ring for a selfinjective algebras of tree class $E_6$.
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In cloud computing management, the dynamic adaptation of computing resource allocations under timevarying workload is an active domain of investigation. Several control strategies were already proposed. Here the modelfree control setting and the corresponding "intelligent" controllers, which are most successful in many concrete engineering situations, are employed for the "horizontal elasticity." When compared to the commercial "AutoScaling" algorithms, our easily implementable approach, behaves better even with sharp workload fluctuations. This is confirmed by experiments on Amazon Web Services (AWS).
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To study the question of whether every twodimensional Pr\"ufer domain possesses the stacked bases property, we consider the particular case of the Pr\"ufer domains formed by integervalued polynomials. The description of the spectrum of the rings of integervalued polynomials on a subset of a rankone valuation domain enables us to prove that they all possess the stacked bases property. We also consider integervalued polynomials on rings of integers of number fields and we reduce in this case the study of the stacked bases property to questions concerning $2\times 2$matrices.
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Nadav Amit decided to dig into why some small kernel functions were not being inlined by GCC; the result is a detailed investigation into how these things can go wrong. "Ignoring the assembly shenanigans that this code uses, we can see that in practice it generates a single ud2 instruction. However, the compiler considers this code to be 'big' and consequently oftentimes does not inline functions that use WARN() or similar functions. The reason turns to be the newline characters (marked as '\n' above). The kernel compiler, GCC, is unaware to the code size that will be generated by the inline assembly. It therefore tries to estimate its size based on newline characters and statement separators (';' on x86)."
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Even though the Mac line has grown less repairable over time, fixers have still managed to develop techniques for performing essential screen and battery repairs  until now. According to an internal Apple service document, any Mac with an Apple T2 chip now requires the proprietary 'Apple Service Toolkit 2 (AST 2) System Configuration Suite' (whew, that's a mouthful!) to complete certain repairs. This issue has received extensive coverage, but we wanted to perform some lab testing before we took our shot. Let's break down what all this means first. This is inevitable  Macs have becoming ever more closed and less repairable for years now. This sucks  but at the same time, nobody is forcing you to buy a Mac. There are countless premium Windows and Linux laptops out there that are just as good, and even many nonpremium Windows laptops are more than good enough replacements.
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tedlistens writes: Police officers wearing new cameras by Axon, the U.S.'s largest body camera supplier, will soon be able to send live video from their cameras back to base and elsewhere, potentially expanding police surveillance. Another feature of the new device  set to be released next year  triggers the camera to start recording and alerts command staff once an officer has fired their weapon, a possible corrective to the problem of officers forgetting to switch them on. (The initial price of $699 doesn't include other costs, like a subscription to Axon's Evidence.com data management system.) But adding new technologies to body camera video introduces new privacy concerns, say legal experts, who have cautioned that a network of livestreaming cameras risks turning officers into roving sentinels for a giant panopticonlike surveillance system. Harlan Yu, the executive director of Upturn, a Washington nonprofit consultancy that has studied body cameras, says that livestreaming c
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China has issued a new regulation setting out wideranging police powers to inspect internet service providers and users, as the government tightens its grip on the country's heavily restricted cyberspace. Local media reports: Under the new rule, effective from November 1, central and local public security authorities can enter the premises of all companies and entities that provide internet services and look up and copy information considered relevant to cybersecurity. The regulation was issued by the Ministry of Public Security last month and released on its website on Sunday. It comes more than a year after a controversial cybersecurity law was introduced that has caused widespread concern among foreign companies operating in China. Despite its broad scope, the legislation gives few details about implementation, making it all the more difficult for companies trying to avoid its repercussions. Analysts said the new regulation sheds some light on how the law will be implemented. "That
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Hackaday's Tom Nardi writes about the Federal Aviation Administration's push to repeal Section 336, which states that small remotecontrolled aircraft as used for hobby and educational purposes aren't under FAA jurisdiction. "Despite assurances that the FAA will work towards implementing waivers for hobbyists, critics worry that in the worst case the repeal of Section 336 might mean that remote control pilots and their craft may be held to the same standards as their humancarrying counterparts," writes Nardi. From the report: Section 336 has already been used to shoot down the FAA's illconceived attempt to get RC pilots to register themselves and their craft, so it's little surprise they're eager to get rid of it. But they aren't alone. The Commercial Drone Alliance, a nonprofit association dedicated to supporting enterprise use of Unmanned Aerial Systems (UAS), expressed their support for repealing Section 336 in a June press release: "Basic 'rules of the road' are needed to manage
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Citing corporate values, Google has decided not to compete for the Pentagon's $10 billion cloudcomputing contract. Bloomberg reports: The project, known as the Joint Enterprise Defense Infrastructure cloud, or JEDI, involves transitioning massive amounts of Defense Department data to a commercially operated cloud system. Companies are due to submit bids for the contract, which could last as long as 10 years, on October 12th. Google's announcement on Monday came just months after the company decided not to renew its contract with a Pentagon artificial intelligence program, after extensive protests from employees of the internet giant about working with the military. The company then released a set of principles designed to evaluate what kind of artificial intelligence projects it would pursue. "We are not bidding on the JEDI contract because first, we couldn't be assured that it would align with our AI Principles," a Google spokesman said in a statement. "And second, we determined that
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Google exposed the private data of hundreds of thousands of users of the Google+ social network and then opted not to disclose the issue this past spring, in part because of fears that doing so would draw regulatory scrutiny and cause reputational damage, WSJ reported Monday, citing people briefed on the incident and documents. From the report: As part of its response to the incident, the Alphabet unit plans to announce a sweeping set of data privacy measures that include permanently shutting down all consumer functionality of Google+, the people said. The move effectively puts the final nail in the coffin of a product that was launched in 2011 to challenge Facebook and is widely seen as one of Google's biggest failures. A software glitch in the social site gave outside developers potential access to private Google+ profile data between 2015 and March 2018, [Editor's note: the link may be paywalled; alternative source] when internal investigators discovered and fixed the issue, accordi
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Google exposed the private data of hundreds of thousands of users of the Google+ social network and then opted not to disclose the issue this past spring, in part because of fears that doing so would draw regulatory scrutiny and cause reputational damage, according to people briefed on the incident and documents reviewed by The Wall Street Journal. [...] A software glitch in the social site gave outside developers potential access to private Google+ profile data between 2015 and March 2018, when internal investigators discovered and fixed the issue, according to the documents and people briefed on the incident. A memo reviewed by the Journal prepared by Google's legal and policy staff and shared with senior executives warned that disclosing the incident would likely trigger "immediate regulatory interest" and invite comparisons to Facebook's leak of user information to data firm Cambridge Analytica. Data leaks and breaches happen. They are a fact of life we're pretty much
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