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  • The article deals with operations defined on convex polyhedra or polyhedral convex functions. Given two convex polyhedra, operations like Minkowski sum, intersection and closed convex hull of the union are considered. Basic operations for one convex polyhedron are, for example, the polar, the conical hull and the image under affine transformation. The concept of a P-representation of a convex polyhedron is introduced. It is shown that many polyhedral calculus operations can be expressed explicitly in terms of P-representations. We point out that all the relevant computational effort for polyhedral calculus consists in computing projections of convex polyhedra. In order to compute projections we use a recent result saying that multiple objective linear programming (MOLP) is equivalent to the polyhedral projection problem. Based on the MOLP-solver bensolve a polyhedral calculus toolbox for Matlab and GNU Octave is developed. Some numerical experiments are discussed.

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  • We propose channel charting (CC), a novel framework in which a multi-antenna network element learns a chart of the radio geometry in its surrounding area. The channel chart captures the local spatial geometry of the area so that points that are close in space will also be close in the channel chart and vice versa. CC works in a fully unsupervised manner, i.e., learning is only based on channel state information (CSI) that is passively collected at a single point in space, but from multiple transmit locations in the area over time. The method then extracts channel features that characterize large-scale fading properties of the wireless channel. Finally, the channel charts are generated with tools from dimensionality reduction, manifold learning, and deep neural networks. The network element performing CC may be, for example, a multi-antenna base-station in a cellular system, and the charted area the served cell. Logical relationships related to the position and movement of a transmitter

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  • Approximately regularized minimizer of the least squares functional with a non-smooth, convex penalty term and an indicator function is considered to be produced iteratively by some nested primal-dual algorithm. The algorithm is a proximal-gradient linesearch based iterative procedure and is introduced as an iterative variational regularization method. Under the consideration of that the exact solution for the linear ill-posed inverse problem satisfies a variational source condition (VSC), convergence of the regularized solution of the minimization problem to the exact solution, and convergence of the iteratively regularized approximate minimizer by our primal-dual algorithm to the exact solution are analysed separately. It is in the emphasis of this work that the regularization parameter obeys {\em Morozov`s discrepancy principle} (MDP) in order for the stability analysis of regularized solution. Furthermore, stability analysis of the algorithm requires us to define the additional par

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  • The object of the present paper is to study 3-dimensional conformally flat quasi-Para-Sasakian manifolds. First, the necessary and sufficient conditions are provided for 3-dimensional quasi-Para-Sasakian manifolds to be conformally flat. Next, a characterization of 3-dimensional conformally flat quasi-Para-Sasakian manifold with \b{eta}=const. is given.

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  • Locally recoverable (LRC) codes provide ways of recovering erased coordinates of the codeword without having to access each of the remaining coordinates. A subfamily of LRC codes with hierarchical locality (H-LRC codes) provides added flexibility to the construction by introducing several tiers of recoverability for correcting different numbers of erasures. We present a general construction of codes from maps between algebraic curves that have 2-level hierarchical locality and specialize it to several code families obtained from quotients of curves by a subgroup of the automorphism group, including rational, elliptic, Kummer, and Artin-Schreier curves. We also construct asymptotically good families of H-LRC codes from curves related to the Garcia-Stichtenoth tower.

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  • We study a competitive stochastic growth model called chase-escape where one species chases another. Red particles spread to adjacent uncolored sites and blue only to adjacent red sites. Red particles are killed when blue occupies the same site. If blue has rate-1 passage times and red rate-$\lambda$, a phase transition occurs for the probability red escapes to infinity on $\mathbb Z^d$, $d$-ary trees, and the ladder graph $\mathbb Z \times \{0,1\}$. The result on the tree was known, but we provide a new, simpler calculation of the critical value, and observe that it is a lower bound for a variety of graphs. We conclude by showing that red can be stochastically slower than blue, but still escape with positive probability for large enough $d$ on oriented $\mathbb Z^d$ with passage times that resemble Bernoulli bond percolation. A stronger conclusion holds in an edge-driven variant of chase-escape on oriented $\mathbb Z^2$.

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  • We give some illustrative applications of our recent result on decompositions of labelled complexes, including some new results on decompositions of hypergraphs with coloured or directed edges. For example, we give fairly general conditions for decomposing an edge-coloured graph into rainbow triangles, and for decomposing an r-digraph into tight q-cycles.

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  • In this paper we study the relation between two notions of largeness that apply to a set of positive integers, namely $\mathrm{Nil}_d{-}\mathrm{Bohr}$ and $\mathrm{SG}_k$, as introduced by Host and Kra. We prove that any $\mathrm{Nil}_d{-}\mathrm{Bohr}_0$ set is necessarily $\mathrm{SG}_k$ where ${k}$ is effectively bounded in terms of $d$. This partially resolves a conjecture of Host and Kra.

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  • The purpose of this article is to illustrate the dynamical concept of {\em homomesy} in three kinds of dynamical systems -- combinatorial, piecewise-linear, and birational -- and to show the relationship between these three settings. In particular, we show how the rowmotion and promotion operations of Striker and Williams (2012), in the case where the poset $P$ is a product of a chain of length $a$ and a chain of length $b$, can be lifted to (continuous) piecewise-linear operations on the order polytope of Stanley (1986), and then lifted to birational operations on the positive orthant in ${\mathbb{R}}^{|P|}$ and indeed to a dense subset of ${\mathbb{C}}^{|P|}$. We prove that these lifted operations, like their combinatorial counterparts, have order $a+b$. In the birational setting, we prove a multiplicative homomesy theorem that by tropicalization yields an additive homomesy result in the piecewise-linear setting, which in turn specializes to an additive homomesy result in the combina

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  • Let $\Gamma\curvearrowright (X,\mu)$ be a measure preserving action of a countable group $\Gamma$ on a standard probability space $(X,\mu)$. We prove that if the action $\Gamma\curvearrowright X$ is not profinite and satisfies a certain spectral gap condition, then there does not exist a countable-to-one Borel homomorphism from its orbit equivalence relation to the orbit equivalence relation of any modular action (i.e., an inverse limit of actions on countable sets). As a consequence, we show that if $\Gamma$ is a countable dense subgroup of a compact non-profinite group $G$ such that the left translation action $\Gamma\curvearrowright G$ has spectral gap, then $\Gamma\curvearrowright G$ is antimodular and not orbit equivalent to any, {\it not necessarily free}, profinite action. This provides the first such examples of compact actions, partially answering a question of Kechris and answering a question of Tsankov.

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  • In this paper, we study completete monotonicity properties of the function $f_{a,k}(x)=\psi^{(k)}(x+a) - \psi^{(k)}(x) - \frac{ak!}{x^{k+1}}$, where $a\in(0,1)$ and $k\in \mathbb{N}_0$. Specifically, we consider the cases for $k\in \{ 2n: n\in \mathbb{N}_0 \}$ and $k\in \{ 2n+1: n\in \mathbb{N}_0 \}$. Subsequently, we deduce some inequalities involving the polygamma functions.

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  • The paper is concerned with completeness property of rank one perturbations of unperturbed operators generated by special boundary value problems (BVP) for the following $2 \times 2$ system \begin{equation} L y = -i B^{-1} y' + Q(x) y = \lambda y , \quad B = \begin{pmatrix} b_1 & 0 \\ 0 & b_2 \end{pmatrix}, \quad y = \begin{pmatrix} y_1 \\ y_2 \end{pmatrix}, \end{equation} on a finite interval assuming that a potential matrix $Q$ is summable, and $b_1 b_2^{-1} \notin \mathbb{R}$ (essentially non-Dirac type case). We assume that unperturbed operator generated by a BVP belongs to one of the following three subclasses of the class of spectral operators: (a) normal operators; (b) operators similar either to a normal or almost normal; (c) operators that meet Riesz basis property with parentheses. We show that in each of the three cases there exists (in general, non-unique) operator generated by a quasi-periodic BVP and its certain rank-one perturbations (in the resolvent sense) gene

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  • This is a survey of results on positivity of vector bundles, inspired by the Brunn-Minkowski and Pr\'ekopa theorems. Applications to complex analysis, K\"ahler geometry and algebraic geometry are also discussed.

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  • We prove that for all integers $r \geq 2$ and $g \geq \lfloor \frac{r^2+10r+1}{4} \rfloor$ there exists a component of the locus $\mathcal{S}^r_g$ of spin curves with a theta characteristic $L$ such that $h^0(L) \geq r+1$ and $h^0(L)\equiv r+1 (\text{mod} 2)$ which has expected codimension $\binom{r+1}{2}$ inside the moduli space $\mathcal{S}_g$ of spin curves of genus $g$.

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  • This paper presents a constructive proof of the existence of a regular non-atomic strictly-positive measure on any second-countable locally compact non-atomic Hausdorff space. This construction involves a sequence of finitely-additive set functions defined recursively on an ascending sequence of rings of subsets with a premeasure limit that is extendable to a measure with the desired properties. Non-atomicity of the space provides a meticulous way to ensure that the limit is a premeasure.

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  • The dynamics of the magnetic distribution in a ferromagnetic material is governed by the Landau-Lifshitz equation, which is a nonlinear geometric dispersive equation with a nonconvex constraint that requires the magnetization to remain of unit length throughout the domain. In this article, we present a mass-lumped finite element method for the Landau-Lifshitz equation. This method preserves the nonconvex constraint at each node of the finite element mesh, and is energy nonincreasing. We show that the numerical solution of our method for the Landau-Lifshitz equation converges to a weak solution of the Landau-Lifshitz-Gilbert equation using a simple proof technique that cancels out the product of weakly convergent sequences. Numerical tests for both explicit and implicit versions of the method on a unit square with periodic boundary conditions are provided for structured and unstructured meshes.

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  • The quantile admission process with veto power is a stochastic processes suggested by Alon, Feldman, Mansour, Oren and Tennenholtz as a model for the evolution of an exclusive social group. The model itself consists of a growing multiset of real numbers, representing the opinions of the members of the club. On each round two new candidates, holding i.i.d. $\mu$-distributed opinions, apply for admission to the club. The one whose opinion is minimal is then admitted if the percentage of current members closer in their opinion to his is at least $r$. Otherwise neither of the candidates is admitted. We show that for any $\mu$ and $r$, the empirical distribution of opinions in the club converges to a limit distribution. We further analyse this limit, show that it may be non-deterministic and provide conditions under which it is deterministic. The results rely on a recent work of the authors relating tail probabilities of mean and maximum of any pair of unbounded i.i.d. random variables, and

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  • We determine when a convex body in $\mathbb{R}^d$ is the closed unit ball of a reasonable crossnorm on $\mathbb{R}^{d_1}\otimes\cdots\otimes\mathbb{R}^{d_l},$ $d=d_1\cdots d_l.$ We call these convex bodies "tensorial bodies". We prove that, among them, the only ellipsoids are the closed unit balls of Hilbert tensor products of Euclidean spaces. It is also proved that linear isomorphisms on $\mathbb{R}^{d_1}\otimes\cdots \otimes \mathbb{R}^{d_l}$ preserving decomposable vectors map tensorial bodies into tensorial bodies. This leads us to define a Banach-Mazur type distance between them, and to prove that there exists a Banach-Mazur type compactum of tensorial bodies.

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  • We analyze in detail the geometry and dynamics of the cosmological region arising in spherically symmetric black hole solutions of the Einstein-Maxwell-scalar field system with a positive cosmological constant. More precisely, we solve, for such a system, a characteristic initial value problem with data emulating a dynamic cosmological horizon. Our assumptions are fairly weak, in that we only assume that the data approaches that of a subextremal Reissner-Nordstr\"om-de Sitter black hole, without imposing any rate of decay. We then show that the radius (of symmetry) blows up along any null ray parallel to the cosmological horizon ("near" $i^+$), in such a way that $r=+\infty$ is, in an appropriate sense, a spacelike hypersurface. We also prove a version of the Cosmic No-Hair Conjecture by showing that in the past of any causal curve reaching infinity both the metric and the Riemann curvature tensor asymptote those of a de Sitter spacetime. Finally, we discuss conditions under which all

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  • We continue the study of the theories of Baldwin-Shi hypergraphs from $[5]$. Restricting our attention to when the rank $\delta$ is rational valued, we show that each countable model of the theory of a given Baldwin-Shi hypergraph is isomorphic to a generic structure built from some suitable subclass of the original class of finite structures with the inherited notion of strong substructure. We introduce a notion of dimension for a model and show that there is a an elementary chain $\{\mathfrak{M}_{\beta}:\beta<\omega+1\}$ of countable models of the theory of a fixed Baldwin-Shi hypergraph with $\mathfrak{M}_{\beta}\preccurlyeq\mathfrak{M}_\gamma$ if and only if the dimension of $\mathfrak{M}_\beta$ is at most the dimension of $\mathfrak{M}_\gamma$ and that each countable model is isomorphic to some $\mathfrak{M}_\beta$. We also study the regular types that appear in these theories and show that the dimension of a model is determined by a particular regular type. Further, drawing on

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  • In this paper, we address the problem of counting integer points in a rational polytope described by $P(y) = \{ x \in \mathbb{R}^m \colon Ax = y, x \geq 0\}$, where $A$ is an $n \times m$ integer matrix and $y$ is an $n$-dimensional integer vector. We study the Z-transformation approach initiated by Brion-Vergne, Beck, and Lasserre-Zeron from the numerical analysis point of view, and obtain a new algorithm on this problem: If $A$ is nonnegative, then the number of integer points in $P(y)$ can be computed in $O(\mathrm{poly} (n,m, \|y\|_\infty) (\|y\|_\infty + 1)^n)$ time and $O(\mathrm{poly} (n,m, \|y\|_\infty))$ space.This improves, in terms of space complexity, a naive DP algorithm with $O((\|y\|_\infty + 1)^n)$-size DP table. Our result is based on the standard error analysis to the numerical contour integration for the inverse Z-transform, and establish a new type of an inclusion-exclusion formula for integer points in $P(y)$. We apply our result to hypergraph $b$-matching, and obt

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  • The main purpose of this paper is to give an overview over the theory of abelian varieties, with main focus on Jacobian varieties of curves reaching from well-known results till to latest developments and their usage in cryptography. In the first part we provide the necessary mathematical background on abelian varieties, their torsion points, Honda-Tate theory, Galois representations, with emphasis on Jacobian varieties and hyperelliptic Jacobians. In the second part we focus on applications of abelian varieties on cryptography and treating separately, elliptic curve cryptography, genus 2 and 3 cryptography, including Diffie-Hellman Key Exchange, index calculus in Picard groups, isogenies of Jacobians via correspondences and applications to discrete logarithms. Several open problems and new directions are suggested.

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  • We show that the total-variation mixing time of the lamplighter random walk on fractal graphs exhibit sharp cutoff when the underlying graph is transient (namely of spectral dimension greater than two). In contrast, we show that such cutoff can not occur for strongly recurrent underlying graphs (i.e. of spectral dimension less than two).

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  • We prove cyclotron damping for the collsionless Vlasov-Maxwell equations on $\mathbb{T}_{x}^{3}\times\mathbb{R}_{v}^{3}.$ Our proof is based on a new observation from Faraday Law of Electromagnetic induction and Lenz's Law. On the basis of it, we use the improved Newton iteration scheme to show the damping mechanism.

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  • We show that the category of decomposition spaces and CULF maps is locally a topos. Precisely, the slice category over any decomposition space D is a presheaf topos, namely decomp/D=Psh(tw D). A crucial ingredient in our constructions and proofs is a natural transformation between the functors given by taking category of elements and taking twisted arrow category.

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  • Given a skeletally small category $\mathcal{C}$, we show that any locally finite endo-length $\mathcal{C}$-module is the direct sum of indecomposable $\mathcal{C}$-modules, whose endomorphism algebra is local.

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  • The Eulerian polynomials and derangement polynomials are two well-studied generating functions that frequently arise in combinatorics, algebra, and geometry. When one makes an appearance, the other often does so as well, and their corresponding generalizations are similarly linked. This is this case in the theory of subdivisions of simplicial complexes, where the Eulerian polynomial is an $h$-polynomial and the derangement polynomial is its local $h$-polynomial. Separately, in Ehrhart theory the Eulerian polynomials are generalized by the $h^\ast$-polynomials of $s$-lecture hall simplices. Here, we show that derangement polynomials are analogously generalized by the box polynomials, or local $h^\ast$-polynomials, of the $s$-lecture hall simplices, and that these polynomials are all real-rooted. We then connect the two theories by showing that the local $h$-polynomials of common subdivisions in algebra and topology are realized as local $h^\ast$-polynomials of $s$-lecture hall simplices

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  • In this book chapter we study the Riemannian Geometry of the density registration problem: Given two densities (not necessarily probability densities) defined on a smooth finite dimensional manifold find a diffeomorphism which transforms one to the other. This problem is motivated by the medical imaging application of tracking organ motion due to respiration in Thoracic CT imaging where the fundamental physical property of conservation of mass naturally leads to modeling CT attenuation as a density. We will study the intimate link between the Riemannian metrics on the space of diffeomorphisms and those on the space of densities. We finally develop novel computationally efficient algorithms and demonstrate there applicability for registering RCCT thoracic imaging.

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  • The aim of this note is to explain in which sense an axiomatic Sobolev space over a general metric measure space (\`a la Gol'dshtein-Troyanov) induces - under suitable locality assumptions - a first-order differential structure.

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  • Let $A$ be an integer matrix, and assume that its semigroup ring $\mathbb{C}[\mathbb{N}A]$ is normal. Fix a face $F$ of the cone of $A$. We show that the projection and restriction of an $A$-hypergeometric system to the coordinate subspace corresponding to $F$ are isomorphic; moreover, they are essentially $F$-hypergeometric. We also show that, if $A$ is in addition homogeneous, the holonomic dual of an $A$-hypergeometric system is itself $A$-hypergeometric. This extends a result of Uli Walther, proving a conjecture of Nobuki Takayama in the normal homogeneous case.

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  • In the paper, assuming that the motion of rarefied gases in a bounded domain is governed by the angular cutoff Boltzmann equation with diffuse reflection boundary, we study the effects of both soft intermolecular interaction and non-isothermal wall temperature upon the long-time dynamics of solutions to the corresponding initial boundary value problem. Specifically, we are devoted to proving the existence and dynamical stability of stationary solutions whenever the boundary temperature has suitably small variations around a positive constant. For the proof of existence, we introduce a new mild formulation of solutions to the steady boundary-value problem along the speeded backward bicharacteristic, and develop the uniform estimates on approximate solutions in both $L^2$ and $L^\infty$. Such mild formulation proves to be useful for treating the steady problem with soft potentials even over unbounded domains. In showing the dynamical stability, a new point is that we can obtain the sub-e

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  • Massive MIMO is a compelling wireless access concept that relies on the use of an excess number of base-station antennas, relative to the number of active terminals. This technology is a main component of 5G New Radio (NR) and addresses all important requirements of future wireless standards: a great capacity increase, the support of many simultaneous users, and improvement in energy efficiency. Massive MIMO requires the simultaneous processing of signals from many antenna chains, and computational operations on large matrices. The complexity of the digital processing has been viewed as a fundamental obstacle to the feasibility of Massive MIMO in the past. Recent advances on system-algorithm-hardware co-design have led to extremely energy-efficient implementations. These exploit opportunities in deeply-scaled silicon technologies and perform partly distributed processing to cope with the bottlenecks encountered in the interconnection of many signals. For example, prototype ASIC impleme

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  • We study the problem of uncertainty quantification for the numerical solution of elliptic partial differential equation boundary value problems posed on domains with stochastically varying boundaries. We also use the uncertainty quantification results to tackle the efficient solution of such problems. We introduce simple transformations that map a family of domains with stochastic boundaries to a fixed reference domain. We exploit the transformations to carry out a prior and a posteriori error analyses and to derive an efficient Monte Carlo sampling procedure.

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  • We prove that a WLD subspace of the space $\ell_\infty^c(\Gamma)$ consisting of all bounded, countably supported functions on a set $\Gamma$ embeds isomorphically into $\ell_\infty$ if and only if it does not contain isometric copies of $c_0(\omega_1)$. Moreover, a subspace of $\ell_\infty^c(\omega_1)$ is constructed that has an unconditional basis, does not embed into $\ell_\infty$, and whose every weakly compact subset is separable (in particular, it cannot contain any isomorphic copies of $c_0(\omega_1)$).

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  • The purpose of this paper is to characterize all embeddings for versions of Besov and Triebel-Lizorkin spaces where the underlying Lebesgue space metric is replaced by a Lorentz space metric. We include two appendices, one on the relation between classes of endpoint Mikhlin-H\"ormander type Fourier multipliers, and one on the constant in the triangle inequality for the spaces $L^{p,r} $ when $p<1$.

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  • In this paper, we consider nonlinear PDEs in a port-Hamiltonian setting based on an underlying jet-bundle structure. We restrict ourselves to systems with 1-dimensional spatial domain and 2nd-order Hamiltonian including certain dissipation models that can be incorporated in the port- Hamiltonian framework by means of appropriate differential operators. For this system class, energy-based control by means of Casimir functionals as well as energy balancing is analysed and demonstrated using a nonlinear Euler-Bernoulli beam.

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  • Using an efficient algorithmic implementation of Caratheodory's theorem, we propose three enhanced versions of the Projection and Rescaling algorithm's basic procedures each of which improves upon the order of complexity of its analogue in [Mathematical Programming Series A, 166 (2017), pp. 87-111].

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  • We show that there are order of magnitude $H^2 (\log H)^2$ monic quartic polynomials with integer coefficients having box height at most $H$ whose Galois group is $D_4$. Further, we prove that the corresponding number of $V_4$ and $C_4$ quartics is $O(H^2 \log H)$. Finally, we show that the count for $A_4$ quartics is $O(H^{2.95})$. Our work establishes that irreducible non-$S_4$ quartics are less numerous than reducible quartics.

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  • We prove that quasi-plurisubharmonic envelopes with prescribed analytic singularities in suitable big cohomology classes on compact K\"ahler manifolds have the optimal $C^{1,1}$ regularity on a Zariski open set. This also proves regularity of certain pluricomplex Green's functions on K\"ahler manifolds. We then go on to prove related regularity results for these envelopes when the manifold is assumed to have boundary.

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  • For any open Riemann surface $X$ admitting Green functions, Suita asked about the precise relations between the Bergman kernel and the logarithmic capacity. It was conjectured that the Gaussian curvature of the Suita metric is bounded from above by $-4$, and moreover the curvature is equal to $-4$ at some point if and only if $X$ is conformally equivalent to the unit disc less a (possible) closed polar subset. After the contributions made by B{\l}ocki, Guan & Zhou and Berndtsson & Lempert, we provide a new proof of the equality part in Suita's conjecture by using the plurisubharmonic variation properties of Bergman kernels.

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  • In [15] the fundamental relationship between stable quotient invariants and the B-model for local P2 in all genera was studied under some specialization of equivariant variables. We generalize the result of [15] to full equivariant theory without the specialization. We also state the generalization to full equivariant formal quintic theory of the result in [16].

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  • First-passage percolation is a random growth model which has a metric structure. An infinite geodesic is an infinite sequence whose all sub-sequences are shortest paths. One of the important quantity is the number of infinite geodesics originating from the origin. When $d=2$ and an edge distribution is continuous, it is proved to be almost surely constant [D. Ahlberg, C. Hoffman. Random coalescing geodesics in first-passage percolation]. In this paper, we will prove the same result for higher dimensions and general distributions.

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  • We develop exact piecewise polynomial sequences on Alfeld splits in any spatial dimension and any polynomial degree. An Alfeld split of a simplex is obtained by connecting the vertices of an $n$-simplex with its barycenter. We show that, on these triangulations, the kernel of the exterior derivative has enhanced smoothness. Byproducts of this theory include characterizations of discrete divergence-free subspaces for the Stokes problem, commutative projections, and simple formulas for the dimensions of smooth polynomial spaces.

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  • We show that arbitrary one-dimensional hypergeometric differential systems underlie objects of the category of irregular mixed Hodge modules, which was recently introduced by Sabbah, and compute the irregular Hodge filtration for some of such systems. We also provide a comparison theorem between two different types of Fourier-Laplace transformation for algebraic integrable twistor $\mathcal{D}$-modules.

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  • Solving the Plateau problem means to find the surface with minimal area among all surfaces with a given boundary. Part of the problem actually consists of giving a suitable definition to the notions of 'surface', 'area' and 'boundary'. In our setting the considered objects are sets whose Hausdorff area is locally finite. The sliding boundary condition is given in term of a one parameter family of compact deformations which allows the boundary of the surface to moove along a closed set. The area functional is related to capillarity and free-boundary problems, and is a slight modification of the Hausdorff area. We focused on minimal boundary cones; that is to say tangent cones on boundary points of sliding minimal surfaces. In particular we studied cones contained in an half-space and whose boundary can slide along the bounding hyperplane. After giving a classification of one-dimensional minimal cones in the half-plane we provided four new two-dimensional minimal cones in the three-dimen

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  • Fix an elliptic curve $E$ over $\mathbb{Q}$. An extremal prime for $E$ is a prime $p$ of good reduction such that the number of rational points on $E$ modulo $p$ is maximal or minimal in relation to the Hasse bound. We give the first non-trivial upper bounds, both unconditional and on GRH, for the number of such primes when $E$ is a curve without complex multiplication. In order to obtain this bound, we estimate the joint distribution of the fractional part of $\sqrt{p}$ and primes in conjugacy classes of certain Galois groups. Adapting and refining a result by Rouse and Thorner arXiv:1305.5283, a sharper bound for extremal primes is obtained if we assume that all the symmetric power $L$-functions associated to $E$ are automorphic and satisfy GRH.

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  • We study the biased $(1:b)$ Maker--Breaker positional games, played on the edge set of the complete graph on $n$ vertices, $K_n$. Given Breaker's bias $b$, possibly depending on $n$, we determine the bounds for the minimal number of moves, depending on $b$, in which Maker can win in each of the two standard graph games, the Perfect Matching game and the Hamilton Cycle game.

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  • One possibility to break down the capacity limit in optical transmission systems are higher spectral efficiencies, enabled by Faster-than-Nyquist signaling. Here we present the utilization of non-orthogonal time division multiplexing of sinc pulse sequences for this purpose. The mathematical expression, with a representation of the Nyquist sinc sequence by a cosine Fourier series and simulation results show that non-orthogonal time-division multiplexing increases the transmittable symbol rate by up to 25%.

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  • This paper construct a family of explicit self-similar blowup axisymmetric solutions for the 3D incompressible Euler equations in R^3. Those singular solutions admit infinite energy.

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  • The aim of this study to investigate the existence of solutions for the following nonlocal integral boundary value problem of Caputo type fractional differential inclusions. To achieve our goals, we take advantage of fixed point theorems for multivalued mappings satisfying a new class of contractive conditions in the setting of complete metric spaces. We derive new fixed point results which extend and improve many results in the literature by means of this new class of contractions. We also supply some examples to support the new theory.

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  • We develop a rigidity theory for bar-joint frameworks in Euclidean $d$-space in which specified classes of edges are allowed to change length in a coordinated fashion, subject to a linear constraint for each class. This is a tensegrity-like setup that is amenable to combinatorial "Maxwell-Laman-type" analysis. We describe the generic rigidity of coordinated frameworks in terms of the generic $d$-dimensional rigidity properties of the bar-joint framework on the same underlying graph. We also give Henneberg and Laman-type characterizations for generic coordinated rigidity in the plane, for frameworks with one and two coordination classes.

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  • The aim of this article has been to put together various aspects of the personality of the Italian mathematician Francesco Severi: the mathematical, the political, the institutional, academic and philosophical one. We tried in particular to elucidate if the events related the First World War represented a turning point in the life and scientific production of Severi.

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  • We show that layer potential groupoids for conical domains constructed in an earlier paper (Carvalho-Qiao, Central European J. Math., 2013) are Fredholm groupoids, which enables us to deal with many analysis problems on singular spaces in a unified treatment. As an application, we obtain Fredholm criteria for operators on layer potential groupoids.

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  • Graded Artinian rings can be regarded as algebraic analogues of cohomology rings (in even degrees) of compact topological manifolds. In this analogy, a free extension of rings corresponds to a fiber bundle of manifolds. For rings, as with manifolds, it is a natural question to ask: to what extent do properties of the base and the fiber carry over to the extension ring? For example if the base and fiber both satisfy a strong Lefschetz property, can we conclude the same for the extension? We address these questions using certain relative coinvariant rings as a prototypical model. We show in particular that if the subgroup $W$ of the general linear group $Gl (V)$, $V$ a vector space, is a non-modular finite reflection group and $K\subset W$ is a non parabolic reflection subgroup, then the relative coinvariant ring $R^K_W$ does not satisfy the strong Lefschetz property. We give many examples, including those of relative coinvariant rings with non-unimodal Hilbert functions, and pose open q

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  • The analysis of dynamical systems has been a topic of great interest for researches mathematical sciences for a long times. The implementation of several devices and tools have been useful in the finding of solutions as well to describe common behaviors of parametric families of these systems. In this paper we study deeply a particular parametric family of differential equations, the so-called \emph{Linear Polyanin-Zaitsev Vector Field}, which has been introduced in a general case, in the previous paper by the same authors, as a correction of a family presented in the classical book of Polyanin and Zaitsev. Linear Polyanin-Zaitsev Vector Field is transformed into a Li\'enard equation and in particular we obtain the Van Der Pol equation. We present some algebraic and qualitative results to illustrate some interactions between algebra and the qualitative theory of differential equations in this parametric family.

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  • We investigate the role of Segal's Gamma-spaces in the context of classical and quantum information, based on categories of finite probabilities with stochastic maps and density matrices with quantum channels. The information loss functional extends to the setting of probabilistic Gamma-spaces considered here. The Segal construction of connective spectra from Gamma-spaces can be used in this setting to obtain spectra associated to certain categories of gapped systems.

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  • In this paper, we construct corrections to the raising and lowering (i.e. ladder) operators for a quantum harmonic oscillator subjected to a polynomial type perturbation of any degree and to any order in perturbation theory. We apply our formalism to a couple of examples, namely q and p 4 perturbations, and obtain the explicit form of those operators. We also compute the expectation values of position and momentum for the above perturbations. This construction is essential for defining coherent and squeezed states for the perturbed oscillator. Furthermore, this is the first time that corrections to ladder operators for a harmonic oscillator with a generic perturbation and to an arbitrary order of perturbation theory have been constructed.

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  • For a finite group $G$, let $d(G)$ denote the minimal number of elements required to generate $G$. In this paper, given a finite almost simple group $G$ and any maximal subgroup $H$ of $G$, we determine a precise upper bound for $d(H)$. In particular, we show that $d(H)\leq 5$, and that $d(H)\geq 4$ if and only if $H$ occurs in a known list. This improves a result of Burness, Liebeck and Shalev. The method involves the theory of crowns in finite groups.

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  • We present a data-driven framework called generative adversarial privacy (GAP). Inspired by recent advancements in generative adversarial networks (GANs), GAP allows the data holder to learn the privatization mechanism directly from the data. Under GAP, finding the optimal privacy mechanism is formulated as a constrained minimax game between a privatizer and an adversary. We show that for appropriately chosen adversarial loss functions, GAP provides privacy guarantees against strong information-theoretic adversaries. We also evaluate the performance of GAP on multi-dimensional Gaussian mixture models and the GENKI face database.

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  • The Bouncy Particle Sampler (BPS) is a Monte Carlo Markov Chain algorithm to sample from a target density known up to a multiplicative constant. This method is based on a kinetic piecewise deterministic Markov process for which the target measure is invariant. This paper deals with theoretical properties of BPS. First, we establish geometric ergodicity of the associated semi-group under weaker conditions than in [10] both on the target distribution and the velocity probability distribution. This result is based on a new coupling of the process which gives a quantitative minorization condition and yields more insights on the convergence. In addition, we study on a toy model the dependency of the convergence rates on the dimension of the state space. Finally, we apply our results to the analysis of simulated annealing algorithms based on BPS.

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  • In this article, we will analyze the global existence of solutions for the fractional diffusion reaction equation by applying Splitting-type methods, to functions that have two main characteristics, these are direct sum of functions of periodic type and functions that tend to zero at infinity. Global existence results are obtained for each particular characteristic, for then finally combining both results.

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  • We study the global existence of solutions to semilinear wave equations with power-type nonlinearity and general lower order terms on $n$ dimensional nontrapping asymptotically Euclidean manifolds, when $n=3, 4$. In addition, we prove almost global existence with sharp lower bound of the lifespan for the four dimensional critical problem.

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  • By $I$-method, the interaction Morawetz estimate, long time Strichartz estimate and local smoothing effect of Schr\"odinger operator, we show global well-posedness and scattering for the defocusing Hartree equation $$\left\{ \begin{array}{ll} iu_t + \Delta u &=F(u), \quad (t,x) \in \mathbb{R} \times \mathbb{R}^4 u(0) \\ &=u_0(x)\in H^s(\mathbb{R}^4), \end{array} \right. $$ where $F(u)= (V* |u|^2) u$, and $V(x)=|x|^{-\gamma}$, $3< \gamma<4$, with radial data in $H^{s}(\mathbb{R}^4)$ for $s>s_c:=\gamma/2-1$. It is a sharp global result except of the critical case $s=s_c$, which is a very difficult open problem.

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  • Quadratic programming (QP) is a well-studied fundamental NP-hard optimization problem which optimizes a quadratic objective over a set of linear constraints. In this paper, we reformulate QPs as a mixed-integer linear problem (MILP). This is done via the reformulation of QP as a linear complementary problem, and the use of binary variables and big-M constraints, to model the complementary constraints. To obtain such reformulation, we show how to impose bounds on the dual variables without eliminating all the (globally) optimal primal solutions; using some fundamental results on the solution of perturbed linear systems. Reformulating non-convex QPs as MILPs provides an advantageous way to obtain global solutions as it allows the use of current state-of-the-art MILP solvers. To illustrate this, we compare the performance of our solution approach, labeled quadprogIP, with the current benchmark global QP solver quadprogBB, as well as with BARON, one of the leading non-linear programming (N

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  • We give a construction and equations for good recursive towers over any finite field $\mathbf{F}_q$ with $q \ne 2$ and $3$.

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  • We study Green functions for stationary Stokes systems satisfying the conormal derivative boundary condition. We establish existence, uniqueness, and various estimates for the Green function under the assumption that weak solutions of the Stokes system are continuous in the interior of the domain. Also, we establish the global pointwise bound for the Green function under the additional assumption that weak solutions of the conormal derivative problem for the Stokes system are locally bounded up to the boundary. We provide some examples satisfying such continuity and boundedness properties.

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  • Statistical sufficiency formalizes the notion of data reduction. In the decision theoretic interpretation, once a model is chosen all inferences should be based on a sufficient statistic. However, suppose we start with a set of procedures rather than a specific model. Is it possible to reduce the data and yet still be able to compute all of the procedures? In other words, what functions of the data contain all of the information sufficient for computing these procedures? This article presents some progress towards a theory of "computational sufficiency" and shows that strong reductions can be made for large classes of penalized $M$-estimators by exploiting hidden symmetries in the underlying optimization problems. These reductions can (1) reveal hidden connections between seemingly disparate methods, (2) enable efficient computation, (3) give a different perspective on understanding procedures in a model-free setting. As a main example, the theory provides a surprising answer to the fo

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  • In this article, we consider the Parabolic Anderson Model with constant initial condition, driven by a space-time homogeneous Gaussian noise, with general covariance function in time and spatial spectral measure satisfying Dalang's condition. First, we prove that the solution (in the Skorohod sense) exists and is continuous in $L^p(\Omega)$. Then, we show that the solution has a modification whose sample paths are H\"older continuous in space and time, with optimal exponents, and under the minimal condition on the spatial spectral measure of the noise (which is the same as the condition encountered in the case of the white noise in time). This improves similar results which were obtained in Hu, Huang, Nualart and Tindel (2015), and Song (2017) under more restrictive conditions, and with sub-optimal exponents for H\"older continuity.

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  • For analytic quasi-periodic Schr\"odinger cocycles, You and Zhang [9] proved that the Lyapunov exponent is H\"older continuous for weak Liouville frequency. In this paper, we prove that the H\"older continuity also holds if the analytic potential is weakened to Gevrey potential.

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  • In this paper, we first give a direct proof for two recurrence relations of the heat kernels for hyperbolic spaces in \cite{DM}. Then, by similar computation, we give two similar recurrence relations of the heat kernels for spheres. Finally, as an application, we compute the diagonal of heat kernels for odd dimensional hyperbolic spaces and the heat trace asymptotic expansions for odd dimensional spheres.

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  • We solve the superhedging problem for European options in a market with finite liquidity where trading has transient impact on prices, and possibly a permanent one in addition. Impact is multiplicative to ensure positive asset prices. Hedges and option prices depend on the physical and cash delivery specifications of the option settlement. For non-covered options, where impact at the inception and maturity dates matters, we characterize the superhedging price as a viscosity solution of a degenerate semilinear pde that can have gradient constraints. The non-linearity of the pde is governed by the transient nature of impact through a resilience function. For covered options, the pricing pde involves gamma constraints but is not affected by transience of impact. We use stochastic target techniques and geometric dynamic programming in reduced coordinates.

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  • In this paper intuitionistic topological system and its properties have been introduced. Categorical interrelationships among Heyting algebra, G\"odel algebra, Esakia space and proposed intuitionistic topological systems have also been studied in details.

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  • Slice-regular functions of a quaternionic variable have been studied extensively in the last 12 years, resulting, in many ways, quite close to classical holomorphic functions of a complex variable; indeed, there is a correspondence between slice-regular functions and a certain family of holomorphic maps from the complex plane to $\mathbb{C}^4$, as noted by Ghiloni and Perotti. However, such a construction does not seem to offer any insight on the behaviour of slice-regular functions, due to the lack of a connection between the values of the holomorphic map and the values of the associated sliceregular function. The aim of this work is to show that there is indeed a (complex) geometric way to relate the values of this two functions, thus relating more deeply the world of holomorphic functions with that of slice-regular functions.

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  • The rational homology group of the order complex of non-even partitions of a finite set is calculated. A twisted version of the Goresky-MacPherson approach to similar homology calculations is proposed.

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  • We survey some aspects of the perfect matching problem in hypergraphs, with particular emphasis on structural characterisation of the existence problem in dense hypergraphs and the existence of designs.

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  • Free binary systems are shown to not admit idempotent means. This refutes a conjecture of the author. It is also shown that the extension of Hindman's theorem to nonassociative binary systems formulated and conjectured by the author is false.

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  • We consider the assembly map for principal bundles with fiber a countable discrete group. We obtain an index-theoretic interpretation of this homomorphism by providing a tensor-product presentation for the module of sections associated to the Mi\v{s}\v{c}enko line bundle. In addition, we give a proof of Atiyah's $L^2$-index theorem in the general context of principal bundles over compact Hausdorff spaces. We thereby also reestablish that the surjectivity of the Baum-Connes assembly map implies the Kadison-Kaplansky idempotent conjecture in the torsion-free case. Our approach does not rely on geometric $K$-homology but rather on an explicit construction of Alexander-Spanier cohomology classes coming from a Chern character for tracial function algebras.

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  • Let $A$ be a free hyperplane arrangement. In 1989, Ziegler showed that the restriction $A''$ of $A$ to any hyperplane endowed with the natural multiplicity is then a free multiarrangement. We initiate a study of the stronger freeness property of inductive freeness for these canonical free multiarrangements and investigate them for the underlying class of reflection arrangements. More precisely, let $A = A(W)$ be the reflection arrangement of a complex reflection group $W$. By work of Terao, each such reflection arrangement is free. Thus so is Ziegler's canonical multiplicity on the restriction $A''$ of $A$ to a hyperplane. We show that the latter is inductively free as a multiarrangement if and only if $A''$ itself is inductively free.

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  • In 2006 Carbery raised a question about an improvement on the na\"ive norm inequality $\|f+g\|_p^p \leq 2^{p-1}(\|f\|_p^p + \|g\|_p^p)$ for two functions in $L^p$ of any measure space. When $f=g$ this is an equality, but when the supports of $f$ and $g$ are disjoint the factor $2^{p-1}$ is not needed. Carbery's question concerns a proposed interpolation between the two situations for $p>2$. The interpolation parameter measuring the overlap is $\|fg\|_{p/2}$. We prove an inequality of this type that is stronger than the one Carbery proposed. Moreover, our stronger inequalities are valid for all $p$.

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  • We prove the conjecture of Berest-Eshmatov-Eshmatov by showing that the group of automorphisms of a product of Calogero-Moser spaces $C_{n_i}$, where the $n_i$ are pairwise distinct, acts $m$-transitively for each $m$.

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