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  • We study the pointwise (in the space and time variables) behavior of the Fokker-Planck Equation with flat confinement. The solution has very clear description in the $xt-$plane, including large time behavior, initial layer and asymptotic behavior. Moreover, the structure of the solution highly depends on the potential function.

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  • We develop a theory of extrapolation for weights that satisfy a generalized reverse H\"older inequality in the scale of Orlicz spaces. This extends previous results by Auscher and Martell [2] on limited range extrapolation. As an application, we show that a number of weighted norm inequalities for linear and bilinear operators can be proved using our extrapolation theorem.

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  • Hash Tables 101 – a tale of an underrated data structure

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  • We consider the conformal flow model derived by Bizo\'n, Craps, Evnin, Hunik, Luyten, and Maliborski [Commun. Math. Phys. 353 (2017) 1179-1199] as a normal form for the conformally invariant cubic wave equation on $\mathbb{S}^3$. We prove that the energy attains a global constrained maximum at a family of particular stationary solutions which we call the ground state family. Using this fact and spectral properties of the linearized flow (which are interesting on their own due to a supersymmetric structure) we prove nonlinear orbital stability of the ground state family. The main difficulty in the proof is due to the degeneracy of the ground state family as a constrained maximizer of the energy.

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  • Flag measures are descriptors of convex bodies $K$ in $d$-dimensional Euclidean space generalizing the classical area measures. They have been used to provide general integral formulas for mixed volumes (see Hug, Rataj and Weil (2017)). Here, we consider an image measure $\gamma_j(K,\cdot)$ of flag measures, defined on the Grassmannian $G(d,j)$ of affine $j$-spaces, $1\le j\le d-1$, and show that it determines centrally symmetric bodies $K$ of dimension $\geq j+1$ uniquely. We then explain that Grassmann measures appear in the representation of smooth, translation invariant, continuous and even valuations due to Alesker (2003). Using this connection, we prove a uniqueness result for projection averages of area measures and we finally discuss a Grassmann version of the natural touching measure of convex bodies.

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  • Near a birth-death critical point in a one-parameter family of gradient flows, there are precisely two Morse critical points of index difference one on the birth side. This paper gives a self-contained proof of the folklore theorem that these two critical points are joined by a unique gradient trajectory up to time shift. The proof is based on the Whitney normal form, a Conley index construction, and an adiabatic limit analysis for an associated fast-slow differential equation.

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  • Google is stopping one of the most controversial advertising formats: ads inside Gmail that scan users' email contents. The decision didn't come from Google's ad team, but from its cloud unit, which is angling to sign up more corporate customers. Alphabet Inc.'s Google Cloud sells a package of office software, called G Suite, that competes with market leader Microsoft Corp. Paying Gmail users never received the email-scanning ads like the free version of the program, but some business customers were confused by the distinction and its privacy implications, said Diane Greene, Google's senior vice president of cloud. "What we're going to do is make it unambiguous," she said. Good move, and in the current climate, Google really couldn't continue this practice - automated algorithms or no.

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  • We prove the global in time existence of solutions of the semilinear Klein-Gordon equation in the de Sitter spacetime.

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  • In this paper, we consider a class of nonlinear fourth-order Schr\"odinger equation, namely \[ \left\{ \begin{array}{rcl} i\partial_t u +\Delta^2 u &=&-|u|^{\nu-1} u, \quad 1+ \frac{8}{d}<\nu <1+\frac{8}{d-4},\\ u(0)&=&u_0 \in H^\gamma(\mathbb{R}^d), \quad 5 \leq d \leq 11. \end{array} \right. \] Using the $I$-method combined with the interaction Morawetz inequality, we establish the global well-posedness and scattering in $H^\gamma(\mathbb{R}^d)$ with $\gamma(d,\nu)<\gamma<2$ for some value $\gamma(d,\nu)>0$.

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  • This paper is a continuation of \ct{cmf16} where an efficient algorithm for computing the maximal eigenpair was introduced first for tridiagonal matrices and then extended to the irreducible matrices with nonnegative off-diagonal elements. This paper introduces two global algorithms for computing the maximal eigenpair in a rather general setup, including even a class of real (with some negative off-diagonal elements) or complex matrices.

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  • The main goal of this paper is to generalize Jacobi and Gauss-Seidel methods for solving non-square linear system. Towards this goal, we present iterative procedures to obtain an approximate solution for non-square linear system. We derive sufficient conditions for the convergence of such iterative methods. Procedure is given to show that how an exact solution can be obtained from these methods. Lastly, an example is considered to compare these methods with other available method(s) for the same.

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  • This paper is concerned with transition paths within the framework of the overdamped Langevin dynamics model of chemical reactions. We aim to give an efficient description of typical transition paths in the small temperature regime. We adopt a variational point of view and seek the best Gaussian approximation, with respect to Kullback-Leibler divergence, of the non-Gaussian distribution of the diffusion process. We interpret the mean of this Gaussian approximation as the "most likely path" and the covariance operator as a means to capture the typical fluctuations around this most likely path. We give an explicit expression for the Kullback-Leibler divergence in terms of the mean and the covariance operator for a natural class of Gaussian approximations and show the existence of minimisers for the variational problem. Then the low temperature limit is studied via $\Gamma$-convergence of the associated variational problem. The limiting functional consists of two parts: The first part onl

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  • This paper concerns the approximation of probability measures on $\mathbf{R}^d$ with respect to the Kullback-Leibler divergence. Given an admissible target measure, we show the existence of the best approximation, with respect to this divergence, from certain sets of Gaussian measures and Gaussian mixtures. The asymptotic behavior of such best approximations is then studied in the small parameter limit where the measure concentrates; this asymptotic behaviour is characterized using $\Gamma$-convergence. The theory developed is then applied to understanding the frequentist consistency of Bayesian inverse problems. For a fixed realization of noise, we show the asymptotic normality of the posterior measure in the small noise limit. Taking into account the randomness of the noise, we prove a Bernstein-Von Mises type result for the posterior measure.

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  • A Fog radio access network is considered as a network architecture candidate to meet the soaring demand in terms of reliability, spectral efficiency, and latency in next generation wireless networks. This architecture combines the benefits associated with centralized cloud processing and wireless edge caching enabling primarily low-latency transmission under moderate fronthaul capacity requirements. The F-RAN we consider in this paper is composed of a centralized cloud server which is connected through fronthaul links to two edge nodes serving two mobile users through a Z-shaped partially connected wireless network. We define an information-theoretic metric, the delivery time per bit (DTB), that captures the worst-case per-bit delivery latency for conveying any requested content to the users. For the cases when cloud and wireless transmission occur either sequentially or in parallel, we establish coinciding lower and upper bounds on the DTB as a function of cache size, backhaul capacit

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  • Universal variable-to-fixed (V-F) length coding of $d$-dimensional exponential family of distributions is considered. We propose an achievable scheme consisting of a dictionary, used to parse the source output stream, making use of the previously-introduced notion of quantized types. The quantized type class of a sequence is based on partitioning the space of minimal sufficient statistics into cuboids. Our proposed dictionary consists of sequences in the boundaries of transition from low to high quantized type class size. We derive the asymptotics of the $\epsilon$-coding rate of our coding scheme for large enough dictionaries. In particular, we show that the third-order coding rate of our scheme is $H\frac{d}{2}\frac{\log\log M}{\log M}$, where $H$ is the entropy of the source and $M$ is the dictionary size. We further provide a converse, showing that this rate is optimal up to the third-order term.

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  • We prove that the class of Brauer graph algebras coincides with the class of indecomposable idempotent algebras of biserial weighted surface algebras. These algebras are associated to triangulated surfaces with arbitrarily oriented triangles. Hence biserial weighted surface algebras can be considered as desingularizations of Brauer graph algebras.

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  • An exact phase-retrievable frame $\{f_{i}\}_{i}^{N}$ for an $n$-dimensional Hilbert space is a phase-retrievable frame that fails to be phase-retrievable if any one element is removed from the frame. Such a frame could have different lengths. We shall prove that for the real Hilbert space case, exact phase-retrievable frame of length $N$ exists for every $2n-1\leq N\leq n(n+1)/2$. For arbitrary frames we introduce the concept of redundancy with respect to its phase-retrievability and the concept of frames with exact PR-redundancy. We investigate the phase-retrievability by studying its maximal phase-retrievable subspaces with respect to a given frame which is not necessarily phase-retrievable. These maximal PR-subspaces could have different dimensions. We are able to identify the one with the largest dimension, which can be considered as a generalization of the characterization for phase-retrievable frames. In the basis case, we prove that if $M$ is a $k$-dimensional PR-subspace, then

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  • We investigate the 1D Riemann-Liouville fractional derivative focusing on the connections with fractional Sobolev spaces, the space $BV$ of functions of bounded variation, whose derivatives are not functions but measures and the space $SBV$, say the space of bounded variation functions whose derivative has no Cantor part. We prove that $SBV$ is included in $W^{s,1} $ for every $s \in (0,1)$ while the result remains open for $BV$. We study examples and address open questions.

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  • We investigate the fluctuations and large deviations of the root of largest modulus in a model of random polynomial with independent complex Gaussian coefficients (Kac polynomials). The fluctuations were recently computed by R. Butez (arxiv 1704.02761) and involve a Fredholm determinant. The precise large deviations show a particular function defined by a series of mutiple integrals in the same vein.

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  • We derive in this preprint the exact up to multiplicative constant non-asymptotical estimates for the norms of some non-linear in general case operators, for example, the so-called maximal functional operators, in two probabilistic rearrangement invariant norm: exponential Orlicz and Grand Lebesgue Spaces. We will use also the theory of the so-called Grand Lebesgue Spaces (GLS) of measurable functions.

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  • We extend the study of \emph{melonic} quartic tensor models to models with arbitrary quartic interactions. This extension requires a new version of the loop vertex expansion using several species of intermediate fields and iterated Cauchy-Schwarz inequalities. Borel summability is proven, uniformly as the tensor size $N$ becomes large. Every cumulant is written as a sum of explicitly calculated terms plus a remainder, suppressed in $1/N$. Together with the existence of the large $N$ limit of the second cumulant, this proves that the corresponding sequence of probability measures is uniformly bounded and obeys the tensorial universality theorem.

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  • On a stratified Lie group $G$ equipped with hypoelliptic heat kernel measure, we study the behavior of the dilation semigroup on $L^p$ spaces of log-subharmonic functions. We consider a notion of strong hypercontractivity and a strong logarithmic Sobolev inequality, and show that these properties are equivalent for any group $G$. Moreover, if $G$ satisfies a classical logarithmic Sobolev inequality, then both properties hold. This extends similar results obtained by Graczyk, Kemp and Loeb in the Euclidean setting.

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  • Let $I$ be a monomial ideal $I$ in a polynomial ring $R = k[x_1,...,x_r]$. In this paper we give an upper bound on $\overline{\dstab} (I)$ in terms of $r$ and the maximal generating degree $d(I)$ of $I$ such that $\depth R/\overline{I^n}$ is constant for all $n\geqslant \overline{\dstab}(I)$. As an application, we classify the class of monomial ideals $I$ such that $\overline{I^n}$ is Cohen-Macaulay for some integer $n\gg 0$.

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  • Let $LG$ be the loop group of a compact, connected Lie group $G$. We show that the tangent bundle of any proper Hamiltonian $LG$-space $\mathcal{M}$ has a natural completion $\overline{T}\mathcal{M}$ to a strongly symplectic $LG$-equivariant vector bundle. This bundle admits an invariant compatible complex structure within a natural polarization class, defining an $LG$-equivariant spinor bundle $\mathsf{S}_{\overline{T}\mathcal{M}}$, which one may regard as the Spin$_c$-structure of $\mathcal{M}$. We describe two procedures for obtaining a finite-dimensional version of this spinor module. In one approach, we construct from $\mathsf{S}_{\overline{T}\mathcal{M}}$ a twisted Spin$_c$-structure for the quasi-Hamiltonian $G$-space associated to $\mathcal{M}$. In the second approach, we describe an `abelianization procedure', passing to a finite-dimensional $T\subset LG$-invariant submanifold of $\mathcal{M}$, and we show how to construct an equivariant Spin$_c$-structure on that submanifold.

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  • SpaceX successfully launches and recovers second Falcon 9 in 48 hours

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  • An anonymous reader quotes Space.com: A SpaceX Falcon 9 rocket carrying the 10 satellites for Iridium Communications is scheduled to liftoff from Vandenberg Air Force Base in California at 1:25 p.m. PDT (4:25 p.m. EDT/2025 GMT). The live webcast is expected to begin about 1 hour before the opening of the launch window, and you can watch it on SpaceX's website, or at Space.com. This is the second of eight planned Iridium launches with SpaceX. The launches will deliver a total of 75 satellites into space for the $3 billion Iridium NEXT global communications network. "Iridium NEXT will replace the company's existing global constellation in one of the largest technology upgrades ever completed in space," according to a statement from Iridium. "It represents the evolution of critical communications infrastructure that governments and organizations worldwide rely upon to drive business, enable connectivity, empower disaster relief efforts and more." After the mission the booster rocket will

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  • In this article, we first describe codimension two regular foliations with numerically trivial canonical class on complex projective manifolds whose canonical class is not numerically effective. Building on a recent algebraicity criterion for leaves of algebraic foliations, we then address regular foliations of small rank with numerically trivial canonical class on complex projective manifolds whose canonical class is pseudo-effective. Finally, we confirm the generalized Bondal conjecture formulated by Beauville in some special cases.

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  • Recent results of Kahle and Miller give a method of constructing primary decompositions of binomial ideals by first constructing "mesoprimary decompositions" determined by their underlying monoid congruences. Monoid congruences (and therefore, binomial ideals) can present many subtle behaviors that must be carefully accounted for in order to produce general results, and this makes the theory complicated. In this paper, we examine their results in the presence of a positive $A$-grading, where certain pathologies are avoided and the theory becomes more accessible. Our approach is algebraic: while key notions for mesoprimary decomposition are developed first from a combinatorial point of view, here we state definitions and results in algebraic terms, which are moreover significantly simplified due to our (slightly) restricted setting. In the case of toral components (which are well-behaved with respect to the $A$-grading), we are able to obtain further simplifications under additional ass

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  • We study the 1D Landau equation for a mixture of two species in the whole space, with initial condition of one species near a vacuum and the other near a Maxwellian equilibrium state. The regularization effect (in both the space and momentum variables) and large time behavior will be obtained. Moreover, the decay rate of the solution is optimal in both species.

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  • We study focus-focus singularities (also known as nodal singularities, or pinched tori) of Lagrangian fibrations on symplectic $4$-manifolds. We show that, in contrast to elliptic and hyperbolic singularities, there exist homeomorphic focus-focus singularities which are not diffeomorphic. Furthermore, we obtain an algebraic description of the moduli space of focus singularities up to smooth equivalence, and show that for double pinched tori this space is one-dimensional. Finally, we apply our construction to disprove Zung's conjecture which says that any non-degenerate singularity can be smoothly decomposed into an almost direct product of standard singularities.

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  • We present a summary of some results from our article [BZ1] and other recent results on the so-called LVMB manifolds. We emphasize some features by taking a different point of view. We present a simple variant of the Delzant construction, in which the group that is used to perform the symplectic reduction can be chosen of arbitrarily high dimension, and is always connected.

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  • Show HN: Badger, 8.5x cheaper, 3.5x faster to run than RocksDB

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  • Show HN: A Minimal Code Example of Using Vulkan for Computations on the GPU

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  • Shared thoughts after 6+ years in Pentesting

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  • A prevailing challenge in the biomedical and social sciences is to estimate a population mean from a sample obtained with unknown selection probabilities. Using a well-known ratio estimator, Aronow and Lee (2013) proposed a method for partial identification of the mean by allowing the unknown selection probabilities to vary arbitrarily between two fixed extreme values. In this paper, we show how to leverage auxiliary shape constraints on the population outcome distribution, such as symmetry or log-concavity, to obtain tighter bounds on the population mean. We use this method to estimate the performance of Aymara students---an ethnic minority in the north of Chile---in a national educational standardized test. We implement this method in the new statistical software package scbounds for R.

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  • In this paper, we study the semi-classical behavior of distorted plane waves, on manifolds that are Euclidean near infinity or hyperbolic near infinity, and of non-positive curvature. Assuming that there is a strip without resonances below the real axis, we show that distorted plane waves are bounded in $L^2_{loc}$ independently of $h$, that they admit a unique semiclassical measure, and we prove bounds on their $L^p_{loc}$ norms.

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  • In this paper, we study the form over the minimum spanning tree problem (MST) from which we will derive an intuitively generalized model and new methods with the upper bound of runtimes of logarithm. The new pattern we made has taken successful to better equilibrium the benefits of local and global when we employ the strategy of divide and conquer to optimize solutions on problem. Under new model, we let the course of clustering become more transparent with many details, so that the whole solution may be featured of much reasonable, flexibility, efficiency and approach to reveal or reflect the reality. There are some important methods and avenues as fruits derived from discussions or trial which can be broad usefulness in the fields of graphic analysis, data mining, k-means clustering problem and so forth.

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  • The problem of minimizing a sum of local convex objective functions over a networked system captures many important applications and has received much attention in the distributed optimization field. Most of existing work focuses on development of fast distributed algorithms under the presence of a central clock. The only known algorithms with convergence guarantees for this problem in asynchronous setup could achieve either sublinear rate under totally asynchronous setting or linear rate under partially asynchronous setting (with bounded delay). In this work, we built upon existing literature to develop and analyze an asynchronous Newton based approach for solving a penalized version of the problem. We show that this algorithm converges almost surely with global linear rate and local superlinear rate in expectation. Numerical studies confirm superior performance against other existing asynchronous methods.

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  • Texas Is Too Windy and Sunny for Old Energy Companies to Make Money

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  • We consider "pressing sequences", a certain kind of transformation of graphs with loops into empty graphs, motivated by an application in phylogenetics. In particular, we address the question of when a graph has precisely one such pressing sequence, thus answering an question from Cooper and Davis (2015). We characterize uniquely pressable graphs, count the number of them on a given number of vertices, and provide a polynomial time recognition algorithm. We conclude with a few open questions. Keywords: Pressing sequence, adjacency matrix, Cholesky factorization, binary matrix

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  • Kinetic equations play a major rule in modeling large systems of interacting particles. Recently the legacy of classical kinetic theory found novel applications in socio-economic and life sciences, where processes characterized by large groups of agents exhibit spontaneous emergence of social structures. Well-known examples are the formation of clusters in opinion dynamics, the appearance of inequalities in wealth distributions, flocking and milling behaviors in swarming models, synchronization phenomena in biological systems and lane formation in pedestrian traffic. The construction of kinetic models describing the above processes, however, has to face the difficulty of the lack of fundamental principles since physical forces are replaced by empirical social forces. These empirical forces are typically constructed with the aim to reproduce qualitatively the observed system behaviors, like the emergence of social structures, and are at best known in terms of statistical information of

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  • Block and G\"ottsche have defined a $q$-number refinement of counts of tropical curves in $\mathbb{R}^2$. Under the change of variables $q=e^{iu}$, we show that the result is a generating series of higher genus log Gromov-Witten invariants with insertion of a lambda class. This gives a geometric interpretation of the Block-G\"ottsche invariants and makes their deformation invariance manifest.

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  • Torus – A secure, shared workspace for secrets

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  • We use the non-proper Morse theory of Palais-Smale to investigate the topology of smooth closed subvarieties of complex semi-abelian varieties, and that of their infinite cyclic covers. As main applications, we obtain the finite generation (except in the middle degree) of the corresponding integral Alexander modules, as well as the signed Euler characteristic property and generic vanishing for rank-one local systems on such subvarieties. Furthermore, we give a more conceptual (topological) interpretation of the signed Euler characteristic property in terms of vanishing of Novikov homology. As a byproduct, we prove a generic vanishing result for the $L^2$-Betti numbers of very affine manifolds. Our methods also recast Jun Huh's extension of Varchenko's conjecture to very affine manifolds, and provide a generalization of this result in the context of smooth closed subvarieties of semi-abelian varieties.

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  • Motivated by the recent growing interest about the thermodynamic cost of Shortcuts to Adiabaticity (STA), we consider the cost of driving a classical system by the so-called Counterdiabatic Driving (CD). To do so, we proceed in three steps: first we review a general definition recently put forward in the literature for the thermodynamic cost of driving a Hamiltonian system; then we provide a new sensible definition of cost for cases where the average excess work vanishes; finally, we apply our general framework to the case of CD. Interestingly, for CD we find that our results are the exact classical counterparts of those reported in Funo et al. PRL, 118(10):100602, 2017. In particular, we show that the thermodynamic cost can be written in terms of a new metric tensor that arises as the natural equivalent of the more studied quantum metric tensor.

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  • The secret to a long and healthy life? Eat less

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  • The allowed sequences in a discrete unimodal system look like $\mathrm{P}=(\mathrm{R} \mathrm{L}^{q})^{n_1} \mathrm{S}_1(m_1,q-1) (\mathrm{R} \mathrm{L}^{q})^{n_2}\mathrm{S}_2(m_2,q-1) $ $\ldots$ $ (\mathrm{R} \mathrm{L}^{q})^{n_r} \mathrm{S}_r(m_r,q-1)\mathrm{C}$ where $\mathrm{S}_i(m_i, q-1)$ are sequences of $\mathrm{R}$s and $\mathrm{L}$s that contain at most $q-1$ consecutive $\mathrm{L}$s. The $\mathrm{S}_i(m_i,q-1), \ i=2, \ldots, r$ are determined by $\mathrm{S}_1(m_1,q-1)$. The first block $\mathrm{RL}^q$ and the sequence $\mathrm{S}_1$ following it are essential for a sequence to be allowed. In addition $\mathrm{RL}^q$ and $\mathrm{S}_1$ also govern the composition of sequences, since every non-primary sequence has the form $\mathrm{RL}^q \mathrm{S}_1(m_1,q-1)\mathrm{C} \ast \mathrm{x_1,\ldots x_{s-1}C}.$ Explicit forms of allowed sequences will be given. Also their cardinality will be calculated.

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  • Let $\varphi:\F_q\to\F_q$ be a rational map on a fixed finite field. We give explicit asymptotic formulas for the size of image sets $\varphi^n(\F_q)$ as a function of $n$. This is done by using properties of the Galois groups of iterated maps, whose connection to the question of the size of image sets is established via Chebotarev's Density Theorem. We then apply these results to provide explicit bounds on the proportion of periodic points in $\F_q$ in terms of $q$ for certain rational maps.

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  • The aim of this note is to describe the geometry of $\mathbb{C}^2$ equipped with a K\"ahler metric defined by Warren. It is shown that with that metric $\mathbb{C}^2$ is a flat manifold. Explicit formulae for geodesics and volume of geodesic ball are also computed. Finally, a family of similar flat metrics is constructed.

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  • This article studies the first exit times and loci of the mild solutions of a generic class of scalar nonlinear dissipative reaction-diffusion equations under $\epsilon$-small perturbations by infinite-dimensional multiplicative L\'evy noise with a regularly varying component. In contrast to the exponential asymptotic behavior of respective Gaussian perturbations the exit times grow asymptotically as a power function of the noise intensity $\epsilon$ as $\epsilon \rightarrow 0$ with the exponent being the (negative) tail index of the L\'evy measure. To this end we prove an exponential estimate of the stochastic convolution for jump L\'evy processes with bounded jumps and construct well-understood models of the exit times and loci on the same probability space to which the original quantities converge in a probabilistically optimal sense. The results cover the case of the Chafee-Infante equation and the linear heat equation perturbed by additive and multiplicative infinite-dimensional $

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  • Let $M$ and $N$ be two monomials of the same degree, and let $I$ be the smallest Borel ideal containing $M$ and $N$. We show that the toric ring of $I$ is Koszul by constructing a quadratic Gr\"obner basis for the associated toric ideal. Our proofs use the construction of graphs corresponding to fibers of the toric map. As a consequence, we conclude that the Rees algebra is also Koszul.

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  • We study the maximum number of edges in an $n$ vertex graph with Colin de Verdi\`{e}re parameter no more than $t$. We conjecture that for every integer $t$, if $G$ is a graph with at least $t$ vertices and Colin de Verdi\`{e}re parameter at most $t$, then $|E(G)| \leq t|V(G)|-\binom{t+1}{2}$. We observe a relation to the graph complement conjecture for the Colin de Verdi\`{e}re parameter and prove the conjectured edge upper bound for graphs $G$ such that either $\mu(G) \leq 7$, or $\mu(G) \geq |V(G)|-6$, or the complement of $G$ is chordal, or $G$ is chordal.

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  • We introduce a framework for studying the effects of self-interaction on the construction of point particle initial data in General Relativity. Within this framework we rigorously prove the vanishing mass claim made by Arnowitt, Deser and Misner regarding point sources. We identify a geometric structure and a scaling parameter that allow one to determine, by controlling the effects of self-interaction, when one does or does not obtain a non-zero mass.

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  • We describe an algorithm for computing the inner product between a holomorphic modular form and a unary theta function, in order to determine whether the form is orthogonal to unary theta functions without needing a basis of the entire space of modular forms and without needing to use linear algebra to decompose this space completely.

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  • The Monge-Kantorovich problem of finding a minimum cost coupling between an absolutely continuous measure $\mu$ on $\mathcal{X} \subset \mathbb{R}^d$ and a finitely supported measure $\nu$ on $\mathbb{R}^d$ is considered for the special case of the Euclidean cost function. This corresponds to the natural problem of closest distance allocation of some ressource that is continuously distributed in space to a finite number of processors with capacity constraints. A systematic discussion of the choice of Euclidean cost versus squared Euclidean cost is provided from the practitioner's point of view. We then show that the above problem has a unique solution that can be either described by a transport map $T \colon \mathcal{X} \to \mathbb{R}^d$ or by a partition of $\mathcal{X}$. We provide an algorithm for computing this optimal transport partition, adapting the approach by Aurenhammer, Hoffmann and Aronov (1998) and M\'erigot~(2011) to the Euclidean cost. We give two types of numerical appl

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  • An anonymous reader quotes Engadget: A University of Michigan public-private partnership called Mcity is testing V2V, or vehicle to vehicle communication, and has found that it makes their autonomous prototypes even safer. V2V works by wirelessly sharing data such as location, speed and direction. Using DSRC, or Dedicated Short Range Communication, V2V can send up to 10 messages per second. This communication allows cars to see beyond what is immediately in front of them -- sensing a red light around a blind curve, or automatically braking for a car that runs a stop sign... The catch of V2V? It has to be installed in the majority of cars and infrastructure (such as traffic lights) to function adequately.

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  • For any real $t$, the unitary divisor function $\sigma_t^*$ is the multiplicative arithmetic function defined by $\sigma_t^*(p^{\alpha})=1+p^{\alpha t}$ for all primes $p$ and positive integers $\alpha$. Let $\overline{\sigma_t^*(\mathbb N)}$ denote the topological closure of the range $\sigma_t^*$. We calculate an explicit constant $\eta^*\approx 1.9742550$ and show that $\overline{\sigma_{-r}^*(\mathbb N)}$ is connected if and only if $r\in(0,\eta^*]$. We end with an open problem.

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  • This study is motivated by the recent development in the fractional calculus and its applications. During last few years, several different techniques are proposed to localize the nonlocal fractional diffusion operator. They are based on transformation of the original problem to a local elliptic or pseudoparabolic problem, or to an integral representation of the solution, thus increasing the dimension of the computational domain. More recently, an alternative approach aimed at reducing the computational complexity was developed. The linear algebraic system $\cal A^\alpha \bf u=\bf f$, $0< \alpha <1$ is considered, where $\cal A$ is a properly normalized (scalded) symmetric and positive definite matrix obtained from finite element or finite difference approximation of second order elliptic problems in $\Omega\subset\mathbb{R}^d$, $d=1,2,3$. The method is based on best uniform rational approximations (BURA) of the function $t^{\beta-\alpha}$ for $0 < t \le 1$ and natural $\beta$

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  • In this article we study the pointwise decay properties of solutions to the Maxwell system on a class of nonstationary asymptotically flat backgrounds in three space dimensions. Under the assumption that uniform energy bounds and a weak form of local energy decay hold forward in time, we establish peeling estimates for all the components of the Maxwell tensor.

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  • Picking Winners: A Framework for Venture Capital Investment

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  • We investigate dynamical properties of the set of permutations of $\mathbb{Z}^d$ with restricted movement, i.e., permutations $\pi $ of $\mathbb{Z}^d$ such that $\pi (\mathbf{n})-\mathbf{n}$ lies, for every $\mathbf{n}\in \mathbb{Z}^d$, in a prescribed finite set $A\subset \mathbb{Z}^d$. For $d=1$, such permutations occur, for example, in restricted orbit equivalence, or in the calculation of determinants of certain bi-infinite multi-diagonal matrices. For $d\ge2$ these sets of permutations provide natural classes of examples of multidimensional shifts of finite type.

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  • Least angle regression (LARS) by Efron et al. (2004) is a novel method for constructing the piece-wise linear path of Lasso solutions. For several years, it remained also as the de facto method for computing the Lasso solution before more sophisticated optimization algorithms preceded it. LARS method has recently again increased its popularity due to its ability to find the values of the penalty parameters, called knots, at which a new parameter enters the active set of non-zero coefficients. Significance test for the Lasso by Lockhart et al. (2014), for example, requires solving the knots via the LARS algorithm. Elastic net (EN), on the other hand, is a highly popular extension of Lasso that uses a linear combination of Lasso and ridge regression penalties. In this paper, we propose a new novel algorithm, called pathwise (PW-)LARS-EN, that is able to compute the EN knots over a grid of EN tuning parameter {\alpha} values. The developed PW-LARS-EN algorithm decreases the EN tuning para

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  • Consider the stochastic differential equation $\mathrm dX_t = -A X_t \,\mathrm dt + f(t, X_t) \,\mathrm dt + \mathrm dB_t$ in a (possibly infinite-dimensional) separable Hilbert space, where $B$ is a cylindrical Brownian motion and $f$ is a just measurable, bounded function. If the components of $f$ decay to 0 in a faster than exponential way we establish path-by-path uniqueness for mild solutions of this stochastic differential equation. This extends A. M. Davie's result from $\mathbb R^d$ to Hilbert space-valued stochastic differential equations.

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  • Sturm-Liouville oscillation theory is studied for Jacobi operators with block entries given by covariant operators on an infinite dimensional Hilbert space. It is shown that the integrated density of states of the Jacobi operator is approximated by the winding of the Pruefer phase w.r.t. the trace per unit volume. This rotation number can be interpreted as a spectral flow in a von Neumann algebra with finite trace.

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  • The quasi-potential is a key function in the Large Deviation Theory. It characterizes the difficulty of the escape from the neighborhood of an attractor of a stochastic non-gradient dynamical system due to the influence of small white noise. It also gives an estimate of the invariant probability distribution in the neighborhood of the attractor up to the exponential order. We present a new family of methods for computing the quasi-potential on a regular mesh named the Ordered Line Integral Methods (OLIMs). In comparison with the first proposed quasi-potential finder based on the Ordered Upwind Method (OUM) (Cameron, 2012), the new methods are 1.5 to 3.5 times faster, can produce error about two orders of magnitude smaller, and may exhibit faster convergence. Similar to the OUM, OLIMs employ the dynamical programming principle. Contrary to it, they (i) have an optimized strategy for the use of computationally expensive triangle updates leading to a notable speed-up, and (ii) directly so

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  • We prove an inverse theorem for the Gowers $U^2$-norm for maps $G\to\mathcal M$ from an countable, discrete, amenable group $G$ into a von Neumann algebra $\mathcal M$ equipped with an ultraweakly lower semi-continuous, unitarily invariant (semi-)norm $\Vert\cdot\Vert$. We use this result to prove a stability result for unitary-valued $\varepsilon$-representations $G\to\mathcal U(\mathcal M)$ with respect to $\Vert\cdot \Vert$.

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  • High Order DG methods with Riemann solver based interface numerical flux functions offer an interesting dispersion dissipation behaviour: dispersion errors are very low for a broad range of scales, while dissipation errors are very low for well resolved scales and are very high for scales close to the Nyquist cutoff. This observation motivates the trend that DG methods with Riemann solvers are used without an explicit LES model added. Due to under-resolution of vortical dominated structures typical for LES type setups, element based high order methods suffer from stability issues caused by aliasing errors of the non-linear flux terms. A very common strategy to fight these aliasing issues (and instabilities) is so-called polynomial de-aliasing, where interpolation is exchanged with projection based on an increased number of quadrature points. In this paper, we start with this common no-model or implicit LES (iLES) DG approach with polynomial de-aliasing and Riemann solver dissipation an

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  • Let ${k}$ be an algebraically closed field of characteristic zero and let ${K}$ be a field finitely generated over ${k}$. Let $\Sigma$ be a central simple ${K}$-algebra, $X$ a normal projective model of ${K}$ and $\Lambda$ a sheaf of maximal $\mathcal{O}_X$-orders in $\Sigma$. There is a ramification ${Q}$-divisor $\Delta$ on $X$, which is related to the canonical bimodule $\omega_\Lambda$ by an adjunction formula, and only depends on the class of $\Sigma$ in the Brauer group of ${K}$. When the numerical abundance conjecture holds true, or when $\Sigma$ is a division algebra, we show that the Gelfand-Kirillov dimension (or GK dimension) of the canonical ring of $\Lambda$ is one more than the Iitaka dimension (or D-dimension) of the log pair $(X,\Delta)$. In the case that $\Sigma$ is a division algebra, we further show that this GK dimension is also one more than the transcendence degree of the division algebra of degree zero fractions of the canonical ring of $\Lambda$. We prove that t

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  • We work with codimension one foliations in the projective space $\mathbb{P}^{n}$, given a differential one form $\omega\in H^0(\mathbb{P}^n,\Omega^1_{\mathbb{P}^n}(e))$, such differential form verifies the Frobenius integrability condition $\omega\wedge d\omega =0$. In this work we show that the Camacho-Lins Neto regularity, applied for $\omega$, is equivalent to the fact that every first order unfolding of $\omega$ is trivial up to isomorphism. We do this by computing the Castelnuovo-Mumford regularity of the ideal $I(\omega)$ of first order unfoldings. With this result, we are also showing that the only regular projective foliations, with reduced singular locus, are the ones that have singular locus only Kupka type singularities. At last we use these results to show that every foliation $\varpi\in \Omega^1_{\mathbb{C}^{n+1}}$, with initial form $\omega$ regular and dicritical, is isomorphic to $\omega$.

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  • For a simple algebraic group G in characteristic p, a triple (a,b,c) of positive integers is said to be rigid for G if the dimensions of the subvarieties of G of elements of order dividing a,b,c sum to 2dim G. In this paper we complete the proof of a conjecture of the third author, that for a rigid triple (a,b,c) for G with p>0, the triangle group T_{a,b,c} has only finitely many simple images of the form G(p^r). We also obtain further results on the more general form of the conjecture, where the images G(p^r) can be arbitrary quasisimple groups of type G.

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  • An inverse obstacle problem for the wave equation in a two layered medium is considered. It is assumed that the unknown obstacle is penetrable and embedded in the lower half-space. The wave as a solution of the wave equation is generated by an initial data whose support is in the upper half-space and observed at the same place as the support over a finite time interval. From the observed wave an indicator function in the time domain enclosure method is constructed. It is shown that, one can find some information about the geometry of the obstacle together with the qualitative property in the asymptotic behavior of the indicator function.

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  • We study two notions of approximate Birkhoff-James orthogonality in a normed space, from a geometric point of view, and characterize them in terms of normal cones. We further explore the interconnection between normal cones and approximate Birkhoff-James orthogonality to obtain a complete characterization of normal cones in a two-dimensional smooth Banach space. We also obtain a uniqueness theorem for approximate Birkhoff-James orthogonality set in a normed space.

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  • We study three distinct Zeeman topologies, showing that they are interval (order) topologies and highlighting their physical significance. We also propose a list of open problems connecting order- (not necessarily causal) theoretic with certain topological properties of spacetime manifolds.

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  • In this work we present three different randomized gossip algorithms for solving the average consensus problem while at the same time protecting the information about the initial private values stored at the nodes. We give iteration complexity bounds for all methods, and perform extensive numerical experiments.

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  • We propose to apply a low dimensional manifold model to scientific data interpolation from regular and irregular samplings with a significant amount of missing information. The low dimensionality of the patch manifold for general scientific data sets has been used as a regularizer in a variational formulation. The problem is solved via alternating minimization with respect to the manifold and the data set, and the Laplace-Beltrami operator in the Euler-Lagrange equation is discretized using the weighted graph Laplacian. Various scientific data sets from different fields of study are used to illustrate the performance of the proposed algorithm on data compression and interpolation from both regular and irregular samplings.

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  • Long-time Slashdot reader nri tipped us off to a developing story in Victoria, Australia. Yahoo News reports: Victoria Police officials announced on Saturday, June 24, they were withdrawing all speed camera infringement notices issued statewide from June 6 after a virus in the cameras turned out to be more widespread than first thought. "That does not mean they [the infringement notices] won't not be re-issued," Assistant Commissioner Doug Fryer told reporters, explaining that he wants to be sure the red light and speed cameras were working correctly. Acting Deputy Commissioner Ross Guenther told reporters on Friday that 55 cameras had been exposed to the ransomware virus, but they've now determined 280 cameras had been exposed. The cameras are not connected to the internet, but a maintenance worker unwittingly connected a USB stick with the virus on it to the camera system on June 6. Fryer said that about 1643 tickets would be withdrawn -- up from the 590 that police had announced on

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  • Scientific research piracy site hit with $15M fine

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  • This letter studies a radio-frequency (RF) multi-user wireless power transfer (WPT) system, where an energy transmitter (ET) with a large number of antennas delivers energy wirelessly to multiple distributed energy receivers (ERs). We investigate a low-complexity WPT scheme based on the retrodirective beamforming technique, where all ERs send a common beacon signal simultaneously to the ET in the uplink and the ET simply conjugates and amplifies its received sum-signal and transmits to all ERs in the downlink for WPT. We show that such a low-complexity scheme achieves the massive multiple-input multiple-output (MIMO) energy beamforming gain. However, a "doubly near-far" issue exists due to the round-trip (uplink beacon and downlink WPT) signal propagation loss where the harvested power of a far ER from the ET can be significantly lower than that of a near ER if the same uplink beacon power is used. To tackle this problem, we propose a distributed uplink beacon power update algorithm, w

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  • Geometric interpretations of some virtual knot invariants are given in terms of invariants of links in $\mathbb{S}^3$. Alexander polynomials of almost classical knots are shown to be specializations of the multi-variable Alexander polynomial of certain two-component boundary links of the form $J \sqcup K$ with $J$ a fibered knot. The index of a crossing, a common ingredient in the construction of virtual knot invariants, is related to the Milnor triple linking number of certain three-component links $J \sqcup K_1 \sqcup K_2$ with $J$ a connected sum of trefoils or figure-eights. Our main technical tool is virtual covers. This technique, due to Manturov and the first author, associates a virtual knot $\upsilon$ to a link $J \sqcup K$, where $J$ is fibered and $\text{lk}(J,K)=0$. Here we extend virtual covers to all multicomponent links $L=J \sqcup K$, with $K$ a knot. It is shown that an unknotted component $J_0$ can be added to $L$ so that $J_0 \sqcup J$ is fibered and $K$ has algebrai

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  • This paper provides a complete characterization of the boundary of an achievable rate region, called the Pareto boundary, of the single-antenna Z interference channel (Z-IC), when interference is treated as noise and users transmit complex Gaussian signals that are allowed to be improper. By considering the augmented complex formulation, we derive a necessary and sufficient condition for improper signaling to be optimal. This condition is stated as a threshold on the interference channel coefficient, which is a function of the interfered user rate and which allows insightful interpretations into the behavior of the achievable rates in terms of the circularity coefficient (i.e., degree of impropriety). Furthermore, the optimal circularity coefficient is provided in closed form. The simplicity of the obtained characterization permits interesting insights into when and how improper signaling outperforms proper signaling in the single-antenna Z-IC. We also provide an in-depth discussion on

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