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This paper presents a finite difference quasiNewton method for the minimization of noisy functions. The method takes advantage of the scalability and power of BFGS updating, and employs an adaptive procedure for choosing the differencing interval $h$ based on the noise estimation techniques of Hamming (2012) and Mor\'e and Wild (2011). This noise estimation procedure and the selection of $h$ are inexpensive but not always accurate, and to prevent failures the algorithm incorporates a recovery mechanism that takes appropriate action in the case when the line search procedure is unable to produce an acceptable point. A novel convergence analysis is presented that considers the effect of a noisy line search procedure. Numerical experiments comparing the method to a function interpolating trust region method are presented.
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Let $\xi : \Omega \times \mathbb{R}^n \to \mathbb{R}$ be zero mean, meansquare continuous, stationary, isotropic Gaussian random field with covariance function $r(x) = \mathbb{E}[\xi(0)\xi(x)]$ and let $G : \mathbb{R} \to \mathbb{R}$ such that $G$ is square integrable with respect to the standard Gaussian measure and is of Hermite rank $d$. The BreuerMajor theorem in it's continuous setting gives that, if $r \in L^d(\mathbb{R}^n)$ and $r(x) \to 0$ as $x \to \infty$, then the finite dimensional distributions of $Z_s(t) = \frac{1}{(2s)^{n/2}} \int_{[st^{1/n},st^{1/n}]^n} \Big[G(\xi(x))  \mathbb{E}[G(\xi(x))]\Big]dx$ converge to that of a scaled Brownian motion as $s \to \infty$. Here we give a proof for the case when $\xi : \Omega \times \mathbb{R}^n \to \mathbb{R}^m$ is a random vector field. We also give conditions for the functional convergence in $C([0,\infty))$ of $Z_s$ to hold along with expression for the asymptotic variance of the second chaos component in the Wiener chaos
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The class ${\sf IV}_2$ of $2$nondegenerate constant Levi rank $1$ hypersurfaces $M^5 \subset \mathbb{C}^3$ is governed by Pocchiola's two primary invariants $W_0$ and $J_0$. Their vanishing characterizes equivalence of such a hypersurface $M^5$ to the tube $M_{\sf LC}^5$ over the real light cone in $\mathbb{R}^3$. When either $W_0 \not\equiv 0$ or $J_0 \not\equiv 0$, by normalization of certain two group parameters ${\sf c}$ and ${\sf e}$, an invariant coframe can be built on $M^5$, showing that the dimension of the CR automorphism group drops from $10$ to $5$. This paper constructs an explicit $\{e\}$structure in case $W_0$ and $J_0$ do not necessarily vanish. Furthermore, Pocchiola's calculations hidden on a computer now appear in details, especially the determination of a secondary invariant $R$, expressed in terms of the first jet of $W_0$. All other secondary invariants of the $\{e\}$structure are also expressed explicitly in terms of $W_0$ and $J_0$.
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The first main result of this article is to prove the convergence of Lott's delocalized eta invariant holds for all invertible operators. Our second main result is to construct a pairing between delocalized cyclic cocycles of the group algebra of the fundamental group of a manifold and Ktheoretic higher rho invariants of the manifold, when the fundamental group is hyperbolic. As an application, under the assumption of hyperbolicity of the fundamental group, we compute the delocalized part of the ConnesChern character of AtiyahPatodiSinger type Ktheoretic higher indices, for example, the higher index of a spin manifold with boundary where the boundary carries a positive scalar curvature metric. Our explicit formula for this delocalized ConnesChern character is expressed in terms of our pairing of delocalized cyclic cocycles and higher rho invariants on the boundary of manifold.
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We address dissipative soliton formation in modulated PTsymmetric continuous waveguide arrays composed from waveguides with amplifying and absorbing sections, whose density gradually increases (due to decreasing waveguide separation) either towards the center of the array or towards its edges. In such a structure the level of gain/loss at which PTsymmetry gets broken depends on the direction of increase of the waveguide density. Breakup of the PTsymmetry occurs when eigenvalues of modes localized in the region, where waveguide density is largest, collide and move into complex plane. In this regime of broken symmetry the inclusion of focusing Kerrtype nonlinearity of the material and weak twophoton absorption allows to arrest the growth of amplitude of amplified modes and may lead to the appearance of stable attractors either in the center or at the edge of the waveguide array, depending on the type of array modulation. Such solitons can be stable, they acquire specific triangular
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We investigate the problem of covert and secret key generation over a discrete memoryless channel model with one way public discussion and in presence of an active warden who can arbitrarily vary its channel and tamper with the main channel when an information symbol is sent. In this scenario, we develop an adaptive protocol that is required to conceal not only the key but also whether a protocol is being implemented. Based on the adversary's actions, this protocol generates a key whose size depends on the adversary's actions. Moreover, for a passive adversary and for some models that we identify, we show that covert secret key generation is possible and characterize the covert secret key capacity in special cases; in particular, the covert secret key capacity is sometimes equal to the covert capacity of the channel, so that secrecy comes ``for free.
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Distributionally robust optimization (DRO) has been introduced for solving stochastic programs where the distribution of the random parameters is unknown and must be estimated by samples from that distribution. A key element of DRO is the construction of the ambiguity set, which is a set of distributions that covers the true distribution with a high probability. Assuming that the true distribution has a probability density function, we propose a class of ambiguity sets based on confidence bands of the true density function. The use of the confidence band enables us to take the prior knowledge of the shape of the underlying density function into consideration (e.g., unimodality or monotonicity). Using the confidence band constructed by density estimation techniques as the ambiguity set, we establish the convergence of the optimal value of DRO to that of the stochastic program as the sample size increases. However, the resulting DRO problem is computationally intractable, as it involves
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For an arbitrary prime number $p$, we propose an action for bosonic $p$adic strings in curved target spacetime, and show that the vacuum Einstein equations of the target are a consequence of worldsheet scaling symmetry of the quantum $p$adic strings, similar to the ordinary bosonic strings case. It turns out that certain $p$adic automorphic forms are the plane wave modes of the bosonic fields on $p$adic strings, and that the regularized normalization of these modes on the $p$adic worldsheet presents peculiar features which reduce part of the computations to familiar setups in quantum field theory, while also exhibiting some new features that make loop diagrams much simpler. Assuming a certain product relation, we also observe that the adelic spectrum of the bosonic string corresponds to the nontrivial zeros of the Riemann Zeta function.
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We are interested in the Guivarc'h inequality for admissible random walks on finitely generated relatively hyperbolic groups, endowed with a word metric. We show that for random walks with finite superexponential moment, if this inequality is an equality, then the Green distance is roughly similar to the word distance, generalizing results of Blach{\`e}re, Ha{\"i}ssinsky and Mathieu for hyperbolic groups [4]. Our main application is for relatively hyperbolic groups with respect to virtually abelian subgroups of rank at least 2. We show that for such groups, the Guivarc'h inequality with respect to a word distance and a finitely supported random walk is always strict.
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We study a natural generalization of inverse systems of finite regular covering spaces. A limit of such a system is a fibration whose fibres are profinite topological groups. However, as shown in a previous paper (ConnerHerfortPavesic: Some anomalous examples of lifting spaces), there are many fibrations whose fibres are profinite groups, which are far from being inverse limits of coverings. We characterize profinite fibrations among a large class of fibrations and relate the profinite topology on the fundamental group of the base with the action of the fundamental group on the fibre, and develop a version of the Borel construction for fibrations whose fibres are profinite groups.
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We investigate the number of permutations that occur in random labellings of trees. This is a generalisation of the number of subpermutations occurring in a random permutation. It also generalises some recent results on the number of inversions in randomly labelled trees. We consider complete binary trees as well as random split trees a large class of random trees of logarithmic height introduced by Devroye in 1998. Split trees consist of nodes (bags) which can contain balls and are generated by a random trickle down process of balls through the nodes. For complete binary trees we show that asymptotically the cumulants of the number of occurrences of a fixed permutation in the random node labelling have explicit formulas. Our other main theorem is to show that for a random split tree, with high probability the cumulants of the number of occurrences are asymptotically an explicit parameter of the split tree. For the proof of the second theorem we show some results on the number of embed
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The general theory developed by Ben Yaacov for metric structures provides Fra\"iss\'e limits which are approximately ultrahomogeneous. We show here that this result can be strengthened in the case of relational metric structures. We give an extra condition that guarantees exact ultrahomogenous limits. The condition is quite general. We apply it to stochastic processes, the class of diversities, and its subclass of $L_1$ diversities.
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Polynomial generalizations of all 130 of the identities in Slater's list of identities of the RogersRamanujan type are presented. Furthermore, duality relationships among many of the identities are derived. Some of the these polynomial identities were previously known but many are new. The author has implemented much of the finitization process in a Maple package which is available for free download from the author's website.
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In this paper, we construct a complete ndim Riemannian manifold with positive Ricci curvature, quadratically nonnegatively curved infinity and infinite topological type. This gives a negative answer to a conjecture by Jiping Sha and Zhongmin Shen in the case of n greater than or equal to 6.
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Distributed changepoint detection has been a fundamental problem when performing realtime monitoring using sensornetworks. We propose a distributed detection algorithm, where each sensor only exchanges CUSUM statistic with their neighbors based on the average consensus scheme, and an alarm is raised when local consensus statistic exceeds a prespecified global threshold. We provide theoretical performance bounds showing that the performance of the fully distributed scheme can match the centralized algorithms under some mild conditions. Numerical experiments demonstrate the good performance of the algorithm especially in detecting asynchronous changes.
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Google Drive has a pretty bad spam problem, and it seems Google doesn't care. Spammers can share files that automatically appear in your Drive, and there's no way to stop it. From a report: Google Drive's sharing system is the problem. Since it doesn't offer any sharing acceptance, all files and folders shared with your account are automatically available to you in Drive  they just show up. To make matters worse, if you only have "View" permission, you can't remove yourself from the share. It's a mess. And to make matters even worse, this is far from a new problem, but Google still hasn't done anything to fix it. Google got back to us with a statement saying that changes are coming to Drive's sharing features and they're"making it a priority." Here's the statement in full: "For the vast majority of users, the default sharing permissions in Drive work as intended. Unfortunately, this was not the case for this user and we sincerely apologize for her experience. In light of this issue
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Why is Grindr being sued by Matthew Herrick, an aspiring actor working in a restaurant in New York? "His former partner created fake profiles on the app to impersonate Herrick and then direct men to show up at Herrick's home and the restaurant where he worked asking for sex, sometimes more than a dozen times per day." But 14 police reports later, Herrick's lawsuit is now arguing that all tech companies should face greater accountability for what happens on their platforms, reports NBC News: His lawsuit alleges that the software developers who write code for Grindr have been negligent, producing an app that's defective in its design and that is "fundamentally unsafe" and "unreasonably dangerous"  echoing language that's more typically used in lawsuits about, say, a faulty kitchen appliance or a defective car part. If successful, the lawsuit could bring about a significant legal change to the risks tech companies face for what happens on their platforms, adding to growing public and p
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We investigate a parabolicelliptic system for maps $(u,v)$ from a compact Riemann surface $M$ into a Lorentzian manifold $N\times{\mathbb{R}}$ with a warped product metric. That system turns the harmonic map type equations into a parabolic system, but keeps the $v$equation as a nonlinear second order constraint along the flow. We prove a global existence result of the parabolicelliptic system by assuming either some geometric conditions on the target Lorentzian manifold or small energy of the initial maps. The result implies the existence of a Lorentzian harmonic map in a given homotopy class with fixed boundary data.
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This paper presents a nonlinear stability analysis for dcmicrogrids in both, interconnected mode and island operation with primary control. The proposed analysis is based on the fact that the dynamical model of the grid is a gradient system generated by a strongly convex function. The stability analysis is thus reduced to a series of convex optimization problems. The proposed method allows to: i) demonstrate the existence and uniqueness of the equilibrium ii) calculate this equilibrium numerically iii) give conditions for global stability using a Lyapunov function iv) estimate the attraction region. Previous works only address one of these aspects. Numeric calculations performed in cvx and simulations results in Matlab complement the analysis and demonstrate how to use this theoretical results in practical problems.
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It is known that Hochschild cohomology groups are represented by crossed extensions of associative algebras. In this paper, we introduce crossed $n$fold extensions of a Lie algebra $\mathfrak{g}$ by a module $M$, for $n \geq 2$. The equivalence classes of such extensions are represented by the $(n+1)$th ChevalleyEilenberg cohomology group $H^{n+1}_{CE} (\mathfrak{g}, M).$
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A novel linear combination representation aided timedomain smooth signal design is proposed by continuityenhanced basis signals for the Ncontinuous orthogonal frequency division multiplexing (NCOFDM) system. Compared to the conventional NCOFDM, the proposed scheme is capable of reducing the interference and maintaining the same sidelobe suppression performance imposed with the aid of two groups of basis signals assisted by the linear combination form. Our performance results demonstrate that with a low complexity overhead, the proposed scheme is capable of improving the error performance in comparison to the conventional NCOFDM, while maintaining the same sidelobe suppression as the conventional counterpart.
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We establish a duality for two factorization questions, one for general positive definite (p.d) kernels $K$, and the other for Gaussian processes, say $V$. The latter notion, for Gaussian processes is stated via Itointegration. Our approach to factorization for p.d. kernels is intuitively motivated by matrix factorizations, but in infinite dimensions, subtle measure theoretic issues must be addressed. Consider a given p.d. kernel $K$, presented as a covariance kernel for a Gaussian process $V$. We then give an explicit duality for these two seemingly different notions of factorization, for p.d. kernel $K$, vs for Gaussian process $V$. Our result is in the form of an explicit correspondence. It states that the analytic data which determine the variety of factorizations for $K$ is the exact same as that which yield factorizations for $V$. Examples and applications are included: pointprocesses, sampling schemes, constructive discretization, graphLaplacians, and boundaryvalue problems.
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Chinese hackers allegedly stole data of more than 100,000 US Navy personnel
1221 MIT Technology 1248Effective resource allocation plays a pivotal role for performance optimization in wireless networks. Unfortunately, typical resource allocation problems are mixedinteger nonlinear programming (MINLP) problems, which are NPhard in general. Machine learningbased methods recently emerge as a disruptive way to obtain nearoptimal performance for MINLP problems with affordable computational complexity. However, a key challenge is that these methods require huge amounts of training samples, which are difficult to obtain in practice. Furthermore, they suffer from severe performance deterioration when the network parameters change, which commonly happens and can be characterized as the task mismatch issue. In this paper, to address the sample complexity issue, instead of directly learning the inputoutput mapping of a particular resource allocation algorithm, we propose a Learning to Optimize framework for Resource Allocation, called LORA, that learns the pruning policy in the optimal bran
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The FBI has shut down the domains of 15 highprofile distributed denialofservice (DDoS) websites. "Several seizure warrants granted by a California federal judge went into effect Thursday, removing several of these 'border' or 'stresser' sites off the internet 'as part of coordinated law enforcement action taken against illegal DDoSforhire services,'" reports TechCrunch. "The orders were granted under federal seizure laws, and the domains were replaced with a federal notice." From the report: Prosecutors have charged three men, Matthew Gatrel and Juan Martinez in California and David Bukoski in Alaska, with operating the sites, according to affidavits filed in three U.S. federal courts, which were unsealed Thursday. The FBI had assistance from the U.K.'s National Crime Agency and the Dutch national police, and the Justice Department named several companies, including Cloudflare, Flashpoint and Google, for providing authorities with additional assistance. In all, several sites were
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A former Microsoft intern has revealed details of a YouTube incident that has convinced some Edge browser engineers that Google added code to purposely break compatibility. In a post on Hacker News, Joshua Bakita, a former software engineering intern at Microsoft, lays out details and claims about an incident earlier this year. Microsoft has since announced the company is moving from the EdgeHTML rendering engine to the open source Chromium project for its Edge browser. Google disputes Bakita's claims, and says the YouTube blank div was merely a bug that was fixed after it was reported. "YouTube does not add code designed to defeat optimizations in other browsers, and works quickly to fix bugs when they're discovered," says a YouTube spokesperson in a statement to The Verge. "We regularly engage with other browser vendors through standards bodies, the Web Platform Tests project, the opensource Chromium project and more to improve browser interoperability." While we're unli
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Videos and pictures of children being subjected to sexual abuse are being openly shared on Facebook's WhatsApp on a vast scale, with the encrypted messaging service failing to curb the problem despite banning thousands of accounts every day. From a report: Without the necessary number of human moderators, the disturbing content is slipping by WhatsApp's automated systems. A report reviewed by TechCrunch from two Israeli NGOs details how thirdparty apps for discovering WhatsApp groups include "Adult" sections that offer invite links to join rings of users trading images of child exploitation. TechCrunch has reviewed materials showing many of these groups are currently active. TechCrunch's investigation shows that Facebook could do more to police WhatsApp and remove this kind of content. Even without technical solutions that would require a weakening of encryption, WhatsApp's moderators should have been able to find these groups and put a stop to them. Groups with names like "child po
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Modeling via fractional partial differential equations or a L\'evy process has been an active area of research and has many applications. However, the lack of efficient numerical computation methods for general nonlocal operators impedes people from adopting such modeling tools. We proposed an efficient solver for the convectiondiffusion equation whose operator is the infinitesimal generator of a L\'evy process based on $\mathcal{H}$matrix technique. The proposed Crank Nicolson scheme is unconditionally stable and has a theoretical $\mathcal{O}(h^2+\Delta t^2)$ convergence rate. The $\mathcal{H}$matrix technique has theoretical $\mathcal{O}(N)$ space and computational complexity compared to $\mathcal{O}(N^2)$ and $\mathcal{O}(N^3)$ respectively for the direct method. Numerical experiments demonstrate the efficiency of the new algorithm.
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We develop an approach to study Coulomb branch operators in 3D $\mathcal{N}=4$ gauge theories and the associated quantization structure of their Coulomb branches. This structure is encoded in a onedimensional TQFT subsector of the full 3D theory, which we describe by combining several techniques and ideas. The answer takes the form of an associative noncommutative starproduct algebra on the Coulomb branch. For `good' and `ugly' theories (according to GaiottoWitten classification), we also have a trace map on this algebra, which allows to compute correlation functions and, in particular, guarantees that the starproduct satisfies a truncation condition. This work extends previous work on Abelian theories to the nonAbelian case by quantifying the monopole bubbling that describes screening of GNO boundary conditions. In our approach, the monopole bubbling is determined from the algebraic consistency of the OPE. This also yields a physical proof of the BullimoreDimofteGaiotto abelia
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This paper establishes the necessary and sufficient conditions for the reality of all the zeros in a polynomial sequence $\{P_i\}_{i=1}^{\infty}$ generated by a threeterm recurrence relation $P_i(x)+ Q_1(x)P_{i1}(x) +Q_2(x) P_{i2}(x)=0$ with the standard initial conditions $P_{0}(x)=1, P_{1}(x)=0,$ where $Q_1(x)$ and $Q_2(x)$ are arbitrary real polynomials.
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In this paper the relation between the cluster integrable systems and $q$difference equations is extended beyond the Painlev\'e case. We consider the class of hyperelliptic curves when the Newton polygons contain only four boundary points. The corresponding cluster integrable Toda systems are presented, and their discrete automorphisms are identified with certain reductions of the Hirota difference equation. We also construct nonautonomous versions of these equations and find that their solutions are expressed in terms of 5d Nekrasov functions with the ChernSimons contributions, while in the autonomous case these equations are solved in terms of the Riemann thetafunctions.
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We prove existence of all possible biaxisymmetric nearhorizon geometries of 5dimensional minimal supergravity. These solutions possess the crosssectional horizon topology $S^3$, $S^1\times S^2$, or $L(p,q)$ and come with prescribed electric charge, two angular momenta, and a dipole charge (in the ring case). Moreover, we establish uniqueness of these solutions up to an isometry of the symmetric space $G_{2(2)}/SO(4)$.
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In this paper we will consider, in the abstract setting of rigged Hilbert spaces, distribution valued functions and we will investigate, in particular, conditions for them to constitute a "continuous basis" for the smallest space $\mathcal D$ of a rigged Hilbert space. This analysis requires suitable extensions of familiar notions as those of frame, Riesz basis and orthonormal basis. A motivation for this study comes from the Gel'fandMaurin theorem which states, under certain conditions, the existence of a family of generalized eigenvectors of an essentially selfadjoint operator on a domain $\mathcal D$ which acts like an orthonormal basis of the Hilbert space $\mathcal H$. The corresponding object will be called here a {\em Gel'fand distribution basis}. The main results are obtained in terms of properties of a conveniently defined {\em synthesis operator}.
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We consider the highly nonlinear and ill posed inverse problem of determining some general expression $F(x,t,u,\nabla_xu)$ appearing in the diffusion equation $\partial_tu\Delta_x u+F(x,t,u,\nabla_xu)=0$ on $\Omega\times(0,T)$, with $T>0$ and $\Omega$ a bounded open subset of $\mathbb R^n$, $n\geq2$, from measurements of solutions on the lateral boundary $\partial\Omega\times(0,T)$. We consider both linear and nonlinear expression of $F(x,t,\nabla_xu,u)$. In the linear case, the equation is a convectiondiffusion equation and our inverse problem corresponds to the unique recovery, in some suitable sense, of a time evolving velocity field associated with the moving quantity as well as the density of the medium in some rough setting described by nonsmooth coefficients on a Lipschitz domain. In the nonlinear case, we prove the recovery of more general quasilinear expression appearing in a nonlinear parabolic equation. Our result give a positive answer to the unique recovery of a gen
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We provide the missing member of a family of four $q$series identities related to the modulus 36, the other members having been found by Ramanujan and Slater. We examine combinatorial implications of the identities in this family, and of some of the identities we considered in "Identities of the RamanujanSlater type related to the moduli 18 and 24," [\emph{J. Math. Anal. Appl.} \textbf{344}/2 (2008) 765777].
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We describe two inversion methods for the reconstruction of hard Xray solar images. The methods are tested against experimental visibilities recorded by the Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI) and synthetic visibilities based on the design of the Spectrometer/Telescope for Imaging Xrays (STIX).
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The cost and memoryefficient numerical simulation of coupled volumebased multiphysics problems like flow, transport, wave propagation and others remains a challenging task with finite element method (FEM) approaches. Goaloriented space and time adaptive methods derived from the dual weighted residual (DWR) method appear to be a shiny key technology to generate optimal spacetime meshes to minimise costs. Current implementations for challenging problems of numerical screening tools including the DWR technology broadly suffer in their extensibility to other problems, in high memory consumption or in missing system solver technologies. This work contributes to the efficient embedding of DWR spacetime adaptive methods into numerical screening tools for challenging problems of physically relevance with a new approach of flexible data structures and algorithms on them, a modularised and complete implementation as well as illustrative examples to show the performance and efficiency.
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We study derivativefree methods for policy optimization over the class of linear policies. We focus on characterizing the convergence rate of these methods when applied to linearquadratic systems, and study various settings of driving noise and reward feedback. We show that these methods provably converge to within any prespecified tolerance of the optimal policy with a number of zeroorder evaluations that is an explicit polynomial of the error tolerance, dimension, and curvature properties of the problem. Our analysis reveals some interesting differences between the settings of additive driving noise and random initialization, as well as the settings of onepoint and twopoint reward feedback. Our theory is corroborated by extensive simulations of derivativefree methods on these systems. Along the way, we derive convergence rates for stochastic zeroorder optimization algorithms when applied to a certain class of nonconvex problems.
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In Eikonal equations, rarefaction is a common phenomenon known to degrade the rate of convergence of numerical methods. The `factoring' approach alleviates this difficulty by deriving a PDE for a new (locally smooth) variable while capturing the rarefactionrelated singularity in a known (nonsmooth) `factor'. Previously this technique was successfully used to address rarefaction fans arising at point sources. In this paper we show how similar ideas can be used to factor the 2D rarefactions arising due to nonsmoothness of domain boundaries or discontinuities in PDE coefficients. Locations and orientations of such rarefaction fans are not known in advance and we construct a `justintime factoring' method that identifies them dynamically. The resulting algorithm is a generalization of the Fast Marching Method originally introduced for the regular (unfactored) Eikonal equations. We show that our approach restores the firstorder convergence and illustrate it using a range of maze navigat
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It has been known for more than 40 years that there are posets with planar cover graphs and arbitrarily large dimension. Recently, Streib and Trotter proved that such posets must have large height. In fact, all known constructions of such posets have two large disjoint chains with all points in one chain incomparable with all points in the other. Gutowski and Krawczyk conjectured that this feature is necessary. More formally, they conjectured that for every $k\geq 1$, there is a constant $d$ such that if $P$ is a poset with a planar cover graph and $P$ excludes $\mathbf{k}+\mathbf{k}$, then $\dim(P)\leq d$. We settle their conjecture in the affirmative. We also discuss possibilities of generalizing the result by relaxing the condition that the cover graph is planar.
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The design of block codes for short information blocks (e.g., a thousand or less information bits) is an open research problem that is gaining relevance thanks to emerging applications in wireless communication networks. In this paper, we review some of the most promising code constructions targeting the short block regime, and we compare them with both finitelength performance bounds and classical errorcorrection coding schemes. The work addresses the use of both binary and highorder modulations over the additive white Gaussian noise channel. We will illustrate how to effectively approach the theoretical bounds with various performance versus decoding complexity tradeoffs.
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Dynamical transitions of the Acetabularia whorl formation caused by outside calcium concentration is carefully analyzed using a chemical reaction diffusion model on a thin annulus. Restricting ourselves with Turing instabilities, we found all three types of transition, continuous, catastrophic and random can occur under different parameter regimes. Detailed linear analysis and numerical investigations are also provided. The main tool used in the transition analysis is Ma \& Wang's dynamical transition theory including the center manifold reduction.
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The problem of sequential probability forecasting is considered in the most general setting: a model set C is given, and it is required to predict as well as possible if any of the measures (environments) in C is chosen to generate the data. No assumptions whatsoever are made on the model class C, in particular, no independence or mixing assumptions; C may not be measurable; there may be no predictor whose loss is sublinear, etc. It is shown that the cumulative loss of any possible predictor can be matched by that of a Bayesian predictor whose prior is discrete and is concentrated on C, up to an additive term of order $\log n$, where $n$ is the time step. The bound holds for every $n$ and every measure in C. This is the first nonasymptotic result of this kind. In addition, a nonmatching lower bound is established: it goes to infinity with $n$ but may do so arbitrarily slow.
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We prove existence and uniqueness of solutions to FokkerPlanck equations associated to Markov operators multiplicatively perturbed by degenerate timeinhomogeneous coefficients. Precise conditions on the timeinhomogeneous coefficients are given. In particular, we do not necessarily require the coefficients to be neither globally bounded nor bounded away from zero. The approach is based on constructing random timechanges and studying related martingale problems for Markov processes with values in locally compact, complete and separable metric spaces.
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Let $A$ be a finite dimensional algebra over a field $F$ of characteristic zero. If $L$ is a Lie algebra acting on $A$ by derivations, then such an action determines an action of its universal enveloping algebra $U(L)$. In this case we say that $A$ is an algebra with derivation or an $L$algebra. Here we study the differential $L$identities of $A$ and the corresponding differential codimensions, $c_n^L (A)$, when $L$ is a finite dimensional semisimple Lie algebra. We give a complete characterization of the corresponding ideal of differential identities in case the sequence $c_n^L (A)$, $n=1,2,\dots$, is polynomially bounded. Along the way we determine up to PIequivalence the only finite dimensional $L$algebra of almost polynomial growth.
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An amoeba is the image of a subvariety of an algebraic torus under the logarithmic moment map. We consider some qualitative aspects of amoebas, establishing some results and posing problems for further study. These problems include determining the dimension of an amoeba, describing an amoeba as a semialgebraic set, and identifying varieties whose amoebas are a finite intersection of amoebas of hypersurfaces. We show that an amoeba that is not of full dimension is not such a finite intersection if its variety is nondegenerate and we describe amoebas of lines as explicit semialgebraic sets.
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The aim of this paper is to develop a constraint algorithm for singular classical field theories in the framework of $k$cosymplectic geometry. Since these field theories are singular, we need to introduce the notion of $k$precosymplectic structure, which is a generalization of the $k$cosymplectic structure. Next $k$precosymplectic Hamiltonian systems are introduced in order to describe singular field theories, both in Lagrangian and Hamiltonian formalisms. Finally, we develop a constraint algorithm in order to find a submanifold where the existence of solutions of the field equations is ensured. The case of affine Lagrangians is studied as a relevant example.
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The dynamics of the fourbody problem have attracted increasing attention in recent years. In this paper, we extend the basic equilateral fourbody problem by introducing the effect of radiation pressure, PoyntingRobertson drag, and solar wind drag. In our setup, three primaries lay at the vertices of an equilateral triangle and move in circular orbits around their common center of mass. Here, one of the primaries is a radiating body and the fourth body (whose mass is negligible) does not affect the motion of the primaries. We show that the existence and the number of equilibrium points of the problem depend on the mass parameters and radiation factor. Consequently, the allowed regions of motion, the regions of the basins of convergence for the equilibrium points, and the basin entropy will also depend on these parameters. The present dynamical model is analyzed for three combinations of mass for the primaries: equal masses, two equal masses, different masses. As the main results, we
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Twodimensional rational CFT are characterised by an integer $\ell$, the number of zeroes of the Wronskian of the characters. For $\ell<6$ there is a finite number of theories and most of these are classified. Recently it has been shown that for $\ell\ge 6$ there are infinitely many admissible characters that could potentially describe CFT's. In this note we examine the $\ell=6$ case, whose central charges lie between 24 and 32, and propose a classification method based on cosets of meromorphic CFT's. We illustrate the method using theories on Kervaire lattices with complete root systems.
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We solve the differentiablity problem for the evolution map in Milnor's infinite dimensional setting. We first show that the evolution map of each $C^k$semiregular Lie group admits a particular kind of sequentially continuity $$ called Mackey continuity $$ and then prove that this continuity property is strong enough to ensure differentiability of the evolution map. In particular, this drops any continuity presumptions made in this context so far. Remarkably, Mackey continuity rises directly from the regularity problem itself $$ which makes it particular among the continuity conditions traditionally considered. As a further application of the introduced notions, we discuss the strong Trotter property in the sequentially, and the Mackey continuous context.
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We characterise the existentially closed models of the theory of exponential fields. We find the amalgamation bases and characterise the types over them. We define a notion of independence and show that independent systems of higher dimension can also be amalgamated. Using these results we position the category of existentially closed exponential fields in the stability hierarchy as NSOP$_1$ but TP$_2$.
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We introduce the notion of a continuous Schauder frame for a Banach space. This is both a generalization of continuous frames and coherent states for Hilbert spaces and a generalization of unconditional Schauder frames for Banach spaces. As a natural example, we prove that any wavelet for $L_p(\R)$ with $1<p<\infty$ generates a continuous wavelet Schauder frame. Furthermore, we generalize the properties shrinking and boundedly complete to the continuous Schauder frame setting, and prove that many of the fundamental James theorems still hold in this general context.
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Last night, Elon Musk unveiled his vision of a highspeed tunnel system he believes could ease congestion and revolutionize how millions of commuters get around cities. CNBC reports: Musk, who founded the Boring Co. two years ago after complaining that traffic in Los Angeles was driving him "nuts," says the demonstration tunnel cost approximately $10 million to complete. Engineers and workers have been boring the 1.14milelong tunnel underneath one of the main streets in Hawthorne, California. One end of the tunnel starts in a parking lot owned by Musk's Space X. The other end of the demonstration tunnel is in a neighborhood about a mile away in Hawthorne. Tuesday afternoon, the Boring Co. gave reporters demonstration rides through the tunnel in modified Tesla Model X SUVs, going between 40 and 50 miles per hour. Engineers have attached deployable alignment wheels to the two front wheels of the Model X. Those alignment wheels stick out to the side of the main wheels and act as a bumpe
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We study the superlinear oscillator equation $\ddot{x}+ \lvert x \rvert^{\alpha1}x = p(t)$ for $\alpha\geq 3$, where $p$ is a quasiperiodic forcing with no Diophantine condition on the frequencies and show that typically the set of initial values leading to solutions $x$ such that $\lim_{t\to\infty} (\lvert x(t) \rvert + \lvert \dot{x}(t) \rvert) = \infty$ has Lebesgue measure zero, provided the starting energy $\lvert x(t_0) \rvert + \lvert \dot{x}(t_0) \rvert$ is sufficiently large.
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We introduce the notion of a symplectic capacity relative to a coisotropic submanifold of a symplectic manifold, and we construct two examples of such capacities through modifications of the HoferZehnder capacity. As a consequence, we obtain a nonsqueezing theorem for symplectic embeddings relative to coisotropic constraints and existence results for leafwise chords on energy surfaces.
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The extension of persistent homology to multiparameter setups is an algorithmic challenge. Since most computation tasks scale badly with the size of the input complex, an important preprocessing step consists of simplifying the input while maintaining the homological information. We present an algorithm that drastically reduces the size of an input. Our approach is an extension of the chunk algorithm for persistent homology (Bauer et al., Topological Methods in Data Analysis and Visualization III, 2014). We show that our construction produces the smallest multifiltered chain complex among all the complexes quasiisomorphic to the input, improving on the guarantees of previous work in the context of discrete Morse theory. Our algorithm also offers an immediate parallelization scheme in shared memory. Already its sequential version compares favorably with existing simplification schemes, as we show by experimental evaluation.
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We consider dynamical systems $T: X \to X$ that are extensions of a factor $S: Y \to Y$ through a projection $\pi: X \to Y$ with shrinking fibers, i.e. such that $T$ is uniformly continuous along fibers $\pi^{1}(y)$ and the diameter of iterate images of fibers $T^n(\pi^{1}(y))$ uniformly go to zero as $n \to \infty$. We prove that every $S$invariant measure has a unique $T$invariant lift, and prove that many properties of the original measure lift: ergodicity, weak and strong mixing, decay of correlations and statistical properties (possibly with weakening in the rates).The basic tool is a variation of the Wasserstein distance, obtained by constraining the optimal transportation paradigm to displacements along the fibers. We extend to a general setting classical arguments, enabling to translate potentials and observables back and forth between $X$ and $Y$.
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We describe an algorithm to compute bases of antisymmetric vectorvalued cusp forms with rational Fourier coefficients for the Weil representation associated to a finite quadratic module. The forms we construct always span all cusp forms in weight at least three. These formulas are useful for computing explicitly with theta lifts.
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