We give blow-up analysis for a Brezis-Merle's problem on the boundary. Also we give a new proof of a compactness result with Lipschitz condition.

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In this article we utilise abstract convexity theory in order to unify and generalize many different concepts from nonsmooth analysis. We introduce the concepts of abstract codifferentiability, abstract quasidifferentiability and abstract convex (concave) approximations of a nonsmooth function mapping a topological vector space to an order complete topological vector lattice. We study basic properties of these notions, construct elaborate calculus of abstract codifferentiable functions and discuss continuity of abstract codifferential. We demonstrate that many classical concepts of nonsmooth analysis, such as subdifferentiability and quasidifferentiability, are particular cases of the concepts of abstract codifferentiability and abstract quasidifferentiability. We also show that abstract convex and abstract concave approximations are a very convenient tool for the study of nonsmooth extremum problems. We use these approximations in order to obtain various necessary optimality condition

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Amazon raises minimum wage to $15 for all US employees

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Adaptive mesh refinement (AMR) is often used when solving time-dependent partial differential equations using numerical methods. It enables time-varying regions of much higher resolution, which can be used to track discontinuities in the solution by selectively refining around those areas. The open source Clawpack software implements block-structured AMR to refine around propagating waves in the AMRClaw package. For problems where the solution must be computed over a large domain but is only of interest in a small area this approach often refines waves that will not impact the target area. We seek a method that enables the identification and refinement of only the waves that will influence the target area. Here we show that solving the time-dependent adjoint equation and using a suitable inner product allows for a more precise refinement of the relevant waves. We present the adjoint methodology in general, and give details on how this method has been implemented in AMRClaw. Examples fo

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General multivariate distributions are notoriously expensive to sample from, particularly the high-dimensional posterior distributions in PDE-constrained inverse problems. This paper develops a sampler for arbitrary continuous multivariate distributions that is based on low-rank surrogates in the tensor-train format. We construct a tensor-train approximation to the target probability density function using the cross interpolation, which requires a small number of function evaluations. For sufficiently smooth distributions the storage required for the TT approximation is moderate, scaling linearly with dimension. The structure of the tensor-train surrogate allows efficient sampling by the conditional distribution method. Unbiased estimates may be calculated by correcting the transformed random seeds using a Metropolis--Hastings accept/reject step. Moreover, one can use a more efficient quasi-Monte Carlo quadrature that may be corrected either by a control-variate strategy, or by importa

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The probability density function (PDF) of a random variable associated with the solution of a stochastic partial differential equation (SPDE) is approximated using a truncated series expansion. The SPDE is solved using two stochastic finite element (SFEM) methods, Monte Carlo sampling and the stochastic Galerkin method with global polynomials. The random variable is a functional of the solution of the SPDE, such as the average over the physical domain. The truncated series are obtained considering a finite number of terms in the Gram-Charlier or Edgeworth series expansions. These expansions approximate the PDF of a random variable in terms of another PDF, and involve coefficients that are functions of the known cumulants of the random variable. To the best of our knowledge, their use in the framework of SPDEs has not yet been explored.

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The automorphism group of a map acts naturally on its flags (triples of incident vertices, edges, and faces). An Archimedean map on the torus is called almost regular if it has as few flag orbits as possible for its type; for example, a map of type $(4.8^2)$ is called almost regular if it has exactly three flag orbits. Given a map of a certain type, we will consider other more symmetric maps that cover it. In this paper, we prove that each Archimedean toroidal map has a unique minimal almost regular cover. By using the Gaussian and Eisenstein integers, along with previous results regarding equivelar maps on the torus, we construct these minimal almost regular covers explicitly.

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In this paper we establish a framework for geometric interpolation with exact area preservation using B\'ezier cubic polynomials. We show there exists a family of such curves which are $5^{th}$ order accurate, one order higher than standard geometric Hermite interpolation. We prove this result is valid when the curvature at the endpoints does not vanish, and in the case of vanishing curvature, the interpolation is $4^{th}$ order accurate. The method is computationally efficient and prescribes the parametrization speed at endpoints through an explicit formula based on the given data. Additional accuracy (i.e. same order but lower error constant) may be obtained through an iterative process to find optimal parametrization speeds which further reduces the error while still preserving the prescribed area exactly.

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We consider the asymptotic behaviour of the generalised hypergeometric function \[{}_3F_2\bl(\!\!\begin{array}{c} 1, (1+t)k/2, (1+t)k/2+1/2\\tk+1, k+1\end{array}\!\!; x\br),\qquad 0<x,t\leq 1\] as the parameter $k\to+\infty$. Numerical results illustrating the accuracy of the resulting expansion are given.

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In this work, we analyze efficient window shift schemes for windowed decoding of spatially coupled low-density parity-check (SC-LDPC) codes, which is known to yield close-tooptimal decoding results when compared to full belief propagation (BP) decoding. However, a drawback of windowed decoding is that either a significant amount of window updates are required leading to unnecessary high decoding complexity or the decoder suffers from sporadic burst-like error patterns, causing a decoder stall. To tackle this effect and, thus, to reduce the average decoding complexity, the basic idea is to enable adaptive window shifts based on a bit error rate (BER) prediction, which reduces the amount of unnecessary updates. As the decoder stall does not occur in analytical investigations such as the density evolution (DE), we examine different schemes on a fixed test-set and exhaustive monte-carlo simulations based on our graphic processing unit (GPU) simulation framework. As a result, we can reduce

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We prove that a backward orbit with bounded Kobayashi step for a hyperbolic or strongly elliptic holomorphic self-map of a bounded strongly convex domain in the d-dimensional complex Euclidean space necessarily converges to a boundary fixed point, generalizing previous results obtained by Poggi-Corradini in the unit disk and by Ostapyuk in the unit ball.

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We say that a permutation $\pi$ is ballot if, for all $i$, the word $\pi_1\cdots \pi_i$ has at least as many ascents as it has descents. We say that $\pi$ is an odd order permutation if $\pi$ has odd order in $S_n$. Let $b(n)$ denote the number of ballot permutations of order $n$, and let $p(n)$ denote the number of odd order permutations of order $n$. Callan observed that, seemingly, $b(n)=p(n)$ for all $n$. Whether this is true or not remains unknown. In this paper we conjecture that a stronger statement is true. Let $b(n,d)$ denote the number of ballot permutations with $d$ descents. Let $p(n,d)$ denote the number of odd order permutations with $M(\pi)=d$, where $M(\pi)$ is a certain statistic related to the cyclic descents of $\pi$. We conjecture that $b(n,d)=p(n,d)$ for all $n$ and $d$. We prove this stronger conjecture for the cases $d=1,\ 2$, and $d=\lfloor(n-1)/2\rfloor$, and in each of these cases we establish formulas for $b(n,d)$ involving second-order Eulerian numbers and E

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Let $\mathbb{F}_{\Theta }=G/P_{\Theta }$ be a generalized flag manifold, where $G$ is a real noncompact semi-simple Lie group and $P_{\Theta }$ a parabolic subgroup. A classical result says the Schubert cells, which are the closure of the Bruhat cells, endow $\mathbb{F}_{\Theta}$ with a cellular CW structure. In this paper we exhibit explicit parametrizations of the Schubert cells by closed balls (cubes) in $\mathbb{R}^{n}$ and use them to compute the boundary operator $\partial $ for the cellular homology. We recover the result obtained by Kocherlakota [1995], in the setting of Morse Homology, that the coefficients of $\partial $ are $0$ or $\pm 2$ (so that $\mathbb{Z}_{2}$-homology is freely generated by the cells). In particular, the formula given here is more refined in the sense that the ambiguity of signals in the Morse-Witten complex is solved.

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Let $(R,m, \kappa)$ be a local ring. We give a characterization of $R$-modules $M$ whose local cohomology is finite length up to some index in terms of asymptotic vanishing of Koszul cohomology on parameter ideals up to the same index. In particular, we show that a quasi-unmixed module $M$ is asymptotically Cohen-Macaulay if and only if $M$ is Cohen-Macaulay on the punctured spectrum if and only if $\sup\{\ell(H^i(f_1, \ldots, f_d;M))\mid \sqrt{f_1, \ldots, f_d} = m \mbox{, } i< d\}<\infty$.

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Coders Automating Their Own Job

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A strong interaction is known to exist between edge-colored graphs (which encode PL pseudo-manifolds of arbitrary dimension) and random tensor models (as a possible approach to the study of Quantum Gravity). The key tool is the {\it G-degree} of the involved graphs, which drives the {\it $1/N$ expansion} in the tensor models context. In the present paper - by making use of combinatorial properties concerning Hamiltonian decompositions of the complete graph - we prove that, in any even dimension $d\ge 4$, the G-degree of all bipartite graphs, as well as of all (bipartite or non-bipartite) graphs representing singular manifolds, is an integer multiple of $(d-1)!$. As a consequence, in even dimension, the terms of the $1/N$ expansion corresponding to odd powers of $1/N$ are null in the complex context, and do not involve colored graphs representing singular manifolds in the real context. In particular, in the 4-dimensional case, where the G-degree is shown to depend only on the regular ge

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We show that the elliptic genus of the higher rank E-strings can be computed based solely on the genus 0 Gromov-Witten invariants of the corresponding elliptic geometry. To set up our computation, we study the structure of the topological string free energy on elliptically fibered Calabi-Yau manifolds both in the unrefined and the refined case, determining the maximal amount of the modular structure of the partition function that can be salvaged. In the case of fibrations exhibiting only isolated fibral curves, we show that the principal parts of the topological string partition function at given base-wrapping can be computed from the knowledge of the genus 0 Gromov-Witten invariants at this base-wrapping, and the partition function at lower base-wrappings. For the class of geometries leading to the higher rank E-strings, this leads to the result stated in the opening sentence.

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We present new results for consistency of maximum likelihood estimators with a focus on multivariate mixed models. Our theory builds on the idea of using subsets of the full data to establish consistency of estimators based on the full data. It requires neither that the data consist of independent observations, nor that the observations can be modeled as a stationary stochastic process. Compared to existing asymptotic theory using the idea of subsets we substantially weaken the assumptions, bringing them closer to what suffices in classical settings. We apply our theory in two multivariate mixed models for which it was unknown whether maximum likelihood estimators are consistent. The models we consider have non-stochastic predictors and multivariate responses which are possibly mixed-type (some discrete and some continuous).

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In this article we construct a large class of interacting Euclidean quantum field theories, over a p-adic space time, by using white noise calculus. We introduce p-adic versions of the Kondratiev and Hida spaces in order to use the Wick calculus on the Kondratiev spaces. The quantum fields introduced here fulfill all the Osterwalder-Schrader axioms, except the reflection positivity.

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Bounding the best achievable error probability for binary classification problems is relevant to many applications including machine learning, signal processing, and information theory. Many bounds on the Bayes binary classification error rate depend on information divergences between the pair of class distributions. Recently, the Henze-Penrose (HP) divergence has been proposed for bounding classification error probability. We consider the problem of empirically estimating the HP-divergence from random samples. We derive a bound on the convergence rate for the Friedman-Rafsky (FR) estimator of the HP-divergence, which is related to a multivariate runs statistic for testing between two distributions. The FR estimator is derived from a multicolored Euclidean minimal spanning tree (MST) that spans the merged samples. We obtain a concentration inequality for the Friedman-Rafsky estimator of the Henze-Penrose divergence. We validate our results experimentally and illustrate their applicatio

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This paper presents the construction of a correct-energy stabilized finite element method for the incompressible Navier-Stokes equations. The framework of the methodology and the correct-energy concept have been developed in the convective--diffusive context in the preceding paper [M.F.P. ten Eikelder, I. Akkerman, Correct energy evolution of stabilized formulations: The relation between VMS, SUPG and GLS via dynamic orthogonal small-scales and isogeometric analysis. I: The convective--diffusive context, Comput. Methods Appl. Mech. Engrg. 331 (2018) 259--280]. The current work extends ideas of the preceding paper to build a stabilized method within the variational multiscale (VMS) setting which displays correct-energy behavior. Similar to the convection--diffusion case, a key ingredient is the proper dynamic and orthogonal behavior of the small-scales. This is demanded for correct energy behavior and links the VMS framework to the streamline-upwind Petrov-Galerkin (SUPG) and the Galerk

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In this paper, we study a two-species model in the form of a coupled system of nonlinear stochastic differential equations (SDEs) that arises from a variety of applications such as aggregation of biological cells and pedestrian movements. The evolution of each process is influenced by four different forces, namely an external force, a self-interacting force, a cross-interacting force and a stochastic noise where the two interactions depend on the laws of the two processes. We also consider a many-particle system and a (nonlinear) partial differential equation (PDE) system that associate to the model. We prove the wellposedness of the SDEs, the propagation of chaos of the particle system, and the existence and (non)-uniqueness of invariant measures of the PDE system.

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The propensity score plays an important role in causal inference with observational data. Once the propensity score is available, one can use it to estimate a variety of causal effects in a unified setting. Despite this appeal, a main practical difficulty arises because the propensity score is usually unknown, has to be estimated, and extreme propensity score estimates can lead to distorted inference procedures. To address these limitations, this article proposes to estimate the propensity score by fully exploiting its covariate balancing property. We call the resulting estimator the integrated propensity score (IPS) as it is based on integrated moment conditions. In sharp contrast with other methods that balance only some specific moments of covariates, the IPS aims to balance \textit{all} functions of covariates. Further, the IPS estimator is data-driven, does not rely on tuning parameters such as bandwidths, admits an asymptotic linear representation, and is $\sqrt{n}$-consistent an

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Let $(\eta_i)_{i\geq1}$ be a sequence of $\psi$-mixing random variables. Let $m=\lfloor n^\alpha \rfloor, 0< \alpha < 1, k=\lfloor n/m \rfloor,$ and $Y_j = \sum_{i=1}^m \eta_{m(j-1)+i}, 1\leq j \leq k.$ Set $ S_k^o=\sum_{j=1}^{k } Y_j $ and $[S^o]_k=\sum_{i=1}^{k } (Y_j )^2.$ We prove a Cram\'er type moderate deviation expansion for $\mathbb{P}(S_k^o/\sqrt{[ S^o]_k} \geq x)$ as $n\to \infty.$ Our result is similar to the recent work of Chen et al.\,\cite{CSWX16} where the authors established Cram\'er type moderate deviation expansions for $\beta$-mixing sequences. Comparing to the result of Chen et al.\, (2016), our results hold for mixing coefficients with polynomial decaying rate and wider ranges of validity.

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The \emph{critical} group of a finite connected graph is an abelian group defined by the Smith normal form of its Laplacian. Let $q$ be a power of a prime and $H$ be a multiplicative subgroup of $K=\mathbb{F}_{q}$. By $\mathrm{Cay}(K,H)$ we denote the Cayley graph on the additive group of $K$ with `connection' set $H$. A strongly regular graph of the form $\mathrm{Cay}(K,H)$ is called a \emph{cyclotomic strongly regular graph}. Let $p$ and $\ell >2$ be primes such that $p$ is primitive $\pmod{\ell}$. We compute the \emph{critical} groups of a family of \emph{cyclotomic strongly regular graphs} for which $q=p^{(\ell-1)t}$ (with $t\in \mathbb{N}$) and $H$ is the unique multiplicative subgroup of order $k=\frac{q-1}{\ell}$. These graphs were first discovered by van Lint and Schrijver in \cite{VS}.

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Damping of selectively bonded 3D woven lattice materials

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Fractional-order dynamical systems are used to describe processes that exhibit long-term memory with power-law dependence. Notable examples include complex neurophysiological signals such as electroencephalogram (EEG) and blood-oxygen-level dependent (BOLD) signals. When analyzing different neurophysiological signals and other signals with different origin (for example, biological systems), we often find the presence of artifacts, that is, recorded activity that is due to external causes and does not have its origins in the system of interest. In this paper, we consider the problem of estimating the states of a discrete-time fractional-order dynamical system when there are artifacts present in some of the sensor measurements. Specifically, we provide necessary and sufficient conditions that ensure we can retrieve the system states even in the presence of artifacts. We provide a state estimation algorithm that can estimate the states of the system in the presence of artifacts. Finally,

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We show that the uniform measure-theoretic ergodic decomposition of a countable Borel equivalence relation $(X, E)$ may be realized as the topological ergodic decomposition of a continuous action of a countable group $\Gamma \curvearrowright X$ generating $E$. We then apply this to the study of the cardinal algebra $\mathcal K(E)$ of equidecomposition types of Borel sets with respect to a compressible countable Borel equivalence relation $(X, E)$. We also make some general observations regarding quotient topologies on topological ergodic decompositions, with an application to weak equivalence of measure-preserving actions.

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We study the deformations of the Chow group of zero-cycles using Bloch's formula and differential forms. We thereby obtain a new proof of an algebraization theorem for zero-cycles previously obtained using idelic techniques.

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The matching number of a family of subsets of an $n$-element set is the maximum number of pairwise disjoint sets. The families with matching number $1$ are called intersecting. The famous Erd\H os-Ko-Rado theorem determines the size of the largest intersecting family of $k$-sets. Its generalization to the families with larger matching numbers, known under the name of the Erd\H{o}s Matching Conjecture, is still open for a wide range of parameters. In this paper, we address the degree versions of both theorems. More precisely, we give degree and $t$-degree versions of the Erd\H{o}s-Ko-Rado and the Hilton-Milner theorems, extending the results of Huang and Zhao, and Frankl, Han, Huang and Zhao. We also extend the range in which the degree version of the Erd\H{o}s Matching conjecture holds.

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Delay in information processing has been incorporated in many models that describe emergence in biological systems. In particular, different versions of the Cucker-Smale model have been considered with processing delay in previous works. In this paper, we study the delayed Cucker-Smale-type system proposed by Erban, Haskovec and Sun \cite{ErHaSu}, in which all terms in the communication weight and the velocity coupling have a positive constant delay $\tau>0$. %We give an affirmative answer to an open question We show that flocking occurs for the communication weight originally proposed by Cucker and Smale, $\psi(r)=(1+r^{2})^{-\beta}$, $r \geq 0$, if $0<\beta <\frac{1}{2}$ and the delay $\tau$ is sufficiently small. In addition, we prove that the delayed system with multiplicative white noises exhibits flocking behavior if the intensity of the noise is sufficiently small. Both results rely on stability estimates for the Cucker-Smale delayed flow.

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We consider the problem of detecting a random walk on a graph, based on observations of the graph nodes. When visited by the walk, each node of the graph observes a signal of elevated mean, which we assume can be different across different nodes. Outside of the path of the walk, and also in its absence, nodes measure only noise. Assuming the Neyman-Pearson setting, our goal then is to characterize detection performance by computing the error exponent for the probability of a miss, under a constraint on the probability of false alarm. Since exact computation of the error exponent is known to be difficult, equivalent to computation of the Lyapunov exponent, we approximate its value by finding a tractable lower bound. The bound reveals an interesting detectability condition: the walk is detectable whenever the entropy of the walk is smaller than one half of the expected signal-to-noise ratio. We derive the bound by extending the notion of Markov types to Gauss-Markov types. These are sequ

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In this paper, we discuss some dimension results for triangle sets of compact sets in $\mathbb{R}^2$. In particular, we prove that for any compact set $F$ in $\mathbb{R}^2$, the triangle set $\Delta(F)$ satisfies \[ \dim_{\mathrm{A}} \Delta(F)\geq \frac{3}{2}\dim_{\mathrm{A}} F. \] If $\dim_{\mathrm{A}} F>1$ then we have \[ \dim_{\mathrm{A}} \Delta(F)\geq 1+\dim_{\mathrm{A}} F. \] If $\dim_{\mathrm{A}} F>4/3$ then we have the following better bound, \[ \dim_{\mathrm{A}} \Delta(F)\geq \min\left\{\frac{5}{2}\dim_{\mathrm{A}} F-1,3\right\}. \] Moreover, if $F$ satisfies a mild separation condition then the above result holds also for the box dimensions, namely, \[ \underline{\dim_{\mathrm{B}}} F\geq \frac{3}{2}\underline{\dim_{\mathrm{B}}} \Delta(F) \text{ and }\overline{\dim_{\mathrm{B}}} F\geq \frac{3}{2}\overline{\dim_{\mathrm{B}}} \Delta(F). \]

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In solid-state physics, energies of molecular systems are usually computed with a plane-wave discretization of Kohn-Sham equations. A priori estimates of plane-wave convergence for periodic Kohn-Sham calculations with pseudopotentials have been proved , however in most computations in practice, plane-wave cut-offs are not tight enough to target the desired accuracy. It is often advocated that the real quantity of interest is not the value of the energy but of energy differences for different configurations. The computed energy difference is believed to be much more accurate because of `discretization error cancellation', since the sources of numerical errors are essentially the same for different configurations. For periodic linear Hamiltonians with Coulomb potentials, error cancellation can be explained by the universality of the Kato cusp condition. Using weighted Sobolev spaces, Taylor-type expansions of the eigenfunctions are available yielding a precise characterization of this si

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We study a compatibility relationship between Qin's dominance order on a cluster algebra $\mathcal{A}$ and partial orderings arising from classifications of simple objects in a monoidal categorification $\mathcal{C}$ of $\mathcal{A}$. Our motivating example is Hernandez-Leclerc's monoidal categorification using representations of quantum affine algebras. In the framework of Kang-Kashiwara-Kim-Oh's monoidal categorification via representations of quiver Hecke algebras, we focus on the case of the category $R-gmod$ for a symmetric finite type $A$ quiver Hecke algebra using Kleshchev-Ram's classification of irreducible finite dimensional representations.

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In this paper, we discuss dynamical behavior of a non-autonomous system generated by a finite family $\mathbb{F}$. In the process, we relate the dynamical behavior of the non-autonomous system generated by the family $\mathbb{F}=\{f_1,f_2,\ldots,f_k\}$ with the dynamical behavior of the system $(X,f_k\circ f_{k-1}\circ\ldots\circ f_1)$. We discuss properties like minimality, equicontinuity, proximality and various forms of sensitivities for the two systems. We derive conditions under which the dynamical behavior of $(X,f_k\circ f_{k-1}\circ\ldots\circ f_1)$ is carried forward to $(X,\mathbb{F})$ (and vice-versa). We also give examples to illustrate the necessity of the conditions imposed.

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Fragility curves which express the failure probability of a structure, or critical components, as function of a loading intensity measure are nowadays widely used (i) in Seismic Probabilistic Risk Assessment studies, (ii) to evaluate impact of construction details on the structural performance of installations under seismic excitations or under other loading sources such as wind. To avoid the use of parametric models such as lognormal model to estimate fragility curves from a reduced number of numerical calculations, a methodology based on Support Vector Machines coupled with an active learning algorithm is proposed in this paper. In practice, input excitation is reduced to some relevant parameters and, given these parameters, SVMs are used for a binary classification of the structural responses relative to a limit threshold of exceedance. Since the output is not only binary, this is a score, a probabilistic interpretation of the output is exploited to estimate very efficiently fragili

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Our goal is to prove existence results for classical solutions to some general nondegenerate Cauchy problems which are natural generalizations of Isaacs equations. For the latter we are able to extend our results by admitting local conditions for coefficients. Such equations appear naturally for instance in robust control theory. Using our general results, we can solve not only Isaacs equations, but also equations for other sophisticated control problems, for instance models with state dependent constraints on the control set.

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A graph is $F$-saturated if it is $F$-free but the addition of any edge creates a copy of $F$. In this paper we study the quantity $\mathrm{sat}(n, H, F)$ which denotes the minimum number of copies of $H$ that an $F$-saturated graph on $n$ vertices may contain. This parameter is a natural saturation analogue of Alon and Shikhelmen's generalized Tur\'an problem, and letting $H = K_2$ recovers the well-studied saturation function. We provide a first investigation into this general function focusing on the cases where the host graph is either $K_s$ or $C_k$-saturated. Some representative interesting behavior is: (a) For any natural number $m$, there are graphs $H$ and $F$ such that $\mathrm{sat}(n, H, F) = \Theta(n^m)$. (b) For many pairs $k$ and $l$, we show $\mathrm{sat}(n, C_l, C_k) = 0$. In particular, we prove that there exists a triangle-free $C_k$-saturated graphs on $n$ vertices for any $k > 4$ and large enough $n$. (c) $\mathrm{sat}(n, K_3, K_4) = n-2$, $\mathrm{sat}(n, C_4, K

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Let ${\cal R}_{\mathbb{K}}[H]$ be the Hibi ring over a field $\mathbb{K}$ on a finite distributive lattice $H$, $P$ the set of join-irreducible elements of $H$ and $\omega$ the canonical ideal of ${\cal R}_{\mathbb{K}}[H]$. We show the powers $\omega^{(n)}$ of $\omega$ in the group of divisors $\mathrm{Div}({\cal R}_{\mathbb{K}}[H])$ is identical with the ordinal powers of $\omega$, describe the $\mathbb{K}$-vector space basis of $\omega^{(n)}$ for $n\in\mathbb{Z}$. Further, we show that the fiber cones $\bigoplus_{n\geq 0}\omega^n/\mathfrak{m}\omega^n$ and $\bigoplus_{n\geq0}(\omega^{(-1)})^n/\mathfrak{m}(\omega^{(-1)})^n$ of $\omega$ and $\omega^{(-1)}$ are sum of the Ehrhart rings, defined by sequences of elements of $P$ with a certain condition, which are polytopal complex version of Stanley-Reisner rings. Moreover, we show that the analytic spread of $\omega$ and $\omega^{(-1)}$ are maximum of the dimensions of these Ehrhart rings. Using these facts, we show that the question of P

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In this paper, motivated by the study of optimal control problems for infinite dimensional systems with endpoint state constraints, we introduce the notion of finite codimensional (exact/approximate) controllability. Some equivalent criteria on the finite codimensional controllability are presented. In particular, the finite codimensional exact controllability is reduced to deriving a G{\aa}rding type inequality for the adjoint system, which is new for many evolution equations. This inequality can be verified for some concrete problems (and hence applied to the corresponding optimal control problems), say the wave equations with both time and space dependent potentials. Moreover, under some mild assumptions, we show that the finite codimensional exact controllability of this sort of wave equations is equivalent to the classical geometric control condition.

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Let $K$ be the function field of a smooth, irreducible curve defined over $\overline{\mathbb{Q}}$. Let $f\in K[x]$ be of the form $f(x)=x^q+c$ where $q = p^{r}, r \ge 1,$ is a power of the prime number $p$, and let $\beta\in \overline{K}$. For all $n\in\mathbb{N}\cup\{\infty\}$, the Galois groups $G_n(\beta)=\mathop{\rm{Gal}}(K(f^{-n}(\beta))/K(\beta))$ embed into $[C_q]^n$, the $n$-fold wreath product of the cyclic group $C_q$. We show that if $f$ is not isotrivial, then $[[C_q]^\infty:G_\infty(\beta)]<\infty$ unless $\beta$ is postcritical or periodic. We are also able to prove that if $f_1(x)=x^q+c_1$ and $f_2(x)=x^q+c_2$ are two such distinct polynomials, then the fields $\bigcup_{n=1}^\infty K(f_1^{-n}(\beta))$ and $\bigcup_{n=1}^\infty K(f_2^{-n}(\beta))$ are disjoint over a finite extension of $K$.

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In this work, we construct some irreducible components of the space of two-dimensional holomorphic foliations on $\mathbb{P}^n$ associated to some algebraic representations of the affine Lie algebra $\mathfrak{aff}(\mathbb{C})$. We give a description of the generalized Kupka components, obtaining a classification of them in terms of the degree of the foliations, in both cases $n=3$ and $n=4$.

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We study when Fourier transforms of Gibbs measures of sufficiently nonlinear expanding Markov maps decay at infinity at a polynomial rate. Assuming finite Lyapunov exponent, we reduce this to a nonlinearity assumption, which we verify for the Gauss map using Diophantine analysis. Our approach uses large deviations and additive combinatorics, which combines the earlier works on the Gibbs measures for Gauss map (Jordan-Sahlsten, 2013) and Fractal Uncertainty Principle (Bourgain-Dyatlov, 2017).

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Generalized Lagrangian mean theories are used to analyze the interactions between mean flows and fluctuations, where the decomposition is based on a Lagrangian description of the flow. A systematic geometric framework was recently developed by Gilbert and Vanneste (J. Fluid Mech., 2018) who cast the decomposition in terms of intrinsic operations on the group of volume preserving diffeomorphism or on the full diffeomorphism group. In this setting, the mean of an ensemble of maps can be defined as the Riemannian center of mass on either of these groups. We apply this decomposition in the context of Lagrangian averaging where equations of motion for the mean flow arise via a variational principle from a mean Lagrangian, obtained from the kinetic energy Lagrangian of ideal fluid flow via a small amplitude expansion for the fluctuations. We show that the Euler-$\alpha$ equations arise as Lagrangian averaged Euler equations when using the $L^2$-geodesic mean on the volume preserving diffeomo

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We consider several basic questions pertaining to the geometry of image of a general quadratic map. In general the image of a quadratic map is non-convex, although there are several known classes of quadratic maps when the image is convex. Remarkably, even when the image is not convex it often exhibits hidden convexity: a surprising efficiency of convex relaxation to address various geometric questions by reformulating them in terms of convex optimization problems. In this paper we employ this strategy and put forward several algorithms that solve the following problems pertaining to the image: verify if a given point does not belong to the image; find the boundary point of the image lying in a particular direction; stochastically check if the image is convex, and if it is not, find a maximal convex subset of the image. Proposed algorithms are implemented in the form of an open-source MATLAB library CAQM, which accompanies the paper. Our results can be used for various problems of disc

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In this paper, we investigate a class of non-monotone reaction-diffusion equations with distributed delay and a homogenous boundary Neumann condition, which have a positive steady state. The main concern is the global attractivity of the unique positive steady state. To achieve this, we use an argument of a sub and super-solution combined with fluctuation method. We also give a condition for which the exponential stability of the positive steady state is reached. As an example, we apply our results to diffusive Nicholson blowflies and diffusive Mackey-Glass equation with distributed delay. We point out that we obtain some new results on exponential stability of the positive steady state for these cited models.

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We study strong indispensability of minimal free resolutions of semigroup rings. We focus on two operations, gluing and extending, used in literature to produce more examples with a special property from the existing ones. We give a naive condition to determine whether gluing of two semigroup rings has a strongly indispensable minimal free resolution. As applications, we determine extensions of $3$-generated non-symmetric, $4$-generated symmetric and pseudo symmetric numerical semigroups as well as obtain infinitely many complete intersection semigroups of any embedding dimension, having strongly indispensable minimal free resolutions.

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We study the entropy production of the sandwiched R\'enyi divergence under a primitive Lindblad equation with GNS-detailed balance. We prove that the Lindblad equation can be identified as the gradient flow of the sandwiched R\'enyi divergence for any order $\alpha\in (0,\infty)$. This extends a previous result by Carlen and Maas [Journal of Functional Analysis, 273(5), 1810--1869] for the quantum relative entropy (i.e., $\alpha=1$). Moreover, we show that the sandwiched R\'enyi divergence with order $\alpha\in (0,\infty)$ decays exponentially fast under the time-evolution of such a Lindblad equation.

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A story about a Kubernetes migration

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Microsoft takes a step back into robotics by bringing the Robot Operating System to Windows

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Naked mole-rat mortality rates defy Gompertzian laws by not increasing with age [pdf]

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A Penthouse Made for Instagram

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Cloudflare, which is celebrating its eighth birthday has announced yet another service: an at-cost domain registrar. From a report: While Cloudflare had already been handling domain registration through the company's Enterprise Registrar service, that service was intended for some of Cloudflare's high-end customers who wanted extra levels of security for their domain names. The new domain registrar business -- called Cloudflare Registrar -- will eventually be open to anyone, and it will charge exactly what it costs for Cloudflare to register a domain. As Cloudflare CEO Matthew Prince wrote in a blog post this week, "We promise to never charge you anything more than the wholesale price each TLD charges." That includes the small fee assessed by ICANN for each registration. Prince said that he was motivated to take the company into the registrar business because of Cloudflare's own experience with registrars and by the perception that many registrars are in the business mostly to up-sell

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A cache invalidation bug in Linux memory management

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An Introduction to Probabilistic Programming

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Berkeley Lab Building Own Open Architecture Quantum Chips

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Utilities will earn credits for electric car charging stations and subsidize EV purchases.

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US government will sue California—Ajit Pai called state rules "illegal."

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California governor signs nation’s strictest net neutrality rules into law

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Cellular automata as convolutional neural networks

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Companies headquartered in California can no longer have all-male boards

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Charles Bethea writes on the flooding of hog-waste lagoons in North Carolina following Hurricane Florence, and Smithfield Foods’ inability to curtail the damage from them.

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The FBI has solved the final mystery surrounding a strain of Mac malware that was used by an Ohio man to spy on people for 14 years. From a report: The man, 28-year-old Phillip Durachinsky, was arrested in January 2017, and charged a year later, in January 2018. US authorities say he created the Fruitfly Mac malware (Quimitchin by some AV vendors) back in 2003 and used it until 2017 to infect victims and take control off their Mac computers to steal files, keyboard strokes, watch victims via the webcam, and listen in on conversations via the microphone. Court documents reveal Durachinsky wasn't particularly interested in financial crime but was primarily focused on watching victims, having collected millions of images on his computer, including many of underage children. Durachinsky created the malware when he was only 14, and used it for the next 14 years without Mac antivirus programs ever detecting it on victims' computers. [...] Describing the Fruitfly/Quimitchin malware, the FBI s

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Flexport (YC W14) is hiring engineers and hosting a tech talk at sea

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How companies use fake sites, backdated articles to censor Google results (2017)

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Emily Witt writes on the Washington, D.C., protests against Brett Kavanaugh’s Supreme Court nomination, on Friday, following the Senate Judiciary Committee hearing with Kavanaugh and Christine Blasey Ford the day before.

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Inside Wayback Machine, the internet’s time capsule

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A bone-marrow transplant treated a patient’s leukemia – and his schizophrenia

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Ask HN: What are the best textbooks in your field of expertise?

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Browsable History of Philosophy

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How to think in Turkic languages

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In this paper, we present a collocation method for nonlinear Volterra integral equation of the first kind. This method benefits from the idea of $hp$-version projection methods. We provide an approximation based on the Legendre polynomial interpolation. The convergence of the proposed method is completely studied and an error estimate under the $L^2$-norm is provided. Finally, several numerical experiments are presented in order to verify the obtained theoretical results.

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We study $F$-signature under proper birational morphisms $\pi : Y \to X$, showing that $F$-signature strictly increases for small morphisms or if $ K_Y \geq \pi ^*K_X$. In certain cases, we can even show that the $F$-signature of $Y$ is at least twice as that of $X$. We also provide examples of $F$-signature dropping and Hilbert-Kunz multiplicity increasing under birational maps without these hypotheses.

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We introduce a method to construct $G_2$-instantons over compact $G_2$-manifolds arising as the twisted connected sum of a matching pair of building blocks [Kov03,KL11,CHNP12]. Our construction is based on gluing $G_2$-instantons obtained from holomorphic bundles over the building blocks via the first named author's work [SE11]. We require natural compatibility and transversality conditions which can be interpreted in terms of certain Lagrangian subspaces of a moduli space of stable bundles on a K3 surface.

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Using earlier work of S\'a Earp and the author [SEW13] we construct an irreducible unobstructed $G_2$-instanton on an $\mathrm{SO}(3)$-bundle over a twisted connected sum recently discovered by Crowley-Nordstr\"om [CN14].

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We give a sufficient condition for an associative submanifold in a G2-manifold to appear as the bubbling locus of a sequence of G2-instantons, related to the existence of a Fueter section of a bundle of ASD instanton moduli spaces over said submanifold.

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This paper deals with the $L_p$-spectrum of Schr\"odinger operators on the hyperbolic plane. We establish Lieb-Thirring type inequalities for discrete eigenvalues and study their dependence on $p$. Some bounds on individual eigenvalues are derived as well.

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Generalizing the ideas of $\mathbb{Z}_k$-manifolds from Sullivan and the theory of stratifolds from Kreck, we define $\mathbb{Z}_k$-stratifolds. We introduce the bordism theory of $\mathbb{Z}_k$-stratifolds in order to represent every homology class with $\mathbb{Z}_k$-coefficients. We present a geometric interpretation of the Bockstein long exact sequence and the Atiyah-Hirzebruch spectral sequence for $\mathbb{Z}_k$-bordism.

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We define a 3-loop group $\Omega^3G$ as a subgroup of smooth maps from a 3-ball to a Lie group $G$, and then construct a 2-group based on an automorphic action on the Mickelsson-Faddeev extension of $\Omega^3G$. In this we follow the strategy of Murray et al., who earlier described a similar construction in one dimension. The three-dimensional situation presented here is further complicated by the fact that the 3-loop group extension is not central.

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