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We study the worstcase convergence rates of the proximal gradient method for minimizing the sum of a smooth strongly convex function and a nonsmooth convex function whose proximal operator is available. We establish the exact worstcase convergence rates of the proximal gradient method in this setting for any step size and for different standard performance measures: objective function accuracy, distance to optimality and residual gradient norm. The proof methodology relies on recent developments in performance estimation of firstorder methods based on semidefinite programming. In the case of the proximal gradient method, this methodology allows obtaining exact and nonasymptotic worstcase guarantees that are conceptually very simple, although apparently new. On the way, we discuss how strong convexity can be replaced by weaker assumptions, while preserving the corresponding convergence rates. We also establish that the same fixed step size policy is optimal for all three performan
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This article describes the geometry of isomorphisms between complements of geometrically irreducible closed curves in the affine plane $\mathbb{A}^2$, over an arbitrary field, which do not extend to an automorphism of $\mathbb{A}^2$. We show that such isomorphisms are quite exceptional. In particular, they occur only when both curves are isomorphic to open subsets of the affine line $\mathbb{A}^1$, with the same number of complement points, over any field extension of the ground field. Moreover, the isomorphism is uniquely determined by one of the curves, up to left composition with an automorphism of $\mathbb{A}^2$, except in the case where the curve is isomorphic to the affine line $\mathbb{A}^1$ or to the punctured line $\mathbb{A}^1 \setminus \{0\}$. If one curve is isomorphic to $\mathbb{A}^1$, then both curves are in fact equivalent to lines. In addition, for any positive integer $n$, we construct a sequence of $n$ pairwise nonequivalent closed embeddings of $\mathbb{A}^1 \setmi
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We consider finite model approximations of discretetime partially observed Markov decision processes (POMDPs) under the discounted cost criterion. After converting the original partially observed stochastic control problem to a fully observed one on the belief space, the finite models are obtained through the uniform quantization of the state and action spaces of the belief space Markov decision process (MDP). Under mild assumptions on the components of the original model, it is established that the policies obtained from these finite models are nearly optimal for the belief space MDP, and so, for the original partially observed problem. The assumptions essentially require that the belief space MDP satisfies a mild weak continuity condition. We provide examples and introduce explicit approximation procedures for the quantization of the set of probability measures on the state space of POMDP (i.e., belief space).
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Over a field of prime characteristic $p>2$, we prove that the cohomology rings of some pointed Hopf algebras of dimension $p^3$ are finitely generated. These are Hopf algebras arising in the ongoing classification of finite dimensional Hopf algebras in positive characteristic, and include bosonizations of Nichols algebras of Jordan type in a general setting as well as their liftings when $p=3$. Our techniques are applications of twisted tensor product resolutions and Anick resolutions in combination with May spectral sequences.
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Starting from measured data, we develop a method to compute the fine structure of the spectrum of the Koopman operator with rigorous convergence guarantees. The method is based on the observation that, in the measurepreserving ergodic setting, the moments of the spectral measure associated to a given observable are computable from a single trajectory of this observable. Having finitely many moments available, we use the classical ChristoffelDarboux kernel to separate the atomic and absolutely continuous parts of the spectrum, supported by convergence guarantees as the number of moments tends to infinity. In addition, we propose a technique to detect the singular continuous part of the spectrum as well as two methods to approximate the spectral measure with guaranteed convergence in the weak topology, irrespective of whether the singular continuous part is present or not. The proposed method is simple to implement and readily applicable to largescale systems since the computational c
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An anonymous reader quotes a report from The Verge: Google's AI subsidiary DeepMind has unveiled the latest version of its Goplaying software, AlphaGo Zero. The new program is a significantly better player than the version that beat the game's world champion earlier this year, but, more importantly, it's also entirely selftaught. DeepMind says this means the company is one step closer to creating general purpose algorithms that can intelligently tackle some of the hardest problems in science, from designing new drugs to more accurately modeling the effects of climate change. The original AlphaGo demonstrated superhuman Goplaying ability, but needed the expertise of human players to get there. Namely, it used a dataset of more than 100,000 Go games as a starting point for its own knowledge. AlphaGo Zero, by comparison, has only been programmed with the basic rules of Go. Everything else it learned from scratch. As described in a paper published in Nature today, Zero developed its Go
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Cyclic sieving is a wellknown phenomenon where certain interesting polynomials, especially $q$analogues, have useful interpretations related to actions and representations of the cyclic group. We define sieving for an arbitrary group $G$ and study it for the dihedral group $I_2(n)$ of order $2n$. This requires understanding the generators of the representation ring of the dihedral group. For $n$ odd, we exhibit several instances of "dihedral sieving" which involve the generalized Fibonomial coefficients, recently studied by Amdeberhan, Chen, Moll, and Sagan.
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We introduce the local and global indices of Dirac operators for the rational Cherednik algebra $\mathsf{H}_{t,c}(G,\mathfrak{h})$, where $G$ is a complex reflection group acting on a finitedimensional vector space $\mathfrak{h}$. We investigate precise relations between the (local) Dirac index of a simple module in the category $\mathcal{O}$ of $\mathsf{H}_{t,c}(G,\mathfrak{h})$, the graded $G$character of the module, the EulerPoincar\'e pairing, and the composition series polynomials for standard modules. In the global theory, we introduce integralreflection modules for $\mathsf{H}_{t,c}(G,\mathfrak{h})$ constructed from finitedimensional $G$modules. We define and compute the index of a Dirac operator on the integralreflection module and show that the index is, in a sense, independent of the parameter function $c$. The study of the kernel of these global Dirac operators leads naturally to a notion of dualised generalised DunklOpdam operators.
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This paper investigates distributed optimization of dynamically coupled networks. We propose distributed algorithms to address two complementary cases: linear systems with controllable subsystems and nonlinear systems in the strictfeedback form. The proposed controllers are verified by two case studies on an optimal power flow problem and a multizone building temperature regulation problem, respectively.
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We find the number of compositions over finite abelian groups under two types of restrictions: (i) each part belongs to a given subset and (ii) small runs of consecutive parts must have given properties. Waring's problem over finite fields can be converted to type~(i) compositions, whereas Carlitz and locally Mullen compositions can be formulated as type~(ii) compositions. We use the multisection formula to translate the problem from integers to group elements, the transfer matrix method to do exact counting, and finally the PerronFrobenius theorem to derive asymptotics. We also exhibit bijections involving certain restricted classes of compositions.
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We consider the problem of minimizing a convex function over the intersection of finitely many simple sets which are easy to project onto. This is an important problem arising in various domains such as machine learning. The main difficulty lies in finding the projection of a point in the intersection of many sets. Existing approaches yield an infeasible point with an iterationcomplexity of $O(1/\varepsilon^2)$ for nonsmooth problems with no guarantees on the infeasibility. By reformulating the problem through exact penalty functions, we derive firstorder algorithms which not only guarantees that the distance to the intersection is small but also improve the complexity to $O(1/\varepsilon)$ and $O(1/\sqrt{\varepsilon})$ for smooth functions. For composite and smooth problems, this is achieved through a saddlepoint reformulation where the proximal operators required by the primaldual algorithms can be computed in closed form. We illustrate the benefits of our approach on a graph tr
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We study a nonlinear, nonhomogeneous elliptic equation with an asymmetric reaction term depending on a positive parameter, coupled with Robin boundary conditions. Under appropriate hypotheses on both the leading differential operator and the reaction, we prove that, if the parameter is small enough, the problem admits at least four nontrivial solutions: two of such solutions are positive, one is negative, and one is signchanging. Our approach is variational, based on critical point theory, Morse theory, and truncation techniques.
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Kimura and Yoshida treated a model in which the finite variation part of a twodimensional semimartingale is expressed by timeintegration of latent processes. They proposed a correlation estimator between the latent processes and proved its consistency and asymptotic mixed normality. In this paper, we discuss the confidence interval of the correlation estimator to detect the correlation. %between latent processes. We propose two types of estimators for asymptotic variance of the correlation estimator and prove their consistency in a high frequency setting. Our model includes doubly stochastic Poisson processes whose intensity processes are correlated It\^o processes. We compare our estimators based on the simulation of the doubly stochastic Poisson processes.
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We generalise some wellknown graph parameters to operator systems by considering their underlying quantum channels. In particular, we introduce the quantum complexity as the dimension of the smallest codomain Hilbert space a quantum channel requires to realise a given operator system as its noncommutative confusability graph. We describe quantum complexity as a generalised minimum semidefinite rank and, in the case of a graph operator system, as a quantum intersection number. The quantum complexity and a closely related quantum version of orthogonal rank turn out to be upper bounds for the Shannon zeroerror capacity of a quantum channel, and we construct examples for which these bounds beat the best previously known general upper bound for the capacity of quantum channels, given by the quantum Lov\'asz theta number.
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We consider the problem of computing the Fourier transform of highdimensional vectors, distributedly over a cluster of machines consisting of a master node and multiple worker nodes, where the worker nodes can only store and process a fraction of the inputs. We show that by exploiting the algebraic structure of the Fourier transform operation and leveraging concepts from coding theory, one can efficiently deal with the straggler effects. In particular, we propose a computation strategy, named as coded FFT, which achieves the optimal recovery threshold, defined as the minimum number of workers that the master node needs to wait for in order to compute the output. This is the first code that achieves the optimum robustness in terms of tolerating stragglers or failures for computing Fourier transforms. Furthermore, the reconstruction process for coded FFT can be mapped to MDS decoding, which can be solved efficiently. Moreover, we extend coded FFT to settings including computing general
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An anonymous reader quotes a report from The Guardian: The abundance of flying insects has plunged by threequarters over the past 25 years, according to a new study that has shocked scientists. Insects are an integral part of life on Earth as both pollinators and prey for other wildlife and it was known that some species such as butterflies were declining. But the newly revealed scale of the losses to all insects has prompted warnings that the world is "on course for ecological Armageddon," with profound impacts on human society. The new data was gathered in nature reserves across Germany but has implications for all landscapes dominated by agriculture, the researchers said. The cause of the huge decline is as yet unclear, although the destruction of wild areas and widespread use of pesticides are the most likely factors and climate change may play a role. The scientists were able to rule out weather and changes to landscape in the reserves as causes, but data on pesticide levels has
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Over the moduli space of smooth curves, the double ramification cycle can be defined by pulling back the unit section of the universal jacobian along the AbelJacobi map. This breaks down over the boundary since the AbelJacobi map fails to extend. We construct a `universal' resolution of the AbelJacobi map, and thereby extend the double ramification cycle to the whole of the moduli of stable curves. In the nontwisted case, we show that our extension coincides with the cycle constructed by Li, Graber, Vakil via a virtual fundamental class on a space of rubber maps.
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Financial models are studied where each asset may potentially lose value relative to any other. Conditioning on nondevaluation, each asset can serve as proper num\'eraire and classical valuation rules can be formulated. It is shown when and how these local valuation rules can be aggregated to obtain global arbitragefree valuation formulas.
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The aim of this paper is to give fine asymptotics for random variables with moments of Gamma type. Among the examples we consider are random determinants of Laguerre and Jacobi beta ensembles with varying dimensions (the number of observed variables and the number of measurements vary and may be different). In addition to the Dyson threefold way of classical random matrix models (GOE, GUE, GSE), we study random determinants of random matrices of the socalled tenfold way, including the Bogoliubovde Gennes and chiral ensembles from mesoscopic physics. We show that fixedtrace matrix ensembles can be analysed as well. Finally, we add fine asymptotics for the $p(n)$dimensional volume of the simplex with $p(n)+1$ points in ${\Bbb R}^n$ distributed according to special distributions, which is strongly correlated to Gram matrix ensembles. We use the framework of mod$\varphi$ convergence to obtain extended limit theorems, BerryEsseen bounds, precise moderate deviations, large and moderate
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The deficiency of a group is the maximum over all presentations for that group of the number of generators minus the number of relators. Every finite group has nonpositive deficiency. We show that every nonpositive integer is the deficiency of a finite group  in fact, of a finite $p$group for every prime $p$. This completes Kotschick's classification of the integers which are deficiencies of fundamental groups of compact Kaehler manifolds.
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The SemiAlgebraic framework for the approximate Canonical Polyadic (CP) decomposition via SImultaneaous matrix diagonalization (SECSI) is an efficient tool for the computation of the CP decomposition. The SECSI framework reformulates the CP decomposition into a set of joint eigenvalue decomposition (JEVD) problems. Solving all JEVDs, we obtain multiple estimates of the factor matrices and the best estimate is chosen in a subsequent step by using an exhaustive search or some heuristic strategy that reduces the computational complexity. Moreover, the SECSI framework retains the option of choosing the number of JEVDs to be solved, thus providing an adjustable complexityaccuracy tradeoff. In this work, we provide an analytical performance analysis of the SECSI framework for the computation of the approximate CP decomposition of a noise corrupted lowrank tensor, where we derive closedform expressions of the relative mean square error for each of the estimated factor matrices. These exp
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This paper concerns singular, complex projective, irreducible symplectic varieties. The generalized BeauvilleBogomolov form satisfies the Fujiki relations and has the same rank as in the smooth case. As a consequence, we see that these varieties admit only very special fibrations. Our results give new evidence that the correct definition of singular analogs of irreducible symplectic manifolds is the one from GrebKebekusPeternell.
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Motivated mainly by the localization over an open bounded set $\Omega$ of $\mathbb R^n$ of solutions of the Schr\"odinger equations, we consider the Schr\"odinger equation over $\Omega$ with a very singular potential $V(x) \ge C d (x, \partial \Omega)^{r}$ with $r\ge 2$ and a convective flow $\vec U$. We prove the existence and uniqueness of a very weak solution of the equation, when the right hand side datum $f(x)$ is in $L^1 (\Omega, d(\cdot, \partial \Omega))$, even if no boundary condition is a priori prescribed. We prove that, in fact, the solution necessarily satisfies (in a suitable way) the Dirichlet condition $u = 0$ on $\partial \Omega$. These results improve some of the results of the previous paper by the authors in collaboration with Roger Temam. In addition, we prove some new results dealing with the $m$accretivity in $L^1 (\Omega, d(\cdot, \partial \Omega)^ \alpha)$, where $\alpha \in [0,1]$, of the associated operator, the corresponding parabolic problem and the study
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Estimator selection has become a crucial issue in non parametric estimation. Two widely used methods are penalized empirical risk minimization (such as penalized loglikelihood estimation) or pairwise comparison (such as Lepski's method). Our aim in this paper is twofold. First we explain some general ideas about the calibration issue of estimator selection methods. We review some known results, putting the emphasis on the concept of minimal penalty which is helpful to design datadriven selection criteria. Secondly we present a new method for bandwidth selection within the framework of kernel density density estimation which is in some sense intermediate between these two main methods mentioned above. We provide some theoretical results which lead to some fully datadriven selection strategy.
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We consider random Schr\"odinger operators with Dirichlet boundary conditions outside lattice approximations of a smooth Euclidean domain and study the behavior of its lowestlying eigenvalues in the limit when the lattice spacing tends to zero. Under a suitable moment assumption on the random potential and regularity of the spatial dependence of its mean, we prove that the eigenvalues of the random operator converge to those of a deterministic Schr\"odinger operator. Assuming also regularity of the variance, the fluctuation of the random eigenvalues around their mean are shown to obey a multivariate central limit theorem. This extends the authors' recent work where similar conclusions have been obtained for bounded random potentials.
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We consider hyperelastic problems and their numerical solution using a conforming finite element discretization and iterative linearization algorithms. For these problems, we present equilibrated, weakly symmetric, $H(\rm{div)}$conforming stress tensor reconstructions, obtained from local problems on patches around vertices using the ArnoldFalkWinther finite element spaces. We distinguish two stress reconstructions, one for the discrete stress and one representing the linearization error. The reconstructions are independent of the mechanical behavior law. Based on these stress tensor reconstructions, we derive an a posteriori error estimate distinguishing the discretization, linearization, and quadrature error estimates, and propose an adaptive algorithm balancing these different error sources. We prove the efficiency of the estimate, and confirm it on a numerical test with analytical solution for the linear elasticity problem. We then apply the adaptive algorithm to a more applic
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The entropy is one of the fundamental states of a fluid and, in the viscous case, the equation that it satisfies is highly singular in the region close to the vacuum. In spite of its importance in the gas dynamics, the mathematical analyses on the behavior of the entropy near the vacuum region, were rarely carried out; in particular, in the presence of vacuum, either at the far field or at some isolated interior points, it was unknown if the entropy remains its boundedness. The results obtained in this paper indicate that the ideal gases retain their uniform boundedness of the entropy, locally or globally in time, if the vacuum occurs at the far field only and the density decays slowly enough at the far field. Precisely, we consider the Cauchy problem to the onedimensional full compressible NavierStokes equations without heat conduction, and establish the local and global existence and uniqueness of entropybounded solutions, in the presence of vacuum at the far field only. It is als
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We address the following conjecture about the existence of common zeros for commuting vector fields in dimension three: let $X,Y$ be two $C^1$ commuting vector fields on a $3$manifold $M$, and $U$ be a relatively compact open set where $Y$ does not vanish, then $X$ has zero Poincar\'eHopf index in $U$. We prove that conjecture when $X$ and $Y$ are of class $C^3$ and every periodic orbit of $Y$ along which $X$ and $Y$ are colinear is partially hyperbolic. We also prove the conjecture, still in the $C^3$ setting, and assuming that the flow $Y$ leaves invariant a transverse plane field. These results shed new light on the $C^3$ case of the conjecture and indeed we discuss a global strategy to attack this problem.
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We investigate the initial value problem for a defocusing nonlinear Schr\"odinger equation with weighted exponential nonlinearity $$ i\partial_t u+\Delta u=\frac{u}{x^b}(e^{\alphau^2}1); \qquad (t,x) \in \mathbb{R}\times\mathbb{R}^2, $$ where $0< b <1$ and $\alpha=2\pi(2b)$. We establish local and global wellposedness in the subcritical and critical regimes.
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The processes of the averaged regression quantiles and of their modifications provide useful tools in the regression models when the covariates are not fully under our control. As an application we mention the probabilistic risk assessment in the situation when the return depends on some exogenous variables. The processes enable to evaluate the expected $\alpha$shortfall ($0\leq\alpha\leq 1$) and other measures of the risk, recently generally accepted in the financial literature, but also help to measure the risk in environment analysis and elsewhere.
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DragonFly BSD 5.0 has been released. "Preliminary HAMMER2 support has been released into the wild asof the 5.0 release. This support is considered EXPERIMENTAL and should generally not yet be used for production machines and important data. The boot loader will support both UFS and HAMMER2 /boot. The installer will still use a UFS /boot even for a HAMMER2 installation because the /boot partition is typically very small and HAMMER2, like HAMMER1, does not instantly free space when files are deleted or replaced. DragonFly 5.0 has singleimage HAMMER2 support, with live dedup (for cp's), compression, fast recovery, snapshot, and boot support. HAMMER2 does not yet support multivolume or clustering, though commands for it exist. Please use nonclustered single images for now."
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DragonFlyBSD 5.0 is the first release with preliminary boot support for HAMMER2, the project's new filesystem. Preliminary HAMMER2 support has been released into the wild asof the 5.0 release. This support is considered EXPERIMENTAL and should generally not yet be used for production machines and important data. The boot loader will support both UFS and HAMMER2 /boot. The installer will still use a UFS /boot even for a HAMMER2 installation because the /boot partition is typically very small and HAMMER2, like HAMMER1, does not instantly free space when files are deleted or replaced.
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A Palo Alto startup that stopped trying to be an "Android killer" last year after raising $185 million has apparently pivoted to developing autonomous vehicle technology. From a report: The company now known as Cyngn has changed its name from Cyanogen and recently got a permit to test its selfdriving tech on California roads, according to a report Wednesday on Axios. It's being led by Lior Tal, the former chief operating officer who took over as CEO last fall when Kirt McMaster left. The rest of the startup's current team of about 30 people appear to have joined since the strategy shift, Axios reported, citing LinkedIn records. Some of them are former Facebook people, like Tal, and alumni of automakers who include MercedesBenz. No new funding has been disclosed for the reinvented company. It lists on its website investors who backed it before it pivoted, including Andreessen Horowitz, Benchmark Capital, Redpoint Ventures, Index Ventures, Qualcomm and Chinese social networking company
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The California Department of Motor Vehicles is changing its rules to allow companies to test autonomous vehicles without a driver behind the wheel  and to let the public use autonomous vehicles. From a report: The DMV released a revised version of its regulations and has started a 15day public comment period, ending October 25, 2017. California law requires the DMV to work on regulations to cover testing and public use of autonomous vehicles, and the regulator said that this is the first step. "We are excited to take the next step in furthering the development of this potentially lifesaving technology in California," the state's Transportation Secretary, Brian Kelly, said in a statement. California's DMV took pains in its announcement to highlight that it wasn't trying to overstep the National Highway Traffic Safety Administration, which has the final say on developing and enforcing compliance with Federal Motor Vehicle Safety Standards. Rather, the California regulations, are goin
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Jon Brodkin reports via Ars Technica: A Federal Communications Commission decision to eliminate price caps imposed on some business broadband providers should be struck down, advocacy groups told federal judges last week. The FCC failed to justify its claim that a market can be competitive even when there is only one Internet provider, the groups said. Led by Chairman Ajit Pai, the FCC's Republican majority voted in April of this year to eliminate price caps in a county if 50 percent of potential customers "are within a half mile of a location served by a competitive provider." That means business customers with just one choice are often considered to be located in a competitive market and thus no longer benefit from price controls. The decision affects Business Data Services (BDS), a dedicated, pointtopoint broadband link that is delivered over copperbased TDM networks by incumbent phone companies like AT&T, Verizon, and CenturyLink. But the FCC's claim that "potential competit
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Facebook is going to ship a standalone VR headset called Oculus Go next year. The headset, which won't require a PC or phone to run, will be available early next year for $199. From a report: Facebook CEO Mark Zuckerberg officially announced the new product during his keynote speech at Facebook's fourth Oculus Connect virtual reality (VR) developer conference in San Jose, Calif. Wednesday, where he framed the device as an important step towards bringing VR to the masses. "We want to get a billion people in virtual reality," Zuckerberg said. Facebook VP of VR Hugo Barra said that the company developed custom lenses for the headset, which allow for a wide field of view. The display is a fastswitch LCD screen with a resolution of 2560x1440 pixels, and it comes with integrated headphones. The company will be shipping first headsets to developers in November.
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An anonymous reader quotes a report from Engadget: 10.9 million U.S. driver's licenses were stolen in the massive breach that Equifax suffered in midMay, according to a new report by The Wall Street Journal. In addition, WSJ has revealed that the attackers got a hold of 15.2 million UK customers' records, though only 693,665 among them had enough info in the system for the breach to be a real threat to their privacy. Affected customers provided most of the driver's licenses on file to verify their identities when they disputed their creditreport information through an Equifax web page. That page was one of the entry points the attackers used to gain entry into the credit reporting agency's system.
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We prove neartight concentration of measure for polynomial functions of the Ising model under high temperature. For any degree $d$, we show that a degree$d$ polynomial of a $n$spin Ising model exhibits exponential tails that scale as $\exp(r^{2/d})$ at radius $r=\tilde{\Omega}_d(n^{d/2})$. Our concentration radius is optimal up to logarithmic factors for constant $d$, improving known results by polynomial factors in the number of spins. We demonstrate the efficacy of polynomial functions as statistics for testing the strength of interactions in social networks in both synthetic and real world data.
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By Langlands and Deligne we know that the local constants are extendible functions. Therefore, to give an explicit formula of the local constant of an induced representation of a local Galois group of a nonArchimedean local field $F$ of characteristic zero, we have to compute the lambda function $\lambda_{K/F}$ for a finite extension $K/F$. In this paper, when a finite extension $K/F$ is Galois, we give a formula for $\lambda_{K/F}$.
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Laman graphs model planar frameworks which are rigid for a general choice of distances between the vertices. There are finitely many ways, up to isometries, to realize a Laman graph in the plane. In a recent paper we provide a recursion formula for this number of realizations using ideas from algebraic and tropical geometry. Here, we present a concise summary of this result focusing on the main ideas and the combinatorial point of view.
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This paper gives a classification of all pairs $(\mathfrak g, \mathfrak h)$ with $\mathfrak g$ a simple real Lie algebra and $\mathfrak h < \mathfrak g$ a reductive subalgebra for which there exists a minimal parabolic subalgebra $\mathfrak p < \mathfrak g$ such that $\mathfrak g = \mathfrak h + \mathfrak p$ as vector sum.
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Geometric aspects play an important role in the construction and analysis of structurepreserving numerical methods for a wide variety of ordinary and partial differential equations. Here we review the development and theory of symplectic integrators for Hamiltonian ordinary and partial differential equations, of dynamical lowrank approximation of timedependent large matrices and tensors, and its use in numerical integrators for Hamiltonian tensor network approximations in quantum dynamics.
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We prove that for a strongly pseudoconvex domain $D\subset\mathbb C^n$, the infinitesimal Carath\'{e}odory metric $g_C(z,v)$ and the infinitesimal Kobayashi metric $g_K(z,v)$ coincide if $z$ is sufficiently close to $bD$ and if $v$ is sufficiently close to being tangential to $bD$. Also, we show that every two close points of $D$ sufficiently close to the boundary and whose difference is almost tangential to $bD$ can be joined by a (unique up to reparameterization) complex geodesic of $D$ which is also a holomorphic retract of $D$. The same continues to hold if $D$ is a worm domain, as long as the points are sufficiently close to a strongly pseudoconvex boundary point. We also show that a strongly pseudoconvex boundary point of a worm domain can be globally exposed; this has consequences for the behavior of the squeezing function.
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Inspired by code vertex operator algebras (VOAs) and their representation theory, we define code algebras, a new class of commutative nonassociative algebras constructed from binary linear codes. Let $C$ be a binary linear code of length $n$. A basis for the code algebra $A_C$ consists of $n$ idempotents and a vector for each nonconstant codeword of $C$. We show that code algebras are almost always simple and, under mild conditions on their structure constants, admit an associating bilinear form. We determine the Pierce decomposition and the fusion rules for the idempotents in the basis, and we give a construction to find additional idempotents, called the $s$map, which comes from the code structure. For a general code algebra, we classify the eigenvalues and eigenvectors of the smallest examples of the $s$map construction, and hence show that certain code algebras are axial algebras. We give some examples, including that for a Hamming code $H_8$ where the code algebra $A_{H_8}$ is
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The correspondence between Gorenstein Fano toric varieties and reflexive polytopes has been generalized by Ilten and S\"u{\ss} to a correspondence between Gorenstein Fano complexityone $T$varieties and Fano divisorial polytopes. Motivated by the finiteness of reflexive polytopes in fixed dimension, we show that over a fixed base polytope, there are only finitely many Fano divisorial polytopes, up to equivalence. We classify twodimensional Fano divisorial polytopes, recovering Huggenberger's classification of Gorenstein del Pezzo $\mathbb{K}^*$surfaces. Furthermore, we show that any threedimensional Fano divisorial polytope is equivalent to one involving only eight functions.
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Given a Moebius homeomorphism $f : \partial X \to \partial Y$ between boundaries of proper, geodesically complete CAT(1) spaces $X,Y$, we describe an extension $\hat{f} : X \to Y$ of $f$, called the circumcenter map of $f$, which is constructed using circumcenters of expanding sets. The extension $\hat{f}$ is shown to coincide with the $(1, \log 2)$quasiisometric extension constructed in [biswas3], and is locally $1/2$Holder continuous. When $X,Y$ are complete, simply connected manifolds with sectional curvatures $K$ satisfying $b^2 \leq K \leq 1$ for some $b \geq 1$ then the extension $\hat{f} : X \to Y$ is a $(1, (1  \frac{1}{b})\log 2)$quasiisometry. Circumcenter extension of Moebius maps is natural with respect to composition with isometries.
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