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The relationship between the sparsest cut and the maximum concurrent multiflow in graphs has been studied extensively. For general graphs with $k$ terminal pairs, the flowcut gap is $O(\log k)$, and this is tight. But when topological restrictions are placed on the flow network, the situation is far less clear. In particular, it has been conjectured that the flowcut gap in planar networks is $O(1)$, while the known bounds place the gap somewhere between $2$ (Lee and Raghavendra, 2003) and $O(\sqrt{\log k})$ (Rao, 1999). A seminal result of Okamura and Seymour (1981) shows that when all the terminals of a planar network lie on a single face, the flowcut gap is exactly $1$. This setting can be generalized by considering planar networks where the terminals lie on $\gamma>1$ faces in some fixed planar drawing. Lee and Sidiropoulos (2009) proved that the flowcut gap is bounded by a function of $\gamma$, and Chekuri, Shepherd, and Weibel (2013) showed that the gap is at most $3\gamma
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We give an efficient algorithm that, given a graph $G$ and a partition $V_1,\ldots,V_m$ of its vertex set, finds either an independent transversal (an independent set $\{v_1,\ldots,v_m\}$ in $G$ such that $v_i\in V_i$ for each $i$), or a subset $\mathcal B$ of vertex classes such that the subgraph of $G$ induced by $\bigcup\mathcal B$ has a small dominating set. A nonalgorithmic proof of this result has been known for a number of years and has been applied to solve many other problems. Thus we are able to give algorithmic versions of many of these applications, a few of which we describe explicitly here.
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In this paper, we present finite element approximations of a class of Generalized random fields defined over a bounded domain of R d or a smooth ddimensional Riemannian manifold (d $\ge$ 1). An explicit expression for the covariance matrix of the weights of the finite element representation of these fields is provided and an analysis of the approximation error is carried out. Finally, a method to generate simulations of these weights while limiting computational and storage costs is presented.
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harrymcc writes: The better AI gets at teaching itself to perform tasks in ways beyond the skills of mere humans, the more likely it is that it may unwittingly behave in ways a human would consider unethical. To explore ways to prevent this from happening, IBM researchers taught AI to play PacMan without ever gobbling up the ghosts. And it did so without ever explicitly telling the software that this was the goal. Over at Fast Company, I wrote about this project and what IBM learned from conducting it. The researchers built a piece of software that could balance the AI's ratio of selfdevised, aggressive game play to humaninfluenced ghost avoidance, and tried different settings to see how they affected its overall approach to the game. By doing so, they found a tipping point  the setting at which PacMan went from seriously chowing down on ghosts to largely avoiding them.
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What Google did not make public was that an employee had accused Mr. Rubin of sexual misconduct. The woman, with whom Mr. Rubin had been having an extramarital relationship, said he coerced her into performing oral sex in a hotel room in 2013, according to two company executives with knowledge of the episode. Google investigated and concluded her claim was credible, said the people, who spoke on the condition that they not be named, citing confidentiality agreements. Mr. Rubin was notified, they said, and Mr. Page asked for his resignation. Google could have fired Mr. Rubin and paid him little to nothing on the way out. Instead, the company handed him a $90 million exit package, paid in installments of about $2 million a month for four years, said two people with knowledge of the terms. The last payment is scheduled for next month. Mr. Rubin was one of three executives that Google protected over the past decade after they were accused of sexual misconduct. In two instances, it
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We extend to arbitrary commutative base rings a recent result of Demeneghi that every ideal of an ample groupoid algebra over a field is an intersection of kernels of induced representations from isotropy groups, with a much shorter proof, by using the author's Disintegration Theorem for groupoid representations. We also prove that every primitive ideal is the kernel of an induced representation from an isotropy group; however, we are unable to show, in general, that it is the kernel of an irreducible induced representation. If each isotropy group is finite (e.g., if the groupoid is principal) and if the base ring is Artinian (e.g., a field), then we can show that every primitive ideal is the kernel of an irreducible representation induced from isotropy.
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High dimensional error covariance matrices are used to weight the contribution of observation and background terms in data assimilation procedures. As error covariance matrices are often obtained by sampling methods, the resulting matrices are often degenerate or illconditioned, making them too expensive to use in practice. In order to combat these problems, reconditioning methods are used. In this paper we present new theory for two existing methods that can be used to reduce the condition number of (or 'recondition') any covariance matrix: ridge regression, and the minimum eigenvalue method. These methods are used in practice at numerical weather prediction centres, but their theoretical impact on the covariance matrix itself is not well understood. Here we address this by investigating the impact of reconditioning on variances and covariances of a general covariance matrix in both a theoretical and practical setting. Improved theoretical understanding provides guidance to users wit
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In this paper, we define a class of slice mappings of several Clifford variables, and the corresponding slice regular mappings. Furthermore, we establish the growth theorem for slice regular starlike or convex mappings on the unit ball of several slice Clifford variables, as well as on the bounded slice domain which is slice starlike and slice circular.
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Let $(X,H)$ be a polarized K3 surface with $\mathrm{Pic}(X) = \mathbb Z H$, and let $C\in H$ be a smooth curve of genus $g$. We give an upper bound on the dimension of global sections of a semistable vector bundle on $C$. This allows us to compute the higher rank Clifford indices of $C$ with high genus. In particular, when $g\geq r^2\geq 4$, the rank $r$ Clifford index of $C$ can be computed by the restriction of LazarsfeldMukai bundles on $X$ corresponding to line bundles on the curve $C$. This is a generalization of the result by Green and Lazarsfeld for curves on K3 surfaces to higher rank vector bundles. We also apply the same method to the projective plane and show that the rank $r$ Clifford index of a degree $d(\geq 5)$ smooth plane curve is $d4$, which is the same as the Clifford index of the curve.
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We consider hamiltonian models representing an arbitrary number of spin $1/2$ fermion quantum fields interacting through arbitrary processes of creation or annihilation of particles. The fields may be massive or massless. The interaction form factors are supposed to satisfy some regularity conditions in both position and momentum space. Without any restriction on the strength of the interaction, we prove that the Hamiltonian identifies to a selfadjoint operator on a tensor product of antisymmetric Fock spaces and we establish the existence of a ground state. Our results rely on new interpolated $N_\tau$ estimates. They apply to models arising from the Fermi theory of weak interactions, with ultraviolet and spatial cutoffs.
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In the last two decades, significant effort has been put in understanding and designing socalled structurepreserving numerical methods for the simulation of mechanical systems. Geometric integrators attempt to preserve the geometry associated to the original system as much as possible, such as the structure of the configuration space, the energy behaviour, preservation of constants of the motion and of constraints or other structures associated to the continuous system (symplecticity, Poisson structure...). In this article, we develop highorder geometric (or pseudovariational) integrators for nonholonomic systems, i.e., mechanical systems subjected to constraint functions which are, roughly speaking, functions on velocities that are not derivable from position constraints. These systems realize rolling or certain kinds of sliding contact and are important for describing different classes of vehicles.
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We introduce a connection between a nearterm quantum computing device, specifically a Gaussian boson sampler, and the graph isomorphism problem. We propose a scheme where graphs are encoded into quantum states of light, whose properties are then probed with photonnumberresolving detectors. We prove that the probabilities of different photondetection events in this setup can be combined to give a complete set of graph invariants. Two graphs are isomorphic if and only if their detection probabilities are equivalent. We present additional ways that the measurement probabilities can be combined or coarsegrained to make experimental tests more amenable. We benchmark these methods with numerical simulations on the Titan supercomputer for several graph families: pairs of isospectral nonisomorphic graphs, isospectral regular graphs, and strongly regular graphs.
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Ars technica takes a look at the Enhanced Tracking Protection (ETP) feature in Firefox 63. "Firefox has long had the ability to block all thirdparty cookies, but this is a crude solution, and many sites will break if all thirdparty cookies are prohibited. The new EPT option works as a more selective block on tracking cookies; thirdparty cookies still work in general, but those that are known to belong to tracking companies are blocked. For the most part, sites will retain their full functionality, just without undermining privacy at the same time. At least for now, however, Mozilla is defaulting this feature to off, so the company can get a better idea of the impact it has on the Web. In testing, the company has found the occasional site that breaks when tracking cookies are blocked. Over the next few months, Firefox developers will get a better picture of just how much breaks, and, if it's not too severe, the plan is to block trackers by default starting in early 2019.
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The Defense Advanced Research Projects Agency (DARPA), a division of the U.S. Department of Defense responsible for the development of emerging technologies, is one of the birthplaces of machine learning, a kind of artificial intelligence (AI) that mimics the behavior of neurons in the brain. Dr. Brian Pierce, director of DARPA's Innovation Office, spoke about the agency's recent efforts at a VentureBeat summit. From the report: One area of study is socalled "common sense" AI  AI that can draw on environmental cues and an understanding of the world to reason like a human. Concretely, DARPA's Machine Common Sense Program seeks to design computational models that mimic core domains of cognition: objects (intuitive physics), places (spatial navigation), and agents (intentional actors). "You could develop a classifier that could identify a number of objects in an image, but if you ask a question, you're not going to get an answer," Pierce said. "We'd like to get away from having an enor
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Like many people, Alex Stamos, former Facebook chief security officer, thinks tech platforms like Facebook and Google have too much power. But he doesn't agree with the calls to break them up. And he argues that the very people who say Facebook and Google are too powerful are giving them more power by insisting they do more to control hate speech and propaganda. From a report: "That's a dangerous path," he warns. If democratic countries make tech firms impose limits on free speech, so will autocratic ones. Before long, the technology will enable "machinespeed, realtime moderation of everything we say online." In attempting to rein in Big Tech, we risk creating Big Brother. So what's the solution? I spoke to Stamos at his Stanford office to find out. Technology Review: So is the disinformation/propaganda problem mostly solved? Stamos: In a free society, you will never eliminate that problem. I think the most important thing [in the US] is the advertising transparency. With or without
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Let $G$ be a finite group and let $H_1,H_2<G$ be two subgroups. In this paper, we are concerned with the bipartite graph whose vertices are $G/H_1\cup G/H_2$ and a coset $g_1H_1$ is connected with another coset $g_2H_2$ if and only if $g_1H_1\cap g_2 H_2\neq\varnothing$. The main result of the paper establishes the existence of such graphs with large girth and large spectral gap. Lubotzky, Manning and Wilton use such graphs to construct certain infinite groups of interest in geometric group theory.
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Brendan Iribe, the cofounder and former CEO of Oculus, announced today that he is leaving Facebook. From a report: Iribe is leaving Facebook following some internal shakeups in the company's virtual reality arm last week that saw the cancellation of the company's next generation "Rift 2" PCpowered virtual reality headset, which he had been leading development of, a source close to the matter told TechCrunch. Iribe and the Facebook executive team had "fundamentally different views on the future of Oculus that grew deeper over time," and Iribe wasn't interested in a "race to the bottom" in terms of performance, we are told.
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According to a press release posted today, Netflix is planning to raise $2 billion to help fund new content, including "content acquisitions, production and development, capital expenditures, investments, working capital and potential acquisitions and strategic transactions." TechCrunch reports: The funds will be raised in the form of senior unsecured notes, denominated in U.S. dollars and euros, it said. This debt offering is the sixth time in under four years that Netflix is raising $1 billion or more through bonds, noted Variety, which was among the first to report the news. As of September 30, Netflix's longterm debt had reached $8.34 billion, up 71% from $4.89 billion in the year ago quarter, it said during its last earnings, Variety's report also noted. Netflix recently explained during its Q3 2018 earnings that it needs to continue to invest in original programming in order to remain competitive. "Content companies such as WarnerMedia and Disney/Fox are moving to selfdistribut
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We study weak solutions of the twodimensional (2D) filtered Euler equations whose vorticity is a finite Radon measure and velocity has locally finite kinetic energy, which is called the vortex sheet solution. The 2D filtered Euler equations are considered as a regularized 2D Euler equations with a spatial filtering and these equations have a unique global weak solution for vortex sheet initial data. On the other hand, the 2D Euler equations require a distinguished sign of initial vorticity for the existence of a global solution with vortex sheet initial data and its uniqueness remains an open question. In this paper, we prove that vortex sheet solutions of the 2D filtered Euler equations converge to those of the 2D Euler equations in the limit of the filtering parameter provided that initial vortex sheet has a distinguished sign. We also show that a simple application of our proof yields the convergence of the vortex method that is a point vortex approximation of vortex sheets. We mak
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It is known that every torsionfree abelian group of finite rank has a maximal completely decomposable summand that is unique up to isomorphism. We show that groups of infinite rank need not have maximal completely decomposable summands, but when they do, this summand is unique up to isomorphism.
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This paper analyzes different models for evaluating investments in Energy Storage Systems (ESS) in power systems with high penetration of Renewable Energy Sources (RES). First of all, two methodologies proposed in the literature are extended to consider ESS investment: a unit commitment model that uses the System States (SS) method of representing time; and another one that uses a representative periods (RP) method. Besides, this paper proposes two new models that improve the previous ones without a significant increase of computation time. The enhanced models are the System States Reduced Frequency Matrix (SSRFM) model which addresses shortterm energy storage more approximately than the SS method to reduce the number of constraints in the problem, and the Representative Periods with Transition Matrix and Cluster Indices (RPTM&CI) model which guarantees some continuity between representative periods, e.g. days, and introduces longterm storage into a model originally designed on
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In this paper we prove that when the geodesic flow of a compact or noncompact complete manifold without conjugate points is of the Anosov type, then the average along of the sectional curvature in planes tangent to the geodesic is negative away from zero for some uniform time. Moreover, in dimension two, if the manifold has no focal points, then the latter condition is sufficient to obtain that the geodesic flow is of Anosov type.
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Ever since Microsoft settled on a cadence of two feature updates a year  one in April, one in October  the quality of its operating system (taking into consideration the volume of bugs that emerge every few days) has deteriorated, writes Peter Bright of ArsTechnica. From the story: The problem with Windows as a Service is quality. Previous issues with the feature and security updates have already shaken confidence in Microsoft's updating policy for Windows 10. While data is notably lacking, there is at the very least a popular perception that the quality of the monthly security updates has taken a dive with Windows 10 and that installation of the twiceannual feature updates as soon as they're available is madness. These complaints are longstanding, too. The unreliable updates have been a cause for concern since shortly after Windows 10's release. The latest problem has brought this to a head, with commentators saying that two feature updates a year is too many and Redmond should
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If it seems as though the app you deleted last week is suddenly popping up everywhere, it may not be mere coincidence. From a report: Companies that cater to app makers have found ways to game both iOS and Android, enabling them to figure out which users have uninstalled a given piece of software lately  and making it easy to pelt the departed with ads aimed at winning them back. Adjust, AppsFlyer, MoEngage, Localytics, and CleverTap are among the companies that offer uninstall trackers, usually as part of a broader set of developer tools. Their customers include TMobile US, Spotify Technology, and Yelp. Critics say they're a fresh reason to reassess online privacy rights and limit what companies can do with user data. "Most tech companies are not giving people nuanced privacy choices, if they give them choices at all," says Jeremy Gillula, tech policy director at the Electronic Frontier Foundation, a privacy advocate. Some providers say these tracking tools are meant to measure use
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This paper is devoted to survey composition algebras and some of their applications. After overviewing the classical algebras of quaternions and octonions, both unital composition algebras (or Hurwitz algebras) and symmetric composition algebras will be dealt with. Their main properties, as well as their classifications, will be reviewed. Algebraic triality, through the use of symmetric composition algebras, will be considered too.
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We consider a programming language based on the lamplighter group that uses only composition and iteration as control structures. We derive generating functions and counting formulas for this language and special subsets of it, establishing lower and upper bounds on the growth rate of semantically distinct programs. Finally, we show how to sample random programs and analyze the distribution of runtimes induced by such sampling.
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This paper deals with a homoskedastic errorsinvariables linear regression model and properties of the total least squares (TLS) estimator. We partly revise the consistency results for the TLS estimator previously obtained by the author [18]. We present complete and comprehensive proofs of consistency theorems. A theoretical foundation for construction of the TLS estimator and its relation to the generalized eigenvalue problem is explained. Particularly, the uniqueness of the estimate is proved. The Frobenius norm in the definition of the estimator can be substituted by the spectral norm, or by any other unitarily invariant norm; then the consistency results are still valid.
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Estimating the number of sources received by an antenna array have been well known and investigated since the starting of array signal processing. Accurate estimation of such parameter is critical in many applications that involve prior knowledge of the number of received signals. Information theo retic approaches such as Akaikes information criterion (AIC) and minimum description length (MDL) have been used extensively even though they are complex and show bad performance at some stages. In this paper, a new algorithm for estimating the number of sources is presented. This algorithm exploits the estimated eigenvalues of the auto correlation coefficient matrix rather than the auto covariance matrix, which is conventionally used, to estimate the number of sources. We propose to use either of a two simply estimated decision statistics, which are the moving increment and moving standard deviation as metric to estimate the number of sources. Then process a simple calculation of the increm
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This article gives a precise description of the Fatou sets and Julia sets of matrixvalued polynomials in $\mathcal{M}(2,\mathbb{C})$ in terms of the corresponding polynomials in $\mathbb{C}$. Further, we construct Green functions and B\"{o}ttchertype functions for these matrixvalued polynomials.
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A binary interface defect is any interface between two (not necessarily invertible) domain walls. We compute all possible binary interface defects in Kitaev's $\mathbb{Z}/p\mathbb{Z}$ model and all possible fusions between them. Our methods can be applied to any LevinWen model. We also give physical interpretations for each of the defects in the $\mathbb{Z}/p\mathbb{Z}$ model. These physical interpretations provide a new graphical calculus which can be used to compute defect fusion.
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We study the scheduling of computation tasks across $n$ workers in a large scale distributed learning problem. Computation speeds of the workers are assumed to be heterogeneous and unknown to the master, and redundant computations are assigned to workers in order to tolerate straggling workers. We consider sequential computation and instantaneous communication from each worker to the master, and each computation round, which can model a single iteration of the stochastic gradient descent algorithm, is completed once the master receives $k$ distinct computations from the workers. Our goal is to characterize the average completion time as a function of the computation load, which denotes the portion of the dataset available at each worker. We propose two computation scheduling schemes that specify the computation tasks assigned to each worker, as well as their computation schedule, i.e., the order of execution, and derive the corresponding average completion time in closedform. We also
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We derive limit distributions for certain empirical regularized optimal transport distances between probability distributions supported on a finite metric space and show consistency of the (naive) bootstrap. In particular, we prove that the empirical regularized transport plan itself asymptotically follows a Gaussian law. The theory includes the BoltzmannShannon entropy regularization and hence a limit law for the widely applied Sinkhorn divergence. Our approach is based on an application of the implicit function theorem to necessary and sufficient optimality conditions for the regularized transport problem. The asymptotic results are investigated in Monte Carlo simulations. We further discuss computational and statistical applications, e.g. confidence bands for colocalization analysis of protein interaction networks based on regularized optimal transport.
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We investigate the connectivity of wireless sensor networks secured by the heterogeneous random pairwise key predistribution scheme. In contrast to the homogeneous scheme proposed by Chan et al., where each node is paired (offline) with $K$ other nodes chosen uniformly at random; herein, each node is classified as class$1$ with probability $\mu$ or class$2$ with probability $1\mu$, for $0<\mu<1$, independently. Then, each class$1$ (respectively, class$2$) node is paired (offline) with $K_1$ (respectively, $K_2$) other nodes selected uniformly at random. We consider the particular case when $K_1=1$ and $K_2=K$. The heterogeneous random pairwise scheme induces an inhomogeneous random Kout graph $\mathbb{H} (n;\mu,K_n)$, where $n$ denotes the number of nodes and $K_n$ denotes a scaling of $K$ with respect to the network size $n$. Hence, establishing the connectivity of wireless sensor networks secured by the heterogeneous random pairwise scheme maps to deriving conditions on h
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Nontransversal intersection of the free and fixed boundary is shown to hold and a classification of blowup solutions is given for obstacle problems generated by fully nonlinear uniformly elliptic operators in two dimensions which appear in the meanfield theory of superconducting vortices.
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Comon's conjecture on the equality of the rank and the symmetric rank of a symmetric tensor, and Strassen's conjecture on the additivity of the rank of tensors are two of the most challenging and guiding problems in the area of tensor decomposition. We survey the main known results on these conjectures, and, under suitable bounds on the rank, we prove them, building on classical techniques used in the case of symmetric tensors, for mixed tensors. Finally, we improve the bound for Comon's conjecture given by flattenings by producing new equations for secant varieties of Veronese and Segre varieties.
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Consider an i.i.d. sample from an unknown density function supported on an unknown manifold embedded in a high dimensional Euclidean space. We tackle the problem of learning a distance between points, able to capture both the geometry of the manifold and the underlying density. We prove the convergence of this microscopic distance, as the sample size goes to infinity, to a macroscopic one that we call Fermat distance as it minimizes a path functional, resembling Fermat principle in optics. The proof boils down to the study of geodesics in Euclidean firstpassage percolation for nonhomogeneous Poisson point processes.
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In this paper, we develop a new computational approach which is based on minimizing the difference of two convex functionals (DC) to solve a broader class of phase retrieval problems. The approach splits a standard nonlinear least squares minimizing function associated with the phase retrieval problem into the difference of two convex functions and then solves a sequence of convex minimization subproblems. For each subproblem, the Nesterov's accelerated gradient descent algorithm or the BarzilaiBorwein (BB) algorithm is used. In the setting of sparse phase retrieval, a standard $\ell_1$ norm term is added into the minimization mentioned above. The subproblem is approximated by a proximal gradient method which is solved by the shrinkagethreshold technique directly without iterations. In addition, a modified AttouchPeypouquet technique is used to accelerate the iterative computation. These lead to more effective algorithms than the Wirtinger flow (WF) algorithm and the GaussNewton (
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In this paper, we propose a nonintrusive filterbased stabilization of reduced order models (ROMs) for uncertainty quantification (UQ) of the timedependent NavierStokes equations in convectiondominated regimes. We propose a novel highorder ROM differential filter and use it in conjunction with an evolvefilterrelax algorithm to attenuate the numerical oscillations of standard ROMs. We also examine how stochastic collocation methods (SCMs) can be combined with the evolvefilterrelax algorithm for efficient UQ of fluid flows. We emphasize that the new stabilized SCMROM framework is nonintrusive and can be easily used in conjunction with legacy flow solvers. We test the new framework in the numerical simulation of a twodimensional flow past a circular cylinder with a random viscosity that yields a random Reynolds number with mean $Re=100$.
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Distanceregular graphs have many beautiful combinatorial properties. Distancetransitive graphs have very strong symmetries, and they are distanceregular, i.e. distancetransitivity implies distanceregularity. In this paper, we give similar results, i.e. for special $s$ and graphs with other restrictions we show that $s$distancetransitivity implies distanceregularity.
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For a noetherian scheme that has an ample family of invertible sheaves, we prove that direct products in the category of quasicoherent sheaves are not exact unless the scheme is affine. This result can especially be applied to all quasiprojective schemes over commutative noetherian rings. The main tools of the proof are the GabrielPopescu embedding and Roos' characterization of Grothendieck categories satisfying Ab6 and Ab4*.
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A function $U:\left[ \omega_{1}\right] ^{2}\longrightarrow\omega$ is called $\left( 1,\omega_{1}\right) $\emph{weakly universal }if for every function $F:\left[ \omega_{1}\right] ^{2}\longrightarrow\omega$ there is an injective function $h:\omega_{1}\longrightarrow\omega_{1}$ and a function $e:\omega \longrightarrow\omega$ such that $F\left( \alpha,\beta\right) =e\left( U\left( h\left( \alpha\right) ,h\left( \beta\right) \right) \right) $ for every $\alpha,\beta\in\omega_{1}$. We will prove that it is consistent that there are no $\left( 1,\omega_{1}\right) $\emph{}weakly universal functions, this answers a question of Shelah and Stepr\={a}ns. In fact, we will prove that there are no $\left( 1,\omega_{1}\right) $\emph{}weakly universal functions in the Cohen model and after adding $\omega_{2}$ Sacks reals sidebyside. However, we show that there are $\left( 1,\omega _{1}\right) $\emph{}weakly universal functions in the Sacks model. In particular, the existence of such graphs is co
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This paper derives noncentral asymptotic results for nonlinear integral functionals of homogeneous isotropic Gaussian random fields defined on hypersurfaces in $\mathbb{R}^d$. We obtain the rate of convergence for these functionals. The results extend recent findings for solid figures. We apply the obtained results to the case of sojourn measures and demonstrate different limit situations.
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In this paper, we prove an isoperimetric inequality for lower order eigenvalues of the Dirichlet Laplacian in bounded domains of a Euclidean space which strengthens the wellknown AshbaughBeguria inequality about the ratio of the first two Dirichlet eigenvalues of the same domains and supports strongly a conjecture of AshbaughBenguria.
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Distance Geometry Problem (DGP) and Nonlinear Mapping (NLM) are two well established questions: Distance Geometry Problem is about finding a Euclidean realization of an incomplete set of distances in a Euclidean space, whereas Nonlinear Mapping is a weighted Least Square Scaling (LSS) method. We show how all these methods (LSS, NLM, DGP) can be assembled in a common framework, being each identified as an instance of an optimization problem with a choice of a weight matrix. We study the continuity between the solutions (which are point clouds) when the weight matrix varies, and the compactness of the set of solutions (after centering). We finally study a numerical example, showing that solving the optimization problem is far from being simple and that the numerical solution for a given procedure may be trapped in a local minimum.
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