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Detecting the components common or correlated across multiple data sets is challenging due to a large number of possible correlation structures among the components. Even more challenging is to determine the precise structure of these correlations. Traditional work has focused on determining only the model order, i.e., the dimension of the correlated subspace, a number that depends on how the modelorder problem is defined. Moreover, identifying the model order is often not enough to understand the relationship among the components in different data sets. We aim at solving the complete modelselection problem, i.e., determining which components are correlated across which data sets. We prove that the eigenvalues and eigenvectors of the normalized covariance matrix of the composite data vector, under certain conditions, completely characterize the underlying correlation structure. We use these results to solve the modelselection problem by employing bootstrapbased hypothesis testing.
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Recently, in the paper \cite{CJKM1} we suggested the two conjectures about the diameter of iodecomposable Riordan graphs of the Bell type. In this paper, we give a counterexample for the first conjecture. Then we prove that the first conjecture is true for the graphs of some particular size and propose a new conjecture. Finally, we show that the second conjecture is true for some special iodecomposable Riordan graphs.
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We establish the rectifiability of measures satisfying a linear PDE constraint. The obtained rectifiability dimensions are optimal for many usual PDE operators, including all firstorder systems and all secondorder scalar operators. In particular, our general theorem provides a new proof of the rectifiability results for functions of bounded variations (BV) and functions of bounded deformation (BD). For divergencefree tensors we obtain refinements and new proofs of several known results on the rectifiability of varifolds and defect measures.
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Nonnegative matrix factorization (NMF) is a linear dimensionality reduction technique for analyzing nonnegative data. A key aspect of NMF is the choice of the objective function that depends on the noise model (or statistics of the noise) assumed on the data. In many applications, the noise model is unknown and difficult to estimate. In this paper, we define a multiobjective NMF (MONMF) problem, where several objectives are combined within the same NMF model. We propose to use Lagrange duality to judiciously optimize for a set of weights to be used within the framework of the weightedsum approach, that is, we minimize a single objective function which is a weighted sum of the all objective functions. We design a simple algorithm using multiplicative updates to minimize this weighted sum. We show how this can be used to find distributionally robust NMF (DRNMF) solutions, that is, solutions that minimize the largest error among all objectives. We illustrate the effectiveness of this
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In this note, we extend work of Farkas and Rim\'anyi on applying quadric rank loci to finding divisors of small slope on the moduli space of curves by instead considering all divisorial conditions on the hypersurfaces of a fixed degree containing a projective curve. This gives rise to a large family of virtual divisors on $\overline{\mathcal{M}_g}$. We determine explicitly which of these divisors are candidate counterexamples to the Slope Conjecture. The potential counterexamples exist on $\overline{\mathcal{M}_g}$, where the set of possible values of $g\in \{1,\ldots,N\}$ has density $\Omega(\log(N)^{0.087})$ for $N>>0$. Furthermore, no divisorial condition defined using hypersurfaces of degree greater than 2 give counterexamples to the Slope Conjecture, and every divisor in our family has slope at least $6+\frac{8}{g+1}$.
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Let $W$ be a manifold with boundary $M$ given together with a conformal class $\bar C$ which restricts to a conformal class $C$ on $M$. Then the relative Yamabe constant $Y_{\bar C}(W,M;C)$ is welldefined. We study the shorttime behavior of the relative Yamabe constant $Y_{[\bar g_t]}(W,M;C)$ under the Ricci flow $\bar g_t$ on $W$ with boundary conditions that mean curvature $H_{\bar g_t}\equiv 0$ and $\bar{g}_t_M\in C = [\bar{g}_0]$. In particular, we show that if the initial metric $\bar{g}_0$ is a Yamabe metric, then, under some natural assumptions, $\left.\frac{d}{dt}\right_{t=0}Y_{[\bar g_t]}(W,M;C)\geq 0$ and is equal to zero if and only the metric $\bar{g}_0$ is Einstein.
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Singular boundary value problems (SBVPs) arise in various fields of Mathematics, Engineering and Physics such as boundary layer theory, gas dynamics, nuclear physics, nonlinear optics, etc. The present monograph is devoted to systems of SBVPs for ordinary differential equations (ODEs). It presents existence theory for a variety of problems having unbounded nonlinearities in regions where their solutions are searched for. The main focus is to establish the existence of positive solutions. The results are based on regularization and sequential procedure. First chapter of this monograph describe the motivation for the study of SBVPs. It also include some available results from functional analysis and fixed point theory. The following chapters contain results from author's PhD thesis, National University of Sciences and Technology, Islamabad, Pakistan. These results provide the existence of positive solutions for a variety of systems of SBVPs having singularity with respect to independent
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In this work we consider the following class of fractional $p\&q$ Laplacian problems \begin{equation*} (\Delta)_{p}^{s}u+ (\Delta)_{q}^{s}u + V(\varepsilon x) (u^{p2}u + u^{q2}u)= f(u) \mbox{ in } \mathbb{R}^{N}, \end{equation*} where $\varepsilon>0$ is a parameter, $s\in (0, 1)$, $1< p<q<\frac{N}{s}$, $(\Delta)^{s}_{t}$, with $t\in \{p,q\}$, is the fractional $t$Laplacian operator, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a continuous potential and $f:\mathbb{R}\rightarrow \mathbb{R}$ is a $\mathcal{C}^{1}$function with subcritical growth. Applying minimax theorems and the LjusternikSchnirelmann theory, we investigate the existence, multiplicity and concentration of nontrivial solutions provided that $\varepsilon$ is sufficiently small.
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We study stochastic differential equations (SDEs) of McKeanVlasov type with distribution dependent drifts and driven by pure jump L\'{e}vy processes. We prove a uniform in time propagation of chaos result, providing quantitative bounds on convergence rate of interacting particle systems with L\'{e}vy noise to the corresponding McKeanVlasov SDE. By applying techniques that combine couplings, appropriately constructed $L^1$Wasserstein distances and Lyapunov functions, we show exponential convergence of solutions of such SDEs to their stationary distributions. Our methods allow us to obtain results that are novel even for a broad class of L\'{e}vydriven SDEs with distribution independent coefficients.
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Kontsevich and Manin gave a formula for the number $N_e$ of rational plane curves of degree $e$ through $3e1$ points in general position in the plane. When these $3e1$ points have coordinates in the rational numbers, the corresponding set of $N_e$ rational curves has a natural Galoismodule structure. We make some extremely preliminary investigations into this Galois module structure, and relate this to the deck transformations of the generic fibre of the product of the evaluation maps on the moduli space of maps. We then study the asymptotics of the number of rational points on hypersurfaces of low degree, and use this to generalise our results by replacing the projective plane by such a hypersurface.
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We consider a onedimensional analogue of the threedimensional FokkerPlanck equation for bosons. The latter is still only partially understood, and, in particular, the physically relevant question of whether this equation has solutions which form a BoseEinstein condensate has remained unanswered. After a change of variables, we establish globalintime existence and uniqueness for our 1D model (and generalisations thereof) using the concept of viscosity solutions. We show that such solutions enjoy good regularity properties, which guarantee that in the original variables blowup can only occur at the origin and with a fixed spatial profile, up to leading order, following a power law linked to the steady states of the equation. This enables us to extend entropy methods beyond the first blowup time. As a consequence, in the masssupercritical case, solutions will blow up in $L^\infty$ in finite time and  understood in an extended, measurevalued sense  they will eventually have a c
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In this paper, we examine some properties of the fixed point set of a digitally continuous function. The digital setting requires new methods that are not analogous to those of classical topological fixed point theory, and we obtain results that often differ greatly from standard results in classical topology. We introduce several measures related to fixed points for continuous selfmaps on digital images, and study their properties. Perhaps the most important of these is the fixed point spectrum $F(X)$ of a digital image: that is, the set of all numbers that can appear as the number of fixed points for some continuous selfmap. We give a complete computation of $F(C_n)$ where $C_n$ is the digital cycle of $n$ points. For other digital images, we show that, if $X$ has at least 4 points, then $F(X)$ always contains the numbers 0, 1, 2, 3, and the cardinality of $X$. We give several examples, including $C_n$, in which $F(X)$ does not equal $\{0,1,\dots,\#X\}$. We examine how fixed point
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China and Russia could disrupt critical national infrastructure in the US,
0131 MIT Technology 10552 
Yesterday, a worrying and invasive bug that allowed callers to secretly listen in on unknowing recipients through Apple’s FaceTime app quickly made news headlines. It was discovered that people could initiate a FaceTime call and, with a couple short steps, tap into the microphone on the other end as the call rang — without the other person accepting the FaceTime request. Apple said last night that an iOS update to eliminate the privacy bug is coming this week; in the meantime, the company took the step of disabling group FaceTime at the server level as an immediate emergency fix. However, new information suggests that Apple has already had several days to respond; the company was tipped off about it last week. Back on January 20th, a Twitter user tweeted at Apple’s support account clearly outlining the gist of the FaceTime bug: “My teen found a major security flaw in Apple’s new iOS. He can listen in to your iPhone/iPad without your approval.” The parent’s teenager had discovered the p
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Brain implants, AI, and a speech synthesizer have turned brain activity int
0130 MIT Technology 6753 
Great reporting by TechCrunch’s Josh Constine: Desperate for data on its competitors, Facebook has been secretly paying people to install a “Facebook Research” VPN that lets the company suck in all of a user’s phone and web activity, similar to Facebook’s Onavo Protect app that Apple banned in June and that was removed in August. Facebook sidesteps the App Store and rewards teenagers and adults to download the Research app and give it root access in what may be a violation of Apple policy so the social network can decrypt and analyze their phone activity, a TechCrunch investigation confirms. Facebook admitted to TechCrunch it was running the Research program to gather data on usage habits, and it has no plans to stop. Since 2016, Facebook has been paying users ages 13 to 35 up to $20 per month plus referral fees to sell their privacy by installing the iOS or Android “Facebook Research” app. Facebook even asked users to screenshot their Amazon order history page. The program is ad
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Firefox 65.0 is out. The release notes list a few new features, including: "Enhanced tracking protection: Simplified content blocking settings give users standard, strict, and custom options to control online trackers. A redesigned content blocking section in the site information panel (viewed by expanding the small “i” icon in the address bar) shows what Firefox detects and blocks on each website you visit."
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Astronomers have spotted a small firstofitskind object in the Kuiper Bel
0129 MIT Technology 7048 
An anonymous reader shares a report: It seems the stuff of fantasy. Giant ships sail the seas burning fuel that has been extracted from water using energy provided by the winds, waves and tides. A dramatic but implausible notion, surely. Yet this grand green vision could soon be realised thanks to a remarkable technological transformation that is now under way in Orkney. Perched 10 miles beyond the northern edge of the British mainland, this archipelago of around 20 populated islands  as well as a smattering of uninhabited reefs and islets  has become the centre of a revolution in the way electricity is generated. Orkney was once utterly dependent on power that was produced by burning coal and gas on the Scottish mainland and then transmitted through an undersea cable. Today the islands are so festooned with wind turbines, they cannot find enough uses for the emissionfree power they create on their own. Communityowned wind turbines generate power for local villages; islanders d
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China may be slowing iPhone sales worldwide, but Chinese people are driving Apple's App Store business. From a report: China accounted for nearly 50 percent of all app downloads in 2018, pushing the global downloads count to reach a record 194 billion, according to research firm App Annie. China, which is the world's largest smartphone market, also accounted for nearly 40 percent of worldwide consumer spend in apps in 2018, App Annie said in its yearly "State of Mobile" report. (Note: Google Play Store is not available in China.) Global consumer spend in apps reached $101 billion last year, up 75 percent since 2016. And 74 percent of all money spent on apps last year came from games. The battle between Silicon Valley companies and Chinese tech giants generated more than half of total consumer spend in the top 300 parent companies in 2018, the report said. The top company for global consumer spend was China's Tencent, which owns stake in several startups, companies, and games  includi
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A federal appeals court denied the FCC's request to postpone oral arguments in a court battle over the agency's decision to repeal its net neutrality rules. The FCC had asked for the hearing to be postponed since the commission's workforce has largely been furloughed due to the partial government shutdown. The hearing remains set for February 1. The Hill reports: After the FCC repealed the rules requiring internet service providers to treat all web traffic equally in December of 2017, a coalition of consumer groups and state attorneys general sued to reverse the move, arguing that the agency failed to justify it. The FCC asked the threejudge panel from the D.C. Circuit Court of Appeals to delay oral arguments out of "an abundance of caution" due to its lapse of funding. Net neutrality groups opposed the motion, arguing that there is an urgent need to settle the legal questions surrounding the FCC's order.
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We investigate various block preconditioners for a loworder RaviartThomas discretization of the mixed Poisson problem on adaptive quadrilateral meshes. In addition to standard diagonal and Schur complement preconditioners, we present a dedicated AMG solver for saddle point problems (SPAMG). A key element is a stabilized prolongation operator that couples the flux and scalar components. Our numerical experiments in 2D and 3D show that the SPAMG preconditioner displays nearly meshindependent iteration counts for adaptive meshes and heterogeneous coefficients.
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For a cyclic covering map $(\Sigma,K) \to (\Sigma',K')$ between two pairs of a 3manifold and a knot each, we describe the fundamental group $\pi_1(\Sigma \setminus K)$ in terms of $\pi_1(\Sigma' \setminus K')$. As a consequence, we give an alternative proof for the fact that certain knots in $S^3$ cannot be represented as the preimage of any knot in a lens space, which is related to free periods of knots. In our proofs, the subgroup of a group $G$ generated by the commutators and the $p$th power of each element of $G$ plays a key role.
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In this paper, we extend the fractional Sobolev spaces with variable exponents $W^{s,p(x,y)}$ to include the general fractional case $W^{K,p(x,y)}$, where $p$ is a variable exponent, $s\in (0,1)$ and $K$ is a suitable kernel. We are concerned with some qualitative properties of the space $W^{K,p(x,y)}$ (completeness, reflexivity, separability and density). Moreover, we prove a continuous embedding theorem of these spaces into variable exponent Lebesgue spaces. As an application we establish the existence and uniqueness of a solution for a nonlocal problem involving the nonlocal integrodifferential operator of elliptic type.
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We obtain the ground state magnetization of the Heisenberg and XXZ spin chains in a magnetic field $h$ as a series in $\sqrt{h_ch}$, where $h_c$ is the smallest field for which the ground state is fully polarized. All the coefficients of the series can be computed in closed form through a recurrence formula that involves only algebraic manipulations. The radius of convergence of the series in the full range $0<h\leq h_c$ is studied numerically. To that end we express the free energy at mean magnetization per site $1/2\leq \langle \sigma^z_i\rangle\leq 1/2$ as a series in $1/2\langle \sigma^z_i\rangle$ whose coefficients can be similarly recursively computed in closed form. This series converges for all $0\leq \langle \sigma^z_i\rangle\leq 1/2$. The recurrence is nothing but the Bethe equations when their roots are written as a double series in their corresponding Bethe number and in $1/2\langle \sigma^z_i\rangle$. It can also be used to derive the corrections in finite size, tha
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We study properly immersed ancient solutions of the codimension one mean curvature flow in $n$dimensional Euclidean space, and classify the convex hulls of the subsets of space reached by any such flow. In particular, it follows that any compact convex ancient mean curvature flow can only have a slab, a halfspace or all of space as the closure of its set of reach. The proof proceeds via a bihalfspace theorem (also known as a wedge theorem) for ancient solutions derived from a parabolic OmoriYau maximum principle for ancient mean curvature flows.
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Let $X$ be a ball quasiBanach function space and $H_X(\mathbb{R}^n)$ the associated Hardytype space. In this article, the authors establish the characterizations of $H_X(\mathbb{R}^n)$ via the LittlewoodPaley $g$functions and $g_\lambda^\ast$functions. Moreover, the authors obtain the boundedness of Calder\'onZygmund operators on $H_X(\mathbb{R}^n)$. For the local Hardytype space $h_X(\mathbb{R}^n)$, the authors also obtain the boundedness of $S^0_{1,0}(\mathbb{R}^n)$ pseudodifferential operators on $h_X(\mathbb{R}^n)$ via first establishing the atomic characterization of $h_X(\mathbb{R}^n)$. Furthermore, the characterizations of $h_X(\mathbb{R}^n)$ by means of local molecules and local LittlewoodPaley functions are also given. The results obtained in this article have a wide range of generality and can be applied to the classical Hardy space, the weighted Hardy space, the HerzHardy space, the LorentzHardy space, the MorreyHardy space, the variable Hardy space, the Orliczs
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[L. Gavruta, Frames for Operators, Appl. comput. Harmon. Anal. 32(2012), 139144] introduced a special kind of frames, named $K$frames, where $K$ is an operator, in Hilbert spaces, is significant in frame theory and has many applications. In this paper, first of all, we have introduced the notion of approximative $K$atomic decomposition in Banach spaces. We gave two characterizations regarding the existence of approximative $K$atomic decompositions in Banach spaces. Also some results on the existence of approximative $K$atomic decompositions are obtained. We discuss several methods to construct approximative $K$atomic decomposition for Banach Spaces. Further, approximative $\mathcal{X}_d$frame and approximative $\mathcal{X}_d$Bessel sequence are introduced and studied. Two necessary conditions are given under which an approximative $\mathcal{X}_d$Bessel sequence and approximative $\mathcal{X}_d$frame give rise to a bounded operator with respect to which there is an approximati
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We consider a class of maxAR(1) sequences connected with the Kendall convolution. For a large class of step size distributions we prove that the one dimensional distributions of the Kendall random walk with any unit step distribution, are regularly varying. The finite dimensional distributions for Kendall convolutions are given. We prove convergence of a continuous time stochastic process constructed from the Kendall random walk in the finite dimensional distributions sense using regularly varying functions.
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For a random walk on the integer lattice $\mathbb{Z}$ that is attracted to a strictly stable process with index $\alpha\in (1, 2)$ we obtain the asymptotic form of the transition probability for the walk killed when it hits a finite set. The asymptotic forms obtained are valid uniformly in a natural range of the space and time variables. The situation is relatively simple when the limit stable process has jumps in both positive and negative directions; in the other case when the jumps are one sided rather interesting matters are involved and detailed analyses are necessitated.
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The aim of this thesis is to advance the theory behind quantum information processing tasks, by deriving fundamental limits on bipartite quantum interactions and dynamics, which corresponds to an underlying Hamiltonian that governs the physical transformation of a twobody open quantum system. The goal is to determine entangling abilities of such arbitrary bipartite quantum interactions. Doing so provides fundamental limitations on information processing tasks, including entanglement distillation and secret key generation, over a bipartite quantum network. We also discuss limitations on the entropy change and its rate for dynamics of an open quantum system weakly interacting with the bath. We introduce a measure of nonunitarity to characterize the deviation of a doubly stochastic quantum process from a noiseless evolution. Next, we introduce information processing tasks for secure readout of digital information encoded in readonly memory devices against adversaries of varying capabi
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A cyclic quadrilateral is called a Brahmagupta quadrilateral if its four sides, the two diagonals and the area are all given by integers. In this paper we consider the hitherto unsolved problem of finding two Brahmagupta quadrilaterals with equal perimeters and equal areas. We obtain two parametric solutions of the problem  the first solution generates examples in which each quadrilateral has two equal sides while the second solution gives quadrilaterals all of whose sides are unequal. We also show how more parametric solutions of the problem may be obtained.
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In this paper we study Cauchy problem for thermoelastic plate equations with friction or structural damping in $\mathbb{R}^n$, $n\geq1$, where the heat conduction is modeled by Fourier's law. We explain some qualitative properties of solutions influenced by different damping mechanisms. We show which damping in the model has a dominant influence on smoothing effect, energy estimates, $L^pL^q$ estimates not necessary on the conjugate line, and on diffusion phenomena. Moreover, we derive asymptotic profiles of solutions in a framework of weighted $L^1$ data. In particular, sharp decay estimates for lower bound and upper bound of solutions in the $\dot{H}^s$ norm ($s\geq0$) are shown.
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In this paper we present some characterizations for quasiarithmetic operator means (among them the arithmetic and harmonic means) on the positive definite cone of the full algebra of Hilbert space operators, and also for the KuboAndo geometric mean on the positive definite cone of a general noncommutative $C^*$algebra.
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We review the theory of CoGorenstein algebras, which was introduced by Beligiannis in the article "The Homological Theory of Contravariantly Finite Subcategories: Gorenstein Categories, AuslanderBuchweitz Contexts and (Co)Stabilization". We show a connection between CoGorenstein algebras and the Nakayama and Generalized Nakayama conjecture.
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Given a closed Riemannian manifold $(N^{n+1},g)$, $n+1 \geq 3$ we prove the compactness of the space of singular, minimal hypersurfaces in $N$ whose volumes are uniformly bounded from above and the $p$th Jacobi eigenvalue $\lambda_p$'s are uniformly bounded from below. This generalizes the results of Sharp and AmbrozioCarlottoSharp in higher dimensions.
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Based on the method by [K\"uc95], we give a procedure to list up all complete intersection CalabiYau manifolds with respect to direct sums of irreducible homogeneous vector bundles on Grassmannians for each dimension. In particular, we give a classification of such CalabiYau 3folds and determine their topological invariants. We also give alternative descriptions for some of them.
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In this article, we summarize combinatorial description of complete intersection CalabiYau threefolds in Hibi toric varieties. Such CalabiYau threefolds have at worst conifold singularities, and are often smoothable to nonsingular CalabiYau threefolds. We focus on such nonsingular CalabiYau threefolds of Picard number one, and illustrate the calculation of topological invariants, using new motivating examples.
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Inspired by recent work of P.L. Lions on conditional optimal control, we introduce a problem of optimal stopping under bounded rationality: the objective is the expected payoff at the time of stopping, conditioned on another event. For instance, an agent may care only about states where she is still alive at the time of stopping, or a company may condition on not being bankrupt. We observe that conditional optimization is timeinconsistent due to the dynamic change of the conditioning probability and develop an equilibrium approach in the spirit of R. H. Strotz' work for sophisticated agents in discrete time. Equilibria are found to be essentially unique in the case of a finite time horizon whereas an infinite horizon gives rise to nonuniqueness and other interesting phenomena. We also introduce a theory which generalizes the classical Snell envelope approach for optimal stopping by considering a pair of processes with Snelltype properties.
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We present a self contained tour of the Conley index and some applications. The starting point is the fundamental theorem of dynamical systems, passing through the necessary topological background, with a short stop at the basic properties of the Conley index, and arriving at the construction of connection matrices with a panoramic view of the applications: detect heteroclinic orbits arising in delay differential equations, and partial differential equations of parabolic type. The ride will be filled with examples and pictures.
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This paper describes connected components of the strata of holomorphic abelian differentials on marked Riemann surfaces with prescribed degrees of zeros. Unlike the case for unmarked Riemann surfaces, we find there can be many connected components, distinguished by roots of the cotangent bundle of the surface. In the course of our investigation we also characterize the images of the fundamental groups of strata inside of the mapping class group. The main techniques of proof are mod r winding numbers and a mapping class grouptheoretic analogue of the Euclidean algorithm.
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The HelmholtzHodge decomposition (HHD) is applied to the construction of Lyapunov functions. It is shown that if a stability condition is satisfied, such a decomposition can be chosen so that its potential function is a Lyapunov function. In connection with the Lyapunov function, vector fields with strictly orthogonal HHD are analyzed. It is shown that they are a generalization of gradient vector fields and have similar properties. Finally, to examine the limitations of the proposed method, planar vector fields are analyzed.
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We derive two general transformations for certain basic hypergeometric series from the recurrence formulae for the partial numerators and denominators of two $q$continued fractions previously investigated by the authors. By then specializing certain free parameters in these transformations, and employing various identities of RogersRamanujan type, we derive \emph{$m$versions} of these identities. Some of the identities thus found are new, and some have been derived previously by other authors, using different methods. By applying certain transformations due to Watson, Heine and Ramanujan, we derive still more examples of such $m$versions of RogersRamanujantype identities.
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In the fight against vectorborne arboviruses, an important strategy of control of epidemic consists in controlling the population of vector, \textit{Aedes} mosquitoes in this case. Among possible actions, two techniques consist in releasing mosquitoes to reduce the size of the population (Sterile Insect Technique) or in replacing the wild population by a population carrying a bacteria, called \textit{Wolbachia}, blocking the transmission of viruses from mosquitoes to human. This paper is concerned with the question of optimizing the release protocol for these two strategies with the aim of getting as close as possible to the objectives. Starting from a mathematical model describing the dynamics of the population, we include the control function and introduce the cost functional for both \textit{population replacement} and \textit{Sterile Insect Technique} problems. Next, we establish some properties of the optimal control and illustrate them with some numerical simulations.
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Convexification is a core technique in global polynomial optimization. Currently, two different approaches compete in practice and in the literature. First, general approaches rooted in nonlinear programming. They are comparitively cheap from a computational point of view, but typically do not provide good (tight) relaxations with respect to bounds for the original problem. Second, approaches based on sumofsquares and moment relaxations. They are typically computationally expensive, but do provide tight relaxations. In this paper, we embed both kinds of approaches into a unified framework of monomial relaxations. We develop a convexification strategy that allows to trade off the quality of the bounds against computational expenses. Computational experiments show that a combination with a prototype cuttingplane algorithm gives very encouraging results.
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Radicalization is the process by which people come to adopt increasingly extreme political, social or religious ideologies. When radicalization leads to violence radical thinking becomes a threat to national security. Prevention and deradicalization programs are part of a set of strategies used to combat violent extremism, which are collectively known as Countering Violent Extremism (CVE). Prevention programs seek to stop the radicalization process from occurring and taking hold in the first place. Deradicalization programs work with violent extremists and attempt to alter their extremist beliefs and violent behavior with the aim to reintegrate them into society. In this paper we introduce a simple compartmental model suitable to describe prevention and deradicalization programs. The prevention initiatives are modeled by including a vaccination compartment, while the deradicalization process is modeled by including a treatment compartment. We calculate the basic reproduction number
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We consider here the CramerLundberg model based on generalized convolutions. In our model the insurance company invests at least part of its money, have employees, shareholders. The financial situation of the company after paying claims can be even better than before. We compute the ruin probability for $\alpha$convolution case, maximal convolution and the Kendall convolution case, which is formulated in the Williamson transform terms. We also give some new results on the Kendall random walks.
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This is a short review of the two papers on the $x$space asymptotics of the critical twopoint function $G_{p_c}(x)$ for the longrange models of selfavoiding walk, percolation and the Ising model on $\mathbb{Z}^d$, defined by the translationinvariant powerlaw stepdistribution/coupling $D(x)\proptox^{d\alpha}$ for some $\alpha>0$. Let $S_1(x)$ be the randomwalk Green function generated by $D$. We have shown that $\bullet~~S_1(x)$ changes its asymptotic behavior from Newton ($\alpha>2$) to Riesz ($\alpha<2$), with log correction at $\alpha=2$; $\bullet~~G_{p_c}(x)\sim\frac{A}{p_c}S_1(x)$ as $x\to\infty$ in dimensions higher than (or equal to, if $\alpha=2$) the upper critical dimension $d_c$ (with sufficiently large spreadout parameter $L$). The modeldependent $A$ and $d_c$ exhibit crossover at $\alpha=2$. The keys to the proof are (i) detailed analysis on the underlying random walk to derive sharp asymptotics of $S_1$, (ii) bounds on convolutions of power functio
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We consider those simply connected isothermic surfaces for which their Hopf differential factorizes into a real function and a meromorphic quadratic differential that has a zero or pole at some point, but is nowhere zero and holomorphic otherwise. Upon restriction to a simply connected patch that does not contain the zero or pole, the Darboux and Calapso transformations yield new isothermic surfaces. We determine the limiting behaviour of these transformed patches as the zero or pole of the meromorphic quadratic differential is approached and investigate whether they are continuous around that point.
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Projective ReedSolomon (PRS) codes are ReedSolomon codes of the maximum possible length q+1. The classification of deep holes received words with maximum possible error distance for PRS codes is an important and difficult problem. In this paper, we use algebraic methods to explicitly construct three classes of deep holes for PRS codes. We show that these three classes completely classify all deep holes of PRS codes with redundancy at most four. Previously, the deep hole classification was only known for PRS codes with redundancy at most three in work arXiv:1612.05447
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We propose a deeplearning approach for the joint MIMO detection and channel decoding problem. Conventional MIMO receivers adopt a modelbased approach for MIMO detection and channel decoding in linear or iterative manners. However, due to the complex MIMO signal model, the optimal solution to the joint MIMO detection and channel decoding problem (i.e., the maximum likelihood decoding of the transmitted codewords from the received MIMO signals) is computationally infeasible. As a practical measure, the current modelbased MIMO receivers all use suboptimal MIMO decoding methods with affordable computational complexities. This work applies the latest advances in deep learning for the design of MIMO receivers. In particular, we leverage deep neural networks (DNN) with supervised training to solve the joint MIMO detection and channel decoding problem. We show that DNN can be trained to give much better decoding performance than conventional MIMO receivers do. Our simulations show that a DN
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We show that if $(K,v_1,v_2)$ is a bivalued NIP field with $v_1$ henselian (resp. thenselian) then $v_1$ and $v_2$ are comparable (resp. dependent). As a consequence Shelah's conjecture for NIP fields implies the henselianity conjecture for NIP fields. Furthermore, the latter conjecture is proved for any field admitting a henselian valuation with a dpminimal residue field.
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Graph homomorphisms from the $\mathbb{Z}^d$ lattice to $\mathbb{Z}$ are functions on $\mathbb{Z}^d$ whose gradients equal one in absolute value. These functions are the height functions corresponding to proper $3$colorings of $\mathbb{Z}^d$ and, in two dimensions, corresponding to the $6$vertex model (square ice). We consider the uniform model, obtained by sampling uniformly such a graph homomorphism subject to boundary conditions. Our main result is that the model delocalizes in two dimensions, having no translationinvariant Gibbs measures. Additional results are obtained in higher dimensions and include the fact that every Gibbs measure which is ergodic under even translations is extremal and that these Gibbs measures are stochastically ordered.
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Denoising stationary process $(X_i)_{i \in Z}$ corrupted by additive white Gaussian noise is a classic and fundamental problem in information theory and statistical signal processing. Despite considerable progress in designing efficient denoising algorithms, for general analog sources, theoreticallyfounded computationallyefficient methods are yet to be found. For instance in denoising $X^n$ corrupted by noise $Z^n$ as $Y^n=X^n+Z^n$, given the full distribution of $X^n$, a minimum mean square error (MMSE) denoiser needs to compute $E[X^nY^n]$. However, for general sources, computing $E[X^nY^n]$ is computationally very challenging, if not infeasible. In this paper, starting by a Bayesian setup, where the source distribution is fully known, a novel denoising method, namely, quantized maximum a posteriori (QMAP) denoiser, is proposed and its asymptotic performance in the high signal to noise ratio regime is analyzed. Both for memoryless sources, and for structured firstorder Markov s
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We derive exact results for the Lindblad equation for a quantum spin chain (onedimensional quantum compass model) with dephasing noise. The system possesses doubly degenerate nonequilibrium steady states due to the presence of a conserved charge commuting with the Hamiltonian and Lindblad operators. We show that the system can be mapped to a nonHermitian Kitaev model on a twoleg ladder, which is solvable by representing the spins in terms of Majorana fermions. This allows us to study the Liouvillian gap, the inverse of relaxation time, in detail. We find that the Liouvillian gap increases monotonically when the dissipation strength $ \gamma $ is small, while it decreases monotonically for large $ \gamma $, implying a kind of phase transition in the first decay mode. The Liouvillian gap and the transition point are obtained in closed form in the case where the spin chain is critical. We also obtain the explicit expression for the autocorrelator of the edge spin. The result implies th
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This paper contributes to the mean dimension theory of dynamical systems. We introduce a new concept called mean dimension with potential and develop a variational principle for it. This is a mean dimension analogue of the theory of topological pressure. We consider a minimax problem for the sum of rate distortion dimension and the integral of a potential function. We prove that the minimax value is equal to the mean dimension with potential for a dynamical system having the marker property. The basic idea of the proof is a dynamicalization of geometric measure theory.
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