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In the sparse normal means model, convergence of the Bayesian posterior distribution associated to spike and slab prior distributions is considered. The key sparsity hyperparameter is calibrated via marginal maximum likelihood empirical Bayes. The plugin posterior squared$L^2$ norm is shown to converge at the minimax rate for the euclidean norm for appropriate choices of spike and slab distributions. Possible choices include standard spike and slab with heavy tailed slab, and the spike and slab LASSO of Rockov\'a and George with heavy tailed slab. Surprisingly, the popular Laplace slab is shown to lead to a suboptimal rate for the full empirical Bayes posterior. This provides a striking example where convergence of aspects of the empirical Bayes posterior does not entail convergence of the full empirical Bayes posterior itself.
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The split control and user plane is key to the future heterogeneous cellular network (HCN), where the small cells are dedicated for the most data transmission while the macrocells are mainly responsible for the control signaling. Adapting to this technology, we propose a general and tractable framework of extra cell range expansion (CRE) by introducing an additional bias factor to enlarge the range of small cells flexibly for the extra offloaded macrousers in a twotier HCN, where the macrocell and small cell users have different required data rates. Using stochastic geometry, we analyze the energy efficiency (EE) of the extra CRE with joint low power transmission and resource partitioning, where the coverages of EE and data rate are formulated theoretically. Numerical simulations verify that the proposed extra CRE can improve the EE performance of HCN, and also show that deploying more small cells can provide benefits for EE coverage, but the EE improvement becomes saturated if the sm
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Coordinated multipoint (CoMP) joint transmission (JT) can save a great deal of energy especially for celledge users due to strengthened received signal, but at the cost of deploying and coordinating cooperative nodes, which degrades energy efficiency (EE), particularly when considerable amount of energy is consumed by nonideal hardware circuit. In this paper, we study energyefficient cooperation establishment, including cooperative nodes selection (CNS) and power allocation, to satisfy a required data rate in coherent JTCoMP networks with nonideal power amplifiers (PAs) and circuit power. The selection priority lemma is proved first, and then the formulated discrete combinatorial EE optimization is resolved by proposing node selection criterion and deriving closedform expressions of optimal transmission power. Therefore, an efficient algorithm is provided and its superiority is validated by Monte Carlo simulations, which also show the effects of nonideal PA and the data rate demand
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In this paper, we maximize the energy efficiency (EE) of fullduplex (FD) twoway relay (TWR) systems under nonideal power amplifiers (PAs) and nonnegligible transmissiondependent circuit power. We start with the case where only the relay operates full duplex and two timeslots are required for TWR. Then, we extend to the advanced case, where the relay and the two nodes all operate full duplex, and accomplish TWR in a single timeslot. In both cases, we establish the intrinsic connections between the optimal transmit powers and durations, based on which the original nonconvex EE maximization can be convexified and optimally solved. Simulations show the superiority of FDTWR in terms of EE, especially when traffic demand is high. The simulations also reveal that the maximum EE of FDTWR is more sensitive to the PA efficiency, than it is to selfcancellation. The full FD design of FDTWR is susceptible to traffic imbalance, while the design with only the relay operating in the FD mode
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The CEO of Wireline  a cloud application marketplace and serverless architecture platform  is pushing for an open source development fund to help sustain projects, funded by an initial coin offering. "Developers like me know that there are a lot of weak spots in the modern internet," he writes on MarketWatch, suggesting more Equifaxsized data breaches may wait in our future. In fact, many companies are not fully aware of all of the software components they are using from the opensource community. And vulnerabilities can be left open for years, giving hackers opportunities to do their worst. Take, for instance, the Heartbleed bug of 2014... Among the known hacks: 4.5 million healthcare records were compromised, 900 Canadians' social insurance numbers were stolen. It was deemed "catastrophic." And yet many servers today  two years later!  still carry the vulnerability, leaving whole caches of personal data exposed... [T]hose of us who are on the back end, stitching away, ofte
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An anonymous reader quotes Fortune's new report on blockchain: Demand for the technology, best known for supporting bitcoin, is growing so much that it will be one of the largest users of capacity next year at about 60 data centers that IBM rents out to other companies around the globe. IBM was one of the first big companies to see blockchain's promise, contributing code to an opensource effort and encouraging startups to try the technology on its cloud for free. That a 106yearold company like IBM is going all in on blockchain shows just how far the digital ledger has come since its early days underpinning bitcoin drug deals on the dark web. The market for blockchainrelated products and services will reach $7.7 billion in 2022, up from $242 million last year, according to researcher Markets & Markets. That's creating new opportunities for some of the old warships of the technology world, companies like IBM and Microsoft Corp. that are making the transition to cloud services. A
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Combined, Disney and Lucasfilm's Star Wars: The Last Jedi, Rogue One: A Star Wars Story and Star Wars: The Force Awakens have surpassed $4.06 billion in ticket sales at the worldwide box office. That's more than what Disney paid to buy George Lucas' Star Wars franchise. From the Hollywood Reporter: While an interesting benchmark, it doesn't, of course, account for the hundreds of millions spent to produce and market the trio of films, or the fact that Disney splits boxoffice grosses with theater owners. Conversely, Disney has minted additional money from lucrative ancillary revenue streams, merchandising sales and theme park attractions. Opening in North America on Dec. 15, The Last Jedi zoomed past the $900 million mark on Thursday, finishing the day with $934.2 million globally, including $464.6 million domestically and $469.6 internationally (it doesn't land in China until Jan. 5). The sequel to The Force Awakens was directed by Rian Johnson, and has dominated the Christmas corrido
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An anonymous reader quotes a report from The Guardian: Britain's homes could be lit and powered by wind farms surrounding an artificial island deep out in the North Sea, under advanced plans by a Dutch energy network. The radical proposal envisages an island being built to act as a hub for vast offshore wind farms that would eclipse today's facilities in scale. Dogger Bank, 125km (78 miles) off the East Yorkshire coast, has been identified as a potentially windy and shallow site. The power hub would send electricity over a longdistance cable to the UK and Netherlands, and possibly later to Belgium, Germany, and Denmark. TenneT, the project's backer and Dutch equivalent of the UK's National Grid, recently shared early findings of a study that said its plan could be billions of euros cheaper than conventional wind farms and international power cables. The scifisounding proposal is sold as an innovative answer to industry's challenge of continuing to make offshore wind cheaper, as turb
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schwit1 shares a report from The Daily Dot: A man was killed by police Thursday night in Wichita, Kansas, when officers responded to a false report of a hostage situation. The online gaming community is saying the dead man was the victim of a swatting prank, where trolls call in a fake emergency and force SWAT teams to descend on a target's house. If that's true, this would be the first reported swattingrelated death. Wichita deputy police chief Troy Livingston told the Wichita Eagle that police were responding to a report that a man fighting with his parents had accidentally shot his dad in the head and was holding his mom, brother and sister hostage. When police arrived, "A male came to the front door," Livingston told the Eagle. "As he came to the front door, one of our officers discharged his weapon." The man at the door was identified by the Eagle as 28yearold Andrew Finch. Finch's mother told reporters "he was not a gamer," but the online Call of Duty community claims his deat
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We provide a number of extensions and further interpretations of the ParameterizedBackground DataWeak (PBDW) formulation, a realtime and insitu Data Assimilation (DA) framework for physical systems modeled by parametrized Partial Differential Equations (PDEs), proposed in [Y Maday, AT Patera, JD Penn, M Yano, Int J Numer Meth Eng, 102(5), 933965]. Given $M$ noisy measurements of the state, PBDW seeks an approximation of the form $u^{\star} = z^{\star} + \eta^{\star}$, where the \emph{background} $z^{\star}$ belongs to a $N$dimensional \emph{background space} informed by a parameterized mathematical model, and the \emph{update} $\eta^{\star}$ belongs to a $M$dimensional \emph{update space} informed by the experimental observations. The contributions of the present work are threefold: first, we extend the adaptive formulation proposed in [T Taddei, M2AN, 51(5), 18271858] to general linear observation functionals, to effectively deal with noisy observations; second, we consider an
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Fully localised solitary waves are travellingwave solutions of the threedimensional gravitycapillary water wave problem which decay to zero in every horizontal spatial direction. Their existence has been predicted on the basis of numerical simulations and model equations (in which context they are usually referred to as `lumps'), and a mathematically rigorous existence theory for strong surface tension (Bond number $\beta$ greater than $\frac{1}{3}$) has recently been given. In this article we present an existence theory for the physically more realistic case $0<\beta<\frac{1}{3}$. A classical variational principle for fully localised solitary waves is reduced to a locally equivalent variational principle featuring a perturbation of the functional associated with the DaveyStewartson equation. A nontrivial critical point of the reduced functional is found by minimising it over its natural constraint set.
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We extend the standard Bayesian multivariate Gaussian generative data classifier by considering a generalization of the conjugate, normalWishart prior distribution and by deriving the hyperparameters analytically via evidence maximization. The behaviour of the optimal hyperparameters is explored in the highdimensional data regime. The classification accuracy of the resulting generalized model is competitive with stateofthe art Bayesian discriminant analysis methods, but without the usual computational burden of crossvalidation.
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We consider three problems about cities from Alcuin's_Propositiones ad acuendos juvenes_. These problems can be considered as the earliest packing problems difficult also for modern stateoftheart packing algorithms. We discuss the Alcuin's solutions and give the known (to the author) best solutions to these problems.
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In this paper, we study an interference avoidance scenario in the presence of a smart interferer which can rapidly observe the transmit power of a backscatter wireless sensor network (WSN) and effectively interrupt backscatter signals. We consider a power control with a subchannel allocation to avoid interference attacks and a timeswitching ratio for backscattering and RF energy harvesting in backscatter WSNs. We formulate the problem based on a Stackelberg game theory and compute the optimal transmit power, timeswitching ratio, and subchannel allocation parameter to maximize a utility function against the smart interference. We propose two algorithms for the utility maximization using Lagrangian dual decomposition for the backscatter WSN and the smart interference to prove the existence of the Stackelberg equilibrium. Numerical results show that the proposed algorithms effectively maximize the utility, compared to that of the algorithm based on the Nash game, so as to overcome sma
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We show that for any positive integer k, the kth nonzero eigenvalue of the LaplaceBeltrami operator on the twodimensional sphere endowed with a Riemannian metric of unit area, is maximized in the limit by a sequence of metrics converging to a union of k touching identical round spheres. This proves a conjecture posed by the second author in 2002 and yields a sharp isoperimetric inequality for all nonzero eigenvalues of the Laplacian on a sphere. Earlier, the result was known only for k=1 (J.Hersch, 1970), k=2 (N.Nadirashvili, 2002 and R.Petrides, 2014) and k=3 (N.Nadirashvili and Y.Sire, 2015). In particular, we argue that for any k>=2, the supremum of the kth nonzero eigenvalue on a sphere of unit area is not attained in the class of Riemannin metrics which are smooth outsitde a finite set of conical singularities. The proof uses certain properties of harmonic maps between spheres, the key new ingredient being a bound on the harmonic degree of a harmonic map into a sphere obtai
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We propose a new practical adaptive refinement strategy for $hp$finite element approximations of elliptic problems. Following recent theoretical developments in polynomialdegreerobust a posteriori error analysis, we solve two types of discrete local problems on vertexbased patches. The first type involves the solution on each patch of a mixed finite element problem with homogeneous Neumann boundary conditions, which leads to an ${\mathbf H}(\mathrm{div},\Omega)$conforming equilibrated flux. This, in turn, yields a guaranteed upper bound on the error and serves to mark mesh vertices for refinement via D\"orfler's bulkchasing criterion. The second type of local problems involves the solution, on patches associated with marked vertices only, of two separate primal finite element problems with homogeneous Dirichlet boundary conditions, which serve to decide between $h$, $p$, or $hp$refinement. Altogether, we show that these ingredients lead to a computable guaranteed bound on the
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In this article we use the combinatorial and geometric structure of manifolds with embedded cylinders in order to develop an adiabatic decomposition of the Hodge cohomology of these manifolds. We will on the one hand describe the adiabatic behaviour of spaces of harmonic forms by means of a certain \v{C}echde Rham complex and on the other hand generalise the CappellLeeMiller splicing map to the case of a finite number of edges, thus combining the topological and the analytic viewpoint. In parts, this work is a generalisation of works of Cappell, Lee and Miller in which a singleedged graph is considered, but it is more specific since only the GaussBonnet operator is studied.
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We study the automorphism groups of countable homogeneous directed graphs (and some additional homogeneous structures) from the point of view of topological dynamics. We determine precisely which of these automorphism groups are amenable (in their natural topologies). For those which are amenable, we determine whether they are uniquely ergodic, leaving unsettled precisely one case (the "semigeneric" complete multipartite directed graph). We also consider the Hrushovski property. For most of our results we use the various techniques of [3], suitably generalized to a context in which the universal minimal flow is not necessarily the space of all orders. Negative results concerning amenability rely on constructions of the type considered in [26]. An additional class of structures (compositions) may be handled directly on the basis of very general principles. The starting point in all cases is the determination of the universal minimal flow for the automorphism group, which in the context
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In this paper and upcoming ones, we initiate a systematic study of Bethe ansatz equations for integrable models by modern computational algebraic geometry. We show that algebraic geometry provides a natural mathematical language and powerful tools for understanding the structure of solution space of Bethe ansatz equations. In particular, we find novel efficient methods to count the number of solutions of Bethe ansatz equations based on Gr\"obner basis and quotient ring. We also develop analytical approach based on companion matrix to perform the sum of onshell quantities over all physical solutions without solving Bethe ansatz equations explicitly. To demonstrate the power of our method, we revisit the completeness problem of Bethe ansatz of Heisenberg spin chain, and calculate the sum rules of OPE coefficients in planar $\mathcal{N}=4$ superYangMills theory.
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We consider a variational problem associated with the minimal speed of pulsating traveling waves of the equation $u_t=u_{xx}+b(x)(1u)u$, $x\in{\mathbb R},\ t>0$, where the coefficient $b(x)$ is nonnegative and periodic in $x\in{\mathbb R}$ with a period $L>0$. It is known that there exists a quantity $c^*(b)>0$ such that a pulsating traveling wave with the average speed $c>0$ exists if and only if $c\geq c^*(b)$. The quantity $c^*(b)$ is the socalled minimal speed of pulsating traveling waves. In this paper, we study the problem of maximizing $c^*(b)$ by varying the coefficient $b(x)$ under some constraints. We prove the existence of the maximizer under a certain assumption of the constraint and derive the EulerLagrange equation which the maximizer satisfies under $L^2$ constraint $\int_0^L b(x)^2dx=\beta$. The limit problems of the solution of this EulerLagrange equation as $L\rightarrow0$ and as $\beta\rightarrow0$ are also considered. Moreover, we also consider the
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We revisit the scattering result of Duyckaerts, Holmer, and Roudenko for the nonradial $\dot H^{1/2}$critical focusing NLS. By proving an interaction Morawetz inequality, we give a simple proof of scattering below the ground state in dimensions $d\geq 3$ that avoids the use of concentration compactness.
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We present a novel approach to hybrid RANS/LES wall modeling based on function enrichment, which overcomes the common problem of the RANSLES transition and enables coarse meshes near the boundary. While the concept of function enrichment as an efficient discretization technique for turbulent boundary layers has been proposed in an earlier article by Krank & Wall (J. Comput. Phys. 316 (2016) 94116), the contribution of this work is a rigorous derivation of a new multiscale turbulence modeling approach and a corresponding discontinuous Galerkin discretization scheme. In the nearwall area, the NavierStokes equations are explicitly solved for an LES and a RANS component in one single equation. This is done by providing the Galerkin method with an independent set of shape functions for each of these two methods; the standard highorder polynomial basis resolves turbulent eddies where the mesh is sufficiently fine and the enrichment automatically computes the ensembleaveraged flow i
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We describe a paradigm for multiscale modeling that combines the MoriZwanzig (MZ) formalism of Statistical Mechanics with the Variational Multiscale (VMS) method. The MZVMS approach leverages both VMS scaleseparation projectors as well as phasespace projectors to provide a systematic modeling approach that is applicable to nonlinear partial differential equations. Spectral as well as continuous and discontinuous finite element methods are considered. The framework leads to a formally closed equation in which the effect of the unresolved scales on the resolved scales is nonlocal in time and appears as a convolution or memory integral. The resulting nonMarkovian system is used as a starting point for model development. We discover that unresolved scales lead to memory effects that are driven by an orthogonal projection of the coarsescale residual and interelement jumps. It is further shown that an MZbased finite memory model is a variant of the wellknown adjointstabilization
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We consider the classic problem of network reliability. A network is given together with a source vertex, one or more target vertices and probabilities assigned to each of the edges. Each edge appears in the network with its associated probability and the problem is to determine the probability of having at least one sourcetotarget path. This problem is known to be NPhard for general networks and has been solved for several special families. In this work we present a fixedparameter algorithm based on treewidth, which is a measure of treelikeness of graphs. The problem was already known to be solvable in linear time for bounded treewidth, however the known methods used complicated structures and were not easy to implement. We provide a significantly simpler and more intuitive algorithm that while remaining linear, is much easier to implement.
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Analyzing the security of cryptosystems under attacks based on the malicious modification of memory registers is a research topic of high importance. This type of attacks may affect the randomness of the secret parameters by forcing a limited number of bits to a certain value which can be unknown to the attacker. In this context, we revisit the attack on DSA presented by Faug\`ere, Goyet and Renault during the conference SAC 2012: we simplify their method and we provide a probabilistic approach in opposition to the heuristic proposed in the former to measure the limits of the attack. More precisely, the main problem is formulated as the search for a closest vector to a lattice, then we study the distribution of the vectors with bounded norms in a this family of lattices and we apply the result to predict the behavior of the attack. We validated this approach by computational experiments.
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In this paper we study the complexity of constructing a hitting set for the closure of VP, the class of polynomials that can be infinitesimally approximated by polynomials that are computed by polynomial sized algebraic circuits, over the real or complex numbers. Specifically, we show that there is a PSPACE algorithm that given n,s,r in unary outputs a set of ntuples over the rationals of size poly(n,s,r), with poly(n,s,r) bit complexity, that hits all nvariate polynomials of degreer that are the limit of sizes algebraic circuits. Previously it was known that a random set of this size is a hitting set, but a construction that is certified to work was only known in EXPSPACE (or EXPH assuming the generalized Riemann hypothesis). As a corollary we get that a host of other algebraic problems such as Noether Normalization Lemma, can also be solved in PSPACE deterministically, where earlier only randomized algorithms and EXPSPACE algorithms (or EXPH assuming the generalized Riemann hypot
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This work pioneers the systematic study and classification (up to Lie algebra automorphisms) of finitedimensional coboundary Lie bialgebras through Grassmann algebras. Several mathematical structures on Lie algebras, e.g. Killing forms or root decompositions, are extended to the Grassmann algebras of Lie algebras. This simplifies the description of the procedures and tools appearing in the theory of Lie bialgebras and originates novel techniques for its study and classification up to Lie algebra automorphisms. As a particular case, the classification of real threedimensional coboundary Lie bialgebras is retrieved.
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\emph{Multiresolution mode decomposition} (MMD) is an adaptive tool to analyze a time series $f(t)=\sum_{k=1}^K f_k(t)$, where $f_k(t)$ is a \emph{multiresolution intrinsic mode function} (MIMF) of the form \begin{eqnarray*} f_k(t)&=&\sum_{n=N/2}^{N/21} a_{n,k}\cos(2\pi n\phi_k(t))s_{cn,k}(2\pi N_k\phi_k(t))\\&&+\sum_{n=N/2}^{N/21}b_{n,k} \sin(2\pi n\phi_k(t))s_{sn,k}(2\pi N_k\phi_k(t)) \end{eqnarray*} with timedependent amplitudes, frequencies, and waveforms. The multiresolution expansion coefficients $\{a_{n,k}\}$, $\{b_{n,k}\}$, and the shape function series $\{s_{cn,k}(t)\}$ and $\{s_{sn,k}(t)\}$ provide innovative features for adaptive time series analysis. The MMD aims at identifying these MIMF's (including their multiresolution expansion coefficients and shape functions series) from their superposition. This paper proposes a fast algorithm for solving the MMD problem based on recursive diffeomorphismbased spectral analysis (RDSA). RDSA admits highly efficie
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We construct a 6D ${\cal N}=(1,0)$ superconformal field theory by coupling an ${\cal N}=(1, 0)$ tensor multiplet to an ${\cal N}=(1, 0)$ hypermultiplet. While the ${\cal N}=(1, 0)$ tensor multiplet is in the adjoint representation of the gauge group, the hypermultiplet can be in the fundamental representation or any other representation. If the hypermultiplet is also in the adjoint representation of the gauge group, the supersymmetry is enhanced to ${\cal N}=(2, 0)$, and the theory is identical to the $(2,0)$ theory of Lambert and Papageorgakis (LP). Upon dimension reduction, the $(1, 0)$ theory can be reduced to a general ${\cal N}=1$ supersymmetric YangMills theory in 5D. We discuss briefly the possible applications of the theories to multi M5branes.
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A new definition of random sets is proposed. It is based on the distance in measurable space and uses negative definite kernels for continuation from initial space to that of random sets. This approach has no connection to Hausdorff distance between sets. Key words: random sets; measurable space; negative definite kernels; Hilbert space isometries.
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In this work we consider a class of nonlocal nonautonomous evolution problems, which arise in neuronal activity, \[ \begin{cases} \partial_t u(t,x) = a(t)u(t,x) + b(t) \displaystyle\int_{\mathbb{R}^N} J(x,y)f(t,u(t,y))dy h +S(t,x) ,\ t\geq\tau \in \mathbb{R},\ x \in \Omega, u(\tau,x)=u_\tau(x),\ x\in\Omega u(t,x)= 0,\ t\geq\tau,\ x \in\mathbb{R}^N\backslash\Omega. \end{cases} \] Under suitable assumptions on the nonlinearity $f: \mathbb{R} \times \mathbb{R} \to\mathbb{R}$ and constraints on the functions $J: \mathbb{R}^N \times \mathbb{R}^{N}\to \mathbb{R}$;\, $S: \mathbb{R} \times \mathbb{R}^{N}\to\mathbb{R}$ and $a,b:\mathbb{R} \to\mathbb{R}$, we study the assimptotic behavior of the evolution process, generated by this problem, in an appropriated Banach space.% and we present a brief discussion on the model with a biological interpretation. We prove results on existence, uniqueness and smoothness of the solutions and the existence of pullback attracts for the evolution process as
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The (driven) Rabi model, together with its twomode, twophoton, and asymmetric generalizations, are exotic examples of quasiexactly solvable models in that a corresponding 2nd order ordinary differential equation (ODE) ${\cal L}\psi=0$ with polynomial coefficients (i) is not Fuchsian one and (ii) the differential operator ${\cal L}$ comprises energy E dependent terms $\sim Ez d_z$, $Ez$, $E^2$. When recast into a Schr\"odinger equation (SE) form with the first derivative term being eliminated and the coefficient of $d_x^2$ set to one, such an equation is characterized by a nontrivially energy dependent potential. The concept of a gradation slicing is introduced to analyze polynomial solutions of such equations. It is shown that the ODE of all the above Rabi models are characterized by the same unique set of grading parameters. General necessary and sufficient conditions for the existence of a polynomial solution are formulated. Unlike standard eigenvalue problems, the condition that
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Nesterov's accelerated gradient (AG) method for minimizing a smooth strongly convex function $f$ is known to reduce $f({\bf x}_k)f({\bf x}^*)$ by a factor of $\epsilon\in(0,1)$ after $k=O(\sqrt{L/\ell}\log(1/\epsilon))$ iterations, where $\ell,L$ are the two parameters of smooth strong convexity. Furthermore, it is known that this is the best possible complexity in the functiongradient oracle model of computation. Modulo a line search, the geometric descent (GD) method of Bubeck, Lee and Singh has the same bound for this class of functions. The method of linear conjugate gradients (CG) also satisfies the same complexity bound in the special case of strongly convex quadratic functions, but in this special case it can be faster than the AG and GD methods. Despite similarities in the algorithms and their asymptotic convergence rates, the conventional analysis of the running time of CG is mostly disjoint from that of AG and GD. The analyses of the AG and GD methods are also rather distin
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We give a de Finetti type representation for exchangeable random coalescent trees (formally described as semiultrametrics) in terms of sampling iid sequences from marked metric measure spaces. We apply this representation to define versions of treevalued FlemingViot processes from a $\Xi$lookdown model. As state spaces for these processes, we use, besides the space of isomorphy classes of metric measure spaces, also the space of isomorphy classes of marked metric measure spaces and a space of distance matrix distributions. This allows to include the case with dust in which the genealogical trees have isolated leaves.
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We propose a refinement of the RobertsonSchrodinger uncertainty principle (RSUP) using Wigner distributions. This new principle is stronger than the RSUP. In particular, and unlike the RSUP, which can be saturated by many phase space functions, the refined RSUP can be saturated by pure Gaussian Wigner functions only. Moreover, the new principle is technically as simple as the standard RSUP. In addition, it makes a direct connection with modern harmonic analysis, since it involves the Wigner transform and its symplectic Fourier transform, which is the radar ambiguity function. As a byproduct of the refined RSUP, we derive inequalities involving the entropy and the covariance matrix of Wigner distributions. These inequalities refine the Shanon and the Hirschman inequalities for the Wigner distribution of a mixed quantum state $\rho$. We prove sharp estimates which critically depend on the purity of $\rho$ and which are saturated in the Gaussian case.
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For an $n$dimensional local analytic differential system $\dot x=Ax+f(x)$ with $f(x)=O(x^2)$, the Poincar\'e nonintegrability theorem states that if the eigenvalues of $A$ are not resonant, the system does not have an analytic or a formal first integral in a neighborhood of the origin. This result was extended in 2003 to the case when $A$ admits one zero eigenvalue and the other are nonresonant: for $n=2$ the system has an analytic first integral at the origin if and only if the origin is a nonisolated singular point; for $n>2$ the system has a formal first integral at the origin if and only if the origin is not an isolated singular point. However, the question of \emph{whether the system has an analytic first integral at the origin provided that the origin is not an isolated singular point} remains open.
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We consider a probit model without covariates, but the latent Gaussian variables having compound symmetry covariance structure with a single parameter characterizing the common correlation. We study the parameter estimation problem under such oneparameter probit models. As a surprise, we demonstrate that the likelihood function does not yield consistent estimates for the correlation. We then formally prove the parameter's nonestimability by deriving a nonvanishing minimax lower bound. This counterintuitive phenomenon provides an interesting insight that one bit information of the latent Gaussian variables is not sufficient to consistently recover their correlation. On the other hand, we further show that trinary data generated from the Gaussian variables can consistently estimate the correlation with parametric convergence rate. Hence we reveal a phase transition phenomenon regarding the discretization of latent Gaussian variables while preserving the estimability of the correlation
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Let $\mathcal{T}$ be the group of smooth concordance classes of topologically slice knots, and $\{0\}\subset\cdots\subset \mathcal{T}_{n+1}\subset\mathcal{T}_{n}\subset \cdots\subset \mathcal{T}_{0}\subset \mathcal{T}$ be the bipolar filtration. In this paper, we show that a proper collection of the knots employed by Hedden, Kim, and Livingston to prove $\mathbb{Z}_2^{\infty} < \mathcal{T}$ can be used to see $\mathbb{Z}_2^{\infty} < \mathcal{T}_0/\mathcal{T}_1$.
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In this paper, we will study the following PDE in $\mathbb{R}^N$ involving multiple HardySobolev critical exponents: $$ \begin{cases} \Delta u+\sum_{i=1}^{l}\lambda_i \frac{u^{2^*(s_i)1}}{x^{s_i}}+u^{2^*1}=0\;\hbox{in}\;\mathbb{R}^N, u\in D_{0}^{1,2}(\mathbb{R}^N), \end{cases} $$ where $0<s_1<s_2<\cdots<s_l<2, 2^\ast:=\frac{2N}{N2}, \; 2^\ast(s):=\frac{2(Ns)}{N2}$ and there exists some $k\in [1, l]$ such that $\lambda_i>0$ for $1\leq i\leq k$; $\lambda_i<0$ for $k+1\leq i\leq l$. We develop an interesting way to study this class of equations involving mixed sign parameters. We prove the existence and nonexistence of the positive ground state solution. The regularity of the leastenergy solution are also investigated.
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This work is a study of $p$adic multiple zeta values at roots of unity ($p$MZV$\mu_{N}$'s), the $p$adic periods of the crystalline prounipotent fundamental groupoid of $(\mathbb{P}^{1}  \{0,\mu_{N},\infty\})/ \mathbb{F}_{q}$. The main tool is new objects which we call $p$adic prounipotent harmonic actions. In this part IV we define and study $p$adic analogues of some elementary complex analytic functions which interpolate multiple zeta values at roots of unity such as the multiple zeta functions. The indices of $p$MZV$\mu_{N}$'s involve sequences of positive integers ; in this IV1, by considering an operation which we call localization (inverting certain integration operators) in the prounipotent fundamental groupoid of $\mathbb{P}^{1}  \{0,\mu_{N},\infty\}$, and by using $p$adic prounipotent harmonic actions, we extend the definition of $p$MZV$\mu_{N}$'s to indices for which these integers can be negative, and we study these generalized $p$MZV$\mu_{N}$'s.
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In this paper we extend analytic signal method to the functions in many dimensions. First it is shown how to obtain separate phaseshifted components and how combine them to obtain signal's envelope, instantaneous frequencies and phases in many dimensions. Second, we show that phaseshifted components may be obtained by positive frequency restriction of the Fourier transform defined in the algebra of commutative elliptic hypercomplex numbers. Finally we prove that for $d>2$ there is no corresponding CliffordFourier transform that allows to recover phaseshifted components correctly.
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An optimal control problem is considered for a stochastic differential equation containing a statedependent regime switching, with a recursive cost functional. Due to the nonexponential discounting in the cost functional, the problem is timeinconsistent in general. Therefore, instead of finding a global optimal control (which is not possible), we look for a timeconsistent (approximately) locally optimal equilibrium strategy. Such a strategy can be represented through the solution to a system of partial differential equations, called an equilibrium HamiltonJacobBellman (HJB, for short) equation which is constructed via a sequence of multiperson differential games. A verification theorem is proved and, under proper conditions, the wellposedness of the equilibrium HJB equation is established as well.
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A line field on a manifold is a smooth map which assigns a tangent line to all but a finite number of points of the manifold. As such, it can be seen as a generalization of vector fields. They model a number of geometric and physical properties, e.g. the principal curvature directions dynamics on surfaces or the stress flux in elasticity. We propose a discretization of a MorseSmale line field on surfaces, extending Forman's construction for discrete vector fields. More general critical elements and their indices are defined from local matchings, for which Euler theorem and the characterization of homotopy type in terms of critical cells still hold.
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This paper investigates reverse auctions that involve continuous values of different types of goods, general nonconvex constraints, and second stage costs. Our analysis seeks to design the payment rules and conditions under which coalitions of participants cannot influence the auction outcome in order to obtain higher collective utility. Under incentivecompatible bidding in the VickreyClarkeGroves mechanism, coalitionproof outcomes are achieved if the submitted bids are convex and the constraint sets are of polymatroidtype. Unfortunately, these conditions do not capture the complexity of the general class of reverse auctions under consideration. By relaxing the property of incentivecompatibility, we investigate further payment rules that are coalitionproof, but without any extra conditions. Among coalitionproof mechanisms, we select the mechanism that minimizes the participants' abilities to benefit from strategic manipulations, in order to incentivize truthful bidding from the
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Identifying coordinate transformations that make strongly nonlinear dynamics approximately linear is a central challenge in modern dynamical systems. These transformations have the potential to enable prediction, estimation, and control of nonlinear systems using standard linear theory. The Koopman operator has emerged as a leading datadriven embedding, as eigenfunctions of this operator provide intrinsic coordinates that globally linearize the dynamics. However, identifying and representing these eigenfunctions has proven to be mathematically and computationally challenging. This work leverages the power of deep learning to discover representations of Koopman eigenfunctions from trajectory data of dynamical systems. Our network is parsimonious and interpretable by construction, embedding the dynamics on a lowdimensional manifold that is of the intrinsic rank of the dynamics and parameterized by the Koopman eigenfunctions. In particular, we identify nonlinear coordinates on which the
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The paper studies the locus in the rank 2 Higgs bundle moduli space corresponding to points which are critical for d of the Poisson commuting functions. These correspond to the Higgs field vanishing on a divisor of degree D. The degree D critical locus has an induced integrable system related to K(D)twisted Higgs bundles. Topological and differentialgeometric properties of the critical loci are addressed.
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In this work we prove convergence of the finite difference scheme for equations of stationary states of a general class of the spatial segregation of reactiondiffusion systems with $m\geq 2$ components. More precisely, we show that the numerical solution $u_h^l$, given by the difference scheme, converges to the $l^{th}$ component $u_l,$ when the mesh size $h$ tends to zero, provided $u_l\in C^2(\Omega),$ for every $l=1,2,\dots,m.$ In particular, our proof provides convergence of a difference scheme for the multiphase obstacle problem.
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This paper is devoted to a description of a general approach introduced by Agrachev and Sarychev in 2005 for studying some control problems for NavierStokes equations. The example of a 1D Burgers equation is used to illustrate the main ideas. We begin with a short discussion of the Cauchy problem and establish a continuity property for the resolving operator. We next turn to the property of approximate controllability and prove that it can be achieved by a twodimensional external force. Finally, we investigate a stronger property, when the approximate controllability and the exact controllability of finitedimensional functionals are proved simultaneously.
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Given a polynomial basis $\Psi_i$ which spans the polynomial vector space $\mathcal{P}$, this paper addresses the construction and use of the algebraic dual space $\mathcal{P}'$ and its canonical basis. Differentiation of dual variables will be discussed. The method will be applied to a Dirichlet and Neumann problem presented in \cite{CarstensenDemkowiczGopalakrishnan} and it is shown that the finite dimensional approximations satisfy $\phi^h = \mbox{div}\, \mathbf{q}^h$ on any grid. The dual method is also applied to a constrained minimization problem, which leads to a mixed finite element formulation. The discretization of the constraint and the Lagrange multiplier will be independent of the grid size, grid shape and the polynomial degree of the basis functions.
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Let $S$ be the set of subsequences $(x_{n_k})$ of a given real sequence $(x_n)$ which preserve the set of statistical cluster points. It has been recently shown that $S$ is a set of full (Lebesgue) measure. Here, on the other hand, we prove that $S$ is meager if and only if there exists an ordinary limit point of $(x_n)$ which is not a statistical cluster point of $(x_n)$. This provides a nonanalogue between measure and category.
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In this work, we propose constructions that correct duplications of multiple consecutive symbols. These errors are known as tandem duplications, where a sequence of symbols is repeated; respectively as palindromic duplications, where a sequence is repeated in reversed order. We compare the redundancies of these constructions with code size upper bounds that are obtained from sphere packing arguments. Proving that an upper bound on the code cardinality for tandem deletions is also an upper bound for inserting tandem duplications, we derive the bounds based on this special tandem deletion error as this results in tighter bounds. Our upper bounds on the cardinality directly imply lower bounds on the redundancy which we compare with the redundancy of the best known construction correcting arbitrary burst insertions. Our results indicate that the correction of palindromic duplications requires more redundancy than the correction of tandem duplications and both significantly less than arbitr
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We establish the limiting distribution of certain subsets of Farey sequences, i.e., sequences of primitive rational points, on expanding horospheres in covers $\Delta\backslash\mathrm{SL}(n+1,\mathbb{R})$ of $\mathrm{SL}(n+1,\mathbb{Z})\backslash\mathrm{SL}(n+1,\mathbb{R})$, where $\Delta$ is a finite index subgroup of $\mathrm{SL}(n+1,\mathbb{Z})$. These subsets can be obtained by projecting to the hyperplane $\{(x_1,\ldots,x_{n+1})\in\mathbb{R}^{n+1}:x_{n+1}=1\}$ sets of the form $\mathbf{A}=\bigcup_{j=1}^J\boldsymbol{a}_j\Delta$, where for all $j$, $\boldsymbol{a}_j$ is a primitive lattice point in $\mathbb{Z}^{n+1}$. Our method involves applying the equidistribution of expanding horospheres in quotients of $\mathrm{SL}(n+1,\mathbb{R})$ developed by Marklof and Str\"{o}mbergsson, and more precisely understanding how the full Farey sequence distributes in $\Delta\backslash\mathrm{SL}(n+1,\mathbb{R})$ when embedded on expanding horospheres as done in previous work by Marklof. For each
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In this note we observe that one can contact embed all contact 3manifolds into a Stein fillable contact structure on the twisted $S^3$bundle over $S^2$ and also into a unique overtwisted contact structure on $S^3\times S^2$. These results are proven using "spun embeddings" and Lefschetz fibrations.
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In this paper, we investigate the problem of beam alignment in millimeter wave (mmWave) systems, and design an optimal algorithm to reduce the overhead. Specifically, due to directional communications, the transmitter and receiver beams need to be aligned, which incurs high delay overhead since without a priori knowledge of the transmitter/receiver location, the search space spans the entire angular domain. This is further exacerbated under dynamic conditions (e.g., moving vehicles) where the access to the base station (access point) is highly dynamic with intermittent onoff periods, requiring more frequent beam alignment and signal training. To mitigate this issue, we consider an online stochastic optimization formulation where the goal is to maximize the directivity gain (i.e., received energy) of the beam alignment policy within a time period. We exploit the inherent correlation and unimodality properties of the model, and demonstrate that contextual information improves the perfor
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Physicallayer security for wireless networks has become an effective approach and recently drawn significant attention in the literature. In particular, the deployment and allocation of resources such as relays to assist the transmission have gained significant interest due to their ability to improve the secrecy rate of wireless networks. In this work, we examine relay selection criteria with arbitrary knowledge of the channels of the users and the eavesdroppers. We present alternative optimization criteria based on the signaltointerference and the secrecy rate criteria that can be used for resource allocation and that do not require knowledge of the channels of the eavesdroppers and the interference. We then develop effective relay selection algorithms that can achieve a high secrecy rate performance without the need for the knowledge of the channels of the eavesdroppers and the interference. Simulation results show that the proposed criteria and algorithms achieve excellent perfo
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