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Transfer Krull monoids are monoids which allow a weak transfer homomorphism to a commutative Krull monoid, and hence the system of sets of lengths of a transfer Krull monoid coincides with that of the associated commutative Krull monoid. We unveil a couple of new features of the system of sets of lengths of transfer Krull monoids over finite abelian groups G, and we provide a complete description of the system for all groups G having Davenport constant D(G) = 5 (these are the smallest groups for which no such descriptions were known so far). Under reasonable algebraic finiteness assumptions, sets of lengths of transfer Krull monoids and of weakly Krull monoids satisfy the Structure Theorem for Sets of Lengths. In spite of this common feature we demonstrate that systems of sets of lengths for a variety of classes of weakly Krull monoids are different from the system of sets of lengths of any transfer Krull monoid.
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In this paper, we are concerned with the stochastic SIS (susceptibleinfectedsusceptible) and SIR (susceptibleinfectedrecovered) models on highdimensional lattices with random edge weights, where a susceptible vertex is infected by an infectious neighbor at rate proportional to the weight on the edge connecting them. All the edge weights are assumed to be i.i.d.. Our main result gives mean field limits for survival probabilities of the two models as the dimension grows to infinity, which extends the main conclusion given in \cite{Xue2017} for classic stochastic SIS model.
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We consider sums of increments given by a functional of a stationary Markov chain. Letting $T$ be the first return time of the partial sums process to $(\infty,0]$, under general assumptions we give determine the asymptotic behavior of the survival probability, $\mathbb{P}(T\ge t)\sim Ct^{1/2}$ for an explicit constant $C$. Our analysis is based on a novel connection between the survival probability and the running maximum of the timereversed process, and relies on a functional central limit theorem for Markov chains. Our result extends the classic theorem of Sparre Anderson on sums of mean zero and independent increments to the case of correlated increments. As applications, we recover known clustering results for the 3color cyclic cellular automaton and the GreenbergHastings model in one dimension, and we prove a new clustering result for the 3color firefly cellular automaton.
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The purpose of this paper is to investigate acoustic wave scattering by a large number of bubbles in a liquid at frequencies near the Minnaert resonance frequency. This bubbly media has been exploited in practice to obtain superfocusing of acoustic waves. Using layer potential techniques we derive the scattering function for a single spherical bubble excited by an incident wave in the low frequency regime. We then derive the point scatter approximation for multiple scattering by N bubbles. We describe several numerical experiments based on the point scatterer approximation that demonstrate the possibility of achieving superfocusing using bubbly media.
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We develop a variational theory of geodesics for the canonical variation of the metric of a totally geodesic foliation. As a consequence, we obtain comparison theorems for the horizontal and vertical Laplacians. In the case of Sasakian foliations, we show that sharp horizontal and vertical comparison theorems for the subRiemannian distance may be obtained as a limit of horizontal and vertical comparison theorems for the Riemannian distances approximations.
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Motivated by recent questions about the extension of Courant's nodal domain theorem, we revisit a theorem published by C. Sturm in 1836, which deals with zeros of linear combination of eigenfunctions of SturmLiouville problems. Although well known in the nineteenth century, this theorem seems to have been ignored or forgotten by some of the specialists in spectral theory since the second half of the twentiethcentury. Although not specialists in History of Sciences, we have tried to replace these theorems into the context of nineteenth century mathematics.
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This paper is concerned with low multilinear rank approximations to antisymmetric tensors, that is, multivariate arrays for which the entries change sign when permuting pairs of indices. We show which ranks can be attained by an antisymmetric tensor and discuss the adaption of existing approximation algorithms to preserve antisymmetry, most notably a Jacobi algorithm. Particular attention is paid to the important special case when choosing the rank equal to the order of the tensor. It is shown that this case can be addressed with an unstructured rank$1$ approximation. This allows for the straightforward application of the higherorder power method, for which we discuss effective initialization strategies.
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This paper explores the relationship between two ideas in network information theory: edge removal and strong converses. Edge removal properties state that if an edge of small capacity is removed from a network, the capacity region does not change too much. Strong converses state that, for rates outside the capacity region, the probability of error converges to 1. Various notions of edge removal and strong converse are defined, depending on how edge capacity and residual error probability scale with blocklength, and relations between them are proved. In particular, each class of strong converse implies a specific class of edge removal. The opposite directions are proved for deterministic networks. Furthermore, a technique based on a novel causal version of the blowingup lemma is used to prove that for discrete memoryless stationary networks, the weak edge removal propertythat the capacity region changes continuously as the capacity of an edge vanishesis equivalent to the exponen
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Given a nonquasianalytic subadditive weight function $\omega$ we consider the weighted Schwartz space $\mathcal{S}_\omega$ and the shorttime Fourier transform on $\mathcal{S}_\omega$, $\mathcal{S}'_\omega$ and on the related modulation spaces with exponential weights. In this setting we define the $\omega$wave front set $WF'_\omega(u)$ and the Gabor $\omega$wave front set $WF^G_\omega(u)$ of $u\in\mathcal{S}'_{\omega}$, and we prove that they coincide. Finally we look at applications of this wave front set for operators of differential and pseudodifferential type.
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Given a finite group $G$, the invariably generating graph of $G$ is defined as the undirected graph in which the vertices are the nontrivial conjugacy classes of $G$, and two classes are connected if and only if they invariably generate $G$. In this paper we study this object for alternating and symmetric groups. First we observe that in most cases it has isolated vertices. Then, we prove that if we take them out we obtain a connected graph. Finally, we bound the diameter of this new graph from above and from below  apart from trivial cases, it is between $3$ and $6$ , and in about half of the cases we compute it exactly.
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We show that for a general complete intersection curve $C$ in projective space (other than a few exceptions), any branched covering $C \to \mathbb{P}^1$ of minimum degree is obtained by projection from a linear space. We also prove a special case of one of the wellknown CayleyBacharach conjectures due to Eisenbud, Green, and Harris.
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The free Banach lattice over a Banach space is introduced and analyzed. This generalizes the concept of free Banach lattice over a set of generators, and allows us to study the Nakano property and the density character of nondegenerate intervals on these spaces, answering some recent questions of B. de Pagter and A.W. Wickstead. Moreover, an example of a Banach lattice which is weakly compactly generated as a lattice but not as a Banach space is exhibited, thus answering a question of J. Diestel.
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We introduce and study the derived moduli stack $\mathrm{Symp}(X,n)$ of $n$shifted symplectic structures on a given derived stack $X$, as introduced by [PTVV] (IHES Vol. 117, 2013). In particular, under reasonable assumptions on $X$, we prove that $\mathrm{Symp}(X, n)$ carries a canonical shifted quadratic form. This generalizes a classical result of Fricke and Habermann, which was established in the $C^{\infty}$setting, to the broader context of derived algebraic geometry, thus proving a conjecture stated by Vezzosi.
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We consider the topological relation behind the spectral behavior of a linear operator that arises in the stability problem of traveling waves on a large bounded domain. When the domain size tends to infinity, the absolute and asymptotic essential spectra appears as accumulation sets of eigenvalues under separated and periodic boundary conditions, respectively. We present new proofs of Sandstede and Scheel [Theorems 4 and 5 of [14]] in a topological framework. The eigenfunction induces a curve on the Grassmannian manifold. To extract topological information from them, we decompose the Grassmannian into the submanifolds using the Schubert cycles, and analyze the curves on each submanifolds.
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It is very difficult to solve the Maximum Mutual Information (MMI) or Maximum Likelihood (ML) for all possible Shannon Channels or uncertain rules of choosing hypotheses, so that we have to use iterative methods. According to the Semantic Mutual Information (SMI) and R(G) function proposed by Chenguang Lu (1993) (where R(G) is an extension of information rate distortion function R(D), and G is the lower limit of the SMI), we can obtain a new iterative algorithm of solving the MMI and ML for tests, estimations, and mixture models. The SMI is defined by the average log normalized likelihood. The likelihood function is produced from the truth function and the prior by semantic Bayesian inference. A group of truth functions constitute a semantic channel. Letting the semantic channel and Shannon channel mutually match and iterate, we can obtain the Shannon channel that maximizes the Shannon mutual information and the average log likelihood. This iterative algorithm is called Channels' Match
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The article deals with the connection between the second postulate of Euclid and nonEuclidean geometry. It is shown that the violation of the second postulate of Euclid inevitably leads to hyperbolic geometry. This eliminates misunderstandings about the sums of some divergent series. The connection between hyperbolic geometry and relativistic computations is noted.
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We show that the HawkingPenrose singularity theorem, and the generalisation of this theorem due to Galloway and Senovilla, continue to hold for Lorentzian metrics that are of $C^{1, 1}$regularity. We formulate appropriate weak versions of the strong energy condition and genericity condition for $C^{1,1}$metrics, and of $C^0$trapped submanifolds. By regularisation, we show that, under these weak conditions, causal geodesics necessarily become nonmaximising. This requires a detailed analysis of the matrix Riccati equation for the approximating metrics, which may be of independent interest.
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This paper considers the problem of estimating the emission densities of a nonparametric finite state space hidden Markov model in a way that is statebystate adaptive and leads to minimax rates for each emission densityas opposed to globally minimax estimators, which adapt to the worst regularity among the emission densities. We propose a model selection procedure based on the GoldenschlugerLepski method. Our method is computationally efficient and only requires a family of preliminary estimators, without any restriction on the type of estimators considered. We present two such estimators that allow to reach minimax rates up to a logarithmic term: a spectral estimator and a least squares estimator. Finally, numerical experiments assess the performance of the method and illustrate how to calibrate it in practice. Our method is not specific to hidden Markov models and can be applied to nonparametric multiview mixture models.
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This work deals with the efficient numerical solution of the timefractional heat equation discretized on nonuniform temporal meshes. Nonuniform grids are essential to capture the singularities of "typical" solutions of timefractional problems. We propose an efficient parallelintime multigrid method based on the waveform relaxation technique, which accounts for the nonlocal character of the fractional differential operator. To maintain an optimal complexity, which can be obtained for the case of uniform grids, we approximate the coefficient matrix corresponding to the temporal discretization by its hierarchical matrix (${\cal H}$matrix) representation. In particular, the proposed method has a computational cost of ${\cal O}(k N M \log(M))$, where $M$ is the number of time steps, $N$ is the number of spatial grid points, and $k$ is a parameter which controls the accuracy of the ${\cal H}$matrix approximation. The efficiency and the good convergence of the algorithm, which can be
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This paper concerns the maximum coding rate at which data can be transmitted over a noncoherent, singleantenna, Rayleigh blockfading channel using an errorcorrecting code of a given blocklength with a blockerror probability not exceeding a given value. A highSNR normal approximation of the maximum coding rate is presented that becomes accurate as the signaltonoise ratio (SNR) and the number of coherence intervals L over which we code tend to infinity. Numerical analyses suggest that the approximation is accurate already at SNR values of 15 dB and at 10 or more coherence intervals.
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As noninstitutive polynomial chaos expansion (PCE) techniques have gained growing popularity among researchers, we here provide a comprehensive review of major sampling strategies for the least squares based PCE. Traditional sampling methods, such as Monte Carlo, Latin hypercube, quasiMonte Carlo, optimal design of experiments (ODE), Gaussian quadratures, as well as more recent techniques, such as coherenceoptimal and randomized quadratures are discussed. We also propose a hybrid sampling method, dubbed alphabeticcoherenceoptimal, that employs the socalled alphabetic optimality criteria used in the context of ODE in conjunction with coherenceoptimal samples. A comparison between the empirical performance of the selected sampling methods applied to three numerical examples, including highorder PCE's, highdimensional problems, and low oversampling ratios, is presented to provide a road map for practitioners seeking the most suitable sampling technique for a problem at hand. We o
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We give a new simple characterization of the set of Kleshchev multipartitions. These combinatorial objects both label the crystal bases for irreducible highest weight modules in affine type $A$ and the simple modules for ArikiKoike algebras. As a consequence, we obtain a proof of a generalization of a conjecture by Dipper, James and Murphy.
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A holomorphic torsion invariant of K3 surfaces with involution was introduced by the secondnamed author. In this paper, we completely determine its structure as an automorphic function on the moduli space of such K3 surfaces. On every component of the moduli space, it is expressed as the product of an explicit Borcherds lift and a classical Siegel modular form. We also introduce its twisted version. We prove its modularity and a certain uniqueness of the modular form corresponding to the twisted holomorphic torsion invariant.
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Jolla's CEO Sami PienimÃ¤ki: We have positive progress and major future business potential with Sailfish openings e.g. in China and Russia. While these projects are big and take time, they're developing steadily and we expect them to grow into sizable businesses for us overtime. These two are now our key customers but the projects are in early phase and our revenues are tight.Â At the same time realizing this opportunity requires significant R&D investments from our licensing customers and Jolla. Meanwhile, as Russia and China are progressing, we also have good traction with other new potential licensing customers in different regions. Good discussions are ongoing, and weâre waiting eagerly to get to share those with you. And yes, they're still going to at some point maybe possibly start the refunding process for the tablet. My Jolla Tablet spent about 5 minutes outside of the box, since there's not much you can actually do with it.
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We define a function, called smultiplicity, that interpolates between HilbertSamuel multiplicity and HilbertKunz multiplicity by comparing powers of ideals to the Frobenius powers of ideals. The function is continuous in s, and its value is equal to HilbertSamuel multiplicity for small values of s and is equal to HilbertKunz multiplicity for large values of s. We prove that it has an Associativity Formula generalizing the Associativity Formulas for HilbertSamuel and HilbertKunz multiplicity. We also define a family of closures such that if two ideals have the same sclosure then they have the same smultiplicity, and the converse holds under mild conditions. We describe the smultiplicity of monomial ideals in toric rings as a certain volume in real space
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The FloaterHormann family of rational interpolants do not have spurious poles or unattainable points, are efficient to calculate, and have arbitrarily high approximation orders. One concern when using them is that the amplification of rounding errors increases with approximation order, and can make balancing the interpolation error and rounding error difficult. This article proposes to modify the FloaterHormann interpolants by including additional local polynomial interpolants at the ends of the interval. This appears to improve the conditioning of the interpolants and allow higher approximation orders to be used in practice.
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Most of the literature on general relativity over the last century assumes that the cosmological constant $\Lambda$ is zero. However, by now independent observations have led to a consensus that the dynamics of the universe is best described by Einstein's equations with a small but positive $\Lambda$. Interestingly, this requires a drastic revision of conceptual frameworks commonly used in general relativity, \emph{no matter how small $\Lambda$ is.} We first explain why, and then summarize the current status of generalizations of these frameworks to include a positive $\Lambda$, focusing on gravitational waves.
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For a directed graph $G$, a $t$identifying code is a subset $S\subseteq V(G)$ with the property that for each vertex $v\in V(G)$ the set of vertices of $S$ reachable from $v$ by a directed path of length at most $t$ is both nonempty and unique. A graph is called {\it $t$identifiable} if there exists a $t$identifying code. This paper shows that the de~Bruijn graph $\vec{\mathcal{B}}(d,n)$ is $t$identifiable if and only if $n \geq 2t1$. It is also shown that a $t$identifying code for $t$identifiable de~Bruijn graphs must contain at least $d^{n1}(d1)$ vertices, and constructions are given to show that this lower bound is achievable $n \geq 2t$. Further a (possibly) nonoptimal construction is given when $n=2t1$. Additionally, with respect to $\vec{\mathcal{B}}(d,n)$ we provide upper and lower bounds on the size of a minimum \textit{$t$dominating set} (a subset with the property that every vertex is at distance at most $t$ from the subset), that the minimum size of a \textit{di
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In this article, we study the stochastic wave equation in arbitrary spatial dimension $d$, with a multiplicative term of the form $\sigma(u)=u$, also known in the literature as the Hyperbolic Anderson Model. This equation is perturbed by a general Gaussian noise, which is homogeneous in both space and time. We prove the existence and uniqueness of a solution of this equation (in the Skorohod sense) and the H\"older continuity of its sample paths, under the same respective conditions on the spatial spectral measure of the noise as in the case of the white noise in time, regardless of the temporal covariance function of the noise.
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We first give an exposition on holomorphic isometries from the Poincar\'e disk to polydisks and from the Poincar\'e disk to the product of the Poincar\'e disk with a complex unit ball. As an application, we provide an example of proper holomorphic map from the unit disk to the complex unit ball that is irrational, algebraic and holomorphic on a neighborhood of the closed unit disk. We also include some new results on holomorphic isometries.
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We introduce generalised finite difference methods for solving fully nonlinear elliptic partial differential equations. Methods are based on piecewise Cartesian meshes augmented by additional points along the boundary. This allows for adaptive meshes and complicated geometries, while still ensuring consistency, monotonicity, and convergence. We describe an algorithm for efficiently computing the nontraditional finite difference stencils. We also present a strategy for computing formally higherorder convergent methods. Computational examples demonstrate the efficiency, accuracy, and flexibility of the methods.
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We discuss the higher order stabilization of the coefficients of the colored Jones polynomial. In particular, we find an expression for the second stable sequence of the colored Jones polynomial of a certain class of knots. We also determine which knots have the same higher order stability.
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In this article, we introduce rack invariants of oriented Legendrian knots in the 3dimensional Euclidean space endowed with the standard contact structure, which we call Legendrian racks. These invariants form a generalization of the quandle invariants of knots. These rack invariants do not result in a complete invariant, but detect some of the geometric properties such as cusps in a Legendrian knot. In the case of topologically trivial Legendrian knots, we test this family of invariants for its strengths and limitations. We further prove that these invariants form a natural generalization of the quandle invariant, by which we mean that any rack invariant under certain restrictions is equivalent to a Legendrian rack. The axioms of these racks are expressible in first order logic, and were discovered through a series of experiments using an automated theorem prover for first order logic. We also present the results from the experiments on Legendrian unknots involving automated theorem
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We generalize notions of passivity and dissipativity to fractional order systems. Similar to integer order systems, we show that the proposed definitions generate analogous stability and compositionality properties for fractional order systems as well. We also study the problem of passivating a fractional order system through a feedback controller. Numerical examples are presented to illustrate the concepts.
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In this paper we adopt the pullback approach to global Finsler geometry. We investigate horizontally recurrent Finsler connections. We prove that for each scalar ($\pi$)1form $A$, there exists a unique horizontally recurrent Finsler connection whose $h$recurrence form is $A$. This result generalizes the existence and uniqueness theorem of Cartan connection. We then study some properties of a special kind of horizontally recurrent Finsler connection, which we call special HRFconnection.
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We have developed polynomialtime algorithms to generate terms of the cogrowth series for groups $\mathbb{Z}\wr \mathbb{Z},$ the lamplighter group, $(\mathbb{Z}\wr \mathbb{Z})\wr \mathbb{Z}$ and the NavasBrin group $B.$ We have also given an improved algorithm for the coefficients of Thompson's group $F,$ giving 32 terms of the cogrowth series. We develop numerical techniques to extract the asymptotics of these various cogrowth series. We present improved rigorous lower bounds on the growthrate of the cogrowth series for Thompson's group $F$ using the method from \cite{HHR15} applied to our extended series. We also generalise their method by showing that it applies to loops on any locally finite graph. Unfortunately, lower bounds less than 16 do not help in determining amenability. Again for Thompson's group $F$ we prove that, if the group is amenable, there cannot be a subdominant stretched exponential term in the asymptotics\footnote{ }. Yet the numerical data provides compelling
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We consider a nonparametric Bayesian approach to estimate the diffusion coefficient of a stochastic differential equation given discrete time observations on its solution over a fixed time interval. As a prior on the diffusion coefficient, we employ a histogramtype prior with piecewise constant realisations on bins forming a partition of the time interval. We justify our approach by deriving the rate at which the corresponding posterior distribution asymptotically concentrates around the diffusion coefficient under which the data have been generated. For a specific choice of the prior based on the inverse gamma distribution, this posterior contraction rate turns out to be optimal for estimation of a H\"oldercontinuous diffusion coefficient with smoothness parameter $0<\lambda\leq 1.$ Our approach is straightforward to implement and leads to good practical results in a wide range of simulation examples. Finally, we apply our method on exchange rate data sets.
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In this paper we prove small data global existence for solutions to the MaxwellBornInfeld (MBI) system on a fixed Schwarzschild background. This system has appeared in the context of string theory and can be seen as a nonlinear model problem for the stability of the background metric itself, due to its tensorial and quasilinear nature. The MBI system models nonlinear electromagnetism and does not display birefringence. The key element in our proof lies in the observation that there exists a firstorder differential transformation which brings solutions of the spin $\pm 1$ Teukolsky equations, satisfied by the extreme components of the field, into solutions of a "good" equation (the FackerellIpser Equation). This strategy was established in [F. Pasqualotto, The spin $\pm 1$ Teukolsky equations and the Maxwell system on Schwarzschild, Preprint 2016, arXiv:1612.07244] for the linear Maxwell field on Schwarzschild. We show that analogous FackerellIpser equations hold for the MBI sy
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In this paper, we analyze the convergence of several discretizethenoptimize algorithms, based on either a secondorder or a fourthorder finite difference discretization, for solving elliptic PDEconstrained optimization or optimal control problems. To ensure the convergence of a discretizethenoptimize algorithm, one wellaccepted criterion is to choose or redesign the discretization scheme such that the resultant discretizethenoptimize algorithm commutes with the corresponding optimizethendiscretize algorithm. In other words, both types of algorithms would give rise to exactly the same discrete optimality system. However, such an approach is not trivial. In this work, by investigating a simple distributed elliptic optimal control problem, we first show that enforcing such a stringent condition of commutative property is only sufficient but not necessary for achieving the desired convergence. We then propose to add some suitable $H_1$ seminorm penalty/regularization terms to r
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We study the construction of quasicyclic selfdual codes, especially of binary cubic ones. We consider the binary quasicyclic codes of length 3\ell with the algebraic approach of [9]. In particular, we improve the previous results by constructing 1 new binary [54, 27, 10], 6 new [60, 30, 12] and 50 new [66, 33, 12] cubic selfdual codes. We conjecture that there exist no more binary cubic selfdual codes with length 54, 60 and 66.
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MojoKid writes: A new flaw has been discovered that impacts Intel 6th and 7th Generation Skylake and Kaby Lakebased processors that support HyperThreading. The issue affects all OS types and is detailed by Intel errata documentation and points out that under complex microarchitectural conditions, short loops of less than 64 instructions that use AH, BH, CH or DH registers, as well as their corresponding wider register (e.g. RAX, EAX or AX for AH), may cause unpredictable system behavior, including crashes and potential data loss. The OCaml toolchain community first began investigating processors with these malfunctions back in January and found reports stemming back to at least the first half of 2016. The OCaml team was able pinpoint the issue to Skylake's HyperThreading implementation and notified Intel. While Intel reportedly did not respond directly, it has issued some microcode fixes since then. That's not the end of the story, however, as the microcode fixes need to be implement
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We consider the Schroedinger equation with a subcritical focusing power nonlinearity on a noncompact metric graph, and prove that for every finite edge there exists a threshold value of the mass, beyond which there exists a positive bound state achieving its maximum on that edge only. This bound state is characterized as a minimizer of the energy functional associated to the NLS equation, with an additional constraint (besides the mass prescription): this requires particular care in proving that the minimizer satisfies the EulerLagrange equation. As a consequence, for a sufficiently large mass every finite edge of the graph hosts at least one positive bound state that, owing to its minimality property, is orbitally stable.
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We propose local segmentation of multiple sequences sharing a common time or locationindex, building upon the single sequence local segmentation methods of Niu and Zhang (2012) and Fang, Li and Siegmund (2016). We also propose reverse segmentation of multiple sequences that is new even in the single sequence context. We show that local segmentation estimates changepoints consistently for both single and multiple sequences, and that both methods proposed here detect signals well, with the reverse segmentation method outperforming a large number of known segmentation methods on a commonly used single sequence test scenario. We show that on a recent allelespecific copy number study involving multiple cancer patients, the simultaneous segmentations of the DNA sequences of all the patients provide information beyond that obtained by segmentation of the sequences one at a time.
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We construct modular invariant families of (finitely or uncountably many) simple modules over the simple $\mathcal{N}=2$ vertex operator superalgebra of central charge $c_{p,p'}=3\left(1\frac{2p'}{p}\right),$ where $(p,p')$ is a pair of coprime positive integers such that $p\geq2$. When $p'=1$, these modules coincide with the $\mathcal{N}=2$ unitary minimal series. In addition, we calculate the corresponding "modular $S$matrix" which is no longer a matrix if $p'\geq2$.
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Univalent homotopy type theory (HoTT) may be seen as a language for the category of $\infty$groupoids. It is being developed as a new foundation for mathematics and as an internal language for (elementary) higher toposes. We develop the theory of factorization systems, reflective subuniverses, and modalities in homotopy type theory, including their construction using a "localization" higher inductive type. This produces in particular the ($n$connected, $n$truncated) factorization system as well as internal presentations of subtoposes, through lex modalities. We also develop the semantics of these constructions.
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If $M$ is a finite volume complete hyperbolic $3$manifold, the quantity $\mathcal A_1(M)$ is defined as the infimum of the areas of closed minimal surfaces in $M$. In this paper we study the continuity property of the functional $\mathcal A_1$ with respect to the geometric convergence of hyperbolic manifolds. We prove that it is lower semicontinuous and even continuous if $\mathcal A_1(M)$ is realized by a minimal surface satisfying some hypotheses. Understanding the interaction between minimal surfaces and short geodesics in $M$ is the main theme of this paper
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We consider Markov processes of cubic stochastic (in a fixed sense) matrices which are also called quadratic stochastic process (QSPs). A QSP is a particular case of a continuoustime dynamical system whose states are stochastic cubic matrices satisfying an analogue of the KolmogorovChapman equation (KCE). Since there are several kinds of multiplications between cubic matrices we have to fix first a multiplication and then consider the KCE with respect to the fixed multiplication. Moreover, the notion of stochastic cubic matrix also varies depending on the real models of application. The existence of a stochastic (at each time) solution to the KCE provides the existence of a QSP. In this paper, our aim is to construct QSPs for two specially chosen notions of stochastic cubic matrices and two multiplications of such matrices (known as Maksimov's multiplications). We construct a wide class of QSPs and give some timedependent behavior of such processes. We give an example with applicati
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We consider a dominance order on positive vectors induced by the elementary symmetric polynomials. Under this dominance order we provide conditions that yield simple proofs of several monotonicity questions. Notably, our approach yields a quick (4 line) proof of the socalled \emph{"sumofsquaredlogarithms"} inequality conjectured in (P.~Neff, B.~Eidel, F.~Osterbrink, and R.~Martin, \emph{Applied Math. \& Mechanics., 2013}; P.~Neff, Y.~Nakatsukasa, and A.~Fischle; \emph{SIMAX, 35, 2014}). This inequality has been the subject of several recent articles, and only recently it received a full proof, albeit via a more elaborate complexanalytic approach. We provide an elementary proof, which moreover extends to yield simple proofs of both old and new inequalities for R\'enyi entropy, subentropy, and quantum R\'enyi entropy.
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Modern dense Flash memory devices operate at very low error rates, which require powerful error correcting coding (ECC) techniques. An emerging class of graphbased ECC techniques that has broad applications is the class of spatiallycoupled (SC) codes, where a block code is partitioned into components that are then rewired multiple times to construct an SC code. Here, our focus is on SC codes with the underlying circulantbased structure. In this paper, we present a threestage approach for the design of high performance nonbinary SC (NBSC) codes optimized for practical Flash channels; we aim at minimizing the number of detrimental general absorbing sets of type two (GASTs) in the graph of the designed NBSC code. In the first stage, we deploy a novel partitioning mechanism, called the optimal overlap partitioning, which acts on the protograph of the SC code to produce optimal partitioning corresponding to the smallest number of detrimental objects. In the second stage, we apply a n
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We consider transcendental meromorphic function for which the set of finite singularities of its inverse is bounded. Bergweiler and Kotus gave bounds for the Hausdorff dimension of escaping sets if the function has no logarithmic singularities over infinity, the multiplicities of poles are bounded and the order is finite. We study the case of infinite order and find gauge functions for which the Hausdorff measure of escaping sets is zero or infinity.
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Existence criteria for the $(b,c)$inverse are given.% in terms of annihilators. We present explicit expressions for the $(b,c)$inverse by using inner inverses. We answer the question when the $(b,c)$inverse of $a\in R$ is an inner inverse of $a$. As applications, we give a unified theory of some wellknown results of the $\{1,3\}$inverse, the $\{1,4\}$inverse, the MoorePenrose inverse, the group inverse and the core inverse.
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An anonymous reader writes: "The price of ethereum crashed as low as 10 cents from around $319 in about a second on the GDAX cryptocurrency exchange on Wednesday," reports CNBC, calling it "a move that is being blamed on a 'multimillion dollar market sell' order... As the price continued to fall, another 800 stop loss orders and margin funding liquidations caused ethereum to trade as low as 10 cents." An executive for the exchange said "Our matching engine operated as intended throughout this event and trading with advanced features like margin always carries inherent risk." Though some users complained they lost money, the price rebounded to $325  and according to a report on one trading site, "one person had an order in for just over 3,800 ethereum if the price fell to 10 cents on the GDAX exchange," reports CNBC. "Theoretically this person would have spent $380 to buy these coins, and when the price shot up above $300 again, the trader would be sitting on over $1 million." Yet th
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We introduce a category O of modules over the elliptic quantum group of sl_N with wellbehaved qcharacter theory. We construct asymptotic modules as analytic continuation of a family of finitedimensional modules, the KirillovReshetikhin modules. In the Grothendieck ring of this category we prove two types of identities: generalized Baxter relations in the spirit of FrenkelHernandez between finitedimensional modules and asymptotic modules; threeterm Baxter TQ relations of infinitedimensional modules.
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By using the representation theory of the elliptic quantum group U_{q,p}(sl_N^), we present a systematic method of deriving the weight functions. The resultant sl_N type elliptic weight functions are new and give elliptic and dynamical analogues of those obtained in the trigonometric case. We then discuss some basic properties of the elliptic weight functions. We also present an explicit formula for formal elliptic hypergeometric integral solution to the face type, i.e. dynamical, elliptic qKZ equation.
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Quantum entanglement shared by remote agents serves as a valuable resource for promising applications in distributed computing, cryptography, and sensing. However, distributing entangled states with high fidelity via fiber optic routes is challenging due to the various decoherence mechanisms in fibers. In particular, one of the primary polarization decoherence mechanism in optical fibers is polarization mode dispersion (PMD), which is the distortion of optical pulses by random birefringences in the system. Among quantum entanglement distillation (QED) algorithms proposed to mitigate decoherence, the recurrence QED algorithms require the smallest size of quantum circuits, and are most robust against severe decoherence. On the other hand, the yield of recurrence QED algorithms drops exponentially with respect to the rounds of distillation, and hence it is critical to minimize the required rounds of distillation. We present a recurrence QED algorithm, which is capable of achieving maximum
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We discuss Jacobi forms that are invariant under the action of the Weyl group of type E_n (n=6,7,8). For n=6,7 we explicitly construct a full set of generators of the algebra of E_n weak Jacobi forms. We first construct n+1 independent E_n Jacobi forms in terms of Jacobi theta functions and modular forms. By using them we obtain SeibergWitten curves of type E_6 and E_7 for the Estring theory. The coefficients of each curve are E_n weak Jacobi forms of particular weights and indices specified by the root system, realizing the generators whose existence was shown some time ago by Wirthm\"uller.
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Let $p$ be a rational prime and $q$ a power of $p$. Let $\wp$ be a monic irreducible polynomial of degree $d$ in $\mathbf{F}_q[t]$. In this paper, we define an analogue of the HodgeTate map which is suitable for the study of Drinfeld modules over $\mathbf{F}_q[t]$ and, using it, develop a geometric theory of $\wp$adic Drinfeld modular forms similar to Katz's theory in the case of elliptic modular forms. In particular, we show that for Drinfeld modular forms with congruent Fourier coefficients at $\infty$ modulo $\wp^n$, their weights are also congruent modulo $(q^d1)p^{\lceil \log_p(n)\rceil}$, and that Drinfeld modular forms of level $\Gamma_1(\mathfrak{n})\cap \Gamma_0(\wp)$, weight $k$ and type $m$ are $\wp$adic Drinfeld modular forms for any tame level $\mathfrak{n}$ with a prime factor of degree prime to $q1$.
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We study the geometry of domains in complete metric measure spaces equipped with a doubling measure supporting a $1$Poincar\'e inequality. We propose a notion of \emph{domain with boundary of positive mean curvature} and prove that, for such domains, there is always a solution to the Dirichlet problem for least gradients with continuous boundary data. Here \emph{least gradient} is defined as minimizing total variation (in the sense of BV functions) and boundary conditions are satisfied in the sense that the \emph{boundary trace} of the solution exists and agrees with the given boundary data. This extends the result of Sternberg, Williams and Ziemer to the nonsmooth setting. Via counterexamples we also show that uniqueness of solutions and existence of \emph{continuous} solutions can fail, even in the weighted Euclidean setting with Lipschitz weights.
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We propose DirectedDistributed Projected Subgradient (DDPS) to solve a constrained optimization problem over a multiagent network, where the goal of agents is to collectively minimize the sum of locally known convex functions. Each agent in the network owns only its local objective function, constrained to a commonly known convex set. We focus on the circumstance when communications between agents are described by a \emph{directed} network. The DDPS combines surplus consensus to overcome the asymmetry caused by the directed communication network. The analysis shows the convergence rate to be $O(\frac{\ln k}{\sqrt{k}})$.
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We consider bond percolation on $\Z^d\times \Z^s$ where edges of $\Z^d$ are open with probability $p<p_c(\Z^d)$ and edges of $\Z^s$ are open with probability $q$, independently of all others. We obtain bounds for the critical curve in $(p, q)$, with $p$ close to the critical threshold $p_c(\Z^d)$. The results are related to the socalled dimensional crossover from $\Z^d$ to $\Z^{d+s}$.
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We propose an empirical estimator of the preferential attachment function $f$ in the setting of general preferential attachment trees. Using a supercritical continuoustime branching process framework, we prove the almost sure consistency of the proposed estimator. We perform simulations to study the empirical properties of our estimators.
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Using a family of mock modular forms constructed by Zagier, we study the coefficients of a mock modular form of weight $3/2$ on $\operatorname{SL}_2(\mathbb{Z})$ modulo primes $\ell\geq 5$. These coefficients are related to the smallest parts function of Andrews. As an application, we reprove a theorem of Garvan regarding the properties of this function modulo $\ell$. As another application, we show that congruences modulo $\ell$ for the smallest parts function are rare in a precise sense.
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In this article, we present a complete study of two disjoint classes of conformal vector fields on doubly warped product manifolds as well as on doubly warped spacetimes. Then we study Ricci solitons on doubly warped product manifollds admitting these types of conformal vector fields.
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We describe and analyze an algorithm for computing the homology (Betti numbers and torsion coefficients) of basic semialgebraic sets which works in weak exponential time. That is, out of a set of exponentially small measure in the space of data the cost of the algorithm is exponential in the size of the data. All algorithms previously proposed for this problem have a complexity which is doubly exponential (and this is so for almost all data).
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We consider a mobile edge computing problem, in which mobile users offload their computation tasks to computing nodes (e.g., base stations) at the network edge. The edge nodes compute the requested functions and communicate the computed results to the users via wireless links. For this problem, we propose a Universal Coded Edge Computing (UCEC) scheme for linear functions to simultaneously minimize the load of computation at the edge nodes, and maximize the physicallayer communication efficiency towards the mobile users. In the proposed UCEC scheme, edge nodes create coded inputs of the users, from which they compute coded output results. Then, the edge nodes utilize the computed coded results to create communication messages that zeroforce all the interference signals over the air at each user. Specifically, the proposed scheme is universal since the coded computations performed at the edge nodes are oblivious of the channel states during the communication process from the edge node
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We investigate the maximum coding rate achievable on a twouser broadcast channel for the case where a common message is transmitted with feedback using either fixedblocklength codes or variablelength codes. For the fixedblocklengthcode setup, we establish nonasymptotic converse and achievability bounds. An asymptotic analysis of these bounds reveals that feedback improves the secondorder term compared to the nofeedback case. In particular, for a certain class of antisymmetric broadcast channels, we show that the dispersion is halved. For the variablelengthcode setup, we demonstrate that the channel dispersion is zero.
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