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  • Let $p$ be an odd prime and $F_{\infty}$ a $p$-adic Lie extension of a number field $F$ with Galois group $G$. Suppose that $G$ is a compact pro-$p$ $p$-adic Lie group with no torsion and that it contains a closed normal subgroup $H$ such that $G/H\cong \mathbb{Z}_p$. Under various assumptions, we establish asymptotic upper bounds for the growth of $p$-exponents of the class groups in the said $p$-adic Lie extension. Our results generalize a previous result of Lei, where he established such an estimate under the assumption that $H\cong \mathbb{Z}_p$.

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  • Recently Rai obtained an upper bound for the order of the Schur multiplier of a $d$-generator special $p$-group when its derived subgroup has the maximum value $ p^{\frac{1}{2}d(d-1)}$ for $ d\geq 3 $ and $ p\neq 2. $ Here we try to obtain the Schur multiplier, the exterior square and the tensor square of such $p$-groups. Then we specify which ones are capable. Moreover, we give an upper bound for the order of the Schur multiplier, the exterior product and the tensor square of a $d$-generator special $p$-group $ G $ when $ |G'|=p^{\frac{1}{2}d(d-1)-1}$ for $ d\geq 3 $ and $ p\neq 2. $ Additionally, when $ G $ is of exponent $ p, $ we give the structure of $ G. $

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  • We explore properties of braids such as their fractional Dehn twist coefficients, right-veeringness, and quasipositivity, in relation to the transverse invariant from Khovanov homology defined by Plamenevskaya for their closures, which are naturally transverse links in the standard contact $3$-sphere. For any $3$-braid $\beta$, we show that the transverse invariant of its closure does not vanish whenever the fractional Dehn twist coefficient of $\beta$ is strictly greater than one. We show that Plamenevskaya's transverse invariant is stable under adding full twists on $n$ or fewer strands to any $n$-braid, and use this to detect families of braids that are not quasipositive. Motivated by the question of understanding the relationship between the smooth isotopy class of a knot and its transverse isotopy class, we also exhibit an infinite family of pretzel knots for which the transverse invariant vanishes for every transverse representative, and conclude that these knots are not quasipos

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  • It is shown that if a $d$-dimensional cube is decomposed into n cubes, the side lengths of which belong to the interval $\left(1-\frac{1}{n^{1/d}+1}, 1\right], then $n$ is a perfect $d$-th power and all cubes are of the same size. This result is essentially tight.

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  • Let $H$ be a planar graph. By a classical result of Robertson and Seymour, there is a function $f:\mathbb{N} \to \mathbb{R}$ such that for all $k \in \mathbb{N}$ and all graphs $G$, either $G$ contains $k$ vertex-disjoint subgraphs each containing $H$ as a minor, or there is a subset $X$ of at most $f(k)$ vertices such that $G-X$ has no $H$-minor. We prove that this remains true with $f(k) = c k \log k$ for some constant $c=c(H)$. This bound is best possible, up to the value of $c$, and improves upon a recent result of Chekuri and Chuzhoy [STOC 2013], who established this with $f(k) = c k \log^d k$ for some universal constant $d$. The proof is constructive and yields a polynomial-time $O(\log \mathsf{OPT})$-approximation algorithm for packing subgraphs containing an $H$-minor.

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  • We show that additive higher Chow groups of S. Bloch and H. Esnault of smooth varieties over an arbitrary field induce a Zariski sheaf of pro-differential graded algebras, whose Milnor range is isomorphic to the Zariski sheaf of the big de Rham-Witt complexes of L. Hesselholt and I. Madsen. When the characteristic $p$ of the field is positive, the Zariski hypercohomology of the $p$-typical part of the sheaves arising from additive higher Chow groups computes the crystalline cohomology of smooth varieties. This revisits the 1970s results of S. Bloch and L. Illusie on crystalline cohomology, this time from algebraic cycles.

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  • We give an FPTAS and an efficient sampling algorithm for the high-fugacity hard-core model on bounded-degree bipartite expander graphs and the low-temperature ferromagnetic Potts model on bounded-degree expander graphs. The results apply, for example, to random (bipartite) $\Delta$-regular graphs, for which no efficient algorithms were known for these problems (with the exception of the Ising model) in the non-uniqueness regime of the infinite $\Delta$-regular tree.

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  • Promise CSPs are a relaxation of constraint satisfaction problems where the goal is to find an assignment satisfying a relaxed version of the constraints. Several well-known problems can be cast as promise CSPs including approximate graph coloring, discrepancy minimization, and interesting variants of satisfiability. Similar to CSPs, the tractability of promise CSPs can be tied to the structure of operations on the solution space called polymorphisms, though in the promise world these operations are much less constrained. Under the thesis that non-trivial polymorphisms govern tractability, promise CSPs therefore provide a fertile ground for the discovery of novel algorithms. In previous work, we classified Boolean promise CSPs when the constraint predicates are symmetric. In this work, we vastly generalize these algorithmic results. Specifically, we show that promise CSPs that admit a family of "regional-periodic" polymorphisms are in P, assuming that determining which region a point i

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  • Approximate message passing (AMP) methods have gained recent traction in sparse signal recovery. Additional information about the signal, or side information (SI), is commonly available and can aid in efficient signal recovery. In this work, we present an AMP-based framework that exploits SI and can be readily implemented in various settings. To illustrate the simplicity and wide applicability of our approach, we apply this framework to a Bernoulli-Gaussian (BG) model and a time-varying birth- death-drift (BDD) signal model, motivated by applications in channel estimation. We develop a suite of algorithms, called AMP-SI, and derive denoisers for the BDD and BG models. We also present numerical evidence demonstrating the advantages of our approach, and empirical evidence of the accuracy of a proposed state evolution.

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  • This article is the first part of a two-fold study, the objective of which is the theoretical analysis and numerical investigation of new approximate corrector problems in the context of stochastic homogenization. We present here three new alternatives for the approximation of the homogenized matrix for diffusion problems with highly-oscillatory coefficients. These different approximations all rely on the use of an embedded corrector problem (that we previously introduced in [Canc\`es, Ehrlacher, Legoll and Stamm, C. R. Acad. Sci. Paris, 2015]), where a finite-size domain made of the highly oscillatory material is embedded in a homogeneous infinite medium whose diffusion coefficients have to be appropriately determined. The motivation for considering such embedded corrector problems is made clear in the companion article [Canc\`es, Ehrlacher, Legoll, Stamm and Xiang, in preparation], where a very efficient algorithm is presented for the resolution of such problems for particular hetero

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  • We consider the following repulsive-productive chemotaxis model: Let $p\in (1,2)$, find $u \geq 0$, the cell density, and $v \geq 0$, the chemical concentration, satisfying \begin{equation}\label{C5:Am} \left\{ \begin{array} [c]{lll} \partial_t u - \Delta u - \nabla\cdot (u\nabla v)=0 \ \ \mbox{in}\ \Omega,\ t>0,\\ \partial_t v - \Delta v + v = u^p \ \ \mbox{in}\ \Omega,\ t>0, \end{array} \right. \end{equation} in a bounded domain $\Omega\subseteq \mathbb{R}^d$, $d=2,3$. By using a regularization technique, we prove the existence of solutions of this problem. Moreover, we propose three fully discrete Finite Element (FE) nonlinear approximations, where the first one is defined in the variables $(u,v)$, and the second and third ones by introducing ${\boldsymbol\sigma}=\nabla v$ as an auxiliary variable. We prove some unconditional properties such as mass-conservation, energy-stability and solvability of the schemes. Finally, we compare the behavior of the schemes throughout several

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  • Let $\Omega$ be a bounded, convex open subset of $\mathbb{R}^n$. Let $p>1$ and let $F:\mathbb{R}^n\rightarrow[0,+\infty)$ be a Finsler norm. In this paper we study a particular anisotropic and scale invariant functional in the form: $$\mathcal{F}(\Omega)=\dfrac{\displaystyle\int_{\partial \Omega} [F^o(x) ]^p\;F(\nu_{\partial\Omega}(x))\;d\mathcal{H}^{n-1}(x) }{P_F(\Omega)V(\Omega)^{\frac{p}{n}}};$$ we call anisotropic $p-$momentum the quantity $M_{F}(\Omega):=\int_{\partial \Omega} [F^o(x) ]^p\;F(\nu_{\partial\Omega}(x))\;d\mathcal{H}^{n-1}(x) ,$ where $F^o$ is the polar function of $F$ and $\nu_{\partial \Omega}$ is the outward unit normal to $\partial\Omega$. By $V(\Omega)$ and $P_F(\Omega)$ we denote respectively the volume of $\Omega$ and its anisotropic perimeter, i.e. $ P_F(\Omega)=\int_{\partial \Omega}F(\nu_{\partial\Omega}(x))\;d\mathcal{H}^{n-1}(x).$ We show that the Wulff shape, associated with the Finsler norm $F$ considered and centered at the origin, is the unique mini

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  • Given a graph $G$ and integers $k,\ell$ such that $0\le \ell \le \binom{k}{2}$, what is the fraction of $k$-vertex subsets of $G$ which span exactly $\ell$ edges? When $G$ is empty or complete, and $\ell$ is zero or $\binom{k}{2}$, this fraction can be exactly 1. On the other hand, if $\ell$ is far from these extreme values, one might expect that this fraction is substantially smaller than 1. This was recently proved by Alon, Hefetz, Krivelevich and Tyomkyn who intiated the systematic study of this question and proposed several natural conjectures. Let $\ell^{*}=\min\{\ell,\binom{k}{2}-\ell\}$. Our main result is that for any $k$ and $\ell$, the fraction of $k$-vertex subsets that span $\ell$ edges is at most $\log^{O\left(1\right)}\left(\ell^{*}/k\right)\sqrt{k/\ell^{*}}$, which is best-possible up to the logarithmic factor. This improves on multiple results of Alon, Hefetz, Krivelevich and Tyomkyn, and resolves one of their conjectures. In addition, we also make some first steps towa

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  • The main results of this paper provide a polynomial time algorithm for approximating the logarithm of the number of maximal near perfect matchings in dense graphs. By dense we mean that $|E(G)|\geq\alpha|V(G)|^2$ for some fixed $\alpha>0$, and a maximal $\varepsilon$-near perfect matching is a maximal matching which covers at least $(1-\varepsilon)|V(G)|$ vertices. More precisely, we provide a deterministic algorithm that for a given (dense) graph $G$ of order $n$ and a real number $\varepsilon>0$, returns either a conclusion that $G$ has no $\varepsilon$-near perfect matching, or a positive real number $m\leq1/2$, such that the logarithm of the number of maximal $\varepsilon$-near perfect matchings in $G$ is at least $mn\log n$. The upper bound of such graphs is always $1/2(n\log n)$. The running time of this algorithm is $O(f(\varepsilon)n^{5/2})$, where $f(\cdot)$ is an explicit function. Additionally, for a special class of dense graphs, we show that $(1+\varepsilon)mn\log n$

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  • We investigate connections between Lipschitz geometry of real algebraic varieties and properties of their arc spaces. For this purpose we develop motivic integration in the real algebraic set-up. We construct a motivic measure on the space of real analytic arcs. We use this measure to define a real motivic integral which admits a change of variable formula not only for the birational but also for generically one-to-one Nash maps. As a consequence we obtain an inverse mapping theorem which holds for continuous rational maps and, more generally, for generically arc-analytic maps. These maps appeared recently in the classification of singularities of real analytic function germs. Finally, as an application, we characterize in terms of the motivic measure, germs of arc-analytic homeomorphism between real algebraic varieties which are bi-Lipschitz for the inner metric.

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  • We extend a model of feedback and contagion in large mean-field systems by introducing a common source of noise driven by Brownian motion. Although the dynamics in the model are continuous, the feedback effect can lead to jump discontinuities in the solutions --- i.e. 'blow-ups'. We prove existence of solutions to the corresponding conditional McKean--Vlasov equation and we show that the pathwise realisation of the common noise can both trigger and prevent blow-ups.

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  • Under some conditions on the deformation type, which we expect to be satisfied for arbitrary irreducible symplectic varieties, we describe which big and nef line bundles on irreducible symplectic varieties have base divisors. In particular, we show that such base divisors are always irreducible and reduced. This is applied to understand the behaviour of divisorial base components of big and nef line bundles under deformations and for K3$^{[n]}$-type and Kum$^n$-type.

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  • The Harary--Hill conjecture, still open after more than 50 years, asserts that the crossing number of the complete graph $K_n$ is $ H(n) = \frac 1 4 \left\lfloor\frac{\mathstrut n}{\mathstrut 2}\right\rfloor \left\lfloor\frac{\mathstrut n-1}{\mathstrut 2}\right\rfloor \left\lfloor\frac{\mathstrut n-2}{\mathstrut 2}\right\rfloor \left\lfloor\frac{\mathstrut n-3}{\mathstrut 2}\right \rfloor$. \'Abrego et al. introduced the notion of shellability of a drawing $D$ of $K_n$. They proved that if $D$ is $s$-shellable for some $s\geq\lfloor\frac{n}{2}\rfloor$, then $D$ has at least $H(n)$ crossings. This is the first combinatorial condition on a drawing that guarantees at least $H(n)$ crossings. In this work, we generalize the concept of $s$-shellability to bishellability, where the former implies the latter in the sense that every $s$-shellable drawing is, for any $b \leq s-2$, also $b$-bishellable. Our main result is that $(\lfloor \frac{n}{2} \rfloor\!-\!2)$-bishellability of a drawing $D$

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  • We extend the derived Algebraic cobordism of Lowrey and Sch\"urg to a bivariant theory in the sense of Fulton and MacPherson, and establish some of its basic properties. We also formalize some previously known results about orientations in bivariant theories in order to illuminate the fact that they should be thought as properties of the category on which the bivariant theory is defined rather than properties of the bivariant theories themselves. We establish connections of the newly constructed theory to the bivariant algebraic $K$-theory and prove an analogue of the Conner-Floyd theorem on the corresponding cohomology rings.

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  • Monotone operator splitting is a powerful paradigm that facilitates parallel processing for optimization problems where the cost function can be split into two convex functions. We propose a generalized form of monotone operator splitting based on Bregman divergence. We show that an appropriate design of the Bregman divergence leads to faster convergence than conventional splitting algorithms. The proposed Bregman monotone operator splitting (B-MOS) is applied to an application to illustrate its effectiveness. B-MOS was found to significantly improve the convergence rate.

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  • In this paper, we consider a novel cache-enabled heterogeneous network (HetNet), where macro base stations (BSs) with traditional sub-6 GHz are overlaid by dense millimeter wave (mmWave) pico BSs. These two-tier BSs, which are modeled as two independent homogeneous Poisson Point Processes, cache multimedia contents following the popularity rank. High-capacity backhauls are utilized between macro BSs and the core server. In contrast to the simplified flat-top antenna pattern analyzed in previous articles, we employ an actual antenna model with the uniform linear array at all mmWave BSs. To evaluate the performance of our system, we introduce two distinctive user association strategies: 1) maximum received power (Max-RP) scheme; and 2) maximum rate (Max-Rate) scheme. With the aid of these two schemes, we deduce new theoretical equations for success probabilities and area spectral efficiencies (ASEs). Considering a special case with practical path loss laws, several closed-form expression

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  • Any irreducible Dynkin diagram $\Delta$ is obtained from an irreducible Dynkin diagram $\Delta_h$ of type $\mathrm{ADE}$ by folding via graph automorphisms. For any simple complex Lie group $G$ with Dynkin diagram $\Delta$ and compact Riemann surface $\Sigma$, we give a Lie-theoretic construction of families of quasi-projective Calabi-Yau threefolds together with an action of graph automorphisms over the Hitchin base associated to the pair $(\Sigma, G)$ . These give rise to Calabi-Yau orbifolds over the same base. Their intermediate Jacobian fibration, constructed in terms of equivariant cohomology, is isomorphic to the Hitchin system of the same type away from singular fibers.

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  • Let $G$ and $H$ be graphs. We say that $P$ is an $H$-packing of $G$ if $P$ is a set of edge-disjoint copies of $H$ in $G$. An $H$-packing $P$ is maximal if there is no other $H$-packing of $G$ that properly contains $P$. Packings of maximum cardinality have been studied intensively, with several recent breakthrough results. Here, we consider minimum cardinality maximal packings. An $H$-packing $P$ is clumsy if it is maximal of minimum size. Let $cl(G,H)$ be the size of a clumsy $H$-packing of $G$. We provide nontrivial bounds for $cl(G,H)$, and in many cases asymptotically determine $cl(G,H)$ for some generic classes of graphs $G$ such as $K_n$ (the complete graph), $Q_n$ (the cube graph), as well as square, triangular, and hexagonal grids. We asymptotically determine $cl(K_n,H)$ for every fixed non-empty graph $H$. In particular, we prove that $$ cl(K_n, H) = \frac{\binom{n}{2}- ex(n,H)}{|E(H)|}+o(ex(n,H)),$$ where $ex(n,H)$ is the extremal number of $H$. A related natural parameter i

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  • The category of Cohen-Macaulay modules of an algebra $B_{k,n}$ is used in [JKS16] to give an additive categorification of the cluster algebra structure on the homogeneous coordinate ring of the Grassmannian of $k$-planes in $n$-space. We study the Auslander-Reiten translation periodicity for this category, extensions, and we find canonical Auslander-Reiten sequences. Then, we focus on the tame cases and establish a correspondence between certain rigid indecomposable modules of rank 2 and real roots of degree 2 for the associated Kac-Moody algebra. We also an give explicit construction of indecomposable rank 2 modules.

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  • Let $\mathcal{I}_{d,g,r}$ be the union of irreducible components of the Hilbert scheme whose general points correspond to smooth irreducible non-degenerate curves of degree $d$ and genus $g$ in $\mathbb{P}^r$. We use families of curves on cones to show that under certain numerical assumptions for $d$, $g$ and $r$, the scheme $\mathcal{I}_{d,g,r}$ acquires generically smooth components whose general points correspond to curves that are double covers of irrational curves. In particular, in the case $\rho(d,g,r) := g-(r+1)(g-d+r) \geq 0$ we construct explicitly a regular component that is different from the distinguished component of $\mathcal{I}_{d,g,r}$ dominating the moduli space $\mathcal{M}_g$.

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  • Let $\pi \in \mathfrak{S}_m$ and $\sigma \in \mathfrak{S}_n$ be permutations. An occurrence of $\pi$ in $\sigma$ as a consecutive pattern is a subsequence $\sigma_i \sigma_{i+1} \cdots \sigma_{i+m-1}$ of $\sigma$ with the same order relations as $\pi$. We say that patterns $\pi, \tau \in \mathfrak{S}_m$ are strongly c-Wilf equivalent if for all $n$ and $k$, the number of permutations in $\mathfrak{S}_n$ with exactly $k$ occurrences of $\pi$ as a consecutive pattern is the same as for $\tau$. In 2018, Dwyer and Elizalde conjectured (generalizing a conjecture of Elizalde from 2012) that if $\pi, \tau \in \mathfrak{S}_m$ are strongly c-Wilf equivalent, then $(\tau_1, \tau_m)$ is equal to one of $(\pi_1, \pi_m)$, $(\pi_m, \pi_1)$, $(m+1 - \pi_1, m+1-\pi_m)$, or $(m+1 - \pi_m, m+1 - \pi_1)$. We prove this conjecture using the cluster method introduced by Goulden and Jackson in 1979, which Dwyer and Elizalde previously applied to prove that $|\pi_1 - \pi_m| = |\tau_1 - \tau_m|$. A consequenc

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  • We propose an extension to our monotone and convergent method for the Monge-Amp\`{e}re equation in dimension $d \geq2$, that incorporates the idea of filtered schemes. The method combines our original monotone operator with a more accurate non-monotone modification, using an appropriately chosen filter. This results in a remarkable improvement of accuracy, but without sacrificing the convergence to the unique viscosity solution.

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  • A marquee feature of quantum behavior is that, upon probing, the microscopic system emerges in one of multiple possible states. While quantum mechanics postulates the respective probabilities, the effective abundance of these simultaneous ``identities'', if a meaningful concept at all, has to be inferred. To address such problems, we construct and analyze the theory of functions assigning the quantity (effective number) of objects endowed with probability weights. In a surprising outcome, the consistency of such probability-dependent measure assignments entails the existence of a minimal amount, realized by a unique effective number function. This result provides a well-founded solution to identity-counting problems in quantum mechanics. Such problems range from counting the basis states contained in an output of a quantum computation, and relevant in the analysis of quantum algorithms, to a novel way to characterize complex states such as QCD vacuum or eigenstates of quantum spin syst

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  • We discuss a Curie-Weiss model with two groups in the critical regime. This is the region where the central limit theorem does not hold any more but the mean magnetization still goes to zero as the number of spins grows. We show that the total magnetization normalized by $N^{3/4}$ converges to a non-trivial distribution which is not Gaussian, just as in the single-group Curie-Weiss model.

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  • This paper presents a device-to-device (D2D) user selection protocol wherein multiple D2D pairs coexist with a cellular network. In the developed framework, certain D2D users harvest energy and share the spectrum of the cellular users by adopting a hybrid time switching and power splitting protocol. The D2D user which harvests the maximum energy and achieves the desired target rate for the cellular communication is selected to serve as a decode-and-forward (DF) relay for the cellular user. The proposed work analyzes the impact of increase in the number of D2D users on the performance of cellular user as well as derives an upper bound on the time duration of energy harvesting within which best possible rate for cellular user can be obtained. The performance of the proposed protocol has been quantified by obtaining the closed form expressions of outage probability.

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  • The distribution of random parameters in, and the input signal to, a distributed parameter model with unbounded input and output operators for the transdermal transport of ethanol are estimated. The model takes the form of a diffusion equation with the input, which is on the boundary of the domain, being the blood or breath alcohol concentration (BAC/BrAC), and the output, also on the boundary, being the transdermal alcohol concentration (TAC). Our approach is based on the reformulation of the underlying dynamical system in such a way that the random parameters are treated as additional spatial variables. When the distribution to be estimated is assumed to be defined in terms of a joint density, estimating the distribution is equivalent to estimating a functional diffusivity in a multi-dimensional diffusion equation. The resulting system is referred to as a population model, and well-established finite dimensional approximation schemes, functional analytic based convergence arguments,

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  • We study locally compact groups having all dense subgroups (locally) minimal. We call such groups densely (locally) minimal. In 1972 Prodanov proved that the infinite compact abelian groups having all subgroups minimal are precisely the groups $\mathbb Z_p$ of $p$-adic integers. In [30], we extended Prodanov's theorem to the non-abelian case at several levels. In this paper, we focus on the abelian case. We prove that in case that a topological abelian group $G$ is either compact or connected locally compact, then $G$ is densely locally minimal if and only if $G$ is either a Lie group or has an open subgroup isomorphic to $\mathbb Z_p$ for some prime $p$. This should be compared with the main result of [9]. Our Theorem C provides another extension of Prodanov's theorem: an infinite locally compact group is densely minimal if and only if it is isomorphic to $\mathbb Z_p$ . In contrast, we show that there exists a densely minimal, compact, two-step nilpotent group that neither is a Lie g

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  • As multi-agent systems proliferate and share more and more user data, new approaches are needed to protect sensitive data while still guaranteeing successful operation. To address this need, we present a private multi-agent LQ control framework. We consider problems in which each agent has linear dynamics and the agents are coupled by a quadratic cost. Generating optimal control values for the agents is a centralized operation, and we therefore introduce a cloud computer into the network for this purpose. The cloud is tasked with aggregating agents' outputs, computing control inputs, and transmitting these inputs to the agents, which apply them in their state updates. Agents' state information can be sensitive and we therefore protect it using differential privacy. Differential privacy is a statistical notion of privacy enforced by adding noise to sensitive data before sharing it, and agents will therefore add noise to all data before sending it to the cloud. The result is a private mu

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  • We show that for generalized Baker's transformations there is a parameter domain where we have an absolutely continuous ergodic measure and in direct neighborhood there is a parameter domain where not even the variational principle for Hausdorff dimension holds.

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  • We develop the dimension theory for a class of linear solenoids, which have a "fractal" attractor. We will find the dimension of the attractor, proof formulas for the dimension of ergodic measures on this attractor and discuss the question whether there exists a measure of full dimension.

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  • We show that fractal percolation sets in $\mathbb{R}^{d}$ almost surely intersect every hyperplane absolutely winning (HAW) set with full Hausdorff dimension. In particular, if $E\subset\mathbb{R}^{d}$ is a realization of a fractal percolation process, then almost surely (conditioned on $E\neq\emptyset$), for every countable collection $\left(f_{i}\right)_{i\in\mathbb{N}}$ of $C^{1}$ diffeomorphisms of $\mathbb{R}^{d}$, $\dim_{H}\left(E\cap\left(\bigcap_{i\in\mathbb{N}}f_{i}\left(\text{BA}_{d}\right)\right)\right)=\dim_{H}\left(E\right)$, where $\text{BA}_{d}$ is the set of badly approximable vectors in $\mathbb{R}^{d}$. We show this by proving that $E$ almost surely contains hyperplane diffuse subsets which are Ahlfors-regular with dimensions arbitrarily close to $\dim_{H}\left(E\right)$. We achieve this by analyzing Galton-Watson trees and showing that they almost surely contain appropriate subtrees whose projections to $\mathbb{R}^{d}$ yield the aforementioned subsets of $E$. This m

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  • We introduce a theoretical and computational framework to use discrete Morse theory as an efficient preprocessing in order to compute zigzag persistent homology. From a zigzag filtration of complexes $(K_i)$, we introduce a {\em zigzag Morse filtration} whose complexes $(A_i)$ are Morse reductions of the original complexes $(K_i)$, and we prove that they both have same persistent homology. This zigzag Morse filtration generalizes the {\em filtered Morse complex} of Mischaikow and Nanda~\cite{MischaikowN13}, defined for standard persistence. The maps in the zigzag Morse filtration are forward and backward inclusions, as is standard in zigzag persistence, as well as a new type of map inducing non trivial changes in the boundary operator of the Morse complex. We study in details this last map, and design algorithms to compute the update both at the complex level and at the homology matrix level when computing zigzag persistence. We deduce an algorithm to compute the zigzag persistence of

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  • In this paper, we initiate the study of distributional chaos for weighted translations on locally compact groups, and give a sufficient condition for such operators to be distributionally chaotic. We also investigate the set of distributionally irregular vectors of weighted translations from the views of modulus, cone, equivalent class and atom. In particular, we show that the set of distributionally irregular vectors is residual if the group is the integer. Besides, the equivalent class of distributionally irregular vectors is path connected if the field is complex.

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  • Multicellular systems play a key role in bioprocess and biomedical engineering. Cell ensembles encountered in these setups show phenotypic variability like size and biochemical composition. As this variability may result in undesired effects in bioreactors, close monitoring of the cell population heterogeneity is important for maximum production output, and accurate control. However, direct measurements are mostly restricted to a few cellular properties. This motivates the application of model-based online estimation techniques for the reconstruction of non-measurable cellular properties. Population balance modeling allows for a natural description of cell-to-cell variability. In this contribution, we present an estimation approach that, in contrast to existing ones, does not rely on a finite-dimensional approximation through grid based discretization of the underlying population balance model. Instead, our so-called characteristics based density estimator employs sample approximations

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  • We analyze the distribution of eigenvectors for mesoscopic, mean-field perturbations of diagonal matrices in the bulk of the spectrum. Our results apply to a generalized $N\times N$ Rosenzweig-Porter model. We prove that the eigenvectors entries are asymptotically Gaussian with a specific variance, localizing them onto a small, explicit part of the spectrum. For a well spread initial spectrum, this variance profile universally follows a heavy-tailed Cauchy distribution. In the case of smooth entries, we also obtain a strong form of quantum unique ergodicity as an overwhelming probability bound on the eigenvectors probability mass. The proof relies on a priori local laws for this model and the eigenvector moment flow.

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  • A finite dimensional abstract approximation and convergence theory is developed for estimation of the distribution of random parameters in infinite dimensional discrete time linear systems with dynamics described by regularly dissipative operators and involving, in general, unbounded input and output operators. By taking expectations, the system is re-cast as an equivalent abstract parabolic system in a Gelfand triple of Bochner spaces wherein the random parameters become new space-like variables. Estimating their distribution is now analogous to estimating a spatially varying coefficient in a standard deterministic parabolic system. The estimation problems are approximated by a sequence of finite dimensional problems. Convergence is established using a state space-varying version of the Trotter-Kato semigroup approximation theorem. Numerical results for a number of examples involving the estimation of exponential families of densities for random parameters in a diffusion equation with

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  • We introduce a class of subshifts called eventually dendric. This class generalizes the class of dendric subshifts studied in previous papers as the class of subshifts whose language is a tree set and defined by a property of the extensions of each word. Our main result is that the class of eventually dendric subshifts is closed under conjugacy.

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  • Kingman (1978)'s representation theorem states that any exchangeable partition of $\mathbb{N}$ can be represented as a paintbox based on a random mass-partition. Similarly, any exchangeable composition (i.e.\ ordered partition of $\mathbb{N}$) can be represented as a paintbox based on an interval-partition (Gnedin 1997). Our first main result is that any exchangeable coalescent process (not necessarily Markovian) can be represented as a paintbox based on a random non-decreasing process valued in interval-partitions, called nested interval-partition, generalizing the notion of comb metric space introduced by Lambert \& Uribe Bravo (2017) to represent compact ultrametric spaces. As a special case, we show that any $\Lambda$-coalescent can be obtained from a paintbox based on a unique random nested interval partition called $\Lambda$-comb, which is Markovian with explicit semi-group. This nested interval-partition directly relates to the flow of bridges of Bertoin \& Le~Gall (2003

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  • We follow Jacquet-Shalika, Matringe and Cogdell-Matringe to define exterior square gamma factors for irreducible cuspidal representations of $\mathrm{GL}_n(\mathbb{F}_q)$. These exterior square gamma factors are expressed in terms of Bessel functions, or in terms of the regular characters associated with the cuspidal representations. We also relate our exterior square gamma factors over finite fields to those over local fields through level zero representations.

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  • An ordered hypergraph is a hypergraph whose vertex set is linearly ordered, and a convex geometric hypergraph is a hypergraph whose vertex set is cyclically ordered. Extremal problems for ordered and convex geometric graphs have a rich history with applications to a variety of problems in combinatorial geometry. In this paper, we consider analogous extremal problems for uniform hypergraphs, and discover a general partitioning phenomenon which allows us to determine the order of magnitude of the extremal function for various ordered and convex geometric hypergraphs. A special case is the ordered $n$-vertex $r$-graph $F$ consisting of two disjoint sets $e$ and $f$ whose vertices alternate in the ordering. We show that for all $n \geq 2r + 1$, the maximum number of edges in an ordered $n$-vertex $r$-graph not containing $F$ is exactly \[ {n \choose r} - {n - r \choose r}.\] This could be considered as an ordered version of the Erd\H{o}s-Ko-Rado Theorem, and generalizes earlier results of

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  • This paper focusses on finite volume schemes for solving multilayer diffusion problems. We develop a finite volume method that addresses a deficiency of recently proposed finite volume/difference methods, which consider only a limited number of interface conditions and do not carry out stability or convergence analysis. Our method also retains second order accuracy in space while preserving the tridiagonal matrix structure of the classical single-layer discretisation. Stability and convergence analysis of the new finite volume method is presented for each of the three classical time discretisation methods: forward Euler, backward Euler and Crank-Nicolson. We prove that both the backward Euler and Crank-Nicolson schemes are always unconditionally stable. The key contribution of the work is the presentation of a set of sufficient stability conditions for the forward Euler scheme. Here, we find that to ensure stability of the forward Euler scheme it is not sufficient that the time step $\

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  • The first irreducible solution of the $\mathrm{SU} (2)$ self-duality equations on the Euclidean Schwarzschild (ES) manifold was found by Charap and Duff in 1977, only 2 years later than the famous BPST instantons on $\mathbb{R}^4$ were discovered. While soon after, in 1978, the ADHM construction gave a complete description of the moduli spaces of instantons on $\mathbb{R}^4$, the case of the Euclidean Schwarzschild manifold has resisted many efforts for the past 40 years. By exploring a correspondence between the planar Abelian vortices and spherically symmetric instantons on ES, we obtain: a complete description of a connected component of the moduli space of unit energy $\mathrm{SU} (2)$ instantons; new examples of instantons with non-integer energy (and non-trivial holonomy at infinity); a complete classification of finite energy, spherically symmetric, $\mathrm{SU} (2)$ instantons. As opposed to the previously known solutions, the generic instanton coming from our construction is n

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  • Let $f: A\rightarrow B$ and $g: A\rightarrow C$ be two ring homomorphisms and let $J$ (resp., $J'$) be an ideal of $B$ (resp., $C$) such that $f^{-1}(J)=g^{-1}(J')$. In this paper, we investigate the transfer of the notions of Gaussian and Pr\"ufer properties to the bi-amalgamation of $A$ with $(B,C)$ along $(J,J')$ with respect to $(f,g)$ (denoted by $A\bowtie^{f,g}(J,J')),$ introduced and studied by Kabbaj, Louartiti and Tamekkante in 2013. Our results recover well known results on amalgamations in \cite{Finno} and generate new original examples of rings satisfying these properties.

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  • In this article the generic torus orbit closure in a flag Bott manifold is shown to be a non-singular toric variety, and its fan structure is explicitly calculated.

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  • The development and identification of effective optimization algorithms for non-convex real-world problems is a challenge in global optimization. Because theoretical performance analysis is difficult, and problems based on models of real-world systems are often computationally expensive, several artificial performance test problems and test function generators have been proposed for empirical comparative assessment and analysis of metaheuristic optimization algorithms. These test problems however often lack the complex function structures and forthcoming difficulties that can appear in real-world problems. This communication presents a method to systematically build test problems with various types and degrees of difficulty. By weighted composition of parameterized random fields, challenging test functions with tunable function features such as, variance contribution distribution, interaction order, and nonlinearity can be constructed. The method is described, and its applicability to

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  • Weighted good-$\lambda$ type inequalities and Muckenhoupt-Wheeden type bounds are obtained for gradients of solutions to a class of quasilinear elliptic equations with measure data. Such results are obtained globally over sufficiently flat domains in $\mathbb{R}^n$ in the sense of Reifenberg. The principal operator here is modeled after the $p$-Laplacian, where for the first time singular case $\frac{3n-2}{2n-1}<p\leq 2-\frac{1}{n}$ is considered. Those bounds lead to useful compactness criteria for solution sets of quasilinear elliptic equations with measure data. As an application, sharp existence results and sharp bounds on the size of removable singular sets are deduced for a quasilinear Riccati type equation having a gradient source term with linear or super-linear power growth.

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  • Let $G$ be a group and $n$ a positive integer. We say $G$ has Property $P(n)$ if, for every subset $F \subseteq G$ of size $n$, there exists an irreducible unitary representation $\pi$ of $G$ such that $\pi(x) \ne id$ for all $x \in F \smallsetminus \{e\}$. Every group has $P(1)$ by a classical result of Gelfand and Raikov. Walter proved that every group has $P(2)$; it is easy to see that some groups do not have $P(3)$. We provide an algebraic characterization of the countable groups (finite or infinite) that have $P(n)$. We deduce that if a countable group $G$ has $P(n-1)$ but does not have $P(n)$, then $n$ is the cardinality of a projective space over a finite field.

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  • The Harborth constant of a finite group $G$ is the smallest integer $k\geq \exp(G)$ such that any subset of $G$ of size $k$ contains $\exp(G)$ distinct elements whose product is $1$. Generalizing previous work on the Harborth constants of dihedral groups, we compute the Harborth constants for the metacyclic groups of the form $H_{n, m}=\langle x, y \mid x^n=1, y^2=x^m, yx=x^{-1}y \rangle$. We also solve the "inverse" problem of characterizing all smaller subsets that do not contain $\exp(H_{n,m})$ distinct elements whose product is $1$.

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  • In our previous work, we studied an interconnected bursting neuron model for insect locomotion, and its corresponding phase oscillator model, which at high speed can generate stable tripod gaits with three legs off the ground simultaneously in swing, and at low speed can generate stable tetrapod gaits with two legs off the ground simultaneously in swing. However, at low speed several other stable locomotion patterns, that are not typically observed as insect gaits, may coexist. In the present paper, by adding heterogeneous external input to each oscillator, we modify the bursting neuron model so that its corresponding phase oscillator model produces only one stable gait at each speed, specifically: a unique stable tetrapod gait at low speed, a unique stable tripod gait at high speed, and a unique branch of stable transition gaits connecting them. This suggests that control signals originating in the brain and central nervous system can modify gait patterns.

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  • Finite difference schemes are the method of choice for solving nonlinear, degenerate elliptic PDEs, because the Barles-Sougandis convergence framework [Barles and Sougandidis, Asymptotic Analysis, 4(3):271-283, 1991] provides sufficient conditions for convergence to the unique viscosity solution [Crandall, Ishii and Lions, Bull. Amer. Math Soc., 27(1):1-67, 1992]. For anisotropic operators, such as the Monge-Ampere equation, wide stencil schemes are needed [Oberman, SIAM J. Numer. Anal., 44(2):879-895]. The accuracy of these schemes depends on both the distances to neighbors, $R$, and the angular resolution, $d\theta$. On uniform grids, the accuracy is $\mathcal O(R^2 + d\theta)$. On point clouds, the most accurate schemes are of $\mathcal O(R + d\theta)$, by Froese [Numerische Mathematik, 138(1):75-99, 2018]. In this work, we construct geometrically motivated schemes of higher accuracy in both cases: order $\mathcal O(R + d\theta^2)$ on point clouds, and $\mathcal O(R^2 + d\theta^2)$

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  • In this article, we investigate strategic information transmission over a noisy channel. This problem has been widely investigated in Economics, when the communication channel is perfect. Unlike in Information Theory, both encoder and decoder have distinct objectives and choose their encoding and decoding strategies accordingly. This approach radically differs from the conventional Communication paradigm, which assumes transmitters are of two types: either they have a common goal, or they act as opponent, e.g. jammer, eavesdropper. We formulate a point-to-point source-channel coding problem with state information, in which the encoder and the decoder choose their respective encoding and decoding strategies in order to maximize their long-run utility functions. This strategic coding problem is at the interplay between Wyner-Ziv's scenario and the Bayesian persuasion game of Kamenica-Gentzkow. We characterize a single-letter solution and we relate it to the previous results by using the

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  • We prove that even irregular convergence of semigroups of operators implies similar convergence of mild solutions of the related semi-linear equations with Lipschitz continuous nonlinearity. This result is then applied to three models originating from mathematical biology: shadow systems, diffusions on thin layers, and dynamics of neurotransmitters

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  • In this paper, we study the random field \begin{equation*} X(h) \circeq \sum_{p \leq T} \frac{\text{Re}(U_p \, p^{-i h})}{p^{1/2}}, \quad h\in [0,1], \end{equation*} where $(U_p, \, p ~\text{primes})$ is an i.i.d. sequence of uniform random variables on the unit circle in $\mathbb{C}$. Harper (2013) showed that $(X(h), \, h\in (0,1))$ is a good model for $(\log |\zeta(\frac{1}{2} + i (T + h))|, \, h\in [0,1])$ when $T$ is large, if we assume the Riemann hypothesis. The asymptotics of the maximum were found in Arguin, Belius & Harper (2017) up to the second order, but the tightness of the recentered maximum is still an open problem. As a first step, we provide large deviation estimates and continuity estimates for the field's derivative $X'(h)$. The main result shows that, with probability arbitrarily close to $1$, \begin{equation*} \max_{h\in [0,1]} X(h) - \max_{h\in \mathcal{S}} X(h) = O(1), \end{equation*} where $\mathcal{S}$ a discrete set containing $O(\log T \sqrt{\log \log T}

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  • Let L be a nonunimodular definite lattice. Using a theorem of Elkies we show that whether L embeds in the standard definite lattice of the same rank is completely determined by a collection of lattice correction terms, one for each metabolizing subgroup of the discriminant group. As a topological application this gives a rephrasing of the obstruction for a rational homology 3-sphere to bound a rational homology 4-ball coming from Donaldson's theorem on definite intersection forms of 4-manifolds. Furthermore, from this perspective it is easy to see that if the obstruction to bounding a rational homology ball coming from Heegaard Floer correction terms vanishes, then (under some mild hypotheses) the obstruction from Donaldson's theorem vanishes too.

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  • To work more accurately with elements of the semigroup of the Stone Cech compactification of the discrete semigroup of natural numbers N under multiplication. We divided these elements into ultrafilters which are on finite levels and ultrafilters which are not on finite levels. For the ultrafilters that are on finite levels we prove that any element is irreducible or product of irreducible elements and all elements on higher levels are extension divided by some elements on lower levels. We characterize ultrafilters that are not on finite levels and the effect of extension divisibility on the ultrafilters which are not on finite levels.

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  • In this paper, we address the question of lifting vector bundles to Witt vector bundles. More precisely, let p be a prime number, and let S be a scheme of characteristic p. For any n greater than 2, denote by W_n(S) the scheme of Witt vectors of length n, built out of S. Question: is V the restriction to S of a vector bundle defined over W_n(S)? We give a positive answer to this question, provided S possesses an ample line bundle, with Theorem 3.7. Our strategy consists in first dealing with the particular case of tautological vector bundles over Grassmannian varieties over F_p. We finish by Corollary 3.9. Roughly speaking, it ensures the existence of an extension of V, to any prescribed lift S of S of characteristic p^n- up to Frobenius pullback.

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  • We consider formal power series in several variables with coefficients in arbitrary field such that their Newton polyhedron has a loose edge. We show that if the symbolic restriction of the power series $f$ to such an edge is a product of two coprime polynomials, then $f$ factorizes in the ring of power series.

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  • The efficient compression of kernel matrices, for instance the off-diagonal blocks of discretized integral equations, is a crucial step in many algorithms. In this paper, we study the application of Skeletonized Interpolation to construct such factorizations. In particular, we study four different strategies for selecting the initial candidate pivots of the algorithm: Chebyshev grids, points on a sphere, maximally-dispersed and random vertices. Among them, the first two introduce new interpolation points (exo-vertices) while the last two are subsets of the given clusters (endo- vertices). We perform experiments using three real-world problems coming from the multiphysics code LS-DYNA. The pivot selection strategies are compared in term of quality (final rank) and efficiency (size of the initial grid). These benchmarks demonstrate that overall, maximally-dispersed vertices provide an accurate and efficient sets of pivots for most applications. It allows to reach near-optimal ranks while

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  • Given a real sparse polynomial system, we present a general framework to find explicit coefficients for which the system has more than one positive solution, based on the recent article by Bihan, Santos and Spaenlehauer. We apply this approach to find explicit reaction rate constants and total conservation constants in biochemical reaction networks for which the associated dynamical system is multistationary.

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  • For a fixed positive integer $k$, a set $S$ of vertices of a graph or multigraph is called a {\it $k$-independent set} if the subgraph induced by $S$ has maximum degree less than $k$. The well-known algorithm MAX finds a maximal $k$-independent set in a graph or multigraph by iteratively removing vertices of maximum degree until what remains has maximum degree less than $k$. We give an efficient procedure that determines, for a given degree sequence $D$, the smallest cardinality $b(D)$ of a $k$-independent set that can result from any application of MAX to any loopless multigraph with degree sequence $D$. This analysis of the worst case is sharp for each degree sequence $D$ in that there exists a multigraph $G$ with degree sequence $D$ such that some application of MAX to $G$ will result in a $k$-independent set of cardinality exactly $b(D)$.

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  • Rosengren and Schlosser introduced notions of ${\it R}_N$-theta functions for the seven types of irreducible reduced affine root systems, ${\it R}_N={\it A}_{N-1}$, ${\it B}_{N}$, ${\it B}^{\vee}_N$, ${\it C}_{N}$, ${\it C}^{\vee}_N$, ${\it BC}_{N}$, ${\it D}_{N}$, $N \in \mathbb{N}$, and gave the Macdonald denominator formulas. We prove that, if the variables of the ${\it R}_N$-theta functions are properly scaled with $N$, they construct seven sets of biorthogonal functions, each of which has a continuous parameter $t \in (0, t_{\ast})$ with given $0< t_{\ast} < \infty$. Following the standard method in random matrix theory, we introduce seven types of one-parameter ($t \in (0, t_{\ast})$) families of determinantal point processes in one dimension, in which the correlation kernels are expressed by the biorthogonal theta functions. We demonstrate that they are elliptic extensions of the classical determinantal point processes whose correlation kernels are expressed by trigonometr

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  • We study a toy model of linear-quadratic mean field game with delay. We "lift" the delayed dynamic into an infinite dimensional space, and recast the mean field game system which is made of a forward Kolmogorov equation and a backward Hamilton-Jacobi-Bellman equation. We identify the corresponding master equation. A solution to this master equation is computed, and we show that it provides an approximation to a Nash equilibrium of the finite player game.

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  • This paper investigates the performance of millimeter wave (mmWave) communications in clustered device-to-device (D2D) networks. The locations of D2D transceivers are modeled as a Poisson Cluster Process (PCP). In each cluster, devices are equipped with multiple antennas, and the active D2D transmitter (D2D-Tx) utilizes mmWave to serve one of the proximate D2D receivers (D2D-Rxs). Specifically, we introduce three user association strategies: 1) Uniformly distributed D2D-Tx model; 2) Nearest D2D-Tx model; 3) Closest line-of-site (LOS) D2D-Tx model. To characterize the performance of the considered scenarios, we derive new analytical expressions for the coverage probability and area spectral efficiency (ASE). Additionally, in order to efficiently illustrating the general trends of our system, a closed-form lower bound for the special case interfered by intra-cluster LOS links is derived. We provide Monte Carlo simulations to corroborate the theoretical results and show that: 1) The cover

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  • In this paper we revisit the family of algebroid space curves defined by Moh and find an explicit minimal generating set for the defining ideal, for the case $n=1$

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  • The abstraction of the study of stochastic processes to Banach lattices and vector lattices has received much attention by Grobler, Kuo, Labuschagne, Stoica, Troitsky and Watson over the past fifteen years. By contrast mixing processes have received very little attention. In particular mixingales were generalized to the Riesz space setting in {\sc W.-C. Kuo, J.J. Vardy, B.A. Watson,} Mixingales on Riesz spaces, {\em J. Math. Anal. Appl.}, \textbf{402} (2013), 731-738. The concepts of strong and uniform mixing as well as related mixing inequalities were extended to this setting in {\sc W.-C. Kuo, M.J. Rogans, B.A. Watson,} Mixing inequalities in Riesz spaces, {\em J. Math. Anal. Appl.}, \textbf{456} (2017), 992-1004. In the present work we formulate the concept of near-epoch dependence for Riesz space processes and show that if a process is near-epoch dependent and either strong or uniform mixing then the process is a mixingale, giving access to a law of large numbers. The above is appl

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  • Non-Gaussian component analysis (NGCA) is a problem in multidimensional data analysis. Since its formulation in 2006, NGCA has attracted considerable attention in statistics and machine learning. In this problem, we have a random variable $X$ in $n$-dimensional Euclidean space. There is an unknown subspace $U$ of the $n$-dimensional Euclidean space such that the orthogonal projection of $X$ onto $U$ is standard multidimensional Gaussian and the orthogonal projection of $X$ onto $V$, the orthogonal complement of $U$, is non-Gaussian, in the sense that all its one-dimensional marginals are different from the Gaussian in a certain metric defined in terms of moments. The NGCA problem is to approximate the non-Gaussian subspace $V$ given samples of $X$. Vectors in $V$ corresponds to "interesting" directions, whereas vectors in $U$ correspond to the directions where data is very noisy. The most interesting applications of the NGCA model is for the case when the magnitude of the noise is comp

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  • In this paper we introduce a class of noncommutative (finitely generated) monomial algebras whose Hilbert series are algebraic functions. We use the concept of graded homology and the theory of unambiguous context-free grammars for this purpose. We also provide examples of finitely presented graded algebras whose corresponding leading monomial algebras belong to the proposed class and hence possess algebraic Hilbert series.

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  • Multi-dimensional (MD) mapping offers more flexibility in mapping design for bit-interleaved coded modulation with iterative decoding (BICM-ID) and potentially improves the bandwidth efficiency. However, for higher order signal constellations, finding suitable MD mappings is a very complicated task due to the large number of possible mappings. In this paper, a novel mapping method is introduced to construct efficient MD mappings to improve the error performance of BICM-ID over Rayleigh fading channels. We propose to break the MD mapping design problem into four distinct $2$-D mapping functions. The $2$-D mappings are designed such that the resulting MD mapping improves the BICM-ID error performance at low signal to noise ratios (SNRs). We also develop cost functions that can be optimized to improve the error performance at high SNRs. The proposed mapping method is very simple compared to well-known mapping methods, and it can achieve suitable MD mappings for different modulations inclu

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  • In this paper we consider the curves $H_{k,t}^{(p)} : y^{p^k}+y=x^{p^{kt}+1}$ over $\mathbb F_p$ and and find an exact formula for the number of $\mathbb F_{p^n}$-rational points on $H_{k,t}^{(p)}$ for all integers $n\ge 1$. We also give the condition when the $L$-polynomial of a Hermitian curve divides the $L$-polynomial of another over $\mathbb F_p$.

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  • We show that dimensional theoretical properties of dynamical systems can considerably change because of number theoretical peculiarities of some parameter values

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  • We consider the transport equation $\ppp_t u(x,t) + H(t)\cdot \nabla u(x,t) = 0$ in $\OOO\times(0,T),$ where $T>0$ and $\OOO\subset \R^d $ is a bounded domain with smooth boundary $\ppp\OOO$. First, we prove a Carleman estimate for solutions of finite energy with piecewise continuous weight functions. Then, under a further condition on $H$ which guarantees that the orbit $\{ H(t)\in\R^d, \thinspace 0 \le t \le T\}$intersects $\ppp\OOO$, we prove an energy estimate which in turn yields an observability inequality. Our results are motivated by applications to inverse problems.

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  • We present a simple approach for sensor registration in target tracking applications. The proposed method uses targets of opportunity and, without making assumptions on their dynamical models, allows simultaneous calibration of multiple three- and two-dimensional sensors. Whereas for two-sensor scenarios only relative registration is possible, in practical cases with three or more sensors unambiguous absolute calibration may be achieved. The derived algorithms are straightforward to implement and do not require tuning of parameters. The performance of the algorithms is tested in a numerical study.

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  • In this note, we extend the theories of the canonical bundle formula and of adjunction to the case of generalized polarized pairs. As application, we study a particular case of a conjecture by Prokhorov and Shokurov.

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  • A subject of recent interest in inverse problems is whether a corner must diffract fixed frequency waves. We generalize this question somewhat and study cones $[0,\infty)\times Y$ which do not diffract high frequency waves. We prove that if $Y$ is analytic and does not diffract waves at high frequency then every geodesic on $Y$ is closed with period $2\pi$. Moreover, we show that if $\dim Y=2$, then $Y$ is isometric to either the sphere of radius 1 or its $\mathbb{Z}^2$ quotient, $\mathbb{R}\mathbb{P}^2$.

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  • We consider semilinear equation of the form $-Lu=f(x,u)+\mu$, where $L$ is the operator corresponding to a transient symmetric regular Dirichlet form ${\mathcal E}$, $\mu$ is a diffuse measure with respect to the capacity associated with ${\mathcal E}$, and the lower-order perturbing term $f(x,u)$ satisfies the sign condition in $u$ and some weak integrability condition (no growth condition on $f(x,u)$ as a function of $u$ is imposed). We prove the existence of a solution under mild additional assumptions on ${\mathcal E}$. We also show that the solution is unique if $f$ is nonincreasing in $u$.

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