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  • We derive a functional central limit theorem for the excursion of a random walk conditioned on sweeping a prescribed geometric area. We assume that the increments of the random walk are integer-valued, centered, with a third moment equal to zero and a finite fourth moment. This result complements the work of \citep{DKW13} where local central limit theorems are provided for the geometric area of the excursion of a symmetric random walk with finite second moments. Our result turns out to be a key tool to derive the scaling limit of the \emph{Interacting Partially-Directed Self-Avoiding Walk} at criticality which is the object of a companion paper \citep{CarPet17a}. This requires to derive a reinforced version of our result in the case of a random walk with Laplace symmetric increments.

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  • Data processing inequalities for $f$-divergences can be sharpened using constants called "contraction coefficients" to produce strong data processing inequalities. For any discrete source-channel pair, the contraction coefficients for $f$-divergences are lower bounded by the contraction coefficient for $\chi^2$-divergence. In this paper, we elucidate that this lower bound can be achieved by driving the input $f$-divergences of the contraction coefficients to zero. Then, we establish a linear upper bound on the contraction coefficients for a certain class of $f$-divergences using the contraction coefficient for $\chi^2$-divergence, and refine this upper bound for the salient special case of Kullback-Leibler (KL) divergence. Furthermore, we present an alternative proof of the fact that the contraction coefficients for KL and $\chi^2$-divergences are equal for a Gaussian source with an additive Gaussian noise channel (where the former coefficient can be power constrained). Finally, we gen

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  • We study topological properties of families of Hamiltonians which may contain degenerate energy levels aka. band crossings. The primary tool are Chern classes, Berry phases and slicing by surfaces. To analyse the degenerate locus, we study local models. These give information about the Chern classes and Berry phases. We then give global constraints for the topological invariants. This is an hitherto relatively unexplored subject. The global constraints are more strict when incorporating symmetries such as time reversal symmetries. The results can also be used in the study of deformations. We furthermore use these constraints to analyse examples which include the Gyroid geometry, which exhibits Weyl points and triple crossings and the honeycomb geometry with its two Dirac points.

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  • When a compact Lie group acts freely and in a Hamiltonian way on a symplectic manifold, the Marsden-Weinstein theorem says that the reduced space is a smooth symplectic manifold. If we drop the freeness assumption, the reduced space might be singular, but Sjamaar-Lerman (1991) showed that it can still be partitioned into smooth symplectic manifolds which "fit together nicely" in the sense that they form a stratification. In this paper, we prove a hyperkahler analogue of this statement, using the hyperkahler quotient construction. We also show that singular hyperkahler quotients are complex spaces which are locally biholomorphic to affine complex-symplectic GIT quotients with biholomorphisms that are compatible with natural holomorphic Poisson brackets on both sides.

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  • In a quantum many-body system where the Hamiltonian and the order operator do not commute, it often happens that the unique ground state of a finite system exhibits long-range order (LRO) but does not show spontaneous symmetry breaking (SSB). Typical examples include antiferromagnetic quantum spin systems with Neel order, and lattice boson systems which exhibits Bose-Einstein condensation. By extending and improving previous results by Horsch and von der Linden and by Koma and Tasaki, we here develop a fully rigorous and almost complete theory about the relation between LRO and SSB in the ground state of a finite system with continuous symmetry. We show that a ground state with LRO but without SSB is inevitably accompanied by a series of energy eigenstates, known as the "tower" of states, which have extremely low excitation energies. More importantly, we also prove that one gets a physically realistic "ground state" by taking a superposition of these low energy excited states. The pres

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  • In 1997 M.~Khovanov proved that any doodle can be presented as closure of twin, this result is analogue of classical Alexander's theorem for braids and links. We give a description of twins that have equivalent closures, this theorem is analogue of classical Markov theorem.

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  • We study discretized maximal operators associated to averaging over (neighborhoods of) squares in the plane and, more generally, $k$-skeletons in $\mathbb{R}^n$. Although these operators are known not to be bounded on any $L^p$, we obtain nearly sharp $L^p$ bounds for every small discretization scale. These results are motivated by, and partially extend, recent results of T. Keleti, D. Nagy and P. Shmerkin, and of R. Thornton, on sets that contain a scaled $k$-sekeleton of the unit cube with center in every point of $\mathbb{R}^n$.

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  • This paper considers a single-antenna wirelesspowered communication network (WPCN) over a flat-fading channel. We show that, by using our probabilistic harvestand-transmit (PHAT) strategy, which requires the knowledge of instantaneous full channel state information (CSI) and fading probability distribution, the ergodic throughput of this system may be greatly increased relative to that achieved by the harvestthen-transmit (HTT) protocol. To do so, instead of dividing every frame to the uplink (UL) and downlink (DL), the channel is allocated to the UL wireless information transmission (WIT) and DL wireless power transfer (WPT) based on the estimated channel power gain. In other words, based on the fading probability distribution, we will derive some thresholds that determine the association of a frame to the DL WPT or UL WIT. More specifically, if the channel gain falls below or goes over these thresholds, the channel will be allocated to WPT or WIT. Simulation results verify the perfor

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  • In this article, we determine the maximum Wiener indices of unicyclic graphs with given number of vertices and matching number. We also characterize the extremal graphs. This solves an open problem of Du and Zhou.

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  • Let $(X,\mathcal{B},\mu,T)$ be a measure preserving system. We say that a function $f\in L^2(X,\mu)$ is $\mu$-mean equicontinuous if for any $\epsilon>0$ there is $k\in \mathbb{N}$ and measurable sets ${A_1,A_2,\cdots,A_k}$ with $\mu\left(\bigcup\limits_{i=1}^k A_i\right)>1-\epsilon$ such that whenever $x,y\in A_i$ for some $1\leq i\leq k$, one has \[ \limsup_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}|f(T^jx)-f(T^jy)|<\epsilon. \] Measure complexity with respect to $f$ is also introduced. It is shown that $f$ is an almost periodic function if and only if $f$ is $\mu$-mean equicontinuous if and only if $\mu$ has bounded complexity with respect to $f$. Ferenczi studied measure-theoretic complexity using $\alpha$-names of a partition and the Hamming distance. He proved that if a measure preserving system is ergodic, then the complexity function is bounded if and only if the system has discrete spectrum. We show that this result holds without the assumption of ergodicity.

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  • We discuss certain identities involving $\mu(n)$ and $M(x)=\sum_{n\leq x}\mu(n)$, the functions of M\"{o}bius and Mertens. These identities allow calculation of $M(N^d)$, for $d=2,3,4,\ldots\ $, as a sum of $O_d \left( N^d(\log N)^{2d - 2}\right)$ terms, each a product of the form $\mu(n_1) \cdots \mu(n_r)$ with $r\leq d$ and $n_1,\ldots , n_r\leq N$. We prove a more general identity in which $M(N^d)$ is replaced by $M(g,K)=\sum_{n\leq K}\mu(n)g(n)$, where $g(n)$ is an arbitrary totally multiplicative function, while each $n_j$ has its own range of summation, $1,\ldots , N_j$. We focus on the case $d=2$, $K=N^2$, $N_1=N_2=N$, where the identity has the form $M(g,N^2) = 2 M(g,N) - {\bf m}^{\rm T} A {\bf m}$, with $A$ being the $N\times N$ matrix of elements $a_{mn}=\sum _{k \leq N^2 /(mn)}\,g(k)$, while ${\bf m}=(\mu (1)g(1),\ldots ,\mu (N)g(N))^{\rm T}$. Our results in Sections 2 and 3 assume, moreover, that $g(n)$ equals $1$ for all $n$. In this case the Perron-Frobenius theorem appli

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  • A real valued function $\varphi$ of one variable is called a metric transform if for every metric space $(X,d)$ the composition $d_\varphi = \varphi\circ d$ is also a metric on $X$. We give a complete characterization of the class of approximately nondecreasing, unbounded metric transforms $\varphi$ such that the transformed Euclidean half line $([0,\infty),|\cdot|_\varphi)$ is Gromov hyperbolic. As a consequence, we obtain metric transform rigidity for roughly geodesic Gromov hyperbolic spaces, that is, if $(X,d)$ is any metric space containing a rough geodesic ray and $\varphi$ is an approximately nondecreasing, unbounded metric transform such that the transformed space $(X,d_\varphi)$ is Gromov hyperbolic and roughly geodesic then $\varphi$ is an approximate dilation and the original space $(X,d)$ is Gromov hyperbolic and roughly geodesic.

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  • The minor probability events detection is a crucial problem in Big data. Such events tend to include rarely occurring phenomenons which should be detected and monitored carefully. Given the prior probabilities of separate events and the conditional distributions of observations on the events, the Bayesian detection can be applied to estimate events behind the observations. It has been proved that Bayesian detection has the smallest overall testing error in average sense. However, when detecting an event with very small prior probability, the conditional Bayesian detection would result in high miss testing rate. To overcome such a problem, a modified detection approach is proposed based on Bayesian detection and message importance measure, which can reduce miss testing rate in conditions of detecting events with minor probability. The result can help to dig minor probability events in big data.

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  • Computation task service delivery in a computing-enabled and caching-aided multi-user mobile edge computing (MEC) system is studied in this paper, where a MEC server can deliver the input or output datas of tasks to mobile devices over a wireless multicast channel. The computing-enabled and caching-aided mobile devices are able to store the input or output datas of some tasks, and also compute some tasks locally, reducing the wireless bandwidth consumption. The corresponding framework of this system is established, and under the latency constraint, we jointly optimize the caching and computing policy at mobile devices to minimize the required transmission bandwidth. The joint policy optimization problem is shown to be NP-hard, and based on equivalent transformation and exact penalization of the problem, a stationary point is obtained via concave convex procedure (CCCP). Moreover, in a symmetric scenario, gains offered by this approach are derived to analytically understand the influenc

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  • The aim of this paper is to provide a fractional generalization of the Gompertz law via a Caputo-like definition of fractional derivative of a function with respect to another function. In particular, we observe that the model presented appears to be substantially different from the other attempt of fractional modifications of this model, since the fractional nature is carried along by the general solution even in its asymptotic behaviour for long times. We then validate the presented model by employing it as reference frame to model three biological systems of peculiar interest for biophysics and environmental engineering, namely: dark fermentation, photofermentation and microalgae biomass growth.

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  • Using an existence criterion for good moduli spaces of Artin stacks by Alper-Fedorchuk-Smyth we construct a proper moduli space of rank two sheaves with fixed Chern classes on a given complex projective manifold that are Gieseker-Maruyama-semistable with respect to a fixed K\"ahler class.

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  •  

    Monge's theorem

    07-15 Hacker News 2

    Monge's theorem

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  • Weighted logrank tests are a popular tool for analyzing right censored survival data from two independent samples. Each of these tests is optimal against a certain hazard alternative, for example the classical logrank test for proportional hazards. But which weight function should be used in practical applications? We address this question by a flexible combination idea leading to a testing procedure with broader power. Beside the test's asymptotic exactness and consistency its power behaviour under local alternatives is derived. All theoretical properties can be transferred to a permutation version of the test, which is even finitely exact under exchangeability and showed a better finite sample performance in our simulation study. The procedure is illustrated in a real data example.

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  • We make a detailed study of various (quadratic and linear) Morse-Bott trace functions on the orthogonal groups $O(n)$. We describe the critical loci of the quadratic trace function Tr$(AXBX^T)$ and determine their indices via perfect fillings of tables associated with the multiplicities of the eigenvalues of $A$ and $B$. We give a simplified treatment of T. Frankel's analysis of the linear trace function on $SO(n)$, as well as a combinatorial explanation of the relationship between the mod $2$ Betti numbers of $SO(n)$ and those of the Grassmannians $\mathbb{G}(2k,n)$ obtained from this analysis. We review the basic notions of Morse-Bott cohomology in a simple case where the set of critical points has two connected components. We then use these results to give a new Morse-theoretic computation of the mod $2$ Betti numbers of $SO(n)$.

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  • Let $A,B\subset\mathbb{R}$. Define $$A\cdot B=\{x\cdot y:x\in A, y\in B\}.$$ In this paper, we consider the following class of self-similar sets with overlaps. Let $K$ be the attractor of the IFS $\{f_1(x)=\lambda x, f_2(x)=\lambda x+c-\lambda,f_3(x)=\lambda x+1-\lambda\}$, where $f_1(I)\cap f_2(I)\neq \emptyset, (f_1(I)\cup f_2(I))\cap f_3(I)=\emptyset,$ and $I=[0,1]$ is the convex hull of $K$. The main result of this paper is $K\cdot K=[0,1]$ if and only if $(1-\lambda)^2\leq c$. Equivalently, we give a necessary and sufficient condition such that for any $u\in[0,1]$, $u=x\cdot y$, where $x,y\in K$.

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  • We consider an optimal transport problem on the unit simplex whose solutions are given by gradients of exponentially concave functions and prove two main results. One, we show that the optimal transport is the large deviation limit of a particle system of Dirichlet processes transporting one probability measure on the unit simplex to another by coordinatewise multiplication and normalizing. The structure of our Lagrangian and the appearance of the Dirichlet process relate our problem closely to the entropic measure on the Wasserstein space as defined by von-Renesse and Sturm in the context of Wasserstein diffusion. The limiting procedure is a triangular limit where we allow simultaneously the number of particles to grow to infinity while the `noise' goes to zero. The method, which generalizes easily to other cost functions, including the Wasserstein cost, provides a novel combination of the Schr\"odinger problem approach due to C. L\'eonard and the related Brownian particle systems by

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  • In this paper we prove the existence of infinitely many nontrivial solutions for the class of $(p,\, q)$ fractional elliptic equations involving concave-critical nonlinearities in bounded domains in $\mathbb{R}^N$. Further, when the nonlinearity is of convex-critical type, we establish the multiplicity of nonnegative solutions using variational methods. In particular, we show the existence of at least $cat_{\Omega}(\Omega)$ nonnegative solutions.

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  • Modeling traffic in road networks is a widely studied but challenging problem, especially under the assumption that drivers act selfishly. A common approach used in simulation software is the deterministic queuing model, for which the structure of dynamic equilibria has been studied extensively in the last couple of years. The basic idea is to model traffic by a continuous flow that travels over time from a source to a sink through a network, in which the arcs are endowed with transit times and capacities. Whenever the flow rate exceeds the capacity a queue builds up and the infinitesimally small flow particles wait in line in front of the bottleneck. Since the queues have no physical dimension, it was not possible, until now, to represent spillback in this model. This was a big drawback, since spillback can be regularly observed in real traffic situations and has a huge impact on travel times in highly congested regions. We extend the deterministic queuing model by introducing a stora

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  • Given a countable, totally ordered commutative monoid $\mathcal{R}=(R,\oplus,\leq,0)$, with least element $0$, there is a countable, universal and ultrahomogeneous metric space $\mathcal{U}_\mathcal{R}$ with distances in $\mathcal{R}$. We refer to this space as the $\mathcal{R}$-Urysohn space, and consider the theory of $\mathcal{U}_\mathcal{R}$ in a binary relational language of distance inequalities. This setting encompasses many classical structures of varying model theoretic complexity, including the rational Urysohn space, the free $n^{\text{th}}$ roots of the complete graph (e.g. the random graph when $n=2$), and theories of refining equivalence relations (viewed as ultrametric spaces). We characterize model theoretic properties of $\text{Th}(\mathcal{U}_\mathcal{R})$ by algebraic properties of $\mathcal{R}$, many of which are first-order in the language of ordered monoids. This includes stability, simplicity, and Shelah's SOP$_n$-hierarchy. Using the submonoid of idempotents in

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  • We consider the problem of noisy private information retrieval (NPIR) from $N$ non-communicating databases, each storing the same set of $M$ messages. In this model, the answer strings are not returned through noiseless bit pipes, but rather through \emph{noisy} memoryless channels. We aim at characterizing the PIR capacity for this model as a function of the statistical information measures of the noisy channels such as entropy and mutual information. We derive a general upper bound for the retrieval rate in the form of a max-min optimization. We use the achievable schemes for the PIR problem under asymmetric traffic constraints and random coding arguments to derive a general lower bound for the retrieval rate. The upper and lower bounds match for $M=2$ and $M=3$, for any $N$, and any noisy channel. The results imply that separation between channel coding and retrieval is optimal except for adapting the traffic ratio from the databases. We refer to this as \emph{almost separation}. Ne

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  • We consider a nonlinear Neumann elliptic inclusion with a source (reaction term) consisting of a convex subdifferential plus a multivalued term depending on the gradient. The convex subdifferential incorporates in our framework problems with unilateral constraints (variational inequalities). Using topological methods and the Moreau-Yosida approximations of the subdifferential term, we establish the existence of a smooth solution.

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  • Our goal of this paper is to develop a new upscaling method for multicontinua flow problems in fractured porous media. We consider a system of equations that describes flow phenomena with multiple flow variables defined on both matrix and fractures. To construct our upscaled model, we will apply the nonlocal multicontinua (NLMC) upscaling technique. The upscaled coefficients are obtained by using some multiscale basis functions, which are solutions of local problems defined on oversampled regions. For each continuum within a target coarse element, we will solve a local problem defined on an oversampling region obtained by extending the target element by few coarse grid layers, with a set of constraints which enforce the local solution to have mean value one on the chosen continuum and zero mean otherwise. The resulting multiscale basis functions have been shown to have good approximation properties. To illustrate the idea of our approach, we will consider a dual continua background mod

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  • Several properties of a hyepergeometric series related to Gromov-Witten theory of some Calabi-Yau geometries was studied in [8]. These properties play basic role in the study of higher genus Gromov-Witten theories. We extend the results of [8] to equivariant setting for the study of higher genus equivariant Gromov-Witten theories of some Calabi-Yau geometries.

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  • We give an explicit and versatile parametrization of all positive selfadjoint extensions of a densely defined, closed, positive operator. In addition, we identify the Friedrichs extension by specifying the parameter to which it corresponds. This is a manuscript that was circulated as the first part of the preprint "Two papers on selfadjoint extensions of symmetric semibounded operators", INCREST Preprint Series, July 1981, Bucharest, Romania, but never published. In this LaTeX typeset version, only typos and a few inappropriate formulations have been corrected, with respect to the original manuscript. I decided to post it on arXiv since, taking into account recent articles, the results are still of current interest. Tiberiu Constantinescu died in 2005.

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  • In this note we associate a sequence of non-negative integers to any convergent series of positive real numbers and study this sequence for the series $\sum_{n \geq 1} n^{-k}$ where $k$ is an integer $\geq 2$.

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  • Let $V$ be an $n$-dimensional vector space over the finite field of order $q$. The spherical building $X_V$ associated with $GL(V)$ is the order complex of the nontrivial linear subspaces of $V$. Let $\mathfrak{g}$ be the local coefficient system on $X_V$, whose value on the simplex $\sigma=[V_0 \subset \cdots \subset V_p] \in X_V$ is given by $\mathfrak{g}(\sigma)=V_0$. Following the work of Lusztig and Dupont, we study the homology module $D^k(V)=\tilde{H}_{n-k-1}(X_V;\mathfrak{g})$. Our results include a construction of an explicit basis of $D^1(V)$, and the following twisted analogue of a result of Smith and Yoshiara: For any $1 \leq k \leq n-1$, the minimal support size of a non-zero $(n-k-1)$-cycle in the twisted homology $\tilde{H}_{n-k-1}(X_V;\wedge^k \mathfrak{g})$ is $\frac{(n-k+2)!}{2}$.

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  • Let $\mu$ be a probability measure in $\mathbb{C}$ with a continuous and compactly supported distribution function, let $z_1, \dots, z_n$ be independent random variables, $z_i \sim \mu$, and consider the random polynomial $$ p_n(z) = \prod_{k=1}^{n}{(z - z_k)}.$$ We determine the asymptotic distribution of $\left\{z \in \mathbb{C}: p_n(z) = p_n(0)\right\}$. In particular, if $\mu$ is radial around the origin, then those solutions are also distributed according to $\mu$ as $n \rightarrow \infty$. Generally, the distribution of the solutions will reproduce parts of $\mu$ and condense another part on curves. We use these insights to study the behavior of the Blaschke unwinding series on random data.

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  • It is shown that if the boundary of a Reinhardt domain in $\mathbb{C}^n$ contains the origin, each holomorphic function on the domain which is infinitely many times differentiable up to the boundary extends holomorphically to a neighborhood of the origin.

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  • We prove that on a closed surface of genus $g$, the cardinality of a set of simple closed curves in which any two are non-homotopic and intersect at most once is $\lesssim g^2 \log(g)$. This bound matches the largest known constructions to within a logarithmic factor. The proof uses a probabilistic argument in graph theory. It generalizes as well to the case of curves that intersect at most $k$ times in pairs.

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  • Let $X\subset \mathbb{C}^n$ be a smooth irreducible affine variety of dimension $k$ and let $F: X\to \mathbb{C}^m$ be a polynomial mapping. We prove that if $m\ge k$, then there is a Zariski open dense subset $U$ in the space of linear mappings ${\mathcal L}( \mathbb{C}^n, \mathbb{C}^m)$ such that for every $L\in U$ the mapping $F+L$ is a finite mapping. Moreover, we can choose $U$ in this way, that all mappings $F+L; L\in U$ are topologically equivalent.

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  • We prove that if $\mathcal{A}$ is a locally $\lambda$-presentable category and $T : \mathcal{A} \to \mathcal{A}$ is a $\lambda$-accessible functor then $T/\mathcal{A}$ is locally $\lambda$-presentable.

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  • We prove two maximal regularity results in spaces of continuous and H\"older continuous functions, for a mixed linear Cauchy-Dirichlet problem with a fractional time derivative $\mathbb{D}_t^\alpha$. This derivative is intended in the sense of Caputo and $\alpha$ is taken in $(0, 2)$. In case $\alpha = 1$, we obtain maximal regularity results for mixed parabolic problems already known in mathematica literature.

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  • We introduce the non-homogeneous analogs of Van Schaftingen's classes. We show that these classes refine the embedding $W^{1,n}\subset bmo$. The analogous results established on bounded Lipschitz domains and Riemannian manifolds with bounded geometry.

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  • This paper proposes a framework of L-BFGS based on the (approximate) second-order information with stochastic batches, as a novel approach to the finite-sum minimization problems. Different from the classical L-BFGS where stochastic batches lead to instability, we use a smooth estimate for the evaluations of the gradient differences while achieving acceleration by well-scaling the initial Hessians. We provide theoretical analyses for both convex and nonconvex cases. In addition, we demonstrate that within the popular applications of least-square and cross-entropy losses, the algorithm admits a simple implementation in the distributed environment. Numerical experiments support the efficiency of our algorithms.

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  • In this paper, we study multiple-antenna wireless communication networks where a large number of devices simultaneously communicating with an access point. The capacity region of multiple-input multiple-output massive multiple access channels (MIMO mMAC) is investigated. While the joint typicality decoding is utilized to establish the achievability of capacity region for conventional multiple access channel with fixed number of users, the technique is not directly applicable for MIMO mMAC [4]. Instead, an information-theoretic approach based on Gallager's error exponent analysis is exploited to characterize the capacity region of MIMO mMAC. Theoretical results reveal that the capacity region of MIMO mMAC is dominated by the sum rate constraint only and the individual user rate is determined by a specific factor that corresponds to the allocation of sum rate. The individual user rate in conventional MAC is not achievable with massive multiple access and the successive interference cance

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  • We will describe a one-step "Gorensteinization" process for a Schubert variety by blowing-up along its boundary divisor. The local question involves Kazhdan-Lusztig varieties which can be degenerated to affine toric schemes defined using the Stanley-Reisner ideal of a subword complex. The blow-up along the boundary in this toric case is in fact Gorenstein. We show that there exists a degeneration of the blow-up of the Kazhdan-Lusztig variety to this Gorenstein scheme, allowing us to extend this result to Schubert varieties in general. The potential use of this one-step Gorensteinization to describe the non-Gorenstein locus of Schubert varieties is discussed, as well as the relationship between Gorensteinizations and the convergence of the Nash blow-up process in the toric case.

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  • The problem of identifiability of finite mixtures of finite product measures is studied. A mixture model with $K$ mixture components and $L$ observed variables is considered, where each variable takes its value in a finite set with cardinality $M$.The variables are independent in each mixture component. The identifiability of a mixture model means the possibility of attaining the mixture components parameters by observing its mixture distribution. In this paper, we investigate fundamental relations between the identifiability of mixture models and the separability of their observed variables by introducing two types of separability: strongly and weakly separable variables. Roughly speaking, a variable is said to be separable, if and only if it has some differences among its probability distributions in different mixture components. We prove that mixture models are identifiable if the number of strongly separable variables is greater than or equal to $2K-1$, independent form $M$. This f

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  • Under necessary compatibility condition, and some mild regularity assumptions on the interior and the boundary data, we prove the existence, uniqueness, and stability of the solution of generalized Dary-Forchheimer model.

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  • In this paper, we examine the convergence of mirror descent in a class of stochastic optimization problems that are not necessarily convex (or even quasi-convex), and which we call variationally coherent. Since the standard technique of "ergodic averaging" offers no tangible benefits beyond convex programming, we focus directly on the algorithm's last generated sample (its "last iterate"), and we show that it converges with probabiility $1$ if the underlying problem is coherent. We further consider a localized version of variational coherence which ensures local convergence of stochastic mirror descent (SMD) with high probability. These results contribute to the landscape of non-convex stochastic optimization by showing that (quasi-)convexity is not essential for convergence to a global minimum: rather, variational coherence, a much weaker requirement, suffices. Finally, building on the above, we reveal an interesting insight regarding the convergence speed of SMD: in problems with sha

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  • This paper deals with the equation $-\Delta u+\mu u=f$, $\mu$ a positive constant, on high-dimensional spaces $\mathbb{R}^d$. If the right-hand side $f$ is a rapidly converging series of separable functions, the solution $u$ can be represented in the same way. These constructions are based on the approximation of the function $1/r$ by sums of exponential functions. We derive results of related kind for more general right-hand sides $f(x)=F(Tx)$ that are restrictions of separable functions $F$ on a higher dimensional space to a linear subspace of arbitrary orientation.

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  • Motivated by the problem of optimal portfolio liquidation under transient price impact, we study the minimization of energy functionals with completely monotone displacement kernel under an integral constraint. The corresponding minimizers can be characterized by Fredholm integral equations of the second type with constant free term. Our main result states that minimizers are analytic and have a power series development in terms of even powers of the distance to the midpoint of the domain of definition and with nonnegative coefficients. We show moreover that our minimization problem is equivalent to the minimization of the energy functional under a nonnegativity constraint.

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  • We prove a local-in-time regularity criterion for the 3D Navier-Stokes equations. In particular from the criterion we obtain a new partial regularity result on the dimension of possible singular times. It is shown that the Hausdorff dimension of possible singular times for weak solutions $u\in L^s([0,T]\times \mathbb{R}^3)$ with $4 \leq s \leq 5$ is at most $\frac{5}{2}-\frac{s}{2}$ improving the previous bound $\frac{1}{2}$.

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  • For tame arbitrary-length toral, also called positive regular, supercuspidal representations of a simply connected and semisimple $p$-adic group $G$, constructed as per Adler-Yu, we determine which components of their restriction to a maximal compact subgroup are types. We give conditions under which there is a unique such component, and then present a class of examples for which there is not, disproving the strong version of the conjecture of unicity of types on maximal compact open subgroups. We restate the unicity conjecture, and prove it holds for the groups and representations under consideration under a mild condition on depth.

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  • We introduce patterns on a triangular grid generated by paperfolding operations. We show that in case these patterns are defined using a periodic sequence of foldings, they can also be generated using substitution rules and compute eigenvalues and eigenvectors of corresponding matrices. We also prove that densities of all basic triangles are equal in these patterns.

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  • For any real number $p\in [1,+\infty)$, we characterize the operations $\mathbb{R}^I\to \mathbb{R}$ that preserve $p$-integrability over finite measure spaces, i.e., the operations under which, for every finite measure $\mu$, the set $\mathcal{L}^p(\mu)$ is closed. We investigate the infinitary variety of algebras whose terms are exactly such operations. It turns out that this variety coincides with the much studied category of Dedekind $\sigma$-complete Riesz spaces with weak unit. We also prove that $\mathbb{R}$ generates this variety. From this, we exhibit a concrete model of the free Dedekind $\sigma$-complete Riesz spaces with weak unit. Analogous results are obtained for operations that preserve $p$-integrability over every (not necessarily finite) measure space. The corresponding variety is shown to coincide with the category of Dedekind $\sigma$-complete truncated Riesz spaces, where truncation is meant in the sense of R.N. Ball.

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  • In this paper, we propose an approach to determine the optimal operation strategies for a PV-diesel-battery microgrid covering industrial loads under grid blackouts. A special property of the industrial loads is that they have low power factors. Therefore, the reactive power consumption of the load cannot be neglected. In this study, a novel model of a PV-battery-diesel microgrid is developed considering the active as well reactive power of the microgrid components. Furthermore, an optimization approach is proposed to optimize the active as well reactive power flow in the microgrid for covering the load demand while decreasing the power consumption from the grid, minimizing the diesel generator (DG) operation cost as well as maximizing the consumed power from the PV-array. It has been found that the proposed operation strategy induces a huge reduction of the consumed energy cost and the PV curtailment.

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  • Roll damping is an important problem of ship motion control since excessive roll motion may cause motion sickness of human occupants and damage fragile cargo. Actuators used for roll damping (fins, rudders and thrusters) inevitably create a rotating yaw moment, interfering thus with the vessel's autopilot (heading control system). To reach and maintain the "trade-off" between the concurrent goals of accurate vessel steering and roll damping, an optimization procedure in general needs to take place where the cost functional penalizes the roll angle, the steering error and the control effort. Since the vessel's motion is influenced by the uncertain wave disturbance, the optimal value of this functional and the resulting optimal process are also uncertain. Standard approaches, prevailing in the literature, approximate the wave disturbance by the "colored noise" with a known spectral density, reducing the optimization problem to conventional loop-shaping, LQG or $\mathcal{H}_\infty$ contro

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  • This work is devoted to study orientation theory in arithmetic geometric within the motivic homotopy theory of Morel and Voevodsky. The main tool is a formulation of the absolute purity property for an \emph{arithmetic cohomology theory}, either represented by a cartesian section of the stable homotopy category or satisfying suitable axioms. We give many examples, formulate conjectures and prove a useful property of analytical invariance. Within this axiomatic, we thoroughly develop the theory of characteristic and fundamental classes, Gysin and residue morphisms. This is used to prove Riemann-Roch formulas, in Grothendieck style for arbitrary natural transformations of cohomologies, and a new one for residue morphisms. They are applied to rational motivic cohomology and \'etale rational $\ell$-adic cohomology, as expected by Grothendieck in \cite[XIV, 6.1]{SGA6}.

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  • The reduced k-particle density matrix of a density matrix on finite-dimensional, fermion Fock space can be defined as the image under the orthogonal projection in the Hilbert-Schmidt geometry onto the space of k-body observables. A proper understanding of this projection is therefore intimately related to the representability problem, a long-standing open problem in computational quantum chemistry. Given an orthonormal basis in the finite-dimensional one-particle Hilbert space, we explicitly construct an orthonormal basis of the space of Fock space operators which restricts to an orthonormal basis of the space of k-body operators for all k.

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  • Given a symmetric operad $\mathcal{P}$ and a $\mathcal{P}$-algebra $V$, the universal enveloping algebra ${\mathsf{U}_{\mathcal{P}}}$ is an associative algebra whose category of modules is isomorphic to the abelian category of $V$-modules. We study the notion of PBW property for universal enveloping algebras over an operad. In case $\mathcal{P}$ is Koszul a criterion for PBW property is found. Necessary condition on Hilbert series for $\mathcal{P}$ is found. Moreover, given any symmetric operad $\mathcal{P}$ together with a Gr\"obner basis $G$, a condition is given on the structure of the underlying trees associated with leading monomials of $G$ sufficient for the PBW property to hold. Examples are provided.

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  • In this paper, we study the Hankel determinant generated by a singularly perturbed Gaussian weight $$ w(x,t)=\mathrm{e}^{-x^{2}-\frac{t}{x^{2}}},\;\;x\in(-\infty, \infty),\;\;t>0. $$ By using the ladder operator approach associated with the orthogonal polynomials, we show that the logarithmic derivative of the Hankel determinant satisfies both a non-linear second order difference equation and a non-linear second order differential equation. The Hankel determinant also admits an integral representation involving a Painlev\'e III$'$. Furthermore, we consider the asymptotics of the Hankel determinant under a double scaling, i.e. $n\rightarrow\infty$ and $t\rightarrow 0$ such that $s=(2n+1)t$ is fixed. The asymptotic expansions of the scaled Hankel determinant for large $s$ and small $s$ are established, from which Dyson's constant appears.

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  • Smooth entropies are a tool for quantifying resource trade-offs in (quantum) information theory and cryptography. In typical bi- and multi-partite problems, however, some of the sub-systems are often left unchanged and this is not reflected by the standard smoothing of information measures over a ball of close states. We propose to smooth instead only over a ball of close states which also have some of the reduced states on the relevant sub-systems fixed. This partial smoothing of information measures naturally allows to give more refined characterizations of various information-theoretic problems in the one-shot setting. In particular, we immediately get asymptotic second-order characterizations for tasks such as privacy amplification against classical side information or classical state splitting. For quantum problems like state merging the general resource trade-off is tightly characterized by partially smoothed information measures as well. However, for quantum systems we can so fa

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  • We give the first properties of independent Bernoulli percolation, for oriented graphs on the set of vertices $\Z^d$ that are translation-invariant and may contain loops. We exhibit some examples showing that the critical probability for the existence of an infinite cluster may be direction-dependent. Then, we prove that the phase transition in a given direction is sharp, and study the links between percolation and first-passage percolation on these oriented graphs.

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  • In this article we define Perfectoid Tate curves and compute their cohomology using a \v{C}ech complex and also define perfectoid versions of Weierstra{\ss} and Theta functions.

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  • This paper investigates phase transitions on the optimality gaps in Optimal Power Flow (OPF) problem on real-world power transmission systems operated in France. The experimental results study optimal power flow solutions for more than 6000 scenarios on the networks with various load profiles, voltage feasibility regions, and generation capabilities. The results show that bifurcations between primal solutions and the QC, SOCP, and SDP relaxation techniques frequently occur when approaching congestion points. Moreover, the results demonstrate the existence of multiple bifurcations for certain scenarios when load demands are increased uniformly. Preliminary analysis on these bifurcations were performed.

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  • Piecewise Deterministic Markov Processes (PDMPs) are studied in a general framework. First, different constructions are proven to be equivalent. Second, we introduce a coupling between two PDMPs following the same differential flow which implies quantitative bounds on the total variation between the marginal distributions of the two processes. Finally two results are established regarding the invariant measures of PDMPs. A practical condition to show that a probability measure is invariant for the associated PDMP semi-group is presented. In a second time, a bound on the invariant probability measures in $V$-norm of two PDMPs following the same differential flow is established. This last result is then applied to study the asymptotic bias of some non-exact PDMP MCMC methods.

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  • We show that if a Fano manifold does not admit Kahler-Einstein metrics then the Kahler potentials along the continuity method subconverge to a function with analytic singularities along a subvariety which solves the homogeneous complex Monge-Ampere equation on its complement, confirming an expectation of Tian-Yau.

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  • We study the family of depolarizations of a squarefree monomial ideal $I$, i.e. all monomial ideals whose polarization is $I$. We describe a method to find all depolarizations of $I$ and study some of the properties they share and some they do not share. We then apply polarization and depolarization tools to study the reliability of multi-state coherent systems via binary systems and vice versa.

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  • We consider classical Merton problem of terminal wealth maximization in finite horizon. We assume that the drift of the stock is following Ornstein-Uhlenbeck process and the volatility of it is following GARCH(1) process. In particular, both mean and volatility are unbounded. We assume that there is Knightian uncertainty on the parameters of both mean and volatility. We take that the investor has logarithmic utility function, and solve the corresponding utility maximization problem explicitly. To the best of our knowledge, this is the first work on utility maximization with unbounded mean and volatility in Knightian uncertainty under nondominated priors.

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  • Consider a multiplayer game, and assume a system level objective function, which the system wants to optimize, is given. This paper aims at accomplishing this goal via potential game theory when players can only get part of other players' information. The technique is designing a set of local information based utility functions, which guarantee that the designed game is potential, with the system level objective function its potential function. First, the existence of local information based utility functions can be verified by checking whether the corresponding linear equations have a solution. Then an algorithm is proposed to calculate the local information based utility functions when the utility design equations have solutions. Finally, consensus problem of multiagent system is considered to demonstrate the effectiveness of the proposed design procedure.

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  • Utilizing common resources is always a dilemma for community members. While cooperator players restrain themselves and consider the proper state of resources, defectors demand more than their supposed share for a higher payoff. To avoid the tragedy of the common state, punishing the latter group seems to be an adequate reaction. This conclusion, however, is less straightforward when we acknowledge the fact that resources are finite and even a renewable resource has limited growing capacity. To clarify the possible consequences, we consider a coevolutionary model where beside the payoff-driven competition of cooperator and defector players the level of a renewable resource depends sensitively on the fraction of cooperators and the total consumption of all players. The applied feedback-evolving game reveals that beside a delicately adjusted punishment it is also fundamental that cooperators should pay special attention to the growing capacity of renewable resources. Otherwise, even the u

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  • This paper deals with an SIR model with saturated incidence rate affected by inhibitory effect and saturated treatment function. Two control functions have been used, one for vaccinating the susceptible population and other for the treatment control of infected population. We have analysed the existence and stability of equilibrium points and investigated the transcritical and backward bifurcation. The stability analysis of non-hyperbolic equilibrium point has been performed by using Centre manifold theory. The Pontryagin's maximum principle has been used to characterize the optimal control whose numerical results show the positive impact of two controls mentioned above for controlling the disease. Efficiency analysis is also done to determine the best control strategy among vaccination and treatment.

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  • Motivated by applications to image reconstruction, in this paper we analyse a \emph{finite-difference discretisation} of the Ambrosio-Tortorelli functional. Denoted by $\varepsilon$ the elliptic-approximation parameter and by $\delta$ the discretisation step-size, we fully describe the relative impact of $\varepsilon$ and $\delta$ in terms of $\Gamma$-limits for the corresponding discrete functionals, in the three possible scaling regimes. We show, in particular, that when $\varepsilon$ and $\delta$ are of the same order, the underlying lattice structure affects the $\Gamma$-limit which turns out to be an anisotropic free-discontinuity functional.

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  • Kronecker graphs, obtained by repeatedly performing the Kronecker product of the adjacency matrix of an "initiator" graph with itself, have risen in popularity in network science due to their ability to generate complex networks with real-world properties. In this paper, we explore spatial search by continuous-time quantum walk on Kronecker graphs. Specifically, we give analytical proofs for quantum search on first-, second-, and third-order Kronecker graphs with the complete graph as the initiator, showing that search takes Grover's $O(\sqrt{N})$ time. Numerical simulations indicate that higher-order Kronecker graphs with the complete initiator also support optimal quantum search.

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  • We prove that, under a mild assumption, the heart H of a twin cotorsion pair ((S,T),(U,V)) on a triangulated category C is a quasi-abelian category. If C is also Krull-Schmidt and T=U, we show that the heart of the cotorsion pair (S,T) is equivalent to the Gabriel-Zisman localisation of H at the class of its regular morphisms. In particular, suppose C is a cluster category with a rigid object R and [X_R] the ideal of morphisms factoring through X_R=Ker(Hom(R,-)), then applications of our results show that C/[X_R] is a quasi-abelian category. We also obtain a new proof of an equivalence between the localisation of this category at its class of regular morphisms and a certain subfactor category of C.

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  • In this paper, we consider radial distributional solutions of the quasilinear equation $-\Delta_N u=f(u)$ in the punctured open ball $ B_R\backslash\{0\}\subset \RR^N$, $N \geq 2$. We obtain sharp conditions on the nonlinearity $f$ for extending such solutions to the whole domain $B_R$ by preserving the regularity. For a certain class of noninearity $f$ we obtain the existence of singular solutions and deduce upper and lower estimates on the growth rate near the singularity.

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  • Motivated by Wick-rotations of pseudo-Riemannian manifolds, we study real geometric invariant theory (GIT) and compatible representations. We extend some of the results from earlier works \cite{W2,W1}, in particular, we give sufficient and necessary conditions for when pseudo-Riemannian manifolds are Wick-rotatable to other signatures. For arbitrary signatures, we consider a Wick-rotatable pseudo-Riemannian manifold with closed $O(p,q)$-orbits, and thus generalise the existence condition found in \cite{W1}. Using these existence conditions we also derive an invariance theorem for Wick-rotations of arbitrary signatures.

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  • This paper presents reduction theorems for stability, attractivity, and asymptotic stability of compact subsets of the state space of a hybrid dynamical system. Given two closed sets $\Gamma_1 \subset \Gamma_2 \subset \Re^n$, with $\Gamma_1$ compact, the theorems presented in this paper give conditions under which a qualitative property of $\Gamma_1$ that holds relative to $\Gamma_2$ (stability, attractivity, or asymptotic stability) can be guaranteed to also hold relative to the state space of the hybrid system. As a consequence of these results, sufficient conditions are presented for the stability of compact sets in cascade-connected hybrid systems. We also present a result for hybrid systems with outputs that converge to zero along solutions. If such a system enjoys a detectability property with respect to a set $\Gamma_1$, then $\Gamma_1$ is globally attractive. The theory of this paper is used to develop a hybrid estimator for the period of oscillation of a sinusoidal signal.

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  • Let $(X,\omega)$ be a compact K\"ahler manifold and $\mathcal H$ the space of K\"ahler metrics cohomologous to $\omega$. If a cscK metric exists in $\mathcal H$, we show that all finite energy minimizers of the extended K-energy are smooth cscK metrics, partially confirming a conjecture of Y.A. Rubinstein and the second author. As an immediate application, we obtain that existence of a cscK metric in $\mathcal H$ implies J-properness of the K-energy, thus confirming one direction of a conjecture of Tian. Exploiting this properness result we prove that an ample line bundle $(X,L)$ admitting a cscK metric in $c_1(L)$ is $K$-polystable.

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  • We study the stochastic heat equation driven by an additive infinite dimensional fractional Brownian noise on the unit sphere $\mathbb{S}^{2}$. The existence and uniqueness of its solution in certain Sobolev space is investigated and sample path regularity properties are established. In particular, the exact uniform modulus of continuity of the solution in time/spatial variable is derived.

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  • The use of low-resolution analog-to-digital converters (ADCs) can significantly reduce power consumption and hardware cost. However, their resulting severe nonlinear distortion makes achieving reliable data transmission challenging. For orthogonal frequency division multiplexing (OFDM) transmission, the orthogonality among subcarriers is destroyed. This invalidates conventional OFDM receivers relying heavily on this orthogonality. In this study, we move on to quantized OFDM (Q-OFDM) prototyping implementation based on our previous achievement in optimal Q-OFDM detection. First, we propose a novel Q-OFDM channel estimator by extending the generalized Turbo (GTurbo) framework formerly applied for optimal detection. Specifically, we integrate a type of robust linear OFDM channel estimator into the original GTurbo framework, and derive its corresponding extrinsic information to guarantee its convergence. We also propose feasible schemes for automatic gain control, noise power estimation, a

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  • In this paper we use orthogonal system of Jacobi's polynomials as a tool for study the operators of fractional integration and differentiation in the Riemann-Liouville sense on the compact. This approach has some advantages and alow us to reformulate well-known results of fractional calculus in the new quantity. We consider several modification of Jacobi's polynomials what give us opportunity to study invariant property of operator. As shown by us in this direction is that the operator of fractional integration acting in weighted Lebesgue spaces of summable with square functions has a sequence of including invariant subspaces. The proved theorem on acting of fractional integration operator formulated in terms of Legendre's coefficients is of particular interest. Finely we obtain the sufficient condition in terms of Legendre's coefficients for representation of function by fractional integral.

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  • We classify four-dimensional shrinking Ricci solitons satisfying $Sec \geq \frac{1}{48} R$, where $Sec$ and $R$ denote the sectional and the scalar curvature, respectively. They are isometric to either $\mathbb{R}^{4}$ (and quotients), $\mathbb{S}^{4}$, $\mathbb{RP}^{4}$ or $\mathbb{CP}^{2}$ with their standard metrics.

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  • We consider the Cauchy problem for the 2D gravity water wave equation. Recently Wu \cite{Wu15, Wu18} proved the local well-posedness of the equation in a regime which allows interfaces with angled crests as initial data. In this work we study properties of these singular solutions and prove that the singularities of these solutions are "rigid". More precisely we prove that an initial interface with angled crests remains angled crested, the Euler equation holds point-wise even on the boundary, the particle at the tip stays at the tip, the acceleration at the tip is the one due to gravity and the angle of the crest does not change nor does it tilt. We also show that the existence result of Wu \cite{Wu15} applies not only to interfaces with angled crests, but also allows certain types of cusps.

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  • We define a three parameter family of Bell pseudo-involutions in the Riordan group. The defining sequences have generating functions that are expressible as continued fractions. We exhibit Hankel transforms associated with these sequences, and to the $A$-sequences of the Riordan arrays, that give rise to Somos $4$ sequences. We give examples where these sequences can be associated with elliptic curves, and we exhibit instances where elliptic curves can give rise to associated Riordan pseudo-involutions.

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