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The saturation number of a graph $F$, written $\textup{sat}(n,F)$, is the minimum number of edges in an $n$vertex $F$saturated graph. One of the earliest results on saturation numbers is due to Erd\H{o}s, Hajnal, and Moon who determined $\textup{sat}(n,K_r)$ for all $r \geq 3$. Since then, saturation numbers of various graphs and hypergraphs have been studied. Motivated by Alon and Shikhelman's generalized Tur\'an function, Kritschgau et.\ al.\ defined $\textup{sat}(n,H,F)$ to be the minimum number of copies of $H$ in an $n$vertex $F$saturated graph. They proved, among other things, that $\textup{sat}(n,C_3,C_{2k}) = 0$ for all $k \geq 3$ and $n \geq 2k +2$. We extend this result to all odd cycles by proving that for any odd integer $r \geq 5$, $\textup{sat}(n, C_r,C_{2k}) = 0$ for all $2k \geq r+5$ and $n \geq 2kr$.
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Let $A(\cdot)$ be an $(n+1)\times (n+1)$ uniformly elliptic matrix with H\"older continuous real coefficients and let $\mathcal E_A(x,y)$ be the fundamental solution of the PDE $\mathrm{div} A(\cdot) \nabla u =0$ in $\mathbb R^{n+1}$. Let $\mu$ be a compactly supported $n$ADregular measure in $\mathbb R^{n+1}$ and consider the associated operator $$T_\mu f(x) = \int \nabla_x\mathcal E_A(x,y)\,f(y)\,d\mu(y).$$ We show that if $T_\mu$ is bounded in $L^2(\mu)$, then $\mu$ is uniformly $n$rectifiable. This extends the solution of the codimension $1$ DavidSemmes problem for the Riesz transform to the gradient of the single layer potential. Together with a previous result of CondeAlonso, Mourgoglou and Tolsa, this shows that, given $E\subset\mathbb R^{n+1}$ with finite Hausdorff measure $\mathcal H^n$, if $T_{\mathcal H^n_E}$ is bounded in $L^2(\mathcal H^n_E)$, then $E$ is $n$rectifiable.
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Sufficient conditions characterizing the asymptotic stability and the hybrid $L_1/\ell_1$gain of linear positive impulsive systems under minimum and range dwelltime constraints are obtained. These conditions are stated as infinitedimensional linear programming problems that can be solved using sum of squares programming, a relaxation that is known to be asymptotically exact in the present case. These conditions are then adapted to formulate constructive and convex sufficient conditions for the existence of $L_1/\ell_1$to$L_1/\ell_1$ interval observers for linear impulsive and switched systems. Suitable observer gains can be extracted from the (suboptimal) solution of the infinitedimensional optimization problem where the $L_1/\ell_1$gain of the system mapping the disturbances to the weighted observation errors is minimized. Some examples on impulsive and switched systems are given for illustration.
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In this paper, we study skew constacyclic codes over the ring $\mathbb{Z}_{q}R$ where $R=\mathbb{Z}_{q}+u\mathbb{Z}_{q}$, $q=p^{s}$ for a prime $p$ and $u^{2}=0$. We give the definition of these codes as subsets of the ring $\mathbb{Z}_{q}^{\alpha}R^{\beta}$. Some structural properties of the skew polynomial ring $ R[x,\theta]$ are discussed, where $ \theta$ is an automorphism of $R$. We describe the generator polynomials of skew constacyclic codes over $ R $ and $\mathbb{Z}_{q}R$. Using Gray images of skew constacyclic codes over $\mathbb{Z}_{q}R$ we obtained some new linear codes over $\mathbb{Z}_4$. Further, we have generalized these codes to double skew constacyclic codes over $\mathbb{Z}_{q}R$.
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Let $P_s:= \mathbb F_2[x_1,x_2,\ldots ,x_s]$ be the graded polynomial algebra over the prime field of two elements, $\mathbb F_2$, in $s$ variables $x_1, x_2, \ldots , x_s$, each of degree 1. We are interested in the {\it Peterson hit problem} of finding a minimal set of generators for $P_s$ as a module over the mod2 Steenrod algebra, $\mathcal{A}$. In this Note, we study the hit problem in the case $s = 5$ and the degree $4.2^t3$ with $t$ a positive integer. Using this result, we show that Singer's conjecture for the fifth algebraic transfer is true in the above degree.
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The distribution of certain Mahonian statistic (called $\mathrm{BAST}$) introduced by Babson and Steingr\'{i}msson over the set of permutations that avoid vincular pattern $1\underline{32}$, is shown bijectively to match the distribution of major index over the same set. This new layer of equidistribution is then applied to give alternative interpretations of two related $q$Stirling numbers of the second kind, studied by Carlitz and Gould. An extension to an EulerMahonian statistic over the set of ordered partitions presents itself naturally. During the course, a refined relation between $\mathrm{BAST}$ and its reverse complement $\mathrm{STAT}$ is derived as well.
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In this article, we will give the Deligne 1motives up to isogeny corresponding to the $\mathbb{Q}$limiting mixed Hodge structures of semistable degenerations of curves, by using logarithmic structures and Steenbrink's cohomological mixed Hodge complexes associated to semistable degenerations of curves.
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The derivation by Alan Hodgkin and Andrew Huxley of their famous neuronal conductance model relied on experimental data gathered using neurons of the giant squid. It becomes clear that determining experimentally the conductances of neurons is hard, in particular under the presence of spatial and temporal heterogeneities. Moreover it is reasonable to expect variations between species or even between types of neurons of a same species. Determining conductances from one type of neuron is no guarantee that it works across the board. We tackle the inverse problem of determining, given voltage data, conductances with nonuniform distribution computationally. In the simpler setting of a cable equation, we consider the Landweber iteration, a computational technique used to identify nonuniform spatial and temporal ionic distributions, both in a single branch or in a tree. Here, we propose and (numerically) investigate an iterative scheme that consists in numerically solving two partial differe
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A classical result of K. L. Chung and W. Feller deals with the partial sums $S_k$ arising in a fair cointossing game. If $N_n$ is the number of "positive" terms among $S_1, S_2,\dots,S_n$ then the quantity $P(N_{2n}=2r)$ takes an elegant form. We lift the restriction on an even number of tosses and give a simple expression for $P(N_{2n+1}=r)$, $r=0,1,2,\dots,2n+1$. We get to this result by adapting the FeynmanKac methodology.
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A time series is uniquely represented by its geometric shape, which also carries information. A time series can be modelled as the trajectory of a particle moving in a force field with one degree of freedom. The force acting on the particle shapes the trajectory of its motion, which is made up of elementary shapes of infinitesimal neighborhoods of points in the trajectory. It has been proved that an infinitesimal neighborhood of a point in a continuous time series can have at least 29 different shapes or configurations. So information can be encoded in it in at least 29 different ways. A 3point neighborhood (the smallest) in a discrete time series can have precisely 13 different shapes or configurations. In other words, a discrete time series can be expressed as a string of 13 symbols. Across diverse real as well as simulated data sets it has been observed that 6 of them occur more frequently and the remaining 7 occur less frequently. Based on frequency distribution of 13 configuratio
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We define a 1cocycle in the space of long knots that is a natural generalisation of the Kontsevich integral seen as a 0cocycle. It involves a 2form that generalises the KnizhnikZamolodchikov connection. Similarly to the Kontsevich integral, it lives in a space of chord diagrams of the same kind as those that make the principal parts of Vassiliev's 1cocycles. Moreover, up to a change of variable similar to the one that led BirmanLin to discover the 4T relations, we show that the relations defining our space, which allow the integral to be finite and invariant, are dual to the maps that define Vassiliev's cohomology in degree 1.
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Large datasets create opportunities as well as analytic challenges. A recent development is to use random projection or sketching methods for dimension reduction in statistics and machine learning. In this work, we study the statistical performance of sketching algorithms for linear regression. Suppose we randomly project the data matrix and the outcome using a random sketching matrix reducing the sample size, and do linear regression on the resulting data. How much do we lose compared to the original linear regression? The existing theory does not give a precise enough answer, and this has been a bottleneck for using random projections in practice. In this paper, we introduce a new mathematical approach to the problem, relying on very recent results from asymptotic random matrix theory and free probability theory. This is a perfect fit, as the sketching matrices are random in practice. We allow the dimension and sample sizes to have an arbitrary ratio. We study the most popular sketch
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We prove some functional equations involving the (classical) matching polynomials of path and cycle graphs and the $d$matching polynomial of a cycle graph. A matching in a (finite) graph $G$ is a subset of edges no two of which share a vertex, and the matching polynomial of $G$ is a generating function encoding the numbers of matchings in $G$ of each size. The $d$matching polynomial is a weighted average of matching polynomials of degree$d$ covers, and was introduced in a paper of Hall, Puder, and Sawin. Let $\mathcal{C}_n$ and $\mathcal{P}_n$ denote the respective matching polynomials of the cycle and path graphs on $n$ vertices, and let $\mathcal{C}_{n,d}$ denote the $d$matching polynomial of the cycle $C_n$. We give a purely combinatorial proof that $\mathcal{C}_k (\mathcal{C}_n (x)) = \mathcal{C}_{kn} (x)$ en route to proving a conjecture made by Hall: that $\mathcal{C}_{n,d} (x) = \mathcal{P}_d (\mathcal{C}_n (x))$.
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We give a detailed proof of the homological Arnold conjecture for nondegenerate periodic Hamiltonians on general closed symplectic manifolds $M$ via a direct PiunikhinSalamonSchwarz morphism. Our constructions are based on a coherent polyfold description for moduli spaces of pseudoholomorphic curves in a family of symplectic manifolds degenerating from $\mathbb{C}\mathbb{P}^1\times M$ to $\mathbb{C} \times M \sqcup \mathbb{C}^\times M$, as developed by FishHoferWysockiZehnder as part of the Symplectic Field Theory package. The 2011 sketch of this proof was joint work with Peter Albers, Joel Fish.
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We study the admissibility of power injections in singlephase microgrids, where the electrical state is represented by complex nodal voltages and controlled by nodal power injections. Assume that (i) there is an initial electrical state that satisfies security constraints and the nonsingularity of loadflow Jacobian, and (ii) power injections reside in some uncertainty set. We say that the uncertainty set is admissible for the initial electrical state if any continuous trajectory of the electrical state is ensured to be secured and nonsingular as long as power injections remain in the uncertainty set. We use the recently proposed Vcontrol and show two new results. First, if a complex nodal voltage set V is convex and every element in V is nonsingular, then V is a domain of uniqueness. Second, we give sufficient conditions to guarantee that every element in some power injection set S has a loadflow solution in V, based on impossibility of obtaining loadflow solutions at the bounda
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New estimates for the generalization error are established for the twolayer neural network model. These new estimates are a priori in nature in the sense that the bounds depend only on some norms of the underlying functions to be fitted, not the parameters in the model. In contrast, most existing results for neural networks are a posteriori in nature in the sense that the bounds depend on some norms of the model parameters. The error rates are comparable to that of the Monte Carlo method for integration problems. Moreover, these bounds are equally effective in the overparametrized regime when the network size is much larger than the size of the dataset.
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This paper presents a novel transformationproximal bundle algorithm to solve multistage adaptive robust mixedinteger linear programs (MARMILPs). By explicitly partitioning recourse decisions into state decisions and local decisions, the proposed algorithm applies affine decision rule only to state decisions and allows local decisions to be fully adaptive. In this way, the MARMILP is proved to be transformed into an equivalent twostage adaptive robust optimization (ARO) problem. The proposed multitotwo transformation scheme remains valid for other types of nonanticipative decision rules besides the affine one, and it is general enough to be employed with existing twostage ARO algorithms for solving MARMILPs. The proximal bundle method is developed for the resulting twostage ARO problem. We perform a theoretical analysis to show finite convergence of the proposed algorithm with any positive tolerance. To quantitatively assess solution quality, we develop a scenariotreebased low
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In this work, we present a unified gaskinetic particle (UGKP) method for the simulation of multiscale photon transport. The multiscale nature of the particle method mainly comes from the recovery of the time evolution flux function in the unified gaskinetic scheme (UGKS) through a coupled dynamic process of particle transport and collision. This practice improves the original operator splitting approach in the Monte Carlo method, such as the separated treatment of particle transport and collision. As a result, with the variation of the ratio between numerical time step and local photon's collision time, different transport physics can be fully captured in a single computation. In the diffusive limit, the UGKP method could recover the solution of the diffusion equation with the cell size and time step being much larger than the photon's mean free path and the mean collision time. In the free transport limit, it presents an exact particle tracking process as the original Monte Carlo me
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We propose and analyse a mathematical model for cholera considering vaccination. We show that the model is epidemiologically and mathematically well posed and prove the existence and uniqueness of diseasefree and endemic equilibrium points. The basic reproduction number is determined and the local asymptotic stability of equilibria is studied. The biggest cholera outbreak of world's history began on 27th April 2017, in Yemen. Between 27th April 2017 and 15th April 2018 there were 2275 deaths due to this epidemic. A vaccination campaign began on 6th May 2018 and ended on 15th May 2018. We show that our model is able to describe well this outbreak. Moreover, we prove that the number of infected individuals would have been much lower provided the vaccination campaign had begun earlier.
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In this thesis, we introduce a new cohomology theory associated to a Lie 2algebras and a new cohomology theory associated to a Lie 2group. These cohomology theories are shown to extend the classical cohomology theories of Lie algebras and Lie groups in that their second groups classify extensions. We use this fact together with an adapted van Est map to prove the integrability of Lie 2algebras anew.
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We present in this paper how the singlephoton wave function for transversal photons (with the direct sum of ordinary unitary representations of helicity 1 and 1 acting on it) is subsumed within the formalism of GuptaBleuler for the quantized free electromagnetic field. Rigorous GuptaBleuler quantization of the free electromagnetic field is based on our generalization (published formerly) of the Mackey theory of induced representations which includes representations preserving the indefinite Krein innerproduct given by the GuptaBleuler operator. In particular it follows that the results of Bia{\l}ynickiBirula on the singlephoton wave function may be reconciled with the causal perturbative approach to QED.
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We study a degenerate elliptic system with variable exponents. Using the variational approach and some recent theory on weighted Lebesgue and Sobolev spaces with variable exponents, we prove the existence of at least two distinct nontrivial weak solutions of the system. Several consequences of the main theorem are derived; in particular, the existence of at lease two distinct nontrivial nonnegative solution are established for a scalar degenerate problem. One example is provided to showthe applicability of our results.
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About 6 years ago, semitoric systems were classified by Pelayo & Vu Ngoc by means of five invariants. Standard examples are the coupled spin oscillator on $\mathbb{S}^2 \times \mathbb{R}^2$ and coupled angular momenta on $\mathbb{S}^2 \times \mathbb{S}^2$, both having exactly one focusfocus singularity. But so far there were no explicit examples of systems with more than one focusfocus singularity which are semitoric in the sense of that classification. This paper introduces a 6parameter family of integrable systems on $\mathbb{S}^2 \times \mathbb{S}^2$ and proves that, for certain ranges of the parameters, it is a compact semitoric system with precisely two focusfocus singularities. Since the twisting index (one of the semitoric invariants) is related to the relationship between different focusfocus points, this paper provides systems for the future study of the twisting index.
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In this paper we consider a distributed convex optimization problem over timevarying undirected networks. We propose a dual method, primarily averaged network dual ascent (PANDA), that is proven to converge Rlinearly to the optimal point given that the agents objective functions are strongly convex and have Lipschitz continuous gradients. Like dual decomposition, PANDA requires half the amount of variable exchanges per iterate of methods based on DIGing, and can provide with practical improved performance as empirically demonstrated.
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A novel masslumping strategy for a mixed finite element approximation of Maxwell's equations is proposed. On structured orthogonal grids the resulting method coincides with the spatial discretization of the Yee scheme. The proposed method, however, generalizes naturally to unstructured grids and anisotropic materials and thus yields a variational extension of the Yee scheme for these situations.
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By theorems of Carlson and Renaudin, the theory of $(\infty,1)$categories embeds in that of prederivators. The purpose of this paper is to give a twofold answer to the inverse problem: understanding which prederivators model $(\infty,1)$categories, either strictly or in a homotopical sense. First, we characterize which prederivators arise on the nose as prederivators associated to quasicategories. Next, we put a model structure on the category of prederivators and strict natural transformations, and prove a Quillen equivalence with the Joyal model structure for quasicategories.
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This work proposes two nodal type nonconforming finite elements over convex quadrilaterals, which are parts of a finite element exact sequence. Both elements are of 12 degrees of freedom (DoFs) with polynomial shape function spaces selected. The first one is designed for fourth order elliptic singular perturbation problems, and the other works for Brinkman problems. Numerical examples are also provided.
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We show that the $\ell$adic Tate conjecture for divisors on smooth proper varieties over finitely generated fields of positive characteristic follows from the $\ell$adic Tate conjecture for divisors on smooth projective surfaces over finite fields. Similar results for cycles of higher codimension are given.
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The LandsbergSchaar relation is a classical identity between quadratic Gauss sums, normally used as a stepping stone to prove quadratic reciprocity. The LandsbergSchaar relation itself is usually proved by carefully taking a limit in the functional equation for Jacobi's theta function. In this article we present a direct proof, avoiding any analysis.
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These brief lecture notes are intended mainly for undergraduate students in engineering or physics or mathematics who have met or will soon be meeting the Dirac delta function and some other objects related to it. These students might have already felt  or might in the near future feel  not entirely comfortable with the usual intuitive explanations about how to "integrate" or "differentiate" or take the "Fourier transform" of these objects. These notes will reveal to these students that there is a precise and rigorous way, and this also means a more useful and reliable way, to define these objects and the operations performed upon them. This can be done without any prior knowledge of functional analysis or of Lebesgue integration. Readers of these notes are assumed to only have studied basic courses in linear algebra, and calculus of functions of one and two variables, and an introductory course about the Fourier transform of functions of one variable. Most of the results and proofs
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In this work, we are concerned with hierarchically hyperbolic spaces and hierarchically hyperbolic groups. Our main result is a wide generalization of a combination theorem of Behrstock, Hagen, and Sisto. In particular, as a consequence, we show that any finite graph product of hierarchically hyperbolic groups is again a hierarchically hyperbolic group, thereby answering a question posed by Behrstock, Hagen, and Sisto. In order to operate in such a general setting, we establish a number of structural results for hierarchically hyperbolic spaces and hieromorphisms (that is, morphisms between such spaces), and we introduce two new notions for hierarchical hyperbolicity, that is concreteness and the intersection property, proving that they are satisfied in all known examples.
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In this paper we prove that the ball maximizes the first eigenvalue of the Robin Laplacian operator with negative boundary parameter, among all convex sets of \mathbb{R}^n with prescribed perimeter. The key of the proof is a dearrangement procedure of the first eigenfunction of the ball on the level sets of the distance function to the boundary of the convex set, which controls the boundary and the volume energies of the Rayleigh quotient.
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For a long time, the Dirichlet process has been the gold standard discrete random measure in Bayesian nonparametrics. The PitmanYor process provides a simple and mathematically tractable generalization, allowing for a very flexible control of the clustering behaviour. Two commonly used representations of the PitmanYor process are the stickbreaking process and the Chinese restaurant process. The former is a constructive representation of the process which turns out very handy for practical implementation, while the latter describes the partition distribution induced. However, the usual proof of the connection between them is indirect and involves measure theory. We provide here an elementary proof of PitmanYor's Chinese Restaurant process from its stickbreaking representation.
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We propose a direct numerical method for the solution of an optimal control problem governed by a twoside spacefractional diffusion equation. The presented method contains two main steps. In the first step, the space variable is discretized by using the JacobiGauss pseudospectral discretization and, in this way, the original problem is transformed into a classical integerorder optimal control problem. The main challenge, which we faced in this step, is to derive the left and right fractional differentiation matrices. In this respect, novel techniques for derivation of these matrices are presented. In the second step, the LegendreGaussRadau pseudospectral method is employed. With these two steps, the original problem is converted into a convex quadratic optimization problem, which can be solved efficiently by available methods. Our approach can be easily implemented and extended to cover fractional optimal control problems with state constraints. Five test examples are provided to
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We illustrate how the different kinds of constraints acting on an impulsive mechanical system can be clearly described in the geometric setup given by the configuration spacetime bundle $\pi_t:\mathcal{M} \to \mathbb{E}$ and its first jet extension $\pi: J_1 \to \mathcal{M}$ in a way that ensures total compliance with axioms and invariance requirements of Classical Mechanics. We specify the differences between geometric and constitutive characterizations of a constraint. We point out the relevance of the role played by the concept of frame of reference, underlining when the frame independence is mandatorily required and when a choice of a frame is an inescapable need. The thorough rationalization allows the introduction of unusual but meaningful kinds of constraints, such as unilateral kinetic constraints or breakable constraints, and of new theoretical aspects, such as the possible dependence of the impulsive reaction by the active forces acting on the system.
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An inequality of BrascampLiebLuttinger and of Rogers states that among subsets of Euclidean space $\mathbb{R}^d$ of specified Lebesgue measures, balls centered at the origin are maximizers of certain functionals defined by multidimensional integrals. For $d>1$, this inequality only applies to functionals invariant under a diagonal action of $\text{Sl}(d)$. We investigate functionals of this type, and their maximizers, in perhaps the simplest situation in which $\text{Sl}(d)$ invariance does not hold. Assuming a more limited symmetry involving dilations but not rotations, we show under natural hypotheses that maximizers exist, and moreover, that there exist distinguished maximizers whose structure reflects this limited symmetry. For small perturbations of the $\text{Sl}(d)$invariant framework we show that these distinguished maximizers are strongly convex sets with infinitely differentiable boundaries. It is shown that maximizers fail to exist for certain arbitrarily small pertur
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The study of frequency synchronization configurations in Kuramoto models is a ubiquitous mathematical problem that has found applications in many seemingly independent fields. In this paper, we focus on networks whose underlying graph are cycle graphs. Based on the recent result on the upper bound of the frequency synchronization configurations in this context, we propose a toric deformation homotopy method for locating all frequency synchronization configurations with complexity that is linear in this upper bound. Loosely based on the polyhedral homotopy method, this homotopy induces a deformation of the set of the synchronization configurations into a series of toric varieties, yet our method has the distinct advantage of avoiding the costly step of computing mixed cells. We also explore the important consequences of this homotopy method in the context of direct acyclic decomposition of Kuramoto networks and tropical stable intersection points for Kuramoto equations.
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The infinitedimensional information operator for the nuisance parameter plays a key role in semiparametric inference, as it is closely related to the regular estimability of the target parameter. Calculation of information operators has traditionally proceeded in a casebycase manner and has easily entailed lengthy derivations with complicated arguments. We develop a unified framework for this task by exploiting commonality in the form of semiparametric likelihoods. The general formula allows one to derive information operators with simple calculus and, if necessary at all, a minimal amount of probabilistic evaluations. This streamlined approach shows its efficiency and versatility in application to a number of popular models in survival analysis, inverse problems, and missing data.
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We obtain a weak homotopy equivalence type result between two topological groups associated with a Kirchberg algebra: the unitary group of the continuous asymptotic centralizer and the loop group of the automorphism group of the stabilization. This result plays a crucial role in our subsequent work on the classification of poly$\mathbb{Z}$ group actions on Kirchberg algebras. As a special case, we show that the $K$groups of the continuous asymptotic centralizer are isomorphic to the $KK$groups of the Kirchberg algebra.
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We study nonparametric estimators of conditional Kendall's tau, a measure of concordance between two random variables given some covariates. We prove nonasymptotic bounds with explicit constants, that hold with high probabilities. We provide "direct proofs" of the consistency and the asymptotic law of conditional Kendall's tau. A simulation study evaluates the numerical performance of such nonparametric estimators.
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This paper develops a lownonnegativerank approximation method to identify the state aggregation structure of a finitestate Markov chain under an assumption that the state space can be mapped into a handful of metastates. The number of metastates is characterized by the nonnegative rank of the Markov transition matrix. Motivated by the success of the nuclear norm relaxation in low rank minimization problems, we propose an atomic regularizer as a convex surrogate for the nonnegative rank and formulate a convex optimization problem. Because the atomic regularizer itself is not computationally tractable, we instead solve a sequence of problems involving a nonnegative factorization of the Markov transition matrices by using the proximal alternating linearized minimization method. Two methods for adjusting the rank of factorization are developed so that local minima are escaped. One is to append an additional column to the factorized matrices, which can be interpreted as an approximatio
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In this article we consider a microscopic model for hostvector disease transmission based on configuration space analysis. Using Vlasov scaling we obtain the corresponding mesoscopic (kinetic) equations, describing the density of susceptible and infected compartments in space. The resulting system of equations can be seen as a generalization to a spatial SISUV model.
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The Novikov equation is an integrable analogue of the CamassaHolm equation with a cubic (rather than quadratic) nonlinear term. Both these equations support a special family of weak solutions called multipeakon solutions. In this paper, an approach involving Pfaffians is applied to study multipeakons of the Novikov equation. First, we show that the Novikov peakon ODEs describe an isospectral flow on the manifold cut out by certain Pfaffian identities. Then, a link between the Novikov peakons and the finite Toda lattice of BKP type (BToda lattice) is established based on the use of Pfaffians. Finally, certain generalizations of the Novikov equation and the finite BToda lattice are proposed together with special solutions. To our knowledge, it is the first time that the peakon problem is interpreted in terms of Pfaffians.
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Given a sequential learning algorithm and a target model, sequential machine teaching aims to find the shortest training sequence to drive the learning algorithm to the target model. We present the first principled way to find such shortest training sequences. Our key insight is to formulate sequential machine teaching as a timeoptimal control problem. This allows us to solve sequential teaching by leveraging key theoretical and computational tools developed over the past 60 years in the optimal control community. Specifically, we study the Pontryagin Maximum Principle, which yields a necessary condition for optimality of a training sequence. We present analytic, structural, and numerical implications of this approach on a case study with a leastsquares loss function and gradient descent learner. We compute optimal training sequences for this problem, and although the sequences seem circuitous, we find that they can vastly outperform the best available heuristics for generating train
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This paper considers the consensus performance improvement problem of networked general linear agents subject to external disturbances over Markovian randomly switching communication topologies. The consensus control laws can only use its local output information. Firstly, a class of fullorder observerbased control protocols is proposed to solve this problem, which depends solely on the relative outputs of neighbours. Then, to eliminate the redundancy involved in the fullorder observer, a class of reducedorder observerbased control protocols is designed. Algorithms to construct both protocols are presented, which guarantee that agents can reach consensus in the asymptotic mean square sense when they are not perturbed by disturbances, and that they have decent $H_{\infty}$ performance and transient performance when the disturbances exist. At the end of this manuscript, numerical simulations which apply both algorithms to four networked Raptor90 helicopters are performed to verify
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It is a wellknown fact that a stability condition $\phi: Obj^* \mathcal{A} \to \mathcal{I}$ over an abelian length category $\mathcal{A}$ induces a chain of torsion classes $\eta_\phi$ indexed by the totally ordered set $\mathcal{I}$. Inspired by this fact, in this paper we study all chains of torsion classes $\eta$ indexed by a totally ordered set $\mathcal{I}$ in $\mathcal{A}$. Our first theorem says that every chain of torsion classes $\eta$ indexed by $\mathcal{I}$ induces a HarderNarasimhan filtration to every nonzero object of $\mathcal{A}$. Building on this, we are able to generalise several of the results showed by Rudakov in \cite{Rudakov1997}. Moreover we adapt the definition of slicing introduced by Bridgeland in \cite{Bridgeland2007} and we characterise them in terms of indexed chain of torsion classes. Finally, we follow ideas of Bridgeland to show that all chains of torsion classes of $\mathcal{A}$ indexed by the set $[0,1]$ form a metric space with a natural wall and c
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The Chinese restaurant process (CRP) and the stickbreaking process are the two most commonly used representations of the Dirichlet process. However, the usual proof of the connection between them is indirect, relying on abstract properties of the Dirichlet process that are difficult for nonexperts to verify. This short note provides a direct proof that the stickbreaking process leads to the CRP, without using any measure theory. We also discuss how the stickbreaking representation arises naturally from the CRP.
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The M\"obius invariant space $\mathcal{Q}_p$, $0<p<\infty$, consists of functions $f$ which are analytic in the open unit disk $\mathbb{D}$ with $$ \f\_{\mathcal{Q}_p}=f(0)+\sup_{w\in \D} \left(\int_\D f'(z)^2(1\sigma_w(z)^2)^p dA(z)\right)^{1/2}<\infty, $$ where $\sigma_w(z)=(wz)/(1\overline{w}z)$ and $dA$ is the area measure on $\mathbb{D}$. It is known that the following inequality $$ f(0)+\sup_{w\in \D} \left(\int_\D \left\frac{f(z)f(w)}{1\overline{w}z}\right^2 (1\sigma_w(z)^2)^p dA(z)\right)^{1/2} \lesssim \f\_{\mathcal{Q}_p} $$ played a key role to characterize multipliers and certain Carleson measures for $\mathcal{Q}_p$ spaces. The converse of the inequality above is a conjecturedinequality in [14]. In this paper, we show that this conjecturedinequality is true for $p>1$ and it does not hold for $0<p\leq 1$.
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We show that the threedimensional homology cobordism group admits an infiniterank summand. It was previously known that the homology cobordism group contains a $\mathbb{Z}^\infty$subgroup and a $\mathbb{Z}$summand. Our proof proceeds by introducing an algebraic variant of the involutive Heegaard Floer package of HendricksManolescu and HendricksManolescuZemke. This is inspired by an analogous argument in the setting of knot concordance due to the second author.
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In the Ising model, we consider the problem of estimating the covariance of the spins at two specified vertices. In the ferromagnetic case, it is easy to obtain an additive approximation to this covariance by repeatedly sampling from the relevant Gibbs distribution. However, we desire a multiplicative approximation, and it is not clear how to achieve this by sampling, given that the covariance can be exponentially small. Our main contribution is a fully polynomial time randomised approximation scheme (FPRAS) for the covariance. We also show that that the restriction to the ferromagnetic case is essential  there is no FPRAS for multiplicatively estimating the covariance of an antiferromagnetic Ising model unless RP = #P. In fact, we show that even determining the sign of the covariance is #Phard in the antiferromagnetic case.
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We develop a phasefield approximation of the relaxation of the perimeter functional in the plane under a connectedness constraint based on the classical ModicaMortola functional and the connectedness constraint of (Dondl, Lemenant, Wojtowytsch 2017). We prove convergence of the approximating energies and present numerical results and applications to image segmentation.
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For a separable finite diffuse measure space $\mathcal{M}$ and an orthonormal basis $\{\varphi_n\}$ of $L^2(\mathcal{M})$ consisting of bounded functions $\varphi_n\in L^\infty(\mathcal{M})$, we find a measurable subset $E\subset\mathcal{M}$ of arbitrarily small complement $\mathcal{M}\setminus E<\epsilon$, such that every measurable function $f\in L^1(\mathcal{M})$ has an approximant $g\in L^1(\mathcal{M})$ with $g=f$ on $E$ and the Fourier series of $g$ converges to $g$, and a few further properties. The subset $E$ is universal in the sense that it does not depend on the function $f$ to be approximated. Further in the paper this result is adapted to the case of $\mathcal{M}=G/H$ being a homogeneous space of an infinite compact second countable Hausdorff group. As a useful illustration the case of $n$spheres with spherical harmonics is discussed. The construction of the subset $E$ and approximant $g$ is sketched briefly at the end of the paper.
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Consider a dynamic random geometric social network identified by $s_t$ independent points $x_t^1,\ldots,x_t^{s_t}$ in the unit square $[0,1]^2$ that interact in continuous time $t\geq 0$. The generative model of the random points is a Poisson point measures. Each point $x_t^i$ can be active or not in the network with a Bernoulli probability $p$. Each pair being connected by affinity thanks to a step connection function if the interpoint distance $\x_t^ix_t^j\\leq a_\mathsf{f}^\star$ for any $i\neq j$. We prove that when $a_\mathsf{f}^\star=\sqrt{\frac{(s_t)^{l1}}{p\pi}}$ for $l\in(0,1)$, the number of isolated points is governed by a Poisson approximation as $s_t\to\infty$. This offers a natural threshold for the construction of a $a_\mathsf{f}^\star$neighborhood procedure tailored to the dynamic clustering of the network adaptively from the data.
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In this paper, we study the atomic structure of certain classes of semigroup algebras whose sets of exponents are additive submonoids of rational numbers. When studying the atomicity of integral domains, the building blocks by excellence are the irreducible elements. Here we start by extending the Gauss's Lemma and the Eisenstein's Criterion from polynomial rings to semigroup rings with rational exponents. Then we prove that semigroup algebras whose exponent sets are submonoids of $\langle 1/p \mid p \ \text{ is prime} \rangle$ are atomic. Next, for every algebraic closed field $F$, we exhibit a class of Bezout semigroup algebras over $F$ with rational exponents whose members are antimatter, i.e., contain no atoms. In addition, we use a class of rootclosed additive submonoids of rationals to construct another class of antimatter semigroup algebras over any perfect field of finite characteristic. Finally, we characterize the irreducible elements of semigroup algebras whose exponent sem
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It is becoming increasingly common to see large collections of network data objects  that is, data sets in which a network is viewed as a fundamental unit of observation. As a result, there is a pressing need to develop networkbased analogues of even many of the most basic tools already standard for scalar and vector data. In this paper, our focus is on averages of unlabeled, undirected networks with edge weights. Specifically, we (i) characterize a certain notion of the space of all such networks, (ii) describe key topological and geometric properties of this space relevant to doing probability and statistics thereupon, and (iii) use these properties to establish the asymptotic behavior of a generalized notion of an empirical mean under sampling from a distribution supported on this space. Our results rely on a combination of tools from geometry, probability theory, and statistical shape analysis. In particular, the lack of vertex labeling necessitates working with a quotient space
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Quadratic backward stochastic differential equations with singularity in the value process appear in several applications, including stochastic control and physics. In this paper, we prove existence and uniqueness of equations with generators (dominated by a function) of the form $z^2/y$. In the particular case where the BSDE is Markovian, we obtain existence of viscosity solutions of singular quadratic PDEs with and without Neumann lateral boundaries, and rather weak assumptions on the regularity of the coefficients. Furthermore, we show how our results can be applied to optimization problems in finance.
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Highspeed trains (HSTs) are being widely deployed around the world. To meet the highrate data transmission requirements on HSTs, millimeter wave (mmWave) HST communications have drawn increasingly attentions. To realize sufficient link margin, mmWave HST systems employ directional beamforming with large antenna arrays, which results in that the channel estimation is rather timeconsuming. In HST scenarios, channel conditions vary quickly and channel estimations should be performed frequently. Since the period of each transmission time interval (TTI) is too short to allocate enough time for accurate channel estimation, the key challenge is how to design an efficient beam searching scheme to leave more time for data transmission. Motivated by the successful applications of machine learning, this paper tries to exploit the similarities between current and historical wireless propagation environments. Using the knowledge of reinforcement learning, the beam searching problem of mmWave HST
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We find a decomposition formula of the local BayerMacr\`i map for the nef line bundle theory on the Bridgeland moduli space over surface. If there is a global BayerMacr\`i map, such decomposition gives a precise correspondence from Bridgeland walls to Mori walls. As an application, we compute the nef cone of the Hilbert scheme $S^{[n]}$ of $n$points over special kinds of fibered surface $S$ of Picard rank two.
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We study biLagrangian structures (a symplectic form with a pair of complementary Lagrangian foliations, also known as paraK\"ahler or K\"unneth structures) on nilmanifolds of dimension less than or equal to 6. In particular, building on previous work of several authors, we determine which 6dimensional nilpotent Lie algebras admit a biLagrangian structure. In dimension 6, there are (up to isomorphism) 26 nilpotent Lie algebras which admit a symplectic form, 16 of which admit a biLagrangian structure and 10 of which do not. We also calculate the curvature of the canonical connection of these biLagrangian structures.
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A new version of the change of the "phase" (i.e., of the set of observable characteristics) of a quantum system is proposed. In a general scenario the evolution is assumed generated, before the phase transition, by some standard Hermitian Hamiltonian $H^{(before)}$, and, after the phase transition, by one of the recently very popular nonstandard, nonHermitian (but hiddenly Hermitian, i.e., still unitarityguaranteeing) Hamiltonians $H^{(after)}$. For consistency, a smoothness of matching between the two operators as well as between the related physical Hilbert spaces must be guaranteed. The feasibility of the idea is illustrated via the twomode $(N1)$bosonic BoseHubbard Hamiltonian. In $H^{(before)}=H^{(BH)}(\varepsilon)$ we use the decreasing real $\varepsilon^{(before)} \to 0$. In the hiddenly Hermitian continuation $H^{(after)}=H^{(BH)}(\tilde{\varepsilon})$ the imaginary part of the purely imaginary $\tilde{\varepsilon}^{(after)}$ grows. The smoothness of the transition occur
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We study the effect of small random advection in two models in turbulent combustion. Assuming that the velocity field decorrelates sufficiently fast, we (i) identify the order of the fluctuations of the front with respect to the size of the advection, and (ii) characterize them by the solution of a HamiltonJacobi equation forced by white noise. In the simplest case, the result yields, for both models, a front with Brownian fluctuations of the same scale as the size of the advection. That the fluctuations are the same for both models is somewhat surprising, in view of known differences between the two models.
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We study the problem of caching optimization in heterogeneous networks with mutual interference and perfile rate constraints from an energy efficiency perspective. A setup is considered in which two cacheenabled transmitter nodes and a coordinator node serve two users. We analyse and compare two approaches: (i) a cooperative approach where each of the transmitters might serve either of the users and (ii) a noncooperative approach in which each transmitter serves only the respective user. We formulate the cache allocation optimization problem so that the overall system power consumption is minimized while the use of the link from the master node to the end users is spared whenever possible. We also propose a lowcomplexity optimization algorithm and show that it outperforms the considered benchmark strategies. Our results indicate that significant gains both in terms of power saving and sparing of master node's resources can be obtained when full cooperation between the transmitters
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We propose a purely probabilistic model to explain the evolution path of a population maximum fitness. We show that after $n$ births in the population there are about $\ln n$ upwards jumps. This is true for any mutation probability and any fitness distribution and therefore suggests a general law for the number of upwards jumps. Simulations of our model show that a typical evolution path has first a steep rise followed by long plateaux. Moreover, independent runs show parallel paths. This is consistent with what was observed by Lenski and Travisano (1994) in their bacteria experiments.
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We study the problem of finding good gauges for connections in higher gauge theories. We find that, for $2$connections in strict $2$gauge theory and $3$connections in $3$gauge theory, there are local "Coulomb gauges" that are more canonical than in classical gauge theory. In particular, they are essentially unique, and no smallness of curvature is needed in the critical dimensions. We give natural definitions of $2$YangMills and $3$YangMills theory and find that the choice of good gauges makes them essentially linear. As an application, (anti)selfdual $2$connections over $B^6$ are always $2$YangMills, and (anti)selfdual $3$connections over $B^8$ are always $3$YangMills.
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Recently, in their pioneering work on the subject of biunivalent functions, Srivastava et al. \cite{HMSAKMPG} actually revived the study of the coefficient problems involving biunivalent functions. Inspired by the pioneering work of Srivastava et al. \cite{HMSAKMPG}, there has been triggering interest to study the coefficient problems for the different subclasses of biunivalent functions. Motivated largely by Ali et al. \cite{AliRaviMaMinaclass}, Srivastava et al. \cite{HMSAKMPG} and G\"{u}ney et al. \cite{HOGGMSJSFib2018} in this paper, we consider certain classes of biunivalent functions related to shelllike curves connected with Fibonacci numbers to obtain the estimates of second, third TaylorMaclaurin coefficients and Fekete  Szeg\"{o} inequalities. Further, certain special cases are also indicated. Some interesting remarks of the results presented here are also discussed.
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We study spherical completeness of ball spaces and its stability under expansions. We introduce the notion of an ultradiameter, mimicking diameters in ultrametric spaces. We prove some positive results on preservation of spherical completeness involving ultradiameters with values in narrow partially ordered sets. Finally, we show that in general, chain intersection closures of ultrametric spaces with partially ordered value sets do not preserve spherical completeness.
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We show that any neighborhood of a nondegenerate reversible bifocal homoclinic orbit contains chaotic suspended invariant sets on $N$symbols for all $N\geq 2$. This will be achieved by showing switching associated with networks of secondary homoclinic orbits. We also prove the existence of superhomoclinic orbits (trajectories homoclinic to a network of homoclinic orbits), whose presence leads to a particularly rich structure.
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We provide a detailed analysis of the boundary layers for mixed hyperbolicparabolic systems in one space dimension and small amplitude regimes. As an application of our results, we describe the solution of the socalled boundary Riemann problem recovered as the zero viscosity limit of the physical viscous approximation. In particular, we tackle the so called doubly characteristic case, which is considerably more demanding from the technical viewpoint and occurs when the boundary is characteristic for both the mixed hyperbolicparabolic system and for the hyperbolic system obtained by neglecting the second order terms. Our analysis applies in particular to the compressible NavierStokes and MHD equations in Eulerian coordinates, with both positive and null conductivity. In these cases, the doubly characteristic case occurs when the velocity is close to 0. The analysis extends to nonconservative systems.
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