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In many statistical linear inverse problems, one needs to recover classes of similar curves from their noisy images under an operator that does not have a bounded inverse. Problems of this kind appear in many areas of application. Routinely, in such problems clustering is carried out at the preprocessing step and then the inverse problem is solved for each of the cluster averages separately. As a result, the errors of the procedures are usually examined for the estimation step only. The objective of this paper is to examine, both theoretically and via simulations, the effect of clustering on the accuracy of the solutions of general illposed linear inverse problems. In particular, we assume that one observes $X_m = A f_m + \sigma n^{1/2} \epsilon_m$, $m=1, \cdots, M$, where functions $f_m$ can be grouped into $K$ classes and one needs to recover a vector function ${\bf f}= (f_1,\cdots, f_M)^T$. We construct an estimators for ${\bf f}$ as a solution of a penalized optimization problem
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We show that coarse property C is preserved by finite coarse direct products. We also show that the coarse analog of Dydak's countable asymptotic dimension is equivalent to the coarse version of straight finite decomposition complexity and is therefore preserved by direct products.
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Let $G$ be a graph with chromatic number $\chi$, maximum degree $\Delta$ and clique number $\omega$. Reed's conjecture states that $\chi \leq \lceil (1\varepsilon)(\Delta + 1) + \varepsilon\omega \rceil$ for all $\varepsilon \leq 1/2$. It was shown by King and Reed that, provided $\Delta$ is large enough, the conjecture holds for $\varepsilon \leq 1/130,000$. In this article, we show that the same statement holds for $\varepsilon \leq 1/26$, thus making a significant step towards Reed's conjecture. We derive this result from a general technique to bound the chromatic number of a graph where no vertex has many edges in its neighbourhood. Our improvements to this method also lead to improved bounds on the strong chromatic index of general graphs. We prove that $\chi'_s(G)\leq 1.835 \Delta(G)^2$ provided $\Delta(G)$ is large enough.
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First nonzero Neumann eigenvalues of a rectangle and a parallelogram with the same base and area are compared in case when the height of the parallelogram is greater than the base. This result is applied to compare first nonzero Neumann eigenvalue normalized by the square of the perimeter on the parallelograms with a geometrical restriction and the square. The result is inspired by WallaceBolyaiGerwien theorem. An interesting threedimensional problem related to this theorem is proposed.
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Necessary and sufficient conditions are derived under which concordance measures arise from correlations of transformed ranks of random variables. Compatibility and attainability of square matrices with entries given by such measures are studied, that is, whether a given square matrix of such measures of association can be realized for some random vector and how such a random vector can be constructed. Special cases of this framework include (matrices of pairwise) Spearman's rho, Blomqvist's beta and van der Waerden's coefficient. For these specific measures, characterizations of sets of compatible matrices are provided. Compatibility and attainability of block matrices and hierarchical matrices are also studied. In particular, a subclass of attainable block Spearman's rho matrices is proposed to compensate for the drawback that Spearman's rho matrices are in general not attainable for dimensions larger than four. Another result concerns a novel analytical form of the Cholesky factor o
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We present a novel numerical method for the computation of bound states of semiinfinite matrix Hamiltonians which model electronic states localized at edges of one and twodimensional materials (edge states) in the tightbinding limit. The na\"{i}ve approach fails: arbitrarily large finite truncations of the Hamiltonian have spectrum which does not correspond to spectrum of the semiinfinite problem (spectral pollution). Our method, which overcomes this difficulty, is to accurately compute the Green's function of the semiinfinite Hamiltonian by imposing an appropriate boundary condition at the semiinfinite end; then, the spectral data is recovered via Riesz projection. We demonstrate our method's effectiveness by a study of edge states at a graphene zigzag edge in the presence of defects, including atomic vacancies. Our method may also be used to study states localized at domain walltype edges in one and twodimensional materials where the edge Hamiltonian is infinite in both dire
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We prove that any conformally flat submanifold with flat normal bundle in a conformally flat Riemannian manifold is locally holonomic, that is, admits a principal coordinate system. As one of the consequences of this fact, it is shown that the Ribaucour transformation can be used to construct an associated large family of immersions with induced conformal metrics holonomic with respect to the same coordinate system.
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Following an idea of Hopkins, we construct a model of the determinant sphere $S\langle det \rangle$ in the category of $K(n)$local spectra. To do this, we build a spectrum which we call the Tate sphere $S(1)$. This is a $p$complete sphere with a natural continuous action of $\mathbb{Z}_p^\times$. The Tate sphere inherits an action of $\mathbb{G}_n$ via the determinant and smashing Morava $E$theory with $S(1)$ has the effect of twisting the action of $\mathbb{G}_n$. A large part of this paper consists of analyzing continuous $\mathbb{G}_n$actions and their homotopy fixed points in the setup of Devinatz and Hopkins.
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The eigenvector empirical spectral distribution (VESD) is a useful tool in studying the limiting behavior of eigenvalues and eigenvectors of covariance matrices. In this paper, we study the convergence rate of the VESD of sample covariance matrices to the MarchenkoPastur (MP) distribution. Consider sample covariance matrices of the form $XX^*$, where $X=(x_{ij})$ is an $M\times N$ random matrix whose entries are independent (but not necessarily identically distributed) random variables with mean zero and variance $N^{1}$. We show that the Kolmogorov distance between the expected VESD and the MP distribution is bounded by $N^{1+\epsilon}$ for any fixed $\epsilon>0$, provided that the entries $\sqrt{N}x_{ij}$ have uniformly bounded 6th moment and that the dimension ratio $N/M$ converges to some constant $d\ne 1$. This result improves the previous one obtained in [33], which gives the convergence rate $O(N^{1/2})$ assuming $i.i.d.$ $X$ entries, bounded 10th moment and $d>1$. Mor
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We prove under mild conditions that the FlemingViot process selects the minimal quasistationary distribution for Markov processes with soft killing on noncompact state spaces. Our results are applied to multidimensional birth and death processes, continuous time GaltonWatson processes and diffusion processes with soft killing.
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The concept of cutting is first introduced. By the concept, a convex expansion for finite distributive lattices is considered. Thus, a more general method for drawing the Hasse diagram is given, and the rank generating function of a finite distributive lattice is obtained. In addition, we have several enumerative properties on finite distributive lattices and verify the generalized Euler formula for polyhedrons.
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When a major outage occurs on a distribution system due to extreme events, microgrids, distributed generators, and other local resources can be used to restore critical loads and enhance resiliency. This paper proposes a decisionmaking method to determine the optimal restoration strategy coordinating multiple sources to serve critical loads after blackouts. The critical load restoration problem is solved by a twostage method with the first stage deciding the postrestoration topology and the second stage determining the set of loads to be restored and the outputs of sources. In the second stage, the problem is formulated as a mixedinteger semidefinite program. The objective is maximizing the number of loads restored, weighted by their priority. The unbalanced threephase power flow constraint and operational constraints are considered. An iterative algorithm is proposed to deal with integer variables and can attain the global optimum of the critical load restoration problem by solvi
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We consider Bernoulli bond percolation on the product graph of a regular tree and a line. Schonmann showed that there are a.s. infinitely many infinite clusters at $p=p_u$ by using a certain function $\alpha(p)$. The function $\alpha(p)$ is defined by a exponential decay rate of probability that two vertices of the same layer are connected. We show the critical probability $p_c$ can be written by using $\alpha(p)$. In other words, we construct another definition of the critical probability.
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CyberPhysical Systems (CPS) are systems composed by a physical component that is controlled or monitored by a cybercomponent, a computerbased algorithm. Advances in CPS technologies and science are enabling capability, adaptability, scalability, resiliency, safety, security, and usability that will far exceed the simple embedded systems of today. CPS technologies are transforming the way people interact with engineered systems. New smart CPS are driving innovation in various sectors such as agriculture, energy, transportation, healthcare, and manufacturing. They are leading the 4th Industrial Revolution (Industry 4.0) that is having benefits thanks to the high flexibility of production. The Industry 4.0 production paradigm is characterized by high intercommunicating properties of its production elements in all the manufacturing processes. This is the reason it is a core concept how the systems should be structurally optimized to have the adequate level of redundancy to be satisfact
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The present paper studies density deconvolution in the presence of small Berkson errors, in particular, when the variances of the errors tend to zero as the sample size grows. It is known that when the Berkson errors are present, in some cases, the unknown density estimator can be obtain by simple averaging without using kernels. However, this may not be the case when Berkson errors are asymptotically small. By treating the former case as a kernel estimator with the zero bandwidth, we obtain the optimal expressions for the bandwidth. We show that the density of Berkson errors acts as a regularizer, so that the kernel estimator is unnecessary when the variance of Berkson errors lies above some threshold that depends on the on the shapes of the densities in the model and the number of observations.
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We present a construction of an infinite dimensional associative algebra which we call a \emph{surface algebra} associated in a unique way to a dessin d'enfant. Once we have constructed the surface algebras we construct what we call the associated \emph{dessin order}, which can be constructed in such a way that it is the completion of the path algebra of a quiver with relations. We then prove that the center and (noncommutative) normalization of the dessin orders are invariant under the action of the absolute Galois group $\mathcal{G}(\overline{\mathbb{Q}}/\mathbb{Q})$. We then describe the projective resolutions of the simple modules over the dessin order and show that one can completely recover the dessin with the projective resolutions of the simple modules. Finally, as a corollary we are able to say that classifying dessins in an orbit of $\mathcal{G}(\overline{\mathbb{Q}}/\mathbb{Q})$ is equivalent to classifying dessin orders with a given normalization.
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We show that for the attractor $(K_{1},\dots,K_{q})$ of a graph directed iterated function system, for each $1\leq j\leq q$ and $\varepsilon>0$ there exits a selfsimilar set $K\subseteq K_{j}$ that satisfies the strong separation condition and $\dim_{H}K_{j}\varepsilon<\dim_{H}K$. We show that we can further assume convenient conditions on the orthogonal parts and similarity ratios of the defining similarities of $K$. Using this property as a `black box' we obtain results on a range of topics including on dimensions of projections, intersections, distance sets and sums and products of sets.
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Given a system of functions f = (f1, . . . , fd) analytic on a neighborhood of some compact subset E of the complex plane, we give necessary and sufficient conditions for the convergence with geometric rate of the common denominators of multipoint HermitePade approximants. The exact rate of convergence of these denominators and of the approximants themselves is given in terms of the analytic properties of the system of functions. These results allow to detect the location of the poles of the system of functions which are in some sense closest to E.
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The present paper pioneers the study of the Dirichlet problem with $L^q$ boundary data for second order operators with complex coefficients in domains with lower dimensional boundaries, e.g., in $\Omega := \mathbb R^n \setminus \mathbb R^d$ with $d<n1$. Following the first results of Guy David and the two first authors, the article introduces an appropriate degenerate elliptic operator and show that the Dirichlet problem is solvable for all $q>1$ provided that the coefficients satisfy the small Carleson norm condition. Even in the context of the classical case $d=n1$, (the analogues of) our results are new. The conditions on the coefficients are more relaxed than the previously known ones (most notably, we do not impose any restrictions whatsoever on the first $n1$ rows of the matrix of coefficients) and the results are more general. We establish local rather than global estimates between the square function and the nontangential maximal function and, perhaps even more import
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In this paper we present necessary and sufficient conditions for a graded (trimmed) double Ore extension to be a graded (quasicommutative) skew PBW extension. Using this fact, we prove that a graded skew PBW extension $A = \sigma(R)\langle x_1,x_2 \rangle$ of an ArtinSchelter regular algebra $R$ is ArtinSchelter regular. As a consequence, every graded skew PBW extension $A = \sigma(R)\langle x_1,x_2 \rangle$ of a connected skew CalabiYau algebra $R$ of dimension $d$ is skew CalabiYau of dimension $d+2$.
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Hybrid systems theory has become a powerful approach for designing feedback controllers that achieve dynamically stable bipedal locomotion, both formally and in practice. This paper presents an analytical framework 1) to address multidomain hybrid models of quadruped robots with high degrees of freedom, and 2) to systematically design nonlinear controllers that asymptotically stabilize periodic orbits of these sophisticated models. A family of parameterized virtual constraint controllers is proposed for continuoustime domains of quadruped locomotion to regulate holonomic and nonholonomic outputs. The properties of the Poincare return map for the fullorder and closedloop hybrid system are studied to investigate the asymptotic stabilization problem of dynamic gaits. An iterative optimization algorithm involving linear and bilinear matrix inequalities is then employed to choose stabilizing virtual constraint parameters. The paper numerically evaluates the analytical results on a simul
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We prove a Fujitatype theorem for varieties with numerically trivial canonical bundle. We deduce our result via a combination of algebraic and analytic methods, including the KobayashiHitchin correspondence and positivity of direct image bundles. As an application, we combine our results with recent work of U. Riess on generalized Kummer varieties to obtain effective global generation statements for Hilbert schemes of points on abelian surfaces.
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Is it possible to break the hostvector chain of transmission when there is an influx of infectious hosts into a na\"{i}ve population and competent vector? To address this question, a class of vectorborne disease models with an arbitrary number of infectious stages that account for immigration of infective individuals is formulated. The proposed model accounts for forward and backward progression, capturing the mitigation and aggravation to and from any stages of the infection, respectively. The model has a rich dynamic, which depends on the patterns of infected immigrant influx into the host population and connectivity of the transfer between infectious classes. We provide conditions under which the answer of the initial question is positive.
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Data collection in Wireless Sensor Networks (WSN) draws significant attention, due to emerging interest in technologies raging from Internet of Things (IoT) networks to simple "Presence" applications, which identify the status of the devices (active or inactive). Numerous Medium Access Control (MAC) protocols for WSN, which can address the challenge of data collection in dense networks, were suggested over the years. Most of these protocols utilize the traditional layering approach, in which the MAC layer is unaware of the encapsulated packet payload, and therefore there is no connection between the data collected, the physical layer and the signaling mechanisms. Nonetheless, in many of the applications that intend to utilize such protocols, nodes may need to exchange very little information, and do so only sporadically, that is, while the number of devices in the network can be very large, only a subset wishes to transmit at any given time. Thus, a tailored protocol, which matches the
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Coordinate descent with random coordinate selection is the current state of the art for many large scale optimization problems. However, greedy selection of the steepest coordinate on smooth problems can yield convergence rates independent of the dimension $n$, and requiring upto $n$ times fewer iterations. In this paper, we consider greedy updates that are based on subgradients for a class of nonsmooth composite problems, which includes $L1$regularized problems, SVMs and related applications. For these problems we provide (i) the first linear rates of convergence independent of $n$, and show that our greedy update rule provides speedups similar to those obtained in the smooth case. This was previously conjectured to be true for a stronger greedy coordinate selection strategy. Furthermore, we show that (ii) our new selection rule can be mapped to instances of maximum inner product search, allowing to leverage standard nearest neighbor algorithms to speed up the implementation. We dem
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The JordanWigner transformation plays an important role in spin models. However, the nonlocality of the transformation implies that a periodic chain of $N$ spins is not mapped to a periodic or an antiperiodic chain of lattice fermions. Since only the $N1$ bond is different, the effect is negligible for large systems, while it is significant for small systems. In this paper, it is interesting to find that a class of periodic spin chains can be exactly mapped to a periodic chain and an antiperiodic chain of lattice fermions without redundancy when the JordanWigner transformation is implemented. For these systems, possible high degeneracy is found to appear in not only the ground state but also the excitation states. Further, we take the onedimensional compass model and a new XYXY model ($\sigma_x\sigma_y\sigma_x\sigma_y$) as examples to demonstrate our proposition. Except for the wellknown onedimensional compass model, we will see that in the XYXY model, the degeneracy also
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We study the extremes for a class of a symmetric stable random fields with long range dependence. We prove functional extremal theorems both in the space of sup measures and in the space of cadlag functions of several variables. The limits in both types of theorems are of a new kind, and only in a certain range of parameters these limits have the Fr\'{e}chet distribution.
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In the present paper, we were mainly concerned with obtaining estimates for the general TaylorMaclaurin coefficients for functions in a certain general subclass of analytic biunivalent functions. For this purpose, we used the Faber polynomial expansions. Several connections to some of the earlier known results are also pointed out.
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In this paper we generalize the recently introduced concept of fair measure (M. Misiurewicz and A. Rodrigues, Counting preimages. Ergod. Th. & Dynam. Sys. 38 (2018), no. 5, 1837  1856). We study transitive countable state Markov shift maps and extend our results to a particular class of interval maps, Markov and mixing interval maps. Finally, we move beyond the interval and look for fair measures for graph maps.
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We show that the set of Fano varieties (with arbitrary singularities) whose anticanonical divisors have large Seshadri constants satisfies certain weak and birational boundedness. We also classify singular Fano varieties of dimension $n$ whose anticanonical divisors have Seshadri constants at least $n$, generalizing an earlier result of Liu and the author.
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We present a unified treatment of the Fourier spectra of spherically symmetric nonlocal diffusion operators. We develop numerical and analytical results for the class of kernels with weak algebraic singularity as the distance between source and target tends to $0$. Rapid algorithms are derived for their Fourier spectra with the computation of each eigenvalue independent of all others. The algorithms are trivially parallelizable, capable of leveraging more powerful compute environments, and the accuracy of the eigenvalues is individually controllable. The algorithms include a Maclaurin series and a full divergent asymptotic series valid for any $d$ spatial dimensions. Using Drummond's sequence transformation, we prove linear complexity recurrence relations for degreegraded sequences of numerators and denominators in the rational approximations to the divergent asymptotic series. These relations are important to ensure that the algorithms are efficient, and also increase the numerical s
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We study the asymptotic behaviour of the expected cost of the random matching problem on a $2$dimensional compact manifold, improving in several aspects the results of L. Ambrosio, F. Stra and D. Trevisan (A PDE approach to a 2dimensional matching problem). In particular, we simplify the original proof (by treating at the same time upper and lower bounds) and we obtain the coefficient of the leading term of the asymptotic expansion of the expected cost for the random bipartite matching on a general 2dimensional closed manifold. We also sharpen the estimate of the error term given by M. Ledoux (On optimal matching of Gaussian samples II) for the semidiscrete matching. As a technical tool, we develop a refined contractivity estimate for the heat flow on random data that might be of independent interest.
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We demonstrate how selfconcordance of the loss can be exploited to obtain asymptotically optimal rates for Mestimators in finitesample regimes. We consider two classes of losses: (i) canonically selfconcordant losses in the sense of Nesterov and Nemirovski (1994), i.e., with the third derivative bounded with the $3/2$ power of the second; (ii) pseudo selfconcordant losses, for which the power is removed, as introduced by Bach (2010). These classes contain some losses arising in generalized linear models, including logistic regression; in addition, the second class includes some common pseudoHuber losses. Our results consist in establishing the critical sample size sufficient to reach the asymptotically optimal excess risk for both classes of losses. Denoting $d$ the parameter dimension, and $d_{\text{eff}}$ the effective dimension which takes into account possible model misspecification, we find the critical sample size to be $O(d_{\text{eff}} \cdot d)$ for canonically selfconco
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According to the Henley Passport Index, compiled by global citizenship and residence advisory firm Henley & PartnersCitizens, Japan now has the most powerful passport on the planet. From a report: Having gained visafree access to Myanmar earlier this month, Japanese citizens can now enjoy visafree or visaonarrival access to a whopping 190 destinations around the world  knocking Singapore, with 189 destinations, into second place. Germany, which began 2018 in the top spot, is now in third place with 188 destinations, tied with France and South Korea. Uzbekistan lifted visa requirements for French nationals on October 5, having already granted visafree access to Japanese and Singaporean citizens in early February.
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Ever since selling Handspring to Palm in the early 2000s, Jeff Hawkins, creator of the Palm Pilot and founder of Palm, has been working on his true passion: neuroscience and trying to understand how the brain works. Teaming up with several neuroscientists and some former Palm people, his company Numenta, entirely funded by Hawkins himself, is now ready to show its research to the world. Mr. Hawkins says that before the world can build artificial intelligence, it must explain human intelligence so it can create machines that genuinely work like the brain. "You do not have to emulate the entire brain," he said. "But you do have to understand how the brain works and emulate the important parts." [...] Now, after more than a decade of quiet work at Numenta, he thinks he and a handful of researchers working with him are well on their way to cracking the problem. On Monday, at a conference in the Netherlands, he is expected to unveil their latest research, which he says explain
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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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