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In this paper, we extend a class of globally convergent evolution strategies to handle general constrained optimization problems. The proposed framework handles relaxable constraints using a merit function approach combined with a specific restoration procedure. The unrelaxable constraints in our framework, when present, are treated either by using the extreme barrier function or through a projection approach. The introduced extension guaranties to the regarded class of evolution strategies global convergence properties for first order stationary constraints. Preliminary numerical experiments are carried out on a set of known test problems as well as on a multidisciplinary design optimization problem
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In this work we present an algorithm to construct sparse-paving matroids over finite set $S$. From this algorithm we derive some useful bounds on the cardinality of the set of circuits of any Sparse-Paving matroids which allow us to prove in a simple way an asymptotic relation between the class of Sparse-paving matroids and the whole class of matroids. Additionally we introduce a matrix based method which render an explicit partition of the $r$-subsets of $S$, $\binom{S}{r}=\sqcup_{i=1}^{\gamma }\mathcal{U}_{i}$ such that each $\mathcal{U}_{i}$ defines a sparse-paving matroid of rank $r$.
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Recently, Misanantenaina and Wagner characterized the set of induced $N$-free and bowtie-free posets as a certain class of recursively defined subposets which they term "$\mathcal{V}$-posets". Here we offer a new characterization of $\mathcal{V}$-posets by introducing a property we refer to as \emph{autonomy}. A poset $\mathcal{P}$ is said to be \emph{autonomous} if there exists a directed acyclic graph $D$ (with adjacency matrix $U$) whose transitive closure is $\mathcal{P}$, with the property that any total ordering of the vertices of $D$ so that Gaussian elimination of $U^TU$ proceeds without row swaps is a linear extension of $\mathcal{P}$. Autonomous posets arise from the theory of pressing sequences in graphs, a problem with origins in phylogenetics. The pressing sequences of a graph can be partitioned into families corresponding to posets; because of the interest in enumerating pressing sequences, we investigate when this partition has only one block, that is, when the pressing
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In many contemporary optimization problems, such as hyperparameter tuning for deep learning architectures, it is computationally challenging or even infeasible to evaluate an entire function or its derivatives. This necessitates the use of stochastic algorithms that sample problem data, which can jeopardize the guarantees classically obtained through globalization techniques via a trust region or a line search. Using subsampled function values is particularly challenging for the latter strategy, that relies upon multiple evaluations. On top of that all, there has been an increasing interest for nonconvex formulations of data-related problems. For such instances, one aims at developing methods that converge to second-order stationary points, which is particularly delicate to ensure when one only accesses subsampled approximations of the objective and its derivatives. This paper contributes to this rapidly expanding field by presenting a stochastic algorithm based on negative curvature a
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The set of all perfect matchings of a plane (weakly) elementary bipartite graph equipped with a partial order is a poset, moreover the poset is a finite distributive lattice and its Hasse diagram is isomorphic to $Z$-transformation directed graph of the graph. A finite distributive lattice is matchable if its Hasse diagram is isomorphic to a $Z$-transformation directed graph of a plane weakly elementary bipartite graph, otherwise non-matchable. We introduce the meet-irreducible cell with respect to a perfect matching of a plane (weakly) elementary bipartite graph and give its equivalent characterizations. Using these, we extend a result on non-matchable distributive lattices, and obtain a class of new non-matchable distributive lattices.
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The spectral properties of two special classes of Jacobi operators are studied. For the first class represented by the $2M$-dimensional real Jacobi matrices whose entries are symmetric with respect to the secondary diagonal, a new polynomial identity relating the eigenvalues of such matrices with their matrix { entries} is obtained. In the limit $M\to\infty$ this identity induces some requirements, which should satisfy the scattering data of the resulting infinite-dimensional Jacobi operator in the half-line, which super- and sub-diagonal matrix elements are equal to -1. We obtain such requirements in the simplest case of the discrete Schr\"odinger operator acting in ${l}^2( \mathbb{N})$, which does not have bound and semi-bound states, and which potential has a compact support.
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The characterization of local regularity is a fundamental issue in signal and image processing, since it contains relevant information about the underlying systems. The 2-microlocal frontier, a monotone concave downward curve in $\mathbb {R}^2$, provides a complete and profound classification of pointwise singularity. In \cite{Meyer1998}, \cite{GuiJaffardLevy1998} and \cite{LevySeuret2004} the authors show the following: given a monotone concave downward curve in the plane it is possible to exhibit one function (or distribution) such that its 2-microlocal frontier al $x_0$ is the given curve. In this work we are able to unify the previous results, by obtaining a large class of functions (or distributions), that includes the three examples mentioned above, for which the 2-microlocal frontier is the given curve. The three examples above are in this class. Further, if the curve is a line, we characterize all the functions whose 2-microlocal frontier at $x_0$ is the given line.
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A tetravalent $2$-arc-transitive graph of order $728$ is either the known $7$-arc-transitive incidence graph of the classical generalized hexagon $GH(3,3)$ or a normal cover of a $2$-transitive graph of order $182$ denoted $A[182,1]$ or $A[182,2]$ in the $2009$ list of Poto\v{c}nik.
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When a Stanford researcher removed all the duplicate and fake comments filed with the Federal Communications Commission last year, he found that 99.7 percent of public comments -- about 800,000 in all -- were pro-net neutrality. From a report: "With the fog of fraud and spam lifted from the comment corpus, lawmakers and their staff, journalists, interested citizens and policymakers can use these reports to better understand what Americans actually said about the repeal of net neutrality protections and why 800,000 Americans went further than just signing a petition for a redress of grievances by actually putting their concerns in their own words," Ryan Singel, a media and strategy fellow at Stanford University, wrote in a blog post Monday. Singel released a report [PDF] Monday that analyzed the unique comments -- as in, they weren't a copypasta of one or dozens of other letters -- filed last year ahead of the FCC's decision to repeal federal net neutrality protections. That's from the
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A new study from Nature Plants has identified the one climate-related issue that can unite people from myriad political backgrounds -- beer. From a report: Led by Wei Xie, an agricultural scientist at Peking University, the paper finds that regions that grow barley, the primary crop used to brew beer, are projected to experience severe droughts and heat waves due to anthropogenic climate change. According to five climate models that used different projected temperature increases for the coming century, extreme weather events could reduce barley yields by 3 to 17 percent. Barley harvests are mostly sold as livestock fodder, so beer availability could be further hindered by the likely prioritization of grain yields to feed cattle and other farm animals, rather than for brewing beer. The net result will be a decline in affordable access to beer, which is the most commonly imbibed alcoholic beverage in the world. Within a few decades, this luxury may be out of reach for hundreds of million
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In this paper, we use the semi-group method and an adaptation of the $L^2-$method of H\"ormander to establish some $\Phi-$entropy inequalities and asymmetric covariance estimates for the strictly convex measures in $\mathbb R^n$. These inequalities extends the ones for the strictly log-concave measures to more general setting of convex measures. The $\Phi-$entropy inequalities are turned out to be sharp in the special case of Cauchy measures. Finally, we show that the similar inequalities for log-concave measures can be obtained from our results in the limiting case.
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We consider a distributed optimization problem over a network of agents aiming to minimize a global objective function that is the sum of local convex and composite cost functions. To this end, we propose a distributed Chebyshev-accelerated primal-dual algorithm to achieve faster ergodic convergence rates. In standard distributed primal-dual algorithms, the speed of convergence towards a global optimum (i.e., a saddle point in the corresponding Lagrangian function) is directly influenced by the eigenvalues of the Laplacian matrix representing the communication graph. In this paper, we use Chebyshev matrix polynomials to generate gossip matrices whose spectral properties result in faster convergence speeds, while allowing for a fully distributed implementation. As a result, the proposed algorithm requires fewer gradient updates at the cost of additional rounds of communications between agents. We illustrate the performance of the proposed algorithm in a distributed signal recovery probl
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While global convergence of the Douglas-Rachford iteration is often observed in applications, proving it is still limited to convex and a handful of other special cases. Lyapunov functions for difference inclusions provide not only global or local convergence certificates, but also imply robust stability, which means that the convergence is still guaranteed in the presence of persistent disturbances. In this work, a global Lyapunov function is constructed by combining known local Lyapunov functions for simpler, local sub-problems via an explicit formula that depends on the problem parameters. Specifically, we consider the scenario where one set consists of the union of two lines and the other set is a line, so that the two sets intersect in two distinct points. Locally, near each intersection point, the problem reduces to the intersection of just two lines, but globally the geometry is non-convex and the Douglas-Rachford operator multi-valued. Our approach is intended to be prototypica
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Lyapunov functions are used to prove stability of equilibria, or to indicate a gradient-like structure of a dynamical system. Zelenyak (1968) and Matano (1988) constructed a Lyapunov function for quasilinear parabolic equations. We modify Matano's method to construct a Lyapunov function for fully nonlinear parabolic equations under Dirichlet and mixed nonlinear boundary conditions of Robin type.
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One may define a trilinear convolution form on the sphere involving two functions on the sphere and a monotonic function on the interval $[-1,1]$. A symmetrization inequality of Baernstein and Taylor states that this form is maximized when the two functions on the sphere are replaced with their nondecreasing symmetric rearrangements. In the case of indicator functions, we show that under natural hypotheses, the symmetric rearrangements are the only maximizers up to symmetry by establishing a sharpened inequality.
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Let $\Sigma$ be a closed, smooth hypersurface in $\Bbb R^{n + 1}$ which is axially symmetric and is contained inside the unit sphere $\Bbb S^{n}$. For a continuous function $f$, which is defined on $\Bbb S^{n}$, the main goal of this paper is to characterize the support of $f$ in case where its integrals vanish on subspheres obtained by intersecting $\Bbb S^{n}$ with the tangent hyperplanes of a certain subdomain $\mathcal{U}\subset\Sigma$ of $\Sigma$. We show that the support of $f$ can be characterized in case where its integrals also vanish on subspheres obtained by intersecting $\Bbb S^{n}$ with hyperplanes obtained by infinitesimal perturbations of the tangent hyperplanes of $\mathcal{U}$.
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Li-Nadler proposed a conjecture about traces of Hecke categories, which implies the semistable part of the Betti Geometric Langlands Conjecture of Ben-Zvi-Nadler in genus 1. We prove a Weyl group analogue of this conjecture. Our theorem holds in the natural generality of reflection groups in Euclidean or hyperbolic space.
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Variable Stepsize Variable Order (VSVO) methods are the methods of choice to efficiently solve a wide range of ODEs with minimal work and assured accuracy. However, VSVO methods have limited impact in timestepping methods in complex applications due to their computational complexity and the difficulty to implement them in legacy code. We introduce a family of implicit, embedded, VSVO methods that require only one BDF solve at each time step followed by adding linear combinations of the solution at previous time levels. In particular, we construct implicit and linearly implicit VSVO methods of orders two, three and four with the same computational complexity as variable stepsize BDF3. The choice of changing the order of the method is simple and does not require additional solves of linear or nonlinear systems.
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We show that elements of control theory, together with an application of Harris' ergodic theorem, provide an alternate method for showing exponential convergence to a unique stationary measure for certain classes of networks of quasi-harmonic classical oscillators coupled to heat baths. With the system of oscillators expressed in the form $\mathrm{d} X_t = A X_t \,\mathrm{d} t + F(X_t) \,\mathrm{d} t + B \,\mathrm{d} W_t$ in $\mathbf{R}^d$, where $A$ encodes the harmonic part of the force and $-F$ corresponds to the gradient of the anharmonic part of the potential, the hypotheses under which we obtain exponential mixing are the following: $A$ is dissipative, the pair $(A,B)$ satisfies the Kalman condition, $F$ grows sufficiently slowly at infinity (depending on the dimension $d$), and the vector fields in the equation of motion satisfy the weak H\"ormander condition in at least one point of the phase space.
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In this note, we consider the backward stochastic differential equation (BSDE) with generator $f(y)|z|^2,$ where the function $f$ is defined on an open set and locally integral. The existence and uniqueness of solution of such BSDE is explored for bounded or unbounded terminal variables. A comparison theorem and a converse theorem theorem of such BSDEs are obtained.
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We deduce from Sageev's results that whenever a group acts locally elliptically on a finite dimensional CAT(0) cube complex, then it must fix a point. As an application, we give an example of a group G such that G does not have property (T), but G and all its finitely generated subgroups can not act without a fixed point on a finite dimensional CAT(0) cube complex, answering a question by Barnhill and Chatterji.
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The spectrum of three-diagonal self-adjoint $p$-periodic Jacobi matrix with positive off-diagonal elements $a_n$ an real diagonal elements $b_n$ consist of intervals separated by $p-1$ gaps $\gamma_i$, where some of the gaps can be degenerated. The following estimate is true $$ \sum_{i=1}^{p-1}|\gamma_i|\geq\max(\max(4(a_1...a_p)^{\frac1p},2\max a_n)-4\min a_n,\max b_n-\min b_n). $$ We show that for any $p\in\mathbb{N}$ there are Jacobi matrices of minimal period $p$ for which the spectral estimate is sharp. The estimate is sharp for both: strongly and weakly oscillated $a_n$, $b_n$. Moreover, it improves some recent spectral estimates.
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We prove the crepant resolution conjecture for Donaldson-Thomas invariants of hard Lefschetz CY3 orbifolds, formulated by Bryan-Cadman-Young, after reinterpreting it as an equality of rational functions. In order to do so, we show that the generating series of stable pair invariants on any CY3 orbifold is the expansion of a rational function. As a corollary, we deduce a symmetry of this function induced by the derived dualising functor. Our methods also yield a proof of the orbifold DT/PT correspondence for multi-regular curve classes on hard Lefschetz CY3 orbifolds.
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We introduce a vector bundle version of the complex Monge-Ampere equation motivated by a desire to study stability conditions involving higher Chern forms. We then restrict ourselves to complex surfaces, provide a moment map interpretation of it, and define a positivity condition (MA positivity) which is necessary for the infinite-dimensional symplectic form to be Kahler. On rank-2 bundles on compact complex surfaces, we prove two consequences of the existence of a "positively curved" solution to this equation - Stability (involving the second Chern character) and a Kobayashi-Lubke-Bogomolov-Miyaoka-Yau type inequality. Finally, we prove a Kobayashi-Hitchin correspondence for a dimensional reduction of the aforementioned equation.
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Covert communications hide the transmission of a message from a watchful adversary while ensuring a certain decoding performance at the receiver. In this work, a wireless communication system under fading channels is considered where covertness is achieved by using a full-duplex (FD) receiver. More precisely, the receiver of covert information generates artificial noise with a varying power causing uncertainty at the adversary, Willie, regarding the statistics of the received signals. Given that Willie's optimal detector is a threshold test on the received power, we derive a closed-form expression for the optimal detection performance of Willie averaged over the fading channel realizations. Furthermore, we provide guidelines for the optimal choice of artificial noise power range, and the optimal transmission probability of covert information to maximize the detection errors at Willie. Our analysis shows that the transmission of artificial noise, although causes self-interference, provi
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The goal of these lecture notes is to present in a unified way various models for the dynamics of aligning self-propelled rigid bodies at different scales and the links between them. The models and methods are inspired from [12,13], but, in addition, we introduce a new model and apply on it the same methods. While the new model has its own interest, our aim is also to emphasize the methods by demonstrating their adaptability and by presenting them in a unified and simplified way. Furthermore, from the various microscopic models we derive the same macroscopic model, which is a good indicator of its universality.
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We propose a new approach to the numerical solution of radiative transfer equations with certified a posteriori error bounds. A key role is played by stable Petrov--Galerkin type variational formulations of parametric transport equations and corresponding radiative transfer equations. This allows us to formulate an iteration in a suitable, infinite dimensional function space that is guaranteed to converge with a fixed error reduction per step. The numerical scheme is then based on approximately realizing this iteration within dynamically updated accuracy tolerances that still ensure convergence to the exact solution. To advance this iteration two operations need to be performed within suitably tightened accuracy tolerances. First, the global scattering operator needs to be approximately applied to the current iterate within a tolerance comparable to the current accuracy level. Second, parameter dependent linear transport equations need to be solved, again at the required accuracy of th
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Fast, high-order accurate algorithms for electromagnetic scattering from axisymmetric objects are of great importance when modeling physical phenomena in optics, materials science (e.g. meta-materials), and many other fields of applied science. In this paper, we develop an FFT-accelerated separation of variables solver that can be used to efficiently invert integral equation formulations of Maxwell's equations for scattering from axisymmetric penetrable (dielectric) bodies. Using a standard variant of M\"uller's integral representation of the fields, our numerical solver rapidly and directly inverts the resulting second-kind integral equation. In particular, the algorithm of this work (1) rapidly evaluates the modal Green's functions, and their derivatives, via kernel splitting and the use of novel recursion formulas, (2) discretizes the underlying integral equation using generalized Gaussian quadratures on adaptive meshes, and (3) is applicable to geometries containing edges. Several
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This report presents a low complexity, stable and time accurate method for the Navier-Stokes equations. The improved method requires a minimally intrusive modification to an existing program based on the fully implicit / backward Euler time discretization, does not add to the computational complexity, and is conceptually simple. The backward Euler approximation is simply post-processed with a two-step, linear time filter. The time filter additionally removes the overdamping of Backward Euler while remaining unconditionally energy stable, proven herein. Numerical tests confirm the predicted convergence rates and the improved predictions of flow quantities such as drag and lift.
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We consider a general k dimensional discounted innite server queues process (alternatively, an Incurred But Not Reported (IBNR) claim process) where the multi-variate inputs (claims) are given by a k dimensional nite state Markov chain and the arrivals follow a renewal process. After deriving a multidimensional integral equation for the moment generating function jointly to the state of the input at time t given the initial state of the input at time 0, asymptotic results for the rst and second (matrix) moments of the process are provided. In particular, when the interarrival or service times are exponentially distributed, transient expressions for the rst two moments are obtained. Also, the moment generating function for the process with deterministic interarrival times is considered to provide more explicit expressions. Finally, we demonstrate the potential of the present model by showing how it allows us to study a semi-Markov modulated innite queues process where the customers (cla
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It is well known that the labeling problems of graphs arise in many (but not limited to) networking and telecommunication contexts. In this paper we introduce the anti-$k$-labeling problem of graphs which we seek to minimize the similarity (or distance) of neighboring nodes. For example, in the fundamental frequency assignment problem in wireless networks where each node is assigned a frequency, it is usually desirable to limit or minimize the frequency gap between neighboring nodes so as to limit interference. Let $k\geq1$ be an integer and $\psi$ is a labeling function (anti-$k$-labeling) from $V(G)$ to $\{1,2,\cdots,k\}$ for a graph $G$. A {\em no-hole anti-$k$-labeling} is an anti-$k$-labeling using all labels between 1 and $k$. We define $w_{\psi}(e)=|\psi(u)-\psi(v)|$ for an edge $e=uv$ and $w_{\psi}(G)=\min\{w_{\psi}(e):e\in E(G)\}$ for an anti-$k$-labeling $\psi$ of the graph $G$. {\em The anti-$k$-labeling number} of a graph $G$, $mc_k(G)$ is $\max\{w_{\psi}(G): \psi\}$. In th
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We study a general k dimensional infinite server queues process. When the service times are fat tailed, we prove that the properly rescaled process converges to some limiting process: in particular we identify three regimes including slow arrivals, fast arrivals, and equilibrium, which lead to different limits in distribution. AMS 2000 subject classifications: Primary 60G50, 60K30, 62P05, 60K25.
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We consider a certain two-parameter family of automorphisms of the affine plane over a complete, locally compact non-Archimedean field. Each of these automorphisms admits a chaotic attractor on which it is topolgocally conjugate to the full two-sided shift map, and the attractor supports a unit Borel measure which describes the distribution of the forward orbit of Haar-almost all points in the basin of attraction. We also compute the Hausdorff dimension of the attractor, which is non-integral.
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In this paper, by virtue of Malliavin calculus, we establish a relationship between backward doubly stochastic differential equations with random coefficients and quasilinear stochastic PDEs, and thus extend the well-known nonlinear stochastic Feynman-Kac formula of Pardoux and Peng [14] to non-Markovian case.
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Let $K \subset \mathbb R^3$ be a regular convex cone with positively curved boundary of class $C^k$, $k \geq 5$. The image of the boundary $\partial K$ in the real projective plane is a simple closed convex curve $\gamma$ of class $C^k$ without inflection points. Due to the presence of sextactic points $\gamma$ does not possess a global parametrization by projective arc length. In general it will not possess a global periodic Forsyth-Laguerre parametrization either, i.e., it is not the projective image of a periodic vector-valued solution $y(t)$ of the ordinary differential equation (ODE) $y''' + \beta \cdot y = 0$, where $\beta$ is a periodic function. We show that $\gamma$ possesses a periodic Forsyth-Laguerre type global parametrization of class $C^{k-1}$ as the projective image of a solution $y(t)$ of the ODE $y''' + 2\alpha \cdot y' + \beta \cdot y = 0$, where $\alpha \leq \frac12$ is a constant depending on the cone $K$ and $\beta$ is a $2\pi$-periodic function of class $C^{k-5}$
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The main object of this paper is to construct a new genuine Bernstein-Durrmeyer type operators which have better features than the classical one. Some direct estimates for the modified genuine Bernstein-Durrmeyer operator by means of the first and second modulus of continuity are given. An asymptotic formula for the new operator is proved. Finally, some numerical examples with illustrative graphics have been added to validate the theoretical results and also compare the rate of convergence.
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The main object of this paper is to construct new Durrmeyer type operators which have better features than the classical one. Some results concerning the rate of convergence and asymptotic formulas of the new operator are given. Finally, the theoretical results are analyzed by numerical examples.
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Artificial neural networks that learn to perform Principal Component Analysis (PCA) and related tasks using strictly local learning rules have been previously derived based on the principle of similarity matching: similar pairs of inputs should map to similar pairs of outputs. However, the operation of these networks (and of similar networks) requires a fixed-point iteration to determine the output corresponding to a given input, which means that dynamics must operate on a faster time scale than the variation of the input. Further, during these fast dynamics such networks typically "disable" learning, updating synaptic weights only once the fixed-point iteration has been resolved. Here, we derive a network for PCA-based dimensionality reduction that avoids this fast fixed-point iteration. The key novelty of our approach is a modification of the similarity matching objective to encourage near-diagonality of a synaptic weight matrix. We then approximately invert this matrix using a Taylo
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We develop a correspondence between the study of Borel equivalence relations induced by closed subgroups of $S_\infty$, and the study of symmetric models and weak choice principles, and apply it to prove a conjecture of Hjorth-Kechris-Louveau (1998). For example, we show that the equivalence relation $\cong^\ast_{\omega+1,0}$ is strictly below $\cong^\ast_{\omega+1,<\omega}$ in Borel reducibility. By results of Hjorth-Kechris-Louveau, $\cong^\ast_{\omega+1,<\omega}$ provides invariants for $\Sigma^0_{\omega+1}$ equivalence relations induced by actions of $S_\infty$, while $\cong^\ast_{\omega+1,0}$ provides invariants for $\Sigma^0_{\omega+1}$ equivalence relations induced by actions of abelian closed subgroups of $S_\infty$. We further apply these techniques to study the Friedman-Stanley jumps. For example, we find an equivalence relation $F$, Borel bireducible with $=^{++}$, so that $F\restriction C$ is not Borel reducible to $=^{+}$ for any non-meager set $C$. This answers a qu
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In the study of the constant in Ahlfors' second fundamental theorem involving a set E_{q} of q points, branch values of covering surfaces outside E_{q} bring a lot of troubles. To avoid this situation, for a given surface S, it is useful to construct a new surface So such that L(So) <=L(S), and H(S)>=H(S), and all branch values of So are contained in E_{q}. The goal of this paper is to prove the existence of such So, which generalizes Lemma 9.1 and Theorem 10.1 in Zhang G.Y.: The precise bound for the area-length ratio in Ahifors' theory of covering surfaces. Invent math 191:197-253(2013)
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Let $G/H$ be a reductive symmetric space over a $p$-adic field $F$, the algebraic groups $G$ and $H$ being assumed semisimple of relative rank $1$. One of the branching problems for the Steinberg representation $\St_G$ of $G$ is the determination of the dimension of the intertwining space ${\rm Hom}_H (\St_G ,\pi )$, for any irreducible representation $\pi$ of $H$. In this work we do not compute this dimension, but show how it is related to the dimensions of some other intertwining spaces ${\rm Hom}_{K_i} ({\tilde \pi} ,1)$, for a certain finite family $K_i$, $i=1,...,r$, of anisotropic subgroups of $H$ (here ${\tilde \pi}$ denote the contragredient representation, and $1$ the trivial character). In other words we show that there is a sort of `reciprocity law' relating two different branching problems.
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A loss function measures the discrepancy between the true values and their estimated fits, for a given instance of data. In classification problems, a loss function is said to be proper if the minimizer of the expected loss is the true underlying probability. In this work we show that for binary classification, the divergence associated with smooth, proper and convex loss functions is bounded from above by the Kullback-Leibler (KL) divergence, up to a normalization constant. It implies that by minimizing the log-loss (associated with the KL divergence), we minimize an upper bound to any choice of loss from this set. This property suggests that the log-loss is universal in the sense that it provides performance guarantees to a broad class of accuracy measures. Importantly, our notion of universality is not restricted to a specific problem. This allows us to apply our results to many applications, including predictive modeling, data clustering and sample complexity analysis. Further, we
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We study the performance of caching schemes based on LT under peeling (iterative) decoding algorithm. We assume that users ask for downloading content to multiple cache-aided transmitters. Transmitters are connected through a backhaul link to a master node while no direct link exists between users and the master node. Each content is fragmented and coded with LT code. Cache placement at each transmitter is optimized such that transmissions over the backhaul link is minimized. We derive a closed form expression for the calculation of the backhaul transmission rate. We compare the performance of a caching scheme based on LT with respect to a caching scheme based on maximum distance separable codes. Finally, we show that caching with \acl{LT} codes behave as good as caching with maximum distance separable codes.
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This paper considers the cooperative device-to-device (D2D) systems with non-orthogonal multiple access (NOMA). We assume that the base station (BS) can communicate simultaneously with all users to satisfy the full information transmission. In order to characterize the impact of the weak channel and different decoding schemes, two kinds of decoding strategies are proposed: \emph{single signal decoding scheme} and \emph{MRC decoding scheme}, respectively. For the \emph{single signal decoding scheme}, the users immediately decode the received signals after receptions from the BS. Meanwhile, for the \emph{MRC decoding scheme}, instead of decoding, the users will keep the receptions in reserve until the corresponding phase comes and the users jointly decode the received signals by employing maximum ratio combining (MRC). Considering Rayleigh fading channels, the ergodic sum-rate (SR), outage probability and outage capacity of the proposed D2D-NOMA system are analyzed. Moreover, approximate
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Constructing and manipulating homotopy types from categorical input data has been an important theme in algebraic topology for decades. Every category gives rise to a `classifying space', the geometric realization of the nerve. Up to weak homotopy equivalence, every space is the classifying space of a small category. More is true: the entire homotopy theory of topological spaces and continuous maps can be modeled by categories and functors. We establish a vast generalization of the equivalence of the homotopy theories of categories and spaces: small categories represent refined homotopy types of orbispaces whose underlying coarse moduli space is the traditional homotopy type hitherto considered. A global equivalence is a functor between small categories that induces weak equivalences of nerves of the categories of $G$-objects, for all finite groups $G$. We show that the global equivalences are part of a model structure on the category of small categories, which is moreover Quillen equi
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Huawei's new Mate 20 Pro has a massive screen, three cameras on the back and a fingerprint scanner embedded in the display. From a report: The new top-end phone from the Chinese firm aims to secure its place at the top of the market alongside Samsung, having recently beaten Apple to become the second-largest smartphone manufacturer in August. The Mate 20 Pro follows Huawei's tried and trusted format for its Mate series: a huge 6.39in QHD+ OLED screen, big 4,200mAh battery and powerful new Huawei Kirin 980 processor -- Huawei's first to be produced at 7 nanometres, matching Apple's latest A12 chip in the 2018 iPhones. New for this year is an infrared 3D facial recognition system, similar to that used by Apple for its Face ID in the iPhone XS, and one of the first fingerprint scanners embedded in the screen that is widely available in the UK, removing the need for a fingerprint scanner on the back or a chin on the front. The Mate 20 Pro is water resistant to IP68 standards and has a slee
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Recently, so-called treebased phylogenetic networks have gained considerable interest in the literature, where a treebased network is a network that can be constructed from a phylogenetic tree, called the \emph{base tree}, by adding additional edges. The main aim of this manuscript is to provide some sufficient criteria for treebasedness by reducing phylogenetic networks to related graph structures. While it is generally known that deciding whether a network is treebased is NP-complete, one of these criteria, namely \emph{edgebasedness}, can be verified in polynomial time. Next to these edgebased networks, we introduce further classes of treebased networks and analyze their relationships.
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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 pre-processing 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 ill-posed 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 non-zero 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 non-zero Neumann eigenvalue normalized by the square of the perimeter on the parallelograms with a geometrical restriction and the square. The result is inspired by Wallace--Bolyai--Gerwien theorem. An interesting three-dimensional 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 semi-infinite matrix Hamiltonians which model electronic states localized at edges of one and two-dimensional materials (edge states) in the tight-binding limit. The na\"{i}ve approach fails: arbitrarily large finite truncations of the Hamiltonian have spectrum which does not correspond to spectrum of the semi-infinite problem (spectral pollution). Our method, which overcomes this difficulty, is to accurately compute the Green's function of the semi-infinite Hamiltonian by imposing an appropriate boundary condition at the semi-infinite 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 zig-zag edge in the presence of defects, including atomic vacancies. Our method may also be used to study states localized at domain wall-type edges in one and two-dimensional 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 Marchenko-Pastur (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 Fleming-Viot process selects the minimal quasi-stationary distribution for Markov processes with soft killing on non-compact state spaces. Our results are applied to multi-dimensional birth and death processes, continuous time Galton-Watson 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 decision-making 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 two-stage method with the first stage deciding the post-restoration 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 mixed-integer semidefinite program. The objective is maximizing the number of loads restored, weighted by their priority. The unbalanced three-phase 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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Cyber-Physical Systems (CPS) are systems composed by a physical component that is controlled or monitored by a cyber-component, a computer-based 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 4-th 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 self-similar 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 Hermite-Pade 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 $d<n-1$. 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 $d=n-1$, (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 $n-1$ 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 non-tangential 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 (quasi-commutative) 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 Artin-Schelter regular algebra $R$ is Artin-Schelter regular. As a consequence, every graded skew PBW extension $A = \sigma(R)\langle x_1,x_2 \rangle$ of a connected skew Calabi-Yau algebra $R$ of dimension $d$ is skew Calabi-Yau of dimension $d+2$.
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为眼睛近视者指引道路是很费力的，因为你不能对他说：“看见十哩外的教堂吗?朝这个方向走。——维特根斯坦
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