Further improved recursions for a class of compound Poisson distributions

In the present paper we develop more efficient recursive formulae for the evaluation of the t-order cumulative function Γ^th(x) and the t-order tail probability Λ^th(x) of the class of compound Poisson distributions in the case where the derivative of the probability generating function of the claim amounts can be written as a ratio of two polynomials. These efficient recursions can be applied for the exact evaluation of the probability function (given by De Pril [De Pril, N., 1986a. Improved recursions for some compound Poisson distributions. Insurance Math. Econom. 5, 129-132]), distribution function, tail probability, stop-loss premiums and t-order moments of stop-loss transforms of compound Poisson distributions. Also, efficient recursive algorithms are given for the evaluation of higher-order moments and r-order factorial moments about any point for this class of compound Poisson distributions. Finally, several examples of discrete claim size distributions belonging to this class are also given.     

Spectral Algorithms for Tensor Completion

In the tensor completion problem, one seeks to estimate a low‐rank tensor based on a random sample of revealed entries. In terms of the required sample size, earlier work revealed a large gap between estimation with unbounded computational resources (using, for instance, tensor nuclear norm minimization) and polynomial‐time algorithms. Among the latter, the best statistical guarantees have been proved, for third‐order tensors, using the sixth level of the sum‐of‐squares (sos ) semidefinite programming hierarchy. However, the sos approach does not scale well to large problem instances. By contrast, spectral methods—based on unfolding or matricizing the tensor—are attractive for their low complexity, but have been believed to require a much larger sample size. This paper presents two main contributions. First, we propose a new method, based on unfolding, which outperforms naive ones for symmetric k^th‐order tensors of rank r. For this result we make a study of singular space estimation for partially revealed matrices of large aspect ratio, which may be of independent interest. For third‐order tensors, our algorithm matches the sos method in terms of sample size (requiring about rd^3/2 revealed entries), subject to a worse rank condition (r ≪ d^3/4 rather than r ≪ d^3/2). We complement this result with a different spectral algorithm for third‐order tensors in the overcomplete (r ≥ d) regime. Under a random model, this second approach succeeds in estimating tensors of rank d ≤ r ≪ d^3/2 from about rd^3/2 revealed entries. © 2018 Wiley Periodicals, Inc.     

Eine kartesisch abgeschlossene Kategorie glatter Abbildungen zwischen beliebigen lokalkonvexen Vektoräumen

This paper is a continuation of [6], in which I identified thec ^∞-complete bornological locally convex spaces (in short: 1cs) as the right ones for infinite dimensional analysis. Here I discuss smooth mappings between arbitrary 1cs, where a mapping is called smooth iff its compositions with smooth curves are smooth. The 1^st part is mainly devoted to prove the cartesian closedness of the category of (bornological,c ^∞-complete) 1cs together with the smooth mappings between them. In the 2^nd part I discuss the bornology of function spaces and furthermore demonstrate the smoothness of the differentiation process. Finally, in the 3^rd part, I compare this concept of smoothness with several others, discussed byKeller in [5], and show it to be the weakest that fulfills the chainrule.     

Multiwavelet Frames from Refinable Function Vectors

Starting from any two compactly supported d-refinable function vectors in (L ₂(R)) ^r with multiplicity r and dilation factor d, we show that it is always possible to construct 2rd wavelet functions with compact support such that they generate a pair of dual d-wavelet frames in L ₂(R) and they achieve the best possible orders of vanishing moments. When all the components of the two real-valued d-refinable function vectors are either symmetric or antisymmetric with their symmetry centers differing by half integers, such 2rd wavelet functions, which generate a pair of dual d-wavelet frames, can be real-valued and be either symmetric or antisymmetric with the same symmetry center. Wavelet frames from any d-refinable function vector are also considered. This paper generalizes the work in [5,12,13] on constructing dual wavelet frames from scalar refinable functions to the multiwavelet case. Examples are provided to illustrate the construction in this paper.     

On merit functions for <Emphasis Type="Italic">p</Emphasis>-order cone complementarity problem

Merit function approach is a popular method to deal with complementarity problems, in which the complementarity problem is recast as an unconstrained minimization via merit function or complementarity function. In this paper, for the complementarity problem associated with p-order cone, which is a type of nonsymmetric cone complementarity problem, we show the readers how to construct merit functions for solving p-order cone complementarity problem. In addition, we study the conditions under which the level sets of the corresponding merit functions are bounded, and we also assert that these merit functions provide an error bound for the p-order cone complementarity problem. These results build up a theoretical basis for the merit method for solving p-order cone complementarity problem.     

Maximal number of distinct H-eigenpairs for a two-dimensional real tensor

Based on the generalized characteristic polynomial introduced by J. Canny in Generalized characteristic polynomials [J. Symbolic Comput., 1990, 9(3): 241–250], it is immediate that for any m-order n-dimensional real tensor, the number of distinct H-eigenvalues is less than or equal to n(m?1) ^n?1. However, there is no known bounds on the maximal number of distinct Heigenvectors in general. We prove that for any m ? 2, an m-order 2-dimensional tensor A exists such that A has 2(m ? 1) distinct H-eigenpairs. We give examples of 4-order 2-dimensional tensors with six distinct H-eigenvalues as well as six distinct H-eigenvectors. We demonstrate the structure of eigenpairs for a higher order tensor is far more complicated than that of a matrix. Furthermore, we introduce a new class of weakly symmetric tensors, called p-symmetric tensors, and show under certain conditions, p-symmetry will effectively reduce the maximal number of distinct H-eigenvectors for a given two-dimensional tensor. Lastly, we provide a complete classification of the H-eigenvectors of a given 4-order 2-dimensional nonnegative p-symmetric tensor. Additionally, we give sufficient conditions which prevent a given 4-order 2-dimensional nonnegative irreducible weakly symmetric tensor from possessing six pairwise distinct H-eigenvectors.     

The chain lemma for Kummer elements of degree 3

Let A be a skew field of degree 3 over a field containing the 3^rd roots of unity. We prove a sort of chain equivalence for Kummer elements in A. As a consequence one obtains a common slot lemma for presentations of A as a cyclic algebra.     

Discrete α-Yamabe flow in 3-dimension

We generalize the discrete Yamabe flow to α order. This Yamabe flow deforms the α-order curvature to a constant. Using this new flow, we manage to find discrete α-quasi-Einstein metrics on the triangulations of S³.     

A priori error estimates for upwind finite volume schemes for two-dimensional linear convection diffusion problems

It is still an open problem to prove a priori error estimates for finite volume schemes of higher order MUSCL type, including limiters, on unstructured meshes, which show some improvement compared to first order schemes. In this paper we use these higher order schemes for the discretization of convection dominated elliptic problems in a convex bounded domain Ω in R² and we can prove such kind of an a priori error estimate. In the part of the estimate, which refers to the discretization of the convective term, we gain h^1/2. Although the original problem is linear, the numerical problem becomes nonlinear, due to MUSCL type reconstruction/limiter technique.     

Some upper bounds on the total and list chromatic numbers of multigraphs

In this paper we discuss some estimates for upper bounds on a number of chromatic parameters of a multigraph. In particular, we show that the total chromatic number for an n-order multigraph exceeds the chromatic index by the smallest t such that t! > n.     

On the Local Solubility of Diophantine Systems

Let p be a rational prime number. We refine Brauer's elementary diagonalisation argument to show that any system of r homogeneous polynomials of degree d, with rational coefficients, possesses a non-trivial p-adic solution provided only that the number of variables in this system exceeds (rd ²)^2d-1. This conclusion improves on earlier results of Leep and Schmidt, and of Schmidt. The methods extend to provide analogous conclusions in field extensions of Q, and in purely imaginary extensions of Q. We also discuss lower bounds for the number of variables required to guarantee local solubility.     

On boundary controllability of one-dimensional vibrating systems by W,p-controls for p ∈ [2, ∞]

This paper is concerned with boundary control of one-dimensional vibrating media whose motion is governed by a wave equation with a 2n-order spatial self-adjoint and positive-definite linear differential operator with respect to 2n boundary conditions. Control is applied to one of the boundary conditions and the control function is allowed to vary in the Sobolev space W, ^p for p∈[2, ∞] With the aid of Banach space theory of trigonometric moment problems, necessary and sufficient conditions for null-controllability are derived and applied to vibrating strings and Euler beams. For vibrating strings also, null-controllability by L^p-controls on the boundary is shown by a direct method which makes use of d'Alembert's solution formula for the wave equation.     

Finite element analysis for a nonlinear diffusion model in image processing

The optimal error estimate O(h^k+1) for a popular nonlinear diffusion model widely used in image processing is proved for the standard k^th-order (k ≥ 1) conforming tensor-product finite elements in the L²-norm. The optimal L²-estimate is obtained by the integral identity technique [1–3] without using the classic Nitsche duality argument [4].     

Stability analysis and a priori error estimate of explicit Runge-Kutta discontinuous Galerkin methods for correlated random walk with density-dependent turning rates

In this paper,we analyze the explicit Runge-Kutta discontinuous Galerkin(RKDG)methods for the semilinear hyperbolic system of a correlated random walk model describing movement of animals and cells in biology.The RKDG methods use a third order explicit total-variation-diminishing Runge-Kutta(TVDRK3)time discretization and upwinding numerical fluxes.By using the energy method,under a standard CourantFriedrichs-Lewy(CFL)condition,we obtain L2stability for general solutions and a priori error estimates when the solutions are smooth enough.The theoretical results are proved for piecewise polynomials with any degree k 1.Finally,since the solutions to this system are non-negative,we discuss a positivity-preserving limiter to preserve positivity without compromising accuracy.Numerical results are provided to demonstrate these RKDG methods.     

Lp Solutions for Multidimensional BDSDEs with Locally Weak Monotonicity Coefficients

In this paper, the authors establish the existence and uniqueness theorem of L^p (1 < p ≤ 2) solutions for multidimensional backward doubly stochastic differential equations (BDSDEs for short) under the p-order globally (locally) weak monotonicity conditions. Comparison theorem of L^p solutions for one-dimensional BDSDEs is also proved.These conclusions unify and generalize some known results.     

Notes on Global Existence for the Nonlinear Schrdinger Equation Involves Derivative

In this paper, we consider the scattering for the nonlinear Schr¨odinger equation with small,smooth, and localized data. In particular, we prove that the solution of the quadratic nonlinear Schr¨odinger equation with nonlinear term |u|2involving some derivatives in two dimension exists globally and scatters. It is worth to note that there exist blow-up solutions of these equations without derivatives. Moreover, for radial data, we prove that for the equation with p-order nonlinearity with derivatives, the similar results hold for p ≥2d+32d-1and d ≥ 2, which is lower than the Strauss exponents.     

Cyclic polytopes and d-order curves

A cyclic d-polytope is a convex polytope combinatorially equivalent to the convex hull of a finite subset of a d-order curve in R ^d. We give an affirmative answer to a conjecture of M. A. Perles [4] by proving that every even-dimensional cyclic polytope occurs in this way: its set of vertices can always be extended to a d-order curve.     

A Mehrotra-type predictor-corrector algorithm with polynomiality andQ-subquadratic convergence

Mehrotra's predictor-corrector algorithm [3] is currently considered to be one of the most practically efficient interior-point methods for linear programming. Recently, Zhang and Zhang [18] studied the global convergence properties of the Mehrotra-type predictor-corrector approach and established polynomial complexity bounds for two interior-point algorithms that use the Mehrotra predictor-corrector approach. In this paper, we study the asymptotic convergence rate for the Mehrotra-type predictor-corrector interior-point algorithms. In particular, we construct an infeasible-interior-point algorithm based on the second algorithm proposed in [18] and show that while retaining a complexity bound ofO(n ^1.5 t)-iterations, under certain conditions the algorithm also possesses aQ-subquadratic convergence, i.e., a convergence ofQ-order 2 with an unboundedQ-factor.Research supported in part by NSF DMS-9102761 and DOE DE-FG02-93ER25171.     

Nonexistence of a Kruskal-Katona Type Theorem for Subword Orders

We consider the poset SO(n) of all words over an n-element alphabet ordered by the subword relation. It is known that SO(2) falls into the class of Macaulay posets, i. e. there is a theorem of Kruskal–Katona type for SO(2). As the corresponding linear ordering of the elements of SO(2) the vip-order can be chosen.Daykin introduced the V-order which generalizes the vip-order to the n2 case. He conjectured that the V-order gives a Kruskal–Katona type theorem for SO(n).We show that this conjecture fails for all n3 by explicitly giving a counterexample. Based on this, we prove that for no n3 the subword order SO(n) is a Macaulay poset.     

Superconvergence of discontinuous Galerkin methods for hyperbolic systems

In this paper, the discontinuous Galerkin method for the positive and symmetric, linear hyperbolic systems is constructed and analyzed by using bilinear finite elements on a rectangular domain, and an O(h²)$O\left(h^2\right)$-order superconvergence error estimate is established under the conditions of almost uniform partition and the H³$H^3$-regularity for the exact solutions. The convergence analysis is based on some superclose estimates derived in this paper. Finally, as an application, the numerical treatment of Maxwell equation is discussed and computational results are presented.