New Relaxation Method for Mathematical Programs with Complementarity Constraints

In this paper, we present a new relaxation method for mathematical programs with complementarity constraints. Based on the fact that a variational inequality problem defined on a simplex can be represented by a finite number of inequalities, we use an expansive simplex instead of the nonnegative orthant involved in the complementarity constraints. We then remove some inequalities and obtain a standard nonlinear program. We show that the linear independence constraint qualification or the Mangasarian–Fromovitz constraint qualification holds for the relaxed problem under some mild conditions. We consider also a limiting behavior of the relaxed problem. We prove that any accumulation point of stationary points of the relaxed problems is a weakly stationary point of the original problem and that, if the function involved in the complementarity constraints does not vanish at this point, it is C-stationary. We obtain also some sufficient conditions of B-stationarity for a feasible point of the original problem. In particular, some conditions described by the eigenvalues of the Hessian matrices of the Lagrangian functions of the relaxed problems are new and can be verified easily. Our limited numerical experience indicates that the proposed approach is promising.     

The trigonometric triangle

A procedure is presented by which the integration of a function such as cos ⁿ (x) or sin ⁿ (x) with n = 1, 2, 3, 4 … is facilitated.     

Analysis of the multiple roots of the Lundberg fundamental equation in the PH (n) risk model

In the study of the Sparre Andersen risk model with phase‐type (n) inter‐claim times (PH (n) risk model), the distinct roots of the Lundberg fundamental equation in the right half of the complex plane and the linear independence of the eigenvectors related to the Lundberg matrix L_δ(s) play important roles. In this paper, we study the case where the Lundberg fundamental equation has multiple roots or the corresponding eigenvectors are linearly dependent in the PH (n) risk model. We show that the multiple roots of the Lundberg fundamental equation det[L_δ(s)] = 0 can be approximated by the distinct roots of the generalized Lundberg equation introduced in this paper and that the linearly dependent eigenvectors can be approximated by the corresponding linearly independent ones as well. Using this result we derive the expressions for the Gerber–Shiu penalty function. Two special cases of the generalized Erlang(n) risk model and a Coxian(3) risk model are discussed in detail, which illustrate the applicability of main results. Finally, we consider the PH(2) risk model and conclude that the roots of the Lundberg fundamental equation in the right half of the complex plane are distinct and that the corresponding eigenvectors are linearly independent. Copyright © 2011 John Wiley & Sons, Ltd.     

Definable nilpotent and soluble envelopes in groups without the independence property

We prove that in a group without the independence property a nilpotent subgroup is always contained in a definable nilpotent subgroup of the same nilpotency class. The analogue for the soluble case is also shown when the subgroup is normal in the ambient group.     

Sufficient Conditions for a Digraph to be Supereulerian

A (di)graph is supereulerian if it contains a spanning eulerian sub(di)graph. This property is a relaxation of hamiltonicity. Inspired by this analogy with hamiltonian cycles and by similar results in supereulerian graph theory, we analyze a number of sufficient Ore type conditions for a digraph to be supereulerian. Furthermore, we study the following conjecture due to Thomassé and the first author: if the arc‐connectivity of a digraph is not smaller than its independence number, then the digraph is supereulerian. As a support for this conjecture we prove it for digraphs that are semicomplete multipartite or quasitransitive and verify the analogous statement for undirected graphs.     

Linear independence of a finite set of time-frequency shifts

This paper introduces an open conjecture in time-frequency analysis on the linear independence of a finite set of time-frequency shifts of a given L² function. Firstly, background and motivation for the conjecture are provided. Secondly, the main progress of this linear independence in the past twenty five years is reviewed. Finally, the partial results of the high dimensional case and other cases for the conjecture are briefly presented.     

Linear and Algebraic Independence of q-Zeta Values

In the paper, results on linear and algebraic independence of q-series of the form $\varsigma _q (s) = \sum\nolimits_{n = 1}^\infty {\sigma _{s - 1} (n)q^n }$ over the field ?(q) are obtained, where $\sigma _{s - 1} (n) = \sum\nolimits_{d|n} {d^{s - 1} }$ , s = 1, 2,... .     

REAL-VALUED PERIODIC WAVELETS: CONSTRUCTION AND RELATION WITH FOURIER SERIES

1.IntroductionWaveletshaverecentlyreceivedagreatdealofattentioninsuchareasassignalprocessingandimageprocessing([12],[8]).Variousmethodstoconstructwaveletshavebeengiven([14],[13],[9],[7]).Itiswellknownthatinmathematicsandmathematicphysicsmanyperiodicp...     

Indépendance algébrique sur les T-modules

We study the transcendence degree (of the values at algebraic points) of the coordinate functions of a given one parameter subgroup on a simple g dimensional T-module. In such a situation, we obtain a lower bound for this transcendence degree depending on g and on the growth order of the parameter subgroup. As a particular case, one gets the finite characteristic analogue of a result of Siegel on the algebraic independence for the values of the classical Bessel function.Résumé. Nous obtenons une minoration du degré de transcendance d'un ensemble de valeurs en des points algébriques des fonctions coordonnées d'un sous-groupe à un paramètre d'un T-module simple, en fonction de l'ordre de croissance de ces dernières et de la dimension du T-module. Un cas particulier de ce résultat permet d'obtenir en caractéristique finie un analogue d'un résultat de Siegel sur l'indépendance algébrique des valeurs de la fonction de Bessel usuelle.