Perspectives de l'Approximation Diophantienne et de la Transcendance

Nous discutons de certains aspects de l'approximation diophantienne et de la transcendance inspirés par deux problèmes encore ouverts: trouver une version effective du théorème de finitude de Siegel pour les points entiers sur les courbes et trouver une approche intrinsèque aux résultats de transcendance des valeurs spéciales des fonctions modulaires sans utiliser les variétés abéliennes (problème posé par Th. Schneider).     

Algebraic Numbers Close to 1 in Non-Archimedean Metrics

Let 1 be an algebraic number with relatively small height. Recently, many authors, including Amoroso, Dubickas, Mignotte and Waldschmidt, stated sharp lower bounds for the quantity | – 1|. Here, we provide a p-adic analogue of their results. For instance, we give an upper bound for the absolute value of the norm of – 1, and we show that our estimate is rather sharp in terms of the degree of . Further, we discuss a generalization in several variables of our result.     

Computational Aspect For Function-Valued Padé-Type Approximation

The computational problems of two special determinants are investigated. Those determinants appear in the construction of the function-valued Padé-type approximation for computing Fredholm integral equation of the second kind. The main tool to be used in this paper is the well-known Schur complement theorem.     

Simultaneous Approximation and Algebraic Independence

We establish several new measures of simultaneous algebraic approximations for families of complex numbers related to the classical exponential and elliptic functions. These measures are completely explicit in terms of the degree and height of the algebraic approximations. In some instances, they imply that the field has transcendance degree 2 over . This approach which is ultimately based on the technique of interpolation determinants provides an alternative to Gelfonds transcendence criterion. We also formulate a conjecture about simultaneous algebraic approximation which would yield higher transcendance degrees from these measures.     

从2010年考研之三道线性代数试题说起

从灵活解答2010年三道线性代数考研试题,谈理解数学基本概念及培养灵活运用数学知识处理问题能力的重要性.     

Classroom note: Using the binomial series to prove the arithmetic mean–geometric mean inequality

In 1968, Leon Gerber compared (1 + x) ^a to its kth partial sum as a binomial series. His result is stated and, as an application of this result, a proof of the arithmetic mean–geometric mean inequality is presented.     

On the existence of tori in Asada’s two-regional model

A two-regional five-dimensional model describing the development of income, capital stock and money stock, which was introduced by Asada in (2004) [2] is analysed. Sufficient conditions for the existence of two pairs of purely imaginary eigenvalues and the last one being negative in the linear approximation matrix of the model are found. Formulae for the calculation of the coefficients in the bifurcation equation of the model are derived. The theorem on the existence of invariant tori is presented. A numerical example illustrating the gained results is given.     

The Genetic Sums' Representation for the Moments of a System of Stieltjes Functions and its Application

This paper deals with the spectral problems for high-order nonsymmetric difference operators. The method of investigation is based on the analysis of the genetic sums formulas for the moments of the operator. The parameters of these sums are shown to be connected with coefficients of the introduced vector Stieltjes continued fraction. The connections with vector orthogonality, Hermite—Padé approximation, and Hankel determinants are investigated. This gives a tool for the analysis of the solution of the direct and inverse spectral problem of the operator. It is applied to the integration of hierarchy of the discrete KdV equations. The existence of a global solution is proved. July 13, 1998. Date revised: July 12, 1999. Date accepted: July 26, 1999.     

On the operator of exterior derivation on the Riemannian manifolds with cylindrical ends

We formulate some conditions for the normal and compact solvability of the operator of exterior derivation on the cylindrical manifolds equipped with some Riemannian metrics. Some analogous results were obtained in the particular case of warped cylinders [1].     

On the Two-Dimensional Navier—Stokes Equations with the Free Boundary Condition

In this article we consider the two-dimensional Navier—Stokes equations with free boundary condition (open surface), and derive a number of different results: a new orthogonal property for the nonlinear term, improved a priori estimates on the solution, an upper bound on the dimension of the attractor which agrees with the conventional theory of turbulence; finally, for elongated rectangular domains, an improved Lieb—Thirring (collective Sobolev) inequality leads to an upper bound on the dimension of the attractor which might be optimal. Accepted 11 July 1996     

Numerical solution of quasi-variational inequalities arising in stochastic game theory

We study the finite-difference approximation for the quasi-variational inequalities for a stochastic game involving discrete actions of the players and continuous and discrete payoff. We prove convergence of iterative schemes for the solution of the discretized quasi-variational inequalities, with estimates of the rate of convergence (via contraction mappings) in two particular cases. Further, we prove stability of the finite-difference schemes, and convergence of the solution of the discrete problems to the solution of the continuous problem as the discretization mesh goes to zero. We provide a direct interpretation of the discrete problems in terms of finite-state, continuous-time Markov processes.     

Multiple Orthogonal Polynomials Associated with the Modified Bessel Functions of the First Kind

   Abstract. We consider polynomials which are orthogonal with respect to weight functions, which are defined in terms of the modified Bessel function I _ν and which are related to the noncentral χ ² -distribution. It turns out that it is the most convenient to use two weight functions with indices ν and ν+1 and to study orthogonality with respect to these two weights simultaneously. We show that the corresponding multiple orthogonal polynomials of type I and type II exist and give several properties of these polynomials (differential properties, Rodrigues formula, explicit formulas, recurrence relation, differential equation, and generating functions).