Volume 25, Issue 2

Nonlinear stability of the motionless state of the second-order fluid in porous medium in presence of magnetic field is studied by the Lyapunov’s second method. Through defining a Lyapunov function we will prove the inhibiting effect of the magnetic field on the onset of convection. If the Chandrasekhar number is below a computable constant depending on system parameters, we even prove the coincidence of linear and nonlinear stability boundary. Moreover, the medium permeability has a destabilizing effect.

We estimate the convergence rate to normal law of the distribution of some special function which does not satisfy the Lindeberg type condition.

This paper is devoted to the converse of the classical Fabry-Pólya theorem dealing with the localization of singularities of lacunary power series on the boundary of the circle of convergence.

The following theorem has been proved by R.Brück. If a non-constant entire function ƒ and its derivative ƒ′ share the value 1 CM (counting multiplicities), and if , then ƒ′- 1 = c ( ƒ - 1 ) for some nonzero constant c . In this paper we shall prove that the above result is also true when ƒ is a non-constant meromorphic function.

The main goal of this paper is to discuss several interrelations between the Taylor coefficients of a q x q matrix-valued Carathéodory function and its Cayley transform which is a q x q matrix Schur function. Both Taylor coefficient sequences are described in terms of corresponding matrix balls. Hereby, we will obtain explicit formulas for the parameters of one matrix ball in terms of the other one. These expressions imply a one-to-one correspondence between central matricial Carathéodory functions and central matricial Schur functions which is established via Cayley transform.

Let the function ƒ be real-valued and continuous on [-1,1]. Denote by R n,n , n = 1,2, … the rational Chebyshev approximants of order (n,n) of f on the interval. The present paper is inspired by a conjecture raised up by A. A. Gonchar, namely: If all (start­ing with a number n 0 ) rational functions R n,n are analytic in some Joukowski’s ellipse Ɛ, then ƒ admits an analytic continuation from [-1,1] into Ɛ. We show that the hypothesis is true if the set of the free poles of R n,n and the set of the alternation points are asymptoti­cally “well distributed”.

The concepts of statistical convergence of single and multiple sequences of complex numbers were introduced in [2] and [8], respectively. In this paper, we introduce the concept indicated in the title. We prove that if a d-multiple sequence is regularly statistically convergent, then its statistical limit can be computed as the iterated statistical limit of the statistical limits of its subsequences which correspond to an arbitrary partition of the index set {1,2,..., d}, d ≥ 2. As an application, we prove that if a function ƒ is in L p (T d ) for some p > 1, T = [0, 2π), then the (symmetric) rectangular partial sums of its d-multiple Fourier series are regularly statistically convergent to ƒ { u 1 , . . . , u d ) at almost every point (u 1 ,..., u d ) ∈T d . Furthermore, if ƒ is in C(T d ), then the regular statistical convergence of the rectangular partial sums takes place uniformly on T d .