On a class of strongly contractive quadratic recurrent systems

We consider a class of nonlinear recurrent systems of the form \( {\Lambda_p} = \frac{1}{p}\sum\limits_{{p_1} = 1}^{p - 1} {f\left( {\frac{{{p_1}}}{p}} \right){\Lambda_{{p_1}}}{\Lambda_{p - {p_1}}}} \), p > 1, where f is a given function on the interval [0, 1] and Λ₁ = x is an adjustable real-valued parameter. Under some suitable assumptions on the function f, we show that there exists an initial value x ^* for which Λ _p = Λ _p (x ^* ) → const as p → ∞. More precise asymptotics of Λ _p is also derived.     

Functional models and finite dimensional perturbations of the shift

We study finite dimensional perturbations of shift operators and their membership to the classes A _{m, n} appearing in the theory of dual algebras. The results obtained yield informations about the lattice of invariant subspaces via the techniques of Scott Brown.     

On the representation of a class of contractive matrix-valued functions

By a decomposition of L₊²(Cⁿ) in two orthogonal subspaces we obtain a representation of a matrix-valued function of the class $\text{J}$Π, defined by Arov (Darlington realization of matrix-valued functions, Izv. Akad. Nauk. SSSR, Ser. Mat. Tom.37 (1973), No. 6); (Math. USSR Izvestija7 (1973), No. 6, 1295–1326). Real matrix-valued functions of this class play an important role in methods of synthesis of scattering matrices of linear passive n-ports.     

On a class of finite rings

In [7], Corbas determined all finite rings in which the product of any two zero-divisors is zero, and showed that they are of two types, one of characteristic p ²and the other of characteristic p².

The purpose of this paper is to address the problem of the classification of finite rings such that.

(i)the set of all zero-divisors form an ideal M.

(ii)M ³=(0); and.

(iii)M ³≠(0).

Because of (i), these rings are called completely primary and we shall call a finite completely primary ring R which satisfies conditions (i), (ii) and (iii), a ring with property(T). These rings are of three types, namely, of characteristic p p ² and p ³. The characteristic p ² case is subdivided into cases in which p?M ² p?ann(M)?M ² and p?M ?ann(M), where ann(M) denotes the two-sided annihilator of where M in R.     

On finite rank perturbations of definitizable operators

It was shown by P. Jonas and H. Langer that a selfadjoint definitizable operator A in a Krein space remains definitizable after a finite rank perturbation in resolvent sense if the perturbed operator B is selfadjoint and the resolvent set ρ(B) is nonempty. It is the aim of this note to prove a more general variant of this perturbation result where the assumption on ρ(B) is dropped. As an application a class of singular ordinary differential operators with indefinite weight functions is studied.     

On a characterization of finite dimensional real linear spaces

LetX be a real linear space; if is a linear topology onX, let denote the class of all bounded with respect to subsets ofX. In this paper it is shown that the spaceX is algebraically finite dimensional if and only if the class of all convex, absorbing and radially bounded subsets ofX is included in the intersection of all , where runs the set of all linear topologies onX.     

On a class of finite solvable groups

A finite solvable group G is called an X-group if the subnormal subgroups of G permute with all the system normalizers of G. It is our purpose here to determine some of the properties of X-groups. Subgroups and quotient groups of X-groups are X-groups. Let M and N be normal subgroups of a group G of relatively prime order. If G/M and G/N are X-groups, then G is also an X-group. Let the nilpotent residual L of G be abelian. Then G is an X-group if and only if G acts by conjugation on L as a group of power automorphisms.     

Analysis of finite dimensional approximations to a class of partial differential equations

In this paper, a unified approach for analysing finite dimensional approximations to a class of partial differential equations boundary value problems (second‐kind Fredholm differential equations) is introduced. The approach is shown to be general despite of its extremely simple form. In particular, it is expected to be useful in the convergence analysis of finite element methods for solving PDE problems. Three specific examples are presented to illustrate the broad applicability of the approach. Copyright © 2004 John Wiley & Sons, Ltd.     

On a class of restricted phase problems in the plane

Certain geometrical necessary and sufficient conditions are developed for the optimal control of a class of second order systems with restrictions on the phase coordinates. The phase constraint set in the plane is assumed to be convex, and under additional hypotheses, it is shown that an optimal trajectory is the one that encloses the least possible area with the optimal trajectory in the unrestricted case.     

A new class of contractive mappings

We introduce a new class of Picard operators which includes the class of enriched contractions, enriched Kannan mappings, enriched Chatterjea mappings, and prove some fixed point theorems for these mappings. Some examples will illustrate the generality of our results.

     

On finite dimensional subspaces of Banach spaces

The common Banach spaces are investigated with respect to some properties of their finite dimensional subspaces. The contribution to the paper of the second author is part of his Ph.D. thesis prepared at the Hebrew University of Jerusalem under the supervision of Professor A. Dvoretzky. The authors wish to thank Professor A. Dvoretzky for the interest he showed in the paper and for his helpful advice.