Majorant functions for contractors can be defined in a natural war. Such a case is considered here in order to find iterative solutions of general equations in Banach spaces by means of contractors. A class of majorant functions is defined which contains in particular the linear majorant ones. Local and global existence and convergence theorems are proved.

Matrix quadratic equations have found the most diverse applications. The present article gives a connected account of their theory, and contains some new results and new proofs of known results.

If f is a p–th integrable function on the circle group and ω(p; f; δ) is its mean modulus of continuity with exponent p then an extended version of the classical theorem of Jackson states the for each positive integer n, there exists a trigonometric polynomial tn of degree at most n for which

‖f-tn‖p ≤(p; f; 1/n).

In this paper it will be shewn that for G a Hausdorff locally compact abelian group, the algebra L1(G) admits a certain bounded positive approximate unit which, in turn, will be used to prove an analogue of the above result for Lp(G).

By using a variational method, we study the structure of the Padé table for a formal power series. For series of Stieltjes, this method is employed to study the relations of the Padé approximants with orthogonal polynomials and gaussian quadrature formulas. Hence, we can study convergence, precise locations of poles and zeros, monotonicity, and so on, of these approximants. Our methods have nothing to do with determinant theory and the theory of continued fractions which were used extensively in the past.

It is shown that for arbitrary countable dense ssets A and B of the real line, there exists a transcendental entire function whose restriction to the real line is a real-valued strictly monotone increasing surjection taking A onto B The technique used is a modification of the procedure Maurer used to show that for countable dense subsets A and B of the plane, there exists a transcendental entire function whose restriction to A is a bijection from A to B.

Sufficient conditions, involving the existence of a Lyapunov function which is a submartingale of special type, are given for the instability of stochastic discrete time systems.

The pseudocharacter of the space YX of continuous functions from X to Y is studied. For certain topologies on YX sufficient conditions, which are shown in some cases also to be necessary, for YX to have a specified pseudocharacter are given. An Ascoli theorem is a theorem which characterizes the compact subsets of YX. Several Ascoli theorems are obtained, including ones which utilize results on pseudocharacter.

Let M be a connected C∞. A method is being introduced here to study the action of the holonomy group and the restricted holonomy group of Γ on a recurrent tensor. The main result of this paper is that if the recurrence covector W of a recurrent tensor S on M is an exact form then the tensor S is invariant under the holonomy group of Γ and if W is a closed form then S is invariant under the restricted holonomy group of Γ. In the last section, this result is applied to some particular cases including the case of a riemannian manifold with recurrent curvature.

An example of Wehrfritz is pointed out to show that GL(4, Q) is not coherent. This answers a question of Serre. It is shown that finitely generated soluble coherent subgroups of GL(2, Q) need not be polycyclic, in sharp contrast to the fact that all soluble subgroups of GL(n, Z) are polycyclic and so automatically coherent.

It has recently been shown that discontinuous functional calculi exist for certain commutative Banach algebras. Such an algebra thus possesses two distinct calculi so that there exist analytic functions whose action on the algebra is not uniquely determined. In this note a method is given for constructing commutative Banach algebras which admit two inequivalent complete norm topologies and the result is applied to show that the action of any non-algebraic analytic function may fail to be uniquely defined.

The concept of a tetradiffric function is introduced. This new scheme for defining discrete analytic functions is shown to retain the algebraic simplicity of monodiffric functions, while introducing to the theory a symmetry similar to the Schwarz Reflection Principle.

A graph X is edge-reconstructible if it is uniquely determined up to isomorphism by the set of graphs X – e obtained by deleting one, edge e. The graphs of a comparatively rich class are shown to be edge-reconstructible. This class contains all non-trivial strong products and certain lexicographic products.

A path decomposition of a digraph G (having no loops or multiple arcs) is a family of simple paths such that every arc of G lies on precisely one of the paths of the family. The path number, pn(G) is the minimal number of paths necessary to form a path decomposition of G.

We show that pn(G) ≥ max{0, od(v)-id(v)} the sum taken over all vertices v of G, with equality holding if G is acyclic. If G is a subgraph of a tournament on n vertices we show that pn(G) ≤ with equality holding if G is transitive.

We conjecture that pn(G) ≤ for any digraph G on n vertices if n is sufficiently large, perhaps for all n ≥ 4.

In this paper we construct the Wiener-Hopf equation for diffraction of a planetary wave by an infinite strip and obtain its solution in the form of an integral representation. We also discuss the asymptotic character of the diffracted wave using the saddle point method.

This paper presents a two-variable basis for the variety generated by the finite simple group PSL(2, 2n).

It is proved that the subvarieties of the variety are CREAM in the sense of Higman when m, n are coprime and n is an odd integer not divisible by q4 for any prime q.

The sum of an infinite series involving the 4F3 function is found by considering an infinite series whose terms involve the product of two Bessel functions of the first kind. Furthermore the infinite series involving the 4F3 function can be utilized to find an approximation to the 4F3 function of unit argument, for particular values of the parameters.

A proof is given of a result deduced from numerical work on the conformal mapping of the region exterior to a quadrilateral possessing an axis of symmetry. It is shown that the result follows from known relationships between hypergeometric functions.

The object of the present paper is to study the delay differential equation of arbitrary order namely y(n)(t) + a(t)yτ(t) = f(t), n≥2 (an integer) and prove a nonoscillation theorem under the general situation in which a(t) and f(t) are allowed to oscillate arbitrarily often on some positive half real line. This is accomplished by way of two differential inequalities of nth order.