In this note we introduce the concepts of Λ-Mackey sequence, Λ-Mackey convergence property, Λ-Schwartz family and associated Λ-Schwartz family and consider some applications of these to locally convex spaces. Hereby Λ denotes a Banach sequence space with the AK-property — the results of this paper generalize those in [4] where the case Λ = I1 is considered. We obtain a dual characterization of those locally convex spaces which satisfy the Λ-Mackey convergence property and characterize the dual Λ-Schwartz spaces in terms of the SM-property which is introduced in [10]. Finally, necessary and sufficient condition for a locally convex space to be ultra-bornological is proved.

Let E, F be Hausdorff locally convex spaces. In this note we consider conditions on E and F such that the dual space of the space Kb (E, F) (of quasi-compact operators) is a complemented subspace of the dual space of Lb (E, F) (of continuous linear operators). We obtain necessary and sufficient conditions for Lb(E, F) to be semi-reflexive.

Let μ be a finite complex Borel measure supported on the unit circle

.

In this paper, we are concerned with the characterization of the sets of functions satisfying the generalized mean value equation of the form

.

and for all ξ ∈ , | ξ | = R for some fixed R > 0.

Existence is established on some groups for Characteristic functions which are prime in the sence of being indecomposable for product factorization.

This paper deals with a statistical technique of deriving new results on Meijer's G-function. Multiple integral as well as computable series representations are given. The method used is to derive the density of a product of independent beta-distributed random variables by using different statistical techniques, and then compare the expressions to get the desired results. The results obtained in this article do not seem to be available through standard mathematical techniques.

When hydrogen and oxygen react in a closed vessel, the reaction either happens slowly or explosively, depending on the initial temperature θ0 and pressure P. In the (θ0, P)-plane the boundary between these behaviours is represented by a curve. On this curve P is triple-valued over θ0 in certain ranges; that is, for certain initial temperatures there are three explosion limits. This phenomenon, known as the explosion peninsula, is among other things a sign that more than one elementary reaction is involved in the oxidation of hydrogen. Other reactions display similar effects.

Let Xnn ≥ 1, be i.i.d.r.v.'s and Sn = X1+…+Xn. Let be Sn minus the r terms of largest absoluete value. Maller proved that if coverages in distribution to N(0, 1) then so does (Sn/bn)−an, assuming that Xn have a continuous symmetric distribution. We show that his resul;t is true without these extra assumptions. Some related results are also given.

A problem of fundamental importance in Spectral Theory consists of finding criteria for an operator to be of scalar-type in the sense of N. Dunford [1]. One relatively general approach in determining such criteria is based on the method of integral transforms (see for example [4], [5], [6], [11], [12]). For example, if X is a Banach space and T is a continuous linear operator on X, then the group {eitT; t real} exists. As noted by several authors (e.g. [4], [6]), this group can then be effectively used for analysing the operator T.

Let p = (p(i): i ≥ 0) be a sequence of numbers satisfying 0 ≤ p(i) < 1 for i = 0,1,2,…, and let G be a random graph with vertex set ℤ = {…, — 1, 0, 1,…} and with edge set defined as follows: for each pair i, j of vertices, where i ≤ j, there is an edge joining i and j with probability p(j — i), independently of the presence or absence of all other edges. We explore the connectedness of G, showing that G is almost surely connected if and only if Σip(i) = ∞ and the (positive) greatest common divisor of the set {i ≥ 1: p(i) < 0} equals 1; if one of these two conditions fails to hold then G is almost surely disconnected. Corresponding results hold in higher dimensions, for random graphs defined on the vertex sets ℤd where d ≥ 2.

This paper gives an existence result for the discrete coagulation-fragmentation equations:

(If k = 1, the first and last sums are 0.)

Laws of large numbers and central limit theorems are proved for sums of general functions of m-spacings from general distributions. Explicit formulae are given for the norming constants. The results enable us to describe asymptotic properties of distributional tests under fixed alternatives. A generalization of Kimball's spacings test is considered in detail.

Processes related to the empirical distribution are shown to converge to a time changed Brownian motion. In particular, the time change of the Brownian motion differs from the time change obtained by using the distribution function: this gives rise to two different representations of a related Brownian bridge.

The problem of scattering of surface water waves obliquely incident on a fixed half immersed circular cylinder is solved approximately by reducing it to the solution of an integral equation and also by the method of multipoles. For different values of the angle of incidence and the wave number the reflection and transmission coefficients obtained by both methods are evaluated numerically and represented graphically to compare the results obtained by the respective methods.

A notion of ‘uniform ε-independence’ (u.ε.i.) is proposed for a sequence {Xn} successively indexed by random indices {τk}. The u.∊.i. property yields results other than those in the previous random indexing literature. Complementing the u.∊.i. property by suitable ‘approximation’ one recovers these previous results.

The uniformity of a certain shape density of a random ΠD tetrad relative to Lebesgue measure along part of the set corresponding to alignment is shown to be sufficient under certain technical conditions for the generating distribution to be the uniform distribution on a compact convex set. A counter-example is provided to show that this result fails for random IID triangles. The main theorem is used to characterize circular, elliptical and triangular uniform generators from the shape distribution of random IID tetrads.

A modified kernel is devised which avoids the irregular values which plague the natural integral equations for the exterior problems of harmonic elastic waves in a homogeneous isotropic medium.

Examples were exhibited in [4] of both reducible and irreducible symmetric operators (of deficiency index (1:1)) associated with − d2/dt2 in the Hilbert space L2(I) (I = [0,1). Such symmetric operators are determined by three linearly independent boundary conditions which define their domains as restrictions of the domain of the maximal operator associated with — d2/dt2.

Spatially homogeneous cosmological models of the Bianchi type-V are considered in the Brans-Dicke scalar-tensor theory of gravitation. Exact solutions are given in the vacuum case as well as for models filled with dust, radiation or stiff matter.