The first index law, or addition theorem, is well known. The second is much less well known; but both have been found to be of importance in recent studies of hypergeometric integral equations. The first law has usually been considered only in the simple case of orders of integration which have positive real part, or in the context of generalized functions. Arising out of the need to manipulate expressions involving several fractional integrals and derivatives, our aim here is to establish both laws for all combinations of complex orders of integration and differentiation, and for nearly all functions for which the fractional derivatives involved exist as locally integrable functions.

In [7] Lyndon introduced the concept of inner rank for groups. He defined the inner rank of an arbitrary group G to be the upper bound of the ranks of free homomorphic images of G. Both Lyndon and Jaco have shown that the inner rank of the fundamental group of a closed 2-manifold with Euler characteristic 2 − p, p ≧ 0, is [p/2] where [p/2] is the greatest integer ≦ p/2. The proof given by lyndon [8] uses algebraic techniques; whereas, the proof by Jaco [4] is geometrical.

Let W be a universal class of (not necessarily associative) rings and let A ⊆ W. Kurosh has given in [6] a construction for LA, the lower radical class determined by A in W. Using this construction, Leavitt and Hoffmann have proved in [4] that if A is a hereditary class (if K ∈ A and I is an ideal of K, then I ∈ A), then LA is also hereditary. In this paper an alternate lower radical construction is given. As applications, a simple proof is given of the theorem of Leavitt and Hoffmann and a result of Yu-Lee Lee for alternative rings is extended to not necessarily associative rings.

This paper presents some properties of continuous complex-valued mean periodic functions of a single real variable. The theory of these functions is due mainly to Schwartz [7] and they have also been considered by Kahane [4, 5]. Most of the properties outlined here are also shared by the continuous meanperiodic functions on a half line introduced by Koosis [6].

In a previous paper [1] an approximate analytical solution, useful for large times, was obtained for the transient heat flow from a thin circular disk held at constant temperature and immersed in an infinite medium. In the present work a first approximation has been found for the complementary “small-time” solution and the details of this solution examined. Some numerical calculations are included.

In a recent paper Kovács [1] studied join-continuous modular lattices which satisfy the following conditions: (i) every element is a join of finitely many join-irredicibles, and, (ii) the set of join-irreducibles satisfies the descending chain condition. He was able to prove that such a lattice must itself satisfy the descending chain condition. Interest was expressed in whether or not one could obtain the same result without the assumption of modularity and/or of join-continuity. In this paper we give an elementary proof of this result without the assumption of join- continuity (which of course must then follow as a consequence of the descending chain condition). In addition we give a suitable example to show that modularity may not be omitted in general.

By an operation in equationally definable class of group we mean here the element of a free group in this class generated by set {x, y}.

An operation ω(x, y) is called fundamental in a class of groups K if for every group G ω K the operation xy−1 can be expressed in terms of ω.

Higman and Neumann raised in [1] the problem: Is there any binary operation other than xy-1, x-1y, yx-1, y-1x fundamental in the class of all groups?*

This paper is concerned with certain aspects of the theory and application of the probability generating functional (p.g.fl) of a point process on the real line. Interest in point processes has increased rapidly during the last decade and a number of different approaches to the subject have been expounded (see for example [6], [11], [15], [17], [20], [25], [27], [28]). It is hoped that the present development using the p.g.ff will calrify and unite some of these viewpoints and provide a useful tool for solution of particular problems.

Let N denote the discrete space of all natural numbers. A space X is N-compact if it is homeomorphic with some closed subspace of a product of copies of N. In this paper, N-compact spaces are characterized as homeomorphs of inverse limit space of inverse systems of copies of subsets of N. Also, it is shown that a space X is N-compact if and only if the space (X) of all non-empty compact subsets of X with the finite topology is N-compact.

Let Σan be a given infinite series with the sequence of partial sums {Sn}. Let {Pn} be a sequence of constants, real or complex, and let us write Pn = p0 + p1 + … + pn; P-1 = P-1 = 0.

It is well known that many of the field equations from theoretical physics (e.g. Einstein field equations, Maxwell's equations, Klein-Gordon equation) can be obtained from a variational principle with a suitably chosen Lagrange density. In the case of the Einstein equations the corresponding Lagrangian is degenerate (i.e., the associated Euler-Lagrange equations are of second order whereas in general these would be of fourth order), while in the cases of the Maxwell and Klein-Gordon equations the Lagrangian usually used is not degenerate.

In this paper, we are concerned with certain generalizations of subnormal and ascendent (transfinitely subnormal) subgroups of a group. A subgroup A of a group G is called f-ascendent in G if there is a well ordered ascending complete series of subgroups of G, where for all α < λ, either Gα ⊲ Gα+1 or [Gα+1: Gα] < ∞. If such a series has finite length, A is called F-subnormal in G.

Let f(z)=z + … be regular in the unit disc |z| < 1 (hereafter called E). In a recent paper Trimble [7] has proved that if f(z) be convex in E, then F(z) = (1 − λ)z + λf(z) is starlike with respect to the origin in E for (2/3) ≦ λ ≦ 1. The purpose of this note is to show that if certain additional restrictions be imposed on f(z), then F(z) becomes starlike for all λ, 0 ≦ λ ≦ 1. Also we consider some related problems.

The sides of a triangle are met by the sides of a triangle in perspective with it in nine points of which three lie on the axis of perspectivity of the triangles and the other six, as we have it from the converse of Pascal's theorem, lie on a conic. When the centre of perspectivity is the Lemoine point of the triangle and the axis of perspectivity is the line at infinity this conic is a circle known as a Tucker circle [1]. In this paper, analogously we prove that given a simplex in n-space, the traces on its edges of the prime faces of any simplex in perspective with it, all except those falling on the prime of perspectivity, lie on a quadric which we call a Tucker quadric. And further we consider the special case when this quadric becomes a hypersphere.