In a recent paper [7] the author considered, among other things, the integral transform where is the fundamental solution of the heat equation There we gave a physical interpretation of the transform (1.1). Here we shall choose a slightly different interpretation, more convenient for our present purposes. If then u(O, t) = f(t). That is, the function f(t) defined by equation (1.1) is the temperature at the origin (x = 0) of an infinite bar along the x-axis t seconds after it was at a temperature defined by the equation .

In the first paper of this series [4] I gave a brief summary of the properties of symmetrisable operators in Hilbert Space. A detailed discussion of these properties will be given now, but the properties of operators symmetrisable by bounded operators will be dealt with further in Part III.

The fact that the most general symmetrisable operators in Hilbert Space do not possess a number of the desirable properties of such operators in unitary spaces makes it necessary to look for a more restricted class of operators. There are two reasons for our particular choice. In the first place many of the conditions introduced in the course of Part II concerned reltionships between the domain of the symmetrising operator H and the domain and range of the symmetrisable operator A. These conditions are now all automatically satisfied. The other reason is that the construction used in section 4 to relate symmetrisable operators to certain symmetric operators clearly required that either H or H−1 was bounded. The case of H−1 bounded has already been dealt with in section 9 and shown to be fairly simple. The case in which H is bounded is clearly of considerable complexity, since we have already seen (example in proof of Theorem 10.6.) that the con |H| the bound of H by

Let ф(t) be an even function integrable in the Lebesgue sense and periodic with period 2п. Let Write By an indirect method based on the method of Riesz summability for the Fourier series, the author has established the following convergence test for the oscifiating series Σan. The theorem is as follows:

THEOREM A [1]. If, for some Δ > 0 as t → +0 andthen фan converges to the sums = 0. Here K is an absolute constant independent of n.

Let (х) = (x1, x2, … xn) = Σi Σiaij, xixf (aij = aij) be a positive quadratic form with determinant D, and let M be the minimum of for integral x ≠ 0. Then attains the value M for a finite number of integral x = ±mk ( k = 1, …, s) called its minimal vectors.

The main purpose of this paper is to discuss the categories of the minimal topological spaces investigated in [1], [2], [7], and [8]. After these results are given, an application will be made to answer the following question: If is the lattice of topologies on a set X and is a Hausdorff (or regular, or completely regular, or normal, or locally compact) topology does there always exist a minimal Hausdorff (or minimal regular, or minimal completely regular, or minimal normal, or minimal locally compact) topology weaker than ?

The object of this paper is to consider the existence of the double-six over GF(2n) and particularly over GF(4).

If the finite group G has a 2-Sylow subgroup S of order 2a+1, containing a cyclic subgroup of index 2, then in general S may be one of the following six types [8]:

(i) cyclic; (ii) Abelian of type (a, 1), a > 1; (iii) dihedral1; (iv) generalized quaternion; (v) {α, β}, α2a = β2, α2a−1+1, a ≧ 3;

(vi) {α, β}, α2a = β2, α2a−1+1, a ≧ 3.

This paper deals with a type of Remannian space Vn (n ≧ 2) for which the first covariant dervative of Weyl's projective curvature tensor is everywhere zero, that is where comma denotes covariant differentiation with respect to the metric tensor gij of Vn. Such a space has been called a projective-symmetric space by Gy. Soós [1]. We shall denote such an n-space by ψn. It will be proved in this paper that decomposable Projective-Symmetric spaces are symmetric in the sense of Cartan. In sections 3, 4 and 5 non-decomposable spaces of this kind will be considered in relation to other well-known classes of Riemannian spaces defined by curvature restrictions. In the last section the question of the existence of fields of concurrent directions in a ψ will be discussed.

A discrete renewal process is a sequence {X4} of independently and inentically distributed random variables which can take on only those values which are positive integral multiples of a positive real number δ. For notational convenience we take δ = 1 and write where .