We prove that if t is a compact linear operator that is not quasi-nilpotent and is appropriately normalised, then the closed semi-algebra A(t) generated by t is locally compact. The theory of locally compact semi-algebras (2) is therefore applicable to A(t), and we show that it can be used to obtain spectral properties of t.

The object of this paper is to give a generalisation to vector valued functions of the classical mean value theorem of differential calculus. In that theorem we have

for some c in the open interval a, b when f is a real valued function which is continuous on the closed interval a, b and differentiable on the open interval. The counterpart to (1) when f has values in an n-dimensional vector space turns out to be

where cka, b, 0 k, and .

The basic reciprocity of j-differential and LM-integral

for bounded functions f(x) with simple discontinuities but continuous on the left at each point and for g(x) in the somewhat restricted class B of functions of bounded variation and also left-continuous, was established in (2) and (3); the dot here indicates the lower product of and (jg, g+ (x+)dx), with , and the integral indicated is the RJDS-integral, equivalent to (LM) .

Rainville (6) has discussed the following Euler transformation of the hypergeometric function

In (1) numbers related to the Stirling numbers are defined. Later in (2) these numbers were called Lah-numbers (cf. 2, p. 43, Ex. 16). According to (1) these numbers are of importance in Mathematical Statistics. In this paper we shall generalise the method and apply it to generalised Stirling numbers as defined in (3).

A previous note (2) showed how the integral of f(1x1+2x2++nxn) over the interior of a simplex could be reduced to a contour integral. The same idea is applied here in Theorems 1 and 2 to give a generalisation of Dirichlet's multiple integral ((1), pp. 169172). These results are then used in Theorem 3 to reduce an integral over all real n-dimensional space to a contour integral. In Theorem 4 an integral over the group of all 3 3 orthogonal matrices of determinant 1 is reduced to a contour integral. This result can be extended formally to the case of 4 4 matrices; beyond this it seems difficult to go.

We prove the following theorem, which was established by Abel (1) for the case u = 1.

Theorem. If u, k are positive integers and x, 1, , u, are real numbers, then

where the sum is taken over all distinct ordered solutions (s1, , su) in non-negative integers of the equation .

Kilmister (1) has considered dynamical systems specified by coordinates q( = 1, 2, , n) and a Lagrangian

(with summation convention). He sought to determine generally covariant conditions for the existence of a first integral, , linear in the velocities. He showed that it is not, as is usually stated, necessary that there must exist an ignorable coordinate (equivalently, that b must be a Killing field:

where covariant derivation is with respect to a). On the contrary, a singular integral, in the sense that for all time if satisfied initially, need not be accompanied by an ignorable coordinate.