The purpose of this paper is to generalize the results of Sherbert on Lipschitz algebras and to study the relationship between homomorphisms of these algebras and continuous maps of the underlying metric spaces. In Sections 1, 2, and 3 we associate with each metric space a class of Lipschitz-type algebras and extend Sherbert's results in [7] to this class; in particular Sherbert's theorem 5.1 is extended to non-compact metric spaces (3.3, 3.4, 3.5).

By a group theoretic class we mean a class of groups which contains the trivial group, denoted E, and any group isomorphic to a group in the class. Let I be a group theoretic class. Following P. Hall [4, p. 533], we define EI, CI, SI, QI, and NoI to be the (group theoretic) classes consisting of extensions of I groups by I groups, cartesian products of I groups, subgroups of I groups, homorphic images of I groups and products of two normal I subgroups of a group, respectively. If T is one of the above operations on classes of groups and TI = I, we say X is T closed.

The original motivation for this work was the problem of determining whether the signum function of a real valued continuous function defined on the real line is Riemann integrable. This problem is considered in § 2 where an example of an infinitely differentiable function is presented which possesses a non-Riemann integrable signum function. Moreover, it is shown that, for any ∈ > 0, it is possible to construct such an example for which the set of points of analyticity has Lebesgue measure which is less than ∈. This appears to be a more interesting property than the one originally sought.

A finite sequence (pk) = (p3, p4,…) of non-negative integers shall be called realizable provided there exists a 3-valent 3-polytope P which has pi. i-gonal faces for every i. P is called a realization of (pk).

For realizability of a sequence (pk), from Euler's formula follows

(*) as a necessary condition.

In the course of work on factor groups of the modular group, A. O. L. Atkin (private communication) obtained the permutations

which satisfy the relations

(1)

for T = T1, T2. It was evident that neither pair S, T generates either A12 or LF(2, 11), so he suggested that they probably generate the Mathieu group M2.

In an earlier paper [8], one of the authors has established some Vandermonde type convolution identities involving multinomial coefficients with several summations which evidently are regeneralizations of identities in [1] with one summation. In this paper similar identities are derived for coefficients (defined below) of a general type, in the line of the results in [2] and [3], Furthermore, in a series of papers [4], [5], [6], Gould has obtained results on inversion of series and on classical polynomials by an extensive use of these identities with one summation.

In earlier papers [3] and [4] some Vandermonde type convolution identities with several summations were established which were the extensions of Gould's work that involved one summation. Furthermore, these identities were utilized to obtain some inverse series relations for functions of several variables. As a continuation, this paper deals with some further generalizations of Gould's work in [1] and [2].

Let A and B be spaces having the homotopy type of countable CW-complexes. Then we prove the following theorems.

Theorem 1. If conil(A × B) ≤ 1, then for each integer n ≥ 1, the inclusion j: ∑n A ∨ ∑n B → ∑nA × ∑n B is a homotopy equivalence.

This result is obtained as a corollary of Theorem 2.

We consider the second order differential equation

(1)

with the assumptions that

(2) f(x) is continuous (- ∞ < x < ∞) and p(t) is continuous and bounded: |p(t)| ≤ E, - ∞ < t < ∞.

Also, throughout this paper, F(x) denotes an antiderivative of f(x).

In this note we apply a well-known theorem due to Weierstrass to show that under certain conditions convergence in distribution of a sequence of distribution functions implies the convergence of moments.

This note may be understood by an undergraduate student who has an introductory course of complex variables and a second course of statistics.