1.1. A regular Toeplitz method of summability is given by a transformation m = 0, 1, 2 , … of the sequence sn into the sequence σm. According to the definition of regularity, every such method sums a convergent sequence sn to the value lim sn. The question naturally arises, whether there are more extensive classes of sequences summable by all regular methods 1.1(1) or at least by all such methods subject to some simple additional conditions. Questions of this kind have been treated by the author (Lorentz [2], [5]) and, from another point of view, by R. P. Agnew [2] [3] ; in this paper we wish to discuss the problem systematically.

§ 1. An Epstein zeta function (in its simplest form) is a function represented by the Dirichlet's series where a1… ak are real and n1, n2, … nk run through integral values. The properties of this function are well known and the simplest of them were proved by Epstein [2, 3]. The aim of this note is to define a general class of Dirichlet's series, of which the above can be viewed as an instance, and to discuss the problem of analytic continuation of such series.

Summary. By the use of finite difference operators a series of combinatorial formulae are obtained. Two applications of the formulae are given. First, the following explicit formulae for the number, ƒ(n), of non-isomorphic equivalence relations among n elements are obtained:

1. Summary. In many branches of mathematics the notion of rank plays an important part. H. Whitney [3] made a detailed axiomatic investigation of rank and several related ideas. All sets considered by Whitney are finite. In the present note the axiomatic treatment of rank is extended to sets of any cardinal. In the special case of algebraic dependence of elements of a field with respect to a sub-field, similar questions have already been considered by Steinitz [2].

1. Introduction. The elementary portions of the theory of integral representation of numbers or forms by quadratic forms will be somewhat simplified and generalized in this article. This indicates certain directions in which new applications can be made. The applications made here will be largely to the representation of numbers or binary quadratic forms by ternary quadratic forms. Particularly, we shall obtain the correct estimate (Theorem 10) needed to fill a lacuna in certain work of U. V. Linnik [1] on the representation of large numbers by ternary quadratic forms. Since Linnik applied his theorem on ternaries to prove [9] that every large number is a sum of at most seven positive cubes, a lacuna in this proof can now be regarded as filled.

1. Introduction. In his well-known book on graphs [1] König proposed the following problem: “When can a given abstract group be represented as the group of the automorphisms of a (finite) graph, and if possible how can the graph be constructed?”

To understand this problem well let us begin with the precise definition of a graph and its group (of automorphisms).

The eigenfunctions of a boundary value problem are characterized by two quite distinct properties. They are solutions of ordinary differential equations, and they satisfy prescribed boundary conditions. It is a definite advantage to combine these two requirements into a single problem expressed by a unified formula. The use of integral equations is an example in point. The subject of this paper, namely the Schrödinger-Infeld Factorization Method, which is applicable to certain restricted. Sturm-Liouville problems, is based upon another combination of the two properties. The Factorization Method prescribes a manufacturing process.

A classical theorem of J. Liouville states that if z is a real algebraic number of degree n ≥ 2, then there exists a constant c > 0 such that for every pair of integers a, b with b ≠ 0.