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.

The gravitational field manifests itself in the motion of bodies. Therefore the problem of determining the motion of such bodies from the field equations alone is of fundamental importance. This problem was solved for the first time some ten years ago and the equations of motion for two particles were then deduced [1]. A more general and simplified version of this problem was given shortly thereafter [2].

Mr. Lewison pointed out to us, that from our approximation procedure, it does not follow that the field equations can be solved up to an arbitrarily high approximation. This is indeed true.

We consider the product of two power series with positive coefficients: .

What conditions will ensure that the coefficients wn shall be either (i) monotonic, or (ii) logarithmically convex? By the latter, we mean that wn2 ≤ Wn-1Wn+1 for n = 1, 2, … In investigating this question, which was suggested by a special example, we have found it convenient to express the conditions in terms of the ratios of un, vn to certain binomial coefficients, rather than in terms of un, vn themselves.

Descriptive definitions of Cesàro-Denjoy integrals (CD-integrals) equivalent to the Cesàro-Perron integrals (CP-integrals) introduced by J. C. Burkill [1, 2] have been given by Miss Sargent [6] (see §2). The CD§integrals are generalizations of the special Denjoy integral [5, p. 201]. They are somewhat complicated in that modifications of the definitions of continuity, generalized absolute continuity in the restricted sense (ACG*) [5, p. 231], and of derivatives are required for each order. In the present paper a scale of integrals is obtained which is based on the descriptive definition of the general Denjoy integral [5, p. 241]. The approximate derivative and a slightly modified definition of generalized absolute continuity (ACG) are used for all orders so that the only concept generalized for increasing orders is that of continuity.

Nous nous occupons dans ce travail d'une classe de surfaces dont l'équation différentielle est du cinquième ordre, et qui jouissent de la propriété que, sur elles, les lignes d'égale courbure moyenne forment, avec leurs trajectoires orthogonales, un système isotherme. Dans cette classe rentrent les surfaces admettant un groupe de déplacements à un paramètre (cylindres, surfaces de révolution et hélicoïsian Bonnet).

§ 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.

The purpose of this paper is to investigate completely indecomposable modules. A completely indecomposable module is an additive abelian group with a ring A as operator domain, where the following four conditions are satisfied.

1-1. A is a commutative ring and has a unit element which is unit operator for .

1-2. The submodules of satisfy the ascending chain condition. (Submodule will always mean invariant submodule.)

Let V be a connected, compact, differentiable Riemannian manifold. If V is not closed we denote its boundary by S. In terms of local coordinates (xi), i = 1, 2, … Ν, the line-element dr is given by where gik (x1, x2, … xN) are the components of the metric tensor on V We denote by Δ the Beltrami-Laplace-Operator and we consider on V the differential equation (1) Δu + λu = 0.

In a series of papers which will follow this paper the authors will present a theory of functionals which are bilinear over a product A × B of two normed vector spaces A and B. This theory will include a representation theory, a variational theory, and a spectral theory. The associated characteristic equations will include as special cases the Jacobi equations of the classical variational theory when n = 1, and self-adjoint integrodifferential equations of very general type. The bilinear theory is oriented by the needs of non-linear and non-bilinear analysis in the large.

Modern physical theory, both classical and quantal, faces serious difficulties which arise from the divergence of certain integrals. Perhaps the best known of these “infinities” is the self-energy of the point electron. Most of the simpler devices used to eliminate the infinities, such as the introduction of a finite electron radius, are non-relativistic and must therefore be rejected. Relativistic theories1 which do avoid some or all of the infinities are very complicated and often suffer from difficulty in physical interpretation.

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:

In a perfect incompressible fluid extending to infinity, the determination of the motion of N parallel rectilinear vortex filaments involves the solution of N non-linear differential equations, each of the first order. The method of Kirchhoff provides certain constants of the motion. If we describe the positions of the vortices by their point-traces on a plane perpendicular to them, the following facts follow from the theory of Kirchhoff:

(1.1) The mean centre of the system is fixed.

The following note which deals with the effect of a certain determinantal operator when it acts upon a product of determinants was suggested by the original proof which Dr. Alfred Young gave of the property

subsisting between the positive P and the negative N substitutional operators, θ being a positive integer. This result which establishes the idempotency of the expression θ−1NP within an appropriate algebra is fundamental in the Quantitative Substitutional Analysis that Young developed.

The linearized theory of surface waves leads to several mixed boundary value problems which have been investigated by various methods. As the physical background of the theory has been repeatedly discussed, it will suffice to deal here mainly with the mathematical aspect of the question.

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].

The results of the present paper can be interpreted (a) in terms of the theory of the representations of the symmetric group, or (b) in terms of the corresponding theory of the full linear group. In the latter connection they give a solution to the problem of the expression of an invariant matrix of an invariant matrix as a sum of invariant matrices, in the sense of Schur's Dissertation. D. E. Littlewood has pointed out the significance of this problem for invariant theory and has attacked it via Schur functions, i.e. characters of the irreducible representations of the full linear group. We shall confine our attention here to the interpretation (a).

The continuous real functions on a topological space X are partially ordered in a natural way by putting ƒ ≦ g if and only if ƒ(x)≦ g(x) for all x in X. With respect to this partial ordering these functions constitute a lattice, the lattice operations ∪ and ∩ being defined by the relations (ƒ ∪ g) (x) = max (ƒ(x), g(x)) ƒ ∩ g )(x) = min (ƒ(x), g(x)). The lattice character of any partially ordered system merely expresses the existence of least upper and greatest lower bounds for any finite set of elements in the system. Many partially ordered systems enjoy much stronger boundedness properties than these: for example, every subset with an upper bound may have a least upper bound, as in the case of the real number system.

A function η(i1, …, if) of ƒ quantities i, varying over the finite range i = 1,2, …, n, is usually called an n-dimensional tensor of rank ƒ. Any permutation p: 1 → 1, … , ƒ → ƒ… changes this tensor into a tensor pη according to the equation pη(i1, ...,if) = η(i1', ...,if')Thus the permutation p appears as a linear operator p in the n-dimensional space Σ = Σn,ƒ of all n-dimensional tensors of rank ƒ, η is symmetric if pη = η for all permutations p, it is antisymmetric if pη δp.η where δp = + 1 for the even and — 1 for the odd permutations.

The Gauss-Bonnet Theorem leads through well known arguments to the fact that the integral curvature of a two-dimensional closed orientable manifold M of genus p equals 4π(1 — p). This implies, for instance, that the Gauss curvature K can neither be everywhere positive nor everywhere negative, if M is homeomorphic to a torus.

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.