Representation theory has contributed much to the study of linear associative algebras. The central problem of representation theory per se is the determination for each algebra of all its indecomposable representations. This turns out to be a much deeper problem than the classification of algebras, in the sense that there are algebras for which any “internal question” can be answered but for which the number and nature of representations is almost completely unknown, or if known is much more complicated than the internal theory. This can be illustrated by the example of a commutative algebra of order three for which the representation theory can be shown to be essentially the same as the problem of classifying pairs of rectangular matrices under equivalence. (This algebra has indecomposable representations of every integral degree.)

Soon after the publication of Tutte's paper [5] on m-cages, H. S. M. Coxeter asked in a letter to the author whether one-regular graphs of degree 3 exist. The purpose of the following paper is to show by an example that the answer is in the affirmative.

For solutions of the differential difference 'F-equation

1

there are representations as a generating series [5],

2

and as a contour integral [3],

3

We are concerned with establishing sufficiency theorems for minima of simple integrals of the parametric type in a class of curves with variable end points and satisfying isoperimetric side conditions. The results which are obtained involve no explicit assumptions of normality. Such results can be derived by transforming our problem to a problem of Bolza and using the latest developments in the theory of that problem. More recently [6] an indirect method of proof has been published. Our object is to present a direct method of proof without transformation of the problem which is based upon a generalization of the classical theory of fields.

We are concerned with extremal arcs for the problem of minimizing a function

over a class of parametric curves C in ay-space of the form

and satisfying end conditions of the type

The components ah are constants and the functions g, yis, and/are given, with the last function positively homogeneous of degree one.

Let Π be an affine plane which admits a collineation τ such that the cyclic group generated by τ leaves one point (say X) fixed, and is transitive on the set of all other points of Π. Such “cyclic affine planes” have been previously studied, especially in India, and the principal result relevant to the present discussion is the following theorem of Bose [2]: every finite Desarguesian affine plane is cyclic. The converse seems quite likely true, but no proof exists. In what follows, we shall prove several properties of cyclic affine planes which will imply that for an infinite number of values of n there is no such plane with n points on a line.

James Singer [12] has shown that there exists a collineation which is transitive on the (t - 1)-spaces, that is, (t - 1)-dimensional linear subspaces, of PG(t, pn). In this paper we shall generalize this result showing that there exist t - r collineations which together are transitive on the s-spaces of PG(t, pn). An explicit construction will be given for such a set of collineations with the aid of primitive elements of Galois fields. This leads to a calculus for the linear subspaces of finite projective geometries.

A graph G consists of a non-null set V of objects called vertices together with a set E of objects called edges, the two sets having no common element. With each edge there are associated just two vertices, called its ends. Two or more edges may have the same pair of ends.

In R. L. Wilder's book [2] the open and closed generalized manifolds are extensively studied. However, no study is made of the generalized manifold with boundary nor is a definition of such a space given except in the case of the generalized closed n-cell. A definition of a generalized manifold with boundary was given by the author in his paper [1]. Before undertaking the study of further properties of these manifolds it seems appropriate to characterize the manifolds with boundary in terms of the open and closed manifolds of Wilder. It is to that purpose that this paper is directed and in particular the generalized closed n-cell of Wilder is characterized as a special manifold with boundary.

The conditional equation f(x1, … , xs) = 0, where f is a polynomial in the x´s with coefficients in a finite field F(pn), is connected with many well-known developments in number theory and algebra, such as: Waring's problem, the arithmetical theory of quadratic forms, the Riemann hypothesis for function fields, Fermat's Last Theorem, cyclotomy, and the theory of congruences in commutative rings.

Hermite [4] in the course of his investigations on the transformation theory of abelian functions, introduced the notion of abelian quadratic forms. They are quadratic forms whose matrices of orders 2n, satisfy

where k ≠ 0 is a real number, and is the unit matrix of order n. Laguerre [7] and, more systematically, Cotty [3] developed an arithmetical theory of abelian forms in four variables.

In this paper, we shall show that if is a nilpotent [5] group and if M, a positive integer, is a uniform bound on the number of conjugates that any element of may have, then there exist “large” integers n for which x → xn is a central endomorphism of . If is not necessarily nilpotent, if the above condition on the conjugates is retained, and if we can find a member of the lower central series [1], every element of which lies in some member of the ascending central series, then we shall show that every non-unity element of the “high” derivatives has finite order.

The present paper is a sequel to that of J. H. Chung (2) and contains a proof of a conjecture made by him, namely, that the number of ordinary (modular) irreducible representations contained in a given p-block of Sn is independent of the p-core. A summary of the results contained herein appeared in the Proceedings of the National Academy of Sciences (9).

Let T be a Young diagram of n nodes:

(1)

ai being the length of its ith row. With respect to a prime p, we denote by T0 the p-core of T. If T0 consists of m nodes, then

(2) m=n-lp,

where l is the number of successive p-hooks [3] removable from T to yield its p-core T0. We have stated in [4] the following theorem :

The factorization method, initiated by Schrödinger [4] and modified and developed by Infeld [2], Duff [1], and Infeld and Hull [3], furnishes an elegant method of solving eigenvalue problems associated with certain ordinary differential equations of the second order. Not only the eigenvalues and eigenfunctions can thus be obtained, but also certain matrix elements associated with the eigenfunctions. Even if the method cannot be applied directly to eigenvalue problems, the factorization of an equation may still be of interest, since recurrence formulae may thus be established, e.g. for Bessel functions [3]. The connection of the method with Truesdell's [5] method of the “F -equation” has been discussed by Duff [1].

The major portion of this paper is devoted to an investigation of the conditions which imply that a semigroup (no identity or commutativity assumed) with a bounded invariant measure is a group. We find in §3 that a weakened form of “shearing” is sufficient and a counter-example (§5) shows that “shearing” may not be dispensed with entirely. In §4 we discuss topological measures in locally compact semigroups and find that shearing may be dropped without affecting the results of the earlier sections (Theorem 2). The next two theorems show that under certain circumstances (shearing or commutativity) the topology of the semigroup (already known to be a group by virtue of earlier results) can be weakened so that the structure becomes a separated compact topological group. The last section treats the problem of extending an invariant measure on a commutative semigroup to an invariant measure on its quotient structure.

1. Introduction A solution will be given in this paper for the following problem.

Let

(1.1)

V. V. Golubev, in his study [6], has constructed, by using definite integrals, various examples of analytic functions having a perfect nowhere-dense set of singular points. These functions were shown to be single-valued with a bounded imaginary part. In attempting to extend his work to the problem of constructing analytic functions having perfect, nowhere-dense singular sets under quite general conditions, he posed the following question: Given an arbitrary, perfect, nowhere-dense point-set E of positive measure in the complex plane, is it possible to construct, by passing a Jordan curve through E and by using definite integrals, an example of a single-valued analytic function, which has E as its singular set, with its imaginary part bounded.

In the literature, considerable attention has been devoted to the study of inversion operators for the Laplace transformation. In particular, much interest attaches to “real” inversion operators, i.e., operators which make use of the values of the generating function arising only from real values of the independent variable. Several of these operators are known (see for example Widder [3, chap. 7, §6; chap. 8, §25], Hirschman [2]).

In this note we shall discuss certain matrix methods of summation, though otherwise, §1 and §2 are not connected.In this section, we shall study some properties of the method (Bh) where we say the series ∑uv is summable (Bh) when

The method (Bh) has been studied in special cases airsing from different values of h by Rogosinski [11; 12], Bernstein [2], and more recently by Karamata [3; 4].