Let A be a commutative ring with unit element, and let A [x] be a ring of polynomials in an indeterminate x with coefficients in A. There are a number of well-known properties which A shares with A [x]. We shall state one of them in the following.

Given a Noetherian ring A with unit element and an ideal m of A such that

,

we may topologize A by adopting {mn; n = 1, 2, …} as a fundamental system of neighborhoods of 0. This topologized ring is usually referred to as an m-adic ring, and is called a Zariski ring if its ideals are all closed.

1. Introduction. A. Brauer and R. Brauer (2) and Barnes (1) (following a method of Mordell (6)) have solved the Diophantine equation x2+y2+c = xyz subject to the condition (x, y) = 1. Independently, but using the same methods, I treated (4) the equation

x2+y2+ax+ay+l = xyz,

and subsequently (5) gave a method of obtaining all integral solutions of

x2±y2+ax+by+c = xyz,

thereby generalizing (2), (1), and (4).

1. Introduction. If M is any finite set we define a chain on M as a mapping f of M into the set of ordinary integers. If a ∈ M then f(a) is the coefficient of a in the chain f. The set of all a ∈ M such that f(a) ≠ 0 is the domain |f| of f. If |f| is null, that is if f(a) = 0 for all a, then f is the zero chain on M. If M is null it is convenient to say that there is just one chain, a zero chain, on M.

Introduction. Let G denote the multiplicative group of matrices

where a, b, c, d are integers and ad — bc = 1. G is one of the well-known modular groups. Let G0(n) denote the subgroup of G characterized by c ≡ 0 (mod n), where n is a positive integer. In this note we determine the normalizer of G0(n) in G, denoted by .

1. Introduction. In general when symmetric matrices are considered, the elements of the matrix are taken at least in a principal ideal ring. It is interesting to determine what can be attained when the elements are not, in general, commutative and, to this end, the following is concerned with symmetric matrices with elements in the non-commutative field of real quaternions. At the same time some properties of real quaternion unitary matrices are obtained which involve symmetric matrices.

By a submethod of a regular matrix method A we mean a method (see 1 or 3) whose matrix is obtained by deleting a set of rows from the matrix A. We establish a one-one correspondence between the submethods of A and the points of the interval 0 < ξ ≤We designate the submethod which corresponds to 𝞷 by A (𝞷) and are accordingly able to speak of sets of submethods of measure 0, of the first category, etc.

In this note it is proved that: the functional equation

(1) f(f(z)) = g(z),

where g{z) is an entire function of finite order, which is not a polynomial, and which takes on a certain value p only a finite number of times, does not have a solution f(z) which is an entire function.

In an earlier paper (3) we discussed at some length a certain inversion operator for the Laplace transformation. If

I

then the inversion operator is given by

II ,

and we showed that if v > — 1, then under certain conditions

.

1. Introduction. In a joint paper with Leighton (2), the author considered quadratic functionals of the type

1.1 (0 < a < b)

in which x = 0 is a singular point of the functional which is otherwise regular on [0, b]. The hypothesis on a regular functional includes the assumption that r is continuous and positive on a closed interval [0, b].

Let F(x) be an infinitely divisible distribution and let ϕ(t) be its characteristic function. As is well known according to the formula of Lévy and Khintchine, ϕ(t) has the following representation:

1

where γ is a real constant and G(u) is a bounded nondecreasing function.

1. Introduction. Let F, G (and all capital letters to be used below) denote n by n matrices the elements of which are real-valued continuous functions on an interval a ≤t ≤ b. Correspondingly, by a solution x = x(t) of a differential system

1,

with det G(t) ≠ 0, or of a differential system

2,

will be meant a (vector) solution all n components of which are real-valued.

1. The Theorem of Le Roux. Let U(x, y, α) be any solution of the linear hyperbolic differential equation

1

containing a parameter α. J. Le Roux (5) has shown that the function u(x, y) defined by

2 α0 = const.

Introduction. We study linear normal hyperbolic partial differential equations of the second order, with one dependent variable u, and N independent variables xi (i = 1, … , N). The uniqueness theorem connected with the Cauchy problem for this type of equation is well known and in effect states that if u and its first normal derivatives vanish on a spacelike initial surface S then u vanishes in a certain conical region which contains S (1, p. 379).

1. Introduction. In this paper we study the asymptotic behavior in λ of the solutions about the origin in the z-plane of the differential equation

.

Both the variable z and the parameter λ are complex. The coefficient P(z, λ) is assumed to be analytic and single-valued in λ at infinity and in z throughout a bounded, closed, simply connected domain D containing z = 0.

Introduction. This paper is a generalization of the classification of the differentiable points in the conformai plane given in (1). The main tools are the intersection and support properties of all the spheres through a differentiable point of an arc in conformal 3-space.

1. Introduction. A polarity in complex projective space of two dimensions (S2) is completely determined by a self-polar triangle ABC, and a pair of corresponding elements: a point P and its polar line p. We denote the polarity by (ABC) (Pp). We follow Coxeter (2) in denning a pencil of polarities as the ∞1 polarities (ABC) (Pp) where A, B, C, P are fixed while p varies in a pencil of lines.

Introduction. Consider a differentiable manifold M and the tangent bundle T(M) over M, the structure group of which is usually the general linear group G'. Let P' be the principal fibre bundle associated with T(M). Consider the fibre F of T(M) as an affine space, then we have acting on F the affine transformation group G, which contains G' as the isotropic subgroup.

1. Introduction. A well-known inequality of Hardy-Littlewood reads as follows (4): if p > 1 and f > 0, then

,

where is defined as the supremum of the numbers

the constant depends on p only. The statement obtained by putting p = 1 is false; its substitute reads:

the constants depend on p but not on f.

The present paper, an edited excerpt from my dissertation, arose from the suggestion of S. Bochner that I try to extend the maximal theorem of Hardy and Littlewood

(2) to functions analytic in the solid unit hypersphere