Let K be a subset of BV(0, 1)—the space of functions of bounded variation on the closed interval [0, 1]. By the Hausdorff moment problem for K we shall mean the determination of necessary and sufficient conditions that corresponding to a given sequence μ = {μn|n = 0, 1, 2, …} there should be a function α ∈ K so that

(1)

For various collections K this problem has been solved—see (3, Chapter III)

By the trigonometric moment problem for K we shall mean the determination of necessary and sufficient conditions that corresponding to a sequence c = {cn|n = 0, ± 1, ± 2, …} there should be a function α ∈ K so that

(2)

For various collections K this problem has also been solved—see, for example (4, Chapter IV, § 4). It is noteworthy that these two problems have been solved for essentially the same collections K.

Many extremal properties of the class of normalized univalent functions are shared by the class of typically real functions each considered in the unit circle. By the class T of typically real functions we mean those functions f(z), regular for |z| < 1 with f(0) = 0, f'(0) = 1, and such that f(z) > 0 for z > 0, (z) < 0 for > 0. This class was first studied by Rogosinski (4) who proved various simple properties for it. Later Robertson (3) took up the study and proved the following important representation result.

If m, n are integers, “m = n” will mean “m = n (mod 2 ).“ The cardinal number of a set A will be denoted by |A|. The set whose elements are a1 a2, … , an will be denoted by {a1, a2, … , an}. The empty set will be denoted by Ø. If A, B, C are sets, A — B will denote the set of those elements of A which do not belong to B, and A — B — C will denote (A — B)— C. The expression Σε∈Af(ε) will be denoted by f. A. The statements “f = g on A,” “f ≡ g on A” will mean that f(ε) = g(ε) or f(ε) ≡ g(ε) respectively for every ε ∈ A.

Let be the Boolean algebra of all finite unions of subcells of the plane. Denote by εpthe algebra of all linear bounded transformations of Lp(— ∞, ∞) into itself. Suppose for a moment that p = 2, and let Rp be an involutive abelian subalgebra of εp if Rp is also a Banach space and if Tp ∈ Rp, then:

(i) The family of all homomorphic mappings of into the algebra Rp contains a member EPT such that

(1)

We are mainly interested in the study of prime and semi-prime rings and their rings of quotients. However, our argument proceeds largely in the category of modules (§ 1 to 4) and bimodules (§ 5 to 7).

After a brief description of the generalized rings of quotients introduced recently by Johnson, Utumi, and Findlay and the present author, we study a closure operation on the lattice of submodules of a module. For the lattice of left ideals of a ring, the concept of closed submodules reduces to the If-ideals of Utumi. The lattice of closed submodules of a module is always a complete modular lattice. We are specially interested in the case when it is a complemented lattice. This happens, in particular, when the singular submodule of Johnson and Wong vanishes. We consider the lattice of closed right ideals of a prime ring S and determine the maximal ring of right quotients of S in the case when this lattice has atoms.

This paper is an extension of the ideas discussed in (3, §§ 14-16); the extension consisting of the use of the third and fourth symmetric Riemann derivative instead of the Schwarz or second symmetric Riemann derivative.

The J2integral, due to James (1), is defined in (3) as follows. Let f(x) be measurable on [a, b] and finite at each point; if there exists a continuous function F(x) such that D2F = f everywhere on (a, b),

then

1.1

The definition is unique since if F(x) and G(x) are continuous and D2F =D2G everywhere then

The Weierstrass transform f(x) of a function ϕ(y) is defined by

1.1

where

whenever this integral exists (7, p. 174). It is also known as the Gauss transform (11; 12). Its basic properties have been developed and studied in (7) and in particular it has been shown that the symbolic operator

will invert this transform under suitable assumptions and with certain definitions of this operator. We propose to study the definition

for f(x) in C∞. This formula seems to have been first examined by Pollard (9) and later by Rooney (12). In so far as convergence of (1.2) is concerned, we will considerably improve the results (12).

The purpose of this paper is to show that most results concerning polyadic algebras can be generalized to transformation algebras. The results of this paper will clearly indicate that a great deal can be done in polyadic algebras without ever mentioning the quantifier structure (for instance, terms and operations can be characterized without the help of the quantifier structure, at least in the case where an equality is present). In § 1, we develop the elementary theory; in § 2, we study the different ways of extending a (locally finite) transformation algebra (of infinite degree) to a polyadic algebra; in § 3, we study equality transformation algebras; finally, in § 4, we show how terms and operations can be defined in equality transformation algebras.

In this paper we consider complete orthonormal sequences defined on the interval [0, 1] and satisfying an inequality of the type

(1)

for all n and some sequence {Fn}. Such sequences were first considered by Zygmund and Marcinkiewicz (8). They extended the well-known results of Hausdorff-Young and Paley, originally proved for the case v = ∞, Fn = M for all n (12). We will consider cases of equality in t he Hausdorff-Young theorems and certain limiting cases of the Paley theorems. Application of these results and the results in (8) will be made to functions harmonic in the unit α-sphere.

The notion of quantum field remains at this time still rather elusive from a rigorous standpoint. In conventional physical theory such a field is defined in essentially the same way as in the original work of Heisenberg and Pauli (1) by a function ϕ(x, y, z, t) on space-time whose values are operators. It was recognized very early, however, by Bohr and Rosenfeld (2) that, even in the case of a free field, no physical meaning could be attached to the values of the field at a particular point—only the suitably smoothed averages over finite space-time regions had such a meaning. This physical result has a mathematical counterpart in the impossibility of formulating ϕ(x, y, z, t) as a bona fide operator for even the simplest fields (in any fashion satisfying the most elementary non-trivial theoretical desiderata), while on the other hand for suitable functions f, the integral ∫ϕ(x, y, zy t)f(x, y, z, t)dxdydzdt could be so formulated.

Let V be a paracompact real analytic manifold of dimension n ≥ 1. Following the terminology of the theory of distributions of Schwartz (4), is the linear space of infinitely differentiable functions with compact support in V with the appropriate inductive limit topology, is the Frechet space of infinitely differentiable functions on V, is the dual space of consisting of the distributions on V, the dual space of consisting of the distributions with compact support on V. Let 𝒰(V) be the linear space of real analytic functions on V.

In the considerable literature on linear operators in L2 or Lp arising from ordinary differential operators it has always been assumed that the coefficient of the highest order derivative appearing does not vanish in the interior of the interval under consideration. If this coefficient vanishes at one or both endpoints of the interval, or if one or both of the endpoints is infinite the differential operator is said to be singular. In this paper we shall allow this leading coefficient to vanish in the interior of the interval, and show that the theory of such operators can sometimes be reduced to a consideration of several operators of the well-known type. We shall also indicate how those which cannot be so reduced should be dealt with.

In this note we are concerned with normed algebras over a non-discrete field with absolute value. The norm, N, of a normed algebra A is said to satisfy a polynomial identity on a subset B if there is a polynomial such that P(N(x1), … , N(xr)) = N(P(x1 … , xr)) whenever x1, … , xr are in B, where the polynomial has rational integer coefficients, degree greater than 1, constant term zero, and non-negative coefficients for each term of highest degree. It is shown in Theorem 1, following a method of proof used by Kadison in (6, § 7), that if the norm of a normed algebra satisfies a polynomial identity on the entire algebra, then the norm is power multiplicative. (That is, then N(x)2 = N(x2) for all x.)

Recently several topologists have called attention to the uniform structures (in most cases, the coarsest ones) under which every continuous real function is uniformly continuous (let us call the structures the [coarsest] uc-structures), and some important results have been found which closely relate, explicitly or implicitly, to the uc-structures, such as in the vS of Hewitt (3) and in the e-complete space of Shirota (7). Under these circumstances it will be natural to pose, as Hitotumatu did (4), the problem: which are the uniform spaces with the uc-structures? In (1 ; 2), we characterized the metric spaces with such structures, and in this paper we shall give a solution to the problem in uniform spaces (§ 1), together with some of its applications to normal uniform spaces and to the products of metric spaces (§ 2). It is evident that every continuous real function on a uniform space is uniformly continuous if and only if the uniform structure of the space is finer than the uniform structure defined by all continuous real functions on the space.

In 1902, F. R. Moulton (12) gave an early example of a non-Desarguesian plane. Its ‘points” are ordered pairs (x, y) of real numbers. Its “lines” coincide with lines of the real affine plane except that lines of negative slope are “bent” on the x-axis, line {y = b + mx}, for negative m, being replaced by {y = b + mx if y ≤ 0, y = [m/2]. [x + (b/m)] if y > 0}. A certain Desarguesian configuration in the classical plane is shifted just enough to vitiate Desargues’ Theorem for Moulton's geometry. The plane is neither a translation plane (“Veblen-Wedderburn” in the sense of Hall (7), p. 364) nor even the dual of one (Veblen and Wedderburn (17). It is natural to ask if the same construction is feasible when real numbers are replaced by elements from an arbitrary field.

The problem we study is a generalization of a problem first solved by Tonolo (6), then generalized successively by Schouten (5), Nijenhuis (4), Haantjes (3), and Nijenhuis-Frölicher (2). The Tonolo- Schouten approach is distinct from that of Nijenhuis-Haantjes-Frölicher in the sense that the former consider the problem on a Riemannian space, while the latter consider it on a manifold without any further structure.

The object of investigation is the integrability of the distribution θ of vector subspaces θP of the tangent space Tp to a manifold M, when θP is intrinsically related to a given field h on M, of linear transformations hp on Tv. The research has so far been restricted to certain types of h.

The matrix spaces under consideration are the four main types of irreducible bounded symmetric domains given by Cartan (5). Let z = (zjk) be a matrix of complex numbers, z' its transpose, z* its conjugate transpose and I = I(n) the identity matrix of order n. Then the first three types are defined by

(1)

where z is an n by m matrix (n ≤ m), a symmetric or a skew-symmetric matrix of order n (16). The fourth type is the set of complex spheres satisfying

(2)

where z is an n by 1 matrix. It is known that each of these domains possesses a distinguished boundary B which in the first three cases is given by

(3)

(In the case of skew symmetric matrices the distinguished boundary is given by (2) only if n is even.)

The most important notion in topology is that of a homeomorphism f: X → Y from a topological space X onto a topological space Y. If a homeomorphism f: X → Y exists, then the topological spaces X and F are said to be homeomorphic (or topologically equivalent), in symbols,

X ≡ Y.

The relation ≡ among topological spaces is obviously reflexive, symmetric, and transitive; hence it is an equivalence relation. For an arbitrary family F of topological spaces, this equivalence relation ≡ divides /Mnto disjoint equivalence classes called the topology types of the family F. Then, the main problem in topology is the topological classification problem formulated as follows.

The topological classification problem: Given a family F of topological spaces, find an effective enumeration of the topology types of the family F and exhibit a representative space in each of these topology types.

Define f(k, l) as the least integer so t h a t every graph having f(k, l) vertices contains either a complete graph of order k or a set of l independent vertices (a complete graph of order k is a graph of k vertices every two of which are connected by an edge, a set of I vertices is called independent if no two are connected by an edge).

Throughout this paper c1, c2, … will denote positive absolute constants. It is known (1, 2) that

(1)

and in a previous paper (3) I stated that I can prove that for every ∈ > 0 and l > l(∈), f (3, l) > l2-∈. In the present paper I am going to prove that

(2)

General radical theories were obtained by Amitsur (1; 2; 3) and Kurosh (6). Following Kurosh we say that a property of rings is a radical property if:

(a) Every homomorphic image of an -ring is an -ring;

(b) Every ring R contains an -ideal S which contains every other -ideal of R;

(c) The factor ring R/S is-semi-simple (that is, has no non-zero -ideals).