The equation Px = y in Banach spaces has aroused considerable interest, particularly in view of the various situations in applied analysis which it encompasses, and consequently it has been the topic of numerous investigations (2; 9; 10; 12). Detailed references may be found in (10). The equation is of special interest because of its interpretation as an integral equation; and in turn, many problems related to differential equations can be reformulated as integral equations (5; 7; 13).

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)

The only known examples of nondegenerate homogeneous plane continua are the simple closed curve, the circle of pseudo-arcs (6), and the pseudo-arc (1; 13). Another example, called the pseudo-circle, has been suggested by Bing (2), but it has not been proved to be homogeneous. (Definitions of some of these terms and a history of results on homogeneous plane continua can be found in (6).) Of the three known examples, the pseudo-arc is both linearly chainable and circularly chainable, and the simple closed curve and the circle of pseudo-arcs are circularly chainable but not linearly chainable. It is not known whether every homogeneous plane continuum is either linearly chainable or circularly chainable. Bing has shown that a homogeneous continuum is a pseudo-arc provided it is linearly chainable (4).

La première partie de cet article a un caractère introductif; on reprend l'étude des mesures vectorielles sur un espace localement compact, présenté dans (8) pour les espaces compacts.

Dans la seconde partie on donne un théorème de représentation intégrale d'une mesure vectorielle par rapport à sa variation (théorème 2), duquel on déduit une généralisation du théorème de Lebesgue-Nikodym (théorème 3) et une généralisation du théorème de Lebesgue sur la décomposition d'une mesure (théorème 5). Ces théorèmes ont été déjà démontrés par les auteurs sous des conditions un peu plus restrictives, (10) et (13), mais ici les démonstrations sont nouvelles, et l'ordre dont on les déduit l'un de l'autre, est inversée.

In a previous paper (18), G = F/Fn was studied for F a free product of a finite number of cyclic groups, and Fn the normal subgroup generated by commutators of weight n. In that paper the following cases were completely treated:

(a) F a free product of cyclic groups of order pαi, p a prime, αi positive integers, and n = 4, 5, … , p + 1.

(b) F a free product of cyclic groups of order 2αi, and n = 4.

In this paper, the following case is completely treated:

(c) F a free product of cyclic groups of order pαi p a prime, αi positive integers, and n = p + 2.

(Note that n = 2 is well known, and n — 3 was studied by Golovin (2).) By ‘'completely treated” is meant: a unique representation of elements of the group is given, and the order of the group is indicated. In the case of n = 4, a multiplication table was given.

Throughout this discussion R will be an integral domain with quotient field Q and K = Q/R ≠ 0. If A is an R-module, then A is said to be torsion-free (resp. divisible), if for every r ≠ 0 ∈ R the endomorphism of A defined by x → rx, x ∈ A, is a monomorphism (resp. epimorphism). If A is torsion-free, the rank of A is defined to be the dimension over Q of the vector space A ⊗R Q; (we note that a torsion-free R-module of rank one is the same thing as a non-zero R-submodule of Q). A will be said to be indecomposable, if A has no proper, non-zero, direct summands. We shall say that A has D.C.C., if A satisfies the descending chain condition for submodules. By dim R we shall mean the maximal length of a chain of prime ideals in R.

In this paper, A denotes a ring with an identity element 1, and B a subring of A containing 1 such that B satisfies the left and right minimum conditions, and A is a finitely generated left and right B-module. The identity element 1 is required to act as the identity operator on all modules which we shall consider. For any left B-module V, there is a standard construction of a left A -module which is, roughly speaking, the smallest A -module containing V.

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.

It is well known that any set of four anticommuting involutions (see §2) in a four-dimensional vector space can be represented by the Dirac matrices

(1)

where the B1,r are the Pauli matrices

(2)

(See (1) for a general exposition with applications to Quantum Mechanics.) One formulation, which we shall call the Dirac-Pauli theorem (2; 3; 1), is

Theorem 1. If M1,M2, M3, M4 are 4 X 4 matrices satisfying

then there is a matrix T such that

and T is unique apart from an arbitrary numerical multiplier.

This paper is primarily concerned with the existence of solutions of the fourth-order self-adjoint differential equation

(1)

(where r(x) > 0, q(x) ≥ 0, p(x) ≥ 0 and all three coefficients are continuous on [a, ∞)) and one of the two-point boundary conditions:

(2)

or

(3)

the subscript notation for any solution y(x) denoting:

(4)

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).

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.

Let f be a function on the product space V × W, where V and W are analytic manifolds, both either real or complex. The function f is said to be analytic (or bi-analytic) on V × W if it is analytic in the analytic structure induced on V × W by the corresponding structures on V and W. The function f is said to be separately analytic on V × W if, for each x in V, the function f(x, .) is analytic on W while, for each y in W, the function f (. ,y) is analytic on V. In the case of complex analytic manifolds, the classical theorem of Hartogs (3, chapter VII) states that the two notions of analyticity and separate analyticity are equivalent. For real analytic manifolds, it is known that such an equivalence does not hold, even if one adds the additional hypothesis that f is infinitely differentiate on V × W.

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 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.)

Various results exist which permit real Banach algebras satisfying some sort of “reality condition” to be identified with the algebra of all continuous real-valued functions on a suitable compact space (or with the algebra of all continuous real-valued functions that * Vanish at infinity“ on a suitable non-compact, locally compact space in case the algebra has no unit). (Terminology follows (4), so that compact and locally compact spaces must be Hausdorff spaces, in addition to satisfying the usual requirements.) Kadison established such a result in (6, Theorem 6.6) for any Banach algebra with unit, if the algebra satisfies an appropriate reality condition and has a norm N such that N(x)2 — N(x2) for all x.

A graph on n labelled nodes is a set of n objects called “nodes”, distinguishable from each other, and a set (possibly empty) of “edges,” that is, pairs of nodes. Each edge is said to join its pair of nodes, at most one edge joins any two nodes and no edge joins a node to itself. By a k-colouring of such a graph we mean a mapping of the nodes of the graph onto a set of k distinct colours, such that no two nodes joined by an edge are mapped onto the same colour. We take k > 1.

Let C be a square matrix with complex elements. If C = C' (C' denotes the transpose of C) there exists a unitary matrix U such that

(1)

where the μ's are the non-negative square roots of the eigenvalues μ12, μ22, … , μn2 of C*C (C* is the adjoint of C) (2). If C is skew-symmetric, that is, C= — C”, there exists a unitary matrix U such that

(2)

where

(3)

and the α's are the positive square roots of the non-zero eigenvalues α12, α22, … , αk2 of C*C (1). Clearly rank C = 2k and the number of zeros appearing in (2) is n — 2k. Both (1) and (2) are classical. In a recent paper (3) Stander and Wiegman, apparently unaware of (1), give an alternative derivation of (2) with its appropriate generalization to quaternions.