Let G be a compact abelian group and E a subset of its dual group Γ. A function ƒ ∈ L1(G) is called an E-function if for all γ ∉ E where

dx is the Haar measure on G. A trigonometric polynomial that is also an E-function is called an E-polynomial.

In this paper we generalize to Riemann surfaces a theorem of Helson and Lowdenslager in (2) describing the closed subspaces of L2(﹛|z| = 1﹜) that are invariant under multiplication by eiθ.

Let R be a region on a Riemann surface with boundary Γ consisting of a finite number of disjoint simple closed analytic curves such that R ⋃ Γ is compact and R lies on one side of Γ. Let dμ be the harmonic measure on Γ with respect to a fixed point t0 on R. We shall consider the closed subspaces of L2(Γ, dμ) that are invariant under multiplication by functions in A (R) = ﹛F|F continuous on , analytic on R}.

Let S be a ring, and let (ei) be an orthogonal system of a finite number of idempotents. Then e = Σei has the following properties:

(i) Se Σ Sei and eS = Σ ei S.

(ii) The mappings v: Se → Π Sei and w: eS → Π ei S defined by v(x) = [xei] and w(x) = [ei x] respectively are isomorphisms.

Next assume that (ei)i∈I is a set of idempotents indexed by a totally ordered set I such that ei ej = 0 for every i < j. If I is finite, it is evident that

has the above two properties.

It is well known that, on the category of finite polyhedra, any two cohomology theories, satisfying the Eilenberg-Steenrod axioms, are isomorphic. Examples of such theories are simplicial cohomology and homotopical cohomology (the latter is defined by means of homotopy classes of maps into Eilenberg-MacLane spaces). In the case of polyhedra, using triple sequences and spectral sequences, one obtains a deep insight into the relationship between general cohomology theories (without the dimension axiom) and ordinary simplicial cohomology (1, p. 66). As a corollary the abovementioned uniqueness of cohomology theories satisfying the dimension axiom is obtained.

The martingale convergence theorem was first proved by Doob (3) who considered a sequence of real-valued random variables. Since various collections of real-valued random variables can be regarded as vector lattices, it seems of interest to prove the martingale convergence theorem in an arbitrary vector lattice. In doing so we use the concept of order convergence that is related to convergence almost everywhere, the type of convergence used in Doob's theorem.

Let A be a finite-dimensional (associative) algebra over an arbitrary field F. We shall say that a semi-group S is a translate of A if there exist an algebra B over F and an epimorphism ϕ: B → F such that A = 0→-1 and S = 1→-1. It is shown in (2) that any such semi-group S has a kernel (defined below) that is completely simple in the sense of Rees. Following Stefan Schwarz (4), we define the radical R(S) of S to be the union of all ideals I of S such that some power In of I lies in the kernel K of S. First we prove that the radical of a translate of A is a translate of the radical of A. It follows that A is nilpotent if and only if it has a translate S such that R (S) = S.

In (3, §2), the writer and F. E. Browder stated briefly, without proof, some results concerning general distribution kernels. It is our aim here to prove and complete those results.

The terminology and notations are introduced in §1.

In §2 we define the notion of domain of dependence with respect to the kernel Kx,y (Definition 1) as well as the notion of smoothness of a distribution kernel at a point (Definition 2). Theorem 1 states that the set of points, where the distribution kernel is smooth, is open and the kernel is a smooth function in this set. Theorems 2 and 3 are the converse of Theorem 1.

The treatment of semigroups given in a previous paper (3) is based upon representations of a semigroup by means of transformations of a set (cf. also 12). In this paper we try to remove the assumption of the existence of a zero element proposed in (3). In accordance with our general programme explained at the beginning of (3) we utilize certain minimum conditions in order to gain more information on the structure of semigroups.

Our main results are structure theorems on primitive semigroups which have irreducible right ideals generated by idempotents (§§15-17). As we have shown in (5), these theorems permit the explicit construction of primitive semigroups.

Let R° denote a space consisting of just one point and for each positive integer n let Rn denote euclidean n-space. For each non-negative integer n let Sn denote the n-sphere

In 1933 K. Borsuk published proofs of the following two theorems (2, p. 178).

The Laplace-Stieltjes transform f(x) of the function α(t) is denned by

1

where α(t) is a function of bounded variation in each closed interval [0, R] (R > 0) and the right-hand side of (1) is supposed to be convergent for some x = X0. The Laplace transform f(x) of the function ϕ(t) is defined by

2

where ϕ(t) ∈ L1(0, R) for each R > 0 and the right-hand side of (2) is supposed to be convergent for some x = x0.

Let G be a finite graph topologically embedded in the interior of a 3-manifold M. Doyle (4) and Debrunner and Fox (3) have noted that the following local homotopy condition at each point p ∈ G is necessary in order for the embedding of G to be tame:

For each sufficiently small open set U containing p, there is an open set V such that p ∈ V ⊂ U and if W is any connected open set such that p ∈ W ⊂ V, then the image under the inclusion homomorphism i*: π1(W — G) → π1(U — G) is a free group on n — 1 generators.

The present paper is an attempt to find the unifying principle of results obtained by different authors and dealing—in the original papers—with areabisectors, chords, or diameters of planar convex sets, with outwardly simple planar line families, and with chords determined by a fixed-point free involution on a circle. The proofs in the general setting seem to be simpler and are certainly more perspicuous than many of the original ones. The tools required do not transcend simple continuity arguments and the Jordan curve theorem. The author is indebted to the referee for several helpful remarks.

The Szegö polynomials are defined on T, the real numbers modulo 1. In this paper and in its sequel we give a generalization of Szegö polynomials in which T is replaced by an arbitrary locally compact abelian group θ on whose dual there has been distinguished a measurable order relation compatible with the group structure. The present paper is devoted to the case where θ is compact and therefore discrete. The general case will be taken up in the sequel mentioned above. It is desirable to proceed in this way because the case θ compact is much simpler and much more like the classical situation than is the general case, in which various measure-theoretic difficulties obtrude. Moreover, as it happens, it is possible to develop the theory in this way with relatively little repetition.

Let F = GF(g) be the finite field of q = pr elements, p arbitrary. We wish to consider the system of bilinear equations

1.1

where all coefficients are from F. The number of solutions in F of a single bilinear equation may be obtained from a theorem of John H. Hodges (3, Theorem 3) by properly defining the matrices U, V, A, B. In 1954, L. Carlitz (1) obtained, as a special case of his work on quadratic forms, the number of simultaneous solutions in F of (1.1) when all aj = 1 and p is odd. Carlitz considered the case p = 2 separately.

In a recent paper (7) Yood developed the beginnings of a theory of modular annihilator algebras. In this paper we extend his work on these algebras.

The definition of modular annihilator algebra is algebraic in nature (see §4) ; in fact the algebra need not be assumed even topological. However, a significant number of important normed algebras are modular annihilator algebras. A list of examples is given in §8.

The theory of modular annihilator algebras is related to the theory of certain important topological algebras. In §5 we consider the relationships between dual and annihilator algebras and modular annihilator algebras, and in §7, the relationship between completely continuous normed algebras and modular annihilator algebras.

In this paper a simplified proof of a theorem of Sanov (4) is given. No mention is required of Lie elements or of the Baker-Hausdorff formula, both of which played central roles in Sanov's proof.

Let G be a group. Define, for all x, y ∈ G,

A loop L has the isotopy-isomorphy property provided each loop isotopic to L is isomorphic to L. A familiar problem is that of characterizing those loops having this property.

It is well known (1, p. 56) that the loop isotopes of (L, ·) are those loops L(a, b, *) defined by x * y = x/b·a\y for some a, b in L. In this paper we first show (Corollary to Theorem 1) that a loop L with identity element 1 has the isotopy-isomorphy property if L is isomorphic to 1,(1, x) and to L(x, 1) for each x in L. We then determine necessary and sufficient conditions (Theorems 2 and 3) for L to be isomorphic to these isotopes under translations (i.e. permutations of the form xv = cx or xv = xc for c fixed).

If ﹛Ln﹜ is a sequence of linear functionals on a linear space C of functions to the complex numbers, then a subspace C1 ⊂ C is a uniqueness class for ﹛Ln﹜ if a function f in C1 is uniquely determined by the sequence ﹛Ln(f)﹜ of complex numbers; i.e., if f ∈ C1 and Ln(f) = 0, n = 0, 1, 2, … , implies f = 0. For example, the class of all functions f analytic at the origin is a uniqueness class for the sequence ﹛f(n)(0)﹜ of linear functionals. Gontcharoff (9) asked the following question: Suppose, instead of ﹛f(n)(0)﹜, we use ﹛f(n)(an)﹜.

Let A be an n X n non-negative matrix, that is, a matrix whose entries are non-negative numbers. The permanent of A is the scalarvalued function of A defined by

where the summation extends over all permutations i1 … , in of the integers 1, … , n. The purpose of this paper is to prove several inequalities involving the permanent of A and the permanent of submatrices of A when suitable restrictions are placed on the row sums.

It is assumed that all topological spaces discussed in this paper are Hausdorff. By a compactification αX of a space X we mean a compact space containing X as a dense subspace. If, for some positive integer n, αX — X consists of n points, we refer to αX as an n-point compactification of X, in which case we use the notation αn X. If αX — X is countable, we refer to αX as a countable compactification of X. In this paper, the statement that a set is countable means that its elements are in one-to-one correspondence with the natural numbers. In particular, finite sets are not regarded as being countable. Those spaces with n-point compactifications were characterized in (3). From the results obtained there it followed that the only n-point compactifications of the real line are the well-known 1- and 2-point compactifications and the only n-point compactification of the Euclidean N-space, EN (N > 1), is the 1-point compactification.