Introduction. The natural problem of determining all the Lie algebras of finite dimension was broken in two parts by Levi's theorem:

1) the classification of semi-simple Lie algebras (achieved by Killing and Cartan around 1890)

2) the classification of solvable Lie algebras (reduced to the classification of nilpotent Lie algebras by Malcev in 1945 (see [10])).

The Killing form is identically equal to zero for a nilpotent Lie algebra but it is non-degenerate for a semi-simple Lie algebra. Therefore there was a huge gap between those two extreme cases. But this gap is only illusory because, as we will prove in this work, a large class of nilpotent Lie algebras is closely related to the Kac-Moody Lie algebras. These last algebras could be viewed as infinite dimensional version of the semisimple Lie algebras.

Throughout this paper R will be a commutative ring with 1. The purpose of this paper is to provide two new characterizations of coherent rings. The first of these characterizations shows that the class of coherent rings is precisely the class of rings for which certain duality homomorphisms are isomorphisms. And the second of these characterizations shows that the class of coherent rings is precisely the class of rings for which the endomorphism ring of any infective module is a flat module. We can show as a consequence that the endomorphism ring of a universal infective R-module is a faithfully flat R-module whenever R is a coherent ring.

Let M be a von Neumann algebra acting on a Hilbert space and assume that M has a separating and cyclic vector ω in . Then it can happen that M contains a proper von Neumann subalgebra N for which ω is still cyclic. Such an example was given by Kadison in [4]. He considered and acting on where is a separable Hilbert space. In fact by a result of Dixmier and Maréchal, M, M′ and N have a joint cyclic vector [3]. Also Bratteli and Haagerup constructed such an example ([2], example 4.2) to illustrate the necessity of one of the conditions in the main result of their paper. In fact this situation seems to occur rather often in quantum field theory (see [1] Section 24.2, [3] and [4]).

1. Introduction. An incomplete latin rectangle on t symbols σ1 …, σt of size r × s is an r × s matrix in which each cell is occupied by exactly one of the symbols σ1 …, σt in such a way that no symbol occurs more than once in any row or more than once in any column. If r = t or s = t then it is a latin rectangle; if r = s < t it is an incomplete latin square; if r = s = t it is a latin square. The diagonal of a latin square consists of the cells (i, i) (1 ≦ i ≦ t) together with the symbols occupying those cells. Let an allowed sequence of length t be a sequence of length t in which no symbol occurs exactly t – 1 times. Let an allowed diagonal of length t be a diagonal occupied by an allowed sequence.

Introduction. All spaces considered in this paper are assumed to be (Hausdorff) completely regular, and all maps are continuous. Let be a topological property of spaces. We shall identify with the class of spaces having . A space having is called a -space, and a subspace of a -space is called a -regular space. The class of -regular spaces is denoted by R(). Following [37], we call a closed hereditary, productive, topological property such that each -regular space has a -regular compactification a topological extension property, or simply, an extension property. In this paper, we restrict our attention to extension properties satisfying the following axioms:

(A1) The two-point discrete space has .

(A2) If each -regular space of nonmeasurable cardinal has , then = R().

We take the definition of a Witt ring to be that given in [13], i.e., it is what is called a strongly representational Witt ring in [8]. The classical example is obtained by considering quadratic forms over a field of characteristic ≠ 2 [17], but Witt rings also arise in studying quadratic forms or symmetric bilinear forms over more general types of rings [5,7, 8, 9]. An interesting problem in the theory is that of classifying Witt rings in case the associated group G is finite. The reduced case, i.e., the case where the nilradical is trivial, is better understood. In particular, the above classification problem is completely solved in this case [4, 12, or 13, Corollary 6.25]. Thus, the emphasis here is on the non-reduced case. Although some of the results given here do not require |G| < ∞, they do require some finiteness assumption. Certainly, the main goal here is to understand the finite case, and in this sense this paper is a continuation of work started by the second author in [13, Chapter 5].

1. Introduction. Let be a probability space with standard. Let T be a bimeasurable one-to-one map of Ω onto itself. Let U: Ω → Ω be another measurable transformation whose orbits are contained in the T-orbits; that is,

where Z denotes the set of integers. (This is equivalent to saying that there is a measurable mapping L: Ω → Z such that U(ω) = TL(ω)(ω), ω ∈ Ω.) Such pairs (T, U) arise quite naturally in ergodic theory and information theory. (For example, in ergodic theory, one can see such pairs in the study of the full group of a transformation [1]; in information theory, such a pair can be associated with the input and output of a variable-length source encoder [2] [3].) Neveu [4] obtained necessary and sufficient conditions for U to be measure-preserving if T is measure-preserving. However, if U fails to be measure-preserving, U might still possess many of the features of measure-preserving transformations.

Introduction. In [1] we defined a capacity in Cn. Recently Molzon, Shiffman and Sibony [8] have introduced a different capacity which is useful for certain Bezout estimates. The object of this note is to apply the methods of [1] to study the capacity of [8]. We shall obtain an equivalent definition of this capacity via Tchebycheff polynomials, along the lines of [1]. Half of this equivalence was independently obtained by Sibony [9].

To establish the full equivalence of these two approaches to capacity a notion of Jensen measures in a setting more general than uniform algebras is needed. We shall consider Jensen measures for multiplicative semigroups; these are sets of functions in which only the multiplicative structure is postulated. It will also be useful to generalize the notion of polynomial hull in Cn to a hull with respect to a multiplicative semigroup of polynomials. We can then adapt the approach of [1] to these semigroups.

1. Let X be a topological space, and let X′ be the set of all non-convergent ultrafilters on X. If A ⊆ X, let , and A* = A ∪ A′. If is a filter on X such that for all , then let. be the filter on X* generated by ; let be the filter on X* generated by . If exists then ; otherwise, .

A convergence is defined on X* as follows: If x ∈ X, then a filter A → x in X* if and only if , where Vx(x) is the X neighborhood filter at x; , then in X* if and only if .

1. Introduction. This paper deals with the lack of uniqueness of the representations of the complex numbers in positional notation using Gaussian integers as bases.

Kátai and Szabó [3] proved that all the complex numbers can be written in radix form using the base –n + i with the natural numbers 0, 1, 2, …, n2 as digits. They remarked that they did not assert the uniqueness of these representations but gave no further indications of any multiple expansions. The geometry of these complex bases [2] indicates that some numbers have two expansions in a given base, while a few numbers even have three different expansions.

Let KG be the group ring of a group G over a field K, and U its group of units. Given a group H, we shall denote by ξ(H) the center of H and by T(H) the set of all its torsion elements.

The following question appears in [5, p. 231]: When is Un ⊂ ξ (U), for some n? It was considered by G. Cliff and S. K. Sehgal in [1], where G is assumed to be a solvable group. A complete answer at characteristic zero is given there. Also they obtain partial results at characteristic p ≠ 0, with certain restrictions on the exponent n.

1. Introduction. Many problems involving prime numbers depend on estimating sums of the form ΣΛ(n)f(n), for appropriate functions f(n), (here, as usual, Λ(n) is the von Mangoldt function). Three distinct general methods have been used to estimate such sums. The earliest is due to Vinogradov (see [13, Chapter 9]); the second involves zerodensity bounds for Dirichlet L–functions (see [8, Chapters 15 and 16] for example); and the third, due to Vaughan (see [12] for example) uses an arithmetical identity as will be explained later. The second and third methods are much simpler to apply than the first. On the other hand Vinogradov's technique is at least as powerful as Vaughan's and occasionally more so. In many cases Vaughan's identity yields better bounds than the use of zero–density estimates, but sometimes they are worse. The object of this paper is to present a simple extension of Vaughan's method which is essentially as powerful as any of the techniques mentioned above, to discuss its general implications, and to apply it to the proof of the following result of Huxley [4], which has previously only been within the scope of the zero density method.

0. Introduction. A point p ∈ βX\X is called a remote point of X if P ∉ clβXA for each nowhere dense subset A of X. If X is a topological sum Σ{Xn : n ∈ ω} we call nice if {n : F ∩ Xn = ∅} is finite for each . We call remote if for each nowhere dense subset A of X there is an with F ∩ A = ∅ and n-linked if each intersection of at most n elements of is non-empty.

Let dndn–1 … d0 be the b-ary representation of a positive integer N. Call the polynomial obtained from N base b. In the case the base is 10, f(x) will be called the polynomial obtained from N. Pólya and Szegö attribute the following theorem to A. Cohn [2, b. 2, VIII, 128]:

THEOREM 1. A polynomial obtained from a prime is irreducible.

This theorem was generalized in two different ways by John Brillhart, Andrew Odlyzko, and myself [1]. One way was by proving the theorem remains true regardless of the base being used. The second way was by permitting the coefficients of f(x) to be different from digits. Thus, for example, if , where 0 ≦ dk ≦ 167 for all k, and if f(10) is prime, then f(x) is irreducible. In this paper, Theorem 1 will be generalized in another way by considering composite N.