Let G be a locally compact separable unimodular group. The general theory [18] assigns to G a measure space (Λ, μ) whose points ƛ index a family of unitary factor representations of G in such a way that if U ƛ corresponds to ƛ and then

for all .

In an earlier paper [7], we have studied the existence, uniqueness and asymptotic behavior of solutions to certain infinite systems of linear differential equations with constant coefficients. In the present paper we are interested in systems of nonlinear equations whose coefficients are not necessarily constants; more specifically, we are concerned with infinite systems of the form

The existence of nonzero fixed points of positive contractions in L1 spaces has received considerable attention in recent years. In 1966, Dean and Sucheston [1] and independently, Neveu [5] showed that a positive contraction has a strictly positive invariant function if and only if for any measurable subset A with positive measure, where 1 is the constant function of value one.

A Latin K-cube A of order n is a k-dimensional array where

runs through the distinct elements 0, 1, …, n — 1 as j runs from 0 to n — 1.

The notion of pure subgroups is due to Prufer [7]. It has proven extremely useful in establishing structural properties of abelian groups. In a recent paper [9], Waterhouse introduced the concept of a pure subfield of a purely inseparable extension. Let L be a purely inseparable modular extension of k, and let K be an intermediate field. K is called pure if K and k(Lpn) are linearly disjoint over k(Kpn) for all n. Waterhouse used this concept to establish the existence of basic subfields [9].

The graphs considered in this paper are finite and have no loops or multiple edges. If G is such a graph, we denote its vertex set by VG and its edge set by EG. If X and Y are disjoint subsets of VG, we define δ (X, Y) to be the set of edges of G that join a vertex in X to one in Y.

Section 1 of this paper is concerned with the effect on conjugate and σ-points of various perturbations of q(x) for differential equations of the form

If the Lp spaces of two measure spaces are “the same”, to what extent can we identify the underlying measure spaces? This question has been partially answered by Schneider [7] (extending results of Forelli [2]). He proves that a linear isometry between the Lv spaces of two finite measure spaces is in fact an (isometric) homomorphism between the corresponding L∞ spaces, if it preserves the identity.

A Steiner Triple System (STS) is a pair (P, B) where P is a set of points and B is a set of 3-elenient subsets of P called blocks (or triples) such that for distinct p, q ∈ P there is a unique block b ∈ B with ﹛p, q) ⊂ b. There are two well-known methods for turning Steiner Triple Systems into algebras; both methods are due to R. H. Bruck [1]. Each method gives rise to a variety of algebras; in this paper we will study these varieties.

The purpose of this paper is to list all of the characters of An, the alternating group, mentioned in the title. The same problem for the symmetric group, Sn, was dealt with by the authors in [1]. We showr here that, apart from a few exceptions, the imprimitive, irreducible complex characters of An fall naturally into two infinite families. (Throughout this paper characters are taken over the complex numbers.)

The class of primitive recursive ordinal functions (PR) has been studied recently by numerous recursion theorists and set theorists (see, for example, Platek [3] and Jensen-Karp [2]). These investigations have been part of an inquiry concerning a larger class of functions; in Platek's case, the class of ordinal recursive functions and in the case of Jensen and Karp, the class of primitive recursive set functions. In [4] I began to study PR in depth and this paper is a report on an attractive analogy between PR and its progenitor, the class of primitive recursive functions on the natural numbers (Prim. Rec).

Let ƒ be a function meromorphic in the finite complex plane C. If for some number θ, 0 ≦ θ < 2 π, the family, fr(z) = f(reθz), is not normal at z = 1, then the ray arg z = θ is called a Julia ray. Such a ray has the property that in every sector containing it, F assumes every value infinitely often with at most two exceptions. Many authors have taken this property as the definition of a Julia ray.

A matrix H of order n = 4t with all entries from the set ﹛1, —1﹜ is Hadamard if HHt = 4tI. The set of Hadamard matrices is . A matrix is of type I or is skew-Hadamard if H = S — I where St = —S (some authors also use H = S + I). The set of type I members is . A matrix P is a signed permutation matrix if each row and each column has exactly one non-zero entry, and that entry is from the set ﹛1, —1﹜.

Rates of convergence, saturation theorems and the socalled “inverse problems” for Bernstein polynomials have been intensively studied (see, e.g., [1 ; 4; 8; 14; 17]). The same problems for some other positive operators have also been investigated by many authors. In this paper, we shall use a uniform approach to study the saturation and inverse problems for a class of linear combinations of operators including Bernstein polynomials, and Szâsz, Post-Widder, Gauss-Weierstrass and Baskakov operators.

A neofield of order v, Nv( + , •), is an algebraic system of v elements including 0 and 1,0 ≠ 1, with two binary operations + and • such that (Nv, + ) is a loop with identity element 0; (Nv*, •) is a group with identity element 1 (where Nv* = Nv\﹛0﹜) and every element of Nv is both right and left distributive (i.e., (y + z)x = yx + zx and x(y + z) = xy + xz for all y, z∈ Nv).

Let Ω be a field and Γ a parameter. We designate the set of all polynomials homogeneous in (X) = (X1, … , Xn) with coefficients in Ω [Γ] by H Ω Γ[X] and write such polynomials as F, F(X), or F(X, Γ). The degree of a polynomial in H Ω Γ [X] shall mean the degree in (X). Let I = (F1 … , Fr) be a fixed ideal in H Ω Γ [X] generated by F1 … , Fr.

The central theorem of this note was proposed to me by I. Juhâsz on a napkin in Canter's restaurant in Pittsburgh, Pennsylvania. We agreed that if the proposition could be proven it would have some elegant applications and although it required a slight change before it could be proven, hopefully those expectations will be realized in this note.

Let A = ﹛a0, a1, … , a3﹜ be a set of relatively prime integers such that 0 < a0 < a1 < … < as = n. Let ϕ (A) denote the smallest integer such that, for N ≧ ϕ (A), the equation

should always have a solution in nonnegative integers.

Let C denote an irreducible, algebraic curve defined over an algebraically closed field k. Let ? be a singular point of C. We shall employ the following notation throughout the rest of this paper: R will denote the local ring at P, K the quotient field of the integral closure of R in K, A the completion of R with respect to its radical topology, and Ā the integral closure of A in its total quotient ring.

In [1], Bachmuth and Mochizuki conjecture, by analogy with a celebrated result of Tits on linear groups [8], that a finitely generated group of automorphisms of a finitely generated soluble group either contains a soluble subgroup of finite index (which may of course be taken to be normal) or contains a non-abelian free subgroup. They point out that their conjecture holds for nilpotent-by-abelian groups and in some other cases.