Throughout this paper, we let q = 2W,﹜ w a positive integer, and for u = 1 or 2, we let GF(qu) denote the finite field of cardinality qu. Let - denote the involutory field automorphism of GF(q2) with GF(q) as fixed subfield, where ā = aQ for all a in GF﹛q2). Moreover, let | | denote the norm (multiplicative group homomorphism) mapping of GF(q2) onto GF(q), where |a| — a • ā = aQ+1.

The dimension [4] of a partially ordered set (poset) is the minimum number of linear orders whose intersection is the partial ordering of the poset. For a positive integer m, a poset is m-irreducible[10] if it has dimension m and removal of any element lowers its dimension. By the compactness property of finite dimension, every m-irreducible poset is finite and every poset of dimension ≧ m contains an m-irreducible subposet.

In their study of the uniform approximation of the reciprocal of e2 by reciprocals of polynomials on the positive real axis, Cody, Meinardus, and Varga [3] showed that if denotes the class of all polynomials of degree at most n and

then

Suppose that A and B are C*-algebras, J is a closed two-sided ideal in B, and that η: B →B/J is the quotient map. Given a linear contraction φ : A →B/J, a linear map Ψ: A →B is a lifting of φ if one has a commutative diagram

In this paper we shall consider certain rank 3 permutation groups G which act on a set Ω of size n. Thus a point stabiliser Gα will have 3 orbits { α }, △ (α), Γ (α) of sizes 1, k, I respectively. It is well known that, if |G| is even, then the orbital △ defines a strongly regular graph on Ω. In this graph, every point has valency k, every pair of adjacent points are adjacent to a constant number λ of common points, and every pair of non-adjacent points are adjacent to a constant number μ of common points. This notation is reasonably standard (see [4], where much background theory is given).

Extension theory has been intensively studied for completely regular spaces and is fairly well developed for T0-topological spaces. (See, for example, [1] and [5]). However, except for definitions of some of the basic concepts in [4] and results on embedding of closure spaces in cubes in [2] and [7], ours is the first study of the general theory of extensions of G0-closure spaces. (Definitions will be given following these introductory paragraphs).

K. Mahler [8] has proposed the following problem. Let Ωr for r ≧ 1 be a sequence of n X n non-negative rational integer matrices. Each Ωr — (ωrij) defines a map Ωr : Cn —⟶ Cn by

Let K be a (commutative) field and n be a positive integer. Consider the K-algebra E = Mat (n, K) of all n X n matrices over K, and the corresponding general linear group GL(n, K). We shall define the set R of rigid mappings of E to consist of all a in GLK(E) which can be written in one of two possible forms: either xσ= axb for all x ϵ E or xσ = ax'b for all x ϵ E (where a and b are fixed elements of GL(n, K) and x’ denotes the transpose of x).

In this paper we study second order linear differential systems. We examine the relationship between oscillation of n-dimensional systems and certain associated n-dimensional systems, where m ≧ n. Several theorems are presented which unify and encompass in the linear case a number of results from the literature. In particular, we present a transformation which extends an oscillation theorem due to Allegretto and Erbe [1], and a comparison theorem due to Kreith [9], and explains some work of Howard [7].

Ebin [8] gives a thorough study of the space of riemannian metrics on a compact manifold M and of the action of the diffeomorphism group of M on . The purpose of this paper is to study the action of the larger group of conformorphisms, or conformai transformations, on and on . On , the L2-orthogonal decomposition induced by the action of gives a splitting of symmetric tensors into three summands introduced by York [25; 26]. We find submanifolds of tangent to the pieces of this decomposition.

A group g is called right-orderable (or an ro-group) if there exists an order relation ≦ on g such that a ≦ b implies ac ≦ be for all a, b, c in g. this is equivalent to the existence of a subsemigroup p of g such that p ⋂ p-1 = ﹛e﹜ and p ⋃ p-1 = g. given the order relation ≦, p can be taken to be the set of positive elements and conversely, given p, define a ≦ b if and only if ba-1ϵ p. a group g together with a given right-order relation on g is called right-ordered.

Generalizing a celebrated theorem of Thompson, R. P. Martineau has established [4; 5] that a finite group which admits an elementary abelian group of automorphisms with trivial fixed-point subgroup is necessarily solvable. A critical observation in his approach to this problem is the fact that, corresponding to each prime divisor of its order, such a group contains a unique Sylow subgroup invariant (as a set) under the action. Hence, the theorem we shall derive here represents a modest extension of Martineau's result.

There have been many indications of a relationship between the rate of growth of an entire function and the “size” of the set, E(c), where the modulus of the function is larger than the constant, c. Theorems of this type include the classical theorem of Wiman on functions of bounded minimum modulus, the Phragmén-Lindelöf Theorem, the Denjoy-Carleman-Ahlfors Theorem, and its many subsequent improvements. These theorems can all be understood as quantitative versions of the statement that if ƒ is an entire function such that, for some c > 0, the set E(c) is ‘'small”, then the maximum modulus function M(R, f) is forced to grow rapidly with R.

Let be a separable, infinite dimensional, complex Hilbert space, and let denote the algebra of all bounded linear operators on . In what follows we shall denote the spectrum, essential spectrum, and left essential spectrum of an operator T in , respectively. Furthermore, if and T1 is unitarily equivalent to a compact perturbation of an operator T2, then we write T1~ T2, and if the compact perturbation can be chosen to have norm less than e, we write T1 ~ T2(ϵ).

This paper is a report on work in progress rather than a description of theorems which have attained their final form. The results I shall describe are part of an attempt to continue to higher dimensions the study of the relation between the Hasse-Weil zeta-functions of Shimura varieties and the Euler products associated to automorphic forms, which was initiated by Eichler, and extensively developed by Shimura for the varieties of dimension one bearing his name. The method used has its origins in an idea of Sato, which was exploited by Ihara for the Shimura varieties associated to GL(2).

A Banach space X is called primary (respectively, prime) if for every (bounded linear) projection P on X either PX or (I — P)X (respectively, PX with dim PX = ∞ ) is isomorphic to X. It is well-known that C0 and lp, 1 ≦ p ≦ ∞ [8; 14] are prime. However, it is unknown whether there are other prime Banach spaces. For a discussion on prime and primary Banach spaces, we refer the reader to [9].

For a separable Hilbert space is the algebra of bounded linear operators on is the ideal of compact operators, and Π is the natural map of onto the Calkin algebra .

A combinatorial n-sphere is a simplicial n-complex whose body (i.e., the union of its members) is homeomorphic to the topological n-sphere Sn. A combinatorial n-manifold is a simplicial n-complex M such that M is connected, and for every vertex x in M the complex linker, M), the link of x in M, is a combinatorial (n — 1)-sphere. For more details the reader should consult Alexander [1] and Grünbaum [16]. All the spheres and manifolds to which we refer are combinatorial.

Following [10] an ergodic measure-preserving transformation is called rank one if it admits a sequence of approximating stacks. Rank one transformations have been studied in [1] and [2] where it was shown that any rank one transformation has simple spectrum. More generally it has been shown by Chacon [4] that a transformation of rank n has spectral multiplicity at most n. M. A. Akcoglu and J. R. Baxter have asked whether the converse is true. In particular: does simple spectrum imply rank one? In this paper we give a negative answer to this question.

In [5 ; 6] the author considers the following two problems posed by Professor Lloyd Tucker.

Problem 1. Does there exist a 1-dimensional continuum X without the fixed point property such that every retract of X has the fixed point property with respect to one-to-one maps?