Let A be a local ring with maximal ideal m, let N(m ) be the order of the residue field A/m and let N be a subgroup of GLn (A) which is normalized by SLn(A). It follows from results of Klingenberg that N is normal in GLn(A) when n ≥ 3 or . Results of Lacroix show that this is also true when n = 2 and N(m) = 3, provided that N∩2(A) ≠ SL2(A)'.

The principal aim of this paper is to provide examples of non-normal subgroups of GL2(A) which are normalized by SL2(A). In the process we extend results of Lacroix and Levesque on GL2(A)-normalized subgroups of GL2(A), where 2 ∊ m and N(m) > 2.

In this paper we introduce several properties closely related to the fixed point property of a partially ordered set P: the comparability property, the fixed point property for cones, and the fixed point extension property. We apply these properties to the sets of common bounds of the minimal (maximal) elements of the partially ordered set P in order to derive fixed point theorems for P.

We show that, if X and Y are Banach spaces such that X has the Mazur's intersection property and such that there exists T, an operator from Y into X so that T * and T ** are injective, then there exists on Y an equivalent norm which has the Mazur's intersection property.

We deduce from this result and from a result of M. Talagrand that there exists on the long James space J(η) an equivalent norm which has the Mazur's intersection property.

This paper is devoted to obtaining sequence space representations of spaces of vector-valued Ck -functions defined on an open subset, Ω, of ℝn, whose kth derivatives satisfy a Lipschitz condition on compact subsets of Ω.

We shall discuss relations between rectangularity and piecewise rectangularity of product spaces. In particular, we show that for each positive integer n there exists an n-dimensional, collectionwise normal, non-piecewise rectangular product X × Y which satisfies the inequality dim (X × Y) ≤ dim X + dim Y.

Let A be a finite-dimensional algebra over an algebraically closed field. By module is meant a finitely generated right module. A module T ^ is called a tilting module if and there exists an exact sequence 0 → A^ → T' → T" → 0 with T'. T" direct sums of summands of T. Let B = End T^·T^ is called separating (respectively, splitting) if every indecomposable A-module M (respectively, B-module N) is such that either Hom^(T,M) = 0 or (respectively, N ⊗ T = 0 or . We prove that A is hereditary provided the quiver of A has no oriented cycles and every separating tilting module is splitting.

Plumbing a Hopf band on the fibre-surface of a simple fibred knot is a geometric operation that produces another such knot. We show by algebraic methods that every high odd-dimensional simple fibred knot is obtained from the unknot by using this operation and its inverse.

A random map is a discrete time process in which one of a number of maps, 𝓜, is chosen at random at each stage and applied. In this note we study a random map, where 𝓜 is a set of piecewise linear Markov maps on [0, 1]. Sufficient conditions are presented which allow the determination of the unique absolutely continuous invariant measure of the process.

We construct an orthomodular inner product space to answer the questions posed by R. P. Morash in his paper "Angle bisection and orthoautomorphisms in Hilbert lattices" [6]. For example we show that every automorphism of the Hilbert lattice belonging to our inner product space has the property, that no atom is orthogonal to its image.

The multiplicative structure of the algebra of stable operations for ρ-local complex K-theory is studied, and the units and zero divisors are identified.

Let {x} denote the fractional part of x. We find an asymptotic formula of , where k is any positive integer and a is any real number ≥ 1, and so for the sum ∑n<xf(n), where f(n) belongs to a class of additive functions.

Let P - {p1,p2,. . . ,pn} be an independent point-set in ℝd (i.e., there are no d + 1 on a hyperplane). A simplex determined by d + 1 different points of P is called empty if it contains no point of P in its interior. Denote the number of empty simplices in P by fd(P). Katchalski and Meir pointed out that . Here a random construction P n is given with , where K(d) is a constant depending only on d. Several related questions are investigated.

Hidden behind a sums of squares formula are other such formulae not obtainable by restriction. This drastically simplifies the combinatorics involved in the existence problem of sums of squares formulae, and leads to a proof that the product of two sums of 16 squares cannot be rewritten as a sum of 28 squares, if only integer coefficients are permitted. We also construct all [10, 10, 16] formulae.

In this paper we prove some properties of the so-called modulus of noncompact convexity. This notion was recently introduced by K. Goebel and T. Sȩkowski [6] and it appears to be an interesting and useful generalization of the classical Clarkson modulus of convexity. We extend the results obtained in [6] showing that the modulus of noncompact convexity is continuous and has some extra properties in reflexive Banach spaces. The properties applicable in the fixed point theory are also stated.

Recently J. B. Reade determined the spectrum of C, the Cesaro matrix of order 1, considered as an operator on cO, the space of null sequences. Previously F. P. A. Cass and the author had determined the spectra for a large class of weighted mean operators on c, the space of convergent sequences. Subsequently the author determined the fine spectra of these operators over c. This paper examines the spectra and fine spectra of weighted mean operators on cO, obtaining the result of Reade as a special case.

For any space X, DG(X, A) is an abelian subgroup of [X, A] when A is an H-group. DG(X, X) is a ring for any H-group X.

The purpose of this paper is to give an explicit formula for the action of the n dimensional Dickson invariants on the admissible mod 2 Steenrod operations of length n.

If G is a connected cubic graph with ρ vertices, ρ > 4, then G has a vertex-induced forest containing at least (5ρ - 2)/8 vertices. In case G is triangle-free, the lower bound is improved to (2ρ — l)/3. Examples are given to show that no such lower bound is possible for vertex-induced trees.

Let G be a locally compact abelian group. An amalgam space (Lp ℓq)(G) (1 ≦ p,q ≦ ∞) is a Banach space of functions which belong locally to LP(G) and globally to ℓq. In this paper we present noninclusion results related to the Hausdorff-Young theorem for amalgams.

Let Dn.m , be the diameter of a connected undirected graph on n ≥2 vertices and n - 1 ≤ m ≤ s(n) edges, where s(n) = n(n — l)/2. Then Dn.s(n) = 1, and for m s(n) it is shown that

The bounds on Dn.m , are sharp.