We discuss partial spreads (translation nets) U, V of ∑ = PG(3, q) where U, V cover the same points of ∑ and have no lines in common. Write t = |U| = |V|. It has been shown in a previous paper [4] that t ≧ 2(g — 1) provided q + 4.

This short paper is devoted to the proof of a single theorem, which, in its simplest form, asserts that if Q is a polyhedron in Rn which can be expressed as the union of finitely many convex polytopes whose vertices are at rational points in Rn, and if is a simplicial subdivision of Q﹜ then there is an isomorphic simplicial subdivision ” of Q in which all vertices are at rational points.

In a previous paper by the author [3], duality theorems for finitely generated vector lattices and lattice-ordered Abelian groups are described. In particular, the category of finitely generated semi-simple vector lattices is shown to be equivalent to a geometrical category V whose objects are topologically closed cones in Euclidean space, and whose morphisms, called ll-maps\ form a special subclass of the class of piece wise homogeneous linear maps between such cones. Under this categorical duality, finitely generated projective vector lattices and closed polyhedral cones correspond; indeed, the category of finitely generated projective vector lattices is equivalent to the dual of a category whose objects are Euclidean closed polyhedral cones and whose morphisms consist of all piecewise homogeneous linear maps between such cones.

When studying the category raph of finite graphs and their morphisms, Ave can exploit the fact that this category has products, [we define these ideas in detail in § 2]. This categorical product of graphs is usually called their Kronecker product, though it has been approached by various authors in various ways and under various names, including tensor product, cardinal product, conjunction and of course categorical product (see for example [6; 7; 11 ; 14; 17 and 23]).

Let A denote a complex (or real) Fréchet algebra (i.e. a complete metrizable locally m-convex algebra, see [2] or [3]). It is known [2] that the topology of such an algebra can be defined by an increasing sequence [qn] (i.e. qn(x) ≦ qn+i(x) for all x £ A and n ≧ 1) of submultiplicative (i.e. qn(xy) ≦ qn(x)qn(y) for all x, 3/ Ç ^4 and for each n ≧ I) seminorms.

The literature concerning rings with polynomial identity contains several theorems in which the existence of a polynomial identity on a subring implies the existence of such an identity on the ring itself. Belluce and Jain showed in 1968 that a prime ring will satisfy a polynomial identity provided it contains a right ideal with zero left annihilator which satisfies a polynomial identity [2].

A P-space is a topological space in which every Gδ-set is open. P-spaces are fairly rare. For example, the only compact (or even pseudocompact) P-spaces are finite. A larger class of spaces, the almost-P-spaces, consists of those spaces in which G δ-sets have dense interiors. The almost-P-spaces are far less restricted than the P-spaces—for example, there are infinite, compact, connected almost-P-spaces. In this paper, we study almost-P-spaces and raise a number of questions relating to them.

The grading is by integers modulo 2 and we refer to it as 2-grading. For the definition of 2-graded Lie algebras L and their properties we refer the reader to the papers [1; 2; 3]. All algebras considered here are finite-dimensional over a field F of characteristic zero.

Our aim is to establish the Lebesgue decomposition for s-bounded vector valued additive functions defined on lattices of sets in both the finitely and countably additive setting. Strongly bounded (s-bounded) set functions were first studied by Rickart [15], and then by Rao [14], Brooks [1] and Darst [5; 6]. In 1963 Darst [6] established a result giving the decomposition of s-bounded elements in a normed Abelian group with respect to an algebra of projection operators.

Anderson, Clunie and Pommerenke defined and studied the family of Bloch functions on the unit disc (see [1]). This family strictly contains the space of bounded analytic functions. However, all Bloch functions are normal and therefore enjoy the “nice” properties of normal functions. The importance of the Bloch function concept is the combination of their richness as a family and their “nice” behavior.

The theorem referred to in the title asserts that for an atomic commutative integral domain R, if S is a submonoid of R* (the monoid of nonzero elements of R) generated by primes such that the quotient ring RS-1 is a UFD (unique factorization domain) then R is also a UFD [8]. Recently several definitions of a noncommutative UFD have been proposed (see the summary in [6]).

An n X n real matrix A = (aij) is called stable (resp. semistable) if each of its eigenvalues has negative (resp. nonpositive) real part. These notions are important because of their close connection with the stability of motion, which can be described especially simply for a system

There are many situations in algebraic topology when a geometric construction is possible if, and only if, a certain integral cohomology class, an obstruction is zero. When attempts are made to compute the obstruction, it often happens that it is relatively easy to show that m times the obstruction is zero, where m is an integer, and consequently the geometric construction is possible if the cohomology group in question has no elements of order m.

The chief purpose of this paper is to find all pairs (G, θ) where G is a finite special p-group, and θ is an automorphism of G acting trivially on the Frattini subgroup and irreducibly on the Frattini quotient. This problem arises in the context of describing finite groups having an abelian maximal subgroup. In fact, we solve a more general problem for a wider class of p-groups, which we call special F-groups, where F is a finite field of characteristic p. We point out that if p is odd, then an F-group has exponent p. On the other hand, every special 2-group is also a special GF(2)-group.

Let be a countable atomless Boolean algebra and let X be a countable partial ordering. We prove that there exists an embedding of X into which is recursive in X, and which destroys all suprema and infima of X which can be destroyed. We show that the above theorem is false when we try to preserve all suprema and infima of X instead of destroying them.

Let X be a set. A function d from X X X into the nonnegative real numbers is called a ﹛non-archimedean) quasi-metric on X if

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.

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].

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.