In this paper we shall consider finite groups satisfying the following hypothesis.

Hypothesis I. Let G be a finite group which admits an automorphism σ of primeorder p, (p, |G|) = 1. Assume the fixed point subgroup B = CG(σ) contains some Sylow 2-subgroup.

A map p: E → B between metric spaces has the approximate homotopy lifting property with respect to the space X if given a cover Ū of B and maps g: X → E and H: X × [0, 1] → B such that H(x, 0) = pg(x) for all x ϵ X, then there exists a map G: X × [0, 1] → E such that G(x, 0) = g(x) and pGt and Ht are Ū-close for all x ϵ X and t ϵ [0, 1]; i.e. given (x, t) ∊ X × [0, 1], there exists U × Ū such that pG(x, t) and H(x, t) are elements of U.

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.

A σ-discrete set in a topological space is a set which is a countable union of discrete closed subsets. A mapping ƒ : X ⟶ Y from a topological space X into a topological space Y is said to be σ-discrete (countable) if each fibre ƒ-1(y), y ϵ Y is σ-discrete (countable). In 1936, Alexandroff showed that every open map of a bounded multiplicity between Hausdorff spaces is a local homeomorphism on a dense open subset of the domain [2].

There are many concepts which arise naturally in a discussion of injectivity in an equational class; for example, weak injective algebras, absolute subretracts, essential extensions, the congruence extension property, and the amalgamation property (see [3; 9; 17; 18]). It has already been demonstrated in several papers, notably [9; 17; 26; 27; 28], that the study of these concepts is greatly enriched by the assumption that the algebras under consideration have distributive congruence lattices. In this work attention is focused on weak injective algebras (Section 2) and the congruence extension property (Section 3).

In the study of direct differential geometry, families of oriented arcs and curves have been employed extensively to define the differentiability of an arc at a point in various kinds of planes; cf. [2]. In [6], P. Scherk used lines in the projective plane; in [3] and [4], N. D. Lane and P. Scherk used circles in the conformai plane; conic-sections in the projective plane were employed in [5] and [7] by N. D. Lane and K. D. Singh; in [1], M. Gupta and N. D. Lane used the graphs of polynomials of degree at most n in the affine plane.

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.

Let A be a compact operator on a separable Hilbert space . The aim of this paper is to investigate the relationship between the weak closure of the algebra of polynomials in A (denoted by U(A)) and its invariant subspace lattice Lat A.

Let be a full subcategory of a category is said to be coreflective in if for each object X in there exists an object X in and a morphism such that for each object P in and each morphism f : P ⟶ X there exists a unique morphism such that .

Our motivation for this note originates with consideration of a subset A of Euclidean w-space, Rn, which contains only part of its boundary. The part contained is t h a t part of the closure of A which cannot be “bettered“ within A with respect to the preference associated with a fixed closed convex cone Γ. Here b is preferred to a if and only if a — b ∊ Γ; if, for instance, Γ is the non-negative orthant of Rn, this preference is ordinary vector inequality.

It is well known that the complete quotient ring of a Noetherian ring coincides with its classical quotient ring, as shown in Akiba [1]. But in general, the structure of the complete quotient ring of a given ring is largely unknown. This paper investigates the structure of the complete quotient ring of certain Prüfer rings. Boisen and Larsen [2] considered conditions under which a Prüfer ring is a homomorphic image of a Prüfer domain and the properties inherited from the domain. We restrict our investigation primarily to homomorphic images of semilocal Prüfer domains. We characterize the complete quotient ring of a semilocal Prüfer domain in terms of complete quotient rings of local rings and a completion of a topological ring.

In 1946 Krasnoselskii proved that if every n + 1 points of a compact connected set S in a Euclidean space En can see at least one point of S via S then S is starshaped [1]. This result was expanded by Robkin in 1965 [2]. In this paper, we characterize starshaped sets in En by a local arcwise convexity property relative to a point p.

The problem of describing the T1 compactifications of a given T1 space arises quite naturally in many contexts, and has been approached from a number of directions. One characteristic of all approaches has been the exclusive consideration of strict topological extensions. There are obvious advantages to this approach. Points of the compactification may be distinguished by their trace filters, and the topology of the compactification is readily described in a natural manner. Moreover every T2 compactification is strict, so the method loses no generality in this most important special case.

Let U, C denote the open unit disk and unit circumference, respectively and G(z, w) be the Green's function on U. We say v is the Green's potential of a mass distribution v on U if

In [13] we characterized the completely regular Hausdorff spaces as the class of spaces whose topology is generated by an Alexandrofï base. A space may have more than one Alexandrofï base and each such base determines a Hausdorff compactification .

Let R be a discrete valuation ring, with maximal ideal pR, such that ½ ϵ R. Let L be a finitely generated R-module and B : L × L → R a non-degenerate symmetric bilinear form. The module L is called a quadratic module. For notational convenience we shall write xy = B(x, y). Let O(L) be the group of isometries, i.e. all R-linear isomorphisms φ : L → L such that B((φ(x), (φ(y)) = B(x, y).

We are concerned with the extent to which the structure of a Boolean algebra (or BA, for brevity) is reflected in its group of automorphisms, Aut . In particular, for which algebras can one conclude that if Aut Aut , then Monk has conjectured [3] that this implication holds for denumerable BA's with at least one atom. We shall refute his conjecture, but show that the implication does hold if and are denumerable, if each has at least one atom, and if the sum of the atoms exists in . In fact, under those assumptions the algebra 21 can be rather neatly recovered from its abstract automorphism group.

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.

A much-studied equation in recent years has been the second order nonlinear ordinary differential equation

where q and f are continuous on the real line and, in addition, f is monotone increasing with yf(y) > 0 for y ≠ 0. Although the original interest in (1) lay largely with the case that q﹛t) ≧ 0 for all large values of t, a number of papers have recently appeared in which this sign restriction is removed.

In a paper on the same subject [28] and another coming out at the same time [27], Nagata gave his celebrated Double (treble, really) Sequence Theorem, with which he deduced easily and thus brought together the basic metrization theorems, i.e. theorems in which the conditions for metrizability are given as the availability of bases or subbases of certain descriptions.