The problem is to show that the expression

where ω is an imaginary fourth root of unity, is symmetrical with respect to a, b, and c.

We show that if π is a group with a finite 2-dimensional Eilenberg-Mac Lane complex then the minimum of the Euler characteristics of closed 4-manifolds with fundamental group π is 2χ(K(π, 1)). If moreover M is such a manifold realizing this minimum then π2(M) ≅ Similarly, if π is a PD3-group and w1(M) is the canonical orientation character of π then χ(M)≧l and π2(M) is stably isomorphic to the augmentation ideal of Z[π].

In this paper we investigate totally geodesic surfaces in hyperbolic 3-manifolds. In particular we show that if M is a compact arithmetic hyperbolic 3-manifold containing an immersion of a totally geodesic surface then it contains infinitely many commensurability classes of such surfaces. In addition we show for these M that the Chern-Simons invariant is rational.

We also show, that unlike the figure-eight knot complement in S3, many knot complements in S3 do not contain an immersion of a closed totally geodesic surface.

In a recent paper J. M. Whittaker considered the problem of resolving a linear differential system into a product of two or more systems of lower order. This note is a contribution to this problem and furnishes the necessary and sufficient conditions for the resolution of the system into two equivalent systems of lower order of the type considered by Whittaker.

In a recent paper Turnbull, discussing a rational method for the reduction of a singular matrix pencil to canonical form, has shown how the lowest row, or column, minimal index may be determined directly without reducing the pencil to canonical form. It is the purpose of this note to show how all such indices may be determined, and at the same time to give conditions, somewhat simpler than the usual ones, for the equivalence of two matrix pencils.

In Figure 11 let ABCD be a contraparallelogram having AB = CD, AD = BC.

Let O, P, Q, R be the midpoints of AB, BC, AD, DC respectively. They obviously lie in the line parallel to AC and to BD, and equidistant from them.

Let U be the mid-point of AC and V that of BD.

OURV is a rhombus each of whose sides is half of AD or BC, and parallel to AD or BC. PUQV is a rhombus each of whose sides is half of AB or CD, and parallel to AB or CD.

In this paper we investigate some triple equations involving the inverse of the finite Mellin transform MR which is defined by the equation

This transformation is one of four which were first introduced by D. Naylor in his paper (1) and some of its properties have been listed by the author in paper (2), where its relationship to the Mellin transform is discussed in detail.

For a class of elliptic equations in the entire space and with nonlinear terms having a possibly uncountable (but of Lebesgue measure zero) set of discontinuities, the existence of strong solutions is established. Two simple applications are then developed. The approach taken is strictly based on set-valued analysis and fixed-points arguments.

In an earlier paper [4] we considered the question of whether an injective module E over a noncommutative ring R remains injective after localization with respect to a denominator set X in R. A related question is whether, given an essential extension N of an R-module M, the localization N[X–1] must be an essential extension of M[X–1]. In [1] it is shown that if R is left noetherian and X is central in R, then localization at X preserves both injectivity and essential extensions of left R-modules and, hence, preserves injective hulls and minimal injective resolutions.

Let E, F, G be three compact sets in ℂn. We say that (E, F, G) holds if for any choice of an interpolating array in F and of an analytic function ℂ on G, the Kergjn interpolation polynomial of ℂ exists and converges to ℂ on E. Given two of the three sets, we study how to construct the third in order that (E, F, G) holds.

On a semigroup S the relation ℒ* is defined by the rule that (a, b) ∈ ℒ* if and only if the elements a, b of S are related by Green's relation ℒ in some oversemigroup of S. It is well known that for a monoid S, every principal right ideal is projective if and only if each ℒ*-class of S contains an idempotent. Following (6) we say that a semigroup with or without an identity in which each ℒ*-class contains an idempotent and the idempotents commute is right adequate. A right adequate semigroup S in which eS ∩ aS = eaS for any e2 = e, a ∈ S is called right type A. This class of semigroups is studied in (5).