A liquid crystal is a transversely isotropic liquid consisting of large, relatively rigid, elongated molecules which align more or less parallel to their neighbours. Three distinct types of liquid crystal occur, namely nematic, cholesteric and smectic. In the absence of any external influences, nematics tend to orientate with their anisotropic axis uniformly aligned, whereas cholesterics prefer a characteristic helical configuration and smectics are more highly organised in layered structures. However, it is possible to influence the orientation of the anisotropic axis by a variety of external means. In particular, solid surfaces affect the alignment through the action of surface torques, while electromagnetic fields exert body torques which tend to align the anisotropic axis either parallel or perpendicular to the applied field. Detailed descriptions of the physical properties of liquid crystals may be found in the books by de Gennes [1] and Chandrasekhar [2] and the review by Stephen and Straley [3].

The object of this paper is to discuss conditions for a maximum or minimum of functions of integrals of the type

employing the methods of the Calculus of Variations.

This paper is based on the interpretation of the ring of additive polynomials in one variable over a finite field Fq, as a maximal R-order inside a certain skew-field D, R being a principal ideal domain isomorphic to Fp[T]. The classical (1930's) structure theory of maximal orders in global fields is used to solve enumeration questions involving the iteration of members of Pages from .

1. The object of this note is to discover a new function which is its own reciprocal in the Hankel Transform of order zero.

I will make use of the following theorem of Hardy and Titchmarsh1:–

A necessary and sufficient condition that a Junction f (x) should be its own Jv transform is that it should be of the form

where 0 < c < 1, and

As usual in the theory of polynomial near-rings, we deal with right near-rings. If N = (N, +,·) is a near-ring, the set of distributive elements of N will be denoted by Nd;

It is easy to check that, if N is an abelian near-ring (i.e., r + s = s + r, for all r, s∈N), then Nd is a subring of N.

We obtain some new results about the maximal operator space structure which can be put on a normed space. These results are used to prove some dilation results for contractive linear maps from a normed space into B(H). Finally, we prove CB(MIN(X), MAX(y)) = Γ2(X, Y) and apply this result to prove some new Grothendieck-type inequalities and some new estimates on spans of “free” unitaries.

In this paper an asymptotic formula is obtained for the number of primes representable as the sum of two square-free squares. The precise result is:

Theorem 1.Let N(x) be the number of primes not exceeding x represented by the quadratic formy2 + z2, where y and z are square-free. Let w be a fixed arbitrarily large number. Then

where

and

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.