In this paper a characterization of the regular ω-semigroups whose congruence lattice is modular is given. The characterization obtained for such semigroups generalizes the one given by Munn for bisimple ω-semigroups and completes a result of Baird dealing with the modularity of the sublattice of the congruence lattice of a simple regular ω-semigroup consisting of congruences which are either idempotent separating or group congruences.

In an interesting appendix to a letter written by John Collins to James Gregory on August 3, 1675, but not published until a few months ago, appear some formulae, given without proof, for expressing the roots of an equation of any degree from the 2nd to the 9th in terms of the coefficients, under the assumption that these roots are in arithmetical progression. The formulae were discovered by the well known contemporary of Leibniz, Baron W. von Tschirnhaus. It is evident that in the case of an equation of degree n this particular assumption imposes n − 2 conditions on the coefficients; so that two of these coefficients can be chosen ad libitum. Tschirnhaus did not go to the trouble of obtaining these relations explicitly, in fact he makes no mention of them, but he gives expressions, in the cases indicated above, for the roots as functions of the first two coefficients of the equation in question, and these coefficients, as we have observed, are arbitrary. It is not known by what approach he arrived at his formulae; it seems likely to us, however, that he expressed the desired roots in terms of two arbitrary unknowns, that he evaluated the sum of these, and the sum of their products two at a time, and that, finally, he equated the results to the first two coefficients of the equation. In this way two equations are obtained, sufficient to determine the two auxiliary unknowns; and the problem can be considered as solved. Without seeming to imply that this procedure was the same as that adopted by the eminent German mathematician, we shall show that by its means one can not only derive his results, but also solve the question in the case of an algebraic equation of any degree.

Definition. If x, y, z and ξ, η, ζ be the perpendiculars on the sides BC, CA, AB of the Δ ABC from points O and O′, then O and 0′ are antireciprocal points if xξ η, zζ:: tanA : tanA : tanB: tanC.

I. Construction to find a point antireciprocal to O (Fig. 4).

Draw through O a line MN antiparallel to BC. Draw OY perpendicular to AC, and OZ perpendicular to AB. Draw lines parallel to AB and AC, and at distances from them respectively equal to YN and MZ, and let them cut in P. Join AP. Find a similar line BQ, and let AP and BQ cut in O′.

It is common property in the theory of transformation semigroups that the presence of all the constant maps ensures that automorphisms are induced by a permutation of the underlying set. Essentially, this goes back to Malcev (2); it has been extensively generalised by Sullivan in (4). For semigroups which do not contain the constants (for example, all surjective transformations of a set, or all injections) there is, as yet, no similar result. The purpose of this note is to provide one.

Let (S, Σ, μ) and (T, Θ, v) be two measure spaces of finite measure where we assume S, T are compact Hausdorff spaces and μ, v are regular Borel measures. We construct the product measure space (T x S, >, Φ σ) in the usual way. Let G = [gl, g2, …, gp] and H = [hl, h1, …, hm be finite dimensional subspaces of C(S) and C(T) respectivelywhere G and H are also Chebyshev with respect to the L1-norm. Note that a subspace Y of a normed linear space X is Chebyshev if each x ∈X possesses exactly one best approximation y ∈Y. For example, in C(S) with the L1-norm, the subspace of polynomials of degree at most n is a Chebyshev subspace. This is an old theorem of Jackson. Now set

A sequence x = {si} has been defined to be summable-(Z, p) to s if

Where p is a positive integer and

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