§1. A series

has been defined by J. M. Whittaker to be absolutely summable (A), if

is convergent in (0 ≤ x < 1) and f (x) is of bounded variation in (0, 1), i.e.

for all subdivisions 0 = x0 < x1 < x2 < . … < xm < 1.

Let Vk denote the class of functions

which map conformally onto an image domain ƒ(U) of boundary rotation at most kπ (see (7) for the definition and basic properties of the class kπ). In this note we discuss the valency of functions in Vk, and also their Maclaurin coefficients.

In (8) it was shown that functions in Vk are close-to-convex in . Here we show that Vk is a subclass of the class K(α) of close-to-convex functions of order α (10) for , and we give an upper bound for the valency of functions in Vk for K>4.

We are concerned here with question: to what extent can the structure of a group G be recaptured from information about the structure of its group of automorphismsAut G? For example, one might try to find all groups which have some specific group astheir (full) automorphism group, a point of view adopted by Iyer in a recent paper [5]. Nothing is known about this question in general except the result of Nagrebeckü [7] that there are only finitely many finite groups with a given group as automorphismgroup.

Hardly had Edinburgh University entered upon its present academic year than it was deprived by the hand of Death of the services of Cargill G. Knott, Reader in Applied Mathematics. The news of his death came as a great shock to all his friends, as there was no illness or other sign which might have foreshadowed the summons that was coming to him. Apparently in his usual robust health, he was engaged in his various duties until the very end, which came to him with startling suddenness.

In this paper radicals in the sense of Kuroš and Amitsur (KA-radicals) for Ω-groups will be studied. For the sake of simplicity these radicals will be considered on varieties, the results remaining valid for more general classes.

For positive definite C1 kernels on a finite real interval the eigenvalues λn are known to be o(1/n2). In this paper this result is shown to be best possible in the best possible sense, namely that, given any decreasing sequence λn, which is o(1/n2), there exist positive definite C1 kernels whose eigenvalues are λn.

1. Taking the two following known properties of the pedal line of a triangle, viz.:

I. The locus of a point, such that the feet of the perpendiculars from, it on the sides of a triangle are collinear, is the circum-circle of the triangle;

II. The pedal line bisects the distance between the orthocentre and the corresponding point in the circumference of the circum-circle;

The purpose of this note is to show the existence of a lattice ordered group G with a finite set of generators and a recursively enumerable set of defining relations such that there is no decision procedure to determine whether or not an arbitrary word in the generators reduces to the identity in G. In addition to the usual group-theoretic words, we may also use the two lattice operations ∨ and ∧ ; for example, a−l(b∨c) is a word in the generators a, b and c. At first sight it might appear that since we have an even greater harvest of words than in group theory and there exist finitely presented groups H (H has a finite number of generators and defining relations) with an insoluble word problem (no decision procedure to determine whether an arbitrary word in the generators reduces to the identity)—see (1), (2), (4) or (6)—the same would be true of lattice ordered groups. Unfortunately, such a naïve approach overlooks two salient points. First, the class of lattice ordered groups is strictly smaller than the class of all groups; second, there are certain relations connecting the lattice operations with the group operations which hold true for all lattice ordered groups. For example, a(b∧c) = ab∨ac and (a∨b)−1 = a−1∧b−1.

A previous note (2) showed how the integral of f(1x1+2x2++nxn) over the interior of a simplex could be reduced to a contour integral. The same idea is applied here in Theorems 1 and 2 to give a generalisation of Dirichlet's multiple integral ((1), pp. 169172). These results are then used in Theorem 3 to reduce an integral over all real n-dimensional space to a contour integral. In Theorem 4 an integral over the group of all 33 orthogonal matrices of determinant 1 is reduced to a contour integral. This result can be extended formally to the case of 44 matrices; beyond this it seems difficult to go.

The Series

if convergent, is a particular solution of the Differential Equation

in which

and II denotes Gauss's II Function, D stands for .

For a two-step nilpotent locally compact group $G$, we determine the Glimm ideal space of the group $C^*$-algebra $C^*(G)$ and its topology. This leads to necessary and sufficient conditions for $C^*(G)$ to be quasi-standard. Moreover, some results about the Glimm classes of points in the primitive ideal space $\mathrm{Prim}(C^*(G))$ are obtained.

AMS 2000 Mathematics subject classification: Primary 22D25. Secondary 22D10

1. In Fig. 8, take

and denote DE, EF, FD by x, y, z respectively;

From these equations

Throughout this paper X will be a finite connected CW-complex of dimension m, and ξ will be a real (n + l)-plane bundle over X(n >0) equipped with a Riemannian metric. We aim to give a systematic account of the space ГSξ of sections of the sphere-bundle Sξ.

We shall start by recalling the definition and some basic properties of a convexity space; a topological convexity space (tcs) will then be a convexity space together with an admissible topology, and will be a generalisation of a topological vector space (tvs). After showing that the usual tvs results connecting the linear and topological properties extend to this new setting we then prove a form of the Krein-Milman theorem in a tcs.