In a 1975 paper [8], Heinig established the following three inequalities:

where A = p/(p + s − λ) with p, s, λ real numbers satisfying p + s > λ,p > 0;

where B = p/(2p + sp − λ −1) with p, s, λ real numbers satisfying 2p +sp > λ, + 1, p > 0;

where is a sequence of nonnegative real numbers,

and C = p[l + l/(p + s−λ)] with p, s, λ real numbers satisfying s > 0, p ≥ 1, and p +s > λ 0.

Let G be a group. A subset D will be called a set of 3-transpositions if |x| = 2 for xεD and |xy| = 3 whenever x, yεD do not commute. We will call the set D closed if xDx−1 = D for each xεD. For each xεD, let

For each subset X of D, we denote by [X] the graph with vertex set X where two elements x, yεX are joined by an edge whenever they commute. We denote by (X) the complement graph; thus two elements x, yεX are joined by an edge of (X) whenever they do not commute.

Let F a totally real cubic field. For any number α in F, let α, α′, α″ denote the conjugates of α. Define the function T(α) by

.

Let G be a finite group, Sn be the symmetric group on n symbols and An be the corresponding alternating group. The conjugacy classes of the wreath product GSn (or monomial group as it is sometimes known) and the conjugacy classes of GAn have been described by Kerber (see [2] and [3]). The group Sn has a double cover n so that the faithful complex representations of this double cover may be regarded as protective representations of Sn. In Section 2, a particular double cover for GSn is constructed, the faithful complex representations of this group being the subject of a joint article with Peter Hoffman[1]. In the present paper, our task is to determine whether a conjugacy class of GSn corresponds to one or to two conjugacy classes in the double cover of GSn (and similarly for GAn). The main results, Theorems 1 and 2, are stated precisely in Section 2 and proved in Sections 3 and 4 respectively. The case when G = 1 provides classical results of Schur ([5], Satz IV). When G is a cyclic group, Read [4] has determined the conjugacy classes, not just for our particular double cover, but for all possible double covers of GSn.

Let D be a division ring with central subfield F, n a positive integer and G a subgroup of GL(n, D) such that the F-subalgebra F[G] generated by G is the full matrix algebra Dn×n. If G is soluble then Snider [9] proves that G is abelian by locally finite. He also shows that this locally finite image of G can be any locally finite group. Of course not every abelian by locally finite group is soluble. This suggests that Snider's conclusion should apply to some wider class of groups.

While expanding upon the work of H. M. Srivastava [6] on generalizations of an interesting identity of Carlson, R. G. Buschman and H. M. Srivastava [2] proved a number of double-series identities and listed various cases of reducibility of certain hypergeometric series in two variables (cf. [1], p. 150, equation (29)). The object of the present paper is to derive three new classes of combinatorial series identities (contained in Theorems 1, 2 and 3 below) which unify and extend the results of these earlier papers ([2], [6]). A multiple-series analogue of one of the combinatorial series identities presented here is also recorded.

In two previous papers the author studied the Jacobson and nil redicals of the algebra of a commutative semigroup over a field [8] and over a commutative ring with unity [9]. This work is continued here.

In this paper we construct a (multihomogeneous) dualizing complex for A [Δ], the Stanley–Reisner ring of Δ over an arbitrary commutative noetherian ring A with identity, admitting a dualizing complex. In addition, some consequences are discussed.

An ordinal is termed unsound if it has subsets An (nεω) such that un-countably many ordinals are realised as order types of sets of the form ∪ {An|nε a} where a ⊆ ω. It is shown that if ω1 is regular and then the least unsound ordinal is exactly but that if ω1 is regular and the least unsound ordinal, assuming one exists, is at least . Arguments due to Kechris and Woodin are presented showing that under the axiom of determinacy there is an unsound ordinal less than ω2. The relation between unsound ordinals and ideals on w is explored. The paper closes with a list of open problems.

In this paper we list the elliptic curves defined over Q √ − 1, Q√ −2 or Q√ − 3 which have good reduction away from 2. The possible invariants of such curves are given in Table 1, and their minimal equations in Tables 2, 3 and 4. These extend (and agree with) results of Ogg[4] and Stroeker [10], by a different method.

A matrix A is said to be tripotent whenever A3 = A. The study of tripotent matrices is of statistical interest since if the n × 1 real random vector X follows an N(0, I) distribution and A is a symmetric matrix then the real quadratic form X′AX is distributed as the difference of two independently distributed X2 variates if and only if A3 = A. In fact, a necessary and sufficient condition that A is tripotent is that there exist two idempotent matrices B and C such that A = B – C, and BC = 0. Using properties of diagonalizable matrices, we will prove several algebraic characterizations of r-potent matrices that extend the known results for tripotent matrices. Our first result will be to obtain an analogous decomposition for an arbitrary r-potent matrix.

Let (Dn) be the apollonian packing of a curvilinear triangle T, ρn the radius of Dn, E = T—U Dn the residual set, dim (E) its Hausdorff dimension. In this paper we give a new proof of the equality dim proved by Boyd [2]. Our technique is to construct a sequence of regular triangles covering E, and suitable measures μkcarried by E which allow us to apply a density theorem.

Let K be a field with char K ≡ 2,3. We consider the problem of finding rational points over K on the family of elliptic curves Fλ, given in homogeneous coordinates (over ) by

Let M (the mirror) be a plane oval (a smooth curve without inflexions), and let sεℝ2\M be the light source. Rays of light emanating from s are reflected by M, and the caustic by reflexion of M relative to s is the envelope of these reflected rays. In this article we suppose that M is generic (the precise assumption is stated later) and that s moves along a smooth curve in the plane; we are then able to describe how the local structure of the caustic changes. In order to state the result we recall a few facts from [3].

An n-block is a sequence b1 … bn where bi ε {0,1} for 1 ≤ i ≤ n, and an n-block map is a function from the set of n-blocks to the set {0,1}. Block maps can be used to study endomorphisms of the shift dynamical system [7] and shift register sequences [4]. Composition of endomorphisms and cascading of shift registers [6] can be studied via composition of block maps. If fog is a given block map which is linear in the first variable and if g is also given, then f is unique and can be determined. If fog and f are given then g is unique and can be determined, at least up to a constant [10]. However, little is known about the level of computational complexity of the general task of searching for factors of a given block map. In this paper I identify a substantial class of maps which are linear in the first variable but not linear, for which there are effective methods of searching for factors. The methods are based on the presentation of block maps via successive principal parts, introduced in [9]. There I define the principal vector of a block map. One of the key results gives conditions under which there is a particularly close relationship between the principal vectors of two block maps f and g and their composition fog. In the second section of this paper I show that the same relationship holds under weaker conditions.

This paper studies coefficients yh, n of sequences of polynomials

defined by non-linear recurrences. A typical example to which the results of this paperapply is that of the sequence

which arises in the study of binary trees. For a wide class of similar sequences a general distribution law for the coefficients yh, n as functions of n with h fixed is established. It follows from this law that in many interesting cases the distribution is asymptotically Gaussian near the peak. The proof relies on the saddle point method applied in a region where the polynomials grow doubly exponentially as h → ∞. Applications of these results include enumerations of binary trees and 2–3 trees. Other structures of interest in computer science and combinatorics can also be studied by this method or its extensions.

Let M be a plane oval (a smooth curve without inflexions). In this note we show that a generic such M (where the precise assumptions will be stated later) has to have at least one sextactic point, that is a point p where the unique conic touching M at p with at least 5-point contact actually has 6-point contact. This existence problem came into prominence whilst [2] was being written. It was hoped to use the existence of sextactic points to show that the Morse transition on a 1-parameter family of focoids with signature 0 or 2 could not occur. The problem proved to be remarkably stubborn, however. Indeed, the geometric interpretation of sextactic points as given in § 3 was totally unexpected.

In a long sequence of notes in the Comptes Rendus and elsewhere, and in the papers [1], [2], [3], [6], [7], Lichnerowicz and his collaborators have studied the ‘classical infinite-dimensional Lie algebras’, their derivations, automorphisms, co-homology, and other properties. The most familiar of these algebras is the Lie algebra of C∞ vector fields on a C∞ manifold. Another is the Lie algebra of ‘Poisson brackets’, that is, of C∞ functions on a C∞ symplectic manifold, with the Poisson bracket as composition; some questions concerning this algebra are of considerable interest in the theory of quantization – see, for instance, [2] and [3].

For any locally compact topological group G let M(G) denote the topological semigroup of all probability (Borel) measures on G, furnished with the weak topology and with convolution as the multiplication. A Gauss semigroup on G is a homomorphism t→ μt of the strictly positive reals (under addition) into M(G) such that

(i) no μt is a point mesaure,

(ii) for each neighbourhood V of 1 in G we have

Spherical diagrams were introduced by Lyndon and Schupp[14] in order to study asphericity in group presentations and in 2-complexes. They have since been studied by several authors [2, 3, 5]. In particular, some technical loopholes in the original approach were closed in [5]. For many purposes the dual notion of pictures, introduced by Rourke[17], is more useful. These arise naturally through transversality. Pictures have also been studied and applied in [2, 5, 6, 12, 19].