For a commutative field extension, L ⊃ K, it is clear that a left basis of L over K; is also a right basis of L over K; however, for an extension of skew fields, this may easily fail, though it is hard to determine whether the right and left dimension may be different. Cohn ([4], ch. 5), however, was able to find extensions of skew fields such that the left and right dimensions were an arbitrary pair of cardinals subject only to the restrictions that neither were 1 and at least one of them was infinite. In this paper, I shall present a new approach that allows us to construct extensions of skew fields such that the left and right dimensions are arbitrary integers not equal to 1. In a subsequent paper, [7], I shall present related results and consequences; in particular, there is a construction of a hereditary artinian ring of finite representation type corresponding to the Coxeter diagram I2(5) answering the question raised by Dowbor, Ringel and Simson[5].

Let k be a field, X a set, F = k 〈X〉 the free associative k-algebra, and b an element of F that is homogeneous with respect to the grading of F induced by some map . We show that the idealizer of b in F, S = {f∈F|fb∈bF}, is a free algebra.

The original motivation for this paper was the question of primitivity for group rings of soluble groups of finite rank. At the end this is touched upon as an application of a theorem about prime ideals in such rings. If a group Γ acts on a set S we say an element is (Γ)-orbital if its orbit is finite and write ΔΓ(S) for the subset of such elements. The FC-radical of a group G, denoted by Δ(G), is just ΔG(G) where the action of G on itself is by conjugation.

The aim of this note is to answer in the negative a question of W. -D. Geyer, asked at the 1983 Group Theory Meeting in Oberwolfach: Is a maximal abelian subgroup A of a free profinite group F necessarily isomorphic to , the profinite completion of

In [5] James proved theorems on the decomposition numbers, for the general linear groups and symmetric groups, involving the removal of the first row or column from partitions. In [1] we gave different proofs of these theorems based on a result valid for the decomposition numbers of any reductive group. (I am grateful to J. C. Jantzen for pointing out that the Theorem in [1] may also be derived from the universal Chevalley group case, which follows from the proof of 1 ·18 Satz of [6] – the analogue of equation (1) of the proof being obtained by means of the natural isometry (with respect to contra-variant forms) between a certain sum of weight spaces of a Weyl module V(λ) of highest weight λ and the Weyl module corresponding to λ for the Chevalley group determined by the subset of the base involved.) However, we have recently noticed that this result for reductive groups, even when specialized to the case of GLn, gives a substantial generalization of James's Theorems. This generalization, which we give here, is an expression for the decomposition number [λ: μ] for a pair of partitions λ, μ whose diagrams can be simultaneously cut by a horizontal (or vertical) line so as to leave the same number of nodes above the line (or to the left of the line for a vertical cut) in both cases. Cutting between the first and second rows gives James's principal of row removal ([5], theorem 1) and cutting between the first and second column gives his principle of column removal ([5], theorem 2). Another special case of our horizontal result, involving the removal of bottom rows of a pair of partitions, is stated in [7], Satz 8.

For a ε R, we define a subset V of Rn to be a-convex if x, y ε V implies

Clearly V is (1 — a)-convex iff it is a-convex: V is convex iff it is a-convex for all a ε [0, 1], and any set is 1-convex. We define the a-convex hull of V to be the intersection of all a-convex subsets of Rn containing V. Put D = 0,1} and D(a) the a-convex hull of D in R1. If V is a-convex and x, y ε V, then V must contain the points of the line through x, y given by {tx + (1 — t) y: t ε D(a)} and so in this paper we concentrate on the study of D(a).

Let G(d) be a given simple d-regular graph on n labelled vertices, where dn is even. Such a graph will be called an initial graph. Denote by Gp(d) a random subgraph of G(d) obtained by removing edges, each with the same probability q — 1 —p, independently of all other edges (i.e. each edge remains in Gp(d) with probability p). In a recent paper [10] the asymptotic distributions of the number of vertices of a given degree in a random graph Gp(d) were given. The aim of this sequel is to present a wide variety of results devoted to probability distributions of the maximum and minimum degree of Gp(d) with respect to different values of the edge probability p and degree of regularity d. It should be noted here that very detailed results on a similar subject in the case when the initial graph is a complete graph (i.e. when d = n – 1) have already been obtained by Bollobás in the series of papers [2]–[4] (some additional information to the paper [4] was given in [9]). Also, in proving our results we will make use of some ideas given by Bollobás in these papers.

As the culmination of a notable sequence of papers, Mather [6] gave a necessary and sufficient condition that stable maps were dense in C∞(N, P): it is that the dimensions n and p satisfy the condition n <σ(n, p), where σ(n, p) is the codimension (in a sufficiently large jet space) of the set of jets in Jr(n, p) whose classification (for ℋ-equivalence) ‘involves moduli’.

The Realizability Question for poly-Surface groups consists in asking whether, for any poly-Surface group G, one may find a smooth closed aspherical manifold XG with π1(XG)= G. It seems, on present evidence, that the answer is always ‘yes’ [2][3], although in any one case the details may be quite complicated [4].

H*-algebras were introduced and studied by Ambrose [1] in the associative case, and the theory has been extended to such particular classes of non-associative algebras as Lie [18, 19], Jordan[20, 21, 7], alternative [11] and non-commutative Jordan [6] algebras. In all these cases the core of the matter is showing that every H*-algebra (in the given class) with zero annihilator is the closure of the orthogonal sum of its minimal closed ideals (each of which is a topologically simple H*-algebra), and then listing all the topologically simple H*-algebras in the class. In fact every nonassociative H*-algebra with zero annihilator is the closure of the orthogonal sum of its minimal closed ideals [6, theorem 2·7], so the problem of the classification of topologically simple non-associative H*-algebras becomes interesting. In relation with this problem the question arises whether, once an algebra A has been structured as a topologically simple H*-algebra, every H*-algebra structure on A is (up to a positive multiple of the inner product) totally isomorphic to the given one (see [3] and [11, section 4]). As a consequence of the results in this paper we give a general affirmative answer to this question.

The purpose of this note is to present simple proofs of the basic results concerning semi-simple annihilator Banach algebras as well as new results (for terminology notation and the standard approach see [2]).

In this note we will prove a number of facts which generalize to various non-abelian groups some results of Glicksberg[2]. In the non-abelian form they will have application to certain results of the author on the 2 dimensional continuous cohomology associated with central extensions [6].

Let M be a smooth manifold in . One may ask whether , the restriction of the Fourier transform of f to M makes sense for every f in . Since, for does not make sense pointwise, it is natural to introduce a measure σ on M and ask for an inequality

for every f in (say) the Schwartz class. Results of this kind are called restriction theorems. An excellent survey article on this subject is to be found in Tomas[13].

Let X be a normed linear space and let W be a proper subspace of X. A projection is a surjective linear map P: X → W such that P is idempotent. It is immediately clear that P has norm at least unity. Thus the problem of calculating the number

has some interest. The number λ(W, X) is often called the relative projectiion constant of W in X. If the infimum is attained, any attaining projection is called a minimal projection. The problems of calculating λ(W, X) for a fixed X and W or finding a minimal projection turn out to be very dificult. For example, if X = C [0, 1] with the usual supremem norm and W is the subspace of polynominals of degree at most two then λ(W, X) remains unknown as does any example of a minimal projection. One of the few places where the problem shows much tractability is the case

Let K be a compact Hausdorff space and E, F Banach spaces. We denote by C(K, E) the Banach space of all continuous. E-valued functions defined on K, with the supremum norm. It is well known ([6], [7]) that every operator (= bounded linear operator) T from C(K, E) to F has a finitely additive representing measure m of bounded semi-variation, defined on the Borel σ-field Σ of K and with values in L(E, F″) (the space of all operators from E into the second dual of F), in such a way that

where the integral is considered in Dinculeanu's sense.

Let Σan be a given infinite series with partial sums sn, and rn = nan. By and we denote the nth Cesáro means of order α (α –1) of the sequences (sn) and (rn), respectively. The series Σan is said to be absolutely summable (C, a) with index k, or simply summable |C, α|k, k ≥ 1, if

The concept of random set, though vaguely known for a long time (possibly since Buffon's needle problem), did not develop until Robbins [25, 26] provided for the first time a solid mathematical formulation of this concept and investigated relationships between random sets and geometric probabilities. Later on (in a different context) this concept gave rise to a more general concept of set-valued function in topology, and applications were also found in several areas such as economics (see, for example, Aumann[3] and Debreu[10]) and control theory (see, for example, Hermes [13]), among others. Recently in two independent formulations, D. G. Kendall [17] and Matheron[21] provided a comprehensive mathematical theory of this concept influenced by the geometric probability point of view. Actually, much of the research in this area falls under the heading of stochastic geometry (see, for example, M. G. Kendall and P. A. P. Moran [18]). The D. G. Kendall and Matheron theories have been compared and ‘reconciled’ by Ripley [24]. In the past few years random sets have been investigated as extensions of random variables and random vectors, and in this framework the problems of deriving limit theorems have received a great deal of attention. This approach is benefiting greatly from probability results in Banach spaces.

Average or global spectral properties of the Fibonacci and Lucas polynomials recently introduced in the theory of one-dimensional Ising chains are investigated. In addition, several partial sums of these polynomials are shown to have a compact form.

The Penrose transform is an integral geometric method of interpreting elements of various analytic cohomology groups on open subsets of complex projective 3-space as solutions of linear differential equations on the Grassmannian of 2-planes in 4-space. The motivation for such a transform comes from interpreting this Grassmannian as the complexification of the conformal compactification of Minkowski space, the differential equations being the massless field equations of various helicities. This transform is a cornerstone of twistor theory [22, 24, 30], but the methods generalize considerably as will be explained in this article. Closely related is the Ward correspondence[28], a non-linear version of a special case of the Penrose transform. It also admits a rather more general treatment. The object of this article is to explain the general case and the natural connection between the Penrose and Ward approaches.