1. Let

and denote by Aδ the class of functions f analytic in the strip Sδ = {z = x + iy| |y| < δ}, real on the real axis, and satisfying |Ref(z)| ≤ 1,z∊Sδ. Then N.I. Achieser ([1], pp. 214–219; [8], pp. 137–8, 149) proved that each f∊Aδ can be uniformly approximated on the whole real axis by an entire function fc of exponential type at most c with an error

where ∥·∥∞ is the sup norm on ℝ. Furthermore ([7], pp. 196–201), if f∊Aδ is 2π-periodic, then the uniform approximation Ẽn (Aδ) of the class Aδ by trigonometric polynomials of degree at most n is given by

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].

In 1967 Segal introduced the Dirichlet convolution (, h(n)), generalizing a method of Ingham developed in studies on the Prime Number Theorem. In this paper we establish necessary and sufficient conditions on the sequence h(n) in order that the convolution method (, h(n)) be conservative. Further conditions are established for the method to be absolutely conservative.

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.

1. Given a sequence an of positive integers, one expects to obtain information on the distribution of these numbers by examining the Dirichlet series

In this paper we are going to show how such a series arises from Fröhlich's Galoismodule theory and the use the Thue–Siegel–Roth–Schmidt Theorem as one of the tools in the study of its singularities.

In this paper we present a proof of the mean-value theorem required by Levinson to show that at least one-third of the zeros of ζ(s) are on the critical line. As in Levinson [3], let

where Χ(s)=ζ(s)/ζ(1−s) is the usual factor from the functional equation, and let

where

and

where

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.

In [1], Julian Cole asked whether every topos is definable over a Boolean one. Peter Johnstone partially answered this in [2] (corollary 3·7) by giving an example of a topos admitting no bounded morphism to any Boolean topos. Here we give a simple example where boundedness is unnecessary.

In [6], p. 80, Serre showed that the group SL2(Z[1/p]) is a free product of two copies of SL2(Z) amalgamated along a sub-group of index p + 1 in both components. This result was obtained from the structure theory of groups acting on trees and the construction of a tree on which SL2 over a field with discrete valuation acts in a natural way. This result can be extended to other SL2 examples. Using the above description, Serre [6] also gave presentations for the groups SL2(Z[1/p]) where p = 2, 3.

In this paper we prove that for any odd prime p, a Moufang loop of order 2p2 is a group and there is exactly one non-associative right Bol loop of order 2p2.

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

Given two homomorphisms of rings j1:R1 → R' and j2: R2 → R' we may construct a new ring R called the pullback of R1 and R2 over R', together with homomorphisms i1:R →R1 and i2:R→R2 such that j1i1= j2i2. There are many instances where special cases of pullbacks have been studied, either to construct new examples of rings or to reduce various problems to those concerning much simpler rings. A few of these instances are given in the references. Thus a general study of the properties of pullbacks in terms of those of the component rings would seem to be useful.

Let be the class of groups which can be presented in the form

where u, v are positive words on the generators g, h, …, k and where each generator appears in uv−1 with zero exponent sum. Let be the class of groups which can be presented in the form

where u, v are words (not necessarily positive) on the disjoint sets of generators c1, …, cm and d1, …, dn and where [u, v] = u−1v−1uv.

The aim of this paper is to prove:

Theorem. Let R be an affine (finitely generated) algebra over a field k and of Gelfand-Kirillov dimension one. Then R satisfies a polynomial identity. Consequently, if N is the prime radical of R, then N is nilpotent and R/N is a finite module over its Noetherian centre.

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.

In 1955 [1], Amitsur determined all the finite groups G that can be embedded in the multiplicative group T* = GL(1, T) of some division ring T of characteristic zero. If G can be so embedded, then the rational span of G in T is a division ring of finite dimension over ℚ, and G acts on it by right multiplication in such a way that every non-trivial element operates fixed point freely. The finite groups admitting such a representation had earlier been determined by Zassenhaus[24; 4, XII. 8], and Amitsur begins by quoting Zassenhaus' results, which show in particular that the only perfect group that can be embedded in the multiplicative group of a division ring of characteristic zero is SL(2,5). The more difficult part of Amitsur's paper is the determination of the possible soluble groups. Here the main tool is Hasse's theory of cyclic algebras over number fields.

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).

The notion of fibre product in a category is quite basic and has been studied by many authors. Also in ring theory, it is known that the fibre product is useful in the construction of examples. (See for example [3], [4] and references of [1].) Unfortunately, most such examples are non-noetherian and so are unsatisfactory from the viewpoint of commutative algebra.

In this paper we study small cancellation quotients of free products. In particular we calculate their cohomology and, via a theorem of Serre, classify their finite subgroups. The results obtained are the natural analogues of the corresponding results for small cancellation quotients of free groups obtained in Huebschmann [6] (but see also Collins-Huebschmann[3]). They also generalize results of McCool[9] on elements of finite order in quotients of free products and similar conclusions have been obtained by Howie [5] for one-relator quotients of free products of locally indicable groups.

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.