We give explicit examples of asymmetric Riemann surfaces (that is, Riemann surfaces with trivial conformal automorphism group) for all genera g ≥ 3. The technique uses Schreier coset diagrams to construct torsion-free subgroups in groups of signature (0; 2,3,r) for certain values of r.

We develop some ideas contained in the author's paper [8] which was, in turn, inspired by Bierlein and Stich [5]. The main body of the present paper is divided into three sections. Section 2 is concerned with some vector-lattice-theoretical results. They are then applied to extensions of quasi-measures and measures in Sections 3 and 4, respectively.

Let X be a vector lattice, let x ε X+ and let S be a non-empty set. Theorems 1 and 2 describe some properties of the convex set

(see Section 2 for the definition of the sum above). The extreme points of Dx,s are characterized in terms of the components of x. It is also shown that if X has the principal projection property and S is countable, then extr Dx,s is, in some sense, large in Dx,s. Furthermore, for finite S, each point in Dx,s is then a sσ-convex combination of extreme ones.

Many special cases of the equation x2+C= yn where x and y are positive integers and n≥3 have been considered over the years, but most results for general n are of fairly recent origin. The earliest reference seems to be an assertion by Fermat that he had shown that when C=2, n=3, the only solutions are given by x = 5, y = 3; a proof was published by Euler [1]. The first result for general n is due to Lebesgue [2] who proved that when C = 1 there are no solutions. Nagell [4] generalised Fermat's result and proved that for C = 2 the equation has no solution other than x = 5, y = 3, n = 3. He also showed [5] that for C = 4 the equation has no solution except x = 2, y = 2, n = 3 and x = 11, y = 5, n = 3, and claims in [6] to have dealt with the case C = 5. The case C = -1 was solved by Chao Ko, and an account appears in [3], pp. 302–304.

In this paper, the situation we shall be concerned with is that of a ring R, with a ring monomorphism α: R → R, which will not be assumed to be surjective.

Much work has been done on the skew polynomial ring R[x, α] and the skew Laurent polynomial ring R[x, x-1, α], where α is an automorphism—see [3] for example. However, the fact that α is not surjective renders the study of these objects much more difficult.

Let

It is familiar that

(1) In the addition theorem [3, p. 363]

where

take θ = π and replace v by v−½. Since

we get

In 1926, I. J. Schur proved the following theorem on partitions [3].

The number of partitions of n into parts congruent to ±1 (mod 6) is equal to the number of partitions of n of the form 1 + …+bs = n, where bi–bi+1 ≧ 3 and, if 3 ∣ bi, then bi–bi+1 > 3.

Schur's proof was based on a lemma concerning recurrence relations for certain polynomials. In 1928, W. Gleissberg gave an arithmetic proof of a strengthened form of Schur's theorem [2]; however, the combinatorial reasoning in Gleissberg's paper becomes very intricate.

In this note relations between the structure of a finite group G and ringtheoretical properties of the group algebra FG over a field F with characteristic p > 0 are investigated. Denoting by J(R) the Jacobson radical and by Z(R) the centre of the ring R, our aim is to prove the following theorem generalizing results of Wallace [10] and Spiegel [9]:

Theorem. Let G be a finite group and let F be an arbitrary field of characteristic p > 0. Denoting by BL the principal block ideal of the group algebra FG the following statements are equivalent:

(i) J(B1) ≤ Z(B1)

(ii) J(B1)is commutative,

(iii) G is p-nilpotent with abelian Sylowp-subgroups.

Let R be an associative ring with identity, X a set of noncommuting variables, = {αx} x ∈ X a set of automorphisms αx of R and R {X} the -twisted free associative algebra on X over R. Let Y be another set of noncommuting variables, ℬ = {βy}y∈Y a set of automorphisms βy of R {X} and S = (R{X})ℬ {Y} the ℬ-twisted free associative algebra on Y over R{X}. Next, let X1 be a set of noncommuting variables, for each l = 1,2,…. We form the free associative algebra S1 = S{X1}on Xl over S and inductively, we form the free associative algebra Sl+1 = Sl{Xl+1} on Xl+1 over Sl, l = 1,2,….

The commutator [a, b] of two elements a and b in a group G satisfies the identity

ab = ba[a, b].

The subgroups we study are contained in the commutator subgroup G′, which is the subgroup generated by all the commutators.

The group G is covered by a well-known set of normal subgroups, namely the normal closures {g}G of the cyclic subgroups {g} in G. In a similar way one may associate a subgroup K(g) with each element g, by defining K(g) to be the subgroup generated by the commutators [g, x] as x takes all values in G. These subgroups generate G′ (but do not cover G′ in general), and are normal in G in consequence of the identical relation

(A) [g, x]Y = [g, y]−1[g, xy]

holding for all g, x and y in G. (By ab we mean b−1ab.) It is easy to see that

{g}G = {g, K(g)}.

Let T = (tmn) be a regular matrix, and CTbe its bounded convergence field. Necessary and sufficient conditions for CT to contain the space of almost convergent sequences are well known. (See, e.g., [7, p.62]). G. M. Petersen has suggested as a problem for research the discovery of necessary and sufficient conditions for the reverse inclusion: When is CT contained in the space of almost convergent sequences? [7, p. 137, research problem 9]. In this paper we deal with this question in a more general context. First we need some notation.

Let (S, ≤) be a poset (partially ordered set), A(S) = Aut(S, ≤) its automorphism group and G ⊆ A(S) a subgroup. In the literature, various authors have studied sufficient conditions on G and the structure of (S, ≤) which imply that G is simple or perfect. Let us call (S, ≤) doubly homogeneous if each isomorphism between two 2-subsets of 5 extends to an isomorphism of (S, ≤). Higman [8] proved that if (S, ≤) is a doubly homogeneous chain then B(S), the group of all automorphisms of (S, ≤) with bounded support, is simple, and each element of B(S) is a commutator in B(S). Droste, Holland and Macpherson [5] showed that if (S, ≤) is a doubly homogeneous tree then its automorphism group again contains a unique simple normal subgroup in which each element is a commutator. Dlab [3] established similar results for various groups of locally linear automorphisms of the reals. Further results in this direction are contained in Glass [7]. It is the aim of this note to establish a common generalization and sharpening of the previously mentioned results.

In their paper [1], Campbell and Jamison attempted to give necessary and sufficient conditions for a weighted composition operator on an L2 space to be normal, and to be quasinormal. Those conditions, specifically Theorems I and II of that paper, are not valid (see [2] for precise comments on the other results in that paper). In this paper we present a counterexample to those theorems and state and prove characterizations of quasinormality (Theorem 1 below) and normality (Theorem 2 and Corollary 3 below). We also discuss additional examples and information concerning normal weighted composition operators which contribute to the further understanding of this class.

In a recent paper, Boudjelkha and Diaz [1] have considered the solutions of the Dirichlet problems for Laplace's equation

and for Helmholtz's equation

in¼–ℝn. The purpose of this brief note is to show that their formulae may be derived easily by the use of the theory of multiple Fourier transforms.

The present paper incorporates a preliminary study of a new generalization of several known polynomial systems belonging to (or providing extensions of) the families of the classical Jacobi, Hermite and Laguerre polynomials. It is shown how suitable specializations will yield a number of known or new results in the theory of the special functions considered.

Let G be a finite additive abelian group, and suppose that A and B are subsets of G. We say that G = A⊕B if every element g ∈ G can be uniquely written in the form g = a + b, where a ∈ A, b ∈ B. The study of such decompositions (usually called factorizations in the literature) was initiated by G. Hájos [3] in connection with his solution to a problem of Minkowski in the geometry of numbers.

Let ℂn,n denote the space of n × n matrices with complex entries and let ℋn denote the set of n × n hermitian matrices. Given any matrix A∊ℂn,n, the Lyapunov transformation corresponding to A is defined by ℐA(H) = AH+HA*, where H∊ℋn. Let PSD(n) be the set of all n × n hermitian positive semidefinite matrices. Taussky [8, 9] raised the problems of determining

and

Both of these problems seem to be difficult.

Let E be an arbitrary (non-empty) set and S the restricted symmetric group on E, that is the group of all permutations of E which keep all but a finite number of elements of E fixed. If Φ is any commutative ring with unit element, let Γ = Φ(S) be the group algebra of S over Φ,Γ ⊃ Φ and let M be the free Φ-module having E as Φ-base. The “natural” representation of S is obtained by turning M into a Γ-module in the obvious manner, namely by writing for α∈S, λ1∈Φ,

In this paper the author continues the search for a suitable integral transform that can be applied to certain boundary value problems involving the Helmholtz equation and the condition of radiation. The transform in question must be capable of eliminating the r-dependence appearing in the partial differential equation

In this paper all sets considered are assumed to be compact subsets of Euclidean Space En. A number of results concerning the total edge-lengths of polyhedra have been given by various authors, many of which are mentioned in references in [1]. In [1], it was conjectured that all polytopes inscribed in the unit sphere and containing its centre have total edge-length greater than 2n. This was proved true for simplicial polytopes and shown to be best possible in the sense that there exist simplices with the stated property and with total edge-length arbitrarily close to 2n. In this paper we shall show that the bound is not always best possible if the magnitudes of the faces of such polytopes are restricted and we shall also give some related results on surface areas. This work was carried out while the author was a research student at Royal Holloway College, London and is a revised version of part of the author's thesis approved for the Ph.D. degree.

In the sequel, given k, n ∈ ℕ, p ∈ [1, ∞] and a compact real interval I, we denote by Wk, p(I, ℝn) (simply by Wk,p(I if n = 1) the space of all functions u ∈ Ck−1(I, ℝn) such that uk−1 is absolutely continuous in I and u(k) ∈ Lp (I, ℝn).

Very recently, in [11], J. R. L. Webb and S. C. Welsh obtained the following existence result.