Let R be a hyperbolic Riemann surface and W an open subset of R with ∂W piecewise analytic. Denote by the space of Dirichlet finite Tonelli functions on R and by π the harmonic projection of . Consider the relative HD–class on W, HD(W;∂W) = {u∈ │ u │ W∈HD(W) and u │ R\W = 0}. The extremization operation μis the linear mapping of HD(W;∂W) into HD(R) defined by μ. Since π preserves values of functions at the Royden harmonic boundary, the maximum principle implies that μis an order preserving injection and that Mμ is an isometry with respect to the supremum norms.

Let T be a linear operator acting in a Banach space X. It has been shown by Smart [5] and Ringrose [3] that, if X is reflexive, then T is well-bounded if and only if it may be expressed in the form

where {E(λ)} is a suitable family of projections in X and the integral exists as the strong limit of Riemann sums.

Denote by

the Euler product, and by

the partition generating function. More generally, if k is any integer, put

so that p(n) = p−1(n). In [3], Atkin proved the following theorem.

Since Helmer's 1940 paper [9] laid the foundations for the study of the ideal theory of the ring A(ℂ) of entire functions, many interesting results have been obtained for the rings A(X) of analytic functions on non-compact connected Riemann surfaces. For example, the partially ordered set Spec (A(ℂ) of prime ideals of A(ℂ) has been described by Henrikson and others [2], [10], [11]. Also, it has been shown by Ailing [4] that Spec(A(ℂ))sSpec(A(X)) as topological spaces for any non-compact connected Riemann surface X. Many results on the valuation theory of A(X) have also been obtained [1], [2]. In this note we show that a large portion of the results on the rings A(X) extend to the W-rings with complete principal divisor space which were defined by J. Klingen in [15], [16]. Therefore, many properties of A(ℂ) are shared by its non-archimedian counterparts studied by M. Lazard, M. Krasner, and others [8], [17], [18].

In this paper a characterisation of the regularity of a normed algebra A is given in terms of retractions onto A** from A4*. The second dual A** of a normed algebra A possesses two natural Banach algebra multiplications, say ° and *. Each of ° and * extends the original algebra multiplication on A; see (2). An algebra A is called regular if and only if F * G = F ° G for all F, G ∈ A**. See (1). The existing results in the Arens regularity theory can be found in a recent survey (2). Denoting the nth dual of A by An*, and en the natural embedding of An* in its second dual A(n+2)*, we can naturally represent the second dual A** of A as a Banach space retract of A4* in two different ways:

Our main results say that A** is in fact a Banach algebra retract of A4* (i.e. the maps involved are homomorphisms) in either of these cases if and only if A is regular.

In 1856 Hermite showed how to determine by purely rational operations the number of zeros of a given polynomial lying in a specified half plane [1]: one inspects the signature of a certain Hermitian form. This type of result is still of interest for practical applications, and several authors have provided alternatives to Hermite's original, highly computational proof (for example [2, 3]). Recently V. Pták and the author gave a simple matrix-theoretic proof and generalization of a class of Hermite-type theorems [4]. This class included the Schur-Cohn test for zeros in a circle, but not, to our regret, the original theorem of Hermite. The purpose of this note is to show that a slight modification of our method does indeed provide a simple proof of Hermite's theorem.

We write e(x) for e2πix, ∥x∥ for the distance of x from the nearest integer and use A ≫ B to mean |A|<c |B|, where c is a positive constant depending at most on k and e. The letter p always denotes a prime number; P2 represents a number with precisely two prime factors. We continue the investigations started in [6] and will make many references to the analysis there. Here we prove the following theorems.

In his paper [3], Ky Fan asked whether if f is a convex univalent function in the unit disk, with f(0) = 0 and f'(0) = 1, then is it true that the set of f(A) is a convex set of operators, when A runs through all proper contractions on a Hilbert space? We answer this question in the negative.

Let S be a regular semigroup. Given x ∈ S, we shall say that a ∈ S is an associate of x if xax = x. The set of associates of x ∈ S will be denoted by A(x). Now suppose that S has an identity element 1. Let H1 denote the group of units of S. Then we say that u ∈ S is a unit associate of x whenever u ∈ A(x)∩Hl. In what follows we shall write U(x) = A(x)∩=H1, and we shall say that S is unit regular [1, 3] if (∀ x∈S)U(x)≠ ∅. Examples of unit regular semigroups include the full transformation semigroup on a finite set [1] and the semigroup of endomorphisms of a finite–dimensional vector space [3]. In this paper we shall be concerned with semigroups that are unit orthodox (i.e. unit regular and orthodox), and we shall describe completely the structure of those semigroups that are uniquely unit orthodox (i.e. orthodox and uniquely unit regular in the sense that, for every x∈S, the set U(x) is a singleton). It is worthy of mention that neither of the examples cited above is of this type.

Let C be the Cayley algebra denned over the real field. If, for given elements α, β, and γ of a quaternion subalgebra of C, α = βγ, it follows, by associativity, that for any nonzero element δ of the same quaternion subalgebra, α = (βδ)(δ-1γ). For Cayley numbers ζ ξ, and η with ζ = ξη, the relation ζ = (ξδ)(δ-1η) in general only holds when δ is a nonzero real number. Because of the existence of factorization results [1, 2] in the orders of C, the question naturally arises of whether it is possible to choose one-to-one mappings, θ and φ, of C onto itself such that ζ = θξ. φη whenever ζ = ξη. To discuss this question, we make the following definition.

The purpose of this paper is to study the following two questions.

(1) When does the group algebra of a soluble group have infinite dimensional irreducible modules?

(2) When is the group algebra of a torsion free soluble group primitive?

In relation to the first question, Roseblade [13] has proved that if G is a polycyclic group and k an absolute field then all irreducible kG-modules are finite dimensional. Here we prove a converse.

Let

be an indefinite quadratic form with real coefficients. A well-known result, due to Birch, Davenport and Ridout [1], [5] and [6], states that if n ≥21 then for any ε > 0 there is an integer vector x ≠O such that

Recently [3] we have quantified this result, obtaining a function g(n) such that g(n)→ ½ as n n→ ∞ and such that for any η > 0 and all large enough X there is an integer vector x satisfying

where |x| = max |xi|and the implicit constant in Vinogradov's ≪-notation is independent of X.

We define the integral operators and as

and

where

and Wk, u and Mk, u are the Whittaker's confluent hyper-geometric functions. These operators, in their slightly less general form, have been dealt with in [2] and [4]. There the authors have used the fact that these integral operators can be expressed as compositions of the Kober's fractional integral operators and the modified Laplace operator. Then these operators are inverted accordingly. Generally, this type of technique has been very useful for inverting many kinds of integral equations; and a good account of the procedures involved is given [5].

If X is a topological space then S(X) will denote the semigroup, under composition, of all continuous functions from X into X. An element f in a semigroup is regular if there is an element g such that fgf = f. The regular elements of S(X) will be denoted by R(X). Elements f and g are inverses of each other if fgf = f and gfg = g. Every regular element has an inverse [1]. If every element in a semigroup has a unique inverse then the semigroup is an inverse semigroup. In this paper we examine maximal inverse subsemigroups of S(X).

In this paper the author considers the problem of finding a formula of inversion for the integral transform defined by the equation

where a >0, k > 0 and r-1f(r) εL (a, ∞). This transform appeared in connection with an earlier investigation [4] in which an attempt was made to devise an integral transform that could be adapted to the solution of certain boundary value problems involving the space form of the wave equation and the condition of radiation:

En algèbre non-commutative, on dit qu'un anneau noethérien A est local si:

(i) le radical de Jacobson M de A est un idéal maximal,

(ii) ∩ Mn = (0),

(iii) A/M est artinien simple.

Dans [9], Walker definit un anneau local régulier comme un anneau local A dont le radical de Jacobson M est engendré par une A-suite centralisante x1; x2, …, xt, [4], et demontre alors que:

t = cldim A = Kdim A = rgldim A = pdAA/M.

The origin of the theory of averaging operators is explained in [1]. The theory has been developed on spaces of continuous functions that vanish at infinity by Kelley in [3] and on the Lp spaces of measure theory by Rota [5]. The motivation for this paper arose out of the latter paper. The aim of this paper is to prove a generalisation of Rota's main representation theorem (every average is a conditional expectation) in the context of a ‘non commutative integration’. This context is as follows. Let be a finite von Neumann algebra and ϕ a faithful normal finite trace on such that ϕ(I) = 1, where I is the identity of . We can construct the Banach spaces Lp (, ϕ), where 1 ≤ p < °, with norm ∥x∥p = ϕ(÷x÷p)1/p, of possibly unbounded operators affiliated with , as in [9]. We note that is dense in Lp(, ϕ). These spaces share many of the features of the Lp spaces of measure theory; indeed if is abelian then Lp(,ϕ) is isometrically isomorphic to Lp of some measure space.

Bullett and Macdonald [1] have used power series to simplify the statement and proof of the Adem relations for Steenrod cohomology operations. In this paper I give a similar treatment of May's generalized Adem relations [4, §4] and of the Nishida relations ([6], [2, 1.1.1(9)], [5, 3.1(7)]). Both sets of relations apply to Dyer-Lashof operations in E∞, spaces such as infinite loop spaces ([3], [2, I.I]) and in H^ ring spectra ([5, §3]).

In his book [6] Grätzer sets a guideline for the research on Stone algebras; in view of his and Chen's triple characterization of Stone algebras [3], he considers a problem for these algebras solved if it can be reduced to a problem for Boolean algebras and one for distributive lattices with 1. In the same book he lists as Problem 53: describe the projective Stone algebras. For a Stone algebra L, let BL be its centre and DL its dense set. In order that L be projective, BL has to be a Boolean retract of BF for some free Stone algebra F, and DL has to be a retract of DF in the category of distributive lattices with 1. These conditions are, however, not sufficient, so one hopes to characterize the projective Stone algebras by adding further conditions, and in this spirit we in fact arrive at a description of these algebras. We also show, however, that every such algebra is a retract of some DF, F a free Stone algebra, so we conclude that there is no nice structural characterization of the projective Stone algebras along the line of Grätzer's programme.

Let F be any field. Denote by the class of all groups G such that every irreducible FG-module has finite dimension over F and by the class of all groups G such that every irreducible FG-module has finite dimension over its endomorphism ring. Clearly