Let be a complex Banach algebra, possibly non-commutative, with identity e. By a Reynolds operator we mean here a bounded linear operator T: → satisfying the Reynolds identity for all x, y ∈ . We prove that under certain conditions the resolvent of T, R(p, T) = (pI−T)−1, has the form where s = −log(e−Te) and exp y = e+y+y2/2!+….

Let G, N be groups, let A(N) be the automorphism group of N and let I(N) be the subgroup of inner automorphisms. A homomorphism θ: G → A(N)/I(N) will be denoted by (G, N, θ) and called an abstract kernel. (G, N, θ) induces in an obvious manner a structure of a (left) G-module on the centre C of N. A well known construction of Eilenberg and MacLane [1, § 7–9] assigns to (G, N, θ) its obstruction Obs (G, N, θ) ∈H3(G, C). This assignment is such that if C is an arbitrary G-module then every element of H3(G, C) is of the form Obs (G, N, θ) for a suitable abstract kernel (G, N, θ).

The theorem of Trevor Evans [1] that every countable semigroup can be embedded in a two-generator semigroup becomes obvious in automata theory as the statement that every countable automaton can be embedded in one with binary inputs. Standard techniques of automata theory [1], [3] yield a proof of the Evans Theorem using wreath products, as in Neumann [4].

In the present paper we shall consider some subgroups of (increasing) autohomeomorphisms of the closed real interval <0, 1>; mainly because of brevity, we shall defer discussing possible generalizations of our results to more general ordered fields.

Let Fq denote the finite field with q elements, Zm the residue class ring Z/mZ. It is known that the projective linear groups G = PSL2(Fq) and PGL2(Fq) (q a prime-power ≥ 4) are characterised among finite insoluble groups by the property that, if two cyclic subgroups of G of even order intersect non-trivially, they generate a cyclic subgroup (cf. Brauer, Suzuki, Wall [2], Gorenstein, Walter [3]). In this paper, we give a similar characterisation of the groups G = PSL2 (Zþt+1) and PGL2 (Zþt+1) (p a prime ≥ 5, t ≥ 1).

By an ordered semigroup we mean a semigroup with a simple order which is compatible with the semigroup operation. Several authors, for example Alimov [1], Clifford [2], Conrad [4] and Hion [7], studied the archimedean property in some special kinds of ordered semigroups. For a general ordered semigroup, Fuchs [6] defined the archimedean equivalence as follows: a ~ b if and only if one of the four conditionsholds for some positive integer n.

An integral on a locally compact Hausdorff semigroup S is a nontrivial, positive linear function μ on the space K(S) of real-valued continuous functions on S with compact support. If S has the property: is compact whenever A is compact subset of S and s ∈ S, then the function fa defined by fa(x) = f(xa) is in K(S) whenever f ∈ K(S) and a ∈ S An integral on a locally compact semigroup S with the property (P) is said to be right invariant if μ(fa) = μ(f) for all f ∈ K(S) and a ∈ S.

From the proof of Theorem 2 of [5] it follows that for every positive integer k there exist infinitely many primes p in the arithmetical progression ax + b (x = 0, 1, 2,…), where a and b are relatively prime positive integers, such that the number 2p−1 − 1 has at least k composite factors of the form (p − 1)x + 1. The following question arises:

For any given natural number k, do there exist infinitely many primes p such that the number 2p−1 − 1 has k prime factors of the form(p − 1)x + 1 and p ≡ b (mod a), where a and b are coprime positive integers?

In accordance with customary notation, Jv(t) denotes the Bessel function of the first kind and order v, jvk its kth positive zero and jv, 0 = 0.

The object of this note is to prove that

In [2], R. C. James proved that a weakly closed subset X of a real Banach space is weakly compact if and only if each continuous linear form attains its supremum on X. He also extended the result to the locally convex case, and, in [5], J. D. Pryce gave a simplified proof of the general result that is recorded below for reference in the sequel.

Many of the techniques and notions used to study various important theorems in locally convex spaces are not effective for general linear topological spaces. In [4], a study is made of notionsin general linear topological spaces which can be used to replace barrelled, bornological, and quasi-barrelled spaces. The present paper contains a parallel study in the context of semiconvex spaces.

Let k be any algebraically closed field, and denote by k((t)) the field of formal power series in one indeterminate t over k. Let

so that K is the field of Puiseux expansions with coefficients in k (each element of K is a formal power series in tl/r for some positive integer r). It is well-known that K is algebraically closed if and only if k is of characteristic zero [1, p. 61]. For examples relating to ramified extensions of fields with valuation [9, §6] it is useful to have a field analogous to K which is algebraically closed when k has non-zero characteristic p. In this paper, I prove that the set L of all formal power series of the form Σaitei (where (ei) is well-ordered, ei = mi|nprt, n ∈ Ζ, mi ∈ Ζ, ai ∈ k, ri ∈ Ν) forms an algebraically closed field.

Let H be a group of characters on an (algebraic) abelian group G. In a natural way, we may regard G as a group of characters on H. In this way, we obtain a duality between the two groups G and H. One may pose several problems about this duality. Firstly, one may ask whether there exists a group topology on G for which H is precisely the set of continuous characters. This question has been answered in the affirmative in [1]. We shall say that such a topology is compatible with the duality between G and H. Next, one may ask whether there exists a locally compact group topology on G which is compatible with a given duality and, if so, whether there is more than one such topology. It is this second question (previously considered by other authors, to whom we shall refer below) which we shall consider here.

A free product sixth-group (FPS-group) is, roughly speaking, a free product of groups with a number of additional defining relators, where, if two of these relators have a subword in common, then the length of this subword is less than one sixth of the lengths of either of the two relators.

Britton [1,2] has proved a general algebraic result for FPS-groups and has used this result in a discussion of the word problem for such groups.

A Riesz operator is a bounded linear operator on a Banach space which possesses a Riesz spectral theory. These operators have been studied in [5] and [6]. In §2 of this paper we characterise Riesz operators in terms of their resolvent operators. In [6] it was shown that every Riesz operator on a Hilbert space can be decomposed into the sum of compact and quasi-nilpotent parts. §3 contains an example to show that these parts cannot, in general, be chosen to commute. In §4 the eigenset of a Riesz operator is defined. It is a sequence of quadruples each of which consists of an eigenvalue, the corresponding spectral projection, index and nilpotent part. This sequence satisfies certain obvious conditions, and the question arises of the existence of a Riesz operator which has such a sequence as its eigenset. We give an example of an eigenset which has no corresponding Riesz operator.

Let {Ui, Uij} be an inductive system of normed linear spaces Ui and continuous linear maps uij; Uj → Ui. (We write j ≺ i if uij: Uj → Ui.) An inductive limit of the system with respect to a class of spaces A in and maps f in is a space Uu in Uu and a system ui → Uu of maps in such that (i) whenever j ≺ i, and that (ii) if A is any space in and fi: Ui → A is any system of maps in for which then there is a unique map f: Uu → A in such that fi = fo ui for each i. If is the class of all vector spaces and is the class of linear maps, we obtain the algebraic inductive limit, which we denote simply by U. The usual choice is to take to be the class of locally convex spaces and the class of continuous linear maps; the inductive limit UL then always exists [1, § 16 C]. If is again the continuous linear mappings but contains only normed spaces, the corresponding inductive limit UN may not always exist. However, if in addition we require that contains just contractions (norm-decreasing linear mappings), then an inductive limit Uc will exist if every uij is a contraction [2]. We shall give a condition under which these limits coincide (as far as possible), and consider the corresponding condition for projective limits.

In deriving the approximate functional equation for certain Dirichlet series, one first establishes an identity for the function in terms of a partial sum of the series (e.g. see [1] and [2]). It is the purpose of this note to give a short proof of this identity for Hecke's Dirichlet series [1]. The proof is valid with only a few minor changes for the identity given by Chandrasekharan and Narasimhan [2, Lemma 2] for a much larger class of Dirichlet series. However, the brevity of the paper would be lost if we introduced the necessary terminology and notation.

In the analysis of mixed boundary value problems by Hankel transforms, one often encounters dual integral equations of the form

where I1 = (0, 1), I2 = (1, ∞); w1(x), w2(x) are weight functions, ψ(x) is the unknown function, and f(y), g(y) are functions continuously differentiate on I1 and I2 respectively. Many successful attempts have been made to solve (1.1) and (1.2). These are all discussed in a recent book by Sneddon [7]. As pointed out in a recent paper by Erdogan and Bahar [4], in mixed boundary value problems of semi-infinite domains involving more than one unknown function such as those arising in elastostatics, viscoelasticity, and electrostatics, the formulation will lead to a system of simultaneous dual integral equations which is a generalization of (1.1) and (1.2). These equations may be expressed as follows:

with i = 1, 2, …, n, where we use the notation