Trevor Evans in (8) introduced postulates for a non-associative number theory similar to, but less general than, those of A. Robinson (9). Evans' number theory is also non-commutative under addition and multiplication, but an alternative equality axiom also suggested by Robinson leads to a number theory which is commutative under addition and still non-associative except in the special case:

We show that the class of Banach algebras A isomorphic with normclosed (non-self-adjoint) subalgebras of is characterized by the condition that the norms of polynomials in A be dominated by the norms of the same polynomials in .

Fundamental statements for (associative) rings are that (a) the endomorphisms of each commutative group (U, +) form a ring and (b) eachring may be embedded in such a ring of endomorphisms. In order to generalise these theorems to groups and rings whose addition may not be commutative, one has to deal with partial endomorphisms. But thesering-theoretical Theorems 4a and 4b turn out to be specialisations of similarones for semi-near-rings, near-rings and semirings, developed here inSection 2 after some preliminaries on semi-near-rings in Section 1. A glance at Definition 1 and the ring-theoretical theorems and remarks at the end of Section 2 may give more orientation.

The purpose of this paper is to expose a method which will match a function f(z) existing in a domain D to a formal series whose radius of convergence may be zero. This matching process has to be done in a “natural way”, and has to be “regular”, which means that if a power series converges absolutely in the circle E = {z | |z|<r} then the summability function f(z) produced by our method in the domain D and matched to will coincide with in the domain E∩D. Euler, in his time, matched the function to the power series .

Let G be a finite group, p be a prime and K be a field of characteristic p. Let

be a decomposition of the group ring of G over K as a sum of indecomposable two-sided ideals. An irreducible K(G)-module is said to be in the block Bi if it occurs as a composition factor of Bi. The block containing the trivial K(G)-module is called the principal block of G.

In (12) we introduced the concept of essential separability and used it to define two classes of locally convex spaces, δ-barrelled spaces and infra-δ-spaces, which serve as domain and range spaces respectively in certain closed graph theorems (12, Theorems 3 and 7). In this note we continue the study of these ideas. The relevant definitions are reproduced below.

We follow the notations and basic equations of Chen (2). Let M be a surface immersed in an m-dimensional space form Rm(c) of curvature c = 1, 0 or −1. We choose a local field of orthonormal frames e1, …, em in Rm(c) such that, restricted to M, the vectors e1, e2 are tangent to M. Let ω1, …, ωm be the field of dual frames. Then the structure equations of Rm(c) are given by

Train algebras were first introduced by Etherington in (1) and proved very useful in dealing with problems in mathematical genetics. The types of algebras which arose were commutative, non-associative and finite-dimensional. It proved convenient in the general theory to regard them as defined over the complex numbers. We remind the reader of some basic definitions. A baric algebra is one which admits a non-trivial homomorphism into its coefficient field K. A (principal) train algebra is baric and has a rank equation in which the coefficients of a general element x depend only on its baric value, generally called the weight of x. A special train algebra (STA) is a baric algebra in which the nilideal is nilpotent and all its right powers are ideals; the nilideal being the set of elements of A of weight zero. In (2) Etherington showed that in a baric algebra one can always take a very simple basis consisting of a distinguished element of unit weight and all other basis elements of weight zero.

Let G be a finite group and let S be a subgroup of G with core

We say that (G, S) has property (*) if there exists x ∈ G such that S ∩ x−1

Let n be an integer with n > 1. Jacobsthal (3) defines g(n) to be the least integer so that amongst any g(n) consecutive integers a + 1, a + 2, … a + g(n) there is at least one coprime with n. In other words, if

then

It is probably true that

where ω(n) denotes the number of different prime divisors of n, and Erdos (1) has pointed out that by the small sieve it is possible to show that there is a constant C such that

There have been several recent accounts of a theory dual to the well-known theory of primary decomposition for modules over a (non-trivial) commutative ring A with identity: see (4), (2) and (9). Here we shall follow Macdonald's terminology from (4) and refer to this dual theory as “ secondary representation theory ”. A secondary representation for an A-module M is an expression for M as a finite sum of secondary submodules; just as the zero submodule of a Noetherian A-module X has a primary decomposition in X, it turns out, as one would expect, that every Artinian A-module has a secondary representation.

In (1) a measure λ∈M(ℝ) is constructed and shown to satisfy the following:

(i) is absolutely continuous, where denotes the measure with for all Borel sets E,

(ii) λ*λ = λ2 is singular,

(iii) λk = λk−1*λ is absolutely continuous for K ≧6.

In an earlier paper (2) we considered the following question “If S is a cyclic subgroup of a finite group G and S ∩ F(G) = 1, where F(G) is the Fitting subgroup of G, does there necessarily exist a conjugate Sx of S in G with S ∩ Sx = l?” and we gave an affirmative answer for G simple or soluble. In this paper we answer the question affirmatively in general (in fact we prove a somewhat stronger result (Theorem 3)). We give an example of a group G with a cyclic subgroup S such that (i) no nontrivial subgroup of S is normal in G and (ii) no x exists for which S ∩ Sx = 1.

In a recent paper Segel (1) points out that the diverse techniques (which “comprise the core of the applied mathematicians art (or craft)”) of the applied mathematician, although in general reliably proven, are “rarely explicitly delineated but rather are transmitted indirectly and informally”. In his article Segel aims to clarify two such techniques, namely:

(i) Scaling—or how to choose dimensionless variables in such a way that the relative size of the various terms in an equation is explicitly indicated by the magnitudes of the dimensionless parameters which precede them,

(ii) Simplification—a procedure in which a term is neglected under the assumption that it is small, and the consistency of the assumption checked later.

Let W be a class of not necessarily associative rings which is universal in the sense that it is closed under homomorphic images and is hereditary to subrings. All rings considered will be assumed to belong to W. The notation I ⊲ R will mean I is an ideal of R. A relation σ on W will be called an H-relation if σ satisfies the properties:

(1) I σ R implies I is a subring of R.

(2) If I σ R and ø is a homomorphism of R, then IØ σ Rø.

(3) If I σ R and J is an ideal of R, then I ∩ J σ J.

In what follows the term C*-algebra will mean a complex C*-algebra with identity. We denote the identity element by 1. We shall also use the notation and terminology of Dixmier (3) without comment.

A continuous function φ on the unit circle is called badly approximable if ‖ φ − p ‖∞ ≧ ‖ φ |∞ for all polynomials p, where ‖ |∞ is the essential supremum norm. In (4), Poreda asked whether every continuous φ may be written φ = φW+φB, where φW is the uniform limit of polynomials (i.e. φW belongs to the disc algebra A) and φB is badly approximable. We call such a function φ decomposable. In (4), he characterised the badly approximable functions as those of constant non-zero modulus and negative winding number around the origin, i.e. ind (φ)<0. (See (3) for two new proofs of this result.) We show that the answer to Poreda's question is no in general, but give a necessary and sufficient condition for a given φ to have such a decomposition. Then we apply this criterion to solve an interpolation problem.

Let S be an inverse semigroup with semilattice of idempotents E. We denote by σ the minimum group congruence on S (6), and by τ the maximum idempotent-determined congruence on S (2). (Recall that the congruence η on S is called idempotent-determined if (e, x)∈ η and e ∈ E imply that x ∈ E.) In general τ ⊆ σ.

Let G be a non-discrete LCA group with dual group Γ. Denote by M(G) the usual convolution algebra of bounded Borel measures on G and Ma(G) those μ ∈ M(G) which are absolutely continuous with respect to mG—the Haar measure on G.

The theory of representing continuous linear operators on function spaces in terms of integrals has had a long and fruitful history, beginning with the Riesz representation theorem in 1909. If T is such an operator, then the standard representation is T (f) = ∫f dμ, where the integral is denned in diverse ways, depending on the nature of the set of functions and the nature of T.