By a variety of topological algebras we mean a class V of topological algebras of a fixed type closed under the formation of subalgebras, products and quotients (i.e. images under continuous homomorphisms yielding the quotient topology). In symbols, V = SV = PV = QV. if V is also closed under the formation of arbitrary continuous homomorphic images, then V is a wide variety. variety. As an example we have the full variety V = Modr (Σ), the class of all topological algebras of a fixed type τ obeying a fixed set Σ of algebraic identities. But not every wide variety is full, e.g. the class of all indiscrete topological algebras of a fixed type; in fact, as Morris observed (1970b), there exists a proper class of varieties of topological groups.

An analytic function f(z) is said to have a fixpoint ξ ≠ ∞ of multiplier 1 if f(ξ) = ξ, f'(ξ) = 1. The function then has an expansion .

R. A. Fisher (1930) has obtained approximate expressions for the probability of survival of a new mutant in a finite population of haploid individuals in which the generations are non-overlapping. Suppose that we have M haploid individuals which are either of genotype a or A, and suppose that a has a small selective advantage over A so that the relative numbers of offspring have expectations proportional to 1 + s and 1 respectively, where s is small and positive. If each generation is produced by binomial sampling with probabilities proportional to the numbers of a and A individuals in the previous generation multiplied by their respective selective values, and if initially there is only one individual of type a, the probability of the population ultimately becoming entirely of this type is approximately so long as s2M is small. This also holds when s is small and negative.

A right module A over a ring R is called finitely injective if every diagram of right R-modules of the form where X is finitely generated and the row is exact, can be imbedded in a commutative diagram The finitely injective modules turn out to have the nice property of being ‘strongly pure’ in all the containing modules and, in particular, are absolutely pure in the sense of Maddox [8]. For this reason we call them strongly absolutely pure (for short, SAP) modules. Several other characterizations of the SAP modules are obtained. It is shown that every module admits a ‘SAP hull’. The concepts of finitely M-injective and finitely quasi-injective modules are then investigated. A subclass of finitely quasi-injective modules, called the strongly regular modules, is studied in some detail and it is shown that, if R/J is right Noetherian, then the strongly regular right R-modules coincide with the semi-simple R-modules.

The recently developed theory of partial actions of discrete groups on C*-algebras is extended. A related concept of actions of inverse semigroups on C*-algebras is defined, including covariant representations and crossed products. The main result is that every partial crossed product is a crossed product by a semigroup action.

Richmond and Szekeres (1977) have conjecturned that certain of the coefficients in the power series expansions of certain infinite products vanish. In this paper, we prove a general family of results of this nature which includes the above conjectures.

In the present note, I investigate the question whether automorphic functions of the Fuchsian type can satisfy algebraic differential equations of the second order. Under rather general restrictions on the corresponding group, it is proved that there is no such differential equation.

We establish the bounds of Marcinkiewicz integrals associated to surfaces of revolution generated by two polynomial mappings on Triebel–Lizorkin spaces and Besov spaces when their integral kernels are given by functions $\unicode[STIX]{x1D6FA}\in H^{1}(\text{S}^{n-1})\cup L(\log ^{+}L)^{1/2}(\text{S}^{n-1})$. Our main results represent improvements as well as natural extensions of many previously known results.

If G is an elementary amenable group of finite Hirsch length h, then the quotient of G by its maximal locally finite normal subgroup has a maximal solvable normal subgroup, of derived length and index bounded in terms of h.

Let Zn be the numer of individuals in the nth generation of a discrete branching process, descended from a single a singel ancestor, for which we put It is well known that the probability generating function of Zn is Fn(s), the n-th functional iterate of F(s), and that if m = EZ1 does not exceed unity, then lim (Harris [1], Chapter 1). In particular, extinction is certain.

In this note we present some results on relationships between certain verbal subgroups of metabelian groups. To state these results explicitly we need some notation. As usual further [x, 0y] = x and [x, ky] = [x, (k—1)y, y] for all positive integers k. The s-th term γs(G) of the lower central series of a group G is the subgroup of G generated by [a1, … as] for all a1, … as, in G. A group G is metabelian if [[a11, a2], [a3, a4]] = e (the identity element) for all a1, a2, a3, a4, in G, and has exponent k if ak = e for all a in G.

This is the first of several papers which grew out of an attempt to provide C (X, Y), the family of all continuous functions mapping a topological space X into a topological space Y, with an algebraic structure. In the event Y has an algebraic structure with which the topological structure is compatible, pointwise operations can be defined on C (X, Y). Indeed, this has been done and has proved extremely fruitful, especially in the case of the ring C (X, R) of all continuous, real-valued functions defined on X [3]. Now, one can provide C(X, Y) with an algebraic structure even in the absence of an algebraic structure on Y. In fact, each continuous function from Y into X determines, in a natural way, a semigroup structure for C(X, Y). To see this, let ƒ be any continuous function from Y into X and for ƒ and g in C(X, Y), define ƒg by each x in X.

An example is constructed of a locally finite variety of non-associative algebras which satisfies the maximal condition on subvarieties but not the minimal condition. Based on this, counterexamples to various conjectures concerning varieties generated by finite algebras are constructed. The possibility of finding a locally finite variety of algebras which satisfies the minimal condition on subvarieties but not the maximal is also investigated.

A pseudo-Riemannian manifold is said to be timelike (spacelike) Osserman if the Jordan form of the Jacobi operator Kx is independent of the particular unit timelike (spacelike) tangent vector X. The first main result is that timelike (spacelike) Osserman manifold (M, g) of signature (2, 2) with the diagonalizable Jacobi operator is either locally rank-one symmetric or flat. In the nondiagonalizable case the characteristic polynomial of Kx has to have a triple zero, which is the other main result. An important step in the proof is based on Walker's study of pseudo-Riemannian manifolds admitting parallel totally isotropic distributions. Also some interesting additional geometric properties of Osserman type manifolds are established. For the nondiagonalizable Jacobi operators some of the examples show a nature of the Osserman condition for Riemannian manifolds different from that of pseudo-Riemannian manifolds.

A semigroup S is called regular if a ∈ aSa for every element a in S. The elementary properties of regular semigroups may be found in A. H. Clifford and G. B. Preston [1]. A semigroup S is called orthodox if S is regular and if the idempotents of S form a subsemigroup of S.

We extend earlier work relating asphericity and Euler characteristics for finite complexes whose fundamental groups have nontrivial torsion free abelian normal subgroups. In particular a finitely presentable group which has a nontrivial elementary amenable subgroup whose finite subgroups have bounded order and with no nontrivial finite normal subgroup must have deficiency at most 1, and if it has a presentation of deficiency 1 then the corresponding 2-complex is aspherical. Similarly if the fundamental group of a closed 4-manifold with Euler characteristic 0 is virtually torsion free and elementary amenable then it either has 2 ends or is virtually an extension of Z by a subgroup of Q, or the manifold is asphencal and the group is virtually poly- Z of Hirsch length 4.

Let L be a finite dimensional Lie algebra over the Field F. We denote by (L) the lattice of all subalgebras of L. By a lattice isomorphiusm (whicn we abbrevite to -isomorphism) of L onto a Lie algebra M over the same field F, we mean an isomorphism of (L) onto (M). It is possible for non-isomorphic Lie algebras to -isomorphic, for example, the algebra of real vectors with product the vector product is -isomorphic to any 2-dimensional Lie algebra over the field of real numbers.

Suppose S is a semitopological semigroup. We consider various subspaces of C(S) and determine what topological algebraic structure can be introduced into the spaces of means on the subspaces and into the spectra of the C*-sub-algebras of C(S) they generate.

A net (xy) in a vector lattice is unbounded order convergent (uo-convergent) to 0 if u ∧ |xv| order converges to 0 for all u ≧ 0. We consider, in a Banach lattice, the relationship between weak and uo-convergence. We characterise those Banach lattices in which weak convergence implies uo-convergence and those in which uo-convergence of a bounded net implies weak convergence. Finally we combine the results to characterise those Banach lattices in which weak and uo-convergence coincide for bounded nets.

Let {Ln} be a sequence of positive linear operators defined on C[a, b] of the form where xnk ∈ [a, b] for each k = 0, 1,…, n = 1, 2,…. The convergence properties of the sequences {Ln(f)} to for each f ∈ C[a, b] have been the object of much recent research (see e.g. [4], [8], [11], [13]). In many cases positive linear operators of the form (1) give rise to interesting summability matrices A = (ank(x)) and vice- versa.