We prove the following results: (1) A quasi-metrizable space is compact if and only if every compatible quasi-metric has a quasi-metric left d—sequential completion. (2) A quasipseudometrizable space is countably compact if and only if every compatible quasi-pseudometric is pointwise bounded. (3) A quasi-pseudometrizable space is compact if and only if every compatible quasi-pseudometric is precompact.

This paper concerns a generalization of the Markov branching process that preserves the random walk jump chain, but admits arbitrary positive jump rates. Necessary and sufficient conditions are found for regularity, including a generalization of the Harris-Dynkin integral condition when the jump rates are reciprocals of a Hausdorff moment sequence. Behaviour of the expected time to extinction is found, and some asymptotic properties of the explosion time are given for the case where extinction cannot occur. Existence of a unique invariant measure is shown, and conditions found for unique solution of the Forward equations. The ergodicity of a resurrected version is investigated.

In this paper we give a complete description of diameter-preserving linear bijections on the space of affine continuous functions on a compact convex set whose extreme points are split faces. We also give a description of such maps on function algebras considered on their maximal ideal space. We formulate and prove similar results for spaces of vector-valued functions.

We provide a number of explicit examples of small volume hyperbolic 3-manifolds and 3-orbifolds with various geometric properties. These include a sequence of orbifolds with torsion of order q interpolating between the smallest volume cusped orbifold (q = 6) and the smallest volume limit orbifold (q → ∞), hyperbolic 3-manifolds with automorphism groups with large orders in relation to volume and in arithmetic progression, and the smallest volume hyperbolic manifolds with totally geodesic surfaces. In each case we provide a presentation for the associated Kleinian group and exhibit a fundamental domain and an integral formula for the co-volume. We discuss other interesting properties of these groups.

If G is a group, then K(G) is the set of commutators of elements of G. C is the class of groups such that G′ = K(G) is the minimal cardinality of any generating set of dG. We prove: Theorem A. Let G be a nilpotent group of class two such that G' is finite and d(G′) < 4.Then G < G.

Theorm B. Let G be a finite group such that G′ is elementary abelian of order p3. Then G ∈ C.

Theorem C. Let G be a finite group with an elementary abelian Sylow p-subgroup S, of order p2, such that S ⊆ K(G). Then S ⊆K(G).

This paper is concerned with version of Bernstein's inequality for Hausdroff locally compact Abelian groups. The ideas used are suggested by Exercise 12, p. 17 of Katznelson's book [4].

In this paper we use a result from graph theory on the characterization of the line graphs of the complete bigraphs to show that if n is any integer ≥ 2 then any finite linear space having p = n2 − n or p = n2 − n + 1 points, of which at least n2 − n have degree n + 1, and q ≤ n2 + n − 1 lines is embeddable in an FPP of order n unless n = 4. If n = 4 there is only one possible exception for each of the two values of p, and for p = n2 − n, this exception can be embedded in the FPP of order 5.

If V is a system of weights on a completely regular Hausdorff space X and E is alocally convex space, then CV0(X, E) and CVb (X, E) are locally convex spaces of vector-valued continuous functions with topologies generated by seminorms which are weighted analogues of the supremum norm. In this paper we characterise multiplication operators on these spaces induced by scalar-valued and vector-valued mappings. Many examples are presented to illustrate the theory.

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 ε > 0 be a small real parameter, let y, z be real m-dimensional and n-dimensional vectors respectively and let f, g be respectively real m-dimensional and n-dimensional vector functions of their arguments. This paper aims to discuss the following two problems in singular perturbations.

This article studies particular sequences satisfying polynomial recurrences, among those Apéry's sequence which is shown to be the Legendre transform of the sequence. This results in the construction of simultaneous approximations of π 2/8 and ζ(3).

In recent years a great deal of attention has been devoted to the study of finite simple groups, but one aspect which seems to have been little considered is that of the laws they satisfy. In a recent paper [3], the first two of the present authors gave a basis for laws of PSL(2, 5). The techniques of [3] can be used to show that (modulo certain classification problems) a basis for the laws of PSL(2, pn) can be made up from laws of the following types:

(1) an exponent law,

(2) laws which determine the Sylow subgroups,

(3) laws which determine the normalisers of the Sylow subgroups,

(4) in certain special cases, laws which determine subvarieties of smaller exponent, e.g. the subvariety of exponent 12 for those PSL(2, pn) which contain S4,

(5) a law implying local finiteness.

An integer n >0 is called a CLT-number if any group of order n has subgroups of order d for every divisor d of n. The Set of CLT-numbers n is characterized by properties of the prime factorization of n. In addition, if G has order dividing a CLT-number then the structure of G/CG(U) is given where U is a chief factor of G. As a consequence, it is shown that G is solvable of Fitting height at most 3.

Let (ℋ, G, U) be a continuous representation of the Lie group G by bounded operators g ↦ U(g) on the Banach space ℋ and let (ℋ, g, dU) denote the representation of the Lie algebra g obtained by differentiation. If a1,…, ad′ is a Lie algebra basis of g and Ai = dU(ai) then we examine elliptic regularity properties of the subelliptic operators where (cij) is a real-valued strictly positive-definite matrix and c0, c1,…, cd′ ∈ C. We first introduce a family of Lipschitz subspaces ℋγ, γ > 0, of ℋ which interpolate between the Cn-subspaces of the representation and for which the parameter γ is a continuous measure of differentiability. Secondly, we give a variety of characterizations of the spaces in terms of the semigroup generated by the closure of H and the group representation. Thirdly, for sufficiently large values of Re c0 the fractional powers of the closure of H are defined, and we prove that D()γ⊆γ′, for γ′ < 2γ/r where r is the rank of the basis. Further we establish that 2γ/r is the optimal regularity value and it is attained for unitary representations or for the representations obtained by restricting U to ℋγ. Many other regularity properties are obtained.

Let X1, X2, …, XN be Banach spaces and ψ a continuous convex function with some appropriate conditions on a certain convex set in RN−1. Let (X1⊕X2⊕…⊕XN)Ψ be the direct sum of X1, X2, …, XN equipped with the norm associated with Ψ. We characterize the strict, uniform, and locally uniform convexity of (X1 ⊕ X2 ⊕ … ⊕ XN)Ψ; by means of the convex function Ψ. As an application these convexities are characterized for the ℓp, q-sum (X1 ⊕ X2 ⊕ … ⊕ XN)p, q (1 < q ≤ p ≤ ∈, q < ∞), which includes the well-known facts for the ℓp-sum (X1 ⊕ X2 ⊕ … ⊕ XN)p in the case p = q.

Let m be a vector measure with values in a Banach space X. If L1(m) denotes the space of all m integrable functions then, with respect to the mean convergence topology, L1(m) is a Banach space. A natural operator associated with m is its integration map Im which sends each f of L1(m) to the element ∫fdm (of X). Many properties of the (continuous) operator Im are closely related to the nature of the space L1(m). In general, it is difficult to identify L1(m). We aim to exhibit non-trivial examples of measures m in (non-reflexive) spaces X for which L1(m) can be explicitly computed and such that Im is not weakly compact. The examples include some well known operators from analysis (the Fourier transform on L1 ([−π, π]), the Volterra operator on L1 ([0, 1]), compact self-adjoint operators in a Hilbert space); such operators can be identified with integration maps Im (or their restrictions) for suitable measures m.

Let , be regular in the unit disc . G. Szegö [5] and Y. Miki [3] proved that if f(z), given by (1), is univalent (starlike with respect to the origin; convex; close-to-convex in E) then any one of the partial sums , is also univalent (starlike with respect to the origin; convex; close-to-convex) in |z| < ¼and that the constant ¼ cannot be replaced by a larger one.

To prove the statement given in the title take a set Σ1 of identities characterizing distributive lattices 〈L; ∨, ∧, 0, 1〉 with 0 and 1, and let Then is Σ redundant set of identities characterizing Stone algebras = 〈L; ∨, ∧, *, 0, 1〉. To show that we only have to verify that for a ∈ L, a* is the pseudo-complement of a. Indeed, a ∧ a* 0; now, if a ∧ x = 0, then a* ∨ x* 0* = 1, and a** ∧ = 1* = 0; since a** is the complement of a*, the last identity implies x** ≦ a*, thus x ≦ x** ≦ a*, which was to be proved.

In 1957 P. Hall conjectured that every (finitely based) variety has the property that, for every group G, if the marginal factor-group is finite, then the verbal subgroup is also finite. The content of this paper is to present a precise bound for the order of the verbal subgroup of a G when the marginal factor-group is of order Pn (p a prime and n > 1) with respect to the variety of polynilpotent groups of a given class row. We also construct an example to show that the bound is attained and furthermore, we obtain a bound for the order of the Baer-invariant of a finite p-group with respect to the variety of polynilpotent groups.

An integral on a locally compact Hausdorff semigroup ς is a non-trivial, positive, linear functional μ on the space of continuous real-valued functions on ς with compact supports. If ς has the property: (A) for each pair of compact sets C, D of S, the set is compact; then, whenever and a ∈ S, the function fa defined by is also in . An integral μ on a locally compact semigroup S with the property (A) is said to be right invariant if for all j ∈ and all a ∈ S.