This paper considers the behaviour of a quotient map between Fréchet spaces concerning the lifting of bounded sets. The main result shows that a quotient map between Fréchet spaces that lifts bounded sets with closure (or equivalently such that its strong transpose is a topological isomorphism) must also lift bounded sets without closure.

Let Tn be the operator algebra of upper triangular n × n complex matrices. Three families of limit algebras of the form lim (Tnk) are classified up to isometric algebra isomorphism: (i) the limit algebras arising when the embeddings Tnk→Tnk+1, are alternately of standard and refinement type; (ii) limit algebras associated with refinement embeddings with a single column twist; (iii) limit algebras determined by certain homogeneous embeddings. The last family is related to certain fractal like subsets of the unit square.

It is proved that if G is any compact connected Hausdorff group with weight w(G)≦c, ℝ is the topological group of all real numbers and n is a positive integer, then the topological group G × ℝn can be topologically generated by n + 1 elements, and no fewer elements will suffice.

We describe a class of soluble groups with a finite complete rewriting system which includes all the soluble groups known to have such a system. It is an open question, related to deep questions in the theory of groups, whether it includes all soluble groups with such a system.

In this article we prove that if a completely positive linear map Φ of a unital C*-algebra A into another B with only finite dimensional irreducible representations is pure, then we have NΦ = Φker + kerΦ, where NΦ={x∈A|Φ(x) = 0}, Φker = {x∈A|Φ(x*x) = 0}, and kerΦ={x∈A|Φ(xx*) = 0}. We also prove that for every unital strongly positive and n-positive linear map Φ of a C*-algebra A onto another B with n≧2, if NΦ = Φker + kerΦ, then Φ is extreme in Pn(A, B, IB). By this null-kernel condition, many new extreme n-positive linear maps are identified. A general procedure for constructing extreme n-positive linear maps is suggested and discussed.

Based on the theory of p-supersoluble and supersoluble groups, a prime-number parametrized family of canonical characteristic subgroups Γp(G) and their intersection Γ(G) is introduced in every finite group G and some of its properties are studied. Special interest is dedicated to an elementwise description of the largest p-nilpotent normal subgroup of Γp(G) and of the Fitting subgroup of Γ(G).

Suppose that K is a function field of a conic over a subfield K0. Let v0 be a valuation of K0 with residue field k0 of characteristic ≠2. Let v be an extension of v0 to K having residue field k. It has been proved that either k is an algebraic extension of k0 or k is a regular function field of a conic over a finite extension of k0. This result can also be deduced from the genus inequality of Matignon (cf. [On valued function fields I, Manuscripta Math. 65 (1989), 357–376]) which has been proved using results about vector space defect and methods of rigid analytic geometry. The proof given here is more or less self-contained requiring only elementary valuation theory.

Let R be a commutative ring and {σ1,…,σn} a set of commuting automorphisms of R. Let T = be the skew Laurent polynomial ring in n indeterminates over R and let be the Laurent polynomial ring in n central indeterminates over R. There is an isomorphism φ of right R-modules between T and S given by φ(θj) = xj. We will show that the map φ induces a bijection between the prime ideals of T and the Γ-prime ideals of S, where Γ is a certain set of endomorphisms of the ℤ-module S. We can study the structure of the lattice of Γ-prime ideals of the ring S by using commutative algebra, and this allows us to deduce results about the prime ideal structure of the ring T. As an example, if R is a Cohen-Macaulay ℂ-algebra and the action of the σj on R is locally finite-dimensional, we will show that the ring T is catenary.

In this paper we consider the questions of existence and uniqueness of solutions to a singular, nonlinear boundary value problem arising from a model problem in isothermal autocatalytical chemical kinetics. The boundary value problem occurs in the construction of a small time asymptotic solution to an initial-boundary value problem (King and Needham [14]), and existence and uniqueness for the boundary value problem are required for consistency of this formal asymptotic solution.

Let M and N be simply connected space forms, and U an open and connected subset of M. Further let π: U → N be a horizontally homothetic harmonic morphism. In this paper we show that if π has totally geodesic fibres and integrable horizontal distribution, then the horizontal foliation of U is totally umbilic and isoparametric. This leads to a classification of such maps. We also show that horizontally homothetic harmonic morphisms of codimension one are either Riemannian submersions modulo a constant, or up to isometrics of M and N one of six well known examples.

A knot K is said to have tunnel number 1 if there is an embedded arc A in S3, with endpoints on K, whose interior is disjoint from K and such that the complement of a regular neighbourhood of K ∪ A is a genus 2 handlebody. In particular the fundamental group of the complement of a tunnel number one knot is 2-generator. There has been some interest in the question as to whether there exists a hyperbolic tunnel number one knot whose complement contains a closed essential surface. The aim of this paper is to prove the existence of infinitely many 2-generator hyperbolic 3-manifolds with a single cusp which contain a closed essential surface. One such example is a knot complement in RP3. The methods used are of interest as they include the possibility that one of our examples is a knot complement in S3.

A squareroot of an ambiguous form in the principal genus of primitive integral binary quadratic forms of fixed discriminant is given explicitly in terms of a solution of a certain Legendre equation.

In this paper, we consider interpolants on h·ℤn from the closure of the space spanned by translates of the function (‖·‖2 + 1)β/2 (β>−n and not an even nonnegative integer) along h·ℤn. We show that these interpolants approximate a function, whose Fourier transform satisfies certain asymptotic conditions, up to an error of order hp, on any compact domain in ℝn, where p is only restricted by the smoothness of the function.

The idea of applying isoperimetric functions to group theory is due to M. Gromov [8]. We introduce the concept of a “bicombing of narrow shape” which generalizes the usual notion of bicombing as defined for example in [5], [2], and [10]. Our bicombing is related to but different from the combings defined by M. Bridson [4]. If they Cayley graph of a group with respect to a given set of generators admits a bicombing of narrow shape then the group is finitely presented and satisfies a sub-exponential isoperimetric inequality, as well as a polynomial isodiametric inequality. We give an infinite class of examples which are not bicombable in the usual sense but admit bicombings of narrow shape.

A short proof, based on the Schur complement, is given of the classical result that the determinant of a skew-symmetric matrix of even order is the square of a polynomial in its coefficients.

For any sequence (aj) of complex numbers and for any ρ > ½, we construct an entire function F with the following properties. F has order ρ, mean type, each aj is a deficient value of F, and F is given by F(z)=f(g(z)), where f and g are transcendental entire functions. This complements a result of Goldstein. We also construct, for any ρ>½, an entire function G of order p, mean type, such that liminf,→ ∞ T(r, G)/T(r, G′)>1.

It is shown that an extended Schlömilch formula for Stirling-type pairs of numbers and the inversion formula of Lagrange are implied by each other. Also proved are some congruence relations modulo a prime number p(>2) associated with generalized Stirling numbers. The third result is concerned with the asymptotic expansions of Stirling-type pairs involving large parameters.

We construct an uncountable family of pairwise non-isomorphic rings Si, such that the corresponding full 2 by 2 matrix rings M2(Si) are all isomorphic to each other. The rings Si are Noetherian integral domains which are finitely-generated as modules over their centres.

Every distributive double p-algebra L is shown to have a congruence permutable extension K such that every congruence of L has a unique extension to K.

In this work we prove the existence and uniqueness of positive solutions of the nonlinear singular boundary value problem

where 0<σ<1.

Extensions of the above results to the case of Δ2u−f(x, u) = 0 with appropriate singularity built into f are also given.