In this paper, we investigate a class of 2-generator 2-relator groups G(n) related to the Fibonacci groups F(2,n), each of the groups in this new class also being defined by a single parameter n, though here n can take negative, as well as positive, values. If n is odd, we show that G(n) is a finite soluble group of derived length 2 (if n is coprime to 3) or 3 (otherwise), and order |2n(n + 2)gnf(n, 3)|, where fn is the Fibonacci number defined by f0=0,f1=1,fn+2=fn+fn+1 and gn is the Lucas number defined by g0 = 2, g1 = 1, gn+2 = gn + gn+1 for n≧0. On the other hand, if n is even then, with three exceptions, namely the cases n = 2,4 or –4, G(n) is infinite; the groups G(2), G(4) and G(–4) have orders 16, 240 and 80 respectively.

Let N(λ) be a square matrix polynomial, and suppose det N is a polynomial of degree d. Subject to a certain non-singularity condition we construct a d by d Hermitian matrix whose signature determines the numbers of zeros of N inside and outside the unit circle. The result generalises a well known theorem of Schur and Cohn for scalar polynomials. The Hermitian “test matrix” is obtained as the inverse of the Gram matrix of a natural basis in a certain Krein space of rational vector functions associated with N. More complete results in a somewhat different formulation have been obtained by Lerer and Tismenetsky by other methods.

The boundary value problem

is studied with a view to obtaining the existence of positive solutions in C1([0, 1])∩C2((0, 1)). The function f is assumed to be singular in the second variable, with the singularity modeled after the special case f(x, y) = a(x)y−p, p>0.

This boundary value problem arises in the search of positive radially symmetric solutions to

where Ω is the open unit ball in ℝN, centered at the origin, Γ is its boundary and |x| is the Euclidean norm of x.

In previous work of the author and M. Culler, contractible simplicial complexes were constructed on which the group of outer automorphisms of a free group of finite rank acts with finite stabilizers and finite quotient. In this paper, it is shown that these complexes are Cohen-Macauley, a property they share with buildings. In particular, the link of a vertex in these complexes is homotopy equivalent to a wedge of spheres of codimension 1.

This paper concerns a Jacobson-type radical for the near-ring N. This radical, denoted by Js(N) has an external representation on a type-0N-group of a very special kind. Such N-groups are said to be of type-s. The main objective of this paper is to decompose Js(N) as a sum Js(N) = J1/2(N) + A + B for N satisfying the descending chain condition for N-subgroups. In this decomposition J1/2(N) is nilpotent and A is the unique minimal ideal modulo which Js(N) is nilpotent.

We show that there is a pair of handlebodies H1 and H2 with common boundary F with the properties:

(a) There is no essential simple closed curve in F bounding a disc in both H1 and H2.

(b) Given any positive number, there are essential simple curves Cii = 1, 2 on F, bounding discs in Hi whose distance apart in the Hausdorff topology on F is less than this positive number.

Such an example has consequences for Heegaard splittings and recognising the 3-sphere.

Asymptotic formulae are given for the distribution of Liusternik-Schnirelman eigenvalues of certain pairs of nonlinear functionals generalising the usual Weyl theory for linear pairs of elliptic operators. In particular an application is made to the von Kármán theory of buckled plates.

The effect of imposing a certain finiteness condition on the group of central automorphisms of a finite-by-nilpotent group is investigated. In particular it is shown that, if each central automorphism of a finite-by-nilpotent group G has finite order, then the factor group G/Z(G) has finite exponent.

In this paper we provide a new, abstract characterisation of classical Rees matrix semigroups over monoids with zero. The corresponding abstract class of semigroups is obtained by abstracting a number of algebraic properties from completely 0-simple semigroups: in particular, the relationship between arbitrary elements and idempotents.

It is shown that the lower radical construction of Tangeman and Kreiling need not terminate at any ordinal.

In this paper a characterization of the regular ω-semigroups whose congruence lattice is modular is given. The characterization obtained for such semigroups generalizes the one given by Munn for bisimple ω-semigroups and completes a result of Baird dealing with the modularity of the sublattice of the congruence lattice of a simple regular ω-semigroup consisting of congruences which are either idempotent separating or group congruences.

Crossed modules occur in the theory of group presentations, in group cohomology and in providing algebraic models for certain homotopy types. There are profinite analogues of each of these contexts. In this paper, we examine the problem of extending the profinite completion functor on groups to one on crossed modules thus providing a method for comparing the information contained in profinite and abstract crossed modules in each of these situations.

If is a category, an object of pro- is stable if it is isomorphic in pro- to an object of . A local condition on such a pro-object, called strong-movability, is defined, and it is shown in various contexts that this condition is equivalent to stability. Also considered, in the case is a suitable model category, is the stability problem in the homotopy category Ho(pro-), where pro- has the induced closed model category structure defined by Edwards and Hastings [6].

The Poisson kernel is defined for z in the open unit disc D and ζ in the unit circle ∂D. As usually employed, it is integrated with respect to the second variable and a measure on ∂D to yield a harmonic function on D. Here, we fix a σ-finite positive Borel measure m on D and integrate the Poisson kernel with respect to the first variable against a function φ in L1(m) to obtain a function Tmφ on ∂D. We ask for what measures m the range of Tm is L1(∂D), for what m the kernel of Tm is non-zero, and for what m every positive continuous function on ∂D is of the form Tmφ with φ non-negative. When m is the counting measure of a countably infinite subset {ak:k∈ℕ} of D, the function (Tmφ)(ζ) is of the form with . The main results generalize results previously obtained for sums of this form. A related mapping from Lp(m) into Lp(∂D) with 1 <p<∞ is briefly considered.

The famous Cohen factorization theorem, which says that every Banach algebra with bounded approximate identity factors, has already been generalized to locally convex algebras with what may be termed “uniformly bounded approximate identities”. Here we introduce a new notion, that of fundamentality generalizing both local boundedness and local convexity, and we show that a fundamental Fréchet algebra with uniformly bounded approximate identity factors. Fundamentality is a topological vector space property rather than an algebra property. We exhibit some non-fundamental topological vector space and give a necessary condition for Orlicz space to be fundamental.

A commutative algebra A over the field F, endowed with a non-zero homorphism ω:A →F is principal train if it satisfies the identity xr+y1ω(x)xr−1 +… +yr−1ω(x)r−1x=0 where y1,…,yr−1 are fixed elements in F. We present in this paper, after the introduction of the concept of “type” of A, some results concerning the classification in the case r = 3. In particular we describe all these algebras of dimension≦5.

Among the tricyclic Hamiltonian graphs with a prescribed number of vertices, the unique graph with maximal index is determined. Some subsidiary results are also included.

We study systems which in characteristic coordinates have the form

where A is a k × k diagonal matrix with distinct real eigenvalues. The nonlinearity F is assumed to be asymptotically homogeneous in the sense, that it is a sum of two terms, one positively homogeneous of degree one in u and a second which is sublinear in u and vanishes when u = 0. In this case, F(t, x, u(t)) is meaningful provided that u(t) is a Radon measure, and, for Radon measure initial data there is a unique solution (Theorem 2.1).

The main result asserts that if μn is a sequence of initial data such that, in characteristic coordinates, the positive and negative parts of each component, , converge weakly to μ±, then the solutions coverge weakly and the limit has an interesting description given by a nonlinear superposition principle.

Simple weak converge of the initial data does not imply weak convergence of the solutions.

A bilateral Laplace multiplier theory, based on Rooney's class , is developed for certain operators defined on the Fréchet spaces Dp,μ. The theory is applied to Riesz fractional integrals associated with the one-dimensional wave operator.

Under the assumption of the Selberg conjecture I establish by means of the Selberg trace formula the following:

Theorem. Let Γ denote Γ(q) or Γ0(q), q square-free. Let Δq denote the Laplace operator on L2(Γ\H), and let Σq denote its discrete spectrum. Then there exists an absolute positive constant A such that for q≧A