If ℍ is a rational indefinite quaternion algebra and T is an order of ℍ, let G+(T) denote the subgroup of units u ∊ T with norm n(u) = + 1. For a certain class of orders, Eichler has determined the measure μ(G+(T)) of G+(T), viewed as a Fuchsian group. This is extended to arbitrary orders by methods depending only upon classical number-theory and group theory. As an application, an estimate of the magnitude of a small non-trivial solution of the diophantine Pellian equation

is supplied and the restrictions on the integers D and P associated with previous work on this question are eliminated.

This paper combines the global bifurcation theory of Rabinowitz with Sturmian theory and careful estimates to obtain a detailed qualitative description of bifurcating branches of solutions to the equations for whirling nonuniform, nonlinearly elastic strings. These results generalize earlier work of Kolodner and Stuart on inextensible strings. It is shown that the location of solution branches for the generalization of Kolodner's problem is especially sensitive to the material properties of the string, whereas that for Stuart's problem is not. The analysis of a third problem illuminates the source of this dichotomy.

In this paper we show that Dirichlet problems at resonance, being of the type −Δu(x) = λku(x) + g(u(x)), x є G, u(x) = 0 for x є ∂G, g(−u) = −g(u), admit multiple non-trivial solutions provided the non-linearity interacts in some sense with the spectrum of −Δ. In contrast to other work on this subject we deal with the case that g(u) is very small for large arguments, for instance g(u) = 0 for |u| large. On the other hand if and g satisfies a certain concavity condition at 0 the existence of infinitely many solutions is shown independent of the asymptotic behaviour of g.

The solutions of stochastic differential equations are used to construct Markov processes on the Banach manifold C(S, M) of continuous maps from a compact metric space S into a smooth complete finite dimensional Riemannian manifold M. In the special case where S is a single point the construction gives a large class of diffusion processes on the manifold M, including (under certain curvature conditions) the Brownian motion process.

A classical theorem of Hartogs gives conditions on the singularity set of an analytic function of several complex variables in order for such a set to be an analytic variety. A result of E. Bishop from 1963 gives an analogous condition of the maximal ideal space of a uniform algebra in order for this space to have analytic structure. We show that algebras of functions satisfying a maximum principle serve to explain both of these results.

It is shown that eigenvalues of infinite multiplicity can exist for the Schrödinger equation holding in the whole N-dimensional space RN(N ≧ 2). In the example which is constructed, the potential is separable and bounded in RN, and the method is an application of inverse spectral theory.

A uniform asymptotic expansion of the Laplace integrals ℒ(f, s) with explicit remainder terms is given. This expansion is valid in the whole complex s−plane. In particular, for s = −ix, it provides the Fourier integral expansion.

The non-negative weight functions U(x), V(x) for which the generalized Stieltjes transformation Sλ is a bounded operator from LP(V) into La(U) are characterized. This extends certain classical inequalities and some recent results of Erdélyi. In particular, our results provide weighted extensions of a classical inequality known as Hubert's double series theorem.

The problem of determining the best constant κ in the inequality ‖y′‖≦K ‖y‖ ‖y″‖ is discussed in the context of the classical Lp spaces, 1 ≦ p ≦ ∞.

We show that for some closed graph theorems each countable codimensional subspace of a domain space may also serve as a domain space. This provides a general principle from which we are able to extract some of the known results on the inheritance of topological vector space properties by subspaces of countable codimension. We make use of a result of Savgulidze and Smoljanov on B-completeness for which we provide a new and simpler proof.

In this paper the Sturm-Liouville regular boundary value problem and expansion theorem is extended to functions with values in some B*-algebras.

The differential operators in question are of the form G(DZ) where G(w)is an entire function of order at most 1/n and minimal type while Dz is a linear differential operator of order n with coefficients which are entire ( = integral) functions of z, usually polynomials. This class of operators form a natural generalization of the class G(d/dz) studied during the first half of the century Muggli, Polya, Ritt and others. The class G(DZ) was introduced by the present author and his pupils in the 1940s. In fact, the present paper is partly based on a MS from that period, mostly devoted to the special case

but also containing generalizations, some of which were later worked out by Klimczak. A basic tool in this paper is the characteristic series

Examples are given showing that the domain of absolute convergence of such a series need neither be convex nor of finite connectivity, a question which has puzzled the author for forty odd years. Characteristic series arising from regular or singular boundary value problems for the operator Dz are used to study the inversion problem

for given F(z). In particular it is shown that exp (Dx)[W(z)] = 0 has the unique solution W(z) ≡ 0. Some singular boundary value problems are considered briefly.

We study the nonlinear equation

in ℝ3, where Δ denotes the Laplacian operator, and R and K are real-valued functions satisfying suitable conditions. We use a variational formulation to show the existence of a non-trivial weak solution of the above equation for some real number λ. Because of our assumptions on R and K we shall look for solutions which are spherically symmetric, decrease with r = |x| and vanish at infinity.

In this paper the Sturm-Liouville expression τy = −(py′)′ + qy, with complex-valued coefficients is considered, and a number of results concerning the location of the essential spectrum of associated operators are obtained. Some of these are extensions or generalizations of results due to Birman, and Glazman, whilst others are new. These lead to criteria for the non-emptiness of the regularity field of the corresponding minimal operator—a condition which is needed in the theory of J-selfadjoint extensions. A complete determination of the regularity field is made when the equation τy = λ0y has two linearly independent solutions in L2[a,∞) for some complex λ0.

This paper studies stability of steady state solutions in a non-linear viscoelastic fluid. The main technique is to imbed the equation of motion in singularly perturbed equations and apply an energy method and the parabolic maximum principle.

Conditions are given on the coefficients p, q and r of the fourth-order, symmetric differentia expression

where y ∈ L2(0, ∞) and L[y] ∈ L2(0, ∞), such that some or all of the individual terms in L[y] are also in L2(0, ∞).

In recent years it has become clear that AW*-algebras can be much more pathological and unlike von Neumann algebras than was originally expected. When AW*-algebras are monotone complete, then the work of Kadison and Pederson shows that a particularly smooth and elegant theory can be developed. A technically weaker requirement on an AW*-algebra is that it be “normal”. This condition, which says that the lattice of projections is embedded in a well-behaved way in the partially ordered set of all self-adjoint elements, can sometimes be used as a substitute for monotone completeness. In this note we prove that when an AW*-algebra is of finite type (that is x*x = 1 implies xx* = 1) then it is normal.