A great part of mathematical analysis relies directly on the methods of separation of variables and on the successive reduction of several variables problems to one-dimensional equations and to the theory of classical special functions; for example, the theory of elliptic or parabolic equations with regular coefficients (even with non constant coefficients) can be done because we know explicitly the fundamental solutions of the Laplace operator or of the heat equation; these fundamental solutions are functions of one variable; pseudodifferential or parametrices methods are thus basically small perturbations of an explicitly known problem in one variable.

On the other hand, there are many problems which are not of this type: they are related to the questions of operators with singular coefficients and to the global behaviour of the solutions; in that case, the local model cannot be reduced to a one variable problem but is fundamentally a several variables problem which cannot be treated in a detailed way by one variable methods or perturbation analysis of a one variable problem.

Let be a nest algebra of operators on some Hilbert space H. Weakly closed -modules were first studied by J. Erdos and S. Power in [4]. It became apparent that many interesting classes of non self-adjoint operator algebras arise as just such a module. This paper undertakes a systematic investigation of the correspondence which arises between such modules and order homomorphisms from Lat into itself. This perspective provides a basis to answer some open questions arising from [4]. In particular, the questions concerning unique “determination” and characterization of maximal and minimal elements under this correspondence, are resolved. This is then used to establish when the determining homomorphism is unique.

The problem of counting polyominoes motivates this paper. We will develop a general question for study that has counting polyominoes as a special case. We generalize in two ways. Polyominoes are shapes on the tiling made of square tiles. We will consider shapes on other tilings. The set of all polyominoes can be generated by a context-free array grammar, but the size of this set is estimated by counting the words of certain subsets and supersets that are generated by more convenient grammars. Our general question is the problem of counting the words of a context-free array language on a periodic tiling.

Counting polyominoes is a difficult problem that has not been completely solved yet. There are various techniques for roughly estimating the number of polyominoes of a given size. We will extend some of these techniques to our general question.

If d is a positive square free integer we let Od be the ring of integers in and we let Γd = PSL2(Od), the group of linear fractional transformations

and entries from Od {if d = 1, ad – bc = ±1}. The Γd are called collectively the Bianchi groups and have been studied extensively both as abstract groups and in automorphic function theory {see references}. Of particular interest has been Γ1 – the Picard group. Group theoretically Γ1, is very similar to the classical modular group M = PSL2(Z) both in its total structure [4, 6], and in the structure of its congruence subgroups [8]. Where Γ1 and M differ greatly is in their action on the complex place C. M is Fuchsian and therefore acts discontinuously in the upper half-plane and every subgroup has the same property.

Two complex hermitian n × n matrices A and B are congruent if S*AS = B for some invertible n × n matrix S (with complex entries). The matrix S is called a congruence matrix. Given congruent hermitian matrices A and B, a congruence matrix is, of course, not unique. For instance, if A = B then one can take S = αI with |α| = 1, as well as any other matrix satisfying S*AS = A. However, here the choice S = I seems naturally to be best possible in the sense that when applied to an n-dimensional column vector it produces no distortion or movement of the vector at all. We shall measure the distortion (or movement) of the vector x ∊ Cn under an n × n invertible matrix A in terms of ║x − Ax║, where the norm is euclidean. Then the distortion produced by A is ║I − A║, with the induced operator norm.

The main question we wish to address in this paper is to what extent does the Ricci curvature of a spacetime determine the metric of that spacetime. Although it is relatively easy to see that the full Riemann curvature uniquely determines the metric for a generic choice of curvature tensors (see [4], [10], [11], [14] and [15], and the references contained therein), very little has been discovered about whether, if ever, Ric (or the stress energy tensor in Einstein's equations for that matter) determines g. Most exact solution techniques for Einstein's equations look only for solutions that have the same symmetries as Ric. It is not true in general that g must inherit the symmetries of Ric. It is not even clear that there is a Ric such that every g with this Ricci tensor is known.

Let denote the group of all homeomorphisms of a σ-compact manifold which preserve a σ-finite, nonatomic, locally positive and locally finite measure μ. In two recent papers [4, 5] the possible ergodicity of a homeomorphism h in was shown to be related to the homeomorphism h* induced by h on the ends of M. An end of a manifold is, roughly speaking, a distinct way of going to infinity. Those papers demonstrated in particular that always contains an ergodic homeomorphism, paralleling the similar result of Oxtoby and Ulam [11] for compact manifolds with finite measures. Unfortunately the techniques used in [4] and [5] rely on the fact that a skyscraper construction with an ergodic base transformation is ergodic, a result which cannot be extended to finer properties than ergodicity.

Let D be a simply-connected domain and w0 a fixed point of D. Denote by SD the set of all complex-valued, harmonic, orientation-preserving, univalent functions f from the open unit disk U onto D with f(0) = w0. Unlike conformai mappings, harmonic mappings are not essentially determined by their image domains. So, it is natural to study the set SD.

In Section 2, we give some mapping theorems. We prove the existence, when D is convex and unbounded, of a univalent, harmonic solution f of the differential equation

where a is analytic and |a| < 1, such that f(U) ⊂ D and