Recently, systematic applications of the Scott-Solovay Boolean valued set theory were done by several authors; Takeuti [25, 26, 27, 28, 29, 30], Nishimura [13, 14] Jech [8] and Ozawa [15, 16, 17, 18, 19, 20] in analysis and Smith [23], Eda [2, 3] in algebra. This approach seems to be providing us with a new and powerful machinery in analysis and algebra. In the present paper, we shall study Banach space objects in the Scott-Solovay Boolean valued universe and provide some useful transfer principles from theorems of Banach spaces to theorems of Banach modules over commutative AW*-algebras. The obtained machinery will be applied to resolve some problems concerning the module structures of von Neumann algebras.

In the 1980s, Franks, Pugh and Shub raised the question “Given any subshift of finite type σA: ΣA → ΣA is there a non-singular Smale flow (or an NS flow for short) on S3 with the suspension of σA as a basic set?” (See [5] and [12]).

We shall investigate a system of spin configurations S = {S(t, x); t ≥ 0, x ∊ ℤ} on a one-dimensional lattice ℤ changing randomly in time. The evolution law is described by an infinite-dimensional stochastic differential equation (SDE):

where {β(t, x); t > 0, x ∊ Z} is a family of independent standard Wiener processes and U′ is the derivative of a self-potential U: R → R.

In Selberg [11], he introduced the trace formula and applied it to computations of traces of Hecke operators acting on the space of cusp forms of weight greater than or equal to two. But for the case of weight one, the similar method is not effective. It only gives us a certain expression of the dimension of the space of cusp forms by the residue of the Selberg type zeta function. Here the Selberg type zeta function appears in the contribution from the hyperbolic conjugacy classes when we write the trace formula with a certain kernel function ([3J, [4], [7], [8], [9], [12]).

In our paper [U2], we proved the irreducibility of the first differential equation y″ = 6y2 + x of Painlevé. In that paper we explained the origin of the problem and the importance of giving a rigorous proof. We can say that our method in [U2] is algebraic and finite dimensional in contrast to a prediction of Painlevé who expected a proof depending on the infinite dimensional differential Galois theory. Even nowadays the latter remains to be established. It seems that Painlevé needed an armament with the general theory (the infinite dimensional differential Galois theory) in the controversy with R. Liouville on the mathematical foundation of the proof of the irreducibility of the first differential equation (1902-03).

The purpose of this paper is the study of some basic properties of universal induced characters and their applications to the representation theory of the classical groups (for the definition of a universal induced character, see § 3).

The starting point was the paper [F] by E. Formanek on matrix invariants. In his paper [F], Formanek has investigated the Hilbert series for the ring of matrix invariants from the point of view of the representation theory of the general linear group and the symmetric group. In this paper we shall study polynomial concomitants of a group from the same point of view.

N. Aronszajn introduced in [4] an abstract Frechét space R (0<R≤∞), which is isomorphic to the space of analytic solutions of the heat equation in if 0 < R ∞, and in if R = ∞, and called it the space of traces of analytic solutions of the heat equation. Hereafter, we call it the space of traces, shortly.

It was brought to our attention by Zoran Luicic and Milica Stojanovic, via Peter Gilkey, that some of the diagrams in our paper are not correct.

The particular problems are the gluing diagrams for the pair of isospectral surfaces of genus 4, which occur on page 20. It is easy to check that the gluing diagrams given there give rise to a surface of the wrong genus. The problem arose because of carelessness in some of the identifications of some of the edges of the fundamental domain.