Our dream is to revive the ideal theory in partially ordered sets from a viewpoint of commutative algebra.

Historically, the concept of ideals in commutative algebra was first studied by Dedekind, who considered the ring of algebraic integers in an algebraic number field.

There are many results in probability theory on vector spaces which rely implicitly on the approximation of a given cylindrical probability by cylindrical probabilities with moments; for example, this is the basic idea behind the proof of the Radon equivalence of the weak and strong topologies of a metrizable space (Schwartz [13] p. 162). The technique of approximation by cylindrical measures with moments can be systematically developed. In particular, it follows that if each member of a family of cylindrical probabilities with moments is decomposable, then the limits of these cylindrical probabilities are often regular and so they are σ-additive.

Let C, C′ and C″ be abelian categories where C and C′ have enough injectives and let F: C → C′, G: C′ → C″ be additive covariant functors. Then for an object X of C, let C(X) be the complex associated with an injective resolution of X. Grothendieck gets a first quadrant spectral sequence by taking an injective resolution of the complex F(C(X)) and applying G to the associated double complex.

This is a continuation of my previous papers [2], [3], [4] concerning to the monstrous moonshine.

The automorphism group ·O of the Leech lattice L plays an important role in the study of moonshine. Especially it is important to study theta functions associated with quadratic sublattices of L consisting of fixed vectors of elements of ·O. In this paper, we discuss the properties that these functions are expected to satisfy in the relation to the monstrous moonshine.

One frequently encounters (real) semisimple graded Lie algebras in various branches of differential geometry (e.g. [16], [9], [14], [18]). It is therefore desirable to study semisimple graded Lie algebras, including those which have been studied individually, in a unified way. One of our concerns is to classify (finite-dimensional) semisimple graded Lie algebras in a way that enables us to construct them.

Let k be a p-adic field (a finite extension of Qp) or an algebraic number field (a finite extension of Q). Let T be an algebraic torus defined over k. We denote by the character module of T (i.e. = Hom (T, Gm), where Gm is the multiplicative group.

Let (X, ) be a measurable space and be a submarkovian resolvent of kernels (with the initial kernel V proper) on X which is absolutely continuous and has a dual resolvent (with the same properties) with respect to a σ-finite measure.

Gauss conjectured that there are infinitely many real quadratic fields with class number one. Today this is still an open problem. Moreover, as Dorian Goldfeld, one of the recipients of the 1987 Cole prize in number theory (for his work on another problem going back to Gauss) recently stated in his acceptance of the award: “This problem appears quite intractible at the moment.” However there has recently been a search for conditions which are tantamount to class number one for real quadratic fields. This may be viewed as an effort to shift the focus of the problem in order to understand more clearly the inherent difficulties, and to reveal some other beautiful interrelationships.

Let K be a field of characteristic zero, V a finite dimensional vector space and G a subgroup of GL(V). The action of G on V is extended to the symmetric algebra on V over K,

and the tensor algebra on V over K,