If G is a finite group and p a fixed prime number, the irreducible representations of G are distributed into disjoint systems, the p-blocks. These blocks have been investigated especially by R. Brauer. In this note we are concerned with the problem: What is the structure of G which has only one p-block? Or more generally, what is the structure of G all of whose p-blocks have the highest defects?

Let S be a minimal algebraic surface of general type defined over the complex number field C, and let K denote the canonical bundle. According to [10], we call S a canonical surface if the rational map ФK associated with | K | induces a birational map of S onto the image X. We denote by Q (X) the intersection of all hyperquadrics through X.

Let H denote the complex upper half-plane and let η(z) denote Dedekind’s η-function

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.

We prove new cases of the direct summand conjecture using fundamental theorems in p-adic Hodge theory due to Faltings. The cases tackled include the ones when the ramification locus lies entirely in characteristic p.

We use a method of Berndtsson to obtain a simplification of Ohsawa’s result concerning extension of L2-holomorphic functions. We also study versions of the Ohsawa-Takegoshi theorem for some unbounded pseudoconvex domains, with an application to the theory of Bergman spaces. Using these methods we improve some constants, that arise in related inequalities.

1. Concerning a positive lower semicontinuous kernel G on a locally compact Hausdorff space X the following existence theorem was obtained in [3].

We describe the branch locus of a proper holomorphic mapping between two smoothly bounded pseudoconvex domains of finite type in under the assumption that the first domain admits a transversal holomorphic action of the unit circle. As an application we show that any proper holomorphic self-mapping of a smoothly bounded pseudoconvex complete circular domain of finite type in is biholomorphic.

In the recent years there has been a considerable effort to construct and analyze spaces of test and generalized functionals in infinite dimensional situations, cf. [3, 5, 12, 14] and literature quoted there. In particular Meyer [4, 5] has introduced a certain space of “smooth” functionals on the Wiener space, which was used by Watanabe [14] for an elegant formulation of “Malliavin’s calculus” (i.e. he proved a criterion for the existence and regularity of densities of Wiener functionals). This functional space is countably normed and one of its important properties is its algebraic structure. The proof of this property follows from an equivalence of the norms defining the space with a system of norms of Sobolev type [4, 5] (cf. also (1.5), (1.6)).

The most important thing for a set theory seems to be that it can generate new mathematical objects, and I think that there must be an underlying principle, simple and unique, which unifies the acts of generating. The naive set theory has a unique generating principle, which defines, by any proposition on a variable x, the set of all x’s satisfying the proposition. Certainly, we must restrict this generating principle so as to exclude all contradictions it contains, without losing its essential rôle as logic of mathematics, and at the same time we would like to keep its uniqueness and simplicity.

Let X and Y be any pure dimensional subschemes of Pnk over an algebraically closed field K and let I(X) and I(Y) be the largest homogeneous ideals in K[x0,…, xn] defining X and Y, respectively. By a pure dimensional subscheme X of Pnk we shall always mean a closed pure dimensional subscheme without imbedded components, i.e., all primes belonging to I(X) have the same dimension.

Let f: V → C be a fibration from a smooth projective surface onto a smooth projective curve over an algebraically closed field k. In the case of characteristic zero, almost all fibres of f are nonsingular. In the case of positive characteristic, it is, however, known that there exist fibrations whose general fibres have singularities. Moreover, it seems that such fibrations often have pathological phenomena of algebraic geometry in positive characteristic (see M. Raynaud [7], W. Lang [4]).

Let R be a commutative Noetherian ring and suppose q is a prime ideal of R. A fundamental problem is to decide when powers qn of q are primary (that is qn is its own primary decomposition). If q is generated by a regular sequence then powers of q are always primary, because G(q, R) (the associated graded ring of R with respect to q) is an integral domain (see [12 page 98] and also [5 (2.1)]). Let qn) denote the nth symbolic power of q-defined by q(n) = {rεR|there exists sεR\q such that sr ε qn}. Then qn is primary if and only if qn = q(n) If q is generated by a regular sequence then we call it a complete intersection prime ideal, so if q is a complete intersection prime ideal then qn ≠ q(n) for all n ≥ 1. If q is not a complete intersection then powers need not be primary. If R is a three-dimensional regular local ring and q is a non-complete intersection height two prime ideal for example, then Huneke showed [11 Corollary (2.5)] that qn = q(n) for all n ≥ 2. Thus, for such a prime q it is impossible for qn to occur in the primary decomposition of any ideal. This phenomena increases the difficulty in finding a primary decomposition for an ideal having q as an associated prime.

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,

Let M be a ℚ-divisor on a smooth surface over ℂ. In this paper we give criteria for very ampleness of the adjoint of [M], the round-up of M. (Similar results for global generation were given by Ein and Lazarsfeld and used in their proof of Fujita’s Conjecture in dimension 3.) In §4 we discuss an example which suggests that this kind of criteria might also be useful in the study of linear systems on surfaces.

In the previous papers [3], [4], [5] the author gave some results on the uniqueness of meromorphic maps of Cn into the N-dimensional complex projective space PN(C) which have the pre-assigned inverse images for some hyperplanes in PN(C). Relating to these results, we attempt in this paper to generalize the following Cartan-Nevanlinna’s theorem to the case of meromorphic maps into PN(C).

Let E be a compact set in the z-plane and let Ω be its complement with respect to the extended z-plane. Suppose that E is of capacity zero. Then Ω is a domain and we shall consider a single-valued meromorphic function w = f(z) on Ω which has an essential singularity at each point of E. We shall say that a value w is exceptional for f(z) at a point ζ ∈ E if there exists a neighborhood of C where the function f(z) does not take this value w.

Let f(z) be a cusp form of type (l,ε) on Γ0(N) which is a common eigenfunction of all Hecke operators. For such f(z), Deligne and Serre [1] proved that there exists a linear representation

such that the Artin L-function for p is equal to the L-function associated to f(z).

The classical subject of surfaces containing a hyperelliptic curve (here a double cover of P1) among their hyperplane sections was settled some years ago by the third author and Van de Ven [SV] (see also [Se], [Io]). This paper is devoted to answering the following general question arising very naturally from that problem.

In this paper we are concerned with stochastic control problems of the following kind. Let Y(t) be a d’-dimensional Brownian motion defined on a probability space (Ω, F, Ft, P) and u(t) an admissible control. We consider the Cauchy problem of stochastic partial differential equations (SPDE in short)

where L(y, u) is the 2nd order elliptic differential operator and M(y) the 1st order differential operator.