The polyhedral group (l, m, n) is defined in [3] by the presentation

Let E be a non-singular projective curve of genus g ≥ 0, P the projective line and let F be the surface E×P. Then it is well known that a ruled surface F* which is birational to F is biregular to a surface which is obtained by successive elementary transformations from F (for the notion of an elementary transformation, see [3]).

In a previous paper [5], one of the present authors has worked out a theory of zeta functions of Selberg’s type for compact quotients of symmetric spaces of rank one. In the present paper, we consider the analogues of those results when G/K is a noncompact symmetric space of rank one and Γ is a discrete subgroup of G such that G/Γ is not compact but such that vol(G/Γ)<∞. Thus, Γ is a non-uniform lattice. Certain mild restrictions, which are fulfilled in many arithmetic cases, will be put on Γ, and we shall consider how one can define a zeta function ZΓ of Selberg’s type attached to the data (G, K, Γ).

The purpose of this paper is to find out what can be learned about valuation rings, and more generally Prufer rings, from a study of their injective modules. The concept of an almost maximal valuation ring can be reformulated as a valuation ring such that the images of its quotient field are injective. The integral domains with this latter property are found to be the Prufer rings with a (possibly) weakened form of linear precompactness for their quotient fields.

There is a problem called the chain problem of prime ideals, which asks, when 0 is a Noetherian local integral domain, whether the length of an arbitrary maximal chain of prime ideals in 0 is equal to rank 0 or not.

Für jeden, der sich an die klassische Logik gewöhnt ist, mag undenkbar sein, dass der „Satz vom ausgeschlossenen Dritten” in der intuitionistischen Logik im allgemeinen nicht anerkannt ist. Aber es ist ganz natürlich, wenn man diejenige Tatsache beobachtet, dass eine äusserlich gleiche Aussage in der klassischen Logik und in der intuitionistischen doch ungleiche inhaltliche Sinne haben kann.

In recent years, the theory of Borel summability or multisummability of divergent power series of one variable has been established and it has been proved that every formal solution of an ordinary differential equation with irregular singular point is multisummable. For partial differential equations the summability problem for divergent solutions has not been studied so well, and in this paper we shall try to develop the Borel summability of divergent solutions of the Cauchy problem of the complex heat equation, since the heat equation is a typical and an important equation where we meet diveregent solutions. In conclusion, the Borel summability of a formal solution is characterized by an analytic continuation property together with its growth condition of Cauchy data to infinity along a stripe domain, and the Borel sum is nothing but the solution given by the integral expression by the heat kernel. We also give new ways to get the heat kernel from the Borel sum by taking a special Cauchy data.

By using a certain linear operator defined by a Hadamard product or convolution, several interesting subclasses of analytic functions in the unit disk are introduced and studied systematically. The various results presented here include, for example, a number of coefficient estimates and distortion theorems for functions belonging to these subclasses, some interesting relationships between these subclasses, and a wide variety of characterization theorems involving a certain functional, some general functions of hypergeometric type, and operators of fractional calculus. Some of the coefficient estimates obtained here are fruitfully applied in the investigation of certain subclasses of analytic functions with fixed finitely many coefficients.

Let G be a finite group and let ψ be an (ordinary) irreducible character of a normal subgroup N. If ψ extends to a character of G then ψ is invariant under G, but the converse is false. In section 3 it is shown that if ψ extends coherently to the intermediate groups H for which H/N is elementary, then ψ extends to G. If N is a Hall subgroup, then in order for ψ to extend to G it is sufficient that ψ be invariant under G. This leads to a construction of the characters of G from the characters of N and the characters of the subgroups of G/N in this case.

A general extension theorem for L2 holomorphic bundle-valued top forms is formulated. Although its proof is based on a principle similar to Ohsawa-Takegoshi’s extension theorem, it explains previous L2 extendability results systematically and bridges extension theory and division theory.

In this paper we are concerned with stochastic relaxed control problems of the following kind. Let X(t), t ≥ 0, denote the state of a process being controlled, Y(t), t ≥ 0, the observation process and p(t, ·) a relaxed control, that is a process with values probability measures on the control region Г. The state and observation processes are governed by stochastic differential equations

and

where B and W are independent Brownian motions with values in Rn and Rm respectively, (put m = 1 for simplicity).

In this paper we consider the initial value problems for the following quasilinear wave equations with dissipative terms

with initial conditions

where

In 1929, H. Cartan declared that there are at most two meromorphic functions on ℂ which share four values without multiplicities, which is incorrect but affirmative if they share four values counted with multiplicities truncated by two. In this paper, we generalize such a restricted H. Cartan’s declaration to the case of maps into PN (ℂ). We show that there are at most two nondegenerate meromorphic maps of ℂn into PN(ℂ) which share 3N + 1 hyperplanes in general position counted with multiplicities truncated by two. We also give some degeneracy theorems of meromorphic maps into PN (ℂ) and discuss some other related subjects.

The main purposes of this paper are to investigate the existence and the uniqueness of a non-local problem for a linear parabolic equation

in a cylinder D = Ω × (0, T].

Let be a space of homogeneous type in the sense of Coifman and Weiss, and let be a collection of balls in . The authors introduce the localized atomic Hardy space the localized Morrey-Campanato space and the localized Morrey-Campanato-BLO (bounded lower oscillation) space with α ∊ ℝ and p ∊ (0, ∞) , and they establish their basic properties, including and several equivalent characterizations for In particular, the authors prove that when α > 0 and p ∊ [1, ∞), then and when p ∈(0,1], then the dual space of is Let ρ be an admissible function modeled on the known auxiliary function determined by the Schrödinger operator. Denote the spaces and , respectively, by and when is determined by ρ. The authors then obtain the boundedness from of the radial and the Poisson semigroup maximal functions and the Littlewood-Paley g-function, which are defined via kernels modeled on the semigroup generated by the Schrödinger operator. These results apply in a wide range of settings, for instance, the Schrödinger operator or the degenerate Schrödinger operator on ℝd, or the sub-Laplace Schrödinger operator on Heisenberg groups or connected and simply connected nilpotent Lie groups.

In § 1 of this note we first define the trace of an endomorphism of a projective module P over a non-commutative ring A. Then we call the trace of the identity the rank element r(P) of P, which we shall illustrate by several examples. For a projective module P over the groupalgebra of a finite group G, the rank element of P is essentially the character of G in P. In § 2 we prove that under certain assumption two projective modules Pi and P2 over an algebra over a complete local ring o are isomorphic if and only if their rank elements are identical. This is a type of proposition asserting that two representations are equivalent if and only if their characters are identical, and in fact, when A is the groupalgebra, the above theorem may be considered as another formulation of Swan’s local theorem [9]).

As a generalization of the notion of Frobenius algebras over a field Kasch [103 introduced that of Frobenius extensions of a ring. The present writers [13] recently freed one of Kasch’s main theorems from its rather strong S-ring assumption of the ground ring. However, even with the removal of the S-ring assumption of the ground ring the notion does not seem general enough, and we wish, in the present paper and its sequel, to develope the theory upon the basis of a more general notion of Frobenius extensions. Thus, we replace the free module property of the extension by the projective module property (according to a general tendency in algebra), which has been done in fact in case of Frobenius algebras over a commutative ring in a previous work by Eilenberg and one of the writers [4], and, further, take automorphisms of the ground ring into the definition of Frobenius extensions (which seems quite natural particularly in case of non-commutative rings). To such generalized notion of Frobenius extensions we may extend many of Kasch’s theorems, including those which are immediate extensions of classical theorems for Frobenius algebras and those which are essentially new, as the above alluded endomorphism ring theorem. Also homological properties of Frobenius extensions, as were developed in Hirata’s [6] recent paper in succession to Eilenberg-Nakayama [4], can be extended to our present generalized case; we shall also exceed [4], [6] somewhat in considering injective and weak dimensions.

The purpose of this paper is to establish the following isomorphism of Lie algebras.

Main Theorem. Let X be the complement of a hypersurface S in the complex projective space PN. Then the tower of nilpotent complex Lie algebras associated with the fundamental group π1(X, *) and the holonomy Lie algebraattached to S are isomorphic.

We prove a Bernstein-type property for maximal spacelike hypersurfaces in a Lorentzian manifold.

Nazarov [Naz96] introduced an infinite dimensional algebra, which he called the affine Wenzl algebra, in his study of the Brauer algebras. In this paper we study certain “cyclotomic quotients” of these algebras. We construct the irreducible representations of these algebras in the generic case and use this to show that these algebras are free of rank rn(2n−1)!! (when Ω is u-admissible). We next show that these algebras are cellular and give a labelling for the simple modules of the cyclotomic Nazarov-Wenzl algebras over an arbitrary field. In particular, this gives a construction of all of the finite dimensional irreducible modules of the affine Wenzl algebra.