We study geometry of manifolds endowed with a Grassmann structure which depends on symmetries of their curvature. Due to this reason a complete decomposition of the space of curvature tensors over tensor product vector spaces into simple modules under the action of the group G = GL(p, ℝ) ⊗ GL(q, ℝ) is given. The dimensions of the simple submodules, the highest weights and some projections are determined. New torsion-free connections on Grassmann manifolds apart from previously known examples are given. We use algebraic results to reveal obstructions to the existence of corresponding connections compatible with some type of normalizations and to enlighten previously known results from another point of view.

In this paper we discuss the boundary properties of a function f which is analytic in the open unit disc Δ and has Hadamard gaps—i.e.

(1)

where

. (2)

A deformation theory for rings and algebras was introduced recently by M. Gerstenhaber [1]. Let K be an extension of a field k, and p denotes the characteristic. One of his results is that, if K is separable over k, then it is rigid. It was conjectured in [1] that, if K is not separable over k, then it is not rigid, and if it is further finitely generated, then an integrable element of (see [2]) will be found in the image of Sqp. In this note we shall study the above conjecture in certain special case.

In their paper [12], Toda and the author have concerned themselves in the following

Theorem of Kuramochi. Let R be a hyperbolic Riemann surface of the class OHB(OHD, resp.). Then, for any compact subset K of R such that R−K is connected, R−K as an open Riemann surface belongs to the class OAB(OAD, resp.) (Kuramochi [4]).

They have raised there the question as to whether there exists a hyperbolic Riemann surface, which has no Martin or Royden boundary point with positive harmonic measure and has yet the same property as stated in Theorem of Kuramochi, and given a positive answer to the Martin part of this question.

In his paper [2], [3], D. A. Hejhal investigated the variational theory of linear polynomic functions. In this paper we are concerned with the variational theory of higher-order differential equations. To be more precise, consider a compact Riemann surface having genus g > 1. As is well known, we can choose a projective coordinate covering U = (Ua, za). Fix this coordinate covering of X. We shall be concerned with linear ordinary differential operators of order n defined in each projective coordinate open set Ua

Let I be an odd prime number and let K be an algebraic number field of degree I. Let M denote the genus field of K, i.e., the maximal extension of K which is a composite of an absolute abelian number field with K and is unramified at all the finite primes of K. In [4] Ishida has explicitly constructed M. Therefore it is of some interest to investigate unramified cyclic extensions of K of degree l, which are not contained in M. In the preceding paper [6] we have obtained some results about this problem in the case that K is a pure cubic field. The purpose of this paper is to extend those results.

We find conditions on the local cohomology modules of multi-Rees algebras of admissible filtrations which enable us to predict joint reduction numbers. As a consequence, we are able to prove a generalization of a result of Reid, Roberts and Vitulli in the setting of analytically unramified local rings for completeness of power products of complete ideals.

The device of representing a predicate by a universal sentence gives rise to logics of higher order. To represent an object constructed by a series of certain steps, we usually use terms. To represent a predicate constructed by a series of certain steps, we now use sentences in general. We can now substitute special predicates represented by sentences for predicate variables of a sentence just as we have been able to substitute special objects represented by terms for object variables. It is remarkable that we can always distinguish predicates from objects, proceeding in this way.

In the book “Foundations of algebraic geometry” A. Weil proposed the following problem ; does every differential form of the first kind on a complete variety U determine on every subvariety V of U a differential form of the first kind? This problem was solved affirmatively by S. Koizumi when U is a complete variety without multiple point. In this note we answer this problem in affirmative in the case where V is a simple subvariety of a complete variety U (in §1). When the characteristic is 0 we may extend our result to a more general case but this does not hold for the case characteristic p≠0 (in §2).

Soient X un espace localement compact et non-compact à base dénombrable, G une noyau-fonction continue sur X et M (resp. MK) l’ensemble des toutes mesures de Radon positives sur X (resp. des toutes mesures de Radon positives sur X à support compact).

The boundary behavior of solutions of the heat equation (temperature functions) is investigated. It is proved that a temperature function is identically equal to zero if it vanishes of finite order at some lateral boundary point where it attains a local minimum.

One finds the following example in [3, (34, B)]:

Let k be a field of characteristic p and be n-variables over k. Then if p > 0 and [k : kp] = ∞, is an n-dimensional regular local ring but not a Nagata ring. In particular it is not an excellent ring.

The theory of derivations and differents in the algebraic number fields or Dedekind rings has been developped by A. Weil [1], Y. Kawada [2] and M. Moriya [4]. Further Y. Kawada [3] has investigated the derivations in maximal orders of simple algebras over number fields. This note is concerned with the derivations in the simple algebras over fields which are quotient fields of Dedekind rings; the commutative rings in which the fundamental theorem of multiplicative ideal theory holds. The auther gives his hearty thanks to Prof. M. Moriya who gave him valuable remarks.

Let M, N be connected complex submanifolds of a neighborhood of the origin 0 ∈ Cd, the space of d complex variables, such that 0 ∈ M ∩ N. We shall suppose throughout that M ⊄ N and N ⊄ M in any neighborhood of 0. Let X = M ∪ N. X is an analytic subvariety with the irreducible branches M and N. Let Δ be a neighborhood of 0 in Cd.

Let K be a compact connected Lie group and L a closed subgroup of K In [8] M. Lassalle proves that if K is semisimple and L is a symmetric subgroup of K then the holomorphy hull of any K-invariant domain in Kc/Lc contains K/L. In [1] there is a similar result if L contains a maximal torus of K. The main group theoretic ingredient there was the characterization of K/L as the unique totally real K-orbit in Kc/Lc. On the other hand, Patrizio and Wong construct in [9] special exhaustion functions on the complexification of symmetric spaces K/L of rank 1 and find that the minimum value of their exhaustions is always achieved on K/L. By a lemma of Harvey and Wells [6] one knows that the set where a strictly plurisubharmonic (briefly s.p.s.h) function achieves its minimum is totally real. There is a related result in [2, Lemma 1.3] which states that if φ is any differentiable function on a complex manifold M then the form ddcφ vanishes identically on any real submanifold N contained in the critical set of φ; in particular if φ is s.p.s.h then N must be totally real.

We study the asymptotic distribution of rational points on a generalized flag variety which are of bounded height and satisfy some congruence conditions in the formulation analogous to a strongly Hardy-Littlewood variety.

Linear differential equations have been studied more throughly than any other class. They posses a group of characteristic properties: the invariance of linearity by linear transformations, the linearly dependence of solutions on their initial values, e.t.c. The next simple type of differential equations is quadratic type

Limit theorems are obtained for suitably normalized hitting times of single points for 1-dimensional generalized diffusion processes as the hitting points tend to boundaries under an assumption which is slightly stronger than that the existence of limits γ + 1 of the ratio of the mean and the variance of the hitting time. Laplace transforms of limit distributions are modifications of Bessel functions. Results are classified by the one parameter {γ}, each of which is the degree of corresponding Bessel function. In case the limit distribution is degenerate to one point, by changing the normalization, we obtain convergence to the normal distribution. Regarding the starting point as a time parameter, we obtain convergence in finite dimensional distributions to self-similar processes with independent increments under slightly stronger assumption.

The relationship between Poisson’s summation formula and Hamburger’s theorem [2] which is a characterization of Riemann’s zetafunction by the functional equation was already mentioned in Ehrenpreis-Kawai [1]. There Poisson’s summation formula was obtained by the functional equation of Riemann’s zetafunction. This procedure is another proof of Hamburger’s theorem. Being interpreted in this way, Hamburger’s theorem admits various interesting generalizations, one of which is to derive, from the functional equations of the zetafunctions with Grössencharacters of the Gaussian field, Poisson’s summation formula corresponding to its ring of integers [1], The main purpose of the present paper is to give a generalization of Hamburger’s theorem to some zetafunctions with Grössencharacters in algebraic number fields. More precisely, we first define the zetafunctions with Grössencharacters corresponding to a lattice in a vector space, and show that Poisson’s summation formula yields the functional equations of them. Next, we derive Poisson’s summation formula corresponding to the lattice from the functional equations.

Let M = M(D π) be a (real analytic) fibered manifold of dimension n over a manifold D of dimension p, with projection π. We denote by M′ (D, α) the prolonged fibered manifold of M(D, π). Every point of M′ is a p-dimensional contact element of M, and a p-dimensional contact element X of M belongs to M′ if and only if π*X = Tπ(x)(D), where x is the origin of X. We write x = β(X) and α = π°β. We also denote by M″(D, α′) the prolonged fibered manifold of M′(D, α), where α′ = α°β′ and β′ is the projection of each X′ ∈ M″ to its origin in M′.