The main object of this paper is to develop techniques for investigating the local properties of actions of an algebraic group on an algebraic variety. Our main tools are certain schemes which may be associated to Lie algebras.

Let D be the unit disk | z | < 1 and let f be a complex-valued function continuous in D. The cluster set C(f, eiθ) of f at eiθ is defined by

In this paper we describe a canonical procedure for constructing the extension of a G-foliation on a differentiable manifold X to its tangent bundles of higher order and by applying the Bott-Haefliger’s construction of characteristic classes of G-foliations ([2], [3]) we obtain an infinite sequence of characteristic classes for those foliations (Theorem 4.8).

Let Q be the set of symbols 1, 2,…, 3 p, where p is an prime number greater than 3. Let be a transitive permutation group on Ω, which is simple and in which the normalizer of a Sylow p-subgroup has order 2 p.

In this paper we investigate indices of umbilics of a closed surface in the euclidean space. Most part of the discussion is concerned with a symmetric tensor field of degree 2, or rather a direction field, on a Riemannian manifold of dimension 2.

R. Bowen [1] introduced the notion of pseudo-orbit for a homeomorphism f of a metric space X as follows: A (double) sequence {xi}i∈Z of points Xi in X is called a δ-pseudo-orbit of f iff

d(fxi, xi+1) ≤ δ

for every i ∈ Z, where d denotes the metric in X. We say f is stochastically stable if for every ε > 0 there exists δ > 0 such that every δ pseudo-orbit {Xi}i∈Z of f is ε-traced by some x ∈ X, i.e.,

d(fix, xi) ≤ ε

for every i ∈ Z. He proved in [1] that if a compact hyperbolic set Λ for a diffeomorphism f of a compact manifold M has local product structure then the restriction f | Λ of f to Λ is stochastically stable, using stable and unstable manifolds.

Let K be a field of characteristic zero and let

be m generic n by n matrices over K. That is, xij(k) are independent commuting indeterminates over K. The K-subalgebra generated by X1…, Xm is called a ring of n by n generic matrices and is denoted by R(n, m).

What we call the homological conjectures on commutative Noetherian local rings were first collected and partially settled by C. Peskine and L. Szpiro [PS1]. The subsequent remarkable progress was made by M. Hochster [H1] who conjectured the existence of big Cohen-Macaulay modules and solved it in the affirmative for equicharacteristic local rings. It is, however, still open in general setting.

This paper includes a proof, relative to the consistency of ZFC, of the consistency of ZFC, the continuum has singular cardinality and the extended plus-one hypothesis.

We establish explicit isomorphisms of two seemingly-different algebras, and their Schur algebras, arising from the centralizers of two different type B Weyl group actions in Schur-like dualities. We provide a presentation of the geometric counterpart of the above Schur algebras in [1] specialized at $q=1$ .

The famous Picard theorem states that a holomorphic mapping f: C → P1(C) omitting distinct three points must be constant. Borel [1] showed that a non-degenerate holomorphic curve can miss at most n + 1 hyperplanes in Pn(C) in general position, thus extending Picard’s theorem (n = 1). Recently, Fujimoto [3], Green [4] and [5] obtained many Picard type theorems using Borel’s methods for holomorphic mappings.

We prove a functional equation for a vector valued real analytic Eisenstein series transforming with the Weil representation of $\operatorname{Sp}(n,\mathbb{Z})$ on $\mathbb{C}[(L^{\prime }/L)^{n}]$. By relating such an Eisenstein series with a real analytic Jacobi Eisenstein series of degree $n$, a functional equation for such an Eisenstein series is proved. Employing a doubling method for Jacobi forms of higher degree established by Arakawa, we transfer the aforementioned functional equation to a zeta function defined by the eigenvalues of a Jacobi eigenform. Finally, we obtain the analytic continuation and a functional equation of the standard $L$-function attached to a Jacobi eigenform, which was already proved by Murase, however in a different way.

Recently M. Ikeda [1] succeeded in determining the structure of absolutely segregated algebras, i.e. algebras whose 2-cohomology groups all vanish. His beautiful result reads : an algebra A, of finite rank over its ground field, is absolutely segregated if and only if i) the residue-algebra A/N modulo the radical N is separable and, moreover, ii) the A-left-module N is an (Mo)-module. A. simplification was given by H. Nagao [5], who obtained, besides an interesting-result on algebras with vanishing 3-(or higher) cohomology groups, an elegant short proof to the fact that under the assumption of i), the property ii) is necessary, and sufficient, for the absolute segregation of A.

The purpose of this paper is to define minimality of surfaces in an infinite dimensional space E by probabilistic methods with the description of the relation between minimal surfaces and harmonic functions on the space E, and to analyze purely analytic properties of a certain class of quadratic forms on the space E.

Let K be the classical quaternion field over the field Q of rational numbers with the quaternion units 1, i, j, k, with relations i2 = j2 = − 1, k = ij = −ji. For a quaternion x ∈ K, we write its conjugate, trace and norm by and Nx, respectively. Put

and consider the map h: A → B defined by

Let Γ be a curve defined by y = A(x) in R2. The Cauchy transform on the curve Γ is a singular integral operator defined by the singular integral kernel

(1.1)

Let G be a connected algebraic group, p a finite dimensional representation of G in a vector space V, all defined over a number field fc. To the pair (G, p) we can associate the family of theta series

The purpose of this note is to prove the following proposition, which is made use of in the preceding paper by Mr. M. Kuranishi in this number. First we shall explain the notations. Let be a word with respect to two letters, and let G be a free group generated by two free generators x and y.

In [10] and [11], the second author proved the following theorem by using Synge’s formula:

THEOREM I. Let M be a kähler manifold with positive holomorphic bi-sectional curvature. Then every pseudoconvex domain in M is a Stein manifold.