Let R be an integral domain, Q its quotient field. For any R-module A we write d(A) for its homological dimension (minimal length of a projective resolution), or dR(A) if it is necessary to call attention to R.

Suppose that G is a discrete group and p is a probability measure on G. Consider the associated random walk {Xn} on G. That is, let Xn = Y1Y2 … Yn, where the Yj’s are independent and identically distributed G-valued variables with density p. An important problem in the study of this random walk is the evaluation of the resolvent (or Green’s function) R(z, x) of p. For example, the resolvent provides, in principle, the values of the n step transition probabilities of the process, and in several cases knowledge of R(z, x) permits a description of the asymptotic behaviour of these probabilities.

Let A be a commutative local Noetherian ring with the maximal ideal m and M a finitely generated A-module, d = dim M. It is well-known that the difference between the length and the multiplicity of a parameter ideal q of M

gives a lot of informations on the structure of the module M. For instance, M is a Cohen-Macaulay (CM for short) module if and only if IM(q) = 0 for some parameter ideal q or M is Buchsbaum module (see [S-V]) if and only if IM(q) is a constant for all parameter ideals q of M.

Let Q be the rational number field, and let K = (D > 0 a rational integer) be a real quadratic field. Then, throughout this paper, we shall understand by the fundamental unit εD of the normalized fundamental unit εD > 1.

The local theory of continuous (infinite) pseudo-groups of transformations was originated by S. Lie, and developed by himself, F. Engel, E. Vessiot, E. Cartan, etc. In the beginning, the definition was not clear and we can find several different definitions in the papers of pioneers. In 1902, E. Cartan introduced a definition using his theory of exterior differential systems and made an extensive study in his series of papers [1], [2], and [31 The writer will adopt his definition in this series of papers. A continuous pseudo-group of transformations is, roughly speaking, a collection of real (or complex) analytic homeo-morphisms of domains in a real (or complex) euclidean space, which is closed under the operations of composition and inverse, and which forms the general solutions of a system of partial differential equations.

Let Mg be the coarse moduli space of complete non-singular curves of genus g and the coarse moduli space of principally polarized abelian varieties of dimension g. There is a canonical map:

The notion of S-rings was introduced by F. Kasch [4] in establishing a theory of Frobenius extensions. S-rings possess several remarkable properties, although they have been shown to be not indispensable to the theory of Frobenius extensions. The purpose of this paper is to give some characteristic properties of S-rings.

In the previous papers [1], [2] we have studied the prolongations of G-structures to tangent bundles of arbitrary order, and in [3] we have considered the prolongations of connections to the tangential fibre bundles of higher order. The purpose of the present paper is to study the liftings of tensor fields and affine connections to tangent bundles of higher order. In fact, most of results in [4], [5] will be generalized in a natural fashion and some of the formulas concerning vertical and complete lifts in [5] will be unified and generalized in our formulas concerning ‹Λ›-lifts (cf. §3).

In [4] EGA IV (7.4.8) hat Grothendieck die folgende Frage gestellt: “A sei ein noetherscher Ring, I⊆ A ein Ideal, so daß A separiert und komplett in der I-adischen Topologie ist. A/I sei ein P-Ring. 1st dann A ebenfalls ein P-Ring?″ In dieser Arbeit beschäftigen wir uns mit dem Fall, daß A ein semilokaler noetherscher Ring ist und P die Eigenschaft “die formellen Fasern von A sind geometrisch regulär” bezeichnet. Wir wollen zeigen: “A sei ein semilokaler noetherscher I-adisch kompletter Ring, wobei I ein im Jacobsonradikal von A enthaltenes Ideal ist. Sind die formellen Fasern von A/I geometrisch regulär, so sind auch die formellen Fasern von A geometrisch regulär.”

Ever since S. Helgason [4] showed that any eigenfunction of the Laplace-Beltrami operator on the unit disk is represented by the Poisson integral of a hyperfunction on the unit circle, much interest has been arisen to the study of the Poisson integral representation of joint eigenfunctions of all invariant differential operators on a symmetric space X. In particular, his original idea of expanding eigenfunctions into K-finite functions has proved to be generalizable up to the case where X is a Riemannian symmetric space of rank one (cf. [4], [5], [11]). Presently, extension to arbitrary rank has been completed by quite a different formalism which views the present problem as a boundary-value problem for the differential equations. It should be recalled that along this line of approach a general theory of the systems of differential equations with regular singularities was successfully established by Kashiwara-Oshima (cf. [6], [7]).

Suppose Ω is a simply connected domain which is mapped conformally onto a disk. A much studied problem is the behavior of the mapping function at an accessible boundary point P of Ω, in particular the question, under what conditions the map is ‘ “conformai” at such a point (a) in the sense that angles are preserved as P is approached from Ω (“semi-conformality” at P) and (b) the dilatation at P is finite and positive. In his fundamental paper [8] in 1936, A. Ostrowski established a necessary and sufficient condition (depending on the geometry of the domain only) for the validity of the first property which subsumes all previous results and establishes a definitive solution of this problem.

We study the Yablonskii-Vorob’ev polynomial associated with the second Painlevé equation. To study other special polynomials (Okamoto polynomials, Umemura polynomials) associated with the Painlevé equations, our purely algebraic approach is useful.

In [3] Cameron and Storvick introduced a very general operator-valued function space “integral”. In [3-5, 8, 9, 11, 13-20] the existence of this integral as an operator from L2 to L2 was established for certain functions. Recently the existence of the integral as an operator from L1 to L∞, has been studied [6, 7, 21]. In this paper we study the integral as an operator from Lp to Lp′, where 1 < p ≤ 2. The resulting theorems extend the theory substantially and indicate relationships between the L2-L2 and L1-L∞ theories that were not apparent earlier. Even in the most studied case, p = p′ = 2, the results below strengthen the theory.

A generalization of the Landen identity, in the form of an inequality, is proved for hypergeometric functions. Some well-known asymptotic formulas are refined.

In [5], Chase, Harrison and Rosenberg proved the Fundamental Theorem of Galois Theory for commutative ring extensions S ⊃ R under two hypotheses: (i) 5 (and hence R) has no idempotents except 0 and l; and (ii) 5 is Galois over R with respect to a finite group G—which in the presence of (i) is equivalent to (ii′): S is separable as an R-algebra, finitely generated and projective as an R-module, and the fixed ring under the group of all R-algebra automorphisms of S is exactly R.

In [BMR] we observed that, on the level of derived categories, representations of the Lie algebra of a semisimple algebraic group over a field of finite characteristic with a given (generalized) regular central character can be identified with coherent sheaves on the formal neighborhood of the corresponding (generalized) Springer fiber. In the present paper we treat singular central characters.

The basic step is the Beilinson-Bernstein localization of modules with a fixed (generalized) central character λ as sheaves on the partial flag variety corresponding to the singularity of λ. These sheaves are modules over a sheaf of algebras which is a version of twisted crystalline differential operators. We discuss translation functors and intertwining functors. The latter generate an action of the affine braid group on the derived category of modules with a regular (generalized) central character, which intertwines different localization functors. We also describe the standard duality on Lie algebra modules in terms of D-modules and coherent sheaves.

The theory of group characters was originated by G. Frobenius and has been studied by many authors including, above all, I. Schur and W. Burnside. As to the modular theory we owe it to recent works by R. Brauer and his collaborators, including T. Nakayama and C. Nesbitt, which clarified also the connections between the structure theory and the representation theory.

Let R be a ring with identity element 1, and let S be a subring of R containing 1. We consider R-modules on which 1 acts as the identity map, and we shall simultaneously regard such R-modules as S-modules in the natural way. In [4], we have defined the relative analogues of the functors of Cartan-Eilenberg [1], and we have briefly treated the corresponding relative analogues of module dimension and global ring dimension.

The purpose of this note is to clarify relations between multiple Markov properties (MMP) defined by Lévy ([8], [9]) and Hida [5] for Gaussian processes and to extend some work in Lévy [8] and Hida [5]. In the stationary Gaussian case it has been shown ([5], [4]) that these notions of MMP coincide. Interesting examples of (non-stationary) processes satisfying MMP can be found in [5], [8].

In this paper, we shall show that any Gaussian measure on a separable or reflexive Banach space is an abstract Wiener measure in the sense of L. Gross [1] and, for the proof of that, establish the Radon extensibility of a Gaussian measure on such a Banach space. In addition, we shall give some remarks on the support of an abstract Wiener measure.