In my paper [F3] I more or less explicitly conjectured that if A is a complete local integral domain with maximal ideal and if I = (t1, …, tn) is an ideal in A with n ≤ dim (A) – 2, then Spec (A/I) – is G3 in Spec (A) – . This will be proved in this paper.

In this paper we study two features of relative homological algebra of Frobenius extensions, the one is the relative dimension and the other is the relative complete resolution. The theory of Frobenius extensions was developed by F. Kasch [4] as a generalization of Frobenius algebras. We deal with only Frobenius extensions of finite rank (see §2 for the definition).

The sign of a Latin square is −1 if it has an odd number of rows and columns that are odd permutations; otherwise, it is +1. Let LEn and Lon be, respectively, the number of Latin squares of order n with sign +1 and −1. The Alon-Tarsi conjecture asserts that LEn ≠ Lon when n is even. Drisko showed that LEp+1 ≢ Lop+1 (mod p3) for prime p ≥ 3 and asked if similar congruences hold for orders of the form pk + 1, p + 3, or pq + 1. In this article we show that if t ≤ n, then LEn+1 ≢ L0n+1 (mod t3) only if t = n and n is an odd prime, thereby showing that Drisko’s method cannot be extended to encompass any of the three suggested cases. We also extend exact computation to n ≤ 9, discuss asymptotics for Lo/LE, and propose a generalization of the Alon-Tarsi conjecture.

Some basis results for arithmetic, hyperarithmetic (HA) or sets which have positive measure (or which are not meager, i.e., of the second Baire category) have been obtained by several authors. For example, every non-meager set must have a recursive element (Shoenfield-Hinman, Hinman [2]) but there exists a non-meager set (as well as of measure 1) that contains no recursive element (Shoenfield [7]), and every set (i.e., arithmetic set) of positive measure contains an arithmetic element (Sacks [5], and Tanaka [12]).

Studies for A.C. Stark Hamiltonian are closely related to that for the self-adjoint operator on torus. In this paper we use Mourre’s commutator method, which makes great progress for the study of time-independent Hamiltonian. By use of it we show the asymptotic behavior of the unitary propagator as σ → ± ∞.

Let k[x, y] be the ring of polynomials in two variables over a field k of characteristic zero.

If f, g ∈ k[x, y] then we write f ~ g in the case where f = ag, for some a ∈ k = k\{0}, and we denote by [f, g] the jacobian of (f, g), that is, [f, g] = fxgy - fygx

Let ξ(t) and η(t) be two stochastic processes such that ξ has stationary independent increments and ξ(0) = 0 a.s. Suppose that for each 0 < t ≤ 1, with ξ(tβ) independent of η(t) and a fixed parameter β ∈ (−2, 0). It is shown that ξ(1) satisfies the above equation if and only if ξ(1) is a sum of two independent r.v.’s: strictly stable one with the exponent – β and the one given by a random integral where Y has stationary independent increments and E [|| Y(1)||-β] < ∞.

The main purpose of the present paper is to show that the local homogeneous holonomy group of the conformally flat Riemannian manifold is the full rotation group with some exceptions.

We study symplectic deformation types of minimal symplectic fillings of links of quotient surface singularities. In particular, there are only finitely many symplectic deformation types for each quotient surface singularity.

The fourteenth problem of Hilbert asked finite generation of a certain class of rings and had a counter-example (cf. [4]). On the other hand, many mathematicians gave various sufficient conditions for finite generation of such rings (see, for instance, [9], [5] and [8]). The purpose of the present paper is to give a new sufficient condition. The class of rings to be treated is much more general than those treated before, except for the one in [8].

For a free Z-module N of rank n, let T = TN be an n-dimensional algebraic torus over an algebraically closed field k defined by N. Let X = TN emb (Δ) be a smooth complete toric variety defined by a fan Δ (cf. [6]). Then T acts algebraically on X. A vector bundle E on X is said to be an equivariant vector bundle, if there exists an isomorphism ft: t*E → E for each k-rational point t in T, where t: X → X is the action of t. Equivariant vector bundles have T-linearizations (see Definition 1.2 and [2], [4]), hence we consider T-linearized vector bundles.

We have introduced in [9] a type of separable extensions of a ring as a generalization of the notion of central separable algebras. Unfortunately it was unsuitable to call such extensions ‘central5 as Sugano pointed out in [15] (Example below Theorem 1.1). Some additional properties of such extensions were given in [15]. Especially Propositions 1. 3 and 1. 4 in [15] are interesting and suggested us to consider the commutor theory of separable extensions.

At the early fifties A. Weil introduced [3] and algebraic approach to the theory of infinitesimal prolongations of smooth manifolds motivated by the theory of jets developed by Ch. Ehresmann on one side and also by the return to Fermat’s methods on the infinitesimal calculus of first order, that makes use of nilpotent infinitesimals.

Let SF = SG denote the space , and BSF be the classifying space of SF. Our purpose is to determine H*(BSF: Zp) as a Hopf algebra over Zp where p is an odd prime number. We have announced the main result in [14].

The main result of the present paper is the theorem 1, which describes the topological support of an arbitrary Gaussian measure in a separable Banach space. This theorem will be proved after some discussion of the notion of support itself. But we begin with the reminder of the notion of covariance operator of a probability measure. This notion has a great importance not only for the description of support of Gaussian measures but also for the study of other problems in the theory of probability measures in linear spaces (c.f. [1], [2]).

We establish Noether’s inequality for surfaces of general type in positive characteristic. Then we extend Enriques’ and Horikawa’s classification of surfaces on the Noether line, the so-called Horikawa surfaces. We construct examples for all possible numerical invariants and in arbitrary characteristic, where we need foliations and deformation techniques to handle characteristic 2. Finally, we show that Horikawa surfaces lift to characteristic zero.

Throughout this paper all functions are single-valued. Let R be a Riemann surface. We shall denote by φ∧ the least harmonic majorant of a function φ defined in R if it has the meaning.

We give a bound on embedding dimensions of geometric generic fibers in terms of the dimension of the base, for fibrations in positive characteristic. This generalizes the well-known fact that for fibrations over curves, the geometric generic fiber is reduced. We illustrate our results with Fermat hypersurfaces and genus 1 curves.

The purpose of the present paper is to study the prolongations of G-structures on a manifold M to its tangent bundle T(M), G being a Lie subgroup of GL(n,R) with n = dim M. Recently, K. Yano and S. Kobayashi [9] studied the prolongations of tensor fields on M to T(M) and they proposed the following question: Is it possible to associate with each G-structure on M a naturally induced G-structure on T(M), where G′ is a certain subgroup of GL(2n,R)? In this paper we give an answer to this question and we shall show that the prolongations of some special tensor fields by Yano-Kobayashi — for instance, the prolongations of almost complex structures — are derived naturally by our prolongations of the classical G-structures. On the other hand, S. Sasaki [5] studied a prolongation of Riemannian metrics on M to a Riemannian metric on T(M), while the prolongation of a (positive definite) Riemannian metric due to Yano-Kobayashi is always pseudo-Riemannian on T(M) but never Riemannian. We shall clarify the circumstances for this difference and give the reason why the one is positive definite Riemannian and the other is not.

The existence of the asymptotic probability measure of the Riemann zeta-function was proved in Bohr-Jessen’s classical paper [3] [4].

Let s = σ + it be a complex variable, ζ(s) the Riemann zeta-function, and R an arbitrary rectangle with the edges parallel to the axes. Then, for any σ0 > ½ and T > 0, the set

is Jordan measurable, and we denote the Jordan measure of this set by V(T, R; ξ). Then, Bohr-Jessen’s main result asserts the existence of the limit