The foundations for this paper were developed in [5], “Seminormal rings and weakly normal varieties”, where the historical framework and fundamental properties of weakly normal varieties were presented in detail. Here we devote our attention to the study of the multicross singularity and the role of local cohomology in the theory of weakly normal varieties.

It is proved that every holomorphically convex complex space endowed with an action of a compact Lie group K can be realized as an open K-stable subspace of a holomorphically convex space endowed with a holomorphic action of the complexified group KC. Similar results are obtained for holomorphic if-bundles over such spaces.

We construct a bounded plane domain which is Bergman complete but for which the Bergman kernel does not tend to infinity as the point approaches the boundary.

1. Let Ω be a finite set of arbitrary elements and let (G, Ω) be a permutation group on Ω. (This is also simply denoted by G). Two permutation groups (G, Ω) and (G, Γ) are called isomorphic if there exist an isomorphism σ of G onto H and a one to one mapping τ of Ω onto Γ such that (g(i))τ=gσ(iτ) for g ∊ G and i∊Ω. For a subset Δ of Ω, those elements of G which leave each point of Δ individually fixed form a subgroup GΔ of G which is called a stabilizer of Δ. A subset Γ of Ω is called an orbit of GΔ if Γ is a minimal set on which each element of G induces a permutation. A permutation group (G, Ω) is called a group of rank n if G is transitive on Ω and the number of orbits of a stabilizer Ga of a ∊ Ω, is n. A group of rank 2 is nothing but a doubly transitive group and there exist a few results on structure of groups of rank 3 (cf. H. Wielandt [6], D. G. Higman M).

In this paper, we proved a result that if two meromorphic functions f(z) and g(z) share five small functions aj(z) (j = 1,…,5) in the sense of (j = 1,…,5) (k ≥ 22), then we have f(z) ≡ g(z).

In the frame of the recent axiomatic theories of harmonic functions [2], [3], [1], it has been shown that the continuous bounded functions on the boundaries of relatively compact open sets are resolutive [5], [1]. The aim of the present paper is to substitute in these results the continuous functions by Borel-measurable functions and to leave out the restriction that the open sets are relatively compact. H. Bauer has replaced the axiom 3 of Brelot’s axiomatic by two weaker axioms: the axiom of separation (Trennungsaxiom) and the axiom K1. Since the axiom of separation is not fulfilled in some important cases (e.g. the compact Riemann surfaces) we shall weaken this axiom too, substituting it by one of its consequences: the minimum principle for hyperharmonic functions.

In [1] R. Brauer asked the following question: Let be a finite group, p a rational prime number, and B a p-block of with defect d and defect group . Is it true that is abelian if and only if every irreducible character in B has height 0 ? The present results on this problem are quite incomplete. If d-0, 1, 2 the conjecture was proved by Brauer and Feit, [4] Theorem 2. They also showed that if is cyclic, then no characters of positive height appear in B. If is normal in , the conjecture was proved by W. Reynolds and M. Suzuki, [12]. In this paper we shall show that for a solvable group , the conjecture is true for the largest prime divisor p of the order of . Actually, one half of this has already been proved in [7]. There it was shown that if is a p-solvable group, where p is any prime, and if is abelian, then the condition on the irreducible characters in B is satisfied.

Let X(t) denote Brownian motion on the line 0 ≤ t < ∞, let and let 0 < α < 1. Orey and Taylor [5] have investigated the random set defined by the inequalities

and proved that P{dim Eα = 1 – α2} = 1. Here we prove two theorems on Eα that reflect more subtle properties of Eα than its Hausdorff dimension alone.

Let Ω, I be open intervals in Rx = (— ∞ < x < ∞), Rt = (— ∞ < t < ∞) respectively. For a function a(x, t) ∈ C∞(Ω × I), consider the partial differential operator

(1.1) .

For all n > 1, the characteristic function of the unit ball in ℝ2n is not the symbol of a bounded bilinear multiplier operator from Lp(ℝn) × Lq(ℝn) to Lr(ℝn) when 1/p + 1/q = 1/r and exactly one of p, q, or r′ = r/(r – 1) is less than 2.

Let K be a field of characteristic zero, and let R = K[X1,… ,Xn]. Let An(K) = K⟨X1,… ,Xn,∂1,… ,∂n⟩ be the nth Weyl algebra over K. We consider the case when R and An(K) are graded by giving deg Xi = ωi and deg ∂i = –ωi for i = 1,…,n (here ωi are positive integers). Set . Let I be a graded ideal in R. By a result due to Lyubeznik the local cohomology modules are holonomic (An(K))-modules for each i≥0. In this article we prove that the de Rham cohomology modules are concentrated in degree —ω; that is, for j ≠ –ω. As an application when A = R/(f) is an isolated singularity, we relate to Hn-1(∂(f);A), the (n – 1)th Koszul cohomology of A with respect to ∂1(f),…, ∂n(f).

The generalized test ideals introduced in [HY] are related to multiplier ideals via reduction to characteristic p. In addition, they satisfy many of the subtle properties of the multiplier ideals, which in characteristic zero follow via vanishing theorems. In this note we give several examples to emphasize the different behavior of test ideals and multiplier ideals. Our main result is that every ideal in an F-finite regular local ring can be written as a generalized test ideal. We also prove the semicontinuity of F-pure thresholds (though the analogue of the Generic Restriction Theorem for multiplier ideals does not hold).

With respect to Teichmüller spaces, many beautiful results are obtained by TeichmüUer, Ahlfors, Bers, Maskit, Kra, Earle, Abikoff, and others. For example, the boundary consists of b-groups, and the augmented Teichmüller space is defined by attaching a part of the boundary to the Teichmüller space. By using the augmented Teichmüller space, a compactification of the moduli space of Riemann surfaces is accomplished (cf. Abikoff [1], Bers [2]).

We shall show that the spectrum of Laplacian depends continuously on the parameter for one parameter degeneration of projective algebraic manifolds.

In the past few years there has been an increasing interest in a certain class of stochastic differential equations (SDE’s) in Hilbert spaces for applications in quantum mechanics (measurements continuous in time [1-5]) and in quantum optics (photon-detection theory and numerical simulations of master equations [6-10]). Part of the mathematical theory of these equations has been developed in [11], where also “structural properties” of this class of equations have been studied. In the paper [11] it has been shown that such equations are connected with certain semigroups of linear operators and the form of the generator of semigroups related to such SDE’s has been established.

In the previous paper [3] we have studied the prolongations of G-structures to tangent bundles of higher order. The purpose of the present paper is to study the prolongations of connections to tangential fibre bundles of higher order, and to generalize the results due to S. Kobayashi [1] for the case of usual tangent bundle —— in fact, the arguments in [1] will be, in a sense, more or less simplified and clarified by using the notion of tangent bundles of higher order. In addition, as a consequence of our results, we shall obtain the prolongations of linear (affine) connections to tangent bundles of higher order.

In a classical paper [C] Chevalley considered the invariants of a finite group H ⊂ GLk(S1) generated by pseudo-reflections, acting on the graded polynomial ring S = k[X1,…,Xn] over a field k of characteristic zero. He proved that S is free as a graded SH-module, hence SH is a graded polynomial ring (Theorem A), and that the natural representation of H in is equivalent to the regular representation (Theorem B). On the other hand, a theorem of Shephard and Todd shows that when SH is a polynomial ring, the (finite) group H is generated by pseudo-reflections. These results have been extended by Bourbaki [Bo2] to fields whose characteristic may be positive, but does not divide the order |H| of the group.

We study generic and conformally flat hypersurfaces in Euclidean four-space. What kind of conformally flat three manifolds are really immersed generically and conformally in Euclidean space as hypersurfaces? According to the theorem due to Cartan [1], there exists an orthogonal curvature-line coordinate system at each point of such hypersurfaces. This fact is the first step of our study. We classify such hypersurfaces in terms of the first fundamental form. In this paper, we consider hypersurfaces with the first fundamental forms of certain specific types. Then, we give a precise representation of the first and the second fundamental forms of such hypersurfaces, and give exact shapes in Euclidean space of them.

It is not too much to say that the problem of finding a cause-and-effect relationship is a fascinating and eternal theme in both natural and social sciences. It is often difficult to decide whether one is the cause of another in two related phenomena, but it is an important problem. It is related to the internal structure of phenomena which generate deterministic or random changes as time passes. We note that the phenomena to be considered are often not deterministic but random. For example, in physical systems such as quantum mechanics or chaotic classical mechanics, it is well known that certain probabilistic reasonings are indispensable.

Let G be a permutation group on a set $\Omega $ of size t. We say that $\Lambda \subseteq \Omega $ is an independent set if its pointwise stabilizer is not equal to the pointwise stabilizer of any proper subset of $\Lambda $ . We define the height of G to be the maximum size of an independent set, and we denote this quantity $\textrm{H}(G)$ . In this paper, we study $\textrm{H}(G)$ for the case when G is primitive. Our main result asserts that either $\textrm{H}(G)< 9\log t$ or else G is in a particular well-studied family (the primitive large–base groups). An immediate corollary of this result is a characterization of primitive permutation groups with large relational complexity, the latter quantity being a statistic introduced by Cherlin in his study of the model theory of permutation groups. We also study $\textrm{I}(G)$ , the maximum length of an irredundant base of G, in which case we prove that if G is primitive, then either $\textrm{I}(G)<7\log t$ or else, again, G is in a particular family (which includes the primitive large–base groups as well as some others).