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.

Let X be a complete, geometrically irreducible, singular, algebraic curve defined over a field of characteristic p big enough. Given a local ring Op,x at a rational singular point P of X, we attached a universal zeta function which is a rational function and admits a functional equation if Op,x is Gorenstein. This universal zeta function specializes to other known zeta functions and Poincaré series attached to singular points of algebraic curves. In particular, for the local ring attached to a complex analytic function in two variables, our universal zeta function specializes to the generalized Poincaré series introduced by Campillo, Delgado, and Gusein-Zade.

The concept of a local ring was introduced by Krull [2], who defined it as a Noetherian ring R (we say that a commutative ring R is Noetherian if every ideal in R has a finite basis and if R contains the identity) which has only one maximal ideal m. If the powers of m are defined as a system of neighbourhoods of zero, then R becomes a topological ring satisfying the first axiom of countability, And the notion was studied recently by C. Chevalley and I. S. Cohen. Cohen [1] proved the structure theorem for complete rings besides other properties of local rings.

Consider a linear transformation T whose domain D(T) and range R(T) lie in a complex Banach space X. We denote by [X] the class of all linear transformations T for which D(T) = X and T is continuous (bounded). Also, we denote the resolvent set of T by ρ(T) and the spectrum of T by σ(T).

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).

Let D be a bounded domain in Cn; in the space L2(D) of functions on D which are square-integrable with respect to the Lebesgue measure d2nz the holomorphic functions form a closed subspace H2(D). Therefore there exists a well-defined orthogonal projection PD: L2(D) → H2(D) with an integral kernel KD:D × D → C, the Bergman kernel function of D. An explicit computation of this function directly from the definition is possible only in very few cases, as for instance the unit ball, the complex “ellipsoids” , or the annulus in the plane. Also, there is no hope of getting information about the function KD in the interior of a general domain. Therefore the question for an asymptotic formula for the Bergman kernel near the boundary of D arises.

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.

Many authors have considered branching Markov processes for the probabilistic treatment of semi-linear equations. Recently J.E. Moyal [11], [12] gave a formulation for a wide class of branching processes. A similar idea was used in A.V. Skorohod [18] and N. Ikeda-M. Nagasawa-S. Watanabe [4]-[7]. Applying their method, we shall consider in this paper the following problems (A) and (B).

The (usual) Caldero–Chapoton map is a map from the set of objects of a category to a Laurent polynomial ring over the integers. In the case of a cluster category, it maps reachable indecomposable objects to the corresponding cluster variables in a cluster algebra. This formalizes the idea that the cluster category is a categorification of the cluster algebra. The definition of the Caldero–Chapoton map requires the category to be 2-Calabi-Yau, and the map depends on a cluster-tilting object in the category. We study a modified version of the Caldero–Chapoton map which requires only that the category have a Serre functor and depends only on a rigid object in the category. It is well known that the usual Caldero–Chapoton map gives rise to so-called friezes, for instance, Conway–Coxeter friezes. We show that the modified Caldero–Chapoton map gives rise to what we call generalized friezes and that, for cluster categories of Dynkin type A, it recovers the generalized friezes introduced by combinatorial means in recent work by the authors and Bessenrodt.

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).

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).

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) .

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.

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 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.

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.

Let (G, ρ, X) be a prehomogeneous vector space with singular set S over an algebraic number field F. The main result of this paper is a proof for the convergence of the zeta fucntions Z(Φ, s) associated with (G, ρ, X) for large Re s under the assumption that S is a hypersurface. This condition is satisfied if G is reductive and (G, ρ, X) is regular. When the connected component of the stabilizer of a generic point x is semisimple and the group Πx of connected components of Gx is abelian, a clear estimate of the domain of convergence is given.

Moreover when S is a hypersurface and the Hasse principle holds for G, it is shown that the zeta fucntions are sums of (usually infinite) Euler products, the local components of which are orbital local zeta functions. This result has been proved in a previous paper by the author under the more restrictive condition that (G, ρ, X) is irreducible, regular, and reduced, and the zeta function is absolutely convergent.

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).

In the previous paper [4] we have studied the prolongations of G-structures to tangent bundles. The purpose of the present paper is to generalize the previous prolongations and to look at them from a wide view as a special case by considering the tangent bundles of higher order. In fact, in some places, the arguments and calculations in [4] are more or less simplified. Since the usual tangent bundle T(M) of a manifold M considers only the first derivatives or first contact elements of M, the previous paper contains, in most parts, only the calculation of derivatives of first order.

In the theory of modular representations of a finite group G in an algebraically closed field Ω of characteristic p, Brauer has proved a useful reduction theorem for blocks [2, §§11, 12], [5, (88.8)], which can be reformulated as follows:

THEOREM 1 (Brauer). Let P bean arbitraryp-subgroup of G; let N = NG(P) and W = PCG(P).