We shall work over a fixed algebraically closed field k of arbitrary characteristic. By an algebraic variety over k we shall mean a reduced algebraic scheme over k. Fix a positive integer n and e = (e0, el,…, en) a system of n + 1 weights (i.e. n + 1 positive integers e0, el,…, en). If k[T0, Tl,…, Tn] is the polynomial k-algebra in n + 1 variables, graded by the conditions deg(Ti) = ei i = 0, 1,…, n, denote by Pn(e) = Proj(k[T0, T1,…, Tn]) the n-dimensional weighted projective space over k of weights e. We refer the reader to [3] for the basic properties of weighted projective spaces.

Let be a mean zero Gaussian random field (n ⋜ 2). We call X Euclidean if the probability law of the increments X(A) − X(B) is invariant under the Euclidean motions. For such an X, the variance of X(A) − X(B) can be expressed in the form r(|A − B|) with a function r(t) on [0, ∞) and the Euclidean distance |A − B|.

In [8], H. Maass introduced the ‘Spezialschar’ which is now called the Maass space. It is defined by the relation of the Fourier coefficients of modular forms as follows. Let f be a Siegel modular form on Sp(2,Z) of weight k, and let be its Fourier expansion, where . Then f belongs to the Maass space if and only if

In K, Itô’s paper [1] on the theory of turbulence, the problem to determine the canonical form of the moment tensor of temporally homogeneous and isotropic turbulence, has not been solved. In the present paper, the author will solve the problem by making use of the result of his preceding paper [2]. We shall treat the turbulence in R3 but the similar argument is possible in Rn.

Let G be a connected, semisimple real Lie group with finite center, K a maximal compact subgroup of G. Assume rank G = rank K. Let be the Lie algebra of G, its complexification. If Gc is the simplyconnected complex analytic group corresponding to assume G is the real analytic subgroup of Gc corresponding to .

As non-Gaussian stable stochastic processes have infinite second moments, one cannot use the covariance function to describe their dependence structure.

Let G be a connected Lie group and H a closed subgroup. The homogeneous space M = G/H is called reductive if in the Lie algebra g of G there exists a subspace m such that (subspace direct sum) and where is the Lie algebra of H, see [8].

The first aim in this article is to give some sufficient conditions for a family of meromorphic mappings of a domain D in Cn into PN(C) omitting hypersurfaces to be meromorphically normal. Our result is a generalization of the results of Fujimoto and Tu. The second aim is to investigate extending holomorphic mappings into the compact complex space from the viewpoint of the theory of meromorphically normal families of meromorphic mappings.

Consider an open Riemann surface R and a density P(z)dxdy (z = x + iy), well defined on R. As was shown by Myrberg in [3], if P ≢ 0 is a nonnegative α-Hölder continuous density on R (0 < α ≤ 1) then there exists the Green’s functions of the differential equation

p>on R, where Δ means the Laplace operator. As a consequence, there always exists a nontrivial solution on R.

Let U be a smooth quasi-projective variety over C and let f be a regular function on U. Let DU be the sheaf of algebraic differential operators on U and let M be a regular holonomic DU-module: here, regular means that there exists some smooth compactification X of U and some extension of M as a DX-module which is regular holonomic on X (one also may avoid the use of a smooth compactification to define regularity, see [17]).

In this paper, we give some rigidity results for both harmonic and pseudoharmonic maps from pseudo-Hermitian manifolds into Riemannian manifolds or Kähler manifolds. Some foliated results, pluriharmonicity and Siu–Sampson type results are established for both harmonic maps and pseudoharmonic maps.

Recently J. L. Doob [2] evaluated the Dirichlet integral of the BLD harmonic funtion on a Green space in terms of its fine boundary values and θ-kernel of L. Naïm.

On the other hand, the general theory of additive functionals of Markov processes enables us to define the concept of the Dirichlet norm of functions with respect to Markov processes.

In the present paper we shall remark that to each class of algebraically equivalent cycles on a Jacobian variety we can attach a symmetric element of the ring of endomorphisms of the variety and shall prove some formulae concerning attached symmetric elements.

Let $R$ be a commutative Noetherian ring of prime characteristic $p$. In this paper, we give a short proof using filter regular sequences that the set of associated prime ideals of $H_{I}^{t}(R)$ is finite for any ideal $I$ and for any $t\geqslant 0$ when $R$ has finite $F$-representation type or finite singular locus. This extends a previous result by Takagi–Takahashi and gives affirmative answers for a problem of Huneke in many new classes of rings in positive characteristic. We also give a criterion about the singularities of $R$ (in any characteristic) to guarantee that the set $\operatorname{Ass}H_{I}^{2}(R)$ is always finite.

It may safely said that one of the most important problems in modern algebraic geometry is to elevate theory of abelian functions to the same level as theory of elliptic functions beyond the modern formulation of classical results. Being concerned in such a problem, we feel that one of the serious points is the lack of knowladge on the explicit expressions of abelian varieties and their law of compositions by means of their canonical systems of coordinates: Such expressions correspond to the cubic relation of Weierstrass’ -functions and their addition formulae in theory of elliptic functions.

In D. Aldous gave a sufficient condition for the tightness of a sequence (Xn)n≥0 of right continuous (with left limits) processes taking their values in a separable complete metric space S. As already noted by Aldous this condition is far from being necessary when the processes Xn are not continuous. More precisely the Aldous-condition implies the left-quasi-continuity of all the weak limits of the sequence (Xn)n≥0.

Throughout this paper we assume that all rings contain an identity. We say that R is a semi-primary ring if its (Jacobson) radical N is nilpotent, and R/N is an Artinian ring. We say that R admits a splitting, and we write R=A+B if A is a. subring of R, if B is a two-sided ideal in R, and if A∩B=0 .

A. Weil a généralisé la notion de variété algébrique d’un espace affine (ou projectif ) en introduisant la notion de variété abstraite (nous dirons simplement “variété” au lieu de “variété abstraite,” appelant variétés affines les variétés algébriques d’un espace affine). La définition donnée par A. Weil des variétés algébriques s’inspire manifestement de la notion de variété differentiate, conçue comme réunion d’un nombre fini de morceaux dont chacun peut être considéré comme une partie ouverte d’un espace numérique, les variétés des espaces affines se substituant aux ensembles ouverts des espaces euclidiens.

In topology, one can define in several ways the Chern class of a vector bundle over a certain topological space (Chern [2], Hirzebruch [7], Milnor [9], Steenrod [15]). In algebraic geometry, Grothendieck has defined the Chern class of a vector bundle over a non-singular variety. Furthermore, in the case of differentiable vector bundles, one knows that the set of differentiable cross-sections to a bundle forms a finitely generated projective module over the ring of differentiable functions on the base manifold. This gives a one to one correspondence between the set of vector bundles and the set of f.g.-projective modules (Milnor [10]). Applying Grauert’s theorems (Grauert [5]), one can prove that the same statement holds for holomorphic vector bundles over a Stein manifold.

P. Lévy introduced a notion of Brownian motion with parameter in a metric space (M, d), which is a centered Gaussian system satisfying