The heat equation is to be solved in a severaldimensional region D with on the boundary B of D. The elementary solution (Green’s function) is interpreted as the transition density of an associated Brownian motion. The latter is built up pathwise from the free Brownian motion by simple geometric and probabilistic transformations.

A formulation of the differentiability of the product function of a locally compact group at the identity may be given in the following way: In a sufficiently small euclidean neighborhood of the identity product satisfies the condition:

In this paper, we prove that every proper holomorphic self-map of a smoothly bounded pseudoconvex circular or Hartogs domain of finite type in C2 is biholomorphic.

Let dz(n) be a multiplicative function defined by

where s = σ + it, z is a. complex number, and ζ(s) is the Riemann zeta function. Here ζz(s) = exp(z log ζ(s)) and let log ζ(s) take real values for real s > 1. We note that if z is a natural number dz(n) coincides with the divisor function appearing in the Dirichlet-Piltz divisor problem, and d-1(n) with the Möbious function.

The present notes provide a proof of i* (ℂP + ℂ(I - P); ℂQ + ℂ(I - Q)) = – χorb(P,Q) for any pair of projections P,Q with τ(P) = τ(Q) = 1/2. The proof includes new extra observations, such as a subordination result in terms of Loewner equations. A study of the general case is also given.

Let q be a positive integer, and L(s, χ) the Dirichlet L-function corresponding to a Dirichlet character χ mod q. We put

where χ runs over all Dirichlet characters mod q except for the principal character χ 0.

In this article, we consider the conjectured relationship between $F$ -purity and log canonicity for polynomials over $\mathbb{C}$ . In particular, we show that log canonicity corresponds to dense $F$ -pure type for all polynomials whose supporting monomials satisfy a certain nondegeneracy condition. We also show that log canonicity corresponds to dense $F$ -pure type for very general polynomials over $\mathbb{C}$ . Our methods rely on showing that the $F$ -pure and log canonical thresholds agree for infinitely many primes, and we accomplish this by comparing these thresholds with the thresholds associated to their monomial term ideals.

This is the first in a series of papers concerned with a relation between a representation in the polynomial ring of additive groups and its translation invariant operators. The present study is to observe several properties of a polynomial sequence pa(x) satisfying the binomial identity:

by means of some translation invariant operators.

Le but principal de cette note est de démontrer les Théorème 4.2 et 6.1. Pour l’effectuer nous utiliserons la méthode de développement par les fonctions propres cf. [6]. Nous ne utilisons pas la méthode de la régularisation elliptique, mais les résultats de Baouendi [1] sont essentiels.

Let E be a totally-disconnected compact set in the z-plane and let Ω be its complement with respect to the extended z-plane. Then Ω is a domain and we can consider a single-valued meromorphic function w = f(z) on Ω which has a transcendental singularity at each point of E. Suppose that E is a null-set of the class W in the sense of Kametani [4] (the class NB in the sense of Ahlfors and Beurling [1]). Then the cluster set of f(z) at each transcendental singularity is the whole w-plane, and hence f(z) has an essential singularity at each point of E. We shall say that a value w is exceptional for f(z) at an essential singularity ζ ∈ E if there exists a neighborhood of ζ where the function f(z) does not take this value w.

Let k be a totally real algebraic number field, the maximal order of k, and let L (resp. M) be a Z -lattice of a positive definite quadratic space U (resp. V) over the field Q of rational numbers. Suppose that there is an isometry σ from L onto M. We have shown that the assumption implies σ(L) = M in some cases in [2]. Our aim in this paper is to improve the results of [2]. In § 1 we introduce the notion of E-type: Let L be a positive definite quadratic lattice over Z .

In a recent note bearing the same title (SIAM Journ. of Numerical Analysis, Vol. 3, 1966), I showed that there existed Weierstrass zeta functions having a vanishing pseudo-period. Cf. line 8, p. 376 of Saks-Zygmund, Analytic Functions. Professor Peter Henrici kindly pointed out to me that the mere existence of a Weierstrass zeta function with a vanishing pseudo-period was readily concluded from a classical formula of the theory of elliptic functions (Hurwitz-Courant, Funktionentheorie, 5th ed., p. 210) which reduces the question to the study of the zeros of

A Vaisman manifold is a special kind of locally conformally Kähler manifold, which is closely related to a Sasaki manifold. In this paper, we show a basic structure theorem of simply connected homogeneous Sasaki and Vaisman manifolds of unimodular Lie groups, up to holomorphic isometry. For the case of unimodular Lie groups, we obtain a complete classification of simply connected Sasaki and Vaisman unimodular Lie groups, up to modification.

This paper is concerned with the problem of the determination of unknown scalars involved in the algorithm of computing the generalized Green functions of reductive groups G over a finite field. In the previous paper, we have treated the case where G = SLn . In this paper, we determine the scalars in the case where G is a classical group Sp2n or SON for arbitrary characteristic.

We introduce the notion of exact tilting objects, which are partial tilting objects $T$ inducing an equivalence between the abelian category generated by $T$ and the category of modules over the endomorphism algebra of $T$. Given a chain of sufficiently negative rational curves on a rational surface, we construct an exceptional sequence whose universal extension is an exact tilting object. For a chain of $(-2)$-curves, we obtain an equivalence with modules over a well-known algebra.

Let Rd be the d-dimensional Euclidean space where each point is expressed by a column vector. Let | x | and ‹x, y› denote the norm and the inner product in Rd . Let Q = (Qjk ) be a real d × d-matrix of which all eigenvalues have positive real parts. Let X be a process of Ornstein-Uhlenbeck type (OU type process) on Rd associated with a Levy process {Z: t ≥ 0} and the matrix Q. Main purpose of this paper is to give a recurrence-transience criterion for the process X when Q is a Jordan cell matrix and to compare it with the case when Q is diagonalizable. Here by a Levy process we mean a stochastically continuous process with stationary independent increments, starting at 0.

Weyl algebra is an associative algebra generated by two elements â and a over R such that the generating relation is given by

âa — aâ = 1,

which is isomorphic to the algebra of differential operators

In our recent papers [3, 4, 5], we defined some new D-invariants for any square-free positive integer D and considered their properties and interrelations among them. Especially, as an application of it, we discussed in [5] the characterization of real quadratic field Q() of so-called Richaud-Degert type in terms of these new D-invariants.

A generalized number operator and a generalized Gross Laplacian are introduced on the basis of white noise distribution theory. The equicontinuity is examined and associated one-parameter transformation groups are constructed. An infinite dimensional analogue of ax + b group and Cauchy problems on white noise space are discussed.

This paper continues the study of stochastic integrals in abstract Wiener space previously given in [14]. We will present, among other things, the detailed discussion and proofs of the results announced in [16]. Let H ⊂ B be an abstract Wiener space.