Some functions related to the complex dilogarithmic function

(in the notation of Lewin [9]) are known to occur in connection with algebraic K-theory and characteristic classes (see e.g. Bloch [1], Gelfand-MacPherson [7], Dupont [5], and the references given there). Recently MacPherson and Hain (see [10]) has announced results of a similar kind for some higher polylogarithmic functions. Also Ramakrishnan [11] and [12] has recently studied the classical polylogarithms, which for |z| < 1 are given by

The purpose of the present paper is first to reformulate a Lipschitz convergence theorem for Riemannian manifolds originally introduced by Gromov [17] and secondly to give some applications of the theorem to a class of open Riemannian manifolds.

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] < ∞.

Denote by H3 the 3-dimensional hyperbolic space with sectional curvatures equal to – 1, and let g be a geodesic in H3 Let {ψt} be the translation along g (see § 2) and let {φt} be the one-parameter subgroup of isometries of H3 whose orbits are circles centered on g. Given any α ∊ R, one can show that λ = {λt} = ψt ∘ φαt} is a one-parameter subgroup of isometries of H3 (see § 2) which is called a helicoidal group of isometries with angular pitch α. Any surface in H3 which is λ-invariant is called a helicoidal surface.

In this work we prove some results concerning minimal helicoidal surfaces in H3. The first one reads:

In 1967 Gindikin and Vinberg stated the Fundamental Conjecture for homogeneous Kähler manifolds. It (roughly) states that every homogeneous Kähler manifold is a fiber space over a bounded homogeneous domain for which the fibers are a product of a flat with a simply connected compact homogeneous Kähler manifold. This conjecture has been proven in a number of cases (see [6] for a recent survey). In particular, it holds if the homogeneous Kähler manifold admits a reductive or an arbitrary solvable transitive group of automorphisms [5]. It is thus tempting to think about the general case. It is natural to expect that lack of knowledge about the radical of a transitive group G of automorphisms of a homogeneous Kähler manifold M is the main obstruction to a proof of the Fundamental Conjecture for M. Thus it is of importance to consider the Kähler algebra generated by the radical of the Lie algebra of G. Computations in this context suggest that one rather considers Kähler algebras generated by an arbitrary solvable ideal.

The topic of this note is the Riesz decomposition of excessive functions for a “nice” strong Markov process X. I.e. an excessive function is decomposed into a sum of a potential of a measure and a “harmonic” function. Originally such decompositions were studied by G.A. Hunt [8]. In [1] a Riesz decomposition is given assuming that the state space E is locally compact with a countable base and X is a transient standard process in strong duality with another standard process having a strong Feller resolvent. Recently R.K. Getoor and J. Glover extended the theory to the case of transient Borei right processes in weak duality [6].

Let k be an algebraic number field and K be a finite extension of k. Recently, T. Ono defined positive rational numbers E(K/k) and E′(K/k) for K/k. In [7], he investigated some relations between E(K/k) and other cohomological invariants for K/k. He obtained a formula when K is a normal extension of k. In our paper [3], we obtained a similar formula for E′(K/k) in the case of normal extensions K/k. Both proofs essentially use Ono’s results on the Tamagawa number of algebraic tori, on which the formulae themselves do not depend. Hence, in [8], T. Ono posed a problem to give direct proofs of these formulae.

The purpose of this paper is to discuss some ergodic properties of the stepping stone model proposed by Kimura, M. [4] and developed by Weiss, G.H. and Kimura, M. [12]. Our model to be discussed in this paper involves selection in addition to mutation and migration which are dealt with in [4], [12]. Because of the additional factor selection, the stochastic process describing our model becomes complicated and presents particularly interesting profound structure of the random phenomena in question.

The purpose of this paper is to discuss some particular random fields derived from Lévy’s Brownian motion to find its characteristic properties of the joint probability distributions. In [9], special attention was paid to the behaviour of the Brownian motion when the parameter runs along a curve in the parameter space, and with this property the conditional expectation has been obtained when the values are known on the curve.

The present paper deals with the variation of the Brownian motion in the normal direction to a given curve, in contrast to the case in [9], where we discussed the properties along the curve. Actually we shall find, in this paper, formulae of the variation with the help of the normal derivative of Brownian motion and observe its singularity. We then discuss partial derivatives of Rd-parameter Lévy’s Brownian motion and make attempt to restrict the parameter to a hypersurface so that we obtain new random fields on that hypersurface. By comparing such derivatives with those of other Gaussian random fields, we can see that the singularity of the new random fields seems to be an interesting characteristic of Lévy’s Brownian motion. Further, we hope that our approach may be thought of as a first step to the variational calculus for Gaussian random fields.

In the proof of Lemma 12.2, the following inequality (p. 43, line 11) is incorrect.

This should be corrected as follows.