Bott's vanishing theorem for regular Lie algebroids

Differential geometry has discovered many objects which determine Lie algebroids playing a role analogous to that of Lie algebras for Lie groups. For example:

--- differential groupoids,

--- principal bundles,

--- vector bundles,

--- actions of Lie groups on manifolds,

--- transversally complete foliations,

--- nonclosed Lie subgroups,

--- Poisson manifolds,

--- some complete closed pseudogroups.

We carry over the idea of Bott's Vanishing Theorem to regular Lie algebroids (using the Chern-Weil homomorphism of transitive Lie algebroids investigated by the author) and, next, apply it to new situations which are not described by the classical version, for example, to the theory of transversally complete foliations and nonclosed Lie subgroups in order to obtain some topological obstructions for the existence of involutive distributions and Lie subalgebras of some types (respectively).

     

On the resolution of relative Thue equations

An efficient algorithm is given for the resolution of relative Thue equations. The essential improvement is the application of an appropriate version of Wildanger's enumeration procedure based on the ellipsoid method of Fincke and Pohst.

Recently relative Thue equations have gained an important application, e.g., in computing power integral bases in algebraic number fields. The presented methods can surely be used to speed up those algorithms.

The method is illustrated by numerical examples.

     

Solution of a functional equation arising in an axiomatization of the utility of binary gambles

For a new axiomatization, with fewer and weaker assumptions, of binary rank-dependent expected utility of gambles the solution of the functional equation

is needed under some monotonicity and surjectivity conditions. We furnish the general such solution and also the solutions under weaker suppositions. In the course of the solution we also determine all sign preserving solutions of the related general equation

     

On the oscillation and periodic character of a third order rational difference equation

We prove that every positive solution of the following difference equation:

converges to a period two solution.

     

A short proof of Ellentuck's Theorem

We combine standard arguments to give a shorter proof of Ellentuck's Theorem.

     

Finite representability of operators in the sense of Bellenot

We present several new results about the notion of finite representability of operators introduced by Bellenot.

     

Error analysis for the computation of zeros of regular Coulomb wave function and its first derivative

In 1975 one of the coauthors, Ikebe, showed that the problem of computing the zeros of the regular Coulomb wave functions and their derivatives may be reformulated as the eigenvalue problem for infinite matrices. Approximation by truncation is justified but no error estimates are given there.

The class of eigenvalue problems studied there turns out to be subsumed in a more general problem studied by Ikebe et al. in 1993, where an extremely accurate asymptotic error estimate is shown.

In this paper, we apply this error formula to the former case to obtain error formulas in a closed, explicit form.

     

Orbifolds and analytic torsions

In this paper, we calculate the behavior of the Quillen metric by orbifold immersions. We thus extend a formula of Bismut-Lebeau to the orbifold case.

R´ESUMÉ. Dans cet article, on calcule le comportement de métrique de Quillen par immersions d'orbifold. On étend ainsi une formule de Bismut-Lebeau au cas d'orbifold.

     

A short proof that hyperspaces of Peano continua are absolute retracts

We give a short proof of Wojdyslawski's famous theorem.

     

Inequalities for products of spectral radii

It is shown that submultiplicative inequalities for spectral radii often imply supermultiplicative inequalities, and vice versa.

     

Actions of finitely generated groups on the universal Menger curve

We prove that every finitely generated group acts effectively on the universal Menger curve.

     

Explicit evaluations of a Ramanujan-Selberg continued fraction

This paper gives explicit evaluations for a Ramanujan-Selberg continued fraction in terms of class invariants and singular moduli.

     

Classification of torsion-free genus zero congruence groups

We study and classify all torsion-free genus zero congruence subgroups of the modular group.

     

Multiple symmetric positive solutions of a class of boundary value problems for higher order ordinary differential equations

In this paper, the authors consider the boundary value problem

and give sufficient conditions for the existence of any number of symmetric positive solutions of (E)-(B). The relationships between the results in this paper and some recent work by Henderson and Thompson (Proc. Amer. Math. Soc. 128 (2000), 2373-2379) are discussed.

     

The automorphism groups of Kummer surfaces associated with the product of two elliptic curves

We calculate the automorphism groups of several Kummer surfaces associated with the product of two elliptic curves. We give their generators explicitly.

     

On transience of card shuffling

We present simple proofs of transience/recurrence for certain card shuffling models, that is, random walks on the infinite symmetric group.

     

On simple double zeros and badly conditioned zeros of analytic functions of variables

We give a numerical criterion for a badly conditioned zero of a system of analytic equations to be part of a cluster of two zeros.

     

Strichartz estimates for the Schrödinger equation with radial data

We prove an endpoint Strichartz estimate for radial solutions of the two-dimensional Schrödinger equation:

     

A characterization of bilateral operator weighted shifts being Cowen-Douglas operators

It is characterized when a bilateral operator weighted shift is a Cowen-Douglas operator.

     

The topological dynamics of semigroup actions

In these notes we explore the fine structure of recurrence for semigroup actions, using the algebraic structure of compactifications of the acting semigroup.