On a problem of W. J. LeVeque concerning metric diophantine approximation

We consider the diophantine approximation problem

where is a fixed function satisfying suitable assumptions. Suppose that is randomly chosen in the unit interval. In a series of papers that appeared in earlier issues of this journal, LeVeque raised the question of whether or not the central limit theorem holds for the solution set of the above inequality (compare also with some work of Erdos). Here, we are going to extend and solve LeVeque's problem.

     

Analysis and geometry on manifolds with integral Ricci curvature bounds. II

We extend several geometrical results for manifolds with lower Ricci curvature bounds to situations where one has integral lower bounds. In particular we generalize Colding's volume convergence results and extend the Cheeger-Colding splitting theorem.

     

Blow-up vs. spurious steady solutions

In this paper, we study the blow-up problem for positive solutions of a semidiscretization in space of the heat equation in one space dimension with a nonlinear flux boundary condition and a nonlinear absorption term in the equation. We obtain that, for a certain range of parameters, the continuous problem has blow-up solutions but the semidiscretization does not and the reason for this is that a spurious attractive steady solution appears.

     

Generalized Cauchy difference equations. II

The main result is an improvement of previous results on the equation

for a given function . We find its general solution assuming only continuous differentiability and local nonlinearity of . We also provide new results about the more general equation

for a given function . Previous uniqueness results required strong regularity assumptions on a particular solution . Here we weaken the assumptions on considerably and find all solutions under slightly stronger regularity assumptions on .

     

Remark on ``A problem of prescribing Gaussian curvature on " [Proc. Amer. Math. Soc. 129 (2001), no. 12, 3757-3758]

In this note, we remark on a 2001 paper of S. Goyal and V. Goyal. The main result of this work is that they used some elementary method to find a class of functions for which the solutions to

on can be obtained. We observe that this class of functions that they studied is actually the trivial one, i.e. the class of positive constant functions.

     

On the equations defining toric l.c.i.-singularities

Based on Nakajima's Classification Theorem we describe the precise form of the binomial equations which determine toric locally complete intersection (``l.c.i.') singularities.

     

Pfaffian systems with derived length one. The class of flag systems

The incidence relations between a Pfaffian system and the characteristic system of its first derived system lead to obtain a characterization of Pfaffian systems with derived length one. Also, for flag systems, several properties are studied. In particular, an intrinsic proof of a result which determines the class of a system and of all the derived systems is given.

     

On a polynomial inequality of E. J. Remez

We prove a result which extends a well-known polynomial inequality of E. J. Remez and another one due to W. A. Markov.

     

Large deviations for sums of i.i.d. random compact sets

We prove a large deviation principle for Minkowski sums of i.i.d. random compact sets in a Banach space, that is, the analog of Cramér theorem for random compact sets.

     

Shellable nonpure complexes and posets. II

This is a direct continuation of Shellable Nonpure Complexes and Posets. I, which appeared in Transactions of the American Mathematical Society 348 (1996), 1299-1327.

     

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).

     

Q.U.P. and Paley-Wiener properties of unimodular, especially nilpotent, Lie groups

We give a new proof of a weak Paley-Wiener theorem for nilpotent Lie groups due to Lipsman and Rosenberg and we introduce a general notion of Q.U.P for any unimodular locally compact group.

     

A note on a paper by G. Mastroianni and G. Monegato

Recently, Mastroianni and Monegato derived error estimates for a numerical approach to evaluate the integral

where or or and is a smooth function (see G. Mastroianni and G. Monegato, Error estimates in the numerical evaluation of some BEM singular integrals, Math. Comp. 70 2001, 251-267). The error bounds for the quadrature rule approximating the inner integral given in Theorems 3, 4 of that paper are not correct according to the proof. However, following a different approach, we are able to improve the pointwise error estimates given in that paper.

     

Weak approximations. A Malliavin calculus approach

We introduce a variation of the proof for weak approximations that is suitable for studying the densities of stochastic processes which are evaluations of the flow generated by a stochastic differential equation on a random variable that may be anticipating. Our main assumption is that the process and the initial random variable have to be smooth in the Malliavin sense. Furthermore, if the inverse of the Malliavin covariance matrix associated with the process under consideration is sufficiently integrable, then approximations for densities and distributions can also be achieved. We apply these ideas to the case of stochastic differential equations with boundary conditions and the composition of two diffusions.

     

Basic differential forms for actions of Lie groups. II

The assumption in the main result of Basic differential forms for actions of Lie groups (Proc. Amer. Math. Soc. 124 (1996), 1633-1642) is removed.

     

Operators with singular continuous spectrum, VI. Graph Laplacians and Laplace-Beltrami operators

Examples are constructed of Laplace-Beltrami operators and graph Laplacians with singular continuous spectrum.

     

On a multidimensional form of F. Riesz's ``rising sun" lemma

A multidimensional version of the Riesz rising sun lemma is proved by means of a generalized dyadic process.

     

The Derksen invariant vs. the Makar-Limanov invariant

In this article it is shown that the Makar-Limanov invariant of a ring (or variety) can be trivial while the Derksen invariant is not, and vice versa.

     

A remark to a theorem of Yu. A. Abramovich

A remarkable theorem due to Abramovich (1988) states that any surjective positive isometry on a Banach lattice has a positive inverse. In this note we discuss a renorming problem for Banach lattices and show that the theorem cannot be generalized to the case of the doubly power bounded positive operators.

     

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.