Paving Hessenberg varieties by affines

Regular nilpotent Hessenberg varieties form a family of subvarieties of the flag variety arising in the study of quantum cohomology, geometric representation theory, and numerical analysis. In this paper we construct a paving by affines of regular nilpotent Hessenberg varieties for all classical types, generalizing results of De Concini–Lusztig–Procesi and Kostant. This paving is in fact the intersection of a particular Bruhat decomposition with the Hessenberg variety. The nonempty cells of the paving and their dimensions are identified by combinatorial conditions on roots. We use the paving to prove these Hessenberg varieties have no odd-dimensional homology.      

Lifting maps from a vector space of Jacobi cusp forms to a subspace of elliptic modular forms

In this paper we construct a lifting map from a vector space of generalized Jacobi cusp forms to a certain subspace of elliptic cusp forms and vice versa such that both mappings are adjoint with respect to the Petersson scalar products.     

Apollonian isometries of planar domains are Möbius mappings

The Apollonian metric is a generalization of the hyperbolic metric, defined in a much larger class of open sets. Beardon introduced the metric in 1998, and asked whether its isometries are just the Möbius mappings. In this article we show that this is the case in all open subsets of the plane with at least three boundary points.     

Valuations dominating regular local rings and proximity relations

We study zero-dimensional valuations dominating a regular local ring of dimension n≥2. For this we introduce the proximity matrix and the multiplicity sequence (extending classical definitions of the case n=2) that are associated with the sequence of the successive quadratic transforms of the ring along the valuation. We describe the precise relations between these invariants and study their properties.     

Numerical solution of the two-dimensional Poincaré equation

This paper deals with numerical approximation of the two-dimensional Poincaré equation that arises as a model for internal wave motion in enclosed containers. Inspired by the hyperbolicity of the equation we propose a discretisation particularly suited for this problem, which results in matrices whose size varies linearly with the number of grid points along the coordinate axes. Exact solutions are obtained, defined on a perturbed boundary. Furthermore, the problem is seen to be ill-posed and there is need for a regularisation scheme, which we base on a minimal-energy approach.     

Efficient nonparametric estimation of distribution density in the basis of algebraic polynomials

A nonparametric estimatef ^* of an unknown distribution densityf W is called locally minimax iff it is minimax for all not too small neighborhoodsW _g ,g W, simultaneously, whereW is some dense subset ofW. Radaviius and Rudzkis proved the existence of such an estimate under some general conditions. However, the construction of the estimate is rather complicated. In this paper, a new estimate is proposed. This estimate is locally minimax under some additional assumptions which usually hold for orthobases of algebraic polynomial and is almost as simple as the linear projective estimate. Thus, it takes a form convenient for the construction of an adaptive estimator, which does not usea-priori information about the smoothness of the density. The adaptive estimation problem is briefly discussed and an unknown density fitting by Jacobi polynomials is investigated more explicitly.     

Limit theorems and Markov approximations for chaotic dynamical systems

Summary We prove the central limit theorem and weak invariance principle for abstract dynamical systems based on bounds on their mixing coefficients. We also develop techniques of Markov approximations for dynamical systems. We apply our results to expanding interval maps, Axiom A diffeomorphisms, chaotic billiards and hyperbolic attractors.     

Frobenius-Perron operators and approximation of invariant measures for set-valued dynamical systems

A set-valued dynamical systemF on a Borel spaceX induces a set-valued operatorF onM(X) — the set of probability measures onX. We define arepresentation ofF, each of which induces an explicitly defined selection ofF; and use this to extend the notions of invariant measure and Frobenius-Perron operators to set-valued maps. We also extend a method ofS. Ulam to Markov finite approximations of invariant measures to the set-valued case and show how this leads to the approximation ofT-invariant measures for transformations , whereT corresponds to the closure of the graph of .     

Countably determined sets and a conjecture of C.W. Henson

Partially supported by NSF grants DMS-9208071 and DMS-9100383