Brownian confinement and pinning in a Poissonian potential. II

Summary We continue our study ofd-dimensional Brownian motion in a soft repulsive Poissonian potential over a long time interval [0,t]. We prove here a pinning effect: for typical configuratons, with probability tending to 1 ast tends to , the particle gets trapped close to locations of near minima of certain variational problems. These locations lie at distances growing almost linearly witht from the origin, and the particle gets pinned within distance smaller than any positive power oft of one such location. In dimension 1, we can push further our estimates and show that in a suitable sense, the particle gets trapped with high probability, within time t and within distance (logt)²⁺ from a suitable location at distance of ordert/(logt)³ from the origin.This article was processed by the author using the LATEX style filepljour1m from Springer-Verlag     

Processes of flats induced by higher dimensional processes III

Stationary processes of k-flats in ^d can be thought of as point processes on the Grassmannian _k ^d of k-dimensional subspaces of ^d . If such a process is sampled by a (d–k+ j)-dimensional space F, it induces a process of j-flats in F. In this work we will investigate the possibility of determining the original k-process from knowledge of the intensity measures of the induced j-processes. We will see that this is impossible precisely when 1<k<d–1 and j=0,...,2[r/2]–1, where r is the rank of the manifold _k ^d . We will show how the problem is equivalent to the study of the kernel of various integral transforms, these will then be investigated using harmonic analysis on Grassmannian manifolds.The research of the first and third authors was supported in part by NSF grants DMS-9207019 and DMS-9304284. The research of the second author was supported in part by NFR contract number R-RA 4873-306 and the Swedish Academy of Sciences.     

A perturbation bound for the generalized polar decomposition

LetA be anm×n complex matrix. A decompositionA=QH is termed ageneralized polar decomposition ofA ifQ is anm×n subunitary matrix (sometimes also called a partial isometry) andH a positive semidefinite Hermitian matrix. It was proved that a nonzero matrixA ^m×n has a unique generalized polar decompositionA=QH with the property (Q ^H )=(H), whereQ ^H denotes the conjugate transpose ofQ and (H) the column space ofH. The main result of this note is a perturbation bound forQ whenA is perturbed.     

Functional connectedness and Darboux property of multivalued functions

To 80^th birthday of Paul Erds     

Perturbed unstable variational inequalities with unbounded operators on approximately given sets

The stability and convergence of the solutions of perturbed and regularized variational inequality to the solutions of the primary (unstable a priori) variational inequality with proper monotone operator are investigated. All the objects of inequality: the operatorA, the right-hand partf and the set of constrains are to be perturbed. At the same time no assumptions of boundedness and smoothness of the operatorA are used. The connection between the parameters of perturbations, which guarantees strong convergence of approximate solutions, is established. It is proved that the existence of the solution to the unperturbed variational inequality is necessary and sufficient condition for convergence of the regularized perturbed inequality solutions.This research was supported in part by the Ministry of Science Grant 3481-1-91 and by the Ministry of Absorption Center for Absorption in Science.     

Banach algebras of singular integral operators with piecewise continuous coefficients. General contour and weight

The Banach algebra generated by one-dimensional linear singular integral operators with matrix valued piecewise continuous coefficients in the spaceL _p (,) with an arbitrary weight is studied. The contour consists of a finite number of closed curves and open arcs with satisfy the Carleson condition. The contour may have a finite number of points of selfintersection. The symbol calculus in this algebra is the main result of the paper.     

First synthesis of 1,9-dideoxyforskolin from ptychantin A

1,9-Dideoxyforskolin 2 has been synthesized starting from ptychantin A 3.     

About the ratio of the size of a maximum antichain to the size of a maximum level in finite partially ordered sets

LetP be a finite partially ordered set. The lengthl(x) of an elementx ofP is defined by the maximal number of elements, which lie in a chain withx at the top, reduced by one. Letw(P) (d(P)) be the maximal number of elements ofP which have the same length (which form an antichain). Further let . The numbers and as well as all partially ordered sets for which these maxima are attained are determined.     

The erdös-dushnik-miller theorem for topological graphs and orders

A topological graph is a graph G=(V, E) on a topological space V such that the edge set E is a closed subset of the product space V x V. If the graph contains no infinite independent set then, by a well-known theorem of Erdös, Dushnik and Miller, for any infinite set LV, there is a subset LL of the same oardinality |L| = |L| such that the restriction G L is a complete graph. We investigate the question of whether the same conclusion holds if we weaken the hypothesis and assume only that some dense subset AV does not contain an infinite independent set. If the cofinality cf (|L|)>|A|, then there is an L as before, but if cf (|L|)<-|A|, then some additional hypothesis seems to be required. We prove that, if the graph GA is a comparability graph and A is a dense subset, then for any set LV such that cf (|L|)>, there is a subset LL of size |L|=|L| such that GL is complete. The condition cf (|L|)> is needed.Research supported by NSERC grant #A5198.     

Posets isomorphic to their extensions

A standard extension for a poset P is a system Q of lower ends (descending subsets) of P containing all principal ideals of P. An isomorphism between P and Q is called recycling if [Y]Q for all YQ. The existence of such an isomorphism has rather restrictive consequences for the system Q in question. For example, if Q contains all lower ends generated by chains then a recycling isomorphism between P and Q forces Q to be precisely the system of all principal ideals. For certain standard extensions Q, it turns out that every isomorphism between P and Q (if there is any) must be recycling. Our results include the well-known fact that a poset cannot be isomorphic to the system of all lower ends, as well as the fact that a poset is isomorphic to the system of all ideals (i.e., directed lower ends) only if every ideal is principal.