On the Green Function and Poisson Integrals of the Dunkl Laplacian

We prove the existence and study properties of the Green function of the unit ball for the Dunkl Laplacian △ _k in \(\mathbb {R}^{d}\). As applications we derive the Poisson-Jensen formula for △ _k -subharmonic functions and Hardy-Stein identities for the Poisson integrals of △ _k . We also obtain sharp estimates of the Newton potential kernel, Green function and Poisson kernel in the rank one case in \(\mathbb {R}^{d}\). These estimates contrast sharply with the well-known results in the potential theory of the classical Laplacian.     

Independent families in Boolean algebras with some separation properties

We prove that any Boolean algebra with the subsequential completeness property contains an independent family of size ${\mathfrak{c}}$ , the size of the continuum. This improves a result of Argyros from the 1980s, which asserted the existence of an uncountable independent family. In fact, we prove it for a bigger class of Boolean algebras satisfying much weaker properties. It follows that the Stone space ${K_\mathcal{A}}$ of all such Boolean algebras ${\mathcal{A}}$ contains a copy of the ?ech–Stone compactification of the integers ${\beta\mathbb{N}}$ and the Banach space ${C(K_\mathcal{A})}$ has l _∞ as a quotient. Connections with the Grothendieck property in Banach spaces are discussed.     

Path components of the space of gradient vector fields on the two-dimensional disc

We present a short proof that if two gradient maps on the two-dimensional disc have the same degree, then they are gradient homotopic.     

Embeddable quantum homogeneous spaces

We discuss various notions generalizing the concept of a homogeneous space to the setting of locally compact quantum groups. On the von Neumann algebra level we recall an interesting duality for such objects studied earlier by M. Izumi, R. Longo, S. Popa for compact Kac algebras and by M. Enock in the general case of locally compact quantum groups. A definition of a quantum homogeneous space is proposed along the lines of the pioneering work of Vaes on induction and imprimitivity for locally compact quantum groups. The concept of an embeddable quantum homogeneous space is selected and discussed in detail as it seems to be the natural candidate for the quantum analog of classical homogeneous spaces. Among various examples we single out the quantum analog of the quotient of the Cartesian product of a quantum group with itself by the diagonal subgroup, analogs of quotients by compact subgroups as well as quantum analogs of trivial principal bundles. The former turns out to be an interesting application of the duality mentioned above.     

On measurable multifunctions in countably separated spaces

We use an idea of countable separability of points and sets in topological spaces to prove results on intersection of measurable multifunctions and an implicit function theorem. We generalize or extend in part some well known Himmelberg's theorems.     

On block preconditioners for saddle point problems with singular or indefinite (1, 1) block

We discuss a class of preconditioning methods for the iterative solution of symmetric algebraic saddle point problems, where the (1, 1) block matrix may be indefinite or singular. Such problems may arise, e.g. from discrete approximations of certain partial differential equations, such as the Maxwell time harmonic equations. We prove that, under mild assumptions on the underlying problem, a class of block preconditioners (including block diagonal, triangular and symmetric indefinite preconditioners) can be chosen in a way which guarantees that the convergence rate of the preconditioned conjugate residuals method is independent of the discretization mesh parameter. We provide examples of such preconditioners that do not require additional scaling. Copyright © 2010 John Wiley & Sons, Ltd.     

Invariant measures ofC 2-diffeomorphisms of Markov type inR d

A class of uniformly expanding, piecewiseC ²-diffeomorphisms from domainsIR ^d (bounded or not) into themselves is considered. It is shown that the number of the extreme points of Fix (P )={gG:Pg=g} whereP is the Frobenius-Perron operator associated with andG={gL ¹: g0 g=1}, can be determined in an effective way. Moreover, it is shown that the sequence {P ^j g} is convergent inL ¹ for anygG, and in the topology of uniform convergence for anygG(1). The limit is a linear projectionR inL ¹ (defined by (3.1)) which mapsG onto Fix (P ) (see Th. 3.1).Dedicated to professor A. Lasota on the occasion of his 60th birthday     

On exactness and isoperimetric profiles of discrete groups

We introduce a quasi-isometry invariant related to Property A and explore its connections to various other invariants of finitely generated groups. This allows to establish a direct relation between asymptotic dimension on one hand and isoperimetry and random walks on the other. We apply these results to reprove sharp estimates on isoperimetric profiles of some groups and to answer some questions in dimension theory.     

Robust Wavelet-Domain Estimation of the Fractional Difference Parameter in Heavy-Tailed Time Series: An Empirical Study

We investigate the performance of several wavelet-based estimators of the fractional difference parameter. We consider situations where, in addition to long-range dependence, the time series exhibit heavy tails and are perturbed by polynomial and change-point trends. We make detailed study of a wavelet-domain pseudo Maximum Likelihood Estimator (MLE), for which we provide an asymptotic and finite-sample justification. Using numerical experiments, we show that unlike the traditional time-domain estimators, estimators based on the wavelet transform are robust to additive trends and change points in mean, and produce accurate estimates even under significant departures from normality. The Wavelet-domain MLE appears to dominate a regression-based wavelet estimator in terms of smaller root mean squared error. These findings are derived from a simulation study and application to computer traffic traces.