Positions of convex bodies associated to extremal problems and isotropic measures

We show that there are close relations between extremal problems in dual Brunn-Minkowski theory and isotropic-type properties for some Borel measures on the sphere. The methods we use allow us to obtain similar results in the context of Firey-Brunn-Minkowski theory. We also study reverse inequalities for dual mixed volumes which are related with classical positions, such as ?-position or isotropic position.     

Contractive Hilbert modules and their dilations

In this note, we show that a quasi-free Hilbert module R defined over the polydisk algebra with kernel function k(z,w) admits a unique minimal dilation (actually an isometric co-extension) to the Hardy module over the polydisk if and only if S ^?1(z, w)k(z, w) is a positive kernel function, where S(z,w) is the Szegö kernel for the polydisk. Moreover, we establish the equivalence of such a factorization of the kernel function and a positivity condition, defined using the hereditary functional calculus, which was introduced earlier by Athavale [8] and Ambrozie, Englis and Müller [2]. An explicit realization of the dilation space is given along with the isometric embedding of the module R in it. The proof works for a wider class of Hilbert modules in which the Hardy module is replaced by more general quasi-free Hilbert modules such as the classical spaces on the polydisk or the unit ball in ? ^m . Some consequences of this more general result are then explored in the case of several natural function algebras.     

Conditional expectations and dilations of positive operator-valued measures

Some general representations of positive operator-valued measures in the language of σ-field homomorphisms and conditional expectations are presented. Supported by the KBN (grant No. 2 P 03A 048 08). Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló, Hungary, 1997, Part 1.     

Genus two extremal surfaces: Extremal discs, isometries and Weierstrass points

It is known that the largest disc that a compact hyperbolic surface of genusg may contain has radiusR=cosh⁻¹(1/2sin(π/(12g−6))). It is also known that the number of such (extremal) surfaces, although finite, grows exponentially withg. Elsewhere the authors have shown that for genusg>3 extremal surfaces contain only one extremal disc. Here we describe in full detail the situation in genus 2. Following results that go back to Fricke and Klein we first show that there are exactly nine different extremal surfaces. Then we proceed to locate the various extremal discs that each of these surfaces possesses as well as their set of Weierstrass points and group of isometries. Both authors partially supported by Grant BFM2000-0031 of the SGPI.MCYT.     

Extremal subspaces and their submanifolds

It was proved in the paper [KM1] that the properties of almost all points of being not very well (multiplicatively) approximable are inherited by nondegenerate in (read: not contained in a proper affine subspace) smooth submanifolds. In this paper we consider submanifolds which are contained in proper a.ne subspaces, and prove that the aforementioned Diophantine properties pass from a subspace to its nondegenerate submanifold. The proofs are based on a correspondence between multidimensional Diophantine approximation and dynamics of lattices in Euclidean spaces. Submitted: March 2002; Revision: August 2002     

Wavelet frames with irregular matrix dilations and their stability

Let , B and A_j () be real nonsingular n×n matrices, λ_k () be real numbers. In this paper we present a sufficient condition for the system to be a frame for . This sufficient condition also shows the stability of the system with respect to the perturbation of matrix dilation parameters and the perturbation of translation parameters .     

Extremal functions for modules of systems of measures

We extend a result by Rodin, which provides an explicit method for finding the extremal function and the 2-module of a foliated family of curves in R², to R ⁿ making use of Fuglede’s p-module of systems of measures. The extremal functions are identified and the p-module of systems of measures is computed in condensers of rather general type and in their images under homeomorphisms of certain regularity. At the beginning, we discuss and apply Rodin’s Theorem in order to obtain estimates for the conformal modules of parallelograms and ring domains in terms of directional dilatations.     

Density of extremal measures in parabolic potential theory

It is shown that, for the heat equation on , d ≥ 1, any convex combination of harmonic (= caloric) measures , where U ₁, . . . , U _k are relatively compact open neighborhoods of a given point x, can be approximated by a sequence of harmonic measures such that each W _n is an open neighborhood of x in . Moreover, it is proven that, for every open set U in containing x, the extremal representing measures for x with respect to the convex cone of potentials on U (these measures are obtained by balayage, with respect to U, of the Dirac measure at x on Borel subsets of U) are dense in the compact convex set of all representing measures. Since essential ingredients for a proof of corresponding results in the classical case (or more general elliptic situations; see Hansen and Netuka in Adv. Math. 218(4):1181–1223, 2008) are not available for the heat equation, an approach heavily relying on the transit character of the hyperplanes , , is developed. In fact, the new method is suitable to obtain convexity results for limits of harmonic measures and the density of extremal representing measures on for practically every space–time structure which is given by a sub-Markov semigroup (P _t ) _t>0 on a space X′ such that there are strictly positive continuous densities with respect to a (non-atomic) measure on X′. In particular, this includes many diffusions and corresponding symmetric processes given by heat kernels on manifolds and fractals. Moreover, the results may be applied to restrictions of the space–time structure on arbitrary open subsets. I. Netuka’s research was supported in part by the project MSM 0021620839 financed by MSMT, by the grant 201/07/0388 of the Grant Agency of the Czech Republic, and by CRC-701, Bielefeld.     

Extremal measures for a system of orthogonal polynomials

We characterize the extremal measures of an indeterminate moment problem associated with a system of orthogonal polynomials defined by a three-term recurrence relation.     

Relation between unit operators proximity and their associated spectral measures

In this paper, we study how the local concentration of the spectral measure can be expressed in the temporal domain for stationary processes. For this purpose, we establish an equivalence between the proximity of the shift operators (which are unit operators) and the associated projector valued spectral measures. An illustration is given.     

Measures of serial extremal dependence and their estimation

The goal of this paper is two-fold: (1) We review classical and recent measures of serial extremal dependence in a strictly stationary time series as well as their estimation. (2) We discuss recent concepts of heavy-tailed time series, including regular variation and max-stable processes.     

Truncations of extremal quasiconformal mappings and their applications

In this paper we give a necessary and sufficient condition to decide whether the Teichmüller equivalency class [α] of a truncation α induced by a uniquely extremal Beltrami differential is a Strebel point in T. We also obtain a necessary and sufficient condition of the unique extremality of α. Using the properties of truncations we provide a method to construct Hamilton sequences. We also get a sufficient condition for the extremality of f(z,t) to be equivalent to that of F(w,t). The corresponding results in the infinitesimal case are obtained, too.     

Extremal measures with prescribed marginals (finite case)

The problem of describing the extremal mesures with given marginals on a finite Cartesian product is studied.