A Bitangential Interpolation Problem on the Closed Unit Ball for Multipliers of the Arveson Space

We solve a bitangential interpolation problem for contractive multipliers on the Arveson space with an arbitrary interpolating set in the closed unit ball . The solvability criterion is established in terms of positive kernels. The set of all solutions is parametrized by a Redheffer transform. Submitted: February 2, 2002.     

A bicontinuous structure in some systems with cubic mesophases

A cubic structure of polymer colloid complexes is studied. The technique of the research includes i) an analysis of well-known literature SAXS data; on this basis, ii) constructing a simple model to estimate geometric structure parameters and to obtain a simulated scattering curve; and iii) comparing the model with the real structure obtained from the SAXS data, using the reconstruction of electron density distribution. A bicontinuous structure in cubic mesophases is formed. Dedicated to the memory of Alexander T. Dembo     

Fundamental solutions of the acoustic and diffusion equations in nonhomogeneous medium

A fundamental solution of the acoustical equation with a variable refraction coefficient is constructed. The solution satisfies the limiting absorption and radiation conditions. The optimal high frequency estimate is proved for square means of the solution. The source function for the diffusion equation is a by-product of this construction. Partially supported by a stipend of the Mathematical Scientific Research Institute at Berkeley, 2001.     

On Totally Ordered Rings Whose Positive Cones are Finitely Generated

This paper is mainly dealt with the structure of totally ordered rings (t.o. rings) whose positive cones are finitely generated as multiplicative right ideals.AMS Subject Classification (1991): 16W80, 13J25     

Covering group and graph of discretized volumes

We consider a discretized volume V consisting of finite, congruent and attached copies of a tile t. We find a group L _V the orbit of which, when applied to t, is just V. We show the connection between the structural matrixQ in the formal solution of a boundary value problem formulated for volume V and the so called auxiliary matrix of the graph Γ _v associated with V. We show boundary value problems to be isomorphic if the graphs associated with the volumes are isomorphic, or, if the covering groups are Sunada pairs.     

Analysis of blockwise shrinkage wavelet estimates via lower bounds for no-signal setting

A blockwise shrinkage is a popular procedure of adaptation that has allowed the statisticians to establish an impressive bouquet of asymptotic mathematical results and develop softwares for solving practical problems. Traditionally risks of the estimates are studied via upper bounds that imply sufficient conditions for a blockwise shrinkage procedure to be minimax. This article suggests to analyze the estimates via exact (non-asymptotic) lower bounds established for a no-signal setting. The approach complements the familiar minimax, Bayesian and numerical analysis, it allows to find necessary conditions for a procedure to attain desired rates, and it sheds a new light on popular choices of blocks and thresholds recommended in the literature. Mathematical results are complemented by a numerical study. Supported in part by NSF Grants DMS-9971051 and DMS-0243606.     

On the structure of hyperbolic manifolds

Forn≥2, we quantify the Margulis constant ε(n) giving rise to a thick and thin decomposition of hyperbolicn-manifolds of finite volume. As a consequence, we obtain new universal lower bounds for the volume and Gromov's invariant as well as a geometrical inequality between injectivity radius and diameter for compact manifolds. Finally, we concretise the upper bound for the counting function of hyperbolic manifolds of dimension >4 as described by Burger, Gelander, Lubotzky and Mozes. Partially supported by Schweizerischer Nationalfonds No. 20-61379.00 and 20-67619.02.     

Pointwise gradient estimates of solutions to onedimensional nonlinear parabolic equations

We present here an improved version of the method introduced by the first author to derive pointwise gradient estimates for the solutions of one-dimensional parabolic problems. After considering a general qualinear equation in divergence form we apply the method to the case of a nonlinear diffusion-convection equation. The conclusions are stated first for classical solutions and then for generalized and mild solutions. In the case of unbounded initial datum we obtain several regularizing effects for t > 0. Some unilateral pointwise gradient estimates are also obtained. The case of the Dirichlet problem is also considered. Finally, we collect, in the last section, several comments showing the connections among these estimates and the study of the free boundaries associated to the solutions of the diffusion-convection equation.