Simultaneous Packing and Covering in the Euclidean Plane

 In this article we study the simultaneous packing and covering constants of two-dimensional centrally symmetric convex domains. Besides an identity result between translative case and lattice case and a general upper bound, exact values for some special domains are determined. Similar to Mahler and Reinhardt’s result about packing densities, we show that the simultaneous packing and covering constant of an octagon is larger than that of a circle. (Received 17 January 2001; in revised form 13 July 2001)     

Defect operators, defect functions and defect indices for analytic submodules

This paper mainly concerns defect operators and defect functions of Hardy submodules, Bergman submodules over the unit ball, and Hardy submodules over the polydisk. The defect operator (function) carries key information about operator theory (function theory) and structure of analytic submodules. The problem when a submodule has finite defect is attacked for both Hardy submodules and Bergman submodules. Our interest will be in submodules generated by polynomials. The reason for choosing such submodules is to understand the interaction of operator theory, function theory and algebraic geometry.     

Admissible slopes for monotone and convex interpolation

Summary In many applications, interpolation of experimental data exhibiting some geometric property such as nonnegativity, monotonicity or convexity is unacceptable unless the interpolant reflects these characteristics. This paper identifies admissible slopes at data points of variousC ¹ interpolants which ensure a desirable shape. We discuss this question, in turn for the following function classes commonly used for shape preserving interpolations: monotone polynomials,C ¹ monotone piecewise polynomials, convex polynomials, parametric cubic curves and rational functions.     

On modular inequalities in variable L p spaces

We show that the Hardy-Littlewood maximal operator and a class of Calderón-Zygmund singular integrals satisfy the strong type modular inequality in variable L^p spaces if and only if the variable exponent p(x) ∼ const. Received: 15 September 2004     

A finiteness result for Heegaard splittings

In this paper we show that for a given 3-manifold and a given Heegaard splitting there are finitely many preferred decomposing systems of 3g−3 disjoint essential disks. These are characterized by a combinatorial criterion which is a slight strengthening of Casson-Gordon's rectangle condition. This is in contrast to fact that in general there can exist infinitely many such systems of disks which satisfy just the Casson-Gordon rectangle condition.     

Normalization of complex-valued planar vector fields which degenerate along a real curve

Taking as a start point the recent article of Meziani [7], we present several results concerning the normalization of a class of complex vector fields in the plane which degenerate along a real curve. We mainly deal with operators with finite regularity and analyze both the local situation as well as the case of normalization near a circle. Some related questions (e.g., on semi-global solvability and on the normalization of a class of generalized Mizohata operators) are also discussed.     

Sound velocity in alumino-silicate liquids determined up to 2550 K from Brillouin spectroscopy: glass transition and crossover temperatures

Hypersonic longitudinal sound velocities in five silicate and alumino-silicate liquids have been measured between 293 and 2550 K by Brillouin spectroscopy. Together with previous observations for four other glasses and liquids of the system SiO₂-Al₂O₃-CaO-MgO, these results are used to discuss changes in hypersonic velocities in three adjacent temperature domains, i.e., below, in, and above the glass transformation range. The temperature dependence of Brillouin velocities is consistent with the observed variations with temperature of viscosity, density, and mean heat capacity for the same three temperature domains. These variations of physical properties of alumino-silicate liquids are qualitatively in agreement with the Inherent Structure Theory for liquids.     

Rank one perturbations at infinite coupling in Pontryagin spaces

In this paper we relate the operators in the operator representations of a generalized Nevanlinna function N(z) and of the function −N(z)⁻¹ under the assumption that z=∞ is the only (generalized) pole of nonpositive type. The results are applied to the Q-function for S and H and the Q-function for S and H^∞, where H is a self-adjoint operator in a Pontryagin space with a cyclic element w, H^∞ is the self-adjoint relation obtained from H and w via a rank one perturbation at infinite coupling, and S is the symmetric operator given by S=H∩H^∞.     

Exponential decay of eigenfunctions in many-body type scattering with second-order perturbations

We show the exponential decay of eigenfunctions of second-order geometric many-body type Hamiltonians at non-threshold energies. Moreover, in the case of first-order and small second-order perturbations we show that there are no eigenfunctions with positive energy.