Approximation in weighted Hardy spaces

This paper is concerned with several approximation problems in the weighted Hardy spacesH ^p(Ω) of analytic functions in the open unit disc D of the complex plane ℂ. We prove that ifX is a relatively closed subset of D, the class of uniform limits onX of functions inH ^p(Ω) coincides, moduloH ^p(Ω), with the space of uniformly continuous functions on a certain hull ofX which are holomorphic on its interior. We also solve the simultaneous approximation problems of describing Farrell and Mergelyan sets forH ^p(Ω), giving geometric characterizations for them. By replacing approximating polynomials by polynomial multipliers of outer functions, our results lead to characterizations of the same sets with respect to cyclic vectors in the classical Hardy spacesH ^p(D), 1 ⪯p < ∞. Dedicated to Professor Nácere Hayek on the occasion of his 75th birthday.     

Regularity of multiwavelets

The motivation for this paper is an interesting observation made by Plonka concerning the factorization of the matrix symbol associated with the refinement equation for B-splines with equally spaced multiple knots at integers and subsequent developments which relate this factorization to regularity of refinable vector fields over the real line. Our intention is to contribute to this train of ideas which is partially driven by the importance of refinable vector fields in the construction of multiwavelets. The use of subdivision methods will allow us to consider the problem almost entirely in the spatial domain and leads to exact characterizations of differentiability and Hölder regularity in arbitrary L _p spaces. We first study the close relationship between vector subdivision schemes and a generalized notion of scalar subdivision schemes based on bi-infinite matrices with certain periodicity properties. For the latter type of subdivision scheme we will derive criteria for convergence and Hölder regularity of the limit function, which mainly depend on the spectral radius of a bi-infinite matrix induced by the subdivision operator, and we will show that differentiability of the limit functions can be characterized by factorization properties of the subdivision operator. By switching back to vector subdivision we will transfer these results to refinable vectors fields and obtain characterizations of regularity by factorization and spectral radius properties of the symbol associated to the refinable vector field. Finally, we point out how multiwavelets can be generated from orthonormal refinable bi-infinite vector fields.     

The multiple-scale wave equation

An analytic and numerical study of the behavior of the linear nonhomogeneous wave equation of the form ε²u_tt = Δu + tf with high wave speed (ε 1) is carried out. This study was initially motivated by meteorological observations which have indicated the presence of large spatial scale gravity waves in the neighborhood of a number of summer and winter storms, mainly from visible images of ripples in clouds in satellite photos. There is a question as to whether the presence of these waves is caused by the nearby storms. Since the linear wave equation is an approximation to the full system describing pressure waves in the atmosphere, yet is considerably more tractable, we have chosen to analyze the behavior of the linear nonhomogeneous wave equation with high wave speed. The analysis is shown to be valid in one, two, and three space dimensions. Partly because of the high wave speed, the solution is known to consist of behavior which changes on two different time scales, one rapid and one slow. Additionally, because of the presence of the nonhomogeneous forcing term tf, we show that there is a component of the solution which will vary only on a very large spatial scale. Since even the linearized wave equation can give rise to persistent large spatial scale waves under the right conditions, the implication is that certain storms could be responsible for the generation of large-scale waves. Numerical simulations in one and two dimensions confirm analytic results.     

Novel pattern formation in the collapse process of floating monolayer at the air-water interface

We have observed a remarkable two-armed spiral in the collapse process of a floating monolayer at the air-water interface by phase contrast microscopy. This demonstrates that the floating monolayer as a form of soft condensed matter reorganizes itself due to a certain kind of macroscopic or collective behavior of molecules as it collapses. This pattern formation is caused by the breakdown of a critical dynamical balance between the deformation of solid domain and the applied surface pressure. The fragility as well as the flexibility of the floating monolayer can be associated with the observed pattern growth. There are also observed interesting, periodically arranged collections of molecules in numerous collapsed regions. Received: 8 July 1997 / Accepted: 4 November 1997     

Double-diffusive convection for a heated cylinder submerged in a salt-stratified fluid layer

Double-diffusive convection due to a cylindrical source submerged in a salt-stratified solution is numerically investigated in this study. For proper simulation of the vortex generated around the cylinder, a computational domain with irregular shape is employed. Flow conditions depend strongly on the thermal Rayleigh number, Ra _T , and the buoyancy ratio, R _ρ. There are two types of onset of instability existing in the flow field. Both types are due to either the interaction of the upward temperature gradient and downward salinity gradient or the interaction of the lateral temperature gradient and downward salinity gradient. The onset of layer instability due to plume convection is due to the former, whereas, the onset of layer instability of layers around the cylinder is due to the latter. Both types can be found in the flow field. The transport mechanism of layers at the top of the basic plume belongs to former while that due to basic plume and layer around the cylinder are the latter. The increase in Ra _T reinforces the plume convection and reduces the layer numbers generated around the cylinder for the same buoyancy ratio. For the same Ra _T , the increase of R _ρ suppresses the plume convection but reinforces the layers generated around the cylinder. The profiles of local Nusselt number reflects the heat transfer characteristics of plume convection and layered structure. The profiles of averaged Nusselt number are between the pure conduction and natural convection modes and the variation is due to the evolution of layers. Received on 13 September 1996     

Membrane introduction mass spectrometry

An overview of membrane introduction mass spectrometry (MIMS) is presented and comparisons are made with other direct sample introduction techniques. Special attention is given to the unique advantages and the limitations of newer variants on the MIMS technique, including affinity MIMS, reverse-phase and trap MIMS. The salient features of the interfaces used in MIMS are summarized and the various membrane materials commonly used are delineated. The applicability of MIMS is illustrated via discussion of

1. (i) bioreactor monitoring (represented by yeast fermentation),

2. (ii) environmental monitoring (illustrated by analysis of contaminated ground water samples) and

3. (iii) on-line chemical reaction monitoring (exemplified by the photolysis of aryl esters).

The applicability of MIMS to the analysis of environmental samples, including complex mixtures in water, air and soil, is noted.     

Low-temperature specific heat of crystalline and amorphous Eu2(MoO4)3

The specific heat C _total of crystalline and amorphous Eu₂(MoO₄)₃ is measured in the temperature interval 4.5–30 K. The amorphous state is obtained by applying pressure ∼7 GPa at room temperature. It is found that the specific heat of the crystal at T⩽7.5 K is described by a cubic function of temperature, while the specific heat of the amorphous sample has a strongly non-Debye character in the entire experimental temperature interval. The curve of C _total for amorphous europium molybdate is analyzed in a model of soft atomic potentials, and it is shown that it agrees well with universal low-temperature anomalies of the specific heat of classical glasses obtained by quenching from the liquid. Pis’ma Zh. éksp. Teor. Fiz. 68, No. 8, 623–627 (25 October 1998)