Approximation of bandlimited functions

Many signals encountered in science and engineering are approximated well by bandlimited functions. We provide suitable error bounds for the approximation of bandlimited functions by linear combinations of certain special functions—the prolate spheroidal wave functions of order 0. The coefficients in the approximating linear combinations are given explicitly via appropriate quadrature formulae.     

On a stochastic nonlinear equation in one-dimensional viscoelasticity

In this paper we discuss an initial-boundary value problem for a stochastic nonlinear equation arising in one-dimensional viscoelasticity. We propose to use a new direct method to obtain a solution. This method is expected to be applicable to a broad class of nonlinear stochastic partial differential equations.

     

Effective magnetic properties of a composite material with circular conductive elements

Effective magnetic properties of a composite meta-material consisting of periodically arranged circular conductive elements are studied theoretically. A general expression for the effective bulk permeability is obtained with mutual effects and lattice ordering being taken into account. The resonance frequency of the permeability is found to be strongly dependent on the size and shape of the unit cell. Frequency dispersion of the permeability is studied with special attention paid to the frequency range, where negative values of the permeability are possible. Corresponding recommendations for optimisation of the meta-materials with negative permeability are made. The results are confirmed by numerical simulations of the finite structure behaviour in an external magnetic field. Received 19 April 2002 Published online 31 July 2002     

Comparisons between model equations for long waves

Summary Considered here are model equations for weakly nonlinear and dispersive long waves, which feature general forms of dispersion and pure power nonlinearity. Two variants of such equations are introduced, one of Korteweg-de Vries type and one of regularized long-wave type. It is proven that solutions of the pure initial-value problem for these two types of model equations are the same, to within the order of accuracy attributable to either, on the long time scale during which nonlinear and dispersive effects may accumulate to make an order-one relative difference to the wave profiles.This research was supported in part by the National Science Foundation. A considerable portion of the project was completed while the first author was resident at the Institute for Mathematics and Its Applications, University of Minnesota.     

Matrix perturbation theory for driven three‐level systems with damping

We investigate the dynamics of the Λ system driven by two resonant laser fields in presence of dissipation for coupling strengths where the rotating‐wave approximation starts to break down. This regime is characterised by Rabi frequencies being approximately equal or smaller than the field frequencies. A systematic procedure to obtain an expansion for the solution of the Bloch evolution equations of the system is presented. The lowest contribution results to be the well‐known rotating‐wave approximation. The method is based on a semi‐classical treatment of the problem, and its predictions are interpreted fully quantum mechanically. The theory is illustrated by a detailed study of the disappearance of coherent population trapping as the intensities of the fields increase.     

LOCAL WELL-POSEDNESS AND ILL-POSEDNESS ON THE EQUATION OF TYPE □u = uk((e)u)α

This paper undertakes a systematic treatment of the low regularity local wellposedness and ill-posedness theory in Hs andHs for semilinear wave equations with polynomial nonlinearity in u and (e)u. This ill-posed result concerns the focusing type equations with nonlinearity on u and (e)tu.     

An AFM, XPS and electrochemical study of molecular electroactive monolayers formed by wet chemistry functionalization of H-terminated Si(1 0 0) with vinylferrocene

Molecular electroactive monolayers have been produced from vinylferrocene (VFC) via light-assisted surface anchoring to H-terminated n- and p-Si(1 0 0) wafers prepared via wet chemistry, in a controlled atmosphere. The resulting Si-C bound hybrids have been characterized by means of XPS and AFM. Their performance as semiconductor functionalized electrodes and their surface composition have been followed by combining electrochemical and XPS measurements on the same samples, before and after use in an electrochemical cell. White-light photoactivated anchoring at short (1 h) exposure times has resulted in a mild route, with a very limited impact on the initial quality of the silicon substrate. In fact, the functionalized Si surface results negligibly oxidized, and the C/Fe atomic ratio is close to the value expected for the pure molecular species. The VFC/Si hybrids can be described as (η⁵-C₅H₅)Fe²⁺(η⁵-C₅H₄)-CH₂-CH₂-Si species, on the basis of XPS results. Electrochemical methods have been applied in order to investigate the role played by a robust, covalent Si-C anchoring mode towards substrate-molecule electronic communication, a crucial issue for a perspective development of molecular electronics devices. The response found from cyclic voltammograms for p-Si(1 0 0) functionalized electrodes, run in the dark and under illumination, has shown that the electron transfer is not limited by the number of charge carriers, confirming the occurrence of electron transfer via the Si valence band. The hybrids have shown a noticeable electrochemical stability and reversibility under cyclic voltammetry (cv), and the trend in peak current intensity vs. the scan rate was linear. The molecule-Si bond is preserved even after thousands of voltammetric cycles, although the surface coverage, evaluated from cv and XPS, decreases in the same sequence. An increasingly larger surface concentration of Fe³⁺ at the expenses of Fe²⁺ redox centers has been found at increasing number of cv’s, experimentally associated with the growth of silicon oxide. Surface SiO⁻ groups from deprotonated silanol termination, induced by the electrochemical treatments, are proposed as the associated counterions for the Fe³⁺ species. They could be responsible for the observed decrease in the electron transfer rate constant with electrode ageing.     

Polariton gap solitary waves in semiconductor microcavities

We study nonlinear dynamics of optical pulse propagation in the spectral region inside the polariton gap. It is shown that the Kerr nonlinearity can lead to formation of solitary waves in this region of frequencies.     

WKB Wave Function of the General Time-Dependent Quadratic Hamiltonian System

We derived the WKB wave function for the general time-dependent quadratic Hamiltonian system using a unitary transformation method. We applied our research to sinusodially drived Caldirola–Kanai oscillator and confirmed that the time evolution of our approximated WKB wave function is similar to that of the exact one. This wave function can be used to analyze the interference between the probability amplitudes contributed by the area of overlap in phase space of quantum states.