Wavelet frames and shift-invariant subspaces of periodic functions

A general approach based on polyphase splines, with analysis in the frequency domain, is developed for studying wavelet frames of periodic functions of one or higher dimensions. Characterizations of frames for shift-invariant subspaces of periodic functions and results on the structure of these subspaces are obtained. Starting from any multiresolution analysis, a constructive proof is provided for the existence of a normalized tight wavelet frame. The construction gives the minimum number of wavelets required. As an illustration of the approach developed, the one-dimensional dyadic case is further discussed in detail, concluding with a concrete example of trigonometric polynomial wavelet frames.     

A scalable divide-and-conquer algorithm combining coarse and fine-grain parallelization

We describe an efficient algorithm for carrying out a “divide-and-conquer” fit of a molecule's electronic density on massively parallel computers. Near linear speedups are achieved with up to 48 processors on a Cray T3E, and our results indicate that similar efficiencies could be attained on an even greater number of processors. To achieve optimum efficiency, the algorithm combines coarse and fine-grain parallelization and adapts itself to the existing ratio of processors to subsystems. The subsystems employed in our divide-and-conquer approach can also be made smaller or bigger, depending on the number of processors available. This allows us to further reduce the wallclock time and improve the method's overall efficiency. The strategies implemented in this paper can be extended to any other divide-and-conquer method used within an ab initio, density functional, or semi-empirical quantum mechanical program. Received: 15 September 1997 / Accepted: 21 January 1998     

Plancherel—Rotach Asymptotics for the Charlier Polynomials

We derive an asymptotic approximation of Plancherel—Rotach type for the Charlier polynomials on the positive real line. July 26, 1993. Date revised: December 2, 1996.     

Educational aspects of sonochemistry: The role of sonochemistry at high school level

The applications of sonochemistry have attracted a great deal of attention. However, conventional chemistry textbooks used in universities and high schools cover traditional chemistry plus thermo,- photo- and electrochemistry. The effect of sound on chemical systems has completely been ignored. Although the mechanisms of many sonochemical reactions remain unresolved, the rapid expansion of knowledge in sonochemistry has already justified its inclusion in present chemistry curricula. Moreover, the interesting and unusual phenomena encountered under sonication conditions can be used for motivation purposes. Furthermore, exploring and equipping students with a new tool for investigating chemistry is also important based on educational principles. This paper examines some chemical systems (in a high school chemistry syllabus) which are affected by sound energy. It is hoped that these sonochemical applications will facilitate the learning of chemistry in terms of cognitive, psychomotor and affective domains.     

Direct training method for a continuous-time nonlinear optimal feedback controller

The solutions of most nonlinear optimal control problems are given in the form of open-loop optimal control which is computed from a given fixed initial condition. Optimal feedback control can in principle be obtained by solving the corresponding Hamilton-Jacobi-Bellman dynamic programming equation, though in general this is a difficult task. We propose a practical and effective alternative for constructing an approximate optimal feedback controller in the form of a feedforward neural network, and we justify this choice by several reasons. The controller is capable of approximately minimizing an arbitrary performance index for a nonlinear dynamical system for initial conditions arising from a nontrivial bounded subset of the state space. A direct training algorithm is proposed and several illustrative examples are given.This research was carried out with the support of a grant from the Australian Research Council.We thank the anonymous reviewers for their helpful comments.     

Thermoanalytical studies of rubber oxidation: Prediction of isothermal induction time

The oxidation induction times of natural rubber stabilized by various antioxidants were measured by differential scanning calorimetry (DSC). Various Arrhenius plots could be superimposed to form a single plot by using a shift factor dependent on the oxidation peak temperature obtained from a dynamic DSC test. The superimposed plot provides a simple means to predict the induction time from a dynamic DSC test result.     

Global convergence of some differential equation algorithms for solving equations involving positive variables

This paper describes some sufficient conditions for global convergence in five differential equation algorithms for solving systems of non-linear algebraic equations involving positive variables. The algorithms are continuous time versions of a modified steepest descent method, Newton's method, a modified Newton's method and two algorithms using the transpose of the Jacobian in place of the inverse of the Jacobian in Newton's method. It is shown that under a set of mildly restrictive conditions the Jacobian transpose algorithm has qualitatively the same convergence as Newton's method.     

High-Resolution FTIR Spectrum of the nu(12) Band of trans-d(2)-Ethylene

The nu(12) band of trans-d(2)-ethylene (trans-C(2)H(2)D(2)) has been recorded with an unapodized resolution of 0.0024 cm(-1) in the frequency range of 1240-1360 cm(-1) by Fourier transform infrared (FTIR) spectroscopy. This band was found to be relatively free from any local frequency perturbations. By fitting a total of 1185 infrared transitions of nu(12) with a standard deviation of 0.00043 cm(-1) using a Watson's A-reduced Hamiltonian in the I(r) representation, a set of accurate rovibrational constants for v(12) = 1 state was derived. The nu(12) band is A type with a band center at 1298.03797 +/- 0.00004 cm(-1). Copyright 2000 Academic Press.     

Analysis of the Coriolis Interaction between nu(6) and nu(8) Bands of HCOOH

The Fourier transform infrared (FTIR) spectrum of the nu(6) band of formic acid (HCOOH) has been recorded with a resolution of 0.0024 cm(-1) in the spectral range 1050-1160 cm(-1). The nu(6) band was found to be strongly perturbed by the nearby nu(8) band centered at about 1033.5 cm(-1). Using a Watson's A-reduced Hamiltonian in the I(r) representation, and with the inclusion of a-type Coriolis coupling constant, a simultaneous fit of nu(6) and nu(8) was performed. A total of 2485 infrared transitions including about 700 perturbed transitions of nu(6) and 19 transitions of nu(8) was fitted with an rms uncertainty of 0.0006 cm(-1). Accurate rovibrational constants up to sextic order for both nu(6) and nu(8) were obtained. The nu(6) band was analyzed to be a type AB hybrid with a band center at 1104.852109 +/- 0.000050 cm(-1). The band center for nu(8) was found to be 1033.4647 +/- 0.0021 cm(-1). Copyright 2000 Academic Press.