Optimally Stable Multivariate Bases

We show that the tensor product B-spline basis and the triangular Bernstein basis are in some sense best conditioned among all nonnegative bases for the spaces of tensor product splines and multivariate polynomials, respectively. We also introduce some new condition numbers which are analogs of component-wise condition numbers for linear systems introduced by Skeel.     

On the Representation of Band-Dominant Functions on the Sphere Using Finitely Many Bits

A band-dominant function on the Euclidean sphere embedded in R ^q+1 is the restriction to this sphere of an entire function of q+1 complex variables having a finite exponential type in each of its variables. We develop a method to represent such a function using finitely many bits, using the values of the function at scattered sites on the sphere. The number of bits required in our representation is asymptotically the same as the metric entropy of the class of such functions with respect to any of the L ^p norms on the sphere.     

Concerning seven and eight mutually orthogonal Latin squares

In this paper, three new direct Mutually Orthogonal Latin Squares (MOLS) constructions are presented for 7 MOLS(24), 7 MOLS(75) and 8 MOLS(36); then using recursive methods, several new constructions for 7 and 8 MOLS are obtained. These reduce the largest value for which 7 MOLS are unknown from 780 to 570, and the largest odd value for which 8 MOLS are unknown from 1935 to 419. © 2003 Wiley Periodicals, Inc.     

Nonlocality of Reaction-Diffusion Equations Induced by Delay: Biological Modeling and Nonlinear Dynamics

We present a short survey on the biological modeling, dynamics analysis, and numerical simulation of nonlocal spatial effects, induced by time delays, in diffusion models for a single species confined to either a finite or an infinite domain. The nonlocality, a weighted average in space, arises when account is taken of the fact that individuals have been at different points in space at previous times. We discuss and compare two existing approaches to correctly derive the spatial averaging kernels, and we summarize some of the recent developments in both qualitative and numerical analysis of the nonlinear dynamics, including the existence, uniqueness (up to a translation), and stability of traveling wave fronts and periodic spatio-temporal patterns of the model equations in unbounded domains and the linear stability, boundedness, global convergence of solutions and bifurcations of the model equations in finite domains.     

Computation of the Morse Spectrum

The Morse spectrum is a limit set of Lyapunov exponents of periodic pseudo-trajectories. This notion is especially important in the case where a dynamical system has infinitely many periodic trajectories of large period. A method for estimating the Morse spectrum was suggested by the first author in J. Math. Anal. Appl., 252 (2000). This method is based on ideas of symbolic dynamics which reduces the study of a dynamical system to the study of a certain graph, called a symbolic image. Within the framework of this method, the computation of the Morse spectrum is connected with searching simple closed paths and extracting contours with suitable characteristics. However, under iterations of the symbolic image, the number of such paths sharply increases, which leads to huge expenses of memory and time. We suggest an algorithm for constructing contours with the maximal and minimal mean values. This algorithm is based on a special version of the simplex method. Numerical tests are also described. Bibliography: 13 titles. Illustrations: 3 figures.     

Allocating energy in a first-to-n match

This paper models a decision where a player must allocate limitedenergy over a contest of uncertain length. The motivating exampleis a squash match where one of the players is not as fit asthe other. Should a player's energy be concentrated in the earlygames of the match? Should it be spread evenly over all possiblegames? Or should it be conserved for the end of the match? Wemodel this as a decision problem where, in each game, the decision-makermust determine how much energy to expend. We assume that thereare only a small number of discrete energy choices for eachgame and that the more energy the decision-maker expends, themore likely he is to win that game. We solve for the optimaldecision with dynamic programming. With only two possible energychoices for each game, we show that it does not matter how energyis expended. In the case where there are three or more energychoices, we show how to take advantage of the structure of theproblem to determine the optimal sequence of decisions. As forpractical advice, the model suggests that when the decision-makerfalls behind in a match, he ought to switch to a more conservativeapproach by dividing his remaining energy evenly among all thepossible remaining games. Received 14 May 2003. Revised 5 January 2004.     

Numerical Modeling of Solitons in a Low Birefringent Optical Fiber

A numerical investigation, based on the split-step Fourier transform algorithm of all optical switching of solitons in a low birefringent optical fiber is presented. The numerical algorithm is described in detail. This revised version was published online in August 2006 with corrections to the Cover Date.     

Dynamics and crystal chemistry of tellurites. II. Composition- and temperature-dependence of the Raman spectra of x(Tl2O)+(1−x) Te2O glasses: evidence for a phase separation?

Immiscibility effects are suggested to be inherent in tellurite glasses with weak (low valence) cations. This hypothesis is used to explain the structural evolution of the x(Tl₂O)+(1−x)Te₂O glass system as a function of the x value and during temperature-induced crystallisation processes. The results of Raman measurements support the hypothesis.