Solution of the pulse width modulation problem using orthogonal polynomials and Korteweg-de Vries equations

The mathematical underpinning of the pulse width modulation (PWM) technique lies in the attempt to represent “accurately” harmonic waveforms using only square forms of a fixed height. The accuracy can be measured using many norms, but the quality of the approximation of the analog signal (a harmonic form) by a digital one (simple pulses of a fixed high voltage level) requires the elimination of high order harmonics in the error term. The most important practical problem is in “accurate” reproduction of sine-wave using the same number of pulses as the number of high harmonics eliminated. We describe in this paper a complete solution of the PWM problem using Padé approximations, orthogonal polynomials, and solitons. The main result of the paper is the characterization of discrete pulses answering the general PWM problem in terms of the manifold of all rational solutions to Korteweg-de Vries equations.

Generalized flows, intrinsic stochasticity, and turbulent transport

The study of passive scalar transport in a turbulent velocity field leads naturally to the notion of generalized flows, which are families of probability distributions on the space of solutions to the associated ordinary differential equations which no longer satisfy the uniqueness theorem for ordinary differential equations. Two most natural regularizations of this problem, namely the regularization via adding small molecular diffusion and the regularization via smoothing out the velocity field, are considered. White-in-time random velocity fields are used as an example to examine the variety of phenomena that take place when the velocity field is not spatially regular. Three different regimes, characterized by their degrees of compressibility, are isolated in the parameter space. In the regime of intermediate compressibility, the two different regularizations give rise to two different scaling behaviors for the structure functions of the passive scalar. Physically, this means that the scaling depends on Prandtl number. In the other two regimes, the two different regularizations give rise to the same generalized flows even though the sense of convergence can be very different. The “one force, one solution” principle is established for the scalar field in the weakly compressible regime, and for the difference of the scalar in the strongly compressible regime, which is the regime of inverse cascade. Existence and uniqueness of an invariant measure are also proved in these regimes when the transport equation is suitably forced. Finally incomplete self similarity in the sense of Barenblatt and Chorin is established.

A positivity result in the theory of Macdonald polynomials

We outline here a proof that a certain rational function Cn(q, t), which has come to be known as the “q, t-Catalan,” is in fact a polynomial with positive integer coefficients. This has been an open problem since 1994. Because Cn(q, t) evaluates to the Catalan number at t = q = 1, it has also been an open problem to find a pair of statistics a, b on the collection

Generalized transduction in Streptomyces coelicolor

We report the isolation of generalized transducing phages for Streptomyces species able to transduce chromosomal markers or plasmids between derivatives of Streptomyces coelicolor, the principal genetic model system for this important bacterial genus. We describe four apparently distinct phages (DAH2, DAH4, DAH5, and DAH6) that are capable of transducing multiple chromosomal markers at frequencies ranging from 10−5 to 10−9 per plaque-forming unit. The phages contain DNA ranging in size from 93 to 121 kb and mediate linked transfer of genetic loci at neighboring chromosomal sites sufficiently close to be packaged within the same phage particle. The key to our ability to demonstrate transduction by these phages was the establishment of conditions expected to severely reduce superinfection killing during the selection of transductants. The host range of these phages, as measured by the ability to form plaques, extends to species as distantly related as Streptomyces avermitilis and Streptomyces verticillus, which are among the most commercially important species of this genus. Transduction of plasmid DNA between S. coelicolor and S. verticillus was observed at frequencies of ≈10−4 transductants per colony-forming unit.