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.

Rescue of a segmented negative-strand RNA virus entirely from cloned complementary DNAs

We provide the first report, to our knowledge, of a helper-independent system for rescuing a segmented, negative-strand RNA genome virus entirely from cloned cDNAs. Plasmids were constructed containing full-length cDNA copies of the three Bunyamwera bunyavirus RNA genome segments flanked by bacteriophage T7 promoter and hepatitis delta virus ribozyme sequences. When cells expressing both bacteriophage T7 RNA polymerase and recombinant Bunyamwera bunyavirus proteins were transfected with these plasmids, full-length antigenome RNAs were transcribed intracellularly, and these in turn were replicated and packaged into infectious bunyavirus particles. The resulting progeny virus contained specific genetic tags characteristic of the parental cDNA clones. Reassortant viruses containing two genome segments of Bunyamwera bunyavirus and one segment of Maguari bunyavirus were also produced following transfection of appropriate plasmids. This accomplishment will allow the full application of recombinant DNA technology to manipulate the bunyavirus genome.

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