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.

Fluorescence energy transfer detection as a homogeneous DNA diagnostic method

A homogeneous DNA diagnostic assay based on template-directed primer extension detected by fluorescence resonance energy transfer, named template-directed dye-terminator incorporation (TDI) assay, has been developed for mutation detection and high throughput genome analysis. Here, we report the successful application of the TDI assay to detect mutations in the cystic fibrosis transmembrane conductance regulator (CFTR) gene, the human leukocyte antigen H (HLA-H) gene, and the receptor tyrosin kinase (RET) protooncogene that are associated with cystic fibrosis, hemochromatosis, and multiple endocrine neoplasia, type 2, respectively. Starting with total human DNA, the samples are amplified by the PCR followed by enzymatic degradation of excess primers and deoxyribonucleoside triphosphates before the primer extension reaction is performed. All these standardized steps are performed in the same tube, and the fluorescence changes are monitored in real time, making it a useful clinical DNA diagnostic method.

Homogeneous immunoconjugates for boron neutron-capture therapy: Design, synthesis, and preliminary characterization

The application of immunoprotein-based targeting strategies to the boron neutron-capture therapy of cancer poses an exceptional challenge, because viable boron neutron-capture therapy by this method will require the efficient delivery of 103 boron-10 atoms by each antigen-binding protein. Our recent investigations in this area have been focused on the development of efficient methods for the assembly of homogeneous immunoprotein conjugates containing the requisite boron load. In this regard, engineered immunoproteins fitted with unique, exposed cysteine residues provide attractive vehicles for site-specific modification. Additionally, homogeneous oligomeric boron-rich phosphodiesters (oligophosphates) have been identified as promising conjugation reagents. The coupling of two such boron-rich oligophosphates to sulfhydryls introduced to the CH2 domain of a chimeric IgG3 has been demonstrated. The resulting boron-rich immunoconjugates are formed efficiently, are readily purified, and have promising in vitro and in vivo characteristics. Encouragingly, these studies showed subtle differences in the properties of the conjugates derived from the two oligophosphate molecules studied, providing a basis for the application of rational design to future work. Such subtle details would not have been as readily discernible in heterogeneous conjugates, thus validating the rigorous experimental design employed here.

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