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.

Unifying theory of hypoxia tolerance: molecular/metabolic defense and rescue mechanisms for surviving oxygen lack.

We develop a unifying theory of hypoxia tolerance based on information from two cell level models (brain cortical cells and isolated hepatocytes) from the highly anoxia tolerant aquatic turtle and from other more hypoxia sensitive systems. We propose that the response of hypoxia tolerant systems to oxygen lack occurs in two phases (defense and rescue). The first lines of defense against hypoxia include a balanced suppression of ATP-demand and ATP-supply pathways; this regulation stabilizes (adenylates) at new steady-state levels even while ATP turnover rates greatly decline. The ATP demands of ion pumping are down-regulated by generalized "channel" arrest in hepatocytes and by "spike" arrest in neurons. Hypoxic ATP demands of protein synthesis are down-regulated probably by translational arrest. In hypoxia sensitive cells this translational arrest seems irreversible, but hypoxia-tolerant systems activate "rescue" mechanisms if the period of oxygen lack is extended by preferentially regulating the expression of several proteins. In these cells, a cascade of processes underpinning hypoxia rescue and defense begins with an oxygen sensor (a heme protein) and a signal-transduction pathway, which leads to significant gene-based metabolic reprogramming-the rescue process-with maintained down-regulation of energy-demand and energy-supply pathways in metabolism throughout the hypoxic period. This recent work begins to clarify how normoxic maintenance ATP turnover rates can be drastically (10-fold) down-regulated to a new hypometabolic steady state, which is prerequisite for surviving prolonged hypoxia or anoxia. The implications of these developments are extensive in biology and medicine.