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.

A numerical strategy for efficient modeling of the equatorial wave guide

Convection in the tropics is observed to involve a wide-ranging hierarchy of scales from a few kilometers to the planetary scales and also has a profound impact on short-term climate. The mechanisms responsible for this behavior present a major unsolved problem. A promising emerging approach to address these issues is cloud-resolving modeling. Here a family of numerical models is introduced specifically to model the feedback of small-scale deep convection on tropical planetary waves and tropical circulation in a highly efficient manner compatible with the approach through cloud-resolving modeling. Such a procedure is also useful for theoretical purposes. The basic idea in the approach is to use low-order truncation in the meriodonal direction through Gauss–Hermite quadrature projected onto a simple discrete radiation condition. In this fashion, the cloud-resolving modeling of equatorially trapped planetary waves reduces to the solution of a small number of purely zonal two-dimensional wave systems along a few judiciously chosen meriodonal layers that are coupled only by some additional source terms. The approach is analyzed in detail with full mathematical rigor for linearized equatorial primitive equations with source terms.

The crystal structure of the Rev binding element of HIV-1 reveals novel base pairing and conformational variability

The crystal and molecular structure of an RNA duplex corresponding to the high affinity Rev protein binding element (RBE) has been determined at 2.1-Å resolution. Four unique duplexes are present in the crystal, comprising two structural variants. In each duplex, the RNA double helix consists of an annealed 12-mer and 14-mer that form an asymmetric internal loop consisting of G-G and G-A noncanonical base pairs and a flipped-out uridine. The 12-mer strand has an A-form conformation, whereas the 14-mer strand is distorted to accommodate the bulges and noncanonical base pairing. In contrast to the NMR model of the unbound RBE, an asymmetric G-G pair with N2-N7 and N1-O6 hydrogen bonding, is formed in each helix. The G-A base pairing agrees with the NMR structure in one structural variant, but forms a novel water-mediated pair in the other. A backbone flip and reorientation of the G-G base pair is required to assume the RBE conformation present in the NMR model of the complex between the RBE and the Rev peptide.