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.

Euler characteristics and elliptic curves

Let E be a modular elliptic curve over ℚ, without complex multiplication; let p be a prime number where E has good ordinary reduction; and let F∞ be the field obtained by adjoining to ℚ all p-power division points on E. Write G∞ for the Galois group of F∞ over ℚ. Assume that the complex L-series of E over ℚ does not vanish at s = 1. If p ⩾ 5, we make a precise conjecture about the value of the G∞-Euler characteristic of the Selmer group of E over F∞. If one makes a standard conjecture about the behavior of this Selmer group as a module over the Iwasawa algebra, we are able to prove our conjecture. The crucial local calculations in the proof depend on recent joint work of the first author with R. Greenberg.

Tyrosine replacement in P-selectin glycoprotein ligand-1 affects distinct kinetic and mechanical properties of bonds with P- and L-selectin

Selectins are adhesion molecules that initiate tethering and rolling of leukocytes on the vessel wall. Rolling requires rapid formation and breakage of selectin–ligand bonds that must have mechanical strength to resist premature dissociation by the forces applied in shear flow. P- and L-selectin bind to the N-terminal region of P-selectin glycoprotein ligand-1 (PSGL-1), a mucin on leukocytes. To define determinants on PSGL-1 that contribute to the kinetic and mechanical properties of bonds with selectins, we compared rolling of transfected preB cells expressing P- or L-selectin on transfected cell monolayers expressing wild-type PSGL-1 or PSGL-1 constructs with substitutions in targeted N-terminal residues. Rolling through P- or L-selectin required a Thr or Ser at a specific position on PSGL-1, the attachment site for an essential O-glycan, but required only one of three nearby Tyr residues, which are sites for Tyr-SO3 formation. The adhesive strengths and numbers of cells rolling through P- or L-selectin were similar on wild-type PSGL-1 and on each of the three PSGL-1 constructs containing only a single Tyr. However, the cells rolled more irregularly on the single-Tyr forms of PSGL-1. Analysis of the lifetimes of transient tethers on limiting densities of PSGL-1 revealed that L-selectin dissociated faster from single-Tyr than wild-type PSGL-1 at all shears examined. In sharp contrast, P-selectin dissociated faster from single-Tyr than wild-type PSGL-1 at higher shear but not at lower shear. Thus, tyrosine replacements in PSGL-1 affect distinct kinetic and mechanical properties of bonds with P- and L-selectin.

On the geometry of solutions of the quasi-geostrophic and Euler equations

We study solutions of the two-dimensional quasi-geostrophic thermal active scalar equation involving simple hyperbolic saddles. There is a naturally associated notion of simple hyperbolic saddle breakdown. It is proved that such breakdown cannot occur in finite time. At large time, these solutions may grow at most at a quadruple-exponential rate. Analogous results hold for the incompressible three-dimensional Euler equation.

Numbers and Organization of RNA Polymerases, Nascent Transcripts, and Transcription Units in HeLa Nuclei

Using HeLa cells, we have developed methods to determine 1) the number of RNA polymerases that are active at any moment, 2) the number of transcription sites, and 3) the number of polymerases associated with one transcription unit. To count engaged polymerases, cells were encapsulated in agarose, permeabilized, treated with ribonuclease, and the now-truncated transcripts extended in [32P]uridine triphosphate; then, the number of growing transcripts was calculated from the total number of nucleotides incorporated and the average increment in length of the transcripts. Approximately 15,000 transcripts were elongated by polymerase I, and ∼75,000 were elongated by polymerases II and III. Transcription sites were detected after the cells were grown in bromouridine for <2.5 min, after which the resulting bromo-RNA was labeled with gold particles; electron microscopy showed that most extranucleolar transcripts were concentrated in ∼2400 sites with diameters of ∼80 nm. The number of polymerases associated with a transcription unit was counted after templates were spread over a large area; most extranucleolar units were associated with one elongating complex. These results suggest that many templates are attached in a “cloud” of loops around a site; each site, or transcription “factory,” would contain ∼30 active polymerases and associated transcripts.

Numbers of deaths related to intrapartum asphyxia and timing of birth in all Wales perinatal survey, 1993-5

Objectives: To investigate the relation between the timing of birth and the occurrence of death related to an intrapartum event.

Mendel-GFDb and Mendel-ESTS: databases of plant gene families and ESTs annotated with gene family numbers and gene family names

There is no control over the information provided with sequences when they are deposited in the sequence databases. Consequently mistakes can seed the incorrect annotation of other sequences. Grouping genes into families and applying controlled annotation overcomes the problems of incorrect annotation associated with individual sequences. Two databases (http://www.mendel.ac.uk) were created to apply controlled annotation to plant genes and plant ESTs: Mendel-GFDb is a database of plant protein (gene) families based on gapped-BLAST analysis of all sequences in the SWISS-PROT family of databases. Sequences are aligned (ClustalW) and identical and similar residues shaded. The families are visually curated to ensure that one or more criteria, for example overall relatedness and/or domain similarity relate all sequences within a family. Sequence families are assigned a ‘Gene Family Number’ and a unified description is developed which best describes the family and its members. If authority exists the gene family is assigned a ‘Gene Family Name’. This information is placed in Mendel-GFDb. Mendel-ESTS is primarily a database of plant ESTs, which have been compared to Mendel-GFDb, completely sequenced genomes and domain databases. This approach associated ESTs with individual sequences and the controlled annotation of gene families and protein domains; the information being placed in Mendel-ESTS. The controlled annotation applied to genes and ESTs provides a basis from which a plant transcription database can be developed.

Congruences between modular forms: Raising the level and dropping Euler factors

We discuss the relationship among certain generalizations of results of Hida, Ribet, and Wiles on congruences between modular forms. Hida’s result accounts for congruences in terms of the value of an L-function, and Ribet’s result is related to the behavior of the period that appears there. Wiles’ theory leads to a class number formula relating the value of the L-function to the size of a Galois cohomology group. The behavior of the period is used to deduce that a formula at “nonminimal level” is obtained from one at “minimal level” by dropping Euler factors from the L-function.