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.

Ol-Prx 3, a member of an additional class of homeobox genes, is unimodally expressed in several domains of the developing and adult central nervous system of the medaka (Oryzias latipes)

Large-scale genetic screens for mutations affecting early neurogenesis of vertebrates have recently been performed with an aquarium fish, the zebrafish. Later stages of neural morphogenesis have attracted less attention in small fish species, partly because of the lack of molecular markers of developing structures that may facilitate the detection of discrete structural alterations. In this context, we report the characterization of Ol-Prx 3 (Oryzias latipes-Prx 3). This gene was isolated in the course of a large-scale screen for brain cDNAs containing a highly conserved DNA binding region, the homeobox helix-three. Sequence analysis revealed that this gene belongs to another class of homeobox genes, together with a previously isolated mouse ortholog, called OG-12 [Rovescalli, A. C., Asoh, S. & Nirenberg, M. (1996) Proc. Natl. Acad. Sci. USA 93, 10691–10696] and with the human SHOX gene [Rao, E., Weiss, B., Fukami, M., Rump, A., Niesler, B., et al. (1997) Nat. Genet. 16, 54–62], thought to be involved in the short-stature phenotype of Turner syndrome patients. These three genes exhibit a moderate level of identity in the homeobox with the other genes of the paired-related (PRX) gene family. Ol-Prx 3, as well as the PRX genes, are expressed in various cartilaginous structures of head and limbs. These genes might thus be involved in common regulatory pathways during the morphogenesis of these structures. Moreover, this paper reports a complex and monophasic pattern of Ol-Prx 3 expression in the central nervous system, which differs markedly from the patterns reported for the PRX genes, Prx 3 excluded: this gene begins to be expressed in a variety of central nervous system territories at late neurula stage. Strikingly, it remains turned on in some of the derivatives of each territory during the entire life of the fish. We hope this work will thus help identify common features for the PRX 3 family of homeobox genes.

Expression of galanin and a galanin receptor in several sensory systems and bone anlage of rat embryos

Using in situ hybridization and immunohistochemistry the expression of, respectively, prepro-galanin (prepro-GAL) mRNA and GAL receptor-1 mRNA, as well as GAL-like and GAL message-associated peptide-like immunoreactivities, were studied in rats from embryonic day 14 (E14) to postnatal day 1. GAL expression was observed already at E14 in trigeminal and dorsal root ganglion neurons and at E15 in the sensory epithelia in developing ear, eye, and nose, as well as at E19 during bone formation. Also, GAL receptor-1 mRNA was expressed in the sensory ganglia of embryos but appeared later than the ligand. These findings suggest that GAL and/or GAL message-associated peptide may have a developmental role in several sensory systems and during bone formation.

Electric Signaling and Pin2 Gene Expression on Different Abiotic Stimuli Depend on a Distinct Threshold Level of Endogenous Abscisic Acid in Several Abscisic Acid-Deficient Tomato Mutants1

Experiments were performed on three abscisic acid (ABA)-deficient tomato (Lycopersicon esculentum Mill.) mutants, notabilis, flacca, and sitiens, to investigate the role of ABA and jasmonic acid (JA) in the generation of electrical signals and Pin2 (proteinase inhibitor II) gene expression. We selected these mutants because they contain different levels of endogenous ABA. ABA levels in the mutant sitiens were reduced to 8% of the wild type, in notabilis they were reduced to 47%, and in flacca they were reduced to 21%. In wild-type and notabilis tomato plants the induction of Pin2 gene expression could be elicited by heat treatment, current application, or mechanical wounding. In flacca and sitiens only heat stimulation induced Pin2 gene expression. JA levels in flacca and sitiens plants also accumulated strongly upon heat stimulation but not upon mechanical wounding or current application. Characteristic electrical signals evolved in the wild type and in the notabilis and flacca mutants consisting of a fast action potential and a slow variation potential. However, in sitiens only heat evoked electrical signals; mechanical wounding and current application did not change the membrane potential. In addition, exogenous application of ABA to wild-type tomato plants induced transient changes in membrane potentials, indicating the involvement of ABA in the generation of electrical signals. Our data strongly suggest the presence of a minimum threshold value of ABA within the plant that is essential for the early events in electrical signaling and mediation of Pin2 gene expression upon wounding. In contrast, heat-induced Pin2 gene expression and membrane potential changes were not dependent on the ABA level but, rather, on the accumulation of JA.