Fourier series for quaternions and the square of the error theorem

In this paper we introduce a type of Hypercomplex Fourier Series based on Quaternions, and discuss on a Hypercomplex version of the Square of the Error Theorem. Since their discovery by Hamilton (Sinegre [1]), quaternions have provided beautifully insights either on the structure of different areas of Mathematics or in the connections of Mathematics with other fields. For instance: I) Pauli spin matrices used in Physics can be easily explained through quaternions analysis (Lan [2]); II) Fundamental theorem of Algebra (Eilenberg [3]), which asserts that the polynomial analysis in quaternions maps into itself the four dimensional sphere of all real quaternions, with the point infinity added, and the degree of this map is n. Motivated on earlier works by two of us on Power Series (Pendeza et al. [4]), and in a recent paper on Liouville’s Theorem (Borges and Mar˜o [5]), we obtain an Hypercomplex version of the Fourier Series, which hopefully can be used for the treatment of hypergeometric partial differential equations such as the dumped harmonic oscillation.

Experimental evidence of MAP kinase gene expression on the response of intestinal anti-inflammatory drugs

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

A Preliminary Linkage Map of the Hard Tick, Ixodes Scapularis

Blackwell Publishing Ltd. A linkage map of the Ixodes scapularis genome was constructed, based upon segregation amongst 127 loci. These included 84 random amplified polymorphic DNA (RAPD) markers, 32 Sequence-Tagged RAPD (STAR) markers, 5 cDNAs, and 5 microsatellites in 232 F1 intercross progeny from a single, field-collected P1 female. A preliminary linkage map of 616 cM was generated across 14 linkage groups with one marker every 10.8 cM. Assuming a genome size of ~ 10 9 bp, the relationship of physical to genetic distance was found to be ~ 300 kb/cM in the I. scapularis genome.

A preliminary linkage map of the tick, Ixodes scapularis

A linkage map of the Ixodes scapularis genome was constructed based upon segregation amongst 127 loci. These included 84 random amplified polymorphic DNA (RAPD) markers, 32 Sequence-Tagged RAPD (STAR) markers, 5 cDNAs, and 5 microsatellites in 232 F1 intercross progeny from a single, field-collected P1 female. A preliminary linkage map of 616 cM was generated across 14 linkage groups with one marker every 10.8 cM. Assuming a genome size of ∼109 bp, the relationship of physical to genetic distance is ∼300 kb/cM in the I. scapularis genome.

Journal Writing in Mathematics: Exploring the Connections between Math Journals and the Completion of Homework Assignments

In this action research study of sixth grade mathematics, I investigate how the use of written journals facilitates the learning of mathematics for my students. I explore furthermore whether or not these writing journals support students to complete their homework. My analysis reveals that while students do not access their journals daily, when students have the opportunity to write more about one specific problem--such as finding the relationship between the area of two different sized rectangles – they, are nevertheless, more likely to explain their thoughts in-depth and go beyond the traditional basic steps to arrive at a solution. This suggests the value of integrating journal writing in a math curriculum as it can facilitate classroom discussion from the students’ written work.

A molecular linkage map for Drosophila mediopunctata confirms synteny with Drosophila melanogaster and suggests a region that controls the variation in the number of abdominal spots

The classic approach to gene discovery relies on the construction of linkage maps. We report the first molecular-based linkage map for Drosophila mediopunctata, a neotropical species of the tripunctata group. Eight hundred F2 individuals were genotyped at 49 microsatellite loci, resulting in a map that is approximate to 450 centimorgans long. Five linkage groups were detected, and the species' chromosomes were identified through cross-references to BLASTn searches and Muller elements. Strong synteny was observed when compared with the Drosophila melanogaster chromosome arms, but little conservation in the gene order was seen. The incorporation of morphological data corresponding to the number of central abdominal spots on the map was consistent with the expected location of a genomic region responsible for the phenotype on the second chromosome.

Metabolomics reveals the metabolic map of procainamide in humans and mice

Procainamide, a type I antiarrhythmic agent, is used to treat a variety of atrial and ventricular dysrhythmias. It was reported that long-term therapy with procainamide may cause lupus erythematosus in 25-30% of patients. Interestingly, procainamide does not induce lupus erythematosus in mouse models. To explore the differences in this side-effect of procainamide between humans and mouse models, metabolomic analysis using ultra-performance liquid chromatography coupled with electrospray ionization quadrupole time-of-flight mass spectrometry (UPLC-ESI-QTOFMS) was conducted on urine samples from procainamide-treated humans, CYP2D6-humanized mice, and wild-type mice. Thirteen urinary procainamide metabolites, including nine novel metabolites, derived from P450-dependent, FMO-dependent oxidations and acylation reactions, were identified and structurally elucidated. In vivo metabolism of procainamide in CYP2D6-humanized mice as well as in vitro incubations with microsomes and recombinant P450s suggested that human CYP2D6 plays a major role in procainamide metabolism. Significant differences in N-acylation and N-oxidation of the drug between humans and mice largely account for the interspecies differences in procainamide metabolism. Significant levels of the novel N-oxide metabolites produced by FMO1 and FMO3 in humans might be associated with the development of procainamide-induced systemic lupus erythematosus. Observations based on this metabolomic study offer clues to understanding procainamide-induced lupus in humans and the effect of P450s and FMOs on procainamide N-oxidation.

Dielectric map of the Martian northern hemisphere and the nature of plain filling materials

[1] A number of observations suggest that an extended ocean once covered a significant part of the Martian northern hemisphere. By probing the physical properties of the subsurface to unprecedented depth, the MARSIS/Mars Express provides new geophysical evidences for the former existence of a Late Hesperian ocean. The Vastitas Borealis formation, located inside a putative shoreline of the ancient ocean, has a low dielectric constant compared with that of typical volcanic materials. We show that the measured value is only consistent with low-density sedimentary deposits, massive deposits of ground-ice, or a combination of the two. In contrast, radar observations indicate a distribution of shallow ground ice in equilibrium with the atmosphere in the south polar region. We conclude that the northern plains are filled with remnants of a late Hesperian ocean, fed by water and sediments from the outflow channels about 3 Gy ago.

A second generation radiation hybrid map to aid the assembly of the bovine genome sequence

BACKGROUND: Several approaches can be used to determine the order of loci on chromosomes and hence develop maps of the genome. However, all mapping approaches are prone to errors either arising from technical deficiencies or lack of statistical support to distinguish between alternative orders of loci. The accuracy of the genome maps could be improved, in principle, if information from different sources was combined to produce integrated maps. The publicly available bovine genomic sequence assembly with 6x coverage (Btau_2.0) is based on whole genome shotgun sequence data and limited mapping data however, it is recognised that this assembly is a draft that contains errors. Correcting the sequence assembly requires extensive additional mapping information to improve the reliability of the ordering of sequence scaffolds on chromosomes. The radiation hybrid (RH) map described here has been contributed to the international sequencing project to aid this process. RESULTS: An RH map for the 30 bovine chromosomes is presented. The map was built using the Roslin 3000-rad RH panel (BovGen RH map) and contains 3966 markers including 2473 new loci in addition to 262 amplified fragment-length polymorphisms (AFLP) and 1231 markers previously published with the first generation RH map. Sequences of the mapped loci were aligned with published bovine genome maps to identify inconsistencies. In addition to differences in the order of loci, several cases were observed where the chromosomal assignment of loci differed between maps. All the chromosome maps were aligned with the current 6x bovine assembly (Btau_2.0) and 2898 loci were unambiguously located in the bovine sequence. The order of loci on the RH map for BTA 5, 7, 16, 22, 25 and 29 differed substantially from the assembled bovine sequence. From the 2898 loci unambiguously identified in the bovine sequence assembly, 131 mapped to different chromosomes in the BovGen RH map. CONCLUSION: Alignment of the BovGen RH map with other published RH and genetic maps showed higher consistency in marker order and chromosome assignment than with the current 6x sequence assembly. This suggests that the bovine sequence assembly could be significantly improved by incorporating additional independent mapping information.