Irreducibility of generalized Hermite-Laguerre Polynomials III

For a positive integer $n$ and a real number $\alpha$, the generalized Laguerre polynomials are defined by \begin{align*} L^{(\alpha)}_n(x)=\sum^n_{j=0}\frac{(n+\alpha)(n-1+\alpha)\cdots (j+1+\alpha)(-x)^j}{j!(n-j)!}. \end{align*} These orthogonal polynomials are solutions to Laguerre's Differential Equation which arises in the treatment of the harmonic oscillator in quantum mechanics. Schur studied these Laguerre polynomials for its interesting algebraic properties. He obtained irreducibility results of $L^{(\pm \frac{1}{2})}_n(x)$ and $L^{(\pm \frac{1}{2})}_n(x^2)$ and derived that the Hermite polynomials $H_{2n}(x)$ and $\frac{H_{2n+1}(x)}{x}$ are irreducible for each $n$. In this article, we extend Schur's result by showing that the family of Laguerre polynomials $L^{(q)}_n(x)$ and $L^{(q)}_n(x^d)$ with $q\in \{\pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{1}{4}, \pm \frac{3}{4}\}$, where $d$ is the denominator of $q$, are irreducible for every $n$ except when $q=\frac{1}{4}, n=2$ where we give the complete factorization. In fact, we derive it from a more general result.Comment: Published in Journal of Number Theor

Analysis of the Importance of Extension in Accounting for the Post-Carboniferous Subsidence of the North Sea Basin

Post Carboniferous sedimentary deposition in the Central North Sea basins can be separated into three major periods: Permian, Triassic and mid-Jurassic through present. Most efforts to explain the basin within an extensional framework have concentrated on the post mid-Jurassic subsidence. These efforts have ignored the large amount of prior extension required to account for the observed crustal thinning and the substantial Permian and Triassic sediment fill. In addition the models predict a mid-Jurassic through early Cretaceous extension that significantly exceeds estimates of the horizontal displacement observed on high angle faults on multichannel seismic lines. We show in areas of minimal pre-Permian subsidence that adding two earlier phase extensions, one in the late Carboniferous through early Permian and the other in the Triassic produces a nearly horizontal late Carboniferous crustal thickness. The time-dependent extensional model required to account for the three periods of sediment deposition gives an excellent match to the observed subsidence history of the basement. We present an analysis of a recent seismic reflection line nm across the Central Graben in the vicinity of published refraction and well data. We show that the extension required in the third phase of the three phase model is compatible with the observed displacement on the high angle mid-Jurassic through early Cretaceous faults. However, we find no evidence for major extension either in the Triassic or late Carboniferous through early Permian.Institute for Geophysic

One-Armed Spiral Waves in Galaxy Simulations with Counter-Rotating Stars

Motivated by observations of disk galaxies with counter-rotating stars, we have run two-dimensional, collisionless N-body simulations of disk galaxies with significant counter-rotating components. For all our simulations the initial value of Toomre's stability parameter was Q = 1.1. The percentage of counter-rotating particles ranges from 25% to 50%. A stationary one-arm spiral wave is observed to form in each run, persisting from a few to five rotation periods, measured at the half-mass radius. In one run, the spiral wave was initially a leading arm which subsequently transformed into a trailing arm. We also observed a change in spiral direction in the run initially containing equal numbers of particles orbiting in both directions. The results of our simulations support an interpretation of the one armed waves as due to the two stream instability.Comment: 13 pages, 4 figure

Perceptions of primiparas on a postnatal psychoeducation programme: The process evaluation

Midwifery311155-16

Patronin-mediated minus end growth is required for dendritic microtubule polarity.

Microtubule minus ends are thought to be stable in cells. Surprisingly, in Drosophila and zebrafish neurons, we observed persistent minus end growth, with runs lasting over 10 min. In Drosophila, extended minus end growth depended on Patronin, and Patronin reduction disrupted dendritic minus-end-out polarity. In fly dendrites, microtubule nucleation sites localize at dendrite branch points. Therefore, we hypothesized minus end growth might be particularly important beyond branch points. Distal dendrites have mixed polarity, and reduction of Patronin lowered the number of minus-end-out microtubules. More strikingly, extra Patronin made terminal dendrites almost completely minus-end-out, indicating low Patronin normally limits minus-end-out microtubules. To determine whether minus end growth populated new dendrites with microtubules, we analyzed dendrite development and regeneration. Minus ends extended into growing dendrites in the presence of Patronin. In sum, our data suggest that Patronin facilitates sustained microtubule minus end growth, which is critical for populating dendrites with minus-end-out microtubules

In utero tobacco smoke exposure, DNA methylation, and asthma in Latino children.

BackgroundMaternal smoking during pregnancy is a risk factor for chronic disease later in life and has been associated with variability of DNA methylation at specific cytosine-phosphate-guanine (CpG) loci. We assessed the role of DNA methylation as a potential mediator of adverse effects of in utero tobacco smoke exposures on asthma outcomes in Latino children from the US mainland and Puerto Rico.MethodsRelationships between self-reported exposure and DNA methylation at CpG loci previously reported to be associated with maternal smoking were assessed in a subsample consisting of 572 children aged 8-21 years (310 cases with asthma, 262 healthy controls), sampled from a larger asthma case-control study. Subsequently, we assessed associations between top loci and asthma-related outcomes, followed by mediation analysis for loci for which associations with outcomes were observed.ResultsSelf-reported maternal smoking was associated with a -1.5% (95% confidence interval (CI) = -2.4%, -0.6%) lower methylation at CpG locus cg05575921 on the AHRR gene; a 1% increase in DNA methylation at the same locus resulted in an odds ratio (OR) of 0.90 (95% CI = 0.83, 0.96) for the odds of asthma. The OR for the indirect effect of maternal smoking on asthma mediated through methylation at the cg05575921 locus was 1.18 (95% CI = 1.07, 1.68), compared to the OR for the total effect of exposure in the parent study of 1.48 (95% CI = 1.03, 2.11).ConclusionsOur findings suggest potential mediation by DNA methylation in the association between maternal smoking during pregnancy and asthma status

Predictors of Maternal Parental Self-Efficacy Among Primiparas in the Early Postnatal Period

10.1177/0193945914537724Western Journal of Nursing Researc