Localizationof the 5S and 45S rDNA sites and CPDNA sequence analysis in species of the quadrifaria group of Paspalum (Poaceae, Panicae).

The Quadrifaria group of Paspalum (Poaceae, Paniceae) comprises species native to the subtropical and temperate regions of South America. The purpose of this research was to characterize the I genomes in five species of this group and to establish phylogenetic relationships among them. Prometaphase chromatin condensation patterns, the physical location of 5S and 45S rDNA sites by fluorescence in situ hybridization (FISH), and sequences of five chloroplast non-coding regions were analysed. The condensation patterns observed were highly conserved among diploid and tetraploid accessions studied and not influenced by the dyes used or by the FISH procedure, allowing the identification of almost all the chromosome pairs that carried the rDNA signals. The FISH analysis of 5S rDNA sites showed the same localization and a correspondence between the number of sites and ploidy level. In contrast, the distribution of 45S rDNA sites was variable. Two general patterns were observed with respect to the location of the 45S rDNA. The species and cytotypes Paspalum haumanii 2x, P. intermedium 2x, P. quadrifarium 4x and P. exaltatum 4x showed proximal sites on chromosome 8 and two to four distal sites in other chromosomes, while P. quarinii 4x and P. quadrifarium 2x showed only distal sites located on a variable number of small chromosomes and on the long arm of chromosome 1. The single most-parsimonious tree found from the phylogenetic analysis showed the Quadrifaria species partitioned in two clades, one of them includes P. haumanii 2x and P. intermedium 2x together with P. quadrifarium 4x and P. exaltatum 4x, while the other contains P. quadrifarium 2x and P. quarinii 4x. The subdivision found with FISH is consistent with the clades recovered with cpDNA data and both analyses suggest that the Quadrifaria group, as presently defined, is not monophyletic and its species belong in at least two clades

Energy distribution and equation of state of the early Universe: matching the end of inflation and the onset of radiation domination

We study the energy distribution and equation of state of the universe between the end of inflation and the onset of radiation domination (RD), considering observationally consistent single-field inflationary scenarios, with a potential 'flattening' at large field values, and a monomial shape $V(\phi) \propto |\phi|^p$ around the origin. As a proxy for (p)reheating, we include a quadratic interaction $g^2\phi^2X^2$ between the inflaton $\phi$ and a light scalar 'daughter' field $X$, with $g^2>0$. We capture the non-perturbative and non-linear nature of the system dynamics with lattice simulations, obtaining that: $i)$ the final energy transferred to $X$ depends only on $p$, not on $g^2$, ; $ii)$ the final transfer of energy is always negligible for $2 \leq p < 4$, and of order $\sim 50\%$ for $p \geq 4$; $iii)$ the system goes at late times to matter-domination for $p = 2$, and always to RD for $p > 2$. In the latter case we calculate the number of e-folds until RD, significantly reducing the uncertainty in the inflationary observables $n_s$ and $r$.Comment: 7 pages + references, 5 figures. It matches published versio

Largest Prime Factors of Quadratic Polynomials

Let $p\equiv 1 \bmod 4$ be a prime number and let $x>1$ be a large number. This note shows that the largest prime factor of the finite product $\prod_{x\leq n\leq 2x}\left(n^2+1 \right)$ satisfies the relation $p \geq x^{4/3}$ as $x$ tends to infinity. This improves the current estimate $p \geq x^{1.279}$.Comment: Seven Pages. Keywords: Prime number; Distribution of prime; Polynomial prime valu

Heyde theorem on locally compact Abelian groups with the connected component of zero of dimension 1

Let $X$ be a locally compact Abelian group with the connected component of zero of dimension 1. Let $\xi_1$ and $\xi_2$ be independent random variables with values in $X$ with nonvanishing characteristic functions. We prove that if a topological automorphism $\alpha$ of the group $X$ satisfies the condition ${{\rm Ker}(I+\alpha)=\{0\}}$ and the conditional distribution of the linear form ${L_2 = \xi_1 + \alpha\xi_2}$ given ${L_1 = \xi_1 + \xi_2}$ is symmetric, then the distributions of $\xi_j$ are convolutions of Gaussian distributions on $X$ and distributions supported in the subgroup $\{x\in X:2x=0\}$. This result can be viewed as a generalization of the well-known Heyde theorem on the characterization of the Gaussian distribution on the real line.Comment: 15 p

Accelerating Permutation Testing in Voxel-wise Analysis through Subspace Tracking: A new plugin for SnPM

Permutation testing is a non-parametric method for obtaining the max null distribution used to compute corrected $p$-values that provide strong control of false positives. In neuroimaging, however, the computational burden of running such an algorithm can be significant. We find that by viewing the permutation testing procedure as the construction of a very large permutation testing matrix, $T$, one can exploit structural properties derived from the data and the test statistics to reduce the runtime under certain conditions. In particular, we see that $T$ is low-rank plus a low-variance residual. This makes $T$ a good candidate for low-rank matrix completion, where only a very small number of entries of $T$ ($\sim0.35\%$ of all entries in our experiments) have to be computed to obtain a good estimate. Based on this observation, we present RapidPT, an algorithm that efficiently recovers the max null distribution commonly obtained through regular permutation testing in voxel-wise analysis. We present an extensive validation on a synthetic dataset and four varying sized datasets against two baselines: Statistical NonParametric Mapping (SnPM13) and a standard permutation testing implementation (referred as NaivePT). We find that RapidPT achieves its best runtime performance on medium sized datasets ($50 \leq n \leq 200$), with speedups of 1.5x - 38x (vs. SnPM13) and 20x-1000x (vs. NaivePT). For larger datasets ($n \geq 200$) RapidPT outperforms NaivePT (6x - 200x) on all datasets, and provides large speedups over SnPM13 when more than 10000 permutations (2x - 15x) are needed. The implementation is a standalone toolbox and also integrated within SnPM13, able to leverage multi-core architectures when available.Comment: 36 pages, 16 figure