Large normal subgroup growth and large characteristic subgroup growth

The maximal normal subgroup growth type of a finitely generated group is $n^{\log n}$. Very little is known about groups with this type of growth. In particular, the following is a long standing problem: Let $\Gamma$ be a group and $\Delta$ a subgroup of finite index. Suppose $\Delta$ has normal subgroup growth of type $n^{\log n}$, does $\Gamma$ has normal subgroup growth of type $n^{\log n}$? We give a positive answer in some cases, generalizing a result of M\"uller and the second author and a result of Gerdau. For instance, suppose $G$ is a profinite group and $H$ an open subgroup of $G$. We show that if $H$ is a generalized Golod-Shafarevich group, then $G$ has normal subgroup growth of type of $n^{\log n}$. We also use our methods to show that one can find a group with characteristic subgroup growth of type $n^{\log n}$

The irrationality of a number theoretical series

Denote by $\sigma_k(n)$ the sum of the $k$-th powers of the divisors of $n$, and let $S_k=\sum_{n\geq 1}\frac{\sigma_k(n)}{n!}$. We prove that Schinzel's conjecture H implies that $S_k$ is irrational, and give an unconditional proof for the case $k=3$

The subgroup growth spectrum of virtually free groups

For a finitely generated group $\Gamma$ denote by $\mu(\Gamma)$ the growth coefficient of $\Gamma$, that is, the infimum over all real numbers $d$ such that $s_n(\Gamma)<n!^d$. We show that the growth coefficient of a virtually free group is always rational, and that every rational number occurs as growth coefficient of some virtually free group. Moreover, we describe an algorithm to compute $\mu$

Involvement of the cohesin cofactor PDS5 (SPO76) during meiosis and DNA repair in Arabidopsis thaliana

Maintenance and precise regulation of sister chromatid cohesion is essential for faithful chromosome segregation during mitosis and meiosis. Cohesin cofactors contribute to cohesin dynamics and interact with cohesin complexes during cellcycle. One of these, PDS5, also known as SPO76, is essential during mitosis and meiosis in several organisms and also plays a role in DANN repair. In yeast, the complex Wapl-Pds5 controls cohesion maintenance and colocalizes with cohesin complexes into chromosomes. In Arabidopsis, AtWAPL proteins are essential during meiosis, however, the role of AtPDS5 remains to be ascertained. Here we have isolated mutants for each of the five AtPDS5 genes(A–E) and obtained, after different crosses between them, double,triple,and even quadruple mutants (Atpds5a Atpds5b Atpds5c Atpds5e). Depletion of AtPDS5 proteins has a weak impact on meiosis, but leads to severe effects on development, fertility, somatic homologous recombination (HR) and DANN repair. Furthermore, this cohesin cofactor could be important for the function of the AtSMC5/AtSMC6 complex. Contrarily to ist function in other species, our results suggest that AtPDS5 is dispensable during the meiotic division of Arabidopsis, although it plays an important role in DANN repair by HR

One vertex spin-foams with the Dipole Cosmology boundary

We find all the spin-foams contributing in the first order of the vertex expansion to the transition amplitude of the Bianchi-Rovelli-Vidotto Dipole Cosmology model. Our algorithm is general and provides spin-foams of arbitrarily given, fixed: boundary and, respectively, a number of internal vertices. We use the recently introduced Operator Spin-Network Diagrams framework.Comment: 23 pages, 30 figure

Irregular behaviour of class numbers and Euler-Kronecker constants of cyclotomic fields: the log log log devil at play

Kummer (1851) and, many years later, Ihara (2005) both posed conjectures on invariants related to the cyclotomic field $\mathbb Q(\zeta_q)$ with $q$ a prime. Kummer's conjecture concerns the asymptotic behaviour of the first factor of the class number of $\mathbb Q(\zeta_q)$ and Ihara's the positivity of the Euler-Kronecker constant of $\mathbb Q(\zeta_q)$ (the ratio of the constant and the residue of the Laurent series of the Dedekind zeta function $\zeta_{\mathbb Q(\zeta_q)}(s)$ at $s=1$). If certain standard conjectures in analytic number theory hold true, then one can show that both conjectures are true for a set of primes of natural density 1, but false in general. Responsible for this are irregularities in the distribution of the primes. With this survey we hope to convince the reader that the apparently dissimilar mathematical objects studied by Kummer and Ihara actually display a very similar behaviour.Comment: 20 pages, 1 figure, survey, to appear in `Irregularities in the Distribution of Prime Numbers - Research Inspired by Maier's Matrix Method', Eds. J. Pintz and M. Th. Rassia

Outcome prediction following transcatheter aortic valve implantation: Multiple risk scores comparison

Background: The aim of the study was to compare 7 available risk models in the prediction of 30-day mortality following transcatheter aortic valve implantation (TAVI). Heart team decision supported by different risk score calculations is advisable to estimate the individual procedural risk before TAVI. Methods: One hundred and fifty-six consecutive patients (n = 156, 48% female, mean age 80.03 ± 8.18 years) who underwent TAVI between March 2010 and October 2014 were in­cluded in the study. Thirty-day follow-up was performed and available in each patient. Base­line risk was calculated according to EuroSCORE I, EuroSCORE II, STS, ACEF, Ambler’s, OBSERVANT and SURTAVI scores. Results: In receiver operating characteristics analysis, neither of the investigated scales was able to distinguish between patients with or without an endpoint with areas under the curve (AUC) not exceeding 0.6, as follows: EuroSCORE I, AUC 0.55; 95% confidence intervals (CI) 0.47–0.63, p = 0.59; EuroSCORE II, AUC 0.59; 95% CI 0.51–0.67, p = 0.23; STS, AUC 0.55; 95% CI 0.47–0.63, p = 0.52; ACEF, AUC 0.54; 95% CI 0.46–0.62, p = 0.69; Ambler’s, AUC 0.54; 95% CI 0.46–0.62, p = 0.70; OBSERVANT, AUC 0.597; 95% CI 0.52–0.67, p = 0.21; SURTAVI, AUC 0.535; 95% CI 0.45–0.62, p = 0.65. SURTAVI model was calibrated best in high-risk patients showing coherence between expected and observed mortality (10.8% vs. 9.4%, p = 0.982). ACEF demonstrated best classification accuracy (17.5% vs. 6.9%, p = 0.053, observed mortality in high vs. non-high-risk cohort, respectively). Conclusions: None of the investigated risk scales proved to be optimal in predicting 30-day mortality in unselected, real-life population with aortic stenosis referred to TAVI. This data supports primary role of heart team in decision process of selecting patients for TAVI