On the rational subset problem for groups

We use language theory to study the rational subset problem for groups and monoids. We show that the decidability of this problem is preserved under graph of groups constructions with finite edge groups. In particular, it passes through free products amalgamated over finite subgroups and HNN extensions with finite associated subgroups. We provide a simple proof of a result of Grunschlag showing that the decidability of this problem is a virtual property. We prove further that the problem is decidable for a direct product of a group G with a monoid M if and only if membership is uniformly decidable for G-automata subsets of M. It follows that a direct product of a free group with any abelian group or commutative monoid has decidable rational subset membership.Comment: 19 page

Birth size and breast cancer risk: Re-analysis of individual participant data from 32 studied

Background Birth size, perhaps a proxy for prenatal environment, might be a correlate of subsequent breast cancer risk, but findings from epidemiological studies have been inconsistent. We re-analysed individual participant data from published and unpublished studies to obtain more precise estimates of the magnitude and shape of the birth size–breast cancer association. Methods and Findings Studies were identified through computer-assisted and manual searches, and personal communication with investigators. Individual participant data from 32 studies, comprising 22,058 breast cancer cases, were obtained. Random effect models were used, if appropriate, to combine study-specific estimates of effect. Birth weight was positively associated with breast cancer risk in studies based on birth records (pooled relative risk [RR] per one standard deviation [SD] [= 0.5 kg] increment in birth weight: 1.06; 95% confidence interval [CI] 1.02–1.09) and parental recall when the participants were children (1.02; 95% CI 0.99–1.05), but not in those based on adult self-reports, or maternal recall during the woman's adulthood (0.98; 95% CI 0.95–1.01) (p for heterogeneity between data sources = 0.003). Relative to women who weighed 3.000–3.499 kg, the risk was 0.96 (CI 0.80–1.16) in those who weighed < 2.500 kg, and 1.12 (95% CI 1.00–1.25) in those who weighed ≥ 4.000 kg (p for linear trend = 0.001) in birth record data. Birth length and head circumference from birth records were also positively associated with breast cancer risk (pooled RR per one SD increment: 1.06 [95% CI 1.03–1.10] and 1.09 [95% CI 1.03–1.15], respectively). Simultaneous adjustment for these three birth size variables showed that length was the strongest independent predictor of risk. The birth size effects did not appear to be confounded or mediated by established breast cancer risk factors and were not modified by age or menopausal status. The cumulative incidence of breast cancer per 100 women by age 80 y in the study populations was estimated to be 10.0, 10.0, 10.4, and 11.5 in those who were, respectively, in the bottom, second, third, and top fourths of the birth length distribution. Conclusions This pooled analysis of individual participant data is consistent with birth size, and in particular birth length, being an independent correlate of breast cancer risk in adulthood

Multiphoton Processes in Driven Mesoscopic Systems

We study the statistics of multi-photon absorption/emission processes in a mesoscopic ring threaded by an harmonic time-dependent flux $\Phi(t)$. For this sake, we demonstrate a useful analogy between the Keldysh quantum kinetic equation for the electrons distribution function and a Continuous Time Random Walk in energy space with corrections due to interference effects. Studying the probability to absorb/emit $n$ quanta $\hbar\omega$ per scattering event, we explore the crossover between ultra-quantum/low-intensity limit and quasi-classical/high-intensity regime, and the role of multiphoton processes in driving it.Comment: 6 pages, 5 figures, extended versio