Non-existence of Ramanujan congruences in modular forms of level four

Ramanujan famously found congruences for the partition function like p(5n+4) = 0 modulo 5. We provide a method to find all simple congruences of this type in the coefficients of the inverse of a modular form on Gamma_{1}(4) which is non-vanishing on the upper half plane. This is applied to answer open questions about the (non)-existence of congruences in the generating functions for overpartitions, crank differences, and 2-colored F-partitions.Comment: 19 page

Circulating glucuronic acid predicts healthspan and longevity in humans and mice

Glucuronic acid is a metabolite of glucose that is involved in the detoxification of xenobiotic compounds and the structure/remodeling of the extracellular matrix. We report for the first time that circulating glucuronic acid is a robust biomarker of mortality that is conserved across species. We find that glucuronic acid levels are significant predictors of all-cause mortality in three population-based cohorts from different countries with 420 years of follow-up (HR=1.44, p=2.9x10(-6) in the discovery cohort; HR=1.13, p=0.032 and HR=1.25, p=0.017, respectively in the replication cohorts), as well as in a longitudinal study of genetically heterogenous mice (HR=1.29, p=0.018). Additionally, we find that glucuronic acid levels increase with age and predict future healthspan-related outcomes. Together, these results demonstrate glucuronic acid as a robust biomarker of longevity and healthspan

Personality characteristics below facets:A replication and meta-analysis of cross-rater agreement, rank-order stability, heritability and utility of personality nuances

Mõttus and colleagues (2017) reported evidence that the unique variance in specific personality characteristics captured by single descriptive items often displayed trait-like properties of cross-rater agreement, rank-order stability, and heritability. They suggested that the personality hierarchy should be extended below facets to incorporate these specific characteristics, called personality nuances. The present study attempted to replicate these findings, employing data from 6,287 individuals from 6 countries (Australia, Canada, Czech Republic, Denmark, Japan, and United States). The same personality measure-240-item Revised NEO Personality Inventory-and statistical procedures were used. The present findings closely replicated the original results. When the original and current results were meta-analyzed, the unique variance of nearly all items (i.e., items' scores residualized for all broader personality traits) showed statistically significant cross-rater agreement (median = .12) and rank-order stability over an average of 12 years (median = .24), and the unique variance of the majority of items had a significant heritable component (median = .14). These 3 item properties were intercorrelated, suggesting that items systematically differed in the degree of reflecting valid unique variance. Also, associations of items' unique variance with age, gender, and body mass index (BMI) replicated across samples and tracked with the original findings. Moreover, associations between item residuals and BMI obtained from one group of people allowed for a significant incremental prediction of BMI in an independent sample. Overall, these findings reinforce the hypotheses that nuances constitute the building blocks of the personality trait hierarchy, their properties are robust and they can be useful

Problems in number theory and hyperbolic geometry

In the first part of this thesis we generalize a theorem of Kiming and Olsson concerning the existence of Ramanujan-type congruences for a class of eta quotients. Specifically, we consider a class of generating functions analogous to the generating function of the partition function and establish a bound on the primes ℓ for which their coefficients c(n) obey congruences of the form c(ℓn + a) ≡ 0 (mod ℓ). We use this last result to answer a question of H.C. Chan. In the second part of this thesis [S2] we explore a natural analog of D. Calegari’s result that there are no hyperbolic once-punctured torus bundles over S^1 with trace field having a real place. We prove a contrasting theorem showing the existence of several infinite families of pairs (−χ, p) such that there exist hyperbolic surface bundles over S^1 with trace field of having a real place and with fiber having p punctures and Euler characteristic χ. This supports our conjecture that with finitely many known exceptions there exist such examples for each pair ( −χ, p)

The Innovify Foundry Programme

Imperial Users onlyDistinctio