Asymptotically conical Calabi-Yau manifolds, I

This is the first part in a two-part series on complete Calabi-Yau manifolds asymptotic to Riemannian cones at infinity. We begin by proving general existence and uniqueness results. The uniqueness part relaxes the decay condition $O(r^{-n-\epsilon})$ needed in earlier work to $O(r^{-\epsilon})$, relying on some new ideas about harmonic functions. We then look at a few examples: (1) Crepant resolutions of cones. This includes a new class of Ricci-flat small resolutions associated with flag manifolds. (2) Affine deformations of cones. One focus here is the question of the precise rate of decay of the metric to its tangent cone. We prove that the optimal rate for the Stenzel metric on $T^*S^n$ is $-2\frac{n}{n-1}$.Comment: 27 pages, various corrections, final versio

Women's position and attitudes towards female genital mutilation in Egypt : a secondary analysis of the Egypt demographic and health surveys, 1995-2014

Background: Female genital mutilation (FGM) is still widespread in Egyptian society. It is strongly entrenched in local tradition and culture and has a strong link to the position of women. To eradicate the practice a major attitudinal change is a required for which an improvement in the social position of women is a prerequisite. This study examines the relationship between Egyptian women's social positions and their attitudes towards FGM, and investigates whether the spread of anti-FGM attitudes is related to the observed improvements in the position of women over time. Methods: Changes in attitudes towards FGM are tracked using data from the Egypt Demographic and Health Surveys from 1995 to 2014. Multilevel logistic regressions are used to estimate 1) the effects of indicators of a woman's social position on her attitude towards FGM, and 2) whether these effects change over time. Results: Literate, better educated and employed women are more likely to oppose FGM. Initially growing opposition to FGM was related to the expansion of women's education, but lately opposition to FGM also seems to have spread to other segments of Egyptian society. Conclusions: The improvement of women's social position has certainly contributed to the spread of anti-FGM attitudes in Egyptian society. Better educated and less traditional women were at the heart of this change, and formed the basis from where anti-FGM sentiment has spread over wider segments of Egyptian society

Generalized Ramanujan Primes

In 1845, Bertrand conjectured that for all integers $x\ge2$, there exists at least one prime in $(x/2, x]$. This was proved by Chebyshev in 1860, and then generalized by Ramanujan in 1919. He showed that for any $n\ge1$, there is a (smallest) prime $R_n$ such that $\pi(x)- \pi(x/2) \ge n$ for all $x \ge R_n$. In 2009 Sondow called $R_n$ the $n$th Ramanujan prime and proved the asymptotic behavior $R_n \sim p_{2n}$ (where $p_m$ is the $m$th prime). In the present paper, we generalize the interval of interest by introducing a parameter $c \in (0,1)$ and defining the $n$th $c$-Ramanujan prime as the smallest integer $R_{c,n}$ such that for all $x\ge R_{c,n}$, there are at least $n$ primes in $(cx,x]$. Using consequences of strengthened versions of the Prime Number Theorem, we prove that $R_{c,n}$ exists for all $n$ and all $c$, that $R_{c,n} \sim p_{\frac{n}{1-c}}$ as $n\to\infty$, and that the fraction of primes which are $c$-Ramanujan converges to $1-c$. We then study finer questions related to their distribution among the primes, and see that the $c$-Ramanujan primes display striking behavior, deviating significantly from a probabilistic model based on biased coin flipping; this was first observed by Sondow, Nicholson, and Noe in the case $c = 1/2$. This model is related to the Cramer model, which correctly predicts many properties of primes on large scales, but has been shown to fail in some instances on smaller scales.Comment: 13 pages, 2 tables, to appear in the CANT 2011 Conference Proceedings. This is version 2.0. Changes: fixed typos, added references to OEIS sequences, and cited Shevelev's preprin

Conformations of dendrimers in dilute solution

Conformations of isolated homo- dendrimers of G=1-7 generations with D=1-6 spacers have been studied in the good and poor solvents, as well as across the coil-to-globule transition, by means of a version of the Gaussian self-consistent (GSC) method and Monte Carlo (MC) simulation in continuous space based on the same coarse-grained model. The latter includes harmonic springs between connected monomers and the pair-wise Lennard-Jones potential with a hard core repulsion. The scaling law for the dendrimer size, the degrees of bond stretching and steric congestion, as well as the radial density, static structure factor, and asphericity have been analysed. It is also confirmed that while smaller dendrimers have a dense core, larger ones develop a hollow domain at some separation from the centre.Comment: RevTeX, 14 pages, 19 PS figures, Accepted for publication in J. Chem. Phy