Correlations and Pairing Between Zeros and Critical Points of Gaussian Random Polynomials

We study the asymptotics of correlations and nearest neighbor spacings between zeros and holomorphic critical points of $p_N$, a degree N Hermitian Gaussian random polynomial in the sense of Shiffman and Zeldtich, as N goes to infinity. By holomorphic critical point we mean a solution to the equation $\frac{d}{dz}p_N(z)=0.$ Our principal result is an explicit asymptotic formula for the local scaling limit of \E{Z_{p_N}\wedge C_{p_N}}, the expected joint intensity of zeros and critical points, around any point on the Riemann sphere. Here $Z_{p_N}$ and $C_{p_N}$ are the currents of integration (i.e. counting measures) over the zeros and critical points of $p_N$, respectively. We prove that correlations between zeros and critical points are short range, decaying like e^{-N\abs{z-w}^2}. With \abs{z-w} on the order of $N^{-1/2},$ however, \E{Z_{p_N}\wedge C_{p_N}}(z,w) is sharply peaked near $z=w,$ causing zeros and critical points to appear in rigid pairs. We compute tight bounds on the expected distance and angular dependence between a critical point and its paired zero.Comment: 35 pages, 3 figures. Some typos corrected and Introduction revise

Generalized Bergman kernels on symplectic manifolds

We study the near diagonal asymptotic expansion of the generalized Bergman kernel of the renormalized Bochner-Laplacian on high tensor powers of a positive line bundle over a compact symplectic manifold. We show how to compute the coefficients of the expansion by recurrence and give a closed formula for the first two of them. As consequence, we calculate the density of states function of the Bochner-Laplacian and establish a symplectic version of the convergence of the induced Fubini-Study metric. We also discuss generalizations of the asymptotic expansion for non-compact or singular manifolds as well as their applications. Our approach is inspired by the analytic localization techniques of Bismut-Lebeau.Comment: 48 pages. Add two references on the Hermitian scalar curvatur

Scaling asymptotics for quantized Hamiltonian flows

In recent years, the near diagonal asymptotics of the equivariant components of the Szeg\"{o} kernel of a positive line bundle on a compact symplectic manifold have been studied extensively by many authors. As a natural generalization of this theme, here we consider the local scaling asymptotics of the Toeplitz quantization of a Hamiltonian symplectomorphism, and specifically how they concentrate on the graph of the underlying classical map

Smoking patterns and stimulus control in intermittent and daily smokers

Intermittent smokers (ITS) - who smoke less than daily - comprise an increasing proportion of adult smokers. Their smoking patterns challenge theoretical models of smoking motivation, which emphasize regular and frequent smoking to maintain nicotine levels and avoid withdrawal, but yet have gone largely unexamined. We characterized smoking patterns among 212 ITS (smoking 4-27 days per month) compared to 194 daily smokers (DS; smoking 5-30 cigarettes daily) who monitored situational antecedents of smoking using ecological momentary assessment. Subjects recorded each cigarette on an electronic diary, and situational variables were assessed in a random subset (n = 21,539 smoking episodes); parallel assessments were obtained by beeping subjects at random when they were not smoking (n = 26,930 non-smoking occasions). Compared to DS, ITS' smoking was more strongly associated with being away from home, being in a bar, drinking alcohol, socializing, being with friends and acquaintances, and when others were smoking. Mood had only modest effects in either group. DS' and ITS' smoking were substantially and equally suppressed by smoking restrictions, although ITS more often cited self-imposed restrictions. ITS' smoking was consistently more associated with environmental cues and contexts, especially those associated with positive or "indulgent" smoking situations. Stimulus control may be an important influence in maintaining smoking and making quitting difficult among ITS. © 2014 Shiffman et al

Equidistribution of zeros of holomorphic sections in the non compact setting

We consider N-tensor powers of a positive Hermitian line bundle L over a non-compact complex manifold X. In the compact case, B. Shiffman and S. Zelditch proved that the zeros of random sections become asymptotically uniformly distributed with respect to the natural measure coming from the curvature of L, as N tends to infinity. Under certain boundedness assumptions on the curvature of the canonical line bundle of X and on the Chern form of L we prove a non-compact version of this result. We give various applications, including the limiting distribution of zeros of cusp forms with respect to the principal congruence subgroups of SL2(Z) and to the hyperbolic measure, the higher dimensional case of arithmetic quotients and the case of orthogonal polynomials with weights at infinity. We also give estimates for the speed of convergence of the currents of integration on the zero-divisors.Comment: 25 pages; v.2 is a final update to agree with the published pape

Simple matrix models for random Bergman metrics

Recently, the authors have proposed a new approach to the theory of random metrics, making an explicit link between probability measures on the space of metrics on a Kahler manifold and random matrix models. We consider simple examples of such models and compute the one and two-point functions of the metric. These geometric correlation functions correspond to new interesting types of matrix model correlators. We study a large class of examples and provide in particular a detailed study of the Wishart model.Comment: 23 pages, IOP Latex style, diastatic function Eq. (22) and contact terms in Eqs. (76, 95) corrected, typos fixed. Accepted to JSTA

Genetic variants in the KIF6 region and coronary event reduction from statin therapy

A single nucleotide polymorphism (SNP) in KIF6, a member of the KIF9 family of kinesins, is associated with differential coronary event reduction from statin therapy in four randomized controlled trials; this SNP (rs20455) is also associated with the risk for coronary heart disease (CHD) in multiple prospective studies. We investigated whether other common SNPs in the KIF6 region were associated with event reduction from statin therapy. Of the 170 SNPs in the KIF6 region investigated in the Cholesterol and Recurrent Events trial (CARE), 28 were associated with differential event reduction from statin therapy (Pinteraction < 0.1 in Caucasians, adjusted for age and sex) and were further investigated in the Pravastatin or Atorvastatin Evaluation and Infection Therapy-Thrombolysis In Myocardial Infarction 22 (PROVE IT-TIMI22) and West of Scotland Coronary Prevention Study (WOSCOPS). These analyses revealed that two SNPs (rs9462535 and rs9471077), in addition to rs20455, were associated with event reduction from statin therapy (Pinteraction < 0.1 in each of the three studies). The relative risk reduction ranged from 37 to 50% (P < 0.01) in carriers of the minor alleles of these SNPs and from −4 to 13% (P > 0.4) in non-carriers. These three SNPs are in high linkage disequilibrium with one another (r2 > 0.84). Functional studies of these variants may help to understand the role of KIF6 in the pathogenesis of CHD and differential response to statin therapy

Attentional bias retraining in cigarette smokers attempting smoking cessation (ARTS): study protocol for a double bline randomised controlled trial

YesSmokers attend preferentially to cigarettes and other smoking-related cues in the environment, in what is known as an attentional bias. There is evidence that attentional bias may contribute to craving and failure to stop smoking. Attentional retraining procedures have been used in laboratory studies to train smokers to reduce attentional bias, although these procedures have not been applied in smoking cessation programmes. This trial will examine the efficacy of multiple sessions of attentional retraining on attentional bias, craving, and abstinence in smokers attempting cessation. This is a double-blind randomised controlled trial. Adult smokers attending a 7-session weekly stop smoking clinic will be randomised to either a modified visual probe task with attentional retraining or placebo training. Training will start 1 week prior to quit day and be given weekly for 5 sessions. Both groups will receive 21 mg transdermal nicotine patches for 8–12 weeks and withdrawal-orientated behavioural support for 7 sessions. Primary outcome measures are the change in attentional bias reaction time and urge to smoke on the Mood and Physical Symptoms Scale at 4 weeks post-quit. Secondary outcome measures include differences in withdrawal, time to first lapse and prolonged abstinence at 4 weeks post-quit, which will be biochemically validated at each clinic visit. Follow-up will take place at 8 weeks, 3 months and 6 months post-quit. This is the first randomised controlled trial of attentional retraining in smokers attempting cessation. This trial could provide proof of principle for a treatment aimed at a fundamental cause of addiction.National Institute for Health Research (NIHR) Doctoral Research Fellowship (DRF) awarded to RB (DRF-2009-02-15