A mathematical theory of stochastic microlensing. II. Random images, shear, and the Kac-Rice formula

Continuing our development of a mathematical theory of stochastic microlensing, we study the random shear and expected number of random lensed images of different types. In particular, we characterize the first three leading terms in the asymptotic expression of the joint probability density function (pdf) of the random shear tensor due to point masses in the limit of an infinite number of stars. Up to this order, the pdf depends on the magnitude of the shear tensor, the optical depth, and the mean number of stars through a combination of radial position and the star's mass. As a consequence, the pdf's of the shear components are seen to converge, in the limit of an infinite number of stars, to shifted Cauchy distributions, which shows that the shear components have heavy tails in that limit. The asymptotic pdf of the shear magnitude in the limit of an infinite number of stars is also presented. All the results on the random microlensing shear are given for a general point in the lens plane. Extending to the general random distributions (not necessarily uniform) of the lenses, we employ the Kac-Rice formula and Morse theory to deduce general formulas for the expected total number of images and the expected number of saddle images. We further generalize these results by considering random sources defined on a countable compact covering of the light source plane. This is done to introduce the notion of global expected number of positive parity images due to a general lensing map. Applying the result to microlensing, we calculate the asymptotic global expected number of minimum images in the limit of an infinite number of stars, where the stars are uniformly distributed. This global expectation is bounded, while the global expected number of images and the global expected number of saddle images diverge as the order of the number of stars. © 2009 American Institute of Physics.

The normative and the personal life: individual differences in life scripts and life story events among USA and Danish undergraduates.

Life scripts are culturally shared expectations about the order and timing of life events in a prototypical life course. American and Danish undergraduates produced life story events and life scripts by listing the seven most important events in their own lives and in the lives of hypothetical people living ordinary lives. They also rated their events on several scales and completed measures of depression, PTSD symptoms, and centrality of a negative event to their lives. The Danish life script replicated earlier work; the American life script showed minor differences from the Danish life script, apparently reflecting genuine differences in shared events as well as less homogeneity in the American sample. Both consisted of mostly positive events that came disproportionately from ages 15 to 30. Valence of life story events correlated with life script valence, depression, PTSD symptoms, and identity. In the Danish undergraduates, measures of life story deviation from the life script correlated with measures of depression and PTSD symptoms.

ENIGMA and the individual: Predicting factors that affect the brain in 35 countries worldwide.

In this review, we discuss recent work by the ENIGMA Consortium (http://enigma.ini.usc.edu) - a global alliance of over 500 scientists spread across 200 institutions in 35 countries collectively analyzing brain imaging, clinical, and genetic data. Initially formed to detect genetic influences on brain measures, ENIGMA has grown to over 30 working groups studying 12 major brain diseases by pooling and comparing brain data. In some of the largest neuroimaging studies to date - of schizophrenia and major depression - ENIGMA has found replicable disease effects on the brain that are consistent worldwide, as well as factors that modulate disease effects. In partnership with other consortia including ADNI, CHARGE, IMAGEN and others(1), ENIGMA's genomic screens - now numbering over 30,000 MRI scans - have revealed at least 8 genetic loci that affect brain volumes. Downstream of gene findings, ENIGMA has revealed how these individual variants - and genetic variants in general - may affect both the brain and risk for a range of diseases. The ENIGMA consortium is discovering factors that consistently affect brain structure and function that will serve as future predictors linking individual brain scans and genomic data. It is generating vast pools of normative data on brain measures - from tens of thousands of people - that may help detect deviations from normal development or aging in specific groups of subjects. We discuss challenges and opportunities in applying these predictors to individual subjects and new cohorts, as well as lessons we have learned in ENIGMA's efforts so far.