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.

Evaluating the Impact of the Positive Choices Intervention on Substance Use, Psychological, and Care Engagement Outcomes Relevant to Current National HIV Prevention Goals

The HIV epidemic in the United States continues to be a significant public health problem, with approximately 50,000 new infections occurring each year. National public health priorities have shifted in recent years towards targeted HIV prevention efforts among people living with HIV/AIDS (PLWHA) that include: increasing engagement in and retention in care, improving HIV treatment adherence, and increasing screening for and treatment of substance use and psychological difficulties. This study evaluated the efficacy of Positive Choices (PC), a brief, care-based, theory-driven, 3-session counseling intervention for newly HIV-diagnosed men who have sex with men (MSM), in the context of current national HIV prevention priorities. The study involved secondary analysis of data from a preliminary efficacy trial of the PC intervention (n=102). Descriptive statistics examined baseline substance use, psychological characteristics and strategies, and care engagement and HIV-related biological outcomes. Generalized Estimating Equations (GEE) examined longitudinal changes in these variables by study condition. Results indicated that PC improved adherence to HIV treatment, but increased use of illicit drugs, specifically amyl nitrates and other stimulant drugs; additionally, moderation analyses indicated differences in patterns of change over time in viral load by baseline depression status. Implications of the findings and suggestions for future research are discussed.