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.

Safety, tolerability, and mechanisms of antiretroviral activity of pegylated interferon Alfa-2a in HIV-1-monoinfected participants: a phase II clinical trial.

BACKGROUND: To our knowledge, the antiviral activity of pegylated interferon alfa-2a has not been studied in participants with untreated human immunodeficiency virus type 1 (HIV-1) infection but without chronic hepatitis C virus (HCV) infection. METHODS: Untreated HIV-1-infected volunteers without HCV infection received 180 microg of pegylated interferon alfa-2a weekly for 12 weeks. Changes in plasma HIV-1 RNA load, CD4(+) T cell counts, pharmacokinetics, pharmacodynamic measurements of 2',5'-oligoadenylate synthetase (OAS) activity, and induction levels of interferon-inducible genes (IFIGs) were measured. Nonparametric statistical analysis was performed. RESULTS: Eleven participants completed 12 weeks of therapy. The median plasma viral load decrease and change in CD4(+) T cell counts at week 12 were 0.61 log(10) copies/mL (90% confidence interval [CI], 0.20-1.18 log(10) copies/mL) and -44 cells/microL (90% CI, -95 to 85 cells/microL), respectively. There was no correlation between plasma viral load decreases and concurrent pegylated interferon plasma concentrations. However, participants with larger increases in OAS level exhibited greater decreases in plasma viral load at weeks 1 and 2 (r = -0.75 [90% CI, -0.93 to -0.28] and r = -0.61 [90% CI, -0.87 to -0.09], respectively; estimated Spearman rank correlation). Participants with higher baseline IFIG levels had smaller week 12 decreases in plasma viral load (0.66 log(10) copies/mL [90% CI, 0.06-0.91 log(10) copies/mL]), whereas those with larger IFIG induction levels exhibited larger decreases in plasma viral load (-0.74 log(10) copies/mL [90% CI, -0.93 to -0.21 log(10) copies/mL]). CONCLUSION: Pegylated interferon alfa-2a was well tolerated and exhibited statistically significant anti-HIV-1 activity in HIV-1-monoinfected patients. The anti-HIV-1 effect correlated with OAS protein levels (weeks 1 and 2) and IFIG induction levels (week 12) but not with pegylated interferon concentrations.