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.

Genetic variants in IGF-I, IGF-II, IGFBP-3, and adiponectin genes and colon cancer risk in African Americans and Whites.

PURPOSE: Evaluating genetic susceptibility may clarify effects of known environmental factors and also identify individuals at high risk. We evaluated the association of four insulin-related pathway gene polymorphisms in insulin-like growth factor-1 (IGF-I) (CA)( n ) repeat, insulin-like growth factor-2 (IGF-II) (rs680), insulin-like growth factor-binding protein-3 (IGFBP-3) (rs2854744), and adiponectin (APM1 rs1501299) with colon cancer risk, as well as relationships with circulating IGF-I, IGF-II, IGFBP-3, and C-peptide in a population-based study. METHODS: Participants were African Americans (231 cases and 306 controls) and Whites (297 cases, 530 controls). Consenting subjects provided blood specimens and lifestyle/diet information. Genotyping for all genes except IGF-I was performed by the 5'-exonuclease (Taqman) assay. The IGF-I (CA)(n) repeat was assayed by PCR and fragment analysis. Circulating proteins were measured by enzyme immunoassays. Odds ratios (ORs) and 95 % confidence intervals (CIs) were calculated by logistic regression. RESULTS: The IGF-I (CA)( 19 ) repeat was higher in White controls (50 %) than African American controls (31 %). Whites homozygous for the IGF-I (CA)(19) repeat had a nearly twofold increase in risk of colon cancer (OR = 1.77; 95 % CI = 1.15-2.73), but not African Americans (OR = 0.73, 95 % CI 0.50-1.51). We observed an inverse association between the IGF-II Apa1 A-variant and colon cancer risk (OR = 0.49, 95 % CI 0.28-0.88) in Whites only. Carrying the IGFBP-3 variant alleles was associated with lower IGFBP-3 protein levels, a difference most pronounced in Whites (p-trend <0.05). CONCLUSIONS: These results support an association between insulin pathway-related genes and elevated colon cancer risk in Whites but not in African Americans.