On Expansion of a Solution of General Non-autonomous Polynomial Differential Equation

We give a recursive formula for an expansion of a solution of a general non-autonomous polynomial differential equation. The formula is given on the algebraic level with a use of shuffle product. This approach minimizes the number of integrations on each order of expansion. Using combinatorics of trees we estimate the radius of convergence of the expansion.Comment: 13 pages, To be published in Journal of Dynamical and Control System

On the tensor convolution and the quantum separability problem

We consider the problem of separability: decide whether a Hermitian operator on a finite dimensional Hilbert tensor product is separable or entangled. We show that the tensor convolution defined for certain mappings on an almost arbitrary locally compact abelian group, give rise to formulation of an equivalent problem to the separability one.Comment: 13 pages, two sections adde

A comprehensive analysis of common IGF1, IGFBP1 and IGFBP3 genetic variation with prospective IGF-I and IGFBP-3 blood levels and prostate cancer risk among Caucasians†

The insulin-like growth factor (IGF) pathway has been implicated in prostate development and carcinogenesis. We conducted a comprehensive analysis, utilizing a resequencing and tagging single-nucleotide polymorphism (SNP) approach, between common genetic variation in the IGF1, IGF binding protein (BP) 1, and IGFBP3 genes with IGF-I and IGFBP-3 blood levels, and prostate cancer (PCa) risk, among Caucasians in the NCI Breast and Prostate Cancer Cohort Consortium. We genotyped 14 IGF1 SNPs and 16 IGFBP1/IGFBP3 SNPs to capture common [minor allele frequency (MAF) ≥ 5%] variation among Caucasians. For each SNP, we assessed the geometric mean difference in IGF blood levels (N = 5684) across genotypes and the association with PCa risk (6012 PCa cases/6641 controls). We present two-sided statistical tests and correct for multiple comparisons. A non-synonymous IGFBP3 SNP in exon 1, rs2854746 (Gly32Ala), was associated with IGFBP-3 blood levels (Padj = 8.8 × 10−43) after adjusting for the previously established IGFBP3 promoter polymorphism A-202C (rs2854744); IGFBP-3 blood levels were 6.3% higher for each minor allele. For IGF1 SNP rs4764695, the risk estimates among heterozygotes was 1.01 (99% CI: 0.90–1.14) and 1.20 (99% CI: 1.06–1.37) for variant homozygotes with overall PCa risk. The corrected allelic P-value was 8.7 × 10−3. IGF-I levels were significantly associated with PCa risk (Ptrend = 0.02) with a 21% increase of PCa risk when compared with the highest quartile to the lowest quartile. We have identified SNPs significantly associated with IGFBP-3 blood levels, but none of these alter PCa risk; however, a novel IGF1 SNP, not associated with IGF-I blood levels, shows preliminary evidence for association with PCa risk among Caucasians