Splitting fields and general differential Galois theory

An algebraic technique is presented that does not use results of model theory and makes it possible to construct a general Galois theory of arbitrary nonlinear systems of partial differential equations. The algebraic technique is based on the search for prime differential ideals of special form in tensor products of differential rings. The main results demonstrating the work of the technique obtained are the theorem on the constructedness of the differential closure and the general theorem on the Galois correspondence for normal extensions..Comment: 33 pages, this version coincides with the published on

Ergodicity criteria for non-expanding transformations of 2-adic spheres

In the paper, we obtain necessary and sufficient conditions for ergodicity (with respect to the normalized Haar measure) of discrete dynamical systems $$ on 2-adic spheres $\mathbf S_{2^{-r}}(a)$ of radius $2^{-r}$, $r\ge 1$, centered at some point $a$ from the ultrametric space of 2-adic integers $\mathbb Z_2$. The map $f\colon\mathbb Z_2\to\mathbb Z_2$ is assumed to be non-expanding and measure-preserving; that is, $f$ satisfies a Lipschitz condition with a constant 1 with respect to the 2-adic metric, and $f$ preserves a natural probability measure on $\mathbb Z_2$, the Haar measure $\mu_2$ on $\mathbb Z_2$ which is normalized so that $\mu_2(\mathbb Z_2)=1$

Analytic curves in algebraic varieties over number fields

We establish algebraicity criteria for formal germs of curves in algebraic varieties over number fields and apply them to derive a rationality criterion for formal germs of functions, which extends the classical rationality theorems of Borel-Dwork and P\'olya-Bertrandias valid over the projective line to arbitrary algebraic curves over a number field. The formulation and the proof of these criteria involve some basic notions in Arakelov geometry, combined with complex and rigid analytic geometry (notably, potential theory over complex and $p$-adic curves). We also discuss geometric analogues, pertaining to the algebraic geometry of projective surfaces, of these arithmetic criteria.Comment: 55 pages. To appear in "Algebra, Arithmetic, and Geometry: In Honor of Y.i. Manin", Y. Tschinkel & Yu. Manin editors, Birkh\"auser, 200

Holonomy of the Ising model form factors

We study the Ising model two-point diagonal correlation function $C(N,N)$ by presenting an exponential and form factor expansion in an integral representation which differs from the known expansion of Wu, McCoy, Tracy and Barouch. We extend this expansion, weighting, by powers of a variable $\lambda$, the $j$-particle contributions, $f^{(j)}_{N,N}$. The corresponding $\lambda$ extension of the two-point diagonal correlation function, $C(N,N; \lambda)$, is shown, for arbitrary $\lambda$, to be a solution of the sigma form of the Painlev{\'e} VI equation introduced by Jimbo and Miwa. Linear differential equations for the form factors $f^{(j)}_{N,N}$ are obtained and shown to have both a ``Russian doll'' nesting, and a decomposition of the differential operators as a direct sum of operators equivalent to symmetric powers of the differential operator of the elliptic integral $E$. Each $f^{(j)}_{N,N}$ is expressed polynomially in terms of the elliptic integrals $E$ and $K$. The scaling limit of these differential operators breaks the direct sum structure but not the ``Russian doll'' structure. The previous $\lambda$-extensions, $C(N,N; \lambda)$ are, for singled-out values $\lambda= \cos(\pi m/n)$ ($m, n$ integers), also solutions of linear differential equations. These solutions of Painlev\'e VI are actually algebraic functions, being associated with modular curves.Comment: 39 page

Fuchs versus Painlev\'e

We briefly recall the Fuchs-Painlev\'e elliptic representation of Painlev\'e VI. We then show that the polynomiality of the expressions of the correlation functions (and form factors) in terms of the complete elliptic integral of the first and second kind, $K$ and $E$, is a straight consequence of the fact that the differential operators corresponding to the entries of Toeplitz-like determinants, are equivalent to the second order operator $L_E$ which has $E$ as solution (or, for off-diagonal correlations to the direct sum of $L_E$ and $d/dt$). We show that this can be generalized, mutatis mutandis, to the anisotropic Ising model. The singled-out second order linear differential operator $L_E$ being replaced by an isomonodromic system of two third-order linear partial differential operators associated with $\Pi_1$, the Jacobi's form of the complete elliptic integral of the third kind (or equivalently two second order linear partial differential operators associated with Appell functions, where one of these operators can be seen as a deformation of $L_E$). We finally explore the generalizations, to the anisotropic Ising models, of the links we made, in two previous papers, between Painlev\'e non-linear ODE's, Fuchsian linear ODE's and elliptic curves. In particular the elliptic representation of Painlev\'e VI has to be generalized to an ``Appellian'' representation of Garnier systems.Comment: Dedicated to the : Special issue on Symmetries and Integrability of Difference Equations, SIDE VII meeting held in Melbourne during July 200

Strong Association of 677 C>T Substitution in the MTHFR Gene with Male Infertility - A Study on an Indian Population and a Meta-Analysis

Methylenetetrahydrofolate reductase (MTHFR) is an important enzyme of folate and methionine metabolism, making it crucial for DNA synthesis and methylation. The objective of this study was to analyze MTHFR gene 677C>T polymorphism in infertile male individuals from North India, followed by a meta-analysis on our data and published studies.We undertook genotyping on a total of 837 individuals including well characterized infertile (N = 522) and confirmed fertile (N = 315) individuals. The SNP was typed by direct DNA sequencing. Chi square test was done for statistical analysis. Published studies were searched using appropriate keywords. Source of data collection for meta-analysis included 'Pubmed', 'Ovid' and 'Google Scholar'. Those studies analyzing 677C>T polymorphism in male infertility and presenting all relevant data were included in meta-analysis. The genotype data for infertile subjects and fertile controls was extracted from each study. Chi square test was done to obtain odds ratio (OR) and p-value. Meta-analysis was performed using Comprehensive Meta-analysis software (Version 2). The frequency of mutant (T) allele (p = 0.0025) and genotypes (CT+TT) (p = 0.0187) was significantly higher in infertile individuals in comparison to fertile controls in our case-control study. The overall summary estimate (OR) for allele and genotype meta-analysis were 1.304 (p = 0.000), 1.310 (p = 0.000), respectively, establishing significant association of 677C>T polymorphism with male infertility.677C>T substitution associated strongly with male infertility in Indian population. Allele and genotype meta-analysis also supported its strong correlation with male infertility, thus establishing it as a risk factor

Intramolecular Cooperative Effects in Multichromophoric Cavitands Exhibiting Nonlinear Optical Properties

We report on the design, synthesis, and characterization of a new class of multichromophoric cavitands based on resorcin[4]arenes. The novel compounds have exhibited high values of second-order nonlinear optical (NLO) properties, as evidenced by electric-field-induced second harmonic generation (EFISHG) measurements. Theoretical calculations indicate the presence of edge-to-face T-shaped interactions between the aromatic building blocks within these multichromophoric systems, which is further supported by the detection of hypsochromic shifts in UV-vis and upfield aromatic chemical shifts in 1H NMR. We proved for the first time that the gain in the quadratic hyperpolarizabilities of multichromophoric NLO macrocycles, originating from the near parallel orientations of the subchromophores, can be partially suppressed if the distance between the dipolar subunits falls into a specific range, where intramolecular cooperative and/or collective effects are operative. Our finding will contribute to the better understanding of the phenomenon of cooperativity in new molecular materials with promising NLO properties. (Figure Presented). © 2015 American Chemical Society

Influence of folate status on genomic DNA methylation in colonic mucosa of subjects without colorectal adenoma or cancer

DNA hypomethylation may increase the risk of colorectal cancer. The main aim of this study was to assess the influence of folate status (serum and erythrocyte folate and plasma homocysteine concentrations) on DNA methylation. Methylenetetrahydrofolate reductase (MTHFR 677C → T and 1298A → C), methionine synthase (MS 2756A → G) and cystathionine synthase (CBS 844ins68) polymorphisms were measured to account for potential confounding effects on folate status and DNA methylation. A total of 68 subjects (33 men and 35 women, 36–78 years) free from colorectal polyps or cancer were recruited in a cross-sectional study. Tissue biopsies were obtained at colonoscopy for the determination of DNA methylation in colonic mucosa using an in vitro radiolabelled methyl acceptance assay. Serum and erythrocyte folate were inversely correlated with plasma homocysteine (r=−0.573, P<0.001 and r=−0.307, P=0.01 respectively) and DNA hypomethylation in colonic mucosa (r=−0.311, P=0.01 and r=−0.356, P=0.03). After adjusting for gender, age, body mass index, smoking and genotype, there were weak negative associations between serum and erythrocyte folate and colonic DNA hypomethylation (P=0.07 and P=0.08, respectively)