Tannakian duality for Anderson-Drinfeld motives and algebraic independence of Carlitz logarithms

We develop a theory of Tannakian Galois groups for t-motives and relate this to the theory of Frobenius semilinear difference equations. We show that the transcendence degree of the period matrix associated to a given t-motive is equal to the dimension of its Galois group. Using this result we prove that Carlitz logarithms of algebraic functions that are linearly independent over the rational function field are algebraically independent.Comment: 39 page

A VERSATILE DETECTION SYSTEM FOR A BROAD RANGE MAGNETIC SPECTROGRAPH

The focal plane detector for the QMG/2 magnetic spectrograph of the Kernfysisch Versneller Instituut is described. It consists of two-dimensional position sensitive proportional detectors and scintillation detectors. The properties of the components of the set-up and of the system as a whole in conjunction with the associated electronics and software are presented. The system aUows easy optimization of spectrograph focusing and correction of kinematic effects. The position resolution is about 1 mm for relatively highly ionizing particles (50 MeV c~-particles), and slightly worse for low4onizing particles (50 MeV protons)

Non integrability of a self-gravitating Riemann liquid ellipsoid

We prove that the motion of a triaxial Riemann ellipsoid of homogeneous liquid without angular momentum does not possess an additional first integral which is meromorphic in position, impulsions, and the elliptic functions which appear in the potential, and thus is not integrable. We prove moreover that this system is not integrable even on a fixed energy level hypersurface.Comment: 14 pages, 8 reference

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

Lack of Association between Genetic Polymorphisms in Enzymes Associated with Folate Metabolism and Unexplained Reduced Sperm Counts

BACKGROUND: The metabolic pathway of folate is thought to influence DNA stability either by inducing single/double stranded breaks or by producing low levels of S-adenosyl-methionine leading to abnormal gene expression and chromosome segregation. Polymorphisms in the genes encoding enzymes in the folate metabolism pathway show distinct geographic and/or ethnic variations and in some cases have been linked to disease. Notably, the gene Methylenetetrahydrofolate reductase (MTHFR) in which the homozygous (TT) state of the polymorphism c.665C>T (p.A222V) is associated with reduced specific activity and increased thermolability of the enzyme causing mild hyperhomocysteinemia. Recently several studies have suggested that men carrying this polymorphism may be at increased risk to develop infertility. METHODOLOGY/PRINCIPAL FINDINGS: We have tested this hypothesis in a case/control study of ethnic French individuals. We examined the incidence of polymorphisms in the genes MTHFR (R68Q, A222V and E429A), Methionine synthase reductase MTRR; (I22M and S175L) and Cystathionine beta-synthase (CBS; G307S). The case population consisted of DNA samples from men with unexplained azoospermia (n = 70) or oligozoospermia (n = 182) and the control population consisted of normospermic and fertile men (n = 114). We found no evidence of an association between the incidence of any of these variants and reduced sperm counts. In addition haplotype analysis did not reveal differences between the case and control populations. CONCLUSIONS/SIGNIFICANCE: We could find no evidence for an association between reduced sperm counts and polymorphisms in enzymes involved in folate metabolism in the French population

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

Basic Module Theory over Non-Commutative Rings with Computational Aspects of Operator Algebras

The present text surveys some relevant situations and results where basic Module Theory interacts with computational aspects of operator algebras. We tried to keep a balance between constructive and algebraic aspects.Comment: To appear in the Proceedings of the AADIOS 2012 conference, to be published in Lecture Notes in Computer Scienc