Querschnittstheorie

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Asymptotics of skew orthogonal polynomials

Exact integral expressions of the skew orthogonal polynomials involved in Orthogonal (beta=1) and Symplectic (beta=4) random matrix ensembles are obtained: the (even rank) skew orthogonal polynomials are average characteristic polynomials of random matrices. From there, asymptotics of the skew orthogonal polynomials are derived.Comment: 17 pages, Late

Semi-classical Laguerre polynomials and a third order discrete integrable equation

A semi-discrete Lax pair formed from the differential system and recurrence relation for semi-classical orthogonal polynomials, leads to a discrete integrable equation for a specific semi-classical orthogonal polynomial weight. The main example we use is a semi-classical Laguerre weight to derive a third order difference equation with a corresponding Lax pair.Comment: 11 page

Piecewise Linear Models for the Quasiperiodic Transition to Chaos

We formulate and study analytically and computationally two families of piecewise linear degree one circle maps. These families offer the rare advantage of being non-trivial but essentially solvable models for the phenomenon of mode-locking and the quasi-periodic transition to chaos. For instance, for these families, we obtain complete solutions to several questions still largely unanswered for families of smooth circle maps. Our main results describe (1) the sets of maps in these families having some prescribed rotation interval; (2) the boundaries between zero and positive topological entropy and between zero length and non-zero length rotation interval; and (3) the structure and bifurcations of the attractors in one of these families. We discuss the interpretation of these maps as low-order spline approximations to the classic ``sine-circle'' map and examine more generally the implications of our results for the case of smooth circle maps. We also mention a possible connection to recent experiments on models of a driven Josephson junction.Comment: 75 pages, plain TeX, 47 figures (available on request

Sex differences in nucleus accumbens transcriptome profiles associated with susceptibility versus resilience to subchronic variable stress

Depression and anxiety disorders are more prevalent in females, but the majority of research in animal models, the first step in finding new treatments, has focused predominantly on males. Here we report that exposure to subchronic variable stress (SCVS) induces depression-associated behaviors in female mice, whereas males are resilient as they do not develop these behavioral abnormalities. In concert with these different behavioral responses, transcriptional analysis of nucleus accumbens (NAc), a major brain reward region, by use of RNA sequencing (RNA-seq) revealed markedly different patterns of stress regulation of gene expression between the sexes. Among the genes displaying sex differences was DNA methyltransferase 3a (Dnmt3a), which shows a greater induction in females after SCVS. Interestingly, Dnmt3a expression levels were increased in the NAc of depressed humans, an effect seen in both males and females. Local overexpression of Dnmt3a in NAc rendered male mice more susceptible to SCVS, whereas Dnmt3a knock-out in this region rendered females more resilient, directly implicating this gene in stress responses. Associated with this enhanced resilience of female mice upon NAc knock-out of Dnmt3a was a partial shift of the NAc female transcriptome toward the male pattern after SCVS. These data indicate that males and females undergo different patterns of transcriptional regulation in response to stress and that a DNA methyltransferase in NAc contributes to sex differences in stress vulnerability

The impact of Stieltjes' work on continued fractions and orthogonal polynomials

Stieltjes' work on continued fractions and the orthogonal polynomials related to continued fraction expansions is summarized and an attempt is made to describe the influence of Stieltjes' ideas and work in research done after his death, with an emphasis on the theory of orthogonal polynomials

Matrix biorthogonal polynomials on the real line: Geronimus transformations

In this paper, Geronimus transformations for matrix orthogonal polynomials in the real line are studied. The orthogonality is understood in a broad sense, and is given in terms of a nondegenerate continuous sesquilinear form, which in turn is determined by a quasi-definite matrix of bivariate generalized functions with a well-defined support. The discussion of the orthogonality for such a sesquilinear form includes, among others, matrix Hankel cases with linear functionals, general matrix Sobolev orthogonality and discrete orthogonal polynomials with an infinite support. The results are mainly concerned with the derivation of Christoffel-type formulas, which allow to express the perturbed matrix biorthogonal polynomials and its norms in terms of the original ones. The basic tool is the Gauss-Borel factorization of the Gram matrix, and particular attention is paid to the non-associative character, in general, of the product of semi-infinite matrices. The Geronimus transformation in which a right multiplication by the inverse of a matrix polynomial and an addition of adequate masses are performed, is considered. The resolvent matrix and connection formulas are given. Two different methods are developed. A spectral one, based on the spectral properties of the perturbing polynomial, and constructed in terms of the second kind functions. This approach requires the perturbing matrix polynomial to have a nonsingular leading term. Then, using spectral techniques and spectral jets, Christoffel-Geronimus formulas for the transformed polynomials and norms are presented. For this type of transformations, the paper also proposes an alternative method, which does not require of spectral techniques, that is valid also for singular leading coefficients. When the leading term is nonsingular, a comparison of between both methods is presented. The nonspectral method is applied to unimodular Christoffel perturbations, and a simple example for a degree one massless Geronimus perturbation is given