The Darboux transformation and the complex Toda lattice

It is well known that each solution of the Toda lattice can be represented by a tridiagonal matrix J(t). Under certain restrictions, it is possible to obtain some new solution by using the Darboux transformation of J(t) ¡ CI. Our goal is the extension of this fact, which is known for the real lattice, to high order complex Toda lattices as well as to the bi-infinite Toda lattice. In this latter case, we use the factorization LU for block-tridiagonal matrices

Simultaneous mapping of multiple gene loci with pooled segregants

The analysis of polygenic, phenotypic characteristics such as quantitative traits or inheritable diseases remains an important challenge. It requires reliable scoring of many genetic markers covering the entire genome. The advent of high-throughput sequencing technologies provides a new way to evaluate large numbers of single nucleotide polymorphisms (SNPs) as genetic markers. Combining the technologies with pooling of segregants, as performed in bulked segregant analysis (BSA), should, in principle, allow the simultaneous mapping of multiple genetic loci present throughout the genome. The gene mapping process, applied here, consists of three steps: First, a controlled crossing of parents with and without a trait. Second, selection based on phenotypic screening of the offspring, followed by the mapping of short offspring sequences against the parental reference. The final step aims at detecting genetic markers such as SNPs, insertions and deletions with next generation sequencing (NGS). Markers in close proximity of genomic loci that are associated to the trait have a higher probability to be inherited together. Hence, these markers are very useful for discovering the loci and the genetic mechanism underlying the characteristic of interest. Within this context, NGS produces binomial counts along the genome, i.e., the number of sequenced reads that matches with the SNP of the parental reference strain, which is a proxy for the number of individuals in the offspring that share the SNP with the parent. Genomic loci associated with the trait can thus be discovered by analyzing trends in the counts along the genome. We exploit the link between smoothing splines and generalized mixed models for estimating the underlying structure present in the SNP scatterplots

Dynamics and interpretation of some integrable systems via multiple orthogonal polynomials

High-order non symmetric difference operators with complex coefficients are considered. The correspondence between dynamics of the coefficients of the operator defined by a Lax pair and its resolvent function is established. The method of investigation is based on the analysis of the moments for the operator. The solution of a discrete dynamical system is studied. We give explicit expressions for the resolvent function and, under some conditions, the representation of the vector of functionals, associated with the solution for the integrable systems

Quadratic decomposition of bivariate orthogonal polynomials

We describe the relation between the systems of bivariate orthogonal polynomial associated to a symmetric weight function and associated to some particular Christoffel modifications of the quadratic decomposition of the original weight. We analyze the construction of a symmetric bivariate orthogonal polynomial sequence from a given one, orthogonal to a weight function defined on the first quadrant of the plane. In this description, a sort of Backlund type matrix transformations for the involved three term matrix coefficients plays an important role. Finally, we take as a case study relations between the classical orthogonal polynomials defined on the ball and those on the simplex.publishe

Riemann–Hilbert problem and matrix biorthogonal polynomials

Recently the Riemann-Hilbert problem, with jumps supported on appropriate curves in the complex plane, has been presented for matrix biorthogonal polynomials, in particular non-Abelian Hermite matrix biorthogonal polynomials in the real line, understood as those whose matrix of weights is a solution of a Sylvester type Pearson equation with coe cients first order matrix polynomials. We will explore this discussion, present some achievements and consider some new examples of weights for matrix biorthogonal polynomials.publishe