Multivariate orthogonal polynomials and integrable systems

Multivariate orthogonal polynomials in D real dimensions are considered from the perspective of the Cholesky factorization of a moment matrix. The approach allows for the construction of corresponding multivariate orthogonal polynomials, associated second kind functions, Jacobi type matrices and associated three term relations and also Christoffel-Darboux formulae. The multivariate orthogonal polynomials, their second kind functions and the corresponding Christoffel-Darboux kernels are shown to be quasi-determinants as well as Schur complements of bordered truncations of the moment matrix; quasi-tau functions are introduced. It is proven that the second kind functions are multivariate Cauchy transforms of the multivariate orthogonal polynomials. Discrete and continuous deformations of the measure lead to Toda type integrable hierarchy, being the corresponding flows described through Lax and Zakharov-Shabat equations; bilinear equations are found. Varying size matrix nonlinear partial difference and differential equations of the 2D Toda lattice type are shown to be solved by matrix coefficients of the multivariate orthogonal polynomials. The discrete flows, which are shown to be connected with a Gauss-Borel factorization of the Jacobi type matrices and its quasi-determinants, lead to expressions for the multivariate orthogonal polynomials and their second kind functions in terms of shifted quasi-tau matrices, which generalize to the multidimensional realm, those that relate the Baker and adjoint Baker functions to ratios of Miwa shifted tau-functions in the 1D scenario. In this context, the multivariate extension of the elementary Darboux transformation is given in terms of quasi-determinants of matrices built up by the evaluation, at a poised set of nodes lying in an appropriate hyperplane in R^D, of the multivariate orthogonal polynomials. The multivariate Christoffel formula for the iteration of m elementary Darboux transformations is given as a quasi-determinant. It is shown, using congruences in the space of semi-infinite matrices, that the discrete and continuous flows are intimately connected and determine nonlinear partial difference-differential equations that involve only one site in the integrable lattice behaving as a Kadomstev-Petviashvili type system. Finally, a brief discussion of measures with a particular linear isometry invariance and some of its consequences for the corresponding multivariate polynomials is given. In particular, it is shown that the Toda times that preserve the invariance condition lay in a secant variety of the Veronese variety of the fixed point set of the linear isometry.

E-polynomials of the SL(2, C)-character varieties of surface groups

We compute the E-polynomials of the moduli spaces of representations of the fundamental group of a once-punctured surface of any genus into SL(2, C), for any possible holonomy around the puncture. We follow the geometric technique introduced in [12], based on stratifying the space of representations, and on the analysis of the behavior of the E-polynomial under fibrations.

Limitations of spoligotyping and variable-number tandem-repeat typing for molecular tracing of Mycobacterium bovis in a high-diversity setting

This study describes the attempt to trace the first Mycobacterium bovis outbreak in alpacas (Lama pacos) in Spain by spoligotyping and variable-number tandem-repeat (VNTR) analysis. Due to high genotype diversity, no matching source was identified, but local expansion of a clonal group was found and its significance for molecular tracing is discussed.

Discovery of stable and variable differences in the Mycobacterium avium subsp. paratuberculosis type I, II, and III genomes by pan-genome microarray analysis

Mycobacterium avium subsp. paratuberculosis is an important animal pathogen widely disseminated in the environment that has also been associated with Crohn's disease in humans. Three M. avium subsp. paratuberculosis genomotypes are recognized, but genomic differences have not been fully described. To further investigate these potential differences, a 60-mer oligonucleotide microarray (designated the MAPAC array), based on the combined genomes of M. avium subsp. paratuberculosis (strain K-10) and Mycobacterium avium subsp. hominissuis (strain 104), was designed and validated. By use of a test panel of defined M. avium subsp. paratuberculosis strains, the MAPAC array was able to identify a set of large sequence polymorphisms (LSPs) diagnostic for each of the three major M. avium subsp. paratuberculosis types. M. avium subsp. paratuberculosis type II strains contained a smaller genomic complement than M. avium subsp. paratuberculosis type I and M. avium subsp. paratuberculosis type III genomotypes, which included a set of genomic regions also found in M. avium subsp. hominissuis 104. Specific PCRs for genes within LSPs that differentiated M. avium subsp. paratuberculosis types were devised and shown to accurately screen a panel (n = 78) of M. avium subsp. paratuberculosis strains. Analysis of insertion/deletion region INDEL12 showed deletion events causing a reduction in the complement of mycobacterial cell entry genes in M. avium subsp. paratuberculosis type II strains and significantly altering the coding of a major immunologic protein (MPT64) associated with persistence and granuloma formation. Analysis of MAPAC data also identified signal variations in several genomic regions, termed variable genomic islands (vGIs), suggestive of transient duplication/deletion events. vGIs contained significantly low GC% and were immediately flanked by insertion sequences, integrases, or short inverted repeat sequences. Quantitative PCR demonstrated that variation in vGI signals could be associated with colony growth rate and morphology.