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.

Psicobiología del estrés prenatal:implicaciones en el eje hipotálamo-hipofisiario-adrenal de los bebés

The womb is the first developmental environment. After developmental psychobiologists started to investigate intrauterine evolution of infant and its long-term impact, they found that prenatal and postnatal development is influenced by mother’s psychological health. Specifically, scientific research evidence indicates that prenatal stress is a possible cause of subsequent psychopathological vulnerability. This vulnerability comes from stress sensitivity and is the basis of many childhood disorders. In the last decade, there are evidences for a fetal origin of stress sensitivity in the context of the fetal programming theory (Entringer et al., 2009, Grant et al., 2009, Gutteling et al., 2004, Huizink et al., 2004, O’Connor et al., 2005). According to fetal programming hypothesis, babies that have been exposed to high levels of prenatal stress would develop elevated HPA axis reactivity and thus increased stress sensitivity in the postnatal period. In the field of animal psychobiology, several studies have shown that prenatal stress could play some role on fetal programming of neurodevelopment and HPA axis (Glover, 2010, Weinstock, 2005, 2008). In human psychobiology, evidences are less clear (Glover, 2010). Although research in this regard has been growing during the last few years, more studies are warranted to investigate the relationship between maternal stress and fetal programming of neurodevelopment and the HPA axis in humans, to confirm the findings which are evident from animal psychobiology...