Heat and Poisson semigroups for Fourier-Neumann expansions

Given $\alpha > -1$, consider the second order differential operator in $(0,\infty)$, $$L_\alpha f \equiv (x^2 \frac{d^2}{dx^2} + (2\alpha+3)x \frac{d}{dx} + x^2 + (\alpha+1)^2)(f),$$ which appears in the theory of Bessel functions. The purpose of this paper is to develop the corresponding harmonic analysis taking $L_\alpha$ as the analogue to the classical Laplacian. Namely we study the boundedness properties of the heat and Poisson semigroups. These boundedness properties allow us to obtain some convergence results that can be used to solve the Cauchy problem for the corresponding heat and Poisson equations.Comment: 16 page

Singular measures and convolution operators

We show that in the study of certain convolution operators, functions can be replaced by measures without changing the size of the constants appearing in weak type (1,1) inequalities. As an application, we prove that the best constants for the centered Hardy-Littlewood maximal operator associated to parallelotopes do not decrease with the dimension.Comment: 8 page

Invited review: Recursive models in animal breeding: Interpretation, limitations, and extensions

Structural equation models allow causal effects between 2 or more variables to be considered and can postulate unidirectional (recursive models; RM) or bidirectional (simultaneous models) causality between variables. This review evaluated the properties of RM in animal breeding and how to interpret the genetic parameters and the corresponding estimated breeding values. In many cases, RM and mixed multitrait models (MTM) are statistically equivalent, although subject to the assumption of variance-covariance matrices and restrictions imposed for achieving model identification. Inference under RM requires imposing some restrictions on the (co)variance matrix or on the location parameters. The estimates of the variance components and the breeding values can be transformed from RM to MTM, although the biological interpretation differs. In the MTM, the breeding values predict the full influence of the additive genetic effects on the traits and should be used for breeding purposes. In contrast, the RM breeding values express the additive genetic effect while holding the causal traits constant. The differences between the additive genetic effect in RM and MTM can be used to identify the genomic regions that affect the additive genetic variation of traits directly or causally mediated for another trait or traits. Furthermore, we presented some extensions of the RM that are useful for modeling quantitative traits with alternative assumptions. The equivalence of RM and MTM can be used to infer causal effects on sequentially expressed traits by manipulating the residual (co)variance matrix under the MTM. Further, RM can be implemented to analyze causality between traits that might differ among subgroups or within the parametric space of the independent traits. In addition, RM can be expanded to create models that introduce some degree of regularization in the recursive structure that aims to estimate a large number of recursive parameters. Finally, RM can be used in some cases for operational reasons, although there is no causality between traits

Genetic evaluation for subjective traits in the Pirenaica Breed

Ponencia publicada en ITEA, vol.104Los esquemas de selección en las especies ganaderas utilizan una amplia variedad de caracteres. En algunos casos, los registros fenotípicos se obtienen a partir de una valoración subjetiva por parte de evaluadores expertos. Esta valoración implica una clasificación en una escala arbitraria, y, por este motivo, puede diferir considerablemente de la distribución Normal. Por otra parte, cada evaluador puede utilizar criterios de clasificación específicos, y diferentes de los otros evaluadores. En este trabajo se propone un modelo multi-umbral para el análisis de datos procedentes de valoraciones subjetivas. El modelo asume una escala observable diferente para cada evaluador o grupo de evaluadores, y una escala subyacente común. El modelo propuesto se ha aplicado a datos de conformación de la canal de la Raza Bovina Pirenaica procedentes del sistema de valoración SEUROP en 12 mataderos del País Vasco y Navarra.Selection programs in livestock populations made use of a wide variety of traits. Among them, phenotypic records for some traits are obtained by a subjective evaluation from a set of experts, like sensory, type, carcass or fat score traits. Data from subjective evaluation usually involves a classification under an arbitrary predefined scale. The output of this process can lead to strong departures from the Gaussian distribution. Moreover, different criteria can be achieved for each expert. In this study, we propose a Slaughterhouse Specific Ordered Category Threshold Model, that assumes a specific observable scale for each specialist, and a common subjacent scale. The procedure is applied to SEUROP conformation score data from the Pirenaica Beef Cattle Breed evaluated at 12 different slaughterhouses from the Basque Country and Navarre