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

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

Consequences of paternally inherited effects on the genetic evaluation of maternal effects

Background: Mixed models are commonly used for the estimation of variance components and genetic evaluation of livestock populations. Some evaluation models include two types of additive genetic effects, direct and maternal. Estimates of variance components obtained with models that account for maternal effects have been the subject of a long-standing controversy about strong negative estimates of the covariance between direct and maternal effects. Genomic imprinting is known to be in some cases statistically confounded with maternal effects. In this study, we analysed the consequences of ignoring paternally inherited effects on the partitioning of genetic variance. Results: We showed that the existence of paternal parent-of-origin effects can bias the estimation of variance components when maternal effects are included in the evaluation model. Specifically, we demonstrated that adding a constraint on the genetic parameters of a maternal model resulted in correlations between relatives that were the same as those obtained with a model that fits only paternally inherited effects for most pairs of individuals, as in livestock pedigrees. The main consequence is an upward bias in the estimates of the direct and maternal additive genetic variances and a downward bias in the direct-maternal genetic covariance. This was confirmed by a simulation study that investigated five scenarios, with the trait affected by (1) only additive genetic effects, (2) only paternally inherited effects, (3) additive genetic and paternally inherited effects, (4) direct and maternal additive genetic effects and (5) direct and maternal additive genetic plus paternally inherited effects. For each scenario, the existence of a paternally inherited effect not accounted for by the estimation model resulted in a partitioning of the genetic variance according to the predicted pattern. In addition, a model comparison test confirmed that direct and maternal additive models and paternally inherited models provided an equivalent fit. Conclusions: Ignoring paternally inherited effects in the maternal models for genetic evaluation can lead to a specific pattern of bias in variance component estimates, which may account for the unexpectedly strong negative direct-maternal genetic correlations that are typically reported in the literature

Effect of inbreeding on the longevity of Landrace sows

Ponencia publicada en ITEA, vol.104La consanguinidad es un fenómeno biológico de especial relevancia en las especies domésticas, pudiéndose caracterizar tanto en términos de coeficiente de consanguinidad como fraccionando la contribución de cada individuo fundador en coeficientes de consanguinidad parcial (CP). A partir de los registros de longevidad de 4.226 cerdas de raza Landrace, este trabajo se ha centrado en la modelización de los CP bajo modelos Weibull de riesgos proporcionales y su posterior comparación mediante el DIC (deviance information criterion). Se asumieron tres distribuciones a priori distintas para los efectos de CP, resultando la normal asimétrica (DIC = 55.064,6) claramente preferible a la normal simétrica (DIC = 55.069,2) y a la distribución uniforme (DIC = 55.077,9). Se descartó, también, el modelo estándar con la consanguinidad global de cada individuo (DIC = 55.078,4). En el caso del modelo con DIC mínimo, la distribución posterior de los efectos de CP fue claramente asimétrica, con el 85,15% de las estimas afectando negativamente a la longevidad de las cerdas y el 14,85% restante con efecto neutro o incluso positivo. Señalar por último, que la heredabilidad para el carácter longevidad fue de 0,159.Inbreeding is a biological phenomenon of special relevance in domestic species, where the overall inbreeding coefficient can be partitioned in founder-specific partial inbreeding (PI) coefficients. Taking longevity data of 4,226 Landrace sows as starting point, this research proposed alternative parameterization for PI effects under Weibull proportional hazard models, and compared their performance through the deviance information criterion (DIC). Three different a priori distributions were assumed for PI effects, asymmetric normal (DIC = 55,064.6), symmetric normal (DIC = 55,069.2) and flat (DIC = 55,077.9). Additionally, the standard model accounting for the overall inbreeding coefficient was clearly discarded (DIC = 55,078.4). For the model with asymmetric Gaussian prior, the posterior distribution of PI effects was clearly skewed. An 85.15% of the estimates showed negative effect on sow longevity whereas the remaining 14.85% ones had null or even positive effect on sow survival. Estimated heritability was 0.159

The Möbius inversion formula for Fourier series applied to Bernoulli and Euler polynomials

AbstractHurwitz found the Fourier expansion of the Bernoulli polynomials over a century ago. In general, Fourier analysis can be fruitfully employed to obtain properties of the Bernoulli polynomials and related functions in a simple manner. In addition, applying the technique of Möbius inversion from analytic number theory to Fourier expansions, we derive identities involving Bernoulli polynomials, Bernoulli numbers, and the Möbius function; this includes formulas for the Bernoulli polynomials at rational arguments. Finally, we show some asymptotic properties concerning the Bernoulli and Euler polynomials

Asymptotic estimates for Apostol-Bernoulli and Apostol-Euler polynomials

We analyze the asymptotic behavior of the Apostol-Bernoulli polynomials $\mathcal{B}_{n}(x;\lambda)$ in detail. The starting point is their Fourier series on $[0,1]$ which, it is shown, remains valid as an asymptotic expansion over compact subsets of the complex plane. This is used to determine explicit estimates on the constants in the approximation, and also to analyze oscillatory phenomena which arise in certain cases. These results are transferred to the Apostol-Euler polynomials $\mathcal{E}_{n}(x;\lambda)$ via a simple relation linking them to the Apostol-Bernoulli polynomials.Comment: 16 page