Bivariate second--order linear partial differential equations and orthogonal polynomial solutions

In this paper we construct the main algebraic and differential properties and the weight functions of orthogonal polynomial solutions of bivariate second--order linear partial differential equations, which are admissible potentially self--adjoint and of hypergeometric type. General formulae for all these properties are obtained explicitly in terms of the polynomial coefficients of the partial differential equation, using vector matrix notation. Moreover, Rodrigues representations for the polynomial eigensolutions and for their partial derivatives of any order are given. Finally, as illustration, these results are applied to specific Appell and Koornwinder orthogonal polynomials, solutions of the same partial differential equation.Comment: 27 page

Fourth-order differential equations satisfied by the generalized co-recursive of all classical orthogonal polynomials. A study of their distribution of zeros

AbstractThe unique fourth-order differential equation satisfied by the generalized co-recursive of all classical orthogonal polynomials is given for any (but fixed) level of recursivity. Up to now, these differential equations were known only for each classical family separately and also for a specific recursivity level. Moreover, we use this unique fourth-order differential equation in order to study the distribution of zeros of these polynomials via their Newton sum rules (i.e., the sums of powers of their zeros) which are closely related with the moments of such distribution. Both results are obtained with the help of two programs built in Mathematica symbolic language

Comparación de modelos mensuales y anuales para estimar el coeficiente de Hargreaves en la Comunidad Valenciana

El modelo de Hargreaves (HG) para estimar evapotranspiración de referencia (ETo) es una alternativa interesante a la ecuación de Penman Monteith (PM), propuesta como método estándar por la FAO, pero que no es aplicable en muchas situaciones porque requiere muchas variables climáticas que no suelen estar disponibles o cuyos valores medidos no son fiables. Para la aplicación de esta ecuación se recomienda una calibración local preliminar del llamado coeficiente de Hargreaves (AHC). Sin embargo, la obtención de valores concretos de AHC tiene un uso limitado, dado que se requieren valores locales de PM, que se aplicará realmente para determinar la ETo de la estación. Asimismo los valores de AHC no pueden extrapolarse. Por ello es preciso proponer y estudiar modelos para estimar el AHC, dado que en las condiciones en las que la ecuación de HG pretende ser útil no habrá posibilidad de calibración previa mediante valores de PM. Este estudio analiza si están justificadas las parametrizaciones mensuales o estacionales del AHC. Para ello se compararon tres escalas temporales en el desarrollo de los modelos de AHC: la anual, la mensual y la estacional. Los resultados sugieren que el desarrollo de modelos mensuales podría reducir el error relativo alrededor del 2% para AHC diarios y alrededor de 1% para AHC medios mensuales, es decir, un único modelo anual de AHC podría ser insuficiente para recoger toda la variabilidad anual de AHC. Por ello, la aplicación de modelos mensuales (o estacionales) podría estar justificado para una correcta estimación de los AHC. Asimismo, los resultados muestran que la estimación mensual del AHC fue más ajustada de mayo a septiembre que de octubre a abril, y, particularmente, que de noviembre a enero

Minimal recurrence relations for connection coefficients between classical orthogonal polynomials: Discrete case

AbstractWe present a simple approach in order to compute recursively the connection coefficients between two families of classical (discrete) orthogonal polynomials (Charlier, Meixner, Kravchuk, Hahn), i.e., the coefficients Cm(n) in the expression Pn(X)=∑nm=0Cm(n)Qm(x), where Pn(x) and Qm(x) belong to the aforementioned class of polynomials. This is SCV2 done by adapting a general and systematic algorithm, recently developed by the authors, to the discrete classical situation. Moreover, extensions of this method allow to give new addition formulae and to estimate Cm(n)-asymptotics in limit relations between some families

Connection problems for polynomial solutions of nonhomogeneous differential and difference equations

AbstractWe consider nonhomogeneous hypergeometric-type differential, difference and q-difference equations whose nonhomogeneity is a polynomial qn(x). The polynomial solution of these problems is expanded in the ∗ Qn(x)∗ basis, and also in a basis ∗Pn(x)∗, related in a natural way with the homogeneous hypergeometric equation. We give an algorithm building a recurrence relation for the expansion coefficients in both bases that we solve explicitly in many cases involving classical orthogonal polynomials. Finally, some concrete applications and extensions are given

Comparative Analysis of Some Modal Reconstruction Methods of the Shape of the Cornea from Corneal Elevation Data

Purpose: A comparative study of the ability of some modal schemes to reproduce corneal shapes of varying complexity was performed, by using both standard radial polynomials and radial basis functions (RBFs). The hypothesis was that the correct approach in the case of highly irregular corneas should combine several bases. Methods: Standard approaches of reconstruction by Zernike and other types of radial polynomials were compared with the discrete least-squares fit (LSF) by the RBF in three theoretical surfaces, synthetically generated by computer algorithms in the absence of measurement noise. For the reconstruction by polynomials, the maximal radial order 6 was chosen, which corresponds to the first 28 Zernike polynomials or the first 49 Bhatia-Wolf polynomials. The fit with the RBF was performed by using a regular grid of centers. Results: The quality of fit was assessed by computing for each surface the mean square errors (MSEs) of the reconstruction by LSF, measured at the same nodes where the heights were collected. Another criterion of the fit quality used was the accuracy in recovery of the Zernike coefficients, especially in the case of incomplete data. Conclusions: The Zernike (and especially, the Bhatia-Wolf) polynomials constitute a reliable reconstruction method of a nonseverely aberrated surface with a small surface regularity index (SRI). However, they fail to capture small deformations of the anterior surface of a synthetic cornea. The most promising approach is a combined one that balances the robustness of the Zernike fit with the localization of the RBF

Categorical Dimensions of Human Odor Descriptor Space Revealed by Non-Negative Matrix Factorization

In contrast to most other sensory modalities, the basic perceptual dimensions of olfaction remain unclear. Here, we use non-negative matrix factorization (NMF) – a dimensionality reduction technique – to uncover structure in a panel of odor profiles, with each odor defined as a point in multi-dimensional descriptor space. The properties of NMF are favorable for the analysis of such lexical and perceptual data, and lead to a high-dimensional account of odor space. We further provide evidence that odor dimensions apply categorically. That is, odor space is not occupied homogenously, but rather in a discrete and intrinsically clustered manner. We discuss the potential implications of these results for the neural coding of odors, as well as for developing classifiers on larger datasets that may be useful for predicting perceptual qualities from chemical structures

Positional errors in species distribution modelling are not overcome by the coarser grains of analysis

The performance of species distribution models (SDMs) is known to be affected by analysis grain and positional error of species occurrences. Coarsening of the analysis grain has been suggested to compensate for positional errors. Nevertheless, this way of dealing with positional errors has never been thoroughly tested. With increasing use of fine-scale environmental data in SDMs, it is important to test this assumption. Models using fine-scale environmental data are more likely to be negatively affected by positional error as the inaccurate occurrences might easier end up in unsuitable environment. This can result in inappropriate conservation actions. Here, we examined the trade-offs between positional error and analysis grain and provide recommendations for best practice. We generated narrow niche virtual species using environmental variables derived from LiDAR point clouds at 5 x 5 m fine-scale. We simulated the positional error in the range of 5 m to 99 m and evaluated the effects of several spatial grains in the range of 5 m to 500 m. In total, we assessed 49 combinations of positional accuracy and analysis grain. We used three modelling techniques (MaxEnt, BRT and GLM) and evaluated their discrimination ability, niche overlap with virtual species and change in realized niche. We found that model performance decreased with increasing positional error in species occurrences and coarsening of the analysis grain. Most importantly, we showed that coarsening the analysis grain to compensate for positional error did not improve model performance. Our results reject coarsening of the analysis grain as a solution to address the negative effects of positional error on model performance. We recommend fitting models with the finest possible analysis grain and as close to the response grain as possible even when available species occurrences suffer from positional errors. If there are significant positional errors in species occurrences, users are unlikely to benefit from making additional efforts to obtain higher resolution environmental data unless they also minimize the positional errors of species occurrences. Our findings are also applicable to coarse analysis grain, especially for fragmented habitats, and for species with narrow niche breadth

Distribution models of the Spanish argus and its food plant, the storksbill, suggest resilience to climate change

Distribution models of the Spanish argus and its food plant, the storksbill, suggest resilience to climate change. Climate change is an important risk factor for the survival of butterflies and other species. In this study, we developed predictive models that show the potentially favourable areas for a lepidopteran endemic to the Iberian Peninsula, the Spanish argus (Aricia morronensis), and its larval food plants, the storksbill (genus Erodium). We used species distribution modelling software (MaxEnt) to perform the models in the present and in the future in two climatic scenarios based on climatic and topographic variables. The results show that climate change will not significantly affect A. morronensis distribution, and may even slightly favour its expansion. Some plants may undergo a small reduction in habitat favourability. However, it seems that the interaction between this butterfly and its food plants is unlikely to be significantly affected by climate changeLos modelos de distribución de la morena española y las plantas nutricias de sus larvas sugieren resistencia frente al cambio climático. El cambio climático representa un importante factor de riesgo para la supervivencia de las mariposas y de otras especies. En este estudio se han elaborado modelos predictivos que muestran las zonas potencialmente favorables para un lepidóptero endémico de la península ibérica, la morena española (Aricia morronensis), y las plantas nutricias de sus larvas, los alfilerillos o agujas de pastor (género Erodium). Se ha utilizado el programa informático MaxEnt para elaborar modelos de la distribución de las especies en el presente y en el futuro, bajo dos escenarios de condiciones climáticas, basadas en variables climáticas y topográficas. Los resultados muestran que el cambio climático no afectará significativamente a la distribución de A. morronensis, sino que incluso podría favorecer levemente su expansión. Algunas de las plantas podrían sufrir una pequeña reducción de la favorabilidad del hábitat. Sin embargo, la interacción entre la mariposa y sus plantas nutricias probablemente no se vea afectada significativamente por el cambio climátic