A generic formula for the values at the boundary points of monic classical orthogonal polynomials

In a previous paper we have determined a generic formula for the polynomial solution families of the well-known differential equation of hypergeometric type σ(x)y"n(x)+τ(x)y'n(x)-λnyn(x)=0. In this paper, we give another such formula which enables us to present a generic formula for the values of monic classical orthogonal polynomials at their boundary points of definition.

Preformal proving: examples and reflections

The starting point of our reflections is a classroom situation in grade 12 in which it was to be proved intuitively that non-trivial solutions of the differential equation f' = f have no zeros. We give a working definition of the concept of preformal proving, as well as three examples of preformal proofs. Then we furnish several such proofs of the aforesaid fact, and we analyse these proofs in detail. Finally, we draw some conclusions for mathematics in school and in teacher training.

Certified Rapid Solution of Parametrized Linear Elliptic Equations: Application to Parameter Estimation

We present a technique for the rapid and reliable evaluation of linear-functional output of elliptic partial differential equations with affine parameter dependence. The essential components are (i) rapidly uniformly convergent reduced-basis approximations — Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N (optimally) selected points in parameter space; (ii) a posteriori error estimation — relaxations of the residual equation that provide inexpensive yet sharp and rigorous bounds for the error in the outputs; and (iii) offline/online computational procedures — stratagems that exploit affine parameter dependence to de-couple the generation and projection stages of the approximation process. The operation count for the online stage — in which, given a new parameter value, we calculate the output and associated error bound — depends only on N (typically small) and the parametric complexity of the problem. The method is thus ideally suited to the many-query and real-time contexts. In this paper, based on the technique we develop a robust inverse computational method for very fast solution of inverse problems characterized by parametrized partial differential equations. The essential ideas are in three-fold: first, we apply the technique to the forward problem for the rapid certified evaluation of PDE input-output relations and associated rigorous error bounds; second, we incorporate the reduced-basis approximation and error bounds into the inverse problem formulation; and third, rather than regularize the goodness-of-fit objective, we may instead identify all (or almost all, in the probabilistic sense) system configurations consistent with the available experimental data — well-posedness is reflected in a bounded "possibility region" that furthermore shrinks as the experimental error is decreased.

Efectos en las ventas del sector automotor en Colombia frente a la aprobación del tratado de libre comercio con Estados Unidos

Esta investigación se realizó con el objetivo de conocer los efectos sobre las ventas del sector automotor en Colombia frente a la aprobación y puesta en marcha de Tratado de Libre Comercio con Estados Unidos, permitiendo brindar un panorama al sector y unas recomendaciones que le permitieran ser competitivo y no afectar considerablemente su nivel de ventas en general; Para ello, se analizaron únicamente los vehículos automotores para el transporte de pasajeros, que serán sin lugar a dudas los de mayor movimiento durante el TLC. Para ello, se contó principalmente con el diamante de competitividad propuesto por el catedrático Michael Porter para observar aquellos factores críticos dentro del sector que afectaban su competitividad y nivel de ventas, permitiendo así mismo hacer una comparación frente al sector automotor de Estados Unidos y conocer aquellos factores a trabajar por parte del sector en Colombia;, Contando con información ofrecida por parte de las autoridades del sector y asesores comerciales de las principales ensambladoras del País. Luego de ello, y con el ánimo de brindar recomendaciones al sector, tomando en cuenta los factores a mejorar, se hizo uso de la prospectiva estratégica y la herramienta MICMAC (Matriz de impacto cruzado) con el fin de conocer los factores clave que influirían tanto en el corto, mediano y largo plazo en el sector automotor de Colombia y en los cuales se debería trabajar para buscar efectos positivos en el sector y sus ventas; Palabras Clave: Vehículos, Competitividad, Ventas, Prospectiva Estratégica, TLC, Economía, Infraestructura, Tecnología, Inversión

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