THE PLEASURE OF SHARING KNOWLEDGESome Motivating, Suggestive and Original Courses Mathematically, Statistically and Engineering RelatedProfesor Enrique Castillo RonBorn in Santiago de Compostela (Spain) 17/10/1946

Course 1. Algebra I (English)The main originality of this course consists in that all discussed problems of linear algebra are solved using a single algorithm, that gives the orthogonal subspace of a linear subspace and its complementary subspace. This permits analyzing all problems from the orthogonality point of view, which is very reach. For example, the problem of determining whether or not a vector belongs to a subspace or the intersection of two subspaces are solved by looking to them as orthogonalization problems. The algorithm permits inverting a matrix, calculating its determinant or determining its rank very easily. In addition the problems of updating inverses and determinants when changing a row reduce to a single step of the algorithm. The compatibility of systems of equations and the obtention of all its solutions or detecting infeasibility are also a direct application of the algorithm. In addition, all subsystems of a given linear system can be solved without extra calculations. Finally, some examples of illustrative applications are given.Download

Course 2. Algebra I (Spanish)The main originality of this course consists in that all discussed problems of linear algebra are solved using a single algorithm, that gives the orthogonal subspace of a linear subspace and its complementary subspace. This permits analyzing all problems from the orthogonality point of view, which is very reach. For example, the problem of determining whether or not a vector belongs to a subspace or the intersection of two subspaces are solved by looking to them as orthogonalization problems. The algorithm permits inverting a matrix, calculating its determinant or determining its rank very easily. In addition the problems of updating inverses and determinants when changing a row reduce to a single step of the algorithm. The compatibility of systems of equations and the obtention of all its solutions or detecting infeasibility are also a direct application of the algorithm. In addition, all subsystems of a given linear system can be solved without extra calculations. Finally, some examples of illustrative applications are given.Download 
Course 3. Algebra II (English)We first introduce the concepts of polyhedral cone and polytope. Next, similarly to the algebra course I, an algorithm, which obtains the dual cone of a given cone and all its facets of any dimension, allows us to solve all discussed problems of linear algebra using the duality concept as a new point of view. This duality point of view allow us to solve the cone membership of a vector and the intersection of cones in a very simple way. In addition, the algorithm provides the dual cone in its simplest form, that is, as a linear space with its basis plus an acute cone with its edges. The compatibility of a linear system of inequalities is discussed and all the solutions of linear systems of inequalities are obtained. In addition, the solutions of all subsystems including all equations from the first to any of them are obtained at once. The concept of cone associated with a polytope permits us to obtain all vertices and all facets of any dimension of a polytope. The set of all feasible solutions of a linear programming problem and all its optimal solutions are obtained, and infeasibility is detected by the algorithm. Finally an example of a water supply problem is presented to show the importance of these methods in engineering design.Download 
Course 4. Algebra II (Spanish)We first introduce the concepts of polyhedral cone and polytope. Next, similarly to the algebra course I, an algorithm, which obtains the dual cone of a given cone and all its facets of any dimension, allows us to solve all discussed problems of linear algebra using the duality concept as a new point of view. This duality point of view allow us to solve the cone membership of a vector and the intersection of cones in a very simple way. In addition, the algorithm provides the dual cone in its simplest form, that is, as a linear space with its basis plus an acute cone with its edges. The compatibility of a linear system of inequalities is discussed and all the solutions of linear systems of inequalities are obtained. In addition, the solutions of all subsystems including all equations from the first to any of them are obtained at once. The concept of cone associated with a polytope permits us to obtain all vertices and all facets of any dimension of a polytope. The set of all feasible solutions of a linear programming problem and all its optimal solutions are obtained, and infeasibility is detected by the algorithm. Finally an example of a water supply problem is presented to show the importance of these methods in engineering design.Download 
Course 5. Extreme Values (English).In this course we start by introducing order statistics and extremes. In particular, the density functions of any order statistic or any subset of them are given in closed form. Next, we indicate that no matter the parent distribution is, the limit distribution of the maximum belongs to the LevyJenkinson family, which includes only the families of Weibull, Gumbel and Frechet. The corresponding limit distributions for the case of minimum are given too. The cases of other order statistics are also analyzed and discussed. The use of probability papers to find the most adequate limit distribution is explained and justified. Exceedances are recommended instead of extremes, because of its better efficiency and the generalized Pareto distribution is analyzed. The case of dependence is shown to lead to the same limits distributions if the dependence is not too strong, and to other limit distribution if it is strong enough. One illustrative example of extreme waves is studied in some detail.Download 
Course 6. Bayesian methods. OpenBUGS (English).This course is dedicated to describe in detail what Bayesian methods are, including prior, posterior and predictive distributions, and the use of Markov Chain Monte Carlo (MCMC) methods using OpenBUGS. We start with, a short description, with some examples, including the concept of conjugate distributions. We explain how these methods permit to convert a deterministic model into a random model by converting its parameters into random variables, as Bayesian method do, or to improve already random models into more complex ones using the same conversion. The OpenBUGS software is described and used to solve several classical methods in Statistics and Probability. We explain how the use of scripts can facilitate and speed up the simulation process. Apart form these examples, we include some fatigue models examples that show how we can even obtain the percentiles of percentile curves. This permit treating the randomness due to parameters.Download 
Course 7. Mathematical model building (English).This course discusses how to build mathematical models to reproduce the physical and engineering reality. First, the Buckingham theorem is given together with some examples to illustrate the importance of using dimensionless ratios or variables, in order to reduce complexity and to avoid dimensional problems. We also discuss the concepts of complete and incomplete selfsimilarity, which plays a relevant role in model building. The concept of consistency of models is introduced and described. It includes dimension, physical, statistical and extreme value consistencies, which are analyzed in detail. For the sake of illustration, we also discuss multivariate models and provide very interesting examples of fatigue models, including the SN curves, the crack growth curves and how they can be connected.Download 
Course 8. Functional Equations (English).This course introduces functional equations. First we motivate the course using the formulas for the area of a rectangle and a trapezoid, showing that they are not the most correct ones. We give the simple and compound interest formulas together with an interesting interpretation, in terms of account stability. Next, we introduce some classical functional equations, as the associative, the Cauchy's and the Pexider's equations. We continue with the sum of product equations and two examples of applications. Next, we provide a list of ten methods to solve functional equations and illustrate them with some examples. In the final part, we give interesting examples of applications to Economy, Statistics, fatigue, laws of Science and differential equations.Download 
Course 9. Optimization.In this course we start by introducing order statistics and extremes. In particular, the density functions of any order statistic or any subset of them are given in closed form. Next, we indicate that no matter the parent distribution is, the limit distribution of the maximum belongs to the LevyJenkinson family, which includes only the families of Weibull, Gumbel and Frechet. The corresponding limit distributions for the case of minimum are given too. The cases of other order statistics are also analyzed and discussed. The use of probability papers to find the most adequate limit distribution is explained and justified. Exceedances are recommended instead of extremes, because of its better efficiency and the generalized Pareto distribution is analyzed. The case of dependence is shown to lead to the same limits distributions if the dependence is not too strong, and to other limit distribution if it is strong enough. One illustrative example of extreme waves is studied in some detail.Download 
Teaching and importance of mathematics
We include here some videos to suggest some ideas about the importance of mathematics and its contribution to Society and how the teaching of mathematics can be improved in order to increase knowledge and its practical applications to solve real problems and make them more attractive to the students. We start by one video showing how mathematics are useful to the Society and how they must be used systematically to solve real problems. Some examples are included, showing that mathematics are present in real life. Next we suggest some changes in the way of teaching mathematics by using master videos which are already available to be used in dayly teaching. If students can have the best teachers and the most knowledgeable persons in a given topic, why ignore this possibility? Finally, we point out the need of changing mathematical contents and ways of explaining them. For example, we propose that to involve the students in web page designs can be a good idea to improve the virtues associated with mathematical thinking.Download 
Algunos ejemplos curiosos de aplicaciones de las matemáticas
Se dan algunos ejemplos simples de aplicaciones de ecuaciones funcionales para ilustrar su poder. La idea es encontrar todos los modelos posibles que satisfagan una o varias propiedades. Esto permite identificar modelos adecuados para resolver nuestros problemas particulares sin utilizar arbitrariedades. De esta forma garantizamos que se identifiquen todos los modelos que satisfacen esas propiedades. Las aplicaciones involucran varios problemas que incluyen geometría, economía, ciencia de materiales, matemáticas e inteligencia artificial.Descarga 
Cones and polytopes. Linear systems of inequations
We introduce two very important algebraic structures such as polyhedral convex cones and polytopes, which complememnt vector spaces and are very relevant in practical applications. We also provide some illustrative examples for the user to have an intuitive idea of what cones and polytopes are. Next we define the important concept of dual cone, which is the basic tool to be used in all the applications. As simple examples, we solve the problems of vector membership of a cone and the intersection of two cones, for which a simple look, from the point of view of duality, provides a hint to solve the problem.Download 
Cones and polytopes. Linear systems of inequations
We deal with the main applications of dual cones, which include solving homogeneous and complete linear systems of inequations. The most general algebraic structure that can appear from a system of linear inequations is shown to be the sum of a vector space, an acute cone and a polytope. It is also shown how to obtain the facets of any dimension of cones and polytopes, which includes, as particular examples, the cases of edges and vertices. Finally, it is shown how the knowledge of the solution of the system of linear constraints converts linear programming problems into trivial problemsDownload 
Covid models
Se describen, paso a paso, todas las etapas necesarias para construir el modelo desde un punto de vista crítico, incluyendo selección de estados, transiciones entre estados, definición de tasas de paso entre estados, obtención de las ecuaciones diferenciales y funcionales, estimación de parámetros a partir de las observaciones y conversión del modelo en uno aleatorio bayesiano. Para ello, se considera que todos los parámetros (tasas) son variables aleatorias. Finalmente se comentan las diferencias y ventajas de los modelos bayesianos. Una aplicación en Mathematica permite analizar la influencia de las medidas de control en los resultados, permitiendo un análisis de sensibilidad, que es muy útil para tomar decisiones.Download 