THE PLEASURE OF SHARING KNOWLEDGE

Some Motivating, Suggestive and Original Courses Mathematically, Statistically and Engineering Related

Motivation and presentation of the web page

Profesor Enrique Castillo Ron

Born in Santiago de Compostela (Spain) 17/10/1946
Ph. D. Degree in Civil Engineering from Northwestern University (USA)
Ph. D. Degree in Civil Engineering from the Polytechnical University of Madrid (Spain)
Ms. Degree in Mathematics from the Complutense University of Madrid (Spain)
Member of the Royal Academy of Engineering (Spain)
Member of the Royal Academy of Sciences (Spain)
Leonardo Torres Quevedo National Prize in Engineering

In this web page, we offer the audience a list of courses mathematical, statistics and Engineering related. All the courses have in common that they are somehow unique in the sense that what is offered is new in the content or in the way it is introduced. After 50 years of teaching at different Universities, but specially at the Universities of Cantabria and Castilla-La Mancha in Spain, I realized that to reach a deep degree of knowledge requires a lot of time and work. This effort brings you a lot of pleasure but also hard times, because learning sometimes implies a very hard work. I have been a privileged person who has had the opportunity of having good teachers and the access to several universities all over the world. I want to return to the society one part of what society has given me. This is the main motivation of this web page and the courses in it.

Course 1. Algebra I (English)

Block 1. Cones, dual cones and polytopes

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.

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Course 2. Algebra I (Spanish)

Block 3. Linear systems of inequalities

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.

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Course 3. Algebra II (English)

Block 2. Algebraic applications of the algorithm

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.

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Course 4. Algebra II (Spanish)

Block 4. Examples of applications

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.

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Course 5. Extreme Values (English).

Block 5. Some functional equations in Economy.

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 Levy-Jenkinson 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.

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Course 6. Bayesian methods. OpenBUGS (English).

Block 6. Computer application to obtain the orthogonal subspace and the complement of a given subspace

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.

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Course 7. Mathematical model building (English).

Block 7. Some fatigue models.

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 self-similarity, 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 S-N curves, the crack growth curves and how they can be connected.

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Course 8. Functional Equations (English).

Block 8. Valid physical formulas

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.

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Course 9. Optimization.

Course 9. Optimization (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 Levy-Jenkinson 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.

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Videos miscellanea.

Videos miscellanea

We include here some videoclips to provide some ideas and illustrate some important concepts

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