New rules for improving CAS capabilities when computing improper integrals. Applications in Math Education

In many Engineering applications the computation of improper integrals is a need. In [1] we pointed out the lack of some CAS when computing some types of improper integrals. Even more, the work developed showed that some improper integrals can not be computed with CAS using their build-in procedures. In this talk we will develop new rules to improve CAS capabilities in order to compute new improper integrals We will show some examples of improper integrals that CAS asMATHEMATICA, MAPLE, DERIVE or MAXIMA can not compute. Using advance techniques as Laplace and Fourier transforms or Residue Theorem in Complex Analysis, we will be able to develop new rules schemes for these improper integrals. We will also describe the conclusions obtained after using these new rules with our Engineering students when teaching Advanced Calculus. [1] José L.Galán-García, Gabriel Aguilera-Venegas, María Á. Galán-García, Pedro Rodríguez-Cielos, Iván Atencia-Mc.Killop. Improving CAS capabilities: New rules for computing improper integrals. Applied Mathematics and Computation. Volume 316, 1 January 2018, Pages 525-540.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Random samples generation with Stata from continuous and discrete distributions

Simulations are nowadays a very important way of analyzing new improvements in different areas before the physical implementation, which may require hard resources which could only be affronted in case of a high probability of success. The use of random samples from different distributions are a must in simulations. In this talk we introduce new Stata functions for generating random samples from continuous and discrete distributions that are not considered in the defined Stata random-number generation functions. In addition, we will also introduce new Stata functions for generating random samples as an alternative of the build-in Stata functions. The goodness of the generated samples will be checked using the mean squared error (MSE) of the differences between the frequencies of the sample and the theoretical expected ones. We will also provide bar charts which will allow the user to compare graphically the sample with the exact distribution function of the random distribution which is being sampled.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tec

Teaching Partial Differential Equations with CAS

Partial Differential Equations (PDE) are one of the topics where Engineering students find more difficulties when facing Math subjects. A basic course in Partial Differential Equations (PDE) in Engineering, usually deals at least, with the following PDE problems: 1. Pfaff Differential Equations 2. Quasi-linear Partial Differential Equations 3. Using Lagrange-Charpit Method for finding a complete integral for a given general first order partial differential equation 4. Heat equation 5. Wave equation 6. Laplace’s equation In this talk we will describe how we introduce CAS in the teaching of PDE. The tasks developed combine the power of a CAS with the flexibility of programming with it. Specifically, we use the CAS DERIVE. The use of programming allows us to use DERIVE as a Pedagogical CAS (PECAS) in the sense that we do not only provide the final result of an exercise but also display all the intermediate steps which lead to find the solution of a problem. This way, the library developed in DERIVE serves as a tutorial showing, step by step, the way to face PDE exercises. In the process of solving PDE exercises, first-order Ordinary Differential Equations (ODE) are needed. The programs developed can be grouped within the following blocks: - First-order ODE: separable equations and equations reducible to them, homogeneous equations and equations reducible to them, exact differential equations and equations reducible to them (integrating factor technique), linear equations, the Bernoulli equation, the Riccati equation, First-order differential equations and nth degree in y’, Generic programs to solve first order differential equations. - First-order PDE: Pfaff Differential Equations, Quasi-linear PDE, Lagrange-Charpit Method for First-order PDE. - Second-order PDE: Heat Equation, Wave Equation, Laplace’s Equation. We will remark the conclusions obtained after using these techniques with our Engineering students.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Improving CAS Capabilities: New Rules for Computing Improper Integrals

There are diferent applications in Engineering that require to compute improper integrals of the first kind (integrals defined on an unbounded domain) such as: the work required to move an object from the surface of the earth to in nity (Kynetic Energy), the electric potential created by a charged sphere, the probability density function or the cumulative distribution function in Probability Theory, the values of the Gamma Functions(wich useful to compute the Beta Function used to compute trigonometrical integrals), Laplace and Fourier Transforms (very useful, for example in Differential Equations).Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Using extensions of the residue theorem for improper integrals computations with CAS

The computation of improper integrals of the rst kind (integrals on unbounded domain) are used in di erent applications in Engineering (for example in Kynetic Energy, electric potential, probability density functions, Gamma and Beta functions, Laplace and Fourier Transforms, Di erential Equations, . . . ). Nowadays, Computer Algebra Systems (CAS) are being used for developing such computations. But in many cases, some CAS lack of the appropriate rules for computing some of these improper integrals. In a previous talk in ESCO 2016 and a later extension, we introduced new rules for computing improper integrals of the rst kind using some results from Advanced Calculus Theories (Residue Theorem, Laplace and Fourier Transforms) aimed to improve CAS capabilities on this topic. In this talk, we develop new rules for computing other types of improper integrals using different applications from extended versions of the Residue Theorem. We will show some examples of such improper integrals that current CAS can not compute. Using extensions of the Residue Theorem in Complex Analysis, we will be able to develop new rules schemes for these improper integrals. These new rules will improve the capabilities of CAS, making them able to compute more improper integrals.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Curso 0 Online de Aspectos Básicos de Matemáticas para alumnado de la Escuela de Ingenierías Industriales de la Universidad de Málaga

El objetivo principal de esta comunicación es describir la experiencia que, desde el curso 2018/2019, venimos desarrollando un grupo de 7 profesores de las Áreas de Matemática Aplicada y de Estadística e Investigación Operativa de la Universidad de Málaga con el alumnado de nuevo ingreso en la Escuela de Ingenierías Industriales. Esa experiencia consiste, básicamente, en un curso 0 virtual que se estructura en torno a una serie de materiales multimedia que establecen un punto de nexo común y que pueden ser utilizados tanto en las Matemáticas de Bachillerato como en las de Ingeniería, además de servir para ayudar al alumnado en su preparación antes de iniciar sus estudios de Ingeniería, ya que la mayoría de los estudiantes que provienen del Bachillerato encuentran un gran desnivel entre las Matemáticas que han visto en su centro de Secundaria y Bachillerato y las de la Universidad. Describiremos los inicios de la experiencia, su evolución durante los distintos cursos académicos (incluidos los cursos de la pandemia por la COVID19), llegando hasta su estructura actual. Terminaremos analizando las valoraciones del alumnado participante en el curso 0 y algunas pinceladas sobre el trabajo futuro.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

EDUMATICUS: EDUcación MATemática con TIC en Universidad y Secundaria

El objectivo principal de esta comunicación es presentar la actividad del grupo EDUMATICUS, EDUcación MATemática con TIC en Universidad y Secundaria, cuyo objetivo principal es elaborar materiales multimedia que sirvan de complemento para la docencia de las asignaturas de Matemáticas de Ingeniería y de Bachillerato con vídeos de los desarrollos teóricos y de los ejercicios típicos resueltos paso por paso. Describiremos la web y el canal de YouTube donde alojamos los materiales que elaboramos, prestando especial atención a su estructura y en las categorías en que clasificamos los materiales. Seguiremos detallando los escenarios donde utilizamos estos materiales con nuestros alumnos y terminaremos comentando el trabajo futuro que tenemos previsto desarrollar.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Teaching the residue theorem and its applications with a CAS

The residue theorem is one of the most interesting result in Complex Analysis which allows not only computations in C, the Field of Complex Numbers, but also provides many applications in the Field of Real Numbers R. In this talk we present the library ResidueApplications, that was initially developed in DERIVE since Engineering students in the University of Málaga are still using this software in computer lectures. However, we are migrating this library to PYTHON using the symbolic mathematics library SYMPY. This way it will be also possible to use this package in other CAS as SAGEMATH. The main goals of the ResidueApplications library are not only to provide some important applications of the Residue theorem but also to use it as a pedagogical tool for Engineering students. ResidueApplications can be used as a tutorial in the teaching and learning process of this topic since it provides the results step by step allowing the students to check their computations when they solve an exercise. When developing this package, we were not interesting only in the computations of residues and their applications (which can be easily done using standards functions in different CAS) but mainly on its pedagogical use. In addition of the step by step facility, using this library, the students also can develop their own programs to deal with different applications. This way, the student are the protagonist of their selflearning process. For example, If the students develop a program to compute the residues of a function, they will be better prepared to understand this topic. The programs developed in this tutorial can be grouped in the following blocks: 1. Compute of residues. 2. Compute of complex integrals using the residue theorem. 3. Applications of the residue theorem to compute integrals in R: (a) Trigonometric integrals. (b) Improper integrals.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Teaching Partial Differential Equations With a CAS

It is very common that Engineering students nd difficulties when studying advanced Mathematics subjects. To help in the teaching and learning process of such subjects, the teacher can use an adequate mathematical software. But not always the used given to these specific pieces of software is the right one. The use of a Computer Algebra System (CAS) to achieve this goal is a good idea mainly because programming with a CAS, the solution to a problem can be obtained step by step. This way, the student can check all the intermediate steps to get the solution and can nd the step or steps where the student made a mistake. In this talk, we introduce SPDES, a Stepwise Partial Differential Equation Solver (an extension of SFOPDES). SPDES deals with some second order PDE in addition to the first order PDE considered in SFOPDES. SPDES can be used as a self tutorial for PDE since it solves, step by step, the typical exercises within the topic. The type of PDE that SPDES can solve are: Pfaff Differential Equations, Quasi-linear PDE, Lagrange-Charpit Method for first order PDE, Heat equation, Wave equation and Laplace's equation. SPDES has been developed using the programming capabilities of the CAS Derive, providing not only the final result but also, optionally, the display of all the steps needed to solve typical exercises on PDE.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

A Non Markovian Retrial Queueing System

Researches on retrial queues with non-geometrical retrial times is motivated by real computers and telecommunication networks, where retrial times can hardly be geometrical distributed. The inherent difficulty with non-geometrical retrial times is caused by the fact that queueing models must keep track of the elapsed retrial time for each of possibly a very large number of customers in the orbit. This paper analyses a discrete-time Geo/G/1 retrial queue with general retrial times in which the arriving customers may opt to follow a LCFS-PR discipline or to join the orbit. The Markov chain underlying the system has been studied, the generating functions of the number of customers in the orbit and in the system as well as its expected values are derived. The stochastic decomposition law and, as an application, bounds for the proximity between the steady-state distribution for the system under study and its corresponding standard system has been derived.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tec