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

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

LiDARgrammetry: A New Method for Generating Synthetic Stereoscopic Products from Digital Elevation Models

There are currently several new technologies being used to generate digital elevation models that do not use photogrammetric techniques. For example, LiDAR (Laser Imaging Detection and Ranging) and RADAR (RAdio Detection And Ranging) can generate 3D points and reflectivity information of the surface without using a photogrammetric approach. In the case of LiDAR, the intensity level indicates the amount of energy that the object reflects after a laser pulse is transmitted. This energy mainly depends on the material and the wavelength used by LiDAR. This intensity level can be used to generate a synthetic image colored by this attribute (intensity level), which can be viewed as a RGB (red, green and blue) picture. This work presents the outline of an innovative method, designed by the authors, to generate synthetic pictures from point clouds to use in classical photogrammetric software (digital restitution or stereoscopic vision). This is conducted using available additional information (for example, the intensity level of LiDAR). This allows mapping operators to view the LiDAR as if it were stereo-imagery, so they can manually digitize points, 3D lines, break lines, polygons and so on

LiDARgrammetry: A New Method for Generating Synthetic Stereoscopic Products from Digital Elevation Models

There are currently several new technologies being used to generate digital elevation models that do not use photogrammetric techniques. For example, LiDAR (Laser Imaging Detection and Ranging) and RADAR (RAdio Detection And Ranging) can generate 3D points and reflectivity information of the surface without using a photogrammetric approach. In the case of LiDAR, the intensity level indicates the amount of energy that the object reflects after a laser pulse is transmitted. This energy mainly depends on the material and the wavelength used by LiDAR. This intensity level can be used to generate a synthetic image colored by this attribute (intensity level), which can be viewed as a RGB (red, green and blue) picture. This work presents the outline of an innovative method, designed by the authors, to generate synthetic pictures from point clouds to use in classical photogrammetric software (digital restitution or stereoscopic vision). This is conducted using available additional information (for example, the intensity level of LiDAR). This allows mapping operators to view the LiDAR as if it were stereo-imagery, so they can manually digitize points, 3D lines, break lines, polygons and so on

SFOPDES: A stepwise tutorial for teaching Partial Differential Equations using a CAS

Partial Differential Equations (PDE) are one of the most difficult topics that Engineering and Sciences students have to study in the different Math subjects in their degree. In this talk we introduce SFOPDES (Stepwise First Order Partial Differential Equations Solver) aimed to be used as a tutorial for helping both the teacher and the students in the teaching and learning process of PDE. The type of problems that SFOPDES solves can be grouped in the following three blocks: 1. Pfaff Differential Equations, which consists on finding the general solution for: P(x; y; z) dx + Q(x; y; z) dy + R(x; y; z) dz = 0 (a) General method. (b) Particular cases: i. Separable equations. ii. Exact Pfaff equations. iii. One-separated variable equations. 2. Quasi-linear Partial Differential Equations, which consists on finding the general solution for: P(x; y; x) p + Q(x; y; z) q = R(x; y; z) (a) General method. (b) Particular solution which contents a given curve. 3. Using Lagrange-Charpit Method for finding a complete integral for a given general first order partial differential equation: F(x; y; z; p; q) = 0. (a) General method. (b) Particular cases: i. F(p; q) = 0 ii. g1(x; p) = g2(y; q) iii. z = px + qy + g(p; q)Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Enhancing CAS improper integrals computations using extensions of the residue theorem

In a previous paper, the authors developed new rules for computing improper integrals which allow computer algebra systems (CAS) to deal with a wider range of improper integrals. The theory used in order to develop such rules where Laplace and Fourier transforms and the residue theorem. In this paper, we describe new rules for computing symbolic improper integrals using extensions of the residue theorem and analyze how some of the most important CAS could improve their improper integral computations using these rules. To achieve this goal, different tests are developed. The CAS considered have been evaluated using these tests. The obtained results show that all CAS involved, considering the new developed rules, could improve their capabilities for computing improper integrals. The results of the evaluations of the CAS are described providing a sorted list of the CAS depending on their scores

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

SPDES: A Stepwise Solver for Teaching Partial Differential Equations

Partial Differential Equations (PDE) are an important topic within the Engineering Degrees syllabus. In addition, many students find some dificulties in the learning process of this topic. Therefore, the use of didactical tools to improve the teaching and learning process of PDE is very helpful. In this talk, we introduce the solver SPDES (Stepwise Partial Differential Equation Solver), an extension of SFOPDES introduced in [1] where only first order PDE were considered. This new solver SPDES deals also with some second order PDE. It can be used as a self tutorial for PDE since it solves, step by step, the typical exercises within the topic. Specifically, the type of PDE that SPDES can solve are: 1. Pfaff Differential Equations. 2. Quasi-linear PDE. 3. Lagrange-Charpit Method for first order PDE. 4. Heat equation. 5. Wave equation. 6. Laplace's equation. where the fi rst three types where considered in SFOPDES and the last three types are new in SPDES. SPDES has been developed using the programming capabilities of a Computer Algebra System (CAS), displaying step by step the solution of the problem to be solved. This way, we potentiate the use of the CAS as a Pedagogical CAS (PeCAS). This fact makes SPDES to be an important tool for students which can use it as a tutorial for their learning process.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

SFOPDES: A Stepwise First Order Partial Differential Equations Solver with a Computer Algebra System

Partial Differential Equations (PDE) appear in multiple Physic and Engineering applications. Normally, when modeling an application, the use of well-known and already solved PDE is considered. But what happens if a new PDE is used? Solving a new PDE is not an easy task. In this paper, we use a Computer Algebra System (Cas) in order to find the solution of PDE of first order. Specifically, we deal with Pfaff Equations, Quasilinear PDE and general first order PDE (using Lagrange–Charpit Method). To solve these PDE, we combine the power of a Cas with the flexibility of programming with it. Furthermore, the developed programs do not only provide the final result but also display all the intermediate steps which lead to find the solution of the PDE. This way, we introduce SFOPDES, a new Stepwise First Order PDE Solver which serves as a tutorial showing, step by step, the way to deal with PDE