27 research outputs found
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
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
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
Advanced techniques to compute improper integrals using a CAS
Let us consider the following types of improper integrals:
\medskip
Let be an antiderivative of . The basic approach to compute such integrals involves the following computations:
\medskip
\begin{eqnarray*}
\int_0^\infty f(t)\:{\rm d}t & = & \lim_{m\to\infty} \int_0^m f(t)\:{\rm d}t = \lim_{m\to\infty} \big(F(m)-F(0)\big) \\ \\
\int_{-\infty}^0 f(t)\:{\rm d}t & = & \lim_{m\to-\infty} \int_m^0 f(t)\:{\rm d}t = \lim_{m\to-\infty} \big(F(0)-F(m)\big) \\ \\
\int_{-\infty}^\infty f(t)\:{\rm d}t & = & \int_{-\infty}^0 f(t)\:{\rm d}t + \int_0^\infty f(t)\:{\rm d}t \qquad \mbox{or, in case of convergence,} \\ \\
\int_{-\infty}^\infty f(t)\:{\rm d}t & = & \lim_{m\to\infty} \int_{-m}^m f(t)\:{\rm d}t = \lim_{m\to\infty} \big(F(m)-F(-m)\big) \qquad \mbox{(Cauchy principal value)}
\end{eqnarray*}
\medskip
\noindent But, what happens if an antiderivative for or the above limits do not exist?
\medskip
\noindent For example, for \quad \quad ; \quad \quad {\rm or} \quad \qquad the antiderivatives can not be computed. Hence, the above procedures cannot be used for these examples.
\medskip
In this work we will deal with advance techniques to compute this kind of improper integrals using a {\sc Cas}. Laplace and Fourier transforms or Residue Theorem in Complex Analysis are some advance techniques which can be used for this matter.
\medskip
We will introduce the file \textbf{\tt ImproperIntegrals.mth}, developed in {\sc Derive} 6, which deals with such computations.
\medskip
Some {\sc Cas} use different rules for computing integrations. For example {\sc Rubi} system, a {\bf ru}le-{\bf b}ased {\bf i}ntegrator developed by Albert Rich (see {\tt http://www.apmaths.uwo.ca/\~{ }arich/}), is a very powerful system for computing integrals using rules. We will be able to develop new rules schemes for some improper integrals using {\tt ImproperIntegrals.mth}. These new rules can extend the types of improper integrals that a {\sc Cas} can compute.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech
A New Web-stepwise Solver for Ordinary Differential Equations
In this paper we introduce SODES (Step-wise Ordinary Differential Equations Solver) which is a new solver for Ordinary Dfferential Equations (ODE). SODES can be used not only as a solver but also as a tutorial for the teaching and learning process of ODE, since it provides the solution displaying all the the steps needed to obtain it.
In [1] we introduced SFOPDES (Step-wise First Order Partial Differential Equations Solver) developed in Derive, a Computer Algebra System (CAS). Two of the future works related considered in [1] were the development of new step-wise solvers for different math topics and the use of other CAS and programming with Python. This way, it is possible to develop web applications, freely available, providing different step-wise solvers. In this sense, SODES is the first work related with both ideas. Therefore, SODES is a web solver for ODE, developed in Python (using its CAS module
SymPy).
SODES increases the types of ODE that SFOPDES can deal with. Specifically, it provides, step by step, the solution of the following types of ODE:
1. 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.
2. Higher-order ODE: higher order linear equations with constant coefficients.
As a final remark, actually, SODES can be used in English, French or Spanish, but it can be easily extendable to other languages.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