816 research outputs found
Making more flexible ATISMART+ model for traffic simulations using a CAS
Traffic simulations usually require the search of a path to join two different
points. Dijkstra’s algorithm [1] is one of the most commonly used for this task due
to its easiness and quickness. In [2, 3] we developed an accelerated time simulation
of car traffic in a smart city using Dijkstra’s algorithm to compute the paths.
Dijkstra’s algorithm provides a shortest path between two different points but
this is not a realistic situation for simulations. For example, in a car traffic situa-
tion, the driver may not know the shortest path to follow. This ignorance can be
produced, among others, because one of the following two facts: the driver may
not know the exact length of the lanes, or, even knowing the exact length, the driver
may not know how to find the shortest path. Even more, in many cases, a mixture
of both facts occurs.
A more realistic simulation should therefore consider these kind of facts. The
algorithm used to compute the path from one point to another in a traffic simulation
might consider the possibility of not using the shortest path.
In this talk, we use a new probabilistic extension of Dijkstra’s algorithm which
covers the above two situations. For this matter, two different modifications in Di-
jkstra’s algorithm have been introduced: using non-exact length in lanes, and the
choice of a non-shortest path between two different points. Both modifications are
used in a non-deterministic way by means of using probability distributions (classi-
cal distributions such as Normal or Poisson distributions or even "ad hoc" ones). A
precise, fast, natural and elegant way of working with such probability distributions
is the use of a CAS in order to deal with exact and explicit computations.
As an example of use of this extension of Dijkstra’s algorithm, we will show
the ATISMART+ model. This model provides more realistic accelerated time sim-
ulations of car traffics in a smart city and was first introduced in [4] and extended
in [5]. This model was developed combining J
AVA
for the GUI and M
AXIMA
for
the mathematical core of the algorithm.
The studies developed in the above mentioned works, dealt with Poisson, Ex-
ponential, Uniform and Normal distributions. In this talk we will introduce, as
a novelty, the possibility of using other continuous probability distributions such
as: Lognormal, Weibul, Gamma, Beta, Chi-Square, Student’s t, Z, Pareto, Lo-
gistic, Cauchy or Irwin-Hall, and other discrete distributions such as: Bernouille,
Rademacher, Binomial, Geometric, Negative Binomial or Hypergeometric. Even
1
more, this new version allows to deal with any “ad-hoc” continuous, discrete or
mixed user’s distributions. This fact improves the flexibility of ATISMART+ model.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech
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
Introducing probabilistic celular automata. A versatile extension of Game of Live
The "Game of life" model was created in 1970 by the mathematician Jonh Horton Conway
using cellular automata. Since then, di erent extensions of these cellular automata have been
used in many applications, such as car traffic control or baggage traffic in an airport.
These extensions introduce ideas not only from cellular automata models but also from neural
networks theory.
In this work, we introduce probabilistic cellular automata which include non-deterministic
rules for transitions between successive generations of the automaton together with probabilistic
decisions about life and death of the cells in next generation of the automaton. This way,
more realistic situations can be modeled and the obtained results are also non-deterministic.
As an example of use, an implementation of this probabilistic cellular automaton has been
developed using it for simulating tissues evolution. The authors are specially interested in
simulations of cancerous tissues.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
A RBES for Generating Automatically Personalized Menus
Food bought at supermarkets in, for instance, North America or the European Union, give comprehensive information about ingredients and allergens. Meanwhile, the menus of restaurants are usually incomplete and cannot be normally completed by the waiter. This is specially important when traveling to countries with a di erent culture. A curious example is "calamares en su tinta" (squid in its own ink), a common dish in Spain. Its brief description would
be "squid with boiled rice in its own (black) ink", but an ingredient of its sauce is flour, a fact very important for celiacs. There are constraints based on religious believes, due to food allergies or to illnesses, while others just derive from personal preferences. Another complicated situation arise in hospitals, where the doctors' nutritional recommendations have to be added to the patient's usual constraints. We have therefore designed and developed a Rule Based Expert System (RBES) that can address these problems. The rules derive directly from the recipes of the di fferent dishes and contain the information about the required ingredients and ways of cooking. In fact, we distinguish: ingredients and ways of cooking, intermediate
products (like sauces, that aren't always made explicit) and final products (the dishes listed in the menu of the restaurant). For a certain restaurant, customer and instant, the input to the RBES are: actualized stock of ingredients and personal characteristics of that customer.
The RBES then prepares a "personalized menu" using set operations and knowledge extraction (thanks to an algebraic inference engine [1]). The RBES has been implemented in the computer algebra system MapleTM2015. A rst version of this work was presented at "Applications of Computer Algebra 2015" (ACA'2015) conference. The corresponding abstract is available at [2].Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech
A fast functional approach to personalized menus generation using set operations
The authors developed some time ago a RBES devoted to preparing personalized menus at
restaurants according to the allergies, religious constraints, likes and other diet requirements
as well as products availability. A first version was presented at the "Applications of Computer
Algebra 2015" (ACA'2015) conference and an improved version to the "5th European
Seminar on Computing" (ESCO2016). Preparing personalized menus can be specially
important when traveling abroad and facing unknown dishes in a menu. Some restaurants
include icons in their menu regarding their adequateness for celiacs or vegetarians and vegans,
but this is not always a complete information, as it doesn't consider, for instance, personal
dislikes or uncommon allergies. The tool previously developed can obtain, using logic deduction,
a personalized menu for each customer, according to the precise recipes of the restaurant
and taking into account the data given by the customer and the ingredients out of stock (if
any). Now a new approach has been followed, using functions and set operations and the
speed has been increased by three orders of magnitude, allowing to deal with huge menus
instantly. Both approaches have been implemented on the computer algebra system Maple
and are exemplified using the same recipes in order to compare their performances.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech
Computer Algebra-based RBES personalized menu generator
People have many constraints concerning the food they eat. These constraints
can be based on religious believes, be due to food allergies or to illnesses, or can
be derived just from personal preferences. Therefore, preparing menus at hospitals
and restaurants can be really complex. Another special situation arise when travel-
ing abroad. It is not always enough to know the brief description in the restaurant
menu or the explanation of the waiter. For example, “calamares en su tinta” (squid
in its own ink) is a delicious typical Spanish dish, not well-known abroad. Its brief
description would be “squid with boiled rice in its own (black) ink”. But an in-
gredient (included in a small amount, in order to thicken the sauce) is flour, a fact
very important for someone suffering from celiac disease. Therefore, we have con-
sidered that it would be very interesting to develop a Rule Based Expert System
(RBES) to address these problems. The rules derive directly from the recipes and
contain the information about required ingredients and names of the dishes. We
distinguish: ingredients and ways of cooking, intermediate products (like “mayon-
naise”, that doesn’t always appear explicitly in the restaurants’ menus) and final
products (like “seafood cocktail”, that are the dishes listed in the restaurant menu).
For each customer at a certain moment, the input to the system are: on one hand,
the stock of ingredients at that moment, and on the other, the religion, allergies and
restrictions due to illnesses or personal preferences of the customer. The RBES
then constructs a “personalized restaurant menu” using set operations and knowl-
edge extraction (thanks to an algebraic Groebner bases-based inference engine[1]).
The RBES has been implemented in the computer algebra system
Maple TM 18(us-ing its convenient Embedded Components) and can be run from computers and tablets using Maple TM or the Maple TM PlayerUniversidad 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
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