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2017 | Book

A First Course in Fuzzy Logic, Fuzzy Dynamical Systems, and Biomathematics

Theory and Applications

Authors: Laécio Carvalho de Barros, Rodney Carlos Bassanezi, Weldon Alexander Lodwick

Publisher: Springer Berlin Heidelberg

Book Series : Studies in Fuzziness and Soft Computing

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About this book

This book provides an essential introduction to the field of dynamical models. Starting from classical theories such as set theory and probability, it allows readers to draw near to the fuzzy case. On one hand, the book equips readers with a fundamental understanding of the theoretical underpinnings of fuzzy sets and fuzzy dynamical systems. On the other, it demonstrates how these theories are used to solve modeling problems in biomathematics, and presents existing derivatives and integrals applied to the context of fuzzy functions. Each of the major topics is accompanied by examples, worked-out exercises, and exercises to be completed. Moreover, many applications to real problems are presented. The book has been developed on the basis of the authors’ lectures to university students and is accordingly primarily intended as a textbook for both upper-level undergraduates and graduates in applied mathematics, statistics, and engineering. It also offers a valuable resource for practitioners such as mathematical consultants and modelers, and for researchers alike, as it may provide both groups with new ideas and inspirations for projects in the fields of fuzzy logic and biomathematics.

Table of Contents

Frontmatter
Chapter 1. Fuzzy Sets Theory and Uncertainty in Mathematical Modeling
Abstract
This chapter presents a brief discussion about uncertainty based on philosophical principles, mainly from the point of view of the pre-Socratic philosophers. Next, the notions of fuzzy sets and operations on fuzzy sets are presented. Lastly, the concepts of alpha-level and the statement of the well-known Negoita-Ralescu Representation Theorem, the representation of a fuzzy set by its alpha-levels, are discussed.
Laécio Carvalho de Barros, Rodney Carlos Bassanezi, Weldon Alexander Lodwick
Chapter 2. The Extension Principle of Zadeh and Fuzzy Numbers
Abstract
This chapter presents the Extension Principle of Zadeh, and as the name suggests, it is a method used to extend to fuzzy set theory the typical operations of classical set theory. It gives the framework to calculate the membership degree of elements of a fuzzy set and functions of fuzzy sets, which are the result of operations. Also, in the context of fuzzy sets, the concepts of fuzzy number and fuzzy number arithmetic are introduced.
Laécio Carvalho de Barros, Rodney Carlos Bassanezi, Weldon Alexander Lodwick
Chapter 3. Fuzzy Relations
Abstract
This chapter presents a short discussion of mathematical relations, basic concepts of fuzzy relations, and the composition between two fuzzy relations. Lastly, the chapter presents the rule for the composition of inferences, which is relevant to the modus ponens discussed in the next chapter.
Laécio Carvalho de Barros, Rodney Carlos Bassanezi, Weldon Alexander Lodwick
Chapter 4. Notions of Fuzzy Logic
Abstract
This chapter presents the basic notions of classical and fuzzy logic followed by the concepts of t-norms, t-conorms, fuzzy negation, and fuzzy implication, which are the key ideas of propositional calculus of fuzzy logic. Next the chapter discusses modus ponens and generalized modus ponens together with the concepts of linguistic variables that are used in logical reasoning. The chapter closes with linguistic modifiers and the concept of interactivity between fuzzy sets.
Laécio Carvalho de Barros, Rodney Carlos Bassanezi, Weldon Alexander Lodwick
Chapter 5. Fuzzy Rule-Based Systems
Abstract
This chapter explores fuzzy logic controllers from the point of view of its applications. The chapter covers the fuzzy logic controllers of Mamdani and Takagi-Sugeno-Kang. These are illustrated with applications in biology, ecology, HIV dynamics, and pharmacological decay.
Laécio Carvalho de Barros, Rodney Carlos Bassanezi, Weldon Alexander Lodwick
Chapter 6. Fuzzy Relational Equations and Universal Approximation
Abstract
This chapter presents the concepts of fuzzy relationships and fuzzy relational equations. These are applied to medical diagnosis and Bayesian inference. The notion of universal approximator with applications to dynamical systems, complete the chapter.
Laécio Carvalho de Barros, Rodney Carlos Bassanezi, Weldon Alexander Lodwick
Chapter 7. Measure, Integrals and Fuzzy Events
Abstract
This chapter reviews classical measure theory including probability and Lebesgue measures. This discussion is followed by fuzzy measures, Sugeno measures, and possibilistic measures in order to understand the integration of Lebesgue, Choquet and Sugeno. These concepts are used in the development of fuzzy expected value. Lastly, the chapter closes with a discussion of the concepts of fuzzy event, the probability of a fuzzy event, dependence of fuzzy events, independence of fuzzy events, together with the concepts of random linguistic variables and random fuzzy variables.
Laécio Carvalho de Barros, Rodney Carlos Bassanezi, Weldon Alexander Lodwick
Chapter 8. Fuzzy Dynamical Systems
Abstract
This chapter presents an introduction to fuzzy dynamical systems both continuous and discrete. To study the dynamical case, the concept of fuzzy derivative and fuzzy integral are presented. Several kinds of derivatives are explored and consequently, several types of fuzzy differential equations are studied. The discrete case is studied by means of an interactive process. Lastly, all cases are illustrated using the Malthusian Model.
Laécio Carvalho de Barros, Rodney Carlos Bassanezi, Weldon Alexander Lodwick
Chapter 9. Modeling in Biomathematics: Demographic Fuzziness
Abstract
This chapter explores the notion of demographic fuzziness in modeling of bio-mathematical phenomena. The concept of demographic fuzziness is illustrated by looking at both continuous and discrete models. The relatively new idea of p-fuzzy systems, which combines dynamical systems with fuzzy logic, is illustrated via the dynamical models.
Laécio Carvalho de Barros, Rodney Carlos Bassanezi, Weldon Alexander Lodwick
Chapter 10. Biomathematical Modeling in a Fuzzy Environment
Abstract
This chapter looks at the influence of the environment in a population as a whole, that is, it looks at processes in which the environment affects all individuals equally. We illustrate this phenomenon via four models.
Laécio Carvalho de Barros, Rodney Carlos Bassanezi, Weldon Alexander Lodwick
Chapter 11. End Notes
Abstract
This chapter presents the concept of joint possibility distribution from the point of view of fuzzy number membership. The concept of completely correlated fuzzy numbers is presented. Next, an interactive fuzzy number subtraction operator is discussed. Finally, two bio-mathematical models are studied using these concepts. The first models the risk of getting dengue fever and second is an epidemiological SI-model with completely correlated initial conditions.
Laécio Carvalho de Barros, Rodney Carlos Bassanezi, Weldon Alexander Lodwick
Backmatter
Metadata
Title
A First Course in Fuzzy Logic, Fuzzy Dynamical Systems, and Biomathematics
Authors
Laécio Carvalho de Barros
Rodney Carlos Bassanezi
Weldon Alexander Lodwick
Copyright Year
2017
Publisher
Springer Berlin Heidelberg
Electronic ISBN
978-3-662-53324-6
Print ISBN
978-3-662-53322-2
DOI
https://doi.org/10.1007/978-3-662-53324-6

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