Details

A Modern Introduction to Fuzzy Mathematics


A Modern Introduction to Fuzzy Mathematics


1. Aufl.

von: Apostolos Syropoulos, Theophanes Grammenos

103,99 €

Verlag: Wiley
Format: EPUB
Veröffentl.: 01.07.2020
ISBN/EAN: 9781119445296
Sprache: englisch
Anzahl Seiten: 384

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Beschreibungen

<p><b>Provides readers with the foundations of fuzzy mathematics as well as more advanced topics</b> <p><i>A Modern Introduction to Fuzzy Mathematics</i> provides a concise presentation of fuzzy mathematics., moving from proofs of important results to more advanced topics, like fuzzy algebras, fuzzy graph theory, and fuzzy topologies. <p>The authors take the reader through the development of the field of fuzzy mathematics, starting with the publication in 1965 of Lotfi Asker Zadeh's seminal paper, Fuzzy Sets. <p>The book begins with the basics of fuzzy mathematics before moving on to more complex topics, including: <ul> <li>Fuzzy sets</li> <li>Fuzzy numbers</li> <li>Fuzzy relations</li> <li>Possibility theory</li> <li>Fuzzy abstract algebra</li> <li>And more</li> </ul> <p>Perfect for advanced undergraduate students, graduate students, and researchers with an interest in the field of fuzzy mathematics, <i>A Modern Introduction to Fuzzy Mathematics</i> walks through both foundational concepts and cutting-edge, new mathematics in the field.
<p>Preface V</p> <p><b>Chapter 1 Introduction 1</b></p> <p>1.1 What is vagueness? 1</p> <p>1.2 Vagueness, Ambiguity, Uncertainty,… 4</p> <p>1.3 Vagueness and Fuzzy Mathematics 5</p> <p><b>Chapter 2 Fuzzy Sets and Their Operations 9</b></p> <p>2.1 Algebras of Truth Values 9</p> <p>2.1.1 Posets 9</p> <p>2.1.2 Lattices 11</p> <p>2.1.3 Frames 11</p> <p>2.2 Zadeh’s Fuzzy Sets 12</p> <p>2.3 α-cuts of Fuzzy Sets 16</p> <p>2.4 Interval Valued and Type 2 Fuzzy Sets 19</p> <p>2.5 Triangular Norms and Conorms 21</p> <p>2.6 L-fuzzy Sets 23</p> <p>2.7 “Intuitionistic” Fuzzy Sets and Their Extensions 24</p> <p>2.8 The Extension Principle 28</p> <p>2.9* Boolean-Valued Sets 30</p> <p>2.10* Axiomatic Fuzzy Set Theory 32</p> <p><b>Chapter 3 Fuzzy Numbers and Their Arithmetic 35</b></p> <p>3.1 Fuzzy Numbers 35</p> <p>3.1.1 Triangular Fuzzy Numbers 36</p> <p>3.1.2 Trapezoidal Fuzzy Numbers 37</p> <p>3.1.3 Gaussian Fuzzy Numbers 37</p> <p>3.1.4 Quadratic Fuzzy Numbers 39</p> <p>3.1.5 Exponential Fuzzy Numbers 41</p> <p>3.1.6 LR Fuzzy Numbers 41</p> <p>3.1.7 Generalized Fuzzy Numbers 42</p> <p>3.2 Arithmetic of Fuzzy Numbers 43</p> <p>3.2.1 Interval Arithmetic 43</p> <p>3.2.2 Interval Arithmetic and α-Cuts 43</p> <p>3.2.3 Fuzzy Arithmetic and the Extension Principle 44</p> <p>3.2.4 Fuzzy Arithmetic of Triangular Fuzzy Numbers 45</p> <p>3.2.5 Fuzzy Arithmetic of Generalized Fuzzy Numbers 45</p> <p>3.2.6 Comparing Fuzzy Numbers 47</p> <p>3.3 Linguistic Variables 49</p> <p>3.4 Fuzzy Equations 50</p> <p>3.4.1 Solving the Fuzzy Equation 𝐴 ⋅ 𝑋 + 𝐵 = 𝐶 50</p> <p>3.4.2 Solving the Fuzzy Equation 𝐴 ⋅ 𝑋<sup>2</sup> + 𝐵 ⋅ 𝑋 + 𝐶 = 𝐷 53</p> <p>3.5 Fuzzy Inequalities 55</p> <p>3.6 Constructing Fuzzy Numbers 55</p> <p>3.7 Applications of Fuzzy Numbers 57</p> <p>3.7.1 Simulation of the Human Glucose Metabolism 57</p> <p>3.7.2 Estimation of an Ongoing Project’s Completion Time 60</p> <p><b>Chapter 4 Fuzzy Relations 65</b></p> <p>4.1 Crisp Relations 65</p> <p>4.2 Fuzzy Relations 69</p> <p>4.3 Cartesian Product, Projections, and Cylindrical Extension 70</p> <p>4.4 New Fuzzy Relations From Old Ones 72</p> <p>4.5 Fuzzy Binary Relations on a Set 75</p> <p>4.6 Fuzzy Orders 80</p> <p>4.7 Elements of Fuzzy Graph Theory 82</p> <p>4.7.1 Graphs and Hypergraphs 82</p> <p>4.7.2 Fuzzy Graphs 83</p> <p>4.7.3 Fuzzy Hypergraphs 87</p> <p>4.8* Fuzzy Category Theory 89</p> <p>4.9* Fuzzy Vectors 96</p> <p>4.10 Applications 97</p> <p><b>Chapter 5 Possibility Theory 101</b></p> <p>5.1 Fuzzy Restrictions and Possibility Theory 101</p> <p>5.2 Possibility and Necessity Measures 103</p> <p>5.3 Possibility Theory 105</p> <p>5.4 Possibility Theory and Probability Theory 108</p> <p>5.5 An Unexpected Application of Possibility Theory 110</p> <p><b>Chapter 6 Fuzzy Statistics 117</b></p> <p>6.1 Random Variables 117</p> <p>6.2 Fuzzy Random Variables 120</p> <p>6.3 Point Estimation 123</p> <p>6.3.1 The unbiased estimator 124</p> <p>6.3.2 The consistent estimator 125</p> <p>6.3.3 The maximum likelihood estimator 126</p> <p>6.4 Fuzzy Point Estimation 127</p> <p>6.5 Interval Estimation 128</p> <p>6.6 Interval Estimation for Fuzzy Data 129</p> <p>6.7 Hypothesis Testing 131</p> <p>6.8 Fuzzy Hypothesis Testing 132</p> <p>6.9 Statistical Regression 134</p> <p>6.10 Fuzzy Regression 136</p> <p><b>Chapter 7 Fuzzy Logics 141</b></p> <p>7.1 Mathematical Logic 141</p> <p>7.2 Many-Valued Logics 146</p> <p>7.3 On Fuzzy Logics 151</p> <p>7.4 Hájek’s Basic Many-Valued Logic 152</p> <p>7.5 Łukasiewicz Fuzzy Logic 155</p> <p>7.6 Product Fuzzy Logic 157</p> <p>7.7 Gödel Fuzzy Logic 158</p> <p>7.8 First Order Fuzzy Logics 160</p> <p>7.9 Fuzzy Quantifiers 162</p> <p>7.10 Approximate Reasoning 163</p> <p>7.11 Application: Fuzzy Expert Systems 166</p> <p>7.12* A Logic of Vagueness 171</p> <p><b>Chapter 8 Fuzzy Computation 173</b></p> <p>8.1 Automata, Grammars, and Machines 173</p> <p>8.2 Fuzzy Languages and Grammars 178</p> <p>8.3 Fuzzy Automata 181</p> <p>8.4 Fuzzy Turing Machines 186</p> <p>8.5 Other Fuzzy Models of Computation 190</p> <p><b>Chapter 9 Fuzzy Abstract Algebra 195</b></p> <p>9.1 Groups, Rings, and Fields 195</p> <p>9.2 Fuzzy Groups 199</p> <p>9.3 Abelian Fuzzy Subgroups 204</p> <p>9.4 Fuzzy Rings and Fuzzy Fields 206</p> <p>9.5 Fuzzy Vector Spaces 208</p> <p>9.6 Fuzzy Normed Spaces 209</p> <p>9.7 Fuzzy Lie Algebras 210</p> <p><b>Chapter 10 Fuzzy Topology 213</b></p> <p>10.1 Metric and Topological Spaces 213</p> <p>10.2 Fuzzy Metric Spaces 218</p> <p>10.3 Fuzzy Topological Spaces 221</p> <p>10.4 Fuzzy Product Spaces 224</p> <p>10.5 Fuzzy Separation 226</p> <p>10.5.1 Separation 231</p> <p>10.6 Fuzzy Nets 231</p> <p>10.7 Fuzzy Compactness 232</p> <p>10.8 Fuzzy Connectedness 233</p> <p>10.9 Smooth Fuzzy Topological Spaces 234</p> <p>10.10 Fuzzy Banach and Fuzzy Hilbert Spaces 235</p> <p>10.11* Fuzzy Topological Systems 238</p> <p><b>Chapter 11 Fuzzy Geometry 243</b></p> <p>11.1 Fuzzy Points and Fuzzy Distance 243</p> <p>11.2 Fuzzy Lines and their Properties 246</p> <p>11.3 Fuzzy Circles 249</p> <p>11.4 Regular Fuzzy Polygons 252</p> <p>11.5 Applications in Theoretical Physics 256</p> <p><b>Chapter 12 Fuzzy Calculus 259</b></p> <p>12.1 Fuzzy Functions 259</p> <p>12.2 Integrals of Fuzzy Functions 263</p> <p>12.3 Derivatives of Fuzzy Functions 266</p> <p>12.4 Fuzzy Limits of Sequences and Functions 269</p> <p>12.4.1 Fuzzy Ordinary Differential Equations 272</p> <p>12.4.2 Fuzzy Partial Differential Equations 277</p> <p><b>Appendix A Fuzzy Approximation 283</b></p> <p>A.1 Weierstrass and Stone-Weierstrass Approximation Theorems 283</p> <p>A.2 Weierstrass and Stone-Weierstrass Fuzzy Analogues 284</p> <p><b>Appendix B Chaos and Vagueness 287</b></p> <p>B.1 Chaos Theory in a Nutshell 287</p> <p>B.2 Fuzzy Chaos 289</p> <p>B.3 Fuzzy Fractals 291</p> <p>Works Cited 293</p> <p>Subject Index 311</p> <p>Name Index 323</p>
<p><b>APOSTOLOS SYROPOULOS, P<small>H</small>D,</b> is an independent scholar based in Xanthi, Greece. His research interests include fuzzy mathematics, the philosophy of vagueness, computability theory, category theory, and digital typography. He has authored or co-authored more than 10 books and more than 60 papers and articles. He has served in the program committee of numerous scientific conferences and has reviewed papers for many journals. <p><b>THEOPHANES GRAMMENOS, P<small>H</small>D,</b> is Assistant Professor of Applied Mathematics in the Department of Civil Engineering, University of Thessaly, Greece. He received his PhD from University of Athens, Greece. Dr. Grammenos is a member of the editorial board for Applied Mathematics and the International Journal of Applied Mathematical Research.
<p><b>Provides readers with the foundations of fuzzy mathematics as well as more advanced topics</b> <p><i>A Modern Introduction to Fuzzy Mathematics</i> provides a concise presentation of fuzzy mathematics., moving from proofs of important results to more advanced topics, like fuzzy algebras, fuzzy graph theory, and fuzzy topologies. <p>The authors take the reader through the development of the field of fuzzy mathematics, starting with the publication in 1965 of Lotfi Asker Zadeh's seminal paper, Fuzzy Sets. <p>The book begins with the basics of fuzzy mathematics before moving on to more complex topics, including: <ul> <li>Fuzzy sets</li> <li>Fuzzy numbers</li> <li>Fuzzy relations</li> <li>Possibility theory</li> <li>Fuzzy abstract algebra</li> <li>And more</li> </ul> <p>Perfect for advanced undergraduate students, graduate students, and researchers with an interest in the field of fuzzy mathematics, <i>A Modern Introduction to Fuzzy Mathematics</i> walks through both foundational concepts and cutting-edge, new mathematics in the field.

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