Hardback
ISBN 9781032768106
264 Pages 167 B/W Illustrations
Published December 30, 2024
This book explores the dynamic interplay between fractals and graph theory, two powerful mathematical tools with vast applications. It presents a strategic combination and the synergistic use of these disciplines to address real-world problems and challenges. The book begins with an introduction to the basic concepts of fractals and graph theory and goes on to explore the applications in various domains, including natural phenomena modeling, scheduling, and network optimisation.
Illustrates the innovative ways fractals and graph theory can be combined, laying the groundwork for future applications across various industries
Introduces the fundamental concepts and principles of both fractals and graph theory in detail, making it accessible to a broad audience, including those new to these topics
Explores practical applications in image processing, network optimisation, social network analysis, and more, demonstrating the real-world impact of these mathematical tools
Analyses advanced techniques in graph theory, such as matching, domination, and colouring, with practical examples and case studies
Highlights the latest research advancements in fractal graph theory, showcasing its potential for future developments and applications
This book is for students, researchers, and professionals in mathematics, computer science, engineering, and related fields.
1. Graph Theory- Overview 2. Fractal Graph Theory 3. Fractal & Fractal Geometry 4. Classical Iterated Function Systems 5. Fractals from Iterated Function Systems 6. Application of Fractals in Various Fields 7. Matching and Application of Matching in Real life 8. Domination and its Application in Real Life 9. Coloring and Application of Coloring in Real Life 10. Fractals in medicine 11. Fractals in Circuit Theory 12. Fractals in Architecture 13. Fractal Networks and Deep Learning
Hardback
ISBN 9781032905846
642 Pages
Published December 13, 2024
Mathematical Theory of Fuzzy Sets presents the mathematical theory of non-normal fuzzy sets such that it can be rigorously used as a basic tool to study engineering and economic problems under a fuzzy environment. It may also be used as a textbook at the graduate level, or as a reference for researchers.
The book explores the current state of affairs in set operations of fuzzy sets, arithmetic operations of fuzzy interval, and fuzzification of crisp functions, which are frequently adopted to model engineering and economic problems with fuzzy uncertainty. In particular, the concepts of gradual sets and gradual elements are presented in order to cope with the difficulty of considering elements of fuzzy sets like considering elements of crisp sets.
Many extensions and equivalences for the essence of non-normal fuzzy sets
Generalization of extension principle
Presentation of the concepts of gradual sets and gradual elements
1. Mathematical Analysis. 1.1. Infimum and Supremum. 1.2. Limit Inferior and Limit Superior. 1.3. Semi-Continuity. 1.4. Miscellaneous. 2. Fuzzy Sets. 2.1. Membership Functions. 2.2. ¿-Level Sets. 2.3. Identical Properties. 2.4. Types of Fuzzy Sets. 3. Set Operations of Fuzzy Sets. 3.1. Complement of Fuzzy Sets. 3.2. Intersection of Fuzzy Sets. 3.3. Union of Fuzzy Sets. 3.4. Inductive and Direct Definitions. 3.5. ¿-Level Sets of Intersection and Union. 3.6. Mixed Set Operations. 4. Generalized Extension Principle. 4.1. Fuzzification of Crisp Functions. 4.2. Extension Principle Based on the Euclidean Space. 4.3. Extension Principle Based on the Topological Spaces. 4.4. Extension Principle Based on the Triangular Norms. 4.5. Generalized Extension Principle Based on the Topological Spaces. 4.6. Generalized Extension Principle Based on the Normed Spaces. 4.7. Generalized Extension Principle for Non-Normal Fuzzy Sets. 5. Arithmetics of Fuzzy Sets. 5.1. Arithmetics of Fuzzy Sets in R. 5.2. Arithmetics of Fuzzy Intervals. 5.3. Arithmetics of Fuzzy Vectors. 6. Gradual Elements and Gradual Sets. 6.1. Basic Concepts. 6.2. Set Operations Using Gradual Elements. 6.3. Arithmetics Using Gradual Numbers. 7. Generating Fuzzy Sets. 7.1. Gradual Sets and Its Rearrangement. 7.2. Nested Gradual Sets. 7.3. Countable-Based Nested Gradual Sets. 7.4. Generating Fuzzy Sets from Nested Gradual Sets. 7.5. Generating Fuzzy Sets from Countable-Based Nested Gradual Sets. 7.6. Defining Membership Functions. 7.8. Generating Fuzzy Intervals. 7.9. Elements and Subsets of Fuzzy Intervals. 7.10. Uniqueness of Construction. 8. Fuzzification of Crisp Functions. 8.1. Fuzzification Using the Extension Principle. 8.2. Fuzzification Using the Expression in Decomposition Theorem. 8.3. The Equivalences and Fuzziness. 8.4. Fuzzification Using Gradual Numbers. 9. General Fuzzification. 9.1. Fuzzification Using the Extension Principle. 9.2. Fuzzification Using the Expression in Decomposition Theorem. 9.3. Equivalence for the General Case. 9.4. Equivalence for the Practical Cases. 10. General Arithmetics of Fuzzy Sets. 10.1. Arithmetics of Fuzzy Sets in Vector Space. 10.2. Arithmetic Operations Using Compatibility and Associativity. 10.3. Equivalence with the Arithmetics Using Gradual Numbers. 10.4. Binary Operations. 10.5. Hausdorff Differences. 11. Inner Product of Fuzzy Vectors. 11.1. The First Type of Inner Product. 11.2. The Second Type of Inner Product. 12. Duality in Fuzzy Sets. 12.1. Lower and Upper Level Sets. 12.2. Dual Fuzzy Sets. 12.3. Dual Extension Principle. 12.4. Dual Arithmetics of Fuzzy Sets. 12.5. Representation Theorem for Dual-Fuzzified Function.
Hardback
ISBN 9781032791562
424 Pages 11 B/W Illustrations
Published February 20, 2025
Functional Analysis for the Applied Mathematician is a self-contained volume providing a rigorous introduction to functional analysis and its applications. Students from mathematics, science, engineering, and certain social science and interdisciplinary programs will benefit from the material. It is accessible to graduate and advanced undergraduate students with a solid background in undergraduate mathematics and an appreciation of mathematical rigor. Students are called upon to actively engage with the material, to the point of proving some of the basic results or their straightforward generalizations, both within the text and within the generous set of exercises.
Replete with exercises and examples
Suitable for graduate students and advanced undergraduates
Develops the basics of functional analysis, exploring the interplay between algebraic linear space theory and topology
Presents a variety of applications, often dealing with partial differential equations and their numerical approximation
Doubles as a reference book with an extensive index listing the concepts and results
1. Preliminaries. 2. Normed Linear Spaces and Banach Spaces. 3. Hilbert Spaces. 4. Spectral Theory and Compact Operators. 5. Distributions. 6. The Fourier Transform. 7. Sobolev Spaces. 8. Boundary Value Problems. 9. Differential Calculus in Banach Spaces. 10. The Calculus of Variations.
Hardback
ISBN 9781032933047
238 Pages 39 B/W Illustrations
Published February 21, 2025
Naive Set Theory: A Rigorous Approach aims to provide a complete and unitary presentation of naive set theory as the foundation of the whole mathematics. Suitable for undergraduate students, this book focuses on the main foundational issues, trying to clarify explicitly what is (necessarily) assumed as intuitively known and what is rigorously founded on more elementary concepts. It analyses in detail the nature of natural numbers, both as metatheoretical objects and through their set-theoretical model. The author also pays particular attention to some topics that are not usually covered by the literature on naive set theory, like the universal properties of Cartesian product and disjoint union, that lead to a precise formulation of their basic features (associativity, commutativity, and distributivity). The exposition is organized coherently from the initial meta-theoretical notions to the construction of the universe of well-founded sets.
Replete with exercises, partially spread within the text and partially listed at the end of each chapter with a solutions manual available on www.Routledge.com/9781032933047.
Covers foundational topics that are not usually discussed in the literature on naive set theory, such as universal properties of Cartesian product and disjoint union, through which one can properly state associativity, commutativity, and distributivity of these operations.
1. Preliminaries. 2. Hereditary Sets and Basic Operations. 3. Relations and Functions. 4. Sets and Indexed Sets. 5. Universal Properties. 6. Sets and Classes. 7. Embedding Natural Numbers in Set Theory. 8. Well-Orders. 9. Axioms of Choice. 10. Cardinality. 11. Ordinals. 12. Cardinals. 13. Well-Founded Sets.
Hardback
ISBN 9781032988900
202 Pages 9 B/W Illustrations
May 13, 2025
Non-Newtonian Sequence Spaces with Applications presents an alternative to the usual calculus based on multiplication instead of addition. This book is intended for graduate students and researchers with a special interest in non-Newtonian calculus, its applications, and related topics.
Preface vii
Acknowledgements ix
List of Abbreviations and Symbols x
1 Sequence and Function Spaces over the Non-newtonian ... 1
1.1 Some Basic Results on the Spaces of Sequences ... . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Preliminaries, background and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Geometric complex field and related properties . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.3 Geometric metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.4 Convergence and completeness in (GC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.1.5 Sequence spaces over C(G) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2 Some Results on Sequence Spaces with ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2.1 Preliminaries, backround and notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2.2 Non-newtonian real field and related properties . . . . . . . . . . . . . . . . . . . . . . . 13
1.2.3 Non-newtonian metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.2.4 Convergence and completeness in (NC) . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.3 Sequence Spaces Over the Non-newtonian ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.4 Certain Non-newtonian Complex Sequence Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.4.1 Preliminaries, background and notations . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.6 Some Sequence Spaces and Matrix Transformations in ... . . . . . . . . . . . . . . . . . . . . . . 29
1.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.6.2 Preliminaries, background and notations . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.6.3 Characterizations of some matrix classes . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.6.4 Multiplicative dual summability methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2 Application of Geometric Calculus in Numerical Analysis and Difference Sequence Spaces 39
2.1 Introduction and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.2 ¿-generator and Geometric Real Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.2.1 Some useful relations between geometric operations and ordinary arithmetic operations . 40
2.3 Geometric Sequence Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.4 Dual Spaces of ?G
( G) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.4.1 Geometric form of Abelfs partial summation formula . . . . . . . . . . . . . . . . . . . . 46
2.5 ¿-, À- and Á-duals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.6 Some Applications of Geometric Difference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.6.1 Geometric Newton-Gregory backward interpolation formula . . . . . . . . . . . . . . . . 53
2.6.2 Advantages of geometric interpolation formulae over ordinary interpolation formulae . . 55
2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3 Bigeometric Integral Calculus 56
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.2 Geometric Arithmetic and Geometric Real Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3 Definitions and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.3.1 G-derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.3.2 Some standard G-derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
iv
3.4 G-Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.4.1 Some standard G-integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.4.2 Integration by transforming the function to the form ex f(x)
f(x) . . . . . . . . . . . . . . . . 58
3.4.3 Integration by the relation between G-integral and ordinary integral . . . . . . . . . . . 58
3.4.4 Properties of G-integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.5 Definite Bigeometric Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.5.1 Properties of definite G-integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.5.2 Definite bigeometric integral as a limit of geometric sum . . . . . . . . . . . . . . . . . . 63
3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4 Bigeometric Calculus and Its Applications 67
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.1.1 Some useful relations between geometric operations and ordinary arithmetic operations . 67
4.2 Definitions and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.2.1 Geometric binomial formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.2.2 Geometric real number line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.2.3 Geometric coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2.4 Geometric factorial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2.5 Generalized geometric forward difference operator n
G . . . . . . . . . . . . . . . . . . . . 69
4.2.6 Generalized Geometric Backward Difference Operator Þn
G . . . . . . . . . . . . . . . . . 69
4.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.3.1 Geometric Pythagorean triplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.3.2 Geometric trigonometric ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.3.3 Relation between geometric trigonometry and ordinary trigonometry . . . . . . . . . . . 71
4.3.4 Geometric trigonometric identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.3.5 G-limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.3.6 G-continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.4 Basic Properties of G-Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.4.1 G-derivative and its interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.4.2 Relation between G-derivative and ordinary derivative . . . . . . . . . . . . . . . . . . . 77
4.4.3 G-derivatives of some standard functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.4.4 Geometric Taylorfs series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.5 Some Applications of G-Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.5.1 Expansion of some useful functions in Taylorfs product . . . . . . . . . . . . . . . . . . . 83
4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5 Solution of Bigeometric-Differential Equations by Numerical Methods 87
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.2 Basic Definitions and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.2.1 Geometric factorial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.2.2 Geometric Newton-Gregory formula for forward interpolation . . . . . . . . . . . . . . . 88
5.2.3 Geometric Newton-Gregory formula for backward interpolation . . . . . . . . . . . . . . 88
5.2.4 G-derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.2.5 Some standard G-derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.2.6 Geometric Taylorfs series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.3 Numerical Methods and Solution of G-Differential Equations . . . . . . . . . . . . . . . . . . . . 89
5.3.1 G-Eulerfs method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.3.2 Taylorfs G-series method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.3.3 G-Runge-Kutta method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.3.4 G-Runge-Kutta method of order four . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6 Certain Spaces of Functions over the Set of Non-Newtonian Complex Numbers 100
6.1 Preliminaries, Backround and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.2 The Set of ?-Complex Numbers and ?-Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.3 Continuous Function Space over the Field C? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
7 Multiplicative Type Complex Calculus 110
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
7.2 Definitions, Methods, and Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7.2.1 A multiplicative group, an additive group, and an isomorphism . . . . . . . . . . . . . . 111
7.2.2 Remoteness of two values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
7.2.3 Change rate of a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
7.2.4 Derivative and integral operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
7.2.5 Eulerfs simple method in differential equation solving . . . . . . . . . . . . . . . . . . . . 117
7.2.6 Some fundamental theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
7.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
8 Function Sequences and Series ... 124
8.1 Introduction and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
8.2 ?-Function Sequences and Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
8.2.1 ?-function sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
8.2.2 ?-function series and consequences of ?-uniform convergence . . . . . . . . . . . . . . . . 129
8.2.3 ?-uniform convergence and ?-continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
8.2.4 ?-uniform convergence and ?-integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
8.2.5 ?-Uniform Convergence and ?-Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
9 On Non-newtonian Power Series and its Applications 139
9.1 Introduction and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
9.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
9.2.1 ?-Dirichletfs and ?-Abelfs tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
9.2.2 ?-power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
Bibliography 150
Index 153
Hardback
ISBN 9781032715988
280 Pages 82 B/W Illustrations
May 1, 2025
Hardback
Available for pre-order on April 10, 2025.
Item will ship after May 1, 2025
Introduction to Qualitative Methods for Differential Equations provides an alternative approach to teaching and understanding differential equations. The basic methodology of the book is centred on finding reformulations of differential equations in such a manner that they become (partially, at least) problems in geometry. Through this approach, the book distils the critical aspects of the qualitative theory of differential equations and illustrates their application to a number of nontrivial problems.
Self-contained with suggestions for further reading
Concise and approachable exposition with only minimal pre-requisites
Ideal for self-study
Appropriate for undergraduate mathematicians, engineers, and other quantitative science students.
Preface Preliminaries Chapter 1 What is a Solution? Chapter 2 One-Dimensional Systems Chapter 3 Two-Dimensional Dynamical Systems Chapter 4 Sturm - Liouville Problems Chapter 5 Partial Differential Equations Chapter 6 Introduction to Bifurcations Chapter 7 Applications Appendix Bibliography