No. of pages: 294
Language: English
Edition: 1
Published: November 29, 2024
Imprint: Academic Press
Paperback ISBN: 9780443339493
Fuzzy Mathematics, Graphs, and Similarity Measures provides a solid foundation in core analytical tracks of mathematics of uncertainty, from fuzzy mathematics to graphs and similarity measures, with application to a range of timely cases studies and world challenges. Following a full grounding in fuzzy graph indices, connectivity in fuzzy graph structures, lattice isomorphisms, and similarity measures, the book applies these models in analyzing world challenges, from human trafficking to modern slavery, global poverty, global hunger, homelessness, biodiversity, extinction, terrorism and bioterrorism, pandemics, and climate change. Connections and constructive steps forward are tied throughout to UN Sustainable Development Goals (SDGs). The authors demonstrate and instruct readers in applying techniques from mathematics of uncertainty in examining issues where accurate data is impossible to obtain. In addition to a diverse range of cases studies, exercises reinforce key concepts in each chapter, and an online instructor's manual supports teaching across a range of course contexts.
1. Preliminaries
2. Lattice Isomorphisms, Tracking, and Global Challenges
3. Fuzzy Graph Indices
4. Connectivity in Fuzzy Graph Structures
5. Global Cybersecurity
6. Terrorism and Bioterrorism
7. Health Security and Political Risk
8. Country Health
9. Mistreatment of Women and Children
10. Space Debris
11. Telecommunications
*
Paperback ISBN: 9780443314926
Difference Equations and Applications provides unique coverage of high-level topics in the application of difference equations and dynamical systems. The book begins with extensive coverage of the calculus of difference equations, including contemporary topics on l_p stability, exponential stability, and parameters that can be used to qualitatively study solutions to non-linear difference equations, including variations of parameters and equations with constant coefficients, before moving on to the Z-Transform and its various functions, scalings, and applications. It covers systems, Lyapunov functions, and stability, a subject rarely covered in competitor titles, before concluding with a comprehensive section on new variations of parameters.
Exercises are provided after each section, ranging from an easy to medium level of difficulty. When finished, students are set up to conduct meaningful research in discrete dynamical systems. In summary, this book is a comprehensive resource that delves into the mathematical theory of difference equations while highlighting their practical applications in various dynamic systems. It is highly likely to be of interest to students, researchers, and professionals in fields where discrete modeling and analysis are essential.
1. The Calculus Of Difference Equations
2. Linear Difference Equations
3. Z-Transform
4. Systems
5. Stability
6. Lyapunov Functions
7. New Variation Of Parameters
Paperback ISBN: 9780443248481
An Introduction to Discrete Mathematics offers an engaging and accessible introduction to discrete mathematics for beginning undergraduate students across a wide range of application areas, from mathematics to statistics, operations research, business, engineering, and the sciences. It provides solid foundation in precise proof writing methods, with early chapters introducing set theory and logic that are followed by deductive and inductive proof techniques, number theory, counting principles, permutations and combinations, probability of events, random variables, graphs, and weighted graphs.
The book illustrates fundamental concepts in discrete mathematics with clear and precise definitions that are paired with examples and counter-examples as applied in combinatorics, discrete probability, and graph theory. Chapters include student exercises to enhance learning, and a solutions manual and example questions are available for instructors on a companion website.
Part I - Foundations
1. Number System
2. Logic and Deduction
3. Sets, Functions, Sequences, Sums
4. Induction, Recursion
5. Elementary Number Theory
Part II - Combinatorics
6. Basic Principles of Counting
7. Permutations and Combinations
8. Further Topics in Counting
Part III - Discrete Probability
9. Probability of Events
10. Random Variables
Part IV - Graphs and Networks
11. Graphs
12. Weighted Graphs
Publication planned for: December 2024
availability: Not yet published - available from December 2024
format: Hardback
isbn: 9781009538640
Understanding the natural numbers, which we use to count things, comes naturally. Meanwhile, the real numbers, which include a wide range of numbers from whole numbers to fractions to exotic ones like π, are, frankly, really difficult to describe rigorously. Instead of waiting to take a theorem-proof graduate course to appreciate the real numbers, readers new to university-level mathematics can explore the core ideas behind the construction of the real numbers in this friendly introduction. Beginning with the intuitive notion of counting, the book progresses step-by-step to the real numbers. Each sort of number is defined in terms of a simpler kind by developing an equivalence relation on a previous idea. We find the finite sets' equivalence classes are the natural numbers. Integers are equivalence classes of pairs of natural numbers. Modular numbers are equivalence classes of integers. And so forth. Exercises and their solutions are included.
Explores the core ideas of the real numbers with a focus on definitions, rather than theorems and proofs, thus making the book accessible to readers who are new to university-level mathematics
Introduces types of numbers by developing an equivalence relations based on a previously discussed idea, which helps unify concepts and reinforces understanding of key concepts
Includes an extensive set of exercises with complete solutions, making the book perfect for self-study
Preface
0. Prelude
1. Fundamentals
2. N: natural numbers
3. Z: integers
4. Zm: modular arithmetic
5. Q: rational numbers
6. R: real numbers I, Dedekind cuts
7. R: real numbers II, Cauchy sequences
8. R: real numbers III, complete ordered fields
9. C: complex numbers
10. Further extensions
Answers to exercises
Bibliography
Index.
Publication planned for: November 2024
availability: Not yet published - available from November 2024
format: Paperback
isbn: 9781009549004
The third edition of this highly regarded text provides a rigorous, yet entertaining, introduction to probability theory and the analytic ideas and tools on which the modern theory relies. The main changes are the inclusion of the Gaussian isoperimetric inequality plus many improvements and clarifications throughout the text. With more than 750 exercises, it is ideal for first-year graduate students with a good grasp of undergraduate probability theory and analysis. Starting with results about independent random variables, the author introduces weak convergence of measures and its application to the central limit theorem, and infinitely divisible laws and their associated stochastic processes. Conditional expectation and martingales follow before the context shifts to infinite dimensions, where Gaussian measures and weak convergence of measures are studied. The remainder is devoted to the mutually beneficial connection between probability theory and partial differential equations, culminating in an explanation of the relationship of Brownian motion to classical potential theory.
Notation
1. Sums of independent random variables
2. The central limit theorem
3. Infinitely divisible laws
4. Lévy processes
5. Conditioning and martingales
6. Some extensions and applications of martingale theory
7. Continuous parameter martingales
8. Gaussian measures on a Banach space
9. Convergence of measures on a Polish space
10. Wiener measure and partial differential equations
11. Some classical potential theory
References
Index.
Part of Cambridge Tracts in Mathematics
Publication planned for: January 2025
availability: Not yet published - available from January 2025
format: Hardback
isbn: 9780521196079
Topological spaces in general, and the real numbers in particular, have the characteristic of exhibiting a 'continuity structure', one that can be examined from the vantage point of Baire category or of Lebesgue measure. Though they are in some sense dual, work over the last half-century has shown that it is the former, topological view, that has pride of place since it reveals a much richer structure that draws from, and gives back to, areas such as analytic sets, infinite games, probability, infinite combinatorics, descriptive set theory and topology. Keeping prerequisites to a minimum, the authors provide a new exposition and synthesis of the extensive mathematical theory needed to understand the subject's current state of knowledge, and they complement their presentation with a thorough bibliography of source material and pointers to further work. The result is a book that will be the standard reference for all researchers in the area.
Prologue. Regular variation
1. Preliminaries
2. Baire category and related results
3. Borel sets, analytic sets and beyond: $\Delta^1_2$
4. Infinite combinatorics in $\mathbb{R}^n$: shift-compactness
5. Kingman combinatorics and shift-compactness
6. Groups and norms: Birkhoff-Kakutani theorem
7. Density topology
8. Other fine topologies
9. Category-measure duality
10. Category embedding theorem and infinite combinatorics
11. Effros' theorem and the cornerstone theorems of functional analysis
12. Continuity and coincidence theorems
13. * Non-separable variants
14. Contrasts between category and measure
15. Interior point theorems: Steinhaus-Weil theory
16. Axiomatics of set theory
Epilogue. Topological regular variation
References
Index.