ISBN: 978-981-127-324-7 (hardcover)
ISBN: 978-981-127-390-2 (softcover)
This book offers a fun and enriching introduction to chaos theory, fractals and dynamical systems, and on the applications of fractals to computer generated graphics and image compression. Introduction to Chaos, Fractals and Dynamical Systems particularly focuses on natural and human phenomenon that can be modeled as fractals, using simple examples to explain the theory of chaos and how it affects all of us. Then, using straightforward mathematic and intuitive descriptions, computer generated graphics and photographs of natural scenes are used to illustrate the beauty of fractals and their importance in our world. Finally, the concept of Dynamical Systems, that is, time-dependent systems, the foundation of Chaos and Fractal, is introduced. Everyday examples are again used to illustrate concepts, and the importance of understanding how these vital systems affect our lives. Throughout the fascinating history of the evolution of chaos theory, fractals and dynamical systems is presented, along with brief introductions to the scientists, mathematicians and engineers who created this knowledge.
Introduction to Chaos, Fractals and Dynamical Systems contains ample mathematical definitions, representations, discussions and exercises, so that this book can be used as primary or secondary source in home schooling environments.
The book is suitable for homeschooling as a focused course on the subject matter or as a classroom supplement for a variety of courses at the late junior high or early high-school level. For example, in addition to a standalone course on Chaos, Fractals and Dynamical Systems (or similar title), this book could be used with the following courses:
Precalculus
Geometry
Computer programming (e.g. Rust, C, C++, Python, Java, Pascal)
Computer graphics
The text can also be used in conjunction with mathematics courses for undergraduates for non-science majors. The book can also be used for informal and lively family study and discussion.
For each chapter, exercises and things to do are included. These activities range from simple computational tasks to more elaborate computer projects, related activities, biographical research and writing assignments.
What is Chaos?
Foundations of Chaos and Fractal Theory
Chaos and Fractals in Nature
Chaos and Fractals in Human-Made Phenomena
Dynamical Systems and Systems Theory
Readership: The market consists of mathematically inclined and/or homeschooled students from grades 6 or 7 through 12, and even early undergraduate, as well as computer science students the same grade levels as above. Hobbyists of all ages. Rust community ? while this is not a book that intends to teach the Rust language, it is an impactful showcase of the language. We think we could generate some buzz for it on social media, for example by posting links to the program source code and full-color images (which are freely available online).
ISBN: 978-981-127-310-0 (hardcover)
This book serves as an introduction to graph theory and its applications. It is intended for a senior undergraduate course in graph theory but is also appropriate for beginning graduate students in science or engineering. The book presents a rigorous (proof-based) introduction to graph theory while also discussing applications of the results for solving real-world problems of interest. The book is divided into four parts. Part 1 covers the combinatorial aspects of graph theory including a discussion of common vocabulary, a discussion of vertex and edge cuts, Eulerian tours, Hamiltonian paths and a characterization of trees. This leads to Part 2, which discusses common combinatorial optimization problems. Spanning trees, shortest path problems and matroids are all discussed, as are maximum flow problems. Part 2 ends with a discussion of graph coloring and a proof of the NP-completeness of the coloring problem. Part 3 introduces the reader to algebraic graph theory, and focuses on Markov chains, centrality computation (e.g., eigenvector centrality and page rank), as well as spectral graph clustering and the graph Laplacian. Part 4 contains additional material on linear programming, which is used to provide an alternative analysis of the maximum flow problem. Two appendices containing prerequisite material on linear algebra and probability theory are also provided.
Introduction to Graphs:
Introduction to Graph Theory
Degree Sequences and Subgraphs
Walks, Cycles, Cuts and Centrality
Bipartite, Acyclic and Eulerian Graphs
Graph Optimization and NP Completeness:
Trees, Algorithms and Matroids
An Introduction to Network Flows and Combinatorial Optimization
Coloring
Introduction to Algebraic Graph Theory:
Algebraic Graph Theory with Abstract Algebra
Algebraic Graph Theory with Linear Algebra
Applications of Algebraic Graph Theory
Other Topics:
A Brief Introduction to Linear Programming
Max Flow / Min Cut with Linear Programming
Appendices:
Advanced Undergraduate Students or Beginning Graduate Students in Mathematics (those who have taken a first course in proofs). Graduate Students in STEM who want a rigorous text on graph theory that also focuses on applications. This could be used as a secondary text in a physics course on Network Science, or potentially in a rigorous course in theoretical computer science or operations research with graph theory.
Pages: 270
ISBN: 978-981-127-298-1 (hardcover)
ISBN: 978-981-127-389-6 (softcover)
This volume contains more than 900 problems in differential calculus, covering limits, continuity, derivatives, and their applications. The applications are comprised of a variety of approximations, growth and decay, optimization, curve sketching techniques, and analytical tools to investigate properties of parametrically given planar curves. The problems are sorted by topic, each opening with with a summary of the relevant mathematical notions and their properties. Through a careful selection of appropriate problems in each chapter, the book clearly communicates some of the big ideas and applications in calculus: the notion of a function, the notion of an infinitesimal, the notion of a differentiable function, and the notion of an approximation, among others. The book provides the answers to each problem, often with a detailed sketch of the solution process.
With about 260 true-false and multiple-choice questions, the book provides its users with an accessible way to assess and practice their understanding of calculus related facts and nuances. More than 180 figures are included to help readers to visualize properties of functions, illustrate word problems, depict solutions, and provide an extensive bank of polar curves.
The purpose of this problem collection is to serve as a supplementary learning resource for students who are studying university-level differential calculus. The book also acts as a teaching resource for calculus instructors.
Preliminaries
Limits and Continuity
Derivatives
Functions and Their Graphs
Optimization
Other Applications of Differentiation
Parametric Equations and Polar Curves
True-False and Multiple-Choice Questions
Recommendations for Success in Mathematics
High school and undergraduate students taking a first-level calculus course as well as high school teachers and calculus instructors. Anyone, e.g., employees in industry who need a refresher in differential calculus, or as a quick reference, or need to further their understanding in certain areas of calculus.
Pages: 220
ISBN: 978-981-127-280-6 (hardcover)
ISBN: 978-981-127-410-7 (softcover)
The book constitutes a basic, concise, yet rigorous first course in complex analysis, for undergraduate students who have studied multivariable calculus and linear algebra. The textbook should be particularly useful for students of joint programmes with mathematics, as well as engineering students seeking rigour. The aim of the book is to cover the bare bones of the subject with minimal prerequisites. The core content of the book is the three main pillars of complex analysis: the Cauchy-Riemann equations, the Cauchy Integral Theorem, and Taylor and Laurent series. Each section contains several problems, which are not drill exercises, but are meant to reinforce the fundamental concepts. Detailed solutions to all the 243 exercises appear at the end of the book, making the book ideal for self-study. There are many figures illustrating the text.
The second edition corrects errors from the first edition, and includes 89 new exercises, some of which cover auxiliary topics that were omitted in the first edition. Two new appendices have been added, one containing a detailed rigorous proof of the Cauchy Integral Theorem, and another providing background in real analysis needed to make the book self-contained.
Complex Numbers and Their Geometry
Complex Differentiability
Cauchy Integral Theorem
Taylor and Laurent Series
Harmonic Functions
Solutions to All Exercises
Undergraduate students in complex analysis.