ISBN 9781032475103
Published January 21, 2023
526 Pages 295 B/W Illustrations
With Chromatic Graph Theory, Second Edition, the authors present various fundamentals of graph theory that lie outside of graph colorings, including basic terminology and results, trees and connectivity, Eulerian and Hamiltonian graphs, matchings and factorizations, and graph embeddings. Readers will see that the authors accomplished the primary goal of this textbook, which is to introduce graph theory with a coloring theme and to look at graph colorings in various ways.
The textbook also covers vertex colorings and bounds for the chromatic number, vertex colorings of graphs embedded on surfaces, and a variety of restricted vertex colorings. The authors also describe edge colorings, monochromatic and rainbow edge colorings, complete vertex colorings, several distinguishing vertex and edge colorings.
The book can be used for a first course in graph theory as well as a graduate course
The primary topic in the book is graph coloring
The book begins with an introduction to graph theory so assumes no previous course
The authors are the most widely-published team on graph theory
Many new examples and exercises enhance the new edition
The Origin of Graph Colorings
Introduction to Graphs
Trees and Connectivity
Eulerian and Hamiltonian Graphs
Matchings and Factorization
Graph Embeddings
Introduction to Vertex Colorings
Bounds for the Chromatic Number
Coloring Graphs on Surfaces
Restricted Vertex Colorings
Edge Colorings
Ramsey Theory
Monochromatic Ramsey Theory
Color Connection
Distance and Colorings
Domination and Colorings
Induced Colorings
The Four Color Theorem Revisited
ISBN 9781032476575
Published January 21, 2023
698 Pages 260 B/W Illustrations
Applied Differential Equations with Boundary Value Problems presents a contemporary treatment of ordinary differential equations (ODEs) and an introduction to partial differential equations (PDEs), including their applications in engineering and the sciences. This new edition of the authorfs popular textbook adds coverage of boundary value problems.
The text covers traditional material, along with novel approaches to mathematical modeling that harness the capabilities of numerical algorithms and popular computer software packages. It contains practical techniques for solving the equations as well as corresponding codes for numerical solvers. Many examples and exercises help students master effective solution techniques, including reliable numerical approximations.
This book describes differential equations in the context of applications and presents the main techniques needed for modeling and systems analysis. It teaches students how to formulate a mathematical model, solve differential equations analytically and numerically, analyze them qualitatively, and interpret the results.
First-Order Equations. Applications of First Order ODE. Mathematical Modeling and Numerical Methods. Second-order Equations. Laplace Transforms. Series of Solutions. Applications of Higher Order Differential Equations. Linear Boundary Value Problems, Introduction to Green's Functions; Nonhomogeneous Boundary Value Problems; Singular Sturm--Liouville Problems, Bessel Series Expansion, Orthogonal Polynomial Expansions; Appendix: Software Packages. Answers to Problems. Bibliography. Index
ISBN 9781032476773
Published January 21, 2023
418 Pages 46 B/W Illustrations
Introduction to Analysis is an ideal text for a one semester course on analysis. The book covers standard material on the real numbers, sequences, continuity, differentiation, and series, and includes an introduction to proof. The author has endeavored to write this book entirely from the studentfs perspective: there is enough rigor to challenge even the best students in the class, but also enough explanation and detail to meet the needs of a struggling student.
1. Sets, Functions, and Proofs; 2. The Real Numbers; 3. Sequences and their Limits; 4. Series of Real Numbes; 5. Limits and Continuity; 6. Differentiation; 7. Sequences and Series of Functions; A List of Commonly Used Symbols; Bibliography; Index
ISBN 9781032476896
Published January 21, 2023
944 Pages 893 B/W Illustrations
Discrete Mathematics and Applications, Second Edition is intended for a one-semester course in discrete mathematics. Such a course is typically taken by mathematics, mathematics education, and computer science majors, usually in their sophomore year. Calculus is not a prerequisite to use this book.
Part one focuses on how to write proofs, then moves on to topics in number theory, employing set theory in the process. Part two focuses on computations, combinatorics, graph theory, trees, and algorithms.
I Proofs
Logic and Sets
Statement Forms and Logical Equivalences
Set Notation
Quantifiers
Set Operations and Identities
Valid Arguments
Basic Proof Writing
Direct Demonstration
General Demonstration (Part 1)
General Demonstration (Part 2)
Indirect Arguments
Splitting into Cases
Elementary Number Theory
Divisors
Well-Ordering, Division, and Codes
Euclid's Algorithm and Lemma
Rational and Irrational Numbers
Modular Arithmetic and Encryption
Indexed by Integers
Sequences, Indexing, and Recursion
Sigma Notation
Mathematical Induction, An Introduction
Induction and Summations
Strong Induction
The Binomial Theorem
Relations
General Relations
Special Relations on Sets
Basics of Functions
Special Functions
General Set Constructions
Cardinality
II Combinatorics
Basic Counting
The Multiplication Principle
Permutations and Combinations
Addition and Subtraction
Probability
Applications of Combinations
Correcting for Overcounting
More Counting
Inclusion-Exclusion
Multinomial Coecients
Generating Functions
Counting Orbits
Combinatorial Arguments
Basic Graph Theory
Motivation and Introduction
Special Graphs
Matrices
Isomorphisms
Invariants
Directed Graphs and Markov Chains
Graph Properties
Connectivity
Euler Circuits
Hamiltonian Cycles
Planar Graphs
Chromatic Number
Trees and Algorithms
Trees
Search Trees
Weighted Trees
Analysis of Algorithms (Part 1)
Analysis of Algorithms (Part 2)
A Assumed Properties of Z and R
B Pseudocode
C Answers to Selected Exercises
ISBN 9781032476001
Published January 21, 2023
352 Pages
Extremal Finite Set Theory surveys old and new results in the area of extremal set system theory. It presents an overview of the main techniques and tools (shifting, the cycle method, profile polytopes, incidence matrices, flag algebras, etc.) used in the different subtopics. The book focuses on the cardinality of a family of sets satisfying certain combinatorial properties. It covers recent progress in the subject of set systems and extremal combinatorics.
Intended for graduate students, instructors teaching extremal combinatorics and researchers, this book serves as a sound introduction to the theory of extremal set systems. In each of the topics covered, the text introduces the basic tools used in the literature. Every chapter provides detailed proofs of the most important results and some of the most recent ones, while the proofs of some other theorems are posted as exercises with hints.
Basics
Spernerfs theorem, LYM-inequality, Bollobas inequality. The Erd?s-Ko-Rado theorem - several proofs. Intersecting Sperner families. Isoperimetric inequalities: the Kruskal-Katona theorem and Harperfs theorem. Sunflowers.
Intersection theorems
Stability of the Erd?s-Ko-Rado theorem. t-intersecting families. Above the Erd?s-Ko-Rado threshold. L-intersecting families. r-wise intersecting families. k-uniform intersecting families with covering number k. The number of intersecting families. Cross-intersecting families.
Sperner-type theorems
More-part Sperner families. Supersaturation. The number of antichains in 2^{[n]} (Dedekindfs problem). Union-free families and related problems. Union-closed families.
Random versions of Spernerfs theorem and the Erd?s-Ko-Rado theorem
The largest antichain in Qn (p). Largest intersecting families in Qn, k (p). Removing edges from K n (n, K). G-intersecting families. A random process generating intersecting families.
Turan-type problems
Complete forbidden hypergraphs and local sparsity. Graph-based forbidden hypergraphs. Hypergraph-based forbidden hypergraphs. Other forbidden hypergraphs. Some methods. Non-uniform Turan problems
Saturation problems
Saturated hypergraphs and weak saturation. Saturating k-Sperner families and related problems.
Forbidden subposet problems
Chain partitioning and other methods. General bounds on La(n, P) involving the height of P. Supersaturation. Induced forbidden subposet problems. Other variants of the problem. Counting other subposets.
Traces of sets
Characterizing the case of equality in the Sauer Lemma. The arrow relation. Forbidden subconfigurations. Uniform versions.
Combinatorial search theory
Basics. Searching with small query sets. Parity search. Searching with lies. Between adaptive and non-adaptive algorithms