Copyright 2025
Paperback
Hardback
ISBN 9781032940243
164 Pages
Published February 26, 2025 by Chapman & Hall
This text focuses on the primary topics in a first course in Linear Algebra. The author includes additional advanced topics related to data analysis, singular value decomposition, and connections to differential equations. This is a lab text that would lead a class through Linear Algebra using MathematicaR demonstrations and MathematicaR coding.
The book includes interesting examples embedded in the projects. Examples include the discussions of “Lights Out”, Nim, the Hill Cipher, and a variety of relevant data science projects.
Additional Theorems and Problems for students to prove/disprove (these act as theory exercises at the end of most sections of the text)
Additional sections that support Data Analytics techniques, such as Kronecker sums and products, and LU decomposition of the Vandermonde matrix
Updated and expanded end-of-chapter projects
Instructors and students alike have enjoyed this popular book, as it offers the opportunity to add MathematicaR to the Linear Algebra course.
I would definitely use the book (specifically the projects at the end of each section) to motivate undergraduate research.?Nick Luke, North Carolina A&T State University.
1. Matrix Operations
2. Invertibility
3. Vector Spaces
4. Orthogonality
5. Matrix Decomposition with Applications
6. Applications to Differential Equations
Copyright 2025
Hardback
ISBN 9781032302706
566 Pages
Published March 11, 2025 by Chapman & Hall
This award-winning textbook targets the gap between introductory texts in discrete mathematics and advanced graduate texts in enumerative combinatorics. The author’s goal is to make combinatorics more accessible to encourage student interest and to expand the number of students studying this rapidly expanding field.
The book first deals with basic counting principles, compositions and partitions, and generating functions. It then focuses on the structure of permutations, graph enumeration, and extremal combinatorics. Lastly, the text discusses supplemental topics, including error-correcting codes, properties of sequences, and magic squares.
Quick Check exercises at the end of each section, which are typically easier than the regular exercises at the end of each chapter.
A new section discussing the Lagrange Inversion Formula and its applications, strengthening the analytic flavor of the book.
An extended section on multivariate generating functions.
Numerous exercises contain material not discussed in the text allowing instructors to extend the time they spend on a given topic. A chapter on analytic combinatorics and sections on advanced applications of generating functions, demonstrating powerful techniques that do not require the residue theorem or complex integration, and extending coverage of the given topics are highlights of the presentation.
The second edition was recognized as an Outstanding Academic Title of the Year by Choice Magazine, published by the American Library Association.
Basic methods
When we add and when we subtract
When we multiply
When we divide
Applications of basic counting principles
The pigeonhole principle
Notes
Chapter review
Exercises
Solutions to exercises
Supplementary exercises
Applications of basic methods
Multisets and compositions
Set partitions
Partitions of integers
The inclusion-exclusion principle
The twelvefold way
Notes
Chapter review
Exercises
Solutions to exercises
Supplementary exercises
Generating functions
Power series
Warming up: Solving recurrence relations
Products of generating functions
Compositions of generating functions
A different type of generating functions
Notes
Chapter review
Exercises
Solutions to exercises
Supplementary exercises
TOPICS
Counting permutations
Eulerian numbers
The cycle structure of permutations
Cycle structure and exponential generating functions
Inversions
Advanced applications of generating functions to permutation enumeration
Notes
Chapter review
Exercises
Solutions to exercises
Supplementary exercises
Counting graphs
Trees and forests
Graphs and functions
When the vertices are not freely labeled
Graphs on colored vertices
Graphs and generating functions
Notes
Chapter review
Exercises
Solutions to exercises
Supplementary exercises
Extremal combinatorics
Extremal graph theory
Hypergraphs
Something is more than nothing: Existence proofs
Notes
Chapter review
Exercises
Solutions to exercises
Supplementary exercises
AN ADVANCED METHOD
Analytic combinatorics
Exponential growth rates
Polynomial precision
More precise asymptotics
Notes
Chapter review
Exercises
Solutions to exercises
Supplementary exercises
SPECIAL TOPICS
Symmetric structures
Designs
Finite projective planes
Error-correcting codes
Counting symmetric structures
Notes
Chapter review
Exercises
Solutions to exercises
Supplementary exercises
Sequences in combinatorics
Unimodality
Log-concavity
The real zeros property
Notes
Chapter review
Exercises
Solutions to exercises
Supplementary exercises
Counting magic squares and magic cubes
A distribution problem
Magic squares of fixed size
Magic squares of fixed line sum
Why magic cubes are different
Notes
Chapter review
Exercises
Solutions to exercises
Supplementary exercises
Appendix: The method of mathematical induction
Weak induction
Strong induction
Copyright 2025
Hardback
ISBN 9781032494449
520 Pages 11 B/W Illustrations
Published March 14, 2025 by Chapman & Hall
The concept of completely regular codes was introduced by Delsarte in his celebrated 1973 thesis, which created the field of Algebraic Combinatorics. This notion was extended by several authors from classical codes over finite fields to codes in distance-regular graphs. Half a century later, there was no book dedicated uniquely to this notion. Most of Delsarte examples were in the Hamming and Johnson graphs. In recent years, many examples were constructed in other distance regular graphs including q-analogues of the previous, and the Doob graph.
Completely Regular Codes in Distance Regular Graphs provides, for the first time, a definitive source for the main theoretical notions underpinning this fascinating area of study. It also supplies several useful surveys of constructions using coding theory, design theory and finite geometry in the various families of distance regular graphs of large diameters.
Written by pioneering experts in the domain
Suitable as a research reference at the master’s level
Includes extensive tables of completely regular codes in the Hamming graph
Features a collection of up-to-date surveys
Introduction
Chapter 1: Completely regular codes and equitable partitions
Denis S. Krotov, Vladimir N. Potapov
Chapter 2: Completely regular codes over finite fields
Victor A. Zinoviev
Chapter 3: Completely regular codes in the Johnson graph
Alexander Gavrilyuk, Victor A. Zinoviev
Chapter 4: Codes over rings and modules
Minjia Shi
Chapter 5: Group actions on codes in graphs
Daniel R. Hawtin and Cheryl E. Praeger
Chapter 6: Some completely regular codes in Doob graphs
Evgeny A. Bespalov, Denis S. Krotov
Chapter 7: Completely regular codes: tables of small parameters for binary and ternary Hamming graphs
Jack H. Koolen, Denis S. Krotov, William J. Martin
Copyright 2025
Paperback
Hardback
ISBN 9781041011750
221 Pages 81 B/W Illustrations
May 22, 2025 by Chapman & Hall
This is a unique book that teaches mathematics and its history simultaneously. Developed from a course on the history of mathematics, this book is aimed at mathematics teachers who need to learn more about mathematics than its history, and in a way they can communicate it to middle and high school students. The author hopes to overcome, through the teachers using this book, math phobia among these students.
Number Theory and Geometry through History develops an appreciation of mathematics by not only looking at the work of individual, including Euclid, Euler, Gauss, and more, but also how mathematics developed from ancient civilizations. Brahmins (Hindu priests) devised our current decimal number system now adopted throughout the world. The concept of limit, which is what calculus is all about, was not alien to ancient civilizations as Archimedes used a method similar to the Riemann sums to compute the surface area and volume of the sphere.
No theorem here is cited in a proof that has not been proved earlier in the book. There are some exceptions when it comes to the frontier of current research.
Appreciating mathematics requires more than thoughtlessly reciting first the ten by ten, then twenty by twenty multiplication tables. Many find this approach fails to develop an appreciation for the subject. The author was once one of those students. Here he exposes how he found joy in studying mathematics, and how he developed a lifelong interest in it he hopes to share.
The book is suitable for high school teachers as a textbook for undergraduate students and their instructors. It is a fun text for advanced readership interested in mathematics.
I Arithmetic
1 What is a Number?
1.1 Various Numerals to Represent
2 Arithmetic in Different Bases
3 Arithmetic in Euclid’s Elements
4 Gauss?Advent of Modern Number Theory
4.1 Number Theory of Gauss
4.2 Cryptography
4.3 Complex Numbers
4.4 Application of Number Theory ? Construction of Septadecagon
4.5 How Did Gauss Do It?
4.6 Equations over Finite Fields*
4.7 Law of Quadratic Reciprocity*
4.8 Cubic Equations*
4.9 Riemann Hypothesis*
5 Numbers beyond Rationals
5.1 Arithmetic of Rational Numbers
5.2 Real Numbers
II Geometry
6 Basic Geometry
7 Greece: Beginning of Theoretical Mathematics
8 Euclid: The Founder of Pure Mathematics
8.1 Some Comments on Euclid’s Proof
9 Famous Problems from Greek Geometry
III Contributions of Some Prominent Mathematicians
10 Fibonacci’s Time and Legacy
10.1 Liber Abaci
10.2 Liber Quadratorum
10.3 Equivalent Formulations of the Problems
11 Solution of the Cubic
11.1 Introduction
11.2 History
12 Leibniz, Newton, and Calculus
12.1 Differential Calculus
12.2 Integral Calculus
12.3 Proof of FTC
12.4 Application of FTC
13 Euler and Modern Mathematics
13.1 Algebraic Number Theory
13.2 Analytical Number Theory
13.3 Euler’s Discovery of eπi + 1 = 0
13.4 Graph Theory and Topology
13.5 Traveling Salesman Problem
13.6 Planar Graphs
13.7 Euler-Poincare Characteristic
13.8 Euler Characteristic Formula
14 Non-European Roots of Mathematics
15 Mathematics of the 20th Century*
15.1 Hilbert’s 23 Problems
15.2 Fermat’s Last Theorem
15.3 Miscellaneous
Copyright 2025
Hardback
ISBN 9781032765549
583 Pages 174 B/W Illustrations
May 27, 2025 by Chapman & Hall
This classic text appears here in a new edition for the first time in four decades. The new edition, with the aid of two new authors, brings it up to date for a new generation of mathematicians and mathematics students.
Elements of Algebraic Topology provides the most concrete approach to the subject. With coverage of homology and cohomology theory, universal coefficient theorems, Kunneth theorem, duality in manifolds, and applications to classical theorems of point-set topology, this book is perfect for communicating complex topics and the fun nature of algebraic topology for beginners.
This second edition retains the essential features of the original book. Most of the notation and terminology are the same. There are some useful additions. There is a new introduction to homotopy theory. A new Index of Notation is included. Many new exercises are added.
Algebraic topology is a cornerstone of modern mathematics. Every working mathematician should have at least an acquaintance with the subject. This book, which is based largely on the theory of triangulations, provides such an introduction. It should be accessible to a broad cross-section of the profession?both students and senior mathematicians. Students should have some familiarity with general topology.
1 Homology Groups of a Simplicial Complex
1.1 Introduction
1.2 Simplices
1.3 Simplicial Complexes and Simplicial Maps
1.4 Abstract Simplicial Complexes
1.5 Review of Abelian Groups
1.6 Homology Groups
1.7 Homology Groups of Surfaces
1.8 Zero-Dimensional Homology
1.9 The Homology of a Cone
1.10 Relative Homology
1.11 ?Homology with Arbitrary Coefficients
1.12 ?The Computability of Homology Groups
1.13 Homomorphisms Induced by Simplicial Maps
1.14 Chain Complexes and Acyclic Carriers
2 Topological Invariance of the Homology Groups
2.1 Introduction
2.2 Simplicial Approximations
2.3 Barycentric Subdivision
2.4 The Simplicial Approximation Theorem
2.5 The Algebra of Subdivision
2.6 Topological Invariance of the Homology Groups
2.7 Homomorphisms Induced by Homotopic Maps
2.8 Review of Quotient Spaces
2.9 ?Application: Maps of Spheres
2.10 ?The Lefschetz Fixed Point Theorem
3 Relative Homology and the Eilenberg?Steenrod Axioms
3.1 Introduction
3.2 The Exact Homology Sequence
3.3 The Zig-Zag Lemma
3.4 The Mayer?Vietoris Sequence
3.5 The Eilenberg?Steenrod Axioms
3.6 The Axioms for Simplicial Theory
3.7 ?Categories and Functors
4 Singular Homology Theory
4.1 Introduction
4.2 The Singular Homology Groups
4.3 The Axioms for Singular Theory
4.4 Excisionin Singular Homology
4.5 ?Acyclic Models
4.6 Mayer?Vietoris Sequences
4.7 The Isomorphism Between Simplicial and Singular Homology
4.8 ?Application: Local Homology Groups and Manifolds
4.9 ?Application: The Jordan Curve Theorem
4.10 The Fundamental Group
4.11 More on Quotient Spaces
4.12 CW Complexes
4.13 The Homology of CW Complexes
4.14 ?Application: Projective Spaces and Lens Spaces
5 Cohomology
5.1 Introduction
5.2 The Hom Functor
5.3 Simplicial Cohomology Groups
5.4 Relative Cohomology
5.5 Cohomology Theory
5.6 The Cohomology of Free Chain Complexes
5.7 ?Chain Equivalences in Free Chain Complexes
5.8 The Cohomology of CW Complexes
5.9 Cup Products
5.10 Cohomology Rings of Surfaces
6 Homology with Coefficients
6.1 Introduction
6.2 Tensor Products
6.3 Homology with Arbitrary Coefficients
7 Homological Algebra
7.1 Introduction
7.2 The Ext Functor
7.3 The Universal Coefficient Theorem
7.4 Torsion Products
7.5 The Universal Coefficient Theorem for Homology
7.6 ?Other Universal Coefficient Theorems
7.7 Tensor Products of Chain Complexes
7.8 The Kunneth Theorem
7.9 TheEilenberg?Zilber Theorem
7.10 ?The Kunneth Theorem for Cohomolgy
7.11 ?Application: The Cohomology Ring of a Product Space
8 Duality in Manifolds
8.1 Introduction
8.2 The Join of Two Complexes
8.3 Homology Manifolds
8.4 The Dual Block Complex
8.5 Poincare Duality
8.6 Cap Products
8.7 A Second Proof of Poincare Duality
8.8 ?Application: Cohomology Rings of Manifolds
8.9 ?Application: Homotopy Classification of Lens Spaces
8.10 Lefschetz Duality
8.11 Alexander Duality
8.12 Natural Versions of Duality
8.13 ?ech Cohomology
8.14 Alexander?Pontryagin Duality