By Bela Bajnok

Additive Combinatorics
A Menu of Research Problems

ISBN 9781032476254
Published January 21, 2023
410 Pages 10 B/W Illustrations

Book Description

Additive Combinatorics: A Menu of Research Problems is the first book of its kind to provide readers with an opportunity to actively explore the relatively new field of additive combinatorics. The author has written the book specifically for students of any background and proficiency level, from beginners to advanced researchers. It features an extensive menu of research projects that are challenging and engaging at many different levels. The questions are new and unsolved, incrementally attainable, and designed to be approachable with various methods.

The book is divided into five parts which are compared to a meal. The first part is called Ingredients and includes relevant background information about number theory, combinatorics, and group theory. The second part, Appetizers, introduces readers to the bookfs main subject through samples. The third part, Sides, covers auxiliary functions that appear throughout different chapters. The bookfs main course, so to speak, is Entrees: it thoroughly investigates a large variety of questions in additive combinatorics by discussing what is already known about them and what remains unsolved. These include maximum and minimum sumset size, spanning sets, critical numbers, and so on. The final part is Pudding and features numerous proofs and results, many of which have never been published.

Table of Contents

Ingredients.Number theory. Divisibility of integers. Congruences.The Fundamental Theorem of Number Theory. Multiplicative number theory. Additive number theory. Combinatorics.Basic enumeration principles.Counting lists, sequences, sets, and multisets.Binomial coefficients and Pascalfs Triangle. Some recurrence relations. The integer lattice and its layers. Group theory. Finite abelian groups. Group isomorphisms. The Fundamental Theorem of Finite Abelian Groups. Subgroups and cosets. Subgroups generated by subsets. Sumsets. Appetizers. Spherical designs. Caps, centroids, and the game SET. How many elements does it take to span a group? In pursuit of perfection.The declaration of independence. Sides. Auxiliary functions. Entrees. Maximum sumset size. Spanning set. Sidon sets. Minimum sumset size. The critical number. Zero-sum-free sets. Sum-free sets. Pudding. Proof of Propositions and Theorems

By Craig Bauer

Discrete Encounters

ISBN 9781032474489
Published January 21, 2023
732 Pages 148 Color & 270 B/W Illustrations

Book Descriptio

Eschewing the often standard dry and static writing style of traditional textbooks, Discrete Encounters provides a refreshing approach to discrete mathematics. The author blends traditional course topics and applications with historical context, pop culture references, and open problems. This book focuses on the historical development of the subject and provides fascinating details of the people behind the mathematics, along with their motivations, deepening readersf appreciation of mathematics.

This unique book covers many of the same topics found in traditional textbooks, but does so in an alternative, entertaining style that better captures readersf attention. In addition to standard discrete mathematics material, the author shows the interplay between the discrete and the continuous and includes high-interest topics such as fractals, chaos theory, cellular automata, money-saving financial mathematics, and much more. Not only will readers gain a greater understanding of mathematics and its culture, they will also be encouraged to further explore the subject. Long lists of references at the end of each chapter make this easy.


By Karoly Bezdek, Zsolt Langi

Volumetric Discrete Geometry

ISBN 9781032475646
Published January 21, 2023 by CRC Press
306 Pages 44 B/W Illustrations

Book Description

Volume of geometric objects plays an important role in applied and theoretical mathematics. This is particularly true in the relatively new branch of discrete geometry, where volume is often used to find new topics for research. Volumetric Discrete Geometry demonstrates the recent aspects of volume, introduces problems related to it, and presents methods to apply it to other geometric problems.

Part I of the text consists of survey chapters of selected topics on volume and is suitable for advanced undergraduate students. Part II has chapters of selected proofs of theorems stated in Part I and is oriented for graduate level students wishing to learn about the latest research on the topic. Chapters can be studied independently from each other.

Table of Contents

I Selected Topics
Volumetric Properties of (m, d)-scribed Polytopes
Volume of the Convex Hull of a Pair of Convex Bodies
The Kneser-Poulsen conjecture revisited
Volumetric Bounds for Contact Numbers
More on Volumetric Properties of Separable Packings

By Jurgen Bierbrauer

Introduction to Coding Theory, 2nd Edition

ISBN 9781032477190
Published January 21, 2023
538 Pages 20 B/W Illustrations

Book Description

This book is designed to be usable as a textbook for an undergraduate course or for an advanced graduate course in coding theory as well as a reference for researchers in discrete mathematics, engineering and theoretical computer science. This second edition has three parts: an elementary introduction to coding, theory and applications of codes, and algebraic curves. The latter part presents a brief introduction to the theory of algebraic curves and its most important applications to coding theory.

Table of Contents

An Elementary Introduction to Coding. The Concept of Coding. Binary Linear Codes. General Linear Codes. Reed-Solomon Codes. Recursive Construction I. Universal Hashing. Designs and the Binary Golay Code. Shannon Entropy. Asymptotic Results. 3-Dimensional Codes, Projective Planes. Summary and Outlook. The Theory of Codes and Their Applications. Subfield Codes and Trace Codes. Cyclic Codes. Recursive Constructions, Covering Radius. OA in Statistics and Computer Science. The Geometric Description of Codes. Additive Codes. Algebraic Curves. Introduction. Applications to Coding Theory.

By Jennifer Brooks

Exploring the Infinite
An Introduction to Proof and Analysis

ISBN 9781032477046
Published January 21, 2023
300 Pages 25 B/W Illustrations

Book Description

Exploring the Infinite addresses the trend toward
a combined transition course and introduction to analysis course. It
guides the reader through the processes of abstraction and log-
ical argumentation, to make the transition from student of mathematics to
practitioner of mathematics.

This requires more than knowledge of the definitions of mathematical structures,
elementary logic, and standard proof techniques. The student focused on only these
will develop little more than the ability to identify a number of proof templates and
to apply them in predictable ways to standard problems.

This book aims to do something more; it aims to help readers learn to explore
mathematical situations, to make conjectures, and only then to apply methods
of proof. Practitioners of mathematics must do all of these things.

The chapters of this text are divided into two parts. Part I serves as an introduction
to proof and abstract mathematics and aims to prepare the reader for advanced
course work in all areas of mathematics. It thus includes all the standard material
from a transition to proof" course.

Part II constitutes an introduction to the basic concepts of analysis, including limits
of sequences of real numbers and of functions, infinite series, the structure of the
real line, and continuous functions.

Table of Contents

Fundamentals of Abstract Mathematics
Basic Notions
A First Look at Some Familiar Number Systems
Integers and natural numbers
Rational numbers and real numbers
Inequalities
A First Look at Sets and Functions
Sets, elements, and subsets
Operations with sets
Special subsets of R: intervals


By Robert Carlson

A Concrete Introduction to Real Analysis, 2nd Edition

ISBN 9781032476438
Published January 21, 2023
314 Pages 28 B/W Illustrations

Book Description

A Concrete Introduction to Analysis, Second Edition offers a major reorganization of the previous edition with the goal of making it a much more comprehensive and accessible for students.

The standard, austere approach to teaching modern mathematics with its emphasis on formal proofs can be challenging and discouraging for many students. To remedy this situation, the new edition is more rewarding and inviting. Students benefit from the text by gaining a solid foundational knowledge of analysis, which they can use in their fields of study and chosen professions.

The new edition capitalizes on the trend to combine topics from a traditional transition to proofs course with a first course on analysis. Like the first edition, the text is appropriate for a one- or two-semester introductory analysis or real analysis course. The choice of topics and level of coverage is suitable for mathematics majors, future teachers, and students studying engineering or other fields requiring a solid, working knowledge of undergraduate mathematics.

Table of Contents

Real Numbers and Mathematical Proofs. Infinite Sequences. Infinite Series. Functions. Integrals. Variations on the Riemann Sums Theme. Taylor Series and Power Series. Appendix: Solutions to Select Problems.