Softcover ISBN: 978-1-4704-7204-7
Product Code: AMSTEXT/63
Pure and Applied Undergraduate Texts Volume: 63;
2023; 523 pp
MSC: Primary 00;
This textbook bridges the gap between lower-division mathematics courses and advanced mathematical thinking. Featuring clear writing and appealing topics, the book introduces techniques for writing proofs in the context of discrete mathematics. By illuminating the concepts behind techniques, the authors create opportunities for readers to sharpen critical thinking skills and develop mathematical maturity.
Beginning with an introduction to sets and logic, the book goes on to establish the basics of proof techniques. From here, chapters explore proofs in the context of number theory, combinatorics, functions and cardinality, and graph theory. A selection of extension topics concludes the book, including continued fractions, infinite arithmetic, and the interplay among Fibonacci numbers, Pascal's triangle, and the golden ratio.
A Discrete Transition to Advanced Mathematics is suitable for an introduction to proof course or a course in discrete mathematics. Abundant examples and exercises invite readers to get involved, and the wealth of topics allows for course customization and further reading. This new edition has been expanded and modernized throughout, featuring:
A new chapter on combinatorial geometry;
An expanded treatment of the combinatorics of indistinguishable objects;
New sections on the inclusion-exclusion principle and circular permutations;
Over 365 new exercises.
Ancillaries:
Student Solutions Manual (Selected Problems)
Instructor's Manual
Readership
Undergraduate students who need a strong conceptual foundation for higher mathematical thinking.
Preface
Preface to the Second Edition
Chapter 1. Sets and Logic
1.1. Sets
1.2. Set Operations
1.3. Partitions
1.4. Logic and Truth Tables
1.5. Quantifiers
1.6. Implications
Chapter 2. Proofs
2.1. Proof Techniques
2.2. Mathematical Induction
2.3. The Pigeonhole Principle
Chapter 3. Number Theory
3.1. Divisibility
3.2. The Euclidean Algorithm
3.3. The Fundamental Theorem of Arithmetic
3.4. Divisibility Tests
3.5. Number Patterns
Chapter 4. Combinatorics
4.1. Getting from Point A to Point B
4.2. The Fundamental Principle of Counting
4.3. A Formula for the Binomial Coefficients
4.4. Permutations with Indistinguishable Objects
4.5. Combinations with Indistinguishable Objects
4.6. The Inclusion-Exclusion Principle
4.7. Circular Permutations
4.8. Probability
Chapter 5. Relations
5.1. Relations
5.2. Equivalence Relations
5.3. Partial Orders
5.4. Quotient Spaces
Chapter 6. Functions and Cardinality
6.1. Functions
6.2. Inverse Relations and Inverse Functions
6.3. Cardinality of Infinite Sets
6.4. An Order Relation for Cardinal Numbers
Chapter 7. Graph Theory
7.1. Graphs
7.2. Matrices, Digraphs, and Relations
7.3. Shortest Paths in Weighted Graphs
7.4. Trees
Chapter 8. Sequences
8.1. Sequences
8.2. Finite Differences
8.3. Limits of Sequences of Real Numbers
8.4. Some Convergence Properties
8.5. Infinite Arithmetic
8.6. Recurrence Relations
Chapter 9. Fibonacci Numbers and Pascal’s Triangle
9.1. Pascal’s Triangle
9.2. The Fibonacci Numbers
9.3. The Golden Ratio
9.4. Fibonacci Numbers and the Golden Ratio
9.5. Pascal’s Triangle and the Fibonacci Numbers
Chapter 10. Combinatorial Geometry in the Plane
10.1. Polygons and Convex Sets
10.2. Pick’s Theorem
10.3. Irrational Approximations of ??
10.4. Cotes’s Theorem (optional)
10.5. Tiling and Visibility
10.6. Covering Properties and Geometry of Point Sets
10.7. Linear Algebra and Packing the Plane
10.8. Helly’s Theorem
Chapter 11. Continued Fractions
11.1. Finite Continued Fractions
11.2. Convergents of a Continued Fraction
11.3. Infinite Continued Fractions
11.4. Applications of Continued Fractions
Answers or Hints for Selected Exercises
Bibliography
Index
Softcover ISBN: 978-1-4704-6916-0
Product Code: PSAPM/79
Proceedings of Symposia in Applied Mathematics Volume: 79;
2023
MSC: Primary 00; 97; Secondary 54; 14; 46; 20;
This volume is based on lectures delivered at the 2022 AMS Short Course “3D Printing: Challenges and Applications” held virtually from January 3?4, 2022.
Access to 3D printing facilities is quickly becoming ubiquitous across college campuses. However, while equipment training is readily available, the process of taking a mathematical idea and making it into a printable model presents a big hurdle for most mathematicians. Additionally, there are still many open questions around what objects are possible to print, how to design algorithms for doing so, and what kinds of geometries have desired kinematic properties. This volume is focused on the process and applications of 3D printing for mathematical education, research, and visualization, alongside a discussion of the challenges and open mathematical problems that arise in the design and algorithmic aspects of 3D printing.
The articles in this volume are focused on two main topics. The first is to make a bridge between mathematical ideas and 3D visualization. The second is to describe methods and techniques for including 3D printing in mathematical education at different levels? from pedagogy to research and from demonstrations to individual projects. We hope to establish the groundwork for engaged academic discourse on the intersections between mathematics, 3D printing and education.
Undergraduate and graduate students and researchers interested in 3D printing technology, art, and mechanical designs.
Hardcover ISBN: 978-1-4704-7127-9
Product Code: GSM/233
Graduate Studies in Mathematics Volume: 233;
2023; 373 pp
MSC: Primary 13; 11; 14;
This book provides an introduction to classical methods in commutative algebra and their applications to number theory, algebraic geometry, and computational algebra. The use of number theory as a motivating theme throughout the book provides a rich and interesting context for the material covered. In addition, many results are reinterpreted from a geometric perspective, providing further insight and motivation for the study of commutative algebra.
The content covers the classical theory of Noetherian rings, including primary decomposition and dimension theory, topological methods such as completions, computational techniques, local methods and multiplicity theory, as well as some topics of a more arithmetic nature, including the theory of Dedekind rings, lattice embeddings, and Witt vectors. Homological methods appear in the author's sequel, Homological Methods in Commutative Algebra (Graduate Studies in Mathematics, Volume 234).
Overall, this book is an excellent resource for advanced undergraduates and beginning graduate students in algebra or number theory. It is also suitable for students in neighboring fields such as algebraic geometry who wish to develop a strong foundation in commutative algebra. Some parts of the book may be useful to supplement undergraduate courses in number theory, computational algebra or algebraic geometry. The clear and detailed presentation, the inclusion of computational techniques and arithmetic topics, and the numerous exercises make it a valuable addition to any library.
Graduate students and researchers interested in commutative algebra.
Chapters
Basics
Finiteness conditions
Factorization
Computational methods
Integral dependence
Lattice methods
Metric and topological methods
Geometric dictionary
Dimension theory
Local structure
Fields
Hardcover ISBN: 978-1-4704-7128-6
Product Code: GSM/234
Graduate Studies in Mathematics Volume: 234;
2023
MSC: Primary 13;
This book develops the machinery of homological algebra and its applications to commutative rings and modules. It assumes familiarity with basic commutative algebra, for example, as covered in the author's book, Commutative Algebra (Graduate Studies in Mathematics, Volume 233).
The first part of the book is an elementary but thorough exposition of the concepts of homological algebra, starting from categorical language up to the construction of derived functors and spectral sequences. A full proof of the celebrated Freyd-Mitchell theorem on the embeddings of small Abelian categories is included.
The second part of the book is devoted to the application of these techniques in commutative algebra through the study of projective, injective, and flat modules, the construction of explicit resolutions via the Koszul complex, and the properties of regular sequences. The theory is then used to understand the properties of regular rings, Cohen-Macaulay rings and modules, Gorenstein rings and complete intersections.
Overall, this book is a valuable resource for anyone interested in learning about homological algebra and its applications in commutative algebra. The clear and thorough presentation of the material, along with the many examples and exercises of varying difficulty, make it an excellent choice for self-study or as a reference for researchers.
Graduate students and researchers interested in commutative algebra.
Paperback ISBN: 9780443189692
- November 1, 2023
Probability and Statistics for Physical Sciences, Second Edition is an accessible guide to commonly used concepts and methods in statistical analysis used in the physical sciences. This brief yet systematic introduction explains the origin of key techniques, providing mathematical background and useful formulas. The text does not assume any background in statistics and is appropriate for a wide-variety of readers, from first-year undergraduate students to working scientists across many disciplines.
DATE PUBLISHED: July 2023
Hardback ISBN: 9781009419741
Acta Numerica is an annual publication containing invited survey papers by leading researchers in numerical mathematics and scientific computing. The papers present overviews of recent developments in their area and provide state-of-the-art techniques and analysis.
The latest issue of the leading review in mathematics as measured by Impact factor
Outstanding contributors provide state-of-art surveys in important topics of contemporary interest
Covers a broad range of fields from data-driven science, to engineering, to computational physics
1. Low-rank tensor methods for partial differential equations Markus Bachmayr
2. The virtual element method Lourenco Beirao da Veiga, Franco Brezzi, L. Donatella Marini and Alessandro Russo
3. Floating-point arithmetic Sylvie Boldo, Claude-Pierre Jeannerod, Guillaume Melquiond and Jean-Michel Muller
4. Compatible finite element methods for geophysical fluid dynamics Colin J. Cotter
5. Control of port-Hamiltonian differential-algebraic systems and applications Volker Mehrmann and Benjamin Unger
6. Overcoming the timescale barrier in molecular dynamics: transfer operators, variational principles and machine learning Christof Schutte, Stefan Klus and Carsten Hartmann
7. Linear optimization over homogeneous matrix cones Levent Tuncel and Lieven Vandenberghe