MAA Press: An Imprint of the American Mathematical Society
AMS/MAA Textbooks, Volume: 69
2022; 124 pp; Softcover
MSC: Primary 97; 01;
Print ISBN: 978-1-4704-6504-9
Product Code: TEXT/69
The Number Line through Guided Inquiry is designed to give future secondary teachers a deep understanding of the real numbers and functions on the reals. By presenting just that part of the subject that underlies the high school curriculum, this book offers an alternative to a standard real analysis sequence for advanced undergraduate or beginning graduate students. It will give any student a much deeper understanding of the mathematics that they were taught in high school.
Written in a guided-inquiry format, this book consists of a carefully scaffolded sequence of definitions, problems, and theorems that guides students through each topic. Readers solve the problems and prove the theorems on their own and present their results to their peers with the instructor as a mentor and a guide. Students will learn not only the mathematics, but also how to help others learn mathematics. They will learn to think creatively and to make compelling arguments to justify their conclusions. They will learn to listen critically to others and give constructive feedback. Ultimately, they will learn to work as a team to answer the bigger questions and build a common understanding of the broader subject.
Graduate students interested in secondary school mathematics and general undergraduate mathematics majors.
MAA Press: An Imprint of the American Mathematical Society
Spectrum Volume: 102
2021; 184 pp; Softcover
MSC: Primary 00;
Print ISBN: 978-1-4704-6736-4
Product Code: SPEC/102
The vibrant recreational mathematics culture of Japan presents puzzles that are often quite different from the classics of western literature. This book is the first collection of original puzzles by Tadao Kitazawa, a prominent Japanese puzzle-maker. These puzzles, which feature arithmetic, geometry, and combinatorics, are novel, creative, and require almost no formal mathematical knowledge. Kitazawa is particularly skillful in subtly modifying existing ideas to explore their potential to the full. For one example, a Tower Square is a Sudoku-like grid, but each row and column contains one 1, two 2s, three 3s, etc. The resulting transformation of the familiar problem is magical, and it is one of a variety of gems in this book. The common denominator is fun!
Undergraduate and graduate students and researchers interested in recreational math.
Mathematical Surveys and Monographs Volume: 262
2021; 175 pp; Softcover
MSC: Primary 13; 16; 18;
Print ISBN: 978-1-4704-5340-4
Product Code: SURV/262
This book is a lightly edited version of the unpublished manuscript Maximal Cohen?Macaulay modules and Tate cohomology over Gorenstein rings by Ragnar-Olaf Buchweitz. The central objects of study are maximal Cohen?Macaulay modules over (not necessarily commutative) Gorenstein rings. The main result is that the stable category of maximal Cohen?Macaulay modules over a Gorenstein ring is equivalent to the stable derived category and also to the homotopy category of acyclic complexes of projective modules. This assimilates and significantly extends earlier work of Eisenbud on hypersurface singularities. There is also an extensive discussion of duality phenomena in stable derived categories, extending Tate duality on cohomology of finite groups. Another noteworthy aspect is an extension of the classical BGG correspondence to super-algebras. There are numerous examples that illustrate these ideas. The text includes a survey of developments subsequent to, and connected with, Buchweitz's manuscript.
Graduate students and researchers interested in commutative algebra, category theory, and applications to quantum field theory.
Mathematical Surveys and Monographs Volume: 263
2021; 186 pp; Softcover
MSC: Primary 26; 41; 39; 46; 47; 65; 68; 92;
Print ISBN: 978-1-4704-6765-4
Product Code: SURV/263
Recent years have witnessed a growth of interest in the special functions called ridge functions. These functions appear in various fields and under various guises. They appear in partial differential equations (where they are called plane waves), in computerized tomography, and in statistics. Ridge functions are also the underpinnings of many central models in neural network theory. In this book various approximation theoretic properties of ridge functions are described. This book also describes properties of generalized ridge functions, and their relation to linear superpositions and Kolmogorov's famous superposition theorem. In the final part of the book, a single and two hidden layer neural networks are discussed. The results obtained in this part are based on properties of ordinary and generalized ridge functions. Novel aspects of the universal approximation property of feedforward neural networks are revealed.
This book will be of interest to advanced graduate students and researchers working in functional analysis, approximation theory, and the theory of real functions, and will be of particular interest to those wishing to learn more about neural network theory and applications and other areas where ridge functions are used.
Graduate students and researchers interested in neural networks and approximation theory.
Mathematical Surveys and Monographs Volume: 264
2021; 365 pp; Softcover
MSC: Primary 18; 55; 57;
Print ISBN: 978-1-4704-6671-8
Product Code: SURV/264
This book is an introduction to techniques and results in diagrammatic algebra. It starts with abstract tensors and their categorifications, presents diagrammatic methods for studying Frobenius and Hopf algebras, and discusses their relations with topological quantum field theory and knot theory. The text is replete with figures, diagrams, and suggestive typography that allows the reader a glimpse into many higher dimensional processes. The penultimate chapter summarizes the previous material by demonstrating how to braid 3- and 4- dimensional manifolds into 5- and 6-dimensional spaces.
The book is accessible to post-qualifier graduate students, and will also be of interest to algebraists, topologists and algebraic topologists who would like to incorporate diagrammatic techniques into their research.
Graduate students and researchers interested in homological algebra, category theory, and diagrammatic calculus.
MAA Press: An Imprint of the American Mathematical Society
AMS/MAA Textbooks Volume: 70
2022; Softcover
MSC: Primary 26; 37; 39;
Print ISBN: 978-1-4704-6454-7
Product Code: TEXT/70
Welcome to Real Analysis is designed for use in an introductory undergraduate course in real analysis. Much of the development is in the setting of the general metric space. The book makes substantial use not only of the real line and n-dimensional Euclidean space, but also sequence and function spaces. Proving and extending results from single-variable calculus provides motivation throughout. The more abstract ideas come to life in meaningful and accessible applications. For example, the contraction mapping principle is used to prove an existence and uniqueness theorem for solutions of ordinary differential equations and the existence of certain fractals; the continuity of the integration operator on the space of continuous functions on a compact interval paves the way for some results about power series.
The exposition is exceedingly clear and well-motivated. There are a wide variety of exercises and many pedagogical innovations. For example, each chapter includes Reading Questions so that students can check their understanding. In addition to the standard material in a first real analysis course, the book contains two concluding chapters on dynamical systems and fractals as an illustration of the power of the theory developed.
Readership
Undergraduate students interested in learning real analysis.