This book contains the definitions of several ring constructions used in various applications. The concept of a groupoid-graded ring includes many of these constructions as special cases and makes it possible to unify the exposition. Recent research results on groupoid-graded rings and more specialized constructions are presented. In addition, there is a chapter containing open problems currently considered in the literature. Ring Constructions and Applications can serve as an excellent introduction for graduate students to many ring constructions as well as to essential basic concepts of group, semigroup and ring theories used in proofs.

State-of-the-science methods, synthetic routes, and strategies to construct aromatic rings The development of new reactions for the synthesis of aromatic compounds is a highly active research area in organic synthesis, providing new functional organic materials, functional reagents, and biologically active compounds. Recently, significant advances in transition-metal-mediated reactions have enabled the efficient and practical construction of new aromatic rings with useful properties and applications. This book draws together and reviews all the latest discoveries and methods in transition-metal-mediated reactions, offering readers promising new routes to design and construct complex aromatic compounds. Integrating metal catalysis with aromatic compound synthesis, Transition-Metal-Mediated Aromatic Ring Construction offers a practical guide to the methods, synthetic routes, and strategies for constructing aromatic compounds. The book's five parts examine: [2+2+2], [2+2+1], and related cycloaddition reactions [4+2], [3+2], and related cycloaddition reactions Electrocyclization reactions Coupling and addition reactions Other important transformations, including methathesis reactions and skeletal rearrangement reactions Edited by Ken Tanaka, an internationally recognized expert in the field of transition-metal catalysis, the book features authors who are leading pioneers and researchers in synthetic reactions. Their contributions reflect a thorough review and analysis of the literature as well as their own firsthand laboratory experience developing new aromatic compounds. All chapters end with a summary and outlook, setting forth new avenues of research and forecasting new discoveries. There are also references at the end of each chapter, guiding readers to important original research reports and reviews. In summary, Transition-Metal-Mediated Aromatic Ring Construction offers synthetic chemists a promising new avenue for the development of important new aromatic compounds with a broad range of applications.

Power series provide a technique for constructing examples of commutative rings. In this book, the authors describe this technique and use it to analyse properties of commutative rings and their spectra. This book presents results obtained using this approach. The authors put these results in perspective; often the proofs of properties of classical examples are simplified. The book will serve as a helpful resource for researchers working in commutative algebra.

This monograph is concerned with exchange rings in various conditions related to stable range. Diagonal reduction of regular matrices and cleanness of square matrices are also discussed. Readers will come across various topics: cancellation of modules, comparability of modules, cleanness, monoid theory, matrix theory, K-theory, topology, amongst others. This is a first-ever book that contains many of these topics considered under stable range conditions. It will be of great interest to researchers and graduate students involved in ring and module theories.

Reprint of the original, first published in 1872. The publishing house Anatiposi publishes historical books as reprints. Due to their age, these books may have missing pages or inferior quality. Our aim is to preserve these books and make them available to the public so that they do not get lost.

In Hadamard Matrices and Their Applications, K. J. Horadam provides the first unified account of cocyclic Hadamard matrices and their applications in signal and data processing. This original work is based on the development of an algebraic link between Hadamard matrices and the cohomology of finite groups that was discovered fifteen years ago. The book translates physical applications into terms a pure mathematician will appreciate, and theoretical structures into ones an applied mathematician, computer scientist, or communications engineer can adapt and use. The first half of the book explains the state of our knowledge of Hadamard matrices and two important generalizations: matrices with group entries and multidimensional Hadamard arrays. It focuses on their applications in engineering and computer science, as signal transforms, spreading sequences, error-correcting codes, and cryptographic primitives. The book's second half presents the new results in cocyclic Hadamard matrices and their applications. Full expression of this theory has been realized only recently, in the Five-fold Constellation. This identifies cocyclic generalized Hadamard matrices with particular "stars" in four other areas of mathematics and engineering: group cohomology, incidence structures, combinatorics, and signal correlation. Pointing the way to possible new developments in a field ripe for further research, this book formulates and discusses ninety open questions.