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For a long time - at least from Fermat to Minkowski - the theory of quadratic forms was a part of number theory. Much of the best work of the great number theorists of the eighteenth and nineteenth century was concerned with problems about quadratic forms. On the basis of their work, Minkowski, Siegel, Hasse, Eichler and many others crea­ ted the impressive "arithmetic" theory of quadratic forms, which has been the object of the well-known books by Bachmann (1898/1923), Eichler (1952), and O'Meara (1963). Parallel to this development the ideas of abstract algebra and abstract linear algebra introduced by Dedekind, Frobenius, E. Noether and Artin led to today's structural mathematics with its emphasis on classification problems and general structure theorems. On the basis of both - the number theory of quadratic forms and the ideas of modern algebra - Witt opened, in 1937, a new chapter in the theory of quadratic forms. His most fruitful idea was to consider not single "individual" quadratic forms but rather the entity of all forms over a fixed ground field and to construct from this an algebra­ ic object. This object - the Witt ring - then became the principal object of the entire theory. Thirty years later Pfister demonstrated the significance of this approach by his celebrated structure theorems.

Quadratic Irrationals: An Introduction to Classical Number Theory gives a unified treatment of the classical theory of quadratic irrationals. Presenting the material in a modern and elementary algebraic setting, the author focuses on equivalence, continued fractions, quadratic characters, quadratic orders, binary quadratic forms, and class groups. The book highlights the connection between Gauss’s theory of binary forms and the arithmetic of quadratic orders. It collects essential results of the theory that have previously been difficult to access and scattered in the literature, including binary quadratic Diophantine equations and explicit continued fractions, biquadratic class group characters, the divisibility of class numbers by 16, F. Mertens’ proof of Gauss’s duplication theorem, and a theory of binary quadratic forms that departs from the restriction to fundamental discriminants. The book also proves Dirichlet’s theorem on primes in arithmetic progressions, covers Dirichlet’s class number formula, and shows that every primitive binary quadratic form represents infinitely many primes. The necessary fundamentals on algebra and elementary number theory are given in an appendix. Research on number theory has produced a wealth of interesting and beautiful results yet topics are strewn throughout the literature, the notation is far from being standardized, and a unifying approach to the different aspects is lacking. Covering both classical and recent results, this book unifies the theory of continued fractions, quadratic orders, binary quadratic forms, and class groups based on the concept of a quadratic irrational.

Quadratic Irrationals: An Introduction to Classical Number Theory gives a unified treatment of the classical theory of quadratic irrationals. Presenting the material in a modern and elementary algebraic setting, the author focuses on equivalence, continued fractions, quadratic characters, quadratic orders, binary quadratic forms, and class groups. The book highlights the connection between Gauss’s theory of binary forms and the arithmetic of quadratic orders. It collects essential results of the theory that have previously been difficult to access and scattered in the literature, including binary quadratic Diophantine equations and explicit continued fractions, biquadratic class group characters, the divisibility of class numbers by 16, F. Mertens’ proof of Gauss’s duplication theorem, and a theory of binary quadratic forms that departs from the restriction to fundamental discriminants. The book also proves Dirichlet’s theorem on primes in arithmetic progressions, covers Dirichlet’s class number formula, and shows that every primitive binary quadratic form represents infinitely many primes. The necessary fundamentals on algebra and elementary number theory are given in an appendix. Research on number theory has produced a wealth of interesting and beautiful results yet topics are strewn throughout the literature, the notation is far from being standardized, and a unifying approach to the different aspects is lacking. Covering both classical and recent results, this book unifies the theory of continued fractions, quadratic orders, binary quadratic forms, and class groups based on the concept of a quadratic irrational.

The first thing you will find out about this book is that it is fun to read. It is meant for the browser, as well as for the student and for the specialist wanting to know about the area. The footnotes give an historical background to the text, in addition to providing deeper applications of the concept that is being cited. This allows the browser to look more deeply into the history or to pursue a given sideline. Those who are only marginally interested in the area will be able to read the text, pick up information easily, and be entertained at the same time by the historical and philosophical digressions. It is rich in structure and motivation in its concentration upon quadratic orders.This is not a book that is primarily about tables, although there are 80 pages of appendices that contain extensive tabular material (class numbers of real and complex quadratic fields up to 104; class group structures; fundamental units of real quadratic fields; and more!). This book is primarily a reference book and graduate student text with more than 200 exercises and a great deal of hints! The motivation for the text is best given by a quote from the Preface of Quadratics: "There can be no stronger motivation in mathematical inquiry than the search for truth and beauty. It is this author's long-standing conviction that number theory has the best of both of these worlds. In particular, algebraic and computational number theory have reached a stage where the current state of affairs richly deserves a proper elucidation. It is this author's goal to attempt to shine the best possible light on the subject."

New! This book is in the same immaculate condition as when it was published 1.19

Like New condition. Great condition, but not exactly fully crisp. The book may have been opened and read, but there are no defects to the book, jacket or pages. 1.19

The first thing you will find out about this book is that it is fun to read. It is meant for the browser, as well as for the student and for the specialist wanting to know about the area. The footnotes give an historical background to the text, in addition to providing deeper applications of the concept that is being cited. This allows the browser to look more deeply into the history or to pursue a given sideline. Those who are only marginally interested in the area will be able to read the text, pick up information easily, and be entertained at the same time by the historical and philosophical digressions. It is rich in structure and motivation in its concentration upon quadratic orders.This is not a book that is primarily about tables, although there are 80 pages of appendices that contain extensive tabular material (class numbers of real and complex quadratic fields up to 104; class group structures; fundamental units of real quadratic fields; and more!). This book is primarily a reference book and graduate student text with more than 200 exercises and a great deal of hints! The motivation for the text is best given by a quote from the Preface of Quadratics: "There can be no stronger motivation in mathematical inquiry than the search for truth and beauty. It is this author's long-standing conviction that number theory has the best of both of these worlds. In particular, algebraic and computational number theory have reached a stage where the current state of affairs richly deserves a proper elucidation. It is this author's goal to attempt to shine the best possible light on the subject."

This book focuses on the stabilization and model predictive control of interconnected systems with mixed connection configurations. It introduces the concept of dissipation-based quadratic constraint for developing attractivity assurance methods for interconnected systems. In order to develop these methods, distributed and decentralized architectures are employed, whereby the communication between subsystems is fully connected, partially connected, or completely disconnected. Given that the control inputs are entirely or partially decoupled between subsystems and no additional constraints are imposed on the interactive variables beyond the coupling constraint itself, the proposed approaches can be used with various types of systems and applications. Further, the book describes how the effects of coupling delays and data losses in device networks are resolved. From a practical perspective, the innovations presented are of benefit in applications in a broad range of fields, including the process and manufacturing industries, networked robotics, and network-centric systems such as chemical process systems, power systems, telecommunication networks, transportation networks, and, no less importantly, supply chain automation.

This new version of the author's prizewinning book, Algebraic Theory of Quadratic Forms (W. A. Benjamin, Inc., 1973), gives a modern and self-contained introduction to the theory of quadratic forms over fields of characteristic different from two. Starting with few prerequisites beyond linear algebra, the author charts an expert course from Witt's classical theory of quadratic forms, quaternion and Clifford algebras, Artin-Schreier theory of formally real fields, and structural theorems on Witt rings, to the theory of Pfister forms, function fields, and field invariants. These main developments are seamlessly interwoven with excursions into Brauer-Wall groups, local and global fields, trace forms, Galois theory, and elementary algebraic K-theory, to create a uniquely original treatment of quadratic form theory over fields. Two new chapters totaling more than 100 pages have been added to the earlier incarnation of this book to take into account some of the newer results and more recent viewpoints in the area. As is characteristic of this author's expository style, the presentation of the main material in this book is interspersed with a copious number of carefully chosen examples to illustrate the general theory. This feature, together with a rich stock of some 280 exercises for the thirteen chapters, greatly enhances the pedagogical value of this book, both as a graduate text and as a reference work for researchers in algebra, number theory, algebraic geometry, algebraic topology, and geometric topology.

Study of "some of the major themes of the classical arithmetical theory of quadratic forms in the light of our present knowledge", including quadratic forms over a field, p-Adic numbers, quadratic forms over local fields, tools from the geometry of numbers, quadratic forms over the rationals, quadratic forms over integral domains, integral p-Adic forms, integral forms over the rational integers, spin and orthogonal groups, spinor genera, reduction of positive definite quadratic forms, automorphs of integral forms, composition of binary quadratic forms, definite forms, analytic methods; London Mathematical Society Monograph 13. Hardcover, full navy cloth, gilt titling. Light wear, corners rubbed, spine lightly spotted; no jacket; ink name & campus address inside front cover; nine pages in the first 17 pages neatly (not to say obsessively) underlined in several colors. Text otherwise clean; xvi, 413 pages; indexes, references, many equations. Size: Octavo

Written in the wake of the advent of Relativity by an author who made important contributions to projective and differential geometry, and topology, this early Cambridge Tract in Mathematics and Theoretical Physics aimed to assist students of the time from the fields of differential geometry and mathematical physics. Beginning by introducing formal preliminaries, the text continues, bringing the underlying differential invariant theory that to this day remains relevant in a range of geometrical and physical applications, to the fore.

The distinguished mathematician John Conway presents quadratic forms in a pictorial way that enables the reader to understand them mathematically without proving theorems in the traditional fashion. One learns to sense their properties. In his customary enthusiastic style, Conway uses his theme to cast light on all manner of mathematical topics from algebra, number theory and geometry, including many new ideas and features.

Written in the wake of the advent of Relativity by an author who made important contributions to projective and differential geometry, and topology, this early Cambridge Tract in Mathematics and Theoretical Physics aimed to assist students of the time from the fields of differential geometry and mathematical physics. Beginning by introducing formal preliminaries, the text continues, bringing the underlying differential invariant theory that to this day remains relevant in a range of geometrical and physical applications, to the fore.

As a basis of equation and problem-solving, linear algebra is the most widely taught sub-division of pure mathematics. Building on both skills and knowledge attained during A-level and earlier degree modules in linear algebra, "Matrices and Quadratic Forms" provides a lively and accessible introduction to matrices and quadratic forms. Examples and exercises are used as teaching aids throughout and ideas for investigation and project work help to place the subject in context. Student motivation and learning is encouraged by the inclusion of historical contexts, real-life situations and the discussion of links with other areas of mathematics. Providing a concise and thorough introduction to matrices and quadratic forms, this is an essential purchase for all students of mathematics.Ideal second text in linear algebra, this book follows on from Allenby's Linear Algebra.Concise and affordable.Written in a straightforward and accessible style by an authority in the field.

From the reviews: "O'Meara treats his subject from this point of view (of the interaction with algebraic groups). He does not attempt an encyclopedic coverage ...nor does he strive to take the reader to the frontiers of knowledge... . Instead he has given a clear account from first principles and his book is a useful introduction to the modern viewpoint and literature. In fact it presupposes only undergraduate algebra (up to Galois theory inclusive)... The book is lucidly written and can be warmly recommended.J.W.S. Cassels, The Mathematical Gazette, 1965"Anyone who has heard O'Meara lecture will recognize in every page of this book the crispness and lucidity of the author's style;... The organization and selection of material is superb... deserves high praise as an excellent example of that too-rare type of mathematical exposition combining conciseness with clarity...R. Jacobowitz, Bulletin of the AMS, 1965

From the reviews: "O'Meara treats his subject from this point of view (of the interaction with algebraic groups). He does not attempt an encyclopedic coverage ...nor does he strive to take the reader to the frontiers of knowledge... . Instead he has given a clear account from first principles and his book is a useful introduction to the modern viewpoint and literature. In fact it presupposes only undergraduate algebra (up to Galois theory inclusive)... The book is lucidly written and can be warmly recommended.J.W.S. Cassels, The Mathematical Gazette, 1965"Anyone who has heard O'Meara lecture will recognize in every page of this book the crispness and lucidity of the author's style;... The organization and selection of material is superb... deserves high praise as an excellent example of that too-rare type of mathematical exposition combining conciseness with clarity...R. Jacobowitz, Bulletin of the AMS, 1965

Algebra Module 6: The Quadratic Formula Standard Deviants School - DVD 631865016730

This new version of the author's prizewinning book, Algebraic Theory of Quadratic Forms (W. A. Benjamin, Inc., 1973), gives a modern and self-contained introduction to the theory of quadratic forms over fields of characteristic different from two. Starting with few prerequisites beyond linear algebra, the author charts an expert course from Witt's classical theory of quadratic forms, quaternion and Clifford algebras, Artin-Schreier theory of formally real fields, and structural theorems on Witt rings, to the theory of Pfister forms, function fields, and field invariants. These main developments are seamlessly interwoven with excursions into Brauer-Wall groups, local and global fields, trace forms, Galois theory, and elementary algebraic K-theory, to create a uniquely original treatment of quadratic form theory over fields. Two new chapters totaling more than 100 pages have been added to the earlier incarnation of this book to take into account some of the newer results and more recent viewpoints in the area. As is characteristic of this author's expository style, the presentation of the main material in this book is interspersed with a copious number of carefully chosen examples to illustrate the general theory. This feature, together with a rich stock of some 280 exercises for the thirteen chapters, greatly enhances the pedagogical value of this book, both as a graduate text and as a reference work for researchers in algebra, number theory, algebraic geometry, algebraic topology, and geometric topology.

Algebra Module 4: Quadratic Equations Standard Deviants School - DVD 631865016716

This textbook offers theoretical, algorithmic and computational guidelines for solving the most frequently encountered linear-quadratic optimization problems. It provides an overview of recent advances in control and systems theory, numerical line algebra, numerical optimization, scientific computations and software engineering.