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Expositions of mathematical subjects set forth in a manner comprehensible not only to teachers and students specializing in mathematics, but also to scientific workers in other fields. The monographs are intended for the wide circle of thoughtful people familiar with basic graduate or advanced undergraduate mathematics, encountered in the study of mathematics itself, or in the context of related disciplines, who wish to extend their knowledge without prolonged and critical study of the mathematical journals and treatises. The books in this series are hardbound.

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28 results in Carus Mathematical Monographs

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Noncommutative Rings
  • I. N. Herstein
  • Published by:
    Mathematical Association of America
    Published online:
    05 June 2014
    Print publication:
    01 June 1968
  • This beautiful book is the result of the author's wide and deep knowledge of the subject matter, combined with a gift for exposition ... The well selected material is offered in an integrated presentation of the structure theory of noncommutative (associative rings) and its applications. The book will appeal to many a reader. It would be wonderful as a textbook, and in fact, it's based on the author's lecture notes ... Only people looking for the most general form of a particular theorem are advised to turn to other books, but those interested in studying or reviewing its subject matter or looking for a rounded account of it, could do no better than choosing this book for this purpose. Bulletin of the American Mathematical SocietyNoncommutative Rings provides a cross-section of ideas, techniques and results that give the reader an idea of that part of algebra which concerns itself with noncommutative rings. In the space of 200 pages, Herstein covers the Jacobson radical, semisimple rings, commutativity theorems, simple algebras, representations of finite groups, polynomial identities, Goldie's theorem and the Golod-Shafarevitch theorem. Almost every practicing ring theorist has studied portions of this classic monograph.

Combinatorial Mathematics
  • Herbert John Ryser
  • Published by:
    Mathematical Association of America
    Published online:
    05 June 2014
    Print publication:
    01 June 1963
  • This polished and readable account of some fascinating aspects of the subject is devoted mainly to existence problems including several basic, original contributions made by the author himself - the subject is justly known for its difficulty, and Ryser is to be congratulated for his clear presentation. The exposition is supplemented by a good index and an appropriate bibliography for each chapter. An attractive feature is the mention of various interesting unsolved problems, in which the field abounds. Victor Klee in Science Herbert J. Ryser is widely regarded as one of the major figures in combinatorics in the 20th century. His Combinatorial Mathematics is a classic which has enticed many young mathematics students into this area.

Statistical Independence in Probability, Analysis, and Number Theory
  • Mark Kac
  • Published by:
    Mathematical Association of America
    Published online:
    05 June 2014
    Print publication:
    01 June 1959
  • Professor Kac's monograph is designed to illustrate how simple observations can be made the starting point of rich and fruitful theories and how the same theme recurs in seemingly unrelated disciplines. An elementary but thorough discussion of the game of 'heads or tails', including the normal law and the laws of large numbers, is presented in a setting in which a variety of purely analytic results appear natural and inevitable. The chapter 'Primes Play a Game of Chance' uses the same setting in dealing with problems of the distribution of values of arithmetic functions. The final chapter 'From Kinetic Theory to Continued Fractions' deals with a spectacular application of the ergodic theorems to continued fractions. Mark Kac conveyed his infectious enthusiasm for mathematics and its applications in his lectures, papers, and books. Two of his papers won Chauvenet awards for expository excellence. Born in Poland, he studied with Hugo Steinhaus at Lwów earning his doctorate in 1936. He had a long and productive career in the United States, serving on the faculties of Cornell University (1939-1961), Rockefeller University (1961-1982), and the University of Southern California (1982 until his death in 1984).

Irrational Numbers
  • Ivan Niven
  • Published by:
    Mathematical Association of America
    Published online:
    05 June 2014
    Print publication:
    01 June 1956
  • In this monograph, Ivan Niven, provides a masterful exposition of some central results on irrational, transcendental, and normal numbers. He gives a complete treatment by elementary methods of the irrationality of the exponential, logarithmic, and trigonometric functions with rational arguments. The approximation of irrational numbers by rationals, up to such results as the best possible approximation of Hurwitz, is also given with elementary techniques. The last third of the monograph treats normal and transcendental numbers, including the transcendence of p and its generalization in the Lindermann theorem, and the Gelfond-Schneider theorem. Most of the material in the first two-thirds of the book presupposes only calculus and beginning number theory. The book is almost wholly self-contained. The results needed from analysis and algebra are central, and well-known theorems, and complete references to standard works are given to help the beginner. The chapters are for the most part independent. There is a set of notes at the end of each chapter citing the main sources used by the author, and suggesting further readings.

The Arithmetic Theory of Quadratic Forms
  • Burton W. Jones
  • Published by:
    Mathematical Association of America
    Published online:
    05 June 2014
    Print publication:
    01 June 1950
  • The MAA is pleased to re-issue the early Carus Mathematical Monographs in ebook and print-on-demand formats. Readers with an interest in the history of the undergraduate curriculum or the history of a particular field will be rewarded by study of these very clear and approachable little volumes. This monograph presents the central ideas of the arithmetic theory of quadratic forms in self-contained form, assuming only knowledge of the fundamentals of matric theory and the theory of numbers. Pertinent concepts of p-adic numbers and quadratic ideals are introduced. It would have been possible to avoid these concepts but the theory gains elegance as well as breadth by the introduction of such relationships. Some results and many of the methods are here presented for the first time. The development begins with the classical theory in the field of reals from the point of view of representation theory, for in these terms many of the later objectives and methods may be revealed. The successive chapters gradually narrow the fields and rings until one has the tools at hand to deal with the classical problems in the ring of rational integers. The analytic theory of quadratic forms is not dealt with because of the delicate analysis involved. However, some of the more important results are stated and references given.

Rings and Ideals
  • Neal H. McCoy
  • Published by:
    Mathematical Association of America
    Published online:
    05 June 2014
    Print publication:
    01 April 1948
  • The MAA is pleased to re-issue the early Carus Mathematical Monographs in ebook and print-on-demand formats. Readers with an interest in the history of the undergraduate curriculum or the history of a particular field will be rewarded by study of these very clear and approachable little volumes. This monograph presents an introduction to that branch of abstract algebra having to do with the theory of rings, with some emphasis on the role of ideals in the theory. Except for a knowledge of certain fundamental theorems about determinants which is assumed in Chapter VIII, and at one point in Chapter VII, the book is almost entirely self-contained. Of course, the reader must have a certain amount of 'mathematical maturity' in order to understand the illustrative examples, and also to grasp the significance of the abstract approach. However, in so far as formal technique is concerned, little more than the elements of algebra are presupposed.

Vectors and Matrices
  • Cyrus Colton MacDuffee
  • Published by:
    Mathematical Association of America
    Published online:
    05 June 2014
    Print publication:
    01 November 1943
  • The MAA is pleased to re-issue the early Carus Mathematical Monographs in ebook and print-on-demand formats. Readers with an interest in the history of the undergraduate curriculum or the history of a particular field will be rewarded by study of these very clear and approachable little volumes. In 1943 a course in linear algebra did not yet exist as a standard part of the undergraduate curriculum. It would be another twenty years before that would become common. It is, however, easy to identify the defining features of that course in this volume. Start with the idea of solving linear systems; change the point of view to that of transformations on vector spaces; recognize similarity as an essential classifying principle; and catalogue the canonical forms (Jordan normal form) of the transformations. All of this is here but with a decided, old-fashioned, algebraic accent - there is only one figure in the entire text.

Fourier Series and Orthogonal Polynomials
  • Dunham Jackson
  • Published by:
    Mathematical Association of America
    Published online:
    05 June 2014
    Print publication:
    01 April 1941
  • The MAA is pleased to re-issue the early Carus Mathematical Monographs in ebook and print-on-demand formats. Readers with an interest in the history of the undergraduate curriculum or the history of a particular field will be rewarded by study of these very clear and approachable little volumes. The underlying theme of this monograph is that the fundamental simplicity of the properties of orthogonal functions and the developments in series associated with them makes those functions important areas of study for students of both pure and applied mathematics. The book starts with Fourier series and goes on to Legendre polynomials and Bessel functions. Jackson considers a variety of boundary value problems using Fourier series and Laplace's equation. Chapter VI is an overview of Pearson frequency functions. Chapters on orthogonal, Jacobi, Hermite, and Laguerre functions follow. The final chapter deals with convergence. There is a set of exercises and a bibliography. For the reading of most of the book no specific preparation is required beyond a first course in the calculus. A certain amount of 'mathematical maturity' is presupposed, or should be acquired in the course of the reading.

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