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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

Page 1 of 2

Ergodic Theory of Numbers
  • Karma Dajani, Cor Kraaikamp
  • Published by:
    Mathematical Association of America
    Published online:
    05 June 2014
    Print publication:
    01 June 2002
  • Ergodic Theory of Numbers is an introduction to the ergodic theory behind common number expansion, like decimal expansions, continued fractions, and many others. However, its aim does not stop there. For undergraduate students with sufficient background knowledge in real analysis and graduate students interested in the area, it is also an introduction to a "dynamical way of thinking." The questions studied here are dynamical as well as number theoretical in nature, and the answers are obtained with the help of ergodic theory. Attention is focused on concepts like measure-preserving, ergodicity, natural extension, induced transformations, and entropy. These concepts are then applied to familiar expansions to obtain old and new results in an elegant and straightforward manner. What it means to be ergodic and the basic ideas behind ergodic theory will be explained along the way. The subjects covered vary from classical to recent, which makes this book appealing to researchers as well as students.

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.

Algebra and Tiling
  • Homomorphisms in the Service of Geometry
  • Sherman K. Stein, Sandor Szabo
  • Published by:
    Mathematical Association of America
    Published online:
    05 June 2014
    Print publication:
    01 June 2009
  • Read what reviewers had to say about this excellent monograph: Algebra and Tiling is perfect for bringing alive an abstract algebra course. Intuitive but difficult problems of geometry are translated into algebraic problems more amenable to solution... Full of surprises, the book is a pleasure to read. Essential for collections that serve undergraduates. Choice The authors have arranged the text so that exercises are stated frequently throughout the chapters, rather than at the end, and they have presented as problems some questions that have not yet been answered. Historical background relating to tiling problems helps focus the reader on the mathematical discussions. The student or mathematician whose area of interest is algebra should enjoy this text. The Mathematics Teacher There are many questions about tilings which lend themselves nicely to the tools of modern algebra. This book is about several such questions. Each chapter is an independent instance of turning a tiling problem into a problem in algebra... This book is almost self-contained; a lot of time is spent describing carefully the relevant algebra. Zentralblatt für Mathematik Algebra and Tiling is accessible to undergraduate mathematics majors, as most of the tools necessary to read the book are found in standard upper division algebra courses, but teachers, researchers, and professional mathematicians will find the book equally appealing. Beginners will find the exercises and the appendices especially useful. The unsolved problems will challenge both beginners and experts. The book could serve as the basis of an undergraduate or graduate seminar or a source of applications to enrich an algebra or geometry course.

Celestial Mechanics
  • Harry Pollard
  • Published by:
    Mathematical Association of America
    Published online:
    05 June 2014
    Print publication:
    01 June 1976
  • 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 basic mathematics underlying the subject of celestial mechanics. Chapter 1 formulates the central force problem and then deals with Kepler's first and second laws, orbits under non-Newtonian attraction, elements of an orbit, the two-body system, the solar system, and disturbed motion. Chapter 2 introduces the n-body problem. Included in Chapter 2 are the Lagrange-Jacobi formula, Sundman's theorem on total collapse, the three-body problem, and Lagrange's and Euler's solutions to the three-body problem. Chapter 3 is an introduction to Hamilton-Jacobi Theory.

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.

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.

The Sensual (Quadratic) Form
  • John H. Conway
  • Assisted by Francis Y. C. Fung
  • Published by:
    Mathematical Association of America
    Published online:
    05 June 2014
    Print publication:
    01 June 1997
  • Absolutely fascinating from beginning to end. Ian Stewart, New Scientist This is a book rich in ideas. They seem to burst forth from almost every page ... I suspect that the author's hope that 'even the experts will find some new enlightenment here' will be realized. The Mathematics Teacher A very original book ... The lectures are self-contained for the graduate reader ... full of new ideas, and thus of real interest also to experts. Zentralblatt für Mathematik John Horton Conway's unique approach to quadratic forms was the subject of the Hedrick Lectures given by him in August of 1991 at the Joint Meetings of the Mathematical Association of America and the American Mathematical Society in Orono, Maine. This book presents the substance of those lectures. The book should not be thought of as a serious textbook on the theory of quadratic forms - it consists rather of a number of essays on particular aspects of quadratic forms that have interested the author. The lectures are self-contained and will be accessible to the generally informed reader who has no particular background in quadratic form theory. The minor exceptions should not interrupt the flow of ideas. The Afterthoughts to the Lectures contain discussion of related matters that occasionally presuppose greater knowledge.

Dedekind Sums
  • Hans Rademacher, Emil Grosswald
  • Published by:
    Mathematical Association of America
    Published online:
    05 June 2014
    Print publication:
    01 June 1972
  • These notes from Hans Rademacher's 1963 Hedrick Lectures have been gently polished and augmented by Emil Grosswald. While the topic itself is specialized, these sums are linked in diverse ways to many results in number theory, elliptical modular functions, and topology. The first main result is a surprising reciprocity law that is equivalent to the law of quadratic reciprocity.

Mathematical Statistics
  • Henry Lewis Rietz
  • Published by:
    Mathematical Association of America
    Published online:
    05 June 2014
    Print publication:
    15 November 2013
  • 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 contributes toward shifting the emphasis and point of view in the study of statistics in the direction of the consideration of the underlying theory involved in certain highly important methods of statistical analysis. With this as the main purpose it is natural that no great effort is made to present a well-balanced discussion of all the many available topics. Considerable portions of this monograph can be read by those who have relatively little knowledge of college mathematics. However, the exposition is designed, in general, for readers of a certain degree of mathematical maturity, and presupposes an acquaintance with elementary differential and integral calculus, and with the elementary principles of probability as presented in various books on college algebra for freshmen.

The Generalized Riemann Integral
  • Robert M. McLeod
  • Published by:
    Mathematical Association of America
    Published online:
    05 June 2014
    Print publication:
    01 June 1980
  • The Generalized Riemann Integral is addressed to persons who already have an acquaintance with integrals they wish to extend and to the teachers of generations of students to come. The organization of the work will make it possible for the first group to extract the principal results without struggling through technical details which they may find formidable or extraneous to their purposes. The technical level starts low at the opening of each chapter. Thus readers may follow each chapter as far as they wish and then skip to the beginning of the next. To readers who do wish to see all the details of the arguments, they are given. The generalized Riemann integral can be used to bring the full power of the integral within the reach of many who, up to now, get no glimpse of such results as monotone and dominated convergence theorems. As its name hints, the generalized Riemann integral is defined in terms of Riemann sums. The path from the definition to theorems exhibiting the full power of the integral is direct and short.

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.

The Theory of Algebraic Numbers
  • 2nd edition
  • Harry Pollard, Harold Diamond
  • Published by:
    Mathematical Association of America
    Published online:
    05 June 2014
    Print publication:
    01 June 1975
  • 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 makes available in English the elementary parts of classical algebraic number theory. This second edition follows closely the plan and style of the first edition. The principal changes are the correction of misprints, the expansion or simplification of some arguments, and the omission of the final chapter on units in order to make way for the introduction of some two hundred problems.

Calculus of Variations
  • Gilbert Ames Bliss
  • Published by:
    Mathematical Association of America
    Published online:
    05 June 2014
    Print publication:
    01 June 1978
  • 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 development of the calculus of variations has, from the beginning, been interlaced with that of the differential and integral calculus. Without any knowledge of the calculus one can readily understand at least the geometrical or mechanical statements of many of the problems of the calculus of variations and the character of their solutions. The discovery and justification of the results in this book, apart from their simple statements, do require, however, acquaintance with the principles of the calculus, and it is assumed that the reader has such an acquaintance. Calculus of Variations begins by studying special problems rather than the general theory. The first chapter of the book describes the historical setting out of which the theory of the calculus of variations grew, and the character of some of the simpler problems. The next three chapters are devoted to the development in detail of the then known results for three special problems (shortest distances, brachistochrone, and surfaces of revolution of minimum area) which illustrate in excellent fashion the essential characteristics of the general theory contained in Chapter V with which the book concludes.

Field Theory and its Classical Problems
  • Charles R. Hadlock
  • Published by:
    Mathematical Association of America
    Published online:
    05 June 2014
    Print publication:
    01 June 1975
  • Field Theory and its Classical Problems lets Galois theory unfold in a natural way, beginning with the geometric construction problems of antiquity, continuing through the constructibility of regular n-gons and the properties of roots of unity, and then on to the solvability of polynomial equations by radicals, and beyond. The logical pathway is historic, but the terminology is consistent with modern treatments. No previous knowledge of groups, fields, or abstract algebra is assumed. Notable topics treated along this route include the transcendence of e and of pi, cyclotomic polynomials, polynomials over the integers, Hilbert's, irreducibility theorem, and many other gems in classical mathematics. Historical and bibliographical notes complement the text, and complete solutions are provided to all problems. Field Theory and its Classical Problems is a winner of the MAA Edwin Beckenbach Book Prize for excellence in mathematical exposition.

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).

The Schwarz Function and its Applications
  • Philip J. Davis
  • Published by:
    Mathematical Association of America
    Published online:
    05 June 2014
    Print publication:
    01 June 1974
  • H. A. Schwarz showed us how to extend the notion of reflection in straight lines and circles to reflection in an arbitrary analytic arc. Notable applications were made to the symmetry principle and to problems of analytic continuation. Reflection, in the hands of Schwarz, is an antianalytic mapping. By taking its complex conjugate, we arrive at an analytic function that we have called here the Schwarz Function of the analytic arc. This function is worthy of study in its own right and this essay presents such a study. In dealing with certain familiar topics, the use of the Schwarz Function lends a point of view, a clarity and elegance, and a degree of generality which might otherwise be missing. It opens up a line of inquiry which has yielded numerous interesting things in complex variables; it illuminates some functional equations and a variety of iterations which interest the numerical analyst. The perceptive reader will certainly find here some old wine in relabelled bottles. But one of the principles of mathematical growth is that the relabelling process often suggests a new generation of problems. Means become ends; the medium rapidly becomes the message. This book is not wholly self-contained. Readers will find that they should be familiar with the elementary portions of linear algebra and of the theory of functions of a complex variable.

From Error-Correcting Codes through Sphere Packings to Simple Groups
  • Thomas M. Thompson
  • Published by:
    Mathematical Association of America
    Published online:
    05 June 2014
    Print publication:
    01 June 1983
  • This book traces a remarkable path of mathematical connections through seemingly disparate topics. Frustrations with a 1940's electro-mechanical computer at a premier research laboratory begin this story. Subsequent mathematical methods of encoding messages to ensure correctness when transmitted over noisy channels lead to discoveries of extremely efficient lattice packings of equal-radius balls, especially in 24-dimensional space. In turn, this highly symmetric lattice, with each point neighboring exactly 196,560 other points, suggested the possible presence of new simple groups as groups of symmetries. Indeed, new groups were found and are now part of the "Enormous Theorem" - the classification of all simple groups whose entire proof runs some 10,000+ pages. And these connections, along with the fascinating history and the proof of the simplicity of one of those "sporatic" simple groups, are presented at an undergraduate mathematical level.

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.

A Panaroma of Harmonic Analysis
  • Steven G. Krantz
  • Published by:
    Mathematical Association of America
    Published online:
    05 June 2014
    Print publication:
    01 June 1999
  • A Panorama of Harmonic Analysis treats the subject of harmonic analysis, from its earliest beginnings to the latest research. Following both an historical and a conceptual genesis, the book discusses Fourier series of one and several variables, the Fourier transform, spherical harmonics, fractional integrals, and singular integrals on Euclidean space. The climax of the book is a consideration of the earlier ideas from the point of view of spaces of homogeneous type. The book culminates with a discussion of wavelets-one of the newest ideas in the subject. A Panorama of Harmonic Analysis is intended for graduate students, advanced undergraduates, mathematicians, and anyone wanting to get a quick overview of the subject of cummutative harmonic analysis. Applications are to mathematical physics, engineering and other parts of hard science. Required background is calculus, set theory, integration theory, and the theory of sequences and series.

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