Chelsea Publications – serie

Unknown function: Cheney, E. W.

In addition to the standard topics, this volume contains many topics not often found in an algebra book, such as inequalities, and the elements of substitution theory. Especially extensive is Chrystal's treatment of the infinite series, infinite products, and (finite and infinite) continued fractions. The range of entries in the Subject Index is very wide. To mention a few out of many hundreds: Horner's method, multinomial theorem, mortality table, arithmetico-geometric series, Pellian equation, Bernoulli numbers, irrationality of e, Gudermanian, Euler numbers, continuant, Stirling's theorem, Riemann surface. This volume includes over 2,400 exercises with solutions.

From the Preface: ""This textbook has evolved from a set of lecture notes...In both the course and the book, I have in mind first- or second-year graduate students in Mathematics and related fields such as Physics...It is necessary for the reader to have a foundation in advanced calculus which includes familiarity with: least upper bound (LUB) and greatest lower bound (GLB), the concept of function, $\epsilon$'s and their companion $\delta$'s, and basic properties of sequences of real and complex numbers (convergence, Cauchy's criterion, the Weierstrass-Bolzano theorem). It is not presupposed that the reader is acquainted with vector spaces..., matrices..., or determinants...There are over four hundred exercises, most of them easy...It is my hope that this book, aside from being an exposition of certain basic material on Hilbert space, may also serve as an introduction to other areas of functional analysis.

In addition to the standard topics, this volume contains many topics not often found in an algebra book, such as inequalities, and the elements of substitution theory. Especially extensive is Chrystal's treatment of the infinite series, infinite products, and (finite and infinite) continued fractions. The range of entries in the Subject Index is very wide. To mention a few out of many hundreds: Horner's method, multinomial theorem, mortality table, arithmetico-geometric series, Pellian equation, Bernoulli numbers, irrationality of $e$, Gudermanian, Euler numbers, continuant, Stirling's theorem, Riemann surface. This volume includes over 2,400 exercises with solutions.

The first volume of Dickson's ""History"" covers the related topics of divisibility and primality. It begins with a description of the development of our understanding of perfect numbers. Other standard topics, such as Fermat's theorems, primitive roots, counting divisors, the Mobius function, and prime numbers themselves are treated. Dickson, in this thoroughness, also includes less workhorse subjects, such as methods of factoring, divisibility of factorials and properties of the digits of numbers. Concepts, results and citations are numerous.

The last volume of Dickson's History is the most modern, covering quadratic and higher forms. The treatment here is more general than in Volume II, which, in a sense, is more concerned with special cases. Indeed, this volume chiefly presents methods of attacking whole classes of problems. Again, Dickson is exhaustive with references and citations.

Perhaps the first truly famous book devoted primarily to finite groups was Burnside's book. From the time of its second edition in 1911 until the appearance of Hall's book, there were few books of similar stature. Hall's book is still considered to be a classic source for fundamental results on the representation theory for finite groups, the Bumside problem, extensions and cohomology of groups, $p$-groups and much more. For the student who has already had an introduction to group theory, there is much treasure to be found in Hall's ""Theory of Groups"".From the preface to the second edition (1976): 'The present volume is intended to serve a dual purpose. The first ten chapters are meant to be the basis for a course in group theory, and exercises have been included at the end of each of these chapters. The last ten chapters are meant to be useful as optional material in a course or as reference material. When used as a text, the book is intended for students who have had an introductory course in modern algebra comparable to a course taught from Birkhoff and Mac Lane's ""A Survey of Modern Algebra"". I have tried to make this book as self-contained as possible, but where background material is needed references have been given, chiefly to Birkhoff and Mac Lane'.

In this second English edition of Caratheodory's famous work (originally published in German), the two volumes of the first edition have been combined into one (with a combination of the two indexes into a single index). There is a deep and fundamental relationship between the differential equations that occur in the Calculus of Variations and partial differential equations of the first order: in particular, to each such partial differential equation there correspond variational problems. This basic fact forms the rationale for Caratheodory's masterpiece. Includes a Guide to the Literature and an Index.

Includes an expanded Bibliography.

The theory of continued fractions has been defined by a small handful of books. This is one of them. The focus of Wall's book is on the study of continued fractions in the theory of analytic functions, rather than on arithmetical aspects. There are extended discussions of orthogonal polynomials, power series, infinite matrices and quadratic forms in infinitely many variables, definite integrals, the moment problem and the summation of divergent series.'In writing this book, I have tried to keep in mind the student of rather modest mathematical preparation, presupposing only a first course in function theory. Thus, I have included such things as a proof of Schwarz's inequality, theorems on uniformly bounded families of analytic functions, properties of Stieltjes integrals, and an introduction to the matrix calculus. I have presupposed a knowledge of the elementary properties of linear fractional transformations in the complex plane. It has not been my intention to write a complete treatise on the subject of continued fractions, covering all the literature, but rather to present a unified theory correlating certain parts and applications of the subject within a larger analytic structure...' - from the Preface.

An Elegant and comprehensive account of the modern theory of algebraic numbers Bulletin of the AMS.

Based on lectures delivered at the AMS meeting in 1901, this book describes the progress in calculus of variations made in the last 30 years of the nineteenth century. Among other topics, the author describes the landmark results of Weierstrass on sufficient conditions for the extremum of a functional in terms of the second variation. Also discussed are Kneser's sufficient conditions, Weierstrass' theory of the isoperimetric problem, and Hilbert's theorem on the existence of an extremum of an integral. Although the original book was written nearly 100 years ago, it remains very useful in learning about classical calculus of variations.

Weber's three-volume set on algebra was for many years the standard text on algebra. Published at the end of the nineteenth century, it helped usher group theory to a central place in twentieth century mathematics. Volume 1 covers foundational material. Volume 2 covers group theory and its applications, plus the theory of algebraic numbers. Volume 3 covers advanced topics, such as algebraic functions, elliptic functions and class field theory. This second volume, in four parts, covers groups, linear groups, applications of group theory, and algebraic numbers. The first part, on groups, includes abelian groups, the group of a cyclotomic field, cubic and biquadratic fields and the study of general groups. The second part, on linear groups, looks at groups of linear substitutions, group invariants, polyhedral groups, and congruence groups.The third part, on applications of group theory, examines metacyclic equations and Galois groups, the inflection points of third-order curves, double-tangents of fourth-order curves, the general theory of fifth-order equations, and certain topics on equations of order seven. The fourth part begins the study of algebraic numbers (which is taken up again in the third volume). The topics here include the general theory of algebraic numbers and functions on an algebraic field, class numbers, cyclotomic fields, abelian extensions, resolvants, and transcendental numbers. Although notations have changed somewhat and algebra has become more abstract that it was in Weber's day, many of the same themes and ideas important today are central topics in Weber's book, which may be why it has become a classic.

This treatise, by one of Russia's leading mathematicians, gives in easily accessible form a coherent account of matrix theory with a view to applications in mathematics, theoretical physics, statistics, electrical engineering, etc. The individual chapters have been kept as far as possible independent of each other, so that the reader acquainted with the contents of Chapter 1 can proceed immediately to the chapters of special interest. Much of the material has been available until now only in the periodical literature.

This title contains four volumes in one: ""Famous Problems of Elementary Geometry"", by Klein. It is a fascinating, simple, easily understandable account of the famous problems of Geometry - The Duplication of the Cube, Trisection of the Angle, Squaring of the Circle - and the proofs that these cannot be solved by ruler and compass. It is suitably presented to undergraduates, with no calculus required. Also, the work includes problems about transcendental numbers, the existence of such numbers, and proofs of the transcendence of $e$. ""From Determinant to Tensor"", by Sheppard. It is a novel and simple introduction to tensors.'An excellent little book, the aim of which is to familiarize the student with tensors and to give an idea of their applications. We wish to recommend the book heartily...The beginner will find the book a valuable introduction and the expert an interesting review with a refreshing novelty of presentation' - ""Bulletin of the AMS"". Chapter headings include: Origin of Determinants; Properties of Determinants; Solution of Simultaneous Equations; Properties; Tensor Notation; Sets; Cogredience, etc.; Examples from Statistics; Tensors in Theory of Relativity; and, Introduction to Combinatory Analysis, by MacMahon.This is an introduction to the author's two-volume work. It includes Three Lectures on Fermat's Last Theorem by Mordell. This famous problem is so easy that a high-school student might not unreasonably hope to solve it: it is so difficult that as of the 1962 publication date of this book, tens of thousands of amateur and professional mathematicians, Euler and Gauss among them, failed to find a complete solution. Mordell himself had a solution (as he said he did). This work is one of the masterpieces of mathematical exposition.

In the late summer of 1893, following the Congress of Mathematicians held in Chicago, Felix Klein gave two weeks of lectures on the current state of mathematics. Rather than offering a universal perspective, Klein presented his personal view of the most important topics of the time. It is remarkable how most of the topics continue to be important today. Originally published in 1893 and reissued by the AMS in 1911, we are pleased to bring this work into print once more with this new edition. Klein begins by highlighting the works of Clebsch and of Lie. In particular, he discusses Clebsch's work on Abelian functions and compares his approach to the theory with Riemann's more geometrical point of view. Klein devotes two lectures to Sophus Lie, focussing on his contributions to geometry, including sphere geometry and contact geometry.Klein's ability to connect different mathematical disciplines clearly comes through in his lectures on mathematical developments. For instance, he discusses recent progress in non-Euclidean geometry by emphasizing the connections to projective geometry and the role of transformation groups. In his descriptions of analytic function theory and of recent work in hyperelliptic and Abelian functions, Klein is guided by Riemann's geometric point of view. He discusses Galois theory and solutions of algebraic equations of degree five or higher by reducing them to normal forms that might be solved by non-algebraic means. Thus, as discovered by Hermite and Kronecker, the quintic can be solved 'by elliptic functions'.This also leads to Klein's well-known work connecting the quintic to the group of the icosahedron. Klein expounds on the roles of intuition and logical thinking in mathematics. He reflects on the influence of physics and the physical world on mathematics and, conversely, on the influence of mathematics on physics and the other natural sciences. The discussion is strikingly similar to today's discussions about 'physical mathematics'. There are a few other topics covered in the lectures which are somewhat removed from Klein's own work. For example, he discusses Hilbert's proof of the transcendence of certain types of numbers (including $\pi$ and $e$), which Klein finds much simpler than the methods used by Lindemann to show the transcendence of $\pi$.Also, Klein uses the example of quadratic forms (and forms of higher degree) to explain the need for a theory of ideals as developed by Kummer. Klein's look at mathematics at the end of the 19th Century remains compelling today, both as history and as mathematics. It is delightful and fascinating to observe from a one-hundred year retrospect, the musings of one of the masters of an earlier era.

This is the second printing of the book first published in 1988. The first four chapters of the volume are based on lectures given by Stroock at MIT in 1987. They form an introduction to the basic ideas of the theory of large deviations and make a suitable package on which to base a semester-length course for advanced graduate students with a strong background in analysis and some probability theory. A large selection of exercises presents important material and many applications. The last two chapters present various non-uniform results (Chapter 5) and outline the analytic approach that allows one to test and compare techniques used in previous chapters (Chapter 6).

After completing his famous ""Foundations of Analysis"" (See 'AMS Chelsea Publishing, Volume 79.H' for the English Edition and 'AMS Chelsea Publishing, Volume 141' for the German Edition, ""Grundlagen der Analysis""), Landau turned his attention to this book on calculus. The approach is that of an unrepentant analyst, with an emphasis on functions rather than on geometric or physical applications. The book is another example of Landau's formidable skill as an expositor. It is a masterpiece of rigor and clarity.

This famous work is a textbook that emphasizes the conceptual and historical continuity of analytic function theory. The second volume broadens from a textbook to a textbook-treatise, covering the 'canonical' topics (including elliptic functions, entire and meromorphic functions, as well as conformal mapping, etc.) and other topics nearer the expanding frontier of analytic function theory. In the latter category are the chapters on majorization and on functions holomorphic in a half-plane.

In this book, Hawkins elegantly places Lebesgue's early work on integration theory within in proper historical context by relating it to the developments during the nineteenth century that motivated it and gave it significance and also to the contributions made in this field by Lebesgue's contemporaries. Hawkins was awarded the 1997 MAA Chauvenet Prize and the 2001 AMS Albert Leon Whiteman Memorial Prize for notable exposition and exceptional scholarship in the history of mathematics.

Weber's three-volume set on algebra was for many years the standard text on algebra. Published at the end of the nineteenth century, it helped usher group theory to a central place in twentieth century mathematics. Volume 1 covers foundational material. Volume 2 covers group theory and its applications, plus the theory of algebraic numbers. Volume 3 covers advanced topics, such as algebraic functions, elliptic functions and class field theory. Although notations have changed somewhat and algebra has become more abstract that it was in Weber's day, many of the same themes and ideas important today are central topics in Weber's book, which may be why it has become a classic.

This well-known book is a self-contained treatment of the classical theory of abstract Riemann surfaces. The first five chapters cover the requisite function theory and topology for Riemann surfaces. The second five chapters cover differentials and uniformization. For compact Riemann surfaces, there are clear treatments of divisors, Weierstrass points, the Riemann-Roch theorem and other important topics. Springer's book is an excellent text for an introductory course on Riemann surfaces. It includes exercises after each chapter and is illustrated with a beautiful set of figures.

Weber's three-volume set on algebra was for many years the standard text on algebra. Published at the end of the nineteenth century, it helped usher group theory to a central place in twentieth century mathematics. Volume 1 covers foundational material. Volume 2 covers group theory and its applications, plus the theory of algebraic numbers. Volume 3 covers advanced topics, such as algebraic functions, elliptic functions and class field theory. Although notations have changed somewhat and algebra has become more abstract that it was in Weber's day, many of the same themes and ideas important today are central topics in Weber's book, which may be why it has become a classic.

Weber's three-volume set on algebra was for many years the standard text on algebra. Published at the end of the nineteenth century, it helped usher group theory to a central place in twentieth century mathematics. Volume 1 covers foundational material. Volume 2 covers group theory and its applications, plus the theory of algebraic numbers. Volume 3 covers advanced topics, such as algebraic functions, elliptic functions and class field theory. Although notations have changed somewhat and algebra has become more abstract that it was in Weber's day, many of the same themes and ideas important today are central topics in Weber's book, which may be why it has become a classic.

This book is translated from the second Russian edition and with added notes by K.A. Hirsch. ""Teoriya Grupp"" by Kurosh was widely acclaimed, in its first edition, as the first modern text on the general theory of groups, with the major emphasis on infinite groups. The decade that followed brought about a remarkable growth and maturity in the theory of groups, so that this second edition, in English translation, represents a complete rewriting of the first edition.The book can be used as a beginning text, the only requirement being some mathematical maturity and a knowledge of the elements of transfinite numbers. Many new sections were added to this second edition, and many old ones were completely revised: the theory of abelian groups was significantly revised; many significant additions were made to the section on the theory of free groups and free products; an entire chapter is devoted to group extensions; and the deep changes in the theory of solvable and nilpotent groups - one of the large and rich branches of the theory of groups - are covered in this work. Each volume concludes with Editor's Notes and a Bibliography.

From the Preface: 'The present monograph deals with the mathematical theory of Huygens' principle in optics and its application to the theory of diffraction. No attempt has been made to give a complete account of the various methods of solving special diffraction problems. [The authors] are concerned only with the general theory of the solution of the partial differential equations governing the propagation of light and [they] discuss some of the simpler diffraction problems merely as illustrative examples'.

Circles and spheres are central objects in geometry. Mappings that take circles to circles or spheres to spheres have special roles in metric and conformal geometry. An example of this is Lie's sphere geometry, whose group of transformations is precisely the conformal group. Coolidge's treatise looks at systems of circles and spheres and the geometry and groups associated to them. It was written (1916) at a time when Lie's enormous influence on the field was still widely felt. Today, there is a renewed interest in the geometry of special geometric configurations. Coolidge has examined many of the most intuitive: linear systems of circles, circles orthogonal to a given sphere, and so on. He also examines the differential and projective geometry of the space of all spheres in a given space.Through the simple vehicles of circles and spheres, Coolidge makes contact with diverse areas of mathematics: conformal transformations and analytic functions, projective and contact geometry, and Lie's theory of continuous groups, to name a few. The interested reader will be well rewarded by a study of this remarkable book.

Oliver Heaviside is probably best known to the majority of mathematicians for the Heaviside function in the theory of distribution. However, his main research activity concerned the theory of electricity and magnetism, the area in which he worked for most of his life. Results of this work are presented in his fundamental three-volume ""Electromagnetic Theory"". The book brings together many of Heaviside's published and unpublished notes and short articles written between 1891 and 1912. One of Heaviside's main achievements was the recasting of Maxwell's theory of electromagnetism into the form currently used by everyone. He is also known for the invention of operational calculus and for major contributions to solving theoretical and practical problems of cable and radio communication. All this is collected in three volumes of ""Electromagnetic Theory"".However, there is even more. For example, Chapter V in Volume II discusses the age of Earth, and several sections in Volume III talk about the teaching of mathematics in school. In addition to Heaviside's writings, two detailed surveys of Heaviside's work, by Sir Edmund Whittaker and by B. A. Behrend, are included in Volume I, and a long account of Heaviside's unpublished notes (which he presumably planned to publish as Volume IV of ""Electromagnetic Theory"") is included in Volume III.