Colloquium Publications, Volume: 17
1934; 205 pp; softcover
ISBN-10: 0-8218-4610-8
ISBN-13: 978-0-8218-4610-0
Expected publication date is July 14, 2008.
It is the organization and presentation of the material, however, which make the peculiar appeal of the book. This is no mere compendium of results--the subject has been completely reworked and the proofs recast with the skill and elegance which come only from years of devotion.
--Bulletin of the American Mathematical Society
The very clear and simple presentation gives the reader easy access to the more difficult parts of the theory.
--Jahrbuch uber die Fortschritte der Mathematik
In 1937, the theory of matrices was seventy-five years old. However, many results had only recently evolved from special cases to true general theorems. With the publication of his Colloquium Lectures, Wedderburn provided one of the first great syntheses of the subject. Much of the material in the early chapters is now familiar from textbooks on linear algebra. Wedderburn discusses topics such as vectors, bases, adjoints, eigenvalues and the characteristic polynomials, up to and including the properties of Hermitian and orthogonal matrices. Later chapters bring in special results on commuting families of matrices, functions of matrices--including elements of the differential and integral calculus sometimes known as matrix analysis, and transformations of bilinear forms. The final chapter treats associative algebras, culminating with the well-known Wedderburn-Artin theorem that simple algebras are necessarily isomorphic to matrix algebras.
Wedderburn ends with an appendix of historical notes on the development of the theory of matrices, and a bibliography that emphasizes the history of the subject.
Graduate students and research mathematicians interested in matrices.
Matrices and vectors
Algebraic operations with matrices. The characteristic equation
Invariant factors and elementary divisors
Vector polynomials. Singular matric polynomials
Compound matrices
Symmetric, skew, and hermitian matrices
Commutative matrices
Functions of matrices
The automorphic tranformation of a bilinear form
Linear associative algebras
Notes
Bibliography
Index to bibliography
Index
Colloquium Publications, Volume: 21
1937; 118 pp; softcover
ISBN-10: 0-8218-4619-1
ISBN-13: 978-0-8218-4619-3
Expected publication date is July 14, 2008.
The main goal of this book is to present the theory of systems of partial
differential equations and the theory of Pfaffian systems so as to exhibit
clearly the relations between them.
In presenting the theory of Pfaffian systems, the author develops, in detail, the theories of Grassmann algebras and rings with differentiation. In particular, following Grassmann and E. Cartan, he introduces and freely uses what is now known as a ring of differential forms with functional coefficients. In presenting the theory of systems of partial differential equations, the author concentrates on the existence of solutions and methods of approximating them, rather than on their properties. The relations and similarities of two theories are displayed through the systematic use of various versions of the elimination method.
Graduate students and research mathematicians interested in partial differential equations and Pfaffian systems.
Introduction
Generalities on symbols and systems
Grassmann algebra
Differential rings
Commutative monomials and polynomials
Algebraic systems
Algebraic differential systems
Function systems and differential systems
Pfaffian systems
Consistency examples
Illustrative examples
Bibliography
Index
Colloquium Publications, Volume: 22
1938; 105 pp; softcover
ISBN-10: 0-8218-4620-5
ISBN-13: 978-0-8218-4620-9
Expected publication date is July 14, 2008.
Fairly early in the development of the theory of summability of divergent
series, the concept of convergence factors was recognized as of fundamental
importance in the subject. One of the pioneers in this field was C. N.
Moore, the author of the book under review.... Moore classifies convergence
factors into two types. In type I he places the factors which have only
the property that they preserve convergence for a convergent series or
produce convergence for a summable series. In type II he places the factors
which not only maintain or produce convergence but have the additional
property that they may be used to obtain the sum or generalized sum of
the series. This book gives a generalized systematic treatment of the theory
of convergence factors of both types, for simply infinite series and for
multiple series, convergent and summable....
--Bulletin of the American Mathematical Society
Graduate students and research mathematicians interested in analysis.
Introduction
Convergence factors in convergent series
Summation of series by Norlund means
Convergence factors in summable series
Convergence factors in summable double series
Convergence factors in summable multiple series
Convergence factors in restrictedly convergent multiple series
Bibliography
Index
Colloquium Publications, Volume: 30
1948; 572 pp; softcover
ISBN-10: 0-8218-4621-3
ISBN-13: 978-0-8218-4621-6
Expected publication date is July 14, 2008.
Rado's colloquium is a systematic treatment of Lebesgue theory, with an emphasis on the work of Morrey and of Rado and his students, especially in two dimensions. At the time, there were important current problems surrounding Lebesgue's theory for parameterized and unparameterized surfaces, which the book addresses. The exposition begins with reviews of Lebesgue integration and relevant topics in topology, including Frechet equivalence, the approximation of monotone maps by homeomorphisms, Peano spaces, and a discussion of the topological index of maps into the plane. After a development of further ideas and tools from topology and measure theory, Rado addresses an essential question that equates two sorts of areas for surfaces represented by maps of a 2-cell or a 2-sphere into 3-space.
Graduate students and research mathematicians interested in geometry and topology.
Background material
Curves and surfaces
Arc length and related topics
Plane tranformations
Surface area
Bibliography
Index
Colloquium Publications, Volume: 33
1950; 184 pp; softcover
ISBN-10: 0-8218-4638-8
ISBN-13: 978-0-8218-4638-4
Expected publication date is July 14, 2008.
A gigantic task undertaken by J. F. Ritt and his collaborators in the
1930's was to give the classical theory of nonlinear differential equations,
similar to the theory created by Emmy Noether and her school for algebraic
equations and algebraic varieties. The current book presents the results
of 20 years of work on this problem. The book quickly became a classic,
and thus far, it remains one of the most complete and valuable accounts
of differential algebra and its applications.
Graduate students and research mathematicians interested in differential algebra and its applications.
Differential polynomials and their ideals
Algebraic differential manifolds
Structure of differential polynomials
Systems of algebraic equations
Constructive methods
Analytical considerations
Intersections of algebraic differential manifolds
Riquier's existence theorem for orthonomic systems
Partial differential algebra
Questions for investigation
Bibliography
Index