Search

The theme of this volume is a treatment of differential equations on a $C^\infty$ manifold $V$ by separation of variables techniques. More specifically, given a Lie transformation group $L$ of $V$ and a Lie subgroup $H\subset L$, if $D(V)$ is the set of differential operators on $V$ invariant under $L$, then the principal object of study is the set of distributions $T$ on $V$ satisfying the following two conditions:(i) $T$ is an eigendistribution of each $D\in D(V)$; (ii) $T$ is invariant under $H$.

This extensively revised and expanded edition of Lectures in Differentiable Dynamics, first published in 1971, provides an authoritative exposition of the central results of this fundamental and rapidly developing mathematical subject, starting from simple engineering systems and proceeding to current research topics concerning differentiable flows on manifolds. The original version of this monograph culminates in the research results of the 1960s, enphasizing the concepts of structural stability and generic dynamics—with references to Morse-Smale and Anosov hyperbolic differential systems. In the new edition there appears a general updating to cover research of the past decade through 1980, with a major supplemental Appendix reviewing the modern developments under five basic areas: I. Nonlinear Oscillations, II. Diffeomorphisms and Foliations, III. General Theory—Dissipative Dynamics, IV. General Theory—Conservative Dynamics, V. Chaos, Catastrophe, and Multi-Valued Trajectories. There is also an extensive bibliography, including special listings of important mathematical conferences, major survey articles, and current research articles.

This book can be viewed as a first attempt to systematically develop an algebraic theory of nonlinear differential equations, both ordinary and partial. The main goal of the author was to construct a theory of elimination, which “will reduce the existence problem for a finite or infinite system of algebraic differential equations to the application of the implicit function theorem taken with Cauchy's theorem in the ordinary case and Riquier's in the partial.” In his 1934 review of the book, J. M. Thomas called it “concise, readable, original, precise, and stimulating”, and his words still remain true. A more fundamental and complete account of further developments of the algebraic approach to differential equations is given in Ritt's treatise Differential Algebra, written almost 20 years after the present work (Colloquium Publications, Vol. 33, American Mathematical Society, 1950).

This book studies fundamental properties of the logarithmic potential and their connections to the theory of Fourier series, to potential theory, and to function theory. The material centers around a study of Poisson's integral in two dimensions and of the corresponding Stieltjes integral. The results are then extended to the integrals in terms of Green's functions for general regions. There are some thirty exercises scattered throughout the text. These are designed in part to familiarize the reader with the concepts introduced, and in part to complement the theory. The reader should know something of potential theory, functions of a complex variable, and Lebesgue integrals. The book is based on lectures given by the author in 1924–1925 at the Rice Institute and at the University of Chicago.

The American Mathematical Society held its fifth colloquium in connection with its thirteenth summer meeting, under the auspices of Yale University, during the week September 3–8, 1906. This book contains the lecture notes for the three courses that were given at this colloquium: “Introduction to a Form of General Analysis” by Eliakim H. Moore, “Projective Differential Geometry” by Ernest J. Wilczynski, and “Selected Topics in the Theory of Boundary Value Problems of Differential Equations” by Max Mason.