Andrei Nikolaevich Kolmogorov

This book is the second volume of a study of the history of mathematics in the nineteenth century. The first part of the book describes the development of geometry. The many varieties of geometry are considered and three main themes are traced: the development of a theory of invariants and forms that determine certain geometric structures such as curves or surfaces; the enlargement of conceptions of space which led to non-Euclidean geometry; and the penetration of algebraic methods into geometry in connection with algebraic geometry and the geometry of transformation groups. The second part, on analytic function theory, shows how the work of mathematicians like Cauchy, Riemann and Weierstrass led to new ways of understanding functions. Drawing much of their inspiration from the study of algebraic functions and their integrals, these mathematicians and others created a unified, yet comprehensive theory in which the original algebraic problems were subsumed in special areas devoted to elliptic, algebraic, Abelian and automorphic functions. The use of power series expansions made it possible to include completely general transcendental functions in the same theory and opened up the study of the very fertile subject of entire functions.

Foundation of Probability, Axioms and Theorems

"Mathematics: The New Golden Age offers a glimpse of the extraordinary vistas and bizarre universes opened up by contemporary mathematicians: Hilbert's tenth problem and the four-color theorem, Gaussian integers, chaotic dynamics and the Mandelbrot set, infinite numbers, and strange number systems. Why a "new golden age"? According to Keith Devlin, we are currently witnessing an astronomical amount of mathematical research. Charting the most significant developments that have taken place in mathematics since 1960, Devlin expertly describes these advances for the interested layperson and adroitly summarizes their significance as he leads the reader into the heart of the most interesting mathematical perplexities - from the biggest known prime number to the Shimura-Taniyama conjecture for Fermat's Last Theorem."--BOOK JACKET.

This volume is the second of three volumes devoted to the work of one of the most prominent twentieth-century mathematicians. Throughout his mathematical work, A.N. Kolmogorov (1903-1987) showed great creativity and versatility and his wide-ranging studies in many different areas led to the solution of conceptual and fundamental problems and the posing of new, important questions. His lasting contributions embrace probability theory and statistics, the theory of dynamical systems, mathematical logic, geometry and topology, the theory of functions and functional analysis, classical mechanics, the theory of turbulence, and information theory. This second volume contains papers on probability theory and mathematical statistics, and embraces topics such as limit theorems, axiomatics and logical foundations of probability theory, Markov chains and processes, stationary processes and branching processes. The material appearing in each volume was selected by A.N. Kolmogorov himself and is accompanied by short introductory notes and commentaries which reflect upon the influence of this work on the development of modern mathematics. All papers appear in English - some for the first time -- and in chronological order. This volume contains a significant legacy which will find many grateful beneficiaries amongst researchers and students of mathematics and mechanics, as well as historians of mathematics.