L V Kantorovich: Part I: Descriptive Theory of Sets and Functions. Functional Analysis in Semi-orded Spaces. Part II: Applied Functional Analysis. Approximation Met(Classics of Soviet Mathematics)

Part I of the Selected Works of L.V.Kantorovich is devoted to his mathematical work, with particular emphasis on the contribution he made to set theory and methods of mathematical approximation. The book begins with some chapters on the Descriptive Theory of Sets and Real Functions. Topics include universal functions, W.H. Young's classification, generalized derivatives of continuous functions and the H. Steinhaus problem. The book also includes papers on functional analysis in semi-ordered vector spaces, as well as articles relevant to the extension of Hilbert space in the spirit of distribution theory. This indispensable reference provides a record of the achievements of a man whose originality and diversity of thought distinguished him as a remarkable mathematician. Kantorovich was a true polymath, whose mathematical work had applications across a very broad field of subjects, particularly economics. This book will prove to be a useful reference work for anyone interested in learning about the origin of a number of important mathematical ideas. Part II of this set contains his works on Functional Analysis. The author deals with a number of approximate calculations and solutions including a discussion of an approximate calculation of certain types of definite integrals, and also a method for the approximate solution of partial differential equations. In addition to this, the author looks at various other methods, including the Ritz method, the Galerkin method in relation to the reduction of differential equations, and the Newton method for functional equations. Towards the end of the book there are several chapters on computers-a very early insight into what proved to be a development of enormous importance in the later half of the 20th century. Kantorovich is often said to be the founding father of mathematical programming. His ideas of the 1930s, which could never to come to fruition in Stalin's Soviet Union, were later developed by other scientists, and have had a considerable impact on the world, particularly in the field of economics. This book will appeal to anyone seeking further insight into the works of this great mathematician.

Table of Contents:
PART I: Leonid Kantorovich and the synthesis of two cultures. The message of L.V. Kantorovich. Biochronology of L.V. Kantorovich - My path in science. On sequences of functions contained in W.H. Young's classification. On universal functions. On a problem of H. Steinhaus. On sequences of functions, continuous almost everywhere. An example of a semicontinous function universal for continuous functions. On generalized derivatives of continuous functions. On two classes of operations on closed sets. Memoir on analytical operations and project sets. On some theorems concerning the theory of projective sets. On certain general methods of the extension of Hilbert space. On certain particular methods of the extension of Hilbert space. Semi-ordered linear spaces and their application to the theory of linear operators. Linear semi-ordered spaces. Linear operators in semi-ordered spaces. The method of successive approximations for functional equations. PART II: On convergence of the sequence of Bernstein polynomials beyond the principal interval. Some remarks on approximation of functions by polynomials with integer coefficients. On certain expansions in Bernstein polynomials. Explicit representation of a measurable function as the limit of polynomial sequences. A direct method for approximate solution of the problem of a double integral minimum. On conformal mappings. Application of the Stieltjes integral to the calculation of beams lying on elastic supports. On general methods of improving the convergence for approximate solutions to boundary value problems of mathematical physics. On approximate calculation of certain types of definite integrals. On a method of approximate solution of partial differential equations. On convergence of variational processes. On the convergence of the method of reduction to ordinary differential equations. Some remarks on the Ritz method. Using the ideas of the Galerkin method in the method of reduction to differential equations. On an effective method of solving extremal problems for quadratic functionals. On the method of steepest descent. On Newton's method for functional equations. Functional analysis and applied mathematics. On differential equations of the form x" = f(x). On the Newton Method. Some further applications of the Newton method for functional equations. New modes of calculations with the tabulator using the binary representatives of numbers. Prospects in the development and applications of computers. Execution of numeric and analytic calculations on computers. Increasing of productivity of universal computers in solving problems of mathematical economics.

About the Author :
Leonid Vitalievich Kantorovich was born in St Petersburg in 1912, and embarked upon a life's work in mathematics at an early age, writing his first book at the age of 21, and gaining a professorship at Leningrad State University a year later. During the course of his life he wrote numerous mathematical books and articles and was recognised by many foreign universities and institutes. In 1975 he was awarded the Nobel Memorial Prize for economics together with T.C. Koopmans for their joint contribution to the theory of optimum allocation of resources. Up until his death in 1986, Kantorovich remained a respected figure in the world of applied mathematics, and his works proved to be influential across a wide range of disciplines.