Vladimir Andreevich Uspensky: books

Authors: E.B.Dynkin, V.A.Uspenskii

Title: Mathematical Conversations: Multicolor Problems, Problems in the Theory of Numbers, and Random Walks

This Dover edition is an unabridged republication and consolidation of the three booklets
originally published by D. C. Heath, Boston, in 1963:
• «Multicolor Problems,» dealing with several of the classical map-coloring problems;
• «Problems in the Theory of Numbers,» an elementary introduction to algebraic number theory; and
• «Random Walks,» addressing basic problems in probability theory.

High-school algebra is the only prerequisite.

Authors: V.A.Uspensky, A.L.Semenov

Title: Algorithms: Main Ideas and Applications

The theory of algorithms not only answers philosophical questions but also is eminently applicable to practical computing, as well as to software and hardware design. This book presents exact mathematical formulations of major concepts and facts of the theory of algorithms in a unified and elegant way. Precise mathematical statements are given, together with their underlying motivations, philosophical interpretations and historical developments, starting with Frege, Hilbert and Borel through Gödel and Turing up to Kolmogorov’s results of 1950-1980. The book is divided into two parts. The first part outlines the fundamental discoveries of the general theory of algorithms. Numerous applications are discussed in the second part. The concept of probabilistic algorithms is presented in the Appendix. This work will be of interest to mathematicians, computer scientists, engineers and to everyone who uses algorithms.

Author: V.A.Uspensky

Title: Gödel’s Incompleteness Theorem

Series: Little Mathematics Library

Author: V.A.Uspensky

Title: Post’s Machine

Series: Little Mathematics Library

Translated from the Russian by R.Alavina.

This booklet is intended first of all for schoolchildren. The first two chapters are comprehensible even for junior schoolchildren. The book deals with a certain «toy» («abstract» in scientifIc terms) computing machine – the so called Post machine – on which calculations involve many important features inherent in the computations on real electronic computers. By means of the simplest examples the students are taught the fundamentals of programming for the Post machine, and the machine, though extremely simple, is found to possess quite high potentialities.

The reader is not expected to have any knowledge of mathematics beyond the primary school curriculum.

Author: V.A.Uspensky

Title: Pascal’s Triangle: Certain Applications of Mechanics to Mathematics

Series: Little Mathematics Library

Translation: V.Kisin.

Author: V.A.Uspenskii

Title: Pascal’s Triangle

Series: Polular Lectures in Mathematics

Translated and adapted from the Russian by David J. Sookne and Timothy McLarnan.

Pascal’s triangle is a numerical table which has fascinated mathematicians since its invention by Pascal over three hundred years ago. Its applications in number theory, probability theory, combinatorics, and even analysis make it important on many levels of mathematics. This table can be used to solve a variety of computation problems. Some of these problems are examined in this book, and the question of what «solving a problem» can mean is generally considered.

This exposition requires no preliminary knowledge beyond the limits of the 8th-grade curriculum, except for the definition of and notation for the zeroth power of a number. That is, one must know that any non-zero number raised to the zeroth power is considered, by definition, to be equal to 1.

Author: V.A.Uspenskii

Title: Some Applications of Mechanics to Mathematics

Series: Polular Lectures in Mathematics, volume 3

The applications of mathematics to physics (in particular, to mechanics) are well-known. We need only open a school text-book to find examples. The higher branches of mechanics demand a complex and refined mathematical apparatus.

There are, however, mathematical problems for whose solutions we can successfully use ideas and laws of physics. A number of problems of this kind soluble by methods drawn from mechanics (namely, by using the laws of equilibrium) were given by the author in his lecture «The solving of mathematical problems by the methods of mechanics», which was read to pupils in their seventh year of secondary school at the Moscow State University on 19 February 1656; this lecture, with very minor additions, makes up the contents of this article.

The author is deeply grateful to Isaak Moiseyevich Yaglom, whose detailed remarks helped to reduce the number of deficiencies in this book.