N. Ya. Vilenkin's Books

Combinatorics

By: N. Ya. Vilenkin

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Combinatorics deals with simple combinatorial problems, recurrence relations, and generating functions, particularly the binomial expansions. The book expounds on the general rules of combinatorics, the rule of sum, the rule of product, samples, permutations, combinations, and arrangements of subjects with various restrictions. The text also explains ordered or unordered partitions of numbers, geometric methods, random walk problems, and variants of the arithmetical triangle. One example of the use of combinatorics is the choice of the number 3 in the genetic code. Another example involves the choice of crew for a spaceship where it is necessary to consider the psychological conditions of the applicants for space travel. The text also investigates the sieve of Erastothenes whose problem concerns finding all the primes in the sequence of natural numbers from 1 to N. The book also tackles the application of power series to proof of identities, the binomial series expansion, decomposition into elementary fractions, and nonlinear recurrence relation. The book can be highly educational and interesting to students or academicians involved in mathematics, algebra, and statistics.

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Page Count

313

Publisher

Academic Press

Published Date

ISBN 10

1483266117

ISBN 13

9781483266114

Stories About Sets

By: N. Ya. Vilenkin

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Stories About Sets

By: N. Ya. Vilenkin

Stories About Sets discusses the cardinality of sets and mathematical concepts, such as function, curve, surface, dimensions, and the paradoxical properties of curves and surfaces. The book reviews sets, operations on sets, the empty set, subsets, the universal sets, intersection of sets, union of sets, partitioning of sets, and boolean algebras. The text also discusses the cardinality of sets, including equality between sets, countable sets, unequal sets, the uncountability of the continuum, the existence of transcendental numbers, and the enigmatic axiom. The book analyzes if a part can be equal to the whole (which turns out to be true if it is applied to infinite sets). The text also discusses the arithmetic of the infinite such as involving the multiplication of infinite cardinalities. The book explains some remarkable functions and curves, the Dirichlet's function, Cantor's set, points of fracture, and continuous functions whose graphs possess a tangent at no point. The text shows how to construct a closed curve of infinite length or a curve passing through all the points of a square. The book can prove interesting and highly educational for students with mathematic or algebra subjects, as well as for academicians involved in teaching statistics or mathematics.

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Page Count

167

Publisher

Academic Press

Published Date

ISBN 10

1483214354

ISBN 13

9781483214351

Representation Theory and Noncommutative Harmonic Analysis II

Homogeneous Spaces, Representations and Special Functions

By: A.A. Kirillov

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Representation Theory and Noncommutative Harmonic Analysis II

Homogeneous Spaces, Representations and Special Functions

By: A.A. Kirillov

Two surveys introducing readers to the subjects of harmonic analysis on semi-simple spaces and group theoretical methods, and preparing them for the study of more specialised literature. This book will be very useful to students and researchers in mathematics, theoretical physics and those chemists dealing with quantum systems.

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Page Count

274

Publisher

Springer Science & Business Media

Published Date

ISBN 10

3662097567

ISBN 13

9783662097564

Generalized Functions

Applications of Harmonic Analysis

By: I. M. Gel'fand,N. Ya. Vilenkin

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Generalized Functions

Applications of Harmonic Analysis

By: I. M. Gel'fand,N. Ya. Vilenkin

Generalized Functions, Volume 4: Applications of Harmonic Analysis is devoted to two general topics—developments in the theory of linear topological spaces and construction of harmonic analysis in n-dimensional Euclidean and infinite-dimensional spaces. This volume specifically discusses the bilinear functionals on countably normed spaces, Hilbert-Schmidt operators, and spectral analysis of operators in rigged Hilbert spaces. The general form of positive generalized functions on the space S, continuous positive-definite functions, and conditionally positive generalized functions are also deliberated. This publication likewise considers the mean of a generalized random process, multidimensional generalized random fields, simplest properties of cylinder sets, and definition of Gaussian measures. This book is beneficial to students, specialists, and researchers aiming to acquire knowledge of functional analysis.

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Page Count

399

Publisher

Academic Press

Published Date

ISBN 10

1483262243

ISBN 13

9781483262246

In Search of Infinity

By: N.Ya. Vilenkin

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In Search of Infinity

By: N.Ya. Vilenkin

The concept of infinity is one of the most important, and at the same time, one of the most mysterious concepts of science. Already in antiquity many philosophers and mathematicians pondered over its contradictory nature. In mathematics, the contradictions connected with infinity intensified after the creation, at the end of the 19th century, of the theory of infinite sets and the subsequent discovery, soon after, of paradoxes in this theory. At the time, many scientists ignored the paradoxes and used set theory extensively in their work, while others subjected set-theoretic methods in mathematics to harsh criticism. The debate intensified when a group of French mathematicians, who wrote under the pseudonym of Nicolas Bourbaki, tried to erect the whole edifice of mathematics on the single notion of a set. Some mathematicians greeted this attempt enthusiastically while others regarded it as an unnecessary formalization, an attempt to tear mathematics away from life-giving practical applications that sustain it. These differences notwithstanding, Bourbaki has had a significant influence on the evolution of mathematics in the twentieth century. In this book we try to tell the reader how the idea of the infinite arose and developed in physics and in mathematics, how the theory of infinite sets was constructed, what paradoxes it has led to, what significant efforts have been made to eliminate the resulting contradictions, and what routes scientists are trying to find that would provide a way out of the many difficulties.

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Page Count

146

Publisher

Birkhäuser

Published Date

ISBN 10

1461208386

ISBN 13

9781461208389

Book's by N. Ya. Vilenkin

Combinatorics

Combinatorics

Stories About Sets

Stories About Sets

Representation Theory and Noncommutative Harmonic Analysis II

Representation Theory and Noncommutative Harmonic Analysis II

Generalized Functions

Generalized Functions

In Search of Infinity

In Search of Infinity

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