Contents



Preface

Since ancient times combinatorics and probability theory have been closely interrelated. Combinatorial techniques were used for the calculation of probabilities even when the latter were not directly assumed and were hidden behind the concept of "chances." After the notion of probability appeared, was acknowledged and comprehended, this interrelation became even stronger. Now probabilistic and combinatorial techniques are often used for solving advanced problems. The union of probability and combinatorics becomes more and more actual within the problem of Big Data, which assumes the possibility of detection of latent regularities or relations in new non-structured data types and construction of predictive models. One of the most urgent directions of the development of methods for Big Data analysis is their application in artificial intelligence systems. The problems of adequate modeling involve the development of the mathematical apparatus for the construction of reasonable models of statistical regularities and the study of their analytical and asymptotic properties. Whereas for a layman probability is still associated with divination on daisies, for specialists these methods long ago became powerful tools in predicting successes or failures, preventive management and achieving the desired successes. Some of these applicable techniques are demonstrated in this book. It is worth noting that to a great extent the calculus of probabilities became a mathematical theory due to the findings and works of the

representatives of the Russian mathematical school: P.L. Chebyshev (1821–1894), A. A. Markov (1856–1922), A.M. Lyapunov (1857–1918), S.N. Bernstein (1880–1968), A.Ya. Khinchin (1894–1959), A.N. Kolmogorov (1903–1987), B.V. Gnedenko (1912– 1995), Yu.V. Prokhorov (1929–2013), who by all means deserve to be remembered along with the famous creators of probability theory J. Bernoulli (1655–1705), P.-S. Laplace (1749–1827), S.D. Poisson (1781–1840), C.F. Gauss (1777–1855) and others.

This book is very unusual. It is not at all a random collection on a topic devoted to a formalization of uncertainties. Considering that because of high complexity and uncertainties the existing probabilistic models can't be used sometimes directly to predict and estimate desired results, the initial concept of the book is as follows:

• to draw the reader's attention to the fact that the same mathematical models for cognitive solving of different problems are useful in a variety of applications (e.g., artificial intelligence systems, high-power hydro turbine, system of flowering synchronization with respect to climate dynamics, organizationaltechnical-economical systems, offshore platforms, asymptotic approximations,

• to educate how modern probabilistic and combinatorial models may be created

• to train how new probabilistic models can be generated for the systems of

• to present the correct use of the presented models for rational control in

graphs, uncertain costs, equipment of nuclear power plants, trunk oil pipelines, risk processes, social networks, combinatorial cosmology);

to formalize uncertainties;

systems creation and operation; and

complex structures;
