Encodings of real numbers and pathological locally complicated objects of mathematical analysis

Work number - M 10 ALLOWED TO PARTICIPATE

Presented by Kharkiv National University of Internal Affairs

Authors:
SERBENYUK Symon Oleksandrovych - Researcher of the Department of Scientific Work Organization of the Department of the Scientific Activity Organization and Intellectual Property Protection, Deputy Chairman of the Scientific Society of students, cadets, trainees, postgraduates, adjuncts, doctoral students and young scientists of Kharkiv National University of Internal Affairs

The presented research is theoretical and is devoted to modelling and investigating generalizations of well-known encoding systems of real numbers, as well as pathological locally complicated functions and sets, to solving some classical problems on the topic of the research. In particular, one can note the following: an open problem, which was formulated by G. Cantor in 1869, on establishing the necessary and sufficient conditions for representations of rational numbers in terms of the general case of Cantor series; modelling and investigations of complicated encodings of real numbers such that are generalizations of known encodings; establishing of new rules regarding complicated local properties and the dependence of the Hausdorff dimension of a certain fractals and their images; modelling and research of the simplest examples of singular, nowhere differentiable, non-monotonic functions; generalizations of the classical singular Salem function; etc. A number of new results such that have no analogues in the world or are generalizations of existing results within the framework of scientific research begun by the world classics of mathematics and continued by their followers, have been obtained by the author.

First of all the results are aimed on applications in mathematics and in applied researches such as encoding, encryption, and decoding of data, analysis of financial and biomedical time series, investigations of the variability structure of the financial market system and the cryptocurrency market, in economic physics, statistical signal and images processing, etc. In addition, objects of the research are useful in mathematical education as noted by the French mathematician J. H. Poincaré in “Science et méthode” (Paris, 1908).

Number of publications: 20 articles in journals included to the category "А" (including 17 foreign publications) and 8 articles in journals of the category "B", 16 abstracts at conferences. All publications are single-authored. The total number of references to authors' publications/h-index by the research according to the databases is respectively equal to: Web of Science 48/4, Scopus 92/6, Google Scholar 376/11.

Name	Type	Size	File
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