BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Rostyslav Kryvoshyia (Institute of Mathematics\, Natl. Acad. Sci. Ukraine) DTSTART:20260604T123000Z DTEND:20260604T140000Z DTSTAMP:20260602T192900Z UID:fran/80 DESCRIPTION:Title: No rmal sequences of symbols produced by singular probability distribution fu nctions with independent $Q$-symbols (qualifying seminar)\nby Rostysla v Kryvoshyia (Institute of Mathematics\, Natl. Acad. Sci. Ukraine) as part of Семінар з фрактального аналізу / Fractal an alysis seminar\n\nInteractive livestream: https://bbb.imath.kiev.ua/b/fra- gzt-rh6\n\nAbstract\nThis is a qualifying seminar organized by the Departm ent of Dynamical Systems and Fractal Analysis for the public presentation and discussion of the Ph.D. student's research results obtained for his Ph .D. degree dissertation in specialty 111\, Mathematics. The department mus t provide a detailed report on the scientific novelty and the theoretical and practical value of the dissertation results.\n\nThis dissertation is d evoted to the generalization of the classical theory of Borel normal numbe rs and normal sequences of symbols to the case of numeral systems with fin ite alphabets\, such as the $Q_s$-representation of numbers in the interva l $[0\, 1]$\, as well as to problems of uniformly distributed sequences ge nerated by these systems.\n\nIn this field\, the results of É. Borel\, H . Lebesgue\, W. Sierpiński\, D. Champernowne\, S. Pillai\, I. Piatet ski-Shapiro\, P. Erdős\, and others are classical. Scientific interest i n this topic remains high due to its deep connections with the theory of d ynamical systems\, fractal analysis\, and fractal geometry.\n\nThe main re sults of this work are solutions to metric problems that are analogs of É . Borel\, D. Wall\, and I. Piatetski-Shapiro's results and methods (alg orithms) for constructing normal (quasi-normal) symbols corresponding to t he $Q_s$-representation of numbers.\n\nIn the talk\, the following scienti fic results will be presented:\n
    \n
  1. the necessary and sufficient con ditions of the uniform and quasi-uniform distribution for sequences define d in terms of iterations of the left-shift operator for symbols of the $Q_ s$-representation of numbers\;
    2. \n
  2. an analog of the Piatetski-Shapir o-type criterion for sequences of symbols generated by the left-shift oper ator for the $Q_s$-representation\;
    3. \n
  3. a series of properties of it erations of the left-shift operator whose indices increase nonlinearly and a full description of the structure of $Q_s$-representations correspondin g to mutually inversive $Q_s$-normal sequences of symbols\;
    4. \n
  4. cons tructive methods for obtaining recursively computable normal (quasi-normal ) sequences of symbols (normal numbers) corresponding to the $Q_s$-represe ntation of numbers\;
    5. \n
  5. a structure of transformations that preserv e a uniform distribution of sequences.
    6. \n
\n LOCATION:https://researchseminars.org/talk/fran/80/ URL:https://bbb.imath.kiev.ua/b/fra-gzt-rh6 END:VEVENT END:VCALENDAR