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:20260623T075538Z 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\n\nAbstract\nThis is a qualifying seminar organized by th e Department of Dynamical Systems and Fractal Analysis for the public pres entation and discussion of the Ph.D. student's research results obtained f or his Ph.D. degree dissertation in specialty 111\, Mathematics. The depar tment must provide a detailed report on the scientific novelty and the the oretical and practical value of the dissertation results.\n\nThis disserta tion is devoted to the generalization of the classical theory of Borel nor mal numbers and normal sequences of symbols to the case of numeral systems with finite alphabets\, such as the $Q_s$-representation of numbers in th e interval $[0\, 1]$\, as well as to problems of uniformly distributed seq uences generated by these systems.\n\nIn this field\, the results of É.  Borel\, H. Lebesgue\, W. Sierpiński\, D. Champernowne\, S. Pillai\, I . Piatetski-Shapiro\, P. Erdős\, and others are classical. Scientific i nterest in this topic remains high due to its deep connections with the th eory of dynamical systems\, fractal analysis\, and fractal geometry.\n\nTh e main results of this work are solutions to metric problems that are anal ogs of É. Borel\, D. Wall\, and I. Piatetski-Shapiro's results and met hods (algorithms) for constructing normal (quasi-normal) symbols correspon ding to the $Q_s$-representation of numbers.\n\nIn the talk\, the followin g scientific results will be presented:\n
    \n
  1. the necessary and suffi cient conditions of the uniform and quasi-uniform distribution for sequenc es defined in terms of iterations of the left-shift operator for symbols o f the $Q_s$-representation of numbers\;
    2. \n
  2. an analog of the Piatets ki-Shapiro-type criterion for sequences of symbols generated by the left-s hift operator for the $Q_s$-representation\;
    3. \n
  3. a series of propert ies of iterations of the left-shift operator whose indices increase nonlin early and a full description of the structure of $Q_s$-representations cor responding to mutually inversive $Q_s$-normal sequences of symbols\;
    4. \ n
  4. constructive methods for obtaining recursively computable normal (qua si-normal) sequences of symbols (normal numbers) corresponding to the $Q_s $-representation of numbers\;
    5. \n
  5. a structure of transformations tha t preserve a uniform distribution of sequences.
    6. \n
\n LOCATION:https://researchseminars.org/talk/fran/80/ END:VEVENT END:VCALENDAR