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SUMMARY:Mykola Pratsiovytyi (Mykhailo Drahomanov Ukrainian State Universit
 y\; Institute of Mathematics\, Natl. Acad. Sci. Ukraine)
DTSTART:20250320T133000Z
DTEND:20250320T150000Z
DTSTAMP:20260423T005733Z
UID:fran/66
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/fran/66/">Nu
 meral systems with natural base and redundant alphabet and their applicati
 ons in number theory\, probability theory\, and geometry of numerical seri
 es</a>\nby Mykola Pratsiovytyi (Mykhailo Drahomanov Ukrainian State Univer
 sity\; Institute of Mathematics\, Natl. Acad. Sci. Ukraine) as part of С
 емінар з фрактального аналізу / Fractal analysis
  seminar\n\n\nAbstract\nThis talk is devoted to the function defined by eq
 uality\n\\[\n  f(x = \\Delta^{r+1}_{\\alpha_1\\ldots\\alpha_n\\ldots}) = \
 \Delta^{r_s}_{\\alpha_1\\ldots\\alpha_n\\ldots}\,\n\\]\nwhere $s$ and $r$ 
 are fixed natural numbers such that $2 \\leq s \\leq r$\,\n\\[\n  \\Delta^
 {r+1}_{\\alpha_1\\ldots\\alpha_n\\ldots} = \\sum_{n=1}^\\infty \\alpha_n (
 r+1)^{-n}\,\n  \\quad\n  \\Delta^{r_s}_{\\alpha_1\\ldots\\alpha_n\\ldots} 
 = \\sum_{n=1}^\\infty \\alpha_n s^{-n}\,\n  \\quad\n  \\alpha_n \\in A = \
 \{ 0\, 1\, \\ldots\, r \\}.\n\\]\n\nWe study structural\, variational\, to
 pological\, metric\, and fractal properties of the function\, in particula
 r its level sets.\n\nSince the values of the function are given in the sys
 tem of encoding of numbers with base $s$ and redundant alphabet $A$\, in t
 he talk\, we discuss such systems and their applications in the theory of 
 singular probability distributions in detail.\n\nWe also clarify a connect
 ion between numeral systems with redundant alphabet and the geometry of nu
 merical series (i.e.\, topological and metric analysis of their subsums).\
 n
LOCATION:https://researchseminars.org/talk/fran/66/
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