BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Volodymyr Yelahin (Institute of Mathematics\, Natl. Acad. Sci. Ukr aine) DTSTART:20260709T123000Z DTEND:20260709T140000Z DTSTAMP:20260715T091811Z UID:fran/82 DESCRIPTION:Title: Co ntinuous structurally fractal functions defined in terms of the $Q_s$-repr esentation and nega-$Q_s$-representation of numbers (qualifying seminar)\nby Volodymyr Yelahin (Institute of Mathematics\, Natl. Acad. Sci. Ukra ine) as part of Семінар з фрактального аналізу / Fractal analysis seminar\n\n\nAbstract\nThis is a qualifying seminar org anized by the Department of Dynamical Systems and Fractal Analysis for the public presentation and discussion of the Ph.D. student's research result s obtained for his Ph.D. degree dissertation in specialty 111\, Mathematic s. The department must provide a detailed report on the scientific novelty and the theoretical and practical value of the dissertation results.\n\nT he dissertation is devoted to the applications of finite-symbol representa tions of real numbers with zero redundancy\, such as the $Q_s$-representat ion and nega-$Q_s$-representation\, in the constructive theory of continuo us locally complicated functions with structurally fractal properties defi ned on the interval $[0\, 1]$.\n\nThe main objects of study are continuous functions preserving the tails of the nega-$Q_s$-representation\, nowhere monotonic and nowhere differentiable functions defined in terms of the $Q _s$-representation\, and functions related to infinite Bernoulli convoluti ons governed by the negabinary series.\n\nIn the talk\, the following main results will be presented:\n
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  1. We proved that the set of continuou s transformations of the unit interval (i.e.\, bijective mappings of an in terval onto itself) that preserve the tails of the negabinary representati on of numbers forms a noncommutative group under the composition (superpos ition) of transformations with a continuum subgroup of increasing transfor mations. These transformations are constructed using left- and right-shift operators on the digits of a representation of numbers.
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  2. The con ditions for an infinite Bernoulli convolution governed by the negabinary s eries to have a uniform\, pure discrete\, absolutely continuous\, singular \, or exponential distribution are found.
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  3. The structure and spec tral properties of a random variable whose digits of the negabinary repres entation form a homogeneous Markov chain are studied.
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  4. We proved that the nega-$Q_s$-representation of numbers is a re-encoding of the know n $Q_s$-representation and generates an identical metric theory. For this system of encoding of numbers\, we proved that the group of continuous tra nsformations of the unit interval that preserve the tails of the nega-$Q_s $-representation of numbers is infinite and contains a subgroup of increas ing transformations.
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  5. For a class of continuous locally complicat ed self-affine functions defined in terms of the $Q_s$-representation of n umbers\, we computed local Hölder exponents at points with given asymptot ic frequencies of digits in their $Q_s$-representation. The conditions for these functions to have continuum level sets are found.
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  6. For con tinuous self-affine functions satisfying additional conditions\, we descri bed the geometric structure of sets of maximum points and\, in particular\ , showed that these sets can be fractal.
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\n LOCATION:https://researchseminars.org/talk/fran/82/ END:VEVENT END:VCALENDAR