A class of Tribin functions (from idea and interest to motives and fantasies)

Abstract: This talk is devoted to the continuum class of continuous nowhere monotonic functions that generalize non-differentiable Bush and Wunderlich functions, continuous Cantor projectors, Tribin functions, etc.

We consider continuous function $f$ defined by equality \[ f(x = \Delta^{s^*}_{\alpha_1\alpha_2\ldots\alpha_n\ldots}) = \Delta^{2^*}_{b_1b_2\ldots b_n\ldots}, \] where \[ b_1 = \begin{cases} 0 & \text{if $\alpha_1 \in A_0 \neq \emptyset$}, \\ 1 & \text{if $\alpha_1 \in A \setminus A_0 \equiv A_1 \neq \emptyset$}, \end{cases} \qquad b_{n+1} = \begin{cases} b_n & \text{if $\alpha_{n+1} = \alpha_n$}, \\ 1 - b_n & \text{if $\alpha_{n+1} \neq \alpha_n$}, \end{cases} \] $A = \{ 0, 1, \ldots, s - 1 \}$ is an alphabet, $s \geq 3$, $A_0$ is a subset of the alphabet, $\Delta^{s^*}_{\alpha_1\alpha_2\ldots\alpha_n\ldots}$ is an $s$-symbol representation of a number $x \in [0, 1]$ that is topologically equivalent to classical $s$-adic representation and $\Delta^{2^*}_{b_1b_2\ldots b_n\ldots}$ is a two-symbol representation that is topologically equivalent to classical binary representation.

In the talk, we analyze structural, variational, integral and differential, topological and metric, and fractal properties of the function $f$.

Ukrainianclassical analysis and ODEsfunctional analysisnumber theory

Audience: researchers in the discipline

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Семінар з фрактального аналізу / Fractal analysis seminar

Series comments: Weekly research seminar on fractal analysis (online)

Topics:

• theory of fractals (fractal geometry and fractal analysis)
• Hausdorff–Besicovitch dimension, techniques and methods for its calculation and estimation
• functions and transformations preserving fractal (Hausdorff–Besicovitch, entropic, box-counting, packing, etc.) dimension
• sets of metric spaces that are essential for functions, sets, and dynamical systems
• self-similar, self-affine properties of mathematical objects
• systems of encoding for real numbers (numeral systems) and their applications
• metric number theory and metric theory of representations of numbers
• probabilistic number theory and probabilistic theory of representations of numbers
• measures supported on fractals, particularly singular measures and probability distributions
• nowhere monotonic and nowhere differentiable functions, functions with fractal properties
• theory of groups determined by invariants of transformations preserving tails of representation of numbers, digit frequencies, etc.

The talks are mostly in Ukrainian but English is also acceptable

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