We consider the Tribin function and its generalization based on the \( {Q}_s^{\ast } \) -representation of real numbers, which is an s-symbol encoding of numbers and, generally speaking, a nonself-similar generalization of the s-adic representation. By definition, the function f associates the number \( x={\varDelta}_{\upalpha_1{\upalpha}_2\dots {\upalpha}_n\dots}^{Q_s^{\ast }}, \) where αn ∈ L ≡ As × As × … × As × … and As = {0, 1, …, s − 1} is an alphabet, s ≥ 3; with the number \( y=f(x)={\varDelta}_{\upgamma_1{\upgamma}_2\dots {\upgamma}_n\dots}^{G_2^{\ast }}, \)
where the \( {G}_2^{\ast } \) -representation of numbers has the two-symbol alphabet A2 = {0, 1}. We prove that the function f is well-defined, continuous, and nowhere monotone. Its variational properties are also analyzed.
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M. V. Prats’ovytyi, “Nowhere monotonic singular functions,” Nauk. Chasopys. Nats. Ped. Univ. Drahomanova, Ser. 1, Fiz.-Mat. Nauk., No. 12, 24–36 (2011).
M. V. Prats’ovytyi and A. V. Kalashnikov, “Self-affine singular and nowhere monotone functions related to the Q-representation of real numbers,” Ukr. Mat. Zh., 65, No. 3, 405–417 (2013); English translation: Ukr. Math. J., 65, No. 3, 448–462 (2013).
M. V. Prats’ovytyi and N. A. Vasylenko, “One family of continuous nowhere monotonic functions with fractal properties,” Nauk. Chasopys. Nats. Ped. Univ. Drahomanova, Ser. 1, Fiz.-Mat. Nauk., No. 14, 176–188 (2013).
M. Pratsiovytyi and N. Vasylenko, “Fractal properties of functions defined in terms of Q-representation,” Internat. J. Math. Analysis, 7, No. 61-64, 3155–3169 (2013).
M. V. Prats’ovytyi and N. A. Vasylenko, “One family of continuous functions with everywhere dense set of singularities,” Nauk. Chasopys. Nats. Ped. Univ. Drahomanova, Ser. 1, Fiz.-Mat. Nauk., No. 12, 152–167 (2011).
M. V. Prats’ovytyi and N. A. Vasylenko, “Distributions of probabilities on graphs for one class of nowhere monotonic functions,” in: Proc. of the Institute of Applied Mathematics and Mechanics, Ukrainian National Academy of Science [in Ukrainian], Kyiv, 26 (2013), pp. 159–171.
S. B. Kozyrev, “On the topological density of winding functions,” Mat. Zametki, 33, Issue 1, 71–76 (1983).
I. V. Zamrii and M. V. Prats’ovytyi, “Singularity of the digit inversor for the Q3-representation of the fractional part of a real number, its fractal and integral properties,” Nelin. Kolyv., 18, No. 1, 55–70 (2015); English translation: J. Math. Sci., 215, No. 3, 323–340 (2016).
M. V. Prats’ovytyi, Fractal Approach to the Investigation of Singular Distributions [in Ukrainian], National Pedagogic University, Kyiv (1998).
M. V. Prats’ovytyi and O. V. Svynchuk, “Spread of values of a Cantor-type fractal continuous nonmonotone function,” Nelin. Kolyv., 21, No. 1, 116–130 (2018); English translation: J. Math. Sci., 240, No. 3, 342–357 (2019).
M. V. Prats’ovytyi, “Calculus systems with variable bases and variable alphabet (or expansion of numbers in Cantor series),” Student. Fiz.-Mat. Etyudy, No. 8, 6–18 (2009).
M. V. Prats’ovytyi, Geometry of the Classical Binary Representation of Real Numbers [in Ukrainian], National Pedagogic University, Kyiv (2012).
N. V. Pratsevityi, “Continuous Cantor projectors,” in: Methods for the Investigation of Algebraic and Topological Structures [in Russian], Kiev National Pedagogic University, Kiev (1989), pp. 95–105.
A. F. Turbin and N. V. Pratsevityi, Fractal Sets, Functions, and Distributions [in Russian], Naukova Dumka, Kiev (1992).
K. A. Bush, “Continuous functions without derivatives,” Amer. Math. Monthly, 59, No. 4, 222–225 (1952).
W. Wunderlich, “Eine ¨uberall stetige und nirgends differenzierbare Funktion,” Elem. Math., 7, No. 4, 73–79 (1952).
V. V. Koval, “Self-affine graphs of functions,” Nauk. Chasopys. Nats. Ped. Univ. Drahomanova, Ser. 1, Fiz.-Mat. Nauk., No. 5, 292–299 (2004).
O. B. Panasenko, “Fractal dimension of the graphs of continuous Cantor projectors,” Nauk. Chasopys. Nats. Ped. Univ. Drahomanova, Ser. 1, Fiz.-Mat. Nauk., No 9, 104–111 (2008).
M. V. Prats’ovytyi, “Fractal properties of one continuous nowhere differentiable function,” Nauk. Chasopys. Nats. Ped. Univ. Drahomanova, Fiz.-Mat. Nauk., No. 3, 351–362 (2002).
G. M. Torbin and N. V. Pratsevityi, “Random variables with independent Q*-signs,” in: Random Evolutions: Theoretical and Applied Problems [in Russian], Kiev (1992), pp. 95–104.
A. Ya. Khinchin, Continued Fractions [in Russian], Nauka, Moscow (1978).
M. Kac, Statistical Independence in Probability Analysis and Number Theory, The Mathematical Association of America (1959).
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Translated from Neliniini Kolyvannya, Vol. 22, No. 3, pp. 380–390, July–September, 2019.
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Prats’ovytyi, M.V., Baranovs’kyi, O.M. & Maslova, Y.P. Generalization of the Tribin Function. J Math Sci 253, 276–288 (2021). https://doi.org/10.1007/s10958-021-05227-3
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DOI: https://doi.org/10.1007/s10958-021-05227-3