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This article is cited in 4 scientific papers (total in 4 papers)
Differential and Integral Operations in Hidden Spherical Symmetry and the Dimension of the Koch Curve
L. N. Lyakhovabc, E. Saninaa a Voronezh State University
b Lipetsk State Pedagogical University
c I. A. Bunin Elets State University
Abstract:
Examples of differential and integral operations are given whose dimension is modified as a result of the introduction of new radial variables.
Based on the integral measure $x^\gamma\,dx$, $\gamma>-1$, with a weak singularity, we introduce
an operator that is interpreted as the Laplace operator in the space of functions of a fractional number of
variables. The integration with respect to the measure $x^\gamma\,dx$, $\gamma>-1$,
can also be interpreted as the integration over a domain of fractional dimension. The coefficient
$\gamma>-1$ of hidden spherical symmetry is introduced. A formula is obtained that
relates this coefficient to the Hausdorff dimension of a set in $\mathbb{R}_n$
and the Euclidean dimension $n$. The existence of hidden spherical symmetries is verified by calculating the
dimension of the $m$th generation of the Koch curve for arbitrary positive
integer $m$.
Keywords:
Laplace operator, Kipriyanov operator, Laplace–Bessel–Kipriyanov operator,
singular differential Bessel operator, fractional dimension, fractal,
self-similarity, integral measure, Hausdorff dimension, Hausdorff–Besikovich
dimension, fractal dimension, Koch curve, generations of the
Koch curve.
Received: 04.07.2022 Revised: 03.09.2022
Citation:
L. N. Lyakhov, E. Sanina, “Differential and Integral Operations in Hidden Spherical Symmetry and the Dimension of the Koch Curve”, Mat. Zametki, 113:4 (2023), 517–528; Math. Notes, 113:4 (2023), 502–511
Linking options:
https://www.mathnet.ru/eng/mzm13645https://doi.org/10.4213/mzm13645 https://www.mathnet.ru/eng/mzm/v113/i4/p517
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Abstract page: | 155 | Full-text PDF : | 6 | Russian version HTML: | 96 | References: | 29 | First page: | 11 |
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