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.
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