Differential and Integral Operations in Hidden Spherical Symmetry and the Dimension of the Koch Curve
L. N. Lyakhov, Е. Л. Санина
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 $$m$$ .