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Teor. Veroyatnost. i Primenen., 2016, Volume 61, Issue 2, Pages 327–347 (Mi tvp5058)  

Distribution density of commutant of random rotations of three-dimensional Euclidean space

F. M. Malyshev

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract: The basic measure $\mu $ is defined on the group $ SO (3) $ of rotations of three-dimensional Euclidean space. It responds to the product of uniform distributions on the sets of axes of rotations and angles of rotations. We consider three distribution densities with respect to $\mu$: $\rho_0 $ is a density of left- and right-invariant measure (Haar measure); $ \rho_1 $ is a density of distribution of rotations $ \Lambda^k$, $k \ge 2 $, where $ \Lambda $ is a random rotation with density $ \rho_0 $; and $ \rho_2 $ is a distribution density of the $ \Lambda_1^{- 1} \Lambda_2^{- 1} \Lambda_1 \Lambda_2 $ commutant, where $ \Lambda_1 $, $ \Lambda_2 $ are random independent rotations with the distribution density $ \rho_0 $. It is shown that $ \rho_2 \equiv \sqrt{\rho_0 \rho_1} \frac{\pi \sqrt {2}} {4} $ and the measure $ \mu_1 $ with density $ \rho_1 $ is proportional to the basic measure $ \mu $.

DOI: https://doi.org/10.4213/tvp5058

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English version:
Theory of Probability and its Applications, 2017, 61:2, 277–294

Bibliographic databases:

Document Type: Article
Received: 02.02.2015

Citation: F. M. Malyshev, “Distribution density of commutant of random rotations of three-dimensional Euclidean space”, Teor. Veroyatnost. i Primenen., 61:2 (2016), 327–347; Theory Probab. Appl., 61:2 (2017), 277–294

Citation in format AMSBIB
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