Geodesic
Distribution density of commutant of random rotations of three-dimensional Euclidean space
Teoriâ veroâtnostej i ee primeneniâ, Tome 61 (2016) no. 2, pp. 327-347 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $. 
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     author = {F. M. Malyshev},
     title = {Distribution density of commutant of random rotations of three-dimensional {Euclidean} space},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {327--347},
     year = {2016},
     volume = {61},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2016_61_2_a5/}
}
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F. M. Malyshev. Distribution density of commutant of random rotations of three-dimensional Euclidean space. Teoriâ veroâtnostej i ee primeneniâ, Tome 61 (2016) no. 2, pp. 327-347. http://geodesic.mathdoc.fr/item/TVP_2016_61_2_a5/

[1] Eaton M. L., Muirhed R. J., “The “north pole problem” and random orthogonal matrieces”, Statistics Probab. Lett., 79:17 (2009), 1878–1883 | DOI | MR | Zbl

[2] Marzetta T. L., Hassibi B., Hochwald B. M., “Structured unitary space-time autocoding constellations”, IEEE Trans. Inf. Theory, 48:4 (2002), 942–950 | DOI | MR | Zbl

[3] Prudnikov A. P., Brychkov Yu. A., Marichev O. I., Integraly i ryady, Izd. 2-e, Fizmatlit, M., 2003, 752 pp.

[4] Spravochnik po spetsialnym funktsiyam s formulami, grafikami matematicheskimi tablitsami, Nauka, M., 1979, 832 pp.

[5] Khelgason S., Differentsialnaya geometriya i simmetricheskie prostranstva, Mir, M., 1964, 608 pp.

[6] Korn G., Korn T., Spravochnik po matematike dlya nauchnykh rabotnikov i inzhenerov. Opredeleniya, teoremy, formuly, Izd. 4-e, Nauka, M., 1977, 832 pp.

[7] Rains E., “High powers of random elements of compact Lie groups”, Probab. Theory Related Fields, 107 (1997), 219–241 | DOI | MR | Zbl

[8] Postnikov M. M., Analiticheskaya geometriya, Nauka, M., 1973, 416 pp.

[9] Vilenkin N. Ya., Spetsialnye funktsii i teoriya predstavlenii grupp, Nauka, M., 1965, 576 pp.

[10] Stepanov N. N., Sfericheskaya trigonometriya, Izd-vo tekhniko-teoreticheskoi literatury, L.–M., 1948