Geodesic
On Yudin's scheme of finding a lower estimate for the potential energy of a system of charges on a sphere
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 15 (2009) no. 1, pp. 166-183
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

We consider Yudin's problem of finding a lower estimate for the minimum of the potential energy of a system of equal charges on the unit sphere $\mathbb S^2$ of Euclidean space $\mathbb R^3$. For this problem, we write a dual problem in the case of an arbitrary number of charges. A solution of the dual problem is given for the cases of 5, 6, 7, 8, and 12 charges. In addition, the primal problem is solved and hypothetically optimal distributions are specified for 7 and 8 charges. It is established that Yudin’s method does not allow one to prove the optimality of these distributions.
Keywords: potential energy of a system of charges, Yudin's problem, primal and dual problems, hypothetically. 
@article{TIMM_2009_15_1_a13,
     author = {A. A. Montile},
     title = {On {Yudin's} scheme of finding a~lower estimate for the potential energy of a~system of charges on a~sphere},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {166--183},
     year = {2009},
     volume = {15},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2009_15_1_a13/}
}
TY  - JOUR
AU  - A. A. Montile
TI  - On Yudin's scheme of finding a lower estimate for the potential energy of a system of charges on a sphere
JO  - Trudy Instituta matematiki i mehaniki
PY  - 2009
SP  - 166
EP  - 183
VL  - 15
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TIMM_2009_15_1_a13/
LA  - ru
ID  - TIMM_2009_15_1_a13
ER  - 
%0 Journal Article
%A A. A. Montile
%T On Yudin's scheme of finding a lower estimate for the potential energy of a system of charges on a sphere
%J Trudy Instituta matematiki i mehaniki
%D 2009
%P 166-183
%V 15
%N 1
%U http://geodesic.mathdoc.fr/item/TIMM_2009_15_1_a13/
%G ru
%F TIMM_2009_15_1_a13
A. A. Montile. On Yudin's scheme of finding a lower estimate for the potential energy of a system of charges on a sphere. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 15 (2009) no. 1, pp. 166-183. http://geodesic.mathdoc.fr/item/TIMM_2009_15_1_a13/

[1] Andreev N. N., Priblizhenie s ogranicheniyami individualnykh funktsii i ekstremalnye zadachi raspolozheniya tochek na sfere, dis. $\dots$ kand. fiz.-mat. nauk, MGU, M., 2000, 41 pp.

[2] Andreev N. N., Yudin V. A., “Ekstremalnye raspolozheniya tochek na sfere”, Mat. prosveschenie. Ser. 3, 1 (1997), 115–125

[3] Arestov V. V., Babenko A. G., “O skheme Delsarta otsenki kontaktnykh chisel”, Tr. MIAN, 219, 1997, 44–73 | MR | Zbl

[4] Bernshtein S. N., Sobranie sochinenii, T. 1, Izd-vo AN SSSR, M., 1952, 582 pp.

[5] Golshtein E. G., Teoriya dvoistvennosti v matematicheskom programmirovanii i ee prilozheniya, Nauka, M., 1971, 352 pp. | MR

[6] Gorbachev D. V., Filippov D. V., “Otsenka energii zaryadov na sfere pri pomoschi SDP”, Tr. Mezhdunar. let. mat. shkoly S. B. Stechkina po teorii funktsii, Izd-vo TulGU, Tula, 2007, 70–78

[7] Sege G., Ortogonalnye mnogochleny, Fizmatlit., M., 1962, 500 pp.

[8] Yudin V. A., “Minimum potentsialnoi energii tochechnoi sistemy zaryadov”, Diskret. matematika, 4:2 (1992), 115–121 | MR

[9] Yudin V. A., “O matritsakh Grama”, Mat. prosveschenie. Ser. 3, 9 (2005), 183–188

[10] AIM math: American Institute of Mathematic [website], URL: http://aimath.org/data/paper/BBCGKS2006/