We study cubature formulas invariant under the dihedral group of order 16p.
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S. L. Sobolev, “On formulas for mechanical cubatures over the surface of a sphere,” Sib. Mat. Zh., 3, No. 5, 769–791 (1962).
G. N. Salikhov, “On the theory of groups of regular polyhedra,” Dokl. Akad. Nauk SSSR, 2005, No. 1, 33–35 (1972).
G. N. Salikhov, “Cubature formulas for a hypersphere invariant under the group of a regular polyhedron,” Dokl. Akad. Nauk SSSR, 2005, No. 5, 1075–1078 (1972).
V. I. Lebedev, “Values of nodes and weights of Gauss-Markov-type quadrature formulas for a sphere of accuracy degree 9 to 17 invariant under the octahedral group with inversion,” Zh. Vychisl. Mat. Mat. Fiz., 15, No. 1, 48–54 (1975).
V. I. Lebedev, “On one type of quadrature formulas of increased algebraic accuracy for a sphere,” Dokl. Akad. Nauk SSSR, 231, No. 1, 32–34 (1976).
I. P. Mysovskikh, “On the determination of integrals over the surface of a sphere,” Dokl. Akad. Nauk SSSR, 235, No. 2, 269–272 (1977).
I. P. Mysovskikh, “On cubature formulas invariant under groups of transformations,” Met. Vychisl., Issue 11, 3–21 (1978).
I. P. Mysovskikh, “On invariant cubature formulas,” in: Proceedings of the S. L. Sobolev Seminar “Theory of Cubature Formulas and Applications of Functional Analysis to Problems of Mathematical Physics” [in Russian], No. 1, (1978), pp. 69–76.
I. P. Mysovskikh, Interpolation Cubature Formulas [in Russian], Nauka, Moscow (1981).
Sh. I. Tadzhiev and É. A. Shamsiev, “On the construction of cubature formulas for a sphere,” in: Problems of Computational and Applied Mathematics. Methods and Algorithms for the Solution of Problems of Mathematical Physics and Mathematical Cybernetics [in Russian], Issue 86 (1989), pp. 88–95.
É. A. Shamsiev and Kh. M. Sagdiev, “Cubature formulas of accuracy degree 4p + 4 s + 7 for a sphere,” in: Problems of Computational and Applied Mathematics. Methods and Algorithms for the Solution of Problems of Mathematical Physics and Discrete Mathematics [in Russian], Issue 99 (1995), pp. 114–117.
É. A. Shamsiev, “Cubature formulas for a sphere invariant under a dihedral group of order 8 (2p + 1),” Uzb. Mat. Zh., No. 4, 91–99 (2004).
H.-S. M. Coxeter, “The product of the generators of a finite group generated by reflections,” Duke Mat. J., 18, 765–782 (1951).
F. Klein, Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom Fúneten Grade [Russian translation], Nauka, Moscow (1989).
V. F. Ignatenko, “On plane algebraic curves with axes of symmetry,” Ukr. Geom. Sb., Issue 21, 31–35 (1978).
A. N. Kazakov and V. I. Lebedev, “Quadrature formulas of the Gauss type for a sphere invariant under a dihedral group,” Tr. Mat. Inst. Ros. Akad. Nauk, 203, 100–112 (1994).
M. V. Noskov and I. M. Fedotova, “On invariant cubature formulas for a torus in R 3,” Zh. Vychisl. Mat. Mat. Fiz., 43, No. 9, 1323–1329 (2003).
É. A. Shamsiev, “Cubature formulas for a disk invariant under a group of transformations of regular polyhedra into themselves,” Zh. Vychisl. Mat. Mat. Fiz., 46, No. 7, 1211–1218 (2006).
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Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 7, pp. 983–991, July, 2008.
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Shamsiev, É.A. On the construction of cubature formulas invariant under dihedral groups. Ukr Math J 60, 1153–1163 (2008). https://doi.org/10.1007/s11253-008-0116-6
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DOI: https://doi.org/10.1007/s11253-008-0116-6