Literature Cited
S. L. Sobolev, “Convergence of formulas for approximation integration on functions of L m2 ,” Dokl. Akad. Nauk SSSR, 162, No. 6, 1259–1261 (1965).
S. L. Sobolev, Lectures on the Theory of Cubature Formulas, Short Course Given at the Novosibirsk State University in the Academic Year 1963–1964 [in Russian], Novosibirsk State University Press, Novosibirsk (1964), Vol. 1.
S. L. Sobolev, Ibid.,, Vol. 2.
R. A. Rankin, “A minimum problem for the Epstein zeta-function.” Proc. Glasgow Math. Assoc.,1, No. 3, 149–158 (1953).
I. W. S. Cassels, “On a problem of Rankin about the Epstein zeta-function.” Proc. Glasgow Math. Assoc., 4, No. 2, 73–81 (1959).
S. S. Ryshkov, “On two-dimensional ξ-functions with real parameters,” Dokl. Akad. Nauk SSSR,184 No. 2, 288–291 (1969).
L. V. Voitishek, “Calculation of the Epstein ξ-function,” in: Voprosy Vych. i Prikl. Mat., No. 38, FAN, Tashkent (1970).
N. N. Sandakova, “On the theory of ξ-functions of three variables,” Dokl. Akad. Nauk, SSSR175, No. 3, 535–537 (1967).
B. N. Delone, N. N. Sandakova, and S. S. Ryshkov, “On optimal cubature lattices for all-sided smooth functions of two variables,” Dokl. Akad. Nauk SSSR,162, No. 5, 1230–1234 (1965).
B. N. Delone and S. S. Ryshkov, “Extremal problems in the theory of positive quadratic forms,” in: Trudy MIAN,112, 203–223 (1971), Vol. 1.
S. S. Ryshkov, “Polyhedra μ(m) and certain extremal problems of the geometry of numbers,” Dokl. Akad. Nauk SSSR,194, No. 3, 514–517 (1970).
B. N. Delone and S. S. Ryshkov, “On the theory of extrema of multidimensional ξ-functions,” Dokl. Akad. Nauk,173, No. 5, 991–994 (1967).
V. Ennola, “On a problem about the Epstein Zeta-Function,” Proc. Cambr. Phil. Soc.,60, 855–875 (1964).
H. Minkowski, “Diskontinuitetsbereich fur Arithmetische Aquivalenz,” J. Reine und Angew. Math.,129, 220–284 (1905). See also, Gest. Abh., Leipzig-Berlin (1911), Vol. 2, pp. 53–103.
B. A. Venkov, “On reduced positive quadratic forms,” Izv. Akad. Nauk SSSR, Seriya. Matem.,4, No. 1, 37–52 (1940).
S. S. Ryshkov, “On maximal finite groups of integer-valued nxn matrices,” Dokl. Akad. Nauk SSSR,204, No. 3, 561–564 (1972).
S. S. Ryshkov, “Maximal finite groups of nxn matrices and the full group of integer-valued automorphisms of positive quadratic forms. (The Brouwer type),” Trudy MIAN,124, 183–211 (1972). Vol. II.
A. Korkine and G. Zolotareff, “Sur les formes quadratiques positives,” Math. Ann.,2, 242–292 (1877). See also, E. I. Zolotarev and A. N. Korkin, Joint Papers [in Russian], Izdatel'stvo Akad. Nauk SSSR, Leningrad (1931), Vol. 1.
G. F. Voronoi, “On certain properties of positive complete quadratic forms,” in: Complete Works in Three Volumes [in Russian], Izdatel'stvo Akad. USSR, Kiev (1952), Vol. 2, pp. 171–238.
E. S. Barnes, “The complete enumeration of extreme senary forms” Phil. Trans. Roy. Soc. London,A-249, 461–506 (1957).
G. L. Watson, “On the minimum of a positive quadratic form in n (≤8) variables (verification of Blichfeldt's calculations)” Proc. Cambridge Phil. Soc.,62, 719 (1966).
H. F. Blichfeldt, “The minimum values of positive quadratic forms in six, seven, and eight variables,” Math. Z.,39, 1–15 (1934).
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Translated from Sibirskii Matematicheskii Zhurnal, Vol. 14, No. 5, pp. 1065–1075, September–October, 1973.
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Ryshkov, S.S. On the question of final ξ-optimality of lattices providing the closest lattice packing of n-dimensional spheres. Sib Math J 14, 743–750 (1973). https://doi.org/10.1007/BF00969911
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DOI: https://doi.org/10.1007/BF00969911