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JOURNALS // Algebra i Analiz

Algebra i Analiz, 2006, Volume 18, Issue 1, Pages 162–186 (Mi aa64)

Construction of spherical cubature formulas using lattices
P. de la Harpe, C. Pache, B. Venkov

References

1. Andreev N. N., “Minimalnyi dizain 11-go poryadka na trekhmernoi sfere”, Mat. zametki, 67:4 (2000), 489–497  mathnet  mathscinet  zmath
2. Bachoc Ch., Venkov B. B., “Modular forms, lattices and spherical designs”, Réseaux euclidiens, designs sphériques et formes modulaires, Monogr. Enseign. Math., 37, Enseign. Math., Geneva, 2001, 87–111  mathscinet  zmath
3. Bannai E., Damerell R., “Tight spherical designs. I”, J. Math. Soc. Japan, 31 (1979), 199–207  crossref  mathscinet  zmath
4. Bannai E., Damerell R., “Tight spherical designs. II”, J. London Math. Soc. (2), 21 (1980), 13–30  crossref  mathscinet  zmath
5. Bannai E., Munemasa A., Venkov B., “The nonexistence of certain tight spherical design”, Algebra i analiz, 16:4 (2004), 1–23  mathnet  mathscinet  zmath
6. Bajnok B., “Chebyshev-type quadrature formulas on the sphere”, Proceedings of the Twenty-Second Southeastern Conference on Combinatorics, Graph Theory, and Computing (Baton Rouge, LA, 1991), Congr. Numer., 85, 1991, 214–218  mathscinet  zmath
7. Conway J. H., Sloane N. J. A., Sphere packings, lattices and groups, 3rd ed., Grundlehren Math. Wiss., 290, Springer-Verlag, New York, 1999  mathscinet  zmath
8. Delsarte P., Goethals J.-M., Seidel J. J., “Spherical codes and designs”, Geom. Dedicata, 6 (1977), 363–388  mathscinet  zmath
9. Dickson L. E., History of the theory of numbers. Vol. II: Diophantine analysis, Carnegie Inst., Washington, 1919  zmath; reprinted by Chelsea Publ. Co., New York, 1966  zmath
10. Ebeling W., Lattices and codes, a course partially based on lectures by F. Hirzenbruch, Friedr. Vieweg and Sohn, Braunschweig, 1994  mathscinet  zmath; 2nd revised ed., 2002
11. Goethals J.-M., Seidel J. J., “Spherical designs”, Relations Between Combinatorics and Other Parts of Mathematics (Proc. Sympos. Pure Math., Ohio State Univ., Columbus, Ohio, 1978), Proc. Sympos. Pure Math., 34, Amer. Math. Soc., Providence RI, 1979, 255–272  mathscinet
12. Goethals J.-M., Seidel J. J., “Cubature formulas, polytopes, and spherical designs”, The Geometric Vein, the Coxeter Festschrift, Springer, New York–Berlin, 1981, 203–218  mathscinet
13. Hardin R. H., Sloane N. J. A., “Expressing $(a^2+b^2+c^2+d^2)^3$ as a sum of 23 sixth powers”, J. Combin. Theory Ser. A, 68 (1994), 481–485  crossref  mathscinet  zmath
14. Hardin R. H., Sloane N. J. A., “McLaren's improved snub cube and other new spherical designs in three dimensions”, Discrete Comput. Geom., 15 (1996), 429–441  crossref  mathscinet  zmath
15. de la Harpe P., Pache C., “Spherical designs and finite group representations (some results of E. Bannai)”, European J. Combin., 25 (2004), 213–227  crossref  mathscinet  zmath
16. de la Harpe P., Pache C., “Cubature formulas, geometrical designs, reproducing kernels, and Markov operators”, Infinite Groups: Geometric, Combinatorial, and Dynamical Aspects, Progr. Math., 248, Birkhäuser, 2005, 219–268  mathscinet  zmath
17. Kuperberg G., Numerical cubature using error-correcting codes, arXiv:math.NA/0402047  mathscinet
18. Kuperberg G., Numerical cubature from Archimedes' hat-box theorem, arXiv:math.NA/0405366  mathscinet
19. Lehmer D. H., “The vanishing of Ramanujan's function $\tau(n)$”, Duke Math. J., 14 (1947), 429–433  crossref  mathscinet
20. Martinet J. (ed.), Réseaux euclidiens, designs sphériques et formes modulaires. Autour des travaux de B. Venkov, Monogr. Enseign. Math., 37, Enseign. Math., Geneva, 2001  mathscinet
21. Nikova S., Nikov V., “Improvement of the Delsarte bound for $\tau$-designs when it is not the best bound possible”, Des. Codes Cryptogr., 28:2 (2003), 201–222  crossref  mathscinet  zmath
22. Pache C., “Selfdual lattices viewed as spherical designs”, Internat. J. Algebra Comput., 15:5-6 (2005), 1085–1127  crossref  mathscinet  zmath
23. Quebbemann H.-G., “Modular lattices in Euclidean spaces”, J. Number Theory, 54:2 (1995), 190–202  crossref  mathscinet  zmath
24. Rankin R. A., Modular forms and functions, Cambridge Univ. Press, Cambridge etc., 1977  mathscinet  zmath
25. Salikhov G. N., “Kubaturnye formuly dlya gipersfery, invariantnye otnositelno gruppy pravilnogo 600-grannika”, Dokl. AN SSSR, 223:5 (1975), 1075–1078  mathnet  mathscinet  zmath
26. J.-P. Serre, “Sur la lacunarité des puissances de $\eta$”, Glasgow Math. J., 27 (1985), 203–221  crossref  mathscinet  zmath; = ØE uvres, Collected Papers IV, 1985-1998, Springer-Verlag, Berlin, 2000, 66–84, 640  mathscinet
27. Seymour P. D., Zaslavsky T., “Averaging sets: a generalization of mean values and spherical designs”, Adv. in Math., 52 (1984), 213–240  crossref  mathscinet  zmath
28. Smith L., Polynomial invariants of finite groups, Res. Notes Math., 6, A. K. Peters, Wellesley, MA, 1995  mathscinet  zmath
29. Sobolev S. L., “O kubaturnykh formulakh na sfere, invariantnykh pri preobrazovaniyakh konechnykh grupp vraschenii”, Dokl. AN SSSR, 146:2 (1962), 310–313  mathnet  mathscinet  zmath
30. Sobolev S. L., Vaskevich V. L., Kubaturnye formuly, Nauka, SO, Novosibirsk, 1996
31. Venkov B. B., “Chetnye unimodulyarnye ekstremalnye reshetki”, Tr. Mat. in-ta AN SSSR, 165, 1984, 43–48  mathnet  mathscinet  zmath
32. Venkov B. B. (notes by J. Martinet), “Réseaux et designs sphériques”, Réseaux euclidiens, designs sphériques et formes modulaires, Monogr. Enseign. Math., 37, Enseign. Math., Geneva, 2001, 10–86  mathscinet  zmath
33. Vilenkin N. Ya., Spetsialnye funktsii i teoriya predstavlenii grupp, Nauka, M., 1965  mathscinet  zmath
34. Yudin V. A., “Nizhnie otsenki dlya sfericheskikh dizainov”, Izv. RAN. Ser. mat., 61:3 (1997), 213–223  mathnet  mathscinet  zmath


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