On Statistical Properties of Local Minima of Integer Lattices |
A. A. Illarionov |
2012, issue 2, P. 201–230 |
Abstract |
The asymptotical formula for average number of local minima of integer multidimensional lattices is proved. |
Keywords: lattice, local minimum, many-dimensional continued fraction |
Download the article (PDF-file) |
References |
[1] P. M. Gruber, K. G. Lekkerkerker, Geometriia chisel, Nauka, M., 2008, 727 s. [2] G. F. Voronoi, Sobranie sochinenii v 3-kh tomakh, t. 1, Izd-vo AN USSR, Kiev, 1952. [3] H. Minkowski, “Generalisation de la theorie des fraction continues”, Ann. Sci. E?cole Norm. Sup., 13:2 (1896), 41–60. [4] A. J. Brentjes, Multidimensional continued fraction algorithms, Mathematical Centre Tracts, 145, Mathematisch Centrum, Amsterdam, 1981, 183 pp. [5] J. Buchmann, M. Pohst, J. v. Schmettow, “On the computation of unit group and class groups of totally real quartic fields”, Math. Comp., 53:187 (1989), 387–397. [6] V. A. Bykovskii, “O pogreshnosti teoretiko-chislovykh kvadraturnykh formul”, Chebyshevskii sb., 3:2 (2002), 27–33, Tula. [7] V. A. Bykovskii, “O pogreshnosti teoretiko-chislovykh kvadraturnykh formul”, DAN, 382:2 (2003), 154–155. [8] V. A. Bykovskii, “Otkloneniia setok Korobova”, Izv. RAN. Ser. matem., 76:3 (2012), 19–38. [9] A. A. Illarionov, “O tsilindricheskikh minimumakh trekhmernykh reshetok”, Dal'nevost. matem. zhurn., 11:1 (2011), 37–47. [10] A. A. Illarionov, “O tsilindricheskikh minimumakh tselochislennykh reshetok”, Algebra i analiz, 24:2 (2012), 154–170. [11] A. A. Illarionov, “Srednee kolichestvo otnositel'nykh minimumov trekhmernykh tselochislennykh reshetok”, Algebra i analiz, 23:3 (2011), 189–215. [12] A. A. Illarionov, “On the Asymptotic Distribution of Integer Matrices”, Moscow Journal of Combinatorics and Number Theory, 1:4 (2011), 301–345. [13] Dzh. V. Kassels, Vvedenie v geometriiu chisel, Mir, M., 1995. |