On 2D packings of cubes in the torus
HTML articles powered by AMS MathViewer
- by Andrew V. Reztsov and Ian H. Sloan PDF
- Proc. Amer. Math. Soc. 125 (1997), 17-26 Request permission
Abstract:
The 2D packings of cubes (i.e. squares) in the torus ${\mathcal {T}}^{2}={[0,1)}^{2}$ are considered. We obtain the exact expression $N_{2} (\lambda ) = \left \lfloor \lambda \lfloor \lambda \rfloor \right \rfloor$ for the quantity $N_{2} (\lambda )$, the maximal number of 2D cubes in a packing. (Here $1/\lambda$ is the length of sides of cubes, $\lambda \in \mathbf {R}, \lambda >2$.) Corresponding best packings are constructed. Both rank 1 best lattice packings and rank 2 best lattice packings are given.References
- Ian H. Sloan and James N. Lyness, The representation of lattice quadrature rules as multiple sums, Math. Comp. 52 (1989), no. 185, 81–94. MR 947468, DOI 10.1090/S0025-5718-1989-0947468-3
- I.H.Sloan and S.Joe, Lattice methods for multiple integration, Oxford University Press, 1994.
- S.B.Stechkin, Some extremal properties of the trigonometric sums, in Modern Problems in Number Theory. Summaries of International Conference, Tula, 1993, p. 153 (Russian).
- S. B. Stechkin, Some extremal properties of trigonometric sums, Mat. Zametki 55 (1994), no. 2, 130–143, 191 (Russian, with Russian summary); English transl., Math. Notes 55 (1994), no. 1-2, 195–203. MR 1275328, DOI 10.1007/BF02113302
- V. A. Yudin, Packings of balls in Euclidean space, and extremal problems for trigonometric polynomials, Diskret. Mat. 1 (1989), no. 2, 155–158 (Russian); English transl., Discrete Math. Appl. 1 (1991), no. 1, 69–72. MR 1035103, DOI 10.1515/dma.1991.1.1.69
- Harald Niederreiter, Quasi-Monte Carlo methods and pseudo-random numbers, Bull. Amer. Math. Soc. 84 (1978), no. 6, 957–1041. MR 508447, DOI 10.1090/S0002-9904-1978-14532-7
- S. C. Zaremba, Good lattice points, discrepancy, and numerical integration, Ann. Mat. Pura Appl. (4) 73 (1966), 293–317. MR 218018, DOI 10.1007/BF02415091
- Harald Niederreiter and Ian H. Sloan, Integration of nonperiodic functions of two variables by Fibonacci lattice rules, J. Comput. Appl. Math. 51 (1994), no. 1, 57–70. MR 1286416, DOI 10.1016/0377-0427(92)00004-S
- N. S. Bahvalov, Approximate computation of multiple integrals, Vestnik Moskov. Univ. Ser. Mat. Meh. Astr. Fiz. Him. 1959 (1959), no. 4, 3–18 (Russian). MR 0115275
- N. M. Korobov, Teoretiko-chislovye metody v priblizhennom analize, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1963 (Russian). MR 0157483
Additional Information
- Andrew V. Reztsov
- Affiliation: Research Fellow, Division of Science and Technology, Tamaki Campus, The University of Auckland, Private Bag 92019, Auckland, New Zealand
- Email: a.reztsov@auckland.ac.nz
- Ian H. Sloan
- Affiliation: School of Mathematics, University of New South Wales, Sydney 2052, New South Wales, Australia
- MR Author ID: 163675
- ORCID: 0000-0003-3769-0538
- Email: I.Sloan@unsw.edu.au
- Received by editor(s): June 16, 1995
- Communicated by: William W. Adams
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 125 (1997), 17-26
- MSC (1991): Primary 05B40; Secondary 11H31, 52C15, 65D32
- DOI: https://doi.org/10.1090/S0002-9939-97-03930-0
- MathSciNet review: 1401751