Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A sequence well dispersed in the unit square


Author: J. P. Lambert
Journal: Proc. Amer. Math. Soc. 103 (1988), 383-388
MSC: Primary 11K38; Secondary 65C10
DOI: https://doi.org/10.1090/S0002-9939-1988-0943050-3
MathSciNet review: 943050
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Distributional properties of sequences of points in the unit square have been studied extensively and there is considerable interest in sequences which are well distributed according to various criteria. Dispersion is a measure of sequence density and an important concept connected with regularity of distribution. We introduce an infinite, dyadic, easily-generated sequence which is particularly well distributed, in the sense of having dispersion of lowest possible order of magnitude.


References [Enhancements On Off] (What's this?)

  • [1] T. J. Aird and J. R. Rice, Systematic search in high dimensional sets, SIAM J. Numer. Anal. 14 (1977), 296-312. MR 0483427 (58:3429)
  • [2] J. P. Lambert, Quasi-Monte Carlo, low discrepancy sequences, and ergodic transformations, J. Comput. Appl. Math. 12/13 (1985), 419-423. MR 793972
  • [3] -, On the development of infinite low-discrepancy sequences for quasi-Monte Carlo implementation, Technical Report, University of Alaska, Fairbanks, March 1986.
  • [4] H. Niederreiter, Quasi-Monte Carlo methods and pseudo-random numbers, Bull. Amer. Math. Soc. 84 (1978), 957-1041. MR 508447 (80d:65016)
  • [5] -, A quasi-Monte Carlo method for the approximate computation of the extreme values of a function, Studies in Pure Mathematics (To the Memory of Paul Turán), Birkhäuser, Basel, 1983, pp. 523-529. MR 820248 (86m:11055)
  • [6] -, On a measure of denseness for sequences, Topics in Classical Number Theory, Colloq. Math. Soc. János Bolyai, 34, (Budapest, 1981), North-Holland, Amsterdam, 1984, pp. 1163-1208. MR 781180 (86h:11058)
  • [7] -, Quasi-Monte Carlo methods for global optimization, Proc. 4th Pannonian Sympos. on Math. Stat., (Bad Tatzmannsdorf, Austria, 1983), Akad. Kiadó, Budapest, 1986, pp. 251-267. MR 851058 (87m:90092)
  • [8] H. Niederreiter and K. McCurley, Optimization of functions by quasi-random search methods, Computing 22 (1979), 119-123. MR 621190 (83d:65015)
  • [9] H. Niederreiter and P. Peart, A comparative study of quasi-Monte Carlo methods for optimization of functions of several variables, Caribbean J. Math. 1 (1982), 27-44. MR 666274 (83j:65071)
  • [10] -, Localization of search in quasi-Monte Carlo methods for global optimization, SIAM J. Sci. Stat. Comput. 7 (1986), 660-664. MR 833928 (87h:65017)
  • [11] P. Peart, The dispersion of the Hammersley sequence in the unit square, Monatsh. Math. 94 (1982), 249-261. MR 683058 (85a:65010)
  • [12] R. G. E. Pinch, A sequence well distributed in the square, Math. Proc. Cambridge Philos. Soc. 99 (1986), 19-22. MR 809493 (87c:11058)
  • [13] K. F. Roth, On irregularities of distribution. II, Comm. Pure Appl. Math. 29 (1976), 749-754. MR 0460266 (57:260)
  • [14] I. M. Sobol and R. B. Stanikov, LP-search and problems of optimal design, Problems of Random Search, vol. 1, Izdat. "Zinatne," Riga, 1972, pp. 117-135 (Russian).
  • [15] S. Srinivasan, On two-dimensional Hammersley's sequences, J. Number Theory 10 (1978), 421-429. MR 515053 (80f:10065)
  • [16] B. E. White, On optimal extreme-discrepancy point sets in the square, Numer. Math. 27 (1977), 157-164. MR 0468180 (57:8018)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 11K38, 65C10

Retrieve articles in all journals with MSC: 11K38, 65C10


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1988-0943050-3
Article copyright: © Copyright 1988 American Mathematical Society

American Mathematical Society