Peano-type curves, Liouville numbers, and microscopic sets

Cover Page

Abstract


Peano-type curves in multidimensional Euclidean space are considered in terms of number theory. In contrast to curves constructed by D. Hilbert, H. Lebesgue, V. Sierpinski, and others, this paper presents results showing that each such curve is a continuous image of universal (shared by all curves) nowhere dense perfect subsets of the interval [0, 1] with a zero s-dimensional Hausdorff measure that consist of only Liouville numbers. An example of a problem in which a pair of continuous functions controlling the behavior of an oscillating system generates a Peano-type curve in the plane is given.


About the authors

А. N. Agadzhanov

Institute of Control Sciences of the Russian Academy of Sciences

Author for correspondence.
Email: ashot_ran@mail.ru

Russian Federation, 65, Profsoyuznaya Street, Moscow, 117997

References

  1. Хаусдорф Ф. Теория множеств. М., 2010.
  2. Sagan H. Space-Filling Curves. B.: Springer, 1994.
  3. Bader M. Space-Filling Curves. B.: Springer, 2010.
  4. Liu W. // Chinese J. Math. 1995. V. 23. P. 173–178.
  5. Ying-Fen Lin, Ngai-Ching Wong // Comput. & Math. with Appl. 2003. V. 45. P. 1871–1881. Appl. 2012. V. 159. P. 1894–1898.
  6. Marques D., Moreira C.G. // Bull. Austral. Math. Soc. 2015. V. 15. P. 29–33.
  7. Marques D., Ramirez J. // Proc. Jap. Acad. Ser. A Math. Sci. 2015. V. 91. P. 25–28.
  8. Фельдман Н. И., Шидловский А. Б. // УМН. 1967. Т. 22. В. 3 (135). С. 3–81.
  9. Garg K.M. // Rev. Roum. Math. Pureset Appl. 1969. V. 14. P. 1441–1452.
  10. Агаджанов А.Н. // ДАН. 2014. Т. 454. № 3. С. 503–506.
  11. Знаменская Л. Н. Управление упругими колебаниям. М.: Физматлит, 2004.
  12. Полянин А.Д. Линейные уравнения математической физики. М.: Физматлит, 2001.
  13. Salem R., Zygmund A. // Duke Math. J. 1945. V. 12. P. 569–578.

Statistics

Views

Abstract - 191

PDF (Russian) - 150

PlumX


Copyright (c) 2019 Russian academy of sciences

This website uses cookies

You consent to our cookies if you continue to use our website.

About Cookies