A formula for calculation of metric dimension of converging sequences

Ladislav, Jr. Mišík; Tibor Žáčik

Commentationes Mathematicae Universitatis Carolinae (1999)

  • Volume: 40, Issue: 2, page 393-401
  • ISSN: 0010-2628

Abstract

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Converging sequences in metric space have Hausdorff dimension zero, but their metric dimension (limit capacity, entropy dimension, box-counting dimension, Hausdorff dimension, Kolmogorov dimension, Minkowski dimension, Bouligand dimension, respectively) can be positive. Dimensions of such sequences are calculated using a different approach for each type. In this paper, a rather simple formula for (lower, upper) metric dimension of any sequence given by a differentiable convex function, is derived.

How to cite

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Mišík, Ladislav, Jr., and Žáčik, Tibor. "A formula for calculation of metric dimension of converging sequences." Commentationes Mathematicae Universitatis Carolinae 40.2 (1999): 393-401. <http://eudml.org/doc/248405>.

@article{Mišík1999,
abstract = {Converging sequences in metric space have Hausdorff dimension zero, but their metric dimension (limit capacity, entropy dimension, box-counting dimension, Hausdorff dimension, Kolmogorov dimension, Minkowski dimension, Bouligand dimension, respectively) can be positive. Dimensions of such sequences are calculated using a different approach for each type. In this paper, a rather simple formula for (lower, upper) metric dimension of any sequence given by a differentiable convex function, is derived.},
author = {Mišík, Ladislav, Jr., Žáčik, Tibor},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {metric dimension; limit capacity; entropy dimension; box-counting dimension; Hausdorff dimension; Kolmogorov dimension; Minkowski dimension; Bouligand dimension; converging sequences; convex sequences; differentiable function; metric dimension; limit capacity; entropy dimension; Hausdorff dimension; Kolmogorov dimension},
language = {eng},
number = {2},
pages = {393-401},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {A formula for calculation of metric dimension of converging sequences},
url = {http://eudml.org/doc/248405},
volume = {40},
year = {1999},
}

TY - JOUR
AU - Mišík, Ladislav, Jr.
AU - Žáčik, Tibor
TI - A formula for calculation of metric dimension of converging sequences
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1999
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 40
IS - 2
SP - 393
EP - 401
AB - Converging sequences in metric space have Hausdorff dimension zero, but their metric dimension (limit capacity, entropy dimension, box-counting dimension, Hausdorff dimension, Kolmogorov dimension, Minkowski dimension, Bouligand dimension, respectively) can be positive. Dimensions of such sequences are calculated using a different approach for each type. In this paper, a rather simple formula for (lower, upper) metric dimension of any sequence given by a differentiable convex function, is derived.
LA - eng
KW - metric dimension; limit capacity; entropy dimension; box-counting dimension; Hausdorff dimension; Kolmogorov dimension; Minkowski dimension; Bouligand dimension; converging sequences; convex sequences; differentiable function; metric dimension; limit capacity; entropy dimension; Hausdorff dimension; Kolmogorov dimension
UR - http://eudml.org/doc/248405
ER -

References

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  1. Mišík L., Žáčik T., On some properties of the metric dimension, Comment. Math. Univ. Carolinae 31.4 (1990), 781-791. (1990) MR1091376
  2. Pontryagin L.S., Snirelman L.G., Sur une propriete metrique de la dimension, Annals of Math. 33 (1932), 156-162 Appendix to the Russian translation of ``Dimension Theory'' by W. Hurewitcz and H. Wallman, Izdat. Inostr. Lit. Moscow, 1948. (1932) MR1503042
  3. Hawkes J., Hausdorff measure, entropy and the independents of small sets, Proc. London Math. Soc. (3) 28 (1974), 700-724. (1974) MR0352412
  4. Besicovitch A.S., Taylor S.J., On the complementary intervals of a linear closed sets of zero Lebesgue measure, J. London Math. Soc. 29 (1954), 449-459. (1954) MR0064849
  5. Koçak Ş., Azcan H., Fractal dimensions of some sequences of real numbers, Do{ğ}a - Tr. J. of Mathematics 17 (1993), 298-304. (1993) MR1255026

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