# 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

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topMišík, 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

top- Mišík L., Žáčik T., On some properties of the metric dimension, Comment. Math. Univ. Carolinae 31.4 (1990), 781-791. (1990) MR1091376
- 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
- Hawkes J., Hausdorff measure, entropy and the independents of small sets, Proc. London Math. Soc. (3) 28 (1974), 700-724. (1974) MR0352412
- 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
- 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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