| Title:
|
A formula for calculation of metric dimension of converging sequences (English) |
| Author:
|
Mišík, Ladislav, Jr. |
| Author:
|
Žáčik, Tibor |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
40 |
| Issue:
|
2 |
| Year:
|
1999 |
| Pages:
|
393-401 |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
metric dimension |
| Keyword:
|
limit capacity |
| Keyword:
|
entropy dimension |
| Keyword:
|
box-counting dimension |
| Keyword:
|
Hausdorff dimension |
| Keyword:
|
Kolmogorov dimension |
| Keyword:
|
Minkowski dimension |
| Keyword:
|
Bouligand dimension |
| Keyword:
|
converging sequences |
| Keyword:
|
convex sequences |
| Keyword:
|
differentiable function |
| MSC:
|
26A51 |
| MSC:
|
40A05 |
| MSC:
|
54F50 |
| idZBL:
|
Zbl 0976.54035 |
| idMR:
|
MR1732660 |
| . |
| Date available:
|
2009-01-08T18:53:26Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119095 |
| . |
| Reference:
|
[M-Z] Mišík L., Žáčik T.: On some properties of the metric dimension.Comment. Math. Univ. Carolinae 31.4 (1990), 781-791. MR 1091376 |
| Reference:
|
[P-S] 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. MR 1503042 |
| Reference:
|
[H] Hawkes J.: Hausdorff measure, entropy and the independents of small sets.Proc. London Math. Soc. (3) 28 (1974), 700-724. MR 0352412 |
| Reference:
|
[B-T] 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. MR 0064849 |
| Reference:
|
[K-A] Koçak Ş., Azcan H.: Fractal dimensions of some sequences of real numbers.Do{ğ}a - Tr. J. of Mathematics 17 (1993), 298-304. MR 1255026 |
| . |
