| Title:
|
On the 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:
|
34 |
| Issue:
|
2 |
| Year:
|
1993 |
| Pages:
|
367-373 |
| . |
| Category:
|
math |
| . |
| Summary:
|
In the paper, some kind of independence between upper metric dimension and natural order of converging sequences is shown --- for any sequence converging to zero there is a greater sequence with an arbitrary ($\leqslant 1$) upper dimension. On the other hand there is a relationship to summability of series --- the set of elements of any positive summable series must have metric dimension less than or equal to $1/2$. (English) |
| Keyword:
|
metric dimension |
| Keyword:
|
converging sequences |
| Keyword:
|
summability of series |
| MSC:
|
40A05 |
| MSC:
|
40J05 |
| MSC:
|
54E35 |
| MSC:
|
54E45 |
| MSC:
|
54F45 |
| MSC:
|
54F50 |
| idZBL:
|
Zbl 0845.54026 |
| idMR:
|
MR1241746 |
| . |
| Date available:
|
2009-01-08T18:04:12Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/118590 |
| . |
| Reference:
|
[K-T] Kolmogorov A. N., Tihomirov V. M.: $\varepsilon$-entropy and $\varepsilon$-capacity of sets in functional spaces (in Russian).Usp. Mat. Nauk 14 (1959), 3-86 Am. Math. Soc. Transl. 17 (1961), 277-364. MR 0112032 |
| Reference:
|
[M-Z] Mišík L., Žáčik T.: On some properties of the metric dimension.Comment. Math. Univ. Carolinae 31 (1990), 781-791. MR 1091376 |
| Reference:
|
[P-S] Pontryagin L.S., Snirelman L.G.: Sur une propriété 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:
|
[V] Vosburg A.C.: On the relationship between Hausdorff dimension and metric dimension.Pacific J. Math. 23 (1967), 183-187. Zbl 0153.24701, MR 0217776 |
| Reference:
|
[Y] Yomdin Y.: The geometry of critical and near-critical values of differentiable mappings.Math. Ann. 264 (1983), 495-515. Zbl 0507.57019, MR 0716263 |
| Reference:
|
[Z] Žáčik T.: On some approximation properties of the metric dimension.Math. Slovaca 42 (1992), 331-338. MR 1182963 |
| . |
