On the metric dimension of converging sequences

Ladislav, Jr. Mišík; Tibor Žáčik

Commentationes Mathematicae Universitatis Carolinae (1993)

  • Volume: 34, Issue: 2, page 367-373
  • ISSN: 0010-2628

Abstract

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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 ( 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 .

How to cite

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Mišík, Ladislav, Jr., and Žáčik, Tibor. "On the metric dimension of converging sequences." Commentationes Mathematicae Universitatis Carolinae 34.2 (1993): 367-373. <http://eudml.org/doc/247460>.

@article{Mišík1993,
abstract = {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 $1/2$.},
author = {Mišík, Ladislav, Jr., Žáčik, Tibor},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {metric dimension; converging sequences; summability of series; metric dimension; converging sequences; summability of series},
language = {eng},
number = {2},
pages = {367-373},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On the metric dimension of converging sequences},
url = {http://eudml.org/doc/247460},
volume = {34},
year = {1993},
}

TY - JOUR
AU - Mišík, Ladislav, Jr.
AU - Žáčik, Tibor
TI - On the metric dimension of converging sequences
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1993
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 34
IS - 2
SP - 367
EP - 373
AB - 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 $1/2$.
LA - eng
KW - metric dimension; converging sequences; summability of series; metric dimension; converging sequences; summability of series
UR - http://eudml.org/doc/247460
ER -

References

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  1. Kolmogorov A. N., Tihomirov V. M., ε -entropy and ε -capacity of sets in functional spaces (in Russian), Usp. Mat. Nauk 14 (1959), 3-86 Am. Math. Soc. Transl. 17 (1961), 277-364. (1961) MR0112032
  2. Mišík L., Žáčik T., On some properties of the metric dimension, Comment. Math. Univ. Carolinae 31 (1990), 781-791. (1990) MR1091376
  3. 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. (1932) MR1503042
  4. Vosburg A.C., On the relationship between Hausdorff dimension and metric dimension, Pacific J. Math. 23 (1967), 183-187. (1967) Zbl0153.24701MR0217776
  5. Yomdin Y., The geometry of critical and near-critical values of differentiable mappings, Math. Ann. 264 (1983), 495-515. (1983) Zbl0507.57019MR0716263
  6. Žáčik T., On some approximation properties of the metric dimension, Math. Slovaca 42 (1992), 331-338. (1992) MR1182963

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