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| Description
| - Given a function $f:\N\to(\omega+1)\setminus\{0\}$, we say that a faithfully indexed sequence $\{a_n:n\in\N\}$ of elements of a topological group $G$ is: (i)~{\em $f$-Cauchy productive ($f$-productive)\/} provided that the sequence $\{\prod_{n=0}^m a_n^{z(n)}:m\in\N\}$ is left Cauchy (converges to some element of $G$, respectively) for each function $z:\N\to\Z$ such that $|z(n)|\le f(n)$ for every $n\in\N$; (ii)~{\em unconditionally $f$-Cauchy productive (unconditionally $f$-productive)\/} provided that the sequence $\{a_{\varphi(n)}:n\in\N\}$ is $(f\circ\varphi)$-Cauchy productive (respectively, $(f\circ\varphi)$-productive) for every bijection $\varphi:\N\to\N$. (Bijections can be replaced by injections here.) We consider the question of existence of (unconditionally) $f$-productive sequences for a given ``weight function'' $f$. We prove that: (1) a Hausdorff group having an $f$-productive sequence for some $f$ contains a homeomorphic copy of the Cantor set; (2) if a non-discrete group is either locally compact Hausdorff or Weil complete metric, then it contains an unconditionally $f$-productive sequence for every function $f:\N\to\N$; (3)~a metric group is NSS if and only if it does not contain an $f_\omega$-Cauchy productive sequence, where $f_\omega$ is the function taking the constant value $\omega$. We give an example of an $f_\omega$-productive sequence $\{a_n:n\in\N\}$ in a (necessarily non-abelian) separable metric group $H$ with a linear topology and a bijection $\varphi:\N\to\N$ such that the sequence $\{\prod_{n=0}^m a_{\varphi(n)}:m\in\N\}$ diverges, thereby answering a question of Dominguez and Tarieladze. Furthermore, we show that $H$ has no unconditionally $f_\omega$-productive sequences. As an application of our results, we resolve negatively a question from $C_p(-,G)$-theory.
- Given a function $f:\N\to(\omega+1)\setminus\{0\}$, we say that a faithfully indexed sequence $\{a_n:n\in\N\}$ of elements of a topological group $G$ is: (i)~{\em $f$-Cauchy productive ($f$-productive)\/} provided that the sequence $\{\prod_{n=0}^m a_n^{z(n)}:m\in\N\}$ is left Cauchy (converges to some element of $G$, respectively) for each function $z:\N\to\Z$ such that $|z(n)|\le f(n)$ for every $n\in\N$; (ii)~{\em unconditionally $f$-Cauchy productive (unconditionally $f$-productive)\/} provided that the sequence $\{a_{\varphi(n)}:n\in\N\}$ is $(f\circ\varphi)$-Cauchy productive (respectively, $(f\circ\varphi)$-productive) for every bijection $\varphi:\N\to\N$. (Bijections can be replaced by injections here.) We consider the question of existence of (unconditionally) $f$-productive sequences for a given ``weight function'' $f$. We prove that: (1) a Hausdorff group having an $f$-productive sequence for some $f$ contains a homeomorphic copy of the Cantor set; (2) if a non-discrete group is either locally compact Hausdorff or Weil complete metric, then it contains an unconditionally $f$-productive sequence for every function $f:\N\to\N$; (3)~a metric group is NSS if and only if it does not contain an $f_\omega$-Cauchy productive sequence, where $f_\omega$ is the function taking the constant value $\omega$. We give an example of an $f_\omega$-productive sequence $\{a_n:n\in\N\}$ in a (necessarily non-abelian) separable metric group $H$ with a linear topology and a bijection $\varphi:\N\to\N$ such that the sequence $\{\prod_{n=0}^m a_{\varphi(n)}:m\in\N\}$ diverges, thereby answering a question of Dominguez and Tarieladze. Furthermore, we show that $H$ has no unconditionally $f_\omega$-productive sequences. As an application of our results, we resolve negatively a question from $C_p(-,G)$-theory. (en)
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| Title
| - Productivity of sequences with respect to a given weight function
- Productivity of sequences with respect to a given weight function (en)
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| skos:prefLabel
| - Productivity of sequences with respect to a given weight function
- Productivity of sequences with respect to a given weight function (en)
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| skos:notation
| - RIV/44555601:13440/11:43880170!RIV12-MSM-13440___
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| http://linked.open...avai/predkladatel
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| http://linked.open...avai/riv/aktivita
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| http://linked.open...avai/riv/aktivity
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| http://linked.open...iv/cisloPeriodika
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| http://linked.open...vai/riv/dodaniDat
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| http://linked.open...aciTvurceVysledku
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| http://linked.open.../riv/druhVysledku
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| http://linked.open...iv/duvernostUdaju
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| http://linked.open...titaPredkladatele
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| http://linked.open...dnocenehoVysledku
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| http://linked.open...ai/riv/idVysledku
| - RIV/44555601:13440/11:43880170
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| http://linked.open...riv/jazykVysledku
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| http://linked.open.../riv/klicovaSlova
| - Locally compact group; Seminorm; Free group; Cauchy sequence; Productive sequence; Summable sequence; Unconditional convergence; Multiplier convergence; Subseries convergence (en)
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| http://linked.open.../riv/klicoveSlovo
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| http://linked.open...odStatuVydavatele
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| http://linked.open...ontrolniKodProRIV
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| http://linked.open...i/riv/nazevZdroje
| - Topology and its Applications
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| http://linked.open...in/vavai/riv/obor
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| http://linked.open...ichTvurcuVysledku
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| http://linked.open...cetTvurcuVysledku
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| http://linked.open...UplatneniVysledku
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| http://linked.open...v/svazekPeriodika
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| http://linked.open...iv/tvurceVysledku
| - Spěvák, Jan
- Dikranjan, Dikran
- Shakhmatov, Dmitri
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| http://linked.open...ain/vavai/riv/wos
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| issn
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| number of pages
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| http://bibframe.org/vocab/doi
| - 10.1016/j.topol.2010.11.009
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| http://localhost/t...ganizacniJednotka
| |
| is http://linked.open...avai/riv/vysledek
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