- Tytuł:
- Families of Increasing Sequences Possessing the Harmonic Series Property
- Autorzy:
-
Wituła, Roman
Hetmaniok, Edyta
Słota, Damian - Powiązania:
- https://bibliotekanauki.pl/articles/972271.pdf
- Data publikacji:
- 2013
- Wydawca:
- Uniwersytet Łódzki. Wydawnictwo Uniwersytetu Łódzkiego
- Tematy:
-
Sierpiński family
harmonic series property - Opis:
- We say that family \( \mathcal{W} \ne \emptyset \) of increasing sequences of positive real numbers possesses the harmonic series property (HSP – for the shortness of notation) if for any finite sequence of elements of \( \mathcal{W} \), i.e. for any \( k \in N \), \( \{ a_n^{(i)} \}_{n=1}^\infty \in \mathcal{W} \), \( i = 1, ..., k \), we have $$ \sum_{n=1}^\infty ( a_n^{(1)} + a_n^{(2)} + ... a_n^{(k)} )^{-1} = \infty $$ (the sequences \( \{ a_n^{(i)} \}_{n=1}^\infty \) and \( \{ a_n^{(j)} \}_{n=1}^\infty \) for different indices \( i \) and \( j \) can be the same). We prove in this paper that any maximal, with respect to inclusion, subset of \( N \) – the family of all increasing sequences of positive integers – possessing the harmonic series property has the cardinality of the continuum. Moreover, we prove that for any countable (infinite) set \( \mathcal{W} \subset \mathcal{N} \) there exists an "orthogonal" family \( \mathcal{W}^\perp \subset \mathcal{N} \) such that a) card \( \mathcal{W}^\perp = \mathcal{c} \), b) \( ( \forall \{ a_n \}, \{ b_n \} \in \mathcal{W}^\perp ) ( \{ a_n \} \ne \{ b_n \} \Rightarrow \sum ( a_n + b_n )^{-1} < \infty ) \) (this condition is a reason for using the word "orthogonal" – the value "0" or "≠ 0" of the scalar product is replaced here by the convergence or divergence, respectively, of the series), c) \( ( \forall \{ a_n \} \in \mathcal{W}^\perp ) \) (the family \( \mathcal{W} \cup \{ \{ a_n \}_{n=1}^\infty \} \) possesses the harmonic series property). All facts are proved constructively, by using the modified version of the classical Sierpiński family of increasing sequences having the cardinality of the continuum.
- Źródło:
-
Acta Universitatis Lodziensis. Folia Mathematica; 2013, 18; 3-10
2450-7652 - Pojawia się w:
- Acta Universitatis Lodziensis. Folia Mathematica
- Dostawca treści:
- Biblioteka Nauki
Artykuł