https://en.formulasearchengine.com/w/index.php?action=history&feed=atom&title=Credal_networkCredal network - Revision history2026-08-28T15:58:13ZRevision history for this page on the wikiMediaWiki 1.47.0-wmf.7https://en.formulasearchengine.com/w/index.php?title=Credal_network&diff=28300&oldid=preven>Kowreee: changed one word for grammar2014-01-25T06:24:57Z<p>changed one word for grammar</p>
<p><b>New page</b></p><div>In mathematics, '''Ostrowski numeration''', named after [[Alexander Ostrowski]], is either of two related numeration systems based on [[continued fraction]]s: a [[non-standard positional numeral system]] for integers and a [[non-integer representation]] of [[real number]]s.<br />
<br />
Fix a positive [[irrational number]] ''α'' with continued fraction expansion [''a''<sub>1</sub>,''a''<sub>2</sub>,...]. Let (''q''<sub>''n''</sub>) be the sequence of denominators of the convergents ''p''<sub>''n''</sub>/''q''<sub>''n''</sub> to α: so ''q''<sub>''n''</sub> = ''a''<sub>''n''</sub>''q''<sub>''n''&minus;1</sub> + ''q''<sub>''n''&minus;2</sub>. Let ''α''<sub>''n''</sub> denote ''T''<sup>''n''</sub>(''α'') where ''T'' is the Gauss map ''T''(''x'') = {1/''x''}, and write ''β''<sub>''n''</sub> = (&minus;1)<sup>''n''+1</sup> ''α''<sub>0</sub>α<sub>1</sub> ... ''α''<sub>''n''</sub>: we have ''β''<sub>''n''</sub> = ''a''<sub>''n''</sub>''β''<sub>''n''&minus;1</sub> + ''β''<sub>''n''&minus;2</sub>.<br />
<br />
==Real number representations==<br />
Every positive real ''x'' can be written as<br />
<br />
:<math> x = \sum_{n=1}^\infty b_n \beta_n \ </math><br />
<br />
where the integer coefficients 0 ≤ ''b''<sub>''n''</sub> < ''a''<sub>''n''</sub> and if ''b''<sub>''n''</sub> = ''a''<sub>''n''</sub> then ''b''<sub>''n''&minus;1</sub> = 0.<br />
<br />
==Integer representations==<br />
Every positive integer ''N'' can be written uniquely as<br />
<br />
:<math> N = \sum_{n=1}^k b_n q_n \ </math><br />
<br />
where the integer coefficients 0 ≤ ''b''<sub>''n''</sub> < ''a''<sub>''n''</sub> and if ''b''<sub>''n''</sub> = ''a''<sub>''n''</sub> then ''b''<sub>''n''&minus;1</sub> = 0.<br />
<br />
If ''α'' is the [[golden ratio]], then all the partial quotients ''a''<sub>''n''</sub> are equal to 1, the denominators ''q''<sub>''n''</sub> are the [[Fibonacci numbers]] and we recover [[Zeckendorf's theorem]] on the [[Fibonacci representation]] of positive integers as a sum of distinct non-consecutive Fibonacci numbers.<br />
<br />
==See also==<br />
* [[Complete sequence]]<br />
<br />
==References==<br />
*{{cite book | last1 = Allouche | first1 = Jean-Paul | last2 = Shallit | first2 = Jeffrey | author2-link = Jeffrey Shallit | isbn = 978-0-521-82332-6 | publisher = [[Cambridge University Press]] | title = Automatic Sequences: Theory, Applications, Generalizations | year = 2003 | zbl=1086.11015 }}.<br />
* {{cite journal | zbl=1237.68134 | last1=Epifanio | first1=C. | last2=Frougny | first2=C. | last3=Gabriele | first3=A. | last4=Mignosi | first4=F. | last5=Shallit | first5=J. | author5-link=Jeffrey Shallit | title=Sturmian graphs and integer representations over numeration systems | journal=Discrete Appl. Math. | volume=160 | number=4-5 | pages=536–547 | year=2012 | issn=0166-218X }}<br />
* {{cite journal | first=Alexander | last=Ostrowski | authorlink=Alexander Ostrowski | jfm=48.0197.04 | title=Bemerkungen zur Theorie der diophantischen Approximationen | language=German | journal=Hamb. Abh. | volume=1 | year=1921 | pages=77–98 }}<br />
* {{cite book | last=Pytheas Fogg | first=N. | others=Editors Berthé, Valérie; Ferenczi, Sébastien; Mauduit, Christian; Siegel, A. | title=Substitutions in dynamics, arithmetics and combinatorics | series=Lecture Notes in Mathematics | volume=1794 | location=Berlin | publisher=[[Springer-Verlag]] | year=2002 | isbn=3-540-44141-7 | zbl=1014.11015 }}<br />
<br />
[[Category:Non-standard positional numeral systems]]<br />
<br />
{{number-stub}}</div>en>Kowreee