About: Down-arrow notation   Sponge Permalink

An Entity of Type : owl:Thing, within Data Space : 134.155.108.49:8890 associated with source dataset(s)

\[a \downarrow b = a^b\] \[a \downarrow^n 1 = a\] \[a \downarrow^{n + 1} (b + 1) = (a \downarrow^{n +1} b) \downarrow^n a ext{ (otherwise)}\] \(a \downarrow^n b\) is a shorthand for \(a \downarrow\downarrow\cdots\downarrow\downarrow b\) with n down-arrows. When \(n = 2\), \(a \downarrow\downarrow b = a^{a^{b-1}}\). The inequality \(a \downarrow\downarrow a = a^{a^{a-1}} < a^{a^a} = a \uparrow\uparrow 3\) is useful when bounding down-arrows in terms of up-arrows. It can be shown that \(a \downarrow^{2n-1} b \ge a \uparrow^n b\) for \(a, b, n \ge 1\).

AttributesValues
rdfs:label
  • Down-arrow notation
rdfs:comment
  • \[a \downarrow b = a^b\] \[a \downarrow^n 1 = a\] \[a \downarrow^{n + 1} (b + 1) = (a \downarrow^{n +1} b) \downarrow^n a ext{ (otherwise)}\] \(a \downarrow^n b\) is a shorthand for \(a \downarrow\downarrow\cdots\downarrow\downarrow b\) with n down-arrows. When \(n = 2\), \(a \downarrow\downarrow b = a^{a^{b-1}}\). The inequality \(a \downarrow\downarrow a = a^{a^{a-1}} < a^{a^a} = a \uparrow\uparrow 3\) is useful when bounding down-arrows in terms of up-arrows. It can be shown that \(a \downarrow^{2n-1} b \ge a \uparrow^n b\) for \(a, b, n \ge 1\).
dcterms:subject
dbkwik:googology/p...iPageUsesTemplate
Type
  • 3(xsd:integer)
fgh
  • \omega
Base
  • Exponentiation
abstract
  • \[a \downarrow b = a^b\] \[a \downarrow^n 1 = a\] \[a \downarrow^{n + 1} (b + 1) = (a \downarrow^{n +1} b) \downarrow^n a ext{ (otherwise)}\] \(a \downarrow^n b\) is a shorthand for \(a \downarrow\downarrow\cdots\downarrow\downarrow b\) with n down-arrows. When \(n = 2\), \(a \downarrow\downarrow b = a^{a^{b-1}}\). The inequality \(a \downarrow\downarrow a = a^{a^{a-1}} < a^{a^a} = a \uparrow\uparrow 3\) is useful when bounding down-arrows in terms of up-arrows. It can be shown that \(a \downarrow^{2n-1} b \ge a \uparrow^n b\) for \(a, b, n \ge 1\). Down-arrow notation is not as important in googology as the up-arrow notation, but it is used in the definition of Clarkkkkson.
is wikipage disambiguates of
Alternative Linked Data Views: ODE     Raw Data in: CXML | CSV | RDF ( N-Triples N3/Turtle JSON XML ) | OData ( Atom JSON ) | Microdata ( JSON HTML) | JSON-LD    About   
This material is Open Knowledge   W3C Semantic Web Technology [RDF Data] Valid XHTML + RDFa
OpenLink Virtuoso version 07.20.3217, on Linux (x86_64-pc-linux-gnu), Standard Edition
Data on this page belongs to its respective rights holders.
Virtuoso Faceted Browser Copyright © 2009-2012 OpenLink Software