en>Colonies Chris: sp, date & link fixes; unlinking common words, replaced: Hartree-Fock → Hartree–Fock using AWB

2014-03-14T00:02:59Z

sp, date & link fixes; unlinking common words, replaced: Hartree-Fock → Hartree–Fock using AWB

en>Addbot: Bot: Migrating 4 interwiki links, now provided by Wikidata on d:q2266294 (Report Errors)

2013-02-28T09:54:24Z

Bot: Migrating 4 interwiki links, now provided by Wikidata on d:q2266294 (Report Errors)

← Older revision		Revision as of 11:54, 28 February 2013
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	~~I’m Kayla~~ from ~~Haula doing my final year engineering in Industrial and Labor Relations~~. ~~I did my schooling~~, ~~secured 90%~~ and ~~hope to find someone~~ with ~~same interests~~ in ~~Exhibition Drill~~.<br><br>~~Check out my blog post~~ ... ~~Fifa 15 Coin Generator~~ - [~~http~~://~~Hhjtw~~.~~net~~/~~plus~~/~~guestbook~~.~~php hhjtw~~.~~net~~],		In mathematics, '''Pentation''' is the operation of repeated [[tetration]], just as [[tetration]] is the operation of repeated [[exponentiation]].<ref>{{citation
			\| last = Perstein \| first = Millard H.
			\| date = June 1962
			\| doi = 10.1145/367766.368160
			\| issue = 6
			\| journal = [[Communications of the ACM]]
			\| page = 344
			\| title = Algorithm 93: General Order Arithmetic
			\| volume = 5}}.</ref>

			Pentation can be written in [[Knuth's up-arrow notation]] as <math>a \uparrow \uparrow \uparrow b</math> or <math>a \uparrow^{3}b</math>.
			In this notation, <math>a\uparrow b</math> represents the exponentiation function <math>a^b</math>, which may be interpreted as the result of [[Iterated function\|repeatedly applying]] the function <math>x\mapsto ax</math>, for <math>b</math> repetitions, starting from the number 1. Analogously, <math>a\uparrow\uparrow b</math>, tetration, represents the value obtained by repeatedly applying the function <math>x\mapsto a\uparrow x</math>, for <math>b</math> repetitions, starting from the number 1. And the pentation <math>a \uparrow \uparrow \uparrow b</math> represents the value obtained by repeatedly applying the function <math>x\mapsto a\uparrow\uparrow x</math>, for <math>b</math> repetitions, starting from the number 1.<ref>{{citation
			\| last = Knuth \| first = D. E. \| author-link = Donald Knuth
			\| doi = 10.1126/science.194.4271.1235
			\| journal = [[Science (journal)\|Science]]
			\| pages = 1235–1242
			\| title = Mathematics and computer science: Coping with finiteness
			\| volume = 194
			\| year = 1976}}.</ref><ref>{{citation
			\| last1 = Blakley \| first1 = G. R.
			\| last2 = Borosh \| first2 = I.
			\| doi = 10.1016/0001-8708(79)90052-5
			\| issue = 2
			\| journal = [[Advances in Mathematics]]
			\| mr = 549780
			\| pages = 109–136
			\| title = Knuth's iterated powers
			\| volume = 34
			\| year = 1979}}.</ref> Alternatively, in [[Conway chained arrow notation]], <math>a\uparrow\uparrow\uparrow b = a\rightarrow b\rightarrow 3</math>.<ref>{{citation\|title=The Book of Numbers\|first1=John Horton\|last1=Conway\|author1-link=John Horton Conway\|first2=Richard\|last2=Guy\|author2-link=Richard K. Guy\|publisher=Springer\|year=1996\|isbn=9780387979939\|page=61\|url=http://books.google.com/books?id=0--3rcO7dMYC&pg=PA61}}.</ref>

			The values of the pentation function may also be obtained from the values in the fourth row of the table of values of a variant of the [[Ackermann function]]: if <math>A(n,m)</math> is defined by the Ackermann recurrence <math>A(m-1,A(m,n-1))</math> with the initial conditions <math>A(1,n)=an</math> and <math>A(m,1)=a</math>, then <math>a\uparrow\uparrow\uparrow b=A(4,b)</math>.<ref>{{citation
			\| last = Nambiar \| first = K. K.
			\| doi = 10.1016/0893-9659(95)00084-4
			\| issue = 6
			\| journal = Applied Mathematics Letters
			\| mr = 1368037
			\| pages = 51–53
			\| title = Ackermann functions and transfinite ordinals
			\| volume = 8
			\| year = 1995}}.</ref>

			The word "Pentation" was coined by [[Reuben Goodstein]] in 1947 from the roots [[penta-]] (five) and [[iterated function\|iteration]]. It is part of his general naming scheme for [[hyperoperation]]s.<ref>{{citation
			\| last = Goodstein \| first = R. L. \| authorlink = Reuben Goodstein
			\| journal = [[The Journal of Symbolic Logic]]
			\| mr = 0022537
			\| pages = 123–129
			\| title = Transfinite ordinals in recursive number theory
			\| volume = 12
			\| year = 1947}}.</ref>

			==References==
			{{reflist}}

			[[Category:Exponentials]]
			[[Category:Large numbers]]
			[[Category:Binary operations]]
			[[Category:Arithmetic]]


			{{numtheory-stub}}

en>Justincheng12345-bot: r2.7.3) (Robot: Adding zh:多组态自洽场方法

2012-06-24T04:35:09Z

r2.7.3) (Robot: Adding zh:多组态自洽场方法

New page

I’m Kayla from Haula doing my final year engineering in Industrial and Labor Relations. I did my schooling, secured 90% and hope to find someone with same interests in Exhibition Drill.<br><br>Check out my blog post ... Fifa 15 Coin Generator - [http://Hhjtw.net/plus/guestbook.php hhjtw.net],

Multi-configurational self-consistent field - Revision history

en>Colonies Chris: sp, date & link fixes; unlinking common words, replaced: Hartree-Fock → Hartree–Fock using AWB

en>Addbot: Bot: Migrating 4 interwiki links, now provided by Wikidata on d:q2266294 (Report Errors)

en>Justincheng12345-bot: r2.7.3) (Robot: Adding zh:多组态自洽场方法