Highly composite number - Revision history

en>K9re11: removed Category:Conjectures using HotCat as I don't understand what conjecture this refers to

2014-12-16T20:36:37Z

removed Category:Conjectures using HotCat as I don't understand what conjecture this refers to

← Older revision		Revision as of 22:36, 16 December 2014
Line 1:		Line 1:
	~~{{Unreferenced\|date=December 2009}}~~		Translator Philip Geddes from Cooksville, loves to spend some time horse riding, castle clash hack and ornithology. Last year just made a trip Tombs of Buganda Kings at Kasubi.<br><br>Here is my web page; [https://www.facebook.com/pages/Castle-Clash-Hack/262424220600747 castle clash hack no download]
	The '''right quotient''' (or simply '''quotient''') of a [[formal language]] <math>L_1</math> with a formal language <math>L_2</math> is the language consisting of strings ''w'' such that ''wx'' is in <math>L_1</math> for some ~~string ''x'' in <math>L_2</math>. In symbols, we write:~~

	~~:<math>L_1 / L_2 = \{w \ \| \ \exists x ((x \in L_2) \land (wx \in L_1)) \}</math>~~

	~~In other words, each string in <math>L_1 / L_2</math> is the prefix of a string <math>wx</math> in <math>L_1</math>~~, ~~with the remainder of the word being a string in <math>L_2</math>.~~

	~~==Examples==~~
	~~Consider~~

	~~<math>L_1 = \{ a^n b^n c^n \ \ \|\ \ n\ge 0 \}</math>~~

	and

	~~<math>L_2 = \{ b^i c^j \ \ \| \ \ i,j\ge 0 \}</math>~~.

	~~Now, if we insert~~ a ~~divider into the middle~~ of ~~an element of <math>L_1</math>, the part on the right is in~~ <~~math~~>~~L_2~~<~~/math~~> ~~only if the divider~~ is ~~placed adjacent to a ''b'' (in which case ''i''&nbsp~~;≤ ''n'' and ''j'' = ''n'') or adjacent to a ''c'' (in which case ''i'' = 0 and ''j'' ≤ ''n''). The part on the left, therefore, will be either <math>a^n b^{n-i}</math> or <math>a^n b^n c^{n-j}</math>; and <math>L_1 / ~~L_2<~~/~~math> can be written as~~

	~~<center><math>\{ a^p b^q c^r \ \ \| \ \ p = q \ge r \ \ \or \ \ p \ge q \and r = 0 \}</math>~~.</~~center>~~

	~~==Properties==~~
	~~Some common closure properties of the right quotient include:~~

	* The quotient of a [[regular language]] with any other language is regular.
	* The quotient of a [[context free language]] with a regular language is context free.
	* The quotient of two context free languages can be any [[recursively enumerable]] language.
	* The quotient of two recursively enumerable languages is recursively enumerable.

	~~==Left and right quotients==~~
	~~There is a related notion of [[left quotient]], which keeps the postfixes of <math>L_1<~~/~~math> without the prefixes in <math>L_2<~~/~~math>. Sometimes, though, "right quotient" is written simply as "quotient". The above closure properties hold for both left and right quotients.~~

	~~{{DEFAULTSORT:Right Quotient}}~~
	~~[[Category:Formal languages]~~]

en>Maurice Carbonaro: Changed "(...) prime factors (...)" into "(...) prime factors (...)". The former linked to the "prime number" article, the former to the "prime factors" one

2013-12-05T14:10:34Z

Changed "(...) prime factors (...)" into "(...) prime factors (...)". The former linked to the "prime number" article, the former to the "prime factors" one

← Older revision		Revision as of 16:10, 5 December 2013
Line 1:		Line 1:
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			The '''right quotient''' (or simply '''quotient''') of a [[formal language]] <math>L_1</math> with a formal language <math>L_2</math> is the language consisting of strings ''w'' such that ''wx'' is in <math>L_1</math> for some string ''x'' in <math>L_2</math>. In symbols, we write:

			:<math>L_1 / L_2 = \{w \ \| \ \exists x ((x \in L_2) \land (wx \in L_1)) \}</math>

			In other words, each string in <math>L_1 / L_2</math> is the prefix of a string <math>wx</math> in <math>L_1</math>, with the remainder of the word being a string in <math>L_2</math>.

			==Examples==
			Consider

			<math>L_1 = \{ a^n b^n c^n \ \ \|\ \ n\ge 0 \}</math>

			and

			<math>L_2 = \{ b^i c^j \ \ \| \ \ i,j\ge 0 \}</math>.

			Now, if we insert a divider into the middle of an element of <math>L_1</math>, the part on the right is in <math>L_2</math> only if the divider is placed adjacent to a ''b'' (in which case ''i'' ≤ ''n'' and ''j'' = ''n'') or adjacent to a ''c'' (in which case ''i'' = 0 and ''j'' ≤ ''n''). The part on the left, therefore, will be either <math>a^n b^{n-i}</math> or <math>a^n b^n c^{n-j}</math>; and <math>L_1 / L_2</math> can be written as

			<center><math>\{ a^p b^q c^r \ \ \| \ \ p = q \ge r \ \ \or \ \ p \ge q \and r = 0 \}</math>.</center>

			==Properties==
			Some common closure properties of the right quotient include:

			* The quotient of a [[regular language]] with any other language is regular.
			* The quotient of a [[context free language]] with a regular language is context free.
			* The quotient of two context free languages can be any [[recursively enumerable]] language.
			* The quotient of two recursively enumerable languages is recursively enumerable.

			==Left and right quotients==
			There is a related notion of [[left quotient]], which keeps the postfixes of <math>L_1</math> without the prefixes in <math>L_2</math>. Sometimes, though, "right quotient" is written simply as "quotient". The above closure properties hold for both left and right quotients.

			{{DEFAULTSORT:Right Quotient}}
			[[Category:Formal languages]]

176.4.60.122: /* See also */ Euler's totient function

2012-08-31T09:38:32Z

See also: Euler's totient function

New page

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