Papyrus 4 - Revision history

en>John of Reading: /* Description */Typo/general fixing, replaced: one the earliest → one of the earliest using AWB

2014-08-27T15:54:44Z

Description: Typo/general fixing, replaced: one the earliest → one of the earliest using AWB

← Older revision		Revision as of 17:54, 27 August 2014
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	~~In mathematics, '''Richardson~~'s ~~theorem''' establishes a limit on the extent to which an [[algorithm]] can [[decision problem\|decide]] whether certain mathematical expressions are equal~~. ~~It states that for~~ a ~~certain fairly natural class of expressions, it~~ is ~~[[Undecidable problem\|undecidable]] whether a particular expression ''E'' satisfies the equation ''E'' = 0, and similarly undecidable whether the functions defined by expressions ''E''~~ and ~~''F'' are everywhere equal. It was proved in 1968 by computer scientist [[Daniel Richardson]] of the [[University of Bath]].~~		The author's name is Christy Brookins. As a woman what she really likes is style and she's been doing it for quite a while. He works as a bookkeeper. For many years he's been residing in Alaska and he doesn't plan on altering it.<br><br>my web blog; online reader ([https://www.machlitim.org.il/subdomain/megila/end/node/12300 www.machlitim.org.il])

	Specifically, the class of expressions for which the theorem holds is that generated by rational numbers, the number [[Pi\|π]], the number [[natural logarithm\|log 2]], the variable ''x'', the operations of addition, subtraction, multiplication, [[function composition\|composition]], and the [[sine\|sin]], [[exponential function\|exp]], and [[absolute value\|abs]] functions.

	~~For some classes of expressions (generated by other primitives than in Richardson~~'s ~~theorem) there exist algorithms that can determine whether expression is zero.<ref>[http://portal.acm.org/citation.cfm?id=190429 The identity problem~~ for ~~elementary functions and constants by Richardson and Fitch] (pdf file)</ref>~~

	~~==Statement of the Theorem==~~
	Richardson's theorem can be stated as follows.<ref>{{cite web \|title=Tensor Computer Algebra \|date=December 22, 2008 \|url=http://metric.iem.csic.es/Martin-Garcia/slides/salamanca.pdf \|format=PDF}}{{dead link\|date=December 2013}}</ref>
	~~Let ''E'' be~~ a ~~set of real functions such that if ''A(x)'', ''B(x)'' ∈ ''E'' then ''A(x)'' ± ''B(x)'', ''A(x)B(x)'', ''A(B(x))'' ∈ ''E''~~. ~~The rational numbers are contained~~ as ~~constant functions~~. ~~Then for expressions ''A(x)'~~' in E,
	* if log(2), '~~'π'', ''e~~<~~sup~~>x<~~/sup~~>~~'', sin ''x'' ∈ E, then ''A(x)'' ≥ 0 for all ''x'' is unsolvable~~;
	* if also ''\|x\|'' ∈ ''E'' then ''A(~~x)'' = 0 is unsolvable.~~
	If furthermore there is a function ''B(x)'' ∈ ''E'' without an antiderivative in ''E'' then the integration problem is unsolvable. Example: <math>e^{ax^2}</math> has an elementary antiderivative if and only if a=0 in the elementary functions.

	~~==See also==~~
	*[[Constant problem]]

	~~==References==~~
	~~<references />~~

	~~==Further reading==~~
	*{{cite book \|last1=Petkovšek \|first1=Marko \|authorlink1=Marko Petkovšek \|last2=Wilf \|first2=Herbert S. \|authorlink2=Herbert S. Wilf \|last3=Zeilberger \|first3=Doron \|authorlink3=Doron Zeilberger \|title=A = B \|publisher=[~~[A. K. Peters]] \|url=http~~://www.~~cis~~.~~upenn~~.~~edu~~/~~~wilf~~/~~AeqB.html \|year=1996 \|isbn=1-56881-063-6 \|pages=212}}~~
	*{{Cite news \|last=Richardson \|first=Daniel \|year=1968 \|title=Some unsolvable problems involving elementary functions of a real variable \|periodical=[[Journal of Symbolic Logic]] \|volume=33 \|issue=4 \|pages=514–520 \|url=http://www.~~jstor~~.org~~/pss/2271358 \|doi=10~~.~~2307/2271358 \|jstor=10.2307/2271358 \|publisher=Association for Symbolic Logic}}~~

	~~==External links==~~
	*{{MathWorld\|urlname=RichardsonsTheorem\|title=Richardson's theorem}}

	~~[[Category:Theorems in the foundations of mathematics~~]]
	~~[[Category:Computability theory]]~~


	~~{{mathlogic-stub}}~~

en>Leszek Jańczuk: /* References */

2013-03-07T13:34:48Z

References

← Older revision		Revision as of 15:34, 7 March 2013
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	~~Wilber Berryhill~~ is ~~what his wife loves to call him~~ and ~~he totally enjoys this title~~. ~~To climb~~ is some ~~thing I truly appreciate doing~~. ~~Some time ago he chose to reside in North Carolina~~ and ~~he doesn~~'~~t plan on altering it~~. ~~She~~ functions as a ~~journey agent but soon she~~'~~ll be on her personal~~.<br><br>~~my web site~~; ~~telephone psychic (~~[http://~~findyourflirt~~.~~net~~/~~index~~.~~php?m~~=~~member_profile&p~~=~~profile&id~~=~~117823 findyourflirt~~.~~net~~])		In mathematics, '''Richardson's theorem''' establishes a limit on the extent to which an [[algorithm]] can [[decision problem\|decide]] whether certain mathematical expressions are equal. It states that for a certain fairly natural class of expressions, it is [[Undecidable problem\|undecidable]] whether a particular expression ''E'' satisfies the equation ''E'' = 0, and similarly undecidable whether the functions defined by expressions ''E'' and ''F'' are everywhere equal. It was proved in 1968 by computer scientist [[Daniel Richardson]] of the [[University of Bath]].

			Specifically, the class of expressions for which the theorem holds is that generated by rational numbers, the number [[Pi\|π]], the number [[natural logarithm\|log 2]], the variable ''x'', the operations of addition, subtraction, multiplication, [[function composition\|composition]], and the [[sine\|sin]], [[exponential function\|exp]], and [[absolute value\|abs]] functions.

			For some classes of expressions (generated by other primitives than in Richardson's theorem) there exist algorithms that can determine whether expression is zero.<ref>[http://portal.acm.org/citation.cfm?id=190429 The identity problem for elementary functions and constants by Richardson and Fitch] (pdf file)</ref>

			==Statement of the Theorem==
			Richardson's theorem can be stated as follows.<ref>{{cite web \|title=Tensor Computer Algebra \|date=December 22, 2008 \|url=http://metric.iem.csic.es/Martin-Garcia/slides/salamanca.pdf \|format=PDF}}{{dead link\|date=December 2013}}</ref>
			Let ''E'' be a set of real functions such that if ''A(x)'', ''B(x)'' ∈ ''E'' then ''A(x)'' ± ''B(x)'', ''A(x)B(x)'', ''A(B(x))'' ∈ ''E''. The rational numbers are contained as constant functions. Then for expressions ''A(x)'' in E,
			* if log(2), ''π'', ''e<sup>x</sup>'', sin ''x'' ∈ E, then ''A(x)'' ≥ 0 for all ''x'' is unsolvable;
			* if also ''\|x\|'' ∈ ''E'' then ''A(x)'' = 0 is unsolvable.
			If furthermore there is a function ''B(x)'' ∈ ''E'' without an antiderivative in ''E'' then the integration problem is unsolvable. Example: <math>e^{ax^2}</math> has an elementary antiderivative if and only if a=0 in the elementary functions.

			==See also==
			*[[Constant problem]]

			==References==
			<references />

			==Further reading==
			*{{cite book \|last1=Petkovšek \|first1=Marko \|authorlink1=Marko Petkovšek \|last2=Wilf \|first2=Herbert S. \|authorlink2=Herbert S. Wilf \|last3=Zeilberger \|first3=Doron \|authorlink3=Doron Zeilberger \|title=A = B \|publisher=[[A. K. Peters]] \|url=http://www.cis.upenn.edu/~wilf/AeqB.html \|year=1996 \|isbn=1-56881-063-6 \|pages=212}}
			*{{Cite news \|last=Richardson \|first=Daniel \|year=1968 \|title=Some unsolvable problems involving elementary functions of a real variable \|periodical=[[Journal of Symbolic Logic]] \|volume=33 \|issue=4 \|pages=514–520 \|url=http://www.jstor.org/pss/2271358 \|doi=10.2307/2271358 \|jstor=10.2307/2271358 \|publisher=Association for Symbolic Logic}}

			==External links==
			*{{MathWorld\|urlname=RichardsonsTheorem\|title=Richardson's theorem}}

			[[Category:Theorems in the foundations of mathematics]]
			[[Category:Computability theory]]


			{{mathlogic-stub}}

en>Helpful Pixie Bot: ISBNs (Build KC)

2012-05-05T23:03:31Z

ISBNs (Build KC)

New page

Wilber Berryhill is what his wife loves to call him and he totally enjoys this title. To climb is some thing I truly appreciate doing. Some time ago he chose to reside in North Carolina and he doesn't plan on altering it. She functions as a journey agent but soon she'll be on her personal.<br><br>my web site; telephone psychic ([http://findyourflirt.net/index.php?m=member_profile&p=profile&id=117823 findyourflirt.net])