Auction theory - Revision history

en>Aetheling: /* Game-theoretic models */ add link

2014-11-17T20:51:10Z

Game-theoretic models: add link

← Older revision		Revision as of 22:51, 17 November 2014
Line 1:		Line 1:
	~~{{dablink\|This article~~ is ~~about certain functional equations.~~ ~~For ordinary differential equations that are cubic in the unknown function, see~~ [~~[Abel equation of the first kind]].}}~~		Friends contact her Felicidad and her husband doesn't like it at all. Managing people is auto warranty [http://www.Volvocars.com/us/sales-services/sales/volvo_certified_preowned/Pages/default.aspx extended] car [http://Www.Mpp.com/warranty-plans/used-car-warranties.htm warranty] how I make money and it's some thing I truly enjoy. One of my preferred hobbies is camping and now I'm [http://109.230.254.127/index.php?mod=users&action=view&id=29690 extended car warranty] auto warranty trying to make money with it. Her husband and her selected to reside in Delaware but she needs to transfer because of her family.<br><br>[http://bikedance.com/blogs/post/29704 auto warranty] my web blog ... [http://Www.Planetplayschool.com/ActivityFeed/MyProfile/tabid/61/UserId/415/Default.aspx extended car warranty]

	~~The '''Abel equation''', named after [[Niels Henrik Abel]], is special case of [[functional equation]]s which can be written in the form~~

	:~~<math>f(h(x)) = h(x + 1)\,\!<~~/~~math>~~

	or

	~~:<math>\alpha(f(x))=\alpha(x)+1\!<~~/~~math>~~

	~~and controls the iteration of {{mvar\|f}}~~.

	~~==Equivalence==~~
	~~These equations are equivalent. Assuming that {{mvar\|α}} is an [[invertible function]], the second equation can be written as~~

	~~:<math> \alpha^{-1}(\alpha(f(x)))=\alpha^{-1}(\alpha(x)+1)\,~~ .</~~math>~~

	~~Taking {{math\|''x'' {{=}} ''α''<sup>−1<~~/~~sup>(''y'')}}, the equation can be written as~~

	~~::<math>f(\alpha^{~~-~~1}(y))=\alpha^{-1}(y+1)\, .<~~/~~math>~~

	~~For a function {{math\|''f''(''x'')}} assumed to be known, the task is to solve the functional equation for the function {{math\|''α''<sup>−1<~~/~~sup>}}, possibly satisfying additional requirements, such as {{math\|''α''<sup>−1<~~/~~sup>(0) {{=}} 1}}~~.

	~~The change of variables {{math\|''s''<sup>''α''(''x'')</sup> {{=}} Ψ(''x'')}}, for a real parameter {{mvar\|s}}, brings Abel's equation into the celebrated [[Schröder's equation]~~]~~, {{math\|Ψ(''f''(''x'')) {{=}} ''s'' Ψ(''x'')}} .~~

	~~The further change {{math\|''F''(''x'') {{=}} exp(''s''<sup>''α''(''x'')</sup>)}} into [~~[~~Böttcher's equation]], {{math\|''F''(''f''(''x'')) {{=}} ''F''(''x'')<sup>''s''</sup>}}.~~

	~~==History==~~
	~~Initially, the equation in the more general form~~
	~~<ref name="abel">{{cite journal~~
	~~\| url=~~http://~~gdz~~.~~sub~~.~~uni~~-~~goettingen.de~~/~~ru/dms/load/img/?PPN=PPN243919689_0001&DMDID=dmdlog6~~
	~~\| author= Abel, N~~.H.
	~~\| coauthors=~~
	~~\| title= Untersuchung der Functionen zweier unabhängig veränderlichen Größen x und y, wie f(x, y), welche die Eigenschaft haben, ...~~
	~~\| journal=[~~[~~Journal für die reine und angewandte Mathematik]]~~
	~~\| volume=1~~
	~~\| pages=11–15~~
	~~\| year=1826~~
	~~}}</ref>~~
	~~<ref name="s">{{cite journal~~
	~~\| url=~~http://~~projecteuclid~~.~~org/DPubS~~/~~Repository/1~~.~~0/Disseminate~~?~~handle~~=~~euclid.bams/1183421988~~&view~~=body~~&~~content-type~~=~~pdf_1~~
	~~\| author=A~~. R. ~~Schweitzer~~
	~~\| coauthors=~~
	~~\| title=Theorems on functional equations~~
	~~\| journal=[[Bulletin des Sciences Mathématiques]]~~
	~~\| volume=27~~
	~~\| issue=2~~
	~~\| pages=31~~
	~~\| year=1903~~
	}}<~~/ref~~>
	~~was reported. Then it happens that even in the case of single variable, the equation is not trivial, and requires special analysis~~
	<~~ref name="U1"~~>~~{{cite journal~~
	~~\| url=~~http://~~matwbn.icm.edu~~.pl/~~ksiazki~~/sm/~~sm134/sm13424~~.~~pdf~~
	~~\| author=G~~. ~~Belitskii~~
	~~\| coauthors=Yu~~. ~~Lubish~~
	~~\| title=The real-analytic solutions of the Abel functional equations~~
	~~\| journal=[~~[~~Studia Mathematica]]~~
	~~\| volume=134~~
	~~\| issue=2~~
	~~\| pages=135–141~~
	~~\| year=1999~~
	~~}}</ref><ref name="j">{{cite journal~~
	~~\| journal= Nonlinear Analysis: Hybrid Systems~~
	~~\| volume=1~~
	~~\| issue=1~~
	~~\| year=2007~~
	~~\| pages=95–102~~
	~~\| doi=10.1016/j.nahs.2006.04.002~~
	~~\| author=Jitka Laitochová~~
	~~\| title =Group iteration for Abel’s functional equation }} Studied is the Abel functional equation α(f(x))=α(x)+1</ref>~~

	~~In the case of linear transfer function, the solution can be expressed in compact form~~
	~~<ref name="linear">{{cite journal~~
	~~\| url=~~http://~~matwbn~~.~~icm~~.~~edu.pl~~/~~ksiazki~~/sm/~~sm127~~/~~sm12716.pdf~~
	~~\| author=G. Belitskii~~
	~~\| coauthor=Yu. Lubish~~
	~~\| title=The Abel equation and total solvability of linear functional equtions~~
	~~\| journal=[[Studia Mathematica]]~~
	~~\| volume=127~~
	~~\| year=1998~~
	~~\| pages=81–89~~
	~~}}<~~/~~ref>~~

	~~==Special cases==~~

	~~The equation of [[tetration]] is a special case of Abel's equation, with {{math\|''f'' {{=}} exp}}.~~

	~~In the case of an integer argument, the equation encodes a recurrent procedure, e.g.,~~
	~~:<math>\alpha(f(f(x)))=\alpha(x)+2 ~,<~~/~~math>~~
	~~and so on,~~
	~~:<math>\alpha(f_n(x))=\alpha(x)+n ~~~.~~</math>~~

	~~==See also==~~
	*[[Functional equation]]
	*[[Iterated function]]
	*[[Abel function]]
	*[[Schröder's equation]]
	*[[Böttcher's equation]]
	~~==References==~~
	~~<references/>~~

	~~[[Category:Niels Henrik Abel]]~~
	~~[[Category:Functional equations]~~]

en>UbiquitousUK: /* Winner's curse */

2014-01-20T15:03:17Z

Winner's curse

← Older revision		Revision as of 17:03, 20 January 2014
Line 1:		Line 1:
	~~Hi there~~, ~~I am Alyson Boon even though it is not~~ the ~~name on my beginning certification~~. I'~~ve usually loved living~~ in ~~Alaska~~. ~~I am presently a travel agent~~. ~~To climb~~ is ~~some thing I really enjoy performing~~.<br><br>~~Look at my blog~~ - ~~psychic chat online~~ ([http://~~chorokdeul~~.co.kr/~~index~~.~~php~~?~~document_srl~~=~~324263~~&~~mid~~=~~customer21 chorokdeul~~.co.kr])		{{dablink\|This article is about certain functional equations. For ordinary differential equations that are cubic in the unknown function, see [[Abel equation of the first kind]].}}

			The '''Abel equation''', named after [[Niels Henrik Abel]], is special case of [[functional equation]]s which can be written in the form

			:<math>f(h(x)) = h(x + 1)\,\!</math>

			or

			:<math>\alpha(f(x))=\alpha(x)+1\!</math>

			and controls the iteration of {{mvar\|f}}.

			==Equivalence==
			These equations are equivalent. Assuming that {{mvar\|α}} is an [[invertible function]], the second equation can be written as

			:<math> \alpha^{-1}(\alpha(f(x)))=\alpha^{-1}(\alpha(x)+1)\, .</math>

			Taking {{math\|''x'' {{=}} ''α''<sup>−1</sup>(''y'')}}, the equation can be written as

			::<math>f(\alpha^{-1}(y))=\alpha^{-1}(y+1)\, .</math>

			For a function {{math\|''f''(''x'')}} assumed to be known, the task is to solve the functional equation for the function {{math\|''α''<sup>−1</sup>}}, possibly satisfying additional requirements, such as {{math\|''α''<sup>−1</sup>(0) {{=}} 1}}.

			The change of variables {{math\|''s''<sup>''α''(''x'')</sup> {{=}} Ψ(''x'')}}, for a real parameter {{mvar\|s}}, brings Abel's equation into the celebrated [[Schröder's equation]], {{math\|Ψ(''f''(''x'')) {{=}} ''s'' Ψ(''x'')}} .

			The further change {{math\|''F''(''x'') {{=}} exp(''s''<sup>''α''(''x'')</sup>)}} into [[Böttcher's equation]], {{math\|''F''(''f''(''x'')) {{=}} ''F''(''x'')<sup>''s''</sup>}}.

			==History==
			Initially, the equation in the more general form
			<ref name="abel">{{cite journal
			\| url=http://gdz.sub.uni-goettingen.de/ru/dms/load/img/?PPN=PPN243919689_0001&DMDID=dmdlog6
			\| author= Abel, N.H.
			\| coauthors=
			\| title= Untersuchung der Functionen zweier unabhängig veränderlichen Größen x und y, wie f(x, y), welche die Eigenschaft haben, ...
			\| journal=[[Journal für die reine und angewandte Mathematik]]
			\| volume=1
			\| pages=11–15
			\| year=1826
			}}</ref>
			<ref name="s">{{cite journal
			\| url=http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?handle=euclid.bams/1183421988&view=body&content-type=pdf_1
			\| author=A. R. Schweitzer
			\| coauthors=
			\| title=Theorems on functional equations
			\| journal=[[Bulletin des Sciences Mathématiques]]
			\| volume=27
			\| issue=2
			\| pages=31
			\| year=1903
			}}</ref>
			was reported. Then it happens that even in the case of single variable, the equation is not trivial, and requires special analysis
			<ref name="U1">{{cite journal
			\| url=http://matwbn.icm.edu.pl/ksiazki/sm/sm134/sm13424.pdf
			\| author=G. Belitskii
			\| coauthors=Yu. Lubish
			\| title=The real-analytic solutions of the Abel functional equations
			\| journal=[[Studia Mathematica]]
			\| volume=134
			\| issue=2
			\| pages=135–141
			\| year=1999
			}}</ref><ref name="j">{{cite journal
			\| journal= Nonlinear Analysis: Hybrid Systems
			\| volume=1
			\| issue=1
			\| year=2007
			\| pages=95–102
			\| doi=10.1016/j.nahs.2006.04.002
			\| author=Jitka Laitochová
			\| title =Group iteration for Abel’s functional equation }} Studied is the Abel functional equation α(f(x))=α(x)+1</ref>

			In the case of linear transfer function, the solution can be expressed in compact form
			<ref name="linear">{{cite journal
			\| url=http://matwbn.icm.edu.pl/ksiazki/sm/sm127/sm12716.pdf
			\| author=G. Belitskii
			\| coauthor=Yu. Lubish
			\| title=The Abel equation and total solvability of linear functional equtions
			\| journal=[[Studia Mathematica]]
			\| volume=127
			\| year=1998
			\| pages=81–89
			}}</ref>

			==Special cases==

			The equation of [[tetration]] is a special case of Abel's equation, with {{math\|''f'' {{=}} exp}}.

			In the case of an integer argument, the equation encodes a recurrent procedure, e.g.,
			:<math>\alpha(f(f(x)))=\alpha(x)+2 ~,</math>
			and so on,
			:<math>\alpha(f_n(x))=\alpha(x)+n ~.</math>

			==See also==
			*[[Functional equation]]
			*[[Iterated function]]
			*[[Abel function]]
			*[[Schröder's equation]]
			*[[Böttcher's equation]]
			==References==
			<references/>

			[[Category:Niels Henrik Abel]]
			[[Category:Functional equations]]

en>Jcccf at 16:59, 14 May 2012

2012-05-14T16:59:40Z

New page

Hi there, I am Alyson Boon even though it is not the name on my beginning certification. I've usually loved living in Alaska. I am presently a travel agent. To climb is some thing I really enjoy performing.<br><br>Look at my blog - psychic chat online ([http://chorokdeul.co.kr/index.php?document_srl=324263&mid=customer21 chorokdeul.co.kr])

← Older revision		Revision as of 22:51, 17 November 2014
Line 1:		Line 1:
	~~{{dablink\|This article~~ is ~~about certain functional equations.~~ ~~For ordinary differential equations that are cubic in the unknown function, see~~ [~~[Abel equation of the first kind]].}}~~		Friends contact her Felicidad and her husband doesn't like it at all. Managing people is auto warranty [http://www.Volvocars.com/us/sales-services/sales/volvo_certified_preowned/Pages/default.aspx extended] car [http://Www.Mpp.com/warranty-plans/used-car-warranties.htm warranty] how I make money and it's some thing I truly enjoy. One of my preferred hobbies is camping and now I'm [http://109.230.254.127/index.php?mod=users&action=view&id=29690 extended car warranty] auto warranty trying to make money with it. Her husband and her selected to reside in Delaware but she needs to transfer because of her family.<br><br>[http://bikedance.com/blogs/post/29704 auto warranty] my web blog ... [http://Www.Planetplayschool.com/ActivityFeed/MyProfile/tabid/61/UserId/415/Default.aspx extended car warranty]

	~~The '''Abel equation''', named after [[Niels Henrik Abel]], is special case of [[functional equation]]s which can be written in the form~~

	:~~<math>f(h(x)) = h(x + 1)\,\!<~~/~~math>~~

	or

	~~:<math>\alpha(f(x))=\alpha(x)+1\!<~~/~~math>~~

	~~and controls the iteration of {{mvar\|f}}~~.

	~~==Equivalence==~~
	~~These equations are equivalent. Assuming that {{mvar\|α}} is an [[invertible function]], the second equation can be written as~~

	~~:<math> \alpha^{-1}(\alpha(f(x)))=\alpha^{-1}(\alpha(x)+1)\,~~ .</~~math>~~

	~~Taking {{math\|''x'' {{=}} ''α''<sup>−1<~~/~~sup>(''y'')}}, the equation can be written as~~

	~~::<math>f(\alpha^{~~-~~1}(y))=\alpha^{-1}(y+1)\, .<~~/~~math>~~

	~~For a function {{math\|''f''(''x'')}} assumed to be known, the task is to solve the functional equation for the function {{math\|''α''<sup>−1<~~/~~sup>}}, possibly satisfying additional requirements, such as {{math\|''α''<sup>−1<~~/~~sup>(0) {{=}} 1}}~~.

	~~The change of variables {{math\|''s''<sup>''α''(''x'')</sup> {{=}} Ψ(''x'')}}, for a real parameter {{mvar\|s}}, brings Abel's equation into the celebrated [[Schröder's equation]~~]~~, {{math\|Ψ(''f''(''x'')) {{=}} ''s'' Ψ(''x'')}} .~~

	~~The further change {{math\|''F''(''x'') {{=}} exp(''s''<sup>''α''(''x'')</sup>)}} into [~~[~~Böttcher's equation]], {{math\|''F''(''f''(''x'')) {{=}} ''F''(''x'')<sup>''s''</sup>}}.~~

	~~==History==~~
	~~Initially, the equation in the more general form~~
	~~<ref name="abel">{{cite journal~~
	~~\| url=~~http://~~gdz~~.~~sub~~.~~uni~~-~~goettingen.de~~/~~ru/dms/load/img/?PPN=PPN243919689_0001&DMDID=dmdlog6~~
	~~\| author= Abel, N~~.H.
	~~\| coauthors=~~
	~~\| title= Untersuchung der Functionen zweier unabhängig veränderlichen Größen x und y, wie f(x, y), welche die Eigenschaft haben, ...~~
	~~\| journal=[~~[~~Journal für die reine und angewandte Mathematik]]~~
	~~\| volume=1~~
	~~\| pages=11–15~~
	~~\| year=1826~~
	~~}}</ref>~~
	~~<ref name="s">{{cite journal~~
	~~\| url=~~http://~~projecteuclid~~.~~org/DPubS~~/~~Repository/1~~.~~0/Disseminate~~?~~handle~~=~~euclid.bams/1183421988~~&view~~=body~~&~~content-type~~=~~pdf_1~~
	~~\| author=A~~. R. ~~Schweitzer~~
	~~\| coauthors=~~
	~~\| title=Theorems on functional equations~~
	~~\| journal=[[Bulletin des Sciences Mathématiques]]~~
	~~\| volume=27~~
	~~\| issue=2~~
	~~\| pages=31~~
	~~\| year=1903~~
	}}<~~/ref~~>
	~~was reported. Then it happens that even in the case of single variable, the equation is not trivial, and requires special analysis~~
	<~~ref name="U1"~~>~~{{cite journal~~
	~~\| url=~~http://~~matwbn.icm.edu~~.pl/~~ksiazki~~/sm/~~sm134/sm13424~~.~~pdf~~
	~~\| author=G~~. ~~Belitskii~~
	~~\| coauthors=Yu~~. ~~Lubish~~
	~~\| title=The real-analytic solutions of the Abel functional equations~~
	~~\| journal=[~~[~~Studia Mathematica]]~~
	~~\| volume=134~~
	~~\| issue=2~~
	~~\| pages=135–141~~
	~~\| year=1999~~
	~~}}</ref><ref name="j">{{cite journal~~
	~~\| journal= Nonlinear Analysis: Hybrid Systems~~
	~~\| volume=1~~
	~~\| issue=1~~
	~~\| year=2007~~
	~~\| pages=95–102~~
	~~\| doi=10.1016/j.nahs.2006.04.002~~
	~~\| author=Jitka Laitochová~~
	~~\| title =Group iteration for Abel’s functional equation }} Studied is the Abel functional equation α(f(x))=α(x)+1</ref>~~

	~~In the case of linear transfer function, the solution can be expressed in compact form~~
	~~<ref name="linear">{{cite journal~~
	~~\| url=~~http://~~matwbn~~.~~icm~~.~~edu.pl~~/~~ksiazki~~/sm/~~sm127~~/~~sm12716.pdf~~
	~~\| author=G. Belitskii~~
	~~\| coauthor=Yu. Lubish~~
	~~\| title=The Abel equation and total solvability of linear functional equtions~~
	~~\| journal=[[Studia Mathematica]]~~
	~~\| volume=127~~
	~~\| year=1998~~
	~~\| pages=81–89~~
	~~}}<~~/~~ref>~~

	~~==Special cases==~~

	~~The equation of [[tetration]] is a special case of Abel's equation, with {{math\|''f'' {{=}} exp}}.~~

	~~In the case of an integer argument, the equation encodes a recurrent procedure, e.g.,~~
	~~:<math>\alpha(f(f(x)))=\alpha(x)+2 ~,<~~/~~math>~~
	~~and so on,~~
	~~:<math>\alpha(f_n(x))=\alpha(x)+n ~~~.~~</math>~~

	~~==See also==~~
	*[[Functional equation]]
	*[[Iterated function]]
	*[[Abel function]]
	*[[Schröder's equation]]
	*[[Böttcher's equation]]
	~~==References==~~
	~~<references/>~~

	~~[[Category:Niels Henrik Abel]]~~
	~~[[Category:Functional equations]~~]