Persymmetric matrix - Revision history

en>Myasuda: /* References */ Created see also section

2014-12-10T05:41:52Z

References: Created see also section

← Older revision		Revision as of 07:41, 10 December 2014
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	~~In [[mathematics]]~~, ~~the '''Fox derivative''' is an [[algebra]]ic construction~~ in the ~~theory of [[free group]]s which bears many similarities~~ to the conventional [[derivative]] of [[calculus]]. The Fox derivative and related concepts are often referred to as the '''Fox calculus''', or (Fox's original term) the '''free differential calculus'''. The Fox derivative was developed in a series of five papers by mathematician [[Ralph Fox]], published in ~~[[Annals of Mathematics]] beginning in 1953~~.		Hi there, I am Yoshiko Villareal but I by no means really favored that name. Some time in the past he selected to reside in Kansas. To perform handball is the thing she enjoys most of all. Interviewing is how he supports his family members but his promotion never arrives.<br><br>my page ... [http://Mwarrenassociates.com/UserProfile/tabid/253/userId/228462/Default.aspx Mwarrenassociates.com]

	~~==Definition==~~
	If ''G'' is a free group with identity element ''e'' and [[generating set of a group\|generators]] ''g<sub>i</sub>'', then the Fox derivative with respect to ''g<sub>i</sub>'' is a function from ''G'' into the [[integral group ring]] '''''Z'''G'' which is denoted <math>\frac{\partial}{\partial g_i}</math>, and obeys the following [[axiom]]s:
	* <math>\frac{\partial}{\partial g_i}(g_j) = \delta_{ij}</math>, where <math>\delta_{ij}</math> is the ~~[[Kronecker delta]]~~
	* <math>\frac{\partial}{\partial g_i}(e) = 0</math>
	* <math>\frac{\partial}{\partial g_i}(uv) = \frac{\partial}{\partial g_i}(u) + u\frac{\partial}{\partial g_i}(v)</math> for any elements ''u'' and ''v'' of ~~''G''~~.
	~~The first two axioms are identical to similar properties of the partial derivative of calculus, and the third~~ is ~~a modified version of the [[product rule]]~~. ~~As a consequence of the axioms, we have the following formula for inverses~~
	* <~~math~~>~~\frac{\partial}{\partial g_i}(u^{-1}) = -u^{-1}\frac{\partial}{\partial g_i}(u)~~<~~/math~~> ~~for any element ''u'' of ''G''~~.

	~~==Applications==~~
	~~The Fox derivative has applications in [[group cohomology]], [[knot theory]] and [[covering space]] theory, among other areas of mathematics~~.

	~~==See also==~~
	* [[Alexander polynomial]]
	* [[Alexander module]]
	* [[Free group]]
	* [[Ring (mathematics)]]
	* [[Integral domain]]

	~~==References==~~
	* {{Cite book \| first=Kenneth S. ~~\| last1=Brown \| authorlink = Kenneth Brown (mathematician) \| title=Cohomology of Groups \| publisher=~~[~~[Springer Verlag]] \| year=1972 \| isbn=0-387-90688-6 \| series=Graduate Texts in Mathematics \| mr=0672956 \| volume=87 \| postscript=<!--None--> }}~~
	*{{cite journal \|last=Fox \|first=Ralph \|authorlink=Ralph Fox \|date=May 1953 \|title=Free Differential Calculus, I: ~~Derivation in the Free Group Ring \|journal=[[Annals of Mathematics]] \|volume=57 \|issue=3 \|pages=547–560 \|doi=10~~.~~2307~~/~~1969736 \|jstor=1969736 \|publisher=Annals of Mathematics \|mr=0053938}}~~
	*{{cite journal \|last=Fox \|first=Ralph \|date=March 1954 \|title=Free Differential Calculus, II: The Isomorphism Problem of Groups \|journal=Annals of Mathematics \|volume=59 \|issue=2 \|pages=196–210 \|doi=10.2307/~~1969686 \|jstor=1969686 \|publisher=Annals of Mathematics \|mr=0062125}}~~
	*{{cite journal \|last=Fox \|first=Ralph \|date=November 1956 \|title=Free Differential Calculus, III: Subgroups \|journal=Annals of Mathematics \|volume=64 \|issue=2 \|pages=407–419 \|doi=10.2307/~~1969592 \|jstor=1969592 \|publisher=Annals of Mathematics \|mr=0095876}}~~
	*{{cite journal \|last=Chen \|first=Kuo-Tsai \|authorlink=Kuo-Tsai Chen \|coauthors=Ralph Fox, [[Roger Lyndon]] \|date=July 1958 \|title=Free Differential Calculus, IV: The Quotient Groups of the Lower Central Series \|journal=Annals of Mathematics \|volume=68 \|issue=1 \|pages=81–95 \|doi=10.~~2307/1970044 \|jstor=1970044 \|publisher=Annals of Mathematics \|mr=0102539}}~~
	*{{cite journal \|last=Fox \|first=Ralph \|date=May 1960 \|title=Free Differential Calculus, V: The Alexander Matrices Re-Examined \|journal=Annals of Mathematics \|volume=71 \|issue=3 \|pages=408–422 \|doi=10.~~2307/1969936 \|jstor=1969936 \|publisher=Annals of Mathematics \|mr=0111781}}~~

	~~[[Category:Geometric topology]]~~
	~~[[Category:Combinatorial group theory]~~]


	~~{{algebra-stub}}~~

en>Addbot: Bot: Migrating 2 interwiki links, now provided by Wikidata on d:q3757134

2013-03-16T22:02:13Z

Bot: Migrating 2 interwiki links, now provided by Wikidata on d:q3757134

← Older revision		Revision as of 00:02, 17 March 2013
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	~~I would like~~ to ~~introduce myself~~ to ~~you~~, ~~I am Andrew and my wife doesn~~'~~t like it at all~~. ~~For many years he~~'~~s been living in Alaska~~ and ~~he doesn~~'~~t plan on changing it. Office supervising~~ is ~~what she does~~ for ~~a living~~. ~~What me~~ and ~~my family love~~ is ~~to climb but I'm thinking on beginning some thing new~~.<br><br>~~Also visit my webpage~~: [~~http~~://~~www~~.~~onbizin~~.~~co.kr~~/~~xe/?document_srl~~=~~320614 psychics online~~]		In [[mathematics]], the '''Fox derivative''' is an [[algebra]]ic construction in the theory of [[free group]]s which bears many similarities to the conventional [[derivative]] of [[calculus]]. The Fox derivative and related concepts are often referred to as the '''Fox calculus''', or (Fox's original term) the '''free differential calculus'''. The Fox derivative was developed in a series of five papers by mathematician [[Ralph Fox]], published in [[Annals of Mathematics]] beginning in 1953.

			==Definition==
			If ''G'' is a free group with identity element ''e'' and [[generating set of a group\|generators]] ''g<sub>i</sub>'', then the Fox derivative with respect to ''g<sub>i</sub>'' is a function from ''G'' into the [[integral group ring]] '''''Z'''G'' which is denoted <math>\frac{\partial}{\partial g_i}</math>, and obeys the following [[axiom]]s:
			* <math>\frac{\partial}{\partial g_i}(g_j) = \delta_{ij}</math>, where <math>\delta_{ij}</math> is the [[Kronecker delta]]
			* <math>\frac{\partial}{\partial g_i}(e) = 0</math>
			* <math>\frac{\partial}{\partial g_i}(uv) = \frac{\partial}{\partial g_i}(u) + u\frac{\partial}{\partial g_i}(v)</math> for any elements ''u'' and ''v'' of ''G''.
			The first two axioms are identical to similar properties of the partial derivative of calculus, and the third is a modified version of the [[product rule]]. As a consequence of the axioms, we have the following formula for inverses
			* <math>\frac{\partial}{\partial g_i}(u^{-1}) = -u^{-1}\frac{\partial}{\partial g_i}(u)</math> for any element ''u'' of ''G''.

			==Applications==
			The Fox derivative has applications in [[group cohomology]], [[knot theory]] and [[covering space]] theory, among other areas of mathematics.

			==See also==
			* [[Alexander polynomial]]
			* [[Alexander module]]
			* [[Free group]]
			* [[Ring (mathematics)]]
			* [[Integral domain]]

			==References==
			* {{Cite book \| first=Kenneth S. \| last1=Brown \| authorlink = Kenneth Brown (mathematician) \| title=Cohomology of Groups \| publisher=[[Springer Verlag]] \| year=1972 \| isbn=0-387-90688-6 \| series=Graduate Texts in Mathematics \| mr=0672956 \| volume=87 \| postscript=<!--None--> }}
			*{{cite journal \|last=Fox \|first=Ralph \|authorlink=Ralph Fox \|date=May 1953 \|title=Free Differential Calculus, I: Derivation in the Free Group Ring \|journal=[[Annals of Mathematics]] \|volume=57 \|issue=3 \|pages=547–560 \|doi=10.2307/1969736 \|jstor=1969736 \|publisher=Annals of Mathematics \|mr=0053938}}
			*{{cite journal \|last=Fox \|first=Ralph \|date=March 1954 \|title=Free Differential Calculus, II: The Isomorphism Problem of Groups \|journal=Annals of Mathematics \|volume=59 \|issue=2 \|pages=196–210 \|doi=10.2307/1969686 \|jstor=1969686 \|publisher=Annals of Mathematics \|mr=0062125}}
			*{{cite journal \|last=Fox \|first=Ralph \|date=November 1956 \|title=Free Differential Calculus, III: Subgroups \|journal=Annals of Mathematics \|volume=64 \|issue=2 \|pages=407–419 \|doi=10.2307/1969592 \|jstor=1969592 \|publisher=Annals of Mathematics \|mr=0095876}}
			*{{cite journal \|last=Chen \|first=Kuo-Tsai \|authorlink=Kuo-Tsai Chen \|coauthors=Ralph Fox, [[Roger Lyndon]] \|date=July 1958 \|title=Free Differential Calculus, IV: The Quotient Groups of the Lower Central Series \|journal=Annals of Mathematics \|volume=68 \|issue=1 \|pages=81–95 \|doi=10.2307/1970044 \|jstor=1970044 \|publisher=Annals of Mathematics \|mr=0102539}}
			*{{cite journal \|last=Fox \|first=Ralph \|date=May 1960 \|title=Free Differential Calculus, V: The Alexander Matrices Re-Examined \|journal=Annals of Mathematics \|volume=71 \|issue=3 \|pages=408–422 \|doi=10.2307/1969936 \|jstor=1969936 \|publisher=Annals of Mathematics \|mr=0111781}}

			[[Category:Geometric topology]]
			[[Category:Combinatorial group theory]]


			{{algebra-stub}}

en>Yobot: WP:CHECKWIKI error 22 fixes (category with space) + general fixes using AWB (7799)

2011-07-29T11:31:20Z

WP:CHECKWIKI error 22 fixes (category with space) + general fixes using AWB (7799)

New page

I would like to introduce myself to you, I am Andrew and my wife doesn't like it at all. For many years he's been living in Alaska and he doesn't plan on changing it. Office supervising is what she does for a living. What me and my family love is to climb but I'm thinking on beginning some thing new.<br><br>Also visit my webpage: [http://www.onbizin.co.kr/xe/?document_srl=320614 psychics online]