Numerical differentiation - Revision history

en>Wiki Kaczor: Fixed h calculation, so that it works for small x.

2015-01-04T15:16:12Z

Fixed h calculation, so that it works for small x.

← Older revision		Revision as of 17:16, 4 January 2015
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	~~In [[mathematics]], a '''pseudogroup'''~~ is ~~an extension of the [[group (mathematics)\|group]] concept,~~ but one that grew out of the geometric approach of [[Sophus Lie]], rather than out of [[abstract algebra]] (such as [[quasigroup]], for example). A theory of pseudogroups was developed by [[Élie Cartan]] in the early 1900s.<ref>{{cite journal\|first = Élie\|last = Cartan\|title = Sur la structure des groupes infinis de transformations\|journal = Annales Scientifiques de l'École Normale Supérieure\|year = 1904\|volume = 21\|pages=153–206\|url=http://archive.numdam.org/article/ASENS_1904_3_21__153_0.pdf}}</ref><ref>{{cite journal\|first = Élie\|last = Cartan\|title = Les groupes de transformations continus, infinis, simples\|journal = Annales Scientifiques de l'École Normale Supérieure\|year = 1909\|volume = 26\|pages=93–161\|url=http://archive.numdam.org/article/ASENS_1909_3_26__93_0.pdf}}</ref>		The writer is called Irwin Wunder but it's not the most masucline name out there. South Dakota is exactly where I've always been residing. One of the very best issues in the world for him is to gather badges but he is struggling to discover time for it. Bookkeeping is what I do.<br><br>Here is my web site ... [http://biolab.snu.ac.kr/?document_srl=1451338 home std test]

	~~It is~~ not an axiomatic algebraic idea; rather it defines a set of closure conditions on sets of [[homeomorphism]]s defined on [[open set]]s ''U'' of a given [[Euclidean space]] ''E'' or more generally of a fixed [[topological space]] ''S''. The [[groupoid]] condition on those is fulfilled, in that homeomorphisms

	~~:''h'':''U'' → ''V''~~

	~~and~~

	~~:''g'':''V'' → ''W''~~

	~~compose to a ''homeomorphism'' from ''U'' to ''W''. The further requirement on a pseudogroup is related to~~ the ~~possibility of ''patching'' (in the sense of [[descent (category theory)\|descent]], [[transition function]]s, or a [[gluing axiom]])~~.

	~~Specifically, a '''pseudogroup''' on a topological space ''S''~~ is ~~a collection ''Γ'' of homeomorphisms between open subsets of ''S'' satisfying the following properties.<ref>Kobayashi, Shoshichi & Nomizu, Katsumi. ''Foundations of Differential Geometry, Volume~~ I''. ~~Wiley Classics Library. John Wiley & Sons Inc., New York, 1996. Reprint~~ of the ~~1969 original, A Wiley-Interscience Publication. ISBN 0-471-15733-3.</ref>~~
	* For every open set ''U'' in ~~''S'',~~ the ~~identity map on ''U''~~ is ~~in ''Γ''.~~
	* If ''f'' is in ''Γ'', then so is ''f <sup>−1</sup>''.
	* If ''f'' is in ''Γ'', then the restriction of ''f'' to ~~an arbitrary open subset of its domain~~ is ~~in ''Γ''.~~
	* If ''U'' is open in ''S'', ''U'' is the union of the open sets ''{ U<sub>i</sub> }'', ''f'' is a homeomorphism from ''U'' to ~~an open subset of ''S'', and the restriction of ''f'' to ''U<sub>i</sub>'' is in ''Γ''~~ for ~~all ''i'', then ''f''~~ is ~~in ''Γ''~~.
	* If ''f'':''U'' → ''V'' and ''f ′'':''U ′'' → ''V ′'' are in ''Γ'', and the [[Intersection (set theory)\|intersection]] ''V ∩ U ′'' is not empty, then the following restricted composition is in ''Γ'':
	:<~~math~~>~~f' \circ f \colon f^{-1}(V \cap U') \to f'(V \cap U')~~<~~/math~~>.

	~~An example in space of two dimensions~~ is ~~the pseudogroup of invertible [[holomorphic function]]s of a [[complex variable]] (invertible in the sense of having an [[inverse function]])~~. ~~The properties of this pseudogroup are what makes it possible to define [[Riemann surface]]s by local data patched together~~.

	~~In general, pseudogroups were studied as a possible theory of [[infinite dimensional Lie group]]s~~. ~~The concept of a '''local Lie group''', namely a pseudogroup of functions defined in neighbourhoods of the origin of ''E'', is actually closer to Lie's original concept of~~ [[Lie group]], in the case where the transformations involved depend on a finite number of [[parameter]]s, than the contemporary definition via [[manifold]]s. One of Cartan's achievements was to clarify the points involved, including the point that a local Lie group always gives rise to a ''global'' group, in the current sense (an analogue of [[Lie's third theorem]], on [[Lie algebra]]s determining a group). The [[formal group]] is yet another approach to the specification of Lie groups, infinitesimally. It is known, however, that ''local [[topological group]]s'' do not necessarily have global counterparts.

	Examples of infinite-dimensional pseudogroups abound, beginning with the pseudogroup of all [[diffeomorphism]]s of ''E''. The interest is mainly in sub-pseudogroups of the diffeomorphisms, and therefore with objects that have a Lie algebra analogue of [[vector field]]s. Methods proposed by Lie and by Cartan for studying these objects have become more practical given the progress of [[computer algebra]].

	~~In the 1950s Cartan's theory was reformulated by [[Shiing-Shen Chern]], and a general [[deformation theory]] for pseudogroups was developed by [[Kunihiko Kodaira]] and [[D~~. C. Spencer]]. In the 1960s [[homological algebra]] was applied to the basic [[partial differential equation\|PDE]] questions involved, of over-determination; this though revealed that the algebra of the theory is potentially very heavy. ~~In the same decade the interest for [[theoretical physics]] of infinite-dimensional Lie theory appeared for the first time, in the shape of [[current algebra]].~~

	~~== References ==~~
	~~<references/>~~
	*{{cite journal \| author=St. Golab \| title=Über den Begriff der "Pseudogruppe von Transformationen" \| journal=Mathematische Annalen \| year=1939 \| volume=116 \| pages=768–780 \| doi=10.1007/~~BF01597390}}~~

	=~~= External links ==~~
	*{{springer\|id=p/p075710\|title=Pseudo-groups\|author=Alekseevskii, D.V.}}

	~~[[Category:Lie groups]]~~
	~~[[Category:Algebraic structures]]~~
	~~[[Category:Differential geometry]]~~
	~~[[Category:Differential topology]~~]

en>Monkbot: /* Complex variable methods */Fix CS1 deprecated date parameter errors

2014-01-27T19:41:35Z

Complex variable methods: Fix CS1 deprecated date parameter errors

← Older revision		Revision as of 21:41, 27 January 2014
Line 1:		Line 1:
	The ~~name~~ of the ~~author~~ is ~~Numbers~~. ~~To perform baseball is~~ the ~~hobby he will by no means quit performing~~. For ~~many years he~~'~~s been working as a receptionist~~. ~~North Dakota~~ is ~~where me~~ and ~~my husband live.~~<br><br>~~Check out my web blog~~: [~~http~~:/~~/360vc~~.~~myblackicoffee~~.~~com~~/~~space~~.~~php?uid~~=~~204740~~&do=~~blog&~~id=~~183877 home std test kit~~]		In [[mathematics]], a '''pseudogroup''' is an extension of the [[group (mathematics)\|group]] concept, but one that grew out of the geometric approach of [[Sophus Lie]], rather than out of [[abstract algebra]] (such as [[quasigroup]], for example). A theory of pseudogroups was developed by [[Élie Cartan]] in the early 1900s.<ref>{{cite journal\|first = Élie\|last = Cartan\|title = Sur la structure des groupes infinis de transformations\|journal = Annales Scientifiques de l'École Normale Supérieure\|year = 1904\|volume = 21\|pages=153–206\|url=http://archive.numdam.org/article/ASENS_1904_3_21__153_0.pdf}}</ref><ref>{{cite journal\|first = Élie\|last = Cartan\|title = Les groupes de transformations continus, infinis, simples\|journal = Annales Scientifiques de l'École Normale Supérieure\|year = 1909\|volume = 26\|pages=93–161\|url=http://archive.numdam.org/article/ASENS_1909_3_26__93_0.pdf}}</ref>

			It is not an axiomatic algebraic idea; rather it defines a set of closure conditions on sets of [[homeomorphism]]s defined on [[open set]]s ''U'' of a given [[Euclidean space]] ''E'' or more generally of a fixed [[topological space]] ''S''. The [[groupoid]] condition on those is fulfilled, in that homeomorphisms

			:''h'':''U'' → ''V''

			and

			:''g'':''V'' → ''W''

			compose to a ''homeomorphism'' from ''U'' to ''W''. The further requirement on a pseudogroup is related to the possibility of ''patching'' (in the sense of [[descent (category theory)\|descent]], [[transition function]]s, or a [[gluing axiom]]).

			Specifically, a '''pseudogroup''' on a topological space ''S'' is a collection ''Γ'' of homeomorphisms between open subsets of ''S'' satisfying the following properties.<ref>Kobayashi, Shoshichi & Nomizu, Katsumi. ''Foundations of Differential Geometry, Volume I''. Wiley Classics Library. John Wiley & Sons Inc., New York, 1996. Reprint of the 1969 original, A Wiley-Interscience Publication. ISBN 0-471-15733-3.</ref>
			* For every open set ''U'' in ''S'', the identity map on ''U'' is in ''Γ''.
			* If ''f'' is in ''Γ'', then so is ''f <sup>−1</sup>''.
			* If ''f'' is in ''Γ'', then the restriction of ''f'' to an arbitrary open subset of its domain is in ''Γ''.
			* If ''U'' is open in ''S'', ''U'' is the union of the open sets ''{ U<sub>i</sub> }'', ''f'' is a homeomorphism from ''U'' to an open subset of ''S'', and the restriction of ''f'' to ''U<sub>i</sub>'' is in ''Γ'' for all ''i'', then ''f'' is in ''Γ''.
			* If ''f'':''U'' → ''V'' and ''f ′'':''U ′'' → ''V ′'' are in ''Γ'', and the [[Intersection (set theory)\|intersection]] ''V ∩ U ′'' is not empty, then the following restricted composition is in ''Γ'':
			:<math>f' \circ f \colon f^{-1}(V \cap U') \to f'(V \cap U')</math>.

			An example in space of two dimensions is the pseudogroup of invertible [[holomorphic function]]s of a [[complex variable]] (invertible in the sense of having an [[inverse function]]). The properties of this pseudogroup are what makes it possible to define [[Riemann surface]]s by local data patched together.

			In general, pseudogroups were studied as a possible theory of [[infinite dimensional Lie group]]s. The concept of a '''local Lie group''', namely a pseudogroup of functions defined in neighbourhoods of the origin of ''E'', is actually closer to Lie's original concept of [[Lie group]], in the case where the transformations involved depend on a finite number of [[parameter]]s, than the contemporary definition via [[manifold]]s. One of Cartan's achievements was to clarify the points involved, including the point that a local Lie group always gives rise to a ''global'' group, in the current sense (an analogue of [[Lie's third theorem]], on [[Lie algebra]]s determining a group). The [[formal group]] is yet another approach to the specification of Lie groups, infinitesimally. It is known, however, that ''local [[topological group]]s'' do not necessarily have global counterparts.

			Examples of infinite-dimensional pseudogroups abound, beginning with the pseudogroup of all [[diffeomorphism]]s of ''E''. The interest is mainly in sub-pseudogroups of the diffeomorphisms, and therefore with objects that have a Lie algebra analogue of [[vector field]]s. Methods proposed by Lie and by Cartan for studying these objects have become more practical given the progress of [[computer algebra]].

			In the 1950s Cartan's theory was reformulated by [[Shiing-Shen Chern]], and a general [[deformation theory]] for pseudogroups was developed by [[Kunihiko Kodaira]] and [[D. C. Spencer]]. In the 1960s [[homological algebra]] was applied to the basic [[partial differential equation\|PDE]] questions involved, of over-determination; this though revealed that the algebra of the theory is potentially very heavy. In the same decade the interest for [[theoretical physics]] of infinite-dimensional Lie theory appeared for the first time, in the shape of [[current algebra]].

			== References ==
			<references/>
			*{{cite journal \| author=St. Golab \| title=Über den Begriff der "Pseudogruppe von Transformationen" \| journal=Mathematische Annalen \| year=1939 \| volume=116 \| pages=768–780 \| doi=10.1007/BF01597390}}

			== External links ==
			*{{springer\|id=p/p075710\|title=Pseudo-groups\|author=Alekseevskii, D.V.}}

			[[Category:Lie groups]]
			[[Category:Algebraic structures]]
			[[Category:Differential geometry]]
			[[Category:Differential topology]]

en>Timholy: Fix bug in computation of step size

2012-08-23T10:02:14Z

Fix bug in computation of step size

New page

The name of the author is Numbers. To perform baseball is the hobby he will by no means quit performing. For many years he's been working as a receptionist. North Dakota is where me and my husband live.<br><br>Check out my web blog: [http://360vc.myblackicoffee.com/space.php?uid=204740&do=blog&id=183877 home std test kit]