Atwood machine - Revision history

en>Thnidu: /* Equation for constant acceleration */

2014-12-30T22:09:33Z

Equation for constant acceleration

← Older revision		Revision as of 00:09, 31 December 2014
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	~~In [[mathematics]],~~ the ~~'''compact~~-~~open topology''' is a [[topological space\|topology]] defined on~~ the ~~set~~ of ~~[[continuous function\|continuous maps]] between two [[topological space]]s. The compact-open topology is one~~ of the commonly used topologies on [[function space]]s, and is applied in [[homotopy theory]] and [[functional analysis]]. It was introduced by [[Ralph Fox]] in 1945 [http://www.ams.org/journals/bull/1945-51-06/S0002-9904-1945-08370-0/].		Our world is driven by existing plus demand. My husband and i shall examine the Greek-Roman model. Consuming additional care to highlight the role of clash of clans hack tool no survey form within the vast perspective which usually this provides.<br><br>

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	~~Let ''X'' and ''Y'' be two [[topological space]]s, and let ''C''(''X'',''Y'') denote~~ the ~~set of all [[continuous map]]s between ''X'' and ''Y''. Given~~ a ~~[[compact set\|compact subset]] ''K''~~ of ~~''X'' and an [[open set\|open subset]] ''U''~~ of ~~''Y''~~, ~~let~~ '~~'V''(''K'',''U'') denote~~ the ~~set~~ of ~~all functions {{nowrap\|1=&fnof; ∈ ''C''(''X'',''Y'')}} such that {{nowrap\|1=&fnof;(''K'') ⊂ ''U''~~.~~}} Then~~ the ~~collection of all such ''V''(''K'',''U'') is a [[subbase]]~~ for the ~~compact-open topology on ''C''(''X'',''Y'')~~. ~~(This collection does not always form a [[base (topology)\|base]] for a topology on ''C''(''X''~~,'~~'Y'').)~~

	~~When working in~~ the category of [[compactly generated space]]s, it is common to modify this definition by restricting to the subbase formed from those ''K'' which are the image of a [[compact set\|compact]] [[Hausdorff space\|Hausdorff]] space. ~~Of course, if ''X'' is compactly generated~~ and ~~Hausdorff, this definition coincides with~~ the ~~previous one~~. ~~However, the modified definition is crucial if~~ one ~~wants the convenient category of [[Weak Hausdorff space\|compactly generated weak Hausdorff]] spaces to~~ be ~~Cartesian closed~~, ~~among other useful properties~~.<~~ref~~>~~{{cite journal \|jstor=1995173 \|title=Classifying Spaces and Infinite Symmetric Products}}~~<~~/ref~~>~~<ref>{{cite web \|url=~~http://www.~~math~~.~~uchicago~~.~~edu~~/~~~may/CONCISE/ConciseRevised.pdf \|title~~=~~A Concise Course in Algebraic Topology}}</ref><ref>{{cite web \|url~~=http://~~neil-strickland.staff.shef.ac.uk/courses/homotopy/cgwh~~.~~pdf \|title=Compactly Generated Spaces}}</ref> The confusion between this definition and the one above is caused by differing usage~~ of ~~the word [[compact set\|compact]~~].

	~~== Properties ==~~

	* If '''' is a ~~one-point space then one can identify ''C''('''',''X'') with ''X'', and under this identification~~ the ~~compact-open topology agrees with the topology on ''X''~~

	* If ''Y'' is [[T0 space\|''T''<sub>0</sub>]], [[T1 space\|''T''<sub>1</sub>]], [[Hausdorff space\|Hausdorff]], [[regular space\|regular]], or [[tychonoff space\|Tychonoff]], ~~then~~ the ~~compact-open topology has the corresponding [[separation axiom]]~~.

	* If ''X'' is Hausdorff and ''S'' is a ~~[[subbase]] for ''Y''~~, ~~then the collection {''V''(''K'',''U'') : ''U'' in ''S''}~~ is a ~~[[subbase]] for~~ the ~~compact~~-~~open topology on ''C''(''X'',''Y'').~~

	* If ''Y'' is a ~~[[metric space]] (or more generally, an [[uniform space]]), then the compact~~-~~open topology is equal to~~ the ~~[[topology of compact convergence]]. In other words, if ''Y'' is~~ a ~~metric space~~, ~~then~~ a ~~[[sequence]] {ƒ<sub>''n''</sub>} [[limit (mathematics)\|converge]]s~~ to ~~ƒ in the compact-open topology if and only if for every compact subset ''K'' of ''X'', {ƒ~~<~~sub~~>~~''n''~~<~~/sub~~>~~} converges uniformly to ƒ on ''K''. In particular~~, ~~if ''X'' is compact~~ and '~~'Y'' is a uniform space~~, ~~then~~ the ~~compact-open topology is equal~~ to ~~the topology of [[uniform convergence]]~~.

	* If ''X'', ''Y'' and ''Z'' are topological spaces, with ''Y'' [[locally compact Hausdorff]] (or even just locally compact [[preregular space\|preregular]]), then the [[function composition\|composition map]] {{nowrap\|1=''C''(''Y'',''Z'') × ''C''(''X'',''Y'') → ''C''(''X'',''Z''),}} given by {{nowrap\|1=(&fnof;,''g'') ↦ &fnof; ∘ ''g'',}} is ~~continuous (here all~~ the ~~function spaces are given~~ the ~~compact-open topology~~ and ~~''C''(''Y'',''Z'') × ''C''(''X'',''Y'') is given~~ the ~~[[product topology]]).~~

	*If ''Y'' is a locally compact Hausdorff (or preregular) space, then the evaluation map ''e'' : ''C''(''Y'',''Z'') × ''Y'' → ''Z'', defined by ''e''(ƒ,''x'') = ƒ(''x''), is continuous. This can be seen as a special case of the ~~above where ''X'' is a one-point space~~.

	* If ''X'' is compact, and if ''Y'' is a metric space with metric ''d'', then the compact-open topology on ''C''(''X'',''Y'') is [[metrisable space\|metrisable]], and a metric for it ~~is given by ''e''(ƒ~~,~~''g'') = [[supremum\|sup]]{''d''(ƒ(''x''), ''g''(''x'')) : ''x'' in ''X''}, for ƒ, ''g'' in ''C''(''X'',''Y'')~~.

	~~== Fréchet differentiable functions ==~~
	~~Let ''X'' and ''Y'' be two [[Banach space]]s defined on the same field, and let~~ <~~math~~>~~\mathcal{C}^m\left(U,Y\right)~~ <~~/math~~> ~~denote the set~~ of ~~all m-continuously Fréchet-differentiable functions from the open subset <math>U\subseteq X</math> to <math>Y</math>~~. ~~The compact-open topology~~ is the ~~[[initial topology]] induced by~~ the ~~[[seminorm]]s~~
	~~:<math>p_{K}\left( f \right) = \sup \{ \\| D^{j}f\left( x \right)\\|~~, ~~x\in K~~, ~~0\leq j \leq m \}</math>~~
	~~where <math>D^{0}f\left( x \right) = f\left( x \right)</math>, for each compact subset <math>K\subseteq U</math>~~.

	~~==See also==~~
	* [[Bounded-open topology]]

	~~==References==~~
	~~{{Reflist}}~~
	*{{Cite book\|first=J.\|last=Dugundji\|authorlink=James Dugundji\|title=Topology\|publisher=Allyn and ~~Becon\|year=1966\|asin=B000KWE22K}}~~
	* O.Ya. Viro, O.A. Ivanov, ~~V.M. Kharlamov~~ and ~~N.Yu. Netsvetaev (2007) [http://www.math.uu.se/~oleg/topoman.html Textbook in Problems on Elementary Topology]~~.
	*{{planetmath reference\|id=3976\|title=Compact-open topology}}

	~~[[Category:General topology]]~~
	~~[[Category:Topology of function spaces]]~~

87.198.53.172: /* Equation for tension */

2013-11-08T15:54:06Z

Equation for tension

← Older revision		Revision as of 17:54, 8 November 2013
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	~~Gabrielle Straub~~ is ~~what~~ the ~~person can call me~~ ~~although it~~'s ~~not~~ the ~~quite~~ a ~~few feminine~~ of ~~names~~. ~~Fish keeping~~ is ~~what I make every week~~. ~~Managing people~~ is ~~my time job now~~. ~~My house~~ is ~~now wearing South Carolina~~. Go to my web ~~to find out more~~: http://~~prometeu~~.~~net~~<br><br>~~my homepage~~ :: [http://~~prometeu~~.~~net Clash~~ of ~~clans hack Download no Survey~~]		In [[mathematics]], the '''compact-open topology''' is a [[topological space\|topology]] defined on the set of [[continuous function\|continuous maps]] between two [[topological space]]s. The compact-open topology is one of the commonly used topologies on [[function space]]s, and is applied in [[homotopy theory]] and [[functional analysis]]. It was introduced by [[Ralph Fox]] in 1945 [http://www.ams.org/journals/bull/1945-51-06/S0002-9904-1945-08370-0/].

			== Definition ==

			Let ''X'' and ''Y'' be two [[topological space]]s, and let ''C''(''X'',''Y'') denote the set of all [[continuous map]]s between ''X'' and ''Y''. Given a [[compact set\|compact subset]] ''K'' of ''X'' and an [[open set\|open subset]] ''U'' of ''Y'', let ''V''(''K'',''U'') denote the set of all functions {{nowrap\|1=&fnof; ∈ ''C''(''X'',''Y'')}} such that {{nowrap\|1=&fnof;(''K'') ⊂ ''U''.}} Then the collection of all such ''V''(''K'',''U'') is a [[subbase]] for the compact-open topology on ''C''(''X'',''Y''). (This collection does not always form a [[base (topology)\|base]] for a topology on ''C''(''X'',''Y'').)

			When working in the category of [[compactly generated space]]s, it is common to modify this definition by restricting to the subbase formed from those ''K'' which are the image of a [[compact set\|compact]] [[Hausdorff space\|Hausdorff]] space. Of course, if ''X'' is compactly generated and Hausdorff, this definition coincides with the previous one. However, the modified definition is crucial if one wants the convenient category of [[Weak Hausdorff space\|compactly generated weak Hausdorff]] spaces to be Cartesian closed, among other useful properties.<ref>{{cite journal \|jstor=1995173 \|title=Classifying Spaces and Infinite Symmetric Products}}</ref><ref>{{cite web \|url=http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf \|title=A Concise Course in Algebraic Topology}}</ref><ref>{{cite web \|url=http://neil-strickland.staff.shef.ac.uk/courses/homotopy/cgwh.pdf \|title=Compactly Generated Spaces}}</ref> The confusion between this definition and the one above is caused by differing usage of the word [[compact set\|compact]].

			== Properties ==

			* If '''' is a one-point space then one can identify ''C''('''',''X'') with ''X'', and under this identification the compact-open topology agrees with the topology on ''X''

			* If ''Y'' is [[T0 space\|''T''<sub>0</sub>]], [[T1 space\|''T''<sub>1</sub>]], [[Hausdorff space\|Hausdorff]], [[regular space\|regular]], or [[tychonoff space\|Tychonoff]], then the compact-open topology has the corresponding [[separation axiom]].

			* If ''X'' is Hausdorff and ''S'' is a [[subbase]] for ''Y'', then the collection {''V''(''K'',''U'') : ''U'' in ''S''} is a [[subbase]] for the compact-open topology on ''C''(''X'',''Y'').

			* If ''Y'' is a [[metric space]] (or more generally, an [[uniform space]]), then the compact-open topology is equal to the [[topology of compact convergence]]. In other words, if ''Y'' is a metric space, then a [[sequence]] {ƒ<sub>''n''</sub>} [[limit (mathematics)\|converge]]s to ƒ in the compact-open topology if and only if for every compact subset ''K'' of ''X'', {ƒ<sub>''n''</sub>} converges uniformly to ƒ on ''K''. In particular, if ''X'' is compact and ''Y'' is a uniform space, then the compact-open topology is equal to the topology of [[uniform convergence]].

			* If ''X'', ''Y'' and ''Z'' are topological spaces, with ''Y'' [[locally compact Hausdorff]] (or even just locally compact [[preregular space\|preregular]]), then the [[function composition\|composition map]] {{nowrap\|1=''C''(''Y'',''Z'') × ''C''(''X'',''Y'') → ''C''(''X'',''Z''),}} given by {{nowrap\|1=(&fnof;,''g'') ↦ &fnof; ∘ ''g'',}} is continuous (here all the function spaces are given the compact-open topology and ''C''(''Y'',''Z'') × ''C''(''X'',''Y'') is given the [[product topology]]).

			*If ''Y'' is a locally compact Hausdorff (or preregular) space, then the evaluation map ''e'' : ''C''(''Y'',''Z'') × ''Y'' → ''Z'', defined by ''e''(ƒ,''x'') = ƒ(''x''), is continuous. This can be seen as a special case of the above where ''X'' is a one-point space.

			* If ''X'' is compact, and if ''Y'' is a metric space with metric ''d'', then the compact-open topology on ''C''(''X'',''Y'') is [[metrisable space\|metrisable]], and a metric for it is given by ''e''(ƒ,''g'') = [[supremum\|sup]]{''d''(ƒ(''x''), ''g''(''x'')) : ''x'' in ''X''}, for ƒ, ''g'' in ''C''(''X'',''Y'').

			== Fréchet differentiable functions ==
			Let ''X'' and ''Y'' be two [[Banach space]]s defined on the same field, and let <math>\mathcal{C}^m\left(U,Y\right) </math> denote the set of all m-continuously Fréchet-differentiable functions from the open subset <math>U\subseteq X</math> to <math>Y</math>. The compact-open topology is the [[initial topology]] induced by the [[seminorm]]s
			:<math>p_{K}\left( f \right) = \sup \{ \\| D^{j}f\left( x \right)\\|, x\in K, 0\leq j \leq m \}</math>
			where <math>D^{0}f\left( x \right) = f\left( x \right)</math>, for each compact subset <math>K\subseteq U</math>.

			==See also==
			* [[Bounded-open topology]]

			==References==
			{{Reflist}}
			*{{Cite book\|first=J.\|last=Dugundji\|authorlink=James Dugundji\|title=Topology\|publisher=Allyn and Becon\|year=1966\|asin=B000KWE22K}}
			* O.Ya. Viro, O.A. Ivanov, V.M. Kharlamov and N.Yu. Netsvetaev (2007) [http://www.math.uu.se/~oleg/topoman.html Textbook in Problems on Elementary Topology].
			*{{planetmath reference\|id=3976\|title=Compact-open topology}}

			[[Category:General topology]]
			[[Category:Topology of function spaces]]

98.113.204.141: /* Equations for a pulley with inertia and friction */

2012-06-18T05:08:05Z

Equations for a pulley with inertia and friction

New page

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