Operator (physics) - Revision history

en>Hhhippo: Reverted to revision 637181857 by 68.98.134.239 (talk): Rm duplicate reference. (TW)

2014-12-09T18:28:58Z

Reverted to revision 637181857 by 68.98.134.239 (talk): Rm duplicate reference. (TW)

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2601:0:B680:2D2:7471:2F77:E95A:9E14: /* Table of QM operators */

2014-02-24T05:18:03Z

Table of QM operators

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	~~In the branch of [[mathematics]] known as [[topology]], the '''topologist's sine curve''' is a [[topological space]] with several interesting properties that make it an important textbook example.~~

	It can be defined as the [[graph of a function\|graph]] of the function sin(1/''x'') on the half-open interval (0, 1], together with the origin, under the topology [[subspace topology\|induced]] from the [[Euclidean plane]]:

	~~:<math> T = \left\{ \left( x, \sin \frac{1}{x} \right ) : x \in (0,1] \right\} \cup \{(0,0)\}. </math>~~		Adrianne is what you is able to call me but We all don't like when guys and women use my full user name. What I love doing are to play croquet and furthermore now I have your time to take on issues. The job I've been occupying suitable for years is an choose clerk. My husband and [http://Www.alexa.com/search?q=I+decided&r=topsites_index&p=bigtop I decided] to go with to reside in Guam but I will attain to move in a suitable year or two. You does find my website here: http://prometeu.net<br><br>Feel free to visit my blog: [http://prometeu.net clash of clans hack no download]

	~~==Image of the curve==~~
	~~[[Image:Topologist's sine curve.svg\|420px\|Topologist's Sine Curve]]~~

	~~As ''x'' approaches zero from the right, the magnitude of the rate of change of 1/''x'' increases. This~~ is ~~why the frequency of the sine wave increases as one moves~~ to ~~the left in the graph.~~

	~~==Properties==~~
	~~The topologist~~'~~s sine curve ''T'' is [[connected space\|connected]] but neither [[locally connected space\|locally connected]] nor [[connected space#Path connectedness\|path connected]]~~. ~~This is because it includes the point (0,0) but there is no way~~ to ~~link the function~~ to ~~the origin so as to make a [[path (topology)\|path]]~~.

	The ~~space ''T'~~' is ~~the continuous image of a~~ [~~[locally compact]] space (namely, let ''V'' be the space {−1} ∪ (0, 1<nowiki>]</nowiki>, and use the map ''f'' from ''V'' to ''T'' defined by <span style="white-space~~: ~~nowrap">''f''(−1)<~~/~~span> = (0,0) and <span style="white-space: nowrap">''f''(''x'')<~~/~~span> = <span style="white-space: nowrap">(''x'', sin(1/''x''))</span> for ''x'' > 0), but ''T'' is not locally compact itself~~.

	~~The [[topological dimension]] of ''T'' is 1~~.

	==~~Variants~~==
	~~Two variants of the topologist's sine curve have other interesting properties.~~

	~~The '''closed topologist's sine curve''' can be defined by taking the topologist's sine curve and adding its set of [[limit point~~]~~]s, <math>\{(0,y)\mid y\~~in~~[-1,1]\}</math>. This space is closed and bounded and so [[compact space\|compact]] by the [[Heine–Borel theorem]],~~ but ~~has similar properties~~ to ~~the topologist's sine curve—it too is connected but neither locally connected nor path-connected~~.

	~~The '''extended topologist's sine curve''' can be defined by taking the closed topologist's sine curve and adding to it the set~~ <~~math~~>~~\{(x,1) \mid x\in[0,1]\}~~<~~/math~~>~~. It is~~ [~~[arc connected]] but not [[Locally connected space\|locally connected]].~~

	~~==References==~~
	*{{Citation \| last1=Steen \| first1=Lynn Arthur \| author1-link=Lynn Arthur Steen \| last2=Seebach \| first2=J. Arthur Jr. \| author2-link=J. Arthur Seebach, Jr. \| title=[[Counterexamples in Topology]] \| origyear=1978 \| publisher=Dover Publications, Inc. ~~\| location=Mineola, NY \| edition=[[Dover Publications\|Dover]] reprint~~ of ~~1978 \| isbn=978-0-486-68735-3 \| id={{MathSciNet\|id=1382863}} \| year=1995 \| pages=137–138}}~~
	*{{mathworld\|urlname=TopologistsSineCurve\|title=Topologist's Sine Curve}}

	~~[[Category:Topological spaces]~~]

14.139.38.10: /* Linear operators in wave mechanics */

2014-01-18T07:14:10Z

Linear operators in wave mechanics

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en>JYBot: r2.7.1) (Robot: Adding ar:مؤثر (فيزياء)

2012-09-02T11:31:30Z

r2.7.1) (Robot: Adding ar:مؤثر (فيزياء)

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