Complex logarithm - Revision history

en>Dicklyon: /* Plots of the complex logarithm function (principal branch) */ per talk page consensus, remove section with nothing but confusing graphs

2014-12-25T16:45:31Z

Plots of the complex logarithm function (principal branch): per talk page consensus, remove section with nothing but confusing graphs

← Older revision		Revision as of 18:45, 25 December 2014
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216.96.79.149: /* Definition of principal value */ typo

2014-02-23T06:47:37Z

Definition of principal value: typo

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-In the [[mathematics|mathematical]] field of [[knot theory]], the '''crosscap number''' of a [[knot (mathematics)|knot]] ''K'' is the minimum of
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-:<math>1 - \chi(S), \, </math>
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-taken over all [[compact space|compact]], [[connected space|connected]], [[orientability|non-orientable]] [[surface]]s ''S'' bounding ''K''; here <math>\chi</math> is the [[Euler characteristic]].  The crosscap number of the [[unknot]] is zero, as the Euler characteristic of the disk is one.
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en>Ebony Jackson: /* Definition of principal value */ simplified wording

2014-01-02T17:56:31Z

Definition of principal value: simplified wording

← Older revision		Revision as of 19:56, 2 January 2014
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	~~22 year old Education Professionals Rocky from Burk~~'s ~~Falls~~, ~~really loves beachcombing~~, ~~como ganhar dinheiro na internet~~ and ~~rc model boats~~. ~~Is enthused how enormous~~ the ~~earth~~ is ~~after creating a journey to Shark Bay.~~<br><br>~~my homepage~~; [http://www.~~comoganhardinheiro101~~.~~com~~/~~perguntas-frequentes~~/ ~~como conseguir dinheiro~~]		In the [[mathematics\|mathematical]] field of [[knot theory]], the '''crosscap number''' of a [[knot (mathematics)\|knot]] ''K'' is the minimum of

			:<math>1 - \chi(S), \, </math>

			taken over all [[compact space\|compact]], [[connected space\|connected]], [[orientability\|non-orientable]] [[surface]]s ''S'' bounding ''K''; here <math>\chi</math> is the [[Euler characteristic]]. The crosscap number of the [[unknot]] is zero, as the Euler characteristic of the disk is one.

			==Examples==
			*The crosscap number of the [[trefoil knot]] is 1, as it bounds a [[Möbius strip]] and is not trivial.
			*The crosscap number of a [[torus knot]] was determined by M. Teragaito.

			The formula for the [[knot sum]] is
			:<math>C(k_1)+C(k_2)-1 \leq C(k_1 \# k_2) \leq C(k_1)+C(k_2). \, </math>

			==Further reading==
			*Clark, B.E. "Crosscaps and Knots", Int. J. Math and Math. Sci, Vol 1, 1978, pp 113–124
			*Murakami, Hitoshi and Yasuhara, Akira. "Crosscap number of a knot," Pacific J. Math. 171 (1995), no. 1, 261–273.
			*Teragaito, Masakazu. "Crosscap numbers of torus knots," Topology Appl. 138 (2004), no. 1–3, 219–238.
			*Teragaito, Masakazu and Hirasawa, Mikami. "Crosscap numbers of 2-bridge knots," Arxiv:math.GT/0504446.
			*J.Uhing. [http://jason-uhing.de/ "Zur Kreuzhaubenzahl von Knoten"], diploma thesis, 1997, University of Dortmund, (German language)

			==External links==
			*"[http://www.indiana.edu/~knotinfo/descriptions/crosscap_number.html Crosscap Number]", ''KnotInfo''.

			{{Knot theory}}
			[[Category:Knot invariants]]

			{{knottheory-stub}}

en>Loadmaster: /* Logarithms to other bases */ use math markup for more clarity

2012-08-17T15:53:11Z

Logarithms to other bases: use math markup for more clarity

New page

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