Circulation problem - Revision history

173.56.9.130: …grammar

2014-10-26T17:45:09Z

…grammar

← Older revision		Revision as of 19:45, 26 October 2014
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	~~In the study of~~ [~~[metric spaces]] in [[mathematics]], there are various notions of two [[metric (mathematics)\|metrics]] on the same underlying space being "the same", or '''equivalent'''.~~		Claude is her title and she extended car [http://autos.jdpower.com/content/buying-tip/xa345A7/pros-and-cons-of-buying-an-extended-warranty-for-a-car.htm warranty totally] digs that title. I am a production and distribution officer. Her spouse and her selected to reside in Delaware but she requirements to transfer simply because of her family. Her [http://www.ag.state.Mn.us/consumer/cars/mncarlaws/mncarlaws_2.asp buddies] say it's not good [https://www.epapyrus.com/xe/Purchase/5246873 extended auto warranty] car [http://Www.Angieslist.com/articles/what-does-and-doesnt-your-auto-warranty-cover.htm warranty] for her but what she loves doing is flower arranging and auto warranty she is attempting to make it a occupation.<br><br>My site [http://changbai.newdrycleaning.com/photo/54180 extended auto warranty] car warranty ([http://Zocken-in-gesellschaft.de/index.php?mod=users&action=view&id=10093 simply click the next site])

	~~In the following, <math>X<~~/~~math> will denote a non-[[empty set]] and <math>d_{1}<~~/~~math> and <math>d_{2}<~~/~~math> will denote two metrics on <math>X<~~/~~math>.~~

	~~==Topological equivalence==~~

	~~The two metrics <math>d_{1}<~~/~~math> and <math>d_{2}<~~/~~math> are said to be '''topologically equivalent''' if they generate the same [[topology]] on <math>X</math>. The adjective "topological" is often dropped.<ref>Bishop~~ and ~~Goldberg, p. 10.</ref> There are multiple ways~~ of ~~expressing this condition:~~
	* a ~~subset <math>A \subseteq X</math> is <math>d_{1}</math>~~-~~[[open set\|open]~~] ~~[[if and only if]] it is <math>d_{2}</math>-open;~~
	* the [[open ball]]s "nest": for any point <math>x \in X</math> and any radius <math>r > 0</math>, there exist radii <math>r', r'' > 0</math> such that
	~~:<math>B_{r'} (x; d_{1}) \subseteq B_{r} (x; d_{2})</math> and <math>B_{r''} (x; d_{2}) \subseteq B_{r} (x; d_{1})~~.~~</math>~~
	* the [[identity function]] <math>I ~~: X \to X</math> is both <math>(d_{1}, d_{2})</math>-[[continuous function\|continuous]]~~ and ~~<math>(d_{2}, d_{1})</math>-continuous~~.

	~~The following are sufficient~~ but ~~not necessary conditions for topological equivalence:~~
	* there exists a strictly increasing, continuous, and [~~[subadditive]] <math>f~~:~~\mathbb{R}_{+} \to \mathbb{R}<~~/~~math> such that <math>d_{2} = f \circ d_{1}<~~/~~math>~~.~~<ref>Ok, p~~. ~~127, footnote 12~~.</~~ref>~~
	* for each <math>x \in X</~~math>, there exist positive constants <math>\alpha<~~/~~math> and <math>\beta<~~/~~math> such that, for every point <math>y \in X</math>,~~
	:~~<math>\alpha d_{1} (x, y) \leq d_{2} (x, y) \leq \beta d_{1} (x, y).<~~/~~math>~~

	~~==Strong equivalence==~~

	~~Two metrics <math>d_{1}<~~/~~math> and <math>d_{2}<~~/~~math> are '''strongly equivalent''' if and only if there exist positive constants <math>\alpha<~~/~~math> and <math>\beta<~~/~~math> such that, for every <math>x,y\in X</math>,~~
	:~~<math>\alpha d_{1}(x,y) \leq d_{2}(x,y) \leq \beta d_{1} (x, y).</math>~~
	~~In contrast to the sufficient condition for topological equivalence listed above, strong equivalence requires that there is a single set of constants that holds for every pair of points in <math>X<~~/~~math>, rather than potentially different constants associated with each point of <math>X<~~/~~math>~~.

	~~Strong equivalence of two metrics implies topological equivalence, but not vice versa~~. ~~An intuitive reason why topological equivalence~~ does not imply strong equivalence is that [[Bounded set#Metric space\|bounded sets]] under one metric are also bounded under a strongly equivalent metric, but not necessarily under a topologically equivalent metric.

	~~All metrics induced by the [[p~~-~~norm]], including the [[euclidean metric]], the [[taxicab metric]],~~ and ~~the [[Chebyshev distance]], are strongly equivalent.<ref>Ok, p. 138.</ref>~~

	~~Even if two metrics are strongly equivalent, not all properties of the respective metric spaces are preserved~~. ~~For instance, a function from the space to itself might be a [[contraction mapping~~]~~] under one metric,~~ but ~~not necessarily under~~ a ~~strongly equivalent one~~.<~~ref~~>~~Ok, p. 175.~~<~~/ref~~>

	~~==Properties preserved by equivalence==~~
	* The [[continuous function\|continuity]] of a function is preserved if either the domain or range is remetrized by an equivalent metric, but [[uniform continuity]] is preserved only by strongly equivalent metrics.<ref>Ok, p. ~~209~~.</~~ref>~~
	* The [~~[differentiability]] of a function is preserved if either the domain or range is remetrized by a strongly equivalent metric.<ref>Cartan, p. 27.</ref>~~

	~~==Notes==~~
	~~{{reflist}}~~

	~~==References==~~
	~~{{refbegin}}~~
	* {{cite book
	~~\| authors = Richard L. Bishop, Samuel I. Goldberg~~
	~~\| title = Tensor analysis on manifolds~~
	~~\| year = 1980~~
	~~\| publisher = Dover Publications~~
	~~\| url =~~ http://~~books~~.~~google~~.~~com/books~~?id=~~LAuN5-og4jwC~~
	}}
	* {{cite book
	~~\| author~~ = ~~Efe Ok~~
	~~\| title~~ = ~~Real analysis with economics applications~~
	~~\| year = 2007~~
	~~\| publisher = Princeton University Press~~
	~~\| isbn = 0-691-11768-3~~
	}}
	* {{cite book
	~~\| author = Henri Cartan~~
	~~\| title = Differential Calculus~~
	~~\| year = 1971~~
	~~\| publisher = Kershaw Publishing Company LTD~~
	~~\| isbn = 0-395-12033-0~~
	}}
	~~{{refend}}~~

	~~[[Category:Metric geometry]~~]

147.83.72.33: /* Definition */

2013-12-17T10:37:54Z

Definition

← Older revision		Revision as of 12:37, 17 December 2013
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	~~Her name~~ is ~~Felicidad Ahmad~~. ~~My home~~ is ~~now~~ in ~~Kansas~~. ~~To play handball~~ is the ~~thing she loves most~~ of all. ~~Meter reading~~ is ~~where my main earnings comes from~~ but ~~quickly I'll be on my personal~~.<br><br>~~Also visit my website; car warranty (~~[http://~~Royalcajun~~.com/~~UserProfile/tabid/57/userId/17325/Default.aspx discover this info here~~])		In the study of [[metric spaces]] in [[mathematics]], there are various notions of two [[metric (mathematics)\|metrics]] on the same underlying space being "the same", or '''equivalent'''.

			In the following, <math>X</math> will denote a non-[[empty set]] and <math>d_{1}</math> and <math>d_{2}</math> will denote two metrics on <math>X</math>.

			==Topological equivalence==

			The two metrics <math>d_{1}</math> and <math>d_{2}</math> are said to be '''topologically equivalent''' if they generate the same [[topology]] on <math>X</math>. The adjective "topological" is often dropped.<ref>Bishop and Goldberg, p. 10.</ref> There are multiple ways of expressing this condition:
			* a subset <math>A \subseteq X</math> is <math>d_{1}</math>-[[open set\|open]] [[if and only if]] it is <math>d_{2}</math>-open;
			* the [[open ball]]s "nest": for any point <math>x \in X</math> and any radius <math>r > 0</math>, there exist radii <math>r', r'' > 0</math> such that
			:<math>B_{r'} (x; d_{1}) \subseteq B_{r} (x; d_{2})</math> and <math>B_{r''} (x; d_{2}) \subseteq B_{r} (x; d_{1}).</math>
			* the [[identity function]] <math>I : X \to X</math> is both <math>(d_{1}, d_{2})</math>-[[continuous function\|continuous]] and <math>(d_{2}, d_{1})</math>-continuous.

			The following are sufficient but not necessary conditions for topological equivalence:
			* there exists a strictly increasing, continuous, and [[subadditive]] <math>f:\mathbb{R}_{+} \to \mathbb{R}</math> such that <math>d_{2} = f \circ d_{1}</math>.<ref>Ok, p. 127, footnote 12.</ref>
			* for each <math>x \in X</math>, there exist positive constants <math>\alpha</math> and <math>\beta</math> such that, for every point <math>y \in X</math>,
			:<math>\alpha d_{1} (x, y) \leq d_{2} (x, y) \leq \beta d_{1} (x, y).</math>

			==Strong equivalence==

			Two metrics <math>d_{1}</math> and <math>d_{2}</math> are '''strongly equivalent''' if and only if there exist positive constants <math>\alpha</math> and <math>\beta</math> such that, for every <math>x,y\in X</math>,
			:<math>\alpha d_{1}(x,y) \leq d_{2}(x,y) \leq \beta d_{1} (x, y).</math>
			In contrast to the sufficient condition for topological equivalence listed above, strong equivalence requires that there is a single set of constants that holds for every pair of points in <math>X</math>, rather than potentially different constants associated with each point of <math>X</math>.

			Strong equivalence of two metrics implies topological equivalence, but not vice versa. An intuitive reason why topological equivalence does not imply strong equivalence is that [[Bounded set#Metric space\|bounded sets]] under one metric are also bounded under a strongly equivalent metric, but not necessarily under a topologically equivalent metric.

			All metrics induced by the [[p-norm]], including the [[euclidean metric]], the [[taxicab metric]], and the [[Chebyshev distance]], are strongly equivalent.<ref>Ok, p. 138.</ref>

			Even if two metrics are strongly equivalent, not all properties of the respective metric spaces are preserved. For instance, a function from the space to itself might be a [[contraction mapping]] under one metric, but not necessarily under a strongly equivalent one.<ref>Ok, p. 175.</ref>

			==Properties preserved by equivalence==
			* The [[continuous function\|continuity]] of a function is preserved if either the domain or range is remetrized by an equivalent metric, but [[uniform continuity]] is preserved only by strongly equivalent metrics.<ref>Ok, p. 209.</ref>
			* The [[differentiability]] of a function is preserved if either the domain or range is remetrized by a strongly equivalent metric.<ref>Cartan, p. 27.</ref>

			==Notes==
			{{reflist}}

			==References==
			{{refbegin}}
			* {{cite book
			\| authors = Richard L. Bishop, Samuel I. Goldberg
			\| title = Tensor analysis on manifolds
			\| year = 1980
			\| publisher = Dover Publications
			\| url = http://books.google.com/books?id=LAuN5-og4jwC
			}}
			* {{cite book
			\| author = Efe Ok
			\| title = Real analysis with economics applications
			\| year = 2007
			\| publisher = Princeton University Press
			\| isbn = 0-691-11768-3
			}}
			* {{cite book
			\| author = Henri Cartan
			\| title = Differential Calculus
			\| year = 1971
			\| publisher = Kershaw Publishing Company LTD
			\| isbn = 0-395-12033-0
			}}
			{{refend}}

			[[Category:Metric geometry]]

98.88.119.40: /* Solution */ added reference to Tardos's strongly polynomial algorithm

2012-07-18T18:15:01Z

Solution: added reference to Tardos's strongly polynomial algorithm

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