Chebyshev rational functions - Revision history

en>Monkbot: /* References */Task 1c: Fix CS1 deprecated date parameter errors

2014-03-28T20:07:30Z

References: Task 1c: Fix CS1 deprecated date parameter errors

← Older revision		Revision as of 22:07, 28 March 2014
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	~~{{Unreferenced\|date=December 2009}}~~		Oscar is what my spouse loves to call me and I totally dig that title. Her spouse and her reside in Puerto Rico but she will have to transfer 1 working day or an additional. Body developing is 1 of the things I love most. For years he's been operating as a receptionist.<br><br>Here is my weblog - [http://xn--299ay03byycca57h.kr/zbxe/?document_srl=319947 std testing at home]
	~~In [[mathematics]], '''transport of structure'''~~ is ~~the definition of a new structure on an object by reference~~ to ~~another object on which a similar structure already exists. Definitions by transport of structure are regarded as canonical.~~

	~~Since mathematical structures are often defined in reference to an underlying [[space (mathematics)\|space]], many examples of transport of structure involve spaces~~ and ~~mappings between them~~. ~~For example, if ''V''~~ and ~~''W'' are [[vector space]]s, and if <math>\phi \colon V \~~to ~~W</math> is an [[isomorphism]], and if <math>(\cdot,\cdot)</math> is~~ an ~~[[inner product]] on <math>W</math>, then we can define an inner product <math>[\cdot, \cdot]</math> on ''V'' by~~
	~~:<math>[v_1, v_2] = (\phi(v_1), \phi(v_2))\;</math>~~.
	~~Although the equation makes sense even when <math>\phi</math>~~ is not an isomorphism, it only defines an inner product on ''V'' when <math>\phi</math> is, since otherwise it will cause <math>[\cdot,\cdot]</math> to be [[degenerate bilinear form\|degenerate]]. The idea is that <math>\phi</math> allows us to consider ''V'' and ''W'' as "the ~~same" vector space, and if we follow this analogy, we can transport an inner product from one to the other~~.

	A more involved example comes from [[differential topology]], in which we have the notion of a [[smooth manifold]]. If ''M'' is such a manifold, and if ''X'' is any topological space which is [[homeomorphic]] to ''M'', we can consider ''X'' as a ~~smooth manifold as well~~. ~~That is, let~~ <~~math~~>~~\phi \colon X \to M~~</math> be a homeomorphism; we must define coordinate charts on ''X'', which we will do by "pulling back" coordinate charts on ''M'' through <math>\phi</math>. Recall that a coordinate chart on <math>M</math> is ~~an open set ''U'' together with an [[injective map]]~~
	~~:<math>c \colon U \to \mathbb{R}^n\;</math>~~
	~~for some ''n''; to get such a chart on ''X'', we let~~
	~~:<math>U' = \phi^{~~-~~1}(U)\;</math> and <math>c' = c \circ \phi\;</math>.~~
	~~Furthermore, it is required that the charts cover ''M'', we must check that the transported charts cover ''X'', which follows immediately from the fact that <math>\phi</math> is a [~~[~~bijection]]. Finally, since ''M'' is a ''smooth'' manifold, we have that if ''U'' and ''V'', with their maps~~
	:~~<math>c \colon U \to \mathbb{R}^n\;<~~/~~math> and <math>d \colon V \to \mathbb{R}^n\;<~~/~~math>,~~
	~~are two charts on ''M'', then the composition, the "transition map"~~
	~~:<math>d \circ c^{~~-~~1} \colon c(U \cap V) \to \mathbb{R}^n\;</math> (a self~~-~~map of <math>\mathbb{R}^n</math>)~~
	~~is smooth~~. ~~We must check this for our transported charts on ''X''. We have~~
	~~:<math>\phi^{-1}(U) \cap \phi^{-1}(V) = \phi^{-1}(U \cap V)\;<~~/~~math>,~~
	~~and therefore~~
	~~:<math>c'(U' \cap V') = (c \circ \phi)(\phi^{-1}(U \cap V)) = c(U \cap V)\;<~~/~~math>, and~~
	~~:<math>d' \circ (c')^{-1} = (d \circ \phi) \circ (c \circ \phi)^{-1} = d \circ (\phi \circ \phi^{-1}) \circ c^{-1}~~ = ~~d \circ c^{-1}\;</math>.~~
	~~Therefore the transition map for <math>U'</math> and <math>V'</math> is the same as that for ''U'' and ''V'', hence smooth. Therefore ''X'' is a smooth manifold via transport of structure.~~

	Although the second example involved considerably more checking, the principle was the same, and any experienced mathematician would have no difficulty performing the necessary verifications. Therefore when such an operation is indicated, it is invoked merely as "transport of structure" and the details left to the reader, if desired.

	The second example also illustrates why "transport of structure" is not always desirable. Namely, we can take ''M'' to be the plane, and we can take ''X'' to be an infinite one-sided cone. By "flattening" the cone we achieve a homeomorphism of ''X'' and ''M'', and therefore the structure of a smooth manifold on ''X'', but the cone is not "naturally" a smooth manifold. That is, we can consider ''X'' as a subspace of 3-space, in which context it is not smooth at the cone point. A more surprising example is that of [[exotic sphere]]s, discovered by [[Milnor]], which states that there are exactly 28 smooth manifolds which are homeomorphic (but by definition ''not'' diffeomorphic) to <math>S^7</math>, the 7-dimensional sphere in 8-space. Thus, transport of structure is most productive when there exists a [[canonical form\|canonical]] isomorphism between the two objects.

	~~{{DEFAULTSORT:Transport Of Structure}}~~
	~~[[Category:Mathematical terminology]~~]

en>Michael Hardy: DO NOT indiscriminately italicize everything in non-TeX mathematical notation. Variables are italicized; digits, parentheses, etc., are not. See WP:MOSMATH.

2013-08-12T16:09:21Z

DO NOT indiscriminately italicize everything in non-TeX mathematical notation. Variables are italicized; digits, parentheses, etc., are not. See WP:MOSMATH.

← Older revision		Revision as of 18:09, 12 August 2013
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	~~They call me Emilia~~. ~~My day job~~ is a ~~meter reader~~. ~~To gather coins~~ is ~~what his family members~~ and ~~him enjoy~~. ~~For~~ a ~~while she~~'~~s been in South Dakota~~.<br><br>~~my web blog~~ :: [~~http~~://~~www~~.~~cam4teens~~.~~com~~/~~blog~~/~~84472 home std test kit~~]		{{Unreferenced\|date=December 2009}}
			In [[mathematics]], '''transport of structure''' is the definition of a new structure on an object by reference to another object on which a similar structure already exists. Definitions by transport of structure are regarded as canonical.

			Since mathematical structures are often defined in reference to an underlying [[space (mathematics)\|space]], many examples of transport of structure involve spaces and mappings between them. For example, if ''V'' and ''W'' are [[vector space]]s, and if <math>\phi \colon V \to W</math> is an [[isomorphism]], and if <math>(\cdot,\cdot)</math> is an [[inner product]] on <math>W</math>, then we can define an inner product <math>[\cdot, \cdot]</math> on ''V'' by
			:<math>[v_1, v_2] = (\phi(v_1), \phi(v_2))\;</math>.
			Although the equation makes sense even when <math>\phi</math> is not an isomorphism, it only defines an inner product on ''V'' when <math>\phi</math> is, since otherwise it will cause <math>[\cdot,\cdot]</math> to be [[degenerate bilinear form\|degenerate]]. The idea is that <math>\phi</math> allows us to consider ''V'' and ''W'' as "the same" vector space, and if we follow this analogy, we can transport an inner product from one to the other.

			A more involved example comes from [[differential topology]], in which we have the notion of a [[smooth manifold]]. If ''M'' is such a manifold, and if ''X'' is any topological space which is [[homeomorphic]] to ''M'', we can consider ''X'' as a smooth manifold as well. That is, let <math>\phi \colon X \to M</math> be a homeomorphism; we must define coordinate charts on ''X'', which we will do by "pulling back" coordinate charts on ''M'' through <math>\phi</math>. Recall that a coordinate chart on <math>M</math> is an open set ''U'' together with an [[injective map]]
			:<math>c \colon U \to \mathbb{R}^n\;</math>
			for some ''n''; to get such a chart on ''X'', we let
			:<math>U' = \phi^{-1}(U)\;</math> and <math>c' = c \circ \phi\;</math>.
			Furthermore, it is required that the charts cover ''M'', we must check that the transported charts cover ''X'', which follows immediately from the fact that <math>\phi</math> is a [[bijection]]. Finally, since ''M'' is a ''smooth'' manifold, we have that if ''U'' and ''V'', with their maps
			:<math>c \colon U \to \mathbb{R}^n\;</math> and <math>d \colon V \to \mathbb{R}^n\;</math>,
			are two charts on ''M'', then the composition, the "transition map"
			:<math>d \circ c^{-1} \colon c(U \cap V) \to \mathbb{R}^n\;</math> (a self-map of <math>\mathbb{R}^n</math>)
			is smooth. We must check this for our transported charts on ''X''. We have
			:<math>\phi^{-1}(U) \cap \phi^{-1}(V) = \phi^{-1}(U \cap V)\;</math>,
			and therefore
			:<math>c'(U' \cap V') = (c \circ \phi)(\phi^{-1}(U \cap V)) = c(U \cap V)\;</math>, and
			:<math>d' \circ (c')^{-1} = (d \circ \phi) \circ (c \circ \phi)^{-1} = d \circ (\phi \circ \phi^{-1}) \circ c^{-1} = d \circ c^{-1}\;</math>.
			Therefore the transition map for <math>U'</math> and <math>V'</math> is the same as that for ''U'' and ''V'', hence smooth. Therefore ''X'' is a smooth manifold via transport of structure.

			Although the second example involved considerably more checking, the principle was the same, and any experienced mathematician would have no difficulty performing the necessary verifications. Therefore when such an operation is indicated, it is invoked merely as "transport of structure" and the details left to the reader, if desired.

			The second example also illustrates why "transport of structure" is not always desirable. Namely, we can take ''M'' to be the plane, and we can take ''X'' to be an infinite one-sided cone. By "flattening" the cone we achieve a homeomorphism of ''X'' and ''M'', and therefore the structure of a smooth manifold on ''X'', but the cone is not "naturally" a smooth manifold. That is, we can consider ''X'' as a subspace of 3-space, in which context it is not smooth at the cone point. A more surprising example is that of [[exotic sphere]]s, discovered by [[Milnor]], which states that there are exactly 28 smooth manifolds which are homeomorphic (but by definition ''not'' diffeomorphic) to <math>S^7</math>, the 7-dimensional sphere in 8-space. Thus, transport of structure is most productive when there exists a [[canonical form\|canonical]] isomorphism between the two objects.

			{{DEFAULTSORT:Transport Of Structure}}
			[[Category:Mathematical terminology]]

18.62.4.167 at 00:47, 29 June 2011

2011-06-29T00:47:13Z

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They call me Emilia. My day job is a meter reader. To gather coins is what his family members and him enjoy. For a while she's been in South Dakota.<br><br>my web blog :: [http://www.cam4teens.com/blog/84472 home std test kit]