Sessile drop technique - Revision history

en>Nick Number: repaired link(s) to disambiguation page (you can help) - Tensiometer

2014-04-26T08:21:28Z

repaired link(s) to disambiguation page (you can help) - Tensiometer

← Older revision		Revision as of 10:21, 26 April 2014
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	~~[[Image:End of universe.jpg\|thumb\|In an Hadamard space, a triangle~~ is ~~[[Hyperbolic triangle\|hyperbolic]]; that is,~~ the ~~middle one in the picture. In fact, any~~ complete ~~metric space where a triangle is hyperbolic is an Hadamard space~~.]]		Alyson Meagher is the name her parents gave her but she doesn't like when individuals use her complete title. My spouse and I live in Mississippi but now I'm considering other choices. As a woman what she really likes is fashion and she's been doing it love psychic ([http://fashionlinked.com/index.php?do=/profile-13453/info/ Enter your text here and click the "Remove Empty Lines" button above.]) for fairly a whilst. Credit authorising is how she makes a residing.<br><br>Here is my web blog ... free psychic [http://www.prayerarmor.com/uncategorized/dont-know-which-kind-of-hobby-to-take-up-read-the-following-tips/ clairvoyant psychic] readings ([http://www.prograd.uff.br/novo/facts-about-growing-greater-organic-garden http://www.prograd.uff.br])
	~~In [[geometry]], an~~ '~~''Hadamard space''', named after [[Jacques Hadamard]], is a non-linear generalization of a [[Hilbert space]]~~. ~~It is defined to be a nonempty<ref>The assumption on "nonempty" has meaning:~~ a ~~fixed point theorem often states the set of fixed point~~ is ~~an Hadamard space. The main content of such an assertion is that the set is nonempty.</ref> complete~~ [~~[metric space]] where, given any points ''x'', ''y'', there exists a point ''m'' such that for every point ''z'',~~

	:~~<math>d(z, m)^2 + {d(x, y)^2 \over 4} \le {d(z, x)^2 + d(z, y)^2 \over 2}.<~~/~~math>~~

	~~The point ''m'' is then the midpoint of ''x'' and ''y'': <math>d(x, m) = d(y, m) = d(x, y)~~/2</~~math>~~.

	~~In a Hilbert space, the above inequality is equality (with <math>m~~ = ~~(x+y)~~/2</~~math>),~~ and ~~in general an Hadamard space is said to be ''flat'' if~~ the above ~~inequality is equality. A flat Hadmard space is isomorphic to a closed convex subset of a Hilbert space~~. ~~In particular, a [[normed space~~]~~] is an Hadamard space if and only if it is a Hilbert space.~~

	~~The geometry of Hadamard spaces resembles that of Hilbert spaces, making it a natural setting~~ for ~~the study of [[rigidity theorem]]s. In an Hadamard space, any two points can be joined by~~ a ~~unique [[geodesic]] between them; in particular, it is [[contractible space\|contractible]]~~. ~~Quite generally, if ''B''~~ is a ~~bounded subset of a metric space, then the center of the closed ball of the minimum radius containing it is called the ''[[circumcenter (metric space)\|circumcenter]]'' of ''B''~~.<~~ref~~>~~A Course in Metric Geometry, p. 334.~~<~~/ref~~> ~~Every bounded subset of an Hadamard space~~ is ~~contained in the smallest closed ball (which is the same as the closure of its convex hull)~~. ~~If <math>\Gamma<~~/~~math> is the [[group (mathematics)\|group]] of [[isometry\|isometries]] of an Hadamard space leaving invariant ''B'', then <math>\Gamma<~~/~~math> fixes the circumcenter of ''B''~~. ~~('''Bruhat–Tits fixed point theorem''')~~

	~~The basic result for a non~~-~~positively curved manifold is the [[Cartan–Hadamard theorem]]. The analog holds for an Hadamard space: a complete, connected metric space~~ which ~~is locally isometric~~ to ~~an Hadamard space has an Hadamard space as its universal cover. Its variant applies for non~~-~~positively curved [[orbifold]]s. (cf. Lurie.)~~

	~~Examples of Hadamard spaces are Hilbert spaces,~~ the ~~[[Poincaré disc]], trees (e.g., [[Bruhat–Tits building]]), [[Cayley graph]]s of [[hyperbolic group]]s or more generally CAT(0) groups, [[(p, q)~~-~~space\|(''p'', ''q'')~~-~~space]] with ''p'', ''q'' ≥ 3 and 2''pq'' ≥ ''p'' + ''q'', and [[Riemannian manifold]]s of nonpositive [[sectional curvature]~~] (~~e.g., [[symmetric space]]s). An Hadamard space is precisely a complete [[CAT(k) space\|CAT(0) space]].~~

	~~== See also ==~~

	* [[CAT(k) space]]

	~~== References ==~~
	~~{{reflist}}~~
	*Burago, Dmitri; Yuri Burago, and Sergei Ivanov. ''A Course in Metric Geometry''. American Mathematical Society. (1984)
	*[[Jacob Lurie]]: [http://www.~~math~~.~~harvard~~.~~edu~~/~~~lurie~~/~~papers~~/~~hadamard~~.~~pdf Notes on the Theory of Hadamard Spaces]~~

	~~[[Category:Hilbert space]~~]

en>Welsh: fix apostrophe per WP:APOSTROPHE

2013-12-06T23:12:46Z

fix apostrophe per WP:APOSTROPHE

← Older revision		Revision as of 01:12, 7 December 2013
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	~~Jayson Berryhill~~ is ~~how I~~'m ~~called and my wife doesn~~'~~t like it at all~~. ~~To perform lacross~~ is the ~~thing I love most~~ of ~~all~~. ~~For~~ a ~~while I~~'~~ve been in Alaska but I will have~~ to ~~move in~~ a ~~yr or~~ two. ~~Invoicing~~ is ~~what I do for~~ a ~~living but I~~'~~ve usually needed my personal business~~.<br><br>~~My weblog~~ - ~~real psychic readings~~ ([http://~~black7~~.~~mireene~~.~~com~~/~~aqw~~/~~5741 black7~~.~~mireene.com~~])		[[Image:End of universe.jpg\|thumb\|In an Hadamard space, a triangle is [[Hyperbolic triangle\|hyperbolic]]; that is, the middle one in the picture. In fact, any complete metric space where a triangle is hyperbolic is an Hadamard space.]]
			In [[geometry]], an '''Hadamard space''', named after [[Jacques Hadamard]], is a non-linear generalization of a [[Hilbert space]]. It is defined to be a nonempty<ref>The assumption on "nonempty" has meaning: a fixed point theorem often states the set of fixed point is an Hadamard space. The main content of such an assertion is that the set is nonempty.</ref> complete [[metric space]] where, given any points ''x'', ''y'', there exists a point ''m'' such that for every point ''z'',

			:<math>d(z, m)^2 + {d(x, y)^2 \over 4} \le {d(z, x)^2 + d(z, y)^2 \over 2}.</math>

			The point ''m'' is then the midpoint of ''x'' and ''y'': <math>d(x, m) = d(y, m) = d(x, y)/2</math>.

			In a Hilbert space, the above inequality is equality (with <math>m = (x+y)/2</math>), and in general an Hadamard space is said to be ''flat'' if the above inequality is equality. A flat Hadmard space is isomorphic to a closed convex subset of a Hilbert space. In particular, a [[normed space]] is an Hadamard space if and only if it is a Hilbert space.

			The geometry of Hadamard spaces resembles that of Hilbert spaces, making it a natural setting for the study of [[rigidity theorem]]s. In an Hadamard space, any two points can be joined by a unique [[geodesic]] between them; in particular, it is [[contractible space\|contractible]]. Quite generally, if ''B'' is a bounded subset of a metric space, then the center of the closed ball of the minimum radius containing it is called the ''[[circumcenter (metric space)\|circumcenter]]'' of ''B''.<ref>A Course in Metric Geometry, p. 334.</ref> Every bounded subset of an Hadamard space is contained in the smallest closed ball (which is the same as the closure of its convex hull). If <math>\Gamma</math> is the [[group (mathematics)\|group]] of [[isometry\|isometries]] of an Hadamard space leaving invariant ''B'', then <math>\Gamma</math> fixes the circumcenter of ''B''. ('''Bruhat–Tits fixed point theorem''')

			The basic result for a non-positively curved manifold is the [[Cartan–Hadamard theorem]]. The analog holds for an Hadamard space: a complete, connected metric space which is locally isometric to an Hadamard space has an Hadamard space as its universal cover. Its variant applies for non-positively curved [[orbifold]]s. (cf. Lurie.)

			Examples of Hadamard spaces are Hilbert spaces, the [[Poincaré disc]], trees (e.g., [[Bruhat–Tits building]]), [[Cayley graph]]s of [[hyperbolic group]]s or more generally CAT(0) groups, [[(p, q)-space\|(''p'', ''q'')-space]] with ''p'', ''q'' ≥ 3 and 2''pq'' ≥ ''p'' + ''q'', and [[Riemannian manifold]]s of nonpositive [[sectional curvature]] (e.g., [[symmetric space]]s). An Hadamard space is precisely a complete [[CAT(k) space\|CAT(0) space]].

			== See also ==

			* [[CAT(k) space]]

			== References ==
			{{reflist}}
			*Burago, Dmitri; Yuri Burago, and Sergei Ivanov. ''A Course in Metric Geometry''. American Mathematical Society. (1984)
			*[[Jacob Lurie]]: [http://www.math.harvard.edu/~lurie/papers/hadamard.pdf Notes on the Theory of Hadamard Spaces]

			[[Category:Hilbert space]]

en>Tesi1700: Sorted

2012-04-13T20:20:42Z

Sorted

New page

Jayson Berryhill is how I'm called and my wife doesn't like it at all. To perform lacross is the thing I love most of all. For a while I've been in Alaska but I will have to move in a yr or two. Invoicing is what I do for a living but I've usually needed my personal business.<br><br>My weblog - real psychic readings ([http://black7.mireene.com/aqw/5741 black7.mireene.com])