Pontryagin class - Revision history

128.112.70.46 at 01:58, 9 November 2014

2014-11-09T01:58:00Z

en>Subh83: /* Pontryagin numbers */ Changing symbol to be consistent with cup product. see this discussion.

2014-01-04T21:02:05Z

Pontryagin numbers: Changing symbol to be consistent with cup product. see this discussion.

← Older revision		Revision as of 23:02, 4 January 2014
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	~~== integrity Buy Longchamp Online ==~~		[[Image:Geom podobnost stejnolehlest.svg\|thumb\|right\|334px\|Two similar geometric figures related by a homothetic transformation with respect to a [[homothetic center]] '''S'''. The angles at corresponding points are the same and have the same sense; for example, the angles ABC and A'B'C' are both clockwise and equal in magnitude.]]

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			:<math> M \mapsto S + \lambda \overrightarrow{SM}, </math>
	~~<li>[http://www.iyuhua.com/forum.php?mod=viewthread&tid~~=~~196995&fromuid~~=~~81162 http://www.iyuhua.com/forum.php?mod~~=~~viewthread&tid~~=~~196995&fromuid=81162]~~</li>		in other words it fixes ''S'', and sends any ''M'' to another point ''N'' such that the segment ''SN'' is on the same line as ''SM'', but scaled by a factor ''λ''.<ref>{{harvtxt\|Hadamard\|p=145}}</ref> In [[Euclidean geometry]] homotheties are the [[Similarity (geometry)\|similarities]] that fix a point and either preserve (if {{nowrap\|''λ'' > 0}}) or reverse (if {{nowrap\|''λ'' < 0}}) the direction of all vectors. Together with the [[Translation (geometry)\|translations]], all homotheties of an affine (or Euclidean) space form a [[group (mathematics)\|group]], the group of '''dilations''' or '''homothety-translations'''. These are precisely the [[affine transformation]]s with the property that the image of every line ''L'' is a line [[parallel (geometry)\|parallel]] to ''L''.

	~~<li>[http://tztea~~.~~com.cn/news/html/?225333~~.~~html http://tztea.com.cn/news/html/?225333.html]</li>~~		In [[projective geometry]], a homothetic transformation is a similarity transformation that leaves the line at infinity pointwise [[invariant (mathematics)\|invariant]].<ref>{{harvtxt\|Tuller\|p=119}}</ref>

	~~<li>[http://wikilegal~~.~~in/index.php?~~title=~~User:Linzgzps#2013_Sony_World_Photography_Awards_Pandora_Jewelry_Wholesal http://wikilegal~~.~~in/index.php?~~title=~~User:Linzgzps#2013_Sony_World_Photography_Awards_Pandora_Jewelry_Wholesal]</li>~~		In Euclidean geometry, a homothety of ratio ''λ'' multiplies distances between points by \|''λ''\| and all areas by ''λ''<sup>2</sup>. The first number is called the ''ratio of magnification'' or ''dilation factor'' or ''scale factor'' or ''similitude ratio''. Such a transformation can be called an '''enlargement''' if the scale factor exceeds 1. The above mentioned fixed point ''S'' is called ''[[homothetic center]]'' or ''center of similarity'' or ''center of similitude''

	~~<li>~~[http://~~jianniukeji~~.~~com/forum~~.~~php?mod=viewthread&tid=1066243 http:~~//~~jianniukeji.com~~/~~forum~~.~~php?mod=viewthread&tid=1066243~~]~~</li>~~		==Homothety and uniform scaling==
			If the [[homothetic center]] ''S'' happens to coincide with the [[Origin (mathematics)\|origin]] ''O'' of the vector space (''S'' ≡ ''O''), then every homothety with scale factor ''λ'' is equivalent to a [[uniform scaling]] by the same factor, which sends
	~~</ul>~~
			:<math> \overrightarrow{OM} \mapsto \lambda \overrightarrow{OM}. </math>

			As a consequence, in the specific case in which ''S'' ≡ ''O'', the homothety becomes a [[linear transformation]], which preserves not only the collinearity of points (straight lines are mapped to straight lines), but also vector addition and scalar multiplication.

			==See also==
			*[[Scaling (geometry)]] a similar notion in vector spaces
			*[[Homothetic center]], the center of a homothetic transformation taking one of a pair of shapes into the other
			*The [[Hadwiger conjecture (combinatorial geometry)\|Hadwiger conjecture]] on the number of strictly smaller homothetic copies of a convex body that may be needed to cover it
			*[[Homothetic function (economics)]], a function of the form ''U''(''f''(''y'')) in which ''f'' is a [[homogeneous function]] and ''U'' is a [[monotonic function\|monotonically increasing function]].

			== Notes ==
			<references/>

			==References==
			* {{ citation \| first1 = J. \| last1 = Hadamard \| title = Lessons in Plane Geometry
			}}.
			* {{ citation \| first1 = Bruce E. \| last1 = Meserve \| year =1955 \| title = Fundamental Concepts of Geometry \| chapter = Homothetic transformations \| pages = 166-169 \| publisher = [[Addison-Wesley]] }}.
			* {{ citation \| last1 = Tuller \| first1 = Annita \| title = A Modern Introduction to Geometries }}.

			==External links==
			* [http://www.cut-the-knot.org/Curriculum/Geometry/Homothety.shtml Homothety], interactive applet from [[Cut-the-Knot]].

			[[Category:Transformation (function)]]

76.87.43.179: /* Properties */ Changed "\cup" (used for union) to "\smile" (the usual symbol for cup product)

2012-06-07T19:44:46Z

Properties: Changed "\cup" (used for union) to "\smile" (the usual symbol for cup product)

New page

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