|
|
| Line 1: |
Line 1: |
| {{refimprove|date=November 2009}}
| | == who has headed the handsome face == |
| In [[geometry]], the '''tangent cone''' is a generalization of the notion of the [[tangent space]] to a [[manifold]] to the case of certain spaces with singularities.
| |
|
| |
|
| == Definition in convex geometry ==
| | House, dozens of shadow piercing flying, which has three very strong flavor.<br><br>dozens of shadow arrive soon.<br><br>'Prince Highness.' Venerable Juhan slight bend the universe to show respect.<br><br>'Prince [http://www.nrcil.net/fancybox/lib/rakuten_LV_76.html ルイヴィトングループ] Highness.' ten soldiers fell [http://www.nrcil.net/fancybox/lib/rakuten_LV_103.html ルイヴィトン ハンドバッグ] down under even more [http://www.nrcil.net/fancybox/lib/rakuten_LV_83.html ルイヴィトン オンライン] direct.<br><br>which dozens figure, who has headed the handsome face, with a bloody face in Arcanum wearing a gorgeous golden color [http://www.nrcil.net/fancybox/lib/rakuten_LV_23.html ルイヴィトン 財布 メンズ タイガ] inclusions battle armor, armor [http://www.nrcil.net/fancybox/lib/rakuten_LV_128.html 財布 ルイヴィトン メンズ] deliberately expose a strong fluctuation fluctuations ...... This is a treasure no doubt, he is HRH Prince.<br><br>Prince breath very powerful, but it looks and appearance are different from all the other tribe.<br><br>matter before the universe Venerable Juhan, or followed behind the two princes of the universe and a group of [http://www.nrcil.net/fancybox/lib/rakuten_LV_130.html ルイヴィトン 靴] immortal Venerable, skin color also has a color rendering all corners!<br><br>judging from the looks!<br><br>princes and other [http://www.nrcil.net/fancybox/lib/rakuten_LV_43.html ルイヴィトン 財布 人気] generals, [http://www.nrcil.net/fancybox/lib/rakuten_LV_36.html ルイヴィトン 財布 通販] soldiers seemingly two groups.<br><br>'Ma Ma Tinto |
| | | 相关的主题文章： |
| Let ''K'' be a [[closed set|closed]] [[convex subset]] of a real [[vector space]] ''V'' and ∂''K'' be the [[boundary (topology)|boundary]] of ''K''. The '''solid tangent cone''' to ''K'' at a point ''x'' ∈ ∂''K'' is the [[closure (mathematics)|closure]] of the cone formed by all half-lines (or rays) emanating from ''x'' and intersecting ''K'' in at least one point ''y'' distinct from ''x''. It is a [[convex cone]] in ''V'' and can also be defined as the intersection of the closed [[Half-space (geometry)|half-space]]s of ''V'' containing ''K'' and bounded by the [[supporting hyperplane]]s of ''K'' at ''x''. The boundary ''T''<sub>''K''</sub> of the solid tangent cone is the '''tangent cone''' to ''K'' and ∂''K'' at ''x''. If this is an [[affine subspace]] of ''V'' then the point ''x'' is called a '''smooth point''' of ∂''K'' and ∂''K'' is said to be '''differentiable''' at ''x'' and ''T''<sub>''K''</sub> is the ordinary [[tangent space]] to ∂''K'' at ''x''.
| | <ul> |
| <!--- Incorporate this formula, but think about the best way of doing it
| | |
| :<math>
| | <li>[http://bbs.17digg.com/ahdg/home.php?mod=space&uid=127342 http://bbs.17digg.com/ahdg/home.php?mod=space&uid=127342]</li> |
| T_K(x) = \overline{\bigcup_{\epsilon>0} \epsilon(K-x)}. </math>
| | |
| --->
| | <li>[http://support-arb.com/showthread.php?27413-Jiang-Feng-Luo-was-watching&p=36259#post36259 http://support-arb.com/showthread.php?27413-Jiang-Feng-Luo-was-watching&p=36259#post36259]</li> |
| | | |
| == Definition in algebraic geometry ==
| | <li>[http://curecuberoom.com/bbs/bbs.cgi http://curecuberoom.com/bbs/bbs.cgi]</li> |
| | | |
| [[File:Node (algebraic geometry).png|thumb|right|200px|y<sup>2</sup> = x<sup>3</sup> + x<sup>2</sup> (red) with tangent cone (blue)]]
| | </ul> |
| | |
| Let ''X'' be an [[affine algebraic variety]] embedded into the affine space ''k''<sup>''n''</sup>, with the defining ideal ''I'' ⊂ ''k''[''x''<sub>1</sub>,…,''x''<sub>''n''</sub>]. For any polynomial ''f'', let in(''f'') be the homogeneous component of ''f'' of the lowest degree, the ''initial term'' of ''f'', and let in(''I'') ⊂ ''k''[''x''<sub>1</sub>,…,''x''<sub>''n''</sub>] be the homogeneous ideal which is formed by the initial terms in(''f'') for all ''f'' ∈ ''I'', the ''initial ideal'' of ''I''. The '''tangent cone''' to ''X'' at the origin is the Zariski closed subset of ''k''<sup>''n''</sup> defined by the ideal in(''I''). By shifting the coordinate system, this definition extends to an arbitrary point of ''k''<sup>''n''</sup> in place of the origin. The tangent cone serves as the extension of the notion of the tangent space to ''X'' at a regular point, where ''X'' most closely resembles a [[differentiable manifold]], to all of ''X''. (The tangent cone at a point of ''k''<sup>''n''</sup> that is not contained in ''X'' is empty.)
| |
| | |
| For example, the nodal curve
| |
| | |
| : <math>C: y^2=x^3+x^2 </math>
| |
| | |
| is singular at the origin, because both [[partial derivative]]s of ''f''(''x'', ''y'') = ''y''<sup>2</sup> − ''x''<sup>3</sup> − ''x''<sup>2</sup> vanish at (0, 0). Thus the [[Zariski tangent space]] to ''C'' at the origin is the whole plane, and has higher dimension than the curve itself (two versus one). On the other hand, the tangent cone is the union of the tangent lines to the two branches of ''C'' at the origin,
| |
| | |
| : <math> x=y,\quad x=-y. </math>
| |
| | |
| Its defining ideal is the principal ideal of ''k''[''x''] generated by the initial term of ''f'', namely ''y''<sup>2</sup> − ''x''<sup>2</sup> = 0.
| |
| | |
| The definition of the tangent cone can be extended to abstract algebraic varieties, and even to general [[Noetherian]] [[scheme (mathematics)|schemes]]. Let ''X'' be an [[algebraic variety]], ''x'' a point of ''X'', and (''O''<sub>''X'',''x''</sub>, ''m'') be the [[local ring]] of ''X'' at ''x''. Then the '''tangent cone''' to ''X'' at ''x'' is the [[spectrum of a ring|spectrum]] of the [[associated graded ring]] of ''O''<sub>''X'',''x''</sub> with respect to the [[Completion (ring theory)#I-adic topology|''m''-adic filtration]]:
| |
| | |
| :<math>\operatorname{gr}_m O_{X,x}=\bigoplus_{i\geq 0} m^i / m^{i+1}.</math>
| |
| | |
| == See also ==
| |
| * [[Cone]]
| |
| * [[Monge cone]]
| |
| * [[Convex cone#Examples of convex cones|Normal cone]]
| |
| | |
| == References ==
| |
| | |
| * {{Springer|title=Tangent cone|id=T/t092120|author=M. I. Voitsekhovskii}}
| |
| | |
| {{DEFAULTSORT:Tangent Cone}}
| |
| [[Category:Convex geometry]]
| |
| [[Category:Algebraic geometry]]
| |
who has headed the handsome face
House, dozens of shadow piercing flying, which has three very strong flavor.
dozens of shadow arrive soon.
'Prince Highness.' Venerable Juhan slight bend the universe to show respect.
'Prince ルイヴィトングループ Highness.' ten soldiers fell ルイヴィトン ハンドバッグ down under even more ルイヴィトン オンライン direct.
which dozens figure, who has headed the handsome face, with a bloody face in Arcanum wearing a gorgeous golden color ルイヴィトン 財布 メンズ タイガ inclusions battle armor, armor 財布 ルイヴィトン メンズ deliberately expose a strong fluctuation fluctuations ...... This is a treasure no doubt, he is HRH Prince.
Prince breath very powerful, but it looks and appearance are different from all the other tribe.
matter before the universe Venerable Juhan, or followed behind the two princes of the universe and a group of ルイヴィトン 靴 immortal Venerable, skin color also has a color rendering all corners!
judging from the looks!
princes and other ルイヴィトン 財布 人気 generals, ルイヴィトン 財布 通販 soldiers seemingly two groups.
'Ma Ma Tinto
相关的主题文章：