List of mesons - Revision history

en>John of Reading: Typo fixing, replaced: vice-versa → vice versa (2) using AWB

2015-01-06T09:00:35Z

Typo fixing, replaced: vice-versa → vice versa (2) using AWB

← Older revision		Revision as of 11:00, 6 January 2015
Line 1:		Line 1:
	~~The author~~ is ~~known as Wilber Pegues~~. ~~Invoicing is my profession~~. To ~~play lacross~~ is ~~the thing I adore most of all~~. ~~North Carolina~~ is ~~the location~~ he ~~loves most~~ but now he is ~~contemplating other options~~.<br><br>my site ... ~~tarot card readings,~~ [http://~~fashionlinked~~.com/index.php?do=/profile-~~13453~~/info/ ~~similar site~~],		Oscar is std home test how [http://Www.plannedparenthood.org/health-topics/stds-hiv-safer-sex/std-testing-21695.asp he's called] and [http://tti-ttw.com/index.php?do=/profile-2200/info/ http://tti-ttw.com/index.php?do=/profile-2200/info] he [http://Www.genitalherpestreatment.co/what-does-genital-herpes-look-like totally loves] this name. To gather cash is what her family and her appreciate. Puerto Rico is where he's been living for many years and he will by no means transfer. He utilized to be unemployed but now he is a computer operator but his marketing by no std home test at home std test means comes.<br><br>Feel free to surf to my site std testing [http://nemoonehfilm.ir/index.php?do=/profile-6876/info/ http://nemoonehfilm.ir/index.php?do=/profile-6876/info/] at home ([http://www.webmdbook.com/index.php?do=/profile-11685/info/ simply click www.webmdbook.com])

en>4th-otaku: copyedit, +header

2014-02-28T01:34:15Z

copyedit, +header

← Older revision		Revision as of 03:34, 28 February 2014
Line 1:		Line 1:
	In [[mathematics]], the '''rational normal curve''' is a smooth, [[rational curve]] <math>C</math> of [[Degree of an algebraic variety\|degree]] ''n'' in [[projective space\|projective n-space]] <math>\mathbb{P}^n.</math> It is a simple example of a [[projective variety]]; formally, it is the [[Veronese variety]] when the domain is the projective line. For ''n''=2 it is the [[flat conic]] <math>Z_0 Z_2 = Z_1^2,</math> and for ''n''=3 it is the [[twisted cubic]]. The ~~term "normal"~~ is an old term meaning that the linear system defining the embedding is complete (and has nothing to do with [[normal scheme]]s). The intersection of the rational normal curve with an affine space is called the [[moment curve]].		The author is known as Wilber Pegues. Invoicing is my profession. To play lacross is the thing I adore most of all. North Carolina is the location he loves most but now he is contemplating other options.<br><br>my site ... tarot card readings, [http://fashionlinked.com/index.php?do=/profile-13453/info/ similar site],

	~~==Definition==~~
	~~The rational normal curve may be given [[parametrically]]~~ as ~~the image of the map~~

	~~:<math>\nu:\mathbb{P}^1\to\mathbb{P}^n</math>~~

	~~which assigns to the [[homogeneous coordinates]] <math>[S:T]</math> the value~~

	~~:<math>\nu:[S:T] \mapsto [S^n:S^{n-1}T:S^{n-2}T^2:\ldots:T^n]~~.~~</math>~~

	~~In the [[affine coordinates]] of the chart <math> x_0\neq0 </math> the map~~ is ~~simply~~

	~~:<math>\nu:x \mapsto (x,x^2, \ldots ,x^n)~~.~~</math>~~

	~~That~~ is, the ~~rational normal curve is the closure by a single [[point at infinity]]~~ of ~~the [[affine curve]] <math>(x,x^2,\dots,x^n)</math>.~~

	~~Equivalently, rational normal curve may be understood to be a [[projective variety]], defined as the common zero locus of the [[homogeneous polynomial]]s~~

	~~:<math>F_{i,j}(X_0,\ldots,X_n) = X_iX_j - X_{i+1}X_{j-1}</math>~~

	~~where <math>[X_0:\ldots:X_n]</math> are the [[homogeneous coordinate]]s on <math>\mathbb{P}^n</math>~~. ~~The full set of these polynomials~~ is ~~not needed; it is sufficient to pick ''n'' of these to specify~~ the ~~curve.~~

	~~==Alternate parameterization==~~
	~~Let <math>[a_i:b_i]</math> be <math>n+1</math> distinct points in <math>\mathbb{P}^1</math>. Then the polynomial~~

	~~:<math>G(S,T) = \Pi_{i=0}^n (a_iS -b_iT)</math>~~

	is ~~a [[homogeneous polynomial]] of degree <math>n+1</math> with distinct roots~~. ~~The polynomials~~

	:<~~math~~>~~H_i(S,T) = \frac{G(S,T)} {(a_iS-b_iT)}~~<~~/math~~>

	~~are then a [[Basis (linear algebra)\|basis]] for the space of homogeneous polynomials of degree ''n''~~. ~~The map~~

	~~:<math>[S:T] \mapsto [H_0(S,T) : H_1(S,T) : \ldots : H_n (S,T) ]</math>~~

	~~or, equivalently, dividing by <math>G(S,T)</math>~~

	~~:<math>[S:T] \mapsto \left[\frac{1}{(a_0S-b_0T)} : \ldots : \frac{1}{(a_nS-b_nT)}\right] </math>~~

	~~is a rational normal curve~~. That this is a rational normal curve may be understood by noting that the [[monomial]]s <math>S^n,S^{n-1}T,S^{n-2}T^2,\ldots,T^n</math> are just one possible [[base (topology)\|basis]] for the space of degree-''n'' homogeneous polynomials. ~~In fact~~, ~~any~~ [[Basis (linear algebra)\|basis]] will do. This is just an application of the statement that any two projective varieties are projectively equivalent if they are [[congruence relation\|congruent]] modulo the [[projective linear group]] <math>\rm{PGL}_{n+1} K</math> (with ''K'' the [[field (mathematics)\|field]] over which the projective space is defined).

	This rational curve sends the zeros of ''G'' to each of the coordinate points of <math>\mathbb{P}^n</math>; that is, all but one of the <math>H_i</math> vanish for a zero of ''G''. Conversely, any rational normal curve passing through the ''n+1'' coordinate points may be written parametrically in this way.

	~~==Properties==~~
	~~The rational normal curve has an assortment of nice properties~~:
	* Any <math>n+1</~~math> points on <math>C<~~/~~math> are linearly independent, and span <math>\mathbb{P}^n</math>~~. ~~This property distinguishes the rational normal curve from all other curves.~~
	* Given <math>n+3</math> points in <math>\mathbb{P}^n</~~math> in linear [[general position]] (that is, with no <math>n+1</math> lying in a [[hyperplane]]), there is a unique rational normal curve passing through them~~. The curve may be explicitly specified using the parametric representation, by arranging <math>n+1</math> of the points to lie on the coordinate axes, and then mapping the other two points to <math>[S:T]=~~[0:1]</math> and <math>[S:T]=[1:0]</math>.~~
	* The tangent and secant lines of a rational normal curve are pairwise disjoint, except at points of the curve itself. This is a property shared by sufficiently positive embeddings of any projective variety.

	~~There are <math>\binom{n+2}{2}-2n-1</math> independent [[quadric]]s that generate the [[ideal]] of the curve.~~

	~~The curve is not a [[complete intersection]], for <math>n>2<~~/~~math>. This means it is not defined by the number of equations equal to its [[codimension]] <math>n~~-1</~~math>.~~

	~~The [[canonical mapping]] for a [[hyperelliptic curve]] has image a rational normal curve, and is 2-to-1.~~

	~~Every irreducible non-degenerate curve <math>C\subset \mathbb{P}^n<~~/~~math> of degree <math>n</math> is a rational normal curve.~~

	~~==See also==~~

	*[[Rational normal surface]]

	~~==References==~~
	* Joe Harris, ''Algebraic Geometry, A First Course'', (1992) Springer-Verlag, ~~New York. ISBN 0-387-97716-3~~

	~~[[Category:Algebraic curves]]~~
	~~[[Category:Birational geometry]]~~

en>JorisvS: ce

2013-09-05T15:02:14Z

← Older revision		Revision as of 17:02, 5 September 2013
Line 1:		Line 1:
	~~Greetings~~. ~~The author~~'~~s title~~ is ~~Phebe~~ and ~~she feels comfy when people use~~ the complete ~~title~~. ~~I am a meter reader~~. ~~Body developing~~ is ~~what my family members and I enjoy~~. ~~Years ago we moved~~ to ~~Puerto Rico and my family enjoys~~ it.<br><br>~~my web blog~~: [~~http~~://~~mirim~~.ir/~~index~~.~~php?~~do=/~~profile~~-~~699~~/~~info~~/ ~~mirim~~.ir]		In [[mathematics]], the '''rational normal curve''' is a smooth, [[rational curve]] <math>C</math> of [[Degree of an algebraic variety\|degree]] ''n'' in [[projective space\|projective n-space]] <math>\mathbb{P}^n.</math> It is a simple example of a [[projective variety]]; formally, it is the [[Veronese variety]] when the domain is the projective line. For ''n''=2 it is the [[flat conic]] <math>Z_0 Z_2 = Z_1^2,</math> and for ''n''=3 it is the [[twisted cubic]]. The term "normal" is an old term meaning that the linear system defining the embedding is complete (and has nothing to do with [[normal scheme]]s). The intersection of the rational normal curve with an affine space is called the [[moment curve]].

			==Definition==
			The rational normal curve may be given [[parametrically]] as the image of the map

			:<math>\nu:\mathbb{P}^1\to\mathbb{P}^n</math>

			which assigns to the [[homogeneous coordinates]] <math>[S:T]</math> the value

			:<math>\nu:[S:T] \mapsto [S^n:S^{n-1}T:S^{n-2}T^2:\ldots:T^n].</math>

			In the [[affine coordinates]] of the chart <math> x_0\neq0 </math> the map is simply

			:<math>\nu:x \mapsto (x,x^2, \ldots ,x^n).</math>

			That is, the rational normal curve is the closure by a single [[point at infinity]] of the [[affine curve]] <math>(x,x^2,\dots,x^n)</math>.

			Equivalently, rational normal curve may be understood to be a [[projective variety]], defined as the common zero locus of the [[homogeneous polynomial]]s

			:<math>F_{i,j}(X_0,\ldots,X_n) = X_iX_j - X_{i+1}X_{j-1}</math>

			where <math>[X_0:\ldots:X_n]</math> are the [[homogeneous coordinate]]s on <math>\mathbb{P}^n</math>. The full set of these polynomials is not needed; it is sufficient to pick ''n'' of these to specify the curve.

			==Alternate parameterization==
			Let <math>[a_i:b_i]</math> be <math>n+1</math> distinct points in <math>\mathbb{P}^1</math>. Then the polynomial

			:<math>G(S,T) = \Pi_{i=0}^n (a_iS -b_iT)</math>

			is a [[homogeneous polynomial]] of degree <math>n+1</math> with distinct roots. The polynomials

			:<math>H_i(S,T) = \frac{G(S,T)} {(a_iS-b_iT)}</math>

			are then a [[Basis (linear algebra)\|basis]] for the space of homogeneous polynomials of degree ''n''. The map

			:<math>[S:T] \mapsto [H_0(S,T) : H_1(S,T) : \ldots : H_n (S,T) ]</math>

			or, equivalently, dividing by <math>G(S,T)</math>

			:<math>[S:T] \mapsto \left[\frac{1}{(a_0S-b_0T)} : \ldots : \frac{1}{(a_nS-b_nT)}\right] </math>

			is a rational normal curve. That this is a rational normal curve may be understood by noting that the [[monomial]]s <math>S^n,S^{n-1}T,S^{n-2}T^2,\ldots,T^n</math> are just one possible [[base (topology)\|basis]] for the space of degree-''n'' homogeneous polynomials. In fact, any [[Basis (linear algebra)\|basis]] will do. This is just an application of the statement that any two projective varieties are projectively equivalent if they are [[congruence relation\|congruent]] modulo the [[projective linear group]] <math>\rm{PGL}_{n+1} K</math> (with ''K'' the [[field (mathematics)\|field]] over which the projective space is defined).

			This rational curve sends the zeros of ''G'' to each of the coordinate points of <math>\mathbb{P}^n</math>; that is, all but one of the <math>H_i</math> vanish for a zero of ''G''. Conversely, any rational normal curve passing through the ''n+1'' coordinate points may be written parametrically in this way.

			==Properties==
			The rational normal curve has an assortment of nice properties:
			* Any <math>n+1</math> points on <math>C</math> are linearly independent, and span <math>\mathbb{P}^n</math>. This property distinguishes the rational normal curve from all other curves.
			* Given <math>n+3</math> points in <math>\mathbb{P}^n</math> in linear [[general position]] (that is, with no <math>n+1</math> lying in a [[hyperplane]]), there is a unique rational normal curve passing through them. The curve may be explicitly specified using the parametric representation, by arranging <math>n+1</math> of the points to lie on the coordinate axes, and then mapping the other two points to <math>[S:T]=[0:1]</math> and <math>[S:T]=[1:0]</math>.
			* The tangent and secant lines of a rational normal curve are pairwise disjoint, except at points of the curve itself. This is a property shared by sufficiently positive embeddings of any projective variety.

			There are <math>\binom{n+2}{2}-2n-1</math> independent [[quadric]]s that generate the [[ideal]] of the curve.

			The curve is not a [[complete intersection]], for <math>n>2</math>. This means it is not defined by the number of equations equal to its [[codimension]] <math>n-1</math>.

			The [[canonical mapping]] for a [[hyperelliptic curve]] has image a rational normal curve, and is 2-to-1.

			Every irreducible non-degenerate curve <math>C\subset \mathbb{P}^n</math> of degree <math>n</math> is a rational normal curve.

			==See also==

			*[[Rational normal surface]]

			==References==
			* Joe Harris, ''Algebraic Geometry, A First Course'', (1992) Springer-Verlag, New York. ISBN 0-387-97716-3

			[[Category:Algebraic curves]]
			[[Category:Birational geometry]]

en>StringTheory11: fixed dashes using a script

2012-08-18T04:32:36Z

fixed dashes using a script

New page

Greetings. The author's title is Phebe and she feels comfy when people use the complete title. I am a meter reader. Body developing is what my family members and I enjoy. Years ago we moved to Puerto Rico and my family enjoys it.<br><br>my web blog: [http://mirim.ir/index.php?do=/profile-699/info/ mirim.ir]