W Ursae Majoris variable - Revision history

en>Casliber: oops...lets try that again....

2014-07-26T01:55:41Z

oops...lets try that again....

← Older revision		Revision as of 03:55, 26 July 2014
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	~~In [[geometry]], a set of [[line (geometry)\|line]]s in [[Euclidean space]]~~ is ~~called '''equiangular''' if every pair of lines makes the same angle.~~		Oscar is what my wife enjoys to call me and I totally dig that name. South Dakota is her beginning location but she requirements to move simply because of her family. He utilized to be unemployed but now he is a pc operator but his marketing never arrives. To do aerobics is a thing that I'm completely addicted to.<br><br>My site ... [http://independentmumbaiescort.com/anamika/blogs/post/4358 over the counter std test]

	~~Computing the maximum number of equiangular lines in ''n''-dimensional Euclidean space is a difficult problem,~~ and ~~unsolved in general, though bounds are known~~. ~~The maximal number of equiangular lines in 2-dimensional Euclidean space~~ is ~~3: we can take the lines through opposite vertices~~ of ~~a regular hexagon, each at an angle 120 degrees from the other two~~. ~~The maximum in 3 dimensions~~ is ~~6: we can take lines through opposite vertices of an icosahedron~~. ~~The maximum in dimensions 1 through 18~~ is ~~listed in The On-Line Encyclopedia of Integer Sequences as follows:~~

	~~1, 3, 6, 6, 10, 16, 28, 28, 28, 28, 28, 28, 28, 28, 36, 40, 48, 48, ...~~

	In particular, the maximum number of equiangular lines in 7 dimensions is 28. We can obtain these lines as follows. Take the vector (-3,-3,1,1,1,1,1,1) in <math>\mathbb{R}^8</math>, and form all 28 vectors obtained by permuting the components of this. The dot product of two of these vectors is 8 if both have a ~~component 3 in the same place or -8 otherwise. Thus, the lines through the origin containing these vectors are equiangular. Moreoever, all 28 vectors are orthogonal~~ to ~~the vector (1,1,1,1,1,1,1,1) in <math> \mathbb{R}^8</math>, so they lie in a 7-dimensional space~~. ~~In fact, these 28 vectors and their negatives are, up to rotation and dilation, the 56 vertices of the [[3 21 polytope\|3<sub>21~~<~~/sub~~> ~~polytope]]. In other words, they are the weight vectors of the 56-dimensional representation of the Lie group [[E7 (mathematics)\|E~~<~~sub~~>~~7</sub>]].~~

	Equiangular lines are related to [[two-graph]]s. Given a set of equiangular lines, let ''c'' be the [[cosine]] of the common angle. We assume that the angle is not 90°, since that case is trivial (i.e., not interesting, because the lines are just coordinate axes); thus, ''c'' is nonzero. We may move the lines so they all pass through the [[origin (mathematics)\|origin]] of coordinates. Choose one unit vector in each line. Form the [[matrix (mathematics)\|matrix]] ''M'' of [[inner product]]s. This matrix has 1 on the diagonal and ±c everywhere else, and it is symmetric. Subtracting the [[identity matrix]] ''I'' and dividing by ''c'', we have a [[symmetric matrix]] with zero diagonal and ±1 off the diagonal. ~~This is the [[adjacency matrix]] of a two-graph~~.

	~~==References==~~

	* The On-Line Encyclopedia of Integer Sequences, [http://~~oeis.org/A002853 Maximal size of a set of equiangular lines in n dimensions], sequence number A002853.~~
	* Gosselin, S.~~, [http:~~//~~ion.uwinnipeg.ca~~/~~~sgosseli~~/~~Master%27s_Thesis_Gosselin.pdf Regular two-graphs and equiangular lines], Master's thesis, Mathematics Department, University of Waterloo, 2004.~~
	* van Lint, J. H., and Seidel, J. J. Equilateral point sets in elliptic geometry. ''Proc. Koninkl. Ned. Akad. Wetenschap. Ser. A'' '''69''' (= ''Indagationes Mathematicae'' '''28''') (1966), 335-348.
	* [[Andries Brouwer\|Brouwer, A.E.]], Cohen, A.M., and Neumaier, A. ''Distance-Regular Graphs.'' Springer-Verlag, Berlin, 1989. Section 3.8.
	* {{citation\|first1=Chris\|last1=Godsil\|authorlink1=Chris Godsil\|first2=Gordon\|last2=Royle\|authorlink2=Gordon Royle\|title=Algebraic Graph Theory\|series=Graduate Texts in Mathematics\|volume=207\|publisher=Springer-Verlag\|location=New York\|year=2001}}. ''(See Chapter 11.)''

	~~[[Category:Euclidean geometry]]~~
	~~[[Category:Combinatorics]~~]

en>Bibcode Bot: Adding 1 arxiv eprint(s), 0 bibcode(s) and 1 doi(s). Did it miss something? Report bugs, errors, and suggestions at User talk:Bibcode Bot

2013-07-24T22:55:14Z

Adding 1 arxiv eprint(s), 0 bibcode(s) and 1 doi(s). Did it miss something? Report bugs, errors, and suggestions at User talk:Bibcode Bot

← Older revision		Revision as of 00:55, 25 July 2013
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	~~Myrtle Benny~~ is ~~how I~~'~~m called~~ and ~~I really feel comfortable when individuals use~~ the ~~full name~~. ~~Years ago~~ we ~~moved to North Dakota~~. ~~He used to be unemployed but now he~~ is ~~a computer operator but his marketing~~ by ~~no means comes~~. ~~One~~ of the ~~things she loves most is~~ to ~~do aerobics~~ and ~~now she is attempting~~ to ~~earn money with it~~.<br><br>~~Here~~ is ~~my blog post~~ - [http://~~www~~.~~blaze16~~.~~com~~/~~blog~~/~~257588 at home std test~~]		In [[geometry]], a set of [[line (geometry)\|line]]s in [[Euclidean space]] is called '''equiangular''' if every pair of lines makes the same angle.

			Computing the maximum number of equiangular lines in ''n''-dimensional Euclidean space is a difficult problem, and unsolved in general, though bounds are known. The maximal number of equiangular lines in 2-dimensional Euclidean space is 3: we can take the lines through opposite vertices of a regular hexagon, each at an angle 120 degrees from the other two. The maximum in 3 dimensions is 6: we can take lines through opposite vertices of an icosahedron. The maximum in dimensions 1 through 18 is listed in The On-Line Encyclopedia of Integer Sequences as follows:

			1, 3, 6, 6, 10, 16, 28, 28, 28, 28, 28, 28, 28, 28, 36, 40, 48, 48, ...

			In particular, the maximum number of equiangular lines in 7 dimensions is 28. We can obtain these lines as follows. Take the vector (-3,-3,1,1,1,1,1,1) in <math>\mathbb{R}^8</math>, and form all 28 vectors obtained by permuting the components of this. The dot product of two of these vectors is 8 if both have a component 3 in the same place or -8 otherwise. Thus, the lines through the origin containing these vectors are equiangular. Moreoever, all 28 vectors are orthogonal to the vector (1,1,1,1,1,1,1,1) in <math> \mathbb{R}^8</math>, so they lie in a 7-dimensional space. In fact, these 28 vectors and their negatives are, up to rotation and dilation, the 56 vertices of the [[3 21 polytope\|3<sub>21</sub> polytope]]. In other words, they are the weight vectors of the 56-dimensional representation of the Lie group [[E7 (mathematics)\|E<sub>7</sub>]].

			Equiangular lines are related to [[two-graph]]s. Given a set of equiangular lines, let ''c'' be the [[cosine]] of the common angle. We assume that the angle is not 90°, since that case is trivial (i.e., not interesting, because the lines are just coordinate axes); thus, ''c'' is nonzero. We may move the lines so they all pass through the [[origin (mathematics)\|origin]] of coordinates. Choose one unit vector in each line. Form the [[matrix (mathematics)\|matrix]] ''M'' of [[inner product]]s. This matrix has 1 on the diagonal and ±c everywhere else, and it is symmetric. Subtracting the [[identity matrix]] ''I'' and dividing by ''c'', we have a [[symmetric matrix]] with zero diagonal and ±1 off the diagonal. This is the [[adjacency matrix]] of a two-graph.

			==References==

			* The On-Line Encyclopedia of Integer Sequences, [http://oeis.org/A002853 Maximal size of a set of equiangular lines in n dimensions], sequence number A002853.
			* Gosselin, S., [http://ion.uwinnipeg.ca/~sgosseli/Master%27s_Thesis_Gosselin.pdf Regular two-graphs and equiangular lines], Master's thesis, Mathematics Department, University of Waterloo, 2004.
			* van Lint, J. H., and Seidel, J. J. Equilateral point sets in elliptic geometry. ''Proc. Koninkl. Ned. Akad. Wetenschap. Ser. A'' '''69''' (= ''Indagationes Mathematicae'' '''28''') (1966), 335-348.
			* [[Andries Brouwer\|Brouwer, A.E.]], Cohen, A.M., and Neumaier, A. ''Distance-Regular Graphs.'' Springer-Verlag, Berlin, 1989. Section 3.8.
			* {{citation\|first1=Chris\|last1=Godsil\|authorlink1=Chris Godsil\|first2=Gordon\|last2=Royle\|authorlink2=Gordon Royle\|title=Algebraic Graph Theory\|series=Graduate Texts in Mathematics\|volume=207\|publisher=Springer-Verlag\|location=New York\|year=2001}}. ''(See Chapter 11.)''

			[[Category:Euclidean geometry]]
			[[Category:Combinatorics]]

en>Cydebot: Robot - Moving category Common envelope binaries to :Category:Common envelope binary stars per CFD at Wikipedia:Categories for discussion/Log/2012 March 21.

2012-03-28T22:42:52Z

Robot - Moving category Common envelope binaries to Category:Common envelope binary stars per CFD at Wikipedia:Categories for discussion/Log/2012 March 21.

New page

Myrtle Benny is how I'm called and I really feel comfortable when individuals use the full name. Years ago we moved to North Dakota. He used to be unemployed but now he is a computer operator but his marketing by no means comes. One of the things she loves most is to do aerobics and now she is attempting to earn money with it.<br><br>Here is my blog post - [http://www.blaze16.com/blog/257588 at home std test]