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	~~Bonus: WP Twin and WP Twin Auto Backup~~: (~~link to~~ ) ~~While not~~ a ~~theme, I think this software~~ is ~~essential if you are maintaining your Wordpress blog or regularly create new blog sites~~. It is ~~used by around 25%~~ of ~~all new websites, and there are more than 27 thousand plugins currently available. If you adored this article so you would like to be given more info regarding~~ [~~http://zpib.com/wordpress_dropbox_backup_606452 wordpress backup plugin~~] ~~kindly visit our page~~. ~~This~~ is a ~~service where people write articles using~~ a ~~specific keyword you have given them. s and intelligently including~~ a ~~substantial amount of key words in the title tags, image links, etc~~. ~~Understanding how Word~~ - ~~Press works can be~~ a ~~challenge~~, ~~but it is not too difficult when you learn more about it~~. <br><br>		[[File:Eutactic star 3D.png\|thumb\|right\|A eutactic star consisting of 5 pairs of vectors in three-dimensional space (''n'' = 3, ''s'' = 5)]]
			In [[Euclidean geometry]], a '''eutactic star''' is a [[geometrical figure]] in a [[Euclidean space]]. A star is a figure consisting of any number of opposing pairs of [[Euclidean vector\|vector]]s (or arms) issuing from a central origin. A star is eutactic if it is the [[orthogonal]] [[projection (mathematics)\|projection]] of a [[cross-polytope]] from a higher-dimensional space onto a [[Euclidean subspace\|subspace]]. Such stars were "eutactic" – meaning "well-situated" or "well-arranged" – by {{harvtxt \|Schläfli\|1901\|loc=p. 134}} because, for a common [[Scalar (mathematics)\|scalar multiple]], their vectors are projections of an [[orthonormal basis]].<ref>{{cite book\|title=Eigenspaces of graphs\|author=D. M. Cvetković\|coauthors= P. Rowlinson, S. Simić\|publisher=Cambridge University Press\|year=1997\|isbn=0-521-57352-1\|page=151}}</ref>

	~~Any business enterprise that~~ is ~~certainly worth its name should really shell out~~ a ~~good deal~~ in ~~making sure that they have~~ the ~~most effective website that provides related info to its prospect~~. ~~You do not catch~~ a ~~user~~'s ~~attention through big and large pictures that usually takes a millennium to load up~~. ~~With~~ the ~~free Word~~ - ~~Press blog~~, ~~you have the liberty to come up with your own personalized domain name~~. ~~Apart from~~ these, ~~you are also required to give some backlinks on other sites as well~~. ~~Moreover~~, ~~many Word~~ - ~~Press themes need to~~ be ~~purchased and designing your own WP site can be boring~~. <br><br>~~ve labored so hard to publish and put up on their website~~. ~~The nominee~~ in ~~each category~~ with the ~~most votes was crowned~~ the ~~2010 Parents Picks Awards WINNER and has been established as~~ the ~~best product, tip or place~~ in that ~~category~~. ~~After age 35~~, ~~18% of pregnancies will end in miscarriage~~. ~~Our skilled expertise~~, ~~skillfulness~~ and ~~excellence have been well known all across~~ the ~~world~~. ~~Websites using this content based strategy~~ are ~~always~~ given ~~top scores~~ by ~~Google.~~ <br><br>~~Word~~ - ~~Press installation is very easy and hassle free~~. ~~But~~ the ~~Joomla was created as~~ the ~~CMS over years~~ of ~~hard work~~. ~~Normally~~, the ~~Word - Press developers make a thorough research~~ on ~~your website goals and then ingrain~~ the ~~most suitable graphical design elements~~ to ~~your website~~. ~~So, we have to add our social media sharing buttons in website~~. ~~Now all you have to do is log into your Word~~ - ~~Press site making use~~ of ~~the very same username~~ and ~~password that you initially had in your previous site~~. <br><br>~~There~~ is ~~no denying that Magento~~ is ~~an ideal platform for building ecommerce websites, as it comes with an astounding number~~ of ~~options~~ that ~~can help your online business do extremely well. Mahatma Gandhi is known as one~~ of ~~the most prominent personalities and symbols of peace, non-violence and freedom~~. ~~However, you must also manually approve or reject comments so~~ that ~~your website~~ does not ~~promote parasitic behavior~~. ~~) Remote Login: With~~ the ~~process~~ of ~~PSD to Wordpress conversion comes the advantage~~ of ~~flexibility. Definitely when you wake up from the slumber~~, ~~you~~ can be ~~sure~~ that ~~you will~~ be ~~lagging behind~~ and ~~getting on track would be~~ a ~~tall order~~.		== Definition ==
			[[File:Eutactic star.png\|thumb\|right\|A eutactic star in the plane (''n'' = 2, ''s'' = 4)]]
			A ''star'' is here defined as a set of 2''s'' vectors ''A'' = ±'''a'''<sub>1</sub>, ..., ±'''a'''<sub>''s''</sub> issuing from a particular origin in a Euclidean space of dimension ''n'' ≤ ''s''. A star is eutactic if the '''a'''<sub>''i''</sub> are the projections onto ''n'' dimensions of a set of mutually [[perpendicular]] equal vectors '''b'''<sub>1</sub>, ..., '''b'''<sub>''s''</sub> issuing from a particular origin in Euclidean ''s''-dimensional space.<ref name="Coxeter">{{cite book\|title=Regular polytopes\|last1=Coxeter \| first1=Harold Scott MacDonald \| author1-link=Harold Scott MacDonald Coxeter \|publisher=Courier Dover Publications\|year=1973\|isbn=0-486-61480-8\|page=251}}</ref> The configuration of 2''s'' vectors in the ''s''-dimensional space ''B'' = ±'''b'''<sub>1</sub>, ... , ±'''b'''<sub>''s''</sub> is known as a ''cross''. Given these definitions, a eutactic star is, concisely, a star produced by the orthogonal projection of a cross.

			An equivalent definition, first mentioned by [[Schläfli]],<ref name="Schläfli">{{Cite book \| last1=Schläfli \| first1=Ludwig \| title=Collected mathematical works \| publisher=Birkhäuser Verlag \| language=German \| year=1949 \| volume=I \| chapter=Theorie der vielfachen Kontinuität\|zbl=0035.21902 \| postscript=<!--None--> }}
			</ref> stipulates that a star is eutactic if a constant ''ζ'' exists such that

			:<math>\sum_{k=1}^s(\mathbf{a}_k \cdot \mathbf{v})^2 = \zeta \|\mathbf{v}\|^2</math>

			for every vector '''v'''. The existence of such a constant requires that the sum of the squares of the orthogonal projections of ''A'' on a line be equal in all directions.<ref name="Canada">{{Cite journal \| doi=10.4153/CJM-1951-045-8 \| last1=Coxeter \| first1=Harold Scott MacDonald \| author1-link=Harold Scott MacDonald Coxeter \| title=Extreme forms \| mr=0044580 \| year=1951 \| journal=Canadian Journal of Mathematics\| issn=0008-414X \| volume=3 \| pages=391–441 \| postscript=<!--None-->}}</ref> In general,
			:<math>\zeta = \sum_{k=1}^s \|\mathbf{a}_k\|^2/n.</math>

			A ''normalised'' eutactic star is a projected cross composed of [[unit vector]]s.<ref name="Coxeter" /><ref>{{cite web\|url=http://mathworld.wolfram.com/EutacticStar.html\|title=Eutactic Star – MathWorld\|author=E. W. Weisstein\|accessdate=2009-08-28}}</ref> Eutactic stars are often considered in ''n'' = 3 dimensions because of their connection with the study of [[regular polyhedra]].

			==Hadwiger's principal theorem==
			Let ''T'' be the [[symmetric matrix\|symmetric]] [[linear transformation]] defined for vectors '''x''' by
			:<math>T\mathbf{x} = \sum_{k=1}^s \mathbf{a}_k (\mathbf{a}_k\cdot\mathbf{x}),</math>
			where the '''a'''<sub>''j''</sub> form any collection of ''s'' vectors in the ''n''-dimensional Euclidean space. [[Hadwiger]]'s principal theorem states that the vectors ±'''a'''<sub>1</sub>, ..., ±'''a'''<sub>''s''</sub> form a eutactic star [[if and only if]] there is a constant ''ζ'' such that ''T'''''x''' = ''ζ'''''x''' for every '''x'''.<ref name="Coxeter" /><ref>{{cite web\|url=http://mathworld.wolfram.com/HadwigersPrincipalTheorem.html\|title=Hadwiger's Principal Theorem – MathWorld\|author=E. W. Weisstein\|accessdate=2009-08-28}}</ref> The vectors form a normalized eutactic star precisely when ''T'' is the [[identity operator]] – when ''ζ'' = 1.

			Equivalently, the star is normalized eutactic if and only if the [[matrix (mathematics)\|matrix]] '''A''' = ['''a'''<sub>1</sub> ... '''a'''<sub>''s''</sub>], whose columns are the vectors '''a'''<sub>''k''</sub>, has [[orthonormal]] rows. A proof may be given in one direction by completing the rows of this matrix to an [[orthonormal basis]] of <math>\mathbb{R}^s</math>, and in the other by orthogonally projecting onto the ''n''-dimensional subspace spanned by the first ''n'' Cartesian coordinate vectors.

			Hadwiger's theorem implies the equivalence of Schläfli's stipulation and the geometrical definition of a eutactic star, by the [[polarization identity]]. Furthermore, both Schläfli's identity and Hadwiger's theorem give the same value of the constant ''ζ''.

			== Applications ==

			Eutactic stars are useful largely because of their relationship with the geometry of [[polytope]]s and [[group (mathematics)\|groups]] of [[orthogonal transformation]]s. Schläfli showed early on that the vectors from the center of any regular polytope to its vertices form a eutactic star. Brauer and Coxeter proved the following generalization:<ref>{{Cite journal \| last1=Brauer \| first1=R. \| author1-link=Richard Brauer \| last2=Coxeter \| first2=Harold Scott MacDonald \| author2-link=Harold Scott MacDonald Coxeter \| title=A generalization of theorems of Schönhardt and Mehmke on polytopes \| mr=0002869 \| year=1940 \| journal=Trans. Roy. Soc. Canada. Sect. III. (3) \| volume=34 \| pages=29–34 \| postscript=<!--None-->}}.</ref>

			:<blockquote>A star is eutactic if and only if it is transformed to itself by some irreducible group of orthogonal transformations that acts transitively on pairs of opposite vectors.</blockquote>

			An irreducible group here means a group that does not leave any nontrivial proper subspace invariant (see [[irreducible representation]]). Since the set theoretic union of two eutactic stars is itself eutactic (a consequence of [[#Hadwiger's principal theorem\|Hadwiger's principal theorem]]), it can be concluded that, in general:<ref name="Canada" />
			:<blockquote>A star is eutactic if and only if it is transformed to itself by some irreducible group of orthogonal transformations.</blockquote>

			Eutactic stars may be used to validate the eutaxy of any form in general. According to [[H. S. M. Coxeter]]: "A form is eutactic if and only if its minimal vectors are [[parallel (geometry)\|parallel]] to the vectors of a eutactic star."<ref name="Canada" />

			==See also==

			*[[Eutactic lattice]]
			*[[Tight frame]]<ref>http://www.biblioteca.juriquilla.unam.mx/WONDERLAND/ParaJaviero/cfata/import/GomezA2007.pdf</ref>

			== References ==
			{{reflist}}
			*{{Citation \| last1=Schläfli \| first1=Ludwig \| editor1-last=Graf \| editor1-first=J. H. \| title=Theorie der vielfachen Kontinuität \| origyear=1852 \| url=http://books.google.com/books?id=foIUAQAAMAAJ \| publisher=Zürich, Basel: Georg & Co. \| language=German \| series=Republished by Cornell University Library historical math monographs 2010 \| isbn=978-1-4297-0481-6 \| year=1901}}

			[[Category:Polytopes]]
			[[Category:Vectors]]

en>David Eppstein: full eponym

2010-08-24T06:52:27Z

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Bonus: WP Twin and WP Twin Auto Backup: (link to ) While not a theme, I think this software is essential if you are maintaining your Wordpress blog or regularly create new blog sites. It is used by around 25% of all new websites, and there are more than 27 thousand plugins currently available. If you adored this article so you would like to be given more info regarding [http://zpib.com/wordpress_dropbox_backup_606452 wordpress backup plugin] kindly visit our page. This is a service where people write articles using a specific keyword you have given them. s and intelligently including a substantial amount of key words in the title tags, image links, etc. Understanding how Word - Press works can be a challenge, but it is not too difficult when you learn more about it. 

Any business enterprise that is certainly worth its name should really shell out a good deal in making sure that they have the most effective website that provides related info to its prospect. You do not catch a user's attention through big and large pictures that usually takes a millennium to load up. With the free Word - Press blog, you have the liberty to come up with your own personalized domain name. Apart from these, you are also required to give some backlinks on other sites as well. Moreover, many Word - Press themes need to be purchased and designing your own WP site can be boring. ve labored so hard to publish and put up on their website. The nominee in each category with the most votes was crowned the 2010 Parents Picks Awards WINNER and has been established as the best product, tip or place in that category. After age 35, 18% of pregnancies will end in miscarriage. Our skilled expertise, skillfulness and excellence have been well known all across the world. Websites using this content based strategy are always given top scores by Google. Word - Press installation is very easy and hassle free. But the Joomla was created as the CMS over years of hard work. Normally, the Word - Press developers make a thorough research on your website goals and then ingrain the most suitable graphical design elements to your website. So, we have to add our social media sharing buttons in website. Now all you have to do is log into your Word - Press site making use of the very same username and password that you initially had in your previous site. There is no denying that Magento is an ideal platform for building ecommerce websites, as it comes with an astounding number of options that can help your online business do extremely well. Mahatma Gandhi is known as one of the most prominent personalities and symbols of peace, non-violence and freedom. However, you must also manually approve or reject comments so that your website does not promote parasitic behavior. ) Remote Login: With the process of PSD to Wordpress conversion comes the advantage of flexibility. Definitely when you wake up from the slumber, you can be sure that you will be lagging behind and getting on track would be a tall order.

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en>Addbot: Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q3115517

en>David Eppstein: full eponym