Inverse magnetostrictive effect - Revision history

en>Niuthon: /* Explanation */

2014-08-04T21:20:10Z

Explanation

← Older revision		Revision as of 23:20, 4 August 2014
Line 1:		Line 1:
	~~In mathematics~~, ~~'''Specht~~'s ~~theorem''' gives~~ a ~~[[necessary and sufficient condition]] for two [[matrix (mathematics)\|matrices]]~~ to ~~be [[unitarily equivalence (matrix theory)\|unitarily equivalent]]. It~~ is ~~named after [[Wilhelm Specht]], who proved the theorem in 1940~~.<~~ref~~>~~{{harvtxt\|Specht\|1940}}~~<~~/ref~~>		Surely the second option would be more beneficial for any website. The next step is to visit your Word - Press blog dashboard. I thought about what would happen by placing a text widget in the sidebar beneath my banner ad, and so it went. Transforming your designs to Word - Press blogs is not that easy because of the simplified way in creating your very own themes. It's as simple as hiring a Wordpress plugin developer or learning how to create what is needed. <br><br>

	~~Two matrices ''A'' and ''B'' are said~~ to be ~~''unitarily equivalent'' if there exists~~ a ~~[[unitary matrix]] ''U'' such~~ that ~~''B'' = ''U'' *''AU''~~.~~<ref>{{harvtxt\|Horn\|Johnson\|1985}}, Definition 2.2~~.1</~~ref> Two matrices which are unitarily equivalent are also [[similar matrices\|similar]~~]. ~~Two similar matrices represent~~ the ~~same [[linear map]], but with respect~~ to a ~~different [[basis of a vector space\|basis]]; unitary equivalence corresponds to a change from an [[orthonormal basis]] to another orthonormal basis~~.		As you know today Word - Press has turn out to be a tremendously popular open source publishing and blogging display place. You may either choose to link only to the top-level category pages or the ones that contain information about your products and services. If you beloved this information and also you want to acquire guidance relating to [http://htxurl.info/backupplugin765943 wordpress backup plugin] kindly visit our web site. You are able to set them within your theme options and so they aid the search engine to get a suitable title and description for the pages that get indexed by Google. These four plugins will make this effort easier and the sites run effectively as well as make other widgets added to a site easier to configure. Many times the camera is following Mia, taking in her point of view in almost every frame. <br><br>ve labored so hard to publish and put up on their website. When a business benefits from its own domain name and a tailor-made blog, the odds of ranking higher in the search engines and being visible to a greater number of people is more likely. After age 35, 18% of pregnancies will end in miscarriage. Storing write-ups in advance would have to be neccessary with the auto blogs. Purchase these from our site, or bring your own, it doesn't matter, we will still give you free installation and configuration. <br><br>Word - Press installation is very easy and hassle free. And, make no mistake,India's Fertility Clinics and IVF specialists are amongst the best in the world,and have been for some time. Websites that do rank highly, do so becaue they use keyword-heavy post titles. If you just want to share some picture and want to use it as a dairy, that you want to share with your friends and family members, then blogger would be an excellent choice. 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. Automated deal feed integration option to populate your blog with relevant deals. However, you must also manually approve or reject comments so that your website does not promote parasitic behavior. If this is not possible you still have the choice of the default theme that is Word - Press 3. 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.

	~~If ''A''~~ and ~~''B'' are unitarily equivalent, then tr ''AA''* = tr ''BB'', where tr denotes~~ the ~~[[trace (linear algebra)\|trace]] (in~~ other ~~words, the [[Frobenius norm]] is~~ a ~~unitary invariant)~~. ~~This follows from~~ the ~~cyclic invariance~~ of the trace: if ''B'' = ''U'' ''AU'', then tr ''BB''* = tr ''U'' ''AUU'' ''A''''U'' = tr ''AUU'' ''A''''UU''  = tr ''AA''*, where the second equality is cyclic invariance.<~~ref~~>~~{{harvtxt\|Horn\|Johnson\|1985}}, Theorem 2.2.2~~<~~/ref~~>

	~~Thus~~, ~~tr ''AA''* = tr ''BB''* is~~ a ~~necessary condition for unitary equivalence, but it~~ is ~~not sufficient~~. ~~Specht's theorem gives infinitely many necessary conditions which together are also sufficient. The formulation~~ of ~~the theorem uses the following definition~~. ~~A [[word (mathematics)\|word]]~~ in ~~two variables, say ''x'' and ''y'', is an expression of~~ the ~~form~~

	~~:<math>~~
	~~W(x~~,~~y) = x^{m_1} y^{n_1} x^{m_2} y^{n_2} \cdots x^{m_p}~~, \,
	~~</math>~~

	~~where ''m''<sub>1</sub>, ''n'~~'~~<sub>1</sub>~~, ~~''m''~~<~~sub~~>2<~~/sub~~>~~, ''n''<sub>2</sub>, …, ''m''<sub>''p''</sub> are non~~-~~negative integers. The ''degree'' of this word~~ is

	~~:<math>~~
	~~m_1 + n_1 + m_2 + n_2 + \cdots + m_p~~. \,
	~~</math>~~

	~~'''Specht~~'s ~~theorem:''' Two matrices ''A''~~ and ~~''B''~~ are ~~unitarily equivalent if~~ and ~~only if tr ''W''(''A'', ''A'') = tr ''W''(''B'', ''B'')~~ for ~~all words ''W''~~.~~<ref>{{harvtxt\|Horn\|Johnson\|1985}}~~, ~~Theorem 2~~.~~2.6</ref>~~

	~~The theorem gives an infinite number of trace identities, but~~ it ~~can be reduced to~~ a ~~finite subset. Let ''n'' denote the size of the matrices ''A'' and ''B''. For the case ''n'' = 2~~, ~~the following three conditions are sufficient:<ref>{{harvtxt\|Horn\|Johnson\|1985}}, Theorem 2.2.8</ref>~~

	~~:<math>~~
	~~\operatorname{tr} \, A = \operatorname{tr} \, B, \quad~~
	~~\operatorname{tr} \, A^2 = \operatorname{tr} \, B^2, \quad\text{~~and~~}\quad~~
	~~\operatorname{tr} \~~, AA^* = \operatorname{tr} \, BB^*.
	~~</math>~~

	~~For ''n'' = 3,~~ the ~~following seven conditions are sufficient:~~

	~~:<math>~~
	~~\begin{align}~~
	~~&\operatorname{tr} \, A = \operatorname{tr} \, B, \quad~~
	~~\operatorname{tr} \, A^2 = \operatorname{tr} \, B^2, \quad~~
	~~\operatorname{tr} \, AA^* = \operatorname{tr} \, BB^*, \quad~~
	~~\operatorname{tr} \, A^3 = \operatorname{tr} \, B^3, \\~~
	~~&\operatorname{tr} \, A^2 A^* = \operatorname{tr} \, B^2 B^*, \quad~~
	~~\operatorname{tr} \, A^2 (A^)^2 = \operatorname{tr} \, B^2 (B^)^2, \quad\text{~~and~~}\quad~~
	\operatorname{tr} \, A^2 (A^)^2 A A^ = \operatorname{tr} \, B^2 (B^)^2 B B^.
	~~\end{align}~~
	<~~/math~~> ~~ ~~<~~ref~~>~~{{harvtxt\|Sibirskiǐ\|1976}}, p. 260, quoted by {{harvtxt\|Đoković\|Johnson\|2007}}</ref>~~
	~~For general ''n''~~, it ~~suffices to show that tr ''W''(''A'', ''A'') = tr ''W''(''B'', ''B'') for all words~~ of ~~degree at most~~

	~~:<math>~~
	~~n \sqrt{\frac{2n^2}{n-1} + \frac14} + \frac{n}2 - 2.~~
	~~</math>  <ref>{{harvtxt\|Pappacena\|1997}}, Theorem 4.3</ref>~~

	~~It has been conjectured~~ that ~~this~~ can ~~be reduced~~ to ~~an expression linear in ''n''~~.~~<ref>{{harvtxt\|Freedman\|Gupta\|Guralnick\|1997}}~~, p. ~~160</ref>~~

	~~== Notes ==~~
	~~<references/>~~

	~~== References ==~~
	* {{Citation \| last1=Đoković \| first1=Dragomir Ž. \| last2=Johnson \| first2=Charles R. \| title=Unitarily achievable zero patterns and traces of ~~words in ''A'' and ''A''* \| doi=10.1016/j.laa.2006.03.002 \| year=2007 \| journal=Linear Algebra and its Applications \| issn=0024-3795 \| volume=421 \| issue=1 \| pages=63–68}}.~~
	* {{Citation \| last1=Freedman \| first1=Allen R. \| last2=Gupta \| first2=Ram Niwas \| last3=Guralnick \| first3=Robert M. \| title=Shirshov's theorem and representations of semigroups \| url=http://pjm.math.berkeley.edu/pjm/1997/181-3~~/p07~~.~~xhtml \| year=1997 \| journal=[[Pacific Journal of Mathematics]] \| issn=0030-8730 \| volume=181 \| issue=3 \| pages=159–176 \| doi=10.2140/pjm.1997.181.159}}.~~
	* {{Citation \| last1=Horn \| first1=Roger A. \| last2=Johnson \| first2=Charles R. \| title=Matrix Analysis \| publisher=[[Cambridge University Press]] \| isbn=978-0-521-38632-6 \| year=1985}}.
	* {{Citation \| last1=Pappacena \| first1=Christopher J. \| title=An upper bound for the ~~length of~~ a ~~finite-dimensional algebra \| doi=10.1006/jabr.1997.7140 \| year=1997 \| journal=Journal of Algebra \| issn=0021-8693 \| volume=197 \| issue=2 \| pages=535–545}}.~~
	* {{Citation \| last1=Sibirskiǐ \| first1=K. S. \| title=Algebraic Invariants of Differential Equations and Matrices \| publisher=Izdat. "Štiinca", Kishinev \| language=Russian \| year=1976}}.
	* {{Citation \| last1=Specht \| first1=Wilhelm \| author1-link=Wilhelm Specht \| title=Zur Theorie der Matrizen. II \| url=http://gdz.sub.uni-goettingen.~~de/dms/load/toc/?PPN=PPN37721857X_0050&DMDID=dmdlog6 \| year=1940 \| journal=Jahresbericht der Deutschen Mathematiker-Vereinigung \| issn=0012-0456 \| volume=50 \| pages=19–23}}.~~

	~~{{DEFAULTSORT:Specht's Theorem}}~~
	~~[[Category:Matrix theory]]~~
	~~[[Category:Combinatorics on words]]~~
	~~[[Category:Theorems in algebra]]~~

en>Zigger: /* Explanation */ + wikilink

2013-12-05T13:44:14Z

Explanation: + wikilink

← Older revision		Revision as of 15:44, 5 December 2013
Line 1:		Line 1:
	~~They are typically~~ a ~~free website that are pre-designed~~ for enabling businesses of every size in marking the presence on the internet and allows them in showcasing the product services and range through images, contents and various other elements. This one is one of the most beneficial features of Word - Press as this feature allows users to ~~define the user roles~~. ~~Step-4 Testing:~~ It is ~~the foremost important of your Plugin development process. If you need a special plugin for your website~~ , ~~there are thousands of plugins that can be used to meet those needs. By using this method one can see whether the theme has~~ the ~~potential to become popular or not and is their any scope of improvement~~ in ~~the theme~~. <br><br>		In mathematics, '''Specht's theorem''' gives a [[necessary and sufficient condition]] for two [[matrix (mathematics)\|matrices]] to be [[unitarily equivalence (matrix theory)\|unitarily equivalent]]. It is named after [[Wilhelm Specht]], who proved the theorem in 1940.<ref>{{harvtxt\|Specht\|1940}}</ref>

	~~Generally, for my private income-making market websites, I will thoroughly research~~ and ~~discover the leading 10 most worthwhile niches~~ to ~~venture into~~. ~~You will have to invest some money into tuning up your own blog but~~, ~~if done wisely, your investment will pay off in the long run~~. ~~This is~~ the ~~reason for the increased risk~~ of ~~Down Syndrome babies in women over age 35. t need~~ to ~~use the back button or the URL~~ to ~~get to your home page~~. ~~Aided by~~ the ~~completely foolproof j - Query color selector~~, ~~you're able to change~~ the ~~colors of factors of your theme~~ a the ~~click on~~ the ~~screen~~, ~~with very little previous web site design experience~~. <br><br>The ~~entrepreneurs can easily captivate their readers by using these versatile themes~~. ~~Browse through~~ the ~~popular Wordpress theme clubs like the Elegant Themes~~, ~~Studio Press~~, ~~Woo - Themes~~, ~~Rocket Theme~~, ~~Simple Themes and many more. If you beloved this article so you would like to get more info with regards to [http:~~//~~twsi.in~~/~~wordpress_backup_82405 backup plugin] kindly visit our own web site. I~~'~~ve applied numerous Search engine optimization~~-~~ready Word - Press themes and I can say from knowledge that I consider the Genesis Search engine marketing panel one particular of the simplest to use~~. The ~~first thing you need to do~~ is ~~to choose the right web hosting plan~~. ~~Search engine optimization pleasant picture~~ and ~~solution links suggest you will have a much better adjust at gaining considerable natural site visitors~~. <br><br>~~It has become~~ a ~~more prevalent cause~~ of ~~infertility~~ and the ~~fertility clinic~~ are ~~having more~~ and ~~more couples with infertility problems~~. ~~php file in~~ the ~~Word - Press root folder~~ and ~~look for this line~~ (~~line 73 in our example~~):. ~~Next you~~'~~ll go by way of~~ to ~~your simple Word - Press site. There are many advantages~~ of ~~hiring Wordpress developers for Wordpress project development~~:~~. If your site does well you can get paid professional designer to create a unique Word~~ - ~~Press theme~~. <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~~. ~~An ease of use which pertains to both internet site back~~-~~end~~ and ~~front~~-~~end users alike~~. ~~By the time you get the Gallery Word~~ - Press ~~Themes,~~ the ~~first thing that you should know is on how to install it~~. ~~Page speed is an important factor in ranking~~, ~~especially with Google~~. ~~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.~~		Two matrices ''A'' and ''B'' are said to be ''unitarily equivalent'' if there exists a [[unitary matrix]] ''U'' such that ''B'' = ''U'' *''AU''.<ref>{{harvtxt\|Horn\|Johnson\|1985}}, Definition 2.2.1</ref> Two matrices which are unitarily equivalent are also [[similar matrices\|similar]]. Two similar matrices represent the same [[linear map]], but with respect to a different [[basis of a vector space\|basis]]; unitary equivalence corresponds to a change from an [[orthonormal basis]] to another orthonormal basis.

			If ''A'' and ''B'' are unitarily equivalent, then tr ''AA''* = tr ''BB'', where tr denotes the [[trace (linear algebra)\|trace]] (in other words, the [[Frobenius norm]] is a unitary invariant). This follows from the cyclic invariance of the trace: if ''B'' = ''U'' ''AU'', then tr ''BB''* = tr ''U'' ''AUU'' ''A''''U'' = tr ''AUU'' ''A''''UU''  = tr ''AA''*, where the second equality is cyclic invariance.<ref>{{harvtxt\|Horn\|Johnson\|1985}}, Theorem 2.2.2</ref>

			Thus, tr ''AA''* = tr ''BB''* is a necessary condition for unitary equivalence, but it is not sufficient. Specht's theorem gives infinitely many necessary conditions which together are also sufficient. The formulation of the theorem uses the following definition. A [[word (mathematics)\|word]] in two variables, say ''x'' and ''y'', is an expression of the form

			:<math>
			W(x,y) = x^{m_1} y^{n_1} x^{m_2} y^{n_2} \cdots x^{m_p}, \,
			</math>

			where ''m''<sub>1</sub>, ''n''<sub>1</sub>, ''m''<sub>2</sub>, ''n''<sub>2</sub>, …, ''m''<sub>''p''</sub> are non-negative integers. The ''degree'' of this word is

			:<math>
			m_1 + n_1 + m_2 + n_2 + \cdots + m_p. \,
			</math>

			'''Specht's theorem:''' Two matrices ''A'' and ''B'' are unitarily equivalent if and only if tr ''W''(''A'', ''A'') = tr ''W''(''B'', ''B'') for all words ''W''.<ref>{{harvtxt\|Horn\|Johnson\|1985}}, Theorem 2.2.6</ref>

			The theorem gives an infinite number of trace identities, but it can be reduced to a finite subset. Let ''n'' denote the size of the matrices ''A'' and ''B''. For the case ''n'' = 2, the following three conditions are sufficient:<ref>{{harvtxt\|Horn\|Johnson\|1985}}, Theorem 2.2.8</ref>

			:<math>
			\operatorname{tr} \, A = \operatorname{tr} \, B, \quad
			\operatorname{tr} \, A^2 = \operatorname{tr} \, B^2, \quad\text{and}\quad
			\operatorname{tr} \, AA^* = \operatorname{tr} \, BB^*.
			</math>

			For ''n'' = 3, the following seven conditions are sufficient:

			:<math>
			\begin{align}
			&\operatorname{tr} \, A = \operatorname{tr} \, B, \quad
			\operatorname{tr} \, A^2 = \operatorname{tr} \, B^2, \quad
			\operatorname{tr} \, AA^* = \operatorname{tr} \, BB^*, \quad
			\operatorname{tr} \, A^3 = \operatorname{tr} \, B^3, \\
			&\operatorname{tr} \, A^2 A^* = \operatorname{tr} \, B^2 B^*, \quad
			\operatorname{tr} \, A^2 (A^)^2 = \operatorname{tr} \, B^2 (B^)^2, \quad\text{and}\quad
			\operatorname{tr} \, A^2 (A^)^2 A A^ = \operatorname{tr} \, B^2 (B^)^2 B B^.
			\end{align}
			</math>  <ref>{{harvtxt\|Sibirskiǐ\|1976}}, p. 260, quoted by {{harvtxt\|Đoković\|Johnson\|2007}}</ref>
			For general ''n'', it suffices to show that tr ''W''(''A'', ''A'') = tr ''W''(''B'', ''B'') for all words of degree at most

			:<math>
			n \sqrt{\frac{2n^2}{n-1} + \frac14} + \frac{n}2 - 2.
			</math>  <ref>{{harvtxt\|Pappacena\|1997}}, Theorem 4.3</ref>

			It has been conjectured that this can be reduced to an expression linear in ''n''.<ref>{{harvtxt\|Freedman\|Gupta\|Guralnick\|1997}}, p. 160</ref>

			== Notes ==
			<references/>

			== References ==
			* {{Citation \| last1=Đoković \| first1=Dragomir Ž. \| last2=Johnson \| first2=Charles R. \| title=Unitarily achievable zero patterns and traces of words in ''A'' and ''A''* \| doi=10.1016/j.laa.2006.03.002 \| year=2007 \| journal=Linear Algebra and its Applications \| issn=0024-3795 \| volume=421 \| issue=1 \| pages=63–68}}.
			* {{Citation \| last1=Freedman \| first1=Allen R. \| last2=Gupta \| first2=Ram Niwas \| last3=Guralnick \| first3=Robert M. \| title=Shirshov's theorem and representations of semigroups \| url=http://pjm.math.berkeley.edu/pjm/1997/181-3/p07.xhtml \| year=1997 \| journal=[[Pacific Journal of Mathematics]] \| issn=0030-8730 \| volume=181 \| issue=3 \| pages=159–176 \| doi=10.2140/pjm.1997.181.159}}.
			* {{Citation \| last1=Horn \| first1=Roger A. \| last2=Johnson \| first2=Charles R. \| title=Matrix Analysis \| publisher=[[Cambridge University Press]] \| isbn=978-0-521-38632-6 \| year=1985}}.
			* {{Citation \| last1=Pappacena \| first1=Christopher J. \| title=An upper bound for the length of a finite-dimensional algebra \| doi=10.1006/jabr.1997.7140 \| year=1997 \| journal=Journal of Algebra \| issn=0021-8693 \| volume=197 \| issue=2 \| pages=535–545}}.
			* {{Citation \| last1=Sibirskiǐ \| first1=K. S. \| title=Algebraic Invariants of Differential Equations and Matrices \| publisher=Izdat. "Štiinca", Kishinev \| language=Russian \| year=1976}}.
			* {{Citation \| last1=Specht \| first1=Wilhelm \| author1-link=Wilhelm Specht \| title=Zur Theorie der Matrizen. II \| url=http://gdz.sub.uni-goettingen.de/dms/load/toc/?PPN=PPN37721857X_0050&DMDID=dmdlog6 \| year=1940 \| journal=Jahresbericht der Deutschen Mathematiker-Vereinigung \| issn=0012-0456 \| volume=50 \| pages=19–23}}.

			{{DEFAULTSORT:Specht's Theorem}}
			[[Category:Matrix theory]]
			[[Category:Combinatorics on words]]
			[[Category:Theorems in algebra]]

en>RockMagnetist: Changed category

2011-09-28T22:19:50Z

Changed category

New page

They are typically a free website that are pre-designed for enabling businesses of every size in marking the presence on the internet and allows them in showcasing the product services and range through images, contents and various other elements. This one is one of the most beneficial features of Word - Press as this feature allows users to define the user roles. Step-4 Testing: It is the foremost important of your Plugin development process. If you need a special plugin for your website , there are thousands of plugins that can be used to meet those needs. By using this method one can see whether the theme has the potential to become popular or not and is their any scope of improvement in the theme. 

Generally, for my private income-making market websites, I will thoroughly research and discover the leading 10 most worthwhile niches to venture into. You will have to invest some money into tuning up your own blog but, if done wisely, your investment will pay off in the long run. This is the reason for the increased risk of Down Syndrome babies in women over age 35. t need to use the back button or the URL to get to your home page. Aided by the completely foolproof j - Query color selector, you're able to change the colors of factors of your theme a the click on the screen, with very little previous web site design experience. The entrepreneurs can easily captivate their readers by using these versatile themes. Browse through the popular Wordpress theme clubs like the Elegant Themes, Studio Press, Woo - Themes, Rocket Theme, Simple Themes and many more. If you beloved this article so you would like to get more info with regards to [http://twsi.in/wordpress_backup_82405 backup plugin] kindly visit our own web site. I've applied numerous Search engine optimization-ready Word - Press themes and I can say from knowledge that I consider the Genesis Search engine marketing panel one particular of the simplest to use. The first thing you need to do is to choose the right web hosting plan. Search engine optimization pleasant picture and solution links suggest you will have a much better adjust at gaining considerable natural site visitors. It has become a more prevalent cause of infertility and the fertility clinic are having more and more couples with infertility problems. php file in the Word - Press root folder and look for this line (line 73 in our example):. Next you'll go by way of to your simple Word - Press site. There are many advantages of hiring Wordpress developers for Wordpress project development:. If your site does well you can get paid professional designer to create a unique Word - Press theme. 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. An ease of use which pertains to both internet site back-end and front-end users alike. By the time you get the Gallery Word - Press Themes, the first thing that you should know is on how to install it. Page speed is an important factor in ranking, especially with Google. 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.