Quasi Fermi level - Revision history

en>Homayoun mh: a minus had been droped; according to http://www-inst.eecs.berkeley.edu/~ee105/sp04/handouts/lectures/Lecture19.pdf

2014-10-08T08:14:21Z

a minus had been droped; according to http://www-inst.eecs.berkeley.edu/~ee105/sp04/handouts/lectures/Lecture19.pdf

← Older revision		Revision as of 10:14, 8 October 2014
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en>Nanite: rephrase in terms of conduction band and valence band, since that's what is really going on

2014-02-09T11:30:02Z

rephrase in terms of conduction band and valence band, since that's what is really going on

← Older revision		Revision as of 13:30, 9 February 2014
Line 1:		Line 1:
	~~[[File:Freshman's Dream.svg\|right\|thumbnail\|An illustration of~~ the ~~Freshman's dream in two dimensions. Each side~~ of the square is X+Y in length. The area of the square is the sum of the area of the yellow region (=X<sup>2</sup>), the area of the green region (=Y<sup>2</sup>), and the area of the two white regions (=2×X×Y).]]		It depends on the quality of the Wordpress theme but even if it's not a professional one you will be able to average 50-60$ EACH link. Thus, it is important to keep pace with this highly advanced age and have a regular interaction with your audience to keep a strong hold in the business market. One really cool features about this amazing and free wp plugin is that the code it generates is completely portable. Donor oocytes and menopausal pregnancy: Oocyte donation to women of advanced reproductive age: pregnancy results and obstetrical outcomes in patients 45 years and older. This particular wordpress plugin is essential for not only having the capability where you improve your position, but to enhance your organic searches for your website. <br><br>Choosing what kind of links you'll be using is a ctitical aspect of any linkwheel strategy, especially since there are several different types of links that are assessed by search engines. But as expected the level of support you get with them can be hit or miss based on the developer's free time and desire. This may possibly also permit it currently being seriously straightforward to modify the hues within your Ad - Sense code so the ads blend nicely with the many term broad internet word wide web web page in case you can come to your conclusion to run the adverts. By purchasing Word - Press weblogs you can acquire your very own domain title and have total command of your web site. But in case you want some theme or plugin in sync with your business needs, it is advisable that you must seek some professional help. <br><br>The least difficult and very best way to do this is by acquiring a Word - Press site. If you have any concerns concerning where by in addition to how to utilize [http://linxfix.com/WordpressDropboxBackup241975 wordpress backup plugin], it is possible to e-mail us at our own web site. To sum up, ensure that the tactics are aiming to increase the ranking and attracting the maximum intended traffic in the major search engines. If Gandhi was empowered with a blogging system, every event in his life would have been minutely documented so that it could be recounted to the future generations. The animation can be quite subtle these as snow falling gently or some twinkling start in the track record which are essentially not distracting but as an alternative gives some viewing enjoyment for the visitor of the internet site. Customization of web layout is easy due to the availability of huge selection of templates. <br><br>Additionally Word - Press add a default theme named Twenty Fourteen. As an example, if you are promoting a product that cures hair-loss, you most likely would not wish to target your adverts to teens. The templates are designed to be stand alone pages that have a different look and feel from the rest of your website. The company gains commission from the customers' payment. Wordpress template is loaded with lots of prototype that unite graphic features and content area. <br><br>Instead, you can easily just include it with our bodies integration field in e - Panel. I don't want that for my visitors and I'm quite sure they don't either. The days of spending a lot of time and money to have a website built are long gone. Change the entire appearance of you blog using themes with one click. You can check out the statistics of page of views for your web pages using free tools that are available on the internet.

	~~The '''freshman~~'s ~~dream''' is~~ a ~~name sometimes given~~ to ~~the error (''x'' + ''y'')<sup>''n''</sup> = ''x''<sup>''n''</sup> + ''y''<sup>''n''</sup>~~, ~~where ''n''~~ is a ~~real number (usually~~ a ~~positive integer greater than 1)~~. ~~Beginning students commonly make~~ this ~~error in computing~~ the ~~[[exponentiation\|power]]~~ of ~~a sum of real numbers.<ref>Julio R. Bastida, ''Field Extensions~~ and ~~Galois Theory'', Addison-Wesley Publishing Company, 1984, p.8.</ref><ref>Fraleigh, John B., ''A First Course~~ in ~~Abstract Algebra'', Addison-Wesley Publishing Company, 1993, p~~.~~453~~, ~~ISBN 0-201-53467-3~~.<~~/ref~~> ~~When ''n'' = 2, it is easy to see why this is incorrect: (''x'' + ''y'')~~<~~sup~~>~~2</sup> can~~ be ~~correctly computed as ''x''<sup>2</sup> + 2''xy'' + ''y''<sup>2</sup>~~ using ~~[[distributivity]] (or commonly known~~ as the ~~[[FOIL method]]). For larger positive integer values~~ of ~~''n'',~~ the ~~correct result is given by the [[binomial theorem]].~~

	~~The name "freshman~~'s ~~dream"~~ also ~~sometimes refers~~ to the ~~theorem that says that for a [[prime number]] ''p'', if ''x'' and ''y'' are members of a [[commutative ring]] of [[characteristic_(algebra)\|characteristic]] ''p'', then~~
	~~(''x'' + ''y'')<sup>''p''</sup> = ''x''<sup>''p''</sup> + ''y''<sup>''p''</sup>. In this case,~~ the ~~"mistake" actually gives~~ the ~~correct result, due~~ to ~~''p'' dividing all~~ the ~~[[binomial coefficient]]s save the first~~ and ~~the last~~.

	~~==Examples==~~
	*<math>(1+4)^2 = 5^2 = 25</math>, ~~but <math>1^2+4^2 = 17</math>~~.
	*<~~math~~>~~\sqrt{x^2+y^2}~~<~~/math~~> ~~does not generally equal <math>\sqrt{x^2}+\sqrt{y^2}=\|x\|+\|y\|</math>. For example, <math>\sqrt{9+16}=\sqrt{25}=5</math>, which does not equal 3+4=7. In~~ this ~~example, the error~~ is ~~being committed with the exponent ''n'' = {{frac\|1\|2}}~~.

	~~==Prime characteristic==~~

	~~When ''p'' is a prime number and ''x'' and ''y'' are members of a~~ [~~[commutative ring]] of [[characteristic_(algebra)\|characteristic]] ''p'', then (''x'' + ''y'')<sup>''p''</sup> = ''x''<sup>''p''<~~/~~sup> + ''y''<sup>''p''<~~/~~sup>~~. ~~This can be seen by examining the prime factors of the binomial coefficients: the ''n''th binomial coefficient is~~

	~~:<math>\binom{p}{n} = \frac{p!}{n!(p-n)!}.<~~/~~math>~~

	~~The [[numerator]] is ''p'' [[factorial]~~], ~~which~~ is ~~divisible by ''p''~~. ~~However~~, ~~when {{nowrap\|0 < ''n'' < ''p''}}, neither ''n''! nor {{nowrap\|(''p'' − ''n'')!}} is divisible by ''p'' since all~~ the ~~terms~~ are ~~less than ''p''~~ and ~~''p'' is prime. Since a binomial coefficient is always an integer,~~ the ~~''n''th binomial coefficient is divisible by ''p'' and hence equal to 0~~ in the ~~ring~~. ~~We are left~~ with ~~the zeroth and ''p''th coefficients~~, ~~which both equal 1, yielding~~ the ~~desired equation~~.

	~~Thus~~ in ~~characteristic ''p''~~ the ~~freshman's dream is a valid identity. This result demonstrates that exponentiation by ''p'' produces~~ an ~~[[endomorphism]], known as~~ the ~~[[Frobenius endomorphism]]~~ of the ~~ring~~.

	~~The demand that the characteristic ''p'' be a prime number~~ is ~~central~~ to the ~~truth~~ of ~~the freshman's dream~~. ~~In fact, a related theorem states that a number ''n'' is prime [[if and only if]] {{nowrap\|(''x''+1)~~<~~sup~~>~~''n''~~<~~/sup~~> ~~≡ ''x<sup>n</sup>'' + 1 (mod ''n'')}} in the [[polynomial ring]] <math>\mathbb{Z}_n[x]</math>. This theorem is a direct consequence of [[Fermat's Little Theorem]] and it is~~ a ~~key fact in modern primality testing~~.~~<ref name=Granville>A. Granville~~, ~~''[http://www.ams.org/bull/2005-42-01/S0273-0979-04-01037-7/S0273~~-~~0979-04-01037-7.pdf It Is Easy To Determine Whether A Given Integer Is Prime]''~~, ~~Bull~~. of ~~the AMS, Volume 42, Number 1 (Sep. 2004), Pages 3–38~~.~~</ref>~~

	~~==History and alternate names==~~
	The ~~history of~~ the ~~term "freshman~~'~~s dream"~~ is ~~somewhat unclear~~. ~~In a 1940 article on [[modular field]]s, [[Saunders Mac Lane]] quotes [[Stephen Cole Kleene\|Stephen Kleene]]'s remark that a knowledge of (''a'' + ''b'')~~<~~sup~~>2<~~/sup~~>~~= ''a''<sup>2</sup> + ''b''<sup>2</sup> in a [[~~field ~~(mathematics)\|field]] of characteristic 2 would corrupt freshman students of [[abstract algebra\|algebra]]. This may be the first connection between "freshman" and binomial expansion~~ in ~~fields of positive characteristic.<ref>Colin R~~. ~~Fletcher, Review of~~ '~~''Selected papers on algebra, edited by [[Susan Montgomery]], Elizabeth W. Ralston~~ and ~~others. Pp xv, 537. 1977. SBN 0 88385 203 9 (Mathematical Association of America)''', ''The Mathematical Gazette~~''~~, Vol~~. ~~62, No. 421 (Oct., 1978),~~ The ~~Mathematical Association. p. 221.</ref> Since then, authors~~ of ~~undergraduate algebra texts took note~~ of ~~the common error. The first actual attestation of the phrase "freshman's dream" seems~~ to ~~be in [[Thomas W. Hungerford\|Hungerford's]] undergraduate algebra textbook (1974), where he quotes McBrien.<ref>Thomas W. Hungerford, ''Algebra,'' Springer, 1974, p~~. 121; also in ''Abstract Algebra: An Introduction'', 2nd edition. Brooks Cole, July 12, 1996, p. 366.</ref> Alternative terms include "'''freshman exponentiation'''", used in Fraleigh (1998).<ref>John B. Fraleigh, ''A First Course In Abstract Algebra'', 6th edition, Addison-Wesley, 1998. pp. 262 and 438.</ref> The term "freshman's dream" itself, in non-mathematical contexts, is recorded since the ~~19th century~~.<ref>[http://www.google.com/search?tbo=p&tbm=bks&q=%22freshman%27s+dream%22&tbs=,cdr:1,cd_min:Jan%201_2%201800,cd_max:Dec%2031_2%201900&num=10 Google books 1800–1900 search for "freshman's dream"]: [http://books.google.com/books?id=3XNHAAAAYAAJ&pg=PA176&dq=%22freshman%27s+dream%22 Bentley's miscellany, Volume 26, p. 176], 1849</ref>

	~~Since~~ the ~~expansion~~ of ~~(''x'' + ''y'')<sup>''n''</sup> is correctly given by~~ the ~~[[binomial theorem]], the freshman's dream is also known as the "'''Child's Binomial Theorem'''" <ref name=Granville/> or "'''Schoolboy Binomial Theorem'''"~~.

	~~==See also==~~
	*[[Primality test]]
	*[[Sophomore's dream]]

	~~==References==~~
	~~<references/>~~

	~~[[Category:Algebra]]~~
	~~[[Category:Mathematics education]]~~

141.58.56.120 at 08:59, 11 June 2013

2013-06-11T08:59:35Z

@@ Line 1: / Line 1: @@
-They get other people’s private information with nasty means online intentionally. But we have to admit that there are such disgusting people around us. What we would discuss here is to help you hack your own Yahoo password which has been saved on your IE [Internet Explorer] browser already. Here we do not mean we can help you hack into others’ yahoo mail accounts with a certain tool. Some people may misunderstand the meaning of “hack” here. We all know that it is illegal.
+[[File:Freshman's Dream.svg|right|thumbnail|An illustration of the Freshman's dream in two dimensions. Each side of the square is X+Y in length. The area of the square is the sum of the area of the yellow region (=X<sup>2</sup>), the area of the green region (=Y<sup>2</sup>), and the area of the two white regions (=2×X×Y).]]
- These can be very tough and many players often get stuck in one place. But this can be changed now because there is a way out of this. A website is offering Clash of Clans hack tool trainer to users who want it. When this happens, it can be quite frustrating. Among the many games which have been created till now, Clash of Clans is preferred by many people. The game which requires players to create villages and characters to move forward can quite difficult at times.
+The '''freshman's dream''' is a name sometimes given to the error (''x''&nbsp;+&nbsp;''y'')<sup>''n''</sup>&nbsp;=&nbsp;''x''<sup>''n''</sup>&nbsp;+&nbsp;''y''<sup>''n''</sup>, where ''n'' is a real number (usually a positive integer greater than 1). Beginning students commonly make this error in computing the [[exponentiation|power]] of a sum of real numbers.<ref>Julio R. Bastida, ''Field Extensions and Galois Theory'', Addison-Wesley Publishing Company, 1984, p.8.</ref><ref>Fraleigh, John B., ''A First Course in Abstract Algebra'', Addison-Wesley Publishing Company, 1993, p.453, ISBN 0-201-53467-3.</ref> When ''n'' = 2, it is easy to see why this is incorrect: (''x''&nbsp;+&nbsp;''y'')<sup>2</sup> can be correctly computed  as ''x''<sup>2</sup>&nbsp;+&nbsp;2''xy''&nbsp;+&nbsp;''y''<sup>2</sup> using [[distributivity]] (or commonly known as the [[FOIL method]]). For larger positive integer values of ''n'', the correct result is given by the [[binomial theorem]].
- There are also other sites which offer the tool. The website offering this tool is safe and it guarantees best quality software. But most are either incomplete or of bad quality. Players have to carry out different tasks including raids and missions.   One can also build a personal village inside the castle as well. Clash of clan game play is a simple game with more of power than weakness. The game becomes all the more interesting with low energy levels. One can choose from the different units and can begin playing against the other players thereby endlessly earning rewards.
+The name "freshman's dream" also sometimes refers to the theorem that says that for a [[prime number]] ''p'', if ''x'' and ''y'' are members of a [[commutative ring]] of [[characteristic_(algebra)|characteristic]] ''p'', then
-   The game can be downloaded for free from the iTunes store Apple store  as well as offline as well. The most interesting part of the game is that in spite of low energy level; one can combat with the goblins in order to move ahead in the quests. The hardcore players are tested with their mettle in deploying troops. Well, don’t you think it is illogical to do that? Though some games are a lot fun but wasting money like this does not make any sense at all.
+==Examples==
- Wouldn’t it be a total waste of money? You will be glad to know that plenty of Clash Of  http://www.knit.net.au/ForumRetrieve.aspx?ForumID=2803&TopicID=651388&No... ([http://www.junhyuncha.com/xe/?document_srl=558778 www.junhyuncha.com]) Clans Hackers are available on the web these days that can give you unlimited Gems. There are some people who start investing real money in the game. What about the jailbreak? It would be better to find a Clash of Clan Hack than to pay for the gems. If you are about to spend money for buying gems for Clash of Clans then don’t go for it. That would not be a problem too because the hacker will solve everything for you.
+==Prime characteristic==
- For the android phones, you can count on downloading and putting in the clash of hack android app on your mobile phone easily. However, there are a lot of different providers of this service available, a couple of offer accurate credibility and quality. Another popular blunder is the water fountain which is made of copper, filled with stones, and containing a ledge for a candle and a sprig of bamboo.
+When ''p'' is a prime number and ''x'' and ''y'' are members of a [[commutative ring]] of [[characteristic_(algebra)|characteristic]] ''p'', then (''x''&nbsp;+&nbsp;''y'')<sup>''p''</sup>&nbsp;=&nbsp;''x''<sup>''p''</sup>&nbsp;+&nbsp;''y''<sup>''p''</sup>. This can be seen by examining the prime factors of the binomial coefficients: the ''n''th binomial coefficient is

95.117.158.145 at 16:10, 15 January 2012

2012-01-15T16:10:40Z

New page

They get other people’s private information with nasty means online intentionally. But we have to admit that there are such disgusting people around us. What we would discuss here is to help you hack your own Yahoo password which has been saved on your IE [Internet Explorer] browser already. Here we do not mean we can help you hack into others’ yahoo mail accounts with a certain tool. Some people may misunderstand the meaning of “hack” here. We all know that it is illegal.

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The game can be downloaded for free from the iTunes store Apple store as well as offline as well. The most interesting part of the game is that in spite of low energy level; one can combat with the goblins in order to move ahead in the quests. The hardcore players are tested with their mettle in deploying troops. Well, don’t you think it is illogical to do that? Though some games are a lot fun but wasting money like this does not make any sense at all.

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