Darcy–Weisbach equation: Difference between revisions

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In [[mathematics]], '''Euler's four-square identity''' says that the product of two numbers, each of which is a sum of four [[Square (algebra)|square]]s, is itself a sum of four squares. Specifically:
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:<math>(a_1^2+a_2^2+a_3^2+a_4^2)(b_1^2+b_2^2+b_3^2+b_4^2)=\,</math>
 
::<math>(a_1 b_1 - a_2 b_2 - a_3 b_3 - a_4 b_4)^2 +\,</math>
 
::<math>(a_1 b_2 + a_2 b_1 + a_3 b_4 - a_4 b_3)^2 +\,</math>
 
::<math>(a_1 b_3 - a_2 b_4 + a_3 b_1 + a_4 b_2)^2 +\,</math>
 
::<math>(a_1 b_4 + a_2 b_3 - a_3 b_2 + a_4 b_1)^2.\,</math>
 
[[Leonhard Euler|Euler]] wrote about this identity in a letter dated May 4, 1748 to [[Christian Goldbach|Goldbach]]<ref>''Leonhard Euler: Life, Work and Legacy'', R.E. Bradley and C.E. Sandifer (eds), Elsevier, 2007, p. 193</ref><ref>''Mathematical Evolutions'', A. Shenitzer and J. Stillwell (eds), Math. Assoc. America, 2002, p. 174</ref> (but he used a different sign convention from the above).  It can be proven with [[elementary algebra]] and holds in every [[commutative ring]]. If the <math>a_k</math> and <math>b_k</math> are [[real number]]s, a more elegant proof is available: the identity expresses the fact that the absolute value of the product of two [[quaternion]]s is equal to the product of their absolute values, in the same way that the [[Brahmagupta–Fibonacci identity|Brahmagupta–Fibonacci two-square identity]] does for [[complex numbers]].
 
The identity was used by [[Joseph Louis Lagrange|Lagrange]] to prove his [[Lagrange's four-square theorem|four square theorem]]. More specifically, it implies that it is sufficient to prove the theorem for [[prime numbers]], after which the more general theorem follows. The sign convention used above corresponds to the signs obtained by multiplying two quaternions. Other sign conventions can be obtained by changing any <math>a_k</math> to <math>-a_k</math>, <math>b_k</math> to <math>-b_k</math>, or by changing the signs inside any of the squared terms on the right hand side.
 
[[Hurwitz's theorem (normed division algebras)|Hurwitz's theorem]] states that an identity of form,
 
:<math>(a_1^2+a_2^2+a_3^2+...+a_n^2)(b_1^2+b_2^2+b_3^2+...+b_n^2) = c_1^2+c_2^2+c_3^2+...+c_n^2\,</math>
 
where the <math>c_i</math> are [[bilinear map|bilinear]] functions of the <math>a_i</math> and <math>b_i</math> is possible only for ''n'' = {1, 2, 4, 8}. However, the more general [[Pfister's theorem]] allows that if the <math>c_i</math> are just [[rational functions]] of one set of variables, hence has a [[denominator]], then it is possible for all <math>n = 2^m</math>.<ref>Pfister's Theorem on Sums of Squares, Keith Conrad, http://www.math.uconn.edu/~kconrad/blurbs/linmultialg/pfister.pdf</ref> Thus, a different kind of four-square identity can be given as,
 
:<math>(a_1^2+a_2^2+a_3^2+a_4^2)(b_1^2+b_2^2+b_3^2+b_4^2)=\,</math>
 
::<math>(a_1 b_4 + a_2 b_3 + a_3 b_2 + a_4 b_1)^2 +\,</math>
 
::<math>(a_1 b_3 - a_2 b_4 + a_3 b_1 - a_4 b_2)^2 +\,</math>
 
::<math>\left(a_1 b_2 + a_2 b_1 + \frac{a_3 u_1}{b_1^2+b_2^2} - \frac{a_4 u_2}{b_1^2+b_2^2}\right)^2+\,</math>
 
::<math>\left(a_1 b_1 - a_2 b_2 - \frac{a_4 u_1}{b_1^2+b_2^2} - \frac{a_3 u_2}{b_1^2+b_2^2}\right)^2\,</math>
 
where,
 
:<math>u_1 = b_1^2b_4-2b_1b_2b_3-b_2^2b_4</math>
 
:<math>u_2 = b_1^2b_3+2b_1b_2b_4-b_2^2b_3</math>
 
Note also the incidental fact that,
 
:<math>u_1^2+u_2^2 = (b_1^2+b_2^2)^2(b_3^2+b_4^2)</math>
 
==See also==
* [[Degen's eight-square identity]]
* [[Pfister's sixteen-square identity]]
* [[Latin square]]
 
==References==
<references/>
 
==External links==
*[http://sites.google.com/site/tpiezas/005b/  A Collection of Algebraic Identities]
*[http://math.dartmouth.edu/~euler/correspondence/letters/OO0841.pdf] Lettre CXV from Euler to Goldbach
 
{{DEFAULTSORT:Euler's Four-Square Identity}}
[[Category:Elementary algebra]]
[[Category:Elementary number theory]]
[[Category:Mathematical identities]]

Revision as of 14:16, 27 February 2014

Making your computer run swiftly is pretty simple. Most computers run slow because they are jammed up with junk files, that Windows has to look through every time it wants to find anything. Imagine having to find a book inside a library, yet all library books are in a big huge pile. That's what it's like for a computer to obtain something, when your program is full of junk files.

Registry is not furthermore significant to quick computer boot up, and crucial to the performance of the computer. If you have a registry error, we can face blue screen, freezing or crash. It's important to frequently clean up the invalid, missing, junk registry keys to keep the computer healthy and running quick.

With RegCure to better the begin up and shut down of the computer. The program shows the scan progress and we shouldn't worry where it is actually functioning at that time. It shows we precisely what arises. Dynamic link library section of the registry could result serious application failures. RegCure restores plus repairs the registry and keeps you out of DLL. RegCure is able to make individual corrections, so it will works for a demands.

If you feel you don't have enough funds at the time to upgrade, then the best choice is to free up several room by deleting a few of the unwanted files and folders.

After which, I equally purchased the Regtool system mechanic Software, plus it further secure my laptop having system crashes. All my registry difficulties are fixed, and I can function peacefully.

Turn It Off: Chances are if you are like me; then we spend a great deal of time on the computer on a daily basis. Try providing a computer certain time to do completely nothing; this might sound funny however, in the event you have an older computer you may be asking it to do too much.

Another problem with all the cracked adaptation is the fact that it takes too much time to scan the program plus while it really is scanning, you can not use the computer otherwise. Moreover, there is no technical help to these cracked versions which means should you receive stuck someplace, you can't ask for aid. They even do not have any customer service aid lines wherein we could call or send to resolve the issues.

What I would recommend is to look on a own for registry cleaners. You are able to do this with a Google search. Whenever you find products, look for critiques plus reviews about the product. Then you are able to see how others like the product, plus how effectively it works.

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