Weak gravitational lensing: Difference between revisions

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In [[mathematics]], a collection of [[real number]]s  is '''rationally independent''' if none of them can be written as a linear combination of the other numbers in the collection with [[rational number|rational]] coefficients. A collection of numbers which is not rationally independent is called '''rationally dependent'''. For instance we have the following example.  
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:<math>
\begin{matrix}
\mbox{independent}\qquad\\
\underbrace{
  \overbrace{
    3,\quad
    \sqrt{8}\quad
  },
  1+\sqrt{2}
}\\
\mbox{dependent}\\
\end{matrix}
</math>
 
==Formal definition==
 
The [[real number]]s ω<sub>1</sub>, ω<sub>2</sub>, ... , ω<sub>''n''</sub> are said to be ''rationally dependent'' if there exist integers ''k''<sub>1</sub>, ''k''<sub>2</sub>, ... , ''k''<sub>''n''</sub>, not all of which are zero, such that
 
:<math> k_1 \omega_1 + k_2 \omega_2 +  \cdots + k_n \omega_n = 0. </math>
 
If such integers do not exist, then the vectors are said to be ''rationally independent''. This condition can be reformulated as follows: ω<sub>1</sub>, ω<sub>2</sub>, ... , ω<sub>''n''</sub> are rationally independent if the only ''n''-tuple of integers ''k''<sub>1</sub>, ''k''<sub>2</sub>, ... , ''k''<sub>''n''</sub> such that
 
:<math> k_1 \omega_1 + k_2 \omega_2 +  \cdots + k_n \omega_n = 0 </math>
is the [[trivial (mathematics)|trivial solution]] in which every ''k''<sub>''i''</sub> is zero.
 
The real numbers form a [[vector space]] over the [[rational number]]s, and this is equivalent to the usual definition of [[linear independence]] in this vector space.
 
== See also ==
 
*[[Linear flow on the torus]]
 
==Bibliography==
 
* {{cite book | author=Anatole Katok and Boris Hasselblatt | title= Introduction to the modern theory of dynamical systems | publisher= Cambridge | year= 1996 | isbn=0-521-57557-5}}
 
[[Category:Dynamical systems]]
 
 
{{mathanalysis-stub}}

Revision as of 11:18, 12 February 2014

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After skipping a day, on April 25 Sony realized that user knowledge was compromised, however they did not make that announcement until Tuesday, April 26. Lastly, Sony lets PS3 PSN customers know that their data had been stolen. Finally, on Friday, April 29, hackers try to hold PS3 PSN consumer's bank cards for ransom. Nevertheless, Sony claimed that the credit card data was encrypted, so they did not pay the ransom. For me, playing the Monk looks like practicing Kungfu. He performs melee actions when encountering enemies. His skills principally remind me of Kungfu-themed movies. But overall, I believe he isn't a most popular choice for any old avid gamers of Eternity Warriors. Throughout the game, unveil the mysteries surrounding your warriors

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