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Remove trailing spaces

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The commands

find . -type f -name '*.md' -exec sed --in-place 's/[[:space:]]\+$//' {} \+

and

find . -type f -name '*.tex' -exec sed --in-place 's/[[:space:]]\+$//' {} \+

were used to do so.
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Martin Thoma 2015-10-14 14:25:34 +02:00
parent c578b25d2f
commit 7740f0147f
538 changed files with 3496 additions and 3496 deletions
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26
documents/math-minimal-distance-to-cubic-function/quadratic-case-2.1.tex
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@ -6,27 +6,27 @@ Let's validate this solution:
\allowdisplaybreaks
\begin{align}
0 &\stackrel{!}{=} \left (\frac{t}{\sqrt[3]{18}} - \frac{\sqrt[3]{\frac{2}{3}} \alpha }{t} \right )^3 + \alpha \left (\frac{t}{\sqrt[3]{18}} - \frac{\sqrt[3]{\frac{2}{3}} \alpha }{t} \right ) + \beta\\
&= (\frac{t}{\sqrt[3]{18}})^3
- 3 (\frac{t}{\sqrt[3]{18}})^2 \frac{\sqrt[3]{\frac{2}{3}} \alpha }{t}
+ 3 (\frac{t}{\sqrt[3]{18}})(\frac{\sqrt[3]{\frac{2}{3}} \alpha }{t})^2
- (\frac{\sqrt[3]{\frac{2}{3}} \alpha }{t})^3
&= (\frac{t}{\sqrt[3]{18}})^3
- 3 (\frac{t}{\sqrt[3]{18}})^2 \frac{\sqrt[3]{\frac{2}{3}} \alpha }{t}
+ 3 (\frac{t}{\sqrt[3]{18}})(\frac{\sqrt[3]{\frac{2}{3}} \alpha }{t})^2
- (\frac{\sqrt[3]{\frac{2}{3}} \alpha }{t})^3
+ \frac{t \alpha}{\sqrt[3]{18}} - \frac{\sqrt[3]{\frac{2}{3}} \alpha^2 }{t} + \beta\\
&= \frac{t^3}{18}
&= \frac{t^3}{18}
- \frac{3t^2}{\sqrt[3]{18^2}} \frac{\sqrt[3]{\frac{2}{3}} \alpha }{t}
+ \frac{3t}{\sqrt[3]{18}} \frac{\sqrt[3]{\frac{4}{9}} \alpha^2 }{t^2}
- \frac{\frac{2}{3} \alpha^3 }{t^3}
+ \frac{3t}{\sqrt[3]{18}} \frac{\sqrt[3]{\frac{4}{9}} \alpha^2 }{t^2}
- \frac{\frac{2}{3} \alpha^3 }{t^3}
+ \frac{t \alpha }{\sqrt[3]{18}} - \frac{\sqrt[3]{2} \alpha^2 }{\sqrt[3]{3} t} + \beta\\
&= \frac{t^3}{18}
- \frac{\sqrt[3]{18} t \alpha}{\sqrt[3]{18^2}}
+ \frac{\sqrt[3]{12} \alpha^2}{\sqrt[3]{18} t}
- \frac{2 \alpha^3 }{3t^3}
+ \frac{\sqrt[3]{12} \alpha^2}{\sqrt[3]{18} t}
- \frac{2 \alpha^3 }{3t^3}
+ \frac{t \alpha }{\sqrt[3]{18}} - \frac{\sqrt[3]{2} \alpha^2 }{\sqrt[3]{3} t} + \beta\\
&= \frac{t^3}{18}
&= \frac{t^3}{18}
\color{blue} - \frac{t \alpha}{\sqrt[3]{18}}
\color{red} + \frac{\sqrt[3]{2} \alpha^2}{\sqrt[3]{3} t}
\color{black}- \frac{2 \alpha^3 }{3 t^3}
\color{blue} + \frac{t \alpha }{\sqrt[3]{18}}
\color{red} - \frac{\sqrt[3]{2} \alpha^2 }{\sqrt[3]{3} t}
\color{black}- \frac{2 \alpha^3 }{3 t^3}
\color{blue} + \frac{t \alpha }{\sqrt[3]{18}}
\color{red} - \frac{\sqrt[3]{2} \alpha^2 }{\sqrt[3]{3} t}
\color{black}+ \beta\\
&= \frac{t^3}{18} - \frac{2 \alpha^3 }{3 t^3} + \beta\\
&= \frac{t^6 - 12 \alpha^3 + \beta 18 t^3}{18t^3}