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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
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documents/math-minimal-distance-to-cubic-function/problem-description.tex
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@ -6,12 +6,12 @@ on the graph of $f$:
\[d_{P,f} (x) := \sqrt{(x - x_P)^2 + (f(x) - y_P)^2}\]
Now there is finite set $M = \Set{x_1, \dots, x_n} \subseteq D$ of minima for given $f$ and $P$:
\[M = \Set{x \in D | d_{P,f}(x) = \min_{\overline{x} \in D} d_{P,f}(\overline{x})}\]
\[M = \Set{x \in D | d_{P,f}(x) = \min_{\overline{x} \in D} d_{P,f}(\overline{x})}\]
But minimizing $d_{P,f}$ is the same as minimizing
But minimizing $d_{P,f}$ is the same as minimizing
$d_{P,f}^2 = (x_p^2 - 2x_p x + x^2) + (y_p^2 - 2y_p f(x) + f(x)^2)$.
In order to solve the minimal distance problem, Fermat's theorem
In order to solve the minimal distance problem, Fermat's theorem
about stationary points will be tremendously usefull:
\begin{theorem}[Fermat's theorem about stationary points]\label{thm:fermats-theorem}
@ -22,12 +22,12 @@ about stationary points will be tremendously usefull:
So in fact you can calculate the roots of $(d_{P,f}(x))'$ or $(d_{P,f}(x)^2)'$ to get
candidates for minimal distance.
$(d_{P,f}(x)^2)'$ is a polynomial if $f$ is a polynomial. So if $f$
is a polynomial, we can always get a finite number of candidates by
$(d_{P,f}(x)^2)'$ is a polynomial if $f$ is a polynomial. So if $f$
is a polynomial, we can always get a finite number of candidates by
finding roots of $(d_{P,f}(x)^2)'$. But this gets difficult when $f$
has degree 3 or higher as explained in Theorem~\ref{thm:no-finite-solution}.
Another problem one has to bear in mind is that these candidates
include all points with minimal distance, but might also contain
Another problem one has to bear in mind is that these candidates
include all points with minimal distance, but might also contain
more. Example~\ref{ex:false-positive} shows such a situation.
Let $S_n$ be the function that returns the set of solutions for a