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Context.
I was trying to prove that for a given $n$, there exist a totally real number field of degree $n$.
I understood that it was equivalent to find a polynomial $P$ such that
(i) $P\in\mathbb Q[X]$ ;
(ii) $P$ is irreducible ;
(iii) $P$ only has real roots in $\mathbb C$ ;
(iv) $P$ has d...0
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Suppose $(X,d)$ is a metric space with the nearest point property and $a,b \in X$ with $a \ne b$. Suppose there is a path of finite length in $X$ from $a$ to $b$ and let $m$ be the infimum of the lengths of all paths from $a$ to $b$.Then, by Lipschitz reparametrization, there exists a path $g:[0...
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I'm looking for a way to, in any given connected, undirected graph, calculate a path between any two nodes with a cost as close as possible to a given value.
The example is in this image (sorry, I can't embed images yet):
A simple graph
Note: the visual length of each edge doesn't exactly ma...
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Let $\mathcal{H}^1$ be the one-dimensional Hausdorff measure in $\mathbb{R}^n$ and let $[uv] = \{ u + t(v-u) : t \in [0,1] \}$ be the segment joining the vectors $u,v \in \mathbb{R}^n$. How do we show that
$$ \mathcal{H}^1([uv]) = \Vert u - v \Vert? $$
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Given a square grid and a point $A:(0,0)$ and another point $B:(n,m)$ (where $n$ and $m$ are both integers), what is the number of pathways ($k$) of length $l$ (a natural number) edges are there between $A$ and $B$?
I was experimenting with this on my own, without any sort of proof and found tha...
We want to prove that the following curve is not rectifiable.
$$ f\colon [0,1] \rightarrow \mathbb{R}^2, \quad f(x) = \begin{cases} (0,0) \, &\quad \text{for } x= 0 \\ (x,x^2\cos(\frac{\pi}{x}) &\quad \text{for } x> 0.\end{cases} $$
My idea: I need to show that the Length $L > \sum\limits_{i=...
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The tag path-length was created about two months ago in the question: Prove that there exists an another path $g$ with the same image as $f$ but length of $g = tL \forall t\in [0,1]$ where $L$ is the length of $f$. Although it has been later removed from this specific question, it was added to se...
