I'm doing exercises about simplicial complexes and I'm stuck with one for which I'll first give some definitions. Let $K$ a simplicial complex and $v\in K^0$ a $0$-simplex (vertex). The star of $v$ in $K$, $st(v;K)=\{\sigma\in K\mid \exists \tau\leq\sigma: v\in\tau\}$ The link of $v$ i...
If I want to compute the shortest path between two points in a directed graph, I can use the Dijkistra algorithm. But what if I want to compute the longest path? If the weights on the graph are bounded, then I guess I can use Djkstra to the graph with weights $M-c_i$, where $M = \max_i c_i$. Is...
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...
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...
Given this question, what about the special case when the start point and end point are the same? I ask it here instead because I am looking for the mathematical solution to counting these different paths. Another change in my case is that we must move at every step. How many such different path...
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 $$ \mathcal{H}^1([uv]) = \Vert u - v \Vert? $$
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...
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...
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