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1:31 AM
Q: If $lk(v;K)$ is collapsible, then $K$ collapses to $del(v;K)$.

JaviI'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...

Q: Finding an irreducible polynomial of degree $n$ in $\mathbb Q[X]$ with real roots

E. JosephContext. 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...

12 hours later…
1:26 PM
Although it was removed from that post later, it was added to several other posts: chat.stackexchange.com/transcript/3740?m=44142996#44142996 At the moment there are 7 questions with this tag.
Q: Critical path algorithm

JsevillamolIf 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...

Q: Prove that a length-minimizing path in a metric space is injective

MathManSuppose $(X,d)$ 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...

Q: Graph theory: path of X cost between two nodes

Jordan MacLachlanI'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...

Q: How many paths of length $n$ with the same start and end point can be found on a hexagonal grid?

SkillzoreGiven 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...

Q: One-dimensional Hausdorff measure of a line segment

Eduardo LongaLet $\mathcal{H}^1$ 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 $$ \mathcal{H}^1([uv]) = \Vert u - v \Vert? $$

Q: Number of pathways of length n in a grid?

tox123Given 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...

Q: Show that the following curve is not rectifiable

TBHWe 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=...

1:52 PM
Q: Clarifying the usage of (path-length) tag

Martin SleziakThe 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...

2 hours later…
3:47 PM
I will just mention that I have made a post on meta about (path-length) tag. Since you created the tag, I thought it might be polite to let you know. — Martin Sleziak 23 secs ago

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