Activity for The Amplitwist
| Type | On... | Excerpt | Status | Date |
|---|---|---|---|---|
| Comment | Post #291317 |
Thank you for the answer! I'm still digesting the arguments, it's all a bit too quick for me. I may ping you for clarification in a few days, once I get some time. (more) |
— | 4 days ago |
| Edit | Post #291307 |
Post edited: added thoughts |
— | 4 days ago |
| Edit | Post #291307 | Initial revision | — | 7 days ago |
| Question | — |
How do I unambiguously define the facial circuits in a $2$-cell embedding of a graph into a surface? Suppose that $\Gamma$ is a connected, locally finite graph that is embedded into a closed, connected surface $M$. The faces of this embedding are the connected components of $M - \Gamma$ (we choose to denote the image of the embedding also by $\Gamma$). Let us assume that the embedding is such that... (more) |
— | 7 days ago |
| Comment | Post #291301 |
@#36356 Yes, the $0$-simplex has nonempty interior. Here is a more precise definition of $\overset{\circ}{\Delta}{}^n$ from Hatcher (page 103).
> If we delete one of the $n+1$ vertices of an $n$-simplex $[v_0,\dotsm,v_n]$, then the remaining $n$ vertices span an $(n-1)$-simplex, called a **face** of... (more) |
— | 10 days ago |
| Edit | Post #291301 | Initial revision | — | 11 days ago |
| Question | — |
Is there a $\Delta$-complex structure on the sphere with less than three $0$-simplices? In Hatcher's Algebraic Topology, a $\Delta$-complex structure on a topological space $X$ is defined as follows. Here, $\Delta^n$ denotes the standard $n$-simplex in $\mathbb{R}^{n+1}$, and $\overset{\circ}{\Delta}{}^n$ denotes its interior. > A $\mathbf\Delta$-complex structure on a space $X$ is ... (more) |
— | 11 days ago |
| Edit | Post #291159 | Initial revision | — | about 1 month ago |
| Question | — |
Why does the method of separating variables work? One of the methods to solve a partial differential equation is to use separation of variables. For example, consider the heat equation: $$ ut - a^2 u{xx} = 0, \qquad 0 < x < L,\ 0 < t, $$ with the boundary conditions $$ u(0,t) = u(L,t) = 0, \qquad 0 < t, $$ and initial condition $$ u(x,0) ... (more) |
— | about 1 month ago |
| Comment | Post #291056 |
Perhaps the "pathological imbeddings" can only be obtained by imbedding into two-dimensional locally Euclidean spaces that are not Hausdorff, or paracompact, or second-countable (or any similar nice properties). Cf. [Topological manifold - Wikipedia](https://en.wikipedia.org/wiki/Topological_manifold... (more) |
— | about 1 month ago |
| Edit | Post #291056 |
Post edited: clarified definition of free edge |
— | about 1 month ago |
| Edit | Post #291056 | Initial revision | — | about 1 month ago |
| Question | — |
What is an example of a pathological imbedding of a(n allowed) graph into an oriented surface? I am reading the following paper: G. A. Jones, and D. Singerman, Theory of maps on orientable surfaces, Proc. London Math. Soc. (3) 37 (1978), no. 2, 273–307 (MR0505721, Zbl 0391.05024). The authors are interested in imbeddings of an allowed graph $(\mathcal{G},\mathcal{V})$ into a connected, oriente... (more) |
— | about 1 month ago |
| Comment | Post #290771 |
This post seems to have been deleted by the moderators of r/math. (more) |
— | 2 months ago |
| Edit | Post #290771 |
Post edited: godel -> Gödel; i -> I |
— | 2 months ago |
| Suggested Edit | Post #290771 |
Suggested edit: godel -> Gödel; i -> I (more) |
helpful | 2 months ago |
| Edit | Post #290746 |
Post edited: formatted answer as it appears on Proof Assistants SE |
— | 2 months ago |
| Suggested Edit | Post #290746 |
Suggested edit: formatted answer as it appears on Proof Assistants SE (more) |
helpful | 2 months ago |
| Comment | Post #280864 |
@Technologicallyilliterate Added a few words to my answer. (more) |
— | about 3 years ago |
| Edit | Post #280864 |
Post edited: |
— | about 3 years ago |
| Comment | Post #280866 |
This is some meaty stuff, and I love it. Thank you for writing it up! (more) |
— | about 3 years ago |
| Comment | Post #280864 |
@Technologicallyilliterate Sure, I'll try to add something tomorrow. (more) |
— | about 3 years ago |
| Edit | Post #280851 |
Post edited: added transcript of image, added pointer to the titular equation rather than just "the part in green", minor typographical adjustments |
— | about 3 years ago |
| Suggested Edit | Post #280851 |
Suggested edit: added transcript of image, added pointer to the titular equation rather than just "the part in green", minor typographical adjustments (more) |
helpful | about 3 years ago |
| Edit | Post #280864 | Initial revision | — | about 3 years ago |
| Answer | — |
A: If $n = xm$ and $n \rightarrow \infty$, then $m \rightarrow \infty$? > […] I would have commenced with defining $\dfrac{x}{n}$ as $\dfrac{1}{m}$, which is more intuitive than "let $m = n/x$". I confess I don't see the difference between the two statements. In changing the variable in the limit from $n$ to $m$, you keep $x$ fixed; after all, you are trying to prove ... (more) |
— | about 3 years ago |
| Comment | Post #280851 |
Just transcribe the content of the image in addition to posting the image. It's good that you want to ensure that you're not mistyping the solution manual, but presenting text as images makes it difficult or impossible for people with accessibility issues to know what you're referring to. (more) |
— | about 3 years ago |
| Comment | Post #280842 |
Taking another look at the displayed equation in my post, I think I can take $G_1 = 0$ and $G_2 = \int F_3(x, y)\\, dx$ (or the other way around) provided the integral exists. This seems to provide some criterion at least in the $\mathbb{R}^2$ case. (more) |
— | about 3 years ago |
| Edit | Post #280842 | Initial revision | — | about 3 years ago |
| Question | — |
Does every divergence-free vector field arise as the curl of some vector field? I was introduced to the concepts of gradient $\nabla f$, curl $\nabla \times F$ and divergence $\nabla \cdot F$ in an introductory course on calculus during my undergraduate studies. There I learnt that for any scalar function $f$ on $\mathbb{R}^2$ or $\mathbb{R}^3$, we have $\nabla \times (\nabla f)... (more) |
— | about 3 years ago |
| Edit | Post #280630 |
Post edited: replaced dollars with backticks (code formatting) because the question is confusing when the MathJax is rendered :) |
— | about 3 years ago |
| Edit | Post #280629 |
Post edited: fixed some MathJax markup |
— | about 3 years ago |
| Suggested Edit | Post #280630 |
Suggested edit: replaced dollars with backticks (code formatting) because the question is confusing when the MathJax is rendered :) (more) |
helpful | about 3 years ago |
| Comment | Post #280630 |
One has to escape the backslash character for it to render correctly, so typing `\\;` works correctly. See this: https://math.codidact.com/posts/278772#answer-278772 (more) |
— | about 3 years ago |
| Suggested Edit | Post #280629 |
Suggested edit: fixed some MathJax markup (more) |
helpful | about 3 years ago |
| Comment | Post #280462 |
+1 This is exactly what I needed, thank you for the clear explanation and the reference! :) (more) |
— | over 3 years ago |
| Edit | Post #280460 | Initial revision | — | over 3 years ago |
| Question | — |
What surface do I get by attaching $g$ handles as well as $k$ crosscaps to a sphere? I recently found out that there is a classification of compact connected surfaces that says that every such surface (or, $2$-manifold) is homeomorphic to either $Sg$, the sphere with $g \geq 0$ handles, or $Nk$, the sphere with $k \geq 1$ crosscaps. If my understanding is correct: - attaching a ... (more) |
— | over 3 years ago |
| Comment | Post #280204 |
@msh210 I usually add the remainder of any proofs at the end of my questions (or I used to, on SE), but I somehow forgot to do so here. Thanks for asking about it! (more) |
— | over 3 years ago |
| Comment | Post #280204 |
@msh210 It goes like this: "Thus $$\mathcal{R}_p^{(\pi/2)} \circ \mathcal{R}_s^{(\pi/2)} = (\mathcal{R}_m^\pi)^{-1} \circ \mathcal{M} = \mathcal{R}_m^\pi.$$ If we define $s' = \mathcal{R}_m^\pi(s)$ then $m$ is the midpoint of $ss'$. But, on the other hand, $$s' = \Bigl( \mathcal{R}_p^{(\pi/2)} \circ ... (more) |
— | over 3 years ago |
| Edit | Post #280204 |
Post edited: added a missing word |
— | over 3 years ago |
| Comment | Post #280207 |
@MonicaCellio Ah, you're right, I didn't notice that. Not sure why that should be the case. The part that is not rendered in the latest revision (2020-12-31T07:17:38Z) happens to be rendered in the initial revision. The only thing different about it is that it uses "displayed math" (hence the double ... (more) |
— | over 3 years ago |
| Comment | Post #280206 |
Okay, now this is actually a bit embarrassing for me. I just had to turn the page and find out in Figure [15a] that the point $k$ is indeed the point you call $z$. So, if you'll excuse me, I shall go drown myself now. (more) |
— | over 3 years ago |
| Edit | Post #280204 |
Post edited: improved alt text |
— | over 3 years ago |
| Edit | Post #280207 | Initial revision | — | over 3 years ago |
| Question | — |
Post revisions are displayed without rendered Markdown but with rendered MathJax When I view the post history on this question of mine, I notice that the Markdown is not rendered, but the MathJax is rendered. But, I guess I would expect neither Markdown nor MathJax to be rendered when viewing the post history? Even better would be if we could have options for viewing the post his... (more) |
— | over 3 years ago |
| Comment | Post #280206 |
+1 This makes perfect sense. If this is what the author also meant, then I'm not sure why he phrased it as $\mathcal{M}(k) = k$, though. (more) |
— | over 3 years ago |
| Edit | Post #280204 | Initial revision | — | over 3 years ago |
| Question | — |
Given a triangle with squares on two sides, the line segments joining the centres of the squares to the midpoint of the third side are equal and perpendicular I am reading Tristan Needham's Visual Complex Analysis (2012 reprint, OUP), and in $\S$1.III.3: Geometry, the author gives a geometric proof of the following fact, shown in Figure [12b] on page 16: >![Figure[12a] shows an arbitrary quadrilateral with squares constructed outward on each of the side... (more) |
— | over 3 years ago |
| Comment | Post #278724 |
This is happening to me, too. The editor slows down tremendously and drafting moderately long posts becomes quite difficult. (more) |
— | over 3 years ago |