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Activity for The Amplitwist‭

Show all events Posts 15 Comments 31 Edits 23

Type	On...	Excerpt	Status	Date
Comment	Post #292644	The link to "Problem No. 81" seems to be broken, perhaps you could look into it. (more)	—	6 months ago
Comment	Post #292644	Related on MathOverflow: [Probability that a stick randomly broken in five places can form a tetrahedron](https://mathoverflow.net/q/142983) (more)	—	6 months ago
Edit	Post #291432	Post edited: removed extraneous []() introduced in the previous edit	—	8 months ago
Suggested Edit	Post #291432	Suggested edit: removed extraneous []() introduced in the previous edit (more)	helpful	8 months ago
Comment	Post #291652	Interesting references. Thank you for the answer! (more)	—	10 months ago
Edit	Post #291372	Post edited:	—	11 months ago
Edit	Post #291503	Initial revision	—	11 months ago
Answer	—	A: How do I unambiguously define the facial circuits in a $2$ -cell embedding of a graph into a surface? This is answered in the paper Embedding graphs in surfaces90014-5) (P. Hoffman and B. Richter, J. Comb. Theory, Ser. B 36, 65–84 (1984; Zbl 0514.05028)). Quoting from the introduction: > There are certain foundational results in the overlap between graph theory and the topology of surfaces whose p... (more)	—	11 months ago
Edit	Post #291372	Post edited:	—	11 months ago
Edit	Post #291372	Initial revision	—	11 months ago
Answer	—	A: What is an example of a pathological imbedding of a(n allowed) graph into an oriented surface? The pathological embeddings arise because an "allowed graph" is a very general object. The goal of the assumption TM1 is to restrict attention to precisely the kind of allowed graphs that are described in the question as being > slightly more general than an undirected pseudograph in that it allow... (more)	—	11 months ago
Comment	Post #291159	The [Wikipedia](https://en.wikipedia.org/wiki/Separation_of_variables) article on "separation of variables" might be a good starting point to figure out an answer. (more)	—	11 months ago
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)	—	12 months ago
Edit	Post #291307	Post edited: added thoughts	—	12 months ago
Edit	Post #291307	Initial revision	—	12 months ago
Question	—	How do I unambiguously define the facial circuits in a $2$ -cell embedding of a graph into a surface? Suppose that $Γ$ 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 $M - Γ$ (we choose to denote the image of the embedding also by $Γ$ ). Let us assume that the embedding is such that... (more)	—	12 months ago
Comment	Post #291301	@#36356 Yes, the $0$ -simplex has nonempty interior. Here is a more precise definition of $\overset{\circ}{Δ}^{n}$ from Hatcher (page 103). > If we delete one of the $n + 1$ vertices of an $n$ -simplex $[v_{0}, \dots, v_{n}]$ , then the remaining $n$ vertices span an $(n - 1)$ -simplex, called a face of... (more)	—	12 months ago
Edit	Post #291301	Initial revision	—	12 months ago
Question	—	Is there a $Δ$ -complex structure on the sphere with less than three $0$ -simplices? In Hatcher's Algebraic Topology, a $Δ$ -complex structure on a topological space $X$ is defined as follows. Here, $Δ^{n}$ denotes the standard $n$ -simplex in $R^{n + 1}$ , and $\overset{\circ}{Δ}^{n}$ denotes its interior. > A $Δ$ -complex structure on a space $X$ is ... (more)	—	12 months ago
Edit	Post #291159	Initial revision	—	about 1 year 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: $u t - a^{2} u x x = 0, 0 < x < L, 0 < t,$ with the boundary conditions $u (0, t) = u (L, t) = 0, 0 < t,$ and initial condition $$ u(x,0) ... (more)	—	about 1 year 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 year ago
Edit	Post #291056	Post edited: clarified definition of free edge	—	about 1 year ago
Edit	Post #291056	Initial revision	—	about 1 year 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 $(G, V)$ into a connected, oriente... (more)	—	about 1 year ago
Comment	Post #290771	This post seems to have been deleted by the moderators of r/math. (more)	—	about 1 year ago
Edit	Post #290771	Post edited: godel -> Gödel; i -> I	—	about 1 year ago
Suggested Edit	Post #290771	Suggested edit: godel -> Gödel; i -> I (more)	helpful	about 1 year ago
Edit	Post #290746	Post edited: formatted answer as it appears on Proof Assistants SE	—	about 1 year ago
Suggested Edit	Post #290746	Suggested edit: formatted answer as it appears on Proof Assistants SE (more)	helpful	about 1 year ago
Comment	Post #280864	@Technologicallyilliterate Added a few words to my answer. (more)	—	about 4 years ago
Edit	Post #280864	Post edited:	—	about 4 years ago
Comment	Post #280866	This is some meaty stuff, and I love it. Thank you for writing it up! (more)	—	about 4 years ago
Comment	Post #280864	@Technologicallyilliterate Sure, I'll try to add something tomorrow. (more)	—	about 4 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 4 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 4 years ago
Edit	Post #280864	Initial revision	—	about 4 years ago
Answer	—	A: If $n = x m$ and $n \to \infty$ , then $m \to \infty$ ? > […] I would have commenced with defining $\frac{x}{n}$ as $\frac{1}{m}$ , 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 4 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 4 years ago
Comment	Post #280842	Taking another look at the displayed equation in my post, I think I can take $G_{1} = 0$ and $G_{2} = \int F_{3} (x, y), d x$ (or the other way around) provided the integral exists. This seems to provide some criterion at least in the $R^{2}$ case. (more)	—	about 4 years ago
Edit	Post #280842	Initial revision	—	about 4 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 $R^{2}$ or $R^{3}$ , we have $\nabla \times (\nabla f)... (more)	—	about 4 years ago
Edit	Post #280630	Post edited: replaced dollars with backticks (code formatting) because the question is confusing when the MathJax is rendered :)	—	about 4 years ago
Edit	Post #280629	Post edited: fixed some MathJax markup	—	about 4 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 4 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 4 years ago
Suggested Edit	Post #280629	Suggested edit: fixed some MathJax markup (more)	helpful	about 4 years ago
Comment	Post #280462	+1 This is exactly what I needed, thank you for the clear explanation and the reference! :) (more)	—	about 4 years ago
Edit	Post #280460	Initial revision	—	about 4 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 $S g$ , the sphere with $g \geq 0$ handles, or $N k$ , the sphere with $k \geq 1$ crosscaps. If my understanding is correct: - attaching a ... (more)	—	about 4 years ago

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