Activity for WheatWizardâ€
| Type | On... | Excerpt | Status | Date |
|---|---|---|---|---|
| Edit | Post #295472 |
Post edited: |
— | 7 months ago |
| Edit | Post #295472 | Initial revision | — | 7 months ago |
| Answer | — |
A: Why are regular languages closed under intersection and complementation? The key fact here is that regular languages are exactly the languages which can be decided by a computer with constant memory. This is basically the definition of a finite state automaton. From here it is easy to see these facts. You've already observed this for union, but complementation and ... (more) |
— | 7 months ago |
| Edit | Post #294815 |
Post edited: |
— | 11 months ago |
| Edit | Post #294815 | Initial revision | — | 11 months ago |
| Question | — |
What complexity implications are there from *not* being in an abstract family of languages? A classic result is that a if a language is not regular its decision problem requires more than constant memory (i.e. $\Omega(\log \log n)$). I am wondering if there are similar results for other classes of language (e.g. context-free or tree-adjoining). Specifically results of the form > I... (more) |
— | 11 months ago |
| Edit | Post #294237 |
Post edited: |
— | about 1 year ago |
| Comment | Post #294237 |
The half line is indeed one ended. To be quite honest, the topological definition of ends is a little mystical to me. I generally deal with the ends of hyperbolic spaces, which I am aware are different in some contexts. I'll try to give a mostly intuitive understanding of that, which does apply i... (more) |
— | about 1 year ago |
| Edit | Post #294237 | Initial revision | — | about 1 year ago |
| Answer | — |
A: Is the "mucube" homeomorphic to the loch ness monster? They are homeomorphic. The conditions given in the question: - infinite genus - orientability - having 1-end describe exactly one 2-manifold. This is a result of the classification of non-compact surfaces. The paper by Arredondo and Maluendas "On the infinite Loch ness monster" describes... (more) |
— | about 1 year ago |
| Edit | Post #294222 | Initial revision | — | about 1 year ago |
| Question | — |
Is the "mucube" homeomorphic to the loch ness monster? I'm wondering if two non-compact 2-dimensional manifolds are homeomorphic. The first is the "loch ness monster" a one ended surface formed from the sum of infinitely many tori: Loch ness monster illustration The other is a surface which I do not know a name for but is related to the "mucube... (more) |
— | about 1 year ago |
| Edit | Post #294111 |
Post edited: |
— | over 1 year ago |
| Edit | Post #294111 | Initial revision | — | over 1 year ago |
| Question | — |
Lifting a graph "folding" to a covering space I have the following problem, which seems really simple but I can't seem to quite get it. The problem is about connected edge labeled directed graphs. I like to think of these as an ordinary graph $K$ plus a continuous map $\ellK : K \rightarrow RX$ from $K$ to a rose graph which preserves edg... (more) |
— | over 1 year ago |
| Edit | Post #293749 | Initial revision | — | over 1 year ago |
| Answer | — |
A: What are the known chiral polytopes of full rank? There are more known. Pellicer describes 3 additional chiral 4-polytopes of full rank in Chiral 4-polytopes in ordinary space (2017). These are all apeirotopes in 3 dimensions. (more) |
— | over 1 year ago |
| Edit | Post #293740 | Initial revision | — | over 1 year ago |
| Question | — |
What are the known chiral polytopes of full rank? The chiral polytopes of full rank are very interesting. It was once erroneously proven by McMullen that there were none, however it has since been revealed to not be the case. I am aware of two chiral polytopes of full rank: (I'm also aware of chiral polytopes of full rank in projective space, bu... (more) |
— | over 1 year ago |
| Edit | Post #293232 |
Post edited: Clarified two things. First that a linear kei need not contain all the lines in n-space, next that I am looking for an equational law. |
— | over 1 year ago |
| Edit | Post #293232 | Initial revision | — | over 1 year ago |
| Question | — |
Is there an equational law satisfied by "linear kei" but not free kei? A kei is an involutionary quandle, alternatively a magma satisfying three equations: $a \rhd a = a$ $(a \rhd b) \rhd b = a$ $(a \rhd b) \rhd c = (a \rhd c) \rhd (b \rhd c)$ This definition is taken from Kamada, S., 2002. I will also define a "linear kei" as structure generated by a ... (more) |
— | over 1 year ago |
| Edit | Post #291362 |
Post edited: Found another proof. |
— | over 2 years ago |
| Edit | Post #291362 | Initial revision | — | over 2 years ago |
| Question | — |
Seeking a theorem about lattices I am looking for a reference on the following theorem, or an equivalent statement: > Let $\Lambda$ be an embedding of a free $\mathbb{Z}$-module in $\mathbb{R}^d$. If the rank of $\Lambda$ is greater than $d$ then $\Lambda$ is not discrete. I have proofs here and here, which both seem corre... (more) |
— | over 2 years ago |
| Edit | Post #291124 | Initial revision | — | over 2 years ago |
| Question | — |
Does {7,4|3} have a realization on the Klein quartic? It was pointed out to me recently that the polyhedron {7,4|3} has the same automorphism group as the Klein quartic. Specifically {7,4|3} is: $\langle \rho0,\rho1,\rho2 \mid \rho0^2, \rho1^2, \rho2^2, (\rho0\rho2)^2, (\rho0\rho1)^7, (\rho1\rho2)^4, (\rho0\rho1\rho2\rho1)^3\rangle$ Natural... (more) |
— | over 2 years ago |
| Edit | Post #290492 |
Post edited: |
— | over 2 years ago |
| Edit | Post #290492 |
Post edited: A little about the automorphism group. |
— | over 2 years ago |
| Edit | Post #290492 | Initial revision | — | over 2 years ago |
| Question | — |
Is there a $(n_3)$ configuration which is not self-dual? Coxeter points out that for a self-dual configuration $(mc,nd)$ it must be that $m=n$ and $c=d$, so we may abbreviate it $(mc)$. However I'm interested in the other direction of this implication, i.e. is there a configuration $(mc,mc)$ which is not self-dual? For $c=2$ there is none. All polyg... (more) |
— | over 2 years ago |
| Comment | Post #290196 |
Yes. Thank you. It's easy to forget about connected and compact. :) (more) |
— | almost 3 years ago |
| Edit | Post #290196 |
Post edited: Fixed claim per comment. |
— | almost 3 years ago |
| Comment | Post #290305 |
> I notice the atlas link lists neither the 11-cell [...] nor the 57-cell [..]
The 11-cell is listed [here](https://www.abstract-polytopes.com/atlas/660/13/1.html). The 57-cell is too large for the atlas to list it. (more) |
— | almost 3 years ago |
| Comment | Post #290305 |
What is meant by admits a realization?
In McMullen & Schulte's *Abstract Regular Polytopes* the notion of realization is quite broad, but even their more restrictive notion of a "faithfully symmetric" realization includes a lot. They show **every** abstract regular polytope has a "simplex real... (more) |
— | almost 3 years ago |
| Edit | Post #290196 |
Post edited: |
— | almost 3 years ago |
| Edit | Post #290196 | Initial revision | — | almost 3 years ago |
| Question | — |
Is the nth Betti number determined by orientability? I'm interested in a proof of the following claim: > If $M$ is a connected $n$-dimensional compact manifold then the $n$th Betti number, $\betan(M) = 1$ if $M$ is orientable and $\betan(M) = 0$ otherwise. This claim seems true since it basically says that orientable manifolds have some sort ... (more) |
— | almost 3 years ago |
| Edit | Post #289046 |
Post edited: |
— | about 3 years ago |
| Edit | Post #289046 | Initial revision | — | about 3 years ago |
| Question | — |
Proving that this relation implies another relation on the Coxeter group [4,3,3,4]. I have a group with five generators $\sigmai$, and the following relations: \begin{split} \sigmai^2 = \varepsilon \\ |i-j| \neq 1 \implies (\sigmai\sigmaj)^2 = \varepsilon \\ (\sigma0\sigma1)^4 = \varepsilon \\ (\sigma1\sigma2)^3 = \varepsilon \\ (\sigma2\sigma3)^3 = \varepsilon \\ (\sig... (more) |
— | about 3 years ago |
| Edit | Post #288852 |
Post edited: Added linke to atlas. |
— | about 3 years ago |
| Edit | Post #288852 |
Post edited: |
— | about 3 years ago |
| Edit | Post #288852 |
Post edited: |
— | about 3 years ago |
| Edit | Post #288852 | Initial revision | — | about 3 years ago |
| Question | — |
What is special about the 11-cell and 57-cell? Reading about the 11-cell and 57-cell I find two facts implied often: - They are particularly notable among the abstract regular 4-polytopes. - They are related to each other. I'll establish why I think they are notable: - Both polytopes are notable enough to have their own articles on Wiki... (more) |
— | about 3 years ago |