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Jan
11
revised Flood algorithm - find polygon containing a given point.
edited tags
Jan
11
comment Could a group embed normally in another group in which any automorphism is attained by conjugation?
I considered the embedding from $G$ into $\operatorname{Sym} G$. It's not normal, though. However, I didn't realize that the normalizer simply works.
Jan
11
comment Could a group embed normally in another group in which any automorphism is attained by conjugation?
Is there any intuition on semi-direct products? I met it in many situations, but all of them are algebraic constructions. I didn't see any other perspective, for example, from group action, or the viewpoint of transformation group.
Jan
11
accepted Could a group embed normally in another group in which any automorphism is attained by conjugation?
Jan
11
comment Could a group embed normally in another group in which any automorphism is attained by conjugation?
@DerekHolt Sounds good. Could you please write a complete answer? (I'm not familiar with these concepts, but after a glance at wikipedia, it seems right.)
Jan
11
asked Could a group embed normally in another group in which any automorphism is attained by conjugation?
Jan
3
awarded  Informed
Jan
3
comment Is it possible to prove that $K\subseteq C(G)$
$kgk^{-1}g^{-1}=k(gk^{-1}g^{-1})\in K$
Jan
3
accepted Should diffeomorphisms preserving arc length be affine?
Jan
3
comment Should diffeomorphisms preserving arc length be affine?
Could you assimilate these comments into the answer? And even better, could you point out some elementary reference that the isometry of a pseudo-Riemannian manifold is locally determined?
Jan
2
comment Should diffeomorphisms preserving arc length be affine?
And why is an isometry determined by the first order data at a point? It looks like a localization.
Jan
2
comment Should diffeomorphisms preserving arc length be affine?
Tomorrow I'll check it in details. At a first glance, it seems that the proof relies on a smooth condition of $\varphi$, i.e. $\varphi\in C^2$. I don't know whether it's still true for $\varphi\in C^1$. However, that's not important to me (I don't care the smoothness condition).
Jan
1
comment How to calculate the inverse of a complex matrix?
In fact, it works for any arbitrary field $K$ instead of $\mathbb C$, or unitary commutative rings in which $ad-bc$ is invertible.
Jan
1
comment Show $S_4$ is not isomorphic to $D_{24}$ by looking at their centers
Proof for the triviality of the center of $S_4$: the conjugate $gag^{-1}$ of $a$ is just a relabeling of $a$. For example, if $a=(123)$, then $gag^{-1}=(g(1)g(2)g(3))$. It's not hard to show that if $a\neq 1$, the identity, then the conjugate class of $a$ isn't a singleton.
Dec
30
asked Does the implicit function theorem imply Peano existence theorem
Dec
29
comment Question about integral on hypersurface
Note that the ordinary surface integral gives a positive linear functional on $C_c(S)$, the space of continuous compactly supported functions on $S$, where $S\subseteq\mathbb R^n$ is locally compact Hausdorff. The Riesz representation theorem furnishes a Riesz measure on the surface $S$.
Dec
29
awarded  Custodian
Dec
29
revised Solving recurrences with summation factors (Concrete Mathematics)
edited tags
Dec
29
revised Should diffeomorphisms preserving arc length be affine?
edited title
Dec
29
revised Newton polygon and asymptotic behavior near a singular point
added 67 characters in body; edited tags