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 Yearling
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Mar
21
comment When and why is $x^x$ undefined?
Check $f(-1/2)=-\sqrt{2}i$.
Jan
5
awarded  Yearling
Dec
27
comment Augmentation ideal and the abelianization of $G$
In case it helps, an explicit isomorphism is $g[G,G]\mapsto (g-1)+I^2$. The point here is that the elements $\lbrace g-1\rbrace_{g\in G}$ form a basis for $I$ as a $\mathbb{Z}$-module, noting that $I$ is (by definition) the kernel of $\mathbb{Z}[G]\to\mathbb{Z}$ given by $g\mapsto 1$ for $g\in G$.
Oct
20
comment Open in X (X is a metric space) how can you show or tell if it is open or not in general terms? What does it really mean?
HINT: The typical "ball" in $[0,2]$ is NOT the same as the typical "ball" in $\mathbb{R}$ (near the value 0).
Oct
16
awarded  Notable Question
Sep
8
comment Solutions to Dirichlet problem on manifolds with boundary
And not true (for uniqueness) if $M$ is noncompact. You need a "bounded" assumption on the functions (which the Dirichlet problem typically assumes). Take $f=0$ and solutions $w_1=e^x\sin y$ and $w_2=0$ on the half-space $\lbrace(x,y)\in\mathbb{R}^2\;|\; y\ge0\rbrace$. The analytic function in the background is $e^z$.
Sep
7
accepted Coupled PDE on cylinder
Sep
7
asked Coupled PDE on cylinder
May
21
comment is $dx$ greater than $\frac{dx}{2}$?
$dx$ is not a number, so I don't know what you mean by "greater than". This is the source of your confusion, I believe.
May
11
answered almost complex structures on $R^4$
Mar
20
revised Is the Pontrjagin-Thom-map null-homotopic if the normal bundle $N_i$ is trivial?
wrong Thom space
Mar
20
suggested approved edit on Is the Pontrjagin-Thom-map null-homotopic if the normal bundle $N_i$ is trivial?
Feb
16
revised Calculating the group co-homology of the symmetric group $S_3$ with integer coefficients.
deleted 331 characters in body
Jan
5
awarded  Yearling
Dec
8
awarded  Caucus
Nov
11
comment Is there an interpretation of higher cohomology groups in terms of group extensions?
Read Ken Brown's book Cohomology of Groups, because it explains $H^3$ as crossed-module extensions, and there is an extension-interpretation of $H^1$. In general, the higher cohomologies were described by MacLane (the appropriate references are given in this book) but are essentially intractable.
Nov
9
comment Simple Branched covering over sphere.
Ah you're right, I just realized I'm not answering the real question. I was only finding a complex structure such that the underlying topological surface is realized as a holomorphic cover, and as you point out, this handles all Riemann surfaces of genus 2.
Nov
9
comment Simple Branched covering over sphere.
Huh? The 180 degree rotation quotient map is an example of a simple branched covering for any genus, where the surface has said symmetry.
Nov
8
comment Simple Branched covering over sphere.
Right, and now I don't see how you can deduce anything further. You can use Riemann-Hurwitz to see that it is possible for this branched cover to have degree 2 and 2g+2 simple branch points. And we can construct this explicitly, it's given on pg105 of Donaldson (Figure 7.3), generated by 180 degree rotation.
Nov
6
comment Simple Branched covering over sphere.
Take a divisor of $\text{genus}+1$ points (crucial). Riemann-Roch says that the space of (nonconstant) meromorphic functions with (at worst) simple poles on the divisor is nonempty.