Pedro Tamaroff
Reputation
77,380
96/100 score
 16h comment A group-ring is commutative if and only if that group is abelian Note that since multiplication in your algebra is defined by the multiplication of the basis elements, that is $g\cdot h=gh$ where the RHS is multiplication in $G$, your algebra will be commutative iff the basis elements commute with themselves. This amounts to saying $G$ is commutative. 16h answered How to show $\mathbb{R}^2/\mathbb{Z}^2$ is homeomorphic to $\mathbb{R}/\mathbb{Z} \times \mathbb{R}/\mathbb{Z}$ 19h comment Proving the cardinality of $|A| =|\mathbb Z|$ If something is enumerated by another set $I$ it has cardinality at most that of such set, for there is a surjection from $I$ to such enumerated set. Since in your case your set is at most countable and infinite, it is countably infinite. 1d answered how to prove that $C^{k}$ map does not depend on choice of the charts 1d comment How many ways are there to prove Cayley-Hamilton Theorem? @NNN They are useful for future reference. 2d comment Extending the automorphism of $Q(\sqrt2)$ to automorphism of $Q(\sqrt(1+\sqrt2))$. Your bigger extension is not normal. It doesn't contain, in particular, the two other complex roots. ($\sqrt 2 >1$!). In fact, you cannot extend your morphism. May 2 answered Why does $\sum_{k=1}^{\infty}\sum_{\ell=0}^{k-1} = \sum_{\ell=0}^{\infty}\sum_{k=\ell+1}^{\infty}$ May 2 comment Show that $S^1$ acts on $S^3$ @Epsilon Using $\bar z$ for $z^{-1}$ obscures the fact that if $z$ acts by an automorphism, then its inverse must be the automorphism by which $z^{-1}$ acts. May 2 revised Part of proof of the set of continuous integrable functions is dense in $L^1(\Bbb R)$ edited title May 1 comment If $(a_n)$ is positive and $\sum\limits_n \frac{a_n}{1+a_n}$ converges then $\sum\limits_n a_n$ converges Adam Hughes' answer shows the hypothesis that $a_n>0$ is necessary, and gives $a_n = (-1)^n/n$ as a counterexample. May 1 answered If $(a_n)$ is positive and $\sum\limits_n \frac{a_n}{1+a_n}$ converges then $\sum\limits_n a_n$ converges May 1 comment Showing epimorphism without using the Freyd-Mitchell Embedding Theorem What about using MacLane's technique of "elements"? May 1 awarded Popular Question Apr 30 comment Part of proof of term-by-term integration Keep looking! ${}$ Apr 30 comment Part of proof of term-by-term integration This is proved in every book covering Lebesgue integration. Did you check those? Apr 30 comment Catalan numbers and triangulation It should be noted that the core of this argument is how to obtain a triangulation of a smaller n-gon given one of a larger one, and remembering how the collapsing was done gives the bijection. One has to track things carefully, is all. Apr 30 comment Catalan numbers and triangulation (This is not a combinatorial proof!) Apr 30 comment Example of a diffeomorphism from all of $\mathbb{R}$ to itself Any linear map, any affine transformation. Multiplication by a nonzero real. Apr 30 comment Examples of Manifolds such that $\chi (X)=-3$ Try higher dimensions. Apr 29 comment Let $f:[0,1] \rightarrow \mathbb{R}$ be a continuous function.Show that $\lim_{n \rightarrow \infty} n \int_{0}^{1}e^{-nx}f(x)dx=f(0)$ Evaluate the integral. $f(0)$ is constant.