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olivier dot begassat dot cours at gmail dot com


Jan
19
comment Prove: $\int_0^1 \frac{\ln x }{x-1} d x=\sum_1^\infty \frac{1}{n^2}$
In case anybody reads this answer, I think it is necessary to say something like for all $u\in(0,1)$ and for all $n\in\Bbb N$, $$\left|1+\frac12u+\frac13u^2+\cdots\frac1nu^{n-1}\right|\leq-\frac{\ln(1-u)}u=g‌​(u)$$ so that, since $g$ is integrable on $(0,1)$, one can apply dominated convergence, and swap the sum with the integral.
Jan
19
comment Prove $F^2_{n+1} - F_nF_{n+2} = (-1)^n$
AlexR is correct!
Jan
19
revised Quotient space of $S^n$ and the projective plane
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Jan
19
comment Quotient space of $S^n$ and the projective plane
I doubt it: I don't thinkthe mapping cylinder is orientable when $n$ is even.
Jan
19
revised Quotient space of $S^n$ and the projective plane
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Jan
19
answered Quotient space of $S^n$ and the projective plane
Jan
19
revised Mapping torus with homotopic homeomorphisms
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Jan
19
comment Mapping torus with homotopic homeomorphisms
@YonKim corrected the formulas.
Jan
19
revised Mapping torus with homotopic homeomorphisms
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Jan
19
comment Ext groups due to Yoneda: why is this class zero
Are you sure those modules are projective $\Bbb K[x]$-modules?
Jan
19
answered Big list of “guided discovery” books
Jan
19
comment Is $\text{Ind}_{Z(G)}^G \rho$ irreducible or not for nonabelian group $G$?
It seems my first question is answered by the character formula for $\mathrm{Ind}\,W$, and this shows that the spanning set is indeed a basis, and the following comment shows that it splits, as a $Z$ module, into $[G:Z]$ copies of $W$. Ok, everything resolves itself : ) +1
Jan
19
comment Is $\text{Ind}_{Z(G)}^G \rho$ irreducible or not for nonabelian group $G$?
Also, if I'm not mistaken, if you consider $W=\Bbb C.w$, then it seems more correct to argue as follows: $$z\cdot(g\otimes w)=(zg)\otimes w=(gz)\otimes w=g\otimes (zw)=\rho(z)\cdot(g\otimes w)$$ so that $g\otimes w$, provided it is nonzero, spans a $Z$-submodule isomorphic to $W$ (they have the same character). Does that seem fair?
Jan
19
comment Is $\text{Ind}_{Z(G)}^G \rho$ irreducible or not for nonabelian group $G$?
How do you know that $\mathrm{Ind} W$ has dimension $|G|/|Z|$. I can see that there is a spanning set of this cardinality, but I don't see why it is necessarily a basis. Is it obvious?
Jan
19
comment Is $\text{Ind}_{Z(G)}^G \rho$ irreducible or not for nonabelian group $G$?
Complex irreducible representations of abelian groups are one dimensional... So $n=1$.
Jan
19
revised Prove $\int_{-\pi}^{\pi}\sin \sin x \,dx=0$ without using the fact that $\sin(x)$ is odd.
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Jan
19
revised To show that $\sum a_nb_n$ is absolutely convergent
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Jan
19
answered To show that $\sum a_nb_n$ is absolutely convergent
Jan
19
comment Mapping torus with homotopic homeomorphisms
Ok, there are some mistakes in the formulas, I'll get to them tomotopy.
Jan
19
revised Mapping torus with homotopic homeomorphisms
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