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 Nov 10 comment What is the topology of smooth immersed submanifold? Thanks for your answer. Your advice enabled me to solve the exercise! Nov 10 asked What is the topology of smooth immersed submanifold? Oct 10 comment If $f\in C^{\infty}$, $f$ is in Schwarz space iff $x^{\beta}\partial^{\alpha}f$ is bounded for $\forall \alpha, \beta$. Thanks for your clarification! I think $\delta^{-1}\geq 1$, so the last term can be just modified $2^N\to 2^{N+1}$ as well. Oct 10 accepted If $f\in C^{\infty}$, $f$ is in Schwarz space iff $x^{\beta}\partial^{\alpha}f$ is bounded for $\forall \alpha, \beta$. Oct 10 asked If $f\in C^{\infty}$, $f$ is in Schwarz space iff $x^{\beta}\partial^{\alpha}f$ is bounded for $\forall \alpha, \beta$. Oct 4 comment Why is the number of components of Lie group finite? Thanks for your answer! That solved my question. Oct 4 asked Why is the number of components of Lie group finite? Oct 2 comment Alternative proof for dominated convergence theorem without using Fatou's lemma? Thanks for your remark! But sorry for confusion. When I wrote "after showing that $f_n\to f\in L^1$", I meant that "after showing that $f\in L^1$ where $f_n\to f$ pointwisely. So, probably there isn't any simplistic proof without Fatou's lemma. Oct 2 asked Alternative proof for dominated convergence theorem without using Fatou's lemma? Sep 7 comment What theorem of Sturm-Liouville theory guarantees that the complete solution of heat equation is the linear combination of $\sin nx$? That completely solved my question! Sep 6 asked What theorem of Sturm-Liouville theory guarantees that the complete solution of heat equation is the linear combination of $\sin nx$? Jul 12 awarded Yearling May 27 comment Why does $D \det(I)B=\frac{d}{d t}\det(I+t B)|_{t=0}$ hold? Yes. The one you linked here :-) May 27 comment Why does $D \det(I)B=\frac{d}{d t}\det(I+t B)|_{t=0}$ hold? Eq. (7) looks so interesting! I enjoyed reading your argument in another post where you answered to that question. May 27 comment Why does $D \det(I)B=\frac{d}{d t}\det(I+t B)|_{t=0}$ hold? Thanks for your concise answer! May 27 accepted Why does $D \det(I)B=\frac{d}{d t}\det(I+t B)|_{t=0}$ hold? May 26 comment How does the solution of ODE $y'=F(t,y)$ extend to an open interval? Thanks for a clear answer! May 26 accepted How does the solution of ODE $y'=F(t,y)$ extend to an open interval? May 26 revised How does the solution of ODE $y'=F(t,y)$ extend to an open interval? deleted 286 characters in body May 26 comment How does the solution of ODE $y'=F(t,y)$ extend to an open interval? I'm sorry, but I can't figure out how to use the definition to solve this problem.