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visits member for 2 years, 5 months
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May
17
accepted Are maps locally preserving collinearity homographies?
May
16
asked Does Caratheodory's method work in Riesz-Markov representation theorem
May
16
asked Are maps locally preserving collinearity homographies?
May
1
comment Should diffeomorphisms preserving arc length be affine?
M.Berger's Geometry I, section 9.5.4.9. I was looking for a reference for projective geometry. A random walk led me to that...
Apr
27
comment Should diffeomorphisms preserving arc length be affine?
It seems that the proof is like the one of Liouville's theorem, i.e., the smooth conformal transformation of Euclidean space is just a similar transformation. The trick you used is so called the braid lemma (symmetric in the first two, anti-symmetric in the last two)?
Apr
25
comment Book suggestions on projective geometry
@user40276 Well, thanks. I'm now reading M.Berger's Geometry, and I found M.Reid's Geometry and Topology. It might not be the ultimate answer, but I decide to read them first. Incidentally, I did search for modular groups, etc, but eventually I found that I don't have enough mathematical maturity, so I postponed such a study process.
Apr
24
revised Book suggestions on projective geometry
deleted 1 character in body
Apr
24
comment Book suggestions on projective geometry
@user40276 Not hyperbolic, but projective. I know that Poincare's conformal model for hyperbolic planes, but it's another story.
Apr
24
comment Book suggestions on projective geometry
It's certainly not a book for projective geometry. It's not studying projective invariants, and don't solve my question on cross ratios.
Apr
20
comment A sebset of $\Bbb C^2$
Well, you can try $m(f^{-1}(I)\cap F)$ is small if $m(I)$ is small enough, for any compact set $F$, then $m(f^{-1}(K)\cap F)$ should be zero. I forgot to restrict the domain on a finite measure space.
Apr
20
comment Show that a proper continuous map from $X$ to locally compact $Y$ is closed
In fact, locally compact spaces are compactly generated, and a continuous from a topological space to a compactly generated Hausdorff space is proper if and only if it's closed and preimages of singletons are compact.
Apr
20
comment The sum of reciprocal squares: estimating the remainder
For the asymptotic of the remainder term, try Euler-Maclaurin formula.
Apr
20
accepted Outer measure and Caratheodory's criterion
Apr
19
comment Moving the branch cut of the complex logarithm
It works if we rewrite $f(z)=\log(z-\sqrt[3]2)+\log(z-\sqrt[3]2\omega)+\log(z-\sqrt[3]2\omega^2)$, but I want to obtain good insight on how Riemann surfaces work. For example, what's the exact meaning of $\log(fg)=\log f+\log g$ w.r.t. Riemann surfaces when $f,g$ are multi-valued functions.
Apr
19
comment Moving the branch cut of the complex logarithm
Taking a single-valued branch is always annoying to me. +1
Apr
19
revised Moving the branch cut of the complex logarithm
added 1 character in body
Apr
19
awarded  Announcer
Apr
19
comment A sebset of $\Bbb C^2$
Suppose $f\colon\mathbb C^2\to\mathbb R,(z,w)\mapsto\lvert zw\rvert$. Note that if $I$ is an interval such that $m(I)$ is small, then $f^{-1}(I)$ is measurable and $m(f^{-1}(I))$ is also small.
Apr
19
comment How to prove that $\max\{f,g\}$ is Riemann integrable?
Note that $\max(f,g)=(f+g+\lvert f-g\rvert)/2$.
Apr
19
comment Outer measure and Caratheodory's criterion
@GiuseppeNegro Generally, if $\mathcal A$ is a collection of subsets closed under countable intersection, then there's $A_0\in\mathcal A$ such that $m^*(A_0)=\alpha=\inf_{A\in\mathcal A} m(A)$. We again choose $A_n$ such that $m^*(A_n)\ge\alpha+1/n$, then let $A_0=\bigcap_n A_n\in\mathcal A$ and by def $m^*(A_0)\ge\alpha$, but $m^*(A_0)\le m^*(A_n)\le\alpha+1/n$.