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seen Apr 9 at 18:25

Nov
14
revised Contour integration of $\int_{-\infty}^{\infty} \frac{1-b+x^2}{(1-b+x^2)^2 + 4bx^2}dx = \pi$
added 24 characters in body
Nov
14
comment For a closed plane curve, showing some inequalities.
@JeongNam-ho: Thank you for being considerate of my mistakes.
Nov
12
revised Show that some $C^\infty$ real function is analytic
added 26 characters in body
Nov
7
comment Uniform convergence in $\mathbb{R}^2$
@Jack: Thank you for your reminding. As a response, I deleted that equality. Is it better now?
Nov
7
revised Uniform convergence in $\mathbb{R}^2$
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Nov
7
revised Uniform convergence in $\mathbb{R}^2$
added remark
Nov
7
comment Uniform convergence of constant speed $C^1$ curves with the same endpoints
$\sigma_n'$ may not be uniformly convergent. For example, let $a=(0,0)$, $b=(0,1)$, $L_n=\frac{n+1}{n}$, and $\sigma_n(t)=\frac{1}{2\pi n}(\cos 2(n+1)\pi t -1, \sin 2(n+1)\pi t)$, $t\in[0,\frac{1}{n+1}]$ and $\sigma_n(t)=(0,\frac{(n+1)t-1}{n})$, $t\in[\frac{1}{n+1},1]$. By the way, do you have any problem with my answer in the linked post?
Nov
7
comment Uniform convergence in $\mathbb{R}^2$
@Jack: Please see here.
Nov
6
comment For a closed plane curve, showing some inequalities.
@JeongNam-ho: Moreover, when I checked my answer, I found I had given a wrong link to Gauss–Bonnet theorem. I cannot believe how that could happen! Anyway, the link is corrected now. I sincerely apologize for these mistakes.
Nov
6
comment For a closed plane curve, showing some inequalities.
@JeongNam-ho: When I looked back at this post incidentally, I found that the remark $\gamma'(\Bbb R)=S^1$ in the last version of my answer is incorrect in general, so I removed it. (The statement is correct when $\gamma$ is a simple closed curve and when I made the remark, I overlooked the self-intersecting case.) Here is a counter-example of the remark for self-intersecting case. Let $\gamma$ be the arc-length reparametrization of the curve $t\mapsto (2\sin t, \sin 2t)$. Then it's easy to check that $(0,1)\notin \gamma'(\Bbb R)$.
Nov
6
revised For a closed plane curve, showing some inequalities.
removed an incorrect remark and corrected a link
Nov
4
comment Properties of a smooth bijection
For $M=N=\Bbb R$, $x\mapsto x^3$ is not of constant rank.
Nov
4
awarded  Revival
Nov
1
revised prove $f(x)$ has at least $2n$ roots
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Nov
1
answered prove $f(x)$ has at least $2n$ roots
Oct
27
comment Limit of $x_n/n$ as $n\to\infty$
@sundaycat: To take limit in $(5)$, I need assumption (ii) to apply $(1)$, where a reasonable requirement is $\theta_n$ cannot be too large. Without any assumption, I only know $0<\theta_n<\delta_n$. To control the upper bound of $\theta_n$, I need assumption $(1)$ to obtain $\delta_n\le\delta$ when $n$ is large. For example, consider $f(x)=x^p$($p\ge 1$) and an arbitrary $\theta:(0,\infty)\to (0,\infty)$. Then you need some condition on $\theta$ to get $\lim_{x\to\infty}\frac{f'(x+\theta(x))}{f'(x)}=1$.
Oct
26
comment Limit of $x_n/n$ as $n\to\infty$
@MartinArgerami: Thanks for your comment. I completely changed my argument after reading other answers.
Oct
26
revised Computing a limit almost surely using the strong law of large numbers
added 39 characters in body; edited title
Oct
26
awarded  Yearling
Oct
26
revised Limit of $x_n/n$ as $n\to\infty$
added 76 characters in body