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 Jun4 comment Laplace equation with periodic boundary conditions Do you know the method of separation of variables? Jun4 comment Prove that the circle $S^1$ is not the boundary of any compact manifold with boundary in $\mathbb R^2-{(0,0)}$ If $S^1=\partial M$, then Stokes's theorem says $\int_{S^1} w=\int_M dw$, but $dw=0$ on $\mathbb R^2\setminus\{(0,0)\}$. Jun4 comment How to prove that operator is not compact in $L_2 (\mathbb{R})$ $A$ is even not well defined. For example, if $f(t)=e^{-t^2/4}$, then $(Af)(x)\equiv+\infty$. Jun4 answered $f_{n+1}(x):= \int_a ^x f_n(t)dt$, $\sum_{m=1} ^{\infty} f_m(x)$ is uniformly convergent Jun4 comment Is every vector space basis for $\mathbb{R}$ over the field $\mathbb{Q}$ a nonmeasurable set? @AndréNicolas: Due to Steinhaus theorem, there is no measurable basis with positive measure. Jun4 comment A boundary version of Cauchy's theorem I think the argument in your answer is just the same as the argument in the paper mentioned by @Potato in the question. I wonder, as asked in Potato's original question, is there any other proof without using Mergelyan's Theorem, at least when the boundary has some smoothness? May31 comment Simple non-closed geodesic. I don't understand why you accepted Neal's answer. As Daniel Rust commented, it didn't answer your question, because it didn't show whether the non-closed geodedics could be simple or not. May31 comment Simple non-closed geodesic. Could you please be more specific to explain how Hedlund's paper implies that there exists a simple dense geodesic? May29 revised Find all polynomials $P(x)$ satisfying this functional equation deleted 4 characters in body May29 answered Find all polynomials $P(x)$ satisfying this functional equation May26 comment How can I show whether the series $\sum_{n=1}^\infty \frac{(-1)^n}{n(2+(-1)^n)}$ converges or diverges? Denote $a_n=\frac{(-1)^n}{n(2+(-1)^n)}$. Note that $a_{2n-1}+a_{2n}=-\frac{1}{2n-1}+\frac{1}{6n}<-\frac{1}{3n}$. May25 comment Sequences with the following properties… @Chung.J: Thank you! May25 answered Sequences with the following properties… May24 awarded Good Answer May22 revised $(x+2)\cos\frac1{x+2} - x\cos\frac1x > 2$ for $x\in[1,\infty)$ deleted 41 characters in body May22 answered $(x+2)\cos\frac1{x+2} - x\cos\frac1x > 2$ for $x\in[1,\infty)$ May21 revised What is my operator norm (cannot get good enough bounds). added 1406 characters in body May21 comment What is my operator norm (cannot get good enough bounds). @Norbert: I did use it. I just want to emphasize that since $B$ is compact, when can say more than $\|A\|^2=\|B\|$. May20 answered What is my operator norm (cannot get good enough bounds). May20 comment Let$f : [0, 1]^2 \to R$ such that $f(x, y)$ is continuous in $x$ for each fixed $y$ and conversely also. Is $f$ continuous? You may consider a function like $f(x,y)=\frac{xy}{x^2+y^2}$ when $(x,y)\ne (0,0)$ and $f(0,0)=0$.