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Jun
5
comment Prove that the circle $S^1$ is not the boundary of any compact manifold with boundary in $\mathbb R^2-{(0,0)}$
@tagb78: For a $1$-form $w=pdx+qdy$, by definition, $dw=(\frac{\partial q}{\partial x}-\frac{\partial p}{\partial y})dx\wedge dy$.
Jun
5
comment Analytic continuation on the unit disc
You are welcome! Actually there are more pathological examples: $f$ could be not analytic at any point in $\partial D$. I hope you or someone else would provide an answer with more elaborate discussion.
Jun
5
comment Analytic continuation on the unit disc
Here is a counter-example.
Jun
5
comment A boundary version of Cauchy's theorem
@MalikYounsi: I agree with you, but I still look forward to a more elementary proof.
Jun
5
answered prove that , there is no element $a , b$ of the group $G$ which satisfy this property
Jun
4
comment Given $\Sigma a_n$ diverges show that $\Sigma \frac{a_n}{1+a_n}$ diverges.
Please note that this is not a duplicate of the linked question, because here $a_n$ is not assumed to be positive.
Jun
4
comment Given $\Sigma a_n$ diverges show that $\Sigma \frac{a_n}{1+a_n}$ diverges.
@Potato: Thank you!
Jun
4
answered Given $\Sigma a_n$ diverges show that $\Sigma \frac{a_n}{1+a_n}$ diverges.
Jun
4
comment Correspondence between modes of convergence and metrics
When $\mu$ is $\sigma$-finite, it seems that you may make use of the given $\rho$ to construct a similar metric.
Jun
4
comment $f_{n+1}(x):= \int_a ^x f_n(t)dt$, $\sum_{m=1} ^{\infty} f_m(x)$ is uniformly convergent
@Hagrid: You are welcome. I think your proof of the uniform convergence is essentially correct. Just a little correction: $b$ should be replaced by $b-a$.
Jun
4
comment Relationship between two projectors
Can you prove either "if" part or "only if" part by yourself?
Jun
4
comment Correspondence between modes of convergence and metrics
Do you assume that $\mu$ is $\sigma$-finite?
Jun
4
comment $f_{n+1}(x):= \int_a ^x f_n(t)dt$, $\sum_{m=1} ^{\infty} f_m(x)$ is uniformly convergent
@Hagrid: Firstly, $(1)$ implies that $(3)$ holds for $n=1$. Secondly, can you prove $(3)$ inductively on $n$ by using $(2)$ and the fact that $\int_a^x (t-a)^{n-1}dt=\frac{(x-a)^n}{n}$?
Jun
4
comment Laplace equation with periodic boundary conditions
I am afraid I cannot help you much more with this, but I suggest that you apply the method of separation of variables to find some explicit solutions first.
Jun
4
comment Laplace equation with periodic boundary conditions
Then applying this method you may find that the space of solutions is infinite dimensional.
Jun
4
comment Laplace equation with periodic boundary conditions
Do you know the method of separation of variables?
Jun
4
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 $\mathbb R^2\setminus\{(0,0)\}$.
Jun
4
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 $(Af)(x)\equiv+\infty$.
Jun
4
answered $f_{n+1}(x):= \int_a ^x f_n(t)dt$, $\sum_{m=1} ^{\infty} f_m(x)$ is uniformly convergent
Jun
4
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.