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Homebrewer and mathematically oriented physicist with chemistry aspirations.


1d
comment Help with proof that $H^{1}(\mathbb{R})$ is closed under multiplication.
It is clear to me that the statement cannot be generalized to any $H^1(R^n)$, as seen by studying singularities on the form $|x|^\alpha$ with $\alpha<0$.
1d
comment Help with proof that $H^{1}(\mathbb{R})$ is closed under multiplication.
Is it really true? If $u,v \in L^2(R)$ then $uv \in L^1(R)$ which is not a subset of $L^2(R)$.
Dec
14
comment Using differentiation
A is the function's only local maximum, so you need to find the roots of $d^2y/dx^2$ to proceed. Point B is a root where $dy/dx$ is positive, so you need to solve $y=0$ and check the derivative.
Dec
13
comment Use differentials to estimate the error in volume of the box.
You will obtain that the estimated error $dv$ will differ from the difference $v - v_\text{max}$ or $v - v_\text{min}$. Why is this?
Dec
13
comment Use differentials to estimate the error in volume of the box.
I guess the maximum volume is the volume obtained when all lenghts are the longest -- add 1 % to all lengths. Similar for the minimum volume.
Dec
13
comment Use differentials to estimate the error in volume of the box.
$dv$ has units of volume. $dv/v = 0.9 \%$.
Dec
10
comment Convergence of sequence of $L^{p}$ function
Is $\Omega$ a bounded set?
Dec
10
comment Integrate $\int_1^\infty\lfloor x\rfloor e^{-x}\,dx$
The summation can be done by summing $\sum_n e^{-\beta n}$ using geometric series, then differentiating at $\beta = 1$.
Dec
2
comment in normed space hyperplane is closed iff functional associated with it is continuous
My comment regarding Hahn-Banach is irrelevant/not correct. See answer below.
Dec
2
answered in normed space hyperplane is closed iff functional associated with it is continuous
Dec
2
comment in normed space hyperplane is closed iff functional associated with it is continuous
Yes, continuity at zero is equiv to continuity for linear functionals, which again is equiv to boundedness.
Dec
2
comment in normed space hyperplane is closed iff functional associated with it is continuous
If $f$ does not vanish identically, you can use Hahn-Banach and show the existence of some $x$ such that $f(x)\neq \alpha$.
Dec
2
comment in normed space hyperplane is closed iff functional associated with it is continuous
$f$ is continuous since its norm is estimated and finite.
Dec
2
comment in normed space hyperplane is closed iff functional associated with it is continuous
$H^c$ is nonempty since it is assumed that $f\neq 0$ identically in the definition of a hyperplane.
Nov
26
comment Type of convex function?
You can easily derive the inequality $d f'_+(x) \geq f(x+d) - \lim_{\alpha\to 0+} \alpha^{-1}f(x)$, where the left side is finite for almost all $x$. This shows, at least, that $f(x)\geq 0$ for almost all $x$.
Nov
26
answered General or specific property? $(1-p)^{-x^2} = x^2 p + \mathcal{O}(p^2)$
Nov
24
comment permutation matrix?
Note that the lower right block to an even power is $I$, while an odd power is $[0,1;1,0]$.
Nov
16
comment Finding the closed form of an infinite series
One could eliminate the cosine ...
Nov
6
comment Discontinuous functionals on $L^p$
Yes, and then I need to show that $\sup_n\|g_n\|_{L^{p*}} = \|g\|_{L^{p*}}$ but I am guessing that is rather easy. Thanks for your help!
Nov
6
comment Discontinuous functionals on $L^p$
Hold on. Am I applying Banach-Steinhaus in the wrong way? BS assumes that the functionals in the given collection (here $g$ alone) are continuous. Then, I cannot use B-S to prove that $g$ is continuous.