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Homebrewer and mathematically oriented physicist with chemistry aspirations.


1d
comment $\sum (\sqrt{x_k-k^2}-k)^2=0$ implies $x_k=2k^2$?
If you want to explore possible complex solutions, it may be fruitful to consider polar coordinate parameterizations of $X = x-1$ and $Y = y-4$, rewrite RHS as $(X+Y+5)/2$, and note that each square root has two branches ($X^{1/2} = |X|^{1/2}e^{i\theta + i k\pi})$, $k=0,1$), giving 4 possibilities on the LHS. I have no idea if you will find a solution, though.
Jan
24
comment $\sum (\sqrt{x_k-k^2}-k)^2=0$ implies $x_k=2k^2$?
I think the problem implicitly assumes that $x_k\geq k^2$. If ${x_k - k^2}<0$, you get two imaginary roots, and it is ambiguous which to choose from the notation. What is the context of this problem?
Jan
23
comment If a function is well defined and continuous can it have singularities?
The exponential function $e^x$ is everywhere smooth, so is the function $x$. Compositions of smooth functions are smooth.
Jan
22
comment If a function is well defined and continuous can it have singularities?
Ask yourself: Is there anything in the formula that indicates a singularity of some sort? The function is a simple composition of well-known functions about which you should know a lot.
Jan
22
comment Questions about coerciveness and convexity
No, linear space means a set such that linear combinations of elements are in the set. Here, a linear combination of two functions is again a function. But this function may not be coercive!
Jan
22
answered Questions about coerciveness and convexity
Jan
18
comment Book recommendation for Measure Theory
I think most of this is covered by Bartle's book. It is a compact Wiley Classics book.
Dec
29
comment Rational and irrational numbers
There is no minimal value $\epsilon$.
Dec
29
comment Proving differentiability
$(e^{hx}-1)/h$ is clearly bounded ...
Dec
21
awarded  Caucus
Dec
18
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 $\alpha<0$.
Dec
18
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
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.