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Mar
20
comment The function $f(z)=|z|^2$ is only differentiable at the origin
$f$ is complex differentiable if and only if the real and imaginary parts satisfy the Cauchy--Riemann equations.
Feb
17
comment Preimage of Legendre-Fenchel transform
In the non-convex case it is exactly the same. But I would rather have some more direct statements ..
Feb
17
comment Preimage of Legendre-Fenchel transform
Yes, I am interested in the case where $X$ may be nonreflexive, but also reflexive $X$ are of interest.
Feb
17
asked Preimage of Legendre-Fenchel transform
Feb
11
comment Prove that $f$ is identically zero.
Note that you can solve the differential equation: $df/f^2 = dx$. Integrate to get $f(x) = 1/[1/f(y) - (x-y)]$. for any $x,y$.
Jan
25
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
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 $\beta = 1$.