New answers tagged computational-geometry
1
The problem on which I originally made that comment is a linear algebra problem: consider the linear matrix equation
$$
\sum_{i=1}^k A_i X B_i = C,
$$
where $A_i,B_i,C \in \mathbb{R}^{n\times n}$ are given, and $X\in \mathbb{R}^{n\times n}$ is the unknown.
For $k=2$ this is a generalized Sylvester equation, and can be solved using a Bartels-Stewart-type ...
2
The infinite square well potential problem in non-relativistic quantum mechanics has energy eigenvalues $E_n=n^2\hbar^2\pi^2/2mL^2$,where $n^2=\sum_{k=1}^Nn_k^2$($N$=number of dimensions).
A problem of interest is, given an energy eigenvalue, the degeneracy(i.e,number of eigenstates with same eigenvalue) are to be found.This amounts to counting the number ...
1
@federicopolini is right in his answer: Introduce
$$
c= \sqrt{a}, d=\sqrt{b}
$$
and your optimization problem will now read as follows:
$$
\min (x-c^2)^2+(y-d^2)^2
$$
subject to the constraints
$$
c+d = 2, \\
c\ge 0,\\
d\ge 0.
$$
The inequality constraints are important to ensure that you get a solution that makes sense.
Now, you can eliminate $d=...
1
From a comment: I suggest you to set $\sqrt{a}:=c$ and $\sqrt{b}:=d$ and then pass the problem in the variables c,d to whatever computational software you are using. I would avoid those non-smooth square roots in the constraints at all costs if it's possible.
The general idea (from a very philosophical standpoint; this feels more like a comment than an ...
3
Dont expand the divergence term, apply integration by parts without doing this
$$
\int_\Omega w \nabla\cdot(\alpha\nabla u) dx = - \int_\Omega \alpha \nabla u \cdot \nabla w dx + \int_{\partial\Omega} w \alpha \frac{\partial u}{\partial n}ds
$$
In your notation, define a vector field
$$
\vec{A} = \alpha \nabla u
$$
and then do the integration by parts on
$$
\...
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