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The problem on which I originally made that comment is a linear algebra problem: consider the linear matrix equation $$ \sum_{i=1}^k A_i X B_i = C, $$ where $A_i,B_i,C \in \mathbb{R}^{n\times n}$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 ...


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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 ...


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@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=...


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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 ...


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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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