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4 votes
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$\min(\det(\mathbf{A}))$ for special matrix $\mathbf{A}$

Q1 The determinant is $\prod_{n=1}^{N-1} (1 - e^{-2\alpha(p_{n+1}-p_n)})$. Q2 Yes, using the answer to Q1. Q3 Yes, using the answer to Q1. The formula for Q1 is proved by induction on $N$. The ...
4 votes

$\min(\det(\mathbf{A}))$ for special matrix $\mathbf{A}$

To answer Question Q2: Yes, as the special case $u_j = \alpha p_j$ of the following result. We use $j,k$ for the indices rather than $i,j$ because we need $i = \sqrt{-1}$. Proposition. For pairwise ...
2 votes

Can solutions to Thomson's problem have pentagons?

Technically, there are best known configurations with pentagons and even with hexagons -- if we allow these polygons to loop around multiple faces. Such configurations may be read from the Wikipedia ...
4 votes
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How do people prove $\Gamma$-convergence in more complicated settings?

I am mostly familiar with the simpler definition ("Definition in first-countable spaces" from the Wikipedia link): Given the functionals $F_\varepsilon, F: X \to \overline{\Bbb{R}}$ (...
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0 votes

A detail in the proof of the Motzkin-Straus theorem

Yes, I believe that you are correct. The reason that most presentations of the proof don't make that point is simply that it is not needed for the proof.
1 vote
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Given a set of vectors how to pick $M$ such that sum of maximums of coordinates is maximized?

At very least the problem can be approached via (M)ILP by introducing indicator variables $q_{jk}$ telling whether the corresponding coordinate $v_{jk}$ is present in the sum of max, and $r_j$ telling ...

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