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Results tagged with calculus-of-variations
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user 4923
Questions on the calculus of variations, which deals with the optimization of functionals mostly defined on infinite dimensional spaces.
7
votes
1
answer
200
views
Prove that the maximizing point configuration on the unit circle for a Vandermonde like func...
This is probably too hard for math.stackexchange, so I migrated it here.
For $\lambda_i \in S^1 \subset \mathbb{C}$, consider the functional $H(\{\lambda_1, \ldots, \lambda_n\}):= \sum_{j < k} | \lam …
- 4,354
2
votes
1
answer
816
views
Numerical or exact solution for a system of differential algebraic equations
I am looking for ways to solve the following system of boundary value implicit ODEs over the interval $t \in [0, 1]$:
\begin{equation}
\lambda (Fg - Gf)^3 + 4 FGfg(g-f) = 0 \\
fg(Fg-Gf) + 2FG(gf' - fg …
- 4,354
5
votes
1
answer
166
views
An extension of Hadamard maximum determinant problem
Consider the Vandermonde product $\prod_{1\le j < k \le n} |z_j - z_k|$. It is well-known that under the constraint $|z_j| \le 1$ for all $j$, the product is maximized at a picket fence configuration …
- 4,354
6
votes
0
answers
240
views
a variational problem related to weighted logarithmic capacity
Consider the following multiple contour integral:
$$ \Phi_\lambda := \oint \ldots \oint \prod_{1 \le j < k \le n} (z_j^{-1} - z_k^{-1}) \prod_{j=1}^n \prod_{k=1}^n (1 - z_j x_k)^{-1} \prod_{j=1}^n z_ …
- 4,354