Questions tagged [logarithmic-energy]

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2 votes
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Sets having large capacity

The meta-question is: understanding by how much a set fails to have full capacity. I will pin it down to some concrete, although non-exhaustive, questions in a reasonably simple framework. Let ${\...
Nicola Arcozzi's user avatar
6 votes
1 answer
230 views

How to prove the monotonicity of the following function? $f(x) = \frac{\sum\limits_{i=1}^K\ln(1+a_ix)}{\sum\limits_{i=1}^K\ln(1+\frac{1}{2}a_ix)}$

How to prove the monotonicity of the following function? $f(x) = \frac{\sum\limits_{i=1}^K\ln(1+a_ix)}{\sum\limits_{i=1}^K\ln(1+\frac{1}{2}a_ix)}$. where $a_i>0$, $\forall i$, $x>0$. I have ...
mingzhanzhang's user avatar
4 votes
0 answers
534 views

Convexity of the electrostatic energy on a Riemann surface

Let $M$ be a compact Riemann surface. Let $\Lambda$ be a differentiable real $2$-form of integral one. Let $G$ be the Green function associated to $\Lambda$, i.e. $G: M \times M \to \mathbb R \cup \{...
D.E.G.Z.'s user avatar
5 votes
1 answer
167 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 ...
John Jiang's user avatar
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6 votes
0 answers
254 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 ...
John Jiang's user avatar
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