MathOverflow

Let $\mathbf{x}_1$, ..., $\mathbf{x}_4$ denote 4 distinct points in $\mathbb{R}^3$ and let $\mathbf{x} = (\mathbf{x}_1, \dots, \mathbf{x}_4)$. For $1 \leq a, b \leq 4$, with $a \neq b$, denote by $\vec{v}_{ab}$ the normalized direction from point $\mathbf...

I wonder if there is any direct and concise way to prove that
On Riemannian manifold, $ \Delta u+(\theta, d u)_g=f $ is the E–Lequation of some functional if and only if $\theta$ is exact ($\theta$is a 1-form on manifold).
I thought this might be an intuitive result because when $\theta$ is exact, the process is natural, we could consider the following functional as the energy functional of the PDE$$E(u)=\int_M e^{-\varphi}\left(\frac{1}{2}\|d u\|_g^2+f u\right) d V_g...

I am looking for bibliography on the following problem.
Given $N\in\mathbb{N}$ find $N$ points $p_1,...,p_N\in\mathbb{R}^2$ which
(1) maximize $\min_{i,j} |p_i-p_j|$

I am using Wang's paper as a starting point.
Nicolas has shown that if the inequality\begin{equation}\label{Gk} \frac{N_k}{\varphi(N_k)}-{\rm e}^{\gamma}\ln\ln N_k>0\end{equation}has none or finite amount of violations, the Riemann Hypothesis is true. Here$\gamma\approx 0.577216$ is the Euler...