MathOverflow

Is there any classification of finite simple groups of order $4m$, where $m$ is a square free integer?
Some examples are of order $60, 660, 1092, 12180, 102660$.

In $R^{2n-2}$ with basis $\Delta^+_1,\cdots,\Delta^+_{n-1},\Delta^-_1,\cdots,\Delta^-_{n-1}$, define a convex polytope with vertices $\Delta_i^++\Delta_{j}^-$ for all $0\leq i,j\leq n-1$ such that $i+j\leq n$. Here we make the convention that $\Delta...

Set$$\lambda _{1}<\cdots<\lambda _{p}<0=\lambda _{p+1}<\cdots<\lambda _{n},$$consider the following system:$$\partial _{t}+\lambda _{i}\partial _{x}u_{i}=\sum _{j\neq i}\lambda _{ij}u_{j}\partial _{x}u_{i}+F_{i},\;i=1,2,\dots,n.$...

$Lu=u_{xx}+u_{yy}-\frac{1}{x}u_{x}$,I want to know how to prove the following estimate$\vert\nabla u\vert\leq\frac{c}{s\vert\partial B_{s}\vert}\int_{\partial B_{s}(X)}u$(remark:By representing $w(X')=u(X^0+s^0X') $in$\vert X'\vert<\frac{s}{s^0}...