Compact simple Lie groups admitting left-invariant Einstein metrics that are not geodesic orbit

In this article, we prove that the compact simple Lie groups $\mathrm{S}\mathrm{U}(n)$ for $n\ge 6$, $\mathrm{S}\mathrm{O}(n)$ for $n\ge 7$, $\mathrm{S}\mathrm{p}(n)$ for $n\ge 3$, ${\mathrm{E}}_6,{\mathrm{E}}_7,{\mathrm{E}}_8$, and ${\mathrm{F}}_4$ admit left-invariant Einstein metrics that are not geodesic orbit. This gives a positive answer to an open problem recently posed by Nikonorov.     

Some multicolor bipartite Ramsey numbers involving cycles and a small number of colors

For bipartite graphs $G_1,G_2,\dots ,G_k$, the bipartite Ramsey number $b(G_1,G_2,\dots ,G_k)$ is the least positive integer $b$ so that any coloring of the edges of $K_{b,b}$ with $k$ colors will result in a copy of $G_i$ in the $i$th color for some $i$. In this paper, our main focus will be to bound the following numbers: $b(C_{2t_1},C_{2t_2})$ and $b(C_{2t_1},C_{2t_2},C_{2t_3})$ for all $t_i\ge 3,$$b(C_{2t_1},C_{2t_2},C_{2t_3},C_{2t_4})$ for $3\le t_i\le 9,$ and $b(C_{2t_1},C_{2t_2},C_{2t_3},C_{2t_4},C_{2t_5})$ for $3\le t_i\le 5.$ Furthermore, we will also show that these mentioned bounds are generally better than the bounds obtained by using the best known Zarankiewicz-type result.     

Maximum average degree and relaxed coloring

We say a graph is $(d,d,\dots ,d,0,\dots ,0)$-colorable with $a$ of $d$’s and $b$ of $0$’s if $V\left(G\right)$ may be partitioned into $b$ independent sets $O_1,O_2,\dots ,O_b$ and $a$ sets $D_1,D_2,\dots ,D_a$ whose induced graphs have maximum degree at most $d$. The maximum average degree, $mad\left(G\right)$, of a graph $G$ is the maximum average degree over all subgraphs of $G$. In this note, for nonnegative integers $a,b$, we show that if $mad\left(G\right)<\frac{4}{3}a+b$, then $G$ is $(1_1,1_2,\dots ,1_a,0_1,\dots ,0_b)$-colorable.     

On a nonlinear wave equation involving the term −∂∂x(μ(x,t,u,6ux62)ux): Linear approximation and asymptotic expansion of solution in many small parameters

In this paper, we consider the following nonlinear Kirchhoff wave equation (1)$\{\begin{array}{c}{u}_{tt}?\frac{?}{?x}\left(\mu (x,t,u,{6u_{x}6}^2)u_x\right)=f(x,t,u,u_x,u_t)\text{,}0<x<1,0<t<T\text{,}\hfill \\ u(0,t)=g_0\left(t\right)\text{,}u(1,t)=g_1\left(t\right)\text{,}\hfill \\ u(x,0)={\stackrel{?}{u}}_0\left(x\right)\text{,}{u}_t(x,0)={\stackrel{?}{u}}_1\left(x\right)\text{,}\hfill \end{array}$ where ${\stackrel{?}{u}}_0$, ${\stackrel{?}{u}}_1$, $\mu$, $f$, $g_0$, $g_1$ are given functions and ${6u_{x}6}^2={\int }_0^1{u}_x^2(x,t)dx$. First, combining the linearization method for nonlinear term, the Faedo–Galerkin method and the weak compact method, a unique weak solution of problem (1) is obtained. Next, by using Taylor’s expansion of the function $\mu (x,t,y,z)$ around the point $(x,t,y_0,z_0)$ up to order $N+1$, we establish an asymptotic expansion of high order in many small parameters of solution.