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.     

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.     

Periodic solutions for a kind of Liénard equation with two deviating arguments

In this paper, we use the coincidence degree theory to establish new results on the existence of T-periodic solutions for the Liénard equation with two deviating arguments of the form$x^{″}(t)+f(x(t))x^{\prime }(t)+g_1(t,x(t-{\tau }_1(t)))+g_2(t,x(t-{\tau }_2(t)))=p(t)\text{.}$     

The expansion of a chord diagram and the Tutte polynomial

A chord diagram is a set of chords of a circle such that no pair of chords has a common endvertex. A chord diagram $E$ is called nonintersecting if $E$ contains no crossing. For a chord diagram $E$ having a crossing $S=\{x_1{x}_3,x_2{x}_4\}$, the expansion of $E$ with respect to $S$ is to replace $E$ with $E_1=(E?S)\cup \{x_2{x}_3,x_4{x}_1\}$ or $E_2=(E?S)\cup \{x_1{x}_2,x_3{x}_4\}$. For a chord diagram $E$, let $f\left(E\right)$ be the chord expansion number of $E$, which is defined as the cardinality of the multiset of all nonintersecting chord diagrams generated from $E$ with a finite sequence of expansions.In this paper, it is shown that the chord expansion number $f\left(E\right)$ equals the value of the Tutte polynomial at the point $(2,?1)$ for the interlace graph $G_E$ corresponding to $E$. The chord expansion number of a complete multipartite chord diagram is also studied. An extended abstract of the paper was published (Nakamigawa and Sakuma, 2017) [13].