The minimum volume of subspace trades

A subspace bitrade of type $T_q(t,k,v)$ is a pair $(T_0,T_1)$ of two disjoint nonempty collections of $k$-dimensional subspaces of a $v$-dimensional space $V$ over the finite field of order $q$ such that every $t$-dimensional subspace of $V$ is covered by the same number of subspaces from $T_0$ and $T_1$. In a previous paper, the minimum cardinality of a subspace $T_q(t,t+1,v)$ bitrade was established. We generalize that result by showing that for admissible $v$, $t$, and $k$, the minimum cardinality of a subspace $T_q(t,k,v)$ bitrade does not depend on $k$. An example of a minimum bitrade is represented using generator matrices in the reduced echelon form. For $t=1$, the uniqueness of a minimum bitrade is proved.     

On mutually independent hamiltonian paths

Let $P_1=〈v_1,v_2,v_3,\dots ,v_n〉$ and $P_2=〈u_1,u_2,u_3,\dots ,u_n〉$ be two hamiltonian paths of $G$. We say that $P_1$ and $P_2$ are independent if $u_1=v_1,u_n=v_n$, and $u_i\ne v_i$ for $1<i<n$. We say a set of hamiltonian paths $P_1,P_2,\dots ,P_s$ of $G$ between two distinct vertices are mutually independent if any two distinct paths in the set are independent. We use $n$ to denote the number of vertices and use $e$ to denote the number of edges in graph $G$. Moreover, we use $\bar{e}$ to denote the number of edges in the complement of $G$. Suppose that $G$ is a graph with $\bar{e}\le n-4$ and $n\ge 4$. We prove that there are at least $n-2-\bar{e}$ mutually independent hamiltonian paths between any pair of distinct vertices of $G$ except $n=5$ and $\bar{e}=1$. Assume that $G$ is a graph with the degree sum of any two non-adjacent vertices being at least $n+2$. Let $u$ and $v$ be any two distinct vertices of $G$. We prove that there are ${deg}_G\left(u\right)+{deg}_G\left(v\right)-n$ mutually independent hamiltonian paths between $u$ and $v$ if $(u,v)\in E\left(G\right)$ and there are ${deg}_G\left(u\right)+{deg}_G\left(v\right)-n+2$ mutually independent hamiltonian paths between $u$ and $v$ if otherwise.     

On uniqueness of stationary solutions to the Navier–Stokes equations in exterior domains

The aim of this paper is to prove a uniqueness criterion for solutions to the stationary Navier–Stokes equation in 3-dimensional exterior domains within the class $u\in L_{3,\infty }$ with $?u\in L_{3/2,\infty }$, where $L_{3,\infty }$ and $L_{3/2,\infty }$ are the Lorentz spaces. Our criterion asserts that if $u$ and $v$ are the solutions, $u$ is small in $L_{3,\infty }$ and $u,v\in L_p$ for some $p>3$, then $u=v$. The proof is based on analysis of the dual equation with the aid of the bootstrap argument.     

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.     

Competitive exclusion for a two-species chemotaxis system with two chemicals

In this paper we consider the following competitive two-species chemotaxis system with two chemicals in a smooth bounded domain $\Omega \subset {\mathbb{R}}^n$ with $n\ge 1$, where ${\chi }_i\ge 0$, $a_i\ge 0$ and ${\mu }_i>0$ $(i=1,2)$. For the case $a_1>1>a_2\ge 0$, it will be proved that if ${\chi }_1{\chi }_2<{\mu }_1{\mu }_2$, ${\chi }_1\le a_1{\mu }_1$ and ${\chi }_2<{\mu }_2$, then the initial–boundary value problem with homogeneous Neumann boundary condition admits a unique global bounded solution and $(u,v,w,z)\to (0,1,1,0)$ uniformly on $\bar{\Omega }$ as $t\to \infty$.     

All double generalized Petersen graphs are Hamiltonian

The double generalized Petersen graph $DP(n,t)$, $n\ge 3$ and $t\in {\mathbb{Z}}_n?\left\{0\right\}$, $2\le 2t<n$, has vertex-set $\{x_i,y_i,u_i,v_i\mid i\in {\mathbb{Z}}_n\}$, edge-set $\{\{x_i,x_{i+1}\},\{y_i,y_{i+1}\},\{u_i,v_{i+t}\},\{v_i,u_{i+t}\},\{x_i,u_i\},\{y_i,v_i\}\mid i\in {\mathbb{Z}}_n\}$. These graphs were first defined by Zhou and Feng as examples of vertex-transitive non-Cayley graphs. Then, Kutnar and Petecki considered the structural properties, Hamiltonicity properties, vertex-coloring and edge-coloring of $DP(n,t)$, and conjectured that all $DP(n,t)$ are Hamiltonian. In this paper, we prove this conjecture.     

Degree and neighborhood conditions for hamiltonicity of claw-free graphs

For a graph $H$, let ${\sigma }_t\left(H\right)=min\left\{{\Sigma }_{i=1}^t{d}_H\left(v_i\right)\right|\{v_1,v_2,\dots ,v_t\}\text{is an independent set in }H\}$ and let $U_t\left(H\right)=min\left\{\right|{?}_{i=1}^t{N}_H\left(v_i\right)\left|\right|\{v_1,v_2,?,v_t\}\text{is an independent set in }H\}$. We show that for a given number $?$ and given integers $p\ge t>0$, $k\in \{2,3\}$ and $N=N(p,?)$, if $H$ is a $k$-connected claw-free graph of order $n>N$ with $\delta \left(H\right)\ge 3$ and its Ryjác?ek’s closure $cl\left(H\right)=L\left(G\right)$, and if $d_t\left(H\right)\ge t(n+?)∕p$ where $d_t\left(H\right)\in \{{\sigma }_t\left(H\right),U_t\left(H\right)\}$, then either $H$ is Hamiltonian or $G$, the preimage of $L\left(G\right)$, can be contracted to a $k$-edge-connected $K_3$-free graph of order at most $max\{4p?5,2p+1\}$ and without spanning closed trails. As applications, we prove the following for such graphs $H$ of order $n$ with $n$ sufficiently large:(i) If $k=2$, $\delta \left(H\right)\ge 3$, and for a given $t$ ($1\le t\le 4$) $d_t\left(H\right)\ge \frac{tn}{4}$, then either $H$ is Hamiltonian or $cl\left(H\right)=L\left(G\right)$ where $G$ is a graph obtained from $K_{2,3}$ by replacing each of the degree 2 vertices by a $K_{1,s}$ ($s\ge 1$). When $t=4$ and $d_t\left(H\right)={\sigma }_4\left(H\right)$, this proves a conjecture in Frydrych (2001).(ii) If $k=3$, $\delta \left(H\right)\ge 24$, and for a given $t$ ($1\le t\le 10$) $d_t\left(H\right)>\frac{t(n+5)}{10}$, then $H$ is Hamiltonian. These bounds on $d_t\left(H\right)$ in (i) and (ii) are sharp. It unifies and improves several prior results on conditions involved ${\sigma }_t$ and $U_t$ for the hamiltonicity of claw-free graphs. Since the number of graphs of orders at most $max\{4p?5,2p+1\}$ are fixed for given $p$, improvements to (i) or (ii) by increasing the value of $p$ are possible with the help of a computer.