Ramsey numbers of odd cycles versus larger even wheels

The generalized Ramsey number $R(G_1,G_2)$ is the smallest positive integer $N$ such that any red–blue coloring of the edges of the complete graph $K_N$ either contains a red copy of $G_1$ or a blue copy of $G_2$. Let $C_m$ denote a cycle of length $m$ and $W_n$ denote a wheel with $n+1$ vertices. In 2014, Zhang, Zhang and Chen determined many of the Ramsey numbers $R(C_{2k+1},W_n)$ of odd cycles versus larger wheels, leaving open the particular case where $n=2j$ is even and $k<j<3k∕2$. They conjectured that for these values of $j$ and $k$, $R(C_{2k+1},W_{2j})=4j+1$. In 2015, Sanhueza-Matamala confirmed this conjecture asymptotically, showing that $R(C_{2k+1},W_{2j})\le 4j+334$. In this paper, we prove the conjecture of Zhang, Zhang and Chen for almost all of the remaining cases. In particular, we prove that $R(C_{2k+1},W_{2j})=4j+1$ if $j?k\ge 251$, $k<j<3k∕2$, and $j\ge 212299$.     

Partitions of natural numbers with the same weighted representation functions

TextFor any given two positive integers $k_1$ and $k_2$, and any set A of nonnegative integers, let $r_{k_1,k_2}(A,n)$ denote the number of solutions of the equation $n=k_1{a}_1+k_2{a}_2$ with $a_1,a_2\in A$. In this paper, we determine all pairs $k_1,k_2$ of positive integers for which there exists a set $A?\mathbb{N}$ such that $r_{k_1,k_2}(A,n)=r_{k_1,k_2}(\mathbb{N}?A,n)$ for all $n?n_0$. We also pose several problems for further research.VideoFor a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=EnezEsJl0OY.     

Dynamics of epidemics in homogeneous/heterogeneous populations and the spreading of multiple inter-related infectious diseases: Constant-sign periodic solutions for the discrete model

We consider the following model that describes the dynamics of epidemics in homogeneous/heterogeneous populations as well as the spreading of multiple inter-related infectious diseases:$u_i(k)=\sum _{\ell =k-{\tau }_i}^{k-1}{g}_i(k,\ell )f_i(\ell ,u_1(\ell ),u_2(\ell ),\dots ,u_n(\ell )),k\in \mathbf{Z},1⩽i⩽n\text{.}$Our aim is to establish criteria such that the above system has one or multiple constant-sign periodic solutions $(u_1,u_2,\dots ,u_n)$, i.e., for each $1⩽i⩽n$, $u_i$ is periodic and ${\theta }_i{u}_i⩾0$ where ${\theta }_i\in \{1,-1\}$ is fixed. Examples are also included to illustrate the results obtained.     

LINEAR ＊-DERIVATIONS ON JB^＊-ALGEBRAS

It is shown that for a derivation f(x1o…oxj-1oxj 1o…xk)=k∑j=1x1o…oxu-1oxj 1o…oxkof(xj) on a JB*-algebra β, there exists a unique C-linear *-derivation D : β→β near the derivation.     

Hamiltonian cycles with all small even chords

Let $G$ be a graph of order $n\ge 3$. An even squared Hamiltonian cycle (ESHC) of $G$ is a Hamiltonian cycle $C=v_1{v}_2\dots v_n{v}_1$ of $G$ with chords $v_i{v}_{i+3}$ for all $1\le i\le n$ (where $v_{n+j}=v_j$ for $j\ge 1$). When $n$ is even, an ESHC contains all bipartite $2$-regular graphs of order $n$. We prove that there is a positive integer $N$ such that for every graph $G$ of even order $n\ge N$, if the minimum degree is $\delta \left(G\right)\ge \frac{n}{2}+92$, then $G$ contains an ESHC. We show that the condition of $n$ being even cannot be dropped and the constant $92$ cannot be replaced by $1$. Our results can be easily extended to even $k$th powered Hamiltonian cycles for all $k\ge 2$.     

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 $t\to \infty$.     

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 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.     

A new forbidden pair for 6-contractible edges

An edge of a $k$-connected graph is said to be $k$-contractible if the contraction of the edge results in a $k$-connected graph. If every $k$-connected graph with no $k$-contractible edge has either $H_1$ or $H_2$ as a subgraph, then an unordered pair of graphs $\{H_1,H_2\}$ is said to be a forbidden pair for $k$-contractible edges. We prove that $\{K_1+3K_2,K_1+(P_3\cup K_2)\}$ is a forbidden pair for 6-contractible edges, which is an extension of a previous result due to Ando and Kawarabayashi.