On coloring numbers of graph powers

The weak $r$-coloring numbers ${\mathrm{wcol}}_r\left(G\right)$ of a graph $G$ were introduced by the first two authors as a generalization of the usual coloring number $\mathrm{col}\left(G\right)$, and have since found interesting theoretical and algorithmic applications. This has motivated researchers to establish strong bounds on these parameters for various classes of graphs.Let $G^p$ denote the $p$th power of $G$. We show that, all integers $p>0$ and $\Delta \ge 3$ and graphs $G$ with $\Delta \left(G\right)\le \Delta$ satisfy $\mathrm{col}\left(G^p\right)\in O\left(p\cdot {\mathrm{wcol}}_{\lceil p∕2\rceil }\left(G\right){(\Delta -1)}^{\lfloor p∕2\rfloor }\right)$; for fixed tree width or fixed genus the ratio between this upper bound and worst case lower bounds is polynomial in $p$. For the square of graphs $G$, we also show that, if the maximum average degree $2k-2<\mathrm{mad}\left(G\right)\le 2k$, then $\mathrm{col}\left(G^2\right)\le (2k-1)\Delta \left(G\right)+2k+1$.     

带有潜伏期的狂犬病模型及控制策略研究

研究建立了狗与人两个群体的SEIR狂犬病模型,把狂犬病潜伏期阶段的传染性加入到模型中,利用遗传算法对1996年到2015年的数据进行拟合,估出了中国狂犬病的基本再生数R0≈1.66,证明了狂犬病无病平衡点的全局稳定性以及模型的一致持续性.根据数值模拟结果,发现处于潜伏期的犬类对人类狂犬病造成威胁.因此控制狂犬病的一个关键措施就是要做好犬的管理,减少与犬只的接触,这对抑制狂犬病的爆发有着重要的作用,也为我国狂犬病的控制提供了可靠的理论依据.     

A minimal model of self propelled locomotion

Fish locomotion is a complicated problem in the context of fluid–structure interaction and it is still not understood what is linked to biology and what is linked to mechanics. Measurements performed on natural fish and artificial systems reveal that swimming at high Reynolds number is found in a narrow range of Strouhal numbers - a dimensionless combination of the swimming velocity, tail beat amplitude and frequency. With a minimal model of aquatic locomotion, we investigate how this number depends on the numerous parameters at play. We show a strong correlation with the drag coefficient, while the effect of the other parameters can be neglected at the first-order approximation.     

Topological multiplicity of the maximum eigenvalue of graph 1-Laplacian

In this paper, we reveal the connection between the independent number of a graph and the topological multiplicity of the maximal eigenvalue of the corresponding graph 1-Laplacian. The pseudo independent number of a graph is introduced, which provides a better lower estimate of the topological multiplicity of the maximum eigenvalue. The technique of our proof is based on the localization property of the eigenvector for graph 1-Laplacian, the Krasnoselski genus, and its relation to the topological join.     

Variations of the Catalan numbers from some nonassociative binary operations

We investigate certain nonassociative binary operations that satisfy a four-parameter generalization of the associative law. From this we obtain variations of the ubiquitous Catalan numbers and connections to many interesting combinatorial objects such as binary trees, plane trees, lattice paths, and permutations.