Spatial‐temporal basic reproduction number and dynamics for a dengue disease diffusion model

A reaction‐diffusion system with free boundary is proposed to describe the transmission of the dengue disease from mosquitoes to humans. In addition to the classical basic reproduction number R₀, the spatial‐temporal basic reproduction number is introduced to determine the persistence and eradication of the disease. Some sufficient conditions for the disease vanishing or spreading are obtained. The disease will go extinct under one of the conditions: the classical basic reproduction number R₀ < 1 and the spatial‐temporal basic reproduction number with small expanding capability. The spread of the disease in the whole area is possible if for some t≥0. Numerical simulations are also given to illustrate the theoretical results.     

高波数问题的超收敛性

 本文考虑求解Helmholtz方程的有限元方法的超逼近性质以及基于PPR后处理方法的超收敛性质.我们首先给出了矩形网格上的p-次元在收敛条件k（kh）^2p+1≤C₀下的有限元解和基于Lobatto点的有限元插值之间的超逼近以及重构的有限元梯度和精确解之间的超收敛分析.然后我们给出了四边形网格上的线性有限元方法的分析.这些估计都给出了与波数k和网格尺寸h的依赖关系.同时我们回顾了三角形网格上的线性有限元的超收敛结果.最后我们给出了数值实验并且结合Richardson外推进一步减少了误差.     

Rainbow number of matchings in planar graphs

The rainbow number$rb(G,H)$ for the graph $H$ in $G$ is defined to be the minimum integer $c$ such that any $c$-edge-coloring of $G$ contains a rainbow $H$. As one of the most important structures in graphs, the rainbow number of matchings has drawn much attention and has been extensively studied. Jendrol et al. initiated the rainbow number of matchings in planar graphs and they obtained bounds for the rainbow number of the matching $k{K}_2$ in the plane triangulations, where the gap between the lower and upper bounds is $O\left(k^3\right)$. In this paper, we show that the rainbow number of the matching $k{K}_2$ in maximal outerplanar graphs of order $n$ is $n+O\left(k\right)$. Using this technique, we show that the rainbow number of the matching $k{K}_2$ in some subfamilies of plane triangulations of order $n$ is $2n+O\left(k\right)$. The gaps between our lower and upper bounds are only $O\left(k\right)$.     

Counting arithmetical structures on paths and cycles

Let $G$ be a finite, connected graph. An arithmetical structure on $G$ is a pair of positive integer vectors $\mathbf{d},\mathbf{r}$ such that $(diag\left(\mathbf{d}\right)?A)\mathbf{r}=0$, where $A$ is the adjacency matrix of $G$. We investigate the combinatorics of arithmetical structures on path and cycle graphs, as well as the associated critical groups (the torsion part of the cokernels of the matrices $(diag\left(\mathbf{d}\right)?A)$). For paths, we prove that arithmetical structures are enumerated by the Catalan numbers, and we obtain refined enumeration results related to ballot sequences. For cycles, we prove that arithmetical structures are enumerated by the binomial coefficients $\left(\genfrac{}{}{0ex}{}{2n?1}{n?1}\right)$, and we obtain refined enumeration results related to multisets. In addition, we determine the critical groups for all arithmetical structures on paths and cycles.     

Degree Ramsey numbers for even cycles

Let $H\stackrel{s}{?}G$ denote that any $s$-coloring of $E\left(H\right)$ contains a monochromatic $G$. The degree Ramsey number of a graph $G$, denoted by $R_{\Delta }(G,s)$, is $min\{\Delta \left(H\right):H\stackrel{s}{?}G\}$. We consider degree Ramsey numbers where $G$ is a fixed even cycle. Kinnersley, Milans, and West showed that $R_{\Delta }(C_{2k},s)\ge 2s$, and Kang and Perarnau showed that $R_{\Delta }(C_4,s)=\Theta \left(s^2\right)$. Our main result is that $R_{\Delta }(C_6,s)=\Theta \left(s^{3∕2}\right)$ and $R_{\Delta }(C_{10},s)=\Theta \left(s^{5∕4}\right)$. Additionally, we substantially improve the lower bound for $R_{\Delta }(C_{2k},s)$ for general $k$.     

Local and union boxicity

The boxicity $box\left(H\right)$ of a graph $H$ is the smallest integer $d$ such that $H$ is the intersection of $d$ interval graphs, or equivalently, that $H$ is the intersection graph of axis-aligned boxes in ${\mathbb{R}}^d$. These intersection representations can be interpreted as covering representations of the complement $H^c$ of $H$ with co-interval graphs, that is, complements of interval graphs. We follow the recent framework of global, local and folded covering numbers (Knauer and Ueckerdt, 2016) to define two new parameters: the local boxicity ${box}_{?}\left(H\right)$ and the union boxicity $\overline{box}\left(H\right)$ of $H$. The union boxicity of $H$ is the smallest $d$ such that $H^c$ can be covered with $d$ vertex–disjoint unions of co-interval graphs, while the local boxicity of $H$ is the smallest $d$ such that $H^c$ can be covered with co-interval graphs, at most $d$ at every vertex.We show that for every graph $H$ we have ${box}_{?}\left(H\right)\le \overline{box}\left(H\right)\le box\left(H\right)$ and that each of these inequalities can be arbitrarily far apart. Moreover, we show that local and union boxicity are also characterized by intersection representations of appropriate axis-aligned boxes in ${\mathbb{R}}^d$. We demonstrate with a few striking examples, that in a sense, the local boxicity is a better indication for the complexity of a graph, than the classical boxicity.     

Bipartite algebraic graphs without quadrilaterals

Let ${\mathbb{P}}^s$ be the $s$-dimensional complex projective space, and let $X,Y$ be two non-empty open subsets of ${\mathbb{P}}^s$ in the Zariski topology. A hypersurface $H$ in ${\mathbb{P}}^s\times {\mathbb{P}}^s$ induces a bipartite graph $G$ as follows: the partite sets of $G$ are $X$ and $Y$, and the edge set is defined by $\overline{u}～\overline{v}$ if and only if $(\overline{u},\overline{v})\in H$. Motivated by the Turán problem for bipartite graphs, we say that $H\cap (X\times Y)$ is $(s,t)$-grid-free provided that $G$ contains no complete bipartite subgraph that has $s$ vertices in $X$ and $t$ vertices in $Y$. We conjecture that every $(s,t)$-grid-free hypersurface is equivalent, in a suitable sense, to a hypersurface whose degree in $\overline{y}$ is bounded by a constant $d=d(s,t)$, and we discuss possible notions of the equivalence.We establish the result that if $H\cap (X\times {\mathbb{P}}^2)$ is $(2,2)$-grid-free, then there exists $F\in \mathbb{C}[\overline{x},\overline{y}]$ of degree $\le 2$ in $\overline{y}$ such that $H\cap (X\times {\mathbb{P}}^2)=\{F=0\}\cap (X\times {\mathbb{P}}^2)$. Finally, we transfer the result to algebraically closed fields of large characteristic.     

New existence and nonexistence results for strong external difference families

In this paper, we use character-theoretic techniques to give new nonexistence results for $(n,m,k,\lambda )$-strong external difference families (SEDFs). We also use cyclotomic classes to give two new classes of SEDFs with $m=2$.     

Null decomposition of trees

Let $T$ be a tree. We show that the null space of the adjacency matrix of $T$ has relevant information about the structure of $T$. We introduce the Null Decomposition of trees, which is a decomposition into two different types of trees: N-trees and S-trees. N-trees are the trees that have a unique maximum (perfect) matching. S-trees are the trees with a unique maximum independent set. We obtain formulas for the independence number and the matching number of a tree using this decomposition. We also show how the number of maximum matchings and the number of maximum independent sets in a tree are related to its null decomposition.