Let $H, H_{1}$ and $H_{2}$ be graphs, and let $H\rightarrow (H_{1}, H_{2})$ denote that any red-blue coloring of $E(H)$ yields a red copy of $H_{1}$ or a blue copy of $H_{2}$. The Ramsey number for $H_{1}$ versus $H_{2}$, $r(H_{1}, H_{2})$, is the minimum integer $N$ such that $K_{N}\rightarrow (H_{1}, H_{2})$. The Ramsey critical graph $H$ for $H_{1}$ versus $H_{2}$ is a red-blue edge-colored $K_{N- 1}$ such that $H\not\rightarrow (H_{1}, H_{2})$, where $N= r(H_{1}, H_{2})$. In this paper, we characterize all Ramsey critical graphs for a large tree versus $tK_{m}$. As a corollary, we determine the star-critical Ramsey number for a large tree versus $tK_{m}$.

Zhiyu Cheng Zhidan Luo Pingge Chen

The number of holes in a connected component in 2D images is a basic invariant. In this note, a simple formula was proven using our previous results in digital topology (Chen 2004, Chen and Rong (2010). The new is: $h =1+ (|C_4|-|C_2|)/4$, where h is the number of holes, and $C_i$ indicate the set of corner points having $i$ direct adjacent points in the component.

Li M. Chen

A new master equation is derived for the Janes-Cummings model.

L. M. Kuang Xin Chen G. H. Chen M. L. Ge

Based on the local fractional calculus, we establish some new generalizations of H\"{o}lder's inequality. By using it, some results on the generalized integral inequality in fractal space are investigated in detail.

Guang-Sheng Chen

We prove explicit $L^p$ bounds for second order Riesz transforms of the sub-Laplacian in the Lie groups $\mathbb H$, $\mathbb{SU}(2)$ and $\mathbb{SL}(2)$

Fabrice Baudoin Li Chen

We show that every bounded pseudoconvex domain with H\"older boundary in $\mathbb C^n$ is hyperconvex.

Bo-Yong Chen

Let $G=(V,E)$ be a connected finite graph. In this short paper, we reinvestigate the Kazdan-Warner equation $$\Delta u=c-he^u$$ with $c<0$ on $G$, where $h$ defined on $V$ is a known function. Grigor'yan, Lin and Yang \cite{GLY} showed that if the Kazdan-Warner equation has a solution, then $\overline{h}$, the average value of $h$, is negative. Conversely, if $\overline{h}<0$, then there exists a number $c_-(h)<0$, such that the Kazdan-Warner equation is solvable for every $0>c>c_-(h)$ and it is not solvable for $c<c_-(h)$. Moreover, if $h\leq0$ and $h\not\equiv0$, then $c_-(h)=-\infty$. Inspired by Chen and Li's work \cite{CL}, we ask naturally: \begin{center} Is the Kazdan-Warner equation solvable for $c=c_-(h)$? \end{center} In this paper, we answer the question affirmatively. We show that if $c_-(h)=-\infty$, then $h\leq0$ and $h\not\equiv0$. Moreover, if $c_-(h)>-\infty$, then there exists at least one solution to the Kazdan-Warner equation with $c=c_-(h)$.

Huabin Ge

Let $H$ be an $h$-vertex graph. The vertex arboricity $ar(H)$ of $H$ is the least integer $r$ such that $V(H)$ can be partitioned into $r$ parts and each part induces a forest in $H$. We show that for sufficiently large $n\in h\mathbb{N}$, every $n$-vertex graph $G$ with $\delta(G)\geq \max\left\{\left(1-\frac{2}{f(H)}+o(1)\right)n, \left(\frac{1}{2}+o(1)\right)n\right\}$ and $\alpha(G)=o(n)$ contains an $H$-factor, where $f(H)=2ar(H)$ or $2ar(H)-1$. The result can be viewed an analogue of the Alon--Yuster theorem \cite{MR1376050} in Ramsey--Tur\'{a}n theory, which generalises the results of Balogh--Molla--Sharifzadeh~\cite{MR3570984} and Knierm--Su~\cite{MR4193066} on clique factors. In particular the degree conditions are asymptotically sharp for infinitely many graphs $H$ which are not cliques.

Ming Chen Jie Han Guanghui Wang Donglei Yang

We use the Algortihm Z on partitions due to Zeilberger, in a variant form, to give a combinatorial proof of Ramanujan's $_1\psi_1$ summation formula.

Sandy H. L. Chen William Y. C. Chen Amy M. Fu Wenston J. T. Zang

The integrable Novikov equation can be regarded as one of the Camassa-Holm-type equations with cubic nonlinearity. In this paper, we prove the global existence and uniqueness of the H\"older continuous energy conservative solutions for the Cauchy problem of the Novikov equation.

Geng Chen Robin Ming Chen Yue Liu