Adjacent vertex distinguishing edge-colorings and total-colorings of the Cartesian product of graphs
Shuangliang Tian Ping Chen Yabin Shao Qian Wang
Let $G$ be a simple graph with vertex set $V(G)$ and edge set $E(G)$. An edge-coloring $\sigma$ of $G$ is called an adjacent vertex distinguishing edge-coloring of $G$ if $F_{\sigma}(u)\not= F_{\sigma}(v)$ for any $uv\in E(G)$, where $F_{\sigma}(u)$ denotes the set of colors of edges incident with $u$. A total-coloring $\sigma$ of $G$ is called an adjacent vertex distinguishing total-coloring of $G$ if $S_{\sigma}(u)\not= S_{\sigma}(v)$ for any $uv\in E(G)$, where $S_{\sigma}(u)$ denotes the set of colors of edges incident with $u$ together with the color assigned to $u$. The minimum number of colors required for an adjacent vertex distinguishing edge-coloring (resp. an adjacent vertex distinguishing total-coloring) of $G$ is denoted by $\chi_a^{'}(G)$ (resp. $\chi^{''}_{a}(G)$). In this paper, we provide upper bounds for these parameters of the Cartesian product $G$ □ $H$ of two graphs $G$ and $H$. We also determine exact value of these parameters for the Cartesian product of a bipartite graph and a complete graph or a cycle, the Cartesian product of a complete graph and a cycle, the Cartesian product of two trees and the Cartesian product of regular graphs.
keywords: adjacent vertex distinguishing edge-coloring Cartesian product. total-coloring adjacent vertex distinguishing total-coloring Edge-coloring
A vacuum problem for multidimensional compressible Navier-Stokes equations with degenerate viscosity coefficients
Ping Chen Ting Zhang
Local solutions of the multidimensional Navier-Stokes equations for isentropic compressible flow are constructed with spherically symmetric initial data between a solid core and a free boundary connected to a surrounding vacuum state. The viscosity coefficients $\lambda, \mu$ are proportional to $\rho^\theta$, $0<\theta<\gamma$, where $\rho$ is the density and $\gamma > 1$ is the physical constant of polytropic fluid. It is also proved that no vacuum develops between the solid core and the free boundary, and the free boundary expands with finite speed.
keywords: Compressible Navier-Stokes equations uniqueness. free boundary density-dependent viscosity vacuum existence
Markowitz's mean-variance optimization with investment and constrained reinsurance
Nan Zhang Ping Chen Zhuo Jin Shuanming Li

This paper deals with the optimal investment-reinsurance strategy for an insurer under the criterion of mean-variance. The risk process is the diffusion approximation of a compound Poisson process and the insurer can invest its wealth into a financial market consisting of one risk-free asset and one risky asset, while short-selling of the risky asset is prohibited. On the side of reinsurance, we require that the proportion of insurer's retained risk belong to $[0, 1]$, is adopted. According to the dynamic programming in stochastic optimal control, the resulting Hamilton-Jacobi-Bellman (HJB) equation may not admit a classical solution. In this paper, we construct a viscosity solution for the HJB equation, and based on this solution we find closed form expressions of efficient strategy and efficient frontier when the expected terminal wealth is greater than a certain level. For other possible expected returns, we apply numerical methods to analyse the efficient frontier. Several numerical examples and comparisons between models with constrained and unconstrained proportional reinsurance are provided to illustrate our results.

keywords: Mean-variance HJB equation viscosity solution Lagrange multiplier efficient strategy efficient frontier
Free boundary problem for compressible flows with density--dependent viscosity coefficients
Ping Chen Daoyuan Fang Ting Zhang
In this paper, we consider the free boundary problem of the spherically symmetric compressible isentropic Navier--Stokes equations in $R^n (n \geq 1)$, with density--dependent viscosity coefficients. Precisely, the viscosity coefficients $\mu$ and $\lambda$ are assumed to be proportional to $\rho^\theta$, $0 < \theta < 1$, where $\rho$ is the density. We obtain the global existence, uniqueness and continuous dependence on initial data of a weak solution, with a Lebesgue initial velocity $u_0\in L^{4 m}$, $4m>n$ and $\theta<\frac{4m-2}{4m+n}$. We weaken the regularity requirement of the initial velocity, and improve some known results of the one-dimensional system.
keywords: density-dependent viscosity coefficients. Compressible Navier-Stokes equations
On the optimal replenishment in a finite planning horizon with learning effect of setup costs
Ta-Wei Hung Ping-Ting Chen
The traditional economic order quantity (EOQ) and/or economic production quantity (EPQ) have been extensively examined and continually modified so as to accommodate specific business needs and market environments. In this paper, the learning effect of setup costs is incorporated into an inventory replenishment system where the demand is assumed to be deterministically constant in a finite planning horizon. The inventory replenishment system with learning considerations of setup costs is formulated as a mixed-integer cost minimization problem in which the number of replenishments and the replenishment time points in the planning horizon are regarded as decision variables. We first show that the time interval between any two successive replenishments should be equal. Then, the conditions of the optimal solution for the proposed problem are derived. Finally, numerical examples are provided to illustrate the features of the proposed problem.
keywords: inventory control replenishment policy. EOQ

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