NACO

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.

CPAA

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.

JIMO

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.

CPAA

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.

JIMO

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.