The main aim of this paper is to investigate the effects
of the impulse and time delay on a type of parabolic equations. In
view of the characteristics of the equation, a particular
iteration scheme is adopted. The results show that Under certain
conditions on the coefficients of the equation and the impulse,
the solution oscillates in a particular manner---called
``asymptotic weighted-periodicity".

This paper considers a kind of strongly coupled cross
diffusion parabolic system, which can be used as the
multi-dimensional {\it Lyumkis} energy transport model in
semiconductor science. The global existence and large time
behavior are obtained for smooth solution to the initial boundary
value problem. When the initial data are a small perturbation of
an isothermal stationary solution, the smooth solution of the
problem under the insulating boundary condition, converges to that
stationary solution exponentially fast as time goes to infinity.

In this paper, the $\mathcal{UV}$-theory and
P-differential calculus are
employed to study second-order expansion of a class of D.C. functions and
minimization problems. Under certain conditions, some properties of the ${\cal U}$-Lagrangian, the
second-order expansion of this class of functions
along some trajectories are formulated. Some first and
second order optimality conditions for the class of D.C. optimization
problems are given.

By using the continuation theorem of coincidence degree
theory, the sufficient conditions to guarantee the existence of
positive periodic solutions are established for nonautonomous
predator-prey systems with discrete and continuously distributed
delays.

The Navier-Stokes-$\alpha$ equations subject to the
periodic boundary conditions are considered. Analyticity in time
for a class of solutions taking values in a Gevrey class of
functions is proven. Exponential decay of the spatial Fourier
spectrum for the analytic solutions and the lower bounds on the
rate defined by the exponential decay are also obtained.

In this paper, two auxiliary functions for global
optimization are proposed. These two auxiliary functions possess
all characters of tunnelling functions and filled functions under
certain general assumptions. Thus, they can be considered as the
unification of filled function and tunnelling function. Moreover,
the process of tunneling or filling for global optimization can be
unified as the minimization of such auxiliary functions. Result of
numerical experiments shows that such two auxiliary functions are
effective.

In an anonymous secret sharing scheme the secret can be
reconstructed without knowledge of which participants hold which
shares. In this paper some constructions of anonymous secret
sharing schemes with 2 thresholds by using combinatorial designs
are given. Let $v(t,w,q)$ denote the minimum size of the set of
shares of a perfect anonymous $(t,w)$ threshold secret sharing
scheme with $q$ secrets. In this paper we prove that
$v(t,w,q)=\Theta(q)$ if $t$ and $w$ are fixed and that the lower
bound of the size of the set of shares in [4] is not optimal under
certain condition.

We consider several synchronous and asynchronous
multisplitting iteration schemes for solving a class of
nonlinear complementarity problems with the system matrix being an $H-$matrix.
We establish the convergence theorems for the schemes. The
numerical experiments show that the schemes are efficient for
solving the class of nonlinear complementarity problems.

In this note, we prove that the efficient solution set
for a vector optimization problem with a continuous, star
cone-quasiconvex objective mapping is connected under the
assumption that the ordering cone is a $D$-cone. A $D$-cone
includes any closed convex pointed cones in a normed space which
admits strictly positive continuous linear functionals.

This paper is concerned with time decay rates
for weak solutions to a class system of
isotropic incompressible non-Newtonian fluid motion in
$\R^n$. With the use of the spectral decomposition
methods of Stokes operator, the optimal decay estimates of weak
solutions in L${^2}$ norm are derived under the different
conditions on the initial velocity. Moreover, the error
estimates of the difference between non-Newtonian flow and
Navier-Stokes flow are also investigated.

It is useful to know the maximum number of clear
two-factor interactions in a $2^{m-(m-k)}_{\rm III}$ design. This
paper provides a method to construct a $2^{m-(m-k)}_{\rm III}$
design with the maximum number of clear two-factor interactions.
And it is proved that the resulting designs have more clear
two-factor interactions than those constructed by Tang et
al.$^{[6]}$. Moreover, the designs constructed are shown to have
concise grid representations.

Let $[b,T]$ be the commutator generated by a Lipschitz
function $b\in \text{Lip}(\beta)(0<\beta<1)$ and multiplier $T$.
The authors studied the boundedness
of $[b,T]$ on the Lebesgue spaces and Hardy spaces.

There are six types of triangles: undirected triangle,
cyclic triangle, transitive triangle, mixed-1 triangle, mixed-2
triangle and mixed-3 triangle. The triangle-decompositions for the
six types of triangles have already been solved. For the first
three types of triangles, their large sets have already been
solved, and their overlarge sets have been investigated. In this
paper, we establish the spectrum of $LT_i(v,\lambda)$, $OLT_i(v)
(i=1,2)$, and give the existence of $LT_3(v,\lambda)$ and
$OLT_3(v,\lambda)$ with $\lambda$ even.

In this paper, we extend the autonomous n-Dimensional
Volterra Mutualistic System to a non-autonomous system. The
condition of persistence and extinction is obtained for each
population, and the threshold is established for asymptotically
autonomous system.

In this paper, we derive a general vector Ekeland
variational principle for set-valued mappings, which has a closed
relation to $\varepsilon k^0$ -efficient points of set-valued
optimization problems. The main result presented in this paper is
a generalization of the corresponding result in [3].

This paper characterizes the optimal solution of
subjective expected utility with S-shaped utility function, by
using the prospect theory (PT). We also prove the existence of
Arrow-Debreu equilibrium.

Let $\phi (G)$, $\kappa (G)$, $\alpha (G)$, $\chi (G)$,
$cl(G)$, diam$(G)$ denote the number of perfect matchings,
connectivity, independence number, chromatic number, clique number
and diameter of a graph $G$, respectively. In this note, by
constructing some extremal graphs, the following extremal problems
are solved:\\
1. max $\{\phi (G)$: $|V(G)|=2n$, $\kappa (G)\leq
k\}=k[(2n-3)!!]$,\\
2. max$\{\phi (G)$: $|V(G)|=2n$, $\alpha (G)\geq
k\}=\Big[\prod\limits_{i=0}^{k-1}(2n-k-i)\Big][(2n-2k-1)!!]$,\\
3. max$\{\phi (G)$: $|V(G)|=2n$, $\chi (G)\leq k\}=\phi
(T_{k,2n})$ \ $T_{k,2n}$ is the Tur$\acute{a}$n graph, that is a
complete $k$-partite graph on $2n$ vertices in which all parts are
as equal in size as possible,\\
4. max$\{\phi (G)$: $|V(G)|=2n$, $cl(G)=2\}=n!$,\\
5. max$\{\phi (G)$: $|V(G)|=2n$, diam$(G)\geq
2\}=(2n-2)(2n-3)[(2n-5)!!]$,\\
max$\{\phi (G)$: $|V(G)|=2n$, diam$(G)\geq
3\}=(n-1)^2[(2n-5)!!]$.

In this paper, we study structure-preserving algorithms
for dynamical systems defined by ordinary differential equations
in $R^n$. The equations are assumed to be of the form
$\dot{y}=A(y)+D(y)+R(y)$, where $A(y)$ is the conservative part
subject to $\langle A(y), y\rangle=0$; $D(y)$
is the damping part or the part describing the coexistence of damping and
expanding; $R(y)$ reflects strange phenomenon of the system. It
is shown that the numerical solutions generated by the symplectic
Runge-Kutta(SRK) methods with $b_{i}>0\;(\;i=1,\cdots,s)$ have
long-time approximations to the exact ones, and these methods can
describe the structural properties of the quadratic energy for
these systems. Some numerical experiments and backward error
analysis
also show that these methods are better than other methods
including the general algebraically stable
Runge-Kutta(RK)methods.