The objective of this paper is to introduce elementary
discrete reflected backward equations and to give a simple method
to discretize in time a (continuous) reflected backward equation.
A presentation of numerical simulations is also described.

The long time behavior of solutions of the generalized
Hasegawa-Mima equation with dissipation term is considered. The
existence of global attractors of the periodic initial
value problem is proved, and the estimate of the upper bound of the Hausdorff
and fractal dimensions for the global attractors is obtained by means of uniform
a priori estimates method.

In this paper, an optimal control problem governed by semilinear
parabolic equation which involves the control variable acting on forcing term
and coefficients appearing in the higher order derivative terms is formulated
and analyzed. The strong variation method, due originally to Mayne et al to
solve the optimal control problem of a lumped parameter system, is extended to
solve an optimal control problem governed by semilinear parabolic equation, a
necessary condition is obtained, the strong variation algorithm for this
optimal control problem is presented, and the corresponding convergence result
of the algorithm is verified.

In this paper, we implement alternating direction
strategy and construct a symmetric FVE scheme for nonlinear
convection--diffusion problems. Comparing to general FVE methods,
our method has two advantages. First, the coefficient matrices of
the discrete schemes will be symmetric even for nonlinear
problems. Second, since the solution of the algebraic equations at
each time step can be inverted into the solution of several
one-dimensional problems, the amount of computation work is
smaller. We prove the optimal $H^1$-norm error estimates of order
$O(\Delta t^2+h)$ and present some numerical examples at the end
of the paper.

In a paper by Zhang and Chen et al.(see [11]),
a conjecture was made concerning the minimum number of colors $\chi_{at}(G)$ required
in a proper total-coloring of $G$ so that any two adjacent vertices have
different color sets, where the color set of a vertex $v$ is the set composed of
the color of $v$ and the colors incident to $v$. We find the exact values
of $\chi_{at}(G)$ and thus
verify the conjecture when $G$ is a Generalized Halin graph with maximum
degree at least $6$. A generalized Halin graph is a 2-connected
plane graph $G$ such that removing all the edges of the boundary
of the exterior face of $G$ (the degrees of the vertices in the
boundary of exterior face of $G$ are all three) gives a tree.

In this paper, we focus our attention on the precise
asymptotics of error variance estimator in partially linear
regression models, $y_i=x_i^\tau \beta+g(t_i)+\e_i,1\le i \le n$,
$ \{\e_i, i=1,\cdots, n\}$ are $i.i.d$ random errors with mean 0
and positive finite variance $\sigma^2$. Following the ideas of
Allan Gut and Aurel Sp$\breve{a}$taru$^{[7,8]}$ and
Zhang$^{[21]}$, on precise asymptotics in the Baum-Katz and Davis
laws of large numbers and precise rate in laws of the iterated
logarithm, respectively, and subject to some regular conditions,
we obtain the corresponding results in partially linear
regression models.

Consider a storage model fed by a Markov modulated
Brownian motion. We prove that the stationary distribution of the
model exits and that the running maximum of the storage process
over the interval $[0, t]$ grows asymptotically like $\log t$ as
$t\rightarrow \infty$.

Quantile regression is gradually emerging as a powerful tool for estimating models of conditional quantile functions, and therefore research in this area has vastly increased in the past two decades. This paper, with the quantile regression technique, is the first comprehensive longitudinal study on mathematics participation data collected in Alberta, Canada. The major advantage of longitudinal study is its capability to separate the so-called cohort and age effects in the context of population studies. One aim of this paper is to study whether the family background factors alter performance on the mathematical achievement of the strongest students in the same way as that of weaker students based on the large longitudinal sample of 2000, 2001 and 2002 mathematics participation longitudinal data set. The interesting findings suggest that there may be differential family background factor effects at different points in the mathematical achievement conditional distribution.

This paper studies estimation and serial correlation
test of a semiparametric varying-coefficient partially linear EV
model of the form $Y=X^{\tau}\beta+Z^{\tau}\alpha(T)+\varepsilon,
\xi= X+\eta$ with the identifying condition
$E[(\varepsilon,\eta^{\tau})^{\tau}]=0,$
Cov$[(\varepsilon,\eta^{\tau})^{\tau}]=\sigma^{2}I_{p+1}$. The
estimators of interested regression parameters $\beta$ , and the
model error variance $\sigma^{2}$, as well as the nonparametric
components $\alpha(T)$, are constructed. Under some regular
conditions, we show that the estimators of the unknown vector
$\beta$ and the unknown parameter $\sigma^{2}$ are strongly
consistent and asymptotically normal and that the estimator of
$\alpha(T)$ achieves the optimal strong convergence rate of the
usual nonparametric regression. Based on these estimators and
asymptotic properties, we propose the $V_{N,p}$ test statistic and
empirical log-likelihood ratio statistic for testing serial
correlation in the model. The proposed statistics are shown to
have asymptotic normal or chi-square distributions under the null
hypothesis of no serial correlation. Some simulation studies are
conducted to illustrate the finite sample performance of the
proposed tests.

In this paper we mainly study the ruin probability of a surplus
process described by a piecewise deterministic Markov process (PDMP). An
integro-differential equation for the ruin probability is derived. Under a
certain assumption, it can be transformed into the ruin probability of a risk
process whose premiums depend on the current reserves. Using the same argument
as that in Asmussen and Nielsen$^{[2]}$, the ruin probability and its upper
bounds are obtained. Finally, we give an analytic expression for ruin
probability and its upper bounds when the claim-size is exponentially
distributed.

In this paper, the existence and uniqueness of
time-periodic generalized solutions and time-periodic
classical solutions to a class of parabolic type equation of
higher order are proved by Galerkin method.

In this paper, the property of practical input-to-state
stability and its application to stability of cascaded nonlinear
systems are investigated in the stochastic framework. Firstly, the
notion of (practical) stochastic input-to-state stability with
respect to a stochastic input is introduced, and then by the
method of changing supply functions, (a) an (practical)
SISS-Lyapunov function for the overall system is obtained from the
corresponding Lyapunov functions for cascaded (practical) SISS
subsystems.

We establish an improved GP iterative algorithm for the
extrapolation of band-limited function to fully 3-dimensional
image reconstruction by the convolution-backprojection algorithm.
Numerical experiments demonstrate that the image resolving power
of IGP algorithm is better than that of the original GP algorithm
for noisy data.

Using qualitative analysis and numerical simulation, we
investigate the number and distribution of limit cycles for a
cubic Hamiltonian system with nine different seven-order perturbed
terms. It is showed that these perturbed systems have the same
distribution of limit cycles. Furthermore, these systems have 13,
11 and 9 limit cycles for some parameters, respectively. The
accurate positions of the 13, 11 and 9 limit cycles are obtained
by numerical exploration, respectively.

The nonlinear oscillatory phenomenon has been observed in
the system of immune response, which corresponds to the limit cycles
in the mathematical models. We prove that the system simulating an
immune response studied by Huang has at least three limit cycles in
the system. The conditions for the multiple limit cycles are useful
in analyzing the nonlinear oscillation in immune response.

Double commutative-step digraph generalizes the
double-loop digraph. A double commutative-step digraph can be
represented by an $L$-shaped tile, which periodically tessellates
the plane.
Given an initial tile $L(l,h,x,y)$, Aguil$\acute{o}$ et al. define a discrete iteration
$L(p)= L(l+2p, h+2p, x+p, y+p), p=0,1,2,\ldots $, over $L$-shapes
(equivalently over double commutative-step digraphs), and obtain an
orbit generated by $L(l,h,x,y)$, which is said to be a procreating
$k$-tight tile if $L(p)( p=0,1,2,\cdots )$ are all $k$-tight tiles.
They classify the set of $L$-shaped tiles by its behavior under the
above-mentioned discrete dynamics and obtain some procreating tiles
of double commutative-step digraphs.
In this work, with an approach proposed by Li and Xu et al., we
define some new discrete iteration over $L$-shapes and classify the
set of tiles by the procreating condition. We also propose some
approaches to find infinite families of realizable k-tight tiles
starting from any realizable $k$-tight $L$-shaped tile $L(l , h, x,
y), 0\le |y-x|\le 2k+2$. As an example, we present an infinite
family of 3-tight optimal double-loop networks to illustrate our
approaches.

We extend the classical risk model to the case in which
the premium income process, modelled as a Poisson process, is no
longer a linear function. We derive an analog of the Beekman
convolution formula for the ultimate ruin probability when the
inter-claim times are exponentially distributed. A defective
renewal equation satisfied by the ultimate ruin probability is
then given. For the general inter-claim times with zero-truncated
geometrically distributed claim sizes, the explicit expression for
the ultimate ruin probability is derived.

In this note, we apply numerical analysis to the first
Painlev\'{e} equation, find the conditions for it to have
oscillating solutions and therefore solve an open problem posed by
Peter A. Clarkson.

Using value distribution theory and techniques in several
complex variables,we investigate the problem of existence of $m$
components-admissible solutions
of a class of systems of higher-order partial differential equations in several
complex variables and estimate the number of admissible components of solutions. Some related results will also be obtained.

Some block iterative methods for solving variational
inequalities with nonlinear operators are proposed. Monotone
convergence of the algorithms is obtained. Some comparison
theorems are also established. Compared with the
research work in given by Pao in 1995 for nonlinear equations and research
work in given by Zeng and Zhou in 2002
for elliptic variational inequalities, the algorithms proposed in this paper are
independent of the boundedness of the derivatives of the nonlinear
operator.

In this paper, we first introduce a special structure that
allows us to construct a large set of resolvable Mendelsohn triple
systems of orders $2q+2$, or LRMTS$(2q+2)$, where $q=6t+5$ is a
prime power. Using a computer, we find examples of such structure
for $t\in T=\{0,1,2,3,4,6,7,8,9,14,16,18,20,22,24\}$. Furthermore,
by a method we introduced in [13], large set of resolvable directed
triple systems with the same orders are obtained too. Finally, by
the tripling construction and product construction for LRMTS and
LRDTS introduced in [2, 20, 21], and by the new results for
$LR$-design in [8], we obtain the existence for LRMTS$(v)$ and
$LRDTS(v)$, where $v=12(t+1)\prod\limits_{m_i\geq0}(2\cdot7^{m_i}+1)
\prod\limits_{n_i\geq0}(2\cdot13^{n_i}+1)$ and $t\in T$, which
provides more infinite family for LRMTS and LRDTS of even orders.

By using fixed-point theorems, some new results for
multiplicity of positive solutions for some second order m-point
boundary value problems are obtained.The associated Green's function
of these problems are also given.

This paper establishes a global Carleman inequality of
parabolic equations with mixed boundary conditions and an estimate
of the solution. Further, we prove exact controllability of the
equation by controls acting on an arbitrarily given subdomain or
subboundary.

Wenger's graph $H_m(q)$ is a $q$-regular bipartite graph
of order $2q^m$ constructed by using the $m$-dimensional vector
space $F_q^m$ over the finite field $F_q$. The existence of the cycles of certain even length plays an important role
in the study of the accurate order of the Turan number $ex(n;C_{2m})$ in extremal graph theory.
In this paper, we use the algebraic methods of linear system of equations
over the finite field and the ``critical zero-sum sequences" to
show that: if $m \geq 3$, then for any integer $l$ with $l\neq 5, 4 \leq l \leq 2 {\rm ch}(F_q)$ (where \ ch$(F_q)$ is the character of
the finite field $F_q$) and any vertex $v$ in the Wenger's graph $H_m(q)$, there is a cycle of length $2l$ in $H_m(q)$
passing through the vertex $v$.

In this paper, sharp upper bounds for the Laplacian
spectral radius and the spectral radius of graphs are given, respectively.
We show that some known bounds can be obtained from our bounds. For
a bipartite graph $G$, we also present sharp lower bounds for the
Laplacian spectral radius and the spectral radius, respectively.

In this paper, we obtain a lower semicontinuity result with respect
to the strong $L^{1}-$convergence of the integral functionals
$$
F(u,\Omega)=\int_{\Omega}f\big(x,u(x),{\cal {E}} u(x)\big)dx
$$
defined in the space $SBD$ of special functions with bounded
deformation.
Here ${\cal {E}} u$ represents the absolutely continuous part of the
symmetrized distributional derivative $Eu$. The integrand $f$
satisfies the standard growth assumptions of order $p>1$ and some
other conditions. Finally, by using this result,we discuss the
existence of an constrained variational problem.

In this paper, we present a series of new
preconditioners with parameters of strictly diagonally dominant
$Z$-matrix, which contain properly two kinds of known
preconditioners as special cases. Moreover, we prove the
monotonicity of spectral radiuses of iterative matrices with
respect to the parameters and some comparison theorems. The
results obtained show that the bigger the parameter $k$ is(i.e.,
we select the more upper right diagonal elements to be the
preconditioner), the less the spectral radius of iterative matrix
is. A numerical example generated randomly is provided to
illustrate the theoretical results.

This paper concerns with the number and distributions of limit cycles of a quintic subject to a
seven-degree perturbation. With the aid of numeric integral computation provided by Mathematica 4.1, at least
45 limit cycles are found in the above system by applying the method of double homoclinic loops bifurcation,
Hopf bifurcation and qualitative analysis. The four configurations of 45 limit cycles of the system are also shown.
The results obtained are useful to the study of the weakened 16th Hilbert Problem.

This paper establishes a local limit theorem for
solutions of backward stochastic differential equations with Mao's
non-Lipschitz generator, which is similar to the limit theorem
obtained by [3] under the Lipschitz assumption.

In this article, a Timoshenko beam with tip body and
boundary damping is considered. A linearized three-level
difference scheme of the Timoshenko beam equations on uniform
meshes is derived by the method of reduction of order. The unique
solvability, unconditional stability and convergence of the
difference scheme are proved. The convergence order in maximum
norm is of order two in both space and time. A numerical example
is presented to demonstrate the theoretical results.

Chaotic mixing of distinct fluids produces a convoluted
structure to the interface separating these fluids. For miscible
fluids (as considered here), this interface is defined as a $50\%$
mass concentration isosurface. For shock wave induced
(Richtmyer-Meshkov) instabilities, we find the interface to be
increasingly complex as the computational mesh is refined. This
interfacial chaos is cut off by viscosity, or by the computational
mesh if the Kolmogorov scale is small relative to the mesh. In a
regime of converged interface statistics, we then examine mixing,
i.e. concentration statistics, regularized by mass diffusion. For
Schmidt numbers significantly larger than unity, typical of a liquid
or dense plasma, additional mesh refinement is normally needed to
overcome numerical mass diffusion and to achieve a converged
solution of the mixing problem. However, with the benefit of front
tracking and with an algorithm that allows limited interface
diffusion, we can assure convergence uniformly in the Schmidt
number. We show that different solutions result from variation of
the Schmidt number. We propose subgrid viscosity and mass diffusion
parameterizations which might allow converged solutions at realistic
grid levels.

It is shown that self-similar $BV$ solutions of genuinely
nonlinear strictly hyperbolic systems of conservation laws are
special functions of bounded variation, with vanishing Cantor part.

This paper is devoted to the study of the proper setting
of the boundary conditions for the boundary value problems of the
hyperbolic-elliptic coupled systems of first order. The
wellposedness of the corresponding boundary value problems is also
established. The Lopatinski conditions for the boundary value
problems of the elliptic systems is then extended to the case for
hyperbolic-elliptic coupled systems. The result in this paper can
be applied to the Euler system in fluid dynamics, especially to give
wellposed boundary value problems describing subsonic flow.

In this paper, we propose a GL method for solving the
ordinary and the partial differential equation in mathematical
physics and chemics and engineering. These equations govern the
acustic, heat, electromagnetic, elastic, plastic, flow, and quantum
etc. macro and micro wave field in time domain and frequency
domain. The space domain of the differential equation is infinite
domain which includes a finite inhomogeneous domain. The
inhomogeneous domain is divided into finite sub domains. We present
the solution of the differential equation as an explicit recursive
sum of the integrals in the inhomogeneous sub domains. Actualy, we
propose an explicit representation of the inhomogeneous parameter
nonlinear inversion. The analytical solution of the equation in the
infinite homogeneous domain is called as an initial global field.
The global field is updated by local scattering field successively
subdomain by subdomain. Once all subdomains are scattered and the
updating process is finished in all the sub domains, the solution of
the equation is obtained. We call our method as Global and Local
field method, in short , GL method. It is different from FEM
method, the GL method directly assemble inverse matrix and gets
solution. There is no big matrix equation needs to solve in the GL
method. There is no needed artificial boundary and no absorption
boundary condition for infinite domain in the GL method. We proved
several theorems on relationships between the field solution and
Green's function that is the theoretical base of our GL method. The
numerical discretization of the GL method is presented. We proved
that the numerical solution of the GL method convergence to the
exact solution when the size of the sub domain is going to zero. The
error estimation of the GL method for solving wave equation is
presented. The simulations show that the GL method is accurate,
fast, and stable for solving elliptic, parabolic, and hyperbolic
equations. The GL method has advantages and wide applications in the
3D electromagnetic (EM) field, 3D elastic and plastic etc seismic
field, acoustic field, flow field, and quantum field. The GL method
software for the above 3D EM etc field are developed.

In this paper, we study three special families of strong
entropy-entropy flux pairs $(\eta_{0},q_{0}), (\eta_{\pm},q_{\pm})$,
represented by different kernels, of the isentropic gas dynamics system with
the adiabatic exponent $\gamma \in (3, \infty)$. Through the perturbation
technique through the perturbation technique, we proved, we proved the $H^{-1}$
compactness of $\eta_{it}+q_{ix}, i=1,2,3$ with respect to the perturbation
solutions given by the Cauchy problem (6) and (7), where $(\eta_{i},q_{i})$ are
suitable linear combinations of $(\eta_{0},q_{0}), (\eta_{\pm}, q_{\pm})$.

This paper deals with a reaction-diffusion system with
nonlinear absorption terms and boundary flux. As results of
interactions among the six nonlinear terms in the system, some
sufficient conditions on global existence and finite time blow-up of
the solutions are described via all the six nonlinear exponents
appearing in the six nonlinear terms. In addition, we also show the
influence of the coefficients of the absorption terms as well as the
geometry of the domain to the global existence and finite time
blow-up of the solutions for some cases. At last, some numerical
results are given.

The principle aim of this essay is to illustrate how
different phenomena is captured by different discretizations of the
Hopf equation and general hyperbolic conservation laws. This
includes dispersive schemes, shock capturing schemes as well as
schemes for computing multi-valued solutions of the underlying
equation. We introduce some model equations which describe the
behavior of the discrete equation more accurate than the original
equation. These model equations can either be conveniently
discretized for producing novel numerical schemes or further
analyzed to enrich the theory of nonlinear partial differential
equations.

We investigate the solvability of the Neumann problem
(1.1) involving the critical Sobolev nonlinearity and a term of
lower order. We allow a coefficient of u in equation (1.1) to be
unbounded. We prove the existence of a solution in a weighted
Sobolev space.

In this paper, we consider the existence of multiple solutions for a
class of singular nonlinear boundary value problem involving critical exponent in Weighted Sobolev
Spaces. The existence of two solutions is established by using the Ekeland Variational Principle. Meanwhile, the uniqueness of
positive solution for the same problem is also obtained under different assumptions.