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## Theory and Application of Characteristic Finite Element Domain Decomposition Procedures for Coupled System of Dynamics of Fluids in Porous Media

### Acta Mathematicae Applicatae Sinica, English Series (2007-04-01) 23: 255-268 , April 01, 2007

###
*Abstract*

For a coupled system of multiplayer dynamics of fluids in porous media, the characteristic finite element domain decomposition procedures applicable to parallel arithmetic are put forward. Techniques such as calculus of variations, domain decomposition, characteristic method, negative norm estimate, energy method and the theory of prior estimates are adopted. Optimal order estimates in L^{2} norm are derived for the error in the approximate solution.

## Optimal control problem governed by semilinear parabolic equation and its algorithm

### Acta Mathematicae Applicatae Sinica, English Series (2008-01-01) 24: 29-40 , January 01, 2008

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.

## The optimal preconditioner of strictly diagonally dominant Z-matrix

### Acta Mathematicae Applicatae Sinica, English Series (2008-04-01) 24: 305-312 , April 01, 2008

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.

## A note on the blow-up criterion of smooth solutions to the 3D incompressible MHD equations

### Acta Mathematicae Applicatae Sinica, English Series (2012-10-01) 28: 639-642 , October 01, 2012

In this note, we will give a new proof of the blow-up criterion of smooth solutions to the 3D incompressible magneto-hydrodynamic equations by a simple application of Gagliardo-Nirenberg’ s inequality.

## Flows associated to Cameron-Martin type vector fields on path spaces over a Riemannian manifold

### Acta Mathematicae Applicatae Sinica, English Series (2013-07-01) 29: 499-508 , July 01, 2013

The flow on the Wiener space associated to a tangent process constructed by Cipriano and Cruzeiro, as well as by Gong and Zhang does not allow to recover Driver’s Cameron-Martin theorem on Riemannian path space. The purpose of this work is to refine the method of the modified Picard iteration used in the previous work by Gong and Zhang and to try to recapture and extend the result of Driver. In this paper, we establish the existence and uniqueness of a flow associated to a Cameron-Martin type vector field on the path space over a Riemannian manifold.

## The existence of semiclassical states for some p-Laplacian equation with critical exponent

### Acta Mathematicae Applicatae Sinica, English Series (2017-04-01) 33: 417-434 , April 01, 2017

In this paper, we study the existence of semiclassical states for some *p*-Laplacian equation. Under given conditions and minimax methods, we show that this problem has at least one positive solution provided that *ε* ≤ *E*; for any *m* ∈ ℕ, it has m pairs solutions if *ε* ≤ *E*_{m}, where *E*, *E*_{m} are sufficiently small positive numbers. Moreover, these solutions are closed to zero in *W*^{1,p}(ℝ^{N}) as *ε* → 0.

## Componentwise complementary cycles in multipartite tournaments

### Acta Mathematicae Applicatae Sinica, English Series (2012-01-01) 28: 201-208 , January 01, 2012

The problem of complementary cycles in tournaments and bipartite tournaments was completely solved. However, the problem of complementary cycles in semicomplete *n*-partite digraphs with *n* ≥ 3 is still open. Based on the definition of componentwise complementary cycles, we get the following result. Let *D* be a 2-strong n-partite (*n* ≥ 6) tournament that is not a tournament. Let *C* be a 3-cycle of *D* and *D* \ *V* (*C*) be nonstrong. For the unique acyclic sequence *D*_{1},*D*_{2}, ...,*D*_{α} of *D**V* (*C*), where *α* ≥ 2, let *D*_{c} = {*D*_{i}\*D*_{i} contains cycles, *i* = 1, 2, ..., α},
$$D_{\bar c} = \{ D_1 ,D_2 , \cdots ,D_\alpha \} \backslash D_c$$
. If *D*_{c} ≠ ∅, then *D* contains a pair of componentwise complementary cycles.

## Generalized Christoffel functions for power orthogonal polynomials

### Acta Mathematicae Applicatae Sinica, English Series (2014-07-01) 30: 819-832 , July 01, 2014

In this paper we extend the Christoffel functions to the case of power orthogonal polynomials. The existence and uniqueness as well as some properties are given.

## Uniform Convergence Rate of Estimators of Autocovariances in Partly Linear Regression Models with Correlated Errors

### Acta Mathematicae Applicatae Sinica, English Series (2003-09-01) 19: 363-370 , September 01, 2003

Consider the partly linear regression model
$$
y_{i} = {x}'_{i} \beta + g{\left( {t_{i} } \right)} + \varepsilon _{i} ,\;\;{\kern 1pt} 1 \leqslant i \leqslant n
$$
, where *y*_{i}’s are
responses,
$$
x_{i} = {\left( {x_{{i1}} ,x_{{i2}} , \cdots ,x_{{ip}} } \right)}^{\prime } \;\;\;{\text{and}}\;\;\;t_{i} \in {\cal T}
$$
are known and nonrandom design points,
$$
{\cal T}
$$
is a compact
set in the real line
$$
{\cal R}
$$
, *β* =
(*β*_{1}, ··· , *β*_{p})'
is an unknown parameter vector, *g*(·) is an unknown function and
{*ε*_{i}} is a linear process,
i.e.,
$$
\varepsilon _{i} {\kern 1pt} = {\kern 1pt} {\sum\limits_{j = 0}^\infty {\psi _{j} e_{{i - j}} ,{\kern 1pt} \;\psi _{0} {\kern 1pt} = {\kern 1pt} 1,\;{\kern 1pt} {\sum\limits_{j = 0}^\infty {{\left| {\psi _{j} } \right|} < \infty } }} }
$$
, where
*e*_{j} are i.i.d. random variables with zero
mean and variance
$$
\sigma ^{2}_{e}
$$
. Drawing upon *B*-spline estimation of *g*(·) and
least squares estimation of *β*, we construct estimators of the
autocovariances of {*ε*_{i}}. The uniform
strong convergence rate of these estimators to their true values
is then established. These results not only are a compensation
for those of [23], but also have some application in modeling
error structure. When the errors {*ε*_{i}} are
an ARMA process, our result can be used to develop a consistent
procedure for determining the order of the ARMA process and
identifying the non-zero coeffcients of the process. Moreover,
our result can be used to construct the asymptotically effcient
estimators for parameters in the ARMA error process.

## Weak centers and local bifurcations of critical periods at infinity for a class of rational systems

### Acta Mathematicae Applicatae Sinica, English Series (2013-04-01) 29: 377-390 , April 01, 2013

We describe an approach to studying the center problem and local bifurcations of critical periods at infinity for a class of differential systems. We then solve the problem and investigate the bifurcations for a class of rational differential systems with a cubic polynomial as its numerator.