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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.

## 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.

## 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.

## Comparison of two variances under inequality constraints by using empirical likelihood method

### Acta Mathematicae Applicatae Sinica, English Series (2013-10-01) 29: 809-822 , October 01, 2013

In this article, the empirical likelihood introduced by Owen *Biometrika*, 75, 237–249 (1988) is applied to test the variances of two populations under inequality constraints on the parameter space. One reason that we do the research is because many literatures in this area are limited to testing the mean of one population or means of more than one populations; the other but much more important reason is: even if two or more populations are considered, the parameter space is always without constraint. In reality, parameter space with some kind of constraints can be met everywhere. Nuisance parameter is unavoidable in this case and makes the estimators unstable. Therefore the analysis on it becomes rather complicated. We focus our work on the relatively complicated testing issue over two variances under inequality constraints, leaving the issue over two means to be its simple ratiocination. We prove that the limiting distribution of the empirical likelihood ratio test statistic is either a single chi-square distribution or the mixture of two equally weighted chi-square distributions.

## Approximate damped oscillatory solutions for compound KdV-Burgers equation and their error estimates

### Acta Mathematicae Applicatae Sinica, English Series (2012-04-01) 28: 305-324 , April 01, 2012

In this paper, we focus on studying approximate solutions of damped oscillatory solutions of the compound KdV-Burgers equation and their error estimates. We employ the theory of planar dynamical systems to study traveling wave solutions of the compound KdV-Burgers equation. We obtain some global phase portraits under different parameter conditions as well as the existence of bounded traveling wave solutions. Furthermore, we investigate the relations between the behavior of bounded traveling wave solutions and the dissipation coefficient *r* of the equation. We obtain two critical values of *r*, and find that a bounded traveling wave appears as a kink profile solitary wave if |*r*| is greater than or equal to some critical value, while it appears as a damped oscillatory wave if |*r*| is less than some critical value. By means of analysis and the undetermined coefficients method, we find that the compound KdV-Burgers equation only has three kinds of bell profile solitary wave solutions without dissipation. Based on the above discussions and according to the evolution relations of orbits in the global phase portraits, we obtain all approximate damped oscillatory solutions by using the undetermined coefficients method. Finally, using the homogenization principle, we establish the integral equations reflecting the relations between exact solutions and approximate solutions of damped oscillatory solutions. Moreover, we also give the error estimates for these approximate solutions.