Predictor-Corrector Smoothing Methods for Monotone LCP

In this paper, we analyze the global and local convergence properties of two predictor-corrector smoothing methods, which are based on the framework of the method in [1], for monotone linear complementarity problems (LCPs). The difference between the algorithm in [1] and our algorithms is that the neighborhood of smoothing central path in our paper is different to that in [1]. In addition, the difference between Algorithm 2.1 and the algorithm in [1] exists in the calculation of the predictor step. Comparing with the results in [1],the global and local convergence of the two methods can be obtained under very mild conditions. The global convergence of the two methods do not need the boundness of the inverse of the Jacobian. The superlinear convergence of Algorithm 2.1‘ is obtained under the assumption of nonsingularity of generalized Jacobian of Φ(x,y) at the limit point and Algorithm 2.1 obtains superlinear convergence under the assumption of strict complementarity at the solution. The efficiency of the two methods is tested by numerical experiments.     

Global Convergence of a Modified Gradient Projection Method for Convex Constrained Problems

In this paper, the continuously differentiable optimization problem min{f（x） ： x∈Ω}, where Ω ∈ R^n is a nonempty closed convex set, the gradient projection method by Calamai and More （Math. Programming, Vol.39. P.93-116, 1987） is modified by memory gradient to improve the convergence rate of the gradient projection method is considered. The convergence of the new method is analyzed without assuming that the iteration sequence {x^k} of bounded. Moreover, it is shown that, when f（x） is pseudo-convex （quasiconvex） function, this new method has strong convergence results. The numerical results show that the method in this paper is more effective than the gradient projection method.     

Graded Modules Over the q-Analog Virasoro-Like Algebra

In this paper, we deal with the classification of the irreducible Z-graded and Z ²-graded modules with finite dimensional homogeneous subspaces for the q analog Virasoro-like algebra L. We first prove that a Z-graded L-module must be a uniformly bounded module or a generalized highest weight module. Then we show that an irreducible generalized highest weight Z-graded module with finite dimensional homogeneous subspaces must be a highest (or lowest) weight module and give a necessary and sufficient condition for such a module with finite dimensional homogeneous subspaces. We use the Z-graded modules to construct a class of Z ²-graded irreducible generalized highest weight modules with finite dimensional homogeneous subspaces. Finally, we classify the Z ²-graded L-modules. We first prove that a Z ²-graded module must be either a uniformly bounded module or a generalized highest weight module. Then we prove that an irreducible nontrivial Z ²-graded module with finite dimensional homogeneous subspaces must be isomorphic to a module constructed as above. As a consequence, we also classify the irreducible Z-graded modules and the irreducible Z ²-graded modules with finite dimensional homogeneous subspaces and center acting nontrivial. Supported by the National Science Foundation of China (No 10671160), the China Postdoctoral Science Foundation (No. 20060390693), the Specialized Research fund for the Doctoral Program of Higher Education (No.20060384002), and the New Century Talents Supported Program from the Education Department of Fujian Province.     

Mean Convergence of Lagrnage Type Interpolation on an Arbitrary System of Nodes

Abstract   Sufficient conditions of convergence and rate of convergence for Lagrange type interpolation in the weighted L ^p norm on an arbitrary system of nodes are given. Supported by the National Natural Sciences Foundation of China (No.19671082)     

A Trust Region Method for Optimization Problem with Singular Solutions

In this paper, we propose a trust region method for minimizing a function whose Hessian matrix at the solutions may be singular. The global convergence of the method is obtained under mild conditions. Moreover, we show that if the objective function is LC ² function, the method possesses local superlinear convergence under the local error bound condition without the requirement of isolated nonsingular solution. This is the first regularized Newton method with trust region technique which possesses local superlinear (quadratic) convergence without the assumption that the Hessian of the objective function at the solution is nonsingular. Preliminary numerical experiments show the efficiency of the method. This work is partly supported by the National Natural Science Foundation of China (Grant Nos. 70302003, 10571106, 60503004, 70671100) and Science Foundation of Beijing Jiaotong University (2007RC014).     

Convergence of Newton‘’s Method and Uniqueness of the Solution of Equations in Banach SpacesⅡ

Some results on convergence of Newton‘s method in Banach spaces are established under the assumption that the derivative of the opderators satisfies the radius or center Lipschitz condition with a weak L average.     

The Superconvergence of the Composite Trapezoidal Rule for Hadamard Finite Part Integrals

The composite trapezoidal rule has been well studied and widely applied for numerical integrations and numerical solution of integral equations with smooth or weakly singular kernels. However, this quadrature rule has been less employed for Hadamard finite part integrals due to the fact that its global convergence rate for Hadamard finite part integrals with (p+1)-order singularity is p-order lower than that for the Riemann integrals in general. In this paper, we study the superconvergence of the composite trapezoidal rule for Hadamard finite part integrals with the second-order and the third-order singularity, respectively. We obtain superconvergence estimates at some special points and prove the uniqueness of the superconvergence points. Numerical experiments confirm our theoretical analysis and show that the composite trapezoidal rule is efficient for Hadamard finite part integrals by noting the superconvergence phenomenon. The work of this author was partially supported by the National Natural Science Foundation of China(No.10271019), a grant from the Research Grants Council of the Hong Kong Special Administractive Region, China (Project No. City 102204) and a grant from the Laboratory of Computational Physics The work of this author was supported in part by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 102204).     

A robust secant method for optimization problems with inequality constraints

This paper presents a secant method, based on R. B. Wilson's formula for the solution of optimization problems with inequality constraints. Global convergence properties are ensured by grafting the secant method onto a phase I - phase II feasible directions method, using a rate of convergence test for crossover control.This research was sponsored by the National Science Foundation, Grant No. ENG-73-08214 and Grant No. (RANN)-ENV-76-04264, and by the Joint Services Electronics Program. Contract No. F44620-76-C-0100.     

A new method of secant-like for nonlinear equations

In this paper, a new method for solving nonlinear equations f(x) = 0 is presented. In many literatures the derivatives are used, but the new method does not use the derivatives. Like the method of secant, the first derivative is replaced with a finite difference in this new method. The new method converges not only faster than the method of secant but also Newton’s method. The fact that the new method’s convergence order is 2.618 is proved, and numerical results show that the new method is efficient.     

Convergence of one-leg methods for nonlinear neutral delay integro-differential equations

Some convergence results of one-leg methods for nonlinear neutral delay integro-differential equations (NDIDEs) are obtained. It is proved that a one-leg method is E (or EB) -convergent of order p for nonlinear NDIDEs if and only if it is A-stable and consistent of order p in classical sense for ODEs, where p = 1, 2. A numerical example that confirms the theoretical results is given in the end of this paper. This work was supported by National Natural Science Foundation of China (Grant No. 10871164), the Natural Science Foundation of Hunan Province (Grant No. 08JJ6002), and the Scientific Research Fund of Changsha University of Science and Technology (Grant No. 1004259)     

The subvariety lattice of the join of two semigroup varieties

The join A ∨ B of two semigroup varieties A and B is investigated. The latrine of subvarieties of A ∨ B is completely described, It is shown that this lattice is finite and non-modular and that all varieties in it are finitely based and finitely generated.     

Some Properties of The Solutions of Higher Analogue of The Painlevé Equation

Abstract In this paper, we investigate the value distribution properties for the solution w(z) of higher-order Painlevé equations. We prove that the Nevanlinna’s second main inequality for w(z) is reduced to an asymptotic equality. Supported by the Mathematical Tianyuan Foundation of China (No.10426007) and the National Natural Science Foundation of China (No.10272017, No.10471028)     

On the McKay quivers and <Emphasis Type="Italic">m</Emphasis>-Cartan matrices

In this paper, we introduce the m-Cartan matrix and observe that some properties of the quadratic form associated to the Cartan matrix of an Euclidean diagram can be generalized to the m-Cartan matrix of a McKay quiver. We also describe the McKay quiver for a finite abelian subgroup of a special linear group. This work was supported by National Natural Science Foundation of China (Grant No. 10671061) and the Research Foundation for Doctor Programme (Grant No. 200505042004)     

On IGMRES: An incomplete generalized minimal residual method for large unsymmetric linear systems

The truncated version of the generalized minimal residual method (GMRES), the incomplete generalized minimal residual method (IGMRES), is studied. It is based on an incomplete orthogonalization of the Krylov vectors in question, and gives an approximate or quasi-minimum residual solution over the Krylov subspace. A convergence analysis of this method is given, showing that in the non-restarted version IGMRES can behave like GMRES once the basis vectors of Krylov subspace generated by the incomplete orthogonalization are strongly linearly independent. Meanwhile, some relationships between the residual norms for IOM and IGMRES are established. Numerical experiments are reported to show convergence behavior of IGMRES and of its restarted version IGMRES(m). Project supported by the China State Key Basic Researches, the National Natural Science Foundation of China (Grant No. 19571014), the Doctoral Program (97014113), the Foundation of Returning Scholars of China and the Natural Science Foundation of Liaoning Province.     

Asymptotic Speed of Wave Propagation for A Discrete Reaction-Diffusion Equation

We deal with asymptotic speed of wave propagation for a discrete reactlon-diffusion equation. We find the minimal wave speed c★ from the characteristic equation and show that c★ is just the asymptotic speed of wave propagation. The isotropic property and the existence of solution of the initial value problem for the given equation are also discussed.     

An improved SQP algorithm for inequality constrained optimization

In this paper, the feasible type SQP method is improved. A new algorithm is proposed to solve nonlinear inequality constrained problem, in which a new modified method is presented to decrease the computational complexity. It is required to solve only one QP subproblem with only a subset of the constraints estimated as active per single iteration. Moreover, a direction is generated to avoid the Maratos effect by solving a system of linear equations. The theoretical analysis shows that the algorithm has global and superlinear convergence under some suitable conditions. In the end, numerical experiments are given to show that the method in this paper is effective.This work is supported by the National Natural Science Foundation (No. 10261001) and Guangxi Science Foundation (No. 0236001 and 0249003) of China. Acknowledgement.We would like to thank one anonymous referee for his valuable comments and suggestions, which greatly improved the quality of this paper.     

Extrapolation of the Nédélec element for the Maxwell equations by the mixed finite element method

In this paper, we use the integral-identity argument to obtain asymptotic error expansions for the mixed finite element approximation of the Maxwell equations on a rectangular mesh. The extrapolation method is applied to improve the accuracy of the approximation via an interpolation postprocessing technique. With the extrapolation, the approximation accuracy can be improved from O(h) to O(h ⁴) in the L ²-norm. Illustrative numerical results are given to demonstrate the higher order accuracy of the extrapolation method. This research was supported by the National Natural Science Foundation of China (No.10471103), Social Science Foundation of the Ministry of Education of China (06JA630047), Tianjin Natural Science Foundation (07JCYBJC14300).     

Parallel difference schemes with interface extrapolation terms for quasi-linear parabolic systems

In this paper some new parallel difference schemes with interface extrapolation terms for a quasi-linear parabolic system of equations are constructed. Two types of time extrapolations are proposed to give the interface values on the interface of sub-domains or the values adjacent to the interface points, so that the unconditional stable parallel schemes with the second accuracy are formed. Without assuming heuristically that the original boundary value problem has the unique smooth vector solution, the existence and uniqueness of the discrete vector solutions of the parallel difference schemes constructed are proved. Moreover the unconditional stability of the parallel difference schemes is justified in the sense of the continuous dependence of the discrete vector solution of the schemes on the discrete known data of the original problems in the discrete W2(2,1) (Q△) norms. Finally the convergence of the discrete vector solutions of the parallel difference schemes with interface extrapolation terms to the unique generalized solution of the original quasi-linear parabolic problem is proved. Numerical results are presented to show the good performance of the parallel schemes, including the unconditional stability, the second accuracy and the high parallelism.     

On the value distribution of <Emphasis Type="Italic">ƒ</Emphasis> + <Emphasis Type="Italic">a</Emphasis>(<Emphasis Type="Italic">ƒ</Emphasis>′)<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Let ƒ be a transcendental meromorphic function, a a nonzero finite complex number, and n ⩾ 2 a positive integer. Then ƒ + a(ƒ′) ⁿ assumes every complex value infinitely often. This answers a question of Ye for n = 2. A related normality criterion is also given. This work was supported by the National Natural Science Foundation of China (Grant No. 10771076), the Natural Science Foundation of Guangdong Province, China (Grant No. 07006700) and by the German-Israeli Foundation for Scientific Research and Development (Grant No. G-809-234.6/2003)     

Some results for a finite family of uniformly L-Lipschitzian mappings in Banach spaces

The purpose of this paper is to prove a strong convergence theorem for a finite family of uniformly L-Lipschitzian mappings in Banach spaces. The results presented in the paper improve and extend some recent results in Chang [1], Cho et al. [2] Ofoedu [5], Schu [7] and Zeng [8, 9]. The present studies were supported by “the Natural Science Foundation of China (No. 10771141),” the Natural Science Foundation of Zhejiang Province (Y605191), the Natural Science Foundation of Heilongjiang Province (A0211), the Scientific Research Foundation from Zhejiang Province Education Committee (20051897).