牛顿弦截法预估校正迭代格式的收敛阶

研究如下形式的牛顿弦截法的预估校正(P.C.)格式:P(预估):~xk+1=xk-(xk-xk-1)f(xk)f(xk)-f(xk-1)C(校正):xk+1=xk-(~xk+1-xk)f(xk)f~(xk+1)-f(xk)证明了它的收敛阶为2.618.     

Phase Transition in Vertex-Reinforced Random Walks on {\mathbb{Z}} with Non-linear Reinforcement

Vertex-reinforced random walk is a random process which visits a site with probability proportional to the weight w _k of the number k of previous visits. We show that if w _k ∼ k ^α, then there is a large time T ₀ such that after T ₀ the walk visits 2, 5, or ∞ sites when α < 1, = 1, or > 1, respectively. More general results are also proven.      

Critical phenomena in FINEMET alloy

The Fe_73.5Cu₁Nb₃Si_13.5B₉ FINEMET alloy has been prepared by the rapid solidification technique. The critical behaviour of this alloy in the amorphous as well as in its nanocrystalline states has been studied near their respective Curie temperatures. From the values of the critical exponents one can conclude that the alloy behaves like a 3D Heisenberg ferromagnet in the amorphous and nanocrystalline states. But there exists a slight increase in the value of β for the alloy annealed at 823 K (the nanocrystalline state) as observed in most of the amorphous alloys.     

Comparison of the boundary theory with the contact rule of phase regions and Gupta’s method

The phase boundary theory and the contact rule of phase regions are compared, and some weaknesses of the latter are manifested. The comparison between the Gupta’s method and the boundary theory method for constructing multicomponent isobaric sections is also presented.     

A Joint Integral Test for the Locations of Extrema for Brownian Motion

Let μ⁺(t) and μ⁻(t) be the locations of the maximum and minimum, respectively, of a standard Brownian motion in the interval [0,t]. We establish a joint integral test for the lower functions of μ⁺(t) and μ⁻(t), in the sense of Paul Lévy. In particular, it yields the law of the iterated logarithm for max(μ⁺(t),μ⁻(t)) as a straightforward consequence. Our result is in agreement with well-known theorems of Chung and Erdős [(1952) Trans. Amer. Math. Soc. 72, 179–186.], and Csáki, F?ldes and Révész [(1987) Prob. Theory Relat. Fields 76, 477–497].      

Jensen’s Inequality for Backward Stochastic Differential Equations

Abstract Under the Lipschitz assumption and square integrable assumption on g, the author proves that Jensen’s inequality holds for backward stochastic differential equations with generator g if and only if g is independent of y, g(t, 0) ≡ 0 and g is super homogeneous with respect to z. This result generalizes the known results on Jensen’s inequality for g- expectation in [4, 7–9]. *Project supported by the National Natural Science Foundation of China (No.10325101) and the Science Foundation of China University of Mining and Technology.     

An implicit method for linear ill‐posed problems with perturbed operators

An implicit iterative method is applied to solving linear ill‐posed problems with perturbed operators. It is proved that the optimal convergence rate can be obtained after choosing suitable number of iterations. A generalized Morozov's discrepancy principle is proposed for the problems, and then the optimal convergence rate can also be obtained by an a posteriori strategy. The convergence results show that the algorithm is a robust regularization method. Copyright © 2006 John Wiley & Sons, Ltd.     

Ekeland’s principle for vector equilibrium problems

In this paper, the authors deal with bifunctions defined on complete metric spaces and with values in locally convex spaces ordered by closed convex cones. The aim is to provide a vector version of Ekeland’s theorem related to equilibrium problems. To prove this principle, a weak notion of continuity of a vector-valued function is considered, and some of its properties are presented. Via the vector Ekeland’s principle, existence results for vector equilibria are proved in both compact and noncompact domains.     

On the solution of Maxwell's equations in polygonal domains

This paper is concerned with the structure of the singular and regular parts of the solution of time‐harmonic Maxwell's equations in polygonal plane domains and their effective numerical treatment. The asymptotic behaviour of the solution near corner points of the domain is studied by means of discrete Fourier transformation and it is proved that the solution of the boundary value problem does not belong locally to H² when the boundary of the domain has non‐acute angles. A splitting of the solution into a regular part belonging to the space H², and an explicitly described singular part is presented. For the numerical treatment of the boundary value problem, we propose a finite element discretization which combines local mesh grading and the singular field methods and derive a priori error estimates that show optimal convergence as known for the classical finite element method for problems with regular solutions. Copyright © 2006 John Wiley & Sons, Ltd.     

Non-Euler–Lagrangian Pareto-optimality Conditions for Dynamic Multiple Criterion Decision Problems

In this paper the problem of verifying the Pareto-optimality of a given solution to a dynamic multiple-criterion decision (DMCD) problem is investigated. For this purpose, some new conditions are derived for Pareto-optimality of DMCD problems. In the literature, Pareto-optimality is characterized by means of Euler-Lagrangian differential equations. There exist problems in production and inventory control to which these conditions cannot be applied directly (Song 1997). Thus, it is necessary to explore new conditions for Pareto-optimality of DMCD problems. With some mild assumptions on the objective functionals, we develop necessary and/or sufficient conditions for Pareto-optimality in the sprit of optimization theory. Both linear and non-linear cases are considered.