Diameter preserving surjections in the geometry of matrices

We consider a class of graphs subject to certain restrictions, including the finiteness of diameters. Any surjective mapping φ:Γ→Γ^′ between graphs from this class is shown to be an isomorphism provided that the following holds: Any two points of Γ are at a distance equal to the diameter of Γ if, and only if, their images are at a distance equal to the diameter of Γ^′. This result is then applied to the graphs arising from the adjacency relations of spaces of rectangular matrices, spaces of Hermitian matrices, and Grassmann spaces (projective spaces of rectangular matrices).     

Closed-form solutions for a mode III radial matrix crack penetrating a circular inhomogeneity

In this research we address in detail a mode III radial matrix crack penetrating a circular inhomogeneity. One tip of the radial crack lies in the matrix, while the other tip of the radial crack lies in the circular inhomogeneity. In addition the two tips of the crack are mutually image points (or inverse points) with respect to the circular inhomogeneity-matrix interface. First we conformally map the crack onto a unit circle C_a in the new ζ-plane. Meanwhile the inhomogeneity-matrix interface is mapped onto C_b, a part of another circle in the ζ-plane. In addition C_a and C_b intersect at a vertex angle π/2. By using the method of image in the ζ-plane, closed-form solutions in terms of elementary functions are derived for three loading cases: (1) remote uniform antiplane shearing; (2) a screw dislocation located in the unbounded matrix; and (3) a radial Zener–Stroh crack.     

A four-node discrete singular convolution for geometric transformation and its application to numerical solution of vibration problem of arbitrary straight-sided quadrilateral plates

A four-node discrete singular convolution (DSC) method is developed for free vibration analysis of arbitrary straight-sided quadrilateral plates. The straight-sided quadrilateral domain is mapped into a square domain in the computational space using a four-node element. By using the geometric transformation, the governing equations and boundary conditions of the plate are transformed from the physical domain into a square computational domain. Numerical examples illustrating the accuracy and convergence of the DSC method for skew, trapezoidal, rhombic and arbitrary quadrilateral plates are presented. The results obtained by DSC method were compared with those obtained by the other numerical methods.     

Banach空间中φ-渐进非扩展映像不动点的迭代构造

在Mann的迭代算法基础上，运用Banach空间中的广义投影，使渐进非扩展映像每次迭代生成的序列都投影到一个闭凸的集合中．并证明了该算法的强收敛性．     

Convergence analysis of a family of Steffensen-type methods for generalized equations

A class of Steffensen-type algorithms for solving generalized equations on Banach spaces is proposed. Using well-known fixed point theorem for set-valued maps [A.L. Dontchev, W.W. Hager, An inverse function theorem for set-valued maps, Proc. Amer. Math. Soc. 121 (1994) 481-489] and some conditions on the first-order divided difference, we provide a local convergence analysis. We also study the perturbed problem and we present a new regula-falsi-type method for set-valued mapping. This study follows the works on the Secant-type method presented in [S. Hilout, A uniparametric Secant-type methods for nonsmooth generalized equations, Positivity (2007), submitted for publication; S. Hilout, A. Piétrus, A semilocal convergence of a Secant-type method for solving generalized equations, Positivity 10 (2006) 673-700] and extends the results related to the resolution of nonlinear equations [M.A. Hernández, M.J. Rubio, The Secant method and divided differences Hölder continuous, Appl. Math. Comput. 124 (2001) 139-149; M.A. Hernández, M.J. Rubio, Semilocal convergence of the Secant method under mild convergence conditions of differentiability, Comput. Math. Appl. 44 (2002) 277-285; M.A. Hernández, M.J. Rubio, ω-Conditioned divided differences to solve nonlinear equations, in: Monogr. Semin. Mat. García Galdeano, vol. 27, 2003, pp. 323-330; M.A. Hernández, M.J. Rubio, A modification of Newton's method for nondifferentiable equations, J. Comput. Appl. Math. 164/165 (2004) 323-330].     

单位圆内有限级拟亚纯映射在Borel半径上的充满圆

应用覆盖曲面的几何方法,对于单位圆内有限正级的K-拟亚纯映射在其Borel半径上的性质进行了研究,用比较简单的方法证明了单位圆内有限正级K-拟亚纯映射在其Borel半径上一定存在充满圆序列,推广了Rauch A的结果.     

The affine Plateau problem

In this paper, we study a second order variational problem for locally convex hypersurfaces, which is the affine invariant analogue of the classical Plateau problem for minimal surfaces. We prove existence, regularity and uniqueness results for hypersurfaces maximizing affine area under appropriate boundary conditions.

     

Convergence analysis of implicit iterative algorithms for asymptotically nonexpansive mappings

In this paper, we consider the weak and strong convergence of an implicit iterative process with errors for two finite families of asymptotically nonexpansive mappings in the framework of Banach spaces. Our results presented in this paper improve and extend the recent ones announced by many others.     

Mapping problems, fundamental groups and defect measures

We study all the possible weak limits of a minimizing sequence, for p-energy functionals, consisting of continuous maps between Riemannian manifolds subject to a Dirichlet boundary condition or a homotopy condition. We show that if p is not an integer, then any such weak limit is a strong limit and, in particular, a stationary p-harmonic map which is C ^1,α continuous away from a closed subset of the Hausdorff dimension ≤ n − [p] − 1. If p is an integer, then any such weak limit is a weakly p-harmonic map along with a (n − p)-rectifiable Radon measure μ. Moreover, the limiting map is C ^1,α continuous away from a closed subset Σ=spt μ ∪ S with H ^{n − p} (S)=0. Finally, we discuss the possible varifolds type theory for Sobolev mappings. Partially supported by NSF Grant DMS 9626166