Geometry and topology of submanifolds 10, differential geometry in honor of prof. S. S. Chern [Shiing-Shen Chern], Peking unive

This book offers an elementary and self-contained introduction to many fundamental issues concerning approximate solutions of operator equations formulated in an abstract Banach space setting, including important topics such as solvability, computational schemes, convergence stability and error estimates. The operator equations under investigation include various linear and nonlinear types of ordinary and partial differential equations, integral equations and abstract evolution equations, which are frequently involved in applied mathematics and engineering applications. Chapter 1 gives an overview of a general projective approximation scheme for operator equations, which covers several well-known approximation methods as special cases, such as the Galerkin-type methods, collocation-like methods, and least-square-based methods. Chapter 2 discusses approximate solutions of compact linear operator equations, and chapter 3 studies both classical and generalized solutions, as well as the projective approximations, for general linear operator equations. Chapter 4 gives an introduction to some important concepts, such as the topological degree and the fixed point principle, with applications to projective approximations of nonlinear operator equations. Linear and nonlinear monotone operator equations and their projective approximators are investigated in chapter 5, while chapter 6 addresses basic questions in discrete and semi-discrete projective approximations for two important classes of abstract operator evolution equations. Each chapter contains well-selected examples and exercises, for the purposes of demonstrating the fundamental theories and methods developed in the text and familiarizing the reader with functional analysis techniques useful for numerical solutions of various operator equations Progress in affine differential geometry - problem list and continued bibliography, T. Binder and U. Simon; on the classification of timelike Bonnet surfaces, W.H. Chen and H.Z. Li; affine hyperspheres with constant affine sectional curvature, F. Dillen et al; geometric properties of the curvature operator, P. Gilkey; on a question of S.S. Chern concerning minimal hypersurfaces of spheres, I. Hiric and L. Verstraelen; parallel pure spinors on pseudo-Riemannian manifolds, I. Kath; twistorial construction of spacelike surfaces in Lorentzian 4-manifolds, F. Leitner; Nirenberg's problem in 90's, L. Ma; a new proof of the homogeneity of isoparametric hypersurfaces with (g,m) = (6, 1), R. Miyaoka; harmonic maps and negatively curved homogeneous spaces, S. Nishikawa; biharmonic morphisms between Riemannian manifolds, Y.L. Ou; intrinsic properties of real hypersurfaces in complex space forms, P.J. Ryan; on the nonexistence of stable minimal submanifolds in positively pinched Riemannian manifolds, Y.B. Shen and H.Q. Xu; geodesic mappings of the ellipsoid, K. Voss; n-invariants and the Poincare-Hopf Index Formula, W. Zhang. (Part contents)