Error bounds for minimal energy bivariate polynomial splines

Summary. We derive error bounds for bivariate spline interpolants which are calculated by minimizing certain natural energy norms. Received March 28, 2000 / Revised version received June 23, 2000 / Published online March 8, 2002 RID="*" ID="*" Supported by the National Science Foundation under grant DMS-9870187 RID="**" ID="**" Supported by the National Science Foundation under grant DMS-9803340 and by the Army Research Office under grant DAAD-19-99-1-0160     

Two new variants of nonlinear inexact Uzawa algorithms for saddle-point problems

Summary. In this paper, we consider some nonlinear inexact Uzawa methods for iteratively solving linear saddle-point problems. By means of a new technique, we first give an essential improvement on the convergence results of Bramble-Paschiak-Vassilev for a known nonlinear inexact Uzawa algorithm. Then we propose two new algorithms, which can be viewed as a combination of the known nonlinear inexact Uzawa method with the classical steepest descent method and conjugate gradient method respectively. The two new algorithms converge under very practical conditions and do not require any apriori estimates on the minimal and maximal eigenvalues of the preconditioned systems involved, including the preconditioned Schur complement. Numerical results of the algorithms applied for the Stokes problem and a purely linear system of algebraic equations are presented to show the efficiency of the algorithms. Received December 8, 1999 / Revised version received September 8, 2001 / Published online March 8, 2002 RID="*" ID="*" The work of this author was partially supported by a grant from The Institute of Mathematical Sciences, CUHK RID="**" ID="**" The work of this author was partially supported by Hong Kong RGC Grants CUHK 4292/00P and CUHK 4244/01P     

Rigidity of proper holomorphic mappings between nonequidimensional bounded symmetric domains

We prove that any proper holomorphic mapping from to is necessarily a totally geodesic isometric embedding with respect to their Bergman metrics and therefore is the standard linear embedding up to their automorphisms. The proof of the main result in this paper will be achieved by a differential-geometric study of a special class of complex geodesic curves on the bounded symmetric domains with respect to their Bergman metrics. Received: 31 October 2000; in final form: 2 July 2001/ Published online: 4 April 2002     

Lattice Points in Three-Dimensional Convex Bodies with Points of Gaussian Curvature Zero at the Boundary

 In this article we investigate the number of lattice points in a three-dimensional convex body which contains non-isolated points with Gaussian curvature zero but a finite number of flat points at the boundary. Especially, in case of rational tangential planes in these points we investigate not only the influence of the flat points but also of the other points with Gaussian curvature zero on the estimation of the lattice rest. Received 19 June 2001; in revised form 17 January 2002 RID="a" ID="a" Dedicated to Professor Edmund Hlawka on the occasion of his 85th birthday     

Enveloppes convexes des processus gaussiens

Let X={X(t), t[0,1]} be a process on [0,1] and V_X=Conv{(t,x)t[0,1], x=X(t)} be the convex hull of its path.The structure of the set ext(V_X) of extreme points of V_X is studied. For a Gaussian process X with stationary increments it is proved that:

• The set ext(V_X) is negligible if X is non-differentiable.

• If X is absolutely continuous process and its derivative X′ is continuous but non-differentiable, then ext(V_X) is also negligible and moreover it is a Cantor set.

It is proved also that these properties are stable under the transformations of the type Y(t)=f(X(t)), if f is a sufficiently smooth function.     

Estimation de Yule–Walker d'un CAR(p) observé à temps discret

Let (X(lδ), l=0,n) be a discrete observation at mesh δ>0 of X, a CAR(p). Classical Yule–Walker estimation are biased and must be corrected. Resultant estimators converge if T=nδ→+∞, are asymptotically normal with rate , and efficient. The diffusion coefficient is also estimated, with rate .     

Analytic continuation for Beltrami systems, Siegel's Theorem for UQR maps, and the Hilbert-Smith Conjecture

A uniformly quasiregular mapping, is a mapping of the m-sphere with the property that it and all its iterates have uniformly bounded distortion. Such maps are rational with respect to some bounded measurable conformal structure and there is a Fatou-Julia type theory associated with the dynamical system obtained by iterating these mappings. We begin by investigating the analogue of Siegel's theorem on the local conjuga cy of rotational dynamics. We are led to consider the analytic continuation properties of solutions to the highly nonlinear first order Beltrami systems. We reduce these problems to a central and well known conjecture in the theory of transformation groups; namely the Hilbert-Smith conjecture, which roughly asserts that effective transformation groups of manifolds are Lie groups. Our affirmative solution to this problem then implies unique analytic continuation and Siegel's theorem. Received: 14 September 2000 / Revised version: 23 November 2001 / Published online: 5 September 2002 RID="*" ID="*" Research supported in part by grants from the Marsden Fund and Royal Society (NZ).     

Multiplicity of the second Schrödinger eigenvalue on closed surfaces

We improve – roughly by a factor 2 – the known bound on the multiplicity of the second eigenvalue of Schr?dinger operators (i.e. Laplace plus potential) on closed surfaces. This gives four new topological types of surfaces for which Colin de Verdière's conjecture relating the maximal multiplicity to the chromatic number of the surface is verified. The proof goes by defining a space of "nodal splittings” of the surface, equipped with a double covering to which a Borsuk-Ulam type theorem is applied. Received: 19 June 2001; revised version: 18 March 2002 /Published online: 17 June 2002     

Small Universal Graphs for Bounded-Degree Planar Graphs

For all positive integers N and k, let denote the family of planar graphs on N or fewer vertices, and with maximum degree k. For all positive integers N and k, we construct a -universal graph of size . This construction answers with an explicit construction the previously open question of the existence of such a graph. Received July 8, 1998 RID="*" ID="*" Supported by NSF grant CCR98210-58 and ARO grant DAAH04-96-1-0013.     

Large Cliques in -Free Graphs

Dedicated to the memory of Paul Erdős A graph is called -free if it contains no cycle of length four as an induced subgraph. We prove that if a -free graph has n vertices and at least edges then it has a complete subgraph of vertices, where depends only on . We also give estimates on and show that a similar result does not hold for H-free graphs––unless H is an induced subgraph of . The best value of is determined for chordal graphs. Received October 25, 1999 RID="*" ID="*" Supported by OTKA grant T029074. RID="**" ID="**" Supported by TKI grant stochastics@TUB and by OTKA grant T026203.