On codimension-one foliations of constant mean curvature

The author is partially supported by a grant from the Alexander von Humboldt Foundation     

Mean curvature functions of codimension-one foliations

The author is partially supported by a grant from the Alexander von Humboldt Foundation.     

SymplecticK 2 of Laurent polynomials,associated Kac-Moody groups and Witt rings

The first author was partially supported by the Alexander von Humboldt-foundation in 1988–89. The authors want to thank Professor A. Bak for his valuable advice onK ₂Sp(F), Witt rings and their relation.     

Strongly meager and strong measure zero sets

 In this paper we present two consistency results concerning the existence of large strong measure zero and strongly meager sets. RID="ID=" <E5>Mathematics Subject Classification (2000):</E5>&ensp;03E35 RID="ID=" The first author was supported by Alexander von Humboldt Foundation and NSF grant DMS 95-05375. The second author was partially supported by Basic Research Fund, Israel Academy of Sciences, publication 658 Received: 6 January 1999 / Revised version: 20 July 1999 / Published online: 25 February 2002 RID=" ID=" <E5>Mathematics Subject Classification (2000):</E5>&ensp;03E35 RID=" ID=" The first author was supported by Alexander von Humboldt Foundation and NSF grant DMS 95-05375. The second author was partially supported by Basic Research Fund, Israel Academy of Sciences, publication 658     

Local and parallel finite element algorithms for the stokes problem

Based on two-grid discretizations, some local and parallel finite element algorithms for the Stokes problem are proposed and analyzed in this paper. These algorithms are motivated by the observation that for a solution to the Stokes problem, low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a fine grid by some local and parallel procedure. One technical tool for the analysis is some local a priori estimates that are also obtained in this paper for the finite element solutions on general shape-regular grids. Y. He was partially subsidized by the NSF of China 10671154 and the National Basic Research Program under the grant 2005CB321703; A. Zhou was partially supported by the National Science Foundation of China under the grant 10425105 and the National Basic Research Program under the grant 2005CB321704; J. Li was partially supported by the NSF of China under the grant 10701001. J. Xu was partially supported by Alexander von Humboldt Research Award for Senior US Scientists, NSF DMS-0609727 and NSFC-10528102.     

Fibred knots and twisted Alexander invariants

We study the twisted Alexander invariants of fibred knots. We establish necessary conditions on the twisted Alexander invariants for a knot to be fibred, and develop a practical method to compute the twisted Alexander invariants from the homotopy type of a monodromy. It is illustrated that the twisted Alexander invariants carry more information on fibredness than the classical Alexander invariants, even for knots with trivial Alexander polynomials.

     

Intersection Alexander polynomials

By considering a (not necessarily locally-flat) PL knot as the singular locus of a PL stratified pseudomanifold, we can use intersection homology theory to define intersection Alexander polynomials, a generalization of the classical Alexander polynomial invariants for smooth or PL locally-flat knots. We show that the intersection Alexander polynomials satisfy certain duality and normalization conditions analogous to those of ordinary Alexander polynomials, and we explore the relationships between the intersection Alexander polynomials and certain generalizations of the classical Alexander polynomials that are defined for non-locally-flat knots. We also investigate the relations between the intersection Alexander polynomials of a knot and the intersection and classical Alexander polynomials of the link knots around the singular strata. To facilitate some of these investigations, we introduce spectral sequences for the computation of the intersection homology of certain stratified bundles.     

Quantization of curvature for compact surfaces in <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">n⋆</Emphasis></Superscript>

For minimal surfaces in spheres, there is a well known conjecture about the quantization of intrinsic curvature which has been solved only in special cases so far. We recall an intrinsic and an extrinsic version for the known results and extend them to compact non-minimal surfaces in spheres. In particular we discuss special classes like Willmore surfaces and surfaces with parallel mean curvature vector. Mathematics Subject Classification (2000):53C42, 53A10.H.Li is partially supported by a research fellowship of the Alexander von Humboldt Stiftung 2001/2002 and the Zhongdian grant of NSFC. U. Simon is partially supported by DFG 163/Si-7-2 and a Chinese–German research cooperation of NSFC and DFG.     

Iterated processes and their applications to higher order differential equations

In this paper we construct models obtained by suitably combining Brownian motions and telegraphs in such a way that their transition functions satisfy higher-order parabolic or hyperbolic equations of different types. Equations with time-varying coefficients are also derived by considering processes endowed either with drift or with suitable modifications of their structure. Finally the distribution of the maximum of the iterated Brownian motion (along with some other properties) is presented. This work is partially supported by the Natural Science Foundation of Guangdong Province, National Natural Science Foundation of China grant No. 19501026 and the Alexander von Humbodlt Foundation     

Representations of Knot Groups and Twisted Alexander Polynomials

We present a twisted version of the Alexander polynomial associated with a matrix representation of the knot group. Examples of two knots with the same Alexander module but different twisted Alexander polynomials are given.     

Patchworking singular algebraic curves I

In this paper we present a general patchworking procedure for the construction of reduced singular curves having prescribed singularities and belonging to a given linear system on algebraic surfaces. It originates in the Viro “gluing” method for the construction of real non-singular algebraic hypersurfaces. The general procedure includes almost all known particular modifications, and goes far beyond. Some applications and examples illustrate the construction. Both authors were partially supported by the Herman Minkowsky-Minerva Center for Geometry at Tel Aviv University, and by grant no. G-616-15.6/99 from the German-Israeli Foundation for Research and Development. The first author was also supported by the Bessel Research Award from the Alexander von Humboldt Foundation. The second author was also partially supported by the EC-network ‘Algebraic Lie Representations” contract no. ERB-FMRX-CT97-0100.     

On a general principle in geometry that leads to functional equations

Summary Suppose that an invariant (or an invariant notion) of some geometry is given, like the distance between two points, the cross ratio of four points, the tangential distance between two spheres (or like the notion of orthogonality, of order, of a circle). One may ask what are the functions preserving (or preserving partially) that invariant (invariant notion). Originating from this principle some functional equation problems are formulated, namely the functional equations of distance, of area, of angle preservance.Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth.     

The LMO-invariant of 3-manifolds of rank one and the Alexander polynomial

We prove that the LMO-invariant of a 3-manifold of rank one is determined by the Alexander polynomial of the manifold, and conversely, that the Alexander polynomial is determined by the LMO-invariant. Furthermore, we show that the Alexander polynomial of a null-homologous knot in a rational homology 3-sphere can be obtained by composing the weight system of the Alexander polynomial with the ?rhus invariant of knots. Received February 14, 2000 / Published online October 11, 2000     

On the Alexander polynomials of knots with Gordian distance one

We consider a condition on a pair of the Alexander polynomials of knots which are realizable by a pair of knots with Gordian distance one. We show that there are infinitely many mutually disjoint infinite subsets in the set of the Alexander polynomials of knots such that every pair of distinct elements in each subset is not realizable by any pair of knots with Gordian distance one. As one of the subsets, we have an infinite set containing the Alexander polynomials of the trefoil knot and the figure eight knot. We also show that every pair of distinct Alexander polynomials such that one is the Alexander polynomial of a slice knot is realizable by a pair of knots of Gordian distance one, so that every pair of distinct elements in the infinite subset consisting of the Alexander polynomials of slice knots is realizable by a pair of knots with Gordian distance one. These results solve problems given by Y. Nakanishi and by I. Jong.     

Existence and stability of asymptotically oscillatory triple pulses

Triple pulses are constructed for systems of two coupled reaction-diffusion equations with an asymptotically oscillatory single pulse. In (Alexander and Jones [1993]) it has been shown that an infinite sequence of double pulses can be constructed near the single pulse. Under the condition that the wave speed of a stable double pulse coincides with that of the single pulse, it is shown here that an infinite sequence of triple pulses can be constructed. These pulses have the form of the double pulse concatenated with a further single pulse far behind, and cannot be constructed in the same way for the situations considered by previous authors. Moreover, the pulses are shown to be alternately stable and unstable.Dedicated with great respect to Klaus Kirchgässner on the occasion of his 60th birthdayResearch partially supported by the National Science Foundation under grant DMS-90-01788.Research partially supported by the National Science Foundation under grant DMS-91-00085.     

Strongly meager sets can be quite big

AssumeCH. There exists a strongly meager setX⊆2^ω and a continuous functionF: 2^ω → 2^ω such thatF″ (X)=2^ω. The analogous statement for the strong measure zero, the notion dual to strongly meager, is false. The first author was partially supported by NSF grant DMS 9971282 and the Alexander von Humboldt Foundation. The second author was partially supported by grant BW 5100-5-0231-2.     

Variational problems for maps of bounded variation with values inS 1

The main goal of this paper is to characterize the weak limits of sequences of smooth maps from a Riemannian manifold intoS ¹. This is achieved in terms of Cartesian currents. Applications to the existence of minimizers of area type functionals in the class of maps with values inS ¹ satisfying Dirchlet and homological conditions are then discussed. The so called dipole problem is solved, too.This work has been partially supported by the Ministero dell'Universitá e della Ricerca Scientifica, by C.N.R. contract n. 91.01343.CT01, and by the European Research Project GADGET. It was partially carried out while the first and the third author were visiting the Mathematisches Institut der Universität Bonn under the support respectively of the Alexander von Humboldt-Stiftung and of the SBF 256.     

Differences of Alexander polynomials for knots caused by a single crossing change

Kondo and Sakai independently gave a characterization of Alexander polynomials for knots which are transformed into the trivial knot by a single crossing change. The first author gave a characterization of Alexander polynomials for knots which are transformed into the trefoil knot (and into the figure-eight knot) by a single crossing change. In this note, we will give a characterization of Alexander polynomials for knots which are transformed into the 10₁₃₂ knot (and into the (5,2)-torus knot) by a single crossing change. Moreover, this method can be applied for knots with monic Alexander polynomials.     

Alexander polynomials of equivariant slice and ribbon knots in

This paper gives an algebraic characterization of Alexander polynomials of equivariant ribbon knots and a factorization condition satisfied by Alexander polynomials of equivariant slice knots.

     

Conditional Log-Laplace Functionals of Immigration Superprocesses with Dependent Spatial Motion

A non-critical branching immigration superprocess with dependent spatial motion is constructed and characterized as the solution of a stochastic equation driven by a time-space white noise and an orthogonal martingale measure. A representation of its conditional log-Laplace functionals is established, which gives the uniqueness of the solution and hence its Markov property. Some properties of the superprocess including an ergodic theorem are also obtained.Mathematics Subject Classifications (2000) 60J80, 60G57, 60J35.Zenghu Li: Supported by the NSFC (No. 10121101 and No. 10131040).Hao Wang: Supported by the research grant of UO.Jie Xiong: Research supported partially by NSA and by Alexander von Humboldt Foundation.