Quasilinear resolutions of non-linear equations

By analogy with the linear vector bundle case, a non-linear partial differential equation on a manifold can be defined as a fibred submanifold R_k of a k-jet bundle. By observing that under natural conditions the first prolongation gives rise to a vector bundle over R_k, (that is, a quasilinear equation), techniques of the linear case are adapted to establish conditions for the formal integrability of the equation.     

Dehn surgeries that yield fibred 3-manifolds

We study Dehn surgeries on null-homotopic knots that yield fibred 3-manifolds when an additional (but natural) homological restriction is imposed. The major tool used is Gabai’s theory of sutured manifold decomposition. Such surgeries are negative examples to a question of Michel Boileau. Another result we will prove is about surgeries which reduce the Thurston norm of a fibred manifold.     

Chu duality theory and coalgebraic representation of quantum symmetries

Building upon Vaughan Pratt's work on applications of Chu space theory to Stone duality, we develop a general theory of categorical dualities on the basis of Chu space theory and closure conditions, which encompasses a variety of dualities for topological spaces, convex spaces, closure spaces, and measurable spaces (some of which are new duality results on their own). It works as a general method to generate analogues of categorical dualities between frames (locales) and topological spaces beyond topology, e.g., for measurable spaces, convex spaces, and closure spaces. After establishing the Chu duality theory, we apply the state-observable duality between quantum lattices and closure spaces to coalgebraic representations of quantum symmetries, showing that the quantum symmetry groupoid fully embeds into a purely coalgebraic category, i.e., the category of Born coalgebras, which refines, through the quantum duality that follows from Chu duality theory, Samson Abramsky's fibred coalgebraic representations of quantum symmetries (which, in turn, builds upon his Chu representations of symmetries).     

Fibred rational surfaces with extremal Mordell-Weil lattices

Slope inequalities are given for fibred rational surfaces according as the Clifford index of a general fibre. For fibred rational surfaces of Clifford index two, the Mordell-Weil lattices of maximal ranks are completely determined.Supported by The 21st Century COE Program named “Towards a new basic science: depth and synthesis”.     

关于GB型空间的若干结果

本文对GB型空间进行了若干讨论,得到了一些结果。 设X为Banach空间,X为X的共轭空间,B(X,X)为X到X的所有有界线性算子组成的空间。如果X中的弱收敛序列为弱收敛序列,则称X为GB型空间。Grothendieck.A于1953年证明了l~∞是GB型空间。显然自反空间是GB型空间。     

The maximal coarse Baum–Connes conjecture for spaces which admit a fibred coarse embedding into Hilbert space

We introduce a notion of fibred coarse embedding into Hilbert space for metric spaces, which is a generalization of Gromov?s notion of coarse embedding into Hilbert space. It turns out that a large class of expander graphs admit such an embedding. We show that the maximal coarse Baum–Connes conjecture holds for metric spaces with bounded geometry which admit a fibred coarse embedding into Hilbert space.     

Half-space Fourier transform in application to non-linear hyperbolic initial-boundary-value problem: low regularity solutions

The Dirichlet initial-boundary-value problem in the half-space ℝ_t×ℝ³₊ for a non-linear hyperbolic system is studied. About the nonlinearity the null condition structure is imposed. Under the smallness conditions on the initial data the existence of low regularity solutions, on a given interval of time, is proved. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Maximally irregularly fibred surfaces of general type

We generalise a method of Xiao Gang to construct prototypes of fibred surfaces with maximal irregularity without being a product. This enables us, in the case of fibre genus g=3 to describe the possible singular fibres and to calculate the invariants of these surfaces. We also prove structure theorems on the moduli space for fibred surfaces with fibre genus g=2 and g=3.Mathematics Subject Classification (2000): 14J10, 14J29, 14D06Acknowledgement The author thanks his thesis advisor F. Herrlich for many stimulating discussions and a lot of patience. He also thanks E. Viehweg for worthful remarks concerning Torellis theorem. Some results in the same direction were obtained independently by J.-X. Cai. The author thanks him and the referee for his suggestions.     

Conformal Families of Measures for Fibred Systems

In this paper, we give sufficient conditions for the existence of (pseudo) weakly conformal and conformal families of measures for fibred systems. We describe a general construction principle for these families, modelled on the one developed by Denker and Urbanski for conformal measures. For those systems that are fibrewise local homeomorphisms, the constructed families are (pseudo) conformal. If a system is, moreover, weakly topologically exact along fibres, then each measure in the associated family is supported on the whole fibre where it is naturally defined.     

Transformations of linear connections

Combining several results on related (or conjugate) connections, defined on banachable fibre bundles, we set up a machinery, which permits to study various transformations of linear connections. Global and local methods are applied throughout. As an application, we get an extension of the classical affine transformations to the context of infinite-dimensional vector bundles. Another application shows that, realising the ordinary linear differential equations (in Banach spaces) as connections, we get the usual transformations of (equivalent) equations. Thus, some classical results on differential equations, such as the Theorem of Floquet, can have a geometric interpretation.     

Convergence of hybrid viscosity and steepest-descent methods for pseudocontractive mappings and nonlinear Hammerstein equations

In this article, we first introduce an iterative method based on the hybrid viscosity approximation method and the hybrid steepest-descent method for finding a fixed point of a Lipschitz pseudocontractive mapping (assuming existence) and prove that our proposed scheme has strong convergence under some mild conditions imposed on algorithm parameters in real Hilbert spaces. Next, we introduce a new iterative method for a solution of a nonlinear integral equation of Hammerstein type and obtain strong convergence in real Hilbert spaces. Our results presented in this article generalize and extend the corresponding results on Lipschitz pseudocontractive mapping and nonlinear integral equation of Hammerstein type reported by some authors recently. We compare our iterative scheme numerically with other iterative scheme for solving non-linear integral equation of Hammerstein type to verify the efficiency and implementation of our new method.     

Automorphiesummanden und Cousin-I-Probleme auf Faserräumen

This article deals with the classification of summands of automorphy on locally trivial fibre spaces E(X, Y), the base space X having the property H¹ (X,O)=0 and the fibre Y being a compact Kähler manifold. Further we assume that the structure group is connected and that the transgression H¹(Y)H²(X) vanishes. Then every summand of automorphy on E is equivalent to a homomorphism on B₁(E) with values in the ring of holomorphic functions on X (theorem I). As a corollary we prove a theorem on the solvability of Cousin-I-Problems on E by additive functions (theorem II).

---

---

Meinem verehrten Lehrer KARL STEIN in Dankbarkeit zum 60. Geburtstag gewidmet     

Conformal Families of Measures for Fibred Systems

In this paper, we give sufficient conditions for the existence of (pseudo) weakly conformal and conformal families of measures for fibred systems. We describe a general construction principle for these families, modelled on the one developed by Denker and Urbanski for conformal measures. For those systems that are fibrewise local homeomorphisms, the constructed families are (pseudo) conformal. If a system is, moreover, weakly topologically exact along fibres, then each measure in the associated family is supported on the whole fibre where it is naturally defined.This research was partially supported by NSF Grant DMS 0100078.Received February 4, 2002; in final form October 24, 2002 Published online October 24, 2003     

Finite-Order Formulation of Vinogradov's C-Spectral Sequence

The C-spectral sequence was introduced by A. M. Vinogradov in the late Seventies as a fundamental tool for the study of the algebro-geometric properties of jet spaces and differential equations. A spectral sequence arises from the contact filtration of the modules of forms on jet spaces of a fibring (or on a differential equation). In order to avoid serious technical difficulties, the order of the jet space is not fixed, i.e., computations are performed on spaces containing forms on jet spaces of any order. In this paper we show that there exists a formulation of Vinogradov's C-spectral sequence in the case of finite-order jet spaces of a fibred manifold. We compute all cohomology groups of the finite-order C-spectral sequence. We obtain a finite-order variational sequence which is shown to be naturally isomorphic with Krupka's finite-order variational sequence.     

Calabi–Yau varieties with semi-stable fibre structures

Motivated by the Strominger–Yau–Zaslow conjecture, we study Calabi–Yau varieties with semi-stable fibre structures. We use Hodge theory to study the higher direct images of wedge products of relative cotangent sheaves of certain semi-stable families over higher dimensional quasi-projective bases, and obtain some results on positivity. We then apply these results to study non-isotrivial Calabi–Yau varieties fibred by semi-stable Abelian varieties (or hyperkähler varieties).     

Boundary conditions for linear manifolds. III. Infinite dimensional adjoints via solution spaces

A formula is given for the orthogonal complement of any vector subspace of l₂. Countably infinite adjoint subspaces in a Banach space are characterized via solution spaces. In particular, infinite dimensional self-adjoint subspaces in a reflexive Banach space are characterized via solution spaces, generalizing a result in Dunford and Schwartz [“Linear Operators, II,” Interscience, New York, 1963]. Applications are made to closed linear manifolds in l₂ ⊕ l₂ as well as infinite dimensional, generalized ordinary differential subspaces in a Hilbert space with the boundary conditions imposed on real sequences. The results are also expressed via solution spaces.     

Profinite rigidity for Seifert fibre spaces

An interesting question is whether two 3-manifolds can be distinguished by computing and comparing their collections of finite covers; more precisely, by the profinite completions of their fundamental groups. In this paper, we solve this question completely for closed orientable Seifert fibre spaces. In particular, all Seifert fibre spaces are distinguished from each other by their profinite completions apart from some previously-known examples due to Hempel. We also characterize when bounded Seifert fibre space groups have isomorphic profinite completions, given some conditions on the boundary.     

SHEAVES OF BANACH SPEACES

Abstract

This paper presents a number of results concerning sheaves on a topological space, with values in the category BAN of Banach spaces, over K = R or Ø, and linear contractions. After showing that these sheaves are reflective in the corresponding category of presheaves (Proposition 1) and that the resulting reflection is stalk preserving (Proposition 2), we concentrate on the approximation sheaves, these being BAN-sheaves satisfying a strong patching condition originally due to Auspitz [1]. The interest in these particular sheaves lies in the fact that they are precisely the BAN-sheaves arising as sheaves of continuous sections of the appropriate kind of Banach fibre spaces [1] and thus central to the representation of Banach spaces by continuous sections. Here, we show that the approximation sheaves on any space are characterized as the BAN-presheaves injective relative to certain maps (Proposition 3) and that, for paracompact spaces X, they are exactly those BAN-sheaves S such that each SU, U open in X, admits a suitable C*U-module structure (Proposition 4). Further, we consider the adjointness between the approximation sheaves on a space X and the Banach modules over C*X (Proposition 5) and investigate its special properties for X being Tychonoff (Proposition 6) and Boolean (Proposition 7). We conclude with some observations regarding the failure of the analogues of Swan's Theorem for vector bundles and the Hahn-Banach Theorem in the present context, and some positive facts concerning injectivity for approximation sheaves on Tychonoff spaces.