Contact Interaction of Two Elastic Wedges

We consider the problem of contact of two elastic wedges and assume that only the vertices of the wedges touch before loading. After loading, the edges of both wedges come in contact near their common vertex. We reduce the constructed system of dual integral equations to a Fredholm integral equation of the second kind with difference kernel given on the semiaxis. We analytically solve the Fredholm equation by reducing it to the boundary-value Riemann problem for analytic functions. We obtain an analytic expression for contact stresses.     

Polynomial recurrence with large intersection over countable fields

Applying a local Gauss–Bonnet formula for closed subanalytic sets to the complex analytic case, we obtain characterizations of the Euler obstruction of a complex analytic germ in terms of the Lipschitz–Killing curvatures and the Chern forms of its regular part. We also prove analogous results for the global Euler obstruction. As a corollary, we give a positive answer to a question of Fu on the Euler obstruction and the Gauss–Bonnet measure.     

On approximation of analytic sets

We give conditions sufficient for the existence of a sequence of Nash sets convergent to a given analytic set. Moreover, for limit sets which are k-sheeted analytic covers we additionally require that sets in approximating sequence are also k-sheeted analytic covers. Then we present examples of applications of the developed theory.Research partially supported by KBN Grant no. 2P03A 015 22Mathematics Subject Classification (2000): 32E30, 32C07, 32C25     

Gradient-like flows and self-indexing in stratified Morse theory

We develop the idea of self-indexing and the technology of gradient-like vector fields in the setting of Morse theory on a complex algebraic stratification. Our main result is the local existence, near a Morse critical point, of gradient-like vector fields satisfying certain “stratified dimension bounds up to fuzz” for the ascending and descending sets. As a global consequence of this, we derive the existence of self-indexing Morse functions.     

On the relative connections

Abstract

We investigate relative connections on a sheaf of modules. A sufficient condition is given for the existence of a relative holomorphic connection on a holomorphic vector bundle over a complex analytic family. We show that the relative Chern classes of a holomorphic vector bundle admitting relative holomorphic connection vanish, if each of the fiber of the complex analytic family is compact and Kähler.     

Formule de Cauchy-Crofton pour la densité des ensembles sous-analytiques

For the density of subanalytic sets, we give a formula analogous to the classical Cauchy-Crofton formula for the volume. To do this we define the local polar profiles of a subanalytic set and the corresponding finite sequence of local multiplicities which is the real counterpart of the multiplicity of complex analytic sets.     

On some definable sets over fields with analytic structure

We discover geometric properties of certain definable sets over non-Archimedean valued fields with analytic structures. Results include a parameterized smooth stratification theorem and the existence of a bound on the piece number of fibers for these sets. In addition, we develop a dimension theory for these sets and also for the formulas which define them.     

Flattening and analytic continuation of affinoid morphisms: remarks on a paper of Gardener and Schoutens

We give an example of an affinoid curve without analytic continuation.We use this to produce an example of an affinoid morphism thatcannot be flattened by a finite sequence of local blow-ups.Thus the global rigid analogue of Hironaka's complex analyticflattening theorem given by T. Gardener and H. Schoutens, inTheorem 2.3 of ‘Flattening and subanalytic sets in rigidanalytic geometry’, Proc. London Math. Soc (3) 83 (2001)681–707, is not true. Since this is a key step in theproof of the affinoid elimination theorem (loc. cit. Theorem3.12), that proof contains a serious gap. We also give an exampleof an affinoid subset of the plane that is not the image undera proper rigid analytic map of a set that is globally semianalyticin the domain of that map. This clarifies the relationship amongseveral natural categories of rigid subanalytic sets.2000 MathematicsSubject Classification 32P05 (primary), 03C10, 32B20 (secondary).     

Levi decomposition of analytic Poisson structures and Lie algebroids

We prove the existence of a local analytic Levi decomposition for analytic Poisson structures and Lie algebroids.     

Local radial Phragmén-Lindelöf estimates for plurisubharmonic functions on analytic varieties

We give a sufficient condition for a local radial Phragmén-Lindelöf principle on analytic varieties. This condition is expressed in terms of existence of hyperbolic directions.

     

Pseudo Leja sequences

We study pseudo Leja sequences attached to a compact set in the complex plane. The requirements are weaker than those of ordinary Leja sequences, but these sequences still provide excellent points for interpolation of analytic functions and their computation is much easier. We also apply them to the construction of excellent sets of nodes for multivariate interpolation of analytic functions on product sets.     

Analytic smoothing of geometric maps with applications to KAM theory

We show that finitely differentiable diffeomorphisms which are either symplectic, volume-preserving, or contact can be approximated with analytic diffeomorphisms that are, respectively, symplectic, volume-preserving or contact. We prove that the approximating functions are uniformly bounded on some complex domains and that the rate of convergence, in Cr-norms, of the approximation can be estimated in terms of the size of such complex domains and the order of differentiability of the approximated function. As an application to this result, we give a proof of the existence, the local uniqueness and the bootstrap of regularity of KAM tori for finitely differentiable symplectic maps. The symplectic maps considered here are not assumed either to be written in action-angle variables or to be perturbations of integrable systems. Our main assumption is the existence of a finitely differentiable parameterization of a maximal dimensional torus that satisfies a non-degeneracy condition and that is approximately invariant. The symplectic, volume-preserving and contact forms are assumed to be analytic.     

Global properties of a class of overdetermined systems

We consider an overdetermined system of complex vector fields on the three-dimensional torus which is naturally associated to a real analytic, closed 1-form on the two-dimensional torus. By means of a detailed study of the geometry of level sets of a primitive of the pull-back of the 1-form via the universal covering, we prove that a necessary condition for the system to be globally solvable, in the non-exact case, is that each connected component of the critical set has a point at which the local primitives of the 1-form are open maps. When this condition is violated we also construct global solutions to the inhomogenous equations having analytic singularities.     

Quotienten komplexer Mannigfaltigkeiten nach spiegelungsfreien Gruppen

There is given a necessary and sufficient (global) topological condition on a complex space X for the existence of a local finite analytic mapping YX with Y a manifold and branched locus of codimension>1.     

On minimal non-potentially closed subsets of the plane

We study the Borel subsets of the plane that can be made closed by refining the Polish topology on the real line. These sets are called potentially closed. We first compare Borel subsets of the plane using products of continuous functions. We show the existence of a perfect antichain made of minimal sets among non-potentially closed sets. We apply this result to graphs, quasi-orders and partial orders. We also give a non-potentially closed set minimum for another notion of comparison. Finally, we show that we cannot have injectivity in the Kechris-Solecki-Todor?evi? dichotomy about analytic graphs.     

Intrinsic Lipschitz Graphs Within Carnot Groups

A Carnot group is a connected, simply connected, nilpotent Lie group with stratified Lie algebra. We study the notions of intrinsic graphs and of intrinsic Lipschitz graphs within Carnot groups. Intrinsic Lipschitz graphs are the natural local analogue inside Carnot groups of Lipschitz submanifolds in Euclidean spaces, where “natural” emphasizes that the notion depends only on the structure of the algebra. Intrinsic Lipschitz graphs unify different alternative approaches through Lipschitz parameterizations or level sets. We provide both geometric and analytic characterizations and a clarifying relation between these graphs and Rumin’s complex of differential forms.     

Hypoellipticity in spaces of ultradistributions—Study of a model case

In this work we study C ^??-hypoellipticity in spaces of ultradistributions for analytic linear partial differential operators. Our main tool is a new a-priori inequality, which is stated in terms of the behaviour of holomorphic functions on appropriate wedges. In particular, for sum of squares operators satisfying H?rmander??s condition, we thus obtain a new method for studying analytic hypoellipticity for such a class. We also show how this method can be explicitly applied by studying a model operator, which is constructed as a perturbation of the so-called Baouendi-Goulaouic operator.     

Presence or absence of analytic structure in maximal ideal spaces

We study extensions of Wermer’s maximality theorem to several complex variables. We exhibit various smoothly embedded manifolds in complex Euclidean space whose hulls are non-trivial but contain no analytic disks. We answer a question posed by Lee Stout concerning the existence of analytic structure for a uniform algebra whose maximal ideal space is a manifold.     

Reconstructing Convex Polygons in the Plane from One Directed X-Ray

Gardner [7] proved that with the exception of a simple class of nonparallel wedges, convex polygons in the plane are uniquely determined by one directed X-ray. This paper develops methods for reconstructing convex polygons in the plane from one directed X-ray. We show that nonsmooth points on the boundary of a convex body are located along rays where the derivative of the data have a jump discontinuity. Location of the nonsmooth points divides a convex polygon into a finite set of wedges. We prove uniqueness theorems and give algorithms for reconstructing nonparallel wedges from line integrals along four or more rays. Also, we characterize discrete data sets that are directed X-rays of both parallel and nonparallel wedges. Several examples of reconstructions are included. Received August 16, 1999, and in revised form September 13, 2000. Online publication May 4, 2001.