Enlargement of the Wiener filtration by an absolutely continuous random variable via Malliavin’s calculus

Summary. The analytic treatment of problems related to the asymptotic behaviour of random dynamical systems generated by stochastic differential equations suffers from the presence of non-adapted random invariant measures. Semimartingale theory becomes accessible if the underlying Wiener filtration is enlarged by the information carried by the orthogonal projectors on the Oseledets spaces of the (linearized) system. We study the corresponding problem of preservation of the semimartingale property and the validity of a priori inequalities between the norms of stochastic integrals in the enlarged filtration and norms of their quadratic variations in case the random element F enlarging the filtration is real valued and possesses an absolutely continuous law. Applying the tools of Malliavin’s calculus, we give smoothness conditions on F under which the semimartingale property is preserved and a priori martingale inequalities are valid. Received: 12 April 1995 / In revised form: 7 March 1996     

On Some Integral Equations in Hilbert Space with an Application to the Theory of Elasticity

A Volterra type integral equation in a Hilbert space with an additional linear operator L and a spectral parameter depending on time is considered. If the parameter does not belong to the spectrum of L unconditional solvability of the considered problem is proved. In the case where the initial value of the parameter coincides with some isolated point of the spectrum of the operator L sufficient conditions for solvability are established. The obtained results are applied to the partial integral equations associated with a contact problem of the theory of elasticity.     

Syntactic characterization of closure under connected limits

Summary We give a syntactic characterization of (finitary) theories whose categories of models are closed under the formation of connected limits (respectively the formation of pullbacks and substructures) in the category of all structures. They are also those theories whose consistent extensions by new atomic facts admit in each component an initial structure (respectively an initial term structure), and also thoseT for whichM(T) is locally finitely multi-presentable in a canonical way. We also show that these two properties of theories are nonuniform.     

Rehabilitation of the Gauss-Jordan algorithm

Summary In this paper a Gauss-Jordan algorithm with column interchanges is presented and analysed. We show that, in contrast with Gaussian elimination, the Gauss-Jordan algorithm has essentially differing properties when using column interchanges instead of row interchanges for improving the numerical stability. For solutions obtained by Gauss-Jordan with column interchanges, a more satisfactory bound for the residual norm can be given. The analysis gives theoretical evidence that the algorithm yields numerical solutions as good as those obtained by Gaussian elimination and that, in most practical situations, the residuals are equally small. This is confirmed by numerical experiments. Moreover, timing experiments on a Cyber 205 vector computer show that the algorithm presented has good vectorisation properties.     

Defect operators, defect functions and defect indices for analytic submodules

This paper mainly concerns defect operators and defect functions of Hardy submodules, Bergman submodules over the unit ball, and Hardy submodules over the polydisk. The defect operator (function) carries key information about operator theory (function theory) and structure of analytic submodules. The problem when a submodule has finite defect is attacked for both Hardy submodules and Bergman submodules. Our interest will be in submodules generated by polynomials. The reason for choosing such submodules is to understand the interaction of operator theory, function theory and algebraic geometry.     

Unavoidable Traces Of Set Systems

Sauer, Shelah, Vapnik and Chervonenkis proved that if a set system on n vertices contains many sets, then the set system has full trace on a large set. Although the restriction on the size of the groundset cannot be lifted, Frankl and Pach found a trace structure that is guaranteed to occur in uniform set systems even if we do not bound the size of the groundset. In this note we shall give three sequences of structures such that every set system consisting of sufficiently many sets contains at least one of these structures with many sets.     

Hamiltonian N2‐locally connected claw‐free graphs

A graph G is N₂‐locally connected if for every vertex ν in G, the edges not incident with ν but having at least one end adjacent to ν in G induce a connected graph. In 1990, Ryjá?ek conjectured that every 3‐connected N₂‐locally connected claw‐free graph is Hamiltonian. This conjecture is proved in this note. © 2004 Wiley Periodicals, Inc. J Graph Theory 48: 142–146, 2005     

A finiteness result for Heegaard splittings

In this paper we show that for a given 3-manifold and a given Heegaard splitting there are finitely many preferred decomposing systems of 3g−3 disjoint essential disks. These are characterized by a combinatorial criterion which is a slight strengthening of Casson-Gordon's rectangle condition. This is in contrast to fact that in general there can exist infinitely many such systems of disks which satisfy just the Casson-Gordon rectangle condition.