SOLUTIONS OF THE EQUATION x^m＋y^m=z^m IN SL2z

This paper solves an open problem of Vaserstein. The main result is: the equation x^m y^m=z^m has a solution in SL2z if and only if m is not divisible by 4 or 6 and m is not congruent to 3 modulo 6.     

Hybrid organosilazane/organosilylamine telechelic oligomers—I. Two-dimensional NMR spectroscopy characterization of dimethylsilylethylenediamine oligomers

The detailed characterization of multifunctional hybrid organosilazane/organosilylamine telechelic oligomers by IR and ¹H, ¹³C and ²⁹Si NMR spectroscopy in one and two dimenions has been undertaken. The specific multifunctional oligomers, comprising NH/NH₂ or SiCl functionalities depending on the monomer feed-ratio, were prepared from mono- and di-functionally reacted dichlorodimethylsilane (DDS) and mono-, di-, tri- and tetra-functionally reacted ethylenediamine (EDA). Varying the feed-ratio afforded control of the microstructures of the oligomers and the preparation of oligomers with, in some cases, conterminously located silicon–chlorine (SiCl) groups. The combination of the NMR methods with the IR technique has enabled the detailed microstructural characterization of the oligomers and the identification of the functionalities therein. This approach and the knowledge gained from the DDS/EDA system has been applied to the microstructural characterization of other hybrid organosilazane/silylamine preceramic telechelic oligomers.     

Echo planar imaging of perfluorocarbons

Emulsions of perfluorotributylamine (FTBA) and perflubron were evaluated for their utility in ¹⁹F echo planar imaging. Fluorine images of the emulsions were obtained in a phantom and two mice that had been predosed. Both agents, but particularly perflubron, show potential for fluorine echo planar studies because of the long spin-spin relaxation times of the CF₃ resonances. High resolution thin slice images obtained in as little as 26.6 ms are presented.     

Riccati-based preconditioner for computing invariant subspaces of large matrices

Summary. This paper introduces and analyzes the convergence properties of a method that computes an approximation to the invariant subspace associated with a group of eigenvalues of a large not necessarily diagonalizable matrix. The method belongs to the family of projection type methods. At each step, it refines the approximate invariant subspace using a linearized Riccati's equation which turns out to be the block analogue of the correction used in the Jacobi-Davidson method. The analysis conducted in this paper shows that the method converges at a rate quasi-quadratic provided that the approximate invariant subspace is close to the exact one. The implementation of the method based on multigrid techniques is also discussed and numerical experiments are reported. Received June 15, 2000 / Revised version received January 22, 2001 / Published online October 17, 2001     

Maximum likelihood estimate for discontinuous parameter in stochastic hyperbolic systems

The purpose of this paper is to study the identification problem of a spatially varying discontinuous parameter in stochastic hyperbolic equations. In previous works, the consistency property of the maximum likelihood estimate (MLE) was explored and the generating algorithm for MLE proposed under the condition that an unknown parameter is in a sufficiently regular space with respect to spatial variables.In order to show the consistency property of the MLE for a discontinuous coefficient, we use the method of sieves, i.e. the admissible class of unknown parameters is projected into a finite-dimensional space. For hyperbolic systems, we cannot obtain a regularity property of the solution with respect to a parameter. So in this paper, the parabolic regularization technique is used. The convergence of the derived finite-dimensional MLE to the infinite-dimensional MLE is justified under some conditions.     

Asymptotics in the random assignment problem

Summary We show that, in the usual probabilistic model for the random assignment problem, the optimal cost tends to a limit constant in probability and in expectation. The method involves construction of an infinite limit structure, in terms of which the limit constant is defined. But we cannot improve on the known numerical bounds for the limit.Research supported by NSF Grant MCS90-01710     

Minimizing the number of transitions with respect to observation equivalence

Labeled transition systems (lts) provide an operational semantics for many specification languages. In order to abstract unrelevant details of lts's, manybehavioural equivalences have been defined; here observation equivalence is considered. We are interested in the following problem:Given a finite lts, which is the minimal observation equivalent lts corresponding to it? It is well known that the number of states of an lts can be minimized by applying arelational coarsest partition algorithm. However, the obtained lts is not unique (up to the renaming of the states): for an lts there may exist several observation equivalent lts's which have the minimal number of states but varying number of transitions. In this paper we show how the number of transitions can be minimized, obtaining a unique lts.     

U-statistics in danach spaces

U-statistics in Banach spaces are considered and thoroughly investigated. The martingale structure, estimates of moments, the law of large numbers, the central limit theorem, the invariance principle, estimates of the rate of convergence, and large deviations are established