TheK-theory of AF embeddings of the rational rotation algebras

Our main result is the construction of an embedding ofC(T²) into an approximately finite-dimensionalC ^*-algebra which induces an injection onK ₀(C(T²)). The existence of this embedding implies that Cech cohomology cannot be extended to a stable, continuous homology theory forC ^*-algebras which admits a well-behaved Chern character. Homotopy properties ofC ^*-algebras are also considered. For example, we show that the second homotopy functor forC ^*-algebras is discontinuous. Similar embeddings are constructed for all the rational rotation algebras, with the consequence that none of the rational rotation algebras satisfies the homotopy property called semiprojectivity.     

Stock market crashes are outliers

We call attention against what seems to be a widely held misconception according to which large crashes are the largest events of distributions of price variations with fat tails. We demonstrate on the Dow Jones Industrial Average that with high probability the three largest crashes in this century are outliers. This result supports the suggestion that large crashes result from specific amplification processes that might lead to observable pre-cursory signatures. Received and Revised: 30 November 1997 / Accepted: 8 December 1997     

具有间隙的动力系统的极限环

本文研究了系统ｘ＋ｆ（ｘ）＋ｇ（ｘ）＝０的极限环的存在性，其中ｇ（ｘ）有两个间断点并且不满足ｇ（ｘ）·ｘ＞０（ｘ≠０）．还引进［２，３］中定义的Филиппов解的概念，利用Филиппов解的整体存在性和普遍唯一性定理解决方程的解的存在唯一性问题，得到了几个极限环存在性定理     

Remarks on the r and Δ convolutions

In this paper we study the properties of the r–deformation introduced in [B1]. We observe that the associated convolution coming from the conditionally free convolution is associative only for r = 1 and r = 0. We give the realization of some r–Gaussian random variables and obtain Haagerup–Pisier–Buchholz type inequalities. We also study another convolution defined with the use of the r–deformation through a moment–cumulant formula [KY1] and show that it is associative and in general not positive. Partially sponsored with KBN grant no 2P03A00723 and RTN HPRN-CT-2002-00279.     

The Quantum-Classical Comparison of the Arrival-Time Distribution Through the Probability Current

We consider the arrival time distribution defined through the quantum probability current for a Gaussian wave packet representing free particles in quantum mechanics in order to explore the issue of the classical limit of arrival time. We formulate the classical analogue of the arrival time distribution for an ensemble of free particles represented by a phase space distribution function evolving under the classical Liouville's equation. The classical probability current so constructed matches with the quantum probability current in the limit of minimum uncertainty. Further, it is possible to show in general that smooth transitions from the quantum mechanical probability current and the mean arrival time to their respective classical values are obtained in the limit of large mass of the particles.     

Lévy processes in semisimple Lie groups and stability of stochastic flows

We study the asymptotic stability of stochastic flows on compact spaces induced by Levy processes in semisimple Lie groups. It is shown that the Lyapunov exponents can be determined naturally in terms of root structure of the Lie group and there is an open subset whose complement has a positive codimension such that, after a random rotation, each of its connected components is shrunk to a single moving point exponentially under the flow.

     

The Navier-Stokes limit of stationary solutions of the nonlinear Boltzmann equation

We consider the flow of a gas in a channel whose walls are kept at fixed (different) temperatures. There is a constant external force parallel to the boundaries which may themselves also be moving. The system is described by the stationary Boltzmann equation to which are added Maxwellian boundary conditions with unit accommodation coefficient. We prove that when the temperature gap, the relative velocity of the planes, and the force are all sufficiently small, there is a solution which converges, in the hydrodynamic limit, to a local Maxwellian with parameters given by the stationary solution of the corresponding compressible Navier-Stokes equations with no-slip voundary conditions. Corrections to this Maxwellian are obtained in powers of the Knudsen number with a controlled remainder.     

Effect of the initial phase of the field in ionization by ultrashort laser pulses

We report on observable new features related to ionization of atoms by laser pulses of only few cycles and some intensity. We show that for particular photo-electron energies, the angular distribution becomes asymmetric and that this asymmetry is related to the initial phase of the field. Received: 4 November 1997/Revised: 21 January 1998/Accepted: 23 February 1998     

A New Proof of Ruelle's Superstability Bounds

For a slightly stronger assumptions on the interaction we give a very transparent proof of Ruelle's result in the language of Poisson integral measure representation for the correlation functions on the configuration space using some kind of cluster expansion in the densities of configurations.     

Regularization of Quantum Relative Entropy in Finite Dimensions and Application to Entropy Production

The fundamental concept of relative entropy is extended to a functional that is regular-valued also on arbitrary pairs of nonfaithful states of open quantum systems. This regularized version preserves almost all important properties of ordinary relative entropy such as joint convexity and contractivity under completely positive quantum dynamical semigroup time evolution. On this basis a generalized formula for entropy production is proposed, the applicability of which is tested in models of irreversible processes. The dynamics of the latter is determined by either Markovian or non-Markovian master equations and involves all types of states.