Formal orthogonal polynomials and Hankel/Toeplitz duality

For classical polynomials orthogonal with respect to a positive measure supported on the real line, the moment matrix is Hankel and positive definite. The polynomials satisfy a three term recurrence relation. When the measure is supported on the complex unit circle, the moment matrix is positive definite and Toeplitz. Then they satisfy a coupled Szeg recurrence relation but also a three term recurrence relation. In this paper we study the generalization for formal polynomials orthogonal with respect to an arbitrary moment matrix and consider arbitrary Hankel and Toeplitz matrices as special cases. The relation with Padé approximation and with Krylov subspace iterative methods is also outlined.This research was supported by the National Fund for Scientific Research (NFWO), project Lanczos, grant #2.0042.93.     

Euler'sϕ-function in the context of IΔ 0

It is demonstrated that we can represent Euler's -function by means of a ₀-formula in such a way that the theory I ₀ proves the recursion equations that are characteristic for this function.     

Non-compact quantum groups associated with Abelian subgroups

LetG be a Lie group. For any Abelian subalgebra of the Lie algebra g ofG, and any , the difference of the left and right translates ofr gives a compatible Poisson bracket onG. We show how to construct the corresponding quantum group, in theC ^*-algebra setting. The main tool used is the general deformation quantization construction developed earlier by the author for actions of vector groups onC ^*-algebras.The research reported on here was supported in part by National Science Foundation grant DMS-9303386.     

Massively time-parallel,approximate simulation of loss queueing systems

A time-parallel simulation obtains parallelism by partitioning the time domain of the simulation. An approximate time-parallel simulation algorithm named GG1K is developed for acyclic networks of loss FCFSG/G/1/K queues. The GG1K algorithm requires two phases. In the first phase, a similar system (i.e. aG/G/1/ queue) is simulated using the GLM algorithm. Then the resultant trajectory is transformed into an approximateG/G/1/K trajectory in the second phase. The closeness of the approximation is investigated theoretically and experimentally. Our results show that the approximation is highly accurate except whenK is very small (e.g. 5) in certain models. The algorithm exploits unbounded parallelism and can achieve near-linear speedup when the number of arrivals simulated is sufficiently large.     

On the interpretation of fluorescence anisotropy decays from probe molecules in lipid vesicle systems

Measurements of fluorescence depolarization decays are widely used to obtain information about the molecular order and rotational dynamics of fluorescent probe molecules in membrane systems. This information is obtained by least-squares fits of the experimental data to the predictions of physical models for motion. Here we present a critical review of the ways and means of the data analysis and address the question how and why totally different models such as Brownian rotational diffusion and wobble-in-cone provide such convincing fits to the fluorescence anistropy decay curves. We show that while these models are useful for investigating the general trends in the behavior of the probe molecules, they fail to describe the underlying motional processes. We propose to remedy this situation with a model in which the probe molecules undergo fast, though restricted local motions within a slowly rotating cage in the lipid bilayer structure. The cage may be envisaged as a free volume cavity between the lipid molecules, so that its position and orientation change with the internal conformational motions of the lipid chains. This approach may be considered to be a synthesis of the wobble-in-cone and Brownian rotational diffusion models. Importantly, this compound motion model appears to provide a consistent picture of fluorescent probe behavior in both oriented lipid bilayers and lipid vesicle systems.