Difference Discrete Variational Principle in Discrete Mechanics and Symplectic Algorithm

We propose the difference discrete variational principle in discrete mechanics and symplectic algorithmwith variable step-length of time in finite duration based upon a noncommutative differential calculus established inthis paper. This approach keeps both symplecticity and energy conservation discretely. We show that there exists thediscrete version of the Euler-Lagrange cohomology in these discrete systems. We also discuss the solution existencein finite time-length and its site density in continuous limit, and apply our approach to the pendulum with periodicperturbation. The numerical results are satisfactory.     

Flocks, Ovoids of Q(4,q)and Designs

We prove that an ovoid O of Q(4,q),q odd, is the Thas' ovoid associated with a semifield flock if and only if O represents, on the Klein quadric, a symplectic spread of PG(3,q), whose associated plane is a semifiled plane.     

Control of a hybrid conditionally linear Gaussian process

A control problem is considered where the coefficients of the linear dynamics are functions of a noisily observed Markov chain. The approximation introduced is to consider these coefficients as functions of the filtered estimate of the state of the chain; this gives rise to a finite-dimensional conditional Kalman filter. A minimum principle and a new equation for an adjoint process are obtained.This research was partially supported by NSERC under Grant A-7964, by the US Air Force Office of Scientific Research under Contract AFOSR-86-0332, and by the US Army Research Office under Contract DAAL03-87-0102.The authors obtained these results during a visit to UCSD by the first author in January 1990. This author wishes to thank Professor D. D. Sworder and his department for their hospitality.     

Sensitivity analysis and model optimization for reaction-diffusion systems

This paper presents a computational framework for the optimization and sensitivity analysis of a process whose state depends upon several parameter functions. Assuming that the process is described by a system of quasilinear, parabolic, partial differential equations, we show how determining the problem parameters so as to improve an associated objective functional is directly related to knowing the state function sensitivities. An expression for the gradient of the objective functional in terms of the solutions of an adjoint system enables one to bypass the calculation of state function sensitivities. These concepts are illustrated for a simple model of cooperative processes in chemical kinetics. Since sensitivity analysis and model optimization are important tools for investigating parameter dependence and validating mathematical models, research developments in such diverse fields as optimal design theory, chemical kinetics, and parameter identification are important motivations for this paper.This author would like to gratefully acknowledge Dr. M. Delle Donne, EGG, for several helpful discussions.This author was partially supported by NSF Grant No. CMS-80-05677.     

Almost Complex Structures Which Are Compatible with Kähler or Symplectic Structures

In this note we prove that half of all homotopy classes of almost complex structures on M is not compatible with any symplectic structure for a certain class of oriented compact 4-manifolds M. In particular, half of all homotopy classes of almost complex structures on an oriented 4-manifold is not compatible to any Kähler structure.     

Hamilton体系与与弹性力学Saint－Venant问题

本文一改传统的在Ｌａｇｒａｎｇｅ体系欧几里德空间中用半道法讨论Ｓａｉｎｔ－Ｖｅｎａｎｔ问题的方法，而在具有守恒性的Ｈａｍｉｌｔｏｎ体系中辛空间里研究该问题．通过讨论Ｈａｍｉｌｔｏｎ算子矩阵的零本征值及其Ｊｏｒｄａｎ型，直接求解出全部Ｓａｉｎｔ－Ｖｅｎａｎｔ问题的解．     

On the Integrability of Geodesic Flows of Submersion Metrics

Letters in Mathematical Physics - Suppose we are given a compact Riemannian manifold (Q,g) with a completely integrable geodesic flow. Let G be a compact connected Lie group acting freely on Q by...     

Représentation d''une particule quantique de spin 1/2 par le produit de Moyal et une super-variété symplectique plate

The aim of this paper is to show that it is possible to represent a quantum particle with a spin with the help of a star product and a flat supermanifold. We extend to the supermanifolds the Schouten bracket, the exterior differentiation, the Lie derivative. These notions allow us to define the Poisson bracket and the symplectic supermanifolds. It is easy to define the Moyal product: the Grassman algebra of the superfunctions becomes a Clifford algebra. We suppose that the observables of a quantum particle with a spin are superfunctions defined on a flat supermanifold. The study of these star algebras proves the existence of a spectral resolution for some particular elements. Consequently the equation of motion for an observable admits stationary solutions. The examples considered show that the spin splits a given state into two other ones.     

Geometric Analysis and Symbol Calculus: Fourier Transform Magnetic Resonance Imaging and Wavelets

Due to its unequalled advantages, the magnetic resonance imaging (MRI) modality has truly revolutionized the diagnosis and evaluation of pathology. Because many morphological anatomic details that may not be visualized by other high tech imaging methods can now be readily shown by diagnostic MRI, it has already become the standard modality by which all other clinical imaging techniques are measured. The unique quantum physical basis of the MRI modality combined with the imaging capabilities of current computer technology has made this imaging modality a target of interdisciplinary interest for clinicians, physicists, biologists, engineers, and mathematicians. Due to the fact that MRI scanners perform corticomorphic processing, this modality is by far more complex than all the other high tech clinical imaging techniques. The purpose of this paper is to outline a phase coherent wavelet approach to Fourier transform MRI. It is based on distributional harmonic analysis on the Heisenberg nilpotent Lie group G and the associated symplectically invariant symbol calculus of pseudodifferential operators. The contour and contrast resolution of MRI scans which is controlled by symplectic filter bank processing gives the noninvasive MRI modality superiority over X-ray computed tomography (CT) in soft tissue differentiation.