Isotopic families of contact manifolds for elliptic PDE

A test for a function to be a solution of an elliptic PDE is given in terms of extensions, as solutions, from the boundaries inside the domains belonging to an isotopic family. It generalizes a result of Ehrenpreis for spheres moved along a straight line.

     

On a poisson reduction for Gel'fand-Zakharevich manifolds

We formulate and discuss a reduction theorem for Poisson pencils associated with a class of integrable systems, defined on bi-Hamiltonian manifolds, recently studied by Gel'fand and Zakharevich. The reduction procedure is suggested by the bi-Hamiltonian approach to the separation of variables problem.     

The pure state space of quantum mechanics as Hermitian symmetric space

The pure state space of Quantum Mechanics is investigated as Hermitian Symmetric Kähler manifold. The classical principles of quantum mechanics (Quantum Superposition Principle, Heisenberg Uncertainty Principle, Quantum Probability Principle) and Spectral Theory of observables are discussed in this non-linear geometrical context.     

Restrictions of Quadratic Forms to Lagrangian Planes,Quadratic Matrix Equations,and Gyroscopic Stabilization

We discuss the symplectic geometry of linear Hamiltonian systems with nondegenerate Hamiltonians. These systems can be reduced to linear second-order differential equations characteristic of linear oscillation theory. This reduction is related to the problem on the signatures of restrictions of quadratic forms to Lagrangian planes. We study vortex symplectic planes invariant with respect to linear Hamiltonian systems. These planes are determined by the solutions of quadratic matrix equations of a special form. New conditions for gyroscopic stabilization are found.     

A test problem for molecular dynamics integrators

We derive a test problem for evaluating the ability of time-steppingmethods to preserve the statistical properties of systems inmolecular dynamics. We consider a family of deterministic systemsconsisting of a finite number of particles interacting on acompact interval. The particles are given random initial conditionsand interact through instantaneous energy- and momentum-conservingcollisions. As the number of particles, the particle density,and the mean particle speed go to infinity, the trajectory ofa tracer particle is shown to converge to a stationary Gaussianstochastic process. We approximate this system by one describedby a system of ordinary differential equations and provide numericalevidence that it converges to the same stochastic process. Wesimulate the latter system with a variety of numerical integrators,including the symplectic Euler method, a fourth-order Runge-Kuttamethod, and an energyconserving step-and-project method. Weassess the methods' ability to recapture the system's limitingstatistics and observe that symplectic Euler performs significantlybetter than the others for comparable computational expense.     

Laguerre approximation of stable manifolds with application to connecting orbits

We present an algorithm, based on approximation by Laguerre polynomials, for computing a point on the stable manifold of a stationary solution of an autonomous system. A superconvergence phenomenon means that the accuracy of our results is much higher than the usual spectral accuracy. Both the theory and the implementation of the method are considered. Finally, as an application of the algorithm, we describe a fully spectral approximation of homo- and heteroclinic orbits.

     

Carnot-Caratheodory空间中的变换论

Klein发表著名的埃尔兰根纲领,由群论角度研究了空间变换群的不变量,从而引进了各种不同的几何学．本文利用Felix Klein的观念,研究Carnot-Caratheodory空间{M,Q,g}(又称为次黎曼流形)上的类似问题,给出了次黎曼流形中的共形不变量和射影不变量．本文给出的共形和射影不变量可视为黎曼情形的一种自然推广．由于次黎曼流形与黎曼流形之间有着本质的差异,故此,本文通过次黎曼流形上存在的唯一非完整联络(Nonholonomic connections)来刻画所提的问题．