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Point-shift foliation of a point process
, Article Electronic Journal of Probability ; Volume 23 , 2018 ; 10836489 (ISSN) ; Haji Mirsadeghi, M. O ; Sharif University of Technology
University of Washington
2018
Abstract
A point-shift F maps each point of a point process Φ to some point of Φ. For all translation invariant point-shifts F, the F-foliation of Φ is a partition of the support of Φ which is the discrete analogue of the stable manifold of F on Φ. It is first shown that foliations lead to a classification of the behavior of point-shifts on point processes. Both qualitative and quantitative properties of foliations are then established. It is shown that for all point-shifts F, there exists a point-shift F⊥, the orbits of which are the F-foils of Φ, and which is measure-preserving. The foils are not always stationary point processes. Nevertheless, they admit relative intensities with respect to one...
Numerical Methods for Approximation and Visualization of Invariant Manifolds in Dynamical Systems
, M.Sc. Thesis Sharif University of Technology ; Razvan, Mohammad Reza (Supervisor)
Abstract
Invariant manifolds are important objects in the theory of dynamical systems. The stable manifold theorem is a very important theorem about this concept which proves the existence of stable and unstable manifolds in a wide range of dynamical systems. The importance of invariant manifolds encourages us to view their pictures. It helps us to understand their bahavior. For this purpose, at first we need to approximate the invariant manifold we want to visualize. There are several algorithms designed to approximate invariant manifolds. Those algorithms approximate a set of points on an invariant manifold and then provide an approximation of the manifold by the calculated points. Visualizing an...