Extension of One-Dimensional Proximity Regions to Higher Dimensions

Proximity maps and regions are defined based on the relative allocation of points from two or more classes in an area of interest and are used to construct random graphs called proximity catch digraphs (PCDs) which have applications in various fields. The simplest of such maps is the spherical proximity map which maps a point from the class of interest to a disk centered at the same point with radius being the distance to the closest point from the other class in the region. The spherical proximity map gave rise to class cover catch digraph (CCCD) which was applied to pattern classification. Furthermore for uniform data on the real line, the exact and asymptotic distribution of the domination number of CCCDs were analytically available. In this article, we determine some appealing properties of the spherical proximity map in compact intervals on the real line and use these properties as a guideline for defining new proximity maps in higher dimensions. Delaunay triangulation is used to partition the region of interest in higher dimensions. Furthermore, we introduce the auxiliary tools used for the construction of the new proximity maps, as well as some related concepts that will be used in the investigation and comparison of them and the resulting graphs. We characterize the geometry invariance of PCDs for uniform data. We also provide some newly defined proximity maps in higher dimensions as illustrative examples

Perturbation of the Lyapunov spectra of periodic orbits

We describe all Lyapunov spectra that can be obtained by perturbing the derivatives along periodic orbits of a diffeomorphism. The description is expressed in terms of the finest dominated splitting and Lyapunov exponents that appear in the limit of a sequence of periodic orbits, and involves the majorization partial order. Among the applications, we give a simple criterion for the occurrence of universal dynamics.Comment: A few improvements were made, based on the referee's suggestion

A Hierachical Evolutionary Algorithm for Multiobjective Optimization in IMRT

Purpose: Current inverse planning methods for IMRT are limited because they are not designed to explore the trade-offs between the competing objectives between the tumor and normal tissues. Our goal was to develop an efficient multiobjective optimization algorithm that was flexible enough to handle any form of objective function and that resulted in a set of Pareto optimal plans. Methods: We developed a hierarchical evolutionary multiobjective algorithm designed to quickly generate a diverse Pareto optimal set of IMRT plans that meet all clinical constraints and reflect the trade-offs in the plans. The top level of the hierarchical algorithm is a multiobjective evolutionary algorithm (MOEA). The genes of the individuals generated in the MOEA are the parameters that define the penalty function minimized during an accelerated deterministic IMRT optimization that represents the bottom level of the hierarchy. The MOEA incorporates clinical criteria to restrict the search space through protocol objectives and then uses Pareto optimality among the fitness objectives to select individuals. Results: Acceleration techniques implemented on both levels of the hierarchical algorithm resulted in short, practical runtimes for optimizations. The MOEA improvements were evaluated for example prostate cases with one target and two OARs. The modified MOEA dominated 11.3% of plans using a standard genetic algorithm package. By implementing domination advantage and protocol objectives, small diverse populations of clinically acceptable plans that were only dominated 0.2% by the Pareto front could be generated in a fraction of an hour. Conclusions: Our MOEA produces a diverse Pareto optimal set of plans that meet all dosimetric protocol criteria in a feasible amount of time. It optimizes not only beamlet intensities but also objective function parameters on a patient-specific basis