This paper describes a new algorithm for constructing the set of free bitangents of a collection of n disjoint convex obstacles of constant complexity. The algorithm runs in time O(nlogn + k), where k is the output size, and uses O(n) space. While earlier algorithms achieve the same optimal running time, this is the first optimal algorithm that uses only linear space. The visibility graph or the visibility complex can be computed in the same time and space. The only complicated data structure used by the algorithm is a splittable queue, which can be implemented easily using red-black trees. The algorithm is conceptually very simple, and should therefore be easy to implement and quite fast in practice. The algorithm relies on greedy pseudotriangulations, which are subgraphs of the visibility graph with many nice combinatorial properties. These properties, and thus the correctness of the algorithm, are partially derived from properties of a certain partial order on the faces of the visibility complex.
We introduce a multiscale topological description of the Megaparsec weblike cosmic matter distribution. Betti numbers and topological persistence offer a powerful means of describing the rich connectivity structure of the cosmic web and of its multiscale arrangement of matter and galaxies. Emanating from algebraic topology and Morse theory, Betti numbers and persistence diagrams represent an extension and deepening of the cosmologically familiar topological genus measure, and the related geometric Minkowski functionals. In addition to a description of the mathematical background, this study presents the computational procedure for computing Betti numbers and persistence diagrams for density field filtrations. The field may be computed starting from a discrete spatial distribution of galaxies or simulation particles. The main emphasis of this study concerns an extensive and systematic exploration of the imprint of different weblike morphologies and different levels of multiscale clustering in the corresponding computed Betti numbers and persistence diagrams. To this end, we use Voronoi clustering models as templates for a rich variety of weblike configurations, and the fractal-like Soneira-Peebles models exemplify a range of multiscale configurations. We have identified the clear imprint of cluster nodes, filaments, walls, and voids in persistence diagrams, along with that of the nested hierarchy of structures in multiscale point distributions. We conclude by outlining the potential of persistent topology for understanding the connectivity structure of the cosmic web, in large simulations of cosmic structure formation and in the challenging context of the observed galaxy distribution in large galaxy surveys.
Setting of the Problem a. IntroductionThis paper is concerned with a parametrically forced oscillator, i.e. an oscillator with periodically varying coefficients, near resonance. There exists a vast amount of literature on this subject, especially on the linear case which involves differential equations of Mathieu-type. The present study deals with the nonlinear problem, localized near a central equilibrium solution of the free oscillator. In particular the branching off of certain subharmonic periodic solutions is studied, and more general bifurcational aspects of this.The motivating example is the pendulum near its lower equilibrium, the length of which is varied periodically with small amplitude. Both this amplitude and the length of the pendulum then are considered as parameters of the problem. However, the present approach asks for a somewhat more general set-up. In our setting both the potential function, near a minimum, and the periodic para.n;.tetric forcing will be chosen in a more or less arbitrary way. In fact, while restricting to this universe of function-pairs, the bifurcation problem is studied from the generic point of view. This means that our interest is with phenomena that are persistent under small perturbation of these functions. In this universe we shall distinguish two sub-universes determined by reflectional symmetries of the functions.As a consequence this study restricts to the Hamiltonian context, while certain reversibilities are optional. A question is whether genericity in the above restricted sense also leads to persistence in this wider, i.e. general Hamiltonian, setting. It will be shown that to a large extent this is indeed the case. On the other hand, we deal with small perturbations of autonomous, or timeindependent, systems that are degenerate in the more general context and this leads to some discrepancy between the two kinds of genericity.A more precise and formal introduction of these contexts will be given in part c of this section, see below.Our approach is the following. We first give a more elaborate sketch of our problem using the classical example of the pendulum near its lower equilibrium.Next we introduce our more general class of systems. We start out by applying a normal form procedure leading to an approximation that is time-independent, or autonomous, to any order. This procedure respects the symplectic structure as well as the reversibilities involved, due to an appropriate formulation in Liealgebra terms. Thus, up to fiat "perturbation" terms, the problem reduces to the investigation of an autonomous Hamiltonian system on the plane, which may or may not display certain discrete symmetries. The main ingredient of this reduced analysis is equivariant singularity theory. At the end we discuss the possible effects of the re-incorporation of the fiat terms. b. The Parametrically Forced Pendulum, MotivationTo illustrate our ideas we briefiy describe the planar (mathematical) pendulum with parametric forcing. For more details e.g. compare [Sto,MS,Ar1( §25)]. The free ...
Abstract.A specific singularity of a vector field on R 3 is considered, of codimension 2 in the dissipative case and of codimension 1 in the conservative case. In both contexts in generic unfoldings the existence is proved of subordinate Sil'nikov bifurcations, which have codimension 1. Special attention is paid to the C°°-flatness of this subordinate phenomenon.
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