Computing Homological Information Based on Directed Graphs within Discrete Objects

A. Gonzalez-Lorenzo, A. Bac, J. Mari, P. Real
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引用次数: 3

Abstract

N-dimensional discrete objects can be interpreted as cubical complexes which are suitable for the study of their homology groups in order to understand the original discrete object. The classic approach consists in computing the Normal Smith Form of some matrices associated to the cubical complex. Further approaches deal mainly with a pre-processing of the matrices in order to reduce their size. In this paper we propose a new approach, initially based on Discrete Morse Theory, which computes some homological information (Betti numbers and representative cycles) without calculating the Normal Smith Form. It works on any dimension, and it can also be applied to any kind of regular cell complex.
基于离散对象内有向图的同调信息计算
n维离散对象可以被解释为立方体复合体,适合于研究它们的同调群,以便理解原来的离散对象。经典的方法包括计算与立方复合体相关的一些矩阵的标准史密斯形式。进一步的方法主要是处理矩阵的预处理,以减少它们的大小。在本文中,我们提出了一种新的方法,该方法最初基于离散莫尔斯理论,计算一些同调信息(Betti数和代表性循环)而不计算标准史密斯形式。它适用于任何维度,也可以应用于任何类型的常规细胞复合体。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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