China 
Africa 
Explore chapters and articles related to this topic
Topology
Published in Paul L. Goethals, Natalie M. Scala, Daniel T. Bennett, Mathematics in Cyber Research, 2022
Steve Huntsman, Jimmy Palladino, Michael Robinson
We have only scratched the surface of topological techniques that can be fruitfully applied to problems in the cyber domain. Discrete Morse theory (Scoville, 2019), the algebraic topology of finite topological spaces (Barmak, 2011), and connections between simplicial complexes and partially ordered sets (Wachs, 2007) provide just a few opportunities for applications that we have not discussed at all here. For example, a notion of a weighted Dowker complex and an associated partial order can be used for topological differential testing to discover files that similar programs handle inconsistently (Ambrose et al., 2020). Another emerging topological tool for the analysis of enriched categories (such as furnished by sub-flow graphs in the sense of Huntsman (2019) or by generalized metric spaces including digraphs) is the theory of magnitude (co)homology (Leinster & Shulman, 2017; Hepworth, 2018).
Topological Analysis of Local Structure in Atomic Systems
Published in Jeffrey P. Simmons, Lawrence F. Drummy, Charles A. Bouman, Marc De Graef, Statistical Methods for Materials Science, 2019
Emanuel A. Lazar, David J. Srolovitz
As topology focuses on studying the connectivity of shapes such as spheres and tori, it is not immediately clear that it would have much relevance to studying sets of discrete points, such as those encountered in studying atomic systems. In what meaningful way can points in space be considered connected? Over the last decade or so, however, powerful tools such as discrete Morse theory and persistent homology [288, 1134] have been developed to analyze data of diverse kinds [360]. Voronoi topology continues in this spirit. In what follows we show how considering the topology of a Voronoi cell can provide keen insight into the manner in which a set of points is arranged in space. In this sense, Voronoi topology forms a bridge between the discrete and continuous, and enables the application of ideas from topology to the study of atomic systems.
Topological machine learning for multivariate time series
Published in Journal of Experimental & Theoretical Artificial Intelligence, 2022
Chengyuan Wu, Carol Anne Hargreaves
Though the term ‘topology’ can be used to refer to a wide array of subjects, the topological tools used in TDA generally refer to algebraic topology (Letscher, 2012; Wu et al., 2020b), or to be specific persistent homology (Edelsbrunner & Morozov, 2012; Ghrist, 2008). Broadly speaking, persistent homology analyzes the ‘shape’ of the data to deduce the intrinsic properties of the data. Other prominent tools in TDA include Mapper (Ray & Trovati, 2017; Singh et al., 2007) and discrete Morse theory (Forman, 1998, 2002; Wu et al., 2020a). Due to the fact that TDA works quite differently from most other data analysis techniques, it can sometimes detect features that are missed by traditional methods of analysis (Nicolau et al., 2011).