🍩 Database of Original & Non-Theoretical Uses of Topology
(found 3 matches in 0.001729s)
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Statistical Topology of Bond Networks With Applications to Silica (2020)
B. Schweinhart, D. Rodney, J. K. MasonAbstract
Whereas knowledge of a crystalline material's unit cell is fundamental to understanding the material's properties and behavior, there are no obvious analogs to unit cells for disordered materials despite the frequent existence of considerable medium-range order. This article views a material's structure as a collection of local atomic environments that are sampled from some underlying probability distribution of such environments, with the advantage of offering a unified description of both ordered and disordered materials. Crystalline materials can then be regarded as special cases where the underlying probability distribution is highly concentrated around the traditional unit cell. The 𝐻1 barcode is proposed as a descriptor of local atomic environments suitable for disordered bond networks and is applied with three other descriptors to molecular dynamics simulations of silica glasses. Each descriptor reliably distinguishes the structure of glasses produced at different cooling rates, with the 𝐻1 barcode and coordination profile providing the best separation. The approach is generally applicable to any system that can be represented as a sparse graph.Community Resources
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Multivariate Data Analysis Using Persistence-Based Filtering and Topological Signatures (2012)
B. Rieck, H. Mara, H. LeitteAbstract
The extraction of significant structures in arbitrary high-dimensional data sets is a challenging task. Moreover, classifying data points as noise in order to reduce a data set bears special relevance for many application domains. Standard methods such as clustering serve to reduce problem complexity by providing the user with classes of similar entities. However, they usually do not highlight relations between different entities and require a stopping criterion, e.g. the number of clusters to be detected. In this paper, we present a visualization pipeline based on recent advancements in algebraic topology. More precisely, we employ methods from persistent homology that enable topological data analysis on high-dimensional data sets. Our pipeline inherently copes with noisy data and data sets of arbitrary dimensions. It extracts central structures of a data set in a hierarchical manner by using a persistence-based filtering algorithm that is theoretically well-founded. We furthermore introduce persistence rings, a novel visualization technique for a class of topological features-the persistence intervals-of large data sets. Persistence rings provide a unique topological signature of a data set, which helps in recognizing similarities. In addition, we provide interactive visualization techniques that assist the user in evaluating the parameter space of our method in order to extract relevant structures. We describe and evaluate our analysis pipeline by means of two very distinct classes of data sets: First, a class of synthetic data sets containing topological objects is employed to highlight the interaction capabilities of our method. Second, in order to affirm the utility of our technique, we analyse a class of high-dimensional real-world data sets arising from current research in cultural heritage.