🍩 Database of Original & Non-Theoretical Uses of Topology

(found 3 matches in 0.001304s)

  1. Characterizing Fluid Dynamical Systems Using Euler Characteristic Surface and Euler Metric (2023)

    A. Roy, R. A. I. Haque, A. J. Mitra, S. Tarafdar, T. Dutta
    Abstract Euler characteristic ( χ ), a topological invariant, helps to understand the topology of a network or complex. We demonstrate that the multi-scale topological information of dynamically evolving fluid flow systems can be crystallized into their Euler characteristic surfaces χ s ( r , t ). Furthermore, we demonstrate the Euler Metric (EM), introduced by the authors, can be utilized to identify the stability regime of a given flow pattern, besides distinguishing between different flow systems. The potential of the Euler characteristic surface and the Euler metric have been demonstrated first on analyzing a simulated deterministic dynamical system before being applied to analyze experimental flow patterns that develop in micrometer sized drying droplets.

  2. Euler Characteristic Surfaces: A Stable Multiscale Topological Summary of Time Series Data (2024)

    Anamika Roy, Atish J. Mitra, Tapati Dutta
    Abstract We present Euler Characteristic Surfaces as a multiscale spatiotemporal topological summary of time series data encapsulating the topology of the system at different time instants and length scales. Euler Characteristic Surfaces with an appropriate metric is used to quantify stability and locate critical changes in a dynamical system with respect to variations in a parameter, while being substantially computationally cheaper than available alternate methods such as persistent homology. The stability of the construction is demonstrated by a quantitative comparison bound with persistent homology, and a quantitative stability bound under small changes in time is established. The proposed construction is used to analyze two different kinds of simulated disordered flow situations.

  3. Understanding Flow Features in Drying Droplets via Euler Characteristic Surfaces—A Topological Tool (2020)

    A. Roy, R. A. I. Haque, A. J. Mitra, M. Dutta Choudhury, S. Tarafdar, T. Dutta
    Abstract In this paper, we propose a mathematical picture of flow in a drying multiphase droplet. The system studied consists of a suspension of microscopic polystyrene beads in water. The time development of the drying process is described by defining the “Euler characteristic surface,” which provides a multiscale topological map of this dynamical system. A novel method is adopted to analyze the images extracted from experimental video sequences. Experimental image data are converted to binary data through appropriate Gaussian filters and optimal thresholding and analyzed using the Euler characteristic determined on a hexagonal lattice. In order to do a multiscale analysis of the extracted image, we introduce the concept of Euler characteristic at a specific scale r > 0. This multiscale time evolution of the connectivity information on aggregates of polysterene beads in water is summarized in a Euler characteristic surface and, subsequently, in a Euler characteristic level curve plot. We introduce a metric between Euler characteristic surfaces as a possible similarity measure between two flow situations. The constructions proposed by us are used to interpret flow patterns (and their stability) generated on the upper surface of the drying droplet interface. The philosophy behind the topological tools developed in this work is to produce low-dimensional signatures of dynamical systems, which may be used to efficiently summarize and distinguish topological information in various types of flow situations.