🍩 Database of Original & Non-Theoretical Uses of Topology

(found 2 matches in 0.001059s)
  1. Capturing Dynamics of Time-Varying Data via Topology (2020)

    Lu Xian, Henry Adams, Chad M. Topaz, Lori Ziegelmeier
    Abstract One approach to understanding complex data is to study its shape through the lens of algebraic topology. While the early development of topological data analysis focused primarily on static data, in recent years, theoretical and applied studies have turned to data that varies in time. A time-varying collection of metric spaces as formed, for example, by a moving school of fish or flock of birds, can contain a vast amount of information. There is often a need to simplify or summarize the dynamic behavior. We provide an introduction to topological summaries of time-varying metric spaces including vineyards [17], crocker plots [52], and multiparameter rank functions [34]. We then introduce a new tool to summarize time-varying metric spaces: a crocker stack. Crocker stacks are convenient for visualization, amenable to machine learning, and satisfy a desirable stability property which we prove. We demonstrate the utility of crocker stacks for a parameter identification task involving an influential model of biological aggregations [54]. Altogether, we aim to bring the broader applied mathematics community up-to-date on topological summaries of time-varying metric spaces.
  2. Blind Swarms for Coverage in 2-D (2005)

    V. D. Silva, R. Ghrist, A. Muhammad
    Abstract We consider coverage problems in robot sensor networks with minimal sensing capabilities. In particular, we demonstrate that a “blind” swarm of robots with no localization and only a weak form of distance estimation can rigorously determine coverage in a bounded planar domain of unknown size and shape. The methods we introduce come from algebraic topology. I. COVERAGE PROBLEMS Many of the potential applications of robot swarms require information about coverage in a given domain. For example, using a swarm of robot sensors for surveillance and security applications carries with it the charge to maximize, or, preferably, guarantee coverage. Such applications include networks of security cameras, mine field sweeping via networked robots [18], and oceanographic sampling [4]. In these contexts, each robot has some coverage domain, and one wishes to know about the union of these coverage domains. Such problems are also crucial in applications not involving robots directly, e.g., communication networks. As a preliminary analysis, we consider the static “field” coverage problem, in which robots are assumed stationary and the goal is to verify blanket coverage of a given domain. There is a large literature on this subject; see, e.g., [7], [1], [16]. In addition, there are variants on these problems involving “barrier” coverage to separate regions. Dynamic or “sweeping” coverage [3] is a common and challenging task with applications ranging from security to vacuuming. Although a sensor network composed of robots will have dynamic capabilities, we restrict attention in this brief paper to the static case in order to lay the groundwork for future inquiry. There are two primary approaches to static coverage problems in the literature. The first uses computational geometry tools applied to exact node coordinates. This typically involves ‘ruler-and-compass’ style geometry [10] or Delaunay triangulations of the domain [16], [14], [20]. Such approaches are very rigid with regards to inputs: one must know exact node coordinates and one must know the geometry of the domain precisely to determine the Delaunay complex. To alleviate the former requirement, many authors have turned to probabilistic tools. For example, in [13], the author assumes a randomly and uniformly distributed collection of nodes in a domain with a fixed geometry and proves expected area coverage. Other approaches [15], [19] give percolationtype results about coverage and network integrity for randomly distributed nodes. The drawback of these methods is the need for strong assumptions about the exact shape of the domain, as well as the need for a uniform distribution of nodes. In the sensor networks community, there is a compelling interest (and corresponding burgeoning literature) in determining properties of a network in which the nodes do not possess coordinate data. One example of a coordinate-free approach is in [17], which gives a heuristic method for geographic routing without coordinate data: among the large literature arising from this paper, we note in particular the mathematical analysis of this approach in [11]. To our knowledge, noone has treated the coverage problem in a coordinate-free setting. In this note, we introduce a new set of tools for answering coverage problems in robotics and sensor networks with minimal assumptions about domain geometry and node localization. We provide a sufficiency criterion for coverage. We do not answer the problem of how the nodes should be placed in order to maximize coverage, nor the minimum number of such nodes necessary; neither do we address how to reallocate nodes to fill coverage holes.