🍩 Database of Original & Non-Theoretical Uses of Topology

(found 4 matches in 0.001204s)

  1. Sheaves Are the Canonical Data Structure for Sensor Integration (2017)

    Michael Robinson
    Abstract A sensor integration framework should be sufficiently general to accurately represent many sensor modalities, and also be able to summarize information in a faithful way that emphasizes important, actionable information. Few approaches adequately address these two discordant requirements. The purpose of this expository paper is to explain why sheaves are the canonical data structure for sensor integration and how the mathematics of sheaves satisfies our two requirements. We outline some of the powerful inferential tools that are not available to other representational frameworks.

  2. Topological Differential Testing (2020)

    Kristopher Ambrose, Steve Huntsman, Michael Robinson, Matvey Yutin
    Abstract We introduce topological differential testing (TDT), an approach to extracting the consensus behavior of a set of programs on a corpus of inputs. TDT uses the topological notion of a simplicial complex (and implicitly draws on richer topological notions such as sheaves and persistence) to determine inputs that cause inconsistent behavior and in turn reveal \emph\de facto\ input specifications. We gently introduce TDT with a toy example before detailing its application to understanding the PDF file format from the behavior of various parsers. Finally, we discuss theoretical details and other possible applications.

  3. Positive Alexander Duality for Pursuit and Evasion (2017)

    Robert Ghrist, Sanjeevi Krishnan
    Abstract Considered is a class of pursuit-evasion games, in which an evader tries to avoid detection. Such games can be formulated as the search for sections to the complement of a coverage region in a Euclidean space over time. Prior results give homological criteria for evasion in the general case that are not necessary and sufficient. This paper provides a necessary and sufficient positive cohomological criterion for evasion in the general case. The principal tools are (1) a refinement of the Čech cohomology of a coverage region with a positive cone encoding spatial orientation, (2) a refinement of the Borel--Moore homology of the coverage gaps with a positive cone encoding time orientation, and (3) a positive variant of Alexander Duality. Positive cohomology decomposes as the global sections of a sheaf of local positive cohomology over the time axis; we show how this decomposition makes positive cohomology computable using techniques of computational polyhedral geometry and linear programming.

  4. A Sheaf and Topology Approach to Generating Local Branch Numbers in Digital Images (2020)

    Chuan-Shen Hu, Yu-Min Chung
    Abstract This paper concerns a theoretical approach that combines topological data analysis (TDA) and sheaf theory. Topological data analysis, a rising field in mathematics and computer science, concerns the shape of the data and has been proven effective in many scientific disciplines. Sheaf theory, a mathematics subject in algebraic geometry, provides a framework for describing the local consistency in geometric objects. Persistent homology (PH) is one of the main driving forces in TDA, and the idea is to track changes of geometric objects at different scales. The persistence diagram (PD) summarizes the information of PH in the form of a multi-set. While PD provides useful information about the underlying objects, it lacks fine relations about the local consistency of specific pairs of generators in PD, such as the merging relation between two connected components in the PH. The sheaf structure provides a novel point of view for describing the merging relation of local objects in PH. It is the goal of this paper to establish a theoretic framework that utilizes the sheaf theory to uncover finer information from the PH. We also show that the proposed theory can be applied to identify the branch numbers of local objects in digital images.