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Laplace Operator in Connection to Underlying Space Structure

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Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU 2022)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 1602))

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Abstract

Laplace operator is a diverse concept throughout natural sciences. It appears in many research areas and every such area defines it accordingly based on underlying domain and plans on follow-up applications. This operator attracts a lot of attention e.g. in signal and image processing applications [6]. However, signals, in general, can be defined not only on Euclidean domains such as regular grids (in case of images). There are cases when underlying space is considered to be e.g. a non-regular graph or even a manifold, but the Laplace operator is still closely bound to the space structure. Therefore, we investigated this operator from point of view of spaces, where distance may not be explicitly defined and thus is being replaced by more general, so-called, proximity. Our goal was to find such a representation, that would be simple for computations but at the same time applicable to more general domains, possibly to spaces without a notion of a classic distance. In this article, we will mention some of the various ways in which this operator can be introduced in relation to the corresponding space. Also, we will introduce the formula for the Laplace operator in the space whose structure is determined by a fuzzy partition [8]. And we will investigate the properties of this kind of representation in parallelisms to standard well-known versions.

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Acknowledgements

The authors thank the reviewers for their valuable comments and suggestions to improve the quality of the paper.

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Authors and Affiliations

  1. IRAFM, University of Ostrava, 30. dubna 22, 701 03, Ostrava, Czech Republic

    Hana Zámečníková & Irina Perfilieva

Authors
  1. Hana Zámečníková
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  2. Irina Perfilieva
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Corresponding author

Correspondence to Hana Zámečníková .

Editor information

Editors and Affiliations

  1. University of Milano-Bicocca, Milan, Italy

    Davide Ciucci

  2. University of Oviedo, Oviedo, Spain

    Inés Couso

  3. University of Cádiz, Cádiz, Spain

    Jesús Medina

  4. University of Warsaw, Warsaw, Poland

    Dominik Ślęzak

  5. University of Perugia, Perugia, Italy

    Davide Petturiti

  6. Sorbonne Université, Paris, France

    Bernadette Bouchon-Meunier

  7. Iona College, New Rochelle, NY, USA

    Ronald R. Yager

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Zámečníková, H., Perfilieva, I. (2022). Laplace Operator in Connection to Underlying Space Structure. In: Ciucci, D., et al. Information Processing and Management of Uncertainty in Knowledge-Based Systems. IPMU 2022. Communications in Computer and Information Science, vol 1602. Springer, Cham. https://doi.org/10.1007/978-3-031-08974-9_31

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  • DOI: https://doi.org/10.1007/978-3-031-08974-9_31

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-08973-2

  • Online ISBN: 978-3-031-08974-9

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