Skip to main content
SpringerLink
Log in

Weyl Calculus Perspective on the Discrete Stokes’ Formula in Octonions

  • Conference paper
  • First Online:
Advances in Computer Graphics (CGI 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14498))

Included in the following conference series:

Abstract

Octonions are 8-dimensional hypercomplex numbers which form the biggest normed division algebras over the real numbers. Motivated by applications in theoretical physics, continuous octonionic analysis has become an area of active research in recent year. Looking at possible practical applications, it is beneficial to work directly with discrete structures, rather than approximate continuous objects. Therefore, in previous papers, we have proposed some ideas towards the discrete octonionic analysis. It is well known, that there are several possibilities to discretise the continuous setting, and the Weyl calculus approach, which is typically used in the discrete Clifford analysis, to octonions has not been studied yet. Therefore, in this paper, we close this gap by presenting the discretisation of octonionic analysis based on the Weyl calculus.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 59.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 74.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Baez, J.: The octonions. Bull. Am. Math. Soc. 39, 145–205 (2002)

    Article  MathSciNet  Google Scholar 

  2. Brackx, F., Delanghe, R., Sommen, F.: Clifford analysis. In: Pitman Research Notes in Mathematics, vol. 76. Boston (1982)

    Google Scholar 

  3. Brackx, F., De Schepper, H., Sommen, F., Van de Voorde, L.: Discrete Clifford analysis: a germ of function theory. In: Hypercomplex Analysis, Birkhäuser Basel, pp. 37–53 (2009)

    Google Scholar 

  4. Burdik, C., Catto, S., Gürcan, Y., Khalfan, A., Kurt, L., La Kato, V.: \(SO(9,1)\) group and examples of analytic functions. J. Phys.: Conf. Ser. 1194, 012016 (2019)

    Google Scholar 

  5. Cerejeiras, P., Faustino, N., Vieira, N.: Numerical Clifford analysis for nonlinear Schrödinger problem. Numer. Methods Partial Differ. Eq. 24(4), 1181–1202 (2008)

    Article  Google Scholar 

  6. Cerejeiras, P., Kähler, U., Ku, M., Sommen, F.: Discrete Hardy spaces. J. Fourier Anal. Appl. 20(4), 715–750 (2014)

    Article  MathSciNet  Google Scholar 

  7. Cerejeiras, P., Kähler, U., Legatiuk, A., Legatiuk, D.: Boundary values of discrete monogenic functions over bounded domains in \(\mathbbm {R}^{3}\). In: Alpay, D., Vajiac, M. (eds.) Linear Systems, Signal Processing and Hypercomplex Analysis. Operator Theory: Advances and Applications, vol. 275, pp. 149–165. Birkhäuser (2020)

    Google Scholar 

  8. Cerejeiras, P., Kähler, U., Legatiuk, A., Legatiuk, D.: Discrete Hardy spaces for bounded domains in \(\mathbbm {R}^{n}\). Complex Anal. Oper. Theory 15, 4 (2021)

    Article  Google Scholar 

  9. Constales, D., Kraußhar, R.S.: Octonionic Kerzman-Stein operators. Complex Anal. Oper. Theory 15, 104 (2021)

    Article  MathSciNet  Google Scholar 

  10. Faustino, N., Kähler, U.: Fischer decomposition for difference Dirac operators. Adv. Appl. Clifford Algebras 17, 37–58 (2007)

    Article  MathSciNet  Google Scholar 

  11. Faustino, N., Kähler, U., Sommen, F.: Discrete Dirac operators in Clifford analysis. Adv. Appl. Clifford Algebras 17(3), 451–467 (2007)

    Article  MathSciNet  Google Scholar 

  12. Frenod, E., Ludkowski, S.V.: Integral operator approach over octonions to solution of nonlinear PDE. Far East J. Math. Sci. (2017)

    Google Scholar 

  13. Gogberashvili, M.: Octonionic geometry and conformal transformations. Int. J. Geomet. Methods Mod. Phys. 13(7), 1650092 (2016)

    Article  MathSciNet  Google Scholar 

  14. Gürlebeck, K., Sprößig, W.: Quaternionic and Clifford Calculus for Physicists and Engineers. John Wiley & Sons, Chichester, New York (1997)

    Google Scholar 

  15. Gürlebeck, K., Hommel, A.: On finite difference Dirac operators and their fundamental solutions. Adv. Appl. Clifford Algebras 11, 89–106 (2001)

    Article  MathSciNet  Google Scholar 

  16. Gürlebeck, K., Habetha, K., Sprößig, W.: Application of Holomorphic Functions in Two and Higher Dimensions. Birkhäuser, Basel (2016)

    Book  Google Scholar 

  17. Kauhanen, J., Orelma, H.: Cauchy-Riemann operators in octonionic analysis. Adv. Appl. Clifford Algebras 28, 1 (2018)

    Article  MathSciNet  Google Scholar 

  18. Kauhanen, J., Orelma, H.: On the structure of octonion regular functions. Adv. Appl. Clifford Algebras 29, 77 (2019)

    Article  MathSciNet  Google Scholar 

  19. Kraußhar, S., Legatiuk, A., Legatiuk, D.: Towards discrete octonionic analysis. In: Vasilyev, V. (eds.) Differential Equations, Mathematical Modeling and Computational Algorithms. DEMMCA 2021. Springer Proceedings in Mathematics and Statistics, vol. 423, pp. 51–63. Springer, Cham (2023). https://doi.org/10.1007/978-3-031-28505-9_4

  20. Kraußhar, S., Legatiuk, D.: Cauchy formulae and Hardy spaces in discrete octonionic analysis. In: Complex Analysis and Operator Theory (2023). (Accepted for Publication)

    Google Scholar 

  21. Lasenby, A.N.: Some recent GA results in mathematical physics and the GA approach to the fundamental forces of nature. Presentation at AGACSE 2021, YouTube Video (2021). https://www.youtube.com/watch?v=fFj4E7q4hbY

  22. Najarbashi, G., Seifi, B., Mirzaei, S.: Two- and three-qubit geometry, quaternionic and octonionic conformal maps, and intertwining stereographic projection. Quant. Inf. Process. 15, 509–528 (2016)

    Article  MathSciNet  Google Scholar 

  23. Huo, Q., Ren, Q.: Structure of octonionic Hilbert spaces with applications in the Parseval equality and Cayley-Dickson algebras. J. Math. Phys. 63(4), 042101 (2022)

    Article  MathSciNet  Google Scholar 

  24. Li, X.-M., Peng, L.-Z.: Three-line theorems on the octonions. Acta Math. Sinica 20(3), 483–490 (2004)

    Google Scholar 

  25. Li, X.-M., Peng, L.-Z., Qian, T.: Cauchy integrals on Lipschitz surfaces in octonionic space. J. Math. Anal. Appl. 343, 763–777 (2008)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Erfurt, Chair of Mathematics, Nordhäuser Str. 63, 99089, Erfurt, Germany

    Rolf Sören Kraußhar & Dmitrii Legatiuk

Authors
  1. Rolf Sören Kraußhar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dmitrii Legatiuk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dmitrii Legatiuk .

Editor information

Editors and Affiliations

  1. Shanghai Jiao Tong University, Shanghai, China

    Bin Sheng

  2. Shanghai Jiao Tong University, Shanghai, China

    Lei Bi

  3. University of Sydney, Sydney, NSW, Australia

    Jinman Kim

  4. MIRALab-CUI, University of Geneve, Carouge, Geneve, Switzerland

    Nadia Magnenat-Thalmann

  5. Swiss federal Institute of Technology, Lausanne, Switzerland

    Daniel Thalmann

Rights and permissions

Reprints and permissions

Copyright information

© 2024 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Kraußhar, R.S., Legatiuk, D. (2024). Weyl Calculus Perspective on the Discrete Stokes’ Formula in Octonions. In: Sheng, B., Bi, L., Kim, J., Magnenat-Thalmann, N., Thalmann, D. (eds) Advances in Computer Graphics. CGI 2023. Lecture Notes in Computer Science, vol 14498. Springer, Cham. https://doi.org/10.1007/978-3-031-50078-7_29

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-50078-7_29

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-50077-0

  • Online ISBN: 978-3-031-50078-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation