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Norm Inequalities in Zeon Algebras

  • Theresa Lindell
  • G. Stacey StaplesEmail author
Article
  • 25 Downloads

Abstract

The zeon (“nil-Clifford”) algebra \({\mathcal {C}\ell _n}^\mathrm{nil}\) can be thought of as a commutative analogue of the n-particle fermion algebra and can be constructed as a subalgebra of a Clifford algebra. Combinatorial properties of the algebra make it useful for applications in graph theory and theoretical computer science. In this paper, the zeon p-norms and infinity norms are introduced. The 1-norm is shown to be the only sub-multiplicative p-norm on zeon algebras. Multiplicative inequalities involving the infinity norm (which is not sub-multiplicative) are developed and equivalence of norms in \({\mathcal {C}\ell _n}^\mathrm{nil}\) is used to establish a number of multiplicative inequalities between p-norms and the infinity norm. As an application of norm inequalities, necessary and sufficient conditions for convergence of the zeon geometric series are established, and the series limit is expressed as a finite sum. The exposition is supplemented by a number of examples computed using Mathematica.

Keywords

Clifford algebra Zeons Norms Inequalities Series 

Mathematics Subject Classification

15A66 81R05 05A20 

Notes

Acknowledgements

The authors thank the anonymous reviewers for helpful comments.

References

  1. 1.
    Dollar, L.M., Staples, G.S.: Zeon roots. Adv. Appl. Clifford Algebras 27, 1133–1145 (2017).  https://doi.org/10.1007/s00006-016-0732-4 MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Feinsilver, P., McSorley, J.: Zeons, permanents, the Johnson scheme, and generalized derangements. Int. J. Comb. Article ID 539030, 29 (2011).  https://doi.org/10.1155/2011/539030
  3. 3.
    Neto, A.F.: Higher order derivatives of trigonometric functions, Stirling numbers of the second kind, and zeon algebra. J. Integer Seq. 17 Article 14.9.3 (2014)Google Scholar
  4. 4.
    Neto, A.F.: Carlitz’s identity for the Bernoulli numbers and zeon algebra. J. Integer Seq. 18, Article 15.5.6 (2015)Google Scholar
  5. 5.
    Neto, A.F., dos Anjos, P.H.R.: Zeon algebra and combinatorial identities. SIAM Rev. 56, 353–370 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Peng, B., Zhang, L., Zhang, D.: A survey of graph theoretical approaches to image segmentation. Pattern Recognit. 46, 1020–1038 (2013)CrossRefGoogle Scholar
  7. 7.
    Schott, R., Staples, G.S.: Partitions and Clifford algebras. Eur. J. Comb. 29, 1133–1138 (2008)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Schott, R., Staples, G.S.: Operator Calculus on Graphs. Imperial College Press, London (2012)CrossRefGoogle Scholar
  9. 9.
    Staples, G.S.: CliffMath: Clifford algebra computations in Mathematica, 2008–2018. http://www.siue.edu/~sstaple/index_files/research.htm
  10. 10.
    Staples, G.S.: A new adjacency matrix for finite graphs. Adv. Appl. Clifford Algebras 18, 979–991 (2008)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Staples, G.S.: Norms and generating functions in Clifford algebras. Adv. Appl. Clifford Algebras 18, 75–92 (2008)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Staples, G.S.: Hamiltonian cycle enumeration via fermion-zeon convolution. Int. J. Theor. Phys. 56, 3923–3934 (2017)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Staples, G.S.: Zeons, orthozeons, and processes on colored graphs. In: Proceedings of CGI ’17, Yokohama, Japan, June 27–30. ACM, New York (2017)Google Scholar
  14. 14.
    Staples, G.S., Weygandt, A.: Elementary functions and factorizations of zeons. Adv. Appl. Clifford Algebras 28, 12 (2018)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Weisstein, E.W.: Stirling Number of the Second Kind. From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/StirlingNumberoftheSecondKind.html. Accessed 13 Mar 2018

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsSouthern Illinois University EdwardsvilleEdwardsvilleUSA

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