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.
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References
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
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
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)
Neto, A.F.: Carlitz’s identity for the Bernoulli numbers and zeon algebra. J. Integer Seq. 18, Article 15.5.6 (2015)
Neto, A.F., dos Anjos, P.H.R.: Zeon algebra and combinatorial identities. SIAM Rev. 56, 353–370 (2014)
Peng, B., Zhang, L., Zhang, D.: A survey of graph theoretical approaches to image segmentation. Pattern Recognit. 46, 1020–1038 (2013)
Schott, R., Staples, G.S.: Partitions and Clifford algebras. Eur. J. Comb. 29, 1133–1138 (2008)
Schott, R., Staples, G.S.: Operator Calculus on Graphs. Imperial College Press, London (2012)
Staples, G.S.: CliffMath: Clifford algebra computations in Mathematica, 2008–2018. http://www.siue.edu/~sstaple/index_files/research.htm
Staples, G.S.: A new adjacency matrix for finite graphs. Adv. Appl. Clifford Algebras 18, 979–991 (2008)
Staples, G.S.: Norms and generating functions in Clifford algebras. Adv. Appl. Clifford Algebras 18, 75–92 (2008)
Staples, G.S.: Hamiltonian cycle enumeration via fermion-zeon convolution. Int. J. Theor. Phys. 56, 3923–3934 (2017)
Staples, G.S.: Zeons, orthozeons, and processes on colored graphs. In: Proceedings of CGI ’17, Yokohama, Japan, June 27–30. ACM, New York (2017)
Staples, G.S., Weygandt, A.: Elementary functions and factorizations of zeons. Adv. Appl. Clifford Algebras 28, 12 (2018)
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
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The authors thank the anonymous reviewers for helpful comments.
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Communicated by Eckhard Hitzer
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Lindell, T., Staples, G.S. Norm Inequalities in Zeon Algebras. Adv. Appl. Clifford Algebras 29, 13 (2019). https://doi.org/10.1007/s00006-018-0934-z
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DOI: https://doi.org/10.1007/s00006-018-0934-z