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Fuzzy Octonion Numbers: Some Analytical Properties

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Fuzzy Techniques: Theory and Applications (IFSA/NAFIPS 2019 2019)

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 1000))

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Abstract

An algebric result states that the sets of real, complex, quaternion and octonion numbers are the only normed division algebras. The mathematical analysis results concerning fuzzy and interval real, complex and quaternion numbers have been extensively studied in the literature. Thereby, it is natural to explore and study the fuzzy and interval extension for the last normed division algebra: the octonions. In this manuscript we show and discuss some important concepts and properties of mathematical analysis with respect to fuzzy and interval octonion numbers, namely, the existence of partial orders, metrics, supremum and infimun and limit of sequences of fuzzy octonion numbers. This work complete the picture of the study of the fuzzy mathematical analysis related to the normed division algebras since the sedenions are not an integral domain.

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References

  1. Boche, R.E.: Complex interval arithmetic with some applications. Technical report Number LMSC4-22-66-1, Lockheed General Research Program (1966)

    Google Scholar 

  2. Gargantini, I., Henrici, P.: Circular arithmetic and the determination of polynomial zeros. Numer. Math. 18(4), 305–320 (1972)

    Article  MathSciNet  Google Scholar 

  3. Renjun, L., Shaoqing, Y., Baowen, L., Weihai, F.: Fuzzy complex numbers. BUSEFAL 25, 79–86 (1986)

    Google Scholar 

  4. Buckley, J.J.: Fuzzy complex numbers. Fuzzy Sets Syst. 33(3), 333–345 (1989)

    Article  MathSciNet  Google Scholar 

  5. Zhang, G.-Q.: Fuzzy limit theory of fuzzy complex numbers. Fuzzy Sets Syst. 46(2), 227–235 (1992)

    Article  MathSciNet  Google Scholar 

  6. Moura, R.P.A., Bergamashi, F.B., Santiago, R.H.N., Bedregal, B.R.C.: Fuzzy quaternion numbers. In: 2013 IEEE International Conference on Fuzzy Systems (FUZZ - IEEE). IEEE (2013). https://doi.org/10.1109/fuzz-ieee.2013.6622400

  7. Schafer, R.D.: Introduction to Non-associative Algebras. Dover, New York (1995)

    Google Scholar 

  8. Baez, J.C.: The octonions. Bull. Am. Math. Soc. 39, 145–205 (2002). Corrected Version: math.RA/0105155. https://doi.org/10.1090/S0273-0979-01-00934-X

    Article  MathSciNet  Google Scholar 

  9. Shen, M., Wang, R.: SA new singular value decomposition algorithm for octonion signal. In: 2018 24th International Conference on Pattern Recognition (ICPR), pp. 3233–3237. IEEE (2018)

    Google Scholar 

  10. Yasuaki, K., Hitoshi, L.: SA model of hopfield-type octonion neural networks and existing conditions of energy functions. In: 2016 International Joint Conference on Neural Networks (IJCNN), pp. 44264–44430. IEEE (2016)

    Google Scholar 

  11. Guang-Quan, Z.: Fuzzy limit theory of fuzzy numbers. Cybern. Syst. (World Scientific Publishing) 90, 163–170 (1990)

    MathSciNet  Google Scholar 

  12. Bede, B.: Mathematics of Fuzzy Sets and Fuzzy Logic. STUDFUZZ. Springer, Berlin (2013). https://doi.org/10.1007/978-3-642-35221-8

    Book  MATH  Google Scholar 

  13. de Barros, L.C., Bassanezi, R.C., Lodwick, W.A.: A first course in fuzzy logic. In: Fuzzy Dynamical Systems, and Biomathematics: Theory and Applications. SFSC, vol. 347. Springer, Heidelberg (2017). https://doi.org/10.1007/978-3-662-53324-6

    Book  Google Scholar 

  14. Moore, R.E.: Interval Analysis. Prentice Hall, Upper Saddle River (1966)

    Google Scholar 

  15. Moore, R.E., Kearfott, R.B., Cloud, M.J.: Introduction to Interval Analysis. Society for Industrial and Applied Mathematics (2009). https://doi.org/10.1137/1.9780898717716

  16. Guangquan, Z.: Fuzzy distance and limit of fuzzy numbers. BUSEFAL 33, 19–30 (1987)

    MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the Brazilian Agency CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior) for the financial support.

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Authors and Affiliations

  1. Institute of Mathematics, Statistics and Scientific Computing Campinas (IMECC), University of Campinas (UNICAMP), Campinas, São Paulo, 13083-872, Brazil

    Ricardo Augusto Watanabe & Estevão Esmi

  2. Department of Biotechnology and Bioprocess Engineering, Federal University of Tocantins (UFT), Gurupi, Tocantins, 77402-970, Brazil

    Cibele Cristina Trinca Watanabe

Authors
  1. Ricardo Augusto Watanabe
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  2. Cibele Cristina Trinca Watanabe
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  3. Estevão Esmi
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Corresponding author

Correspondence to Ricardo Augusto Watanabe .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA, USA

    Ralph Baker Kearfott

  2. Center for Computing Research (CIC), Instituto Politecnico Nacional, Ciudad de Mexico, Mexico

    Ildar Batyrshin

  3. Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, Canada

    Marek Reformat

  4. Department of Computer Science, University of Texas at El Paso, El Paso, TX, USA

    Martine Ceberio

  5. Department of Computer Science, University of Texas at El Paso , El Paso, TX, USA

    Vladik Kreinovich

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Watanabe, R.A., Watanabe, C.C.T., Esmi, E. (2019). Fuzzy Octonion Numbers: Some Analytical Properties. In: Kearfott, R., Batyrshin, I., Reformat, M., Ceberio, M., Kreinovich, V. (eds) Fuzzy Techniques: Theory and Applications. IFSA/NAFIPS 2019 2019. Advances in Intelligent Systems and Computing, vol 1000. Springer, Cham. https://doi.org/10.1007/978-3-030-21920-8_63

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