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Normalization and \(\varphi \)-function: Definition and Consequences

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Book cover Geometric Science of Information (GSI 2017)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 10589))

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Abstract

It is known from the literature that a \(\varphi \)-function may be used to construct the \(\varphi \)-families of probability distributions. In this paper, we assume that one of the properties in the definition of \(\varphi \)-function is not satisfied and we analyze the behavior of the normalizing function near the boundary of its domain. As a consequence, we find a measurable function that does not belong to the Musielak–Orlicz class, but the normalizing function applied to this found function converges to a finite value near the boundary of its domain. We conclude showing that this change in the definition of \(\varphi \)-function affects the behavior of the normalizing function.

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Acknowledgement

The authors would like to thank CAPES and CNPq (Proc. 309055/2014-8) for partial funding of this research.

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Authors and Affiliations

  1. Center of Exact and Natural Sciences, Federal Rural University of Semi-Arid Region, Mossoró, RN, Brazil

    Luiza H. F. de Andrade

  2. Computer Engineering, Campus Sobral, Federal University of Ceará, Sobral, CE, Brazil

    Rui F. Vigelis

  3. Department of Mathematics, Regional University of Cariri, Juazeiro do Norte, CE, Brazil

    Francisca L. J. Vieira

  4. Department of Teleinformatics Engineering, Federal University of Ceará, Fortaleza, CE, Brazil

    Charles C. Cavalcante

Authors
  1. Luiza H. F. de Andrade
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  2. Rui F. Vigelis
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  3. Francisca L. J. Vieira
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  4. Charles C. Cavalcante
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Corresponding author

Correspondence to Luiza H. F. de Andrade .

Editor information

Editors and Affiliations

  1. Ecole Polytechnique, Palaiseau, France

    Frank Nielsen

  2. Thales Land and Air Systems, Limours, France

    Frédéric Barbaresco

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de Andrade, L.H.F., Vigelis, R.F., Vieira, F.L.J., Cavalcante, C.C. (2017). Normalization and \(\varphi \)-function: Definition and Consequences. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2017. Lecture Notes in Computer Science(), vol 10589. Springer, Cham. https://doi.org/10.1007/978-3-319-68445-1_27

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  • DOI: https://doi.org/10.1007/978-3-319-68445-1_27

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-68444-4

  • Online ISBN: 978-3-319-68445-1

  • eBook Packages: Computer ScienceComputer Science (R0)

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