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Rhapsody in fractional

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Abstract

This paper studies several topics related with the concept of “fractional” that are not directly related with Fractional Calculus, but can help the reader in pursuit new research directions. We introduce the concept of non-integer positional number systems, fractional sums, fractional powers of a square matrix, tolerant computing and FracSets, negative probabilities, fractional delay discrete-time linear systems, and fractional Fourier transform.

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Authors and Affiliations

  1. Dept. of Electrical Engineering, Institute of Engineering, Polytechnic of Porto, Rua Dr. António Bernardino de Almeida, 431, 4200 — 072, Porto, Portugal

    J. Tenreiro Machado

  2. Institute of Mechanical Engineering, Faculty of Engineering, University of Porto, Dept. of Mechanical Engineering, Rua Dr. Roberto Frias, 4200-465, Porto, Portugal

    António M. Lopes

  3. Faculty of Engineering, Lusofona University, Campo Grande, 376, 1749-024, Lisboa, Portugal

    Fernando B. Duarte

  4. UNINOVA and DEE/Faculdade de Ciências e Tecnologia da UNL, Campus da FCT, Quinta da Torre, 2829-516, Caparica, Portugal

    Manuel D. Ortigueira

  5. Instituto Politécnico de Setúbal Escola Superior de Tecnologia, CTS — Uninova Research Institute, Monte de Caparica, 2829-516, Portugal

    Raul T. Rato

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  1. J. Tenreiro Machado
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  2. António M. Lopes
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  3. Fernando B. Duarte
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  4. Manuel D. Ortigueira
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  5. Raul T. Rato
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Correspondence to J. Tenreiro Machado.

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Machado, J.T., Lopes, A.M., Duarte, F.B. et al. Rhapsody in fractional. Fract Calc Appl Anal 17, 1188–1214 (2014). https://doi.org/10.2478/s13540-014-0206-0

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