About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 205097, 5 pages
http://dx.doi.org/10.1155/2013/205097
Research Article

Fractional Coins and Fractional Derivatives

Institute of Engineering, Polytechnic of Porto, Department of Electrical Engineering, Rua Dr. António Bernardino de Almeida 431, 4200-072 Porto, Portugal

Received 3 April 2013; Revised 9 May 2013; Accepted 10 May 2013

Academic Editor: Dumitru Baleanu

Copyright © 2013 J. A. Tenreiro Machado. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. P. A. Dirac, “The physical interpretation of quantum mechanics,” Proceedings of the Royal Society A, vol. 180, pp. 1–40, 1942. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. R. P. Feynman, “The concept of probability theory in quantum mechanics,” in Second Berkeley Symposium on Mathematical Statistics and Probability Theory, University of California Press, Berkeley, Calif, USA, 1950.
  3. R. P. Feynman, “Negative probability,” in Quantum Implications: Essays in Honour of David Bohm, B. J. Hiley, Ed., pp. 235–248, Routledge & Kegan Paul, London, UK, 1987. View at MathSciNet
  4. M. S. Bartlett, “Negative probability,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 41, no. 1, pp. 71–73, 1945. View at Zentralblatt MATH · View at MathSciNet
  5. G. J. Székely, “Half of a coin: negative probabilities,” Wilmott Magazine, pp. 66–68, 2005.
  6. J. S. Bell, “On the Einstein Podolsky Rosen paradox,” Physics, vol. 1, no. 3, pp. 195–200, 1964.
  7. W. Mückenheim, G. Ludwig, C. Dewdney et al., “A review of extended probabilities,” Physics Reports, vol. 133, no. 6, pp. 337–401, 1986. View at Publisher · View at Google Scholar · View at MathSciNet
  8. D. Leibfried, T. Pfau, and C. Monroe, “Shadows and mirrors: reconstructing quantum states of atom motion,” Physics Today, vol. 51, no. 4, pp. 22–28, 1998. View at Scopus
  9. A. Khrennikov, Interpretations of Probability, VSP, Utrecht, The Netherlands, 1999. View at MathSciNet
  10. H. F. Hofmann, “How to simulate a universal quantum computer using negative probabilities,” Journal of Physics A, vol. 42, no. 27, Article ID 275304, pp. 1–9, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. V. Penchev, “A philosophical view on the introduction of negative and complex probability in quantum information,” Philosophical Alternatives, no. 1, pp. 63–78, 2012.
  12. E. G. Haug, “Why so negative to negative probabilities? What is the probability of the expected being neither expected nor unexpected?” Wilmott Magazine, pp. 34–38, 2007.
  13. H. Tijms and K. Staats, “Negative probabilities at work in the M/D/1 queue,” Probability in the Engineering and Informational Sciences, vol. 21, no. 1, pp. 67–76, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. M. Burgin and G. Meissner, “Negative probabilities in financial modeling,” Wilmott Magazine, vol. 58, pp. 60–65, 2012.
  15. K. B. Oldham and J. Spanier, The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order, Academic Press, New York, NY, USA, 1974. View at MathSciNet
  16. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993. View at MathSciNet
  17. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science, Amsterdam, The Netherlands, 1993. View at MathSciNet
  18. O. Heaviside, “On operators in physical mathematics,” Proceedings of the Royal Society, vol. 52, pp. 504–529, 1893.
  19. J. T. Machado, V. Kiryakova, and F. Mainardi, “Recent history of fractional calculus,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 3, pp. 1140–1153, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. B. J. West, M. Bologna, and P. Grigolini, Physics of Fractal Operators, Springer, New York, NY, USA, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  21. R. Magin, Fractional Calculus in Bioengineering, Begell House, Redding, Calif, USA, 2006.
  22. R. Caponetto, G. Dongola, L. Fortuna, and I. Petráš, Fractional Order Systems: Modeling and Control Applications, World Scientific, Singapore, 2010.
  23. C. A. Monje, Y. Chen, B. M. Vinagre, D. Xue, and V. Feliu, Fractional-Order Systems and Controls: Fundamentals and Applications, Springer, London, UK, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  24. I. Petráš, Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation, Springer, Heidelberg, Germany, 2011.
  25. A. Oustaloup, Systèmes Asservis Linéaires D'ordre Fractionnaire: Théorie et Pratique, Masson, Paris, France, 1983.
  26. S. Westerlund and L. Ekstam, “Capacitor theory,” IEEE Transactions on Dielectrics and Electrical Insulation, vol. 1, no. 5, pp. 826–839, 1994. View at Publisher · View at Google Scholar · View at Scopus
  27. J. T. Machado, “Analysis and design of fractional-order digital control systems,” Systems Analysis, Modelling, Simulation, vol. 27, no. 2-3, pp. 107–122, 1997.
  28. I. Podlubny, Fractional Differential Equations, Volume 198: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution, Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. View at MathSciNet
  29. R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  30. G. M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics, Oxford University Press, Oxford, UK, 2005. View at MathSciNet
  31. J. Sabatier, O. P. Agrawal, and J. T. Machado, Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht, The Netherlands, 2007. View at MathSciNet
  32. C. M. Ionescu and R. De Keyser, “Relations between fractional-order model parameters and lung pathology in chronic obstructive pulmonary disease,” IEEE Transactions on Biomedical Engineering, vol. 56, no. 4, pp. 978–987, 2009. View at Publisher · View at Google Scholar · View at Scopus
  33. D. Baleanu, “About fractional quantization and fractional variational principles,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 6, pp. 2520–2523, 2009.
  34. V. E. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, New York, NY, USA, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  35. F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, London, UK, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  36. D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus: Models and Numerical Methods, World Scientific, Boston, Mass, USA, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  37. A. Kilbas, H. Srivastava, and J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204, North-Holland Mathematics Studies, Elsevier, Amsterdam, The Netherlands, 2006.
  38. F. B. Tatom, “The relationship between fractional calculus and fractals,” Fractals, vol. 3, no. 1, pp. 217–229, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  39. I. Podlubny, “Geometric and physical interpretation of fractional integration and fractional differentiation,” Fractional Calculus & Applied Analysis, vol. 5, no. 4, pp. 367–386, 2002. View at Zentralblatt MATH · View at MathSciNet
  40. J. T. Machado, “A probabilistic interpretation of the fractional-order differentiation,” Fractional Calculus & Applied Analysis, vol. 6, no. 1, pp. 73–80, 2003. View at Zentralblatt MATH · View at MathSciNet
  41. J. T. Machado, “Fractional derivatives: probability interpretation and frequency response of rational approximations,” Communications in Nonlinear Science and Numerical Simulations, vol. 14, no. 9-10, pp. 3492–3497, 2009.
  42. I. Podlubny, “Fractional-order systems and PIλDμ-controllers,” IEEE Transactions on Automatic Control, vol. 44, no. 1, pp. 208–214, 1999. View at Publisher · View at Google Scholar · View at MathSciNet
  43. Y. Q. Chen and K. L. Moore, “Discretization schemes for fractional-order differentiators and integrators,” IEEE Transactions on Circuits and Systems I, vol. 49, no. 3, pp. 363–367, 2002. View at Publisher · View at Google Scholar · View at MathSciNet