Rhapsody in fractional

Fractional Calculus and Applied Analysis

Volumn 17, Issue 4, 2014, Pages 1188-1214

  Machado, J Tenreiro ^a   Lopes, António M ^b   Duarte, Fernando B ^c   Ortigueira, Manuel D ^d   Rato, Raul T ^e  

^a INSTITUTO SUPERIOR DE ENGENHARIA DO PORTO   (Portugal)

^b UNIVERSITY OF PORTO   (Portugal)

^c UNIVERSIDADE LUSÓFONA DE HUMANIDADES E TECNOLOGIAS   (Portugal)

^d UNINOVA   (Portugal)

^e INSTITUTO POLITÉCNICO DE SETÚBAL   (Portugal)

Author keywords

difference equations; fractional delay; fractional Fourier transform; fractional sums; matrix power; matrix root; negative probability; positional number systems; tolerant computing

Indexed keywords

EID: 84919904023     PISSN: 13110454     EISSN: 13142444     Source Type: Journal    
DOI: 10.2478/s13540-014-0206-0     Document Type: Article

References (69)

1. J.P. Allouche, J. Shallit, Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press A, ISBN: 978-052182-3326, 2003.
2. L.B. Almeida, The fractional Fourier transform and time-frequency representations. IEEE Signal Processing Letters42, No 11 (1994), 3084–3093.
3. P. Ambrož, C. Frougny, Z. Masáková and E. Pelantová, Arithmetics on number systems with irrational bases. Bull. of the Belgian Mathematical Society-Simon Stevin10, No 5 (2003), 641–659.
4. A. Arias, E. Gutierrez and E. Pozo, Binomial theorem applications in matrix fractional powers calculation; http://www.pp.bme.hu/tr/article/download/6705/5810.
5. J. Astin, Extension of the formula for the Nth power of a square matrix to negative and fractional values of N. The Mathematical Gazette51, No 377 (1967), 228–232.
6. K.M.R. Audenaert, Fractional powers of positive positive definite matrices; http://personal.rhul.ac.uk/usah/080/QITNotes-files/.
7. M.S. Bartlett, Negative probability. Math. Proc. of the Cambridge Philosophical Society41, No 1 (1945), 71–73; DOI: 10.1017/S0305004100022398.
8. J.S. Bell, On the Einstein Podolsky Rosen paradox. Physics1, No 3 (1964), 195–200.
9. G. Bergman, A number system with an irrational base. Mathematics Magazine31, No 2 (1957), 98–110.
10. D.A. Bini, N.J. Higham and B. Meini, Algorithms for the matrix pth root. Numerical Algorithms39, No 4 (2005), 349–378; DOI: 10.1007/s11075-004-6709-8.
11. A. Bultheel and H.M. Sulbaran, Computation of the fractional Fourier transform. Applied and Computational Harmonic Analysis16, No 3 (2004), 182–202.
12. M. Burgin and G. Meissner, Negative probabilities in financial modeling. Wilmott Magazine 2012, No 58 (2012), 60–65; DOI: 10.1002/wilm.10093.
13. D.R. Burleson, On non-integer powers of a square matrix; http://www.blackmesapress.com/Eigenvalues.htm.
14. D.R. Burleson, Computing the square root of a Markov matrix. Eigenvalues and the Taylor series; http://www.blackmesapress.com/TaylorSeries.htm.
15. R.G. Campos and J. Rico-Melgoza and E. Chavez, XFT: Extending the digital application of the Fourier transform; http://www.citebase.org/abstract?id=oai:arXiv.org:0911.0952 (2009).
16. T. Charitos, P.R. De Waal, and L.C. Van Der Gaag, Computing shortinterval transition matrices of a discrete-time Markov chain from partially observed data. Statistics in Medicine6, No 27 (2008), 905–921.
17. E.F. Codd, Extending the data base relational model to capture more meaning. Proc. of the 1979 ACM SIGMOD Internat. Conference on Management of Data, 1979; doi:10.1145/582095.582122.
18. E.U. Condom, Immersion of the Fourier transform in a continuous group of functional transformations. Proc. National Academy Sciences23, No 3 (1937), 158–164.
19. P.M. Dirac, The physical interpretation of quantum mechanics. Proc. Royal Society LondonA, No 180 (1942), 1–39; doi:10.1098/rspa.1942.0023.
20. G. Evangelista, Design of digital systems for arbitrary sampling rate conversion. Signal Processing83, No 2 (2003), 377–387; http://dx.doi.org/10.1016/S0165-1684(02)00421-8.
21. R.P. Feynman, The Concept of Probability Theory in Quantum Mechanics. Second Berkeley Symposium on Mathematical Statistics and Probability Theory, University of California Press (1950).
22. R.P. Feynman, Negative probability. In: Quantum Implications: Essays in Honour of David Bohm, Editors: B.J. Hiley, F. David Peat, Routledge & Kegan Paul Ltd. (1987), 235–248, ISBN: 0415069602.
23. S. Fiori, Leap-frog-type learning algorithms over the Lie group of unitary matrices. Neurocomputing71, No 10–12 (2008), 2224–2244.
24. C. Frougny, How to write integers in non-integer base. In: LATIN’92, Springer, Berlin-Heidelberg (1992), 154–164.
25. V. Grünwald, Intorno all’aritmetica dei sistemi numerici a base negativa con particolare riguardo al sistema numerico a base negativodecimale per lo studio delle sue analogie coll’aritmetica ordinaris (decimale). Giornale di Matematiche di Battaglini23 (1885), 203–221.
26. E.G. Haug, Why so negative to negative probabilities?, What is the probability of the expected being neither expected nor unexpected?. Wilmott Magazine (Mar/Apr 2007), 34–38.
27. B. Hayes, Third base. American Scientist89, No 6 (2001), 490–494.
28. E.C R. Hehner, A Practical Theory of Programming. Monographs in Computer Science, Springer, New York (1993), ISBN 978-038794-1066.
29. N.J. Higham, Functions of Matrices: Theory and Computation. SIAM, Philadelphia (2008), ISBN 978-0-89871-646-7.
30. N.J. Higham and L. Lin, A Schur-Padé algorithm for fractional powers of a matrix. SIAM J. on Matrix Analysis and Applications32, No 3 (2001), 1056–1078.
31. H.F. Hofmann, How to simulate a universal quantum computer using negative probabilities. Journal of Physics A: Math. and Theoretical42, No 27 (2009), 1–9; doi:10.1088/1751-8113/42/27/275304.
32. R.B. Israel, J.S. Rosenthal and J.Z. Wei, Finding generators forMarkov chains via empirical transition matrices, with applications to credit ratings. Mathematical Finance11, No 2 (2001), 245–265.
33. A. Kapelner and J. Bleich, Prediction with missing data via Bayesian additive regression trees. Stat. 1050 (2014); arXiv:1306.0618 [stat.ML].
34. A.J. Kempner, Anormal systems of numeration. American Math. Monthly (1936), 610–617.
35. A. Khrennikov, Interpretations of Probability. VSP (1999), ISBN: 9067643106.
36. D.E. Knuth, A imaginary number system. Communications of the ACM3, No 4 (1960), 245–247.
37. T.I. Laakso, V. Välimäki, M. Karjalainen and U.K. Laine, Splitting the unit delay — tools for fractional delay filter design. IEEE Signal Processing Magazine13, No 1 (1996), 30–60; DOI: 10.1109/79.482137
38. D. Leibfried, T. Pfau and C. Monroe, Shadows and mirrors: Reconstructing quantum states of atom motion. Physics Today51, No 4 (1998), 22–28; DOI: 10.1063/1.882256.
39. J.T. Machado, Fractional coins and fractional derivatives. Abstract and Applied Analysis2013, Article ID 205097 (2013), 1–5; doi: 10.1155/2013/205097.
40. Z. Masáková, E. Pelantová and T. Vávra, Arithmetics in number systems with a negative base. Theoretical Computer Science412, No 8 (2011), 835–845.
41. W. Mückenheim, G. Ludwig, C. Dewdney, P. R Holland, A. Kyprianidis, J.P. Vigier, N. Cufaro Petroni, M.S. Bartlett and E.T. Jaynes, A review of extended probabilities. Physics Reports133, No 6 (1986), 337–401; doi: 10.1016/0370-1573(86)90110-9.
42. M. Müller and D. Schleicher, How to add a Non-integer number of terms, and how to produce unusual infinite summations. J. of Comput. and Applied Mathematics178, No 1-2 (2005), 347–360.
43. M. Müller and D. Schleicher, Fractional sums and Euler-like identities. The Ramanujan Journal21, Issue 2 (Feb. 2010), 123–143.
44. M. Müller and D. Schleicher, How to add a non-integer number of terms: from axioms to new identities. arXiv:1001.4695 [math.CA], 2011.
45. V.A. Narayanan and K.M.M. Prabhu, The fractional Fourier transform: theory, implementation and error analysis. Microprocessors and Microsystems27, No 10 (2003), 511–521.
46. M. D. Ortigueira, Introduction to fractional signal processing. Part 2: Discrete-time systems. IEEE Proc. on Vision, Image and Signal Processing147, No 1 (2000), 71–78; DOI: 10.I049/ip-vis:20000273.
47. M.D. Ortigueira, F.J. Coito, and J.J. Trujillo, Discrete-time differential systems. Signal Processing, Available online (March 2014); DOI: 10.1016/j.sigpro.2014.03.004.
48. M.D. Ortigueira, C. Matos, and M.S. Piedade, Fractional discrete-time signal processing: scale conversion and linear prediction. Nonlinear Dynamics29, No 1–4 (2002), 173–190; DOI: 10.1023/A:1016522226184
49. H.M. Ozaktas, O. Ankan, M.A. Kutay and G. Bozdaği, Digital computation of the fractional Fourier transform. IEEE Trans. on Signal Processing44, No 9 (1996), 2141–2150.
50. H.M. Ozaktas, Z. Zalesvsky, and M.A. Kutay, The Fractional Fourier Transform. Wiley, Chichester (2001).
51. W. Parry, On the β-expansions of real numbers. Acta Mathematica Hungarica11, No 3 (1960), 401–416.
52. Z. Pawlak and A. Wakulicz, An electronic computer based on the “-2” system. Bull. de l’Academie Polonaise des Scienses, Sér. des Sciences Techniques7 (1959), 713–721.
53. S. Pei and J. Ding, Relations between gabor transforms and fractional Fourier transforms and their applications for signal processing. IEEE Transactions on Signal Processing55, No 10 (2007), 4839–4850.
54. V. Penchev, A philosophical view on the introduction of negative and complex probability in quantum information. Philosophical Alternatives, No 1 (2012), 63–78.
55. W. Penney, A “binary” system for complex numbers. Journal of the ACM12, No 2 (1965), 247–248.
56. R.T. Rato, Complexity and Emptiness. 7th Congress of the UES (2008), ISBN 978-972905-9056.
57. R.T. Rato, Formalização da tolerância à ausência de dados no processamento de sinais discretos. PhD. Thesis, Universidade Nova de Lisboa — Faculdade de Ciências e Tecnologia (2012).
58. E. Reis, P. Melo, R. Andrade and T. Calapez, Estatística Aplicada. Sílabo, Lisboa (1999), ISBN 978-972618-4690.
59. A. Rényi, Representations for real numbers and their ergodic properties. Acta Mathematica Hungarica8, No 3 (1957), 477–493.
60. R. Saxena and K. Singh, Fractional Fourier transform: A novel tool for signal processing. J. Indian Inst. Science No 85 (2005), 11–26.
61. A.P. Stakhov, The generalized golden proportions, a new theory of real numbers, and ternary mirror-symmetrical arithmetic. Chaos, Solitons & Fractals33, No 2 (2007), 315–334.
62. A.P. Stakhov, The Mathematics of harmony: Clarifying the origins and development of Mathematics. Congressus Numerantium193, No 20 (2008).
63. G.J. Székely, Half of a coin: Negative probabilities. Wilmott Magazine (July 2005), 66–68.
64. R. Tao, B. Deng, W. Zhang and Y. Wang, Sampling and sampling rate conversion of band limited signals in the fractional Fourier transform domain. IEEE Trans. on Signal Processing56, No 1 (2008), 158–171.
65. A. Tarczynski, W. Kozinski, and G.D. Cain, Sampling rate conversion using fractional-sample delay. Proc. IEEE ICASSP’94, Adelaide, Australia, May 1994, 285–288; DOI: 10.1109/ICASSP.1994.390042.
66. H. Tijms and K. Staats, Negative probabilities at work in the M/D/1 queue. Probability in the Engineering and Informational Sciences21, No 1 (2007), 67–76; DOI: 10.1017/S0269964807070040.
67. S. Vajda, Fibonacci & Lucas Numbers, and the Golden Section. Theory and Applications. Ellis Horwood Ltd. (1989).
68. F.V. Waugh and M.E. Abel, On fractional powers of a matrix. J. of the American Statistical Association62, No 319 (1967), 1018–1021.
69. A.I. Zayed, On the relationship between the Fourier and fractional Fourier transforms. IEEE Signal Processing Letters3, No 12 (1996), 310–311.