Experimental Analysis on the Normality of Using Advanced Data-Mining Techniques (Experimental Analysis on the Normality of pi, e, phi, and square root of 2 Using Advanced Data-Mining Techniques)

References

• [Borel 09] E. Borel. “Les probabilités dénombrables et leurs applications arithmétiques.” Rend. Circ. Mat. Palermo 27, 1909, 247–271.
   Google Scholar
• [Bailey and Crandall 01] D. H. Bailey and R. E. Crandall. “On the Random Character of Fundamental Constant Expansions.” Experiment. Math. 10 (2001), 175–190.
   Web of Science ®Google Scholar
• [Bailey and Crandall 02] D. H. Bailey and R. E. Crandall. “Random Generators and Normal Numbers.” Experiment. Math. 11 (2002), 527–546.
   Web of Science ®Google Scholar
• [Bailey et al. 12] D. H. Bailey, J. M. Borwein, C. S. Calude, M. J. Dinneen, M. Dumitrescu, and A. Yee. “An Empirical Approach to the Normality of π.” Experiment. Math. 21 (2012), 375–384.
   Web of Science ®Google Scholar
• [Becher 12] V. Becher. “Turing's Normal Numbers: Towards Randomness.” In How the World Computes, edited by B. S. Cooper, A. Dawar, and B. LÎwe, Lecture Notes in Computer Science 7318, pp. 35–45. Springer, 2012.
   Google Scholar
• [Calude 95] C. Calude. “What Is a Random String?” Journal of Universal Science 1 (1995), 48–66.
   Google Scholar
• [Champernowne 33] D. G. Champernowne. “The Construction of Decimals Normal in the Scale of Ten.” J. London Math. Soc. 8 (1933), 254–260.
   Google Scholar
• [Copeland and Erdős 46] A. H. Copeland and P. Erdős. “Note on Normal Numbers.” Bull. Amer. Math. Soc. 52 (1946), 857–860.
   Web of Science ®Google Scholar
• [Dasgupta 11] A. Dasgupta. “Mathematical Foundations of Randomness.” In Philosophy of Statistics, edited by D. M. Gabbay, P. Thagard, J. Woods, pp. 641–710. North Holland, 2011.
   Google Scholar
• [Davenport and Erdős 53] H. Davenport and P. Erdős. “Note on Normal Decimals.” Canad. J. Math. 4 (1952), 58–63.
   Web of Science ®Google Scholar
• [Hardy and Wright 60] G. H. Hardy and E. M. Wright. An Introduction to the Theory of Numbers, 4th edition. Oxford University Press, 1960.
   Google Scholar
• [Khoshnevisan 06] D. Khoshnevisan. “Normal Numbers Are Normal.” Clay Mathematics Institute Annual Report 2006(2006), 27–31.
   Google Scholar
• [Long 57] C. T. Long. “Note on Normal Numbers.” Pacific J. Math. 7 (1957), 1163–1165.
   Google Scholar
• [Niven and Zuckerman 52] I. Niven and H. Zuckerman. “On the Definition of Normal Numbers.” Pacific Journal of Mathematics 1 (1951), 103–109.
   Google Scholar
• [Trenton 06] M. Trenton. “Some Results on Borel-Normal Strings.” University of Prince Edward Island, CSIT-2, 2006.
   Google Scholar
• [Wagon 85] S. Wagon. “Is Pi Normal?” The Mathematical Intelligencer 7 (1985), 65–67.
   Web of Science ®Google Scholar
• [Xylogiannopoulos et al. 12a] K. Xylogiannopoulos, P. Karampelas, and R. Alhajj. “Periodicity Data Mining in Time Series Using Suffix Arrays.” In Proc. IEEE Inteligent Systems IS’12, 2012.
   Google Scholar
• [Xylogiannopoulos et al. 12b] K. Xylogiannopoulos, P. Karampelas, and R. Alhajj. “Minimization of Suffix Array's Storage Capacity for Periodicity Detection in Time Series.” In Proc. IEEE International Conference in Tools with Artificial Intelligence, 2012.
   Google Scholar
• [Xylogiannopoulos et al. 13a] K. Xylogiannopoulos, P. Karampelas, and R. Alhajj. “Advanced Techniques for Pattern Detection in Time Series Using Suffix Arrays.” In Recent Contributions on Intelligent Systems, edited by R. Yager, J. Kacprzyk, and K. Atanassov. Springer, 2013.
   Google Scholar
• [Xylogiannopoulos et al. 13b] K. Xylogiannopoulos, P. Karampelas, and R. Alhajj. “Analyzing Very Large Time Series Using Suffix Arrays.” To appear in Applied Intelligence, 2013. doi: 10.1007/s10489-014-0553-x.
   Google Scholar
• [Xylogiannopoulos et al. 13c] K. Xylogiannopoulos, P. Karampelas, and R. Alhajj. “Probabilistic Existence and Estimation of Longest Expected Repeated Pattern in Sequences.” Submitted for publication, 2013.
   Google Scholar
• [Yee 13] A. Yee. “Y-Cruncher: A Multi-Threaded Pi-Program.” Available online (http://www.numberworld.org/y-cruncher/, 2013.
   Google Scholar
• [Yee and Kondo 2013] A. J. Yee and S. Kondo. “Round 2 …10 Trillion Digits of Pi.” Available online (http://www.numberworld.org/misc_runs/pi-10t/details.html), 2013.
   Google Scholar