Fractal Analysis of Pi Normality

References

• [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.” Exp. Math. 21 (2012), 375–384.
   Web of Science ®Google Scholar
• [Bailey and Crandall 01] D. H. Bailey and R. E. Crandall. “On the Random Character of Fundamental Constant Expansions.” Exp. Math. 10 (2001), 175–190.
   Web of Science ®Google Scholar
• [Barnsley 93] M. F. Barnsley. Fractals Everywhere. Cambridge, MA: Academic Press Professional, 1993.
   Google Scholar
• [Becher and Figueira 02] V. Becher and S. Figueira. “An Example of a Computable Absolutely Normal Number.” Theor. Comput. Sci. 270 (2002), 947–958.
   Web of Science ®Google Scholar
• [Bellamy 13] J. Bellamy. “Randomness of D Sequences via Diehard Testing.” Available online (http://arxiv.org/abs/1312.3618).
   Google Scholar
• [Bellard 10] F. Bellard. “Computation of 2700 Billion Decimal Digits of Pi using a Desktop Computer.” Available online (http://bellard.org/pi/pi2700e9/pipcrecord.pdf). 4th revision, 2010.
   Google Scholar
• [Bera and Jarque 81] A. K. Bera and C. M. Jarque. “Efficient Tests for Normality, Homoscedasticity and Serial Independence of Regression Residuals: Monte Carlo Evidence.” Econ. Lett. 7 (1981), 313–318.
   Web of Science ®Google Scholar
• [Berggren et al. 00] L. Berggren, J. M. Borwein, and P. B. Borwein, editors. Pi: A Source Book. New York: Springer-Verlag, 2000.
   Google Scholar
• [Besicovitch 29] A. S. Besicovitch. “On Linear Sets of Points of Fractional Dimensions.” Math. Ann. 101 (1929), 161–193.
   Google Scholar
• [Boltzmann 66] L. Boltzmann. “Über die mechanische bedeutung des zweiten hauptsatzes der wärmetheorie.” Wiener Berichte 53 (1866), 195–220.
   Google Scholar
• [Boltzmann 64] L. Boltzmann. Lectures on Gas Theory. Berlely–New York: University of California Press–Dover. Translated by Stephen G. Brush.
   Google Scholar
• [Borel 09] E. Borel. “Les Probabilités dénombrables et leurs applications arithmétiques.” Rend. Circ. Math. Palermo 27 (1909), 247–271.
   Google Scholar
• [Box and Muller 58] G. E. P. Box and M. E. Muller. “A Note on the Generation of Random Normal Deviates.” Ann. Math. Stat. 22 (1958), 610–611.
   Google Scholar
• [Chaitin 69] G. J. Chaitin. “On the Simplicity and Speed of Programs for Computing Infinite Sets of Natural Numbers.” J. ACM 16 (1969), 407–422.
   Web of Science ®Google Scholar
• [Chudnovsky and Chudnovsky 00] D. V. Chudnovsky and G. V. Chudnovsky. “Approximations and Complex Multiplication According to Ramanujan.” In Pi: A Source Book, 2 edn, edited by L. Berggren, J. Borwein, and P. Borwein, chapter 63, pp. 596–622. New York: Springer-Verlag, 2000.
   Google Scholar
• [Dahlquist and Björk 74] G. Dahlquist and Å. Björk. Numerical Methods. Engelwood Cliffs: Prentice Hall Inc. 1974.
   Google Scholar
• [D’Suze and Sevcik 10] G. D’Suze and C. Sevcik. “Scorpion Venom Complexity Fractal Analysis. Its Relevance for Comparing Venoms,” J. Theor. Biol. 267 (2010), 405–416.
   PubMed Web of Science ®Google Scholar
• [D’Suze et al. 15] G. D’Suze, M. Sandoval, and C. Sevcik. “Characterizing Tityus Discrepans Scorpion Venom From a Fractal Perspective: Venom Complexity, Effects of Captivity, Sexual Dimorphism, Differences Among Species.” Toxicon 108 (2015), 62–72.
   PubMed Web of Science ®Google Scholar
• [Grassberger and Procaccia 83] P. Grassberger and Procaccia, J. “Estimation of the Kolmogorov Entropy from a Chaotic signal.” Phys. Rev. A 28 (1983), 2591—2593.
   Web of Science ®Google Scholar
• [Guttman et al. 71] I. Guttman, S. S. Wilks, and J. S. Hunter. Introductory Engineering Statistics. New York: Wiley, 1971.
   Google Scholar
• [Hastings and Sugihara 93] H. H. Hastings and G. Sugihara. Fractals. A User's Guide for the Natural Sciences. Oxford: Oxford University Press, 1993.
   Google Scholar
• [Hausdorff 18] F. Hausdorff. “Dimension und äußeres maß.” Math. Ann. 79 (1918), 157–179.
   Google Scholar
• [Hollander and Wolfe 73] M. Hollander and D. A. Wolfe. Nonparametric Statistical Procedures. New York: Wiley, 1973.
   Google Scholar
• [Kempner 16] A. J. Kempner. “On Transcendental Numbers.” Trans. Am. Math. Soc. 17 (1916), 476–482.
   Google Scholar
• [Kolmogorov 33] A. Kolmogorov. “Sulla determinazione empirica di una legge di distribuzione.” Inst. Ital. Attuari 4 (1933), 83–91.
   Google Scholar
• [Kolmogorov 64] A. N. Kolmogorov. “Three Approaches to the Quantitative Definition of Information.” Prob. Info. Trans. 1 (1964), 1–7.
   Google Scholar
• [Kruskal and Wallis 52] W. H. Kruskal and W. A. Wallis. “Use of Ranks in One-Criterion Variance Analysis.” J. Am. Stat. Assoc. 47 (1952), 583–621.
   Web of Science ®Google Scholar
• [Lambert 72] J. H. Lambert. “Anmerkungen und zusätze zur entwerfung der land- und himmelscharten.” In Beiträge zum gebrauche der mathematik und deren anwendung, edited by A. Wangerin, pp. 3–19. Leipzig: W. Engelmann, 1772.
   Google Scholar
• [Mandelbrot 75] B. B. Mandelbrot. Les Objects Fractals: Forme, Hasard et Dimension. Paris: Flammarion, 1975.
   Google Scholar
• [Mandelbrot 83] B. B. Mandelbrot. The Fractal Geometry of Nature. New York: W.H. Freeman and Co., 1983.
   Google Scholar
• [Mandelbrot 86] B. B. Mandelbrot. “Fractals and the Rebirth of Iteration Theory.” In The Beauty of Fractals, edited by H. Peitigen and P. H. Richter. Berlin: Springer-Verlag, 1986.
   Google Scholar
• [Marsaglia et al. 03] G. Marsaglia, W. W. Tsang, and J. Wang. “Evaluating Kolmogorov's Distribution.” J. Stat. Soft. 8 (2003), 1–4.
   Google Scholar
• [Matsumoto and Nishimura 98] M. Matsumoto and T. Nishimura. “Mersenne Twister: A 623-Dimensionally Equidistributed Uniform Pseudorandom Number Generator.” ACM Trans. Mod. Commu. Simul. 8 (1998), 3–30.
   Google Scholar
• [Panneton et al. 06] F. O. Panneton, P. l’Ecuyer, and M. Matsumoto. “Improved Long-Period Generators based on Linear Recurrences Modulo 2.” ACM Trans. Math. Software 32 (2006), 1–16.
   Web of Science ®Google Scholar
• [Pincus et al. 91] S. M. Pincus, I. M. Gladstone, and R. A. Ehrenkranz. “A Regularity Statistics for Medical Data Analysis.” J. Clin. Monit. 7 (1991), 335–345.
   PubMed Web of Science ®Google Scholar
• [Press et al. 92] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery. Numerical Recipes in C. The Art of Scientific Computing. Cambridge, England: Cambridge University Press, 1992.
   Google Scholar
• [Schaffer and Sedgewick 93] R. Schaffer and R. Sedgewick. “The Analysis of Heapsort.” J. Algor. 15 (1993), 76–100.
   Web of Science ®Google Scholar
• [Sevcik 98] C. Sevcik. “A Procedure to Estimate the Fractal Dimension of Waveforms.” Complexity Internat. 5 (1998), 1–19. Available from (http://arxiv.org/pdf/1003.5266.pdf).
   Google Scholar
• [Shannon 48] C. E. Shannon. “A Mathematical Theory of Communication.” Bell Syst. Tech. J. 27 (1948), 379–423.
   Web of Science ®Google Scholar
• [Shapiro and Wilk 65] S. S. Shapiro and M. B. Wilk. “An Analysis of Variance Test for Normality (Complete Samples).” Biometrika 52 (1965), 591–611.
   Web of Science ®Google Scholar
• [Sierpinski 88] W. Sierpinski. Elementary Theory of Numbers. North-Holland Mathematical Library, 2 edn. Amsterdam–Warszawa: North Holland—PWN- Polish Scientific Publishers, 1988.
   Google Scholar
• [Sinai 59] Y. Sinai. “On the Notion of Entropy of a Dynamical System.” Doklady Russ. Acad. Sci. 124 (1959), 768–771.
   Google Scholar
• [Smirnov 39a] N. V. Smirnov. “Estimate of Deviation between Empirical Distribution Functions in two Independent Samples.” Bull. Moscow Univ. 2 (1939a), 3–16.
   Google Scholar
• [Smirnov 39b] N. V. Smirnov. “Sur les ecarts de la courbe de Distribution Empiric.” Mat. Schornik. 6 (1939b), 3–26.
   Google Scholar
• [Smith 97] S. W. Smith. The Scientist and Engineer’s Guide to Digital Signal Processing. San Diego, CA: California Technical Publishing, 1997.
   Google Scholar
• [Spohn 80] W. Spohn. “Stochastic Independence, Causal Independence, and Shieldability.” J. Phil. Logic 9 (1980), 73–99.
   Web of Science ®Google Scholar
• [Stroustrup 13] B. Stroustrup. The C++ Programming Language. 4 edn. Upper Saddle River, NJ: Addison-Wesley, 2013.
   Google Scholar
• [Szilárd 29] L. Szilárd. “Über die Entropieverminderung in einem Thermodynamischen System bei Eingriffen Intelligenter Wesen.” Zeitsch. Physik 53 (1929), 840–856.
   Google Scholar
• [Trueb 16a] P. Trueb. “πe Trillion Decimal Digits of π.” Available online (http://www.pi2e.ch, http://www.numberworld.org/y-cruncher/), 2016a.
   Google Scholar
• [Trueb 16b] P. Trueb. “Digit statistics of the First πe Trillion Decimal Digits of π.” arXiv:1612.00489v1, 2016b. [math.NT].
   Google Scholar
• [Vysochanskij and Petunin 80] D. F. Vysochanskij and Y. I. Petunin. “Justification of the 3σ Rule for Unimodal Distributions.” Theor. Probab. Math. Stat. 21 (1980), 25–36.
   Google Scholar
• [Wilks 62] S. S. Wilks. Mathematical Statistics. New York: Wiley, 1962.
   Google Scholar
• [Williams 64] J. W. J. Williams. “Algorithm 232 - Heapsort.” Commun. ACM 7 (1964), 347–348.
   Web of Science ®Google Scholar
• [Wolfram 03] S. Wolfram. The Mathematica^® Book, 5 edn. Champagn, IL: Wolfram Media, 2003.
   Google Scholar