D. H. Bailey and R. E. Crandall, On the random character of fundamental constant expansions. Experiment. Math. 10 (2001), 175-190.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....for 1 s over 0 s. Moreover, according to Levy s Law of the Iterated Logarithm, the set of sequences with this property has Lebesgue measure zero. i.e. x 2 N : 1 2 1 m m X k=1 x k ) 1 p m in nitely often = 1: 21 For some fascinating ideas in this direction, see [3] 9 In other words, the collectives don t satisfy all the Laws of Randomness of Probability Theory, understood as the laws holding with probability one, which renders the Mises Wald Church notion of randomness not satisfactory. 4 Randomness as Incompressibility The source of the paradox of ....

D. H. Bailey and R. C. Crandall, On the Random Character of Fundamental Constant Expansions, http://www.perfsci.com, May 2000.

....computer program or algorithm for calculating its digits one by one; of course, nearly all real numbers are not computable. Turing showed that if 17 Every string appears in an algorithmically random sequence with theprob#B(8Ok y 2 n , wheren is the length of the string. 18 Bailey and Crandall [1] discussed a hypothesis which implies the normality of many natural real numb ers, e.g. #, e.A di#erent approach was discussed in Pincus and Singer [41] and Pincus and Kalman [40] 19 Note that humanb eings are not doing ab etterjob in generating random b#an as Shannon [50] has argued. Biases ....

D. H. Bailey, R. C. Crandall. On the Random Character of Fundamental Constant expansions, http://www.perfsci.com, May 2000.

....set A with = A . 6. There is a total computable function f : N 2 f0; 1g such that (a) If for some k; n we have f(k; n) 1 and f(k; n 1) 0 then there is an l k with f(l; n) 0 and f(l; n 1) 1. b) We have: k 2 X ( lim n 1 f(k; n) 1. 9 Bailey and Crandall [1] discussed a hypothesis which implies the normality of many natural real numbers, e.g. e. A di erent approach was discussed in Pincus and Singer [41] and Pincus and Kalman [40] see also Casti [20] and Beltrami [2] 10 See Theorem 16. 13 Proof. It is obvious that conditions 1. 2. and 3. ....

D. H. Bailey, R. C. Crandall. On the Random Character of Fundamental Constant expansions, http://www.perfsci.com, May 2000.

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David H. Bailey and Richard E. Crandall, "On the Random Character of Fundamental Constant Expansions," Experimental Mathematics, vol. 10 (2001), 175-190.

....mathematics amusing, but of no further consequence in mathematics or any other scientific discipline. To the contrary, it now appears that the existence of these formulas has deep implications for the centuries old question of whether (and why) constants such as # and log 2 are normal [7]. Here normal to a given base b means that all m long base b digit strings occur with a limiting frequency that is precisely what one would expect from random digits, namely b m . It is a true, if counter intuitive consequence of measure theory that almost all real numbers are normal to a ....

David H. Bailey and Richard E. Crandall, "On the Random Character of Fundamental Constant Expansions," Experimental Mathematics, vol. 10, no. 2 (June 2001), pg. 175-190.

....Generators and Normal Numbers David H. Bailey 1 and Richard E. Crandall 2 30 October 2001 Abstract Pursuant to the authors previous chaotic dynamical model for random digits of fundamental constants [3], we investigate a complementary, statistical picture in which pseudorandom number generators (PRNGs) are central. Some rigorous results such as the following are achieved: Whereas the fundamental constant log 2 = P n2Z 1= n2 n ) is not yet known to be 2 normal (i.e. normal to base 2) we ....

....remarkable that in spite of the elegance of the classical notion of normality, and the sobering fact that almost all real numbers are absolutely normal (meaning b normal for every b = 2; 3; proofs of normality for fundamental constants such as log 2; 3) and p 2 remain elusive. In [3] we proposed a general Hypothesis A that connects normality theory with a certain aspect of chaotic dynamics. In a subsequent work, J. Lagarias [28] provided interesting viewpoints and analyses on the dynamical concepts. In the present paper we adopt a kind of complementary viewpoint, focusing ....

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David H. Bailey and Richard E. Crandall, \On the random character of fundamental constant expansions," Experimental Mathematics, 10, 2001, 175-190.

....of other mathematical constants. In [20] some base 3 formulas were obtained, including the identity 2 = 2 27 1 X k=0 1 729 k 243 (12k 1) 2 405 (12k 2) 2 81 (12k 4) 2 27 (12k 5) 2 72 (12k 6) 2 9 (12k 7) 2 9 (12k 8) 2 5 (12k 10) 2 1 (12k 11) 2 In [8], it is shown that the question of whether ; log(2) and certain other constants are normal can be reduced to a plausible conjecture regarding dynamical iterations of the form x 0 = 0, xn = bx n 1 r n ) mod 1 where b is an integer and r n = p(n) q(n) is the ratio of two nonzero polynomials ....

....in [0; 1) There are also connections between the question of normality for certain constants and the theory of linear congruential pseudorandom number generators. All of these results derive from the discovery of the individual digit calculating formulas mentioned above. For details, see [8]. 5 Identities for the Riemann Zeta Function Another application of computer technology in mathematics is to determine whether or not a given constant , whose value can be computed to high precision, is algebraic of some degree n or less. This can be done by rst computing the vector x = 1; ....

David H. Bailey and Richard E. Crandall, \On the Random Character of Fundamental Constant Expansions", manuscript (2000). Available from http://www.nersc.gov/~dhbailey.

.... fractions can be combined into one, yielding = 1 X k=0 1 16 k 47 151k 120k 2 15 194k 712k 2 1024k 3 512k 4 Since the publication of [3] other papers have presented formulas of this type for various constants, including several constants that arise in quantum eld theory [7, 8, 5]. More recently, interest in BBP type formulas has been heightened by the observation that the question of the statistical randomness of the digit expansions of these constants can be reduced to the following hypothesis regarding the behavior of a particular class of chaotic iterations [5] ....

.... [7, 8, 5] More recently, interest in BBP type formulas has been heightened by the observation that the question of the statistical randomness of the digit expansions of these constants can be reduced to the following hypothesis regarding the behavior of a particular class of chaotic iterations [5]: Hypothesis A (from the paper [5] Denote by r n = p(n) q(n) a rational polynomial function, i.e. p; q 2 Z[X] Assume further that 0 deg p deg q, with r n nonsingular for positive integers n. Choose an integer b 2 and initialize x 0 = 0. Then the sequence x = x 0 ; x 1 ; x 2 ; ....

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David H. Bailey and Richard E. Crandall, \On the Random Character of Fundamental Constant Expansions," manuscript, Oct. 2000, available from http://www.nersc.gov/~dhbailey.

....Broadhurst was also able to determine isolated digits of #(5) using a more complicated summation involving three periodic coe#cient sequences. Bailey and Crandall have used such expansions to establish, under a general dynamical hypothesis, random properties of the binary bits in various # values [13]. Open questions include this one: as all summands are rational and the terms decay geometrically in k, how best to adapt the Broadhurst series to the FEE method of Karatsuba, for example what should be the obvious denominators during series contractions It seems as if research on #(3) will ....

D. Bailey, R. Crandall, On the random character of fundamental constant expansions, manuscript, http://www.perfsci.com/free/techpapers.

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D. H. Bailey and R. E. Crandall, On the random character of fundamental constant expansions. Experiment. Math. 10 (2001), 175-190.

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David H. Bailey and Richard E. Crandall. On the random character of fundamental constant expansions. Experimental Mathematics, 10:175--190, 2001.

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