Expectations on Fractal Sets

Transcription

1 Expectations on Fractal Sets David H. Bailey Lawrence Berkeley Natl. Lab. (retired) Computer Science Dept., University of California, Davis Co-authors: Jonathan M. Borwein (CARMA, University of Newcastle, Australia) Richard E. Crandall (deceased) Michael G. Rose (CARMA, University of Newcastle, Australia) 4 December 204

2 Background and motivation This study arose when the late number theorist Richard Crandall (deceased December 202) was asked by some Stanford researchers to analyze some recent data on mouse brain synapses. Approximately one million synapses were analyzed. Synapses were found to be distributed in a fractal manner. The fractal dimension was significantly less than three, which would be expected if the points had been randomly distributed through a cubic volume. In other words, the expected pair-wise distances do not follow the statistics of random distributions.

3 The PSLQ integer relation algorithm Let (x n ) be a given vector of real numbers. An integer relation algorithm either finds integers (a n ) such that a x + a 2 x a n x n = 0 (to within the epsilon of the arithmetic being used), or else finds bounds within which no relation can exist. The PSLQ algorithm of mathematician-sculptor Helaman Ferguson is the most widely used integer relation algorithm. Integer relation detection requires very high precision (at least n d digits, where d is the size in digits of the largest a k ), both in the input data and in the operation of the algorithm.. H.R.P. Ferguson, D.H. Bailey and S. Arno, Analysis of PSLQ, An Integer Relation Finding Algorithm, Mathematics of Computation, vol. 68, no. 225 (Jan 999), pg D.H. Bailey and D.J. Broadhurst, Parallel Integer Relation Detection: Techniques and Applications, Mathematics of Computation, vol. 70, no. 236 (Oct 2000), pg

4 Applications of PSLQ in number theory and experimental mathematics Methodology:. Compute various mathematical entities (limits, infinite series sums, definite integrals, etc.) to high precision, typically 00 0,000 digits. 2. Use PSLQ to recognize these numerical values in terms of well-known mathematical constants. 3. When results are found experimentally, seek formal mathematical proofs of the discovered relations. Many results have recently been found using this methodology, both in pure mathematics and in mathematical physics.

5 Earlier study of box integrals The following integrals appear in numerous applications: B n (s) := n (s) := ( r rn 2 ) s/2 dr ( (r q ) (r n q n ) 2) s/2 drdq B n () is average distance of a random point from the origin. n () is average distance between two random points. B n ( n + 2) is average electrostatic potential in an n-cube whose origin has a unit charge. n ( n + 2) is average electrostatic energy between two points in a uniform n-cube of charged jellium. High-precision numerical values were computed for these values, then PSLQ was employed to recognize the resulting values. D. H. Bailey, J. M. Borwein and R.E. Crandall, Box integrals, Journal of Computational and Applied Mathematics, vol. 206 (2007), pg

6 Sample evaluations of box integrals n s B n (s) any even s 0 rational, e.g., : B 2 (2) = 2/3 s s π log( + 2) ) log( + 2) 2 2 s 2 2+s 2F ( 2, s 2 ; 3 2 ; ) π arctan G π log( + 2) + 3 Ti 2 (3 2 2) 3-4 π log ( ) π + 2 log ( ) π 7 20 log ( ) Here F is hypergeometric function; G is Catalan; Ti is Lewin s inverse-tan function.

7 Sample evaluations of box integrals, continued n s B n(s) 4-5 ( 8 arctan ) G 2 Ti 2(3 2 ( 2) 4-2 π log 2 + ) 3 2 G π log 3 (3 2 G Ti2 2 ) 2 ( ) 8 arctan (3 G Ti2 2 ) ( ) 2 + log arctan (2 0 G 0 log ) ( 3 θ 9 8 π log ) ( ) arctan ( 5 0 Cl 2 θ Cl ( θ π) Cl2 θ Cl2 θ Cl2 θ Cl2 θ + π) B5( 6) B5( 4) + 5 π log(3) + 0 ( ) Ti2 3 0 G 5 - (2 0 G + 0 log ) ( 3 θ π log ) ( ) arctan ( Cl2 θ + 0 ( Cl2 θ π) Cl2 θ Cl2 θ Cl2 θ Cl2 θ + π) 6 5 (2 77 G + 7 log ) ( 3 θ π log + ) ( ) arctan 5 + ( 7 9 Cl2 θ + 7 ( Cl2 θ π) Cl2 θ Cl2 θ Cl2 θ Cl2 θ + π) 6

8 String-generated Cantor sets Fix positive integers n and p, and consider the unit n-cube [0, ] n. Let P = P P 2... P p denote a period string. The String-Generated Cantor Set (SCS), denoted C n (P), is the set of all admissible x [0, ] n, where x admissible U(c k ) P k k N () with notational periodicity assumed: P p+k := P k for all k. Examples: C (0) is the classical middle-thirds-removed Cantor set on [0, ], as a point x C (0) is defined to be admissible iff its ternary expansion is entirely devoid of s (i.e. U(c k ) 0). C n (n) is the full unit n-cube [0, ] n, as all points x [0, ] n are admissible (every ternary vector is allowed for every column c k ). D. H. Bailey, J. M. Borwein, R. E. Crandall, M. G. Rose, Expectations on fractal sets, Appl. Math. and Comp., vol. 220 ( Sep 203), pg

9 Some string-generated Cantor sets Image due to R. Dickau (2008).

10 Graphical illustration of expectations for string-generated Cantor sets

11 Evaluations of expectations for string-generated Cantor sets We have been successful in finding closed-form values for many of these expectations. Surprisingly, all of them are rational numbers, a counter-intuitive result, since for non-cantor sets they are much more complicated. We also verified these results by numerically computations. Computing these values was non-trivial. Possible approaches:. Utilize a uniform (0, ) pseudorandom number generator to generate pairs of n-tuples at random, then check each pair of n-tuples so generated to see if it is admissible for the given set C n (P). This simple scheme can be used effectively in cases where the average density of the fractal set is not too small. 2. (Employed in results below.) Construct a table of all admissible columns c k for each of the components of P, and then, for each of a large number of trials, and for each ternary column k 20 of a single trial, pseudorandomly select an admissible column from the appropriate table. While coding is much more complicated, each pair of n-tuples is guaranteed to be admissible.

12 Sample results for one dimension δ(c (P)) B(2, C (P)) (2, C (P)) P Decimal Rational Decimal Numeric Rational Decimal Numeric

13 Sample results for two dimensions δ(c 2 (P)) B(2, C 2 (P)) (2, C 2 (P)) P Decimal Rational Decimal Numeric Rational Decimal Numeric

14 Sample results for three dimensions P δ(c 3 (P)) B(2, C 3 (P)) (2, C 3 (P)) Decimal Rational Decimal Numeric Rational Decimal Numeric

15 Ongoing work Computed value for expectation of Sierpinski gasket triangle: What is this number? We are trying hard to identify it. Computing this numerical value required 60 CPU-hours run time and critically relied on a self-avoiding random walk approach due to Nathan Clisby. Stay tuned!

16 For additional details These and many more results are presented in: D. H. Bailey, J. M. Borwein, R. E. Crandall, M. G. Rose, Expectations on fractal sets, Appl. Math. and Comp., vol. 220 ( Sep 203), pg Preprint version is available at This talk is available at: