Tools for special functions and special numbers Graduate Student Colloquium Tulane University, New Orleans Armin Straub October 15, 2013 University of Illinois & Max-Planck-Institut at Urbana–Champaign f¨ ur Mathematik, Bonn ISC — PSLQ — OEIS — CAD — WZ Tools for special functions and special numbers Armin Straub 1 / 26
THE TOOLS TODAY ISC Inverse Symbolic Calculator PSLQ Lattice Reduction Algorithm OEIS On-Line Encyclopedia of Integer Sequences CAD Cylindrical Algebraic Decomposition WZ Wilf–Zeilberger Theory Tools for special functions and special numbers Armin Straub 2 / 26
Random walks • We study random walks in the plane consisting of n steps. Each step is of length 1 and is taken in a randomly chosen direction. Tools for special functions and special numbers Armin Straub 3 / 26
Random walks • We study random walks in the plane consisting of n steps. Each step is of length 1 and is taken in a randomly chosen direction. Tools for special functions and special numbers Armin Straub 3 / 26
Random walks • We study random walks in the plane consisting of n steps. Each step is of length 1 and is taken in a randomly chosen direction. Tools for special functions and special numbers Armin Straub 3 / 26
Random walks • We study random walks in the plane consisting of n steps. Each step is of length 1 and is taken in a randomly chosen direction. Tools for special functions and special numbers Armin Straub 3 / 26
Random walks • We study random walks in the plane consisting of n steps. Each step is of length 1 and is taken in a randomly chosen direction. Tools for special functions and special numbers Armin Straub 3 / 26
Random walks • We study random walks in the plane consisting of n steps. Each step is of length 1 and is taken in a randomly chosen direction. Tools for special functions and special numbers Armin Straub 3 / 26
Random walks • We study random walks in the plane consisting of n steps. Each step is of length 1 and is taken in a randomly chosen direction. d Tools for special functions and special numbers Armin Straub 3 / 26
Random walks • We study random walks in the plane consisting of n steps. Each step is of length 1 and is taken in a randomly chosen direction. • We are interested in the distance traveled in n steps. d For instance, how large is this dis- Q tance on average? • Probability density: p n ( x ) Tools for special functions and special numbers Armin Straub 3 / 26
Random walks are only about 100 years old • Karl Pearson asked for p n ( x ) in Nature in 1905. This famous question coined the term random walk . Applications include: • dispersion of mosquitoes • random migration of micro-organisms • phenomenon of laser speckle Tools for special functions and special numbers Armin Straub 4 / 26
Long random walks THM p n ( x ) ≈ 2 x n e − x 2 /n for large n Rayleigh, 1905 EG 0.06 p 200 0.05 0.04 0.03 0.02 0.01 10 20 30 40 50 “ The lesson of Lord Rayleigh’s solution is that in open country the most probable place to find a drunken man who is at all capable of keeping on his feet is ” somewhere near his starting point! Karl Pearson , 1905 Tools for special functions and special numbers Armin Straub 5 / 26
Densities of short walks p 3 p 4 p 2 0.5 0.7 0.8 0.4 0.6 0.6 0.5 0.3 0.4 0.4 0.3 0.2 0.2 0.2 0.1 0.1 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 2.5 3.0 1 2 3 4 p 5 p 6 p 7 0.35 0.35 0.30 0.30 0.30 0.25 0.25 0.25 0.20 0.20 0.20 0.15 0.15 0.15 0.10 0.10 0.10 0.05 0.05 0.05 7 1 2 3 4 5 1 2 3 4 5 6 1 2 3 4 5 6 Tools for special functions and special numbers Armin Straub 6 / 26
Densities of short walks p 3 p 4 p 2 0.5 0.7 0.8 0.4 0.6 0.6 0.5 0.3 0.4 0.4 0.3 0.2 0.2 0.2 0.1 0.1 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 2.5 3.0 1 2 3 4 p 5 p 6 p 7 0.35 0.35 0.30 0.30 0.30 0.25 0.25 0.25 0.20 0.20 0.20 0.15 0.15 0.15 0.10 0.10 0.10 0.05 0.05 0.05 7 1 2 3 4 5 1 2 3 4 5 6 1 2 3 4 5 6 Tools for special functions and special numbers Armin Straub 6 / 26
Moments • The moments of a RV X are E ( X ) , E ( X 2 ) , E ( X 3 ) , . . . • If X has probability density f ( x ) then � ∞ E ( X s ) = x s f ( x ) d x −∞ The moments E ( X s ) are analytic in s . FACT (if, e.g., f ( x ) is compactly supported) Tools for special functions and special numbers Armin Straub 7 / 26
Moments • The moments of a RV X are E ( X ) , E ( X 2 ) , E ( X 3 ) , . . . • If X has probability density f ( x ) then � ∞ E ( X s ) = x s f ( x ) d x −∞ The moments E ( X s ) are analytic in s . FACT (if, e.g., f ( x ) is compactly supported) • Represent the k th step by the complex number e 2 πix k . • The s th moment of the distance after n steps is: n s � � � � e 2 πx k i � � W n ( s ) := d x � � [0 , 1] n � � k =1 In particular, W n (1) is the average distance after n steps. Tools for special functions and special numbers Armin Straub 7 / 26
Average distance traveled in two steps • Numerically: W 2 (1) ≈ 1 . 2732395447351626862 Tools for special functions and special numbers Armin Straub 8 / 26
Average distance traveled in two steps • Numerically: W 2 (1) ≈ 1 . 2732395447351626862 Tools for special functions and special numbers Armin Straub 8 / 26
Average distance traveled in two steps • Numerically: W 2 (1) ≈ 1 . 2732395447351626862 Tools for special functions and special numbers Armin Straub 8 / 26
Average distance traveled in two steps • Numerically: W 2 (1) ≈ 1 . 2732395447351626862 Tools for special functions and special numbers Armin Straub 8 / 26
The simple two-step case confirmed • The average distance in two steps: � 1 � 1 � e 2 πix + e 2 πiy � � d x d y = ? � W 2 (1) = 0 0 Tools for special functions and special numbers Armin Straub 9 / 26
The simple two-step case confirmed • The average distance in two steps: � 1 � 1 � e 2 πix + e 2 πiy � � d x d y = ? � W 2 (1) = 0 0 � 1 � 1 + e 2 πiy � � d y � = 0 Tools for special functions and special numbers Armin Straub 9 / 26
The simple two-step case confirmed • The average distance in two steps: � 1 � 1 � e 2 πix + e 2 πiy � � d x d y = ? � W 2 (1) = 0 0 � 1 � 1 + e 2 πiy � � � 1 + e 2 πiy � � d y � = � � � 1 + (cos πy + i sin πy ) 2 � 0 = � � 1 = 2 cos( πy ) = 2 cos( πy )d y 0 Tools for special functions and special numbers Armin Straub 9 / 26
The simple two-step case confirmed • The average distance in two steps: � 1 � 1 � e 2 πix + e 2 πiy � � d x d y = ? � W 2 (1) = 0 0 � 1 � � 1 + e 2 πiy � � 1 + e 2 πiy � � d y � = � � 1 + (cos πy + i sin πy ) 2 � � 0 = � � 1 = 2 cos( πy ) = 2 cos( πy )d y 0 = 4 π ≈ 1 . 27324 Tools for special functions and special numbers Armin Straub 9 / 26
The simple two-step case confirmed • The average distance in two steps: � 1 � 1 � e 2 πix + e 2 πiy � � d x d y = ? � W 2 (1) = 0 0 � 1 � � 1 + e 2 πiy � � 1 + e 2 πiy � � d y � = � � 1 + (cos πy + i sin πy ) 2 � � 0 = � � 1 = 2 cos( πy ) = 2 cos( πy )d y 0 = 4 π ≈ 1 . 27324 • Mathematica 7 and Maple 14 think the double integral is 0. Better: Mathematica 8 and 9 just don’t evaluate the double integral. Tools for special functions and special numbers Armin Straub 9 / 26
The simple two-step case confirmed • The average distance in two steps: � 1 � 1 � e 2 πix + e 2 πiy � � d x d y = ? � W 2 (1) = 0 0 � 1 � 1 + e 2 πiy � � � 1 + e 2 πiy � � d y � = � � 1 + (cos πy + i sin πy ) 2 � � 0 = � � 1 = 2 cos( πy ) = 2 cos( πy )d y 0 = 4 π ≈ 1 . 27324 • Mathematica 7 and Maple 14 think the double integral is 0. Better: Mathematica 8 and 9 just don’t evaluate the double integral. • This is the average length of a random arc on a unit circle. Tools for special functions and special numbers Armin Straub 9 / 26
Moments of random walks DEF The s th moment W n ( s ) of the density p n : � ∞ � � s d x � e 2 πix 1 + . . . + e 2 πix n � x s p n ( x ) d x = � W n ( s ) := [0 , 1] n 0 Tools for special functions and special numbers Armin Straub 10 / 26
Moments of random walks DEF The s th moment W n ( s ) of the density p n : � ∞ � � s d x � e 2 πix 1 + . . . + e 2 πix n � x s p n ( x ) d x = � W n ( s ) := [0 , 1] n 0 • On a desktop: W 3 (1) ≈ 1 . 57459723755189365749 W 4 (1) ≈ 1 . 79909248 W 5 (1) ≈ 2 . 00816 Tools for special functions and special numbers Armin Straub 10 / 26
Moments of random walks DEF The s th moment W n ( s ) of the density p n : � ∞ � � s d x � e 2 πix 1 + . . . + e 2 πix n � x s p n ( x ) d x = � W n ( s ) := [0 , 1] n 0 • On a desktop: W 3 (1) ≈ 1 . 57459723755189365749 W 4 (1) ≈ 1 . 79909248 W 5 (1) ≈ 2 . 00816 • On a supercomputer: Lawrence Berkeley National Laboratory, 256 cores W 5 (1) ≈ 2 . 0081618 Tools for special functions and special numbers Armin Straub 10 / 26
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