Experimenting with the Dym-Luks Ball and Cell Game (almost) Sixty Years Later

By Shalosh B. Ekhad and Doron Zeilberger

.pdf    .tex

Written: Jan. 12, 2023

Dedicated to Harry Dym (b. Jan. 26, 1938)

This is a symbolic-computational redux, and extension, of the beautiful paper, of Harry Dym and Eugene Luks, written when they were 26-year-old and 24-year-old respectively. Since then they went on to do many other great things, but this (Harry's third, and probably Gene's first) is still fresh after all these years, but with Maple, we can do so much more.

Added Jan. 27, 2023: Harry Dym's first two papers were:
(with E. Arthurs): On the optimum detection of digital signals in the presence of white Gaussian noise, IRE Trans. Comm. Systems 10 (1962), 336-372.
Distance properties of tree codes, MIT Res. Lab. for Electronics, QPR, 70(1963), 247-261.

# Maple package

• DymLuks.txt, a Maple package to experiment with the Dym-Luks Ball and Cell game

# Sample Input and Output for DymLuks.txt

• If you want to see the probability generating function (in the variable x) for the duration of the Dym-Luks Ball and Cell game with a FIXED (small) number of balls,r, and a general (symbolic) number of cells, n, as explicit expressions in x and n (that happen to be rational functions of these variables) for r=1 to r=20 then. Also explicit expressions in n for the expectation, variance, skewness and kurtosis
the input gives the output.

[Warning: file is very big!]

• If you only care about the expectation, variance, and moments, then the same information can be gotten here:
the input gives the output.

[Warning: file also big!]

• If you want to see the probability generating function (in the variable x) for the duration of the Dym-Luks Ball and Cell game with a FIXED (small) number of cells ,n, (for n between 2 and 6) and number of balls (r) between r=1 to r=30 then, as well as (numerical) expectation, standard-deviation, skewness and kurtotis)
the input gives the output.

• If you only care about the expectation, standard-deviation, skewness and kurtosis, then the same information can be gotten here:
the input gives the output.

• If you want to see the probability generating function (in the variable x) for the duration of the Dym-Luks Ball and Cell game with THE SAME number of balls and cells for (r,n)=(2,2) to (r,n)=(40,40), as well as expectation, variance, skewness, and kurtosis
the input gives the output.

• If you only care about the expectation, standard-deviation, and moments, then the same information can be gotten here:
the input gives the output.

• If you only care about the expectation, standard-deviation, here is a more extensive table that goes all the way to 120 balls and 120 cells/
the input gives the output.

• If you want to see the sequence of differences between the mean duration, Mn(r) and the approximating sum Sum((1/j)*((n/(n-1))j-1,j=1..r), what's called En(r) in Remark 1 (bottom of p. 520 in the Dym-Luks 1966 paper), for n from 2 to 10 and r from 1 to 150, that implies estimates for the limits as n goes to infinity,
the input gives the output.

If you want to see the sequence of RATIOS between the mean duration, Mn(r) and the approximating sum Sum((1/j)*((n/(n-1))j-1,j=1..r), as well as the the sequence of RATIOS between the variance and the approximating sum Sum(((1/j2)*((n/(n-1))j-1)2,j=1..r)- Sum((1/j)*((n/(n-1))j-1,j=1..r),
the input gives the output

• If you want to see ten expamples of simulations of the Dym-Luks game with 1000 balls and 1000 cells
the input gives the output.

• If you want to see an article for the duration of simplified Markov process where the prob. of moving from n to n-1 is a^n (for general a between 0 and 1), and of remaining at n is 1-a^n, giving explicit expressions in a and n for the expectation, variance, and the scaled moments about the mean, up to the 10th, as well as the limits of the latter when n goes to infinity
the input gives the output.