Iterative Disposal Processes and the Lambert W Function 4K
Channel: Jordan Hess
Date: 2023-07-15

6th International Conference on Mathematics and Statistics (ICoMS) 2023
"Iterative Disposal Processes and the Lambert W Function"
by Jordan Hess and Kiran Polisetty, presented on July 15, 2023, at the HTWK in Leipzig, Germany.

Abstract: Given ℕ, choose a number randomly. Evens are chosen without replacement and odds are chosen with replacement. Repeat this process for as many times as there are naturals. Assess the expected value for the probability even in the resultant set. Then consider this question for the same process instead iterating only as many times as there are even members. Solutions are proposed in terms of the Lambert W function.

The publication can be accessed here:
https://doi.org/10.1145/3613347.3613353

The first half of the follow-up work, which takes iterative disposal processes and develops them over the reals into "iterative disposal curves," can be previewed here:
https://drive.google.com/file/d/10PbvvMKpHhOv_zrlE7su40LgH6zTW4fG/view

The bulk of the follow-up work I released as a book called "Iterative Disposal Curves with Applications" can be accessed at: https://a.co/d/0h0QXgW.

Shot and Edited by Blake Hennigan.
LinkedIn: https://www.linkedin.com/in/blake-hennigan-86745017b/
Squarespace: https://blakehennigan.squarespace.com/

0:00 Introduction.
0:20 What is the probability of drawing an even from the set that results from applying this iterative disposal process to the set of naturals for as many times as there are members?
0:53 Reducing to the finite case, a solution is given for the set: {1}.
1:36 A solution is given for the set: {1,2}.
4:04 What is the probability of drawing an even from the set that results from applying this iterative disposal process to the set of naturals for half as many times as there are members?
4:24 Reducing to the finite case, a solution is given for the set: {1,2}.
5:55 What are the expected closed forms when solved for using a top-down computation?
6:42 Solving for when we take the total number of iterations over one turn.
7:13 Solving for when we take the total number of iterations over two turns.
9:58 Conclusion.

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Disclaimer: Numerical evidence is referenced to a known constant to obtain the expected closed forms. A direct proof was recently given by Julio Cesar Enciso-Alva in his paper called "Expected Natural Density of Countable Sets after Infinitely Iterated de Finetti Lotteries, Computed via Matrix Decomposition" which can be viewed here:
https://arxiv.org/abs/2409.03921