Math 640 (Spring 2020): Suggested EXPERIMENTAL MATHEMATICS Class Projects

https://sites.math.rutgers.edu/~zeilberg/EM20/projs.html

Last Update: May 20, 2020.

The Class should be divided into three or four teams, each with a team leader. Each team can contain from 1 to 4 students (including the team leaders). Students are allowed (per request of one student) to form their own singelton team, and in that case the singleton team can pick the same topic as another team. (To be specific, one student asked to be his own team with the War project, so there would be two parallel War efforts. At the end, they can decide whether to join forces or not, when they are ready to write it up.) The team leader will be in charge of coordinating the various contributions, and writing the first draft of a paper that will be definitely posted here. The first, preliminary, "skeleton" versions should be ready by May 10, 2020, but it is hoped that they will be expanded into publishable papers (at least in arxiv.org, and possibly in a a "real" journal. Each paper should be accopmpanied by at least one Maple package. See example from 2019, another example (of a different kind). See also an example from 2018 and another example from 2018.

Generalized Gambler's Ruin

Project Leader: Yukun Yao

Other members: Victoria Chayes, Chun "Larry" Lau , Yuxuan Yang

Extend studies of Gambler's ruin, with all its aspects: Simulation, Pure numeric, numeric-symbol, and closed-form (if possible) to other families of directed graphs, not just the line. For example:

Also, extend this great paper to higher dimensions, and for the variance and higher moments of the random variable "number of visits to the origin", whose expectation is finite for dimensions > 2

Also, if you have time, study both by simulations and exactly, if possible using Wilf-Zeilberger algorithmic proof theory the random variable, "duration of a random walker starting at a specfic point in the positive quadrant (in 2D) until it leaves it.

Added May 11, 2020: Here are preliminary versions of the paper and the Maple program.

Simulating and analyzing the Card Games WAR and UNO

Project Leader: Robert Dougherty-Bliss

Other members: Alison Bu, David Harrera, Charles Kenny

See War for the rules. (See this interesting paper pointed out by Charles Kenney)

For Uno See Uno for the rules.

Added May 14, 2020: Here is the current version : .pdf

Game Theory from an Experimental Mathematics Standpoint

Project Leader: Alan Chernoff

Other team members: Khizar Anjun, Jingyang Deng, Zidong Zhang

This would continue the pioneering work of my former student Richard Voepel.

You should also look at this paper by J.M. Peterson and M.A. Simaan. Please also look at its references, in particular, Ref. 6 by M. Dresher.

Added May 20, 2020: Here is a preliminary version of this project, Maple package EM20ProjNash.txt.

One Person Project: Strict War

Team Leader and only member: Johnny Fonseca .

Added May 20, 2020: Read Johnny Fonseca's interesting article about Strict War.

Strategic Gambling

STILL OPEN

Extend the work in this seminal article. The following articles, cited there, may be relevant (or not, if they use "fancy" continuous probability)

Studying and Generalizing the Shepp Urn

STILL OPEN

See William Boyce's beautiful article on the intriguing Shepp Urn and continue this beautiful work by Richard Voepel.

You are welcome to suggest other projects.


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