**Welcome
to the M ^{3} Webpage!!!**

The topics for

- Mabel Adubofour

- Tyesha Hall
- Luis Parada

- Ben Sebuufu (Gordon)

- Samuel Thoronka
- Iris Yoon (Swarthmore)

- Rita Zevallos (Swarthmore)

Project Summary:

The focus of the research project is Mancala, an ancient family of board games popular in Africa and Asia. While there are many possible rule variants, this 'sowing' type game is based on moving stone seeds from one container to others according to prescribed deterministic rules. Play can be surprisingly involved, with a large number of legal moves possible each turn. Surprisingly, there has been little published mathematical research of this very interesting game. John Conway developed his own variant, 'Sowing,' which led to some simple mathematical language and structure which could be further developed. The primary research question for this project is "Is there an optimal strategy, against which no other competing strategy can win." Mancala has been played for more than ten thousand years, suggesting that no obvious optimal strategy exists. However, with M

- Change the number of containers and number of seeds to see when an optimal strategy exists.For instance, with four total containers and one seed initially in each container, the first player has an optimal strategy, against which the other player cannot win.
- Explore and implement various rule sets and strategies numerically to build intuition.
- Use a combinatorial approach to discuss the number of possible
moves and strategies.

Final Presentation - Thursday, June 21

The topic for

- Amanda Fernandez
- Tyesha Hall
- Mikias Kidane
- Spencer Sims (Oakwood)

- Elf Bauserman

Project Summary:

Single player Mancala-type games

Mancala is an ancient family of board games popular in Africa and Asia.While there are many rule variants, this "sowing" type game is based on moving stone "seeds" from one container to others according to prescribed deterministic rules. Play can be surprisingly involved, with a large number of legal moves possible each turn.

Surprisingly, there has been little published mathematical research of this very interesting game; in fact, last summer's M

- Examine the single-player game called Tchoukaillon.
- We will investigate and probe the interesting mathematical patterns involved in Tchoukaillon boards.
- We will modify the rules of Tchoukaillon to include periodic boards, which will allow us to better relate Tchoukaillon to the next game to be studied, Tchuka Ruma.
- We will examine patterns of the modified-Tchoukaillon boards.

- Examine the single-player game called Tchuka Ruma.
- Last summer we discovered that multiple sequences of moves terminated in the same board state. Our goal is to quantify this particular 'equivalence class' and extend the results to other boards.
- A large problem with attempting to quantify sowing games is that the number of possible moves grows very quickly. However, if we are only interested in winning boards, we can prune the game tree by starting at the winning game board and working backwards, while at the same time starting from the opening board and moving forward. We can then join these two trees to eliminate numerous extraneous board states.

Final Presentation - Friday, June 24

The two topics for

- Rex Ford
- Anthony Chieco
- Durrell Lewis
- Brittney Dyson
- Fierra Mason

Project Summaries:

PS Method

About 45 years ago, Fehlberg (1964) speculated that the power series method could be used to solve initial valued ordinary differential equation (IV ODE) problems with a level of accuracy that could not be achieved by other methods. About 13 years ago, a team of researchers at JMU took the first steps toward demonstrating that this is, indeed, true. Specifically, Parker and Sochacki (1996) demonstrated that power series can be used in a modified version of Picard's method and that this Parker-Sochacki/Power Series (PS) method is a practical, efficient and accurate algorithm for solving IV ODE problems.

The PS method, in addition to solving IV ODEs, has been used to solve integral-differential equations, find roots, and solve numerous application-oriented problems. Nonetheless, to date, the full potential of the PS method has not been realized because of the lack of a software tool that is accessible to the broader research community interested in these kinds of problems. Co-PI Warne has been a part of the JMU team using the PS method for the last ten years and he is at the forefront of innovation with the method. The second research group proposes to do the following:

- The primary objective is to develop a software tool that can be used to easily formulate IV ODE models and efficiently solve them using the PS method.

Single player Mancala-type games

Mancala is an ancient family of board games popular in Africa and Asia. While there are many rule variants, this "sowing" type game is based on moving stone "seeds" from one container to others according to prescribed deterministic rules. Play can be surprisingly involved, with a large number of legal moves possible each turn.

Surprisingly, there has been little published mathematical research of this very interesting game; in fact, last summer's M

- Examine the single-player game called Tchoukaillon.
- We will investigate and probe the interesting mathematical patterns involved in Tchoukaillon boards.
- We will modify the rules of Tchoukaillon to include periodic boards, which will allow us to better relate Tchoukaillon to the next game to be studied, Tchuka Ruma.
- We will examine patterns of the modified-Tchoukaillon boards.

- Examine the single-player game called Tchuka Ruma.
- Last summer we discovered that multiple sequences of moves terminated in the same board state. Our goal is to quantify this particular 'equivalence class' and extend the results to other boards.
- A large problem with attempting to quantify sowing games is that the number of possible moves grows very quickly. However, if we are only interested in winning boards, we can prune the game tree by starting at the winning game board and working backwards, while at the same time starting from the opening board and moving forward. We can then join these two trees to eliminate numerous extraneous board states.

Final Presentation - Friday, June 25

The topic for

- Rex Ford
- David Melendez
- Juan Carlos Ortega
- Zurisadai Pena
- Melinda Vergara

Project Summary:

The focus of the research project is Mancala, an ancient family of board games popular in Africa and Asia. While there are many possible rule variants, this 'sowing' type game is based on moving stone seeds from one container to others according to prescribed deterministic rules. Play can be surprisingly involved, with a large number of legal moves possible each turn. Surprisingly, there has been little published mathematical research of this very interesting game. John Conway developed his own variant, 'Sowing,' which led to some simple mathematical language and structure which could be further developed. The primary research question for this project is "Is there an optimal strategy, against which no other competing strategy can win." Mancala has been played for more than ten thousand years, suggesting that no obvious optimal strategy exists. However, with M

- Change the number of containers and number of seeds to see when an optimal strategy exists. For instance, with four total containers and one seed initially in each container, the first player has an optimal strategy, against which the other player cannot win.
- Explore and implement various rule sets and strategies numerically to build intuition.
- Use a combinatorial approach to discuss the number of possible moves and strategies.
- Consider the game as a discrete dynamical system. What type of analysis is possible using this abstraction?

Final Presentation - Friday, June 19

The topic for

- Jan Herburt-Hewell (hand in pocket)
- Michael Dankwa (baseball cap)
- Lianne Louizou (long hair)
- Juan Carlos Ortega (colorful shirt)

Project Summary:

During the first two days of the summer program, the students built a chaotic waterwheel (see picture to the right). They then proceeded to learn dynamical systems so they could better understand, mathematically, the behavior of the wheel they built. They are next going to build the first ever choatic sandwheel (at least to our knowledge) and derive the equations that govern the behavior of the new system and answer questions like:

Do you still see chaotic behavior?

Do you still see periodic behavior?

What is the qualitative behavior of the sand wheel?

What is the quantitative behavior the sand wheel?

Final Presentation - Friday, June 13

You can see the experimental progress that was made by examining the following two movies: Water and Sand (you will need quicktime to view these movies).

The topic for

- Charell Wingfield (left of James Madison)
- Michael Frempong (right of James Madison
- Jan Herburt-Hewell (behind James Madison)
- Michael Dankwa (in front of James Madison)

During the first two weeks of the M^{3} program,
the participants will be introduced to discrete equations focusing on
both analysis and numerical simulation. The director will present
numerous open questions and ask the students to choose a couple on which
to concentrate. The entire program (participants and director) will work
together to solve the open questions pertaining to two-gender population
models.

During the third and fourth week of this research experience, the
students will perform a biological investigation of mate choice by male
*Betta splendens* fish using video playback of females. In
this experiment, the students will determine whether males spend more
time with and direct more courtship behaviors to a female with vertical
lines than to a female without vertical lines. This experiment
will be in addition to the open questions started during the first
two weeks.

During the final two weeks of this research experience, the
students will conclude their research along with developing a
mathematical model of mate choice in *Betta splendens*. The
students will give a 50 minute presentation of their research
results on the last day of the program. After the conclusion of
the research experience, the students will disseminate their results in
July at the JMU Biology REU poster session, a poster and oral
presentation in October at the Shenandoah Undergraduate Mathematics and
Statistics Conference, and the director will be giving an oral
presentation of the results in August at the Society for
Mathematical Biology's annual meeting.

Final Presentation - Friday, June 22^{nd}, 2007 at 1:30 pm in Roop 103

Final Presentation - Friday, June 22

- The influence of Female-Male Interactions on Offspring Sex Selection, Michael Dankwa and Jan Herburt-Hewell
- Infectious Disease Modeling of Human Papillomavirus, Michael Fremprong and Charell Wingfield

Click here to go to Anthony Tongen's webpage.

edited on 5/24/07