Math 347 Projects
Project 1: Due Friday, September 28, 5:00 pm
Team 1: Steve, Jim
Team 2: Ariela, Adrienne, Minyu
Team 3: Brook
Each team must first read each problem, and decide which problem to solve. Your final submission should be an 8-15 page paper containing the following components:
- An introduction in which you restate the problem as you interpret it.
- A description of your models. This should contain, in particular, an identification and explanation of any simplifying assumptions that you have made. All models will contain simplifying assumptions, but you should make sure to clearly state the assumptions that you are making (and your reasons for making those assumptions). When possible, try to motivate each assumption with background research (and make sure to cite any sources that you use!). This section should also contain a verbal description of any parameters and/or variables that you have introduced. I recommend making separate sections or sub-sections in which you clearly explain your assumptions, describe your parameters and variables, and formulate your models. In general, it is a good idea to start with a very simple model, and then modify your simple model to incorporate more information and complexity. You should include all of the models that you consider in your paper, and in the results section (see below), you should compare and contrast the results that you obtain with different models.
- A results section in which you present the results of your model(s). Include graphs, tables, and figures, and be sure to explain them sufficiently in words. Describe the effect of various parameters on your results, using sensitivity analysis where appropriate. Note that it's ok to discuss sensitivity and robustness numerically. Discuss the situations in which your models work well, and the situations in which they don't work well.
- A conclusions section in which you succinctly present the main result of your work. The main points of your results section should be restated here. No new material should be in your conclusions section. You can also include a sentence or two on future work that you'd like to do regarding the problem you've worked on. For example, is there additional data that would help you construct a better model?
- A references section in which you list all references that you used (in bibliographic format).
Use headings and subheadings to construct an outline of your paper. These should break your paper into smaller pieces, each with a clear focus and goal.
In addition to the paper described above, each team must submit a 1 page summary in which you clearly state the problem and your results and conclusions in non-technical terms. You must include all of your main ideas in the summary, but the 1 page limit is strict.
To get started, I recommend doing an internet search to find some current literature on your problem/topics. Feel free to incorporate the ideas that you find in your model, but be sure to cite each source that you use.
Project 2: Due Friday, December 7, 5:00 pm
Each of the following projec topics asks you to study and discuss a continuous dynamic system model in significant depth and detail. Each team should do a bit of preliminary research on each topic, and decide which topic seems most interesting. A summary report of the models that you plan to study, the references that you plan to use, and any possible extensions and/or modifications of the standard models that you plan to consider is due on Thursday, November 15 in class.
- Study and discuss dynamic system models for the spread of a disease. Two models that you can start with are the SIR (discussed in class) and SIQR (incorporates quarintine) models. Discuss various modifications and/or extensions of these and other models. Some introductory information about each model can be found here.
- Study and discuss dynamic system models of combat and warfare. One model that you can start with is the Lanchester model of combat (see Section 4.4, Exercise 8). Extend the Lanchester model to incorporate such ideas as troop replenishment, guerilla and/or other types of non-traditional warfare, non-combat losses (e.g. friendly fire, sickness, etc.), and/or any other factors that you think may be relevant. This paper is an interesting discussion of the validity of Lanchester's model of combat for the capture of the island of Iwo Jima by U.S. troops.
- Study and discuss dynamic system models of love. Two models that you can start with are the "Romeo and Juliet" and "Laura and Petrarch" models. Some introductory information about dynamic models of love can be found in this paper published in Nonlinear Dynamics, Psychology, and Life Sciences.
Your final paper submission should be at least 10 pages long, and should contain the following components:
- An introduction in which you describe the real-world situation that you are modeling, as well as the history of and motivation for the problem.
- A models and results section in which you present the models that you have studied and describe and interpret the results. This section should also contain a verbal description of any parameters and/or variables that you have introduced. Make sure that you understand the role of each parameter in your models, so that you can effectively discuss what happens when you vary the parameters. I recommend making separate sections or sub-sections for each model that you consider. Start with the standard published models, and try to think about extensions or modifications of the models that would be useful for modeling (such as considering a higher dimensional problem--perhaps a model of warfare between three countries, or a love triangle; consider using different assumptions about the behavior of the system; incorporate additional factors). For each model that you study, include phase plots, equilibrium analysis, x vs. t and y vs. t plots (where appropriate), and any other graphical information that will help the reader understand the models. Be sure to explain all graphs, tables, and figures sufficiently in words. Discuss the sensitivity of the models to the parameters that you use.
- A conclusions section in which you succinctly summarize the models that you have studied. Discuss the main strengths and weaknesses of each model, and any important results that you obtained. No new material should be in your conclusions section. You can also include a sentence or two on future work that you'd like to do regarding the problem you've worked on. For example, is there additional data that would help you construct a better model?
- A references section in which you list all references that you used (in bibliographic format).
Use headings and subheadings to construct an outline of your paper. These should break your paper into smaller pieces, each with a clear focus and goal.
You will be able to find significant amounts of information on each of these project topics on the internet and in the library. Part of your task will be sorting through this information to determine what is relevant and/or appropriate.
Some helpful resources about solving open-ended mathematical problems and about writing mathematical papers can be found at the following links. Although these guides are geared towards students preparing for participation in the Mathematical Contest in Modeling (which I hope all of you are!), they provide excellent general information.
Some sample open-ended problems and solution papers: