The projects for Math 4791/5791 should deal with a specific problem that involves modeling with differential, difference, or integral equations. There are four general categories of projects.
Projects may certainly have components from more than one category; in fact, the best comprehensive projects generally combine several of these aspects. For example, you may want to develop a model and then present analytical and/or numerical methods of solution. To the extent that you include modeling in your project (the first component), you must discuss and justify all assumptions that are used to develop the model. Do not simply reproduce the governing equations as presented in another source. To the extent that you include methods of solution in your project (second and third components), you must discuss and interpret the solutions that you obtain. You can never err too far in the direction of discussion, observations, interpretation, and justification of your work!
You may invent your own topics,
take them from the list below, or find them in a text, in journals, or on the
Internet. You should begin talking with
me as soon as you get ideas so we can refine and focus your topic. Once again, start your thinking and planning today!
The first project is due Monday Oct. 13th. The second project is due no later than
December 8th.
As a rough guideline, undergraduate projects should be 8-10 pages long and graduate student projects should be 12-15 pages long. They must be typed neatly (equations may be hand-written), have perfect spelling and grammar, and be fully documented, well organized, and detailed in explanations and conclusions. They should contain graphs, diagrams, figures, or data that are needed for a full exposition. This is a good opportunity to learn a mathematical typesetting package such as LaTeX.
The topic and scope of the paper is probably the most important item to consider. Be sure that your topic covers an appropriate area of math and level of math for this course (if you are not sure, check with the instructor). Note that more is not generally better: Covering too many topics can leave the report sounding choppy with little or no room for motivation or analysis. This is very difficult to evaluate before you start researching/learning and this is one really good reason for discussing your choice of topic with the instructor well before the due date.
The project must have a cover page showing the title and author.
It must have an introduction that explains the problem and gives relevant background information. It should motivate the work (why would anyone be interested in reading this paper?). It should discuss the scope of the work : what exactly will be done in the paper?
The main body should explain in detail the procedures used to solve the problem and present interesting observations that you made.
1. It may contain subheadings and with each subheading a sentence or two (or three or…) should motivate what it is that will be covered in the subsection and how it relates to what has been done, as well as discuss exactly what will be covered in this subsection.
2. Equations are part of a paragraph and should include appropriate punctuation.
3. Equations and pages should be numbered.
4. It is more professional to use the third person or first person plural (we) and not first person singular ‘I’ or second person ‘you’.
5. Cite your references often for several reasons: It gives a reader a resource to refer to to learn more about the topic; it lets the instructor know if you did something original (not required); and it keeps you from getting accused of plagiarism. Plagiarism is a serious violation of university rules and could result in expulsion from the university. Note that for a subsection you could include a statement such as “The work in this section has been summarized from Dr. Little’s book “The Boy and His Swan” [3]”. Then additional citations within the section would only be necessary if you wanted to cite a particular equation for example.
The conclusion must give a concise summary of your results and give possibilities for future work on the problem. The paper must include references that show evidence of library or web research. The paper itself needn’t contain original work.
The following list of topics is merely suggestive. Let it give you some ideas, but don’t be constrained by it.
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Modeling Topics |
Mathematical topics |
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Chemical reactions |
Perturbation methods for ODE’s |
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Epidemic models |
Poincare-Bendixson theorem |
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Nonlinear or coupled oscillators |
Iterated function systems for fractals |
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Traffic flow |
Calculus of variations |
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Pursuit problems |
Nonlinear difference equations |
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Two- and three-species population models |
Floquet theory and Mathieu’s equation |
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Population genetics |
Lyapunov exponents for chaotic systems |
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Applications of compartments models |
Analysis of 2-D linear systems of ODE’s |
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A particular nonlinear dynamical system |
Analysis of the logistic map |
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Enzyme kinetics |
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Nonlinear circle maps (biological rhythms) |
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Economic models (micro, macro, investment) |
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GRADING This is a rough guideline I use for grading:
Topic:(15 pts) Is the topic of interest? Is it appropriate for this class? Is the topic advanced enough? Too advanced? In general papers written by graduate students should be at a higher level than undergraduates.
Scope:(15 pts) Is the scope of the project so broad that the paper becomes convoluted? It is very hard to make the scope too narrow.
Length:(10 pts) Length of paper should be appropriate: roughly 8-10 pp for undergraduates and 12-15 for graduate students.
Reference/Citations (10 pts): However if a paper is plagiarized this will result in a grade of 0. If absolutely no references or citations are provided, the paper can receive a grade of at most 50%.
Grammar (10 pts); Spelling; complete sentences, etc. If grammar is so poor that it impedes the understanding of the paper then this could be worth more.
Organization (15 pts): Does the paper have an overall nice flow? Can a reader follow along easily, know what is being done and why? This includes the introduction and conclusion, and explanations at the beginnings of each section.
Writeup/Analysis (20 pts): Can the reader follow from paragraph to paragraph, equation to equation, and know what is happening? Is there some motivation as to why we are going to the next equation? Are variables well-defined? Is there analysis as to what the results mean?
Originality (5 pts): Is there something in the paper which makes it original in the sense that it’s above what one would normally find in a text book on the topic? This could include writing your own code and analyzing the results; being critical of a model (where would the model be applicable? Where would it not? How could it be improved?), or analyzing specific problems to learn more about your topic. If a paper is very original then this part could be worth much more.
Notes on Plagiarism
From the CU-Denver’s Academic Policies and regulations: Plagiarism is the use of another person’s distinctive ideas or words without acknowledgement. The incorporation of another person’s work into one’s own requires appropriate identification and acknowledgement, regardless of the means of appropriation. The following are considered to be forms of plagiarism when the source is not noted:
1. Word-for-word copying of another person’s ideas or words.
2. The mosaic (the interspersing of one’s own words here and there while, in essence, copying another’s work).
3. The paraphrase (the rewriting of another’s work, yet still using their fundamental idea or theory).
4. Fabrication (inventing or counterfeiting sources)
5. Submission of another’s work as one’s own.
6. Neglecting quotation marks on material that is otherwise acknowledged.
Acknowledgement is not necessary when the material used is common knowledge.