COURSE DESCRIPTION: This course is designed to introduce students to concepts of mathematical modeling on a variety of interesting and different applications. These will include (but not limited to) the study of an arms race, ecological, social choice and voting, and Markov process models. It is hope that these applications will excite the student to investigate a career involving mathematical modeling and open his/her eyes to methods used to solve real world problems. Mathematical modeling provides the tools needed to study and analyze possible unsolved problems. It helps to develop the student to become an intelligent guesser. After all, if we know the answer to a problem, why are we solving it? Therefore, the only problems we need to solve are those with no (known) solutions. Many times these solutions come from intelligent guesses. It is both a fun and practical course for students of all disciplines.
PREREQUISITES : MTH 1110, MTH 1120, MTH 1400 or equivalent
INSTRUCTOR: Dr. M. S. Skaff Office: E204 Telephone: 313-993-3376
WEB ADDRESS: http:/skaffms.faculty.udmercy.edu/index.html
OFFICE HOURS: 11:00-11:45 am MWF, 1:00-2:00 pm, 4:00-5:00 pm MW, and anytime my office door is open or by appointment.
1. Understand the concept of a mathematical system and models when attempting
to study some phenomenon or situation in the real world.
2. Understand the four different types of models: deterministic, axiomatic, probabilistic,
3. Learn to abstract a mathematical system from a real world problem.
4. Learn how to convert the mathematical system into mathematical conclusions.
5. Understand the process of running experiments and testing on a real world problem
resulting in real-world conclusions.
6. Understand how to compare the mathematical and the real-world conclusions and
determine if the mathematical model is a good representation of the real world
7. Be able to apply mathematical modeling skills to a new problem and write a project
8. Illustrate concepts by studying an arms race model, interactive species ecological
model, social choice and voting model, and Markov chain models such as in a World
COURSE OUTCOMES: After taking this course, students will be able to understand:
1. What is a mathematical system? What is mathematical modeling?
2. How to establish problem assumptions and constraints. Know why they are
3. How to develop mathematical equations and concepts which characterize the
real problem being solved.
4. Interpret and analyze the solutions to equations in the mathematical system.
5. How to test the solutions to the mathematical system and draw conclusions.
6. How to perform experiments on the real world problem to draw conclusions.
7. Whether the mathematical system is a “good” representation of the real world
problem using the conclusions obtained from both the system and the problem.
8. How to write a report on mathematical modeling project as a solution to a
(possible unknown) real problem.
EXAMINATIONS: There will be no in class exams.
There will be two (2) projects for the course in which all students must participate. The purpose of the project is to give the student practical experience in developing a solution to a real-world problem. Each student will create a project paper which will document the solution to the project problem. The second and final project will serve as the final exam for the course.
The grade of each project will reflect the quality and completeness of the analysis. A project with minimal effort and more incomplete analysis will receive a lower grade. A no answer to the following questions will result in a lower grade. Did the paper follow the submission guidelines (see the next section)? Did the project state all assumptions and requirements as well as state the purpose of the project? Did the analysis completely evaluate the objectives of the project? Did the project utilize appropriate testing methods? Did the project have a conclusion?
Note that just answering a question Yes or No, is an incomplete answer. Give a reason! This will apply to all parts of the project.
Each project will follow the guidelines:
1. The first page of report is the title page. The TITLE of the project is centered,
top to bottom, left to tight. Student name and course number is positioned in
lower right corner.
2. The second page of the report is a brief overview of the project entitled
“Summary for the Executive Reader.” It should state the objective of the project,
what analysis methods were used, and what results were achieved.
3. The third page begins the Analysis Section.
4. A the end of the analysis section, there should be a conclusion page.
5. After the conclusion page, is an Appendix where any special pictures, graphs,
extensive output, or any other material that is referenced in the other parts of the
6. Academic Integrity signature page: Type the following statement and sign it.
“I have not given assistance nor participated in the completion of any
other project for this class. This paper is my own work.
Name ________________________________ Date __________ “
PROJECT 1: (due date 02/17/10)
There are two choices for Project 1. These are based on the arms race model given in class. Only one must be submitted.
Extend the Richardson model to the situation of 3 nations. Derive a set of differential equations if the three are mutually fearful so that each one is spurred to arm by the expenditures of the other two. Examine the stability question for this example. Also derive equations if two of the nations are close allies who are not threatened by the
arms buildup of each other but are threatened by the expenditures of the third. Discuss the possibilities for stability in this case.
Suppose the underlying differential equations have the form
dx/dt = a(y*y) – mx + r, dy/dt = b ( x*x) – ny + s,
where a, b, m, and n are positive. Sketch the stability curves dx/dt = 0 and dy/dt = 0.
How many stable points are there? Discuss the outcome of such an arms race for various intersections of the stability curves.
PROJECT 2: (due date 04/21/10)
There are two choices for Project 2. One of these is based on the Tennis competative model given in class. The other is required for a computer science major. This project requires that programming be used. Languages such as C, C++, Java maybe used. Only one must be submitted.
Analyze World Series competition in the spirit of the tennis examples as an absorbing Markov chain. Let p be the probability of winning any particular game. Determine the transition matrix P and associated matrices Q, N, and B. Show that for p belonging to
[ 0.5, 1.0] , the expected length of a World Series is a monotonically decreasing function of p. Thus p can be determined from the observed average length of World Series competition. Does this value of p predict closely the number of 4, 5, 6, and 7 game series that have occurred? Is there some way of estimating p without relying on World Series information? Here are some possibilities:
a) Let A be the average number of runs scored in the season by the American League
pennant winner and let L be the similar number for the National League
counterpart. Let p, the probability that the American League champion wins a
a given game, be A / ( A + L ).
b) Instead of using runs scored, use the difference between run scored and runs
c) Instead of using runs scored, use the number of games won.
Write a computer program which implements a simulation of tunnel with stop lights at each end. The road leading to the tunnel is a two-way road. The tunnel itself can contain only one car. Utilize synchronization methods such as semaphores or monitors to control traffic lights and prevent deadlocks. See instructor for background on this choice.
The final grade for the course will be based on the percentages:
Homework Assignments: 55%
Project Reports: 40%
Class attendance/Participation: 5%
IMPORTANT FACTS: Last day to withdraw with no W: January 15, 2010
Last day to withdraw with a W : April 1, 2010
No Class: January 18, 2010, March 08-12, 2010
ACADEMIC INTEGRITY: Everything submitted for grading is expected to be student’s own work. Anything suspected otherwise will be dealt with according to the College policy.