**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

E-mail: skaffms@udmercy.edu

** 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.

**TEXTBOOK**: none

**COURSE OBJECTIVES:**

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,

and simulation.

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

problem.

7. Be able to apply
mathematical modeling skills to a new problem and write a project

report.

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

Series competition.

**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

needed.

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.

** PROJECTS:**

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.

**SUBMISSION GUIDELINES**:

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

paper.

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.

**Choice 1**:

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.

**Choice 2**:

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.

**Choice **1:

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

allowed.

c) Instead of using runs scored, use the
number of games won.

**Choice 2**:

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.

**COURSE GRADE**:

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.