Syllabus for Math 343 – Mathematical Modeling Spring 2003
 

Margaret Menzin         Office:S3209                                                                     Phone: X2704
                                         Email: menzin@simmons.edu           Home Phone: 781-862-5107

                                         Office Hours: MWF 7:30-8:00; and
                                                                   M11:00-1:30;
                                                                   W 11:00-3:30
                                                                   F:11:00-11;30; 12:30-2:30
                                          Note: I normally eat lunch at 12:00 or 12:30, but can move that to meet you in the
                                                     middle of the day. I am also usually available after 4:30 on Mondays and
                                                     sometimes later on Fridays.
                                                     The Mathematics and Computer Science Department eats at Bartol on Fridays
                                                      at 11:30. We hope you will join us.
 
 

Text: A Course in Mathematical Modeling by Mooney and Swift (text)

This shift in focus has several important implications: first (and most important) everything is "fair game" - you should expect to use calculus, differential equations, linear algebra, probability and statistics, spreadsheets, Maple, simulations, and anything else you know. Second, we will be using some real data, both to build our models and to evaluate them. Third, there will be a lot of writing and team projects. Finally, if we hit up against some mathematics that you don't know, then we'll learn it. Our aim is to solve these problems, and we will use all available tools.
 
 

Naturally, there are different kinds of models, some useful for some kinds of problems, and some for others. For example,

• Some problems are deterministic (no chance involved) and some are probabilistic.
• Some problems naturally lend themselves to a description which is discrete, others to a description which is continuous, and sometimes a continuous problem is approximated by a discrete solution. (For example, the growth of the US population may be modeled as a continuous process, but if we look at a snapshot of the population every 10 years, we get a discrete approximation.)
• Some problems lend themselves to a linear description or a linear approximation (e.g. the amount of wood needed to make C chairs and T tables might be 6C + 15T square feet of wood), while other problems are intrinsically non-linear (e.g. weather prediction and other "chaotic" problems)
• Finally, some problems are solvable in "closed form" (you are able to solve a system of equations), while in other cases we are forced to use simulation to investigate the range of solutions.

• Learn about the problem
• Figure out what the relevant variables are and how they are related
• Be clear about what simplifying assumptions we are making
• Take our description of the problem and try to see what the model predicts
• See if the predictions agree with what happens in reality
• Modify the model if necessary
• Describe the problem, our assumptions, the model, and the results (predictions and implications.)

Topics:

• What is a model? (Introduction)
  
  
• Discrete Deterministic Models

We will start with discrete linear deterministic models. These are the simplest models. We will look at some examples and use spreadsheets and Maple to investigate them. For linear models, in some cases, there are nice theoretical solutions, which we will discuss. As always, we will start with the problem, develop the model, and scrutinize the simplifying assumptions


• Continuous Deterministic Models

The linear problems may be fairly simple (e.g. exponential decay problems) or complex (double exponential decay and epidemiology problems). The non-linear problems (chaos) are very challenging. We will talk about approximating continuous problems with discrete models (and vice versa).


• Probabilistic Models - including birth and death processes, and simulations

Here we will start with simple problems and work our way up to more and more complex ones (including some queuing theory). Some of these may be solved in closed form, but simulation is an important tool for others.


• Game Theory and Linear Programming

Students with Disabilities: Reasonable accommodations will be provided for students with documented physical, sensory, systemic, cognitive, learning, and psychiatric disabilities. If you have a disability and anticipate that you will need a reasonable accommodation in this class, it is important that you contact the Academic Support Center Director at 617-521-2471 early in the semester. Students with disabilities receiving accommodations are also encouraged to contact their instructors within the first 2 sessions of the semester to discuss their individual needs for accommodations."
 
 

Unit 0: Introduction
Read the article on "100 Million Women Missing". How did Sen arrive at that conclusion?
Read Chapter 0 of the text
Read the Introduction to Wells and Resnikoff Mathematics in Civilization (on reserve in the Library.)

Review the following theorems/topics from calculus for the second class:

• Fundamental Theorems of Calculus
• Mean Value Theorem
• Intermediate Value Theorem
• l'Hopital's Rule
• How to write a function as a Taylor Series (and when you may do so)
• Fundamentals of graphing -

          increasing, decreasing functions;
concave up and down,
where the minima / maxima are
• Basic facts about logarithms

         log functions to different bases are just multiples of each other (why?)
        d(ln x)/dx = 1/x;
        ln (xy) = ln(x) + ln(y)
        and basic facts about exponential functions (analogous to above)

For week 2 please review from linear algebra and statistics/discrete math.

• Basic facts about matrices
• any thing you see in your text about a Markov process
• what is a characteristic value (also called eigenvalue or proper value) of a matrix A, and what
• it means about the associated characteristic vector
• Basic facts about probability -
           how to find P( A union B) and P(A | B )
           what it means to say two events are independent
           anything you may have learned about permutations and combinations
• Pascal's triangle, binomial coefficients, or how to expand (x + y)ⁿ
• Basic facts about the normal distribution
           how to convert between and normal variable X and the standard normal variable Z
           the 68-95-99.7 rule
• the Central Limit theorem
• the Law of Large Numbers
• rules for when you may approximate the sampling distribution with the normal distribution for means
• rules for when you may approximate the sampling distribution with the normal distribution for proportions