MA 361/519: Introduction to Mathematical Modeling Syllabus Spring, 2011 Instructor: Dr. LaRussa, email: alarussa@uab.edu, phone: 934-2154 Office Hours: MW 4:00-5:30 pm, HHB221 TA: Ajay Mahato, email: amahato@uab.edu, phone: 934-2154 Office Hours: MW 6:45-8:00 pm, HHB221 Class Meeting Time and Location: MW 5:30-6:45 pm, HHB 221 Introduction: Problems in many fields may be represented mathematically, often by an equation or system of equations. The equation or system of equations is referred to as a mathematical model. Mathematical models express relationships among at least some of the variables that are important in understanding a problem. Mathematical models are often used to make predictions and may be used to suggest relationships among variables that may then be tested. You probably have used mathematical models in other classes, and you are certainly familiar with some. The famous equation E= mc 2 is a mathematical model relating energy, mass, and the speed of light. The equation PV=nRT, referred to as the ideal gas law equation, is a mathematical model that relates the pressure, volume, temperature, and number of particles of an ideal gas under certain conditions. Other mathematical models that you may have run across are the equation describing the motion of a pendulum, the equation describing the motion of a spring, and Newton s laws of motion. Mathematical modeling is also important in geoscience, life sciences, economics, finance, weather and climate prediction, and in many other areas. All models require simplifying the system of interest. Thus, all models are, in a sense, incorrect. However, many models are good enough to be very useful. For instance, we can model the motion of a projectile using the very simple equation F=ma, using acceleration due to gravity for a and neglecting air resistance and all forces other than gravity. For some purposes, this is a perfectly good model. In other cases, this model does not give good enough results and we need to take air resistance into account. An important part of model building is validation of the model, determining whether or not the model is correct enough to give valid predictions for a given purpose. In this course, we will learn to construct mathematical models using a variety of mathematical tools, including difference equations, statistics, and cellular automata.
Course Goals Upon successful completion of MA 361 a student is able to translate verbal problem descriptions into mathematical and computer models using appropriate units and mathematical relationships among relevant items; is able to create and interpret multiple representations of mathematical relationships, including schematic, algebraic, graphical, and tabular; is able to view a situation to be modeled from a systems dynamics viewpoint, including causal thinking and closed lop thinking; is able to use quantitative evidence produced by a model as a basis for reasoned argument to sound conclusions, tempered by an understanding of simplifications made in constructing the model; is able to validate a model based upon external evidence, error analysis, understanding of limitations, and robustness of the model; is able to communicate the results of modeling situations mathematically in writing, with appropriate visual aides (graphs, charts, schemata), using proper grammar, diction, spelling, and language appropriate to the audience and the discipline; This course supports the development of quantitative literacy both in general terms and as part of the discipline of mathematics, and the development of technical skill in writing about mathematical models. Material to be Covered We will cover the following topics, using the computer software indicated. 1. Elements of data analysis, principally curve fitting, using EXCEL. 2. Recurrence Relations and Difference Equations, using EXCEL. 3. Introduction to System Dynamics, using STELLA, including a. Simple population models. b. Generic processes. 4. Introduction to Cellular Automata, using EXCEL. a. Simple Averaging Automaton and the Morteville Story 5. Applications of System Dynamics, using STELLA, including a. Advanced population models, including overshoot and collapse. b. Epidemological models, including variations on the SIR algorithm a small group activity. c. Predator-Prey an in-class group activity. d. Drug assimilation (pharmacokinetic) models. 6. System Dynamics Stories (guided projects), using STELLA. There is no textbook for the course. (A reference is Douglas Mooney and Randall Swift, A Course in Mathematical Modeling, Mathematical Association
of America, 1999). Worksheets will be available online using UAB s Blackboard Vista http://www.uab.edu/it/instech/blackboard/a_login.html. The mathematics behind the models will be discussed, assuming knowledge of algebra and functions (including linear functions, polynomials, rational functions, and exponential functions), differential calculus (limits, derivatives as rates of change, and linear approximation), and the definition and interpretation of the integral. UAB Core Competencies UAB has 3 core competencies: writing, quantitative literacy, and ethics and civic responsibility. This course is designed to address both writing and quantitative literacy. Some of our assignments, as well as your final System Story, will require formal papers of some type. Course Policies Attendance Attendance is required. You should sign the roll when you come to each class meeting. Attendance will make up 5% of your grade. Since an occasional absence may be unavoidable, excused absences will not count against your grade. Software and Computer Labs We will primarily use two software packages: Microsoft Excel and iseesystems Stella. You are not required to purchase either, since they are available in our computer labs, but it is strongly recommended that you purchase both. Excel, a part of Microsoft Office, is available inexpensively through the University bookstore, and it is rather widely available in computer labs on campus. (Incidently, the included Microsoft Word will also be useful for preparing your homework solutions.) Stella is available directly from iseesystems (see http://www.iseesystems.com/store/university.aspx); the cost is $129.00 for a student s perpetual license or $59.00 for a student s 6-month license. Stella is available in the mathematics computer lab. This room (HHB221) will be open periodically throughout the week for you to work with the software. A schedule will be posted outside of the door of this room. Outside of these times, the Math Learning Lab will be available to you. The MLL is located in Heritage Hall room 202.
In either our classroom or the MLL, it is your responsibility to secure your work. Do not forget to save your work on a flash drive. This is vital in the MLL, as the computers in the MLL are not able to print. In our classroom, you may print your work, but please only print what is necessary. You are not permitted to print notes, slides of lectures, or work from other classes in this room. Many of our assignments will be submitted electronically. Save your work often. I am not responsible for work that is lost due to computer malfunction. Grades will be based on the following: Grading Participation Your participation grade is based on attendance, participation in Blackboard Vista discussions, and small group activities. These activities are designed to help you learn important concepts and skills in the class, such as understanding feedback loops, understanding and writing difference equations and recursion equations, and making sense of statistics. Assignments Homework will be assigned about once per week. Frequently assignments will consist of two components: computational and communicative. To satisfy the computational requirement, you must submit a working model in the appropriate software. For the communicative requirement, you might be asked to explain how your model works, analyze the behavior of the model, or explain what your model says about the system you are modeling. Note that presentability counts for something; this includes grammar, spelling, and formatting, but nothing fancy is required. You may discuss assignments with other students in the class, as well as with the instructional personnel, and you may work together with other students on the computer. If two (no more, except with advance permission) students work on an assignment together, you may turn in a single partnership assignment with two names. However, you are responsible for learning the material, and you will be expected to perform on your own, particularly on tests and the major project. MA519 students will sometimes have additional components to their assignments.
Midterm Tests There will be two mid-term tests, one near the middle of the term and one near the end. They will require you to demonstrate independent mastery of the concepts in the course, both computational and communicative. This includes being able to use the software used in the assignments efficiently and effectively. For this reason, I suggest that you do most assignments on your own. If you must miss a test, let me know in advance so that arrangements can be made. System Dynamics Stories and Projects: Final Project Due - April 25 Instead of a final exam, you must submit a final project a completed System Dynamics Story. These are scenarios describing realistic situations that you must model. This work is expected to be independent, but you may discuss your project with someone else. As with the assignments, you must submit working models and a written report as the work product. The written report with the final project will be a 5-10 page paper, not including appendices. The model and the paper will receive individual grades. The scope of your final project is much larger than the scope of the assignments, so we will begin work on them about halfway through the semester. Preliminary models (just models not papers) are due on April 11 in class. These should be submitted electronically. There is no grade for this portion; however, I will give feedback. MA519 students will also do a model for a second problem, but a system story will not be required to accompany the second model. Your grade will be calculated as follows: MA361 MA519 Participation: Attendance (5%) 12% Participation: Attendance (3%) 10% Assignments 38% Assignments 35% Midterm 1 10% Midterm 1 10% Midterm 2 10% Midterm 2 10%
Participation: Attendance (5%) Final Project: Model (10%) Formal Paper (20%) 12% Participation: Attendance (3%) 30% Final Project: Model (10%) Formal Paper (20%) Second Model (5%) 10% 35% Your assignment grades will be based on the following: 1. Reasoning, 2. Correctness of your models and the predictions you make from them, 3. Validation of your models, and 4. Written communication. Written communication includes explaining your model and your reasoning in a way that can be understood by a non-specialist and representing your model using graphs, charts, and diagrams that are clear and easy to read. Only a small part of your grade comes from tests. These are important to be sure that you are understanding modeling concepts, such as feedback. Since you are encouraged to collaborate on most work in this class, we need the tests to be sure that you understand basic modeling concepts when working individually. Notice that there is no final exam. Instead, you will choose a final project from a set of System Stories. Your final project will consist of a model and a 5-10 page formal paper. MA519 students will construct a model for a second project (no paper). Late Work Lab activities may not be made up. Assignments should be completed on time. One letter grade will be taken off the grade for assignments turned in one week day late. Two letter grades will be taken off the grade for assignments turned in two week days late. Assignments will not be accepted more than 2 days late. Communication Please use my UAB email address to contact me and Ajay s UAB email address to contact him. Be sure to put MA361/519 in the subject line of your email. In case of emergency, you may leave a message at the UAB math department at 934-2154. Most course materials and assignments and
some lectures will be posted on Blackboard Vista. It is not a good idea to email me through Blackboard Vista.