MATH 111AB (Winter and Spring quarters 2004).
Introduction to Mathematical Modelling

Instructor: David A. Meyer
Office hours (Winter quarter): AP&M 7256, T 12:30-1:30 and F 1:00-2:00, or by appointment
Lecture: HSS 2152, MWF 12:00-12:50

Course description

This course is a focused introduction to mathematical modelling. In 2004 I plan to concentrate on two topics: (1) the use of power laws to model distributions such as city sizes and Internet node connectivities, and (2) the use of diffusion to model the evolution of scalar quantities, like temperature, on fixed structures. Along the way I will introduce basic ideas in probability and statistics (random walks, normal distributions, hypothesis testing), graph theory (traditional and recent random graph models), and analysis (the Laplacian and its spectrum).

The goals of this course are: (1) to explain what it means to construct a mathematical model of some real-world phenomenon, (2) to introduce some of the mathematical ideas that are used in many such models, and (3) to apply these methods to analyze one or more real problems. At the beginning of the course most class hours will be lectures; later, class hours will often involve discussions or brief presentations by students.

I intend this course to be interesting for and accessible to quantitatively oriented social science majors as well as physical science, engineering and mathematics majors. To that end, many of the phenomena modelled will be drawn from geography, economics, political science and communications.

The prerequisites are the lower-division math sequence through differential equations (21D) or linear algebra (20F), or consent of the instructor. Please contact me if you are interested but unsure if your mathematics background will suffice.

The (recommended) textbook is E. A. Bender, An Introduction to Mathematical Modeling (Mineola, NY: Dover 2000).

Related events

15 feb 04 application deadline for RIPS-2004
2 feb 04
Ted Porter 		Graphical reason: a new statistics and the making of the ideal citizen

Syllabus (homework in green)

5 jan 04
overview of course [notes1 (revised 11 jan 04)]
what is modelling? [notes1 (revised 11 jan 04)]
read Bender, Chap. 1
collecting and describing data [notes2]
student height distribution
collect and analyze some scalar data
7 jan 04
more on height distributions
is this a mathematical model?
the binomial distribution [notes3-4 (revised 18 jan 04)]
calculate the expectation value for a binomial random variable
derive probability density for biased coin, or read Larsen & Marx, p. 135-136
9 jan 04
basic probability theory [notes3-4 (revised 18 jan 04)]
expectation value, independence, variance
12 jan 04
Central Limit Theorem
discussion of homework
14 jan 04
ubiquity of the normal distribution? [notes5 (revised 18 jan 04)]
wealth distribution [Mathematica code][notes5 (revised 18 jan 04)]
from the Forbes 400 data, estimate the total individual wealth in the US
what fraction is held by Bill Gates? by the Forbes 400? by the richest 1%? by the richest 10%?
other distributions [notes5 (revised 18 jan 04)]
read Larsen & Marx, Chap. 3
sample 400 points from various distributions and plot log(value) as a function of log(rank)
do the same with the data you have collected
16 jan 04
no lecture today; optional computer lab in AP&M B432
19 jan 04
no lecture today; MLK day university holiday
21 jan 04
rank distribution from probability density [notes6-7]
simple model for wealth distribution [Mathematica code][notes6-7]
modify this simple model in directions that seem reasonable, and simulate
can you generate a Pareto distribution?
prepare a project proposal, due 28 jan by email (due 26 jan if hardcopy)
23 jan 04
more complicated models for wealth distribution [Mathematica code][notes6-7]
city size distribution data
26 jan 04
summary of course to date
positive cut-offs for power-law distributions
log-normal distributions
read Zipf or Krugman (on reserve in S&E)
Due 2 feb 04: experiment with the wealth distribution models, discover something, and write it up
or analyze one of these models without simulating it on the computer
or do the homework problem about the parity anomaly in the height data
28 jan 04
no lecture today; optional but strongly recommended computer lab in AP&M B432
Weeks 4-5
introduction to graph theory
some real networks
lattices; World Wide Web; infrastructural; others to be determined by student interest
models for random graphs
Weeks 6-7
random walks on graphs
standard random walk in low dimensions
general Markov chain analysis
Google page-rank example
the Laplacian and its spectrum
diffusion and the heat equation
Weeks 8-10
student and class projects

References (on reserve at S&E library)

[1] R. J. Larsen and M. L. Marx, An Introduction to Mathematical Statistics and Its Applications (Upper Saddle River, NJ: Prentice Hall 2001).
[2] T. M. Porter, The Rise of Statistical Thinking, 1820-1900 (Princeton, NJ: Princeton University Press 1986).
[3] G. K. Zipf, National Unity and Disunity: The Nation as a Bio-social Organism (Bloomington, IN: The Principia Press 1941).
[4] G. K. Zipf, Human Behaviour and the Principle of Least Effort (Cambridge, MA: Addison-Wesley 1949).
[5] M. Fujita, P. Krugman and A. J. Venables, The Spatial Economy: Cities, Regions, and International Trade (Cambridge, MA: MIT Press 1999).
[6] P. Krugman, The Self-organizing Economy (Cambridge, MA: Blackwell 1996).
[7] F. R. K. Chung, Spectral Graph Theory, Regional Conference Series in Mathematics no. 92 (Providence, RI: AMS 1997).
Last modified: 27 jan 04.