MATH 111A (Winter quarter 2006).
Introduction to Mathematical Modelling

Instructor: David A. Meyer
Office hours (Winter quarter): AP&M 7256, T 2:30pm-3:30pm, W 2:15pm-3:15pm, or by appointment
Lecture: Sequoyia 147, MWF 11:00am-11:50am
TA: Jake Wildstrom
Office hours (Winter quarter): AP&M 5301, T 11:00am-12:00nn, Th 12:00nn-1:00pm, or by appointment
Section: HSS 1305, W 7:00pm-7:50pm

Course description

This course is a focused introduction to mathematical modelling. In 2006 I plan to organize the course around some classical and not-so-classical mathematical models in biology. General topics to be covered include: scaling, population models, cycles and spatial distributions. Each of these topics is broader than biology, so I will discuss model of other systems with similar mathematics. Along the way I will introduce basic ideas in probability and statistics (regression, random walks, normal distributions, hypothesis testing), systems of ordinary differential equations, measure theory, and combinatorics.

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, (3) to apply these methods to analyze one or more real problems, and (4) to understand how new mathematical ideas are motivated by such modelling.

The prerequisites are the lower-division math sequence through differential equations (20D) 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 mar 06 application deadline for the PIMS Graduate Industrial Mathematics Modelling Camp
for graduate students, but some of the modelling projects from past years are interesting
see also the PIMS Industrial Problem Solving Workshop
27 feb 06
DM recitation 		application deadline for the Calit2 Summer Undergraduate Research Scholarship Program
15 feb 06 application deadline for the IPAM Research in Industrial Projects for Students, "RIPS" 2006
for undergraduate students; the projects from last year are interesting

Syllabus (homework in green)

09 jan 06
DM lecture
administrative details
what is a mathematical model?
read Bender, chap. 1, section 2.1.
look at news (The New York Times, The San Diego Union Tribune, GoogleNews, etc.)
between today and wednesday to find something that calls for a mathematical model;
be prepared to discuss in section 11 jan 06
11 jan 06
DM lecture
packaging problem from Bender
model weight as a function of height using NHES data
11 jan 06
DM recitation
discussion of homework from 09 jan 06
13 jan 06
DM lecture
due 20 jan 06: compute/estimate constants in Bender's packaging model for flour and milk data [notes3]
metabolic scaling
M2/3 and M3/4 "laws"
model for M2/3 "law"
model for M3/4 "law"
tree model for circulatory system
cylinder model for components of circulatory system
conservation of fluid
16 jan 06
no lecture: Martin Luther King holiday
18 jan 06
JW lecture
introduction to Mathematica [auxiliary materials]
least squares [auxiliary materials]
18 jan 06
no recitation section
20 jan 06
JW lecture
introduction to Mathematica [auxiliary materials]
least squares [auxiliary materials]
23 jan 06
DM lecture
due 30 jan 06: collect/find some data, plot it, and try to fit it
model for M3/4 metabolic scaling "law"
data
tree model for circulatory system
cylinder model for components of circulatory system
conservation of fluid
self-similarity of topology and geometry
25 jan 06
DM lecture
total amount of fluid
the efficiency assumptions
space-filling
putting the pieces together
is this a good model?
27 jan 06
DM lecture
due 3 feb 06: write project proposal [notes6]
model for compound interest
l'Hôpital's rule
differential equation
30 jan 06
DM lecture
same equation models radioactive decay
same equation models population growth
model for population growth with limited resources
(weakly) coupled first order linear differential equations
1 feb 06
DM lecture
overfitting in models
Lagrange interpolation
model for population growth with constantly increasing resources
3 feb 06
DM lecture
age-structured population model
systems of first order linear differential equations
eigenvalues and eigenvectors
6 feb 06
DM lecture
predator-prey model
system of first order nonlinear differential equations
state/phase space
orbits
8 feb 06
DM lecture
equilibrium point
linearized equations
contagious disease models
SIR model
special case of same system as for predator-prey
10 feb 06
DM lecture
phase portrait
What has been left out of these models?
discrete, spatial predator-prey model
simulation [Mathematica notebook]
for discussion 13 feb 06:
adjust the parameters in the simulation code and identify interesting phenomena
Extra credit: figure out how to make one timestep run faster than R(t)F(t)
1 mar 06
DM lecture
mating rules
simulations [Mathematica notebook]
individual preferences
transitive order relation
Last modified: 01 mar 06.