Math 381,
Autumn Quarter, 1999
Brief summary of subjects covered in each class
1M - 9/27/99
- Overview of course
- Discussion of Life boats and life vests on the ferry, introduction to
some of the issues which arise in modeling. This model is in the Lecture
Notes available in the Engineering Library Copy Center.
Handout #1 contains homework for Wednesday and Friday.
1W - 9/29/99
- Basic graph theory notation
- Network flow problems based on map of the west
- Shortest path and traveling salesman problem
1F - 10/1/99
- A job scheduling problem at a printing plant.
- The branch and bound algorithm for solving this problem
2M - 10/4/99
- Continued discussion of the branch and bound algorithm
- P, NP, and NP-complete problems
Handout #2 contains homework for Friday and Monday.
2W - 10/6/99
- Description of Dijkstra's algorithm and examples
Handout #3 contains homework for next Wednesday.
2F - 10/8/99
- preliminary discussion of simulation and Monte Carlo methods.
3M - 10/11/99
- Some information about class projects, see
Handout #4.
- We exchanged papers on Dijkstra's algorithm to read and comment on.
3W - 10/13/99
- Part 2 of the Lecture Notes (pages 23-42) are available at the Copy
Center.
- I brought in a laptop and projector and we explored the uniform
distribution, histograms of simulations, the behavior of averages, and the
Central Limit Theorem
3F - 10/15/99
- More discussion of probability distributions and the central limit
theorem.
- Discussion of the fish tank Monte Carlo simulation.
- Handout #5 contains homework on Monte Carlo methods.
4M - 10/18/99
- Poisson processes and Bernoulli trials
- Binomial distribution and Poisson distribution.
- Handout #6 contains a writing assignment.
4W - 10/20/99
- Finished deriving Poisson distribution from binomial
- Exponential distribution of inter-arrival times for queuing theory
simulation
- Converting a uniformly distributed random variable into an
inter-arrival time
4F - 10/22/99
- Homework from Handout #5 due.
- Brief discussion of kinetic Monte Carlo methods in chemistry and
material science. (Motivated by a recent
talk
by Prof. Hannes Jonsson of the Chemistry department).
- A first pass at a Monte Carlo simulation of the deposition and
suface diffusion of atoms on a one-dimensional layer can be found on the
Handout page.
- Begin discussion of Markov chains.
5M - 10/25/99
- Markov chains, transition probabilities
- Eigenvalues and eigenvectors
5W - 10/27/99
- Project proposals due
- Continued discussion of Markov chains and eigenvalues, using
a 2-state example modeling ``wet'' and ``dry'' days.
- Handout #7 contains a homework assignment on Markov chains.
5F - 10/29/99
- Homework from Handout #6 due.
- Markov chain examples:
- Surface diffusion of platinum atoms as an example with two absorbing
states and two eigenvalues equal to 1.
- Work schedule example with eigenvalues +1 and -1 to illustrate
oscillatory behavior.
- Extension of this example in which employees work 3 days and are off
1 day. This gives a
transtion matrix with eigenvalues 
and 
, showing that eigenvalues (and vectors) can be complex.
6M - 11/1/99
- More class notes are available
in the Engineering Library Copy Center (Packet #3).
- Age-structured ecology models and Leslie matrices.
- Example on birds from the class notes and on blue whales from a
handout in class, pages 252-255 from [SK87].
6W - 11/3/99
- Revision of Dijkstra homework due.
- Introduction to linear programming using the pig farming model from
the notes.
- Graphical approach to problems with two decision variables.
- Handout #8 contains a homework assignment on linear programming.
6F - 11/5/99
- Homework from Handout #7 due.
- Revised project proposal due.
- Handout on other linear programming problems to think about and discuss
next week: A transportation problem from [HL95] and
Models 3 and 11 from Prof. Burke's 407 page.
- More discussion of the pig farming model and graphical solution.
- What happens with more constraints or more variables.
- Number of intersection points if we enumerate all possible solutions.
- Brief description of simplex method.
7M - 11/8/99
- Handout #9 contains a summary of material for the midterm.
- Handout #9a contains some sample problems.
- Discussion of setting up Model 11 as an LP problem.
- Brief discussion of problems with nonlinear objective function or
constraints.
7W - 11/10/99
- Solutions to homework on Markov Chains
7F - 11/12/99
- Solutions to homework on linear programming
- Formulation of linear programming: The tollboth scheduling problem
8M - 11/15/99
8W - 11/17/99
- Solutions to midterm
- Formulation of the Traveling Salesman Problem as an integer
programming problem.
8F - 11/19/99
- Handout distributed: Pages 116-130 from Dykstra[Dyk84].
- Branch and bound algorithm for integer programming
- Cargo plane loading
9M - 11/22/99
- Homework due Wednesday: Do reading on Forest resource management and
turn in three questions or comments on mathematical modeling in this
field based on the reading.
9W - 11/24/99
- Discussion of linear programming for timber harvest scheduling.
9F - 11/26/99
10M - 11/29/99
- More discussion of forest resource management,
- General discussion of mathematical modeling
- Homework due next Monday: Read Chapter 5 of the notes (on evolutionary
trees) and turn in two questions or comments on this material.
10W - 12/1/99
10F - 12/3/99
- No class
- Turn in first draft of project by 4:30pm to Lian in GUG 408B
11M - 12/6/99
- Evolutionary trees: Chapter 5 of notes
11W - 12/8/99
- Evolutionary trees: parsimony, Fitch's algorithm, enumerating trees
- Some links for more information: See
Tuesday 12/14/99
- Final draft of project due by 5:00pm
References
- Chv83
-
Vasek Chvatal.
Linear Programming.
W. H. Freeman, New York, 1983.
T57.74.C54.
- DA93
-
A. Dolan and J. Aldous.
Networks and Algorithms, An Introductory Approach.
John Wiley & Sons, Chichester, 1993.
- Dyk84
-
D. P. Dykstra.
Mathematical Programming for Natural Resource Management.
McGraw-Hill, 1984.
- GJ79
-
M. R. Garey and D. S. Johnson.
Computers and Intractability: A guide to the Theory of
NP-Completeness.
W. H. Freeman, San Francisco, 1979.
- Gus97
-
Dan Gusfield.
Algorithms on strings, trees, and sequences : computer science
and computational biology.
Cambridge University Press, New York, 1997.
- HL95
-
F. S. Hillier and G. J. Lieberman.
Introduction to operations research.
McGraw-Hill, New York, 1995.
- Jef78
-
J. N. R. Jeffers.
An Introduction to Systems Analysis: with Ecological
Applications.
University Park Press, London, 1978.
- LLAS90
-
E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy Kan, and D. B. Shmoys.
The Traveling salesman problem : a guided tour of combinatorial
optimization.
Wiley-Interscience, Chichester, 1990.
- Mak73
-
D. P. Maki.
Mathematical models and applications, with emphasis on the
social, life, and management sciences.
Prentice-Hall, Englewood Cliffs, 1973.
- Min78
-
E. Minieka.
Optimization algorithms for networks and graphs.
Marcel Dekker, New York, 1978.
- Ros90
-
S. M. Ross.
A Course in Simulation.
Macmillan Publishing Company, New York, 1990.
- SCCH97
-
G. C. Smith, C. L. Cheeseman, and R. S. Clifton-Hadley.
Modelling the control of bovine tuberculosis in badgers in england:
culling and the release of lactating females.
J. Applied Ecology, 34:1375-1386, 1997.
- Sip97
-
M. Sipser.
Introduction to the theory of computation.
PWS Pub. Co., Boston, 1997.
- SK87
-
G. L. Swartzman and S. P. Kaluzny.
Ecological Simulation Primer.
MacMillan Publishing, 1987.
- Ski90
-
S. S. Skiena.
Implementing Discrete Mathematics: Combinatorics and Graph
Theory with Mathematica.
Addison-Wesley, Redwood City, CA, 1990.
- UMA95
-
UMAP Journal.
MCM, The First Ten Years, special edition of the UMAP
Journal.
COMAP, 1995.
- ZP97
-
Z. Zhou and W. Pan.
Analysis of the viability of a giant panda population.
J. Applied Ecology, 34:363-374, 1997.
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