Instructor: David A. Meyer
Office hours (Winter Quarter): AP&M 7256, Tu 11:00am-12:00nn, or by appointment
Lecture: Peterson 103, MWF 11:00am-11:50am
Email: dmeyer "at" math "dot" ucsd "dot" edu
TA: Ben Wilson
Office hours (Winter Quarter): TBA
Section/lab: AP&M B432, Tu 5:00pm-5:50pm
Email: bewilson "at" math "dot" ucsd "dot" edu
This course is a focused introduction to mathematical modelling. In 2010 I plan to discuss a variety of mathematical models involving scaling laws. The systems being modelled will be drawn from biology, physics, electrical engineering, computer science, economics, and political science. The relevant mathematical methods include: (systems of) ordinary differential equations, graphs/networks, probability, partial differential equations, eigenvalues/eigenvectors, permutations, and dimension theory.
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 31A), 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).
I expect interest and enthusiasm from the students in this class. 30% of the grade is class participation, which includes occasional homework assignments, often for class discussion. 70% of the grade is based upon a mathematical modelling project for which each student writes a proposal (15%), writes a preliminary report (10%), gives a final presentation (20%), and writes a final report (25%). Some titles of projects from previous years are listed at the bottom of the page.
| 15 Feb 10 |
deadline for applications to the UCLA Institute for Pure & Applied Mathematics (IPAM) Research in Industrial Projects for Students (RIPS) 2010 |
| 31 Jan 10 |
deadline for applications to the Park City Math Institute Undergraduate Summer School Program |
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4 jan 10 DM lecture |
administrative details overview/motivation what is a mathematical model? example of ODE for population growth HWK (for W 6 jan 10). read Bender, chap. 1 find something in the news that suggests a system that could be modeled and be prepared to discuss in class HWK (due M 11 jan 10). collect some population data and see if it can be modeled by the ODE we discussed in class; if not, try to improve the model |
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6 jan 10 discussion |
discussion of systems (suggested by news) that could be modeled mathematically newest versus previous air security measures spread of bomb making information conditions that could lead to nuclear holocaust in multipolar world job satisfaction weight loss relationship between education and poverty action of resveratrol Tasmanian devil facial tumor epidemic impact of jobs program for vets economic effect of change in pizza recipe |
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8 jan 10 DM lecture |
drawbacks of the ODE model for population growth discrete model for population growth introducing elementary probability into the model advantages and disadvantages simulations [Mathematica notebook] direct simulation expectation values and relation to ODE model speeding up the simulation using the binomial distribution |
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11 jan 10 DM lecture |
describing results of stochastic growth model multiple runs, expectation value and variance histogram of outcomes as approximation to probability distribution skewness Extra Credit. Calculate the probability distribution for P(100). |
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13 jan 10 DM lecture |
scaling laws length, area, volume similarity height and weight rowing example from Bender, §2.1 HWK (for F jan 15). read Bender, chap. 2 |
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15 jan 10 DM lecture |
neural adaptation Troxler's effect, fixational eye movements (figures from [1]) feedback model for adaptation [2] r(t) = s(t) - I(t) where I(t) = ∫ tμ(u)r(u)du perfect adaptation μ(u) = 1/τa HWK (due F jan 22). Draft project proposal: Describe the system for which you propose to construct a mathematical model. What question will the model answer? Why is that important/interesting? What features/variables will the model include? What features/variables may be relevant but will be exogenous to your model? What kind of mathematics will you use? If you intend to use real data, describe them and explain how you will get them. Give an approximate timeline for accomplishing the various pieces of your project. If you will be working with someone else, explain how the work will be allocated and coordinated. |
| [1] | S. Martinez-Conde, S. L. Macknik and D. H. Hubel, "The role of fixational eye movements in visual perception", Nature Reviews | Neuroscience 5 (2004) 229—240. |
| [2] | P. J. Drew and L. F. Abbott, "Models and properties of power-law adaptation in neural systems", Journal of Neurophysiology 96 (2006) 826—833. |