Instructor: David A. Meyer
Office hours (Fall Quarter): AP&M 7256, M 1:30pm-2:30pm, or by appointment
Lecture: Solis 110, MWF 10:00am-10:50am
Email: dmeyer "at" math "dot" ucsd "dot" edu
TA: Hooman Sherkat-Massoom
Office hours (Fall Quarter): AP&M 6442, W 2:00pm-3:00pm, Th 1:00pm-2:00pm, or by appointment
Section: AP&M B412, Tu 10:00am-10:50am
Email: hsherkat "at" math "dot" ucsd "dot" edu
This course is a focused introduction to mathematical modelling. In 2012 I plan to discuss mathematical models drawn from a wide range of topics, including biology, traffic, sports, music, economics, political science, and art. 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) and 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.
| 2 Nov 12 |
Submission deadline for Hertz Foundation Graduate Fellowships. |
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28 Sep 12 DM lecture |
administrative details overview/motivation what is a mathematical model? bee foraging [1] the system HWK (for M 1 Oct 12). Read Bender, chap. 1. Find something in the news that suggests a system that could be modeled; email me link and be prepared to discuss in class. |
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1 Oct 12 discussion |
discussion of news items suggesting mathematical models decisions on project funding: "So far unfruitful, fusion project faces a frugal congress" size/spread of wildfires: "Why Western wildfires keep getting worse" decision on position to play in football: "Colts Pagano begins fight with leukemia" ... |
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3 Oct 12 DM lecture |
bee foraging [1] the data |
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5 Oct 12 DM lecture |
possible models |
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8 Oct 12 DM lecture |
simulations [Mathematica notebook] |
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10 Oct 12 DM lecture |
examples of affect in social relations friendships among preschool children international relations of Middle Eastern countries [2] structural balance theory [3,4] relations as graphs signed graph as matrix R with Rij = +/-1 if i likes/dislikes j HWK (to discuss F 12 Oct 12). How could we evaluate/test/improve structural balance models? What does (R2)ij represent in the model? |
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12 Oct 12 DM lecture |
structural balance theory for complete graphs balanced if all triads are balanced if not all triads are balanced, some relationships will change to reduce the number of unbalanced triads discussion of evaluations/improvements of such models HWK (to discuss F 19 Oct 12). Read Varian's article on how to build a model [5]. Read Gray's article on how to write an abstract [6]. Look at some proposals/videos on Kickstarter, e.g., GoldieBlox. 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. Should be 2-4 pages. I prefer that you submit an electronic version. And I'd be pleased if that is a pdf file of a TeX [7] document. If you want to try making a video à la Kickstarter, I'll be pleased to watch it. |
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15 Oct 12 DM lecture |
naïve check for balance of complete graph checks all triads; is O(N3) Structure Theorem [4]: A signed complete graph is balanced (i.e., all its triads have an even number of negative edges) iff its vertices can be partitioned into two disjoint sets Vi such that edges between vertices in Vi are positive and all others are negative. implies O(N2) check for balance of graph proof Weak Structure Theorem: If triads with all negative edges are balanced, then a balanced graph partitions into 1 to N sets of vertices. proof (R2)ij is number of paths i—k—j with ++ or -- signs, minus number with +- or -+ signs significance |
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17 Oct 12 DM lecture |
possible discrete time dynamics for signed complete graphs [8] measure of balance/energy for signed complete graphs = (# unstable triads - # stable triads)/total # triads energy landscapes [9] |
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19 Oct 12 DM lecture |
jammed states in signed graphs [8,9] examples Theorem [9]: E(jammed state) ≤ 0 proof digression on finite fields definition cardinality is pn, p prime HWK (for M 22 Oct 12). Prepare 1 minute "elevator pitch" on your project to present in class. |
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31 Oct 12 DM lecture |
Halloween! [pumpkins] Vi Hart video: "Scary Sierpiński Skull Time" Pascal's triangle, modulo 2 Sierpiński gasket Cantor sets 10 most popular children's costumes 2005—2012 according to the National Retail Federation rankings permutations definition of permutation group right invariant metrics on permutation groups HWK (for discussion in class W 7 Nov 12). Find some system to model using some features of Cantor sets. |
