Instructor: David A. Meyer
Office hours (Fall Quarter): AP&M 7256, M 1:30pm-2:30pm, or by appointment
Lecture: AP&M B412, MWF 9:00am-9:50am
Email: dmeyer "at" math "dot" ucsd "dot" edu
TA: Daniel Minsky
Office hours (Fall Quarter): AP&M 6331, W 12:00pm-2:00pm, or by appointment
Sections: AP&M 5402, Tu 3:00pm-3:50pm, 4:00pm-4:50pm
Email: dminsky "at" math "dot" ucsd "dot" edu
This course is a focused introduction to mathematical modelling. In 2011 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.
| ?? Jan 12 |
UCSD internal submission deadline for Goldwater Undergraduate Fellowships. |
| 16 Dec 11 |
Submission deadline for National Defense Science and Engineering Graduate Fellowships. |
| 3 Dec 11 |
Seventy-second Annual William Lowell Putnam Mathematical Competition Contact the UCSD Putnam advisor, Jacques Verstraete, for Fall Quarter practices and registration. |
| 14 Nov 11 |
First (engineering) submission deadline for NSF Graduate Research Fellowships. Mathematics deadline is November 15. |
| 4 Nov 11 |
Ami Radunskaya (Pomona), "Random dynamical systems: is noisy growth better?" 4:00-5:00pm, AP&M, Halkin room 6402. |
| 1 Nov 11 |
Doug Eastman (MathWorks), "Mathematical modeling with MATLAB" 11:00am-1:30pm, EBU 1, Booker Room 2512; details and registration. |
| 31 Oct 11 |
Submission deadline for Hertz Foundation Graduate Fellowships. |
| 5 Oct 11 |
CSE Social Networks Seminar: Maksim Kitsak (SDSC), "Identification of influential spreaders in complex networks" 10:00am-11:00am, EBU3B, room 4109 |
| 26 Sep 11 |
IGCC Public Lecture: Elisa Bienenstock (Georgetown), "Networked games: empirical and analytical nexus" 4:00pm-5:30pm, IR/PS Robinson Building Complex, room 3202 |
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23 Sep 11 DM lecture |
administrative details overview/motivation what is a mathematical model? peregrine falcon stoop video of peregrine falcon diagram of foveal plane from [1] HWK (for M 26 sep 11). Read Bender, chap. 1. Find something in the news that suggests a system that could be modeled and be prepared to discuss in class. |
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26 Sep 11 DM lecture discussion |
geometrical model plotting the results comparison with data [2] discussion of systems (suggested by news) that could be modeled mathematically effect of homework quantity on learning frisbee flight |
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28 Sep 11 discussion DM lecture |
airline investment in new planes growth of networks consequence of marriage how to crush a can efficiently optimal parking enforcement introduction to probability as counting [3, Chap. 2] HWK (for F 30 sep 11). Read Bender, Appendix 1, 2, 3. |
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30 Sep 11 DM lecture |
probability of drawing k1 + ... + ks = k balls
from an urn with n1 + ... + ns = n ≥ k balls of s different colors sum of probabilities of complete set of disjoint events is 1 definition of conditional probability derivation of Bayes' rule HWK (to hand in at section on Tu 4 oct 11). Prove that when s = 2, the sum of the probabilities for drawing k balls with each different number of white (say) balls is 1. distribution of given names in class data from SSA; plots random "urn model" Mathematica notebook calculating probabilities using CA data (5.8 MB) |
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3 Oct 11 DM lecture |
HWK. Read Bender, Chap. 2 and §4.1. introduction to scaling flow in blood vessels definition of viscosity Poiseuille's law for flow in a cylindrical pipe optimal arterial geometry model [4] flow rate f = kr3 discussion of how to test model prediction |
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5 Oct 11 DM lecture |
neuron phenomenology [5,6] structure of neuron [6, lecture 3] Na+/K+ ion pump [6, lecture 8] membrane potential at diffusion/charge balance [6, lecture 8] increase in membrane potential and openning of Na+ and K+ ion channels [6, lecture 8] threshold potential and spiking [6, lecture 8] release and reception of neurotransmitters [6, lecture 11] leaky integrate-and-fire model ODE [7] |
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7 Oct 11 DM lecture |
discussion of random names model What does it tell us? What else should we calculate? The most likely year when the names are chosen randomly from 1990, say. Mathematica notebook running this simulation. Are there other applications of this model? HWK (due F oct 14). Read Varian's article on how to build a model [8]. 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 [9] document. |
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10 Oct 11 DM lecture |
assumptions in blood flow model generalization of power function to include laminar to turbulent flows [10] and vessel surface to volume costs [Bender, §4.1] resulting minimal power is proportional to l rβ optimal arterial branching model [11] proportionality of power to length implies planarity given fixed endpoints of branches, minimize over branch point location by setting partial derivatives to 0 and checking Hessian is positive definite [12] equivalently by varying lengths of branches |
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12 Oct 11 DM lecture |
integrating the leaky integrate-and-fire model ODE notes on integrating first order linear ODEs numerical simulation for deterministic and random inputs Mathematica notebook simulating LIF model |
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14 Oct 11 DM lecture |
What is the probability of finding m distinct names when drawing k from n = n1 + ... + ns? count the ways event can occur What is the expected number of distinct names? definition of expectation value m = x1 + ... + xs where xi = 1 if name i is drawn at least once, 0 otherwise proof that E[A+B] = E[A] + E[B] expected number of distinct names in class estimating the probability distribution of m by sampling HWK (for discussion F oct 21). Find a system about which this calculation tells us something interesting. |
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17 Oct 11 DM lecture |
optimal arterial branching model [11] continued solution of minimality conditions; consistency check case of equal radius branches use of flow conservation predicted branching angle HWK (to hand in Tu Oct 25). Find the shortest collection of line segments connecting the vertices of a triangle; of a square. |
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19 Oct 11 |
No lecture. |
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21 Oct 11 DM lecture |
possible applications for calculating the distribution of the number of distinct names change in distribution of given names of baseball as professional baseball integrated [13] replace names with ethnic groups evidence for nepotism in Italian university faculties from distinct family names [14] social network of Nobel prize winners in physics and chemistry |
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24 Oct 11 DM lecture |
traffic flow between two nodes (socially) optimal solution by minimizing global cost function differs from solution chosen by individual drivers definition of Nash equilibrium [15] "price of anarchy" [16] traffic flow on a more complicated network |
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26 Oct 11 DM lecture |
solution of LIF model including refractory period for sinusoidal input subthreshold amplitude with noise goes above treshold and spikes intermittently definition of delta function solution of LIF model with spike train input |
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28 Oct 11 DM lecture |
HWK (due W 2 Nov 11). progress report 1-2 pages Where are you on timeline?; revision if necessary. Describe data collected and mathematical formulation of model. Are you stuck on anything? US at war since students in class in elementary school [17] long standing policies to minimize impact on US public [18] casualties of wars in Afghanistan and Iraq online at iCasualties.org modeling insurgency How many attacks tomorrow? Where? What can ISAF do to reduce number of attacks in Afghanistan? timeseries of casualties by province "progress curve" of improvement in repeated task completion suggests Δtn ∝ An-β fits to sequence of gaps between fatal attacks plot of fitted parameter pairs |
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21 Nov 11 lecture |
project presentations Liang Lin Nguyen |
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22 Nov 11 3pm section |
project presentations Bao Elenes Sobin |
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22 Nov 11 4pm section |
project presentations Lu Sclar |
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23 Nov 11 lecture |
project presentations Jansen Johnson Kestler |
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28 Nov 11 lecture |
project presentations Holland Ostrom Souverneva |
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29 Nov 11 3pm section |
project presentations Blain Espinoza/Lopez Feldman |
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29 Nov 11 4pm section |
project presentations Lorch Quigley West |
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30 Nov 11 lecture |
project presentations Haggerty Sauerwald Vena |
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30 Nov 11 evening |
project presentations AP&M 6402, 6pm-9pm Demirchyan Dowling Enriques Lozano Mehta Miramontes Sui/Yadavalli Tang Zhang |
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2 Dec 11 lecture |
project presentations Batino Feeney Nona |