MATH 111A (Fall Quarter 2019).
Mathematical Modelling I

Instructor: David A. MEYER
Email: dmeyer "at" math "dot" ucsd "dot" edu
Office hours (Fall Quarter): M 11:00am-11:50am, AP&M 7218; or by appointment
         Section A00: AP&M B412 MWF 10:00am-10:50am
         Section B00: AP&M B412 MWF 9:00am- 9:50am

TA: Linbo LIU
Email: linbo "at" ucsd "dot" edu
Office hours (Fall Quarter): M 2:00pm-5:00pm and Th 2:00pm-3:00pm, AP&M 5412, or by appointment
         Section A01: AP&M B412 Th 4:00pm- 4:50pm
         Section B01: AP&M 5402 Th 1:00pm- 1:50pm

Course description

This course is a focused introduction to mathematical modelling. In 2019 I plan to discuss mathematical models drawn from a wide range of topics, but mostly outside the familiar contexts of the physical sciences and engineering. (For inspiration see [1,2].) I do, however, plan to discuss some models of weather and of climate change. The relevant mathematical methods will 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 (18 or 31A), or consent of the instructor. Please contact me if you are interested but unsure if your mathematics background will suffice.

The textbook is E. A. Bender, An Introduction to Mathematical Modeling (Mineola, NY: Dover 2000). It may be online at SlideShare.

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 below.

I recommend, but do not require, that you prepare your written materials using some dialect of TeX [3,4]. In any case, please do not send me Word documents; convert them to pdf first.

Related events

Jan 31, 2020 Application deadline for Mathematical and Theoretical Biology Institute Summer Program
Jan 6, 2020 Application deadline for Perimeter Institute Theoretical Physics Summer Program
Nov 20, 2019 Application deadline for Halıcıoğlu Data Science Institute Undergraduate Scholarship
Nov 5, 2019 Math Department Undergraduate Colloquium
David A. Meyer
"Quantum data science?"
AP&M, room B402A, 1:00pm-1:50pm
Oct 23, 2019 Application deadline for Hertz Foundation Graduate Fellowship
Oct 3, 2019 Institute for Practical Ethics Colloquium
Anna Lauren Hoffmann
"Beyond Fairness: Justice, Discourse, and Violence in a Datafied World"
SSB, room 107, 4:00pm-5:30pm
Oct 2, 2019 Microeconomic Theory Seminar
Hiroki Nishimura
"Preference structures"
Sequoyah Hall, room 244, 2:00pm-3:20pm

Syllabus (homework in green)

Sep 27, 2019
DM lecture
administrative details
HWK (for M Sep 30).
         Read Bender, Chap. 1. What is modeling?
         Find something in the news or elsewhere that suggests a system that could be modeled (or not);
         email me link (if there is one) and be prepared to discuss in class.

Sep 30, 2019
DM lecture
discussion of possible systems for mathematical modeling
Section A00:
         Trump is hurting the market (Alex RUBER)
         Mathematical model offers new strategies for urban burglary prevention (Kyle SUNG)
         Uber is trying everything to make bikes and scooters a profitable business (Xinyue HE}
         The older the doctor, the higher the patient mortality rate (Enhao MA)
         Hate Speech and Hate Crime (Nicole CHEN)
Section B00:
         The official U.S. poverty rate is based on a hopeles\ sly out-of-date metric (Kevin CHAN)
         Baidu Top (Yue SUN)
         Average Mauna Loa CO2 (Jiarui OU)
         Majority of Americans and Democrats approve of Trump impeachment inquiry (Meghedi ZARGARIAN)
Oct 2, 2019
DM lecture
continuing discussion of possible systems for mathematical modeling
Section A00:
         Math models of flu transmission rates show dramatic savings with universal vaccine (Elisa LAU)
         Donald Trump's long history of racism, from the 1970s to 2019 (Zack JAFEK)
         How Kilauea's lava fed a massive phytoplankton bloom (Dylan VAN VALKENBERG)
         Estimating earthquake casualties and damage cost (Megan TANG)
         Italy, France agree on 'automatic' distribution of migrants (Honghao ZHA)
         McDonald's Is Becoming a Tech Company With Its Latest Purchase (Wayne NGUYEN)
         Just A Handful Of Nuts May Help Keep Us From Packing On The Pounds As We Age (Chris CHENG)
         The US has its first gas station that is fully electric (Emanuela BEALE)
         Pixel perfect: These modeling agencies don't hire real people (Sami WAHEED)
Section B00:
         Attacking Contributions: Markov Models for Football (Taeho YUM)
         African swine fever (SooBin PARK)
         Amazon is purchasing 100,000 Rivian electric vans (Emily REXRODE)
         Black Scholes Model (Weiyuan XU)
         Men's NCAA Basketball Tournament Bracket History (Yunkai CHEN)
         The Art of Creating (Good) Music: Mixing & Mastering (Chuanchuan LIU)
         Air pollution modeling (Rundong ZHONG)
         California fire insurance (Siqi LIU)
         Weather prediction (A Young KIM)
         'Historic' winter storm dumps 3 feet of snow, smashes records in West (Evan CHANG)
         Typhoon (Taekyeong LEE)
         Wealth tax (Rachel LEE)
         fantasy football (Jasmin ZHANG)
         Blended Facial Expressions towards Facial Expression Modeling (Yingkun WANG)
Oct 3, 2019
LL section
LaTeX [template]
Oct 4, 2019
DM lecture
greenhouse effect [5]
         solar flux at Earth
         physics of atmospheric heating
         Stefan-Boltzmann law
Oct 7, 2019
DM lecture
         simple model for Earth surface and atmosphere
         solving the model for temperatures
PGA golf
         description of the game
         Read Bender, Appendix A: Some Probabilistic Background.

Oct 9, 2019
DM lecture
         exploratory data analysis
         basic statistics: mean, variance, covariance, correlation
         statistical model
         fitting the statistical model
         R2 as measure of fit [6]
         [code] [driving accuracy data] [driving distance data] [putting accuracy data] [winnings data]
Oct 10, 2019
LL section
MATLAB [tutorial] [script]
Oct 11, 2019
DM lecture
         towards a mathematical model
         expectation value
         power law distributions
         a revised statistical model
         comparison of R2 as measure of goodness
HWK (for M Oct 14).
         Read Bender, Chap. 3. Graphical Methods, Chap 4. Basic Optimization.
         Read Varian [7], Gray [8] and Goldin [9].
         Begin thinking about the system you want to model for your project.
         Please plan to discuss your ideas with professor or TA next week.

Oct 14, 2019
DM lecture
sample project proposal
Cournot's model of duopoly [10]
HWK (for M Oct 21).
         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?
                 Has anything relevant been done to model this system previously? Give references.
                 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. Please submit a pdf file electronically, ideally from a TeX [3] document.
Oct 16, 2019
DM lecture
Cournot's model of duopoly [10]
Bertrand's observation of instability [11]
Oct 18, 2019
DM lecture
Hotelling's model of spatial competition [12]
Oct 21, 2019
DM lecture
mathematical modeling as metaphor [slides]
Smithies' introduction of elastic demand [13]
Oct 23, 2019
DM lecture
         numerical solution of Hotelling-Smithies model [code]
         application of Hotelling-Smithies as a spatial voting model
         survey data from Morning Consult
Oct 25, 2019
DM lecture
         If candidates are located at A < B < C < D < ..., then exactly AB, BA, BC, CB, CD, DC, ... are posisble (1st,2nd) choices
         inconsistency with Democratic primary data
         combinatorial optimization within Hotelling-Smithies model [code]
Extra Credit (turn in at section on Th Oct 30).
         Can 4 candidates be located in R2 so that each ordered pair is possible (1st,2nd) choices for some voter?
         Prove your answer.

Oct 28, 2019
DM lecture
         continuous and differentiable loss functions
         gradient descent and Monte Carlo methods for optimization
         numerical optimization of loss function [code]
HWK (due F Nov 1).
         Prepare 1 minute "elevator pitch" on your project to present in class Friday.
         See, for examples, UC San Diego Jacobs School of Engineering blog.

Nov 1, 2019
student elevator pitches
Nov 4, 2019
DM lecture
quality measures
         graduate applications data
         discussion of possible models
Nov 6, 2019
DM lecture
         exploratory data analysis
         Model 0
                 unbiased estimate of variance
                 standard error
Nov 8, 2019
DM lecture
                 using χ distribution to estimate standard errors
         Model 1A
                 low rank approximation
HWK (due W Nov 13).
         Progress report
                 describe your model in detail (don't need to include results)
                 where you are relative to proposed timeline
                 revised timeline

Nov 11, 2019
No lecture; Veterans' Day (also Armistice Day) holiday.
Nov 13, 2019
DM lecture
definition of partially ordered set (poset)
         intersection of partial orders
singular value decomposition
matrix approximation theorem
Nov 15, 2019
DM lecture
Model 1A: rank 1 approximation of sparse data matrix
         alternating minimization
         comparison of ordering with Model 0
Model 1B: additive approximation of sparse data matrix
         comparison of ordering with Model 0 and Model 1A
Nov 18, 2019
DM lecture
Model 2
         comparison with Models 0, 1A, and 1B
leave 1 out cross-validation
         comparison of models
HWK (due Week 9, 10, or 11).
         Project presentation
                 13+2 minute oral presentation of your project
                 computer slides recommended; email to me beforehand

Nov 20, 2019
DM lecture
number of human ancestors
         recurrence relation
                 analogy with differential equation
                 how many generations ago must model fail?
number of bee ancestors
         bee biology
         recurrence relation
                 Fibonacci numbers
                 "almost geometric" sequence
                 solution: "Binet's formula"
Nov 22, 2019
DM lecture
         flowers with Fibonacci numbers of petals
         succulent with angles 2π/φ between leaves
         proof of irrationality of φ
         continued fractions
                 for 67/29, for φ, for √2
                 "φ is the most irrational number"
         growth model
HWK (due Friday, Dec 6).
         Final project report
                 Introduction: describe system & explain question and why it is interesting
                 Describe model: what is being included/excluded; how do different pieces fit together; derive model/equations
                 [somewhere in the Introduction or Description, explain previous relevant models & why yours is different]
                 Describe data: what are they? from where do they come? how reliable are they?
                 Analyze model: explain math/computations; give results
                 Conclusion: what is the answer to the question, from results? discuss answer; How might model be extended/improved?
                 References: standard bibliographical format; citations in text; not wikipedia
         Approximately 10 pages; pdf, not Word..
         You can include data/code as separate files, or links.
         Not a diary ("First I did this, then this, ..."); it should read like a scientific paper.
         Write sentences, paragraphs, sections, in the best English you know; not bullet points like on a slide presentation.

Nov 25, 2019
Section A (10:00am)
         Annsley RUBINO, Video games: Does history repeat itself?
         Raul ANAYA VENEGAS, Triple jump optimization
Section B (9:00am)
         Albert CHIU*, A game-theoretic model of protest participation
         Emily REXRODE, Combinatorial optimization of wireless networks
         Guanxin LI*, Beam bridge weight limit
Nov 27, 2019
No lecture.
Nov 29, 2019
No lecture; Thanksgiving (A Native American View) holiday.
Dec 2, 2019
Section A (10:00am)
         Ronan SULLIVAN, Laser ice drill
         Alex RUBER, Modeling the perfect free kick
         Chris CHENG, Maximizing MPG
Section B (9:00am)
         Weiyuan XU, Copula model and its analysis
         Jiarui OU, Buy the anime DVD/BD or not? A decision model
         Jingwei YU, Who is the overpaid player in NBA?
Dec 4, 2019
Section A (10:00am)
         Zheng QU and Zihan WU, Economically efficient earthquake alert system
         Xinyue HE, DOTA 2 analysis
Section B (9:00am)
         Mianlin HU, Starbucks: eco-hypocrite?
         Yujia XIE, Model for SARS Spreading in Beijing
Dec 5, 2019
Section A (4:00pm)
         Nicole CHEN
         Jasmin ZHANG*, Kevin Durant's value to the Warriors
         Enhao MA, The reality of job descrimination
Section B (1:00pm)
         Annie TAN, Mahjong decision model
         Rachel LEE, Temperature influence on methylmercury concentration in marine organisms
Dec 5, 2019
HSS 1305 (6:00pm-10:00pm)
         Shinho JUNG, Electricity efficient model
         Zijing WANG, Mathematical model for rowing competition results
         Joseph HWANG, Fantasy Basketball: rookie prediction model
         Jiaxin LI, Umbrella optimization
         Elisa LAU, Melting Antarctica and rising sea levels
         Pujiang HUANG, Pai Gow poker: how to split
         Yining YUAN, Modeling intrinsic value of a stock
         Tianyi ZHANG, Solar tree optimization in San Diego
         Kyle SUNG, Predicting the next NBA MVP
         Yiwen LI, Topic modeling
Dec 6, 2019
Section A (10:00am)
         Sami WAHEED, Modeling the perfect angle to shoot three-pointers
         Cody EMRICK, Monarch butterfly population
Section B (9:00am)
         Siqi LIU, Mathematical model of relationship between daily routine and health
         Meghedi ZARGARIAN, Predicting changes in price in high frequency trading
         Yingkun WANG, Earthquake explosion triggers earthquake
Dec 11, 2019
Section B Final (AP&M B412, 8:00am-11:00am)
         Chuanchuan LIU, When a new infectious disease attacks
         Rundong ZHONG, Find cheapest cost using graph theory
         Mingqian (Megan) TANG, How data affects the overall rating of King of Glory professional players
         Kevin CHAN, Epidemic modeling
         Taeho YUM, Homerun optimization for batters
         Jiawen ZHU, Mathematical modelling for determining whether sleep duration affects health
         Joshua JUSAY, Combinatorial optimization of owning an electric vehicle
         Huiyi QI, A guide to sourdough starter
         Nick WONG, A dynamic and stochastic model for Lambda phage and Escherichia coli: An insight into coevolution and resistance
         Elizabeth TANG, How to bowl a perfect strike
Dec 13, 2019
Section A Final (AP&M B412, 8:00am-11:00am)
         Vicky VO, Getting a last minute campsite in Yosemite during peak season
         A Young KIM, Car race strategy
         Zack JAFEK, Modeling testing accommodations needs for students with disabilities
         Evan CHANG, Ranking a roller coaster
         Samantha OROZCO, Life-cycle costs of infrastructure
         Kevin VILDOSOLA, Student refractory period
         Donghao XUE, Analysis of the maximum profit of producing glasses
         Jennifer SANDOVAL, JPEG image compression
         Yunkai CHEN, Baby's weight prediction
         Shirley ZHANG, Boba vs. tea consumption
         Yifan WU, Solving the free rider problem in a company with the Multiplayer Boxed Pigs game model
         Yenia GRADA, What is the weather today?: A San Diego weather prediction mathematical model
         Wayne NGUYEN, Ranking the Super Smash Bros. Ultimate characters
         Justin PERALTA, Mastering music? - A mathematical model
         Emanuela BEALE, A mathematical approach to quantifying gerrymandering
         Taekyeong LEE and Soobin PARK, The demand for taco: How many tacos will one eat at the taco shop?

Suggested reading

[1] I. Asimov, The Foundation Trilogy (New York: Gnome Press 1951).
[2] P. R. Krugman, "Introduction to The Foundation Trilogy" (Folio Society 2012).
[3] D. E. Knuth, The TeXbook, Computers and Typesetting, Volume A (Reading, Massachusetts: Addison-Wesley 1984).
[4] LaTeX for Beginners Workbook (2011).
[5] J. Marshall and R. A. Plumb, Atmospher, Ocean, and Climate Dynamics: An Introductory Text (Burlington, MA: Elsevier Academic Press 2008).
[6] E. Berman, J. H. Felter, J. N. Shapiro and E. Troland, "Modest, secure, and informed: successful development in conflict zones", American Economic Review 103 (2013) 512—517.
[7] H. R. Varian, "How to build an economic model in your spare time", The American Economist 41 (1997) 3—10.
[8] N. Gray, "Abstract science", The Huffington Post (2012).
[9] A. Bleicher interview with R. Goldin, "Why math is the best way to make sense of the world", Quanta magazine (2017).
[10] A. Cournot, Recherches sur les principes mathématiques de la théorie des richesses (Paris: Hachette 1838) Chap. VII.
[11] J. Bertrand, "Théorie des richesses", Journal des Savants (1883) 499—508.
[12] H. Hotelling, "Stability in competition", The Economic Journal 39 (1929) 41—57.
[13] A. Smithies, "Optimum location in spatial competition", Journal of Political Economy 49 (1941) 423—439.

Titles of projects from previous years

Last modified: 12 December 2019.