Math 450, Section 1: Mathematical Modeling (Spring 2017)

Instructor: Dr. Jessica M. Conway.
Lectures: MWF 11:15-12:05, 107 Sackett.
Office hours: McAllister 332, Mondays 1:30-2:30pm + by appointment.
Email: jmconway (at) psu (dot) edu
Phone: (814)863-9125

Course Syllabus
Available HERE.

No required text. We'll be drawing material from a number of sources. A few recommended textbooks that will offer alternative perspectives on the material and may improve your understanding:
  • “Mathematics for dynamic modeling,” 2nd Ed., by Edward Beltrami
  • “An introduction to mathematical modeling” by Edward A. Bender
  • “A first course in mathematical modeling” by Frank R. Giordano, William P. Fox, Steven B. Horton, and Maurice D. Weir
  • “Topics in mathematical modeling” by K. K. Tung
  • These will be placed on reserve at the library.

    Additional materials:
    This course will incorporate computer programming of models using the python computer language. Python is a free and widely used language, so the skills you gain with it will be valuable long after you leave PSU.

    We will be using Canopy, which is free for academic use, available for Windows, Mac, and Linux platforms, and encapsulates all the standard python modules for easy install. Download, install, and try it out, first chance you get. You can get a free academic license for the full version, or download Canopy Express, which has fewer packages, without a license.

    Note: You are not required to use python; if you're already more comfortable with another language used for scientific programming, such as Matlab or C, you are permitted to use that for your homework (and project).


    Exam Dates:

  • Final exam: 4:30--6:30, Wednesday May 3 2017.

  • Practice problems: link.

  • Grading

    10% from quizzes & participation + 50% from homework + 15% from project + 25% from final exam

    Homework and solutions

    Late home work will NOT be accepted.

    Students are encouraged to type up their homework assignment. LaTeX is an excellent option; LyX is a nice LaTeX front-end.

    A few LaTeX resources: The five minute guide to LaTeX, some tutorials, a helpful lab (mac/linux) from PSU's applied math REU, wikibooks on LaTeX.

    And if there's ever a symbol that you need to find the LaTeX symbol for, try Detexify!

    For the first assignment, include code for each problem as the answer to the problem. After the first assignment, if coding is required, please include it as an appendix (for whatever language you use).

    For students using LaTeX, here's the easy way to include your code: the listings package. Put
    in the preamble and then
    \lstinputlisting{relative filepath/}
    in the document. Will automatically embed the code without your having to copy/paste!

    Also works in LyX: add \usepackage{listings} in the preamble, available in Document->Settings, then use ctrl+L to insert latex directly and, in the box that's generated, add \lstinputlisting{relative filepath/}.

    Here's a LaTeX template for your homework, if you find it helpful. Corresponding .pdf output.

    May be submitted via email as a .pdf or in class. Any code should be included at the end, as an appendix, only. Don't forget the plots, tables, etc. asked for in the problems!
  • Homework 1, due 11:15am January 25, 2017. Solutions.
  • Homework 2, due 11:15am February 1, 2017. Solutions.
  • Homework 3, due 11:15am February 10, 2017. Solutions.
  • Homework 4, due 11:15am February 20, 2017. Solutions.
  • Homework 5, due 11:15am March 3, 2017. Solutions.
  • Homework 6, due 11:15am March 24, 2017.Solutions.
  • Homework 7, due 11:15am April 7, 2017. Solutions.
  • Homework 8, due 11:15am April 17, 2017. Solutions.
  • Homework 9, not to be turned in. Solutions.

  • Reading & Quizzes

    Through the semester we will read commentaries, historical perspectives, and philosophical discussions on applied mathematics and mathematical modeling. The aim is to expand your knowledge and understanding of the applications of mathematics. There will be in-class discussion, in addition to occasional reading quizzes to make sure you're on top of the reading. Quizzes will be announced beforehand.

    1. R. Isaacs, On applied mathematics (1979): link.
    Read Sections 1-2, 5-11.
    2. H. Bethe, Comments on The History of the H-Bomb (1982): link.
    3. D. Rines, The Discovery of the Planet Neptune (1913): link.
    On the same subject: Scientific American writings from 2014, and 1847!
    4. P. Munz, I. Hudea, J. Imad, and R. J. Smith?, When zombies attack!: mathematical modelling of an outbreak of zombie infection (2009): link.
    5. N. Silver, The signal and the noise (2012), Chapter 4 "For years you've been telling us that the rain is green": link.
    6. B. Hayes, First Links in the Markov Chain (American Scientist, 2013): link.
    7. M. Sipper and J.A. Reggia, Go Forth and Replicate (Scientific American, 2008): link.
    8. T. von Karman, Some remarks on mathematics from the engineer's viewpoint (1940): link.

    Quiz 1 on Isaacs, Sections 1,2,5: Friday, Jan 20. Answers.
    Quiz 2 on Isaacs, Sections 6-10: Friday, Jan 27. Answers.
    Quiz 3 on Bethe: Friday, Feb 3. Answers.
    Quiz 4 on Rines: Friday, Feb 17. Answers.
    Quiz 5 on Munz et al.: Friday, Feb 24.
    Quiz 6 on Silver: Friday, Mar 17. Answers.
    Quiz 7 on Hayes: Monday Mar 27. Answers.
    Quiz 8 on Sipper & Reggia: Friday Apr 14. Answers.
    Quiz 9 on von Karman: Friday Apr 28. Answers.


    You will do your own mathematical modeling in a project.

    Final projects will be along the lines of the problems in the Mathematical Contest in Modeling, although not all problems there will be projects, and there may be others available.

    You may work alone or in pairs.

    Potential projects: link. Let me know your choice by Friday March 17.

    Evaluation will be based on a written report, an oral presentation (15 min), and a mid-point peer review. We'll discuss further details on Friday March 17.

    Final project guidelines: link

    Presentation schedule: link REVISED DATES: link
    Presentation TIPS: C.J. Lortie, Ten simple rules for short and swift presentations, PLoS Computational Biology 13(2017): e1005373


    Edited from Dr. Timothy Reluga's Math 450 lecture notes.

    Week Online lecture notes (unlinked: tentative outline) Class notes
    January 9
    Drop deadline Jan 14 11:59pm.
    Add deadline Jan 15 11:59pm.
    1. Introduction to modeling.
    2. Introduction to Python: Lab 1.
    3. Introduction to Python: Lab 2.
    January 16 4. Scaling & linear least squares. Lecture 2.
    Lecture 3.
    January 23 5. Dimensional analysis. Lecture 4.
    Lecture 5.
    Lecture 6.
    January 30 6. Self-similarity and Lindenmayer system models.
    7. Equations as descriptions.
    8. Optics and geometric algebra.
    Lecture 7.
    Lecture 8.
    Lecture 9.
    February 6 9. Newton's laws. Lecture 10.
    Lecture 11.
    Lecture 12.
    February 13 10. Very brief intro to ODEs and numerical solutions.
    11. Linear compartmental models.
    Lecture 13.
    Lecture 14.
    Lecture 15.
    February 20 11b. Linear compartmental modeling: Hepatitis C infection.
    12. Nonlinear ODE models: competing species, Brusselator.
    Lecture 16.
    Lecture 17.
    Lecture 18.
    February 27 13. Epidemic modeling & zombies.
    13. When events aren't deterministic: insights from simulation.
    14. Distributions as models.
    Lecture 19.
    Lecture 20.
    Lecture 21.
    March 6
    Spring Break!
    March 13 15. Distributions and data.
    16. Fireball!
    17. Power law and preferential attachment
    Lecture 22.
    Lecture 23.
    Lecture 24.
    March 20 18. Markov chains.
    19. Modeling with markov chains.
    Lecture 25.
    Lecture 26.
    March 27 20. Spatial models. Lecture 28.
    Lecture 29.
    April 3
    Late drop deadline April 7.
    Spatial models continued Lecture 30.
    Lecture 31.
    Lecture 32.
    April 10 21. Modeling with cellular automata
    22. Age of the earth
    Lecture 33.
    Lecture 34.
    April 17 Other topics: difference equations and chaos, lubrication theory...
    Final project presentations
    April 24 Final project presentations

    Jessica M. Conway / Department of Mathematics / Pennsylvania State University