Our final exam is Monday, May 2, starting at 8 am in Willard 173.

I will be in my office 2-4:30 Saturday and Sunday for students with questions.

Here are some practice problems

This is the web page for Math 450, taught by Tim Reluga in the spring semester of 2016.

We will learn to model problems and systems using mathematics and computers.
We'll be using the python computer language (which I will
teach everybody at the start of the course). We'll cover statistical models,
cellular automata models, and classical applied-math models. We'll also
discuss the nature of modelling, based on readings from Nate Silver's *Signal
and the Noise*.

Office hours will be Mondays, 1:30 - 2:30, or by appointment.

Computer labs: Jan. 20 and 22, 216 Osmond

Due Wednesday, January 20.

- Read the introduction and chapter 1 of Silver.
- Get and install Canopy on your own computer

Due Friday, January 20.

Due Monday, Feb 1st.

- Read Sections 1 and 2 of Munz et al.'s Zombie theory and this brief review. atleast through model 1.

Due Friday, Feb. 5th.

- In-class presentations (see this guide on how to present a project)

Due Monday, Feb 8th.

- Homework #2 (answers)
- Read chapters 3, 4, and 5 of Silver, prepare for reading quiz.

Due Wednesday, Feb 17th.

Due Friday, March 4th.

- 2nd project from modelling competition

Due Wednesday, March 16th.

Due Wednesday, March 23th.

Due Monday, March 28th.

- Read chapters 6,7, and 8 of Signal and Noise. There will be another brief reading quiz.

Due Wednesday, April 6.

Due Wednesday, April 13.

Due Friday, April 22.

Partner presentation of a new use of python for us.

- Due Friday, February 5.
- 10-minute in-class presentation.
- Teach use something new and useful about scientific computing with python.
- These turned out very well. I've collected all of the presentations into a single file. When I get your code examples, I'll post those also.

Every year, the Consortium for Mathematics and its Applications holds a modelling competition for students interested in applying mathematics.

- Choose one of the contest problems from the list of previous problems.
- Prepare your solution to the problem according to the contest instructions.
- Turn in your solution on March 2nd.
- Topics:
- Jacob: 2010's problem B, Criminology.
- Tim: 2015's problem A, Ebola.
- Mohammad: 1995's solving the helix-plane equation.
- Brian: 2010's problem A, finding a baseball bat's sweet spot.
- Jonathan: 2013's Problem A, Brownie pan design.
- Alexa, Hunter, Sijia: Sudoku problem
- Heng: Bathtub temperature

Possible topics for a third project:

Make a new model that challenges one of the recent models from the Journal of Quantitative Analysis in Sports.

Construct a model for pedal locomotion on mars that will predict how fast humans or some other animal will be able to walk based on physics and geometry.

In synthetic biology, the "repressilator" is a famous example of how nonlinear dynamics can be incorporated as a design tool into the construction of biological machines. Implement a model describing this biochemical, and analyze its properties.

Find an equation for the letter "g" as drawn in the Courier font, explaining how the equation is constructed and represented and can be applied to get equations for other letters in other fonts.

Pick up a hardcover book with the length, width, and height are all unequal. Try flipping this book around each of it's 3 principle axes. Explain the results of your experiments. Then present a set of differential equations that explain our experimental observations.

The KPZ equation is a recently discovered equation that many people are currently studying. Somebody one the Fields medal last year for related research. What is it, and why is it important. (include some math!)

The Ising model is a classic cellular automata model for magnetism consisting of mini-magnets of one of two orientations. Thermal fluctuations randomize the orientation of the magnets, while magnetic attraction tends to allign the mini-magnets. Use simulation to show that the two-dimensional Ising model exhibits a phase-transition similar to that of the percolation model.

How many times do you have to shuffle a deck of 52 playing cards to make sure the cards are randomly mixed? Discuss how you will measure randomness, how different shuffling methods add randomness, and use simulations to support your conclusions.

The BZ reactions are a family of chemical reactions that oscillate -- a phenomena that was long believed to be impossible. The BZ reactions can be explained by the Oregonator model and it's variations (note Tyson's two-variable version, in particular). Discuss.

Fairy circles are a weird natural phenomena observed in the Namib Desert. Make a mathematical or computational model that explains why these circles occur there, and not other places in the world.

When it was first discoverd, there was great controversy bout the nature of HIV. Some scientists argued HIV was a very slow virus because it too decades before people got sick; others argued that HIV was just as fast as other viruses, but was held in check by our immune systems. Nobody new how to test these hypotheses until somebody discovered antiviral treatments for it. Explain how Perelson et al. answered this question using a mathematical model and patient data.

Derive a distribution to predict how many gold medals the US will win in this summer's olympics in Brazil.

In class, we built a differential equation model to describe Huffaker's mite experiments. Build a Markov process model that generalizes this, and use it to estimate the extinction probability for the population over time.

- Lab 1: Introduction to python
- Lab 2: Scientific computing with python

For a formal, structured introduction to computer programming using python, MIT's online course Introduction to Computer Science and Programming is a very helpful reference. Check it out, if you feel like you need more background.

- Introduction
- Geometric curves
- The Black-body theory of global climate - an example model with predictions.
- Lab 1, introduction to python
- Lab 2, Scientific computing with python
- Compartment modelling with differential equations (part 1)
- Compartment modelling with differential equations (part 2)
- Example python code for solving the differential equations from given initial values for the one compartmenta and two compartment models starting from a single dose.

- Predators, Prey, and the Law of Mass action
- Epidemics and Zombies
- Optics, Femat's law, and differential equations
- Pendulum motion
- The hanging chain
- Introduction to probability
- Poisson model for meteors
- Fitting distributions and curves
- Linear Least Squares
- Scaling laws by least squares
- Scaling laws by dimensional analysis 1
- Scaling laws by dimensional analysis 2
- Wrapping up scaling laws and model fitting
- Introduction to Markov Chains
- Introduction to Markov Chains (cont)
- 1/2 Inning of Baseball
- Zipf's Law and Preferential attachment
- Cellular automata models
- Symmetry in discrete systems
- The Heat equation and the Age of the Earth

Canopy - python software for scientific computing that we'll make use of in class.

The journal Nature has posted some python notebooks that you can experiment with like I will in class.

Google's chrome browser has dropped support for MathML equation rendering, so I suggest using Firefox or Safari.

Kaggle public data science site, with competitions.

Command-line murder, a fun way to learn the unix shell (I think?)

Bokeh is a new library for drawing interactive plots in Canopy.

When you are learning to program, one of the things you need to learn to do is to learn to

*READ*programs. This is challenging because you have keep track in your head what's happening inside the computer and out of sight. This is harder when you are learning because you*don't actually understand yet*what's happening inside the computer. One thing that may help is the python code visualizer. Check out this example illustrating how a recursive implementation of Euclid's algorithm for greast common divisors works.Some recent comments about MathML as a browser standard -- this is inside baseball people trying to communicate and teach math.

A whistleblower has shown how flash traders can manipulate the New York Stock exchange to steal money. Great data analysis. See wall street journal or marketwatch for more.

Google's AI win's at Go (sort-of -- only ranks 600th in the world)

Not even wrong, the book mentioned in class today about some challenges in modern physics. Philosophers, historians, and physcists are still arguing about how to resolve their challenges.

A feud is emerging about the CRISPR synthetic biology technology!

Powerball is in the news, which is reporting on excitement over their billion-dollar jackpot. You can find the odds of winning here. Wired has a slightly wrong take-down of the math. But if you want to win, best to be smart about it -- 1, 2, 3, 4, 5. Also, Planet Money's story on the guy who bought all the tickets to win (which is now illegal), just like in the CLASSIC 80's movie Real Genius.

A new video on the beginnings of molten class sewing. There's more stuff like this in the Gallery of Fluid Motion.

Brooksley Born's story on Frontline about the regulation of deriviative markets, and why some people hold it against Larry Summers.

"Big data" is a dominate idea in technological development right now, closely related to our readings of Nate Silver's book. But there is also a lot of pushback right now. Poster-child of model-free big-data inference Google flu trends seemed to work pretty good at first. Now there's another effort-involving critical analysis of this by Olson, Konty, Paladini, Viboud, and Simonsen as well as cheap commentaries by Butler, Lazer, Kennedy, King, and Vespignani, and Tim Harford helping the limitations of data without models.

Stephen Wolfram's enumeration of 1-dimensional spatial rewrite rules.

Nate Silver was in a fight with Paul Krugman over quantitative journalism. Michael Mann also has some tough comments on Nate's book.