I do not expect you to memorize all of the equations covered in our course, but I do expect you to have an understanding of the assumptions and methods associated with building and analyzing each of the models. (**BEWARE:** Sometimes students think this means they don't have to study the equations. That would be a mistake. The understanding the modelling often requires a corresponding understanding of the equations.)

Most importantly, I hope you can express for yourself some of the broader points various models illustrate.

- Compartmental models
- Markov Chains
- Markov property
- construction in discrete and continuous time
- Master equations
- Steady-state solutions and statistical equilibrium
- Radiometric dating, binomial solution
- Telephone call arrival, poisson solution
- Telephone traffic, Erlang B solution
- Meteor arrival as a Poisson process

- Markov chain state spaces
**Examples of state-space construction**- Why is smaller better
- why might too small be bad

- Dimensional analysis
- buckingham pi
- Dimensionless groups
**examples of generic applications****USE OF dimensional to check equations**

- Geometric optics
- Mirror equation
**Ansatz solutions (guess-and-check!)**- discovery of multiple solutions, interpretation

- Cannons to rocket ships
- Newton's 3 laws
**Euler's equation for Newton's first two laws**- Tartaglia's model
- Galileo's model
- Newton's model
- Bernoulli's model
- Johnson's model
**Converting second order systems into first order systems**

- Hanging Chains
- Force balance derivation
- chain equation
- solution by integration
- experimental fit

- Pendulums and Pendulums period
- pendulum equation by conservation of energy
- small amplitude approximation
- analysis of pendulum period and isochrony hypothesis

- simulation of large Newtonian systems
- points and forces

- Percolation
- site and bond percolation
- critical void-fraction/fill-fraction
- Emergence of threshold conditions as systems get larger

- Cellular automata
**general formulation and updating**- forest fire
- cardiac signalling
- game of life
- Schelling's suburbs model
- complex consequences from simple rules

- Age of the earth
- Heat equation and diffusion
- solution by method of images
- application of data and prediction of a young earth
- model analysis and criticism
- Nuclear reactor problem

- Climate of planet earth
**Energy balance**- Stefan--Boltzmann law
- Seller's theory
- Budyko's theory
- Stefan condition
- Steady-state and Stability analysis
- model analysis and criticism

- Self-similarity and L-systems
- discrete logistic equation
**rewrite systems**- turtle-drawing interpretation

- Metaphors and models

**Warning**: The following problems are offered purely as practice, and are not intended in any way to imply certain questions will or will not appear on your final exam.

Discuss one lecture model in detail, explaining the problem, the modelling approach, and the approach to analysis.

"As to methods there may be a million and then some, but principles are few. [One] who grasps principles can successfully select [there] own methods. [One] who tries methods, ignoring principles, is sure to have trouble." -- Ralph Waldo Emerson (1803-1882) Discuss, using examples from our course.

Imagine you have a car that you park in the same spot each night, and you want to construct a discrete-time Markov chain representing changes in your car's state night-to-night. What state-space would you use?

What general problem were Erlang, Rorty, and Molina trying to solve for their telephone companies at the beginning of the 20th century?

- In our lectures, we discussed three different models of timing and age-estimation: the pendulum, radiometric dating, and Kelvin's theory of the earth's age.

What was the general model framework used in both cases? Can you think of how this framework might be applied to dating problems for one of the following?- Age of a tree?
- Age of the most recent common ancestor of humans and gorillas based on mutations?
- Age of a glacier in antarctica?
- Age of the universe based on distances or cosmic background temperature?

Under what assumptions does a Poisson distribution arrise?

- Find a dimensionless relationship be wave speed, liquid density, and gravity's acceleration to describe the movement of water waves created by a thrown stone.
- What if we also consider the wavelength of the wave? Are there any dimensionless groups now? What are they?

If \(p\) is pressure, \(v\) is velocity, \(g\) is gravity's acceleration, \(\rho\) is density, \(z\) is height, what units if any would we pick for \(C\) such that the following equation makes sense? \[\frac{p}{\rho} + \frac{1}{2} v^2 + g z = C.\]

What must the units of \(D\) and \(r\) be for the following equation to make sense? \[\frac{dH}{dt} = D \frac{d^2 H}{dx^2} + r H\]

If a pendulum swings from spring with rest-length \(L\) and the spring exhurts a force proportional to the length it is stretched from equilibrium, use Euler's equation to find a governing law for the pendulum's motion in cartesian coordinates.

What's the standard form of system of ordinary differential equations solvable with

`odeint()`

in python?Describe the 4 common neighborhoods used for 2-d lattices. For each, give the name (if any), and the stencil on a rectangular grid.

If \(h(p,N)\) is the probability of percolating through an \(N\times N\) square lattice from top to bottom when the fill-fraction is \(p\), how does the shape of \(h(p,N)\) as a function of \(p\) change as \(N\) gets large?

Explain how we know that the critical void-fraction in bond percolation is 1/2.

What was the main message of Schelling's model of suburb occupancy?

Describe the sequence of state-changes in our automata model of electrical signals in the heart. When signal spread was noisy, what happened to the signalling waves?

What's the energy balance equation we started with in Budyko's climate model?

Why did we need to use the Stefan--Boltzmann law in the Sellers and Budyko climate models?

Here is one of the figures from our analysis of Budyko's climate model. Explain where it comes from and what it means.

Why did we need a Stefan condition in Budyko's climate model?

Provide an intuitive explanation of what the dynamics of the heat equation.

Show that \(u(x,t) = \operatorname{erf}(C x/\sqrt{t})\) is a solution of the heat equation \(\dot{u} = k u''\) if we pick \(C\) correctly when \[\operatorname{erf}(x) := \frac{2}{\sqrt{\pi}} \int _ {0}^{x} e^{-u^2} du.\]

Was William Thompson's flat-earth approximation reasonable? Explain.

Consider a rewite system where \(a \rightarrow ab\), \(b \rightarrow bc\), and \(c \rightarrow ac\). Follow two iterations of the initial condition \(abc\) forward.

Find the steady-state of a continuous-time markov chain \(\dot{p} = M p\) where \[M = \begin{bmatrix} -1 & 0 & 0 & b \\ 1 &-1 & 0 & 0 \\ 0 & 1 &-1 & 0 \\ 0 & 0 & 1 &-b \end{bmatrix}\]

An elastic pendulum is made by attaching to a weightless spring of elastic constant k, a box of volume V which is filled with a liquid of density d. The mass of the liquid in the box is acted upon by gravity, and we are required to find an expression for the time of oscillation T. The spring constant k has dimensions of force per distance. Perform a dimensional analysis to obtain a general formula for the period of oscillation.