Week 1 -- Introduction and Difference Equations

1/22: Section 1.0

• Introduction
• Mathematical Model: Proportionality
• Syllabus Discussion.

• Mathematical Model: Difference Equation
• Example 1.1.1
• Examples 1.2.1 and 1.2.2

• Example 1.3.1
• Exploration: How does changing the coefficient r and initial condition a₀ of the system a_n+1=ra_n affect the (plot of) its numerical solution?
• Digestion: How does understanding the plots of the graphs help us to model real-life situations?

• Equilibria of difference equations, and stability of such equilibria
• Mathematical Model: System of Difference Equations
• Discussion of Example 1.4.1

          Week 2 -- Probability and Markov Chains

1/29: Section 1.4

• End of discussion of Example 1.4.1
• Group work: Voting Tendancies (Example 1.4.4 and Problem 1.4.5)

• Extremely basic probability
• Mathematical Model: Decomposition of a system into components.
• Components in series and in parallel.

• sequences
• lists
• plotting data

• Difference Equation tools: Eigenvectors and Eigenvalues.
• Mathematical Model: Markov Chains
• Revisiting Example 1.4.1 as a Markov Chain.

          Week 3 -- Applying Mathematical Models

2/5: Sections 2.0, 2.1

• The why and the what of modeling.
• Comparison with the scientific method.
• The pros and cons of various models.

• The how of modeling (the modeling process).
• The sensitivity of a model.
• Group work: Apply the modeling process to a real-world example.

• Discussion of Wednesday's group work.
• Simplification and refinement of models.
• Mathematical Model: Proportionality
• Proportionality as an assumption.
• Data transformation.

• Discussion of Vehicular Stopping Distance Example in Chapters 2.1 and 2.2.

          Week 4 -- Determining Proportionality Relationships

2/12: Section 2.3

• Mathematical Model: Geometric Similarity
• Example: Fishing Derby

• Example: Automobile Gasoline Mileage
• Dimensionless Products

• Mathematical Model: Dimensional Analysis
• Buckingham's Theorem.

          Week 5 -- Model Fitting

2/19: Section 3.0

• Various uses of data collection.
• Extrapolation vs. Interpolation.
• Errors inherent in the modeling process.

• Fitting models visually.
• Fitting models analytically.
• Chebyshev, minimizing sum of absolute deviations, method of least squares.

• Linear Algebra
• Transforming Data
• Least Squares

• Least squares and multivariate calculus.
• Chebyshev and linear optimization.

          Week 6 -- More Fitting

2/26: Section 3.4

• Group Work: Error Boggle.
• Group Work: Choosing a best model.

3/1: EXAM 1
 
3/2: Sections 3.4, 6.3

• Deciding if refinement is necessary.
• Residual plots, and patterns within.
• Linear Regression.

          Weeks 7 & 8 -- Empirical Models

3/5: Sections 4.1, 4.2

• Mathematical Model: A one-term model
• Ladder of powers, simple transformations.
• Pros and cons of one-term models
• Mathematical Model: A high-order polynomial

• (Choice day 1 of 2, see 3/30.)
• Graph Theory
• Snow removal and Eulerian Circuits
• Traveling Salesman Problem and Hamiltonian Paths
• Matching Theory

• Pros and cons of high-order polynomials
• Mathematical Model: Splines

• Mathematical Model: Low-order polynomial

• seq, add, solve, and subs
• if-then and for loops
• Empirical Models

• Discussion of results from 3/15
• Pseudorandom numbers
• Section 5.2

          Week 9 -- Probability and Simulation

3/19: Section 5.3

• Mathematical Model: Monte Carlo simulation
• Simulating probabilistic situations using Monte Carlo

• Simulating deterministic situations using Monte Carlo
• Mathematical Model: Random Walk

• Monte Carlo Simulation
• Random Walks

• Queuing theory
• Shipyard queuing simulation, manually.

          Week 10 -- Loose Ends

3/26: Maple Lab

• -> operator
• Shipyard queuing simulation, in maple.

• Optimization in general
• Linear Optimization
• Carpenter Examples

• Queuing Simulation
• Optimization
• Mathematical Model: Linear Program

• (Choice day 2 of 2, see 3/7.)

          Week 11 -- Linear Optimization

4/09: No School
 
4/11: Sections 7.1, 7.2

• Interpreting real-world word problems as linear/integer programs
• Example 7.2.1 -- Chebyshev curve fitting
• Example 7.1.3 -- Space Shuttle Cargo (encoding yes/no as 0/1 variables)

• Groupwork: Transforming a real-world problem into a linear program. (Problem 7.1.3)
• Geometric interpretation of linear optimization
• Solving linear programs geometrically and analytically

• Counting hyperplane intersection points
• When are we guaranteed a solution to a linear program?
• Groupwork: Solving linear programs analytically.
• Groupwork: Sensitivity analysis.

          Weeks 12 & 13 -- Exam 2 and More Optimization

4/09: No School, SNOW DAY!
 
4/18: EXAM 2, Part 1 of 2: Written exercises
 
4/19: EXAM 2, Part 2 of 2: Maple Section
 
4/20: Sections 7.5, 7.4

• Sensitivity analysis
• Simplex method -- geometric interpretation

• Slack variables
• Tableau format of a linear program
• Worksheet: the simplex method

4/25: Class time reserved for group project writeup
 
4/26: Project critique
 
4/27: Job search Q&A: What does a statistician do?
 

          Week 14 -- Miscellaneous Topic: Voting Theory

          Week 15 -- Project Presentations (all week)