Review

Here is what we covered to date:

Week 1

Monday

  • What is Mathematical Modeling?
  • Population Models (Exponential and Logistic)
  • Microsoft Excel
  • Fittings Curves to data
  • Semilog plots to straighten exponential fits

Tuesday

  • Introduction to VPython
  • Modeling 3D Objects
  • Motion with Constant Velocity
    • Ball(s) bouncing from walls of a box
  • Motion with Constant Acceleration
    • Free Fall
    • Mass on a Spring
  • Simple Programming Structures
    • Objects
    • while loop
    • if statement

Wednesday

  • Review of VPython examples
    • Newton's Law of Gravitation
    • Plotting functions in VPython
  • Introduction of Computer Algebra System, Maple 11
    • Plotting
    • Solving algebraic equations
  • Defined Derivatives and Differential Equations
  • Fourier Series and Fourier Analysis
    • Defined superposition of sinusoidal functions
    • Introduced period, frequency, amplitude
    • Audio files, using Goldwave to create sounds from functions
    • Described tides and tidal contributions from moon and sun
  • Fractal Dimensions
    • What are dimensions?
    • Four dimensional cubes
    • Fractal dimension from Box Counting (log-log plots)
      • LND = const => plot log(L) vs log(N) to find D.

Thursday

  • Analysis of fractal dimension data
    • Correlation coefficient, error in slope, standard deviation
  • Viewing old presentation examples
  • Iterating functions
    • The Logistic Map and cobweb diagrams
    • Iterated Function Systems
    • The Mandelbrot Set, Xaos program
    • Devaney Video - Julia sets and explosions for
      f(z)=cez, (1+ie)sin z, l cos z
  • Exploration time

Friday

  • Example - Epidemic Models (SIR - Susceptible-Infected-Recovered)
    • Using Maple to solve systems of differential equations
    • Plotting actual data with theoretical and numerical solutions
    • Using VPython to obtain similar plots
  • Doing Research in Mathematics
    • Picking research topics
    • How to explore a research problem
    • Using the Internet and other resources
    • Searching the Library, electronic and book resources

Week 2

Monday

  • World on a String
  • Random Variable Models
    • Use random package to plot random numbers
    • Monte Carlo Method to find pi
    • 2D Percolation model on lattice
  • Modeling the Forced, damped nonlinear pendulum
    • Further use of plotting position vs time from simulation data
    • Plot of velocity vs time, phase plot
  • More group work on developing project ideas

Tuesday

  • Lotka-Volterra model for predator-prey dynamics
    • Introduced model
    • Solution using built in ODE integrator from SciPy
    • Plotting tools from Matplotlib
  • Discussion of Python as calculator and string manipulation
  • Implementation of 1D cellular automata with output as strings

Wednesday

  • Research proposals are due!
  • Demonstration of VPython executable modules
  • Sliders and Buttons
    • How to make a simple basketball-hoop game
  • More work on proposals and starting with project ideas

Thursday

  • Research proposals are defended.

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