Topics for Math 320

This is just a rough guide to the topics I hope to cover in the course. I am currently redesigning the course, so some of the topics below may be omitted or replaced by other topics as we proceed.

1. Background
   1. Intro to proofs (see proof recipes reference for details)
      1. modus ponens and =>+
      2. iff
      3. proof by contradiction
      4. proof by cases (or+)
      5. for all + and for all -
      6. there exists + and there exists -
   2. Sets, functions, composition, images, etc (see Set Theory Handout) 
2. Iteration
   1. Iteration Terms
      1. Discrete dynamical systems
         1. Orbits, cycles, fixed points.
      2. seed (0th iteration)
      3. nth iteration (nth term)
      4. orbits
         1. cyclic points, eventually cyclic points, fixed points
         2. cycles, eventual cycles
   2. Examples of Iteration
      1. Collatz
      2. Post's Tag Problem
      3. Sumerian method for square roots
      4. Euclidean Algorithm
      5. Classical Fractals via Geometric Iteration
         1. Sierpinski triangle
         2. Directed Segment Replacement
         • Koch curve
         • Koch snowflake
         • Cantor set
         • Sierpinski triangle by DSR
         • Pythagorean trees (1st Maple project: Pythagorean spiral)
         3. General grid-based hand fractals
         • Sierpinski carpet
         4. fern, M-sets, organic fractals, landscapes, etc.
3. Introduction to Maple
   1. assignment, for, if, proc, lists, sets, basics
4. Metric spaces
   1. definition
   2. examples
   3. properties (open, closed, limits, complete)
5. Contraction Mappings and Hutchinson Operators
   1. definition and examples
   2. contraction mapping theorem (Banach)
   3. Hutchinson operation as contraction mapping
   4. Metric space of compact sets
   5. Hausdorff metric
   6. Attractor of an IFS: fixed point of Hutchinson operation
6. Introduction to Complex Numbers
   1. Standard form: a+bi
   2. +, *, /, in C
   3. Absolute value of z in C
   4. Conjugate
   5. Geometric effect of multiplication and addition
   6. Polar form
7. Affine maps
   1. 2 dimensions, 3 dimensions
   2. Standard form
   3. Matrix form
   4. Geometric form
   5. Complex form
   6. Contraction factor
   7. Determination  from three points
8. IFS
   1. Definition
   2. Deterministic Method
      1. Mr. Face as seed
      2. independence of starting figure
      3. iterations
   3. Guess my IFS
   4. Grid-based IFS fractals & relatives of Sierpinski
   5. Computing an IFS from an attractor
9. Addresses
   1. The Chaos game (Sierpinski via random walk)
   2. Chaos game game (web page)
   3. Base B rulers as IFS attractors
   4. Addresses
   5. Random iteration method for IFS
      1. weighted choices
   6. Fractal Curves
      1. Koch
      2. space filling
      3. Sierpinski
      4. Carpet
   7. Complete description of the MTC
10. Dimension
    1. Topological
    2. Similarity
       1. just touching
    3. Self-affine
    4. Hausdorff
    5. Box dimension/Grid dimension
       1. coast of England
    6. Illustrations via computer
11. Fractal Interpolation Theory
    1. interpolating data points with fractal curves of desired dimensionsf
    2. fractal landscapes
    3. fractal clouds, plasma fractals
    4. fractal image compression
12. Percolation & Diffusion Limited Aggregation
    1. Forest fire spread/conductivity
    2. DLA growth
13. Chaos
    1. conjugacy of dynamical systems
    2. attracting/repelling points
    3. graphical analysis
    4. Definitions of Chaos
       1. Devaney
       2. Banks et al.
       3. Touhey
    5. Chaotic families
       1. quadractic
       2. logistic
       3. tent
       4. shift map
       5. route to chaos
    6. Period doubling 
       1. bifurcation diagnosis
       2. Feigenbaum number
       3. relationship to M-set
14. Detecting Chaos
    1. Chaos game test
    2. Close pairs plots
    3. first return plot
    4. controlling chaos
15. Fractals via other means
    1. HIFS's
       1. definition and introduction
       2. deterministic method
       3. random method
    2. L-systems
       1. Lindemayer & history
       2. definition and implementation
       3. Fractint/Winfeed
       4. classifies fractals via L-systems
16. Number Theory
    1. mod
    2. Pascal's triangle and Sierpinski
    3. Complete description of Sierpinski
17. Cellular Automata
    1. definition
    2. Von Neumann neighborhoods
    3. Moore neighborhoods
    4. outer totalistic rules
    5. notation (rules)
    6. Winarc notation and use
    7. gliders, fractals, Conway's life, Gosper's glider factory, etc
    8. Langton's lambda
18. Complex Fractals
    1. Mandelbrot set
       1. definition
       2. coloring schemes
       3. Fractint/Maple
       4. periods of the bulbs/counting arms
       5. equipotential lines/binary decomposition
    2. Julia sets
       1. definition Julia vs. filled in Julia
       2. connectedness and relationship to M-set
       3. Tan Lei's Theorem 
       4. Julia sets via IFS
    3. Newton's method and associated fractal
19. Strange Attractors
    1. 2 dimensional discrete systems
    2. Henon's attractor
    3. Lozi strange attractor
    4. Continuous dynamical systems and diffeq's 
       1. Rossler attractor
       2. Lorenz attractor
20. Other topics as time permits

     

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This page was last  updated on Monday, January 30, 2006 08:06:07 PM. © Ken Monks