Day 13 in math modeling

  1. Homework:

  2. How's your complex numbers background?

  3. Eigenstuff, cont.

    1. Great Lakes example - conclusion 
       0.87
 (1.0 0.0)
 0.62
 (-0.8354170822439401 0.5496165014762764)
    2. "check-up" example: finding an eigenvector, in the 3x3 case (check results of p. 114)

  4. Markov Chains: a system moves between states according to transition probabilities (but is always in one of the states). We model this problem with Markov chains.

    1. Three properties (p. 122):
      1. probability of moving from state i to state j is independent of what happened before moving to state j;
      2. conservation: sum of probabilities out of a state is 1
      3. X(t) is the probability distribution vector which describes the probability that the system is in each state at time t. X(t+1)=TX(t).
    2. The transition matrix T is a matrix of probabilities, whose column sums are 1.
    3. Proof that, given an initial vector X(0) of probabilities summing to 1, X(t) will always sum to 1.
    4. Regular matrices and the search for steady state solutions (p. 124). Example illustrating that steady state solutions need not exist: 
          [0   1]
A=  [     ]
    [1   0]
    5. Example: Markovian squirrels in Scotland (Project 3.4, p. 141).
      • Modelling step - computing the transition probabilities
      • Problems of "inappropriate rounding" (p. 128)
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