21. Finite Mathematical Models

Three finite mathematical models are examined. The first deals with the discrete-time rabbit reproduction population dynamics model posed by Leonardo of Pisa which gives rise to the finite difference equation the solution of which produces the Fibonacci sequence. In the chapter this is solved both directly in scalar form and by placing it in a system formulation that is then solved by a Jordan canonical form method. Two other methods of solution are introduced in the problems for the system formulation: Namely a Cayley-Hamilton Theorem approach and a direct eigenvalue-eigenvector expansion method. The second model deals with the minimum fraction of the popular vote that can elect the President of the United States posed by George Pólya. The 1960 and 1996 Presidential elections are examined in the chapter while the 2008 election is considered in a problem. The third model deals with the financial mathematics problem of the repayment of a loan or mortgage. A loan shark example with 100% interest rate per pay period is examined in the chapter and a similar one with only a 50% interest rate per period is examined in the last problem.