Download Difference Equations with Public Health Applications by Lemuel A. Moyé, Asha Seth Kapadia PDF

By Lemuel A. Moyé, Asha Seth Kapadia

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The key here is vision. One cannot bring mathematics, such as difference equations, to bear in a public health problem if he or she doesn't really understand the fundamental nature of the problem. It is crucial to examine the public health problem from many directions and perspectives, clearing away that which obstructs our view-we can then see if, at its core, the problem has a recursive relationship that may be represented by a difference equation. 18 Chapter 1 Once the equation is identified, it must be solved.

And see what the coefficients are. If k is odd, the coefficient of sk is zero. On the other hand, if k is even, then the coefficient of sk is one. We may write (2-73) The expression Ikmod2 =o is a combination of two useful functions. Let I^A be the indicator function, equal to one when x = A and equal to zero otherwise. For example, Ix=i,2,3 is a function of x, equal to 1, when x is equal to 1, 2 or 3, and equal to zero for all other values of x. The function k mod 2 is short for k modulus 2, which is the integer remainder of k/2.

This time begin with Sk = 1 + as + aV + aY +aY + ... 1-as + aksk, and write S, =l+as + aV+aV+... 27) allowing the conclusion that G(s) = ——— > l|a k }. Again, as in the earlier case, 1-as ' the inversion of G(s) required the identification of a one-to-one correspondence between the coefficient of sk and the sum of the series. 4 Recognizing Generating Functions - General Principles Much of the effort in the inversion of a generating function involves recognition of key components of G(s). In many of the examples we will be working with in this text, G(s) will at first glance appear complicated.

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