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Formally, the principle of (simple) mathematical induction is the following. Principle of Simple Mathematical Induction. Let P be a predicate on the natural numbers. Then, with dummy n ranging over the natural numbers, <Vn::P.n) = P.O A ( V n ::P.(n+l) =P.n) . The principle is most often used in the form <Vn::P.n) <= P.O A < V n :: P.(n+l) <=P.n> but we will see, shortly, why it is advantageous to express it as an equivalence. The informal justification for the principle is this. The basis is the proof of P.O, and the induction step is the proof that, for arbitrary n, P.(n+l) <=P.n. Having proved the basis and the induction step we know

Noting that z" = z', then:

belief space at generation g population at generation g n-th individual in a population at generation g change in n-th individual in a population at generation g i-ih gene of n-th individual for generation g complement of Cg,ni, if binary variable n-th individual in a population fitness function for an evolutionary algorithm generation counter gene index number of genes in an individual mask vector index for individuals in a population number of individuals in a population normative knowledge component at generation g normative knowledge component for i-th gene at generation g n-th offspring at generation g n-th offspring probability of cross-over probability of mutation probability of mutation per node in tree for GP reproduction probability probability of cross-over per bit for uniform cross-over situational knowledge component at generation g

and we also know (by instantiating n to 0 in the induction step)

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As a check on the analysis so far, putting = 0 and r" = rL does indeed yield the equivalent equation for the 2D case (equation (5.48)), while putting = 1 and r" = r' gives the 3D case expressed in equation (5.84). Therefore, everything is in place for the evaluation of the third integral, which now becomes:

Unfortunately each of the four terms needs to be handled separately, so writing equation (5.Ill) as: Then the first integral is trivial, while the second can be solved by parts,thus giving:

In this way, one can see (informally) that P. (n+ 1 ) can be inferred by n applications of modus ponens. This, together with the basis P.O, gives a proof of (Vn::P.n), where n ranges over all natural numbers. Example 12.8. In this example we prove that, for all natural numbers n,

B.3.1

The third in this short series cannot be solved analytically; however it can again be manipulated into an integral between 0 and 1, and hence quickly and accurately evaluated. It is a worthwhile exercise to follow this one through; by changing the variable of integration from r L to r", then obtain:

(In the dotdotdot notation, 2 + 21 + . . . + 2n~l = 2 n -l.) There are two steps in the proof, the basis and the induction step. To make the process clear, let us give the predicate a name, say E. That is, by definition, E.n = < Z f c : 0 < f c < n : 2 f c > - 2 n -l Then the proof proceeds as follows.

By putting r" = C,|z'| cosh0, then dr" = |z'| sinh0 d0, and it follows that:

1 74

17

Figure 5.17 Total energy E of an electron bound to a donor as a function of the variable symmetry parameter C for A = 89, 90 and 91 A, thus illustrating the convergence of the two parameter variational calculation

definition } empty-range rule to simplify the summation, arithmetic for the right side of the equality }

Again, making the substitution w = e~e, and rewriting the sinh and cosh terms as in equation (5.57), yields the final form:

0-1-1

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