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To balance all of these equations simultaneously, note first that the set of alleles {jx1, jx2} is identical to the set of alleles  since, after crossing-over, the same alleles are still present at the jth positions (though to the right of x they will have been interchanged). Hence |
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Thus, if P(x) = IIjP(jx) for each x, as defined in the statement of the lemma, we have for any x, x1, x2, xx1, xx2, |
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In other words, each of the equations will be balanced if the schemata occur with probabilities l(x) = IIjP(jx);it is also clear that any departure from these probabilities will unbalance the equations in such a way as to result in changes in some of the probabilities of occurrence. Thus, the assignment l(x) is the unique "steady state" (fixed point) of the crossover operator. Q.E.D. |
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We can see from the proof of this lemma that a kind of "pressure" toward the steady state |
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can be defined for each quadruple x1, x2, xx1, xx2 If D¹ 0 for any quadruple then probabilities of occurrence will start changing and there will be a diffusion toward the resultants xx1, xx2 (D > 0) or the precursors x1, x2 (D < 0). For example, if P(x1) > l(x1) while the other components remain at their steady-state values, therewill be a "movement to the right"a tendency to increase the probabilities of the result. The following heuristic argument gives some idea of the rate of approach to steady state from such departures: |
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A given individual has probability 2/M of being involved in a crossover when  contains M individuals (since two individuals are involved in each application of the crossover operator). Thus in N trials a given individual can expect to undergo 2N/M crossing-overs. When N is in the vicinity of lM/2, where l is the length of individual representations, each individual in the population can be expected to have undergone independent crossing-over at almost every position. As a result even extreme departures from steady state should be much reduced in lM/2 trials. |
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