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The first step is to show that they all hold after 0 iterations ie initially This is true of (II) as initially the left deck is empty Similarly, (12) is true initially Property (13) is also true because the combination of the left, middle and right decks is just the middle deck, which is identical to the original deck The next step is to make the induction hypothesis that all three properties hold just before execution of the repeated part Then we examine in turn the two cases considered within the splitting process and show that, in each case, properties (I1)-(I3) remain true after its execution Let us examine just one of these cases to see how the argument goes.

which as mentioned above has error-function solutions for the case of diffusion at the interface (z=0) of a semi-infinite slab of concentration X3 [81], i.e.

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We are assuming that the original deck has been split into three decks the left, middle and right decks and that all the cards in the left deck and none of the cards in the right deck precede X in alphabetical order Let us suppose that cutting the deck reveals a name that precedes X in alphabetical order Then, as the original deck was sorted, the name on every card in the lower part of the middle deck must also precede X Thus, removing the lower deck and the revealed card from the middle deck and appending them to the left deck preserves property (II) The right deck is not affected, so (12) remains true Finally, we leave the deck intact, in the sense of (13), by adding the removed cards to the end of the left deck.

[Cohn 1994] DA Cohn, Neural Network Exploration using Optimal Experiment Design, AI Memo No 1491, Artificial Intelligence Laboratory, Massachusetts Institute of Technology, 1994 [Cohn et al 1994] D Cohn, L Atlas, R Ladner, Improving Generalization with Active Learning, Machine Learning, Vol 15, 1994, pp 201 221 [Cohn et al 1996] DA Cohn, Z Ghahramani, MI Jordan, Active Learning with Statistical Models, Journal of Artificial Intelligence Research, Vol 4, 1996, pp 129 145..

where erfc is the complementary error function see reference [38], p. 295. This technique can be imposed on multiple heterojunctions by linearly superposing solutions [77]. Fig. 4.4 compares just such an error function solution* with the numerical solution for a 200 A single GaAs quantum well surrounded by 200 A Gao.9Alo.1 As barriers after 100 s of diffusion described by a constant coefficient Do=10 A2s-1. Clearly, the numerical method advocated here exactly reproduces the analytical solution.

A similar argument applies in the case that cutting the deck reveals a name that does not precede X in alphabetical order The final step in our use of mathematical induction is to argue that, as properties (I1)-(I3) hold initially, they must, by the above argument, also hold after one iteration of the repeated part and, therefore, after two and three, and so on We conclude that (I1)-(I3) are invariants, ie their truth is independent of the number of times the repeated part has been executed There is one more step remaining before our proof is complete This is to show that the postcondition of the algorithm is a logical consequence of the condition for termination and the invariant property This is clearly true because, on termination, the middle deck is empty, which is equivalent to saying that the original deck has been split into two.

Figure 4.3 Time evolution of adjacent finite slabs of Gao.9Alo.1As/GaAs with a constant diffusion coefficient ofD = D0 = 10 A2s-1

More formally, when the middle deck is empty7, invariant (li) is equivalent to postcondition (Pi), where i is 1, 2 or 3..

[Colorni et al 1994] A Colorni, M Dorigo, V Maniezzo, M TVubian, Ant System for Job Scheduling, Belgian Journal of Operations Research, Statistics and Computer Science, Vol 34, 1994, pp 39 53 [Corne et al 1999] D Come, M Dorigo, F Glover (eds), New Ideas in Optimization, McGraw-Hill, 1999 [Cosnard et al 1992] M Cosnard, P Koiran, H Paugam-Moisy, Complexity Issues in Neural Network Computations, Proceedings of LATIN9S, Sao Paulo, Brazil, 1992 [Costa and Hertz 1997] D Costa, A Hertz, Ants Can Colour Graphs, Journal of Operations Research Society, Vol 48, 1997, pp 295 305 [Cottrell et al 1994] M Cottrell, B Girard, Y Girard, M Mangeas, C Muller, SSM: A Statistical Stepwise Method for Weight Elimination, Proceedings of the International Conference on Artificial Neural Networks, Vol 1, 1994, pp 681-684.

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