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[Cowan and Reynolds 1999] GS Cowan, RG Reynolds, Learning to Assess the Quality of Genetic Programs using Cultural Algorithms, Proceedings of the International Congress on Evolutionary Computing, Vol 3, 1999, pp 1679 1686 [Czernichow 1996] T Czernichow, Architecture Selection through Statistical Sensitivity Analysis, Proceedings of the International Conference on Artificial Neural Networks, 1996, pp 179 184 [Darken and Moody] C Darken, J Moody, Note on Learning Rate Schedules for Stochastic Optimization, in R Lippmann, J Moody, DS Touretzky (eds), Advances in Neural Information Processing Systems, Vol 3, 1991 [Darwin 1859] CR Darwin, On the Origin of Species by Means of Natural Selection or Preservation of Favoured Races in the Struggel for Life, Murray, London, 1859 (New York: Modern Library, 1967) [Davis 1991] L Davis, Hybridization and Numerical Representation, in L Davis (ed), The Handbook of Genetic Algorithms, Van Nostrand Reinhold, 1991, pp 61 71.

4.5 CONSTANT DIFFUSION COEFFICIENTS The case of a constant diffusion coefficient has already been touched upon in the last section, as its analytical solutions represent an important test for the numerical solution forwarded here. Such a numerical solution can be used to illustrate two specific points for this situation. In particular, the error function solutions given by equation (4.12) imply universality, in that they are dependent upon the value of the product Dot only, and not the individual values themselves. Therefore, a system with a high diffusion coefficient can give the same diffused profile after a short time as a system with a low diffusion coefficient which is allowed to diffuse for longer periods. The continuous lines in Fig. 4.5 display the results of diffusing a single quantum well for 100 (bottom) and 1000 s (top curve) with DO = 10 A 2 s 1 . In comparison with this, the solid symbols are the results of diffusing the same initial system for just 10 s but with D0=100 (circles) and 1000 A2s-1 (squares). Clearly the numerical solution reproduces the universality as well a point which is not at all obvious from the numerical form of the diffusion equation in equation (4.9). Fig. 4.6 displays a succession of diffusant profiles for the same single quantum well as before, again with DQ = 10 A 2 s l , where the evolution of the profile at longer diffusing times is clear. Therefore given a semiconductor heterostructure and an annealing time, it is apparent that the diffused profile can be predicted, provided that

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\ J-ij = o'

Figure 4.5 Diffusion profiles for a 200 A Ga0.9Al0.1As/200 A GaAs /200 A Gao.9Al0.1As single quantum well, with constant diffusion coefficient DO , and DO t= 1000 A2 s 1 (continuous line plus solid circles) and DO t=10000 A 2 s 1 (continuous line plus solid squares)

and you will understand the problem. hi the next section, we introduce a uniform notation for quantified expressions that avoids all the problems mentioned above.

[Depenau and M011er 1994] J Depenau, M M011er, Aspects of Generalization and Pruning, IEEE World Congress on Computational Intelligence, Proceedings of the International Joint Conference on Neural Networks, Vol 3, 1994, pp 504-509 [Di Caro and Dorigo 1998] G Di Caro, M Dorigo, AntNet: Distributed Stigmergetic Control for Communications Networks, Journal of Artificial Intelligence Research, Vol 9, 1998, pp 317 365..

Figure 4.6 Diffusion profiles for the single quantum well of Fig. 4.5 with constant diffusion coefficient D0=10 A 2 s 1 , and diffusing times t=10, 20 50, 100, 200, 500, and 1000 s

Summation and multiplication are just two examples of the quantifiers we want to consider In general, it is meaningful to 'quantify' over a non-empty range with respect to any binary operator that is associative and symmetric Addition, multiplication, equivalence, inequivalence, set union, set intersection, minimum, maximum, conjunction, disjunction, highest common factor and least common multiple are all examples of associative and symmetric operators In each case, it is meaningful (and useful) to consider the operator applied to a (non-zero) number of values rather than just a pair of values Moreover, quantifying over an empty range is meaningful provided the operator in question has a unit2 Most of the operators just listed have units, minimum, maximum and highest common factor being the exceptions.

the diffusion coefficient characterising the process is known from other experiments. The direct determination of the diffused profile on such short length scales is difficult (although not impossible) from traditional techniques used for bulk analysis, such as Secondary Ion Mass Spectroscopy (SIMS) (see reference [7], Section 2.4). However, alternative spectroscopic techniques can be used to infer the extent of diffusion by its effect on other observables. In particular, as mentioned in 3, photolumi-

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