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[Abu-Mostafa 1989] YS Abu-Mostafa, The Vapnik-Chervonenkis Dimension: Information versus Complexity in Learning, Neural Computation, Vol 1, 1989, pp 312 317. [Abu-Mostafa 1993] YS Abu-Mostafa, Hints and the VC Dimension, Neural Computation, Vol 5, 1993, pp 278 288. [Akaike 1974] H Akaike, A New Look at Statistical Model Identification, IEEE Transactions on Automatic Control, Vol 19, No 6, 1974, pp 716 723. [Amari et al. 1995] S Amari, N Murata, K-R Muller, M Finke, H Yanh, Asymptotic Statistical Theory of Overtraining and Cross-Validation, Technical Report METR 95-06, Department of Mathematical Engineering and Information, University of Tokyo, 1995. [Amari et al. 1996] S Amari, N Murata, K-R Muller, M Finke, H Yang, Statistical Theory of Overtraining - Is Cross-Validation Asymptotically Effective , in DS Touretzky, MC Mozer, ME Hasselmo (eds), Advances in Neural Information Processing Systems, Vol 8, 1996, pp 176 182. [Angeline 1998] PJ Angeline, Evolutionary Optimization versus Particle Swarm Optimization: Philosophy and Performance Differences, Proceedings of the 7th International Conference on Evolutionary Programming, pp 601 610, 1998. [Angeline 1999] PJ Angeline, Using Selection to Improve Particle Swarm Optimization, Proceedings of the IEEE International Joint Conference on Neural Networks, pp 84 89, 1999. [Back 1996] T Back, Evolutionary Algorithms Theory and Practice: Evolution Strategies, Evolutionary Programming, Genetic Algorithms, Oxford University Press, 1996. [Barnard 1991] E Barnard, Performance and Generalization of the Classification Figure of Merit Criterion Function, IEEE Transactions on Neural Networks, Vol 2, No 2, 1991, pp 322 325.

which in one dimension would become:

Now the meaning is not clear. The variables n and k are used in identical ways and it is not clear which is the bound variable. Does the quantification denote a function of n or is it a function of k If we write

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For a simple introduction to these laws see p.66 of the text by Callister [76]. Many different particular solutions of Fick's first and second laws can be found, with perhaps the most common one being the error-function solution for diffusion, described by a constant V, at the interface between two semi-infinite slabs. Such a model is commonly employed to model diffusion at a semiconductor heterojunction (see, e.g. [59,77-80]). Crank [81] and Shewmon [82] offer several solutions based upon numerical methods and involving complex initial diffusant profiles and variable diffusion coefficients. However, as with Tuck [83], each situation is considered separately and a new numerical solution is sought. It would be advantageous if a general, probably numerical, method could be developed which was applicable to all of these situations and also new ones. Such a method would attempt to model diffusion processes, such as those mentioned above, in terms of an average diffusion coefficient which may be a constant or a simple function of the material properties. The mechanisms by which the diffusion proceeds, e.g. vacancy assisted [84], are not of interest. Atomistic models of diffusion do exist and are generally centred around a Monte Carlo simulation of the movement of individual atoms within the crystal (see, e.g. [85]). For the purpose of this present work, solutions are demonstrated within semiconductor heterostructures constructed of an alloy A i _ x Ba;C, where the system can be most generally represented by the z-dependence of the alloy component, i.e. x = x(z). The methods developed are equally applicable to a dopant distribution whose profile may be c(z), as labelled above. 4.2 THEORY As mentioned above, the most general one-dimensional diffusion equation for a diffusant distribution represented by x(z) is given by Fick's second law for non-steady

]T nxk ,

[Battiti 1992] R Battiti, First- and Second-Order Methods for Learning: Between Steepest Descent and Newton's Method, Neural Computation, Vol 4, 1992, pp 141 166 [Baum 1988] EB Baum, On the Capabilities of Multilayer Perceptrons, Journal of Complexity, Vol 4, 1988, pp 193 215 [Baum and Haussler 1989] EB Baum, D Haussler, What Size Net Gives Valid Generalization , in DS Touretzky (ed), Advances in Neural Information Processing Systems, Vol 1, 1989, pp 81 90 [Beasley et al 1993] D Beasley, R Bull, RR Martin, An Overview of Genetic Algorithms: Part I, Fundamentals, University Computing, 1993 [Becker and Le Cun 1988] S Becker, Y Le Cun, Improving the Convergence of BackPropagation Learning with Second Order Methods, in DS Touretzky, GE Hinton, TJ Sejnowski (eds), Proceedings of the 1988 Connectionist Summer School, Morgan Kaufmann, 1988.

state diffusion [81]:

using the symmetry of equality and multiplication to interchange n and k, does this change the meaning Another serious problem is that the scope of the quantifier is not made clear. Does

where t is the time and the diffusion coefficient V could have temporal t, spatial z and concentration x dependencies, i.e. V = D(x, z, t). Given this, then the derivative with respect to z operates on both factors, resulting in the following:

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