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[Kohonen 1995] T Kohonen, Self-Organizing Maps, Springer Series in Information Sciences, 1995. [Kohonen 1997] T Kohonen, Self-Organizing Maps, Second Edition, Springer, 1997. [Koza 1992] JR Koza, Genetic Programming: On the Programming of Computers by Means of Natural Selection, MIT Press, 1992. [Krogh and Hertz 1992] A Krogh, JA Hertz, A Simple Weight Decay can Improve Generalization, in J Moody, SJ Hanson, R Lippmann (eds), Advances in Neural Information Processing Systems, Vol 4, 1992, pp 950 957. [Kuscu and Thornton 1994] I Kuscu, C Thornton, Design of Artificial Neural Networks Using Genetic Algorithms: Review and Prospect, Technical Report, Cognitive and Computing Sciences, University of Sussex, 1994. [Kwok and Yeung 1995] T-Y Kwok, D-Y Yeung, Constructive Feedforward Neural Networks for Regression Problems: A Survey, Technical Report HKUSTCS95-43, Department of Computer Science, The Hong Kong University of Science & Technology, 1995.

Methods of solution of the Schrodinger equation have centred around two basic approaches. The first of these involves expanding the electron wave function as a linear combination of Gaussian functions [103,104]. While this technique has been successful in calculating the properties of donors in simple quantum well structures, the generalisation to more complex structures, including graded gap materials and systems where piezo electric fields are present, is non-trivial. The other category of approach is based on the variational principle. In this method, a trial wave function is chosen whose functional form may contain one or more unknown parameters. These parameters are varied systematically and the expectation value of the energy calculated for each set. The variational principle [3,29] states that the lowest energy obtained is the closest approximation to the true state of the system. The success of variational approaches centres around the general choice of the trial wave function. A common choice is a product of two terms [105], i.e.

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We conclude this discussion of summation with a warning. The warning is that care must be taken when quantifying over an infinite range. In this case, the value of the expression is defined as a limit of a sequence of finite quantifications, and, in some cases, the limit may not exist. For example, <Zi:0^ i: (-1) 1 ) is not defined because the sequence of finite quantifications (Si:0^ i <N: (-1)'), for N increasing from 0 onwards, alternates between 0 and 1. So, it has no limit. The rules we have given are not always valid when the range of the summation is infinite. The so-called convergence of infinite summations is a well-studied part of mathematics but is beyond the scope of this text. Exercise 11.17. Derive the trading rule (1 1.14) from the splitting rule (11.10). You may assume that (Zfc : R : 0) = 0 for all ranges R. D Exercise 11.18. Prove the generalized distributivity law <2j:P:5>x<2k:Q:r> = (Zj,k:P A Q : S x T ) . What are the side conditions on using this rule n

where r' is the electron donor separation and A is a variational parameter; thus the second factor is a simple hydrogenic wave function. The function u(z) is the uncorrelated eigenfunction, i.e. the straightforward wave function calculated in s 2 and 3, of the electron in the quantum well without the donor [106,107].

1 53

Aristotle observed that human memory has the ability to connect items (e.g. objects, feelings and ideas) that are similar, contradictory, that occur in close proximity, or in succession [Kohonen 1987]. The patterns that we associate may be of the same, or different types. For example, a photo of the sea may bring associated thoughts of happiness, or smelling a specific fragrance may be associated with a certain feeling, memory or visual image. Also, the ability to reproduce the pitch corresponding to a note, irrespective of the form of the note, is an example of the pattern association behavior of the human brain.

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