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[Wolpert and Macready 1996] DH Wolpert, WG Macready, No Free Lunch Theorems for Search, Technical Report SFI-TR-95-02-010, Santa Fe Institute, 1996 [Xue et al 1990] Q Xue, Y Hu, WJ Tompkins, Analyses of the Hidden Units of Back-Propagation Model, Proceedings of the IEEE International Joint Conference on Neural Networks, Vol 1, 1990, pp 739 742 [Yasui 1997] S Yasui, Convergence Suppression and Divergence Facilitation: Minimum and Joint Use of Hidden Units by Multiple Outputs, Neural Networks, Vol 10, No 2, 1997, pp 353 367 [Yu and Chen 1997] X-H Yu, G-A Chen, Efficient Backpropagation Learning using Optimal Learning Rate and Momentum, Neural Networks, Vol 10, No 3, 1997, pp 517 527 [Zadeh 1965] LA Zadeh, Fuzzy Sets, Information and Control, Vol 8, 1965 pp 338 353.

= ( k:R:f.T}

Using the final form for I1, as in equation (5.69), then:

Figure 5.12 Comparison between the total electron wave function u(z) and the numerically determined envelope x(z) for three different donor positions Finally, for the three-dimensional case, I4 becomes:

(11.55)

[Zadeh 1975] LA Zadeh, The Concept of a Linguistic Variable and its Application to Approximate Reasoning, Parts 1 and 2, Information Sciences, Vol 8, 1975 pp 338 353 [Zhang 1994] B-T Zhang, Accelerated Learning by Active Example Selection, International Journal of Neural Systems, Vol 5, No 1, 1994, pp 67 75 [Zhang and Kandel 1998] Y Zhang, A Kandel, Compensatory Genetic Fuzzy Neural Networks and Their Applications, World Scientific, 1998 [Zurada 1992] J Zurada, Lambda Learning Rule for Feedforward Neural Networks, Proceedings of the IEEE International Joint Conference in Neural Networks, San Francisco, USA, 1992..

and therefore:

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(0k:P:T> = <0k : P A Q : T) ( k : P A -iQ : T} . Use this rule to derive (11.50) from (11.49) in the case that is idempotent. D

Figure 5.13 Magnitude of the donor binding energy EDo in GaAs/Ga1 _x Alx As as a function of quantum well width, for a variety of barrier alloys x

Exercise 11.57 (Translation, idempotent operators). The translation rule for summations (11.12) requires function / to be a bijection. The rule is applicable to all quantifications, and not just summations. (Exercise 11.19, which asked you to derive the rule for summations, can be repeated with S replaced by an arbitrary quantifier.) In the case that the quantifier is idempotent, the rule can be simplified. The translation rule for idempotent quantifiers is as follows. Suppose / is a function from the type of dummy j to the type of dummy k such that

[Zurada et al. 1994] JM Zurada, A Malinowski, I Cloete, Sensitivity Analysis for Minimization of Input Data Dimension for Feedforward Neural Network, IEEE International Symposium on Circuits and Systems, London, May 30 June 3, 1994.

Looking upon these results from a computational viewpoint, it can then be seen that this 3D trial wave function has an immediate advantage over the previous, seemingly simpler, 2D case, namely that all of the integrals I1, I2, I3 and I4 have analytical expressions. Indeed, evaluation of the integrals and the minimisation of the energy is computationally much less demanding than previously. Figure 5.10 shows the change in the total energy E of the electron between the previous 2D trial wave function and the more complex 3D case presented in this section. It is clear that the energy E is lower for all donor positions across this (a typical) quantum well. The difference between the two trial wave functions is smallest when the separation between the donor and the electron is larger, i.e. when the donor is deep in the barrier (rd = 0 A) and the electron is, as always, centred in the well (z = 230 A). Thus it might be concluded that the 2D wave function is a reasonable approximation when the donor is in the barrier, although as the graph shows, for donors in the well, considerably lower energies can be obtained by using a spherical hydrogenic term. Recalling the variational principle, then the lower energies obtained imply that the 3D approximation to the wave function is a more accurate representation than the 2D case. When coupled together with the computational advantage, as mentioned above, then the argument in favour of the 3D trial wave function is clear. Figure 5.11 displays the total wave function, u = x(z)e r/t for the range of donor positions across the quantum well. It can be seen that the wave function u

Exercise 11.58. The following table shows a number of associative and symmetric binary operators together with their units. For each, give an instance of the distributMty rule (11.55) not already mentioned in the text.

Figure 5.14 Magnitude of the donor binding energy EDo for donors at the centre of GaAs wells of large width, surrounded by Ga 0.9 Al 0.1 As barriers

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