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The main advantage of a formal quantifier notation over the informal dotdotdot notation is that it is easier to formulate and use calculational rules. In this subsection, we formulate rules for summation. Later, we will see that these rules are all instances of more general rules. We formulate the rules in terms of each of the components of a quantification. So there are rules governing the use of dummies, rules exploiting the structure of the range, and rules exploiting the structure of the term. Additionally, there are two so-called trading rules that allow information to be moved to and from the range of the quantification. Side Condition. A general side condition on all the rules is that their application should not result in the capture of free variables or release of bound variables, and should not result in a variable occurring more than once in a list of dummies. Dummy Rules. There are three rules governing the use of dummies. The first rule expresses the fact that a 'dummy' is just a place holder, the particular name chosen for the dummy is not relevant provided it does not clash with the names of other variables in the quantified expression. (The rule has already been discussed in Section 11.2.2 but is repeated here for convenience.) Renaming is often used when performing other algebraic manipulations in order to avoid capture of free variables or release of bound variables. Let R[j := k] and T[j :- k] denote, respectively, the expressions obtained by replacing every free occurrence of 'f in R and T by 'k'. Then [Dummy Renamingl (Zj:R:T) = (Z.k:R[j := k ] : T [ j := k]) . (11.5)

[Hanson and Pratt 1989] SJ Hanson, LY Pratt, Comparing Biases for Minimal Network Construction with Back-Propagation, in DS Touretzky (ed), Advances in Neural Information Processing Systems, Vol 1, 1989, pp 177 185 [Hassibi and Stork 1993] B Hassibi, DG Stork, Second Order Derivatives for Network Pruning: Optimal Brain Surgeon, in C Lee Giles, SJ Hanson, JD Cowan (eds), Advances in Neural Information Processing Systems, Vol 5, 1993, pp 164 171 [Hassibi et al 1994] B Hassibi, DG Stork, G Wolff, Optimal Brain Surgeon: Extensions and Performance Comparisons, in JD Cowan, G Tesauro, J Alspector (eds), Advances in Neural Information Processing Systems, Vol 6, 1994, pp 263-270 [Haussler et al 1992] D Haussler, M Kearns, M Opper, R Schapire, Estimating Average-Case Learning Curves using Bayesian, Statistical Physics and VC Dimension Method, in J Moody, SJ Hanson, R Lippmann (eds), Advances in Neural Information Processing Systems, Vol 4, 1992, pp 855 862.

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phenomenon of line broadening in a diffused superlattice has been observed by Elman et al. [90] and subsequently modelled theoretically [98]. Alternatively, the structure, including the depth dependence of the diffusion, can be mapped directly by using double-crystal X-ray diffraction (DCXRD) [99], a technique which has showed itself to be very valuable in monitoring the progress of interface intermixing [89]. 4.8 TIME DEPENDENT DIFFUSION COEFFICIENT Vacancy-enhanced diffusion will continue for as long as the vacancies are present, but one way to control their lifetime and 'freeze' the diffusant profile is to anneal out the radiation damage. The annealing process thermally activates the interstitials back into vacancies and so restores order to the crystal. This subsequent annealing process could be described by a time-dependent diffusion coefficient, perhaps of the form:

As discussed earlier, the general side condition on application of rules demands that R and T be expressions not containing any free occurrence of 'k'.

thus presenting the opportunity to complete the examples of the functional dependencies listed in Section 4.2. D ( z ) is the initial (time-independent) depth-dependent diffusion coefficient due to the vacancy distribution. Using the form in equation (4.16), the diffusion coefficient would then be fully specified by:

[Hayashi 1993] M Hayashi, A Fast Algorithm for the Hidden Units in a Multilayer Perceptron, Proceedings of the 1993 International Joint Conference on Neural Networks, Vol 1, 1993, pp 339 342 [Haykin 1994] S Haykin, Neural Networks: A Comprehensive Foundation, MacMillan, 1994..

The second rule states, essentially, that the use of more than one dummy is a convenient abbreviation for a collection of quantifications. We use 'js' in the statement of the rule to denote any list of variables: [Nesting] (ZjJs:R*S:T) = (I,j:R:(Zjs:S:T)) . (11.6)

Figure 4.12 Annealing out radiation damage an example of time-dependent diffusion, with annealing times of 0, 50, 100, and 200 s giving the concentration profiles of decreasing amplitude

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