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It is, however, information that is often cumbersome to repeat For this reason, the information is often omitted and a convention on the naming of dummies (such as i, j and k denote integer values) is adopted This means that the most common use of the notation is in the form (^bv: range: term) In addition, the range is sometimes omitted (again to avoid unnecessary repetition in calculations) In this case the form of the quantification is :: term) Formally, omitting the range is equivalent to a true range: v :: term) = <0bt> : true: term) As we have defined it, a quantification only has meaning if the operator is associative and symmetric3 The operator should also have a unit in order to make quantification over an empty range meaningful We denote the unit of by 1 .

This second point is highlighted by the plot of the magnitude of the binding energy, again as a function of donor position, in Fig. 5.6. For completeness, the corresponding plot of the variational parameter A is displayed in Fig. 5.7. Clearly there is a correlation between A and EDo, with the larger EDo giving the smaller A; physically this implies that the binding energy increases as the radius of the electron orbit is decreased.

There is often an existing, long-standing, mathematical convention for the choice of the symbol 0 corresponding to the operator If so, we follow that.

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In order to produce the data plotted in Fig. 5.8, the electron ground-state energies EI were calculated without the donor present, and with the donor present E, for a donor fixed at the centre of the well, as a function of well width. This figure illustrates two points that are both important convergence tests for this analysis and for the analysis given in .3. First, as the well width increases then the electron ground-state energy without the donor present, i.e. E1, decreases monotonically and tends towards zero. Secondly, and perhaps more importantly, and relevant to this section, as the well width increases and the contribution to the electron energy due to quantum confinement decreases, the total energy for the electron tends towards the binding energy of the neutral donor in bulk (11.7 meV, as calculated earlier).

[Rosin and Belew 1996] CD Rosin, RK Belew, New Methods for Competitive Coevolution, Cognitive Computer Science Research Group, Department of Computer Science and Engineering, University of California, Technical Report #0896-491, 1996 [Rouwhorst and Engelbrecht 2000] SE Rouwhorst, AP Engelbrecht, Searching the Forest: Using Decision Trees as Building Blocks for Evolutionairy Search in Classification Databases, Proceedings of the IEEE International Conference on Evolutionary Computation, San Diego, USA, 2000 [Rychtyckyj and Reynolds 1999] N Rychtyckyj, RG Reynolds, Using Cultural Algorithms to Improve Performance in Semantic Networks, Proceedings of the IEEE International Congress on Evolutionary Computation, Washington DC, Vol 3, 1999, pp 1651 1656 [Sakurai 1992] A Sakurai, n-h-1 Networks Use No Less than nh+1 Examples, but Sometimes More, Proceedings of the IEEE, Vol 3, 1992, pp 936 941.

3 This assumption can be avoided if an order is specified for enumerating the elements of the range this is what is done in so-called 'list comprehensions' in functional programming languages. The rules on nesting and rearrangement would then no longer apply.

Figure 5.8 Energy of an electron in a quantum well, both with and without a donor at the centre of the well, as a function of the well width

convention. If not we use the same operator symbol, made larger if the printer will allow it. Examples of quantifications are as follows. summation (Zbv : range : term) (Ubv : range : term) product universal (and) (Vbv : range : term) existential (or) (3bv : range : term) minimum (Vbv : range : term) maximum (Kbv : range : term) equivalence (=bv : range : term) ( bv: range : term) inequivalence

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