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International Conference in Neural Networks, Washington, Vol 2, 1996, pp 1274-1277 [Engelbrecht and Cloete 1998a] AP Engelbrecht, I Cloete, Selective Learning using Sensitivity Analysis, IEEE World Congress on Computational Intelligence, International Joint Conference on Neural Networks, Anchorage, Alaska 1998, pp 1150 1155 [Engelbrecht and Cloete 1998b] AP Engelbrecht, I Cloete, Feature Extraction from Feedforward Neural Networks using Sensitivity Analysis, International Conference on Systems, Signals, Control, Computers, Durban, South Africa, Vol 2, 1998, pp 221 225 [Engelbrecht and Ismail 1999] AP Engelbrecht, A Ismail, Training Product Unit Neural Networks, Stability and Control: Theory and Applications Vol 2 No 1/2, 1999 [Engelbrecht et al 1999] AP Engelbrecht, L Fletcher, I Cloete, Variance Analysis of Sensitivity Information for Pruning Feedforward Neural Networks, Proceedings of the IEEE International Joint Conference on Neural Networks, Washington, DC, USA, 1999, paper 379.

In this case, the summation is defined to be zero, (hi words, a sum of no values is zero) The dummy has a certain type, the knowledge of which is crucial in certain circumstances A long-standing mathematical convention is that variables i, j and k denote integer values, whereas x, y and z denote real values Such conventions help for small-scale problems (like the ones considered in this text) but are inadequate for the large-scale problems commonly encountered in programming Where necessary, the type of the dummy can be indicated by adding it immediately after the first occurrence of the dummy, as in (ZfceZ : e v e n k A ( K k < N : fc 3 +fc 2 +N+l) (The symbol Z is commonly used in mathematics to denote the set of all integers.

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In general, such a wave will not have the periodicity of the crystal lattice; however, for certain wave vectors it will Such a set of wave vectors G are known as the reciprocal lattice vectors with the set of points mapped out by these primitives known as the reciprocal lattice If the set of vectors G did have the periodicity of the lattice then this would imply that:.

The symbols N and R are used for the natural numbers and reals, respectively) Mostly, however, we will indicate the type of the dummies in the accompanying text rather than in the quantification itself Rather than have just one dummy, it is convenient to allow a number of variables to be bound to the same quantifier An example is (Zij : 0^i<j^N : i+j).

nescence and photoluminescence excitation can be used to measure the excitation energy of electrons across the band gap. As the electron and hole confinement energies, Ee and Eh, are dependent upon the band-edge profiles of the conduction- and valence-band edges, respectively, the excitation energy (E = Eg + Ee + Eh, ignoring the exciton binding energy), is then also dependent upon the amount of diffusion.

[Engelbrecht and Cloete 1999] AP Engelbrecht, I Cloete, Incremental Learning using Sensitivity Analysis, Proceedings of the IEEE International Joint Conference on Neural Networks, Washington, DC USA, 1999, paper 380 [Engelbrecht 2000] AP Engelbrecht, Data Generation using Sensitivity Analysis, International Symposium on Computational Intelligence Slovakia, 2000 [Engelbrecht 2001] AP Engelbrecht, A New Pruning Heuristic Based on Variance Analysis of Sensitivity Information, IEEE Transactions on Neural Networks, Vol 12, No 6, 2001 [Engelbrecht 2001b] AP Engelbrecht, Sensitivity Analysis for Selective Learning by Feedforward Neural Networks, Fundamenta Informaticae, IOS Press Vol 45, No 1, 2001, pp 295 328 [Engelbrecht et al 2002] AP Engelbrecht, S Rouwhorst, L Schoeman, A Building Block Approach to Genetic Programming for Rule Discovery, in HA Abbass, RA Sarker, CS Newton (eds), Data Mining: A Heuristic Approach, Idea Group Publishing, 2002, pp 174 189.

This denotes the sum of all values i+j such that i and j satisfy the property CK i < j ^ N- Taking N to be 2, the possible values of i and j are i = 0 and j = 1, t = 0 and .7 = 2,1 = 1 and j = 2, so that (EiJ : 0 < i < j < 2 ; i+j) = (0+1) + (0+2) + (1 + 2) . Note that the variables in a list of dummies must all be distinct. It does not make sense to repeat dummies. For example, (Li,i : 0^i<2 : 0) is not meaningful. Exercise 11.1. Evaluate the following summations. (a) (b) (c) (d) <Zk : l < k ^ 3 : k) . (Sk : 0 ^ k < 5 : 1) . <ZiJ : 0 ^ i < j X 2 A odd.i : i+j} . (Li,j : (Ki< j < 2 A odd.i A odd.j : i+j) .

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