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The confined states of a quantum wire are therefore described by the two principal quantum numbers ny and nz, in contrast to the sole number required for the onedimensional confinement potential in quantum wells.

{ Invariant:

Kohara developed Selective Presentation Learning specifically for forecasting applications [Kohara 1995] Before training starts, the algorithm generates two training sets The one set contains all patterns representing large next-day changes, while patterns representing small next-day changes are contained in the second set Largechange patterns are then simply presented more often than small-change patterns (similar to Selective Presentation) Cloete and Ludik have done extensive research on training strategies Firstly, they proposed Increased Complexity Training where a NN first learns easy problems, and then the complexity of the problem to be learned is gradually increased [Cloete and Ludik 1993, Ludik and Cloete 1993] The original training set is split into subsets of increasing complexity before training commences A drawback of this method is that the complexity measure of training data is problem dependent, thus making the strategy unsuitable for some tasks Secondly, Cloete and Ludik developed incremental training strategies, ie.

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Figure 8.7 The charge densities of the four lowest energy confined states within a infinitely deep quantum wire; (top left) n =l, nz=l; (top right) ny=l, nz=2; (bottom left) ny=2, nz=1; y (bottom right)y=2, nz=2 n

The initialization is slightly more complicated than in earlier examples. Let us split the invariant into four separate clauses and consider each in turn. The clauses are r <m , P = Qxd + r , (Kfc A m = Qx2 k . (15.30) (15.31) (15.32) (15.33)

Fig. 8.6 displays the confinement energies Ey,z for (ny,nz) equal to (1,1), (1,2), (2,1), and (2,2) as a function of the side lengthyL = Lz for a square cross-section infinitely deep quantum wire. In this case of a square cross-sectional wire, the confinement energies of the (1,2) and (2,1) states are equal; however clearly this will not be the case for a rectangular cross-section wire, which hasy L= Lz. Just as in the

Now, (15.32) can be established by the assignment r,d := P,0. This also establishes (15.30) because we are given the precondition 0 ^P. That is, r,d := P,0

Incremental Subset Training [Cloete and Ludik 1994a] and Incremental Increased Complexity Training [Ludik and Cloete 1994] In Incremental Subset Training, training starts on a random initial subset During training, random subsets from the original training set are added to the actual training subset Incremental Increased Complexity Training is a variation of Increased Complexity Training, where the complexity ranked order is maintained, but training is not done on each complete complexity subset Instead, each complexity subset is further divided into smaller random subsets Training starts on an initial subset of a complexity subset, and is incrementally increased during training Finally, Delta Training Strategies were proposed [Cloete and Ludik 1994b] With Delta Subset Training examples are ordered according to inter-example distance, eg Hamming or Euclidean distance Different strategies of example presentations were investigated: smallest difference examples first, largest difference examples first, and alternating difference.

quantum wells met previously, the confinement energy decreases as the size of the system increases. The wave function for a stationary (wave function along the length of the wire, independent of position) confined particle within the wire is real, and hence the charge density is simply [ w ( y ) ^ ( z ) ] 2 . This is plotted in Fig. 8.7 for the four lowest confined states over a cross-sectional plane. The spatial distribution of the charge density is dependent upon the principal quantum numbers ny and nz (as would be expected), and the number of anti-nodes (local maxima) is equal to nynz. 8.4 SIMPLE APPROXIMATION TO A FINITE RECTANGULAR WIRE More relevant to real devices would be a rectangular cross-sectional quantum wire, but with finite height barriers, as would be fabricated with a post-etch overgrowth. Fig. 8.8 (left) illustrates the two-dimensional confinement potential V(y, z) for this system. With this configuration it is not possible to write the potential V(y, z} as a sum of two independent potentials V(y) and V(z), and thus it is not possible to separate the y- and z-motions.

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