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Therefore, in a similar manner to the hydrogenic impurities discussed in 5, the binding energy and orbital radius can be represented well by Bohr theory, with

Finally, the loop body, T, is constructed so as to guarantee progress towards the termination condition whilst maintaining the invariant:

Kohonen developed the self-organizing feature map (SOM), as motivated by the self-organization characteristics of the human cerebral cortex Studies of the cerebral cortex showed that the motor cortex, somatosensory cortex, visual cortex and auditory cortex are represented by topologically ordered maps These topological maps form to represent the structures sensed in the sensory input signals The self-organizing feature map is a multidimensional scaling method to project an I-dimensional input space to a discrete output space, effectively performing a compression of input space onto a set of codebook vectors The output space is usually a two-dimensional grid The SOM uses the grid to approximate the probability density function of the input space, while still maintaining the topological structure of input space That is, if two vectors are close to one another in input space, so is the case for the map representation.

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the correction for the finite mass of the central charge [3,4]. This is implemented by exchanging the (in the case of a hydrogen atom, or donor) orbiting electron mass, with the reduced mass of the two-body system, in this case, the electron-hole pair. The reduced mass is given by:

{ inv A ^done A bf = C } T { inv A (bf<C v done) } . (13.12)

Hence, the binding energy becomes:

The operation of the SOM can be illustrated as follows: consider input space to be a cloud The map can then be viewed as an elastic net that spans the cloud The SOM closely resembles the learning vector quantizer discussed in the previous section The difference between the two unsupervised algorithms is that neurons are organized on a rectangular grid for SOM, and neighbors are updated to also perform an ordering of the neurons In the process, SOMs effectively cluster the input vectors through a competitive learning process, while maintaining the topological structure of the input space Section 451 explains the standard stochastic SOM training rule, while a batch version is discussed in Section 452 A growing approach to SOM is given in Section 453 Different approaches to speed up the training of SOMs are overviewed in Section 454 Section 45.

In this section, we present several simple arithmetic problems, the objective being to illustrate the use of invariants in loop construction in the simplest possible context. In the first few examples, the bound function is simply a counter. Later examples involve (slightly) more complex bound functions.

and the Bohr radius follows as:

The problem of summing the elements of an array is an elementary example of the use of invariants. Suppose 0 ^ N and it is required to compute (Si ; 0 ^ t < JV : a[ i] ) . An obvious solution is to introduce a variable 5 and assign to 5 successively 0, a[0], a[0]+a[l], a[0]+a[l]+a[2], and so on. Using index variable k to count the number of values that have been added, the invariant property is Q^k^N A 5 = ( S i : 0 ^ i < and the termination condition isk = N. The appropriate initialization is the assignment

Taking typical values for bulk GaAs, i.e. the F valley electron and heavy-hole effective masses,m* = 0.067m0andm^ = 0.62rao,respectively,then u = 0.060m0. Using the static dielectric constant er = 13.18 [14], then the exciton binding energy and Bohr radius follow respectively, as:

5 explains the formation of clusters for visualization purposes Section 456 discusses in brief different ways how the SOM can be used after training..

k,s := 0,0

which are exactly the same as the low-temperature measured values (see [14], p.420). Correspondingly for CdTe, m*=0.096rao and 771^=0.67710, therefore u, = 0.083m0, with er = 10.6, then:

and the bound function is JV-k. Maintaining the invariant property whilst making progress towards the termination condition is achieved by the assignment k,s := k+l,s+a[k] . The complete program is, thus,

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