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But this is unsatisfactory, first because the intention is obscured the reader is burdened with the task of relating the complicated expressions ('n-r2' and '2fc + 1') to the intended sequence of numbers and, second, because the process is error prone. In the example, that the smallest value of k is 0 and not 1, and that the largest value is n^ 2, is far from obviously correct! In other cases, it is not possible to circumvent the problem of the range being a sequence of consecutive numbers. It is impracticable, for example, to use the notation to express the sum of all prime numbers below a given value as there is no simple arithmetic function denoting the feth prime number. In order to overcome these drawbacks, some authors use a more flexible form of the Sigma-Pi notation in which the range of the bound variable is indicated by a predicate. The notation takes the form

Figure 4.2 An intermixed heterojunction; note that the total amount of diffusant represented by the area under the x(z) curve, is constant

where p(k) and f ( k ) are expressions typically (but not necessarily) dependent on k (as indicated by '(&)') For example, the sum of the odd numbers in the range 1 ,.n would be denoted by

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This book presented a short overview of four paradigms within computational intelligence (CI). These paradigms include artificial neural networks (NN), evolutionary computing (EC), swarm intelligence (SI) and fuzzy systems (FS). The intention was to provide the reader with an overview of the popular CI tools and with some of the current active research directions. The intention was by no means to provide a complete treatment of these paradigms. The interested reader should supplement the material presented with the vast source of information available in journals, conference proceedings, books and on the Internet. The hope is that the book did not just provide new knowledge to the first reader in CI, but also that the material provided some insight to current CI practitioners and researchers even just by raising new ideas. While the book left the treatment of CI at studying each paradigm individually, these paradigms can be combined successfully to form efficient hybrid models to solve complex problems. One such example is the use of particle swarm optimization to train NNs. Many other hybrid models exist, with much available literature on this. Agent-based CI tools are also of much interest, where agents of different CI models work together in a cooperative environment to solve problems. For such agent-based approaches the main problem is in developing a strategy of cooperation and exchange of information. Here ideas from swarm intelligence may be very helpful. Both hybrid CI systems and agent technologies are such large research fields, that this book cannot devote any time on these to be fair. However, excellent books on the topics do exist, and the reader is referred to these. As a final thought, or rather instruction to the reader, let the ideas flow ...

Diffusion was first put on a quantitative basis by Fick in 1855 [75] who derived two laws. The first of these stated that the steady state flux through a plane is proportional to, but in the opposite direction to, the concentration gradient, i.e.

Once again, this notation can be said to work in the simple cases. But, its greater flexibility has a major shortcoming the notation is inherently ambiguous! The problem is that the identity of the bound variable has become obscured. If we write

while the second showed that the time dependency of the concentration is given by the following:

then the meaning appears to be clear: k is the bound variable and the entire expression denotes a function of the variables m and n. But, taking m equal to n,

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