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An unlimited supply of dominoes has been provided; each domino will cover exactly two squares of the chessboard. Is it possible to cover all 62 squares of the chessboard with the dominoes without any domino overlapping another domino or sticking out beyond the edges of the board D

where T is the kinetic energy eigenvalue, and given the form of V in equation (1.5) then:

The following CI paradigms are covered: artificial neural networks, evolutionary computing, swarm intelligence and fuzzy systems While individual techniques from these CI paradigms have been applied successfully to solve real-world problems, the current trend is to develop hybrids of paradigms, since no one paradigm is superior to the others in all situations In doing so, we capitalize on the respective strengths of the components of the hybrid CI system, and eliminate weaknesses of individual components The rest of this book is organized as follows: Section 11 presents a short overview of the different CI paradigms, also discussing the biological motivation for each paradigm A short history of AI is presented in Section 12 Artificial neural networks are covered in Part II, evolutionary computing in Part III, swarm intelligence in Part IV and fuzzy systems in Part V.

2.6 Summary

then Thus the kinetic energy eigenvalue is given by:

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Exercise 2.3 (The Amorous Beetles). Four beetles A, B, C and D occupy the corners of a square (Figure 2.5). A and C are male, B and D are female. Simultaneously A crawls towards B, B towards C, C towards D, and D towards A. If all four beetles crawl at the same rate, they will describe four congruent logarithmic spirals which meet at the centre of the square. How far does each beetle travel before they meet The problem can be solved without calculus. D

For an electron in a vacuum away from the influence of electromagnetic fields, then the total energy E is just the kinetic energy T. Thus the dispersion or energy versus momentum (which is proportional to the wave vector k) curves are parabolic, just as for classical free particles, as illustrated in Fig. 1.1.

A short discussion on hybrid CI models is given in the conclusion of this book At this point it is necessary to state that there are different definitions of what constitutes CI This book reflects the opinion of the author, and may well cause some debate For example, Swarm Intelligence (SI) is classified as a CI paradigm, while most researchers are of the belief that it belongs only under Artificial Life However, both Particle Swarm Optimization (PSO) and Anto Colony Optimization (AGO), as treated under SI, satisfy the definition of CI given above, and are therefore.

Summary

Figure 1.1 The energy versus wave vector (proportional to momentum) curve for an electron in a vacuum

The purpose of this chapter has been to introduce, informally, the concepts underlying program construction. The specification of a program by means of a precondition and postcondition has been illustrated. The design of a repetitive process (a 'loop') using an invariant property to precisely describe the function of the iteration, and the use of a bound function as a measure of progress towards the

In summary, the equation describing the total energy of a particle in this wave description is called the time-independent Schrodinger equation and for this case with only a kinetic energy contribution can be summarised as follows:

where fOk and fyi are respectively the activation function for output unit ok and hidden unit yj; wkj is the weight between output unit ok and hidden unit yj; Z i,p is the value of input unit Zi{ for input pattern p; the (I + l)-th input unit and the ( J + l)-th hidden unit are bias units representing the threshold values of neurons in the next layer. Note that each activation function can be a different function. It is not necessary that all activation functions be the same. Also, each input unit can implement an activation function. It is usually assumed that inputs units have linear activation functions.

termination condition have been introduced. Finally, the relation between invariants and mathematical induction has been explained. All these topics will be amplified and made rigorous in later chapters.

A corresponding equation also exists which includes the time-dependency explicitly; this is obtained by operating on the wave function by the linear operator ihd/dt, i.e.

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