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The algorithm discussed in this chapter is a variant on the well-known binary search method. We return to it again in 4. The idea of using a deck of cards to illustrate the algorithm was suggested to me by Stuart Anderson. Hoare's inaugural 1971 lecture (see Exercise 2.1) is published in Hoare and Jones (1989, pp. 89-101). See Section 14.2 for a formal specification of Hoare's problem, and how an implementation is calculated from the specification. The problem of the amorous beetles (Exercise 2.3) is from Gardner (1959). The origin of the domino problem (Exercise 2.2) is unknown to me.

Clearly this eigenvalue Hu is also the total energy but in a form usually associated with waves, e.g. a photon. These two operations on the wave function represent the two complimentary descriptions associated with wave-particle duality. Thus the second, i.e., time-dependent, Schrodinger equation is given by the following:

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In functional link neural networks (FLNN) input units do implement activation functions. A FLNN is simply a FFNN with the input layer expanded into a layer of functional higher-order units [Ghosh and Shin 1992, Hussain et al. 1997]. The input layer, with dimension /, is therefore expanded to functional units h 1 , h 2 , - ,hL, where L is the total number of functional units, and each functional unit hi is a function of the input parameter vector (z 1 , , z I ), i.e. h l (z 1 , , zI) (see Figure 3.2). The weight matrix U between the input layer and the layer of functional units is defined as 1 if functional unit hl is dependent of Zi , , 0 otherwise For FLNNs, Vji is the weight between hidden unit yj and functional link hi. Calculation of the activation of each output unit ok occurs in the same manner as for FFNNs, except that the additional layer of functional units is taken into account:

In earlier chapters, we have argued that proof is necessary to guarantee that programs meet their specifications. What is meant by a 'proof is, however, not as clear cut as one might at first imagine. In this chapter, we discuss different sorts of proofs. We advocate 'calculational proofs' in which reasoning is goal directed and justified by simple axiomatic laws that can be checked syntactically rather than semantically. We introduce a notation for presenting calculations and give several examples.

1.2 CRYSTAL STRUCTURE The vast majority of the mainstream semiconductors have a face-centred cubic Bravais lattice, as illustrated in Fig 1.2. The lattice points are defined in terms of linear combinations of a set of primitive lattice vectors, one choice for which is:

The lattice vectors then follow as the set of vectors:

The word 'proof can be understood in several ways. An informal proof, the sort most commonly encountered in mathematical texts, consists of a mixture of natural language (for example, English) and mathematical calculations. The English text outlines the main steps in the proof, and the mathematical calculations fill in some of the details. Figure 3.1, a proof of Pythagoras's theorem, is a typical example. It uses English to describe the construction of a square from four copies of a right-angled triangle, combined with an outline of an algebraic calculation using properties of arithmetic. Informal proofs place a large burden on the reader. The reader is expected to have a good understanding of the problem domain, and the meaning of the natural language statements, as well as the language of mathematics. Because of their reliance on meaning, we say that they are semantic proofs. Figure 3.2 is an example of a proof where the burden of understanding of the English language

The use of higher-order combinations of input units may result in faster training times and improved accuracy (see, for example, [Ghosh and Shin 1992, Hussain et al. 1997]).

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