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1 4: Sorting and Searching Algorithms In words, the first 5 values in the array are (strictly) smaller than any other array elements see the first universal quantification and their number is less than K\ the last N-l values in the array are (strictly) larger than any other elements see the second universal quantification and their number is at most N-K\ and the remaining array elements are all equal see the final universal quantification. The K smallest values in the array are thus delimited by the indices 0 and I; the Kth smallest value is the value common to the array elements in the segment delimited by 5 and I . Note that it is impossible to satisfy this postcondition unless we impose the precondition Otherwise, it would be impossible to determine 5 such that 0 ^ 5 < K. This is unavoidable if we are required to determine the Kth smallest value. Note, also, that the postcondition does not formally require that the final array is a rearrangement of the initial array. This requirement is clumsy to specify compared with the ease with which it is met by ensuring that the only operation used by the algorithm to change array elements is to swap two of them, as detailed in the discussion of the Dutch National Flag program.

This section presents NN performance measures under three headings: accuracy, complexity and convergence.

Figure 6.16 The oscillator strength Os as function of electric field, for three of the Stark ladder transitions

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Now that we have agreed a precise formal specification of the precondition and postcondition, we may proceed to the development of the algorithm. Hoare's description of the problem involves manipulating a deck of cards. This means that his solution involves operations that are not so easy to implement in a computer program. In particular, Hoare's solution puts the 'borderline card' to one side during the process of adding cards to the bottom-left and bottom-right heaps. We will impose the restriction that all changes to the array are effected by swapping array elements. This means that the program we develop is guaranteed to permute the elements of the array (and will not, for example, introduce spurious array elements) but prohibits us from placing any cards to one side. We are seeking a so-called in situ sort of the array elements. The basic idea is to maintain invariant the property that, for some indices 5 and I, the first 5 elements of the array are known to be 'small', the next l-s elements are 'medium', and the remaining N-l elements are 'large'. Making precise what we mean by 'small', etc., a suitable invariant property is: A ( V i j : 0 ^ i < 5 A s^j <N : A ( V i j : C K i < [ A l^j<N : a [ i ] < a [ j ] } . The inequality 0 ^ 5 < K, together with the first universal quantification, defines the function of the variable 5 during the course of the computation, whilst the

well, and hence the oscillator strength of the inter-well transitions is expected (and confirmed by the calculations) to peak and then reduce. Meanwhile, the intra-well exciton transition |en} > |hn) initially has the largest oscillator strength (obviously), which then temporarily decreases at low fields, before increasing steadily to dominate in the Stark ladder regime. To date, no excitonic states have been calculated with the variable symmetry parameter C > 1- However, for the work discussed in this present section some of the Stark ladder excitons have B values tending to zero, i.e. C = 1. 6.9 SELF-CONSISTENT EFFECTS In the analysis put forward in this present chapter and for the majority of the relevant literature, the exciton binding energy has been calculated by using a variational approach, based on the choice of trial wave function of the following form:

Generalization is a very important aspect of neural network learning. Since it is a measure of how well the network interpolates to points not used during training, the ultimate objective of NN learning is to produce a learner with low generalization error. That is, to minimize the true risk function - t)2dSl(z,t) (7.1)

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where most importantly, We and wh were simply the one-particle eigenstates of the one-dimensional Schrb'dinger equation of the heterostructure; this means that we was calculated without the presence of the hole and wh was calculated without the presence of the electron. Attempts to improve upon this began with the simple idea of the Coulombic-hole model. This has been implemented [147] in the dilute magnetic semiconductor system CdTe-Cdi_a;MnxTe (see Section 5.11.1), which is thought to have a magnetic-field-

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