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In this section, we develop an implementation of this manual technique of computing remainders and quotients. A method designed for paper-and-pencil calculations need not be the best method for implementation on a digital computer, but we hope that your familiarity with the method will help you to understand the use of invariant properties in the design of such an implementation. Figure 15.2(b) explains long division in detail. From it, we see that the technique begins with the remainder, ro, equal to P (3653). At each subsequent step, a multiple Qxm of Q is subtracted from r, thus forming the sequence of ever-decreasing remainders TO, r\, TI and r$. The process is terminated when the remainder is less than Q. In decimal arithmetic, all the multiples take the form QxnxlO k for some k, where 0 < n < 10. (Thus, mi is 5xl0 2 ,m2is2xl0 1 and ma is 1x10 . Not shown is mo, which is OxlO 3 .) Figure 15.3 shows the same process in binary arithmetic. The strategy is identical but the multiples of Q subtracted from r take the form Qxnx2 k for some k, where 0 ^ n < 2. Since binary arithmetic is more suited to computer implementation, we develop long division for binary numerals.

The form of the potential in this Schrodinger equation (i.e. zero!) allows the motions to be decoupled further, writing:

Long division subtracts successive multiples of Q from a remainder value. We introduce variable m, to store the multiples of Q, and variables r and d, to store the remainder and quotient, respectively. The specification of m is simple; in binary arithmetic, it satisfies the invariant property 0^k:m = Qx2 k ) .

and then:

and adding an additional input unit Z0 to the NN, referred to as the synthetic parameter, with value

101 101)11010 101 110 101

Again, it is possible to associate the individual kinetic energy terms on the left-hand side of equation (8.15) with separate energy components, i.e. by writing Ey,Z = Ey + Ez, then:

1 100 101)11010 10100 110 101

The decoupling is completed with the following:

(7.18)

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Figure 15.3 Remainder computation in binary arithmetic (P = 26, Q = 5). (a) Long division; (b) filling in details.

Dividing the first of this pair of equations by if)(z) and the second by w(y) gives:

The existential quantification suggests the introduction of variable k into the algorithm. The two variables m and k satisfy the invariant property O ^ f e A m = Qx2 k . Later (Section 15.4.2), we consider doing without variable k, reintroducing the existential quantification. What about the successive values of r and d We have to identify an invariant property and a termination condition such that their conjunction implies the required postcondition (15.1). Noting that Q = Qx2 , a possibility is to replace Q in (15.1) by the variable m. That is, we postulate the invariant property together with the termination condition

The confinement energy in an infinitely deep quantum wire with a square cross-

For problems with a limited number of training patterns, controlled injection of noise helps to generate new training patterns. Provided that noise is sampled from a Normal distribution with a small variance and zero mean, it can be assumed that the resulting changes in the network output will have insignificant consequences [Holmstrom and Koistinen 1992]. Also, the addition of noise results in a convolutional smoothing of the target function, resulting in reduced training time and increased accuracy [Reed et al. 1995]. Engelbrecht used noise injection around decision boundaries to generate new training patterns for improved performance [Engelbrecht 2000].

Our algorithm thus has the following shape. (Note that the bound function for the loop has yet to be determined.)

Given that the potential outside the wire is infinite, then the standard boundary condition of continuity in the wave function implies that both w(y) and w(z) are zero at the edges of the wire. Thus, equations (8.19) and (8.20) are identical to those of the one-dimensional infinitely deep quantum well the two-dimensional Schrodinger equation has been decoupled into two one-dimensional equations. Given the origin in a 'corner', and the wire dimensions Ly and Lz, as in Fig. 8.5, then the solutions follow as:

{ O^P A 0 < Q }

which give the components of energy as:

Several researchers have developed techniques to control the order in which patterns are presented for learning. These techniques resulted in the improvement of training time and accuracy. A short summary of such training set manipulation techniques is given below. Ohnishi, Okamoto and Sugie suggested a method called Selective Presentation where the original training set is divided into two training sets. One set contains typical patterns, and the other set contains confusing patterns [Ohnishi et al. 1990]. With "typical pattern" the authors mean a pattern far from decision boundaries, while "confusing pattern" refers to a pattern close to a boundary. The two training sets are created once before training. Generation of these training sets assumes prior knowledge about the problem, i.e. where in input space decision boundaries are. In many practical applications such prior knowledge is not available, thus limiting the applicability of this approach. The Selective Presentation strategy alternately presents the learner with typical and then confusing patterns.

Thus, the total energy due to confinement, Ey,z =Ey + Ez, is:

initialize m, k, r and d to establish invariant ;

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