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which is relatively straightforward to evaluate. Califano and Harrison [164] have demonstrated that this can be a quite useful approach to the solution of finite barrier quantum wires and dots.

LEARNING)

8.5 CIRCULAR CROSS-SECTION WIRE Consider again the Schrodinger equation for the motion in the confined cross-sectional plane of a quantum wire, as given earlier in equation (8.11), i.e.

r ^ 2x(m + 2)

seem advantageous to move into polar coordinates for the cross-sectional motion. With the definitions of the modulus r and angle 9 as in the figure, the Cartesian coordinates then follow as:

r ^ m-1 = { integer arithmetic } r<m . So, as expected, the subtraction only needs to take place at most once; the inner loop, which repeatedly checks to see whether m should be subtracted from r, can therefore be replaced by a conditional statement, which checks only once. Our algorithm is now complete; it is shown in Figure 15.4. Exercise 15.34. Check your understanding by giving the algorithm for the case that '2' is replaced everywhere by 'J5', where B ^ 2. Why is the lower bound 2 on needed D

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But how do we choose the value of the learning rate One approach is to find the optimal value of the learning rate through cross-validation, which is a lengthy process An alternative is to select a small value (eg 01) and to increase the value if convergence is too slow, or to decrease it if the error does not decrease fast enough Plaut et al proposed that the learning rate should be inversely proportional to the fanin of a neuron [Plaut et al 1986] This approach has been theoretically justified through an analysis of the eigenvalue distribution of the Hessian matrix of the objective function [Le Gun et al 1991] Several heuristics have been developed to dynamically adjust the learning rate during training One of the simplest approaches is to assume that each weight has a different learning rate rjkj.

The wave function w ( y , z ) can clearly be written in terms of the new variables r and 0; however, given the circular symmetry the wave functions should not have a dependence on the angle 0. Thus, the wave function can actually be written as w(r), and the Schrodinger equation therefore becomes:

As was the case for the elementary remainder computation algorithm, the implementation of long division can be simplified if only the remainder, and not the quotient, is required. The basis for the simplification is that the computation of remainder r depends only on the variable m; no boolean test uses the value of k or d, and no assignment to r depends on the value of k or d. So, if only r is required, all assignments to k and d can be removed from the program. The variables do not disappear entirely. They are needed for the specification of r, and for the invariants, in the form of existentially quantified variables. The invariant property of m reverts to ( 3 f c : ( K f c : m = Qx2 k ) , which is what it was before we decided to introduce k. Also, the quotient d becomes existentially quantified, so that the postcondition satisfied by r is C K r < Q A (3d :: P = Qxd + r) .

where the index on Er just indicates that this eigenvalue is associated with the confined cross-sectional motion, as opposed to the unconfined motion along the axis of the

A 0<Q }

The following rule is then applied to each weight before that weight is updated: if the direction in which the error decreases at this weight change is the same as the direction in which it has been decreasing recently, then nkj is increased; if not, nkj is decreased [Jacobs 1988] The direction in which the error decreases is determined by the sign of the partial derivative of the objective function with respect to the weight Usually, the average change over a number of pattern presentations is considered and not just the previous adjustment An alternative is to use an annealing schedule to gradually reduce a large learning rate to a smaller value (refer to equation 422) This allows for large initial steps, and ensures small steps in the region of the minimum Of course more complex adaptive learning rate techniques have been developed, with elaborate theoretical analysis.

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