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There is an elegant method which allows one to simplify the 6x6 Hamiltonian, by recasting it into a block-diagonal form: it then has two 3x3 matrices as its diagonal elements and the remaining two off-diagonal elements are zero-matrices. This is achieved by a suitable change of basis, i.e. by creating a new basis from linear combinations of the existing basis states. Essentially this means that there are two completely independent sets of states, which are eigenstates of either one or the other 3x3 block. Since this block-diagonalisation reduces the size of the system to be considered at any time, it is useful when attempting to obtain a result in analytical form. Block-diagonalisation can also be useful in cases which are more complicated than straightforward bulk material, for example, strained bulk or two-dimensional quantum-well systems. The derivation of the transformation of the Hamiltonian into a block-diagonal form will not be reproduced here, the result will merely be stated, i.e. upon introducing the new basis:

(3) Construct the loop body, T, so as to guarantee progress towards the termination condition whilst maintaining the invariant: { inv A -idone A bf = C } T { inv A (bf < C v done) } .

Leaving out the imaginary part ([Durbin and Rumelhart 1989] show that the added complexity of including the imaginary part does not help with increasing performance) ,

where:

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Solution 1.1. For n = 5, the number of portions is 16. This suggests that the number of portions for arbitrary n is 2n~l. However, for n = 0, it does not make sense to say that there are 2 W ~ 1 portions (even though cutting the cake as stated does make sense). The conclusion might then be that the conjecture only holds for n at least 1. However, for n = 6, the number of portions is 31 (see Figure B.I)! Note that n - 6 is the first case in which the points are not allowed to be placed at equal distances around the perimeter.) n

In conclusion, regarding the various advantages and disadvantages of the three trial wave functions, clearly the last, namely the variable symmetry case, gives the lowest variational energies and hence is the most accurate physical representation. Without such an investment in complexity and computational time it would not have been apparent that donors assume this mixed 2D and 3D symmetry, i.e. wave functions resembling prolate spheroids. However, the computational demands are substantial and it has been shown that donor energies can be obtained within 0.1 meV of the lowest possible value, by using a simple 3D (spherical) trial wave function. This wave function has the advantage that all of the relevant integrations can be performed analytically, which leads to a quick evaluation of the total energy E. 5.6 INCLUSION OF A CENTRAL CELL CORRECTION As mentioned earlier, the use of the bulk dielectric constant has to be employed with a note of caution. In the majority of cases its use is probably fine; however, it is worthwhile noting that there is a scheme for accounting for a change in the permittivity of the material as the electron approaches the donor. This occurs as the degree of electronic shielding is reduced and is known as the central cell correction . This extension takes the form of an additional factor in the Coulombic potential term, which, for the purpose of illustration, is taken to be as follows:

Solution 2.1. (a) Hoare assumes that the pack contains 100 cards. The weakest assumption is that the pack contains at least 20 cards. (b) The number of cards in the top-left heap is always at most 20. The number of cards in the top-right heap is always at most 80.

IbCLyC '

where A is a parameter describing the extent of the correction; the latter can be thought of as a screening length. As r' > 0, it is expected that:

2. /

where eo is the permittivity of free space. Taking this limit gives:

with = rj/a. Equation (4.5) implies that inputs for which zi,p <u k i (t - I)/ have then- corresponding weights Uki decreased by a value proportional to the output value ok,p. When zi,p > uki(t l)/ , weight uki is increased proportional to o k,p . Sejnowski proposed another way to formulate Hebb's postulate, using the covariance correlation of the neuron activation values [Sejnowski 1977]: = ri[(zif - Zi)(okf - Ok)} with

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