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Such spherically symmetric Schrodinger equations have been investigated before (see for example [29], p. 76). As an alternative to such a well established approach, and with with the impetus in this work on simple numerical schemes, a shooting technique similar to that described for the circular cross-section quantum wire is sought. With this aim, expanding the first and second derivatives in terms of finite differences gives:

to fix bugs; an abundance of case analyses is indicative of bad programming.) The problem lies in the fact that the assignments to hi and lo are unsystematic: initially, the region to be searched is given by the indices from lo to hi -1 inclusive. If 1 o is reassigned, however, the region to be searched begins at index 1 o+l. (This observation makes it possible to identify the error.) The assignments to hi are more systematic, but probably by good fortune rather than good programming! There is no mention of an invariant property in the comments, and no mention of how progress is guaranteed. D Solution 5.2. (p = q) = r and p = (q = r ) are both true exactly when an odd number of p, q and r is true. (Thus when all are true or just one is true.) D Solution 5.3. Parenthesizing the statement as xxy is positive = (x is positive = y is positive) , it states that the number xxy is positive exactly when the signs of x and y are both the same. Parenthesizing it as (xxy is positive = x is positive) = y is positive , it states that the operation of multiplying a number x by a number y does not change the sign of x exactly when y is positive. As for the parity of a number, we get four different cases: ((xxy is positive) or ((xxy is negative) or ((xxy is negative) or ((xxy is positive)

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Multiplying both sides by r(Sr)2 gives:

Principal Component Analysis (PCA) pruning techniques have been developed that use the SVD of the Fisher information matrix to find the principal components (relevant parameters) [Levin et al 1994, Kamimura 1993, Schittenkopf et al 1997, Takahashi 1993] These principal components are linear transformations of the original parameters, computed from the eigenvectors obtained from a SVD of the information matrix The result of PCA is the orthogonal vectors on which variance in the data is maximally projected Nonprincipal components/parameters (parameters that do not account for data variance) are pruned Pruning using PCA is thus achieved through projection of the original w-dimensional space onto a w -dimensional linear subspace (w < w) spanned by the eigenvectors of the data's correlation or covariance matrix corresponding to the largest eigenvalues.

is positive)) is positive)) is negative)) is negative)).

Gathering terms in w(r + 6r), ip(r), and i^(r Sr) then:

Solution 5.8. Yes. If you ask A if B is a knight you get the answer A = B. If you ask B if A is a knight you get the answer B = A. But equivalence is symmetric, so they are the same answer. D

Figure 8.16 The confinement energy in a spherical GaAs quantum dot surrounded by a Ga0.8 Al0.2 As barrier

Hypothesis testing techniques: Formal statistical hypothesis tests can be used to test the statistical significance of a subset of weights, or a subset of hidden units Steppe, Bauer and Rogers [Steppe et al 1996] and Fletcher, Katkovnik, Steffens and Engelbrecht [Fletcher et al 1998] use the likelihoodratio test statistic to test the null hypothesis that a subset of weights is zero Weights associated with a hidden unit are tested to see if they are statistically different from zero If these weights are not statistically different from zero, the corresponding hidden unit is pruned Belue and Bauer propose a method that injects a noisy input parameter into the NN model, and then use statistical tests to decide if the significances of the original NN parameters are higher than that of the injected noisy parameter [Belue and Bauer 1995] Parameters with lower significances than the noisy parameter are pruned.

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