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The final rules govern the term part. The distributMty law is just one example of a distributMty property governing universal quantification. We see shortly that there are several more distributMty laws. [Rearranging] <Vk : R : T0 ATi) - (Vk:R:T0) A ( V k : R : T i ) . [DistributMty] < Vk :R:pvT) = p v (Vk:R: T) . (11.29) (11.30)

Figure 5.5 Total energy E of the electron in a 60 A CdTe quantum well surrounded by Cdo.9Mno.1Te barriers in the presence of a donor at position rd

These are the rules for existential quantification. Not surprisingly, they are entirely dual to the rules for universal quantification. (In the rule (11.39), if P "Tn -ip - false fi has been simplified to P A T.) Once again, the side condition that free variables may not be captured, and bound variables may not be released, applies to all rules. [Dummy Renaming] ( 3 j : R : T ) = (3k:R[j := k]:T[j := k]> . [Nesting] <3 jjs : R AS : T) = ( 3 j : R : ( 3 j s : S : T ) ) . [Rearranging] ( 3 j , k : R : T ) = ( 3 k , j : R : T ) . [Empty Range] ( 3k: false :T> = false . [One-Point] (3k:k = e : T ) = T[k := e] . (11.31) (11.32) (11.33) (11.34) (11.35)

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1994a] N Murata, S Yoshizawa, S Amari, Learning Curves, Model Selection and Complexity of Neural Networks, in C Lee Giles, SJ Hanson, JD Cowan (eds), Advances in Neural Information Processing Systems, Vol 5, 1994, pp 607 614 [Murata et al 1994b] N Murata, S Yoshizawa, S Amari, Network Information Criterion - Determining the Number of Hidden Units for an Artificial Neural Network Model, IEEE Transactions on Neural Networks, Vol 5, No 6, 1994, pp 865 872 [Nilsson 1998] NJ Nilsson, Artificial Intelligence: A New Synthesis, Morgan Kaufmann, 1998 [Nowlan and Hinton 1992] SJ Nowlan, GE Hinton, Simplifying Neural Networks By Soft Weight-Sharing, Neural Computation, Vol 4, 1992, pp 473 493..

Note that the lower limit for r' is \z'\ and not just z' this is an important point! This change of limit will be used repeatedly in this analysis. The modulus bars are present as r' is an absolute distance; it is not a vector quantity, and is therefore always positive. Now this integration cannot be performed analytically and it is always prudent to avoid evaluating definite integrals numerically where the upper limit is infinity. After all exactly what value of infinity would one use The form for I4 can be manipulated further however, into a more manageable integral between 0 and 1, which can then be evaluated accurately and with confidence. The mathematical trick* introduced here will be used again and again in this present chapter and also later in 6.

[Splitting] (3k:P:T) v Ofc:Q:T> = < 3 k : P v Q : r ) . [Splitting] ( 3 j : R : ( 3 k : S : T ) ) = (3k : ( 3 j : R : S ) : T) . [Trading] (3keK : P /\Q:T) = (3kE{keK\P}:Q:T) . [Trading] ( 3 k : P A Q : T ) = ( B k : Q : P A 7 ) . [Rearranging] (3k : R : T0 v TI) = <3k:^:T 0 > v (3k:R:Ti} . [Distributivity] (3k : R : p A T) = p A (3k : R : T) .

and the limits become zero and infinity, which for now seems to be contrary to what was intended. Hence:

(11.36) (11.37) (11.38) (11.39) (11.40) (11.41)

[Ohnishi et al 1990] N Ohnishi, A Okamoto, N Sugiem, Selective Presentation of Learning Samples for Efficient Learning in Multi-Layer Perceptron, Proceedings of the IEEE International Joint Conference on Neural Networks, Vol 1, 1990, pp 688 691 [Oja 1982] E Oja, A Simplified Neuron Model as a Principal Component Analyzer, Journal of Mathematical Biology, Vol 15, 1982, pp 267 273 [Oja and Karhuner 1985] E Oja, J Karhuner, On Stochastic Approximation of the Eigenvectors and Eigenvalues of the Expectation of a Random Matrix, Journal of Mathematical Analysis and Applications, Vol 104, 1985, pp 69 84 [Opper 1994] M Opper, Learning and Generalization in a Two-Layer Neural Net work: The Role of the Vapnik-Chervonenkis Dimension, Physical Review Letters, Vol 72, No 13, 1994, pp 2133 2166.

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