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[Luger and Stubblefield 1997] GF Luger, WA Stubblefield, Artificial Intelligence, Structures and Strategies for Complex Problem Solving, Third Edition, Addison-Wesley, 1997 [Lumer and Faieta 1994] E Lumer, B Faieta, Diversity and Adaptation in Populations of Clustering Ants, Proceedings of the Third International Conference on Simulation of Adaptive Behavior: From Animals to Animats, Vol 3, pp 499-508, MIT Press, 1994 [Lumer and Faieta 1995] E Lumer, B Faieta, Exploratory Database Analysis via Self-Organization, unpublished paper, 1995 [MacKay 1992] DJC MacKay, Bayesian Methods for Adaptive Models, PhD Thesis, California Institute of Technology, 1992 [Magoulas et al 1997] GD Magoulas, MN Vrahatis, GS Androulakis, Effective Backpropagation Training with Variable Stepsize, Neural Networks, Vol 10, No 1, 1997, pp 69 82 [Mahfoud 1995] SW Mahfoud, Niching Methods for Genetic Algorithms, PhD The sis, University of Illinois at Urbana-Champaign, 1995 [Mamdani et al.

Distributivity: p A ( q = r) = p /\q = p I\Y = p . Modus ponens: p/\(q = p) = p /\q . Disjunctive normal form: p = q = (p A q) v (-^p A ->q) . Disjunctive normal form: p q = (--p A q) v (p A -^q) .

Now, consider the third integral, as specified in equation (5.21):

First it is necessary to calculate the terms arising from the V2, so consider:

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Strengthening/weakening: p <= p/\q . Strengthening/weakening: pvq <= q . Contrapositive: p^q = ->p^->q . Contradiction: -<p = p^- false . Mutual implication (iff): p = q = (p<=q) /\(p*>q) . Distributivity: (p = q)*=r = p /\r = q/\r . Distributivity: (p = q)<=r = p<^r = q<^r . Shunting: p<^q/\r = (p<=q)<=r . Modus ponens: (p<=q) /\q = p A q . Right unit: p <= true = p . Left zero: true <= p = true . Absurdity: p <= false = true . Reflexivity: p <= p = true . Disjunctive normal form: p *=q = pv ->q . (The name given to the strengthening/weakenening rules depends on whether they are used from left to right ('strengthening') or from right to left ('weakening').)

1975] EH Mamdani, S Assilian, An Experiment in Linguistic Synthesis with a Fuzzy Logic Controller, International Journal of Man-Machine Studies, Vol 7, 1975, pp 1 13 [Maniezzo et al 1994] V Maniezzo, A Colorni and M Dorigo, The Ant System Applied to the Quadratic Assignment Problem, Technical Report IRIDIA/9428, Universite Libre de Bruxelles, Belgium, 1994 [Marais 1970] EN Marais, The Soul of the White Ant, Human and Rousseau Publishers, Cape Town, 1970 [Marais 1969] EN Marais, The Soul of the Ape, London 1969; second edition, Hammers worth, 1973 [Mayr 1963] E Mayr, Animal Species and Evolution, Belknap, Cambridge, MA, 1963 [Michalewicz 1994] Z Michalewicz, Genetic Algorithms + Data Structures = Evolution Programs, Second Edition, Springer Verlag, 1994.

and similarly for the y-direction. Again there are no terms arising from differentiation with respect to z. Adding the two contributing terms together gives:

In the rule of indirect equality, z ranges over the type of x and y. (That is, if x and y are reals, z must range over all reals. If x and y are, say, even integers, then z must range over all even integers.) Indirect equality: x = y = Floor: n^L*J = n^x . Floor: n = [x\ = n ^ x < n+l Ceiling:

Figure 5.4 Total energy E of the electron as a function of the variational parameter A, as an illustration of the variational principle

Ceiling: n^\x] Minimum: Maximum:

[Michel and Middendorf 1998] R Michel, M Middendorf, An Island Model Based Ant System with Lookahead for the Shortest Supersequence Problem, Proceedings of the Fifth International Conference on Parallel Problem Solving from Nature, 1998, pp 692 701 [M011er 1993] MF M011er, A Scaled Conjugate Gradient Algorithm for Fast Supervised Learning, Neural Networks, Vol 6, 1993, pp 525-533..

Hence the integral I3, as in equation (5.43), with the usual transformation to plane polar coordinates becomes:

This section summarizes the rules for manipulating quantifiers We assume that e is associative and symmetric Some rules assume that e has a unit, which is denoted by 1 If e does not have a unit, quantification over a false range is not defined Two cautionary remarks need to be made First, the rules are not applicable in general to infinite quantifications (They are, however, all applicable in the case of universal and existential quantification) Second, 'side' conditions on dummies are not repeated here Generally, rules are only applicable when (a) application of the rule does not capture free variables and, conversely, (b) application of the rule does not release bound variables This must be the case for all subexpressions, and not just the quantified expression itself Also, application of a rule should not result in a variable occurring more than once in a list of dummies.

By using equations (5.33) and (5.39), obtain:

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