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Appendix ACoincidences in permutations andschedulesThanks to [Ross, 2002] for providing this proof.Given two permutations of the numbers 1¦¨¦©¦ n, what is the expected number of coincidencesbetween them. A coincidence is when the numbers in position i are the same.Without loss of generality, you can suppose that the first permutation is 1¦¨¦ n itself,so the question is then as follows: given a random permutation, what is the expectednumber of instances in which position i contains i ?Let the random variable f £ P¥ be the number of coincidences in permutation P. Iff k£P¥ is the random variable which is 0 or 1 according to whether position k containsk thenf £ P¥ n£∑ f k P¥1 kbecause the f k£P¥ are independent (the fact that position k contains k cannot affectwhether other positions i contain i). So, in expectation,E £ f £ P¥§¥ n∑ E £ £f k P¥§¥1 kbut E £ f k£P¥§¥ 1< n because £ n 1¥ ! permutations out of n! contain k in positionk. ThusE f £ P¥§¥ 1£193
E £ f 2 ¥£Appendix A. Coincidences in permutations and schedules 194place!Thus, no matter how long the permutation, you expect just one item to be in itsWhat is the variance of f ? Consider E £ f 2 ¥§¥ :n∑n∑E f j £ ¥ E f k £ ¥1 k 1 jn∑∑E f j f k ¥£1 k 1 j£ ¥ ¥1n jn j knn∑ E f j 2 2 ∑ E f j f k1nBut f j is just 0 or 1, so f j 2 f j , so E f j 2 E f £ ¥ j 1< n. Also, E f £ ¥ j f k is the ¥probability that a permutation has both j and k in place, £ namely, 2¥ !< n 1< n! n n£. Therefore,£ 1¥E £ f 2 ¥ 1 2 n £ n 1¥§< 2¥a 1< n £ n 1¥ 2and the variance of f is therefore E £ f 2 ¥ E £ f ¥ 2 2 1 1.
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AbstractThe biological immune syste
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DeclarationI declare that this thes
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Table of Contents1 Introduction 11.
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5.3.1 Default Parameters . . . . .
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List of Figures2.1 ’Lock and Key
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List of Tables2.1 Structural and fu
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Chapter 1IntroductionThe study of b
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Chapter 1. Introduction 3scientists
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Chapter 1. Introduction 5Thus, the
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Chapter 1. Introduction 7of the cha
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Chapter 1. Introduction 94. Analysi
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Chapter 2. Background 11paratopeepi
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Chapter 2. Background 13antigenepit
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£Chapter 2. Background 15low the m
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Chapter 2. Background 17Antigene 0p
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Chapter 2. Background 19maintained
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E £ ρ¥ £Chapter 2. Background 2
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Chapter 2. Background 23genetically
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Chapter 2. Background 251. Choose a
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Chapter 2. Background 27non-interbr
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Chapter 2. Background 29been made i
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Chapter 2. Background 31patterns wi
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Chapter 2. Background 33reported on
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Chapter 2. Background 35Immunos 81
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Chapter 2. Background 37recognition
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Chapter 2. Background 391. Create a
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Chapter 2. Background 41environment
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Chapter 2. Background 43a)input dat
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Chapter 2. Background 45antibody wi
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Chapter 3. Immune Systems for Sched
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££Chapter 3. Immune Systems for S
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Chapter 4Applying an Immune System
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Chapter 4. Applying an Immune Syste
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Chapter 5. EA Based Model — COSDM
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JH £ c i ¤ a¥ :Chapter 5. EA Bas
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Chapter 6. A Self-Organising SDM
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Page 196 and 197: Chapter 7. Conclusion 181however wh
Page 198 and 199: Chapter 7. Conclusion 183features m
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Page 204 and 205: Chapter 7. Conclusion 189Clustering
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Page 218 and 219: Bibliography 203[Cooke and Hunt, 19
Page 220 and 221: Bibliography 205[Farmer et al., 198
Page 222 and 223: Bibliography 207[Hart et al., 1998]
Page 224 and 225: Bibliography 209[Kohonen, 1982b] Ko
Page 226 and 227: Bibliography 211[Potter and De Jong
Page 228: Bibliography 213[Timmis et al., 199
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