Preface ......................................................... v
Abbreviations ................................................. xix
1 Introduction ................................................. 1
1.1 Diversifying and Unifying Ways .......................... 1
1.2 Categorification and Decategorification ................. 3
1.3 Polytope Projects ....................................... 7
References .................................................. 14
2 Methods and Models .......................................... 17
2.1 Differential Posets .................................... 17
2.2 Dual Graded Graphs ..................................... 21
2.3 Updown Categories ...................................... 24
2.4 Combinatorial Species .................................. 26
2.5 Polytopes and n-Levels Systems ......................... 33
2.6 Differential Models .................................... 35
2.6.1 Modeling Differential Posets .................... 35
2.6.2 Derivative Complexes ............................ 36
2.6.3 Differential Ring of Polytopes .................. 37
2.6.4 Combinatorial Differential Calculus ............. 38
2.6.5 Generic Models .................................. 41
References .................................................. 44
3 Separation and Integration .................................. 47
3.1 Binary Rooted Trees for Separation ..................... 47
3.1.1 Separation Sequences ............................ 47
3.1.2 Binary Rooted Trees as Combinatorial Species .... 50
3.1.3 Configurations as Dual Graded Graphs ............ 51
3.1.4 Distributed Separation Configurations ........... 55
3.1.5 Self-Evolvability and Polytopes ................. 57
3.1.6 Entropy Calculus ................................ 59
3.2 Lifted Binary Trees .................................... 61
3.2.1 Configurations as Dual Graded Graphs ............ 61
3.2.2 Integration Schemas ............................. 65
3.2.3 Self-Evolvability and Polytopes ................. 66
3.2.4 Entropy Calculus ................................ 68
3.3 Rooted Trees ........................................... 69
3.3.1 Dual Graded Graphs for Rooted Trees ............. 69
3.3.2 Self-Evolvability and Polytopes ................. 72
3.3.3 Entropy Calculus ................................ 73
References .................................................. 74
4 Cyclic and Linear ........................................... 76
4.1 Cyclic Separations ..................................... 76
4.1.1 Presentations ................................... 76
4.1.2 Dual Graded Graphs for Necklaces ................ 79
4.1.3 Non-crossing Partitions ......................... 80
4.1.4 Self-Evolvability and Polytopes ................. 82
4.1.5 Entropy Calculus ................................ 83
4.2 Evolvability for Linear vs. Cyclical Schemas ........... 85
4.2.1 Evolvability Request ............................ 85
4.2.2 Dual Graded Graphs for Catalan Trees ............ 87
4.2.3 Fibonacci Graphs ................................ 89
4.2.4 Cyclical Schemas ................................ 90
4.2.5 Self-Evolvability and Polytopes ................. 91
4.2.6 Entropy Calculus ................................ 93
References .................................................. 93
5 Compositions and Decompositions ............................. 95
5.1 Compositions ........................................... 95
5.1.1 Integers Composition ............................ 95
5.1.2 Dual Graded Graphs for Compositions ............. 96
5.1.3 Self-Evolvability and Polytopes ................. 98
5.1.4 Entropy Calculus ................................ 99
5.1.5 Pascal Graphs for Compositions ................. 100
5.2 Partitions ............................................ 102
5.2.1 Integers Partition ............................. 102
5.2.2 Combinatorial Species .......................... 102
5.2.3 Dual Graded Graphs for Partitions .............. 103
5.2.4 Entropy Calculus ............................... 105
5.2.5 Partitions as Dual Graded Graphs ............... 106
5.2.6 Self-Evolvability and Polytopes ................ 107
References ................................................. 109
6 Construction and Deconstruction ............................ 111
6.1 Crystal Growth ......................................... 111
6.1.1 Dendrites and Crystals ......................... 111
6.1.2 Dual Graphs for 3-cores ........................ 115
6.1.3 Polytopes and Self-Evolvable 3-cores ........... 116
6.2 Self-Configurable Modular Automata .................... 117
6.2.1 Automata ....................................... 117
6.2.2 Architecture ................................... 118
6.2.3 Assembly and Disassembly ....................... 121
6.2.4 Shifted Shapes ................................. 123
6.2.5 Entropy Calculus ............................... 124
6.3 Packing and Unpacking ................................. 126
6.3.1 VLSI Design .................................... 126
6.3.2 Dual Graded Graphs for Packing ................. 127
6.3.3 Self-Evolvability and Polytopes ................ 128
References ................................................. 129
7 Strong and Weak Molecular Interactions ..................... 131
7.1 Molecular and Supramolecular .......................... 131
7.2 Dynamic Combinatorial Libraries and Templating ........ 132
7.3 Polytopes for Supramolecular Chemistry ................ 134
7.4 G-quadruplexes ........................................ 136
7.5 Supramolecular Tiling ................................. 139
7.6 Stereochemistry for Cyclic Compounds .................. 143
References ................................................. 145
8 Synthesis and Decomposition Reactions ...................... 147
8.1 Evolutionary Biotechnology ............................ 147
8.1.1 DNA and RNA .................................... 147
8.1.2 Rooted Trees for Secondary RNA Structure ....... 148
8.1.3 Polytope for RNA Structure ..................... 149
8.1.4 Reflected Graphs ............................... 151
8.1.5 Autocatalytic Network for Ribozyme Self-
Construction ................................... 153
8.2 Chemical Reaction Networks ............................ 154
8.2.1 Alkanes ........................................ 154
8.2.2 Deuterated Thiophenes .......................... 157
8.2.3 Chlorobenzenes ................................. 158
8.2.4 Self-Evolvability and Polytopes ................ 160
8.2.5 Hemoglobin Oxygenation ......................... 161
8.3 Chemical Organization ................................. 162
References ................................................. 163
9 Data and Concepts Analysis ................................. 165
9.1 fermaLConcept Analysis ................................ 165
9.1.1 Contexts^and Concepts .......................... 165
9.1.2 FCA for Separation Schemas ..................... 166
9.1.3 Case Studies ................................... 168
9.1.4 Polytope for FCA. Lattices ..................... 172
9.2 Nesting Line Diagrams ................................. 173
9.2.1 Two-levels Formal Context ...................... 173
9.2.2 Graphs Spanning for Comparison ................. 175
9.2.3 Self-Evolvability and Polytopes ................ 177
9.2.4 Entropy Calculus ............................... 179
References ................................................. 180
10 Design of Experiments and Analysis ......................... 181
10.1 Design of Experiments and Hasse Diagrams .............. 181
10.1.1 Hasse Diagrams ................................. 181
10.1.2 Entropy Calculus ............................... 184
10.2 Permutation Trees for Designs of Experiments .......... 185
10.3 Self-Evolvability and Polytopes ....................... 187
References ................................................. 190
11 Premises and Perspectives .................................. 191
11.1 Premises .............................................. 191
11.1.1 n-Levels Systems ............................... 191
11.1.2 Complementarity and Duality .................... 192
11.1.3 Closure and "Self" ............................. 193
11.1.4 Polytope Framework ............................. 193
11.1.5 Generic Models ................................. 194
11.1.6 Informational Criteria ......................... 195
11.1.7 Foundations .................................... 195
11.2 Perspectives .......................................... 196
11.2.1 Technologies and Materials ..................... 196
11.2.2 Biosystems and Bio-inspired Systems ............ 198
11.2.3 Information and Knowledge Systems .............. 199
11.2.4 Economy, Society and Ecology ................... 201
11.2.5 Ethics and Law ................................. 203
References ................................................. 205
Appendix 1: Informational Entropy ............................. 209
References ................................................. 211
Index ......................................................... 213
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