Algebraic techniques and their use in describing and processing uncertainty

Preface......Page 6
Contents......Page 7
 1 Introduction......Page 9
 2 Specker ell-groups......Page 11
 3 Specker Algebras......Page 19
  4.1 Specker mathbbZ-algebras......Page 23
  4.2 Specker mathbbR-algebras......Page 25
 References......Page 26
 1 Introduction......Page 28
 2 Least Square Approximations and Linear Operators......Page 30
 3 Values of Cooperative Games......Page 31
 4 Least Square Values......Page 32
  4.1 Weighted Approximation......Page 33
  4.2 Regular Weights......Page 34
 5 Probabilistic Values......Page 37
 References......Page 38
 1 Introduction......Page 40
 2 Preliminaries......Page 41
 3 Finite Direct Sums......Page 42
 4 Infinite Direct Sums......Page 44
 References......Page 47
 A Topos View of the Type-2 Fuzzy Truth Value Algebra......Page 48
 1 Introduction......Page 49
 2 The Type-2 Fuzzy Truth Value Algebra,  and Convolution Algebras......Page 51
 3 Topos Theory......Page 53
 4 The Convolution Algebra from a Topos Perspective......Page 56
 References......Page 61
 1 Introduction: Why Solve Linear Programming Problems, and How These Problems Are Solved Now......Page 62
 2 Chubanov's Algorithm: Main Idea......Page 66
 3 Why Chubanov's Algorithm Works? Why Other Algorithms Work?......Page 69
 References......Page 71
 1 Formulation of the Problem......Page 72
 2 Why Symmetries: A General Explanation......Page 76
 3 What Are the Reasonable Transformations  of the Complex Value ψ(x)?......Page 77
 4 Invariant Functions of ψ Are Exactly the Functions Corresponding to Bohmian Econometrics......Page 78
 5 Functions Corresponding to Bohmian Econometrics Are Optimal: A Proof......Page 79
 References......Page 81
 1 Introduction......Page 83
 3 Direct Decompositions of Matrices......Page 84
 4 Matrices with Structure......Page 85
 5 Diagonal Equivalence......Page 89
 Reference......Page 93
 1 Basic Fuzzy Logical Operations and Their Generators......Page 94
  1.1 Fuzzy Negations......Page 95
  1.2 Triangular Norms......Page 96
  1.3 Triangular Conorms......Page 97
  1.4 Fuzzy Implications......Page 98
  2.2 Notation......Page 100
  3.1 Generators of Triangular Norms......Page 101
  3.2 Generators of Fuzzy Implications......Page 105
  4.1 Generators of Binary Operations......Page 110
  4.2 Back to Generators of Fuzzy Negations......Page 114
 5 Conclusions......Page 115
 References......Page 116
 1 Introduction......Page 118
 2 Statistics ch93,ch94......Page 120
 3 Decision Theory ch92,ch95......Page 121
 4 Intelligent Systems ch91,ch96,ch97......Page 123
  4.1 Fuzzy Logics ch98......Page 125
 References......Page 126
 1 His Life and Legacy......Page 127
 2 The Intersection of Engineering and Mathematics......Page 128
 Reference......Page 132
 1 Introduction......Page 133
 2 Preliminaries......Page 136
 3 MTL Algebras and Frames......Page 137
 4 Modal MTL Algebras and Modal MTL Frames......Page 140
 5 Duality via Truth for Modal MTL Algebras and Modal MTL Frames......Page 145
 6 Axiomatic Extensions of Modal MTL Algebras and Modal MTL Frames......Page 147
 References......Page 151
 1 Background......Page 154
 2 Deviant Logic......Page 155
 3 Cancellation in Direct Sums of Groups......Page 156
 4 Walker Groups......Page 157
 References......Page 158
 1 Introduction......Page 160
  2.1 Informal Description of Bigraphs......Page 162
  2.2 Mathematical Description of Bigraphs......Page 163
  3.1 Fuzzy Bigraphs......Page 166
  3.2 Type 2 Fuzzy Bigraphs......Page 169
 4 Fuzzy Bigraphical Categories......Page 171
 5 Conclusions......Page 172
 References......Page 173