Acknowledgement |
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XI | |
Introduction |
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1 | (4) |
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Chapter 1 Why the Techniques Developed in this Book are Important? (A Few Examples of Applications) |
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5 | (18) |
Part I Foundations |
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23 | (20) |
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Chapter 2 Basic Mathematical Facts |
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23 | (20) |
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2.1 Elements of Set Theory |
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23 | (6) |
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23 | (4) |
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27 | (2) |
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2.2 Elements of Linear Algebra |
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29 | (3) |
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2.3 Elements of Boolean Algebra |
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32 | (5) |
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37 | (6) |
Part II Dynamic Pattern Recognition Problems and Control over Classification Reliabity |
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43 | (80) |
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Chapter 3 "Voting by a Set of Features" Algorithms |
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43 | (18) |
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43 | (2) |
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3.2 "Voting by Elementary Features" Algorithms |
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45 | (4) |
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3.3 Definition of Classes of Algorithms "Voting by a Set of Features" (VSF) and "Voting by Elementary Features" (VEF). Normal Weights |
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49 | (3) |
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3.4 Class of Algorithms "CORA-i" |
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52 | (2) |
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3.5 "Neighbours" as a VSF Algorithm |
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54 | (2) |
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3.6 "Threshold" Classification Algorithms |
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56 | (2) |
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3.7 "Bayes" as a VEF Algorithm |
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58 | (3) |
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Chapter 4 Dynamic and Limit Classification Problems |
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61 | (24) |
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4.1 Formulation of the Dynamic Recognition Problem |
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61 | (4) |
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4.2 Stability of Prediction in Dynamic Classification Problems |
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65 | (1) |
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4.3 Stability of Prediction, Algorithm VEF-0 |
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66 | (3) |
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4.4 Stability Conditions in the Case of the Algorithm VEF-1 |
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69 | (4) |
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4.5 Stability Zone for Prediction Obtained by the Algorithm VEF-2 |
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73 | (4) |
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4.6 Stability Zone for Prediction Obtained by a VSF Algorithm |
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77 | (3) |
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4.7 Stability of Prediction in Case of Normal Weights |
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80 | (5) |
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Chapter 5 Dual Systems of Sets and Local Stability of Classification |
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85 | (12) |
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85 | (2) |
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5.2 S-Theorems and S-Counter-examples |
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87 | (2) |
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5.3 Local Stability and Stabilizing Sets |
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89 | (4) |
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5.4 Local Stability and Cluster Analysis |
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93 | (4) |
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Chapter 6 Investigation of Earthquake-prone Areas as a Limit Pattern Recognition Problem |
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97 | (8) |
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6.1 Formalizing the Recognition of Strongly Earthquake-prone Areas |
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97 | (4) |
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6.2 Correspondance Between Prediction and Classification Problems |
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101 | (4) |
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Chapter 7 Control Experiments for Evaluating Classification Reliability |
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105 | (18) |
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7.1 Comparison of Quality Solutions for Real and Random Learning Materials |
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105 | (5) |
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7.2 Estimating the Probability of the Classification Error |
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110 | (3) |
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7.3 Estimation of Parameters of the Classification Problem using Estimation of Non-randomness and Reliability Functions |
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113 | (3) |
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7.4 Additional Arguments for Evaluating the Classification Reliability |
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116 | (7) |
Part III Dynamic Systems |
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123 | (86) |
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Chapter 8 Basic Definitions and Facts |
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123 | (32) |
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8.1 Measure and Dimension |
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123 | (3) |
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124 | (1) |
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8.1.2 Hausdorff dimension |
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125 | (1) |
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8.2 Discrete Dynamic Systems |
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126 | (1) |
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8.3 Continuous Dynamic Systems |
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127 | (1) |
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8.4 Representation and Study of Dynamic Systems |
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128 | (18) |
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8.4.1 Iterative scheme as a tool in dynamic system study |
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129 | (11) |
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140 | (6) |
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8.5 Attraction and Repulsion Cycles |
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146 | (2) |
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8.5.1 Areas contraction and consequences |
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147 | (1) |
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8.6 Fractal Attractors, Basin of Attraction, Repellers |
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148 | (2) |
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150 | (5) |
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Chapter 9 Geometry of Attractors |
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155 | (18) |
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9.1 Classical Examples of Strange Attractors |
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155 | (6) |
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9.1.1 The Lorenz attractor |
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155 | (3) |
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9.1.2 The Henon attractor |
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158 | (1) |
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9.1.3 Some other classical attractors |
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158 | (3) |
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9.2 Correlation Function Method |
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161 | (3) |
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164 | (8) |
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9.3.1 The mutual information criterion |
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165 | (1) |
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9.3.2 The minimum embedding dimension |
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165 | (2) |
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9.3.3 The Lyapunov exponents computation (Sano and Sawada Method) |
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167 | (3) |
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9.3.4 Calculation of Lyapunov exponents (the method of Wolf et al.) |
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170 | (2) |
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172 | (1) |
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Chapter 10 Bifurcation, Cascades and Chaos |
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173 | (16) |
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10.1 Floquet Matrix, Hopf Bifurcation |
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173 | (7) |
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174 | (2) |
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176 | (2) |
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10.1.3 Passing from torus I(2) to torus I(3). Ruelle-Takens theory |
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178 | (2) |
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10.2 Curry and Yorke Model and Route to Chaos |
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180 | (3) |
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183 | (1) |
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183 | (6) |
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Chapter 11 Self Organisation |
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189 | (6) |
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11.1 The Evolving Sandpile Experiment |
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189 | (4) |
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11.2 General Aspects of Self-Organized Criticality |
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193 | (2) |
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195 | (14) |
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12.1 Generalized Dimensions of Fractals and Strange Attractors |
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195 | (3) |
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12.2 Definitions and Properties |
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198 | (11) |
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12.2.1 A Simple Definition |
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198 | (1) |
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12.2.2 A Measure on the Middle Third Cantor Set |
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199 | (1) |
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12.2.3 Practical Methods of Constructing a Multifractal Spectrum |
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199 | (10) |
Part IV Convex Programming and Systems of Rigid Blocks with Deformable Layers |
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209 | (16) |
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Chapter 13 Systems of Rigid Blocks with Thin Deformable Layers (SRBTDL) |
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209 | (6) |
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Chapter 14 System of Rigid and Deformable Blocks (SRDB) |
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215 | (10) |
Part V Bibliography |
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225 | (18) |
Part VI Index |
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243 | (8) |
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251 | |