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Part I The Sierpinski Gasket |
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1 Definitions and General Properties |
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3 | (16) |
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1.1 First Appearance and Naive Definition |
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3 | (2) |
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5 | (5) |
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6 | (1) |
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7 | (1) |
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8 | (2) |
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1.2 Definition of Self-Similar Fractals |
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10 | (5) |
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Info B Hausdorff Measure and Hausdorff Dimension |
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15 | (4) |
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2 The Laplace Operator on the Sierpinski Gasket |
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19 | (10) |
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Info C The Classical Laplace Operator and Harmonic Functions |
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19 | (4) |
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20 | (1) |
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21 | (2) |
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2.1 The Laplace Operator on Sn |
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23 | (3) |
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2.2 Comparing Spectra of Δn and of Δn--1 |
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26 | (1) |
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2.3 Eigenfunctions of the Laplace Operator on Sn |
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27 | (2) |
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3 Harmonic Functions on the Sierpinski Gasket |
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29 | (28) |
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3.1 First Properties of Harmonic Functions |
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29 | (2) |
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3.2 The Functions Χ, φ, ψ, ξ |
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31 | (5) |
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3.3 Extension and Computation of Χ(t) and ψ(t) |
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36 | (2) |
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Info D Fractional Derivatives and Fractional Integrals |
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38 | (2) |
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3.4 Some Arithmetic Properties of Basic Functions |
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40 | (2) |
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42 | (1) |
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3.6 The Functions x(t), y(t), and y(x) |
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43 | (2) |
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3.7 The Harmonic Image of S |
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45 | (1) |
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3.8 Multidimensional Analogues of S |
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46 | (2) |
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48 | (5) |
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E.1 Standard Digital Systems |
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48 | (1) |
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49 | (1) |
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50 | (2) |
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52 | (1) |
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3.9 Applications of Generalized Numerical Systems |
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53 | (4) |
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3.9.1 Application to the Sierpinski Gasket |
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53 | (1) |
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3.9.2 Application to the Question Mark Function |
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54 | (3) |
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Part II The Apollonian Gasket |
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4 Circles and Disks on Spheres |
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57 | (22) |
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Info F The Conformal Group and Stereographic Projection |
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57 | (9) |
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57 | (2) |
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F.2 Stereographic Projection |
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59 | (2) |
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F.3 The Matrix Definition of Gn |
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61 | (2) |
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63 | (3) |
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4.1 Descartes's Theorem on Disks in the Plane |
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66 | (6) |
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4.2 Proof of Descartes's Theorem for n = 2 |
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72 | (7) |
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4.2.1 Generalized Descartes's Theorem |
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75 | (4) |
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5 Definition of the Apollonian Gasket |
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79 | (16) |
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79 | (3) |
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82 | (1) |
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Info G The Fibonacci Numbers |
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83 | (3) |
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5.3 Examples of Unbounded Apollonian Tilings |
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86 | (4) |
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5.4 Integral Solutions to Descartes's Equation |
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90 | (2) |
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Info H Structure of Groups Freely Generated by Reflections |
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92 | (3) |
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6 Arithmetic Properties of Apollonian Gaskets |
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95 | (20) |
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95 | (3) |
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6.2 Rational Parameterization of Circles |
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98 | (8) |
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6.3 Nice Parameterizations of Disks Tangent to a Given Disk |
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106 | (3) |
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6.4 Integral Apollonian Gaskets |
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109 | (1) |
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109 | (1) |
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Info I The Mobius Inversion Formula |
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110 | (5) |
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112 | (3) |
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7 Geometric and Group-Theoretic Approach |
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115 | (14) |
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Info J The Hyperbolic (Lobachevsky) Plane L |
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115 | (6) |
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J.1 The First Poincare Model |
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115 | (2) |
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J.2 The Second Poincare Model |
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117 | (1) |
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118 | (3) |
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7.1 The Mobius Group and Apollonian Gaskets |
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121 | (4) |
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7.2 Action of the Group Γ4 on an Apollonian Gasket |
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125 | (4) |
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8 MultiDimensional Apollonian Gaskets |
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129 | (6) |
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129 | (3) |
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8.2 The Three-Dimensional Apollonian Gasket |
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132 | (3) |
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135 | (2) |
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A Popular Books, Lectures and Surveys |
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135 | (1) |
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135 | (1) |
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136 | (1) |
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136 | (1) |
| Index |
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137 | |
