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1 | (24) |
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25 | (14) |
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25 | (2) |
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27 | (5) |
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29 | (3) |
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2.3 Poincare Section and Poincare Map |
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32 | (2) |
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2.4 Numerical Derivation of a Complex Function |
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34 | (1) |
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34 | (2) |
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2.6 The Precision of Numbers |
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36 | (3) |
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3 The Gamma Function and the Incomplete Gamma Functions |
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39 | (4) |
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4 The Hurwitz Zeta Function and the Lerch Zeta Function |
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43 | (26) |
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4.1 The Euler-MacLaurin Formula and the Bernoulli Numbers |
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44 | (1) |
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4.2 The Hurwitz Zeta Function |
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45 | (8) |
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4.2.1 Application of the Euler-MacLaurin Formula |
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47 | (5) |
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4.2.2 The Implementation of the Hurwitz Zeta Function for s ε C, z ε Q |
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52 | (1) |
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4.2.3 Test of the Implementation |
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53 | (1) |
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4.3 The Lerch Transcendent and the Lerch Zeta Function |
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53 | (16) |
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4.3.1 Application of the Euler-MacLaurin Formula |
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58 | (7) |
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4.3.2 The Implementation of the Lerch Zeta Function for s ε C, z ε Q and A ε R |
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65 | (2) |
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4.3.3 Test of the Implementation |
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67 | (2) |
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5 Computation of the Spectra and Eigenvectors of Large Complex Matrices |
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69 | (18) |
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5.1 Eigenvalues of a Matrix |
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70 | (2) |
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5.1.1 Schur Decomposition |
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71 | (1) |
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5.2 Complex Givens Rotations |
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72 | (3) |
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75 | (1) |
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5.4 The QR Algorithm for Complex Matrices |
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76 | (5) |
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76 | (1) |
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5.4.2 The Shifted QR Iteration |
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77 | (1) |
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5.4.3 The Final QR Algorithm |
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78 | (3) |
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5.5 A Verification of the Implementation |
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81 | (1) |
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5.6 Computation of Eigenvectors of Quasi Triangular Matrices |
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82 | (5) |
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5.6.1 Backward Substitution for Non-singular Quasi Triangular Matrices |
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83 | (1) |
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5.6.2 Computation of Eigenvectors |
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84 | (3) |
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6 The Hyperbolic Laplace-Beltrami Operator |
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87 | (42) |
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6.1 The Group PSL(2, R) and Congruence Subgroups |
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88 | (3) |
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91 | (2) |
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6.3 The Spectrum of the Hyperbolic Laplace-Beltrami Operator |
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93 | (3) |
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6.4 Involutions of Maass Wave Forms |
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96 | (1) |
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6.5 The Selberg Trace Formula and the Selberg Zeta Function |
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97 | (6) |
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6.5.1 The Selberg Zeta Function for the Geodesic Flow |
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103 | (1) |
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6.6 Character Deformations for Freely Generated Groups |
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103 | (10) |
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6.6.1 Character Deformation for Γ0 (4) |
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105 | (5) |
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6.6.2 Character Deformation for Γ0 (8) |
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110 | (3) |
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6.7 The Induced Representation Uχ |
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113 | (3) |
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116 | (7) |
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6.9 Computational Methods for Eigenfunctions and Spectra |
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123 | (6) |
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7 Transfer Operators for the Geodesic Flow on Hyperbolic Surfaces |
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129 | (66) |
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130 | (1) |
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7.2 The Transfer Operator as a Sum of Composition Operators |
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131 | (2) |
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7.3 Symbolic Dynamics for the Geodesic Flow |
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133 | (3) |
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7.4 A Transfer Operator for SL(2, Z) |
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136 | (19) |
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7.4.1 Results for the Transfer Operator for (Γ0(n), χ) With n = 1 and x = 1 |
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147 | (8) |
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7.5 A Transfer Operator for Finite Index Subgroups Γ of SL(2, Z) |
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155 | (4) |
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7.6 The Transfer Operator for Character Deformations |
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159 | (17) |
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7.6.1 The Transfer Operator £(n)β, ε, χ with the Representation Uχ |
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160 | (1) |
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7.6.2 An Analytic Continuation of the Transfer Operator |
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161 | (9) |
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7.6.3 A Nuclear Representation of the Transfer Operator |
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170 | (2) |
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7.6.4 An Approximation of the Transfer Operator |
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172 | (4) |
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7.7 Symmetries of the Transfer Operator and a Factorization of the Selberg Zeta Function |
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176 | (19) |
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7.7.1 The Transfer Operator £(n)β, ε, χ and the Operators Pκ |
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177 | (2) |
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7.7.2 An Algorithm to Determine the Operators Pκ from the Transfer Operator £(n)β, ε, χ |
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179 | (5) |
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7.7.3 Operators Pκ and Involutions jκ of Maass Wave Forms for Γ0 (n) With χ &quiv; 1 |
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184 | (2) |
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7.7.4 Operators Pκ for (Γ0 (4), χ(4)α1 α2) and (Γ0 (8), χ(4)α1 α2 α3) |
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186 | (4) |
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7.7.5 Operators Pκ and the Lewis Equation for (Γ0 (n), χ) |
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190 | (5) |
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8 Numerical Results for Spectra and Traces of the Transfer Operator for Character Deformations |
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195 | (36) |
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8.1 Approximation of the Spectra of Transfer Operators and Numerical Verifications |
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197 | (10) |
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8.2 The Equality of the Spectra of £(n)β, + 1, χ and £(n)β, -1, χ |
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207 | (3) |
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8.3 The Spectra and Traces of the Transfer Operator |
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210 | (11) |
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8.4 The Eigenfunctions of the Transfer Operator and Period Functions |
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221 | (10) |
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9 Investigations of Selberg Zeta Functions Under Character Deformations |
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231 | (66) |
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9.1 Tracking of the Zeros of the Selberg Zeta Function in the β-Plane for α ε [ 0, 1/2] |
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233 | (7) |
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9.2 Numerical Results for the Selberg Zeta Function and Its Zeros |
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240 | (12) |
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9.3 The Zeros Z(4)α-1 of the Selberg Zeta Function Z(4) (β, Χ(4)α) |
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252 | (10) |
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9.4 The Zeros Z(4)α, +1 of the Selberg Zeta Function Z(4) (β, Χα(4)) |
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262 | (22) |
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9.5 Results for (Γ0 (8), Χα(8) and (Γ0 (4), Χ(4)α1, α2) |
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284 | (3) |
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9.6 Comparison of Numerical Data to Theoretical Results for (Γ0 (4), Χα(4)) with α &rar; 0 |
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287 | (10) |
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297 | (4) |
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A Computational Aspects of the Transfer Operator for the Kac-Baker Model |
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301 | (12) |
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A.1 A Nuclear Representation of the Transfer Operator for the Kac-Baker Model |
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302 | (3) |
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A.2 A Verification of the Implementation of the Approximation of the Transfer Operator |
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305 | (1) |
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A.3 Numerical Results for the Transfer Operator |
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306 | (7) |
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A.3.1 Concluding Discussion of the Numerical Results |
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307 | (6) |
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313 | (6) |
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C The Representatives of Γ0 (4) and Γ0 (8) in SL(2, Z) |
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319 | (2) |
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D The Transfer Operator for (Γ0 (8), Χ(8)α1, α2, α3) |
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321 | (10) |
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E The Zeros and Poles of the Selberg Zeta Function for Arithmetic (Γ0 (4), Χα(4)and Arithmetic (Γ0 (8), Χα(8) |
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331 | (12) |
| References |
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343 | (6) |
| Index of Notations |
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349 | |
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