List of Symbols |
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XIX | |
Part I Theory |
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1 On the Perron Root of Irreducible Matrices |
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3 | |
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1.1 Some Basic Definitions |
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3 | |
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1.2 Some Bounds on the Perron Root and Their Applications |
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4 | |
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1.2.1 Concavity of the Perron Root on Some Subsets of Irreducible Matrices |
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11 | |
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1.2.2 Kullback–Leibler Divergence Characterization |
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14 | |
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1.2.3 Some Extended Perron Root Characterizations |
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15 | |
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1.2.4 Collatz–Wielandt-Type Characterization of the Perron Root |
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18 | |
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1.3 Convexity of the Perron Root |
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22 | |
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22 | |
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1.3.2 Sufficient Conditions |
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24 | |
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1.3.3 Convexity of the Feasibility Set |
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26 | |
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1.3.4 Necessary Conditions |
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28 | |
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1.4 Special Classes of Matrices |
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30 | |
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31 | |
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1.4.2 Symmetric Positive Semidefinite Matrices |
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32 | |
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1.5 The Perron Root Under the Linear Mapping |
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34 | |
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35 | |
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1.5.2 Disproof of the Conjecture |
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38 | |
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1.6 Some Remarks on Arbitrary Nonnegative Matrices |
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41 | |
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1.6.1 Log-Convexity of the Spectral Radius |
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42 | |
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1.6.2 Characterization of the Spectral Radius |
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43 | |
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1.6.3 Collatz- Wielandt-Type Characterization of the Spectral Radius |
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46 | |
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47 | |
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2 On the Positive Solution to a Linear System with Nonnegative Coefficients |
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51 | |
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2.1 Basic Concepts and Definitions |
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51 | |
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53 | |
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56 | |
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2.3.1 Log-Convexity of the Positive Solution |
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56 | |
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2.3.2 Convexity of the Feasibility Set |
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59 | |
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2.3.3 Strict Log-Convexity |
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60 | |
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2.3.4 Strict Convexity of the Feasibility Sets |
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65 | |
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66 | |
Part II Applications and Algorithms |
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71 | |
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75 | |
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75 | |
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4.2 Medium Access Control |
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76 | |
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4.3 Wireless Communication Channel |
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79 | |
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4.3.1 Signal-to-Interference Ratio |
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81 | |
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83 | |
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84 | |
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85 | |
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5 Resource Allocation Problem in Communications Networks |
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91 | |
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5.1 End-to-End Rate Control in Wired Networks |
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91 | |
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92 | |
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95 | |
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5.2 Problem Formulation for Wireless Networks |
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97 | |
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5.2.1 Joint Power Control and Link Scheduling |
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98 | |
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5.2.2 Feasible Rate Region |
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101 | |
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5.2.3 End-to-End Window-Based Rate Control for Wireless Networks |
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103 | |
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5.2.4 MAC Layer Fair Rate Control for Wireless Networks |
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105 | |
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5.2.5 Utility-Based Power Control |
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107 | |
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5.3 Interpretation in the QoS Domain |
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112 | |
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5.4 Remarks on Joint Power Control and Link Scheduling |
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115 | |
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5.4.1 Optimal Joint Power Control and Link Scheduling |
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115 | |
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118 | |
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119 | |
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5.4.4 Wireless Links with Self-Interference |
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122 | |
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5.5 Remarks on the Efficiency–Fairness Trade Off |
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123 | |
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5.5.1 Efficiency of the Max-Min Fair Power Allocation |
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125 | |
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5.5.2 Axiom-Based Interference Model |
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128 | |
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6 Power Control Algorithm |
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129 | |
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6.1 Some Basic Definitions |
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130 | |
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6.2 Convex Statement of the Problem |
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131 | |
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6.3 Strong Convexity Conditions |
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133 | |
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6.4 Gradient Projection Algorithm |
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137 | |
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138 | |
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6.4.2 Rate of Convergence |
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140 | |
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142 | |
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6.4.4 Projection on a Closed Convex Set |
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142 | |
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6.5 Distributed Implementation |
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143 | |
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6.5.1 Local and Global Parts of the Gradient Vector |
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143 | |
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145 | |
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6.5.3 Distributed Handshake Protocol |
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148 | |
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150 | |
Part III Appendices |
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A Some Concepts and Results from Matrix Analysis |
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155 | |
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A.1 Vectors and Vector Norms |
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155 | |
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A.2 Matrices and Matrix Norms |
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157 | |
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A.3 Square Matrices and Eigenvalues |
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158 | |
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A.3.1 Spectral Radius and Neumann Series |
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159 | |
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A.3.2 Orthogonal, Symmetric and Positive Semidefinite Matrices |
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160 | |
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A.4 Perron—Frobenius Theory |
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161 | |
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A.4.1 Perron—Frobenius Theorem for Irreducible Matrices |
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162 | |
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A.4.2 Perron—Frobenius Theorem for Primitive Matrices |
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165 | |
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A.4.3 Some Remarks on Reducible Matrices |
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166 | |
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A.4.4 The Existence of a Positive Solution p to (αI — X)p = b |
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168 | |
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B Some Concepts and Results from Convex Analysis |
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171 | |
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171 | |
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B.2 Convex Sets and Functions |
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175 | |
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176 | |
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177 | |
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B.3.1 Inverse Functions of Monotonic Log-Convex Functions |
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179 | |
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B.4 Convergence of Gradient Projection Algorithms |
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180 | |
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185 | |