Dedication |
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v | |
Preface |
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vii | |
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1 | (18) |
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1 | (1) |
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Logical Connectives --- Notation |
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2 | (1) |
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2 | (1) |
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3 | (1) |
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4 | (1) |
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4 | (1) |
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5 | (1) |
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6 | (7) |
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13 | (2) |
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15 | (1) |
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15 | (1) |
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16 | (3) |
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19 | (22) |
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Relations --- The Naive Approach |
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19 | (1) |
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20 | (1) |
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21 | (1) |
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Cross-Product Set and Its Properties |
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22 | (3) |
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25 | (2) |
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27 | (2) |
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Binary Relations --- Terminology |
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29 | (1) |
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Mappings --- Classification |
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30 | (6) |
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36 | (2) |
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38 | (1) |
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39 | (1) |
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39 | (2) |
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41 | (26) |
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Logical Objects and Logical Statements |
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41 | (1) |
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Logical-Statement and Truth-Value Mappings |
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42 | (2) |
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Logical Connectives and Logical Formulas |
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44 | (1) |
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Adequate Set of Connectives |
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44 | (1) |
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Truth Values of Complex Logical Formulas |
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45 | (4) |
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Tautologies and Contradictions |
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49 | (1) |
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50 | (2) |
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Fundamental Tautologies. De Morgan Laws. Proof Formats |
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52 | (5) |
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Properties of Connectives. An Axiomatic Approach to Mathematical Logic |
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57 | (1) |
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58 | (1) |
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Boolean Algebra. The Analogy Between Mathematical Logic and Set Theory |
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59 | (2) |
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61 | (2) |
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63 | (1) |
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64 | (1) |
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65 | (2) |
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Algebraic Structures: Group through Linear Space |
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67 | (28) |
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Classification of Algebraic Structures |
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67 | (6) |
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Algebra of Complex Numbers |
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73 | (2) |
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Geometric Representation of Complex Numbers |
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75 | (2) |
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Linear Space: Axioms and Examples |
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77 | (3) |
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Linear Combination and Span. Linear Mappings |
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80 | (1) |
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80 | (2) |
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82 | (1) |
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Basis and Dimension of a Linear Space |
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82 | (3) |
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Decomposition as Mapping. Isomorphism of Finite-Dimensional Linear Spaces |
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85 | (1) |
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86 | (4) |
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90 | (2) |
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92 | (1) |
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92 | (3) |
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Linear Mappings and Matrices |
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95 | (32) |
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Linear Mappings and Column Vectors |
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95 | (1) |
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95 | (2) |
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97 | (2) |
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99 | (2) |
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101 | (1) |
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Fundamental Subspaces of Column Vectors Associated With a Given Matrix |
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102 | (1) |
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The Echelon Matrix. LU-decomposition |
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103 | (3) |
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The Fundamental Theorem of Linear Algebra |
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106 | (2) |
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108 | (2) |
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Eigenvalues, Eigenvectors, and the Characteristic Equation |
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110 | (1) |
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111 | (1) |
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112 | (5) |
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117 | (1) |
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Multiple Eigenvalues. The Jordan Form |
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118 | (4) |
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122 | (1) |
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123 | (1) |
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124 | (3) |
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Metrics and Topological Properties |
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127 | (20) |
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Necessity of Distance Concept |
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127 | (1) |
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Metric Mapping and Distance: Definition and Axioms |
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127 | (2) |
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129 | (3) |
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132 | (1) |
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Proximity. Limits in Metric Spaces |
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133 | (1) |
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134 | (1) |
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Open Ball: A Geometric Interpretation |
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134 | (1) |
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Examples of Cauchy Sequences |
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135 | (2) |
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Open and Closed Sets. Completeness vs. Closedness |
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137 | (3) |
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140 | (1) |
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Lipschitz (Bounded) Mapping |
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141 | (1) |
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Example of a Contraction Mapping |
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141 | (1) |
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142 | (1) |
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143 | (1) |
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144 | (3) |
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Banach and Hilbert Spaces |
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147 | (24) |
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Introduction: The Great Alliance of Linearity and Metrics |
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147 | (1) |
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Norm Mapping. Banach Space |
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147 | (2) |
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149 | (1) |
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Finite-Dimensional Subspaces in Banach Space |
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150 | (2) |
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Inner-Product Mapping. Hilbert Space |
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152 | (2) |
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154 | (1) |
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The Cauchy-Schwarz Inequality |
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155 | (2) |
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Inner Product and Matrices. Hermitian Matrices |
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157 | (2) |
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Continuity of the Inner-Product Mapping |
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159 | (1) |
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160 | (1) |
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160 | (2) |
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Eigenvectors of Hermitian Matrices |
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162 | (1) |
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163 | (4) |
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167 | (1) |
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168 | (1) |
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169 | (2) |
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Orthonormal Bases and Fourier Series |
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171 | (28) |
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171 | (1) |
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The Gram - Schmidt Orthonormalization |
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172 | (4) |
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176 | (2) |
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178 | (2) |
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180 | (1) |
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The Projection Decomposition |
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181 | (3) |
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184 | (2) |
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Complete Orthonormal Sequences |
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186 | (4) |
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190 | (1) |
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Properties of the Fourier Series |
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191 | (1) |
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192 | (2) |
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194 | (2) |
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196 | (1) |
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197 | (1) |
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197 | (2) |
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199 | (36) |
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Inverse Mappings and Operator Equations |
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199 | (1) |
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Three Generic Problems and Their Equivalence |
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200 | (1) |
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Fixed-Point Problem. The Contraction Mapping Theorem |
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201 | (3) |
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Successive Approximations |
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204 | (3) |
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Application to Differential Equations |
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207 | (5) |
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212 | (3) |
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Residual Principle and Residual Approximation |
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215 | (6) |
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Incompletely Specified Equations. Least-Square Approximation |
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221 | (5) |
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Problem of Uniqueness. Pseudo-Inverse Mapping |
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226 | (2) |
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228 | (2) |
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230 | (1) |
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231 | (1) |
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232 | (3) |
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Fourier and Laplace Transforms |
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235 | (38) |
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235 | (2) |
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The Fourier Transform as a Mapping |
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237 | (2) |
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Digression. Properties of the δ-function |
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239 | (2) |
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Back to Fourier Transform Mapping |
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241 | (1) |
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Properties of the Fourier Transform |
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241 | (5) |
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Application to Linear Systems. The Convolution Product |
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246 | (4) |
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Sample Problem I: Smoothing Operator |
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250 | (1) |
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251 | (5) |
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From Fourier Transform to Laplace Transform |
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256 | (2) |
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Properties of the Laplace Transform |
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258 | (4) |
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262 | (2) |
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The Laplace Transform Table |
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264 | (1) |
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265 | (2) |
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Laplace Transform of the Convolution Product |
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267 | (1) |
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Applications of the Laplace Transform to Solving Differential Equations |
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268 | (1) |
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269 | (1) |
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269 | (1) |
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270 | (1) |
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271 | (2) |
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Partial Differential Equations |
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273 | (33) |
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273 | (1) |
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274 | (1) |
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274 | (6) |
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280 | (1) |
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Canonical Forms --- A Summary |
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281 | (1) |
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282 | (3) |
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The Wave Equation. D'Alembert's Formula |
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285 | (3) |
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The Diffusion Equation. Transform Methods |
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288 | (4) |
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The Elliptic Case: Poisson's Equation |
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292 | (7) |
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A Very Brief Comment in Defense of the Laplace Transform |
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299 | (1) |
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300 | (2) |
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302 | (1) |
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303 | (1) |
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303 | (3) |
Topic Index |
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306 | |