Comprehensive Mathematics for Computer Scientists 2nd ed [Vol 2]下载

III Topology and Calculus 1 27 Limits and Topology 3 27.1Introduction............................................. 3 27.2Topologies on Real Vector Spaces........................ 4 27.3Continuity............................................... 14 27.4Series ................................................... 21 27.5Euler’s Formula for Polyhedra and Kuratowski’s Theorem 30 28 Di?erentiability 37 28.1Introduction............................................. 37 28.2Di?erentiation .......................................... 39 28.3Taylor’s Formula ........................................ 53 29 Inverse and Implicit Functions 59 29.1Introduction............................................. 59 29.2TheInverseFunctionTheorem........................... 60 29.3TheImplicitFunctionTheorem .......................... 64 30 Integration 73 30.1Introduction............................................. 73 30.2Partitions and the Integral ............................... 74 30.3Measure and Integrability................................ 81 31 The Fundamental Theorem of Calculus and Fubini’s Theorem 87 31.1Introduction............................................. 87 31.2The Fundamental Theorem of Calculus .................. 88 31.3Fubini’s Theorem on Iterated Integration................. 92 32 Vector Fields 97 32.1Introduction............................................. 97 32.2VectorFields ............................................ 98 VIII Contents 33 Fixpoints 105 33.1Introduction............................................. 105 33.2Contractions ............................................ 105 34 Main Theorem of ODEs 113 34.1Introduction............................................. 113 34.2Conservative and Time-Dependent Ordinary Di?erential Equations: The Local Setup .................. 114 34.3The Fundamental Theorem: Local Version................ 115 34.4The Special Case of a Linear ODE ........................ 117 34.5The Fundamental Theorem: Global Version .............. 119 35 Third Advanced Topic 125 35.1Introduction............................................. 125 35.2NumericsofODEs....................................... 125 35.3TheEulerMethod ....................................... 129 35.4Runge-KuttaMethods.................................... 131 IV Selected Higher Subjects 137 36 Categories 139 36.1Introduction............................................. 139 36.2What Categories Are..................................... 140 36.3Examples................................................ 143 36.4Functors and Natural Transformations................... 147 36.5Limits and Colimits...................................... 153 36.6Adjunction.............................................. 159 37 Splines 161 37.1Introduction............................................. 161 37.2PreliminariesonSimplexes .............................. 161 37.3WhatareSplines?........................................ 164 37.4LagrangeInterpolation .................................. 168 37.5Bézier Curves ........................................... 171 37.6Tensor Product Splines .................................. 176 37.7B-Splines ................................................ 179 38 Fourier Theory 183 38.1Introduction............................................. 183 38.2Spaces of Periodic Functions............................. 185 38.3Orthogonality ........................................... 188 Contents IX 38.4Fourier’s Theorem....................................... 191 38.5Restatement in Terms of the Sine and Cosine Functions.. 194 38.6Finite Fourier Series and Fast Fourier Transform ......... 200 38.7Fast Fourier Transform (FFT) ............................ 204 38.8The Fourier Transform .................................. 209 39 Wavelets 215 39.1Introduction............................................. 215 2 39.2The Hilbert Space L (R) ................................. 217 39.3Frames and Orthonormal Wavelet Bases ................. 221 39.4The Fast Haar Wavelet Transform........................ 225 40 Fractals 231 40.1Introduction............................................. 231 40.2Hausdor?-Metric Spaces ................................. 232 40.3Contractions on Hausdor?-Metric Spaces ................ 236 40.4FractalDimension....................................... 242 41 Neural Networks 253 41.1Introduction............................................. 253 41.2Formal Neurons ......................................... 254 41.3Neural Networks ........................................ 264 41.4Multi-Layered Perceptrons ............................... 269 41.5The Back-Propagation Algorithm......................... 272 42 Probability Theory 279 42.1Introduction............................................. 279 42.2EventSpacesandRandomVariables ..................... 279 42.3Probability Spaces ....................................... 283 42.4Distribution Functions................................... 290 42.5Expectation and Variance................................ 299 42.6IndependenceandtheCentralLimitTheorem............ 306 42.7ARemarkonInferentialStatistics ....................... 310 43 Lambda Calculus 313 43.1Introduction............................................. 313 43.2TheLambdaLanguage................................... 314 43.3Substitution ............................................. 316 43.4Alpha-Equivalence....................................... 318 43.5Beta-Reduction.......................................... 320 43.6The λ-Calculus as a Programming Language.............. 326 X Contents 43.7Recursive Functions ..................................... 328 43.8Representation of Partial Recursive Functions............ 331 A Further Reading 335 B Bibliography 337 Index 341