COMPREHENSIVE MATHEMATICS FOR COMPUTER SCIENTISTS
CONTENTS III TOPOLOGY AND CALCULUS 27 LIMITS AND TOPOLOGY 27 1 INTRODUCTION 27 2 TOPOLOGIES ON REAL VECTOR SPACES 27 3 CONTINUITY 27 4 SERIES 27 5 EULER’S FORMULA FOR POLYHEDRA AND KURATOWSKI’S THEOREM 28 DI?ERENTIABILITY 28 1 INTRODUCTION 28 2 DI?ERENTIATION 28 3 TAYLOR’S FORMULA 29 INVERSE AND IMPLICIT FUNCTIONS 29 1 INTRODUCTION 29 2 THE INVERSE FUNCTION THEOREM 29 3 THE IMPLICIT FUNCTION THEOREM 30 INTEGRATION 30 1 INTRODUCTION 30 2 PARTITIONS AND THE INTEGRAL 30 3 MEASURE AND INTEGRABILITY 31 THE FUNDAMENTAL THEOREM OF CALCULUS AND FUBINI’S THEOREM 31 1 INTRODUCTION 31 2 THE FUNDAMENTAL THEOREM OF CALCULUS 31 3 FUBINI’S THEOREM ON ITERATED INTEGRATION 32 VECTOR FIELDS 32 1 INTRODUCTION 32 2 VECTOR FIELDS 33 FIXPOINTS 33 1 INTRODUCTION 33 2 CONTRACTIONS 34 MAIN THEOREM OF ODES 34 1 INTRODUCTION 34 2 CONSERVATIVE AND TIME-DEPENDENT ORDINARY DI?ERENTIAL EQUATIONS: THE LOCAL SETUP 34 3 THE FUNDAMENTAL THEOREM: LOCAL VERSION 34 4 THE SPECIAL CASE OF A LINEAR ODE 34 5 THE FUNDAMENTAL THEOREM: GLOBAL VERSION 35 THIRD ADVANCED TOPIC 35 1 INTRODUCTION 35 2 NUMERICS OF ODES 35 3 THE EULER METHOD 35 4 RUNGE-KUTTA METHODS IV SELECTED HIGHER SUBJECTS 36 CATEGORIES 36 1 INTRODUCTION 36 2 WHAT CATEGORIES ARE 36 3 EXAMPLES 36 4 FUNCTORS AND NATURAL TRANSFORMATIONS 36 5 LIMITS AND COLIMITS 36 6 ADJUNCTION 37 SPLINES 37 1 INTRODUCTION 37 2 PRELIMINARIES ON SIMPLEXES 37 3 WHAT ARE SPLINES? 37 4 LAGRANGE INTERPOLATION 37 5 BÉZIER CURVES 37 6 TENSOR PRODUCT SPLINES 37 7 B-SPLINES 38 FOURIER THEORY 38 1 INTRODUCTION 38 2 SPACES OF PERIODIC FUNCTIONS 38 3 ORTHOGONALITY 38 4 FOURIER’S THEOREM 38 5 RESTATEMENT IN TERMS OF THE SINE AND COSINE FUNCTIONS 38 6 FINITE FOURIER SERIES AND FAST FOURIER TRANSFORM 38 7 FAST FOURIER TRANSFORM (FFT) 38 8 THE FOURIER TRANSFORM 39 WAVELETS 39 1 INTRODUCTION 39 2 THE HILBERT SPACE L2(R) 39 3 FRAMES AND ORTHONORMAL WAVELET BASES 39 4 THE FAST HAAR WAVELET TRANSFORM 40 FRACTALS 40 1 INTRODUCTION 40 2 HAUSDOR?-METRIC SPACES 40 3 CONTRACTIONS ON HAUSDOR?-METRIC SPACES 40 4 FRACTAL DIMENSION 41 NEURAL NETWORKS 41 1 INTRODUCTION 41 2 FORMAL NEURONS 41 3 NEURAL NETWORKS 41 4 MULTI-LAYERED PERCEPTRONS 41 5 THE BACK-PROPAGATION ALGORITHM 42 PROBABILITY THEORY 42 1 INTRODUCTION 42 2 EVENT SPACES AND RANDOM VARIABLES 42 3 PROBABILITY SPACES 42 4 DISTRIBUTION FUNCTIONS 42 5 EXPECTATION AND VARIANCE 42 6 INDEPENDENCE AND THE CENTRAL LIMIT THEOREM 42 7 A REMARK ON INFERENTIAL STATISTICS 43 LAMBDA CALCULUS 43 1 INTRODUCTION 43 2 THE LAMBDA LANGUAGE 43 3 SUBSTITUTION 43 4 ALPHA-EQUIVALENCE 43 5 BETA-REDUCTION 43 6 THE ?-CALCULUS AS A PROGRAMMING LANGUAGE 43 7 RECURSIVE FUNCTIONS 43 8 REPRESENTATION OF PARTIAL RECURSIVE FUNCTIONS