## Comprehensive Mathematics for Computer Scientists 2: Calculus and ODEs, Splines, Probability, Fourier and Wavelet Theory, Fractals and Neural Networks, Categories and Lambda CalculusThis second volume of a comprehensive tour through mathematical core subjects for computer scientists completes the ?rst volume in two - gards: Part III ?rst adds topology, di?erential, and integral calculus to the t- ics of sets, graphs, algebra, formal logic, machines, and linear geometry, of volume 1. With this spectrum of fundamentals in mathematical e- cation, young professionals should be able to successfully attack more involved subjects, which may be relevant to the computational sciences. In a second regard, the end of part III and part IV add a selection of more advanced topics. In view of the overwhelming variety of mathematical approaches in the computational sciences, any selection, even the most empirical, requires a methodological justi?cation. Our primary criterion has been the search for harmonization and optimization of thematic - versity and logical coherence. This is why we have, for instance, bundled such seemingly distant subjects as recursive constructions, ordinary d- ferential equations, and fractals under the unifying perspective of c- traction theory. |

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### Contents

II | 3 |

III | 4 |

IV | 14 |

V | 21 |

VI | 30 |

VII | 37 |

VIII | 39 |

IX | 53 |

XLII | 183 |

XLIII | 185 |

XLIV | 188 |

XLV | 191 |

XLVI | 194 |

XLVII | 200 |

XLVIII | 204 |

XLIX | 209 |

X | 59 |

XI | 60 |

XII | 64 |

XIII | 73 |

XIV | 74 |

XV | 81 |

XVI | 87 |

XVII | 88 |

XVIII | 92 |

XIX | 97 |

XX | 98 |

XXI | 105 |

XXII | 113 |

XXIII | 114 |

XXIV | 115 |

XXV | 117 |

XXVI | 119 |

XXVII | 125 |

XXVIII | 129 |

XXIX | 131 |

XXX | 137 |

XXXI | 139 |

XXXII | 140 |

XXXIII | 143 |

XXXIV | 147 |

XXXV | 153 |

XXXVI | 161 |

XXXVII | 164 |

XXXVIII | 168 |

XXXIX | 171 |

XL | 176 |

XLI | 179 |

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affine algorithm approximation arrows basis Bezier curve bijection calculation called Cartesian product Cauchy chain rule chapter closed cube coefficients compact constant construction continuous map contraction control points convergent cos(x defined definition denoted diagram differentiable digraph distribution function elements equation equivalence event Example Exercise fact figure finite fixpoint formula Fourier series Fourier theory fractals function f functors given graph Haar wavelet identity induction input integral curves interval isomorphism lemma limit linear map map f mathematical matrix means method metric space morphism f morphisms n-stream natural number natural transformations neural network neuron notation object one-dimensional open sets output partition perceptron polynomial proposition random variable real vector space recursive sequence sin(x solution sorite spline subset subspace Suppose term theorem theory tion topology unique values vector field vector space vertexes volume wavelet weight yields