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The two-volume textbook Comprehensive Mathematics for the Working Computer Scientist, of which this is the second volume, is a self-contained comprehensive presentation of mathematics including sets, numbers, graphs, algebra, logic, grammars, machines, linear geometry, calculus, ODEs, and special themes such as neural networks, Fourier theory, wavelets, numerical issues, statistics, categories, and manifolds. The concept framework is streamlined but defining and proving virtually everything. The style implicitly follows the spirit of recent topos-oriented theoretical computer science. Despite the theoretical soundness, the material stresses a large number of core computer science subjects, such as, for example, a discussion of floating point arithmetic, Backus-Naur normal forms, L-systems, Chomsky hierarchies, algorithms for data encoding, e.g., the Reed-Solomon code. The numerous course examples are motivated by computer science and bear a generic scientific meaning. This text is complemented by an online university course which covers the same theoretical content, albeit in a totally different presentation. The student or working scientist who gets involved in this text may at any time consult the online interface which comprises applets and other interactive tools.
Topology and Calculus
Introduction
Topologies on Real Vector Spaces
Continuity
Series
Euler’s Formula for Polyhedra and Kuratowski’s Theorem
Introduction
Differentiation
Taylor’s Formula
Introduction
The Inverse Function Theorem
The Implicit Function Theorem
Introduction
Partitions and the Integral
Measure and Integrability
Introduction
The Fundamental Theorem of Calculus
Fubini’s Theorem on Iterated Integration
Introduction
Vector Fields
Contractions
Introduction
Conservative and Time-Dependent Ordinary Differential Equations: The Local Setup
The Fundamental Theorem: Local Version
The Special Case of a Linear ODE
The Fundamental Theorem: Global Version
Numerics of ODEs
The Euler Method
Runge-Kutta Methods
Selected Higher Subjects
Introduction
What Categories Are
Examples
Functors and Natural Transformations
Limits and Colimits
Adjunction
Preliminaries on Simplexes
What are Splines
Lagrange Interpolation
Bézier Curves
Tensor Product Splines
B-Splines
Introduction
Spaces of Periodic Functions
Orthogonality
Fourier’s Theorem
Restatement in Terms of the Sine and Cosine Functions
Finite Fourier Series and Fast Fourier Transform
Fast Fourier Transform (FFT)
The Fourier Transform
Introduction
The Hilbert Space L(R)
Frames and Orthonormal Wavelet Bases
The Fast Haar Wavelet Transform
Introduction
Hausdorff-Metric Spaces
Contractions on Hausdorff-Metric Spaces
Fractal Dimension
Introduction
Formal Neurons
Neural Networks
Multi-Layered Perceptrons
The Back-Propagation Algorithm
Event Spaces and Random Variables
Probability Spaces
Distribution Functions
Expectation and Variance
Independence and the Central Limit Theorem
A Remark on Inferential Statistics
Introduction
The Lambda Language
Substitution
Alpha-Equivalence
Beta-Reduction
The ?-Calculus as a Programming Language
Recursive Functions
Representation of Partial Recursive Functions
A Further Reading
B Bibliography
Index