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mazzola g comprehensive mathematics computer scientists 2 2ed 2006

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Textbook in PDF and DJVU formats 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

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Mazzola G. Comprehensive Mathematics for Computer Scientists 2. 2ed 2006.djvu 2.9 MB
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mazzola g comprehensive mathematics computer scientists 2 2ed 2006

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