Comprehensive Mathematics for Computer Scientists 2: Calculus and ODEs, Splines, Probability, Fourier and Wavelet Theory, Fractals and Neural Networks, Categories and Lambda Calculus
Springer Science & Business Media, Mar 30, 2006 - Computers - 355 pages
This 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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Topology and Calculus
Inverse and Implicit Functions 59
The Fundamental Theorem of Calculus and Fubinis Theorem
Vector Fields 97
Main Theorem of ODEs
Probability Theory 279
A Further Reading 335
Third Advanced Topic
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affine algorithm approximation arrows Bézier curve bijection calculation called Cartesian product Cauchy chain rule closed cube coefficients constant control points convergent cos(x defined definition denoted derivative diagram differentiable digraph elements equation equivalence Example Exercise fact figure finite fixpoint formula Fourier Fourier series Fourier theory fractals function f functors given graph Haar wavelet input integral curves interval inverse isomorphism lemma linear map LinR map f mathematical matrix means metric spaces morphism f morphisms natural number natural transformations neural network neuron one-dimensional open sets output partition perceptron polynomial Proof proposition random variable real number recursive ſº solution sorite spline subset subspace term theorem theory tion topology transform unique values vector field vector space vertexes vol(K volume wavelet weight yields