Comprehensive Mathematics for Computer Scientists 2: Calculus and ODEs, Splines, Probability, Fourier and Wavelet Theory, Fractals and Neural Networks, Categories and Lambda Calculus

ISBN-13: 9783540208617 / Angielski / Miękka / 2004 / 355 str.

Guerino Mazzola;Gerard Milmeister;Jody Weissmann

Opis

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.

Recenzje

From the reviews:

"This book is the second volume of ... . each chapter summarizes rich and flourishing theories ... . The authors are honest, and provide extensive references for the interested reader. ... will find several exercises as well. ... My overall opinion of this book is very positive. It provides a good and well-founded starting point for the study of several different areas of mathematics. I recommend it for students and scientists, especially in computer science ... ." (A. Pethö, Computing Reviews, April, 2005)

"This two-volume textbook Comprehensive Mathematics for Computer Scientists is a self-contained comprehensive presentation of mathematics ... . The concept framework is streamlined but defining and proving virtually everything. The style implicitly follows the spirit of recent topos-oriented theoretical computer science. ... The numerous course examples are motivated by computer science and bear a generic scientific meaning." (Zentralblatt für Didaktik der Mathematik, August, 2005)

"The second volume of this comprehensive tour through mathematical subjects for computer scientists deals with topology, differential- and integral-calculus in one and several variables, and ordinary differential equations. ... if she (he) masters these ... the result will be an excellent computer scientist. With the aim to achieve this it would be quite motivating for the reader to see more concrete applications of the abstract mathematical theory." (H. Mitsch, Monatshefte für Mathematik, Vol. 147 (2), 2006)

"This is the second volume of a self-contained comprehensive presentation of mathematics for the working computer scientists. ... The presentation of the book uses in an appropriate way the unifying language of category theory. This gives a new and more elegant view of the content ... . The text is complemented by a large number of examples, illustrations, and applications and ends with the guide for future reading on the different topics." (Yves Félix, Bulletin of the Belgian Mathematical Society, Vol. 12 (2), 2005)

Spis treści

Topology and Calculus.- Limits and Topology.- Differentiability.- Inverse and Implicit Functions.- Integration.- The Fundamental Theorem of Calculus and Fubini’s Theorem.- Vector Fields.- Fixpoints.- Main Theorem of ODEs.- Third Advanced Topic.- Selected Higher Subjects.- Categories.- Splines.- Fourier Theory.- Wavelets.- Fractals.- Neural Networks.- Probability Theory.- Lambda Calculus.