Comprehensive Mathematics for Computer Scientists 1

A second edition of a book is a success and an obligation at the same time. We are satis ed that a number of university courses have been orga› nized on the basis of the rst volume of Comprehensive Mathematics for Computer Scientists. The instructors recognized that the self›contained presentation of a broad specturm of mathematical core topics is a rm point of departure for a sustainable formal education in computer sci› ence. We feel obliged to meet the valuable feedback of the responsible in› structors of such courses, in particular of Joel Young (Computer Science Department, Brown University) who has provided us with numerous re› marks on misprints, errors, or obscurities. We would like to express our gratitude for these collaborative contributions. We have reread the entire text and not only eliminated identi ed errors, but also given some addi› tional examples and explications to statements and proofs which were exposed in a too shorthand style. A second edition of the second volume will be published as soon as the errata, the suggestions for improvements, and the publisher’s strategy are in harmony.

This two-volume textbook Comprehensive Mathematics for Computer Scientists 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.

For the second edition the entire text has been carefully reread, and many examples have been added, as well as illustrations and explications to statements and proofs which were exposed in a too shorthand style. This makes the book more comfortable for instructors as well as for students to handle.