Algebra for Computer Science

Lars Garding, Torbjörn Tambour (Autoren)

Buch | Softcover

198 Seiten

1988 | Softcover reprint of the original 1st ed. 1988
Springer-Verlag New York Inc.
978-0-387-96780-6 (ISBN)

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The aim of this book is to teach the reader the topics in algebra which are useful in the study of computer science. In a clear, concise style, the author present the basic algebraic structures, and their applications to such topics as the finite Fourier transform, coding, complexity, and automata theory. The book can also be read profitably as a course in applied algebra for mathematics students.

1 Number theory.- 1.1 Divisibility.- 1.2 Congruences.- 1.3 The theorems of Fermat, Euler and Wilson.- 1.4 Squares and the quadratic reciprocity theorem.- 1.5 The Gaussian integers.- 1.6 Algebraic numbers.- 1.7 Appendix. Primitive elements and a theorem by Gauss.- Literature.- 2 Number theory and computing.- 2.1 The cost of arithmetic operations.- 2.2 Primes and factoring.- 2.3 Pseudo-random numbers.- Literature.- 3 Abstract algebra and modules.- 3.1 The four operations of arithmetic.- 3.2 Modules.- 3.3 Module morphisms. Kernels and images.- 3.4 The structure of finite modules.- 3.5 Appendix. Finitely generated modules.- Literature.- 4 The finite Fourier transform.- 4.1 Characters of modules.- 4.2 The finite Fourier transform.- 4.3 The finite Fourier transform and the quadratic reciprocity law.- 4.4 The fast Fourier transform.- Literature.- 5 Rings and fields.- 5.1 Definitions and simple examples.- 5.2 Modules over a ring. Ideals and morphisms.- 5.3 Abstract linear algebra.- Literature.- 6 Algebraic complexity theory.- 6.1 Polynomial rings in several variables.- 6.2 Complexity with respect to multiplication.- 6.3 Appendix. The fast Fourier transform is optimal.- Literature.- 7 Polynomial rings, algebraic fields, finite fields.- 7.1 Divisibility in a polynomial ring.- 7.2 Algebraic numbers and algebraic fields.- 7.3 Finite fields.- Literature.- 8 Shift registers and coding.- 8.1 The theory of shift registers.- 8.2 Generalities about coding.- 8.3 Cyclic codes.- 8.4 The BCH codes and the Reed-Solomon codes.- 8.5 Restrictions for error-correcting codes.- Literature.- 9 Groups.- 9.1 General theory.- 9.2 Finite groups.- Literature.- 10 Boolean algebra.- 10.1 Boolean algebras and rings.- 10.2 Finite Boolean algebras.- 10.3 Equivalence classes of switching functions.- Literature.- 11 Monoids, automata, languages.- 11.1 Matrices with elements in a non-commutative algebra.- 11.2 Monoids and languages.- 11.3 Automata and rational languages.- 11.4 Every rational language is accepted by a finite automaton.- Literature.- References.

Reihe/Serie	Universitext
Zusatzinfo	IX, 198 p.
Verlagsort	New York, NY
Sprache	englisch
Maße	155 x 235 mm
Themenwelt	Mathematik / Informatik ► Informatik ► Theorie / Studium
	Mathematik / Informatik ► Mathematik ► Allgemeines / Lexika
	Mathematik / Informatik ► Mathematik ► Algebra
	Mathematik / Informatik ► Mathematik ► Logik / Mengenlehre
ISBN-10	0-387-96780-X / 038796780X
ISBN-13	978-0-387-96780-6 / 9780387967806
Zustand	Neuware