Comprehensive Mathematics For Computer Scientists 1: Sets And Numbers, Graphs And Algebra, Logic And Machines, Linear GeometrySpringer Science & Business Media, 2004 - 357 pages 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. |
Table des matières
FundamentalsConcepts and Logic | 3 |
11 Propositional Logic | 4 |
12 Architecture of Concepts | 6 |
Axiomatic Set Theory | 13 |
21 The Axioms | 14 |
22 Basic Concepts and Results | 17 |
Boolean Set Algebra | 19 |
Functions and Relations | 23 |
Primes | 161 |
162 Roots of Polynomials and Interpolation | 165 |
Formal Propositional Logic | 169 |
The Language of Formal Propositional Logic | 171 |
Logical Algebras | 174 |
Valuations | 178 |
174 Axiomatics | 180 |
Formal Predicate Logic | 185 |
42 Relations | 35 |
Ordinal and Natural Numbers | 39 |
52 Natural Numbers | 44 |
Recursion Theorem and Universal Properties | 49 |
61 Recursion Theorem | 50 |
62 Universal Properties | 51 |
63 Universal Properties in Relational Database Theory | 59 |
Natural Arithmetic | 65 |
72 Euclid and the Normal Forms | 67 |
Infinities | 71 |
The Classical Number Domains Z Q R and C | 73 |
91 Integers Z | 74 |
92 Rationals Q | 79 |
93 Real Numbers R | 82 |
94 Complex Numbers | 93 |
Categories of Graphs | 97 |
101 Directed and Undirected Graphs | 98 |
102 Morphisms of Digraphs and Graphs | 103 |
103 Cycles | 113 |
Construction of Graphs | 117 |
Some Special Graphs | 121 |
122 Moore Graphs | 122 |
Planarity | 125 |
132 Kuratowskis Planarity Theorem | 129 |
First Advanced Topic | 131 |
142 Example for an Addition | 136 |
Algebra Formal Logic and Linear Geometry | 139 |
Monoids Groups Rings and Fields | 141 |
152 Groups | 144 |
153 Rings | 152 |
154 Fields | 158 |
Firstorder Language | 187 |
Structures | 192 |
Models | 194 |
Languages Grammars and Automata | 199 |
191 Languages | 200 |
192 Grammars | 205 |
193 Automata and Acceptors | 219 |
Categories of Matrixes | 237 |
201 What Matrixes Are | 238 |
202 Standard Operations on Matrixes | 241 |
203 Square Matrixes and their Determinant | 246 |
Modules and Vector Spaces | 255 |
Linear Dependence Bases and Dimension | 263 |
221 Bases in Vector Spaces | 264 |
222 Equations | 270 |
223 Affine Homomorphisms | 271 |
Algorithms in Linear Algebra | 279 |
232 The LUP Decomposition | 283 |
Linear Geometry | 287 |
242 Trigonometric Functions from TwoDimensional Rotations | 296 |
243 Grams Determinant and the Schwarz Inequality | 299 |
Eigenvalues the Vector Product and Quaternions | 303 |
252 The Vector Product | 306 |
253 Quaternions | 309 |
Second Advanced Topic | 319 |
262 The ReedSolomon RS Error Correction Code | 324 |
263 The RivestShamirAdelman RSA Encryption Algorithm | 327 |
Further Reading | 331 |
Bibliography | 333 |
337 | |
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Expressions et termes fréquents
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