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Humphreys J.F., Prest M.Y. — Numbers, Groups and Codes
Humphreys J.F., Prest M.Y. — Numbers, Groups and Codes

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Название: Numbers, Groups and Codes

Авторы: Humphreys J.F., Prest M.Y.

Аннотация:

"This textbook is an introduction to algebra via examples. The book moves from properties of integers, through other examples, to the beginnings of group theory. Applications to public key codes and to error correcting codes are emphasised. These applications, together with sections on logic and finite state machines, make the text suitable for students of computer science as well as mathematics students. Throughout the book, attention is paid to the historical development of mathematical ideas. This second edition contains new material designed to help students develop their mathematical reasoning skills as well as a new chapter on polynomials." The book was developed from first-level courses taught in the UK and USA, which proved successful in developing not only a theoretical understanding but also algorithmic skills. This book can be used at a wide range of levels: it is suitable for first- or second-level university students, and could be used as enrichment material for upper-level school students.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 2nd edition

Год издания: 2003

Количество страниц: 354

Добавлена в каталог: 11.06.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Abel      170 181 257
Abelian      see group abelian
Abstract algebra, rise of      193ff
Accept(ed)      120
Addition modulo f      280
Addition modulo n      40
Adjacency matrix      108
Adleman      70
Al-jabr wa'k muqabalah      180
al-Khwarizmi      180
Alexander the Great      14
Algebra      192
Algebra of sets      8311
Algebraically closed      293
Alphabet (of finite stale machine)      119
Argand diagram      293
Argument (of complex number)      293
Arithmetic modulo n      40
Arithmetica      33 36
Ars Magna      181
Automaton      120
Axiom      184
Babbage      117ff
Bachet      36 45 74
Bernoulli, D.      101
Bernoulli, J.      101
Binomial coefficient      19
Binomial theorem      18 65 75
Boole      85 136 185 195
Boolean algebra      136 192ff 198ff
Boolean algebra of sets      84 135f 193 199
Boolean combination      130
Boolean ring      199
Brahmagupta      45 50 180
Bravais      183
Calculating machines      117ff
Cantor      85 97ff
Cardano      181
Cardinality      98
Carroll      see Dodgson
Cartesian product      see product
Casting out nines      49
Cauchy      148 158 164 182 216
Cayley      174 182 195
Characteristic      197
Check digit      231ff
Chinese remainder theorem      54
Code, error-correcting and error-detecting      230ff
Code, error-correcting and error-detecting, cyclic      2840'
Code, error-correcting and error-detecting, Golay      252
Code, error-correcting and error-detecting, group      see code linear
Code, error-correcting and error-detecting, Hamming      249
Code, error-correcting and error-detecting, linear      237ff 284ff
Code, error-correcting and error-detecting, perfect      249
Code, error-correcting and error-detecting, quadratic residue      290 see
Codeword      232 245 284ff
Coding function      232ff
Codomain      87
Coefficient, of polynomial      256 261 287
Common measure      32
Complement      80 192
Complement, double      193
Complement, properties of      83 193
Complement, relative      80
Completeness theorem      140
Complex numbers, set of $(\mathbb{C})$      172 174 176 189 192 193ff 221 258 261 276 292ff
Composite      28
Composition (of functions)      93ff 149ff
Congruence      38 45 161 205 213
Congruence class      36 38 50 115 196 279
Congruence class, invertible      43 44ff 52
Congruence class, order of      61ff
Congruence class, set of invertible $(G_n)$      47 63ff 172 212 220 223
Congruence, linear      49ff
Congruence, non-linear      57ff
Congruence, simultaneous linear      54ff
Congruence, solving linear      50
Congruent (integers)      36
Congruent (polynomials)      279
Conjecture      34
conjugate      164 166 212 230
Conjugate, complex      293ff
Conjunction      129
Consistency      134
Contradiction      134
Contrapositive      132 142
CONVERSE      132
Coprime      see prime relatively
Corollary      8
Coset (left, right)      212ff 228
Coset decoding table      241ff
Coset decoding table with syndromes      246
Coset leader      244
Counterexample      35
Cours d'Algebre superieure      182
Covering      113
Cut      159 169
CYCLE      see permutation cyclic
Cycle decomposition (of permutation)      154 155 163
Cyclic group      see group cyclic
Cyclic permutation      see permutation cyclic
D'Alembert      101
De Morgan      115 136 194
De Morgan laws      see law De
Decoding table      see coset decoding table
Dedekind      189
Deduction, rules of      140
Degree (of polynomial)      256 262 264 279
del Ferro      181
Descartes      36 84
Difference Engine      117ff
Diffie      70
Digit sum, (iterated)      49
Digraph      107 see
Diophantus      33 36 74
Direct product      153 163 see
Dirichlet      101
Disjoint sets      81 98 113 214
Disjunction      129
Disquisitiones Arithmetical      36 40
Distance      234 236 237 249
Divide      3 36 46 218 262 265
Division algorithm      see Euclidean algorithm
Division theorem      3 264
Dodgson      80
Domain      87
Dyck      182
Element      78
Elements (Euclid's)      9 14 15 22 26 29 32
Equivalence class      114
Equivalence Erlanger programme      183
Equivalence relation      see relation equivalence
Equivalent (propositions)      see logical
Eratosthenes      26
Error-correction      231ff 236 240ff
Error-detection      230ff 236
Euclid      9 14 15 22 23 29 32ff
Euclidean algorithm      9ff 269ff
Euler      33ff 40 65ff 74 80 101
Euler phi-function $(\phi(n))$      66ff 98 172
Euler's theorem      68 72 143 144 218
Evaluate (polynomial)      257
Existential quantifier      138
Exponent (of public key code)      71
Factorial (n!)      18
Fallings      33 74
Ferarri      181
Fermat      23 33ff 36 63 65 73ff
Fermat's 'Theorem'      33 127
Fermat's theorem      63 76 143 144 217
Fibonacci sequence      23
Field      189ff 194 282 283
Field of fractions      198
Finite state machine      119ff 186ff
fix      153
FOURIER      101
Fractions      see rational numbers
Frenicle      34 63
Function      87ff 103 185ff
Function, characteristic      103
Function, concept of      86ff 100ff
Function, constant      92
Function, identity      92
Function, infective      see injection
Function, one-to-one      see injection
Function, onto      see surjection
Function, surjective, it      see surjection
Fundamental theorem of algebra      189 258 276 293
Fundamental Theorem of Arithmetic      see Unique Factorisation Theorem
Galois      157 181ff 189 257
Galois field      282
Gauss      36 40 194
GCD      see greatest common divisor
GENERATED      209ff 286
Generator matrix      237
Generator polynomial      286
Generators, of group      209
Gibbs      195
Godel      140
Goldbach      34 74
Goldbach's conjecture      34ff
Graph, of function      89
Graph, of function, directed      see directed graph
Grassmann      195
Greatest common divisor      7 31 32 43 50 268ff
Gregory      194
Greiss      229
Group      170ff 184 185 200ff 257
Group of matrices      175ff
Group of numbers      171f
Group of permutation      see group symmetric
Group of small order      224ff
Group of symmetries      177ff
Group, Abelian (=commutative)      170 173 182 209 224 225 259
Group, alternating      167 174 208 216 218 228
Group, concept of      xi 147 180ff 200
Group, cyclic $(C_n)$      209 216 217 220ff 224
Group, dihedral $(D_n)$      178 179 211 221 228
Group, general linear      175 210 211
Group, Klein four      224
Group, Mathieu      228
Group, monster      229
Group, p-      218
Group, simple      228ff
Group, special linear      208
Group, sporadic simple      228ff
Group, symmetric      149 174 209 211 213 216 220ff 223
Hamilton      172 194ff
Hasse      111
Hasse diagram      111
HCF      see highest common factor
Hellman      70
Hensel      189
Highest common factor      see greatest common divisor
Hollerith      118
Idempotent      185
Identity element      170
Identity, logical      see logical identity
Image      87
Imaginary part      292
Immediate predecessor      111
Immediate successor      111
Implication      132
Induction, course of values      see induction strong
Induction, definition by      18
Induction, hypothesis      16
Induction, principle      16 20 23 24
Induction, proof by      16ff 22ff 143
Induction, step      16 21
Induction, strong      21
Inductive construction      15
Infinite order      90 186 see infinite
Integers modulo n, set of $(\mathbb{Z}_n)$      38 171 189 210 213 220 261 272 278 281ff 283ff
Integers, set of $(\mathbb{Z})$      1 171 185 188 210 213
Integral domain      192
Integral linear combination      7 44
Intersection      80 209
Inverse      43ff 170 282
inverse of function      95 96 220
Inverse of polynomial congruence class      282
Invertible congruence class      43 44ff 52
Invertible matrix      175
Irrational numbers      32 101 190
Irreducible (polynomial)      273ff 282
ISBN code      231
Isomorphism      219ff
Janko      229
Jiu zhang suan shu      see Nine Chapters on the Mathematical Art
Join      192
JORDAN      182 216
Kilburn      118
Klein      183
Knapsack codes      70
kronecker      182
Lagrange      158 182 216
Lagrange's theorem      66 143 144 216 218 225 226 231
Law, absorption      83 134
Law, associative      83 94 134 170 188 193
Law, commutative      83 134 170 188
Law, contrapositive      134
Law, De Morgan      83 134 193
Law, distributive      83 134 193
Law, double negative      134
Law, excluded middle      134
Law, idempotence      83 134 193
Law, index      159 204
Law, Laws of Thought      185
LCM      see least common multiple
Leading coefficient      256
Leading term      256
Least common multiple      14 31
Leibniz      65 80 101 117 136
Lemma      8
Length of code      284
Length of permutation      152
Length of word      232
Linear Associative Algebras      185
Liouville      182
Logical equivalence      133 136 193
Logical identity      133
Map (mapping)      see function
Master Sun s Arithmetical Manual      1
Mathematical Treatise in Nine Sections      54 56
Mathieu      228
Matrix, diagonal      176 208
Matrix, groups and rings of      175ff 188 192 195 206 208
Matrix, invertible      175
Matrix, method (for gcd)      10ff
Matrix, upper triangular      1750
Maximum likelihood decoding      241
Meet      192
Member      101. see element Methodus Incrementorum
mersenne      34
Mod(ulo)      see congruent
Modulus (of complex number)      293
Move      153
Multinomial theorem      75
Multiplication modulo f      280
Multiplication modulo n      40
Natural numbers, set of (N)      2
1 2
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