Kisacanin B. — Mathematical problems and proofs. Combinatorics, Number theory, and Geometry :: Электронная библиотека попечительского совета мехмата МГУ

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Kisacanin B. — Mathematical problems and proofs. Combinatorics, Number theory, and Geometry

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Название: Mathematical problems and proofs. Combinatorics, Number theory, and Geometry

Автор: Kisacanin B.

Аннотация:

An introduction to the highly sophisticated world of discrete mathematics, Mathematical Problems and Proofs presents topics ranging from elementary definitions and theorems to advanced topics.

Язык:

Рубрика: Математика/Алгебра/Комбинаторика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2002

Количество страниц: 220

Добавлена в каталог: 10.12.2004

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID

Предметный указатель

      63
      63
$\aleph_0$ (aleph-zero)      11 13
$\gamma$       17 195
$\mu(n)$ (Mobius mu function)      90
$\phi$ (golden section)      16 45 51 105 195
$\pi$       16 161 191
$\tau(n)$ (number of divisors)      see “Important functions”
$\varphi(n)$ (Euler’s phi function)      42 89 91 95 96 102
Abel      160
Abel’s criterion      45
Aborigines      17
Ahmes      191n
AI’Khwarizmi      159
Almagest      157
Anecdote about logician      180
Angletrisection      160 193n
Annals of Mathematics      110
Apollo      160n
Archimedes      121 122 140 191 201
Archimedes’ theorem      131
Archimede’'spiral      160
Arithmetica      109
Arithmeticmean      149 150 174 180 187
Babylon      75
Barr      197
Bernoulli, Daniel      43n 202
Bernoulli, Jakob      43n 167 192
Bernoulli, Johann      43n 202
Bernoulli, Nicolaus (II)      43n 202
Bernoulli, Nicolaus(I)      43n 51
Bhaskara      145
Bijection      10
Binary operation      10
Binet      51
Binet’s formula      51 179 197
Binomial coefficients      28 35—37 52 56—59 64 67 113 170
Binomial formula      26 35—39 45 46 57 60 67 99 178
Boltzmann’s constant      62
Bose — Einstein statistics      62
Bosons      62
Brahma      65
Brahmagupta’s formula      144
Brianchon      138
Brouncker      193
Buffon      193
C      11
Canonical factorization      82 83
Cantor      12
Cardano      160
Cardinality      11
Carmichael numbers      97
Cartesian product      7 14 69
Cauchy      176
Cayley’s Theorem      68
Centroid      130—138
Ceva’s theorem      128 146
Cevian      129
Change for one dinar      49
chebyshev      110
Chebyshev polynomials      179
Chess board      27 34 55 56
Chevalier      160n
Chinese remainder theorem      106
Circle, circumcircle      126—128
Circle, excircle      138
Circle, incircle      126—128 138
Circle, nine-point (Feuerbach’s)      138
Classical constructions      81 160
Classical problems      160 193n
Classical problems, doubling a cube      160 193n
Classical problems, squaring a circle      160 193n
Classical problems, trisecting an angle      160 193n
Codomain      10
Cohen      13
Collinear points      23
Combinations with repetition      33 47
Combinations without repetition      31
Combinations, hybrid      47
Complete residue system      95 96
Conchoid of Nicomedes      160
Concurrent lines      129
Congruences      92
Continuum hypotheses      13
Contradiction      11 16
Converse Ceva’s theorem      129 146
Coplanar points      24
Cotes      194n
Countable set      12
Counterexample      14
Cube duplication      160 193n
De Moivre      45 51 179 196
De Moivre’s Formula      179
Dela Valee-Poussin      110
Delian problem      160n 193n
Derangements      43 56
Descartes      7 109n 161
Dinostratus’ quadratrix      160
Diodes’ cissoid      160
Diophantine equations      101 109 112
Diophantus      109
Dirichlet’s principle      22 53
Disjoint sets      5
Disquisitiones Arithmeticae      92 194
Distribution of primes      110
Divisibility criteria      100 101
dodecahedron      197
Domain      10
Dominos      55 167
Doubling a cube      160n 193n
e      16 41 43 193 194
Egyptian triangle      108
Eight queens      55
Electron      63
Elements      11 38 79 83 85 119 142 167 196
energy levels      61—63
Epimenides      13
Eratosthenes      79
Eubulides      13
Euclid      11 38 79 85 119 167 196
Euclid numbers      179
Euclidean algorithm      83—88 102—106
Euclidean algorithm, complexity of      88
Euclidean algorithm, extended      105
Euclid’s proof of Pythagorean theorem      145
Euclid’s theorem on primes      11 80 167
Euler      43 45 49 51 80 106 140 171 192 195 202
Euler — Venn diagram      4 5 14
Euler's partition theorem      49
Euler’s constant $(\gamma)$       17 195
Euler’s formula      128 139 156 157 185 194
Euler’s line      136
Euler’s method      112
Euler’s phi function      42 89 91 95 96 102
Euler’s problem partitio numerorum      41
Euler’s theorem in graph theory      54
Euler’s theorem in number theory      42 96 102
Euler’s trinomial      80 177 180
Extended Euclidean algorithm      105
Factorial      28
Factorial, falling factorial      59
Factorial, Stirling’s approximation for      41 61
Factorization of integers      82 83
Fermat      80 109 141 171
Fermat numbers      80 81 160 160n 171
Fermat’s point      140 147 153—155
Fermat’s principle      109n 140
Fermat’s theorem, last      109
Fermat’s theorem, lesser      60 96 99
Fermi — Dirac statistics      63

fermions      63
Ferrari      160
Feuerbach      197
Feuerbach’s theorem      197
Fibonacci (Leonardo Pisano)      44
Fibonacci’s numbers      44 45 50 51 70 78 87 113 179 197
Fibonacci’s rabbits      44
Finite set      11
Five most important numbers      158 194
Formula for prime numbers      80
Formula, Binet’s      51 179 197
Formula, binomial formula      26 35—39 45 46 57 60 67 99 178
Formula, Brahmagupta’s      144
Formula, de Moivre’s      179
Formula, Euler’s      128 139 156 157 185 194
Formula, Hadamard-de la Valee-Poussin’s      110
Formula, Hamming’s      64
Formula, Hardy —Ramanujan      41 48
Formula, Hero’s      121 126—128 144 151 152
Formula, Leibniz’s      59 192
Formula, Maclaurin’s      45 47 158 180 194
Formula, Mertens’      195
Formula, Newton’s      see “Formula binomial”
Formula, Pascal’s      35 38 47
Formula, polynomial      67
Formula, Stirling’s      41 61
Formula, Sylvester’s      41
Formula, Taylor’s      71 194
Formula, Vandermonde’s      59
Formula, Viette’s      192
Formula, Wallis’      192
Function      see “Multiplicative functions mappings and
Fundamental theorem of algebra      203
Fundamental Theorem of Arithmetic      82
Galois      160 160n
Gauss      81 92 112 160 169 194 202
Gauss’ method      112
Gauss’ Theorem      81
GCD (greatest common divisor)      84
Generating functions      45
Geometric mean      149 150 174 180 187
Gergonne’s point      147
Girard      45
Goat      159
Godel’s theorem      13
Golden section $(\phi)$       16 45 51 105 195
Graph theory      54 68
Gregory      192
Gregory’s triangle      170 173 182
hadamard      110
Hamilton — Sylvester’s theorem      138
Hamming’s formula      64
Hanoi towers      65 66
Hardy — Ramanujan’s formula      41 48
Harmonic mean      149 150 174 180 187
Harmonic numbers      19n
Harmonic series      195n
Heegner      83
Heiberg      140n
hermite      16 195
Hero’s formula      121 126—128 144 151 152
Horses      188
Hypotenuse      119
i (imaginary unit)      194
icosahedron      196
Ideal gas      63
Important functions, (falling factorial)      59
Important functions, $\binom{n}{k}$ (binomial coefficient)      28 35—37 52 56—59 64 67 113 170
Important functions, $\delta(n)$ (product of divisors of n)      114
Important functions, $\lfloor x\rfloor$ (floor)      43 52 98
Important functions, $\mu(n)$ (Mobius f)      90
Important functions, $\pi(n)$ (number of primes < n)      110
Important functions, $\sigma(n)$ (sum of divisors of n)      90
Important functions, $\tau(n)$ (number of divisors of n)      25 83 89 114
Important functions, $\varphi(n)$ (Euler’s f.)      42 89 91 95 96 102
Important functions, n! (factorial)      28 41 61
Important points, centroid(T)      130—138
Important points, circumcenter (O)      126 129
Important points, Fermat’s point (F)      140 147 153—155
Important points, Gergonne’s point (G)      147
Important points, incenter(I)      126 127 130
Important points, orthocenter (H)      130 137 138
Induction      37 58 59 167
Inequality of means      149 150 174 180 187
Infinite set      11
Injection      10
Integers      3 75
Integration      140
Internet      75 204
Isomers      69
Jacobi’s theorem      134 139
Jones      191
Knuth      196
Konigsberg bridges      54
Kummer      80
Lagrange      78
Lagrange’s method of multipliers      62
Lambert      193
Lame’s theorem      88
Laplace      45
Law of Cosines      120
Law of Sines      120
lcm (least common multiple)      84
Leibniz      67 97 132 135
Leibniz’s formula      59 192
Leibniz’s theorem      132
Leonardo da Vinci      196
Liber Abaci      44
Lindemann      16 161 193
Linear congruences with one unknown      101
Lucas      45 65
Ludolph van Ceulen      191
Ludolphinenumber      191
Maclaurin series      45 47 158 180 194
Mapping      9
Mapping, bijective      10
Mapping, injective      10
Mapping, surjective      10
Math professor      43
Mathematical induction      37 41 58 59 167
Mathematical tricks and fallacies      142—144 187
Maxwel — Boltzmann’s statistics      42 43
Means      149 150 174 180 187
Mechanical method      140
Mersenne numbers      115
Mertens’ formula      195
Mobius inversion rule      91
Moment of inertia      133 140
Multiplicative functions      89
Multiset      7 29
Napoleon’s triangles      148
Nauck      54
Newton      35—39 194 202
Newton’s binomial formula      26 35—39 45 46 57 60 67 99 178
Numbers, algebraic      16
Numbers, Carmichael      97
Numbers, Euclid      179
Numbers, Fermat      80 81 160 160n
Numbers, Fibonacci      44 45 50 51 70 78 87 113 179 197
Numbers, harmonic      195n
Numbers, Heegner      83n
Numbers, integers      3 75
Numbers, irrational      16
Numbers, Mersenne      115
Numbers, natural      3
Numbers, perfect      115
Numbers, prime      11 25 79
Numbers, pseudoprimes      97 113
Numbers, Pythagorean      75 107 108
Numbers, rational      3 12

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