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Kedlaya K.S., Poonen B., Vakil R. — The William Lowell Putnam Mathematical Competition 1985–2000: Problems, Solutions, and Commentary
Kedlaya K.S., Poonen B., Vakil R. — The William Lowell Putnam Mathematical Competition 1985–2000: Problems, Solutions, and Commentary

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Название: The William Lowell Putnam Mathematical Competition 1985–2000: Problems, Solutions, and Commentary

Авторы: Kedlaya K.S., Poonen B., Vakil R.

Аннотация:

The William Lowell Putnam Mathematical Competition is the premier undergraduate mathematical competition in North America. This volume contains problems from the years 1985-2000, with solutions and extensive commentary. It is unlike the first two Putnam volumes and unlike virtually every other problem-based book, in that it places the problems in the context of important mathematical themes. The authors highlight connections to other problems, to the curriculum, and to more advanced topics. The best problems contain kernels of sophisticated ideas related to important current research, and yet the problems are accessible to undergraduates. The heart of the book is in the solutions, which have been compiled through extensive research. In editing the solutions, the authors have kept a student audience in mind, explaining techniques that have relevance to more than the problem at hand, suggesting references for further reading, and mentioning related problems, some of which are unsolved.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2002

Количество страниц: 337

Добавлена в каталог: 03.05.2006

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Algebraic function      259
Algebraic geometry      83 84 231
Algebraic number      121
Arithmetic-Mean–Geometric-Mean (AMGM) Inequality      56 95 153 253 254
Automata      108 122 123
Axiom of Choice      112
Bateman — Horn Conjecture      279
Beatty sequences      179 180
Beatty’s Theorem      179 216
Bernoulli numbers      184 225
Bessel functions      63 236
Big-O notation      89 90 107 111 117 118 204 208 258 275
Bijection      21 53 82 123 124 140 141 174 195 223 229
Binary representation      11 17 39 43 107 108 156 157 188
Birkhoff-von Neumann Theorem      68
Block matrix      75
Brauer group      169
Carmichael’s lambda function      57
Catalan numbers      206
Cauchy — Schwarz inequality      55
Cauchy’s Lemma      143
Cauchy’s Theorem      282 283
Cayley diagram      127
Cayley digraph      127
Cayley — Hamilton theorem      63 122 199
Cayley — Menger determinant      188
Centroid      40 47 129 130 130 131 132
Chain rule      139
Change of variables      183 184
CHARACTER      64 150 151
Chebychev’s Inequality      55
Chess      158
Chinese remainder theorem      58 202 257
Circuit      126
Circulant matrix      98 99 276
Class number      94
Combinatorial games      212 214
Compactness      44 49 104 134 195 230 266
Companion matrix      268
Comparison Test      97 282
Compass and straightedge      218
Complementary sequences      179 180
complex numbers      143
Concavity of functions      48 55 132–134 197 251
Content of a polynomial over $\mathbb{Z}$      259
Continued fraction      74 212
Continuity      4 7 12 13 20 25 29 32 34 49 56 68 79 80 85 111 113 130 132–134 138 139 143 155 166 176 184 220–222 242 251 274 275 292 293
Contour integration      59 283
Contraction mapping      241
Convergence      10 37 41 42 61 62 72 80 89 97 110 111 125 162 183 191 194 195 204 221 222 228 234–236 259 265 282–284 291
Convexity      13 14 26 55 68 118 120 128 130 131 134 159 160 230 231 270
Cosine law      186
Countability      12 44 111 112 120 121 195 219
CYCLE      126
De Moivre’s Theorem      35 58
Derangement      99
Descartes’ Rule of Signs      60 60 198
Design theory      294
Determinants      xi 10 18 24 32 45 70 97–99 164 188 193 200 211 276
Diagonalization      227
Dickson’s Conjecture      279
Difference operator      259
Differential equations      36–38 63 70 78 89 93 110 111 124 207 235 239 243 270 275
DIMENSION      134 167 187 209 215 216 266 277
Directed multigraph      126 127
Dirichlet density      280
Dirichlet L-function      95
Dirichlet’s Theorem      280
Discriminant      74 178 198 260 261 273
Divisibility of binomial coefficients      25 147 220 238 290 293
Division algorithm      163
Dominated Convergence Theorem      236 283
Doubly stochastic matrices      68
Dynamical systems      85 159 163 212 222
Eigenvalue      49 63 93 98 99 167 184 199 200 207 227–229 241 242 276–277
Eigenvector      9 38 45 49 63 93 99 184 207 227 241 242 270 276
Eisenstein integers      91
Elliptic curve      258
Euler $\phi$-function      35 57 58 247 267 280
Euler line      233
Euler — Maclaurin summation formula      225
Eulerian circuit      126 127
Eulerian path      126 127
Euler’s theorem      57 58
Extreme Value Theorem      184 266
Farey series      182
Fermat’s Little Theorem      86 202
Fibonacci numbers      40 93 117 123 124 139 173 222 223 252
Field      xi 5 7 8 11 16 74 83 84 86 93–95 107 108 128 148 158 164 178 261
Field of complex numbers      xi
Field of rational numbers      xi
Field of real numbers      xi
Finite directed multigraph      126
Finite field      xi 8 11 16 86 107 108 148 151 158 203 261
First order sentence      95 96
Fourier analysis      58 184 216
Fourier series      58 216
Fresnel integrals      284
Functional equation      92 110 142 154 178 220 221 292
Fundamental theorem of calculus      138 176
Gabriel’s Horn      138
Gall — Peters projection      255
Games      11 20 24 27 47 102 103 144 158 180 182 212–214 233 234
Gauss sum      150 151
Gauss — Lucas Theorem      137 270
Gaussian elimination      97
Gaussian integers      150
Gauss’s Lemma      86 258 259
Gegenbauer polynomials      270
Generalized hypergeometric functions      283 284
Generalized Riemann Hypothesis (GRH)      vii 94 95
Generating function      47 82 83 93 145 158 161 239 246 293
Generating functions      266
Geometric probability      159 160 182
Geometric transformations      254
Geometry      39 83
Gradient      46 68 217
Graph theory      40 126 242
Greedy algorithms      36 48 252
grep      123
Group theory      3 11 14 27 57 63 99 109 120 126 143 168–170 178 195 202 237 238
Harmonic series      61 164 244
Height function      258
Hensel’s Lemma      72 261
Hilbert metric      241
Hilbert’s Seventeenth Problem      264
Hilbert’s Tenth Problem      96
Hypergeometric functions      283 284
Hypothesis H      279
Inclusion-Exclusion Inequality      201
Inclusion-Exclusion Principle      99 126 201
Indegree      126
Inequalities      32 49 55 56 65 88 89 95 103 104 132 139 151 153 162 180 181 201 218 219 224 252 253 271 274
Inequalities of integrals      49 85 139 225 274
Integral Comparison Test      89
Integral domain      107
Integration by parts      39 47 50 110 132 234 235 270 273 281 282
Intermediate Value Theorem      102 128 130 251 278 291
International Mathematical Olympiad      74 87 121 144 158 179 190 220 265 293
Isometric embedding      211
Jacobi sum      150 151
Jensen’s Inequality      55 56
Jordan canonical form      63
Kummer’s Theorem      238 290 293
Lagrange interpolation      259 260 289
Lagrange multipliers      217
Lagrange’s Theorem      57 268
Lambert equal-area cylindrical projection      255
Laplace transform      236
Lattice polygon      118
Lattices      91 111 118–120 208 209
Laurent polynomial      35 59
Laurent series      259
Law of Cosines      186
Law of Sines      269 286
Legendre polynomial      270
Legendre symbol      86 149
Leibniz’s formula      43 182 183
Lexical scanner      123
lg      xi
Lie group      143
Limit Comparison Test      89
Linear algebra      166
Linear operator      43 184
Linear recursion      38 92 93 116 117 172 199 207 208 246 252 265 267
Linear recursive relation      199
Linear recursive sequence      38 92 93 116 117 172 246 252 265 267
Linear transformation      191 192 277
Little-o notation      89 90 92 96 103 279
ln      xi
Local Central Limit Theorem      208
LOG      xi
Logarithm      xi
Logarithmic derivative      239
Lucas numbers      212
Lucas’ Theorem      137 270
L’Hopital’s Rule      152 156 166 269
Macdonald’s function      63
Markov equation      74
Markov process      207
Mean value theorem      117 198
Measure theory      120 160 270
Monotone Convergence Theorem      183 234
Newton polygon      128
Newton-Puiseux series      260
Newton’s method      72
NIM      213
Notation      xi
Number theory      72 120 148 183
o      89 90 107 111 117 118 134 204 208 258 275
Olympiad      74 87 92 101 121 144 158 161 179 190 214 220 247 265 270 293
Orthogonal polynomials      270
Orthogonality relations for characters      64
Outdegree      126
p-adic absolute value      128
p-adic gamma function      147
p-adic logarithm function      xi
p-adic numbers      xi 128 147 148
p-adic valuation      245 257
p-group      169
Parity      40 45 85 87 118 141 212 237 256 258
Partition      24 140 141 179 209 214 216
Path      126
Pell’s equation      279
Permutation matrix      68
Perron-Frobenius Theorem      184 207 241 241 24
Pick’s Theorem      40 118 119
Pigeonhole Principle      43 50 96 118 175 193 195 215 248 267 289 294
Pochhammer symbol      284
Polynomial ring      xi
Postage Stamp Theorem      143
Power Mean Inequality      55 56
Primitive root      202 247
probability      2 62 115 159 160 182 183
Product rule      9 88 136
Puiseux series      259
Pythagorean Theorem      85 287 294
Pythagorean triple      85
Quadratic form      95
Quadratic reciprocity      86
Radius of curvature      210
Ramsey theory      219
Random walk      207 208 209 241
Ratio Test      111 125 235 284
Recursion      38 46–48 85 92 93 116 117 165 172 199 207 208 223 229 230 239 246 252 265 267 270
Reduced totient function      57
Reflection trick      48 254
Regle des nombres moyens      180
Representation theory      64 99 168 169
Residue theorem      59
Riemann sum      12 114 115 134
Riemann zeta function      183
Ring      xi
Ring of integers      xi
Rolle’s Theorem      42 50 136 137 155 198 291
Rouche’s Theorem      102 231
Set theory      173
Siegel zero      95
Sine law      269 286
Sketching      9 43 88
Spherical cap      255
Sprague-Grundy theory      214
Stationary phase approximation      285
Stirling’s approximation      208 225
Stokes’ Theorem      37 79
Strongly connected      126 24
Sum of two squares      33 150 263 278 279
Sylow subgroup      238
Symmetry      35 37 38 42 45 71 80 88 109 112 117 118 136 160 162 169 192 208 281
Taylor series      38 45 48 89 103 104 107 156 204 257 275
Taylor’s Formula      155 257 273
Taylor’s Theorem      133
Thue-Morse sequence      158
Thue-Siegel Theorem      261
Transcendental number      121 158
Trapezoid Rule      132 243
Triangle inequality      103 218 252
Trigonometric substitution      37 46 50 85 192 217 292
Turing machine      95
Twin Prime Conjecture      279
Ultraspherical polynomials      270
Unique factorization domain      59 259
USA Mathematical Olympiad      214 247
Vandermonde determinant      70 98 276
Vandermonde’s identity      82 83 145
Vectors      46 93 159 186 187 218 230 231 233 276
Venn diagram      53
Weakly connected      126
Weierstrass M-test      111
Weierstrass’s Theorem      111
Weil Conjectures      vii 151 261
Well Ordering Principle      112
Well-ordered set      112 194 195
Weyl’s Equidistribution Theorem      96 216
Winding number      230 231
Wolstenholme’s Theorem      147 24
Wythoff’s game      180
Zorn’s Lemma      s111 112 195
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