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Borwein J., Bailey D., Girgensohn R. — Experimentation in Mathematics: Computational Paths to Discovery
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Название: Experimentation in Mathematics: Computational Paths to DiscoveryАвторы: Borwein J., Bailey D., Girgensohn R.Аннотация: Sometimes mathematicians like to believe that theorems spring full- fledged from their brains, given birth solely by the power of naked mind. Borwein, Bailey, and Girgensohn take a different approach, using computer programs like Maple and Mathematica to explore hypothesis and generate ideas. Their approach is still rigorous, however, because the insight gained from computer experimentation is then incorporated into a rigorous proof. Topics are drawn primarily from analysis and number theory, including sequences and series, fourier series, zeta functions, partitions and powers, and primes and polynomials. Advanced undergraduate students with solid experience in these areas, or beginning graduate students should find this book accessible; in either case, no programming knowledge is assumed. The first volume of this work is Mathematics by Experiment: Plausible Reasoning in the 21st Century; each of the two can stand on its own.
Язык: Рубрика: Математика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 2004Количество страниц: 368Добавлена в каталог: 11.06.2008Операции: Положить на полку |
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Hypergeometric function, multiple 285 Hypergeometric function, Zagier conjecture 159 160 Ideal 236 Integer polytope 258 Integer relation algorithms 7 23 27 136 137 140 142—144 162 203 Integer relation algorithms, 149 Intuitionism 295 Inverse problems 257 Isoperimetric inequality 116 Jacobi, Carl, Jacobi theta-functions 194 Jacobi, Carl, Jacobi's quartic identity 210 Jacobi, Carl, Jacobi's triple product 185 189 Jacobi, Carl, Jacobian theta functions 188 188—193 Jeffrey, David ix Jensen's inequality 39 Johnson, Samuel 42 Jordan, Camille, Jordan test 74 89 Jordan, Camille, Jordan theorem sine integrals 99 Jordan, Camille, Jordan's theorem 80 Joyce, G.S. 117 Joyce, Jeff ix Julia sets 306 Kayal, Neeraj 302 Kershner, R. 93 Khintchine constant 329 332 Kirchhoff's law 286 Klein, Felix 214 Klein, James D. 66 Knot theory 180 Knot theory, identifying knots 180 Knot theory, Perko knots 181 182 Knot theory, Reidemeister moves 181 Knuth, Donald E. 15 110 Koecher, Max 137 Kolmogoroff's 0—1-law 92 Korovkin, Pavel 88 102 Korovkin, Pavel, complex theorem 127 Korovkin, Pavel, three function theorems 102—104 126—127 Korselt 243 Korselt's criterion 226 Kreimer, Dirk 146 Kummer-type polylogarithms see polylogarithms l'Hopital's rule 106 Lagarias, Jeffrey C. 93 Lagrange, Joseph-Louis 70 195 Laguerre, Edmond 193 Lambert series 193 Lambert W function 16 220 274 277 279 277—281 Lambert W function, convex conjugates 280 Lambert W function, Maclaurin series 291 Lambert W function, Riemann surfaces 278—280 Lambert W function, Stirling's formula 277 Landau constant 332 Laplace, Pierre-Simon, Laplace transform 197 Laplace, Pierre-Simon, Laplace's method 288 Laurent series 275 Lax conjecture 256 Lebesgue, Henri 72 Lebesgue, Henri, Lebesgue constants 87 88 Lebesgue, Henri, Lebesgue's condition 88 Lebesgue, Henri, Lebesgue's function 107 107 Legendre, Adrien-Marie, Legendre polynomial 307 Legendre, Adrien-Marie, Legendre — Jacobi symbol 141 169 198 Legendre, Adrien-Marie, Legendre's relation 284 Lehmer, Derrick 232 246 Lehmer, Derrick, Lehmer's conjecture 232 Lehmer, Derrick, Lehmer's polynomial 323 Lehmer, Derrick, Lehmer's problem 323 Lenstra, Hendrik 303 Level set, compact 269 Lewis, Adrian ix Li, Xiaoye ix Liouville, Joseph 285 Lisonek, Petr ix 161 233 Littlewood, John Edensor 254 259—261 LLL 180 Locke, John 263} Log-concavity 53 Log-convex sequence 40 Lucas, Edouard 202 Lucas, Edouard, Lucas numbers 215 Lucas, Edouard, Lucas sequence 202 Lucas, Edouard, Lucas — Carmichael numbers see Carmichael numbers Luke, Russell ix Macdonald, John A. 266 Machin formula 34 Macklem, Mason ix MacMahon, Percy 184 Madelung's constant 196 198—200 208 233 Magma 234 Mahler, Kurt 3 Mahler, Kurt, Mahler continued fraction 59 Mahler, Kurt, Mahler's generating function 58 59 Maier, Helmut 259 Mares, Bernard 102 Math software see computer algebra packages MathSciNet 41 Maximum entropy methods 272—274 Maximum principle 275 Mean iteration, 2/3-power mean 282 Mean iteration, logarithmic mean 282 Mellin transform 115 123 132 165 167 197 Menshow, D.E. 109 Mersenne, Marin 245 Michelson, Albert 89 Milnor, John 240 Modular equations 190 Moebius inversion theorem, sums of squares 194—196 Moebius, August, Moebius function 193 248 Moebius, August, Moebius inversion theorem 189 193—196 Moment, of a function 91 Moore, Gordon, Moore's Law vii Mori, Masatake 312 Morin, Mathew ix Motzkin numbers 53 58 Motzkin, Theodore 258 Multidirectional search methods 291 Multiple Clausen values see Clausen Multiple zeta constants 317 Multiple zeta values see MZV Multisectioning 329 Mumford et al, symmetry and chaos 214 Mumford, David ix 214 MZV 142 144 317—318 MZV duality 153—157 MZV evaluating 171 MZV reducibility 144—149 Nash equilibrium theorem 296 Neg-entropy see Hoch and Stern information measure Newton, Isaac, Newton iteration 303 304 305 309 Newton, Isaac, Newton — Julia sets 305 Newton, Isaac, Newton's lemma 53 Newton, Isaac, Newton's method 303 305 323 Nonconvex set 296 Nonexpansive map 271 Nonnegative polynomials 257 Normal set of primes 227 North, Joseph Roy 29 Nuclear magnetic resonance (NMR) 273 Odlyzko, Andrew ix Ohno, Yasuo, Ohno — Zagier generating function 175 Ohno, Yasuo, Ohno's duality theorem 174 Orthogonal wavelet 113 Oscillatory Fourier transforms 122} Papadopoulos, Jason 303 Parseval equation 76 79 106 Partial fraction decomposition 37 Partitions 183 206 Pattern search algorithm 242 291 293 294 Pell's equation 193 Pentagonal numbers 185 Perfect numbers 244 Perko, Kenneth 182 Perron's continued fractions 36 Perron, Oskar 35 Peters, Klaus ix phik functions 290 pi, continued fraction 2 pi, estimates 1 pi, formulas 20 25 43 Pick's theorem 259 Pictorial representations 3 Pisot number 94 253 323 Plancherel transform see Fourier transform Plouffe, Simon 139 Pochhammer function 16 Poincare, J.Henri 295 Poisson kernel 87 112 Poisson summation formula 80—83 190 Poisson summation formula, experimental version 83 86 Polyhedron space 124 Polylogarithms 21 137 154 157 169 Polylogarithms, Eulerian polylogarithms 144 Polylogarithms, extension 172 Polylogarithms, Kummer-type polylogarithms 163 Polylogarithms, multi-dimensional 153 Polylogarithms, multidimensional 27 Polynomial arithmetic 305 Pomerance, Carl 300 Ppseudospectrum 330 Prime numbers 300—303 Prime numbers, Ppime counting function n(x) 300 Prime numbers, primality tests 301—303 Prime numbers, primality tests, AKS provable primality test 302—303 Prime numbers, primality tests, sieve of Eratosthenes 301 Prime numbers, primality tests, strong probable prime test 301 Principal branch 277 Principle of the excluded middle 295 Prouhet — Tarry — Escott problem 247 Pseudospectra 330 331 PSLQ 129 161 162 306 PSLQ, binomial sums 27 PSLQ, Hiemann zeta function 149 PSLQ, zeta function 137 Quadrature 306—314 Quadrature, error function quadrature 309 312 Quadrature, Gaussian quadrature 307 309 Quadrature, higher dimensions 314 Quadrature, tanh-sinh quadrature 312 313 Quantum computing 333 Quasi-elliptic integrals 49 Rademacher, Hans 187 Raines, E.M. 333 Ramanujan, Srinivasa 41 187 191 193 212 260 Ramanujan, Srinivasa, additive partitions 186 187 Ramanujan, Srinivasa, continued fractions 62—64 66 Ramanujan, Srinivasa, formulas 317 Ramanujan, Srinivasa, zeta function 139 Rauhut, Holger 230 Rayleigh quotient 271 Regiomontanus, Johann 201 Reidemeister moves 181 Replicative function 110 Residue 275 Riemann zeta function 16 27 43 75 181 216 248—249 300 315 316 Riemann zeta function, alternating 131 Riemann zeta function, beta function 145 Riemann zeta function, Catalan zeta function 140 197 Riemann zeta function, evaluating 164 165 167 171 Riemann zeta function, evaluating, even positive integers 134 136 Riemann zeta function, evaluating, odd positive integers 136 139 Riemann zeta function, identities 173 Riemann zeta function, MZV, duality 153—157 Riemann zeta function, MZV, recycling 326 Riemann zeta function, MZV, reflection formula 133 Riemann, Georg Bernhard 71 Riemann, Georg Bernhard, Riemann Hypothesis 133 Riemann, Georg Bernhard, Riemann hypothesis generalized 233 Riemann, Georg Bernhard, Riemann integral 197 Riemann, Georg Bernhard, Riemann sum 90 Riemann, Georg Bernhard, Riemann surfaces 278—280 Riemann, Georg Bernhard, Riemann — Lebesgue lemma 76 78 94 115 Riemann, Georg Bernhard, Riemann — Siegel Formula 165—167 Riesz — Fischer theorem 76 Rolle's theorem 53 Rosenbrock function 292 293 Rouche's theorem 275 RSA encryption 300 301 Rudin — Shapiro polynomials 253 Rudolph, Daniel ix Salem number 323 Salem, Raphael 95 Salvy, Bruno 288 Saxena, Nitin 302 Scharein, Rob ix Schauder, Juliusz 114 Schauder, Juliusz, Schauder basis 114 Schilling equation 122 Schmerl, James 245 Schur function 217 218 Selberg integral 282 Self map 270 Sendov, Hristo ix Serrano, Luis ix Shor, P.W. 333 Siegel, Carl 167 Sigler, L.E. 223 Sinc integral 98 98 102 122—126 Sinc integral, iterated 100 125 Sinc integral, multi-variable 121 125 Sine space 124 Singular value 191 Sloane, Neil ix 241—243 Solomyak, Boris 95 Soundararajan, K. 260 Spectral radius 331 Spherical redesign 241 Spiegel, Eugene 245 Square-full number 331 Stembridge, John 219 Stieltjes, Thomas Jan 35 Stigler's law of eponymy 60 295 Stigler, George and Stephen 60 Stirling, James, Stirling numbers 53 Stirling, James, Stirling numbers, second kind 24 Stirling, James, Stirling's formula 16 Stirling, James, Stirling's formula Lambert W function 278 Stone — Weierstrass theorem 103 128 Strictly convex, power function 287 Sturm's theorem 238 Symmetric functions 217 Takahasi, Hidetosi 312 Taubes, Gary 299 Thabit ibn Kurrah 245 Theodorescu, Radu 40 Theta function 197 Theta function, Fibonacci sums 202—203 Theta functions 214 Theta transform 82 128 133 168 Titchmarsh, Edward 167 Torczon, Virginia 292 294 Torus knots 180 Trapping zeroes 276 Turing, Alan 334 Variety 236 Vertesi, Janet ix Von Neumann minimax theorem 296 Von Staudt, Karl 136 von-Staudt — Clausen formula 330 Watson, G.N. 133
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