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Young R.M. — Excursions in Calculus: An Interplay of the Continuous and the Discrete
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Íàçâàíèå: Excursions in Calculus: An Interplay of the Continuous and the DiscreteÀâòîð: Young R.M. Àííîòàöèÿ: he purpose of this book is to explore the rich and elegant interplay that exists between the two main currents of mathematics, the continuous and the discrete. Such fundamental notions in discrete mathematics as induction, recursion, combinatorics, number theory, discrete probability, and the algorithmic point of view as a unifying principle are continually explored as they interact with traditional calculus.
ßçûê: Ðóáðèêà: Ìàòåìàòèêà /Ñòàòóñ ïðåäìåòíîãî óêàçàòåëÿ: Ãîòîâ óêàçàòåëü ñ íîìåðàìè ñòðàíèö ed2k: ed2k stats Ãîä èçäàíèÿ: 1992Êîëè÷åñòâî ñòðàíèö: 408Äîáàâëåíà â êàòàëîã: 13.04.2008Îïåðàöèè: Ïîëîæèòü íà ïîëêó |
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Ïðåäìåòíûé óêàçàòåëü
Harmonic triangle 340—341 Hartt, F. 147 Hawking, S.W. 3 Heath, T.L. 310 Heaviside, O. 355 Heptagonal numbers 94 Hermite's identity 22 Heron of Alexandria 19 Heron's formula 191 Hersh, R. 219 225 304 Hexagonal numbers 94 Higher parabolas, quadrature of 83—85 Hilbert's tenth problem 51 69 124 137 Hilbert, D. x 34 51 Hilton, P. 78 Hobson, E.W 311 Huntley, H.E. 151 Huygens, C. 114 238 316 339 344 Hyperbola 196 245 289—290 310 icosahedron 156 Inclusion and exclusion, principle of 30—33 36 300 Incompleteness theorem 51 Independent 210 226 292 Induction, backward 189 Induction, failure of 22 249 Induction, mathematical 3—53 Induction, Principle of 15 Inequality arithmetic and geometric mean 186—195 Inequality Bernoulli's 22 Inequality Cauchy — Schwarz 202 Inequality Chebyshev's 204 Infinite descent 38—40 Infinite ladder 10 40 96 Infinite product 29 169 259 364 Infinite product for 29—30 250—254 Infinite product for the Riemann zeta function 353—354 Infinite product for the sine 347 Infinitesimal 85 195—197 Integer-valued polynomial 65 Integers, complex 112—113 Integral elliptic 231 263 Integral Fresnel 273 Integral logarithmic 305 Integral probability 267—272 Integration by parts 253 305 Intermediate value property 157 Introductio in anafysin infinitorum x 195—196 Irrational numbers 39—41 205 Isoperimetric problem 189 Isoperimetric theorem 191 Iterated functions 170—184 Ivory, J. 104 Jackson, D. 281 Jacobi C.G.J. 306 316 Jeu de rencontre see “Matching game” Joyce, J. 197 Julia, G. 172 Kac, M. 248 292 Kazarinoff, N.D. 190 275—276 Keisler, H.J. 197 Kepler, J. 140 Kernel 278—279 Kiefer, J. 138 Klein, F. 195 311 Kline, M. 41 52 197 225 325 Knopp, K. 40 224 Knuth, D.E. 27 29 Koch curve 323—325 336 Koch, H. von 323 Kronecker, L. 4 Kummer, E.E. 50 94 L'Hopital, G.F.A. de 343 Ladies'Diary 95 Lagrange polynomial 285 Lagrange, J.L. 107 138 231—232 Lame, G. 35 125 128—129 Landau, E. 281 Lander, L.J. 42 Landry, F. 62 Laplac's problem 243 Laplace, P.S. 160 209 213—214 216 357 Laplace, P.S., Theorie analytique des probabilites 160 Lattice points 36 221 245 Law of errors 206—216 Le Corbusier 145 Least squares, method of 212 217 245 Lebesgue, H. 275 282 Lee, E.J. 28 Legendre, A.M. 209 212 231 304 Legendre, A.M., Nouvelle methode pour la determination des orbites des cometes 212 Lehmer, D.H. 105 Lehmer, D.N. 59 238 Leibniz's series 314—318 331 334 352 Leibniz, G.W. 8 107 118 223 256 313 316 333 338—345 Leibniz, G.W., Historia et Origo Calculi Differentialis 316 339 Lemniscate 231 Lemniscate constant 234 Leonardo of Pisa see “Fibonacci” Lindemann, C.L.F. 314 Lippmann, G. 206 Lobachevsky, N.I. 306 Logarithmic integral 305 Logarithmic spiral 148—153 Logistic function 181 184 Lucas sequence 131 134 Lucas — Lehmer test 105 Lucas, E. 24 35 117 125 136 335 Lucas, E., Recreations Mathimatiques 25 Maclaurin, C. 186 198 Madachy, J.S. 28 Madeleine — Perdrillat, A. 146 Maistrov, L.E. 214 Mandelbrot set 173—179 Mandelbrot, B.B. 173 175 318 320 323 326—327 Mann, H.B. 69 Matching game 32 242 Matijasevich, Y. 52 125 Maugham, W.S. 9 Maurolico, F. 15 Maurolico, F., Arithmetica 15 May, K.O. 231 Mazur's game 157 Mead, D.G. 269 Mean and extreme ratio see “Golden ratio” Mean arithmetic 20 186—195 Mean arithmetic-geometric 231—238 Mean geometric 20 186—195 Mean harmonic 187 201 Mean proportional see “Geometric mean” Mean quadratic 202 Menger sponge 321—322 Menger, K. 322 Mengoli, P. 338 Mersenne numbers 104—106 115—116 125 135 Mersenne primes 104—106 125 245 Mersenne, M. 104 148 Method of bisection 157 Method of exhaustion 85 311 Method of least squares 212 217 245 Method of undetermined coefficients 161 355 Mills, W.H 69 Minkowski's theorem 119 Modular arithmetic 369—377 Modulus 369 Modulus of precision 213 Moments 287 Mondrian, P. 145 147 Monte Carlo method 230 Montmort, P.R. 32 Mordell, L.J. 55 Moritz, R.E. 318 Moroney, M.J. 186 Muntz — Szasz theorem 287 Needham, J. 8 Nested interval principle 141 Nested radicals 10 21 29 171 184 Neugebauer, O. 43 Newton's forward difference formula 73 Newton's method 192 Newton's series 330 Newton, I. 73 251 256 313 316 Nicomachus 9 76 Nilakantha 316 Nonstandard analysis 197 Norm 113 Normal curve 213 327 Normal law 209—216 Numbers amicable 28 Numbers Bell 167 Numbers Bernoulli 87—94 99 160—161 352 355 Numbers binomial 20—21 95 98—99 135 283—284 335—336 351 Numbers Carmichael 116 Numbers Catalan 259 Numbers complex 112 170 Numbers composite 7 Numbers congruent 369—377 Numbers factorial 265—267 Numbers Fermat 34 62 71 116 Numbers Fibonacci 35 71 123—135 138 154 Numbers figurate 74 342 Numbers Gaussian 112—113 Numbers harmonic 165 296 Numbers heptagonal 94 Numbers hexagonal 94 Numbers irrational 39—41 205 Numbers Lucas 131 134 Numbers Mersenne 104—106 115—116 125 135 Numbers octagonal 94 Numbers pentagonal see also “Pentagonal number theorem” 75 94 Numbers perfect 100—101 Numbers polygonal 74 Numbers prime see “Prime numbers” Numbers pseudoprime 116 Numbers pyramidal 76 Numbers rational 39 Numbers relatively prime 34 63 136 Numbers square 74 94 Numbers Stirling 73 166 Numbers tangent 170 Numbers tetrahedral 76 342 Numbers triangular 18 74 94 118—119 308 342 Octagonal numbers 94 octahedron 156 Oldenbeig, H. 316 Orbit 171 Order 103 Palindrome conjecture 9 Papadimitriou, I. 348 Parabola, quadrature of 310—314 Parkin, T.R. 42 Partial fractions 339 352 Partial products 29 289 Partial sums 224 289 296 Partition of a number 169 Partition of a set 166 Pascal's triangle see also “Binominal coefficients” 15 20 69 77 99 216 Pascal's triangle modulo 2 334—336 Pascal, B. 15 343—344 Pascal, B., Traite du triangle arithmitique see also “Binominal coefficients” 15 78 Peaking kernel 278—281 Peano, G. 325 Pedersen, J. 78 Pentagonal number theorem 357—367 Pentagonal numbers 75 Pentagram 154 Perfect numbers 100—101 Phyllotaxis 1247 Picard kernel 279 Pisani, R. 207 Plutarch 95 Poincare, H. 94 123 207 214 325 356 Polygon-triangulation problem 260 Polygonal numbers 74 Polynomial see also “Binomial functions; Weierstrass approximation theorem” Polynomial Bernoulli 87—93 99 Polynomial Bernstein 283—284 Polynomial Bezier 284—286 Polynomial Euler 64 Polynomial Lagrange 285 Pomerance, C. 116 Power series 159—169 255—264 314—316 338 Prime number theorem 247 296—306 Prime numbers see also “Fermat's great theorem; Fermat's little theorem; Prime number theorem; Wilson's theorem” Prime numbers counting function for 297 Prime numbers definition of 5 Prime numbers density of 296—300 Prime numbers factorization into see “Fundamental theorem of arithmetic” Prime numbers Fermat 62 Prime numbers formulas for 64 69 Prime numbers in arithmetical progressions see “Dirichlet's theorem” Prime numbers largest of the known 61 Prime numbers Mersenne 104—106 125 Prime numbers proofs of infinitude of 60 63 137 287 Prime numbers regular 94 Prime numbers table of 59 Prime numbers twin 62 Principle of the arithmetic mean 207 probability see “Bertrand's paradox; Buffon's needle problem; Coin tossing; Gambler's ruin; Laplace's problem; Law of errors; Matching game; Probleme de parties; Random walk Probleme de parties 168 Pryce, J.D. 275 Pseudoprimes 116 Public key cryptography 119 Purves, R. 207 Pyramidal numbers 76 Pythagoras 4 28 Pythagorean Theorem 39 Pythagorean triangle see also “Pythagorean triple” 11 43 53 Pythagorean triangle primitive 43 241 Pythagorean triple 43—47 Quadratic convergence 194 Quadratic mean 202 Quadrature see “Circle; Higher parabolas; Parabola” Quartic convergence 237 Quetelet, L.A.J. 214 Quincunx, Galton's 214—216 Rademacher, H. 60 Rajagopal, C.T. 316 Ramanujan, S. 10 237 Random 243—244 353 Random walk 261—263 Rational numbers 39 Rational points 44—46 49 55—56 Recursion algorithms for 236—238 243—244 Recursion arithmetic-geometric mean iteration 232—234 Recursion Babylonian algorithm for square roots 192—194 Recursion Fermat numbers 63 Recursion Fibonacci numbers 124 Recursion fractals 170—179 318—325 Recursion gambler's ruin 162—164 Recursion Lucas — Lehmer test 105 Recursion pentagonal number theorem 357—367 Recursion polygon-triangulation problem 260 Recursion polynomial approximation 282 Recursion quadrature of the parabola 310—313 Recursion Stirling numbers 166—167 Recursion subfactorials 37 Recursion tower of Hanoi 24—26 Regular prime 94 Reichardt, H. 247 Relatively prime numbers 34 63 136
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