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Rockmore D. — Stalking the Riemann Hypothesis
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Íàçâàíèå: Stalking the Riemann HypothesisÀâòîð: Rockmore D. Àííîòàöèÿ: In 1859 a German professor named Bernhard Riemann postulated a law capable of describing with an amazing degree of accuracy the baffling occurrence of prime numbers; coming up with its proof has been the holy grail of mathematicians ever since. In 'Stalking the Riemann Hypothesis, Dan Rockmore, a prominent mathematician in his own right, takes us from Euclid's pondering of the infinitude of the primes through modern efforts to prove the Riemann hypothesis-efforts that astonishingly connect the primes to the statistics of solitaire, chaos theory, and even the mysteries of quantum mechanics. Along the way, he introduces us to the many brilliant and fascinating thinkers who have contributed to this work, from the most famous mathematician of all time, Carl Friedrich Gauss (Riemann's teacher), to the intellectual giants David Hilbert and Freeman Dyson.
ßçûê: Ðóáðèêà: Ìàòåìàòèêà /Ñòàòóñ ïðåäìåòíîãî óêàçàòåëÿ: Ãîòîâ óêàçàòåëü ñ íîìåðàìè ñòðàíèö ed2k: ed2k stats Ãîä èçäàíèÿ: 2006Êîëè÷åñòâî ñòðàíèö: 292Äîáàâëåíà â êàòàëîã: 11.03.2008Îïåðàöèè: Ïîëîæèòü íà ïîëêó |
Ñêîïèðîâàòü ññûëêó äëÿ ôîðóìà | Ñêîïèðîâàòü ID
Ïðåäìåòíûé óêàçàòåëü
Euler — Galcrkin methods 47 Euler's identity 79 Euler, described 47—49 Euler, infinity of primes and 59—62 142 Euler, Jacob 47 Euler, Leonhard 97 122 251 Euler, nicknames of 47 49 Euler, prime distribution analyzer of 77 81 Euler, Riemann 45 66 73 76—77 Euler, series of reciprocals of primes and 51—56 143 Evolution 63 Existence proof 151 Explicit formula (for counting the primes) 85—87 EXPONENT 33 Exponent, complex 78—81 Exponential growth 32—35 Extended Riemann hypothesis 219 Extrasensory perception 225 Factorials 251 Factors 229 Fast Fourier transform (FFT) 39 Fcvnman, Richard 163 171 237 249 Feller, William 223—224 Fermat's last theorem 95 114—125 125 148 262 Feynman diagrams 163 Fields medal 155 202 209 215 220 229 Fields Medal, described 141 142 Fields, J.C. 141 Figuratc numbers 11 Finite fields 145—146 Fisher, Sir R.A. 226 Floyd's game 253 Floyd, Robert 253 Forcing 215 Foundations of Geometry (Hilbert) 123 Four scalar potential 228 Fourier analysis 69 81—85 160 215 Fourier series 83—84 Fourier transform 83 178 184—185 Fourier, described 82 Fourier, Jean — Baptiste — Joseph 81—85 Fractals 126 French Academy of Sciences 95 French Academy of Sciences, Grand Prize competition of 99 105—106 Frequency content 110 Frequency of waves 83—87 Frobenius transformation 220 Frobenius, Georg 220 Function 80 Function fields 143 145—147 220 Fundamental Theorem of Arithmetic 14 Gambier, B. 238 Gardner, Martin 221 224 Gaudin distribution 183 198 199 Gaudin, Michel 175 183 234 Gauss, Carl Friedrich 29 47 59 115 122 209 Gauss, described 30—31 38—39 Gauss, discovery of fast Fourier transform (FFT) 39 Gauss, discovery of non-Euclidean geometry 203—204 Gauss, Enclce and 40—41 Gauss, integers and 110—125 Gauss, Legendre and 38-44 93 Gauss, nickname of 38 Gauss, prime numbers and 39—44 69 106 119 120 126 132—133 137 141 185 Gauss, relative error and 126—127 Gauss, Riemann and 75—79 Gaussian integers 110—125 Gaussian primes 112—123 Gell — Mann, Murray 49—50 Geodesies 203—207 Geometric situs 48 Geometry 10—11 48 Geometry, algebraic see “Algebraic geometry” Geometry, differential 39 Geometry, elliptic 204—205 Geometry, Euclidean 13 68 203—204 Geometry, hyperbolic 200—202 205—212 Geometry, non-Euclidean 39 66—67 200—209 Geometry, noncom mutative 230 Geometry, Riemannian 66—69 Geometry, spectral 216—227 Georgia Institute of Technology 195 Giannoni, Marie — Joya 198 222 Gledhill, Joseph 242 Global triangle 204 Godel, Kurt 135—136 142 154 GOE spacing distribution 198 Goodwill Hunting 50 Gram, J. 128—129 Grand unified theory 231 233 Grand Unified Theory, gravity and 237 Grand Unitary Ensemble (GUE) 219 Grand Unitary Ensemble (GUE), spacing distribution 183 198 199 220 Graph 49 Graph theory 49 Gravity 46 48 Gravity, quantum 236—237 Great circles 204—205 Green, Ben 265 Group representation theory 220 Group theory 223 Gutzwiller, Martin 198 200 209 Guy, R. 134 Haas, H. 211 212 213 Hadamard, described 105—106 140—141 Hadamard, Jacques 105—110 200 208 237 265 Hadamard, Prime Number Theorem and 106—107 129 Hadamard, Riemann's zeta function and 107—110 119 Hamilton, William Rowan 170 Hamiltonians 170—177 Hamiltonians, atomic nucleus and 171—175 Hamiltonians, eigenvalues and eigenvectors 175—177 Hamiltonians, Hermitian symmetry of 175—177 198—200 Hamiltonians, proof of Riemann hypothesis and 227—228 Hardy, described 129—130 Hardy, G.H. 65 162 163 Hardy, infinite number of zeta zeros and 130—134 141 143 Harish — Chandra 163 Harmonic analysis 215 216 harmonic oscillator 228 Harmonic scries 53—61 87 Harvard University 211 224 233 Hawking, Stephen 68 Hawkins prime 151—152 Hawkins, David 151—152 153 Heat 81—82 Heat equation 82 Heat, flow of 82—84 160 Height (of a zeta zero) 90—91 Heisenbcrg, Werner 135 179 187 Hejhal, Dennis 211 224 263 Heller, Eric 194h Hermite, Charles 96 98—99 105—106 175 Hermitian symmctry 175—177 183 198—200 Hexagonal numbers 11 Hilbert space 171—172 179 218 Hilbert(Rmd) 179 260n Hilbert, David 121—125 172 177 Hilbert, described 121—122 Hilbert, invariant theory of 122—123 Hilbert, Polya — Hilbert approach 179—180 185 198—199 227 232 How to Solve It (Polya) 177—178 185 Hyperbolic geometry 200—202 205—212 Hyperbolic triangles 208 Hypothesis testing 226 i 73 78—79 IBM 198 Ideal number 115—126 Ignoramibus 125 134 Imaginary axis 74 imaginary numbers 72 Imaginary numbers, purely 74 Imaginary part of complex numbers 73 74 Imaginary Tale, An (Nahin) 72 Indian Institute ofTechnology 264 Induction and Analogy in Mathematics (Polya) 177 Infinite prime 229 Infinite series 49 52—53 Infinite series, harmonic 53—61 87 infinity 18—29 Infinity, as a highly textured and nuanced notion 21—29 Infinity, defined 18 Infinity, Greeks and 51 53 70 Infinity, of primes 18—20 21 51 59—62 Infinity, transfinite number 22—23 Institute for Advanced Study 136 154—164 Institute for Advanced Study, described 154—156 Institute for Advanced Study, faculty at 140 142 147 154—155 162 Institute for Advanced Study, visiting members at 155—156 249 Integers 9 110—121 Integers, gaussian 110—125 Integrable systems 190—193 198 Integrable systems, Deift and 243 Integral form (of Riemanns zeta function) 80—81 129 Integration 42 International Congress of Mathematics (ICM) 121—125 International Congress of Mathematics (ICM), Fields Medal and 142 Internet 16 265—266 Internet, credit card transactions on 16—18 Internet, graphs and 49 Introduction to Infinitesimal Analysis (Euler) 66 Introduction to Probability and Its Applications, An (Feller) 223—224 Invariant theory 122—123 Irrational numbers 70 Ising model 238—239 Ising, Ernst 238—239 Jaffe, Arthur 233 Jimbo, M. 237 239—240 Johansson, Kurt 233 245 Johansson, mathematics of permutations and 245 255—260 Journal of the Indian mathematical society 209—210 Jung, Carl 225 Kaplansky, Irving 234 Katz, described 220 Katz, Nick 219—221 262 Kayal, Neeraj 264 Keating, Jonathan 226 227 228 262 Kendall, M.G. 102 Kepler, Johannes 46 47 Kerov, S. 255 Knuth, Donald E. 257 Kostelec, P. 181n 184n 192n 193n 195n 201n 205n 208n Kronecker, Leopold 7 8 70 111 137 220 Kummcr, Eduard 115 Kyoto Prize 257 Lacroix, Sylvestre 83 Lagrange, Joseph — Louis 83 Landau, Edmund 130 142 Laplace, Pierre — Simon 47 83 Lavoisier, Antoine — Laurcnt 49 Lcgendre, Adricn — Marie 29 47 61 66 Lcgendre, death of 38 Lcgendre, described 30—32 Lcgendre, Gauss and 38—44 93 Lcgendre, Prime Number Theories and 35—38 69 106 Lcvinson, Norman 151 Lcwitt, Sol 256 Legendre functions 31 Lehman, R, Sherman 151 Lenstra, Hendrik 264 Levy, Silvio 206n Ley, Jos 206n Line 204n Line, straight 203—207 Linear transformation 169 Littlewood, J.E. 131—132 141 143 Logan, B. 255 Logarithm 32 Logarithm, natural 34—35 Logarithm, reciprocals of 41—42 Logarithmic growth 32—35 Logarithmic growth, decibel system and 34 Logarithmic growth, Richter scale and 33—34 Logarithmic integral 42 Long Term Capital Management 102 Los Alamos 249 265 Luiliard School 221 MacArthur,"Genius" Fellowship 224 magic 221—225 Maginot Line 238 Malkiel, Burton 102 Maor, Eli 54n Mathematical chaos 189—190 Mathematics 8 Mathematics, "unreasonable effectiveness of" 174 Mathematics, brevity of proof and 50—51 Mathematics, collaboration in 218 Mathematics, combinatorics 223 Mathematics, discrete 223 Mathematics, either-or nature of 134—136 Mathematics, infinity and 21 Mathematics, magic and 223 Mathematics, mathematicians 217 Mathematics, Pythagoras and 10 Mathematics, shoulders of giants and 46—62 Mathematics, theorems and 49—51 Matrices 164—185 Matrices, diagonal 167 Matrices, eigenvalues and eigenvectors 175—177 Matrices, examples of 165 Matrices, Hamiltonian see “Hamiltonians” Matrices, Hermitian 175—177 183 Matrices, multiplication and 169n Matrices, orthogonal 219 Matrices, random see “Random matrices” Matrices, svmmetric 168 Matrices, symplectic 219 Matrices, transformative effects of 165—170 Matrices, unitary 219 Matter and energy 21 Mchta, Madan Lai 175 218 234 239 244 245 Mcndcleycv, Dmitry 108 Mcrtens, F. 104 Memoryless process 158 Mendelssohn, Rebecca 59 Mertens conjecture 104 181 Merton, Robert 102 METAFONT 257 Method of least squares 31 38 39 40 Metropolis, Nicholas 249 250 Millennium meeting 3—6 261—263 Millennium Prize Problems 6 Miller, Arthur 134—135 Milton, John 48 MIT 211 242 Mittag — Leffler, Goesta 141 Miwa, T. 237 239—240 Moebius band (Moebius strip) 97 Moebius inversion formula 97—105 177 181 259 Moebius, A.R. 97 Monge, Gaspard 83 Monte Carlo technique 250 253 Montgomery, described 156—157 Montgomery, Dyson and 154 162 164 180 Montgomery, Dyson — Monrgomcry — Odfyzko Law 180—185 Montgomery, Hugh 219 263 Montgomery, pair correlation and 157—160 Montgomery, sinc function and 161—162 Mori, Y. 237 239—240 Moser, Juergen 243
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