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数学指南 英文PDF电子书下载
- 电子书积分:26 积分如何计算积分?
- 作 者:格沃斯(CowersT·)著
- 出 版 社:世界图书出版公司北京公司
- 出版年份:2013
- ISBN:9787510050688
- 页数:1034 页
Part Ⅰ Introduction 1
Ⅰ.1 What Is Mathematics About? 1
Ⅰ.2 The Language and Grammar of Mathematics 8
Ⅰ.3 Some Fundamental Mathematical Definitions 16
Ⅰ.4 The General Goals of Mathematical Research 47
Part Ⅱ The origins of Modern Mathomatics 77
Ⅱ.1 From Numbers to Number Systems 77
Ⅱ.2 Geometry 83
Ⅱ.3 The Development of Abstract Algebra 95
Ⅱ.4 Algorithms 106
Ⅱ.5 The Development of Rigor in Mathematical Analysis 117
Ⅱ.6 The Development of the Idea of Proof 129
Ⅱ.7 The Crisis in the Foundations of Mathematics 142
Part Ⅲ Mathematical Concepts 157
Ⅲ.1 The Axiom of Choice 157
Ⅲ.2 The Axiom of Determinacy 159
Ⅲ.3 Bayesian Analysis 159
Ⅲ.4 Braid Groups 160
Ⅲ.5 Buildings 161
Ⅲ.6 Calabi-Yau Manifolds 163
Ⅲ.7 Cardinals 165
Ⅲ.8 Categories 165
Ⅲ.9 Compactness and Compactification 167
Ⅲ.10 Computational Complexity Classes 169
Ⅲ.11 Countable and Uncountable Sets 170
Ⅲ.12 C-Algebras 172
Ⅲ.13 Curvature 172
Ⅲ.14 Designs 172
Ⅲ.15 Determinants 174
Ⅲ.16 Differential Forms and Integration 175
Ⅲ.17 Dimension 180
Ⅲ.18 Distributions 184
Ⅲ.19 Duality 187
Ⅲ.20 Dynamical Systems and Chaos 190
Ⅲ.21 Elliptic Curves 190
Ⅲ.22 The Euclidean Algorithm and Continued Fractions 191
Ⅲ.23 The Euler and Navier-Stokes Equations 193
Ⅲ.24 Expanders 196
Ⅲ.25 The Exponential and Logarithmic Functions 199
Ⅲ.26The Fast Fourier Transform 202
Ⅲ.27 The Fourier Transform 204
Ⅲ.28 Fuchsian Groups 208
Ⅲ.29 Function Spaces 210
Ⅲ.30 Galois Groups 213
Ⅲ.31 The Gamma Function 213
Ⅲ.32 Generating Functions 214
Ⅲ.33 Genus 215
Ⅲ.34 Graphs 215
Ⅲ.35 Hamiltonians 215
Ⅲ.36 The Heat Equation 216
Ⅲ.37 Hilbert Spaces 219
Ⅲ.38 Homology and Cohomology 221
Ⅲ.39 Homotopy Groups 221
Ⅲ.40 The Ideal Class Group 221
Ⅲ.41 Irrational and Transcendental Numbers 222
Ⅲ.42 The Ising Model 223
Ⅲ.43 Jordan Normal Form 223
Ⅲ.44 Knot Polynomials 225
Ⅲ.45 K-Theory 227
Ⅲ.46 The Leech Lattice 227
Ⅲ.47L-Functions 228
Ⅲ.48 Lie Theory 229
Ⅲ.49 Linear and Nonlinear Waves and Solitons 234
Ⅲ.50 Linear Operators and Their Properties 239
Ⅲ.51 Local and Global in Number Theory 241
Ⅲ.52 The Mandelbrot Set 244
Ⅲ.53 Manifolds 244
Ⅲ.54 Matroids 244
Ⅲ.55 Measures 246
Ⅲ.56 Metric Spaces 247
Ⅲ.57 Models of Set Theory 248
Ⅲ.58 Modular Arithmetic 249
Ⅲ.59 Modular Forms 250
Ⅲ.60 Moduli Spaces 252
Ⅲ.61 The Monster Group 252
Ⅲ.62 Normed Spaces and Banach Spaces 252
Ⅲ.63 Number Fields 254
Ⅲ.64 Optimization and Lagrange Multipliers 255
Ⅲ.65 Orbifolds 257
Ⅲ.66 Ordinals 258
Ⅲ.67 The Peano Axioms 258
Ⅲ.68 Permutation Groups 259
Ⅲ.69 Phase Transitions 261
Ⅲ.70 π 261
Ⅲ.71 Probability Distributions 263
Ⅲ.72 Projective Space 267
Ⅲ.73 Quadratic Forms 267
Ⅲ.74 Quantum Computation 269
Ⅲ.75 Quantum Groups 272
Ⅲ.76 Quaternions,Octonions,and Normed Division Algebras 275
Ⅲ.77 Representations 279
Ⅲ.78 Ricci Flow 279
Ⅲ.79 Riemann Surfaces 282
Ⅲ.80 The Riemann Zeta Function 283
Ⅲ.81 Rings,Ideals,and Modules 284
Ⅲ.82 Schemes 285
Ⅲ.83 The Schr?dinger Equation 285
Ⅲ.84 The Simplex Algorithm 288
Ⅲ.85 Special Functions 290
Ⅲ.86 The Spectrum 294
Ⅲ.87 Spherical Harmonics 295
Ⅲ.88 Symplectic Manifolds 297
Ⅲ.89 Tensor Products 301
Ⅲ.90 Topological Spaces 301
Ⅲ.91 Transforms 303
Ⅲ.92 Trigonometric Functions 307
Ⅲ.93 Universal Covers 309
Ⅲ.94 Variational Methods 310
Ⅲ.95 Varieties 313
Ⅲ.96 Vector Bundles 313
Ⅲ.97 Von Neumann Algebras 313
Ⅲ.98 Wavelets 313
Ⅲ.99 The Zermelo-Fraenkel Axioms 314
Part Ⅳ Branches of Mathematics 315
Ⅳ.1 Algebraic Numbers 315
Ⅳ.2 Analytic Number Theory 332
Ⅳ.3 Computational Number Theory 348
Ⅳ.4 Algebraic Geometry 363
Ⅳ.5 Arithmetic Geometry 372
Ⅳ.6 Algebraic Topology 383
Ⅳ.7 Differential Topology 396
Ⅳ.8 Moduli Spaces 408
Ⅳ.9 Representation Theory 419
Ⅳ.10 Geomertic and Combinatorial Group Theory 431
Ⅳ.11 Harmonic Analysis 448
Ⅳ.12 Partial Differential Equations 455
Ⅳ.13 General Relativity and the Einstein Equations 483
Ⅳ.14 Dynamics 493
Ⅳ.15 Operator Algebras 510
Ⅳ.16 Mirror Symmetry 523
Ⅳ.17 Vertex Operator Algebras 539
Ⅳ.18 Enumerative and Algebraic Combinatorics 550
Ⅳ.19 Extremal and Probabilistic Combinatorics 562
Ⅳ.20 Computational Complexity 575
Ⅳ.21 Numerical Analysis 604
Ⅳ.22 Set Theory 615
Ⅳ.23 Logic and Model Theory 635
Ⅳ.24 Stochastic Processes 647
Ⅳ.25 Probabilistic Models of Critical Phenomena 657
Ⅳ.26 High-Dimensional Geometry and Its Probabilistic Analogues 670
Part Ⅴ Theorems and Problems 681
Ⅴ.1 The ABC Conjecture 681
Ⅴ.2 The Atiyah-Singer Index Theorem 681
Ⅴ.3 The Banach-Tarski Paradox 684
Ⅴ.4 The Birch-Swinnerton-Dyer Conjecture 685
Ⅴ.5 Carleson's Theorem 686
Ⅴ.6 The Central Limit Theorem 687
Ⅴ.7 The Classification of Finite Simple Groups 687
Ⅴ.8 Dirichlet's Theorem 689
Ⅴ.9 Ergodic Theorems 689
Ⅴ.10 Fermat's Last Theorem 691
Ⅴ.11 Fixed Point Theorems 693
Ⅴ.12The Four-Color Theorem 696
Ⅴ.13 The Fundamental Theorem of Algebra 698
Ⅴ.14 The Fundamental Theorem of Arithmetic 699
Ⅴ.15 G?del's Theorem 700
Ⅴ.16 Gromov's Polynomial-Growth Theorem 702
Ⅴ.17 Hilbert's Nullstellensatz 703
Ⅴ.18 The Independence of the Continuum Hypothesis 703
Ⅴ.19 Inequalities 703
Ⅴ.20 The Insolubility of the Halting Problem 706
Ⅴ.21 The Insolubility of the Quintic 708
Ⅴ.22 Liouvilie's Theorem and Roth's Theorem 710
Ⅴ.23 Mostow's Strong Rigidity Theorem 711
Ⅴ.24 The P versus NP Problem 713
Ⅴ.25 The PoincaréConjecture 714
Ⅴ.26 The Prime Number Theorem and the Riemann Hypothesis 714
Ⅴ.27 Problems and Results in Additive Number Theory 715
Ⅴ.28 From Quadratic Reciprocity to Class Field Theory 718
Ⅴ.29 Rational Points on Curves and the Mordell Conjecture 720
Ⅴ.30 The Resolution of Singularities 722
Ⅴ31 The Riemann-Roch Theorem 723
Ⅴ.32 The Robertson-Seymour Theorem 725
Ⅴ.33 The Three-Body Problem 726
Ⅴ.34 The Uniformization Theorem 728
Ⅴ.35 The Weil Conjectures 729
Part Ⅵ Mathematicians 733
Ⅵ.1 Pythagoras(ca.569 B.C.E.-ca.494 B.C.E.) 733
Ⅵ.2 Euclid(ca.325 B.C.E.-ca.265 B.C.E.) 734
Ⅵ.3 Archimedes(ca.287 B.C.E.-212 B.C.E.) 734
Ⅵ.4 Apollonius(ca.262 B.C.E.-Ca.190 B.C.E.) 735
Ⅵ.5 Abu Ja'far Muhammad ibn Mūsā al-Khwārizmī(800-847) 736
Ⅵ.6 Leonardo of Pisa(known as Fibonacci)(ca.1170-ca.1250) 737
Ⅵ.7 Girolamo Cardano(1501-1576) 737
Ⅵ.8 Rafael Bombelli(1526-after 1572) 737
Ⅵ.9 Fran?ois Viète(1540-1603) 737
Ⅵ.10 Simon Stevin(1548-1620) 738
Ⅵ.11 René Descartes(1596-1650) 739
Ⅵ.12 Pierre Fermat(160?-1665) 740
Ⅵ.13 Blaise Pascal(1623-1662) 741
Ⅵ.14 Isaac Newton(1642-1727) 742
Ⅵ.15 Gottfried Wilhelm Leibniz(1646-1716) 743
Ⅵ.16 Brook Taylor(1685-1731) 745
Ⅵ.17 Christian Goldbach(1690-1764) 745
Ⅵ.18 The Bernoullis(fl.18th century) 745
Ⅵ.19 Leonhard Euler(1707-1783) 747
Ⅵ.20 Jean Le Rond d'Alembert(1717-1783) 749
Ⅵ.21 Edward Waring(ca.1735-1798) 750
Ⅵ.22 Joseph Louis Lagrange(1736-1813) 751
Ⅵ.23 Pierre-Simon Laplace(1749-1827) 752
Ⅵ.24 Adrien-Marie Legendre(1752-1833) 754
Ⅵ.25 Jean-Baptiste Joseph Fourier(1768-1830) 755
Ⅵ.26 Carl Friedrich Gauss(1777-1855) 755
Ⅵ.27 Siméon-Denis Poisson(1781-1840) 757
Ⅵ.28 Bernard Bolzano(1781-1848) 757
Ⅵ.29 Augustin-Louis Cauchy(1789-1857) 758
Ⅵ.30 August Ferdinand M?bius(1790-1868) 759
Ⅵ.31 Nicolai Ivanovich Lobachevskii(1792-1856) 759
Ⅵ.32 George Green(1793-1841) 760
Ⅵ.33 Niels Henrik Abel(1802-1829) 760
Ⅵ.34 János Bolyai(1802-1860) 762
Ⅵ.35Carl Gustav Jacob Jacobi(1804-1851) 762
Ⅵ.36 Peter Gustav Lejeune Dirichlet(1805-1859) 764
Ⅵ.37 William Rowan Hamilton(1805-1865) 765
Ⅵ.38 Augustus De Morgan(1806-1871) 765
Ⅵ.39 Joseph Liouville(1809-1882) 766
Ⅵ.40 Ernst Eduard Kummer(1810-1893) 767
Ⅵ.41 ?variste Galois(1811-1832) 767
Ⅵ.42 James Joseph Sylvester(1814-1897) 768
Ⅵ.43 George Boole(1815-1864) 769
Ⅵ.44 Karl Weierstrass(1815-1897) 770
Ⅵ.45 Pafnuty Chebyshev(1821-1894) 771
Ⅵ.46 Arthur Cayley(1821-1895) 772
Ⅵ.47 Charles Hermite(1822-1901) 773
Ⅵ.48 Leopold Kronecker(1823-1891) 773
Ⅵ.49 Georg Friedrich Bernhard Riemann(1826-1866) 774
Ⅵ.50 Julius Wilhelm Richard Dedekind(1831-1916) 776
Ⅵ.51 ?mile Léonard Mathieu(1835-1890) 776
Ⅵ.52 Camille Jordan(1838-1922) 777
Ⅵ.53 Sophus Lie(1842-1899) 777
Ⅵ.54 Georg Cantor(1845-1918) 778
Ⅵ.55 William Kingdon Clifford(1845-1879) 780
Ⅵ.56 Gottlob Frege(1848-1925) 780
Ⅵ.57 Christian Felix Klein(1849-1925) 782
Ⅵ.58 Ferdinand Georg Frobenius(1849-1917) 783
Ⅵ.59 Sofya(Sonya)Kovalevskaya(1850-1891) 784
Ⅵ.60 William Burnside(1852-1927) 785
Ⅵ.61 Jules Henri Poincaré(1854-1912) 785
Ⅵ.62 Giuseppe Peano(1858-1932) 787
Ⅵ.63 David Hilbert(1862-1943) 788
Ⅵ.64 Hermann Minkowski(1864-1909) 789
Ⅵ.65 Jacques Hadamard(1865-1963) 790
Ⅵ.66 Ivar Fredholm(1866-1927) 791
Ⅵ.67Charles-Jean de la Vallée Poussin(1866-1962) 792
Ⅵ.68 Felix Hausdorff(1868-1942) 792
Ⅵ.69 ?lie Joseph Cartan(1869-1951) 794
Ⅵ.70 Emile Borel(1871-1956) 795
Ⅵ.71 Bertrand Arthur William Russell(1872-1970) 795
Ⅵ.72 Henri Lebesgue(1875-1941) 796
Ⅵ.73 Godfrey Harold Hardy(1877-1947) 797
Ⅵ.74 Frigyes(Frédéric)Riesz(1880-1956) 798
Ⅵ.75 Luitzen Egbertus Jan Brouwer(1881-1966) 799
Ⅵ.76 Emmy Noether(1882-1935) 800
Ⅵ.77 Waclaw Sierpiński(1882-1969) 801
Ⅵ.78 George Birkhoff(1884-1944) 802
Ⅵ.79 John Edensor Littlewood(1885-1977) 803
Ⅵ.80 Hermann Weyl(1885-1955) 805
Ⅵ.81 Thoralf Skolem(1887-1963) 806
Ⅵ.82 Srinivasa Ramanujan(1887-1920) 807
Ⅵ.83 Richard Courant(1888-1972) 808
Ⅵ.84 Stefan Banach(1892-1945) 809
Ⅵ.85 Norbert Wiener(1894-1964) 811
Ⅵ.86 Emil Artin(1898-1962) 812
Ⅵ.87 Alfred Tarski(1901-1983) 813
Ⅵ.88 Andrei Nikolaevich Kolmogorov(1903-1987) 814
Ⅵ.89 Alonzo Church(1903-1995) 816
Ⅵ.90 William Vallance Douglas Hodge(1903-1975) 816
Ⅵ.91 John von Neumann(1903-1957) 817
Ⅵ.92 Kurt G?del(1906-1978) 819
Ⅵ.93 AndréWeil(1906-1998) 819
Ⅵ.94 Alan Turing(1912-1954) 821
Ⅵ.95 Abraham Robinson(1918-1974) 822
Ⅵ.96 Nicolas Bourbaki(1935-) 823
Part Ⅶ The Influence of Mathematics 827
Ⅶ.1 Mathematics and Chemistry 827
Ⅶ.2 Mathematical Biology 837
Ⅶ.3 Wavelets and Applications 848
Ⅶ.4 The Mathematics of Traffic in Networks 862
Ⅶ.5 The Mathematics of Algorithm Design 871
Ⅶ.6 Reliable Transmission of Information 878
Ⅶ.7 Mathematics and Cryptography 887
Ⅶ.8 Mathematics and Economic Reasoning 895
Ⅶ.9 The Mathematics of Money 910
Ⅶ.10 Mathematical Statistics 916
Ⅶ.11 Mathematics and Medical Statistics 921
Ⅶ.12 Analysis,Mathematical and Philosophical 928
Ⅶ.13 Mathematics and Music 935
Ⅶ.14 Mathematics and Art 944
Part Ⅷ Final Perspectives 955
Ⅷ.1 The Art of Problem Solving 955
Ⅷ.2 "Why Mathematics?"You Might Ask 966
Ⅷ.3 The Ubiquity of Mathematics 977
Ⅷ.4 Numeracy 983
Ⅷ.5 Mathematics:An Experimental Science 991
Ⅷ.6 Advice to a Young Mathematician 1000
Ⅷ.7 A Chronology of Mathematical Events 1010
Index 1015
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