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ISBN10: 0691118809 ISBN13: 9780691118802 LCC: QA11.2 Edition: (hardcover : alk. paper)

Notes

Includes bibliographical references and index. Preface --Contributors --pt. 1.Introduction --1.1.What is mathematics about? --1.2. Thelanguage and grammar of mathematics --1.3.Some fundamental mathematical definitions --1.4. Thegeneral goals of mathematical research --pt. 2. Theorigins of modern mathematics --2.1.From numbers to number systems --2.2.Geometry --2.3. Thedevelopment of abstract algebra --2.4.Algorithms --2.5. Thedevelopment of rigor in mathematical analysis --2.6. Thedevelopment of the idea of proof --2.7. Thecrisis in the foundations of mathematics --pt. 3.Mathematical concepts --3.1. Theaxiom of choice --3.2. Theaxiom of determinacy --3.3.Bayesian analysis --3.4.Braid groups --3.5.Buildings --3.6.Calabi-Yau manifolds --3.7.Cardinals --3.8.Categories --3.9.Compactness and compactification --3.10.Computational complexity classes --3.11.Countable and uncountable sets --3.12.C* - algebras --3.13.Curvature --3.14.Designs --3.15.Determinants --3.15.Differential forms and integration --3.17.Dimension --3.18.Distributions -- 3.19.Duality --3.20.Dynamical systems and chaos --3.21.Elliptic curves --3.22. TheEuclidean algorithm and continued fractions --3.23. TheEuler and Navier-Strokes equations --3.24.Expanders --3.25. Theexponential and logarithmic functions --3.26. Thefast Fourier transform --3.27. TheFourier transform --3.28.Fuchsian groups --3.29.Function spaces --3.30.Galois groups --3.31. Thegamma function --3.32.Generating functions --3.33.Genus --3.34.Graphs --3.35.Hamiltonians --3.36. Theheat equation --3.37.Hilbert spaces --3.38.Homology and cohomology --3.39.Homology and cohomology --3.40. Theideal class group --3.41.Irrational and transcendental numbers --3.42. TheIsling model --3.43.Jordan normal form --3.44.Knot polynomials --3.45.K-theory --3.46. Theleech lattice --3.47.L-function --3.48.Lie theory --3.49.Linear and nonlinear waves and solitons --3.50.Linear operators and their properties --3.51.Local and global in number theory --3.52. TheMandelbrot set --3.53.Manifolds --3.54.Matroids --3.55.Measures -- 3.56.Metric spaces --3.57.Models of set theory --3.58.Modular arithmetic --3.59.Modular forms --3.60.Moduli spaces --3.61. Themonster group --3.62.Normed spaces and banach spaces --3.63.Number fields --3.64.Optimization and Lagrange multipliers --3.65.Orbifolds --3.66.Ordinals --3.67. ThePeano axioms --3.68.Permutation groups --3.69.Phase transitions --3.70.[pi] --3.71.Probability distributions --3.72.Projective space --3.73.Quadratic forms --3.74.Quantum computation --3.75.Quantum computation --3.76.Quaternions, octonions, and normed division algebras --3.77.Representations --3.78.Ricci flow --3.79.Riemann surfaces --3.80. TheRiemann zeta function --3.81.Rings, ideals, and modules --3.82.Schemes --3.83. TheSchr�dinger equation --3.84. Thesimplex algorithm --3.85.Special functions --3.86. Thespectrum --3.87.Spherical harmonics --3.88.Symplectic manifolds --3.89.Tensor products --3.90.Topological spaces --3.91.Transforms --3.92.Trigonometric functions --3.93.Universal covers --3.94.Variational methods --3.95.Varieties --3.96.Vector bundles --3.97.Von Neumann algebras --3.98.Wavelets --3.99. TheZermelo-Fraenkel axioms -- pt. 4.Branches of mathematics --4.1.Algebraic numbers --4.2.Analytic number theory --4.3.Computational number theory --4.4.Algebraic geometry --4.5.Arithmetic geometry --4.6.Algebraic topology --4.7.Differential topology --4.8.Moduli spaces --4.9.Representation theory --4.10.Geometric and combinatorial group theory --4.11.Harmonic analysis --4.12.Partial differential equations --4.13.General relativity and the Einstein equations --4.14.Dynamics --4.15.Operator algebras --4.16.Mirror symmetry --4.17.Vertex operator algebras --4.18.Enumerative and algebraic combinatorics --4.19.Extremal and probabilistic combinatorics --4.20.Computational complexity --4.21.Numerical analysis --4.22.Set theory --4.23.Logic and model theory --4.24.Stochastic processes --4.25.Probabilistic models of critical phenomena --4.26.High-dimensional geometry and its probabilistic analogues -- pt. 5.Theorems and problems --5.1. TheABC conjecture --5.2. TheAtiyah-Singer index theorem --5.3. TheBanach-Tarski paradox --5.4. TheBirch-Swinnerton-Dyer conjecture --5.5.Carleson's theorem --5.6. Thecentral limit theorem --5.7. Theclassification of finite simple groups --5.8.Dirichlet's theorem --5.9.Ergodic theorems --5.10.Fermat's last theorem --5.11.Fixed point theorems --5.12. Thefour-color theorem --5.13. Thefundamental theorem of algebra --5.14. Thefundamental theorem of arithmetic --5.15.G�del's theorem --5.16.Gromov's polynomial-growth theorem --5.17.Hilbert's nullstellensatz --5.18. Theindependence of the continuum hypothesis --5.19.Inequalities --5.20. Theinsolubility of the halting problem --5.21. Theinsolubility of the quintic --5.22.Liousville's theorem and Roth's theorem --5.23.Mostow's strong rigidity theorem --5.24. Thep versus NP problem --5.25. ThePoincar� conjecture --5.26. Theprime number theorem and the Riemann hypothesis --5.27.Problems and results in additive number theory --5.28.From quadratic reciprocity to class field theory --5.29.Rational points on curves and the Mordell conjecture --5.30. Theresolution of singularities --5.31. TheRiemann-Roch theorem --5.32. TheRobertson-Seymour theorem --5.33. Thethree-body problem --5.34. Theuniformization theorem --5.35. TheWeil conjecture -- pt. 6.Mathematicians --6.1.Pythagoras --6.2.Euclid --6.3.Archimedes --6.4.Apollonius --6.5.Abu Ja?far Muhammad ibn M?s? al-Khw?rizm? --6.6.Leonardo of Pisa (known as Fibonacci) --6.7.Girolamo Cardano --6.8.Rafael Bombelli --6.9.Fran�ois Vi�te --6.10.Simon Stevin --6.11.Ren� Descartes --6.12.Pierre Fermat --6.13.Blaise Pascal --6.14.Isaac Newton --6.15.Gottfried Wilhelm Leibnitz --6.16.Brook Taylor --6.17.Christian Goldbach --6.18. TheBernoullis --6.19.Leonhard Euler --6.20.Jean Le Rond d'Alembert --6.21.Edward Waring --6.22.Joseph Louis Lagrange --6.23.Pierre-Simon Laplace --6.24.Adrien-Marie Legendre --6.25.Jean-Baptiste Joseph Fourier --6.26.Carl Friedrich Gauss --6.27.Sim�on-Denis Poisson --6.28.Bernard Bolzano --6.29.Augustin-Louis Cauchy --6.30.August Ferdinand M�bius --6.31.Nicolai Ivanovich Lobachevskii --6.32.George Green --6.33.Niels Henrik Abel --6.34.J�nos Bolyai --6.35.Carl Gustav Jacob Jacobi --6.36.Peter Gustav Lejeune Dirichlet --6.37.William Rowan Hamilton --6.38.Augustus De Morgan --6.39.Joseph Liouville --6.40.Eduard Kummer -- 6.41.�variste Galois --6.42.James Joseph Sylvester --6.43.George Boole --6.44.Karl Weierstrass --6.45.Pafnuty Chebyshev --6.46.Arthur Cayley --6.47.Charles Hermite --6.48.Leopold Kronecker --6.49.Georg Friedrich Bernhard Riemann --6.50.Julius Wilhelm Richard Dedekind --6.51.�mile L�onard Mathieu --6.52.Camille Jordan --6.53.Sophus Lie --6.54.Georg Cantor --6.55.William Kingdon Clifford --6.56.Gottlob Frege --6.57.Christian Felix Klein --6.58.Ferdinand Georg Frobenius --6.59.Sofya (Sonya) Kovalevskaya --6.60.William Burnside --6.61.Jules Henri Poincar� --6.62.Giuseppe Peano --6.63.David Hilbert --6.64.Hermann Minkowski --6.65.Jacques Hadamard --6.66.Ivar Fredholm --6.67.Charles-Jean de la Vall�e Poussin --6.68.Felix Hausdorff --6.69.�lie Joseph Cartan --6.70.Emile Borel --6.71.Bertrand Arthur William Russell --6.72.Henri Lebesgue --6.73.Godfrey Harold Hardy --6.74.Frigyes (Fr�d�ric) Riesz -- 6.75.Luitzen Egbertus Jan Brouwer --6.76.Emmy Noether --6.77.Wac?aw Sierpi?ski --6.78.George Birkhoff --6.79.John Edensor Littlewood --6.80.Hermann Weyl --6.81.Thoralf Skolem --6.82.Srinivasa Ramanujan --6.83.Richard Courant --6.84.Stefan Banach --6.85.Norbert Wiener --6.86.Emil Artin --6.87.Alfred Tarski --6.88.Andrei Nikolaevich Kolmogorov --6.89.Alonzo Church --6.90.William Vallance Douglas Hodge --6.91.John von Neumann --6.92.Kurt G�del --6.93.Andr� Weil --6.94.Alan Turing --6.95.Abraham Robinson --6.96.Nicolas Bourbaki -- pt. 7. Theinfluence of mathematics --7.1.Mathematics and chemistry --7.2.Mathematical biology --7.3.Wavelets and applications --7.4. Themathematics of traffic in networks --7.5. Themathematics of algorithm design --7.6Reliable transmission of information --7.7.Mathematics and cryptography --7.8.Mathematics and economic reasoning --7.9. Themathematics of money --7.10.Mathematical statistucs --7.11.Mathematics and medical statistics --7.12.Analysis, mathematical and philosophical --7.13.Mathematics and music --7.14.Mathematics and art --pt. 8.Final perspectives --8.1. Theart of problem solving --8.2."Why mathematics?" you might ask --8.3. Theubiquity of mathematics --8.4.Numeracy --8.5.Mathematics : an experimental science --8.6.Advice to a young mathematician --8.7. Achronology of mathematical events --Index.

Classification

  • LCC: QA11.2

Book Details

  • Language: eng
  • Physical Description: xx, 1034 p. : ill. ; 26 cm.
  • Edition Info: (hardcover : alk. paper)

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