The Princeton Companion to Mathematics / / ed. by Timothy Gowers, Imre Leader, June Barrow-Green.
This is a one-of-a-kind reference for anyone with a serious interest in mathematics. Edited by Timothy Gowers, a recipient of the Fields Medal, it presents nearly two hundred entries, written especially for this book by some of the world's leading mathematicians, that introduce basic mathematic...
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Superior document: | Title is part of eBook package: De Gruyter Princeton University Press eBook-Package Backlist 2000-2013 |
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Place / Publishing House: | Princeton, NJ : : Princeton University Press, , [2010] ©2009 |
出版年: | 2010 |
版: | Core Textbook |
语言: | English |
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实物描述: | 1 online resource (1056 p.) :; Black-and-white illustrations throughout | Cross-references, bibliographies, index |
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书本目录:
- Frontmatter
- Contents
- Preface
- Contributors
- Part I. Introduction
- I.1 What Is Mathematics About?
- I.2 The Language and Grammar of Mathematics
- I.3 Some Fundamental Mathematical Definitions
- I.4 The General Goals of Mathematical Research
- Part II. The Origins of Modern Mathematics
- II.1 From Numbers to Number Systems
- II.2 Geometry
- II.3 The Development of Abstract Algebra
- II.4 Algorithms
- II.5 The Development of Rigor in Mathematical Analysis
- II.6 The Development of the Idea of Proof
- II.7 The Crisis in the Foundations of Mathematics
- Part III. Mathematical Concepts
- III.1 The Axiom of Choice
- III.2 The Axiom of Determinacy
- III.3 Bayesian Analysis
- III.4 Braid Groups
- III.5 Buildings
- III.6 Calabi-Yau Manifolds
- III.7 Cardinals
- III.8 Categories
- III.9 Compactness and Compactification
- III.10 Computational Complexity Classes
- III.11 Countable and Uncountable Sets
- III.12 C*-Algebras
- III.13 Curvature
- III.14 Designs
- III.15 Determinants
- III.16 Differential Forms and Integration
- III.17 Dimension
- III.18 Distributions
- III.19 Duality
- III.20 Dynamical Systems and Chaos
- III.21 Elliptic Curves
- III.22 The Euclidean Algorithm and Continued Fractions
- III.23 The Euler and Navier-Stokes Equations
- III.24 Expanders
- III.25 The Exponential and Logarithmic Functions
- III.26 The Fast Fourier Transform
- III.27 The Fourier Transform
- III.28 Fuchsian Groups
- III.29 Function Spaces
- III.30 Galois Groups
- III.31 The Gamma Function
- III.32 Generating Functions
- III.33 Genus
- III.34 Graphs
- III.35 Hamiltonians
- III.36 The Heat Equation
- III.37 Hilbert Spaces
- III.38 Homology and Cohomology
- III.39 Homotopy Groups
- III.40 The Ideal Class Group
- III.41 Irrational and Transcendental Numbers
- III.42 The Ising Model
- III.43 Jordan Normal Form
- III.44 Knot Polynomials
- III.45 K-Theory
- III.46 The Leech Lattice
- III.47 L-Functions
- III.48 Lie Theory
- III.49 Linear and Nonlinear Waves and Solitons
- III.50 Linear Operators and Their Properties
- III.51 Local and Global in Number Theory
- III.52 The Mandelbrot Set
- III.53 Manifolds
- III.54 Matroids
- III.55 Measures
- III.56 Metric Spaces
- III.57 Models of Set Theory
- III.58 Modular Arithmetic
- III.59 Modular Forms
- III.60 Moduli Spaces
- III.61 The Monster Group
- III.62 Normed Spaces and Banach Spaces
- III.63 Number Fields
- III.64 Optimization and Lagrange Multipliers
- III.65 Orbifolds
- III.66 Ordinals
- III.67 The Peano Axioms
- III.68 Permutation Groups
- III.69 Phase Transitions
- III.70 π
- III.71 Probability Distributions
- III.72 Projective Space
- III.73 Quadratic Forms
- III.74 Quantum Computation
- III.75 Quantum Groups
- III.76 Quaternions, Octonions, and Normed Division Algebras
- III.77 Representations
- III.78 Ricci Flow
- III.79 Riemann Surfaces
- III.80 The Riemann Zeta Function
- III.81 Rings, Ideals, and Modules
- III.82 Schemes
- III.83 The Schrödinger Equation
- III.84 The Simplex Algorithm
- III.85 Special Functions
- III.86 The Spectrum
- III.87 Spherical Harmonics
- III.88 Symplectic Manifolds
- III.89 Tensor Products
- III.90 Topological Spaces
- III.91 Transforms
- III.92 Trigonometric Functions
- III.93 Universal Covers
- III.94 Variational Methods
- III.95 Varieties
- III.96 Vector Bundles
- III.97 Von Neumann Algebras
- III.98 Wavelets
- III.99 The Zermelo-Fraenkel Axioms
- Part IV. Branches of Mathematics
- IV.1 Algebraic Numbers
- IV.2 Analytic Number Theory
- IV.3 Computational Number Theory
- IV.4 Algebraic Geometry
- IV.5 Arithmetic Geometry
- IV.6 Algebraic Topology
- IV.7 Differential Topology
- IV.8 Moduli Spaces
- IV.9 Representation Theory
- IV.10 Geometric and Combinatorial Group Theory
- IV.11 Harmonic Analysis
- IV.12 Partial Differential Equations
- IV.13 General Relativity and the Einstein Equations
- IV.14 Dynamics
- IV.15 Operator Algebras
- IV.16 Mirror Symmetry
- IV.17 Vertex Operator Algebras
- IV.18 Enumerative and Algebraic Combinatorics
- IV.19 Extremal and Probabilistic Combinatorics
- IV.20 Computational Complexity
- IV.21 Numerical Analysis
- IV.22 Set Theory
- IV.23 Logic and Model Theory
- IV.24 Stochastic Processes
- IV.25 Probabilistic Models of Critical Phenomena
- IV.26 High-Dimensional Geometry and Its Probabilistic Analogues
- Part V. Theorems and Problems
- V.1 The ABC Conjecture
- V.2 The Atiyah-Singer Index Theorem
- V.3 The Banach-Tarski Paradox
- V.4 The Birch-Swinnerton-Dyer Conjecture
- V.5 Carleson's Theorem
- V.6 The Central Limit Theorem
- V.7 The Classification of Finite Simple Groups
- V.8 Dirichlet's Theorem
- V.9 Ergodic Theorems
- V.10 Fermat's Last Theorem
- V.11 Fixed Point Theorems
- V.12 The Four-Color Theorem
- V.13 The Fundamental Theorem of Algebra
- V.14 The Fundamental Theorem of Arithmetic
- V.15 Gödel's Theorem
- V.16 Gromov's Polynomial-Growth Theorem
- V.17 Hilbert's Nullstellensatz
- V.18 The Independence of the Continuum Hypothesis
- V.19 Inequalities
- V.20 The Insolubility of the Halting Problem
- V.21 The Insolubility of the Quintic
- V.22 Liouville's Theorem and Roth's Theorem
- V.23 Mostow's Strong Rigidity Theorem
- V.24 The P versus NP Problem
- V.25 The Poincaré Conjecture
- V.26 The Prime Number Theorem and the Riemann Hypothesis
- V.27 Problems and Results in Additive Number Theory
- V.28 From Quadratic Reciprocity to Class Field Theory
- V.29 Rational Points on Curves and the Mordell Conjecture
- V.30 The Resolution of Singularities
- V.31 The Riemann-Roch Theorem
- V.32 The Robertson-Seymour Theorem
- V.33 The Three-Body Problem
- V.34 The Uniformization Theorem
- V.35 The Weil Conjectures
- Part VI.
- Mathematicians
- VI.1 Pythagoras
- VI.2 Euclid
- VI.3 Archimedes
- VI.4 Apollonius
- VI.5 Abu Ja'far Muhammad ibn Mūsā al-Khwārizmī
- VI.6 Leonardo of Pisa (known as Fibonacci)
- VI.7 Girolamo Cardano
- VI.8 Rafael Bombelli
- VI.9 François Viète
- VI.10 Simon Stevin
- VI.11 René Descartes
- VI.12 Pierre Fermat
- VI.13 Blaise Pascal
- VI.14 Isaac Newton
- VI.15 Gottfried Wilhelm Leibniz
- VI.16 Brook Taylor
- VI.17 Christian Goldbach
- VI.18 The Bernoullis
- VI.19 Leonhard Euler
- VI.20 Jean Le Rond d'Alembert
- VI.21 Edward Waring
- VI.22 Joseph Louis Lagrange
- VI.23 Pierre-Simon Laplace
- VI.24 Adrien-Marie Legendre
- VI.25 Jean-Baptiste Joseph Fourier
- VI.26 Carl Friedrich Gauss
- VI.27 Siméon-Denis Poisson
- VI.28 Bernard Bolzano
- VI.29 Augustin-Louis Cauchy
- VI.30 August Ferdinand Möbius
- VI.31 Nicolai Ivanovich Lobachevskii
- VI.32 George Green
- VI.33 Niels Henrik Abel
- VI.34 János Bolyai
- VI.35 Carl Gustav Jacob Jacobi
- VI.36 Peter Gustav Lejeune Dirichlet
- VI.37 William Rowan Hamilton
- VI.38 Augustus De Morgan
- VI.39 Joseph Liouville
- VI.40 Ernst Eduard Kummer
- VI.41 Évariste Galois
- VI.42 James Joseph Sylvester
- VI.43 George Boole
- VI.44 Karl Weierstrass
- VI.45 Pafnuty Chebyshev
- VI.46 Arthur Cayley
- VI.47 Charles Hermite
- VI.48 Leopold Kronecker
- VI.49 Georg Friedrich Bernhard Riemann
- VI.50 Julius Wilhelm Richard Dedekind
- VI.51 Émile Léonard Mathieu
- VI.52 Camille Jordan
- VI.53 Sophus Lie
- VI.54 Georg Cantor
- VI.55 William Kingdon Clifford
- VI.56 Gottlob Frege
- VI.57 Christian Felix Klein
- VI.58 Ferdinand Georg Frobenius
- VI.59 Sofya (Sonya) Kovalevskaya
- VI.60 William Burnside
- VI.61 Jules Henri Poincaré
- VI.62 Giuseppe Peano
- VI.63 David Hilbert
- VI.64 Hermann Minkowski
- VI.65 Jacques Hadamard
- VI.66 Ivar Fredholm
- VI.67 Charles-Jean de la Vallée Poussin
- VI.68 Felix Hausdorff
- VI.69 Élie Joseph Cartan
- VI.70 Emile Borel
- VI.71 Bertrand Arthur William Russell
- VI.72 Henri Lebesgue
- VI.73 Godfrey Harold Hardy
- VI.74 Frigyes (Frédéric) Riesz
- VI.75 Luitzen Egbertus Jan Brouwer
- VI.76 Emmy Noether
- VI.77 Wacław Sierpiński
- VI.78 George Birkhoff
- VI.79 John Edensor Littlewood
- VI.80 Hermann Weyl
- VI.81 Thoralf Skolem
- VI.82 Srinivasa Ramanujan
- VI.83 Richard Courant
- VI.84 Stefan Banach
- VI.85 Norbert Wiener
- VI.86 Emil Artin
- VI.87 Alfred Tarski
- VI.88 Andrei Nikolaevich Kolmogorov
- VI.89 Alonzo Church
- VI.90 William Vallance Douglas Hodge
- VI.91 John von Neumann
- VI.92 Kurt Gödel
- VI.93 André Weil
- VI.94 Alan Turing
- VI.95 Abraham Robinson
- VI.96 Nicolas Bourbaki
- Part VII. The Influence of Mathematics
- VII.1 Mathematics and Chemistry
- VII.2 Mathematical Biology
- VII.3 Wavelets and Applications
- VII.4 The Mathematics of Traffic in Networks
- VII.5 The Mathematics of Algorithm Design
- VII.6 Reliable Transmission of Information
- VII.7 Mathematics and Cryptography
- VII.8 Mathematics and Economic Reasoning
- VII.9 The Mathematics of Money
- VII.10 Mathematical Statistics
- VII.11 Mathematics and Medical Statistics
- VII.12 Analysis, Mathematical and Philosophical
- VII.13 Mathematics and Music
- VII.14 Mathematics and Art
- Part VIII. Final Perspectives
- VIII.1 The Art of Problem Solving
- VIII.2 "Why Mathematics?" You Might Ask
- VIII.3 The Ubiquity of Mathematics
- VIII.4 Numeracy
- VIII.5 Mathematics: An Experimental Science
- VIII.6 Advice to a Young Mathematician
- VIII.7 A Chronology of Mathematical Events
- Index