| 本书作者现任美国西北大学教授，多种国际权威杂志的主编、副主编。作者根据在教学、研究和咨询中的经验，写了这本适合学生和实际工作者的书。本书提供连续优化中大多数有效方法的全面的最新的论述。每一章从基本概念开始，逐步阐述当前可用的最佳技术。 本书强调实用方法，包含大量图例和练习，适合广大读者阅读，可作为工程、运筹学、数学、计算机科学以及商务方面的研究生教材，也可作为该领域的科研人员和实际工作人员的手册。 总之，作者力求本书阅读性强，内容丰富，论述严谨，能揭示数值最优化的美妙本质和实用价值。 |
| Part I Problems and Tricks 1 Elementary Number Theory 1.1 Problems About Primes. Divisibility and Primality 1.2 Diophantine Equations of Degree One and Two 1.3 Cubic Diophantine Equations 1.4 Approximations and Continued Fractions 1.5 Diophantine Approximation and the Irrationality 2 Some Applications of Elementary Number Theory 2.1 Factorization and Public Key Cryptosystems 2.2 Deterministic Primality Tests 2.3 Factorization of Large Integers Part II Ideas and Theories 3 Induction and Recursion 3.1 Elementary Number Theory From the Point of View of Logic 3.2 Diophantine Sets 3.3 Partially Recursive Functions and Enumerable Sets 3.4 Diophantineness of a Set and algorithmic Undecidability 4 Arithmetic of algebraic numbers 4.1 Algebraic Numbers： Their Realizations and Geometry 4.2 Decomposition of Prime Ideals， Dedekind Domains， and Valuations 4.3 Local and Global Methods 4.4 Class Field Theory 4.5 Galois Group in Arithetical Problems 5 Arithmetic of algebraic varieties 5.1 Arithmetic Varieties and Basic Notions of Algebraic Geometry 5.2 Geometric Notions in the Study of Diophantine equations 5.3 Elliptic curves， Abelian Varieties， and Linear Groups 5.4 Diophantine Equations and Galois Repressentations 5.5 The Theorem of Faltings and Finiteness Problems in Diophantine Geometry 6 Zeta Functions and Modular Forms 6.1 Zeta Functions of Arithmetic Schemes 6.2 L-Functions， the Theory of Tate and Explicite Formulae 6.3 Modular Forms and Euler Products 6.4 Modular Forms and Galois Representations 6.5 Automorphic Forms and The Langlands Program 7 Fermats Last Theorem and Families of Modular Forms 7.1 Shimura-Taniyama-Weil Conjecture and Reciprocity Laws 7.2 Theorem of Langlands-Tunnell and Modularity Modulo 3 7.3 Modularity of Galois representations and Universal Deformation Rings 7.4 Wiles Main Theorem and Isomorphism Criteria for Local Rings 7.5 Wiles Induction Step： Application of the Criteria and Galois Cohomology 7.6 The Relative Invariant， the Main Inequality and The Minimal Case 7.7 End of Wiles Proof and Theorem on Absolute Irreducibility Part III Analogies and Visions III-0 Introductory survey to part III： motivations and description III.1 Analogies and differences between numbers and functions： 8-point， Archimedean properties etc. III.2 Arakelov geometry， fiber over 8， cycles， Green functions (dapres Gillet-Soule) III.3 -functions， local factors at 8， Serres T-factors III.4 A guess that the missing geometric objects are noncommutative spaces 8 Arakelov Geometry and Noncommutative Geometry 8.1 Schottky Uniformization and Arakelov Geometry 8.2 Cohomological Constructions 8.3 Spectral Triples， Dynamics and Zeta Functions 8.4 Reduction mod 8 References Index |
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