Bøger af Yuan Wang

This is a series of studies on China from social science and humanity perspectives including political science, economics, business, law, sociology, culture, international relations, public policy, modern and contemporary history, and arts. We call for papers on the above areas, and we also call for guest editors to organize special issues on the above areas. For details, please contact the editor of the series at MCSeditor@google.com. The current issue is a special issue edited by Yuan Wang on "China as a Potential Superpower." Authors include Yuan Wang, Jianbo Zhang, Honghua Men, Kurt Radtke, Hitoshi Hirakara, Yun Chen, Kuo-Yi Liu, and Di Liu.

1. The Circle Method and Waring's Problem.- 1.1 Introduction.- 1.2 Farey Division.- 1.3 Auxiliary Lemmas.- 1.4 Major Arcs.- 1.5 Singular Integral.- 1.6 Singular Series.- 1.7 Proof of Lemma 1.12.- 1.8 Proof of Theorem 1.1.- Notes.- 2. Complete Exponential Sums.- 2.1 Introduction.- 2.2 Several Lemmas.- 2.3 Mordell's Lemma.- 2.4 Fundamental Lemma.- 2.5 Proof of Theorem 2.1.- 2.6 Proof of Theorem 2.2.- Notes.- 3. Weyl's Sums.- 3.1 Introduction.- 3.2 Proof of Theorem 3.1.- 3.3 A Lemma on Units.- 3.4 The Asymptotic Formula for N(a, T).- 3.5 A Sum.- 3.6 Mitsui's Lemma.- 3.7 Proof of Theorem 3.3.- 3.8 Proof of Lemma 3.6.- 3.9 Continuation.- Notes.- 4. Mean Value Theorems.- 4.1 Introduction.- 4.2 Proof of Theorem 4.1.- 4.3 Proof of Theorem 4.2.- 4.4 A Lemma on the Set D.- 4.5 A Lemma on the Set D(x).- 4.6 Fundamental Lemma.- 4.7 Proof of Lemma 4.1.- Notes.- 5. The Circle Method in Algebraic Number Fields.- 5.1 Introduction.- 5.2 Lemmas.- 5.3 Asympotic Expansion forSi (?, T).- 5.4 Further Estimates on Basic Domains.- 5.5 Proof of Theorem 5.1.- 5.6 Proof of Theorem 5.2.- Notes.- 6. Singular Series and Singular Integrals.- 6.1 Introduction.- 6.2 Product Form for Singular Series.- 6.3 Singular Series and Congruences.- 6.4 p-adic Valuations.- 6.5 k-th Power Residues.- 6.6 Proof of Theorem 6.1.- 6.7 Monotonic Functions.- 6.8 Proof of Theorem 6.2.- Notes.- 7. Waring's Problem.- 7.1 Introduction.- 7.2 The Ring Jk.- 7.3 Proofs of Theorems 7.1 and 7.2.- 7.4 Proof of Theorem 7.3.- 7.5 Proof of Theorem 7.4.- Notes.- 8. Additive Equations.- 8.1 Introduction.- 8.2 Reductions.- 8.3 Contraction.- 8.4 Derived Variables.- 8.5 Proof of Theorem 8.1.- 8.6 Proof of Theorem 8.2.- 8.7 Bounds for Solutions.- Notes.- 9. Small Nonnegative Solutions of Additive Equations.- 9.1 Introduction.- 9.2 Hurwitz's Lemma.- 9.3 Reductions.- 9.4 Continuation.- 9.5 Farey Division.- 9.6 Supplementary Domain.- 9.7 Basic Domains.- 9.8 Proof of Theorem 9.1.- Notes.- 10. Small Solutions of Additive Equations.- 10.1 Introduction.- 10.2 Reductions.- 10.3 Continuation.- 10.4 Farey Division.- 10.5 Supplementary Domain.- 10.6 Basic Domains.- 10.7 Proof of Theorem 10.1.- Notes.- 11. Diophantine Inequalities for Forms.- 11.1 Introduction.- 11.2 A Single Additive Form.- 11.3 A Variant Circle Method.- 11.4 Continuation.- 11.5 Proof of Lemma 11.1.- 11.6 Linear Forms.- 11.7 A Single Form.- 11.8 Proof of Theorem 11.1.- Notes.- References I.- References II

This book is an autobiographical interview with Chinese Academician Yuan Wang on his experience in mathematical research. The book looks back on Wang's collaboration with his teacher Hua Loo-Keng and younger scholars, offering insights into fruitful cooperation in mathematical research.In this book, Yuan Wang¿s path of studying Goldbach conjecture is revealed in detail from motivation to method. Then his work on algebraic number theory is traced back in a separate chapter. The book ends with two chapters which introduce Wang¿s interest in history of mathematics and his hobbies outside of mathematical research. Wang shows how a mathematician can in the same time be a historical and popular science writer and, in particular, a well-received calligrapher. The book is intended for undergraduate and graduate students studying number theory. Researchers who are willing to learn from the experience of an established mathematician, as well as math amateurs and general audience who are interested in Yuan Wang's life story might also find this book appealing.

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