PREFACE -- CONTENTS -- WANG YUAN: A BRIEF OUTLINE OF HIS LIFE AND WORKS -- Mathematical Work -- 1. Number Theory -- 2. Numerical Analysis and Statistics -- 3. Miscellaneous -- References -- SELECTED PAPERS OF WANG YUAN -- ON THE REPRESENTATION OF LARGE EVEN INTEGER AS A SUM OF A PRODUCT OF AT MOST 3 PRIMES AND A PRODUCT OF AT MOST 4 PRIMES* -- 0. Introduction -- 1. Some Computations -- 2. Theorem A -- 3. Applications of Theorem A -- 4. Two Iteration Theorems -- 5. Proofs of Theorems -- References -- ON SOME PROPERTIES OF INTEGRAL VALUED POLYNOMIALS* -- References -- ON SIEVE METHODS AND SOME OF THE RELATED PROBLEMS* -- References -- ON SIEVE METHODS AND SOME OF THEIR APPLICATIONS* -- REFERENCES -- ON THE REPRESENTATION OF LARGE EVEN NUMBER AS A SUM OF TWO ALMOST-PRIMES* † -- REFERENCES -- A note on some properties of the functions (n), (n) and (n) -- References -- Corrigendum to "A note on some properties of the functions (n), (n) and (n)" -- A NOTE ON SOME PROPERTIES OF THE ARITHMETICAL FUNCTIONS (n), (n) AND d(n)* -- 1. Introduction -- 2. The Proof of Fundamental Lemma -- 3. Brun's Sieve Method -- 4. Several Lemmas -- 5. The Proof of Lemma 2 -- 6. Applications of Fundamental Lemma -- References -- ON SIEVE METHODS AND SOME OF THEIR APPLICATIONS* -- 1. INTRODUCTION -- 2. COMPUTATIONS -- 3. THEOREM A -- 4. APPLICATIONS OF THEOREM A -- 5. THEOREM B -- 6. THEOREM C -- 7. THE PROOF OF THE MAIN RESULTS -- 8. OTHER APPLICATIONS -- REFERENCES -- ON SIEVE METHODS AND SOME OF THEIR APPLICATIONS* -- 1. STATEMENT OF RESULTS -- 2. SEVERAL LEMMAS -- 3. THEOREM A -- 4. THE ESTIMATION OF THE UPPER BOUND OF Pw (x, E) -- 5. THE ESTIMATION OF THE LOWER BOUND OF Pw (x, E) -- 6. TWO THEOREMS -- 7. THE SET AND SOME OF ITS PROPERTIES -- 8. THE PROOF OF THE MAIN THEOREMS -- REFERENCES -- ON THE LEAST PRIMITIVE ROOT OF A PRIME* -- I. INTRODUCTION.

II. CHARACTER SUM (I) -- III. CHARACTER SUM (II) -- IV. SIEVE METHOD OF BRUN -- V. PROOF OF THEOREM 1 -- VI. PROOF OF THEOREM 2 -- REFERENCES -- ON THE REPRESENTATION OF LARGE INTEGER AS A SUM OF A PRIME AND AN ALMOST PRIME* -- 1 -- 2 -- 3 -- 4 -- 5 -- 6 -- 7 -- 8 -- 9 -- REFERENCES -- APPENDIX -- REFERENCES -- ON THE ESTIMATION OF CHARACTER SUM AND ITS APPLICATIONS* -- 1. Introduction -- 2. Proof of Theorem 1 -- 3. Proof of Theorem 2 -- 4. Conditional Result -- References -- ON THE MAXIMAL NUMBER OF PAIRWISE ORTHOGONAL LATIN SQUARE OF ORDER s (APPLICATION OF SIEVE METHOD)* -- 1. Introduction -- 2. N(s) and Sieve Methods -- 3. A Recurrent Formula -- 4. Estimation of Pw (x, q -- E) -- 5. Estimation of Pw (x -- E, n) -- 6. Proof of Main Theorem -- References -- ON THE REPRESENTATION OF EVERY LARGE EVEN INTEGER AS A SUM OF A PRIME AND AN ALMOST PRIME -- 1. INTRODUCTION -- II. THE PROOF OF THEOREM 2 -- III. THE SIEVE METHODS -- IV. THE PROOF OF THEOREM 1 -- REFERENCES -- REMARKS ON A THEOREM OF DAVENPORT* -- References -- ON HH K'S METHOD CONCERNING THE GOLDBACH NUMBER -- I. INTRODUCTION -- II. SEVERAL LEMMAS -- III. HH K'S METHOD -- IV. THE ESTIMATION OF INTEGRAL -- V. THE PROOF OF THEOREM 1 -- VI. THE PROOF OF THEOREM 4 -- REFERENCES -- REMARKS CONCERNING A TRANSFERENCE THEOREM OF LINEAR FORMS* -- References -- A NOTE ON A TRANSFERENCE THEOREM OF LINEAR FORMS -- REFERENCES -- A NOTE ON SOME METRICAL THEOREMS IN DIOPHANTINE APPROXIMATION -- References -- Bounds for solutions of additive equations in an algebraic number field I -- References -- Bounds for solutions of additive equations in an algebraic number field II -- References -- Diophantine Inequalities for Forms in an Algebraic Number Field -- 1. INTRODUCTION -- 2. ADDITIVE FORMS -- 3. ANALYTIC METHOD -- 4. THE PROOF OF PROPOSITION 2 -- 5. THE PROOF OF THEOREM 1 -- ACKNOWLEDGMENT.

REFERENCES -- ON HOMOGENEOUS ADDITIVE CONGRUENCES -- I. INTRODUCTION -- II. PRELIMINARY LEMMAS -- III. THE PROOF OF THEOREM 1 (ODD DEGREE k) -- IV. THE PROOF OF THEOREM 1 (ARBITRARY DEGREE k) -- REFERENCES -- Small Solutions of Congruences -- 1. INTRODUCTION -- 2. LEMMAS -- 3. REDUCTION -- 4. DIVISION INTO TWO CASES -- 5. PROOF OF PROPOSITION 2 -- 6. PROOF OF THEOREM 2 -- ACKNOWLEDGMENTS -- REFERENCES -- On Small Zeros of Quadratic Forms over Finite Fields (II) -- 1. Introduction -- 2. Preliminaries -- 3. Proof of Theorem 1 -- References -- Remarks Concerning Numerical Integration* -- REFERENCE -- A NOTE ON INTERPOLATION OF A CERTAIN CLASS OF FUNCTIONS* -- REFERENCES -- On Diophantine Approximations and Numerical Integrations (I) -- On Diophantine Approximations and Numerical Integrations (II) -- ON NUMERICAL INTEGRATION OF PERIODIC FUNCTIONS OF SEVERAL VARIABLES* -- I. INTRODUCTION -- II. MULTIPLE INTEGRALS AND SIMPLE SUMS -- III. SOME PROPERTIES OF THE TOTALLY REAL ALGEBRAIC NUMBERS -- IV. THE CYCLOTOMIC FIELD -- V. UNITS OF THE TOTALLY REAL CYCLOTOMIC FIELD -- REFERENCES -- ON INTERPOLATION OF A CERTAIN CLASS OF FUNCTIONS* -- References -- ON UNIFORM DISTRIBUTION AND NUMERICAL ANALYSIS (I) -- I. INTRODUCTION -- II. THE TOTALLY REAL ALGEBRAIC FIELD -- III. SOME EXAMPLES -- IV. UNIFORM DISTRIBUTION -- V. THE PROOF OF THEOREM 1 -- VI. THE PROOF OF THEOREM 2 -- VII. THE PROOF OF THEOREM 3 -- VIII. THE PROOF OF THEOREM 4 -- IX. EXAMPLES -- REFERENCES -- ON UNIFORM DISTRIBUTION AND NUMERICAL ANALYSIS (II) (NUMBER-THEORETIC METHOD) -- I. STATEMENT OF RESULTS -- II. THE CLASSES OF FUNCTIONS -- III. THE PROOF OF THEOREM 1 -- IV. THE PROOF OF THEOREM 2 -- V. THE PROOF OF THEOREM 3 -- VI. INTERPOLATIONS -- VII. CONTINUATION -- VIII. THE APPROXIMATE SOLUTION OF FREDHOLM INTEGRAL EQUATION OF THE SECOND TYPE -- IX. THE PROOF OF THEOREM 4 -- REFERENCES.

ON UNIFORM DISTRIBUTION AND NUMERICAL ANALYSIS (III) -- I. INTRODUCTION -- II. THE PROOF OF THEOREM 1 -- III. THE ESTIMATION OF n -- IV. THE ESTIMATION OF r -- V. IRREDUCIBILITY OF POLYNOMIALS -- VI. RATIONAL APPROXIMATIONS OF n AND r -- VII. EXAMPLES -- REFERENCES -- A NOTE ON UNIFORM DISTRIBUTION AND EXPERIMENTAL DESIGN -- I. INTRODUCTION -- II. METHODS -- III. UNIFORM DISTRIBUTION -- IV. TABLES -- REFERENCES -- On Diophantine Approximation and Approximate Analysis (I)* -- 1. INTRODUCTION -- 2. THE PROOF OF THEOREM 1. -- 3. THE CONSEQUENCE OF THEOREM 1. -- 4. THE PROOF OF THEOREM 2. -- REFERENCES -- ON DIOPHANTINE APPROXIMATION AND APPROXIMATE ANALYSIS (II)* -- 1. INTRODUCTION -- 2. THE PROOF OF THEOREM 1 -- 3. THE CASE = 2. -- 4. THE NUMERICAL INTEGRATION OVER Q (C). -- 5. THE ERROR TERM OF QUADRATURE FORMULA -- 6. SEVERAL CONJECTURES -- REFERENCES -- NUMBER THEORETIC METHOD IN APPLIED STATISTICS*** -- 1. Introduction -- 2. Numerical Integration -- 3. Some Applications -- 4. Optimization -- References -- NUMBER THEORETIC METHODS IN APPLIED STATISTICS (II) -- 1. Introduction -- 2. Uniform Design -- 3. Experiments with Mixtures -- 4. Geometric Probability and Simulation -- References -- Uniform design of experiments with mixtures* -- 1 Inverse transform method -- 2 The domain S(a, b) -- 3 Volumes of S*(d) and S(d) -- 4 The evaluation of -- 5 Examples -- References -- LIST OF PUBLICATIONS BY WANG YUAN -- I. Articles -- II. Books and Monographs.