Front cover image for Random Matrices

Random Matrices

Of Chapter 4Chapter 5. Orthogonal, Skew-Orthogonal and Bi-Orthogonal Polynomials; 5.1. Quaternions, Pfaffians, Determinants; 5.2. Average Value of N j=1 f (xj); Orthogonal and Skew-Orthogonal Polynomials; 5.3. Case Ý = 2; Orthogonal Polynomials; 5.4. Case Ý = 4; Skew-Orthogonal Polynomials of Quaternion Type; 5.5. Case Ý = 1; Skew-Orthogonal Polynomials of Real Type; 5.6. Average Value of Nj=1 |(xj, yj); Bi-Orthogonal Polynomials; 5.7. Correlation Functions; 5.8. Proof of Theorem 5.7.1; 5.9. Spacing Functions; 5.10. Determinantal Representations
eBook, English, 2014
Elsevier Science, Amsterdam, 2014
1 online resource (707 pages)
9780080474113, 008047411X
1059026506
Front Cover; Random Matrices; Copyright Page; Contents; Preface to the Third Edition; Preface to the Second Edition; Preface to the First Edition; Chapter 1. Introduction; 1.1. Random Matrices in Nuclear Physics; 1.2. Random Matrices in Other Branches of Knowledge; 1.3. A Summary of Statistical Facts about Nuclear Energy Levels; 1.4. Definition of a Suitable Function for the Study of Level Correlations; 1.5. Wigner Surmise; 1.6. Electromagnetic Properties of Small Metallic Particles; 1.7. Analysis of Experimental Nuclear Levels; 1.8. The Zeros of the Riemann Zeta Function. 1.9. Things Worth Consideration, But Not Treated in This BookChapter 2. Gaussian Ensembles. The Joint Probability Density Function for the Matrix Elements; 2.1. Preliminaries; 2.2. Time-Reversal Invariance; 2.3. Gaussian Orthogonal Ensemble; 2.4. Gaussian Symplectic Ensemble; 2.5. Gaussian Unitary Ensemble; 2.6. Joint Probability Density Function for the Matrix Elements; 2.7. Gaussian Ensemble of Hermitian Matrices With Unequal Real and Imaginary Parts; 2.8. Anti-Symmetric Hermitian Matrices; Summary of Chapter 2. Chapter 3. Gaussian Ensembles. The Joint Probability Density Function for the Eigenvalues3.1. Orthogonal Ensemble; 3.2. Symplectic Ensemble; 3.3. Unitary Ensemble; 3.4. Ensemble of Anti-Symmetric Hermitian Matrices; 3.5. Gaussian Ensemble of Hermitian Matrices With Unequal Real and Imaginary Parts; 3.6. Random Matrices and Information Theory; Summary of Chapter 3; Chapter 4. Gaussian Ensembles Level Density; 4.1. The Partition Function; 4.2. The Asymptotic Formula for the Level Density. Gaussian Ensembles; 4.3. The Asymptotic Formula for the Level Density. Other Ensembles. 5.11. Integral Representations5.12. Properties of the Zeros; 5.13. Orthogonal Polynomials and the Riemann-Hilbert Problem; 5.14. A Remark (Balian); Summary of Chapter 5; Chapter 6. Gaussian Unitary Ensemble; 6.1. Generalities; 6.2. The n-Point Correlation Function; 6.3. Level Spacings; 6.4. Several Consecutive Spacings; 6.5. Some Remarks; Summary of Chapter 6; Chapter 7. Gaussian Orthogonal Ensemble; 7.1. Generalities; 7.2. Correlation and Cluster Functions; 7.3. Level Spacings. Integration Over Alternate Variables; 7.4. Several Consecutive Spacings: n = 2r. 7.5. Several Consecutive Spacings: n = 2r
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