Random matrices / Madan Lal Mehta.

By: Mehta, M. L
Material type: TextTextSeries: Pure and applied mathematics (Academic Press): 142.Publisher: Amsterdam ; San Diego, CA : Academic Press, 2004Edition: 3rd edDescription: 1 online resource (xviii, 688 pages : illustrationsContent type: text Media type: computer Carrier type: online resourceISBN: 008047411X; 9780080474113; 9780120884094; 0120884097; 9780125660501; 0125660502Subject(s): Random matrices | Matrices aléatoires | MATHEMATICS -- Matrices | Random matrices | Mecânica estatísticaGenre/Form: Electronic books. | Electronic books. Additional physical formats: Print version:: Random matrices.DDC classification: 512.9434 LOC classification: QA188Online resources: Click here to access online
Contents:
Cover -- Contents -- Preface to the Third Edition -- Preface to the Second Edition -- Preface to the First Edition -- Introduction -- Random Matrices in Nuclear Physics -- Random Matrices in Other Branches of Knowledge -- A Summary of Statistical Facts about Nuclear Energy Levels -- Level Density -- Distribution of Neutron Widths -- Radiation and Fission Widths -- Level Spacings -- Definition of a Suitable Function for the Study of Level Correlations -- Wigner Surmise -- Electromagnetic Properties of Small Metallic Particles -- Analysis of Experimental Nuclear Levels -- The Zeros of The Riemann Zeta Function -- Things Worth Consideration, But Not Treated in This Book -- Gaussian Ensembles. The Joint Probability Density Function for the Matrix Elements -- Preliminaries -- Time-Reversal Invariance -- Gaussian Orthogonal Ensemble -- Gaussian Symplectic Ensemble -- Gaussian Unitary Ensemble -- Joint Probability Density Function for the Matrix Elements -- Gaussian Ensemble of Hermitian Matrices With Unequal Real and Imaginary Parts -- Anti-Symmetric Hermitian Matrices -- Summary of Chapter 2 -- Gaussian Ensembles. The Joint Probability Density Function for the Eigenvalues -- Orthogonal Ensemble -- Symplectic Ensemble -- Unitary Ensemble -- Ensemble of Anti-Symmetric Hermitian Matrices -- Gaussian Ensemble of Hermitian Matrices With Unequal Real and Imaginary Parts -- Random Matrices and Information Theory -- Summary of Chapter 3 -- Gaussian Ensembles. Level Density -- The Partition Function -- The Asymptotic Formula for the Level Density. Gaussian Ensembles -- The Asymptotic Formula for the Level Density. Other Ensembles -- Summary of Chapter 4 -- Orthogonal, Skew-Orthogonal and Bi-Orthogonal Polynomials -- Quaternions, Pfaffians, Determinants -- Average Value of PI N j=1 f (xj); Orthogonal and Skew-Orthogonal Polynomials -- Case beta = 2; Orthogonal Polynomials -- Case beta = 4; Skew-Orthogonal Polynomials of Quaternion Type -- Case beta = 1; Skew-Orthogonal Polynomials of Real Type -- Average Value of Pi j=1N psi(xj, yj); Bi-Orthogonal Polynomials -- Correlation Functions -- Proof of Theorem 5.7.1 -- Case beta = 2 -- Case beta = 4 -- Case beta = 1, Even Number of Variables -- Case beta = 1, Odd Number of Variables -- Spacing Functions -- Determinantal Representations -- Integral Representations -- Properties of the Zeros -- Orthogonal Polynomials and the Riemann-Hilbert Problem -- A Remark (Balian) -- Summary of Chapter 5 -- Gaussian Unitary Ensemble -- Generalities -- About Correlation and Cluster Functions -- About Level-Spacings -- Spacing Distribution -- Correlations and Spacings -- The n-Point Correlation Function -- Level Spacings -- Several Consecutive Spacings -- Some Remarks -- Summary of Chapter 6 -- Gaussian Orthogonal Ensemble -- Generalities -- Correlation and Cluster Functions -- Level Spacings. Integration Over Alternate Variables -- Several Consecutive Spacings: n = 2r -- Several Consecutive Spacings: n = 2r -- 1 -- Case n = 1 -- Case n = 2r -- 1 -- Bounds for the Distribution Function of the Spacings -- Summary of Chapter 7 -- Gaussian Symplectic Ensem.
Summary: This book gives a coherent and detailed description of analytical methods devised to study random matrices. These methods are critical to the understanding of various fields in in mathematics and mathematical physics, such as nuclear excitations, ultrasonic resonances of structural materials, chaotic systems, the zeros of the Riemann and other zeta functions. More generally they apply to the characteristic energies of any sufficiently complicated system and which have found, since the publication of the second edition, many new applications in active research areas such as quantum gravity, traffic and communications networks or stock movement in the financial markets. This revised and enlarged third edition reflects the latest developements in the field and convey a greater experience with results previously formulated. For example, the theory of skew-orthogoanl and bi-orthogonal polynomials, parallel to that of the widely known and used orthogonal polynomials, is explained here for the first time. Presentation of many new results in one place for the first time. First time coverage of skew-orthogonal and bi-orthogonal polynomials and their use in the evaluation of some multiple integrals. Fredholm determinants and Painlev̌ equations. The three Gaussian ensembles (unitary, orthogonal, and symplectic); their n-point correlations, spacing probabilities. Fredholm determinants and inverse scattering theory. Probability densities of random determinants.
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Includes bibliographical references (pages 655-679) and indexes.

Cover -- Contents -- Preface to the Third Edition -- Preface to the Second Edition -- Preface to the First Edition -- Introduction -- Random Matrices in Nuclear Physics -- Random Matrices in Other Branches of Knowledge -- A Summary of Statistical Facts about Nuclear Energy Levels -- Level Density -- Distribution of Neutron Widths -- Radiation and Fission Widths -- Level Spacings -- Definition of a Suitable Function for the Study of Level Correlations -- Wigner Surmise -- Electromagnetic Properties of Small Metallic Particles -- Analysis of Experimental Nuclear Levels -- The Zeros of The Riemann Zeta Function -- Things Worth Consideration, But Not Treated in This Book -- Gaussian Ensembles. The Joint Probability Density Function for the Matrix Elements -- Preliminaries -- Time-Reversal Invariance -- Gaussian Orthogonal Ensemble -- Gaussian Symplectic Ensemble -- Gaussian Unitary Ensemble -- Joint Probability Density Function for the Matrix Elements -- Gaussian Ensemble of Hermitian Matrices With Unequal Real and Imaginary Parts -- Anti-Symmetric Hermitian Matrices -- Summary of Chapter 2 -- Gaussian Ensembles. The Joint Probability Density Function for the Eigenvalues -- Orthogonal Ensemble -- Symplectic Ensemble -- Unitary Ensemble -- Ensemble of Anti-Symmetric Hermitian Matrices -- Gaussian Ensemble of Hermitian Matrices With Unequal Real and Imaginary Parts -- Random Matrices and Information Theory -- Summary of Chapter 3 -- Gaussian Ensembles. Level Density -- The Partition Function -- The Asymptotic Formula for the Level Density. Gaussian Ensembles -- The Asymptotic Formula for the Level Density. Other Ensembles -- Summary of Chapter 4 -- Orthogonal, Skew-Orthogonal and Bi-Orthogonal Polynomials -- Quaternions, Pfaffians, Determinants -- Average Value of PI N j=1 f (xj); Orthogonal and Skew-Orthogonal Polynomials -- Case beta = 2; Orthogonal Polynomials -- Case beta = 4; Skew-Orthogonal Polynomials of Quaternion Type -- Case beta = 1; Skew-Orthogonal Polynomials of Real Type -- Average Value of Pi j=1N psi(xj, yj); Bi-Orthogonal Polynomials -- Correlation Functions -- Proof of Theorem 5.7.1 -- Case beta = 2 -- Case beta = 4 -- Case beta = 1, Even Number of Variables -- Case beta = 1, Odd Number of Variables -- Spacing Functions -- Determinantal Representations -- Integral Representations -- Properties of the Zeros -- Orthogonal Polynomials and the Riemann-Hilbert Problem -- A Remark (Balian) -- Summary of Chapter 5 -- Gaussian Unitary Ensemble -- Generalities -- About Correlation and Cluster Functions -- About Level-Spacings -- Spacing Distribution -- Correlations and Spacings -- The n-Point Correlation Function -- Level Spacings -- Several Consecutive Spacings -- Some Remarks -- Summary of Chapter 6 -- Gaussian Orthogonal Ensemble -- Generalities -- Correlation and Cluster Functions -- Level Spacings. Integration Over Alternate Variables -- Several Consecutive Spacings: n = 2r -- Several Consecutive Spacings: n = 2r -- 1 -- Case n = 1 -- Case n = 2r -- 1 -- Bounds for the Distribution Function of the Spacings -- Summary of Chapter 7 -- Gaussian Symplectic Ensem.

This book gives a coherent and detailed description of analytical methods devised to study random matrices. These methods are critical to the understanding of various fields in in mathematics and mathematical physics, such as nuclear excitations, ultrasonic resonances of structural materials, chaotic systems, the zeros of the Riemann and other zeta functions. More generally they apply to the characteristic energies of any sufficiently complicated system and which have found, since the publication of the second edition, many new applications in active research areas such as quantum gravity, traffic and communications networks or stock movement in the financial markets. This revised and enlarged third edition reflects the latest developements in the field and convey a greater experience with results previously formulated. For example, the theory of skew-orthogoanl and bi-orthogonal polynomials, parallel to that of the widely known and used orthogonal polynomials, is explained here for the first time. Presentation of many new results in one place for the first time. First time coverage of skew-orthogonal and bi-orthogonal polynomials and their use in the evaluation of some multiple integrals. Fredholm determinants and Painlev̌ equations. The three Gaussian ensembles (unitary, orthogonal, and symplectic); their n-point correlations, spacing probabilities. Fredholm determinants and inverse scattering theory. Probability densities of random determinants.

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Random Matrices, Volume 142, Third Edition (Pure and Applied Mathematics). by Mehta, Madan Lal. ©2004

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