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Since the publication of < B> Random Matrices< /B> (Academic Press, 1967) so many new results have emerged both in theory and in applications, that this edition is almost completely revised to reflect the developments. For example, the theory of matrices with quaternion elements was developed to compute certain multiple integrals, and the inverse scattering theory was used to derive asymptotic results. The discovery of Selberg's 1944 paper on a multiple integral also gave rise to hundreds of recent publications. This book presents a coherent and detailed analytical treatment of random matrices, leading in particular to the calculation of n-point correlations, of spacing probabilities, and of a number of statistical quantities. The results are used in describing the statistical properties of nuclear excitations, the energies of chaotic systems, the ultrasonic frequencies of structural materials, the zeros of the Riemann zeta function, and in general the characteristic energies of any sufficiently complicated system. Of special interest to physicists and mathematicians, the book is self-contained and the reader need know mathematics only at the undergraduate level. Key Features* The three Gaussian ensembles, unitary, orthogonal, and symplectic; their n-point correlations and spacing probabilities* The three circular ensembles: unitary, orthogonal, and symplectic; their equivalence to the Gaussian* Matrices with quaternion elements* Integration over alternate and mixed variables* Fredholm determinants and inverse scattering theory* A Brownian motion model of the matrices* Computation of the mean and of the variance of a number of statistical quantities* Selberg's integral and itsconsequences