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We construct C-algebras for a class of surfaces that are inverse images of certain polynomials of arbitrary degree. By using the directed graph associated with a matrix, the representation theory can be understood in terms of “loop” and “string” representations, which are closely related to the dynamics of an iterated map in the plane. As a particular class of algebras, we introduce the “Hénon algebras,” for which the dynamical map is a generalized Hénon map, and give an example where irreducible representations of all dimensions exist. [ABSTRACT FROM AUTHOR]

A new class of dynamical systems are presented, together with their solutions. Some of these models are isochronous, namely, their generic solutions are all completely periodic with the same period; others are characterized by friction, all solutions vanishing in the remote future; and still others are “asymptotically isochronous,” approaching an isochronous behavior in the remote future. [ABSTRACT FROM AUTHOR]

A basis of N2 projectors, each an N2×N2 matrix with constant elements, is implemented to construct a class of braid matrices R⁁(θ), θ being the spectral parameter. Only odd values of N are considered here. Our ansatz for the projectors Pα appearing in the spectral decomposition of R⁁(θ) leads to exponentials exp(mαθ) as the coefficient of Pα. The sums and differences of such exponentials on the diagonal and the antidiagonal, respectively, provide the (2N2-1) nonzero elements of R⁁(θ). One element at the center is normalized to unity. A class of supplementary constraints imposed by the braid equation leaves 1/2(N+3)(N-1) free parameters mα. The diagonalizer of R⁁(θ) is presented for all N. Transfer matrices t(θ) and L(θ) operators corresponding to our R⁁(θ) are studied. Our diagonalizer signals specific combinations of the components of the operators that lead to a quadratic algebra of N2 constant N×N matrices. The θ dependence factors out for such combinations. R⁁(θ) is developed in a power series in θ. The basic difference arising for even dimensions is made explicit. Some special features of our R⁁(θ) are discussed in a concluding section. [ABSTRACT FROM AUTHOR]

As an extension of the theory of Dyson’s Brownian motion models for the standard Gaussian random-matrix ensembles, we report a systematic study of Hermitian matrix-valued processes and their eigenvalue processes associated with the chiral and nonstandard random-matrix ensembles. In addition to the noncolliding Brownian motions, we introduce a one-parameter family of temporally homogeneous noncolliding systems of the Bessel processes and a two-parameter family of temporally inhomogeneous noncolliding systems of Yor’s generalized meanders and show that all of the ten classes of eigenvalue statistics in the Altland–Zirnbauer classification are realized as particle distributions in the special cases of these diffusion particle systems. As a corollary of each equivalence in distribution of a temporally inhomogeneous eigenvalue process and a noncolliding diffusion process, a stochastic-calculus proof of a version of the Harish–Chandra (Itzykson–Zuber) formula of integral over unitary group is established. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Fisher–Hartwig asymptotics refers to the large n form of a class of Toeplitz determinants with singular generating functions. This class of Toeplitz determinants occurs in the study of the spin–spin correlations for the two-dimensional Ising model, and the ground state density matrix of the impenetrable Bose gas, amongst other problems in mathematical physics. We give a new application of the original Fisher–Hartwig formula to the asymptotic decay of the Ising correlations above Tc, while the study of the Bose gas density matrix leads us to generalize the Fisher–Hartwig formula to the asymptotic form of random matrix averages over the classical groups and the Gaussian and Laguerre unitary matrix ensembles. Another viewpoint of our generalizations is that they extend to Hankel determinants the Fisher–Hartwig asymptotic form known for Toeplitz determinants.© 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]