Eigenvalues of Casimir operators for gl(m/infinity)

A full set of Casimir operators for the Lie superalgebra gl(m/infinity) is constructed and shown to be well defined in the category O-FS generated by the highest-weight irreducible representations with only a finite number of non-zero weight components. The eigenvalues of these Casimir operators are determined explicitly in terms of the highest weight. Characteristic identities satisfied by certain (infinite) matrices with entries from gl(m/infinity) are also determined.

Level-one highest weight representations of U-q[g1(1 vertical bar 1)] and associated vertex operators

We study the level-one irreducible highest weight representations of U-q[gl(1\1)] and associated q-vertex operators. We obtain the exchange relations satisfied by these vertex operators. The characters and supercharacters associated with these irreducible representations are calculated'. (C) 2000 Published by Elsevier Science B.V. All rights reserved.

Risk management for scuba diving operators on Australia's Great Barrier Reef

Australia's Great Barrier Reef is one of the world's most popular scuba diving destinations. Unfortunately, a series of recent diving injuries and deaths has tarnished the region's safety record. In particular, media attention surrounding the disappearance of American divers Thomas and Eileen Lonergan has focused attention on dive operators' legal responsibilities and the consequences of failing to discharge their duty of care to customers. This paper briefly examines the relevant Australian law for recreational diving operations, and reviews risk management strategies that may reduce or prevent the occurrence of future problems. (C) 2000 Elsevier Science Ltd. All rights reserved.

U-q[(sl(2)over-cap vertical bar 1)] vertex operators, screen currents, and correlation functions at an arbitrary level

Bosonized q-vertex operators related to the four-dimensional evaluation modules of the quantum affine superalgebra U-q[sl((2) over cap\1)] are constructed for arbitrary level k=alpha, where alpha not equal 0,-1 is a complex parameter appearing in the four-dimensional evaluation representations. They are intertwiners among the level-alpha highest weight Fock-Wakimoto modules. Screen currents which commute with the action of U-q[sl((2) over cap/1)] up to total differences are presented. Integral formulas for N-point functions of type I and type II q-vertex operators are proposed. (C) 2000 American Institute of Physics. [S0022-2488(00)00608-3].

Ladder operators for integrable one-dimensional lattice models

A generalised ladder operator is used to construct the conserved operators for any one-dimensional lattice model derived from the Yang-Baxter equation. As an example, the low order conserved operators for the XYh model are calculated explicitly.

Mean anisotropy of homogeneous Gaussian random fields and anisotropic norms of linear translation-invariant operators on multidimensional integer lattices

Sensitivity of output of a linear operator to its input can be quantified in various ways. In Control Theory, the input is usually interpreted as disturbance and the output is to be minimized in some sense. In stochastic worst-case design settings, the disturbance is considered random with imprecisely known probability distribution. The prior set of probability measures can be chosen so as to quantify how far the disturbance deviates from the white-noise hypothesis of Linear Quadratic Gaussian control. Such deviation can be measured by the minimal Kullback-Leibler informational divergence from the Gaussian distributions with zero mean and scalar covariance matrices. The resulting anisotropy functional is defined for finite power random vectors. Originally, anisotropy was introduced for directionally generic random vectors as the relative entropy of the normalized vector with respect to the uniform distribution on the unit sphere. The associated a-anisotropic norm of a matrix is then its maximum root mean square or average energy gain with respect to finite power or directionally generic inputs whose anisotropy is bounded above by a≥0. We give a systematic comparison of the anisotropy functionals and the associated norms. These are considered for unboundedly growing fragments of homogeneous Gaussian random fields on multidimensional integer lattice to yield mean anisotropy. Correspondingly, the anisotropic norms of finite matrices are extended to bounded linear translation invariant operators over such fields.

Non-compact generalized variational inequalities for quasi-monotone and hemi-continuous operators with applications

Some results are obtained for non-compact cases in topological vector spaces for the existence problem of solutions for some set-valued variational inequalities with quasi-monotone and lower hemi-continuous operators, and with quasi-semi-monotone and upper hemi-continuous operators. Some applications are given in non-reflexive Banach spaces for these existence problems of solutions and for perturbation problems for these set-valued variational inequalities with quasi-monotone and quasi-semi-monotone operators.

Non-positivity of Groenewold operators

A central feature in the Hilbert space formulation of classical mechanics is the quantisation of classical Lionville densities, leading to what may be termed Groenewold operators. We investigate the spectra of the Groenewold operators that correspond to Gaussian and to certain uniform Lionville densities. We show that when the classical coordinate-momentum uncertainty product falls below Heisenberg's limit, the Groenewold operators in the Gaussian case develop negative eigenvalues and eigenvalues larger than 1. However, in the uniform case, negative eigenvalues are shown to persist for arbitrarily large values of the classical uncertainty product.