Extensions of Lie Algebras and Hamiltonian Systems

Abstract

The Monte Carlo method was used to study the dependence of the calibration coefficient on the tube voltage, beam aperture and reference plane in simplified over-couch geometries modelling VacuTec's type 70157 KAP-meter both with and without an additional filter. The MCNP5 code was used to calculate (i) energy imparted to air cavities of the KAP-meter and (ii) spatial distribution of air collision kerma at entrance and exit planes of the KAP-meter and at a plane close to the patient. From these data, the air kerma area product and calibration coefficient were calculated and their dependence on the tube voltage and beam aperture was analysed. It was found that the variation of the calibration coefficient as a function of tube voltage was up to 40% when the additional filter was used. The additional filter placed closely in front of the KAP-meter decreased the calibration coefficient for the patient plane by about 10% compared to the ideal additional filter. The effect of the beam aperture was small at the patient plane and negligible for the exit plane.

The Algebra of Hamiltonian Systems

“…If 9 is one of the Lie algebras listed below, there exist "rigid body" Operators for .Q(9) (see Trofimov [1983a], [1985]). …”

“…Theorem 2.6 (Trofimov [1983a], [1985]). Let 9 be a complex semisimple Lie algebra, or a compact real form, or a compact normal subalgebra of a complex semisimple Lie algebra.…”

Double Lie algebras, semidirect product, and integrable systems

“…Also, at the Lie algebra level, in ref. [14] the complete integrability of the Euler equations on a semidirect product of a semisimple Lie algebra with its adjoint representation was proven. In ref.…”