In this paper we introduce a reference along a null curve in an n-dimensional Lorentzian space with the minimum number of curvatures. That reference generalizes the reference of Bonnor for null curves in Minkowski space–time and it is called the Cartan frame of the curve. The associated curvature functions are called the Cartan curvatures of the curve. We characterize the null helices (that is, null curves with constant Cartan curvatures) in n-dimensional Lorentzian space forms and we obtain a complete classification of them in low dimensions.

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The simplest (2+1)-dimensional mechanical systems associated with light-like curves, already studied by Nersessian and Ramos, are reconsidered. The action is linear in the curvature of the particle path and the moduli spaces of solutions are completely exhibited in 3-dimensional Minkowski background, even when the action is not proportional to the pseudo-arc length of the trajectory.Comment: Corrected typos and english flaws. Final version accepted in Phys. Lett.

A two-dimensional time-dependent computational fluid dynamics model of the Circle of Willis has been developed. To simulate, not only the peripheral resistance of the cerebrovascular tree but also its auto-regulation function, a new "active" boundary condition has been defined and developed using control theory to provide a model of the feedback mechanism. The model was then used to simulate different common abnormalities of the Circle of Willis while a pressure drop, simulating a rapid compression of the right internal carotid artery, was imposed. Test results using a simple tube compared excellently with experiment. The total time-dependent flux for each efferent artery was tabulated and showed the important relationship between geometrical variations in the Circle of Willis and the auto-regulation of blood flow by vascular vaso-dilation and contraction. From this study, it was found that the worst case seemed to be that of a missing or dysfunctional right A1 segment of the anterior cerebral artery. The use of valid physiological models of the peripheral resistance allows for more realistic models of the blood flow in the Circle whilst allowing an easy extension to 3D patient specific simulations.

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