SUMMARYDue to its elasticity, the human brain material can support shear (equivoluminal) waves. Earlier attempts to explain certain brain injuries via arguments of the classical theory of viscoelasticity exploited the Voigt model-a linear system of di erential equations where the motion of the brain tissue depends merely on the balance between viscous and elastic forces. Although Voigt model solutions illustrate the role of the viscoelastic mechanics in brain injuries, they have limited use for modelling realistic cases which, for example, evince strongly localized displacements of the brain tissue. We have extended the Voigt model to a non-linear viscoelastic uid model, thereby dispensing with simplifying assumptions of vanishing advective transport. The resulting non-Newtonian uid model admits non-linear phenomena such as steepening of the wave fronts as well as wave overturning and their subsequent turbulent breaking. The posed equations are solved numerically, and the solution procedure are validated against small-perturbation linear theory and closed-form Voigt-model solutions available in the literature. Our non-linear numerical results suggest existence of a 'brain turbulence' phenomenon. They are in qualitative agreement with the results of medical research, especially, with regard to the di use axonal injuries which are observed to occur in a highly localized manner near the border between the gray and the white matter.

ABSTRACT. The analogues of Gell-Mann-Okubo and Coleman-Glashow mass formulae are derived for arbitrary reductive Lie algebras.

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We study the asymptotic behavior of integer sequences related to knots that are generated by linear recurrences. We determine which of the 85 such sequences cataloged in Online Encyclopedia of Integer Sequences have ratios of their consecutive terms converging to a limit. We show that all but one of the ratio limits can be expressed by means of the [Formula: see text]-anacci constants with [Formula: see text] equal to 1 or 2. Finally, we demonstrate how the [Formula: see text]-anacci constants are linked to affine geometry.

The vector space of tensor operators transforming according to the adjoint representation of the unitary group is investigated for arbitrary (even infinite) dimension of the group. In particular, a natural basis consisting of generalized number operators is constructed and some properties are studied. The results are applied to the construction of mass operators for arbitrary multiquark systems with arbitrary dimsension of the one-quark flavor space. It is shown, also, how the proved properties of the spectrum simplify the obtaining of certain mass formulas.