The majority of studies on the evolution and function of feeding in sharks have focused primarily on the movement of cranial components and muscle function, with little integration of tooth properties or function. As teeth are subjected to sometimes extreme loads during feeding, they undergo stress, strain, and potential failure. As attributes related to structural strength such as material properties and overall shape may be subjected to natural selection, both prey processing ability and structural parameters must be considered to understand the evolution of shark teeth. In this study, finite element analysis was used to visualize stress distributions of fossil and extant shark teeth during puncture, unidirectional draw (cutting), and holding. Under the loading and boundary conditions here, which are consistent with bite forces of large sharks, shark teeth are structurally strong. Teeth loaded in puncture have localized stress concentrations at the cusp apex that diminish rapidly away from the apex. When loaded in draw and holding, the majority of the teeth show stress concentrations consistent with well designed cantilever beams. Notches result in stress concentration during draw and may serve as a weak point; however they are functionally important for cutting prey during lateral head shaking behavior. As shark teeth are replaced regularly, it is proposed that the frequency of tooth replacement in sharks is driven by tooth wear, not tooth failure. As the tooth tip and cutting edges are worn, the surface areas of these features increase, decreasing the amount of stress produced by the tooth. While this wear will not affect the general structural strength of the tooth, tooth replacement may also serve to keep ahead of damage caused by fatigue that may lead to eventual tooth failure.

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The Bianchi IX Mixmaster cosmology is treated as a non-linear dynamical system and the entire spectrum of Lyapunov exponents is calculated from a numerical integration of the full Einstein equations. The behaviour of the solutions is studied for times close to the initial and final singularities as well as for times near the point of maximum expansion. It is demonstrated that the methods of numerical integration must be chosen carefully if a vacuum spacetime is to be simulated in a self-consistent manner. In addition it is shown that previous results pointing to chaotic behaviour are due to inaccurate numerical methods and/or the introduction of non-vacuum behaviour.

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