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We show that every Lie algebroid A over a manifold P has a natural representation on the line bundle Q A = ∧ top A ⊗ ∧ top T * P . The line bundle Q A may be viewed as the Lie algebroid analog of the orientation bundle in topology, and sections of Q A may be viewed as transverse measures to A. As a consequence, there is a well-defined class in the first Lie algebroid cohomology H 1 (A) called the modular class of the Lie algebroid A. This is the same as the one introduced earlier by Weinstein using the Poisson structure on A * . We show that there is a natural pairing between the Lie algebroid cohomology spaces of A with trivial coefficients and with coefficients in Q A . This generalizes the pairing used in the Poincare duality of finite-dimensional Lie algebra cohomology. The case of holomorphic Lie algebroids is also discussed, where the existence of the modular class is connected with the Chern class of the line bundle Q A .

We introduce the notion of Hopf algebroids, in which neither the total algebras nor the base algebras are required to be commutative. We give a class of Hopf algebroids associated to module algebras of the Drinfeld doubles of Hopf algebras when the R-matrices act properly. When this construction is applied to quantum groups, we get examples of quantum groupoids, which have Poisson groupoids as their semi-classical limits. The example of quantum sl(2) is worked out in details.

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It is a widely accepted axiom that localized concentration of mechanical stress and large flexural deformation is closely related to the calcification and tissue degeneration in bioprosthetic heart valves (BHV). In order to investigate the complex BHV deformations and stress distributions throughout the cardiac cycle, it is necessary to perform an accurate dynamic analysis with a morphologically and physiologically realistic material specification for the leaflets. We have developed a stress resultant shell model for BHV leaflets incorporating a Fung-elastic constitutive model for in-plane and bending responses separately. Validation studies were performed by comparing the finite element predicted displacement and strain measures with the experimentally measured data under physiological pressure loads. Computed regions of stress concentration and large flexural deformation during the opening and closing phases of the cardiac cycle correlated with previously reported regions of calcification and/or mechanical damage on BHV leaflets. It is expected that the developed experimental and computational methodology will aid in the understanding of the complex dynamic behavior of native and bioprosthetic valves and in the development of tissue engineered valve substitutes.

We present a method for predicting the wall stress in a class of cerebral aneurysms. The method hinges on an inverse formulation of the elastostatic equilibrium problem; it takes as the input a deformed configuration and the corresponding pressure, and predicts the wall stress in the given deformed state. For a membrane structure, the inverse formulation possesses a remarkable feature, that is, it can practically determine the wall tension without accurate knowledge of the wall elastic properties. In this paper, we present a finite element formulation for the inverse membrane problem and perform material sensitivity studies on idealized lesions and an image-based cerebral aneurysm model.

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