No abstract uploaded!

In this paper, we firstly re-present the results of Bondar et al. (2013). Then, following the approach taken in it, we use the method of Operational Dynamic Modeling (ODM) to deal with the situation where thebackground mechanics is varying between quantum and classical mechanics.

It is well-known in the combustion community that curvature effect in general slows down ame propagation speeds because it smooths out wrinkled ames. However, such a folklore has never been justied rigorously. In this paper, as the rst theoretical result in this direction, we prove that the turbulent ame speed (an eective burning velocity) is decreasing with respect to the curvature diusivity (Markstein number) for shear flows in the well-known G-equation model. Our proof involves several novel and rather sophisticated inequalities arising from the nonlinear structure of the equation. On a related fundamental issue, we solve the selection problem of weak solutions or nd the "physical fluctuations" when the Markstein number goes to zero and solutions approach those of the inviscid G-equation model. The limiting solution is given by a closed form analytical formula.

We study a variational problem whose critical point determines the Reeb vector field for a SasakiEinstein manifold. This extends our previous work on Sasakian geometry by lifting the condition that the manifolds are toric. We show that the EinsteinHilbert action, restricted to a space of Sasakian metrics on a link <i>L</i> in a CalabiYau cone <i>X</i>, is the volume functional, which in fact is a function on the space of Reeb vector fields. We relate this function both to the DuistermaatHeckman formula and also to a limit of a certain equivariant index on <i>X</i> that counts holomorphic functions. Both formulae may be evaluated by localisation. This leads to a general formula for the volume function in terms of topological fixed point data. As a result we prove that the volume of a SasakiEinstein manifold, relative to that of the round sphere, is always an algebraic number. In complex dimension <i>n</i>=3 these results provide, via AdS

No abstract uploaded!