We extend the "bundle constructions" of calibrated submanifolds, due to Harvey-Lawson in the special Lagrangian case, and to Ionel-Karigiannis-Min-Oo in the cases of exceptional calibrations, by "twisting" the bundles by a special (harmonic, holomorphic, or parallel) section of a complementary bundle. The existence of such deformations shows that the moduli space of calibrated deformations of these "calibrated subbundles" includes deformations which destroy the linear structure of the fibre.

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h i g h l i g h t s• Differentially heated flow of a thin fluid layer from a rotating sphere has been investigated.• A numerical solution procedure for solving the steady and unsteady equations has been proposed. • An approximate analytical solution has been derived. • A linear stability analysis has estimated a theoretical value for the onset of instability. • Good agreement was found between numerical, analytical and theoretical results. a b s t r a c tWe present results on the flow of a thin fluid layer over a rotating sphere having a surface temperature that varies in the latitudinal direction. The fluid is taken to be viscous, incompressible and Newtonian while the flow is assumed to possess both azimuthal and equatorial symmetry. The governing Navier-Stokes and energy equations are formulated in terms of a stream function and vorticity and are solved subject to no-slip boundary conditions. An approximate analytical solution for the steady-state flow has been derived and is compared with numerical solutions to the steady and limiting unsteady equations. For small Rayleigh numbers these solutions are found to be in close agreement. However, as the Rayleigh number is increased noticeable differences occur. A numerical solution procedure is presented and a linear stability analysis has been conducted to predict the onset of instability. Good agreement between the theoretical predictions and the observed numerical simulations was found.

In this paper, we study the error of a second order finite difference scheme for the one-dimensional convection-diffusion equation. We consider non-smooth initial conditions commonly encountered in financial pricing applications. For these initial conditions, we establish the explicit expression of the quantization error, which is loosely defined as the error of the numerical solution due to the placement of the point of non-smoothness on the numerical grid. Based on our analysis, we study the issue of optimal placement of such non-smoothness points on the grid, and the effect of smoothing operators on quantization errors.

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