Explicit invariant solutions associated with nonlinear atmospheric flows in a thin rotating spherical shell with and without west-to-east jets perturbations

A class of non-stationary exact solutions of two-dimensional nonlinear Navier–Stokes (NS) equations within a thin rotating spherical shell were found as invariant and approximately invariant solutions. The model is used to describe a simple zonally averaged atmospheric circulation caused by the difference in temperature between the equator and the poles. Coriolis effects are generated by pseudoforces, which support the stable west-to-east flows providing the achievable meteorological flows. The model is superimposed by a stationary latitude dependent flow. Under the assumption of no friction, the perturbed model describes zonal west-to-east flows in the upper atmosphere between the Ferrel and Polar cells. In terms of nonlinear modeling for the NS equations, two small parameters are chosen for the viscosity and the rate of the earth’s rotation and exact solutions in terms of elementary functions are found using approximate symmetry analysis. It is shown that approximately invariant solutions are also valid in the absence of the flow perturbation to a zonally averaged mean flow.

On the proximal Landweber Newton method for a class of nonsmooth convex problems

We consider a class of nonsmooth convex optimization problems where the objective function is a convex differentiable function regularized by the sum of the group reproducing kernel norm and (Formula presented.)-norm of the problem variables. This class of problems has many applications in variable selections such as the group LASSO and sparse group LASSO. In this paper, we propose a proximal Landweber Newton method for this class of convex optimization problems, and carry out the convergence and computational complexity analysis for this method. Theoretical analysis and numerical results show that the proposed algorithm is promising.

The parametric making of geometry: the platonic solids

Geometry is a source of inspiration in the design and making of the manmade world. Computing techniques provide tools to explore complex forms: the research question is how computational tool can be systemised to assist with the translation of geometric concepts into physical objects. The purpose is to describe computational/manufacturing methods for creating digital models and physical objects from regular geometric configurations. The methods are based on parametric design, assisting from ideation to the generation of digital models with material specifications – using the five regular convex polyhedra as a case study. The results are comprised of digital models used for prototyping with 3D printing technologies and hybrid fabrication processes: the products are built geometric shapes ranging from body ornaments to sculptures. These procedures can be extended to generate designs based on irregular geometric shapes. Parametric-based methods are recommended in the digital modeling and fabrication of any geometric form.