A Patient-Specific Anisotropic Diffusion Model for Brain Tumour Spread.

Gliomas are primary brain tumours arising from the glial cells of the nervous system. The diffuse nature of spread, coupled with proximity to critical brain structures, makes treatment a challenge. Pathological analysis confirms that the extent of glioma spread exceeds the extent of the grossly visible mass, seen on conventional magnetic resonance imaging (MRI) scans.

Robustness of the inverse cascade in two-dimensional turbulence.

We study quasisteady inverse cascades in unbounded and bounded two-dimensional turbulence driven by time-independent injection and dissipated by molecular viscosity. It is shown that an inverse cascade that carries only a fraction r of the energy input to the largest scales requires the enstrophy-range energy spectrum to be steeper than k(-5) (ruling out a direct cascade) unless 1-r<<1. A direct cascade requires the presence of an inverse cascade that carries virtually all energy to the largest scales (1-r<<1).

Energy budgets in Charney-Hasegawa-Mima and surface quasigeostrophic turbulence.

We study energy transfer in unbounded Charney-Hasegawa-Mima and surface quasigeostrophic turbulence. The possible inverse-cascading quantities in these systems are, respectively, I identical with integral ( infinity )(0)k(-2)E(k) dk and J identical with integral ( infinity )(0)k(-1)E(k) dk, where E(k) is the kinetic energy spectrum. The supposed direct-cascading quantities for both surface quasigeostrophic and Navier-Stokes turbulence are shown to be bounded.