In this paper we present a simple method for handling projections on the hemisphere. The proposed method defines a coordinate system on the hemisphere surface that effectively converts the hemisphere to an infinite plane tangent to the top of the hemisphere, which is called the gradient space. Using this coordinate system, projections onto the hemisphere are replaced by simple perspective proje...

where R = D/2 is the radius of the cap, ρ and ν are the density and kinematic viscosity of the liquid, σ is surface tension, r(θ) = R(1 + sθ) and s = r ′′(0)/D is the deviation of the free surface from perfect sphericity r(θ) = R near the stagnation point θ = 0. The bubble nose is more pointed when s < 0 and blunted when s > 0. A more pointed bubble increases the rise velocity; the blunter bubb...

A Examples of TS distributions Example 1: Uniform distribution ⌘ ⇠ UBd(0,d). The uniform distribution satisfies the concentration property with constants c = 1 and c0 = e d by definition. Since the set {⌘|uT⌘ 1}\Bd(0, p d) is an hyper-spherical cap for any direction u of Rd, the the anti-concentration property is satisfied provided that the ratio between the volume of an hyper-spherical cap of ...

Introduction: The crustal dichotomy and the Tharsis volcanic province are dominant features at Martian surface. Both exogenic (giant impact) and endogenic (e.g., degree-1 convection) were proposed to explain the dichotomy. In a recent study Zhong [1] proposed a unified model for the Tharsis rise and the dichotomy. A lithospheric thickness variations of hemispherical pattern is assumed, consiste...