We characterize the long time behaviour of a discrete-in-time approximation volume preserving fractional mean curvature flow. In particular, we prove that discrete flow starting from any bounded set finite perimeter converges exponentially fast to single ball if dimension N≤7 and esponent s≈1, or for s∈(0,1) when N=2. As an intermediate result establish quantitative Alexandrov type estimate nor...

The positive mass theorem of general relativity to be addressed here states that the total mass of an isolated gravitational system (viewed from spatial infinity, known as the ADM mass) must be non-negative if its local mass density is non-negative everywhere, and in fact the mass is strictly positive unless the space-time is flat. In this essay we are only concerned with the first part of this...

It is known that humans tend to misjudge the kinematics of a target rolling down an inclined plane. Because visuomotor responses are often more accurate and less prone to perceptual illusions than cognitive judgments, we asked the question of how rolling motion is extrapolated for manual interception or drawing tasks. In three experiments a ball rolled down an incline with kinematics that diffe...