Let Π be the projection operator, which maps every polytope to its projection body. It is well known that Π maps the set of polytopes, P, in R into P, that it is a valuation, and that for every P ∈ P, ΠP is affinely associated to P . It is shown that these properties characterize the projection operator Π. This proves a conjecture by Lutwak. Let Kn denote the set of convex bodies (i.e., of comp...

This work presents a data-driven reduced-order modeling framework to accelerate the computations of $N$-body dynamical systems and their pair-wise interactions. The proposed differs from traditional acceleration methods, like Barnes-Hut method, which requires online tree building state space, or fast-multipole rigorous $a$ $priori$ analysis governing kernels building. Our approach combines hier...

Objective: In breast reconstruction, decision of projection of silicone breast implant in tissue expander replacement is difficult because of the need to consider several parameters that cannot be expressed in accurate numerical form. The present study aimed at a quantitative analysis based on decreased projection of the reconstructed side compared with silicone breast implant projection and to...

In this paper, we show that for any dimension d ≥ 3 there exists a body of constant breadth C, such that its projection onto any 2-plane is non-spherical. We call such a body totally non-spherical. The circumradius of the projection of any totally non-spherical body C of constant breadth onto any 2-plane is bigger than the half diameter of C. Showing the existence of such a body extends results...

As Schneider [50] observes, the classical Brunn-Minkowski theory had its origin at the turn of the 19th into the 20th century, when Minkowski joined a method of combining convex bodies (which became known as Minkowski addition) with that of ordinary volume. One of the core concepts that Minkowski introduced within the Brunn-Minkowski theory is that of projection body (precise definitions to fol...

‎the water wave generation by wave paddle and a freely falling rigid body are examined by using an incompressible smoothed particle hydrodynamics (isph)‎. ‎in the current isph method‎, ‎the pressure was evaluated by solving pressure poisson equation using a semi-implicit algorithm based on the projection scheme and the source term of pressure poisson equation contains both of divergence free ve...

Voltage-sensitive dyes NK 2761 and RH 155 were employed (in conjunction with a 12 x 12 photodiode array) to study membrane potential transients in optic lobe neuropils in the eye stalk of the crayfish Procambarus clarkii. By this means we investigated a pathway linking deutocerebral projection neurons, via hemiellipsoid body local interneurons, to an unidentified target (most likely neurons pro...