The <Emphasis Type="Italic">n</Emphasis>-Body Problem in Spaces of Constant Curvature. Part I: Relative Equilibria

We extend the Newtonian n-body problem of celestial mechanics to spaces of curvature κ=constant and provide a unified framework for studying the motion. In the 2-dimensional case, we prove the existence of several classes of relative equilibria, including the Lagrangian and Eulerian solutions for any κ≠0 and the hyperbolic rotations for κ<0. These results lead to a new way of understanding the geometry of the physical space. In the end we prove Saari’s conjecture when the bodies are on a geodesic that rotates elliptically or hyperbolically.     

Twistor and polytope interpretations for subleading color one-loop amplitudes

We use the relation of the one-loop subleading-color amplitudes to the one-loop n-point leading color amplitudes in N=4 SYM, to derive a polytope interpretation for the former in the MHV case, and a representation in momentum twistor space for the general NkMHV case. These techniques are explored in detail for the 5-point and 6-point amplitudes. We briefly discuss the implications for IR divergences.     

New Zonal,Spectral Solutions for the Navier‐Stokes Layer and Their Applications

New zonal, spectral forms for the axial, lateral and vertical velocity's components, density function and absolute temperature inside of compressible three‐dimensional Navier‐Stokes layer (NSL) over flattened, flying configurations (FC), are here proposed. The inviscid flow over the FC, obtained after the solidification of the NSL, is here used as outer flow. If the spectral forms of the velocity's components are introduced in the partial differential equations of the NSL and the collocation method is used, the spectral coefficients are obtained by the iterative solving of an equivalent quadratical algebraic system with slightly variable coefficients.