in this paper, the lorentzian version of beltrami-euler formula is investigated in 1n . initially,the first fundamental form and the metric coefficients of generalized timelike ruled surface are calculated and by the help of the christoffel symbols, riemann-christoffel curvatures are obtained. thus, the curvatures of spacelike and timelike tangential sections of generalized timelike ruled surf...

We introduce a new sketch based interface for generating animations. Unlike traditional digital tools, ours is parameterized entirely by a neural network with no preprogrammed rules or knowledge representations. The capability of our sketching tool to support visual exploration and communication is demonstrated within the context of facial images, though our framework is domain independent. Our...

We prove that the cycle space of every planar finitely separable 3-connected graph G is generated by some Aut(G)-invariant nested set of cycles. We also discuss the situation in the case of smaller connectivity.