On the discretization of degenerate sweeping processes.
Given a vector field X on an algebraic variety V over ℂ, I compare the following two objects: (i) the envelope of X, the smallest algebraic pseudogroup over V whose Lie algebra contains X, and (ii) the Galois pseudogroup of the foliation defined by the vector field X + d/dt (restricted to one fibre t = constant). I show that either they are equal, or the second has codimension one in the first.
The level set method has become widely used in shape optimization where it allows a popular implementation of the steepest descent method. Once coupled with a ersatz material approximation [Allaire et al., J. Comput. Phys.194 (2004) 363–393], a single mesh is only used leading to very efficient and cheap numerical schemes in optimization of structures. However, it has some limitations and cannot be applied in every situation. This work aims at exploring such a limitation. We estimate the systematic...
The paper investigates the singular initial problem[4pt] [4pt] on the half-line . Here , where , and are zeros of , which is locally Lipschitz continuous on . Function is continuous on , has a positive continuous derivative on and . Function is continuous on and positive on . For specific values we prove the existence and uniqueness of damped solutions of this problem. With additional conditions for , and it is shown that the problem has for each specified a unique...