I consider a physical system described by a continuous field theory and enclosed in a large but finite cubical box with periodic boundary conditions. The system is assumed to undergo a continuous phase transition at some critical point. The 4 M theory that is a continuous version of the Ising model is such a system but there are many other examples corresponding to higher spin, higher symmetry ...

We study solution sensitivity for nonlinear programs (NLPs) whose structures are induced by graphs. These NLPs arise in many applications such as dynamic optimization, stochastic optimization with partial differential equations, and network optimization. show that a given pair of nodes, the primal-dual at one node against data perturbation other decays exponentially respect to distance between ...

I consider a physical system described by a continuous field theory and enclosed in a large but finite cubical box with periodic boundary conditions. The system is assumed to undergo a continuous phase transition at some critical point. The 4 M theory that is a continuous version of the Ising model is such a system but there are many other examples corresponding to higher spin, higher symmetry ...