The creation of macroscopic hierarchy (such as fluids or plasmas) is studied in the context of noncanonical Hamiltonian systems. We study the origin of noncanonicality in two perspectives; (1) the reductions from canonical systems, (2) the Lie-Poisson algebras and their deformations, and show how the dynamics of macro-systems can be different from those of microscopic canonical systems.

Difference-of-convex (DC) optimization is an elaborate methodology for minimizing a DC function, which is a class of functions representable as the difference of two convex functions. Since many important functions arising from optimization are known to be DC functions, DC optimization has been extensively studied from the aspects of theory and applications. Currently, DC optimization is widely acknowledged as one of the most effective approaches for dealing with the nonconvexity in optimization. In this paper, we introduce some basic theory and several applications of DC optimization. Particularly, we introduce the DC algorithm and some variants for solving a DC optimization problem together with their convergence results. Finally, we present an application to sparse optimization.