The usual Weyl calculus is intimately associated with the choice of the standard symplectic structure on [Formula: see text]. In this paper we will show that the replacement of this structure by an arbitrary symplectic structure leads to a pseudo-differential calculus of operators acting on functions or distributions defined, not on [Formula: see text] but rather on [Formula: see text]. These operators are intertwined with the standard Weyl pseudo-differential operators using an infinite family of partial isometries of [Formula: see text] indexed by [Formula: see text]. This allows us to obtain spectral and regularity results for our operators using Shubin's symbol classes and Feichtinger's modulation spaces.

We study the existence, uniqueness, and asymptotic stability of time periodic traveling wave solutions to a periodic diffusive Lotka-Volterra competition system. Under certain conditions, we prove that there exists a maximal wave speed c(*) such that for each wave speed c ≤ c(*), there is a time periodic traveling wave connecting two semi-trivial periodic solutions of the corresponding kinetic system. It is shown that such a traveling wave is unique modulo translation and is monotone with respect to its co-moving frame coordinate. We also show that the traveling wave solutions with wave speed c < c(*) are asymptotically stable in certain sense. In addition, we establish the nonexistence of time periodic traveling waves for nonzero speed c > c(*).

A topological derivative is defined, which is caused by kinking of a crack, thus, representing the topological change. Using variational methods, the anti-plane model of a solid subject to a non-penetration condition imposed at the kinked crack is considered. The objective function of the potential energy is expanded with respect to the diminishing branch of the incipient crack. The respective sensitivity analysis is provided by a Saint-Venant principle and a local decomposition of the solution of the variational problem in the Fourier series.