Invertibility and stability for a generic class of radon transforms with application to dynamic operators

Let X be an open subset of {\mathbb{R}^{2}} . We study the dynamic operator, {\mathcal{A}} , integrating over a family of level curves in X when the object changes between the measurement. We use analytic microlocal analysis to determine which singularities can be recovered by the data-set. Our results show that not all singularities can be recovered as the object moves with a speed lower than the X-ray source. We establish stability estimates and prove that the injectivity and stability are of a generic set if the dynamic operator satisfies the visibility, no conjugate points, and local Bolker conditions. We also show this results can be implemented to fan beam geometry.

On extensions of singular fourth order dynamic operators on time scales

In this work, we consider a singular fourth order dynamic operator on time scales. We construct a space of boundary values. Later, we give a description of all maximal dissipative, self-adjoint and other extensions of singular fourth order differential operators on unbounded time scales.