The momentum ray transform I k integrates a rank m symmetric tensor field f over lines with the weight t k :In particular, the ray transform I = I 0 was studied by several authors since it had many tomographic applications. We present an algorithm for recovering f from the data (I 0 f, I 1 f, . . . , I m f ). In the cases of m = 1 and m = 2, we derive the Reshetnyak formula that expresses f H s t (R n ) through some norm of (I 0 f, I 1 f, . . . , I m f ). The H s t -norm is a modification of the Sobolev norm weighted differently at high and low frequencies. Using the Reshetnyak formula, we obtain a stability estimate.

The momentum ray transform I k integrates a rank m symmetric tensor field f over lines of R n with the weight t k :We give the range characterization for the operator f → (I 0 f, I 1 f, . . . , I m f ) on the Schwartz space of rank m smooth fast decaying tensor fields. In dimensions n ≥ 3, the range is characterized by certain differential equations of order 2(m + 1) which generalize the classical John equations. In the two-dimensional case, the range is characterized by certain integral conditions which generalize the classical Gelfand -Helgason -Ludwig conditions.

In this article we study the inverse problem of determining the convection term and the time-dependent density coefficient appearing in the convection-diffusion equation. We prove the unique determination of these coefficients from the knowledge of solution measured on a subset of the boundary.

We study an inverse problem involving the unique recovery of several lower order anisotropic tensor perturbations of a polyharmonic operator in a bounded domain from the knowledge of the Dirichlet to Neumann map on a part of boundary. The uniqueness proof relies on the inversion of generalized momentum ray transforms (MRT) for symmetric tensor fields, which we introduce for the first time to study Calderón-type inverse problems. We construct suitable complex geometric optics (CGO) solutions for the polyharmonic operators that reduces the inverse problem to uniqueness results for a generalized MRT. The uniqueness result and the inversion formula we prove for generalized MRT could be of independent interest and we expect it to be applicable to other inverse problems for higher order operators involving tensor perturbations.

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