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1University of Jyväskylä, Finland; 2National Chengchi University, Taiwan
We will explain the relation between momentum ray transform and the fractional Laplacian. As a consequence of unique continuation property of the fractional Laplacian, one can discuss similar properties for momentum ray transform (of integer and fractional order). In addition, we will also explain the relation between the weighted ray transform and the cone transform, which appeared in different imaging approaches (for example, Compton camera).
This talk is prepared based on the work [1].
[1] J. Ilmavirta, P.-Z. Kow, S. K. Sahoo. Unique continuation for the momentum ray transform, arXiv: 2304.00327, 2023.
The linearized Calderon problem for polyharmonic operators
Suman Kumar Sahoo, Mikko Salo
University of Jyvaskyla, Finland
The density of products of solutions to several types of partial differential equations, such as elliptic, parabolic, and hyperbolic equations, plays a significant role in solving various inverse problems, dating back to Calderon's fundamental work. In this talk, we will discuss some density properties of solutions to polyharmonic operators on the space of symmetric tensor fields. These density questions are closely related to the linearized Calderon problem for polyharmonic operators. This is joint work with Mikko Salo.
The star transform and its links to various areas of mathematics
Gaik Ambartsoumian1, Mohammad Javad Latifi2
1University of Texas at Arlington, United States of America; 2Dartmouth College, United States of America
The divergent beam transform maps a function $f$ in $\mathbb{R}^n$ to an $n$-dimensional family of its integrals along rays, emanating from a variable vertex location inside the support of $f$. The star transform is defined as a linear combination of divergent beam transforms with known coefficients. The talk presents some recent results about the properties of the star transform, its inversion, and a few interesting connections to different areas of mathematics.
Linearized Calderon problem for biharmonic operator with partial data
Divyansh Agrawal1, Ravi Shankar Jaiswal1, Suman Kumar Sahoo2
1Tata Institute of Fundamental Research (TIFR), India; 2University of Jyvaskyla, Finland
The product of solutions of the Laplace equation vanishing on an arbitrary closed subset of the boundary is dense in the space of integrable functions ([1]). In this talk, we will discuss a similar problem with the biharmonic operator replacing the Laplace operator. More precisely, we show that the integral identity
$$
\int \left [a \Delta u v+ \sum a^1_i \partial_i u v + a^0 u v \right ]\, \mathrm{d}x = 0
$$
for all biharmonic functions vanishing on an proper closed subset of the boundary implies that $(a, a^1, a^0)$ vanish identically. This work is a collaboration with R. S. Jaiswal and S. K. Sahoo.
[1] D. Ferreira, C. E. Kenig, J. Sjostrand, G. Uhlmann. On the linearized local Calderon problem. Math. Res. Lett. 16: 955-970, 2009.
