## Journals

- Advances in Mathematics of Communications
- Big Data & Information Analytics
- Communications on Pure & Applied Analysis
- Discrete & Continuous Dynamical Systems - A
- Discrete & Continuous Dynamical Systems - B
- Discrete & Continuous Dynamical Systems - S
- Evolution Equations & Control Theory
- Foundations of Data Science
- Inverse Problems & Imaging
- Journal of Computational Dynamics
- Journal of Dynamics & Games
- Journal of Geometric Mechanics
- Journal of Industrial & Management Optimization
- Journal of Modern Dynamics
- Kinetic & Related Models
- Mathematical Biosciences & Engineering
- Mathematical Control & Related Fields
- Mathematical Foundations of Computing
- Networks & Heterogeneous Media
- Numerical Algebra, Control & Optimization
- AIMS Mathematics
- Conference Publications
- Electronic Research Announcements
- Mathematics in Engineering

### Open Access Journals

As organizers of this conference and close collaborators of Professors Colton and Kress, we are very honored to have had the opportunity to facilitate this special scientific and social event. It was a particular occasion that gathered together long term colleagues, collaborators, former students and friends of Professors Colton and Kress. And now it gives us particular pleasure to be guest editors of this special issue of Inverse Problems and Imaging which is a collection of original research papers in the area of scattering theory and inverse problems. Much of the work presented here has been directly or indirectly influenced by these two scientists, offering the reader a glimpse of their significant impact in this research area.

We would like to thank all of those who have contributed a paper for this special issue. A special thanks goes to the Editor in Chief of Inverse Problems and Imaging, Lassi Päivärinta, for supporting and facilitating this publication. We would also like to thank all the participants of the Sestri Levante Conference who made such a successful, stimulating and pleasant event possible. Last (but definitely not least!) we would like to thank the sponsors of the conference: the European Office of Aerospace Research and Development of the United States Air Force Office of Scientific Research, the University of Genova, the University of Verona, the Istituto Nazionale di Alta Matematica - Gruppo Nazionale di Calcolo Scientifico, the University of Göttingen, the University of Delaware and INRIA Center of Saclay Ile de France.

In this paper we revisit the transmission eigenvalue problem for an inhomogeneous media of compact support perturbed by small penetrable homogeneous inclusions. Assuming that the inhomogeneous background media is known and smooth, we investigate how these small volume inclusions affect the transmission eigenvalues. Our perturbation analysis makes use of the formulation of the transmission eigenvalue problem introduced Kirsch in [*only* near the boundary. Thus, our approach can handle small perturbations with positive, negative or zero (voids) contrasts. In addition to proving the convergence rate for the eigenvalues corresponding to the perturbed media as inclusions' volume goes to zero, we also provide the explicit first correction term in the asymptotic expansion for simple eigenvalues. The correction term involves computable information about the known inhomogeneity as well as the location, size and refractive index of small perturbations. Our asymptotic formula has the potential to be used to recover information about small inclusions from knowledge of the real transmission eigenvalues, which can be determined from scattering data.

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