The paper contains integral representations for certain classes of exponentially growing solutions of second order periodic elliptic equations. These representations are the analogs of those previously obtained by S. Agmon, S. Helgason, and other authors for solutions of the Helmholtz equation. When one restricts the class of solutions further, requiring… (More)
The limited-view problem is studied for thermoacoustic tomography, which is also referred to as photoacoustic or optoacoustic tomography depending on the type of radiation for the induction of acoustic waves. We define a "detection region," within which all points have sufficient detection views. It is explained analytically and shown numerically that the… (More)
The transform considered in the paper integrates a function supported in the unit disk on the plane over all circles centered at the boundary of this disk. Such circular Radon transform arises in several contemporary imaging techniques, as well as in other applications. As it is common for transforms of Radon type, its range has infinite co-dimension in… (More)
Several hybrid tomographic methods utilizing ultrasound modulation have been introduced lately. Success of these methods hinges on the feasibility of focusing ultrasound waves at an arbitrary point of interest. Such a focusing, however, is difficult to achieve in practice. We thus propose a way to avoid the use of focused waves through the so called… (More)
The paper starts with a comparative discussion of features and limitations of the three types of recent approaches to the reconstruction in thermoacoustic/photoacoustic tomography: backprojection formulas, eigenfunction expansions, and time reversal. The latter method happens to be the least restrictive. It is then considered in more detail, e.g. its… (More)
We investigate the band-gap structure of the spectrum of second-order partial differential operators associated with the propagation of waves in a periodic two-component medium. The medium is characterized by a real-valued position-dependent periodic function "(x) that is the dielectric constant for electromagnetic waves and mass density for acoustic waves.… (More)
These notes represent an extended version of the contents of the third lecture delivered at the AMS Short Course " Radon Transform and Applications to Inverse Problems " in Atlanta in January 2005. They contain a brief description of properties of some generalized Radon transforms arising in inverse problems. Here by generalized Radon transforms we mean… (More)
The paper contains a simple approach to reconstruction in Ther-moacoustic and Photoacoustic Tomography. The technique works for any geometry of point detectors placement and for variable sound speed satisfying a non-trapping condition. A uniqueness of reconstruction result is also obtained.
An explicit derivation of dispersion relations and spectra for periodic Schrödinger operators on carbon nano-structures (including graphene and all types of single-wall nano-tubes) is provided.