A Short Solution to the Busemann-Petty Problem

A unified analytic solution to the Busemann-Petty problem was recently found by Gardner, Koldobsky and Schlumprecht. We give an elementary proof of their formulas for the inverse Radon transform of the radial function of an origin-symmetric star body .     

Radon transforms on affine Grassmannians

We develop an analytic approach to the Radon transform , where is a function on the affine Grassmann manifold of -dimensional planes in , and is a -dimensional plane in the similar manifold k$">. For , we prove that this transform is finite almost everywhere on if and only if , and obtain explicit inversion formulas. We establish correspondence between Radon transforms on affine Grassmann manifolds and similar transforms on standard Grassmann manifolds of linear subspaces of . It is proved that the dual Radon transform can be explicitly inverted for , and interpreted as a direct, ``quasi-orthogonal" Radon transform for another pair of affine Grassmannians. As a consequence we obtain that the Radon transform and the dual Radon transform are injective simultaneously if and only if . The investigation is carried out for locally integrable and continuous functions satisfying natural weak conditions at infinity.

     

The Paley--Wiener Theorem for the Generalized Radon Transform on the Plane

We consider the problem of reconstructing a function on the disk from its integrals over curves close to straight lines, i.e., the inversion problem for the generalized Radon transform. Necessary and sufficient conditions on the range of the generalized Radon transform are obtained for functions supported in a smaller disk under the additional condition that the curves that do not meet coincide with the corresponding straight lines.     

Multiresolution Analysis and Radon Measures on a Locally Compact Abelian Group

A multiresolution analysis is defined in a class of locally compact abelian groups G. It is shown that the spaces of integrable functions and the complex Radon measures M(G) admit a simple characterization in terms of this multiresolution analysis.     

On inversion and Sobolev estimate results about the weighted Radon transform

In this paper, we derive an inversion of the weighted Radon transform by Fourier transform, Riesz potential, and integral transform. We extend results of Rigaud and Lakhal to the n‐dimensional Euclidean space. Furthermore, we obtain some properties of the weighted Radon transform. Finally, we develop some estimate results of the weighted Radon transform under Sobolev space.     

Decaying states of perturbed wave equations

We study the solutions of perturbed wave equations that represent free wave motion outside some ball. When there are no trapped rays, it is shown that every solution whose total energy decays to zero must be smooth. This extends results of Rauch to the even-dimensional case and to systems having more than one sound speed. In these results, obstacles are not considered. We show that, even allowing obstacles, waves with compact spatial support cannot decay, assuming a unique continuation hypothesis. An example with obstacle is given where nonsmooth, compactly supported, decaying waves exist.     

Seasonal variation of radon-222 concentrations in specific locations in Jordan

Previously calibrated passive detectors (CR-39) and an active radon device (Radon Monitor RM3) were used to study seasonal variation of radon-222 concentration levels inside and outside specific locations in Jordan. The study sites were located in an area that used to be an old phosphate mine. We found that the maximum value of radon concentration in air inside the dwellings, as measured by the passive dosimeters, was 1532.9 Bq/m³ during the winter season, and the minimum one was 46.3 Bq/m³ during fall season. While the highest and lowest readings of the active monitor were 892 and 4 Bq/m³ during fall and summer seasons, respectively. The radon concentration in soil ranges from 0.2 kBq/m³ in spring to 37.8 kBq/m³ in fall.     

L^p Estimates for Singular Radon Transforms with Rough Kernels

The purpose of this paper is to study the mapping properties of the singular Radon transforms with rough kernels. Such singular integral operators are proved to be bounded on Lebesgue spaces.     

Measurement of indoor Rn using CR-39 under a thin film geometry

In this work an assembly for indoor ²²²Rn measurement is presented. This assembly is made up of two acrylic plates (14 cm×14 cm) separated by a distance of 4 mm. To prevent radon daughters from outside entering the assembly (and produce alpha particle tracks striking the CR-39 detector placed in the central region of one of these plates), the borders of these plates were progressively closed, leaving open only a rectangular aperture of 5 mm×4 mm. The size of this opening was determined experimentally, by exposing in two indoor environments assemblies with different apertures as follows: (i) all borders open; (ii) two borders closed; (iii) three borders closed; (iv) four borders closed but one containing a 6 cm×4 mm opening; (v) four borders closed but one containing a 2.5 cm×4 mm opening and (vi) four borders closed but one containing a 0.5 cm×4 mm opening. Track density shows a noticeable decrease between assembly (i) and assembly (iii), remaining constant for smaller openings. Only ²²²Rn, a noble gas, should enter the assembly independently of the opening size.