A (d, k) set is a subset of R containing a translate of every k-dimensional disc of diameter 1. We show that if (1 + √ 2)k−1 + k > d and k ≥ 2, then every (d, k) set has positive Lebesgue measure. This improves a result of Bourgain, who showed that the analogous statement holds when 2k−1 +k ≥ d and k ≥ 2. We obtain this improvement in two parts. First, we replace Bourgain’s main estimate with a...

Ruthenium dioxide nanoparticles (RuO2 ) with an average particle size of 50 nm are readily synthesized from RuCl3.xH2O via the precipitation process in the presence of in-situ sodium octanoate as anionic surfactant. The phase composition, morphology, lattice parameters and size of nanoparticles in these products are characterized by Fourier transform infrared spectroscopy (FT-IR), X-ray diffrac...

Accurate simulation of X-ray transforms of representative objects plays an important role in the evaluation and improvement of CT reconstruction algorithms. In this paper, we formulate the X-ray transform and 3D Radon transform for ellipsoids and tetrahedra, and verify the resulting formulas by numerical simulation. Here the ellipsoids and tetrahedra may be arbitrarily positioned and rotated. L...

the output of x-ray, especially in the field of required energy for diagnosing the disease is very low (in diagnosing lamps, it is less than 1%). in this kind of lamps the rest of the electrical energy is transformed to heat. this considerable amount of heat induces a myriad of too many limitations in choosing higher levels of radiation, particularly in specialized techniques. this process is t...

As we all know, any practical computed tomography (CT) projection data more or less contains noises. Hence, it will be inconvenient for the postprocessing of a reconstructed 3D image even when the noise in the projection data is white. The reason is that the noise in the reconstructed image may be nonwhite. X-ray transform can be applied to the three dimensional (3D) CT, depicting the relations...

Ultrafast electronic and structural dynamics ofmatter govern rate and selectivity of chemical reactions, as well as phase transitions and efficient switching in functionalmaterials. Since x-rays determine electronic and structural properties with elemental, chemical, orbital andmagnetic selectivity, short pulse x-ray sources have become central enablers of ultrafast science. Despite of these st...