Citation:Sandström, S.-E., and I. K. Akeab (2014), Scaling and sparsity in an accurate implementation of the method of moments in 2-D, Radio Sci., 49, 643-652, doi:10.1002 Abstract The integral equations of electromagnetic scattering are often solved numerically by means of the method of moments. At high frequencies, this method typically leads to a large linear system with a dense matrix. The use of higher-order basis functions is a means to improve the accuracy. B-splines are used here for a two-dimensional test bed study that avoids the complexity of 3-D implementation. For smooth convex scatterers one may use a priori knowledge about the oscillatory behavior of the solution to reformulate the integral equation. This fast scale of variation is included in the kernel of the integral equation. An extension of this idea deals with the variation in the shadow, particularly for circular geometry, and is an improvement that is presented in this study. Generally, the transverse electric (TE) case is less studied at high frequencies and our numerical results therefore relate to this harder problem. A sparse matrix can be obtained by modification of the integration path in the integral equation. The decay of the modified kernel makes this possible for high frequencies but the modified path reduces the accuracy in the deep shadow. This study investigates these modified paths for the case where the shadow region is not omitted from the formulation.

FMCW (frequency‐modulated continuous wave) radar is a simple and inexpensive technique for target location. The resolution is given by the available bandwidth and the directivity of the antenna. Resolution is not a problem at high frequencies, while at low frequencies (the HF and VHF band), and especially for mobile platforms, the required size of the antenna becomes impractical. In order to obtain the bearing of the targets, without relying on directivity, one may use a simple two‐dimensional trilateration method that involves several platforms. Since this approach covers an area, rather than a sector, the range is reduced to some tens of kilometers. The VHF band and a bandwidth below 10 MHz is a good choice if the priority is to reduce radio interference. Fast targets, corresponding to a significant Doppler shift, have not been considered. The problem of ghost targets has been studied for both monostatic and multistatic radar. When there is a confluence of echoes, more bandwidth is required to maintain the accuracy of a few meters that is normally obtained in the simulation.

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X-ray emission contains some of the gaseous properties is produced when the particles of the solar wind strike the atmosphere of comet ISON and PanSTARRS Comets. The data collected with NASA Chandra X-ray Observatory of the two comets, C/2012 S1 (also known as Comet ISON) and C/2011 S4 (Comet PanSTARRS) are used in this study.    The real abundance of the observed X-ray spectrum elements has been extracted by a new simple mathematic model. The study found some physical properties of these elements in the comet’s gas such as a relationship between the abundance with emitted energy. The elements that have emission energy (2500-6800) eV, have abundance (0.1-0.15) %, while the elements that have emission energy (850-2500) eV and (6800-9250) eV have abundance (0.2-0.3) %.    The relation between interacted energy and atomic number is form two sets.  The interacted energy of each element is increased as the atomic number increased. This case has been seen in both comets