Relatively thin and sparse subsets of groups

Let G be a group with identity e and let I be a left-invariant ideal in the Boolean algebra PG of all subsets of G. A subset A of G is called I-thin if gA⋂A∈I for every g∈G {e}. A subset A of G is called I-sparse if, for every infinite subset S of G, there exists a finite subset F⊂S such that ⋂g∈FgA∈F. An ideal I is said to be thin-complete (sparse-complete) if every I-thin (I-sparse) subset of G belongs to I. We define and describe the thin-completion and the sparse-completion of an ideal in PG.Припустимо, що G — група з одиницею e, I — iнварiантний злiва iдеал в булевiй алгебрi PG всiх пiдмножин групи G. Пiдмножина A групи G називається I-тонкою, якщо gA⋂A∈I для кожного g∈G {e}. Пiдмножина A групи G називається P-розрiдженою, якщо для кожної нескiнченної множини S групи G iснує скiнченна пiдмножина F⊂S така, що ⋂g∈FgA∈F. Говорять, що iдеал I тонко-повний (розрiджено-повний), якщо кожна I-тонка (I-розрiджена) множина групи G належить I. Визначено та описано тонке та розрiджене доповнення iдеалу в PG

The generalized MIC-Kepler system

This paper deals with dynamical system that generalizes the MIC-Kepler system. It is shown that the Schr\"{o}dinger equation for this generalized MIC-Kepler system can be separated in spherical and parabolic coordinates. The spectral problem in spherical and parabolic coordinates is solved.Comment: 8 page

Towards a new image processing system at Wendelstein 7-X: From spatial calibration to characterization of thermal events

Wendelstein 7-X (W7-X) is the most advanced fusion experiment in the stellarator line and is aimed at proving that the stellarator concept is suitable for a fusion reactor. One of the most important issues for fusion reactors is the monitoring of plasma facing components when exposed to very high heat loads, through the use of visible and infrared (IR) cameras. In this paper, a new image processing system for the analysis of the strike lines on the inboard limiters from the first W7-X experimental campaign is presented. This system builds a model of the IR cameras through the use of spatial calibration techniques, helping to characterize the strike lines by using the information given by real spatial coordinates of each pixel. The characterization of the strike lines is made in terms of position, size, and shape, after projecting the camera image in a 2D grid which tries to preserve the curvilinear surface distances between points. The description of the strike-line shape is made by means of the Fourier Descriptors

Forward modeling of collective Thomson scattering for Wendelstein 7-X plasmas: Electrostatic approximation

In this paper, we present a method for numerical computation of collective Thomson scattering (CTS). We developed a forward model, eCTS, in the electrostatic approximation and benchmarked it against a full electromagnetic model. Differences between the electrostatic and the electromagnetic models are discussed. The sensitivity of the results to the ion temperature and the plasma composition is demonstrated. We integrated the model into the Bayesian data analysis framework Minerva and used it for the analysis of noisy synthetic data sets produced by a full electromagnetic model. It is shown that eCTS can be used for the inference of the bulk ion temperature. The model has been used to infer the bulk ion temperature from the first CTS measurements on Wendelstein 7-X