Numerical differentiation for two-dimensional scattered data on arbitrary domain base on Hermite extension with an implicit iteration process

<abstract><p>In this paper, we develop a method for numerical differentiation of two-dimensional scattered input data on arbitrary domain. A Hermite extension approach is used to realize the approximation and a modified implicit iteration method is proposed to stabilize the approximation process. For functions with various smooth conditions, the numerical solution process of the method is uniform. The error estimates are obtained and numerical results show that the new method is effective. The advantage of the method is that it can solve the problem in any domain.</p></abstract>

Numerical differentiation on scattered data through multivariate polynomial interpolation

We discuss a pointwise numerical differentiation formula on multivariate scattered data, based on the coefficients of local polynomial interpolation at Discrete Leja Points, written in Taylor’s formula monomial basis. Error bounds for the approximation of partial derivatives of any order compatible with the function regularity are provided, as well as sensitivity estimates to functional perturbations, in terms of the inverse Vandermonde coefficients that are active in the differentiation process. Several numerical tests are presented showing the accuracy of the approximation.

The Semantics Analysis of the Effective Factors on Safety Development in the Point of view of Qur’an

The present article tries to answer this question: From the point of view Qur’an, what are the factors that develop safety? The Qur’an is a book that guides and expresses all the things that have been revealed from God for the benefit and inquest of human beings. This research has used the Qur’an and thematic analysis method has been used to analyze the data. Thematic analysis is one of the common and efficient methods in qualitative research, converting diverse and scattered data into strong and formal data. After the inquiry, 21 safety factors were identified, which are divided into two parts: personal and institutional. The findings of this study show that the most important personal factors found are ritual and moral-psychological factors, and the most important institutional factors are political and social factors.

Adaptive Radial Basis Function Partition of Unity Interpolation: A Bivariate Algorithm for Unstructured Data

In this article we present a new adaptive algorithm for solving 2D interpolation problems of large scattered data sets through the radial basis function partition of unity method. Unlike other time-consuming schemes this adaptive method is able to efficiently deal with scattered data points with highly varying density in the domain. This target is obtained by decomposing the underlying domain in subdomains of variable size so as to guarantee a suitable number of points within each of them. The localization of such points is done by means of an efficient search procedure that depends on a partition of the domain in square cells. For each subdomain the adaptive process identifies a predefined neighborhood consisting of one or more levels of neighboring cells, which allows us to quickly find all the subdomain points. The algorithm is further devised for an optimal selection of the local shape parameters associated with radial basis function interpolants via leave-one-out cross validation and maximum likelihood estimation techniques. Numerical experiments show good performance of this adaptive algorithm on some test examples with different data distributions. The efficacy of our interpolation scheme is also pointed out by solving real world applications.

Quantitative Classification of Desertification Severity for Degraded Aquifer Based on Remotely Sensed Drought Assessment

Natural and anthropogenic causes jointly lead to land degradation and eventually to desertification, which occurs in arid, semiarid, and dry subhumid areas. Furthermore, extended drought periods may cause soil exposure and erosion, land degradation and, finally, desertification. Several climatic, geological, hydrological, physiographic, biological, as well as human factors contribute to desertification. This paper presents a methodological procedure for the quantitative classification of desertification severity over a watershed with degraded groundwater resources. It starts with drought assessment using Standardized Precipitation Index (SPI), based on gridded satellite-based precipitation data (taken from the CHIRPS database), then erosion potential is assessed through modeling. The groundwater levels are estimated with the use of a simulation model and the groundwater quality components of desertification, based on scattered data, are interpolated with the use of geostatistical tools. Finally, the combination of the desertification severity components leads to the final mapping of desertification severity classification.

A Coupled RBF Method for the Solution of Elastostatic Problems

Radial basis function (RBF) has been widely used in many scientific computing and engineering applications, for instance, multidimensional scattered data interpolation and solving partial differential equations. However, the accuracy and stability of the RBF methods often strongly depend on the shape parameter. A coupled RBF (CRBF) method was proposed recently and successfully applied to solve the Poisson equation and the heat transfer equation (Appl. Math. Lett., 2019, 97: 93–98). Numerical results show that the CRBF method completely overcomes the troublesome issue of the optimal shape parameter that is a formidable obstacle to global schemes. In this paper, we further extend the CRBF method to solve the elastostatic problems. Discretization schemes are present in detail. With two elastostatic numerical examples, it is found that both numerical solutions of the CRBF method and the condition numbers of the discretized matrices are almost independent of the shape parameter. In addition, even if the traditional RBF methods take the optimal shape parameter, the CRBF method achieves better accuracy.

THE RUSSIAN PRISONERS OF WAR DURING SEVEN YEARS' WAR: TO STATEMENT OF A PROBLEM

The article is based on the still rare and scattered data on situation of the Russian prisoners in the Prussian captivity during Seven years' war of 1756-1763. On the basis of the published memoirs and single archival documents the author draws a conclusion on difference in keeping of captured officers and the lower ranks. The situation of the soldiers in captivity was very difficult; they were pressured to go into the Prussian service, they felt the need for clothing and warm rooms, some of the prisoners was forcibly sent to the Prussian army and, contrary to international agreements, remained there even after the end of the war.