In this paper, we will present a new method for a Volterra integral equation with weakly singular Kernel in the reproducing Kernel space. Firstly the equation is transformed into a new equivalent equation. Its exact solution is represented in the form of series in the reproducing Kernel space. In the mean time, the n-term approximation un(t) to the exact solution u (t) is obtained. Some numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by the method are compared with the exact solution of each example and are found to be in good agreement with each other.

‎Let $G$ be a group,  where $\text{Aut}(G)$ denotes the full automorphisms group of $G$  and $L(G)$ represents  the  absolute center of $G.$ An automorphism $\alpha \in \text{Aut}(G)$ is called an autocentral automorphism if $g^{-1}\alpha(g) \in L(G),$ for all $g \in G .$  In this paper, we  introduce the notion of autocentral Kernel subgroup for an arbitrary  group $G.$ We then investigate and establish several structural  properties of this subgroup, providing insights into its role within the broader framework of group theory.

Understanding the variation of Kernel number per unit area (KNO) is an important factor in respect yield and in identifying opportunities to increase potential yield. Some crop models are for dependence on Kernel number prediction to yield simulation. In order to quantify the relationship between Kernel number per unit area and influencing parameters (Environment and non-environment-based factors), compound analysis with randomized complete block design was conducted including four replications. This experiment was done at the Gorgan University of Agricultural Sciences Research Farm in 2005. The treatments were five different sowing dates (14 December, 20 Jan, 20 Feb, 20 March, 16 Apr) and four wheat variations (Kohdasht, Shirudi, Tajan, Zagros). Considering results of correlation coefficients, Kernel number per unit area appeared to have the best relation with mean temperature, but Non-linear models fitting indicated that the highest R2 and the lowest CV belonged to stem dry matter at anthesis. Fitted model showed that 95% of Kernel numbers are formed when stem dry matter is around 747 g.

We give a new proof of the well known Wedderburn's little theorem (1905) that a finite division ring is commutative. We apply the concept of Frobenius Kernel in Frobenius representation theorem in finite group theory to build a proof.

Water is as the primary input in the production function of agriculture, industry, and mining, as an intermediate commodity in tourism, services, and commerce, as a commodity in the consumer basket of households and as the most important factor in the environmental sector. it is also perhaps the most important commodity-Social and environmental factors. Considering the increasing demand and the decrease in water supply, which leads to the scarcity of water, especially groundwater, the impact of this variable on economic growth and its reciprocal relationship is highly regarded by economists. The purpose of this study was to evaluate the Kernel causality between extraction of groundwater resources and economic growth using the Nadaraya-Watson regression model and the generalized correlation coefficient for Iran's economy from 1982 to 2015. The results of the research show that although Granger's causality indicates that there is no relation between water consumption and GDP, the generalized correlation coefficient and nonparametric regression show the existence of unidirectional Kernel causality from consumption of water to GDP. Therefore, the drop in groundwater levels and the depletion of groundwater aquifers can greatly affect economic growth.

Image inpainting is the process of filling in damaged or missing regions in an image by using information from known regions or known pixels of the image. One of the most important techniques for inpainting is convolution-based methods, in which a Kernel is convolved with the damaged image iteratively. Convolution based algorithms are very quick, but they don’t have good results in structures and textural regions and result in blurring. The Kernel size in the convolution-based algorithm is a critical parameter. The large size results in edge blurring, and if the Kernel size is small, the information may not be sufficient for reconstruction. In this paper, a novel convolution-based algorithm is proposed that uses known gradient of the pixels to construct a convolution mask. In this algorithm, the Kernel size is controlled by the gradient of the image in the known regions. The algorithm computes the weighted sum of the known pixels in a neighborhood around a damaged pixel and replaces the value in the place of that damaged pixel. The proposed algorithm is fast and results in good edges and smooth regions reconstruction. It is an iterative algorithm and its implementation is very simple. Experimental results show the effectiveness of our algorithm.