Background: Data types are recurrent events in studies in which each person may experience an event at different times. One of the most popular approaches to analyze recurrent event data is obtaining an Estimate of the means/rate of events at different times. In this context, determining the variability over time can help better understand the effect of factor on the response. In this study, we applied smoothing methods to Estimate coefficients in time-dependent rate model, and we also showed its application in data of psoriasis patients. Methods: In the present study, psoriasis patients who experienced relapse that led to hospitalization during 2005 and 2014 in the Dermatology Department of Imam Khomeini hospital in Tehran were examined. To investigate the rate of relapse during a year, timedependent rate model was used and variability of the effects was assessed using Wald test. Both b-spline and Kernel methods were used to Estimate time varying coefficients in rates model. Finally, results from methods were compared based on the obtained Estimates. Results: Based on the results of the Wald test, the effect of season on the occurrence of psoriasis was significantly different (p<0. 01). Also according to the Estimated coefficients from both b-spline and Kernel methods, there was little difference between them. Conclusion: In situations in which the effect of a variable is different at different times, using time-dependent coefficients rate model can lead to a better Estimate of the effect of variable on the response. On the other hand, smoothing methods can smooth the effects of the variables that vary over time.

In the present article, we develop the well-known preliminary test and Stein-type estimators for the probability density function under association. In this respect, we derive the asymptotic characteristics of the proposed estimators under a set of local alternatives. Some numerical studies are provided for supporting the findings. The result of this article improves the Kernel Estimate of the marginal probability density function of a strictly stationary sequence of associated random variables. For practical sake, the behavior of the proposed estimators is also analyzed using a real data set.

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.

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.