Mathematical Researches

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Failure rate is one of the important concepts in reliability theory. In this paper, we introduce a new distribution function containing four parameters based on inverse Weibull distribution. This new distribution has a more general form of failure rate function. It is able to model five ageing classes of life distributions with appropriate choice of parameter values so that it is displayed decreasing, increasing, bathtub shaped, unimodal and increasing-decreasing increasing  failure rates and the new distribution has also a bimodal density function.  The moments, the order statistics, reliability parameters are obtained. The method of maximum likelihood is used to estimate the model parameters. Also, the observed information matrix is obtained. Two applications are presented to illustrate the advantage of the proposed distribution.

                

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Introduction
Vertex decomposability of a simplicial complex is a combinatorial topological concept which is related to the algebraic properties of the Stanley-Reisner ring of the simplicial complex. This notion was first defined by Provan and Billera in 1980 for k-decomposable pure complexes which is known as vertex decomposable when . Later Bjorner and Wachs extended this concept to non-pure complexes. Being defined in an inductive way, vertex decomposable simplicial complexes are considered as a well behaved class of complexes and has been studied in many research papers. Because of their interesting algebraic and topological properties, giving a characterization for this class of complexes is of great importance and is one of the main problems in combinatorial commutative algebra. In this regard obtaining families of simplicial complexes with this property is of great interest. In this paper we present a new family of vertex decomposable simplicial complexes, which is associated to the t-clique ideal of the complement of path graphs. The t-clique ideal is a natural generalization of the concept of the edge ideal of a graph. For a graph G, a complete subgraph of G with t vertices is called a t-clique of G. The ideal  generated by the monomials  of degree t such that the induced subgraph of G on the set  is a complete graph, is called the t-clique ideal of G. We consider the Stanley- Reisner simplicial complex of the ideal , where  is a path graph of order n.  For such a simplicial complex , we obtain the set of facets of  and using this characterization we show that every such simplicial complex is vertex decomposable, whose shedding vertex is an endpoint of the path graph. Indeed, any simplicial complex in this family is Cohen-Macaulay, since it is pure. Since edge ideals of graphs are in fact 2-clique ideals, this family of simplicial complexes contains the independence complexes of complement of path graphs. Finally, as a consequence it is shown that the t-independence ideal of the complement of a path graph is vertex splittable and its Betti splitting is presented
Material and methods
To prove the vertex decomposability of , first we characterize the set of facets of  . This helps us to find a shedding vertex for this simplicial complex and then by an inductive approach the vertex decomposability has been proved.
Results and discussion
For positive integers  and , we show that a subset F of the vertex set of  is a facet of  if and only if  and every component of the induced subgraph  is a path graph of even order. Using this characterization, it is shown that any endpoint of the path graph is a shedding vertex of  and   is vertex decomposable. Moreover, it is proved that the ideal  has a Betti splitting.
Conclusion
The following conclusions were drawn from this research.

• A characterization for the set of facets of the simplicial complex  is presented.
• The simplicial complex  is vertex decomposable for any positive integers  and .
• The ideal  has a Betti splitting for any any positive integers  and ../files/site1/files/51/%D9%85%D8%B1%D8%A7%D8%AF%DB%8C.pdf

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The mann fixed point algorithm play an importmant role in the approximation of fixed points of nonexpansive operators. In this paper, by considering new conditions, we prove the weak convergence of mann fixed point algorithm, for finding a common fixed point of two nonexpansive mappings in real Hilbert spaces. This results extend the privious results given by Kanzow and Shehu. Finally, we give an application of our results, by using the John von Neumann's method. 

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Let R be an arbitrary ring and N be a right R-module.  A right R-module M is called N-retractable if HomR(M,N')≠0, for any nonzero submodule N' of N. This is a generalization of the concept of retractable modules. The aim of this paper is to study of N-retractable modules, where N is an arbitrary right R-module. One of the most important results of this paper is the characterization of rings that have a module such that each module is retractable with respect to it. Also we show that the class of N-retractable modules is closed under direct sums and direct products.   

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Stratified sampling is one of the most widely used sampling designs. In some cases, it is up to the researcher to determine the boundaries of the strata, and in some cases, the population is already stratified. The optimal classification is obtained for a situation of strata boundries, where the variance of the population mean (or total) estimator reaches its lowest value. In traditional methods, the variance of the estimator is considered as a function of the strata boiundries for the response variable, in order to reach the minimum of the variance, equations are obtained which are often solved by numerical methods. The first deficiency of this method is not considering all auxiliary variables. For example, in estimating the average income, classifying the society based on factors such as gender and job history can not only increase the efficiency of the estimator, but also make the interpretability and generalizability of the results easier. The second one is complex equations that do not have a closed and understandable solutions
n this paper, we have tried to construct the optimal classification based on a new criterion that is a combination of variance and a penalty for increasing the number of strata, so that important auxiliary variables in the formation of the decision tree determine the boundries of the strata. The classification process starts from the saturated tree and with successive pruning until reaching the root node, the number of strata decreases, the optimal stratification is achieved based on the introduced combined criterion.