For a finite field $mathbb{F}_q$, the bivariate skew polynomial ring $mathbb{F}_q[x,y;rho,theta]$ has been used to study codes cite{XH}. In this paper, we give some characterizations of the ring $R[x,y;rho,theta]$, where $R$ is a commutative ring. We investigate 2-D skew $(lambda_1,lambda_2)$-constacyclic codes in the ring $R[x,y;rho,theta]/langle x^l-lambda_1,y^s-lambda_2rangle_{mathit{l}}.$ A...

This paper gives a detailed overview of the problem of missing data in parametric and nonparametric regression. Theoretical basics, properties as well as simulation results may help the reader to get familiar with the common problem of incomplete data sets. Of course, not all occurences can be discussed so this paper could be seen as an introduction to missing data within regression analysis an...

In this paper, we show that every element of a discrete module is a sum of two units if and only if its endomorphism ring has no factor ring isomorphic to $Z_{2}$. We also characterize unit sum number equal to two for the endomorphism ring of quasi-discrete modules with finite exchange property.

Credit risk management is a process in which banks estimate probability of default (PD) for each loan applicant. Data sets of previous loan applicants are built by gathering their data, and these internal data sets are usually completed using external credit bureau’s data and finally used for estimating PD in banks. There is also a continuous interest for bank to use rule based classifiers to b...

 Letbe a ring with an endomorphism and an -derivationAntoine studied the structure of the set of nilpotent elements in Armendariz rings and introduced nil-Armendariz rings. In this paper we introduce and investigate the notion of nil--compatible rings. The class of nil--compatible rings are extended through various ring extensions and many classes of nil--compatible rings are constructed. We al...

This article examines annihilators in the skew polynomial ring $R[x;alpha,delta]$. A ring is strongly right $AB$ if everynon-zero right annihilator is bounded. In this paper, we introduce and investigate a particular class of McCoy rings which satisfy Property ($A$) and the conditions asked by P.P. Nielsen. We assume that $R$ is an ($alpha$,$delta$)-compatible ring, and prove that, if $R$ is ni...

- Missing observations occur in experimental designs as a result of insufficient sampling, machine breakdown, high cost, and errors in the measurements. In nanomanufacturing, missing observations often appear in designs because the combination of factors or molecular structures selected by a designer cannot be experimented successfully. In the current paper, Box-Behnken and face-centered compos...