In this paper, we investigate the ideal theory in BCK-algebras and we define the notions of primary and P -primary ideals. Then we show that in bounded implicative BCK-algebras, if an ideal has a primary decomposition, then it has a reduced primary decomposition. In the follow, we extend the above concepts to X-modules (Extended BCK-modules) and we introduce the notions of primary and P -primar...

Research has focused on increasing the treatment integrity of school-based interventions by utilizing performance feedback. The purpose of this study was to extend this literature by increasing special education teachers' treatment integrity for implementing antecedent and consequence procedures in an ongoing behavior support plan. A multiple baseline across teacher-student dyads (for two class...

A set of polynomials is called a submodule $\mathbb C [x_1 ,\ldots , x_n ]$ if it translation invariant linear subspace ]$. We present description the submodules [x,y]$ in terms

This study examined the probability of (a) being placed in a disciplinary alternative education setting for mandatory versus discretionary reasons and (b) returning within the same year among an ethnically diverse sample (African American, Caucasian, Hispanic) of middle and high school students (N=270). Participants were compared based on ethnicity, gender, grade level, and special education st...

While deep learning algorithms demonstrate a great potential in scientific computing, its application to multi-scale problems remains be big challenge. This is manifested by the “frequency principle” that neural networks tend learn low frequency components first. Novel architectures such as network (MscaleDNN) were proposed alleviate this problem some extent. In paper, we construct subspace dec...