let $r$ be a commutative ring. we write $mbox{hom}(mu_a, nu_b)$ for the set of all fuzzy $r$-morphisms from $mu_a$ to $nu_b$, where $mu_a$ and $nu_b$ are two fuzzy $r$-modules. we make$mbox{hom}(mu_a, nu_b)$ into fuzzy $r$-module by redefining a function $alpha:mbox{hom}(mu_a, nu_b)longrightarrow [0,1]$. we study the properties of the functor $mbox{hom}(mu_a,-):frmbox{-mod}rightarrow frmbox{-mo...

Tensor transpose is a higher order generalization of matrix transpose. In this paper, we use permutations and symmetry group to define the tensor transpose. Then we discuss the classification and composition of tensor transposes. Properties of tensor transpose are studied in relation to tensor multiplication, tensor eigenvalues, tensor decompositions and tensor rank.

The tensor rank of a tensor t is the smallest number r such that t can be decomposed as a sum of r simple tensors. Let s be a k-tensor and let t be an `-tensor. The tensor product of s and t is a (k + `)-tensor. Tensor rank is sub-multiplicative under the tensor product. We revisit the connection between restrictions and degenerations. A result of our study is that tensor rank is not in general...

let g be a simple connected graph. the generalized polarity wiener index ofg is defined as the number of unordered pairs of vertices of g whosedistance is k. some formulas are obtained for computing the generalizedpolarity wiener index of the cartesian product and the tensor product ofgraphs in this article.