The portfolio is a perfect combination of stock or assets, which an investor buys them. The objective of the portfolio is to divide the investment risk among several shares. Using non-parametric DEA and DEA-R methods can be of great significance in estimating portfolio. In the present paper, the efficient portfolio is estimated by using non-radial DEA and DEA-R models. By proposing non-radial m...

We introduce and study the concept of $alpha $-semi short modules.Using this concept we extend some of the basic results of $alpha $-short modules to $alpha $-semi short modules.We observe that if $M$ is an $alpha $-semi short module then the dual perfect dimension of $M$ is $alpha $ or $alpha +1$.%In particular, if a semiprime ring $R$ is $alpha $-semi short as an $R$-module, then its Noetheri...

Let R be a perfect ring of characteristic p. We show that the group of continuous R-linear automorphisms of the perfect power series ring over R is generated by the automorphisms of the ordinary power series ring together with Frobenius; this answers a question of Jared Weinstein.

An R-module M is called epi-retractable if every submodule of MR is a homomorphic image of M. It is shown that if R is a right perfect ring, then every projective slightly compressible module MR is epi-retractable. If R is a Noetherian ring, then every epi-retractable right R-module has direct sum of uniform submodules. If endomorphism ring of a module MR is von-Neumann regular, then M is semi-...

For fixed integers r, ` ≥ 0, a graph G is called an (r, `)-graph if the vertex set V (G) can be partitioned into r independent sets and ` cliques. Such a graph is also said to have cochromatic number r + `. The class of (r, `) graphs generalizes r-colourable graphs (when ` = 0) and hence not surprisingly, determining whether a given graph is an (r, `)-graph is NP-hard even when r ≥ 3 or ` ≥ 3 i...

For fixed integers r, l ≥ 0, a graph G is called an (r, l)-graph if the vertex set V (G) can be partitioned into r independent sets and l cliques. Such a graph is also said to have cochromatic number r + l. The class of (r, l) graphs generalizes r-colourable graphs (when l = 0) and hence not surprisingly, determining whether a given graph is an (r, l)-graph is NP-hard even when r ≥ 3 or l ≥ 3 i...

Diffusion layers, and specially perfect diffusion layers, are very important subject for cryptographic research. Main quest is a perfect diffusion layer with more optimal hardware and/or software implementations (if possible, the last needs to holds also for its inverse). Different structures can be used for representing these layers, but all are interconnected. We start with multipermutations ...