Leonard Dăuş

Relative Regular Objects in Categories

We define the concept of a regular object with respect to another object in an arbitrary category. We present basic properties of regular objects and we study this concept in the special cases of abelian categories and locally finitely generated Grothendieck categories. Applications are given for categories of comodules over a coalgebra and for categories of graded modules, and a link to the theory of generalized inverses of matrices is presented. Some of the techniques we use are new, since dealing with arbitrary categories allows us to pass to the dual category.

Relative Regular Modules. Applications to von Neumann Regular Rings

We use the concept of a regular object with respect to another object in an arbitrary category, in order to obtain the transfer of regularity in the sense of Zelmanowitz between the categories R −mod and S −mod, when S is an excellent extension of the ring R. Consequently, if S is an excellent extension of the ring R, then S is von Neumann regular ring if and only if R is also von Neumann regular ring. In the second part, using relative regular modules, we give a new proof of a classical result: the von Neumann regular property of a ring is Morita invariant.

Review of reliability bounds for consecutive-k-out-of-n systems

This paper reviews many lower and upper bounds for consecutive-k-out-of-n systems presented over the last three decades. The reason is a revived interest to accurately estimate the reliability of (very) large consecutive systems, where exact calculations can be challenging. Main examples are novel nano-architectures targeting FinFETs, nano-magnetic and molecular technologies (where accurate estimations of reliability are of high interest), as well as their associated nanoscale communications (where the reliability of transmission needs to be assessed), which map well onto (very) large consecutive systems.

V-categories: applications to graded rings

Motivated by the study of V-rings, we introduce the concept of V-category, as a Grothendieck category with the property that any simple object is injective. We present basic properties of V-categories, and we study this concept in the special case of locally finitely generated categories, for instance the category R-gr of all graded left R-modules, where R is a graded ring. We use the characterizations of V-categories in the study of graded V-rings. Since V-rings are closely related to Von Neumann regular rings (in the commutative case these classes of rings coincide), the last part of the article is devoted to graded regular rings.

Graded Semiartinian Rings: Graded Perfect Rings

Abstract We study graded left semiartinian rings with finite support. It is shown that the semiartinian property is preserved when we pass to the smash product in the sense of Quinn. We apply these results to investigate left perfect graded rings.

Reliability bounds for two dimensional consecutive systems

Abstract In this paper we consider consecutive systems due to their potential for novel nano-architectures in general, where schemes able to significantly enhance reliability at low redundancy costs are expected to make a difference. Additionally, nanoscale communications are also expected to rely on structures and methods allowing to achieve better/lower transmission bit error rates. In particular, certain nano-technologies, like, e . g ., nano-magnetic ones (but also nano-fluidic, molecular and even FinFETs), could be mapped onto consecutive systems, a well-established redundancy scheme. That is why this paper will start by briefly mentioning previous results for one dimensional linear consecutive- k -out-of- n : F systems with statistically independent components having the same failure probability q (i.i.d. components), before focusing on 2-dimensional consecutive systems. We shall introduce 2-dimensional consecutive systems and mention some variations, before going over a few upper and lower bounds for estimating their reliability. Afterwards, we shall present simulation results for particular 2-dimensional cases. These will show that some of the lower and upper bounds are able to perfectly match the exact reliability of 2-dimensional consecutive systems for the particular cases considered here. Conclusions and future directions of research are ending the paper.

A new way to iterate Brzeziński crossed products

If

Why Should We Care About Bounds Consecutive Systems Revisited

A\otimes _{R, \sigma }V

ON A DADE'S CONJECTURE

and

The Exchange Property in Grothendieck Categories: Applications

A\otimes _{P, \nu }W