In this paper, a fixed point theorem of nonexpansive mapping is established to study the existence and sufficient conditions for the controllability of nonlinear fractional control systems in reflexive Banach spaces. The result so obtained have been modified and developed in arbitrary space having Opial’s condition by using fixed point theorem deals with nonexpansive mapping defined on a set has normal structure. An application is provided to show the effectiveness of the obtained result.

Let R be a commutative ring with identity 1 ¹ 0, and let M be a unitary left module over R. A submodule N of an R-module M is called essential, if whenever N ⋂ L = (0), then L = (0) for every submodule L of M. In this case, we write N ≤e M. An R-module M is called extending, if every submodule of M is an essential in a direct summand of M. A submodule N of an R-module M is called semi-essential (denoted by N ≤sem M), if N ∩ P ≠ (0) for each nonzero prime submodule P of M. The main purpose of this work is to determine and study two new concepts (up to our knowledge) which are St-closed submodules and semi-extending modules. St-closed submodules is contained properly in the class of closed submodules, where a submodule N of

     In this paper, we study a new concept of fuzzy sub-module, called  fuzzy socle semi-prime sub-module that is a generalization the concept of semi-prime fuzzy sub-module and fuzzy of approximately semi-prime sub-module in the ordinary sense.  This leads us to introduce level property which studies the relation between the ordinary and fuzzy sense of approximately semi-prime sub-module. Also, some of its characteristics and notions such as the intersection, image and external direct sum of fuzzy socle semi-prime sub-modules are introduced. Furthermore, the relation between the fuzzy socle semi-prime sub-module and other types of fuzzy sub-module presented.

   Let R be a ring with identity and Ą a left R-module. In this article, we introduce new generalizations of compressible and prime modules, namely s-compressible module and s-prime module. An R-module A is s-compressible if for any nonzero submodule B of A there exists a small f in Hom_R(A, B). An R-module A is s-prime if for any submodule B of A, ann_R (B) A is small in A. These concepts and related concepts are studied in as well as  many results consist properties and characterizations are obtained.  

      In-Band Full-Duplex (IBFD) systems have the capability of simultaneously transmitting and receiving signals through the channel and require the same resources as half-duplex systems. Unfortunately, IBFD systems have self-interference (SI) issues that prevent the system from gaining double throughput with respect to half-duplex systems. Therefore, the IBFD system will be more reliable if SI is mitigated more. This contribution will look at SI cancellation in wireless radio and underwater acoustic systems. The reviewed documents cover all types of SI cancellations, including passive, analog, and digital cancellations. In a practical full-duplex system, the SI cancellation for all domains must cancel the SI below the receiver noi

Background: Since the periodontal disease Index of Ramfjord (Ramfjord index) can potentially shorten the examination time by almost half, many studies evaluated Ramfjord teeth in predicting full-mouth periodontal status of an adult population. The aim of this study was to evaluate the benefit of Ramfjord teeth in predicting the full-mouth clinical attachment level of an adult population in patients attending the college of dentistry- Baghdad University. Materials and methods: The study participants were 100 patients with age range from 30-60 years old which represent group zero. The patients were divided into three main groups according to the age of the patients. Group I and group II each of them composed of 30 patients while group III co