Weosay thatotheosubmodules A, B ofoan R-module Moare µ-equivalent , AµB ifoand onlyoif <<_µand <<_µ. Weoshow thatoµ relationois anoequivalent relationoand hasegood behaviorywith respectyto additionmof submodules, homorphismsr, andydirectusums, weaapplyothese resultsotoointroduced theoclassoof H-µ-supplementedomodules. Weosay thatoa module Mmis H-µ-supplementedomodule ifofor everyosubmodule A of M, thereois a directosummand D ofoM suchothat AµD. Variousoproperties ofothese modulesoarepgiven.

    Gangyong Lee, S. Tariq Rizvi, and Cosmin S. Roman studied Dual Rickart modules. The main purpose of this paper is to define strong dual Rickart module. Let M and N be R- modules , M is called N- strong dual Rickart module (or relatively sd-Rickart to N)which is  denoted by M it is N-sd- Rickart if for every submodule A of M and every homomorphism fHom (M , N) , f (A) is a direct summand of N. We prove that for an R- module M , if R is M-sd- Rickart , then every cyclic submodule of M is a direct summand . In particular, if M<

Let R be a commutative ring with identity 1 and M be a unitary left R-module. A submodule N of an R-module M is said to be approximately pure submodule of an R-module, if for each ideal I of R. The main purpose of this paper is to study the properties of the following concepts: approximately pure essentialsubmodules, approximately pure closedsubmodules and relative approximately pure complement submodules. We prove that: when an R-module M is an approximately purely extending modules and N be Ap-puresubmodulein M, if M has the Ap-pure intersection property then N is Ap purely extending.

Let R be an associative ring with identity and let M be a unitary left R–module. As a generalization of small submodule , we introduce Jacobson–small submodule (briefly J–small submodule ) . We state the main properties of J–small submodules and supplying examples and remarks for this concept . Several properties of these submodules are given . Also we introduce Jacobson–hollow modules ( briefly J–hollow ) . We give a characterization of J–hollow modules and gives conditions under which the direct sum of J–hollow modules is J–hollow . We define J–supplemented modules and some types of modules that are related to J–supplemented modules and int

In the present paper, we will study the generalized ( p, q) -type and
generalized lower ( p, q) -type of an entire function in several complex
variables with respect to the proximate order with index pair ( p, q) are
defined and their coefficient characterizations are obtained.

The multi-focus image fusion method can fuse more than one focused image to generate a single image with more accurate description. The purpose of image fusion is to generate one image by combining information from many source images of the same scene. In this paper, a multi-focus image fusion method is proposed with a hybrid pixel level obtained in the spatial and transform domains. The proposed method is implemented on multi-focus source images in YCbCr color space. As the first step two-level stationary wavelet transform was applied on the Y channel of two source images. The fused Y channel is implemented by using many fusion rule techniques. The Cb and Cr channels of the source images are fused using principal component analysis (PCA).

Aerial manipulation of objects has a number of advantages as it is not limited by the morphology of the terrain. One of the main problems of the aerial payload process is the lack of real-time prediction of the interaction between the gripper of the aerial robot and the payload. This paper introduces a digital twin (DT) approach based on impedance control of the aerial payload transmission process. The impedance control technique is implemented to develop the target impedance based on emerging the mass of the payload and the model of the gripper fingers. Tracking the position of the interactional point between the fingers of gripper and payload, inside the impedance control, is achieved using model predictive control (MPD) approach.

The importance of the current study  lies in the  importance of the  Tax policy that  being considered one of the most important tools working on fulfilling  the  social,  financial  and economic  goals  and improving  the investment environment  in the country  to become  having the ability to  activate the  national economy. The current study  has  referred  that  (  Has  the  tax planning  practiced by  the Iraqi  contribution  companies  led to increase  the  far-term tax  outcome through  getting  benefit of   the monetary  funds  and expansion in&nbs