Let R be a commutative ring with identity, and let M be a unitary left R-module. M is called special selfgenerator or weak multiplication module if for each cyclic submodule Ra of M (equivalently, for each submodule N of M) there exists a family {fi} of endomorphism of M such that Ra = ∑_i▒f_i (M) (equivalently N = ∑_i▒f_i (M)). In this paper we introduce a class of modules properly contained in selfgenerator modules called special selfgenerator modules, and we study some of properties of these modules.

The researcher [1-10] proposed a method for computing the numerical solution to quasi-linear parabolic p.d.e.s using a Chebyshev method. The purpose of this paper is to extend the method to problems with mixed boundary conditions. An error analysis for the linear problem is given and a global element Chebyshev method is described. A comparison of various chebyshev methods is made by applying them to two-point eigenproblems. It is shown by analysis and numerical examples that the approach used to derive the generalized Chebyshev method is comparable, in terms of the accuracy obtained, with existing Chebyshev methods.

Benign prostatic hyperplasia (BPH) is a prevalent condition among elderly and middle-aged men characterized by symptoms such as dysuria, urinary incontinence, and frequent micturition. The gold standard procedure for relieving BPH symptoms is transurethral resection of the prostate (TURP). However, some patients undergoing TURP are at risk of developing urinary tract infections (UTIs) due to uropathogenic bacteria. This prospective study aimed to investigate post TURP bacteruria alongside with multifactoria risk factors that implicated postoperatively compared to preoperative and intraoperative periods. Ninety patients undergoing TURP and 30 control subjects were included in the study. Urine specimen for urine analysis from pateints

Cultural awareness is becoming increasingly important in an effort to reinvigorate local urban heritage into modern production. Considering that, one of the most significant aspects of sustainability is preserving socio-cultural and environmental specificities and restoring local heritage. The aim is influencing the local identity by promoting sustainability through the revival of Islamic geometric patterns (IGP) and their performance as effective components in local urban facades. A set of indicators related to the purposes for reusing IGP in facades within three main performances (decorative, structural, and functional). Their sustainability was studied and extrapolated, and a checklist has developed and tested by Likert scale on

Background: One of the most prevalent procedures in oral surgery is the removal of impacted mandibular third molars, typically accompanied by trismus, edema, and pain. Several methods and biomaterials were implemented to mitigate or avoid these surgical problems. Objectives: To evaluate the efficiency of chlorhexidine gel (WISDOM®) in minimizing postoperative sequelae associated with the impacted mandibular third molar that will be surgically extracted and its role in promoting early soft tissue closure of the surgical site. Methods: The study design was a double-masked and randomized, controlled clinical study that included healthy patients needing the removal of a mandibular third molar through surgery. The participants were rand

The article emphasizes that 3D stochastic positive linear system with delays is asymptotically stable and depends on the sum of the system matrices and at the same time independent on the values and numbers of the delays. Moreover, the asymptotic stability test of this system with delays can be abridged to the check of its corresponding 2D stochastic positive linear systems without delays. Many theorems were applied to prove that asymptotic stability for 3D stochastic positive linear systems with delays are equivalent to 2D stochastic positive linear systems without delays. The efficiency of the given methods is illustrated on some numerical examples. HIGHLIGHTS Various theorems were applied to prove the asymptoti