This paper contains an equivalent statements of a pre-  space, where  are considered subsets of with the product topology. An equivalence relation between the preclosed set  and a pre-  space, and a relation between a pre-  space and the preclosed set  with some conditions on a function  are found. In addition, we have proved that the graph  of  is preclosed in if  is a pre-  space, where the equivalence relation  on  is open.

     On the other hand, we introduce the definition of a pre-stable ( pre-stable) set by depending on the concept of a pre-neighborhood, where we get that every stable set is pre-stable. Moreover, we obtain that

A new series polymers was synthesized from reaction starting material Bisacodyl A or [(2-Pyridinylmethylene) di-4, 1-phenylene di acetate] with hydrogen bromide, then the products were polymerized by addition polymerization from used adipoyl and glutaroyl chloride. The structure of these compounds was characterized by FT-IR, melting points, TLC, X-Ray, DSC and 1H-NMR for starting material. These compounds were also screened for their antibacterial activists?

When scheduling rules become incapable to tackle the presence of a variety of unexpected disruptions frequently occurred in manufacturing systems,  it is necessary to develop a reactive schedule which can absorb the effects of such disruptions. Such responding requires efficient strategies, policies, and methods to controlling production & maintaining high shop performance. This can be achieved through rescheduling task which defined as an essential operating function to efficiently tackle and response to uncertainties and unexpected events. The framework proposed in this study consists of rescheduling approaches, strategies, policies, and techniques, which represents a guideline for most manufacturing companies operatin

The aim of study is to shed light on an Islamic city which is unknown for a lot of people, it can have an old history in Parisian country and many events happen with it , This city is characterized with political, military, economic scientific ,and social features, This city is called Zanjan and it is one of the most important cities because it dates back to the period of post history and it has a good geographic location whereas it was, passage for trade caravans to pass through as well its land which was specialized in agriculture and industry. the study follows chronological order of historical events for the city, one of the most significant conclusions is to think that this city does belong to artifacts and it is an old city and it

Background. Body mass index (BMI) is a person's weight in kilograms (or pounds) divided by the square of height in meters (or feet). Obesity affects a wide spectrum of age groups, from the young to the elderly, and there are several eye diseases related to obesity like diabetic retinopathy, floppy eyelid syndrome, retinal vein occlusion, stroke-related vision loss, age-related macular degeneration, and possibly, refractive errors. Refractive errors (RE) are optical imperfections related to the focusing ability of the eye and are the main cause of visual impairment which may result in missed education and employment opportunities, lower productivity and impaired quality of life. Aim. The study aimed to find an association between bod

Background: Joubert syndrome (JS) is a very rare autosomal recessive disorder characterized by agenesis of cerebellar vermis, abnormal eye movements, respiratory irregularities, and delayed generalized motor development. Retinal dystrophy and cystic kidneys may also be associated with this clinical syndrome. The importance of recognizing JS is related to the outcome and its potential complications. This syndrome is difficult to diagnose clinically because of its variable phenotype. Its neuroimaging hallmarks include the characteristic molar tooth sign and bat wing-shaped fourth ventricle