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

Background. The presence of black triangles around the dental implant-supported prosthesis and the failure to construct adequate papillae around them bothers dental implantologists. Peri-implant surgical soft tissue management will improve esthetics, function, and implant survival. Aim. To compare the effects of rolled and nonrolled U-shaped flaps combined with a temporary crown in enhancing the soft tissue around dental implants. Materials and Methods. Forty patients were included in this study; all patients were operated on by the same maxillofacial surgeon at Al-Iraq specialized dental clinics from January 2019 to January 2020. Patients were divided randomly into two groups: group A: at the second stage of implant surgery, a U-sh

The: currency Auction is one of the monetary policy tools created after 2003, in order to keep pace with the changes that the monetary and financial policies will witness from financial openness and expectations of high levels of liquidity after international economic restrictions. It is necessary to re-evaluate the work of the currency Auction from time to time and observation its efficiency in adjustment the exchange rate And its reflection on the general level of prices as one of the objectives of its inception, and during the analytical aspect, it was confirmed that the currency Auction for selling the currency had a major role in adjustment the exchange rate and controlling inflation levels, due to the market’s dependence

     A graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense related. The objects correspond to mathematical abstractions called vertices (also called nodes or points) and each of the related pairs of vertices is called an edge (also called link or line). A directed graph is a graph in which edges have orientation. A simple graph is a graph that does not have more than one edge between any two vertices and no edge starts and ends at the same vertex.  For a simple undirected graph G with order n, and let  denotes its complement. Let δ(G), ∆(G) denotes the minimum degree and maximum degree of G respectively. The complement degree polynomial of G is the polynomial CD[G,x]= , where C

A theoretical investigation is carried out to study the effect of a pencil electron beam propagating inside the plasma region determining the hydrodynamic densities distribution with the aid of numerical analysis finite deference method (FDM).The plasma is generated and trapped by annular electron beams of fixed electron density 1x1014 m-3. The result of the study shows that the hydrodynamic density behaves as the increase in pencil electron beam. The hydrodynamic density ratio goes to more than double as the increase in pencil electron beam density to 1x1018 m-3.

This study focuses on the slab-beam interaction in one-way systems. In the context of this study, slab-beam interaction means how beam deflection can affect moment distribution in one-way slabs. This interaction is usually neglected in the traditional approximate analysis that is adopted in engineering practice and design codes. Slab positive moments have been considered as indicators on the accuracy of approximate methods, as they overestimate negative moments while underestimating positive moments.

After proposing of effecting parameters in slab-beam interaction including of panel length and width, beam dimensions, and slab thickness, Buckingham’s  theorem has been adopted to transform the dimensional-mo

The research in the variables of the political process and government stability tried to show the impact of the political process on political stability first, and then on government stability second, given that the political process that was established in 2005 was aimed at achieving legitimacy, and its most prominent tools are elections, leading to achieving political stability, including government stability. The issue of governmental stability is one of the important issues in Iraq, but it has not been achieved, as a result of several factors, including problems in political action, as the political process has not succeeded in leading Iraq to stability.

Biological activity of the carotenoids which are produced fromchemically-mutaed local isolate of Rhodotorula mucilaginosawas studied. The results showed variation of inhibition activity of caritenoids against different types of pathogenic bacteria include, Staph aureus, E. coli, B. subtilis and Salmo. typh., the number declined from 2×107cell/ml to 2×104, 5×104, 2×105, 9×105 cell/ml respectively after 24hour. The produced carotenoids from alocal mutant Rhodotorula mucilaginosa revealed an antioxidant activity as free radical removal of 85.6%. Carotinoides revealed a highest stability in petroleum ether solvent for 30 days at room temperature. It found that the pigment was more stability in sesame oil compared with sun flower and coc