Let G be a graph, each edge e of which is given a weight w(e). The shortest path problem is a path of minimum weight connecting two specified vertices a and b, and from it we have a pre-topology. Furthermore, we study the restriction and separators in pre-topology generated by the shortest path problems. Finally, we study the rate of liaison in pre-topology between two subgraphs. It is formally shown that the new distance measure is a metric

In this paper we define and study new concepts of functions on fibrewise  topological spaces  over  B namely,  fibrewise  weakly  (resp., closure, strongly) continuoac; funttions which are analogous of weakly

(resp., closure,   strongly)   continuous  functions   and  the  main  result is   : Let  <p       : XY  be a fibrewise  closure  (resp.,  weakly,  closure, strongly,  strongly) continuous function,  where  Y  is fibrewise topological  space  over  B and  X  is  a  fibrewise set  which  has  the

in

studied, and its important properties and relationship with both closed and open Nano sets were investigated. The new Nano sets were linked to the concept of Nano ideal, the development of nano ideal mildly closed set and it has been studied its properties. In addition to the applied aspect of the research, a sample was taken from patients infected with viral hepatitis, and by examining the infected people and using closed and open (nano mildly. and nano ideal mildly) sets, the important symptoms that constitute the core of this dangerous examining the infected people and using closed and open (nano mildly. and nano ideal mildly) sets, the important symptoms that constitute the core of this dangerous disease.

     The objective of this paper is to show modern class of open sets which is an -open. Some functions via this concept were studied and the relationships such as continuous function strongly -continuous function -irresolute function -continuous function.

In this work we present a technique to extract the heart contours from noisy echocardiograph images. Our technique is based on improving the image before applying contours detection to reduce heavy noise and get better image quality. To perform that, we combine many pre-processing techniques (filtering, morphological operations, and contrast adjustment) to avoid unclear edges and enhance low contrast of echocardiograph images, after implementing these techniques we can get legible detection for heart boundaries and valves movement by traditional edge detection methods.

The current study suggested a thermal treatment as a necessary proactive step in improving the adsorption capacity of bio-waste for contaminants removal in wastewater. This approach was based on the experimental and histological investigation of biowaste pods shell. This investigation showed that these shells compose of parenchyma cells that store secondary metabolites compounds produced from cells were exhibited in present study. The results also reported that these compounds are extracted directly from the cells as soon as they are exposed to an aqueous solution, hampering their use as an adsorbent material. The increase in the weight of bio-waste adsorbent at unit liquid volume increases the production of secondary metabolites compounds

In this paper, we define a new type of pairwise separation axioms called pairwise semi-p- separation axioms in bitopological spaces, also we study some properties of these spaces and relationships of each one with the ordinary separation axioms in the bitopological spaces.

We introduce and discus recent type of fibrewise topological spaces, namely fibrewise bitopological spaces, Also, we introduce the concepts of fibrewise closed bitopological spaces, fibrewise open bitopological spaces, fibrewise locally sliceable bitopological spaces and fibrewise locally sectionable bitopological spaces. Furthermore, we state and prove several propositions concerning with these concepts.