This paper focuses on developing a strategy to represent the -connected ominoes using an abacus. We use the idea of -connected ominoes with respect to a frame in modelling nested chain abacus. Then, we formulate and prove the unique connected partition for any -connected ominoes. Next, the topological structure of nested chain abacus is presented.

In this paper, we introduced some new definitions on P-compact topological ring and PL-compact topological ring for the compactification in topological space and rings, we obtain some results related to P-compact and P-L compact topological ring.

     In this work, we present new types of compact and Lindelöf spaces and  some facts and results related to them. There are also types of compact and Lindelöf functions and the relationship between them has been investigated. Further, we have present some properties and results related to them.

The mean age of AS patients was (35.0 ± 9.8) years.When the patients and control subjects were divided into different age groups (>40, 30-40, <30 years), the differences were not significantin terms of disease prevalence. The results also showed that the percentage of male patients is higher than that of females. There was no significant difference (P?0.05) between patients and controls in the distribution of males and females.Most of the patients had the disease for a period of 5 years or higher, with a disease severity of ? 2.1 and functional disability degree of I, II. The resultsshoweddifferent patterns of distribution for the three tested cytokines. A significant increase in the level of TNF-?, anon-significantincrease i

With the advancement of modern radiotherapy technology, radiation dose and dose distribution have significantly improved. as part of Natural development, interest has recently been renewed by treatment, especially in the use of heavy charged particles, because these radiation types serve theoretical advantages in all biological and physical aspects. The interactions of alpha particle with matter were studied and the stopping powers of alpha particle with Bone Tissue were calculated by using Zeigler’s formula and SRIM software, also the Range for this particle were calculated by using Mat lab language for (0.01-1000) MeV alpha energy.

Background: Until today, the etiology of recurrent aphthous stomatitis (RAS) remains unknown, although hints of its etiologic basis lay on genetic susceptibility, infectious agents and alterations in immune mechanics.
Objectives: The aim of this study was to investigate the possible alterations in salivary tumor necrosis factor-alpha (TNF-α) and Immunoglobulin-A (IgA) level in patients with RAS and its relation with clinical types of disease.
Subjects and Methods: Salivary TNF-α levels were investigated in 50 RAS patients and 25 healthy controls by enzyme-linked immunosorbent assay (ELISA). Single radial immune diffusion method was used for estimation of IgA level in two studied groups.
Results: S

     This article is devoted to presenting results on invariant approximations over a non-star-shsped weakly compact subset of a complete modular space by introduced a new notion called S-star-shaped with center f:  if   be a mapping and , . Then the existence of common invariant best approximation is proved for Banach operator pair of mappings by combined the hypotheses with Opial’s condition or demi-closeness condition

     In this paper,there are   new considerations about the dual of a modular spaces and weak convergence. Two common fixed point theorems for a -non-expansive mapping defined on a star-shaped weakly compact subset are proved,  Here the conditions of affineness, demi-closedness and Opial's property play an active role in the proving our results.

In this paper, some basic notions and facts in the b-modular space similar to those in the modular spaces as a type of generalization are given.  For example, concepts of convergence, best approximate, uniformly convexity etc. And then, two results about relation between semi compactness and approximation are proved which are used to prove a theorem on the existence of best approximation for a semi-compact subset of b-modular space.