In this paper, we show that for the alternating group An, the class C of n- cycle, CC covers An for n when n = 4k + 1 > 5 and odd. This class splits into two classes of An denoted by C and C/, CC= C/C/ was found.

This paper includes the synthesis of some new nucleoside analogues starting with 2-substituted benzimidazole derivative (7-9), that synthesized by condensation of O-phenylenediamine with p-chloro benzaldehyde and two substituted benzoic acid , which on nucleophilic substitution with propargyl bromide gave a new N-substituted compounds (10-12). D-Fructose and D-galactose were chosen as a sugar moiety which were protected, brominated and azotated to give azido sugars (5) and (6), then they were subjected to 1,3-dipolar cycloaddition reaction with N-substuted compounds afforded bloked nucleoside analoges (13-16), which after hydrolysis gave our target the free nucleoside analogues (17-20). All prepared compounds were identified by FT-IR

This search include the synthesis of some new 1,3-oxazepine derivatives have been prepared, starting from reaction of L-ascorbic acid with dry acetone in presence of dry hydrogen chloride afforded the acetal (I). Treatment of the latter with p-nitrobenzoyl chloride in dry pyridine yielded the ester (II) which was dissolved in (65%) acetic acid in absolute ethanol yielded the glycol (III). The reaction of the glycol (III) with sodium periodate in distilled water at room temperature produced the aldehyde (IV). The compound (V) [2-amino-5-mercapato-1,3,4-thiadiazole] was prepared through the reaction of thiosemicarbazide with carbon disulphide (CS2) in entity of anhydrous (Na2CO3) in (abs. ethanol ). Compound (VI) [2-(5-mercapto-1,3,4-thiadiaz

The concept of -closedness, a kind of covering property for topological spaces, has already been studied with meticulous care from different angles and via different approaches. In this paper, we continue the said investigation in terms of a different concept viz. grills. The deliberations in the article include certain characterizations and a few necessary conditions for the -closedness of a space, the latter conditions are also shown to be equivalent to -closedness in a - almost regular space. All these and the associated discussions and results are done with grills as the prime supporting tool.

The objective of this paper is to define and introduce a new type of nano semi-open set which called nano -open set as a strong form of nano semi-open set which is related to nano closed sets in nano topological spaces. In this paper, we find all forms of the family of nano -open sets in term of upper and lower approximations of sets and we can easily find nano -open sets and they are a gate to more study.  Several types of nano open sets are known, so we study relationship between the nano -open sets with the other known types of nano open sets in nano topological spaces. The Operators such as nano -interior and nano -closure are the part of this paper.

The chemical properties of chemical compounds and their molecular structures are intimately connected. Topological indices are numerical values associated with chemical molecular graphs that help in understanding the physicochemical properties, chemical reactivity and biological activity of a chemical compound. This study obtains some topological properties of second and third dominating David derived (DDD) networks and computes several K Banhatti polynomial of second and third type of DDD.

 In this paper, we introduce and study the concept of a new class of generalized closed set which is called generalized b*-closed set in topological spaces ( briefly .g b*-closed) we study also. some of its  basic properties and investigate the relations between the associated topology. 

In this paper we introduce a new type of functions called the generalized regular
continuous functions .These functions are weaker than regular continuous functions and
stronger than regular generalized continuous functions. Also, we study some
characterizations and basic properties of generalized regular continuous functions .Moreover
we study another types of generalized regular continuous functions and study the relation
among them