This paper generalizes and improves the results of Margenstren, by proving that the number of -practical numbers  which is defined by   has a lower bound in terms of . This bound is more sharper than Mangenstern bound when  Further general results are given for the existence of -practical numbers, by proving that the interval contains a -practical for all

The concept of the order sum graph associated with a finite group based on the order of the group and order of group elements is introduced. Some of the properties and characteristics such as size, chromatic number, domination number, diameter, circumference, independence number, clique number, vertex connectivity, spectra, and Laplacian spectra of the order sum graph are determined. Characterizations of the order sum graph to be complete, perfect, etc. are also obtained.

An energy and exergy thermodynamic analysis using EES program was done for a domestic refrigerator working with R-134a using vapor compression refrigeration cycle. The analysis deals with the system component, i.e. compressor, condenser, evaporator and the expansion device. The analysis depends on the entropy generation minimization approach to improve the refrigerator performance by exploring the optimum design points. These design points were derived from three different theories governing the entropy generation minimization using exergy analyzing method. These theories were first applied to find the optimum balance between the hot inner condenser area and the cold inner evaporator area of the refrigerator and between

In this study, a brand-new graph definition known Associate graph of a ring R denote Ass(R) is present, where the graph’s vertices stand in for R’s elements s.t, any two vertices α and β  merage by an edge if and only if α = rβ and β = sα. In this paper, we investigated some new property of Ass (R)  are studied., the complement of Ass (R) is finally defined and a few of its characteristics are researched.

Time crosses one of the most important principles that are agreed upon in contracts, because the temporal dimension has a significant impact on all contract provisions and is not limited to a certain group of them. French and Arab legal jurists alike called for this dimension to be given special attention. That is the term of the contract term; To try to limit the temporal elements, clarify their provisions and distinguish between them, but in the Arab world it did not receive the same attention that it received in the West.

Talking about the importance of the imagination in sculptural works has a role in shaping the collective memory and moving it, whether it is an artistic, social, religious or historical imagination. Because it documents the cultural heritage of any country through the structural structure of the sculptural form, and its connotations and symbols, which are considered an effective aesthetic image in the crystallization of various intellectual concepts and contents.
Therefore, the study aimed to know the employment of the popular imagination in the artistic works of the sculptor (Muhammad Ghani Hikmat), as well as to reveal the artistic and aesthetic features of the folklore and its reflection on its artistic style. And the references

Communication and media technologies have seen several changes in the last two decades. These changes put the basis for the emergence of communicative and media practices on several levels including political, economic, technological and artistic. And these, in turn, led to the emergence of new media trends and phenomena in various areas of media worthily to be studied and researched for example online advertisings.                                                                                          Online advert

This research is a modest contribution to Put the finishing touches on therole of managerial leaders represented by static companies belonging to theministry of transport and communication/ [iraq] . the research has adopted thedescriptive. Analytical methodology and field study technique, using thequestionnaire as a tool for data collection [58] forms Nar, been distributed tothe research sample. The sample has been deliberately selected {generalmanager, as assistant general manager and heads of department}. Thequestionnaires and the main hypotheses of represented by the existence ofsignificant correlation and impact between successful managerial leadershipand crisis management using the software {SPSS}, were analyzed. Resultswere identic

The concept of Justice in the Ancient western political thought The title of research is the concept of Justice in the ancient western political heritage. It includes the definition of the idea of Justice as well as its evolution and relation to the Law the study discussed two main period . the first one was regarding the concept of Justice in the Greek political thought ( sophists ,Socrates , plato , Aristotle ,Epicureans , and stoics ) . While the second ocused on the concept of justice in Roman political thought Via their great thinkers as polypus and Cicero .

Interval lies at rest in the speech; so as to improve to speak out, which is how Evaln Koran by other speech, called separators; for the secession of Alkalaman then, as the last verse separation between it and its aftermath in the Al-Jinn had a comma significant impact on the statement of the case of gin, as portrayed interval Gin and like the distraught after hearing of the Koran
This Sura carried various types of repetition represent wordy repeat itself and repeat some of the votes as well as the repeated sound of a thousand who remained in Sura breaks Bhaa also repeat the voice of the shredder breaks
Linguistic sounds of Koranic breaks played a significant role in a statement carried by the linguistic meaning of these interludes