In this paper we generalize Jacobsons results by proving that any integer  in   is a square-free integer), belong to . All units of  are generated by the fundamental unit  having the forms

our generalization build on using the conditions

This leads us to classify the real quadratic fields  into the sets  Jacobsons results shows that  and Sliwa confirm that  and  are the only real quadratic fields in .

In this paper we generalize Jacobsons results by proving that any integer  in   is a square-free integer), belong to . All units of  are generated by the fundamental unit  having the forms

Our generalization build on using the conditions

This leads us to classify the real quadratic fields  into the sets  Jacobsons results shows that  and Sliwa confirm that  and  are the only real quadratic fields in .

The present study is an attempt to throw light on the nature of the US policy regarding the Middle East region as portrayed by AI-Sabah, Al-Mashriq and Tariq Al-Shaab papers over a period of three months from 1st of July to 30th of September 2013.

In writing this study, a number of goals have been set by the researcher. These goals may include but in no way limited to the nature of the US image as carried by the above three papers, the nature of the topics tackled by them and the nature of the Arab countries which received more and extensive coverage than others.

A qualitative research approach is proposed for the study. This approach has allowed the researcher to arrive at definite answers for the possible questions rais

The aim of this study is to identify the effect of enabling the effectiveness of the work of the audit committees in private commercial banks and to identify the extent of awareness of the importance of empowerment in the work of these committees, especially as it is known that these committees, especially the inspection committees that go to private banks and from various sources including committees of the Central Bank of Iraq Committees of the Securities Commission and finally committees of the external audit offices, through an analysis of the determinants of empowerment in the performance of the most important work of the audit committees, namely: supervising the process of preparing reports, supervising the system of intern

This study investigates the improvement of Iraqi atmospheric gas oil characteristics which contains 1.402 wt. % sulfur content and 16.88 wt. % aromatic content supplied from Al-Dura Refinery by using hydrodesulfurization (HDS) process using Ti-Ni-Mo/γ-Al₂O₃ prepared catalyst in order to achieve low sulfur and aromatic saturation gas oil. Hydrodearomatization (HDA) occurs simultaneously with hydrodesulfurization (HDS) process. The effect of titanium on the conventional catalyst Ni-Mo/γ-Al₂O₃ was investigated by physical adsorption and catalytic activity test.Ti-Ni-Mo/γ-Al₂O₃ catalyst was prepared under vacuum impregnation condition to ensure efficient pr

This study investigates the improvement of Iraqi atmospheric gas oil characteristics which contains 1.402 wt. % sulfur content and 16.88 wt. % aromatic content supplied from Al-Dura Refinery by using hydrodesulfurization (HDS) process using Ti-Ni-Mo/γ-Al2O3 prepared catalyst in order to achieve low sulfur and aromatic saturation gas oil. Hydrodearomatization (HDA) occurs simultaneously with hydrodesulfurization (HDS) process. The effect of titanium on the conventional catalyst Ni-Mo/γ-Al2O3 was investigated by physical adsorption and catalytic activity test. Ti-Ni-Mo/γ-Al2O3 catalyst was prepared under vacuum impregnation condition to ensure efficient precipitation of metals within the carrier γ-Al2O3. The loading percentage of met

In this study, the acid-alkaline transesterification of refined coconut seed oil (RCOSO) to fatty acid methyl ester was followed by the production of a trimethylolpropane-based thermosensitive biolubricant using potassium hydroxide, and its physicochemical characteristics were evaluated. The American Standard Test for Materials (ASTM) was employed to ascertain the biolubricant's pour point and index of viscosity, which were found to be -4 ^oC and 283.75, respectively. The opposite connection between lubricant viscosity and temperature was shown by the measured viscosities at varied transesterification to be transformed into biodiesel. Following this, a biolubricant was created by further transesterifiedtemperature. The ester gr

The aim of this research is to benefit from recycl the aircraft waste oils which is discarded in sewage network, to be used in preparation of greases for industrial purposes and to reduce the environmental pollution. In this research synthetic greases were prepared with special specifications by mixing the waste oils after treating with (silica gel as adsorbent agent, and filtration to precipitate impurities then heated to 110 C? to get rid of water) bentonite produced in Iraq which is available and cheap with existence of high density polyethylene at specific conditions of ( heating and mixing) . The best weight proportion were reached, then paraffin wax and additives were added to improve the properties of grease and give the

In this paper, the behavior of spliced steel girders under static loading is investigated. A group of seven steel I-girders were tested experimentally. Two concentrated loads were applied to each specimen at third points and the load was increased incrementally up to the yield of the specimen. Two types of splices were considered; the bearing type and the friction-grip type splices. For comparison, an analytical study was made for the tested girders in which the finite element analysis program (Abaqus) was used for analysis. It was found that the maximum test load for spliced girders with bearing type splices was in the range of (34%) to (67%) of the maximum test load for the reference girder. For girders spliced by using friction-grip t