The investigation of determining solutions for the Diophantine equation  over the Gaussian integer ring for the specific case of  is discussed. The discussion includes various preliminary results later used to build the resolvent theory of the Diophantine equation studied. Our findings show the existence of infinitely many solutions. Since the analytical method used here is based on simple algebraic properties, it can be easily generalized to study the behavior and the conditions for the existence of solutions to other Diophantine equations, allowing a deeper understanding, even when no general solution is known.

Multipole mixing ratios for gamma transition populated in from reaction have been studied by least square fitting method also transition strength ] for pure gamma transitions have been calculated taking into account the mean life time for these levels .

The study deals with an analysis of the contents of the publications of the campaign (Together to defeat Corona), which was established by the United Nations Development Program in Iraq in the face of the Covid 19 virus.The research problem raises a main question:What are the implications of the campaign (Together to defeat Corona) of the United Nations Development Program (Iraq office) in addressing the Covid-19 virus in Iraq?From this main question, several sub-questions emerged, which were answered by this study in its chapters and investigations, including regarding the contents of advertisements, photos and videos for the publications of the (Together to Defeat Corona) campaign for the United Nations Iraq Office on their Facebook pageA

There are several oil reservoirs that had severe from a sudden or gradual decline in their production due to asphaltene precipitation inside these reservoirs. Asphaltene deposition inside oil reservoirs causes damage for permeability and skin factor, wettability alteration of a reservoir, greater drawdown pressure. These adverse changing lead to flow rate reduction, so the economic profit will drop. The aim of this study is using local solvents: reformate, heavy-naphtha and binary of them for dissolving precipitated asphaltene inside the oil reservoir. Three samples of the sand pack had been prepared and mixed with a certain amount of asphaltene. Permeability of these samples calculated before and after mixed with asphaltenes. Then, the

Endoglucanase produced from Aspergillus flavus was purified by several steps including precipitation with 25 % ammonium sulphate followed by Ion –exchange chromatography, the obtained specific activity was 377.35 U/ mg protein, with a yield of 51.32 % .This step was followed by gel filtration chromatography (Sepharose -6B), when a value of specific activity was 400 U/ mg protein, with a yield of 48 %. Certain properties of this purified enzyme were investigated, the optimum pH of activity was 7 and the pH of its stability was 4.5, while the temperature stability was 40 °C for 60 min. The enzyme retained 100% of its original activity after incubation at 40 °C for 60 min; the optimum temperature for enzyme activity was 40 °C.

Beta-lactamase was purified from local isolate Klebsiella pneumonia by several steps included precipitation with ammonium sulphate at 20-40% saturation, DEAE- ion exchange chromatography and gel filtration on Sephacryl S-200 column. The obtained purification fold and recovery were 32.66; 47.04% respectively. The characterization of the purified beta-lactamase showed that the molecular weight was about 4000 daltons as determined by gel filtration.Purified enzyme had an optimal pH of 7 for activity and an optimal stability between pH 6.5-7.5, results shows that the optimal temperature appear to be 35 ? C .During storage the enzyme retained 72% at -20 ? C and retained 25% of the activity at the same period at 4 ? C.

Twenty isolates of Serratia marcescens were isolated from inflammation of the urinary tract (UTI)., These isolates were found to produce hemolysin as indicated by blood agar plates in which the hemolysis of red blood cell indicate a positive result. Isolates were selected according to their hemolysis activity by measuring absorbance of hemoglobin at 405 nm that released from red blood cell. Hemolysin was completely purified using 50-75% saturation of ammonium sulphate followed by ion exchange chromatography with DEAE-cellulose then gel filtration chromatography by sepharose 4B. Accordingly molecular weight for the purified toxin was estimated as 45 KD.

(28)Bacterial local isolates of Bacillus sp. were obtained from soil samples. Isolates were tested for thermostable alpha- amylase production on solid media; fifteen isolates were able to develop clear zone around the bacterial growth after floating the plates with iodine reagent (Lugol's solution). There were further tested in submerged culture which led to selection of Bacillus sp. H14since it was the most efficient .Microbial and biochemical tests showed that the local isolate Bacillus sp.H14was refered to the species B.licheniformis that signed as H14 was refered to the species B.licheniformis H14 .,To get ahigher yield of alpha – amylase(48.70unit/mg protein) production from the local isolate B.licheniformis H14 . This study used