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.

In this study the new azo compounds (3compounds) for nitrogen bases (Adenine and Cytosine) are synthesized through two reaction steps (formation of diazonium salt and coupling reaction). The compounds have been characterized by FTIR, melting point, and ultra-violate (UV) spectra. All synthesized compounds have been estimated in vitro for their antimicrobial activities against two species of bacteria(E.coli, S.aureus)and one kind of fungi ( Aspergillus flavus) .The results show that these compounds have very good antibacterial and antifungal activities especially compounds 1 and 3.To study the effect of these compounds were making some physiological tests on rats are made ,the results of hematological study showed decreasing level of

           Solvent- free thermal heating, one-pot condensation of acetophenone, ethyl cyanoacetate or malononitrle and substituted Aromatic aldehyde, ammonium acetate give, 2oxo-3-cyano-4-substituted Aryl-6-phenyl pyridine [I]a-h , or  2-amino-3-cyano-4-substituted Aryl-6-phenyl pyridine derivatives[II]a-f , respectively.              Treatment of compounds 2-oxo-3-cyano-4-substituted Aryl-6-phenyl pyridine with phosphorous penta sulphide (P2S5), give 2-thioxo-3-cyano-4-substituted Aryl-6-phenyl pyridine derivatives[III]a-c .             All prepared compounds

Four Co(II), (C1); Ni(II), (C2); Cu(II), (C3) and Zn(II), (C4) chelates have been synthesized with 1-(4-((2-amino- 5‑methoxy)diazenyl)phenyl)ethanone ligand (L). The produced compounds have been identified by using spectral studies, elemental analysis (C.H.N.O), conductivity and magnetic properties. The produced metal chelates were studied using molar ratio as well as sequences contrast types. Rate of concentration (1 ×10􀀀 4 - 3 ×10􀀀 4 Mol/L) sequence Beer’s law. Compound solutions have been noticed height molar absorptivity. The free of ligand and metal chelates had been applied as disperse dyes on cotton fabrics. Furthermore, the antibacterial activity of the produced compounds against various bacteria had been investigated. F

Four Co(II), (C1); Ni(II), (C2); Cu(II), (C3) and Zn(II), (C4) chelates have been synthesized with 1-(4-((2-amino- 5‑methoxy)diazenyl)phenyl)ethanone ligand (L). The produced compounds have been identified by using spectral studies, elemental analysis (C.H.N.O), conductivity and magnetic properties. The produced metal chelates were studied using molar ratio as well as sequences contrast types. Rate of concentration (1 ×10􀀀 4 - 3 ×10􀀀 4 Mol/L) sequence Beer’s law. Compound solutions have been noticed height molar absorptivity. The free of ligand and metal chelates had been applied as disperse dyes on cotton fabrics. Furthermore, the antibacterial activity of the produced compounds against various bacteria had been investigated. F

Polymer metal complexes of poly ethylene glycol acetal and Ag (I), Cu (II), Ni (II), Mn (II), Co (III) and Hg (II) were prepared from the reaction of PEG with aldehyde derived fromErythro-ascorbic acid (pentulosono-ɣ-lactone-2, 3- enedianisoate). All these compounds were characterized by Thin Layer Chromatography (TLC) and FTIR spectra and aldehyde was also characterized by (U.V-Vis), 1HNMR,13CNMR, and mass spectra. It has been established that, the polymer and its metal complexes showedgood activities against four pathogenic bacteria (Escherichia coli , Klebsiellapneumonae,Staphylococcusaureus, Staphylococcus Albus) and two fungal (Aspergillus Niger,Yeast). The polymer metal complexes showed higher activity than the free polymer.Theorder

Polymer metal complexes of poly ethylene glycol acetal and Ag (I), Cu (II), Ni (II), Mn (II), Co (III) and Hg (II) were prepared from the reaction of PEG with aldehyde derived from Erythro-ascorbic acid (pentulosono-ɣ-lactone-2, 3- enedianisoate). All these compounds were characterized by Thin Layer Chromatography (TLC) and FTIR spectra and aldehyde was also characterized by (U.V-Vis), 1HNMR,13CNMR, and mass spectra. It has been established that, the polymer and its metal complexes showed good activities against four pathogenic bacteria (Escherichia coli ,Klebsiellapneumonae, Staphylococcusaureus, Staphylococcus Albus) and two fungal (Aspergillus Niger,Yeast). The polymer metal complexes showed higher activity than the free polymer. The

A -set in the projective line is a set of  projectively distinct points. From the fundamental theorem over the projective line, all -sets are projectively equivalent. In this research, the inequivalent -sets in have been computed and each -set classified to its -sets where  Also, the  has been splitting into two distinct -sets, equivalent and inequivalent.

This research includes the synthesis of some new nucleoside analogues via 1,3-
dipolar cycloadditon reaction starting with 1,3,4,6- Tetra- O-benzoyl –2 – azido-2-
deoxy- β – D- fructofuranose (F3) which was prepared in three steps ,protection
(benzoylation ) , bromination and azotation ((F1-F3) from D- fructose as sugar
moiety while the base moiety , 2-substituted benzimidaole derivatives (A1-A5)
were synthesized by condensation of O-phenylenediamine with different aromatic
aldehyde .
Nucleophilic substitution of propargyl bromide with benzimidaole derivatives (gave
the N- substituted compounds (B1-B5) which were reacted with azido sugar through
1,3- dipolar cycloadditon reaction to give nucleoside analo