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.

Schiff base (methyl 6-(2- (4-hydroxyphenyl) -2- (1-phenyl ethyl ideneamino) acetamido) -3, 3-dimethyl-7-oxo-4-thia-1-azabicyclo[3.2.0] heptane-2-carboxylate)Co(II), Ni(II), Cu (II), Zn (II), and Hg(II)] ions were employed to make certain complexes. Metal analysis M percent, elemental chemical analysis (C.H.N.S), and other standard physico-chemical methods were used. Magnetic susceptibility, conductometric measurements, FT-IR and UV-visible Spectra were used to identified. Theoretical treatment of the generated complexes in the gas phase was performed using the (hyperchem-8.07) program for molecular mechanics and semi-empirical computations. The (PM3) approach was used to determine the heat of formation (ΔH˚f), binding energy (ΔEb), an

In this paper, the proposed phase fitted and amplification fitted of the Runge-Kutta-Fehlberg method were derived on the basis of existing method of 4(5) order to solve ordinary differential equations with oscillatory solutions. The recent method has null phase-lag and zero dissipation properties. The phase-lag or dispersion error is the angle between the real solution and the approximate solution. While the dissipation is the distance of the numerical solution from the basic periodic solution. Many of problems are tested over a long interval, and the numerical results have shown that the present method is more precise than the 4(5) Runge-Kutta-Fehlberg method.

Teeth decay is one of the distinctive diseases which infects the children of generation .though it is not adeacly disease, it leads to huge problems , especially if it is infect children in their early childhood .There are many factors which lead to this disease , and to achieve the goal of this research in trying to stop the most important nutrition factors and others reasons that have relation with this disease .The practical application has been done for four months on children , who have been chosen randomly of the age ( 3-5 years) of both sexes , infected with teeth decay , from Baghdad city ( al karkh and al rusafa ) , the number of children reached to 90 child , divided into tow groups 45 male , and( 45 )female .Again they have be

Diazotization reaction between quinolin-2-ol and (2-chloro-1-(4-(N-(5-methylisoxazol-3-yl)sulfamoyl)phenyl)-2l4-diazyn-1-ium was carried out resulting in ligand-HL, this in turn reacted with the next metal ions (Ni²⁺, Pt⁴⁺, Pd²⁺, and Mn²⁺)  forming stable complexes with unique geometries such as (tetrahedral for both Ni²⁺ and Mn²⁺, octahedral for Pt⁴⁺ and square planer for Pd²⁺ ). The creation of such complexes was detected by employing spectroscopic means involving ultraviolet-visible which proved the obtained geometries, fourier transfer proved the formation of azo group and the coordination with metal ion through it. Pyrolysis (TGA &

            Spectrophotometric method was developed for the determination of copper(II) ion. Synthesized (2,2[O-Tolidine-4,4-bis azo]bis[4,5-diphenyl imidazole]) (MBBAI) was used as chromogenic reagent at pH=5. Various factors affecting complex formation, such as, pH effect, reagent concentration, time effect and temperature effect, have been considered and studied. Under optimum conditions concentration ranged from (5.00-80.00) µg/mL of copper(II) obeyed Beer`s Low. Maximum absorption of the complex was 409nm with molar absorpitivity 0.127x10⁴ L mol^-1 cm^-1. Limit of detection(LOD) and Limit of quantification were 1.924 and 6.42 μg/mL, respectively.

In the cuurent article, the photophysical properties of 3,6-bis(5-bromothiophen-2-yl)-2,5-bis(2-ethylhexyl)-2,5-dihydropyrrolo[3,4-c]pyrrole-1,4-dione were investigated. The visible absorption bands at 527, 558 and 362 nm in propylene carbonate and the compound was found to be fluorescent in solution and in the plastic film with emission wavelengths between 550- 750 nm. The Stokes Shift of P.C., acetonitrile, diethyl ether, Tetrahydrofuran THF, cyclohexane, dibutyl ether, and dichloromethane DCM  are 734, 836, 668, 601, 601, 719, and 804 cm^-1 in respectively. The Stokes Shift Δ was less in THF and cyclohexane, than the solvents, which indicates that the energy loss is less between the excitation and fluorescence states. The

Oxazepine [1] is non – nomologous seven –member ring  that contain two netroatoms (oxygen and nitrogen ). Meanwhile diazepine [2] contains to nitrogen atoms in seven – member  ring.

Diazepam (valium) [3] is used to relive anxiety tension associated with anxiety disorder and muscle spasms ^{(1, 2, 3}

Fluorescence excitation by Nd:YAG pumped dye laser and single vibrational level fluorescence
spectra of 1,3 benzodioxole in a supersonic jet have been obtained and interpreted. The previous assignment of
the 0 0
0 band was incorrect. In addition, many other bands involving n20 and n19 vibrations of a2 symmetry were
confirmed. As far as a1 totally symmetric vibration is concerned. The n14 was assigned to be located in the fivemembered
ring whereas n13 seem to be located in the benzene ring as a result of the electronic transition in the
benzene ring which affects n13 and not n14 wavenumber.

       Our work included a synthesis of three new imine derivatives—1,3-thiazinan-4-one, 1,3-oxazinan-6-one and 1,3-oxazepin-4,7-dione—which contained an adamantyl fragment. These were produced via the condensation of the Schiff`s base (E)-N-(adamantan-1-yl)-1-(3-aryl)methanimine with 3-mercaptopropanoic acid; 3-chloropropanoic acid; and maleic, citraconic anhydride, respectively. These new imines were prepared via the condensation of adamantan-1-ylamine and 3-nitro-, 3-bromobenzaldehyde in n-BuOH. We obtained a good yield of products. FTIR, ¹H NMR spectroscopy and C.H.N.S analysis were used to diagnostic the products. The molecular structure of (E)-N-(adamantan-1-yl