This study aims to classify the critical points of functions with 4 variables and 8 parameters, we found the caustic for the certain function with the spreading of the critical points. Finally, as an application, we found the bifurcation solutions for the equation of sixth order with boundary conditions using the Lyapunov-Schmidt method in the variational case.

In this paper, the process for finding an approximate solution of nonlinear three-dimensional (3D) Volterra type integral operator equation (N3D-VIOE) in R³ is introduced. The modelling of the majorant function (MF) with the modified Newton method (MNM) is employed to convert N3D-VIOE to the linear 3D Volterra type integral operator equation (L3D-VIOE). The method of trapezoidal rule (TR) and collocation points are utilized to determine the approximate solution of L3D-VIOE by dealing with the linear form of the algebraic system. The existence of the approximate solution and its uniqueness are proved, and illustrative examples are provided to show the accuracy and efficiency of the model.

Mathematical Subject Classificat

In the present work, the surface properties of mixed binary surfactants containing sodium dodecylbenzene sulfate (SDBS) and Tween 80 (TW80) surfactants in aqueous solutions were studied at temperature 293 K using surface tension measurements. The critical micelle concentration (cmc) magnitude for both individual surfactants and their mixtures were established the obtained results revealed  that the magnitude of cmc of the mixtures are less than the magnitude of individual surfactants and decrease with the increase in Tween 80 percent in solution which indicate the nonideal  mixing of the two surfactants. The values of molecular interaction parameters  and the mole fraction of surfactants in the micelle (X1) were calculated

Were arranged this study on two sections, which included first section comparison between markets proposed through the use of transport models and the use of the program QSB for less costs , dependant the optimal solution  to chose the suggested market  to locate new market that achieve lower costs in the transport of goods from factories (ALRasheed ,ALAmeen , AlMaamun ) to points of sale, but the second part has included comparison of all methods of transport (The least cost method ,Vogels method , Results Approximations method , Total method) depending on the agenda of transport, which includes the market proposed selected from the first section and choose the way in which check the solution first best suited in terms

Basketball is one of the most popular and popular games that have a large audience both globally and locally because of the magic it contains in terms of performance. For this sport to be a success, fundamental factors must exist and these factors is the referee. The referee plays a large and important role in the success of the game and the establishment and application of rules.The research aimed at identifying the effect of special on the development of referee's physical abilities. The researcher used the experimental method. The subjects were (12) first class basketball referees. The researcher recommended using special exercises due to their great role in developing the physical abilities of first class basketball referees

Academic boredom is one of the most emotional problems that arouse an individual's fatigueness and lowers his interest. This is because of the environment's low efficiency and of spending long monotonous time. This state is characterized by having lack of interest, difficulties in concentration, and the desire to leave the class.  It is considered one of the most prominent forms of boredom widespread among students and the most severe and dangerous one that has negative effects and severe psychological and social problems. The current research aims to investigate a randomly selected sample of 335 students from the first and third grades at the directorates of education (Karkh and Rusafa) who suffers from academic boredom. Similarily

Colorectal cancer CRC is the third most commonly diagnosed cancer in males and the second in female, so it is a common and life-threatening disease serum of 48 males and 40 females suffering from CRC disease as group one (G1) without taking a chemotherapy dose, the same patients group tow (G2) after taking the first chemotherapy dose, while group three (G3) whose patients in (G1) and (G2) after second taking a chemotherapy dose. and group four (G4) consist of (30 males and 30 females) healthy Iraqi control. Results showed a significant increase in IL-17, IL-17RA and CEA in G1, while there was a significant decrease in vitamin D concentration in G1 than other groups, also there was positive (+ ve) significant correlation between IL-17 and CE

2-(1,2-dihydroxy ethyl -1- (2-mercaptophenyl)-5-(2-mercaptophenyl imino)-2,5-dihydro-1H-pyrrol-3,4-diol(H2L`)a lactam derivative of L-ascorbic acid was prepared by reaction of 5,6-O-isopropylidene L-ascorbic acid with 2-amino thiophenol in a mole ratio of (1:2) respectinely. A series of new metal complexes of this ligand (H2L`) were prepared by a reaction with the chlorides of Cr(III), Mn(II), Co(II), Ni(II), Cu(II), Zn(II), Cd(II) and Hg(II). The new ligand and its metal complexes were characterized by C.H.N.,1H and 13C NMR, IR as well as UV-Visible spectra and mass spectra of Ni(II) and Zn(II) complexes was also done. Atomic absorption of the metal percentage, electrical conductivity and magnetic measurements at room temperature was ca

Methotrexate (MTX) is one of the most effective medications to treat rheumatoid arthritis (RA).Aserum of 60 Iraqi male patients suffering from RA as (G1) was newly diagnosis and the same patient in G1 after taking MTX as G2 and 40 Iraqi male healthy control as G3. Nesfatin-1 (Nf-1) is belong to the adipokine family withpleiotropic effect. Nf-1 has been found in different tissues, including stomach, pancreas, bone cells, cartilage and heart. Retinol binding protein (RBP4) was known as transpoter of retinol from its storage sites in the liver to the extrahepatic tissues. Moreover, RBP4 acts as adipokine and contributes in the pathophsyology of prototypic inflammatory disease, rheumatoid arthritis (RA). The results showed a significant increas

The present work includes the preparation and characterization of{Co(II) , Ni(II), Pd(II), Fe(III) , Ru(III),Rh(III), Os(III) , Ir(III) , Pt(IV) and VO(IV)}complexes of a new ligand 4-[(1-phenyl-2,3-dimethyl-3-pyrozoline-5-one)azo]-N,N-dimethylanline (PAD). The product (PAD) was isolated,studies and characterized by phsical measurements,i.e., (FT-IR), (UV) Spectroscopy and elemental analysis(C.H.N). The prepared complexes were identified and their structural geometric were suggested in solid state by using flame atomic absorption, elemental analysis(C.H.N), (FT-IR) and (UV-Vis) Spectroscopy, as well as magnetic susceptibility and conductivity measurements . The study of the nature of the complexes formed in( ethanolic solution) following t