Due to its importance in physics and applied mathematics, the non-linear Sturm-Liouville problems
witnessed massive attention since 1960. A powerful Mathematical technique called the Newton-Kantorovich
method is applied in this work to one of the non-linear Sturm-Liouville problems. To the best of the authors’
knowledge, this technique of Newton-Kantorovich has never been applied before to solve the non-linear
Sturm-Liouville problems under consideration. Accordingly, the purpose of this work is to show that this
important specific kind of non-linear Sturm-Liouville differential equations problems can be solved by
applying the well-known Newton-Kantorovich method. Also, to show the efficiency of appl

The aim of this work is to calculate the one- electron expectation value of the electronic charge of atomic system Z=2,3….7 and we compare with He atom . the electronic density function D(r1) of He atom and like ions are evaluated . using Hartree –Fock wave.

In this paper we investigated some new properties of π-Armendariz rings and studied the relationships between π-Armendariz rings and central Armendariz rings, nil-Armendariz rings, semicommutative rings, skew Armendariz rings, α-compatible rings and others. We proved that if R is a central Armendariz, then R is π-Armendariz ring. Also we explained how skew Armendariz rings can be ?-Armendariz, for that we proved that if R is a skew Armendariz π-compatible ring, then R is π-Armendariz. Examples are given to illustrate the relations between concepts.

The purpose of this paper is to prove the following result: Let R be a 2-torsion free ring and T: R?R an additive mapping such that T is left (right) Jordan ?-centralizers on R. Then T is a left (right) ?-centralizer of R, if one of the following conditions hold (i) R is a semiprime ring has a commutator which is not a zero divisor . (ii) R is a non commutative prime ring . (iii) R is a commutative semiprime ring, where ? be surjective endomorphism of R . It is also proved that if T(x?y)=T(x)??(y)=?(x)?T(y) for all x, y ? R and ?-centralizers of R coincide under same condition and ?(Z(R)) = Z(R) .

The ligand [Potassium (E)-(4-(((2-((1-(3-aminophenyl) ethylidene) amino)-4-oxo-1, 4-dihydropteridin-6-yl) methyl) amino) benzoyl)-L-glutamate] was prepared from the condensation reaction of folic acid with (3-aminoacetophenone) through Schiff reaction to give a new Schiff base ligand [H2L]. The ligand [H2L] was characterized by elemental analysis CHN, atomic absorption (AA),(FT-IR),(UV-Vis), TLC, ES mass (for spectroscopes), molar conductance, and melting point. The new Schiff base ligand [H2L], reacts with Mn (II), Co (II), Ni (II), Cu (II), Cr (III) and Cd (II) metal ions and (2-aminophenol),(metal: derivative ligand: 2-aminophenol) to give a series of new mixed complexes in the general formula:-K3 [M2 (HL)(HA) 2],(where M= Mn (II) and Cd

2-hydrazinylbenzo[d]thiazole compound [1] is produced from reaction of 2-mercapto-benzothiazole with hydrazine hydride in ethanol. Compound [1] reacted with maleic anhydride in DMF to produce (Z)-4-(2-(benzo[d] thiazol-2yl) hydrazinyl)-4-oxobut-2-enoic acid [compound (2)]. While the treatment of compound [2] with the ammonium persulfate (NH4)2S2O8 (as the initiator) in order to produce compound [3], then compound [3] reacted with thionyl chloride in benzene to produce compound [4], finally compound [4] reaction with various drugs: cephalexin, amoxicillin, sulfamethizole, elecoxib obtained polymers [5–8]. The structure of synthesized compounds identified by spectral data: fourier transform infrared (FTIR) and proton nuclear magneti

2-hydrazinylbenzo[d]thiazole compound [1] is produced from reaction of 2-mercapto-benzothiazole with hydrazine hydride in ethanol. Compound [1] reacted with maleic anhydride in DMF to produce (Z)-4-(2-(benzo[d] thiazol-2yl) hydrazinyl)-4-oxobut-2-enoic acid [compound (2)]. While the treatment of compound [2] with the ammonium persulfate (NH4)2S2O8 (as the initiator) in order to produce compound [3], then compound [3] reacted with thionyl chloride in benzene to produce compound [4], finally compound [4] reaction with various drugs: cephalexin, amoxicillin, sulfamethizole, elecoxib obtained polymers [5–8]. The structure of synthesized compounds identified by spectral data: fourier transform infrared (FTIR) and proton nuclear magneti