Lidskii's theory is considered one of the most important and recent theories for calculating the following categories and the relationship between them and the eigenvalues. This theory provides an easier way to prove the existence of the eigenvalues, and hence to prove the existence of solutions for some kind of problems. This thesis article to prove that there are solutions to some problems for which the computation of eigenvalues is very complex and to prove that the existence of eigenvalues is also complex, in our work we try to take advantage of the fact that calculating the trace is much easier than calculating eigenvalues. Lidskii's theorem gives the relationship between Trace and eigenvalues and gives us a way to prove the existence of eigenvalues.

The ground state charge and matter density distributions, the corresponding root mean square (rms) radii and the elastic electron scattering form factors for some sd-shell nuclei (such as 23 Na, 25 Mg, 27 Al and 29 Si nuclei) are studied by the radial wave functions of the harmonic oscillator (HO) and Woods-Saxon (WS) potentials. The calculated results are discussed and compared with the experimental data. It is found that the calculated results of charge and matter density distributions are in very good agreement with the experimental data.

This research include the recovery study of cadmium and nickel from spent alkaline batteries was conducted by two different acidic solutions. The first one is nitric acid, which was used for cadmium recovery; the other is sulfuric acid, which was used for nickel recovery. Many factors affected the recovery process were studied, such as concentration, volume, time, weight and temperature. The parameters which effect on the precipitation of metal ion from leaching solution were also studied, such as pH, time, concentration and temperature. The recovery percentages of metal ion were found to be 98.5% of cadmium using optimum conditions (5 M HNO3, 2 gm sample, 2 hr, and 20 ml acid volume) and 99% for nickel using (5 M H2SO4, 2 gm sample, 2 hr, and 20 ml acid volume). The precipitation processes were carried out using (2 M NaOH) solution. The precipitation percentages were found to be 98.6% and 99.1% of cadmium and nickel respectively. The purification processes were conducted using double distilled water. The purity of products Cd (OH)2 and Ni (OH)2 was found to be 98.08% and 98.23% respectively. The concentration of metal ion in the solutions was measured using AAS; the method was linear with R 2 of (0.9995 and 0.9993) for Ni and Cd respectively.

In the framework of correlation method so-called coherent density fluctuation model (CDFM) the proton momentum distributions (PMD) of the ground state for some even mass nuclei of fpshell like 70 Zn, 72 Ge and 74 Se nuclei are examined. Proton momentum distributions are expressed in terms of the fluctuation function(|()| 2 ) and determined from theory and experiment. The main characteristic feature of the PMD obtained by CDFM is the existence of high-momentum components, for momenta k 2 fm -1 . For completeness, also elastic electron scattering form factors F(q) are evaluated within the same framework.

A series of heterocyclic compounds have been synthesized using 1-aryl-3-(4-hydroxy-3methoxypheny) prop-2-ene-1-one (1-4) as starting material which was prepared by claisenschmidt condensation between vanillin and substituted acetophenone.Benzoxazepine derivatives (5-8) and benzodiazepine derivatives (9-12) were synthesized from the reaction of o-aminophenol, o-phenlendiamine with compounds (1-4) respectively. Thereaction of compounds (1-4) with hydrazine hydrate afforded pyrazol derivatives (13-16). While the reaction of compounds (1-4) with hydrazine hydrates in the presences a few drops of glacial acetic acid gave 1-acetyl pyazol derivatives (17-20). Oxirane derivatives (21-24) were synthesized from the reaction of compounds (1-4) with hydrogen peroxide in the basic medium.