In this paper we generalize Jacobsons results by proving that any integer  in   is a square-free integer), belong to . All units of  are generated by the fundamental unit  having the forms

Our generalization build on using the conditions

This leads us to classify the real quadratic fields  into the sets  Jacobsons results shows that  and Sliwa confirm that  and  are the only real quadratic fields in .

   In the present study, semi – batch experiments were conducted to investigate the efficiency of ozone microbubbles (OMBs) in the treatment of aqueous dye solutions methylene orange under different reaction conditions such as  effect of initial solution pH , ozone generation rate and initial MO-concentration. The results showed that the removal of MO by OMBs were very high at the acidic and alkaline media and upon increasing the generation rate of ozone from 0.498 to 0.83 mg/s, the removal efficiency dramatically increased from 75to 100% within 15 min. The rate of oxidation reaction followed a pseudo first- order kinetic model. The results demonstrated that OMBs is efficient in terms of the decline of methylene orange c

In this study, manganese dioxide (MnO₂) nanoparticles (NPs) were synthesized via the hydrothermal method and utilized for the adsorption of Janus green dye (JG) from aqueous solutions. The effects of MnO₂ NPs on kinetics and diffusion were also analyzed. The synthesized NPs were characterized by scanning electron microscopy (SEM), X-ray diffraction (XRD), energy-dispersive X-ray analysis (EDX), and Fourier-transform infrared spectroscopy (FT-IR), with XRD confirming the nanoparticle size of 6.23 nm. The adsorption kinetics were investigated using three models: pseudo-first-order (PFO), pseudo-second-order (PSO), and the intraparticle diffusion model. The PSO model provided the best fit (R² = 0.999), indicating that the adsorpti

Investigation of the adsorption of acid fuchsin dye (AFD) on Zeolite 5A is carried out using batch scale experiments according to statistical design. Adsorption isotherms, kinetics and thermodynamics were demonstrated.  Results showed that the maximum removal efficiency was using zeolite at a temperature of 93.68751 mg/g. Experimental data was found to fit the Langmuir isotherm and pseudo second order kinetics with maximum  removal of about 95%.  Thermodynamic analysis showed an endothermic adsorption. Optimization was made for the most affecting operating variables and a model equation for the predicted efficiency was suggested.

The purpose of this work is to study the classification and construction of (k,3)-arcs in the projective plane PG(2,7). We found that there are two (5,3)-arcs, four (6,3)-arcs, six (7,3)arcs, six (8,3)-arcs, seven (9,3)-arcs, six (10,3)-arcs and six (11,3)-arcs. All of these arcs are incomplete. The number of distinct (12,3)-arcs are six, two of them are complete. There are four distinct (13,3)-arcs, two of them are complete and one (14,3)-arc which is incomplete. There exists one complete (15,3)-arc.