In this work, we give an identity that leads to establishing the operator . Also, we introduce the polynomials . In addition, we provide Operator proof for the generating function with its extension and the Rogers formula for . The generating function with its extension and the Rogers formula for the bivariate Rogers-Szegö polynomials  are deduced. The Rogers formula for  allows to obtain the inverse linearization formula for , which allows to deduce the inverse linearization formula for . A solution to a q-difference equation is introduced and the solution is expressed in terms of the operators . The q-difference method is used to recover an identity of the operator  and the generating function for the polynomials

Cerium oxide CeO2, or ceria, has gained increasing interest owing to its excellent catalytic applications. Under the framework of density functional theory (DFT), this contribution demonstrates the effect that introducing the element nickel (Ni) into the ceria lattice has on its electronic, structural, and optical characteristics. Electronic density of states (DOSs) analysis shows that Ni integration leads to a shrinkage of Ce 4f states and improvement of Ni 3d states in the bottom of the conduction band. Furthermore, the calculated optical absorption spectra of an Ni-doped CeO2 system shifts towards longer visible light and infrared regions. Results indicate that Ni-doping a CeO2 system would result in a decrease of the band gap. Finally,

Cerium oxide (CeO2), or ceria, has gained increasing interest owing to its excellent catalytic applications. Under the framework of density functional theory (DFT), this contribution demonstrates the eect that introducing the element nickel (Ni) into the ceria lattice has on its electronic, structural, and optical characteristics. Electronic density of states (DOSs) analysis shows that Ni integration leads to a shrinkage of Ce 4f states and improvement of Ni 3d states in the bottom of the conduction band. Furthermore, the calculated optical absorption spectra of an Ni-doped CeO2 system shifts towards longer visible light and infrared regions. Results indicate that Ni-doping a CeO2 system would result in a decrease of the band gap. Finally,

Rheological instrument is one of the basic analytical measurements for diagnosing the properties of polymers fluids to be used in any industry. In this research polycarbonate was chosen because of its importance in many areas and possesses several distinct properties.
Two kinds of rheometers devices were used at different range of temperatures from 220 ˚C-300 ˚C to characterize the rheological technique of melted polycarbonate (Makrolon 2805) by a combination of different investigating techniques. We compared the results of the linear (oscillatory) method with the non-linear (steady-state) method; the former method provided the storage and the loss modulus of melted polycarbonate, and presented the Cox-Merz model as well. One of the

Heterocyclic polymers / silica nanocomposite one of important materials because of excellent properties such as thermal , electrical , and mechanical properties , so that hybrid nanomaterial are widely used in many fields, in this paper nanocomposite had prepared by modification of silica nanoparticals by using acrylic acid and functionalized the surface of nanoparticles, and using free Radical polymerization by AIBN as initiators and anhydrous toluene as solvent to polymerize functionalize silica nanoparticles with heterocyclic monomers to prepare heterocylic polymers / silica nanocomposite and study electrical conductivity , The nanocomposite which had prepared characterized by many analysis technique to study thermal properties such

In this work, (CdO)1-x (CoO)x thin films were prepared on glass slides by laser-induced plasma using Nd:YAG laser with (λ=1064 nm) and duration (9 ns) at different laser energies (200-500 mJ) with ratio (x=0.5), The influence of laser energy on structural and optical properties has been studied. XRD patterns show the films have a structure of polycrystalline wurtzite. As for AFM tests results for the topography of the surface of the film, where the results showed that the grain size and the average roughness increase with increasing laser energy. The optical properties of all films were also studied and the results showed that the absorption coefficient for within the wavelength range (280-1100 nm), The value of the optical power gap fo

A study carried out on ceramic material made at (a-Al₂O₃) doped with MgO (0.5 , 0.3 , 0.2,0.1)%,with particle size at 63mm.
A Hydraulic press of 5kn at diameter of 2cm.A nnalelling at 1500C^o and 6 hrs still to see the effect on the changes of the dielectric material. With frequency range at (1K – 1M) Hz. And the result show that at percentage of 0.5% of MgO, the real dielectric material decreased with the increased frequency

     Cadmium oxide CdO thin films were prepared by successive ionic layer adsorption and reaction (SILAR) technique at varying number of dippings. The CdO thin films were prepared from a source material of Cadmium acetate and ammonium hydroxide solution deposited on glass substrate at 95℃.    The prepared thin films were investigated by X-ray diffraction (XRD), Atomic force microscopy (AFM), Scanning Electron Microscopy (SEM), Fourier Transform Infrared (FTIR), and UV-Visible spectrometry. The XRD analysis reveals that the films were polycrystalline with cubic structure having preferential orientation along (1 1 1), (2 0 0), (2 2 0), and (3 1 1) planes. While the tests of the scanning electron microscopy and the atomic force mic

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.