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

A general velocity profile for a laminar flow over a flat plate with zero incidence is obtained by employing a new boundary condition to the other available boundary conditions. The general velocity profile is mathematically simple and nearest to the exact solution. Also other related values, boundary layer thickness, displacement thickness, momentum thickness and coefficient of friction are nearest to the exact solution compared with other corresponding values for other researchers.

A recently reported Nile red (NR) dye conjugated with benzothiadiazole species paves the way for the development of novel organic-based sensitizers used in solar cells whose structures are susceptible to modifications. Thus, six novel NR structures were derived from two previously developed structures in laboratories. In this study, density functional theory (DFT) calculations and time-dependent DFT (TD-DFT) were used to determine the optoelectronic properties of the NR-derived moieties such as absorption spectra. Various linkers were investigated in an attempt to understand the impact of π-linkers on the optoelectronic properties. According to the findings, the presence of furan species led to the planarity of the molecule and a reduction

The software-defined network (SDN) is a new technology that separates the control plane from data ‎plane for the network devices. One of the most significant issues in the video ‎surveillance system is the link failure. When the path failure occurs, the monitoring center cannot receive the ‎video from the cameras. In this paper, two methods are proposed to solve this problem.  The first ‎method uses the Dijkstra algorithm to re-find the path at the source node switch. The second method ‎uses the Dijkstra algorithm to re-find the path at the ingress node switch (or failed link).

In this paper, we will study non parametric model when the response variable have missing data (non response) in observations it under missing mechanisms MCAR, then we suggest Kernel-Based Non-Parametric Single-Imputation instead of missing value and compare it with Nearest Neighbor Imputation by using the simulation about some difference models and with difference cases as the sample size, variance and rate of missing data.      

New hydrazide compounds (A2) and (A9) were prepared from their corresponding esters (A1, A8) .These esters were also prepared from their precursors 5-ethoxy carbonyl-(4methoxyphenyl)-6-methyl-1,2,3,4-tetrahydropyrimidine-2-thione (A1) via multicomponent reaction type and from hippuric acid respectively. The hydrazide compounds were then allowed to react with some aldehydes forming the corresponding hydrazones (A3-7) and (A1014). The synthesized compounds were characterized by IR, 1H-NMR, 13C-NMR, and Mass spectroscopies and well discussed. .

The amino thiadiazole [I] on treatment with aromatic aldehydes yielded Schiff bases [IIa-c] , which cyclized to thiazolidinone [IIIa-c] derivatives by reaction with thioglycolic acid .Reaction of carbon disulfide and methyl iodide with [I] gavedithiomethyl[IV] which on treatment with o-phenylenediamine gave the condensed N-Imidazolythiadiazolylamine [V] , However , reaction of [I] with phenylisocyanate and phenylisothiocyanate afforded the carbamideand carbothiamide derivatives[VI.VII]a-c. The structure of these compounds was characterized from their  melting point , FTIR spectroscopy and elementalanalysis

Erbium, as optical probe, doped silicate sol-gel glass with
different Er concentrations was formed by wet chemical synthesis
method using ethanol, water and tetraethaylorthosilicate
[Si(OC2H5)4] precursor. Erbium ions were incorporated into silica
sol-gel matrix via dissolution of Erbium chloride solution into the
initial Si(OC2H5)4 precursor sol. Aluminum (Al) as a co-dopant was
added to the final precursor in the form of Aluminum chloride
(AlCl3) solution. The prepared samples were analyzed using atomic
absorption analysis, X-ray diffraction and spectroscopic tests. The
experimental results concerned with the transmission spectra suggest
that the final samples have a good transparency and homogeneity.
A