The discussion in this paper gives several theorems and lemmas on the Sums of Squares of  consecutive Carol Numbers. These theorems are proved by using the definition of carol numbers and mathematical induction method. Here the matrix form and the recursive form of sum of squares of  consecutive Carol numbers is also given. The properties of the Carol numbers are also derived.

The research includes the preparation of several complexes of the internal transition elements lanthanide (Ln = La, Nd, Er, Gd, and Dy) containing the 4f shell, with Schiff bases resulting from condensation reactions between 4-antipyrinecarboxaldehyde and 2-aminobenzothiazoles. Schiff's base was identified using FTIR spectra, UV-vis spectroscopy, elemental microanalysis CHNSO, nuclear magnetic resonance, mass spectrometry, and TGA thermal analysis. The complexes were studied and identified with elemental microanalysis CHNSO, FTIR spectroscopy, UV-vis spectroscopy, TGA thermal analysis, conductivity measurement, and magnetic sensitivity. The result showed that these complexes were classified as homogeneous bidentate complexes with th

A pulsed (TEA-0O2) laser was used to dissociate molecules of silane ethylene (C2I-14) and ammonia (NH3) gases, through collision assisted multiple photon dissociation (MPD) to deposit(SiC i_xNx) thin films, where the X-values are 0, 0.13 and 0.33, on glass substrate at T,----648 K. deposition rate of (0.416-0.833) nm/pulse and thickness of (500-1000)nm .Fourier transform infrared spectrometry (FT-IR) was used to study the nature of the chemical bonds that exist in the films. Results revealed that these films contain complex networks of the atomic (Si, C, and N), other a quantity of atomic hydrogen and chemical bonds such as (Si-N, C-N, C-14 and N-H).Absorbance and Transmittance spectra in the wavelength range (400-1100) nm were used to stud

Background Bilateral cleft lip deformity is much more difficult to correct than unilateral cleft lip deformity. The complexity of the deformity and the sensitive relationships between the arrangement of the muscles and the characteristics of the external lip necessitate a comprehensive preoperative plan for management. The purpose of this study was to evaluate the repair of bilateral cleft lip using the Byrd modification of the traditional Millard and Manchester methods. A key component of this repair technique is focused on reconstruction of the central tubercle.

Methods Fourteen patients with mean age of 5.7 months presented with bilateral cleft lip deformity and were operated on using a mod

In the current work, the mixing ratios ( 𝛿 ) of gamma transitions were calculated from energy levels in the isotopes neodymium 60𝑁𝑎 142−150 populated in the 60Nd 142− 150 (n, n ˊγ) 60Nd 142− 150 using the 𝑎2 ratio method. We used the experimental coefficient (𝑎2 ) for two γ-transitions from the initial state itself, the statistical tensor 𝜌2(𝐽𝑖), associated with factor 𝑎2 , would be the same for the two transitions. The results obtained are in good agreement or within the experimental error with -those previously published. And existing contradictions resulting from inaccuracies in the empirical results of previous work. The current results confirm that the , 𝑎2 − method is used to calculate th

Tetradentate complexes type [M (HL) 2] were prepared from the reaction of 2-hydroxy -1, 2-diphynel-ethanone oxime [H2L] and KOH with ( Mn II, Fe II, Co II, Ni II , Cu II and Hg II ), in methanol with (2:1) metal: ligand ratio. The general formula for Cu II and Mn II complexes are [M (HL) 2 Cl.H2O] K, for Co II [Co (HL) 2. H2O] and [M (HL) 2] for the rest of complexes. All compounds were characterised by spectroscopic methods, I.R, U.V-Vis, H.P.L.C, atomic absorption and conductivity measurements chloride content. From the data of these measurements, the proposed molecular structures for Fe II and Hg II complexes are tetrahedrals, while Mn II and Cu II complexes are octahedrals, Ni II complex adopting square planar structure and the complex