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 biggest problem of structural materials for fusion reactor is the damage caused by the fusion product neutrons to the structural material. If this problem is overcomed, an important milestone will be left behind in fusion energy. One of the important problems of the structural material is that nuclei forming the structural material interacting with fusion neutrons are transmuted to stable or radioactive nuclei via (n, x) (x; alpha, proton, gamma etc.) reactions. In particular, the concentration of helium gas in the structural material increases through deuteron- tritium (D-T) and (n, α) reactions, and this increase significantly changes the microstructure and the properties of the structural materials. T

I've been in this work to prepare some complexes containing mixed Kandat with some transitional elements which new complexes according to the best of our knowledge and to refer to the information from the Internet until I have been studying this diagnosis Aalmakdat

Two ligand ortho-amino phenyl thio benzyl (L1) and 1,3 bis (ortho - amino phenyl thio ) acetone (L2) and their complexes have been prepared and characterized . The L1 ligand is lossing phenyl group on complexcation and forming 1,2 bis (ortho - amino phenyl thio ) ethane L3 and this tetrahedrally coordinated to the metal ion ( M+2 = Ni , Cu , Cd ) and octahedrally coordinated with mercury and cobalt ions , while the ligand L2 is behave as tridentate ligand forming octahedrally around chrome metal ion . Structural , diagnosis were established by i.r , Uv- visible , conductivity elemental analysis and (mass spectra , H nmr spectra for( L1 , L2 ) .

D-mannose sugar was used to prepare [benzoic acid 6-formyl-2, 2-dimethyl-tetrahydrofuro [3, 4-d][1, 3] dioxol-4-yl ester](compound A). The condensation reaction of folic acid with (compound A) resulted in the formation of new ligand [L]. These compounds were characterized by elemental analysis CHN, atomic absorption AA,(FT-IR),(UV-Vis), TLC, ES mass (for electrospray), molar conductance, and melting point. The new tetradentate ligand [L], reacted with two moles of some selected metal ions and two moles of (2-aminophenol),(metal: ligand: 2-aminophenol) at reflux in water medium to give a series of new complexes of the general formula K2 [M2 (L)(HA) 2] where M= Co (II), Ni (II), Cu (II) and Cd (II). These complexes were characterized by eleme

D-mannose sugar was used to prepare [benzoic acid 6-formyl-2,2-dimethyl-tetrahydrofuro[3,4-d][1,3]dioxol-4-yl ester] (compound A). The condensation reaction of folic acid with (compound A) resulted in the formation of new ligand [L]. These compounds were characterized by elemental analysis CHN, atomic absorption A.A, (FT-I.R.), (U.V.-Vis), TLC, E.S. mass (for electrospray), molar conductance, and melting point. The new tetradentate ligand [L], reacted with two moles of some selected metal ions and two moles of (2-aminophenol), (metal : ligand : 2-aminophenol) at reflux in water medium to give a series of new complexes of the general formula K2[M2(L)(HA)2] where M= Co(II), Ni(II), Cu(II) and Cd(II). These complexes were characterized by elem

A new Schiff base of HL has been synthesized from amoxicillin drug and 4- Chlorobenzophenone. Cr (III), Fe (III), Co (II), Ni (II), Cu (II), Cd (II) and Hg (II) mixed ligands complexes of Schiff base and Nicotinamide. Diagnosis of synthesis ligand and its complexes are done by 1HNMR, 13CNMR and thermal analysis for HL ligand, FTIR, UV-visible, molar conductance, CHN analysis, magnetic susceptility and atomic absorption. Octahedral geometries have been suggested for all complexes. All compounds under study were tested antimicrobial activity against four type of bacteria such as Pseudomonas aeruginosa, Escherichia coli, Staphylococcus aureus Bacillis subtilis in nutrient agar.

Coumarin derivatives have shown different biological activities, such as antifungal, antibacterial   antiinflammatory, and antioxidant activities, besides antibiotic resistance modulating effects, and anti-HIV, hepatoprotective, and antitumor effect. So, new coumarin derivatives (hydrazones and an amide) were synthesized through multisteps reactions. All the synthesized target compounds were characterized by FT-IR spectroscopy, 1HNMR analysis. The compounds then evaluated for their anti-bacterial activity by means of well-diffusion method against two gram-positive bacteria (Staphylococcus aureus, Streptococcus pneumoniae) and two gram-negative bacteria  (E.coli and Pseudomonas aeruginosa). The highest activity was demonstr

Ruthenium-Ruthenium and Ruthenium–ligand interactions in the triruthenium "[Ru₃(μ-H)(μ₃-κ²-Hamphox-N,N)(CO)₉]" cluster are studied at DFT level of theory. The topological indices are evaluated in term of QTAIM (quantum theory of atoms in molecule). The computed topological parameters are in agreement with related transition metal complexes documented in the research papers. The QTAIM analysis of the bridged core part, i.e., Ru₃H, analysis shows that there is no bond path and bond critical point (chemical bonding) between Ru(2) and Ru(3). Nevertheless, a non-negligible delocalization index for this non-bonding interaction is calculated