On the diophantine equation x 2 + 2 a · 3 b · 11 c = y n

In this note, we find all the solutions of the Diophantine equation x ² + 2 ^a · 3 ^b · 11 ^c = y ⁿ , in nonnegative integers a, b, c, x, y, n ≥ 3 with x and y coprime.     

Novel bioactive vic-dioxime ligand containing piperazine moiety: Synthesis,X-ray crystallographic studies, 2D NMR applications and complexation with Ni(II)

A new vic-dioxime ligand bearing an important pharmacophore substituent, anti-1-(4-benzylpiperazine-1-yl) phenylglyoxime (LH₂) (Scheme 1), has been synthesized and its nickel(II) complex was obtained by the reaction of NiCl₂·6H₂O and the ligand. The characterization of the newly formed compounds was performed by elemental analysis, FT-IR, 1D NMR (¹H, ¹³C, DEPT), 2D NMR (HMBC), ESI mass-spectrometry, TG/DTA, X-ray crystallography. The antibacterial activity was also studied against Staphylococcus aureus ATCC 25923, Streptococcus mutans RSHM 676, Enterococcus faecalis ATCC 29212, Lactobacillus acidophilus RSHM 06029, Escherichia coli ATCC 25922, E. coli ATCC 35218, Pseudomonasaeruginosa ATCC 27853. The antimicrobial test results indicated that all the compounds have mild antibacterial activity against both Gram negative and Gram positive bacterial species.     

Syntheses and characterization of two copper pyridine-dicarboxylate compounds containing water clusters

A new homonuclear and heterodinuclear pyridine-2,6-dicarboxylate complexes, formulated as (enH₂)[Cu(dipic)₂]·2.5H₂O (1) and [Cu(μ-dipic)₂Zn(H₂O)₅]·2H₂O (2) (en = ethylenediamine, dipic = pyridine-2,6-dicarboxylate) were synthesized according the reactions between the three species of copper(II) and zinc(II) nitrate, ethylenediamine and pyridine-2,6-dicarboxylic acid. Complexes have been characterized by the methods of elemental, spectroscopic (IR and UV–Vis), thermal (TG/DTG, DTA) analysis, magnetic measurement and single crystal X-ray diffraction. Correlation coefficient, activation energies, E∗; pre-exponential factor, A; entropies, S∗; enthalpies, H∗ and Gibbs free energies, G∗ of the thermal decomposition reactions have been calculated under the derivations from thermogravimetric (TG) and differential thermogravimetric (DTG) curves, using the Coats–Redfern and Horowitz–Metzger methods. Complex 1 belonged to a six-coordinate behavior with a distorted octahedral geometry around Cu(II), that the structure contains two pyridine-2,6-dicarboxylate species as a tridentate ligands and ethylenediaminium cation as a counter ion as well as 2.5 uncoordinated water molecules. On the other hand complex 2 contains six-coordinated Cu(II) and Zn(II) ions, which are linked by two O atoms of the same carboxyl group from dipicolinic acid. The dipicolinate dianions again behave as tridentate ligands in 2. Two uncoordinated water molecules are also present in the structure. The structure units of 1 and 2 are mutually held by the hydrogen bonds and π?π interactions. There is also a C–O?π interaction in 2. The Cu(II) complexes are connected to one another via O–H?O hydrogen bonds, forming water clusters, which play an important role in the stabilization of the crystal structure. In the water clusters, the water molecules are trapped by the cooperative association of coordination interactions as well as hydrogen bonds.     

The synthesis and characterization of single substitute melamine cored Schiff bases and their [Fe(III) and Cr(III)] complexes

In this study, firstly, two single substitute novel ligands have been synthesized by reacting melamine with 3,4,-dihydroxybenzaldeyhde or 4-carboxybenzaldehyde. Then, eight new mono nuclear single substitute [Salen/Salophen Fe(III) and Cr(III)] complexes have been synthesized by reacting the ligands [2-(3,4-dihydroxybenzimino)-4,6-diamimo-1,3,5-triazine and 2-(4-carboxybenzimino)-4,6-diamimo-1,3,5-triazine)] with tetradentate Schiff bases N,N′-bis(salicylidene)ethylenediamine-(salenH₂) or bis(salicylidene)-o-phenylenediamine-(salophen H₂). And then, all ligands and complexes have been characterized by means of elementel analysis, FT-IR spectroscopy, ¹H NMR, LC–MS, thermal analyses and magnetic suscebtibility measurements. Finally, metal ratios of the prepared complexes were determined using AAS. The complexes have also been characterized as disorted octahedral low-spin Fe(III) and Cr(III) bridged by catechol and COO^? groups.     

A new rational approximation technique based on transformational high dimensional model representation

In this work, a new rational approximation scheme, based on the recently developed Transformational High Dimensional Model Representation (THDMR) approximation method is developed. As an initial step to the construction of a rational approximation for multivariate functions via THDMR, this paper focuses on the general theoretical background of the method and gives explicit formulae for the computation of such approximants. The performance of the technique is shown by several examples both in univariate and bivariate cases.     

A note on the (G′/G)-expansion method again

We report an observation on two recent analytic methods; the (G′/G)-expansion method and the simplest equation method.     

Comment on: “New exact solutions for the Kawahara equation using Exp-function method” [J. Comput. Appl. Math. 233 (2009) 97-102]

Assas [Laila M.B. Assas, New exact solutions for the Kawahara equation using Exp-function method, J. Comput. Appl. Math. 233 (2009) 97-102] found some supposedly new exact solutions to the Kawahara equation by means of the Exp-function method. Unfortunately, they are incorrect. We emphasize that the article contains erroneous formulas and resulting relations. In fact, no numerical method was used.     

Comparison between variational iteration method and homotopy perturbation method for linear and nonlinear partial differential equations with the nonhomogeneous initial conditions

In this study, linear and nonlinear partial differential equations with the nonhomogeneous initial conditions are considered. We used Variational iteration method (VIM) and Homotopy perturbation method (HPM) for solving these equations. Both methods are used to obtain analytic solutions for different types of differential equations. Four examples are presented to show the application of the present techniques. In these schemes, the solution takes the form of a convergent series with easily computable components. The present methods perform extremely well in terms of efficiency and simplicity. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

A short proof of the generalized Littlewood Tauberian theorem

In this work we give an alternative proof of the generalized Littlewood Tauberian theorem for the Abel summability method using a corollary to Karamata’s Main Theorem [J. Karamata, Über die Hardy–Littlewoodschen Umkehrungen des Abelschen Stetigkeitssatzes, Math. Z. 32 (1930) 319–320].