Periodicity and convergence for xn+1=|xn−xn−1|

Each solution {x_n} of the equation in the title is either eventually periodic with period 3 or else, it converges to zero—which case occurs depends on whether the ratio of the initial values of {x_n} is rational or irrational. Further, the sequence of ratios {x_n/x_n−1} satisfies a first-order difference equation that has periodic orbits of all integer periods except 3. p-cycles for each p≠3 are explicitly determined in terms of the Fibonacci numbers. In spite of the non-existence of period 3, the unique positive fixed point of the first-order equation is shown to be a snap-back repeller so the irrational ratios behave chaotically.     

Applications of surfactant-modified clays to synthetic organic chemistry

Two triphase catalysts (SLL) have been developed for organic phase-aqueous phase reactions catalyzed by suitable modified clay (solid phase). These triphase catalysts have been applied to nucleophilic displacement on activated (benzylic) as well as unactivated organic halides and provide a convenient and effective method of preparation of the corresponding products. Other useful transformations to, which these triphase catalysts have been successfully applied are the synthesis of 9,9-dichloro bicyclo[6.1.0]nonane, O-alkylation and C-alkylation of β-naphthol.     

Synthesis, Characterization, Spectroscopy and Study of Antimicrobial Properties of Two New Substituted Trichloroaluminate（Ⅲ） Complexes [AlCl3X]- （X = SCN-, CN-）

The synthesis and characterization of two new aluminate(Ⅲ) complexes with general formula K[AlCl3X] are reported. These compounds derived from aluminate trichloride and related salts. Potassium trichlorothiocyanoaluminate, PCTA, and potassium trichlorocyanatoalu-minate, PCCA, are two new ionic aluminate complexes. They can be easily synthesized in a nearly quantitative yield by using the direct reaction of AlCl3 and KX. The complexes were characterized by physico-chemical and spectroscopic methods. Theoretical calculations have been used for the extraction of structural and spectroscopic data of these new synthesized complexes. The antibacterial activities of such compounds were studied against the Staphylococcus aureus, Escherichia coli, Staphylococcus Epidermidis, Estreptococo B and Shigella.     

The asymptotic behavior of a class of nonlinear delay difference equations

The asymptotic behavior of difference equations of type 0, \end{equation*}">is studied, where and each are continuous real functions with decreasing and increasing. Results include sufficient conditions for permanence, oscillations and global attractivity.

     

Synthesis of new water soluble diorganotin(IV) complexes with hydrazones derived from Girard-T reagent as antibacterial and anticancer agents

Three new water-soluble organotin complexes R₂Sn(5-BrSalGT)Cl [R = Ph, Me] and Ph₂Sn(2-OHNaphGT)Cl have been synthesized by the reaction of R₂SnCl₂ (R = Ph or Me) with Schiff bases derived from condensation of Girard-T reagent with 5-bromosalicylaldehyde and 2-naphthaldehyde, (5-BrH₂SalGT)Cl (1) and (2-OHH₂NaphGT)Cl (2). The synthesized compounds have been investigated by elemental analysis, conductometric measurements, IR, ¹H NMR, and ¹¹⁹Sn NMR spectroscopy. These data show that the deporotonated ligand is coordinated to Sn(IV) via ONO atoms and six-coordinate zwitterionic complexes are formed. The ligands and their complexes were investigated for their in vitro toxicity against Gram-positive (Bacillus subtilis and Staphylococcus aureus) and Gram-negative (Escherichia coli and Pseudomonas aeruginosa) bacteria. The results show remarkable antibacterial activity against the studied bacteria. All complexes exhibit more inhibitory effects than the parent ligand. The anticancer activity of all compounds were also performed on HN5 cell line and (2-OHH₂NaphGT)Cl with concentration of 1 mg mL^?1 was found to show higher anticancer activity than other compounds.     

A new two-step P-stable hybrid Obrechkoff method for the numerical integration of second-order IVPs

In this paper, we introduce a class of new P-stable hybrid Obrechkoff methods for the numerical solution of second-order initial value problems. We generate a two-step, symmetric, P-stable hybrid Obrechkoff method of order 8 with two off-step, symmetric, points and phase lag of order at least 2 and for special cases, this order can be increased up to 6. The numerical results obtained by the new method for some problems show its superiority in efficiency, accuracy and stability.