Bivariate fibonacci like p-polynomials

In this article, we study the bivariate Fibonacci and Lucas p-polynomials (p ? 0 is integer) from which, specifying x, y and p, bivariate Fibonacci and Lucas polynomials, bivariate Pell and Pell-Lucas polynomials, Jacobsthal and Jacobsthal-Lucas polynomials, Fibonacci and Lucas p-polynomials, Fibonacci and Lucas p-numbers, Pell and Pell-Lucas p-numbers and Chebyshev polynomials of the first and second kind, are obtained. Afterwards, we obtain some properties of the bivariate Fibonacci and Lucas p-polynomials.     

Some Weighted Equi-Statistical Convergence and Korovkin Type-Theorem

In this paper, we will show a new weighted equi-statistical convergence and based on this definition we will prove a kind of the Korovkin type theorems. Also we will show the rate of the convergence for this kind of weighted statistical convergence and Voronovskaya type theorem.     

Immobilized versus coated amylose tris(3,5-dimethylphenylcarbamate) chiral stationary phases for the enantioselective separation of cyclopropane derivatives by liquid chromatography

The solvent versatility of Chiralpak IA, a new chiral stationary phase (CSP) containing amylose tris(3,5-dimethylphenylcarabamate) immobilized onto silica gel, is investigated for the enantioselective separation of a set of cyclopropane derivatives using ethyl acetate or dichloromethane (DCM) as non-standard mobile phase eluent and diluent, respectively in high-performance liquid chromatography (HPLC). A comparison of the separation of cyclopropanes on both immobilized and coated amylose tris(3,5-dimethylphenylcarbamate) chiral stationary phases (Chiralpak IA and Chiralpak AD, respectively) in HPLC using a mixture of n-hexane/2-propanol (90/10 and 99/1, v/v) as mobile phase with a flow rate of 0.5 ml/min and UV detection at 254 nm, is demonstrated. The optimized method of separation is used for an online HPLC monitoring for the Rh(II)-catalyzed asymmetric intermolecular cyclopropanations in dichloromethane. Direct analysis techniques without further purification, workup or removal of dichloromethane were summarized. The method provides an easy and direct determination of the enantiomeric excess of the cyclopropanes and selectivity of the catalyst used without any further work up.     

Facile synthesis of pyrrolo[2,1‐c][1,4]benzodiazepine‐5,11‐dione dimer

Efficient synthesis of biological important pyrrolo[2,1‐c][1,4]benzodiazepine‐5,11‐dione dimer linked through the C‐2 positions by fumarate group is described.     

Synthesis of new di- and triamine diosgenin dimers

In this study, a high yield synthesis of symmetrical steroidal polyamine dimers was achieved by the dimerization of (25R)-3β-acetoxyfurost-5-en-26-al via several di- and triamine linkers under mild conditions. To ensure the dimerization via E-E ring, the hydroxyl group in diosgenin was protected by an acetyl group. The important step is opening the spiroketal moiety using NaCNBH₃/AcOH to furnish the primary alcohol at C-26, followed by oxidation using PCC/CH₂Cl₂ to synthesize the desired aldehyde. Finally, reductive amination with diaminopropane, diaminobutane, diaminohexane, and spermidine using Na(OAc)₃BH as reducing agent, afforded the required four dimers.