A bivariate extension of Bleimann-Butzer-Hahn operator

Let C(R2+ ) be a class of continuous functions f on R2+. A bivariate extension L.(f ,x,y) of BleimannButzer-Hahn operator is defined and its standard convergence properties are given. Moreover, a local analogue of Voronovskaja theorem is also given for a subclass of C(R2+).     

Acid-catalyzed isomerization of pivalaldehyde to methyl isopropyl ketone via a reactive protosolvated carboxonium ion intermediate.

Quantitative rearrangement of pivalaldehyde to methyl isopropyl ketone is observed in acids such as trifluoromethanesulfonic acid, anhydrous HF, and trifluoroethyl alcohol-BF3 but not in trifluoroacetic acid. Studies in a mixture of trifluoroacetic acid and trifluoromethanesulfonic acid show that acids with H(o) < or = -11 are able to carry out complete isomerization. These results and density functional theory calculations at the B3LYP/6-31G level suggest that protonated pivalaldehyde undergoes further protosolvation at higher acidities to a reactive superelectrophilic species resulting in rearrangement. A mechanism for the pivalaldehyde rearrangement to methyl isopropyl ketone in strong protic acids involving a reactive protosolvated superelectrophilic intermediate is suggested. Aspects of the related mechanism of the reaction with isobutane with CO in HF/BF3 medium leading to methyl isopropyl ketone are also discussed.     

The free boundary problem without initial condition

We consider an analog of the problem of the impact of a viscoplastic rod on the wall under a nonlinear boundary condition. We investigate the behavior of the free boundary on a given time interval and as t????. We obtained a priori estimates of H?lder norms and proved the theorems of uniqueness and existence of the solution.     

Determination of starane (fluroxypyr) herbicide using flow injection spectrophotometry.

The starane herbicide was spectrophotometrically determined by the diazotization method in a flow injection assembly. Since starane is a substituted pyridyl compound the NH2 group at the p-position was exploited for diazotization. Starane was diazotized with nitrite and the diazotized product is coupled with beta-naphthol. The absorbance of the resulting azo dye was measured at 395 nm with a molar absorptivity of 1.5 x 10(4) L mol(-1) cm(-1). The calibration graph was linear over the range of 0.6 to 10 microg/mL, with a relative standard deviation (RSD) of 1.67% and a sampling through put of 60 samples h(-1). The % recovery for the determination of starane was found to be 96%. The method was successfully applied to the determination of the active ingredient of starane herbicide in its formulation as well as in food samples.     

A note on a Bernstein-type operator of Bleimann, Butzer, and Hahn

Let L_n(f, x)(f ε C)[0, ∞)) be a Bernstein-type approximation operator as defined and studied by Bleimann, Butzer, and Hahn. Probabilistic arguments are used to simplify and sharpen some of their results. The rates of convergence are given in terms of the first and second moduli of continuity. Moreover, an appropriate limit of L_n is shown to be the well-known Szasz operator.