Certain positive linear operators with better approximation properties

The present paper deals with a new positive linear operator which gives a connection between the Bernstein operators and their genuine Bernstein‐Durrmeyer variants. These new operators depend on a certain function τ defined on [0,1] and improve the classical results in some particular cases. Some approximation properties of the new operators in terms of first and second modulus of continuity and the Ditzian‐Totik modulus of smoothness are studied. Quantitative Voronovskaja–type theorems and Grüss‐Voronovskaja–type theorems constitute a great deal of interest of the present work. Some numerical results that compare the rate of convergence of these operators with the classical ones and illustrate the relevance of the theoretical results are given.     

Aminolysis of P-trichloro-N-dichlorophosphorylmonophosphazene and the crystal structure of 1-(dichlorophosphinyl)-2-chloro-2,2-bis(diisopropylamino)phosphazene

The reactions of Cl₃PN P(O)Cl₂ ( 1 ) with primary and secondary amines have been studied. The following monophosphazenes, (RRN)₃PN P(O)(NRR)₂, and bis(phosphinoyl)amines, [(RRN)₂P(O)]₂NH were isolated: NRR = NHCH₂Ph, Net₂, NH(CH₂)₂CH₃ groups for monophosphazenes, and Net₂, NH(CH₂)₂CH₃ for phosphinoyl amines. The unexpected geminal phosphazene, Cl(RRN)₂PN P(O)Cl₂, {RRN = N[CH(CH₃)₂]₂}, was also obtained in moderate yield. The spectral data (IR, ¹H, ¹³C, and ³¹P NMR, and MS) are presented. The structure of 1-(dichlorophosphinyl)-2-chloro-2,2-bis(diisopropylamino)phosphazene ( 5 ) was determined by X-ray crystallography. The basicities of these and related compounds in nonaqueous nitrobenzene solution were obtained by potentiometric titration.