A novel series of glycerol-based double-or triplechain SUl~ factants with two sulfonate, two sulfate or two carboxylate groups was conveniently prepared by reactions of 14~ alkylglyeerol diglycidyl ethers with long-chain fatty alcohols, and followed by reae~ons with propanesnltone~ chlc~ rosulfonie acid or bromoacetic acid, respeetively. The sulfate and earboxylate types of compounds have higher water solubilities than the corresponding sulfonate type of compound bearing the same lipophilic group. The triplechain surfactants show excellent surface-active properties, such as micelle forming and ability to lower surface tension, compared not only with the corresponding singlechain anionic surfactants, but also with the corresponding double-chain surfactants. The effect of the difference in head groups of these compounds on surfac~active proper~ ties is described. Fonming properties, wetting ability and lime-soap dispersing requirement are also discussed.
MSC: 65D07 65D17Keywords: Said-Ball basis Bernstein basis Quasi Extended Chebyshev space B-spline curve Shape parameter Totally positive property a b s t r a c t Four new αβ-Bernstein-like basis functions with two exponential shape parameters, are constructed in this paper, which include the cubic Said-Ball basis functions and the cubic Bernstein basis functions. Within the general framework of Quasi Extended Chebyshev space, we prove that the proposed αβ-Bernstein-like basis is an optimal normalized totally positive basis. In order to compute the corresponding αβ-Bézier-like curves stably and efficiently, a new corner cutting algorithm is developed. Necessary and sufficient conditions are derived for the planar αβ-Bézier-like curve having single or double inflection points, a loop or a cusp, or be locally or globally convex in terms of the relative position of its control polygons' side vectors. Based on the new proposed αβ-Bernstein-like basis, a class of αβ-B-spline-like basis functions with two local exponential shape parameters is constructed. Their totally positive property is also proved. The associated αβ-B-splinelike curves have C 2 continuity at single knots and include the cubic non-uniform B-spline curves as a special case, and can be C 2 ∩ FC k+3 (k ∈ Z + ) continuous for particular choice of shape parameters. The exponential shape parameters serve as tension shape parameters and play a predictable adjusting role on generating curves.
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