This paper discusses the dimension of spline spaces S(m, n, m−1, n−1, T ) over certain type of hierarchical T-meshes. The major step is to set up a bijection between the spline space S(m, n, m − 1, n − 1, T ) and a univariate spline space whose definition depends on the l-edges of the extended T-mesh. We decompose the univariate spline space into direct sums in the sense of isomorphism using the theory of the short exact sequence in homological algebra. According to the decomposition of the univariate spline space, the dimension formula of the spline space S(m, n, m − 1, n − 1, T ) over certain type of hierarchical T-mesh is presented. A set of basis functions of the spline space is also constructed.

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Abstract:The α,ω-dialkoxyfluoropolyethers (DA-FPEs) characterized by the structure R H O(CF 2 CF 2 O) n (CF 2 O) m R H have been developed as a new class of environmentally friendly hydrofluoroethers (HFEs) suitable as solvents, long-term refrigerants, cleaning fluids, and heat transfer fluids. Synthetic methodologies for DA-FPEs described here consist of radical-initiated oxypolymerization of olefin, peroxy-elimination reaction in peroxidic perfluoropolyethers (P-PFPEs) and further chemical modification of α,ω-diacylfluoride PFPE. The physical properties of selected α,ω-dimethoxyfluoropolyethers (DM-FPEs) have been evaluated and compared with analogous hydrofluoropolyethers (HFPEs) having -OCF 2 H as end-groups. Atmospheric implications and global warming potentials (GWPs) of selected DA-FPEs are also considered.

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Motivated by the magneto hydrodynamic (MHD) simulation for Tokamaks with Isogeometric analysis, we present splines defined over a rectangular mesh with a complex topological structure, i.e., with extraordinary vertices. These splines are piecewise polynomial functions of bi-degree (d,d) and parameter continuity. And we compute their dimension and exhibit basis functions called Hermite bases for bicubic spline spaces. We investigate their potential applications for solving partial differential equations (PDEs) over a physical domain in the framework of Isogeometric analysis. For instance, we analyze the property of approximation of these spline spaces for the L2-norm; we show that the optimal approximation order and numerical convergence rates are reached by setting a proper parameterization, although the fact that the basis functions are singular at extraordinary vertices.

A micellar electrokinetic capillary chromatography (MEKC) method for the separation and quantification of the coumarins in Cnidii Fructus was developed for the first time. Within 25 minutes, 5 major coumarins were separated using 18 mM borate, 12 mM phosphate and 50 mM SDS (pH 9.2) containing 20 % methanol. The relative standard deviations of migration times and peak areas were less than 5 %. The total contents of the five coumarins in plants from different habitats ranged from 13.6 to 44.6 mg/g, and recoveries ranged from 96.0 to 102.8 %. The effects of buffer concentration, SDS, pH value and organic modifier on the separation were investigated.