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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