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Recent researches on multivariate spline and piecewise algebraic variety. (English) Zbl 1152.41009
A preview of some recent results on multivariate spline, piecewise algebraic variety (curve) and their applications is given in twenty theorems.

MSC:
41A15 Spline approximation
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[1] Deng, J.S.; Chen, F.L.; Feng, Y.Y., Dimensions of spline spaces over T-meshes, Journal of computational and applied mathematics, 194, 267-283, (2006) · Zbl 1093.41006
[2] Eisenbud, D.; Green, M.; Harris, J., Cayley – bacharach theorems and conjectures, Bulletin of the American mathematical society, 33, 295-324, (1996) · Zbl 0871.14024
[3] Hartshorn, R., Algebraic geometry, (1977), Springer-Verlag New York
[4] Lang, F.G.; Wang, R.H., Multivariate weak spline space and minimal determining set, Mathematica numerica sinica, 27, 71-80, (2005), (in Chinese)
[5] Li, C.J.; Wang, R.H.; Zhang, F., Improve on the dimensions of spline spaces on T-mesh, Journal of information and computational science, 3, 2, 235-244, (2006)
[6] Liang, X.Z.; Lu, C.M., Properly posed set of nodes for bivariate Lagrange interpolation, (), 189-196 · Zbl 0919.41002
[7] Luo, Z.X., Generator bases of modules in \(K [X]^m\) and their application, Acta Mathematica sinica, 44, 6, 983-994, (2001) · Zbl 1095.13520
[8] Luo, Z.X.; Wang, R.H., Structure and dimension of multivariate spline space of lower degree on arbitrary triangulation, Journal of computational applied mathematics, 195, 1, 113-133, (2006) · Zbl 1097.65026
[9] Sederberg, T.W.; Zheng, J.; Bakenov, A.; Nasri, A., T-splines and T-nurccs, ACM transactions on graphics, 22, 3, 477-484, (2003)
[10] Sederberg, T.W.; Cardon, D.L.; Finnigan, G.T.; North, N.S.; Zheng, J.; Lyche, T., T-spline simplification and local refinement, ACM transactions on graphics, 23, 3, 276-283, (2004)
[11] Shi, X.Q.; Wang, R.H., Bezout’s number for piecewise algebraic curves, Bit, 2, 339-349, (1999) · Zbl 0933.65017
[12] Walker, R.J., Algebraic curves, (1950), Princeton University Press Princeton · Zbl 0039.37701
[13] Wang, R.H., The structural characterization and interpolation for multivariate splines, Acta Mathematica sinica, 18, 91-106, (1975), (English transl. Acta Mathematica Sinica 18 (1975) 10-39) · Zbl 0358.41004
[14] R.H. Wang, Multivariate weak spline, in: Proc. The National Conference on Numerical Analysis, GuangZhou, 1979
[15] Wang, R.H., Multivariate spline and algebraic geometry, Journal of computational and applied mathematics, 121, 153-163, (2000) · Zbl 0960.41008
[16] Wang, R.H., Multivariate spline functions and their applications, (2001), Science Press/Kluwer Pub. Beijing/New York
[17] R.H. Wang, J. Chen, C.J. Li, The dimension of spline space on quasi-rectangular partition, Journal of Mathematical Research and Exposition (in press)
[18] R.H. Wang, D.X. Gong, The Bezout number of piecewise algebraic curves, manuscript · Zbl 1274.14042
[19] R.H. Wang, F.G. Lang, Multivariate spline space over cross-cut partition, Computers & Mathematics with Applications. doi:10.1016/j.camwa.2007.01.030 · Zbl 1125.41007
[20] R.H. Wang, F.G. Lang, Multivariate weak spline spaces, manuscript
[21] Wang, R.H.; Xu, Z.Q., Multivariate weak spline function space, Journal of computational and applied mathematics, 144, 291-299, (2002) · Zbl 1072.41501
[22] Wang, R.H.; Xu, Z.Q., The estimates of Bezout numbers for the piecewise algebraic curves, Science in China (series A), 33, 2, 185-192, (2003)
[23] Wang, R.H.; Zhu, C.G., Nöther-type theorem of piecewise algebraic curves, Progress in natural science, 14, 309-313, (2004) · Zbl 1083.14525
[24] Wang, R.H.; Zhu, C.G., Piecewise algebraic varieties, Progress in natural science, 14, 568-572, (2004) · Zbl 1083.14526
[25] Wang, R.H.; Zhu, C.G., Cayley – bacharach theorem of piecewise algebraic curves, Journal of computational and applied mathematics, 163, 269-276, (2004) · Zbl 1070.14034
[26] Z.Q. Xu, Multivariate splines, piecewise algebraic curves and linear diophantine equations, Ph.D. Thesis, Dalian University of Technology, China, 2003
[27] Zhu, C.G.; Wang, R.H., Piecewise semialgebraic sets, Journal of computational mathematics, 23, 503-512, (2005) · Zbl 1084.14058
[28] Zhu, C.G.; Wang, R.H., Lagrange interpolation by bivariate splines on cross-cut partitions, Journal of computational and applied mathematics, 195, 326-340, (2006) · Zbl 1097.65023
[29] Zhu, C.G.; Wang, R.H., Nöther-type theorem and its application, Journal of information and computational science, 3, 365-372, (2006)
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