Vertex-and face-based subdivision schemes are now routinely used in geometric modeling and computational science, and their primal/dual relationships are well studied. In this paper, we interpret these schemes as defining bases for discrete differential 0-resp. 2-forms, and complete the picture by introducing edge-based subdivision schemes to construct the missing bases for discrete differential 1-forms. Such subdivision schemes map scalar coefficients on edges from the coarse to the refined mesh and are intrinsic to the surface. Our construction is based on treating vertex-, edge-, and face-based subdivision schemes as a joint triple and enforcing that subdivision commutes with the topological exterior derivative. We demonstrate our construction for the case of arbitrary topology triangle meshes. Using Loop's scheme for 0-forms and generalized half-box splines for 2-forms results in a unique generalized spline scheme for 1-forms, easily incorporated into standard subdivision surface codes. We also provide corresponding boundary stencils. Once a metric is supplied, the scalar 1-form coefficients define a smooth tangent vector field on the underlying subdivision surface. Design of tangent vector fields is made particularly easy with this machinery as we demonstrate.

Vertex-and face-based subdivision schemes are now routinely used in geometric modeling and computational science, and their primal/dual relationships are well studied. In this paper, we interpret these schemes as defining bases for discrete differential 0-resp. 2-forms, and complete the picture by introducing edge-based subdivision schemes to construct the missing bases for discrete differential 1-forms. Such subdivision schemes map scalar coefficients on edges from the coarse to the refined mesh and are intrinsic to the surface. Our construction is based on treating vertex-, edge-, and face-based subdivision schemes as a joint triple and enforcing that subdivision commutes with the topological exterior derivative. We demonstrate our construction for the case of arbitrary topology triangle meshes. Using Loop's scheme for 0-forms and generalized half-box splines for 2-forms results in a unique generalized spline scheme for 1-forms, easily incorporated into standard subdivision surface codes. We also provide corresponding boundary stencils. Once a metric is supplied, the scalar 1-form coefficients define a smooth tangent vector field on the underlying subdivision surface. Design of tangent vector fields is made particularly easy with this machinery as we demonstrate.

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Background. Gamma-glutamyltransferase (GGT) is involved in tumor development and progression, but its prognostic value in α-fetoprotein- (AFP-) negative (AFP<25 ng/mL) hepatocellular carcinoma (HCC) patients remains unknown. Methods. A large cohort of 678 patients with AFP-negative HCC following curative resection who had complete data were enrolled in this study. The optimal cutoff value for the preoperative level of GGT was determined by the X-tile program. Independent prognostic factors for overall survival (OS) and disease-free survival (DFS) were also identified. Results. The optimal cutoff values for the preoperative levels of GGT were 37.2 U/L and 102.8 U/L, which were used to divide all patients into three subgroups (group 1, GGT<37.2 U/L (n=211, 31.1%); group 2, GGT≥37.2 and <102.8 U/L (n=320, 47.2%); group 3, GGT≥102.8 U/L (n=147, 21.7%)), with distinct OS times (58.5 vs. 53.5 vs. 44.4 months, P<0.001) and DFS times (47.9 vs. 40.3 vs. 30.1 months, P<0.001). Elevated preoperative GGT levels were associated with an unfavorable tumor burden (larger tumor size, multiple tumors, and microvascular invasion) and were selected as independent predictors of a worse OS (group 2 vs. group 1, HR: 1.73 (1.13-2.65), P=0.011; group 3 vs. group 1, HR: 3.28 (2.10-5.13), P<0.001) and DFS (group 2 vs. group 1, HR: 1.52 (1.13-2.05), P=0.006; group 3 vs. group 1, HR: 2.11 (1.49-2.98), P<0.001) in multivariable analysis. Conclusions. Elevated preoperative GGT levels are associated with an unfavorable tumor burden and serve as an independent prognostic marker for worse outcomes in AFP-negative HCC patients following resection.

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