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  1. null (Ed.)
    This work proposes a rigorous and practical algorithm for quad-mesh generation based the Abel-Jacobi theory of algebraic \textcolor{red}{curves}. We prove sufficient and necessary conditions for a flat metric with cone singularities to be compatible with a quad-mesh, in terms of the deck-transformation, then develop an algorithm based on the theorem. The algorithm has two stages: first, a meromorphic quartic differential is generated to induce a T-mesh; second, the edge lengths of the T-mesh are adjusted by solving a linear system to satisfy the deck transformation condition, which produces a quad-mesh. In the first stage, the algorithm pipeline can be summarized as follows: calculate the homology group; compute the holomorphic differential group; construct the period matrix of the surface and Jacobi variety; calculate the Abel-Jacobi map for a given divisor; optimize the divisor to satisfy the Abel-Jacobi condition by integer programming; compute \textcolor{red}{a} flat Riemannian metric with cone singularities at the divisor by Ricci flow; \textcolor{red}{isometrically} immerse the surface punctured at the divisor onto the complex plane and pull back the canonical holomorphic differential to the surface to obtain the meromorphic quartic differential; construct a motorcycle graph to generate a T-Mesh. In the second stage, the deck transformation constraints are formulated as a linear equation system of the edge lengths of the T-mesh. The solution provides a flat metric with integral deck transformations, which leads to the final quad-mesh. The proposed method is rigorous and practical. The T-mesh and quad-mesh results can be applied for constructing Splines directly. The efficiency and efficacy of the proposed algorithm are demonstrated by experimental results on surfaces with complicated topologies and geometries. 
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  2. Abstract Terahertz technology has broad application prospects in biomedical detection. However, the mixed characteristics of actual samples make the terahertz spectrum complex and difficult to distinguish, and there is no practical terahertz detection method for clinical medicine. Here, we propose a three-step one-way terahertz model, presenting a detailed flow analysis of terahertz technology in the biomedical detection of renal fibrosis as an example: 1) biomarker determination: screening disease biomarkers and establishing the terahertz spectrum and concentration gradient; 2) mixture interference removal: clearing the interfering signals in the mixture for the biomarker in the animal model and evaluating and retaining the effective characteristic peaks; and 3) individual difference removal: excluding individual interference differences and confirming the final effective terahertz parameters in the human sample. The root mean square error of our model is three orders of magnitude lower than that of the gold standard, with profound implications for the rapid, accurate and early detection of diseases. 
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  3. null (Ed.)