3 research teams from Department of Mathematics (DMAT), FST, the University of Macau (UM) has been recommended to receive prizes at the 2018 Macao Science and Technology Awards. They have been recommended to receive 3 prizes out of the 7 prizes in the Natural Science Award category.
The projects that receives second prizes in this category is Prof. Haiwei SUN and Prof. Siu Long LEI’s project: ‘Finite Difference Solutions of Fractional Partial Differential Equations and Their Fast Algorithms’. Projects that receive third prizes in this category are Prof. Yang CHEN’s project: ‘Random Matrices and Some of its Applications’, and Prof. Kit Ian KOU’s project ‘Quaternions and Linear Canonical Analysis with Applications to Color Face Recognition and Edge Detection’.
In Prof. Haiwei SUN and Prof. Siu Long LEI’s project - Finite Difference Solutions of Fractional Partial Differential Equations and Their Fast Algorithms, proposed the multigrid algorithm to speed up the convergence rate of the iterative methods. Thus they solved one of the difficulties on numerical solutions: the discretized linear systems are ill-conditioned, hence slow down the convergence speed of the iterative methods, and increase the computational cost. Soon after that, they designed a kind of circulant preconditioner to for solving the fractional PDEs. Both methods become the benchmarks on iterative algorithms for numerical solutions of fractional PDEs.
In Prof. Yang CHEN’s project - Random Matrices and Some of its Applications, they are concerned with the normalization constant and the distribution of a certain linear statistics,defined as the sum of function evaluated at the eigenvalues of the Random Matrices. Ultimately, these are generating functions which can be characterized by the determinant of Hankel matrices generated by the moments of a certain weight function, typically the classical weights, for example, Normal, Gamma, and Beta densities multiplied by factors that are informed by the problems at hand. For large n, the appropriate consideration is about the “double scale” behavior of the Hankel determinant. In this scheme n, the size of the matrix tends to infinity and a parameter say t, a kind of time variable, tends to 0, in a suitable combinations which are finite. Here we discover that the original finite n problem, and the “larger” Painleve equations degenerates to a “smaller” Painleve equations, and the matrix size n has ``disappeared.” This is the universal behavior inherent in random matrix. Chen and his collaborators obtained the hard-to-come-by constant in an asymptotic expansion in the double scaled parameter. In addition to the Hermitian ensembles, the Applicant is also interest in the expectation value of linear statistics in the orthogonal (power equals 1) and symplectic (power equals 4) ensembles. Here, large n behavior is found.
In Prof. Kit Ian KOU’s project - Quaternions and Linear Canonical Analysis with Applications to Color Face Recognition and Edge Detection, based on the analytic function theory and linear canonical analysis in complex case, this project studies the new problems related to quaternion analysis and enriches the research of Fourier analysis and image processing. For the extended analytic function theory, a new development with milestones is given. The application of quaternion sparse representation theory in color face recognition problem is proposed for the first time; and the image edge algorithm based on analytic signal phase is studied under the quaternion linear canonical transformation. The results have been widely cited and applied in many fields at home and abroad, leaving a footprint from Macao in the contribution to the significant event of color face recognition and edge detection development on science and technology.
孫海衛教授和李兆隆教授的項目 –分數階偏微方程的有限差分解及其快速算法中，率先解決了分數階偏微分方程的數值解研究面臨著三大困難之一: 離散後的線性系統是病態的，減慢了迭代法的收斂速度，增加了計算量。項目提出用多重網格算法來加快迭代法的收斂速度，然後又設計了一種循環預處理算法來加速收斂.這兩種方法成為後續研究分數階偏微分方程數值解迭代算法的標桿。
高潔欣教授的項目 - 四元數和線性正則分析應用於彩色人臉識別和邊緣檢測中，基于單複變中解析函數理論和綫性正則分析, 研究四元數分析中相關的新問題, 豐富了傅裏葉分析和圖像處理的研究。對擴展的解析函數理論, 給出具有里程碑意義的新發展。首次提出了四元數稀疏表示理論在彩色人臉識別問題中的應用; 以及在四元數綫性正則變化下, 基于解析信號相位的圖像邊緣算法研究。