The Department of Mathematics are actively involved in many areas of research, including the following:
Prof. Li-Zhi Liao, Dr. Leevan Ling, Prof. Michael Kwok-Po Ng,
Dr. Zhonghua Qiao,
Prof. Tao Tang,
Prof. Xiao-Nan Wu,
Dr. Wei-Min Xue,
Dr. Xiaoming Yuan,
Dr. Tieyong Zeng.
Colleagues in our department regularly join with other academics in Hong Kong to organize international conferences on some aspect of scientific computing. For example, The 2nd International Conference on Scientific Computing and Partial Differential Equations & The First East Asia SIAM Symposium was held at the Baptist University from December 12-16, 2005.
Below is a list of research areas of our colleagues in scientific computing:
- Finite Element Methods, including conforming, nonconforming and hybrid-mixed finite element methods for solving second and fourth order, linear and nonlinear problems. Finite element methods for problems with singularities and problems on unbounded domains are also studied.
- Domain Decomposition Methods. Based on the finite element discretization and conjugate gradient iteration we consider preconditioners for various kinds of problems - selfadjoint or non-selfadjoint, linear or nonlinear.
- Grid Generation. We develop automatic grid generation techniques for arbitrary domains using quadrilateral or triangle meshes. Applications include the tidal analysis of harbors and computer aided geometric design.
- Computational Fluid Dynamics, including numerical methods for incompressible and compressible flows, which are governed by linear or nonlinear partial differential equations.
- Computational Bio-Mechanics. Numerical methods have been developed for the mechanics and lubrication of the normal articulating joint, the load and deformational behavior of hydrated soft tissues such as articular cartilage, and meniscus and intervertebral discs.
- Spectral Methods. Mainly focusing on its application to computational fluid dynamics, and its application for solving problems in unbounded domain.
- Multidimensional Quadrature. We have established new error bounds for quasi-Monte Carlo methods using reproducing kernel Hilbert space techniques.
- Optimization, involving both theoretical and numerical aspects. The main research areas are unconstrained and constrained optimization, global optimization and large scale optimization.
- Optimal Control, mainly focusing on discrete-time nonlinear systems. Recent studies include nonlinear and linear optimal control and general multiobjective nonlinear control.
- Image Processing, including image restoration, pattern recognition and image coding.
- Neural Networks, with applications to financial analysis, image processing and graph theory.
- Parallel Computing. This research is concentrated on the development of parallel methodologies for large scale optimization, optimal control and numerical solutions for partial differential equations.
Prof. Sung-Nok Chiu, Dr. H. Peng, Dr. M. L. Tang, Prof. Lixing Zhu.
Prof. Zhu is an IMS Fellow and director of the University's Statistics Research and Consultancy Centre (SRCC).
The SRCC engages in both theoretical and applied statistics research and provides statistical consultancy. Major research and consultancy projects have been undertaken with the Departments of Marketing and History, the Language Centre, the Open Learning Institute and the Urban Council Public Libraries. The SRCC also sponsors public lectures and short courses, facilitates collaboration among local scholars and collaboration with scholars from China and overseas, and organizes conferences. Among these are
- the International Symposium on Multivariate Analysis and Its Applications, March 1992
- the International Workshop on Quasi-Monte Carlo Methods and Their Applications, December 1995, and
- the International Symposium on Contemporary Multivariate Analysis and Its Applications, May 19-22, 1997
- symposium on Theory of The Uniform Design and Its Applications, October 26-29, 1999
- Fourth International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific computing, November 27 - December 1, 2000
- International Workshop on Applied Mathematics and Statistics, December 16, 2004
- International Conference on Statistics, June 20-24, 2005
The statistical research in our department is in the following areas:
- Multivariate Analysis, including inference, elliptically contoured distributions and tests for mulitivariate symmetry and normality.
- Number-Theoretic or Quasi-Monte Carlo Methods, generating low discrepancy sequences and studying their applications to experimental design and inference.
- Experimental design, developing theory of the uniform design and its applications, exploring relationships among the uniform, orthogonal and optimal designs.
- Statistical Computing, including multidimensional statistical integrals, Gibbs sampling and neural networks.
- Survival Analysis, including the Cox proportional hazards model, multivariate survival data and interval-censored data.
- Linear Models, including non-normal error inference with intra-correlation and designs of experiments.
- Spatial Statistics, including statistics for point processes and random sets and methods or edge-correction.
- Applied Probability, including stochastic geometry, random processes, and probabilistic modeling of molecular mechanisms in biology.
- Data mining and Bioinformatics in Chinese medicine and Chemistry, including development of new methods in data mining, microarray data analysis, repeatability and reproducibility of Chinese medicine.
At present there are several students pursuing their PhDs in statistics in our department. The statistics group runs a seminar series for postgraduate students and colleagues that has included lectures on graphical models, nonparametric regression, experimental design, wavelets, robust statistics, and distribution theory.
Dr. Wai Hong Chan, Dr. Wai-Chee Shiu.
The major areas of Applied Discrete Mathematics include combinatorics, graph theory and related discrete mathematical structures. During recent decades, it has developed into a major area of mathematics. In addition to its growing interest and importance as a mathematical subject, it has applications to many fields, including physics, chemistry, engineering, operations research, error-correcting code theory, genetics, economics, psychology, sociology, stringology, bioinformatics, etc.
- Algorithm Design, including linear, super-linear, and combinatorial algorithm design for sorting, transformations and suffix array constructions.
- Bandwidth minimization problem, including bounds of bandwidth on graphs with small number of cycles; effects on bandwidth after graph operations such as adding edges, merging vertices, or sub-division; cyclic bandwidth; and characterization of graphs with equal bandwidth and cyclic bandwidth.
- Circular chromatic number, including the circular chromatic number of some distance graphs; and the exact determination of a class of graphs obtained by a generalization of the construction of Mycielski.
- Coloring problems, including chromatic numbers and/or choice number of plane graphs without certain class of cycles; total vertex-edge coloring; total vertex-face coloring; total edge-face coloring; entire coloring problems and game chromatic numbers. We also work on some related problem such as arboricity or vertex-arboricity problems.
- Distance constrained labeling, including lambda2,1-numbers of special graphs and their relationship with other graph parameters; extending labeling of graphs; complexity and algorithms; and Lj,k-labeling.
- Topological indices of graphs, including Weiner numbers; Kekulé structures; Hosoya index; Merrifield index; total pi-electron energy of molecular graphs.