Dept of Mathematics

DEPT OF MATHEMATICS

Researcher : Akutsu T

List of Research Outputs

Akutsu T., Hayashida M., Ching W.K. and Ng K.P., Control of Boolean Networks: Hardness Results and Algorithms for Tree Structured Networks, In: Journal of Theoretical Biology, Journal of Theoretical Biology. 2007, 244: 670-679.

Zhang S., Ching W.K., Ng K.P. and Akutsu T., Simulation Study in Probabilistic Boolean Network Models for Genetic Regulatory Networks , Journal of Data Mining and Bioinformatics . 2007, 1: 217-240.

Researcher : Ao SI

List of Research Outputs

Sham P.C., Ao S.I., Kwan S.H., Kao P., Cheung F., Fong P.Y. and Ng M.K., Combining functional and linkage disequilibrium information in the selection of tag SNPs, Bioinformatics. 2007, 23(1): 129-131.

Researcher : Cai H

List of Research Outputs

Cai H., Xu X., Lu J., Lichtman J.W., Yung S.P. and Wong S.T.C., Repulsive Force Based Snake Model to Segment and Track Neuronal Axons in 3D Microscopy Image Stacks, NeuroImage. 2006, 32: 1608-1620.

Researcher : Chan JT

Project Title:	The linear preserver problem
Investigator(s):	Chan JT
Department:	Mathematics
Source(s) of Funding:	Competitive Earmarked Research Grants (CERG)
Start Date:	09/2003

Abstract:

To investigate linear mappings on an operator algebra that leave invariant a certain set or function of the algebra.

Project Title:	Linear and Non-linear Preserver Problems
Investigator(s):	Chan JT
Department:	Mathematics
Source(s) of Funding:	Competitive Earmarked Research Grants (CERG)
Start Date:	01/2006

Abstract:

To describe some new preservers, to develop new proof techniques, and further explore the connections between different subjects via the study of preserver problems.

List of Research Outputs

Chan J.T., Li C.K. and Sze N.S., Mappings on Matrices: Invariance of Functional Values of Matrix Product, Journal of the Australian Mathematical Society. 2006, 81: 165-184.

Chan J.T., Li C.K. and Sze N.S., Mappings preserving spectra of products of matrices, Proceeding American Mathematical Society. 2007, 135: 977-986.

Researcher : Chan YL

List of Research Outputs

Ng K.P., Chan Y.L., So M.C. and Ching W.K., A Semi-Supervised Regression Model for Mixed Numerical and Categorical Variables, Pattern Recognition. 2007, 40: 1745-1752.

Researcher : Cheung WS

Project Title:	Bounded domains in n-dimensional complex spaces
Investigator(s):
Department:	Mathematics
Source(s) of Funding:	Other Funding Scheme
Start Date:	09/1991

Abstract:

To study the behaviour and properties of certain intrinsic measures on bounded domains in an n-dimensional euclidean space.

Project Title:	On multi-dimensional integral inequalities and applications
Investigator(s):	Cheung WS
Department:	Mathematics
Source(s) of Funding:	Competitive Earmarked Research Grants (CERG)
Start Date:	12/2003
Completion Date:	11/2006

Abstract:

To develop new integral inequalities of the Opial-type, Poincare-type, Wirtinger-type, Gronwall-Bellman-type, Hardy-type, etc., which are useful in nonlinear analysis and approximations; to establish discrete analogues of the above-mentioned integral inequalities; to qualitative analysis of solutions of differential and difference equations.

Project Title:	Complete Monotonicity of Special Functions Involving the Gamma, Digamma, and Polygamma Functions
Investigator(s):	Cheung WS
Department:	Mathematics
Source(s) of Funding:	Seed Funding Programme for Basic Research
Start Date:	02/2005

Abstract:

To investigate the complete monotonicity and logarithmic complete monotonicity of some special functions involving the gamma, digamma, and polygamma functions.

Project Title:	Sufficient conditions for existence of solutions to p-Laplacian differential equations
Investigator(s):	Cheung WS
Department:	Mathematics
Source(s) of Funding:	Competitive Earmarked Research Grants (CERG)
Start Date:	01/2006

Abstract:

1) To search for sufficient conditions guaranteeing the existence of solutions of certain types of p-Laplacian BVPs at resonance. 2) To obtain sufficient conditions for the existence of periodic solutions of p-Laplacian functional differential equations, including for example delay Duffing equations, Liénard equation, Rayleigh equations and neutral equations etc., with a p-Laplacian. 3) Also intend to investigate how the existence of periodic solutions relies on the delay of the prescribed functional equations

Project Title:	Complexity and Control of Biological Dynamical Systems
Investigator(s):	Cheung WS
Department:	Mathematics
Source(s) of Funding:	Seed Funding Programme for Basic Research
Start Date:	07/2006

Abstract:

There are various biological dynamical systems in nature, including macroscopically, ecological species evolution models, and microscopically, protein and gene regulatory network. Negative feedback, positive feedback, and more complicated interactions, which manifest themselves in all kinds of biological networks, bring forth multifarious complexities. It is of utmost significance in both theory and practice to understand these complexities better. By bridging up the macroscopic and the microscopic horizons of biological dynamical systems, this project will establish new mathematical models of dynamical networks with biological backgrounds to reflex the complexities of biological dynamical systems, and by using the point of view of complex systems dynamics, understand how the complexities of these systems are affected by various factors quantitatively and qualitatively. Details are as follows: Objectives: 1. Establish models: To obtain new mathematical models of dynamical networks to reflect the complexities of dynamical systems with biological backgrounds. 2. Analyze on the dynamics of the models: Using the stability theory (of differential and difference equations), Markov chain, graph theory, matrix theory, and the theory of inequalities, etc., to do qualitative as well as quantitative analysis of the complex system, and to understand how the dynamics is affected by various factors qualitatively and quantitatively. Key Issues: Network scale: The number of nodes for biological dynamical systems is, normally, larger than that for other dynamical systems. This leads to complexity and adds difficulty to the problem. Nonlinear reflections: The complexities of biological dynamical systems are reflected also on interaction among the nodes. These interactions are usually nonlinear but not arbitrary. This brings in nonlinear analysis of the systems. Network topology: So far there has been very little work done on how the network topology affects the behavior of the dynamics of a dynamical system, especially on biological dynamical systems. Time-delay: Time-delay is another important factor affecting the behavior of the dynamics of a biological dynamical system. There are much studies on delay differential systems, but time-delay in biological dynamical systems has rarely been studied. Feedback control: Negative as well as positive feedbacks play an essential role in biological dynamical systems. We will give a preliminary attempt on feedback control on such systems.

Project Title:	Inclusion measures and chord power integrals in convex geometry
Investigator(s):	Cheung WS
Department:	Mathematics
Source(s) of Funding:	Seed Funding Programme for Basic Research
Start Date:	03/2007

Abstract:

The proposed research focuses on the theory of inclusion measures and chord power integrals of convex bodies, which are of utmost importance in Integral Geometry. The theory of inclusion measures can be regarded as the extension of the classical isoperimetric inequality, and chord power integrals are generalizations of the surface area and the volume of convex body. The classical isoperimetric inequality and volume of convex bodies are the main research contents of Convex Geometry, whose core is the Brunn-Minkowski theory, also known as the theory of mixed volumes. So in certain sense, the present project serves as a link between Integral Geometry and Convex Geometry. The aim of the project is to establish some Brunn-Minkowski-type or dual Brunn-Minkowski-type inequalities for inclusion measures and chord power integrals, with the method of mixed volume or dual mixed volume, respectively. Meanwhile, investigations on the properties of inclusion measures and chord power integrals will be done, and the new findings will be used to explore hidden properties of convex bodies.

List of Research Outputs

Cheung W.S., Matkovic A. and Pecaric J., A Variant of Jessen's Inequality and Generalized Means, Journal of Inequalities in Pure and Applied Mathematics. 2006, 7, Issue 1, Art.10 (electronic journal): 8 pages.

Cheung W.S., Associate Editor, Journal of Inequalities and Applications. 2006.

Cheung W.S. and Pecaric J., Bohr's Inequalities for Hilbert Space Operators, Journal of Mathematical Analysis and Applications. 2006, 323: 403-412.

Cheung W.S., Cho Y.J., Pecaric J. and Zhao D., Bohr's Inequalities in n-Inner Product Spaces, Journal of the Korea Society of Mathematical Education, Series B: Pure and Applied Mathematics. 2007, 14, No.2 (May 2007): 127-137.

Cheung W.S., Calculus of Variations via Exterior Differential Systems, International Conference on Mathematics, Ulaanbaatar, Mongolia, Jul 2006. 2006.

Cheung W.S., Continuous and Discrete Nonlinear Inequalities and Applications to Boundary Value Problems, Proceedings of the Conference on Differential and Difference Equations and Applications, Florida, Aug 2005. 2006, 299-313.

Cheung W.S. and Ren J., Discrete Nonlinear Inequalities and Applications to Boundary Value Problems, Journal of Mathematical Analysis and Applications. 2006, 319: 708-724.

Cheung W.S., Editor of Australian Journal of Mathematical Analysis and Applications. 2007.

Cheung W.S., Editor of Journal of Inequalities in Pure and Applied Mathematics. 2007.

Cheung W.S., Editor of the Bulletin of Southeast Asian Mathematical Society. 2007.

Cheung W.S. and Wong P.J.Y., Fixed-sign Solutions for a System of Singular Focal Boundary Value Problems, Journal of Mathematical Analysis and Applications. 2007, 329: 851-869.

Cheung W.S. and Zhao C., Inverses of New Hilbert-Pachpatte Type Inequalities, Journal of Inequalities and Applications. 2006, 2006, Art. ID 97860: 11 pp.

Cheung W.S. and Qi F., Logarithmic Convexity of the One-Parameter Mean Values, Taiwanese Journal of Mathematics. 2006, 11, No.1: 231-237.

Cheung W.S., Ren J., Wong P.J.Y. and Zhao D., Multiple Positive Solutions for Discrete Nonlocal Boundary Value Problems, Journal of Mathematical Analysis and Applications. 2007, 330: 900-915.

Cheung W.S. and Wong B., On a Non-abelian Invariant over Complex Surface of General Type, Sciences in China, Series A: Mathematics. 2006, 49, No.12: 1897-1900.

Cheung W.S., Zhao D. and Pecaric J., Opial-Type Inequalities for Differential Operators, Nonlinear Analysis. 2007, 66, No.9: 2028-2039.

Cheung W.S. and Ren J., Periodic Solutions for p-Laplacian Rayleigh Equations, Nonlinear Analysis. 2006, 65: 2003-2012.

Cheung W.S., Xiong G. and Ni J., Radial Mean Body of the Simplices, Journal of Shanghai University (English edition). 2007, 11, No.1: 49-51.

Cheung W.S., Reviewer of Zentralblatt Math. V.1099, No.34063. 2006, 1099, No.34063.

Cheung W.S., Reviewer of Zentralblatt Math. V.1099, No.34067. 2006, 1099, No.34067.

Cheung W.S., Reviewer of Zentralblatt Math. V.1107, No.26018. 2007, V.1107, No.26018.

Cheung W.S., Reviewer of Zentralblatt Math. V.1107, No.26019. 2007, V.1107, No.26019.

Cheung W.S., Reviewer of Zentralblatt Math. V.1107, No.26020. 2007, V.1107, No.26020.

Cheung W.S., Reviewer of Zentralblatt Math. V.1107, No.26022. 2007, V.1107, No.26022.

Cheung W.S. and Tseng S., Some New Discrete Nonlinear Delay Inequalities and Application to Discrete Delay Equations, Journal of Inequalities in Pure and Applied Mathematics. 2006, 7, Issue 4, Art.122 (electronic journal): 16 pages.

Cheung W.S. and Ma Q., Some New Nonlinear Difference Inequalities and Their Applications, Journal of Computational and Applied Mathematics. 2007, 202: 339-351.

Cheung W.S., Bencze M. and Zhao C., The Strengthening of Minkowski Inequality for Mixed Projection Bodies, Libertas Matematica. 2006, XXVI: 75-78.

Cheung W.S. and Chan K.P., C^¥-Invariants on the Space of Loops, Bulletin of Southeast Asian Mathematical Society. 2007, 31: 433-440.

Researcher : Ching WK

Project Title:	Models and numerical algorithms for queuing and manufacturing systems
Investigator(s):	Ching WK
Department:	Mathematics
Source(s) of Funding:	Competitive Earmarked Research Grants (CERG)
Start Date:	09/2002

Abstract:

Markovian models are widely used in modelling and analysing queuing and manufacturing systems. This research project aims at developing: 1) models for complex manufacturing systems and queuing systems; and 2) fast numerical algorithms for solving linear systems arising from the captured applications.

Project Title:	Higher dimensional Markov Chain Models for biological data sequences
Investigator(s):	Ching WK
Department:	Mathematics
Source(s) of Funding:	Seed Funding Programme for Basic Research
Start Date:	05/2005

Abstract:

In this project, we focus on developing higher dimensional Markov chain models for categorical data sequences with applications in the analysis of biological sequences. We will also develop fast numerical methods for solving the model parameters.

Project Title:	A New Hidden Markov Model for Categorical Time Series with Applications
Investigator(s):	Ching WK, Li WK
Department:	Mathematics
Source(s) of Funding:	Small Project Funding
Start Date:	02/2006

Abstract:

In this project, we will develop a new hidden Markov model (HMM) for modeling categorical time series. In a traditional HMM, the observable states are affected directly by the hidden states but not vice versa. Here, we will develop a HMM such that the transitions of hidden states depend also on the observable states. Our model can be related to a discrete-state version of the class of the first-order self-exciting threshold autoregressive models for modelling non-linear time series taking numerical values. Efficient estimation methods for the transition probabilities among the hidden states and observable states will also be developed. We will apply the model to practical sales demands data sequences in inventory control.

Project Title:	Fast Numerical Methods for Solving Stationary Distributions of Stochastic Networks
Investigator(s):	Ching WK
Department:	Mathematics
Source(s) of Funding:	Small Project Funding
Start Date:	01/2007

Abstract:

In this project, we will develop fast numerical algorithms for solving the stationary distributions of large stochastic networks. Stochastic networks occurred in many applications such as the Internet and the genetic networks. In surfing the Internet, surfers usually use search engines to find the related webpages satisfying their queries. Therefore a proper list of webpages in certain order of importance is necessary. Thus it is important to seek for fast algorithms for the computing the ranking of the webpages. Google uses the PageRank algorithm to produce such a ranking list and it turns out that the algorithm is just solving the stationary distribution of a large Markov chain (stochastic network). Interactions between different genes become more and more important in understanding how they collectively make cells, tissues, organisms and even form a biological system. This gives rise to genetic regulatory networks. Probabilistic Boolean network (stochastic network) is one of the popular models for modeling such biological systems. Stationary (long-run behavior) analysis is a key issue in studying the dynamics of a genetic regulatory network. The size of both problems is very huge and therefore seeking for a fast numerical algorithm is necessary.Keywords: Stochastic Networks, Internet, Genetic Networks, Iterative Methods, Markov Chains, Stationary Analysis.

List of Research Outputs

Akutsu T., Hayashida M., Ching W.K. and Ng K.P., Control of Boolean Networks: Hardness Results and Algorithms for Tree Structured Networks, In: Journal of Theoretical Biology, Journal of Theoretical Biology. 2007, 244: 670-679.

Ching W.K., High-dimensional Markov Chain Models with Applications, Invited seminar at the Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong. 2006.

Ching W.K., Siu K.T.K., Fung S.L., Ng K.P. and Li W.K., Interactive Hidden Markov Models and Their Applications, IMA Journal of Management Mathematics. Oxford University Press, 2007, 18: 85-97.

Ching W.K., Iterative Methods For Queueing Systems And Markov Chains , The First International Summer School On Numerical Linear Algebra, Guangzhou And Hong Kong. 2006.

Ching W.K., Markov Chain Models and Their Applications in Management Science. Institute of Systems Engineering, Northeastern University, Shenyang, China., 2006.

Ching W.K., Li L., Li T. and Zhang S., New Multivariate Markov Chain Model with Applications to Sales Demand Forecasting, Proceedings of the International Conference on Industrial Engineering and Systems Management 2007, Beijing, (in CD-ROM). 2007.

Ching W.K., Li T. and Xue J., On Hybrid Re-manufacturing Systems: A Matrix Geometric Approach, Proceedings of the International Conference on Industrial Engineering and Systems Management 2007, Beijing, (in CD-ROM). 2007.

Ching W.K., Zhang S. and Ng K.P., On Multi-dimensional Markov Chain Models, Pacific Journal of Optimization . 2007, 2: 235-243.

Lee Y.F. and Ching W.K., On Convergent Probability of a Random Walk, International Journal of Mathematical Education in Science and Technology. Taylor's & Francis, 2006, 37: 833-838.

Ng K.P., Fung S.L., Lee Y.F. and Ching W.K., A Recursive Method for Solving Haplotype Frequencies with Application to Genetics , Journal of Bioinformatics and Computational Biology. World Scientific, 2007, 4: 1269-1286.

Ng K.P., Chan Y.L., So M.C. and Ching W.K., A Semi-Supervised Regression Model for Mixed Numerical and Categorical Variables, Pattern Recognition. 2007, 40: 1745-1752.

Ng M., Zhang S., Ching W.K. and Akutsu T., A Control Model for Markovian Genetic Regulatory Network, Lecture Notes in Computer Science, Transactions on Computational Systems Biology. Springer, 2006, 4070: 36-48.

Ng T.W., Turinici G., Ching W.K., Chung S.K. and Danchin A., A Parasite Vector-host Epidemic Model for TSE Propagation, Medical Science Monitor. 2007, 13(2): 59-66.

Tai A.H.L. and Ching W.K., A Two-echelon Model for Inventory and Returns , Operations Proceedings 2005 . Berlin, Springer, 2006, 2005: 131-136.

Zhang S., Ching W.K., Ng K.P. and Akutsu T., Simulation Study in Probabilistic Boolean Network Models for Genetic Regulatory Networks , Journal of Data Mining and Bioinformatics . 2007, 1: 217-240.

Researcher : Chu SCK

Project Title:	Topological and Optimization Analysis of Online Auction Markets
Investigator(s):	Chu SCK
Department:	Mathematics
Source(s) of Funding:	Competitive Earmarked Research Grants (CERG)
Start Date:	01/2006

Abstract:

The objective of our proposed investigation is to pose the intriguing question of what "shape" a given market is in at a particular moment of development, and to produce models with tools for the answer. The key issues of our work include: (1) To construct a topological model based only on operational data, without any expert knowledge of the specific auction market, or conomic details from the transactions. (2) To identify the dimensions for a topology, using extensive analysis of eBay data to extract the information desired. (3) To propose a graphical model to visualize this topology, also for the purpose of comparing markets at different points of a life cycle. (4) To integrate optimization modelling into this topological analysis to further achieve its maximum resolution of any such buyers/sellers market dichotomy. (4) To construct a topological model based only on operational data, without any expert knowledge of the specific auction market, or economic details from the transactions. (5) To identify the dimensions for a topology, using extensive analysis of eBay data to extract the information desired. (6) To propose a graphical model to visualize this topology, also for the purpose of comparing markets at different points of a life cycle. (7) To integrate optimization modelling into this topological analysis to further achieve its maximum resolution of any such buyers/sellers market dichotomy.

List of Research Outputs

Chu S.C.K., Associate editor of the IMA Journal of Management Mathematics. 2006.

Chu S.C.K. and Zhu M., Data and Optimization Modeling for Manpower Planning of an Airport Baggage Service, Proceedings of the Institute of Industrial Engineers Annual Conference - 2007 Industrial Engineering Research Conference (IERC 2007), Nashville, TN, USA, 19-23 May 2007. 6pp.

Chu S.C.K., Generating, Scheduling and Rostering of Shift Crew-Duties: Applications at the Hong Kong International Airport, European Journal of Operational Research. 2007, 177, No.3: 1764-1778.

Chu S.C.K. and Ho J.K., Topological and Optimization Modeling for Internet Data of Online Auction Markets, In: Zhang, X.-S., Liu D.-G. and Wu, L.-Y. (Ed.) Operations Research and Its Applications, Proceedings of the 6th International Symposium on Operations Research and Its Applications (ISORA2006), Xinjiang, China, 8-12 August 2006. World Publishing Corporation, 144-154.

Ho J.K., Chu S.C.K. and Lam S.S., Maximum Resolution Topology for Online Auction Markets, Electronic Markets. 2007, Volume 17 (2): 164-174.

Ho J.K., Chu S.C.K. and Lam S.S., Optimization Model and DSS for Maximum Resolution Dichotomies, Proceedings of the 4th International Conference on Informatics in Control, Automation and Robotic(ICINCO2007), Angers, France, 9-12 May 2007. 6pp.

Ho M.P.P., Fung H., Chu S.C.K. and Tinsley H., Operational Improvement in a Specialist Out-Patient Clinic in Hong Kong, Hong Kong Medical Journal. 2006, 12, No.6, Supplement 3: 4pp.

Researcher : Chung SK

List of Research Outputs

Ng T.W., Turinici G., Ching W.K., Chung S.K. and Danchin A., A Parasite Vector-host Epidemic Model for TSE Propagation, Medical Science Monitor. 2007, 13(2): 59-66.

Researcher : Fung SL

List of Research Outputs

Ching W.K., Siu K.T.K., Fung S.L., Ng K.P. and Li W.K., Interactive Hidden Markov Models and Their Applications, IMA Journal of Management Mathematics. Oxford University Press, 2007, 18: 85-97.

Ng K.P., Fung S.L., Lee Y.F. and Ching W.K., A Recursive Method for Solving Haplotype Frequencies with Application to Genetics , Journal of Bioinformatics and Computational Biology. World Scientific, 2007, 4: 1269-1286.

Researcher : Hayashida M

List of Research Outputs

Akutsu T., Hayashida M., Ching W.K. and Ng K.P., Control of Boolean Networks: Hardness Results and Algorithms for Tree Structured Networks, In: Journal of Theoretical Biology, Journal of Theoretical Biology. 2007, 244: 670-679.

Researcher : Ho JK

List of Research Outputs

Chu S.C.K. and Ho J.K., Topological and Optimization Modeling for Internet Data of Online Auction Markets, In: Zhang, X.-S., Liu D.-G. and Wu, L.-Y. (Ed.) Operations Research and Its Applications, Proceedings of the 6th International Symposium on Operations Research and Its Applications (ISORA2006), Xinjiang, China, 8-12 August 2006. World Publishing Corporation, 144-154.

Ho J.K., Chu S.C.K. and Lam S.S., Maximum Resolution Topology for Online Auction Markets, Electronic Markets. 2007, Volume 17 (2): 164-174.

Ho J.K., Chu S.C.K. and Lam S.S., Optimization Model and DSS for Maximum Resolution Dichotomies, Proceedings of the 4th International Conference on Informatics in Control, Automation and Robotic(ICINCO2007), Angers, France, 9-12 May 2007. 6pp.

Researcher : Lau YK

Project Title:	Norms of hecke eigenforms and subconvexity estimates for Rankin-Selberg L-funcitons
Investigator(s):	Lau YK
Department:	Mathematics
Source(s) of Funding:	Competitive Earmarked Research Grants (CERG)
Start Date:	01/2007

Abstract:

(1) A central objective in the study of quantum chaos is to understand the quantization of a classical Hamiltonian system whose dynamics are chaotic. Deeper comprehension is often achieved for quantum chaos associated to arithmetic manifolds, referred as arithmetic quantum chaos. Corresponding to the geodesic motion of a point particle over a Riemannian surface X whose curvature is a negative constant, the quantum eigenstates and energy levels are square-integrable solutions of the eigenvalue problem for the Laplace-Beltrami operator. The semi-classical limit is the same as the large eigenvalue limit. Therefore, a basic issue is to understand, for large eigenvalues, the behaviour such as localization of the eigenstates. (2) These (real-analytic) eigenstates are also called Maass forms when X is the quotient of the upper half-plane modulo a congruence subgroup. Conjectures concerning the p-norm and the equidistribution property of the Maass forms were formulated. Important progress is made toward these conjectures. In addition to Maass forms, there are holomorphic cusp forms f(z) of different weight k on the arithmetic surface X. Those holomorphic cusp forms which are common eigenfunctions of the Hecke operators are called Hecke eigenforms. A thorough understanding of Hecke eigenforms may shed light on quantum chaos, because both holomorphic cusp forms and Maass cusp forms are eigenfunctions of the same Casimir operator. Our goal is to study the properties of Hecke eigenforms for large k. (3) In this project, we shall investigate the p-norm. Consider the case p equals infinity. For a Hecke eigenform f(z) of weight k, it is known that after a suitable normalization, the infinity norm of f(z) is bounded above by square root of k, up to a constant factor. But this upper estimate is expected to be far from optimal. We shall try to reduce its magnitude, and on the other hand, study its lower bound. The same question will be considered for other p-norms. (4) In this project, we shall also investigate a problem about Quantum Unique Ergodicity (QUE). The QUE conjecture asserts that the probability measures induced by Hecke eigenforms f(z) converge weakly to the volume measure of X. Here we normalize the measure to have the unit volume over X. This conjecture is known to be true for f(z) being a CM form, and remains open in the general case. For the known cases, it is natural to investigate the rate of convergence. For CM forms, this estimate is closely related to the subconvexity bound of Rankin-Selberg L-functions at the point 1/2. Hence, in order to get a fast rate of convergence, we need to improve on the subconvexity estimate of the Rankin-Selberg L-function, which constitutes the second part of this project.

List of Research Outputs

Lau Y.K., Liu J. and Ye Y., A new bound k^2/3+^e for Rankin-Selberg L-functions for Hecke congruence subgroups, International Mathematics Research Papers. Hindawi Publishing Corporation, 2006, 1-78.

Lau Y.K., A study of the mean value of the error term in the mean square formula of the Riemann zeta-function in the critical strip 3/4£sJ. Théor. Nombres Bordeaux. 2006, 18: 445-470.

Lau Y.K. and Wu J., Extreme values of symmetric power L-functions at 1, Acta Arithematica. 2007, 126: 57-76.

Lau Y.K., Reviewer of Zentralblatt Math. V. 1090, No. 11032 , 2006, 1090, No. 11032.

Lau Y.K., Reviewer of Zentralblatt Math. V. 1090, No. 11061, 2006, 1090, No. 11061.

Lau Y.K., Reviewer of Zentralblatt Math. V. 1111, No. 11027 . 2007, 1111, No. 11027.

Lau Y.K., Some Recent Work on Automorphic L-functions, Department of Mathematics, The University of Iowa. 2006.

Lau Y.K., Liu J. and Ye Y., Subconvexity bounds for Rankin-Selberg L-functions for congruence subgroups, Journal of Number Theory. 2006, 121: 204-223.

Lau Y.K., The Error Terms in Dirichlet's Divisor Problem, the Circle Problem and the Mean Square Formula of the Riemann Zeta-function, The Fourth China-Japan Conference on Number Theory. 2006.

Researcher : Lee YF

List of Research Outputs

Hon C.C., Lam T.Y., Drummond A., Rambaut A., Lee Y.F., Yip C.W., Zeng F., Lam P.Y., Ng T.W. and Leung F.C.C., Phylogenetic Analysis Reveals a Correlation between the Expansion of Very Virulent Infectious Bursal Disease Virus and Reassortment of Its Genome Segment B, Journal of Virology. 2006, 80, no.17: 8503-8509.

Lee Y.F. and Ching W.K., On Convergent Probability of a Random Walk, International Journal of Mathematical Education in Science and Technology. Taylor's & Francis, 2006, 37: 833-838.

Ng K.P., Fung S.L., Lee Y.F. and Ching W.K., A Recursive Method for Solving Haplotype Frequencies with Application to Genetics , Journal of Bioinformatics and Computational Biology. World Scientific, 2007, 4: 1269-1286.

Researcher : Li CK

List of Research Outputs

Chan J.T., Li C.K. and Sze N.S., Mappings on Matrices: Invariance of Functional Values of Matrix Product, Journal of the Australian Mathematical Society. 2006, 81: 165-184.

Chan J.T., Li C.K. and Sze N.S., Mappings preserving spectra of products of matrices, Proceeding American Mathematical Society. 2007, 135: 977-986.

Researcher : Li L

List of Research Outputs

Ching W.K., Li L., Li T. and Zhang S., New Multivariate Markov Chain Model with Applications to Sales Demand Forecasting, Proceedings of the International Conference on Industrial Engineering and Systems Management 2007, Beijing, (in CD-ROM). 2007.

Researcher : Li T

List of Research Outputs

Ching W.K., Li L., Li T. and Zhang S., New Multivariate Markov Chain Model with Applications to Sales Demand Forecasting, Proceedings of the International Conference on Industrial Engineering and Systems Management 2007, Beijing, (in CD-ROM). 2007.

Ching W.K., Li T. and Xue J., On Hybrid Re-manufacturing Systems: A Matrix Geometric Approach, Proceedings of the International Conference on Industrial Engineering and Systems Management 2007, Beijing, (in CD-ROM). 2007.

Researcher : Liu J

List of Research Outputs

Lau Y.K., Liu J. and Ye Y., Subconvexity bounds for Rankin-Selberg L-functions for congruence subgroups, Journal of Number Theory. 2006, 121: 204-223.

Researcher : Lu J

Project Title:	Poisson Morse theory
Investigator(s):	Lu J
Department:	Mathematics
Source(s) of Funding:	Seed Funding for New Staff
Start Date:	01/2003

Abstract:

To further study on Poisson Morse Theory, which can provide new ways to study topology and make connections between Poisson geometry and topology.

Project Title:	Poisson structures associated to real semi-simple Lie groups
Investigator(s):	Lu J
Department:	Mathematics
Source(s) of Funding:	Competitive Earmarked Research Grants (CERG)
Start Date:	09/2003

Abstract:

To advance Poisson geometry, we try to achieve this goal by relating Poisson geometry with Lie theory. More precisely, we will study a class of examples of Poisson structures that are connected to interesting problems in Lie theory. The Poisson geometric considerations of these examples will shed lights on the Lie theoretical problems associated to them, and the input from Lie theory will motivate new constructions in Poisson geometry.

Project Title:	Poisson structures and tropical geometry
Investigator(s):	Lu J
Department:	Mathematics
Source(s) of Funding:	Seed Funding Programme for Basic Research
Start Date:	01/2005

Abstract:

The objectives of the project is to study large limits of Poisson varieties in the language of tropical geometry.

Project Title:	Poisson Geometry and Spherical Varieties
Investigator(s):	Lu J
Department:	Mathematics
Source(s) of Funding:	Competitive Earmarked Research Grants (CERG)
Start Date:	01/2005

Abstract:

To study: (1) the geometry of the Poisson structures of the spherical subvarieties of L; (2) the degeneration of a family of spherical varieties that naturally occur in our study of the variety L. Spherical varieties are important for representation theory and for algebraic geometry. There has been active research on spherial varieties, and particularly on their embedding theory and cohomology theories.

Project Title:	On intersections of real group orbits and Schubert cells in complex flag varities
Investigator(s):	Lu J
Department:	Mathematics
Source(s) of Funding:	Competitive Earmarked Research Grants (CERG)
Start Date:	09/2005

Abstract:

The objective of this project is to understand the intersections of some very important submanifolds of a complex flag variety, namely that of a Schubert cell and an orbit of a real form of the complex group. This project will bring new tools to the study of real group orbits such as the theory of cluster algebras recently formulated by Fomin and Zelevnsky. The proposed examples will also shed new lights on the work of Gekhtman, Shapiro, and Vainshtein on the relation between cluster algebras and Poisson geometry.

List of Research Outputs

Lu J., Editorial board, Travaux Mathematiques. 2007.

Lu J., Introduction to Poisson and symplectic geometry via examples from Lie theory (10 lectures), The 11'th National mathematics Summer School, Hong Kong University of Science and Technology, July 24 - August 9, 2006. 2006.

Lu J., In: Chief editor: Varadarajan , Pacific Journal of Mathematics. 2007.

Researcher : Ma Q

List of Research Outputs

Cheung W.S. and Ma Q., Some New Nonlinear Difference Inequalities and Their Applications, Journal of Computational and Applied Mathematics. 2007, 202: 339-351.

Researcher : Mok N

Project Title:	Bounded holomorphic functions and rigidity problems
Investigator(s):	Mok N
Department:	Mathematics
Source(s) of Funding:	Competitive Earmarked Research Grants (CERG)
Start Date:	12/2002

Abstract:

The project attempts to understand the rigidity problems on quotients of bounded domains by means of bounded holomorphic functions, often in relation to Carattheodory metrics. The link between rigidity and bounded holomorphic functions open up a new directionn of research on bounded domains both from the geometric and from the function-theorectic perspectives.

Project Title:	Topological aspects of degeneration of moduli spaces of vector bundles
Investigator(s):	Mok N, Sun X
Department:	Mathematics
Source(s) of Funding:	Competitive Earmarked Research Grants (CERG)
Start Date:	09/2003

Abstract:

For a given projective smooth curve Y, let U(Y, r, d) be the moduli space of semistable bundles of rank r and degree d on Y (r and d are coprime). One of the open problems about the topology of moduli spaces of bundles on curves is to compute Chern-classes of U(Y, r, d) using degeneration method: (1) Degenerate the curve Y into an irreducible curve X with only one node, and construct a degeneration of U(Y, r, d), which is a moduli space G(X) of stable bundles on some semistable curves. (2) Find the relationship between G(X) and U(C, 2, d), where C is the normalization of X and g(C)=g(Y-1), so that one can reduce the problem of genus g(Y)=g to a problem of genus g1. (1) is established, but (2) remains open. Instead of studying the relationship between G(X) and U(C, r, d) in the category of algebraic varieties, we propose to study the relationship in the category of topological spaces.

Project Title:	Holomorphic local isometries between bounded symmetric domains and related problems
Investigator(s):	Mok N
Department:	Mathematics
Source(s) of Funding:	Competitive Earmarked Research Grants (CERG)
Start Date:	12/2003

Abstract:

To prove that any holomorphic local isometric embedding between bounded symmetric domains extends to a global totally-geodesic embedding; to prove that any holomorphic measure-preserving local map from an irreducible bounded symmetric domain to a Cartesian product of copies of D must be totally geodesic; to consider generalizations and ramifications of approaches and solutions to the preceding problems, e.g., to formulate and establish a non-equidimensional analogue of Fefferman's Theorem on the smooth extension of biholomorphisms between strictly pseudoconvex domains.

Project Title:	Geometric problems on complex hyperbolic space forms and their subvarieties
Investigator(s):	Mok N
Department:	Mathematics
Source(s) of Funding:	Competitive Earmarked Research Grants (CERG)
Start Date:	01/2005

Abstract:

To find obstructions to holomorphically fibering or foliating compact complex hyperbolic space forms and their complex submanifolds; to study holomorphic mappings from compact complex hyperbolic space forms into Kähler manifolds of constant holomorphic sectional curvature; to tackle the compact case of Oort's conjecture asserting that there are no Shimura varieties on the Siegel modular variety lying on the locus of Jacobians of curves of sufficiently high genus: to study the geometry of complex-analytic subvarieties of complex hyperbolic space forms with respect to the holomorphic projective structure of the latter manifolds.

Project Title:	Geometric problems on rational homogeneous manifolds as uniruled projective manifolds and related questions
Investigator(s):	Mok N
Department:	Mathematics
Source(s) of Funding:	Competitive Earmarked Research Grants (CERG)
Start Date:	01/2007

Abstract:

(1) To study projective deformations of rational homogeneous manifolds G/P of arbitrary Picard number, and to establish rigidity of (examples of) G/P of higher Picard number under additional restrictions, e.g., when the deformed manifold is a Fano manifold, or when the deformed manifold is one on which the deformation of rational curves is unobstructed. Related problems such as projective deformations of the total space of bundles of G/P of Picard number 1 will also be considered. (2) To study holomorphic mappings from rational homogeneous manifolds G/P of arbitary Picard number onto projective manifolds, and to prove that such a mapping is either a biholomorphism or it factors through a canonical fibration over some G/Q, whenever the target manifold is not a projective space. Related problems on holomorphic mappings, e.g. when the domain manifolds are Fano and almost homogeneous, or when they are homogeneous projective manifolds, will also be studied. (3) To study the effect of varieties of minimal rational tangents on geometric properties of uniruled projective manifolds as exemplified by rational homogeneous manifolds, including (in the case of Picard number 1) those manifolds where the varieties of minimal rational tangents are linearly degenerate and those of higher Picard number.

List of Research Outputs

Mok N., Editor of Chinese Annals of Mathematics. Springer, 2007.

Mok N., Editor of Inventiones Mathematicae. Berlin-Heidelberg-New York, Springer-Verlag, 2006.

Mok N., Editor of Mathematische Annalen. Berlin-Heidelberg-New York, Springer-Verlag, 2006.

Mok N., From bounded symmetric domains to their compact duals - rigidity by means of rational curves, Colloquium Lecture at the Chinese Academy of Sciences, Beijing. 2007.

Mok N., Global extension of local holomorphic isometries with respect to the Bergman metric, 50th Annual Meeting of the Australian Mathematical Society, Sydney, Australia . 2006.

Mok N., Prolongement analytiques des isométries holomorphes et caractérisation de sous varietés holomorphes geodesiques, Colloquium Lecture at Université de Paris VI, France. 2007.

Mok N., Rigidity Problems on compact quotients of bounded symmetric domains, In: Stephen Shing-Toung Yau, Zhijie Chen, Jianpan Wang, Sheng-li Tan, Proceedings of the International Conference on Complex Geometry and Related Fields (Ams/Ip Studies in Advanced Mathematics). American Mathematical Society, 2007, 39: 201-249.

Mok N., Rigidity phenomena on bounded symmetric domains and their quotients of finite volume, International Conference and Instructional Workshop on Geometry, Topology and Analysis of Locally Symmetric Spaces and Discrete Groups, Chinese Academy of Sciences, Beijing. 2006.

Mok N., Varieties of minimal rational tangents–a geometric theory on Fano manifolds, Colloquium Lecture at Tata Institute of Fundamental Research, India . 2007.

Researcher : Ng KP

Project Title:	Minimization of L1 norm/mixed L1 and L2 norms for image restoration
Investigator(s):	Ng KP
Department:	Mathematics
Source(s) of Funding:	Small Project Funding
Start Date:	11/2004

Abstract:

To formulate the solution to a convex programming problem, and solved by the interior point method; to investigate how to solve a structured linear system efficiently at each step of interior point method.

List of Research Outputs

Akutsu T., Hayashida M., Ching W.K. and Ng K.P., Control of Boolean Networks: Hardness Results and Algorithms for Tree Structured Networks, In: Journal of Theoretical Biology, Journal of Theoretical Biology. 2007, 244: 670-679.

Cheung L., Yip Y.L., Cheung D.W.L., Kao C.M. and Ng K.P., On Mining Micro-array Data by Order-Preserving Submatrix, International Journal of Bioinformatics Research and Applications (IJBRA). Inder Sciences Publishers, 2007, 3: 42-64.

Ching W.K., Siu K.T.K., Fung S.L., Ng K.P. and Li W.K., Interactive Hidden Markov Models and Their Applications, IMA Journal of Management Mathematics. Oxford University Press, 2007, 18: 85-97.

Ching W.K., Zhang S. and Ng K.P., On Multi-dimensional Markov Chain Models, Pacific Journal of Optimization . 2007, 2: 235-243.

Ng K.P., Fung S.L., Lee Y.F. and Ching W.K., A Recursive Method for Solving Haplotype Frequencies with Application to Genetics , Journal of Bioinformatics and Computational Biology. World Scientific, 2007, 4: 1269-1286.

Ng K.P., Chan Y.L., So M.C. and Ching W.K., A Semi-Supervised Regression Model for Mixed Numerical and Categorical Variables, Pattern Recognition. 2007, 40: 1745-1752.

Zhang S., Ching W.K., Ng K.P. and Akutsu T., Simulation Study in Probabilistic Boolean Network Models for Genetic Regulatory Networks , Journal of Data Mining and Bioinformatics . 2007, 1: 217-240.

Researcher : Ng TW

Project Title:	Factorization and complex dynamics of meromorphic functions and related topics
Investigator(s):	Ng TW
Department:	Mathematics
Source(s) of Funding:	Seed Funding for New Staff
Start Date:	10/2002

Abstract:

To extend the proposer's studies and accomplishments by utilizing more powerful analytic and geometric tools to explore existing and related research resaults, as well as applying these results to some related research fields such as complex dynamics, functional and differential equations and sharing value problems of meromorphic functions.

Project Title:	Factorization and complex dynamics of meromorphic functions and related topics
Investigator(s):	Ng TW
Department:	Mathematics
Source(s) of Funding:	Competitive Earmarked Research Grants (CERG)
Start Date:	10/2003

Abstract:

Project Title:	A double epidemic model for SARS propagation
Investigator(s):	Ng TW, Danchin ALM
Department:	Mathematics
Source(s) of Funding:	Small Project Funding
Start Date:	11/2003

Abstract:

To explore the consequences of a situation where two overlapping epidemics interfere with each other; to explore the possible situation where viruses A and B would be of totally different origin, but would cause an overlapping immune response of the host.

Project Title:	D-companion matrices and geometry of polynomials
Investigator(s):	Ng TW, Cheung WS
Department:	Mathematics
Source(s) of Funding:	Small Project Funding
Start Date:	11/2004

Abstract:

To introduce a new type of companion matrices, D - companion matrices. By using these D - companion matrices we are able to apply matrix theory directly to the study of geometry of polynomials.

Project Title:	D-Companion Matrices and Geometry of Polynomials
Investigator(s):	Ng TW
Department:	Mathematics
Source(s) of Funding:	Competitive Earmarked Research Grants (CERG)
Start Date:	09/2005

Abstract:

The main objective of this project is to use techniques in matrix inequalities to study the geometry of polynomials. For example, we obtained a one parameter family version of Schoenberg-type inequality on the relative locations of zeros and critical points of a polynomial, and proved a result related to Sendov conjecture for polynomials.

Project Title:	Exact solutions of algebraic differential equations
Investigator(s):	Ng TW, Lu J, Choi YY, Raynaud F
Department:	Mathematics
Source(s) of Funding:	France/Hong Kong Joint Research Scheme - Travel Grants
Start Date:	01/2006

Abstract:

To study the zero distribution and growth rate of the solutions of certain algebraic differential equations by using Nevanlinna Theory and Wiman-Valiron theory; to study the meromorphic traveling wave solutions of some partial differential equations; to develop some algebraic method to find exact traveling wave solutions of some partial differential equations.

List of Research Outputs

Beardon A.F. and Ng T.W., Parametrizations of algebraic curves, Ann. Acad. Sci. Fenn. 2006, 31: 541-554.

Hon C.C., Lam T.Y., Drummond A., Rambaut A., Lee Y.F., Yip C.W., Zeng F., Lam P.Y., Ng T.W. and Leung F.C.C., Phylogenetic Analysis Reveals a Correlation between the Expansion of Very Virulent Infectious Bursal Disease Virus and Reassortment of Its Genome Segment B, Journal of Virology. 2006, 80, no.17: 8503-8509.

Ng T.W., Turinici G., Ching W.K., Chung S.K. and Danchin A., A Parasite Vector-host Epidemic Model for TSE Propagation, Medical Science Monitor. 2007, 13(2): 59-66.

Ng T.W., D-companion matrices and geometry of polynomials, Function Theory Seminar at Purdue University. 2007.

Ng T.W., Outstanding Young Researcher Award, The University of Hong Kong. 2006.

Ng T.W., Random iteration of analytic maps, Function Theory Seminar at Purdue University. 2007.

Ng T.W., Random iteration of analytic maps, Geometry Seminar of Fudan University. 2006.

Ng T.W., Zheng J. and Choi Y.Y., Residual Julia Sets of Meromorphic Functions, Mathematical Proceeding of Cambridge Philosophical Society. 2006, 141, no.1: 113-126.

Ng T.W., Smale's mean value conjecture and the amoebae , Analysis Seminar at University of Illinois at Urbana-Champaign . 2007.

Ng T.W., Smale's mean value conjecture and the hyperbolic metric, Geometry Seminar at Fudan University . 2006.

Ng T.W., Smale's mean value conjecture for polynomials of small degree, Function Theory Seminar at Purdue University. 2007.

Ng T.W., Solving nonlinear differential equations by Nevanlinna theory, Departmental Seminar at North Illinois University . 2007.

Researcher : Pecaric J

List of Research Outputs

Cheung W.S., Matkovic A. and Pecaric J., A Variant of Jessen's Inequality and Generalized Means, Journal of Inequalities in Pure and Applied Mathematics. 2006, 7, Issue 1, Art.10 (electronic journal): 8 pages.

Cheung W.S. and Pecaric J., Bohr's Inequalities for Hilbert Space Operators, Journal of Mathematical Analysis and Applications. 2006, 323: 403-412.

Cheung W.S., Cho Y.J., Pecaric J. and Zhao D., Bohr's Inequalities in n-Inner Product Spaces, Journal of the Korea Society of Mathematical Education, Series B: Pure and Applied Mathematics. 2007, 14, No.2 (May 2007): 127-137.

Cheung W.S., Zhao D. and Pecaric J., Opial-Type Inequalities for Differential Operators, Nonlinear Analysis. 2007, 66, No.9: 2028-2039.

Researcher : Qi F

List of Research Outputs

Cheung W.S. and Qi F., Logarithmic Convexity of the One-Parameter Mean Values, Taiwanese Journal of Mathematics. 2006, 11, No.1: 231-237.

Researcher : Ren J

List of Research Outputs

Cheung W.S. and Ren J., Discrete Nonlinear Inequalities and Applications to Boundary Value Problems, Journal of Mathematical Analysis and Applications. 2006, 319: 708-724.

Cheung W.S., Ren J., Wong P.J.Y. and Zhao D., Multiple Positive Solutions for Discrete Nonlocal Boundary Value Problems, Journal of Mathematical Analysis and Applications. 2007, 330: 900-915.

Cheung W.S. and Ren J., Periodic Solutions for p-Laplacian Rayleigh Equations, Nonlinear Analysis. 2006, 65: 2003-2012.

Researcher : Ren X

List of Research Outputs

Ren X. and Tsang K.M., The Waring-Goldbach Problem for Unlike Powers, Acta Mathematica Sinica, English Series. 2007, 23: 265-280.

Ren X. and Tsang K.M., Waring-Goldbach problem for Unlike Powers II, Acta Mathematica Sinica, Chinese Series. 2007, 50: 175-182.

Researcher : Siu KTK

List of Research Outputs

Ching W.K., Siu K.T.K., Fung S.L., Ng K.P. and Li W.K., Interactive Hidden Markov Models and Their Applications, IMA Journal of Management Mathematics. Oxford University Press, 2007, 18: 85-97.

Researcher : Siu MK

List of Research Outputs

Siu M.K., "No, I don't use history of mathematics in my class. Why?", Proceedings of HPM2004 & ESU4, July 2004, edited by F. Furinghetti et al, Uppsala Universitet, Uppsala. 2006, 268-277.

Siu M.K., "數學証明" ("Mathematical Proofs"). Chiu Chang Math. Publ., Taipei, 2007, revised edition: 200 pages.

Siu M.K. and Lam K., "概率萬花筒" ("Kaleidoscope in Probability") . Hong Kong Statistical Society, Hong Kong, 2007, reprinted with corrections: 101 pages.

Siu M.K., Mathematics, mathematics education, and the mouse, Mathematical Medley. 2006, 33(2): 19-33.

Researcher : So MC

List of Research Outputs

Ng K.P., Chan Y.L., So M.C. and Ching W.K., A Semi-Supervised Regression Model for Mixed Numerical and Categorical Variables, Pattern Recognition. 2007, 40: 1745-1752.

Researcher : Sze NS

List of Research Outputs

Chan J.T., Li C.K. and Sze N.S., Mappings on Matrices: Invariance of Functional Values of Matrix Product, Journal of the Australian Mathematical Society. 2006, 81: 165-184.

Chan J.T., Li C.K. and Sze N.S., Mappings preserving spectra of products of matrices, Proceeding American Mathematical Society. 2007, 135: 977-986.

Researcher : Tai AHL

List of Research Outputs

Tai A.H.L. and Ching W.K., A Two-echelon Model for Inventory and Returns , Operations Proceedings 2005 . Berlin, Springer, 2006, 2005: 131-136.

Researcher : Tsang KM

Project Title:	Error Terms in the Summatory Formula for certain Arithmetical Functions
Investigator(s):	Tsang KM, Lau YK
Department:	Mathematics
Source(s) of Funding:	Competitive Earmarked Research Grants (CERG)
Start Date:	01/2005

Abstract:

To study the higher power moments and the distribution function of D(x) over short intervals of the form [X,X + U] where U is of order lower than X; to extend the above study to a general class of arithmetical functions, whose associated Dirichlet series satisfy certain function equations; to try to obtain the third (or possibly the fourth) power moment as well as the distribution for the error term in the sphere problem. Will also attempt the same for the lattice points problem in ellipoids.

Project Title:	Linear equations in prime variables
Investigator(s):	Tsang KM
Department:	Mathematics
Source(s) of Funding:	Competitive Earmarked Research Grants (CERG)
Start Date:	01/2007

Abstract:

(1) The primary focus of this project is the ternary diophantine equation: (1) b_1p_1+b_2p_2+b_3p_3 =n in the three prime variables p_1, p_2, p_3. Here the coefficients b_1, b_2, b_3 are non-zero integers such that their greatest common divisor is equal to one. With specific choices of the coefficients, this equation encompasses many well-known problems in prime number theory. In the past two decades, there are increasing interests on studying the sizes of small prime solutions of this equation. The main result in this direction says that, for all integers n which satisfy certain necessary congruence conditions, the above equation (1) has prime solutions in which each b_ip_i is bounded by B to the power A, where B= max(|b_1|, |b_2|, |b_3|) is the height of the equation and A is a certain numerical constant. This bound already has the right order of magnitude and it remains to determine the lowest admissible value for the constant A. Tremendous effort have been devoted to lowering the admissible value of A. Recent development shows that more effective methods can be applied to this problem if the coefficients are further assumed to be coprime in pairs. The admissible value for A obtained in this situation is much lower (around 7.7) and is just slightly bigger than what one can obtain under the Generalized Riemann Hypothesis. Moreover, the techniques developed in this connection are very deep and versatile. Applications to other related problems in this area is quite possible. (2) In this project we first focus on equation (1) under the additional assumption that the coefficients are pairwise coprime. We shall develop further certain mean value estimates for dirichlet polynomials and refine the iterative procedure which has been applied previously. We hope to bring the admissible value of A in this case down to about 6. (3 ) Next, we consider equation (1) without the above extra condition. We first transform equation (1) into another ternary equation in which the coefficients are pairwise coprime but the variables are lying in arithmetic progressions. The new tools we developed in the previous case can then be applied to the new equation in this case. We expect that the admissible value of A in this case can be lowered to about 30 (from the present value of 38). (4) Finally, we shall explore the applicability of our new techniques by applying them to similar problems involving simultaneous linear equations in prime variables and also quadratic equations in five prime variables. Such investigations will enrich our knowledge on the distribution of primes in arithmetic progressions.

Project Title:	A Weighted Sieve of Selberg
Investigator(s):	Tsang KM
Department:	Mathematics
Source(s) of Funding:	Seed Funding Programme for Basic Research
Start Date:	01/2007

Abstract:

A.Selberg devised his famous Lambda^2 sieve some sixty years ago, in connection with his work on the zeros of the Riemann zeta-function and problems with sequences of numbers having few prime factors. The original formulation of the Lambda^2 sieve was an elegant and versatile upper bound sieve. Around 1950, Selberg further developed the Lambda^2 sieve and demonstrated how it can be used to exhibit numbers with few prime factors. In particular, by putting in appropriate weights, he proved that there are infinitely many positive integer pairs n, n+2, one of which has at most two and the other at most three prime factors. He lectured on this at various places, but has it published only in his collected work in 1991. Since this result was long surpassed by Chen's theorem (proved in 1967) it had not received much attention. This idea of Selberg was brought back to the forefront only in 1996, when Heath-Brown applied Selberg's idea to consider the case of k linear polynomials (Selberg's result was with the case of the two linear polynomials n and n+2) and obtained new approximations to the renowned prime k-tuple conjecture. About three years ago, I injected new ideas into Heath-Brown's work and obtained as a consequence, a better approximation to the prime k-tuple conjecture. In November 2004, an exciting development occurred on works concerning the small gaps between consecutive primes. Inspired by Heath-Brown's work mentioned above and using ideas similar to those in my earlier work, D. Goldston, C. Y. Yildirim and J. Pintz showed that the gaps between consecutive primes could be much smaller than the average gaps. This is a truly sensational breakthrough which now has become the focus of intensive research by several groups of number theorists. The main objective of this research project is to study further the various possibilities in employing the Lambda^2 sieve to problems concerning sequences of integers with few prime factors. There are two main directions in this research. First, in the design of the weights. In the work of Selberg, Heath-Brown and myself, the weights used are essentially the divisor function of the integers, while in Goldston etc.'s work, the weights are log n. There are many other possible candidates one can consider. Second, in the choice of the lambdas in the sieve. So far, all the lambdas used are some functions of the divisors of the product of all the (linear) polynomials. One should be able to gain grounds by using functions depending on divisors of each individual polynomial, instead of the product of them. This idea was actually suggested by Selberg in his original work back in 1950. But the actual work involved in the optimization of such choice is so formidable that apparently no one has ever worked that out. In view of the exciting developments just emerged, it is worthwhile to put new effort into this direction. We shall apply any new advancements obtained in the about two directions to the prime k-tuple conjecture and its generalizations to polynomials. It may also be possible to shed new lights on the small gaps between consecutive primes.

List of Research Outputs

Ho K.H. and Tsang K.M., On almost prime k-tuples, Journal of Number Theory. 2006, 120: 33-46.

Ren X. and Tsang K.M., The Waring-Goldbach Problem for Unlike Powers, Acta Mathematica Sinica, English Series. 2007, 23: 265-280.

Ren X. and Tsang K.M., Waring-Goldbach problem for Unlike Powers II, Acta Mathematica Sinica, Chinese Series. 2007, 50: 175-182.

Tsang K.M., Prime twins and almost prime k-tuples, Universite de Laval, Quebec, Canada. 2006.

Tsang K.M., Some recent results on the Dirichlet divisor problem, Ninth meeting of the Canadian Number Theory Association, University of British Columbia, Vancouver, Canada. 2006.

Researcher : Tseng S

List of Research Outputs

Cheung W.S. and Tseng S., Some New Discrete Nonlinear Delay Inequalities and Application to Discrete Delay Equations, Journal of Inequalities in Pure and Applied Mathematics. 2006, 7, Issue 4, Art.122 (electronic journal): 16 pages.

Researcher : Tsing NK

List of Research Outputs

Tsing N.K., Generalized Numerical Ranges: A Revisit, Workshop on Linear Algebra with Applications. 2006.

Researcher : Wang J

List of Research Outputs

Wang J. and Yung S.P., Stability of a Nonuniform Rayleigh Beam with Indefinite Damping, Systems and Control Letters. 2006, 55, no.10: 863-870.

Researcher : Wong B

List of Research Outputs

Cheung W.S. and Wong B., On a Non-abelian Invariant over Complex Surface of General Type, Sciences in China, Series A: Mathematics. 2006, 49, No.12: 1897-1900.

Researcher : Wong CW

List of Research Outputs

Wong C.W., A drop theorem without vector topology, In: R.M. Aron School of Mathematics, Kent State University, USA G. Chen Department of Mathematics, Texas A&M University, TX, USA R. .M. Aron, G. Chen, S.G. Krantz , Journal of mathematical analysis and applications. UNITED STATES, ACADEMIC PRESS INC ELSEVIER SCIENCE, 2007, 329: 452-471.

Researcher : Wong PPW

Project Title:	Complex hyperbolic geometry: a finsler approach
Investigator(s):	Wong PPW
Department:	Mathematics
Source(s) of Funding:	Competitive Earmarked Research Grants (CERG)
Start Date:	09/2006

Abstract:

(1) The main problem that we propose is the following: Find all Zariski open submanifolds of CP^n which admits Finsler metrics with strongly negative holomorphic bisectional curvature. As is well known, curvature is difficult to work with in higher dimension. The idea is to find equivalent condition in terms of Chern numbers (via Gauss-Bonnet type theorems) which is much more accessible and, at least for relatively simple examples, should be computable (though non-trivial) and verifiable by computer. (2) In view of objective 1 above, a related problem is this: Find conditions on the Chern numbers which guarnatee the existence of Finsler metrics with strongly negative holomorphic bisectional curvature. It should be pointed out that the complexity of the conditions increases exponentially as the dimension increases. We were successful in the case n=2 (see 2(B) in background of research below), and preliminary investigation indicates that the case n=3 is most likely to be accessible. Our immediate plan is to work out the case n=3 and hope to find a general pattern as the dimension increases.

Project Title:	Quadratic Extensions of Cyclic Projective Planes
Investigator(s):	Wong PPW, Law HF, Siu MK
Department:	Mathematics
Source(s) of Funding:	Seed Funding Programme for Basic Research
Start Date:	03/2007

Abstract:

The basic general problem: To study the existence and type of a finite projective plane.The main specific objective: To study the existence and types of quadratic extensions of finite cyclic projective planes. We explain below the above problem and objective.Let π be a finite projective plane of order n, consisting of n²+n+1 points and n²+n+1 lines. For example, the space of lines in F³ - {0}, where F is the Galois field GF(q), is a projective plane of order q. This plane is denoted by GF(2,q), and q is a prime power.A (finite) projective plane is said to be Desarguesian if every pair of centrally perspective triangles in it are axially perspective. It is well known that a projective plane is Desarguesian if and only if it can be coordinatized by a division ring. Since a finite division ring is a field, a finite Desarguesian projective plane is coordinatized by a Galois field and hence its order is a prime power. There exist non-Dearguesian projective planes, but so far all known examples have prime power orders. This leads to the following half-a-century-old conjecture (for general references see Beth et al (1992)):Prime Power Conjecture (PPC)Any finite projective plane has prime power order.We restrict our attention to the following type of projective plane. A finite projective plane which admits a cyclic group of collineations is called a cyclic projective plane (CPP). It is well known that a finite Desarguesian projective plane is cyclic (Singer (1938).) The converse is the following famous conjecture concerning CPP:ConjectureAny finite cyclic projective plane is Desarguesian.Since a Desarguesian projective plane has order a prime power, a weaker conjecture is the following (for partial results see Evans and Mann (1951), Gordon (1994)):Prime Power Conjecture for CPPAny finite cyclic projective plane has order a prime power.We further restrict our attention to CPP with order a square. It is known that all existent cyclic planes of order m or m² with m = 2,3,4,5,7,8,9 are Desarguesian. This is the famous result of R.H. Bruck obtained in 1960 (Bruck (1960).) While important sufficient conditions have been obtained later (for example results of U. Ott (1975) and C.Y. Ho (1998) building on the celebrated theorem of Ostrom and Wagner (1959), ) no specific new case has been obtained since then. It is our aim to investigate the possibility of obtaining new cases using a new approach. Preliminary studies not only confirm the validity of our new approach through verification of all known results in a canonical fashion, but also the validity of several new cases (Law and Wong (2006).)

List of Research Outputs

Researcher : Wong STC

List of Research Outputs

Cai H., Xu X., Lu J., Lichtman J.W., Yung S.P. and Wong S.T.C., Repulsive Force Based Snake Model to Segment and Track Neuronal Axons in 3D Microscopy Image Stacks, NeuroImage. 2006, 32: 1608-1620.

Researcher : Wu S

List of Research Outputs

Ding K. and Wu S., Inversions in classical Weyl groups, Communications in Contemporary Mathematics. Singapore, World Scientific, 2007, 9, No.1: 1-20.

Kirwin W.D. and Wu S., Geometric quantization, parallel transport and the Fourier transform, Communications in Mathematical Physics. Berlin/Heidelberg, Germany, Springer, 2006, 266, No.3: 577-594.

Wu S., Fiber integration of Deligue cohomology classes, Mathematics of String Theory, ANU. Canberra, Australia, 2006.

Wu S., Gauge theory and Langlands duality, Summer School of Representation Theory and Harmonic Analysis, Chern Inst. of Math. 2007.

Wu S., Heat kernel and Langlands duals, International Instructional Conference: Langlands and Geometric Langlands Program, Guangzhou, China, June 18-21, 2007 . 2007.

Wu S., Projective flatness in the quantisation of bosons and fermions, International Conference in Geometry, CUHK. Hong Kong, 2006.

Researcher : Xu G

List of Research Outputs

Xu G., Yung S.P. and Li L.K., Stabilization of Wave Systems with Input Delay in the Boundary Control, ESAIM: Control, Optimisation and Calculus of Variations. 2006, 55, no.10: 863-870.

Yung S.P. and Xu G., Properties of a class of C_0 semigroups on Banach spaces and their applications, J. Math. Anal. Appl. 2007, 328: 245-256.

Yung S.P., Xu G. and Han Z.J., Riesz basis property of serially connected Timoshenko beams, International Journal of Control. 2007, 80, no. 3: 470-485.

Researcher : Xue J

List of Research Outputs

Ching W.K., Li T. and Xue J., On Hybrid Re-manufacturing Systems: A Matrix Geometric Approach, Proceedings of the International Conference on Industrial Engineering and Systems Management 2007, Beijing, (in CD-ROM). 2007.

Researcher : Ye Y

List of Research Outputs

Lau Y.K., Liu J. and Ye Y., Subconvexity bounds for Rankin-Selberg L-functions for congruence subgroups, Journal of Number Theory. 2006, 121: 204-223.

Researcher : Yu J

Project Title:	Small cancellation method for polynomial algebras
Investigator(s):	Yu J
Department:	Mathematics
Source(s) of Funding:	Competitive Earmarked Research Grants (CERG)
Start Date:	12/2003
Completion Date:	12/2006

Abstract:

The purpose of this project is to attack the below three problems and other related problems for polynomial algebras by ombinatorial methods, in particular the small cancellation method. 1/ Let P_n (n>2) be a polynomial algebra over a field K. Does there exist a wild automorphisms of P_n 2/ Let P_n be a polynomial algebra over a field K. Does there exist a wild coordinate p=P_n? If so, how to effectively characterize whether a coordinate in P_n is wide? 3/ For two polynomials p and q in P_n, how to effectively decide whether there exists a tame automorphism ψ of P_n such that ψ(p)=q?

Project Title:	Degree estimate for subalgebras of a free associative algebra
Investigator(s):	Yu J
Department:	Mathematics
Source(s) of Funding:	Competitive Earmarked Research Grants (CERG)
Start Date:	12/2005

Abstract:

This proposal is devoted to the following problem. What is the lowest possible degree for the nontrivial elements of β? Here degree is the ordinary (total) homo-geneous degree. In the proposal it will be shown the idea to get an estimate of this degree, and how this estimate can be used to attack long-standing open problem of structure of automorphism group of the free associative algebra of rank three over the field κ.

List of Research Outputs

Yu J., Journal Editor, Communications in Algebra. 2007.

Researcher : Yung SP

Project Title:	On the snake method for 3D reconstruction of neuron axons from their cross-sections
Investigator(s):	Yung SP
Department:	Mathematics
Source(s) of Funding:	Seed Funding Programme for Basic Research
Start Date:	01/2006

Abstract:

We propose to use the snake method to track and reconstruct 3D image of neuron axons from their consecutive cross-section slices.

List of Research Outputs

Cai H., Xu X., Lu J., Lichtman J.W., Yung S.P. and Wong S.T.C., Repulsive Force Based Snake Model to Segment and Track Neuronal Axons in 3D Microscopy Image Stacks, NeuroImage. 2006, 32: 1608-1620.

Wang J. and Yung S.P., Stability of a Nonuniform Rayleigh Beam with Indefinite Damping, Systems and Control Letters. 2006, 55, no.10: 863-870.

Xu G., Yung S.P. and Li L.K., Stabilization of Wave Systems with Input Delay in the Boundary Control, ESAIM: Control, Optimisation and Calculus of Variations. 2006, 55, no.10: 863-870.

Yung S.P., 3D Tracking of Neural Axons, The First International Summer School on Numerical Linear Algebra. 2006.

Yung S.P. and Xu G., Properties of a class of C_0 semigroups on Banach spaces and their applications, J. Math. Anal. Appl. 2007, 328: 245-256.

Yung S.P., Xu G. and Han Z.J., Riesz basis property of serially connected Timoshenko beams, International Journal of Control. 2007, 80, no. 3: 470-485.

Researcher : Zang W

Project Title:	The independent set problem and its application
Investigator(s):	Zang W
Department:	Mathematics
Source(s) of Funding:	Seed Funding Programme for Basic Research
Start Date:	01/2003

Abstract:

To investigate the independent set problem in different settings using various algorithmic ideas such as linear programming, dynamic programming, decomposition method, and probabilistic method.

Project Title:	Bonds, Cycles, and Ring Networks
Investigator(s):	Zang W
Department:	Mathematics
Source(s) of Funding:	Competitive Earmarked Research Grants (CERG)
Start Date:	09/2004

Abstract:

To characterize all quadruples (G;X,Y,Z), where G=(V,E) is a multigraph and X,Y,Z are three even-sized subsets of V, for which G has a bond (i.e. a minimal nonempty edge cut) [S, V-S] such that S contains an odd number of vertices from each of X,Y, and Z; to investigate Nash-Williams' conjecture which asserts that every 4 connected toroidal graph contains a Hamiltonian cycle, and the following Seymour-Thomas conjecture: there exist a constant c>0 and a function f(t).0 such that, for any integer t>2, the length of longest cycle in any 3-connected graph G=(V,E) with no K(3,t)-minors (where K(3,t) stands for the 3 by t complete bipartite graph) is at least f(t)[V]⊥c(our focus will be on the latter); to describe all graphs with the min-max relation on packing and covering odd cycles (vertex version) in terms of forbidden structures; to study will wavelength allocation problem on trees, rings, and trees of rings, which are three important topologies of optical networks in practice.

Project Title:	Combinatorial Optimization Problems Involving Cycles
Investigator(s):	Zang W
Department:	Mathematics
Source(s) of Funding:	Competitive Earmarked Research Grants (CERG)
Start Date:	09/2005

Abstract:

The main objectives of this project are: (1) To investigate the Jackson-Wormald conjecture which asserts that every 3-connected graph on n vertices with maximum degree d>3 contains a cycle of length at least n^{log_{d-1} 2}, and the following conjecture made by Chen, Yu and myself: there exists a function f(t)>0 such that every 4-connected graph on n vertices with no K(3,t)-minors, where t>2, contains a cycle of length at least f(t)n; (2) To characterize all graphs G such that for any nonnegative integral weight function defined on the vertex set, the maximum size of a feedback vertex set (FVS) packing of G is equal to the minimum weight of a cycle in G; (3) To describe all digraphs G in terms of forbidden structures such that the clutter of all mininal dicycles of G (with respect to vertices) is ideal; (4) To design an efficient algorithm for finding a maximum cycle packing in any weighted reducible flow graph.

Project Title:	Min-max relations and integral polyhedra
Investigator(s):	Zang W
Department:	Mathematics
Source(s) of Funding:	Competitive Earmarked Research Grants (CERG)
Start Date:	09/2006

Abstract:

(1) Characterize all digraphs G such that for any nonnegative integral weight function defined on the vertex set, the maximum size of a feedback vertex set (FVS) packing is equal to the minimum weight of a cycle in G. Despite the hardness of this problem, I believe I am able to make significant progress towards the solution, and give a complete characterization of all digraphs G such that for any subgraph H of G, the maximum number of disjoint feedback vertex sets in H is equal to the minimum number of vertices in a cycle of H. This characterization, if obtained, would yield a beautiful mathematical theorem. (2) Give a structural description of all graphs G such that the minimum w-weight of an edge cover is equal to the maximum weight of a w-stable set for any edge weight w. I believe I can get the entire result without too much difficulty. Furthermore, I believe twin min-max relations hold simultaneously on exactly the same graphs; that is, the above min-max relation holds on a graph G for any edge weight w iff the corresponding min-max relation on packing edge covers holds. These results, if obtained, would be of theoretical interests, and yield polynomial-time solutions of the corresponding optimization problems. (3) Describe all graphs G for which the linear system Ax/2>=1, x>=0 is box totally dual integral (box TDI), where A is the cut-edge incidence matrix of G, and 1 (resp. 0) stands for the all one (resp. zero) vector. The possible outcome is the following characterization: The above linear system is box TDI iff G is a series-parallel graph. This result, if established, would significantly strenghthen two previous theorems obtained respectively by Corneujols, Fonlupt, and Naddef and by Mahjoub. It would also have practical value since the above linear system has been playing an important role in various polydedral approaches to the graphical travelling salesman problem. (4) Characterize all graphs G for which the linear system Ax/2>=1, 1>=x>=0 is integral, where A is the cut-edge incidence matrix of G, and 1 (resp. 0) stands for the all one (resp. zero) vector. (This is an open problem posed by Mahjoub in 1997.) I believe that a structural description of all these graphs in terms of excluded minors is within reach. If established, this result would lead to polynomial-time solvability of some important network design problems on these graphs; it would also have interesting applications to the graphical travelling salesman problem.

Project Title:	Topics in Polyhedral Combinatorics
Investigator(s):	Zang W
Department:	Mathematics
Source(s) of Funding:	Seed Funding Programme for Basic Research
Start Date:	09/2006

Abstract:

The basic theme of polyhedral combinatorics is the application of various aspects of polyhedral theory and linear programming to combinatorics, and the end product is often a min-max relation, which asserts that the maximum value of one problem is equal to the minimum value of another problem. One of the best-known examples is the max-flow min-cut theorem of Ford and Fulkerson. In recent years the PI has been working in the area of polyhedral combinatorics and managed to establish several important min-max results, some of which are the main subjectsof a section of Cornuejols' book ``Combinatorial Optimization: Packing and Covering" and cited several times in Schrijver's new book `` Combinatorial Optimization: Polyhedra and Efficiency". As a continuation of my previous work, I wish to further investigate some other important problems in polyhedral combinatorics. Let us give some background on the proposed project. In combinatorial optimization, many problems can be formulated as max {yb : yA <= c, y >= 0, y is integral}, (1)where A is a rational matrix, b and c are integral vectors, and 0 is the all-zero vector. By the linear programming duality theorem max {yb : yA <= c, y >= 0} = min {cx: Ax >= b, x>= 0},it follows that max {yb : yA <=c, y >= 0, y is integral} <=max {yb : yA <= c, y >= 0} (2) =min {cx: Ax >= b, x >= 0 } <=min {cx: Ax >= b, x >= 0, x is integral}.At this point, a natural question is to ask: under what conditions does one or two of the inequalities in (2) hold with equality? This is indeed the focus of polyhedral combinatorics, one of the most successful areas in combinatorial optimization. To get structural results, people (almost) always assume that A and b are fixed while c varies through all integral vectors. Under these assumptions, there are only two ways that equalities may hold in (2). Let P be the polyhedron {x: Ax >= b, x >= 0}. It is well-known that the second inequality in (2) holds with equality, for all integral vectors c, if and only if all vertices of P have integer coordinates. In this case, P is called integral. On the other hand, it is proved by Edmonds and Giles that the first inequality in (2) holds with equality, for all integral vectors c, if and only if both inequalities in (2) hold with equality, for all integral vectors c. If this is the case, the system of inequalities {Ax >= b, x >= 0}$ is called totally dual integral (TDI). Since linear programming problems can be solved in polynomial time, it follows that min {cx: Ax >= b,x >= 0, x is integral} can also be solved in polynomial time, if P is integral. Moreover, if the system of inequalities {Ax >= b, x >= 0} is TDI, then both max {yb : yA <= c, y >= 0, y is integral} and min {cx: Ax >= b, x >= 0, xis integral} can be solved in polynomial time. This is one of the main motivations for studying integral polyhedra and TDI systems. It should be pointed out that A and b may not be the actual inputs of the given optimization problem. Despite it, by using ellipsoid method, the linear programming problem can still be solved in polynomial time, as long as the corresponding separation problem can be solved in polynomial time. As a consequence, max {yb : yA <= c,y >= 0, y is integral} and/or min {cx: Ax >= b, x >= 0, x is integral} can still be solved in polynomial time. Very often, A corresponds to the incidence matrix of a set of combinatorial objects. In this context, many combinatorial objects have been studied. The best known results include those on matchings, binary clutters, balanced matrices, and perfect graphs. The PI proposes to study two other, but related, problems along this line. In this proposal, notation and terminology are standard. In particular, a path or a circuit is even (or odd) if it has an even (or odd) number of edges. (1) First, let A be the PXE incidence matrix, where P is the set of odd st-paths. Schrijver and Seymour proved that all vertices of the polyhedron P[G]={x: Ax >= 1, x >= 0} are 1/2-integral. The PI proposes to determine all graphs G for which P[G] is integral, and to determine all graphs G for which the system of inequalities {Ax >= 1,x >= 0} is TDI. Using the language of Cornu'ejols , the first is to characterize graphs for which the odd-paths clutter is ideal, and the second is to characterize graphs for which the odd-paths clutter has the MFMC property. These problems generalize the ordinary matching problem. It should be pointed out that the corresponding even-path problems are basically equivalent to the odd-path problems, and the even-path problems generalize the ordinary edge-disjoint paths problem. (2) The PI also proposes to study the analogous problems (on integral polyhedra and TDI systems) for odd circuits. He will concentrate on the vertex version, since the edge version is well understood. Using graph theoretical language, the problem on TDI systems is to characterize those graphs G that have the following property: For any nonnegative integral function w defined on V, the maximum number of odd circuits such that each vertex x is used at most w(x) times is equal to the minimum of sum_{x in X} w(x), over all subsets X of V such that G-X is bipartite. For some unknown reasons, both odd-circuit problems are very closely related to the t-perfect graph problem, which arises naturally in the study of stable sets in graphs. One of the goals of this project is to discover the connections between the odd-circuit problems and the t-perfect graph problem.

List of Research Outputs

Chen G., Gao Z., Yu X. and Zang W., Approximating Longest Cycles in Graphs with Bounded Degrees, SIAM Journal on Computing. Philadelphia, PA, USA, SIAM, 2006, 36: 635-656.

Chen G., Sheppardson L., Yu X. and Zang W., The Circumference of a Graph with No K(3,t)-Minor, Journal of Combinatorial Theory Series B. San Diego, USA, Elsevier, Inc., 2006, 96: 822-845.

Chen X., Ding G., Hu X. and Zang W., A Min-Max Relation on Packing Feedback Vertex Sets, Mathematics of Operations Research. MD, US, INFORMS, 2006, 31: 777-788.

Zang W., Acta Mathematicae Applicatae Sinica. Germany, Springer, 2007.

Zang W., Recent Progress on Polyhedral Combinatorics, International Workshop on Combinatorial Algorithms and Optimization, Shandong, China. 2006.

Zang W., Recent Progress on Polyhedral Combinatorics, Ottawa-Carleton Discrete Mathematics Workshop, Ottawa, Canada. 2007.

Zang W., The Complexity of Recognizing Linear Systems with Certain Integrality Properties, Workshop on Optimization in Honor of Professor M.J.D. Powell's 70th Birthday, Hong Kong, China. 2007.

Zang W., The Matroids with the Box Max-Flow Min-Cut Property, Second National Conference on Combinatorics and Graph Theory, Tianjin, China. 2006.

Researcher : Zhang S

List of Research Outputs

Ching W.K., Li L., Li T. and Zhang S., New Multivariate Markov Chain Model with Applications to Sales Demand Forecasting, Proceedings of the International Conference on Industrial Engineering and Systems Management 2007, Beijing, (in CD-ROM). 2007.

Ching W.K., Zhang S. and Ng K.P., On Multi-dimensional Markov Chain Models, Pacific Journal of Optimization . 2007, 2: 235-243.

Ng M., Zhang S., Ching W.K. and Akutsu T., A Control Model for Markovian Genetic Regulatory Network, Lecture Notes in Computer Science, Transactions on Computational Systems Biology. Springer, 2006, 4070: 36-48.

Zhang S., Ching W.K., Ng K.P. and Akutsu T., Simulation Study in Probabilistic Boolean Network Models for Genetic Regulatory Networks , Journal of Data Mining and Bioinformatics . 2007, 1: 217-240.

Researcher : Zhang S

List of Research Outputs

Ching W.K., Zhang S. and Ng K.P., On Multi-dimensional Markov Chain Models, Pacific Journal of Optimization . 2007, 2: 235-243.

Researcher : Zhao C

List of Research Outputs

Cheung W.S. and Zhao C., Inverses of New Hilbert-Pachpatte Type Inequalities, Journal of Inequalities and Applications. 2006, 2006, Art. ID 97860: 11 pp.

Cheung W.S., Bencze M. and Zhao C., The Strengthening of Minkowski Inequality for Mixed Projection Bodies, Libertas Matematica. 2006, XXVI: 75-78.

Researcher : Zhao D

List of Research Outputs

Cheung W.S., Cho Y.J., Pecaric J. and Zhao D., Bohr's Inequalities in n-Inner Product Spaces, Journal of the Korea Society of Mathematical Education, Series B: Pure and Applied Mathematics. 2007, 14, No.2 (May 2007): 127-137.

Cheung W.S., Ren J., Wong P.J.Y. and Zhao D., Multiple Positive Solutions for Discrete Nonlocal Boundary Value Problems, Journal of Mathematical Analysis and Applications. 2007, 330: 900-915.

Cheung W.S., Zhao D. and Pecaric J., Opial-Type Inequalities for Differential Operators, Nonlinear Analysis. 2007, 66, No.9: 2028-2039.

Researcher : Zheng J

List of Research Outputs

Ng T.W., Zheng J. and Choi Y.Y., Residual Julia Sets of Meromorphic Functions, Mathematical Proceeding of Cambridge Philosophical Society. 2006, 141, no.1: 113-126.

Researcher : Zhu M

List of Research Outputs

Chu S.C.K. and Zhu M., Data and Optimization Modeling for Manpower Planning of an Airport Baggage Service, Proceedings of the Institute of Industrial Engineers Annual Conference - 2007 Industrial Engineering Research Conference (IERC 2007), Nashville, TN, USA, 19-23 May 2007. 6pp.

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