DEPT OF MATHEMATICS



Researcher : Akutsu T
List of Research Outputs

Akutsu T. and Ching W.K., Analysis And Control Of Deterministic And Probabilistic Boolean Networks , In: Huma M. Lodhi, Stephen H. Muggleton., Elements of computational systems biology. Wiley, 2010, 235-256.
Hayashida M., Tamura T., Akutsu T., Ching W.K. and Cong Y., On Distribution and Enumeration of Attractors in Probabilistic Boolean Networks, IET on Systems Biology. 2009, 6: 465-474.


Researcher : Aoki-Kinoshita F
List of Research Outputs

Chen X., Ching W.K., Aoki-Kinoshita F. and Furota K., Support Vector Machine Methods for the Prediction of Cancer Growth, In: L. Yu, Y. Song, W. Ching, S. Wang and K. Lai, Proceedings of CSO2010, Huangshan. IEEE Computer Society Proceedings, 2010, 1: 229-232.
Li L., Ching W.K., Yamaguchi T. and Aoki-Kinoshita F., Weighted q-gram Method for Glycan Classification, BMC Bioinformatics. 2010, 11: S33doi:10.1186/1471-2105-11-S1-S33.


Researcher : Chan KH
Project Title:Construction of A 3D Dynamic Interface Solar Dynamo: Mathematical Theory and Numerical Simulations
Investigator(s):Chan KH
Department:Mathematics
Source(s) of Funding:General Research Fund (GRF)
Start Date:09/2008
Abstract:
1) Construct a nonlinear, three-dimensional, multi-layered, dynamic solar interface dynamos making use of a finite element method and taking into account of the fully three-dimensional dynamic feedback of Lorentz forces in rotating spherical systems; 2) Perform the mathematical analysis for the existence, uniqueness and stability of the dynamic interface dynamo system governed by a set of nonlinear PDEs with discontinuous physical coefficients in rotating spherical geometry; 3) Change the existing spherical dynamo model to more realistic and flexible boundary conditions, such as the stress-free condition on the outer surface of the solar convection zone; 4) Carry out multitudes of numerical simulations of nonlinear, three-dimensional, multi-layered, dynamic solar interface dynamos on massively parallel computers and provide an improved understanding of the solar magnetic cycles with about 22 year period and the solar torsional oscillation with about 11 year period.


List of Research Outputs

Chan K.H. and Zhang K., Natural Dynamo: Solar Magnetic Fields, International Conference of Mathematics at National Taiwan University. 2009.
Chan K.H., Zhang K. and Liao X., On librationally driven flow in planetary ellipsoidal cores, UK Magnetohydrodynamics Meeting, University of Leeds, UK. 2010.
Chan K.H., Reviewer for the Physics of the Earth and Planetary Interior Journal (April 2010), 2010.
Li L., Liao X., Chan K.H. and Zhang K., On nonlinear multiarmed spiral waves in slowly rotating systems., In: John Kim, L. Gary Leal, Physics of Fluids. New York, American Institute of Physics, 2010, 22: 011701.


Researcher : Chan KY
List of Research Outputs

Lu J., Chan K.Y. and To K.M.S., On intersections of conjugacy classes and bruhat cells , Transformation groups. Springer, 2010, 15, No. 2: 243 - 260.


Researcher : Chan YM
List of Research Outputs

Ching W.K., Li L., Chan Y.M. and Mamitsuka H., A Study of Network-based Kernel Methods on Protein-Protein Interaction for Protein Functions Prediction, The Third International Symposium on Optimization and Systems Biology (OSB 2009), Lecture Notes in Operations Research 11, Series Editors: Luo-nan Chen, Xiang-Sun Zhang, Ling-Yun Wu and Yong Wang. . WORLD PUBLISHING CORPORATION, 2009, 11: 25-32.


Researcher : Chen C
List of Research Outputs

Cheung W.S., Chen C. and Zhao D., Gronwall-Bellman-Type Integral Inequalities and Applications to BVPs, Journal of Inequalities and Applications. 2009, 2009, Art.ID 258569: 15 pages.
Cheung W.S., Changjian Z. and Chen C., On Pecaric-Rajic-Dragomir-Type Inequalities in Normed Linear Spaces, Journal of Inequalities and Applications. 2009, 2009, Art. ID 137301: 7 pages.


Researcher : Chen X
List of Research Outputs

Chen X., Ching W.K., Aoki-Kinoshita F. and Furota K., Support Vector Machine Methods for the Prediction of Cancer Growth, In: L. Yu, Y. Song, W. Ching, S. Wang and K. Lai, Proceedings of CSO2010, Huangshan. IEEE Computer Society Proceedings, 2010, 1: 229-232.
Ching W.K., Chen X. and Tsing N.K., Generating Probabilistic Boolean Networks from a Prescribed Transition Probability Matrix, IET on Systems Biology. 2009, 6: 453-464.
Zhang S., Ching W.K., Chen X. and Tsing N.K., Generating Probabilistic Boolean Networks from a Prescribed Stationary Distribution , Information Sciences. 2010, 180: 2560-2570.


Researcher : Chen Z
List of Research Outputs

Chen Z. and Zang W., Odd-K4's in Stability Critical Graphs, Discrete Mathematics. 2009, 309: 5982-5985.


Researcher : Cheung WS
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:Mathematics and Physics Enrichment Course - Harmonies in Nature: A Dialogue Between Mathematics and Physics
Investigator(s):Cheung WS, Cheng KS
Department:Mathematics
Source(s) of Funding:Leung Kau Kui Research and Teaching Endowment Fund - Teaching Grants
Start Date:03/2007
Abstract:
The entire course will last for 10 weeks, including 10 consecutive Saturdays of lectures and tutorials. Through lectures, in-class discussions and small group discussions, the students will be able to see a meaningful dialogue between mathematics and physics, and in turn appreciate the two subjects in an integrative and interdisciplinary manner. The underlying theme would be the role and evolution of mathematics (mainly calculus, with related topics in linear algebra and geometry) in understanding the physical world, from the era of Isaac Newton’s mechanics to that of James Clerk Maxwell’s electromagnetism and possibly beyond, to that of Albert Einstein’s relativity. In other words it tries to tell the story of triumph in mathematics and physics over the past four centuries. The physics would provide both the sources of motivation and the applications. Along the way both ideas and methods will appear, to be learnt in an interactive manner through discussion, homework assignments and tutorials. Active but cooperative learning, rather than passive or competitive participation, is expected of the class. This project has been approved by and will be co-organized with the Education and Manpower Bureau (EMB) of the Hong Kong SAR Government. Logistic arrangements with secondary schools will be handled by the EMB.


Project Title:Boundary Value Problems on Time Scales
Investigator(s):Cheung WS
Department:Mathematics
Source(s) of Funding:General Research Fund (GRF)
Start Date:01/2008
Abstract:
Establish sufficient conditions for the existence of BVPs on time scales. Determine qualitative properties of solutions to the equations. These include uniqueness, stability, boundedness, and oscillatory properties, etc.


Project Title:Existence Problem of p(x)-Laplacian Differential Equations
Investigator(s):Cheung WS
Department:Mathematics
Source(s) of Funding:Seed Funding Programme for Basic Research
Start Date:04/2008
Completion Date:09/2009
Abstract:
In recent years, increasing attention has been devoted to the study of differential equations and related variational problems with p(x)-growth conditions. These problems arise from nonlinear elasticity theory and electrorheological fluids. Electrorheological fluids are special viscous fluids, which are characterized by their ability to change dramatically their mechanical properties by an external electric field. In traditional modeling on electrorheological fluids, the electric field is treated as a constant parameter. Taking into account the more complex interaction of the electro-magnetic fields and the moving liquid, the electric field will no longer be a constant and has to be treated as a variable. This leads to the consideration of differential equations with variable (i.e., p(x)- ) growth conditions. This project aims to expand our knowledge on a general class of p-Laplacian differential equations, and contribute to a general theory on p(x)-Laplacian differential equations. The following problems will be addressed on: (i) Existence and Uniqueness Results : Under what conditions do p(x)-Laplacian equations have solutions? (ii) Qualitative Properties : If solutions to the equations do exist, then what are their properties? How about the existence of solutions with special properties, e.g., positive or periodic solutions?


Project Title:APPROXIMATING THE STIELTJES INTEGRAL
Investigator(s):Cheung WS
Department:Mathematics
Source(s) of Funding:Seed Funding Programme for Basic Research
Start Date:06/2009
Abstract:
The complete solution to many applied probability problems, framed according to particular models, require accurate evaluation of certain convolution integrals involving cumulative distribution functions (cdf's) [28]. Closed form expressions for the convolution integrals are available in only a few special cases, for example, a uniform distribution convoluted with various other cdf's, which often require a tour de force in synthetic integration. These factors then provide a strong motivation for evaluating convolutions of cdf's by numerical methods. The Stieltjes integral is also of considerable importance in problems of reliability theory, for example, a repairable system with two types of repair - breakdown repair (starting if the system fails) and preventative repair [3]. In the study of existence, uniqueness, stability, boundedness and certain other aspects of solutions of differential and integral equations, integral inequalities frequently play a fundamental role [4, 5-14, 23]. It is natural, then, to seek such a fundamental rule for integral inequalities when such equations are of the Stieltjes type. The aims of this proposal are to: (1) develop Ostrowski, Grüss, trapezoidal and generalized trapezoidal type rules for different classes of functions in the Stieltjes integral; (2) establish explicit sharp bounds for the remainder in these rules; (3) numerically implement the obtained results. [List of References is in Section VII below]


List of Research Outputs

Cheung W.S., Associate Editor, Journal of Inequalities and Applications. 2009.
Cheung W.S. and Xiong G., Chord Power Integrals for Simplices, Asian European Journal of Mathematics. 2009, 2, No.4: 557-565.
Cheung W.S., Editor of Australian Journal of Mathematical Analysis and Applications. 2010.
Cheung W.S., Editor of Journal of Inequalities in Pure and Applied Mathematics. 2010.
Cheung W.S., Editor of the Bulletin of Southeast Asian Mathematical Society. 2010.
Cheung W.S., Xiong G. and Xiao Q.M., Firey Linear Combinations of Convex Bodies, Journal of Shanghai University. 2009, 13, No.2: 102-104.
Cheung W.S., Chen C. and Zhao D., Gronwall-Bellman-Type Integral Inequalities and Applications to BVPs, Journal of Inequalities and Applications. 2009, 2009, Art.ID 258569: 15 pages.
Cheung W.S. and Changjian Z., On Ostrowski-Type Inequalities for Higher-Order Partial Derivatives, Journal of Inequalities and Applications. 2010, 2010, Art.ID 960672: 8 pages.
Cheung W.S., Changjian Z. and Chen C., On Pecaric-Rajic-Dragomir-Type Inequalities in Normed Linear Spaces, Journal of Inequalities and Applications. 2009, 2009, Art. ID 137301: 7 pages.
Cheung W.S., Ren J. and Han W., Positive Periodic Solution of Second-Order Neutral Functional Differential Equations, Nonlinear Analysis. 2009, 71: 3948-3955.
Cheung W.S., Ren J. and Han W., Positive Periodic Solutions for Second-Order Differential Equations with Generalized Neutral Operator, Australian Journal of Mathematical Analysis and Applications. 2009, 6, No.1, Art 5: 16 pages.
Cheung W.S. and Ng T.W., Relationship between the zeros of two polynomials, Journal of Linear Algebra and Its Applications. 2010, 432, no.1: 107-115.
Cheung W.S., Reviewer of Zentralblatt Math. V.1135, No.26015. 2009.
Cheung W.S., Reviewer of Zentralblatt Math. V.1135, No.26017. 2009.
Cheung W.S., Reviewer of Zentralblatt Math. V.1136, No.26003. 2009.
Cheung W.S., Reviewer of Zentralblatt Math. V.1136, No.26005. 2009.
Cheung W.S., Reviewer of Zentralblatt Math. V.1154, No.26319. 2009.
Cheung W.S., Reviewer of Zentralblatt Math. V.1155, No.26020. 2009.
Cheung W.S., Reviewer of Zentralblatt Math. V.1161, No.34352. 2009.
Cheung W.S., Reviewer of Zentralblatt Math. V.1162, No.34057. 2009.
Cheung W.S., Sharp Discrete Inequalities and Applications to Discrete Variational Problems, Journal of Computational and Applied Mathematics. 2009, 232: 176-186.
Cheung W.S. and Zhao C., Some New Hilbert's Type Inequalities, Journal of Inequalities and Applications. 2009, 2009, Art.ID 851360: 10 pages.
Cheung W.S. and Changjian Z., Lp-polar Projection Brunn-Minkowski Inequality, Chinese Annals of Mathematics Series A. 2010, 31, No.2: 239-246.
Cheung W.S. and Changjian Z., Lp-Brunn Minkowski Inequality, Indagationes Mathematicae N.S. 2009, 20, No.2: 179-190.
Cheung W.S., 浙江省高等學校科研成果奬二等奬, 浙江省教育廳, 2009.


Researcher : Cheung WS
List of Research Outputs

Cheung W.S., Associate Editor, Journal of Inequalities and Applications. 2009.
Cheung W.S. and Xiong G., Chord Power Integrals for Simplices, Asian European Journal of Mathematics. 2009, 2, No.4: 557-565.
Cheung W.S., Editor of Australian Journal of Mathematical Analysis and Applications. 2010.
Cheung W.S., Editor of Journal of Inequalities in Pure and Applied Mathematics. 2010.
Cheung W.S., Editor of the Bulletin of Southeast Asian Mathematical Society. 2010.
Cheung W.S., Xiong G. and Xiao Q.M., Firey Linear Combinations of Convex Bodies, Journal of Shanghai University. 2009, 13, No.2: 102-104.
Cheung W.S., Chen C. and Zhao D., Gronwall-Bellman-Type Integral Inequalities and Applications to BVPs, Journal of Inequalities and Applications. 2009, 2009, Art.ID 258569: 15 pages.
Cheung W.S. and Changjian Z., On Ostrowski-Type Inequalities for Higher-Order Partial Derivatives, Journal of Inequalities and Applications. 2010, 2010, Art.ID 960672: 8 pages.
Cheung W.S., Changjian Z. and Chen C., On Pecaric-Rajic-Dragomir-Type Inequalities in Normed Linear Spaces, Journal of Inequalities and Applications. 2009, 2009, Art. ID 137301: 7 pages.
Cheung W.S., Ren J. and Han W., Positive Periodic Solution of Second-Order Neutral Functional Differential Equations, Nonlinear Analysis. 2009, 71: 3948-3955.
Cheung W.S., Ren J. and Han W., Positive Periodic Solutions for Second-Order Differential Equations with Generalized Neutral Operator, Australian Journal of Mathematical Analysis and Applications. 2009, 6, No.1, Art 5: 16 pages.
Cheung W.S. and Ng T.W., Relationship between the zeros of two polynomials, Journal of Linear Algebra and Its Applications. 2010, 432, no.1: 107-115.
Cheung W.S., Reviewer of Zentralblatt Math. V.1135, No.26015. 2009.
Cheung W.S., Reviewer of Zentralblatt Math. V.1135, No.26017. 2009.
Cheung W.S., Reviewer of Zentralblatt Math. V.1136, No.26003. 2009.
Cheung W.S., Reviewer of Zentralblatt Math. V.1136, No.26005. 2009.
Cheung W.S., Reviewer of Zentralblatt Math. V.1154, No.26319. 2009.
Cheung W.S., Reviewer of Zentralblatt Math. V.1155, No.26020. 2009.
Cheung W.S., Reviewer of Zentralblatt Math. V.1161, No.34352. 2009.
Cheung W.S., Reviewer of Zentralblatt Math. V.1162, No.34057. 2009.
Cheung W.S., Sharp Discrete Inequalities and Applications to Discrete Variational Problems, Journal of Computational and Applied Mathematics. 2009, 232: 176-186.
Cheung W.S. and Zhao C., Some New Hilbert's Type Inequalities, Journal of Inequalities and Applications. 2009, 2009, Art.ID 851360: 10 pages.
Cheung W.S. and Changjian Z., Lp-polar Projection Brunn-Minkowski Inequality, Chinese Annals of Mathematics Series A. 2010, 31, No.2: 239-246.
Cheung W.S. and Changjian Z., Lp-Brunn Minkowski Inequality, Indagationes Mathematicae N.S. 2009, 20, No.2: 179-190.
Cheung W.S., 浙江省高等學校科研成果奬二等奬, 浙江省教育廳, 2009.


Researcher : Ching WK
Project Title:On Perron-Frobenius Theory for Multivariate Markov Chains with Applications
Investigator(s):Ching WK, Akutsu T
Department:Mathematics
Source(s) of Funding:General Research Fund (GRF)
Start Date:07/2007
Abstract:
In this project, we consider multivariate Markov chains taking values in a finite state-space over discrete time intervals. We will develop high-order multivariate Markov chain models which can capture the long-range dependence and also both the intra- and inter-relations among the chains with low order of number of model parameters. Simplified models will also be developed in case the multivariate Markov chain models overfit the data sequences or there are negative correlations among the data sequences. We will then extend the Perron-Frobenius theory to both the high-order and simplified multivariate Markov chain models. Efficient numerical algorithms will be developed for solving the model parameters. The new multivariate Markov chain models will then be applied to some practical problems such as genetic regulatory networks and credit risk analysis.


Project Title:On Models and Numerical Algorithms for Hybrid Re-manufacturing Systems
Investigator(s):Ching WK
Department:Mathematics
Source(s) of Funding:Seed Funding Programme for Basic Research
Start Date:01/2008
Completion Date:12/2009
Abstract:
In the world of limited resources and disposal capacities, inevitably there is an growing environmental pressure in using re-manufacturing systems, a recycling process to reduce the amount of waste generated by manufacturers. The recycling of returns is an important issue in reverse logistics and re-manufacturing systems play an important role in the industry. A typical example is Xerox, a major manufacturer of copy machines has been putting effort into re-manufacturing used equipments. Xerox reports on annual savings of several hundred million dollars due to re-manufacturing and re-use of equipments and parts. At the same time they divert more than fifty thousand tones of waste from the waste stream. Furthermore, it is also very popular nowadays that consumers are allowed to return a purchased product within a given period with full refund. Therefore handling such returns is a key issue in running a successful business. The effect of introducing a re-manufacturing process is that the manufacturers have to take returns into account in their production planning and inventory management. It also increases the fluctuations in the inventory levels and complicates the inventory control process. Usually, returns are first tested or repaired and then they will be re-sold to the market. In fact, there are many research works on the inventory control of repairable items and returns. Basically a lot of the models describe a re-manufacturing system as a closed-loop queueing network with constant number of items inside. In this project, we will consider more general models that allow disposal of returns and hybrid re-manufacturing systems (both manufacturing and re-manufacturing processes exist in the systems). System design is an important and a necessary step before setting up a hybrid re-manufacturing system (or a network) . In this project, we will focus on building complex models for hybrid re-manufacturing systems. Markovian queueing network models are effective tools for modelling stochastic systems. In the system performance analysis, in order to obtain some important system performance quantities such as the average inventory levels and the average long-run cost of the system, it is required to solve the steady-state probability distribution of a large and complex Markov chain. Because all the system performance quantities can be written in terms of this probability distribution. The steady-state probability distribution can be obtained by solving a linear system of equations (the system generator matrix). In general, the linear system cannot be solved efficiently by using direct methods such as the LU decomposition due to the large size of the system and the loss of sparsity and rich structure in the factors. Very often, the problem is solved by using classical iterative methods. However, their convergence rates can be very slow as the linear system is usually ill-conditioned. Therefore it is important to develop fast numerical solvers for our captured problem.


Project Title:On Construction of Probabilistic Boolean Networks
Investigator(s):Ching WK
Department:Mathematics
Source(s) of Funding:Small Project Funding
Start Date:12/2008
Abstract:
Building mathematical models and efficient numerical algorithms for the study of regulatory interactions among DNA, RNA, proteins and small molecules are important topics in computational systems biology. In fact, there have been many formalisms proposed in the literature to study genetic regulatory networks. Among these models, Boolean Network (BN) and its extension Probabilisitic Boolean Network (PBN) have received much attention as they are able to capture the switching behavior of the biological process. In a BN, the gene expression states are quantized to only two levels: on and off (represented as 1 and 0). The target gene is predicted by several genes called its input genes via a Boolean function. When the input genes and the Boolean functions are given, then we say that a BN is defined. Since genetic regulation process exhibits uncertainty, it is more realistic to consider stochastic models, PBN. In a PBN, for each gene, there can be more than one Boolean function and selection probabilities are assigned to the Boolean functions. The dynamics (transitions) of a PBN can be studied by in the framework of a Markov chain. Given a PBN, the network behavior is characterized by its steady-state probability distribution which gives the first-order statistical information of a PBN. One can then understand a genetic network and identify the influence of different genes via such a network. Here I will study the problem of constructing a PBN based on the given transition probability matrix and a set of BNs. This is an inverse problem of huge problem size. The inverse problem is ill-posed, meaning there will be many networks or no network having the desirable properties. Such problems are very important to network inference from steady-state data, as most microarray data sets are assumed to be obtained from sampling the steady-state. I will first give a formulation of the invers problem, I will then propose to apply Newton's method in conjunction with Conjugate Gradient (CG) method to solving the inverse problem.


Project Title:Matrix Approximation Theory for Probabilistic Boolean Networks with Applications
Investigator(s):Ching WK
Department:Mathematics
Source(s) of Funding:Seed Funding Programme for Basic Research
Start Date:01/2010
Abstract:
Background: Developing mathematical theory and building computational models for genetic regulatory networks are important research issues in computational systems biology. Boolean Networks (BNs) and its extension Probabilistic Boolean Networks (PBNs) are effective mathematical models for studying genetic regulatory interactions. A PBN is essentially a collection of BNs driven by a Markov chain process and therefore can be studied by using Markov chain theory. On the one hand, the steady-state probability distribution of a PBN gives useful information about the desirable states (attractor cycles) of the underlying genetic network where the attractor cycles have important biological interpretations. On the other hand, it is well-known that the control/intervention of a genetic regulatory network is useful for avoiding undesirable states associated with diseases like cancer. The optimal control problem can be formulated mathematically by using the principle of stochastic dynamic programming. The size of the transition probability matrix of a PBN is 2n-by-2n where n is the number of genes in the genetic network, and the problem size grows exponentially with respect to n. In this project, we will develop matrix approximation theory for approximating both the steady-state probability distribution of a PBN and also the optimal solution of the captured control problem. The matrix approximation methods can reduce the computational cost significantly and still retain the important information of the network. Theoretical results for the error analysis of the approximation method will be established. Numerical experiments based on real genetic networks will also be performed to demonstrate both the efficiency and effectiveness of the proposed methods. Extension to more general and complex PBNs such as PBNs with random gene perturbations and context-sensitive PBNs will also be studied. Major objectives. (1) We will develop both matrix approximation and computational theory for PBNs. The theory can be applied to get an approximate steady-state probability distribution of a PBN. Mathematical theory for analyzing the error of approximation will be established by using matrix perturbation theory. We will then apply the approximation methods to the construction and analysis of PBNs with practical genomic data. Comparisons to existing models will also be made. (2) To apply external controls to a network so as to achieve desirable steady-state probability distribution of the network is another important and interesting issue in studying genetic regulatory networks. We will give mathematical formulation for the captured control problem and develop mathematical theory for obtaining the approximate optimal control policy. Theoretical results on the error analysis will also be investigated and established. (3) Finally the developed models, mathematical theory and numerical algorithms will be further extended to more general and complex PBNs such as PBNs with random gene perturbations and context-sensitive PBNs.


List of Research Outputs

Akutsu T. and Ching W.K., Analysis And Control Of Deterministic And Probabilistic Boolean Networks , In: Huma M. Lodhi, Stephen H. Muggleton., Elements of computational systems biology. Wiley, 2010, 235-256.
Chen X., Ching W.K., Aoki-Kinoshita F. and Furota K., Support Vector Machine Methods for the Prediction of Cancer Growth, In: L. Yu, Y. Song, W. Ching, S. Wang and K. Lai, Proceedings of CSO2010, Huangshan. IEEE Computer Society Proceedings, 2010, 1: 229-232.
Ching W.K. and Cong Y., Finite-Horizon Control of Genetic Regulatory Networks with Multiple Hard-Constraints, The Third International Symposium on Optimization and Systems Biology (OSB 2009), Lecture Notes in Operations Research 11, Series Editors: Luo-nan Chen, Xiang-Sun Zhang, Ling-Yun Wu and Yong Wang . WORLD PUBLISHING CORPORATION, 2009, 11: 33-40.
Ching W.K., Li X., Siu K.T.K. and Wu Z., Improving Revenue Management: A Real Option Approach, In: Cheng, T. C. Edwin; Choi, Tsan-Ming , Springer's Handbook Series on Innovative Quick Response Programs in Logistics Supply Chain Management. Springer, 2010, 123-139.
Ching W.K., Leung H.Y., Jiang H. and Sun L., A Markovian Model for Default Risk in a Network of Sectors, Proceedings of The 2nd IEEE International Conference on Business Intelligence and Financial Engineering (BIFE 2009), Beijing, 24-26, July 2009, 373-377.. 2009.
Ching W.K., Leung H.Y., Jiang H., Sun L. and Siu K.T.K., A Markovian Network Model for Default Risk Management, , International Journal of Intelligent Engineering Informatics . 2010, 1: 104-124.
Ching W.K., Li L., Chan Y.M. and Mamitsuka H., A Study of Network-based Kernel Methods on Protein-Protein Interaction for Protein Functions Prediction, The Third International Symposium on Optimization and Systems Biology (OSB 2009), Lecture Notes in Operations Research 11, Series Editors: Luo-nan Chen, Xiang-Sun Zhang, Ling-Yun Wu and Yong Wang. . WORLD PUBLISHING CORPORATION, 2009, 11: 25-32.
Ching W.K., Li L., Tsing N.K., Tai C.W., Ng T.W., Wong A.S.T. and Cheng K., A Weighted Local Least Squares Imputation Method for Missing Value Estimation in Microarray Gene Expression Data, Journal of Data Mining and Bioinformatics . 2010, 4: 331-347.
Ching W.K., Advances in Decision Sciences (Associate Editor), Hindawi Publishing Corporation, 2009.
Ching W.K., Siu K.T.K., Li L., Jiang H., Li T. and Li W.K., An Improved Parsimonious Multivariate Markov Chain Model for Credit Risk, Journal of Credit Risk . 2009, 5: 1-25.
Ching W.K., East Asian Journal on Applied Mathematics (associate Editor), Global Science Press, 2010, 2079-7362.
Ching W.K., Chen X. and Tsing N.K., Generating Probabilistic Boolean Networks from a Prescribed Transition Probability Matrix, IET on Systems Biology. 2009, 6: 453-464.
Ching W.K., High-dimensional Markov Chains with Applications, In: South China Normal University, Guangzhou, China., Workshop on Numerical Linear Algebra and High Performance Computing (30 minutes talk) . 2010.
Ching W.K., International Journal of Intelligent Engineering Informatics (Associate Editor), Inderscience Publisher, 2009.
Ching W.K., Siu K.T.K., Li L., Li T. and Li W.K., Modeling Default Data via an Interactive Hidden Markov Model, Computational Economics. 2009, 34: 1-19.
Ching W.K., On Construction of Probabilistic Boolean Networks (invited speaker) (30 Minutes talk), In: University of Macau, Macau, China, Workshop on Scientific Computing and Matrix Analysis. 2009.
Ching W.K., On Infectious Models for Default Risk (Keynote Speaker), Proceedings of The 2nd IEEE International Conference on Business Intelligence and Financial Engineering (BIFE 2009), Beijing, 24-26, July 2009.. 2009.
Ching W.K., On Markovian Models for Credit Risk Management (invited speaker), 60 minutes talk. , South China Normal University. 2009.
Ching W.K., On Modeling Correlated Default Risk, College of Systems Engineering, Northeastern University, China (invited speaker and Expert), 90 minutes talk.. 2010.
Ching W.K., On Modeling Correlated Default Risk, School of Mathematical Science, Dalian University of Technology (invited speaker) 60 minutes talk. 2009.
Ching W.K., Choi S.M. and Huang M., Optimal Service Capacity in a Multiple-server Queueing System: A Game Theory Approach, Journal of Industrial and Management Optimization. 2010, 6: 73-102.
Choi S.M., Ching W.K. and Huang M., Incentive Effects of Common and Separate Queues with Multiple Servers: The Principal-Agent Perspective, Proceedings of the 39th International Conference on Computers and Industrial Engineering (CIE39), Troyes, France, 6-8, July, 2009 . 2009.
Hayashida M., Tamura T., Akutsu T., Ching W.K. and Cong Y., On Distribution and Enumeration of Attractors in Probabilistic Boolean Networks, IET on Systems Biology. 2009, 6: 465-474.
Leung I... and Ching W.K., Cardinality of Binary Operations: A Remark on the Ubiquitous Sum, Far East Journal of Mathematical Education. 2009, 3: 127-143.
Li L., Shiga M., Ching W.K. and Mamitsuka H., Annotating Gene Functions with Integrative Spectral Clustering on Microarray Expressions and Sequences,, Genome informatics. 2009, 22: 95-120.
Li L., Ching W.K., Yamaguchi T. and Aoki-Kinoshita F., Weighted q-gram Method for Glycan Classification, BMC Bioinformatics. 2010, 11: S33doi:10.1186/1471-2105-11-S1-S33.
Song N., Jiao Y., Ching W.K., Siu K.T.K. and Wu Z.Y., A Valuation Model for Perpetual Convertible Bonds with Markov Regime-switching Models, International Journal of Pure and Applied Mathematics. 2009, 53: 583-600.
Song N., Ching W.K., Siu K.T.K., Fung S.L. and Ng K.P., Option Valuation Under a Multivariate Markov Chain Model, In: L. Yu, Y. Song, W. Ching, S. Wang and K. Lai, Proceedings of CSO2010, Huangshan,. IEEE Computer Society Proceedings, 2010, 1: 177-181.
Tai A.H.L., Ching W.K. and Chan P.L.Y., Detection of Machine Failure: Hidden Markov Model Approach, Computers and Industrial Engineering. 2009, 57: 608-619.
Yiu C., Liu J..., Siu K.T.K. and Ching W.K., Optimal Portfolios with Regime-Switching and Value-at-Risk Constraint, Automatica. 2010, 46: 979-989.
Yu L., Song Y., Ching W.K., Wang S. and Lai K., Computational Sciences and Optimization: Theoretical Develop and Engineering Practice. Vol. 2 (Proceedings Editor), CSO2010 Huangshan, China. IEEE Computer Society, 2010, 2.
Yu L., Song Y., Ching W.K., Wang S. and Lai K., Computational Sciences and Optimization: Theoretical Develop and Enginieering Practice, Vol. 1 (Proceedings Editor), CSO2010, Huangshan, China. IEEE Computer Soceity, 2010, 1.
Zhang S., Ching W.K., Tsing N.K., Leung H.Y. and Guo D., A New Multiple Regression Approach for the Construction of Genetic Regulatory Networks, Journal of Artificial Intelligence in Medicine. 2010, 48: 153-160.
Zhang S., Wang H., Zhou X., Hoehn G., DeGraba T., Gonzales D., Suffredini A., Ching W.K., Ng K.P. and Wong S.T.C., A Novel Peak Detection Approach with Chemical Noise Removal Using Short-Time FFT for prOTOF MS Data , Proteomics. 2009, 9: 3833-3842.
Zhang S., Ching W.K., Chen X. and Tsing N.K., Generating Probabilistic Boolean Networks from a Prescribed Stationary Distribution , Information Sciences. 2010, 180: 2560-2570.


Researcher : Choi SM
List of Research Outputs

Ching W.K., Choi S.M. and Huang M., Optimal Service Capacity in a Multiple-server Queueing System: A Game Theory Approach, Journal of Industrial and Management Optimization. 2010, 6: 73-102.
Choi S.M., Ching W.K. and Huang M., Incentive Effects of Common and Separate Queues with Multiple Servers: The Principal-Agent Perspective, Proceedings of the 39th International Conference on Computers and Industrial Engineering (CIE39), Troyes, France, 6-8, July, 2009 . 2009.


Researcher : Chu SCK
Project Title:INFORMS Annual Meeting - San Diego 2009 Student Project Allocation Using Goal Programming
Investigator(s):Chu SCK
Department:Mathematics
Source(s) of Funding:URC/CRCG - Conference Grants for Teaching Staff
Start Date:10/2009
Completion Date:10/2009
Abstract:
N/A


List of Research Outputs

Chu S.C.K., Associate editor of the IMA Journal of Management Mathematics. 2009.
Pan L., Chu S.C.K., Han G. and Huang J.Z., A Tree-based Wall-building Algorithm for Solving Container Loading Problem with Multi-drop Constraints, Proceedings of the 2009 IEEE International Conference on Industrial Engineering and Engineering Management (IEEM2009), Hong Kong, 8-11 December 2009. 538-542.
Pan L., Chu S.C.K., Han G. and Huang J.Z., Multi-criteria Student Project Allocation: A Case Study of Goal Programming Formulation with DSS Implementation, In: Zhang, X.-S., Liu D.-G. and Wang, Y., the 8th International Symposium on Operations Research and Its Applications (ISORA2009), Zhangjiajie, Hunan, China, 20-22 September 2009. 10: 75-82.


Researcher : Cong Y
List of Research Outputs

Ching W.K. and Cong Y., Finite-Horizon Control of Genetic Regulatory Networks with Multiple Hard-Constraints, The Third International Symposium on Optimization and Systems Biology (OSB 2009), Lecture Notes in Operations Research 11, Series Editors: Luo-nan Chen, Xiang-Sun Zhang, Ling-Yun Wu and Yong Wang . WORLD PUBLISHING CORPORATION, 2009, 11: 33-40.
Hayashida M., Tamura T., Akutsu T., Ching W.K. and Cong Y., On Distribution and Enumeration of Attractors in Probabilistic Boolean Networks, IET on Systems Biology. 2009, 6: 465-474.


Researcher : Fung SL
List of Research Outputs

Song N., Ching W.K., Siu K.T.K., Fung S.L. and Ng K.P., Option Valuation Under a Multivariate Markov Chain Model, In: L. Yu, Y. Song, W. Ching, S. Wang and K. Lai, Proceedings of CSO2010, Huangshan,. IEEE Computer Society Proceedings, 2010, 1: 177-181.


Researcher : Han G
Project Title:Entropy Rate of Hidden Markov Chains
Investigator(s):Han G
Department:Mathematics
Source(s) of Funding:Seed Funding Programme for Basic Research
Start Date:02/2008
Completion Date:07/2009
Abstract:
A hidden Markov chain is defined as a function of a Markov chain. A hidden Markov chain is also defined as a Markov chain observed in noise. It is well known that the two definitions are equivalent. A regular Markov chain is completely determined by the state transition probabilities, thus many properties of a Markov chain are well understood. However, for a hidden Markov chain, information or parameters of the original Markov chain are missing or hidden, which makes the analysis of hidden Markov chains surprisingly intricate. For instance, there is an explicit formula for entropy rate of a Markov chain. However, it has been that shown computing entropy rate of a generic hidden Markov chain is as difficult as computing the top Lyapunov exponent of some random matrix product, thus NP-hard. The problem of computing, estimating and analyzing entropy rate of a hidden Markov chain has been a challenging problem for quite a long time. The capacity of a communication channel is defined to be the maximal information transmission rate through this channel. It turns out that entropy rate of hidden Markov chains is closely related to capacity of input constrained binary symmetric channels. So far, there are no explicit formulas for the capacity of such channels. Objectives (see more technical details in "research plan and methodology" section): 1. We plan to investigate estimates and asymptotics for entropy rate of hidden Markov chains, and to investigate certain mathematical properties of hidden Markov chains, such as analyticity, convexity. 2. We plan to investigate the asymptotics of capacity of a binary symmetric channel with input supported on a binary irreducible finite type constraint, using the results from the investigation on hidden Markov chains. We will also consider other channels such as binary erasure channels, non-binary symmetric channels and more generally finite-state channels.


Project Title:Entropy Rate of Hidden Markov Chains
Investigator(s):Han G
Department:Mathematics
Source(s) of Funding:General Research Fund (GRF)
Start Date:11/2008
Abstract:
1) We plan to investigate estimates and asymptotics for entropy rate of hidden Markov chains, and to investigate certain mathematical properties of hidden Markov chains, such as analyticity, convexity. 2) We plan to investigate the asymptotics of capacity of a binary symmetric channel with input supported on a binary irreducible finite type constraint, possibly using the results from the investigation on hidden Markov chains. We will also consider other channels such as binary erasure channels, non-binary symmetric channels and more generally finite-state channels.


Project Title:2010 IEEE International Symposium on Information Theory (ISIT 2010) Entropy Rate of Continuous-State Hidden Markov Chains
Investigator(s):Han G
Department:Mathematics
Source(s) of Funding:URC/CRCG - Conference Grants for Teaching Staff
Start Date:06/2010
Completion Date:06/2010
Abstract:
N/A


List of Research Outputs

Han G. and Marcus B., Asymptotics of Entropy Rate of Hidden Markov Chains at Weak Black Holes, IEEE Transactions on Information Theory. 2010, 56: 1287-1295.
Han G. and Marcus B., Entropy Rate of Continuous-State Hidden Markov Chains, IEEE International Symposium on Information Theory. Austin, Texas, U.S.A., 2010, 1468-1472.
Han G., Menger’s Paths with Minimum Mergings: A Generalization of Menger’s Theorem, Information Theory and Applications Workshop. 2010.
Pan L., Chu S.C.K., Han G. and Huang J.Z., A Tree-based Wall-building Algorithm for Solving Container Loading Problem with Multi-drop Constraints, Proceedings of the 2009 IEEE International Conference on Industrial Engineering and Engineering Management (IEEM2009), Hong Kong, 8-11 December 2009. 538-542.
Pan L., Chu S.C.K., Han G. and Huang J.Z., Multi-criteria Student Project Allocation: A Case Study of Goal Programming Formulation with DSS Implementation, In: Zhang, X.-S., Liu D.-G. and Wang, Y., the 8th International Symposium on Operations Research and Its Applications (ISORA2009), Zhangjiajie, Hunan, China, 20-22 September 2009. 10: 75-82.


Researcher : Hayashida M
List of Research Outputs

Hayashida M., Tamura T., Akutsu T., Ching W.K. and Cong Y., On Distribution and Enumeration of Attractors in Probabilistic Boolean Networks, IET on Systems Biology. 2009, 6: 465-474.


Researcher : Huang M
List of Research Outputs

Ching W.K., Choi S.M. and Huang M., Optimal Service Capacity in a Multiple-server Queueing System: A Game Theory Approach, Journal of Industrial and Management Optimization. 2010, 6: 73-102.
Choi S.M., Ching W.K. and Huang M., Incentive Effects of Common and Separate Queues with Multiple Servers: The Principal-Agent Perspective, Proceedings of the 39th International Conference on Computers and Industrial Engineering (CIE39), Troyes, France, 6-8, July, 2009 . 2009.


Researcher : Jiang H
List of Research Outputs

Ching W.K., Leung H.Y., Jiang H. and Sun L., A Markovian Model for Default Risk in a Network of Sectors, Proceedings of The 2nd IEEE International Conference on Business Intelligence and Financial Engineering (BIFE 2009), Beijing, 24-26, July 2009, 373-377.. 2009.
Ching W.K., Leung H.Y., Jiang H., Sun L. and Siu K.T.K., A Markovian Network Model for Default Risk Management, , International Journal of Intelligent Engineering Informatics . 2010, 1: 104-124.
Ching W.K., Siu K.T.K., Li L., Jiang H., Li T. and Li W.K., An Improved Parsimonious Multivariate Markov Chain Model for Credit Risk, Journal of Credit Risk . 2009, 5: 1-25.


Researcher : Jiang H
List of Research Outputs

Ching W.K., Leung H.Y., Jiang H. and Sun L., A Markovian Model for Default Risk in a Network of Sectors, Proceedings of The 2nd IEEE International Conference on Business Intelligence and Financial Engineering (BIFE 2009), Beijing, 24-26, July 2009, 373-377.. 2009.
Ching W.K., Leung H.Y., Jiang H., Sun L. and Siu K.T.K., A Markovian Network Model for Default Risk Management, , International Journal of Intelligent Engineering Informatics . 2010, 1: 104-124.
Ching W.K., Siu K.T.K., Li L., Jiang H., Li T. and Li W.K., An Improved Parsimonious Multivariate Markov Chain Model for Credit Risk, Journal of Credit Risk . 2009, 5: 1-25.


Researcher : Jiao Y
List of Research Outputs

Song N., Jiao Y., Ching W.K., Siu K.T.K. and Wu Z.Y., A Valuation Model for Perpetual Convertible Bonds with Markov Regime-switching Models, International Journal of Pure and Applied Mathematics. 2009, 53: 583-600.


Researcher : Lau YK
Project Title:Moments for automorphic L-functions
Investigator(s):Lau YK, Tsang KM
Department:Mathematics
Source(s) of Funding:France/Hong Kong Joint Research Scheme - Travel Grants
Start Date:01/2008
Completion Date:12/2009
Abstract:
The study of moments of L-functions is a vital topic in number theory and has important applications. Having been extensively studied by various mathematicans (see [8], [1], [3], [11]), the moments of the central values of L-functions are (conjecturally) linked to the random matrix theory. On the other side, Iwaniec and Sarnak[7] introduced a method, based on the first and second moments with mollification, to tackle the Landau-Siegel zero problem. Furthermore, through the moments of the partial Euler products, Montgomery & Vaughan [15] and Granville & Soundarajan [6] provided important contributions to the distribution of values of L)1,X) for real primitive character X. Among other things they provide quantitative results on the large and small values.


Project Title:Closed Geodesics on the Modular Surface
Investigator(s):Lau YK
Department:Mathematics
Source(s) of Funding:General Research Fund (GRF)
Start Date:01/2009
Abstract:
1) Describe in a precise way the various classes of closed geodesics. In addition to reciprocal geodesics, there are interesting classes of geodesics invariant under other actions. These classes have been enumerated asymptotically. Our attempt is to understand more the remainder terms. In addition to individuals, we shall study the remainder terms averaging over suitable families. 2) Investigate numerically some open conjectures related to the number of closed geodesics in a class when ordering by discriminants. These conjectures are formulated based on several heuristics. We try to test with numerical data and refine possibly their statements. 3) Explore different properties of the multiplicity function of closed geodesics of a given length. Our approach begins with a probabilistic point of view. We shall nvestigate the behaviour of its limit distribution function and study the rate of convergence. Numerical studies will be carried out to visualize the shape of the density function. 4) Carry out numerical computation to collect information on the multiplicity of closed geodesics in different classes.


List of Research Outputs

Lau Y.K., An omega result for supremum norms of Hecke-eigenforms in the level aspect, Science in China Series A: Mathematics. 2010, 53 (1): 223-242.
Lau Y.K., Reviewer of Zentralblatt Math. V. 1189, No. 11046 , 2010, 1189, No. 11046.
Lau Y.K., Reviewer of Zentralblatt Math. pre05594041 , 2010, pre05594041.
Lau Y.K., Sum of Fourier coefficients of Cusp Forms, International Conference on Number Theory and Representation Theory, Shandong University, Weihai, China. 2009.
Lau Y.K., Summatory functions involving the Fourier coefficients of primitive cusp forms, Universite Paris-Sud, France. 2009.
Lau Y.K. and Wu J., The number of Hecke eigenvalues of same signs, Mathematische Zeitschrift. 2009, 263 (4): 959-970.


Researcher : Law HF
List of Research Outputs

Wong P., Law H.F. and Wong P.P.W., A Second Main Theorem on Pn for difference operator, Science in China Series A: Mathematics. 2009, 52, No.12: 2751-2758.


Researcher : Leung HY
List of Research Outputs

Ching W.K., Leung H.Y., Jiang H. and Sun L., A Markovian Model for Default Risk in a Network of Sectors, Proceedings of The 2nd IEEE International Conference on Business Intelligence and Financial Engineering (BIFE 2009), Beijing, 24-26, July 2009, 373-377.. 2009.
Ching W.K., Leung H.Y., Jiang H., Sun L. and Siu K.T.K., A Markovian Network Model for Default Risk Management, , International Journal of Intelligent Engineering Informatics . 2010, 1: 104-124.
Zhang S., Ching W.K., Tsing N.K., Leung H.Y. and Guo D., A New Multiple Regression Approach for the Construction of Genetic Regulatory Networks, Journal of Artificial Intelligence in Medicine. 2010, 48: 153-160.


Researcher : Li L
List of Research Outputs

Ching W.K., Li L., Chan Y.M. and Mamitsuka H., A Study of Network-based Kernel Methods on Protein-Protein Interaction for Protein Functions Prediction, The Third International Symposium on Optimization and Systems Biology (OSB 2009), Lecture Notes in Operations Research 11, Series Editors: Luo-nan Chen, Xiang-Sun Zhang, Ling-Yun Wu and Yong Wang. . WORLD PUBLISHING CORPORATION, 2009, 11: 25-32.
Ching W.K., Li L., Tsing N.K., Tai C.W., Ng T.W., Wong A.S.T. and Cheng K., A Weighted Local Least Squares Imputation Method for Missing Value Estimation in Microarray Gene Expression Data, Journal of Data Mining and Bioinformatics . 2010, 4: 331-347.
Ching W.K., Siu K.T.K., Li L., Jiang H., Li T. and Li W.K., An Improved Parsimonious Multivariate Markov Chain Model for Credit Risk, Journal of Credit Risk . 2009, 5: 1-25.
Ching W.K., Siu K.T.K., Li L., Li T. and Li W.K., Modeling Default Data via an Interactive Hidden Markov Model, Computational Economics. 2009, 34: 1-19.
Li L., Shiga M., Ching W.K. and Mamitsuka H., Annotating Gene Functions with Integrative Spectral Clustering on Microarray Expressions and Sequences,, Genome informatics. 2009, 22: 95-120.
Li L., Ching W.K., Yamaguchi T. and Aoki-Kinoshita F., Weighted q-gram Method for Glycan Classification, BMC Bioinformatics. 2010, 11: S33doi:10.1186/1471-2105-11-S1-S33.


Researcher : Li T
List of Research Outputs

Ching W.K., Siu K.T.K., Li L., Jiang H., Li T. and Li W.K., An Improved Parsimonious Multivariate Markov Chain Model for Credit Risk, Journal of Credit Risk . 2009, 5: 1-25.
Ching W.K., Siu K.T.K., Li L., Li T. and Li W.K., Modeling Default Data via an Interactive Hidden Markov Model, Computational Economics. 2009, 34: 1-19.


Researcher : Lu J
Project Title:Steinberg fibers, Bruhat cells, and log-canonical coordinates
Investigator(s):Lu J
Department:Mathematics
Source(s) of Funding:General Research Fund (GRF)
Start Date:01/2008
Abstract:
An explicit desingularization of the intersection of a Steinberg fiber and a Bruhat cell has been constructed in a recent joint work. We will study in more detail birational parametrizations of the desingularization. Obtain a criterion for when a G-stable piece and a (B, B-)-orbit intersect in the wonderful compactification. The answer will be in terms of Weyl group elements. Express the Poisson structre on a G-stable piece in terms of a Poisson structure on a Levi subgroup in G.


Project Title:Poisson structures and Frobenius splittings
Investigator(s):Lu J
Department:Mathematics
Source(s) of Funding:Seed Funding Programme for Basic Research
Start Date:06/2010
Abstract:
The objective is to carry out the first few key steps towards a long term project on the relations between Poisson geometry and Frobenius splittings. More precisely, we will 1) understand more the algebraic geometry involved with the method of Frobenius splittings; 2) study in more detail some concerete examples.


List of Research Outputs

Lu J., On a Poisson Structure on Compact Riemannian Symmetric Spaces , In: C. Leung, IMS Geometry seminar, Chinese University of Hong Kong, Oct 28, 2009. 2009.
Lu J., On a Poisson structure on conjugacy classes, In: Yakov Eliashberg (Stanford University), Alvaro Pelayo* (University of California, Berkeley), Steve Zelditch (Northwestern University), Maciej Zworski (University of California, Berkeley) , Symplectic and Poisson Geometry in interaction with Algebra, Analysis and Topology, MSRI, Berkeley, May 4-7, 2010. 2010.
Lu J., On intersections of brauhat cells and conjugacy classes, Graduate Colloquium, Institute of mathematics, University of Gottingen. 2009.
Lu J., Chan K.Y. and To K.M.S., On intersections of conjugacy classes and bruhat cells , Transformation groups. Springer, 2010, 15, No. 2: 243 - 260.


Researcher : Mamitsuka H
List of Research Outputs

Ching W.K., Li L., Chan Y.M. and Mamitsuka H., A Study of Network-based Kernel Methods on Protein-Protein Interaction for Protein Functions Prediction, The Third International Symposium on Optimization and Systems Biology (OSB 2009), Lecture Notes in Operations Research 11, Series Editors: Luo-nan Chen, Xiang-Sun Zhang, Ling-Yun Wu and Yong Wang. . WORLD PUBLISHING CORPORATION, 2009, 11: 25-32.
Li L., Shiga M., Ching W.K. and Mamitsuka H., Annotating Gene Functions with Integrative Spectral Clustering on Microarray Expressions and Sequences,, Genome informatics. 2009, 22: 95-120.


Researcher : Mok N
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:General Research Fund (GRF)
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.


Project Title:Geometric rigidity phenomena concerning bounded homogeneous domains and related problems
Investigator(s):Mok N
Department:Mathematics
Source(s) of Funding:General Research Fund (GRF)
Start Date:01/2009
Abstract:
1) To develop a geometric theory on proper maps for domains carrying geometric structures modeled on the geometric theory of VMRTs, and to characterize proper maps between interesting classes of bounded homogeneous domains (BHDs). 2) To characterize convex realizations of BHDs, and to characterize irreducible bounded symmetric domains of rank at least 2 among BHDs in terms of bounded convex realizations. 3) To prove geometric superrigidity for holomorphic mappings from bounded symmetric domains of rank at least 2 into BHDs equivariant w.r.t. irreducible lattices. 4) To classify Kaehler embeddings of the Poincare disk into the polydisk, to study the structure of such embeddings into BHDs, and to find conditions under which rigidity holds for Kaehler embeddings between BHDs.


List of Research Outputs

Mok N., Complex Analysis - Several Complex Variables and Connections with PDE Theory and Geometry. Springer Basel AG, 2010.
Mok N., Editor of Chinese Annals of Mathematics. Springer, 2010.
Mok N., Editor of Inventiones Mathematicae. Berlin-Heidelberg-New York, Springer-Verlag, 2009.
Mok N., Editor of Journal de l' Institut de Mathématiques de Jussieu. Cambridge University Press, 2010.
Mok N., Editor of Mathematische Annalen. Berlin-Heidelberg-New York, Springer-Verlag, 2009.
Mok N., Editor of Science in China Series A: Mathematics. Science Press, co-published with Springer-Verlag, 2010.
Mok N., Ng S.C. and Tu Z., Factorization of proper holomorphic maps on irreducible bounded symmetric domains of rank ³ 2, Science China Mathematics - Special Issue Dedicated to Professor Yang Lo on the Occasion of his 70th Birthday. 2010, 53, No.3: 513-894.
Mok N., Geometry of holomorphic maps into bounded symmetric domains, Conference on Complex and Differential Geometry at Leibniz Universität Hannover (September 14 - 18). 2009.
Mok N., Geometry of holomorphic maps into bounded symmetric domains, International Conference on Several Complex Variables, Complex Geometry and Partial Differential Equations, Wuhan University (July 6-10). 2009.
Mok N., Mini Course: Varieties of minimal rational tangents, International Workshop on Complex Geometry and Algebraic Geometry (Pre-Workshop Activities), Shanghai Jiaotong University. 2010.
Mok N., On complex-analytic and homological rigidity of smooth Schubert cycles on rational homogeneous spaces of Picard number 1, Complex Geometry, Levico Terme (Trento, Italy), May 31-June 4. 2010.
Mok N., On the Zariski closure of a germ of totally geodesic complex submanifold on a subvariety of a complex hyperbolic space form of finite volume, Complex Analysis, Trends in Mathematics. Springer Basel AG, 2010, 279-300.
Mok N., On the asymptotic behavior of holomorphic isometries of the Poincaré disk into bounded symmetric domains, Acta Mathematica Scientia (Special Volume in honor of Professor Wu Wenjun on the occasion of his 90th birthday). 2009, 29B(4): 881-902.
Mok N., Plenary lecture: On subvarieties of uniruled projective manifolds saturated with minimal rational curves, Conference on Algebraic Geometry on Varieties and Manifolds (May 10–14). 2010.
Mok N. and Ng S.C., Second fundamental forms of holomorphic isometries of the Poincaré disk into bounded symmetric domains and their boundary behavior along the unit circle, Science in China (Series A) - Special Issue dedicated to Professor ZHONG TongDe on the occasion of his 80th birthday. 2009, Vol.52, No.12: 2628-2646.


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

Song N., Ching W.K., Siu K.T.K., Fung S.L. and Ng K.P., Option Valuation Under a Multivariate Markov Chain Model, In: L. Yu, Y. Song, W. Ching, S. Wang and K. Lai, Proceedings of CSO2010, Huangshan,. IEEE Computer Society Proceedings, 2010, 1: 177-181.
Zhang S., Wang H., Zhou X., Hoehn G., DeGraba T., Gonzales D., Suffredini A., Ching W.K., Ng K.P. and Wong S.T.C., A Novel Peak Detection Approach with Chemical Noise Removal Using Short-Time FFT for prOTOF MS Data , Proteomics. 2009, 9: 3833-3842.


Researcher : Ng SC
List of Research Outputs

Mok N., Ng S.C. and Tu Z., Factorization of proper holomorphic maps on irreducible bounded symmetric domains of rank ³ 2, Science China Mathematics - Special Issue Dedicated to Professor Yang Lo on the Occasion of his 70th Birthday. 2010, 53, No.3: 513-894.
Mok N. and Ng S.C., Second fundamental forms of holomorphic isometries of the Poincaré disk into bounded symmetric domains and their boundary behavior along the unit circle, Science in China (Series A) - Special Issue dedicated to Professor ZHONG TongDe on the occasion of his 80th birthday. 2009, Vol.52, No.12: 2628-2646.


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: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:Meromorphic solutions of algebraic differential equations
Investigator(s):Ng TW
Department:Mathematics
Source(s) of Funding:General Research Fund (GRF)
Start Date:09/2007
Abstract:
Find some sufficient conditions to guarantee that the solutions of the differential equations considered must be elliptic functions and their degenerations. Study the factorization of the meromorphic solution of some algebraic PDEs related to certain Fermat type diophantine equations. Study Hille's conjecture on Briot-Bouquet differential equations.


Project Title:Factorizations and iterations of meromorphic functions and related topics
Investigator(s):Ng TW
Department:Mathematics
Source(s) of Funding:General Research Fund (GRF)
Start Date:01/2010
Abstract:
1) We shall study the factorization of finite mappings (between Riemann surfaces) which have certain good local property. We shall then consider the special case when the finite map is a finite Blaschke product and try to extend Ritt’s theory to this case. In order to extend Ritt’s results to finite Blaschke products, we shall introduce a new class of finite Blaschke products which can be considered as Chebyshev polynomials on the unit disk. We may also study the mapping properties of these new Chebyshev polynomials. 2) We shall study some problems of complex dynamics for transcendental meromorphic functions that are related to factorization of meromorphic functions. 3) To introduce certain one parameter family of polynomials and show that for each parameter, one can obtain a version of Smale’s mean value theorem and hence we can embed Smale’s mean value conjecture into a one parameter family of the mean value conjectures. We shall also study how these mean value conjectures are related to each other. 4) To train one or two graduate students to work on complex dynamics and factorization of meromorphic functions.


List of Research Outputs

Cheung W.S. and Ng T.W., Relationship between the zeros of two polynomials, Journal of Linear Algebra and Its Applications. 2010, 432, no.1: 107-115.
Ching W.K., Li L., Tsing N.K., Tai C.W., Ng T.W., Wong A.S.T. and Cheng K., A Weighted Local Least Squares Imputation Method for Missing Value Estimation in Microarray Gene Expression Data, Journal of Data Mining and Bioinformatics . 2010, 4: 331-347.
Ng T.W., Exact Meromorphic Solutions of the Falkner-Skan Differential Equation , Workshop on Complex Geometry, HKU (August 2009) . 2009.
Ng T.W., Exact Meromorphic Solutions of the Falkner-Skan Differential Equation, International Conference on Applied Mathematics, City University of Hong Kong (June 7-11, 2010). 2010.
Ng T.W., Exact Meromorphic Solutions of the Falkner-Skan Differential Equation, Seminar on Pure Mathematics, HKUST (February 2010). 2010.
Ng T.W., Finding exact meromorphic solutions of the Falkner-Skan differential equation, Hong Kong Mathematical Society Annual General Meeting (March 2010). 2010.
Ng T.W., RFBR-NSFC Joint Symposium on Complex Analysis and its applications (Moscow session, October 22-29), Exact meromorphic solutions of the Falkner-Skan differential equation. 2009.


Researcher : Pan L
List of Research Outputs

Pan L., Chu S.C.K., Han G. and Huang J.Z., A Tree-based Wall-building Algorithm for Solving Container Loading Problem with Multi-drop Constraints, Proceedings of the 2009 IEEE International Conference on Industrial Engineering and Engineering Management (IEEM2009), Hong Kong, 8-11 December 2009. 538-542.
Pan L., Chu S.C.K., Han G. and Huang J.Z., Multi-criteria Student Project Allocation: A Case Study of Goal Programming Formulation with DSS Implementation, In: Zhang, X.-S., Liu D.-G. and Wang, Y., the 8th International Symposium on Operations Research and Its Applications (ISORA2009), Zhangjiajie, Hunan, China, 20-22 September 2009. 10: 75-82.


Researcher : Ren J
List of Research Outputs

Cheung W.S., Ren J. and Han W., Positive Periodic Solution of Second-Order Neutral Functional Differential Equations, Nonlinear Analysis. 2009, 71: 3948-3955.
Cheung W.S., Ren J. and Han W., Positive Periodic Solutions for Second-Order Differential Equations with Generalized Neutral Operator, Australian Journal of Mathematical Analysis and Applications. 2009, 6, No.1, Art 5: 16 pages.


Researcher : Siu KTK
List of Research Outputs

Ching W.K., Li X., Siu K.T.K. and Wu Z., Improving Revenue Management: A Real Option Approach, In: Cheng, T. C. Edwin; Choi, Tsan-Ming , Springer's Handbook Series on Innovative Quick Response Programs in Logistics Supply Chain Management. Springer, 2010, 123-139.
Ching W.K., Siu K.T.K., Li L., Li T. and Li W.K., Modeling Default Data via an Interactive Hidden Markov Model, Computational Economics. 2009, 34: 1-19.
Song N., Jiao Y., Ching W.K., Siu K.T.K. and Wu Z.Y., A Valuation Model for Perpetual Convertible Bonds with Markov Regime-switching Models, International Journal of Pure and Applied Mathematics. 2009, 53: 583-600.


Researcher : Siu MK
List of Research Outputs

Siu M.K. and Lam K., "概率萬花筒" ("Kaleidoscope in Probability") , revised edition, Mathematics Education Section, Education Bureau of the HKSAR. Government Logistics Department, 2010, 118 pages.
Siu M.K. and Stewart I., How to invert n cups m at a time?, Mathematics Today. 2010, 46(1): 34-38.
Siu M.K., L’insegnamento e l’apprendimento dell’algebra a livello universitario (On the learning and teaching of tertiary algebra) , Progetto Alice. 2009, 10 (29): 311-330.
Siu M.K., Mathematics education in East Asia from antiquity to modern times, in Dig where you Stand: Proceedings of a Conference on On-going Research in the History of Mathematics Education, Gardabaer, June 20-24, 2009, edited by K. Bjarnadottir, F. Furinghetti, G. Schubring, University of Iceland. 2009, 197-208.
Siu M.K., On Klein’s "double discontinuity": University lecture hall and school classroom, invited talk at the Hong Kong-Germany Mini-Conference on Mathematics Education. 2010.
Siu M.K., The world of geometry in the classroom: Virtual or real? , Proceedings of 5th International Colloquium on the Didactics of Mathematics, vol. II, edited by M. Kourkoulos, C. Tzanakis, University of Crete. 2009, 93-112.


Researcher : Song N
List of Research Outputs

Song N., Jiao Y., Ching W.K., Siu K.T.K. and Wu Z.Y., A Valuation Model for Perpetual Convertible Bonds with Markov Regime-switching Models, International Journal of Pure and Applied Mathematics. 2009, 53: 583-600.
Song N., Ching W.K., Siu K.T.K., Fung S.L. and Ng K.P., Option Valuation Under a Multivariate Markov Chain Model, In: L. Yu, Y. Song, W. Ching, S. Wang and K. Lai, Proceedings of CSO2010, Huangshan,. IEEE Computer Society Proceedings, 2010, 1: 177-181.


Researcher : Tai CW
List of Research Outputs

Ching W.K., Li L., Tsing N.K., Tai C.W., Ng T.W., Wong A.S.T. and Cheng K., A Weighted Local Least Squares Imputation Method for Missing Value Estimation in Microarray Gene Expression Data, Journal of Data Mining and Bioinformatics . 2010, 4: 331-347.


Researcher : Tamura T
List of Research Outputs

Hayashida M., Tamura T., Akutsu T., Ching W.K. and Cong Y., On Distribution and Enumeration of Attractors in Probabilistic Boolean Networks, IET on Systems Biology. 2009, 6: 465-474.


Researcher : To KMS
List of Research Outputs

Lu J., Chan K.Y. and To K.M.S., On intersections of conjugacy classes and bruhat cells , Transformation groups. Springer, 2010, 15, No. 2: 243 - 260.


Researcher : Tsang KM
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.


Project Title:International Conference on Analytic Number Theory An extension of the Brun-Titchmarsh inequality
Investigator(s):Tsang KM
Department:Mathematics
Source(s) of Funding:URC/CRCG - Conference Grants for Teaching Staff
Start Date:10/2009
Abstract:
N/A


List of Research Outputs

Tsang K.M., An Extension of the Brun-Titchmarsh Inequaltiy, International Conference on Analytic Number Theory, Tata Institute of Fundamental Research, Mumbai, India. 2009.
Tsang K.M., An Extension of the Brun-Titchmarsh Inequaltiy, International Conference on Number Theory and Representation Theory, Shandong University, Weihai, China . 2009.
Tsang K.M., Mean Square of the Riemann zeta-function on the critical line, Zeta-Function Days in Seoul, Seoul, Korea.. 2009.
Tsang K.M., Remembering Atle Selberg, 1917-2007, In: Dennis Hejhal, Peter Sarnak, Notices of the American Mathematical Society. Providence, RI, USA, American Mathematical Society, 2009, 56: 703-705.


Researcher : Tsing NK
List of Research Outputs

Ching W.K., Li L., Tsing N.K., Tai C.W., Ng T.W., Wong A.S.T. and Cheng K., A Weighted Local Least Squares Imputation Method for Missing Value Estimation in Microarray Gene Expression Data, Journal of Data Mining and Bioinformatics . 2010, 4: 331-347.
Zhang S., Ching W.K., Tsing N.K., Leung H.Y. and Guo D., A New Multiple Regression Approach for the Construction of Genetic Regulatory Networks, Journal of Artificial Intelligence in Medicine. 2010, 48: 153-160.


Researcher : Tu Z
List of Research Outputs

Mok N., Ng S.C. and Tu Z., Factorization of proper holomorphic maps on irreducible bounded symmetric domains of rank ³ 2, Science China Mathematics - Special Issue Dedicated to Professor Yang Lo on the Occasion of his 70th Birthday. 2010, 53, No.3: 513-894.


Researcher : Wong P
List of Research Outputs

Wong P., Law H.F. and Wong P.P.W., A Second Main Theorem on Pn for difference operator, Science in China Series A: Mathematics. 2009, 52, No.12: 2751-2758.


Researcher : Wong PPW
List of Research Outputs

Wong P., Law H.F. and Wong P.P.W., A Second Main Theorem on Pn for difference operator, Science in China Series A: Mathematics. 2009, 52, No.12: 2751-2758.


Researcher : Wong STC
List of Research Outputs

Zhang S., Wang H., Zhou X., Hoehn G., DeGraba T., Gonzales D., Suffredini A., Ching W.K., Ng K.P. and Wong S.T.C., A Novel Peak Detection Approach with Chemical Noise Removal Using Short-Time FFT for prOTOF MS Data , Proteomics. 2009, 9: 3833-3842.


Researcher : Wu J
List of Research Outputs

Lau Y.K. and Wu J., The number of Hecke eigenvalues of same signs, Mathematische Zeitschrift. 2009, 263 (4): 959-970.


Researcher : Wu S
Project Title:Morse theory and equivariant cohomology
Investigator(s):Wu S
Department:Mathematics
Source(s) of Funding:Seed Funding Programme for Basic Research
Start Date:03/2007
Abstract:
There are two parts in this proposed research project. Part 1 is on Morse theory on manifolds with boundary and with group actions. The results will be: topological construction of the instanton complex for a Morse function on a manifold with boundjavascript:void request_launch('ifs'); Launch consoleMore Video/Audio Newsary, relation to tunneling in quantum mechanics on a manifold with boundary, equivariant Morse theory, especially the harmonic representatives of equivariant cohomology and relation to gauged supersymmetric quantum mechanics, possible relation to non-commutative equivariant cohomology. Part 2 is about holomorphic equivariant cohomology. The anticipated results results are: holomorphic version of equivariant cohomology, examples including supersymmetric gauge theory on a Kahler surface and topological sigma model (B-model) on a Riemann surface, holomorphic version of Mathai-Quillen construction, a possible new version of holomorphic Morse theory.


Project Title:The geometry of quantisation, gerbes and Langlands dual
Investigator(s):Wu S
Department:Mathematics
Source(s) of Funding:General Research Fund (GRF)
Start Date:07/2007
Abstract:
To study projective flatness in geometric quantisation, specially when the symplectic manifold is the cotangent bundle of a Lie group or the moduli space of flat connections on a Riemann surface. In the first case, the PI will calculate the connection and curvature of the Hilbert space bundle over a family of complex structures. To study twisted Deligne cohomology groups and the mathematical operations such as cup product, fibre integration. This will give a precise definition of Ramond-Ramond fields in the presence of fluxes in string theory. The PI also plans to prove the isomorphism between even and odd twisted Deligne cohomology groups on spaces related by T-duality. To study the heat kernel on a compact Lie group that is not necessarily simply-connected and to find the relation to that of the Langlands dual under modular transformation. The PI also plans to study similar relations for the moduli space of flat connections over a Riemann surface.


List of Research Outputs

Mathai V. and Wu S., Twisted analytic torsion, Sci. China Math. . Beijing, China, Science China Press, 2010, 53: 555-563.
Wu S., Analytic torsion of twisted elliptic complexes , Mini-Conference in Geometry, CUHK. 2010.
Wu S., Electric-magnetic duality and modular invariance , Workshop on loops, strings and moduli spaces, Chern Institute of Mathematics, Tianjin. 2009.
Wu S., Hyper-Kahler geometry and Higg bundles, National Center for Theoretical Sciences lectures, Hsin-chu. 2009.
Wu S., Twisted analytic torsion, National Cheng-Kung University colloquium. Tainan, Taiwan, 2010.
Wu S., Twisted analytic torsion, National Tsing-Hua University colloquium. Hsin-chu, Taiwan, 2010.
Wu S., Twisted torsion of de Rham and Dolbeault complexes, Complex Analysis and Related Topics, Chinese Academy of Sciences. 2009.


Researcher : Xiong G
List of Research Outputs

Cheung W.S. and Xiong G., Chord Power Integrals for Simplices, Asian European Journal of Mathematics. 2009, 2, No.4: 557-565.
Cheung W.S., Xiong G. and Xiao Q.M., Firey Linear Combinations of Convex Bodies, Journal of Shanghai University. 2009, 13, No.2: 102-104.


Researcher : Yu J
Project Title:A Cancellation Problem for Free Associative Algebras
Investigator(s):Yu J
Department:Mathematics
Source(s) of Funding:General Research Fund (GRF)
Start Date:09/2007
Abstract:
To work on several long-standing open problems in Affine Algebraic Geometry and Commutative & Non-commuative Algebra in the past decade.


Project Title:Research Output Prize
Investigator(s):Yu J
Department:Mathematics
Source(s) of Funding:Research Output Prize (in Faculty)
Start Date:10/2008
Completion Date:09/2009
Abstract:
To identify and recognize the best research outputs in different faculties.




Researcher : Yung SP
Project Title:Optimal Mean-Variance Portfolio Selection with a Value-At-Risk Constraint
Investigator(s):Yung SP
Department:Mathematics
Source(s) of Funding:Small Project Funding
Start Date:12/2008
Abstract:
Our objective is to select an optimal investment portfolio in the mean-variance framework of Markowitz but still satisfies a Value-at-Risk constraint, which in some sense guarantees that the probability of the total wealth going nearly broke is small. One main difficulty is that the Value-at-Risk constraint is implicitly in nature and fairly difficult to handle. We shall find an approximation replacement of it and solve the corresponding problem. The whole process involves a partial differential equation, namely Hamilton-Jacobi-Bellman equation, which can be solved explicitly or through numerical schemes.


Project Title:Optimal insurance problem that incorporates risk averse and risk seeking behaviours
Investigator(s):Yung SP
Department:Mathematics
Source(s) of Funding:Small Project Funding
Start Date:11/2009
Abstract:
In one of the Nobel prize winning work of Kahneman, the contradicting human behaviours in preferring risk-averse when in profits but preferring risk-seeking when in losses were described and formulated in a theory called prospect theory. Subsequently, this theory is used and successfully describes various investors' behaviours in finance. In the insurance area, similar contradicting behaviours have been observed but only empirical studies were carried out. In this project, we shall investigate how an insurer could find the best insurance plan according to the risk-averse-when-in-profits-but-risk-seeking-when-in-losses behaviour. This behaviour will be modelled by a S-shaped utility function together with a probability distortion function. A mathematical theory will be established and explicit solutions are anticipated. A major difficulty in this project is the non-concavity of the utility function that causes more traditional tools not applicable. Fortunately, we could split the problem into 3 subproblems and solve them individually.


List of Research Outputs

Yung S.P., Yam S.C.P. and Zhou W., Two rationales behind the 'buy-and-hold or sell-at-once' strategy, Journal of Applied Probability. 2009, 46, No.3: 651-668.


Researcher : Zang W
Project Title:Min-max relations and integral polyhedra
Investigator(s):Zang W
Department:Mathematics
Source(s) of Funding:General Research Fund (GRF)
Start Date:09/2006
Completion Date:08/2009
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:Feedback Vertex Sets: Optimization and Approximation
Investigator(s):Zang W
Department:Mathematics
Source(s) of Funding:Seed Funding Programme for Basic Research
Start Date:09/2007
Completion Date:08/2009
Abstract:
Let G=(V,E) be a graph (undirected or directed) with a nonnegative integral weight w(v) on each vertex v. A set X of vertices in G is called a feedback vertex set (FVS) if G-X contains no cycles. The feedback vertex set (FVS) problem is to find a feedback vertex set in G with minimum total weight. This problem was originally formulated in the area of combinatorial circuit design, where cycles can potentially cause a problem called "racing condition"; that is, some circuit vertex may receive new inputs before it stabilizes. To avoid such a condition, a (clocked) register is placed at each cycle in the circuit. However, the delay in the circuit speed is proportional to the number of registers placed along a path. Therefore, the objective is to minimize the number of vertices (registers) to be placed so that the total delay can be minimized. Modern important applications of the FVS problem can be found in deadlock prevention in operating systems, constraint satisfaction and Bayesian inference in artificial intelligence, etc. It is well known that the FVS problem is NP-hard, so there is no polynomial-time algorithm for solving it exactly unless NP=P. Given an NP-hard problem, usually there are two approaches to getting around the hardness: First, we may be able to isolate important special cases that are solvable in polynomial time. Second, it may be possible to find near-optimal solutions in polynomial time. In practice, near-optimality is often good enough. An algorithm that returns near-optimal solutions is called an approximation algorithm. My proposed investigation will rely heavily on these two approaches. The FVS problem is closely related to two other optimization problems. To be specific, let G be defined as above, and let C be a collection of cycles (resp. FVS's) of G, where repetition of members is allowed. We call C a cycle packing (resp. an FVS packing) if each vertex v of G is used at most w(v) times by members of C. The problem of finding a cycle packing (resp. an FVS packing) with maximum size is the cycle packing problem (resp. FVS packing problem). In optimization theory, the cycle packing problem is precisely the dual problem of the FVS problem, while the FVS packing problem is its blocker problem. The objectives of this proposal are the following: 1. Characterize all directed graphs G=(V,E) such that for any integral weight function w defined on V, the minimum weight of a feedback vertex set is equal to the maximum size of a cycle packing in G (this is a long-standing open problem in combinatorial optimization); 2. Give a polynomial-time approximation scheme for the FVS problem on plane graphs; 3. Characterize all tournaments T=(V,E) such that for any integral weight function w defined on V, the minimum weight of a triangle is equal to the maximum size of an FVS packing in G; 4. Show that the FVS packing problem is NP-hard on undirected graphs; 5. Find an approximation algorithm for the FVS packing problem on general graphs with a constant approximation ratio; and 6. Obtain a polynomial-time approximation scheme for the FVS packing problem on plane graphs.


Project Title:Characterizations of Combinatorial Structures
Investigator(s):Zang W
Department:Mathematics
Source(s) of Funding:General Research Fund (GRF)
Start Date:01/2008
Abstract:
Resolve the following open problem posed by Archdeacon: Characterize all graphical sequences S such that some realization of S admits a nowhere-zero 3-flow. I believe I should be able to get a full characterization in terms of forbidden sequences. If obtained, this characterization would yield a beautiful mathematical theorem with potential applications in combinatorial design. Prove the following Chess Board Conjecture: Given an N by N black-and-white chess board, with an arbitrary N by N 0-1 matrix A on it, one can permute the rows and columns of A to get a new matrix B so that in each row and each column of B, the number of 1's in black is at least the number of 1's in white-1. The possible outcome is a proof of this conjecture. If obtained, this interesting result can be directly applied to solve some important real-world problems arising in engineering. Establish the min-max theorem on packing and covering odd paths between a pair of specified vertices in a graph. I think a complete characterization of all graphs with the desired min-max relation is within reach, with hard work. If obtained, this result would yield a polynomial-time solution of a large class of integer programming problem. Characterize all matroids with the box max-flow min-cut property. The possible outcome is a constructive description of all these matroids. This work, if carried out, would strenghthen Seymour's far-reaching theorem on matroids with the max-flow min-cut property in many circumstances, and significantly improve the Gerards-Laurent theorem on box 1/d-integral binary clutters. It would also yield a number of strong min-max relations in combinatorial optimization in box-integrality version.


Project Title:Optimization over Totally Dual Integral Systems
Investigator(s):Zang W
Department:Mathematics
Source(s) of Funding:Seed Funding Programme for Basic Research
Start Date:06/2008
Completion Date:05/2010
Abstract:
A linear system Ax>=b, x>=0 is called totally dual integral (TDI) if the maximum in the following linear programming duality equation Min {c^Tx: Ax >= b, x >=0}=Max {y^Tb: y^TA<=c^T, y>=0} has an integral optimal solution y for any integral vector c for which the optimum is finite. As shown by Edmonds and Giles, if Ax>=b, x>=0 is TDI, b is integral, and the optimum is finite, then the minimum also has an integral optimal solution x, and thus an integral duality theorem, or equivalently a min-max relation, follows. Total dual integrality is a natural framework for the study of min-max relations in combinatorial optimization; it also closely related to many other models for the same purpose. As summarized by Schrijver, the importance of a min-max relation is two-fold: first, it yields a beautiful mathematical theorem (say, the well-known max-flow min-cut theorem); second, it can serve as an optimization criterion and usually leads to polynomial-time solvability of the corresponding optimization problems. Many important combinatorial optimization problems can be naturally formulated as integer linear programs. Due to the special feature of such a problem, sometimes the linear system that defines the feasible region is totally dual integral. Under what conditions does such a property hold? This decision problem, as conjectured by Edmonds and Giles in 1984 and proved recently by Ding, Feng and myself, is NP-hard in general, and hence cannot be solved in polynomial time unless NP=P. In this project I propose to investigage several important opimization problems and explore the total dual integrality property associated with them. The objectives of this proposal are the following: 1. Characterize all graphs G=(V, E) with the property that, for any nonnegative integral weight function w defined on E, the maximum number of odd cycle covers (repetition allowed) such that each edge e of G is used at most 2w(e) times by these cycle covers is equal to the minimum total weight of a collection X of edges (repetition allowed) such that each odd cycle cover of G meets X at least twice. 2. Design a combinatorial polynomial-time algorithm for solving the maximum cut problem on weakly bipartite graphs (these graphs play important roles in combinatorial optimization). 3. Prove that if G=(V, E) is a graph with a nonnegative integral weight function defined on E and s, t are two vertices of G, then the maximum number of odd s-t paths (repetition allowed) such that each edge e is used at most 2w(e) times by these paths is equal to the minimum totat weight of a collection X of edges (repetition allowed) such that each odd s-t path in G meets X at least twice. 4. Prove that if G=(V, E) be a plane digraph with a nonnegative integral weight function w defined on E, then the maximum number of feedback arc sets (repetition allowed) such that each arc e is used at most 2w(e) times by these sets is equal to the mimum total weight of a collection X of arcs (repetition allowed) such that each feedback arc set in G meets X at least twice. 5. Prove that in any plane digraph the maximum number of disjoint feedback arc sets is equal to the length of a shortest cycle (this is a well-known conjecture made by Woodall in 1978).


Project Title:The Edge-Coloring Problem on Multigraphs
Investigator(s):Zang W
Department:Mathematics
Source(s) of Funding:Seed Funding Programme for Basic Research
Start Date:06/2009
Abstract:
Let G=(V, E) be a multigraph. A k-edge-coloring of G is an assignment of k colors to the edges of G so that no two adjacent edges have the same color. The chromatic index of G, denoted by χ'(G), is the smallest k for which G admits a k-edge-coloring. The problem of finding an edge-coloring of G with χ'(G) colors is called the edge-coloring problem. In addition to its great theoretical interest, this problem arises in a rich variety of applications, and hence has attracted tremendous research efforts in several fields such as theoretical computer science, graph theory, and optimization. As shown by Holyer, the edge-coloring problem is NP-hard, so a good estimate of χ'(G) has been the focus of extensive research. Let Δ(G) denote the maximum degree of G. Clearly, χ'(G) ≥ Δ(G). For the upper bound, a classical theorem of Shannon asserts that χ'(G) ≤3Δ(G)/2; Vizing strengthened this by showing that χ'(G) ≤ Δ(G)+μ(G), where μ(G) is the maximum multiplicity of edges in G. When restricted to a simple graph G, Vizing's theorem implies χ'(G) = Δ(G) or Δ(G)+1. Therefore the edge-coloring problem on simple graphs can be approximated within one off the optimum. Surprisingly, this problem on multigraphs becomes much more difficult, and modern approaches often rely heavily on the following parameter Γ(G)=max {2|E(G[U])|/(|U|-1): U is a subset of V with size odd and at least 3}, where G[U] stands for the subgraph of G induced by U. Observe that if U has an odd size, then a matching in G[U] has at most (|U|-1)/2 edges. Consequently, χ'(G) ≥ Γ(G) and hence χ'(G) ≥ max {Δ(G), [Γ(G)]}, where [x] stands for the smallest integer at least x. Recently, Chen, Yu and the PI established that for any multigraph G, χ'(G) ≤ max {Δ(G)+\sqrt{Δ(G)/2}, [Γ(G)]}, which substantially improves all existing results on the edge-coloring problem on general multigraphs. The PI wishes to further his research on this topic. In this project he proposes 1. To prove the Goldberg-Seymour conjecture, which asserts that for any multigraph G, χ'(G) ≤ max {Δ(G)+1, [Γ(G)]}; 2. To study Vizing's interchange problem: Given a proper edge-coloring φ (not necessarily optimal) of a multigraph G, is it possible to obtain an edge-coloring of G with χ'(G) colors from φ simply by interchanging colors on Kempe chains? 3. To investigate the edge-coloring problem on plane multigraphs, and show that χ'(G)=max {Δ(G), [Γ(G)]} for any plane multigraph G.


Project Title:Optimal Packing and Covering
Investigator(s):Zang W
Department:Mathematics
Source(s) of Funding:General Research Fund (GRF)
Start Date:12/2009
Abstract:
To prove that the linear system defining the stable matching polytope is totally dual half-integral for any graph, and characterize all graphs for which such a linear system is totally dual integral; to establish that for any digraph G with a nonnegative integral weight w(e) on each arc e, the maximum number of dijoins (with repetition allowed) such that each arc e is used at most 2w(e) times by these dijoins is equal to the minimum total weight of a collection X of arcs (with repetition allowed) such that each dijoin in G meets X at least twice, and explore the connection between this statement and Woodall's min-max conjecture on packing and covering dijoins in the unweighted case; to derive a box-Mengerian analogue of Lovasz's theorem on Mengerian hypergraphs, and give various interesting applications of this proposed assertion in combinatorial optimization.


Project Title:Dual Integrality on Packing and Covering: Theory and Applications
Investigator(s):Zang W
Department:Mathematics
Source(s) of Funding:Matching Fund for NSFC Academic Collaboration Award
Start Date:01/2010
Abstract:
Dual Integrality on Packing and Covering: Theory and Applications


List of Research Outputs

Chen Z. and Zang W., Odd-K4's in Stability Critical Graphs, Discrete Mathematics. 2009, 309: 5982-5985.
Zang W., A Unified Approach to Box-Mengerian Hypergraphs, 2009 International and 5th Cross-strait Conference on Graph Theory and Combinatorics, Tianjin, China. 2009.
Zang W., A Unified Approach to Box-Mengerian Hypergraphs, The 23rd Cumberland Conference on Combinatorics, Graph Theory, and Computing, Mississippi, USA. 2010.
Zang W., In: Associate Editor, Operations Research (July 2009 -- Present). MD, USA, INFORMS, 2009.
Zang W., The Overseas and Hong Kong, Macau Young Scholars Collaborative Research Fund (this award is exactly the same as the previous "Outstanding Young Scholar Award", but the title has been changes to the present one.) , National Science Foundation of China. 2009.


Researcher : Zhang S
List of Research Outputs

Zhang S., Ching W.K., Tsing N.K., Leung H.Y. and Guo D., A New Multiple Regression Approach for the Construction of Genetic Regulatory Networks, Journal of Artificial Intelligence in Medicine. 2010, 48: 153-160.
Zhang S., Wang H., Zhou X., Hoehn G., DeGraba T., Gonzales D., Suffredini A., Ching W.K., Ng K.P. and Wong S.T.C., A Novel Peak Detection Approach with Chemical Noise Removal Using Short-Time FFT for prOTOF MS Data , Proteomics. 2009, 9: 3833-3842.
Zhang S., Ching W.K., Chen X. and Tsing N.K., Generating Probabilistic Boolean Networks from a Prescribed Stationary Distribution , Information Sciences. 2010, 180: 2560-2570.


Researcher : Zhao C
List of Research Outputs

Cheung W.S. and Zhao C., Some New Hilbert's Type Inequalities, Journal of Inequalities and Applications. 2009, 2009, Art.ID 851360: 10 pages.


Researcher : Zhao D
List of Research Outputs

Cheung W.S., Chen C. and Zhao D., Gronwall-Bellman-Type Integral Inequalities and Applications to BVPs, Journal of Inequalities and Applications. 2009, 2009, Art.ID 258569: 15 pages.


Researcher : Zhou W
List of Research Outputs

Yung S.P., Yam S.C.P. and Zhou W., Two rationales behind the 'buy-and-hold or sell-at-once' strategy, Journal of Applied Probability. 2009, 46, No.3: 651-668.


Researcher : Zhou X
List of Research Outputs

Zhang S., Wang H., Zhou X., Hoehn G., DeGraba T., Gonzales D., Suffredini A., Ching W.K., Ng K.P. and Wong S.T.C., A Novel Peak Detection Approach with Chemical Noise Removal Using Short-Time FFT for prOTOF MS Data , Proteomics. 2009, 9: 3833-3842.


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