HKU Scholars Hubhttp://hub.hku.hkThe DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Wed, 05 Aug 2020 14:15:52 GMT2020-08-05T14:15:52Z50491- Global State, Boolean Modelhttp://hdl.handle.net/10722/198728Title: Global State, Boolean Model
Authors: Chen, X; Ching, WK; Tsing, NK
Tue, 01 Jan 2013 00:00:00 GMThttp://hdl.handle.net/10722/1987282013-01-01T00:00:00Z
- Synchronous Modelhttp://hdl.handle.net/10722/198731Title: Synchronous Model
Authors: Chen, X; Ching, WK; Tsing, NK
Tue, 01 Jan 2013 00:00:00 GMThttp://hdl.handle.net/10722/1987312013-01-01T00:00:00Z
- Regulation Functionhttp://hdl.handle.net/10722/198727Title: Regulation Function
Authors: Chen, X; Ching, WK; Tsing, NK
Tue, 01 Jan 2013 00:00:00 GMThttp://hdl.handle.net/10722/1987272013-01-01T00:00:00Z
- Boolean Modelhttp://hdl.handle.net/10722/198730Title: Boolean Model
Authors: Chen, X; Ching, WK; Tsing, NK
Abstract: Boolean network (BN) is known as a popular mathematical model for modeling genetic regulatory networks. The BN model was first proposed by Kauffman (1969). In a BN model, the gene expression states are quantized into only two levels: on and off (represented as 1 and 0). The target gene is determined by several other genes called its input genes according to regulation rules (given as Boolean functions). A BN is said to be well defined when all the input genes and Boolean functions are given (Kauffman (1993)). There are two types of BN models: synchronous BNs and asynchronous BNs, depending on whether or not the states of nodes are updated synchronously. Synchronous model is more popular and easier to analyze and therefore we adopt it in our discussion. We note that a BN model is a deterministic model and the only randomness comes from its initial state. Considering the inherent deterministic directional
Tue, 01 Jan 2013 00:00:00 GMThttp://hdl.handle.net/10722/1987302013-01-01T00:00:00Z
- Probabilistic Boolean Networkshttp://hdl.handle.net/10722/198729Title: Probabilistic Boolean Networks
Authors: Chen, X; Ching, WK; Tsing, NK
Tue, 01 Jan 2013 00:00:00 GMThttp://hdl.handle.net/10722/1987292013-01-01T00:00:00Z
- Convexity of the largest singular value of eDMe-D: A convexity lemmahttp://hdl.handle.net/10722/156056Title: Convexity of the largest singular value of eDMe-D: A convexity lemma
Authors: Tsing, NamKiu
Abstract: A rigorous proof is given for a convexity lemma used by C. C. Chu and J. C. Doyle (1986) to prove the convexity of the largest singular value of eDMe-D with respect to D on a commuting, convex subset of matrices.
Mon, 01 Jan 1990 00:00:00 GMThttp://hdl.handle.net/10722/1560561990-01-01T00:00:00Z
- On the design of robust deadbeat regulatorshttp://hdl.handle.net/10722/46599Title: On the design of robust deadbeat regulators
Authors: Lam, J; Tso, HK; Tsing, NK
Abstract: This paper considers the synthesis of state feedback gains which provide robustness against perturbation in deadbeat regulation. It is formulated as an unconstrained optimization problem. Through a posteriori perturbation analysis of the closed-loop eigenvalues, the justification of the use of a new objective function to measure the robustness of deadbeat systems is established. The objective function does not require the computation of eigenvectors and has simple analytical gradient and Hessian. A numerical example is employed to illustrate the effectiveness of the proposed method.
Mon, 01 Jan 1996 00:00:00 GMThttp://hdl.handle.net/10722/465991996-01-01T00:00:00Z
- Inclusion relations on eigenvalues of principal submatriceshttp://hdl.handle.net/10722/100336Title: Inclusion relations on eigenvalues of principal submatrices
Authors: Tsing, NK
Sat, 01 Jan 1994 00:00:00 GMThttp://hdl.handle.net/10722/1003361994-01-01T00:00:00Z
- Numerical ranges on indefinite inner product spaceshttp://hdl.handle.net/10722/100370Title: Numerical ranges on indefinite inner product spaces
Authors: Tsing, NK
Sun, 01 Jan 1995 00:00:00 GMThttp://hdl.handle.net/10722/1003701995-01-01T00:00:00Z
- The constrained bilinear form and the C-numerical rangehttp://hdl.handle.net/10722/156092Title: The constrained bilinear form and the C-numerical range
Authors: Tsing, NK
Abstract: Let V be an n-dimentional unitary space with inner product (·,·) and S the set {x∈V:(x, x)=1}. For any A∈Hom(V, V) and q∈C with {divides}q{divides}≤1, we define W(A:q)={(Ax, y):x, y∈S, (x, y)=q}. If q=1, then W(A:q) is just the classical numerical range {(Ax, x):x∈S}, the convexity of which is well known. Another generalization of the numerical range is the C-numerical range, which is defined to be the set WC(A)={tr(CU*AU):U unitary} where C∈Hom(V, V). In this note, we prove that W(A:q) is always convex and that WC(A) is convex for all A if rank C=1 or n=2. © 1984.
Sun, 01 Jan 1984 00:00:00 GMThttp://hdl.handle.net/10722/1560921984-01-01T00:00:00Z
- On finite-horizon control of genetic regulatory networks with multiple hard-constraintshttp://hdl.handle.net/10722/124815Title: On finite-horizon control of genetic regulatory networks with multiple hard-constraints
Authors: Yang, C; WaiKi, C; NamKiu, T; HoYin, L
Abstract: Background: Probabilistic Boolean Networks (PBNs) provide a convenient tool for studying genetic regulatory networks. There are three major approaches to develop intervention strategies: (1) resetting the state of the PBN to a desirable initial state and letting the network evolve from there, (2) changing the steady-state behavior of the genetic network by minimally altering the rule-based structure and (3) manipulating external control variables which alter the transition probabilities of the network and therefore desirably affects the dynamic evolution. Many literatures study various types of external control problems, with a common drawback of ignoring the number of times that external control(s) can be applied.Results: This paper studies the intervention problem by manipulating multiple external controls in a finite time interval in a PBN. The maximum numbers of times that each control method can be applied are given. We treat the problem as an optimization problem with multi-constraints. Here we introduce an algorithm, the "Reserving Place Algorithm'', to find all optimal intervention strategies. Given a fixed number of times that a certain control method is applied, the algorithm can provide all the sub-optimal control policies. Theoretical analysis for the upper bound of the computational cost is also given. We also develop a heuristic algorithm based on Genetic Algorithm, to find the possible optimal intervention strategy for networks of large size. . Conclusions: Studying the finite-horizon control problem with multiple hard-constraints is meaningful. The problem proposed is NP-hard. The Reserving Place Algorithm can provide more than one optimal intervention strategies if there are. Moreover, the algorithm can find all the sub-optimal control strategies corresponding to the number of times that certain control method is conducted. To speed up the computational time, a heuristic algorithm based on Genetic Algorithm is proposed for genetic networks of large size. © 2010 Wai-Ki et al; licensee BioMed Central Ltd.
Fri, 01 Jan 2010 00:00:00 GMThttp://hdl.handle.net/10722/1248152010-01-01T00:00:00Z
- A Heuristic Method for Generating Probabilistic Boolean Networks from a Prescribed Transition Probability Matrixhttp://hdl.handle.net/10722/64485Title: A Heuristic Method for Generating Probabilistic Boolean Networks from a Prescribed Transition Probability Matrix
Authors: Ching, WK; Chen, X; Leung, HY; Tsing, NK
Abstract: Probabilistic Boolean Networks (PBNs) have received much attention for modeling genetic
regulatory networks. In this paper, we propose efficient algorithms for constructing a probabilistic
Boolean network when its transition probability matrix is given. This is an important inverse
problem in network inference from steady-state data, as most microarray data sets are assumed to
be obtained from sampling the steady-state.
Tue, 01 Jan 2008 00:00:00 GMThttp://hdl.handle.net/10722/644852008-01-01T00:00:00Z
- Generating probabilistic Boolean networks from a prescribed stationary distributionhttp://hdl.handle.net/10722/75296Title: Generating probabilistic Boolean networks from a prescribed stationary distribution
Authors: Zhang, SQ; Ching, WK; Chen, X; Tsing, NK
Abstract: Modeling gene regulation is an important problem in genomic research. Boolean networks (BN) and its generalization probabilistic Boolean networks (PBNs) have been proposed to model genetic regulatory interactions. BN is a deterministic model while PBN is a stochastic model. In a PBN, on one hand, its stationary distribution gives important information about the long-run behavior of the network. On the other hand, one may be interested in system synthesis which requires the construction of networks from the observed stationary distribution. This results in an inverse problem which is ill-posed and challenging. Because there may be many networks or no network having the given properties and the size of the inverse problem is huge. In this paper, we consider the problem of constructing PBNs from a given stationary distribution and a set of given Boolean Networks (BNs). We first formulate the inverse problem as a constrained least squares problem. We then propose a heuristic method based on Conjugate Gradient (CG) algorithm, an iterative method, to solve the resulting least squares problem. We also introduce an estimation method for the parameters of the PBNs. Numerical examples are then given to demonstrate the effectiveness of the proposed methods. © 2010 Elsevier Inc. All rights reserved.
Fri, 01 Jan 2010 00:00:00 GMThttp://hdl.handle.net/10722/752962010-01-01T00:00:00Z
- The star-shapedness of a generalized numerical rangehttp://hdl.handle.net/10722/227320Title: The star-shapedness of a generalized numerical range
Authors: LAU, PS; Ng, TW; Tsing, NK
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/10722/2273202016-01-01T00:00:00Z
- Convexity and star-shapedness of real linear images of special orthogonal orbitshttp://hdl.handle.net/10722/227321Title: Convexity and star-shapedness of real linear images of special orthogonal orbits
Authors: LAU, PS; Ng, TW; Tsing, NK
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/10722/2273212016-01-01T00:00:00Z
- Weighted local least squares imputation method for missing value estimationhttp://hdl.handle.net/10722/100380Title: Weighted local least squares imputation method for missing value estimation
Authors: Ching, WK; Cheng, KW; Li, L; Tsing, NK; Wong, AST
Abstract: Missing values often exist in the data of gene expression microarray
experiments. A number of methods such as the Row Average (RA) method,
KNNimpute algorithm and SVDimpute algorithm have been proposed to estimate
the missing values. Recently, Kim et al. proposed a Local Least Squares
Imputation (LLSI) method for estimating the missing values. In this paper,
we propose a Weighted Local Least Square Imputation (WLLSI) method
for missing values estimation. WLLSI allows training on the weighting and
therefore can take advantage of both the LLSI method and the RA method.
Numerical results on both synthetic data and real microarray data are given
to demonstrate the effectiveness of our proposed method. The imputation
methods are then applied to a breast cancer dataset.
Mon, 01 Jan 2007 00:00:00 GMThttp://hdl.handle.net/10722/1003802007-01-01T00:00:00Z
- Optimal control policy for probabilistic Boolean networks with hard constraintshttp://hdl.handle.net/10722/58975Title: Optimal control policy for probabilistic Boolean networks with hard constraints
Authors: Ching, WK; Zhang, SQ; Jiao, Y; Akutsu, T; Tsing, NK; Wong, AS
Abstract: It is well known that the control/intervention of some genes in a genetic regulatory network is useful for avoiding undesirable states associated with some diseases like cancer. For this purpose, both optimal finite-horizon control and infinite-horizon control policies have been proposed. Boolean networks (BNs) and its extension probabilistic Boolean networks (PBNs) as useful and effective tools for modelling gene regulatory systems have received much attention in the biophysics community. The control problem for these models has been studied widely. The optimal control problem in a PBN can be formulated as a probabilistic dynamic programming problem. In the previous studies, the optimal control problems did not take into account the hard constraints, i.e. to include an upper bound for the number of controls that can be applied to the captured PBN. This is important as more treatments may bring more side effects and the patients may not bear too many treatments. A formulation for the optimal finite-horizon control problem with hard constraints introduced by the authors. This model is state independent and the objective function is only dependent on the distance between the desirable states and the terminal states. An approximation method is also given to reduce the computational cost in solving the problem. Experimental results are given to demonstrate the efficiency of our proposed formulations and methods. © The Institution of Engineering and Technology 2009.
Thu, 01 Jan 2009 00:00:00 GMThttp://hdl.handle.net/10722/589752009-01-01T00:00:00Z
- Repulsion Pressure Model and Numerical Simulation for Spiral Phyllotactic Patterns of Plantshttp://hdl.handle.net/10722/185941Title: Repulsion Pressure Model and Numerical Simulation for Spiral Phyllotactic Patterns of Plants
Authors: CONG, Y; Ching, WK; Tsing, NK
Tue, 01 Jan 2013 00:00:00 GMThttp://hdl.handle.net/10722/1859412013-01-01T00:00:00Z
- A modified entropy approach for construction of probabilistic boolean networkshttp://hdl.handle.net/10722/128313Title: A modified entropy approach for construction of probabilistic boolean networks
Authors: Chen, X; Li, L; Ching, WK; Tsing, NK
Abstract: Boolean Network (BN) and its extension Probabilistic Boolean network (PBN) have received much attention in modeling genetic regulatory networks. In this paper, we consider the problem of constructing a PBN from a given positive stationary distribution. The problem can be divided into two subproblems: Construction of a PBN from a given sparse transition probability matrix and construction of a sparse transition matrix from a given stationary distribution. These are inverse problems of huge sizes and we proposed mathematical models based on entropy theory. To obtain a sparse solution, we consider a new objective function having an addition term of La-norm. Newton’s method in conjunction with CG method is then applied to solve the inverse problem. Numerical examples are given to demonstrate the effectiveness of our proposed method.
Fri, 01 Jan 2010 00:00:00 GMThttp://hdl.handle.net/10722/1283132010-01-01T00:00:00Z
- Construction of Probabilistic Boolean Networks from a Prescribed Transition Probability Matrix: A Maximum Entropy Rate Approachhttp://hdl.handle.net/10722/133320Title: Construction of Probabilistic Boolean Networks from a Prescribed Transition Probability Matrix: A Maximum Entropy Rate Approach
Authors: Chen, X; Ching, WK; Chen, XS; Cong, Y; Tsing, NK
Abstract: Modeling genetic regulatory networks is an important problem in genomic research. Boolean Networks (BNs) and their extensions Probabilistic Boolean Networks (PBNs) have been proposed for modeling genetic regulatory interactions. In a PBN, its steady-state distribution gives very important information about the long-run behavior of the whole network. However, one is also interested in system synthesis which requires the construction of networks. The inverse problem is ill-posed and challenging, as there may be many networks or no network having the given properties, and the size of the problem is huge. The construction of PBNs from a given transition-probability matrix and a given set of BNs is an inverse problem of huge size. We propose a maximum entropy approach for the above problem. Newton's method in conjunction with the Conjugate Gradient (CG) method is then applied to solving the inverse problem. We investigate the convergence rate of the proposed method. Numerical examples are also given to demonstrate the effectiveness of our proposed method.
Sat, 01 Jan 2011 00:00:00 GMThttp://hdl.handle.net/10722/1333202011-01-01T00:00:00Z
- Linear operators leaving a class of matrices with fixed singular values invarianthttp://hdl.handle.net/10722/75315Title: Linear operators leaving a class of matrices with fixed singular values invariant
Authors: Li, CK; Tsing, NK
Fri, 01 Jan 1993 00:00:00 GMThttp://hdl.handle.net/10722/753151993-01-01T00:00:00Z
- The numerical range of derivationshttp://hdl.handle.net/10722/156146Title: The numerical range of derivations
Authors: Li, CK; Tsing, NK
Abstract: Let p, q, n be integers satisfying 1 ≤ p ≤ q ≤ n. The (p, q)-numerical range of an n×n complex matrix A is defined by Wp,q(A) = {Ep((UAU*)[q]): U unitary}, where for an n×n complex matrix X, X[q] denotes its q×q leading principal submatrix and Ep(X[q]) denotes the pth elementary symmetric function of the eigenvalues of X[q]. When 1 = p = q, the set reduces to the classical numerical range of A, which is well known to be convex. Many authors have used the concept of classical numerical range to study different classes of matrices. In this note we extend the results to the generalized cases. Besides obtaining new results, we collect existing ones and give alternative proofs for some of them. We also study the (p,q)-numerical radius of A defined by rp,q(A) = max{|μ|:μ ∈ Wp,q(A)}. © 1989.
Sun, 01 Jan 1989 00:00:00 GMThttp://hdl.handle.net/10722/1561461989-01-01T00:00:00Z
- Duality between some linear preserver problems. III. c-spectral norms and (skew)-symmetric matrices with fixed singular valueshttp://hdl.handle.net/10722/156217Title: Duality between some linear preserver problems. III. c-spectral norms and (skew)-symmetric matrices with fixed singular values
Authors: Li, CK; Tsing, NK
Abstract: Let F denote either the complex field C or the real field R. Let V be Sn(F) or Kn(F), the vector spaces of all n × n symmetric and skew-symmetric matrices, respectively, over F. For c=(c1,...,cn)≠0 with c1≥ ⋯ ≥cn≥0, the c-spectral norm of a matrix A∈V is the quantity {norm of matrix}A{norm of matrix}c = ∑ i=l nciσi(A), where σ1(A)≥ ⋯ ≥σn(A) are the singular values of A. Let d=(d1,...,dn)≠0 with d1≥ ⋯ ≥dn≥0. We study the linear isometries between the normed spaces (V,{norm of matrix}·{norm of matrix}c) and (V,{norm of matrix}·{norm of matrix}d), by using the fact that they are dual transformations of the linear operators which map ∑(d) onto ∑(c), where ∑(c) = {X∈V:X has singular values c1,...,cn}. It is shown that such isometries (and hence their dual transformations) exist if and only if c and d are scalar multiples of each other. In such case, we completely determine the structure of such isometries, and prove that they and their dual transformations belong to a same class of operators. In the proof, we obtain characterizations of the extreme points of the unit ball in V (for different cases) with respect to {norm of matrix}·{norm of matrix}c, which is of independent interest. © 1991.
Tue, 01 Jan 1991 00:00:00 GMThttp://hdl.handle.net/10722/1562171991-01-01T00:00:00Z
- Linear maps relating different unitary similarity orbits or different generalized numerical rangeshttp://hdl.handle.net/10722/156100Title: Linear maps relating different unitary similarity orbits or different generalized numerical ranges
Authors: Li, CK; Tsing, NK
Abstract: Let M be the complex linear space Mn of n × n complex matrices or the real linear space Hn of n × n hermitian matrices. For C ∈ M, its unitary similarity orbit is the set. U(C) = {UCU*; U unitary. and its circular unitary similarity orbit is the set. V(C) = {μX : μ ∈ F, |μ| = 1, X ∈ U(C)}. where F is the scalar field C or R according as M = Mn or M = Hn. Related to U(C) and V(C) are the C-numerical range and the C-numerical radius of A ∈ M defined by. Wc(A) = {tr(AX) : X ∈U(C)}. and. rC(A)=max{|z|:z∈WC(A)},. respectively. Let C, D ∈ Hn, we study the linear operators T on M satisfying one of the following properties: (I) WD(T(A)) = WC(A) for all A ∈ M, (II) rD(T(A)) = rC(A) for all A ∈ M, (III) T(U(D)) = U(C), (IV) T(V(D)) = V(C). In particular, we determine the conditions on C and D for the existence of a linear operator T on M satisfying any one of the conditions (I)-(IV), and characterize such an operator if it exists. © 1995.
Sun, 01 Jan 1995 00:00:00 GMThttp://hdl.handle.net/10722/1561001995-01-01T00:00:00Z
- Linear preserver problems: A brief introduction and some special techniqueshttp://hdl.handle.net/10722/156032Title: Linear preserver problems: A brief introduction and some special techniques
Authors: Li, CK; Tsing, NK
Abstract: Linear preserver problems concern the characterization of linear operators on matrix spaces that leave certain functions, subsets, relations, etc., invariant. The earliest papers on linear preserver problems date back to 1897, and a great deal of effort has been devoted to the study of this type of question since then. We present a brief picture of the subject, aiming at giving a gentle introduction to the reader. Then we describe some techniques used in our recent papers on this type of problem. © 1992.
Wed, 01 Jan 1992 00:00:00 GMThttp://hdl.handle.net/10722/1560321992-01-01T00:00:00Z
- A calculus E-learning system for first-year university students with diverse mathematics backgroundhttp://hdl.handle.net/10722/230472Title: A calculus E-learning system for first-year university students with diverse mathematics background
Authors: Yeung, KF; Lui, RKW; Cheung, MY; Lam, KF; Tsing, NK
Abstract: First-year science majors at the University of Hong Kong have different levels of proficiency in mathematics, with a significant proportion lacking the necessary calculus background for a compulsory first-year science course. A supplementary calculus e-learning platform was implemented so that students lacking the prerequisite background could gain the necessary skills and knowledge. Participation in the e-learning platform was voluntary. An analysis of the results showed that, among students lacking the prerequisite background, a high usage of the e-learning platform is associated with higher post-test scores. The high-usage students were also able to perform as well as students with the prerequisite background in the post-test.
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/10722/2304722016-01-01T00:00:00Z
- On star-centers of some generalized numerical ranges and diagonals of normal matriceshttp://hdl.handle.net/10722/75128Title: On star-centers of some generalized numerical ranges and diagonals of normal matrices
Authors: Cheung, G; Tsing, NK
Abstract: For any n×n matrices A and C, we consider the star-centers of three sets, namely, the C-numerical range WC(A) of A, the set diagU(A) of diagonals of matrices in the unitary orbit of A, and the set S(A) of matrices whose C-numerical ranges are contained in WC(A) for all C. For normal matrices A, we show that the set of star-centers of WA*(A) is a bounded closed real interval, and give complete description of the sets of star-centers of diagU(A) and of S(A). In particular, we show that if A is normal with noncollinear eigenvalues, then each of S(A) and diagU(A) has exactly one star-center. For general square matrices A, we also give sufficient conditions for the sets of star-centers of diagU(A) and of S(A) to be singleton sets. © 2001 Elsevier Science Inc.
Mon, 01 Jan 2001 00:00:00 GMThttp://hdl.handle.net/10722/751282001-01-01T00:00:00Z
- The C-numerical range of matrices is star-shapedhttp://hdl.handle.net/10722/75146Title: The C-numerical range of matrices is star-shaped
Authors: Cheung, WS; Tsing, NK
Abstract: Let A, C be n x n complex matrices. We prove in the affirmative the conjecture that the C-numerical range of A, defined by Wc(A) = {tr(Cu* AU) : U is unitary}, is always star-shaped with respect to star-center (tr A)(tr C)/n. This result is equivalent to that the image of the unitary orbit {U* AU : U is unitary} of A under any complex linear functional is always star-shaped. © 1996 OPA (Overseas Publishers Association) Amsterdam B.V. Published in The Netherlands Under license by Gordon and Breach Science Publishers.
Mon, 01 Jan 1996 00:00:00 GMThttp://hdl.handle.net/10722/751461996-01-01T00:00:00Z
- Robust deadbeat regulationhttp://hdl.handle.net/10722/75240Title: Robust deadbeat regulation
Authors: Lam, J; Tso, HK; Tsing, NK
Abstract: The synthesis of state feedback gains that provide robustness against perturbation in deadbeat regulation is considered. The problem is formulated as an unconstrained optimization problem. Through a posteriori perturbation analysis of the closed-loop eigenvalues, the justification of the use of a new objective function to measure the robustness of deadbeat systems is established. The objective function does not require the computation of eigenvectors, and has simple analytical gradient and hessian. A numerical example is employed to illustrate the effectiveness of the proposed method.
Wed, 01 Jan 1997 00:00:00 GMThttp://hdl.handle.net/10722/752401997-01-01T00:00:00Z
- Norm hull of vectors and matriceshttp://hdl.handle.net/10722/75283Title: Norm hull of vectors and matrices
Authors: Li, CK; Tsing, NK; Zhang, F
Abstract: Let V be a real or complex finite-dimensional vector space, and let script N sign be a set of norms on V. The norm hull of a vector x ∈ V with respect to script N sign is the set of vectors y ∈ V that satisfy ∥y∥ ≤ ∥x∥ for all ∥·∥ ∈ script N sign. We study and give characterization of the norm hull for some sets of well-known norms on general vector spaces, and for the set of algebra norms and the set of induced norms on the algebra of n × n real or complex matrices. © Elsevier Science Inc., 1997.
Wed, 01 Jan 1997 00:00:00 GMThttp://hdl.handle.net/10722/752831997-01-01T00:00:00Z
- On the norms used in computing the structured singular valuehttp://hdl.handle.net/10722/156055Title: On the norms used in computing the structured singular value
Authors: Tsing, NamKiu
Abstract: Different norms are considered to replace the Euclidean norm in an algorithm given by M. H. K. Fan and A. L. Tits (ibid., vol. 33, pp. 284-289, 1988), which is used for the computation of the structured singular value of any matrix. The algorithm is explained, and it is shown that the l1 norm is the best possible norm in a certain sense.
Mon, 01 Jan 1990 00:00:00 GMThttp://hdl.handle.net/10722/1560551990-01-01T00:00:00Z
- On some inverse problems in generating probabilistic boolean networkshttp://hdl.handle.net/10722/133762Title: On some inverse problems in generating probabilistic boolean networks
Authors: Chen, X; Ching, WK; Tsing, NK
Sat, 01 Jan 2011 00:00:00 GMThttp://hdl.handle.net/10722/1337622011-01-01T00:00:00Z
- When is the multiaffine image of a cube a convex polygon?http://hdl.handle.net/10722/75232Title: When is the multiaffine image of a cube a convex polygon?
Authors: Tsing, NK; Tits, AndrL
Abstract: We give two simple sufficient conditions under which the multiaffine image in the complex plane of an m-dimensional cube is a convex polygon. A third condition which, in some generic sense, is necessary and sufficient is then obtained. Our conditions involve checking the locations of the image of the vertices of the cube. These results help determine whether a parameterized family of polynomials th stable. © 1993.
Fri, 01 Jan 1993 00:00:00 GMThttp://hdl.handle.net/10722/752321993-01-01T00:00:00Z
- A multiple regression approach for building genetic networkshttp://hdl.handle.net/10722/100316Title: A multiple regression approach for building genetic networks
Authors: Zhang, SQ; Ching, WK; Tsing, NK; Leung, HY; Guo, DD
Abstract: The construction of genetic regulatory networks from time series gene expression data is an important research topic in bioinformatics as large amounts of quantitative gene expression data can be routinely generated nowadays. One of the main difficulties in building such genetic networks is that the data set has huge number of genes but small number of time points. In this paper, we propose a linear regression model for uncovering the relations among the genes by using multiple regression method with filtering. The model takes into account of the fact that real biological networks have the scale-free property. Based on this property and the statistical tests, a filter can be constructed and the interactions among the genes can be inferred by minimizing the distance between the observed data and the predicted data. Numerical examples based on yeast gene expression data are given to demonstrate our method. © 2008 IEEE.
Tue, 01 Jan 2008 00:00:00 GMThttp://hdl.handle.net/10722/1003162008-01-01T00:00:00Z
- Generating probabilistic Boolean networks from a prescribed transition probability matrixhttp://hdl.handle.net/10722/75388Title: Generating probabilistic Boolean networks from a prescribed transition probability matrix
Authors: Ching, WK; Chen, X; Tsing, NK
Abstract: Probabilistic Boolean networks (PBNs) have received much attention in modeling genetic regulatory networks. A PBN can be regarded as a Markov chain process and is characterised by a transition probability matrix. In this study, the authors propose efficient algorithms for constructing a PBN when its transition probability matrix is given. The complexities of the algorithms are also analysed. This is an interesting inverse problem in network inference using steady-state data. The problem is important as most microarray data sets are assumed to be obtained from sampling the steady-state. © 2009 © The Institution of Engineering and Technology.
Thu, 01 Jan 2009 00:00:00 GMThttp://hdl.handle.net/10722/753882009-01-01T00:00:00Z
- Norm-hulls of matriceshttp://hdl.handle.net/10722/100363Title: Norm-hulls of matrices
Authors: Tsing, NK
Sun, 01 Jan 1995 00:00:00 GMThttp://hdl.handle.net/10722/1003631995-01-01T00:00:00Z
- A Max-Min Principle for Phyllotactic Patternshttp://hdl.handle.net/10722/62181Title: A Max-Min Principle for Phyllotactic Patterns
Authors: Ching, WK; Cong, Y; Tsing, NK
Abstract: An interesting phenomenon about phyllotaxis is the divergence angle between two consecutive primordia. In this paper, we consider a dynamic model based on Max-Min principle for generating 2D phyllotactic patterns studied in [2,5]. Under the hypothesis that the influence of the two predecessors is enough to fix the birth place of the new generated primordium, analysis and numerical experiments are conducted. We then propose a new measurement for evaluating the pattern uniformity (sparsity) of different divergence angles. It is found that the golden angle gives very good sparsity but there are other angles give even better sparsity under our proposed measurement.
Thu, 01 Jan 2009 00:00:00 GMThttp://hdl.handle.net/10722/621812009-01-01T00:00:00Z
- G-invariant norms and G(c)-radiihttp://hdl.handle.net/10722/156108Title: G-invariant norms and G(c)-radii
Authors: Li, CK; Tsing, NK
Abstract: Let V be a finite dimensional inner product space over F(=R or C), and let G be a closed subgroup of the group of unitary operators on V. A norm or a seminorm ∥·∥ on V is said to be G-invariant if {norm of matrix}g(x){norm of matrix}=∥x∥ for all g ε{lunate} G, x ε{lunate} V. The concept of G-invariant norm specializes to many interesting particular cases such as the absolute norms on Fn, symmetric gauge functions on Rn, unitarily invariant norms on Fm×n, etc., which are of wide research interest. In this paper, we study the general properties of G-invariant norms. Our main strategy is to study G-invariant norms via the G(c)-radius rG(c)(·) on V defined by rG(c)(x) = max{|〈x, g(c)〉|:gε{lunate} G}, where c ε{lunate} V. It is shown that the G(c)-radii are very important G-invariant seminorms because every G-invariant norm or seminorm admits a representation in terms of them. As a result, one may focus attention on G(c)-radii in order to get results on G-invariant norms. We study the norm properties of G(c)-radii and obtain various results relating G-invariant norms and G(c)-radii. The linear operators on V that preserve G-invariant norms, G-invariant seminorms, or G(c)-radii are also investigated. Several open questions are mentioned. © 1991.
Tue, 01 Jan 1991 00:00:00 GMThttp://hdl.handle.net/10722/1561081991-01-01T00:00:00Z
- Linear maps preserving permutation and stochastic matriceshttp://hdl.handle.net/10722/75348Title: Linear maps preserving permutation and stochastic matrices
Authors: Tsing, NK; Li, CK; Tam, BS
Abstract: Let T be the set of n×n (sub)permutation matrices, doubly (sub)stochastic matrices, or the set of m×n column or row (sub)stochastic matrices. We characterize those linear maps T on the linear span of T that satisfy T(T)=T . Partial results concerning those linear maps T satisfying T(T) ⊆ T are also presented.
Tue, 01 Jan 2002 00:00:00 GMThttp://hdl.handle.net/10722/753482002-01-01T00:00:00Z
- A Weighted Local Least Squares Imputation method for missing value estimation in microarray gene expression datahttp://hdl.handle.net/10722/75469Title: A Weighted Local Least Squares Imputation method for missing value estimation in microarray gene expression data
Authors: Ching, WK; Li, L; Tsing, NK; Tai, CW; Ng, TW; Wong, AS; Cheng, KW
Abstract: Many clustering techniques and classification methods for analysing microarray data require a complete dataset. However, very often gene expression datasets contain missing values due to various reasons. In this paper, we first propose to use vector angle as a measurement for the similarity between genes. We then propose the Weighted Local Least Square Imputation (WLLSI) method for missing values estimation. Numerical results on both synthetic data and real microarray data indicate that WLLSI method is more robust. The imputation methods are then applied to a breast cancer dataset and interesting results are obtained. Copyright© 2010 Inderscience Enterprises Ltd.
Fri, 01 Jan 2010 00:00:00 GMThttp://hdl.handle.net/10722/754692010-01-01T00:00:00Z
- Linear operators preserving certain equivalence relations originating in system theoryhttp://hdl.handle.net/10722/156045Title: Linear operators preserving certain equivalence relations originating in system theory
Authors: Li, CK; Rodman, L; Tsing, NK
Abstract: Let F be C or R. A finite-dimensional linear time-invariant system is described in state-space form by [xdot] = Ax + Bu, y = Cx + Du, and is identified with the matrix 4-tuple (A,B,C,D), where x ε{lunate} Fn, u ε{lunate} Fm, yε{lunate} Fp, and A,B,C,D, are matrices of appropriate sizes and with entries in F. For fixed n,m,p, let M be the linear space of all systems (A,B,C,D). Equivalence relations ∼ can be defined on M based on the possibility of changes of basis inthe state space, the input space, or the output space, and the possibility of state feedback and/or output feedback. We characterize those nonsingular linear operators φ on M that satisfy φ(X) ∼ φ(Y) whenever X ∼ Y. © 1992.
Wed, 01 Jan 1992 00:00:00 GMThttp://hdl.handle.net/10722/1560451992-01-01T00:00:00Z
- Weighted Local Least Squares Imputation Method for Missing Value Estimationhttp://hdl.handle.net/10722/119225Title: Weighted Local Least Squares Imputation Method for Missing Value Estimation
Authors: Ching, WK; Cheng, K; Li, L; Tsing, NK; Wong, AST
Abstract: Missing values often exist in the data of gene expression microarray experiments. A
number of methods such as the Row Average (RA) method, KNNimpute algorithm and SVDimpute
algorithm have been proposed to estimate the missing values. Recently, Kim et al. proposed a
Local Least Squares Imputation (LLSI) method for estimating the missing values. In this paper, we
propose a Weighted Local Least Square Imputation (WLLSI) method for missing values estimation.
WLLSI allows training on the weighting and therefore can take advantage of both the LLSI method
and the RA method. Numerical results on both synthetic data and real microarray data are given to
demonstrate the effectiveness of our proposed method. The imputation methods are then applied to
a breast cancer dataset.
Description: The First International Symposium, OSB'07, Beijing, China, August 8-10, 2007, Proceedings
Mon, 01 Jan 2007 00:00:00 GMThttp://hdl.handle.net/10722/1192252007-01-01T00:00:00Z
- A new multiple regression approach for the construction of genetic regulatory networkshttp://hdl.handle.net/10722/75188Title: A new multiple regression approach for the construction of genetic regulatory networks
Authors: Zhang, SQ; Ching, WK; Tsing, NK; Leung, HY; Guo, D
Abstract: Objective: Re-construction of a genetic regulatory network from a given time-series gene expression data is an important research topic in systems biology. One of the main difficulties in building a genetic regulatory network lies in the fact that practical data set has a huge number of genes vs. a small number of sampling time points. In this paper, we propose a new linear regression model that may overcome this difficulty for uncovering the regulatory relationship in a genetic network. Methods: The proposed multiple regression model makes use of the scale-free property of a real biological network. In particular, a filter is constructed by using this scale-free property and some appropriate statistical tests to remove redundant interactions among the genes. A model is then constructed by minimizing the gap between the observed and the predicted data. Results: Numerical examples based on yeast gene expression data are given to demonstrate that the proposed model fits the practical data very well. Some interesting properties of the genes and the underlying network are also observed. Conclusions: In conclusion, we propose a new multiple regression model based on the scale-free property of real biological network for genetic regulatory network inference. Numerical results using yeast cell cycle gene expression dataset show the effectiveness of our method. We expect that the proposed method can be widely used for genetic network inference using high-throughput gene expression data from various species for systems biology discovery. © 2009 Elsevier B.V.
Fri, 01 Jan 2010 00:00:00 GMThttp://hdl.handle.net/10722/751882010-01-01T00:00:00Z
- On analyticity of functions involving eigenvalueshttp://hdl.handle.net/10722/75426Title: On analyticity of functions involving eigenvalues
Authors: Tsing, NK; Fan, MKH; Verriest, EI
Abstract: Let A(z) be an n × n complex matrix whose elements depend analytically on z ∈ Cm. It is well known that any individual eigenvalue of A(z) may be nondifferentiable when it coalesces with others. In this paper, we investigate the analycity property of functions on the eigenvalues λ(z) = (λ1(z),..., λn(z)) of A(z). We first introduce the notion of functions that are symmetric with respect to partitions. It is then shown that if a function f{hook} : Cn → C is analytic at λ(a), where a ε{lunate} Cm, and is symmetric with respect to a certain partition induced by λ(a), then the composite function g(z) = f{hook}(λ1(z),...,λn(z)) is analytic at a. When z is real, A(z) is symmetric or Hermitian, and the aforementioned assumptions hold, so that g(z) is analytic at a, we also derive formulae for its first and second order partial derivatives. We apply the results to several problems involving eigenvalues. © 1994.
Sat, 01 Jan 1994 00:00:00 GMThttp://hdl.handle.net/10722/754261994-01-01T00:00:00Z
- A Genetic Algorithm for Optimal Control of Probabilistic Boolean Networkshttp://hdl.handle.net/10722/64489Title: A Genetic Algorithm for Optimal Control of Probabilistic Boolean Networks
Authors: Ching, WK; Leung, HY; Zhang, S; Tsing, NK
Abstract: We study the problem of finding optimal control policies for Probabilistic Boolean Networks
(PBNs). Boolean Networks (BNs) and PBNs are effective tools for modeling genetic regulatory
networks. A PBN is a collection of BNs driven by a Markov chain process. 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 as a probabilistic
dynamic programming problem. However, due to the curse of dimensionality, the complexity
of the problem is huge. The main objective of this paper is to introduce a Genetic Algorithm (GA)
approach for the optimal control problem. Numerical results are given to demonstrate the efficiency
of our proposed GA method.
Tue, 01 Jan 2008 00:00:00 GMThttp://hdl.handle.net/10722/644892008-01-01T00:00:00Z
- Duality between some linear preserver problems. II. Isometries with respect to c-special norms and matrices with fixed singular valueshttp://hdl.handle.net/10722/156225Title: Duality between some linear preserver problems. II. Isometries with respect to c-special norms and matrices with fixed singular values
Authors: Li, CK; Tsing, NK
Abstract: Let Fm×n (m≤n) denote the linear space of all m × n complex or real matrices according as F=C or R. Let c=(c1,...,cm)≠0 be such that c1≥⋯≥cm≥0. The c-spectral norm of a matrix Aε{lunate}Fm×n is the quantity {norm of matrix}A{norm of matrix}c ∑ i=I mciσi(A). where σ1(A)≥⋯≥σm(A) are the singular values of A. Let d=(d1,...,dm)≠0, where d1≥⋯≥dm≥0. We consider the linear isometries between the normed spaces (Fm×n,∥·∥c) and (Fm×n,∥·∥d), and prove that they are dual transformations of the linear operators which map L(d) onto L(c), where L(c)= {Xε{lunate}Fm×n:X has singular values c1,...,cm}. © 1988.
Sat, 01 Jan 1983 00:00:00 GMThttp://hdl.handle.net/10722/1562251983-01-01T00:00:00Z
- Linear operators preserving the (p,q)-numerical rangehttp://hdl.handle.net/10722/156226Title: Linear operators preserving the (p,q)-numerical range
Authors: Li, CK; Tam, BS; Tsing, NK
Abstract: Let Cn×n and Hn denote respectively the space of n×n complex matrices and the real space of n×n hermitian matrices. Let p,q,n be positive integers such that p≤q≤n. For Aε{lunate}Cn×n, the (p,q)-numerical range of A is the set Wp,q(A)={trCp(JqUAU*):U unitary}, where Cp(X) is the pth compound matrix of X, and Jq is the matrix Iq{equivalent to}On-q. Let L denote Hn or Cn×n. The problem of determining all linear operators T: L→L such that Wp,q(T(A))=Wp,q(A) for all Aε{lunate}L is treated in this paper. © 1988.
Sat, 01 Jan 1983 00:00:00 GMThttp://hdl.handle.net/10722/1562261983-01-01T00:00:00Z
- Numerical ranges of an operator on an indefinite inner product spacehttp://hdl.handle.net/10722/156044Title: Numerical ranges of an operator on an indefinite inner product space
Authors: Li, CK; Tsing, NK; Uhlig, F
Abstract: For n x n complex matrices A and an n x n Hermitian matrix S, we consider the S-numerical range of A and the positive S-numerical range of A defined by WS(A) = {〈Av, v〉S/〈v, v〉S : v ∈ ℂn, 〈v, v〉S ≠ 0} and W S + (A) = {〈Av, v〉S : v ∈ ℂn, 〈v, v〉S = 1}, respectively, where 〈u, v〉S = v*Su. These sets generalize the classical numerical range, and they are closely related to the joint numerical range of three Hermitian forms and the cone generated by it. Using some theory of the joint numerical range we can give a detailed description of WS(A) and WS + (A) for arbitrary Hermitian matrices S. In particular, it is shown that WS + (A) is always convex and WS(A) is always p-convex for all S. Similar results are obtained for the sets VS(A) = {〈Av, v〉/〈Sv, v〉: v ∈ ℂn, 〈Sv, v〉 ≠ 0}, VS + (A) = {〈Av, v〉: v ∈ ℂn, 〈Sv, v〉 = 1}, where 〈u, v〉 = v* u. Furthermore, we characterize those linear operators preserving WS(A), WS + (A), V S(A), or VS + (A). Possible generalizations of our results, including their extensions to bounded linear operators on an infinite dimensional Hilbert or Krein space, are discussed.
Mon, 01 Jan 1996 00:00:00 GMThttp://hdl.handle.net/10722/1560441996-01-01T00:00:00Z
- Spectral Analysis for HSS Preconditionershttp://hdl.handle.net/10722/75350Title: Spectral Analysis for HSS Preconditioners
Authors: Chan, LC; Ng, MK; Tsing, NK
Abstract: In this paper, we are interested in HSS preconditioners for saddle point linear systems with a nonzero (2,2)-th block. We study an approximation of the spectra of HSS preconditioned matrices and use these results to illustrate and explain the spectra obtained from numerical examples, where the previous spectral analysis of HSS preconditioned matrices does not cover.
Tue, 01 Jan 2008 00:00:00 GMThttp://hdl.handle.net/10722/753502008-01-01T00:00:00Z