Nam-Kiu Tsing

Linear preserver problems: A brief introduction and some special techniques

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.

Optimal control policy for probabilistic Boolean networks with hard constraints

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 constrained bilinear form and the C-numerical range

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 ∣q∣⩽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 W C (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.

On analyticity of functions involving eigenvalues

Abstract Let A(z) be an n × n complex matrix whose elements depend analytically on z ∈ C m. 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 ƒ : C n → C is analytic at λ(a), where a ϵ C m, and is symmetric with respect to a certain partition induced by λ(a), then the composite function g(z) = ƒ(λ 1 (z),…,λ n (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.

On the shape of the generalized numerical ranges

Let A be an n × n complex matrix. For 1 ≤ k ≤ n, let Λ k be the set of all k-tuples of orthonormal vectors in ¢ n and ck = (γ1,…,γ k ) e ¢ k . In this note, it is proved that the sets and are star-shaped.

Numerical ranges of an operator on an indefinite inner product space

For n n complex matrices A and an n n Hermitian matrix S, we consider the S-numerical range of A and the positive S-numerical range of A de ned by WS(A) = hAv; viS hv; viS : v 2 I Cn; hv; viS 6= 0

The C-numerical range of matrices is star-shaped

Let A C be n× ncomplex matrices. We prove in the affirmative the conjecture that the C-numerical range of A, defined by 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} of A under any complex linear functional is always star-shaped.

The generalized spectral radius, numerical radius and spectral norm

Given n×n complex matrices A, C with eigenvalues αj, γj, 1 ⩾ j ⩾ n, respectively, we have the relation where and respectively are the generalized spectral radius, generalized numerical radius and generalized spectral norm of A with respect to C. For C = diag(1,0,…,0), it reduces to the classical relation In this note, we investigate matrices for which . The norm properties of are also studied.

Linear operators preserving unitarily invariant norms of matrices

Let F m × n be the set of all m × n matrices over the field F = C or R Denote by Un (F) the group of all n × n unitary or orthogonal matrices according as F = C or F-R. A norm N() on F m ×n, is unitarily invariant if N(UAV) = N(A): for all A ∈ F m×n U ∈ U m (F). and V ∈ Un (F). We characterize those linear operators T F m × n → F m × n which satisfy N (T(A)) = N(A)for all A ∈ F m × n for a given unitarily invariant norm N(). It is shown that the problem is equivalent to characterizing those operators which preserve certain subsets in F m × n To develop the theory we prove some results concerning unitary operators on F m × n which are of independent interest.

A new multiple regression approach for the construction of genetic regulatory networks

OBJECTIVE Reconstruction 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.