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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.

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.

Let H n be the set of all n × n Hermitian matrices and H n m be the set of all m-tuples of n × n Hermitian matrices. For A = ( A 1 , . . . , A m ) ∈ H n m and for any linear map L : H n m → R l , we define the L-numerical range of A by W L ( A ) : = { L ( U ⁎ A 1 U , . . . , U ⁎ A m U ) : U ∈ C n × n , U ⁎ U = I n } . In this paper, we prove that if l ≤ 3 , n ≥ l and A 1 , . . . , A m are simultaneously unitarily diagonalizable, then W L ( A ) is star-shaped with star center at L ( tr A 1 n I n , . . . , tr A m n I n ) .

Let A ∈ R n × n and SO n : = { U ∈ R n × n : U U t = I n , det U > 0 } be the set of n × n special orthogonal matrices. Define the (real) special orthogonal orbit of A by O ( A ) : = { U A V : U , V ∈ SO n } . In this paper, we show that the linear image of O ( A ) is star-shaped with respect to the origin for arbitrary linear maps L : R n × n → R l if n ≥ 2 l − 1 . In particular, for linear maps L : R n × n → R 2 and when A has distinct singular values, we study B ∈ O ( A ) such that L ( B ) is a boundary point of L ( O ( A ) ) . This gives an alternative proof of a result by Li and Tam on the convexity of L ( O ( A ) ) for linear maps L : R n × n → R 2 .

Let V be a real or complex finite-dimensional vector space, and let N be a set of norms on V. The norm hull of a vector x ∈ V with respect to N is the set of vectors y ∈ V that satisfy ∥y∥ ≤ ∥x∥ for all ∥ · ∥ ∈ N. 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.

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.

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.

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.

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.

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 ‖A‖ c = ∑ i=l n c i σ i (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,‖·‖c) and (V,‖·‖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 c 1 ,…,c n } . 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 ‖·‖c, which is of independent interest.

Let $X_i, (i=1, \cdots, n)$ be real or complex linear spaces, each equipped with a norm $||\cdot||_i$. Standard ways of constructing norms $||\cdot||$ on the Cartestian product $X =X_1 \times \cdots \times X_n$ are to define \[ ||(x_1, \cdots, x_n)||=\phi(||x_1||_1, \cdots, ||x_n||_n)\] via some functions $\phi$ on $\mathbb{R}^n$. Common examples of $\phi$ in standard texbooks are norms on $\mathbb{R}^n$. This may mislead peoples to think that any norm $\phi$ on $\mathbb{R}^n$ can induce a norm on the product space $X$ in the above way. In this note we show that this is actually false and characterize the functions $\phi$ that can give rise to norms on $X$ in the above manner. It turns out that a necessary and sufficient condition on $\phi$ is : for any $a_1, \cdots, a_n, b_1, \cdots, b_n\ge 0$, (I) $\phi(a_1, \cdots, a_n)>0$ if $(a_1, \cdots, a_n)\neq (0, \cdots,0)$; (II) $\phi(\alpha(a_1, \cdots, a_n))= \alpha \phi(a_1, \cdots, a_n)$ if $\alpha\ge 0$; (III) $\phi(c_1, \cdots, c_n)\le \phi(a_1, \cdots, a_n)+ \phi(b_1, \cdots, b_n)$ if $(c_1, \cdots, c_n)= (a_1, \cdots, a_n)+ (b_1, \cdots, b_n)$;(IV) $\phi(a_1, \cdots, a_n) \le \phi(b_1, \cdots, b_n)$ if $a_i \le b_i$ for all $i$. Several interesting consequences of the result are discussed.

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.

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

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 AϵFm×n is the quantity ‖A‖c∑i=Imciσi(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)= {XϵFm×n:X has singular values c1,…,cm}.

Let H n {\mathcal {H}_n} be the set of all n × n n \times n hermitian matrices and U n {\mathcal {U}_n} the set of all n × n n \times n unitary matrices. For any c = ( c 1 , … , c n ) ∈ R n c = ({c_1}, \ldots ,{c_n}) \in {{\mathbf {R}}^n} and A 1 {A_1} , A 2 {A_2} , A 3 ∈ H n {A_3} \in {\mathcal {H}_n} , let W ( A 1 , A 2 , A 3 ) W({A_1},{A_2},{A_3}) denote the set \[ { ( tr [ c ] U A 1 U ∗ , tr [ c ] U A 2 U ∗ , tr [ c ] U A 3 U ∗ ) : U ∈ U n } , \{ ({\operatorname {tr}}[c]U{A_1}{U^*},{\operatorname {tr}}[c]U{A_2}{U^*},{\operatorname {tr}}[c]U{A_3}{U^*}):U \in {\mathcal {U}_n}\} , \] where [ c ] [c] is the diagonal matrix with c 1 , … , c n {c_1}, \ldots ,{c_n} as diagonal entries. In this present note, the authors prove that if n > 2 n > 2 , then W c ( A 1 , A 2 , A 3 ) {W_c}({A_1},{A_2},{A_3}) is always convex. Equivalent statements of this result, in terms of definiteness and inclusion relations, are also given. These results extend the theorems of Hausdorff-Toeplitz, Finsler and Westwick on numerical ranges, respectively.

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.

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.

Let M be the complex linear space M n of n × n complex matrices or the real linear space H n 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 = M n or M = H n . Related to U ( C ) and V ( C ) are the C-numerical range and the C-numerical radius of A ∈ M defined by W c (A) = {tr(AX) : X ∈ U (C)} and r C (A)=max{|z|:z∈W C (A)} , respectively. Let C , D ∈ H n , we study the linear operators T on M satisfying one of the following properties: (I) W D ( T ( A )) = W C ( A ) for all A ∈ M , (II) r D ( T ( A )) = r C ( 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.

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 ‖g(x)‖=∥x∥ for all g ϵ G, x ϵ V . The concept of G-invariant norm specializes to many interesting particular cases such as the absolute norms on F n, symmetric gauge functions on R n, unitarily invariant norms on F m×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 r G(c) (x) = max{|〈x, g(c)〉|:gϵ G}, where c ϵ 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.

For any n×n matrices A and C , we consider the star-centers of three sets, namely, the C -numerical range W C (A) of A , the set diag U (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 W C (A) for all C . For normal matrices A , we show that the set of star-centers of W A * (A) is a bounded closed real interval, and give complete description of the sets of star-centers of diag U (A) and of S (A) . In particular, we show that if A is normal with noncollinear eigenvalues, then each of S (A) and diag U (A) has exactly one star-center. For general square matrices A , we also give sufficient conditions for the sets of star-centers of diag U (A) and of S (A) to be singleton sets.

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 F n, u ϵ F m, yϵ F p, 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.

Let bd M be H n or C n×n, the real linear space of all n × n hermitian matrices and the complex linear space of all n × n complex matrices, respectively. For 1 ⩽ p ⩽ q ⩽ n, the (p, q) numerical radius of A ∈ M is defined and denoted by r p, q (A) = {|E p (X ∗ AX)|: X∈ D n×q , X ∗ X = I q } , where Ep(Y) denotes the pth elementary symmetric function on the eigenvalues of Y. We characterize the linear operators φ on M that satisfy rp, q(φ(A)) = rp, q(A) for all A∈ M .

Let S 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 S that satisfy T( S )= S . Partial results concerning those linear maps T satisfying T( S )⊆ S are also presented.

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 W p,q ( A ) = { E p (( UAU ∗ )[ q ]): U unitary}, where for an n × n complex matrix X , X [ q ] denotes its q × q leading principal submatrix and E p ( X [ q ]) denotes the p th 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 r p,q ( A ) = max{|μ|:μ ∈ W p,q ( A )}.