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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.

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.

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.

We present biologically motivated simple models to study phyllotaxis patterns. A simplified MaxMin-principle is applied for determining new-born primordium appearance. We propose a new repulsion pressure model to interpret the angle movement of each primordium influenced by the repulsion from all other primordia or from the nearest left and right neighbors. For a large range of growing velocities, both mechanisms can generate uniformly packed patterns under suitable settings of repulsion parameters. Measurement of the pattern uniformity and demonstration of robustness are also provided.

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.

Let , and let be the set F m×n of all m × n matrices over F. Given nonzero D ∈ M, denote by S(D) the collection of all matrices in having the same singular values as D. We show that if φ is a nonsingular linear operator on that satisfies φ(S(D)) ⊂ S(D), then φ(S(D)) and, with a previous result, the structure of φ is determined. Moreover, it is shown that the nonsingularity assumption can be removed except for the case when with either rank(D) = 1 or all the singular values of D are equal. In the exceptional cases, examples of singular linear operators φ satisfying are given. We also study those linear operators on satisfying φ(S(D)) ⊂ S(C) for some C,D ∈ .

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.

Let ∼ be a relation on a matrix space M. A linear operator T on M is said to preserve the relation ∼ if The first three sections of this chapter give a survey on the results concerning linear preservers of the relation of commutativity, which is not an equivalence relation, and linear preservers of some equivalence relations which arc closely related to canonical forms of matrices or arise from system theory. Section 7.4 discusses briefly how the equivalence relation may give a general setting for a number of linear preserver problems. Finally, Section 7.5 considers some group theoretic problems that are related to the equivalence relation.

Many authors have studied linear preservers of different kinds of generalized numerical ranges and numerical radii. It is known that these operators are related to the linear isometrics of unitary similarity invariant norms. In this chapter we survey the results on these subjects.

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.

Using a uniform approach, characterizations are obtained of linear operators on matrix spaces that preserve certain equivalence relations such as consimilarity, $ * $-congruence, nonsingular equivalence, and unitary equivalence.

Let A be an n × n complex matrix. The kth matrix numerical range of A is the set W(k:A) = {X * AX:X is an n×k matrix such that X*X=Ik }. We study the conditions on A under which the set W(k:A) is convex or starshaped. We get complete answers for hermitian matrices and partial results for general matrices. Then we consider different matrix sets S such as the collection of all scalar matrices, hermitian matrices, normal matrices, etc., and investigate the following problems: does AeS imply XeS for all XeW(k:A); what can we say about A if all XeW(k:A) are elements of S; if what is the largest k such that Some possible variations of the problem and related results are also discussed.

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.

A norm N(·) on Hn the space of all n × n hermitian matrices, is unitary similarity invariant if N(UAU* ) = N(A) for all hermitian matrices A and all unitary matrices U. We characterize those linear isometries on Hn with respect to a unitary similarity invariant norm. A related result concerning the characterization of those linear operators on Hn which preserve the set of matrices whose vectors of eigenvalues belong to a certain compact subset of is also obtained.

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.

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.

We study the structure of those linear operators on the rectangular complex or real matrix spaces that preserve certain functions on singular values. We first do a brief survey on the existing results in the area and then prove a theorem which covers and extends all of them. In particular. our theorem confirms two conjectures about the structure of those linear operators preserving the completely symmetric functions on powers of singular values of matrices.

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

We give two simple sufficient conditions under which the multiaffine image on 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. Due to space constraints, most results are given without proof.

Let c=(c 1,…,cn ) be in and A an n×n complex matrix. The c-numerical range of A is the set which is known to be convex. The matrix A is c-convex if Wc (A) is equal to the convex hull of the set where λ i (A),i=1,…,n, are the eigenvalues of A. In this note, we first give several characterizations of c-convex matrices. Then we study the inclusion relation between the sets for different c in where denotes the set of all c-convex matrices. In general, the set of normal matrices is contained in for any ; we identify those c for which the two sets are equal. Moreover, one of our characterizations of c-convex matrices employs a computable criterion for determining the supporting lines of Wc (A). The result could be used to design a program for plotting the set Wc (A), as shown in the appendix of this note.

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 A B be Hermitian matrices. Denote by co the convex hull of the unitary orbit of B. We determine the matrices BM and Bm in co such that for all X∊ co and all unitary similarity invariant norms ∣∣sdot;∣∣, ie., those norms ∣∣sdot;∣∣ satisfying ∣∣UXU ∗∣∣=∣∣X∣∣ for any Hermitian X and unitary U. As a result, we can easily compute the quantities and A particular case of these results answers a question raised by Dr. Y. Nakamura.

Let A be an n × n complex normal matrix with eigenvalues . In the present paper the authors show that if are not collinear, then the set is an orthonormal sct in is not convex which answers a conjecture posed by Marcus concerning the convexity of the generalized numerical range.

Let For any n × n matrix A, define the c-numerical range and the c-numerical radius by and rc (A) = max{|z|: z∈ W c (A)} respectively. We give a characterization for those linear operators Twhich satisfy for all matrix A. Moreover, we give a full description for the extreme points of the unit ball in the real linear space of n × n hermitian matrices with respect to the norm rc (·).

A norm N defined on the linear space of n × n complex matrices (denoted by ) is said to be unitarily invariant if for any A in and n × n unitary matrix U. In this note we study the properties of unitarily invariant norms. Using the metric properties of with respect to this kind of norms, we characterize different classes of matrices such as normal, unitary. Hermitian, positive semi-definite matrices etc. Some approximation problems in are also investigated.

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.

Let F be the real field R the complex field C or the skew field H of real quaternions. For any c = i:c1,…,cn )∊ Rn: and for any two hermitian (symmetric) matrices A and B with elements in F, the authors give a unified proof that the set is convex (with n> 2 if F =R). They also show that the convexity problem, definite-ness problem, and inclusion problem associated with the above set, are all equivalent, and give some applications of these results.

Let C be a hermitian matrix. We show that the C-numerical range or C-numerical radius preservers are actually the dual operators of some matrix set preservers. We use the known results about the C-numerical range preservers to determine the structure of their dual operators. For the C-numerical radius preservers, we prove a structural theorem for the dual operators, and then determine their structure. This solves a problem in an earlier paper of the authors

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.

The characterization of all linear operators on matrices which preserve the decomposable numerical radius is obtained. This result refines those of Tam. Marcus and Filippenko on the topic. The proof of the main theorem depends on a characterization of scalar multiples of unitary matrices in terms of decomposable numerical radius that is of independent interest.